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[STATEMENT] lemma ent_disjI2_direct[simp]: "B \<Longrightarrow>\<^sub>A A \<or>\<^sub>A B" [PROOF STATE] proof (prove) goal (1 subgoal): 1. B \<Longrightarrow>\<^sub>A A \<or>\<^sub>A B [PROOF STEP] by (simp add: entails_def)
Load LFindLoad. From lfind Require Import LFind. From QuickChick Require Import QuickChick. From adtind Require Import goal33. Derive Show for natural. Derive Arbitrary for natural. Instance Dec_Eq_natural : Dec_Eq natural. Proof. dec_eq. Qed. Lemma conj3eqsynthconj4 : forall (lv0 : natural) (lv1 : natural), (@eq natural (plus lv0 lv1) (plus lv0 lv1)). Admitted. QuickChick conj3eqsynthconj4.
module ZOOclient using Base.Dates.now export Dimension, dim_print export Objective, get_history_bestsofar, obj_clean_history export zoo_min export Parameter, print_population, get_positive_data, get_negative_data export Solution, sol_print, sol_equal, sol_write, write_population export zoolog export rng, my_precision, set_seed, set_precision export construct_init_sample include("ZOOclient/utils/tool_function.jl") include("ZOOclient/utils/zoo_global.jl") include("ZOOclient/dimension.jl") include("ZOOclient/objective.jl") include("ZOOclient/parameter.jl") include("ZOOclient/solution.jl") include("ZOOclient/algos/racos/racos_classification.jl") include("ZOOclient/algos/racos/racos_common.jl") include("ZOOclient/algos/racos/racos.jl") include("ZOOclient/algos/racos/sracos.jl") include("ZOOclient/algos/racos/racos_optimization.jl") include("ZOOclient/algos/asracos/asracos.jl") include("ZOOclient/algos/asracos/asracos_opt.jl") include("ZOOclient/algos/pracos/pracos.jl") include("ZOOclient/algos/pracos/pracos_opt.jl") include("ZOOclient/algos/psracos/psracos.jl") include("ZOOclient/algos/psracos/psracos_opt.jl") include("ZOOclient/algos/pposs/pposs_opt.jl") include("ZOOclient/optimize.jl") end
import SOAS.Syntax.Signature as Sig open import SOAS.Families.Core -- Term syntax for a signature module SOAS.Syntax.Term {T : Set}(open Sig T) {O : Set}(S : Signature O) where open import SOAS.Syntax.Arguments {T} open import SOAS.Metatheory.Syntax {T} open import SOAS.Common open import SOAS.Context {T} open import SOAS.Variable open import SOAS.Construction.Structure open import SOAS.ContextMaps.Inductive open import SOAS.Abstract.Hom open import Categories.Object.Initial open import Data.List.Base using (List ; [] ; [_] ; _∷_) open import Data.Unit open Signature S private variable α β τ : T Γ Δ Π : Ctx module _ (𝔛 : Familyₛ) where open import SOAS.Metatheory.MetaAlgebra ⅀F 𝔛 -- Grammar of terms for a (⅀,𝔛)-meta-algebra data 𝕋 : Familyₛ where con : ⅀ 𝕋 τ Γ → 𝕋 τ Γ var : ℐ τ Γ → 𝕋 τ Γ mvar : 𝔛 τ Π → Sub 𝕋 Π Γ → 𝕋 τ Γ Tmᵃ : MetaAlg 𝕋 Tmᵃ = record { 𝑎𝑙𝑔 = con ; 𝑣𝑎𝑟 = var ; 𝑚𝑣𝑎𝑟 = λ 𝔪 ε → mvar 𝔪 (tabulate ε) } -- 𝕋 is the initial meta-algebra 𝕋:Init : Initial 𝕄etaAlgebras 𝕋:Init = record { ⊥ = 𝕋 ⋉ Tmᵃ ; ⊥-is-initial = record { ! = λ{ {𝒜 ⋉ 𝒜ᵃ} → (𝕤𝕖𝕞 𝒜ᵃ) ⋉ 𝕤𝕖𝕞ᵃ⇒ 𝒜ᵃ } ; !-unique = λ { {𝒜 ⋉ 𝒜ᵃ}(g ⋉ gᵃ⇒) {x = t} → 𝕤𝕖𝕞! 𝒜ᵃ gᵃ⇒ t } } } where module _ {𝒜 : Familyₛ}(𝒜ᵃ : MetaAlg 𝒜) where open MetaAlg 𝒜ᵃ 𝕤𝕖𝕞 : 𝕋 ⇾̣ 𝒜 𝔸 : (as : List (Ctx × T)) → Arg as 𝕋 Γ → Arg as 𝒜 Γ 𝔸 [] tt = tt 𝔸 (_ ∷ []) t = 𝕤𝕖𝕞 t 𝔸 (_ ∷ a ∷ as) (t , ts) = (𝕤𝕖𝕞 t , 𝔸 (a ∷ as) ts) 𝕊 : Sub 𝕋 Π Γ → Π ~[ 𝒜 ]↝ Γ 𝕊 (t ◂ σ) new = 𝕤𝕖𝕞 t 𝕊 (t ◂ σ) (old v) = 𝕊 σ v 𝕤𝕖𝕞 (con (o ⋮ a)) = 𝑎𝑙𝑔 (o ⋮ 𝔸 (Arity o) a) 𝕤𝕖𝕞 (var v) = 𝑣𝑎𝑟 v 𝕤𝕖𝕞 (mvar 𝔪 ε) = 𝑚𝑣𝑎𝑟 𝔪 (𝕊 ε) 𝕤𝕖𝕞ᵃ⇒ : MetaAlg⇒ Tmᵃ 𝒜ᵃ 𝕤𝕖𝕞 𝕤𝕖𝕞ᵃ⇒ = record { ⟨𝑎𝑙𝑔⟩ = λ{ {t = (o ⋮ a)} → cong (λ - → 𝑎𝑙𝑔 (o ⋮ -)) (𝔸-Arg₁ (Arity o) a) } ; ⟨𝑣𝑎𝑟⟩ = refl ; ⟨𝑚𝑣𝑎𝑟⟩ = λ{ {𝔪 = 𝔪}{ε} → cong (𝑚𝑣𝑎𝑟 𝔪) (dext (𝕊-tab ε)) } } where 𝔸-Arg₁ : (as : List (Ctx × T))(a : Arg as 𝕋 Γ) → 𝔸 as a ≡ Arg₁ as 𝕤𝕖𝕞 a 𝔸-Arg₁ [] tt = refl 𝔸-Arg₁ (_ ∷ []) t = refl 𝔸-Arg₁ (_ ∷ a ∷ as) (t , ap) = cong (_ ,_) (𝔸-Arg₁ (a ∷ as) ap) 𝕊-tab : (ε : Π ~[ 𝕋 ]↝ Γ)(v : ℐ α Π) → 𝕊 (tabulate ε) v ≡ 𝕤𝕖𝕞 (ε v) 𝕊-tab ε new = refl 𝕊-tab ε (old v) = 𝕊-tab (ε ∘ old) v module _ {g : 𝕋 ⇾̣ 𝒜}(gᵃ⇒ : MetaAlg⇒ Tmᵃ 𝒜ᵃ g) where open MetaAlg⇒ gᵃ⇒ 𝕤𝕖𝕞! : (t : 𝕋 α Γ) → 𝕤𝕖𝕞 t ≡ g t 𝕊-ix : (ε : Sub 𝕋 Π Γ)(v : ℐ α Π) → 𝕊 ε v ≡ g (index ε v) 𝕊-ix (x ◂ ε) new = 𝕤𝕖𝕞! x 𝕊-ix (x ◂ ε) (old v) = 𝕊-ix ε v 𝔸-Arg₁ : (as : List (Ctx × T))(ar : Arg as 𝕋 Γ) → 𝔸 as ar ≡ Arg₁ as g ar 𝔸-Arg₁ [] tt = refl 𝔸-Arg₁ (_ ∷ []) t = 𝕤𝕖𝕞! t 𝔸-Arg₁ (_ ∷ a ∷ as) (t , ap) = cong₂ _,_ (𝕤𝕖𝕞! t) (𝔸-Arg₁ (a ∷ as) ap) 𝕤𝕖𝕞! (con (o ⋮ a)) rewrite 𝔸-Arg₁ (Arity o) a = sym ⟨𝑎𝑙𝑔⟩ 𝕤𝕖𝕞! (var v) = sym ⟨𝑣𝑎𝑟⟩ 𝕤𝕖𝕞! (mvar 𝔪 ε) rewrite cong (𝑚𝑣𝑎𝑟 𝔪) (dext (𝕊-ix ε)) = trans (sym ⟨𝑚𝑣𝑎𝑟⟩) (cong (g ∘ mvar 𝔪) (tab∘ix≈id ε)) -- Syntax instance for a term grammar 𝕋:Syn : Syntax 𝕋:Syn = record { ⅀F = ⅀F ; ⅀:CS = ⅀:CompatStr ; 𝕋:Init = 𝕋:Init ; mvarᵢ = mvar }
lemma (in bounded_linear) isUCont: "isUCont f"
[GOAL] X Y B : Profinite f : X ⟶ B g : Y ⟶ B ⊢ fst f g ≫ f = snd f g ≫ g [PROOFSTEP] ext ⟨_, h⟩ [GOAL] case w.mk X Y B : Profinite f : X ⟶ B g : Y ⟶ B val✝ : ↑X.toCompHaus.toTop × ↑Y.toCompHaus.toTop h : val✝ ∈ {xy \| ↑f xy.fst = ↑g xy.snd} ⊢ ↑(fst f g ≫ f) { val := val✝, property := h } = ↑(snd f g ≫ g) { val := val✝, property := h } [PROOFSTEP] exact h [GOAL] X Y B : Profinite f : X ⟶ B g : Y ⟶ B Z : Profinite a : Z ⟶ X b : Z ⟶ Y w : a ≫ f = b ≫ g z : ↑Z.toCompHaus.toTop ⊢ (↑a z, ↑b z) ∈ {xy \| ↑f xy.fst = ↑g xy.snd} [PROOFSTEP] apply_fun (· z) at w [GOAL] X Y B : Profinite f : X ⟶ B g : Y ⟶ B Z : Profinite a : Z ⟶ X b : Z ⟶ Y z : ↑Z.toCompHaus.toTop w : ↑(a ≫ f) z = ↑(b ≫ g) z ⊢ (↑a z, ↑b z) ∈ {xy \| ↑f xy.fst = ↑g xy.snd} [PROOFSTEP] exact w [GOAL] X Y B : Profinite f : X ⟶ B g : Y ⟶ B Z : Profinite a : Z ⟶ X b : Z ⟶ Y w : a ≫ f = b ≫ g ⊢ Continuous fun z => { val := (↑a z, ↑b z), property := (_ : ↑(a ≫ f) z = ↑(b ≫ g) z) } [PROOFSTEP] apply Continuous.subtype_mk [GOAL] case h X Y B : Profinite f : X ⟶ B g : Y ⟶ B Z : Profinite a : Z ⟶ X b : Z ⟶ Y w : a ≫ f = b ≫ g ⊢ Continuous fun x => (↑a x, ↑b x) [PROOFSTEP] rw [continuous_prod_mk] [GOAL] case h X Y B : Profinite f : X ⟶ B g : Y ⟶ B Z : Profinite a : Z ⟶ X b : Z ⟶ Y w : a ≫ f = b ≫ g ⊢ (Continuous fun x => ↑a x) ∧ Continuous fun x => ↑b x [PROOFSTEP] exact ⟨a.continuous, b.continuous⟩ [GOAL] X Y B : Profinite f : X ⟶ B g : Y ⟶ B Z : Profinite a b : Z ⟶ pullback f g hfst : a ≫ fst f g = b ≫ fst f g hsnd : a ≫ snd f g = b ≫ snd f g ⊢ a = b [PROOFSTEP] ext z [GOAL] case w X Y B : Profinite f : X ⟶ B g : Y ⟶ B Z : Profinite a b : Z ⟶ pullback f g hfst : a ≫ fst f g = b ≫ fst f g hsnd : a ≫ snd f g = b ≫ snd f g z : (forget Profinite).obj Z ⊢ ↑a z = ↑b z [PROOFSTEP] apply_fun (· z) at hfst hsnd [GOAL] case w X Y B : Profinite f : X ⟶ B g : Y ⟶ B Z : Profinite a b : Z ⟶ pullback f g z : (forget Profinite).obj Z hfst : ↑(a ≫ fst f g) z = ↑(b ≫ fst f g) z hsnd : ↑(a ≫ snd f g) z = ↑(b ≫ snd f g) z ⊢ ↑a z = ↑b z [PROOFSTEP] apply Subtype.ext [GOAL] case w.a X Y B : Profinite f : X ⟶ B g : Y ⟶ B Z : Profinite a b : Z ⟶ pullback f g z : (forget Profinite).obj Z hfst : ↑(a ≫ fst f g) z = ↑(b ≫ fst f g) z hsnd : ↑(a ≫ snd f g) z = ↑(b ≫ snd f g) z ⊢ ↑(↑a z) = ↑(↑b z) [PROOFSTEP] apply Prod.ext [GOAL] case w.a.h₁ X Y B : Profinite f : X ⟶ B g : Y ⟶ B Z : Profinite a b : Z ⟶ pullback f g z : (forget Profinite).obj Z hfst : ↑(a ≫ fst f g) z = ↑(b ≫ fst f g) z hsnd : ↑(a ≫ snd f g) z = ↑(b ≫ snd f g) z ⊢ (↑(↑a z)).fst = (↑(↑b z)).fst [PROOFSTEP] exact hfst [GOAL] case w.a.h₂ X Y B : Profinite f : X ⟶ B g : Y ⟶ B Z : Profinite a b : Z ⟶ pullback f g z : (forget Profinite).obj Z hfst : ↑(a ≫ fst f g) z = ↑(b ≫ fst f g) z hsnd : ↑(a ≫ snd f g) z = ↑(b ≫ snd f g) z ⊢ (↑(↑a z)).snd = (↑(↑b z)).snd [PROOFSTEP] exact hsnd [GOAL] X Y B : Profinite f : X ⟶ B g : Y ⟶ B ⊢ pullback.fst f g = (pullbackIsoPullback f g).hom ≫ Limits.pullback.fst [PROOFSTEP] dsimp [pullbackIsoPullback] [GOAL] X Y B : Profinite f : X ⟶ B g : Y ⟶ B ⊢ pullback.fst f g = (Limits.IsLimit.conePointUniqueUpToIso (pullback.isLimit f g) (Limits.limit.isLimit (Limits.cospan f g))).hom ≫ Limits.pullback.fst [PROOFSTEP] simp only [Limits.limit.conePointUniqueUpToIso_hom_comp, pullback.cone_pt, pullback.cone_π] [GOAL] X Y B : Profinite f : X ⟶ B g : Y ⟶ B ⊢ pullback.snd f g = (pullbackIsoPullback f g).hom ≫ Limits.pullback.snd [PROOFSTEP] dsimp [pullbackIsoPullback] [GOAL] X Y B : Profinite f : X ⟶ B g : Y ⟶ B ⊢ pullback.snd f g = (Limits.IsLimit.conePointUniqueUpToIso (pullback.isLimit f g) (Limits.limit.isLimit (Limits.cospan f g))).hom ≫ Limits.pullback.snd [PROOFSTEP] simp only [Limits.limit.conePointUniqueUpToIso_hom_comp, pullback.cone_pt, pullback.cone_π] [GOAL] α : Type inst✝ : Fintype α X : α → Profinite B : Profinite e : (a : α) → X a ⟶ B ⊢ Continuous fun x => match x with \| { fst := a, snd := x } => ↑(e a) x [PROOFSTEP] apply continuous_sigma [GOAL] case hf α : Type inst✝ : Fintype α X : α → Profinite B : Profinite e : (a : α) → X a ⟶ B ⊢ ∀ (i : α), Continuous fun a => match { fst := i, snd := a } with \| { fst := a, snd := x } => ↑(e a) x [PROOFSTEP] intro a [GOAL] case hf α : Type inst✝ : Fintype α X : α → Profinite B : Profinite e : (a : α) → X a ⟶ B a : α ⊢ Continuous fun a_1 => match { fst := a, snd := a_1 } with \| { fst := a, snd := x } => ↑(e a) x [PROOFSTEP] exact (e a).continuous [GOAL] α : Type inst✝ : Fintype α X : α → Profinite B : Profinite f g : finiteCoproduct X ⟶ B h : ∀ (a : α), ι X a ≫ f = ι X a ≫ g ⊢ f = g [PROOFSTEP] ext ⟨a, x⟩ [GOAL] case w.mk α : Type inst✝ : Fintype α X : α → Profinite B : Profinite f g : finiteCoproduct X ⟶ B h : ∀ (a : α), ι X a ≫ f = ι X a ≫ g a : α x : ↑(X a).toCompHaus.toTop ⊢ ↑f { fst := a, snd := x } = ↑g { fst := a, snd := x } [PROOFSTEP] specialize h a [GOAL] case w.mk α : Type inst✝ : Fintype α X : α → Profinite B : Profinite f g : finiteCoproduct X ⟶ B a : α x : ↑(X a).toCompHaus.toTop h : ι X a ≫ f = ι X a ≫ g ⊢ ↑f { fst := a, snd := x } = ↑g { fst := a, snd := x } [PROOFSTEP] apply_fun (· x) at h [GOAL] case w.mk α : Type inst✝ : Fintype α X : α → Profinite B : Profinite f g : finiteCoproduct X ⟶ B a : α x : ↑(X a).toCompHaus.toTop h : ↑(ι X a ≫ f) x = ↑(ι X a ≫ g) x ⊢ ↑f { fst := a, snd := x } = ↑g { fst := a, snd := x } [PROOFSTEP] exact h [GOAL] α : Type inst✝ : Fintype α X : α → Profinite s : Limits.Cocone (Discrete.functor X) m : (cocone X).pt ⟶ s.pt hm : ∀ (j : Discrete α), NatTrans.app (cocone X).ι j ≫ m = NatTrans.app s.ι j a : α ⊢ ι (fun a => X a) a ≫ m = ι (fun a => X a) a ≫ (fun s => desc (fun a => X a) fun a => NatTrans.app s.ι { as := a }) s [PROOFSTEP] specialize hm ⟨a⟩ [GOAL] α : Type inst✝ : Fintype α X : α → Profinite s : Limits.Cocone (Discrete.functor X) m : (cocone X).pt ⟶ s.pt a : α hm : NatTrans.app (cocone X).ι { as := a } ≫ m = NatTrans.app s.ι { as := a } ⊢ ι (fun a => X a) a ≫ m = ι (fun a => X a) a ≫ (fun s => desc (fun a => X a) fun a => NatTrans.app s.ι { as := a }) s [PROOFSTEP] ext t [GOAL] case w α : Type inst✝ : Fintype α X : α → Profinite s : Limits.Cocone (Discrete.functor X) m : (cocone X).pt ⟶ s.pt a : α hm : NatTrans.app (cocone X).ι { as := a } ≫ m = NatTrans.app s.ι { as := a } t : (forget Profinite).obj (X a) ⊢ ↑(ι (fun a => X a) a ≫ m) t = ↑(ι (fun a => X a) a ≫ (fun s => desc (fun a => X a) fun a => NatTrans.app s.ι { as := a }) s) t [PROOFSTEP] apply_fun (· t) at hm [GOAL] case w α : Type inst✝ : Fintype α X : α → Profinite s : Limits.Cocone (Discrete.functor X) m : (cocone X).pt ⟶ s.pt a : α t : (forget Profinite).obj (X a) hm : ↑(NatTrans.app (cocone X).ι { as := a } ≫ m) t = ↑(NatTrans.app s.ι { as := a }) t ⊢ ↑(ι (fun a => X a) a ≫ m) t = ↑(ι (fun a => X a) a ≫ (fun s => desc (fun a => X a) fun a => NatTrans.app s.ι { as := a }) s) t [PROOFSTEP] exact hm [GOAL] α : Type inst✝ : Fintype α X : α → Profinite a : α ⊢ Limits.Sigma.ι X a ≫ (coproductIsoCoproduct X).inv = finiteCoproduct.ι X a [PROOFSTEP] simp only [coproductIsoCoproduct, Limits.colimit.comp_coconePointUniqueUpToIso_inv, finiteCoproduct.cocone_pt, finiteCoproduct.cocone_ι, Discrete.natTrans_app] [GOAL] α : Type inst✝ : Fintype α X : α → Profinite a : α ⊢ Function.Injective ↑(ι X a) [PROOFSTEP] intro x y hxy [GOAL] α : Type inst✝ : Fintype α X : α → Profinite a : α x y : (forget Profinite).obj (X a) hxy : ↑(ι X a) x = ↑(ι X a) y ⊢ x = y [PROOFSTEP] exact eq_of_heq (Sigma.ext_iff.mp hxy).2 [GOAL] α : Type inst✝ : Fintype α X : α → Profinite B : Profinite π : (a : α) → X a ⟶ B a : α ⊢ ∀ (x : (forget Profinite).obj (X a)), ↑(desc X π) (↑(ι X a) x) = ↑(π a) x [PROOFSTEP] intro x [GOAL] α : Type inst✝ : Fintype α X : α → Profinite B : Profinite π : (a : α) → X a ⟶ B a : α x : (forget Profinite).obj (X a) ⊢ ↑(desc X π) (↑(ι X a) x) = ↑(π a) x [PROOFSTEP] change (ι X a ≫ desc X π) _ = _ [GOAL] α : Type inst✝ : Fintype α X : α → Profinite B : Profinite π : (a : α) → X a ⟶ B a : α x : (forget Profinite).obj (X a) ⊢ ↑(ι X a ≫ desc X π) x = ↑(π a) x [PROOFSTEP] simp only [ι_desc]
[STATEMENT] lemma diff_union_cancelR: "M + N - N = (M::'a multiset)" [PROOF STATE] proof (prove) goal (1 subgoal): 1. M + N - N = M [PROOF STEP] by (fact add_diff_cancel_right')
The imaginary part of $\iota z$ is equal to the real part of $z$.
Promoted to the rank of brigadier , after the German surrender in North Africa , Wheeler was sent to Algiers where he was part of the staff committee planning the invasion of Italy . There , he learned that the India Office had requested that the army relieve him of his duties to permit him to be appointed Director General of Archaeology in India . Although he had never been to the country , he agreed that he would take the job on the condition that he be permitted to take part in the invasion of Italy first . As intended , Wheeler and his 12th Anti @-@ Aircraft Brigade then took part in the invasion of Sicily and then mainland Italy , where they were ordered to use their anti @-@ aircraft guns to protect the British 10th Corps . As the Allies advanced north through Italy , Wheeler spent time in Naples and then Capri , where he met various aristocrats who had anti @-@ fascist sympathies .
(* Title: HOL/Library/Product_Plus.thy Author: Brian Huffman *) section \<open>Additive group operations on product types\<close> theory Product_Plus imports Main begin subsection \<open>Operations\<close> instantiation prod :: (zero, zero) zero begin definition zero_prod_def: "0 = (0, 0)" instance .. end instantiation prod :: (plus, plus) plus begin definition plus_prod_def: "x + y = (fst x + fst y, snd x + snd y)" instance .. end instantiation prod :: (minus, minus) minus begin definition minus_prod_def: "x - y = (fst x - fst y, snd x - snd y)" instance .. end instantiation prod :: (uminus, uminus) uminus begin definition uminus_prod_def: "- x = (- fst x, - snd x)" instance .. end lemma fst_zero [simp]: "fst 0 = 0" unfolding zero_prod_def by simp lemma snd_zero [simp]: "snd 0 = 0" unfolding zero_prod_def by simp lemma fst_add [simp]: "fst (x + y) = fst x + fst y" unfolding plus_prod_def by simp lemma snd_add [simp]: "snd (x + y) = snd x + snd y" unfolding plus_prod_def by simp lemma fst_diff [simp]: "fst (x - y) = fst x - fst y" unfolding minus_prod_def by simp lemma snd_diff [simp]: "snd (x - y) = snd x - snd y" unfolding minus_prod_def by simp lemma fst_uminus [simp]: "fst (- x) = - fst x" unfolding uminus_prod_def by simp lemma snd_uminus [simp]: "snd (- x) = - snd x" unfolding uminus_prod_def by simp lemma add_Pair [simp]: "(a, b) + (c, d) = (a + c, b + d)" unfolding plus_prod_def by simp lemma diff_Pair [simp]: "(a, b) - (c, d) = (a - c, b - d)" unfolding minus_prod_def by simp lemma uminus_Pair [simp, code]: "- (a, b) = (- a, - b)" unfolding uminus_prod_def by simp subsection \<open>Class instances\<close> instance prod :: (semigroup_add, semigroup_add) semigroup_add by standard (simp add: prod_eq_iff add.assoc) instance prod :: (ab_semigroup_add, ab_semigroup_add) ab_semigroup_add by standard (simp add: prod_eq_iff add.commute) instance prod :: (monoid_add, monoid_add) monoid_add by standard (simp_all add: prod_eq_iff) instance prod :: (comm_monoid_add, comm_monoid_add) comm_monoid_add by standard (simp add: prod_eq_iff) instance prod :: (cancel_semigroup_add, cancel_semigroup_add) cancel_semigroup_add by standard (simp_all add: prod_eq_iff) instance prod :: (cancel_ab_semigroup_add, cancel_ab_semigroup_add) cancel_ab_semigroup_add by standard (simp_all add: prod_eq_iff diff_diff_eq) instance prod :: (cancel_comm_monoid_add, cancel_comm_monoid_add) cancel_comm_monoid_add .. instance prod :: (group_add, group_add) group_add by standard (simp_all add: prod_eq_iff) instance prod :: (ab_group_add, ab_group_add) ab_group_add by standard (simp_all add: prod_eq_iff) lemma fst_sum: "fst (\<Sum>x\<in>A. f x) = (\<Sum>x\<in>A. fst (f x))" proof (cases "finite A") case True then show ?thesis by induct simp_all next case False then show ?thesis by simp qed lemma snd_sum: "snd (\<Sum>x\<in>A. f x) = (\<Sum>x\<in>A. snd (f x))" proof (cases "finite A") case True then show ?thesis by induct simp_all next case False then show ?thesis by simp qed lemma sum_prod: "(\<Sum>x\<in>A. (f x, g x)) = (\<Sum>x\<in>A. f x, \<Sum>x\<in>A. g x)" proof (cases "finite A") case True then show ?thesis by induct (simp_all add: zero_prod_def) next case False then show ?thesis by (simp add: zero_prod_def) qed end
lemma eventually_at_left_field: "eventually P (at_left x) \<longleftrightarrow> (\<exists>b<x. \<forall>y>b. y < x \<longrightarrow> P y)" for x :: "'a::{linordered_field, linorder_topology}"
(* Title: HOL/Auth/n_germanSymIndex_lemma_inv__3_on_rules.thy Author: Yongjian Li and Kaiqiang Duan, State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences Copyright 2016 State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences ) header{The n_germanSymIndex Protocol Case Study} theory n_germanSymIndex_lemma_inv__3_on_rules imports n_germanSymIndex_lemma_on_inv__3 begin section{All lemmas on causal relation between inv__3*} lemma lemma_inv__3_on_rules: assumes b1: "r \<in> rules N" and b2: "(f=inv__3 )" shows "invHoldForRule s f r (invariants N)" proof - have c1: "(\<exists> i d. i\<le>N\<and>d\<le>N\<and>r=n_Store i d)\<or> (\<exists> i. i\<le>N\<and>r=n_SendReqS i)\<or> (\<exists> i. i\<le>N\<and>r=n_SendReqE__part__0 i)\<or> (\<exists> i. i\<le>N\<and>r=n_SendReqE__part__1 i)\<or> (\<exists> i. i\<le>N\<and>r=n_RecvReqS N i)\<or> (\<exists> i. i\<le>N\<and>r=n_RecvReqE N i)\<or> (\<exists> i. i\<le>N\<and>r=n_SendInv__part__0 i)\<or> (\<exists> i. i\<le>N\<and>r=n_SendInv__part__1 i)\<or> (\<exists> i. i\<le>N\<and>r=n_SendInvAck i)\<or> (\<exists> i. i\<le>N\<and>r=n_RecvInvAck i)\<or> (\<exists> i. i\<le>N\<and>r=n_SendGntS i)\<or> (\<exists> i. i\<le>N\<and>r=n_SendGntE N i)\<or> (\<exists> i. i\<le>N\<and>r=n_RecvGntS i)\<or> (\<exists> i. i\<le>N\<and>r=n_RecvGntE i)" apply (cut_tac b1, auto) done moreover { assume d1: "(\<exists> i d. i\<le>N\<and>d\<le>N\<and>r=n_Store i d)" have "invHoldForRule s f r (invariants N)" apply (cut_tac b2 d1, metis n_StoreVsinv__3) done } moreover { assume d1: "(\<exists> i. i\<le>N\<and>r=n_SendReqS i)" have "invHoldForRule s f r (invariants N)" apply (cut_tac b2 d1, metis n_SendReqSVsinv__3) done } moreover { assume d1: "(\<exists> i. i\<le>N\<and>r=n_SendReqE__part__0 i)" have "invHoldForRule s f r (invariants N)" apply (cut_tac b2 d1, metis n_SendReqE__part__0Vsinv__3) done } moreover { assume d1: "(\<exists> i. i\<le>N\<and>r=n_SendReqE__part__1 i)" have "invHoldForRule s f r (invariants N)" apply (cut_tac b2 d1, metis n_SendReqE__part__1Vsinv__3) done } moreover { assume d1: "(\<exists> i. i\<le>N\<and>r=n_RecvReqS N i)" have "invHoldForRule s f r (invariants N)" apply (cut_tac b2 d1, metis n_RecvReqSVsinv__3) done } moreover { assume d1: "(\<exists> i. i\<le>N\<and>r=n_RecvReqE N i)" have "invHoldForRule s f r (invariants N)" apply (cut_tac b2 d1, metis n_RecvReqEVsinv__3) done } moreover { assume d1: "(\<exists> i. i\<le>N\<and>r=n_SendInv__part__0 i)" have "invHoldForRule s f r (invariants N)" apply (cut_tac b2 d1, metis n_SendInv__part__0Vsinv__3) done } moreover { assume d1: "(\<exists> i. i\<le>N\<and>r=n_SendInv__part__1 i)" have "invHoldForRule s f r (invariants N)" apply (cut_tac b2 d1, metis n_SendInv__part__1Vsinv__3) done } moreover { assume d1: "(\<exists> i. i\<le>N\<and>r=n_SendInvAck i)" have "invHoldForRule s f r (invariants N)" apply (cut_tac b2 d1, metis n_SendInvAckVsinv__3) done } moreover { assume d1: "(\<exists> i. i\<le>N\<and>r=n_RecvInvAck i)" have "invHoldForRule s f r (invariants N)" apply (cut_tac b2 d1, metis n_RecvInvAckVsinv__3) done } moreover { assume d1: "(\<exists> i. i\<le>N\<and>r=n_SendGntS i)" have "invHoldForRule s f r (invariants N)" apply (cut_tac b2 d1, metis n_SendGntSVsinv__3) done } moreover { assume d1: "(\<exists> i. i\<le>N\<and>r=n_SendGntE N i)" have "invHoldForRule s f r (invariants N)" apply (cut_tac b2 d1, metis n_SendGntEVsinv__3) done } moreover { assume d1: "(\<exists> i. i\<le>N\<and>r=n_RecvGntS i)" have "invHoldForRule s f r (invariants N)" apply (cut_tac b2 d1, metis n_RecvGntSVsinv__3) done } moreover { assume d1: "(\<exists> i. i\<le>N\<and>r=n_RecvGntE i)" have "invHoldForRule s f r (invariants N)" apply (cut_tac b2 d1, metis n_RecvGntEVsinv__3) done } ultimately show "invHoldForRule s f r (invariants N)" by satx qed end
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Mario Carneiro -/ import data.finset.sort /-! # Finite sets This file defines predicates `finite : set α → Prop` and `infinite : set α → Prop` and proves some basic facts about finite sets. -/ open set function universes u v w x variables {α : Type u} {β : Type v} {ι : Sort w} {γ : Type x} namespace set /-- A set is finite if the subtype is a fintype, i.e. there is a list that enumerates its members. -/ inductive finite (s : set α) : Prop \| intro : fintype s → finite lemma finite_def {s : set α} : finite s ↔ nonempty (fintype s) := ⟨λ ⟨h⟩, ⟨h⟩, λ ⟨h⟩, ⟨h⟩⟩ /-- A set is infinite if it is not finite. -/ def infinite (s : set α) : Prop := ¬ finite s /-- The subtype corresponding to a finite set is a finite type. Note that because `finite` isn't a typeclass, this will not fire if it is made into an instance -/ noncomputable def finite.fintype {s : set α} (h : finite s) : fintype s := classical.choice $ finite_def.1 h /-- Get a finset from a finite set -/ noncomputable def finite.to_finset {s : set α} (h : finite s) : finset α := @set.to_finset _ _ h.fintype @[simp] lemma not_infinite {s : set α} : ¬ s.infinite ↔ s.finite := by simp [infinite] @[simp] theorem finite.mem_to_finset {s : set α} (h : finite s) {a : α} : a ∈ h.to_finset ↔ a ∈ s := @mem_to_finset _ _ h.fintype _ @[simp] theorem finite.to_finset.nonempty {s : set α} (h : finite s) : h.to_finset.nonempty ↔ s.nonempty := show (∃ x, x ∈ h.to_finset) ↔ (∃ x, x ∈ s), from exists_congr (λ _, h.mem_to_finset) @[simp] lemma finite.coe_to_finset {s : set α} (h : finite s) : ↑h.to_finset = s := @set.coe_to_finset _ s h.fintype @[simp] lemma finite.coe_sort_to_finset {s : set α} (h : finite s) : (h.to_finset : Type) = s := by rw [← finset.coe_sort_coe _, h.coe_to_finset] @[simp] lemma finite_empty_to_finset (h : finite (∅ : set α)) : h.to_finset = ∅ := by rw [← finset.coe_inj, h.coe_to_finset, finset.coe_empty] @[simp] lemma finite.to_finset_inj {s t : set α} {hs : finite s} {ht : finite t} : hs.to_finset = ht.to_finset ↔ s = t := by simp [←finset.coe_inj] lemma subset_to_finset_iff {s : finset α} {t : set α} (ht : finite t) : s ⊆ ht.to_finset ↔ ↑s ⊆ t := by rw [← finset.coe_subset, ht.coe_to_finset] @[simp] lemma finite_to_finset_eq_empty_iff {s : set α} {h : finite s} : h.to_finset = ∅ ↔ s = ∅ := by simp [←finset.coe_inj] theorem finite.exists_finset {s : set α} : finite s → ∃ s' : finset α, ∀ a : α, a ∈ s' ↔ a ∈ s \| ⟨h⟩ := by exactI ⟨to_finset s, λ _, mem_to_finset⟩ theorem finite.exists_finset_coe {s : set α} (hs : finite s) : ∃ s' : finset α, ↑s' = s := ⟨hs.to_finset, hs.coe_to_finset⟩ /-- Finite sets can be lifted to finsets. -/ instance : can_lift (set α) (finset α) := { coe := coe, cond := finite, prf := λ s hs, hs.exists_finset_coe } theorem finite_mem_finset (s : finset α) : finite {a \| a ∈ s} := ⟨fintype.of_finset s (λ _, iff.rfl)⟩ theorem finite.of_fintype [fintype α] (s : set α) : finite s := by classical; exact ⟨set_fintype s⟩ theorem exists_finite_iff_finset {p : set α → Prop} : (∃ s, finite s ∧ p s) ↔ ∃ s : finset α, p ↑s := ⟨λ ⟨s, hs, hps⟩, ⟨hs.to_finset, hs.coe_to_finset.symm ▸ hps⟩, λ ⟨s, hs⟩, ⟨↑s, finite_mem_finset s, hs⟩⟩ lemma finite.fin_embedding {s : set α} (h : finite s) : ∃ (n : ℕ) (f : fin n ↪ α), range f = s := ⟨_, (fintype.equiv_fin (h.to_finset : set α)).symm.as_embedding, by simp⟩ lemma finite.fin_param {s : set α} (h : finite s) : ∃ (n : ℕ) (f : fin n → α), injective f ∧ range f = s := let ⟨n, f, hf⟩ := h.fin_embedding in ⟨n, f, f.injective, hf⟩ /-- Membership of a subset of a finite type is decidable. Using this as an instance leads to potential loops with `subtype.fintype` under certain decidability assumptions, so it should only be declared a local instance. -/ def decidable_mem_of_fintype [decidable_eq α] (s : set α) [fintype s] (a) : decidable (a ∈ s) := decidable_of_iff _ mem_to_finset instance fintype_empty : fintype (∅ : set α) := fintype.of_finset ∅ $ by simp theorem empty_card : fintype.card (∅ : set α) = 0 := rfl @[simp] theorem empty_card' {h : fintype.{u} (∅ : set α)} : @fintype.card (∅ : set α) h = 0 := eq.trans (by congr) empty_card @[simp] theorem finite_empty : @finite α ∅ := ⟨set.fintype_empty⟩ instance finite.inhabited : inhabited {s : set α // finite s} := ⟨⟨∅, finite_empty⟩⟩ /-- A `fintype` structure on `insert a s`. -/ def fintype_insert' {a : α} (s : set α) [fintype s] (h : a ∉ s) : fintype (insert a s : set α) := fintype.of_finset ⟨a ::ₘ s.to_finset.1, multiset.nodup_cons_of_nodup (by simp [h]) s.to_finset.2⟩ $ by simp theorem card_fintype_insert' {a : α} (s : set α) [fintype s] (h : a ∉ s) : @fintype.card _ (fintype_insert' s h) = fintype.card s + 1 := by rw [fintype_insert', fintype.card_of_finset]; simp [finset.card, to_finset]; refl @[simp] theorem card_insert {a : α} (s : set α) [fintype s] (h : a ∉ s) {d : fintype.{u} (insert a s : set α)} : @fintype.card _ d = fintype.card s + 1 := by rw ← card_fintype_insert' s h; congr lemma card_image_of_inj_on {s : set α} [fintype s] {f : α → β} [fintype (f '' s)] (H : ∀x∈s, ∀y∈s, f x = f y → x = y) : fintype.card (f '' s) = fintype.card s := by haveI := classical.prop_decidable; exact calc fintype.card (f '' s) = (s.to_finset.image f).card : fintype.card_of_finset' _ (by simp) ... = s.to_finset.card : finset.card_image_of_inj_on (λ x hx y hy hxy, H x (mem_to_finset.1 hx) y (mem_to_finset.1 hy) hxy) ... = fintype.card s : (fintype.card_of_finset' _ (λ a, mem_to_finset)).symm lemma card_image_of_injective (s : set α) [fintype s] {f : α → β} [fintype (f '' s)] (H : function.injective f) : fintype.card (f '' s) = fintype.card s := card_image_of_inj_on $ λ _ _ _ _ h, H h section local attribute [instance] decidable_mem_of_fintype instance fintype_insert [decidable_eq α] (a : α) (s : set α) [fintype s] : fintype (insert a s : set α) := if h : a ∈ s then by rwa [insert_eq, union_eq_self_of_subset_left (singleton_subset_iff.2 h)] else fintype_insert' _ h end @[simp] theorem finite.insert (a : α) {s : set α} : finite s → finite (insert a s) \| ⟨h⟩ := ⟨@set.fintype_insert _ (classical.dec_eq α) _ _ h⟩ lemma to_finset_insert [decidable_eq α] {a : α} {s : set α} (hs : finite s) : (hs.insert a).to_finset = insert a hs.to_finset := finset.ext $ by simp @[simp] lemma insert_to_finset [decidable_eq α] {a : α} {s : set α} [fintype s] : (insert a s).to_finset = insert a s.to_finset := by simp [finset.ext_iff, mem_insert_iff] @[elab_as_eliminator] theorem finite.induction_on {C : set α → Prop} {s : set α} (h : finite s) (H0 : C ∅) (H1 : ∀ {a s}, a ∉ s → finite s → C s → C (insert a s)) : C s := let ⟨t⟩ := h in by exactI match s.to_finset, @mem_to_finset _ s _ with \| ⟨l, nd⟩, al := begin change ∀ a, a ∈ l ↔ a ∈ s at al, clear _let_match _match t h, revert s nd al, refine multiset.induction_on l _ (λ a l IH, _); intros s nd al, { rw show s = ∅, from eq_empty_iff_forall_not_mem.2 (by simpa using al), exact H0 }, { rw ← show insert a {x \| x ∈ l} = s, from set.ext (by simpa using al), cases multiset.nodup_cons.1 nd with m nd', refine H1 _ ⟨finset.subtype.fintype ⟨l, nd'⟩⟩ (IH nd' (λ _, iff.rfl)), exact m } end end @[elab_as_eliminator] theorem finite.dinduction_on {C : ∀s:set α, finite s → Prop} {s : set α} (h : finite s) (H0 : C ∅ finite_empty) (H1 : ∀ {a s}, a ∉ s → ∀ h : finite s, C s h → C (insert a s) (h.insert a)) : C s h := have ∀ h : finite s, C s h, from finite.induction_on h (λ h, H0) (λ a s has hs ih h, H1 has hs (ih _)), this h instance fintype_singleton (a : α) : fintype ({a} : set α) := unique.fintype @[simp] theorem card_singleton (a : α) : fintype.card ({a} : set α) = 1 := fintype.card_of_subsingleton _ @[simp] theorem finite_singleton (a : α) : finite ({a} : set α) := ⟨set.fintype_singleton _⟩ lemma subsingleton.finite {s : set α} (h : s.subsingleton) : finite s := h.induction_on finite_empty finite_singleton lemma finite_is_top (α : Type) [partial_order α] : finite {x : α \| is_top x} := (subsingleton_is_top α).finite lemma finite_is_bot (α : Type) [partial_order α] : finite {x : α \| is_bot x} := (subsingleton_is_bot α).finite instance fintype_pure : ∀ a : α, fintype (pure a : set α) := set.fintype_singleton theorem finite_pure (a : α) : finite (pure a : set α) := ⟨set.fintype_pure a⟩ instance fintype_univ [fintype α] : fintype (@univ α) := fintype.of_equiv α $ (equiv.set.univ α).symm theorem finite_univ [fintype α] : finite (@univ α) := ⟨set.fintype_univ⟩ /-- If `(set.univ : set α)` is finite then `α` is a finite type. -/ noncomputable def fintype_of_univ_finite (H : (univ : set α).finite ) : fintype α := @fintype.of_equiv _ (univ : set α) H.fintype (equiv.set.univ _) lemma univ_finite_iff_nonempty_fintype : (univ : set α).finite ↔ nonempty (fintype α) := begin split, { intro h, exact ⟨fintype_of_univ_finite h⟩ }, { rintro ⟨_i⟩, exactI finite_univ } end theorem infinite_univ_iff : (@univ α).infinite ↔ _root_.infinite α := ⟨λ h₁, ⟨λ h₂, h₁ $ @finite_univ α h₂⟩, λ ⟨h₁⟩ h₂, h₁ (fintype_of_univ_finite h₂)⟩ theorem infinite_univ [h : _root_.infinite α] : infinite (@univ α) := infinite_univ_iff.2 h theorem infinite_coe_iff {s : set α} : _root_.infinite s ↔ infinite s := ⟨λ ⟨h₁⟩ h₂, h₁ h₂.fintype, λ h₁, ⟨λ h₂, h₁ ⟨h₂⟩⟩⟩ theorem infinite.to_subtype {s : set α} (h : infinite s) : _root_.infinite s := infinite_coe_iff.2 h /-- Embedding of `ℕ` into an infinite set. -/ noncomputable def infinite.nat_embedding (s : set α) (h : infinite s) : ℕ ↪ s := by { haveI := h.to_subtype, exact infinite.nat_embedding s } lemma infinite.exists_subset_card_eq {s : set α} (hs : infinite s) (n : ℕ) : ∃ t : finset α, ↑t ⊆ s ∧ t.card = n := ⟨((finset.range n).map (hs.nat_embedding _)).map (embedding.subtype _), by simp⟩ lemma infinite.nonempty {s : set α} (h : s.infinite) : s.nonempty := let a := infinite.nat_embedding s h 37 in ⟨a.1, a.2⟩ instance fintype_union [decidable_eq α] (s t : set α) [fintype s] [fintype t] : fintype (s ∪ t : set α) := fintype.of_finset (s.to_finset ∪ t.to_finset) $ by simp theorem finite.union {s t : set α} : finite s → finite t → finite (s ∪ t) \| ⟨hs⟩ ⟨ht⟩ := ⟨@set.fintype_union _ (classical.dec_eq α) _ _ hs ht⟩ lemma finite.sup {s t : set α} : finite s → finite t → finite (s ⊔ t) := finite.union lemma infinite_of_finite_compl [_root_.infinite α] {s : set α} (hs : sᶜ.finite) : s.infinite := λ h, set.infinite_univ (by simpa using hs.union h) lemma finite.infinite_compl [_root_.infinite α] {s : set α} (hs : s.finite) : sᶜ.infinite := λ h, set.infinite_univ (by simpa using hs.union h) instance fintype_sep (s : set α) (p : α → Prop) [fintype s] [decidable_pred p] : fintype ({a ∈ s \| p a} : set α) := fintype.of_finset (s.to_finset.filter p) $ by simp instance fintype_inter (s t : set α) [fintype s] [decidable_pred (∈ t)] : fintype (s ∩ t : set α) := set.fintype_sep s t /-- A `fintype` structure on a set defines a `fintype` structure on its subset. -/ def fintype_subset (s : set α) {t : set α} [fintype s] [decidable_pred (∈ t)] (h : t ⊆ s) : fintype t := by rw ← inter_eq_self_of_subset_right h; apply_instance theorem finite.subset {s : set α} : finite s → ∀ {t : set α}, t ⊆ s → finite t \| ⟨hs⟩ t h := ⟨@set.fintype_subset _ _ _ hs (classical.dec_pred t) h⟩ lemma finite.union_iff {s t : set α} : finite (s ∪ t) ↔ finite s ∧ finite t := ⟨λ h, ⟨h.subset (subset_union_left _ _), h.subset (subset_union_right _ _)⟩, λ ⟨hs, ht⟩, hs.union ht⟩ lemma finite.diff {s t u : set α} (hs : s.finite) (ht : t.finite) (h : u \ t ≤ s) : u.finite := begin refine finite.subset (ht.union hs) _, exact diff_subset_iff.mp h end theorem finite.inter_of_left {s : set α} (h : finite s) (t : set α) : finite (s ∩ t) := h.subset (inter_subset_left _ _) theorem finite.inter_of_right {s : set α} (h : finite s) (t : set α) : finite (t ∩ s) := h.subset (inter_subset_right _ _) theorem finite.inf_of_left {s : set α} (h : finite s) (t : set α) : finite (s ⊓ t) := h.inter_of_left t theorem finite.inf_of_right {s : set α} (h : finite s) (t : set α) : finite (t ⊓ s) := h.inter_of_right t protected theorem infinite.mono {s t : set α} (h : s ⊆ t) : infinite s → infinite t := mt (λ ht, ht.subset h) lemma infinite.diff {s t : set α} (hs : s.infinite) (ht : t.finite) : (s \ t).infinite := λ h, hs ((h.union ht).subset (s.subset_diff_union t)) instance fintype_image [decidable_eq β] (s : set α) (f : α → β) [fintype s] : fintype (f '' s) := fintype.of_finset (s.to_finset.image f) $ by simp instance fintype_range [decidable_eq α] (f : ι → α) [fintype (plift ι)] : fintype (range f) := fintype.of_finset (finset.univ.image $ f ∘ plift.down) $ by simp [(@equiv.plift ι).exists_congr_left] theorem finite_range (f : ι → α) [fintype (plift ι)] : finite (range f) := by haveI := classical.dec_eq α; exact ⟨by apply_instance⟩ theorem finite.image {s : set α} (f : α → β) : finite s → finite (f '' s) \| ⟨h⟩ := ⟨@set.fintype_image _ _ (classical.dec_eq β) _ _ h⟩ theorem infinite_of_infinite_image (f : α → β) {s : set α} (hs : (f '' s).infinite) : s.infinite := mt (finite.image f) hs lemma finite.dependent_image {s : set α} (hs : finite s) (F : Π i ∈ s, β) : finite {y : β \| ∃ x (hx : x ∈ s), y = F x hx} := begin letI : fintype s := hs.fintype, convert finite_range (λ x : s, F x x.2), simp only [set_coe.exists, subtype.coe_mk, eq_comm], end theorem finite.of_preimage {f : α → β} {s : set β} (h : finite (f ⁻¹' s)) (hf : surjective f) : finite s := hf.image_preimage s ▸ h.image _ instance fintype_map {α β} [decidable_eq β] : ∀ (s : set α) (f : α → β) [fintype s], fintype (f <$> s) := set.fintype_image theorem finite.map {α β} {s : set α} : ∀ (f : α → β), finite s → finite (f <$> s) := finite.image /-- If a function `f` has a partial inverse and sends a set `s` to a set with `[fintype]` instance, then `s` has a `fintype` structure as well. -/ def fintype_of_fintype_image (s : set α) {f : α → β} {g} (I : is_partial_inv f g) [fintype (f '' s)] : fintype s := fintype.of_finset ⟨_, @multiset.nodup_filter_map β α g _ (@injective_of_partial_inv_right _ _ f g I) (f '' s).to_finset.2⟩ $ λ a, begin suffices : (∃ b x, f x = b ∧ g b = some a ∧ x ∈ s) ↔ a ∈ s, by simpa [exists_and_distrib_left.symm, and.comm, and.left_comm, and.assoc], rw exists_swap, suffices : (∃ x, x ∈ s ∧ g (f x) = some a) ↔ a ∈ s, {simpa [and.comm, and.left_comm, and.assoc]}, simp [I _, (injective_of_partial_inv I).eq_iff] end theorem finite_of_finite_image {s : set α} {f : α → β} (hi : set.inj_on f s) : finite (f '' s) → finite s \| ⟨h⟩ := ⟨@fintype.of_injective _ _ h (λ a : s, ⟨f a.1, mem_image_of_mem f a.2⟩) $ λ a b eq, subtype.eq $ hi a.2 b.2 $ subtype.ext_iff_val.1 eq⟩ theorem finite_image_iff {s : set α} {f : α → β} (hi : inj_on f s) : finite (f '' s) ↔ finite s := ⟨finite_of_finite_image hi, finite.image _⟩ theorem infinite_image_iff {s : set α} {f : α → β} (hi : inj_on f s) : infinite (f '' s) ↔ infinite s := not_congr $ finite_image_iff hi theorem infinite_of_inj_on_maps_to {s : set α} {t : set β} {f : α → β} (hi : inj_on f s) (hm : maps_to f s t) (hs : infinite s) : infinite t := ((infinite_image_iff hi).2 hs).mono (maps_to'.mp hm) theorem infinite.exists_ne_map_eq_of_maps_to {s : set α} {t : set β} {f : α → β} (hs : infinite s) (hf : maps_to f s t) (ht : finite t) : ∃ (x ∈ s) (y ∈ s), x ≠ y ∧ f x = f y := begin contrapose! ht, exact infinite_of_inj_on_maps_to (λ x hx y hy, not_imp_not.1 (ht x hx y hy)) hf hs end theorem infinite.exists_lt_map_eq_of_maps_to [linear_order α] {s : set α} {t : set β} {f : α → β} (hs : infinite s) (hf : maps_to f s t) (ht : finite t) : ∃ (x ∈ s) (y ∈ s), x < y ∧ f x = f y := let ⟨x, hx, y, hy, hxy, hf⟩ := hs.exists_ne_map_eq_of_maps_to hf ht in hxy.lt_or_lt.elim (λ hxy, ⟨x, hx, y, hy, hxy, hf⟩) (λ hyx, ⟨y, hy, x, hx, hyx, hf.symm⟩) theorem infinite_range_of_injective [_root_.infinite α] {f : α → β} (hi : injective f) : infinite (range f) := by { rw [←image_univ, infinite_image_iff (inj_on_of_injective hi _)], exact infinite_univ } theorem infinite_of_injective_forall_mem [_root_.infinite α] {s : set β} {f : α → β} (hi : injective f) (hf : ∀ x : α, f x ∈ s) : infinite s := by { rw ←range_subset_iff at hf, exact (infinite_range_of_injective hi).mono hf } theorem finite.preimage {s : set β} {f : α → β} (I : set.inj_on f (f⁻¹' s)) (h : finite s) : finite (f ⁻¹' s) := finite_of_finite_image I (h.subset (image_preimage_subset f s)) theorem finite.preimage_embedding {s : set β} (f : α ↪ β) (h : s.finite) : (f ⁻¹' s).finite := finite.preimage (λ _ _ _ _ h', f.injective h') h lemma finite_option {s : set (option α)} : finite s ↔ finite {x : α \| some x ∈ s} := ⟨λ h, h.preimage_embedding embedding.some, λ h, ((h.image some).insert none).subset $ λ x, option.cases_on x (λ _, or.inl rfl) (λ x hx, or.inr $ mem_image_of_mem _ hx)⟩ instance fintype_Union [decidable_eq α] [fintype (plift ι)] (f : ι → set α) [∀ i, fintype (f i)] : fintype (⋃ i, f i) := fintype.of_finset (finset.univ.bUnion (λ i : plift ι, (f i.down).to_finset)) $ by simp theorem finite_Union [fintype (plift ι)] {f : ι → set α} (H : ∀i, finite (f i)) : finite (⋃ i, f i) := ⟨@set.fintype_Union _ _ (classical.dec_eq α) _ _ (λ i, finite.fintype (H i))⟩ /-- A union of sets with `fintype` structure over a set with `fintype` structure has a `fintype` structure. -/ def fintype_bUnion [decidable_eq α] {ι : Type} {s : set ι} [fintype s] (f : ι → set α) (H : ∀ i ∈ s, fintype (f i)) : fintype (⋃ i ∈ s, f i) := by rw bUnion_eq_Union; exact @set.fintype_Union _ _ _ _ _ (by rintro ⟨i, hi⟩; exact H i hi) instance fintype_bUnion' [decidable_eq α] {ι : Type} {s : set ι} [fintype s] (f : ι → set α) [H : ∀ i, fintype (f i)] : fintype (⋃ i ∈ s, f i) := fintype_bUnion _ (λ i _, H i) theorem finite.sUnion {s : set (set α)} (h : finite s) (H : ∀t∈s, finite t) : finite (⋃₀ s) := by rw sUnion_eq_Union; haveI := finite.fintype h; apply finite_Union; simpa using H theorem finite.bUnion {α} {ι : Type} {s : set ι} {f : Π i ∈ s, set α} : finite s → (∀ i ∈ s, finite (f i ‹_›)) → finite (⋃ i∈s, f i ‹_›) \| ⟨hs⟩ h := by rw [bUnion_eq_Union]; exactI finite_Union (λ i, h _ _) instance fintype_lt_nat (n : ℕ) : fintype {i \| i < n} := fintype.of_finset (finset.range n) $ by simp instance fintype_le_nat (n : ℕ) : fintype {i \| i ≤ n} := by simpa [nat.lt_succ_iff] using set.fintype_lt_nat (n+1) lemma finite_le_nat (n : ℕ) : finite {i \| i ≤ n} := ⟨set.fintype_le_nat _⟩ lemma finite_lt_nat (n : ℕ) : finite {i \| i < n} := ⟨set.fintype_lt_nat _⟩ lemma infinite.exists_nat_lt {s : set ℕ} (hs : infinite s) (n : ℕ) : ∃ m ∈ s, n < m := let ⟨m, hm⟩ := (hs.diff $ set.finite_le_nat n).nonempty in ⟨m, by simpa using hm⟩ instance fintype_prod (s : set α) (t : set β) [fintype s] [fintype t] : fintype (set.prod s t) := fintype.of_finset (s.to_finset.product t.to_finset) $ by simp lemma finite.prod {s : set α} {t : set β} : finite s → finite t → finite (set.prod s t) \| ⟨hs⟩ ⟨ht⟩ := by exactI ⟨set.fintype_prod s t⟩ /-- `image2 f s t` is finitype if `s` and `t` are. -/ instance fintype_image2 [decidable_eq γ] (f : α → β → γ) (s : set α) (t : set β) [hs : fintype s] [ht : fintype t] : fintype (image2 f s t : set γ) := by { rw ← image_prod, apply set.fintype_image } lemma finite.image2 (f : α → β → γ) {s : set α} {t : set β} (hs : finite s) (ht : finite t) : finite (image2 f s t) := by { rw ← image_prod, exact (hs.prod ht).image _ } /-- If `s : set α` is a set with `fintype` instance and `f : α → set β` is a function such that each `f a`, `a ∈ s`, has a `fintype` structure, then `s >>= f` has a `fintype` structure. -/ def fintype_bind {α β} [decidable_eq β] (s : set α) [fintype s] (f : α → set β) (H : ∀ a ∈ s, fintype (f a)) : fintype (s >>= f) := set.fintype_bUnion _ H instance fintype_bind' {α β} [decidable_eq β] (s : set α) [fintype s] (f : α → set β) [H : ∀ a, fintype (f a)] : fintype (s >>= f) := fintype_bind _ _ (λ i _, H i) theorem finite.bind {α β} {s : set α} {f : α → set β} (h : finite s) (hf : ∀ a ∈ s, finite (f a)) : finite (s >>= f) := h.bUnion hf instance fintype_seq [decidable_eq β] (f : set (α → β)) (s : set α) [fintype f] [fintype s] : fintype (f.seq s) := by { rw seq_def, apply set.fintype_bUnion' } instance fintype_seq' {α β : Type u} [decidable_eq β] (f : set (α → β)) (s : set α) [fintype f] [fintype s] : fintype (f <> s) := set.fintype_seq f s theorem finite.seq {f : set (α → β)} {s : set α} (hf : finite f) (hs : finite s) : finite (f.seq s) := by { rw seq_def, exact hf.bUnion (λ f _, hs.image _) } theorem finite.seq' {α β : Type u} {f : set (α → β)} {s : set α} (hf : finite f) (hs : finite s) : finite (f <> s) := hf.seq hs /-- There are finitely many subsets of a given finite set -/ lemma finite.finite_subsets {α : Type u} {a : set α} (h : finite a) : finite {b \| b ⊆ a} := ⟨fintype.of_finset ((finset.powerset h.to_finset).map finset.coe_emb.1) $ λ s, by simpa [← @exists_finite_iff_finset α (λ t, t ⊆ a ∧ t = s), subset_to_finset_iff, ← and.assoc] using h.subset⟩ lemma exists_min_image [linear_order β] (s : set α) (f : α → β) (h1 : finite s) : s.nonempty → ∃ a ∈ s, ∀ b ∈ s, f a ≤ f b \| ⟨x, hx⟩ := by simpa only [exists_prop, finite.mem_to_finset] using h1.to_finset.exists_min_image f ⟨x, h1.mem_to_finset.2 hx⟩ lemma exists_max_image [linear_order β] (s : set α) (f : α → β) (h1 : finite s) : s.nonempty → ∃ a ∈ s, ∀ b ∈ s, f b ≤ f a \| ⟨x, hx⟩ := by simpa only [exists_prop, finite.mem_to_finset] using h1.to_finset.exists_max_image f ⟨x, h1.mem_to_finset.2 hx⟩ theorem exists_lower_bound_image [hα : nonempty α] [linear_order β] (s : set α) (f : α → β) (h : s.finite) : ∃ (a : α), ∀ b ∈ s, f a ≤ f b := begin by_cases hs : set.nonempty s, { exact let ⟨x₀, H, hx₀⟩ := set.exists_min_image s f h hs in ⟨x₀, λ x hx, hx₀ x hx⟩ }, { exact nonempty.elim hα (λ a, ⟨a, λ x hx, absurd (set.nonempty_of_mem hx) hs⟩) } end theorem exists_upper_bound_image [hα : nonempty α] [linear_order β] (s : set α) (f : α → β) (h : s.finite) : ∃ (a : α), ∀ b ∈ s, f b ≤ f a := begin by_cases hs : set.nonempty s, { exact let ⟨x₀, H, hx₀⟩ := set.exists_max_image s f h hs in ⟨x₀, λ x hx, hx₀ x hx⟩ }, { exact nonempty.elim hα (λ a, ⟨a, λ x hx, absurd (set.nonempty_of_mem hx) hs⟩) } end end set namespace finset variables [decidable_eq β] variables {s : finset α} lemma finite_to_set (s : finset α) : set.finite (↑s : set α) := set.finite_mem_finset s @[simp] lemma coe_bUnion {f : α → finset β} : ↑(s.bUnion f) = (⋃x ∈ (↑s : set α), ↑(f x) : set β) := by simp [set.ext_iff] @[simp] lemma finite_to_set_to_finset {α : Type} (s : finset α) : (finite_to_set s).to_finset = s := by { ext, rw [set.finite.mem_to_finset, mem_coe] } end finset namespace set /-- Finite product of finite sets is finite -/ lemma finite.pi {δ : Type} [fintype δ] {κ : δ → Type} {t : Π d, set (κ d)} (ht : ∀ d, (t d).finite) : (pi univ t).finite := begin lift t to Π d, finset (κ d) using ht, classical, rw ← fintype.coe_pi_finset, exact (fintype.pi_finset t).finite_to_set, end /-- A finite union of finsets is finite. -/ lemma union_finset_finite_of_range_finite (f : α → finset β) (h : (range f).finite) : (⋃ a, (f a : set β)).finite := begin rw ← bUnion_range, exact h.bUnion (λ y hy, y.finite_to_set) end lemma eq_finite_Union_of_finite_subset_Union {ι} {s : ι → set α} {t : set α} (tfin : finite t) (h : t ⊆ ⋃ i, s i) : ∃ I : set ι, (finite I) ∧ ∃ σ : {i \| i ∈ I} → set α, (∀ i, finite (σ i)) ∧ (∀ i, σ i ⊆ s i) ∧ t = ⋃ i, σ i := let ⟨I, Ifin, hI⟩ := finite_subset_Union tfin h in ⟨I, Ifin, λ x, s x ∩ t, λ i, tfin.subset (inter_subset_right _ _), λ i, inter_subset_left _ _, begin ext x, rw mem_Union, split, { intro x_in, rcases mem_Union.mp (hI x_in) with ⟨i, _, ⟨hi, rfl⟩, H⟩, use [i, hi, H, x_in] }, { rintros ⟨i, hi, H⟩, exact H } end⟩ /-- An increasing union distributes over finite intersection. -/ lemma Union_Inter_of_monotone {ι ι' α : Type} [fintype ι] [linear_order ι'] [nonempty ι'] {s : ι → ι' → set α} (hs : ∀ i, monotone (s i)) : (⋃ j : ι', ⋂ i : ι, s i j) = ⋂ i : ι, ⋃ j : ι', s i j := begin ext x, refine ⟨λ hx, Union_Inter_subset hx, λ hx, _⟩, simp only [mem_Inter, mem_Union, mem_Inter] at hx ⊢, choose j hj using hx, obtain ⟨j₀⟩ := show nonempty ι', by apply_instance, refine ⟨finset.univ.fold max j₀ j, λ i, hs i _ (hj i)⟩, rw [finset.fold_op_rel_iff_or (@le_max_iff _ _)], exact or.inr ⟨i, finset.mem_univ i, le_rfl⟩ end instance nat.fintype_Iio (n : ℕ) : fintype (Iio n) := fintype.of_finset (finset.range n) $ by simp /-- If `P` is some relation between terms of `γ` and sets in `γ`, such that every finite set `t : set γ` has some `c : γ` related to it, then there is a recursively defined sequence `u` in `γ` so `u n` is related to the image of `{0, 1, ..., n-1}` under `u`. (We use this later to show sequentially compact sets are totally bounded.) -/ lemma seq_of_forall_finite_exists {γ : Type} {P : γ → set γ → Prop} (h : ∀ t, finite t → ∃ c, P c t) : ∃ u : ℕ → γ, ∀ n, P (u n) (u '' Iio n) := ⟨λ n, @nat.strong_rec_on' (λ _, γ) n $ λ n ih, classical.some $ h (range $ λ m : Iio n, ih m.1 m.2) (finite_range _), λ n, begin classical, refine nat.strong_rec_on' n (λ n ih, _), rw nat.strong_rec_on_beta', convert classical.some_spec (h _ _), ext x, split, { rintros ⟨m, hmn, rfl⟩, exact ⟨⟨m, hmn⟩, rfl⟩ }, { rintros ⟨⟨m, hmn⟩, rfl⟩, exact ⟨m, hmn, rfl⟩ } end⟩ lemma finite_range_ite {p : α → Prop} [decidable_pred p] {f g : α → β} (hf : finite (range f)) (hg : finite (range g)) : finite (range (λ x, if p x then f x else g x)) := (hf.union hg).subset range_ite_subset lemma finite_range_const {c : β} : finite (range (λ x : α, c)) := (finite_singleton c).subset range_const_subset lemma range_find_greatest_subset {P : α → ℕ → Prop} [∀ x, decidable_pred (P x)] {b : ℕ}: range (λ x, nat.find_greatest (P x) b) ⊆ ↑(finset.range (b + 1)) := by { rw range_subset_iff, intro x, simp [nat.lt_succ_iff, nat.find_greatest_le] } lemma finite_range_find_greatest {P : α → ℕ → Prop} [∀ x, decidable_pred (P x)] {b : ℕ} : finite (range (λ x, nat.find_greatest (P x) b)) := (finset.range (b + 1)).finite_to_set.subset range_find_greatest_subset lemma card_lt_card {s t : set α} [fintype s] [fintype t] (h : s ⊂ t) : fintype.card s < fintype.card t := fintype.card_lt_of_injective_not_surjective (set.inclusion h.1) (set.inclusion_injective h.1) $ λ hst, (ssubset_iff_subset_ne.1 h).2 (eq_of_inclusion_surjective hst) lemma card_le_of_subset {s t : set α} [fintype s] [fintype t] (hsub : s ⊆ t) : fintype.card s ≤ fintype.card t := fintype.card_le_of_injective (set.inclusion hsub) (set.inclusion_injective hsub) lemma eq_of_subset_of_card_le {s t : set α} [fintype s] [fintype t] (hsub : s ⊆ t) (hcard : fintype.card t ≤ fintype.card s) : s = t := (eq_or_ssubset_of_subset hsub).elim id (λ h, absurd hcard $ not_le_of_lt $ card_lt_card h) lemma subset_iff_to_finset_subset (s t : set α) [fintype s] [fintype t] : s ⊆ t ↔ s.to_finset ⊆ t.to_finset := by simp @[simp, mono] lemma finite.to_finset_mono {s t : set α} {hs : finite s} {ht : finite t} : hs.to_finset ⊆ ht.to_finset ↔ s ⊆ t := begin split, { intros h x, rw [←finite.mem_to_finset hs, ←finite.mem_to_finset ht], exact λ hx, h hx }, { intros h x, rw [finite.mem_to_finset hs, finite.mem_to_finset ht], exact λ hx, h hx } end @[simp, mono] lemma finite.to_finset_strict_mono {s t : set α} {hs : finite s} {ht : finite t} : hs.to_finset ⊂ ht.to_finset ↔ s ⊂ t := begin rw [←lt_eq_ssubset, ←finset.lt_iff_ssubset, lt_iff_le_and_ne, lt_iff_le_and_ne], simp end lemma card_range_of_injective [fintype α] {f : α → β} (hf : injective f) [fintype (range f)] : fintype.card (range f) = fintype.card α := eq.symm $ fintype.card_congr $ equiv.of_injective f hf lemma finite.exists_maximal_wrt [partial_order β] (f : α → β) (s : set α) (h : set.finite s) : s.nonempty → ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a' := begin classical, refine h.induction_on _ _, { exact λ h, absurd h empty_not_nonempty }, intros a s his _ ih _, cases s.eq_empty_or_nonempty with h h, { use a, simp [h] }, rcases ih h with ⟨b, hb, ih⟩, by_cases f b ≤ f a, { refine ⟨a, set.mem_insert _ _, λ c hc hac, le_antisymm hac _⟩, rcases set.mem_insert_iff.1 hc with rfl \| hcs, { refl }, { rwa [← ih c hcs (le_trans h hac)] } }, { refine ⟨b, set.mem_insert_of_mem _ hb, λ c hc hbc, _⟩, rcases set.mem_insert_iff.1 hc with rfl \| hcs, { exact (h hbc).elim }, { exact ih c hcs hbc } } end lemma finite.card_to_finset {s : set α} [fintype s] (h : s.finite) : h.to_finset.card = fintype.card s := by { rw [← finset.card_attach, finset.attach_eq_univ, ← fintype.card], congr' 2, funext, rw set.finite.mem_to_finset } lemma infinite.exists_not_mem_finset {s : set α} (hs : s.infinite) (f : finset α) : ∃ a ∈ s, a ∉ f := let ⟨a, has, haf⟩ := (hs.diff f.finite_to_set).nonempty in ⟨a, has, λ h, haf $ finset.mem_coe.1 h⟩ section decidable_eq lemma to_finset_compl {α : Type} [fintype α] [decidable_eq α] (s : set α) [fintype (sᶜ : set α)] [fintype s] : sᶜ.to_finset = (s.to_finset)ᶜ := by ext; simp lemma to_finset_inter {α : Type} [decidable_eq α] (s t : set α) [fintype (s ∩ t : set α)] [fintype s] [fintype t] : (s ∩ t).to_finset = s.to_finset ∩ t.to_finset := by ext; simp lemma to_finset_union {α : Type} [decidable_eq α] (s t : set α) [fintype (s ∪ t : set α)] [fintype s] [fintype t] : (s ∪ t).to_finset = s.to_finset ∪ t.to_finset := by ext; simp lemma to_finset_ne_eq_erase {α : Type} [decidable_eq α] [fintype α] (a : α) [fintype {x : α \| x ≠ a}] : {x : α \| x ≠ a}.to_finset = finset.univ.erase a := by ext; simp lemma card_ne_eq [fintype α] (a : α) [fintype {x : α \| x ≠ a}] : fintype.card {x : α \| x ≠ a} = fintype.card α - 1 := begin haveI := classical.dec_eq α, rw [←to_finset_card, to_finset_ne_eq_erase, finset.card_erase_of_mem (finset.mem_univ _), finset.card_univ, nat.pred_eq_sub_one], end end decidable_eq section variables [semilattice_sup α] [nonempty α] {s : set α} /--A finite set is bounded above.-/ protected lemma finite.bdd_above (hs : finite s) : bdd_above s := finite.induction_on hs bdd_above_empty $ λ a s _ _ h, h.insert a /--A finite union of sets which are all bounded above is still bounded above.-/ lemma finite.bdd_above_bUnion {I : set β} {S : β → set α} (H : finite I) : (bdd_above (⋃i∈I, S i)) ↔ (∀i ∈ I, bdd_above (S i)) := finite.induction_on H (by simp only [bUnion_empty, bdd_above_empty, ball_empty_iff]) (λ a s ha _ hs, by simp only [bUnion_insert, ball_insert_iff, bdd_above_union, hs]) end section variables [semilattice_inf α] [nonempty α] {s : set α} /--A finite set is bounded below.-/ protected lemma finite.bdd_below (hs : finite s) : bdd_below s := @finite.bdd_above (order_dual α) _ _ _ hs /--A finite union of sets which are all bounded below is still bounded below.-/ lemma finite.bdd_below_bUnion {I : set β} {S : β → set α} (H : finite I) : (bdd_below (⋃i∈I, S i)) ↔ (∀i ∈ I, bdd_below (S i)) := @finite.bdd_above_bUnion (order_dual α) _ _ _ _ _ H end end set namespace finset /-- A finset is bounded above. -/ protected lemma bdd_above [semilattice_sup α] [nonempty α] (s : finset α) : bdd_above (↑s : set α) := s.finite_to_set.bdd_above /-- A finset is bounded below. -/ protected lemma bdd_below [semilattice_inf α] [nonempty α] (s : finset α) : bdd_below (↑s : set α) := s.finite_to_set.bdd_below end finset namespace fintype variables [fintype α] {p q : α → Prop} [decidable_pred p] [decidable_pred q] @[simp] lemma card_subtype_compl : fintype.card {x // ¬ p x} = fintype.card α - fintype.card {x // p x} := begin classical, rw [fintype.card_of_subtype (set.to_finset pᶜ), set.to_finset_compl p, finset.card_compl, fintype.card_of_subtype (set.to_finset p)]; intros; simp; refl end /-- If two subtypes of a fintype have equal cardinality, so do their complements. -/ lemma card_compl_eq_card_compl (h : fintype.card {x // p x} = fintype.card {x // q x}) : fintype.card {x // ¬ p x} = fintype.card {x // ¬ q x} := by simp only [card_subtype_compl, h] end fintype /-- If a set `s` does not contain any elements between any pair of elements `x, z ∈ s` with `x ≤ z` (i.e if given `x, y, z ∈ s` such that `x ≤ y ≤ z`, then `y` is either `x` or `z`), then `s` is finite. -/ lemma set.finite_of_forall_between_eq_endpoints {α : Type} [linear_order α] (s : set α) (h : ∀ (x ∈ s) (y ∈ s) (z ∈ s), x ≤ y → y ≤ z → x = y ∨ y = z) : set.finite s := begin by_contra hinf, change s.infinite at hinf, rcases hinf.exists_subset_card_eq 3 with ⟨t, hts, ht⟩, let f := t.order_iso_of_fin ht, let x := f 0, let y := f 1, let z := f 2, have := h x (hts x.2) y (hts y.2) z (hts z.2) (f.monotone $ by dec_trivial) (f.monotone $ by dec_trivial), have key₁ : (0 : fin 3) ≠ 1 := by dec_trivial, have key₂ : (1 : fin 3) ≠ 2 := by dec_trivial, cases this, { dsimp only [x, y] at this, exact key₁ (f.injective $ subtype.coe_injective this) }, { dsimp only [y, z] at this, exact key₂ (f.injective $ subtype.coe_injective this) } end
[GOAL] I : Type u f : I → Type v x y : (i : I) → f i i : I g : I → Type u_1 r : (i : I) → Semiring (f i) m : (i : I) → AddCommMonoid (g i) inst✝ : (i : I) → Module (f i) (g i) ⊢ ∀ (r_1 s : (i : I) → f i) (x : (i : I) → g i), (r_1 + s) • x = r_1 • x + s • x [PROOFSTEP] intros [GOAL] I : Type u f : I → Type v x y : (i : I) → f i i : I g : I → Type u_1 r : (i : I) → Semiring (f i) m : (i : I) → AddCommMonoid (g i) inst✝ : (i : I) → Module (f i) (g i) r✝ s✝ : (i : I) → f i x✝ : (i : I) → g i ⊢ (r✝ + s✝) • x✝ = r✝ • x✝ + s✝ • x✝ [PROOFSTEP] ext1 [GOAL] case h I : Type u f : I → Type v x y : (i : I) → f i i : I g : I → Type u_1 r : (i : I) → Semiring (f i) m : (i : I) → AddCommMonoid (g i) inst✝ : (i : I) → Module (f i) (g i) r✝ s✝ : (i : I) → f i x✝¹ : (i : I) → g i x✝ : I ⊢ ((r✝ + s✝) • x✝¹) x✝ = (r✝ • x✝¹ + s✝ • x✝¹) x✝ [PROOFSTEP] apply add_smul [GOAL] I : Type u f : I → Type v x y : (i : I) → f i i : I g : I → Type u_1 r : (i : I) → Semiring (f i) m : (i : I) → AddCommMonoid (g i) inst✝ : (i : I) → Module (f i) (g i) ⊢ ∀ (x : (i : I) → g i), 0 • x = 0 [PROOFSTEP] intros [GOAL] I : Type u f : I → Type v x y : (i : I) → f i i : I g : I → Type u_1 r : (i : I) → Semiring (f i) m : (i : I) → AddCommMonoid (g i) inst✝ : (i : I) → Module (f i) (g i) x✝ : (i : I) → g i ⊢ 0 • x✝ = 0 [PROOFSTEP] ext1 -- Porting note: not sure why `apply zero_smul` fails here. [GOAL] case h I : Type u f : I → Type v x y : (i : I) → f i i : I g : I → Type u_1 r : (i : I) → Semiring (f i) m : (i : I) → AddCommMonoid (g i) inst✝ : (i : I) → Module (f i) (g i) x✝¹ : (i : I) → g i x✝ : I ⊢ (0 • x✝¹) x✝ = OfNat.ofNat 0 x✝ [PROOFSTEP] rw [zero_smul]
subroutine rpn2(ixy,maxm,meqn,mwaves,maux,mbc,mx,ql,qr,auxl,auxr,wave,s,amdq,apdq) ! Riemann solver for the elasticity equations in 2d, with varying ! material properties rho, lambda, and mu, in a mapped grid ! ! This Riemann solver is for mapped grids. It implements the Riemann solver ! in the normal direction (it doesn't do a rotation of coordinates) ! ! Note that although there are 5 eigenvectors, one eigenvalue ! is always zero and so we only need to compute 4 waves. ! ! solve Riemann problems along one slice of data. ! ! On input, ql contains the state vector at the left edge of each cell ! qr contains the state vector at the right edge of each cell ! ! Note that the i'th Riemann problem has left state qr(:,i-1) ! and right state ql(:,i) ! From the basic clawpack routines, this routine is called with ql = qr ! ! This data is along a slice in the x-direction if ixy=1 ! or the y-direction if ixy=2. ! ! Contents of ql and qr: ! ! q(1,:) = sigma^{11} if ixy=1 or sigma^{22} if ixy=2 ! q(2,:) = sigma^{22} if ixy=1 or sigma^{11} if ixy=2 ! q(3,:) = sigma^{12} = sigma^{21} ! q(4,:) = u if ixy=1 or v if ixy=2 ! q(5,:) = v if ixy=1 or u if ixy=2 ! ! auxl and auxr hold corresponding slice of the aux array: ! Here it is assumed that auxl=auxr gives the cell values ! for this slice. ! ! auxl(1,i) = rho, density ! auxl(2,i) = lambda ! auxl(3,i) = mu ! auxl(4,i) = cp, P-wave speed ! auxl(5,i) = cs, S-wave speed ! ! ! On output, wave contains the waves, ! s the speeds, ! amdq the left-going flux difference A^- \Delta q ! apdq the right-going flux difference A^+ \Delta q ! ! Note that the waves are not in order of increasing lambda. ! Instead the 1- and 2-waves are the P-waves and the 3- and 4-waves ! are the S-waves. (The 5th wave has speed zero and is not used.) implicit none integer, intent(in) :: ixy, maxm, meqn, mwaves, mbc, mx, maux double precision, intent(in) :: ql, qr, auxl, auxr double precision, intent(out) :: wave, s, amdq, apdq dimension wave( meqn, mwaves, 1-mbc:maxm+mbc) dimension s(mwaves, 1-mbc:maxm+mbc) dimension ql(meqn, 1-mbc:maxm+mbc) dimension qr(meqn, 1-mbc:maxm+mbc) dimension apdq(meqn, 1-mbc:maxm+mbc) dimension amdq(meqn, 1-mbc:maxm+mbc) dimension auxl(maux, 1-mbc:maxm+mbc) dimension auxr(maux, 1-mbc:maxm+mbc) integer :: ksig11, ksig22, ku, kv, i, m double precision :: dsig11, dsig22, dsig12, du, dv double precision :: alamr, amur, bulkr, cpr, csr double precision :: alaml, amul, bulkl, cpl, csl double precision :: det, a1, a2, a3, a4 ! Variables required for mapped grid version integer :: map, mw double precision :: nx, ny, nx2, ny2, nxy ! set ku to point to the component of the system that corresponds ! to velocity in the direction of this slice, kv to the orthogonal ! velocity. Similarly ksig11 and ksig22 point to normal stresses. ! 3rd component is always shear stress sig12. if (ixy.eq.1) then ksig11 = 1 ksig22 = 2 ku = 4 kv = 5 map = 6 else ksig11 = 2 ksig22 = 1 ku = 5 kv = 4 map = 9 endif ! note that notation for u and v reflects assumption that the ! Riemann problems are in the x-direction with u in the normal ! direciton and v in the orthogonal direcion, but with the above ! definitions of ku and kv the routine also works with ixy=2 ! split the jump in q at each interface into waves ! The jump is split into leftgoing waves traveling at speeds -cp, -cs ! relative to the material properties to the left of the interface, ! and rightgoing waves traveling at speeds +cp, +cs ! relative to the material properties to the right of the interface, do i = 2-mbc, mx+mbc !Define direction of normal to grid edge normals nx = auxl(map,i) ny = auxl(map+1,i) nx2 = nxnx ny2 = nyny nxy = nxny dsig11 = ql(1,i) - qr(1,i-1) dsig22 = ql(2,i) - qr(2,i-1) dsig12 = ql(3,i) - qr(3,i-1) du = ql(4,i) - qr(4,i-1) dv = ql(5,i) - qr(5,i-1) ! material properties in cells i (on right) and i-1 (on left): alamr = auxl(2,i) amur = auxl(3,i) bulkr = alamr + 2.d0amur cpr = auxl(4,i) csr = auxl(5,i) alaml = auxr(2,i-1) amul = auxr(3,i-1) bulkl = alaml + 2.d0amul cpl = auxr(4,i-1) csl = auxr(5,i-1) ! P-wave strengths: det = bulklcpr + bulkrcpl if (det.eq.0.d0) then write(6,) 'det=0 in rpn2' stop endif a1 = (cpr(dsig11nx2 + dsig22ny2 + 2nxydsig12) + bulkr(nxdu + nydv)) / det a2 = (cpl(dsig11nx2 + dsig22ny2 + 2nxydsig12) - bulkl(nxdu + nydv)) / det ! S-wave strengths: det = amulcsr + amurcsl if (det.eq.0.d0) then ! no s-waves a3 = 0.d0 a4 = 0.d0 else a3 = (csr(dsig12(nx2 - ny2) + nxy(dsig22 - dsig11)) + amur(nxdv - nydu)) / det a4 = (csl(dsig12(nx2 - ny2) + nxy(dsig22 - dsig11)) + amul(nydu - nxdv)) / det endif ! 5th wave has velocity 0 so is not computed or propagated. ! Compute the waves. wave(:,1,i) = 0.d0 wave(1,1,i) = a1 * (alaml + 2amulnx2) wave(2,1,i) = a1 * (alaml + 2amulny2) wave(3,1,i) = a1 * (2amulnxy) wave(4,1,i) = a1 * cpl * nx wave(5,1,i) = a1 * cpl * ny s(1,i) = -cpl wave(:,2,i) = 0.d0 wave(1,2,i) = a2 * (alamr + 2amurnx2) wave(2,2,i) = a2 * (alamr + 2amurny2) wave(3,2,i) = a2 * (2amurnxy) wave(4,2,i) = - a2 * cpr * nx wave(5,2,i) = - a2 * cpr * ny s(2,i) = cpr wave(:,3,i) = 0.d0 wave(1,3,i) = - a3 * (2nxyamul) wave(2,3,i) = a3 * (2nxyamul) wave(3,3,i) = a3 * amul(nx2 - ny2) wave(4,3,i) = - a3 csl * ny wave(5,3,i) = a3 * csl * nx s(3,i) = -csl wave(:,4,i) = 0.d0 wave(1,4,i) = - a4 * (2nxyamur) wave(2,4,i) = a4 * (2nxyamur) wave(3,4,i) = a4 * amur(nx2 - ny2) wave(4,4,i) = a4 csr * ny wave(5,4,i) = -a4 * csr * nx s(4,i) = csr ! Scales speed by relative length of edge of mapped grid do mw=1,mwaves s(mw,i) = s(mw,i)auxl(map+2,i) end do ! compute the leftgoing and rightgoing flux differences: ! Note s(i,1),s(i,3) < 0 and s(i,2),s(i,4) > 0. do m=1,meqn amdq(m,i) = s(1,i)wave(m,1,i) + s(3,i)wave(m,3,i) apdq(m,i) = s(2,i)wave(m,2,i) + s(4,i)*wave(m,4,i) enddo enddo return end subroutine rpn2
lemma open_ball: "open {y. dist x y < d}"
Require Import Coq.Logic.Classical_Prop. Require Import Logic.lib.Ensembles_ext. Require Import Logic.lib.Bijection. Require Import Logic.lib.Countable. Require Import Logic.GeneralLogic.Base. Require Import Logic.GeneralLogic.KripkeModel. Require Import Logic.GeneralLogic.ProofTheory.BasicSequentCalculus. Require Import Logic.GeneralLogic.Semantics.Kripke. Require Import Logic.GeneralLogic.Complete.ContextProperty. Require Import Logic.GeneralLogic.Complete.ContextProperty_Kripke. Require Import Logic.GeneralLogic.Complete.Lindenbaum. Require Import Logic.GeneralLogic.Complete.Lindenbaum_Kripke. Require Import Logic.GeneralLogic.Complete.Canonical_Kripke. Require Import Logic.GeneralLogic.Complete.Complete_Kripke. Require Import Logic.MinimunLogic.Syntax. Require Import Logic.MinimunLogic.ProofTheory.Minimun. Require Import Logic.MinimunLogic.Semantics.Kripke. Require Import Logic.MinimunLogic.Complete.ContextProperty_Kripke. Require Import Logic.MinimunLogic.Complete.Lindenbaum_Kripke. Require Import Logic.MinimunLogic.Complete.Truth_Kripke. Require Import Logic.PropositionalLogic.Syntax. Require Import Logic.PropositionalLogic.ProofTheory.Intuitionistic. Require Import Logic.PropositionalLogic.ProofTheory.DeMorgan. Require Import Logic.PropositionalLogic.ProofTheory.GodelDummett. Require Import Logic.PropositionalLogic.ProofTheory.Classical. Require Import Logic.PropositionalLogic.Semantics.Kripke. Require Import Logic.PropositionalLogic.Complete.ContextProperty_Kripke. Require Import Logic.PropositionalLogic.Complete.Lindenbaum_Kripke. Require Import Logic.PropositionalLogic.Complete.Truth_Kripke. Require Import Logic.PropositionalLogic.Complete.Canonical_Kripke. Require Import Logic.SeparationLogic.Syntax. Require Import Logic.SeparationLogic.ProofTheory.SeparationLogic. Require Import Logic.SeparationLogic.ProofTheory.RewriteClass. Require Import Logic.SeparationLogic.ProofTheory.DerivedRules. Require Import Logic.SeparationLogic.Model.SeparationAlgebra. Require Import Logic.SeparationLogic.Model.OrderedSA. Require Import Logic.SeparationLogic.Semantics.FlatSemantics. Require Import Logic.SeparationLogic.Complete.ContextProperty_Flat. Require Import Logic.SeparationLogic.Complete.Lindenbaum_Flat. Require Import Logic.SeparationLogic.Complete.Truth_Flat. Require Import Logic.SeparationLogic.Complete.Canonical_Flat. Require Import Logic.SeparationLogic.DeepEmbedded.Parameter. Require Logic.SeparationLogic.DeepEmbedded.SeparationEmpLanguage. Require Logic.SeparationLogic.DeepEmbedded.FlatSemantics. Require Logic.SeparationLogic.DeepEmbedded.ParametricSeparationLogic. Local Open Scope logic_base. Local Open Scope syntax. Local Open Scope kripke_model. Local Open Scope kripke_model_class. Import PropositionalLanguageNotation. Import SeparationLogicNotation. Import KripkeModelFamilyNotation. Import KripkeModelNotation_Intuitionistic. Import KripkeModelClass. Section Complete. Context {Sigma: SeparationEmpLanguage.PropositionalVariables} {CV: Countable SeparationEmpLanguage.Var} (SLP: SL_Parameter). Existing Instances SeparationEmpLanguage.L SeparationEmpLanguage.minL SeparationEmpLanguage.pL SeparationEmpLanguage.sL SeparationEmpLanguage.s'L. Existing Instances ParametricSeparationLogic.G ParametricSeparationLogic.AX ParametricSeparationLogic.minAX ParametricSeparationLogic.ipG ParametricSeparationLogic.sG ParametricSeparationLogic.eG ParametricSeparationLogic.ParG. Existing Instances Axiomatization2SequentCalculus_SC Axiomatization2SequentCalculus_bSC Axiomatization2SequentCalculus_fwSC Axiomatization2SequentCalculus_minSC Axiomatization2SequentCalculus_ipSC Axiomatization2SequentCalculus_cpSC. Existing Instances FlatSemantics.MD FlatSemantics.kMD FlatSemantics.R FlatSemantics.J FlatSemantics.SM FlatSemantics.kminSM FlatSemantics.kpSM FlatSemantics.fsSM FlatSemantics.feSM. Definition cP : context -> Prop := Intersection _ (Intersection _ derivable_closed orp_witnessed) consistent. Lemma AL_DC: at_least derivable_closed cP. Proof. solve_at_least. Qed. Lemma AL_OW: at_least orp_witnessed cP. Proof. solve_at_least. Qed. Lemma AL_CONSI: at_least consistent cP. Proof. solve_at_least. Qed. Lemma LIN_CD: forall x: expr, Lindenbaum_constructable (cannot_derive x) cP. Proof. intros. apply Lindenbaum_constructable_suffice; auto. + apply SeparationEmpLanguage.formula_countable; auto. + apply Lindenbaum_preserves_cannot_derive. + unfold cP. repeat apply Lindenbaum_ensures_by_conjunct. - apply Lindenbaum_cannot_derive_ensures_derivable_closed. - apply Lindenbaum_cannot_derive_ensures_orp_witnessed. - apply Lindenbaum_cannot_derive_ensures_consistent. Qed. Lemma LIN_SL: forall (Phi: context) (Psi: sig cP), Lindenbaum_constructable (context_sepcon_included_l Phi (proj1_sig Psi)) cP. Proof. intros. apply Lindenbaum_constructable_suffice; auto. + apply SeparationEmpLanguage.formula_countable; auto. + apply Lindenbaum_preserves_context_sepcon_included_l. + unfold cP. repeat apply Lindenbaum_ensures_by_conjunct. - apply Lindenbaum_context_sepcon_included_l_ensures_derivable_closed. - apply Lindenbaum_context_sepcon_included_l_ensures_orp_witnessed. * apply AL_DC, (proj2_sig Psi). * apply AL_OW, (proj2_sig Psi). - apply Lindenbaum_context_sepcon_included_l_ensures_consistent. apply AL_CONSI, (proj2_sig Psi). Qed. Lemma LIN_SR: forall (Phi: context) (Psi: sig cP), Lindenbaum_constructable (context_sepcon_included_r Phi (proj1_sig Psi)) cP. Proof. intros. eapply Lindenbaum_constructable_Same_set. + symmetry. apply context_sepcon_included_equiv. apply AL_DC, (proj2_sig Psi). + apply LIN_SL. Qed. Definition canonical_frame: FlatSemantics.frame := FlatSemantics.Build_frame (sig cP) (fun a b => Included _ (proj1_sig a) (proj1_sig b)) (fun a b c => Included _ (context_sepcon (proj1_sig a) (proj1_sig b)) (proj1_sig c)). Definition canonical_eval: SeparationEmpLanguage.Var -> FlatSemantics.sem canonical_frame := fun p a => proj1_sig a (SeparationEmpLanguage.varp p). Definition canonical_Kmodel: @Kmodel FlatSemantics.MD FlatSemantics.kMD := FlatSemantics.Build_Kmodel canonical_frame canonical_eval. Definition rel: bijection (Kworlds canonical_Kmodel) (sig cP) := bijection_refl. Definition H_R: forall m n Phi Psi, rel m Phi -> rel n Psi -> (m <= n <-> Included _ (proj1_sig Phi) (proj1_sig Psi)). Proof. intros. change (m = Phi) in H. change (n = Psi) in H0. subst; reflexivity. Qed. Definition H_J: forall m1 m2 m Phi1 Phi2 Phi, rel m1 Phi1 -> rel m2 Phi2 -> rel m Phi -> (join m1 m2 m <-> Included _ (context_sepcon (proj1_sig Phi1) (proj1_sig Phi2)) (proj1_sig Phi)). Proof. intros. change (m = Phi) in H1. change (m1 = Phi1) in H. change (m2 = Phi2) in H0. subst; reflexivity. Qed. Lemma TRUTH: forall x: expr, forall m Phi, rel m Phi -> (KRIPKE: canonical_Kmodel, m \|= x <-> proj1_sig Phi x). Proof. induction x. + exact (truth_lemma_andp cP rel AL_DC x1 x2 IHx1 IHx2). + exact (truth_lemma_orp cP rel AL_DC AL_OW x1 x2 IHx1 IHx2). + exact (truth_lemma_impp cP rel H_R AL_DC LIN_CD x1 x2 IHx1 IHx2). + exact (truth_lemma_sepcon cP rel H_J AL_DC LIN_SL LIN_SR x1 x2 IHx1 IHx2). + exact (truth_lemma_wand cP rel H_J AL_DC LIN_CD LIN_SR x1 x2 IHx1 IHx2). + exact (truth_lemma_emp cP rel H_R H_J AL_DC LIN_CD LIN_SR). + exact (truth_lemma_falsep cP rel AL_CONSI). + intros; change (m = Phi) in H; subst; reflexivity. Qed. Context (SAP: SA_Parameter). Hypothesis PC: Parameter_coincide SLP SAP. Theorem ParametricCompleteness: strongly_complete (ParametricSeparationLogic.G SLP) FlatSemantics.SM (KripkeModelClass _ (FlatSemantics.Kmodel_Monotonic + FlatSemantics.Kmodel_PreOrder + FlatSemantics.Kmodel_SeparationAlgebra + FlatSemantics.Kmodel_UpwardsClosed + FlatSemantics.Kmodel_DownwardsClosed + FlatSemantics.Kmodel_Unital + FlatSemantics.Parametric_Kmodel_Class SAP)). Proof. apply (@general_completeness _ _ _ _ _ _ _ _ cP rel LIN_CD TRUTH). split; [split; [split; [split; [split; [split \|] \|] \|] \|] \|]. + hnf; intros. exact (denote_monotonic cP rel H_R (SeparationEmpLanguage.varp v) (TRUTH (SeparationEmpLanguage.varp v))). + exact (po_R cP rel H_R). + exact (SA cP rel H_J AL_DC LIN_SR). + exact (uSA cP rel H_R H_J AL_DC). + exact (dSA cP rel H_R H_J AL_DC). + exact (unitSA cP rel H_R H_J AL_DC LIN_SR TRUTH). + inversion PC. constructor; intros HH; rewrite HH in *. - pose proof ParametricSeparationLogic.Parametric_C H. exact (classical_canonical_ident cP rel H_R AL_DC AL_OW AL_CONSI). - pose proof ParametricSeparationLogic.Parametric_GD H0. exact (GodelDummett_canonical_no_branch cP rel H_R AL_DC AL_OW). - pose proof ParametricSeparationLogic.Parametric_DM H1. exact (DeMorgan_canonical_branch_join cP rel H_R AL_DC AL_OW AL_CONSI LIN_CD). - pose proof ParametricSeparationLogic.Parametric_GC H2. exact (garbage_collected_canonical_increaing cP rel H_R H_J AL_DC). - pose proof ParametricSeparationLogic.Parametric_NE H3. exact (nonsplit_canonical_split_smaller cP rel H_R H_J AL_DC TRUTH). - pose proof ParametricSeparationLogic.Parametric_ED H4. exact (dup_canonical_incr_join cP rel H_J AL_DC TRUTH). Qed. End Complete.
(* This file requires: - An Isabelle Snapshot from no earlier than October 5, 2007 ) ( This contains occurence typing with: - variables as tests - eta-expansion ) theory TypedSchemePreOr imports Nominal begin ML {ResAtp.set_prover "vampire"} ( ML {* ThmDeps.enable() } ) (* datatype definitions ) atom_decl name datatype ty = Top \| Int \| TT \| FF \| Arr "ty" "ty" "latent_eff" ("_ \<rightarrow> _ : _" [100,100] 100) \| Union "ty list" and latent_eff = NE \| Latent ty constdefs "BoolTy == Union [TT , FF]" primrec (unchecked perm_ty) "pi\<bullet>(ty.Top) = ty.Top" "pi\<bullet>(ty.Int) = ty.Int" "pi\<bullet>(TT) = TT" "pi\<bullet>(FF) = FF" "pi\<bullet>(\<tau> \<rightarrow> \<sigma> : L) = ((pi\<bullet>\<tau>) \<rightarrow> (pi\<bullet>\<sigma>) : (pi\<bullet>L))" "pi\<bullet>(Union l) = Union (pi\<bullet>l)" "pi\<bullet>latent_eff.NE = latent_eff.NE" "pi\<bullet>(Latent S) = Latent (pi\<bullet>S)" declare perm_ty.simps[eqvt] lemma perm_ty_latent_eff_ty_list[simp]: fixes pi ::"name prm" and \<tau> ::"ty" and Ts ::"ty list" and l ::"latent_eff" shows "pi\<bullet>\<tau> = \<tau>" and "pi\<bullet>l = l" and "pi\<bullet>Ts = Ts" by (induct \<tau> and l rule: ty_latent_eff.inducts) auto instance ty :: pt_name by intro_classes auto instance ty :: fs_name by intro_classes (simp add: supp_def) instance latent_eff :: pt_name by intro_classes auto instance latent_eff :: fs_name by intro_classes (simp add: supp_def) fun size_ty :: "ty \<Rightarrow> nat" and size_le :: "latent_eff \<Rightarrow> nat" where "size_ty (ty.Int) = 1" \| "size_ty (TT) = 1" \| "size_ty (FF) = 1" \| "size_ty (Top) = 1" \| "size_ty (Union []) = 1" \| "size_ty (Union (t#ts)) = size_ty t + size_ty (Union ts)" \| "size_ty (T1 \<rightarrow> T2 : L) = (size_ty T1) + (size_ty T2) + (size_le L)" \| "size_le (Latent L) = size_ty L" \| "size_le (latent_eff.NE) = 1" lemma size_ty_pos: "size_ty T > 0" proof (induct T ) fix l show "0 < size_ty (Union l)" by (induct l) auto qed (auto) nominal_datatype eff = NE \| TE "ty" "name" \| VE "name" \| TT \| FF nominal_datatype builtin = Add1 \| NumberP \| BooleanP \| Nott \| ProcP nominal_datatype trm = Var "name" \| App "trm" "trm" \| Abs "\<guillemotleft>name\<guillemotright>trm" "ty" \| Iff "trm" "trm" "trm" \| Num "nat" \| Bool "bool" \| BI "builtin" abbreviation "lam" :: "name \<Rightarrow> ty \<Rightarrow> trm \<Rightarrow> trm" ("Lam [_:_]._" [100,100,100] 100) where "Lam[x:T].b \<equiv> trm.Abs x b T" ( lemmas about names, types, effects ) lemma trm_finite_supp: fixes x::"trm" shows "finite ((supp x)::name set)" using fs_name_class.axioms[of x] by simp lemma pt_trm_inst: "pt TYPE(trm) TYPE(name)" using pt_name_inst by auto lemma fs_trm_inst: "fs TYPE(trm) TYPE(name)" using fs_name_inst by auto lemma perm_ty_latent[simp]: fixes T::"ty" and le::"latent_eff" and pi::"name prm" shows "pi\<bullet>le = le \<and> pi\<bullet>T = T" by simp lemma perm_ty: fixes T::"ty" and le::"latent_eff" and pi::"name prm" shows "pi\<bullet>T = T" by simp lemma perm_builtin[simp]: fixes e::"builtin" and pi::"name prm" shows "pi\<bullet>e = e" by (induct rule: builtin.weak_induct) (simp_all) lemma fresh_ty[simp]: fixes x::"name" and T::"ty" shows "x\<sharp>T" by (simp add: fresh_def supp_def) lemma fresh_latent_eff[simp]: fixes x::"name" and T::"latent_eff" shows "x\<sharp>T" by (simp add: fresh_def supp_def) lemma fresh_builtin[simp]: fixes x::"name" and b::"builtin" shows "x\<sharp>b" by (simp add: fresh_def supp_def) lemma supp_latent_eff_ty: fixes T::ty and le::latent_eff shows " supp le = ({}::name set) \<and> supp T = ({}::name set)" by (simp add: supp_def) text { size of a term } instance trm :: size .. nominal_primrec "size (Var x) = 1" "size (BI b) = 1" "size (Bool b) = 1" "size (Num b) = 1" "size (App t1 t2) = (max (size t1) (size t2)) + 1" "size (Iff t1 t2 t3) = (max (size t1) (max (size t2) (size t3))) + 1" "size (Lam [a:T].t) = (size t) + 1" by (auto simp add: fresh_nat, finite_guess+, fresh_guess+) abbreviation "smaller_than_abb" :: "trm \<Rightarrow> trm \<Rightarrow> bool" ("_ \<guillemotleft> _" [80,80] 80) where "a \<guillemotleft> b == size a < size b" text { complete induction on terms } lemma trm_comp_induct[consumes 0, case_names Var App Lam BI Num Bool Iff]: fixes P::"'a::fs_name \<Rightarrow> trm \<Rightarrow> bool" and t::"trm" and x::"'a::fs_name" assumes a1:"!! n z. (!! x t. (t \<guillemotleft> Var n) \<Longrightarrow> P x t) \<Longrightarrow> P z (Var n)" and a2:"!! t1 t2 z. (!! x t. (t \<guillemotleft> App t1 t2) \<Longrightarrow> P x t) \<Longrightarrow> (!! x. P x t1) \<Longrightarrow> (!! x . P x t2) \<Longrightarrow> P z (App t1 t2)" and a3:"!! a b z T. \<lbrakk>a \<sharp> z ; (!! x t. (t \<guillemotleft> Lam[a:T].b) \<Longrightarrow> P x t)\<rbrakk> \<Longrightarrow> (!! x . P x b) \<Longrightarrow> P z (Lam[a:T].b)" and a4:"!! b z. (!! x t. (t \<guillemotleft> BI b) \<Longrightarrow> P x t) \<Longrightarrow> P z (BI b)" and a5:"!! n z. (!! x t. (t \<guillemotleft> Num n) \<Longrightarrow> P x t) \<Longrightarrow> P z (Num n)" and a6:"!! b z. (!! x t. (t \<guillemotleft> Bool b) \<Longrightarrow> P x t) \<Longrightarrow> P z (Bool b)" and a7:"!! t1 t2 t3 z. (!! x t. t \<guillemotleft> (Iff t1 t2 t3) \<Longrightarrow> P x t) \<Longrightarrow> (!! x. P x t1) \<Longrightarrow> (!! x . P x t2) \<Longrightarrow> (!! x. P x t3) \<Longrightarrow> P z (Iff t1 t2 t3)" shows "P x t" proof (induct t arbitrary: x taking:"(% t :: trm. size t)" rule: measure_induct_rule) case (less s x) thus ?case -- "This would go through automatically, but I'm skeptical of that sort of thing" proof (nominal_induct s avoiding: x rule: trm.induct) case (Var v) thus ?case using a1 by auto next case (App t1 t2) thus ?case using a2 by auto next case (Abs a b T) thus ?case using a3 by auto next case (Iff t1 t2 t3) thus ?case using a7 by auto next case (BI b) thus ?case using a4 by auto next case (Num n) thus ?case using a5 by auto next case (Bool b) thus ?case using a6 by auto qed qed text { closed terms } constdefs fv :: "trm \<Rightarrow> name set" fv_def[simp]:"fv e == ((supp e):: name set)" constdefs closed :: "trm \<Rightarrow> bool" closed_def: "closed e == (fv e = {})" lemma closed_lam: --"A statement about the free variables of lam bodies" assumes "closed (Lam[x:T].b)" (is "closed ?e") shows "fv b <= {x}" proof - have "(supp ([x].b)::name set) = supp b - {x}" using fs_name_class.axioms abs_fun_supp[of b x] pt_trm_inst at_name_inst by auto hence "supp ?e = ((((supp b):: name set) - {x}) :: name set)" using supp_latent_eff_ty trm.supp by simp thus ?thesis using prems closed_def by auto qed lemma closed_eqvt[eqvt]: fixes pi::"name prm" shows "closed e \<Longrightarrow> closed (pi\<bullet>e)" proof - have A:"(pi\<bullet>fv e) = fv (pi\<bullet>e)" using pt_perm_supp[of pi e] at_name_inst pt_trm_inst by auto assume "closed e" hence "fv e = {}" using closed_def by simp hence "(pi\<bullet>fv e) = {}" using empty_eqvt[of pi] by auto hence "closed (pi\<bullet>e)" using A closed_def by auto thus ?thesis . qed text { capture-avoiding substitution } consts subst :: "trm \<Rightarrow> name \<Rightarrow> trm \<Rightarrow> trm" ("_[_::=_]" [100,100,100] 100) nominal_primrec "(Var x)[y::=t'] = (if x=y then t' else (Var x))" "(App t1 t2)[y::=t'] = App (t1[y::=t']) (t2[y::=t'])" "x\<sharp>(y,t',T) \<Longrightarrow> (Lam[x:T].t)[y::=t'] = Lam[x:T].(t[y::=t'])" "(Iff tst thn els)[y::=t'] = (Iff (tst[y::=t']) (thn[y::=t']) (els[y::=t']))" "(BI c)[y::=t'] = (BI c)" "(Num c)[y::=t'] = (Num c)" "(Bool c)[y::=t'] = (Bool c)" by (finite_guess+, auto simp add: abs_fresh, fresh_guess+) lemma subst_eqvt[simp, eqvt]: fixes pi:: "name prm" and t1:: "trm" and t2:: "trm" and a :: "name" shows "pi\<bullet>(t1[b::=t2]) = (pi\<bullet>t1)[(pi\<bullet>b)::=(pi\<bullet>t2)]" by (nominal_induct t1 avoiding: b t2 rule: trm.induct) (auto simp add: perm_bij fresh_prod fresh_atm fresh_bij) lemma subst_rename[rule_format]: shows "c\<sharp>t1 \<longrightarrow> (t1[a::=t2] = ([(c,a)]\<bullet>t1)[c::=t2])" by (nominal_induct t1 avoiding: a c t2 rule: trm.induct) (auto simp add: calc_atm fresh_atm abs_fresh fresh_nat trm.inject perm_nat_def perm_bool) lemma forget: assumes a: "a\<sharp>t1" shows "t1[a::=t2] = t1" using a by (nominal_induct t1 avoiding: a t2 rule: trm.induct) (auto simp add: abs_fresh fresh_atm) lemma subst_removes_var: assumes "e1[x::=e0] = e2" and "x \<sharp> e0" shows "x \<sharp> e2" using prems proof (nominal_induct e1 avoiding: e0 x e2 rule: trm.induct) case (Var v e0' x' e2') thus ?case using at_fresh[of x' v] at_name_inst by (cases "x' = v") auto next case App thus ?case by auto next case Iff thus ?case by auto next case BI thus ?case by auto next case Num thus ?case by (auto simp add: fresh_nat) next case (Bool b) thus ?case by (auto simp add: fresh_def supp_bool trm.supp) next case (Abs v e1' T e0' x' e2') let ?body = "(e1'[x'::=e0'])" have a:"finite ((supp ?body) :: name set)" using fs_name1 by auto have f:"x' \<sharp> (e1'[x'::=e0'])" using Abs by auto hence "v \<sharp> (x',e0',T)" using Abs by auto hence "(Abs v e1' T)[x'::=e0'] = Abs v (e1'[x'::=e0']) T" by auto hence "\<dots> = e2'" using Abs by auto have "v \<noteq> x'" using `v \<sharp> x'` at_fresh[of v x'] at_name_inst by auto hence "x' \<sharp> v" using at_fresh[of x' v] at_name_inst by auto have "x' \<sharp> T" using fresh_def supp_latent_eff_ty by auto have "x' \<sharp> [v].(e1'[x'::=e0'])" using `v ~= x'` fresh_abs_funI1[of ?body x' v ] pt_trm_inst at_name_inst a f by auto hence "x' \<sharp> Abs v (e1'[x'::=e0']) T" using f trm.fresh Abs by auto thus ?case using Abs by auto qed lemma fv_lam: fixes name shows "fv (Lam[name:T].body) = fv body - {name}" proof - have sT:"supp T = ({} :: name set)" using supp_latent_eff_ty by auto have "fv (Lam[name:T].body) = (supp ([name].body):: name set) \<union> supp T" using trm.supp by auto also have "\<dots> = (supp ([name].body):: name set)" using sT by auto also have "\<dots> = supp body - ({name} :: name set)" using abs_fun_supp[of body name] at_name_inst pt_trm_inst fs_name1[of body] by simp also have "\<dots> = fv body - {name}" by simp finally show "fv (Lam[name:T].body) = fv body - {name}" by simp qed lemma subst_closed: assumes "e1[x::=e0] = e2" and "closed e0" shows "fv e2 <= fv e1" using prems proof (nominal_induct e1 avoiding: e0 x e2 rule: trm.induct) case (Var v e0' x' e2') thus ?case using at_fresh[of x' v] at_name_inst closed_def by (cases "x' = v") auto next case (App rator rand e0' x' e2') let ?subrator = "rator[x'::=e0']" let ?subrand = "rand[x'::=e0']" have a:"e2' = App ?subrator ?subrand" using App by simp have s1:"fv ?subrator <= fv rator" using App by simp have s2:"fv ?subrand <= fv rand" using App by simp have b:"fv e2' = fv ?subrator \<union> fv ?subrand" using trm.supp App a by simp have d:"fv (App rator rand) = fv rator \<union> fv rand" using trm.supp by simp show ?case using d s1 s2 b by auto next case BI thus ?case by auto next case Num thus ?case by (auto simp add: fresh_nat) next case (Bool b) thus ?case by (auto simp add: fresh_def supp_bool trm.supp) next case (Iff tst thn els e0' x' e2') let ?subtst = "tst[x'::=e0']" let ?subthn = "thn[x'::=e0']" let ?subels = "els[x'::=e0']" have a:"e2' = Iff ?subtst ?subthn ?subels" using Iff by simp have s1:"fv ?subtst <= fv tst" using Iff by simp have s2:"fv ?subthn <= fv thn" using Iff by simp have s3:"fv ?subels <= fv els" using Iff by simp have b:"fv e2' = fv ?subtst \<union> fv ?subthn \<union> fv ?subels" using trm.supp Iff a by auto have d:"fv (Iff tst thn els) = fv tst \<union> fv thn \<union> fv els" using trm.supp by auto show ?case using d s1 s2 s3 b by auto next case (Abs name body T e0' x' e2') have aa:"fv (body[x'::=e0']) \<subseteq> fv body" using Abs by auto have a:"fv (Lam[name:T].body) = fv body - {name}" using fv_lam by simp have b:"fv (Lam[name:T].(body[x'::=e0'])) = fv (body[x'::=e0']) - {name}" using fv_lam by simp have "name \<sharp> (e0',T,x')" using Abs by auto hence c:"(Lam[name:T].(body[x'::=e0'])) = (Lam[name:T].(body))[x'::=e0']" by simp hence d:"fv e2' = fv (body[x'::=e0']) - {name}" using b Abs by auto thus ?case using a aa by auto qed lemma subst_only_var_closed: assumes "closed e0" and "fv e1 <= {x}" shows "closed (e1[x::=e0])" proof - let ?e2 = "(e1[x::=e0])" have a:"fv ?e2 <= {x}" using prems subst_closed[of e1 x e0 ?e2] by auto have "x \<sharp> e0" using prems fresh_def[of x e0] closed_def[of e0] by auto hence "x \<sharp> ?e2" using subst_removes_var[of e1 x e0 ?e2] by auto hence b:"x \<notin> fv ?e2" using fresh_def[of x ?e2] closed_def[of ?e2] by auto from a b have "fv ?e2 = {}" by auto thus ?thesis using closed_def by simp qed lemma beta_closed_closed: assumes "closed (Lam[x:T].b)" and "closed v" shows "closed (b[x::=v])" using prems closed_lam subst_only_var_closed by auto text { values } inductive values :: "trm \<Rightarrow> bool" ("_ : values" [80]) where abs_value[simp]: "Lam[x:t].b : values" \| bi_value[simp]: "BI c : values" \| num_value[simp]: "Num n : values" \| bool_value[simp]: "Bool b : values" equivariance values abbreviation "in_values" :: "trm \<Rightarrow> bool" ("_ \<in> values" [100] 100) where "e \<in> values \<equiv> (e : values)" abbreviation "not_in_values" :: "trm \<Rightarrow> bool" ("_ \<notin> values" [100] 100) where "e \<notin> values \<equiv> (~ e : values)" lemma not_false_eqvt: fixes pi :: "name prm" and t :: "trm" assumes a:"t ~= trm.Bool False" and b:"t : values" shows "(pi\<bullet>t) ~= trm.Bool False" using b a by induct (auto simp add: perm_bool) inductive_cases iff_value:"Iff a b c : values" inductive_cases app_value:"App a b : values" inductive_cases var_value:"Var a : values" nominal_inductive values by (simp add: abs_fresh) text { Typing Constants } consts \<Delta>\<^isub>\<tau> :: "builtin \<Rightarrow> ty" nominal_primrec "\<Delta>\<^isub>\<tau> NumberP = (Top \<rightarrow> BoolTy : Latent ty.Int)" "\<Delta>\<^isub>\<tau> BooleanP = (Top \<rightarrow> BoolTy : Latent BoolTy)" "\<Delta>\<^isub>\<tau> ProcP = (Top \<rightarrow> BoolTy : Latent (Union [] \<rightarrow> Top : latent_eff.NE))" "\<Delta>\<^isub>\<tau> Add1 = (ty.Int \<rightarrow> ty.Int : latent_eff.NE)" "\<Delta>\<^isub>\<tau> Nott = (Top \<rightarrow> BoolTy : latent_eff.NE)" by simp_all lemma delta_t_eqvt[eqvt]: fixes pi :: "name prm" shows "pi \<bullet> \<Delta>\<^isub>\<tau> b = \<Delta>\<^isub>\<tau> (pi \<bullet> b)" by (nominal_induct b rule: builtin.induct) auto ( Delta Function ) consts \<Delta> :: "builtin \<Rightarrow> trm \<Rightarrow> trm option" add1_fun :: "trm \<Rightarrow> trm option" nott_fun :: "trm \<Rightarrow> trm option" numberp_fun :: "trm \<Rightarrow> bool" booleanp_fun :: "trm \<Rightarrow> bool" procp_fun :: "trm \<Rightarrow> bool" procp_bi_fun :: "builtin \<Rightarrow> bool" nominal_primrec "add1_fun (Num n) = Some (Num (n+1))" "add1_fun (Lam[x:ty].b) = None" "add1_fun (Iff a b c) = None" "add1_fun (App a b) = None" "add1_fun (Bool a) = None" "add1_fun (BI a) = None" "add1_fun (Var a) = None" by (auto, finite_guess+, fresh_guess+) nominal_primrec "nott_fun (Num n) = (Some (Bool False))" "nott_fun (Lam[x:ty].b) = (Some (Bool False))" "nott_fun (Iff a b c) = (Some (Bool False))" "nott_fun (App a b) = (Some (Bool False))" "nott_fun (Bool b) = Some (Bool (~b))" "nott_fun (BI a) = (Some (Bool False))" "nott_fun (Var a) = (Some (Bool False))" by (auto, finite_guess+, fresh_guess+) nominal_primrec "booleanp_fun (Bool b) = True" "booleanp_fun (Num n) = False" "booleanp_fun (Abs a b c) = False" "booleanp_fun (App a b) = False" "booleanp_fun (BI c) = False" "booleanp_fun (Var v) = False" "booleanp_fun (Iff a b c) = False" by (auto, finite_guess+, fresh_guess+) nominal_primrec "procp_fun (Bool b) = False" "procp_fun (Num n) = False" "procp_fun (Abs a b c) = True" "procp_fun (App a b) = False" "procp_fun (BI c) = True" "procp_fun (Var v) = False" "procp_fun (Iff a b c) = False" by (auto, finite_guess+, fresh_guess+) nominal_primrec "numberp_fun (Bool b) = False" "numberp_fun (Num n) = True" "numberp_fun (Abs a b c) = False" "numberp_fun (App a b) = False" "numberp_fun (BI c) = False" "numberp_fun (Var v) = False" "numberp_fun (Iff a b c) = False" by (auto, finite_guess+, fresh_guess+) nominal_primrec "\<Delta> Add1 t = add1_fun t" "\<Delta> Nott t = nott_fun t" "\<Delta> BooleanP t = Some (Bool (booleanp_fun t))" "\<Delta> NumberP t = Some (Bool (numberp_fun t))" "\<Delta> ProcP t = Some (Bool (procp_fun t))" by simp_all lemma delta_eqvt: fixes pi :: "name prm" and b :: builtin and t :: "trm" shows "\<Delta> (pi\<bullet>b) (pi\<bullet>t) = \<Delta> b t" proof - have A:"(pi\<bullet>b) = b" by (nominal_induct b rule: builtin.induct) auto have B:"\<Delta> b (pi\<bullet>t) = \<Delta> b t" proof (nominal_induct rule: builtin.induct) case Add1 thus ?case by (nominal_induct t rule: trm.induct) (auto simp add: perm_nat_def) next case Nott thus ?case by (nominal_induct t rule: trm.induct) (auto simp add: perm_bool) next case BooleanP thus ?case by (nominal_induct t rule: trm.induct) (auto simp add: perm_bool) next case NumberP thus ?case by (nominal_induct t rule: trm.induct) (auto simp add: perm_bool) next case ProcP thus ?case by (nominal_induct t rule: trm.induct) (auto simp add: perm_bool) qed from A B show ?thesis by auto qed lemma delta_eqvt2[eqvt]: fixes pi :: "name prm" and b :: builtin and t :: "trm" shows "(pi\<bullet>(\<Delta> b t)) = \<Delta> (pi\<bullet>b) (pi\<bullet>t)" proof - have A:"(pi\<bullet>b) = b" by (nominal_induct b rule: builtin.induct) auto have B:"\<Delta> b (pi\<bullet>t) = (pi\<bullet>(\<Delta> b t))" proof (nominal_induct rule: builtin.induct) case Add1 thus ?case by (nominal_induct t rule: trm.induct) (auto simp add: perm_nat_def) next case Nott thus ?case by (nominal_induct t rule: trm.induct) (auto simp add: perm_bool) next case BooleanP thus ?case by (nominal_induct t rule: trm.induct) (auto simp add: perm_bool) next case NumberP thus ?case by (nominal_induct t rule: trm.induct) (auto simp add: perm_bool) next case ProcP thus ?case by (nominal_induct t rule: trm.induct) (auto simp add: perm_bool) qed from A B show ?thesis by auto qed lemma delta_closed: fixes b :: builtin and t::trm assumes "\<Delta> b t = Some v" shows "closed v" using prems proof (nominal_induct b rule: builtin.induct) case Add1 thus ?case by (nominal_induct t rule: trm.induct) (auto simp add: supp_nat closed_def trm.supp) next case Nott thus ?case by (nominal_induct t rule: trm.induct) (auto simp add: supp_def perm_bool closed_def trm.supp) next case BooleanP thus ?case by (nominal_induct t rule: trm.induct) (auto simp add: supp_def perm_bool closed_def trm.supp) next case NumberP thus ?case by (nominal_induct t rule: trm.induct) (auto simp add: supp_def perm_bool closed_def trm.supp) next case ProcP thus ?case by (nominal_induct t rule: trm.induct) (auto simp add: supp_def perm_bool closed_def trm.supp) qed lemma delta_value: fixes b :: builtin and t::trm assumes "\<Delta> b t = Some v" shows "v : values" using prems proof (nominal_induct b rule: builtin.induct) case Add1 thus ?case by (nominal_induct t rule: trm.induct) auto next case Nott thus ?case by (nominal_induct t rule: trm.induct) auto next case BooleanP thus ?case by (nominal_induct t rule: trm.induct) auto next case NumberP thus ?case by (nominal_induct t rule: trm.induct) auto next case ProcP thus ?case by (nominal_induct t rule: trm.induct) auto qed text { Evaluation contexts } inductive_set ctxt :: "(trm \<Rightarrow> trm) set" where C_Hole[simp, intro]: "(%t. t) \<in> ctxt" \| C_App1[simp, intro]: "E : ctxt \<Longrightarrow> (%t . (App (E t) u)) : ctxt" \| C_App2[simp, intro]: "E : ctxt \<Longrightarrow> v : values \<Longrightarrow> (%t . (App v (E t))) : ctxt" \| C_Iff[simp, intro]: "E : ctxt \<Longrightarrow> (%t . (Iff (E t) thn els)) : ctxt" lemma ctxt_compose: assumes a:"E1 : ctxt" and b:"E2 : ctxt" shows "(%t. E1 (E2 t)) : ctxt" using a b by (induct E1) auto constdefs closed_ctxt :: "(trm \<Rightarrow> trm) \<Rightarrow> bool" closed_ctxt_def[simp]:"closed_ctxt C == (C : ctxt \<and> closed (C (Num 3)))" --"3 is a trivially closed term" lemma ctxt_closed: assumes "closed_ctxt C" shows "closed (C e) = closed e" using prems proof - have "C : ctxt" using prems by simp thus ?thesis using prems by (induct) (auto simp add: closed_def trm.supp) qed lemma closed_in_ctxt_closed_ctxt: assumes "closed e" and a:"C : ctxt" and "e = C L" shows "closed L \<and> closed_ctxt C" using a prems proof (induct C arbitrary: L e rule: ctxt.induct) case C_Hole thus ?case by (auto simp add: closed_def trm.supp supp_nat) next case (C_App1 E arg L' e') have IH:"!!e L. \<lbrakk>closed e; E \<in> ctxt; e = E L\<rbrakk> \<Longrightarrow> closed L \<and> closed_ctxt E" using prems by blast have cl:"closed (App (E L') arg)" using `e' = (App (E L') arg)` `closed e'` by simp from cl have "closed arg"by (auto simp add: closed_def trm.supp) from cl have "closed (E L')" by (auto simp add: closed_def trm.supp) thus ?case using IH[of "(E L')" L'] `E : ctxt` `closed arg` by (auto simp add: trm.supp closed_def) next case (C_App2 E rator L' e') have IH:"!!e L. \<lbrakk>closed e; E \<in> ctxt; e = E L\<rbrakk> \<Longrightarrow> closed L \<and> closed_ctxt E" using prems by blast have cl:"closed (App rator (E L'))" using prems by blast from cl have "closed rator" by (auto simp add: closed_def trm.supp) from cl have "closed (E L')" by (auto simp add: closed_def trm.supp) thus ?case using IH[of "(E L')" L'] `E : ctxt` `closed rator` by (auto simp add: trm.supp closed_def) next case (C_Iff E thn els L' e') let ?trm = "Iff (E L') thn els" have IH:"!!e L. \<lbrakk>closed e; E \<in> ctxt; e = E L\<rbrakk> \<Longrightarrow> closed L \<and> closed_ctxt E" using prems by blast have cl:"closed ?trm" using prems by blast from cl have "closed thn" and "closed els" by (auto simp add: closed_def trm.supp) from cl have "closed (E L')" by (auto simp add: closed_def trm.supp) thus ?case using IH[of "(E L')" L'] `E : ctxt` `closed thn` `closed els` by (auto simp add: trm.supp closed_def) qed lemma closed_in_ctxt: assumes "closed (C L)" and "C : ctxt" shows "closed L" using `C : ctxt` `closed (C L)` by (induct C) (auto simp add: closed_def trm.supp) text{ Reduction } inductive reduce :: "trm \<Rightarrow> trm \<Rightarrow> bool" ("_ \<hookrightarrow> _" [200,200] 200) where e_beta[simp]: "v : values \<Longrightarrow> x \<sharp> v \<Longrightarrow> (App (Lam[x:t].b) v) \<hookrightarrow> (b[x::=v])" \| e_if_false[simp]: "Iff (Bool False) e1 e2 \<hookrightarrow> e2" \| e_if_true[simp]: "v ~= Bool False \<Longrightarrow> v : values \<Longrightarrow> Iff v e1 e2 \<hookrightarrow> e1" \| e_delta[simp]: "\<lbrakk>v : values; \<Delta> p v = Some e\<rbrakk> \<Longrightarrow> App (BI p) v \<hookrightarrow> e" equivariance reduce nominal_inductive reduce by (simp_all add: abs_fresh subst_removes_var) inductive "step" :: "trm\<Rightarrow>trm\<Rightarrow>bool" (" _ \<longrightarrow> _" [80,80]80) where step_one[intro]:"\<lbrakk>E : ctxt; L \<hookrightarrow> R\<rbrakk> \<Longrightarrow> E L \<longrightarrow> E R" inductive step_multi :: "trm \<Rightarrow> trm \<Rightarrow> bool" (" _ \<longrightarrow>\<^sup> _" [80,80] 80) where sm_refl:"a \<longrightarrow>\<^sup>* a" \| sm_trans:"a \<longrightarrow> b \<Longrightarrow> b \<longrightarrow>\<^sup>* c \<Longrightarrow> a \<longrightarrow>\<^sup>* c" (* doesn't work ) ( equivariance step ) constdefs reduce_forever :: "trm \<Rightarrow> bool" "reduce_forever e == \<forall>e' . (e \<longrightarrow>\<^sup> e') \<longrightarrow> (EX e''. e' \<longrightarrow> e'')" (* reduction examples ) lemma "(App (Lam [x:t].(Var x)) (Num 4)) \<hookrightarrow> Num 4" proof - have "Num 4 : values" "x \<sharp> Num 4" by (auto simp add: fresh_nat) hence h:"(App (Lam [x:t].(Var x)) (Num 4)) \<hookrightarrow> ((Var x)[x::=(Num 4)])" by (rule e_beta) have "((Var x)[x::=(Num 4)]) = Num 4" by auto thus ?thesis using h by auto qed ( some lemmas about reduction ) lemma if_val_reduces: assumes a:"tst : values" shows "Iff tst thn els \<hookrightarrow> thn \<or> Iff tst thn els \<hookrightarrow> els" using a proof (nominal_induct tst rule: trm.induct) case (Bool b) thus ?case using e_if_true e_if_false by (cases b) (auto simp add: trm.inject) qed (auto) ( a helper lemma - whee abstraction ) lemma ex_help: assumes a:"e = E t \<and> E : ctxt \<and> t \<hookrightarrow> t'" shows "\<exists>E t t' . e = E t \<and> E \<in> ctxt \<and> t \<hookrightarrow> t'" using a by blast lemma reduce_in_ctxt: fixes e :: trm assumes ct:"C : ctxt" and ih:"(EX E L R. e = E L \<and> E : ctxt \<and> L \<hookrightarrow> R)" shows "(EX E L R. C e = E L \<and> E : ctxt \<and> L \<hookrightarrow> R)" proof - from ih ct obtain Enew tnew t'new where "e = Enew tnew" and 1:"Enew \<in> ctxt" and g1:"tnew \<hookrightarrow> t'new" by auto let ?E="(%t . C (Enew t))" have g3:"?E tnew = C e" using `e = Enew tnew` by auto thus "\<exists>E t t' . C e = E t \<and> E \<in> ctxt \<and> t \<hookrightarrow> t'" using ctxt_compose[OF ct 1] g1 g3 ex_help[of "C e" ?E tnew] by auto qed inductive_cases iff_bi_red : "(Iff (Const (BI bi)) thn els) \<hookrightarrow> e" inductive_cases iff_red : "(Iff tst thn els) \<hookrightarrow> e" lemma reduce_closed: assumes "closed L" and "L \<hookrightarrow> R" shows "closed R" using `L \<hookrightarrow> R` `closed L` proof (induct) case (e_beta v x t b) have a:"closed (Abs x b t)" using e_beta closed_def trm.supp by simp have b:"closed v" using e_beta closed_def trm.supp by simp from a b show ?case using e_beta beta_closed_closed by simp next case e_if_true thus ?case using closed_def trm.supp by auto next case e_if_false thus ?case using closed_def trm.supp by auto next case e_delta thus ?case using delta_closed by auto qed lemma step_closed: assumes A:"closed e" and B:"(e :: trm) \<longrightarrow> e'" shows "closed e'" proof - from B obtain E L R where C:"E : ctxt" "e = E L" "L \<hookrightarrow> R" "e' = E R" by induct auto hence D:"closed L" "closed_ctxt E" using A closed_in_ctxt_closed_ctxt[of e E L] by auto hence "closed R" using C reduce_closed[of L R] by auto hence "closed e'" using C D ctxt_closed[of E R] by auto thus ?thesis . qed lemma multi_step_closed: assumes A:"closed e" and B:"e \<longrightarrow>\<^sup> e'" shows "closed e'" using B A step_closed by induct auto text {* "partial order" (not really) between effects } inductive subeff :: "eff \<Rightarrow> eff \<Rightarrow> bool"("\<turnstile> _ <e: _" [60,60] 60) where SE_Refl[intro]: "\<turnstile> F <e: F" \| SE_FF[intro]: "F \<noteq> TT \<Longrightarrow> \<turnstile> FF <e: F" \| SE_TT[intro]: "F \<noteq> FF \<Longrightarrow> \<turnstile> TT <e: F" \| SE_VE[intro]: "\<turnstile> NE <e: VE x" \| SE_TE[intro]: "\<turnstile> NE <e: TE S x" equivariance subeff inductive_cases ne_case:"\<turnstile> F1 <e: eff.NE" inductive_cases ne_case_rev:"\<turnstile> eff.NE <e: F1" inductive_cases tt_case:"\<turnstile> F1 <e: eff.TT" inductive_cases tt_case_rev:"\<turnstile> eff.TT <e: F1" inductive_cases ff_case:"\<turnstile> F1 <e: eff.FF" inductive_cases ff_case_rev:"\<turnstile> eff.FF <e: F1" lemma no_sub_FF: "\<lbrakk>\<turnstile> T <e: T' ; T' = FF \<rbrakk> \<Longrightarrow> T = FF" by (induct rule: subeff.induct) auto lemma no_sub_TT: "\<lbrakk>\<turnstile> T <e: T' ; T' = TT \<rbrakk> \<Longrightarrow> T = TT" by (induct rule: subeff.induct) auto constdefs simple_eff :: "eff \<Rightarrow> bool" simple_eff_def[simp]:"simple_eff e == e = eff.NE \<or> e = FF \<or> e = TT" lemma simple_eff_cases[consumes 1, case_names NE FF TT]: fixes F::eff and P :: "eff \<Rightarrow> bool" assumes a:"simple_eff F" and a1:"P NE" and a2:"P FF" and a3:"P TT" shows "P F" using prems by (nominal_induct F rule: eff.induct) auto lemma simple_eff_below_ne: assumes "simple_eff F" shows "\<turnstile> F <e: NE" using prems by (nominal_induct F rule: eff.induct) auto lemma SE_Trans[intro]: assumes"\<turnstile> F1 <e: F2 "and "\<turnstile> F2 <e: F3 " and "simple_eff F1" and "simple_eff F2" and "simple_eff F3" shows " \<turnstile> F1 <e: F3" using `simple_eff F3` prems proof (induct F3 rule: simple_eff_cases) case NE thus ?case using simple_eff_below_ne by auto next case FF thus ?case using no_sub_FF by auto next case TT thus ?case using no_sub_TT by auto qed text{ subtyping } inductive subtype :: "ty \<Rightarrow> ty \<Rightarrow> bool" ("\<turnstile> _ <: _" [60,60] 60) where S_Refl[intro]: "\<turnstile> T <: T" \| S_Fun[intro]: "\<lbrakk>\<turnstile> S1 <: T1 ; \<turnstile> T2 <: S2 ; eff = eff' \<or> eff' = latent_eff.NE\<rbrakk> \<Longrightarrow> \<turnstile> (T1 \<rightarrow> T2 : eff) <: (S1 \<rightarrow> S2 : eff')" \| S_Top[intro]: "\<turnstile> T <: Top" \| S_UnionAbove[intro]: "\<lbrakk>T : set Ts ; \<turnstile> S <: T\<rbrakk> \<Longrightarrow> \<turnstile> S <: Union Ts" \| S_UnionBelow[intro]: "\<lbrakk>!! T.( T : set Ts \<Longrightarrow> \<turnstile> T <: S)\<rbrakk> \<Longrightarrow> \<turnstile> Union Ts <: S" equivariance subtype nominal_inductive subtype done inductive_cases sub_arr_cases: "\<turnstile> T <: S1 \<rightarrow> S2 : L" lemma subtype_arr_elim: "\<lbrakk>\<turnstile> T <: S ; S = S0 \<rightarrow> S1 : le\<rbrakk> \<Longrightarrow> (EX T0 T1. T = T0 \<rightarrow> T1 : le \<and> \<turnstile> S0 <: T0 \<and> \<turnstile> T1 <: S1) \<or> (EX T0 T1 le'. T = T0 \<rightarrow> T1 : le' \<and> le = latent_eff.NE \<and> \<turnstile> S0 <: T0 \<and> \<turnstile> T1 <: S1) \<or> (EX Ts. T = Union Ts)" proof (induct arbitrary: S0 S1 rule: subtype.induct) qed (auto) lemma S_TopE: assumes a: "\<turnstile> Top <: T" shows "T = Top \<or> (EX Ts T'. T = Union Ts \<and> T' : set Ts \<and> \<turnstile> Top <: T')" using a by (cases, auto) lemma S_ArrowE_left: assumes a: "\<turnstile> S\<^isub>1 \<rightarrow> S\<^isub>2 : L <: T" shows "T = Top \<or> (\<exists>T\<^isub>1 T\<^isub>2. T = T\<^isub>1 \<rightarrow> T\<^isub>2 : L \<and> \<turnstile> T\<^isub>1 <: S\<^isub>1 \<and> \<turnstile> S\<^isub>2 <: T\<^isub>2) \<or> (\<exists>T\<^isub>1 T\<^isub>2. T = T\<^isub>1 \<rightarrow> T\<^isub>2 : latent_eff.NE \<and> \<turnstile> T\<^isub>1 <: S\<^isub>1 \<and> \<turnstile> S\<^isub>2 <: T\<^isub>2) \<or> (EX Ts T1. T = Union Ts \<and> T1 : set Ts \<and> \<turnstile> S\<^isub>1 \<rightarrow> S\<^isub>2 : L <: T1)" using a by (cases, auto simp add: ty.inject) lemma union_size_ty: assumes "T : set Ts" shows "size_ty T < size_ty (Union Ts)" using prems size_ty_pos by (induct Ts) auto fun size_ty3 :: "tytyty \<Rightarrow> nat" where size_ty3_def[simp]:"size_ty3 (a,b,c) = size_ty a + size_ty b + size_ty c" inductive_cases union_sub_cases[consumes 1, case_names 1 2 3 4]:"\<turnstile> Union Ts <: S" lemma union_sub_elim: assumes A:"\<turnstile> Union Ts <: T " (is "\<turnstile> ?S <: T") and B:" T1 : set Ts " shows "\<turnstile> T1 <: T" using prems proof (induct "X"=="(T1,?S,T)" arbitrary: T1 Ts T taking: "size_ty3" rule: measure_induct_rule) case (less X) show " \<turnstile> T1 <: T" using `\<turnstile> Union Ts <: T` less proof (induct rule: union_sub_cases) case 1 thus ?case by auto next case 2 thus ?case by auto next case (3 T' Ts') have X_inst:"X = (T1, ty.Union Ts, T)" . have "size_ty T' < size_ty T" using 3 union_size_ty by auto hence "\<turnstile> T1 <: T'" using X_inst 3(4)[OF _ ` \<turnstile> ty.Union Ts <: T'` `T1 : set Ts`] by auto thus ?case using 3 by auto next case 4 thus ?case by auto qed qed lemma S_Trans[intro]: assumes "\<turnstile>S<:Q" and " \<turnstile>Q<:T" shows "\<turnstile>S<:T" using prems proof (induct "X"=="(S,Q,T)" arbitrary: S Q T taking: "size_ty3" rule: measure_induct_rule) case (less X S Q T) show " \<turnstile> S <: T" using `\<turnstile> S <: Q` less proof (induct S Q\<equiv>Q rule: subtype.induct) case S_Refl thus ?case by auto next case (S_Top A) have X_inst:"X = (A,Q,T)" . show ?case proof - { assume "EX Ts T'. T = Union Ts \<and> T' : set Ts \<and> \<turnstile> Q <: T'" then obtain Ts T' where "T = Union Ts "" T' : set Ts "" \<turnstile> Q <: T'" by auto hence "size_ty T' < size_ty T" using union_size_ty by auto hence "size_ty3 (A,Q,T') < size_ty3 (A,Q,T)" by auto hence "\<turnstile> A <: T'" using `\<turnstile> A <: Q` `\<turnstile> Q <: T'` using less(1)[of "(A,Q,T')" A Q T'] X_inst by auto hence ?thesis using S_Top prems by auto } thus ?thesis using S_TopE S_Top by auto qed next case (S_Fun Q1 S1 S2 Q2 L L') hence rh_drv: " \<turnstile> Q1 \<rightarrow> Q2 : L' <: T" by simp have X_inst:"X = (S1 \<rightarrow> S2 : L, Q1 \<rightarrow> Q2 : L', T)" using S_Fun by auto note `Q1 \<rightarrow> Q2 : L' = Q` hence Q12_less: "size_ty Q1 < size_ty Q" "size_ty Q2 < size_ty Q" using size_ty_pos by auto have lh_drv_prm1: " \<turnstile> Q1 <: S1" by fact have lh_drv_prm2: " \<turnstile> S2 <: Q2" by fact have "T=Top \<or> (\<exists>T1 T2 LL. T=T1\<rightarrow>T2 : LL \<and> \<turnstile>T1<:Q1 \<and> \<turnstile>Q2<:T2 \<and> LL = L') \<or> (\<exists>T1 T2. T=T1\<rightarrow>T2 : latent_eff.NE \<and> \<turnstile>T1<:Q1 \<and> \<turnstile>Q2<:T2) \<or> (EX Ts T1. T = Union Ts \<and> T1 : set Ts \<and> \<turnstile> Q1 \<rightarrow> Q2 : L' <: T1)" using S_ArrowE_left[OF rh_drv] by auto moreover { assume "\<exists>T1 T2 LL. T=T1\<rightarrow>T2:LL \<and> \<turnstile>T1<:Q1 \<and> \<turnstile>Q2<:T2 \<and> LL = L'" then obtain T1 T2 LL where T_inst: "T = T1 \<rightarrow> T2 : L'" and rh_drv_prm1: " \<turnstile> T1 <: Q1" and rh_drv_prm2: " \<turnstile> Q2 <: T2" and LL': "LL = L'" by auto from X_inst T_inst have X_inst':"X = (S1 \<rightarrow> S2 : L, Q1 \<rightarrow> Q2 : L', T1 \<rightarrow> T2 : L')" by simp hence "size_ty3 (T1, Q1, S1) < size_ty3 X" using size_ty_pos by auto from X_inst' lh_drv_prm1 rh_drv_prm1 have " \<turnstile> T1 <: S1" using S_Fun(6)[of _ T1 Q1 S1] size_ty_pos by auto moreover from X_inst' lh_drv_prm2 rh_drv_prm2 have " \<turnstile> S2 <: T2" using S_Fun(6)[of _ S2 Q2 T2] size_ty_pos by auto ultimately have " \<turnstile> S1 \<rightarrow> S2 : L <: T1 \<rightarrow> T2 : LL" using LL' S_Fun by (simp add: subtype.S_Fun) hence " \<turnstile> S1 \<rightarrow> S2 : L <: T" using T_inst LL' by simp } moreover { assume "EX Ts T1. T = Union Ts \<and> T1 : set Ts \<and> \<turnstile> Q1 \<rightarrow> Q2 : L' <: T1" then obtain Ts T1 where T_inst: "T = Union Ts" and elem: "T1 : set Ts" and sub:"\<turnstile> Q1 \<rightarrow> Q2 : L' <: T1" by auto have sub':"\<turnstile> S1 \<rightarrow> S2 : L <: Q1 \<rightarrow> Q2 : L'" using S_Fun by simp have sz:"size_ty T1 < size_ty T" using T_inst elem union_size_ty by auto from X_inst T_inst have X_inst':"X = (S1 \<rightarrow> S2 : L, Q1 \<rightarrow> Q2 : L', Union Ts)" by simp from sub sub' X_inst' have " \<turnstile> S1 \<rightarrow> S2 : L <: T1" using S_Fun(6)[OF _ sub' sub] sz T_inst by auto hence " \<turnstile> S1 \<rightarrow> S2 : L <: T" using T_inst elem S_UnionAbove by auto } moreover { assume "\<exists>T1 T2. T=T1\<rightarrow>T2:latent_eff.NE \<and> \<turnstile>T1<:Q1 \<and> \<turnstile>Q2<:T2 " then obtain T1 T2 LL where T_inst: "T = T1 \<rightarrow> T2 : latent_eff.NE" and rh_drv_prm1: " \<turnstile> T1 <: Q1" and rh_drv_prm2: " \<turnstile> Q2 <: T2" by auto from X_inst T_inst have X_inst':"X = (S1 \<rightarrow> S2 : L, Q1 \<rightarrow> Q2 : L', T1 \<rightarrow> T2 : latent_eff.NE)" by simp hence "size_ty3 (T1, Q1, S1) < size_ty3 X" using size_ty_pos by auto from X_inst' lh_drv_prm1 rh_drv_prm1 have " \<turnstile> T1 <: S1" using S_Fun(6)[of _ T1 Q1 S1] size_ty_pos by auto moreover from X_inst' lh_drv_prm2 rh_drv_prm2 have " \<turnstile> S2 <: T2" using S_Fun(6)[of _ S2 Q2 T2] size_ty_pos by auto ultimately have " \<turnstile> S1 \<rightarrow> S2 : L <: T1 \<rightarrow> T2 : latent_eff.NE" using S_Fun by (simp add: subtype.S_Fun) hence " \<turnstile> S1 \<rightarrow> S2 : L <: T" using T_inst by simp } ultimately show " \<turnstile> S1 \<rightarrow> S2 : L <: T" by blast next case (S_UnionAbove T1 Ts S) have sub1:"\<turnstile> S <: T1" . hence sub2:"\<turnstile> T1 <: T" using S_UnionAbove union_sub_elim[of Ts T T1] by auto have sz:"size_ty T1 < size_ty Q" using S_UnionAbove union_size_ty by auto hence "\<turnstile> S <: T" using S_UnionAbove(4)[OF _ sub1 sub2] sz S_UnionAbove(7) by auto thus ?case . next case (S_UnionBelow Ts S) have "!! T0. T0 : set Ts \<Longrightarrow> \<turnstile> T0 <: T" proof - fix T0 assume "T0 : set Ts" hence sz:"size_ty T0 < size_ty (Union Ts)" using union_size_ty by auto have s1:"\<turnstile> T0 <: S" using S_UnionBelow `T0 : set Ts` by auto have s2:"\<turnstile> S <: T" using S_UnionBelow by auto note S_UnionBelow(6) thus "\<turnstile> T0 <: T" using S_UnionBelow(3)[OF _ s1 s2] sz `S = Q` by auto qed thus ?case .. qed qed text { type environments } types varEnv = "(namety) list" text {* valid contexts } inductive valid :: "(name\<times>ty) list \<Rightarrow> bool" where v1[intro]: "valid []" \| v2[intro]: "\<lbrakk>valid \<Gamma>;a\<sharp>\<Gamma>\<rbrakk>\<Longrightarrow> valid ((a,\<sigma>)#\<Gamma>)" equivariance valid nominal_inductive valid done lemma fresh_context[rule_format]: fixes \<Gamma> :: "(name\<times>ty)list" and a :: "name" assumes a: "a\<sharp>\<Gamma>" shows "\<not>(\<exists>\<tau>::ty. (a,\<tau>)\<in>set \<Gamma>)" using a by (induct \<Gamma>) (auto simp add: fresh_prod fresh_list_cons fresh_atm) lemma valid_elim: fixes \<Gamma> :: "(name\<times>ty)list" and pi:: "name prm" and a :: "name" and \<tau> :: "ty" shows "valid ((a,\<tau>)#\<Gamma>) \<Longrightarrow> valid \<Gamma> \<and> a\<sharp>\<Gamma>" by (ind_cases "valid ((a,\<tau>)#\<Gamma>)") simp lemma valid_unicity[rule_format]: assumes a: "valid \<Gamma>" and b: "(c,\<sigma>)\<in>set \<Gamma>" and c: "(c,\<tau>)\<in>set \<Gamma>" shows "\<sigma>=\<tau>" using a b c by (induct \<Gamma>) (auto dest!: valid_elim fresh_context) declare fresh_list_cons[simp] declare fresh_list_nil[simp] ( environment operations ) consts env_plus :: "eff \<Rightarrow> varEnv => varEnv" env_minus :: "eff \<Rightarrow> varEnv => varEnv" ( original type is the SECOND argument ) constdefs less_ty :: "((ty ty) * ty * ty) set" "less_ty == {((a,b),c,d) . (size_ty b) < (size_ty d)}" function (sequential) restrict :: "ty \<Rightarrow> ty \<Rightarrow> ty" where restrict_union: "restrict r (Union (ls :: ty list)) = (if (\<turnstile> r <: Union ls) then r else (if (\<turnstile> Union ls <: r) then (Union ls) else Union (map (restrict r) ls)))" \| restrict_top:"restrict r Top = r" \| restrict_other:"restrict r T = (if (\<turnstile> r <: T) then r else T)" by pat_completeness auto termination using union_size_ty by (relation "measure (% (a,b). size_ty b)") auto lemma restrict_eqvt[eqvt]: fixes pi::"name prm" shows "pi\<bullet>(restrict T1 T2) = restrict (pi\<bullet>T1) (pi\<bullet>T2)" by (induct T2) (auto) text {* this is the key lemma in the whole soundness proof } fun simple_ty :: "ty \<Rightarrow> bool" where "simple_ty ty.Int = True" \| "simple_ty ty.TT = True" \| "simple_ty ty.FF = True" \| "simple_ty (A1 \<rightarrow> A2 : L) = True" ( \| "simple_ty (Union [ty.TT,ty.FF]) = True" ) \| "simple_ty T = False" lemma ty_cases[consumes 0, case_names Top Int TT FF Arr Union]: fixes P :: "ty \<Rightarrow> bool" and T :: ty assumes a1:"P Top" and a2:"P ty.Int" and a3:"P ty.TT" and a3':"P ty.FF" and a4:"!! t1 t2 L. P (t1 \<rightarrow> t2 : L)" and a5:"!! Ts . P (Union Ts)" shows "P T" using ty_latent_eff.induct[of P "(%e . True)" "(%e . True)" T] prems by auto inductive_cases tt_below_union: "\<turnstile> ty.TT <: ty.Union Ts" inductive_cases ff_below_union: "\<turnstile> ty.FF <: ty.Union Ts" ( I believe this to be true without C, but it's easier to prove this way, and that's all we need ) lemma restrict_soundness: assumes A:"\<turnstile> T0 <: T" and B:"\<turnstile> T0 <: S" and C:"simple_ty T0" shows "\<turnstile> T0 <: restrict S T" using prems proof (induct T arbitrary: S T0 taking:"size_ty" rule: measure_induct_rule) case (less T S T0) thus ?case proof (induct T==T rule: ty_cases) case Top thus ?case by auto next case Int thus ?case by auto next case TT thus ?case by auto next case FF thus ?case by auto next case Arr thus ?case by auto next case (Union Ts) have r:"restrict S T = (if (\<turnstile> S <:T) then S else (if (\<turnstile> T <: S) then T else Union (map (restrict S) Ts)))" using prems restrict_union[of S Ts] by auto thus ?case proof - { assume "\<turnstile> S <: T" hence "restrict S T = S" using r by simp hence ?thesis using prems by auto } moreover { assume "\<turnstile> T <: S" "~ (\<turnstile> S <: T)" hence "restrict S T = T" using r by simp hence ?thesis using prems by auto } moreover { assume "~ (\<turnstile> T <: S)" "~ (\<turnstile> S <: T)" hence req:"restrict S T = Union (map (restrict S) Ts)" using r by auto have T:"\<turnstile> T0 <: Union Ts" using prems by simp have "?this \<Longrightarrow> ?thesis" proof (ind_cases "\<turnstile> T0 <: Union Ts") assume 0:"Union Ts = T0" thus ?thesis using `simple_ty T0` by (induct T0 rule: simple_ty.induct) auto next fix Ts' assume "T0 = ty.Union Ts'" thus ?thesis using `simple_ty T0` by (induct T0 rule: simple_ty.induct) auto ( hence 1:"T0 = Union [ty.TT,ty.FF]" using `simple_ty T0` by (induct T0 rule: simple_ty.induct) auto have "\<turnstile> ty.TT <: T0" using 1 by auto hence "\<turnstile> ty.TT <: Union Ts" "\<turnstile> ty.TT <: T" "\<turnstile> ty.TT <: S" using T `\<turnstile> T0 <: S` `\<turnstile> T0 <: T` by auto hence "EX \<sigma> . \<sigma> : set Ts \<and> \<turnstile> ty.TT <: \<sigma>" using tt_below_union by auto then guess \<sigma> by auto hence "size_ty \<sigma> < size_ty T" using prems union_size_ty by auto note prems(4)[OF `size_ty \<sigma> < size_ty T` `\<turnstile> ty.TT <: \<sigma>` `\<turnstile> ty.TT <: S`, simplified] hence A_tt:"\<turnstile> ty.TT <: restrict S \<sigma>" . have mem:"restrict S \<sigma> : set (map (restrict S) Ts)" using `\<sigma> : set Ts` by auto hence "\<turnstile> ty.TT <: Union (map (restrict S) Ts)" using A_tt S_UnionAbove by auto hence tt_sub:"\<turnstile> ty.TT <: restrict S T" using req by auto have "\<turnstile> ty.FF <: T0" using 1 by auto hence "\<turnstile> ty.FF <: Union Ts" "\<turnstile> ty.FF <: T" "\<turnstile> ty.FF <: S" using T `\<turnstile> T0 <: S` `\<turnstile> T0 <: T` by auto hence "EX \<sigma> . \<sigma> : set Ts \<and> \<turnstile> ty.FF <: \<sigma>" using ff_below_union by auto then guess \<sigma> by auto hence "size_ty \<sigma> < size_ty T" using prems union_size_ty by auto note prems(4)[OF `size_ty \<sigma> < size_ty T` `\<turnstile> ty.FF <: \<sigma>` `\<turnstile> ty.FF <: S`, simplified] hence A_ff:"\<turnstile> ty.FF <: restrict S \<sigma>" . have mem:"restrict S \<sigma> : set (map (restrict S) Ts)" using `\<sigma> : set Ts` by auto hence "\<turnstile> ty.FF <: Union (map (restrict S) Ts)" using A_ff S_UnionAbove by auto hence ff_sub:"\<turnstile> ty.FF <: restrict S T" using req by auto from ff_sub tt_sub have "\<turnstile> Union [ty.TT, ty.FF] <: restrict S T" by auto thus ?thesis using 1 by auto ) next fix T' assume "T' : set Ts" "\<turnstile> T0 <: T'" have 1:"\<turnstile> T0 <: restrict S T'" using union_size_ty prems by auto have 2:"set (map (restrict S) Ts) = (restrict S) ` set Ts" by auto have 3:"T' : set Ts" using prems by auto have 4:"restrict S T' : set (map (restrict S) Ts)" using 2 3 by auto hence "\<turnstile> T0 <: Union (map (restrict S) Ts)" using subtype.S_UnionAbove[OF 4 1] by auto thus ?thesis using req by auto qed hence ?thesis using T by simp } ultimately show ?thesis by auto qed qed qed function (sequential) remove :: "ty \<Rightarrow> ty \<Rightarrow> ty" where remove_union: "remove r (Union (ls :: ty list)) = (if (\<turnstile> Union ls <: r) then (Union []) else Union (map (remove r) ls))" \| remove_other:"remove r T = (if (\<turnstile> T <: r) then (Union []) else T)" by pat_completeness auto termination using union_size_ty by (relation "measure (% (a,b). size_ty b)") auto lemma remove_eqvt[eqvt]: fixes pi::"name prm" shows "pi\<bullet>(remove T1 T2) = remove (pi\<bullet>T1) (pi\<bullet>T2)" by (induct T2) (auto) lemma remove_soundness: assumes A:"\<turnstile> T0 <: T" and B:"~ (\<turnstile> T0 <: S)" and C:"simple_ty T0" shows "\<turnstile> T0 <: remove S T" using prems proof (induct T arbitrary: S T0 taking:"size_ty" rule: measure_induct_rule) case (less T S T0) thus ?case proof (induct T==T rule: ty_cases) case (Union Ts) have r:"remove S T = (if (\<turnstile> T <:S) then (Union []) else Union (map (remove S) Ts))" using prems remove_union[of S Ts] by auto thus ?case proof - { assume "\<turnstile> T <: S" hence "remove S T = (Union [])" using r by simp hence ?thesis using prems by auto } moreover { assume "~ (\<turnstile> T <: S)" hence req:"remove S T = Union (map (remove S) Ts)" using r by auto have T:"\<turnstile> T0 <: Union Ts" using prems by simp have "?this \<Longrightarrow> ?thesis" proof (ind_cases "\<turnstile> T0 <: Union Ts") assume 0:"Union Ts = T0" thus ?thesis using `simple_ty T0` by auto next fix Ts' assume "T0 = ty.Union Ts'" thus ?thesis using `simple_ty T0` by auto next fix T' assume "T' : set Ts" "\<turnstile> T0 <: T'" have 1:"\<turnstile> T0 <: remove S T'" using union_size_ty prems by auto have 2:"set (map (remove S) Ts) = (remove S) ` set Ts" by auto have 3:"T' : set Ts" using prems by auto have 4:"remove S T' : set (map (remove S) Ts)" using 2 3 by auto hence "\<turnstile> T0 <: Union (map (remove S) Ts)" using subtype.S_UnionAbove[OF 4 1] by auto thus ?thesis using req by auto qed hence ?thesis using T by simp } ultimately show ?thesis by auto qed qed (auto) qed lemma restrict_remove_soundness: assumes A:"\<turnstile> T0 <: T" and B:"simple_ty T0" shows "(\<turnstile> T0 <: S \<and> \<turnstile> T0 <: restrict S T) \<or> (~ (\<turnstile> T0 <: S) \<and> \<turnstile> T0 <: remove S T)" using restrict_soundness[OF A _ B] remove_soundness[OF A _ B] by auto fun replace :: "ty \<Rightarrow> ty \<Rightarrow> ty" where "replace t u = t" fun mapfun :: "(ty \<Rightarrow> ty \<Rightarrow> ty) \<Rightarrow> ty \<Rightarrow> name \<Rightarrow> (namety) \<Rightarrow> (name * ty)" where mapfun_def: "mapfun f T x (v,S) = (if (x = v) then (v, f T S) else (v,S))" constdefs envop :: "(ty \<Rightarrow> ty \<Rightarrow> ty) \<Rightarrow> name \<Rightarrow> ty \<Rightarrow> (namety) list \<Rightarrow> (namety) list" envop_def[simp]:"envop f n t G == map (% (v,ty). (if (n = v) then (v,f t ty) else (v,ty))) G" lemma envop_mapfun: shows "map (mapfun f T x) \<Gamma> = envop f x T \<Gamma> " using mapfun_def by auto lemma envop_fresh: fixes v::name assumes a:"v \<sharp> \<Gamma>" and c:"valid \<Gamma>" shows "v \<sharp> (envop f n t \<Gamma>)" using c a by (induct \<Gamma> rule: valid.induct) auto lemma envop_valid: assumes "valid \<Gamma>" shows "valid (envop f n t \<Gamma>)" using assms envop_fresh by induct auto lemma envop_forget: assumes "valid \<Gamma>" and "x \<sharp> \<Gamma>" shows "envop f x T \<Gamma> = \<Gamma>" using prems proof (induct rule: valid.induct) case v1 thus ?case by auto next case (v2 \<Gamma>' a S) have "x ~= a" and "x \<sharp> \<Gamma>'" using v2 fresh_list_cons fresh_atm[of x a] by auto hence A:"envop f x T ((a,S)#\<Gamma>') = (a,S)# (envop f x T \<Gamma>')" by auto thus ?case using v2 `x \<sharp> \<Gamma>'` by auto qed nominal_primrec "env_plus (NE) G = G" "env_plus (TE T x) G = envop restrict x T G" "env_plus (VE x) G = envop remove x (ty.FF) G" "env_plus (TT) G = G" "env_plus (FF) G = G" by auto lemma map_eqvt[eqvt]: fixes pi::"name prm" and l::"('a::pt_name) list" shows "pi\<bullet>(map f l) = map (pi\<bullet>f) (pi\<bullet>l)" by (induct l) perm_simp+ lemma env_plus_eqvt[eqvt]: fixes pi::"name prm" shows "pi\<bullet>(env_plus X G ) = env_plus (pi\<bullet>X) (pi\<bullet>G)" proof (nominal_induct X rule: eff.induct) case (TE T x) thus ?case by (perm_simp add: eqvts split_def) auto next case (VE x) thus ?case by (perm_simp add: eqvts split_def) auto qed(auto) nominal_primrec "env_minus (NE) G = G" "env_minus (TE T x) G = envop remove x T G" "env_minus (VE x) G = envop replace x ty.FF G" "env_minus (TT) G = G" "env_minus (FF) G = G" by auto lemma env_minus_eqvt[eqvt]: fixes pi::"name prm" shows "pi\<bullet>(env_minus X G) = env_minus (pi\<bullet>X) (pi\<bullet>G)" proof (nominal_induct X rule: eff.induct) case (TE T x) thus ?case by (perm_simp add: eqvts perm_fun_def split_def) auto case (VE x) thus ?case by (perm_simp add: eqvts perm_fun_def split_def) qed(auto) abbreviation env_plus_syn :: "varEnv \<Rightarrow> eff \<Rightarrow> varEnv" ("_ \|+ _" [70,70] 70) where "(G \|+ eff) == env_plus eff G" abbreviation env_minus_syn :: "varEnv \<Rightarrow> eff \<Rightarrow> varEnv" ("_ \|- _" [70,70] 70) where "(G \|- eff) == env_minus eff G" --"Induction principle for type envs" lemma env_induct[case_names Nil Cons]: fixes \<Gamma> :: varEnv assumes a1:"P []" and a2:"!!G v T. P G \<Longrightarrow> P ((v,T)#G)" shows "P \<Gamma>" using a1 a2 by (induct \<Gamma>) auto lemma envop_eqvt: fixes pi::"name prm" shows "envop f (pi\<bullet>n) T (pi\<bullet>\<Gamma>) = (pi\<bullet> (envop f n T \<Gamma>))" proof (induct \<Gamma> rule: env_induct) case Nil thus ?case by auto next case (Cons G v T0) thus ?case using pt_bij4[of pi n v] pt_name_inst at_name_inst by auto qed lemma env_plus_eqvt: fixes pi::"name prm" shows "(pi\<bullet>\<Gamma>) \|+ pi\<bullet>eff = pi\<bullet>(\<Gamma> \|+ eff)" by (nominal_induct eff avoiding: \<Gamma> rule: eff.induct) (auto simp add: eff.eqvts envop_eqvt simp del: envop_def) lemma env_minus_eqvt: fixes pi::"name prm" shows "(pi\<bullet>\<Gamma>) \|- pi\<bullet>eff = pi\<bullet>(\<Gamma> \|- eff)" by (nominal_induct eff avoiding: \<Gamma> rule: eff.induct) (auto simp add: eff.eqvts envop_eqvt simp del: envop_def) lemma env_plus_simple_eff: assumes "simple_eff eff" shows "\<Gamma> \|+ eff = \<Gamma>" using prems by (induct eff rule: simple_eff_cases) auto lemma env_minus_simple_eff: assumes "simple_eff eff" shows "\<Gamma> \|- eff = \<Gamma>" using prems by (induct eff rule: simple_eff_cases) auto fun comb_eff :: "eff \<Rightarrow> eff \<Rightarrow> eff \<Rightarrow> eff" where "comb_eff F1 F2 F3 = eff.NE" lemma comb_eff_eqvt[eqvt]: fixes pi :: "name prm" shows "(pi\<bullet> (comb_eff F1 F2 F3)) = comb_eff (pi\<bullet>F1) (pi\<bullet>F2) (pi\<bullet>F3) " using comb_eff.simps by auto text {* type judgments } inductive typing :: "varEnv \<Rightarrow> trm \<Rightarrow> ty \<Rightarrow> eff \<Rightarrow> bool" (" _ \<turnstile> _ : _ ; _ " [60,60,60,60] 60) where T_Var[intro]: "\<lbrakk>valid \<Gamma>; (v,T)\<in>set \<Gamma>\<rbrakk>\<Longrightarrow> \<Gamma> \<turnstile> Var v : T ; VE v" \| T_Const[intro]: "valid \<Gamma> \<Longrightarrow> \<Delta>\<^isub>\<tau> b = T \<Longrightarrow> \<Gamma> \<turnstile> (BI b) : T ; TT" \| T_Num[intro]: "valid \<Gamma> \<Longrightarrow> \<Gamma> \<turnstile> (Num n) : ty.Int ; TT" \| T_True[intro]: "valid \<Gamma> \<Longrightarrow> \<Gamma> \<turnstile> (Bool True) : ty.TT ; TT" \| T_False[intro]: "valid \<Gamma> \<Longrightarrow> \<Gamma> \<turnstile> (Bool False) : ty.FF ; FF" \| T_Abs[intro]: "\<lbrakk>x \<sharp> \<Gamma>; ((x,T1)#\<Gamma>) \<turnstile> b : T2; eff\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Lam [x:T1].b : (T1\<rightarrow>T2 : latent_eff.NE) ; TT" \| T_App[intro]: "\<lbrakk>(\<Gamma> \<turnstile> e1 : U ; eff1) ; \<turnstile> U <: (T0 \<rightarrow> T1 : le); (\<Gamma> \<turnstile> e2 : T; eff2) ; \<turnstile> T <: T0\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> App e1 e2 : T1 ; NE" \| T_AppPred[intro]: "\<lbrakk>\<Gamma> \<turnstile> e1 : U; eff1; \<turnstile> U <: (T0 \<rightarrow> T1 : Latent S); \<Gamma> \<turnstile> e2 : T; VE x ; \<turnstile> T <: T0\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> App e1 e2 : T1 ; TE S x" \| T_If[intro]: "\<lbrakk>\<Gamma> \<turnstile> e1 : T1; eff1; (\<Gamma> \|+ eff1) \<turnstile> e2 : T2; eff2; (\<Gamma> \|- eff1) \<turnstile> e3 : T3; eff3; \<turnstile> T2 <: T; \<turnstile> T3 <: T\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> (Iff e1 e2 e3) : T ; comb_eff eff1 eff2 eff3" \| T_AppPredTrue[intro]: "\<lbrakk>\<Gamma> \<turnstile> e1 : U; eff1; \<turnstile> U <: (T0 \<rightarrow> T1 : Latent S); \<Gamma> \<turnstile> e2 : T; eff2 ; \<turnstile> T <: T0; \<turnstile> T <: S\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> App e1 e2 : T1 ; TT" \| T_AppPredFalse[intro]: "\<lbrakk>\<Gamma> \<turnstile> e1 : U; eff1; \<turnstile> U <: (T0 \<rightarrow> T1 : Latent S); \<Gamma> \<turnstile> e2 : T; eff2 ; \<turnstile> T <: T0; ~(\<turnstile> T <: S) ; e2 : values ; closed e2\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> App e1 e2 : T1 ; FF" \| T_IfTrue[intro]: "\<lbrakk>\<Gamma> \<turnstile> e1 : T1 ; TT ; \<Gamma> \<turnstile> e2 : T2 ; eff; \<turnstile> T2 <: T\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> (Iff e1 e2 e3) : T ; NE" \| T_IfFalse[intro]: "\<lbrakk>\<Gamma> \<turnstile> e1 : T1 ; FF ; \<Gamma> \<turnstile> e3 : T3 ; eff; \<turnstile> T3 <: T\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> (Iff e1 e2 e3) : T ; NE" \| T_AbsPred[intro]: "\<lbrakk>x \<sharp> \<Gamma>; ((x,T1)#\<Gamma>) \<turnstile> b : T2; TE S x\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Lam [x:T1].b : (T1\<rightarrow>T2 : Latent S) ; TT" equivariance typing nominal_inductive typing by (auto simp add: abs_fresh) text { then we begin on preservation } abbreviation "sub" :: "(name\<times>ty) list \<Rightarrow> (name\<times>ty) list \<Rightarrow> bool" (" _ \<lless> _ " [80,80] 80) where "\<Gamma>1 \<lless> \<Gamma>2 \<equiv> \<forall>a \<sigma>. (a,\<sigma>)\<in>set \<Gamma>1 \<longrightarrow> (a,\<sigma>)\<in>set \<Gamma>2" lemma envplus_empty: shows "env_plus eff [] = []" by (nominal_induct rule: eff.induct) auto lemma envminus_empty: shows "env_minus eff [] = []" by (nominal_induct rule: eff.induct) auto lemma w_lem1: fixes \<Gamma> \<Gamma>' assumes "\<Gamma> \<lless> \<Gamma>'" and a:"valid \<Gamma>'" shows "map f \<Gamma> \<lless> map f \<Gamma>'" proof - let ?mapfun = f from prems have "set \<Gamma> <= set \<Gamma>'" by auto hence "set (map ?mapfun \<Gamma>) <= set (map ?mapfun \<Gamma>')" by auto hence "(map ?mapfun \<Gamma>) \<lless> (map ?mapfun \<Gamma>')" by blast thus ?thesis . qed lemma weakening_envplus: assumes b:"\<Gamma> \<lless> \<Gamma>'" and a:"valid \<Gamma>'" shows "env_plus eff \<Gamma> \<lless> env_plus eff \<Gamma>'" using a prems w_lem1[of \<Gamma> \<Gamma>'] by (nominal_induct eff avoiding: \<Gamma> \<Gamma>' rule: eff.induct) auto lemma weakening_envminus: assumes "\<Gamma> \<lless> \<Gamma>'" and a:"valid \<Gamma>'" and b:"valid \<Gamma>" shows "env_minus eff \<Gamma> \<lless> env_minus eff \<Gamma>'" using a prems w_lem1[of \<Gamma> \<Gamma>'] by (nominal_induct eff avoiding: \<Gamma> \<Gamma>' rule: eff.induct) auto lemma envplus_valid: assumes "valid \<Gamma>" shows "valid (\<Gamma> \|+ F)" using prems proof (induct rule: valid.induct) case v1 thus ?case using envplus_empty by auto next case (v2 \<Gamma>' a T) from v2 show ?case proof (nominal_induct rule: eff.induct) case (TE S x) let ?op = "restrict" let ?G = "((a, T) # \<Gamma>')" let ?mapfun = "(% (v,ty). (if (x = v) then (v,?op S ty) else (v,ty)))" have C:"valid (map ?mapfun \<Gamma>')" using v2 `valid (\<Gamma>' \|+ TE S x)` by auto hence D:"a \<sharp> (map ?mapfun \<Gamma>')" using v2 envop_fresh `valid (\<Gamma>' \|+ TE S x)` by auto thus ?case proof (cases "a = x") case False from D have E:"valid ((a,T)# (map ?mapfun \<Gamma>'))" using C by auto from False have B:"?G \|+ TE S x = ((a,T)# (map ?mapfun \<Gamma>'))" by auto thus ?thesis using E by auto next case True hence B:"?G \|+ TE S x = ((a,?op S T)# (map ?mapfun \<Gamma>'))" by auto from D have E:"valid ((a,?op S T)# (map ?mapfun \<Gamma>'))" using C by auto thus ?thesis using B by auto qed next case (VE x) thus ?case proof - let ?op = "remove" let ?G = "((a, T) # \<Gamma>')" let ?mapfun = "(% (v,ty). (if (x = v) then (v,?op ty.FF ty) else (v,ty)))" have C:"valid (map ?mapfun \<Gamma>')" using v2 `valid (\<Gamma>' \|+ VE x)` by auto hence D:"a \<sharp> (map ?mapfun \<Gamma>')" using v2 envop_fresh `valid (\<Gamma>' \|+ VE x)` by auto thus ?case proof (cases "a = x") case False from D have E:"valid ((a,T)# (map ?mapfun \<Gamma>'))" using C by auto from False have B:"?G \|+ VE x = ((a,T)# (map ?mapfun \<Gamma>'))" by auto thus ?thesis using E by auto next case True hence B:"?G \|+ VE x = ((a,?op ty.FF T)# (map ?mapfun \<Gamma>'))" by auto from D have E:"valid ((a,?op ty.FF T)# (map ?mapfun \<Gamma>'))" using C by auto thus ?thesis using B by auto qed qed qed (auto) qed lemma envminus_valid: assumes "valid \<Gamma>" shows "valid (\<Gamma> \|- F)" using prems proof (induct rule: valid.induct) case v1 thus ?case using envminus_empty by auto next case (v2 \<Gamma>' a T) thus ?case proof (nominal_induct rule: eff.induct) case (TE S x) let ?op = "remove" let ?G = "((a, T) # \<Gamma>')" let ?mapfun = "(% (v,ty). (if (x = v) then (v,?op S ty) else (v,ty)))" have C:"valid (map ?mapfun \<Gamma>')" using v2 `valid (\<Gamma>' \|- TE S x)` by auto hence D:"a \<sharp> (map ?mapfun \<Gamma>')" using v2 envop_fresh `valid (\<Gamma>' \|- TE S x)` by auto thus ?case proof (cases "a = x") case False from D have E:"valid ((a,T)# (map ?mapfun \<Gamma>'))" using C by auto from False have B:"?G \|- TE S x = ((a,T)# (map ?mapfun \<Gamma>'))" by auto thus ?thesis using E by auto next case True hence B:"?G \|- TE S x = ((a,?op S T)# (map ?mapfun \<Gamma>'))" by auto from D have E:"valid ((a,?op S T)# (map ?mapfun \<Gamma>'))" using C by auto thus ?thesis using B by auto qed next case (VE x) thus ?case proof - let ?op = "replace" let ?G = "((a, T) # \<Gamma>')" let ?mapfun = "(% (v,ty). (if (x = v) then (v,?op ty.FF ty) else (v,ty)))" have C:"valid (map ?mapfun \<Gamma>')" using v2 `valid (\<Gamma>' \|- VE x)` by auto hence D:"a \<sharp> (map ?mapfun \<Gamma>')" using v2 envop_fresh `valid (\<Gamma>' \|- VE x)` by auto thus ?case proof (cases "a = x") case False from D have E:"valid ((a,T)# (map ?mapfun \<Gamma>'))" using C by auto from False have B:"?G \|- VE x = ((a,T)# (map ?mapfun \<Gamma>'))" by auto thus ?thesis using E by auto next case True hence B:"?G \|- VE x = ((a,?op ty.FF T)# (map ?mapfun \<Gamma>'))" by auto from D have E:"valid ((a,?op ty.FF T)# (map ?mapfun \<Gamma>'))" using C by auto thus ?thesis using B by auto qed qed qed (auto) qed lemma weakening: assumes a: "\<Gamma>1 \<turnstile> t : \<sigma> ; F" and b: "valid \<Gamma>2" and c: "\<Gamma>1 \<lless> \<Gamma>2" and d:"valid \<Gamma>1" shows "\<Gamma>2 \<turnstile> t:\<sigma> ; F" using a b c d proof (nominal_induct \<Gamma>1 t \<sigma> F avoiding: \<Gamma>2 rule: typing.strong_induct) case (T_If \<Gamma> t1 T1 F1 t2 T2 F2 t3 T3 F3 T \<Gamma>2) have A:"valid (\<Gamma> \|+ F1)" using T_If envplus_valid by auto have B:"valid (\<Gamma> \|- F1)" using T_If envminus_valid by auto have A':"valid (\<Gamma>2 \|+ F1)" using T_If envplus_valid by auto have B':"valid (\<Gamma>2 \|- F1)" using T_If envminus_valid by auto have C:"(\<Gamma> \|+ F1) \<lless> (\<Gamma>2 \|+ F1)" using T_If weakening_envplus by auto have D:"(\<Gamma> \|- F1) \<lless> (\<Gamma>2 \|- F1)" using T_If weakening_envminus by auto show ?case using T_If A B C D A' B' by blast qed (auto \| atomize)+ ( FIXME: before using meta-connectives and the new induction ) ( method, this was completely automatic ) ( need weakening lemmas about env+/- ) lemma "[] \<turnstile> (Lam[x:Top]. (Iff (App (BI NumberP) (Var x)) (App (BI Add1) (Var x)) (Num 12))) : (Top \<rightarrow> ty.Int : latent_eff.NE) ; TT" apply (rule T_Abs) apply (auto simp add: fresh_def supp_def perm_fun_def) apply (rule T_If) apply auto apply (rule T_AppPred) apply (auto simp add: valid.intros) apply (rule T_App) apply (rule T_Const) defer apply ( simp add: valid.intros) apply (rule S_Refl) apply (rule T_Var) apply (simp add: valid.intros)+ apply (rule S_Refl) apply (simp add: valid.intros)+ done ( inductive cases about typing ) inductive_cases iff_t_cases : "G \<turnstile> Iff tst thn els : t ; e" inductive_cases app_bi_cases : "G \<turnstile> App (Const (BI b)) rand : t ; e" inductive_cases type_arr_case_num: "\<Gamma> \<turnstile> ((Num n)) : (T1 \<rightarrow> T2 : eff) ; eff'" inductive_cases type_arr_case_bool: "\<Gamma> \<turnstile> ((Bool b)) : (T1 \<rightarrow> T2 : eff) ; eff'" inductive_cases type_bi_case: "\<Gamma> \<turnstile> ((BI b)) : t ; eff" inductive_cases type_add1_case: "\<Gamma> \<turnstile> ((BI Add1)) : t ; eff" inductive_cases bi_typing_cases: "\<Gamma> \<turnstile> (BI b) : t ; eff" inductive_cases abs_ty_int: "\<Gamma> \<turnstile> (Abs x body t) : ty.Int ; eff'" inductive_cases abs_ty_bool: "\<Gamma> \<turnstile> (Abs x body t) : BoolTy ; eff'" inductive_cases const_ty_int: "\<Gamma> \<turnstile> e : ty.Int ; eff'" inductive_cases const_ty_bool: "\<Gamma> \<turnstile> e : BoolTy ; eff'" inductive_cases iff_false_ty: "\<Gamma> \<turnstile> Iff (Bool False) thn els : t ; eff" inductive_cases app_bi_ty: "\<Gamma> \<turnstile> App (BI b) arg : t ; eff" ( Typing Values) lemma false_ty_elim[rule_format]: "\<Gamma> \<turnstile> (trm.Bool False) : \<sigma> ; eff \<Longrightarrow> \<sigma> = ty.FF \<and> eff = FF \<and> valid \<Gamma>" apply (ind_cases "\<Gamma> \<turnstile> (trm.Bool False) : \<sigma> ; eff") apply (auto simp add: trm.inject) done lemma num_ty_elim[rule_format]: "\<Gamma> \<turnstile> (Num n) : \<sigma> ; eff \<Longrightarrow> \<sigma> = ty.Int \<and> eff = TT \<and> valid \<Gamma>" by (ind_cases "\<Gamma> \<turnstile> (Num n) : \<sigma> ; eff") auto lemma true_ty_elim[rule_format]: "\<Gamma> \<turnstile> (trm.Bool True) : \<sigma> ; eff \<Longrightarrow> \<sigma> = ty.TT \<and> eff = TT \<and> valid \<Gamma> " by (ind_cases "\<Gamma> \<turnstile> (trm.Bool True) : \<sigma> ; eff") (auto simp add: trm.inject) lemma bool_ty_elim[rule_format]: "\<Gamma> \<turnstile> (trm.Bool b) : \<sigma> ; eff \<Longrightarrow> (\<sigma> = ty.TT \<or> \<sigma> = ty.FF) \<and> valid \<Gamma> " apply (ind_cases "\<Gamma> \<turnstile> (trm.Bool b) : \<sigma> ; eff") apply (auto simp add: trm.inject) done lemma bi_ty_elim[rule_format]: "\<Gamma> \<turnstile> (BI b) : \<sigma> ; eff \<Longrightarrow> eff = TT \<and> \<sigma> = \<Delta>\<^isub>\<tau> b \<and> valid \<Gamma>" apply (ind_cases "\<Gamma> \<turnstile> (BI b) : \<sigma> ; eff") apply (auto simp add: trm.inject) done inductive_cases bool_sub_int: "\<turnstile> BoolTy <: ty.Int" inductive_cases tt_sub_int: "\<turnstile> ty.TT <: ty.Int" inductive_cases ff_sub_int: "\<turnstile> ty.FF <: ty.Int" thm bool_sub_int inductive_cases arr_sub_int: "\<turnstile> A\<rightarrow>B:L <: ty.Int" inductive_cases int_sub_tt: "\<turnstile> ty.Int <: ty.TT" inductive_cases int_sub_ff: "\<turnstile> ty.Int <: ty.FF" inductive_cases arr_sub_tt: "\<turnstile> A\<rightarrow>B:L <: ty.TT" inductive_cases arr_sub_ff: "\<turnstile> A\<rightarrow>B:L <: ty.FF" inductive_cases int_sub_bool_cases: "\<turnstile> ty.Int <: Union [ty.TT, ty.FF]" inductive_cases arr_sub_bool_cases: "\<turnstile> A\<rightarrow>B:L <: Union [ty.TT, ty.FF]" declare BoolTy_def[simp] lemma int_sub_bool: assumes "\<turnstile> ty.Int <: BoolTy" shows "P" using prems proof - have "\<turnstile> ty.Int <: ty.TT \<or> \<turnstile> ty.Int <: ty.FF" using prems int_sub_bool_cases by auto thus ?thesis using int_sub_ff int_sub_tt by auto qed lemma arr_sub_bool: assumes "\<turnstile> A\<rightarrow>B:L <: BoolTy" shows "P" using prems proof - have "\<turnstile> A\<rightarrow>B:L <: ty.TT \<or> \<turnstile> A\<rightarrow>B:L <: ty.FF" using prems arr_sub_bool_cases[of A B L] by auto thus ?thesis using arr_sub_ff[of A B L] arr_sub_tt[of A B L] by auto qed thm int_sub_bool inductive_cases abs_ty_elim2[consumes 1, case_names 1]: "\<Gamma> \<turnstile> Lam[x:S].b : T ; eff" lemma int_value: assumes a:"v : values" and b:"\<Gamma> \<turnstile> v : T ; F" and c:"\<turnstile> T <: ty.Int" shows "EX n. v = (Num n)" using prems proof (induct rule: values.induct) case num_value thus ?case by auto next case (bool_value b) hence "T = ty.TT \<or> T = ty.FF" using bool_ty_elim by auto thus ?case using bool_value tt_sub_int ff_sub_int by auto next case (abs_value b S x) thus ?case using arr_sub_int by (induct rule: abs_ty_elim2) auto next case (bi_value b) thus ?case using bi_ty_elim[of \<Gamma> b T F] using arr_sub_int by (nominal_induct b rule: builtin.induct) auto qed lemma bool_value: assumes a:"v : values" and b:"\<Gamma> \<turnstile> v : T ; F" and c:"\<turnstile> T <: BoolTy" shows "EX b. v = (Bool b)" using prems proof (induct rule: values.induct) case num_value thus ?case using num_ty_elim[of \<Gamma> _ T F] int_sub_bool by auto next case (bool_value b) thus ?case by auto next case (abs_value b S x) thus ?case using arr_sub_bool by (induct rule: abs_ty_elim2) auto next case (bi_value b) thus ?case using bi_ty_elim[of \<Gamma> b T F] using arr_sub_bool by (nominal_induct b rule: builtin.induct) auto qed lemma value_int_ty: assumes a:"\<Gamma> \<turnstile> e : ty.Int ; eff" and b: "e : values" shows "EX n. e = (Num n)" using b a int_value[of e _ ty.Int] by auto lemma value_bool_ty: assumes a:"\<Gamma> \<turnstile> e : BoolTy ; eff" and b: "e : values" shows "EX b. e = (Bool b)" using b a bool_value[OF b a] by auto lemma typing_bi: assumes a:"\<Gamma> \<turnstile> (BI b) : t ; eff" shows "t = \<Delta>\<^isub>\<tau> b" using a bi_typing_cases[of \<Gamma> b t eff "t = \<Delta>\<^isub>\<tau> b"] by (simp add: trm.inject) inductive_cases arr_sub_arr_cases:"\<turnstile> A1 \<rightarrow> A2 : L <: B1 \<rightarrow> B2 : L'" lemma typed_prim_reduce: assumes a:"\<Gamma> \<turnstile> (BI b) : U ; eff" and b:"\<Gamma> \<turnstile> v : T ; eff'" and c:"v : values" and sub:"\<turnstile> T <: T0" and d: "\<turnstile> U <: T0 \<rightarrow> T1 : le" shows "EX v'. \<Delta> b v = Some v'" using a b c d sub proof (nominal_induct b rule: builtin.induct) case Add1 have "U = \<Delta>\<^isub>\<tau> Add1" using Add1 typing_bi[of \<Gamma> Add1 "U" eff] by simp hence "U = ty.Int \<rightarrow> ty.Int : latent_eff.NE" by auto hence "\<turnstile> T <: ty.Int " and "le = latent_eff.NE" using sub d arr_sub_arr_cases[of ty.Int ty.Int latent_eff.NE T0 T1 le] by auto hence "EX n. v = (Num n)" using c b int_value by auto then obtain n where "v = (Num n)" by auto thus ?case by (auto simp add: \<Delta>.simps) next case Nott show ?case using `v : values` Nott by (induct v rule: values.induct) auto next case NumberP thus ?case by auto next case BooleanP thus ?case by auto next case ProcP thus ?case by auto qed text { Progress together with necessary lemmas } ( first some lemmas about decomposing various kinds of syntax ) lemma if_decomp: assumes b:"closed tst \<Longrightarrow> (\<exists>E L R. tst = E L \<and> E \<in> ctxt \<and> L \<hookrightarrow> R) \<or> tst : values" and c:"closed (Iff tst thn els)" shows "(EX E L R. (Iff tst thn els) = E L \<and> E : ctxt \<and> L \<hookrightarrow> R) \<or> (Iff tst thn els) : values" proof - { assume "tst : values" hence "EX E L R. Iff tst thn els = E L \<and> E : ctxt \<and> (L \<hookrightarrow> R)" using if_val_reduces[of tst thn els] ex_help[of "Iff tst thn els" "(%t. t)"] by auto } moreover { assume asm:"~ tst : values" have cl:"closed tst" using `closed (Iff tst thn els)` by (auto simp add: closed_def trm.supp) hence ih:"\<exists>E t t'. tst = E t \<and> E \<in> ctxt \<and> t \<hookrightarrow> t'" using b asm by auto hence "\<exists>E t t' . Iff tst thn els = E t \<and> E \<in> ctxt \<and> t \<hookrightarrow> t'" using reduce_in_ctxt[of "(%t . (Iff t thn els))"] ih by auto } ultimately show ?thesis by auto qed inductive_cases tt_sub_arr_cases: "\<turnstile> ty.TT <:A1 \<rightarrow> A2 : L" inductive_cases ff_sub_arr_cases: "\<turnstile> ty.FF <:A1 \<rightarrow> A2 : L" inductive_cases num_sub_arr_cases: "\<turnstile> ty.Int <:A1 \<rightarrow> A2 : L" lemma app_decomp: assumes a:" G \<turnstile> rator : U ; eff1" and a':"\<turnstile> U <: T0 \<rightarrow> T1 : le" and aa:" G \<turnstile> rand : T ; eff" and b:"closed rator \<Longrightarrow> (\<exists>E L R. rator = E L \<and> E \<in> ctxt \<and> L \<hookrightarrow> R) \<or> rator \<in> values" and bb:"closed rand \<Longrightarrow> (\<exists>E L R. rand = E L \<and> E \<in> ctxt \<and> L \<hookrightarrow> R) \<or> rand \<in> values" and c:"closed (App rator rand)" and sub:"\<turnstile> T <: T0" shows "(EX E L R. (App rator rand) = E L \<and> E : ctxt \<and> L \<hookrightarrow> R) \<or> (App rator rand) : values" proof - have well_typed:"G \<turnstile> (App rator rand) : T1 ; eff.NE" using T_App[of G rator U eff1 T0 T1 le rand T eff] a a' aa sub by auto have "closed rator" using c by (auto simp add: closed_def trm.supp) hence ih1:"(\<exists>E L R. rator = E L \<and> E \<in> ctxt \<and> L \<hookrightarrow> R) \<or> rator \<in> values" using b by simp have "closed rand" using c by (auto simp add: closed_def trm.supp) hence ih2:"(\<exists>E L R. rand = E L \<and> E \<in> ctxt \<and> L \<hookrightarrow> R) \<or> rand \<in> values" using bb by simp { assume asm1:"rator \<in> values" and asm2:"rand \<in> values" hence "(EX E L R. (App rator rand) = E L \<and> E : ctxt \<and> L \<hookrightarrow> R)" using asm1 a aa a' sub well_typed proof (nominal_induct avoiding: rand rule: values.strong_induct ) case (abs_value x t b) let ?E = "(%t. t)" let ?L = "App (Abs x b t) rand" have "?L \<hookrightarrow> (b[x::=rand])" by (rule e_beta) thus ?case using ex_help[of ?L ?E ?L] by auto next case (bool_value b) hence "U = ty.TT \<or> U = ty.FF" using true_ty_elim false_ty_elim by (cases b) auto thus ?case using tt_sub_arr_cases[of T0 T1 le] ff_sub_arr_cases[of T0 T1 le] `\<turnstile> U <: T0 \<rightarrow> T1 : le` by blast next case (num_value b) hence "U = ty.Int" using num_ty_elim by auto thus ?case using num_sub_arr_cases[of T0 T1 le] num_value by auto next case (bi_value b) let ?E = "(%t. t)" let ?L = "App ((BI b)) rand" have h:"\<And>v. (\<Delta> b rand) = (Some v) \<Longrightarrow> App ((BI b)) rand \<hookrightarrow> v" using bi_value by auto have "EX v . (\<Delta> b rand) = (Some v)" using bi_value typed_prim_reduce[of G b U] by auto then obtain v' where "(\<Delta> b rand) = (Some v')" by auto then show ?case using h[of v'] ex_help[of ?L ?E] by auto qed } moreover { assume asm1:"rator \<in> values" and asm2:"rand \<notin> values" have "\<exists>E t t' . App rator rand = E t \<and> E \<in> ctxt \<and> t \<hookrightarrow> t'" using asm1 asm2 reduce_in_ctxt[of "(%t . (App rator t))"] ih2 by auto } moreover { assume asm:"rator \<notin> values" have "\<exists>E t t' . App rator rand = E t \<and> E \<in> ctxt \<and> t \<hookrightarrow> t'" using asm reduce_in_ctxt[of "(%t . (App t rand))"] ih1 by auto } ultimately show ?thesis by auto qed ( then the main lemma, that every well typed term can be decomposed into a context and a redex ) lemma decomposition: assumes a:"closed e" and b:"\<Gamma> \<turnstile> e : t ; eff" shows "(EX E L R. e = E L \<and> E : ctxt \<and> L \<hookrightarrow> R) \<or> e : values" using b a proof (induct rule: typing.induct) case T_Var thus ?case using closed_def by (auto simp add: at_supp at_name_inst trm.supp) next case T_If thus ?case using if_decomp by auto next case T_IfTrue thus ?case using if_decomp by auto next case T_IfFalse thus ?case using if_decomp by auto next case T_App thus ?case using app_decomp by auto next case T_AppPred thus ?case using app_decomp by auto next case T_AppPredTrue thus ?case using app_decomp by auto next case T_AppPredFalse thus ?case using app_decomp by auto qed (simp_all) ( The value cases are automatic ) ( Now we conclude progress ) theorem progress: assumes typed:"\<Gamma> \<turnstile> e : t ; eff" and cl:"closed e" shows "e : values \<or> (EX e'. e \<longrightarrow> e')" proof (cases "e : values") case False hence "(\<exists>E L R. e = E L \<and> E \<in> ctxt \<and> L \<hookrightarrow> R)" using decomposition[OF cl typed] by auto thus ?thesis by auto qed (simp) ( Some useful lemmas for deriving facts from typing derivations ) inductive_cases app_ty_elim2[consumes 1, case_names 1 2 3 4]: "\<Gamma> \<turnstile> App t1 t2 : \<sigma> ; eff" thm app_ty_elim2 inductive_cases iff_ty_elim2[consumes 1, case_names 1 2 3]: "\<Gamma> \<turnstile> Iff t1 t2 t3 : T ; eff" thm abs_ty_elim2 ( slow ) lemma app_ty_elim[rule_format]: "\<Gamma> \<turnstile> App t1 t2 : \<sigma> ; eff \<Longrightarrow> \<exists> T0 T0' T1 le eff' eff'' U. (\<Gamma> \<turnstile> t1 : U ; eff' \<and> \<Gamma> \<turnstile> t2 : T0' ; eff'' \<and> \<turnstile> U <: T0 \<rightarrow> T1 : le \<and> \<turnstile> T0' <: T0 \<and> T1 = \<sigma>)" apply (ind_cases "\<Gamma> \<turnstile> App t1 t2 : \<sigma> ; eff") apply (auto simp add: trm.inject ty.inject) by metis+ lemma abs_ty_elim_eff[rule_format]: "\<lbrakk>\<Gamma> \<turnstile> Lam[a:T0].b : \<sigma> ; eff\<rbrakk> \<Longrightarrow> eff = eff.TT" by (ind_cases "\<Gamma> \<turnstile> Lam[a:T0].b : \<sigma> ; eff") (auto simp add: trm.inject) lemma abs_ty_elim[rule_format]: "\<lbrakk>\<Gamma> \<turnstile> Lam[a:T0].b : \<sigma> ; eff ; a \<sharp> \<Gamma>\<rbrakk> \<Longrightarrow> \<exists> T1 eff' L S. ((a,T0)#\<Gamma> \<turnstile> b : T1 ; eff' \<and> \<sigma> = (T0 \<rightarrow> T1 : L) \<and> eff = eff.TT \<and> ((eff' = TE S a \<and> L = Latent S) \<or> L = latent_eff.NE))" apply (ind_cases "\<Gamma> \<turnstile> Lam[a:T0].b: \<sigma> ; eff") apply(auto simp add: trm.distinct trm.inject alpha) apply(drule_tac pi="[(a,x)]::name prm" in typing.eqvt) apply(auto) apply(subgoal_tac "([(a,x)]::name prm)\<bullet>\<Gamma> = \<Gamma>")(A) apply(force simp add: calc_atm) (A) apply(force intro!: pt_fresh_fresh[OF pt_name_inst, OF at_name_inst]) apply(drule_tac pi="[(a,x)]::name prm" in typing.eqvt) apply(auto) apply(subgoal_tac "([(a,x)]::name prm)\<bullet>\<Gamma> = \<Gamma>")(A) apply(force simp add: calc_atm) (A) apply(force intro!: pt_fresh_fresh[OF pt_name_inst, OF at_name_inst]) done inductive_cases abs_ty_cases[consumes 1, case_names 1 2]:"\<Gamma> \<turnstile> Lam[a:T0].b: \<sigma> ; eff" thm abs_ty_cases ( lemma app_abs_ty_elim_eff[rule_format]: "\<lbrakk>\<Gamma> \<turnstile> App (Abs x b T) t2 : \<sigma> ; eff ; x \<sharp> \<Gamma>\<rbrakk> \<Longrightarrow> eff = eff.NE" proof (ind_cases "\<Gamma> \<turnstile> App (Abs x b T) t2 : \<sigma> ; eff", auto simp add: trm.inject abs_ty_elim) fix T0 S eff1 U assume "\<Gamma> \<turnstile> Lam [x:T].b : U ; eff1 "" \<turnstile> U <: T0 \<rightarrow> \<sigma> : Latent S" "x \<sharp> \<Gamma>" then obtain T1 L where "U = T \<rightarrow> T1 : L" using abs_ty_elim[of \<Gamma> x b T "U" L] by auto thus False using prems arr_sub_arr_cases[of T _ latent_eff.NE T0 \<sigma> "Latent S"] by (auto simp add: ty.inject) next fix T0 S eff1 U assume "\<Gamma> \<turnstile> Lam [x:T].b : U ; eff1 "" \<turnstile> U <: T0 \<rightarrow> \<sigma> : Latent S" "x \<sharp> \<Gamma>" then obtain T1 where "U = T \<rightarrow> T1 : latent_eff.NE" using abs_ty_elim[of \<Gamma> x b T "U"] by auto thus False using prems arr_sub_arr_cases[of T _ latent_eff.NE T0 \<sigma> "Latent S"] by (auto simp add: ty.inject) next fix T0 S eff1 U assume "\<Gamma> \<turnstile> Lam [x:T].b : U ; eff1 "" \<turnstile> U <: T0 \<rightarrow> \<sigma> : Latent S" "x \<sharp> \<Gamma>" then obtain T1 where "U = T \<rightarrow> T1 : latent_eff.NE" using abs_ty_elim[of \<Gamma> x b T "U"] by auto thus False using prems arr_sub_arr_cases[of T _ latent_eff.NE T0 \<sigma> "Latent S"] by (auto simp add: ty.inject) qed ) lemma if_ty_elim_weak[rule_format]: "\<Gamma> \<turnstile> Iff t1 t2 t3: \<sigma> ; eff \<Longrightarrow> \<exists> T0 eff'. (\<Gamma> \<turnstile> t1 : T0 ; eff') \<and> eff = NE" by (ind_cases "\<Gamma> \<turnstile> Iff t1 t2 t3: \<sigma> ; eff") (auto simp add: trm.inject) lemma if_ty_elim[rule_format]: "\<Gamma> \<turnstile> Iff t1 t2 t3: \<sigma> ; eff \<Longrightarrow> (\<exists> T1 T2 T3 F1 F2 F3. (\<Gamma> \<turnstile> t1 : T1 ; F1) \<and> \<Gamma> \|+ F1 \<turnstile> t2 : T2 ; F2 \<and> \<Gamma> \|- F1 \<turnstile> t3 : T3 ; F3 \<and> \<turnstile> T2 <: \<sigma> \<and> \<turnstile> T3 <: \<sigma> \<and> eff = NE) \<or> (\<exists> T1 T3 F3. (\<Gamma> \<turnstile> t1 : T1 ; FF) \<and> \<Gamma> \<turnstile> t3 : T3 ; F3 \<and> \<turnstile> T3 <: \<sigma> \<and> eff = NE) \<or> (\<exists> T1 T2 F2. (\<Gamma> \<turnstile> t1 : T1 ; TT) \<and> \<Gamma> \<turnstile> t2 : T2 ; F2 \<and> \<turnstile> T2 <: \<sigma> \<and> eff = NE)" proof (ind_cases "\<Gamma> \<turnstile> Iff t1 t2 t3: \<sigma> ; eff") fix e1 T1 eff1 e2 T2 eff2 e3 T3 eff3 assume "Iff t1 t2 t3 = Iff e1 e2 e3"" eff = eff.NE"" \<Gamma> \<turnstile> e1 : T1 ; eff1"" env_plus eff1 \<Gamma> \<turnstile> e2 : T2 ; eff2 " "env_minus eff1 \<Gamma> \<turnstile> e3 : T3 ; eff3"" \<turnstile> T2 <: \<sigma>"" \<turnstile> T3 <: \<sigma>" hence A:"t1 = e1" "t2 = e2" "t3 = e3" using trm.inject by auto thus ?thesis using prems by blast next fix e1 T1 e2 T2 effa e3 assume "Iff t1 t2 t3 = Iff e1 e2 e3"" eff = eff.NE"" \<Gamma> \<turnstile> e1 : T1 ; TT "" \<Gamma> \<turnstile> e2 : T2 ; effa "" \<turnstile> T2 <: \<sigma>" thus ?thesis by (auto simp add: trm.inject) next fix e1 T1 e2 T3 effa e3 assume "Iff t1 t2 t3 = Iff e1 e2 e3"" eff = eff.NE"" \<Gamma> \<turnstile> e1 : T1 ; FF "" \<Gamma> \<turnstile> e3 : T3 ; effa "" \<turnstile> T3 <: \<sigma>" thus ?thesis by (auto simp add: trm.inject) qed inductive_cases iff_false_ty_cases: "\<Gamma> \<turnstile> Iff (trm.Bool False) t2 t3: \<sigma> ; eff" inductive_cases ff_eff_cases: "\<Gamma> \<turnstile> e : T; FF" inductive_cases tt_eff_cases: "\<Gamma> \<turnstile> e : T; TT" inductive_cases ne_eff_cases: "\<Gamma> \<turnstile> e : T; NE" lemma if_true_ty_elim[rule_format]: "\<lbrakk>\<Gamma> \<turnstile> Iff v t2 t3: \<sigma> ; eff ; v : values; v ~= Bool False\<rbrakk> \<Longrightarrow> \<exists> T0 eff'. ((\<Gamma> \<turnstile> t2 : T0 ; eff') \<and> \<turnstile> T0 <: \<sigma> \<and> eff = NE)" proof (ind_cases "\<Gamma> \<turnstile> Iff (v) t2 t3: \<sigma> ; eff") fix eff1 eff2 T1 T2 e1 e2 e3 assume "v : values" "env_plus eff1 \<Gamma> \<turnstile> e2 : T2 ; eff2" "Iff v t2 t3 = Iff e1 e2 e3" "\<turnstile> T2 <: \<sigma>" "\<Gamma> \<turnstile> e1 : T1 ; eff1" "eff = NE" have "\<Gamma> \<turnstile> v : T1 ; eff1" using prems by (simp add: trm.inject) have "eff1 = eff.NE \<or> eff1 = FF \<or> eff1 = TT" using `v : values` `\<Gamma> \<turnstile> v : T1 ; eff1` proof (induct rule: values.induct) case (abs_value b T x) thus ?case using abs_value abs_ty_elim_eff by auto next case (num_value n) thus ?case using num_ty_elim by auto next case (bool_value n) thus ?case using false_ty_elim true_ty_elim by (cases n) auto next case bi_value thus ?case using bi_ty_elim by auto qed hence "env_plus eff1 \<Gamma> = \<Gamma>" by auto hence "\<Gamma> \<turnstile> e2 : T2 ; eff2 \<and> \<turnstile> T2 <: \<sigma>" using prems by auto thus ?thesis using prems by (auto simp add: trm.inject) next fix e1 T2 e2 e3 effa assume "Iff v t2 t3 = Iff e1 e2 e3" "\<Gamma> \<turnstile> e2 : T2 ; effa" "\<turnstile> T2 <: \<sigma>" "eff = NE" thus ?thesis by (auto simp add: trm.inject) next fix e1 T1 e3 T3 effa e2 assume "v : values" "v \<noteq> trm.Bool False" "Iff v t2 t3 = Iff e1 e2 e3" "\<Gamma> \<turnstile> e1 : T1 ; FF" "eff = NE" have "v = e1" using prems trm.inject by auto hence tv:"\<Gamma> \<turnstile> v : T1 ; FF" by simp have "v = Bool False" using `v : values` tv proof (induct v rule: values.induct) case (abs_value x T b) thus ?case using abs_ty_elim_eff[of \<Gamma> x b T T1 FF] by auto next case (num_value n) thus ?case using num_ty_elim[of _ _ _ FF] by auto next case (bool_value n) thus ?case using true_ty_elim[of _ _ FF] by (cases n) auto next case bi_value thus ?case using bi_ty_elim[of _ _ _ FF] by auto qed show ?thesis using `v = Bool False` `v ~= Bool False` by auto qed lemma if_false_ty_elim[rule_format]: "\<Gamma> \<turnstile> Iff (trm.Bool False) t2 t3: \<sigma> ; eff \<Longrightarrow> \<exists> T0 eff'. ((\<Gamma> \<turnstile> t3 : T0 ; eff') \<and> \<turnstile> T0 <: \<sigma> \<and> eff = eff.NE)" proof (ind_cases "\<Gamma> \<turnstile> Iff (trm.Bool False) t2 t3: \<sigma> ; eff") fix e1 e2 e3 T1 assume "Iff (trm.Bool False) t2 t3 = Iff e1 e2 e3" and "\<Gamma> \<turnstile> e1 : T1 ; TT" hence "\<Gamma> \<turnstile> (trm.Bool False) : T1 ; TT" by (simp add: trm.inject) hence "TT = FF" using false_ty_elim[of \<Gamma> T1 TT] by simp thus ?thesis by (simp) next fix e1 e2 e3 T1 T3 effa assume "Iff (trm.Bool False) t2 t3 = Iff e1 e2 e3" and " eff = eff.NE"and" \<Gamma> \<turnstile> e1 : T1 ; FF" and " \<Gamma> \<turnstile> e3 : T3 ; effa"and " \<turnstile> T3 <: \<sigma>" thus ?thesis by (auto simp add: trm.inject) next fix e1 e2 e3 T1 T3 eff1 eff3 assume "Iff (trm.Bool False) t2 t3 = Iff e1 e2 e3" and "eff = eff.NE" and f:"\<Gamma> \<turnstile> e1 : T1 ; eff1" and g:"env_minus eff1 \<Gamma> \<turnstile> e3 : T3 ; eff3" and "\<turnstile> T3 <: \<sigma>" hence "e1 = Bool False" and "t3 = e3" by (auto simp add: trm.inject) hence "eff1 = FF" using f false_ty_elim by auto hence "env_minus eff1 \<Gamma> = \<Gamma>" by simp hence "\<Gamma> \<turnstile> e3 : T3 ; eff3" using g by simp thus ?thesis using `t3 = e3` `\<turnstile> T3 <: \<sigma>` `eff = eff.NE` by auto qed lemma var_ty_elim: "\<Gamma> \<turnstile> Var v : \<sigma> ; eff \<Longrightarrow> (v,\<sigma>) : set \<Gamma> \<and> eff = VE v \<and> valid \<Gamma>" by (ind_cases "\<Gamma> \<turnstile> Var v : \<sigma> ; eff") (auto simp add: trm.inject) inductive_cases app_ty_ff:"\<Gamma> \<turnstile> App e arg : T' ; FF" lemma app_ty_ff_elim: "\<Gamma> \<turnstile> App rator rand : T ; FF \<Longrightarrow> EX S T0 le F1 T0' F2 U. \<Gamma> \<turnstile> rator : U ; F1 \<and> \<Gamma> \<turnstile> rand : T0' ; F2 \<and> \<turnstile> T0' <: T0 \<and> \<turnstile> U <: T0 \<rightarrow> T : le \<and> le = Latent S \<and> (~ (\<turnstile> T0' <: S)) \<and> rand : values \<and> closed rand" proof (ind_cases "\<Gamma> \<turnstile> App rator rand : T ; FF") fix e1 T0 S eff1 e2 Ta eff2 U assume "App rator rand = App e1 e2 "" \<Gamma> \<turnstile> e1 : U ; eff1 " "\<turnstile> U <: T0 \<rightarrow> T : Latent S" "\<Gamma> \<turnstile> e2 : Ta ; eff2 ""\<turnstile> Ta <: T0""~ (\<turnstile> Ta <: S)" "e2 : values" "closed e2" have a:" \<Gamma> \<turnstile> rator : U ; eff1 " using prems trm.inject by auto have b:"\<Gamma> \<turnstile> rand : Ta ; eff2" using prems trm.inject by auto have c:"rand : values" "closed rand" using prems trm.inject by auto from a b c prems show ?thesis by blast qed lemma app_ty_tt_elim: "\<Gamma> \<turnstile> App rator rand : T ; TT \<Longrightarrow> EX S T0 le F1 T0' F2 U. \<Gamma> \<turnstile> rator : U ; F1 \<and> \<Gamma> \<turnstile> rand : T0' ; F2 \<and> \<turnstile> T0' <: T0 \<and> \<turnstile> U <: T0 \<rightarrow> T : le \<and> le = Latent S \<and> \<turnstile> T0' <: S " proof (ind_cases "\<Gamma> \<turnstile> App rator rand : T ; TT") fix e1 T0 S eff1 e2 Ta eff2 U assume "App rator rand = App e1 e2 "" \<Gamma> \<turnstile> e1 : U ; eff1 " "\<turnstile> U <: T0 \<rightarrow> T : Latent S" "\<Gamma> \<turnstile> e2 : Ta ; eff2 ""\<turnstile> Ta <: T0"" \<turnstile> Ta <: S" have a:" \<Gamma> \<turnstile> rator : U ; eff1 " using prems trm.inject by auto have b:"\<Gamma> \<turnstile> rand : Ta ; eff2" using prems trm.inject by auto from a b prems show ?thesis by blast qed text { values don't have union types } lemma value_simple_type: assumes A:"\<Gamma> \<turnstile> v : T ; F" and B:"v : values" shows "simple_ty T" using B A proof (nominal_induct v avoiding: \<Gamma> rule: values.strong_induct) case (bi_value b) thus ?case using bi_ty_elim[of \<Gamma> b T F] by (nominal_induct b rule: builtin.induct) (auto simp add: trm.inject) next case num_value thus ?case using num_ty_elim[OF num_value] by auto next case bool_value thus ?case using bool_ty_elim[OF bool_value] by auto next case (abs_value a T' b \<Gamma>') thus ?case using abs_ty_elim[of \<Gamma>' a b T' T F] by auto qed text { lemmas about the effects of closed terms } lemma ve_not_closed: "\<Gamma> \<turnstile> e : T ; eff.VE x \<Longrightarrow> x : supp e" by (ind_cases "\<Gamma> \<turnstile> e : T ; eff.VE x") (auto simp add: eff.inject trm.supp at_supp supp_atm) lemma te_not_closed: "\<Gamma> \<turnstile> e : T ; eff.TE T' x \<Longrightarrow> x : supp e" proof (ind_cases "\<Gamma> \<turnstile> e : T ; eff.TE T' x") fix e1 T0 S eff1 e2 Ta xa assume "e = App e1 e2" "TE T' x = TE S xa" "\<Gamma> \<turnstile> e2 : Ta ; VE xa" have "x = xa" using prems eff.inject by auto hence "\<Gamma> \<turnstile> e2 : Ta ; VE x" using prems eff.inject by auto hence "x : supp e2" using ve_not_closed by auto thus "x : supp e" using prems trm.supp by auto qed lemma closed_eff: assumes "closed e" and "\<Gamma> \<turnstile> e : T ; eff" shows "simple_eff eff" using prems proof (nominal_induct eff rule: eff.induct) case (VE name) thus ?case using ve_not_closed[of _ e _ name] closed_def by auto next case (TE _ name) thus ?case using te_not_closed[of _ e _ _ name] closed_def by auto qed (auto) lemma closed_eff_below_NE: assumes "closed e" and "\<Gamma> \<turnstile> e : T ; eff" shows "\<turnstile> eff <e: eff.NE" using closed_eff simple_eff_below_ne prems by auto inductive_cases const_ty_int_case: "\<Gamma> \<turnstile> (Num n) : ty.Int ; eff" inductive_cases trm_ty_int_case: "\<Gamma> \<turnstile> e : ty.Int ; eff" inductive_cases const_ty_bool_case: "\<Gamma> \<turnstile> (Bool b) : BoolTy ; eff" lemma add1_eff_ne: "\<Gamma> \<turnstile> (App (BI Add1) v) : T1 ; eff1 \<Longrightarrow> eff1 = eff.NE" proof (ind_cases " \<Gamma> \<turnstile> (App (BI Add1) v) : T1 ; eff1") assume "eff1 = eff.NE" thus ?thesis by simp next fix T0 T1 S eff1a e1 e2 U assume "\<Gamma> \<turnstile> e1 : U ; eff1a" "App (BI Add1) v = App e1 e2" "\<turnstile> U <: T0 \<rightarrow> T1 : Latent S" hence "\<Gamma> \<turnstile> (BI Add1) : U ; eff1a" by (simp add: trm.inject) hence "\<Delta>\<^isub>\<tau> Add1 = U" using type_add1_case[of \<Gamma> "U"] by (auto simp add: trm.inject) hence "\<dots> = ty.Int \<rightarrow> ty.Int : latent_eff.NE" by auto thus ?thesis using prems arr_sub_arr_cases[of ty.Int ty.Int latent_eff.NE] by (auto simp add: ty.inject) next fix T0 T1 S eff1a e1 e2 U assume "\<Gamma> \<turnstile> e1 : U ; eff1a" "App (BI Add1) v = App e1 e2" "\<turnstile> U <: T0 \<rightarrow> T1 : Latent S" hence "\<Gamma> \<turnstile> (BI Add1) : U ; eff1a" by (simp add: trm.inject) hence "\<Delta>\<^isub>\<tau> Add1 = U" using type_add1_case[of \<Gamma> "U"] by (auto simp add: trm.inject) hence "\<dots> = ty.Int \<rightarrow> ty.Int : latent_eff.NE" by auto thus ?thesis using prems arr_sub_arr_cases[of ty.Int ty.Int latent_eff.NE] by (auto simp add: ty.inject) next fix T0 T1 S eff1a e1 e2 U assume "\<Gamma> \<turnstile> e1 : U ; eff1a" "App (BI Add1) v = App e1 e2" "\<turnstile> U <: T0 \<rightarrow> T1 : Latent S" hence "\<Gamma> \<turnstile> (BI Add1) : U ; eff1a" by (simp add: trm.inject) hence "\<Delta>\<^isub>\<tau> Add1 = U" using type_add1_case[of \<Gamma> "U"] by (auto simp add: trm.inject) hence "\<dots> = ty.Int \<rightarrow> ty.Int : latent_eff.NE" by auto thus ?thesis using prems arr_sub_arr_cases[of ty.Int ty.Int latent_eff.NE] by (auto simp add: ty.inject) qed inductive_cases type_nott_case: "\<Gamma> \<turnstile> (BI Nott) : t ; e" lemma nott_eff_ne: "\<Gamma> \<turnstile> (App (BI Nott) v) : T1 ; eff1 \<Longrightarrow> eff1 = eff.NE" proof (ind_cases " \<Gamma> \<turnstile> (App (BI Nott) v) : T1 ; eff1") assume "eff1 = eff.NE" thus ?thesis by simp next fix T0 T1 S eff1a e1 e2 U assume "\<Gamma> \<turnstile> e1 : U ; eff1a" "App (BI Nott) v = App e1 e2" "\<turnstile> U <: T0 \<rightarrow> T1 : Latent S" hence "\<Gamma> \<turnstile> (BI Nott) : U ; eff1a" by (simp add: trm.inject) hence "\<Delta>\<^isub>\<tau> Nott = U" using type_nott_case[of \<Gamma> "U"] by (auto simp add: trm.inject) hence "\<dots> = Top \<rightarrow> BoolTy : latent_eff.NE" by auto thus ?thesis using prems arr_sub_arr_cases[of Top BoolTy latent_eff.NE] by (auto simp add: ty.inject) next fix T0 T1 S eff1a e1 e2 U assume "\<Gamma> \<turnstile> e1 : U ; eff1a" "App (BI Nott) v = App e1 e2" "\<turnstile> U <: T0 \<rightarrow> T1 : Latent S" hence "\<Gamma> \<turnstile> (BI Nott) : U ; eff1a" by (simp add: trm.inject) hence "\<Delta>\<^isub>\<tau> Nott = U" using type_nott_case[of \<Gamma> "U"] by (auto simp add: trm.inject) hence "\<dots> = Top \<rightarrow> BoolTy : latent_eff.NE" by auto thus ?thesis using prems arr_sub_arr_cases[of Top BoolTy latent_eff.NE] by (auto simp add: ty.inject) next fix T0 T1 S eff1a e1 e2 U assume "\<Gamma> \<turnstile> e1 : U ; eff1a" "App (BI Nott) v = App e1 e2" "\<turnstile> U <: T0 \<rightarrow> T1 : Latent S" hence "\<Gamma> \<turnstile> (BI Nott) : U ; eff1a" by (simp add: trm.inject) hence "\<Delta>\<^isub>\<tau> Nott = U" using type_nott_case[of \<Gamma> "U"] by (auto simp add: trm.inject) hence "\<dots> = Top \<rightarrow> BoolTy : latent_eff.NE" by auto thus ?thesis using prems arr_sub_arr_cases[of Top BoolTy latent_eff.NE] by (auto simp add: ty.inject) qed inductive_cases type_booleanp_case: "\<Gamma> \<turnstile> (BI BooleanP) : t ; e" lemma value_eff: assumes "v : values" and "\<Gamma> \<turnstile> v : T ; eff" shows "simple_eff eff" using prems proof (induct v rule :values.induct) case abs_value thus ?case using abs_ty_elim_eff by auto next case num_value thus ?case using num_ty_elim by auto next case (bool_value b) thus ?case using true_ty_elim false_ty_elim by (cases b) auto next case bi_value thus ?case using bi_ty_elim by auto qed lemma booleanp_eff_simple: "\<lbrakk>\<Gamma> \<turnstile> (App (BI BooleanP) v) : T1 ; eff1 ; v : values\<rbrakk> \<Longrightarrow> simple_eff eff1" proof (ind_cases " \<Gamma> \<turnstile> (App (BI BooleanP) v) : T1 ; eff1") fix T0 T1 S eff1a e1 e2 xa T assume "\<Gamma> \<turnstile> e2 : T ; VE xa" "App (BI BooleanP) v = App e1 e2" "v : values" hence "VE xa = eff.NE \<or> VE xa = eff.TT \<or> VE xa = eff.FF " using value_eff[of v \<Gamma> T "VE xa"] by (auto simp add: trm.inject) thus ?thesis using ty.distinct by auto qed (auto) lemma numberp_eff_simple: "\<lbrakk>\<Gamma> \<turnstile> (App (BI NumberP) v) : T1 ; eff1 ; v : values\<rbrakk> \<Longrightarrow> simple_eff eff1" proof (ind_cases " \<Gamma> \<turnstile> (App (BI NumberP) v) : T1 ; eff1") fix T0 T1 S eff1a e1 e2 xa T assume "\<Gamma> \<turnstile> e2 : T ; VE xa" "App (BI NumberP) v = App e1 e2" "v : values" hence "VE xa = eff.NE \<or> VE xa = eff.TT \<or> VE xa = eff.FF " using value_eff[of v \<Gamma> T "VE xa"] by (auto simp add: trm.inject) thus ?thesis using ty.distinct by auto qed (auto) lemma procp_eff_simple: "\<lbrakk>\<Gamma> \<turnstile> (App (BI ProcP) v) : T1 ; eff1 ; v : values\<rbrakk> \<Longrightarrow> simple_eff eff1" proof (ind_cases " \<Gamma> \<turnstile> (App (BI ProcP) v) : T1 ; eff1") fix T0 T1 S eff1a e1 e2 xa T assume "\<Gamma> \<turnstile> e2 : T ; VE xa" "App (BI ProcP) v = App e1 e2" "v : values" hence "VE xa = eff.NE \<or> VE xa = eff.TT \<or> VE xa = eff.FF " using value_eff[of v \<Gamma> T "VE xa"] by (auto simp add: trm.inject) thus ?thesis using ty.distinct by auto qed (auto) inductive_cases app_boolp_ff: "\<Gamma> \<turnstile> (App (BI BooleanP) v) : T1 ; FF" lemma booleanp_FF_preserved: assumes "\<Gamma> \<turnstile> (App (BI BooleanP) v) : T1 ; FF" and "v : values" and "\<Delta> BooleanP v = Some u" shows "u = Bool False" using `v : values` prems proof (induct v rule: values.induct) case (bool_value b) let ?P = "\<Gamma> \<turnstile> App (BI BooleanP) (trm.Bool b) : T1 ; FF" have "?P ==> ?case" proof (ind_cases ?P) fix S T e1 e2 T0 eff1 eff2 U assume "App (BI BooleanP) (trm.Bool b) = App e1 e2" "\<Gamma> \<turnstile> e1 : U ; eff1 " "\<Gamma> \<turnstile> e2 : T ; eff2" "~ (\<turnstile> T <: S)" "e2 : values" "\<turnstile> U <: T0 \<rightarrow> T1 : Latent S" hence a:"\<Gamma> \<turnstile> (BI BooleanP) : U ; eff1 " using trm.inject by auto from prems have b:"\<Gamma> \<turnstile> Bool b : T ; eff2" using trm.inject by auto have "U = Top \<rightarrow> BoolTy : Latent BoolTy" using a bi_ty_elim[of \<Gamma> BooleanP "U"] by auto hence c:"S = BoolTy" using prems arr_sub_arr_cases[of Top BoolTy _ T0 T1 "Latent S"] latent_eff.inject by auto from b have d:"T = ty.TT \<or> T = ty.FF" using true_ty_elim false_ty_elim by (cases b) auto from c d have "\<turnstile> T <: S" by auto thus ?thesis using prems by auto qed thus ?case using bool_value by auto qed (auto) lemma typing_valid: assumes "\<Gamma> \<turnstile> e : T ; F" shows "valid \<Gamma>" using prems proof (induct) case (T_Abs a \<Gamma> T b T' F') thus ?case using valid_elim[of a T \<Gamma>] by auto next case (T_AbsPred a \<Gamma> T b T' F') thus ?case using valid_elim[of a T \<Gamma>] by auto qed (auto) lemma booleanp_TT_preserved: assumes "\<Gamma> \<turnstile> (App (BI BooleanP) v) : T1 ; TT" and "v : values" and "\<Delta> BooleanP v = Some u" shows "u = Bool True" proof - note app_ty_tt_elim[of \<Gamma> "(BI BooleanP)" v T1] then obtain A1 U T0 T1 S eff1 eff2 where A:"\<Gamma> \<turnstile> (BI BooleanP) : U ; eff1" "\<turnstile> U <: A1 \<rightarrow> T1 : Latent S" "\<Gamma> \<turnstile> v : T0 ; eff2" "\<turnstile> T0 <: A1" "\<turnstile> T0 <: S" using prems by auto hence B:"U = Top \<rightarrow> BoolTy : Latent BoolTy " using bi_ty_elim[of \<Gamma> BooleanP U] by auto hence C:"\<turnstile> T0 <: BoolTy" using prems arr_sub_arr_cases[of Top BoolTy "Latent BoolTy" _ _ "Latent S"] latent_eff.inject A by auto then obtain b where D:"v = Bool b" using A bool_value[of v \<Gamma> T0] prems by auto thus ?thesis using prems by auto qed lemma booleanp_soundness_eff: assumes "\<Gamma> \<turnstile> App (BI BooleanP) v : T ; F" and "v : values" and "\<Delta> BooleanP v = Some u" and "\<Gamma> \<turnstile> u : T' ; F'" shows "\<turnstile> F' <e: F" proof - have cl:"closed u" using prems delta_closed[of BooleanP v u] by auto hence se:"simple_eff F'" using prems closed_eff by auto have or:"simple_eff F" using booleanp_eff_simple prems by auto thus ?thesis using prems proof (induct rule: simple_eff_cases) case NE thus ?case using se simple_eff_below_ne by auto next case FF thus ?case using false_ty_elim[of \<Gamma> T' F'] booleanp_FF_preserved by auto next case TT thus ?case using true_ty_elim[of \<Gamma> T' F'] booleanp_TT_preserved by auto qed qed inductive_cases app_nump_ff: "\<Gamma> \<turnstile> (App (BI NumberP) v) : T1 ; FF" thm app_nump_ff lemma numberp_FF_preserved: assumes "\<Gamma> \<turnstile> (App (BI NumberP) v) : T1 ; FF" and "v : values" and "\<Delta> NumberP v = Some u" shows "u = Bool False" using `v : values` prems proof (induct v rule: values.induct) case (bool_value b) thus ?case by (cases b) auto next case (num_value b) let ?P = "\<Gamma> \<turnstile> App (BI NumberP) (trm.Num b) : T1 ; FF" have "?P ==> ?case" proof (ind_cases ?P) fix S T e1 e2 T0 eff1 eff2 U assume "App (BI NumberP) (trm.Num b) = App e1 e2" "\<Gamma> \<turnstile> e1 : U ; eff1 " "\<Gamma> \<turnstile> e2 : T ; eff2" "~ (\<turnstile> T <: S)" "e2 : values" "\<turnstile> U <: T0 \<rightarrow> T1 : Latent S" hence a:"\<Gamma> \<turnstile> (BI NumberP) : U ; eff1 " using trm.inject by auto from prems have b:"\<Gamma> \<turnstile> Num b : T ; eff2" using trm.inject by auto have "U = Top \<rightarrow> BoolTy : Latent ty.Int" using a bi_ty_elim[of \<Gamma> NumberP "U"] by auto hence c:"S = ty.Int" using prems arr_sub_arr_cases[of Top _ _ T0 T1 "Latent S"] latent_eff.inject by auto from b have d:"T = ty.Int" using num_ty_elim by auto from c d have "\<turnstile> T <: S" by auto thus ?thesis using prems by auto qed thus ?case using num_value by auto qed (auto) lemma numberp_TT_preserved: assumes "\<Gamma> \<turnstile> (App (BI NumberP) v) : T1 ; TT" and "v : values" and "\<Delta> NumberP v = Some u" shows "u = Bool True" proof - note app_ty_tt_elim[of \<Gamma> "(BI NumberP)" v T1] then obtain A1 U T0 T1 S eff1 eff2 where A:"\<Gamma> \<turnstile> (BI NumberP) : U ; eff1" "\<turnstile> U <: A1 \<rightarrow> T1 : Latent S" "\<Gamma> \<turnstile> v : T0 ; eff2" "\<turnstile> T0 <: A1" "\<turnstile> T0 <: S" using prems by auto hence B:"U = Top \<rightarrow> BoolTy : Latent ty.Int " using bi_ty_elim[of \<Gamma> NumberP U] by auto hence C:"\<turnstile> T0 <: ty.Int" using prems arr_sub_arr_cases[of Top BoolTy "Latent ty.Int" _ _ "Latent S"] latent_eff.inject A by auto then obtain b where D:"v = Num b" using A int_value[of v \<Gamma> T0] prems by auto thus ?thesis using prems by auto qed lemma numberp_soundness_eff: assumes "\<Gamma> \<turnstile> App (BI NumberP) v : T ; F" and "v : values" and "\<Delta> NumberP v = Some u" and "\<Gamma> \<turnstile> u : T' ; F'" shows "\<turnstile> F' <e: F" proof - have cl:"closed u" using prems delta_closed[of NumberP v u] by auto hence se:"simple_eff F'" using prems closed_eff by auto have or:"simple_eff F" using numberp_eff_simple prems by auto thus ?thesis using prems proof (induct rule: simple_eff_cases) case NE thus ?case using se simple_eff_below_ne by auto next case FF thus ?case using false_ty_elim[of \<Gamma> T' F'] numberp_FF_preserved by auto next case TT thus ?case using true_ty_elim[of \<Gamma> T' F'] numberp_TT_preserved by auto qed qed lemma empty_un_bot: fixes T :: ty shows "\<turnstile> Union [] <: T" by auto lemma all_fun_ty_below: fixes S T :: ty shows "\<turnstile> S \<rightarrow> T : F <: (Union []) \<rightarrow> Top : latent_eff.NE" by (rule S_Fun) auto inductive_cases below_latent_ne_cases: "\<turnstile> S \<rightarrow> T : F <: S' \<rightarrow> T' : latent_eff.NE" thm below_latent_ne_cases inductive_cases app_procp_ff: "\<Gamma> \<turnstile> (App (BI ProcP) v) : T1 ; FF" thm app_nump_ff thm app_procp_ff lemma procp_FF_preserved: assumes "\<Gamma> \<turnstile> (App (BI ProcP) v) : T1 ; FF" and "v : values" and "\<Delta> ProcP v = Some u" shows "u = Bool False" using `v : values` prems proof (nominal_induct v avoiding: \<Gamma> rule: values.strong_induct) case (bool_value b) thus ?case by (cases b) auto next case (num_value b) thus ?case by auto next case (abs_value x t b) let ?P = "\<Gamma> \<turnstile> App (BI ProcP) (Lam[x:t].b) : T1 ; FF" have "?P ==> ?case" proof (ind_cases ?P) fix e1 U eff1 T0 S e2 T eff2 assume "App (BI ProcP) (Lam [x:t].b) = App e1 e2 "" \<Gamma> \<turnstile> e1 : U ; eff1 "" \<turnstile> U <: T0 \<rightarrow> T1 : Latent S " "\<Gamma> \<turnstile> e2 : T ; eff2 "" \<turnstile> T <: T0 "" \<not> \<turnstile> T <: S "" e2 \<in> values "" closed e2" hence a:"\<Gamma> \<turnstile> (BI ProcP) : U ; eff1 " using trm.inject by auto from prems have b:"\<Gamma> \<turnstile> Lam[x:t].b : T ; eff2" using trm.inject by auto have "U = Top \<rightarrow> BoolTy : Latent (Union [] \<rightarrow> Top : latent_eff.NE)" using a bi_ty_elim[of \<Gamma> ProcP "U"] by auto hence c:"S = Union [] \<rightarrow> Top : latent_eff.NE" using prems arr_sub_arr_cases[of Top _ _ T0 T1 "Latent S"] latent_eff.inject by auto have "EX A1 A2 L. T = A1 \<rightarrow> A2 : L" using abs_ty_elim[OF b `x \<sharp> \<Gamma>`] by auto then obtain A1 A2 L where d:"T = A1 \<rightarrow> A2 : L" by auto from c d have "\<turnstile> T <: S" using all_fun_ty_below by auto thus ?thesis using prems by auto qed thus ?case using abs_value by auto next case (bi_value c) let ?P = "\<Gamma> \<turnstile> App (BI ProcP) (BI c) : T1 ; FF" have "?P ==> ?case" proof (ind_cases ?P) fix e1 U eff1 T0 S e2 T eff2 assume "App (BI ProcP) (BI c) = App e1 e2 "" \<Gamma> \<turnstile> e1 : U ; eff1 "" \<turnstile> U <: T0 \<rightarrow> T1 : Latent S " "\<Gamma> \<turnstile> e2 : T ; eff2 "" \<turnstile> T <: T0 "" \<not> \<turnstile> T <: S "" e2 \<in> values "" closed e2" hence a:"\<Gamma> \<turnstile> (BI ProcP) : U ; eff1 " using trm.inject by auto from prems have b:"\<Gamma> \<turnstile> (BI c) : T ; eff2" using trm.inject by auto have "U = Top \<rightarrow> BoolTy : Latent (Union [] \<rightarrow> Top : latent_eff.NE)" using a bi_ty_elim[of \<Gamma> ProcP "U"] by auto hence c:"S = Union [] \<rightarrow> Top : latent_eff.NE" using prems arr_sub_arr_cases[of Top _ _ T0 T1 "Latent S"] latent_eff.inject by auto have b':"T = \<Delta>\<^isub>\<tau> c" using bi_ty_elim[OF b] by auto hence "EX A1 A2 FA. T = A1 \<rightarrow> A2 : FA" by (nominal_induct c rule: builtin.induct) auto then obtain A1 A2 F where d:"T = A1 \<rightarrow> A2 : F" by auto hence "\<turnstile> T <: S" using c d all_fun_ty_below by auto thus ?thesis using prems by auto qed thus ?case using bi_value by auto qed lemma proc_value: assumes "\<Gamma> \<turnstile> v : T ; F" and "v : values" and "\<turnstile> T <: (A1 \<rightarrow> A2 : latent_eff.NE)" shows "EX x b t. v = Lam[x:t].b \<or> (EX c. v = BI c)" using `v : values` prems proof (nominal_induct v avoiding: \<Gamma> rule: values.strong_induct) case abs_value thus ?case by auto next case (bool_value b) hence "T = ty.TT \<or> T = ty.FF" using bool_ty_elim by auto thus ?case using sub_arr_cases[of T A1 A2 latent_eff.NE] using bool_value by auto next case (num_value b) hence "T = ty.Int" using num_ty_elim by auto thus ?case using sub_arr_cases[of T A1 A2 latent_eff.NE] using num_value by auto next case (bi_value c) thus ?case by auto qed lemma procp_TT_preserved: assumes "\<Gamma> \<turnstile> (App (BI ProcP) v) : T1 ; TT" and "v : values" and "\<Delta> ProcP v = Some u" shows "u = Bool True" proof - note app_ty_tt_elim[of \<Gamma> "(BI ProcP)" v T1] then obtain A1 U T0 T1 S eff1 eff2 where A:"\<Gamma> \<turnstile> (BI ProcP) : U ; eff1" "\<turnstile> U <: A1 \<rightarrow> T1 : Latent S" "\<Gamma> \<turnstile> v : T0 ; eff2" "\<turnstile> T0 <: A1" "\<turnstile> T0 <: S" using prems by auto hence B:"U = Top \<rightarrow> BoolTy : Latent (Union [] \<rightarrow> Top : latent_eff.NE) " using bi_ty_elim[of \<Gamma> ProcP U] by auto hence C:"\<turnstile> T0 <: (Union [] \<rightarrow> Top : latent_eff.NE)" using prems arr_sub_arr_cases[of Top BoolTy "Latent (Union [] \<rightarrow> Top : latent_eff.NE)" _ _ "Latent S"] latent_eff.inject A by auto moreover { assume D:"EX x b t. v = Lam[x:t].b" then obtain x b t where "v = Lam[x:t].b" by auto hence ?thesis using prems by auto } moreover { assume "EX c. v = BI c" then obtain c where "v = BI c" by auto hence ?thesis using prems by (nominal_induct c rule: builtin.induct) auto } ultimately show ?thesis using prems proc_value[of \<Gamma> v T0 eff2 "Union []" Top] by auto qed lemma procp_soundness_eff: assumes "\<Gamma> \<turnstile> App (BI ProcP) v : T ; F" and "v : values" and "\<Delta> ProcP v = Some u" and "\<Gamma> \<turnstile> u : T' ; F'" shows "\<turnstile> F' <e: F" proof - have cl:"closed u" using prems delta_closed[of ProcP v u] by auto hence se:"simple_eff F'" using prems closed_eff by auto have or:"simple_eff F" using procp_eff_simple prems by auto thus ?thesis using prems proof (induct rule: simple_eff_cases) case NE thus ?case using se simple_eff_below_ne by auto next case FF thus ?case using false_ty_elim[of \<Gamma> T' F'] procp_FF_preserved by auto next case TT thus ?case using true_ty_elim[of \<Gamma> T' F'] procp_TT_preserved by auto qed qed text { soundness of the \<Delta> rule } lemma bool_sub_boolty[simp]: "\<turnstile> ty.FF <: BoolTy" "\<turnstile> ty.TT <: BoolTy" by auto lemma delta_soundness: assumes "\<Delta>\<^isub>\<tau> b = (T0 \<rightarrow> T1 : le)" and "v : values" and "\<Gamma> \<turnstile> v : T ; eff1" and "\<turnstile> T <: T0" and "\<Gamma> \<turnstile> (App (BI b) v) : T1' ; eff" and "\<turnstile> T1 <: T1'" and "\<Delta> b v = Some u" and "valid \<Gamma>" shows "EX eff' T1'. \<Gamma> \<turnstile> u : T1' ; eff' \<and> \<turnstile> eff' <e: eff \<and> \<turnstile> T1' <: T1" using prems proof (nominal_induct b rule: builtin.induct) case Add1 hence a:"eff = NE" using add1_eff_ne[of \<Gamma> v T1' eff] by auto have "EX eff'. \<Gamma> \<turnstile> u : T1 ; eff'" using `v : values` `valid \<Gamma>` Add1 by (induct v rule: values.induct) (auto simp add: ty.inject) then obtain eff' where b:"\<Gamma> \<turnstile> u : T1 ; eff'" by auto have "simple_eff eff'" using delta_closed[of Add1 v u] closed_eff b Add1 by auto hence c:"\<turnstile> eff' <e: eff" using a simple_eff_below_ne by auto from b c show ?case by auto next case Nott hence a:"eff = NE" using nott_eff_ne[of \<Gamma> v T1' eff] by auto have "EX eff' T1'. \<Gamma> \<turnstile> u : T1' ; eff' \<and> \<turnstile> T1' <: T1" using `v : values` `valid \<Gamma>` Nott proof (induct v rule: values.induct) case (bool_value b) thus ?case using bool_value bool_sub_boolty by (cases b) auto next case num_value thus ?case using bool_sub_boolty by auto next case bi_value thus ?case using bool_sub_boolty by auto next case abs_value thus ?case using bool_sub_boolty by auto qed then obtain eff' T1' where b:"\<Gamma> \<turnstile> u : T1' ; eff'" and b':"\<turnstile> T1' <: T1" by auto have "simple_eff eff'" using delta_closed[of Nott v u] closed_eff b Nott by auto hence c:"\<turnstile> eff' <e: eff" using a simple_eff_below_ne by auto from b b' c show ?case by auto next case BooleanP have "valid \<Gamma>" using typing_valid prems by auto have "T1 = BoolTy" using BooleanP by (simp add: ty.inject) then obtain bb where veq:"\<Delta> BooleanP v = Some (Bool bb)" by (nominal_induct v rule: trm.induct) (auto) hence "EX T1' eff'. \<Gamma> \<turnstile> u : T1' ; eff' \<and> \<turnstile> T1' <: T1" using `T1 = BoolTy` BooleanP bool_sub_boolty proof (cases bb) case True hence "u = Bool True" using veq BooleanP by auto hence "\<Gamma> \<turnstile> u : ty.TT; TT" using `valid \<Gamma>` by auto thus ?thesis using `T1 = BoolTy` bool_sub_boolty by auto next case False hence "u = Bool False" using veq BooleanP by auto hence "\<Gamma> \<turnstile> u : ty.FF; FF" using `valid \<Gamma>` by auto thus ?thesis using `T1 = BoolTy` bool_sub_boolty by auto qed then obtain T1a eff' where a:"\<Gamma> \<turnstile> u : T1a ; eff'" and a':"\<turnstile> T1a <: T1" by auto have c:"simple_eff eff'" using delta_closed[of BooleanP v u] closed_eff prems by auto have b:"simple_eff eff" using BooleanP booleanp_eff_simple by auto hence "\<turnstile> eff' <e: eff" using b prems proof (induct rule: simple_eff_cases) case NE thus ?case using simple_eff_below_ne c by auto next case FF hence "u = Bool False" using booleanp_FF_preserved `v : values` `\<Gamma> \<turnstile> App (BI BooleanP) v : T1' ; eff` `\<Delta> BooleanP v = Some u` by auto hence "eff'= FF" using a false_ty_elim by auto thus ?case using FF by auto next case TT hence "u = Bool True" using booleanp_TT_preserved `v : values` `\<Gamma> \<turnstile> App (BI BooleanP) v : T1' ; eff` `\<Delta> BooleanP v = Some u` by auto hence "eff'= TT" using a true_ty_elim by auto thus ?case using TT by auto qed thus ?case using a a' by auto next case NumberP have "valid \<Gamma>" using typing_valid prems by auto have "T1 = BoolTy" using NumberP by (simp add: ty.inject) then obtain bb where veq:"\<Delta> NumberP v = Some (Bool bb)" by (nominal_induct v rule: trm.induct) (auto) hence "EX T1' eff'. \<Gamma> \<turnstile> u : T1' ; eff' \<and> \<turnstile> T1' <: T1" using `T1 = BoolTy` NumberP bool_sub_boolty proof (cases bb) case True hence "u = Bool True" using veq NumberP by auto hence "\<Gamma> \<turnstile> u : ty.TT; TT" using `valid \<Gamma>` by auto thus ?thesis using `T1 = BoolTy` bool_sub_boolty by auto next case False hence "u = Bool False" using veq NumberP by auto hence "\<Gamma> \<turnstile> u : ty.FF; FF" using `valid \<Gamma>` by auto thus ?thesis using `T1 = BoolTy` bool_sub_boolty by auto qed then obtain T1a eff' where a:"\<Gamma> \<turnstile> u : T1a ; eff'" and a':"\<turnstile> T1a <: T1" by auto have c:"simple_eff eff'" using delta_closed[of NumberP v u] closed_eff prems by auto have b:"simple_eff eff" using NumberP numberp_eff_simple by auto hence "\<turnstile> eff' <e: eff" using b prems proof (induct rule: simple_eff_cases) case NE thus ?case using simple_eff_below_ne c by auto next case FF hence "u = Bool False" using numberp_FF_preserved `v : values` `\<Gamma> \<turnstile> App (BI NumberP) v : T1' ; eff` `\<Delta> NumberP v = Some u` by auto hence "eff'= FF" using a false_ty_elim by auto thus ?case using FF by auto next case TT hence "u = Bool True" using numberp_TT_preserved `v : values` `\<Gamma> \<turnstile> App (BI NumberP) v : T1' ; eff` `\<Delta> NumberP v = Some u` by auto hence "eff'= TT" using a true_ty_elim by auto thus ?case using TT by auto qed thus ?case using a a' by auto next case ProcP have "valid \<Gamma>" using typing_valid prems by auto have "T1 = BoolTy" using ProcP by (simp add: ty.inject) then obtain bb where veq:"\<Delta> ProcP v = Some (Bool bb)" by (nominal_induct v rule: trm.induct) (auto) hence "EX T1' eff'. \<Gamma> \<turnstile> u : T1' ; eff' \<and> \<turnstile> T1' <: T1" using `T1 = BoolTy` ProcP bool_sub_boolty proof (cases bb) case True hence "u = Bool True" using veq ProcP by auto hence "\<Gamma> \<turnstile> u : ty.TT; TT" using `valid \<Gamma>` by auto thus ?thesis using `T1 = BoolTy` bool_sub_boolty by auto next case False hence "u = Bool False" using veq ProcP by auto hence "\<Gamma> \<turnstile> u : ty.FF; FF" using `valid \<Gamma>` by auto thus ?thesis using `T1 = BoolTy` bool_sub_boolty by auto qed then obtain T1a eff' where a:"\<Gamma> \<turnstile> u : T1a ; eff'" and a':"\<turnstile> T1a <: T1" by auto have c:"simple_eff eff'" using delta_closed[of ProcP v u] closed_eff prems by auto have b:"simple_eff eff" using ProcP procp_eff_simple by auto hence "\<turnstile> eff' <e: eff" using b prems proof (induct rule: simple_eff_cases) case NE thus ?case using simple_eff_below_ne c by auto next case FF hence "u = Bool False" using procp_FF_preserved `v : values` `\<Gamma> \<turnstile> App (BI ProcP) v : T1' ; eff` `\<Delta> ProcP v = Some u` by auto hence "eff'= FF" using a false_ty_elim by auto thus ?case using FF by auto next case TT hence "u = Bool True" using procp_TT_preserved `v : values` `\<Gamma> \<turnstile> App (BI ProcP) v : T1' ; eff` `\<Delta> ProcP v = Some u` by auto hence "eff'= TT" using a true_ty_elim by auto thus ?case using TT by auto qed thus ?case using a a' by auto qed lemma simple_eff_below_ve: assumes "simple_eff F" shows "\<turnstile> F <e: VE x" using prems by (induct F rule: simple_eff_cases) auto lemma below_ne_simple: assumes "\<turnstile> F <e: G" and "G = NE" shows "simple_eff F" using prems no_sub_TT no_sub_FF by induct auto lemma below_ve_simple: assumes "\<turnstile> F <e: G" and "G = VE x" shows "simple_eff F \<or> F = VE x" using prems by induct auto consts remove_env :: "'a list => 'a => 'a list" ("_ - _" [100,100] 100) primrec "[] - x = []" "(y#xs) - x = (if x=y then (xs - x) else y#(xs - x))" lemma fresh_remove: fixes a::"name" and \<Gamma>::"(name\<times>ty) list" assumes a: "a\<sharp>\<Gamma>" shows "a\<sharp>(\<Gamma> - (x,T))" using a by (induct \<Gamma>) (auto simp add: fresh_list_cons) lemma valid_remove: fixes pair::"name\<times>ty" assumes a: "valid \<Gamma>" shows "valid (\<Gamma> - (x,T))" using a by (induct \<Gamma>) (auto simp add: fresh_remove) lemma set_remove: assumes a: "(a,T)\<in>set \<Gamma>" and b: "a\<noteq>x" shows "(a,T)\<in>set (\<Gamma> - (x,T'))" using a b by (induct \<Gamma>) (auto) lemma closed_elim:"closed e \<Longrightarrow> (supp e = ({}::name set))" using closed_def by auto lemma set_remove_comm: shows "set (l - a) = (set l) - {a}" by (induct l) auto lemma envplus_set: shows "set (\<Gamma> \|+ TE T x) = (mapfun restrict T x) ` set \<Gamma>" proof - have A:"!! a b. a = b \<Longrightarrow> set a = set b" by auto have "\<Gamma> \|+ TE T x = map (mapfun restrict T x) \<Gamma>" by auto hence "set (\<Gamma> \|+ TE T x) = set (map (mapfun restrict T x) \<Gamma>)" using A[of "(\<Gamma> \|+ TE T x)" "map (mapfun restrict T x) \<Gamma>"] by auto also have "\<dots> = (mapfun restrict T x) ` set \<Gamma>" by auto ultimately show ?thesis by auto qed lemma envminus_set: shows "set (\<Gamma> \|- TE T x) = (mapfun remove T x) ` set \<Gamma>" proof - have A:"!! a b. a = b \<Longrightarrow> set a = set b" by auto have "\<Gamma> \|- TE T x = map (mapfun remove T x) \<Gamma>" by auto hence "set (\<Gamma> \|- TE T x) = set (map (mapfun remove T x) \<Gamma>)" using A[of "(\<Gamma> \|- TE T x)" "map (mapfun remove T x) \<Gamma>"] by auto also have "\<dots> = (mapfun remove T x) ` set \<Gamma>" by auto ultimately show ?thesis by auto qed lemma envplus_set_ve: shows "set (\<Gamma> \|+ VE x) = (mapfun remove ty.FF x) ` set \<Gamma>" proof - have A:"!! a b. a = b \<Longrightarrow> set a = set b" by auto have "\<Gamma> \|+ VE x = map (mapfun remove ty.FF x) \<Gamma>" by auto hence "set (\<Gamma> \|+ VE x) = set (map (mapfun remove ty.FF x) \<Gamma>)" using A[of "(\<Gamma> \|+ VE x)" "map (mapfun remove ty.FF x) \<Gamma>"] by auto also have "\<dots> = (mapfun remove ty.FF x) ` set \<Gamma>" by auto ultimately show ?thesis by auto qed lemma envminus_set_ve: shows "set (\<Gamma> \|- VE x) = (mapfun replace ty.FF x) ` set \<Gamma>" proof - have A:"!! a b. a = b \<Longrightarrow> set a = set b" by auto have "\<Gamma> \|- VE x = map (mapfun replace ty.FF x) \<Gamma>" by auto hence "set (\<Gamma> \|- VE x) = set (map (mapfun replace ty.FF x) \<Gamma>)" using A[of "(\<Gamma> \|- VE x)" "map (mapfun replace ty.FF x) \<Gamma>"] by auto also have "\<dots> = (mapfun replace ty.FF x) ` set \<Gamma>" by auto ultimately show ?thesis by auto qed lemma fresh_weakening: assumes a:"x\<sharp>e" and b:"\<Gamma> \<turnstile> e : T ; F" and c: "valid \<Gamma>" shows "(\<Gamma> - (x,T')) \<turnstile> e : T ; F" using b a c proof (nominal_induct \<Gamma> e T F avoiding: x T' rule: typing.strong_induct) case T_Var thus ?case by(force simp add: fresh_atm abs_fresh set_remove valid_remove fresh_remove)+ next case (T_App \<Gamma>' _ _ t1 t2) thus ?case by(force simp add: fresh_atm abs_fresh set_remove valid_remove fresh_remove) next case T_Abs thus ?case by(force simp add: fresh_atm abs_fresh set_remove valid_remove fresh_remove)+ next case T_AbsPred thus ?case by(force simp add: fresh_atm abs_fresh set_remove valid_remove fresh_remove)+ next case T_AppPred thus ?case by(force simp add: fresh_atm abs_fresh set_remove valid_remove fresh_remove)+ next case T_IfTrue thus ?case by(force simp add: fresh_atm abs_fresh set_remove valid_remove fresh_remove)+ next case T_IfFalse thus ?case by(force simp add: fresh_atm abs_fresh set_remove valid_remove fresh_remove)+ next case T_True thus ?case by(force simp add: fresh_atm abs_fresh set_remove valid_remove fresh_remove)+ next case T_False thus ?case by(force simp add: fresh_atm abs_fresh set_remove valid_remove fresh_remove)+ next case T_Num thus ?case by(force simp add: fresh_atm abs_fresh set_remove valid_remove fresh_remove)+ next case T_Const thus ?case by(force simp add: fresh_atm abs_fresh set_remove valid_remove fresh_remove)+ next case (T_AppPredTrue \<Gamma>' e1 U F1 T0 T1 S e2 T F2 x T') have A:"x \<sharp> e1" "x \<sharp> e2" using T_AppPredTrue by auto hence "\<Gamma>' - (x,T') \<turnstile> e1 : U; F1" using T_AppPredTrue by auto also have "\<Gamma>' - (x,T') \<turnstile> e2 : T ; F2" using T_AppPredTrue A by auto ultimately show ?case using T_AppPredTrue(3) `\<turnstile> T <: T0` `\<turnstile> T <: S` by auto next case (T_AppPredFalse \<Gamma>' e1 U F1 T0 T1 S e2 T F2 x T') have A:"x \<sharp> e1" "x \<sharp> e2" using T_AppPredFalse by auto hence "\<Gamma>' - (x,T') \<turnstile> e1 : U ; F1" using T_AppPredFalse by auto also have "\<Gamma>' - (x,T') \<turnstile> e2 : T ; F2" using T_AppPredFalse A by auto ultimately show ?case using `~ (\<turnstile> T <: S)` `\<turnstile> T <: T0` `e2 : values` `closed e2` T_AppPredFalse(3) by auto next case (T_If \<Gamma>' e1 T1 F1 e2 T2 F2 e3 T3 F3 T x) have A:"x \<sharp> e1" "x \<sharp> e2" "x \<sharp> e3" using T_If by auto have "\<Gamma>' - (x,T') \<turnstile> e1 : T1 ; F1" using T_If A by auto thus ?case using T_If proof (nominal_induct "F1" rule: eff.induct) case NE from NE have 1:"\<Gamma>' - (x, T') \<turnstile> e1 : T1 ; eff.NE" by auto from NE have 2:"(\<Gamma>' - (x, T') \|+ eff.NE) \<turnstile> e2 : T2 ; F2" by auto from NE have 3:"(\<Gamma>' - (x, T') \|- eff.NE) \<turnstile> e3 : T3 ; F3" by auto from 1 2 3 show ?case using `\<turnstile> T2 <: T` `\<turnstile> T3 <: T` .. next case TT from TT have 1:"\<Gamma>' - (x, T') \<turnstile> e1 : T1 ; eff.TT" by auto from TT have 2:"(\<Gamma>' - (x, T') \|+ eff.TT) \<turnstile> e2 : T2 ; F2" by auto from 1 2 show ?case using `\<turnstile> T2 <: T` by auto next case FF from FF have 1:"\<Gamma>' - (x, T') \<turnstile> e1 : T1 ; eff.FF" by auto from FF have 3:"(\<Gamma>' - (x, T') \|+ eff.FF) \<turnstile> e3 : T3 ; F3" by auto from 1 3 show ?case using `\<turnstile> T3 <: T` by auto next case (VE z) from VE have 1:"\<Gamma>' - (x, T') \<turnstile> e1 : T1 ; VE z" by auto have val1:"valid (\<Gamma>' \|+ VE z)" using VE envplus_valid[of \<Gamma>' "VE z"] by auto have val2:"valid (\<Gamma>' \|- VE z)" using VE envminus_valid[of \<Gamma>' "VE z"] by auto have "valid (\<Gamma>' \|- VE z)" using VE envminus_valid[of \<Gamma>' "VE z"] by auto have A0:"!!T0 . (\<Gamma>' \|- VE z) - (x, T0) \<turnstile> e3 : T3 ; F3" using VE(7) A `valid (\<Gamma>' \|- VE z)` by auto have A:"!!T0 . (\<Gamma>' \|+ VE z) - (x, T0) \<turnstile> e2 : T2 ; F2" using VE(5) A `valid (\<Gamma>' \|+ VE z)` by auto let ?op = "replace" let ?mapfun = "(% (v,ty). (if (x = v) then (v,?op ty.FF ty) else (v,ty)))" have B:"!! p. set ((\<Gamma>' \|+ VE z) - p) = ((mapfun remove ty.FF z) ` (set \<Gamma>')) - {p}" using envplus_set_ve set_remove_comm[of "(\<Gamma>' \|+ VE z)"] by auto have image_lem:"!! f S v. (f ` S) - {(f v)} <= (f ` (S - {v}))" by auto note image_lem[of "mapfun remove ty.FF z" "set \<Gamma>'" "(x,T0)"] have eq1:"!! p. mapfun remove ty.FF z ` set \<Gamma>' - {mapfun remove ty.FF z p} = set ((\<Gamma>' \|+ VE z) - (mapfun remove ty.FF z p))" using envplus_set_ve set_remove_comm[of "(\<Gamma>' \|+ VE z)"] by auto have eq2:"!! p. mapfun remove ty.FF z ` (set \<Gamma>' - {p}) = set (\<Gamma>' - p \|+ VE z)" proof - fix p show "mapfun remove ty.FF z ` (set \<Gamma>' - {p}) = set (\<Gamma>' - p \|+ VE z)" using envplus_set_ve[of _ "\<Gamma>' - p"] set_remove_comm[of \<Gamma>'] by auto qed have eq3:"!! x T0. mapfun remove ty.FF z (x,T0) = (x, (if (x = z) then (remove ty.FF T0) else (T0)))" by auto let ?removeT' = "(if (x = z) then (remove ty.FF T') else (T'))" have goal:"((\<Gamma>' \|+ VE z) - (x,?removeT')) \<lless> ((\<Gamma>' - (x,T')) \|+ VE z)" proof - have " mapfun remove ty.FF z ` set \<Gamma>' - {mapfun remove ty.FF z (x,T')} <= mapfun remove ty.FF z ` (set \<Gamma>' - {(x,T')})" using image_lem[of "mapfun remove ty.FF z" "set \<Gamma>'" "(x,T')"] by auto hence " mapfun remove ty.FF z ` set \<Gamma>' - {mapfun remove ty.FF z (x,T')} <= set ((\<Gamma>' - ((x,T'))) \|+ VE z)" using eq2[of " (x, T')"] by auto hence "set ((\<Gamma>' \|+ VE z) - (mapfun remove ty.FF z (x,T'))) <= set ((\<Gamma>' - ((x,T'))) \|+ VE z)" using eq1[of "(x,T')"] by auto hence "set ((\<Gamma>' \|+ VE z) - (x,?removeT')) <= set ((\<Gamma>' - (x,T')) \|+ VE z)" using eq3 by auto thus ?thesis by auto qed have val3:"valid ((\<Gamma>' \|+ VE z) - (x,?removeT'))" using val1 valid_remove by auto have "valid ((\<Gamma>' - (x, T')))" using `valid \<Gamma>'` valid_remove envplus_valid by auto hence val4:"valid ((\<Gamma>' - (x, T')) \|+ VE z)" using envplus_valid[of "\<Gamma>' - (x, T')" "VE z"] by auto from A have "(\<Gamma>' \|+ VE z) - (x,?removeT') \<turnstile> e2 : T2 ; F2" by auto hence 2:"(\<Gamma>' - (x, T')) \|+ VE z \<turnstile> e2 : T2 ; F2" using goal val3 val4 weakening[of "(\<Gamma>' \|+ VE z) - (x,?removeT')" e2 T2 F2 "\<Gamma>' - (x,T') \|+ VE z"] by auto have eq4:"!! p. mapfun replace ty.FF z ` set \<Gamma>' - {mapfun replace ty.FF z p} = set ((\<Gamma>' \|- VE z) - (mapfun replace ty.FF z p))" using envminus_set_ve[of _ \<Gamma>'] set_remove_comm[of "(\<Gamma>' \|- VE z)"] by auto have eq5:"!! p. mapfun replace ty.FF z ` (set \<Gamma>' - {p}) = set (\<Gamma>' - p \|- VE z)" proof - fix p show "mapfun replace ty.FF z ` (set \<Gamma>' - {p}) = set (\<Gamma>' - p \|- VE z)" using envminus_set_ve[of _ "\<Gamma>' - p" ] set_remove_comm[of \<Gamma>'] by auto qed have eq6:"!! x T0. mapfun replace ty.FF z (x,T0) = (x, (if (x = z) then (replace ty.FF T0) else (T0)))" by auto let ?replaceT' = "(if (x = z) then (replace ty.FF T') else (T'))" have goal':"((\<Gamma>' \|- VE z) - (x,?replaceT')) \<lless> ((\<Gamma>' - (x,T')) \|- VE z)" proof - have " mapfun replace ty.FF z ` set \<Gamma>' - {mapfun replace ty.FF z (x,T')} <= mapfun replace ty.FF z ` (set \<Gamma>' - {(x,T')})" using image_lem[of "mapfun replace ty.FF z" "set \<Gamma>'" "(x,T')"] by auto hence " mapfun replace ty.FF z ` set \<Gamma>' - {mapfun replace ty.FF z (x,T')} <= set ((\<Gamma>' - ((x,T'))) \|- VE z)" using eq5[of " (x, T')"] by auto hence "set ((\<Gamma>' \|- VE z) - (mapfun replace ty.FF z (x,T'))) <= set ((\<Gamma>' - ((x,T'))) \|- VE z)" using eq4[of "(x,T')"] by auto hence "set ((\<Gamma>' \|- VE z) - (x,?replaceT')) <= set ((\<Gamma>' - (x,T')) \|- VE z)" using eq6 by auto thus ?thesis by auto qed have val5:"valid ((\<Gamma>' \|- VE z) - (x,?replaceT'))" using val2 valid_remove by auto have "valid ((\<Gamma>' - (x, T')))" using `valid \<Gamma>'` valid_remove envminus_valid by auto hence val6:"valid ((\<Gamma>' - (x, T')) \|- VE z)" using envminus_valid[of "\<Gamma>' - (x, T')" "VE z"] by auto from A0 have "(\<Gamma>' \|- VE z) - (x,?replaceT') \<turnstile> e3 : T3 ; F3" by auto hence 3:"(\<Gamma>' - (x, T')) \|- VE z \<turnstile> e3 : T3 ; F3" using goal' val5 val6 weakening[of "(\<Gamma>' \|- VE z) - (x,?replaceT')" e3 T3 F3 "\<Gamma>' - (x,T') \|- VE z"] by auto from 1 2 3 show ?case using `\<turnstile> T2 <: T` `\<turnstile> T3 <: T` .. next case (TE U z) from TE have 1:"\<Gamma>' - (x, T') \<turnstile> e1 : T1 ; TE U z" by auto have val1:"valid (\<Gamma>' \|+ TE U z)" using TE envplus_valid[of \<Gamma>' "TE U z"] by auto have val2:"valid (\<Gamma>' \|- TE U z)" using TE envminus_valid[of \<Gamma>' "TE U z"] by auto have "valid (\<Gamma>' \|- TE U z)" using TE envminus_valid[of \<Gamma>' "TE U z"] by auto have A0:"!!T0 . (\<Gamma>' \|- TE U z) - (x, T0) \<turnstile> e3 : T3 ; F3" using TE(7) A `valid (\<Gamma>' \|- TE U z)` by auto have A:"!!T0 . (\<Gamma>' \|+ TE U z) - (x, T0) \<turnstile> e2 : T2 ; F2" using TE(5) A `valid (\<Gamma>' \|+ TE U z)` by auto let ?op = "remove" let ?mapfun = "(% (v,ty). (if (x = v) then (v,?op S ty) else (v,ty)))" have B:"!! p. set ((\<Gamma>' \|+ TE U z) - p) = ((mapfun restrict U z) ` (set \<Gamma>')) - {p}" using envplus_set set_remove_comm[of "(\<Gamma>' \|+ TE U z)"] by auto have image_lem:"!! f S v. (f ` S) - {(f v)} <= (f ` (S - {v}))" by auto note image_lem[of "mapfun restrict U z" "set \<Gamma>'" "(x,T0)"] have eq1:"!! p. mapfun restrict U z ` set \<Gamma>' - {mapfun restrict U z p} = set ((\<Gamma>' \|+ TE U z) - (mapfun restrict U z p))" using envplus_set set_remove_comm[of "(\<Gamma>' \|+ TE U z)"] by auto have eq2:"!! p. mapfun restrict U z ` (set \<Gamma>' - {p}) = set (\<Gamma>' - p \|+ TE U z)" proof - fix p show "mapfun restrict U z ` (set \<Gamma>' - {p}) = set (\<Gamma>' - p \|+ TE U z)" using envplus_set[of _ _ "\<Gamma>' - p"] set_remove_comm[of \<Gamma>'] by auto qed have eq3:"!! x T0. mapfun restrict U z (x,T0) = (x, (if (x = z) then (restrict U T0) else (T0)))" by auto let ?restrictT' = "(if (x = z) then (restrict U T') else (T'))" have goal:"((\<Gamma>' \|+ TE U z) - (x,?restrictT')) \<lless> ((\<Gamma>' - (x,T')) \|+ TE U z)" proof - have " mapfun restrict U z ` set \<Gamma>' - {mapfun restrict U z (x,T')} <= mapfun restrict U z ` (set \<Gamma>' - {(x,T')})" using image_lem[of "mapfun restrict U z" "set \<Gamma>'" "(x,T')"] by auto hence " mapfun restrict U z ` set \<Gamma>' - {mapfun restrict U z (x,T')} <= set ((\<Gamma>' - ((x,T'))) \|+ TE U z)" using eq2[of " (x, T')"] by auto hence "set ((\<Gamma>' \|+ TE U z) - (mapfun restrict U z (x,T'))) <= set ((\<Gamma>' - ((x,T'))) \|+ TE U z)" using eq1[of "(x,T')"] by auto hence "set ((\<Gamma>' \|+ TE U z) - (x,?restrictT')) <= set ((\<Gamma>' - (x,T')) \|+ TE U z)" using eq3 by auto thus ?thesis by auto qed have val3:"valid ((\<Gamma>' \|+ TE U z) - (x,?restrictT'))" using val1 valid_remove by auto have "valid ((\<Gamma>' - (x, T')))" using `valid \<Gamma>'` valid_remove envplus_valid by auto hence val4:"valid ((\<Gamma>' - (x, T')) \|+ TE U z)" using envplus_valid[of "\<Gamma>' - (x, T')" "TE U z"] by auto from A have "(\<Gamma>' \|+ TE U z) - (x,?restrictT') \<turnstile> e2 : T2 ; F2" by auto hence 2:"(\<Gamma>' - (x, T')) \|+ TE U z \<turnstile> e2 : T2 ; F2" using goal val3 val4 weakening[of "(\<Gamma>' \|+ TE U z) - (x,?restrictT')" e2 T2 F2 "\<Gamma>' - (x,T') \|+ TE U z"] by auto have eq4:"!! p. mapfun remove U z ` set \<Gamma>' - {mapfun remove U z p} = set ((\<Gamma>' \|- TE U z) - (mapfun remove U z p))" using envminus_set[of _ _ \<Gamma>'] set_remove_comm[of "(\<Gamma>' \|- TE U z)"] by auto have eq5:"!! p. mapfun remove U z ` (set \<Gamma>' - {p}) = set (\<Gamma>' - p \|- TE U z)" proof - fix p show "mapfun remove U z ` (set \<Gamma>' - {p}) = set (\<Gamma>' - p \|- TE U z)" using envminus_set[of _ _ "\<Gamma>' - p" ] set_remove_comm[of \<Gamma>'] by auto qed have eq6:"!! x T0. mapfun remove U z (x,T0) = (x, (if (x = z) then (remove U T0) else (T0)))" by auto let ?removeT' = "(if (x = z) then (remove U T') else (T'))" have goal':"((\<Gamma>' \|- TE U z) - (x,?removeT')) \<lless> ((\<Gamma>' - (x,T')) \|- TE U z)" proof - have " mapfun remove U z ` set \<Gamma>' - {mapfun remove U z (x,T')} <= mapfun remove U z ` (set \<Gamma>' - {(x,T')})" using image_lem[of "mapfun remove U z" "set \<Gamma>'" "(x,T')"] by auto hence " mapfun remove U z ` set \<Gamma>' - {mapfun remove U z (x,T')} <= set ((\<Gamma>' - ((x,T'))) \|- TE U z)" using eq5[of " (x, T')"] by auto hence "set ((\<Gamma>' \|- TE U z) - (mapfun remove U z (x,T'))) <= set ((\<Gamma>' - ((x,T'))) \|- TE U z)" using eq4[of "(x,T')"] by auto hence "set ((\<Gamma>' \|- TE U z) - (x,?removeT')) <= set ((\<Gamma>' - (x,T')) \|- TE U z)" using eq6 by auto thus ?thesis by auto qed have val5:"valid ((\<Gamma>' \|- TE U z) - (x,?removeT'))" using val2 valid_remove by auto have "valid ((\<Gamma>' - (x, T')))" using `valid \<Gamma>'` valid_remove envminus_valid by auto hence val6:"valid ((\<Gamma>' - (x, T')) \|- TE U z)" using envminus_valid[of "\<Gamma>' - (x, T')" "TE U z"] by auto from A0 have "(\<Gamma>' \|- TE U z) - (x,?removeT') \<turnstile> e3 : T3 ; F3" by auto hence 3:"(\<Gamma>' - (x, T')) \|- TE U z \<turnstile> e3 : T3 ; F3" using goal' val5 val6 weakening[of "(\<Gamma>' \|- TE U z) - (x,?removeT')" e3 T3 F3 "\<Gamma>' - (x,T') \|- TE U z"] by auto from 1 2 3 show ?case using `\<turnstile> T2 <: T` `\<turnstile> T3 <: T` .. qed qed lemma fresh_weakening_cons: assumes "valid ((a,S)#\<Gamma>)" (is "valid ?\<Gamma>") and "(a,S)# \<Gamma> \<turnstile> e : T ; F" and "a \<sharp> e" shows "\<Gamma> \<turnstile> e : T ; F" proof - have v1:"valid \<Gamma>" using prems valid_elim by blast hence v2:"valid (\<Gamma> - (a,S))" using valid_remove by auto have A:"?\<Gamma> - (a,S) \<turnstile> e : T ; F" using prems fresh_weakening[of a e ?\<Gamma> T F S] by auto have "?\<Gamma> - (a,S) = \<Gamma> - (a,S)" by auto hence B:"\<Gamma> - (a,S) \<lless> \<Gamma>" by (induct \<Gamma>) auto thus ?thesis using A weakening[of _ e T F \<Gamma>] v1 v2 by auto qed lemma closed_empty_env: assumes "closed e" and "\<Gamma> \<turnstile> e : T ; F" and "valid \<Gamma>" shows "[] \<turnstile> e : T ; F" using `valid \<Gamma>` prems proof (induct \<Gamma> rule: valid.induct) case v1 thus ?case by simp next case (v2 \<Gamma>' a S) have "a \<sharp> e" using `closed e` closed_def fresh_def[of a e] by auto thus ?case using v2 fresh_weakening_cons by auto qed lemma closed_any_env: assumes "closed e" and "\<Gamma> \<turnstile> e : T ; F" and val1:"valid \<Gamma>" and val2:"valid \<Gamma>'" shows "\<Gamma>' \<turnstile> e : T ; F" using prems closed_empty_env weakening proof - have A:"[] \<turnstile> e : T ; F" using prems closed_empty_env by auto note weakening have B:"[] \<lless> \<Gamma>'" by auto from A B val1 val2 have "\<Gamma>' \<turnstile> e : T ; F" using weakening by blast thus ?thesis by simp qed inductive_cases ve_ty_elim: "\<Gamma> \<turnstile> e : T ; VE x" lemma te_ty_elim: "\<Gamma> \<turnstile> t1 : T ; TE U z \<Longrightarrow> EX f A A1 Fn S B. t1 = (App f (Var z)) \<and> \<Gamma> \<turnstile> f : B ; Fn \<and> \<turnstile> B <: A1 \<rightarrow> T : Latent U \<and> \<Gamma> \<turnstile> (Var z) : A ; VE z \<and> \<turnstile> A <: A1 " proof (ind_cases "\<Gamma> \<turnstile> t1 : T ; TE U z") fix e1 T0 eff1 e2 Ta x S B assume "t1 = App e1 e2" "TE U z = TE S x" "\<Gamma> \<turnstile> e1 : B ; eff1 "" \<Gamma> \<turnstile> e2 : Ta ; VE x " "\<turnstile> B <: T0 \<rightarrow> T : Latent S" "\<turnstile> Ta <: T0" from prems have A:"e2 = Var x" using ve_ty_elim[of \<Gamma> e2 Ta x] eff.inject by auto hence B:"t1 = App e1 (Var z)" using prems trm.inject eff.inject by auto have C:" \<Gamma> \<turnstile> (Var z) : Ta ; VE z" using prems A trm.inject eff.inject by auto have D:" \<Gamma> \<turnstile> e1 : B ; eff1" using ty.inject latent_eff.inject prems eff.inject by auto have E:"\<turnstile> B <: T0 \<rightarrow> T : Latent U" using ty.inject latent_eff.inject prems eff.inject by auto from `\<turnstile> Ta <: T0` B C D E show ?thesis by blast qed lemma unique_var_typing: assumes "(x,T)#\<Gamma> \<turnstile> Var x : T; VE x" and "(x,T)#\<Gamma> \<turnstile> Var x : T' ; VE x" shows "T = T'" proof (rule ccontr) assume "T ~= T'" have "(x,T) : set ((x,T)#\<Gamma>)" using var_ty_elim[of _ x T "VE x"] prems by auto have "(x,T') : set ((x,T)#\<Gamma>)" "valid ((x,T)#\<Gamma>)" using var_ty_elim[of "((x,T)#\<Gamma>)" x T' "VE x"] prems by auto hence A:"(x,T') : set \<Gamma>" "valid \<Gamma>" "x \<sharp> \<Gamma>" using prems valid_elim[of x T \<Gamma>] by auto have "x : supp \<Gamma>" using `valid \<Gamma>` A proof (induct \<Gamma> rule: valid.induct) case v1 thus ?case by auto next case (v2 \<Gamma>' a S) hence "x \<sharp> (a,S)" "x \<sharp> \<Gamma>'" using fresh_list_cons by auto hence "x \<sharp> a" by auto hence "x ~= a" using fresh_atm by auto hence "(x,T') : set \<Gamma>'" using v2 by auto have "valid \<Gamma>'" using v2 valid_elim[of a S \<Gamma>'] by auto hence "x : supp \<Gamma>'" using v2 `(x,T') : set \<Gamma>'` `x \<sharp> \<Gamma>'` by auto hence "x : supp (a,S) \<union> supp \<Gamma>'" by auto hence "x : (supp ((a,S)#\<Gamma>') :: name set)" using supp_list_cons[of "(a,S)" \<Gamma>'] by force thus ?case by simp qed hence "~ (x \<sharp> \<Gamma>)" using fresh_def[of x \<Gamma>] by blast thus False using `x \<sharp> \<Gamma>` by auto qed inductive_cases tt_sub_ff:"\<turnstile> ty.TT <: ty.FF" inductive_cases ff_sub_tt:"\<turnstile> ty.FF <: ty.TT" lemma value_effect_tt_or_ff: assumes "v : values" and "\<Gamma> \<turnstile> v : T ; F" shows "F = FF \<and> v = Bool False \<or> F = TT \<and> v ~= Bool False" using prems proof (nominal_induct v avoiding: \<Gamma> rule: values.strong_induct) case num_value thus ?case using num_ty_elim by auto next case bi_value thus ?case using bi_ty_elim by auto next case abs_value thus ?case using abs_ty_elim[of \<Gamma> _ _ _ T F] by auto next case (bool_value b) thus ?case using true_ty_elim false_ty_elim by (cases b) auto qed lemma remove_fresh_env: assumes A:"y \<sharp> \<Gamma>" and B:"valid \<Gamma>" shows "\<Gamma> - (y,T) = \<Gamma>" using B A proof (induct \<Gamma> rule: valid.induct) case v1 thus ?case by auto next case (v2 \<Gamma>' a S) have "y \<noteq> a" "y \<sharp> \<Gamma>'" using `y \<sharp> ((a, S) # \<Gamma>')` using fresh_atm[of y a] fresh_list_cons by auto hence "((a, S) # \<Gamma>') - (y, T) = (a, S) # (\<Gamma>' - (y, T))" by auto thus ?case using v2 `y \<sharp> \<Gamma>'` by auto qed lemma subst_preserve_TE_app: assumes tapp:"(y,T0)#\<Gamma> \<turnstile> App e1 e2 : T ; TE S x" and neq:"y \<noteq> x" and val:"valid ((y,T0)#\<Gamma>)" and ih: "!! t bc bf . \<lbrakk>t \<guillemotleft> App e1 e2; (y, T0) # \<Gamma> \<turnstile> t : bc ; bf \<rbrakk> \<Longrightarrow> \<exists>T' F'. \<Gamma> \<turnstile> t[y::=v] : T' ; F' \<and> \<turnstile> T' <: bc \<and> \<turnstile> F' <e: bf" shows "\<Gamma> \<turnstile> (App e1 e2)[y::=v] : T ; TE S x" proof - note te_ty_elim[OF tapp] then obtain f A A1 Fn Sa B where sz:"App e1 e2 = App f (Var x)" and tf:"(y, T0) # \<Gamma> \<turnstile> f : B ; Fn "" \<turnstile> B <: A1 \<rightarrow> T : Latent S" and tvx:"(y, T0) # \<Gamma> \<turnstile> Var x : A ; VE x" and sub:" \<turnstile> A <: A1" by auto hence "e1 = f" and "e2 = Var x" using trm.inject by auto hence "e2[y::=v] = Var x" using `y \<noteq> x` forget by auto hence tsub:" (y, T0)#\<Gamma> \<turnstile> e2[y::=v] : A ; VE x" using `e2=Var x` tvx by auto have fr:"y \<sharp> e2[y::=v]" using `e2 = Var x``e2[y::=v] = Var x` `y \<noteq> x` using fresh_atm by auto have "y \<sharp> \<Gamma>" "valid \<Gamma>" using valid_elim[OF val] by auto hence "((y, T0) # \<Gamma>) - (y, T0) = \<Gamma>" using remove_fresh_env[OF `y \<sharp> \<Gamma>` `valid \<Gamma>`] by auto hence te2:"\<Gamma> \<turnstile> e2[y::=v] : A ; VE x" using fresh_weakening[OF fr tsub `valid ((y,T0)#\<Gamma>)`, of T0] by auto have szf:"f \<guillemotleft> App e1 e2" using sz by auto note ih[OF szf tf(1)] then obtain T' F' where tfsub:"\<Gamma> \<turnstile> f[y::=v] : T' ; F' "" \<turnstile> T' <: B " "\<turnstile> F' <e: Fn" by auto hence " \<turnstile> T' <: A1 \<rightarrow> T : Latent S " using `\<turnstile> B <: A1 \<rightarrow> T : Latent S` by auto hence "\<Gamma> \<turnstile> (App f e2)[y::=v] : T ; TE S x" using te2 tfsub `\<turnstile> A <: A1` by auto thus ?thesis using sz using trm.inject by auto qed inductive_cases te_elim_auto: "\<Gamma> \<turnstile> e : T ; TE S x" thm te_elim_auto lemma subst_preserve_TE: fixes v assumes tapp:"(y,T0)#\<Gamma> \<turnstile> e : T ; TE S x" and neq:"y \<noteq> x" and val:"valid ((y,T0)#\<Gamma>)" and ih: "!! t bc bf . \<lbrakk>t \<guillemotleft> e; (y, T0) # \<Gamma> \<turnstile> t : bc ; bf \<rbrakk> \<Longrightarrow> \<exists>T' F'. \<Gamma> \<turnstile> t[y::=v] : T' ; F' \<and> \<turnstile> T' <: bc \<and> \<turnstile> F' <e: bf" shows "\<Gamma> \<turnstile> e[y::=v] : T ; TE S x" proof - obtain e1 e2 where "e = App e1 e2" using te_elim_auto[OF tapp, of thesis] by auto hence A:"(y,T0)#\<Gamma> \<turnstile> App e1 e2 : T ; TE S x" using tapp by auto note subst_preserve_TE_app[OF A neq val , of v] have "!! t bc bf . \<lbrakk>t \<guillemotleft> App e1 e2; (y, T0) # \<Gamma> \<turnstile> t : bc ; bf \<rbrakk> \<Longrightarrow> \<exists>T' F'. \<Gamma> \<turnstile> t[y::=v] : T' ; F' \<and> \<turnstile> T' <: bc \<and> \<turnstile> F' <e: bf" using ih `e = App e1 e2` by auto hence "\<Gamma> \<turnstile> App e1 e2[y::=v] : T ; TE S x" using subst_preserve_TE_app[OF A neq val , of v] by auto thus ?thesis using `e = App e1 e2` by auto qed inductive_cases lam_latent_eff_elim_auto: "\<Gamma> \<turnstile> Lam[x:T].b : S1 \<rightarrow> S2 : Latent U ; F" thm lam_latent_eff_elim_auto lemma preserve_subst: assumes "(x,T0)#\<Gamma> \<turnstile> e : T ; F" and "\<Gamma> \<turnstile> e' : T1 ; G" and "\<turnstile> T1 <: T0" and "valid ((x,T0)#\<Gamma>)" and "closed e'" and "e' : values" shows "EX T' F'. \<Gamma> \<turnstile> e[x::=e'] : T' ; F' \<and> \<turnstile> T' <: T \<and> \<turnstile> F' <e: F" using prems proof (nominal_induct e avoiding: \<Gamma> x e' T T1 T0 F G rule: trm_comp_induct) case (Var v) have a1: "\<Gamma> \<turnstile>e':T1;G" by fact have a2: "((x,T0)#\<Gamma>) \<turnstile> Var v:T;F" by fact hence a21: "(v,T)\<in>set((x,T0)#\<Gamma>)" and a22: "valid((x,T0)#\<Gamma>)" and a22b:"F = VE v" using var_ty_elim[of "(x, T0) # \<Gamma>"] by auto from a22 have a23: "valid \<Gamma>" and a24: "x\<sharp>\<Gamma>" by (auto dest: valid_elim) from a24 have a25: "\<not>(\<exists>\<tau>. (x,\<tau>)\<in>set \<Gamma>)" by (rule fresh_context) show ?case proof (cases "v=x") assume case1: "v=x" hence "(Var v)[x::=e'] = e'" by simp hence A:"\<Gamma> \<turnstile> (Var v)[x::=e'] : T1; G" using Var by auto have "simple_eff G" using closed_eff `closed e'` prems by auto hence C:"\<turnstile> G <e: F" using a22b simple_eff_below_ve by auto have B:"T = T0" proof (rule ccontr) assume a3:"T ~= T0" from a3 a21 have "(v,T)\<in>set \<Gamma>" by force with case1 a25 show False by force qed hence D:"\<turnstile> T1 <: T" using `\<turnstile> T1 <: T0` by auto have "~ (\<exists>S y. F = TE S y)" using a22b by auto thus ?thesis using A a22b C D by blast next assume case2: "v\<noteq>x" with a21 have a26: "(v,T)\<in>set \<Gamma>" by force have a27:" Var v[x::=e'] = Var v" using case2 by simp from a23 a26 a27 have "\<Gamma> \<turnstile> Var v:T;VE v" by force thus ?thesis using a27 a22b by auto qed next case (Num n) have A:"(Num n)[x::=e'] = Num n" by auto have B:"F = eff.TT" using Num num_ty_elim by auto have "T = ty.Int" using num_ty_elim Num by auto hence "\<Gamma> \<turnstile> (Num n)[x::=e'] : T ; eff.TT" using Num A valid_elim[of x T0 \<Gamma>] by auto thus ?case using Num B by auto next case (Bool b) have v:"valid \<Gamma>" using prems typing_valid by auto have A:"(Bool b)[x::=e'] = Bool b" by auto thus ?case proof (cases b) case True hence B:"T = ty.TT" and "F = TT" using Bool true_ty_elim by auto thus ?thesis using A v True by auto next case False hence B:"T = ty.FF" and "F = FF" using Bool false_ty_elim by auto thus ?thesis using A v False by auto qed next case (BI b) have A:"(BI b)[x::=e'] = (BI b)" by auto have B:"F = eff.TT" using BI bi_ty_elim by auto have "T = \<Delta>\<^isub>\<tau> b" using bi_ty_elim BI by auto hence "\<Gamma> \<turnstile> (BI b)[x::=e'] : T ; eff.TT" using BI A valid_elim[of x T0 \<Gamma>] by auto thus ?case using BI B by auto next case (App s1 s2 \<Gamma>' x' e'' T' T1' T0' F' G') have ih_s1: "\<And>c \<sigma> T F T' F' e' \<Gamma>. ((c,\<sigma>)#\<Gamma>) \<turnstile> s1:T; F \<Longrightarrow> closed e' \<Longrightarrow> e' : values \<Longrightarrow> valid ((c,\<sigma>)#\<Gamma>) \<Longrightarrow> \<Gamma>\<turnstile> e': T' ; F' \<Longrightarrow> \<turnstile> T' <: \<sigma> \<Longrightarrow> EX S G. \<Gamma> \<turnstile> s1[c::=e']:S ; G \<and> \<turnstile> S <: T \<and> \<turnstile> G <e: F" . have ih_s2: "\<And>c \<sigma> T F T' F' e' \<Gamma>. ((c,\<sigma>)#\<Gamma>) \<turnstile> s2:T; F \<Longrightarrow> closed e' \<Longrightarrow> e' : values \<Longrightarrow> valid ((c,\<sigma>)#\<Gamma>) \<Longrightarrow> \<Gamma>\<turnstile> e': T' ; F' \<Longrightarrow> \<turnstile> T' <: \<sigma> \<Longrightarrow> EX S G. \<Gamma> \<turnstile> s2[c::=e']:S ; G \<and> \<turnstile> S <: T \<and> \<turnstile> G <e: F" . have appty:"((x',T0')#\<Gamma>')\<turnstile>App s1 s2 : T'; F'" . from appty have elim1:"\<exists>T0 T0'a T1 le eff' eff'' U.(x',T0')#\<Gamma>' \<turnstile> s1 :U;eff' \<and> (x',T0')# \<Gamma>'\<turnstile> s2 : T0'a;eff'' \<and> \<turnstile> U <: T0\<rightarrow>T1:le \<and> \<turnstile> T0'a <: T0 \<and> T1 = T'" using app_ty_elim by auto from appty have elim2:"(x', T0') # \<Gamma>' \<turnstile> App s1 s2 : T' ; FF \<Longrightarrow> \<exists>T0 T0'a leS eff' eff'' le U.(x',T0')#\<Gamma>' \<turnstile> s1 :U;eff' \<and> (x',T0')# \<Gamma>'\<turnstile> s2 : T0'a;eff'' \<and> \<turnstile> T0'a <: T0\<and> \<turnstile> U <: T0\<rightarrow>T':le \<and> le = Latent leS \<and> ~ (\<turnstile> T0'a <: leS) \<and> s2 : values \<and> closed s2" using app_ty_ff_elim[of "((x',T0')#\<Gamma>')" s1 s2 T'] by blast have elim3:"(x', T0') # \<Gamma>' \<turnstile> App s1 s2 : T' ; TT \<Longrightarrow> \<exists>T0 T0'a leS eff' eff'' U le.(x',T0')#\<Gamma>' \<turnstile> s1 :U;eff' \<and> (x',T0')# \<Gamma>'\<turnstile> s2 : T0'a;eff'' \<and> \<turnstile> T0'a <: T0 \<and> \<turnstile> U <: T0\<rightarrow>T':le \<and> le = Latent leS \<and> \<turnstile> T0'a <: leS " using app_ty_tt_elim[of "((x',T0')#\<Gamma>')" s1 s2 T'] by blast from elim1 obtain T0 T0'a T1 le eff' eff'' U where P:"(x',T0')#\<Gamma>' \<turnstile> s1 :U;eff'"" (x',T0')# \<Gamma>'\<turnstile> s2 : T0'a;eff''"" \<turnstile> T0'a <: T0 "" T1 = T'" "\<turnstile> U <: T0\<rightarrow>T1:le" by auto hence "EX S1 G1. \<Gamma>' \<turnstile> s1[x'::=e''] : S1 ; G1 \<and> \<turnstile> S1 <: U \<and> \<turnstile> G1 <e: eff'" using ih_s1 App by auto then obtain S1 G1 where Q:" \<Gamma>' \<turnstile> s1[x'::=e''] : S1 ; G1 "" \<turnstile> S1 <: U" "\<turnstile> G1 <e: eff'" by auto have sub1:"\<turnstile>S1 <: T0\<rightarrow>T1:le" using P Q by auto ( then apply ih_s2, get something about the substition of s2, and put it all back together. ) from P have "EX S G. \<Gamma>' \<turnstile> s2[x'::=e'']:S ; G \<and> \<turnstile> S <: T0'a \<and> \<turnstile> G <e: eff''" using ih_s2[of x' T0' \<Gamma>' T0'a eff''] App by auto then obtain S2 G2 where S:"\<Gamma>' \<turnstile> s2[x'::=e'']:S2 ; G2 "" \<turnstile> S2 <: T0'a" "\<turnstile> G2 <e: eff''" by auto let ?ns1 = "s1[x'::=e'']" and ?ns2 = "s2[x'::=e'']" have sub2:"\<turnstile> S2 <: T0" using P S by auto have L1:"\<Gamma>' \<turnstile> App ?ns1 ?ns2 : T1 ; NE" using Q S sub1 sub2 by auto have L2:"T1 = T'" . show ?case using appty proof (nominal_induct F' rule: eff.induct) case NE thus ?case using L1 L2 by auto next case VE thus ?case using L1 L2 by auto next case (TE ty var) thus ?case using L1 L2 by auto next case TT from elim3 TT obtain T0 T0'a T1 le leS eff' eff'' U where P:"(x',T0')#\<Gamma>' \<turnstile> s1 :U;eff'"" (x',T0')# \<Gamma>'\<turnstile> s2 : T0'a;eff''"" \<turnstile> T0'a <: T0 "" T1 = T'" "\<turnstile> U <: T0\<rightarrow>T1:le" "le = Latent leS" "\<turnstile> T0'a <: leS" by auto hence "EX S1 G1. \<Gamma>' \<turnstile> s1[x'::=e''] : S1 ; G1 \<and> \<turnstile> S1 <: U \<and> \<turnstile> G1 <e: eff'" using ih_s1 App by blast then obtain S1 G1 where Q:" \<Gamma>' \<turnstile> s1[x'::=e''] : S1 ; G1 "" \<turnstile> S1 <: U" "\<turnstile> G1 <e: eff'" by auto hence R:"\<turnstile> S1 <: T0\<rightarrow>T1:le" using P by auto ( then apply ih_s2, get something about the substition of s2, and put it all back together. ) from P have "EX S G. \<Gamma>' \<turnstile> s2[x'::=e'']:S ; G \<and> \<turnstile> S <: T0'a \<and> \<turnstile> G <e: eff''" using ih_s2[of x' T0' \<Gamma>' T0'a eff''] App by auto then obtain S2 G2 where S:"\<Gamma>' \<turnstile> s2[x'::=e'']:S2 ; G2 "" \<turnstile> S2 <: T0'a" "\<turnstile> G2 <e: eff''" by auto let ?ns1 = "s1[x'::=e'']" and ?ns2 = "s2[x'::=e'']" have noover: "\<turnstile> S2 <: leS" using `\<turnstile> S2 <: T0'a` `\<turnstile> T0'a <: leS` by auto have L1:"\<Gamma>' \<turnstile> App ?ns1 ?ns2 : T1 ; TT" using P Q R S noover T_AppPredFalse[of \<Gamma>' " s1[x'::=e'']" U] by auto have L2:"T1 = T'" . from L1 L2 show ?case by auto next case FF from elim2 FF obtain T0 T0'a T1 le leS eff' eff'' U where P:"(x',T0')#\<Gamma>' \<turnstile> s1 :U;eff'"" (x',T0')# \<Gamma>'\<turnstile> s2 : T0'a;eff''"" \<turnstile> T0'a <: T0 "" T1 = T'" "\<turnstile> U <: T0\<rightarrow>T1:le" "le = Latent leS" "~ (\<turnstile> T0'a <: leS)" "s2 :values" "closed s2" by auto hence "EX S1 G1. \<Gamma>' \<turnstile> s1[x'::=e''] : S1 ; G1 \<and> \<turnstile> S1 <: U \<and> \<turnstile> G1 <e: eff'" using ih_s1 App by auto then obtain S1 G1 where Q:" \<Gamma>' \<turnstile> s1[x'::=e''] : S1 ; G1 "" \<turnstile> S1 <: U" "\<turnstile> G1 <e: eff'" by auto hence R:"\<turnstile> S1 <: T0\<rightarrow>T1:le" using P by auto ( then apply ih_s2, get something about the substition of s2, and put it all back together. ) from P have "EX S G. \<Gamma>' \<turnstile> s2[x'::=e'']:S ; G \<and> \<turnstile> S <: T0'a \<and> \<turnstile> G <e: eff''" using ih_s2[of x' T0' \<Gamma>' T0'a eff''] App by auto then obtain S2 G2 where S:"\<Gamma>' \<turnstile> s2[x'::=e'']:S2 ; G2 "" \<turnstile> S2 <: T0'a" "\<turnstile> G2 <e: eff''" by auto let ?ns1 = "s1[x'::=e'']" and ?ns2 = "s2[x'::=e'']" have "x' \<sharp> s2" using closed_def fresh_def[of x' s2] `closed s2` by auto hence "s2 = ?ns2" using forget by auto hence S':"\<Gamma>' \<turnstile> ?ns2 : T0'a ; eff''" using `(x',T0')#\<Gamma>' \<turnstile> s2 : T0'a ; eff''` fresh_weakening_cons `valid ((x',T0')#\<Gamma>')` `x' \<sharp> s2` by auto have noover: "~(\<turnstile> T0'a <: leS)" . have T:"closed ?ns2" "?ns2 : values" using `s2 = ?ns2` `closed s2` `s2 : values` by auto have L1:"\<Gamma>' \<turnstile> App ?ns1 ?ns2 : T1 ; FF" using P Q R S' T noover by auto have L2:"T1 = T'" by fact from L1 L2 show ?case by auto qed next case (Lam a body \<Gamma>' x' e'' T' T1' T0' F' G' ty ) hence f1: "a\<sharp>\<Gamma>'" and f2: "a\<noteq>x'" and f2': "x'\<sharp>a" and f3: "a\<sharp>e''" and f4: "a\<sharp>((x',T0')#\<Gamma>')" by (auto simp add: fresh_atm fresh_prod fresh_list_cons) have c1: "((x',T0')#\<Gamma>')\<turnstile>Lam [a:ty].body : T' ; F'" by fact ( hence "\<exists>\<tau>2 eff L S. ((a,ty)#(x',T0')#\<Gamma>') \<turnstile> body : \<tau>2 ; eff \<and> T'=ty\<rightarrow>\<tau>2:L \<and> F' = TT" using f4 abs_ty_elim by auto ) then obtain \<tau>2 eff L S where c11: "T'=ty\<rightarrow>\<tau>2:L" and c12: "((a,ty)#(x',T0')#\<Gamma>') \<turnstile> body : \<tau>2 ; eff" and "F' = TT" and c13:"L = latent_eff.NE \<or> (eff = TE S a \<and> L = Latent S)" using f4 abs_ty_elim by blast from c12 have "valid ((a,ty)#(x',T0')#\<Gamma>')" using Lam by auto hence ca: "valid \<Gamma>'" and cb: "a\<sharp>\<Gamma>'" and cc: "x'\<sharp>\<Gamma>'" by (auto dest: valid_elim simp add: fresh_list_cons) have c2: "((a,ty)#(x',T0')#\<Gamma>') \<lless> ((x',T0')#(a,ty)#\<Gamma>')" by force have c3: "valid ((x',T0')#(a,ty)#\<Gamma>')" by (rule v2, rule v2, auto simp add: fresh_list_cons fresh_prod ca cb cc f2' fresh_ty) from c12 c2 c3 have c14: "((x',T0')#(a,ty)#\<Gamma>') \<turnstile> body : \<tau>2 ; eff" using `valid ((a, ty) # (x', T0') # \<Gamma>')` by (force intro: weakening) let ?inner\<Gamma> = "(a,ty)#\<Gamma>'" have validig:"valid ?inner\<Gamma>" using `a \<sharp> \<Gamma>'` `valid \<Gamma>'` by auto have c15:"\<Gamma>' \<lless> ?inner\<Gamma>" by auto hence c16:"?inner\<Gamma> \<turnstile> e'' : T1' ; G'" using weakening[of \<Gamma>' _ _ _ ?inner\<Gamma>] `\<Gamma>' \<turnstile> e'' : T1' ; G'` validig `valid \<Gamma>'` by simp have "EX TA0 FA0. ?inner\<Gamma> \<turnstile> body[x'::=e''] : TA0 ; FA0 \<and> \<turnstile> TA0 <: \<tau>2 \<and> \<turnstile> FA0 <e: eff" using c16 Lam(10)[of x' T0' ?inner\<Gamma> \<tau>2 eff e'' T1' G'] ` \<turnstile> T1' <: T0'` `valid ((x', T0') # (a, ty) # \<Gamma>')` c14 `closed e''` `e'' : values` by auto then obtain TA0 FA0 where body_ty:"?inner\<Gamma> \<turnstile> body[x'::=e''] : TA0 ; FA0 "" \<turnstile> TA0 <: \<tau>2" by auto hence L11:"\<Gamma>' \<turnstile> (Lam[a:ty].(body[x'::=e''])) : ty \<rightarrow> TA0 : latent_eff.NE; eff.TT" using `a \<sharp> \<Gamma>'` by auto note Lam(9)[OF _ _ `(a, ty)# \<Gamma>' \<turnstile> e'' : T1' ; G'` `\<turnstile> T1' <: T0'` `valid ((x', T0')# (a, ty) # \<Gamma>')` `closed e''` `e'' : values`] hence "!! t bc bf . \<lbrakk>t \<guillemotleft> Lam [a:ty].body; (x', T0') #(a, ty)# \<Gamma>' \<turnstile> t : bc ; bf \<rbrakk> \<Longrightarrow> \<exists>T' F'. (a, ty)#\<Gamma>' \<turnstile> t[x'::=e''] : T' ; F' \<and> \<turnstile> T' <: bc \<and> \<turnstile> F' <e: bf" . hence ih_body:"!! t bc bf . \<lbrakk>t \<guillemotleft> body; (x', T0') # (a, ty)#\<Gamma>' \<turnstile> t : bc ; bf \<rbrakk> \<Longrightarrow> \<exists>T' F'. (a, ty)#\<Gamma>' \<turnstile> t[x'::=e''] : T' ; F' \<and> \<turnstile> T' <: bc \<and> \<turnstile> F' <e: bf" by auto hence L12:"L = Latent S \<Longrightarrow> \<Gamma>' \<turnstile> (Lam[a:ty].(body[x'::=e''])) : ty \<rightarrow> \<tau>2 : Latent S; eff.TT" proof - assume "L = Latent S" hence "eff = TE S a" using c13 by auto hence c12':"(a, ty) # (x', T0') # \<Gamma>' \<turnstile> body : \<tau>2 ; TE S a" using c12 by simp have c12'':" (x', T0') # (a, ty) # \<Gamma>' \<turnstile> body : \<tau>2 ; TE S a" using weakening[OF c12' _ c2 ] using `valid ((x', T0') # (a, ty) # \<Gamma>')` `valid ((a, ty) # (x', T0') # \<Gamma>')` by auto have f2':"x' \<noteq> a" using f2 by auto note body_ty subst_preserve_TE[OF c12'' f2' `valid ((x', T0') # (a, ty) # \<Gamma>')` , of e''] hence "(a, ty) # \<Gamma>' \<turnstile> body[x'::=e''] : \<tau>2 ; TE S a" using ih_body by auto hence "?inner\<Gamma> \<turnstile> body[x'::=e''] : \<tau>2 ; TE S a " . thus ?thesis using `a \<sharp> \<Gamma>'` by auto qed ( from L11 L12 have L1:"\<Gamma>' \<turnstile> (Lam[a:ty].(body[x'::=e''])) : ty \<rightarrow> TA0 : L; eff.TT" using c13 by auto ) have L21:"\<turnstile> ty \<rightarrow> TA0 : L <: T'" using c11 ` \<turnstile> TA0 <: \<tau>2` by auto have L22: "\<turnstile> ty \<rightarrow> \<tau>2 : L <: T'" using c11 by auto have L3:"(Lam[a:ty].body)[x'::=e''] = (Lam[a:ty].(body[x'::=e'']))" using Lam by auto have L4:"\<turnstile> eff.TT <e: F'" using `F' = TT` by auto thm Lam have L5:"!! Env a ty body T S x. Env \<turnstile> (Lam [a:ty].body) : T ; TE S x \<Longrightarrow> False" proof - fix Env a ty body T S x assume "Env \<turnstile> (Lam [a:ty].body) : T ; TE S x" have " Env \<turnstile> (Lam [a:ty].body) : T ; TE S x \<Longrightarrow> False" by (ind_cases " Env \<turnstile> (Lam [a:ty].body) : T ; TE S x") thus False using prems by auto qed from L11 L12 L21 L22 L3 L4 L5[of \<Gamma>' a _ ty ] c13 show ?case by auto next case (Iff t1 t2 t3 \<Gamma>' x' e'' T' T0' T1' F' G') let ?\<Gamma> = "(x', T1') # \<Gamma>'" have A:"(\<exists> T1 T2 T3 F1 F2 F3. (?\<Gamma> \<turnstile> t1 : T1 ; F1) \<and> ?\<Gamma> \|+ F1 \<turnstile> t2 : T2 ; F2 \<and> ?\<Gamma> \|- F1 \<turnstile> t3 : T3 ; F3 \<and> \<turnstile> T2 <: T' \<and> \<turnstile> T3 <: T' \<and> F' = NE) \<or> (\<exists> T1 T3 F3. (?\<Gamma> \<turnstile> t1 : T1 ; FF) \<and> ?\<Gamma> \<turnstile> t3 : T3 ; F3 \<and> \<turnstile> T3 <: T' \<and> F' = NE) \<or> (\<exists> T1 T2 F2. (?\<Gamma> \<turnstile> t1 : T1 ; TT) \<and> ?\<Gamma> \<turnstile> t2 : T2 ; F2 \<and> \<turnstile> T2 <: T' \<and> F' = NE)" using Iff if_ty_elim by auto thus ?case proof - { assume "(\<exists> T1 T2 T3 F1 F2 F3. (?\<Gamma> \<turnstile> t1 : T1 ; F1) \<and> ?\<Gamma> \|+ F1 \<turnstile> t2 : T2 ; F2 \<and> ?\<Gamma> \|- F1 \<turnstile> t3 : T3 ; F3 \<and> \<turnstile> T2 <: T' \<and> \<turnstile> T3 <: T' \<and> F' = NE)" then obtain T1 T2 T3 F1 F2 F3 where "?\<Gamma> \<turnstile> t1 : T1 ; F1 "" ?\<Gamma> \|+ F1 \<turnstile> t2 : T2 ; F2 "" ?\<Gamma> \|- F1 \<turnstile> t3 : T3 ; F3 "" \<turnstile> T2 <: T'""\<turnstile> T3 <: T'""F' = NE" by auto hence ?thesis proof (nominal_induct F1 rule: eff.induct) case NE from NE have "\<exists>S1 G1. \<Gamma>' \<turnstile> t1[x'::=e''] : S1 ; G1 \<and> \<turnstile> S1 <: T1 \<and> \<turnstile> G1 <e: NE" using Iff by auto then obtain S1 G1 where A:"\<Gamma>' \<turnstile> t1[x'::=e''] : S1 ; G1 "" \<turnstile> S1 <: T1 "" \<turnstile> G1 <e: NE" by auto have simple:"simple_eff G1" using `\<turnstile> G1 <e: NE` below_ne_simple by auto have p1:"(?\<Gamma> \<turnstile> t2 : T2 ; F2)" using prems by auto have p2:"(?\<Gamma> \<turnstile> t3 : T3 ; F3)" using prems by auto from p1 have "\<exists>S2 G2. \<Gamma>' \<turnstile> t2[x'::=e''] : S2 ; G2 \<and> \<turnstile> S2 <: T2 \<and> \<turnstile> G2 <e: F2" using Iff by auto then obtain S2 G2 where B:"\<Gamma>' \<turnstile> t2[x'::=e''] : S2 ; G2 "" \<turnstile> S2 <: T2 "" \<turnstile> G2 <e: F2" by auto from p2 have "\<exists>S3 G3. \<Gamma>' \<turnstile> t3[x'::=e''] : S3 ; G3 \<and> \<turnstile> S3 <: T3 \<and> \<turnstile> G3 <e: F3" using Iff by auto then obtain S3 G3 where C:"\<Gamma>' \<turnstile> t3[x'::=e''] : S3 ; G3 "" \<turnstile> S3 <: T3 "" \<turnstile> G3 <e: F3" by auto have B':"\<Gamma>' \|+ G1 \<turnstile> t2[x'::=e''] : S2 ; G2" using B simple by auto have C':"\<Gamma>' \|- G1 \<turnstile> t3[x'::=e''] : S3 ; G3" using C simple by auto have D:"\<turnstile> S2 <: T'" using prems B by auto have E:"\<turnstile> S3 <: T'" using prems C by auto from A B' C' D E have " \<Gamma>' \<turnstile> Iff t1 t2 t3[x'::=e''] : T' ; comb_eff G1 G2 G3" by (auto simp del: comb_eff.simps) thus ?case using `F' = NE` by auto next case TT from TT have "\<exists>S1 G1. \<Gamma>' \<turnstile> t1[x'::=e''] : S1 ; G1 \<and> \<turnstile> S1 <: T1 \<and> \<turnstile> G1 <e: TT" using Iff by auto then obtain S1 G1 where A:"\<Gamma>' \<turnstile> t1[x'::=e''] : S1 ; G1 "" \<turnstile> S1 <: T1 "" \<turnstile> G1 <e: TT" by auto have simple:"G1 = TT" using A no_sub_TT by blast have p1:"(?\<Gamma> \<turnstile> t2 : T2 ; F2)" using prems by auto have p2:"(?\<Gamma> \<turnstile> t3 : T3 ; F3)" using prems by auto from p1 have "\<exists>S2 G2. \<Gamma>' \<turnstile> t2[x'::=e''] : S2 ; G2 \<and> \<turnstile> S2 <: T2 \<and> \<turnstile> G2 <e: F2" using Iff by auto then obtain S2 G2 where B:"\<Gamma>' \<turnstile> t2[x'::=e''] : S2 ; G2 "" \<turnstile> S2 <: T2 "" \<turnstile> G2 <e: F2" by auto from p2 have "\<exists>S3 G3. \<Gamma>' \<turnstile> t3[x'::=e''] : S3 ; G3 \<and> \<turnstile> S3 <: T3 \<and> \<turnstile> G3 <e: F3" using Iff by auto then obtain S3 G3 where C:"\<Gamma>' \<turnstile> t3[x'::=e''] : S3 ; G3 "" \<turnstile> S3 <: T3 "" \<turnstile> G3 <e: F3" by auto have B':"\<Gamma>' \|+ G1 \<turnstile> t2[x'::=e''] : S2 ; G2" using B simple by auto have C':"\<Gamma>' \|- G1 \<turnstile> t3[x'::=e''] : S3 ; G3" using C simple by auto have D:"\<turnstile> S2 <: T'" using prems B by auto have E:"\<turnstile> S3 <: T'" using prems C by auto from A B' C' D E have " \<Gamma>' \<turnstile> Iff t1 t2 t3[x'::=e''] : T' ; comb_eff G1 G2 G3" by (auto simp del: comb_eff.simps) thus ?case using `F' = NE` by auto next case FF from FF have "\<exists>S1 G1. \<Gamma>' \<turnstile> t1[x'::=e''] : S1 ; G1 \<and> \<turnstile> S1 <: T1 \<and> \<turnstile> G1 <e: FF" using Iff by auto then obtain S1 G1 where A:"\<Gamma>' \<turnstile> t1[x'::=e''] : S1 ; G1 "" \<turnstile> S1 <: T1 "" \<turnstile> G1 <e: FF" by auto have simple:"G1 = FF" using A no_sub_FF by blast have p1:"(?\<Gamma> \<turnstile> t2 : T2 ; F2)" using prems by auto have p2:"(?\<Gamma> \<turnstile> t3 : T3 ; F3)" using prems by auto from p1 have "\<exists>S2 G2. \<Gamma>' \<turnstile> t2[x'::=e''] : S2 ; G2 \<and> \<turnstile> S2 <: T2 \<and> \<turnstile> G2 <e: F2" using Iff by auto then obtain S2 G2 where B:"\<Gamma>' \<turnstile> t2[x'::=e''] : S2 ; G2 "" \<turnstile> S2 <: T2 "" \<turnstile> G2 <e: F2" by auto from p2 have "\<exists>S3 G3. \<Gamma>' \<turnstile> t3[x'::=e''] : S3 ; G3 \<and> \<turnstile> S3 <: T3 \<and> \<turnstile> G3 <e: F3" using Iff by auto then obtain S3 G3 where C:"\<Gamma>' \<turnstile> t3[x'::=e''] : S3 ; G3 "" \<turnstile> S3 <: T3 "" \<turnstile> G3 <e: F3" by auto have B':"\<Gamma>' \|+ G1 \<turnstile> t2[x'::=e''] : S2 ; G2" using B simple by auto have C':"\<Gamma>' \|- G1 \<turnstile> t3[x'::=e''] : S3 ; G3" using C simple by auto have D:"\<turnstile> S2 <: T'" using prems B by auto have E:"\<turnstile> S3 <: T'" using prems C by auto from A B' C' D E have " \<Gamma>' \<turnstile> Iff t1 t2 t3[x'::=e''] : T' ; comb_eff G1 G2 G3" by (auto simp del: comb_eff.simps) thus ?case using `F' = NE` by auto next case (VE z) hence A:"t1 = (Var z) " "?\<Gamma> \<turnstile> Var z : T1 ; VE z" using ve_ty_elim[OF `?\<Gamma> \<turnstile> t1 : T1; VE z`] using eff.inject by auto have size1:"(Var z\<guillemotleft>Iff t1 t2 t3)" using A by simp note Iff(1)[of "Var z"] hence ih_f:"!! c \<sigma> \<Gamma> T F e' T0 F0 . (c,\<sigma>)#\<Gamma> \<turnstile> (Var z) : T ; F \<Longrightarrow> \<Gamma> \<turnstile> e' : T0 ; F0 \<Longrightarrow> \<turnstile> T0 <: \<sigma> \<Longrightarrow> valid ((c,\<sigma>)#\<Gamma>) \<Longrightarrow> closed e' \<Longrightarrow> e' : values \<Longrightarrow> EX T' F' . \<Gamma> \<turnstile> (Var z)[c::=e'] : T' ; F' \<and> \<turnstile> T' <: T \<and> \<turnstile> F' <e: F" using size1 by auto have "EX A' Fn' . \<Gamma>' \<turnstile> (Var z)[x'::=e''] : A' ; Fn' \<and> \<turnstile> A' <: T1 \<and> \<turnstile> Fn' <e: VE z" using ih_f A Iff by auto then obtain A' Fn' where B:"\<Gamma>' \<turnstile> (Var z)[x'::=e''] : A' ; Fn' \<and> \<turnstile> A' <:T1" by auto let ?mapfun = "(%f. (% (v,ty). (if (z = v) then (v,f ty.FF ty) else (v,ty))))" let ?\<Gamma>1 = "(map (?mapfun remove) \<Gamma>')" let ?\<Gamma>2 = "(map (?mapfun replace) \<Gamma>')" have "valid \<Gamma>'" using `valid ?\<Gamma>` using valid_elim[of x' T1' \<Gamma>'] by auto show ?case proof (cases "x' = z") case True from A True have "(Var z)[x'::=e''] = e''" by auto hence D:"\<Gamma>' \<turnstile> (Var z)[x'::=e''] : T0' ; G'" "closed ((Var z)[x'::=e''])" "((Var z)[x'::=e'']) : values" using Iff by auto have "\<turnstile> T0' <: T1'" . note var_ty_elim[of ?\<Gamma> z _ "VE z"] hence "(x', T1') : set ?\<Gamma>" using A True by auto have "?\<Gamma> \<turnstile> (Var x') : T1' ; VE x'" using `valid ?\<Gamma>` by auto have "simple_ty T0'" using `\<Gamma>' \<turnstile> e'' : T0' ; G'` `e'' : values` value_simple_type by auto let ?mapfun = "(%f. (% (v,ty). (if (z = v) then (v,f ty.FF ty) else (v,ty))))" let ?\<Gamma>1 = "(map (?mapfun remove) \<Gamma>')" let ?\<Gamma>2 = "(map (?mapfun replace) \<Gamma>')" have "?\<Gamma>1 = envop remove z ty.FF \<Gamma>'" by auto have "?\<Gamma>2 = envop replace z ty.FF \<Gamma>'" by auto have "x' \<sharp> \<Gamma>'" using Iff valid_elim[of x' T1' \<Gamma>'] by auto hence "?\<Gamma>1 = \<Gamma>'" using True envop_forget `valid \<Gamma>'` by auto hence GA:"?\<Gamma> \|+ VE z = (x',remove ty.FF T1') # \<Gamma>'" using True by auto hence G1:"\<dots> \<turnstile> t2 : T2; F2" using `(?\<Gamma> \|+ VE z)\<turnstile> t2 : T2 ; F2` by auto have "?\<Gamma>2 = \<Gamma>'" using `x' \<sharp> \<Gamma>'` True envop_forget `valid \<Gamma>'` by auto hence "?\<Gamma> \|- VE z = (x',replace ty.FF T1') # \<Gamma>'" using True by auto hence GB:"?\<Gamma> \|- VE z = (x',ty.FF) # \<Gamma>'" by auto hence G2:"\<dots> \<turnstile> t3 : T3; F3" using `(?\<Gamma> \|- VE z)\<turnstile> t3 : T3 ; F3` by auto have "valid (?\<Gamma> \|+ VE z)" using `valid ?\<Gamma>` envplus_valid[of ?\<Gamma> "VE z"] by auto hence G3:"valid ((x',remove ty.FF T1') # \<Gamma>')" using GA by auto have "valid (?\<Gamma> \|- VE z)" using `valid ?\<Gamma>` envminus_valid[of ?\<Gamma> "VE z"] by auto hence G4:"valid ((x',ty.FF) # \<Gamma>')" using GB by auto show ?thesis proof (cases "e'' = Bool False") case True hence "(Var z)[x'::=e''] = Bool False" using `x' = z` by auto hence "t1[x'::=e''] = Bool False" using `t1 = Var z` by auto hence X1:"?\<Gamma> \<turnstile> t1[x'::=e''] : ty.FF ; FF" using `valid ?\<Gamma>` by auto have "\<Gamma>' \<turnstile> e'' : ty.FF ; FF" using True `valid \<Gamma>'` by auto note `?\<Gamma> \|- VE z = (x', ty.FF) # \<Gamma>'` hence "valid ((x', ty.FF) # \<Gamma>')" using G4 by auto have X2:"(x', ty.FF) # \<Gamma>' \<turnstile> t3 : T3 ; F3" using VE `?\<Gamma> \|- VE z = (x', ty.FF) # \<Gamma>'` by auto note Iff(4)[ OF X2 `\<Gamma>' \<turnstile> e'' : ty.FF ; FF ` _ `valid ((x', ty.FF) # \<Gamma>')`] then obtain T'' F'' where X3:"\<Gamma>' \<turnstile> t3[x'::=e''] : T'' ; F'' " and X4:" \<turnstile> T'' <: T3 "" \<turnstile> F'' <e: F3" using `closed e''` `e'' : values` by auto hence "\<turnstile> T'' <: T'" using `\<turnstile> T3 <: T'` by auto hence "\<Gamma>' \<turnstile> Iff (Bool False) (t2[x'::=e'']) (t3[x'::=e'']) : T'' ; NE" using `valid \<Gamma>'` X3 by auto hence "\<Gamma>' \<turnstile> (Iff t1 t2 t3)[x'::=e''] : T'' ; NE" using ` t1[x'::=e''] = Bool False` by auto thus ?thesis using `\<turnstile> T'' <: T'` `F' = NE` by auto next case False hence "(Var z)[x'::=e''] = e''" using `x' = z` by auto hence "t1[x'::=e''] = e''" using `t1 = Var z` by auto hence X1:"\<Gamma>' \<turnstile> t1[x'::=e''] : T0' ; G'" by auto have "\<Gamma>' \<turnstile> e'' : T0' ; G'" . hence "G' = TT" using value_effect_tt_or_ff[OF `e'' : values` `\<Gamma>' \<turnstile> e'' : T0' ; G'`] False by auto hence X2:"\<Gamma>' \<turnstile> t1[x'::=e''] : T0' ; TT" using X1 by auto have "\<turnstile> T0' <: T1'" . have X3:"\<not> \<turnstile> T0' <: ty.FF" proof (rule ccontr) assume "\<not>\<not> \<turnstile> T0' <: ty.FF" hence "\<turnstile> T0' <: ty.FF" by simp hence "\<turnstile> T0' <: BoolTy" using BoolTy_def by auto hence "EX b. e'' = Bool b" using bool_value `e'' : values ` `\<Gamma>' \<turnstile> e'' : T0' ; G'` by auto then obtain b where A:"e'' = Bool b" by auto thus False proof (cases b) case False thus ?thesis using A `e'' ~= Bool False` by auto next case True hence "T0' = ty.TT" using `\<Gamma>' \<turnstile> e'' : T0' ; G'` A true_ty_elim by auto thus ?thesis using `\<turnstile> T0' <: ty.FF` tt_sub_ff by auto qed qed let ?rty = "remove ty.FF T1'" note `?\<Gamma> \|+ VE z = (x', ?rty) # \<Gamma>'` hence "valid ((x', ?rty) # \<Gamma>')" using G3 by auto have X2:"(x', ?rty) # \<Gamma>' \<turnstile> t2 : T2 ; F2" using VE `?\<Gamma> \|+ VE z = (x', ?rty) # \<Gamma>'` by auto note Iff(3)[ OF X2 `\<Gamma>' \<turnstile> e'' : T0' ; G' ` _ `valid ((x', ?rty) # \<Gamma>')`] then obtain T'' F'' where X3:"\<Gamma>' \<turnstile> t2[x'::=e''] : T'' ; F'' " and X4:" \<turnstile> T'' <: T2 "" \<turnstile> F'' <e: F2" using `closed e''` `e'' : values` remove_soundness[OF `\<turnstile> T0' <: T1'` X3 `simple_ty T0'`] by auto hence "\<turnstile> T'' <: T'" using `\<turnstile> T2 <: T'` by auto hence "\<Gamma>' \<turnstile> Iff e'' (t2[x'::=e'']) (t3[x'::=e'']) : T'' ; NE" using `valid \<Gamma>'` X3 `\<Gamma>' \<turnstile> e'' : T0' ; G' ` `G' = TT` by auto hence "\<Gamma>' \<turnstile> (Iff t1 t2 t3)[x'::=e''] : T'' ; NE" using ` t1[x'::=e''] = e''` by auto thus ?thesis using `\<turnstile> T'' <: T'` `F' = NE` by auto qed next case False from A False have "(Var z)[x'::=e''] = (Var z)" by auto hence D:"\<Gamma>' \<turnstile> (Var z)[x'::=e''] : T1 ; VE z" using False A proof - have q1:"?\<Gamma> \<turnstile> Var z : T1 ; VE z" using A by auto have "x' \<sharp> Var z" using trm.fresh False fresh_atm by auto hence "\<Gamma>' \<turnstile> Var z : T1 ; VE z" using q1 fresh_weakening_cons `valid ((x',T1')# \<Gamma>')` by auto thus ?thesis using `(Var z)[x'::=e''] = Var z` by auto qed hence typetst: "\<Gamma>' \<turnstile> t1[x'::=e''] : T1 ; VE z" using A by auto have F:"?\<Gamma> \|+ VE z = (x',T1') # ?\<Gamma>1" using False by auto hence H:"(x',T1') # ?\<Gamma>1 \<turnstile> t2 : T2 ; F2" using `?\<Gamma> \|+ VE z \<turnstile> t2 : T2 ; F2` by auto hence K:"valid ?\<Gamma>1" using envop_valid `valid \<Gamma>'` by auto have J:"?\<Gamma>1 \<turnstile> e'' : T0' ; G'" using K `valid \<Gamma>'` closed_any_env `closed e''` Iff by blast have "x' \<sharp> ?\<Gamma>1" using Iff valid_elim[of x' T1' \<Gamma>'] envop_fresh[of x' \<Gamma>' remove] `valid \<Gamma>'` by auto hence "valid ((x',T1')#?\<Gamma>1)" using `valid ?\<Gamma>1` by auto hence ex1:"\<exists>S2 G2. ?\<Gamma>1 \<turnstile> t2[x'::=e''] : S2 ; G2 \<and> \<turnstile> S2 <: T2 \<and> \<turnstile> G2 <e: F2" using H J K Iff by auto have G:"?\<Gamma> \|- VE z = (x',T1') # ?\<Gamma>2" using False by auto hence H:"(x',T1') # ?\<Gamma>2 \<turnstile> t3 : T3 ; F3" using `?\<Gamma> \|- VE z \<turnstile> t3 : T3 ; F3` by auto hence K:"valid ?\<Gamma>2" using envop_valid `valid \<Gamma>'` by auto have J:"?\<Gamma>2 \<turnstile> e'' : T0' ; G'" using K `valid \<Gamma>'` closed_any_env `closed e''` Iff by blast have "x' \<sharp> ?\<Gamma>2" using Iff valid_elim[of x' T1' \<Gamma>'] envop_fresh[of x' \<Gamma>' _ z _] `valid \<Gamma>'` by auto hence "valid ((x',T1')#?\<Gamma>2)" using `valid ?\<Gamma>2` by auto hence ex2:"\<exists>S3 G3. ?\<Gamma>2 \<turnstile> t3[x'::=e''] : S3 ; G3 \<and> \<turnstile> S3 <: T3 \<and> \<turnstile> G3 <e: F3" using H J K Iff by auto from ex1 obtain S2 G2 where 1:"?\<Gamma>1 \<turnstile> t2[x'::=e''] : S2 ; G2"" \<turnstile> S2 <: T'" using `\<turnstile> T2 <: T'` by auto from ex2 obtain S3 G3 where 2:"?\<Gamma>2 \<turnstile> t3[x'::=e''] : S3 ; G3"" \<turnstile> S3 <: T'" using `\<turnstile> T3 <: T'` by auto have 4:"?\<Gamma>1 = \<Gamma>' \|+ (VE z)" by auto have 5:"?\<Gamma>2 = \<Gamma>' \|- (VE z)" by auto have X:"!! G G' e T F. G = G' \<Longrightarrow> G \<turnstile> e : T ; F \<Longrightarrow> G' \<turnstile> e : T ; F" by auto from 1 4 have 6:"\<Gamma>' \|+ (VE z) \<turnstile> t2[x'::=e''] : S2 ; G2" using X[of ?\<Gamma>1 "\<Gamma>' \|+ (VE z)" " t2[x'::=e'']" S2 G2] by blast from 2 5 have 7:"\<Gamma>' \|- (VE z) \<turnstile> t3[x'::=e''] : S3 ; G3" using X[of ?\<Gamma>2 "\<Gamma>' \|- (VE z)" " t3[x'::=e'']" S3 G3] by blast from typetst 1 2 6 7 have "\<Gamma>' \<turnstile> (Iff t1 t2 t3)[x'::=e''] : T' ; comb_eff (VE z) G2 G3" by (auto simp del: comb_eff.simps) thus ?thesis using `F' = NE` by auto qed next case (TE U z) have "EX f A A1 Fn B. t1 = (App f (Var z)) \<and> (x', T1') # \<Gamma>' \<turnstile> f : B ; Fn \<and> \<turnstile> B <: A1 \<rightarrow> T1 : Latent U \<and> (x', T1') # \<Gamma>' \<turnstile> (Var z) : A ; VE z \<and> \<turnstile> A <: A1" using TE te_ty_elim[of ?\<Gamma> t1 T1 U z] by auto then obtain f A1 A Fn B where A:"t1 = (App f (Var z)) "" (x', T1') # \<Gamma>' \<turnstile> f : B ; Fn" "\<turnstile> B <: A1 \<rightarrow> T1 : Latent U" "(x', T1') # \<Gamma>' \<turnstile> (Var z) : A ; VE z "" \<turnstile> A <: A1" by auto have size1:"(f\<guillemotleft>Iff t1 t2 t3)" using A(1) by (simp ) note Iff(1)[of f] hence ih_f:"!! c \<sigma> \<Gamma> T F e' T0 F0 . (c,\<sigma>)#\<Gamma> \<turnstile> f : T ; F \<Longrightarrow> \<Gamma> \<turnstile> e' : T0 ; F0 \<Longrightarrow> \<turnstile> T0 <: \<sigma> \<Longrightarrow> valid ((c,\<sigma>)#\<Gamma>) \<Longrightarrow> closed e' \<Longrightarrow> e' : values \<Longrightarrow> EX T' F' . \<Gamma> \<turnstile> f[c::=e'] : T' ; F' \<and> \<turnstile> T' <: T \<and> \<turnstile> F' <e: F" using size1 by auto have "EX A' Fn' . \<Gamma>' \<turnstile> f[x'::=e''] : A' ; Fn' \<and> \<turnstile> A' <: B \<and> \<turnstile> Fn' <e: Fn" using ih_f A Iff by auto then obtain A' Fn' where B:"\<Gamma>' \<turnstile> f[x'::=e''] : A' ; Fn' \<and> \<turnstile> A' <:B" by auto let ?mapfun = "(%f. (% (v,ty). (if (z = v) then (v,f U ty) else (v,ty))))" let ?\<Gamma>1 = "(map (?mapfun restrict) \<Gamma>')" let ?\<Gamma>2 = "(map (?mapfun remove) \<Gamma>')" have "valid \<Gamma>'" using `valid ?\<Gamma>` using valid_elim[of x' T1' \<Gamma>'] by auto show ?case proof (cases "x' = z") case True from A True have "(Var z)[x'::=e''] = e''" by auto hence D:"\<Gamma>' \<turnstile> (Var z)[x'::=e''] : T0' ; G'" "closed ((Var z)[x'::=e''])" "((Var z)[x'::=e'']) : values" using Iff by auto have "\<turnstile> T0' <: T1'" . note var_ty_elim[of ?\<Gamma> z A "VE z"] hence "(x', A) : set ?\<Gamma>" using A True by auto have "?\<Gamma> \<turnstile> (Var x') : T1' ; VE x'" using `valid ?\<Gamma>` by auto hence "T1' = A" using A unique_var_typing[of ] True by auto have "\<turnstile> T0' <: T1'" . hence "\<turnstile> T0' <: A" using `T1' = A` by simp have "simple_ty T0'" using `\<Gamma>' \<turnstile> e'' : T0' ; G'` `e'' : values` value_simple_type by auto have or:" (\<turnstile> T0' <: U \<and> \<turnstile> T0' <: restrict U T1') \<or> (~ (\<turnstile> T0' <: U) \<and> \<turnstile> T0' <: remove U T1')" using `\<Gamma>' \<turnstile> e'' : T0' ; G'` `e'' : values` `\<turnstile> T0' <: T1'` `closed e''` `simple_ty T0'` restrict_remove_soundness by auto have "?\<Gamma>1 = envop restrict z U \<Gamma>'" by auto have "?\<Gamma>2 = envop remove z U \<Gamma>'" by auto have "x' \<sharp> \<Gamma>'" using Iff valid_elim[of x' T1' \<Gamma>'] by auto hence "?\<Gamma>1 = \<Gamma>'" using True envop_forget `valid \<Gamma>'` by auto hence GA:"?\<Gamma> \|+ TE U z = (x',restrict U T1') # \<Gamma>'" using True by auto hence G1:"\<dots> \<turnstile> t2 : T2; F2" using `(?\<Gamma> \|+ TE U z)\<turnstile> t2 : T2 ; F2` by auto have "?\<Gamma>2 = \<Gamma>'" using `x' \<sharp> \<Gamma>'` True envop_forget `valid \<Gamma>'` by auto hence GB:"?\<Gamma> \|- TE U z = (x',remove U T1') # \<Gamma>'" using True by auto hence G2:"\<dots> \<turnstile> t3 : T3; F3" using `(?\<Gamma> \|- TE U z)\<turnstile> t3 : T3 ; F3` by auto have "valid (?\<Gamma> \|+ TE U z)" using `valid ?\<Gamma>` envplus_valid[of ?\<Gamma> "TE U z"] by auto hence G3:"valid ((x',restrict U T1') # \<Gamma>')" using GA by auto have "valid (?\<Gamma> \|- TE U z)" using `valid ?\<Gamma>` envminus_valid[of ?\<Gamma> "TE U z"] by auto hence G4:"valid ((x',remove U T1') # \<Gamma>')" using GB by auto show ?thesis proof - { assume "\<turnstile> T0' <: U "" \<turnstile> T0' <: restrict U T1'" have 2:"(x', restrict U T1') # \<Gamma>' \<turnstile> t2 : T2 ; F2 " using `?\<Gamma> \|+ TE U z \<turnstile> t2 : T2 ; F2` GA by auto note Iff(2)[of x' "restrict U T1'" \<Gamma>' T2 F2 e'' T0' G'] hence "\<exists>S2 G2. \<Gamma>' \<turnstile> t2[x'::=e''] : S2 ; G2 \<and> \<turnstile> S2 <: T2 \<and> \<turnstile> G2 <e: F2" using G3 prems `closed e''` 2 by auto then obtain S2 G2 where L1:"\<Gamma>'\|+ eff.TT \<turnstile> t2[x'::=e''] : S2 ; G2 "" \<turnstile> S2 <: T2 "" \<turnstile> G2 <e: F2" by auto have "\<turnstile> T0' <: A1" "\<turnstile> A' <: A1 \<rightarrow> T1 : Latent U" using A B D `\<turnstile> T0' <: A` by auto hence "\<Gamma>' \<turnstile> (App f (Var z))[x'::=e''] : T1 ; eff.TT" using A B D `\<turnstile> T0' <: A` `\<turnstile> T0' <: U` using T_AppPredTrue[of \<Gamma>' _ A' Fn' A1 T1 U _ T0' G'] by auto hence L3:"\<Gamma>' \<turnstile> t1[x'::=e''] : T1 ; eff.TT" using `t1 = App f (Var z)` by auto have "\<turnstile> S2 <: T'" using L1 `\<turnstile> T2 <: T'` by auto hence "\<Gamma>'\<turnstile> (Iff t1 t2 t3)[x'::=e''] : T' ; eff.NE" using L3 ` \<Gamma>'\|+eff.TT \<turnstile> t2[x'::=e''] : S2 ; G2` by auto hence ?thesis using `F' = NE` by auto } moreover { assume "~ (\<turnstile> T0' <: U) "" \<turnstile> T0' <: remove U T1'" have 3:"(x', remove U T1') # \<Gamma>' \<turnstile> t3 : T3 ; F3 " using `?\<Gamma> \|- TE U z \<turnstile> t3 : T3 ; F3` GB by auto note Iff(3)[of x' "remove U T1'" \<Gamma>' T3 F3 e'' T0' G'] hence "\<exists>S3 G3. \<Gamma>' \<turnstile> t3[x'::=e''] : S3 ; G3 \<and> \<turnstile> S3 <: T3 \<and> \<turnstile> G3 <e: F3" using G4 prems `closed e''` 3 by auto then obtain S3 G3 where L1:"\<Gamma>'\|+ eff.FF \<turnstile> t3[x'::=e''] : S3 ; G3 "" \<turnstile> S3 <: T3 "" \<turnstile> G3 <e: F3" by auto have "\<turnstile> T0' <: A1" using A B D `\<turnstile> T0' <: A` by auto hence "\<Gamma>' \<turnstile> (App f (Var z))[x'::=e''] : T1 ; eff.FF" using A B D `\<turnstile> T0' <: A` `~ (\<turnstile> T0' <: U)` using T_AppPredFalse[of \<Gamma>' _ A' Fn' A1 T1 U _ T0' G'] by auto hence L3:"\<Gamma>' \<turnstile> t1[x'::=e''] : T1 ; eff.FF" using `t1 = App f (Var z)` by auto have "\<turnstile> S3 <: T'" using L1 `\<turnstile> T3 <: T'` by auto hence "\<Gamma>'\<turnstile> (Iff t1 t2 t3)[x'::=e''] : T' ; eff.NE" using L3 ` \<Gamma>'\|+eff.FF \<turnstile> t3[x'::=e''] : S3 ; G3` by auto hence ?thesis using `F' = NE` by auto } ultimately show ?thesis using or by auto qed next case False from A False have "(Var z)[x'::=e''] = (Var z)" by auto hence D:"\<Gamma>' \<turnstile> (Var z)[x'::=e''] : A ; VE z" using False A proof - have q1:"?\<Gamma> \<turnstile> Var z : A ; VE z" . have "x' \<sharp> Var z" using trm.fresh False fresh_atm by auto hence "\<Gamma>' \<turnstile> Var z : A ; VE z" using q1 fresh_weakening_cons `valid ((x',T1')# \<Gamma>')` by auto thus ?thesis using `(Var z)[x'::=e''] = Var z` by auto qed from A B D have "\<Gamma>' \<turnstile> App (f[x'::=e'']) ((Var z)[x'::=e'']) : T1 ; TE U z" by auto hence typetst: "\<Gamma>' \<turnstile> t1[x'::=e''] : T1 ; TE U z" using A by auto have F:"?\<Gamma> \|+ TE U z = (x',T1') # ?\<Gamma>1" using False by auto hence H:"(x',T1') # ?\<Gamma>1 \<turnstile> t2 : T2 ; F2" using `?\<Gamma> \|+ TE U z \<turnstile> t2 : T2 ; F2` by auto hence K:"valid ?\<Gamma>1" using envop_valid `valid \<Gamma>'` by auto have J:"?\<Gamma>1 \<turnstile> e'' : T0' ; G'" using K `valid \<Gamma>'` closed_any_env `closed e''` Iff by blast have "x' \<sharp> ?\<Gamma>1" using Iff valid_elim[of x' T1' \<Gamma>'] envop_fresh[of x' \<Gamma>' restrict z U] `valid \<Gamma>'` by auto hence "valid ((x',T1')#?\<Gamma>1)" using `valid ?\<Gamma>1` by auto hence ex1:"\<exists>S2 G2. ?\<Gamma>1 \<turnstile> t2[x'::=e''] : S2 ; G2 \<and> \<turnstile> S2 <: T2 \<and> \<turnstile> G2 <e: F2" using H J K Iff by auto have G:"?\<Gamma> \|- TE U z = (x',T1') # ?\<Gamma>2" using False by auto hence H:"(x',T1') # ?\<Gamma>2 \<turnstile> t3 : T3 ; F3" using `?\<Gamma> \|- TE U z \<turnstile> t3 : T3 ; F3` by auto hence K:"valid ?\<Gamma>2" using envop_valid `valid \<Gamma>'` by auto have J:"?\<Gamma>2 \<turnstile> e'' : T0' ; G'" using K `valid \<Gamma>'` closed_any_env `closed e''` Iff by blast have "x' \<sharp> ?\<Gamma>2" using Iff valid_elim[of x' T1' \<Gamma>'] envop_fresh[of x' \<Gamma>' remove z U] `valid \<Gamma>'` by auto hence "valid ((x',T1')#?\<Gamma>2)" using `valid ?\<Gamma>2` by auto hence ex2:"\<exists>S3 G3. ?\<Gamma>2 \<turnstile> t3[x'::=e''] : S3 ; G3 \<and> \<turnstile> S3 <: T3 \<and> \<turnstile> G3 <e: F3" using H J K Iff by auto from ex1 obtain S2 G2 where 1:"?\<Gamma>1 \<turnstile> t2[x'::=e''] : S2 ; G2"" \<turnstile> S2 <: T'" using `\<turnstile> T2 <: T'` by auto from ex2 obtain S3 G3 where 2:"?\<Gamma>2 \<turnstile> t3[x'::=e''] : S3 ; G3"" \<turnstile> S3 <: T'" using `\<turnstile> T3 <: T'` by auto have 4:"?\<Gamma>1 = \<Gamma>' \|+ (TE U z)" by auto have 5:"?\<Gamma>2 = \<Gamma>' \|- (TE U z)" by auto have X:"!! G G' e T F. G = G' \<Longrightarrow> G \<turnstile> e : T ; F \<Longrightarrow> G' \<turnstile> e : T ; F" by auto from 1 4 have 6:"\<Gamma>' \|+ (TE U z) \<turnstile> t2[x'::=e''] : S2 ; G2" using X[of ?\<Gamma>1 "\<Gamma>' \|+ (TE U z)" " t2[x'::=e'']" S2 G2] by blast from 2 5 have 7:"\<Gamma>' \|- (TE U z) \<turnstile> t3[x'::=e''] : S3 ; G3" using X[of ?\<Gamma>2 "\<Gamma>' \|- (TE U z)" " t3[x'::=e'']" S3 G3] by blast from typetst 1 2 6 7 have "\<Gamma>' \<turnstile> (Iff t1 t2 t3)[x'::=e''] : T' ; comb_eff (TE U z) G2 G3" by (auto simp del: comb_eff.simps) thus ?thesis using `F' = NE` by auto qed qed } moreover { assume "\<exists> T1 T3 F3. (?\<Gamma> \<turnstile> t1 : T1 ; FF) \<and> ?\<Gamma> \<turnstile> t3 : T3 ; F3 \<and> \<turnstile> T3 <: T' \<and> F' = NE" then obtain T1 T3 F3 where "?\<Gamma> \<turnstile> t1 : T1 ; FF" "?\<Gamma> \<turnstile> t3 : T3 ; F3" "\<turnstile> T3 <: T'" "F' = NE" by auto hence "\<exists>S1 G1. \<Gamma>' \<turnstile> t1[x'::=e''] : S1 ; G1 \<and> \<turnstile> S1 <: T1 \<and> \<turnstile> G1 <e: FF" using Iff by auto then obtain S1 G1 where B:"\<Gamma>' \<turnstile> t1[x'::=e''] : S1 ; G1 "" \<turnstile> S1 <: T1 "" \<turnstile> G1 <e: FF" by auto hence C:"\<Gamma>' \<turnstile> t1[x'::=e''] : S1 ; FF " using B no_sub_FF by auto from prems have "\<exists>S3 G3. \<Gamma>' \<turnstile> t3[x'::=e''] : S3 ; G3 \<and> \<turnstile> S3 <: T3 \<and> \<turnstile> G3 <e: F3" using Iff by auto then obtain S3 G3 where D:"\<Gamma>' \<turnstile> t3[x'::=e''] : S3 ; G3 ""\<turnstile> S3 <: T3" by auto from B C D have "\<Gamma>' \<turnstile> (Iff (t1[x'::=e'']) (t2[x'::=e'']) (t3[x'::=e''])) : S3 ; eff.NE" by auto hence ?thesis using `\<turnstile> T3 <: T'` `\<turnstile> S3 <: T3` `F' = NE` by auto } moreover { assume "\<exists> T1 T2 F2. (?\<Gamma> \<turnstile> t1 : T1 ; TT) \<and> ?\<Gamma> \<turnstile> t2 : T2 ; F2 \<and> \<turnstile> T2 <: T' \<and> F' = NE" then obtain T1 T2 F2 where "?\<Gamma> \<turnstile> t1 : T1 ; TT" "?\<Gamma> \<turnstile> t2 : T2 ; F2" "\<turnstile> T2 <: T'" "F' = NE" by auto hence "\<exists>S1 G1. \<Gamma>' \<turnstile> t1[x'::=e''] : S1 ; G1 \<and> \<turnstile> S1 <: T1 \<and> \<turnstile> G1 <e: TT" using Iff by auto then obtain S1 G1 where B:"\<Gamma>' \<turnstile> t1[x'::=e''] : S1 ; G1 "" \<turnstile> S1 <: T1 "" \<turnstile> G1 <e: TT" by auto hence C:"\<Gamma>' \<turnstile> t1[x'::=e''] : S1 ; TT " using B no_sub_TT by auto from prems have "\<exists>S2 G2. \<Gamma>' \<turnstile> t2[x'::=e''] : S2 ; G2 \<and> \<turnstile> S2 <: T2 \<and> \<turnstile> G2 <e: F2" using Iff by auto then obtain S2 G2 where D:"\<Gamma>' \<turnstile> t2[x'::=e''] : S2 ; G2 ""\<turnstile> S2 <: T2" by auto from B C D have "\<Gamma>' \<turnstile> (Iff (t1[x'::=e'']) (t2[x'::=e'']) (t3[x'::=e''])) : S2 ; eff.NE" by auto hence ?thesis using `\<turnstile> T2 <: T'` `\<turnstile> S2 <: T2` `F' = NE` by auto } ultimately show ?thesis using A by blast qed qed lemma subst_produces_TT: assumes ty:"(x,T0)#\<Gamma> \<turnstile> e : T ; TE S x" and vty:"\<Gamma> \<turnstile> v : T0' ; F" and A:"\<turnstile> T0' <: T0" and B:"valid ((x,T0)#\<Gamma>)" and C:"closed v" and D:"v : values" and E:"\<turnstile> T0' <: S" shows "EX T'. \<Gamma> \<turnstile> e[x::=v] : T' ; TT \<and> \<turnstile> T' <: T" proof - obtain f A A1 Fn Sa B where eq:"e = App f (Var x)" and fty:"(x, T0) # \<Gamma> \<turnstile> f : B ; Fn " and bsub:" \<turnstile> B <: A1 \<rightarrow> T : Latent S " and xty:" (x, T0) # \<Gamma> \<turnstile> Var x : A ; VE x " and asub:" \<turnstile> A <: A1" using te_ty_elim[OF ty] by auto note preserve_subst[OF fty vty A B C D] then obtain T' F' where X1:"\<Gamma> \<turnstile> f[x::=v] : T' ; F' " and " \<turnstile> T' <: B "" \<turnstile> F' <e: Fn" by auto hence X2:"\<turnstile> T' <: A1 \<rightarrow> T : Latent S" using bsub by auto have "(Var x)[x::=v] = v" by simp hence X3:"\<Gamma> \<turnstile> (Var x)[x::=v] : T0' ; F" using vty by auto have "(x, T0) # \<Gamma> \<turnstile> Var x : T0 ; VE x" using `valid ((x, T0) # \<Gamma>)` by auto hence "T0 = A" using xty unique_var_typing by auto hence "\<turnstile> T0' <: A1" using A asub by auto from X1 X2 X3 show ?thesis using T_AppPredTrue[OF X1 X2 X3 `\<turnstile> T0' <: A1` E] using asub E eq by auto qed lemma subst_produces_FF: assumes ty:"(x,T0)#\<Gamma> \<turnstile> e : T ; TE S x" and vty:"\<Gamma> \<turnstile> v : T0' ; F" and A:"\<turnstile> T0' <: T0" and B:"valid ((x,T0)#\<Gamma>)" and C:"closed v" and D:"v : values" and E:"~ \<turnstile> T0' <: S" shows "EX T'. \<Gamma> \<turnstile> e[x::=v] : T' ; FF \<and> \<turnstile> T' <: T" proof - obtain f A A1 Fn Sa B where eq:"e = App f (Var x)" and fty:"(x, T0) # \<Gamma> \<turnstile> f : B ; Fn " and bsub:" \<turnstile> B <: A1 \<rightarrow> T : Latent S " and xty:" (x, T0) # \<Gamma> \<turnstile> Var x : A ; VE x " and asub:" \<turnstile> A <: A1" using te_ty_elim[OF ty] by auto note preserve_subst[OF fty vty A B C D] then obtain T' F' where X1:"\<Gamma> \<turnstile> f[x::=v] : T' ; F' " and " \<turnstile> T' <: B "" \<turnstile> F' <e: Fn" by auto hence X2:"\<turnstile> T' <: A1 \<rightarrow> T : Latent S" using bsub by auto have veq:"(Var x)[x::=v] = v" by simp hence X3:"\<Gamma> \<turnstile> (Var x)[x::=v] : T0' ; F" using vty by auto have "(x, T0) # \<Gamma> \<turnstile> Var x : T0 ; VE x" using `valid ((x, T0) # \<Gamma>)` by auto hence "T0 = A" using xty unique_var_typing by auto hence "\<turnstile> T0' <: A1" using A asub by auto from X1 X2 X3 show ?thesis using T_AppPredFalse[OF X1 X2 X3 `\<turnstile> T0' <: A1` E] using asub E eq C D veq by auto qed inductive_cases beta_cases:"App (Abs x b T) v \<hookrightarrow> e " inductive_cases beta_TT_cases:"\<Gamma> \<turnstile> App (Abs x b T) v : T' ; TT" inductive_cases beta_FF_cases:"\<Gamma> \<turnstile> App (Abs x b T) v : T' ; FF" lemma preserve_red: assumes typed:"\<Gamma> \<turnstile> e : t ; eff" and cl:"closed e" and red:"e \<hookrightarrow> e'" and val:"valid \<Gamma>" shows "EX t' eff'. \<Gamma> \<turnstile> e' : t' ; eff' \<and> \<turnstile> t' <: t \<and> \<turnstile> eff' <e: eff " using red typed cl val red proof (nominal_induct e e' avoiding: \<Gamma> t rule: reduce.strong_induct) case (e_delta v' p v \<Gamma> T) thm app_ty_elim[of \<Gamma> "(BI p)" v' T eff] hence "\<exists>T0 T0' T1 le eff' eff'' U. \<Gamma> \<turnstile> BI p : U ; eff' \<and> \<Gamma> \<turnstile> v' : T0' ; eff'' \<and> \<turnstile> U <: T0 \<rightarrow> T1 : le \<and> \<turnstile> T0' <: T0 \<and> T1 = T" using app_ty_elim[of \<Gamma> "(BI p)" v' T eff] by simp then obtain T0 T0' T1 le eff' eff'' U where A1:" \<Gamma> \<turnstile> BI p : U ; eff'" and A2:"\<Gamma> \<turnstile> v' : T0' ; eff''" and A3:"\<turnstile> T0' <: T0" and A4:"T1 = T" and A5:"\<turnstile> U <: T0 \<rightarrow> T1 : le" by auto hence "U = \<Delta>\<^isub>\<tau> p" using e_delta typing_bi[of \<Gamma> p _ eff'] by simp then obtain A0 A1 LA where "\<Delta>\<^isub>\<tau> p = A0 \<rightarrow> A1 : LA" "U = A0 \<rightarrow> A1 : LA" by (nominal_induct p rule: builtin.induct) auto hence "\<turnstile> A0 \<rightarrow> A1 : LA <: T0 \<rightarrow> T1 : le" using `\<turnstile> U <: T0 \<rightarrow> T1 : le` by auto hence B:"le = LA \<or> le = latent_eff.NE" "\<turnstile> T0 <: A0" "\<turnstile> A1 <: T1" using arr_sub_arr_cases[of A0 A1 LA T0 T1 le] by auto have C1:" \<Gamma> \<turnstile> App (BI p) v' : T1 ; eff" using prems `T1 = T` by auto have C2:"\<turnstile> T0' <: A0" and C3:"\<turnstile> A1 <: T " using B A3 A4 by auto have C4:"valid \<Gamma>" . note delta_soundness[OF `\<Delta>\<^isub>\<tau> p = A0 \<rightarrow> A1 : LA` `v' : values` `\<Gamma> \<turnstile> v' : T0' ; eff''` C2 e_delta(3) C3 `\<Delta> p v' = Some v` C4] then obtain A1' eff' where "\<Gamma> \<turnstile> v : A1' ; eff' "" \<turnstile> eff' <e: eff" "\<turnstile> A1' <: A1" by auto thus ?case using C3 by auto next case (e_if_false thn els \<Gamma>' t') have "eff = NE" using if_false_ty_elim[of _ _ _ _ eff] e_if_false by auto have " \<exists>T0 eff'. \<Gamma>' \<turnstile> els : T0 ; eff' \<and> \<turnstile> T0 <: t' " using if_false_ty_elim[of \<Gamma>' thn els t' eff] e_if_false by auto then obtain T0 eff' where f:"\<Gamma>' \<turnstile> els : T0 ; eff'" and g:"\<turnstile> T0 <: t'" by auto have "closed els" using e_if_false closed_def trm.supp by auto hence "simple_eff eff'" using closed_eff f by auto hence h:"\<turnstile> eff' <e: eff" using simple_eff_below_ne `eff = NE` by auto thus ?case using f g by auto next case (e_if_true v thn els \<Gamma>' t') have "eff = NE" using if_true_ty_elim[of \<Gamma>' v thn els _ eff] e_if_true by auto have " \<exists>T0 eff'. \<Gamma>' \<turnstile> thn : T0 ; eff' \<and> \<turnstile> T0 <: t' " using if_true_ty_elim[of \<Gamma>' v thn els t' eff] e_if_true by auto then obtain T0 eff' where f:"\<Gamma>' \<turnstile> thn : T0 ; eff'" and g:"\<turnstile> T0 <: t'" by auto have "closed thn" using e_if_true closed_def trm.supp by auto hence "simple_eff eff'" using closed_eff f by auto hence h:"\<turnstile> eff' <e: eff" using `eff = NE` by auto thus ?case using f g by auto next case (e_beta v x T b \<Gamma>' T') hence "simple_eff eff" using closed_eff by auto thm app_ty_elim[of \<Gamma>' "(Lam[x:T].b)" v t eff] ( hence "eff = NE" using app_abs_ty_elim_eff by auto ) from e_beta have "\<exists>T0 T0' T1 le eff' eff'' U. \<Gamma>' \<turnstile> Abs x b T : U ; eff' \<and> \<Gamma>' \<turnstile> v : T0' ; eff'' \<and> \<turnstile> T0' <: T0 \<and> T1 = T' \<and> \<turnstile> U <: T0 \<rightarrow> T1 : le" using app_ty_elim[of \<Gamma>' "Abs x b T" v T' eff] by blast then obtain T0 T0' T1 le eff' eff'' U where " \<Gamma>' \<turnstile> Lam[x:T].b :U; eff'" and "\<Gamma>' \<turnstile> v : T0' ; eff''" and "\<turnstile> T0' <: T0" and "T1 = T'" and usub:"\<turnstile> U <: T0 \<rightarrow> T1 : le " by auto hence "\<exists>T1a eff2 L S. (x,T)#\<Gamma>' \<turnstile> b : T1a ; eff2 \<and> U = T \<rightarrow> T1a : L \<and> (L = Latent S \<and> eff2 = TE S x \<or> L = latent_eff.NE)" using abs_ty_elim[of \<Gamma>' x b T "U" eff'] e_beta `x \<sharp> \<Gamma>'` by auto then obtain T1a eff2 L S where bty:"(x,T)#\<Gamma>' \<turnstile> b : T1a ; eff2" and ueq:"U = T \<rightarrow> T1a : L" and "(L = Latent S \<and> eff2 = TE S x \<or> L = latent_eff.NE)" by auto have "closed v" using e_beta closed_def trm.supp by auto have "v : values" using e_beta beta_cases[of _ _ _ v _ "v : values"] trm.inject by auto have "\<turnstile> T0 <: T" using usub ueq arr_sub_arr_cases[of T T1a L T0 T1 le] by auto have "\<turnstile> T1a <: T1" using usub ueq arr_sub_arr_cases[of T T1a L T0 T1 le] by auto hence "\<turnstile> T1a <: T'" using `T1 = T'` by simp have "\<turnstile> T0' <: T" using ` \<turnstile> T0' <: T0` `\<turnstile>T0<:T` by auto have " \<exists>T'a F'. \<Gamma>' \<turnstile> b[x::=v] : T'a ; F' \<and> \<turnstile> T'a <: T1a" using preserve_subst[of x T \<Gamma>' b T1a eff2 v T0' eff''] `\<Gamma>' \<turnstile> v : T0' ; eff''` bty `\<turnstile> T0' <: T` `x \<sharp> \<Gamma>'` `valid \<Gamma>'` `closed v` `v : values` by auto then obtain T'a F' where a:"\<Gamma>' \<turnstile> b[x::=v] : T'a ; F'" and b:"\<turnstile> T'a <: T1a" by auto have "closed (b[x::=v])" using e_beta reduce_closed by blast hence c:"simple_eff F'" using a closed_eff by auto have ?case using `simple_eff eff` e_beta a b proof (induct eff rule: simple_eff_cases) case NE thus ?case using simple_eff_below_ne[of F'] c a b `\<turnstile> T1a <: T'` by auto next case TT obtain U T0 Ta S eff1 eff2 where X1:"\<Gamma>' \<turnstile> (Lam [x:T].b) : U ; eff1 " and X2:"\<turnstile> U <: T0 \<rightarrow> T' : Latent S" and X3:" \<Gamma>' \<turnstile> v : Ta ; eff2 " "\<turnstile> Ta <: T0"" \<turnstile> Ta <: S" using trm.inject beta_TT_cases[OF TT(5), of thesis] by auto note abs_ty_elim[OF X1 `x \<sharp> \<Gamma>'`] then obtain T1' eff' L S' where f: "(x, T) # \<Gamma>' \<turnstile> b : T1' ; eff' " " U = T \<rightarrow> T1' : L " " eff1 = eff.TT " " (eff' = TE S' x \<and> L = Latent S' \<or> L = latent_eff.NE)" by auto hence "eff' = TE S x" "\<turnstile> T0 <: T" "\<turnstile> T1' <: T'"using `U = T \<rightarrow> T1' : L` X2 using arr_sub_arr_cases[of T T1' L T0 T' "Latent S"] by auto hence X4:"(x, T) # \<Gamma>' \<turnstile> b : T1' ; TE S x" using f by auto have valcons:"valid ((x,T)#\<Gamma>')" using `valid \<Gamma>'` `x \<sharp> \<Gamma>'` by auto have "\<turnstile> Ta <: T" using `\<turnstile> Ta <: T0``\<turnstile> T0 <: T` by auto have "EX T2'. \<Gamma>' \<turnstile> b[x::=v] : T2' ; TT \<and> \<turnstile> T2' <: T1'" using X3 using subst_produces_TT[OF X4 `\<Gamma>' \<turnstile> v : Ta ; eff2` `\<turnstile> Ta <: T` valcons ` closed v`` v \<in> values` `\<turnstile> Ta <: S`] by auto thus ?case using `\<turnstile> T1' <: T'` by auto next case FF obtain U T0 Ta S eff1 eff2 where X1:"\<Gamma>' \<turnstile> (Lam [x:T].b) : U ; eff1 " and X2:"\<turnstile> U <: T0 \<rightarrow> T' : Latent S" and X3:" \<Gamma>' \<turnstile> v : Ta ; eff2 " "\<turnstile> Ta <: T0""~ \<turnstile> Ta <: S" using trm.inject beta_FF_cases[OF FF(5), of thesis] by auto note abs_ty_elim[OF X1 `x \<sharp> \<Gamma>'`] then obtain T1' eff' L S' where f: "(x, T) # \<Gamma>' \<turnstile> b : T1' ; eff' " " U = T \<rightarrow> T1' : L " " eff1 = eff.TT " " (eff' = TE S' x \<and> L = Latent S' \<or> L = latent_eff.NE)" by auto hence "eff' = TE S x" "\<turnstile> T0 <: T" "\<turnstile> T1' <: T'" using `U = T \<rightarrow> T1' : L` X2 using arr_sub_arr_cases[of T T1' L T0 T' "Latent S"] by auto hence X4:"(x, T) # \<Gamma>' \<turnstile> b : T1' ; TE S x" using f by auto have valcons:"valid ((x,T)#\<Gamma>')" using `valid \<Gamma>'` `x \<sharp> \<Gamma>'` by auto have "\<turnstile> Ta <: T" using `\<turnstile> Ta <: T0``\<turnstile> T0 <: T` by auto hence "EX T2'. \<Gamma>' \<turnstile> b[x::=v] : T2' ; FF \<and> \<turnstile> T2' <: T1'" using X3 using subst_produces_FF[OF X4 `\<Gamma>' \<turnstile> v : Ta ; eff2` `\<turnstile> Ta <: T` valcons ` closed v`` v \<in> values` _] by auto thus ?case using `\<turnstile> T1' <: T'` by auto qed thus ?case . qed lemma value_no_ctxt: assumes "v : values" and "v = E t" and "E : ctxt" shows "E = (% t . t)" using prems proof (induct) case bi_value show ?case using `E : ctxt` bi_value by (induct E rule: ctxt.induct) (auto simp add: trm.inject) next case num_value show ?case using `E : ctxt` num_value by (induct E rule: ctxt.induct) (auto simp add: trm.inject) next case abs_value show ?case using `E : ctxt` abs_value by (induct E rule: ctxt.induct) (auto simp add: trm.inject) next case bool_value show ?case using `E : ctxt` bool_value by (induct E rule: ctxt.induct) (auto simp add: trm.inject) qed lemma subst_in_ctxt_preserves_type: assumes a:"\<Gamma> \<turnstile> u : T ; F" and b:"C : ctxt" and c:"u = C e" and "~ (e : values)" and "closed (C e)" and "closed (C e')" and "!! T0 F0 . \<Gamma> \<turnstile> e : T0 ; F0 \<Longrightarrow> EX T1 F1. \<Gamma> \<turnstile> e' : T1 ; F1 \<and> \<turnstile> T1 <: T0 \<and> \<turnstile> F1 <e: F0" shows "EX S G. \<Gamma> \<turnstile> C e' : S ; G \<and> \<turnstile> S <: T \<and> \<turnstile> G <e: F" using b prems proof (induct C arbitrary: u e e' T F rule: ctxt.induct) case C_Hole hence "\<Gamma> \<turnstile> e : T ; F" using C_Hole by simp hence "EX S G. \<Gamma> \<turnstile> e' : S; G \<and> \<turnstile> S <: T \<and> \<turnstile> G <e: F" using C_Hole by auto thus ?case by simp next case (C_App1 E arg u' t t' T' F') have A:"closed (E t)" and B:"closed (E t')" using C_App1 closed_def trm.supp by auto have C:"\<Gamma> \<turnstile> App (E t) arg : T' ; F'" using C_App1 by auto hence D:"simple_eff F'" using C_App1 closed_eff by auto thus ?case using C_App1 A B C proof (induct rule: simple_eff_cases) case NE hence "EX T0 T0' T1 le eff' eff'' U. \<Gamma> \<turnstile> E t :U ; eff' \<and> \<Gamma> \<turnstile> arg : T0' ; eff'' \<and> \<turnstile> U <: T0 \<rightarrow> T1 : le \<and> \<turnstile> T0' <: T0 \<and> T1 = T'" using app_ty_elim by auto then obtain T0 T0' le eff' eff'' U where a:"\<Gamma> \<turnstile> E t : U ; eff'" and b:"\<Gamma> \<turnstile> arg : T0' ; eff''" and c:"\<turnstile> T0' <: T0" and d:"\<turnstile> U <: T0 \<rightarrow> T' : le" by auto have "\<exists>S G. \<Gamma> \<turnstile> E t' : S ; G \<and> \<turnstile> S <: U \<and> \<turnstile> G <e: eff'" using C_App1(2)[of "E t" _ eff' t t'] `E : ctxt` `closed (E t)` `closed (E t')` C_App1(8) a `t \<notin> values` C_App1 by auto then obtain S G where et'ty:"\<Gamma> \<turnstile> E t' : S ; G " and subarr:" \<turnstile> S <: U " and "\<turnstile> G <e: eff'" by auto hence "\<Gamma> \<turnstile> App (E t') arg : T' ; NE" using b c d by auto thus ?case by auto next case FF have "EX S T0 T0' le F1 F2 U. \<Gamma> \<turnstile> E t : U ; F1 \<and> \<Gamma> \<turnstile> arg : T0' ; F2 \<and> \<turnstile> T0' <: T0 \<and> le = Latent S \<and> ~ (\<turnstile> T0' <: S) \<and> arg : values \<and> closed arg \<and> \<turnstile> U <: T0 \<rightarrow> T' : le" using `\<Gamma> \<turnstile> App (E t) arg : T' ; FF` app_ty_ff_elim[of \<Gamma> "E t" arg T'] by blast then obtain S T0 T0' F1 F2 U where a:"\<Gamma> \<turnstile> E t :U ; F1" and b:"\<Gamma> \<turnstile> arg : T0' ; F2 " and c:"\<turnstile> T0' <: T0 " and d:" ~ (\<turnstile> T0' <: S)" "closed arg" "arg : values" " \<turnstile> U <: T0 \<rightarrow> T' : Latent S" by auto have "\<exists>S' G. \<Gamma> \<turnstile> E t' : S' ; G \<and> \<turnstile> S' <: U \<and> \<turnstile> G <e: F1" using C_App1(2)[of "E t" U F1 t t'] `E : ctxt` `closed (E t)` `closed (E t')` C_App1(8) a C_App1 by auto then obtain S' G where et'ty:"\<Gamma> \<turnstile> E t' : S' ; G " and subarr:" \<turnstile> S' <: U" and "\<turnstile> G <e: F1" by auto hence "\<Gamma> \<turnstile> App (E t') arg : T' ; FF" using b c d by auto thus ?case by auto next case TT have "EX S T0 T0' le F1 F2 U. \<Gamma> \<turnstile> E t : U ; F1 \<and> \<Gamma> \<turnstile> arg : T0' ; F2 \<and> \<turnstile> T0' <: T0 \<and> le = Latent S \<and> \<turnstile> T0' <: S \<and> \<turnstile> U <: T0 \<rightarrow> T' : le" using `\<Gamma> \<turnstile> App (E t) arg : T' ; TT` app_ty_tt_elim[of \<Gamma> "E t" arg T'] by blast then obtain S T0 T0' F1 F2 U where a:"\<Gamma> \<turnstile> E t : U ; F1" and b:"\<Gamma> \<turnstile> arg : T0' ; F2 " and c:"\<turnstile> T0' <: T0 " and d:"\<turnstile> T0' <: S" "\<turnstile> U <: T0 \<rightarrow> T' : Latent S" by auto have "\<exists>S' G. \<Gamma> \<turnstile> E t' : S' ; G \<and> \<turnstile> S' <: U \<and> \<turnstile> G <e: F1" using C_App1(2)[of "E t" U F1 t t'] `E : ctxt` `closed (E t)` `closed (E t')` C_App1 a by auto then obtain S' G where et'ty:"\<Gamma> \<turnstile> E t' : S' ; G " and subarr:" \<turnstile> S' <: U " and "\<turnstile> G <e: F1" by auto hence "\<Gamma> \<turnstile> App (E t') arg : T' ; TT" using b c d by auto thus ?case by auto qed next case (C_App2 E v u' t t' T' F') have A:"closed (E t)" and B:"closed (E t')" using C_App2 closed_def trm.supp by auto have C:"\<Gamma> \<turnstile> App v (E t) : T' ; F'" using C_App2 by auto hence D:"simple_eff F'" using C_App2 closed_eff by auto thus ?case using C_App2 A B C proof (induct rule: simple_eff_cases) case NE have "\<exists>T0 T0' T1 le eff' eff'' U. \<Gamma> \<turnstile> v : U ; eff' \<and> \<Gamma> \<turnstile> E t : T0' ; eff'' \<and> \<turnstile> T0' <: T0 \<and> T1 = T' \<and> \<turnstile> U <: T0 \<rightarrow> T1 : le" using app_ty_elim[of \<Gamma> v "E t" T' F'] `\<Gamma> \<turnstile> App v (E t) : T' ; F'` by blast then obtain T0 T0' le eff' eff'' U where a:"\<Gamma> \<turnstile> v : U ; eff'" " \<Gamma> \<turnstile> E t : T0' ; eff'' " "\<turnstile> T0' <: T0" "\<turnstile> U <: T0 \<rightarrow> T' : le" by auto hence "\<exists>S G. \<Gamma> \<turnstile> E t' : S ; G \<and> \<turnstile> S <: T0' \<and> \<turnstile> G <e: eff''" using NE(2)[of "E t" T0' eff'' t t'] NE by auto then obtain S G where "\<Gamma> \<turnstile> E t' : S ; G "" \<turnstile> S <: T0'" by auto hence "\<Gamma> \<turnstile> App v (E t') : T' ; eff.NE" using a `\<turnstile> T0' <: T0` by auto thus ?case by auto next case FF have "\<exists>S T0 T0' le eff' eff'' U. \<Gamma> \<turnstile> v : U ; eff' \<and> \<Gamma> \<turnstile> E t : T0' ; eff'' \<and> \<turnstile> T0' <: T0 \<and> le = Latent S \<and> ~ (\<turnstile> T0' <: S) \<and> E t : values \<and> closed (E t) \<and> \<turnstile> U <: T0 \<rightarrow> T' : le" using app_ty_ff_elim[of \<Gamma> v "E t" T'] `\<Gamma> \<turnstile> App v (E t) : T' ; FF` by blast then obtain S T0 T0' le eff' eff'' U where "\<Gamma> \<turnstile> v : U ; eff' "" \<Gamma> \<turnstile> E t : T0' ; eff'' "" \<turnstile> T0' <: T0 " " ~ (\<turnstile> T0' <: S) " " E t : values "" closed (E t)" by auto hence "E = (% t. t)" using value_no_ctxt[of "E t" E t] `E : ctxt` by simp hence "t : values" using `E t : values` by simp thus ?case using `t \<notin> values` by auto next case TT have "\<exists>S T0 T0' le eff' eff'' U. \<Gamma> \<turnstile> v : U ; eff' \<and> \<Gamma> \<turnstile> E t : T0' ; eff'' \<and> \<turnstile> T0' <: T0 \<and> le = Latent S \<and> \<turnstile> T0' <: S \<and> \<turnstile> U <: T0 \<rightarrow> T' : le" using app_ty_tt_elim[of \<Gamma> v "E t" T'] `\<Gamma> \<turnstile> App v (E t) : T' ; TT` by blast then obtain S T0 T0' le eff' eff'' U where a:"\<Gamma> \<turnstile> v : U ; eff' "" \<Gamma> \<turnstile> E t : T0' ; eff'' "" \<turnstile> T0' <: T0 " " \<turnstile> T0' <: S" "\<turnstile> U <: T0 \<rightarrow> T' : Latent S" by auto hence "\<exists>S' G. \<Gamma> \<turnstile> E t' : S' ; G \<and> \<turnstile> S' <: T0' \<and> \<turnstile> G <e: eff''" using TT(2)[of "E t" T0' eff'' t t'] TT by auto then obtain S' G where b:"\<Gamma> \<turnstile> E t' : S' ; G "" \<turnstile> S' <: T0'" by auto have "\<turnstile> S' <: S" using ` \<turnstile> S' <: T0'` `\<turnstile> T0' <: S` by auto have "\<turnstile> S' <: T0" using `\<turnstile> S' <: T0'` `\<turnstile> T0' <: T0` by auto hence "\<Gamma> \<turnstile> App v (E t') : T' ; TT" using b T_AppPredTrue[OF a(1) a(5) b(1) `\<turnstile> S' <: T0` ] using `\<turnstile> S' <: S` by auto thus ?case by auto qed next case (C_Iff E thn els u' t t' T' F') have A:"closed (E t)" and B:"closed (E t')" using C_Iff closed_def trm.supp by auto have C:"\<Gamma> \<turnstile> Iff (E t) thn els: T' ; F'" using C_Iff by auto hence bigor: "(\<exists>T1 T2 T3 F1 F2 F3. \<Gamma> \<turnstile> E t : T1 ; F1 \<and> \<Gamma> \|+ F1 \<turnstile> thn : T2 ; F2 \<and> \<Gamma> \|- F1 \<turnstile> els : T3 ; F3 \<and> \<turnstile> T2 <: T' \<and> \<turnstile> T3 <: T' \<and> F' = eff.NE) \<or> (\<exists>T1 T3 F3. \<Gamma> \<turnstile> E t : T1 ; FF \<and> \<Gamma> \<turnstile> els : T3 ; F3 \<and> \<turnstile> T3 <: T' \<and> F' = eff.NE) \<or> (\<exists>T1 T2 F2. \<Gamma> \<turnstile> E t : T1 ; TT \<and> \<Gamma> \<turnstile> thn : T2 ; F2 \<and> \<turnstile> T2 <: T' \<and> F' = eff.NE)" using if_ty_elim[of \<Gamma> "(E t)" thn els T' F'] by auto thus ?case proof - { assume "(\<exists>T1 T2 T3 F1 F2 F3. \<Gamma> \<turnstile> E t : T1 ; F1 \<and> \<Gamma> \|+ F1 \<turnstile> thn : T2 ; F2 \<and> \<Gamma> \|- F1 \<turnstile> els : T3 ; F3 \<and> \<turnstile> T2 <: T' \<and> \<turnstile> T3 <: T' \<and> F' = eff.NE)" then obtain T1 T2 T3 F1 F2 F3 where P:"\<Gamma> \<turnstile> E t : T1 ; F1""\<Gamma> \|+ F1 \<turnstile> thn : T2 ; F2""\<Gamma> \|- F1 \<turnstile> els : T3 ; F3""\<turnstile> T2 <: T'""\<turnstile> T3 <: T'""F' = eff.NE" by auto have "closed (E t)" and "closed (E t')" using prems trm.supp closed_def by auto hence "EX T1' F1'. \<Gamma> \<turnstile> E t' : T1' ; F1' \<and> \<turnstile> T1' <: T1 \<and> \<turnstile> F1' <e: F1" using C_Iff(2)[of "E t" T1 F1 t t'] C_Iff P by auto then obtain T1' F1' where Q:"\<Gamma> \<turnstile> E t' : T1' ; F1' "" \<turnstile> T1' <: T1 "" \<turnstile> F1' <e: F1" by auto have "simple_eff F1'" and "simple_eff F1" using `closed (E t)` `closed (E t')` P Q closed_eff by auto hence "\<Gamma> \|+ F1 = \<Gamma>" "\<Gamma> \|- F1 = \<Gamma>" "\<Gamma> \|+ F1' = \<Gamma>" "\<Gamma> \|- F1' = \<Gamma>" by (auto simp add: env_plus_simple_eff) hence "\<Gamma> \|+ F1' \<turnstile> thn : T2 ; F2 " "\<Gamma> \|- F1' \<turnstile> els : T3 ; F3" using P by auto hence "\<Gamma> \<turnstile> Iff (E t') thn els : T'; comb_eff F1' F2 F3" using `\<Gamma> \<turnstile> E t' : T1' ; F1'` ` \<turnstile> T2 <: T' `` \<turnstile> T3 <: T'` by (auto simp del: comb_eff.simps) hence ?thesis using `F' = NE` by auto } moreover { assume "(\<exists>T1 T3 F3. \<Gamma> \<turnstile> E t : T1 ; FF \<and> \<Gamma> \<turnstile> els : T3 ; F3 \<and> \<turnstile> T3 <: T' \<and> F' = eff.NE)" then obtain T1 T3 F3 where P:"\<Gamma> \<turnstile> E t : T1 ; FF "" \<Gamma> \<turnstile> els : T3 ; F3 "" \<turnstile> T3 <: T'"" F' = eff.NE" by auto have "closed (E t)" and "closed (E t')" using prems trm.supp closed_def by auto hence "EX T1' F1'. \<Gamma> \<turnstile> E t' : T1' ; F1' \<and> \<turnstile> T1' <: T1 \<and> \<turnstile> F1' <e: FF" using C_Iff(2)[of "E t" T1 FF t t'] C_Iff P by auto then obtain T1' F1' where Q:"\<Gamma> \<turnstile> E t' : T1' ; F1' "" \<turnstile> T1' <: T1 "" \<turnstile> F1' <e: FF" by auto have "F1' = FF" using Q no_sub_FF[of F1' FF] by simp hence "\<Gamma> \<turnstile> E t' : T1' ; FF " using Q by auto hence ?thesis using P by auto } moreover { assume "(\<exists>T1 T2 F2. \<Gamma> \<turnstile> E t : T1 ; TT \<and> \<Gamma> \<turnstile> thn : T2 ; F2 \<and> \<turnstile> T2 <: T' \<and> F' = eff.NE)" then obtain T1 T2 F2 where P:"\<Gamma> \<turnstile> E t : T1 ; TT "" \<Gamma> \<turnstile> thn : T2 ; F2 "" \<turnstile> T2 <: T'"" F' = eff.NE" by auto have "closed (E t)" and "closed (E t')" using prems trm.supp closed_def by auto hence "EX T1' F1'. \<Gamma> \<turnstile> E t' : T1' ; F1' \<and> \<turnstile> T1' <: T1 \<and> \<turnstile> F1' <e: TT" using C_Iff(2)[of "E t" T1 TT t t'] C_Iff P by auto then obtain T1' F1' where Q:"\<Gamma> \<turnstile> E t' : T1' ; F1' "" \<turnstile> T1' <: T1 "" \<turnstile> F1' <e: TT" by auto have "F1' = TT" using Q no_sub_TT[of F1' TT] by simp hence "\<Gamma> \<turnstile> E t' : T1' ; TT " using Q by auto hence ?thesis using P by auto } ultimately show ?thesis using bigor by blast qed qed lemma typing_ctxt: assumes a:"\<Gamma> \<turnstile> C L : T ; eff" and b:"C : ctxt" shows "EX T' eff'. \<Gamma> \<turnstile> L : T' ; eff'" using b a proof(induct C arbitrary: T eff rule: ctxt.induct ) case C_Hole thus ?case by auto next case (C_App1 C' arg S) hence ih: "!! T0 eff. \<Gamma> \<turnstile> C' L : T0 ; eff \<Longrightarrow> \<exists>T' a. \<Gamma> \<turnstile> L : T' ; a" by simp obtain T0 T0' T1 le eff' eff'' U where "\<Gamma> \<turnstile> C' L : U ; eff'" "\<Gamma> \<turnstile> arg : T0' ; eff''" "\<turnstile> T0' <: T0 \<and> T1 = S" "\<turnstile> U <: T0 \<rightarrow> T1 : le" using app_ty_elim[of \<Gamma> "C' L" arg S eff] ` \<Gamma> \<turnstile> App (C' L) arg : S ; eff` by blast thus ?case using ih by auto next case (C_App2 C' rator S F) hence ih: "!! T0 eff. \<Gamma> \<turnstile> C' L : T0 ; eff \<Longrightarrow> \<exists>T' a. \<Gamma> \<turnstile> L : T' ; a" by simp obtain T0 T0' T1 le eff' eff'' U where "\<Gamma> \<turnstile> rator : U ; eff'" "\<Gamma> \<turnstile> C' L : T0' ; eff''" "\<turnstile> T0' <: T0 \<and> T1 = S" "\<turnstile> U <: T0 \<rightarrow> T1 : le" using app_ty_elim[of \<Gamma> rator "C' L" S F] ` \<Gamma> \<turnstile> App rator (C' L) : S ; F` by blast thus ?case using ih by auto next case (C_Iff C' thn els S F) hence ih: "!! T0 eff. \<Gamma> \<turnstile> C' L : T0 ; eff \<Longrightarrow> \<exists>T' a. \<Gamma> \<turnstile> L : T' ; a" by simp obtain T0 eff' where "\<Gamma> \<turnstile> C' L : T0 ; eff'" using if_ty_elim[of \<Gamma> "C' L" thn els S F] ` \<Gamma> \<turnstile> Iff (C' L) thn els : S ; F` by auto thus ?case using ih by auto qed inductive_cases step_cases: "(e::trm) \<longrightarrow> e'" inductive_cases bi_reduce:"BI b \<hookrightarrow> x" inductive_cases bool_reduce:"Bool b \<hookrightarrow> x" inductive_cases abs_reduce:"(Lam[a:T].b) \<hookrightarrow> x" inductive_cases num_reduce:"Num n \<hookrightarrow> x" lemma value_reduce_step: assumes A:"v : values" and B:"v \<longrightarrow> (v'::trm)" shows "v \<hookrightarrow> v'" using B A proof(induct) fix E L R assume "E : ctxt" "L \<hookrightarrow> R" "E L \<in> values" hence "E L = L" and "E R = R" using value_no_ctxt by auto thus "E L \<hookrightarrow> E R" using prems by auto qed lemma value_not_step: assumes "v : values" shows "~ (EX v'. v \<hookrightarrow> v')" proof(rule ccontr, simp) assume "\<exists>v'. v \<hookrightarrow> v'" then obtain v' where A:"v \<hookrightarrow> v'" by auto show False using `v : values` A proof (induct v rule: values.induct) case (bi_value b) thus ?case using bi_reduce by auto next case num_value thus ?case using num_reduce by auto next case abs_value thus ?case using abs_reduce by blast next case bool_value thus ?case using bool_reduce by auto qed qed lemma value_not_reduce: fixes v v' :: trm assumes "v : values" shows "~ (EX v'. v \<longrightarrow> v')" proof (rule ccontr) assume "\<not> \<not> (\<exists>v'. v \<longrightarrow> v')" then obtain v' where "v \<longrightarrow> v'" by auto hence A:"v \<hookrightarrow> v'" using value_reduce_step prems by auto show False using `v : values` A value_not_step by auto qed theorem preservation: fixes e e' :: trm assumes typed:"\<Gamma> \<turnstile> e : t ; eff" and cl:"closed e" and red:"e \<longrightarrow> e'" shows "EX t' eff'. \<Gamma> \<turnstile> e' : t' ; eff' \<and> \<turnstile> t' <: t \<and> \<turnstile> eff' <e: eff" using red typed cl proof - have val:"valid \<Gamma>" using typing_valid typed by auto obtain C L R where "e = C L" "e' = C R" and "L \<hookrightarrow> R" and "C : ctxt" using red step_cases[of e e' thesis] by auto hence f:"EX T F. \<Gamma> \<turnstile> L : T ; F" using typed typing_ctxt by auto have "L \<notin> values" using `L \<hookrightarrow> R` value_not_step by auto have "closed L" and "closed_ctxt C" using closed_in_ctxt_closed_ctxt[of e C L] `C : ctxt` cl `e = C L` by auto hence "closed R" using reduce_closed[of L R] `L \<hookrightarrow> R` by auto hence "closed (C R)" and "closed (C L)" using `closed_ctxt C` ctxt_closed[of C L] ctxt_closed[of C R] `closed L` by auto have " \<And>T0 F0. \<Gamma> \<turnstile> L : T0 ; F0 \<Longrightarrow> \<exists>T1 F1. \<Gamma> \<turnstile> R : T1 ; F1 \<and> \<turnstile> T1 <: T0 \<and> \<turnstile> F1 <e: F0" proof - fix T0 F0 show "\<Gamma> \<turnstile> L : T0 ; F0 \<Longrightarrow> (\<exists>T1 F1. \<Gamma> \<turnstile> R : T1 ; F1 \<and> \<turnstile> T1 <: T0 \<and> \<turnstile> F1 <e: F0)" using `e = C L` `C : ctxt` `L \<hookrightarrow> R` closed_in_ctxt[of C L] cl preserve_red[of \<Gamma> L T0 F0 R] `closed L` val by auto qed hence "\<exists>S G. \<Gamma> \<turnstile> C R : S ; G \<and> \<turnstile> S <: t \<and> \<turnstile> G <e: eff" using `C : ctxt` subst_in_ctxt_preserves_type[of \<Gamma> e t eff C L R] typed `e = C L` `closed (C L)` `closed (C R)` `L \<notin> values`by auto thus ?thesis using `e' = C R` by auto qed text { soundness } lemma soundness_finite: fixes e e' e'' :: trm assumes A:"\<Gamma> \<turnstile> e : T ; F" and B:"e \<longrightarrow>\<^sup> e'" and C:"~ (EX e''. e' \<longrightarrow> e'')" and E:"closed e" shows "EX T' F'. (e' : values \<and> \<Gamma> \<turnstile> e' : T' ; F' \<and> \<turnstile> T' <: T \<and> \<turnstile> F' <e: F)" using B prems proof (induct arbitrary: \<Gamma> T F rule: step_multi.induct) case (sm_refl v) have "v : values" using sm_refl progress[of \<Gamma> v T F] by auto thus ?case using sm_refl by auto next case (sm_trans a b c) have "closed b" "closed c" using `closed a` `b \<longrightarrow>\<^sup>* c` `a \<longrightarrow> b` step_closed[of a b] multi_step_closed[of b c] by auto then obtain T' F' where 1:"\<Gamma> \<turnstile> b : T' ; F'" "\<turnstile> T' <: T" "\<turnstile> F' <e: F" using preservation[of \<Gamma> a T F b] sm_trans by auto then obtain T'' F'' where 2:"\<Gamma> \<turnstile> c : T'' ; F''" "\<turnstile> T'' <: T'" "\<turnstile> F'' <e: F'" "c : values" using sm_trans(3)[of \<Gamma> T' F'] sm_trans `closed b` by blast have "\<turnstile> T'' <: T" using 1 2 by auto have 3:"simple_eff F" using prems closed_eff by auto have 4:"simple_eff F'" using 1 prems closed_eff `closed b` by auto have 5:"simple_eff F''" using prems closed_eff `closed c` by auto from 3 4 5 have "\<turnstile> F'' <e: F" using SE_Trans[of F'' F' F] 1 2 by auto thus ?case using 2 `\<turnstile> T'' <: T` by auto qed text {* interesting fact: let e = (Iff True 3 x) then [] \<turnstile> e : Int ; NE but e is not closed } theorem soundness: assumes A:"\<Gamma> \<turnstile> e : T ; F" and E:"closed e" shows "reduce_forever e \<or> (EX v T' F'. (v : values \<and> e \<longrightarrow>\<^sup> v \<and> \<Gamma> \<turnstile> v : T' ; F' \<and> \<turnstile> T' <: T \<and> \<turnstile> F' <e: F))" proof - { assume "~(reduce_forever e)" hence "EX e'. (e \<longrightarrow>\<^sup>* e') \<and> ~(EX e''. e' \<longrightarrow> e'')" by (auto simp add: reduce_forever_def) then obtain e'::trm where B:"e \<longrightarrow>\<^sup>* e'" and C:"~ (EX e''. e' \<longrightarrow> e'')" by auto hence ?thesis using soundness_finite[OF A B C E] by auto } moreover { assume "reduce_forever e" hence ?thesis by simp } ultimately show ?thesis by auto qed text {* simpler type system, without silly rules } inductive typing2 :: "varEnv \<Rightarrow> trm \<Rightarrow> ty \<Rightarrow> eff \<Rightarrow> bool" (" _ \<turnstile>\<^isub>2 _ : _ ; _ " [60,60,60,60] 60) where T2_Var[intro]: "\<lbrakk>valid \<Gamma>; (v,T)\<in>set \<Gamma>\<rbrakk>\<Longrightarrow> \<Gamma> \<turnstile>\<^isub>2 Var v : T ; VE v" \| T2_Const[intro]: "valid \<Gamma> \<Longrightarrow> \<Delta>\<^isub>\<tau> b = T \<Longrightarrow> \<Gamma> \<turnstile>\<^isub>2 (BI b) : T ; TT" \| T2_Num[intro]: "valid \<Gamma> \<Longrightarrow> \<Gamma> \<turnstile>\<^isub>2 (Num n) : ty.Int ; TT" \| T2_True[intro]: "valid \<Gamma> \<Longrightarrow> \<Gamma> \<turnstile>\<^isub>2 (Bool True) : ty.TT ; TT" \| T2_False[intro]: "valid \<Gamma> \<Longrightarrow> \<Gamma> \<turnstile>\<^isub>2 (Bool False) : ty.FF ; FF" \| T2_Abs[intro]: "\<lbrakk>x \<sharp> \<Gamma>; ((x,T1)#\<Gamma>) \<turnstile>\<^isub>2 b : T2; eff\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile>\<^isub>2 Lam [x:T1].b : (T1\<rightarrow>T2 : latent_eff.NE) ; TT" \| T2_App[intro]: "\<lbrakk>\<Gamma> \<turnstile>\<^isub>2 e1 : U ; eff1 ; \<turnstile> U <: (T0 \<rightarrow> T1 : le); \<Gamma> \<turnstile>\<^isub>2 e2 : T; eff2 ; \<turnstile> T <: T0\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile>\<^isub>2 App e1 e2 : T1 ; NE" \| T2_AppPred[intro]: "\<lbrakk>\<Gamma> \<turnstile>\<^isub>2 e1 : U; eff1; \<turnstile> U <: (T0 \<rightarrow> T1 : Latent S); \<Gamma> \<turnstile>\<^isub>2 e2 : T; VE x ; \<turnstile> T <: T0\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile>\<^isub>2 App e1 e2 : T1 ; TE S x" \| T2_If[intro]: "\<lbrakk>\<Gamma> \<turnstile>\<^isub>2 e1 : T1; eff1; (\<Gamma> \|+ eff1) \<turnstile>\<^isub>2 e2 : T2; eff2; (\<Gamma> \|- eff1) \<turnstile>\<^isub>2 e3 : T3; eff3; \<turnstile> T2 <: T; \<turnstile> T3 <: T\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile>\<^isub>2 (Iff e1 e2 e3) : T ; comb_eff eff1 eff2 eff3" \| T2_AbsPred[intro]: "\<lbrakk>x \<sharp> \<Gamma>; ((x,T1)#\<Gamma>) \<turnstile>\<^isub>2 b : T2; TE S x\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile>\<^isub>2 Lam [x:T1].b : (T1\<rightarrow>T2 : Latent S) ; TT" lemma typing2_typing: assumes "\<Gamma> \<turnstile>\<^isub>2 e : T ; F" shows "\<Gamma> \<turnstile> e : T ; F" using prems by induct (auto simp del: comb_eff.simps) lemma typing2_soundness1: assumes A:"\<Gamma> \<turnstile>\<^isub>2 e : T ; F" and E:"closed e" shows "reduce_forever e \<or> (EX v T' F'. (v : values \<and> e \<longrightarrow>\<^sup> v \<and> \<Gamma> \<turnstile> v : T' ; F' \<and> \<turnstile> T' <: T \<and> \<turnstile> F' <e: F))" using A E soundness typing2_typing by auto lemma typing2_soundness_int: assumes A:"\<Gamma> \<turnstile>\<^isub>2 e : ty.Int ; F" and E:"closed e" shows "reduce_forever e \<or> (EX v F'. (v : values \<and> (e \<longrightarrow>\<^sup>* v) \<and> \<Gamma> \<turnstile>\<^isub>2 v : ty.Int ; F' \<and> \<turnstile> F' <e: F))" proof - from typing2_soundness1[OF A E] have B:"reduce_forever e \<or> (EX v T' F'. (v : values \<and> (e \<longrightarrow>\<^sup>* v) \<and> \<Gamma> \<turnstile> v : T' ; F' \<and> \<turnstile> T' <: ty.Int \<and> \<turnstile> F' <e: F))" by auto thus ?thesis proof assume "EX v T' F'. (v : values \<and> (e \<longrightarrow>\<^sup>* v) \<and> \<Gamma> \<turnstile> v : T' ; F' \<and> \<turnstile> T' <: ty.Int \<and> \<turnstile> F' <e: F)" then guess v T' F' by auto then obtain n where "v = Num n" using int_value by blast hence "\<Gamma> \<turnstile>\<^isub>2 v : ty.Int ; F'" using num_ty_elim[of \<Gamma> _ T' F'] `\<Gamma> \<turnstile> v : T' ; F'` by auto thus ?thesis using `\<turnstile> F' <e: F` `v : values` `(e \<longrightarrow>\<^sup>* v)` by auto qed (auto) qed lemma typing2_soundness_help: assumes A:"\<Gamma> \<turnstile>\<^isub>2 e : \<sigma> ; F" and E:"closed e" and B: "((\<exists>v T' F'. v \<in> values \<and> \<Gamma> \<turnstile> v : T' ; F' \<and> \<turnstile> T' <: \<sigma> \<and> \<turnstile> F' <e: F)) \<Longrightarrow> (reduce_forever e \<or> (\<exists>v F'. v \<in> values \<and> \<Gamma> \<turnstile>\<^isub>2 v : \<sigma> ; F' \<and> \<turnstile> F' <e: F))" shows "(reduce_forever e \<or> (\<exists>v F'. v \<in> values \<and> \<Gamma> \<turnstile>\<^isub>2 v : \<sigma> ; F' \<and> \<turnstile> F' <e: F))" using A B E typing2_soundness1 by blast lemma typing2_soundness_bool: assumes A:"\<Gamma> \<turnstile>\<^isub>2 e : T ; F" and E:"closed e" and sub:"\<turnstile> T <: BoolTy" shows "reduce_forever e \<or> (EX v F' T'. (v : values \<and> (e \<longrightarrow>\<^sup>* v) \<and> \<Gamma> \<turnstile>\<^isub>2 v : T' ; F' \<and> \<turnstile> F' <e: F \<and> \<turnstile> T' <: T))" proof - from typing2_soundness1[OF A E] have B:"reduce_forever e \<or> (EX v T' F'. (v : values \<and> (e \<longrightarrow>\<^sup>* v) \<and> \<Gamma> \<turnstile> v : T' ; F' \<and> \<turnstile> T' <: T \<and> \<turnstile> F' <e: F))" by blast thus ?thesis proof assume "EX v T' F'. (v : values \<and> (e \<longrightarrow>\<^sup>* v) \<and> \<Gamma> \<turnstile> v : T' ; F' \<and> \<turnstile> T' <: T \<and> \<turnstile> F' <e: F)" then guess v T' F' by auto then obtain b where "v = Bool b" using bool_value[OF `v : values` `\<Gamma> \<turnstile> v : T' ; F'`] sub by auto hence "\<Gamma> \<turnstile>\<^isub>2 Bool b : T' ; F'" using `\<Gamma> \<turnstile> v : T' ; F'` true_ty_elim[of \<Gamma> T' F'] false_ty_elim[of \<Gamma> T' F'] by (cases b) auto thus ?thesis using `v = Bool b` `e \<longrightarrow>\<^sup>* v` `v : values` `\<turnstile> F' <e: F` `\<turnstile> T' <: T` by blast qed (auto) qed constdefs ground_type :: "ty \<Rightarrow> bool" "ground_type t == t = ty.Int \<or> t = BoolTy \<or> t = ty.TT \<or> t = ty.FF" lemma supp_env: "(a,b) : set (\<Gamma> :: varEnv) \<Longrightarrow> a : supp \<Gamma>" by (induct \<Gamma>) (auto simp add: supp_list_cons supp_prod supp_atm) lemma envop_supp: assumes "valid \<Gamma>" shows "x : (supp (envop f n t \<Gamma>)) \<Longrightarrow> x : (supp \<Gamma> :: name set)" proof - assume A:"x : (supp (envop f n t \<Gamma>))" have "!! a. a \<notin> (supp \<Gamma> :: name set) \<Longrightarrow> a \<notin> supp (envop f n t \<Gamma>)" proof - fix a show "a \<notin> (supp \<Gamma> :: name set) \<Longrightarrow> a \<notin> supp (envop f n t \<Gamma>)" using fresh_def[of a "envop f n t \<Gamma>"] envop_fresh[OF _ `valid \<Gamma>`] fresh_def[of a \<Gamma>] by auto qed hence "(supp (envop f n t \<Gamma>)) <= (supp \<Gamma> :: name set)" by blast thus ?thesis using A by auto qed declare envop_def[simp del] lemma envplus_supp: assumes "valid \<Gamma>" shows "(supp (\<Gamma> \|+ F) :: name set) <= (supp \<Gamma> :: name set)" using prems apply (nominal_induct F rule: eff.induct) apply (auto simp add: envop_supp) done lemma envminus_supp: assumes "valid \<Gamma>" shows "(supp (\<Gamma> \|- F) :: name set) <= (supp \<Gamma> :: name set)" using prems apply (nominal_induct F rule: eff.induct) apply (auto simp add: envop_supp) done lemma env_supp_typing2: assumes "\<Gamma> \<turnstile>\<^isub>2 e : T ; F" shows "fv e <= supp \<Gamma>" using prems proof (induct \<Gamma> e T F rule: typing2.induct) case T2_Var thus ?case by (auto simp add: trm.supp supp_atm supp_env) next case T2_App thus ?case by (auto simp add: trm.supp) next case T2_AppPred thus ?case by (auto simp add: trm.supp) next case (T2_If \<Gamma> _ _ F) have "valid \<Gamma>" using typing_valid typing2_typing[OF T2_If(1)] by auto thus ?case using T2_If envminus_supp[OF `valid \<Gamma>`, of F] envplus_supp[OF `valid \<Gamma>`, of F] by (auto simp add: trm.supp) next case T2_Abs thus ?case by (auto simp add: trm.supp fv_lam abs_supp supp_list_cons supp_prod supp_latent_eff_ty supp_atm) next case T2_AbsPred thus ?case by (auto simp add: trm.supp fv_lam abs_supp supp_list_cons supp_prod supp_latent_eff_ty supp_atm) next case (T2_Const \<Gamma> b) thus ?case by (nominal_induct b rule: builtin.induct) (auto simp add: trm.supp builtin.supp) qed (auto simp add: trm.supp supp_nat supp_bool) lemma empty_env_typing2_closed: assumes "[] \<turnstile>\<^isub>2 e : T ; F" shows "closed e" using env_supp_typing2[OF prems] closed_def prems supp_list_nil by auto theorem ground_type_soundness: assumes A:"[] \<turnstile>\<^isub>2 e : T ; F" and B:"ground_type T" shows "reduce_forever e \<or> (EX v T' F'. (v : values \<and> (e \<longrightarrow>\<^sup>* v) \<and> [] \<turnstile>\<^isub>2 v : T' ; F' \<and> \<turnstile> T' <: T \<and> \<turnstile> F' <e: F))" proof - have E:"closed e" using empty_env_typing2_closed[OF A] by simp { assume "T = ty.Int" hence "reduce_forever e \<or> (EX v F'. (v : values \<and> (e \<longrightarrow>\<^sup>* v) \<and> [] \<turnstile>\<^isub>2 v : ty.Int ; F' \<and> \<turnstile> F' <e: F))" using A E typing2_soundness_int by auto hence ?thesis using prems by auto } moreover { assume "T ~= ty.Int" hence "\<turnstile> T <: BoolTy" using B by (auto simp add: ground_type_def) hence "reduce_forever e \<or> (EX v F' T'. (v : values \<and> (e \<longrightarrow>\<^sup>* v) \<and> [] \<turnstile>\<^isub>2 v : T' ; F' \<and> \<turnstile> F' <e: F \<and> \<turnstile> T' <: T))" using typing2_soundness_bool[OF A E] by auto hence ?thesis using prems by auto } ultimately show ?thesis using B ground_type_def by auto qed end lemma unique_decomposition: assumes a:"closed e" shows "\<lbrakk>E : ctxt; E t = e; E' : ctxt; E' t' = e\<rbrakk> \<Longrightarrow> E = E'" using a proof (nominal_induct e rule: trm.induct) case (Var v) have f1:"E = (%t. t)" using Var by cases auto have f2:"E'= (%t. t)" using `E' : ctxt` Var by cases auto from f1 f2 show ?case by simp next case (Bool c) have f1:"E = (%t. t)" using Bool by cases auto have f2:"E'= (%t. t)" using `E' : ctxt` Bool by cases auto from f1 f2 show ?case by simp next case (Num c) have f1:"E = (%t. t)" using Num by cases auto have f2:"E'= (%t. t)" using `E' : ctxt` Num by cases auto from f1 f2 show ?case by simp next case Abs have f1:"E = (%t. t)" using `E : ctxt` Abs by cases auto have f2:"E'= (%t. t)" using `E' : ctxt` Abs by cases auto from f1 f2 show ?case by simp next case (Iff tst thn els) { assume "tst \<notin> values" hence "EX E L R. tst = E L \<and> E \<in> ctxt \<and> L \<hookrightarrow> R" using decomposition have f1:"E = (%t. t)" using `E : ctxt` Iff apply cases apply (auto simp add: trm.inject) have f2:"E'= (%t. t)" using `E' : ctxt` Iff by cases auto from f1 f2 have ?case by simp { oops lemma fresh_fact: fixes a::"name" assumes a: "a\<sharp>t1" and b: "a\<sharp>t2" shows "a\<sharp>(t1[b::=t2])" using a b by (nominal_induct t1 avoiding: a b t2 rule: trm.induct) (auto simp add: abs_fresh fresh_atm) lemma id_subs: "t[x::=Var x] = t" by (nominal_induct t avoiding: x rule: trm.induct) (simp_all add: fresh_atm) lemma random_eqvt[simp]: fixes pi :: "name prm" shows "\<forall>T. T \<in> set Ts \<longrightarrow> \<turnstile> T <: S \<and> \<turnstile> pi \<bullet> T <: pi \<bullet> S \<Longrightarrow> \<forall>T. T \<in> set (pi \<bullet> Ts) \<longrightarrow> \<turnstile> T <: pi \<bullet> S" proof - assume 0:"\<forall>T. T \<in> set Ts \<longrightarrow> \<turnstile> T <: S \<and> \<turnstile> pi \<bullet> T <: pi \<bullet> S" hence 1:"!! T. T \<in> set Ts \<Longrightarrow> \<turnstile> T <: S \<and> \<turnstile> pi \<bullet> T <: pi \<bullet> S" by auto have A:"(pi \<bullet> Ts) = Ts" by (induct Ts) auto have B:"pi \<bullet> S = S" by auto have "!! T. T \<in> set (pi \<bullet> Ts) \<Longrightarrow> \<turnstile> T <: pi \<bullet> S" proof - fix T assume "T \<in> set (pi \<bullet> Ts)" hence "T : set Ts" using A by auto hence "\<turnstile> T <: S" using 1 by auto thus "\<turnstile> T <: pi \<bullet> S" using B by auto qed thus ?thesis by auto qed text {* complete induction on typing derivations *} lemma typing_induct_complete[consumes 1, case_names T_Var T_Const T_Num T_True T_False T_App T_Lam T_AppPred T_If T_AppPredTrue T_AppPredFalse T_IfTrue T_IfFalse]: fixes P :: "'a::fs_name\<Rightarrow>(name\<times>ty) list \<Rightarrow> trm \<Rightarrow> ty \<Rightarrow> eff \<Rightarrow> bool" and \<Gamma> :: "(name\<times>ty) list" and t :: "trm" and T :: "ty" and F :: "eff" and x :: "'a::fs_name" assumes a: "\<Gamma> \<turnstile> t : T ; F" and a1: "\<And>\<Gamma> (a::name) \<tau> x. valid \<Gamma> \<Longrightarrow> (a,\<tau>) \<in> set \<Gamma> \<Longrightarrow> (!! x t T \<Gamma> F. (t\<guillemotleft>Var a) \<Longrightarrow> \<Gamma> \<turnstile> t : T ; F \<Longrightarrow> P x \<Gamma> t T F) \<Longrightarrow> valid \<Gamma> \<Longrightarrow> P x \<Gamma> (Var a) \<tau> (VE a)" and a2: "!! \<Gamma> b T x. \<Delta>\<^isub>\<tau> b = T \<Longrightarrow> (!! x t T \<Gamma> F. (t\<guillemotleft>BI b) \<Longrightarrow> \<Gamma> \<turnstile> t : T ; F \<Longrightarrow> P x \<Gamma> t T F) \<Longrightarrow> valid \<Gamma> \<Longrightarrow> P x \<Gamma> (BI b) T NE" and a3: "!! \<Gamma> n x. (!! x t T \<Gamma> F. (t\<guillemotleft>Num n) \<Longrightarrow> \<Gamma> \<turnstile> t : T ; F \<Longrightarrow> P x \<Gamma> t T F) \<Longrightarrow> valid \<Gamma> \<Longrightarrow> P x \<Gamma> (Num n) ty.Int NE" and a4: "!! \<Gamma> x. (!! x t T \<Gamma> F. (t\<guillemotleft>Bool True) \<Longrightarrow> \<Gamma> \<turnstile> t : T ; F \<Longrightarrow> P x \<Gamma> t T F) \<Longrightarrow> valid \<Gamma> \<Longrightarrow> P x \<Gamma> (Bool True) BoolTy TT" and a5: "!! \<Gamma> x. (!! x t T \<Gamma> F. (t\<guillemotleft>Bool False) \<Longrightarrow> \<Gamma> \<turnstile> t : T ; F \<Longrightarrow> P x \<Gamma> t T F) \<Longrightarrow> valid \<Gamma> \<Longrightarrow> P x \<Gamma> (Bool False) BoolTy FF" and a6: "\<And>\<Gamma> \<tau> \<sigma> t1 t2 x F1 F2 le \<tau>0 U. (!! x t T \<Gamma> F. (t\<guillemotleft>App t1 t2) \<Longrightarrow> \<Gamma> \<turnstile> t : T ; F \<Longrightarrow> P x \<Gamma> t T F) \<Longrightarrow> \<Gamma> \<turnstile> t1 : U ; F1 \<Longrightarrow> \<turnstile> U <: \<tau>\<rightarrow>\<sigma>:le \<Longrightarrow> (\<And>z. P z \<Gamma> t1 U F1) \<Longrightarrow> \<Gamma> \<turnstile> t2 : \<tau>0 ; F2 \<Longrightarrow> (\<And>z. P z \<Gamma> t2 \<tau>0 F2) \<Longrightarrow> \<turnstile> \<tau>0 <: \<tau> \<Longrightarrow> P x \<Gamma> (App t1 t2) \<sigma> NE" and a7: "\<And>a \<Gamma> \<tau> \<sigma> t x F0. a\<sharp>x \<Longrightarrow> a\<sharp>\<Gamma> \<Longrightarrow> ((a,\<tau>) # \<Gamma>) \<turnstile> t : \<sigma> ; F0 \<Longrightarrow> (\<And>z. P z ((a,\<tau>)#\<Gamma>) t \<sigma> F0) \<Longrightarrow> (!! x t' T \<Gamma> F. (t'\<guillemotleft>Lam[a:\<tau>].t) \<Longrightarrow> \<Gamma> \<turnstile> t' : T ; F \<Longrightarrow> P x \<Gamma> t' T F) \<Longrightarrow> P x \<Gamma> (Lam [a:\<tau>].t) (\<tau>\<rightarrow>\<sigma>:latent_eff.NE) NE" and a8: "\<And>\<Gamma> \<tau> \<sigma> t1 t2 x F1 \<tau>0 S v U. (!! x t T \<Gamma> F. t \<guillemotleft> App t1 t2 \<Longrightarrow> \<Gamma> \<turnstile> t : T ; F \<Longrightarrow> P x \<Gamma> t T F) \<Longrightarrow> \<Gamma> \<turnstile> t1 : U ; F1 \<Longrightarrow> \<turnstile> U <: (\<tau>\<rightarrow>\<sigma>:Latent S) \<Longrightarrow> (\<And>z. P z \<Gamma> t1 U F1) \<Longrightarrow> \<Gamma> \<turnstile> t2 : \<tau>0 ; VE v \<Longrightarrow> (\<And>z. P z \<Gamma> t2 \<tau>0 (VE v)) \<Longrightarrow> \<turnstile> \<tau>0 <: \<tau> \<Longrightarrow> P x \<Gamma> (App t1 t2) \<sigma> (TE S v)" and a9: "!! \<Gamma> e1 e2 e3 T1 T2 T3 T eff1 eff2 eff3 x. \<lbrakk>\<Gamma> \<turnstile> e1 : T1; eff1; !!z. P z \<Gamma> e1 T1 eff1; (\<Gamma> \|+ eff1) \<turnstile> e2 : T2; eff2; !!z. P z (\<Gamma>\|+ eff1) e2 T2 eff2; (!! x t T \<Gamma> F. t \<guillemotleft> Iff e1 e2 e3 \<Longrightarrow> \<Gamma> \<turnstile> t : T ; F \<Longrightarrow> P x \<Gamma> t T F); (\<Gamma> \|- eff1) \<turnstile> e3 : T3; eff3; !!z. P z (\<Gamma>\|- eff1) e3 T3 eff3; \<turnstile> T2 <: T; \<turnstile> T3 <: T\<rbrakk> \<Longrightarrow> P x \<Gamma> (Iff e1 e2 e3) T NE" and a10: "!! \<Gamma> e1 e2 T0 T1 T S eff1 eff2 x U. \<lbrakk>\<Gamma> \<turnstile> e1 : U; eff1; \<turnstile> U <: (T0 \<rightarrow> T1 : Latent S) ; !!z. P z \<Gamma> e1 U eff1; (!! x t T \<Gamma> F. t \<guillemotleft> App e1 e2 \<Longrightarrow> \<Gamma> \<turnstile> t : T ; F \<Longrightarrow> P x \<Gamma> t T F); \<Gamma> \<turnstile> e2 : T; eff2 ; !! z. P z \<Gamma> e2 T eff2; \<turnstile> T <: T0; \<turnstile> T <: S\<rbrakk> \<Longrightarrow> P x \<Gamma> (App e1 e2) T1 TT" and a11: "!! \<Gamma> e1 e2 T0 T1 T S eff1 eff2 x U. \<lbrakk>\<Gamma> \<turnstile> e1 : U; eff1; \<turnstile> U <: (T0 \<rightarrow> T1 : Latent S) ; !!z. P z \<Gamma> e1 U eff1; (!! x t T \<Gamma> F. t \<guillemotleft> App e1 e2 \<Longrightarrow> \<Gamma> \<turnstile> t : T ; F \<Longrightarrow> P x \<Gamma> t T F); \<Gamma> \<turnstile> e2 : T; eff2 ; !! z. P z \<Gamma> e2 T eff2; \<turnstile> T <: T0; ~(\<turnstile> T <: S) ; e2 : values ; closed e2\<rbrakk> \<Longrightarrow> P x \<Gamma> (App e1 e2) T1 FF" and a12: "!! \<Gamma> e1 e2 e3 T T1 T2 eff x. \<lbrakk>\<Gamma> \<turnstile> e1 : T1 ; TT ; !! z. P z \<Gamma> e1 T1 TT; (!! x t T \<Gamma> F. t \<guillemotleft> Iff e1 e2 e3 \<Longrightarrow> \<Gamma> \<turnstile> t : T ; F \<Longrightarrow> P x \<Gamma> t T F); \<Gamma> \<turnstile> e2 : T2 ; eff; !!z .P z \<Gamma> e2 T2 eff; \<turnstile> T2 <: T\<rbrakk> \<Longrightarrow> P x \<Gamma> (Iff e1 e2 e3) T NE" and a13: "!! \<Gamma> e1 e2 e3 T T1 T3 eff x. \<lbrakk>\<Gamma> \<turnstile> e1 : T1 ; FF ; !! z. P z \<Gamma> e1 T1 FF; (!! x t T \<Gamma> F. t \<guillemotleft> Iff e1 e2 e3 \<Longrightarrow> \<Gamma> \<turnstile> t : T ; F \<Longrightarrow> P x \<Gamma> t T F); \<Gamma> \<turnstile> e3 : T3 ; eff; !!z .P z \<Gamma> e3 T3 eff; \<turnstile> T3 <: T\<rbrakk> \<Longrightarrow> P x \<Gamma> (Iff e1 e2 e3) T NE" shows "P x \<Gamma> t T F" using a proof (nominal_induct t avoiding: x \<Gamma> T F rule: trm_comp_induct) case (Var v) thus ?case using a1 var_ty_elim[of \<Gamma> v T F] by auto next case (App t1 t2 x \<Gamma> T) show ?case using App(4) proof (induct rule: app_ty_elim2) case 1 thus ?thesis using a6 App trm.inject ty.inject by auto next case 2 thus ?thesis using a8 App trm.inject ty.inject by auto next case 3 thus ?thesis using a10 App trm.inject ty.inject by auto next case 4 thus ?thesis using a11 App trm.inject ty.inject by auto qed next case Iff show ?case using Iff(5) proof (induct rule: iff_ty_elim2) case 1 thus ?thesis using a9 Iff trm.inject ty.inject by auto next case 2 thus ?thesis using a12 Iff trm.inject ty.inject by auto next case 3 thus ?thesis using a13 Iff trm.inject ty.inject by auto qed next case (Lam v b x \<Gamma> S1 F S2) show ?case using Lam abs_ty_elim[of \<Gamma> v b S2 S1 F] a7 by (auto simp add: trm.inject ty.inject) next case (BI b) thus ?case using bi_ty_elim[of \<Gamma> b T F] trm.inject a2 by auto next case (Num n) thus ?case using num_ty_elim[of \<Gamma> n T F] trm.inject a3 by auto next case (Bool b) thus ?case using true_ty_elim[of \<Gamma> T F] false_ty_elim[of \<Gamma> T F] trm.inject a4 a5 by (cases b) auto qed
Monument 27 is a carved step depicting K 'awiil Mo ' , a lord from Palenque , as an elderly prisoner , bound and lying on his back with his profile positioned in such a way as to be trodden on time and again .
The modulus of the complex conjugate of a complex number is equal to the modulus of the complex number.
State Before: α : Type u_1 E : α → Type u_2 p q : ℝ≥0∞ inst✝² : (i : α) → NormedAddCommGroup (E i) inst✝¹ : (i : α) → StarAddMonoid (E i) inst✝ : ∀ (i : α), NormedStarGroup (E i) f : (i : α) → E i hf : Memℓp f p ⊢ Memℓp (star f) p State After: case inl α : Type u_1 E : α → Type u_2 q : ℝ≥0∞ inst✝² : (i : α) → NormedAddCommGroup (E i) inst✝¹ : (i : α) → StarAddMonoid (E i) inst✝ : ∀ (i : α), NormedStarGroup (E i) f : (i : α) → E i hf : Memℓp f 0 ⊢ Memℓp (star f) 0 case inr.inl α : Type u_1 E : α → Type u_2 q : ℝ≥0∞ inst✝² : (i : α) → NormedAddCommGroup (E i) inst✝¹ : (i : α) → StarAddMonoid (E i) inst✝ : ∀ (i : α), NormedStarGroup (E i) f : (i : α) → E i hf : Memℓp f ⊤ ⊢ Memℓp (star f) ⊤ case inr.inr α : Type u_1 E : α → Type u_2 p q : ℝ≥0∞ inst✝² : (i : α) → NormedAddCommGroup (E i) inst✝¹ : (i : α) → StarAddMonoid (E i) inst✝ : ∀ (i : α), NormedStarGroup (E i) f : (i : α) → E i hf : Memℓp f p hp : 0 < ENNReal.toReal p ⊢ Memℓp (star f) p Tactic: rcases p.trichotomy with (rfl \| rfl \| hp) State Before: case inl α : Type u_1 E : α → Type u_2 q : ℝ≥0∞ inst✝² : (i : α) → NormedAddCommGroup (E i) inst✝¹ : (i : α) → StarAddMonoid (E i) inst✝ : ∀ (i : α), NormedStarGroup (E i) f : (i : α) → E i hf : Memℓp f 0 ⊢ Memℓp (star f) 0 State After: case inl.hf α : Type u_1 E : α → Type u_2 q : ℝ≥0∞ inst✝² : (i : α) → NormedAddCommGroup (E i) inst✝¹ : (i : α) → StarAddMonoid (E i) inst✝ : ∀ (i : α), NormedStarGroup (E i) f : (i : α) → E i hf : Memℓp f 0 ⊢ Set.Finite {i \| star f i ≠ 0} Tactic: apply memℓp_zero State Before: case inl.hf α : Type u_1 E : α → Type u_2 q : ℝ≥0∞ inst✝² : (i : α) → NormedAddCommGroup (E i) inst✝¹ : (i : α) → StarAddMonoid (E i) inst✝ : ∀ (i : α), NormedStarGroup (E i) f : (i : α) → E i hf : Memℓp f 0 ⊢ Set.Finite {i \| star f i ≠ 0} State After: no goals Tactic: simp [hf.finite_dsupport] State Before: case inr.inl α : Type u_1 E : α → Type u_2 q : ℝ≥0∞ inst✝² : (i : α) → NormedAddCommGroup (E i) inst✝¹ : (i : α) → StarAddMonoid (E i) inst✝ : ∀ (i : α), NormedStarGroup (E i) f : (i : α) → E i hf : Memℓp f ⊤ ⊢ Memℓp (star f) ⊤ State After: case inr.inl.hf α : Type u_1 E : α → Type u_2 q : ℝ≥0∞ inst✝² : (i : α) → NormedAddCommGroup (E i) inst✝¹ : (i : α) → StarAddMonoid (E i) inst✝ : ∀ (i : α), NormedStarGroup (E i) f : (i : α) → E i hf : Memℓp f ⊤ ⊢ BddAbove (Set.range fun i => ‖star f i‖) Tactic: apply memℓp_infty State Before: case inr.inl.hf α : Type u_1 E : α → Type u_2 q : ℝ≥0∞ inst✝² : (i : α) → NormedAddCommGroup (E i) inst✝¹ : (i : α) → StarAddMonoid (E i) inst✝ : ∀ (i : α), NormedStarGroup (E i) f : (i : α) → E i hf : Memℓp f ⊤ ⊢ BddAbove (Set.range fun i => ‖star f i‖) State After: no goals Tactic: simpa using hf.bddAbove State Before: case inr.inr α : Type u_1 E : α → Type u_2 p q : ℝ≥0∞ inst✝² : (i : α) → NormedAddCommGroup (E i) inst✝¹ : (i : α) → StarAddMonoid (E i) inst✝ : ∀ (i : α), NormedStarGroup (E i) f : (i : α) → E i hf : Memℓp f p hp : 0 < ENNReal.toReal p ⊢ Memℓp (star f) p State After: case inr.inr.hf α : Type u_1 E : α → Type u_2 p q : ℝ≥0∞ inst✝² : (i : α) → NormedAddCommGroup (E i) inst✝¹ : (i : α) → StarAddMonoid (E i) inst✝ : ∀ (i : α), NormedStarGroup (E i) f : (i : α) → E i hf : Memℓp f p hp : 0 < ENNReal.toReal p ⊢ Summable fun i => ‖star f i‖ ^ ENNReal.toReal p Tactic: apply memℓp_gen State Before: case inr.inr.hf α : Type u_1 E : α → Type u_2 p q : ℝ≥0∞ inst✝² : (i : α) → NormedAddCommGroup (E i) inst✝¹ : (i : α) → StarAddMonoid (E i) inst✝ : ∀ (i : α), NormedStarGroup (E i) f : (i : α) → E i hf : Memℓp f p hp : 0 < ENNReal.toReal p ⊢ Summable fun i => ‖star f i‖ ^ ENNReal.toReal p State After: no goals Tactic: simpa using hf.summable hp
(* This file is an automatic translation, the licence of the source can be found here: ) ( https://github.com/herd/herdtools7/blob/master/LICENSE.txt ) ( Translation of model Compatibility ) From Coq Require Import Relations Ensembles String. From RelationAlgebra Require Import lattice prop monoid rel kat. From Catincoq.lib Require Import Cat proprel. Section Model. Variable c : candidate. Definition events := events c. Definition R := R c. Definition W := W c. Definition IW := IW c. Definition FW := FW c. Definition B := B c. Definition RMW := RMW c. Definition F := F c. Definition rf := rf c. Definition po := po c. Definition int := int c. Definition ext := ext c. Definition loc := loc c. Definition addr := addr c. Definition data := data c. Definition ctrl := ctrl c. Definition amo := amo c. Definition rmw := rmw c. Definition unknown_set := unknown_set c. Definition unknown_relation := unknown_relation c. Definition M := R ⊔ W. Definition emptyset : set events := empty. Definition classes_loc : set events -> Ensemble (Ensemble events) := partition loc. Definition tag2events := unknown_relation "tag2events". Definition emptyset_0 : set events := domain 0. Definition partition := classes_loc. Definition tag2instrs := tag2events. Definition po_loc := po ⊓ loc. Definition rfe := rf ⊓ ext. Definition rfi := rf ⊓ int. Definition co0 := loc ⊓ ([IW] ⋅ top ⋅ [(W ⊓ !IW)] ⊔ [(W ⊓ !FW)] ⋅ top ⋅ [FW]). Definition toid (s : set events) : relation events := [s]. Definition fencerel (B : set events) := (po ⊓ [top] ⋅ top ⋅ [B]) ⋅ po. Definition ctrlcfence (CFENCE : set events) := (ctrl ⊓ [top] ⋅ top ⋅ [CFENCE]) ⋅ po. Definition imply (A : relation events) (B : relation events) := !A ⊔ B. Definition nodetour (R1 : relation events) (R2 : relation events) (R3 : relation events) := R1 ⊓ !(R2 ⋅ R3). Definition singlestep (R : relation events) := nodetour R R R. ( Definition of map already included in the prelude ) Definition LKW := (failed: try LKW with emptyset_0) emptyset_0. Definition A := ((failed: try X with emptyset_0) emptyset_0) ⊔ ((failed: try A with emptyset_0) emptyset_0). Definition P := M ⊓ !A. Definition WW r := r ⊓ [W] ⋅ top ⋅ [W]. Definition WR r := r ⊓ [W] ⋅ top ⋅ [R]. Definition RW r := r ⊓ [R] ⋅ top ⋅ [W]. Definition RR r := r ⊓ [R] ⋅ top ⋅ [R]. Definition RM r := r ⊓ [R] ⋅ top ⋅ [M]. Definition MR r := r ⊓ [M] ⋅ top ⋅ [R]. Definition WM r := r ⊓ [W] ⋅ top ⋅ [M]. Definition MW r := r ⊓ [M] ⋅ top ⋅ [W]. Definition MM r := r ⊓ [M] ⋅ top ⋅ [M]. Definition AA r := r ⊓ [A] ⋅ top ⋅ [A]. Definition AP r := r ⊓ [A] ⋅ top ⋅ [P]. Definition PA r := r ⊓ [P] ⋅ top ⋅ [A]. Definition PP r := r ⊓ [P] ⋅ top ⋅ [P]. Definition AM r := r ⊓ [A] ⋅ top ⋅ [M]. Definition MA r := r ⊓ [M] ⋅ top ⋅ [A]. Definition noid r : relation events := r ⊓ !id. Definition atom := [A]. Definition witness_conditions := True. Definition model_conditions := True. End Model. Hint Unfold events R W IW FW B RMW F rf po int ext loc addr data ctrl amo rmw unknown_set unknown_relation M emptyset classes_loc tag2events emptyset_0 partition tag2instrs po_loc rfe rfi co0 toid fencerel ctrlcfence imply nodetour singlestep LKW A P WW WR RW RR RM MR WM MW MM AA AP PA PP AM MA noid atom witness_conditions model_conditions : cat. Definition valid (c : candidate) := True. ( End of translation of model Compatibility *)
From Hammer Require Import Hammer. Require Import FunInd. Require Import Zwf. From compcert Require Import Coqlib. From compcert Require Import Maps. From compcert Require Import Zbits. From compcert Require Import Integers. From compcert Require Import Floats. From compcert Require Import Lattice. From compcert Require Import Compopts. From compcert Require Import AST. From compcert Require Import Values. From compcert Require Import Memory. From compcert Require Import Globalenvs. From compcert Require Import Builtins. From compcert Require Import Events. From compcert Require Import Registers. From compcert Require Import RTL. Inductive block_class : Type := \| BCinvalid \| BCglob (id: ident) \| BCstack \| BCother. Definition block_class_eq: forall (x y: block_class), {x=y} + {x<>y}. Proof. hammer_hook "ValueDomain" "ValueDomain.block_class_eq". decide equality. apply peq. Defined. Record block_classification : Type := BC { bc_img :> block -> block_class; bc_stack: forall b1 b2, bc_img b1 = BCstack -> bc_img b2 = BCstack -> b1 = b2; bc_glob: forall b1 b2 id, bc_img b1 = BCglob id -> bc_img b2 = BCglob id -> b1 = b2 }. Definition bc_below (bc: block_classification) (bound: block) : Prop := forall b, bc b <> BCinvalid -> Plt b bound. Lemma bc_below_invalid: forall b bc bound, ~Plt b bound -> bc_below bc bound -> bc b = BCinvalid. Proof. hammer_hook "ValueDomain" "ValueDomain.bc_below_invalid". intros. destruct (block_class_eq (bc b) BCinvalid); auto. elim H. apply H0; auto. Qed. Hint Extern 2 (_ = _) => congruence : va. Hint Extern 2 (_ <> _) => congruence : va. Hint Extern 2 (_ < _) => xomega : va. Hint Extern 2 (_ <= _) => xomega : va. Hint Extern 2 (_ > _) => xomega : va. Hint Extern 2 (_ >= _) => xomega : va. Section MATCH. Variable bc: block_classification. Inductive abool := \| Bnone \| Just (b: bool) \| Maybe (b: bool) \| Btop. Inductive cmatch: option bool -> abool -> Prop := \| cmatch_none: cmatch None Bnone \| cmatch_just: forall b, cmatch (Some b) (Just b) \| cmatch_maybe_none: forall b, cmatch None (Maybe b) \| cmatch_maybe_some: forall b, cmatch (Some b) (Maybe b) \| cmatch_top: forall ob, cmatch ob Btop. Hint Constructors cmatch : va. Definition club (x y: abool) : abool := match x, y with \| Bnone, Bnone => Bnone \| Bnone, (Just b \| Maybe b) => Maybe b \| (Just b \| Maybe b), Bnone => Maybe b \| Just b1, Just b2 => if eqb b1 b2 then x else Btop \| Maybe b1, Maybe b2 => if eqb b1 b2 then x else Btop \| Maybe b1, Just b2 => if eqb b1 b2 then x else Btop \| Just b1, Maybe b2 => if eqb b1 b2 then y else Btop \| _, _ => Btop end. Lemma cmatch_lub_l: forall ob x y, cmatch ob x -> cmatch ob (club x y). Proof. hammer_hook "ValueDomain" "ValueDomain.cmatch_lub_l". intros. unfold club; inv H; destruct y; try constructor; destruct (eqb b b0) eqn:EQ; try constructor. replace b0 with b by (apply eqb_prop; auto). constructor. Qed. Lemma cmatch_lub_r: forall ob x y, cmatch ob y -> cmatch ob (club x y). Proof. hammer_hook "ValueDomain" "ValueDomain.cmatch_lub_r". intros. unfold club; inv H; destruct x; try constructor; destruct (eqb b0 b) eqn:EQ; try constructor. replace b with b0 by (apply eqb_prop; auto). constructor. replace b with b0 by (apply eqb_prop; auto). constructor. replace b with b0 by (apply eqb_prop; auto). constructor. Qed. Definition cnot (x: abool) : abool := match x with \| Just b => Just (negb b) \| Maybe b => Maybe (negb b) \| _ => x end. Lemma cnot_sound: forall ob x, cmatch ob x -> cmatch (option_map negb ob) (cnot x). Proof. hammer_hook "ValueDomain" "ValueDomain.cnot_sound". destruct 1; constructor. Qed. Inductive aptr : Type := \| Pbot \| Gl (id: ident) (ofs: ptrofs) \| Glo (id: ident) \| Glob \| Stk (ofs: ptrofs) \| Stack \| Nonstack \| Ptop. Definition eq_aptr: forall (p1 p2: aptr), {p1=p2} + {p1<>p2}. Proof. hammer_hook "ValueDomain" "ValueDomain.eq_aptr". intros. generalize ident_eq, Ptrofs.eq_dec; intros. decide equality. Defined. Inductive pmatch (b: block) (ofs: ptrofs): aptr -> Prop := \| pmatch_gl: forall id, bc b = BCglob id -> pmatch b ofs (Gl id ofs) \| pmatch_glo: forall id, bc b = BCglob id -> pmatch b ofs (Glo id) \| pmatch_glob: forall id, bc b = BCglob id -> pmatch b ofs Glob \| pmatch_stk: bc b = BCstack -> pmatch b ofs (Stk ofs) \| pmatch_stack: bc b = BCstack -> pmatch b ofs Stack \| pmatch_nonstack: bc b <> BCstack -> bc b <> BCinvalid -> pmatch b ofs Nonstack \| pmatch_top: bc b <> BCinvalid -> pmatch b ofs Ptop. Hint Constructors pmatch: va. Inductive pge: aptr -> aptr -> Prop := \| pge_top: forall p, pge Ptop p \| pge_bot: forall p, pge p Pbot \| pge_refl: forall p, pge p p \| pge_glo_gl: forall id ofs, pge (Glo id) (Gl id ofs) \| pge_glob_gl: forall id ofs, pge Glob (Gl id ofs) \| pge_glob_glo: forall id, pge Glob (Glo id) \| pge_ns_gl: forall id ofs, pge Nonstack (Gl id ofs) \| pge_ns_glo: forall id, pge Nonstack (Glo id) \| pge_ns_glob: pge Nonstack Glob \| pge_stack_stk: forall ofs, pge Stack (Stk ofs). Hint Constructors pge: va. Lemma pge_trans: forall p q, pge p q -> forall r, pge q r -> pge p r. Proof. hammer_hook "ValueDomain" "ValueDomain.pge_trans". induction 1; intros r PM; inv PM; auto with va. Qed. Lemma pmatch_ge: forall b ofs p q, pge p q -> pmatch b ofs q -> pmatch b ofs p. Proof. hammer_hook "ValueDomain" "ValueDomain.pmatch_ge". induction 1; intros PM; inv PM; eauto with va. Qed. Lemma pmatch_top': forall b ofs p, pmatch b ofs p -> pmatch b ofs Ptop. Proof. hammer_hook "ValueDomain" "ValueDomain.pmatch_top'". intros. apply pmatch_ge with p; auto with va. Qed. Definition plub (p q: aptr) : aptr := match p, q with \| Pbot, _ => q \| _, Pbot => p \| Gl id1 ofs1, Gl id2 ofs2 => if ident_eq id1 id2 then if Ptrofs.eq_dec ofs1 ofs2 then p else Glo id1 else Glob \| Gl id1 ofs1, Glo id2 => if ident_eq id1 id2 then q else Glob \| Glo id1, Gl id2 ofs2 => if ident_eq id1 id2 then p else Glob \| Glo id1, Glo id2 => if ident_eq id1 id2 then p else Glob \| (Gl _ _ \| Glo _ \| Glob), Glob => Glob \| Glob, (Gl _ _ \| Glo _) => Glob \| (Gl _ _ \| Glo _ \| Glob \| Nonstack), Nonstack => Nonstack \| Nonstack, (Gl _ _ \| Glo _ \| Glob) => Nonstack \| Stk ofs1, Stk ofs2 => if Ptrofs.eq_dec ofs1 ofs2 then p else Stack \| (Stk _ \| Stack), Stack => Stack \| Stack, Stk _ => Stack \| _, _ => Ptop end. Lemma plub_comm: forall p q, plub p q = plub q p. Proof. hammer_hook "ValueDomain" "ValueDomain.plub_comm". intros; unfold plub; destruct p; destruct q; auto. destruct (ident_eq id id0). subst id0. rewrite dec_eq_true. destruct (Ptrofs.eq_dec ofs ofs0). subst ofs0. rewrite dec_eq_true. auto. rewrite dec_eq_false by auto. auto. rewrite dec_eq_false by auto. auto. destruct (ident_eq id id0). subst id0. rewrite dec_eq_true; auto. rewrite dec_eq_false; auto. destruct (ident_eq id id0). subst id0. rewrite dec_eq_true; auto. rewrite dec_eq_false; auto. destruct (ident_eq id id0). subst id0. rewrite dec_eq_true; auto. rewrite dec_eq_false; auto. destruct (Ptrofs.eq_dec ofs ofs0). subst ofs0. rewrite dec_eq_true; auto. rewrite dec_eq_false; auto. Qed. Lemma pge_lub_l: forall p q, pge (plub p q) p. Proof. hammer_hook "ValueDomain" "ValueDomain.pge_lub_l". unfold plub; destruct p, q; auto with va. - destruct (ident_eq id id0). destruct (Ptrofs.eq_dec ofs ofs0); subst; constructor. constructor. - destruct (ident_eq id id0); subst; constructor. - destruct (ident_eq id id0); subst; constructor. - destruct (ident_eq id id0); subst; constructor. - destruct (Ptrofs.eq_dec ofs ofs0); subst; constructor. Qed. Lemma pge_lub_r: forall p q, pge (plub p q) q. Proof. hammer_hook "ValueDomain" "ValueDomain.pge_lub_r". intros. rewrite plub_comm. apply pge_lub_l. Qed. Lemma pmatch_lub_l: forall b ofs p q, pmatch b ofs p -> pmatch b ofs (plub p q). Proof. hammer_hook "ValueDomain" "ValueDomain.pmatch_lub_l". intros. eapply pmatch_ge; eauto. apply pge_lub_l. Qed. Lemma pmatch_lub_r: forall b ofs p q, pmatch b ofs q -> pmatch b ofs (plub p q). Proof. hammer_hook "ValueDomain" "ValueDomain.pmatch_lub_r". intros. eapply pmatch_ge; eauto. apply pge_lub_r. Qed. Lemma plub_least: forall r p q, pge r p -> pge r q -> pge r (plub p q). Proof. hammer_hook "ValueDomain" "ValueDomain.plub_least". intros. inv H; inv H0; simpl; try constructor. - destruct p; constructor. - unfold plub; destruct q; repeat rewrite dec_eq_true; constructor. - rewrite dec_eq_true; constructor. - rewrite dec_eq_true; constructor. - rewrite dec_eq_true. destruct (Ptrofs.eq_dec ofs ofs0); constructor. - destruct (ident_eq id id0). destruct (Ptrofs.eq_dec ofs ofs0); constructor. constructor. - destruct (ident_eq id id0); constructor. - destruct (ident_eq id id0); constructor. - destruct (ident_eq id id0); constructor. - destruct (ident_eq id id0). destruct (Ptrofs.eq_dec ofs ofs0); constructor. constructor. - destruct (ident_eq id id0); constructor. - destruct (ident_eq id id0); constructor. - destruct (ident_eq id id0); constructor. - destruct (Ptrofs.eq_dec ofs ofs0); constructor. Qed. Definition pincl (p q: aptr) : bool := match p, q with \| Pbot, _ => true \| Gl id1 ofs1, Gl id2 ofs2 => peq id1 id2 && Ptrofs.eq_dec ofs1 ofs2 \| Gl id1 ofs1, Glo id2 => peq id1 id2 \| Glo id1, Glo id2 => peq id1 id2 \| (Gl _ _ \| Glo _ \| Glob), Glob => true \| (Gl _ _ \| Glo _ \| Glob \| Nonstack), Nonstack => true \| Stk ofs1, Stk ofs2 => Ptrofs.eq_dec ofs1 ofs2 \| Stk ofs1, Stack => true \| Stack, Stack => true \| _, Ptop => true \| _, _ => false end. Lemma pincl_ge: forall p q, pincl p q = true -> pge q p. Proof. hammer_hook "ValueDomain" "ValueDomain.pincl_ge". unfold pincl; destruct p, q; intros; try discriminate; auto with va; InvBooleans; subst; auto with va. Qed. Lemma pincl_ge_2: forall p q, pge p q -> pincl q p = true. Proof. hammer_hook "ValueDomain" "ValueDomain.pincl_ge_2". destruct 1; simpl; auto. - destruct p; auto. - destruct p; simpl; auto; rewrite ! proj_sumbool_is_true; auto. - rewrite ! proj_sumbool_is_true; auto. Qed. Lemma pincl_sound: forall b ofs p q, pincl p q = true -> pmatch b ofs p -> pmatch b ofs q. Proof. hammer_hook "ValueDomain" "ValueDomain.pincl_sound". intros. eapply pmatch_ge; eauto. apply pincl_ge; auto. Qed. Definition padd (p: aptr) (n: ptrofs) : aptr := match p with \| Gl id ofs => Gl id (Ptrofs.add ofs n) \| Stk ofs => Stk (Ptrofs.add ofs n) \| _ => p end. Lemma padd_sound: forall b ofs p delta, pmatch b ofs p -> pmatch b (Ptrofs.add ofs delta) (padd p delta). Proof. hammer_hook "ValueDomain" "ValueDomain.padd_sound". intros. inv H; simpl padd; eauto with va. Qed. Definition psub (p: aptr) (n: ptrofs) : aptr := match p with \| Gl id ofs => Gl id (Ptrofs.sub ofs n) \| Stk ofs => Stk (Ptrofs.sub ofs n) \| _ => p end. Lemma psub_sound: forall b ofs p delta, pmatch b ofs p -> pmatch b (Ptrofs.sub ofs delta) (psub p delta). Proof. hammer_hook "ValueDomain" "ValueDomain.psub_sound". intros. inv H; simpl psub; eauto with va. Qed. Definition poffset (p: aptr) : aptr := match p with \| Gl id ofs => Glo id \| Stk ofs => Stack \| _ => p end. Lemma poffset_sound: forall b ofs1 ofs2 p, pmatch b ofs1 p -> pmatch b ofs2 (poffset p). Proof. hammer_hook "ValueDomain" "ValueDomain.poffset_sound". intros. inv H; simpl poffset; eauto with va. Qed. Definition cmp_different_blocks (c: comparison) : abool := match c with \| Ceq => Maybe false \| Cne => Maybe true \| _ => Bnone end. Lemma cmp_different_blocks_none: forall c, cmatch None (cmp_different_blocks c). Proof. hammer_hook "ValueDomain" "ValueDomain.cmp_different_blocks_none". intros; destruct c; constructor. Qed. Lemma cmp_different_blocks_sound: forall c, cmatch (Val.cmp_different_blocks c) (cmp_different_blocks c). Proof. hammer_hook "ValueDomain" "ValueDomain.cmp_different_blocks_sound". intros; destruct c; constructor. Qed. Definition pcmp (c: comparison) (p1 p2: aptr) : abool := match p1, p2 with \| Pbot, _ \| _, Pbot => Bnone \| Gl id1 ofs1, Gl id2 ofs2 => if peq id1 id2 then Maybe (Ptrofs.cmpu c ofs1 ofs2) else cmp_different_blocks c \| Gl id1 ofs1, Glo id2 => if peq id1 id2 then Btop else cmp_different_blocks c \| Glo id1, Gl id2 ofs2 => if peq id1 id2 then Btop else cmp_different_blocks c \| Glo id1, Glo id2 => if peq id1 id2 then Btop else cmp_different_blocks c \| Stk ofs1, Stk ofs2 => Maybe (Ptrofs.cmpu c ofs1 ofs2) \| (Gl _ _ \| Glo _ \| Glob \| Nonstack), (Stk _ \| Stack) => cmp_different_blocks c \| (Stk _ \| Stack), (Gl _ _ \| Glo _ \| Glob \| Nonstack) => cmp_different_blocks c \| _, _ => Btop end. Lemma pcmp_sound: forall valid c b1 ofs1 p1 b2 ofs2 p2, pmatch b1 ofs1 p1 -> pmatch b2 ofs2 p2 -> cmatch (Val.cmpu_bool valid c (Vptr b1 ofs1) (Vptr b2 ofs2)) (pcmp c p1 p2). Proof. hammer_hook "ValueDomain" "ValueDomain.pcmp_sound". intros. assert (DIFF: b1 <> b2 -> cmatch (Val.cmpu_bool valid c (Vptr b1 ofs1) (Vptr b2 ofs2)) (cmp_different_blocks c)). { intros. simpl. rewrite dec_eq_false by assumption. destruct Archi.ptr64. apply cmp_different_blocks_none. destruct (valid b1 (Ptrofs.unsigned ofs1) && valid b2 (Ptrofs.unsigned ofs2)); simpl. apply cmp_different_blocks_sound. apply cmp_different_blocks_none. } assert (SAME: b1 = b2 -> cmatch (Val.cmpu_bool valid c (Vptr b1 ofs1) (Vptr b2 ofs2)) (Maybe (Ptrofs.cmpu c ofs1 ofs2))). { intros. subst b2. simpl. destruct Archi.ptr64. constructor. rewrite dec_eq_true. destruct ((valid b1 (Ptrofs.unsigned ofs1) \|\| valid b1 (Ptrofs.unsigned ofs1 - 1)) && (valid b1 (Ptrofs.unsigned ofs2) \|\| valid b1 (Ptrofs.unsigned ofs2 - 1))); simpl. constructor. constructor. } unfold pcmp; inv H; inv H0; (apply cmatch_top \|\| (apply DIFF; congruence) \|\| idtac). - destruct (peq id id0). subst id0. apply SAME. eapply bc_glob; eauto. auto with va. - destruct (peq id id0); auto with va. - destruct (peq id id0); auto with va. - destruct (peq id id0); auto with va. - apply SAME. eapply bc_stack; eauto. Qed. Lemma pcmp_sound_64: forall valid c b1 ofs1 p1 b2 ofs2 p2, pmatch b1 ofs1 p1 -> pmatch b2 ofs2 p2 -> cmatch (Val.cmplu_bool valid c (Vptr b1 ofs1) (Vptr b2 ofs2)) (pcmp c p1 p2). Proof. hammer_hook "ValueDomain" "ValueDomain.pcmp_sound_64". intros. assert (DIFF: b1 <> b2 -> cmatch (Val.cmplu_bool valid c (Vptr b1 ofs1) (Vptr b2 ofs2)) (cmp_different_blocks c)). { intros. simpl. rewrite dec_eq_false by assumption. destruct Archi.ptr64; simpl. destruct (valid b1 (Ptrofs.unsigned ofs1) && valid b2 (Ptrofs.unsigned ofs2)); simpl. apply cmp_different_blocks_sound. apply cmp_different_blocks_none. apply cmp_different_blocks_none. } assert (SAME: b1 = b2 -> cmatch (Val.cmplu_bool valid c (Vptr b1 ofs1) (Vptr b2 ofs2)) (Maybe (Ptrofs.cmpu c ofs1 ofs2))). { intros. subst b2. simpl. destruct Archi.ptr64. rewrite dec_eq_true. destruct ((valid b1 (Ptrofs.unsigned ofs1) \|\| valid b1 (Ptrofs.unsigned ofs1 - 1)) && (valid b1 (Ptrofs.unsigned ofs2) \|\| valid b1 (Ptrofs.unsigned ofs2 - 1))); simpl. constructor. constructor. constructor. } unfold pcmp; inv H; inv H0; (apply cmatch_top \|\| (apply DIFF; congruence) \|\| idtac). - destruct (peq id id0). subst id0. apply SAME. eapply bc_glob; eauto. auto with va. - destruct (peq id id0); auto with va. - destruct (peq id id0); auto with va. - destruct (peq id id0); auto with va. - apply SAME. eapply bc_stack; eauto. Qed. Lemma pcmp_none: forall c p1 p2, cmatch None (pcmp c p1 p2). Proof. hammer_hook "ValueDomain" "ValueDomain.pcmp_none". intros. unfold pcmp; destruct p1; try constructor; destruct p2; try (destruct (peq id id0)); try constructor; try (apply cmp_different_blocks_none). Qed. Definition pdisjoint (p1: aptr) (sz1: Z) (p2: aptr) (sz2: Z) : bool := match p1, p2 with \| Pbot, _ => true \| _, Pbot => true \| Gl id1 ofs1, Gl id2 ofs2 => if peq id1 id2 then zle (Ptrofs.unsigned ofs1 + sz1) (Ptrofs.unsigned ofs2) \|\| zle (Ptrofs.unsigned ofs2 + sz2) (Ptrofs.unsigned ofs1) else true \| Gl id1 ofs1, Glo id2 => negb(peq id1 id2) \| Glo id1, Gl id2 ofs2 => negb(peq id1 id2) \| Glo id1, Glo id2 => negb(peq id1 id2) \| Stk ofs1, Stk ofs2 => zle (Ptrofs.unsigned ofs1 + sz1) (Ptrofs.unsigned ofs2) \|\| zle (Ptrofs.unsigned ofs2 + sz2) (Ptrofs.unsigned ofs1) \| (Gl _ _ \| Glo _ \| Glob \| Nonstack), (Stk _ \| Stack) => true \| (Stk _ \| Stack), (Gl _ _ \| Glo _ \| Glob \| Nonstack) => true \| _, _ => false end. Lemma pdisjoint_sound: forall sz1 b1 ofs1 p1 sz2 b2 ofs2 p2, pdisjoint p1 sz1 p2 sz2 = true -> pmatch b1 ofs1 p1 -> pmatch b2 ofs2 p2 -> b1 <> b2 \/ Ptrofs.unsigned ofs1 + sz1 <= Ptrofs.unsigned ofs2 \/ Ptrofs.unsigned ofs2 + sz2 <= Ptrofs.unsigned ofs1. Proof. hammer_hook "ValueDomain" "ValueDomain.pdisjoint_sound". intros. inv H0; inv H1; simpl in H; try discriminate; try (left; congruence). - destruct (peq id id0). subst id0. destruct (orb_true_elim _ _ H); InvBooleans; auto. left; congruence. - destruct (peq id id0); try discriminate. left; congruence. - destruct (peq id id0); try discriminate. left; congruence. - destruct (peq id id0); try discriminate. left; congruence. - destruct (orb_true_elim _ _ H); InvBooleans; auto. Qed. Inductive aval : Type := \| Vbot \| I (n: int) \| Uns (p: aptr) (n: Z) \| Sgn (p: aptr) (n: Z) \| L (n: int64) \| F (f: float) \| FS (f: float32) \| Ptr (p: aptr) \| Ifptr (p: aptr). Definition Vtop := Ifptr Ptop. Definition eq_aval: forall (v1 v2: aval), {v1=v2} + {v1<>v2}. Proof. hammer_hook "ValueDomain" "ValueDomain.eq_aval". intros. generalize zeq Int.eq_dec Int64.eq_dec Float.eq_dec Float32.eq_dec eq_aptr; intros. decide equality. Defined. Definition is_uns (n: Z) (i: int) : Prop := forall m, 0 <= m < Int.zwordsize -> m >= n -> Int.testbit i m = false. Definition is_sgn (n: Z) (i: int) : Prop := forall m, 0 <= m < Int.zwordsize -> m >= n - 1 -> Int.testbit i m = Int.testbit i (Int.zwordsize - 1). Inductive vmatch : val -> aval -> Prop := \| vmatch_i: forall i, vmatch (Vint i) (I i) \| vmatch_Uns: forall p i n, 0 <= n -> is_uns n i -> vmatch (Vint i) (Uns p n) \| vmatch_Uns_undef: forall p n, vmatch Vundef (Uns p n) \| vmatch_Sgn: forall p i n, 0 < n -> is_sgn n i -> vmatch (Vint i) (Sgn p n) \| vmatch_Sgn_undef: forall p n, vmatch Vundef (Sgn p n) \| vmatch_l: forall i, vmatch (Vlong i) (L i) \| vmatch_f: forall f, vmatch (Vfloat f) (F f) \| vmatch_s: forall f, vmatch (Vsingle f) (FS f) \| vmatch_ptr: forall b ofs p, pmatch b ofs p -> vmatch (Vptr b ofs) (Ptr p) \| vmatch_ptr_undef: forall p, vmatch Vundef (Ptr p) \| vmatch_ifptr_undef: forall p, vmatch Vundef (Ifptr p) \| vmatch_ifptr_i: forall i p, vmatch (Vint i) (Ifptr p) \| vmatch_ifptr_l: forall i p, vmatch (Vlong i) (Ifptr p) \| vmatch_ifptr_f: forall f p, vmatch (Vfloat f) (Ifptr p) \| vmatch_ifptr_s: forall f p, vmatch (Vsingle f) (Ifptr p) \| vmatch_ifptr_p: forall b ofs p, pmatch b ofs p -> vmatch (Vptr b ofs) (Ifptr p). Lemma vmatch_ifptr: forall v p, (forall b ofs, v = Vptr b ofs -> pmatch b ofs p) -> vmatch v (Ifptr p). Proof. hammer_hook "ValueDomain" "ValueDomain.vmatch_ifptr". intros. destruct v; constructor; auto. Qed. Lemma vmatch_top: forall v x, vmatch v x -> vmatch v Vtop. Proof. hammer_hook "ValueDomain" "ValueDomain.vmatch_top". intros. apply vmatch_ifptr. intros. subst v. inv H; eapply pmatch_top'; eauto. Qed. Hint Extern 1 (vmatch _ _) => constructor : va. Lemma is_uns_mon: forall n1 n2 i, is_uns n1 i -> n1 <= n2 -> is_uns n2 i. Proof. hammer_hook "ValueDomain" "ValueDomain.is_uns_mon". intros; red; intros. apply H; omega. Qed. Lemma is_sgn_mon: forall n1 n2 i, is_sgn n1 i -> n1 <= n2 -> is_sgn n2 i. Proof. hammer_hook "ValueDomain" "ValueDomain.is_sgn_mon". intros; red; intros. apply H; omega. Qed. Lemma is_uns_sgn: forall n1 n2 i, is_uns n1 i -> n1 < n2 -> is_sgn n2 i. Proof. hammer_hook "ValueDomain" "ValueDomain.is_uns_sgn". intros; red; intros. rewrite ! H by omega. auto. Qed. Definition usize := Int.size. Definition ssize (i: int) := Int.size (if Int.lt i Int.zero then Int.not i else i) + 1. Lemma is_uns_usize: forall i, is_uns (usize i) i. Proof. hammer_hook "ValueDomain" "ValueDomain.is_uns_usize". unfold usize; intros; red; intros. apply Int.bits_size_2. omega. Qed. Lemma is_sgn_ssize: forall i, is_sgn (ssize i) i. Proof. hammer_hook "ValueDomain" "ValueDomain.is_sgn_ssize". unfold ssize; intros; red; intros. destruct (Int.lt i Int.zero) eqn:LT. - rewrite <- (negb_involutive (Int.testbit i m)). rewrite <- (negb_involutive (Int.testbit i (Int.zwordsize - 1))). f_equal. generalize (Int.size_range (Int.not i)); intros RANGE. rewrite <- ! Int.bits_not by omega. rewrite ! Int.bits_size_2 by omega. auto. - rewrite ! Int.bits_size_2 by omega. auto. Qed. Lemma is_uns_zero_ext: forall n i, is_uns n i <-> Int.zero_ext n i = i. Proof. hammer_hook "ValueDomain" "ValueDomain.is_uns_zero_ext". intros; split; intros. Int.bit_solve. destruct (zlt i0 n); auto. symmetry; apply H; auto. omega. rewrite <- H. red; intros. rewrite Int.bits_zero_ext by omega. rewrite zlt_false by omega. auto. Qed. Lemma is_sgn_sign_ext: forall n i, 0 < n -> (is_sgn n i <-> Int.sign_ext n i = i). Proof. hammer_hook "ValueDomain" "ValueDomain.is_sgn_sign_ext". intros; split; intros. Int.bit_solve. destruct (zlt i0 n); auto. transitivity (Int.testbit i (Int.zwordsize - 1)). apply H0; omega. symmetry; apply H0; omega. rewrite <- H0. red; intros. rewrite ! Int.bits_sign_ext by omega. f_equal. transitivity (n-1). destruct (zlt m n); omega. destruct (zlt (Int.zwordsize - 1) n); omega. Qed. Lemma is_zero_ext_uns: forall i n m, is_uns m i \/ n <= m -> is_uns m (Int.zero_ext n i). Proof. hammer_hook "ValueDomain" "ValueDomain.is_zero_ext_uns". intros. red; intros. rewrite Int.bits_zero_ext by omega. destruct (zlt m0 n); auto. destruct H. apply H; omega. omegaContradiction. Qed. Lemma is_zero_ext_sgn: forall i n m, n < m -> is_sgn m (Int.zero_ext n i). Proof. hammer_hook "ValueDomain" "ValueDomain.is_zero_ext_sgn". intros. red; intros. rewrite ! Int.bits_zero_ext by omega. transitivity false. apply zlt_false; omega. symmetry; apply zlt_false; omega. Qed. Lemma is_sign_ext_uns: forall i n m, 0 <= m < n -> is_uns m i -> is_uns m (Int.sign_ext n i). Proof. hammer_hook "ValueDomain" "ValueDomain.is_sign_ext_uns". intros; red; intros. rewrite Int.bits_sign_ext by omega. apply H0. destruct (zlt m0 n); omega. destruct (zlt m0 n); omega. Qed. Lemma is_sign_ext_sgn: forall i n m, 0 < n -> 0 < m -> is_sgn m i \/ n <= m -> is_sgn m (Int.sign_ext n i). Proof. hammer_hook "ValueDomain" "ValueDomain.is_sign_ext_sgn". intros. apply is_sgn_sign_ext; auto. destruct (zlt m n). destruct H1. apply is_sgn_sign_ext in H1; auto. rewrite <- H1. rewrite (Int.sign_ext_widen i) by omega. apply Int.sign_ext_idem; auto. omegaContradiction. apply Int.sign_ext_widen; omega. Qed. Hint Resolve is_uns_mon is_sgn_mon is_uns_sgn is_uns_usize is_sgn_ssize : va. Lemma is_uns_1: forall n, is_uns 1 n -> n = Int.zero \/ n = Int.one. Proof. hammer_hook "ValueDomain" "ValueDomain.is_uns_1". intros. destruct (Int.testbit n 0) eqn:B0; [right\|left]; apply Int.same_bits_eq; intros. rewrite Int.bits_one. destruct (zeq i 0). subst i; auto. apply H; omega. rewrite Int.bits_zero. destruct (zeq i 0). subst i; auto. apply H; omega. Qed. Definition provenance (x: aval) : aptr := if va_strict tt then Pbot else match x with \| Ptr p \| Ifptr p \| Uns p _ \| Sgn p _ => poffset p \| _ => Pbot end. Definition ntop : aval := Ifptr Pbot. Definition ntop1 (x: aval) : aval := Ifptr (provenance x). Definition ntop2 (x y: aval) : aval := Ifptr (plub (provenance x) (provenance y)). Definition uns (p: aptr) (n: Z) : aval := if zle n 1 then Uns p 1 else if zle n 7 then Uns p 7 else if zle n 8 then Uns p 8 else if zle n 15 then Uns p 15 else if zle n 16 then Uns p 16 else Ifptr p. Definition sgn (p: aptr) (n: Z) : aval := if zle n 8 then Sgn p 8 else if zle n 16 then Sgn p 16 else Ifptr p. Lemma vmatch_uns': forall p i n, is_uns (Z.max 0 n) i -> vmatch (Vint i) (uns p n). Proof. hammer_hook "ValueDomain" "ValueDomain.vmatch_uns'". intros. assert (A: forall n', n' >= 0 -> n' >= n -> is_uns n' i) by (eauto with va). unfold uns. destruct (zle n 1). auto with va. destruct (zle n 7). auto with va. destruct (zle n 8). auto with va. destruct (zle n 15). auto with va. destruct (zle n 16). auto with va. auto with va. Qed. Lemma vmatch_uns: forall p i n, is_uns n i -> vmatch (Vint i) (uns p n). Proof. hammer_hook "ValueDomain" "ValueDomain.vmatch_uns". intros. apply vmatch_uns'. eauto with va. Qed. Lemma vmatch_uns_undef: forall p n, vmatch Vundef (uns p n). Proof. hammer_hook "ValueDomain" "ValueDomain.vmatch_uns_undef". intros. unfold uns. destruct (zle n 1). auto with va. destruct (zle n 7). auto with va. destruct (zle n 8). auto with va. destruct (zle n 15). auto with va. destruct (zle n 16); auto with va. Qed. Lemma vmatch_sgn': forall p i n, is_sgn (Z.max 1 n) i -> vmatch (Vint i) (sgn p n). Proof. hammer_hook "ValueDomain" "ValueDomain.vmatch_sgn'". intros. assert (A: forall n', n' >= 1 -> n' >= n -> is_sgn n' i) by (eauto with va). unfold sgn. destruct (zle n 8). auto with va. destruct (zle n 16); auto with va. Qed. Lemma vmatch_sgn: forall p i n, is_sgn n i -> vmatch (Vint i) (sgn p n). Proof. hammer_hook "ValueDomain" "ValueDomain.vmatch_sgn". intros. apply vmatch_sgn'. eauto with va. Qed. Lemma vmatch_sgn_undef: forall p n, vmatch Vundef (sgn p n). Proof. hammer_hook "ValueDomain" "ValueDomain.vmatch_sgn_undef". intros. unfold sgn. destruct (zle n 8). auto with va. destruct (zle n 16); auto with va. Qed. Hint Resolve vmatch_uns vmatch_uns_undef vmatch_sgn vmatch_sgn_undef : va. Lemma vmatch_Uns_1: forall p v, vmatch v (Uns p 1) -> v = Vundef \/ v = Vint Int.zero \/ v = Vint Int.one. Proof. hammer_hook "ValueDomain" "ValueDomain.vmatch_Uns_1". intros. inv H; auto. right. exploit is_uns_1; eauto. intuition congruence. Qed. Inductive vge: aval -> aval -> Prop := \| vge_bot: forall v, vge v Vbot \| vge_i: forall i, vge (I i) (I i) \| vge_l: forall i, vge (L i) (L i) \| vge_f: forall f, vge (F f) (F f) \| vge_s: forall f, vge (FS f) (FS f) \| vge_uns_i: forall p n i, 0 <= n -> is_uns n i -> vge (Uns p n) (I i) \| vge_uns_uns: forall p1 n1 p2 n2, n1 >= n2 -> pge p1 p2 -> vge (Uns p1 n1) (Uns p2 n2) \| vge_sgn_i: forall p n i, 0 < n -> is_sgn n i -> vge (Sgn p n) (I i) \| vge_sgn_sgn: forall p1 n1 p2 n2, n1 >= n2 -> pge p1 p2 -> vge (Sgn p1 n1) (Sgn p2 n2) \| vge_sgn_uns: forall p1 n1 p2 n2, n1 > n2 -> pge p1 p2 -> vge (Sgn p1 n1) (Uns p2 n2) \| vge_p_p: forall p q, pge p q -> vge (Ptr p) (Ptr q) \| vge_ip_p: forall p q, pge p q -> vge (Ifptr p) (Ptr q) \| vge_ip_ip: forall p q, pge p q -> vge (Ifptr p) (Ifptr q) \| vge_ip_i: forall p i, vge (Ifptr p) (I i) \| vge_ip_l: forall p i, vge (Ifptr p) (L i) \| vge_ip_f: forall p f, vge (Ifptr p) (F f) \| vge_ip_s: forall p f, vge (Ifptr p) (FS f) \| vge_ip_uns: forall p q n, pge p q -> vge (Ifptr p) (Uns q n) \| vge_ip_sgn: forall p q n, pge p q -> vge (Ifptr p) (Sgn q n). Hint Constructors vge : va. Lemma vge_top: forall v, vge Vtop v. Proof. hammer_hook "ValueDomain" "ValueDomain.vge_top". destruct v; constructor; constructor. Qed. Hint Resolve vge_top : va. Lemma vge_refl: forall v, vge v v. Proof. hammer_hook "ValueDomain" "ValueDomain.vge_refl". destruct v; auto with va. Qed. Lemma vge_trans: forall u v, vge u v -> forall w, vge v w -> vge u w. Proof. hammer_hook "ValueDomain" "ValueDomain.vge_trans". induction 1; intros w V; inv V; eauto using pge_trans with va. Qed. Lemma vmatch_ge: forall v x y, vge x y -> vmatch v y -> vmatch v x. Proof. hammer_hook "ValueDomain" "ValueDomain.vmatch_ge". induction 1; intros V; inv V; eauto using pmatch_ge with va. Qed. Definition vlub (v w: aval) : aval := match v, w with \| Vbot, _ => w \| _, Vbot => v \| I i1, I i2 => if Int.eq i1 i2 then v else if Int.lt i1 Int.zero \|\| Int.lt i2 Int.zero then sgn Pbot (Z.max (ssize i1) (ssize i2)) else uns Pbot (Z.max (usize i1) (usize i2)) \| I i, Uns p n \| Uns p n, I i => if Int.lt i Int.zero then sgn p (Z.max (ssize i) (n + 1)) else uns p (Z.max (usize i) n) \| I i, Sgn p n \| Sgn p n, I i => sgn p (Z.max (ssize i) n) \| I i, (Ptr p \| Ifptr p) \| (Ptr p \| Ifptr p), I i => if va_strict tt \|\| Int.eq i Int.zero then Ifptr p else Vtop \| Uns p1 n1, Uns p2 n2 => Uns (plub p1 p2) (Z.max n1 n2) \| Uns p1 n1, Sgn p2 n2 => sgn (plub p1 p2) (Z.max (n1 + 1) n2) \| Sgn p1 n1, Uns p2 n2 => sgn (plub p1 p2) (Z.max n1 (n2 + 1)) \| Sgn p1 n1, Sgn p2 n2 => sgn (plub p1 p2) (Z.max n1 n2) \| F f1, F f2 => if Float.eq_dec f1 f2 then v else ntop \| FS f1, FS f2 => if Float32.eq_dec f1 f2 then v else ntop \| L i1, L i2 => if Int64.eq i1 i2 then v else ntop \| Ptr p1, Ptr p2 => Ptr(plub p1 p2) \| Ptr p1, Ifptr p2 => Ifptr(plub p1 p2) \| Ifptr p1, Ptr p2 => Ifptr(plub p1 p2) \| Ifptr p1, Ifptr p2 => Ifptr(plub p1 p2) \| (Ptr p1 \| Ifptr p1), (Uns p2 _ \| Sgn p2 _) => Ifptr(plub p1 p2) \| (Uns p1 _ \| Sgn p1 _), (Ptr p2 \| Ifptr p2) => Ifptr(plub p1 p2) \| _, (Ptr p \| Ifptr p) \| (Ptr p \| Ifptr p), _ => if va_strict tt then Ifptr p else Vtop \| _, _ => Vtop end. Lemma vlub_comm: forall v w, vlub v w = vlub w v. Proof. hammer_hook "ValueDomain" "ValueDomain.vlub_comm". intros. unfold vlub; destruct v; destruct w; auto. - rewrite Int.eq_sym. predSpec Int.eq Int.eq_spec n0 n. congruence. rewrite orb_comm. destruct (Int.lt n0 Int.zero \|\| Int.lt n Int.zero); f_equal; apply Z.max_comm. - f_equal. apply plub_comm. apply Z.max_comm. - f_equal. apply plub_comm. apply Z.max_comm. - f_equal; apply plub_comm. - f_equal; apply plub_comm. - f_equal. apply plub_comm. apply Z.max_comm. - f_equal. apply plub_comm. apply Z.max_comm. - f_equal; apply plub_comm. - f_equal; apply plub_comm. - rewrite Int64.eq_sym. predSpec Int64.eq Int64.eq_spec n0 n; congruence. - rewrite dec_eq_sym. destruct (Float.eq_dec f0 f). congruence. auto. - rewrite dec_eq_sym. destruct (Float32.eq_dec f0 f). congruence. auto. - f_equal; apply plub_comm. - f_equal; apply plub_comm. - f_equal; apply plub_comm. - f_equal; apply plub_comm. - f_equal; apply plub_comm. - f_equal; apply plub_comm. - f_equal; apply plub_comm. - f_equal; apply plub_comm. Qed. Lemma vge_uns_uns': forall p n, vge (uns p n) (Uns p n). Proof. hammer_hook "ValueDomain" "ValueDomain.vge_uns_uns'". unfold uns; intros. destruct (zle n 1). auto with va. destruct (zle n 7). auto with va. destruct (zle n 8). auto with va. destruct (zle n 15). auto with va. destruct (zle n 16); auto with va. Qed. Lemma vge_uns_i': forall p n i, 0 <= n -> is_uns n i -> vge (uns p n) (I i). Proof. hammer_hook "ValueDomain" "ValueDomain.vge_uns_i'". intros. apply vge_trans with (Uns p n). apply vge_uns_uns'. auto with va. Qed. Lemma vge_sgn_sgn': forall p n, vge (sgn p n) (Sgn p n). Proof. hammer_hook "ValueDomain" "ValueDomain.vge_sgn_sgn'". unfold sgn; intros. destruct (zle n 8). auto with va. destruct (zle n 16); auto with va. Qed. Lemma vge_sgn_i': forall p n i, 0 < n -> is_sgn n i -> vge (sgn p n) (I i). Proof. hammer_hook "ValueDomain" "ValueDomain.vge_sgn_i'". intros. apply vge_trans with (Sgn p n). apply vge_sgn_sgn'. auto with va. Qed. Hint Resolve vge_uns_uns' vge_uns_i' vge_sgn_sgn' vge_sgn_i' : va. Lemma usize_pos: forall n, 0 <= usize n. Proof. hammer_hook "ValueDomain" "ValueDomain.usize_pos". unfold usize; intros. generalize (Int.size_range n); omega. Qed. Lemma ssize_pos: forall n, 0 < ssize n. Proof. hammer_hook "ValueDomain" "ValueDomain.ssize_pos". unfold ssize; intros. generalize (Int.size_range (if Int.lt n Int.zero then Int.not n else n)); omega. Qed. Lemma vge_lub_l: forall x y, vge (vlub x y) x. Proof. hammer_hook "ValueDomain" "ValueDomain.vge_lub_l". assert (IFSTRICT: forall (cond: bool) x1 x2 y, vge x1 y -> vge x2 y -> vge (if cond then x1 else x2) y). { destruct cond; auto with va. } unfold vlub; destruct x, y; eauto using pge_lub_l with va. - predSpec Int.eq Int.eq_spec n n0. auto with va. destruct (Int.lt n Int.zero \|\| Int.lt n0 Int.zero). apply vge_sgn_i'. generalize (ssize_pos n); xomega. eauto with va. apply vge_uns_i'. generalize (usize_pos n); xomega. eauto with va. - destruct (Int.lt n Int.zero). apply vge_sgn_i'. generalize (ssize_pos n); xomega. eauto with va. apply vge_uns_i'. generalize (usize_pos n); xomega. eauto with va. - apply vge_sgn_i'. generalize (ssize_pos n); xomega. eauto with va. - destruct (Int.lt n0 Int.zero). eapply vge_trans. apply vge_sgn_sgn'. apply vge_trans with (Sgn p (n + 1)); eauto with va. eapply vge_trans. apply vge_uns_uns'. eauto with va. - eapply vge_trans. apply vge_sgn_sgn'. apply vge_trans with (Sgn p (n + 1)); eauto using pge_lub_l with va. - eapply vge_trans. apply vge_sgn_sgn'. eauto with va. - eapply vge_trans. apply vge_sgn_sgn'. eauto using pge_lub_l with va. - eapply vge_trans. apply vge_sgn_sgn'. eauto using pge_lub_l with va. - destruct (Int64.eq n n0); constructor. - destruct (Float.eq_dec f f0); constructor. - destruct (Float32.eq_dec f f0); constructor. Qed. Lemma vge_lub_r: forall x y, vge (vlub x y) y. Proof. hammer_hook "ValueDomain" "ValueDomain.vge_lub_r". intros. rewrite vlub_comm. apply vge_lub_l. Qed. Lemma vmatch_lub_l: forall v x y, vmatch v x -> vmatch v (vlub x y). Proof. hammer_hook "ValueDomain" "ValueDomain.vmatch_lub_l". intros. eapply vmatch_ge; eauto. apply vge_lub_l. Qed. Lemma vmatch_lub_r: forall v x y, vmatch v y -> vmatch v (vlub x y). Proof. hammer_hook "ValueDomain" "ValueDomain.vmatch_lub_r". intros. rewrite vlub_comm. apply vmatch_lub_l; auto. Qed. Definition aptr_of_aval (v: aval) : aptr := match v with \| Ptr p => p \| Ifptr p => p \| _ => if va_strict tt then Pbot else Nonstack end. Lemma match_aptr_of_aval: forall b ofs av, vmatch (Vptr b ofs) av -> pmatch b ofs (aptr_of_aval av). Proof. hammer_hook "ValueDomain" "ValueDomain.match_aptr_of_aval". unfold aptr_of_aval; intros. inv H; auto. Qed. Definition vplub (v: aval) (p: aptr) : aptr := match v with \| Ptr q => plub q p \| Ifptr q => plub q p \| _ => p end. Lemma vmatch_vplub_l: forall v x p, vmatch v x -> vmatch v (Ifptr (vplub x p)). Proof. hammer_hook "ValueDomain" "ValueDomain.vmatch_vplub_l". intros. unfold vplub; inv H; auto with va; constructor; eapply pmatch_lub_l; eauto. Qed. Lemma pmatch_vplub: forall b ofs x p, pmatch b ofs p -> pmatch b ofs (vplub x p). Proof. hammer_hook "ValueDomain" "ValueDomain.pmatch_vplub". intros. assert (DFL: pmatch b ofs (if va_strict tt then p else Ptop)). { destruct (va_strict tt); auto. eapply pmatch_top'; eauto. } unfold vplub; destruct x; auto; apply pmatch_lub_r; auto. Qed. Lemma vmatch_vplub_r: forall v x p, vmatch v (Ifptr p) -> vmatch v (Ifptr (vplub x p)). Proof. hammer_hook "ValueDomain" "ValueDomain.vmatch_vplub_r". intros. apply vmatch_ifptr; intros; subst v. inv H. apply pmatch_vplub; auto. Qed. Definition vpincl (v: aval) (p: aptr) : bool := match v with \| Ptr q \| Ifptr q \| Uns q _ \| Sgn q _ => pincl q p \| _ => true end. Lemma vpincl_ge: forall x p, vpincl x p = true -> vge (Ifptr p) x. Proof. hammer_hook "ValueDomain" "ValueDomain.vpincl_ge". unfold vpincl; intros. destruct x; constructor; apply pincl_ge; auto. Qed. Lemma vpincl_sound: forall v x p, vpincl x p = true -> vmatch v x -> vmatch v (Ifptr p). Proof. hammer_hook "ValueDomain" "ValueDomain.vpincl_sound". intros. apply vmatch_ge with x; auto. apply vpincl_ge; auto. Qed. Definition vincl (v w: aval) : bool := match v, w with \| Vbot, _ => true \| I i, I j => Int.eq_dec i j \| I i, Uns p n => Int.eq_dec (Int.zero_ext n i) i && zle 0 n \| I i, Sgn p n => Int.eq_dec (Int.sign_ext n i) i && zlt 0 n \| Uns p n, Uns q m => zle n m && pincl p q \| Uns p n, Sgn q m => zlt n m && pincl p q \| Sgn p n, Sgn q m => zle n m && pincl p q \| L i, L j => Int64.eq_dec i j \| F i, F j => Float.eq_dec i j \| FS i, FS j => Float32.eq_dec i j \| Ptr p, Ptr q => pincl p q \| (Ptr p \| Ifptr p \| Uns p _ \| Sgn p _), Ifptr q => pincl p q \| _, Ifptr _ => true \| _, _ => false end. Lemma vincl_ge: forall v w, vincl v w = true -> vge w v. Proof. hammer_hook "ValueDomain" "ValueDomain.vincl_ge". unfold vincl; destruct v; destruct w; intros; try discriminate; try InvBooleans; try subst; auto using pincl_ge with va. - constructor; auto. rewrite is_uns_zero_ext; auto. - constructor; auto. rewrite is_sgn_sign_ext; auto. Qed. Definition genv_match (ge: genv) : Prop := (forall id b, Genv.find_symbol ge id = Some b <-> bc b = BCglob id) /\(forall b, Plt b (Genv.genv_next ge) -> bc b <> BCinvalid /\ bc b <> BCstack). Lemma symbol_address_sound: forall ge id ofs, genv_match ge -> vmatch (Genv.symbol_address ge id ofs) (Ptr (Gl id ofs)). Proof. hammer_hook "ValueDomain" "ValueDomain.symbol_address_sound". intros. unfold Genv.symbol_address. destruct (Genv.find_symbol ge id) as [b\|] eqn:F. constructor. constructor. apply H; auto. constructor. Qed. Lemma vmatch_ptr_gl: forall ge v id ofs, genv_match ge -> vmatch v (Ptr (Gl id ofs)) -> Val.lessdef v (Genv.symbol_address ge id ofs). Proof. hammer_hook "ValueDomain" "ValueDomain.vmatch_ptr_gl". intros. unfold Genv.symbol_address. inv H0. - inv H3. replace (Genv.find_symbol ge id) with (Some b). constructor. symmetry. apply H; auto. - constructor. Qed. Lemma vmatch_ptr_stk: forall v ofs sp, vmatch v (Ptr(Stk ofs)) -> bc sp = BCstack -> Val.lessdef v (Vptr sp ofs). Proof. hammer_hook "ValueDomain" "ValueDomain.vmatch_ptr_stk". intros. inv H. - inv H3. replace b with sp by (eapply bc_stack; eauto). constructor. - constructor. Qed. Definition unop_int (sem: int -> int) (x: aval) := match x with I n => I (sem n) \| _ => ntop1 x end. Lemma unop_int_sound: forall sem v x, vmatch v x -> vmatch (match v with Vint i => Vint(sem i) \| _ => Vundef end) (unop_int sem x). Proof. hammer_hook "ValueDomain" "ValueDomain.unop_int_sound". intros. unfold unop_int; inv H; auto with va. Qed. Definition binop_int (sem: int -> int -> int) (x y: aval) := match x, y with I n, I m => I (sem n m) \| _, _ => ntop2 x y end. Lemma binop_int_sound: forall sem v x w y, vmatch v x -> vmatch w y -> vmatch (match v, w with Vint i, Vint j => Vint(sem i j) \| _, _ => Vundef end) (binop_int sem x y). Proof. hammer_hook "ValueDomain" "ValueDomain.binop_int_sound". intros. unfold binop_int; inv H; auto with va; inv H0; auto with va. Qed. Definition unop_long (sem: int64 -> int64) (x: aval) := match x with L n => L (sem n) \| _ => ntop1 x end. Lemma unop_long_sound: forall sem v x, vmatch v x -> vmatch (match v with Vlong i => Vlong(sem i) \| _ => Vundef end) (unop_long sem x). Proof. hammer_hook "ValueDomain" "ValueDomain.unop_long_sound". intros. unfold unop_long; inv H; auto with va. Qed. Definition binop_long (sem: int64 -> int64 -> int64) (x y: aval) := match x, y with L n, L m => L (sem n m) \| _, _ => ntop2 x y end. Lemma binop_long_sound: forall sem v x w y, vmatch v x -> vmatch w y -> vmatch (match v, w with Vlong i, Vlong j => Vlong(sem i j) \| _, _ => Vundef end) (binop_long sem x y). Proof. hammer_hook "ValueDomain" "ValueDomain.binop_long_sound". intros. unfold binop_long; inv H; auto with va; inv H0; auto with va. Qed. Definition unop_float (sem: float -> float) (x: aval) := match x with F n => F (sem n) \| _ => ntop1 x end. Lemma unop_float_sound: forall sem v x, vmatch v x -> vmatch (match v with Vfloat i => Vfloat(sem i) \| _ => Vundef end) (unop_float sem x). Proof. hammer_hook "ValueDomain" "ValueDomain.unop_float_sound". intros. unfold unop_float; inv H; auto with va. Qed. Definition binop_float (sem: float -> float -> float) (x y: aval) := match x, y with F n, F m => F (sem n m) \| _, _ => ntop2 x y end. Lemma binop_float_sound: forall sem v x w y, vmatch v x -> vmatch w y -> vmatch (match v, w with Vfloat i, Vfloat j => Vfloat(sem i j) \| _, _ => Vundef end) (binop_float sem x y). Proof. hammer_hook "ValueDomain" "ValueDomain.binop_float_sound". intros. unfold binop_float; inv H; auto with va; inv H0; auto with va. Qed. Definition unop_single (sem: float32 -> float32) (x: aval) := match x with FS n => FS (sem n) \| _ => ntop1 x end. Lemma unop_single_sound: forall sem v x, vmatch v x -> vmatch (match v with Vsingle i => Vsingle(sem i) \| _ => Vundef end) (unop_single sem x). Proof. hammer_hook "ValueDomain" "ValueDomain.unop_single_sound". intros. unfold unop_single; inv H; auto with va. Qed. Definition binop_single (sem: float32 -> float32 -> float32) (x y: aval) := match x, y with FS n, FS m => FS (sem n m) \| _, _ => ntop2 x y end. Lemma binop_single_sound: forall sem v x w y, vmatch v x -> vmatch w y -> vmatch (match v, w with Vsingle i, Vsingle j => Vsingle(sem i j) \| _, _ => Vundef end) (binop_single sem x y). Proof. hammer_hook "ValueDomain" "ValueDomain.binop_single_sound". intros. unfold binop_single; inv H; auto with va; inv H0; auto with va. Qed. Definition shl (v w: aval) := match w with \| I amount => if Int.ltu amount Int.iwordsize then match v with \| I i => I (Int.shl i amount) \| Uns p n => uns p (n + Int.unsigned amount) \| Sgn p n => sgn p (n + Int.unsigned amount) \| _ => ntop1 v end else ntop1 v \| _ => ntop1 v end. Lemma shl_sound: forall v w x y, vmatch v x -> vmatch w y -> vmatch (Val.shl v w) (shl x y). Proof. hammer_hook "ValueDomain" "ValueDomain.shl_sound". intros. assert (DEFAULT: vmatch (Val.shl v w) (ntop1 x)). { destruct v; destruct w; simpl; try constructor. destruct (Int.ltu i0 Int.iwordsize); constructor. } destruct y; auto. simpl. inv H0. unfold Val.shl. destruct (Int.ltu n Int.iwordsize) eqn:LTU; auto. exploit Int.ltu_inv; eauto. intros RANGE. inv H; auto with va. - apply vmatch_uns'. red; intros. rewrite Int.bits_shl by omega. destruct (zlt m (Int.unsigned n)). auto. apply H1; xomega. - apply vmatch_sgn'. red; intros. zify. rewrite ! Int.bits_shl by omega. rewrite ! zlt_false by omega. rewrite H1 by omega. symmetry. rewrite H1 by omega. auto. - destruct v; constructor. Qed. Definition shru (v w: aval) := match w with \| I amount => if Int.ltu amount Int.iwordsize then match v with \| I i => I (Int.shru i amount) \| Uns p n => uns p (n - Int.unsigned amount) \| _ => uns (provenance v) (Int.zwordsize - Int.unsigned amount) end else ntop1 v \| _ => ntop1 v end. Lemma shru_sound: forall v w x y, vmatch v x -> vmatch w y -> vmatch (Val.shru v w) (shru x y). Proof. hammer_hook "ValueDomain" "ValueDomain.shru_sound". intros. assert (DEFAULT: vmatch (Val.shru v w) (ntop1 x)). { destruct v; destruct w; simpl; try constructor. destruct (Int.ltu i0 Int.iwordsize); constructor. } destruct y; auto. inv H0. unfold shru, Val.shru. destruct (Int.ltu n Int.iwordsize) eqn:LTU; auto. exploit Int.ltu_inv; eauto. intros RANGE. change (Int.unsigned Int.iwordsize) with Int.zwordsize in RANGE. assert (DEFAULT2: forall i, vmatch (Vint (Int.shru i n)) (uns (provenance x) (Int.zwordsize - Int.unsigned n))). { intros. apply vmatch_uns. red; intros. rewrite Int.bits_shru by omega. apply zlt_false. omega. } inv H; auto with va. - apply vmatch_uns'. red; intros. zify. rewrite Int.bits_shru by omega. destruct (zlt (m + Int.unsigned n) Int.zwordsize); auto. apply H1; omega. - destruct v; constructor. Qed. Definition shr (v w: aval) := match w with \| I amount => if Int.ltu amount Int.iwordsize then match v with \| I i => I (Int.shr i amount) \| Uns p n => sgn p (n + 1 - Int.unsigned amount) \| Sgn p n => sgn p (n - Int.unsigned amount) \| _ => sgn (provenance v) (Int.zwordsize - Int.unsigned amount) end else ntop1 v \| _ => ntop1 v end. Lemma shr_sound: forall v w x y, vmatch v x -> vmatch w y -> vmatch (Val.shr v w) (shr x y). Proof. hammer_hook "ValueDomain" "ValueDomain.shr_sound". intros. assert (DEFAULT: vmatch (Val.shr v w) (ntop1 x)). { destruct v; destruct w; simpl; try constructor. destruct (Int.ltu i0 Int.iwordsize); constructor. } destruct y; auto. inv H0. unfold shr, Val.shr. destruct (Int.ltu n Int.iwordsize) eqn:LTU; auto. exploit Int.ltu_inv; eauto. intros RANGE. change (Int.unsigned Int.iwordsize) with Int.zwordsize in RANGE. assert (DEFAULT2: forall i, vmatch (Vint (Int.shr i n)) (sgn (provenance x) (Int.zwordsize - Int.unsigned n))). { intros. apply vmatch_sgn. red; intros. rewrite ! Int.bits_shr by omega. f_equal. destruct (zlt (m + Int.unsigned n) Int.zwordsize); destruct (zlt (Int.zwordsize - 1 + Int.unsigned n) Int.zwordsize); omega. } assert (SGN: forall q i p, is_sgn p i -> 0 < p -> vmatch (Vint (Int.shr i n)) (sgn q (p - Int.unsigned n))). { intros. apply vmatch_sgn'. red; intros. zify. rewrite ! Int.bits_shr by omega. transitivity (Int.testbit i (Int.zwordsize - 1)). destruct (zlt (m + Int.unsigned n) Int.zwordsize). apply H0; omega. auto. symmetry. destruct (zlt (Int.zwordsize - 1 + Int.unsigned n) Int.zwordsize). apply H0; omega. auto. } inv H; eauto with va. - destruct v; constructor. Qed. Definition and (v w: aval) := match v, w with \| I i1, I i2 => I (Int.and i1 i2) \| I i, Uns p n \| Uns p n, I i => uns p (Z.min n (usize i)) \| I i, x \| x, I i => uns (provenance x) (usize i) \| Uns p1 n1, Uns p2 n2 => uns (plub p1 p2) (Z.min n1 n2) \| Uns p n, _ => uns (plub p (provenance w)) n \| _, Uns p n => uns (plub (provenance v) p) n \| Sgn p1 n1, Sgn p2 n2 => sgn (plub p1 p2) (Z.max n1 n2) \| _, _ => ntop2 v w end. Lemma and_sound: forall v w x y, vmatch v x -> vmatch w y -> vmatch (Val.and v w) (and x y). Proof. hammer_hook "ValueDomain" "ValueDomain.and_sound". assert (UNS_l: forall i j n, is_uns n i -> is_uns n (Int.and i j)). { intros; red; intros. rewrite Int.bits_and by auto. rewrite (H m) by auto. apply andb_false_l. } assert (UNS_r: forall i j n, is_uns n i -> is_uns n (Int.and j i)). { intros. rewrite Int.and_commut. eauto. } assert (UNS: forall i j n m, is_uns n i -> is_uns m j -> is_uns (Z.min n m) (Int.and i j)). { intros. apply Z.min_case; auto. } assert (SGN: forall i j n m, is_sgn n i -> is_sgn m j -> is_sgn (Z.max n m) (Int.and i j)). { intros; red; intros. rewrite ! Int.bits_and by auto with va. rewrite H by auto with va. rewrite H0 by auto with va. auto. } intros. unfold and, Val.and; inv H; eauto with va; inv H0; eauto with va. Qed. Definition or (v w: aval) := match v, w with \| I i1, I i2 => I (Int.or i1 i2) \| I i, Uns p n \| Uns p n, I i => uns p (Z.max n (usize i)) \| Uns p1 n1, Uns p2 n2 => uns (plub p1 p2) (Z.max n1 n2) \| Sgn p1 n1, Sgn p2 n2 => sgn (plub p1 p2) (Z.max n1 n2) \| _, _ => ntop2 v w end. Lemma or_sound: forall v w x y, vmatch v x -> vmatch w y -> vmatch (Val.or v w) (or x y). Proof. hammer_hook "ValueDomain" "ValueDomain.or_sound". assert (UNS: forall i j n m, is_uns n i -> is_uns m j -> is_uns (Z.max n m) (Int.or i j)). { intros; red; intros. rewrite Int.bits_or by auto. rewrite H by xomega. rewrite H0 by xomega. auto. } assert (SGN: forall i j n m, is_sgn n i -> is_sgn m j -> is_sgn (Z.max n m) (Int.or i j)). { intros; red; intros. rewrite ! Int.bits_or by xomega. rewrite H by xomega. rewrite H0 by xomega. auto. } intros. unfold or, Val.or; inv H; eauto with va; inv H0; eauto with va. Qed. Definition xor (v w: aval) := match v, w with \| I i1, I i2 => I (Int.xor i1 i2) \| I i, Uns p n \| Uns p n, I i => uns p (Z.max n (usize i)) \| Uns p1 n1, Uns p2 n2 => uns (plub p1 p2) (Z.max n1 n2) \| Sgn p1 n1, Sgn p2 n2 => sgn (plub p1 p2) (Z.max n1 n2) \| _, _ => ntop2 v w end. Lemma xor_sound: forall v w x y, vmatch v x -> vmatch w y -> vmatch (Val.xor v w) (xor x y). Proof. hammer_hook "ValueDomain" "ValueDomain.xor_sound". assert (UNS: forall i j n m, is_uns n i -> is_uns m j -> is_uns (Z.max n m) (Int.xor i j)). { intros; red; intros. rewrite Int.bits_xor by auto. rewrite H by xomega. rewrite H0 by xomega. auto. } assert (SGN: forall i j n m, is_sgn n i -> is_sgn m j -> is_sgn (Z.max n m) (Int.xor i j)). { intros; red; intros. rewrite ! Int.bits_xor by xomega. rewrite H by xomega. rewrite H0 by xomega. auto. } intros. unfold xor, Val.xor; inv H; eauto with va; inv H0; eauto with va. Qed. Definition notint (v: aval) := match v with \| I i => I (Int.not i) \| Uns p n => sgn p (n + 1) \| Sgn p n => Sgn p n \| _ => ntop1 v end. Lemma notint_sound: forall v x, vmatch v x -> vmatch (Val.notint v) (notint x). Proof. hammer_hook "ValueDomain" "ValueDomain.notint_sound". assert (SGN: forall n i, is_sgn n i -> is_sgn n (Int.not i)). { intros; red; intros. rewrite ! Int.bits_not by omega. f_equal. apply H; auto. } intros. unfold Val.notint, notint; inv H; eauto with va. Qed. Definition rol (x y: aval) := match y, x with \| I j, I i => I(Int.rol i j) \| I j, Uns p n => uns p (n + Int.unsigned j) \| I j, Sgn p n => if zlt n Int.zwordsize then sgn p (n + Int.unsigned j) else ntop1 x \| _, _ => ntop1 x end. Lemma rol_sound: forall v w x y, vmatch v x -> vmatch w y -> vmatch (Val.rol v w) (rol x y). Proof. hammer_hook "ValueDomain" "ValueDomain.rol_sound". intros. assert (DEFAULT: forall p, vmatch (Val.rol v w) (Ifptr p)). { destruct v; destruct w; simpl; constructor. } unfold rol; destruct y; try apply DEFAULT; auto. inv H0. unfold Val.rol. inv H; auto with va. - apply vmatch_uns. red; intros. rewrite Int.bits_rol by auto. generalize (Int.unsigned_range n); intros. rewrite Z.mod_small by omega. apply H1. omega. omega. - destruct (zlt n0 Int.zwordsize); auto with va. apply vmatch_sgn. red; intros. rewrite ! Int.bits_rol by omega. generalize (Int.unsigned_range n); intros. rewrite ! Z.mod_small by omega. rewrite H1 by omega. symmetry. rewrite H1 by omega. auto. - destruct (zlt n0 Int.zwordsize); auto with va. Qed. Definition ror (x y: aval) := match y, x with \| I j, I i => I(Int.ror i j) \| _, _ => ntop1 x end. Lemma ror_sound: forall v w x y, vmatch v x -> vmatch w y -> vmatch (Val.ror v w) (ror x y). Proof. hammer_hook "ValueDomain" "ValueDomain.ror_sound". intros. assert (DEFAULT: forall p, vmatch (Val.ror v w) (Ifptr p)). { destruct v; destruct w; simpl; constructor. } unfold ror; destruct y; try apply DEFAULT; auto. inv H0. unfold Val.ror. inv H; auto with va. Qed. Definition rolm (x: aval) (amount mask: int) := and (rol x (I amount)) (I mask). Lemma rolm_sound: forall v x amount mask, vmatch v x -> vmatch (Val.rolm v amount mask) (rolm x amount mask). Proof. hammer_hook "ValueDomain" "ValueDomain.rolm_sound". intros. replace (Val.rolm v amount mask) with (Val.and (Val.rol v (Vint amount)) (Vint mask)). apply and_sound. apply rol_sound. auto. constructor. constructor. destruct v; auto. Qed. Definition neg := unop_int Int.neg. Lemma neg_sound: forall v x, vmatch v x -> vmatch (Val.neg v) (neg x). Proof. hammer_hook "ValueDomain" "ValueDomain.neg_sound". exact ((unop_int_sound Int.neg)). Qed. Definition add (x y: aval) := match x, y with \| I i, I j => I (Int.add i j) \| Ptr p, I i \| I i, Ptr p => Ptr (if Archi.ptr64 then poffset p else padd p (Ptrofs.of_int i)) \| Ptr p, _ \| _, Ptr p => Ptr (poffset p) \| Ifptr p, I i \| I i, Ifptr p => Ifptr (if Archi.ptr64 then poffset p else padd p (Ptrofs.of_int i)) \| Ifptr p, Ifptr q => Ifptr (plub (poffset p) (poffset q)) \| Ifptr p, _ \| _, Ifptr p => Ifptr (poffset p) \| _, _ => ntop2 x y end. Lemma add_sound: forall v w x y, vmatch v x -> vmatch w y -> vmatch (Val.add v w) (add x y). Proof. hammer_hook "ValueDomain" "ValueDomain.add_sound". intros. unfold Val.add, add. destruct Archi.ptr64. - inv H; inv H0; constructor. - inv H; inv H0; constructor; ((apply padd_sound; assumption) \|\| (eapply poffset_sound; eassumption) \|\| idtac). apply pmatch_lub_r. eapply poffset_sound; eauto. apply pmatch_lub_l. eapply poffset_sound; eauto. Qed. Definition sub (v w: aval) := match v, w with \| I i1, I i2 => I (Int.sub i1 i2) \| Ptr p, I i => if Archi.ptr64 then Ifptr (poffset p) else Ptr (psub p (Ptrofs.of_int i)) \| Ptr p, _ => Ifptr (poffset p) \| Ifptr p, I i => if Archi.ptr64 then Ifptr (plub (poffset p) (provenance w)) else Ifptr (psub p (Ptrofs.of_int i)) \| Ifptr p, _ => Ifptr (plub (poffset p) (provenance w)) \| _, _ => ntop2 v w end. Lemma sub_sound: forall v w x y, vmatch v x -> vmatch w y -> vmatch (Val.sub v w) (sub x y). Proof. hammer_hook "ValueDomain" "ValueDomain.sub_sound". intros. unfold Val.sub, sub. destruct Archi.ptr64. - inv H; inv H0; eauto with va. - inv H; inv H0; try (destruct (eq_block b b0)); eauto using psub_sound, poffset_sound, pmatch_lub_l with va. Qed. Definition mul := binop_int Int.mul. Lemma mul_sound: forall v x w y, vmatch v x -> vmatch w y -> vmatch (Val.mul v w) (mul x y). Proof. hammer_hook "ValueDomain" "ValueDomain.mul_sound". exact ((binop_int_sound Int.mul)). Qed. Definition mulhs := binop_int Int.mulhs. Lemma mulhs_sound: forall v x w y, vmatch v x -> vmatch w y -> vmatch (Val.mulhs v w) (mulhs x y). Proof. hammer_hook "ValueDomain" "ValueDomain.mulhs_sound". exact ((binop_int_sound Int.mulhs)). Qed. Definition mulhu := binop_int Int.mulhu. Lemma mulhu_sound: forall v x w y, vmatch v x -> vmatch w y -> vmatch (Val.mulhu v w) (mulhu x y). Proof. hammer_hook "ValueDomain" "ValueDomain.mulhu_sound". exact ((binop_int_sound Int.mulhu)). Qed. Definition divs (v w: aval) := match w, v with \| I i2, I i1 => if Int.eq i2 Int.zero \|\| Int.eq i1 (Int.repr Int.min_signed) && Int.eq i2 Int.mone then if va_strict tt then Vbot else ntop else I (Int.divs i1 i2) \| _, _ => ntop2 v w end. Lemma divs_sound: forall v w u x y, vmatch v x -> vmatch w y -> Val.divs v w = Some u -> vmatch u (divs x y). Proof. hammer_hook "ValueDomain" "ValueDomain.divs_sound". intros. destruct v; destruct w; try discriminate; simpl in H1. destruct (Int.eq i0 Int.zero \|\| Int.eq i (Int.repr Int.min_signed) && Int.eq i0 Int.mone) eqn:E; inv H1. inv H; inv H0; auto with va. simpl. rewrite E. constructor. Qed. Definition divu (v w: aval) := match w, v with \| I i2, I i1 => if Int.eq i2 Int.zero then if va_strict tt then Vbot else ntop else I (Int.divu i1 i2) \| _, _ => ntop2 v w end. Lemma divu_sound: forall v w u x y, vmatch v x -> vmatch w y -> Val.divu v w = Some u -> vmatch u (divu x y). Proof. hammer_hook "ValueDomain" "ValueDomain.divu_sound". intros. destruct v; destruct w; try discriminate; simpl in H1. destruct (Int.eq i0 Int.zero) eqn:E; inv H1. inv H; inv H0; auto with va. simpl. rewrite E. constructor. Qed. Definition mods (v w: aval) := match w, v with \| I i2, I i1 => if Int.eq i2 Int.zero \|\| Int.eq i1 (Int.repr Int.min_signed) && Int.eq i2 Int.mone then if va_strict tt then Vbot else ntop else I (Int.mods i1 i2) \| _, _ => ntop2 v w end. Lemma mods_sound: forall v w u x y, vmatch v x -> vmatch w y -> Val.mods v w = Some u -> vmatch u (mods x y). Proof. hammer_hook "ValueDomain" "ValueDomain.mods_sound". intros. destruct v; destruct w; try discriminate; simpl in H1. destruct (Int.eq i0 Int.zero \|\| Int.eq i (Int.repr Int.min_signed) && Int.eq i0 Int.mone) eqn:E; inv H1. inv H; inv H0; auto with va. simpl. rewrite E. constructor. Qed. Definition modu (v w: aval) := match w, v with \| I i2, I i1 => if Int.eq i2 Int.zero then if va_strict tt then Vbot else ntop else I (Int.modu i1 i2) \| I i2, _ => uns (provenance v) (usize i2) \| _, _ => ntop2 v w end. Lemma modu_sound: forall v w u x y, vmatch v x -> vmatch w y -> Val.modu v w = Some u -> vmatch u (modu x y). Proof. hammer_hook "ValueDomain" "ValueDomain.modu_sound". assert (UNS: forall i j, j <> Int.zero -> is_uns (usize j) (Int.modu i j)). { intros. apply is_uns_mon with (usize (Int.modu i j)); auto with va. unfold usize, Int.size. apply Zsize_monotone. generalize (Int.unsigned_range_2 j); intros RANGE. assert (Int.unsigned j <> 0). { red; intros; elim H. rewrite <- (Int.repr_unsigned j). rewrite H0. auto. } exploit (Z_mod_lt (Int.unsigned i) (Int.unsigned j)). omega. intros MOD. unfold Int.modu. rewrite Int.unsigned_repr. omega. omega. } intros. destruct v; destruct w; try discriminate; simpl in H1. destruct (Int.eq i0 Int.zero) eqn:Z; inv H1. assert (i0 <> Int.zero) by (generalize (Int.eq_spec i0 Int.zero); rewrite Z; auto). unfold modu. inv H; inv H0; auto with va. rewrite Z. constructor. Qed. Definition shrx (v w: aval) := match v, w with \| I i, I j => if Int.ltu j (Int.repr 31) then I(Int.shrx i j) else ntop \| _, _ => ntop1 v end. Lemma shrx_sound: forall v w u x y, vmatch v x -> vmatch w y -> Val.shrx v w = Some u -> vmatch u (shrx x y). Proof. hammer_hook "ValueDomain" "ValueDomain.shrx_sound". intros. destruct v; destruct w; try discriminate; simpl in H1. destruct (Int.ltu i0 (Int.repr 31)) eqn:LTU; inv H1. unfold shrx; inv H; auto with va; inv H0; auto with va. rewrite LTU; auto with va. Qed. Definition shift_long (sem: int64 -> int -> int64) (v w: aval) := match w with \| I amount => if Int.ltu amount Int64.iwordsize' then match v with \| L i => L (sem i amount) \| _ => ntop1 v end else ntop1 v \| _ => ntop1 v end. Lemma shift_long_sound: forall sem v w x y, vmatch v x -> vmatch w y -> vmatch (match v, w with \| Vlong i, Vint j => if Int.ltu j Int64.iwordsize' then Vlong (sem i j) else Vundef \| _, _ => Vundef end) (shift_long sem x y). Proof. hammer_hook "ValueDomain" "ValueDomain.shift_long_sound". intros. assert (DEFAULT: vmatch (match v, w with \| Vlong i, Vint j => if Int.ltu j Int64.iwordsize' then Vlong (sem i j) else Vundef \| _, _ => Vundef end) (ntop1 x)). { destruct v; try constructor; destruct w; try constructor. destruct (Int.ltu i0 Int64.iwordsize'); constructor. } unfold shift_long. destruct y; auto. destruct (Int.ltu n Int64.iwordsize') eqn:LT; auto. destruct x; auto. inv H; inv H0. rewrite LT. constructor. Qed. Definition shll := shift_long Int64.shl'. Lemma shll_sound: forall v w x y, vmatch v x -> vmatch w y -> vmatch (Val.shll v w) (shll x y). Proof. hammer_hook "ValueDomain" "ValueDomain.shll_sound". exact ((shift_long_sound Int64.shl')). Qed. Definition shrl := shift_long Int64.shr'. Lemma shrl_sound: forall v w x y, vmatch v x -> vmatch w y -> vmatch (Val.shrl v w) (shrl x y). Proof. hammer_hook "ValueDomain" "ValueDomain.shrl_sound". exact ((shift_long_sound Int64.shr')). Qed. Definition shrlu := shift_long Int64.shru'. Lemma shrlu_sound: forall v w x y, vmatch v x -> vmatch w y -> vmatch (Val.shrlu v w) (shrlu x y). Proof. hammer_hook "ValueDomain" "ValueDomain.shrlu_sound". exact ((shift_long_sound Int64.shru')). Qed. Definition andl := binop_long Int64.and. Lemma andl_sound: forall v x w y, vmatch v x -> vmatch w y -> vmatch (Val.andl v w) (andl x y). Proof. hammer_hook "ValueDomain" "ValueDomain.andl_sound". exact ((binop_long_sound Int64.and)). Qed. Definition orl := binop_long Int64.or. Lemma orl_sound: forall v x w y, vmatch v x -> vmatch w y -> vmatch (Val.orl v w) (orl x y). Proof. hammer_hook "ValueDomain" "ValueDomain.orl_sound". exact ((binop_long_sound Int64.or)). Qed. Definition xorl := binop_long Int64.xor. Lemma xorl_sound: forall v x w y, vmatch v x -> vmatch w y -> vmatch (Val.xorl v w) (xorl x y). Proof. hammer_hook "ValueDomain" "ValueDomain.xorl_sound". exact ((binop_long_sound Int64.xor)). Qed. Definition notl := unop_long Int64.not. Lemma notl_sound: forall v x, vmatch v x -> vmatch (Val.notl v) (notl x). Proof. hammer_hook "ValueDomain" "ValueDomain.notl_sound". exact ((unop_long_sound Int64.not)). Qed. Definition rotate_long (sem: int64 -> int64 -> int64) (v w: aval) := match v, w with \| L i, I amount => L (sem i (Int64.repr (Int.unsigned amount))) \| _, _ => ntop1 v end. Lemma rotate_long_sound: forall sem v w x y, vmatch v x -> vmatch w y -> vmatch (match v, w with \| Vlong i, Vint j => Vlong (sem i (Int64.repr (Int.unsigned j))) \| _, _ => Vundef end) (rotate_long sem x y). Proof. hammer_hook "ValueDomain" "ValueDomain.rotate_long_sound". intros. assert (DEFAULT: vmatch (match v, w with \| Vlong i, Vint j => Vlong (sem i (Int64.repr (Int.unsigned j))) \| _, _ => Vundef end) (ntop1 x)). { destruct v; try constructor. destruct w; constructor. } unfold rotate_long. destruct x; auto. destruct y; auto. inv H; inv H0. constructor. Qed. Definition roll := rotate_long Int64.rol. Lemma roll_sound: forall v w x y, vmatch v x -> vmatch w y -> vmatch (Val.roll v w) (roll x y). Proof. hammer_hook "ValueDomain" "ValueDomain.roll_sound". exact ((rotate_long_sound Int64.rol)). Qed. Definition rorl := rotate_long Int64.ror. Lemma rorl_sound: forall v w x y, vmatch v x -> vmatch w y -> vmatch (Val.rorl v w) (rorl x y). Proof. hammer_hook "ValueDomain" "ValueDomain.rorl_sound". exact ((rotate_long_sound Int64.ror)). Qed. Definition negl := unop_long Int64.neg. Lemma negl_sound: forall v x, vmatch v x -> vmatch (Val.negl v) (negl x). Proof. hammer_hook "ValueDomain" "ValueDomain.negl_sound". exact ((unop_long_sound Int64.neg)). Qed. Definition addl (x y: aval) := match x, y with \| L i, L j => L (Int64.add i j) \| Ptr p, L i \| L i, Ptr p => Ptr (if Archi.ptr64 then padd p (Ptrofs.of_int64 i) else poffset p) \| Ptr p, _ \| _, Ptr p => Ptr (poffset p) \| Ifptr p, L i \| L i, Ifptr p => Ifptr (if Archi.ptr64 then padd p (Ptrofs.of_int64 i) else poffset p) \| Ifptr p, Ifptr q => Ifptr (plub (poffset p) (poffset q)) \| Ifptr p, _ \| _, Ifptr p => Ifptr (poffset p) \| _, _ => ntop2 x y end. Lemma addl_sound: forall v w x y, vmatch v x -> vmatch w y -> vmatch (Val.addl v w) (addl x y). Proof. hammer_hook "ValueDomain" "ValueDomain.addl_sound". intros. unfold Val.addl, addl. destruct Archi.ptr64. - inv H; inv H0; constructor; ((apply padd_sound; assumption) \|\| (eapply poffset_sound; eassumption) \|\| idtac). apply pmatch_lub_r. eapply poffset_sound; eauto. apply pmatch_lub_l. eapply poffset_sound; eauto. - inv H; inv H0; constructor. Qed. Definition subl (v w: aval) := match v, w with \| L i1, L i2 => L (Int64.sub i1 i2) \| Ptr p, L i => if Archi.ptr64 then Ptr (psub p (Ptrofs.of_int64 i)) else Ifptr (poffset p) \| Ptr p, _ => Ifptr (poffset p) \| Ifptr p, L i => if Archi.ptr64 then Ifptr (psub p (Ptrofs.of_int64 i)) else Ifptr (plub (poffset p) (provenance w)) \| Ifptr p, _ => Ifptr (plub (poffset p) (provenance w)) \| _, _ => ntop2 v w end. Lemma subl_sound: forall v w x y, vmatch v x -> vmatch w y -> vmatch (Val.subl v w) (subl x y). Proof. hammer_hook "ValueDomain" "ValueDomain.subl_sound". intros. unfold Val.subl, subl. destruct Archi.ptr64. - inv H; inv H0; try (destruct (eq_block b b0)); eauto using psub_sound, poffset_sound, pmatch_lub_l with va. - inv H; inv H0; eauto with va. Qed. Definition mull := binop_long Int64.mul. Lemma mull_sound: forall v x w y, vmatch v x -> vmatch w y -> vmatch (Val.mull v w) (mull x y). Proof. hammer_hook "ValueDomain" "ValueDomain.mull_sound". exact ((binop_long_sound Int64.mul)). Qed. Definition mullhs := binop_long Int64.mulhs. Lemma mullhs_sound: forall v x w y, vmatch v x -> vmatch w y -> vmatch (Val.mullhs v w) (mullhs x y). Proof. hammer_hook "ValueDomain" "ValueDomain.mullhs_sound". exact ((binop_long_sound Int64.mulhs)). Qed. Definition mullhu := binop_long Int64.mulhu. Lemma mullhu_sound: forall v x w y, vmatch v x -> vmatch w y -> vmatch (Val.mullhu v w) (mullhu x y). Proof. hammer_hook "ValueDomain" "ValueDomain.mullhu_sound". exact ((binop_long_sound Int64.mulhu)). Qed. Definition divls (v w: aval) := match w, v with \| L i2, L i1 => if Int64.eq i2 Int64.zero \|\| Int64.eq i1 (Int64.repr Int64.min_signed) && Int64.eq i2 Int64.mone then if va_strict tt then Vbot else ntop else L (Int64.divs i1 i2) \| _, _ => ntop2 v w end. Lemma divls_sound: forall v w u x y, vmatch v x -> vmatch w y -> Val.divls v w = Some u -> vmatch u (divls x y). Proof. hammer_hook "ValueDomain" "ValueDomain.divls_sound". intros. destruct v; destruct w; try discriminate; simpl in H1. destruct (Int64.eq i0 Int64.zero \|\| Int64.eq i (Int64.repr Int64.min_signed) && Int64.eq i0 Int64.mone) eqn:E; inv H1. inv H; inv H0; auto with va. simpl. rewrite E. constructor. Qed. Definition divlu (v w: aval) := match w, v with \| L i2, L i1 => if Int64.eq i2 Int64.zero then if va_strict tt then Vbot else ntop else L (Int64.divu i1 i2) \| _, _ => ntop2 v w end. Lemma divlu_sound: forall v w u x y, vmatch v x -> vmatch w y -> Val.divlu v w = Some u -> vmatch u (divlu x y). Proof. hammer_hook "ValueDomain" "ValueDomain.divlu_sound". intros. destruct v; destruct w; try discriminate; simpl in H1. destruct (Int64.eq i0 Int64.zero) eqn:E; inv H1. inv H; inv H0; auto with va. simpl. rewrite E. constructor. Qed. Definition modls (v w: aval) := match w, v with \| L i2, L i1 => if Int64.eq i2 Int64.zero \|\| Int64.eq i1 (Int64.repr Int64.min_signed) && Int64.eq i2 Int64.mone then if va_strict tt then Vbot else ntop else L (Int64.mods i1 i2) \| _, _ => ntop2 v w end. Lemma modls_sound: forall v w u x y, vmatch v x -> vmatch w y -> Val.modls v w = Some u -> vmatch u (modls x y). Proof. hammer_hook "ValueDomain" "ValueDomain.modls_sound". intros. destruct v; destruct w; try discriminate; simpl in H1. destruct (Int64.eq i0 Int64.zero \|\| Int64.eq i (Int64.repr Int64.min_signed) && Int64.eq i0 Int64.mone) eqn:E; inv H1. inv H; inv H0; auto with va. simpl. rewrite E. constructor. Qed. Definition modlu (v w: aval) := match w, v with \| L i2, L i1 => if Int64.eq i2 Int64.zero then if va_strict tt then Vbot else ntop else L (Int64.modu i1 i2) \| _, _ => ntop2 v w end. Lemma modlu_sound: forall v w u x y, vmatch v x -> vmatch w y -> Val.modlu v w = Some u -> vmatch u (modlu x y). Proof. hammer_hook "ValueDomain" "ValueDomain.modlu_sound". intros. destruct v; destruct w; try discriminate; simpl in H1. destruct (Int64.eq i0 Int64.zero) eqn:E; inv H1. inv H; inv H0; auto with va. simpl. rewrite E. constructor. Qed. Definition shrxl (v w: aval) := match v, w with \| L i, I j => if Int.ltu j (Int.repr 63) then L(Int64.shrx' i j) else ntop \| _, _ => ntop1 v end. Lemma shrxl_sound: forall v w u x y, vmatch v x -> vmatch w y -> Val.shrxl v w = Some u -> vmatch u (shrxl x y). Proof. hammer_hook "ValueDomain" "ValueDomain.shrxl_sound". intros. destruct v; destruct w; try discriminate; simpl in H1. destruct (Int.ltu i0 (Int.repr 63)) eqn:LTU; inv H1. unfold shrxl; inv H; auto with va; inv H0; auto with va. rewrite LTU; auto with va. Qed. Definition rolml (x: aval) (amount: int) (mask: int64) := andl (roll x (I amount)) (L mask). Lemma rolml_sound: forall v x amount mask, vmatch v x -> vmatch (Val.rolml v amount mask) (rolml x amount mask). Proof. hammer_hook "ValueDomain" "ValueDomain.rolml_sound". intros. replace (Val.rolml v amount mask) with (Val.andl (Val.roll v (Vint amount)) (Vlong mask)). apply andl_sound. apply roll_sound. auto. constructor. constructor. destruct v; auto. Qed. Definition offset_ptr (v: aval) (n: ptrofs) := match v with \| Ptr p => Ptr (padd p n) \| Ifptr p => Ifptr (padd p n) \| _ => ntop1 v end. Lemma offset_ptr_sound: forall v x n, vmatch v x -> vmatch (Val.offset_ptr v n) (offset_ptr x n). Proof. hammer_hook "ValueDomain" "ValueDomain.offset_ptr_sound". intros. unfold Val.offset_ptr, offset_ptr. inv H; constructor; apply padd_sound; assumption. Qed. Definition negf := unop_float Float.neg. Lemma negf_sound: forall v x, vmatch v x -> vmatch (Val.negf v) (negf x). Proof. hammer_hook "ValueDomain" "ValueDomain.negf_sound". exact ((unop_float_sound Float.neg)). Qed. Definition absf := unop_float Float.abs. Lemma absf_sound: forall v x, vmatch v x -> vmatch (Val.absf v) (absf x). Proof. hammer_hook "ValueDomain" "ValueDomain.absf_sound". exact ((unop_float_sound Float.abs)). Qed. Definition addf := binop_float Float.add. Lemma addf_sound: forall v x w y, vmatch v x -> vmatch w y -> vmatch (Val.addf v w) (addf x y). Proof. hammer_hook "ValueDomain" "ValueDomain.addf_sound". exact ((binop_float_sound Float.add)). Qed. Definition subf := binop_float Float.sub. Lemma subf_sound: forall v x w y, vmatch v x -> vmatch w y -> vmatch (Val.subf v w) (subf x y). Proof. hammer_hook "ValueDomain" "ValueDomain.subf_sound". exact ((binop_float_sound Float.sub)). Qed. Definition mulf := binop_float Float.mul. Lemma mulf_sound: forall v x w y, vmatch v x -> vmatch w y -> vmatch (Val.mulf v w) (mulf x y). Proof. hammer_hook "ValueDomain" "ValueDomain.mulf_sound". exact ((binop_float_sound Float.mul)). Qed. Definition divf := binop_float Float.div. Lemma divf_sound: forall v x w y, vmatch v x -> vmatch w y -> vmatch (Val.divf v w) (divf x y). Proof. hammer_hook "ValueDomain" "ValueDomain.divf_sound". exact ((binop_float_sound Float.div)). Qed. Definition negfs := unop_single Float32.neg. Lemma negfs_sound: forall v x, vmatch v x -> vmatch (Val.negfs v) (negfs x). Proof. hammer_hook "ValueDomain" "ValueDomain.negfs_sound". exact ((unop_single_sound Float32.neg)). Qed. Definition absfs := unop_single Float32.abs. Lemma absfs_sound: forall v x, vmatch v x -> vmatch (Val.absfs v) (absfs x). Proof. hammer_hook "ValueDomain" "ValueDomain.absfs_sound". exact ((unop_single_sound Float32.abs)). Qed. Definition addfs := binop_single Float32.add. Lemma addfs_sound: forall v x w y, vmatch v x -> vmatch w y -> vmatch (Val.addfs v w) (addfs x y). Proof. hammer_hook "ValueDomain" "ValueDomain.addfs_sound". exact ((binop_single_sound Float32.add)). Qed. Definition subfs := binop_single Float32.sub. Lemma subfs_sound: forall v x w y, vmatch v x -> vmatch w y -> vmatch (Val.subfs v w) (subfs x y). Proof. hammer_hook "ValueDomain" "ValueDomain.subfs_sound". exact ((binop_single_sound Float32.sub)). Qed. Definition mulfs := binop_single Float32.mul. Lemma mulfs_sound: forall v x w y, vmatch v x -> vmatch w y -> vmatch (Val.mulfs v w) (mulfs x y). Proof. hammer_hook "ValueDomain" "ValueDomain.mulfs_sound". exact ((binop_single_sound Float32.mul)). Qed. Definition divfs := binop_single Float32.div. Lemma divfs_sound: forall v x w y, vmatch v x -> vmatch w y -> vmatch (Val.divfs v w) (divfs x y). Proof. hammer_hook "ValueDomain" "ValueDomain.divfs_sound". exact ((binop_single_sound Float32.div)). Qed. Definition zero_ext (nbits: Z) (v: aval) := match v with \| I i => I (Int.zero_ext nbits i) \| Uns p n => uns p (Z.min n nbits) \| _ => uns (provenance v) nbits end. Lemma zero_ext_sound: forall nbits v x, vmatch v x -> vmatch (Val.zero_ext nbits v) (zero_ext nbits x). Proof. hammer_hook "ValueDomain" "ValueDomain.zero_ext_sound". assert (DFL: forall nbits i, is_uns nbits (Int.zero_ext nbits i)). { intros; red; intros. rewrite Int.bits_zero_ext by omega. apply zlt_false; auto. } intros. inv H; simpl; auto with va. apply vmatch_uns. red; intros. zify. rewrite Int.bits_zero_ext by omega. destruct (zlt m nbits); auto. apply H1; omega. Qed. Definition sign_ext (nbits: Z) (v: aval) := if zle nbits 0 then Uns (provenance v) 0 else match v with \| I i => I (Int.sign_ext nbits i) \| Uns p n => if zlt n nbits then Uns p n else sgn p nbits \| Sgn p n => sgn p (Z.min n nbits) \| _ => sgn (provenance v) nbits end. Lemma sign_ext_sound: forall nbits v x, vmatch v x -> vmatch (Val.sign_ext nbits v) (sign_ext nbits x). Proof. hammer_hook "ValueDomain" "ValueDomain.sign_ext_sound". assert (DFL: forall p nbits i, 0 < nbits -> vmatch (Vint (Int.sign_ext nbits i)) (sgn p nbits)). { intros. apply vmatch_sgn. apply is_sign_ext_sgn; auto with va. } intros. unfold sign_ext. destruct (zle nbits 0). - destruct v; simpl; auto with va. constructor. omega. rewrite Int.sign_ext_below by auto. red; intros; apply Int.bits_zero. - inv H; simpl; auto with va. + destruct (zlt n nbits); eauto with va. constructor; auto. eapply is_sign_ext_uns; eauto with va. + destruct (zlt n nbits); auto with va. + apply vmatch_sgn. apply is_sign_ext_sgn; auto with va. apply Z.min_case; auto with va. Qed. Definition zero_ext_l (s: Z) := unop_long (Int64.zero_ext s). Lemma zero_ext_l_sound: forall s v x, vmatch v x -> vmatch (Val.zero_ext_l s v) (zero_ext_l s x). Proof. hammer_hook "ValueDomain" "ValueDomain.zero_ext_l_sound". intros s. exact (unop_long_sound (Int64.zero_ext s)). Qed. Definition sign_ext_l (s: Z) := unop_long (Int64.sign_ext s). Lemma sign_ext_l_sound: forall s v x, vmatch v x -> vmatch (Val.sign_ext_l s v) (sign_ext_l s x). Proof. hammer_hook "ValueDomain" "ValueDomain.sign_ext_l_sound". intros s. exact (unop_long_sound (Int64.sign_ext s)). Qed. Definition longofint (v: aval) := match v with \| I i => L (Int64.repr (Int.signed i)) \| _ => ntop1 v end. Lemma longofint_sound: forall v x, vmatch v x -> vmatch (Val.longofint v) (longofint x). Proof. hammer_hook "ValueDomain" "ValueDomain.longofint_sound". unfold Val.longofint, longofint; intros; inv H; auto with va. Qed. Definition longofintu (v: aval) := match v with \| I i => L (Int64.repr (Int.unsigned i)) \| _ => ntop1 v end. Lemma longofintu_sound: forall v x, vmatch v x -> vmatch (Val.longofintu v) (longofintu x). Proof. hammer_hook "ValueDomain" "ValueDomain.longofintu_sound". unfold Val.longofintu, longofintu; intros; inv H; auto with va. Qed. Definition singleoffloat (v: aval) := match v with \| F f => FS (Float.to_single f) \| _ => ntop1 v end. Lemma singleoffloat_sound: forall v x, vmatch v x -> vmatch (Val.singleoffloat v) (singleoffloat x). Proof. hammer_hook "ValueDomain" "ValueDomain.singleoffloat_sound". intros. assert (DEFAULT: vmatch (Val.singleoffloat v) (ntop1 x)). { destruct v; constructor. } destruct x; auto. inv H. constructor. Qed. Definition floatofsingle (v: aval) := match v with \| FS f => F (Float.of_single f) \| _ => ntop1 v end. Lemma floatofsingle_sound: forall v x, vmatch v x -> vmatch (Val.floatofsingle v) (floatofsingle x). Proof. hammer_hook "ValueDomain" "ValueDomain.floatofsingle_sound". intros. assert (DEFAULT: vmatch (Val.floatofsingle v) (ntop1 x)). { destruct v; constructor. } destruct x; auto. inv H. constructor. Qed. Definition intoffloat (x: aval) := match x with \| F f => match Float.to_int f with \| Some i => I i \| None => if va_strict tt then Vbot else ntop end \| _ => ntop1 x end. Lemma intoffloat_sound: forall v x w, vmatch v x -> Val.intoffloat v = Some w -> vmatch w (intoffloat x). Proof. hammer_hook "ValueDomain" "ValueDomain.intoffloat_sound". unfold Val.intoffloat; intros. destruct v; try discriminate. destruct (Float.to_int f) as [i\|] eqn:E; simpl in H0; inv H0. inv H; simpl; auto with va. rewrite E; constructor. Qed. Definition intuoffloat (x: aval) := match x with \| F f => match Float.to_intu f with \| Some i => I i \| None => if va_strict tt then Vbot else ntop end \| _ => ntop1 x end. Lemma intuoffloat_sound: forall v x w, vmatch v x -> Val.intuoffloat v = Some w -> vmatch w (intuoffloat x). Proof. hammer_hook "ValueDomain" "ValueDomain.intuoffloat_sound". unfold Val.intuoffloat; intros. destruct v; try discriminate. destruct (Float.to_intu f) as [i\|] eqn:E; simpl in H0; inv H0. inv H; simpl; auto with va. rewrite E; constructor. Qed. Definition floatofint (x: aval) := match x with \| I i => F(Float.of_int i) \| _ => ntop1 x end. Lemma floatofint_sound: forall v x w, vmatch v x -> Val.floatofint v = Some w -> vmatch w (floatofint x). Proof. hammer_hook "ValueDomain" "ValueDomain.floatofint_sound". unfold Val.floatofint; intros. destruct v; inv H0. inv H; simpl; auto with va. Qed. Definition floatofintu (x: aval) := match x with \| I i => F(Float.of_intu i) \| _ => ntop1 x end. Lemma floatofintu_sound: forall v x w, vmatch v x -> Val.floatofintu v = Some w -> vmatch w (floatofintu x). Proof. hammer_hook "ValueDomain" "ValueDomain.floatofintu_sound". unfold Val.floatofintu; intros. destruct v; inv H0. inv H; simpl; auto with va. Qed. Definition intofsingle (x: aval) := match x with \| FS f => match Float32.to_int f with \| Some i => I i \| None => if va_strict tt then Vbot else ntop end \| _ => ntop1 x end. Lemma intofsingle_sound: forall v x w, vmatch v x -> Val.intofsingle v = Some w -> vmatch w (intofsingle x). Proof. hammer_hook "ValueDomain" "ValueDomain.intofsingle_sound". unfold Val.intofsingle; intros. destruct v; try discriminate. destruct (Float32.to_int f) as [i\|] eqn:E; simpl in H0; inv H0. inv H; simpl; auto with va. rewrite E; constructor. Qed. Definition intuofsingle (x: aval) := match x with \| FS f => match Float32.to_intu f with \| Some i => I i \| None => if va_strict tt then Vbot else ntop end \| _ => ntop1 x end. Lemma intuofsingle_sound: forall v x w, vmatch v x -> Val.intuofsingle v = Some w -> vmatch w (intuofsingle x). Proof. hammer_hook "ValueDomain" "ValueDomain.intuofsingle_sound". unfold Val.intuofsingle; intros. destruct v; try discriminate. destruct (Float32.to_intu f) as [i\|] eqn:E; simpl in H0; inv H0. inv H; simpl; auto with va. rewrite E; constructor. Qed. Definition singleofint (x: aval) := match x with \| I i => FS(Float32.of_int i) \| _ => ntop1 x end. Lemma singleofint_sound: forall v x w, vmatch v x -> Val.singleofint v = Some w -> vmatch w (singleofint x). Proof. hammer_hook "ValueDomain" "ValueDomain.singleofint_sound". unfold Val.singleofint; intros. destruct v; inv H0. inv H; simpl; auto with va. Qed. Definition singleofintu (x: aval) := match x with \| I i => FS(Float32.of_intu i) \| _ => ntop1 x end. Lemma singleofintu_sound: forall v x w, vmatch v x -> Val.singleofintu v = Some w -> vmatch w (singleofintu x). Proof. hammer_hook "ValueDomain" "ValueDomain.singleofintu_sound". unfold Val.singleofintu; intros. destruct v; inv H0. inv H; simpl; auto with va. Qed. Definition longoffloat (x: aval) := match x with \| F f => match Float.to_long f with \| Some i => L i \| None => if va_strict tt then Vbot else ntop end \| _ => ntop1 x end. Lemma longoffloat_sound: forall v x w, vmatch v x -> Val.longoffloat v = Some w -> vmatch w (longoffloat x). Proof. hammer_hook "ValueDomain" "ValueDomain.longoffloat_sound". unfold Val.longoffloat; intros. destruct v; try discriminate. destruct (Float.to_long f) as [i\|] eqn:E; simpl in H0; inv H0. inv H; simpl; auto with va. rewrite E; constructor. Qed. Definition longuoffloat (x: aval) := match x with \| F f => match Float.to_longu f with \| Some i => L i \| None => if va_strict tt then Vbot else ntop end \| _ => ntop1 x end. Lemma longuoffloat_sound: forall v x w, vmatch v x -> Val.longuoffloat v = Some w -> vmatch w (longuoffloat x). Proof. hammer_hook "ValueDomain" "ValueDomain.longuoffloat_sound". unfold Val.longuoffloat; intros. destruct v; try discriminate. destruct (Float.to_longu f) as [i\|] eqn:E; simpl in H0; inv H0. inv H; simpl; auto with va. rewrite E; constructor. Qed. Definition floatoflong (x: aval) := match x with \| L i => F(Float.of_long i) \| _ => ntop1 x end. Lemma floatoflong_sound: forall v x w, vmatch v x -> Val.floatoflong v = Some w -> vmatch w (floatoflong x). Proof. hammer_hook "ValueDomain" "ValueDomain.floatoflong_sound". unfold Val.floatoflong; intros. destruct v; inv H0. inv H; simpl; auto with va. Qed. Definition floatoflongu (x: aval) := match x with \| L i => F(Float.of_longu i) \| _ => ntop1 x end. Lemma floatoflongu_sound: forall v x w, vmatch v x -> Val.floatoflongu v = Some w -> vmatch w (floatoflongu x). Proof. hammer_hook "ValueDomain" "ValueDomain.floatoflongu_sound". unfold Val.floatoflongu; intros. destruct v; inv H0. inv H; simpl; auto with va. Qed. Definition longofsingle (x: aval) := match x with \| FS f => match Float32.to_long f with \| Some i => L i \| None => if va_strict tt then Vbot else ntop end \| _ => ntop1 x end. Lemma longofsingle_sound: forall v x w, vmatch v x -> Val.longofsingle v = Some w -> vmatch w (longofsingle x). Proof. hammer_hook "ValueDomain" "ValueDomain.longofsingle_sound". unfold Val.longofsingle; intros. destruct v; try discriminate. destruct (Float32.to_long f) as [i\|] eqn:E; simpl in H0; inv H0. inv H; simpl; auto with va. rewrite E; constructor. Qed. Definition longuofsingle (x: aval) := match x with \| FS f => match Float32.to_longu f with \| Some i => L i \| None => if va_strict tt then Vbot else ntop end \| _ => ntop1 x end. Lemma longuofsingle_sound: forall v x w, vmatch v x -> Val.longuofsingle v = Some w -> vmatch w (longuofsingle x). Proof. hammer_hook "ValueDomain" "ValueDomain.longuofsingle_sound". unfold Val.longuofsingle; intros. destruct v; try discriminate. destruct (Float32.to_longu f) as [i\|] eqn:E; simpl in H0; inv H0. inv H; simpl; auto with va. rewrite E; constructor. Qed. Definition singleoflong (x: aval) := match x with \| L i => FS(Float32.of_long i) \| _ => ntop1 x end. Lemma singleoflong_sound: forall v x w, vmatch v x -> Val.singleoflong v = Some w -> vmatch w (singleoflong x). Proof. hammer_hook "ValueDomain" "ValueDomain.singleoflong_sound". unfold Val.singleoflong; intros. destruct v; inv H0. inv H; simpl; auto with va. Qed. Definition singleoflongu (x: aval) := match x with \| L i => FS(Float32.of_longu i) \| _ => ntop1 x end. Lemma singleoflongu_sound: forall v x w, vmatch v x -> Val.singleoflongu v = Some w -> vmatch w (singleoflongu x). Proof. hammer_hook "ValueDomain" "ValueDomain.singleoflongu_sound". unfold Val.singleoflongu; intros. destruct v; inv H0. inv H; simpl; auto with va. Qed. Definition floatofwords (x y: aval) := match x, y with \| I i, I j => F(Float.from_words i j) \| _, _ => ntop2 x y end. Lemma floatofwords_sound: forall v w x y, vmatch v x -> vmatch w y -> vmatch (Val.floatofwords v w) (floatofwords x y). Proof. hammer_hook "ValueDomain" "ValueDomain.floatofwords_sound". intros. unfold floatofwords; inv H; simpl; auto with va; inv H0; auto with va. Qed. Definition longofwords (x y: aval) := match y, x with \| I j, I i => L(Int64.ofwords i j) \| _, _ => ntop2 x y end. Lemma longofwords_sound: forall v w x y, vmatch v x -> vmatch w y -> vmatch (Val.longofwords v w) (longofwords x y). Proof. hammer_hook "ValueDomain" "ValueDomain.longofwords_sound". intros. unfold longofwords; inv H0; inv H; simpl; auto with va. Qed. Definition loword (x: aval) := match x with \| L i => I(Int64.loword i) \| _ => ntop1 x end. Lemma loword_sound: forall v x, vmatch v x -> vmatch (Val.loword v) (loword x). Proof. hammer_hook "ValueDomain" "ValueDomain.loword_sound". destruct 1; simpl; auto with va. Qed. Definition hiword (x: aval) := match x with \| L i => I(Int64.hiword i) \| _ => ntop1 x end. Lemma hiword_sound: forall v x, vmatch v x -> vmatch (Val.hiword v) (hiword x). Proof. hammer_hook "ValueDomain" "ValueDomain.hiword_sound". destruct 1; simpl; auto with va. Qed. Definition cmp_intv (c: comparison) (i: Z * Z) (n: Z) : abool := let (lo, hi) := i in match c with \| Ceq => if zlt n lo \|\| zlt hi n then Maybe false else Btop \| Cne => Btop \| Clt => if zlt hi n then Maybe true else if zle n lo then Maybe false else Btop \| Cle => if zle hi n then Maybe true else if zlt n lo then Maybe false else Btop \| Cgt => if zlt n lo then Maybe true else if zle hi n then Maybe false else Btop \| Cge => if zle n lo then Maybe true else if zlt hi n then Maybe false else Btop end. Definition zcmp (c: comparison) (n1 n2: Z) : bool := match c with \| Ceq => zeq n1 n2 \| Cne => negb (zeq n1 n2) \| Clt => zlt n1 n2 \| Cle => zle n1 n2 \| Cgt => zlt n2 n1 \| Cge => zle n2 n1 end. Lemma zcmp_intv_sound: forall c i x n, fst i <= x <= snd i -> cmatch (Some (zcmp c x n)) (cmp_intv c i n). Proof. hammer_hook "ValueDomain" "ValueDomain.zcmp_intv_sound". intros c [lo hi] x n; simpl; intros R. destruct c; unfold zcmp, proj_sumbool. - destruct (zlt n lo). rewrite zeq_false by omega. constructor. destruct (zlt hi n). rewrite zeq_false by omega. constructor. constructor. - constructor. - destruct (zlt hi n). rewrite zlt_true by omega. constructor. destruct (zle n lo). rewrite zlt_false by omega. constructor. constructor. - destruct (zle hi n). rewrite zle_true by omega. constructor. destruct (zlt n lo). rewrite zle_false by omega. constructor. constructor. - destruct (zlt n lo). rewrite zlt_true by omega. constructor. destruct (zle hi n). rewrite zlt_false by omega. constructor. constructor. - destruct (zle n lo). rewrite zle_true by omega. constructor. destruct (zlt hi n). rewrite zle_false by omega. constructor. constructor. Qed. Lemma cmp_intv_None: forall c i n, cmatch None (cmp_intv c i n). Proof. hammer_hook "ValueDomain" "ValueDomain.cmp_intv_None". unfold cmp_intv; intros. destruct i as [lo hi]. destruct c. - destruct (zlt n lo). constructor. destruct (zlt hi n); constructor. - constructor. - destruct (zlt hi n). constructor. destruct (zle n lo); constructor. - destruct (zle hi n). constructor. destruct (zlt n lo); constructor. - destruct (zlt n lo). constructor. destruct (zle hi n); constructor. - destruct (zle n lo). constructor. destruct (zlt hi n); constructor. Qed. Definition uintv (v: aval) : Z * Z := match v with \| I n => (Int.unsigned n, Int.unsigned n) \| Uns _ n => if zlt n Int.zwordsize then (0, two_p n - 1) else (0, Int.max_unsigned) \| _ => (0, Int.max_unsigned) end. Lemma uintv_sound: forall n v, vmatch (Vint n) v -> fst (uintv v) <= Int.unsigned n <= snd (uintv v). Proof. hammer_hook "ValueDomain" "ValueDomain.uintv_sound". intros. inv H; simpl; try (apply Int.unsigned_range_2). - omega. - destruct (zlt n0 Int.zwordsize); simpl. + rewrite is_uns_zero_ext in H2. rewrite <- H2. rewrite Int.zero_ext_mod by omega. exploit (Z_mod_lt (Int.unsigned n) (two_p n0)). apply two_p_gt_ZERO; auto. omega. + apply Int.unsigned_range_2. Qed. Lemma cmpu_intv_sound: forall valid c n1 v1 n2, vmatch (Vint n1) v1 -> cmatch (Val.cmpu_bool valid c (Vint n1) (Vint n2)) (cmp_intv c (uintv v1) (Int.unsigned n2)). Proof. hammer_hook "ValueDomain" "ValueDomain.cmpu_intv_sound". intros. simpl. replace (Int.cmpu c n1 n2) with (zcmp c (Int.unsigned n1) (Int.unsigned n2)). apply zcmp_intv_sound; apply uintv_sound; auto. destruct c; simpl; auto. unfold Int.ltu. destruct (zle (Int.unsigned n1) (Int.unsigned n2)); [rewrite zlt_false\|rewrite zlt_true]; auto; omega. unfold Int.ltu. destruct (zle (Int.unsigned n2) (Int.unsigned n1)); [rewrite zlt_false\|rewrite zlt_true]; auto; omega. Qed. Lemma cmpu_intv_sound_2: forall valid c n1 v1 n2, vmatch (Vint n1) v1 -> cmatch (Val.cmpu_bool valid c (Vint n2) (Vint n1)) (cmp_intv (swap_comparison c) (uintv v1) (Int.unsigned n2)). Proof. hammer_hook "ValueDomain" "ValueDomain.cmpu_intv_sound_2". intros. rewrite <- Val.swap_cmpu_bool. apply cmpu_intv_sound; auto. Qed. Definition sintv (v: aval) : Z * Z := match v with \| I n => (Int.signed n, Int.signed n) \| Uns _ n => if zlt n Int.zwordsize then (0, two_p n - 1) else (Int.min_signed, Int.max_signed) \| Sgn _ n => if zlt n Int.zwordsize then (let x := two_p (n-1) in (-x, x-1)) else (Int.min_signed, Int.max_signed) \| _ => (Int.min_signed, Int.max_signed) end. Lemma sintv_sound: forall n v, vmatch (Vint n) v -> fst (sintv v) <= Int.signed n <= snd (sintv v). Proof. hammer_hook "ValueDomain" "ValueDomain.sintv_sound". intros. inv H; simpl; try (apply Int.signed_range). - omega. - destruct (zlt n0 Int.zwordsize); simpl. + rewrite is_uns_zero_ext in H2. rewrite <- H2. assert (Int.unsigned (Int.zero_ext n0 n) = Int.unsigned n mod two_p n0) by (apply Int.zero_ext_mod; omega). exploit (Z_mod_lt (Int.unsigned n) (two_p n0)). apply two_p_gt_ZERO; auto. intros. replace (Int.signed (Int.zero_ext n0 n)) with (Int.unsigned (Int.zero_ext n0 n)). rewrite H. omega. unfold Int.signed. rewrite zlt_true. auto. assert (two_p n0 <= Int.half_modulus). { change Int.half_modulus with (two_p (Int.zwordsize - 1)). apply two_p_monotone. omega. } omega. + apply Int.signed_range. - destruct (zlt n0 (Int.zwordsize)); simpl. + rewrite is_sgn_sign_ext in H2 by auto. rewrite <- H2. exploit (Int.sign_ext_range n0 n). omega. omega. + apply Int.signed_range. Qed. Lemma cmp_intv_sound: forall c n1 v1 n2, vmatch (Vint n1) v1 -> cmatch (Val.cmp_bool c (Vint n1) (Vint n2)) (cmp_intv c (sintv v1) (Int.signed n2)). Proof. hammer_hook "ValueDomain" "ValueDomain.cmp_intv_sound". intros. simpl. replace (Int.cmp c n1 n2) with (zcmp c (Int.signed n1) (Int.signed n2)). apply zcmp_intv_sound; apply sintv_sound; auto. destruct c; simpl; rewrite ? Int.eq_signed; auto. unfold Int.lt. destruct (zle (Int.signed n1) (Int.signed n2)); [rewrite zlt_false\|rewrite zlt_true]; auto; omega. unfold Int.lt. destruct (zle (Int.signed n2) (Int.signed n1)); [rewrite zlt_false\|rewrite zlt_true]; auto; omega. Qed. Lemma cmp_intv_sound_2: forall c n1 v1 n2, vmatch (Vint n1) v1 -> cmatch (Val.cmp_bool c (Vint n2) (Vint n1)) (cmp_intv (swap_comparison c) (sintv v1) (Int.signed n2)). Proof. hammer_hook "ValueDomain" "ValueDomain.cmp_intv_sound_2". intros. rewrite <- Val.swap_cmp_bool. apply cmp_intv_sound; auto. Qed. Definition cmpu_bool (c: comparison) (v w: aval) : abool := match v, w with \| I i1, I i2 => Just (Int.cmpu c i1 i2) \| Ptr _, I i => if Int.eq i Int.zero then cmp_different_blocks c else Btop \| I i, Ptr _ => if Int.eq i Int.zero then cmp_different_blocks c else Btop \| Ptr p1, Ptr p2 => pcmp c p1 p2 \| _, I i => club (cmp_intv c (uintv v) (Int.unsigned i)) (cmp_different_blocks c) \| I i, _ => club (cmp_intv (swap_comparison c) (uintv w) (Int.unsigned i)) (cmp_different_blocks c) \| _, _ => Btop end. Lemma cmpu_bool_sound: forall valid c v w x y, vmatch v x -> vmatch w y -> cmatch (Val.cmpu_bool valid c v w) (cmpu_bool c x y). Proof. hammer_hook "ValueDomain" "ValueDomain.cmpu_bool_sound". intros. assert (IP: forall i b ofs, cmatch (Val.cmpu_bool valid c (Vint i) (Vptr b ofs)) (cmp_different_blocks c)). { intros. simpl. destruct Archi.ptr64. apply cmp_different_blocks_none. destruct (Int.eq i Int.zero && (valid b (Ptrofs.unsigned ofs) \|\| valid b (Ptrofs.unsigned ofs - 1))). apply cmp_different_blocks_sound. apply cmp_different_blocks_none. } assert (PI: forall i b ofs, cmatch (Val.cmpu_bool valid c (Vptr b ofs) (Vint i)) (cmp_different_blocks c)). { intros. simpl. destruct Archi.ptr64. apply cmp_different_blocks_none. destruct (Int.eq i Int.zero && (valid b (Ptrofs.unsigned ofs) \|\| valid b (Ptrofs.unsigned ofs - 1))). apply cmp_different_blocks_sound. apply cmp_different_blocks_none. } unfold cmpu_bool; inversion H; subst; inversion H0; subst; auto using cmatch_top, cmp_different_blocks_none, pcmp_none, cmatch_lub_l, cmatch_lub_r, pcmp_sound, cmpu_intv_sound, cmpu_intv_sound_2, cmp_intv_None. - constructor. - destruct (Int.eq i Int.zero); auto using cmatch_top. - simpl; destruct (Int.eq i Int.zero); auto using cmatch_top, cmp_different_blocks_none. - destruct (Int.eq i Int.zero); auto using cmatch_top. - simpl; destruct (Int.eq i Int.zero); auto using cmatch_top, cmp_different_blocks_none. Qed. Definition cmp_bool (c: comparison) (v w: aval) : abool := match v, w with \| I i1, I i2 => Just (Int.cmp c i1 i2) \| _, I i => cmp_intv c (sintv v) (Int.signed i) \| I i, _ => cmp_intv (swap_comparison c) (sintv w) (Int.signed i) \| _, _ => Btop end. Lemma cmp_bool_sound: forall c v w x y, vmatch v x -> vmatch w y -> cmatch (Val.cmp_bool c v w) (cmp_bool c x y). Proof. hammer_hook "ValueDomain" "ValueDomain.cmp_bool_sound". intros. unfold cmp_bool; inversion H; subst; inversion H0; subst; auto using cmatch_top, cmp_intv_sound, cmp_intv_sound_2, cmp_intv_None. - constructor. Qed. Definition cmplu_bool (c: comparison) (v w: aval) : abool := match v, w with \| L i1, L i2 => Just (Int64.cmpu c i1 i2) \| Ptr _, L i => if Int64.eq i Int64.zero then cmp_different_blocks c else Btop \| L i, Ptr _ => if Int64.eq i Int64.zero then cmp_different_blocks c else Btop \| Ptr p1, Ptr p2 => pcmp c p1 p2 \| _, _ => Btop end. Lemma cmplu_bool_sound: forall valid c v w x y, vmatch v x -> vmatch w y -> cmatch (Val.cmplu_bool valid c v w) (cmplu_bool c x y). Proof. hammer_hook "ValueDomain" "ValueDomain.cmplu_bool_sound". intros. assert (IP: forall i b ofs, cmatch (Val.cmplu_bool valid c (Vlong i) (Vptr b ofs)) (cmp_different_blocks c)). { intros. simpl. destruct Archi.ptr64; simpl. destruct (Int64.eq i Int64.zero && (valid b (Ptrofs.unsigned ofs) \|\| valid b (Ptrofs.unsigned ofs - 1))). apply cmp_different_blocks_sound. apply cmp_different_blocks_none. apply cmp_different_blocks_none. } assert (PI: forall i b ofs, cmatch (Val.cmplu_bool valid c (Vptr b ofs) (Vlong i)) (cmp_different_blocks c)). { intros. simpl. destruct Archi.ptr64; simpl. destruct (Int64.eq i Int64.zero && (valid b (Ptrofs.unsigned ofs) \|\| valid b (Ptrofs.unsigned ofs - 1))). apply cmp_different_blocks_sound. apply cmp_different_blocks_none. apply cmp_different_blocks_none. } unfold cmplu_bool; inversion H; subst; inversion H0; subst; auto using cmatch_top, cmp_different_blocks_none, pcmp_none, cmatch_lub_l, cmatch_lub_r, pcmp_sound_64. - constructor. - destruct (Int64.eq i Int64.zero); auto using cmatch_top. - simpl; destruct (Int64.eq i Int64.zero); auto using cmatch_top, cmp_different_blocks_none. - destruct (Int64.eq i Int64.zero); auto using cmatch_top. - simpl; destruct (Int64.eq i Int64.zero); auto using cmatch_top, cmp_different_blocks_none. Qed. Definition cmpl_bool (c: comparison) (v w: aval) : abool := match v, w with \| L i1, L i2 => Just (Int64.cmp c i1 i2) \| _, _ => Btop end. Lemma cmpl_bool_sound: forall c v w x y, vmatch v x -> vmatch w y -> cmatch (Val.cmpl_bool c v w) (cmpl_bool c x y). Proof. hammer_hook "ValueDomain" "ValueDomain.cmpl_bool_sound". intros. unfold cmpl_bool; inversion H; subst; inversion H0; subst; auto using cmatch_top. - constructor. Qed. Definition cmpf_bool (c: comparison) (v w: aval) : abool := match v, w with \| F f1, F f2 => Just (Float.cmp c f1 f2) \| _, _ => Btop end. Lemma cmpf_bool_sound: forall c v w x y, vmatch v x -> vmatch w y -> cmatch (Val.cmpf_bool c v w) (cmpf_bool c x y). Proof. hammer_hook "ValueDomain" "ValueDomain.cmpf_bool_sound". intros. inv H; try constructor; inv H0; constructor. Qed. Definition cmpfs_bool (c: comparison) (v w: aval) : abool := match v, w with \| FS f1, FS f2 => Just (Float32.cmp c f1 f2) \| _, _ => Btop end. Lemma cmpfs_bool_sound: forall c v w x y, vmatch v x -> vmatch w y -> cmatch (Val.cmpfs_bool c v w) (cmpfs_bool c x y). Proof. hammer_hook "ValueDomain" "ValueDomain.cmpfs_bool_sound". intros. inv H; try constructor; inv H0; constructor. Qed. Definition maskzero (x: aval) (mask: int) : abool := match x with \| I i => Just (Int.eq (Int.and i mask) Int.zero) \| Uns p n => if Int.eq (Int.zero_ext n mask) Int.zero then Maybe true else Btop \| _ => Btop end. Lemma maskzero_sound: forall mask v x, vmatch v x -> cmatch (Val.maskzero_bool v mask) (maskzero x mask). Proof. hammer_hook "ValueDomain" "ValueDomain.maskzero_sound". intros. inv H; simpl; auto with va. predSpec Int.eq Int.eq_spec (Int.zero_ext n mask) Int.zero; auto with va. replace (Int.and i mask) with Int.zero. rewrite Int.eq_true. constructor. rewrite is_uns_zero_ext in H1. rewrite Int.zero_ext_and in * by auto. rewrite <- H1. rewrite Int.and_assoc. rewrite Int.and_commut in H. rewrite H. rewrite Int.and_zero; auto. destruct (Int.eq (Int.zero_ext n mask) Int.zero); constructor. Qed. Definition of_optbool (ab: abool) : aval := match ab with \| Just b => I (if b then Int.one else Int.zero) \| _ => Uns Pbot 1 end. Lemma of_optbool_sound: forall ob ab, cmatch ob ab -> vmatch (Val.of_optbool ob) (of_optbool ab). Proof. hammer_hook "ValueDomain" "ValueDomain.of_optbool_sound". intros. assert (DEFAULT: vmatch (Val.of_optbool ob) (Uns Pbot 1)). { destruct ob; simpl; auto with va. destruct b; constructor; try omega. change 1 with (usize Int.one). apply is_uns_usize. red; intros. apply Int.bits_zero. } inv H; auto. simpl. destruct b; constructor. Qed. Definition resolve_branch (ab: abool) : option bool := match ab with \| Just b => Some b \| Maybe b => Some b \| _ => None end. Lemma resolve_branch_sound: forall b ab b', cmatch (Some b) ab -> resolve_branch ab = Some b' -> b' = b. Proof. hammer_hook "ValueDomain" "ValueDomain.resolve_branch_sound". intros. inv H; simpl in H0; congruence. Qed. Definition add_undef (x: aval) := match x with \| Vbot => ntop \| I i => if Int.lt i Int.zero then sgn Pbot (ssize i) else uns Pbot (usize i) \| L _ \| F _ \| FS _ => ntop \| _ => x end. Lemma add_undef_sound: forall v x, vmatch v x -> vmatch v (add_undef x). Proof. hammer_hook "ValueDomain" "ValueDomain.add_undef_sound". destruct 1; simpl; auto with va. destruct (Int.lt i Int.zero). apply vmatch_sgn; apply is_sgn_ssize. apply vmatch_uns; apply is_uns_usize. Qed. Lemma add_undef_undef: forall x, vmatch Vundef (add_undef x). Proof. hammer_hook "ValueDomain" "ValueDomain.add_undef_undef". destruct x; simpl; auto with va. destruct (Int.lt n Int.zero); auto with va. Qed. Lemma add_undef_normalize: forall v x ty, vmatch v x -> vmatch (Val.normalize v ty) (add_undef x). Proof. hammer_hook "ValueDomain" "ValueDomain.add_undef_normalize". intros. destruct (Val.lessdef_normalize v ty); auto using add_undef_sound, add_undef_undef. Qed. Definition select (ab: abool) (x y: aval) := match ab with \| Bnone => ntop \| Just b \| Maybe b => add_undef (if b then x else y) \| Btop => add_undef (vlub x y) end. Lemma select_sound: forall ob v w ab x y ty, cmatch ob ab -> vmatch v x -> vmatch w y -> vmatch (Val.select ob v w ty) (select ab x y). Proof. hammer_hook "ValueDomain" "ValueDomain.select_sound". unfold Val.select, select; intros. inv H. - auto with va. - apply add_undef_normalize; destruct b; auto. - apply add_undef_undef. - apply add_undef_normalize; destruct b; auto. - destruct ob as [b\|]. + apply add_undef_normalize. destruct b; [apply vmatch_lub_l\|apply vmatch_lub_r]; auto. + apply add_undef_undef. Qed. Definition vnormalize (chunk: memory_chunk) (v: aval) := match chunk, v with \| _, Vbot => Vbot \| Mint8signed, I i => I (Int.sign_ext 8 i) \| Mint8signed, Uns p n => if zlt n 8 then Uns (provenance v) n else Sgn (provenance v) 8 \| Mint8signed, Sgn p n => Sgn (provenance v) (Z.min n 8) \| Mint8signed, _ => Sgn (provenance v) 8 \| Mint8unsigned, I i => I (Int.zero_ext 8 i) \| Mint8unsigned, Uns p n => Uns (provenance v) (Z.min n 8) \| Mint8unsigned, _ => Uns (provenance v) 8 \| Mint16signed, I i => I (Int.sign_ext 16 i) \| Mint16signed, Uns p n => if zlt n 16 then Uns (provenance v) n else Sgn (provenance v) 16 \| Mint16signed, Sgn p n => Sgn (provenance v) (Z.min n 16) \| Mint16signed, _ => Sgn (provenance v) 16 \| Mint16unsigned, I i => I (Int.zero_ext 16 i) \| Mint16unsigned, Uns p n => Uns (provenance v) (Z.min n 16) \| Mint16unsigned, _ => Uns (provenance v) 16 \| Mint32, (I _ \| Uns _ _ \| Sgn _ _ \| Ifptr _) => v \| Mint32, Ptr p => if Archi.ptr64 then Ifptr p else v \| Mint64, (L _ \| Ifptr _) => v \| Mint64, (Uns p _ \| Sgn p _) => Ifptr p \| Mint64, Ptr p => if Archi.ptr64 then v else Ifptr p \| Mfloat32, FS f => v \| Mfloat64, F f => v \| Many32, (I _ \| Uns _ _ \| Sgn _ _ \| FS _ \| Ifptr _) => v \| Many32, Ptr p => if Archi.ptr64 then Ifptr p else v \| Many64, _ => v \| _, _ => Ifptr (provenance v) end. Lemma vnormalize_sound: forall chunk v x, vmatch v x -> vmatch (Val.load_result chunk v) (vnormalize chunk x). Proof. hammer_hook "ValueDomain" "ValueDomain.vnormalize_sound". unfold Val.load_result, vnormalize; generalize Archi.ptr64; intros ptr64; induction 1; destruct chunk; auto with va. - destruct (zlt n 8); constructor; auto with va. apply is_sign_ext_uns; auto. apply is_sign_ext_sgn; auto with va. - constructor. xomega. apply is_zero_ext_uns. apply Z.min_case; auto with va. - destruct (zlt n 16); constructor; auto with va. apply is_sign_ext_uns; auto. apply is_sign_ext_sgn; auto with va. - constructor. xomega. apply is_zero_ext_uns. apply Z.min_case; auto with va. - destruct (zlt n 8); auto with va. - destruct (zlt n 16); auto with va. - constructor. xomega. apply is_sign_ext_sgn; auto with va. apply Z.min_case; auto with va. - constructor. omega. apply is_zero_ext_uns; auto with va. - constructor. xomega. apply is_sign_ext_sgn; auto with va. apply Z.min_case; auto with va. - constructor. omega. apply is_zero_ext_uns; auto with va. - destruct ptr64; auto with va. - destruct ptr64; auto with va. - destruct ptr64; auto with va. - destruct ptr64; auto with va. - destruct ptr64; auto with va. - destruct ptr64; auto with va. - constructor. omega. apply is_sign_ext_sgn; auto with va. - constructor. omega. apply is_zero_ext_uns; auto with va. - constructor. omega. apply is_sign_ext_sgn; auto with va. - constructor. omega. apply is_zero_ext_uns; auto with va. - destruct ptr64; auto with va. - destruct ptr64; auto with va. - destruct ptr64; auto with va. Qed. Lemma vnormalize_cast: forall chunk m b ofs v p, Mem.load chunk m b ofs = Some v -> vmatch v (Ifptr p) -> vmatch v (vnormalize chunk (Ifptr p)). Proof. hammer_hook "ValueDomain" "ValueDomain.vnormalize_cast". intros. exploit Mem.load_cast; eauto. exploit Mem.load_type; eauto. destruct chunk; simpl; intros. - rewrite H2. destruct v; simpl; constructor. omega. apply is_sign_ext_sgn; auto with va. - rewrite H2. destruct v; simpl; constructor. omega. apply is_zero_ext_uns; auto with va. - rewrite H2. destruct v; simpl; constructor. omega. apply is_sign_ext_sgn; auto with va. - rewrite H2. destruct v; simpl; constructor. omega. apply is_zero_ext_uns; auto with va. - auto. - auto. - destruct v; try contradiction; constructor. - destruct v; try contradiction; constructor. - destruct Archi.ptr64; auto. - auto. Qed. Remark poffset_monotone: forall p q, pge p q -> pge (poffset p) (poffset q). Proof. hammer_hook "ValueDomain" "ValueDomain.poffset_monotone". destruct 1; simpl; auto with va. Qed. Remark provenance_monotone: forall x y, vge x y -> pge (provenance x) (provenance y). Proof. hammer_hook "ValueDomain" "ValueDomain.provenance_monotone". unfold provenance; intros. destruct (va_strict tt). constructor. inv H; auto using poffset_monotone with va. Qed. Lemma vnormalize_monotone: forall chunk x y, vge x y -> vge (vnormalize chunk x) (vnormalize chunk y). Proof with (auto using provenance_monotone with va). hammer_hook "ValueDomain" "ValueDomain.vnormalize_monotone". intros chunk x y V; unfold vnormalize; generalize Archi.ptr64; intro ptr64; inversion V; subst; destruct chunk eqn:C; simpl... - destruct (zlt n 8); constructor... apply is_sign_ext_uns... apply is_sign_ext_sgn... - constructor... apply is_zero_ext_uns... apply Z.min_case... - destruct (zlt n 16); constructor... apply is_sign_ext_uns... apply is_sign_ext_sgn... - constructor... apply is_zero_ext_uns... apply Z.min_case... - unfold provenance; destruct (va_strict tt)... - destruct (zlt n1 8). rewrite zlt_true by omega... destruct (zlt n2 8)... - destruct (zlt n1 16). rewrite zlt_true by omega... destruct (zlt n2 16)... - constructor... apply is_sign_ext_sgn... apply Z.min_case... - constructor... apply is_zero_ext_uns... - constructor... apply is_sign_ext_sgn... apply Z.min_case... - constructor... apply is_zero_ext_uns... - unfold provenance; destruct (va_strict tt)... - destruct (zlt n2 8); constructor... - destruct (zlt n2 16); constructor... - destruct ptr64... - destruct ptr64... - destruct ptr64... - destruct ptr64... - destruct ptr64... - destruct ptr64... - constructor... apply is_sign_ext_sgn... - constructor... apply is_zero_ext_uns... - constructor... apply is_sign_ext_sgn... - constructor... apply is_zero_ext_uns... - unfold provenance; destruct (va_strict tt)... - unfold provenance; destruct (va_strict tt)... - unfold provenance; destruct (va_strict tt)... - unfold provenance; destruct (va_strict tt)... - unfold provenance; destruct (va_strict tt)... - unfold provenance; destruct (va_strict tt)... - unfold provenance; destruct (va_strict tt)... - unfold provenance; destruct (va_strict tt)... - destruct (zlt n 8)... - destruct (zlt n 16)... Qed. Definition val_of_aval (a: aval) : val := match a with \| I n => Vint n \| L n => Vlong n \| F f => Vfloat f \| FS f => Vsingle f \| _ => Vundef end. Definition aval_of_val (v: val) : option aval := match v with \| Vint n => Some (I n) \| Vlong n => Some (L n) \| Vfloat f => Some (F f) \| Vsingle f => Some (FS f) \| _ => None end. Lemma val_of_aval_sound: forall v a, vmatch v a -> Val.lessdef (val_of_aval a) v. Proof. hammer_hook "ValueDomain" "ValueDomain.val_of_aval_sound". destruct 1; simpl; auto. Qed. Corollary list_val_of_aval_sound: forall vl al, list_forall2 vmatch vl al -> Val.lessdef_list (map val_of_aval al) vl. Proof. hammer_hook "ValueDomain" "ValueDomain.list_val_of_aval_sound". induction 1; simpl; constructor; auto using val_of_aval_sound. Qed. Lemma aval_of_val_sound: forall v a, aval_of_val v = Some a -> vmatch v a. Proof. hammer_hook "ValueDomain" "ValueDomain.aval_of_val_sound". intros v a E; destruct v; simpl in E; inv E; constructor. Qed. Inductive acontent : Type := \| ACval (chunk: memory_chunk) (av: aval). Definition eq_acontent : forall (c1 c2: acontent), {c1=c2} + {c1<>c2}. Proof. hammer_hook "ValueDomain" "ValueDomain.eq_acontent". intros. generalize chunk_eq eq_aval. decide equality. Defined. Record ablock : Type := ABlock { ab_contents: ZTree.t acontent; ab_summary: aptr }. Local Notation "a ## b" := (ZTree.get b a) (at level 1). Definition ablock_init (p: aptr) : ablock := {\| ab_contents := ZTree.empty _; ab_summary := p \|}. Definition chunk_compat (chunk chunk': memory_chunk) : bool := match chunk, chunk' with \| (Mint8signed \| Mint8unsigned), (Mint8signed \| Mint8unsigned) => true \| (Mint16signed \| Mint16unsigned), (Mint16signed \| Mint16unsigned) => true \| Mint32, Mint32 => true \| Mfloat32, Mfloat32 => true \| Mint64, Mint64 => true \| Mfloat64, Mfloat64 => true \| Many32, Many32 => true \| Many64, Many64 => true \| _, _ => false end. Definition ablock_load (chunk: memory_chunk) (ab: ablock) (i: Z) : aval := match ab.(ab_contents)##i with \| None => vnormalize chunk (Ifptr ab.(ab_summary)) \| Some (ACval chunk' av) => if chunk_compat chunk chunk' then vnormalize chunk av else vnormalize chunk (Ifptr ab.(ab_summary)) end. Definition ablock_load_anywhere (chunk: memory_chunk) (ab: ablock) : aval := vnormalize chunk (Ifptr ab.(ab_summary)). Function inval_after (lo: Z) (hi: Z) (c: ZTree.t acontent) { wf (Zwf lo) hi } : ZTree.t acontent := if zle lo hi then inval_after lo (hi - 1) (ZTree.remove hi c) else c. Proof. hammer_hook "ValueDomain" "ValueDomain.ablock_load_anywhere". intros; red; omega. apply Zwf_well_founded. Qed. Definition inval_if (hi: Z) (lo: Z) (c: ZTree.t acontent) := match c##lo with \| None => c \| Some (ACval chunk av) => if zle (lo + size_chunk chunk) hi then c else ZTree.remove lo c end. Function inval_before (hi: Z) (lo: Z) (c: ZTree.t acontent) { wf (Zwf_up hi) lo } : ZTree.t acontent := if zlt lo hi then inval_before hi (lo + 1) (inval_if hi lo c) else c. Proof. hammer_hook "ValueDomain" "ValueDomain.inval_if". intros; red; omega. apply Zwf_up_well_founded. Qed. Definition ablock_store (chunk: memory_chunk) (ab: ablock) (i: Z) (av: aval) : ablock := {\| ab_contents := ZTree.set i (ACval chunk av) (inval_before i (i - 7) (inval_after (i + 1) (i + size_chunk chunk - 1) ab.(ab_contents))); ab_summary := vplub av ab.(ab_summary) \|}. Definition ablock_store_anywhere (chunk: memory_chunk) (ab: ablock) (av: aval) : ablock := ablock_init (vplub av ab.(ab_summary)). Definition ablock_loadbytes (ab: ablock) : aptr := ab.(ab_summary). Definition ablock_storebytes (ab: ablock) (p: aptr) (ofs: Z) (sz: Z) := {\| ab_contents := inval_before ofs (ofs - 7) (inval_after ofs (ofs + sz - 1) ab.(ab_contents)); ab_summary := plub p ab.(ab_summary) \|}. Definition ablock_storebytes_anywhere (ab: ablock) (p: aptr) := ablock_init (plub p ab.(ab_summary)). Definition smatch (m: mem) (b: block) (p: aptr) : Prop := (forall chunk ofs v, Mem.load chunk m b ofs = Some v -> vmatch v (Ifptr p)) /\(forall ofs b' ofs' q i, Mem.loadbytes m b ofs 1 = Some (Fragment (Vptr b' ofs') q i :: nil) -> pmatch b' ofs' p). Remark loadbytes_load_ext: forall b m m', (forall ofs n bytes, Mem.loadbytes m' b ofs n = Some bytes -> n >= 0 -> Mem.loadbytes m b ofs n = Some bytes) -> forall chunk ofs v, Mem.load chunk m' b ofs = Some v -> Mem.load chunk m b ofs = Some v. Proof. hammer_hook "ValueDomain" "ValueDomain.loadbytes_load_ext". intros. exploit Mem.load_loadbytes; eauto. intros [bytes [A B]]. exploit Mem.load_valid_access; eauto. intros [C D]. subst v. apply Mem.loadbytes_load; auto. apply H; auto. generalize (size_chunk_pos chunk); omega. Qed. Lemma smatch_ext: forall m b p m', smatch m b p -> (forall ofs n bytes, Mem.loadbytes m' b ofs n = Some bytes -> n >= 0 -> Mem.loadbytes m b ofs n = Some bytes) -> smatch m' b p. Proof. hammer_hook "ValueDomain" "ValueDomain.smatch_ext". intros. destruct H. split; intros. eapply H; eauto. eapply loadbytes_load_ext; eauto. eapply H1; eauto. apply H0; eauto. omega. Qed. Lemma smatch_inv: forall m b p m', smatch m b p -> (forall ofs n, n >= 0 -> Mem.loadbytes m' b ofs n = Mem.loadbytes m b ofs n) -> smatch m' b p. Proof. hammer_hook "ValueDomain" "ValueDomain.smatch_inv". intros. eapply smatch_ext; eauto. intros. rewrite <- H0; eauto. Qed. Lemma smatch_ge: forall m b p q, smatch m b p -> pge q p -> smatch m b q. Proof. hammer_hook "ValueDomain" "ValueDomain.smatch_ge". intros. destruct H as [A B]. split; intros. apply vmatch_ge with (Ifptr p); eauto with va. apply pmatch_ge with p; eauto with va. Qed. Lemma In_loadbytes: forall m b byte n ofs bytes, Mem.loadbytes m b ofs n = Some bytes -> In byte bytes -> exists ofs', ofs <= ofs' < ofs + n /\ Mem.loadbytes m b ofs' 1 = Some(byte :: nil). Proof. hammer_hook "ValueDomain" "ValueDomain.In_loadbytes". intros until n. pattern n. apply well_founded_ind with (R := Zwf 0). - apply Zwf_well_founded. - intros sz REC ofs bytes LOAD IN. destruct (zle sz 0). + rewrite (Mem.loadbytes_empty m b ofs sz) in LOAD by auto. inv LOAD. contradiction. + exploit (Mem.loadbytes_split m b ofs 1 (sz - 1) bytes). replace (1 + (sz - 1)) with sz by omega. auto. omega. omega. intros (bytes1 & bytes2 & LOAD1 & LOAD2 & CONCAT). subst bytes. exploit Mem.loadbytes_length. eexact LOAD1. change (Z.to_nat 1) with 1%nat. intros LENGTH1. rewrite in_app_iff in IN. destruct IN. * destruct bytes1; try discriminate. destruct bytes1; try discriminate. simpl in H. destruct H; try contradiction. subst m0. exists ofs; split. omega. auto. * exploit (REC (sz - 1)). red; omega. eexact LOAD2. auto. intros (ofs' & A & B). exists ofs'; split. omega. auto. Qed. Lemma smatch_loadbytes: forall m b p b' ofs' q i n ofs bytes, Mem.loadbytes m b ofs n = Some bytes -> smatch m b p -> In (Fragment (Vptr b' ofs') q i) bytes -> pmatch b' ofs' p. Proof. hammer_hook "ValueDomain" "ValueDomain.smatch_loadbytes". intros. exploit In_loadbytes; eauto. intros (ofs1 & A & B). eapply H0; eauto. Qed. Lemma loadbytes_provenance: forall m b ofs' byte n ofs bytes, Mem.loadbytes m b ofs n = Some bytes -> Mem.loadbytes m b ofs' 1 = Some (byte :: nil) -> ofs <= ofs' < ofs + n -> In byte bytes. Proof. hammer_hook "ValueDomain" "ValueDomain.loadbytes_provenance". intros until n. pattern n. apply well_founded_ind with (R := Zwf 0). - apply Zwf_well_founded. - intros sz REC ofs bytes LOAD LOAD1 IN. exploit (Mem.loadbytes_split m b ofs 1 (sz - 1) bytes). replace (1 + (sz - 1)) with sz by omega. auto. omega. omega. intros (bytes1 & bytes2 & LOAD3 & LOAD4 & CONCAT). subst bytes. rewrite in_app_iff. destruct (zeq ofs ofs'). + subst ofs'. rewrite LOAD1 in LOAD3; inv LOAD3. left; simpl; auto. + right. eapply (REC (sz - 1)). red; omega. eexact LOAD4. auto. omega. Qed. Lemma storebytes_provenance: forall m b ofs bytes m' b' ofs' b'' ofs'' q i, Mem.storebytes m b ofs bytes = Some m' -> Mem.loadbytes m' b' ofs' 1 = Some (Fragment (Vptr b'' ofs'') q i :: nil) -> In (Fragment (Vptr b'' ofs'') q i) bytes \/ Mem.loadbytes m b' ofs' 1 = Some (Fragment (Vptr b'' ofs'') q i :: nil). Proof. hammer_hook "ValueDomain" "ValueDomain.storebytes_provenance". intros. assert (EITHER: (b' <> b \/ ofs' + 1 <= ofs \/ ofs + Z.of_nat (length bytes) <= ofs') \/ (b' = b /\ ofs <= ofs' < ofs + Z.of_nat (length bytes))). { destruct (eq_block b' b); auto. destruct (zle (ofs' + 1) ofs); auto. destruct (zle (ofs + Z.of_nat (length bytes)) ofs'); auto. right. split. auto. omega. } destruct EITHER as [A \| (A & B)]. - right. rewrite <- H0. symmetry. eapply Mem.loadbytes_storebytes_other; eauto. omega. - subst b'. left. eapply loadbytes_provenance; eauto. eapply Mem.loadbytes_storebytes_same; eauto. Qed. Lemma store_provenance: forall chunk m b ofs v m' b' ofs' b'' ofs'' q i, Mem.store chunk m b ofs v = Some m' -> Mem.loadbytes m' b' ofs' 1 = Some (Fragment (Vptr b'' ofs'') q i :: nil) -> v = Vptr b'' ofs'' /\ (chunk = Mint32 \/ chunk = Many32 \/ chunk = Mint64 \/ chunk = Many64) \/ Mem.loadbytes m b' ofs' 1 = Some (Fragment (Vptr b'' ofs'') q i :: nil). Proof. hammer_hook "ValueDomain" "ValueDomain.store_provenance". intros. exploit storebytes_provenance; eauto. eapply Mem.store_storebytes; eauto. intros [A\|A]; auto. left. generalize (encode_val_shape chunk v). intros ENC; inv ENC. - split; auto. rewrite <- H1 in A; destruct A. + congruence. + exploit H5; eauto. intros (j & P & Q); congruence. - rewrite <- H1 in A; destruct A. + congruence. + exploit H3; eauto. intros [byte P]; congruence. - rewrite <- H1 in A; destruct A. + congruence. + exploit H2; eauto. congruence. Qed. Lemma smatch_store: forall chunk m b ofs v m' b' p av, Mem.store chunk m b ofs v = Some m' -> smatch m b' p -> vmatch v av -> smatch m' b' (vplub av p). Proof. hammer_hook "ValueDomain" "ValueDomain.smatch_store". intros. destruct H0 as [A B]. split. - intros chunk' ofs' v' LOAD. destruct v'; auto with va. exploit Mem.load_pointer_store; eauto. intros [(P & Q & R & S) \| DISJ]. + subst. apply vmatch_vplub_l. auto. + apply vmatch_vplub_r. apply A with (chunk := chunk') (ofs := ofs'). rewrite <- LOAD. symmetry. eapply Mem.load_store_other; eauto. - intros. exploit store_provenance; eauto. intros [[P Q] \| P]. + subst. assert (V: vmatch (Vptr b'0 ofs') (Ifptr (vplub av p))). { apply vmatch_vplub_l. auto. } inv V; auto. + apply pmatch_vplub. eapply B; eauto. Qed. Lemma smatch_storebytes: forall m b ofs bytes m' b' p p', Mem.storebytes m b ofs bytes = Some m' -> smatch m b' p -> (forall b' ofs' q i, In (Fragment (Vptr b' ofs') q i) bytes -> pmatch b' ofs' p') -> smatch m' b' (plub p' p). Proof. hammer_hook "ValueDomain" "ValueDomain.smatch_storebytes". intros. destruct H0 as [A B]. split. - intros. apply vmatch_ifptr. intros bx ofsx EQ; subst v. exploit Mem.load_loadbytes; eauto. intros (bytes' & P & Q). destruct bytes' as [ \| byte1' bytes']. exploit Mem.loadbytes_length; eauto. intros. destruct chunk; discriminate. generalize (decode_val_shape chunk byte1' bytes'). rewrite <- Q. intros DEC; inv DEC; try contradiction. assert (v = Vptr bx ofsx). { destruct H5 as [E\|[E\|[E\|E]]]; rewrite E in H4; destruct v; simpl in H4; try congruence; destruct Archi.ptr64; congruence. } exploit In_loadbytes; eauto. eauto with coqlib. intros (ofs' & X & Y). subst v. exploit storebytes_provenance; eauto. intros [Z \| Z]. apply pmatch_lub_l. eauto. apply pmatch_lub_r. eauto. - intros. exploit storebytes_provenance; eauto. intros [Z \| Z]. apply pmatch_lub_l. eauto. apply pmatch_lub_r. eauto. Qed. Definition bmatch (m: mem) (b: block) (ab: ablock) : Prop := smatch m b ab.(ab_summary) /\ forall chunk ofs v, Mem.load chunk m b ofs = Some v -> vmatch v (ablock_load chunk ab ofs). Lemma bmatch_ext: forall m b ab m', bmatch m b ab -> (forall ofs n bytes, Mem.loadbytes m' b ofs n = Some bytes -> n >= 0 -> Mem.loadbytes m b ofs n = Some bytes) -> bmatch m' b ab. Proof. hammer_hook "ValueDomain" "ValueDomain.bmatch_ext". intros. destruct H as [A B]. split; intros. apply smatch_ext with m; auto. eapply B; eauto. eapply loadbytes_load_ext; eauto. Qed. Lemma bmatch_inv: forall m b ab m', bmatch m b ab -> (forall ofs n, n >= 0 -> Mem.loadbytes m' b ofs n = Mem.loadbytes m b ofs n) -> bmatch m' b ab. Proof. hammer_hook "ValueDomain" "ValueDomain.bmatch_inv". intros. eapply bmatch_ext; eauto. intros. rewrite <- H0; eauto. Qed. Lemma ablock_load_sound: forall chunk m b ofs v ab, Mem.load chunk m b ofs = Some v -> bmatch m b ab -> vmatch v (ablock_load chunk ab ofs). Proof. hammer_hook "ValueDomain" "ValueDomain.ablock_load_sound". intros. destruct H0. eauto. Qed. Lemma ablock_load_anywhere_sound: forall chunk m b ofs v ab, Mem.load chunk m b ofs = Some v -> bmatch m b ab -> vmatch v (ablock_load_anywhere chunk ab). Proof. hammer_hook "ValueDomain" "ValueDomain.ablock_load_anywhere_sound". intros. destruct H0. destruct H0. unfold ablock_load_anywhere. eapply vnormalize_cast; eauto. Qed. Lemma ablock_init_sound: forall m b p, smatch m b p -> bmatch m b (ablock_init p). Proof. hammer_hook "ValueDomain" "ValueDomain.ablock_init_sound". intros; split; auto; intros. unfold ablock_load, ablock_init; simpl. rewrite ZTree.gempty. eapply vnormalize_cast; eauto. eapply H; eauto. Qed. Lemma ablock_store_anywhere_sound: forall chunk m b ofs v m' b' ab av, Mem.store chunk m b ofs v = Some m' -> bmatch m b' ab -> vmatch v av -> bmatch m' b' (ablock_store_anywhere chunk ab av). Proof. hammer_hook "ValueDomain" "ValueDomain.ablock_store_anywhere_sound". intros. destruct H0 as [A B]. unfold ablock_store_anywhere. apply ablock_init_sound. eapply smatch_store; eauto. Qed. Remark inval_after_outside: forall i lo hi c, i < lo \/ i > hi -> (inval_after lo hi c)##i = c##i. Proof. hammer_hook "ValueDomain" "ValueDomain.inval_after_outside". intros until c. functional induction (inval_after lo hi c); intros. rewrite IHt by omega. apply ZTree.gro. unfold ZTree.elt, ZIndexed.t; omega. auto. Qed. Remark inval_after_contents: forall chunk av i lo hi c, (inval_after lo hi c)##i = Some (ACval chunk av) -> c##i = Some (ACval chunk av) /\ (i < lo \/ i > hi). Proof. hammer_hook "ValueDomain" "ValueDomain.inval_after_contents". intros until c. functional induction (inval_after lo hi c); intros. destruct (zeq i hi). subst i. rewrite inval_after_outside in H by omega. rewrite ZTree.grs in H. discriminate. exploit IHt; eauto. intros [A B]. rewrite ZTree.gro in A by auto. split. auto. omega. split. auto. omega. Qed. Remark inval_before_outside: forall i hi lo c, i < lo \/ i >= hi -> (inval_before hi lo c)##i = c##i. Proof. hammer_hook "ValueDomain" "ValueDomain.inval_before_outside". intros until c. functional induction (inval_before hi lo c); intros. rewrite IHt by omega. unfold inval_if. destruct (c##lo) as [[chunk av]\|]; auto. destruct (zle (lo + size_chunk chunk) hi); auto. apply ZTree.gro. unfold ZTree.elt, ZIndexed.t; omega. auto. Qed. Remark inval_before_contents_1: forall i chunk av lo hi c, lo <= i < hi -> (inval_before hi lo c)##i = Some(ACval chunk av) -> c##i = Some(ACval chunk av) /\ i + size_chunk chunk <= hi. Proof. hammer_hook "ValueDomain" "ValueDomain.inval_before_contents_1". intros until c. functional induction (inval_before hi lo c); intros. - destruct (zeq lo i). + subst i. rewrite inval_before_outside in H0 by omega. unfold inval_if in H0. destruct (c##lo) as [[chunk0 v0]\|] eqn:C; try congruence. destruct (zle (lo + size_chunk chunk0) hi). rewrite C in H0; inv H0. auto. rewrite ZTree.grs in H0. congruence. + exploit IHt. omega. auto. intros [A B]; split; auto. unfold inval_if in A. destruct (c##lo) as [[chunk0 v0]\|] eqn:C; auto. destruct (zle (lo + size_chunk chunk0) hi); auto. rewrite ZTree.gro in A; auto. - omegaContradiction. Qed. Lemma max_size_chunk: forall chunk, size_chunk chunk <= 8. Proof. hammer_hook "ValueDomain" "ValueDomain.max_size_chunk". destruct chunk; simpl; omega. Qed. Remark inval_before_contents: forall i c chunk' av' j, (inval_before i (i - 7) c)##j = Some (ACval chunk' av') -> c##j = Some (ACval chunk' av') /\ (j + size_chunk chunk' <= i \/ i <= j). Proof. hammer_hook "ValueDomain" "ValueDomain.inval_before_contents". intros. destruct (zlt j (i - 7)). rewrite inval_before_outside in H by omega. split. auto. left. generalize (max_size_chunk chunk'); omega. destruct (zlt j i). exploit inval_before_contents_1; eauto. omega. tauto. rewrite inval_before_outside in H by omega. split. auto. omega. Qed. Lemma ablock_store_contents: forall chunk ab i av j chunk' av', (ablock_store chunk ab i av).(ab_contents)##j = Some(ACval chunk' av') -> (i = j /\ chunk' = chunk /\ av' = av) \/ (ab.(ab_contents)##j = Some(ACval chunk' av') /\ (j + size_chunk chunk' <= i \/ i + size_chunk chunk <= j)). Proof. hammer_hook "ValueDomain" "ValueDomain.ablock_store_contents". unfold ablock_store; simpl; intros. destruct (zeq i j). subst j. rewrite ZTree.gss in H. inv H; auto. right. rewrite ZTree.gso in H by auto. exploit inval_before_contents; eauto. intros [A B]. exploit inval_after_contents; eauto. intros [C D]. split. auto. omega. Qed. Lemma chunk_compat_true: forall c c', chunk_compat c c' = true -> size_chunk c = size_chunk c' /\ align_chunk c <= align_chunk c' /\ type_of_chunk c = type_of_chunk c'. Proof. hammer_hook "ValueDomain" "ValueDomain.chunk_compat_true". destruct c, c'; intros; try discriminate; simpl; auto with va. Qed. Lemma ablock_store_sound: forall chunk m b ofs v m' ab av, Mem.store chunk m b ofs v = Some m' -> bmatch m b ab -> vmatch v av -> bmatch m' b (ablock_store chunk ab ofs av). Proof. hammer_hook "ValueDomain" "ValueDomain.ablock_store_sound". intros until av; intros STORE BIN VIN. destruct BIN as [BIN1 BIN2]. split. eapply smatch_store; eauto. intros chunk' ofs' v' LOAD. assert (SUMMARY: vmatch v' (vnormalize chunk' (Ifptr (vplub av ab.(ab_summary))))). { exploit smatch_store; eauto. intros [A B]. eapply vnormalize_cast; eauto. } unfold ablock_load. destruct ((ab_contents (ablock_store chunk ab ofs av)) ## ofs') as [[chunk1 av1]\|] eqn:C; auto. destruct (chunk_compat chunk' chunk1) eqn:COMPAT; auto. exploit chunk_compat_true; eauto. intros (U & V & W). exploit ablock_store_contents; eauto. intros [(P & Q & R) \| (P & Q)]. - subst. assert (v' = Val.load_result chunk' v). { exploit Mem.load_store_similar_2; eauto. congruence. } subst v'. apply vnormalize_sound; auto. - assert (Mem.load chunk' m b ofs' = Some v'). { rewrite <- LOAD. symmetry. eapply Mem.load_store_other; eauto. rewrite U. auto. } exploit BIN2; eauto. unfold ablock_load. rewrite P. rewrite COMPAT. auto. Qed. Lemma ablock_loadbytes_sound: forall m b ab b' ofs' q i n ofs bytes, Mem.loadbytes m b ofs n = Some bytes -> bmatch m b ab -> In (Fragment (Vptr b' ofs') q i) bytes -> pmatch b' ofs' (ablock_loadbytes ab). Proof. hammer_hook "ValueDomain" "ValueDomain.ablock_loadbytes_sound". intros. destruct H0. eapply smatch_loadbytes; eauto. Qed. Lemma ablock_storebytes_anywhere_sound: forall m b ofs bytes p m' b' ab, Mem.storebytes m b ofs bytes = Some m' -> (forall b' ofs' q i, In (Fragment (Vptr b' ofs') q i) bytes -> pmatch b' ofs' p) -> bmatch m b' ab -> bmatch m' b' (ablock_storebytes_anywhere ab p). Proof. hammer_hook "ValueDomain" "ValueDomain.ablock_storebytes_anywhere_sound". intros. destruct H1 as [A B]. apply ablock_init_sound. eapply smatch_storebytes; eauto. Qed. Lemma ablock_storebytes_contents: forall ab p i sz j chunk' av', (ablock_storebytes ab p i sz).(ab_contents)##j = Some(ACval chunk' av') -> ab.(ab_contents)##j = Some (ACval chunk' av') /\ (j + size_chunk chunk' <= i \/ i + Z.max sz 0 <= j). Proof. hammer_hook "ValueDomain" "ValueDomain.ablock_storebytes_contents". unfold ablock_storebytes; simpl; intros. exploit inval_before_contents; eauto. clear H. intros [A B]. exploit inval_after_contents; eauto. clear A. intros [C D]. split. auto. xomega. Qed. Lemma ablock_storebytes_sound: forall m b ofs bytes m' p ab sz, Mem.storebytes m b ofs bytes = Some m' -> length bytes = Z.to_nat sz -> (forall b' ofs' q i, In (Fragment (Vptr b' ofs') q i) bytes -> pmatch b' ofs' p) -> bmatch m b ab -> bmatch m' b (ablock_storebytes ab p ofs sz). Proof. hammer_hook "ValueDomain" "ValueDomain.ablock_storebytes_sound". intros until sz; intros STORE LENGTH CONTENTS BM. destruct BM as [BM1 BM2]. split. eapply smatch_storebytes; eauto. intros chunk' ofs' v' LOAD'. assert (SUMMARY: vmatch v' (vnormalize chunk' (Ifptr (plub p ab.(ab_summary))))). { exploit smatch_storebytes; eauto. intros [A B]. eapply vnormalize_cast; eauto. } unfold ablock_load. destruct (ab_contents (ablock_storebytes ab p ofs sz))##ofs' as [[chunk av]\|] eqn:C; auto. destruct (chunk_compat chunk' chunk) eqn:COMPAT; auto. exploit chunk_compat_true; eauto. intros (U & V & W). exploit ablock_storebytes_contents; eauto. intros [A B]. assert (Mem.load chunk' m b ofs' = Some v'). { rewrite <- LOAD'; symmetry. eapply Mem.load_storebytes_other; eauto. rewrite U. rewrite LENGTH. rewrite Z_to_nat_max. right; omega. } exploit BM2; eauto. unfold ablock_load. rewrite A. rewrite COMPAT. auto. Qed. Definition bbeq (ab1 ab2: ablock) : bool := eq_aptr ab1.(ab_summary) ab2.(ab_summary) && ZTree.beq (fun c1 c2 => proj_sumbool (eq_acontent c1 c2)) ab1.(ab_contents) ab2.(ab_contents). Lemma bbeq_load: forall ab1 ab2, bbeq ab1 ab2 = true -> ab1.(ab_summary) = ab2.(ab_summary) /\ (forall chunk i, ablock_load chunk ab1 i = ablock_load chunk ab2 i). Proof. hammer_hook "ValueDomain" "ValueDomain.bbeq_load". unfold bbeq; intros. InvBooleans. split. - unfold ablock_load_anywhere; intros; congruence. - assert (A: forall i, ZTree.get i (ab_contents ab1) = ZTree.get i (ab_contents ab2)). { intros. exploit ZTree.beq_sound; eauto. instantiate (1 := i). destruct (ab_contents ab1)##i, (ab_contents ab2)##i; intros; try contradiction. InvBooleans; subst; auto. auto. } intros. unfold ablock_load. rewrite A, H. destruct (ab_contents ab2)##i; auto. Qed. Lemma bbeq_sound: forall ab1 ab2, bbeq ab1 ab2 = true -> forall m b, bmatch m b ab1 <-> bmatch m b ab2. Proof. hammer_hook "ValueDomain" "ValueDomain.bbeq_sound". intros. exploit bbeq_load; eauto. intros [A B]. unfold bmatch. rewrite A. intuition. rewrite <- B; eauto. rewrite B; eauto. Qed. Definition combine_acontents (c1 c2: option acontent) : option acontent := match c1, c2 with \| Some (ACval chunk1 v1), Some (ACval chunk2 v2) => if chunk_eq chunk1 chunk2 then Some(ACval chunk1 (vlub v1 v2)) else None \| _, _ => None end. Definition blub (ab1 ab2: ablock) : ablock := {\| ab_contents := ZTree.combine combine_acontents ab1.(ab_contents) ab2.(ab_contents); ab_summary := plub ab1.(ab_summary) ab2.(ab_summary) \|}. Lemma smatch_lub_l: forall m b p q, smatch m b p -> smatch m b (plub p q). Proof. hammer_hook "ValueDomain" "ValueDomain.smatch_lub_l". intros. destruct H as [A B]. split; intros. change (vmatch v (vlub (Ifptr p) (Ifptr q))). apply vmatch_lub_l. eapply A; eauto. apply pmatch_lub_l. eapply B; eauto. Qed. Lemma smatch_lub_r: forall m b p q, smatch m b q -> smatch m b (plub p q). Proof. hammer_hook "ValueDomain" "ValueDomain.smatch_lub_r". intros. destruct H as [A B]. split; intros. change (vmatch v (vlub (Ifptr p) (Ifptr q))). apply vmatch_lub_r. eapply A; eauto. apply pmatch_lub_r. eapply B; eauto. Qed. Lemma bmatch_lub_l: forall m b x y, bmatch m b x -> bmatch m b (blub x y). Proof. hammer_hook "ValueDomain" "ValueDomain.bmatch_lub_l". intros. destruct H as [BM1 BM2]. split; unfold blub; simpl. - apply smatch_lub_l; auto. - intros. assert (SUMMARY: vmatch v (vnormalize chunk (Ifptr (plub (ab_summary x) (ab_summary y)))) ). { exploit smatch_lub_l; eauto. instantiate (1 := ab_summary y). intros [SUMM _]. eapply vnormalize_cast; eauto. } exploit BM2; eauto. unfold ablock_load; simpl. rewrite ZTree.gcombine by auto. unfold combine_acontents; destruct (ab_contents x)##ofs as [[chunkx avx]\|], (ab_contents y)##ofs as [[chunky avy]\|]; auto. destruct (chunk_eq chunkx chunky); auto. subst chunky. destruct (chunk_compat chunk chunkx); auto. intros. eapply vmatch_ge; eauto. apply vnormalize_monotone. apply vge_lub_l. Qed. Lemma bmatch_lub_r: forall m b x y, bmatch m b y -> bmatch m b (blub x y). Proof. hammer_hook "ValueDomain" "ValueDomain.bmatch_lub_r". intros. destruct H as [BM1 BM2]. split; unfold blub; simpl. - apply smatch_lub_r; auto. - intros. assert (SUMMARY: vmatch v (vnormalize chunk (Ifptr (plub (ab_summary x) (ab_summary y)))) ). { exploit smatch_lub_r; eauto. instantiate (1 := ab_summary x). intros [SUMM _]. eapply vnormalize_cast; eauto. } exploit BM2; eauto. unfold ablock_load; simpl. rewrite ZTree.gcombine by auto. unfold combine_acontents; destruct (ab_contents x)##ofs as [[chunkx avx]\|], (ab_contents y)##ofs as [[chunky avy]\|]; auto. destruct (chunk_eq chunkx chunky); auto. subst chunky. destruct (chunk_compat chunk chunkx); auto. intros. eapply vmatch_ge; eauto. apply vnormalize_monotone. apply vge_lub_r. Qed. Definition romem := PTree.t ablock. Definition romatch (m: mem) (rm: romem) : Prop := forall b id ab, bc b = BCglob id -> rm!id = Some ab -> pge Glob ab.(ab_summary) /\ bmatch m b ab /\ forall ofs, ~Mem.perm m b ofs Max Writable. Lemma romatch_store: forall chunk m b ofs v m' rm, Mem.store chunk m b ofs v = Some m' -> romatch m rm -> romatch m' rm. Proof. hammer_hook "ValueDomain" "ValueDomain.romatch_store". intros; red; intros. exploit H0; eauto. intros (A & B & C). split; auto. split. - exploit Mem.store_valid_access_3; eauto. intros [P _]. apply bmatch_inv with m; auto. + intros. eapply Mem.loadbytes_store_other; eauto. left. red; intros; subst b0. elim (C ofs). apply Mem.perm_cur_max. apply P. generalize (size_chunk_pos chunk); omega. - intros; red; intros; elim (C ofs0). eauto with mem. Qed. Lemma romatch_storebytes: forall m b ofs bytes m' rm, Mem.storebytes m b ofs bytes = Some m' -> romatch m rm -> romatch m' rm. Proof. hammer_hook "ValueDomain" "ValueDomain.romatch_storebytes". intros; red; intros. exploit H0; eauto. intros (A & B & C). split; auto. split. - apply bmatch_inv with m; auto. intros. eapply Mem.loadbytes_storebytes_disjoint; eauto. destruct (eq_block b0 b); auto. subst b0. right; red; unfold Intv.In; simpl; red; intros. elim (C x). apply Mem.perm_cur_max. eapply Mem.storebytes_range_perm; eauto. - intros; red; intros; elim (C ofs0). eauto with mem. Qed. Lemma romatch_ext: forall m rm m', romatch m rm -> (forall b id ofs n bytes, bc b = BCglob id -> Mem.loadbytes m' b ofs n = Some bytes -> Mem.loadbytes m b ofs n = Some bytes) -> (forall b id ofs p, bc b = BCglob id -> Mem.perm m' b ofs Max p -> Mem.perm m b ofs Max p) -> romatch m' rm. Proof. hammer_hook "ValueDomain" "ValueDomain.romatch_ext". intros; red; intros. exploit H; eauto. intros (A & B & C). split. auto. split. apply bmatch_ext with m; auto. intros. eapply H0; eauto. intros; red; intros. elim (C ofs). eapply H1; eauto. Qed. Lemma romatch_free: forall m b lo hi m' rm, Mem.free m b lo hi = Some m' -> romatch m rm -> romatch m' rm. Proof. hammer_hook "ValueDomain" "ValueDomain.romatch_free". intros. apply romatch_ext with m; auto. intros. eapply Mem.loadbytes_free_2; eauto. intros. eauto with mem. Qed. Lemma romatch_alloc: forall m b lo hi m' rm, Mem.alloc m lo hi = (m', b) -> bc_below bc (Mem.nextblock m) -> romatch m rm -> romatch m' rm. Proof. hammer_hook "ValueDomain" "ValueDomain.romatch_alloc". intros. apply romatch_ext with m; auto. intros. rewrite <- H3; symmetry. eapply Mem.loadbytes_alloc_unchanged; eauto. apply H0. congruence. intros. eapply Mem.perm_alloc_4; eauto. apply Mem.valid_not_valid_diff with m; eauto with mem. apply H0. congruence. Qed. Record amem : Type := AMem { am_stack: ablock; am_glob: PTree.t ablock; am_nonstack: aptr; am_top: aptr }. Record mmatch (m: mem) (am: amem) : Prop := mk_mem_match { mmatch_stack: forall b, bc b = BCstack -> bmatch m b am.(am_stack); mmatch_glob: forall id ab b, bc b = BCglob id -> am.(am_glob)!id = Some ab -> bmatch m b ab; mmatch_nonstack: forall b, bc b <> BCstack -> bc b <> BCinvalid -> smatch m b am.(am_nonstack); mmatch_top: forall b, bc b <> BCinvalid -> smatch m b am.(am_top); mmatch_below: bc_below bc (Mem.nextblock m) }. Definition minit (p: aptr) := {\| am_stack := ablock_init p; am_glob := PTree.empty _; am_nonstack := p; am_top := p \|}. Definition mbot := minit Pbot. Definition mtop := minit Ptop. Definition load (chunk: memory_chunk) (rm: romem) (m: amem) (p: aptr) : aval := match p with \| Pbot => if va_strict tt then Vbot else Vtop \| Gl id ofs => match rm!id with \| Some ab => ablock_load chunk ab (Ptrofs.unsigned ofs) \| None => match m.(am_glob)!id with \| Some ab => ablock_load chunk ab (Ptrofs.unsigned ofs) \| None => vnormalize chunk (Ifptr m.(am_nonstack)) end end \| Glo id => match rm!id with \| Some ab => ablock_load_anywhere chunk ab \| None => match m.(am_glob)!id with \| Some ab => ablock_load_anywhere chunk ab \| None => vnormalize chunk (Ifptr m.(am_nonstack)) end end \| Stk ofs => ablock_load chunk m.(am_stack) (Ptrofs.unsigned ofs) \| Stack => ablock_load_anywhere chunk m.(am_stack) \| Glob \| Nonstack => vnormalize chunk (Ifptr m.(am_nonstack)) \| Ptop => vnormalize chunk (Ifptr m.(am_top)) end. Definition loadv (chunk: memory_chunk) (rm: romem) (m: amem) (addr: aval) : aval := load chunk rm m (aptr_of_aval addr). Definition store (chunk: memory_chunk) (m: amem) (p: aptr) (av: aval) : amem := {\| am_stack := match p with \| Stk ofs => ablock_store chunk m.(am_stack) (Ptrofs.unsigned ofs) av \| Stack \| Ptop => ablock_store_anywhere chunk m.(am_stack) av \| _ => m.(am_stack) end; am_glob := match p with \| Gl id ofs => let ab := match m.(am_glob)!id with Some ab => ab \| None => ablock_init m.(am_nonstack) end in PTree.set id (ablock_store chunk ab (Ptrofs.unsigned ofs) av) m.(am_glob) \| Glo id => let ab := match m.(am_glob)!id with Some ab => ab \| None => ablock_init m.(am_nonstack) end in PTree.set id (ablock_store_anywhere chunk ab av) m.(am_glob) \| Glob \| Nonstack \| Ptop => PTree.empty _ \| _ => m.(am_glob) end; am_nonstack := match p with \| Gl _ _ \| Glo _ \| Glob \| Nonstack \| Ptop => vplub av m.(am_nonstack) \| _ => m.(am_nonstack) end; am_top := vplub av m.(am_top) \|}. Definition storev (chunk: memory_chunk) (m: amem) (addr: aval) (v: aval): amem := store chunk m (aptr_of_aval addr) v. Definition loadbytes (m: amem) (rm: romem) (p: aptr) : aptr := match p with \| Pbot => if va_strict tt then Pbot else Ptop \| Gl id _ \| Glo id => match rm!id with \| Some ab => ablock_loadbytes ab \| None => match m.(am_glob)!id with \| Some ab => ablock_loadbytes ab \| None => m.(am_nonstack) end end \| Stk _ \| Stack => ablock_loadbytes m.(am_stack) \| Glob \| Nonstack => m.(am_nonstack) \| Ptop => m.(am_top) end. Definition storebytes (m: amem) (dst: aptr) (sz: Z) (p: aptr) : amem := {\| am_stack := match dst with \| Stk ofs => ablock_storebytes m.(am_stack) p (Ptrofs.unsigned ofs) sz \| Stack \| Ptop => ablock_storebytes_anywhere m.(am_stack) p \| _ => m.(am_stack) end; am_glob := match dst with \| Gl id ofs => let ab := match m.(am_glob)!id with Some ab => ab \| None => ablock_init m.(am_nonstack) end in PTree.set id (ablock_storebytes ab p (Ptrofs.unsigned ofs) sz) m.(am_glob) \| Glo id => let ab := match m.(am_glob)!id with Some ab => ab \| None => ablock_init m.(am_nonstack) end in PTree.set id (ablock_storebytes_anywhere ab p) m.(am_glob) \| Glob \| Nonstack \| Ptop => PTree.empty _ \| _ => m.(am_glob) end; am_nonstack := match dst with \| Gl _ _ \| Glo _ \| Glob \| Nonstack \| Ptop => plub p m.(am_nonstack) \| _ => m.(am_nonstack) end; am_top := plub p m.(am_top) \|}. Theorem load_sound: forall chunk m b ofs v rm am p, Mem.load chunk m b (Ptrofs.unsigned ofs) = Some v -> romatch m rm -> mmatch m am -> pmatch b ofs p -> vmatch v (load chunk rm am p). Proof. hammer_hook "ValueDomain" "ValueDomain.load_sound". intros. unfold load. inv H2. - destruct (rm!id) as [ab\|] eqn:RM. eapply ablock_load_sound; eauto. eapply H0; eauto. destruct (am_glob am)!id as [ab\|] eqn:AM. eapply ablock_load_sound; eauto. eapply mmatch_glob; eauto. eapply vnormalize_cast; eauto. eapply mmatch_nonstack; eauto; congruence. - destruct (rm!id) as [ab\|] eqn:RM. eapply ablock_load_anywhere_sound; eauto. eapply H0; eauto. destruct (am_glob am)!id as [ab\|] eqn:AM. eapply ablock_load_anywhere_sound; eauto. eapply mmatch_glob; eauto. eapply vnormalize_cast; eauto. eapply mmatch_nonstack; eauto; congruence. - eapply vnormalize_cast; eauto. eapply mmatch_nonstack; eauto. congruence. congruence. - eapply ablock_load_sound; eauto. eapply mmatch_stack; eauto. - eapply ablock_load_anywhere_sound; eauto. eapply mmatch_stack; eauto. - eapply vnormalize_cast; eauto. eapply mmatch_nonstack; eauto. - eapply vnormalize_cast; eauto. eapply mmatch_top; eauto. Qed. Theorem loadv_sound: forall chunk m addr v rm am aaddr, Mem.loadv chunk m addr = Some v -> romatch m rm -> mmatch m am -> vmatch addr aaddr -> vmatch v (loadv chunk rm am aaddr). Proof. hammer_hook "ValueDomain" "ValueDomain.loadv_sound". intros. destruct addr; simpl in H; try discriminate. eapply load_sound; eauto. apply match_aptr_of_aval; auto. Qed. Theorem store_sound: forall chunk m b ofs v m' am p av, Mem.store chunk m b (Ptrofs.unsigned ofs) v = Some m' -> mmatch m am -> pmatch b ofs p -> vmatch v av -> mmatch m' (store chunk am p av). Proof. hammer_hook "ValueDomain" "ValueDomain.store_sound". intros until av; intros STORE MM PM VM. unfold store; constructor; simpl; intros. - assert (DFL: bc b <> BCstack -> bmatch m' b0 (am_stack am)). { intros. apply bmatch_inv with m. eapply mmatch_stack; eauto. intros. eapply Mem.loadbytes_store_other; eauto. left; congruence. } inv PM; try (apply DFL; congruence). + assert (b0 = b) by (eapply bc_stack; eauto). subst b0. eapply ablock_store_sound; eauto. eapply mmatch_stack; eauto. + assert (b0 = b) by (eapply bc_stack; eauto). subst b0. eapply ablock_store_anywhere_sound; eauto. eapply mmatch_stack; eauto. + eapply ablock_store_anywhere_sound; eauto. eapply mmatch_stack; eauto. - rename b0 into b'. assert (DFL: bc b <> BCglob id -> (am_glob am)!id = Some ab -> bmatch m' b' ab). { intros. apply bmatch_inv with m. eapply mmatch_glob; eauto. intros. eapply Mem.loadbytes_store_other; eauto. left; congruence. } inv PM. + rewrite PTree.gsspec in H0. destruct (peq id id0). subst id0; inv H0. assert (b' = b) by (eapply bc_glob; eauto). subst b'. eapply ablock_store_sound; eauto. destruct (am_glob am)!id as [ab0\|] eqn:GL. eapply mmatch_glob; eauto. apply ablock_init_sound. eapply mmatch_nonstack; eauto; congruence. eapply DFL; eauto. congruence. + rewrite PTree.gsspec in H0. destruct (peq id id0). subst id0; inv H0. assert (b' = b) by (eapply bc_glob; eauto). subst b'. eapply ablock_store_anywhere_sound; eauto. destruct (am_glob am)!id as [ab0\|] eqn:GL. eapply mmatch_glob; eauto. apply ablock_init_sound. eapply mmatch_nonstack; eauto; congruence. eapply DFL; eauto. congruence. + rewrite PTree.gempty in H0; congruence. + eapply DFL; eauto. congruence. + eapply DFL; eauto. congruence. + rewrite PTree.gempty in H0; congruence. + rewrite PTree.gempty in H0; congruence. - assert (DFL: smatch m' b0 (vplub av (am_nonstack am))). { eapply smatch_store; eauto. eapply mmatch_nonstack; eauto. } assert (STK: bc b = BCstack -> smatch m' b0 (am_nonstack am)). { intros. apply smatch_inv with m. eapply mmatch_nonstack; eauto; congruence. intros. eapply Mem.loadbytes_store_other; eauto. left. congruence. } inv PM; (apply DFL \|\| apply STK; congruence). - eapply smatch_store; eauto. eapply mmatch_top; eauto. - erewrite Mem.nextblock_store by eauto. eapply mmatch_below; eauto. Qed. Theorem storev_sound: forall chunk m addr v m' am aaddr av, Mem.storev chunk m addr v = Some m' -> mmatch m am -> vmatch addr aaddr -> vmatch v av -> mmatch m' (storev chunk am aaddr av). Proof. hammer_hook "ValueDomain" "ValueDomain.storev_sound". intros. destruct addr; simpl in H; try discriminate. eapply store_sound; eauto. apply match_aptr_of_aval; auto. Qed. Theorem loadbytes_sound: forall m b ofs sz bytes am rm p, Mem.loadbytes m b (Ptrofs.unsigned ofs) sz = Some bytes -> romatch m rm -> mmatch m am -> pmatch b ofs p -> forall b' ofs' q i, In (Fragment (Vptr b' ofs') q i) bytes -> pmatch b' ofs' (loadbytes am rm p). Proof. hammer_hook "ValueDomain" "ValueDomain.loadbytes_sound". intros. unfold loadbytes; inv H2. - destruct (rm!id) as [ab\|] eqn:RM. exploit H0; eauto. intros (A & B & C). eapply ablock_loadbytes_sound; eauto. destruct (am_glob am)!id as [ab\|] eqn:GL. eapply ablock_loadbytes_sound; eauto. eapply mmatch_glob; eauto. eapply smatch_loadbytes; eauto. eapply mmatch_nonstack; eauto with va. - destruct (rm!id) as [ab\|] eqn:RM. exploit H0; eauto. intros (A & B & C). eapply ablock_loadbytes_sound; eauto. destruct (am_glob am)!id as [ab\|] eqn:GL. eapply ablock_loadbytes_sound; eauto. eapply mmatch_glob; eauto. eapply smatch_loadbytes; eauto. eapply mmatch_nonstack; eauto with va. - eapply smatch_loadbytes; eauto. eapply mmatch_nonstack; eauto with va. - eapply ablock_loadbytes_sound; eauto. eapply mmatch_stack; eauto. - eapply ablock_loadbytes_sound; eauto. eapply mmatch_stack; eauto. - eapply smatch_loadbytes; eauto. eapply mmatch_nonstack; eauto with va. - eapply smatch_loadbytes; eauto. eapply mmatch_top; eauto with va. Qed. Theorem storebytes_sound: forall m b ofs bytes m' am p sz q, Mem.storebytes m b (Ptrofs.unsigned ofs) bytes = Some m' -> mmatch m am -> pmatch b ofs p -> length bytes = Z.to_nat sz -> (forall b' ofs' qt i, In (Fragment (Vptr b' ofs') qt i) bytes -> pmatch b' ofs' q) -> mmatch m' (storebytes am p sz q). Proof. hammer_hook "ValueDomain" "ValueDomain.storebytes_sound". intros until q; intros STORE MM PM LENGTH BYTES. unfold storebytes; constructor; simpl; intros. - assert (DFL: bc b <> BCstack -> bmatch m' b0 (am_stack am)). { intros. apply bmatch_inv with m. eapply mmatch_stack; eauto. intros. eapply Mem.loadbytes_storebytes_other; eauto. left; congruence. } inv PM; try (apply DFL; congruence). + assert (b0 = b) by (eapply bc_stack; eauto). subst b0. eapply ablock_storebytes_sound; eauto. eapply mmatch_stack; eauto. + assert (b0 = b) by (eapply bc_stack; eauto). subst b0. eapply ablock_storebytes_anywhere_sound; eauto. eapply mmatch_stack; eauto. + eapply ablock_storebytes_anywhere_sound; eauto. eapply mmatch_stack; eauto. - rename b0 into b'. assert (DFL: bc b <> BCglob id -> (am_glob am)!id = Some ab -> bmatch m' b' ab). { intros. apply bmatch_inv with m. eapply mmatch_glob; eauto. intros. eapply Mem.loadbytes_storebytes_other; eauto. left; congruence. } inv PM. + rewrite PTree.gsspec in H0. destruct (peq id id0). subst id0; inv H0. assert (b' = b) by (eapply bc_glob; eauto). subst b'. eapply ablock_storebytes_sound; eauto. destruct (am_glob am)!id as [ab0\|] eqn:GL. eapply mmatch_glob; eauto. apply ablock_init_sound. eapply mmatch_nonstack; eauto; congruence. eapply DFL; eauto. congruence. + rewrite PTree.gsspec in H0. destruct (peq id id0). subst id0; inv H0. assert (b' = b) by (eapply bc_glob; eauto). subst b'. eapply ablock_storebytes_anywhere_sound; eauto. destruct (am_glob am)!id as [ab0\|] eqn:GL. eapply mmatch_glob; eauto. apply ablock_init_sound. eapply mmatch_nonstack; eauto; congruence. eapply DFL; eauto. congruence. + rewrite PTree.gempty in H0; congruence. + eapply DFL; eauto. congruence. + eapply DFL; eauto. congruence. + rewrite PTree.gempty in H0; congruence. + rewrite PTree.gempty in H0; congruence. - assert (DFL: smatch m' b0 (plub q (am_nonstack am))). { eapply smatch_storebytes; eauto. eapply mmatch_nonstack; eauto. } assert (STK: bc b = BCstack -> smatch m' b0 (am_nonstack am)). { intros. apply smatch_inv with m. eapply mmatch_nonstack; eauto; congruence. intros. eapply Mem.loadbytes_storebytes_other; eauto. left. congruence. } inv PM; (apply DFL \|\| apply STK; congruence). - eapply smatch_storebytes; eauto. eapply mmatch_top; eauto. - erewrite Mem.nextblock_storebytes by eauto. eapply mmatch_below; eauto. Qed. Lemma mmatch_ext: forall m am m', mmatch m am -> (forall b ofs n bytes, bc b <> BCinvalid -> n >= 0 -> Mem.loadbytes m' b ofs n = Some bytes -> Mem.loadbytes m b ofs n = Some bytes) -> Ple (Mem.nextblock m) (Mem.nextblock m') -> mmatch m' am. Proof. hammer_hook "ValueDomain" "ValueDomain.mmatch_ext". intros. inv H. constructor; intros. - apply bmatch_ext with m; auto with va. - apply bmatch_ext with m; eauto with va. - apply smatch_ext with m; auto with va. - apply smatch_ext with m; auto with va. - red; intros. exploit mmatch_below0; eauto. xomega. Qed. Lemma mmatch_free: forall m b lo hi m' am, Mem.free m b lo hi = Some m' -> mmatch m am -> mmatch m' am. Proof. hammer_hook "ValueDomain" "ValueDomain.mmatch_free". intros. apply mmatch_ext with m; auto. intros. eapply Mem.loadbytes_free_2; eauto. erewrite <- Mem.nextblock_free by eauto. xomega. Qed. Lemma mmatch_top': forall m am, mmatch m am -> mmatch m mtop. Proof. hammer_hook "ValueDomain" "ValueDomain.mmatch_top'". intros. constructor; simpl; intros. - apply ablock_init_sound. apply smatch_ge with (ab_summary (am_stack am)). eapply mmatch_stack; eauto. constructor. - rewrite PTree.gempty in H1; discriminate. - eapply smatch_ge. eapply mmatch_nonstack; eauto. constructor. - eapply smatch_ge. eapply mmatch_top; eauto. constructor. - eapply mmatch_below; eauto. Qed. Definition mbeq (m1 m2: amem) : bool := eq_aptr m1.(am_top) m2.(am_top) && eq_aptr m1.(am_nonstack) m2.(am_nonstack) && bbeq m1.(am_stack) m2.(am_stack) && PTree.beq bbeq m1.(am_glob) m2.(am_glob). Lemma mbeq_sound: forall m1 m2, mbeq m1 m2 = true -> forall m, mmatch m m1 <-> mmatch m m2. Proof. hammer_hook "ValueDomain" "ValueDomain.mbeq_sound". unfold mbeq; intros. InvBooleans. rewrite PTree.beq_correct in H1. split; intros M; inv M; constructor; intros. - erewrite <- bbeq_sound; eauto. - specialize (H1 id). rewrite H4 in H1. destruct (am_glob m1)!id eqn:G; try contradiction. erewrite <- bbeq_sound; eauto. - rewrite <- H; eauto. - rewrite <- H0; eauto. - auto. - erewrite bbeq_sound; eauto. - specialize (H1 id). rewrite H4 in H1. destruct (am_glob m2)!id eqn:G; try contradiction. erewrite bbeq_sound; eauto. - rewrite H; eauto. - rewrite H0; eauto. - auto. Qed. Definition combine_ablock (ob1 ob2: option ablock) : option ablock := match ob1, ob2 with \| Some b1, Some b2 => Some (blub b1 b2) \| _, _ => None end. Definition mlub (m1 m2: amem) : amem := {\| am_stack := blub m1.(am_stack) m2.(am_stack); am_glob := PTree.combine combine_ablock m1.(am_glob) m2.(am_glob); am_nonstack := plub m1.(am_nonstack) m2.(am_nonstack); am_top := plub m1.(am_top) m2.(am_top) \|}. Lemma mmatch_lub_l: forall m x y, mmatch m x -> mmatch m (mlub x y). Proof. hammer_hook "ValueDomain" "ValueDomain.mmatch_lub_l". intros. inv H. constructor; simpl; intros. - apply bmatch_lub_l; auto. - rewrite PTree.gcombine in H0 by auto. unfold combine_ablock in H0. destruct (am_glob x)!id as [b1\|] eqn:G1; destruct (am_glob y)!id as [b2\|] eqn:G2; inv H0. apply bmatch_lub_l; eauto. - apply smatch_lub_l; auto. - apply smatch_lub_l; auto. - auto. Qed. Lemma mmatch_lub_r: forall m x y, mmatch m y -> mmatch m (mlub x y). Proof. hammer_hook "ValueDomain" "ValueDomain.mmatch_lub_r". intros. inv H. constructor; simpl; intros. - apply bmatch_lub_r; auto. - rewrite PTree.gcombine in H0 by auto. unfold combine_ablock in H0. destruct (am_glob x)!id as [b1\|] eqn:G1; destruct (am_glob y)!id as [b2\|] eqn:G2; inv H0. apply bmatch_lub_r; eauto. - apply smatch_lub_r; auto. - apply smatch_lub_r; auto. - auto. Qed. End MATCH. Lemma genv_match_exten: forall ge (bc1 bc2: block_classification), genv_match bc1 ge -> (forall b id, bc1 b = BCglob id <-> bc2 b = BCglob id) -> (forall b, bc1 b = BCother -> bc2 b = BCother) -> genv_match bc2 ge. Proof. hammer_hook "ValueDomain" "ValueDomain.genv_match_exten". intros. destruct H as [A B]. split; intros. - rewrite <- H0. eauto. - exploit B; eauto. destruct (bc1 b) eqn:BC1. + intuition congruence. + rewrite H0 in BC1. intuition congruence. + intuition congruence. + erewrite H1 by eauto. intuition congruence. Qed. Lemma romatch_exten: forall (bc1 bc2: block_classification) m rm, romatch bc1 m rm -> (forall b id, bc2 b = BCglob id <-> bc1 b = BCglob id) -> romatch bc2 m rm. Proof. hammer_hook "ValueDomain" "ValueDomain.romatch_exten". intros; red; intros. rewrite H0 in H1. exploit H; eauto. intros (A & B & C). split; auto. split; auto. assert (PM: forall b ofs p, pmatch bc1 b ofs p -> pmatch bc1 b ofs (ab_summary ab) -> pmatch bc2 b ofs p). { intros. assert (pmatch bc1 b0 ofs Glob) by (eapply pmatch_ge; eauto). inv H5. assert (bc2 b0 = BCglob id0) by (rewrite H0; auto). inv H3; econstructor; eauto with va. } assert (VM: forall v x, vmatch bc1 v x -> vmatch bc1 v (Ifptr (ab_summary ab)) -> vmatch bc2 v x). { intros. inv H3; constructor; auto; inv H4; eapply PM; eauto. } destruct B as [[B1 B2] B3]. split. split. - intros. apply VM; eauto. - intros. apply PM; eauto. - intros. apply VM; eauto. Qed. Definition bc_incr (bc1 bc2: block_classification) : Prop := forall b, bc1 b <> BCinvalid -> bc2 b = bc1 b. Section MATCH_INCR. Variables bc1 bc2: block_classification. Hypothesis INCR: bc_incr bc1 bc2. Lemma pmatch_incr: forall b ofs p, pmatch bc1 b ofs p -> pmatch bc2 b ofs p. Proof. hammer_hook "ValueDomain" "ValueDomain.pmatch_incr". induction 1; assert (bc2 b = bc1 b) by (apply INCR; congruence); econstructor; eauto with va. rewrite H0; eauto. Qed. Lemma vmatch_incr: forall v x, vmatch bc1 v x -> vmatch bc2 v x. Proof. hammer_hook "ValueDomain" "ValueDomain.vmatch_incr". induction 1; constructor; auto; apply pmatch_incr; auto. Qed. Lemma smatch_incr: forall m b p, smatch bc1 m b p -> smatch bc2 m b p. Proof. hammer_hook "ValueDomain" "ValueDomain.smatch_incr". intros. destruct H as [A B]. split; intros. apply vmatch_incr; eauto. apply pmatch_incr; eauto. Qed. Lemma bmatch_incr: forall m b ab, bmatch bc1 m b ab -> bmatch bc2 m b ab. Proof. hammer_hook "ValueDomain" "ValueDomain.bmatch_incr". intros. destruct H as [B1 B2]. split. apply smatch_incr; auto. intros. apply vmatch_incr; eauto. Qed. End MATCH_INCR. Definition inj_of_bc (bc: block_classification) : meminj := fun b => match bc b with BCinvalid => None \| _ => Some(b, 0) end. Lemma inj_of_bc_valid: forall (bc: block_classification) b, bc b <> BCinvalid -> inj_of_bc bc b = Some(b, 0). Proof. hammer_hook "ValueDomain" "ValueDomain.inj_of_bc_valid". intros. unfold inj_of_bc. destruct (bc b); congruence. Qed. Lemma inj_of_bc_inv: forall (bc: block_classification) b b' delta, inj_of_bc bc b = Some(b', delta) -> bc b <> BCinvalid /\ b' = b /\ delta = 0. Proof. hammer_hook "ValueDomain" "ValueDomain.inj_of_bc_inv". unfold inj_of_bc; intros. destruct (bc b); intuition congruence. Qed. Lemma pmatch_inj: forall bc b ofs p, pmatch bc b ofs p -> inj_of_bc bc b = Some(b, 0). Proof. hammer_hook "ValueDomain" "ValueDomain.pmatch_inj". intros. apply inj_of_bc_valid. inv H; congruence. Qed. Lemma vmatch_inj: forall bc v x, vmatch bc v x -> Val.inject (inj_of_bc bc) v v. Proof. hammer_hook "ValueDomain" "ValueDomain.vmatch_inj". induction 1; econstructor. eapply pmatch_inj; eauto. rewrite Ptrofs.add_zero; auto. eapply pmatch_inj; eauto. rewrite Ptrofs.add_zero; auto. Qed. Lemma vmatch_list_inj: forall bc vl xl, list_forall2 (vmatch bc) vl xl -> Val.inject_list (inj_of_bc bc) vl vl. Proof. hammer_hook "ValueDomain" "ValueDomain.vmatch_list_inj". induction 1; constructor. eapply vmatch_inj; eauto. auto. Qed. Lemma mmatch_inj: forall bc m am, mmatch bc m am -> bc_below bc (Mem.nextblock m) -> Mem.inject (inj_of_bc bc) m m. Proof. hammer_hook "ValueDomain" "ValueDomain.mmatch_inj". intros. constructor. constructor. - intros. exploit inj_of_bc_inv; eauto. intros (A & B & C); subst. rewrite Z.add_0_r. auto. - intros. exploit inj_of_bc_inv; eauto. intros (A & B & C); subst. apply Z.divide_0_r. - intros. exploit inj_of_bc_inv; eauto. intros (A & B & C); subst. rewrite Z.add_0_r. set (mv := ZMap.get ofs (PMap.get b1 (Mem.mem_contents m))). assert (Mem.loadbytes m b1 ofs 1 = Some (mv :: nil)). { Local Transparent Mem.loadbytes. unfold Mem.loadbytes. rewrite pred_dec_true. reflexivity. red; intros. replace ofs0 with ofs by omega. auto. } destruct mv; econstructor. destruct v; econstructor. apply inj_of_bc_valid. assert (PM: pmatch bc b i Ptop). { exploit mmatch_top; eauto. intros [P Q]. eapply pmatch_top'. eapply Q; eauto. } inv PM; auto. rewrite Ptrofs.add_zero; auto. - intros. unfold inj_of_bc. erewrite bc_below_invalid; eauto. - intros. exploit inj_of_bc_inv; eauto. intros (A & B & C); subst. apply H0; auto. - red; intros. exploit inj_of_bc_inv. eexact H2. intros (A1 & B & C); subst. exploit inj_of_bc_inv. eexact H3. intros (A2 & B & C); subst. auto. - intros. exploit inj_of_bc_inv; eauto. intros (A & B & C); subst. rewrite Z.add_0_r. split. omega. apply Ptrofs.unsigned_range_2. - intros. exploit inj_of_bc_inv; eauto. intros (A & B & C); subst. rewrite Z.add_0_r in H2. auto. Qed. Lemma inj_of_bc_preserves_globals: forall bc ge, genv_match bc ge -> meminj_preserves_globals ge (inj_of_bc bc). Proof. hammer_hook "ValueDomain" "ValueDomain.inj_of_bc_preserves_globals". intros. destruct H as [A B]. split. intros. apply inj_of_bc_valid. rewrite A in H. congruence. split. intros. apply inj_of_bc_valid. apply B. rewrite Genv.find_var_info_iff in H. eapply Genv.genv_defs_range; eauto. intros. exploit inj_of_bc_inv; eauto. intros (P & Q & R). auto. Qed. Lemma pmatch_inj_top: forall bc b b' delta ofs, inj_of_bc bc b = Some(b', delta) -> pmatch bc b ofs Ptop. Proof. hammer_hook "ValueDomain" "ValueDomain.pmatch_inj_top". intros. exploit inj_of_bc_inv; eauto. intros (A & B & C). constructor; auto. Qed. Lemma vmatch_inj_top: forall bc v v', Val.inject (inj_of_bc bc) v v' -> vmatch bc v Vtop. Proof. hammer_hook "ValueDomain" "ValueDomain.vmatch_inj_top". intros. inv H; constructor. eapply pmatch_inj_top; eauto. Qed. Lemma mmatch_inj_top: forall bc m m', Mem.inject (inj_of_bc bc) m m' -> mmatch bc m mtop. Proof. hammer_hook "ValueDomain" "ValueDomain.mmatch_inj_top". intros. assert (SM: forall b, bc b <> BCinvalid -> smatch bc m b Ptop). { intros; split; intros. - exploit Mem.load_inject. eauto. eauto. apply inj_of_bc_valid; auto. intros (v' & A & B). eapply vmatch_inj_top; eauto. - exploit Mem.loadbytes_inject. eauto. eauto. apply inj_of_bc_valid; auto. intros (bytes' & A & B). inv B. inv H4. inv H8. eapply pmatch_inj_top; eauto. } constructor; simpl; intros. - apply ablock_init_sound. apply SM. congruence. - rewrite PTree.gempty in H1; discriminate. - apply SM; auto. - apply SM; auto. - red; intros. eapply Mem.valid_block_inject_1. eapply inj_of_bc_valid; eauto. eauto. Qed. Module AVal <: SEMILATTICE_WITH_TOP. Definition t := aval. Definition eq (x y: t) := (x = y). Definition eq_refl: forall x, eq x x := (@eq_refl t). Definition eq_sym: forall x y, eq x y -> eq y x := (@eq_sym t). Definition eq_trans: forall x y z, eq x y -> eq y z -> eq x z := (@eq_trans t). Definition beq (x y: t) : bool := proj_sumbool (eq_aval x y). Lemma beq_correct: forall x y, beq x y = true -> eq x y. Proof. hammer_hook "ValueDomain" "ValueDomain.AVal.beq_correct". unfold beq; intros. InvBooleans. auto. Qed. Definition ge := vge. Lemma ge_refl: forall x y, eq x y -> ge x y. Proof. hammer_hook "ValueDomain" "ValueDomain.AVal.ge_refl". unfold eq, ge; intros. subst y. apply vge_refl. Qed. Lemma ge_trans: forall x y z, ge x y -> ge y z -> ge x z. Proof. hammer_hook "ValueDomain" "ValueDomain.AVal.ge_trans". unfold ge; intros. eapply vge_trans; eauto. Qed. Definition bot : t := Vbot. Lemma ge_bot: forall x, ge x bot. Proof. hammer_hook "ValueDomain" "ValueDomain.AVal.ge_bot". intros. constructor. Qed. Definition top : t := Vtop. Lemma ge_top: forall x, ge top x. Proof. hammer_hook "ValueDomain" "ValueDomain.AVal.ge_top". intros. apply vge_top. Qed. Definition lub := vlub. Lemma ge_lub_left: forall x y, ge (lub x y) x. Proof. hammer_hook "ValueDomain" "ValueDomain.AVal.ge_lub_left". exact (vge_lub_l). Qed. Lemma ge_lub_right: forall x y, ge (lub x y) y. Proof. hammer_hook "ValueDomain" "ValueDomain.AVal.ge_lub_right". exact (vge_lub_r). Qed. End AVal. Module AE := LPMap(AVal). Definition aenv := AE.t. Section MATCHENV. Variable bc: block_classification. Definition ematch (e: regset) (ae: aenv) : Prop := forall r, vmatch bc e#r (AE.get r ae). Lemma ematch_ge: forall e ae1 ae2, ematch e ae1 -> AE.ge ae2 ae1 -> ematch e ae2. Proof. hammer_hook "ValueDomain" "ValueDomain.ematch_ge". intros; red; intros. apply vmatch_ge with (AE.get r ae1); auto. apply H0. Qed. Lemma ematch_update: forall e ae v av r, ematch e ae -> vmatch bc v av -> ematch (e#r <- v) (AE.set r av ae). Proof. hammer_hook "ValueDomain" "ValueDomain.ematch_update". intros; red; intros. rewrite AE.gsspec. rewrite PMap.gsspec. destruct (peq r0 r); auto. red; intros. specialize (H xH). subst ae. simpl in H. inv H. unfold AVal.eq; red; intros. subst av. inv H0. Qed. Fixpoint einit_regs (rl: list reg) : aenv := match rl with \| r1 :: rs => AE.set r1 (Ifptr Nonstack) (einit_regs rs) \| nil => AE.top end. Lemma ematch_init: forall rl vl, (forall v, In v vl -> vmatch bc v (Ifptr Nonstack)) -> ematch (init_regs vl rl) (einit_regs rl). Proof. hammer_hook "ValueDomain" "ValueDomain.ematch_init". induction rl; simpl; intros. - red; intros. rewrite Regmap.gi. simpl AE.get. rewrite PTree.gempty. constructor. - destruct vl as [ \| v1 vs ]. + assert (ematch (init_regs nil rl) (einit_regs rl)). { apply IHrl. simpl; tauto. } replace (init_regs nil rl) with (Regmap.init Vundef) in H0 by (destruct rl; auto). red; intros. rewrite AE.gsspec. destruct (peq r a). rewrite Regmap.gi. constructor. apply H0. red; intros EQ; rewrite EQ in H0. specialize (H0 xH). simpl in H0. inv H0. unfold AVal.eq, AVal.bot. congruence. + assert (ematch (init_regs vs rl) (einit_regs rl)). { apply IHrl. eauto with coqlib. } red; intros. rewrite Regmap.gsspec. rewrite AE.gsspec. destruct (peq r a). auto with coqlib. apply H0. red; intros EQ; rewrite EQ in H0. specialize (H0 xH). simpl in H0. inv H0. unfold AVal.eq, AVal.bot. congruence. Qed. Fixpoint eforget (rl: list reg) (ae: aenv) {struct rl} : aenv := match rl with \| nil => ae \| r1 :: rs => eforget rs (AE.set r1 Vtop ae) end. Lemma eforget_ge: forall rl ae, AE.ge (eforget rl ae) ae. Proof. hammer_hook "ValueDomain" "ValueDomain.eforget_ge". unfold AE.ge; intros. revert rl ae; induction rl; intros; simpl. apply AVal.ge_refl. apply AVal.eq_refl. destruct ae. unfold AE.get at 2. apply AVal.ge_bot. eapply AVal.ge_trans. apply IHrl. rewrite AE.gsspec. destruct (peq p a). apply AVal.ge_top. apply AVal.ge_refl. apply AVal.eq_refl. congruence. unfold AVal.eq, Vtop, AVal.bot. congruence. Qed. Lemma ematch_forget: forall e rl ae, ematch e ae -> ematch e (eforget rl ae). Proof. hammer_hook "ValueDomain" "ValueDomain.ematch_forget". intros. eapply ematch_ge; eauto. apply eforget_ge. Qed. End MATCHENV. Lemma ematch_incr: forall bc bc' e ae, ematch bc e ae -> bc_incr bc bc' -> ematch bc' e ae. Proof. hammer_hook "ValueDomain" "ValueDomain.ematch_incr". intros; red; intros. apply vmatch_incr with bc; auto. Qed. Module VA <: SEMILATTICE. Inductive t' := Bot \| State (ae: aenv) (am: amem). Definition t := t'. Definition eq (x y: t) := match x, y with \| Bot, Bot => True \| State ae1 am1, State ae2 am2 => AE.eq ae1 ae2 /\ forall bc m, mmatch bc m am1 <-> mmatch bc m am2 \| _, _ => False end. Lemma eq_refl: forall x, eq x x. Proof. hammer_hook "ValueDomain" "ValueDomain.AVal.eq_refl". destruct x; simpl. auto. split. apply AE.eq_refl. tauto. Qed. Lemma eq_sym: forall x y, eq x y -> eq y x. Proof. hammer_hook "ValueDomain" "ValueDomain.AVal.eq_sym". destruct x, y; simpl; auto. intros [A B]. split. apply AE.eq_sym; auto. intros. rewrite B. tauto. Qed. Lemma eq_trans: forall x y z, eq x y -> eq y z -> eq x z. Proof. hammer_hook "ValueDomain" "ValueDomain.AVal.eq_trans". destruct x, y, z; simpl; try tauto. intros [A B] [C D]; split. eapply AE.eq_trans; eauto. intros. rewrite B; auto. Qed. Definition beq (x y: t) : bool := match x, y with \| Bot, Bot => true \| State ae1 am1, State ae2 am2 => AE.beq ae1 ae2 && mbeq am1 am2 \| _, _ => false end. Lemma beq_correct: forall x y, beq x y = true -> eq x y. Proof. hammer_hook "ValueDomain" "ValueDomain.VA.beq_correct". destruct x, y; simpl; intros. auto. congruence. congruence. InvBooleans; split. apply AE.beq_correct; auto. intros. apply mbeq_sound; auto. Qed. Definition ge (x y: t) : Prop := match x, y with \| _, Bot => True \| Bot, _ => False \| State ae1 am1, State ae2 am2 => AE.ge ae1 ae2 /\ forall bc m, mmatch bc m am2 -> mmatch bc m am1 end. Lemma ge_refl: forall x y, eq x y -> ge x y. Proof. hammer_hook "ValueDomain" "ValueDomain.VA.ge_refl". destruct x, y; simpl; try tauto. intros [A B]; split. apply AE.ge_refl; auto. intros. rewrite B; auto. Qed. Lemma ge_trans: forall x y z, ge x y -> ge y z -> ge x z. Proof. hammer_hook "ValueDomain" "ValueDomain.VA.ge_trans". destruct x, y, z; simpl; try tauto. intros [A B] [C D]; split. eapply AE.ge_trans; eauto. eauto. Qed. Definition bot : t := Bot. Lemma ge_bot: forall x, ge x bot. Proof. hammer_hook "ValueDomain" "ValueDomain.VA.ge_bot". destruct x; simpl; auto. Qed. Definition lub (x y: t) : t := match x, y with \| Bot, _ => y \| _, Bot => x \| State ae1 am1, State ae2 am2 => State (AE.lub ae1 ae2) (mlub am1 am2) end. Lemma ge_lub_left: forall x y, ge (lub x y) x. Proof. hammer_hook "ValueDomain" "ValueDomain.VA.ge_lub_left". destruct x, y. apply ge_refl; apply eq_refl. simpl. auto. apply ge_refl; apply eq_refl. simpl. split. apply AE.ge_lub_left. intros; apply mmatch_lub_l; auto. Qed. Lemma ge_lub_right: forall x y, ge (lub x y) y. Proof. hammer_hook "ValueDomain" "ValueDomain.VA.ge_lub_right". destruct x, y. apply ge_refl; apply eq_refl. apply ge_refl; apply eq_refl. simpl. auto. simpl. split. apply AE.ge_lub_right. intros; apply mmatch_lub_r; auto. Qed. End VA. Hint Constructors cmatch : va. Hint Constructors pmatch: va. Hint Constructors vmatch: va. Hint Resolve cnot_sound symbol_address_sound shl_sound shru_sound shr_sound and_sound or_sound xor_sound notint_sound ror_sound rolm_sound neg_sound add_sound sub_sound mul_sound mulhs_sound mulhu_sound divs_sound divu_sound mods_sound modu_sound shrx_sound shll_sound shrl_sound shrlu_sound andl_sound orl_sound xorl_sound notl_sound roll_sound rorl_sound negl_sound addl_sound subl_sound mull_sound mullhs_sound mullhu_sound divls_sound divlu_sound modls_sound modlu_sound shrxl_sound offset_ptr_sound negf_sound absf_sound addf_sound subf_sound mulf_sound divf_sound negfs_sound absfs_sound addfs_sound subfs_sound mulfs_sound divfs_sound zero_ext_sound sign_ext_sound longofint_sound longofintu_sound zero_ext_l_sound sign_ext_l_sound singleoffloat_sound floatofsingle_sound intoffloat_sound intuoffloat_sound floatofint_sound floatofintu_sound intofsingle_sound intuofsingle_sound singleofint_sound singleofintu_sound longoffloat_sound longuoffloat_sound floatoflong_sound floatoflongu_sound longofsingle_sound longuofsingle_sound singleoflong_sound singleoflongu_sound longofwords_sound loword_sound hiword_sound cmpu_bool_sound cmp_bool_sound cmplu_bool_sound cmpl_bool_sound cmpf_bool_sound cmpfs_bool_sound maskzero_sound : va.
The frontier of the intersection of two sets is a subset of the union of the frontiers of the two sets.
inductive Foo : Bool → Type \| Z : Foo false \| O : Foo false → Foo true \| E : Foo true → Foo false open Foo def toNat : {b : Bool} → Foo b → Nat \| _, Z => 0 \| _, O n => toNat n + 1 \| _, E n => toNat n + 1 example : toNat (E (O Z)) = 2 := rfl example : toNat Z = 0 := rfl example (a : Foo false) : toNat (O a) = toNat a + 1 := rfl example (a : Foo true) : toNat (E a) = toNat a + 1 := rfl
State Before: M : Type ?u.97023 α : Type ?u.97026 β : Type ?u.97029 G₀ : Type u_2 inst✝⁹ : TopologicalSpace α inst✝⁸ : GroupWithZero G₀ inst✝⁷ : MulAction G₀ α inst✝⁶ : ContinuousConstSMul G₀ α inst✝⁵ : TopologicalSpace β f : β → α b : β c✝ : G₀ s✝ : Set β E : Type u_1 inst✝⁴ : Zero E inst✝³ : MulActionWithZero G₀ E inst✝² : TopologicalSpace E inst✝¹ : T1Space E inst✝ : ContinuousConstSMul G₀ E c : G₀ s : Set E ⊢ closure (c • s) = c • closure s State After: case inl M : Type ?u.97023 α : Type ?u.97026 β : Type ?u.97029 G₀ : Type u_2 inst✝⁹ : TopologicalSpace α inst✝⁸ : GroupWithZero G₀ inst✝⁷ : MulAction G₀ α inst✝⁶ : ContinuousConstSMul G₀ α inst✝⁵ : TopologicalSpace β f : β → α b : β c : G₀ s✝ : Set β E : Type u_1 inst✝⁴ : Zero E inst✝³ : MulActionWithZero G₀ E inst✝² : TopologicalSpace E inst✝¹ : T1Space E inst✝ : ContinuousConstSMul G₀ E s : Set E ⊢ closure (0 • s) = 0 • closure s case inr M : Type ?u.97023 α : Type ?u.97026 β : Type ?u.97029 G₀ : Type u_2 inst✝⁹ : TopologicalSpace α inst✝⁸ : GroupWithZero G₀ inst✝⁷ : MulAction G₀ α inst✝⁶ : ContinuousConstSMul G₀ α inst✝⁵ : TopologicalSpace β f : β → α b : β c✝ : G₀ s✝ : Set β E : Type u_1 inst✝⁴ : Zero E inst✝³ : MulActionWithZero G₀ E inst✝² : TopologicalSpace E inst✝¹ : T1Space E inst✝ : ContinuousConstSMul G₀ E c : G₀ s : Set E hc : c ≠ 0 ⊢ closure (c • s) = c • closure s Tactic: rcases eq_or_ne c 0 with (rfl \| hc) State Before: case inl M : Type ?u.97023 α : Type ?u.97026 β : Type ?u.97029 G₀ : Type u_2 inst✝⁹ : TopologicalSpace α inst✝⁸ : GroupWithZero G₀ inst✝⁷ : MulAction G₀ α inst✝⁶ : ContinuousConstSMul G₀ α inst✝⁵ : TopologicalSpace β f : β → α b : β c : G₀ s✝ : Set β E : Type u_1 inst✝⁴ : Zero E inst✝³ : MulActionWithZero G₀ E inst✝² : TopologicalSpace E inst✝¹ : T1Space E inst✝ : ContinuousConstSMul G₀ E s : Set E ⊢ closure (0 • s) = 0 • closure s State After: case inl.inl M : Type ?u.97023 α : Type ?u.97026 β : Type ?u.97029 G₀ : Type u_2 inst✝⁹ : TopologicalSpace α inst✝⁸ : GroupWithZero G₀ inst✝⁷ : MulAction G₀ α inst✝⁶ : ContinuousConstSMul G₀ α inst✝⁵ : TopologicalSpace β f : β → α b : β c : G₀ s : Set β E : Type u_1 inst✝⁴ : Zero E inst✝³ : MulActionWithZero G₀ E inst✝² : TopologicalSpace E inst✝¹ : T1Space E inst✝ : ContinuousConstSMul G₀ E ⊢ closure (0 • ∅) = 0 • closure ∅ case inl.inr M : Type ?u.97023 α : Type ?u.97026 β : Type ?u.97029 G₀ : Type u_2 inst✝⁹ : TopologicalSpace α inst✝⁸ : GroupWithZero G₀ inst✝⁷ : MulAction G₀ α inst✝⁶ : ContinuousConstSMul G₀ α inst✝⁵ : TopologicalSpace β f : β → α b : β c : G₀ s✝ : Set β E : Type u_1 inst✝⁴ : Zero E inst✝³ : MulActionWithZero G₀ E inst✝² : TopologicalSpace E inst✝¹ : T1Space E inst✝ : ContinuousConstSMul G₀ E s : Set E hs : Set.Nonempty s ⊢ closure (0 • s) = 0 • closure s Tactic: rcases eq_empty_or_nonempty s with (rfl \| hs) State Before: case inl.inl M : Type ?u.97023 α : Type ?u.97026 β : Type ?u.97029 G₀ : Type u_2 inst✝⁹ : TopologicalSpace α inst✝⁸ : GroupWithZero G₀ inst✝⁷ : MulAction G₀ α inst✝⁶ : ContinuousConstSMul G₀ α inst✝⁵ : TopologicalSpace β f : β → α b : β c : G₀ s : Set β E : Type u_1 inst✝⁴ : Zero E inst✝³ : MulActionWithZero G₀ E inst✝² : TopologicalSpace E inst✝¹ : T1Space E inst✝ : ContinuousConstSMul G₀ E ⊢ closure (0 • ∅) = 0 • closure ∅ State After: no goals Tactic: simp State Before: case inl.inr M : Type ?u.97023 α : Type ?u.97026 β : Type ?u.97029 G₀ : Type u_2 inst✝⁹ : TopologicalSpace α inst✝⁸ : GroupWithZero G₀ inst✝⁷ : MulAction G₀ α inst✝⁶ : ContinuousConstSMul G₀ α inst✝⁵ : TopologicalSpace β f : β → α b : β c : G₀ s✝ : Set β E : Type u_1 inst✝⁴ : Zero E inst✝³ : MulActionWithZero G₀ E inst✝² : TopologicalSpace E inst✝¹ : T1Space E inst✝ : ContinuousConstSMul G₀ E s : Set E hs : Set.Nonempty s ⊢ closure (0 • s) = 0 • closure s State After: case inl.inr M : Type ?u.97023 α : Type ?u.97026 β : Type ?u.97029 G₀ : Type u_2 inst✝⁹ : TopologicalSpace α inst✝⁸ : GroupWithZero G₀ inst✝⁷ : MulAction G₀ α inst✝⁶ : ContinuousConstSMul G₀ α inst✝⁵ : TopologicalSpace β f : β → α b : β c : G₀ s✝ : Set β E : Type u_1 inst✝⁴ : Zero E inst✝³ : MulActionWithZero G₀ E inst✝² : TopologicalSpace E inst✝¹ : T1Space E inst✝ : ContinuousConstSMul G₀ E s : Set E hs : Set.Nonempty s ⊢ closure 0 = 0 Tactic: rw [zero_smul_set hs, zero_smul_set hs.closure] State Before: case inl.inr M : Type ?u.97023 α : Type ?u.97026 β : Type ?u.97029 G₀ : Type u_2 inst✝⁹ : TopologicalSpace α inst✝⁸ : GroupWithZero G₀ inst✝⁷ : MulAction G₀ α inst✝⁶ : ContinuousConstSMul G₀ α inst✝⁵ : TopologicalSpace β f : β → α b : β c : G₀ s✝ : Set β E : Type u_1 inst✝⁴ : Zero E inst✝³ : MulActionWithZero G₀ E inst✝² : TopologicalSpace E inst✝¹ : T1Space E inst✝ : ContinuousConstSMul G₀ E s : Set E hs : Set.Nonempty s ⊢ closure 0 = 0 State After: no goals Tactic: exact closure_singleton State Before: case inr M : Type ?u.97023 α : Type ?u.97026 β : Type ?u.97029 G₀ : Type u_2 inst✝⁹ : TopologicalSpace α inst✝⁸ : GroupWithZero G₀ inst✝⁷ : MulAction G₀ α inst✝⁶ : ContinuousConstSMul G₀ α inst✝⁵ : TopologicalSpace β f : β → α b : β c✝ : G₀ s✝ : Set β E : Type u_1 inst✝⁴ : Zero E inst✝³ : MulActionWithZero G₀ E inst✝² : TopologicalSpace E inst✝¹ : T1Space E inst✝ : ContinuousConstSMul G₀ E c : G₀ s : Set E hc : c ≠ 0 ⊢ closure (c • s) = c • closure s State After: no goals Tactic: exact ((Homeomorph.smulOfNeZero c hc).image_closure s).symm
If $f$ and $g$ converge to $a$ and $b$, respectively, then $f \cdot g$ converges to $a \cdot b$.
{-# OPTIONS --cubical --no-import-sorts --no-exact-split --safe #-} {- This file shows that the property of the natural numbers being a homotopy-initial algebra of the functor (1 + _) is equivalent to fulfilling a closely related inductive elimination principle. Proofing the latter is trivial, since the typechecker does the work for us. For details see the paper [Homotopy-initial algebras in type theory](https://arxiv.org/abs/1504.05531) by Steve Awodey, Nicola Gambino and Kristina Sojakova. -} module Cubical.Data.Nat.Algebra where open import Cubical.Core.Everything open import Cubical.Foundations.Prelude open import Cubical.Foundations.Function open import Cubical.Foundations.HLevels open import Cubical.Foundations.GroupoidLaws open import Cubical.Foundations.Path open import Cubical.Foundations.Isomorphism hiding (section) open import Cubical.Foundations.Transport open import Cubical.Foundations.Univalence open import Cubical.Reflection.StrictEquiv open import Cubical.Data.Nat.Base private variable ℓ ℓ' : Level record NatAlgebra ℓ : Type (ℓ-suc ℓ) where field Carrier : Type ℓ alg-zero : Carrier alg-suc : Carrier → Carrier record NatMorphism (A : NatAlgebra ℓ) (B : NatAlgebra ℓ') : Type (ℓ-max ℓ ℓ') where open NatAlgebra field morph : A .Carrier → B .Carrier comm-zero : morph (A .alg-zero) ≡ B .alg-zero comm-suc : morph ∘ A .alg-suc ≡ B .alg-suc ∘ morph record NatFiber (N : NatAlgebra ℓ') ℓ : Type (ℓ-max ℓ' (ℓ-suc ℓ)) where open NatAlgebra N field Fiber : Carrier → Type ℓ fib-zero : Fiber alg-zero fib-suc : ∀ {n} → Fiber n → Fiber (alg-suc n) record NatSection {N : NatAlgebra ℓ'} (F : NatFiber N ℓ) : Type (ℓ-max ℓ' ℓ) where open NatAlgebra N open NatFiber F field section : ∀ n → Fiber n sec-comm-zero : section alg-zero ≡ fib-zero sec-comm-suc : ∀ n → section (alg-suc n) ≡ fib-suc (section n) isNatHInitial : NatAlgebra ℓ' → (ℓ : Level) → Type (ℓ-max ℓ' (ℓ-suc ℓ)) isNatHInitial N ℓ = (M : NatAlgebra ℓ) → isContr (NatMorphism N M) isNatInductive : NatAlgebra ℓ' → (ℓ : Level) → Type (ℓ-max ℓ' (ℓ-suc ℓ)) isNatInductive N ℓ = (S : NatFiber N ℓ) → NatSection S module AlgebraPropositionality {N : NatAlgebra ℓ'} where open NatAlgebra N isPropIsNatHInitial : isProp (isNatHInitial N ℓ) isPropIsNatHInitial = isPropΠ (λ _ → isPropIsContr) -- under the assumption that some shape is nat-inductive, the type of sections over any fiber -- is propositional module SectionProp (ind : isNatInductive N ℓ) {F : NatFiber N ℓ} (S T : NatSection F) where open NatFiber open NatSection ConnectingFiber : NatFiber N ℓ Fiber ConnectingFiber n = S .section n ≡ T .section n fib-zero ConnectingFiber = S .sec-comm-zero ∙∙ refl ∙∙ sym (T .sec-comm-zero) fib-suc ConnectingFiber {n} sntn = S .sec-comm-suc n ∙∙ (λ i → F .fib-suc (sntn i)) ∙∙ sym (T .sec-comm-suc n) open NatSection (ind ConnectingFiber) renaming (section to α ; sec-comm-zero to ζ ; sec-comm-suc to σ) squeezeSquare : ∀{a}{A : Type a}{w x y z : A} (p : w ≡ x) {q : x ≡ y} (r : z ≡ y) → (P : w ≡ z) → (sq : P ≡ p ∙∙ q ∙∙ sym r) → I → I → A squeezeSquare p {q} r P sq i j = transport (sym (PathP≡doubleCompPathʳ p P q r)) sq i j S≡T : S ≡ T section (S≡T i) n = α n i sec-comm-zero (S≡T i) j = squeezeSquare (S .sec-comm-zero) (T .sec-comm-zero) (α alg-zero) ζ j i sec-comm-suc (S≡T i) n j = squeezeSquare (S .sec-comm-suc n) (T .sec-comm-suc n) (α (alg-suc n)) (σ n) j i isPropIsNatInductive : isProp (isNatInductive N ℓ) isPropIsNatInductive a b i F = SectionProp.S≡T a (a F) (b F) i module AlgebraHInd→HInit {N : NatAlgebra ℓ'} (ind : isNatInductive N ℓ) (M : NatAlgebra ℓ) where open NatAlgebra open NatFiber ConstFiberM : NatFiber N ℓ Fiber ConstFiberM _ = M .Carrier fib-zero ConstFiberM = M .alg-zero fib-suc ConstFiberM = M .alg-suc morph→section : NatMorphism N M → NatSection ConstFiberM morph→section x = record { section = morph ; sec-comm-zero = comm-zero ; sec-comm-suc = λ i n → comm-suc n i } where open NatMorphism x section→morph : NatSection ConstFiberM → NatMorphism N M section→morph x = record { morph = section ; comm-zero = sec-comm-zero ; comm-suc = λ n i → sec-comm-suc i n } where open NatSection x Morph≡Section : NatSection ConstFiberM ≡ NatMorphism N M Morph≡Section = ua e where unquoteDecl e = declStrictEquiv e section→morph morph→section isContrMorph : isContr (NatMorphism N M) isContrMorph = subst isContr Morph≡Section (inhProp→isContr (ind ConstFiberM) (AlgebraPropositionality.SectionProp.S≡T ind)) open NatAlgebra open NatFiber open NatSection open NatMorphism module AlgebraHInit→Ind (N : NatAlgebra ℓ') ℓ (hinit : isNatHInitial N (ℓ-max ℓ' ℓ)) (F : NatFiber N (ℓ-max ℓ' ℓ)) where ΣAlgebra : NatAlgebra (ℓ-max ℓ' ℓ) Carrier ΣAlgebra = Σ (N .Carrier) (F .Fiber) alg-zero ΣAlgebra = N .alg-zero , F .fib-zero alg-suc ΣAlgebra (n , fn) = N .alg-suc n , F .fib-suc fn -- the fact that we have to lift the Carrier obstructs readability a bit -- this is the same algebra as N, but lifted into the correct universe LiftN : NatAlgebra (ℓ-max ℓ' ℓ) Carrier LiftN = Lift {_} {ℓ} (N .Carrier) alg-zero LiftN = lift (N .alg-zero) alg-suc LiftN = lift ∘ N .alg-suc ∘ lower _!_ : ∀ {x y} → x ≡ y → F .Fiber x → F .Fiber y _!_ = subst (F .Fiber) -- from homotopy initiality of N we get -- 1) an algebra morphism μ from N → Σ N F together with proofs of commutativity with the algebras -- 2) projecting out the first component after μ, called α, will turn out to be the identity function -- 3) witnesses that μ respects the definitions given in ΣAlgebra -- a) at zero the witnesses are ζ and ζ-h -- b) at suc the witnesses are σ and σ-h open NatMorphism (hinit ΣAlgebra .fst) renaming (morph to μ ; comm-zero to μ-zc ; comm-suc to μ-sc) module _ n where open Σ (μ n) public renaming (fst to α ; snd to α-h) -- module _ i where open Σ (μ-zc i) public renaming (fst to ζ ; snd to ζ-h) ζ : α (N .alg-zero) ≡ N .alg-zero ζ i = μ-zc i .fst ζ-h : PathP (λ i → F .Fiber (ζ i)) (α-h (N .alg-zero)) (F .fib-zero) ζ-h i = μ-zc i .snd -- module _ n i where open Σ (μ-sc i n) public renaming (fst to σ ; snd to σ-h) σ : ∀ n → α (N .alg-suc n) ≡ N .alg-suc (α n) σ n i = μ-sc i n .fst σ-h : ∀ n → PathP (λ i → F .Fiber (σ n i)) (α-h (N .alg-suc n)) (F .fib-suc (α-h n)) σ-h n i = μ-sc i n .snd -- liftMorph would be the identity morphism if it weren't for size issues liftMorph : NatMorphism N LiftN liftMorph = record { morph = lift ; comm-zero = refl ; comm-suc = refl } -- instead of abstractly defining morphism composition and a projection algebra morphism -- from Σ N F → N, define the composite directly. comm-zero and comm-suc thus are -- defined without path composition fst∘μ : NatMorphism N LiftN morph fst∘μ = lift ∘ α comm-zero fst∘μ i = lift (ζ i) comm-suc fst∘μ i n = lift (σ n i) fst∘μ≡id : fst∘μ ≡ liftMorph fst∘μ≡id = isContr→isProp (hinit LiftN) _ _ -- we get a proof that the index is preserved uniformly P : ∀ n → α n ≡ n P n i = lower (fst∘μ≡id i .morph n) -- we also have proofs that α cancels after the algebra of N Q-zero : α (N .alg-zero) ≡ N .alg-zero Q-zero = ζ Q-suc : ∀ n → α (N .alg-suc n) ≡ N .alg-suc n Q-suc n = σ n ∙ cong (N .alg-suc) (P n) -- but P and Q are the same up to homotopy P-zero : P (N .alg-zero) ≡ Q-zero P-zero i j = hcomp (λ k → λ where (i = i0) → lower (fst∘μ≡id j .comm-zero (~ k)) (i = i1) → ζ (j ∨ ~ k) (j = i0) → ζ (~ k) (j = i1) → N .alg-zero ) (N .alg-zero) P-suc : ∀ n → P (N .alg-suc n) ≡ Q-suc n P-suc n i j = hcomp (λ k → λ where (i = i0) → lower (fst∘μ≡id j .comm-suc (~ k) n) (i = i1) → compPath-filler' (σ n) (cong (N .alg-suc) (P n)) k j (j = i0) → σ n (~ k) (j = i1) → N .alg-suc n ) (N .alg-suc (P n j)) Fsection : NatSection F section Fsection n = P n ! α-h n sec-comm-zero Fsection = P (N .alg-zero) ! α-h (N .alg-zero) ≡[ i ]⟨ P-zero i ! α-h _ ⟩ Q-zero ! α-h (N .alg-zero) ≡⟨ fromPathP ζ-h ⟩ F .fib-zero ∎ sec-comm-suc Fsection n = P (N .alg-suc n) ! α-h (N .alg-suc n) ≡[ i ]⟨ P-suc n i ! α-h _ ⟩ Q-suc n ! α-h (N .alg-suc n) ≡⟨ substComposite (F .Fiber) (σ n) (cong (N .alg-suc) (P n)) _ ⟩ cong (N .alg-suc) (P n) ! (σ n ! α-h (N .alg-suc n)) ≡[ i ]⟨ cong (N .alg-suc) (P n) ! fromPathP (σ-h n) i ⟩ cong (N .alg-suc) (P n) ! (F .fib-suc (α-h n)) ≡⟨ substCommSlice (F .Fiber) (F .Fiber ∘ N .alg-suc) (λ _ → F .fib-suc) (P n) (α-h n) ⟩ F .fib-suc (P n ! α-h n) ∎ isNatInductive≡isNatHInitial : {N : NatAlgebra ℓ'} (ℓ : Level) → isNatInductive N (ℓ-max ℓ' ℓ) ≡ isNatHInitial N (ℓ-max ℓ' ℓ) isNatInductive≡isNatHInitial {_} {N} ℓ = hPropExt isPropIsNatInductive isPropIsNatHInitial ind→init init→ind where open AlgebraPropositionality open AlgebraHInit→Ind N ℓ renaming (Fsection to init→ind) open AlgebraHInd→HInit renaming (isContrMorph to ind→init) -- given two homotopy initial algebras there is a path between the algebras -- this implies moreover that the carrier types are isomorphic -- according to 5.16 in the paper this could be strengthened to isContr (N ≡ M) isNatHInitial→algebraPath : {N M : NatAlgebra ℓ} → (hinitN : isNatHInitial N ℓ) (hinitM : isNatHInitial M ℓ) → N ≡ M isNatHInitial→algebraPath {N = N} {M} hinitN hinitM = N≡M where open Σ (hinitN M) renaming (fst to N→M) open Σ (hinitM N) renaming (fst to M→N) idN : NatMorphism N N idN = record { morph = λ x → x ; comm-zero = refl ; comm-suc = refl } idM : NatMorphism M M idM = record { morph = λ x → x ; comm-zero = refl ; comm-suc = refl } N→M→N : NatMorphism N N morph N→M→N = morph M→N ∘ morph N→M comm-zero N→M→N = (λ i → morph M→N (comm-zero N→M i)) ∙ comm-zero M→N comm-suc N→M→N = (λ i → morph M→N ∘ comm-suc N→M i) ∙ (λ i → comm-suc M→N i ∘ morph N→M) nmn≡idn : N→M→N ≡ idN nmn≡idn = isContr→isProp (hinitN N) _ _ M→N→M : NatMorphism M M morph M→N→M = morph N→M ∘ morph M→N comm-zero M→N→M = (λ i → morph N→M (comm-zero M→N i)) ∙ comm-zero N→M comm-suc M→N→M = (λ i → morph N→M ∘ comm-suc M→N i) ∙ (λ i → comm-suc N→M i ∘ morph M→N) mnm≡idm : M→N→M ≡ idM mnm≡idm = isContr→isProp (hinitM M) _ _ carrier≡ : N .Carrier ≡ M .Carrier carrier≡ = isoToPath (iso (N→M .morph) (M→N .morph) (λ x i → mnm≡idm i .morph x) (λ x i → nmn≡idn i .morph x)) zero≡ : PathP (λ i → carrier≡ i) (N .alg-zero) (M .alg-zero) zero≡ = toPathP λ i → transportRefl (N→M .comm-zero i) i suc≡ : PathP (λ i → carrier≡ i → carrier≡ i) (N .alg-suc) (M .alg-suc) suc≡ = toPathP ( transport refl ∘ N→M .morph ∘ N .alg-suc ∘ M→N .morph ∘ transport refl ≡[ i ]⟨ transportReflF i ∘ N→M .morph ∘ N .alg-suc ∘ M→N .morph ∘ transportReflF i ⟩ N→M .morph ∘ N .alg-suc ∘ M→N .morph ≡[ i ]⟨ N→M .comm-suc i ∘ M→N .morph ⟩ M .alg-suc ∘ N→M .morph ∘ M→N .morph ≡[ i ]⟨ M .alg-suc ∘ mnm≡idm i .morph ⟩ M .alg-suc ∎) where transportReflF : transport refl ≡ (λ x → x) transportReflF = funExt transportRefl N≡M : N ≡ M Carrier (N≡M i) = carrier≡ i alg-zero (N≡M i) = zero≡ i alg-suc (N≡M i) = suc≡ i -- the payoff, it is straight forward to define the algebra and show inductiveness of ℕ NatAlgebraℕ : NatAlgebra ℓ-zero Carrier NatAlgebraℕ = ℕ alg-zero NatAlgebraℕ = zero alg-suc NatAlgebraℕ = suc isNatInductiveℕ : isNatInductive NatAlgebraℕ ℓ section (isNatInductiveℕ F) = nat-sec where nat-sec : ∀ n → F .Fiber n nat-sec zero = F .fib-zero nat-sec (suc n) = F .fib-suc (nat-sec n) sec-comm-zero (isNatInductiveℕ F) = refl sec-comm-suc (isNatInductiveℕ F) n = refl isNatHInitialℕ : isNatHInitial NatAlgebraℕ ℓ isNatHInitialℕ = transport (isNatInductive≡isNatHInitial _) isNatInductiveℕ
#ifndef PROJECTION_HPP #define PROJECTION_HPP #include <Eigen/Core> class Projection { public: Eigen::Matrix4d getTransformation(); double getLeft(); double getRight(); double getBottom(); double getTop(); double getNear(); double getFar(); protected: virtual void generateTransformation() = 0; Eigen::Matrix4d transformation; double left; double right; double bottom; double top; double near; double far; }; #endif // PROJECTION_HPP
module OutsideIn.Prelude where open import Data.Nat public open import Relation.Binary.PropositionalEquality public renaming ([_] to iC) open import Relation.Nullary public open import Function public hiding (case_of_) cong₃ : ∀ {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} (f : A → B → C → D) {x y u v x′ y′} → x ≡ y → u ≡ v → x′ ≡ y′ → f x u x′ ≡ f y v y′ cong₃ f refl refl refl = refl cong₄ : ∀ {a b c d e} {A : Set a} {B : Set b} {C : Set c} {D : Set d} {E : Set e} (f : A → B → C → D → E) {x y u v x′ y′ u′ v′} → x ≡ y → u ≡ v → x′ ≡ y′ → u′ ≡ v′ → f x u x′ u′ ≡ f y v y′ v′ cong₄ f refl refl refl refl = refl import Level postulate extensionality : Extensionality Level.zero Level.zero module Shapes where -- Used throughout to show structure preserving operations preserve structure, -- in order to maintain structural recursion. data Shape : Set where Nullary : Shape Unary : Shape → Shape Binary : Shape → Shape → Shape module Functors where import Data.Vec as V open V using (_∷_; Vec; []) isIdentity : ∀ {A} → (A → A) → Set isIdentity {A} f = ∀ {x} → f x ≡ x id-is-id : ∀ {A} → isIdentity {A} id id-is-id = refl record Functor (X : Set → Set) : Set₁ where field map : ∀ {A B} → (A → B) → X A → X B _<$>_ : ∀ {A B} → (A → B) → X A → X B _<$>_ = map field identity : ∀ {A : Set}{f : A → A} → isIdentity f → isIdentity (map f) field composite : ∀ {A B C : Set} {f : A → B} {g : B → C} → {x : X A} → ( (g ∘ f) <$> x ≡ g <$> (f <$> x)) Pointed : (Set → Set) → Set₁ Pointed X = ∀ {a} → a → X a id-is-functor : Functor id id-is-functor = record { map = id; identity = id; composite = refl } vec-is-functor : ∀ {n} → Functor (λ A → Vec A n) vec-is-functor {n} = record { map = V.map; identity = ident {n} ; composite = composite } where ident : {n : ℕ}{A : Set} {f : A → A} → isIdentity f →{x : Vec A n} → V.map f x ≡ x ident isid {[]} = refl ident isid {x ∷ xs} = cong₂ _∷_ isid (ident isid) composite : {A B C : Set}{n : ℕ} {f : A → B} {g : B → C} {x : Vec A n} → V.map (g ∘ f) x ≡ V.map g (V.map f x) composite {x = []} = refl composite {x = x ∷ xs} = cong₂ _∷_ refl composite private module F = Functor ⦃ ... ⦄ combine-composite′ : {X Y : Set → Set}{A B C : Set} ⦃ F2 : Functor Y ⦄ {V : X (Y A)}{f : A → B}{g : B → C} → (f1map : ∀ {a b} → (a → b) → (X a → X b)) → (f1comp : f1map (F.map g ∘ F.map f) V ≡ f1map (F.map g) (f1map (F.map f) V)) → f1map (F.map ⦃ F2 ⦄ (g ∘ f)) V ≡ f1map (F.map ⦃ F2 ⦄ g) (f1map (F.map ⦃ F2 ⦄ f) V) combine-composite′ ⦃ F2 ⦄ {V} f1map f1comp = trans (cong (λ t → f1map t V) (extensionality (λ x → F.composite ⦃ F2 ⦄))) f1comp combine-composite : {X Y : Set → Set}{A B C : Set} ⦃ F1 : Functor X ⦄ ⦃ F2 : Functor Y ⦄ {V : X (Y A)}{f : A → B}{g : B → C} → F.map ⦃ F1 ⦄ (F.map ⦃ F2 ⦄ (g ∘ f)) V ≡ F.map ⦃ F1 ⦄ (F.map ⦃ F2 ⦄ g) (F.map ⦃ F1 ⦄ (F.map ⦃ F2 ⦄ f) V) combine-composite {X}{Y} ⦃ F1 ⦄ ⦃ F2 ⦄ {V} = combine-composite′ {X}{Y} (F.map ⦃ F1 ⦄) (F.composite ⦃ F1 ⦄) infixr 6 _∘f_ _∘f_ : {X Y : Set → Set} → ( F1 : Functor X )( F2 : Functor Y ) → Functor (X ∘ Y) F1 ∘f F2 = record { map = F.map ⦃ F1 ⦄ ∘ F.map ⦃ F2 ⦄ ; composite = combine-composite ⦃ F1 ⦄ ⦃ F2 ⦄ ; identity = F.identity ⦃ F1 ⦄ ∘ F.identity ⦃ F2 ⦄ } module StupidEquality where open import Data.Bool public using (Bool; true; false) -- This equality doesn't place any proof demands -- because we don't actually care what equality is used. -- This is just for the initial base of type variables, where the user provides -- their own type equality relation. We don't care if it says Int ∼ Bool - this just -- provides a way for users to get some equality information threaded through the -- simplifier Eq : Set → Set Eq X = ∀ (a b : X) → Bool module Monads where open Functors record Monad (X : Set → Set) : Set₁ where field ⦃ is-functor ⦄ : Functor X field ⦃ point ⦄ : Pointed X open Functor is-functor field join : ∀ {a} → X (X a) → X a unit : ∀ {a} → a → X a unit = point _>>=_ : ∀ {a b} → X a → (a → X b) → X b _>>=_ a b = join (b <$> a) _>>_ : ∀ {a b} → X a → X b → X b _>>_ a b = a >>= λ _ → b _>=>_ : ∀ {a b c : Set} → (b → X c) → (a → X b) → (a → X c) _>=>_ a b = λ v → b v >>= a field is-left-ident : ∀ {a b}{x : a → X b}{v} → (point >=> x) v ≡ x v field is-right-ident : ∀ {a b}{x : a → X b}{v} → (x >=> point) v ≡ x v field >=>-assoc : ∀{p}{q}{r}{s}{a : r → X s}{b : q → X r} {c : p → X q}{v} → (a >=> (b >=> c)) v ≡ ((a >=> b) >=> c) v abstract <$>-unit : ∀ {A B}{g : A → B}{x} → g <$> (unit x) ≡ unit (g x) <$>-unit {A}{B}{g}{x} = begin g <$> (unit x) ≡⟨ sym (is-left-ident {x = _<$>_ g}) ⟩ join (unit <$> (g <$> (unit x))) ≡⟨ cong join (sym (composite)) ⟩ join ((λ x → unit (g x)) <$> (unit x)) ≡⟨ is-right-ident ⟩ unit (g x) ∎ where open ≡-Reasoning abstract <$>-bind : ∀ {A B C}{f : A → B}{g : B → X C}{x : X A} → (f <$> x) >>= g ≡ x >>= (λ z → g (f z)) <$>-bind = cong join (sym (composite)) natural-trans : ∀ {A B}{f : A → B}{x : X( X A)} → f <$> (join x) ≡ join ((_<$>_ f) <$> x) natural-trans {A}{B}{f}{x} = begin f <$> (join x) ≡⟨ sym (is-left-ident {x = _<$>_ f}) ⟩ join (unit <$> (f <$> (join x))) ≡⟨ † ⟩ join (unit <$> (f <$> (join (id <$> x)))) ≡⟨ <$>-bind ⟩ join ((λ v → unit (f v)) <$> (join (id <$> x))) ≡⟨ >=>-assoc { c = λ _ → x}{0} ⟩ join ((λ x → join ((unit ∘ f) <$> x)) <$> x ) ≡⟨ sym (<$>-bind) ⟩ ((_<$>_ (λ y → unit (f y))) <$> x ) >>= join ≡⟨ <$>-bind ⟩ x >>= (λ x → x >>= (λ y → unit (f y))) ≡⟨ * ⟩ join ((_<$>_ f) <$> x) ∎ where open ≡-Reasoning † : join (unit <$> (f <$> (join x))) ≡ join (unit <$> (f <$> (join (id <$> x)))) † = cong (λ t → join (unit <$> (f <$> join t))) (sym (identity id-is-id )) * : x >>= (λ x → x >>= (λ y → unit (f y))) ≡ join ((_<$>_ f) <$> x) * = cong (_>>=_ x) (extensionality (λ y → trans (sym <$>-bind) (is-left-ident {x = _<$>_ f}{v = y}))) id-is-monad : Monad id id-is-monad = record { is-functor = id-is-functor ; point = id ; join = id ; >=>-assoc = refl ; is-left-ident = refl ; is-right-ident = refl } record MonadHomomorphism {M₁ M₂ : Set → Set}(h : ∀ {x : Set} → M₁ x → M₂ x) ⦃ M₁-m : Monad M₁ ⦄ ⦃ M₂-m : Monad M₂ ⦄ : Set₁ where open Monad M₁-m using () renaming (unit to unit₁; join to join₁; is-functor to is-functor₁) open Monad M₂-m using () renaming (unit to unit₂; join to join₂; is-functor to is-functor₂) open Functor is-functor₁ using () renaming (map to map₁) open Functor is-functor₂ using () renaming (map to map₂) field h-return : ∀ {A}{x : A} → h (unit₁ x) ≡ unit₂ x field h-fmap : {A B : Set} {f : A → B} {x : M₁ A} → h (map₁ f x) ≡ map₂ f (h x) field h-join : ∀{τ}{x : M₁ (M₁ τ)} → h (join₁ x) ≡ join₂ (h (map₁ h x)) record MonadTrans (X : (Set → Set) → Set → Set) : Set₁ where field produces-monad : ∀ {m} → Monad m → Monad (X m) field lift : ∀ {m}⦃ mm : Monad m ⦄{a} → m a → X m a field is-homomorphism : ∀ {m} → (mm : Monad m) → MonadHomomorphism {m} {X m} (lift {m}) ⦃ mm ⦄ ⦃ produces-monad mm ⦄ module Ⓢ-Type where open Functors open Monads open StupidEquality data Ⓢ (τ : Set) : Set where suc : τ → Ⓢ τ zero : Ⓢ τ cata-Ⓢ : {a b : Set} → b → (a → b) → Ⓢ a → b cata-Ⓢ nil something zero = nil cata-Ⓢ nil something (suc n) = something n sequence-Ⓢ : ∀ {m}{b} → ⦃ monad : Monad m ⦄ → Ⓢ (m b) → m (Ⓢ b) sequence-Ⓢ ⦃ m ⦄ (suc n) = map suc n where open Functor (Monad.is-functor m) sequence-Ⓢ ⦃ m ⦄ (zero) = unit zero where open Monad (m) private fmap-Ⓢ : ∀ {a b} → (a → b) → Ⓢ a → Ⓢ b fmap-Ⓢ f zero = zero fmap-Ⓢ f (suc n) = suc (f n) abstract fmap-Ⓢ-id : ∀ {A} → {f : A → A} → isIdentity f → isIdentity (fmap-Ⓢ f) fmap-Ⓢ-id isid {zero} = refl fmap-Ⓢ-id isid {suc x} = cong suc isid fmap-Ⓢ-comp : ∀ {A B C : Set} {f : A → B} {g : B → C} → ∀ {x} → fmap-Ⓢ (g ∘ f) x ≡ fmap-Ⓢ g (fmap-Ⓢ f x) fmap-Ⓢ-comp {x = zero} = refl fmap-Ⓢ-comp {x = suc n} = refl Ⓢ-is-functor : Functor Ⓢ Ⓢ-is-functor = record { map = fmap-Ⓢ ; identity = fmap-Ⓢ-id ; composite = fmap-Ⓢ-comp } Ⓢ-eq : ∀ {x} → Eq x → Eq (Ⓢ x) Ⓢ-eq x zero zero = true Ⓢ-eq x (suc n) zero = false Ⓢ-eq x zero (suc m) = false Ⓢ-eq x (suc n) (suc m) = x n m private join-Ⓢ : ∀ {x} → Ⓢ (Ⓢ x) → Ⓢ x join-Ⓢ (zero) = zero join-Ⓢ (suc τ) = τ test-join : ∀ {A B}{f : A → B}{x : Ⓢ( Ⓢ A)} → fmap-Ⓢ f (join-Ⓢ x) ≡ join-Ⓢ (fmap-Ⓢ (fmap-Ⓢ f) x) test-join {x = zero} = refl test-join {x = suc n} = refl Ⓢ-is-monad : Monad Ⓢ Ⓢ-is-monad = record { is-functor = Ⓢ-is-functor ; point = suc ; join = join-Ⓢ ; is-left-ident = left-id ; is-right-ident = refl ; >=>-assoc = λ { {c = c}{v} → assoc {τ = c v} } } where left-id : ∀ {τ : Set}{v : Ⓢ τ} → join-Ⓢ (fmap-Ⓢ suc v) ≡ v left-id {v = zero } = refl left-id {v = suc v} = refl assoc : ∀ {q r s : Set} {a : r → Ⓢ s} {b : q → Ⓢ r}{τ : Ⓢ q} → join-Ⓢ (fmap-Ⓢ a (join-Ⓢ (fmap-Ⓢ b τ))) ≡ join-Ⓢ (fmap-Ⓢ (λ v′ → join-Ⓢ (fmap-Ⓢ a (b v′))) τ) assoc {τ = zero} = refl assoc {τ = suc v} = refl Ⓢ-Trans : (Set → Set) → Set → Set Ⓢ-Trans m x = m (Ⓢ x) private lift : ∀ {m : Set → Set}⦃ mm : Monad m ⦄{x} → m x → m (Ⓢ x) lift {m}⦃ mm ⦄{x} v = suc <$> v where open Monad mm open Functor is-functor module MonadProofs {m : Set → Set}⦃ mm : Monad m ⦄ where open Monad mm open Functor is-functor functor : Functor (Ⓢ-Trans m) functor = record { map = λ f v → (fmap-Ⓢ f) <$> v ; identity = λ p → identity (fmap-Ⓢ-id p) ; composite = λ { {x = x} → trans (cong (λ t → t <$> x) (extensionality ext)) composite } } where open ≡-Reasoning ext : ∀ {A B C : Set} {f : A → B} {g : B → C} → (x' : Ⓢ A) → fmap-Ⓢ (g ∘ f) x' ≡ (fmap-Ⓢ g ∘ fmap-Ⓢ f) x' ext (zero) = refl ext (suc n) = refl module Trans = Functor functor private cata-Ⓢ-u0 : ∀ {a b} → (a → m (Ⓢ b)) → Ⓢ a → m (Ⓢ b) cata-Ⓢ-u0 = cata-Ⓢ (unit zero) abstract right-id : {a b : Set} {x : a → Ⓢ-Trans m b} {v : a} → Trans.map x (lift {m} (unit v)) >>= cata-Ⓢ-u0 id ≡ x v right-id {a}{b}{x}{v} = begin Trans.map x (suc <$> (unit v)) >>= cata-Ⓢ-u0 id ≡⟨ * ⟩ (fmap-Ⓢ x <$> (unit (suc v))) >>= cata-Ⓢ-u0 id ≡⟨ † ⟩ join (cata-Ⓢ-u0 id <$> (unit (suc (x v)))) ≡⟨ cong join <$>-unit ⟩ join (unit (x v)) ≡⟨ cong join (sym(identity id-is-id)) ⟩ join (id <$> unit (x v)) ≡⟨ is-right-ident {x = id} ⟩ x v ∎ where open ≡-Reasoning * : Trans.map x (suc <$> (unit v)) >>= cata-Ⓢ-u0 id ≡ (fmap-Ⓢ x <$> (unit (suc v))) >>= cata-Ⓢ-u0 id * = cong (λ t → Trans.map x t >>= cata-Ⓢ-u0 id) <$>-unit † : (fmap-Ⓢ x <$> (unit (suc v))) >>= cata-Ⓢ-u0 id ≡ join (cata-Ⓢ-u0 id <$> (unit (suc (x v)))) † = cong (λ x → x >>= cata-Ⓢ-u0 id) <$>-unit left-id : {b : Set} {t : Ⓢ-Trans m b} → Trans.map (λ x' → lift {m} (unit x')) t >>= cata-Ⓢ-u0 id ≡ t left-id {b}{t} = trans <$>-bind (subst (λ q → t >>= q ≡ t) (sym (extensionality h≗unit)) (is-left-ident {x = λ _ → t} {v = 0})) where h : ∀ {A} → Ⓢ A → m (Ⓢ A) h x = cata-Ⓢ-u0 id (fmap-Ⓢ (λ x' → suc <$> (unit x')) x) h≗unit : ∀ {A} → h {A} ≗ unit h≗unit zero = refl h≗unit (suc y) = <$>-unit assoc : ∀ {p q r s : Set} {a : r → Ⓢ-Trans m s}{b : q → Ⓢ-Trans m r}{c : p → Ⓢ-Trans m q} {v : p} → Trans.map a (Trans.map b (c v) >>= cata-Ⓢ-u0 id) >>= cata-Ⓢ-u0 id ≡ Trans.map (λ v' → Trans.map a (b v') >>= cata-Ⓢ-u0 id) (c v) >>= cata-Ⓢ-u0 id assoc {p}{q}{r}{s}{a}{b}{c}{v} = let †₀ = cata-fmap †₁ = cong (λ x → x >>= cata-Ⓢ (unit zero) a) cata-fmap †₂ = >=>-assoc {c = λ _ → c v} {v = 0} †₃ = cong (_>>=_ (c v)) (extensionality ext) †₄ = cong (λ x → c v >>= cata-Ⓢ (unit zero) x) (extensionality (λ x → sym cata-fmap)) †₅ = sym (cata-fmap) in begin Trans.map a (Trans.map b (c v) >>= cata-Ⓢ-u0 id) >>= cata-Ⓢ-u0 id ≡⟨ †₀ ⟩ ((fmap-Ⓢ b <$> c v) >>= cata-Ⓢ-u0 id) >>= cata-Ⓢ-u0 a ≡⟨ †₁ ⟩ (c v >>= cata-Ⓢ-u0 b) >>= cata-Ⓢ-u0 a ≡⟨ †₂ ⟩ c v >>= (λ cv → cata-Ⓢ-u0 b cv >>= cata-Ⓢ-u0 a) ≡⟨ †₃ ⟩ c v >>= cata-Ⓢ-u0 (λ v' → b v' >>= cata-Ⓢ-u0 a) ≡⟨ †₄ ⟩ c v >>= cata-Ⓢ-u0 (λ v' → (fmap-Ⓢ a <$> b v') >>= cata-Ⓢ-u0 id) ≡⟨ †₅ ⟩ Trans.map (λ v' → Trans.map a (b v') >>= cata-Ⓢ-u0 id) (c v) >>= cata-Ⓢ-u0 id ∎ where open ≡-Reasoning ext : (x : Ⓢ q) → cata-Ⓢ-u0 b x >>= cata-Ⓢ-u0 a ≡ cata-Ⓢ-u0 (λ v' → b v' >>= cata-Ⓢ-u0 a) x ext zero = begin join ((cata-Ⓢ-u0 a) <$> unit zero) ≡⟨ cong join <$>-unit ⟩ join (unit (unit zero)) ≡⟨ cong join (sym (identity id-is-id)) ⟩ join (id <$> unit (unit zero)) ≡⟨ is-right-ident ⟩ unit zero ∎ ext (suc n) = refl cata-fmap : ∀{A B C}{a : A → B}{x : m (Ⓢ A)}{n : m C}{j : B → m C} → (fmap-Ⓢ a <$> x) >>= cata-Ⓢ n j ≡ x >>= cata-Ⓢ n (λ x → j ( a x)) cata-fmap {A}{B}{C}{a}{x}{n}{j} = trans <$>-bind (cong (_>>=_ x) (extensionality ext′)) where ext′ : (x' : Ⓢ A) → cata-Ⓢ n j (fmap-Ⓢ a x') ≡ cata-Ⓢ n (λ x0 → j (a x0)) x' ext′ zero = refl ext′ (suc n) = refl produces-monad = record { point = λ x → lift ⦃ mm ⦄ (unit x) ; is-functor = functor ; join = λ v → v >>= cata-Ⓢ (unit zero) id ; is-left-ident = left-id ; is-right-ident = right-id ; >=>-assoc = λ {_}{_}{_}{_}{a}{b}{c}{v} → assoc {a = a}{b}{c}{v} } module HomomorphismProofs {m : Set → Set}⦃ mm : Monad m ⦄ where open Monad mm open Functor is-functor open ≡-Reasoning cata-Ⓢ-u0 : ∀ {a b} → (a → m (Ⓢ b)) → Ⓢ a → m (Ⓢ b) cata-Ⓢ-u0 = cata-Ⓢ (unit zero) fmap-p : ∀ {A B : Set} {f : A → B} {x} → lift {m} (f <$> x) ≡ (fmap-Ⓢ f) <$> (lift {m} x) fmap-p {A}{B}{f}{x} = begin suc <$> (f <$> x) ≡⟨ sym (composite) ⟩ (λ t → suc (f t)) <$> x ≡⟨ refl ⟩ (λ t → fmap-Ⓢ f (suc t)) <$> x ≡⟨ composite ⟩ (fmap-Ⓢ f) <$> (suc <$> x) ∎ join-p : ∀{τ}{x : m (m (τ))} → suc <$> (join x) ≡ (suc <$> (_<$>_ suc <$> x)) >>= cata-Ⓢ-u0 id join-p {_}{x} = begin suc <$> (join x) ≡⟨ natural-trans ⟩ join ((_<$>_ suc) <$> x) ≡⟨ refl ⟩ x >>= (λ z → cata-Ⓢ-u0 id (suc (suc <$> z))) ≡⟨ sym (<$>-bind) ⟩ ((λ z → suc (suc <$> z)) <$> x) >>= cata-Ⓢ-u0 id ≡⟨ † ⟩ (suc <$> (_<$>_ suc <$> x)) >>= cata-Ⓢ-u0 id ∎ where † = cong (λ x → x >>= cata-Ⓢ-u0 id) (composite) is-homomorphism : MonadHomomorphism (lift {m}) ⦃ mm ⦄ ⦃ MonadProofs.produces-monad ⦃ mm ⦄ ⦄ is-homomorphism = record { h-return = refl ; h-fmap = fmap-p ; h-join = join-p } Ⓢ-Trans-is-trans : MonadTrans (Ⓢ-Trans) Ⓢ-Trans-is-trans = record { produces-monad = λ mm → MonadProofs.produces-monad ⦃ mm ⦄ ; lift = λ{m} → lift {m} ; is-homomorphism = λ mm → HomomorphismProofs.is-homomorphism ⦃ mm ⦄ } module PlusN-Type where open Ⓢ-Type open Monads open Functors open StupidEquality PlusN : (n : ℕ) → Set → Set PlusN zero = id PlusN (suc n) = Ⓢ-Trans (PlusN n) PlusN-eq : ∀ {n}{x} → Eq x → Eq (PlusN n x) PlusN-eq {zero} eq = eq PlusN-eq {suc n} eq = PlusN-eq {n} (Ⓢ-eq eq) PlusN-is-monad : ∀ {n} → Monad (PlusN n) PlusN-is-monad {zero} = id-is-monad PlusN-is-monad {suc n} = MonadTrans.produces-monad Ⓢ-Trans-is-trans (PlusN-is-monad {n}) _⨁_ = flip PlusN sequence-PlusN : ∀ {m}{n}{b} → ⦃ monad : Monad m ⦄ → (m b) ⨁ n → m (b ⨁ n) sequence-PlusN {n = zero} x = x sequence-PlusN {n = suc n} ⦃ m ⦄ x = sequence-PlusN {n = n}⦃ m ⦄ (PlusN-f.map (sequence-Ⓢ ⦃ m ⦄) x) where module PlusN-f = Functor (Monad.is-functor (PlusN-is-monad {n})) PlusN-collect : ∀ {n}{a b} → n ⨁ (a + b) ≡ (n ⨁ a) ⨁ b PlusN-collect {n}{zero} = refl PlusN-collect {n}{suc a}{b} = PlusN-collect {Ⓢ n}{a}{b} open Ⓢ-Type public open PlusN-Type public open Functors public open Monads public open Shapes public open StupidEquality public
= = = Cold War = = =
!------------------------------------------------------------------------------- ! !+ Dynamical step ! !------------------------------------------------------------------------------- module mod_dynstep !----------------------------------------------------------------------------- ! !++ Description: ! This module is for the dynamical step ! ! !++ Current Corresponding Author : H.Tomita ! !++ History: ! Version Date Comment ! ----------------------------------------------------------------------- ! 0.00 04-02-17 Imported from igdc-4.34 ! 06-04-17 Add IN_LARGE_STEP2 ! 06-08-11 Add the option for tracer advection. ! 07-01-26 Add flag [rayleigh_damp_only_w] ! in numfilter_rayleigh_damping. ! 07-05-08 H.Tomita : Change the treatment of I_TKE. ! 08-01-24 Y.Niwa: add revised MIURA2004 for tracer advection ! old: 'MIURA2004OLD', revised: 'MIURA2004' ! 08-01-30 Y.Niwa: add rho_pl = 0.D0 ! 08-04-12 T.Mitsui save memory(prgvar, frcvar, rhog0xxxx) ! 08-05-24 T.Mitsui fix miss-conditioning for frcvar ! 08-09-09 Y.Niwa move nudging routine here ! 08-10-05 T.Mitsui all_phystep_post is already needless ! 09-09-08 S.Iga frhog and frhog_pl in ndg are deleted ( suggested by ES staff) ! 10-05-06 M.Satoh: define QV_conv only if CP_TYPE='TDK' .or. 'KUO' ! 10-07-16 A.T.Noda: bug fix for TDK ! 10-08-16 A.T.Noda: Bug fix (Qconv not diveded by density) ! 10-08-20 A.T.Noda: Bug fix (Qconv should be TEND, and not be multiplied by DT) ! 10-11-29 A.T.Noda: Introduce the Smagorinsky model ! 11-08-16 M.Satoh: bug fix for TDK: conv => TEND ! qv_dyn_tend = v grad q ! = ( div(rho v q) - div(rho v)q )/rho ! 11-08-16 M.Satoh: move codes related to CP_TYPE below the tracer calculation ! 11-11-28 Y.Yamada: Merge Terai-san timer into the original code. ! 12-03-09 S.Iga: tuned (phase4-1) ! 12-04-06 T.yamaura: optimized for K ! 12-05-30 T.Yashiro: Change arguments from character to index/switch ! 12-10-22 R.Yoshida : add papi instructions ! 13-06-13 R.Yoshida : add tracer advection mode ! ----------------------------------------------------------------------- ! !----------------------------------------------------------------------------- ! !++ Used modules ! use mod_debug use mod_adm, only: & ADM_LOG_FID !----------------------------------------------------------------------------- implicit none private !----------------------------------------------------------------------------- ! !++ Public procedure ! public :: dynstep !----------------------------------------------------------------------------- ! !++ Public parameters & variables ! !----------------------------------------------------------------------------- ! !++ Private procedures ! !----------------------------------------------------------------------------- ! !++ Private parameters & variables ! integer, private, parameter :: I_RHOG = 1 ! Density x G^{1/2} x gamma^2 integer, private, parameter :: I_RHOGVX = 2 ! Density x G^{1/2} x gamma^2 x Horizontal velocity (X-direction) integer, private, parameter :: I_RHOGVY = 3 ! Density x G^{1/2} x gamma^2 x Horizontal velocity (Y-direction) integer, private, parameter :: I_RHOGVZ = 4 ! Density x G^{1/2} x gamma^2 x Horizontal velocity (Z-direction) integer, private, parameter :: I_RHOGW = 5 ! Density x G^{1/2} x gamma^2 x Vertical velocity integer, private, parameter :: I_RHOGE = 6 ! Density x G^{1/2} x gamma^2 x Internal Energy integer, private, parameter :: I_RHOGETOT = 7 ! Density x G^{1/2} x gamma^2 x Total Energy !----------------------------------------------------------------------------- contains !----------------------------------------------------------------------------- subroutine dynstep use mod_debug use mod_adm, only: & ADM_prc_me, & ADM_prc_pl, & ADM_gall, & ADM_gall_pl, & ADM_lall, & ADM_lall_pl, & ADM_kall, & ADM_gall_1d, & ADM_gmax, & ADM_gmin, & ADM_kmax, & ADM_kmin, & ADM_log_fid, & ! R.Yoshida 13/06/13 [add] ADM_proc_stop ! R.Yoshida 13/06/13 [add] use mod_cnst, only: & CNST_RAIR, & CNST_RVAP, & CNST_CV use mod_time, only: & TIME_INTEG_TYPE, & TIME_SSTEP_MAX, & TIME_DTL, & TIME_DTS, & TIME_SPLIT use mod_grd, only: & GRD_afac, & GRD_bfac use mod_vmtr, only: & VMTR_GSGAM2, & VMTR_GSGAM2_pl, & VMTR_GSGAM2H, & VMTR_GSGAM2H_pl, & VMTR_GZXH, & VMTR_GZXH_pl, & VMTR_GZYH, & VMTR_GZYH_pl, & VMTR_GZZH, & VMTR_GZZH_pl, & VMTR_PHI, & VMTR_PHI_pl, & VMTR_C2Wfact, & VMTR_C2Wfact_pl use mod_comm, only: & COMM_data_transfer use mod_runconf, only: & TRC_VMAX, & I_QV, & I_TKE, & NQW_STR, & NQW_END, & CVW, & NDIFF_LOCATION, & TRC_ADV_TYPE, & FLAG_NUDGING, & ! Y.Niwa add 08/09/09 CP_TYPE, & ! 2010.5.11 M.Satoh [add] TB_TYPE, & ! [add] 10/11/29 A.Noda THUBURN_LIM ! R.Yoshida 13/06/13 [add] use mod_bsstate, only: & pre_bs, pre_bs_pl, & tem_bs, tem_bs_pl, & rho_bs, rho_bs_pl use mod_bndcnd, only: & bndcnd_all use mod_prgvar, only: & prgvar_set, & prgvar_get, & prgvar_get_noq use mod_diagvar, only: & diagvar, & diagvar_pl, & I_RHOGQV_CONV, & I_QV_DYN_TEND ! 2011.08.16 M.Satoh use mod_thrmdyn, only: & thrmdyn_th, & thrmdyn_eth use mod_src, only: & src_advection_convergence_momentum, & src_advection_convergence, & I_SRC_default use mod_vi, only : & vi_small_step use mod_trcadv_thuburn, only: & src_update_tracer use mod_numfilter, only: & NUMFILTER_DOrayleigh, & ! [add] H.Yashiro 20120530 NUMFILTER_DOverticaldiff, & ! [add] H.Yashiro 20120530 numfilter_rayleigh_damping, & numfilter_hdiffusion, & numfilter_vdiffusion !cx remove nudging function !cx use mod_ndg, only: & ! Y.Niwa add 08/09/09 !cx ndg_nudging_uvtp, & !cx ndg_update_var !cx remove tb_smg_driver !cx use mod_tb_smg, only: & ! [add] 10/11/29 A.Noda !cx tb_smg_driver use mod_forcing_driver, only: & updating ! R.Yoshida 13/06/13 [add] implicit none integer, parameter :: nmax_TEND = 7 integer, parameter :: nmax_PROG = 6 integer, parameter :: nmax_v_mean_c = 5 real(8) :: g_TEND (ADM_gall, ADM_kall,ADM_lall, nmax_TEND) !--- tendency real(8) :: g_TEND_pl (ADM_gall_pl,ADM_kall,ADM_lall_pl,nmax_TEND) real(8) :: g_TENDq (ADM_gall, ADM_kall,ADM_lall, TRC_VMAX) !--- tendency of q real(8) :: g_TENDq_pl(ADM_gall_pl,ADM_kall,ADM_lall_pl,TRC_VMAX) real(8) :: f_TEND (ADM_gall, ADM_kall,ADM_lall, nmax_TEND) !--- forcing tendency real(8) :: f_TEND_pl (ADM_gall_pl,ADM_kall,ADM_lall_pl,nmax_TEND) real(8) :: f_TENDq (ADM_gall, ADM_kall,ADM_lall, TRC_VMAX) !--- forcing tendency of q real(8) :: f_TENDq_pl(ADM_gall_pl,ADM_kall,ADM_lall_pl,TRC_VMAX) real(8) :: PROG0 (ADM_gall, ADM_kall,ADM_lall, nmax_PROG) !--- prognostic variables (save) real(8) :: PROG0_pl (ADM_gall_pl,ADM_kall,ADM_lall_pl,nmax_PROG) real(8) :: PROGq0 (ADM_gall, ADM_kall,ADM_lall, TRC_VMAX) !--- tracer variables (save) real(8) :: PROGq0_pl (ADM_gall_pl,ADM_kall,ADM_lall_pl,TRC_VMAX) real(8) :: PROG (ADM_gall, ADM_kall,ADM_lall, nmax_PROG) !--- prognostic variables real(8) :: PROG_pl (ADM_gall_pl,ADM_kall,ADM_lall_pl,nmax_PROG) real(8) :: PROGq (ADM_gall, ADM_kall,ADM_lall, TRC_VMAX) !--- tracer variables real(8) :: PROGq_pl (ADM_gall_pl,ADM_kall,ADM_lall_pl,TRC_VMAX) real(8) :: PROG_split (ADM_gall, ADM_kall,ADM_lall, nmax_PROG) !--- prognostic variables (split) real(8) :: PROG_split_pl(ADM_gall_pl,ADM_kall,ADM_lall_pl,nmax_PROG) real(8) :: v_mean_c (ADM_gall, ADM_kall,ADM_lall ,nmax_v_mean_c) real(8) :: v_mean_c_pl(ADM_gall_pl,ADM_kall,ADM_lall_pl,nmax_v_mean_c) !--- density ( physical ) real(8) :: rho (ADM_gall, ADM_kall,ADM_lall ) real(8) :: rho_pl(ADM_gall_pl,ADM_kall,ADM_lall_pl) !--- horizontal velocity_x ( physical ) real(8) :: vx (ADM_gall, ADM_kall,ADM_lall ) real(8) :: vx_pl(ADM_gall_pl,ADM_kall,ADM_lall_pl) !--- horizontal velocity_y ( physical ) real(8) :: vy (ADM_gall, ADM_kall,ADM_lall ) real(8) :: vy_pl(ADM_gall_pl,ADM_kall,ADM_lall_pl) !--- horizontal velocity_z ( physical ) real(8) :: vz (ADM_gall, ADM_kall,ADM_lall ) real(8) :: vz_pl(ADM_gall_pl,ADM_kall,ADM_lall_pl) !--- vertical velocity ( physical ) real(8) :: w (ADM_gall, ADM_kall,ADM_lall ) real(8) :: w_pl(ADM_gall_pl,ADM_kall,ADM_lall_pl) !--- [IN]ternal energy ( physical ) real(8) :: ein (ADM_gall, ADM_kall,ADM_lall ) real(8) :: ein_pl(ADM_gall_pl,ADM_kall,ADM_lall_pl) !--- mass concentration of water substance ( physical ) real(8) :: q (ADM_gall, ADM_kall,ADM_lall, TRC_VMAX) real(8) :: q_pl(ADM_gall_pl,ADM_kall,ADM_lall_pl,TRC_VMAX) !--- enthalpy ( physical ) real(8) :: eth (ADM_gall, ADM_kall,ADM_lall ) real(8) :: eth_pl(ADM_gall_pl,ADM_kall,ADM_lall_pl) !--- pressure ( physical ) real(8) :: pre (ADM_gall, ADM_kall,ADM_lall ) real(8) :: pre_pl(ADM_gall_pl,ADM_kall,ADM_lall_pl) !--- temperature ( physical ) real(8) :: tem (ADM_gall, ADM_kall,ADM_lall ) real(8) :: tem_pl(ADM_gall_pl,ADM_kall,ADM_lall_pl) !--- potential temperature ( physical ) real(8) :: th (ADM_gall, ADM_kall,ADM_lall ) real(8) :: th_pl(ADM_gall_pl,ADM_kall,ADM_lall_pl) !--- density deviation from the base state ( G^{1/2} X gamma2 ) real(8) :: rhogd (ADM_gall, ADM_kall,ADM_lall ) real(8) :: rhogd_pl(ADM_gall_pl,ADM_kall,ADM_lall_pl) !--- pressure deviation from the base state ( G^{1/2} X gamma2 ) real(8) :: pregd (ADM_gall, ADM_kall,ADM_lall ) real(8) :: pregd_pl(ADM_gall_pl,ADM_kall,ADM_lall_pl) !--- temperature deviation from the base state ( physical ) real(8) :: temd (ADM_gall, ADM_kall,ADM_lall ) real(8) :: temd_pl(ADM_gall_pl,ADM_kall,ADM_lall_pl) !--- temporary variables real(8) :: qd (ADM_gall, ADM_kall,ADM_lall ) real(8) :: qd_pl(ADM_gall_pl,ADM_kall,ADM_lall_pl) real(8) :: cv (ADM_gall, ADM_kall,ADM_lall ) real(8) :: cv_pl(ADM_gall_pl,ADM_kall,ADM_lall_pl) real(8), parameter :: TKE_MIN = 0.D0 real(8) :: TKEg_corr integer :: small_step_ite real(8) :: small_step_dt logical :: ndg_TEND_out logical, save :: iflag = .true. integer, save :: num_of_iteration_lstep ! number of large steps ( 2-4 ) integer, save :: num_of_iteration_sstep(4) ! number of small steps in each of large steps integer :: g, k ,l, nq, nl integer :: i, j, suf suf(i,j) = ADM_gall_1d ((j)-1) + (i) !--------------------------------------------------------------------------- #ifdef PAPI_OPS ! <-- [add] PAPI R.Yoshida 20121022 !call PAPIF_flips( PAPI_real_time_i, PAPI_proc_time_i, PAPI_flpins, PAPI_mflins, PAPI_check ) call PAPIF_flops( PAPI_real_time_o, PAPI_proc_time_o, PAPI_flpops, PAPI_mflops, PAPI_check ) #endif call DEBUG_rapstart('++Dynamics') if ( iflag ) then iflag = .false. select case(trim(TIME_INTEG_TYPE)) case('RK2') num_of_iteration_lstep = 2 num_of_iteration_sstep(1) = TIME_SSTEP_MAX / 2 num_of_iteration_sstep(2) = TIME_SSTEP_MAX case('RK3') num_of_iteration_lstep = 3 num_of_iteration_sstep(1) = TIME_SSTEP_MAX / 3 num_of_iteration_sstep(2) = TIME_SSTEP_MAX / 2 num_of_iteration_sstep(3) = TIME_SSTEP_MAX case('RK4') num_of_iteration_lstep = 4 num_of_iteration_sstep(1) = TIME_SSTEP_MAX / 4 num_of_iteration_sstep(2) = TIME_SSTEP_MAX / 3 num_of_iteration_sstep(3) = TIME_SSTEP_MAX / 2 num_of_iteration_sstep(4) = TIME_SSTEP_MAX case('TRCADV') ! R.Yoshida 13/06/13 [add] num_of_iteration_lstep = 1 num_of_iteration_sstep(1) = 1 num_of_iteration_sstep(2) = 1 case default write(,) 'Msg : Sub[sub_dynstep]' write(,) ' --- Error : invalid TIME_INTEG_TYPE=', TIME_INTEG_TYPE endselect endif !--- get from prg0 call prgvar_get( PROG(:,:,:,I_RHOG), PROG_pl(:,:,:,I_RHOG), & !--- [OUT] PROG(:,:,:,I_RHOGVX), PROG_pl(:,:,:,I_RHOGVX), & !--- [OUT] PROG(:,:,:,I_RHOGVY), PROG_pl(:,:,:,I_RHOGVY), & !--- [OUT] PROG(:,:,:,I_RHOGVZ), PROG_pl(:,:,:,I_RHOGVZ), & !--- [OUT] PROG(:,:,:,I_RHOGW), PROG_pl(:,:,:,I_RHOGW), & !--- [OUT] PROG(:,:,:,I_RHOGE), PROG_pl(:,:,:,I_RHOGE), & !--- [OUT] PROGq(:,:,:,:), PROGq_pl(:,:,:,:), & !--- [OUT] 0 ) !--- [IN] !--- save PROG0 (:,:,:,:) = PROG (:,:,:,:) PROG0_pl(:,:,:,:) = PROG_pl(:,:,:,:) if ( TRC_ADV_TYPE == 'DEFAULT' ) then if ( trim(TIME_INTEG_TYPE) == 'TRCADV' ) then write(ADM_LOG_FID,) 'Tracer Advection Test Mode' write(ADM_LOG_FID,) 'does not support current setting. STOP.' call ADM_proc_stop endif PROGq0 (:,:,:,:) = PROGq (:,:,:,:) PROGq0_pl(:,:,:,:) = PROGq_pl(:,:,:,:) endif !--------------------------------------------------------------------------- ! !> Start large time step integration ! !--------------------------------------------------------------------------- do nl = 1, num_of_iteration_lstep if ( trim(TIME_INTEG_TYPE) /= 'TRCADV' ) then ! TRC-ADV Test Bifurcation !---< Generate diagnostic values and set the boudary conditions rho(:,:,:) = PROG(:,:,:,I_RHOG ) / VMTR_GSGAM2(:,:,:) vx (:,:,:) = PROG(:,:,:,I_RHOGVX) / PROG(:,:,:,I_RHOG) vy (:,:,:) = PROG(:,:,:,I_RHOGVY) / PROG(:,:,:,I_RHOG) vz (:,:,:) = PROG(:,:,:,I_RHOGVZ) / PROG(:,:,:,I_RHOG) ein(:,:,:) = PROG(:,:,:,I_RHOGE ) / PROG(:,:,:,I_RHOG) do nq = 1, TRC_VMAX q(:,:,:,nq) = PROGq(:,:,:,nq) / PROG(:,:,:,I_RHOG) enddo cv(:,:,:) = 0.D0 qd(:,:,:) = 1.D0 do nq = NQW_STR, NQW_END cv(:,:,:) = cv(:,:,:) + q(:,:,:,nq) * CVW(nq) qd(:,:,:) = qd(:,:,:) - q(:,:,:,nq) enddo cv(:,:,:) = cv(:,:,:) + qd(:,:,:) * CNST_CV tem(:,:,:) = ein(:,:,:) / cv(:,:,:) pre(:,:,:) = rho(:,:,:) * tem(:,:,:) * ( qd(:,:,:)CNST_RAIR + q(:,:,:,I_QV)CNST_RVAP ) do l = 1, ADM_lall do k = ADM_kmin+1, ADM_kmax do g = 1, ADM_gall w(g,k,l) = PROG(g,k,l,I_RHOGW) & / ( VMTR_GSGAM2H(g,k,l) * 0.5D0 * ( GRD_afac(k) * rho(g,k ,l) & + GRD_bfac(k) * rho(g,k-1,l) ) ) enddo enddo !--- boundary conditions call bndcnd_all( ADM_gall, & !--- [IN] rho (:,:,l), & !--- [INOUT] vx (:,:,l), & !--- [INOUT] vy (:,:,l), & !--- [INOUT] vz (:,:,l), & !--- [INOUT] w (:,:,l), & !--- [INOUT] ein (:,:,l), & !--- [INOUT] tem (:,:,l), & !--- [INOUT] pre (:,:,l), & !--- [INOUT] PROG(:,:,l,I_RHOG), & !--- [INOUT] PROG(:,:,l,I_RHOGVX), & !--- [INOUT] PROG(:,:,l,I_RHOGVY), & !--- [INOUT] PROG(:,:,l,I_RHOGVZ), & !--- [INOUT] PROG(:,:,l,I_RHOGW), & !--- [INOUT] PROG(:,:,l,I_RHOGE), & !--- [INOUT] VMTR_GSGAM2 (:,:,l), & !--- [IN] VMTR_GSGAM2H(:,:,l), & !--- [IN] VMTR_PHI (:,:,l), & !--- [IN] VMTR_C2Wfact(:,:,:,l) ) !--- [IN] call thrmdyn_th( ADM_gall, th(:,:,l), tem(:,:,l), pre(:,:,l) ) call thrmdyn_eth( ADM_gall, eth(:,:,l), ein(:,:,l), pre(:,:,l), rho(:,:,l) ) enddo ! region LOOP !--- perturbations ( pred, rhod, temd ) pregd(:,:,:) = ( pre(:,:,:) - pre_bs(:,:,:) ) * VMTR_GSGAM2(:,:,:) rhogd(:,:,:) = ( rho(:,:,:) - rho_bs(:,:,:) ) * VMTR_GSGAM2(:,:,:) temd (:,:,:) = tem(:,:,:) - tem_bs(:,:,:) if ( ADM_prc_me == ADM_prc_pl ) then rho_pl(:,:,:) = PROG_pl(:,:,:,I_RHOG ) / VMTR_GSGAM2_pl(:,:,:) vx_pl (:,:,:) = PROG_pl(:,:,:,I_RHOGVX) / PROG_pl(:,:,:,I_RHOG) vy_pl (:,:,:) = PROG_pl(:,:,:,I_RHOGVY) / PROG_pl(:,:,:,I_RHOG) vz_pl (:,:,:) = PROG_pl(:,:,:,I_RHOGVZ) / PROG_pl(:,:,:,I_RHOG) ein_pl(:,:,:) = PROG_pl(:,:,:,I_RHOGE ) / PROG_pl(:,:,:,I_RHOG) do nq = 1, TRC_VMAX q_pl(:,:,:,nq) = PROGq_pl(:,:,:,nq) / PROG_pl(:,:,:,I_RHOG) enddo cv_pl(:,:,:) = 0.D0 qd_pl(:,:,:) = 1.D0 do nq = NQW_STR, NQW_END cv_pl(:,:,:) = cv_pl(:,:,:) + q_pl(:,:,:,nq) * CVW(nq) qd_pl(:,:,:) = qd_pl(:,:,:) - q_pl(:,:,:,nq) enddo cv_pl(:,:,:) = cv_pl(:,:,:) + qd_pl(:,:,:) * CNST_CV tem_pl(:,:,:) = ein_pl(:,:,:) / cv_pl(:,:,:) pre_pl(:,:,:) = rho_pl(:,:,:) * tem_pl(:,:,:) * ( qd_pl(:,:,:)CNST_RAIR + q_pl(:,:,:,I_QV)CNST_RVAP ) do l = 1, ADM_lall_pl do k = ADM_kmin+1, ADM_kmax do g = 1, ADM_gall_pl w_pl(g,k,l) = PROG_pl(g,k,l,I_RHOGW) & / ( VMTR_GSGAM2H_pl(g,k,l) * 0.5D0 * ( GRD_afac(k) * rho_pl(g,k ,l) & + GRD_bfac(k) * rho_pl(g,k-1,l) ) ) enddo enddo !--- boundary conditions call bndcnd_all( ADM_gall_pl, & !--- [IN] rho_pl (:,:,l), & !--- [INOUT] vx_pl (:,:,l), & !--- [INOUT] vy_pl (:,:,l), & !--- [INOUT] vz_pl (:,:,l), & !--- [INOUT] w_pl (:,:,l), & !--- [INOUT] ein_pl (:,:,l), & !--- [INOUT] tem_pl (:,:,l), & !--- [INOUT] pre_pl (:,:,l), & !--- [INOUT] PROG_pl(:,:,l,I_RHOG), & !--- [INOUT] PROG_pl(:,:,l,I_RHOGVX), & !--- [INOUT] PROG_pl(:,:,l,I_RHOGVY), & !--- [INOUT] PROG_pl(:,:,l,I_RHOGVZ), & !--- [INOUT] PROG_pl(:,:,l,I_RHOGW), & !--- [INOUT] PROG_pl(:,:,l,I_RHOGE), & !--- [INOUT] VMTR_GSGAM2_pl (:,:,l), & !--- [IN] VMTR_GSGAM2H_pl(:,:,l), & !--- [IN] VMTR_PHI_pl (:,:,l), & !--- [IN] VMTR_C2Wfact_pl(:,:,:,l) ) !--- [IN] call thrmdyn_th( ADM_gall_pl, th_pl(:,:,l), tem_pl(:,:,l), pre_pl(:,:,l) ) call thrmdyn_eth( ADM_gall_pl, eth_pl(:,:,l), ein_pl(:,:,l), pre_pl(:,:,l), rho_pl(:,:,l) ) enddo pregd_pl(:,:,:) = ( pre_pl(:,:,:) - pre_bs_pl(:,:,:) ) * VMTR_GSGAM2_pl(:,:,:) rhogd_pl(:,:,:) = ( rho_pl(:,:,:) - rho_bs_pl(:,:,:) ) * VMTR_GSGAM2_pl(:,:,:) temd_pl (:,:,:) = tem_pl(:,:,:) - tem_bs_pl(:,:,:) else rho_pl(:,:,:) = 0.D0 vx_pl (:,:,:) = 0.D0 vy_pl (:,:,:) = 0.D0 vz_pl (:,:,:) = 0.D0 w_pl (:,:,:) = 0.D0 ein_pl(:,:,:) = 0.D0 q_pl (:,:,:,:) = 0.D0 tem_pl(:,:,:) = 0.D0 pre_pl(:,:,:) = 0.D0 th_pl (:,:,:) = 0.D0 eth_pl(:,:,:) = 0.D0 pregd_pl(:,:,:) = 0.D0 rhogd_pl(:,:,:) = 0.D0 temd_pl (:,:,:) = 0.D0 endif !------------------------------------------------------------------------ !> LARGE step !------------------------------------------------------------------------ call DEBUG_rapstart('+++Large step') !--- calculation of advection tendency including Coriolis force call src_advection_convergence_momentum( vx, vx_pl, & !--- [IN] vy, vy_pl, & !--- [IN] vz, vz_pl, & !--- [IN] w, w_pl, & !--- [IN] PROG (:,:,:,I_RHOG ), PROG_pl (:,:,:,I_RHOG ), & !--- [IN] PROG (:,:,:,I_RHOGVX), PROG_pl (:,:,:,I_RHOGVX), & !--- [IN] PROG (:,:,:,I_RHOGVY), PROG_pl (:,:,:,I_RHOGVY), & !--- [IN] PROG (:,:,:,I_RHOGVZ), PROG_pl (:,:,:,I_RHOGVZ), & !--- [IN] PROG (:,:,:,I_RHOGW ), PROG_pl (:,:,:,I_RHOGW ), & !--- [IN] g_TEND(:,:,:,I_RHOGVX), g_TEND_pl(:,:,:,I_RHOGVX), & !--- [OUT] g_TEND(:,:,:,I_RHOGVY), g_TEND_pl(:,:,:,I_RHOGVY), & !--- [OUT] g_TEND(:,:,:,I_RHOGVZ), g_TEND_pl(:,:,:,I_RHOGVZ), & !--- [OUT] g_TEND(:,:,:,I_RHOGW ), g_TEND_pl(:,:,:,I_RHOGW ) ) !--- [OUT] g_TEND (:,:,:,I_RHOG) = 0.D0 g_TEND (:,:,:,I_RHOGE) = 0.D0 g_TEND (:,:,:,I_RHOGETOT) = 0.D0 g_TEND_pl(:,:,:,I_RHOG) = 0.D0 g_TEND_pl(:,:,:,I_RHOGE) = 0.D0 g_TEND_pl(:,:,:,I_RHOGETOT) = 0.D0 !---< numerical diffusion term if ( NDIFF_LOCATION == 'IN_LARGE_STEP' ) then if ( nl == 1 ) then ! only first step f_TEND (:,:,:,:) = 0.D0 f_TEND_pl (:,:,:,:) = 0.D0 f_TENDq (:,:,:,:) = 0.D0 f_TENDq_pl(:,:,:,:) = 0.D0 !------ numerical diffusion call numfilter_hdiffusion( rho, rho_pl, & !--- [IN] vx, vx_pl, & !--- [IN] vy, vy_pl, & !--- [IN] vz, vz_pl, & !--- [IN] w, w_pl, & !--- [IN] temd, temd_pl, & !--- [IN] q, q_pl, & !--- [IN] f_TEND (:,:,:,I_RHOG ), f_TEND_pl (:,:,:,I_RHOG ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGVX ), f_TEND_pl (:,:,:,I_RHOGVX ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGVY ), f_TEND_pl (:,:,:,I_RHOGVY ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGVZ ), f_TEND_pl (:,:,:,I_RHOGVZ ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGW ), f_TEND_pl (:,:,:,I_RHOGW ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGE ), f_TEND_pl (:,:,:,I_RHOGE ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGETOT), f_TEND_pl (:,:,:,I_RHOGETOT), & !--- [INOUT] f_TENDq(:,:,:,:), f_TENDq_pl(:,:,:,:) ) !--- [INOUT] if ( NUMFILTER_DOverticaldiff ) then ! numerical diffusion (vertical) call numfilter_vdiffusion( rho, rho_pl, & !--- [IN] vx, vx_pl, & !--- [IN] vy, vy_pl, & !--- [IN] vz, vz_pl, & !--- [IN] w, w_pl, & !--- [IN] temd, temd_pl, & !--- [IN] q, q_pl, & !--- [IN] f_TEND (:,:,:,I_RHOG ), f_TEND_pl (:,:,:,I_RHOG ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGVX ), f_TEND_pl (:,:,:,I_RHOGVX ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGVY ), f_TEND_pl (:,:,:,I_RHOGVY ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGVZ ), f_TEND_pl (:,:,:,I_RHOGVZ ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGW ), f_TEND_pl (:,:,:,I_RHOGW ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGE ), f_TEND_pl (:,:,:,I_RHOGE ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGETOT), f_TEND_pl (:,:,:,I_RHOGETOT), & !--- [INOUT] f_TENDq(:,:,:,:), f_TENDq_pl(:,:,:,:) ) !--- [INOUT] endif if ( NUMFILTER_DOrayleigh ) then ! rayleigh damping call numfilter_rayleigh_damping( rho, rho_pl, & !--- [IN] vx, vx_pl, & !--- [IN] vy, vy_pl, & !--- [IN] vz, vz_pl, & !--- [IN] w, w_pl, & !--- [IN] f_TEND (:,:,:,I_RHOGVX ), f_TEND_pl (:,:,:,I_RHOGVX ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGVY ), f_TEND_pl (:,:,:,I_RHOGVY ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGVZ ), f_TEND_pl (:,:,:,I_RHOGVZ ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGW ), f_TEND_pl (:,:,:,I_RHOGW ) ) !--- [INOUT] endif endif elseif( NDIFF_LOCATION == 'IN_LARGE_STEP2' ) then f_TEND (:,:,:,:) = 0.D0 f_TEND_pl (:,:,:,:) = 0.D0 f_TENDq (:,:,:,:) = 0.D0 f_TENDq_pl(:,:,:,:) = 0.D0 !------ numerical diffusion call numfilter_hdiffusion( rho, rho_pl, & !--- [IN] vx, vx_pl, & !--- [IN] vy, vy_pl, & !--- [IN] vz, vz_pl, & !--- [IN] w, w_pl, & !--- [IN] temd, temd_pl, & !--- [IN] q, q_pl, & !--- [IN] f_TEND (:,:,:,I_RHOG ), f_TEND_pl (:,:,:,I_RHOG ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGVX ), f_TEND_pl (:,:,:,I_RHOGVX ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGVY ), f_TEND_pl (:,:,:,I_RHOGVY ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGVZ ), f_TEND_pl (:,:,:,I_RHOGVZ ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGW ), f_TEND_pl (:,:,:,I_RHOGW ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGE ), f_TEND_pl (:,:,:,I_RHOGE ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGETOT), f_TEND_pl (:,:,:,I_RHOGETOT), & !--- [INOUT] f_TENDq(:,:,:,:), f_TENDq_pl(:,:,:,:) ) !--- [INOUT] if ( NUMFILTER_DOverticaldiff ) then ! numerical diffusion (vertical) call numfilter_vdiffusion( rho, rho_pl, & !--- [IN] vx, vx_pl, & !--- [IN] vy, vy_pl, & !--- [IN] vz, vz_pl, & !--- [IN] w, w_pl, & !--- [IN] temd, temd_pl, & !--- [IN] q, q_pl, & !--- [IN] f_TEND (:,:,:,I_RHOG ), f_TEND_pl (:,:,:,I_RHOG ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGVX ), f_TEND_pl (:,:,:,I_RHOGVX ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGVY ), f_TEND_pl (:,:,:,I_RHOGVY ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGVZ ), f_TEND_pl (:,:,:,I_RHOGVZ ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGW ), f_TEND_pl (:,:,:,I_RHOGW ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGE ), f_TEND_pl (:,:,:,I_RHOGE ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGETOT), f_TEND_pl (:,:,:,I_RHOGETOT), & !--- [INOUT] f_TENDq(:,:,:,:), f_TENDq_pl(:,:,:,:) ) !--- [INOUT] endif if ( NUMFILTER_DOrayleigh ) then ! rayleigh damping call numfilter_rayleigh_damping( rho, rho_pl, & !--- [IN] vx, vx_pl, & !--- [IN] vy, vy_pl, & !--- [IN] vz, vz_pl, & !--- [IN] w, w_pl, & !--- [IN] f_TEND (:,:,:,I_RHOGVX ), f_TEND_pl (:,:,:,I_RHOGVX ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGVY ), f_TEND_pl (:,:,:,I_RHOGVY ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGVZ ), f_TEND_pl (:,:,:,I_RHOGVZ ), & !--- [INOUT] f_TEND (:,:,:,I_RHOGW ), f_TEND_pl (:,:,:,I_RHOGW ) ) !--- [INOUT] endif endif ! Smagorinksy-type SGS model [add] A.Noda 10/11/29 if ( TB_TYPE == 'SMG' ) then !cx write(0,'(a)') "* Error. Smagorinksy model was removed in mini-version." stop !cx endif !--- Nudging routines [add] Y.Niwa 08/09/09 if ( FLAG_NUDGING ) then !cx write(0,'(a)') "* Error. Nudging function was removed in mini-version." stop !cx endif !--- sum the large step TEND ( advection + coriolis + num.diff.,SGS,nudge ) g_TEND (:,:,:,:) = g_TEND (:,:,:,:) + f_TEND (:,:,:,:) g_TEND_pl(:,:,:,:) = g_TEND_pl(:,:,:,:) + f_TEND_pl(:,:,:,:) call DEBUG_rapend ('+++Large step') !------------------------------------------------------------------------ !> SMALL step !------------------------------------------------------------------------ call DEBUG_rapstart('+++Small step') if ( nl /= 1 ) then ! update split values PROG_split (:,:,:,:) = PROG0 (:,:,:,:) - PROG (:,:,:,:) PROG_split_pl(:,:,:,:) = PROG0_pl(:,:,:,:) - PROG_pl(:,:,:,:) else PROG_split (:,:,:,:) = 0.D0 PROG_split_pl(:,:,:,:) = 0.D0 endif !------ Core routine for small step !------ 1. By this subroutine, prognostic variables !------ ( rho,.., rhoge ) are calculated through !------ the 'num_of_iteration_sstep(nl)'-th times small step. !------ 2. grho, grhogvx, ..., and grhoge has the large step !------ tendencies initially, however, !------ they are re-used in this subroutine. !------ if ( TIME_SPLIT ) then small_step_ite = num_of_iteration_sstep(nl) small_step_dt = TIME_DTS else small_step_ite = 1 small_step_dt = TIME_DTL / (num_of_iteration_lstep+1-nl) endif call vi_small_step( PROG(:,:,:,I_RHOG ), PROG_pl(:,:,:,I_RHOG ), & !--- [INOUT] prognostic variables PROG(:,:,:,I_RHOGVX), PROG_pl(:,:,:,I_RHOGVX), & !--- [INOUT] PROG(:,:,:,I_RHOGVY), PROG_pl(:,:,:,I_RHOGVY), & !--- [INOUT] PROG(:,:,:,I_RHOGVZ), PROG_pl(:,:,:,I_RHOGVZ), & !--- [INOUT] PROG(:,:,:,I_RHOGW ), PROG_pl(:,:,:,I_RHOGW ), & !--- [INOUT] PROG(:,:,:,I_RHOGE ), PROG_pl(:,:,:,I_RHOGE ), & !--- [INOUT] vx, vx_pl, & !--- [IN] diagnostic value vy, vy_pl, & !--- [IN] vz, vz_pl, & !--- [IN] eth, eth_pl, & !--- [IN] rhogd, rhogd_pl, & !--- [IN] pregd, pregd_pl, & !--- [IN] g_TEND(:,:,:,I_RHOG ), g_TEND_pl(:,:,:,I_RHOG ), & !--- [IN] large step TEND g_TEND(:,:,:,I_RHOGVX ), g_TEND_pl(:,:,:,I_RHOGVX ), & !--- [IN] g_TEND(:,:,:,I_RHOGVY ), g_TEND_pl(:,:,:,I_RHOGVY ), & !--- [IN] g_TEND(:,:,:,I_RHOGVZ ), g_TEND_pl(:,:,:,I_RHOGVZ ), & !--- [IN] g_TEND(:,:,:,I_RHOGW ), g_TEND_pl(:,:,:,I_RHOGW ), & !--- [IN] g_TEND(:,:,:,I_RHOGE ), g_TEND_pl(:,:,:,I_RHOGE ), & !--- [IN] g_TEND(:,:,:,I_RHOGETOT), g_TEND_pl(:,:,:,I_RHOGETOT), & !--- [IN] PROG_split(:,:,:,I_RHOG ), PROG_split_pl(:,:,:,I_RHOG ), & !--- [INOUT] split value PROG_split(:,:,:,I_RHOGVX), PROG_split_pl(:,:,:,I_RHOGVX), & !--- [INOUT] PROG_split(:,:,:,I_RHOGVY), PROG_split_pl(:,:,:,I_RHOGVY), & !--- [INOUT] PROG_split(:,:,:,I_RHOGVZ), PROG_split_pl(:,:,:,I_RHOGVZ), & !--- [INOUT] PROG_split(:,:,:,I_RHOGW ), PROG_split_pl(:,:,:,I_RHOGW ), & !--- [INOUT] PROG_split(:,:,:,I_RHOGE ), PROG_split_pl(:,:,:,I_RHOGE ), & !--- [INOUT] v_mean_c, v_mean_c_pl, & !--- [OUT] mean value small_step_ite, & !--- [IN] small_step_dt ) !--- [IN] call DEBUG_rapend ('+++Small step') else ! TRC-ADV Test Bifurcation !--- Make v_mean_c ![add] 20130613 R.Yoshida !--- save point(old) is mean here (although it is not exactly valid for rho) v_mean_c(:,:,:,I_rhog) = PROG0(:,:,:,I_rhog); v_mean_c_pl(:,:,:,I_rhog) = PROG0_pl(:,:,:,I_rhog) v_mean_c(:,:,:,I_rhogvx)= PROG0(:,:,:,I_rhogvx); v_mean_c_pl(:,:,:,I_rhogvx)= PROG0_pl(:,:,:,I_rhogvx) v_mean_c(:,:,:,I_rhogvy)= PROG0(:,:,:,I_rhogvy); v_mean_c_pl(:,:,:,I_rhogvy)= PROG0_pl(:,:,:,I_rhogvy) v_mean_c(:,:,:,I_rhogvz)= PROG0(:,:,:,I_rhogvz); v_mean_c_pl(:,:,:,I_rhogvz)= PROG0_pl(:,:,:,I_rhogvz) v_mean_c(:,:,:,I_rhogw) = PROG0(:,:,:,I_rhogw); v_mean_c_pl(:,:,:,I_rhogw) = PROG0_pl(:,:,:,I_rhogw) endif ! TRC-ADV Test Bifurcation !------------------------------------------------------------------------ !> Tracer advection !------------------------------------------------------------------------ call DEBUG_rapstart('+++Tracer Advection') if ( TRC_ADV_TYPE == 'MIURA2004' ) then if ( nl == num_of_iteration_lstep ) then call src_update_tracer( TRC_VMAX, & !--- [IN] PROGq(:,:,:,:), PROGq_pl(:,:,:,:), & !--- [INOUT] PROG0(:,:,:,I_RHOG), PROG0_pl(:,:,:,I_RHOG), & !--- [IN] v_mean_c(:,:,:,I_rhog), v_mean_c_pl(:,:,:,I_rhog), & !--- [IN] v_mean_c(:,:,:,I_rhogvx), v_mean_c_pl(:,:,:,I_rhogvx), & !--- [IN] v_mean_c(:,:,:,I_rhogvy), v_mean_c_pl(:,:,:,I_rhogvy), & !--- [IN] v_mean_c(:,:,:,I_rhogvz), v_mean_c_pl(:,:,:,I_rhogvz), & !--- [IN] v_mean_c(:,:,:,I_rhogw), v_mean_c_pl(:,:,:,I_rhogw), & !--- [IN] f_TEND (:,:,:,I_RHOG), f_TEND_pl (:,:,:,I_RHOG), & !--- [IN] TIME_DTL, & !--- [IN] THUBURN_LIM ) !--- [IN] ![add] 20130613 R.Yoshida if( TIME_INTEG_TYPE /= 'TRCADV' ) PROGq(:,:,:,:) = PROGq(:,:,:,:) + TIME_DTL * f_TENDq(:,:,:,:) ! update rhogq by viscosity PROGq(:,ADM_kmin-1,:,:) = 0.D0 PROGq(:,ADM_kmax+1,:,:) = 0.D0 if ( ADM_prc_pl == ADM_prc_me ) then if( TIME_INTEG_TYPE /= 'TRCADV' ) PROGq_pl(:,:,:,:) = PROGq_pl(:,:,:,:) + TIME_DTL * f_TENDq_pl(:,:,:,:) PROGq_pl(:,ADM_kmin-1,:,:) = 0.D0 PROGq_pl(:,ADM_kmax+1,:,:) = 0.D0 endif ! [comment] H.Tomita: I don't recommend adding the hyperviscosity term because of numerical instability in this case. endif ! Last large step only elseif( TRC_ADV_TYPE == 'DEFAULT' ) then !This scheme isn't supported in TRC-ADV Test (20130612 R.Yoshida) do nq = 1, TRC_VMAX call src_advection_convergence( v_mean_c(:,:,:,I_rhogvx), v_mean_c_pl(:,:,:,I_rhogvx), & !--- [IN] v_mean_c(:,:,:,I_rhogvy), v_mean_c_pl(:,:,:,I_rhogvy), & !--- [IN] v_mean_c(:,:,:,I_rhogvz), v_mean_c_pl(:,:,:,I_rhogvz), & !--- [IN] v_mean_c(:,:,:,I_rhogw), v_mean_c_pl(:,:,:,I_rhogw), & !--- [IN] q(:,:,:,nq), q_pl(:,:,:,nq), & !--- [IN] g_TENDq(:,:,:,nq), g_TENDq_pl(:,:,:,nq), & !--- [OUT] I_SRC_default ) !--- [IN] [mod] H.Yashiro 20120530 PROGq(:,:,:,:) = PROGq0(:,:,:,:) & + ( num_of_iteration_sstep(nl) * TIME_DTS ) * ( g_TENDq(:,:,:,:) + f_TENDq(:,:,:,:) ) PROGq(:,ADM_kmin-1,:,:) = 0.D0 PROGq(:,ADM_kmax+1,:,:) = 0.D0 if ( ADM_prc_pl == ADM_prc_me ) then PROGq_pl(:,:,:,:) = PROGq0_pl(:,:,:,:) & + ( num_of_iteration_sstep(nl) * TIME_DTS ) & * ( g_TENDq_pl(:,:,:,:) + f_TENDq_pl(:,:,:,:) ) PROGq_pl(:,ADM_kmin-1,:,:) = 0.D0 PROGq_pl(:,ADM_kmax+1,:,:) = 0.D0 endif enddo ! tracer LOOP endif call DEBUG_rapend ('+++Tracer Advection') if ( trim(TIME_INTEG_TYPE) /= 'TRCADV' ) then ! TRC-ADV Test Bifurcation !--- TKE fixer ( TKE >= 0.D0 ) ! 2011/08/16 M.Satoh [comment] need this fixer for every small time steps if ( I_TKE >= 0 ) then if ( TRC_ADV_TYPE == 'DEFAULT' .OR. nl == num_of_iteration_lstep ) then do l = 1, ADM_lall do k = 1, ADM_kall do g = 1, ADM_gall TKEg_corr = TKE_MIN * VMTR_GSGAM2(g,k,l) - PROGq(g,k,l,I_TKE) if ( TKEg_corr >= 0.D0 ) then PROG (g,k,l,I_RHOGE) = PROG (g,k,l,I_RHOGE) - TKEg_corr PROGq(g,k,l,I_TKE) = PROGq(g,k,l,I_TKE) + TKEg_corr endif enddo enddo enddo if ( ADM_prc_pl == ADM_prc_me ) then do l = 1, ADM_lall_pl do k = 1, ADM_kall do g = 1, ADM_gall_pl TKEg_corr = TKE_MIN * VMTR_GSGAM2_pl(g,k,l) - PROGq_pl(g,k,l,I_TKE) if ( TKEg_corr >= 0.D0 ) then PROG_pl (g,k,l,I_RHOGE) = PROG_pl (g,k,l,I_RHOGE) - TKEg_corr PROGq_pl(g,k,l,I_TKE) = PROGq_pl(g,k,l,I_TKE) + TKEg_corr endif enddo enddo enddo endif endif endif !------ Update if ( nl /= num_of_iteration_lstep ) then ! communication call COMM_data_transfer( PROG, PROG_pl ) PROG(suf(ADM_gall_1d,1),:,:,:) = PROG(suf(ADM_gmax+1,ADM_gmin),:,:,:) PROG(suf(1,ADM_gall_1d),:,:,:) = PROG(suf(ADM_gmin,ADM_gmax+1),:,:,:) endif endif ! TRC-ADV Test Bifurcation enddo !--- large step if ( trim(TIME_INTEG_TYPE) == 'TRCADV' ) then call updating( PROG0(:,:,:,:), PROG0_pl(:,:,:,:), & !--- [IN] PROG (:,:,:,:), PROG_pl (:,:,:,:) ) !--- [INOUT] endif call prgvar_set( PROG(:,:,:,I_RHOG), PROG_pl(:,:,:,I_RHOG), & !--- [IN] PROG(:,:,:,I_RHOGVX), PROG_pl(:,:,:,I_RHOGVX), & !--- [IN] PROG(:,:,:,I_RHOGVY), PROG_pl(:,:,:,I_RHOGVY), & !--- [IN] PROG(:,:,:,I_RHOGVZ), PROG_pl(:,:,:,I_RHOGVZ), & !--- [IN] PROG(:,:,:,I_RHOGW), PROG_pl(:,:,:,I_RHOGW), & !--- [IN] PROG(:,:,:,I_RHOGE), PROG_pl(:,:,:,I_RHOGE), & !--- [IN] PROGq(:,:,:,:), PROGq_pl(:,:,:,:), & !--- [IN] 0 ) !--- [IN] call DEBUG_rapend ('++Dynamics') #ifdef PAPI_OPS ! <-- [add] PAPI R.Yoshida 20121022 !call PAPIF_flips( PAPI_real_time_i, PAPI_proc_time_i, PAPI_flpins, PAPI_mflins, PAPI_check ) call PAPIF_flops( PAPI_real_time_o, PAPI_proc_time_o, PAPI_flpops, PAPI_mflops, PAPI_check ) #endif return end subroutine dynstep end module mod_dynstep
Formal statement is: lemma translation_galois: fixes a :: "'a::ab_group_add" shows "T = ((\<lambda>x. a + x) ` S) \<longleftrightarrow> S = ((\<lambda>x. (- a) + x) ` T)" Informal statement is: For any set $S$ and any element $a$, the set $T = \{a + x \mid x \in S\}$ is equal to the set $\{-a + x \mid x \in S\}$.
#' Compute metacell using a native implementation of a graph cover k-means-like approach #' #' @param mc_id id of metacell object to be added #' @param graph_id a knn graph object id into scdb #' @param mat_id a matrix object id #' @param min_mc_size target minimum metacell size. This is only an approximation and smaller MC may be returned by the algorithm #' mcell_add_mc_from_graph = function(mc_id, graph_id, mat_id, min_mc_size) { old_seed = .set_seed(get_param("mc_rseed")) tgs_clust_cool = get_param("scm_tgs_clust_cool") tgs_clust_burn = get_param("scm_tgs_clust_burn_in") graph = scdb_cgraph(graph_id) if(is.null(graph)) { stop("MC-ERR: cell graph id ", graph_id, " is missing when running add_mc_from_graph") } mat = scdb_mat(mat_id) if(is.null(mat)) { stop("MC-ERR: mat id ", mat_id, " is missing when running add_mc_from_graph") } message("running graph clustering now - one iteration no bootstrap") edges = graph@edges colnames(edges) = c("col1", "col2", "weight") node_clust = tgs_graph_cover(edges, min_mc_size, cooling = tgs_clust_cool, burn_in = tgs_clust_burn) f_outlier = (node_clust$cluster == 0) outliers = colnames(mat@mat)[node_clust$node[f_outlier]] mc = as.integer(as.factor(node_clust$cluster[!f_outlier])) names(mc) = colnames(mat@mat)[!f_outlier] message("building metacell object, #mc ", max(mc)) cell_names = colnames(mat@mat) scdb_add_mc(mc_id, tgMCCov(mc, outliers, mat)) message("reordering metacells by hclust and most variable two markers") .restore_seed(old_seed) mcell_mc_reorder_hc(mc_id) }
[STATEMENT] lemma fresh_subst: fixes z::"name" assumes a: "z\<sharp>t\<^sub>1" "z\<sharp>t\<^sub>2" shows "z\<sharp>t\<^sub>1[y::=t\<^sub>2]" [PROOF STATE] proof (prove) goal (1 subgoal): 1. z \<sharp> t\<^sub>1[y::=t\<^sub>2] [PROOF STEP] using a [PROOF STATE] proof (prove) using this: z \<sharp> t\<^sub>1 z \<sharp> t\<^sub>2 goal (1 subgoal): 1. z \<sharp> t\<^sub>1[y::=t\<^sub>2] [PROOF STEP] by (auto simp add: fresh_subst' abs_fresh)
\chapter{Project Management} \label{chap:final-project-management} In this chapter, we evaluate the overall project management during the thesis. Indeed, as defined in the initial specifications, the milestones were meant to be adjustable based on the project iterations. Our primary constraint for the Master's Thesis was the time: indeed, the project is formally framed to start on the 17th of September 2019 and end on 7th of February 2020, for a total of amount of 900 hours. To evaluate the project management, we plan three steps; first, we take a high-level overview and comment on the two project phases, the \glsfirst{sota} research and GraphQA as our research contribution. Secondly, we will review and reflect on the initial specification, and finally conclude on the overall management. \section{High-Level Overview} Initially defined as \textit{Back to Level} and \textit{Diving into the Subject}, the two phases had the same meaning overall to, what we believe to be, our academic vision defined as Research and Contribution. In this chapter, we take a step back to visualize the entire work done as a whole to summarise the exciting adventure of our first academic research. \subsection{State-of-the-Art Research} Our first step to avoid being overwhelmed with knowledge from the most advanced \gls{nlp} papers in the field of \gls{qa} systems and \glspl{gs} was to plan the research. As we did not have the tools to understand the papers properly, we decided to define a workflow to gather valuable information, such as the initial tools to get started with more complicated techniques. The listing below shows our procedure. \begin{itemize} \setlength\itemsep{0em} \item Get up to date with the \gls{nlp} technologies used at our lab, \textit{iCoSys}. \item Explore community-made curated lists\footnote{\textit{Awesome} NLP lists from \url{github.com}}. \item Subscribe to various specilized social medias to stay informed of the latest \gls{nlp} breakthroughs \footnote{Examples from \url{reddit.com} /r/MachineLearning, /r/LanguageTechnology, /r/deeplearning}. \item Read reviews and article summarises of recent papers\footnote{Particularly from community based \url{medium.com} articles}. \item Deeply analyse the latest breakthrough papers and read all the mentioned paper. \item Filter and read the latest preprints \footnote{Most of the articles are coming from \url{arxiv.com} and \url{aclweb.org}}. \end{itemize} Using this workflow, we could in 6 weeks read about 40 papers and gently examine 40 others, which we believe gave us an approximately fair overview of the \gls{nlp} field, and particularly of the \gls{qa} systems and \glspl{gs}. \subsection{Research Contribution} Based on the accumulated knowledge from the \gls{sota} research, we could analyze the current techniques used and their applications in the field of \gls{nlp}, in particular, for \gls{qa} systems and \glspl{gs}. Which helped us define a scope for the project that would make sense in the scope of a Master's Thesis to contribute to \gls{nlp}. Gladly, our constraint to use \gls{wikidata} \gls{kb} could sharpen the possible contributions. As stated in our analysis \ref{chap:analysis}, we went through multiple brainstorming and project iterations to get to GraphQA. From a management point of view, it appears that we respected the initial planning and honored the objectives defined in the initial project specification. Finally, we believe that our work could contribute to \gls{nlp}, making the second phase as a success. \section{Specification Review} In this section, we review the original specifications by adding comments as a retrospective approach. We keep the structure and often paraphrase the original content. To improve reading, when content is reused or paraphrased, we set it in \textit{italic} and in \textbf{bold} for comments. Additionally, for use the \checkmark and $\times$ bullets to mark items as realized or not realized. \subsection{Intrinsic Objectives} \subsubsection{Primaries} In this section we presented the tasks that we believed to be essential to get started with the master's thesis. \begin{itemize}[noitemsep] \item[\checkmark] \textit{Propose a project specification and planning.} %\textbf{We added the initial brainstormed tasks to the annexes.} \item[\checkmark] \textit{Analyze the \gls{sota} of existing technologies and technics of \gls{qa} systems and \gls{generative} \gls{ai}.} \item[\checkmark] \textit{Overview digital transformation in journalism} \textbf{Even if we did the study, we did not include the search as the project shifted toward chatbots and \gls{nlp}.} \item[\checkmark] \textit{Review the current status of the AI-News project.} \item[\checkmark] \textit{Document the study and write the thesis.} \end{itemize} %\todo{add the annexes numbering to for the brainstormed tasks} \subsection{Fact-based Question-Answering Chatbot Objectives} \textit{The first objective is to make, based on the \glsfirst{sota}, an algorithm that takes a question as input and outputs a response, as illustrated on Figure~\ref{fig:management_qa}} \begin{figure} \centering \includegraphics[width=\textwidth,height=2.1cm,keepaspectratio=true]{intro_qa} \caption{Suggested \gls{qa} diagram} \label{fig:management_qa} \end{figure} \subsubsection{Primaries} \begin{itemize}[noitemsep] \item[\checkmark] \textit{Select existing papers and projects treating the subject as a starting point.} \item[\checkmark] \textit{Identify relevant datasets.} \item[\checkmark] \textit{Develop one or more \gls{poc}.} \item[\checkmark] \textit{Test and evaluate solutions.} \item[\checkmark] \textit{Suggest improvements, possible continuation, and future outcomes.} \end{itemize} \subsubsection{Secondaries} \begin{itemize}[noitemsep] \item[$\times$] \textit{Extend the \gls{qa} chatbot using "tailored" knowledge, e.g., \gls{model-ft} with press content.} \textbf{As mentioned in the final notes from the GraphQA chapter \ref{chap:graphqa}, this item can be extrapolated to GraphQA by adding a fine-tuned pretrain language model for a multi-brains approach to reach a consensus-based answer.} \end{itemize} \subsection{Natural Language Question Answering Chatbot Objectives} \textit{The second objective was to extend the output from the \gls{qa} system, from the first objective, by enhancing the answers and generate human-like sentences from the enhanced answers. The initial vision for this objective is as illustrated in Figure~\ref{fig:planning_qa_gen}, a two parts system. The \textit{Enricher} enriches the answer from the \gls{qa} system, e.g. using a knowledge base\footnote{Wikidata.org, a Freebase-based \autocite{paper:bollacker2008} knowledge base or Google's Knowledge Graphs \autocite{blog:intro_knowledge_graph}}. The \textit{Generator} aims at creating readable text from the enriched answer. Besides, we could also use user profiles\footnote{Fictive profiles in the context of the thesis} as input to those two parts.} \begin{figure}[ht!] \centering \includegraphics[width=\textwidth,keepaspectratio=true]{intro_qa_gen} \caption{Suggested \gls{generative} \gls{qa} diagram} \label{fig:planning_qa_gen} \end{figure} \paragraph{Primaries} \begin{itemize}[noitemsep] \item[\checkmark] \textit{Investigate a rule-based system for keyword enrichment.} \item[\checkmark] \textit{Generate sentences with keywords.} \item[\checkmark] \textit{Identify relevant datasets.} \item[\checkmark] \textit{Develop one or more \gls{poc}.} \item[\checkmark] \textit{Test and evaluate solutions.} \item[\checkmark] \textit{Suggest improvements, possible continuation, and future outcomes.} \end{itemize} \paragraph{Secondaries} \begin{itemize}[noitemsep] \item[\checkmark] \textit{Use advanced strategies to enrich keywords.} \item[\checkmark] \textit{Use advanced text generation technics such as GTP-2\footnote{OpenAI's GTP-2 Algorithm \autocite{papers:gpt2}}.} \item[$\times$] \textit{Use user profiles to customize the outputs.} \textbf{This item is mentioned in the final notes from the GraphQA chapter \ref{chap:graphqa}. GraphQA could build long-term context graphs for each user to hold their preferences. It could hold particular interests (entities), injected for the user each time a new Sub-Knowledge Graph is generated. The result would be that GraphQA will try to find a path to the answer using the users injected interests.} \end{itemize} \subsection{Objectives Retrospective} For the objectives, we indeed honored the primary functions and even could add secondary functions to the GraphQA. Even if initial planned, we did not build two distinct \glspl{poc}. Indeed, we started with an hybrid model combining the required features into a single \gls{poc} (see Figure \ref{fig:fig_planning_qa_gen_hybrid}). \subsection{Methodologies} \textit{For consistency, the project was separated into two methodological parts. In the first third, as the project targets information gathering and self-study, we used a standard sequential project management methodology. For the next two-thirds of the project, we used an agile methodology to perform incremental progress while exploring.} \subsubsection{Back to level Milestones} \textit{First third of the study, from 16.09.19 to 25.10.19 (6 weeks).} \begin{enumerate} \setlength\itemsep{0em} \item[\checkmark M1.] \textit{Initial \gls{mt} plan and project specification} \item[\checkmark M2.] \textit{Review the \gls{sota} for the \gls{nlp} and \gls{nlu} technologies and refine the plan if needed.} \end{enumerate} \subsubsection{Diving into the subject Milestones} \textit{From 28.10.19 to 07.02.20 (13 weeks), the following two-third of the work is composed of 6 sprints of two weeks each and one week to finalize the thesis.} \begin{itemize} \setlength\itemsep{0em} \item[\checkmark M3.] \textit{Basic \gls{qa} Chatbot} \item[\checkmark M4.] \textit{Evaluation of basic \gls{qa} Chatbot} \item[\checkmark M5.] \textit{Basic generative \gls{qa} Chatbot} \item[\checkmark M6.] \textit{Evaluation of basic generative \gls{qa} Chatbot} \end{itemize} \subsection{Initial Gantt} \textit{The Figure~\ref{fig:gantt-initial} represents the chart for the initial plan.} \subsection{Methologies Retrospective} The two phases split were respected from a methodological and temporal point of view; however, the objectives hybridization (see Figure \ref{fig:fig_planning_qa_gen_hybrid}) made the milestones slightly altered as the evaluation milestones M4 and M6 are combined. \section{Management Conclusion} We believe that it is important to note that even if the objectives and the results are positives, it is difficult from a management point of view to validate the statement that the end results justify the means, which we think happened in the scope of our master's thesis. Indeed, even if we enjoyed every minute, we did massive overtime for the project to reach our objectives, which means that either the initial project scope or the post-analysis redefined scope was too large for our time constraint. We blame the rescoping as the project shifted toward an understudied field of \gls{nlp} and \gls{qa} systems, which made us notice that we could define a potential new field of the \gls{nlp} research. On a final management note, even if from an industrial point of view, the current overtime would not be easily accepted. In our case; however, from an academic point of view, we justify our overflow as passionate dedication and as a fair attitude to contribute to research. \subsection{Final Milestones} \begin{itemize} \setlength\itemsep{0em} \item[M1.] Initial \gls{mt} plan and project specification \item[M2.] Review the \gls{sota} for the \gls{nlp} and \gls{nlu} technologies and refine the plan if needed. \item[M3.] GraphQA 1 (see Chapter \ref{graphqa:graphqa1}) \item[M4.] GraphQA 2 (see Chapter \ref{graphqa:graphqa2}) \item[M5.] GraphQA 3 (see Chapter \ref{graphqa:graphqa3}) \item[M6.] Evaluation of basic generative \gls{qa} Chatbot \item[M7.] Turn in Master's Thesis \end{itemize} \subsection{Effective Gantt} The Figure~\ref{fig:gantt-final} represents the chart for for the effective plan. \begin{figure} \centering \includegraphics[width=\textwidth,keepaspectratio=true]{fig_planning_qa_gen_hybrid} \caption{Suggested \gls{generative} \gls{qa} diagram} \label{fig:fig_planning_qa_gen_hybrid} \end{figure} \newganttchartelement*{project-milestone}{ project-milestone/.style={ shape=isosceles triangle, inner sep=0pt, draw=cyan, top color=white, bottom color=cyan!50 }, project-milestone incomplete/.style={ /pgfgantt/project-milestone, draw=yellow, bottom color=yellow!50 }, project-milestone label font=\slshape, project-milestone left shift=0pt, project-milestone right shift=0pt } \newgantttimeslotformat{stardate}{ \def\decomposestardate##1.##2\relax{ \def\stardateyear{##1}\def\stardateday{##2} } \decomposestardate#1\relax \pgfcalendardatetojulian{\stardateyear-01-01}{#2} \advance#2 by-1\relax \advance#2 by\stardateday\relax } \begin{figure}%[h]%[htbp] \centering \begin{ganttchart}[vgrid, hgrid]{1}{19} \gantttitle{Sep}{2} \gantttitle{Oct}{5} \gantttitle{Nov}{4} \gantttitle{Dec}{3} \gantttitle{Jan}{4} \gantttitle{Feb}{1}\\ \gantttitlelist{1,...,19}{1}\\ %part 1 \ganttgroup{Back to level}{1}{6} \\ \ganttmilestone{M1, M2}{3} \ganttmilestone{}{6}\\ %part 2 \ganttgroup{Diving}{7}{18} \\ \ganttbar{Sprint 1}{7}{8} \\ \ganttbar{Sprint 2}{9}{10} \\ \ganttmilestone{M3}{10}\\ \ganttbar{Sprint 3}{11}{12} \\ \ganttmilestone{M4}{12}\\ \ganttbar{Sprint 4}{13}{14} \\ \ganttbar{Sprint 5}{15}{16} \\ \ganttmilestone{M5}{16}\\ \ganttbar{Sprint 6}{17}{18} \\ \ganttmilestone{M6}{18}\\ %\ganttlink{elem6}{elem7} %\ganttlink{elem8}{elem9} %part 3 \ganttgroup{Wrap up}{19}{19} \\ \end{ganttchart} \caption{Initial Gantt Chart} \label{fig:gantt-initial} \end{figure} \begin{figure}%[h]%[htbp] \centering \begin{ganttchart}[vgrid, hgrid]{1}{19} \gantttitle{Sep}{2} \gantttitle{Oct}{5} \gantttitle{Nov}{4} \gantttitle{Dec}{3} \gantttitle{Jan}{4} \gantttitle{Feb}{1}\\ \gantttitlelist{1,...,19}{1}\\ %part 1 \ganttgroup{State-of-the-Art Research}{1}{6} \\ \ganttmilestone{M1, M2}{3} \ganttmilestone{}{6}\\ %part 2 \ganttgroup{Research Contribution}{7}{19} \\ \ganttbar{Sprint 1}{7}{8} \\ \ganttbar{Sprint 2}{9}{10} \\ \ganttmilestone{M3}{10}\\ \ganttbar{Sprint 3}{11}{12} \\ \ganttmilestone{M4}{12}\\ \ganttbar{Sprint 4}{13}{14} \\ \ganttbar{Sprint 5}{15}{16} \\ \ganttmilestone{M5}{16}\\ \ganttbar{Sprint 6}{17}{18} \\ \ganttmilestone{M6}{18}\\ \ganttmilestone{M7}{19}\\ %\ganttlink{elem6}{elem7} %\ganttlink{elem8}{elem9} %part 3 \ganttgroup{Writing Thesis}{18}{19} \\ \end{ganttchart} \caption{Effective Gantt Chart} \label{fig:gantt-final} \end{figure}
C ********************************************************* C * * C * TEST NUMBER: 04.02.05.04/02 * C * TEST TITLE : Setting entries of various kinds * C * * C * PHIGS Validation Tests, produced by NIST * C * * C ********************************************************* COMMON /GLOBNU/ CTLHND, ERRSIG, ERRFIL, IERRCT, UNERR, 1 TESTCT, IFLERR, PASSSW, ERRSW, MAXLIN, 2 CONID, MEMUN, WKID, WTYPE, GLBLUN, INDLUN, 3 DUMINT, DUMRL INTEGER CTLHND, ERRSIG, ERRFIL, IERRCT, UNERR, 1 TESTCT, IFLERR, PASSSW, ERRSW, MAXLIN, 2 CONID, MEMUN, WKID, WTYPE, GLBLUN, INDLUN, 3 DUMINT(20), ERRIND REAL DUMRL(20) COMMON /GLOBCH/ PIDENT, GLBERR, TSTMSG, FUNCID, 1 DUMCH CHARACTER PIDENT40, GLBERR60, TSTMSG900, FUNCID80, 1 DUMCH(20)20 INTEGER MAXPL C define a number greater than the number of initialized pattern entries PARAMETER (MAXPL=300) C PHIGS parameter types INTEGER MPAI, PAI, DIMX, DIMY, ISC, ISR, DX, DY, DX2, DY2 PARAMETER (DIMX = 100, DIMY = 100) INTEGER COLIA(DIMX,DIMY), COLIA2(DIMX, DIMY) INTEGER IDUM1, IDUM2, IDUM3, IDUM4, IDUM5, IDUM6, IDUM7 C interior style INTEGER PHOLLO, PSOLID, PPATTR, PHATCH, PISEMP PARAMETER (PHOLLO=0, PSOLID=1, PPATTR=2, PHATCH=3, PISEMP=4) C type of returned value INTEGER PSET, PREALI PARAMETER (PSET = 0, PREALI = 1) LOGICAL ARREQ, PATAVL C used to control tests INTEGER NIE, AII, LISTEL, LISTOT, DEFI(MAXPL) INTEGER LUI, DPI, CNDE, NNDE, I, J CHARACTER MSG300 C parameters for <inquire workstation connection and type> INTEGER SPECWT, SPECON DATA DEFI / MAXPL0 / CALL INITGL ('04.02.05.04/02') C open PHIGS CALL XPOPPH (ERRFIL, MEMUN) C open workstation CALL POPWK (WKID, CONID, WTYPE) CALL PQWKC (WKID, ERRIND, SPECON, SPECWT) CALL CHKINQ ('pqwkc', ERRIND) IF (.NOT. PATAVL(SPECWT)) THEN CALL INMSG ('Skipping all tests because pattern interior ' // 1 'style is not supported on this workstation.') GOTO 666 ENDIF C determine mpai = max # pattern entries CALL PQWKSL (SPECWT, ERRIND, IDUM1, IDUM2, IDUM3, IDUM4, 1 IDUM5, MPAI, IDUM6, IDUM7) CALL CHKINQ ('pqwksl', ERRIND) C * * * * initialized entries * * * * * C There should be at least 1 predefined pattern - C This statement is NOT meant to test the requirement of C 1 predefined pattern; that is done elsewhere. CALL SETMSG ('3', '<Inquire list of pattern indices> should ' // 1 'return initialized indices.') C nie = number of initialized pattern entries C aii = an initialized index (the first) CALL PQEPAI (WKID, 1, ERRIND, NIE, AII) IF (ERRIND .NE. 0) THEN CALL FAIL GOTO 200 ELSE CALL PASS ENDIF C get initialized attributes CALL PQPAR (WKID, AII, PSET, DIMX, DIMY, ERRIND, DX, DY, COLIA) C check errind for overflow IF (ERRIND .EQ. 2001) THEN WRITE (MSG, '(A,A,2I5,A)') 'PQPAR returned Fortran error ', 1 '2001: declared array size of (', DIMX, DIMY, 2 ') insufficient.' CALL INMSG (MSG) GOTO 200 ENDIF CALL CHKINQ ('pqpar', ERRIND) C set different values for attributes DO 80 I = 1, DX DO 75 J = 1, DY COLIA2(I,J) = COLIA(I,J) +1 75 CONTINUE 80 CONTINUE ISC = 1 ISR = 1 DX2 = DX DY2 = DY CALL PSPAR (WKID, AII, DIMX, DIMY, ISC, ISR, DX2, DY2, COLIA2) C get current number of defined entries CALL PQEPAI (WKID, 0, ERRIND, CNDE, IDUM1) CALL CHKINQ ('pqepai', ERRIND) CALL SETMSG ('1 3', 'Redefining a system-initialized entry ' // 1 'should not change the number of defined entries.') CALL IFPF (CNDE .EQ. NIE) C check that aii is still among defined entries CALL SETMSG ('1 3', 'Initialized table entry should be ' // 1 'reported as defined after being set.') DO 150 LISTEL = 1, NIE CALL PQEPAI (WKID, LISTEL, ERRIND, IDUM1, PAI) CALL CHKINQ ('pqepai', ERRIND) IF (PAI .EQ. AII) THEN CALL PASS GOTO 160 ENDIF 150 CONTINUE CALL FAIL GOTO 200 160 CONTINUE CALL PQPAR (WKID, AII, PSET, DIMX, DIMY, ERRIND, DX, DY, COLIA) CALL CHKINQ ('pqpar', ERRIND) CALL SETMSG ('2 5', 'New values should be set for ' // 1 'initialized table entry.') CALL IFPF (DX2 .EQ. DX .AND. 1 DY2 .EQ. DY .AND. 2 ARREQ(DIMX, DIMY, ISC, ISR, DX2, DY2, COLIA2, COLIA)) C * * * * * undefined entries * * * * * 200 CONTINUE C get lowest undefined index C get current number defined entries CALL PQEPAI (WKID, 0, ERRIND, CNDE, IDUM1) CALL CHKINQ ('pqepai', ERRIND) C defi is originally an array of zeros DO 250 LISTEL = 1, CNDE CALL PQEPAI (WKID, LISTEL, ERRIND, IDUM1, PAI) CALL CHKINQ ('pqepai', ERRIND) C mark as defined any index .le. cnde+1 C This will leave at least one UNmarked IF (PAI .LE. CNDE+1) DEFI (PAI) = 1 250 CONTINUE C first remaining zero indicates undefined entry DO 260 PAI = 1, CNDE+1 IF (DEFI (PAI) .EQ. 0) THEN C determine lowest undefined index (not in defined list) = lui LUI = PAI GOTO 270 ENDIF 260 CONTINUE CALL UNMSG 1 ('Cannot find undefined index - error in test algorithm.') C established lui as lowest undefined index 270 CONTINUE C inquire undefined table entry as REALIZED CALL PQPAR (WKID, LUI, PREALI, DIMX, DIMY, ERRIND, DX, DY, COLIA) C check errind for overflow IF (ERRIND .EQ. 2001) THEN WRITE (MSG, '(A,A,2I5,A)') 'PQPAR returned Fortran error ', 1 '2001: declared array size of (', DIMX, DIMY, 2 ') insufficient.' CALL INMSG (MSG) GOTO 350 ENDIF CALL CHKINQ ('pqpar', ERRIND) C inquire pattern index #1 as REALIZED CALL PQPAR (WKID, 1, PREALI, DIMX, DIMY, ERRIND, DX2,DY2, COLIA2) C check errind for overflow IF (ERRIND .EQ. 2001) THEN WRITE (MSG, '(A,A,2I5,A)') 'PQPAR returned Fortran error ', 1 '2001: declared array size of (', DIMX, DIMY, 2 ') insufficient.' CALL INMSG (MSG) GOTO 350 ENDIF CALL CHKINQ ('pqpar', ERRIND) CALL SETMSG ('3 7', '<Inquire pattern representation> ' // 1 'should return attributes for pattern index #1 ' // 2 'when inquiring an undefined entry as REALIZED.') C should have same attributes CALL IFPF (DX2 .EQ. DX .AND. 1 DY2 .EQ. DY .AND. 2 ARREQ(DIMX, DIMY, 1,1, DX, DY, COLIA, COLIA2)) 350 CONTINUE C If number of initialized entries >= implementation max then skip test C of setting undefined entry - goto redefined_test. IF (NIE .GE. MPAI) THEN CALL INMSG ('Skipping undefined-test because all pattern ' // 1 'table entries are initialized.') GOTO 400 ENDIF C Check that program allows enough room IF (NIE .GE. MAXPL) THEN CALL INMSG ('Skipping undefined-test because number ' // 1 'of initialized entries exceed program limits.') GOTO 400 ENDIF C set values for attributes ISC = 4 ISR = 5 DX2 = 2 DY2 = 2 COLIA2(4,5) = 33 COLIA2(5,5) = 53 COLIA2(4,6) = 32 COLIA2(5,6) = 48 CALL PSPAR (WKID, LUI, DIMX, DIMY, ISC, ISR, DX2, DY2, COLIA2) CALL PQEPAI (WKID, 0, ERRIND, LISTOT, IDUM1) CALL CHKINQ ('pqepai', ERRIND) CALL SETMSG ('1 3', 'Defining an undefined entry should ' // 1 'increment the number of defined entries.') CALL IFPF (LISTOT .EQ. CNDE + 1) C check that lui is now among defined entries CALL SETMSG ('1 3', 'Undefined table entry should be ' // 1 'reported as defined after set.') DO 380 LISTEL = 1, LISTOT CALL PQEPAI (WKID, LISTEL, ERRIND, IDUM1, PAI) CALL CHKINQ ('pqepai', ERRIND) IF (PAI .EQ. LUI) THEN CALL PASS GOTO 390 ENDIF 380 CONTINUE CALL FAIL GOTO 400 390 CONTINUE CALL PQPAR (WKID, LUI, PSET, DIMX, DIMY, ERRIND, DX, DY, COLIA) CALL CHKINQ ('pqpar', ERRIND) CALL SETMSG ('2 5', 'New values should be set for ' // 1 'undefined table entry.') CALL IFPF (DX .EQ. DX2 .AND. 1 DY .EQ. DY2 .AND. 2 ARREQ(DIMX, DIMY, ISC, ISR, DX2, DY2, COLIA2, COLIA)) C * * * * * redefined entries * * * * * 400 CONTINUE C get current number defined entries CALL PQEPAI (WKID, 0, ERRIND, CNDE, IDUM1) CALL CHKINQ ('pqepai', ERRIND) C let dpi = a defined pattern index from middle of list CALL PQEPAI (WKID, (CNDE+1)/2, ERRIND, IDUM1, DPI) CALL CHKINQ ('pqepai', ERRIND) C set to first set of values ISC = 1 ISR = 1 DX = 3 DY = 2 COLIA(1,1) = 8 COLIA(2,1) = 22 COLIA(3,1) = 36 COLIA(1,2) = 19 COLIA(2,2) = 53 COLIA(3,2) = 89 CALL PSPAR (WKID, DPI, DIMX, DIMY, ISC, ISR, DX, DY, COLIA) C re-set to different values DX2 = 2 DY2 = 2 COLIA2(1,1) = 3 COLIA2(2,1) = 19 COLIA2(1,2) = 9 COLIA2(2,2) = 47 CALL PSPAR (WKID, DPI, DIMX, DIMY, ISC, ISR, DX2, DY2, COLIA2) C get list size - nnde = new number of defined entries CALL PQEPAI (WKID, 0, ERRIND, NNDE, IDUM1) CALL CHKINQ ('pqepai', ERRIND) CALL SETMSG ('1 3', 'Redefining a user-defined entry should ' // 1 'not change number of defined entries.') CALL IFPF (NNDE .EQ. CNDE) C check that dpi is among defined entries CALL SETMSG ('1 3', 'Redefined table entry should be ' // 1 'reported as defined after set.') DO 420 LISTEL = 1, NNDE CALL PQEPAI (WKID, LISTEL, ERRIND, IDUM1, PAI) CALL CHKINQ ('pqepai', ERRIND) IF (PAI .EQ. DPI) THEN CALL PASS GOTO 430 ENDIF 420 CONTINUE CALL FAIL GOTO 499 430 CONTINUE CALL PQPAR (WKID, DPI, PSET, DIMX, DIMY, ERRIND, DX, DY, COLIA) CALL CHKINQ ('pqpar', ERRIND) CALL SETMSG ('2 5', 'New values should be set for ' // 1 'redefined table entry.') CALL IFPF (DX .EQ. DX2 .AND. 1 DY .EQ. DY2 .AND. 2 ARREQ(DIMX, DIMY, ISC, ISR, DX2, DY2, COLIA2, COLIA)) 499 CONTINUE C * * * * * access all defined entries * * * * * CALL SETMSG ('3 4', '<Inquire list of pattern indices> and ' // 1 '<inquire pattern representation> should detect ' // 2 'all defined entries.') C get listel-th element of defined pattern indices DO 510 LISTEL = 1, NNDE CALL PQEPAI (WKID, LISTEL, ERRIND, IDUM1, PAI) IF (ERRIND .NE. 0) THEN CALL FAIL GOTO 515 ENDIF CALL PQPAR (WKID, PAI, PSET, DIMX, DIMY, ERRIND, DX, DY, COLIA) C check errind for overflow IF (ERRIND .EQ. 2001 .OR. ERRIND .EQ. 0) THEN C OK so far ELSE CALL FAIL GOTO 515 ENDIF 510 CONTINUE CALL PASS 515 CONTINUE 666 CONTINUE CALL ENDIT END
section "Tries via Search Trees" theory Trie_Map imports RBT_Map Trie_Fun begin text \<open>An implementation of tries based on maps implemented by red-black trees. Works for any kind of search tree.\<close> text \<open>Implementation of map:\<close> type_synonym 'a mapi = "'a rbt" datatype 'a trie_map = Nd bool "('a * 'a trie_map) mapi" text \<open>In principle one should be able to given an implementation of tries once and for all for any map implementation and not just for a specific one (RBT) as done here. But because the map (@{typ "'a rbt"}) is used in a datatype, the HOL type system does not support this. However, the development below works verbatim for any map implementation, eg \<open>Tree_Map\<close>, and not just \<open>RBT_Map\<close>, except for the termination lemma \<open>lookup_size\<close>.\<close> term size_tree lemma lookup_size[termination_simp]: fixes t :: "('a::linorder * 'a trie_map) rbt" shows "lookup t a = Some b \<Longrightarrow> size b < Suc (size_tree (\<lambda>ab. Suc(Suc (size (snd(fst ab))))) t)" apply(induction t a rule: lookup.induct) apply(auto split: if_splits) done definition empty :: "'a trie_map" where [simp]: "empty = Nd False Leaf" fun isin :: "('a::linorder) trie_map \<Rightarrow> 'a list \<Rightarrow> bool" where "isin (Nd b m) [] = b" \| "isin (Nd b m) (x # xs) = (case lookup m x of None \<Rightarrow> False \| Some t \<Rightarrow> isin t xs)" fun insert :: "('a::linorder) list \<Rightarrow> 'a trie_map \<Rightarrow> 'a trie_map" where "insert [] (Nd b m) = Nd True m" \| "insert (x#xs) (Nd b m) = Nd b (update x (insert xs (case lookup m x of None \<Rightarrow> empty \| Some t \<Rightarrow> t)) m)" fun delete :: "('a::linorder) list \<Rightarrow> 'a trie_map \<Rightarrow> 'a trie_map" where "delete [] (Nd b m) = Nd False m" \| "delete (x#xs) (Nd b m) = Nd b (case lookup m x of None \<Rightarrow> m \| Some t \<Rightarrow> update x (delete xs t) m)" subsection "Correctness" text \<open>Proof by stepwise refinement. First abstract to type @{typ "'a trie"}.\<close> fun abs :: "'a::linorder trie_map \<Rightarrow> 'a trie" where "abs (Nd b t) = Trie_Fun.Nd b (\<lambda>a. map_option abs (lookup t a))" fun invar :: "('a::linorder)trie_map \<Rightarrow> bool" where "invar (Nd b m) = (M.invar m \<and> (\<forall>a t. lookup m a = Some t \<longrightarrow> invar t))" lemma isin_abs: "isin t xs = Trie_Fun.isin (abs t) xs" apply(induction t xs rule: isin.induct) apply(auto split: option.split) done lemma abs_insert: "invar t \<Longrightarrow> abs(insert xs t) = Trie_Fun.insert xs (abs t)" apply(induction xs t rule: insert.induct) apply(auto simp: M.map_specs RBT_Set.empty_def[symmetric] split: option.split) done lemma abs_delete: "invar t \<Longrightarrow> abs(delete xs t) = Trie_Fun.delete xs (abs t)" apply(induction xs t rule: delete.induct) apply(auto simp: M.map_specs split: option.split) done lemma invar_insert: "invar t \<Longrightarrow> invar (insert xs t)" apply(induction xs t rule: insert.induct) apply(auto simp: M.map_specs RBT_Set.empty_def[symmetric] split: option.split) done lemma invar_delete: "invar t \<Longrightarrow> invar (delete xs t)" apply(induction xs t rule: delete.induct) apply(auto simp: M.map_specs split: option.split) done text \<open>Overall correctness w.r.t. the \<open>Set\<close> ADT:\<close> interpretation S2: Set where empty = empty and isin = isin and insert = insert and delete = delete and set = "set o abs" and invar = invar proof (standard, goal_cases) case 1 show ?case by (simp add: isin_case split: list.split) next case 2 thus ?case by (simp add: isin_abs) next case 3 thus ?case by (simp add: set_insert abs_insert del: set_def) next case 4 thus ?case by (simp add: set_delete abs_delete del: set_def) next case 5 thus ?case by (simp add: M.map_specs RBT_Set.empty_def[symmetric]) next case 6 thus ?case by (simp add: invar_insert) next case 7 thus ?case by (simp add: invar_delete) qed end
$ifndef _CLASSIFY_HOLDER_ $define _CLASSIFY_HOLDER_ $include "Utils.mpl" ClassifyHolder:=module() local ieqCode,sols,unsolvedSols; export reset, # 重置状态 addSol, # 新增解 getSols, # 获取解 addUnsolvedSol, getUnsolvedSols, getIeqCode; # 获取不变量方程的编号 reset:=proc() ieqCode:=0; sols:={}; unsolvedSols:={}; return; end proc: addSol:=proc(s::InvSol) flogf[1]("添加代表元"); flog[1](s:-rep); sols:=sols union {s}; return; end proc: getSols:=proc() local res:=sols; res:=uniqueObj(res,InvSol:-uniqueKey); res:=sort(res,'key'=(x->x:-ieqCode)); return res; end proc: getIeqCode:=proc() ieqCode:=ieqCode+1; return ieqCode; end proc: addUnsolvedSol:=proc(s) unsolvedSols:=unsolvedSols union {s}; return; end proc: getUnsolvedSols:=proc() return unsolvedSols; end proc: end module: $endif
\chapter{Introduction} Hey Everyone! Thanks for checking out the wikibook-project. This project is very much still in beta but has been released to everyone, so we can iterate on it this semester and fill in any gaps for the coming semesters. A few things to note \begin{enumerate} \item You'll be responsible for learning everything in the original wikibook. This is meant as a supplement for that to understand concepts and read more as a book \item This book strives for being more readable than compatible with the lectures, meaning that one lecture can go over many different sections of the book at various levels of resolution. The book is meant to be read through and make sense on a read through. \item This is still a wikibook, if you have thoughts, comments, suggestions, and fixes: please let us know! \end{enumerate}
If $f$ converges to $l$ and $f$ is eventually within $e$ of $a$, then $l$ is within $e$ of $a$.
```python import holoviews as hv hv.extension('bokeh') hv.opts.defaults(hv.opts.Curve(width=500), hv.opts.Points(width=500), hv.opts.Image(width=500, colorbar=True, cmap='Viridis')) ``` ```python import numpy as np import scipy.signal import scipy.linalg ``` # Estimadores adaptivos parte I Hasta ahora hemos estudiando sistemas lineales donde: - sus coeficientes quedan fijos luego del diseño y son constantes en el tiempo - hacen supuestos sobre los estadísticos de la señal/ruido ¿Qué podemos hacer si - no podemos hacer supuestos sobre los estadísticos? - los estadísticos de la señal/ruido cambian en el tiempo? - estamos en un escenario donde los datos llegan continuamente (data streaming)? Cuando los estadísticos cambian en el tiempo decimos que la señal es no estacionaria. En estos casos necesitamos un estimador de tipo adaptivo, es decir sistemas y filtros cuyos coeficientes se pueden adaptar a medida que llegan nuevos datos. Estos estimadores se diseñan de acuerdo a un método de optimización que es online La siguiente figura muestra algunos ejemplos de aplicaciones de sistemas adaptivos El método de optimización online que utilizaremos principalmente en este curso es el gradiente descendente estocástico. Revisemos a continuación los fundamentos. ## Gradiente descendente Sea un vector de pesos $w$ de largo $L+1$ que guarda los coeficientes de un estimador Sea ahora una función de costo que mapea el vector de pesos a un número real $$ J(w): \mathbb{R}^{L+1} \to \mathbb{R} $$ La función de costo debe ser tal que a menor $J$ menor sea el error del estimador Para entrenar un estimador o filtro adaptivo se tienen los siguientes pasos conceptuales 1. Partimos de una solución inicial $w_0$ 1. Modificamos iterativamente $w$ tal que $J(w_{t+1}) < J(w_t)$ 1. Nos detenemos al cumplir un cierto criterio Para modificar iterativamete y eficientemente los pesos utilizaremos la regla del gradiente descendente (GD) $$ w_{t+1} = w_t - \mu \frac{dJ(w)}{dw}, $$ donde $\mu$ se conoce como tasa de aprendizaje o "paso" - Imaginemos $J$ como una superficie de $L+1$ dimensiones - En cada punto el gradiente negativo de $J$ nos indica hacia donde está el descenso más abrupto - La tasa $\mu$ nos da el largo del salto entre $w_t$ y $w_{t+1}$ Observando la expansión de Taylor de primer orden de $J$ en $w_{t}$ $$ \begin{align} J(w_{t+1}) &= J(w_t) + \frac{dJ(w_t)}{dw} (w_{t+1} - w_{t}) \nonumber \\ &= J(w_t) -\mu \left \\| \frac{dJ(w_t)}{dw} \right \\|^2 \leq J(w_t) \nonumber \end{align} $$ es decir que usando la regla GD con $\mu>0$ y asumiendo que $J$ es convexo entonces se cumple que $J$ siempre decrece monotónicamente. La siguiente gráficas interactivas muestran una superficie de costo no convexa para un parámetro unidimensional. Cada punto representa una solución que parte desde una posición inicial distinta. Las flechas corresponden a la derivada multiplicada por la tasa de aprendizaje. Estudie la evolución de las tres soluciones en cada caso. En primer lugar se utiliza $\mu=0.05$ ```python J = lambda w : (w-1)*2 + 0.2np.sin(2np.piw) # Función de costo gradJ = lambda w : 2(w-1) + 0.22np.pinp.cos(2np.piw) # Gradiente mu = 0.05 # Tasa de aprendizaje iteraciones = 15 wt = np.zeros(shape=(iteraciones, 3)) wt[0, :] = np.array([0.05, 0.4, 1.9]) # Solución inicial w_plot = np.linspace(0, 2, num=100) for k in range(1, iteraciones): wt[k, :] = wt[k-1, :] - mugradJ(wt[k-1, :]) ``` ```python loss_surface = hv.Curve((w_plot, J(w_plot)), 'w', 'J') hMap = hv.HoloMap(kdims='Iteración') for k in range(iteraciones): dots = hv.Points((wt[k, :], J(wt[k, :]))).opts(size=10, color='k') mag = mugradJ(wt[k, :]) angle = np.pi/2 - np.sign(-mag)np.pi/2 mag = np.abs(mag) arrows = hv.VectorField((wt[k, :], J(wt[k, :]), angle, mag)).opts(pivot='tail', magnitude=hv.dim('Magnitude'), rescale_lengths=False) hMap[k] = dots arrows loss_surface * hMap ``` :::{warning} Dependiendo de donde partimos la solución final es distinta. El gradiente descedente puede quedarse "atorado" en un mínimo local o en un punto silla ::: Ahora observe como evolucionan las tres soluciones con $\mu=0.5$, es decir 10 veces más grande que el caso anterior ```python J = lambda w : (w-1)*2 + 0.2np.sin(2np.piw) # Función de costo gradJ = lambda w : 2(w-1) + 0.22np.pinp.cos(2np.piw) # Gradiente mu = 0.5 # Tasa de aprendizaje iteraciones = 15 wt = np.zeros(shape=(iteraciones, 3)) wt[0, :] = np.array([0.05, 0.4, 1.9]) # Solución inicial w_plot = np.linspace(0, 2, num=100) for k in range(1, iteraciones): wt[k, :] = wt[k-1, :] - mugradJ(wt[k-1, :]) ``` ```python loss_surface = hv.Curve((w_plot, J(w_plot)), 'w', 'J') hMap = hv.HoloMap(kdims='Iteración') for k in range(iteraciones): dots = hv.Points((wt[k, :], J(wt[k, :]))).opts(size=10, color='k') mag = mugradJ(wt[k, :]) angle = np.pi/2 - np.sign(-mag)np.pi/2 mag = np.abs(mag) arrows = hv.VectorField((wt[k, :], J(wt[k, :]), angle, mag)).opts(pivot='tail', magnitude=hv.dim('Magnitude'), rescale_lengths=False) hMap[k] = dots arrows loss_surface * hMap ``` :::{warning} Si la tasa de aprendizaje es muy alta, los pasos son muy largos y podríamos no converger a un punto estacionario ::: Los ejemplos anteriores nos han mostrado algunas de las limitaciones del algoritmo de gradiente descendente. Es importante tenerlas en cuenta cuando lo utilicemos en nuestras aplicaciones ## Gradiente descendente en el filtro de Wiener Para el filtro de Wiener teníamos que $$ J(h) = \sigma_d^2 - 2 \textbf{h}^T R_{ud} + \textbf{h}^T R_{uu} \textbf{h}, $$ por ende $$ \frac{dJ(h)}{dh} = -2 R_{ud} + 2 R_{uu} \textbf{h} $$ y finalmente $$ \textbf{h}_{t+1} = \textbf{h}_{t} (I - 2 \mu R_{uu}) + 2\mu R_{ud} $$ En este caso la condición para una convergencia estable es $$ 0 < \mu < \frac{1}{\lambda_{\text{max}}}, $$ donde $\lambda_{\text{max}}$ es valor propio más grande de $R_{uu}$ (La prueba de esto puede encontrarse en Haykin, "Adaptive filter theory", Sección 4.3) ## Gradiente descendente estocástico (SGD) El filtro de Wiener es óptimo pero no adaptivo: - Requiere de $N$ muestras de $u$ y $d$ para estimar $R_{ud}$ y $R_{uu}$ - Los pesos se adaptan luego de haber presentado las $N$ muestras: Es una estrategia de tipo batch - Asume que la señal es estacionaria Si nuestros son no estacionarios significa que debemos adaptar el filtro a medida que nuevas muestras son observadas . Para lograr esto podemos usar la versión estocástica del GD: SGD En SGD: - los pesos se adaptan luego de haber presentado una sola muestra o un conjunto pequeño de muestras (mini-batch) - no hay garantía de llegar al óptimo en un problema convexo, pero es más eficiente computacionalmente que GD El siguiente esquema muestra una comparación entre la trayectoria de $w$ cuando se usa GD (negro) y SGD (rojo). En general la trayectoria de SGD será más ruidosa y también podría requerir más pasos, pero cada paso es mucho más económico ## Algoritmo Least Mean Square (LMS) Podemos extender el filtro de Wiener al caso no-estacionario usando SGD, el resultado es un algoritmo simple que además es robusto: El algoritmo LMS - Fue fue inventado en 1960 por [Bernard Widrow](https://en.wikipedia.org/wiki/Bernard_Widrow) y Ted Hoff - A diferencia del filtro de Wiener no se requiere conocimiento estadístico del proceso. Tampoco se requiere calcular e invertir la matriz de correlación - El algoritmo LMS se ajusta o entrena de manera recursiva y online Consideremos la función de costo estocástica para la arquitectura FIR que utilizamos para el filtro de Wiener $$ \begin{align} J^s_n(\textbf{w}) &= e_n^2 \nonumber \\ &= (d_n - y_n)^2 \nonumber \\ &= (d_n - \textbf{w}^T \textbf{u}_n )^2 \nonumber \\ &= (d_n - \sum_{k=0}^{L} w_{n, k} u_{n-k} )^2 \nonumber \end{align} $$ donde definimos $\textbf{u}_n = [u_n, u_{n-1}, \ldots, u_{n-L}]$. :::{note} A diferencia del filtro de Wiener no aplicamos el valor esperado al error cuadrático. Se usa el error cuadrático instantaneo ::: Para continuar calculamos el gradiente en función del peso $w_{n, k}$ $$ \frac{d J^s_n (\textbf{w})}{d w_{n, k}} = - 2 e_n u_{n-k} $$ Luego, usando la regla SGD llegamos a $$ w_{n+1, k} = w_{n, k} + 2 \mu e_n u_{n-k}, k=0, 1, \ldots, L $$ y que en forma matricial es $$ \begin{align} \textbf{w}_{n+1} &= \textbf{w}_{n} + 2 \mu e_n \textbf{u}_{n}\nonumber \\ &= \textbf{w}_{n} + 2 \mu (d_n - \textbf{w}_{n}^T \textbf{u}_{n}) \textbf{u}_{n}, \nonumber \end{align} $$ que se conoce como la regla de Widrow-Hoff :::{important} El algoritmo LMS estima el error instantaneo y actualiza los pesos recursivamente ::: La complejidad de este algoritmo es $L+1$. ### Convergencia del algoritmo LMS (Haykin 6.5) El algoritmo LMS tiende en la media al valor óptimo $$ \mathbb{E}[\textbf{w}_n] \to \textbf{w}^* $$ para $n\to \infty$ Además convergence en la media cuadrada: La varianza de $\textbf{w}_n - \textbf{w}^$ tiene al valor mínimo de $J$ para $n\to \infty$ Esto se cumple si $$ 0 < \mu < \frac{2}{\text{Tr}[R_{uu}]} $$ donde $R_{uu} = \mathbb{E}[\textbf{u}_n \textbf{u}_n^T ]$ es la matriz de autocorrelación y $\text{Tr}[]$ el operador que calcula la traza de una matriz ### Algoritmo Normalized LMS (NLMS) Tenemos la siguiente regla iterativa $$ \begin{align} \textbf{w}_{n+1} &= \textbf{w}_{n} + 2 \mu (d_n - \textbf{w}_{n}^T \textbf{u}_{n}) \textbf{u}_{n} \nonumber \\ & = \textbf{w}_{n} + \Delta \textbf{w}_n \nonumber \end{align} $$ que se puede interpretar graficamente como (donde $\textbf{x}(k)$ y $\textbf{w}(k)$ corresponden a $\textbf{u}_n$ y $\textbf{w}_n$ en nuestra notación, respectivamente) :::{note} Los cambios en el vector de peso $\Delta \textbf{w}_n$ son paralelos a $\textbf{u}_{n}$. Además estos cambios podrían estar dominados por $$ \max \textbf{u}_{n} = [u_n, u_{n-1}, \ldots, u_{n-L}] $$ ::: El algoritmo Normalized LMS* (NLMS) corrige este problema ponderando por la varianza de $\textbf{u}_{n}$ $$ \textbf{w}_{n+1} = \textbf{w}_{n} + 2 \mu (d_n - \textbf{w}_{n}^T \textbf{u}_{n}) \frac{\textbf{u}_{n}}{\left(\\|\textbf{u}_{n}\\|^2 + \delta\right)} $$ donde la constante $\delta$ es un valor pequeño que se usa para evitar divisiones por cero. En lo que sigue usaremos NLMS para revisar algunas aplicaciones ## Implementación del filtro NLMS en Python Podemos implementar las ecuaciones del filtro NLMS como se muestra a continuación ```python class Filtro_NLMS: def __init__(self, L, mu, delta=1e-6, winit=None): self.L = L self.w = np.zeros(shape=(L+1, )) self.mu = mu self.delta = delta def update(self, un, dn): # Asumiendo que un = [u[n], u[n-1], ..., u[n-L]] unorm = np.dot(un, un) + self.delta yn = np.dot(self.w, un) self.w += 2self.mu(dn - yn)(un/unorm) return yn ``` - El filtro recibe como entrada el orden $L$ y la tasa de aprendizaje $\mu$ - Se asume un vector cero para los pesos iniciales, pero también en la práctica podemos partir de una solución anterior si esta existiera - Para actualizar el vector de pesos es necesario entregar el vector $\textbf{u}_n \in \mathbb{R}^{L+1}$ y la salida deseada $d_n \in \mathbb{R}$. La función `update` retorna la salida predicha por el filtro $y_n = w_n^T \textbf{u}_n $ A continuación probaremos este filtro con una aplicación conocida como Adaptive line enhancement* (ALE). ALE se refiere a un sistema adaptivo para eliminar ruido blanco aditivo de una señal. El sistema aprende un filtro pasabanda en torno a la frecuencia de interés En ALE usamos como señal deseada $$ d_n = u_n = \textbf{u}_n[0] $$ El valor predicho por el filtro será la señal $u$ pero libre de ruido blanco. Esto se debe a que el ruido blanco no tiene correlación y por ende el filtro adaptivo no lo puede predecir ```python # Digamos que u = s + n # El objetivo es limpiar u para obtener s # s es una señal determínista y n es ruido blanco Fs, f0 = 100, 5 t = np.arange(0, 4, 1/Fs) s = np.sin(2.0np.pitf0) n = 0.5np.random.randn(len(t)) s[t>2.0] += 5 # Simulemos un cambio abrupto en la media de la señal #s += s(0.5 + 0.5np.cos(2.0np.pit/2)) # Tremolo (AM) u = s + n ``` A diferencia de un filtro estático (como el filtro de Wiener) es posible filtrar incluso ante cambios bruscos en la señal. Estudie como cambia el resultado del filtro con distintos valores de $\mu$ ```python L = 20 u_preds = {} for mu in np.logspace(-2, 0, num=10): myfilter = Filtro_NLMS(L=L, mu=mu) u_preds[mu] = np.zeros(shape=(len(u),)) for k in range(L+1, len(u)): u_preds[mu][k] = myfilter.update(u[k-L-1:k][::-1], u[k]) ``` ```python hMap = hv.HoloMap(kdims='mu') for mu, u_pred in u_preds.items(): s1 = hv.Curve((t, s), 'Tiempo', 'Señal', label='Limpia') s2 = hv.Scatter((t, u), 'Tiempo', 'Señal', label='Contaminada') s3 = hv.Curve((t, u_pred), 'Tiempo', 'Señal', label='Filtrada') hMap[mu] = hv.Overlay([s1, s2, s3]).opts(hv.opts.Overlay(legend_position='top'), hv.opts.Curve(ylim=(-5, 10), height=350)) hMap ``` :::{important} La tasa de aprendizaje $\mu$ controla la velocidad de adaptación. Pero una tasa demasiado grande provoca que el filtro sea inestable. En general el valor óptimo de $\mu$ depende del problema y del valor de $L$ ::: La siguiente figura muestra la respuesta en frecuencia del filtro en función del tiempo para $\mu=0.02$ Observe como a medida que se adapta el filtro se concentra en la frecuencia fundamental de la señal, que en este caso es 5 Hz ```python L = 20 u_preds = {} myfilter = Filtro_NLMS(L=L, mu=0.02) H_history = np.zeros(shape=(512, len(u))) for k in range(L+1, len(u)): myfilter.update(u[k-L-1:k][::-1], u[k]) fk, Hk = scipy.signal.freqz(b=myfilter.w, a=1, fs=Fs) H_history[:, k] = np.abs(Hk) ``` ```python hv.Image((t, fk, H_history), kdims=['Tiempo [s]', 'Frecuencia [Hz]']).opts(cmap='Blues') ``` ## Comparación entre Filtro de Wiener/GD y algoritmo LMS/SGD - Supuestos: Wiener requiere un ambiente estacionario lo cual nos permite calcular $R_{uu}$ y $R_{ud}$. En LMS la señal puede ser no estacionaria. - Aprendizaje: En el filtro de Wiener el aprendizaje es determinista. En LMS el aprendizaje viene promediando a nivel de los estimadores de $w$. En LMS el aprendizaje es estadístico. - Optimalidad: Wiener es óptimo en cambio LMS es sub-óptimo (localmente óptimo). LMS tiende a la solución de Wiener - Costo: LMS se actualiza online y tiene costo $L$. Wiener se entrena offline y tiene costo $L^2$ A continuación se muestra un diagrama que compara el filtro de Wiener y el algoritmo LMS ```python ```
[STATEMENT] lemma floorlog_leD: "floorlog b x \<le> w \<Longrightarrow> b > 1 \<Longrightarrow> x < b ^ w" [PROOF STATE] proof (prove) goal (1 subgoal): 1. \<lbrakk>floorlog b x \<le> w; 1 < b\<rbrakk> \<Longrightarrow> x < b ^ w [PROOF STEP] by (metis floorlog_bounds leD linorder_neqE_nat order.strict_trans power_strict_increasing_iff zero_less_one zero_less_power)
module Main import Data.Strings data MyEither a b = MyLeft a \| MyRight b data MyDecEq : Type -> Type where MyYes : a -> MyDecEq a MyNo : (a -> Void) -> MyDecEq a myDec1 : MyDecEq Char myDec1 = MyYes 'a' myDec2 : MyDecEq Char myDec2 = MyNo ?absurd test : Int test = ?test1 test3 : Int -> Int -> Int test3 x y = x data MyStream : Type -> Type where MkMyStream : Lazy a -> Inf (MyStream a) -> MyStream a mkStream : a -> MyStream a mkStream x = MkMyStream x (mkStream x) headS : MyStream a -> a headS (MkMyStream x _) = x tailS : MyStream a -> MyStream a tailS (MkMyStream _ xs) = xs data MyStream2 : Type -> Type where MkMyStream2 : a -> MyStream2 a -> MyStream2 a MkMyStream2N : MyStream2 a myStream2 : a -> MyStream2 a myStream2 x = MkMyStream2 x (myStream2 x) heads2 : MyStream2 a -> Maybe a heads2 MkMyStream2N = Nothing heads2 (MkMyStream2 x _) = Just x %foreign "stg:prim__consoleLog = base:Prelude.putStrLn" prim__consoleLog : String -> PrimIO () export consoleLog : HasIO io => String -> io () consoleLog x = primIO $ prim__consoleLog x %foreign "stg:prim__applyFnIO = base:Prelude.somewhat" prim__applyFnIO : String -> Int -> (String -> Int -> PrimIO String) -> PrimIO String applyFnIO : HasIO io => String -> Int -> (String -> Int -> IO String) -> io String applyFnIO c i f = primIO $ prim__applyFnIO c i (\s, i => toPrim $ f s i) data MyBuffer : Type -> Type where [external] %foreign "stg:prim__myBuffer = base:Prelude.myBuffer" prim__myBuffer : PrimIO (MyBuffer Int) myBuffer : HasIO io => io (MyBuffer Int) myBuffer = primIO prim__myBuffer data ThreadID1 : Type where [external] %foreign "stg:prim__fork = base:Something.something" prim__fork : (1 prog : PrimIO ()) -> PrimIO ThreadID1 fork : (1 prog : IO ()) -> IO ThreadID1 fork act = fromPrim (prim__fork (toPrim act)) partial main : IO () main = do name <- getLine let nm1 = the (MyEither String String) (MyLeft name) let md1 = the (MyDecEq Char) (MyYes 'a') putStrLn $ strTail $ "!Hello " ++ name case nm1 of MyLeft n => printLn $ fastUnpack n MyRight n => printLn $ fastUnpack n case md1 of MyYes a => printLn $ show a MyNo _ => printLn $ "Noooo!" printLn $ test3 1 test consoleLog "BLAH!" str <- applyFnIO "hello" 1 (\s, i => do { printLn (s,i); pure s}) bfr <- myBuffer tid <- Main.fork (putStrLn "Hello Fork!") printLn $ headS $ tailS $ mkStream 4 printLn $ heads2 $ myStream2 5 pure () {- Main.prim__applyFnIO = Foreign call ["stg:prim__applyFnIO = base:Prelude.somewhat"] [String, Int, String -> Int -> %World -> IORes String, %World] -> IORes Main.{applyFnIO:1} = [0, 1, 2, 3, 4, 5, 6, 7]: %let v8 = (<Main.{applyFnIO:0} underapp 2>(v7, v0, v1, v2, v3, v4, v5, v6)) in (Main.prim__applyFnIO(v4, v5, v8, v7)) Main.applyFnIO = [0, 1, 2, 3, 4]: %case v1 of { %conalt Prelude.IO.HasIO at Prelude/IO.idr:33:1--35:32(v5, v6) => %let v7 = (v6 @ ([__])) in (%let v8 = (<Main.{applyFnIO:1} underapp 1>(v5, v6, v0, v1, v2, v3, v4)) in (v7 @ (v8))) Nothing } -}
module Relation.Path where open import Relation.Path.Operation
(* Title: ZF/UNITY/GenPrefix.thy Author: Sidi O Ehmety, Cambridge University Computer Laboratory Copyright 2001 University of Cambridge \<langle>xs,ys\<rangle>:gen_prefix(r) if ys = xs' @ zs where length(xs) = length(xs') and corresponding elements of xs, xs' are pairwise related by r Based on Lex/Prefix ) section\<open>Charpentier's Generalized Prefix Relation\<close> theory GenPrefix imports ZF begin definition (really belongs in ZF/Trancl) part_order :: "[i, i] \<Rightarrow> o" where "part_order(A, r) \<equiv> refl(A,r) \<and> trans[A](r) \<and> antisym(r)" consts gen_prefix :: "[i, i] \<Rightarrow> i" inductive ( Parameter A is the domain of zs's elements ) domains "gen_prefix(A, r)" \<subseteq> "list(A)list(A)" intros Nil: "<[],[]>:gen_prefix(A, r)" prepend: "\<lbrakk>\<langle>xs,ys\<rangle>:gen_prefix(A, r); \<langle>x,y\<rangle>:r; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> <Cons(x,xs), Cons(y,ys)>: gen_prefix(A, r)" append: "\<lbrakk>\<langle>xs,ys\<rangle>:gen_prefix(A, r); zs:list(A)\<rbrakk> \<Longrightarrow> <xs, ys@zs>:gen_prefix(A, r)" type_intros app_type list.Nil list.Cons definition prefix :: "i\<Rightarrow>i" where "prefix(A) \<equiv> gen_prefix(A, id(A))" definition strict_prefix :: "i\<Rightarrow>i" where "strict_prefix(A) \<equiv> prefix(A) - id(list(A))" (* less or equal and greater or equal over prefixes ) abbreviation pfixLe :: "[i, i] \<Rightarrow> o" (infixl \<open>pfixLe\<close> 50) where "xs pfixLe ys \<equiv> \<langle>xs, ys\<rangle>:gen_prefix(nat, Le)" abbreviation pfixGe :: "[i, i] \<Rightarrow> o" (infixl \<open>pfixGe\<close> 50) where "xs pfixGe ys \<equiv> \<langle>xs, ys\<rangle>:gen_prefix(nat, Ge)" lemma reflD: "\<lbrakk>refl(A, r); x \<in> A\<rbrakk> \<Longrightarrow> \<langle>x,x\<rangle>:r" apply (unfold refl_def, auto) done ( preliminary lemmas ) lemma Nil_gen_prefix: "xs \<in> list(A) \<Longrightarrow> <[], xs> \<in> gen_prefix(A, r)" by (drule gen_prefix.append [OF gen_prefix.Nil], simp) declare Nil_gen_prefix [simp] lemma gen_prefix_length_le: "\<langle>xs,ys\<rangle> \<in> gen_prefix(A, r) \<Longrightarrow> length(xs) \<le> length(ys)" apply (erule gen_prefix.induct) apply (subgoal_tac [3] "ys \<in> list (A) ") apply (auto dest: gen_prefix.dom_subset [THEN subsetD] intro: le_trans simp add: length_app) done lemma Cons_gen_prefix_aux: "\<lbrakk><xs', ys'> \<in> gen_prefix(A, r)\<rbrakk> \<Longrightarrow> (\<forall>x xs. x \<in> A \<longrightarrow> xs'= Cons(x,xs) \<longrightarrow> (\<exists>y ys. y \<in> A \<and> ys' = Cons(y,ys) \<and> \<langle>x,y\<rangle>:r \<and> \<langle>xs, ys\<rangle> \<in> gen_prefix(A, r)))" apply (erule gen_prefix.induct) prefer 3 apply (force intro: gen_prefix.append, auto) done lemma Cons_gen_prefixE: "\<lbrakk><Cons(x,xs), zs> \<in> gen_prefix(A, r); \<And>y ys. \<lbrakk>zs = Cons(y, ys); y \<in> A; x \<in> A; \<langle>x,y\<rangle>:r; \<langle>xs,ys\<rangle> \<in> gen_prefix(A, r)\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" apply (frule gen_prefix.dom_subset [THEN subsetD], auto) apply (blast dest: Cons_gen_prefix_aux) done declare Cons_gen_prefixE [elim!] lemma Cons_gen_prefix_Cons: "(<Cons(x,xs),Cons(y,ys)> \<in> gen_prefix(A, r)) \<longleftrightarrow> (x \<in> A \<and> y \<in> A \<and> \<langle>x,y\<rangle>:r \<and> \<langle>xs,ys\<rangle> \<in> gen_prefix(A, r))" apply (auto intro: gen_prefix.prepend) done declare Cons_gen_prefix_Cons [iff] (* Monotonicity of gen_prefix ) lemma gen_prefix_mono2: "r<=s \<Longrightarrow> gen_prefix(A, r) \<subseteq> gen_prefix(A, s)" apply clarify apply (frule gen_prefix.dom_subset [THEN subsetD], clarify) apply (erule rev_mp) apply (erule gen_prefix.induct) apply (auto intro: gen_prefix.append) done lemma gen_prefix_mono1: "A<=B \<Longrightarrow>gen_prefix(A, r) \<subseteq> gen_prefix(B, r)" apply clarify apply (frule gen_prefix.dom_subset [THEN subsetD], clarify) apply (erule rev_mp) apply (erule_tac P = "y \<in> list (A) " in rev_mp) apply (erule_tac P = "xa \<in> list (A) " in rev_mp) apply (erule gen_prefix.induct) apply (simp (no_asm_simp)) apply clarify apply (erule ConsE)+ apply (auto dest: gen_prefix.dom_subset [THEN subsetD] intro: gen_prefix.append list_mono [THEN subsetD]) done lemma gen_prefix_mono: "\<lbrakk>A \<subseteq> B; r \<subseteq> s\<rbrakk> \<Longrightarrow> gen_prefix(A, r) \<subseteq> gen_prefix(B, s)" apply (rule subset_trans) apply (rule gen_prefix_mono1) apply (rule_tac [2] gen_prefix_mono2, auto) done (* gen_prefix order **) ( reflexivity ) lemma refl_gen_prefix: "refl(A, r) \<Longrightarrow> refl(list(A), gen_prefix(A, r))" apply (unfold refl_def, auto) apply (induct_tac "x", auto) done declare refl_gen_prefix [THEN reflD, simp] ( Transitivity ) ( A lemma for proving gen_prefix_trans_comp ) lemma append_gen_prefix [rule_format (no_asm)]: "xs \<in> list(A) \<Longrightarrow> \<forall>zs. <xs @ ys, zs> \<in> gen_prefix(A, r) \<longrightarrow> \<langle>xs, zs\<rangle>: gen_prefix(A, r)" apply (erule list.induct) apply (auto dest: gen_prefix.dom_subset [THEN subsetD]) done ( Lemma proving transitivity and more) lemma gen_prefix_trans_comp [rule_format (no_asm)]: "\<langle>x, y\<rangle>: gen_prefix(A, r) \<Longrightarrow> (\<forall>z \<in> list(A). \<langle>y,z\<rangle> \<in> gen_prefix(A, s)\<longrightarrow>\<langle>x, z\<rangle> \<in> gen_prefix(A, s O r))" apply (erule gen_prefix.induct) apply (auto elim: ConsE simp add: Nil_gen_prefix) apply (subgoal_tac "ys \<in> list (A) ") prefer 2 apply (blast dest: gen_prefix.dom_subset [THEN subsetD]) apply (drule_tac xs = ys and r = s in append_gen_prefix, auto) done lemma trans_comp_subset: "trans(r) \<Longrightarrow> r O r \<subseteq> r" by (auto dest: transD) lemma trans_gen_prefix: "trans(r) \<Longrightarrow> trans(gen_prefix(A,r))" apply (simp (no_asm) add: trans_def) apply clarify apply (rule trans_comp_subset [THEN gen_prefix_mono2, THEN subsetD], assumption) apply (rule gen_prefix_trans_comp) apply (auto dest: gen_prefix.dom_subset [THEN subsetD]) done lemma trans_on_gen_prefix: "trans(r) \<Longrightarrow> trans[list(A)](gen_prefix(A, r))" apply (drule_tac A = A in trans_gen_prefix) apply (unfold trans_def trans_on_def, blast) done lemma prefix_gen_prefix_trans: "\<lbrakk>\<langle>x,y\<rangle> \<in> prefix(A); \<langle>y, z\<rangle> \<in> gen_prefix(A, r); r<=AA\<rbrakk> \<Longrightarrow> \<langle>x, z\<rangle> \<in> gen_prefix(A, r)" unfolding prefix_def apply (rule_tac P = "\<lambda>r. \<langle>x,z\<rangle> \<in> gen_prefix (A, r) " in right_comp_id [THEN subst]) apply (blast dest: gen_prefix_trans_comp gen_prefix.dom_subset [THEN subsetD])+ done lemma gen_prefix_prefix_trans: "\<lbrakk>\<langle>x,y\<rangle> \<in> gen_prefix(A,r); \<langle>y, z\<rangle> \<in> prefix(A); r<=AA\<rbrakk> \<Longrightarrow> \<langle>x, z\<rangle> \<in> gen_prefix(A, r)" unfolding prefix_def apply (rule_tac P = "\<lambda>r. \<langle>x,z\<rangle> \<in> gen_prefix (A, r) " in left_comp_id [THEN subst]) apply (blast dest: gen_prefix_trans_comp gen_prefix.dom_subset [THEN subsetD])+ done (* Antisymmetry ) lemma nat_le_lemma [rule_format]: "n \<in> nat \<Longrightarrow> \<forall>b \<in> nat. n #+ b \<le> n \<longrightarrow> b = 0" by (induct_tac "n", auto) lemma antisym_gen_prefix: "antisym(r) \<Longrightarrow> antisym(gen_prefix(A, r))" apply (simp (no_asm) add: antisym_def) apply (rule impI [THEN allI, THEN allI]) apply (erule gen_prefix.induct, blast) apply (simp add: antisym_def, blast) txt\<open>append case is hardest\<close> apply clarify apply (subgoal_tac "length (zs) = 0") apply (subgoal_tac "ys \<in> list (A) ") prefer 2 apply (blast dest: gen_prefix.dom_subset [THEN subsetD]) apply (drule_tac psi = "<ys @ zs, xs> \<in> gen_prefix (A,r) " in asm_rl) apply simp apply (subgoal_tac "length (ys @ zs) = length (ys) #+ length (zs) \<and>ys \<in> list (A) \<and>xs \<in> list (A) ") prefer 2 apply (blast intro: length_app dest: gen_prefix.dom_subset [THEN subsetD]) apply (drule gen_prefix_length_le)+ apply clarify apply simp apply (drule_tac j = "length (xs) " in le_trans) apply blast apply (auto intro: nat_le_lemma) done (* recursion equations **) lemma gen_prefix_Nil: "xs \<in> list(A) \<Longrightarrow> <xs, []> \<in> gen_prefix(A,r) \<longleftrightarrow> (xs = [])" by (induct_tac "xs", auto) declare gen_prefix_Nil [simp] lemma same_gen_prefix_gen_prefix: "\<lbrakk>refl(A, r); xs \<in> list(A)\<rbrakk> \<Longrightarrow> <xs@ys, xs@zs>: gen_prefix(A, r) \<longleftrightarrow> \<langle>ys,zs\<rangle> \<in> gen_prefix(A, r)" unfolding refl_def apply (induct_tac "xs") apply (simp_all (no_asm_simp)) done declare same_gen_prefix_gen_prefix [simp] lemma gen_prefix_Cons: "\<lbrakk>xs \<in> list(A); ys \<in> list(A); y \<in> A\<rbrakk> \<Longrightarrow> <xs, Cons(y,ys)> \<in> gen_prefix(A,r) \<longleftrightarrow> (xs=[] \| (\<exists>z zs. xs=Cons(z,zs) \<and> z \<in> A \<and> \<langle>z,y\<rangle>:r \<and> \<langle>zs,ys\<rangle> \<in> gen_prefix(A,r)))" apply (induct_tac "xs", auto) done lemma gen_prefix_take_append: "\<lbrakk>refl(A,r); \<langle>xs,ys\<rangle> \<in> gen_prefix(A, r); zs \<in> list(A)\<rbrakk> \<Longrightarrow> <xs@zs, take(length(xs), ys) @ zs> \<in> gen_prefix(A, r)" apply (erule gen_prefix.induct) apply (simp (no_asm_simp)) apply (frule_tac [!] gen_prefix.dom_subset [THEN subsetD], auto) apply (frule gen_prefix_length_le) apply (subgoal_tac "take (length (xs), ys) \<in> list (A) ") apply (simp_all (no_asm_simp) add: diff_is_0_iff [THEN iffD2] take_type) done lemma gen_prefix_append_both: "\<lbrakk>refl(A, r); \<langle>xs,ys\<rangle> \<in> gen_prefix(A,r); length(xs) = length(ys); zs \<in> list(A)\<rbrakk> \<Longrightarrow> <xs@zs, ys @ zs> \<in> gen_prefix(A, r)" apply (drule_tac zs = zs in gen_prefix_take_append, assumption+) apply (subgoal_tac "take (length (xs), ys) =ys") apply (auto intro!: take_all dest: gen_prefix.dom_subset [THEN subsetD]) done (NOT suitable for rewriting since [y] has the form y#ys) lemma append_cons_conv: "xs \<in> list(A) \<Longrightarrow> xs @ Cons(y, ys) = (xs @ [y]) @ ys" by (auto simp add: app_assoc) lemma append_one_gen_prefix_lemma [rule_format]: "\<lbrakk>\<langle>xs,ys\<rangle> \<in> gen_prefix(A, r); refl(A, r)\<rbrakk> \<Longrightarrow> length(xs) < length(ys) \<longrightarrow> <xs @ [nth(length(xs), ys)], ys> \<in> gen_prefix(A, r)" apply (erule gen_prefix.induct, blast) apply (frule gen_prefix.dom_subset [THEN subsetD], clarify) apply (simp_all add: length_type) ( Append case is hardest ) apply (frule gen_prefix_length_le [THEN le_iff [THEN iffD1]]) apply (frule gen_prefix.dom_subset [THEN subsetD], clarify) apply (subgoal_tac "length (xs) :nat\<and>length (ys) :nat \<and>length (zs) :nat") prefer 2 apply (blast intro: length_type, clarify) apply (simp_all add: nth_append length_type length_app) apply (rule conjI) apply (blast intro: gen_prefix.append) apply (erule_tac V = "length (xs) < length (ys) \<longrightarrow>u" for u in thin_rl) apply (erule_tac a = zs in list.cases, auto) apply (rule_tac P1 = "\<lambda>x. <u(x), v>:w" for u v w in nat_diff_split [THEN iffD2]) apply auto apply (simplesubst append_cons_conv) apply (rule_tac [2] gen_prefix.append) apply (auto elim: ConsE simp add: gen_prefix_append_both) done lemma append_one_gen_prefix: "\<lbrakk>\<langle>xs,ys\<rangle>: gen_prefix(A, r); length(xs) < length(ys); refl(A, r)\<rbrakk> \<Longrightarrow> <xs @ [nth(length(xs), ys)], ys> \<in> gen_prefix(A, r)" apply (blast intro: append_one_gen_prefix_lemma) done (* Proving the equivalence with Charpentier's definition ) lemma gen_prefix_imp_nth_lemma [rule_format]: "xs \<in> list(A) \<Longrightarrow> \<forall>ys \<in> list(A). \<forall>i \<in> nat. i < length(xs) \<longrightarrow> \<langle>xs, ys\<rangle>: gen_prefix(A, r) \<longrightarrow> <nth(i, xs), nth(i, ys)>:r" apply (induct_tac "xs", simp, clarify) apply simp apply (erule natE, auto) done lemma gen_prefix_imp_nth: "\<lbrakk>\<langle>xs,ys\<rangle> \<in> gen_prefix(A,r); i < length(xs)\<rbrakk> \<Longrightarrow> <nth(i, xs), nth(i, ys)>:r" apply (cut_tac A = A in gen_prefix.dom_subset) apply (rule gen_prefix_imp_nth_lemma) apply (auto simp add: lt_nat_in_nat) done lemma nth_imp_gen_prefix [rule_format]: "xs \<in> list(A) \<Longrightarrow> \<forall>ys \<in> list(A). length(xs) \<le> length(ys) \<longrightarrow> (\<forall>i. i < length(xs) \<longrightarrow> <nth(i, xs), nth(i,ys)>:r) \<longrightarrow> \<langle>xs, ys\<rangle> \<in> gen_prefix(A, r)" apply (induct_tac "xs") apply (simp_all (no_asm_simp)) apply clarify apply (erule_tac a = ys in list.cases, simp) apply (force intro!: nat_0_le simp add: lt_nat_in_nat) done lemma gen_prefix_iff_nth: "(\<langle>xs,ys\<rangle> \<in> gen_prefix(A,r)) \<longleftrightarrow> (xs \<in> list(A) \<and> ys \<in> list(A) \<and> length(xs) \<le> length(ys) \<and> (\<forall>i. i < length(xs) \<longrightarrow> <nth(i,xs), nth(i, ys)>: r))" apply (rule iffI) apply (frule gen_prefix.dom_subset [THEN subsetD]) apply (frule gen_prefix_length_le, auto) apply (rule_tac [2] nth_imp_gen_prefix) apply (drule gen_prefix_imp_nth) apply (auto simp add: lt_nat_in_nat) done ( prefix is a partial order: *) lemma refl_prefix: "refl(list(A), prefix(A))" unfolding prefix_def apply (rule refl_gen_prefix) apply (auto simp add: refl_def) done declare refl_prefix [THEN reflD, simp] lemma trans_prefix: "trans(prefix(A))" unfolding prefix_def apply (rule trans_gen_prefix) apply (auto simp add: trans_def) done lemmas prefix_trans = trans_prefix [THEN transD] lemma trans_on_prefix: "trans[list(A)](prefix(A))" unfolding prefix_def apply (rule trans_on_gen_prefix) apply (auto simp add: trans_def) done lemmas prefix_trans_on = trans_on_prefix [THEN trans_onD] ( Monotonicity of "set" operator WRT prefix ) lemma set_of_list_prefix_mono: "\<langle>xs,ys\<rangle> \<in> prefix(A) \<Longrightarrow> set_of_list(xs) \<subseteq> set_of_list(ys)" unfolding prefix_def apply (erule gen_prefix.induct) apply (subgoal_tac [3] "xs \<in> list (A) \<and>ys \<in> list (A) ") prefer 4 apply (blast dest: gen_prefix.dom_subset [THEN subsetD]) apply (auto simp add: set_of_list_append) done (* recursion equations *) lemma Nil_prefix: "xs \<in> list(A) \<Longrightarrow> <[],xs> \<in> prefix(A)" unfolding prefix_def apply (simp (no_asm_simp) add: Nil_gen_prefix) done declare Nil_prefix [simp] lemma prefix_Nil: "<xs, []> \<in> prefix(A) \<longleftrightarrow> (xs = [])" apply (unfold prefix_def, auto) apply (frule gen_prefix.dom_subset [THEN subsetD]) apply (drule_tac psi = "<xs, []> \<in> gen_prefix (A, id (A))" in asm_rl) apply (simp add: gen_prefix_Nil) done declare prefix_Nil [iff] lemma Cons_prefix_Cons: "<Cons(x,xs), Cons(y,ys)> \<in> prefix(A) \<longleftrightarrow> (x=y \<and> \<langle>xs,ys\<rangle> \<in> prefix(A) \<and> y \<in> A)" apply (unfold prefix_def, auto) done declare Cons_prefix_Cons [iff] lemma same_prefix_prefix: "xs \<in> list(A)\<Longrightarrow> <xs@ys,xs@zs> \<in> prefix(A) \<longleftrightarrow> (\<langle>ys,zs\<rangle> \<in> prefix(A))" unfolding prefix_def apply (subgoal_tac "refl (A,id (A))") apply (simp (no_asm_simp)) apply (auto simp add: refl_def) done declare same_prefix_prefix [simp] lemma same_prefix_prefix_Nil: "xs \<in> list(A) \<Longrightarrow> <xs@ys,xs> \<in> prefix(A) \<longleftrightarrow> (<ys,[]> \<in> prefix(A))" apply (rule_tac P = "\<lambda>x. \<langle>u, x\<rangle>:v \<longleftrightarrow> w(x)" for u v w in app_right_Nil [THEN subst]) apply (rule_tac [2] same_prefix_prefix, auto) done declare same_prefix_prefix_Nil [simp] lemma prefix_appendI: "\<lbrakk>\<langle>xs,ys\<rangle> \<in> prefix(A); zs \<in> list(A)\<rbrakk> \<Longrightarrow> <xs,ys@zs> \<in> prefix(A)" unfolding prefix_def apply (erule gen_prefix.append, assumption) done declare prefix_appendI [simp] lemma prefix_Cons: "\<lbrakk>xs \<in> list(A); ys \<in> list(A); y \<in> A\<rbrakk> \<Longrightarrow> <xs,Cons(y,ys)> \<in> prefix(A) \<longleftrightarrow> (xs=[] \| (\<exists>zs. xs=Cons(y,zs) \<and> \<langle>zs,ys\<rangle> \<in> prefix(A)))" unfolding prefix_def apply (auto simp add: gen_prefix_Cons) done lemma append_one_prefix: "\<lbrakk>\<langle>xs,ys\<rangle> \<in> prefix(A); length(xs) < length(ys)\<rbrakk> \<Longrightarrow> <xs @ [nth(length(xs),ys)], ys> \<in> prefix(A)" unfolding prefix_def apply (subgoal_tac "refl (A, id (A))") apply (simp (no_asm_simp) add: append_one_gen_prefix) apply (auto simp add: refl_def) done lemma prefix_length_le: "\<langle>xs,ys\<rangle> \<in> prefix(A) \<Longrightarrow> length(xs) \<le> length(ys)" unfolding prefix_def apply (blast dest: gen_prefix_length_le) done lemma prefix_type: "prefix(A)<=list(A)list(A)" unfolding prefix_def apply (blast intro!: gen_prefix.dom_subset) done lemma strict_prefix_type: "strict_prefix(A) \<subseteq> list(A)list(A)" unfolding strict_prefix_def apply (blast intro!: prefix_type [THEN subsetD]) done lemma strict_prefix_length_lt_aux: "\<langle>xs,ys\<rangle> \<in> prefix(A) \<Longrightarrow> xs\<noteq>ys \<longrightarrow> length(xs) < length(ys)" unfolding prefix_def apply (erule gen_prefix.induct, clarify) apply (subgoal_tac [!] "ys \<in> list(A) \<and> xs \<in> list(A)") apply (auto dest: gen_prefix.dom_subset [THEN subsetD] simp add: length_type) apply (subgoal_tac "length (zs) =0") apply (drule_tac [2] not_lt_imp_le) apply (rule_tac [5] j = "length (ys) " in lt_trans2) apply auto done lemma strict_prefix_length_lt: "\<langle>xs,ys\<rangle>:strict_prefix(A) \<Longrightarrow> length(xs) < length(ys)" unfolding strict_prefix_def apply (rule strict_prefix_length_lt_aux [THEN mp]) apply (auto dest: prefix_type [THEN subsetD]) done (Equivalence to the definition used in Lex/Prefix.thy) lemma prefix_iff: "\<langle>xs,zs\<rangle> \<in> prefix(A) \<longleftrightarrow> (\<exists>ys \<in> list(A). zs = xs@ys) \<and> xs \<in> list(A)" unfolding prefix_def apply (auto simp add: gen_prefix_iff_nth lt_nat_in_nat nth_append nth_type app_type length_app) apply (subgoal_tac "drop (length (xs), zs) \<in> list (A) ") apply (rule_tac x = "drop (length (xs), zs) " in bexI) apply safe prefer 2 apply (simp add: length_type drop_type) apply (rule nth_equalityI) apply (simp_all (no_asm_simp) add: nth_append app_type drop_type length_app length_drop) apply (rule nat_diff_split [THEN iffD2], simp_all, clarify) apply (drule_tac i = "length (zs) " in leI) apply (force simp add: le_subset_iff, safe) apply (subgoal_tac "length (xs) #+ (i #- length (xs)) = i") apply (subst nth_drop) apply (simp_all (no_asm_simp) add: leI split: nat_diff_split) done lemma prefix_snoc: "\<lbrakk>xs \<in> list(A); ys \<in> list(A); y \<in> A\<rbrakk> \<Longrightarrow> <xs, ys@[y]> \<in> prefix(A) \<longleftrightarrow> (xs = ys@[y] \| \<langle>xs,ys\<rangle> \<in> prefix(A))" apply (simp (no_asm) add: prefix_iff) apply (rule iffI, clarify) apply (erule_tac xs = ysa in rev_list_elim, simp) apply (simp add: app_type app_assoc [symmetric]) apply (auto simp add: app_assoc app_type) done declare prefix_snoc [simp] lemma prefix_append_iff [rule_format]: "zs \<in> list(A) \<Longrightarrow> \<forall>xs \<in> list(A). \<forall>ys \<in> list(A). (<xs, ys@zs> \<in> prefix(A)) \<longleftrightarrow> (\<langle>xs,ys\<rangle> \<in> prefix(A) \| (\<exists>us. xs = ys@us \<and> \<langle>us,zs\<rangle> \<in> prefix(A)))" apply (erule list_append_induct, force, clarify) apply (rule iffI) apply (simp add: add: app_assoc [symmetric]) apply (erule disjE) apply (rule disjI2) apply (rule_tac x = "y @ [x]" in exI) apply (simp add: add: app_assoc [symmetric], force+) done (Although the prefix ordering is not linear, the prefixes of a list are linearly ordered.) lemma common_prefix_linear_lemma [rule_format]: "\<lbrakk>zs \<in> list(A); xs \<in> list(A); ys \<in> list(A)\<rbrakk> \<Longrightarrow> \<langle>xs, zs\<rangle> \<in> prefix(A) \<longrightarrow> \<langle>ys,zs\<rangle> \<in> prefix(A) \<longrightarrow>\<langle>xs,ys\<rangle> \<in> prefix(A) \| \<langle>ys,xs\<rangle> \<in> prefix(A)" apply (erule list_append_induct, auto) done lemma common_prefix_linear: "\<lbrakk>\<langle>xs, zs\<rangle> \<in> prefix(A); \<langle>ys,zs\<rangle> \<in> prefix(A)\<rbrakk> \<Longrightarrow> \<langle>xs,ys\<rangle> \<in> prefix(A) \| \<langle>ys,xs\<rangle> \<in> prefix(A)" apply (cut_tac prefix_type) apply (blast del: disjCI intro: common_prefix_linear_lemma) done ( pfixLe, pfixGe \<in> properties inherited from the translations ) (* pfixLe *) lemma refl_Le: "refl(nat,Le)" apply (unfold refl_def, auto) done declare refl_Le [simp] lemma antisym_Le: "antisym(Le)" unfolding antisym_def apply (auto intro: le_anti_sym) done declare antisym_Le [simp] lemma trans_on_Le: "trans[nat](Le)" apply (unfold trans_on_def, auto) apply (blast intro: le_trans) done declare trans_on_Le [simp] lemma trans_Le: "trans(Le)" apply (unfold trans_def, auto) apply (blast intro: le_trans) done declare trans_Le [simp] lemma part_order_Le: "part_order(nat,Le)" by (unfold part_order_def, auto) declare part_order_Le [simp] lemma pfixLe_refl: "x \<in> list(nat) \<Longrightarrow> x pfixLe x" by (blast intro: refl_gen_prefix [THEN reflD] refl_Le) declare pfixLe_refl [simp] lemma pfixLe_trans: "\<lbrakk>x pfixLe y; y pfixLe z\<rbrakk> \<Longrightarrow> x pfixLe z" by (blast intro: trans_gen_prefix [THEN transD] trans_Le) lemma pfixLe_antisym: "\<lbrakk>x pfixLe y; y pfixLe x\<rbrakk> \<Longrightarrow> x = y" by (blast intro: antisym_gen_prefix [THEN antisymE] antisym_Le) lemma prefix_imp_pfixLe: "\<langle>xs,ys\<rangle>:prefix(nat)\<Longrightarrow> xs pfixLe ys" unfolding prefix_def apply (rule gen_prefix_mono [THEN subsetD], auto) done lemma refl_Ge: "refl(nat, Ge)" by (unfold refl_def Ge_def, auto) declare refl_Ge [iff] lemma antisym_Ge: "antisym(Ge)" unfolding antisym_def Ge_def apply (auto intro: le_anti_sym) done declare antisym_Ge [iff] lemma trans_Ge: "trans(Ge)" unfolding trans_def Ge_def apply (auto intro: le_trans) done declare trans_Ge [iff] lemma pfixGe_refl: "x \<in> list(nat) \<Longrightarrow> x pfixGe x" by (blast intro: refl_gen_prefix [THEN reflD]) declare pfixGe_refl [simp] lemma pfixGe_trans: "\<lbrakk>x pfixGe y; y pfixGe z\<rbrakk> \<Longrightarrow> x pfixGe z" by (blast intro: trans_gen_prefix [THEN transD]) lemma pfixGe_antisym: "\<lbrakk>x pfixGe y; y pfixGe x\<rbrakk> \<Longrightarrow> x = y" by (blast intro: antisym_gen_prefix [THEN antisymE]) lemma prefix_imp_pfixGe: "\<langle>xs,ys\<rangle>:prefix(nat) \<Longrightarrow> xs pfixGe ys" unfolding prefix_def Ge_def apply (rule gen_prefix_mono [THEN subsetD], auto) done ( Added by Sidi \<in> prefix and take *) lemma prefix_imp_take: "\<langle>xs, ys\<rangle> \<in> prefix(A) \<Longrightarrow> xs = take(length(xs), ys)" unfolding prefix_def apply (erule gen_prefix.induct) apply (subgoal_tac [3] "length (xs) :nat") apply (auto dest: gen_prefix.dom_subset [THEN subsetD] simp add: length_type) apply (frule gen_prefix.dom_subset [THEN subsetD]) apply (frule gen_prefix_length_le) apply (auto simp add: take_append) apply (subgoal_tac "length (xs) #- length (ys) =0") apply (simp_all (no_asm_simp) add: diff_is_0_iff) done lemma prefix_length_equal: "\<lbrakk>\<langle>xs,ys\<rangle> \<in> prefix(A); length(xs)=length(ys)\<rbrakk> \<Longrightarrow> xs = ys" apply (cut_tac A = A in prefix_type) apply (drule subsetD, auto) apply (drule prefix_imp_take) apply (erule trans, simp) done lemma prefix_length_le_equal: "\<lbrakk>\<langle>xs,ys\<rangle> \<in> prefix(A); length(ys) \<le> length(xs)\<rbrakk> \<Longrightarrow> xs = ys" by (blast intro: prefix_length_equal le_anti_sym prefix_length_le) lemma take_prefix [rule_format]: "xs \<in> list(A) \<Longrightarrow> \<forall>n \<in> nat. <take(n, xs), xs> \<in> prefix(A)" unfolding prefix_def apply (erule list.induct, simp, clarify) apply (erule natE, auto) done lemma prefix_take_iff: "\<langle>xs,ys\<rangle> \<in> prefix(A) \<longleftrightarrow> (xs=take(length(xs), ys) \<and> xs \<in> list(A) \<and> ys \<in> list(A))" apply (rule iffI) apply (frule prefix_type [THEN subsetD]) apply (blast intro: prefix_imp_take, clarify) apply (erule ssubst) apply (blast intro: take_prefix length_type) done lemma prefix_imp_nth: "\<lbrakk>\<langle>xs,ys\<rangle> \<in> prefix(A); i < length(xs)\<rbrakk> \<Longrightarrow> nth(i,xs) = nth(i,ys)" by (auto dest!: gen_prefix_imp_nth simp add: prefix_def) lemma nth_imp_prefix: "\<lbrakk>xs \<in> list(A); ys \<in> list(A); length(xs) \<le> length(ys); \<And>i. i < length(xs) \<Longrightarrow> nth(i, xs) = nth(i,ys)\<rbrakk> \<Longrightarrow> \<langle>xs,ys\<rangle> \<in> prefix(A)" apply (auto simp add: prefix_def nth_imp_gen_prefix) apply (auto intro!: nth_imp_gen_prefix simp add: prefix_def) apply (blast intro: nth_type lt_trans2) done lemma length_le_prefix_imp_prefix: "\<lbrakk>length(xs) \<le> length(ys); \<langle>xs,zs\<rangle> \<in> prefix(A); \<langle>ys,zs\<rangle> \<in> prefix(A)\<rbrakk> \<Longrightarrow> \<langle>xs,ys\<rangle> \<in> prefix(A)" apply (cut_tac A = A in prefix_type) apply (rule nth_imp_prefix, blast, blast) apply assumption apply (rule_tac b = "nth (i,zs)" in trans) apply (blast intro: prefix_imp_nth) apply (blast intro: sym prefix_imp_nth prefix_length_le lt_trans2) done end
[STATEMENT] lemma ide_exp: assumes "ide b" and "ide c" shows "ide (exp b c)" [PROOF STATE] proof (prove) goal (1 subgoal): 1. ide (exp b c) [PROOF STEP] using assms exp_def hf_to_ide_mapsto ide_to_hf_mapsto [PROOF STATE] proof (prove) using this: ide b ide c exp ?b ?c = hf_to_ide (hexp (ide_to_hf ?b) (ide_to_hf ?c)) hf_to_ide \<in> UNIV \<rightarrow> Collect ide ide_to_hf \<in> Collect ide \<rightarrow> UNIV goal (1 subgoal): 1. ide (exp b c) [PROOF STEP] by auto
[STATEMENT] lemma c_smj_symm: "symmetric (c_smj cf f)" [PROOF STATE] proof (prove) goal (1 subgoal): 1. symmetric (c_smj cf f) [PROOF STEP] by (simp add: symmetric_def)
[GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ ⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε ⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] have nneg : ∀ x, 0 ≤ ‖x - x₀‖⁻¹ := fun x => inv_nonneg.mpr (norm_nonneg _) [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ ⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] set b : α → ℝ := fun a => \|bound a\| [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| ⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] have b_int : Integrable b μ := bound_integrable.norm [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b ⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] have b_nonneg : ∀ a, 0 ≤ b a := fun a => abs_nonneg _ [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a ⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] replace h_lipsch : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ [GOAL] case h_lipsch α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a ⊢ ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] exact h_lipsch.mono fun a ha x hx => (ha x hx).trans <\| mul_le_mul_of_nonneg_right (le_abs_self _) (norm_nonneg _) [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] have hF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ := fun x x_in ↦ by have : ∀ᵐ a ∂μ, ‖F x₀ a - F x a‖ ≤ ε * b a := by simp only [norm_sub_rev (F x₀ _)] refine' h_lipsch.mono fun a ha => (ha x x_in).trans _ rw [mul_comm ε] rw [mem_ball, dist_eq_norm] at x_in exact mul_le_mul_of_nonneg_left x_in.le (b_nonneg _) exact integrable_of_norm_sub_le (hF_meas x x_in) hF_int (bound_integrable.norm.const_mul ε) this [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ x : H x_in : x ∈ ball x₀ ε ⊢ Integrable (F x) [PROOFSTEP] have : ∀ᵐ a ∂μ, ‖F x₀ a - F x a‖ ≤ ε * b a := by simp only [norm_sub_rev (F x₀ _)] refine' h_lipsch.mono fun a ha => (ha x x_in).trans _ rw [mul_comm ε] rw [mem_ball, dist_eq_norm] at x_in exact mul_le_mul_of_nonneg_left x_in.le (b_nonneg _) [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ x : H x_in : x ∈ ball x₀ ε ⊢ ∀ᵐ (a : α) ∂μ, ‖F x₀ a - F x a‖ ≤ ε * b a [PROOFSTEP] simp only [norm_sub_rev (F x₀ _)] [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ x : H x_in : x ∈ ball x₀ ε ⊢ ∀ᵐ (a : α) ∂μ, ‖F x a - F x₀ a‖ ≤ ε * \|bound a\| [PROOFSTEP] refine' h_lipsch.mono fun a ha => (ha x x_in).trans _ [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ x : H x_in : x ∈ ball x₀ ε a : α ha : ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ b a * ‖x - x₀‖ ≤ ε * \|bound a\| [PROOFSTEP] rw [mul_comm ε] [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ x : H x_in : x ∈ ball x₀ ε a : α ha : ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ b a * ‖x - x₀‖ ≤ \|bound a\| * ε [PROOFSTEP] rw [mem_ball, dist_eq_norm] at x_in [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ x : H x_in : ‖x - x₀‖ < ε a : α ha : ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ b a * ‖x - x₀‖ ≤ \|bound a\| * ε [PROOFSTEP] exact mul_le_mul_of_nonneg_left x_in.le (b_nonneg _) [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ x : H x_in : x ∈ ball x₀ ε this : ∀ᵐ (a : α) ∂μ, ‖F x₀ a - F x a‖ ≤ ε * b a ⊢ Integrable (F x) [PROOFSTEP] exact integrable_of_norm_sub_le (hF_meas x x_in) hF_int (bound_integrable.norm.const_mul ε) this [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) ⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] have hF'_int : Integrable F' μ := have : ∀ᵐ a ∂μ, ‖F' a‖ ≤ b a := by apply (h_diff.and h_lipsch).mono rintro a ⟨ha_diff, ha_lip⟩ refine' ha_diff.le_of_lip' (b_nonneg a) (mem_of_superset (ball_mem_nhds _ ε_pos) <\| ha_lip) b_int.mono' hF'_meas this [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) ⊢ ∀ᵐ (a : α) ∂μ, ‖F' a‖ ≤ b a [PROOFSTEP] apply (h_diff.and h_lipsch).mono [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) ⊢ ∀ (x : α), (HasFDerivAt (fun x_1 => F x_1 x) (F' x) x₀ ∧ ∀ (x_1 : H), x_1 ∈ ball x₀ ε → ‖F x_1 x - F x₀ x‖ ≤ b x * ‖x_1 - x₀‖) → ‖F' x‖ ≤ b x [PROOFSTEP] rintro a ⟨ha_diff, ha_lip⟩ [GOAL] case intro α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) a : α ha_diff : HasFDerivAt (fun x => F x a) (F' a) x₀ ha_lip : ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ ‖F' a‖ ≤ b a [PROOFSTEP] refine' ha_diff.le_of_lip' (b_nonneg a) (mem_of_superset (ball_mem_nhds _ ε_pos) <\| ha_lip) [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' ⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] refine' ⟨hF'_int, _⟩ [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' ⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] have h_ball : ball x₀ ε ∈ 𝓝 x₀ := ball_mem_nhds x₀ ε_pos [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ ⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] have : ∀ᶠ x in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖((∫ a, F x a ∂μ) - ∫ a, F x₀ a ∂μ) - (∫ a, F' a ∂μ) (x - x₀)‖ = ‖∫ a, ‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀)) ∂μ‖ := by apply mem_of_superset (ball_mem_nhds _ ε_pos) intro x x_in; simp only rw [Set.mem_setOf_eq, ← norm_smul_of_nonneg (nneg _), integral_smul, integral_sub, integral_sub, ← ContinuousLinearMap.integral_apply hF'_int] exacts [hF_int' x x_in, hF_int, (hF_int' x x_in).sub hF_int, hF'_int.apply_continuousLinearMap _] [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ ⊢ ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ [PROOFSTEP] apply mem_of_superset (ball_mem_nhds _ ε_pos) [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ ⊢ ball x₀ ε ⊆ {x \| (fun x => ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖) x} [PROOFSTEP] intro x x_in [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ x : H x_in : x ∈ ball x₀ ε ⊢ x ∈ {x \| (fun x => ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖) x} [PROOFSTEP] simp only [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ x : H x_in : x ∈ ball x₀ ε ⊢ x ∈ {x \| ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖} [PROOFSTEP] rw [Set.mem_setOf_eq, ← norm_smul_of_nonneg (nneg _), integral_smul, integral_sub, integral_sub, ← ContinuousLinearMap.integral_apply hF'_int] [GOAL] case hf α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ x : H x_in : x ∈ ball x₀ ε ⊢ Integrable fun a => F x a case hg α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ x : H x_in : x ∈ ball x₀ ε ⊢ Integrable fun a => F x₀ a case hf α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ x : H x_in : x ∈ ball x₀ ε ⊢ Integrable fun a => F x a - F x₀ a case hg α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ x : H x_in : x ∈ ball x₀ ε ⊢ Integrable fun a => ↑(F' a) (x - x₀) [PROOFSTEP] exacts [hF_int' x x_in, hF_int, (hF_int' x x_in).sub hF_int, hF'_int.apply_continuousLinearMap _] [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ ⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] rw [hasFDerivAt_iff_tendsto, tendsto_congr' this, ← tendsto_zero_iff_norm_tendsto_zero, ← show (∫ a : α, ‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - (F' a) (x₀ - x₀)) ∂μ) = 0 by simp] [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ ⊢ ∫ (a : α), ‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - ↑(F' a) (x₀ - x₀)) ∂μ = 0 [PROOFSTEP] simp [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ ⊢ Tendsto (fun x => ∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ) (𝓝 x₀) (𝓝 (∫ (a : α), ‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - ↑(F' a) (x₀ - x₀)) ∂μ)) [PROOFSTEP] apply tendsto_integral_filter_of_dominated_convergence [GOAL] case hF_meas α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ ⊢ ∀ᶠ (n : H) in 𝓝 x₀, AEStronglyMeasurable (fun a => ‖n - x₀‖⁻¹ • (F n a - F x₀ a - ↑(F' a) (n - x₀))) μ [PROOFSTEP] filter_upwards [h_ball] with _ x_in [GOAL] case h α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ a✝ : H x_in : a✝ ∈ ball x₀ ε ⊢ AEStronglyMeasurable (fun a => ‖a✝ - x₀‖⁻¹ • (F a✝ a - F x₀ a - ↑(F' a) (a✝ - x₀))) μ [PROOFSTEP] apply AEStronglyMeasurable.const_smul [GOAL] case h.hf α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ a✝ : H x_in : a✝ ∈ ball x₀ ε ⊢ AEStronglyMeasurable (fun a => F a✝ a - F x₀ a - ↑(F' a) (a✝ - x₀)) μ [PROOFSTEP] exact ((hF_meas _ x_in).sub (hF_meas _ x₀_in)).sub (hF'_meas.apply_continuousLinearMap _) [GOAL] case h_bound α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ ⊢ ∀ᶠ (n : H) in 𝓝 x₀, ∀ᵐ (a : α) ∂μ, ‖‖n - x₀‖⁻¹ • (F n a - F x₀ a - ↑(F' a) (n - x₀))‖ ≤ ?bound a [PROOFSTEP] refine mem_of_superset h_ball fun x hx ↦ ?_ [GOAL] case h_bound α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ x : H hx : x ∈ ball x₀ ε ⊢ x ∈ {x \| (fun n => ∀ᵐ (a : α) ∂μ, ‖‖n - x₀‖⁻¹ • (F n a - F x₀ a - ↑(F' a) (n - x₀))‖ ≤ ?bound a) x} [PROOFSTEP] apply (h_diff.and h_lipsch).mono [GOAL] case h_bound α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ x : H hx : x ∈ ball x₀ ε ⊢ ∀ (x_1 : α), (HasFDerivAt (fun x => F x x_1) (F' x_1) x₀ ∧ ∀ (x : H), x ∈ ball x₀ ε → ‖F x x_1 - F x₀ x_1‖ ≤ b x_1 * ‖x - x₀‖) → ‖‖x - x₀‖⁻¹ • (F x x_1 - F x₀ x_1 - ↑(F' x_1) (x - x₀))‖ ≤ ?bound x_1 case bound α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ ⊢ α → ℝ [PROOFSTEP] rintro a ⟨-, ha_bound⟩ [GOAL] case h_bound.intro α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ x : H hx : x ∈ ball x₀ ε a : α ha_bound : ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))‖ ≤ ?bound a case bound α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ ⊢ α → ℝ [PROOFSTEP] show ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀))‖ ≤ b a + ‖F' a‖ [GOAL] case h_bound.intro α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ x : H hx : x ∈ ball x₀ ε a : α ha_bound : ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))‖ ≤ b a + ‖F' a‖ [PROOFSTEP] replace ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ := ha_bound x hx [GOAL] case h_bound.intro α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ x : H hx : x ∈ ball x₀ ε a : α ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))‖ ≤ b a + ‖F' a‖ [PROOFSTEP] calc ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀))‖ = ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a) - ‖x - x₀‖⁻¹ • F' a (x - x₀)‖ := by rw [smul_sub] _ ≤ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a)‖ + ‖‖x - x₀‖⁻¹ • F' a (x - x₀)‖ := (norm_sub_le _ _) _ = ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a‖ + ‖x - x₀‖⁻¹ * ‖F' a (x - x₀)‖ := by rw [norm_smul_of_nonneg, norm_smul_of_nonneg] <;> exact nneg _ _ ≤ ‖x - x₀‖⁻¹ * (b a * ‖x - x₀‖) + ‖x - x₀‖⁻¹ * (‖F' a‖ * ‖x - x₀‖) := by gcongr; exact (F' a).le_op_norm _ _ ≤ b a + ‖F' a‖ := ?_ [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ x : H hx : x ∈ ball x₀ ε a : α ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))‖ = ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a) - ‖x - x₀‖⁻¹ • ↑(F' a) (x - x₀)‖ [PROOFSTEP] rw [smul_sub] [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ x : H hx : x ∈ ball x₀ ε a : α ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a)‖ + ‖‖x - x₀‖⁻¹ • ↑(F' a) (x - x₀)‖ = ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a‖ + ‖x - x₀‖⁻¹ * ‖↑(F' a) (x - x₀)‖ [PROOFSTEP] rw [norm_smul_of_nonneg, norm_smul_of_nonneg] [GOAL] case ht α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ x : H hx : x ∈ ball x₀ ε a : α ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ 0 ≤ ‖x - x₀‖⁻¹ [PROOFSTEP] exact nneg _ [GOAL] case ht α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ x : H hx : x ∈ ball x₀ ε a : α ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ 0 ≤ ‖x - x₀‖⁻¹ [PROOFSTEP] exact nneg _ [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ x : H hx : x ∈ ball x₀ ε a : α ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a‖ + ‖x - x₀‖⁻¹ * ‖↑(F' a) (x - x₀)‖ ≤ ‖x - x₀‖⁻¹ * (b a * ‖x - x₀‖) + ‖x - x₀‖⁻¹ * (‖F' a‖ * ‖x - x₀‖) [PROOFSTEP] gcongr [GOAL] case h₂.h α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ x : H hx : x ∈ ball x₀ ε a : α ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ ‖↑(F' a) (x - x₀)‖ ≤ ‖F' a‖ * ‖x - x₀‖ [PROOFSTEP] exact (F' a).le_op_norm _ [GOAL] case h_bound.intro α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ x : H hx : x ∈ ball x₀ ε a : α ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ ‖x - x₀‖⁻¹ * (b a * ‖x - x₀‖) + ‖x - x₀‖⁻¹ * (‖F' a‖ * ‖x - x₀‖) ≤ b a + ‖F' a‖ [PROOFSTEP] simp only [← div_eq_inv_mul] [GOAL] case h_bound.intro α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ x : H hx : x ∈ ball x₀ ε a : α ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ \|bound a\| * ‖x - x₀‖ / ‖x - x₀‖ + ‖F' a‖ * ‖x - x₀‖ / ‖x - x₀‖ ≤ \|bound a\| + ‖F' a‖ [PROOFSTEP] apply_rules [add_le_add, div_le_of_nonneg_of_le_mul] [GOAL] case h_bound.intro.h₁.hb α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ x : H hx : x ∈ ball x₀ ε a : α ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ 0 ≤ ‖x - x₀‖ [PROOFSTEP] first \| rfl \| positivity [GOAL] case h_bound.intro.h₁.hb α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ x : H hx : x ∈ ball x₀ ε a : α ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ 0 ≤ ‖x - x₀‖ [PROOFSTEP] rfl [GOAL] case h_bound.intro.h₁.hb α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ x : H hx : x ∈ ball x₀ ε a : α ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ 0 ≤ ‖x - x₀‖ [PROOFSTEP] positivity [GOAL] case h_bound.intro.h₁.h α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ x : H hx : x ∈ ball x₀ ε a : α ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ \|bound a\| * ‖x - x₀‖ ≤ \|bound a\| * ‖x - x₀‖ [PROOFSTEP] first \| rfl \| positivity [GOAL] case h_bound.intro.h₁.h α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ x : H hx : x ∈ ball x₀ ε a : α ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ \|bound a\| * ‖x - x₀‖ ≤ \|bound a\| * ‖x - x₀‖ [PROOFSTEP] rfl [GOAL] case h_bound.intro.h₂.hb α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ x : H hx : x ∈ ball x₀ ε a : α ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ 0 ≤ ‖x - x₀‖ [PROOFSTEP] first \| rfl \| positivity [GOAL] case h_bound.intro.h₂.hb α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ x : H hx : x ∈ ball x₀ ε a : α ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ 0 ≤ ‖x - x₀‖ [PROOFSTEP] rfl [GOAL] case h_bound.intro.h₂.hb α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ x : H hx : x ∈ ball x₀ ε a : α ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ 0 ≤ ‖x - x₀‖ [PROOFSTEP] positivity [GOAL] case h_bound.intro.h₂.hc α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ x : H hx : x ∈ ball x₀ ε a : α ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ 0 ≤ ‖F' a‖ [PROOFSTEP] first \| rfl \| positivity [GOAL] case h_bound.intro.h₂.hc α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ x : H hx : x ∈ ball x₀ ε a : α ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ 0 ≤ ‖F' a‖ [PROOFSTEP] rfl [GOAL] case h_bound.intro.h₂.hc α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ x : H hx : x ∈ ball x₀ ε a : α ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ 0 ≤ ‖F' a‖ [PROOFSTEP] positivity [GOAL] case h_bound.intro.h₂.h α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ x : H hx : x ∈ ball x₀ ε a : α ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ ‖F' a‖ * ‖x - x₀‖ ≤ ‖F' a‖ * ‖x - x₀‖ [PROOFSTEP] first \| rfl \| positivity [GOAL] case h_bound.intro.h₂.h α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ x : H hx : x ∈ ball x₀ ε a : α ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ ⊢ ‖F' a‖ * ‖x - x₀‖ ≤ ‖F' a‖ * ‖x - x₀‖ [PROOFSTEP] rfl [GOAL] case bound_integrable α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ ⊢ Integrable fun a => b a + ‖F' a‖ [PROOFSTEP] exact b_int.add hF'_int.norm [GOAL] case h_lim α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ ⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun n => ‖n - x₀‖⁻¹ • (F n a - F x₀ a - ↑(F' a) (n - x₀))) (𝓝 x₀) (𝓝 (‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - ↑(F' a) (x₀ - x₀)))) [PROOFSTEP] apply h_diff.mono [GOAL] case h_lim α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ ⊢ ∀ (x : α), HasFDerivAt (fun x_1 => F x_1 x) (F' x) x₀ → Tendsto (fun n => ‖n - x₀‖⁻¹ • (F n x - F x₀ x - ↑(F' x) (n - x₀))) (𝓝 x₀) (𝓝 (‖x₀ - x₀‖⁻¹ • (F x₀ x - F x₀ x - ↑(F' x) (x₀ - x₀)))) [PROOFSTEP] intro a ha [GOAL] case h_lim α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ a : α ha : HasFDerivAt (fun x => F x a) (F' a) x₀ ⊢ Tendsto (fun n => ‖n - x₀‖⁻¹ • (F n a - F x₀ a - ↑(F' a) (n - x₀))) (𝓝 x₀) (𝓝 (‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - ↑(F' a) (x₀ - x₀)))) [PROOFSTEP] suffices Tendsto (fun x => ‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀))) (𝓝 x₀) (𝓝 0) by simpa [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this✝ : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ a : α ha : HasFDerivAt (fun x => F x a) (F' a) x₀ this : Tendsto (fun x => ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))) (𝓝 x₀) (𝓝 0) ⊢ Tendsto (fun n => ‖n - x₀‖⁻¹ • (F n a - F x₀ a - ↑(F' a) (n - x₀))) (𝓝 x₀) (𝓝 (‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - ↑(F' a) (x₀ - x₀)))) [PROOFSTEP] simpa [GOAL] case h_lim α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ a : α ha : HasFDerivAt (fun x => F x a) (F' a) x₀ ⊢ Tendsto (fun x => ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))) (𝓝 x₀) (𝓝 0) [PROOFSTEP] rw [tendsto_zero_iff_norm_tendsto_zero] [GOAL] case h_lim α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ a : α ha : HasFDerivAt (fun x => F x a) (F' a) x₀ ⊢ Tendsto (fun e => ‖‖e - x₀‖⁻¹ • (F e a - F x₀ a - ↑(F' a) (e - x₀))‖) (𝓝 x₀) (𝓝 0) [PROOFSTEP] have : (fun x => ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a - F' a (x - x₀)‖) = fun x => ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀))‖ := by ext x rw [norm_smul_of_nonneg (nneg _)] [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ a : α ha : HasFDerivAt (fun x => F x a) (F' a) x₀ ⊢ (fun x => ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a - ↑(F' a) (x - x₀)‖) = fun x => ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))‖ [PROOFSTEP] ext x [GOAL] case h α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ a : α ha : HasFDerivAt (fun x => F x a) (F' a) x₀ x : H ⊢ ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a - ↑(F' a) (x - x₀)‖ = ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))‖ [PROOFSTEP] rw [norm_smul_of_nonneg (nneg _)] [GOAL] case h_lim α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ x₀_in : x₀ ∈ ball x₀ ε nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹ b : α → ℝ := fun a => \|bound a\| b_int : Integrable b b_nonneg : ∀ (a : α), 0 ≤ b a h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x) hF'_int : Integrable F' h_ball : ball x₀ ε ∈ 𝓝 x₀ this✝ : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖ a : α ha : HasFDerivAt (fun x => F x a) (F' a) x₀ this : (fun x => ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a - ↑(F' a) (x - x₀)‖) = fun x => ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))‖ ⊢ Tendsto (fun e => ‖‖e - x₀‖⁻¹ • (F e a - F x₀ a - ↑(F' a) (e - x₀))‖) (𝓝 x₀) (𝓝 0) [PROOFSTEP] rwa [hasFDerivAt_iff_tendsto, this] at ha [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ h_lip : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε) bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ ⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] obtain ⟨δ, δ_pos, hδ⟩ : ∃ δ > 0, ∀ x ∈ ball x₀ δ, AEStronglyMeasurable (F x) μ ∧ x ∈ ball x₀ ε [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ h_lip : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε) bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ ⊢ ∃ δ, δ > 0 ∧ ∀ (x : H), x ∈ ball x₀ δ → AEStronglyMeasurable (F x) μ ∧ x ∈ ball x₀ ε case intro.intro α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ h_lip : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε) bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ δ : ℝ δ_pos : δ > 0 hδ : ∀ (x : H), x ∈ ball x₀ δ → AEStronglyMeasurable (F x) μ ∧ x ∈ ball x₀ ε ⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] exact eventually_nhds_iff_ball.mp (hF_meas.and (ball_mem_nhds x₀ ε_pos)) [GOAL] case intro.intro α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ h_lip : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε) bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ δ : ℝ δ_pos : δ > 0 hδ : ∀ (x : H), x ∈ ball x₀ δ → AEStronglyMeasurable (F x) μ ∧ x ∈ ball x₀ ε ⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] choose hδ_meas hδε using hδ [GOAL] case intro.intro α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ h_lip : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε) bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ δ : ℝ δ_pos : δ > 0 hδ_meas : ∀ (x : H), x ∈ ball x₀ δ → AEStronglyMeasurable (F x) μ hδε : ∀ (x : H), x ∈ ball x₀ δ → x ∈ ball x₀ ε ⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] replace h_lip : ∀ᵐ a : α ∂μ, ∀ x ∈ ball x₀ δ, ‖F x a - F x₀ a‖ ≤ \|bound a\| * ‖x - x₀‖ [GOAL] case h_lip α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ h_lip : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε) bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ δ : ℝ δ_pos : δ > 0 hδ_meas : ∀ (x : H), x ∈ ball x₀ δ → AEStronglyMeasurable (F x) μ hδε : ∀ (x : H), x ∈ ball x₀ δ → x ∈ ball x₀ ε ⊢ ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ δ → ‖F x a - F x₀ a‖ ≤ \|bound a\| * ‖x - x₀‖ case intro.intro α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ δ : ℝ δ_pos : δ > 0 hδ_meas : ∀ (x : H), x ∈ ball x₀ δ → AEStronglyMeasurable (F x) μ hδε : ∀ (x : H), x ∈ ball x₀ δ → x ∈ ball x₀ ε h_lip : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ δ → ‖F x a - F x₀ a‖ ≤ \|bound a\| * ‖x - x₀‖ ⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] exact h_lip.mono fun a lip x hx => lip.norm_sub_le (hδε x hx) (mem_ball_self ε_pos) [GOAL] case intro.intro α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ δ : ℝ δ_pos : δ > 0 hδ_meas : ∀ (x : H), x ∈ ball x₀ δ → AEStronglyMeasurable (F x) μ hδε : ∀ (x : H), x ∈ ball x₀ δ → x ∈ ball x₀ ε h_lip : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ δ → ‖F x a - F x₀ a‖ ≤ \|bound a\| * ‖x - x₀‖ ⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] replace bound_integrable := bound_integrable.norm [GOAL] case intro.intro α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ δ : ℝ δ_pos : δ > 0 hδ_meas : ∀ (x : H), x ∈ ball x₀ δ → AEStronglyMeasurable (F x) μ hδε : ∀ (x : H), x ∈ ball x₀ δ → x ∈ ball x₀ ε h_lip : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ δ → ‖F x a - F x₀ a‖ ≤ \|bound a\| * ‖x - x₀‖ bound_integrable : Integrable fun a => ‖bound a‖ ⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] apply hasFDerivAt_integral_of_dominated_loc_of_lip' δ_pos [GOAL] case intro.intro.hF_meas α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ δ : ℝ δ_pos : δ > 0 hδ_meas : ∀ (x : H), x ∈ ball x₀ δ → AEStronglyMeasurable (F x) μ hδε : ∀ (x : H), x ∈ ball x₀ δ → x ∈ ball x₀ ε h_lip : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ δ → ‖F x a - F x₀ a‖ ≤ \|bound a\| * ‖x - x₀‖ bound_integrable : Integrable fun a => ‖bound a‖ ⊢ ∀ (x : H), x ∈ ball x₀ δ → AEStronglyMeasurable (fun a => F x a) μ [PROOFSTEP] assumption [GOAL] case intro.intro.hF_int α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ δ : ℝ δ_pos : δ > 0 hδ_meas : ∀ (x : H), x ∈ ball x₀ δ → AEStronglyMeasurable (F x) μ hδε : ∀ (x : H), x ∈ ball x₀ δ → x ∈ ball x₀ ε h_lip : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ δ → ‖F x a - F x₀ a‖ ≤ \|bound a\| * ‖x - x₀‖ bound_integrable : Integrable fun a => ‖bound a‖ ⊢ Integrable fun a => F x₀ a [PROOFSTEP] assumption [GOAL] case intro.intro.hF'_meas α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ δ : ℝ δ_pos : δ > 0 hδ_meas : ∀ (x : H), x ∈ ball x₀ δ → AEStronglyMeasurable (F x) μ hδε : ∀ (x : H), x ∈ ball x₀ δ → x ∈ ball x₀ ε h_lip : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ δ → ‖F x a - F x₀ a‖ ≤ \|bound a\| * ‖x - x₀‖ bound_integrable : Integrable fun a => ‖bound a‖ ⊢ AEStronglyMeasurable F' μ [PROOFSTEP] assumption [GOAL] case intro.intro.h_lipsch α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ δ : ℝ δ_pos : δ > 0 hδ_meas : ∀ (x : H), x ∈ ball x₀ δ → AEStronglyMeasurable (F x) μ hδε : ∀ (x : H), x ∈ ball x₀ δ → x ∈ ball x₀ ε h_lip : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ δ → ‖F x a - F x₀ a‖ ≤ \|bound a\| * ‖x - x₀‖ bound_integrable : Integrable fun a => ‖bound a‖ ⊢ ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ δ → ‖F x a - F x₀ a‖ ≤ ?m.181944 a * ‖x - x₀‖ [PROOFSTEP] assumption [GOAL] case intro.intro.bound_integrable α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ δ : ℝ δ_pos : δ > 0 hδ_meas : ∀ (x : H), x ∈ ball x₀ δ → AEStronglyMeasurable (F x) μ hδε : ∀ (x : H), x ∈ ball x₀ δ → x ∈ ball x₀ ε h_lip : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ δ → ‖F x a - F x₀ a‖ ≤ \|bound a\| * ‖x - x₀‖ bound_integrable : Integrable fun a => ‖bound a‖ ⊢ Integrable fun a => \|bound a\| [PROOFSTEP] assumption [GOAL] case intro.intro.h_diff α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ δ : ℝ δ_pos : δ > 0 hδ_meas : ∀ (x : H), x ∈ ball x₀ δ → AEStronglyMeasurable (F x) μ hδε : ∀ (x : H), x ∈ ball x₀ δ → x ∈ ball x₀ ε h_lip : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ δ → ‖F x a - F x₀ a‖ ≤ \|bound a\| * ‖x - x₀‖ bound_integrable : Integrable fun a => ‖bound a‖ ⊢ ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀ [PROOFSTEP] assumption [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : H → α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable (F' x₀) μ h_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → HasFDerivAt (fun x => F x a) (F' x a) x ⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' x₀ a ∂μ) x₀ [PROOFSTEP] letI : NormedSpace ℝ H := NormedSpace.restrictScalars ℝ 𝕜 H [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : H → α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable (F' x₀) μ h_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → HasFDerivAt (fun x => F x a) (F' x a) x this : NormedSpace ℝ H := NormedSpace.restrictScalars ℝ 𝕜 H ⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' x₀ a ∂μ) x₀ [PROOFSTEP] have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : H → α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable (F' x₀) μ h_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → HasFDerivAt (fun x => F x a) (F' x a) x this : NormedSpace ℝ H := NormedSpace.restrictScalars ℝ 𝕜 H x₀_in : x₀ ∈ ball x₀ ε ⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' x₀ a ∂μ) x₀ [PROOFSTEP] have diff_x₀ : ∀ᵐ a ∂μ, HasFDerivAt (fun x => F x a) (F' x₀ a) x₀ := h_diff.mono fun a ha => ha x₀ x₀_in [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : H → α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable (F' x₀) μ h_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → HasFDerivAt (fun x => F x a) (F' x a) x this : NormedSpace ℝ H := NormedSpace.restrictScalars ℝ 𝕜 H x₀_in : x₀ ∈ ball x₀ ε diff_x₀ : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' x₀ a) x₀ ⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' x₀ a ∂μ) x₀ [PROOFSTEP] have : ∀ᵐ a ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε) := by apply (h_diff.and h_bound).mono rintro a ⟨ha_deriv, ha_bound⟩ refine' (convex_ball _ _).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x x_in => (ha_deriv x x_in).hasFDerivWithinAt) fun x x_in => _ rw [← NNReal.coe_le_coe, coe_nnnorm, Real.coe_nnabs] exact (ha_bound x x_in).trans (le_abs_self _) [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : H → α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable (F' x₀) μ h_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → HasFDerivAt (fun x => F x a) (F' x a) x this : NormedSpace ℝ H := NormedSpace.restrictScalars ℝ 𝕜 H x₀_in : x₀ ∈ ball x₀ ε diff_x₀ : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' x₀ a) x₀ ⊢ ∀ᵐ (a : α) ∂μ, LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε) [PROOFSTEP] apply (h_diff.and h_bound).mono [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : H → α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable (F' x₀) μ h_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → HasFDerivAt (fun x => F x a) (F' x a) x this : NormedSpace ℝ H := NormedSpace.restrictScalars ℝ 𝕜 H x₀_in : x₀ ∈ ball x₀ ε diff_x₀ : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' x₀ a) x₀ ⊢ ∀ (x : α), ((∀ (x_1 : H), x_1 ∈ ball x₀ ε → HasFDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1) ∧ ∀ (x_1 : H), x_1 ∈ ball x₀ ε → ‖F' x_1 x‖ ≤ bound x) → LipschitzOnWith (↑Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x₀ ε) [PROOFSTEP] rintro a ⟨ha_deriv, ha_bound⟩ [GOAL] case intro α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : H → α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable (F' x₀) μ h_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → HasFDerivAt (fun x => F x a) (F' x a) x this : NormedSpace ℝ H := NormedSpace.restrictScalars ℝ 𝕜 H x₀_in : x₀ ∈ ball x₀ ε diff_x₀ : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' x₀ a) x₀ a : α ha_deriv : ∀ (x : H), x ∈ ball x₀ ε → HasFDerivAt (fun x => F x a) (F' x a) x ha_bound : ∀ (x : H), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a ⊢ LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε) [PROOFSTEP] refine' (convex_ball _ _).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x x_in => (ha_deriv x x_in).hasFDerivWithinAt) fun x x_in => _ [GOAL] case intro α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : H → α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable (F' x₀) μ h_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → HasFDerivAt (fun x => F x a) (F' x a) x this : NormedSpace ℝ H := NormedSpace.restrictScalars ℝ 𝕜 H x₀_in : x₀ ∈ ball x₀ ε diff_x₀ : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' x₀ a) x₀ a : α ha_deriv : ∀ (x : H), x ∈ ball x₀ ε → HasFDerivAt (fun x => F x a) (F' x a) x ha_bound : ∀ (x : H), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a x : H x_in : x ∈ ball x₀ ε ⊢ ‖F' x a‖₊ ≤ ↑Real.nnabs (bound a) [PROOFSTEP] rw [← NNReal.coe_le_coe, coe_nnnorm, Real.coe_nnabs] [GOAL] case intro α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : H → α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable (F' x₀) μ h_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → HasFDerivAt (fun x => F x a) (F' x a) x this : NormedSpace ℝ H := NormedSpace.restrictScalars ℝ 𝕜 H x₀_in : x₀ ∈ ball x₀ ε diff_x₀ : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' x₀ a) x₀ a : α ha_deriv : ∀ (x : H), x ∈ ball x₀ ε → HasFDerivAt (fun x => F x a) (F' x a) x ha_bound : ∀ (x : H), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a x : H x_in : x ∈ ball x₀ ε ⊢ ‖F' x a‖ ≤ \|bound a\| [PROOFSTEP] exact (ha_bound x x_in).trans (le_abs_self _) [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : H → α → E F' : H → α → H →L[𝕜] E x₀ : H bound : α → ℝ ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable (F' x₀) μ h_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → HasFDerivAt (fun x => F x a) (F' x a) x this✝ : NormedSpace ℝ H := NormedSpace.restrictScalars ℝ 𝕜 H x₀_in : x₀ ∈ ball x₀ ε diff_x₀ : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' x₀ a) x₀ this : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε) ⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' x₀ a ∂μ) x₀ [PROOFSTEP] exact (hasFDerivAt_integral_of_dominated_loc_of_lip ε_pos hF_meas hF_int hF'_meas this bound_integrable diff_x₀).2 [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : 𝕜 → α → E F' : α → E x₀ : 𝕜 ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound : α → ℝ h_lipsch : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε) bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x => F x a) (F' a) x₀ ⊢ Integrable F' ∧ HasDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] set L : E →L[𝕜] 𝕜 →L[𝕜] E := ContinuousLinearMap.smulRightL 𝕜 𝕜 E 1 [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : 𝕜 → α → E F' : α → E x₀ : 𝕜 ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound : α → ℝ h_lipsch : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε) bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x => F x a) (F' a) x₀ L : E →L[𝕜] 𝕜 →L[𝕜] E := ↑(ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1 ⊢ Integrable F' ∧ HasDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] replace h_diff : ∀ᵐ a ∂μ, HasFDerivAt (fun x => F x a) (L (F' a)) x₀ := h_diff.mono fun x hx => hx.hasFDerivAt [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : 𝕜 → α → E F' : α → E x₀ : 𝕜 ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound : α → ℝ h_lipsch : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε) bound_integrable : Integrable bound L : E →L[𝕜] 𝕜 →L[𝕜] E := ↑(ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1 h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (↑L (F' a)) x₀ ⊢ Integrable F' ∧ HasDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] have hm : AEStronglyMeasurable (L ∘ F') μ := L.continuous.comp_aestronglyMeasurable hF'_meas [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : 𝕜 → α → E F' : α → E x₀ : 𝕜 ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound : α → ℝ h_lipsch : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε) bound_integrable : Integrable bound L : E →L[𝕜] 𝕜 →L[𝕜] E := ↑(ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1 h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (↑L (F' a)) x₀ hm : AEStronglyMeasurable (↑L ∘ F') μ ⊢ Integrable F' ∧ HasDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] cases' hasFDerivAt_integral_of_dominated_loc_of_lip ε_pos hF_meas hF_int hm h_lipsch bound_integrable h_diff with hF'_int key [GOAL] case intro α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : 𝕜 → α → E F' : α → E x₀ : 𝕜 ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound : α → ℝ h_lipsch : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε) bound_integrable : Integrable bound L : E →L[𝕜] 𝕜 →L[𝕜] E := ↑(ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1 h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (↑L (F' a)) x₀ hm : AEStronglyMeasurable (↑L ∘ F') μ hF'_int : Integrable (↑L ∘ F') key : HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), (↑L ∘ F') a ∂μ) x₀ ⊢ Integrable F' ∧ HasDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] replace hF'_int : Integrable F' μ [GOAL] case hF'_int α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : 𝕜 → α → E F' : α → E x₀ : 𝕜 ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound : α → ℝ h_lipsch : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε) bound_integrable : Integrable bound L : E →L[𝕜] 𝕜 →L[𝕜] E := ↑(ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1 h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (↑L (F' a)) x₀ hm : AEStronglyMeasurable (↑L ∘ F') μ hF'_int : Integrable (↑L ∘ F') key : HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), (↑L ∘ F') a ∂μ) x₀ ⊢ Integrable F' [PROOFSTEP] rw [← integrable_norm_iff hm] at hF'_int [GOAL] case hF'_int α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : 𝕜 → α → E F' : α → E x₀ : 𝕜 ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound : α → ℝ h_lipsch : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε) bound_integrable : Integrable bound L : E →L[𝕜] 𝕜 →L[𝕜] E := ↑(ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1 h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (↑L (F' a)) x₀ hm : AEStronglyMeasurable (↑L ∘ F') μ hF'_int : Integrable fun a => ‖(↑L ∘ F') a‖ key : HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), (↑L ∘ F') a ∂μ) x₀ ⊢ Integrable F' [PROOFSTEP] simpa only [(· ∘ ·), integrable_norm_iff, hF'_meas, one_mul, norm_one, ContinuousLinearMap.comp_apply, ContinuousLinearMap.coe_restrict_scalarsL', ContinuousLinearMap.norm_restrictScalars, ContinuousLinearMap.norm_smulRightL_apply] using hF'_int [GOAL] case intro α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : 𝕜 → α → E F' : α → E x₀ : 𝕜 ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound : α → ℝ h_lipsch : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε) bound_integrable : Integrable bound L : E →L[𝕜] 𝕜 →L[𝕜] E := ↑(ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1 h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (↑L (F' a)) x₀ hm : AEStronglyMeasurable (↑L ∘ F') μ key : HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), (↑L ∘ F') a ∂μ) x₀ hF'_int : Integrable F' ⊢ Integrable F' ∧ HasDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] refine' ⟨hF'_int, _⟩ [GOAL] case intro α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : 𝕜 → α → E F' : α → E x₀ : 𝕜 ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound : α → ℝ h_lipsch : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε) bound_integrable : Integrable bound L : E →L[𝕜] 𝕜 →L[𝕜] E := ↑(ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1 h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (↑L (F' a)) x₀ hm : AEStronglyMeasurable (↑L ∘ F') μ key : HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), (↑L ∘ F') a ∂μ) x₀ hF'_int : Integrable F' ⊢ HasDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀ [PROOFSTEP] simp_rw [hasDerivAt_iff_hasFDerivAt] at h_diff ⊢ [GOAL] case intro α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F : 𝕜 → α → E F' : α → E x₀ : 𝕜 ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable F' μ bound : α → ℝ h_lipsch : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε) bound_integrable : Integrable bound L : E →L[𝕜] 𝕜 →L[𝕜] E := ↑(ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1 h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (↑(↑(ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1) (F' a)) x₀ hm : AEStronglyMeasurable (↑L ∘ F') μ key : HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), (↑L ∘ F') a ∂μ) x₀ hF'_int : Integrable F' ⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (ContinuousLinearMap.smulRight 1 (∫ (a : α), F' a ∂μ)) x₀ [PROOFSTEP] simpa only [(· ∘ ·), ContinuousLinearMap.integral_comp_comm _ hF'_int] using key [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F F' : 𝕜 → α → E x₀ : 𝕜 ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable (F' x₀) μ bound : α → ℝ h_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x ⊢ Integrable (F' x₀) ∧ HasDerivAt (fun n => ∫ (a : α), F n a ∂μ) (∫ (a : α), F' x₀ a ∂μ) x₀ [PROOFSTEP] have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F F' : 𝕜 → α → E x₀ : 𝕜 ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable (F' x₀) μ bound : α → ℝ h_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x x₀_in : x₀ ∈ ball x₀ ε ⊢ Integrable (F' x₀) ∧ HasDerivAt (fun n => ∫ (a : α), F n a ∂μ) (∫ (a : α), F' x₀ a ∂μ) x₀ [PROOFSTEP] have diff_x₀ : ∀ᵐ a ∂μ, HasDerivAt (fun x => F x a) (F' x₀ a) x₀ := h_diff.mono fun a ha => ha x₀ x₀_in [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F F' : 𝕜 → α → E x₀ : 𝕜 ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable (F' x₀) μ bound : α → ℝ h_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x x₀_in : x₀ ∈ ball x₀ ε diff_x₀ : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x => F x a) (F' x₀ a) x₀ ⊢ Integrable (F' x₀) ∧ HasDerivAt (fun n => ∫ (a : α), F n a ∂μ) (∫ (a : α), F' x₀ a ∂μ) x₀ [PROOFSTEP] have : ∀ᵐ a ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x : 𝕜 => F x a) (ball x₀ ε) := by apply (h_diff.and h_bound).mono rintro a ⟨ha_deriv, ha_bound⟩ refine' (convex_ball _ _).lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x x_in => (ha_deriv x x_in).hasDerivWithinAt) fun x x_in => _ rw [← NNReal.coe_le_coe, coe_nnnorm, Real.coe_nnabs] exact (ha_bound x x_in).trans (le_abs_self _) [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F F' : 𝕜 → α → E x₀ : 𝕜 ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable (F' x₀) μ bound : α → ℝ h_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x x₀_in : x₀ ∈ ball x₀ ε diff_x₀ : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x => F x a) (F' x₀ a) x₀ ⊢ ∀ᵐ (a : α) ∂μ, LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε) [PROOFSTEP] apply (h_diff.and h_bound).mono [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F F' : 𝕜 → α → E x₀ : 𝕜 ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable (F' x₀) μ bound : α → ℝ h_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x x₀_in : x₀ ∈ ball x₀ ε diff_x₀ : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x => F x a) (F' x₀ a) x₀ ⊢ ∀ (x : α), ((∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → HasDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1) ∧ ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → ‖F' x_1 x‖ ≤ bound x) → LipschitzOnWith (↑Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x₀ ε) [PROOFSTEP] rintro a ⟨ha_deriv, ha_bound⟩ [GOAL] case intro α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F F' : 𝕜 → α → E x₀ : 𝕜 ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable (F' x₀) μ bound : α → ℝ h_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x x₀_in : x₀ ∈ ball x₀ ε diff_x₀ : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x => F x a) (F' x₀ a) x₀ a : α ha_deriv : ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x ha_bound : ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a ⊢ LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε) [PROOFSTEP] refine' (convex_ball _ _).lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x x_in => (ha_deriv x x_in).hasDerivWithinAt) fun x x_in => _ [GOAL] case intro α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F F' : 𝕜 → α → E x₀ : 𝕜 ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable (F' x₀) μ bound : α → ℝ h_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x x₀_in : x₀ ∈ ball x₀ ε diff_x₀ : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x => F x a) (F' x₀ a) x₀ a : α ha_deriv : ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x ha_bound : ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a x : 𝕜 x_in : x ∈ ball x₀ ε ⊢ ‖F' x a‖₊ ≤ ↑Real.nnabs (bound a) [PROOFSTEP] rw [← NNReal.coe_le_coe, coe_nnnorm, Real.coe_nnabs] [GOAL] case intro α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F F' : 𝕜 → α → E x₀ : 𝕜 ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable (F' x₀) μ bound : α → ℝ h_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x x₀_in : x₀ ∈ ball x₀ ε diff_x₀ : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x => F x a) (F' x₀ a) x₀ a : α ha_deriv : ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x ha_bound : ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a x : 𝕜 x_in : x ∈ ball x₀ ε ⊢ ‖F' x a‖ ≤ \|bound a\| [PROOFSTEP] exact (ha_bound x x_in).trans (le_abs_self _) [GOAL] α : Type u_1 inst✝⁷ : MeasurableSpace α μ : Measure α 𝕜 : Type u_2 inst✝⁶ : IsROrC 𝕜 E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : NormedSpace 𝕜 E inst✝² : CompleteSpace E H : Type u_4 inst✝¹ : NormedAddCommGroup H inst✝ : NormedSpace 𝕜 H F F' : 𝕜 → α → E x₀ : 𝕜 ε : ℝ ε_pos : 0 < ε hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ hF_int : Integrable (F x₀) hF'_meas : AEStronglyMeasurable (F' x₀) μ bound : α → ℝ h_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a bound_integrable : Integrable bound h_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x x₀_in : x₀ ∈ ball x₀ ε diff_x₀ : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x => F x a) (F' x₀ a) x₀ this : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε) ⊢ Integrable (F' x₀) ∧ HasDerivAt (fun n => ∫ (a : α), F n a ∂μ) (∫ (a : α), F' x₀ a ∂μ) x₀ [PROOFSTEP] exact hasDerivAt_integral_of_dominated_loc_of_lip ε_pos hF_meas hF_int hF'_meas this bound_integrable diff_x₀
From Test Require Import tactic. Section FOFProblem. Variable Universe : Set. Variable UniverseElement : Universe. Variable wd_ : Universe -> Universe -> Prop. Variable col_ : Universe -> Universe -> Universe -> Prop. Variable col_swap1_1 : (forall A B C : Universe, (col_ A B C -> col_ B A C)). Variable col_swap2_2 : (forall A B C : Universe, (col_ A B C -> col_ B C A)). Variable col_triv_3 : (forall A B : Universe, col_ A B B). Variable wd_swap_4 : (forall A B : Universe, (wd_ A B -> wd_ B A)). Variable col_trans_5 : (forall P Q A B C : Universe, ((wd_ P Q /\ (col_ P Q A /\ (col_ P Q B /\ col_ P Q C))) -> col_ A B C)). Theorem pipo_6 : (forall A B C P Q X : Universe, ((wd_ A B /\ (wd_ B C /\ (wd_ A C /\ (wd_ X A /\ (wd_ X C /\ (wd_ P B /\ (wd_ P C /\ (wd_ Q P /\ (wd_ A P /\ (wd_ B X /\ (wd_ X P /\ (col_ A C Q /\ (col_ X A C /\ (col_ P B C /\ col_ A C P)))))))))))))) -> col_ A B C)). Proof. time tac. Qed. End FOFProblem.
[GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β f : β → Measure α h : ∀ (s : Set α), MeasurableSet s → Measurable fun b => ↑↑(f b) s s : Set α hs : MeasurableSet s ⊢ borel ℝ≥0∞ ≤ MeasurableSpace.map (fun μ => ↑↑μ s) (MeasurableSpace.map f inst✝) [PROOFSTEP] rw [MeasurableSpace.map_comp] [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β f : β → Measure α h : ∀ (s : Set α), MeasurableSet s → Measurable fun b => ↑↑(f b) s s : Set α hs : MeasurableSet s ⊢ borel ℝ≥0∞ ≤ MeasurableSpace.map ((fun μ => ↑↑μ s) ∘ f) inst✝ [PROOFSTEP] exact h s hs [GOAL] α✝ : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α✝ inst✝ : MeasurableSpace β α : Type u_3 m : MeasurableSpace α ⊢ MeasurableAdd₂ (Measure α) [PROOFSTEP] refine' ⟨Measure.measurable_of_measurable_coe _ fun s hs => _⟩ [GOAL] α✝ : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α✝ inst✝ : MeasurableSpace β α : Type u_3 m : MeasurableSpace α s : Set α hs : MeasurableSet s ⊢ Measurable fun b => ↑↑(b.fst + b.snd) s [PROOFSTEP] simp_rw [Measure.coe_add, Pi.add_apply] [GOAL] α✝ : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α✝ inst✝ : MeasurableSpace β α : Type u_3 m : MeasurableSpace α s : Set α hs : MeasurableSet s ⊢ Measurable fun b => ↑↑b.fst s + ↑↑b.snd s [PROOFSTEP] refine' Measurable.add _ _ [GOAL] case refine'_1 α✝ : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α✝ inst✝ : MeasurableSpace β α : Type u_3 m : MeasurableSpace α s : Set α hs : MeasurableSet s ⊢ Measurable fun b => ↑↑b.fst s [PROOFSTEP] exact (Measure.measurable_coe hs).comp measurable_fst [GOAL] case refine'_2 α✝ : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α✝ inst✝ : MeasurableSpace β α : Type u_3 m : MeasurableSpace α s : Set α hs : MeasurableSet s ⊢ Measurable fun b => ↑↑b.snd s [PROOFSTEP] exact (Measure.measurable_coe hs).comp measurable_snd [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β f : α → β hf : Measurable f ⊢ Measurable fun μ => map f μ [PROOFSTEP] refine' measurable_of_measurable_coe _ fun s hs => _ [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β f : α → β hf : Measurable f s : Set β hs : MeasurableSet s ⊢ Measurable fun b => ↑↑(map f b) s [PROOFSTEP] simp_rw [map_apply hf hs] [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β f : α → β hf : Measurable f s : Set β hs : MeasurableSet s ⊢ Measurable fun b => ↑↑b (f ⁻¹' s) [PROOFSTEP] exact measurable_coe (hf hs) [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β ⊢ Measurable dirac [PROOFSTEP] refine' measurable_of_measurable_coe _ fun s hs => _ [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β s : Set α hs : MeasurableSet s ⊢ Measurable fun b => ↑↑(dirac b) s [PROOFSTEP] simp_rw [dirac_apply' _ hs] [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β s : Set α hs : MeasurableSet s ⊢ Measurable fun b => indicator s 1 b [PROOFSTEP] exact measurable_one.indicator hs [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β f : α → ℝ≥0∞ hf : Measurable f ⊢ Measurable fun μ => ∫⁻ (x : α), f x ∂μ [PROOFSTEP] simp only [lintegral_eq_iSup_eapprox_lintegral, hf, SimpleFunc.lintegral] [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β f : α → ℝ≥0∞ hf : Measurable f ⊢ Measurable fun μ => ⨆ (n : ℕ), ∑ x in SimpleFunc.range (SimpleFunc.eapprox (fun x => f x) n), x * ↑↑μ (↑(SimpleFunc.eapprox (fun x => f x) n) ⁻¹' {x}) [PROOFSTEP] refine' measurable_iSup fun n => Finset.measurable_sum _ fun i _ => _ [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β f : α → ℝ≥0∞ hf : Measurable f n : ℕ i : ℝ≥0∞ x✝ : i ∈ SimpleFunc.range (SimpleFunc.eapprox (fun x => f x) n) ⊢ Measurable fun μ => i * ↑↑μ (↑(SimpleFunc.eapprox (fun x => f x) n) ⁻¹' {i}) [PROOFSTEP] refine' Measurable.const_mul _ _ [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β f : α → ℝ≥0∞ hf : Measurable f n : ℕ i : ℝ≥0∞ x✝ : i ∈ SimpleFunc.range (SimpleFunc.eapprox (fun x => f x) n) ⊢ Measurable fun μ => ↑↑μ (↑(SimpleFunc.eapprox (fun x => f x) n) ⁻¹' {i}) [PROOFSTEP] exact measurable_coe ((SimpleFunc.eapprox f n).measurableSet_preimage _) [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure (Measure α) ⊢ (fun s x => ∫⁻ (μ : Measure α), ↑↑μ s ∂m) ∅ (_ : MeasurableSet ∅) = 0 [PROOFSTEP] simp only [measure_empty, lintegral_const, zero_mul] [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure (Measure α) ⊢ ∀ ⦃f : ℕ → Set α⦄ (h : ∀ (i : ℕ), MeasurableSet (f i)), Pairwise (Disjoint on f) → (fun s x => ∫⁻ (μ : Measure α), ↑↑μ s ∂m) (⋃ (i : ℕ), f i) (_ : MeasurableSet (⋃ (b : ℕ), f b)) = ∑' (i : ℕ), (fun s x => ∫⁻ (μ : Measure α), ↑↑μ s ∂m) (f i) (_ : MeasurableSet (f i)) [PROOFSTEP] intro f hf h [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure (Measure α) f : ℕ → Set α hf : ∀ (i : ℕ), MeasurableSet (f i) h : Pairwise (Disjoint on f) ⊢ (fun s x => ∫⁻ (μ : Measure α), ↑↑μ s ∂m) (⋃ (i : ℕ), f i) (_ : MeasurableSet (⋃ (b : ℕ), f b)) = ∑' (i : ℕ), (fun s x => ∫⁻ (μ : Measure α), ↑↑μ s ∂m) (f i) (_ : MeasurableSet (f i)) [PROOFSTEP] simp_rw [measure_iUnion h hf] [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure (Measure α) f : ℕ → Set α hf : ∀ (i : ℕ), MeasurableSet (f i) h : Pairwise (Disjoint on f) ⊢ ∫⁻ (μ : Measure α), ∑' (i : ℕ), ↑↑μ (f i) ∂m = ∑' (i : ℕ), ∫⁻ (μ : Measure α), ↑↑μ (f i) ∂m [PROOFSTEP] apply lintegral_tsum [GOAL] case hf α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure (Measure α) f : ℕ → Set α hf : ∀ (i : ℕ), MeasurableSet (f i) h : Pairwise (Disjoint on f) ⊢ ∀ (i : ℕ), AEMeasurable fun a => ↑↑a (f i) [PROOFSTEP] intro i [GOAL] case hf α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure (Measure α) f : ℕ → Set α hf : ∀ (i : ℕ), MeasurableSet (f i) h : Pairwise (Disjoint on f) i : ℕ ⊢ AEMeasurable fun a => ↑↑a (f i) [PROOFSTEP] exact (measurable_coe (hf i)).aemeasurable [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β ⊢ join 0 = 0 [PROOFSTEP] ext1 s hs [GOAL] case h α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β s : Set α hs : MeasurableSet s ⊢ ↑↑(join 0) s = ↑↑0 s [PROOFSTEP] simp only [hs, join_apply, lintegral_zero_measure, coe_zero, Pi.zero_apply] [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β s : Set α hs : MeasurableSet s ⊢ Measurable fun b => ↑↑(join b) s [PROOFSTEP] simp only [join_apply hs] [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β s : Set α hs : MeasurableSet s ⊢ Measurable fun b => ∫⁻ (μ : Measure α), ↑↑μ s ∂b [PROOFSTEP] exact measurable_lintegral (measurable_coe hs) [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure (Measure α) f : α → ℝ≥0∞ hf : Measurable f ⊢ ∫⁻ (x : α), f x ∂join m = ∫⁻ (μ : Measure α), ∫⁻ (x : α), f x ∂μ ∂m [PROOFSTEP] simp_rw [lintegral_eq_iSup_eapprox_lintegral hf, SimpleFunc.lintegral, join_apply (SimpleFunc.measurableSet_preimage _ _)] [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure (Measure α) f : α → ℝ≥0∞ hf : Measurable f ⊢ ⨆ (n : ℕ), ∑ x in SimpleFunc.range (SimpleFunc.eapprox f n), x * ∫⁻ (μ : Measure α), ↑↑μ (↑(SimpleFunc.eapprox f n) ⁻¹' {x}) ∂m = ∫⁻ (μ : Measure α), ⨆ (n : ℕ), ∑ x in SimpleFunc.range (SimpleFunc.eapprox f n), x * ↑↑μ (↑(SimpleFunc.eapprox f n) ⁻¹' {x}) ∂m [PROOFSTEP] suffices ∀ (s : ℕ → Finset ℝ≥0∞) (f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞), (∀ n r, Measurable (f n r)) → Monotone (fun n μ => ∑ r in s n, r * f n r μ) → ⨆ n, ∑ r in s n, r * ∫⁻ μ, f n r μ ∂m = ∫⁻ μ, ⨆ n, ∑ r in s n, r * f n r μ ∂m by refine' this (fun n => SimpleFunc.range (SimpleFunc.eapprox f n)) (fun n r μ => μ (SimpleFunc.eapprox f n ⁻¹' { r })) _ _ · exact fun n r => measurable_coe (SimpleFunc.measurableSet_preimage _ _) · exact fun n m h μ => SimpleFunc.lintegral_mono (SimpleFunc.monotone_eapprox _ h) le_rfl [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure (Measure α) f : α → ℝ≥0∞ hf : Measurable f this : ∀ (s : ℕ → Finset ℝ≥0∞) (f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞), (∀ (n : ℕ) (r : ℝ≥0∞), Measurable (f n r)) → (Monotone fun n μ => ∑ r in s n, r * f n r μ) → ⨆ (n : ℕ), ∑ r in s n, r * ∫⁻ (μ : Measure α), f n r μ ∂m = ∫⁻ (μ : Measure α), ⨆ (n : ℕ), ∑ r in s n, r * f n r μ ∂m ⊢ ⨆ (n : ℕ), ∑ x in SimpleFunc.range (SimpleFunc.eapprox f n), x * ∫⁻ (μ : Measure α), ↑↑μ (↑(SimpleFunc.eapprox f n) ⁻¹' {x}) ∂m = ∫⁻ (μ : Measure α), ⨆ (n : ℕ), ∑ x in SimpleFunc.range (SimpleFunc.eapprox f n), x * ↑↑μ (↑(SimpleFunc.eapprox f n) ⁻¹' {x}) ∂m [PROOFSTEP] refine' this (fun n => SimpleFunc.range (SimpleFunc.eapprox f n)) (fun n r μ => μ (SimpleFunc.eapprox f n ⁻¹' { r })) _ _ [GOAL] case refine'_1 α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure (Measure α) f : α → ℝ≥0∞ hf : Measurable f this : ∀ (s : ℕ → Finset ℝ≥0∞) (f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞), (∀ (n : ℕ) (r : ℝ≥0∞), Measurable (f n r)) → (Monotone fun n μ => ∑ r in s n, r * f n r μ) → ⨆ (n : ℕ), ∑ r in s n, r * ∫⁻ (μ : Measure α), f n r μ ∂m = ∫⁻ (μ : Measure α), ⨆ (n : ℕ), ∑ r in s n, r * f n r μ ∂m ⊢ ∀ (n : ℕ) (r : ℝ≥0∞), Measurable ((fun n r μ => ↑↑μ (↑(SimpleFunc.eapprox f n) ⁻¹' {r})) n r) [PROOFSTEP] exact fun n r => measurable_coe (SimpleFunc.measurableSet_preimage _ _) [GOAL] case refine'_2 α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure (Measure α) f : α → ℝ≥0∞ hf : Measurable f this : ∀ (s : ℕ → Finset ℝ≥0∞) (f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞), (∀ (n : ℕ) (r : ℝ≥0∞), Measurable (f n r)) → (Monotone fun n μ => ∑ r in s n, r * f n r μ) → ⨆ (n : ℕ), ∑ r in s n, r * ∫⁻ (μ : Measure α), f n r μ ∂m = ∫⁻ (μ : Measure α), ⨆ (n : ℕ), ∑ r in s n, r * f n r μ ∂m ⊢ Monotone fun n μ => ∑ r in (fun n => SimpleFunc.range (SimpleFunc.eapprox f n)) n, r * (fun n r μ => ↑↑μ (↑(SimpleFunc.eapprox f n) ⁻¹' {r})) n r μ [PROOFSTEP] exact fun n m h μ => SimpleFunc.lintegral_mono (SimpleFunc.monotone_eapprox _ h) le_rfl [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure (Measure α) f : α → ℝ≥0∞ hf : Measurable f ⊢ ∀ (s : ℕ → Finset ℝ≥0∞) (f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞), (∀ (n : ℕ) (r : ℝ≥0∞), Measurable (f n r)) → (Monotone fun n μ => ∑ r in s n, r * f n r μ) → ⨆ (n : ℕ), ∑ r in s n, r * ∫⁻ (μ : Measure α), f n r μ ∂m = ∫⁻ (μ : Measure α), ⨆ (n : ℕ), ∑ r in s n, r * f n r μ ∂m [PROOFSTEP] intro s f hf hm [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure (Measure α) f✝ : α → ℝ≥0∞ hf✝ : Measurable f✝ s : ℕ → Finset ℝ≥0∞ f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞ hf : ∀ (n : ℕ) (r : ℝ≥0∞), Measurable (f n r) hm : Monotone fun n μ => ∑ r in s n, r * f n r μ ⊢ ⨆ (n : ℕ), ∑ r in s n, r * ∫⁻ (μ : Measure α), f n r μ ∂m = ∫⁻ (μ : Measure α), ⨆ (n : ℕ), ∑ r in s n, r * f n r μ ∂m [PROOFSTEP] rw [lintegral_iSup _ hm] [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure (Measure α) f✝ : α → ℝ≥0∞ hf✝ : Measurable f✝ s : ℕ → Finset ℝ≥0∞ f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞ hf : ∀ (n : ℕ) (r : ℝ≥0∞), Measurable (f n r) hm : Monotone fun n μ => ∑ r in s n, r * f n r μ ⊢ ⨆ (n : ℕ), ∑ r in s n, r * ∫⁻ (μ : Measure α), f n r μ ∂m = ⨆ (n : ℕ), ∫⁻ (a : Measure α), ∑ r in s n, r * f n r a ∂m α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure (Measure α) f✝ : α → ℝ≥0∞ hf✝ : Measurable f✝ s : ℕ → Finset ℝ≥0∞ f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞ hf : ∀ (n : ℕ) (r : ℝ≥0∞), Measurable (f n r) hm : Monotone fun n μ => ∑ r in s n, r * f n r μ ⊢ ∀ (n : ℕ), Measurable fun μ => ∑ r in s n, r * f n r μ [PROOFSTEP] swap [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure (Measure α) f✝ : α → ℝ≥0∞ hf✝ : Measurable f✝ s : ℕ → Finset ℝ≥0∞ f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞ hf : ∀ (n : ℕ) (r : ℝ≥0∞), Measurable (f n r) hm : Monotone fun n μ => ∑ r in s n, r * f n r μ ⊢ ∀ (n : ℕ), Measurable fun μ => ∑ r in s n, r * f n r μ [PROOFSTEP] exact fun n => Finset.measurable_sum _ fun r _ => (hf _ _).const_mul _ [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure (Measure α) f✝ : α → ℝ≥0∞ hf✝ : Measurable f✝ s : ℕ → Finset ℝ≥0∞ f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞ hf : ∀ (n : ℕ) (r : ℝ≥0∞), Measurable (f n r) hm : Monotone fun n μ => ∑ r in s n, r * f n r μ ⊢ ⨆ (n : ℕ), ∑ r in s n, r * ∫⁻ (μ : Measure α), f n r μ ∂m = ⨆ (n : ℕ), ∫⁻ (a : Measure α), ∑ r in s n, r * f n r a ∂m [PROOFSTEP] congr [GOAL] case e_s α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure (Measure α) f✝ : α → ℝ≥0∞ hf✝ : Measurable f✝ s : ℕ → Finset ℝ≥0∞ f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞ hf : ∀ (n : ℕ) (r : ℝ≥0∞), Measurable (f n r) hm : Monotone fun n μ => ∑ r in s n, r * f n r μ ⊢ (fun n => ∑ r in s n, r * ∫⁻ (μ : Measure α), f n r μ ∂m) = fun n => ∫⁻ (a : Measure α), ∑ r in s n, r * f n r a ∂m [PROOFSTEP] funext n [GOAL] case e_s.h α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure (Measure α) f✝ : α → ℝ≥0∞ hf✝ : Measurable f✝ s : ℕ → Finset ℝ≥0∞ f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞ hf : ∀ (n : ℕ) (r : ℝ≥0∞), Measurable (f n r) hm : Monotone fun n μ => ∑ r in s n, r * f n r μ n : ℕ ⊢ ∑ r in s n, r * ∫⁻ (μ : Measure α), f n r μ ∂m = ∫⁻ (a : Measure α), ∑ r in s n, r * f n r a ∂m [PROOFSTEP] rw [lintegral_finset_sum (s n)] [GOAL] case e_s.h α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure (Measure α) f✝ : α → ℝ≥0∞ hf✝ : Measurable f✝ s : ℕ → Finset ℝ≥0∞ f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞ hf : ∀ (n : ℕ) (r : ℝ≥0∞), Measurable (f n r) hm : Monotone fun n μ => ∑ r in s n, r * f n r μ n : ℕ ⊢ ∑ r in s n, r * ∫⁻ (μ : Measure α), f n r μ ∂m = ∑ b in s n, ∫⁻ (a : Measure α), b * f n b a ∂m [PROOFSTEP] simp_rw [lintegral_const_mul _ (hf _ _)] [GOAL] case e_s.h α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure (Measure α) f✝ : α → ℝ≥0∞ hf✝ : Measurable f✝ s : ℕ → Finset ℝ≥0∞ f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞ hf : ∀ (n : ℕ) (r : ℝ≥0∞), Measurable (f n r) hm : Monotone fun n μ => ∑ r in s n, r * f n r μ n : ℕ ⊢ ∀ (b : ℝ≥0∞), b ∈ s n → Measurable fun a => b * f n b a [PROOFSTEP] exact fun r _ => (hf _ _).const_mul _ [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β f : α → Measure β ⊢ bind 0 f = 0 [PROOFSTEP] simp [bind] [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure α ⊢ bind m 0 = 0 [PROOFSTEP] ext1 s hs [GOAL] case h α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure α s : Set β hs : MeasurableSet s ⊢ ↑↑(bind m 0) s = ↑↑0 s [PROOFSTEP] simp only [bind, hs, join_apply, coe_zero, Pi.zero_apply] [GOAL] case h α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure α s : Set β hs : MeasurableSet s ⊢ ∫⁻ (μ : Measure β), ↑↑μ s ∂map 0 m = 0 [PROOFSTEP] rw [lintegral_map (measurable_coe hs) measurable_zero] [GOAL] case h α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure α s : Set β hs : MeasurableSet s ⊢ ∫⁻ (a : α), ↑↑(OfNat.ofNat 0 a) s ∂m = 0 [PROOFSTEP] simp only [Pi.zero_apply, coe_zero, lintegral_const, zero_mul] [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure α f : α → Measure β s : Set β hs : MeasurableSet s hf : Measurable f ⊢ ↑↑(bind m f) s = ∫⁻ (a : α), ↑↑(f a) s ∂m [PROOFSTEP] rw [bind, join_apply hs, lintegral_map (measurable_coe hs) hf] [GOAL] α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β γ : Type u_3 inst✝ : MeasurableSpace γ m : Measure α f : α → Measure β g : β → Measure γ hf : Measurable f hg : Measurable g ⊢ bind (bind m f) g = bind m fun a => bind (f a) g [PROOFSTEP] ext1 s hs [GOAL] case h α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β γ : Type u_3 inst✝ : MeasurableSpace γ m : Measure α f : α → Measure β g : β → Measure γ hf : Measurable f hg : Measurable g s : Set γ hs : MeasurableSet s ⊢ ↑↑(bind (bind m f) g) s = ↑↑(bind m fun a => bind (f a) g) s [PROOFSTEP] erw [bind_apply hs hg, bind_apply hs ((measurable_bind' hg).comp hf), lintegral_bind hf ((measurable_coe hs).comp hg)] [GOAL] case h α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β γ : Type u_3 inst✝ : MeasurableSpace γ m : Measure α f : α → Measure β g : β → Measure γ hf : Measurable f hg : Measurable g s : Set γ hs : MeasurableSet s ⊢ ∫⁻ (a : α), ∫⁻ (x : β), ((fun μ => ↑↑μ s) ∘ g) x ∂f a ∂m = ∫⁻ (a : α), ↑↑(((fun m => bind m g) ∘ f) a) s ∂m [PROOFSTEP] conv_rhs => enter [2, a]; erw [bind_apply hs hg] [GOAL] α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β γ : Type u_3 inst✝ : MeasurableSpace γ m : Measure α f : α → Measure β g : β → Measure γ hf : Measurable f hg : Measurable g s : Set γ hs : MeasurableSet s \| ∫⁻ (a : α), ↑↑(((fun m => bind m g) ∘ f) a) s ∂m [PROOFSTEP] enter [2, a]; erw [bind_apply hs hg] [GOAL] α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β γ : Type u_3 inst✝ : MeasurableSpace γ m : Measure α f : α → Measure β g : β → Measure γ hf : Measurable f hg : Measurable g s : Set γ hs : MeasurableSet s \| ∫⁻ (a : α), ↑↑(((fun m => bind m g) ∘ f) a) s ∂m [PROOFSTEP] enter [2, a]; erw [bind_apply hs hg] [GOAL] α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β γ : Type u_3 inst✝ : MeasurableSpace γ m : Measure α f : α → Measure β g : β → Measure γ hf : Measurable f hg : Measurable g s : Set γ hs : MeasurableSet s \| ∫⁻ (a : α), ↑↑(((fun m => bind m g) ∘ f) a) s ∂m [PROOFSTEP] enter [2, a] [GOAL] case h α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β γ : Type u_3 inst✝ : MeasurableSpace γ m : Measure α f : α → Measure β g : β → Measure γ hf : Measurable f hg : Measurable g s : Set γ hs : MeasurableSet s a : α \| ↑↑(((fun m => bind m g) ∘ f) a) s [PROOFSTEP] erw [bind_apply hs hg] [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β f : α → Measure β hf : Measurable f a : α ⊢ bind (dirac a) f = f a [PROOFSTEP] ext1 s hs [GOAL] case h α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β f : α → Measure β hf : Measurable f a : α s : Set β hs : MeasurableSet s ⊢ ↑↑(bind (dirac a) f) s = ↑↑(f a) s [PROOFSTEP] erw [bind_apply hs hf, lintegral_dirac' a ((measurable_coe hs).comp hf)] [GOAL] case h α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β f : α → Measure β hf : Measurable f a : α s : Set β hs : MeasurableSet s ⊢ ((fun μ => ↑↑μ s) ∘ f) a = ↑↑(f a) s [PROOFSTEP] rfl [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure α ⊢ bind m dirac = m [PROOFSTEP] ext1 s hs [GOAL] case h α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β m : Measure α s : Set α hs : MeasurableSet s ⊢ ↑↑(bind m dirac) s = ↑↑m s [PROOFSTEP] simp only [bind_apply hs measurable_dirac, dirac_apply' _ hs, lintegral_indicator 1 hs, Pi.one_apply, lintegral_one, restrict_apply, MeasurableSet.univ, univ_inter] [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β μ : Measure (Measure α) ⊢ join μ = bind μ id [PROOFSTEP] rw [bind, map_id] [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β f : α → β hf : Measurable f μ : Measure (Measure α) ⊢ join (map (map f) μ) = map f (join μ) [PROOFSTEP] ext1 s hs [GOAL] case h α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β f : α → β hf : Measurable f μ : Measure (Measure α) s : Set β hs : MeasurableSet s ⊢ ↑↑(join (map (map f) μ)) s = ↑↑(map f (join μ)) s [PROOFSTEP] rw [join_apply hs, map_apply hf hs, join_apply (hf hs), lintegral_map (measurable_coe hs) (measurable_map f hf)] [GOAL] case h α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β f : α → β hf : Measurable f μ : Measure (Measure α) s : Set β hs : MeasurableSet s ⊢ ∫⁻ (a : Measure α), ↑↑(map f a) s ∂μ = ∫⁻ (μ : Measure α), ↑↑μ (f ⁻¹' s) ∂μ [PROOFSTEP] simp_rw [map_apply hf hs] [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β μ : Measure (Measure (Measure α)) ⊢ join (map join μ) = join (join μ) [PROOFSTEP] show bind μ join = join (join μ) [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β μ : Measure (Measure (Measure α)) ⊢ bind μ join = join (join μ) [PROOFSTEP] rw [join_eq_bind, join_eq_bind, bind_bind measurable_id measurable_id] [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β μ : Measure (Measure (Measure α)) ⊢ bind μ join = bind μ fun a => bind (id a) id [PROOFSTEP] apply congr_arg (bind μ) [GOAL] α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β μ : Measure (Measure (Measure α)) ⊢ join = fun a => bind (id a) id [PROOFSTEP] funext ν [GOAL] case h α : Type u_1 β : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β μ : Measure (Measure (Measure α)) ν : Measure (Measure α) ⊢ join ν = bind (id ν) id [PROOFSTEP] exact join_eq_bind ν
\chapter{Algebraic Integers} \setcounter{page}{1} \pagenumbering{arabic} \section{Integral closure and algebraic integers} Here we present fundamental properties of algebraic integers and related theories. It is not our intention to do a treatise of commutative algebra and Galois theory, so for example there would not be the fundamental theorem of Galois theory, or the definition of Dedekind domain. However, if some theorems are explicitly useful in the future, you will still see the theorem and the proof. \subsection{Definition and examples} Problems in solving polynomial equations give rise to a lot of concepts in algebra and geometry. If we are specifically interested in $\mds{Z}$, we have the concept of \textbf{algebraic integers}. \begin{definition} A finite extension $K$ of the rational number $\mds{Q}$ is called a \textbf{number field}. The integral closure of $\mds{Z}$ in $K$ is called the ring of \textbf{algebraic integers} of $K$, and is denoted by $\OK$. To be precise, every element $x \in \OK$ is a zero of a monic polynomial $f \in \Z[X].$ \end{definition} For this concept we have a lot of classic examples: \begin{example} If $K=\Q$, then $\OK$ is simply $\Z$. This is intuitive because suppose $x=a/b \in \Q$ is integral over $\Z$ where $(a,b)=1$, then \[ x^n+c_1x^{n-1}+\cdots+c_n = 0 \] where $c_i \in \Z$. Multiplying by $b^n$ yields \[ a^n+c_1a^{n-1}b+\cdots+c_nb = 0 \] Hence $b$ divides $a^n$. But we also have $(a^n,b)=1$, hence $b=\pm 1$, which is to say $x \in \Z$. There is an more general setting. Since $\Z$ is a unique factorial domain (UFD), and UFD is integrally closed \href{https://proofwiki.org/wiki/Unique_Factorization_Domain_is_Integrally_Closed}{[proof]}, we have $\Z=\OK$. \end{example} \begin{example} The Gaussian rational $\Q(i)=K$. Indeed it is natural to consider Gaussian integer $\Z[i]$ first. For any $z=m+ni \in \Z[i]$, we have \[ z^2-2mz+m^2+n^2=0 \] Hence $\Z[i] \subset \OK$. The converse is similar to our proof when $K=\Q$. \end{example} \begin{example} Quadratic field $K=\Q(\sqrt{d})$, where $d$ is a square-free integer $>1$. This time the algebraic integer ring is different from what you may have thought: $\OK = \Z[\omega]$ where \[ \omega = \begin{cases} \frac{1+\sqrt{d}}{2}, &\quad d = 4k+1, \\ \sqrt{d}, &\quad \text{otherwise}. \end{cases} \] \end{example} It turns out we are studying polynomials such as \begin{itemize} \item $x^2+1=0$. \item $x^2-d=0$. \end{itemize} It also turns out that many properties are not restricted to $\Z$, but to a specific class of rings. Hence we will investigate some properties in the sense of commutative ring theory. The next example deserves more discussion. \subsubsection{The Cyclotomic Fields} Let $\omega = e^{2\pi i/m}$. This is the $m$-th root of $1$. If $\lambda$ is a conjugate of $\omega$, i.e. they both are roots of the same irreducible polynomial over $\Q$, then $\lambda$ is also an $m$-th root of $1$ and is not an $n$-th root of $1$ whenever $n<m$. If we find these $\lambda$, we have a better understanding of $\Q(\omega)$, which is called the cyclotomic field. We will do this in a general setting. \begin{definition} Let $k$ be a field. By a \textbf{root of unity} in $k$ we shall mean an element $\xi \in k$ such that $\xi^n=1$ for some integer $n \ge 1$. Let $n$ be a integer $\ge 2$ and not divisible by the characteristic. The generator for the cyclic group $\bm{\mu}_n$ of $n$-th root of unity is called \textbf{primitive}. \end{definition} \begin{remark} We shall not ignore the characteristic of $k$. Indeed, if the characteristic is $p$, then the equation \[ X^{p^m}-1=0 \] has only one root, namely $1$, and hence there is no $p^m$-th root of unity except $1$. \end{remark} \begin{remark} $\bm{\mu}_n$ has $n$ elements. The derivative of $X^n-1$ is $nX^{n-1} \ne 0$, and the only root of the derivative is $0$, so there is no common root. In the algebraic closure of $k$, the polynomial $X^n-1$ has $n$ distinct roots, which forms $\bm{\mu}_n$. \end{remark} The field extension in terms of root of unity can always be characterised for $(\Z/n\Z)^\ast$ as follows: \begin{theorem}\label{gal-cyclotomic} Let $k$ be any field, let $n$ be not divisible by the characteristic $p$. Let $\omega$ be a primitive $n$-th root of unity in the algebraic closure $k^\mathrm{a}$, then $\gal(k(\omega)/k)$ is a subgroup of $(\Z/n\Z)^\ast$. \end{theorem} \begin{proof} Let $\sigma$ be an embedding of $k(\omega)$ in $k^\mathrm{a}$ over $k$. Then \[ (\sigma\omega)^n = \sigma(\omega^n)=1 \] hence $\sigma\omega$ is also an $n$-th root of unity. It follows that $\sigma\omega=\omega^i$ for some $i=i(\sigma)$ as $\omega$ is primitive. $i(\sigma)$ is uniquely determined mod $n$. It follows that $\sigma$ maps $k(\omega)$ into itself, and hence $k(\omega)$ is normal over $k$. If $\tau$ is another automorphism of $k(\omega)$ over $k$, then \[ \sigma\tau\omega = \omega^{i(\sigma)i(\tau)}. \] Since $\sigma$ and $\tau$ are isomorphisms, it follows that $i(\sigma)$ and $i(\tau)$ are prime to $n$ (otherwise, $\sigma\omega$ would have a period smaller than $n$). This yields a homomorphism of $\gal(k(\omega)/k)$ and is clearly injective because $i(\sigma)$ is uniquely determined by $\sigma$ mod $n$, and the effect of $\sigma$ on $k(\omega)$ is determined by its effect on $\omega$. \end{proof} For a specific field $k$, the question arises whether the image of $\gal(K(\omega)/K)$ in $(\Z/n\Z)^\ast$ is all of $(\Z/n\Z)^\ast$. Looking at $k=\mathds{R}$ or $k=\mathds{C}$, this is not always the case. We give an example on when it is the case. \begin{theorem} Let $\omega$ be a primitive $n$-th root of unity in $\Q^\mathrm{a}$, then \[ [\Q(\omega):\Q] = \varphi(n) \] where $\varphi$ is the Euler function. \end{theorem} \begin{proof} According to theorem \ref{gal-cyclotomic}, $[\Q(\omega):\Q] \le \varphi(n)$, hence it suffices to prove the opposite. Let $f(X)$ be the minimal polynomial of $\omega$ over $\Q$. Then $f(X)$ divides $X^n-1$, and we can write $X^n-1=f(X)h(X)$, where both $f$ and $h$ have leading coefficient $1$. By the Gauss lemma, $f$ and $h$ have integral coefficients. We shall now prove that if $p$ is a prime number not dividing $n$, then $\omega^p$ is also a root of $f$. Suppose $\phi^p$ is not a root of $f$, then it is a root of $h$, and $\phi$ is a root of $h(X^p)$. Hence $f$ divides $h(X^p)$ as well, so we write \[ h(X^p)=f(X)g(X). \] Note $g$ has integral coefficients as well. Since $a^p \equiv a \mod p$ for any integer $a$, we have \[ h(X^p) \equiv h(X)^p \mod p, \] and therefore \[ h(X)^p \equiv f(X)g(X) \mod p. \] Let $\overline{f}$ and $\overline{h}$ be the canonical image of $f$ and $h$ in $\Z/p\Z[X]$, we see that $f$ and $h$ are not relatively prime, hence have common factor(s). But if follows that \[ X^n-\overline{1}=\overline{f}(X)\overline{g}(X) \] has multiple rules, which contradicts the remark we have made at the beginning. Since $\omega^p$ is also a primitive $n$-th root of unity, and any primitive $n$-th root of unity can be obtained by raising $\omega$ to a succession of prime powers with primes not dividing $n$, this implies that all primitive $n$-th roots of unity are roots of $f$, which forces the degree of $f$ to be not less than $\varphi(n).$ \end{proof} It follows immediately that \begin{corollary} We have an isomorphism \[ \gal(\Q(\omega):\Q) \cong (\Z/n\Z)^\ast. \] \end{corollary} \begin{corollary} If $n,m$ are relative prime integers $\ge 1$, then \[ \Q(\omega_n) \cap \Q(\omega_m) = \Q. \] \end{corollary} We will prove that the cyclotomic integers (i.e. the algebraic integer with respect to $\Q(\omega)/\Q$) is actually $\Z[\omega]$. This is a highly non-trivial result and we can only deliver the proof after further study on prime ideals. % TODO: Finish this section and the study of Z[\omega]. Source can be found on Daniel A. Marcus. \subsection{Algebraic extension and integral closure} First of all we show that being algebraic almost implies being integral. \begin{lemma}\label{alg-int} Let $A$ be a domain, $K$ its quotient field, and $x$ algebraic over $K$. Then there exists an element $c \ne 0$ of $A$ such that $cx$ is integral over $A$. \end{lemma} \begin{proof} Since $x$ is algebraic, we have an equation \[ a_nx^n+\cdots+a_0=0 \] with $a_i \in A$ and $a_n \ne 0$. Hence \[ a_n^{n-1}(a_nx^n+\cdots+a_0)=(a_nx)^n+\cdots+a_0a_n^{n-1}=0 \] which is to say $a_nx$ is integral over $A$. \end{proof} Since finite extensions are algebraic, we are always free to use this lemma for the topic of number field. \begin{theorem}\label{closure-f-g} Let $A$ be an integrally closed Noetherian ring. Let $L$ be a finite separable extension of its quotient field $K$. Then the integral closure of $A$ in $L$ is finitely generated over $A$. \end{theorem} By being integrally closed we mean the ring is integrally closed in its quotient field. Some mathematicians also say it is being normal, but I think \textit{normal} does not carry a lot of information. \begin{proof} Since $A$ is Noetherian, all submodules of a finitely generated module over $A$ is finitely generated. Hence it suffices to prove that the integral closure of $A$ is contained in a finitely generated $A$-module.\\ Let $w_1,\dots,w_n$ be a basis of $L$ over $K$. After multiplying each $w_i$ by some suitable element of $A$ (see lemma \ref{alg-int}), we may assume without loss of generality that the $w_i$ are integral over $A$. To study the integral closure of $A$ in $L$, we pick an arbitrary element $z = b_1w_1+\cdots+b_nw_n$ and study its coefficients. \\ Since $L/K$ is separable, the \href{https://stacks.math.columbia.edu/tag/0BIF}{field trace} form \[ Q_{L/K}:L \times L \to K, \quad (x,y) \mapsto \tr_{L/K}(xy) \] is non-degenerate \href{https://stacks.math.columbia.edu/tag/0BIL}{[proof]}, so we claim that $L^\ast$ is isomorphic to $L$ under $Q_{L/K}$. Indeed, one can define a $K$-linear map \[ d:L \to L^\ast, \quad x \mapsto (y \mapsto Q_{L/K}(x,y)=\tr(xy)). \] This map is injective because $Q_{L/K}$ is non-degenerate. Since $L$ and $L^\ast$ has the same dimension, $d$ has to be surjective.\\ Let $w^1,\dots,w^n$ be the dual basis of $w_1,\dots,w_n$. If we put $v_i=d^{-1}(w^i)$, we have \[ \tr(v_i w_j) = \delta_{ij}. \] Let $c \ne 0$ be an element of $A$ such that $cv_i$ is integral over $A$, then $cv_iz$ is integral and so is $\tr(cv_iz)$. Since $\tr$ is a $K$-valued function, we have \[ \tr(czv_i)=c\tr(v_iz)=c d(v_i)(z) = cb_i \in A \implies b_i \in Ac^{-1}. \] Hence \[ z \in Ac^{-1}w_1+\cdots+Ac^{-1}w_n \] which is to say $z$ is finitely generated. Since $z$ is arbitrarily picked, the closure itself is contained in a finitely generated $A$-module, which finishes the proof. \end{proof} Note $Z$ is itself a Noetherian ring and integrally closed. $\Q$ is the fraction ring of $\Z$, and finite extensions of $\Q$ are always separable. It follows (non-trivially) that \begin{corollary} $\OK$ is finitely generated over $\Z$. \end{corollary} Next we study the rank of $\OK$ over $\Z$. Being finitely generated is not exactly what we want. \begin{theorem} Let $A$ be a principal ideal ring, and $L$ a finite separable extension of its quotient field $K$, of degree $n$. Let $B$ be the integral closure of $A$ in $L$. Then $B$ is a free module of rank $n$ over $A$. \end{theorem} \begin{proof} Since $A$ is contained in $K$, $B$ is contained in $L$, whenever $ab=0$ with $a \in A$, $b \in B$, we have $a=0$ or $b=0$. Hence $B$ is torsion-free. Therefore as a finitely generated (theorem \ref{closure-f-g}) torsion-free module, $B$ is a free module over $A$ \href{http://du.ac.in/du/uploads/departments/mathematics/study-material/MMATH18-201\%20_MT_PID.pdf}{[Theorem 2.7]}. Since $L$ is a $n$-dimensional vector space over $K$, for $y \in L$ we have \[ y = c_1e_1+\cdots+c_ne_n \] where $e_1,\dots,e_n$ is a basis and $c_1,\cdots,c_n \in K$. When $y \in B$, we must have $c_1,\cdots,c_n \in A$, which is to say $B$ has rank $[L:K]=n$. \end{proof} Hence the rank of $\OK$ over $\Z$ is determined by $[K:\Q]$. \subsection{Localisation} \begin{theorem}\label{int-clo-loc} Let $A \subset B$ be rings, and $S$ a multiplicatively closed subset of $A$. If $B$ is integral over $A$, then $S^{-1}B$ is integrally closed in $S^{-1}A$. If $C$ is the integral closure of $A$ in $B$, then $S^{-1}C$ is the integral closure of $S^{-1}A$. \end{theorem} \begin{proof} First we assume $B$ is integral over $A$. Pick $x/s \in S^{-1}B$ with $x \in B$ and $s \in S$. By definition we have \[ x^n + a_1x^{n-1}+\cdots + a_n = 0 \] with $a_i \in A$. Multiplying by $(1/s)^n$ gives \[ (x/s)^n + (a_1/s)(x/s)^{n-1}+\cdots+a_n/s^n = 0 \] which shows that $x/s$ is integral over $S^{-1}A$. Hence the first statement is proved. \\ Now we assume $C$ is the integral closure of $A$ in $B$. By the first statement we see $S^{-1}C$ is integral over $S^{-1}A$. Suppose $b/s \in S^{-1}B$ is integral over $S^{-1}A$, we have an equation of the form \[ (b/s)^n+(a_1/s_1)(b/s)^{n-1}+\cdots+a_n/s_n=0. \] Multiplying by $(st)^n$ where $t=s_1\cdots s_n$ gives an equation of integral independence for $bt$ over $A$. Hence $bt \in C$. But $b/s = bt/st$, hence $b/s \in S^{-1}C$ and we are done. \end{proof} If $S$ happens to be a complement of a prime ideal $\mfk{p}$, we have a satisfying result \begin{corollary}\label{int-loc} If $B$ is integral over $A$, then $B_\mfk{p}$ is integral over $A_\mfk{p}$. \end{corollary} If $B$ is replaced by a field extension $L$ of the quotient field of $A$, and $C$ is replaced by the integral closure of $A$, we have the following corollary: \begin{corollary} If $B$ is the integral closure of $A$ in some field extension $L$ of the quotient field of $A$, then $S^{-1}B$ is the integral closure of $S^{-1}A$ in $L$. \end{corollary} Being integrally closed is also a local property. And we will use it to prove that the algebraic integers in the field $\Q(\omega)$ is $\Z[\omega]$. \begin{corollary} Let $A$ be an integral domain. Then the following are equivalent: \begin{enumerate} \item $A$ is integrally closed. \item $A_\mfk{p}$ is integrally closed for each prime ideal $\mfk{p}$. \item $A_\mfk{m}$ is integrally closed for each maximal ideal $\mfk{m}$. \end{enumerate} \end{corollary} \begin{proof} Let $K$ be the quotient field of $A$, let $C$ be the integral closure of $A$ in $K$, and let $f:A \to C$ be the canonical embedding. Then $A$ is integrally closed if and only if $f$ is surjective. On the other hand, by theorem \ref{int-clo-loc}, $A_\mfk{p}$ (respectively $A_\mfk{m}$) is integrally closed if and only if $f_\mfk{p}$ (respectively $f_\mfk{m}$) is surjective. However, a $A$-module homomorphism $\phi:M \to N$ being surjective is a local property \href{https://www.maths.usyd.edu.au/u/de/AGR/CommutativeAlgebra/pp600-610.pdf}{[proof]}. Therefore we have \[ \begin{aligned} A \text{ is integrally closed } &\iff f \text{ is surjective } \\ &\iff f_\mfk{p} \text{ is surjective } \iff A_\mfk{p} \text{ is integrally closed } \\ &\iff f_\mfk{m} \text{ is integrally closed } \iff A_\mfk{m} \text{ is integrally closed.} \end{aligned} \] \end{proof} % TODO: ADD SOME EXAMPLES \subsection{Prime Ideals} By theorem \ref{closure-f-g}, $\OK$ is a finitely-generated $\Z$-module, hence is a Noetherian domain. By transitivity of integral closures, $\OK$ is integrally closed. We are now interested in the Krull dimension of $\OK$. To do this, we investigate more of the prime ideal with respect to integral closure. \begin{definition} Let $B$ be a ring containing a ring $A$. Let $\mfk{p}$ be a prime ideal of $A$ and $\mfk{P}$ a prime ideal of $B$. We say that $\mfk{P}$ \textbf{lies above} $\mfk{p}$ if $\mfk{P} \cap A = \mfk{p}$ and we then write $\mfk{P}\|\mfk{p}$. \end{definition} If $\mfk{P}\|\mfk{p}$, we have a commutative diagram: \[ \begin{tikzcd} B \arrow[r, "\pi'"] & B/\mathfrak{P} \\ A \arrow[r, "\pi"] \arrow[u, "i"] & A/\mathfrak{p} \arrow[u, "i'"'] \end{tikzcd} \] where $i$ and $i'$ are inclusions, $\pi$ and $\pi'$ are canonical homomorphisms. \\ If $B$ is integral over $A$, then $B/\mfk{P}$ is integral over $A/\mfk{p}$, this is because of the following lemma if we take $\sigma$ to be $\pi$: \begin{lemma} Let $A \subset B$ be rings, and $\sigma:B \to C$ be a homomorphism. If $B$ is integral over $A$, then $\sigma(B)$ is integral over $\sigma(A)$. \end{lemma} \begin{proof} If $B$ is integral over $A$, then for any $x \in B$ there is an equation \[ x^n + a_{n-1}x^{n-1}+\cdots+a_0 = 0. \] Therefore \[ \begin{aligned} \sigma(x^n+a_{n-1}x^{n-1}+\cdots+a_0) &= \sigma(x^n)+\sigma(a_{n-1}x^{n-1})+\cdots+\sigma(a_0) \\ &= \sigma(x)^n + \sigma(a_{n-1})\sigma(x)^{n-1}+\cdots+\sigma(\sigma) \\ &= 0. \end{aligned} \] Hence $\sigma(x)$ is integral in $\sigma(A)$. \end{proof} We want to show that prime ideals of $\OK$ is maximal, and they should be corresponded to prime ideals in $\Z$, which is maximal. For this reason we show the existence of lying-above prime ideals. \begin{theorem}\label{lying-above} Let $A$ be a ring, $\mfk{p}$ a prime ideal, and $B \supset A$ integral over $A$. Then $\mfk{p}B \ne B$, and there exists a prime ideal $\mfk{P}$ of $B$ lying above $\mfk{p}$. \end{theorem} \begin{proof} We know that $B_\mfk{p}$ is integral over $A_\mfk{p}$ (corollary \ref{int-loc}) and that $A_\mfk{p}$ is local with maximal ideal $\mfk{m}_\mfk{p}=\mfk{p}A_\mfk{p}$. It follows that \[ \mfk{p}B_\mfk{p}=\mfk{p}A_\mfk{p}B = \mfk{p}A_\mfk{p}B_\mfk{p}=\mfk{m}_\mfk{p}B_\mfk{p}. \] Hence it suffices to prove our assertion when $A$ is local. If $\mfk{p}B=B$, we have an equation \[ 1 = a_1b_1+\cdots+a_nb_n \] with $a_i \in \mfk{p}$ and $b_i \in B$. Let $B_0 = A[b_1,\cdots,b_n]$. Then $\mfk{p}B_0=B_0$ and $B_0$ is a finitely generated $A$-module. Hence by Nakayama's lemma, $B_0=0$, which is absurd. To prove the existence of $\mfk{P}$, consider the following commutative diagram: \[ \begin{tikzcd} B \arrow[r] & B_\mathfrak{p} \\ A \arrow[r] \arrow[u] & A_\mathfrak{p} \arrow[u] \end{tikzcd} \] where all arrows are natural inclusions. As is proved, $\mfk{m}_\mfk{p}B_\mfk{p} \ne B_\mfk{p}$. Hence $\mfk{m}_\mfk{p}B_\mfk{p}$ is contained in a maximal ideal $\mfk{M}$ of $\mfk{p}$, and therefore $\mfk{M} \cap A_\mfk{p}$ contains $\mfk{m}_\mfk{p}$. And we pick $\mfk{P}=\mfk{M} \cap B$. Then $\mfk{P}$ is a prime ideal of $B$, and taking intersection with $A$ going both ways around our diagram shows that $\mfk{M} \cap A = \mfk{p}$, so that \[ \mfk{P} \cap A = \mfk{p}, \] as was to be shown. \end{proof} Now we proceed to the crucial theorem to determine whether a prime lying above is maximal. \begin{theorem}\label{lie-above-maximal} Let $A$ be a subring of $B$, and assume $B$ is integral over $A$. Let $\mfk{P}$ be a prime ideal of $B$ lying over a prime ideal $\mfk{p}$ of $A$. Then $\mfk{P}$ is maximal $\iff$ $\mfk{p}$ is maximal. \end{theorem} \begin{proof} $\implies$: Note $B/\mfk{P}$ is a field and is integral over the ring $A/\mfk{p}$. Were $A/\mfk{p}$ not a field, there would be a non-trivial ideal $\mfk{m}$ of it, and $B/\mfk{P}$ would have a prime ideal $\mfk{M}$ lying above $\mfk{m}$, by theorem \ref{lying-above}. A contradiction. \\ $\impliedby$: Note $A/\mfk{p}$ is a field. It suffices to prove that a ring $R$ which is integral over a field $k$ is a field. If $k$ is a field and non-zero $x \in R$ is integral over $k$, we have a minimal polynomial \[ x^n+c_{n-1}y^{n-1}+\cdots+c_0=0 \] with $c_i \in k$. Since $R$ is integral, we have $c_0 \ne 0$. We can clearly write \[ x^{-1}=-c_0^{-1}(x^{n-1}+c_{n-1}y_{n-2}+\cdots+c_1) \in R, \] which is to say $R$ is integral, and the theorem is therefore proved. \end{proof} By using local properties, we can show the stability of prime ideals lying above: \begin{corollary} Let $A \subset B$ be rings, $B$ integral over $A$; Let $\mfk{P}$ and $\mfk{P}'$ be prime ideals of $B$ such that $\mfk{P} \subset \mfk{P}'$ and both $\mfk{P}$ and $\mfk{P}'$ lie above a prime ideal $\mfk{p}$ of $A$, then $\mfk{P}=\mfk{P}'$. \end{corollary} \begin{proof} By corollary \ref{int-loc}, $B_\mfk{p}$ is integral over $A_\mfk{p}$. Let $\mfk{m}$ be the extension of $\mfk{p}$ in $A_\mfk{p}$ and $\mfk{M},\mfk{M}'$ be the extensions of $\mfk{P}$ and $\mfk{P}'$ respectively in $B_\mfk{p}$. Then $\mfk{m}$ is the maximal ideal of $A_\mfk{p}$; $\mfk{M} \subset \mfk{M}'$, and $\mfk{M}$, $\mfk{M}'$ lies above $\mfk{m}$. Hence by theorem \ref{lie-above-maximal}, $\mfk{M}$ and $\mfk{M}'$ are both maximal, hence equal. This reduces to $\mfk{P}=\mfk{P}'$. \end{proof} If the context is exactly localisation, we have a finer result: \begin{theorem}\label{loc-corr} Let $A$ be a commutative ring, $S \subset A$ be a multiplicatively closed set. We have a $1-1$-correspondence of prime ideals $\mfk{p}$ do not intersect $S$ and prime ideals of $S^{-1}A$: \[ \mfk{p} \mapsto S^{-1}\mfk{p}, \quad \mfk{P} \mapsto \mfk{P} \cap A. \] \end{theorem} \begin{proof} To prove that $S^{-1}\mfk{p}=\mfk{P}$ is prime, suppose that $(a/s)(b/t)=ab/st = p/u \in \mfk{P}$. By definition there exists some $r \in S$ such that $r(abu-stp)=0$. It follows that $rabu=rstp \in \mfk{p}$. But $r,u$ are not in $\mfk{p}$, hence we can only have $ab \in \mfk{p}$, which gives that $a \in \mfk{p}$ or $b \in \mfk{p}$, and as a result $a/s \in \mfk{P}$ or $b/t \in \mfk{P}$. Besides we have $\mfk{p} = \mfk{P} \cap A$. $\subset$ inclusion is clear, but for the converse inclusion, note that if $p/s=a/1 \in \mfk{P} \cap A$, we have $up=uas$ for some $u \in S$, which forces $a$ to be an element of $\mfk{p}$. (Note we implicitly used the fact that $\mfk{p}$ does not intersect $S$.) For the converse, first note that $\mfk{p}=\mfk{P} \cap A$ is indeed a prime ideal. For any $(a/1)(b/1) \in \mfk{p} \subset \mfk{P}$, we must have $a/1 \in \mfk{P}$ or $b/1 \in \mfk{P}$, but both $a/1$ and $b/1$ are in $A$. Also, $\mfk{p}$ does not intersect $S$ because if $s \in S \cap \mfk{p}$, then $(1/s)s=1\in \mfk{p}$, which is absurd. It remains to show that $\mfk{P} = S^{-1}\mfk{p}$. It is clear that $S^{-1}\mfk{p} \subset \mfk{P}$. For the converse, pick any $a/s \in \mfk{P}$, we have $a/1=(a/s)(s/1) \in \mfk{p}$, and therefore $a/s=a(1/s) \in S^{-1}\mfk{p}$. \end{proof} In particular, we can also study the localised ring. \begin{corollary} Let $\mfk{m}_\mfk{p}=\mfk{p}A_\mfk{p}$ be the maximal ideal of $A_\mfk{p}$. We have a canonical embedding \[ A/\mfk{p} \hookrightarrow A_\mfk{p}/\mfk{m}_\mfk{p} \cong K(A/\mfk{p}) \] where $K(B)$ is the field of fractions of $B$. In particular, if $\mfk{p}$ is maximal, one has an isomorphism for all $n \ge 1$: \[ A/\mfk{p}^n \cong A_\mfk{p}/\mfk{m}_\mfk{p}^n. \] \end{corollary} \begin{proof} Since $\mfk{p} = \mfk{m}_\mfk{p} \cap A$, the canonical map \[ a \mod \mfk{p} \mapsto a/1 \mod \mfk{m}_\mfk{p} \] is injective. $A_\mfk{p}/\mfk{m}_\mfk{p}$ can be identified as the field of fraction of $A/\mfk{p}$ because given any non-zero $a/1 \mod \mfk{m}_\mfk{p}$, we see $a \in A \setminus \mfk{p}$, hence $1/a \mod \mfk{m}_\mfk{p}$ is exactly the inverse of $a/1 \mod \mfk{m}_\mfk{p}$. Next we assume that $\mfk{p}$ is maximal and consider the canonical map \[ \varphi:a \mod \mfk{p}^n \mapsto a/1 \mod \mfk{m}_\mfk{p}^n. \] If $n=1$, then both $A/\mfk{p}$ and $A_\mfk{p}/\mfk{m}_\mfk{p}$ are fields, and the field of fractions of a field is itself. To prove it for all $n \ge 0$, we need the fact that $s \mod \mfk{p}^n$ is a unit whenever $n \ge 0$ and $s \in A \setminus \mfk{p}$, which will be shown later as a lemma. $\varphi$ is injective because if $a/1 \in \mfk{m}_\mfk{p}^n$, i.e. $a/1 = b/s$ where $b \in \mfk{p}^n$ and $s \not \in \mfk{p}$, then there exits some $u \not\in \mfk{p}$ such that $uas = ub \in \mfk{p}^n$. Since both $ u \mod \mfk{p}^n$ and $s \mod \mfk{p}^n$ are units, we see $a \mod \mfk{p}^n$ is zero in $A/\mfk{p}^n$. $\varphi$ is surjective because given $a/s \in A_\mfk{p}$ with $a \in A$ and $s \not\in \mfk{p}$, there exists some $a' \in A$ such that $a \equiv a's \mod \mfk{p}^n$. Indeed, let $t \mod \mfk{p}^n$ be the inverse of $s \mod \mfk{p}^n$, we can put $a' = at$. Therefore $a/s \equiv a' \mod \mfk{p}^nA_\mfk{p}$ (by theorem \ref{loc-corr}), which is equivalent to say that $a/s \mod \mfk{m}_\mfk{p}^n$ lies in the image of $\varphi$. \end{proof} \begin{lemma} Let $A$ be a ring and $\mfk{p}$ is a maximal ideal. Then $\mfk{p}^n+sA=A$ for all $n \ge 1$ and $s \in A \setminus \mfk{p}$. Besides, this implies that $s \mod \mfk{p}^n$ is a unit in $A/\mfk{p}^n$. \end{lemma} \begin{proof} When $n=1$, note $\mfk{p} \subsetneqq \mfk{p}+sA$, while $\mfk{p}$ is maximal, we must have $\mfk{p}+sA=A$. When $n \ge 2$, suppose it holds for $n-1$, then we have \[ A = \mfk{p}^{n-1}+sA \implies \mfk{p}=\mfk{p}A = \mfk{p}(\mfk{p}^{n-1}+sA) \subsetneqq \mfk{p}^n+sA, \] which forces $\mfk{p}^n+sA$ to be $A$ itself. Now we consider $A$-modules \[ \mfk{p}^n+sA = A. \] By taking the quotient, we obtain \[ \overline{s}A/\mfk{p}^n=A/\mfk{p}^n. \] It follows that $s \mod \mfk{p}^n$ is a unit in $A/\mfk{p}^n$. \end{proof} And now we are more than ready to prove that $\OK$ is Dedekind. \begin{theorem}\label{o_k-dedekind} Every prime ideal $\mfk{P}$ in $\OK$ is maximal. Hence $\OK$ is of Krull dimension $1$ and is therefore Dedekind. \end{theorem} \begin{proof} Note it suffices to prove that every prime ideal $\mfk{P}$ of $\OK$ lies above some prime ideal of $\Z$, since $\Z$ has Krull dimension $1$, and the proof follows from theorem \ref{lie-above-maximal}. All we need to do is to prove that $\mfk{P} \cap \Z$ is non-zero: since the inverse image of a prime ideal is prime, we are done. For each $x \in \mfk{P}$, we have a minimal polynomial $f \in \Z[X]$ such that \[ f(x) = x^n+c_{n-1}x^{n-1}+\cdots+c_0=0 \] with $c_i \in \Z$ and $c_0 \ne 0$. It follows that \[ c_0=-(x^n+c_{n-1}x^{n-1}+\cdots+c_1x) \in \mfk{P} \cap \Z, \] which is to say $\mfk{P} \cap \Z$ is indeed non-zero. This concludes the proof. \end{proof} \begin{example} As a classic example, consider $K=\Q(\sqrt{-5})$ and $\OK = \Z[\sqrt{-5}]$. This ring is not a unique factorial domain because we have \[ 6 = 2 \cdot 3 = (1-\sqrt{-5}) \cdot (1+\sqrt{-5}). \] But if we view it in the sense of product of ideals, we still have some uniqueness. Let $\mfk{m}$ be the maximal ideal containing $6$, then \[ \mfk{m} = (2,1-\sqrt{-5})(2,1+\sqrt{-5}) \] is unique. Note two ideals on the right hand side are indeed maximal (hence prime) because \[ \begin{aligned} \frac{\Z[\sqrt{-5}]}{(2,1-\sqrt{-5})} &\cong \frac{\Z[X]/(X^2+5)}{(2,1-X,X^2+5)/(X^2+5)} \\ &\cong \frac{\Z[X]}{(2,1-X,X^2+5)} \cong \frac{\Z_2[X]}{(1-X,X^2-1)} \cong \frac{\Z_2[X]}{(1-X)} \cong \Z_2. \end{aligned} \] Likewise, \[ \begin{aligned} \frac{\Z[\sqrt{-5}]}{(3,1+\sqrt{-5})} &\cong \frac{\Z[X]/(X^2+5)}{(3,1+X,X^2+5)/(X^2+5)} \\ &\cong \frac{\Z[X]}{(3,1+X,X^2+5)} \cong \frac{\Z_3[X]}{(1+X,X^2-1)} \cong \frac{\Z_3[X]}{(1+X)} \cong \Z_3. \end{aligned} \] \end{example} We will see when a Dedekind domain is UFD in the future. \section{Galois extensions} \subsection{Special subgroups of the Galois group} In this subsection we study the behaviour of Galois group with respect to integral closure, which can of course help us study number field and algebraic integers if we are interested in the extension itself. If $K$ is a Galois extension of $\Q$, then the Galois group allows us to transform amongst prime ideals in a natural way. This is because of the following theorem. \[ \begin{tikzcd} \mathfrak{P} \arrow[rr, "\exists \sigma \in G"] & & \mathfrak{Q} \\ & \mathfrak{p} \arrow[lu] \arrow[ru] & \end{tikzcd} \] \begin{theorem}\label{galois-lie-above} Let $A$ be a ring, integrally closed in its quotient field $K$. Let $L$ be a finite Galois extension of $K$ with group $G$. Let $\mfk{p}$ be a maximal ideal of $A$, and let $\mfk{P}$, $\mfk{Q}$ be prime ideals of the integral closure of $A$ in $L$ lying above $\mfk{p}$. Then there exists $\sigma \in G$ such that $\sigma\mfk{P} = \mfk{Q}$. \end{theorem} \begin{proof} Suppose that $\mfk{P}=\sigma\mfk{Q}$ for all $\sigma \in G$. By the Chinese remainder theorem, we have some $x \in B$ such that \[ \begin{aligned} x &\equiv 0 \mod \mfk{P} \\ x &\equiv 1 \mod \sigma\mfk{Q}, \quad \forall \sigma \in G. \end{aligned} \] Then the norm \[ N_K^L(x) = \prod_{\sigma \in G}\sigma{x} \] lies in $B \cap K = A$ since $A$ is integrally closed, and lies in $\mfk{P} \cap A = \mfk{p}=\mfk{Q} \cap A \subset \mfk{Q}$. But we also have $\sigma{x} \not\in \mfk{Q}$ for all $\sigma \in G$, hence $N_K^L(x) \not \in \mfk{Q}$, a contradiction. \end{proof} If one localise, the consideration on whether a prime ideal is maximal is not required. Besides, if $A$ is of Krull dimension $1$, then one has no need to consider as well. Since we have shown that $\OK$ is a Dedekind domain, this theorem can be applied as well. Next we show the finiteness of prime ideals lying above. \begin{corollary} Let $A$ be an integrally closed domain whose field of fraction is $K$. Let $E$ be a finite separable extension of $K$, and $B$ the integral closure of $A$ in $E$. Let $\mfk{p}$ be a maximal ideal of $A$. Then there exists only a finite number of prime ideals of $B$ lying above $\mfk{p}$. \end{corollary} \begin{proof} If $E$ is Galois over $K$, then by theorem \ref{galois-lie-above}, $\sigma\mfk{P}_1 = \mfk{P}_2$ for some $\sigma \in \gal(E/K)$. Suppose $\mfk{P}_1\|\mfk{p}$, then the set of prime ideals lying above $\mfk{p}$ is contained in the set \[ \{\mfk{Q} \subset B: \mfk{Q}=\sigma\mfk{P}_1,\sigma\in\gal(E/K)\}, \] hence is finite because $\gal(E/K)$ is finite. If $E$ is not necessarily Galois, we can pick the smallest Galois extension $L/K$ containing $E$, which is a finite extension as well. Let $C$ be the integral closure of $A$ in $L$. Suppose $\mfk{P},\mfk{Q} \in \spec(B)$ are two distinct prime ideals lying above $\mfk{p}$, and $\mfk{P}',\mfk{Q}' \in \spec(C)$ lying above $\mfk{P}$ and $\mfk{Q}$ respectively. Note $\mfk{P}' \ne \mfk{Q}'$ because if not then $\mfk{P}=\mfk{Q}$, a contradiction. Therefore the distinct prime ideals of $B$ lying above $\mfk{p}$ are less than the distinct prime ideals of $C$ lying above $\mfk{p}$, which proves our assertion. \[ \begin{tikzcd} \mathfrak{P}' & & \mathfrak{Q}' & C \\ \mathfrak{P} \arrow[u] & & \mathfrak{Q} \arrow[u] & B \arrow[u, dashed] \\ & \mathfrak{p} \arrow[lu] \arrow[ru] & & A \arrow[u, dashed] \end{tikzcd} \] \end{proof} \begin{example}\label{gauss-int-1} Now take $K=\Q(i)$, $\OK=\Z[i]$ and $\mfk{p}=(5)$. Since $\gal(K/\Q)=\{1,\sigma\}$ where $\sigma$ is the complex conjugation, there are at most two prime ideals lying above $\mfk{p}$. First of all $5\Z[i]$ is not prime because \[ 5+0i = (2-i)(2+i). \] Since $\Z[i]$ is a PID, we only need to consider Gaussian integers that divide $5$. Keeping in mind that $\Z[i]$ is also an Euclidean domain, we have two non-trivial solutions up to units in $\Z[i]$: \[ 5 = (2-i)(2+i). \] As ideals, we have $(2-i)=(1+2i)=(-2+i)$. Since $2-i$ is irreducible in $\Z[i]$, we have found a prime ideal $\mfk{P}=(2-i)$ lies above $\mfk{p}$, and $\mfk{P}'=\sigma\mfk{P}=(2+i)$ has to be the remaining one. Note we have also established the factorisation of $5\Z[i]$ in the Dedekind domain $\OK$. \end{example} This example also shows that the norm defined in theorem \ref{galois-lie-above} makes sense, because we have \[ N_{\Q}^{K}(a+bi)=\prod_{\sigma \in \gal(K/\Q)}\sigma(a+bi) = (a+bi)(a-bi)=a^2+b^2. \] We will study norm extensively in the future. %TODO: add reference if it is needed. Let $A$ be an integrally closed ring with quotient field $K$, and $B$ its integral closure in a finite Galois extension $L$. Then firstly $\sigma{B}=B$ for all $\sigma \in \gal(L/K)$ (Proof: $\sigma{B} \subset L$ is integral over $\sigma{A}=A$, hence has to be $B$ itself). Automorphisms fixing base field give rise to Galois group, and now we are interested in automorphisms that also fix ideals. \subsubsection{Decomposition groups and fields} \begin{definition} Notations being above, let $\mfk{p}$ be a maximal ideal of $A$, $\mfk{P}$ maximal in $B$ lying above $\mfk{p}$. Then the subgroup \[ G_\mfk{P} = \{\sigma \in \gal(L/K):\sigma\mfk{P}=\mfk{P}\} \] is called the \textbf{decomposition group} of $\mfk{P}$. Its fixed field will be denoted by $L^d$, and will be called the \textbf{decomposition field} of $\mfk{P}$. \end{definition} Two non-trivial examples are given, abelian and non-abelian. We will frequently return to these examples after new concepts are introduced. % TODO: expand details of these two examples. \begin{example} Consider $K = \Q(\sqrt{-1},\sqrt{2},\sqrt{5})$, whose Galois group is isomorphic to $\Z/2\Z \times \Z/2\Z \times \Z/2\Z$. The decomposition field of $(5)$ is \[ K^d = \Q(\sqrt{-1},\sqrt{2}). \] \end{example} \begin{example} Consider $K = \Q(\sqrt[3]{19},\omega)$ where $\omega = e^{2\pi i/3}$. The Galois group is $S_3$. The decomposition fields of $(3)$ are \[ \Q(\sqrt[3]{19}),\quad \Q(\omega\sqrt[3]{19}),\quad\Q(\omega^2\sqrt[3]{19}). \] At this point we can only compute the field manually but after introducing ramification index things are much easier. \end{example} It is natural to think whether a decomposition group is normal, when the Galois group is non-abelian. For this question we have an pretty elegant result. \begin{theorem} Notation still being above, the decomposition group of $\sigma\mfk{P}$ where $\sigma \in \gal(L/K)$ is $\sigma G_\mfk{P}\sigma^{-1}$, i.e. \[ G_{\sigma\mfk{P}} = \sigma G_\mfk{P}\sigma^{-1}. \] \end{theorem} This theorem says, the Galois group acting on itself by conjugation yields all decomposition groups of primes of $B$ lying above a certain prime of $A$. \begin{proof} We can write $\gal(L/K) = \bigcup\sigma_jG_\mfk{P}$ as a coset decomposition. We claim this decomposition determines distinct prime ideals lying above $\mfk{p}$. Note \[ \sigma\mfk{P}=\tau\mfk{P} \iff \tau^{-1}\sigma\mfk{P}=\mfk{P} \iff \tau^{-1}\sigma \in G_\mfk{P} \] which is equivalent to say $\tau$ and $\sigma$ lie in the same coset mod $G_\mfk{P}$. This claim actually proves this theorem. On one hand, pick $\lambda \in G_\mfk{P}$, then \[ \sigma\lambda\mfk{P} = \sigma\lambda\sigma^{-1}\sigma\mfk{P} = \sigma\mfk{P} \] Hence $\sigma G_{\mfk{P}}\sigma^{-1} \subset G_{\sigma\mfk{P}}$. Note we have used the fact that $\sigma\lambda,\sigma \in \sigma G_\mfk{P}$. On the other hand, if $\lambda\sigma\mfk{P} = \sigma\mfk{P}$, we have $\lambda\sigma \in \sigma G_\mfk{P}$, hence $\lambda \in \sigma G_\mfk{P}\sigma^{-1}$, which is equivalent to say $G_{\sigma\mfk{P}} \subset \sigma G_\mfk{P} \sigma^{-1}$. \end{proof} As for the field, we have a pretty concrete result: \begin{theorem} Assume $L/K$ is Galois and finite. The field $L^d$ is the smallest $E$ subfield of $L$ containing $K$ such that $\mfk{P}$ is the only prime of $B$ lying above $\mfk{P} \cap E$. \end{theorem} \begin{proof} Let $E$ be the smallest subfield of $L$ satisfying the property above, and write $\mfk{q} = \mfk{P} \cap E$. By the fundamental theorem of Galois theory ($L/E$ is Galois) and theorem \ref{galois-lie-above}, prime ideals of $B$ lying above $\mfk{q}$ differs by an element of $\gal(L/E)$. But by assumption there is only one such prime $\mfk{P}$, hence $H \subset G_\mfk{P}$ and therefore $E \supset L^d$. On the other hand, let $B^d$ be the integral closure of $A$ in $L^d$ (which is Dedekind as well), and let $\mfk{Q}=\mfk{P} \cap B^d$. By theorem \ref{galois-lie-above}, $\mfk{P}$ is the only prime of $B$ lying above $\mfk{Q}$ in $B^d$. Hence $E \subset L^d$, which proves the theorem. \end{proof} We are not done yet: the result can be even much sharpener: \begin{theorem}\label{residue-iso} Notation being above, the canonical injection $\varphi:A/\mfk{p} \to B^d/\mfk{Q}$ is an isomorphism. \end{theorem} \begin{proof} First of all we clarify what we mean by canonical injection: \[ \varphi:x + \mfk{p} \mapsto x + \mfk{Q}. \] This is indeed injective because if if $\varphi(x+\mfk{p})=\mfk{Q}$, we have $x \in \mfk{Q}$. This gives \[ x \in \mfk{Q} \cap A = (\mfk{P} \cap A) \cap B^d = \mfk{p} \cap B^d = \mfk{p}. \] It remains to prove that $\varphi$ is surjective. Given $x + \mfk{Q} \in B^d/\mfk{Q}$, we need to find an element $z \in A$ such that $\varphi(z) = x + \mfk{Q}$. Pick $\sigma \in \gal(L/K) \setminus G_\mfk{P}$. Let \[ \mfk{Q}_\sigma = \sigma^{-1}\mfk{P} \cap B^d. \] Then $\mfk{Q}_\sigma \ne \mfk{Q}$ (note $\mfk{P}$ is assumed to be maximal hence so are $\mfk{Q}$ and $\mfk{Q}_\sigma$). Let $x$ be an element of $B^d$. By Chinese remainder theorem, there exists an element $y \in B^d$ such that \[ \begin{aligned} y &\equiv x \mod \mfk{Q}, \\ y &\equiv 1 \mod \mfk{Q}_\sigma. \end{aligned} \] Hence in particular, \[ \begin{aligned} y &\equiv x \mod \mfk{P}, \\ y &\equiv 1 \mod \sigma^{-1}\mfk{P}. \end{aligned} \] The second congruence gives \[ \sigma{y} \equiv 1 \mod \mfk{P}. \] Note this holds for all $\sigma \not \in G_\mfk{P}$. For $\lambda \in G_\mfk{P}$, we have $\sigma{y}=y \equiv x \mod \mfk{Q}$. Thus we obtain \[ z=N_K^L(y) \equiv x \mod \mfk{P}. \] First of all $z \in K$ by definition of norm. Since $z$ is integral over $A$, it has to be an element in $A$. We also have % TODO: 'definition of norm' deserves more explanation. \[ z \equiv x \mod \mfk{Q} \] because both $z$ and $x$ lie in $B^d$. Hence we obtain $\varphi(z) = x + \mfk{P}$ as we wanted. \end{proof} % TODO: examples. \subsubsection{Inertia groups and fields} We are concerned about the homomorphism induced by the decomposition group. \begin{theorem} Notation still being above, $\overline{B}=B/\mfk{P}$ is a normal extension of $\overline{A}=A/\mfk{p}$, and the map $\sigma \to \overline{\sigma}$ induces a homomorphism of $G_\mfk{P}$ onto $\gal(\overline{B}/\overline{A})$ \end{theorem} \begin{proof} Any element of $\overline{B}$ can be written as $\overline{x}$ for some $x \in B$. Let $\overline{x}$ generate a separable subextension of $\overline{A}$, and let $f$ be the irreducible polynomial for $x$ over $K$. The coefficient of $f$ lie in $A$ because $x$ is integral over $A$, and all the roots of $f$ are integral over $A$. Thus \[ f(X) = \prod_{i=1}^{m}(X-x_i) \] splits into linear factors in $B$. Since \[ \bar{f}(X)=\prod_{i=1}^{m}(X-\overline{x}_i) \] and all the $\overline{x}_i$ lie in $\overline{B}$, it follows that $\bar{f}$ splits into linear factors in $\overline{B}$. We observe that $f(x)=0 \implies \bar{f}(x)=0$. Hence $\overline{B}$ is normal over $\overline{A}$, and \[ [\overline{A}(\overline{x}):\overline{A}] \le [K(x):K] \le [L:K]. \] This implies that the maximal separable subextension of $\overline{A}$ in $\overline{B}$ is of finite degree over $A$ by the primitive element theorem, and in fact is always bounded by $[L:K]$. It remains to prove that $\sigma \mapsto \overline{\sigma}$ is actually a surjective homomorphism. By theorem \ref{residue-iso}, it suffices to take $K=L^d$ and $\gal(L/K)=G_\mfk{P}$. Take a generator $\overline{x}$ of the maximal subextension of $\overline{B}$ over $\overline{A}$, for some $x \in B$. Let $f$ be the irreducible polynomial of $x$ over $K$. Any automorphism of $\overline{B}$ is determined by its effect on $\overline{x}$, and maps $\overline{x}$ to some root of $\bar{f}$. Suppose that $x=x_1$. Given any root $x_i$ of $f$, there exists an element $\sigma$ of $G_\mfk{P}$ such that $\sigma x = x_i$. Hence $\overline{\sigma}\overline{x}=\overline{x}_i$. Hence the automorphism of $\overline{B}$ over $\overline{A}$ is induced by elements of $G$ operate transitively on the root of $\bar{f}$. Hence they give us all automorphisms of the residue class field, and the proof is complete. \end{proof} This enables us to work on algebraic closure with some ease: \begin{corollary} Let $\phi:A \to A/\mfk{p}$ be the canonical homomorphism, and let $\psi_1$, $\psi_2$ be two homomorphisms of $B$ extending $\varphi$ in a given algebraic closure of $A/\mfk{p}$. Then there exists an automorphism $\sigma$ of $L$ over $K$ such that \[ \psi_1 = \psi_2 \circ \sigma \] \end{corollary} \begin{proof} The kernels of $\psi_1$ and $\psi_2$ are prime ideals of $B$ and according to theorem \ref{galois-lie-above} they differ by an automorphism. Hence there exists $\tau \in \gal(L/K)$ such that $\psi_1$ and $\psi_2$ have the same kernel $\mfk{P}$. Hence there exists an automorphism $\omega$ of $\psi_1(B)$ onto $\psi_2(B)$ such that $\omega \circ \psi_1 = \psi_2$. There exists an element $\sigma$ of $G_\mfk{P}$ such that $\omega \circ \psi_1 = \psi_1 \circ \sigma$, which proves what we want. \end{proof} \begin{definition} Let $\overline{G}_\mfk{P}$ be the automorphism group of $B/\mfk{P}$. The kernel of the map \[ G_\mfk{P} \to \overline{G}_\mfk{P} \] is called the inertia group. That is, these elements induce the trivial automorphism in $B/\mfk{P}$. \end{definition} \subsection{Automorphisms} \section{Dedekind domain} \subsection{Operations of ideals} % TODO: Daniel Chapter 3. \subsection{Ramification index} \subsection{The norm and trace} We used the concept of norm and trace at the very beginning. Here is a good chance to study them extensively. Recall for $\mathds{C}$ and $\mathds{R}$ we have a natural faithful representation \[ \begin{aligned} \rho:\mathds{C} &\to End(\mathds{R}^2) \\ a+bi &\mapsto \begin{pmatrix} a & -b \\ b & a \end{pmatrix}. \end{aligned} \] Note $\|\rho(a+bi)\|=a^2+b^2$, which is pretty close to the norm in analysis. It is not a good idea to take square root in the sense of algebra so we directly use the determinant. We will also be using trace of such matrices. \begin{definition} Let $E$ be a finite extension of $k$, which we view as a finite dimensional vector space over $k$. Each $\alpha \in E$ induces a linear map by multiplication: \[ \begin{aligned} m_\alpha : E &\to E \\ x &\mapsto \alpha{x}. \end{aligned} \] We define the norm and trace from $E$ to $k$ by \[ \det(m_\alpha)=N_k^E(\alpha), \quad \tr(m_\alpha)=\tr_k^E(\alpha). \] \end{definition} Note this has nothing to do with norm in the sense of topology. The reason is, the norm lies in $k$, which is an arbitrary field. And a field alone has nothing to do with real numbers. Besides, consider the group field $\Q(\sqrt{2})/\Q$. In this field, the we have the matrix representation of $1+\sqrt{2}$ to be \[ 1+\sqrt{2} \mapsto \begin{pmatrix} 1 & 2 \\ 1 & 1 \end{pmatrix} \] whose determinant is $-1$. It makes little sense to compare this to norm in topology. % TODO: formula of norm and trace. % TODO: further properties. \subsection{Discrete valuation rings} A \textbf{discrete valuation ring} can be considered as a localisation of Dedekind domain. Indeed, if $A$ is a discrete valuation ring, then $A$ is Noetherian and of Krull dimension $1$, and is integrally closed, hence Dedekind. If $A$ is local and Dedekind, then $A$ is a discrete valuation ring. In general, a Noetherian domain $A$ of Krull dimension one is Dedekind if and only if the localisation $A_\mfk{p}$ is a discrete valuation ring for all prime $\mfk{p}$. With respect to localisation we have a natural result: \begin{theorem} Let $A$ be a Dedekind ring and $M,N$ two modules over $A$. If $M_\mfk{p} \subset N_\mfk{p}$ for all prime $\mfk{p}$, then $M \subset N$. \end{theorem} \begin{proof} Let $a \in M$. For each $\mfk{p}$ we can find $x_\mfk{p} \in N$ and $s_\mfk{p} \in A \setminus \mfk{p}$ such that $a = x_\mfk{p}/s_\mfk{p}$. Let $\mfk{b}$ be the ideal generated by the $s_\mfk{p}$, ranging through all $\mfk{p} \in \spec(A)$. Then $\mfk{b}$ is the unit ideal $A$, and we can write \[ 1 = \sum_{\mfk{p} \in \spec(A)} y_\mfk{p}s_\mfk{p} \] with elements $y_\mfk{p} \in A$ all but a finite number of which are $0$. This yields \[ a = \sum_{\mfk{p} \in \spec(A)} y_\mfk{p}s_\mfk{p}a = \sum_{\mfk{p} \in \spec(A)} y_\mfk{p}x_\mfk{p} \in N \] as desired. \end{proof} Now we study torsion-free modules over a discrete valuation ring. If $A$ is a discrete valuation ring, then in particular, $A$ is a principal ideal ring, and any finitely generated torsion-free module $M$ over $A$ is free. If its rank is $n$, and if $\mfk{p}$ is the maximal idea, then $M/\mfk{p}M$ is a free module of rank $n$. Further, we have \begin{theorem} Let $A$ be a local ring and $M$ a free module of rank $n$ over $A$. Let $\mfk{p}$ be the maximal ideal of $A$. Then $M/\mfk{p}M$ is a vector space of dimension $n$ over $A/\mfk{p}$. \end{theorem} \begin{proof} Let $\{x_1,\dots,x_n\}$ be a basis of $M$ over $A$, then \[ M \cong \bigoplus_{i}Ax_i \] and \[ M/\mfk{p}M \cong \bigoplus_{i}(A/\mfk{p})\overline{x}_i \] where $\overline{x}_i$ is the residue class of $x_i$ mod $\mfk{p}$. \end{proof}
If $y$ and $x$ are real numbers, then $\i y = x$ if and only if $x=0$ and $y=0$.
% Created 2020-07-13 lun 12:51 % Intended LaTeX compiler: pdflatex \documentclass[letterpaper,fleqn]{scrartcl} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{graphicx} \usepackage{grffile} \usepackage{longtable} \usepackage{wrapfig} \usepackage{rotating} \usepackage[normalem]{ulem} \usepackage{amsmath} \usepackage{textcomp} \usepackage{amssymb} \usepackage{capt-of} \usepackage{hyperref} \usepackage{khpreamble} \usepackage{tabularx} \usepackage{geometry} \usepackage{pgfplots} \pgfplotsset{compat=1.13} \geometry{top=20mm, bottom=20mm, left=24mm, right=18mm} \author{Kjartan Halvorsen} \date{} \title{Polynomial design (RST) exercise} \hypersetup{ pdfauthor={Kjartan Halvorsen}, pdftitle={Polynomial design (RST) exercise}, pdfkeywords={}, pdfsubject={}, pdfcreator={Emacs 26.3 (Org mode 9.3.6)}, pdflang={English}} \begin{document} \maketitle \section{Determine the order of the controller} \label{sec:org6e78179} \begin{center} \includegraphics[width=0.7\linewidth]{../../figures/2dof-block-explicit} \end{center} In each of the cases determine the order of the feedback controller $F_{b}(z)=\frac{S(z)}{R(z)}$ and write out the $R(z)$ and $S(z)$ polynomials. Determine also the order of the observer polynomial $A_o(z)$. You don't have to solve for the controller coefficients. \subsection{Case 1} \label{sec:orgb815efe} Plant is $H(z) = \frac{b_0z+b_1}{z^3 + a_1z^2 + a_2z}$, desired response to reference signal $H_c(z) = \frac{0.2^2}{z(z-0.8)(z-0.8)}$, observer poles in the origin. \vspace{27mm} \subsection{Case 2} \label{sec:org6948edd} Plant is $H(z) = \frac{b_0z+b_1}{z^3 + a_1z^2 + a_2z}$, desired response to reference signal $H_c(z) = \frac{0.2^2}{(z-0.8)^3}$, observer poles in the origin and integral action in the feedback controller (incremental controller). \vspace{27mm} \subsection{Case 3} \label{sec:orgef90a48} Plant is $H(z) = \frac{b_0z+b_1}{z^2 + a_1z + a_2}$ and there is a delay of 2 sampling periods in the feedback path. The desired response to reference signal $H_c(z) = \frac{0.2^2}{(z-0.8)(z-0.8)}$, observer poles in the origin and integral action in the feedback controller (incremental controller). \end{document}
// // $Id: SpectrumWorkerThreads.hpp 6585 2014-08-07 22:49:28Z chambm $ // // // Original author: William French <william.r.frenchwr .@. vanderbilt.edu> // // Copyright 2014 Vanderbilt University - Nashville, TN 37232 // // Licensed under the Apache License, Version 2.0 (the "License"); // you may not use this file except in compliance with the License. // You may obtain a copy of the License at // // http://www.apache.org/licenses/LICENSE-2.0 // // Unless required by applicable law or agreed to in writing, software // distributed under the License is distributed on an "AS IS" BASIS, // WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. // See the License for the specific language governing permissions and // limitations under the License. // #ifndef _SPECTRUMWORKERTHREADS_HPP_ #define _SPECTRUMWORKERTHREADS_HPP_ #include "pwiz/data/msdata/MSData.hpp" #include <boost/smart_ptr.hpp> namespace pwiz { namespace msdata { class SpectrumWorkerThreads { public: SpectrumWorkerThreads(const SpectrumList& sl); ~SpectrumWorkerThreads(); SpectrumPtr processBatch(size_t index, bool getBinaryData = true); private: class Impl; boost::scoped_ptr<Impl> impl_; }; } // namespace msdata } // namespace pwiz #endif // _SPECTRUMWORKERTHREADS_HPP_
lemma exists_component_superset: "\<lbrakk>t \<subseteq> s; s \<noteq> {}; connected t\<rbrakk> \<Longrightarrow> \<exists>c. c \<in> components s \<and> t \<subseteq> c"
[STATEMENT] lemma length_upto_enum_one: fixes x :: "'a :: len word" assumes lt1: "x < y" and lt2: "z < y" and lt3: "x \<le> z" shows "[x , y .e. z] = [x]" [PROOF STATE] proof (prove) goal (1 subgoal): 1. [x , y .e. z] = [x] [PROOF STEP] unfolding upto_enum_step_def [PROOF STATE] proof (prove) goal (1 subgoal): 1. (if z < x then [] else map (\<lambda>xa. x + xa * (y - x)) [0 .e. (z - x) div (y - x)]) = [x] [PROOF STEP] proof (subst upto_enum_red, subst if_not_P [OF leD [OF lt3]], clarsimp, rule conjI) [PROOF STATE] proof (state) goal (2 subgoals): 1. unat ((z - x) div (y - x)) = 0 2. (z - x) div (y - x) * (y - x) = 0 [PROOF STEP] show "unat ((z - x) div (y - x)) = 0" [PROOF STATE] proof (prove) goal (1 subgoal): 1. unat ((z - x) div (y - x)) = 0 [PROOF STEP] proof (subst unat_div, rule div_less) [PROOF STATE] proof (state) goal (1 subgoal): 1. unat (z - x) < unat (y - x) [PROOF STEP] have syx: "unat (y - x) = unat y - unat x" [PROOF STATE] proof (prove) goal (1 subgoal): 1. unat (y - x) = unat y - unat x [PROOF STEP] by (rule unat_sub [OF order_less_imp_le]) fact [PROOF STATE] proof (state) this: unat (y - x) = unat y - unat x goal (1 subgoal): 1. unat (z - x) < unat (y - x) [PROOF STEP] moreover [PROOF STATE] proof (state) this: unat (y - x) = unat y - unat x goal (1 subgoal): 1. unat (z - x) < unat (y - x) [PROOF STEP] have "unat (z - x) = unat z - unat x" [PROOF STATE] proof (prove) goal (1 subgoal): 1. unat (z - x) = unat z - unat x [PROOF STEP] by (rule unat_sub) fact [PROOF STATE] proof (state) this: unat (z - x) = unat z - unat x goal (1 subgoal): 1. unat (z - x) < unat (y - x) [PROOF STEP] ultimately [PROOF STATE] proof (chain) picking this: unat (y - x) = unat y - unat x unat (z - x) = unat z - unat x [PROOF STEP] show "unat (z - x) < unat (y - x)" [PROOF STATE] proof (prove) using this: unat (y - x) = unat y - unat x unat (z - x) = unat z - unat x goal (1 subgoal): 1. unat (z - x) < unat (y - x) [PROOF STEP] using lt2 lt3 unat_mono word_less_minus_mono_left [PROOF STATE] proof (prove) using this: unat (y - x) = unat y - unat x unat (z - x) = unat z - unat x z < y x \<le> z ?a < ?b \<Longrightarrow> unat ?a < unat ?b \<lbrakk>?y < ?z; ?x \<le> ?y\<rbrakk> \<Longrightarrow> ?y - ?x < ?z - ?x goal (1 subgoal): 1. unat (z - x) < unat (y - x) [PROOF STEP] by blast [PROOF STATE] proof (state) this: unat (z - x) < unat (y - x) goal: No subgoals! [PROOF STEP] qed [PROOF STATE] proof (state) this: unat ((z - x) div (y - x)) = 0 goal (1 subgoal): 1. (z - x) div (y - x) * (y - x) = 0 [PROOF STEP] then [PROOF STATE] proof (chain) picking this: unat ((z - x) div (y - x)) = 0 [PROOF STEP] show "(z - x) div (y - x) * (y - x) = 0" [PROOF STATE] proof (prove) using this: unat ((z - x) div (y - x)) = 0 goal (1 subgoal): 1. (z - x) div (y - x) * (y - x) = 0 [PROOF STEP] by (simp add: unat_div) (simp add: word_arith_nat_defs(6)) [PROOF STATE] proof (state) this: (z - x) div (y - x) * (y - x) = 0 goal: No subgoals! [PROOF STEP] qed
/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import analysis.special_functions.exp_log import data.set.intervals.infinite import algebra.quadratic_discriminant import ring_theory.polynomial.chebyshev import analysis.calculus.times_cont_diff /-! # Trigonometric functions ## Main definitions This file contains the following definitions: * π, arcsin, arccos, arctan * argument of a complex number * logarithm on complex numbers ## Main statements Many basic inequalities on trigonometric functions are established. The continuity and differentiability of the usual trigonometric functions are proved, and their derivatives are computed. * `polynomial.chebyshev.T_complex_cos`: the `n`-th Chebyshev polynomial evaluates on `complex.cos θ` to the value `n * complex.cos θ`. ## Tags log, sin, cos, tan, arcsin, arccos, arctan, angle, argument -/ noncomputable theory open_locale classical topological_space filter open set filter namespace complex /-- The complex sine function is everywhere strictly differentiable, with the derivative `cos x`. -/ lemma has_strict_deriv_at_sin (x : ℂ) : has_strict_deriv_at sin (cos x) x := begin simp only [cos, div_eq_mul_inv], convert ((((has_strict_deriv_at_id x).neg.mul_const I).cexp.sub ((has_strict_deriv_at_id x).mul_const I).cexp).mul_const I).mul_const (2:ℂ)⁻¹, simp only [function.comp, id], rw [sub_mul, mul_assoc, mul_assoc, I_mul_I, neg_one_mul, neg_neg, mul_one, one_mul, mul_assoc, I_mul_I, mul_neg_one, sub_neg_eq_add, add_comm] end /-- The complex sine function is everywhere differentiable, with the derivative `cos x`. -/ lemma has_deriv_at_sin (x : ℂ) : has_deriv_at sin (cos x) x := (has_strict_deriv_at_sin x).has_deriv_at lemma times_cont_diff_sin {n} : times_cont_diff ℂ n sin := (((times_cont_diff_neg.mul times_cont_diff_const).cexp.sub (times_cont_diff_id.mul times_cont_diff_const).cexp).mul times_cont_diff_const).div_const lemma differentiable_sin : differentiable ℂ sin := λx, (has_deriv_at_sin x).differentiable_at lemma differentiable_at_sin {x : ℂ} : differentiable_at ℂ sin x := differentiable_sin x @[simp] lemma deriv_sin : deriv sin = cos := funext $ λ x, (has_deriv_at_sin x).deriv @[continuity] lemma continuous_sin : continuous sin := differentiable_sin.continuous lemma continuous_on_sin {s : set ℂ} : continuous_on sin s := continuous_sin.continuous_on lemma measurable_sin : measurable sin := continuous_sin.measurable /-- The complex cosine function is everywhere strictly differentiable, with the derivative `-sin x`. -/ lemma has_strict_deriv_at_cos (x : ℂ) : has_strict_deriv_at cos (-sin x) x := begin simp only [sin, div_eq_mul_inv, neg_mul_eq_neg_mul], convert (((has_strict_deriv_at_id x).mul_const I).cexp.add ((has_strict_deriv_at_id x).neg.mul_const I).cexp).mul_const (2:ℂ)⁻¹, simp only [function.comp, id], ring end /-- The complex cosine function is everywhere differentiable, with the derivative `-sin x`. -/ lemma has_deriv_at_cos (x : ℂ) : has_deriv_at cos (-sin x) x := (has_strict_deriv_at_cos x).has_deriv_at lemma times_cont_diff_cos {n} : times_cont_diff ℂ n cos := ((times_cont_diff_id.mul times_cont_diff_const).cexp.add (times_cont_diff_neg.mul times_cont_diff_const).cexp).div_const lemma differentiable_cos : differentiable ℂ cos := λx, (has_deriv_at_cos x).differentiable_at lemma differentiable_at_cos {x : ℂ} : differentiable_at ℂ cos x := differentiable_cos x lemma deriv_cos {x : ℂ} : deriv cos x = -sin x := (has_deriv_at_cos x).deriv @[simp] lemma deriv_cos' : deriv cos = (λ x, -sin x) := funext $ λ x, deriv_cos @[continuity] lemma continuous_cos : continuous cos := differentiable_cos.continuous lemma continuous_on_cos {s : set ℂ} : continuous_on cos s := continuous_cos.continuous_on lemma measurable_cos : measurable cos := continuous_cos.measurable /-- The complex hyperbolic sine function is everywhere strictly differentiable, with the derivative `cosh x`. -/ lemma has_strict_deriv_at_sinh (x : ℂ) : has_strict_deriv_at sinh (cosh x) x := begin simp only [cosh, div_eq_mul_inv], convert ((has_strict_deriv_at_exp x).sub (has_strict_deriv_at_id x).neg.cexp).mul_const (2:ℂ)⁻¹, rw [id, mul_neg_one, sub_eq_add_neg, neg_neg] end /-- The complex hyperbolic sine function is everywhere differentiable, with the derivative `cosh x`. -/ lemma has_deriv_at_sinh (x : ℂ) : has_deriv_at sinh (cosh x) x := (has_strict_deriv_at_sinh x).has_deriv_at lemma times_cont_diff_sinh {n} : times_cont_diff ℂ n sinh := (times_cont_diff_exp.sub times_cont_diff_neg.cexp).div_const lemma differentiable_sinh : differentiable ℂ sinh := λx, (has_deriv_at_sinh x).differentiable_at lemma differentiable_at_sinh {x : ℂ} : differentiable_at ℂ sinh x := differentiable_sinh x @[simp] lemma deriv_sinh : deriv sinh = cosh := funext $ λ x, (has_deriv_at_sinh x).deriv @[continuity] lemma continuous_sinh : continuous sinh := differentiable_sinh.continuous lemma measurable_sinh : measurable sinh := continuous_sinh.measurable /-- The complex hyperbolic cosine function is everywhere strictly differentiable, with the derivative `sinh x`. -/ lemma has_strict_deriv_at_cosh (x : ℂ) : has_strict_deriv_at cosh (sinh x) x := begin simp only [sinh, div_eq_mul_inv], convert ((has_strict_deriv_at_exp x).add (has_strict_deriv_at_id x).neg.cexp).mul_const (2:ℂ)⁻¹, rw [id, mul_neg_one, sub_eq_add_neg] end /-- The complex hyperbolic cosine function is everywhere differentiable, with the derivative `sinh x`. -/ lemma has_deriv_at_cosh (x : ℂ) : has_deriv_at cosh (sinh x) x := (has_strict_deriv_at_cosh x).has_deriv_at lemma times_cont_diff_cosh {n} : times_cont_diff ℂ n cosh := (times_cont_diff_exp.add times_cont_diff_neg.cexp).div_const lemma differentiable_cosh : differentiable ℂ cosh := λx, (has_deriv_at_cosh x).differentiable_at lemma differentiable_at_cosh {x : ℂ} : differentiable_at ℂ cos x := differentiable_cos x @[simp] lemma deriv_cosh : deriv cosh = sinh := funext $ λ x, (has_deriv_at_cosh x).deriv @[continuity] lemma continuous_cosh : continuous cosh := differentiable_cosh.continuous lemma measurable_cosh : measurable cosh := continuous_cosh.measurable end complex section /-! ### Simp lemmas for derivatives of `λ x, complex.cos (f x)` etc., `f : ℂ → ℂ` -/ variables {f : ℂ → ℂ} {f' x : ℂ} {s : set ℂ} /-! #### `complex.cos` -/ lemma measurable.ccos {α : Type} [measurable_space α] {f : α → ℂ} (hf : measurable f) : measurable (λ x, complex.cos (f x)) := complex.measurable_cos.comp hf lemma has_strict_deriv_at.ccos (hf : has_strict_deriv_at f f' x) : has_strict_deriv_at (λ x, complex.cos (f x)) (- complex.sin (f x) f') x := (complex.has_strict_deriv_at_cos (f x)).comp x hf lemma has_deriv_at.ccos (hf : has_deriv_at f f' x) : has_deriv_at (λ x, complex.cos (f x)) (- complex.sin (f x) * f') x := (complex.has_deriv_at_cos (f x)).comp x hf lemma has_deriv_within_at.ccos (hf : has_deriv_within_at f f' s x) : has_deriv_within_at (λ x, complex.cos (f x)) (- complex.sin (f x) * f') s x := (complex.has_deriv_at_cos (f x)).comp_has_deriv_within_at x hf lemma deriv_within_ccos (hf : differentiable_within_at ℂ f s x) (hxs : unique_diff_within_at ℂ s x) : deriv_within (λx, complex.cos (f x)) s x = - complex.sin (f x) * (deriv_within f s x) := hf.has_deriv_within_at.ccos.deriv_within hxs @[simp] lemma deriv_ccos (hc : differentiable_at ℂ f x) : deriv (λx, complex.cos (f x)) x = - complex.sin (f x) * (deriv f x) := hc.has_deriv_at.ccos.deriv /-! #### `complex.sin` -/ lemma measurable.csin {α : Type} [measurable_space α] {f : α → ℂ} (hf : measurable f) : measurable (λ x, complex.sin (f x)) := complex.measurable_sin.comp hf lemma has_strict_deriv_at.csin (hf : has_strict_deriv_at f f' x) : has_strict_deriv_at (λ x, complex.sin (f x)) (complex.cos (f x) f') x := (complex.has_strict_deriv_at_sin (f x)).comp x hf lemma has_deriv_at.csin (hf : has_deriv_at f f' x) : has_deriv_at (λ x, complex.sin (f x)) (complex.cos (f x) * f') x := (complex.has_deriv_at_sin (f x)).comp x hf lemma has_deriv_within_at.csin (hf : has_deriv_within_at f f' s x) : has_deriv_within_at (λ x, complex.sin (f x)) (complex.cos (f x) * f') s x := (complex.has_deriv_at_sin (f x)).comp_has_deriv_within_at x hf lemma deriv_within_csin (hf : differentiable_within_at ℂ f s x) (hxs : unique_diff_within_at ℂ s x) : deriv_within (λx, complex.sin (f x)) s x = complex.cos (f x) * (deriv_within f s x) := hf.has_deriv_within_at.csin.deriv_within hxs @[simp] lemma deriv_csin (hc : differentiable_at ℂ f x) : deriv (λx, complex.sin (f x)) x = complex.cos (f x) * (deriv f x) := hc.has_deriv_at.csin.deriv /-! #### `complex.cosh` -/ lemma measurable.ccosh {α : Type} [measurable_space α] {f : α → ℂ} (hf : measurable f) : measurable (λ x, complex.cosh (f x)) := complex.measurable_cosh.comp hf lemma has_strict_deriv_at.ccosh (hf : has_strict_deriv_at f f' x) : has_strict_deriv_at (λ x, complex.cosh (f x)) (complex.sinh (f x) f') x := (complex.has_strict_deriv_at_cosh (f x)).comp x hf lemma has_deriv_at.ccosh (hf : has_deriv_at f f' x) : has_deriv_at (λ x, complex.cosh (f x)) (complex.sinh (f x) * f') x := (complex.has_deriv_at_cosh (f x)).comp x hf lemma has_deriv_within_at.ccosh (hf : has_deriv_within_at f f' s x) : has_deriv_within_at (λ x, complex.cosh (f x)) (complex.sinh (f x) * f') s x := (complex.has_deriv_at_cosh (f x)).comp_has_deriv_within_at x hf lemma deriv_within_ccosh (hf : differentiable_within_at ℂ f s x) (hxs : unique_diff_within_at ℂ s x) : deriv_within (λx, complex.cosh (f x)) s x = complex.sinh (f x) * (deriv_within f s x) := hf.has_deriv_within_at.ccosh.deriv_within hxs @[simp] lemma deriv_ccosh (hc : differentiable_at ℂ f x) : deriv (λx, complex.cosh (f x)) x = complex.sinh (f x) * (deriv f x) := hc.has_deriv_at.ccosh.deriv /-! #### `complex.sinh` -/ lemma measurable.csinh {α : Type} [measurable_space α] {f : α → ℂ} (hf : measurable f) : measurable (λ x, complex.sinh (f x)) := complex.measurable_sinh.comp hf lemma has_strict_deriv_at.csinh (hf : has_strict_deriv_at f f' x) : has_strict_deriv_at (λ x, complex.sinh (f x)) (complex.cosh (f x) f') x := (complex.has_strict_deriv_at_sinh (f x)).comp x hf lemma has_deriv_at.csinh (hf : has_deriv_at f f' x) : has_deriv_at (λ x, complex.sinh (f x)) (complex.cosh (f x) * f') x := (complex.has_deriv_at_sinh (f x)).comp x hf lemma has_deriv_within_at.csinh (hf : has_deriv_within_at f f' s x) : has_deriv_within_at (λ x, complex.sinh (f x)) (complex.cosh (f x) * f') s x := (complex.has_deriv_at_sinh (f x)).comp_has_deriv_within_at x hf lemma deriv_within_csinh (hf : differentiable_within_at ℂ f s x) (hxs : unique_diff_within_at ℂ s x) : deriv_within (λx, complex.sinh (f x)) s x = complex.cosh (f x) * (deriv_within f s x) := hf.has_deriv_within_at.csinh.deriv_within hxs @[simp] lemma deriv_csinh (hc : differentiable_at ℂ f x) : deriv (λx, complex.sinh (f x)) x = complex.cosh (f x) * (deriv f x) := hc.has_deriv_at.csinh.deriv end section /-! ### Simp lemmas for derivatives of `λ x, complex.cos (f x)` etc., `f : E → ℂ` -/ variables {E : Type} [normed_group E] [normed_space ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} {s : set E} /-! #### `complex.cos` -/ lemma has_strict_fderiv_at.ccos (hf : has_strict_fderiv_at f f' x) : has_strict_fderiv_at (λ x, complex.cos (f x)) (- complex.sin (f x) • f') x := (complex.has_strict_deriv_at_cos (f x)).comp_has_strict_fderiv_at x hf lemma has_fderiv_at.ccos (hf : has_fderiv_at f f' x) : has_fderiv_at (λ x, complex.cos (f x)) (- complex.sin (f x) • f') x := (complex.has_deriv_at_cos (f x)).comp_has_fderiv_at x hf lemma has_fderiv_within_at.ccos (hf : has_fderiv_within_at f f' s x) : has_fderiv_within_at (λ x, complex.cos (f x)) (- complex.sin (f x) • f') s x := (complex.has_deriv_at_cos (f x)).comp_has_fderiv_within_at x hf lemma differentiable_within_at.ccos (hf : differentiable_within_at ℂ f s x) : differentiable_within_at ℂ (λ x, complex.cos (f x)) s x := hf.has_fderiv_within_at.ccos.differentiable_within_at @[simp] lemma differentiable_at.ccos (hc : differentiable_at ℂ f x) : differentiable_at ℂ (λx, complex.cos (f x)) x := hc.has_fderiv_at.ccos.differentiable_at lemma differentiable_on.ccos (hc : differentiable_on ℂ f s) : differentiable_on ℂ (λx, complex.cos (f x)) s := λx h, (hc x h).ccos @[simp] lemma differentiable.ccos (hc : differentiable ℂ f) : differentiable ℂ (λx, complex.cos (f x)) := λx, (hc x).ccos lemma fderiv_within_ccos (hf : differentiable_within_at ℂ f s x) (hxs : unique_diff_within_at ℂ s x) : fderiv_within ℂ (λx, complex.cos (f x)) s x = - complex.sin (f x) • (fderiv_within ℂ f s x) := hf.has_fderiv_within_at.ccos.fderiv_within hxs @[simp] lemma fderiv_ccos (hc : differentiable_at ℂ f x) : fderiv ℂ (λx, complex.cos (f x)) x = - complex.sin (f x) • (fderiv ℂ f x) := hc.has_fderiv_at.ccos.fderiv lemma times_cont_diff.ccos {n} (h : times_cont_diff ℂ n f) : times_cont_diff ℂ n (λ x, complex.cos (f x)) := complex.times_cont_diff_cos.comp h lemma times_cont_diff_at.ccos {n} (hf : times_cont_diff_at ℂ n f x) : times_cont_diff_at ℂ n (λ x, complex.cos (f x)) x := complex.times_cont_diff_cos.times_cont_diff_at.comp x hf lemma times_cont_diff_on.ccos {n} (hf : times_cont_diff_on ℂ n f s) : times_cont_diff_on ℂ n (λ x, complex.cos (f x)) s := complex.times_cont_diff_cos.comp_times_cont_diff_on hf lemma times_cont_diff_within_at.ccos {n} (hf : times_cont_diff_within_at ℂ n f s x) : times_cont_diff_within_at ℂ n (λ x, complex.cos (f x)) s x := complex.times_cont_diff_cos.times_cont_diff_at.comp_times_cont_diff_within_at x hf /-! #### `complex.sin` -/ lemma has_strict_fderiv_at.csin (hf : has_strict_fderiv_at f f' x) : has_strict_fderiv_at (λ x, complex.sin (f x)) (complex.cos (f x) • f') x := (complex.has_strict_deriv_at_sin (f x)).comp_has_strict_fderiv_at x hf lemma has_fderiv_at.csin (hf : has_fderiv_at f f' x) : has_fderiv_at (λ x, complex.sin (f x)) (complex.cos (f x) • f') x := (complex.has_deriv_at_sin (f x)).comp_has_fderiv_at x hf lemma has_fderiv_within_at.csin (hf : has_fderiv_within_at f f' s x) : has_fderiv_within_at (λ x, complex.sin (f x)) (complex.cos (f x) • f') s x := (complex.has_deriv_at_sin (f x)).comp_has_fderiv_within_at x hf lemma differentiable_within_at.csin (hf : differentiable_within_at ℂ f s x) : differentiable_within_at ℂ (λ x, complex.sin (f x)) s x := hf.has_fderiv_within_at.csin.differentiable_within_at @[simp] lemma differentiable_at.csin (hc : differentiable_at ℂ f x) : differentiable_at ℂ (λx, complex.sin (f x)) x := hc.has_fderiv_at.csin.differentiable_at lemma differentiable_on.csin (hc : differentiable_on ℂ f s) : differentiable_on ℂ (λx, complex.sin (f x)) s := λx h, (hc x h).csin @[simp] lemma differentiable.csin (hc : differentiable ℂ f) : differentiable ℂ (λx, complex.sin (f x)) := λx, (hc x).csin lemma fderiv_within_csin (hf : differentiable_within_at ℂ f s x) (hxs : unique_diff_within_at ℂ s x) : fderiv_within ℂ (λx, complex.sin (f x)) s x = complex.cos (f x) • (fderiv_within ℂ f s x) := hf.has_fderiv_within_at.csin.fderiv_within hxs @[simp] lemma fderiv_csin (hc : differentiable_at ℂ f x) : fderiv ℂ (λx, complex.sin (f x)) x = complex.cos (f x) • (fderiv ℂ f x) := hc.has_fderiv_at.csin.fderiv lemma times_cont_diff.csin {n} (h : times_cont_diff ℂ n f) : times_cont_diff ℂ n (λ x, complex.sin (f x)) := complex.times_cont_diff_sin.comp h lemma times_cont_diff_at.csin {n} (hf : times_cont_diff_at ℂ n f x) : times_cont_diff_at ℂ n (λ x, complex.sin (f x)) x := complex.times_cont_diff_sin.times_cont_diff_at.comp x hf lemma times_cont_diff_on.csin {n} (hf : times_cont_diff_on ℂ n f s) : times_cont_diff_on ℂ n (λ x, complex.sin (f x)) s := complex.times_cont_diff_sin.comp_times_cont_diff_on hf lemma times_cont_diff_within_at.csin {n} (hf : times_cont_diff_within_at ℂ n f s x) : times_cont_diff_within_at ℂ n (λ x, complex.sin (f x)) s x := complex.times_cont_diff_sin.times_cont_diff_at.comp_times_cont_diff_within_at x hf /-! #### `complex.cosh` -/ lemma has_strict_fderiv_at.ccosh (hf : has_strict_fderiv_at f f' x) : has_strict_fderiv_at (λ x, complex.cosh (f x)) (complex.sinh (f x) • f') x := (complex.has_strict_deriv_at_cosh (f x)).comp_has_strict_fderiv_at x hf lemma has_fderiv_at.ccosh (hf : has_fderiv_at f f' x) : has_fderiv_at (λ x, complex.cosh (f x)) (complex.sinh (f x) • f') x := (complex.has_deriv_at_cosh (f x)).comp_has_fderiv_at x hf lemma has_fderiv_within_at.ccosh (hf : has_fderiv_within_at f f' s x) : has_fderiv_within_at (λ x, complex.cosh (f x)) (complex.sinh (f x) • f') s x := (complex.has_deriv_at_cosh (f x)).comp_has_fderiv_within_at x hf lemma differentiable_within_at.ccosh (hf : differentiable_within_at ℂ f s x) : differentiable_within_at ℂ (λ x, complex.cosh (f x)) s x := hf.has_fderiv_within_at.ccosh.differentiable_within_at @[simp] lemma differentiable_at.ccosh (hc : differentiable_at ℂ f x) : differentiable_at ℂ (λx, complex.cosh (f x)) x := hc.has_fderiv_at.ccosh.differentiable_at lemma differentiable_on.ccosh (hc : differentiable_on ℂ f s) : differentiable_on ℂ (λx, complex.cosh (f x)) s := λx h, (hc x h).ccosh @[simp] lemma differentiable.ccosh (hc : differentiable ℂ f) : differentiable ℂ (λx, complex.cosh (f x)) := λx, (hc x).ccosh lemma fderiv_within_ccosh (hf : differentiable_within_at ℂ f s x) (hxs : unique_diff_within_at ℂ s x) : fderiv_within ℂ (λx, complex.cosh (f x)) s x = complex.sinh (f x) • (fderiv_within ℂ f s x) := hf.has_fderiv_within_at.ccosh.fderiv_within hxs @[simp] lemma fderiv_ccosh (hc : differentiable_at ℂ f x) : fderiv ℂ (λx, complex.cosh (f x)) x = complex.sinh (f x) • (fderiv ℂ f x) := hc.has_fderiv_at.ccosh.fderiv lemma times_cont_diff.ccosh {n} (h : times_cont_diff ℂ n f) : times_cont_diff ℂ n (λ x, complex.cosh (f x)) := complex.times_cont_diff_cosh.comp h lemma times_cont_diff_at.ccosh {n} (hf : times_cont_diff_at ℂ n f x) : times_cont_diff_at ℂ n (λ x, complex.cosh (f x)) x := complex.times_cont_diff_cosh.times_cont_diff_at.comp x hf lemma times_cont_diff_on.ccosh {n} (hf : times_cont_diff_on ℂ n f s) : times_cont_diff_on ℂ n (λ x, complex.cosh (f x)) s := complex.times_cont_diff_cosh.comp_times_cont_diff_on hf lemma times_cont_diff_within_at.ccosh {n} (hf : times_cont_diff_within_at ℂ n f s x) : times_cont_diff_within_at ℂ n (λ x, complex.cosh (f x)) s x := complex.times_cont_diff_cosh.times_cont_diff_at.comp_times_cont_diff_within_at x hf /-! #### `complex.sinh` -/ lemma has_strict_fderiv_at.csinh (hf : has_strict_fderiv_at f f' x) : has_strict_fderiv_at (λ x, complex.sinh (f x)) (complex.cosh (f x) • f') x := (complex.has_strict_deriv_at_sinh (f x)).comp_has_strict_fderiv_at x hf lemma has_fderiv_at.csinh (hf : has_fderiv_at f f' x) : has_fderiv_at (λ x, complex.sinh (f x)) (complex.cosh (f x) • f') x := (complex.has_deriv_at_sinh (f x)).comp_has_fderiv_at x hf lemma has_fderiv_within_at.csinh (hf : has_fderiv_within_at f f' s x) : has_fderiv_within_at (λ x, complex.sinh (f x)) (complex.cosh (f x) • f') s x := (complex.has_deriv_at_sinh (f x)).comp_has_fderiv_within_at x hf lemma differentiable_within_at.csinh (hf : differentiable_within_at ℂ f s x) : differentiable_within_at ℂ (λ x, complex.sinh (f x)) s x := hf.has_fderiv_within_at.csinh.differentiable_within_at @[simp] lemma differentiable_at.csinh (hc : differentiable_at ℂ f x) : differentiable_at ℂ (λx, complex.sinh (f x)) x := hc.has_fderiv_at.csinh.differentiable_at lemma differentiable_on.csinh (hc : differentiable_on ℂ f s) : differentiable_on ℂ (λx, complex.sinh (f x)) s := λx h, (hc x h).csinh @[simp] lemma differentiable.csinh (hc : differentiable ℂ f) : differentiable ℂ (λx, complex.sinh (f x)) := λx, (hc x).csinh lemma fderiv_within_csinh (hf : differentiable_within_at ℂ f s x) (hxs : unique_diff_within_at ℂ s x) : fderiv_within ℂ (λx, complex.sinh (f x)) s x = complex.cosh (f x) • (fderiv_within ℂ f s x) := hf.has_fderiv_within_at.csinh.fderiv_within hxs @[simp] lemma fderiv_csinh (hc : differentiable_at ℂ f x) : fderiv ℂ (λx, complex.sinh (f x)) x = complex.cosh (f x) • (fderiv ℂ f x) := hc.has_fderiv_at.csinh.fderiv lemma times_cont_diff.csinh {n} (h : times_cont_diff ℂ n f) : times_cont_diff ℂ n (λ x, complex.sinh (f x)) := complex.times_cont_diff_sinh.comp h lemma times_cont_diff_at.csinh {n} (hf : times_cont_diff_at ℂ n f x) : times_cont_diff_at ℂ n (λ x, complex.sinh (f x)) x := complex.times_cont_diff_sinh.times_cont_diff_at.comp x hf lemma times_cont_diff_on.csinh {n} (hf : times_cont_diff_on ℂ n f s) : times_cont_diff_on ℂ n (λ x, complex.sinh (f x)) s := complex.times_cont_diff_sinh.comp_times_cont_diff_on hf lemma times_cont_diff_within_at.csinh {n} (hf : times_cont_diff_within_at ℂ n f s x) : times_cont_diff_within_at ℂ n (λ x, complex.sinh (f x)) s x := complex.times_cont_diff_sinh.times_cont_diff_at.comp_times_cont_diff_within_at x hf end namespace real variables {x y z : ℝ} lemma has_strict_deriv_at_sin (x : ℝ) : has_strict_deriv_at sin (cos x) x := (complex.has_strict_deriv_at_sin x).real_of_complex lemma has_deriv_at_sin (x : ℝ) : has_deriv_at sin (cos x) x := (has_strict_deriv_at_sin x).has_deriv_at lemma times_cont_diff_sin {n} : times_cont_diff ℝ n sin := complex.times_cont_diff_sin.real_of_complex lemma differentiable_sin : differentiable ℝ sin := λx, (has_deriv_at_sin x).differentiable_at lemma differentiable_at_sin : differentiable_at ℝ sin x := differentiable_sin x @[simp] lemma deriv_sin : deriv sin = cos := funext $ λ x, (has_deriv_at_sin x).deriv @[continuity] lemma continuous_sin : continuous sin := differentiable_sin.continuous lemma continuous_on_sin {s} : continuous_on sin s := continuous_sin.continuous_on lemma measurable_sin : measurable sin := continuous_sin.measurable lemma has_strict_deriv_at_cos (x : ℝ) : has_strict_deriv_at cos (-sin x) x := (complex.has_strict_deriv_at_cos x).real_of_complex lemma has_deriv_at_cos (x : ℝ) : has_deriv_at cos (-sin x) x := (complex.has_deriv_at_cos x).real_of_complex lemma times_cont_diff_cos {n} : times_cont_diff ℝ n cos := complex.times_cont_diff_cos.real_of_complex lemma differentiable_cos : differentiable ℝ cos := λx, (has_deriv_at_cos x).differentiable_at lemma differentiable_at_cos : differentiable_at ℝ cos x := differentiable_cos x lemma deriv_cos : deriv cos x = - sin x := (has_deriv_at_cos x).deriv @[simp] lemma deriv_cos' : deriv cos = (λ x, - sin x) := funext $ λ _, deriv_cos @[continuity] lemma continuous_cos : continuous cos := differentiable_cos.continuous lemma continuous_on_cos {s} : continuous_on cos s := continuous_cos.continuous_on lemma measurable_cos : measurable cos := continuous_cos.measurable lemma has_strict_deriv_at_sinh (x : ℝ) : has_strict_deriv_at sinh (cosh x) x := (complex.has_strict_deriv_at_sinh x).real_of_complex lemma has_deriv_at_sinh (x : ℝ) : has_deriv_at sinh (cosh x) x := (complex.has_deriv_at_sinh x).real_of_complex lemma times_cont_diff_sinh {n} : times_cont_diff ℝ n sinh := complex.times_cont_diff_sinh.real_of_complex lemma differentiable_sinh : differentiable ℝ sinh := λx, (has_deriv_at_sinh x).differentiable_at lemma differentiable_at_sinh : differentiable_at ℝ sinh x := differentiable_sinh x @[simp] lemma deriv_sinh : deriv sinh = cosh := funext $ λ x, (has_deriv_at_sinh x).deriv @[continuity] lemma continuous_sinh : continuous sinh := differentiable_sinh.continuous lemma measurable_sinh : measurable sinh := continuous_sinh.measurable lemma has_strict_deriv_at_cosh (x : ℝ) : has_strict_deriv_at cosh (sinh x) x := (complex.has_strict_deriv_at_cosh x).real_of_complex lemma has_deriv_at_cosh (x : ℝ) : has_deriv_at cosh (sinh x) x := (complex.has_deriv_at_cosh x).real_of_complex lemma times_cont_diff_cosh {n} : times_cont_diff ℝ n cosh := complex.times_cont_diff_cosh.real_of_complex lemma differentiable_cosh : differentiable ℝ cosh := λx, (has_deriv_at_cosh x).differentiable_at lemma differentiable_at_cosh : differentiable_at ℝ cosh x := differentiable_cosh x @[simp] lemma deriv_cosh : deriv cosh = sinh := funext $ λ x, (has_deriv_at_cosh x).deriv @[continuity] lemma continuous_cosh : continuous cosh := differentiable_cosh.continuous lemma measurable_cosh : measurable cosh := continuous_cosh.measurable /-- `sinh` is strictly monotone. -/ lemma sinh_strict_mono : strict_mono sinh := strict_mono_of_deriv_pos differentiable_sinh (by { rw [real.deriv_sinh], exact cosh_pos }) end real section /-! ### Simp lemmas for derivatives of `λ x, real.cos (f x)` etc., `f : ℝ → ℝ` -/ variables {f : ℝ → ℝ} {f' x : ℝ} {s : set ℝ} /-! #### `real.cos` -/ lemma measurable.cos {α : Type} [measurable_space α] {f : α → ℝ} (hf : measurable f) : measurable (λ x, real.cos (f x)) := real.measurable_cos.comp hf lemma has_strict_deriv_at.cos (hf : has_strict_deriv_at f f' x) : has_strict_deriv_at (λ x, real.cos (f x)) (- real.sin (f x) * f') x := (real.has_strict_deriv_at_cos (f x)).comp x hf lemma has_deriv_at.cos (hf : has_deriv_at f f' x) : has_deriv_at (λ x, real.cos (f x)) (- real.sin (f x) * f') x := (real.has_deriv_at_cos (f x)).comp x hf lemma has_deriv_within_at.cos (hf : has_deriv_within_at f f' s x) : has_deriv_within_at (λ x, real.cos (f x)) (- real.sin (f x) * f') s x := (real.has_deriv_at_cos (f x)).comp_has_deriv_within_at x hf lemma deriv_within_cos (hf : differentiable_within_at ℝ f s x) (hxs : unique_diff_within_at ℝ s x) : deriv_within (λx, real.cos (f x)) s x = - real.sin (f x) * (deriv_within f s x) := hf.has_deriv_within_at.cos.deriv_within hxs @[simp] lemma deriv_cos (hc : differentiable_at ℝ f x) : deriv (λx, real.cos (f x)) x = - real.sin (f x) * (deriv f x) := hc.has_deriv_at.cos.deriv /-! #### `real.sin` -/ lemma measurable.sin {α : Type} [measurable_space α] {f : α → ℝ} (hf : measurable f) : measurable (λ x, real.sin (f x)) := real.measurable_sin.comp hf lemma has_strict_deriv_at.sin (hf : has_strict_deriv_at f f' x) : has_strict_deriv_at (λ x, real.sin (f x)) (real.cos (f x) f') x := (real.has_strict_deriv_at_sin (f x)).comp x hf lemma has_deriv_at.sin (hf : has_deriv_at f f' x) : has_deriv_at (λ x, real.sin (f x)) (real.cos (f x) * f') x := (real.has_deriv_at_sin (f x)).comp x hf lemma has_deriv_within_at.sin (hf : has_deriv_within_at f f' s x) : has_deriv_within_at (λ x, real.sin (f x)) (real.cos (f x) * f') s x := (real.has_deriv_at_sin (f x)).comp_has_deriv_within_at x hf lemma deriv_within_sin (hf : differentiable_within_at ℝ f s x) (hxs : unique_diff_within_at ℝ s x) : deriv_within (λx, real.sin (f x)) s x = real.cos (f x) * (deriv_within f s x) := hf.has_deriv_within_at.sin.deriv_within hxs @[simp] lemma deriv_sin (hc : differentiable_at ℝ f x) : deriv (λx, real.sin (f x)) x = real.cos (f x) * (deriv f x) := hc.has_deriv_at.sin.deriv /-! #### `real.cosh` -/ lemma measurable.cosh {α : Type} [measurable_space α] {f : α → ℝ} (hf : measurable f) : measurable (λ x, real.cosh (f x)) := real.measurable_cosh.comp hf lemma has_strict_deriv_at.cosh (hf : has_strict_deriv_at f f' x) : has_strict_deriv_at (λ x, real.cosh (f x)) (real.sinh (f x) f') x := (real.has_strict_deriv_at_cosh (f x)).comp x hf lemma has_deriv_at.cosh (hf : has_deriv_at f f' x) : has_deriv_at (λ x, real.cosh (f x)) (real.sinh (f x) * f') x := (real.has_deriv_at_cosh (f x)).comp x hf lemma has_deriv_within_at.cosh (hf : has_deriv_within_at f f' s x) : has_deriv_within_at (λ x, real.cosh (f x)) (real.sinh (f x) * f') s x := (real.has_deriv_at_cosh (f x)).comp_has_deriv_within_at x hf lemma deriv_within_cosh (hf : differentiable_within_at ℝ f s x) (hxs : unique_diff_within_at ℝ s x) : deriv_within (λx, real.cosh (f x)) s x = real.sinh (f x) * (deriv_within f s x) := hf.has_deriv_within_at.cosh.deriv_within hxs @[simp] lemma deriv_cosh (hc : differentiable_at ℝ f x) : deriv (λx, real.cosh (f x)) x = real.sinh (f x) * (deriv f x) := hc.has_deriv_at.cosh.deriv /-! #### `real.sinh` -/ lemma measurable.sinh {α : Type} [measurable_space α] {f : α → ℝ} (hf : measurable f) : measurable (λ x, real.sinh (f x)) := real.measurable_sinh.comp hf lemma has_strict_deriv_at.sinh (hf : has_strict_deriv_at f f' x) : has_strict_deriv_at (λ x, real.sinh (f x)) (real.cosh (f x) f') x := (real.has_strict_deriv_at_sinh (f x)).comp x hf lemma has_deriv_at.sinh (hf : has_deriv_at f f' x) : has_deriv_at (λ x, real.sinh (f x)) (real.cosh (f x) * f') x := (real.has_deriv_at_sinh (f x)).comp x hf lemma has_deriv_within_at.sinh (hf : has_deriv_within_at f f' s x) : has_deriv_within_at (λ x, real.sinh (f x)) (real.cosh (f x) * f') s x := (real.has_deriv_at_sinh (f x)).comp_has_deriv_within_at x hf lemma deriv_within_sinh (hf : differentiable_within_at ℝ f s x) (hxs : unique_diff_within_at ℝ s x) : deriv_within (λx, real.sinh (f x)) s x = real.cosh (f x) * (deriv_within f s x) := hf.has_deriv_within_at.sinh.deriv_within hxs @[simp] lemma deriv_sinh (hc : differentiable_at ℝ f x) : deriv (λx, real.sinh (f x)) x = real.cosh (f x) * (deriv f x) := hc.has_deriv_at.sinh.deriv end section /-! ### Simp lemmas for derivatives of `λ x, real.cos (f x)` etc., `f : E → ℝ` -/ variables {E : Type} [normed_group E] [normed_space ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} {s : set E} /-! #### `real.cos` -/ lemma has_strict_fderiv_at.cos (hf : has_strict_fderiv_at f f' x) : has_strict_fderiv_at (λ x, real.cos (f x)) (- real.sin (f x) • f') x := (real.has_strict_deriv_at_cos (f x)).comp_has_strict_fderiv_at x hf lemma has_fderiv_at.cos (hf : has_fderiv_at f f' x) : has_fderiv_at (λ x, real.cos (f x)) (- real.sin (f x) • f') x := (real.has_deriv_at_cos (f x)).comp_has_fderiv_at x hf lemma has_fderiv_within_at.cos (hf : has_fderiv_within_at f f' s x) : has_fderiv_within_at (λ x, real.cos (f x)) (- real.sin (f x) • f') s x := (real.has_deriv_at_cos (f x)).comp_has_fderiv_within_at x hf lemma differentiable_within_at.cos (hf : differentiable_within_at ℝ f s x) : differentiable_within_at ℝ (λ x, real.cos (f x)) s x := hf.has_fderiv_within_at.cos.differentiable_within_at @[simp] lemma differentiable_at.cos (hc : differentiable_at ℝ f x) : differentiable_at ℝ (λx, real.cos (f x)) x := hc.has_fderiv_at.cos.differentiable_at lemma differentiable_on.cos (hc : differentiable_on ℝ f s) : differentiable_on ℝ (λx, real.cos (f x)) s := λx h, (hc x h).cos @[simp] lemma differentiable.cos (hc : differentiable ℝ f) : differentiable ℝ (λx, real.cos (f x)) := λx, (hc x).cos lemma fderiv_within_cos (hf : differentiable_within_at ℝ f s x) (hxs : unique_diff_within_at ℝ s x) : fderiv_within ℝ (λx, real.cos (f x)) s x = - real.sin (f x) • (fderiv_within ℝ f s x) := hf.has_fderiv_within_at.cos.fderiv_within hxs @[simp] lemma fderiv_cos (hc : differentiable_at ℝ f x) : fderiv ℝ (λx, real.cos (f x)) x = - real.sin (f x) • (fderiv ℝ f x) := hc.has_fderiv_at.cos.fderiv lemma times_cont_diff.cos {n} (h : times_cont_diff ℝ n f) : times_cont_diff ℝ n (λ x, real.cos (f x)) := real.times_cont_diff_cos.comp h lemma times_cont_diff_at.cos {n} (hf : times_cont_diff_at ℝ n f x) : times_cont_diff_at ℝ n (λ x, real.cos (f x)) x := real.times_cont_diff_cos.times_cont_diff_at.comp x hf lemma times_cont_diff_on.cos {n} (hf : times_cont_diff_on ℝ n f s) : times_cont_diff_on ℝ n (λ x, real.cos (f x)) s := real.times_cont_diff_cos.comp_times_cont_diff_on hf lemma times_cont_diff_within_at.cos {n} (hf : times_cont_diff_within_at ℝ n f s x) : times_cont_diff_within_at ℝ n (λ x, real.cos (f x)) s x := real.times_cont_diff_cos.times_cont_diff_at.comp_times_cont_diff_within_at x hf /-! #### `real.sin` -/ lemma has_strict_fderiv_at.sin (hf : has_strict_fderiv_at f f' x) : has_strict_fderiv_at (λ x, real.sin (f x)) (real.cos (f x) • f') x := (real.has_strict_deriv_at_sin (f x)).comp_has_strict_fderiv_at x hf lemma has_fderiv_at.sin (hf : has_fderiv_at f f' x) : has_fderiv_at (λ x, real.sin (f x)) (real.cos (f x) • f') x := (real.has_deriv_at_sin (f x)).comp_has_fderiv_at x hf lemma has_fderiv_within_at.sin (hf : has_fderiv_within_at f f' s x) : has_fderiv_within_at (λ x, real.sin (f x)) (real.cos (f x) • f') s x := (real.has_deriv_at_sin (f x)).comp_has_fderiv_within_at x hf lemma differentiable_within_at.sin (hf : differentiable_within_at ℝ f s x) : differentiable_within_at ℝ (λ x, real.sin (f x)) s x := hf.has_fderiv_within_at.sin.differentiable_within_at @[simp] lemma differentiable_at.sin (hc : differentiable_at ℝ f x) : differentiable_at ℝ (λx, real.sin (f x)) x := hc.has_fderiv_at.sin.differentiable_at lemma differentiable_on.sin (hc : differentiable_on ℝ f s) : differentiable_on ℝ (λx, real.sin (f x)) s := λx h, (hc x h).sin @[simp] lemma differentiable.sin (hc : differentiable ℝ f) : differentiable ℝ (λx, real.sin (f x)) := λx, (hc x).sin lemma fderiv_within_sin (hf : differentiable_within_at ℝ f s x) (hxs : unique_diff_within_at ℝ s x) : fderiv_within ℝ (λx, real.sin (f x)) s x = real.cos (f x) • (fderiv_within ℝ f s x) := hf.has_fderiv_within_at.sin.fderiv_within hxs @[simp] lemma fderiv_sin (hc : differentiable_at ℝ f x) : fderiv ℝ (λx, real.sin (f x)) x = real.cos (f x) • (fderiv ℝ f x) := hc.has_fderiv_at.sin.fderiv lemma times_cont_diff.sin {n} (h : times_cont_diff ℝ n f) : times_cont_diff ℝ n (λ x, real.sin (f x)) := real.times_cont_diff_sin.comp h lemma times_cont_diff_at.sin {n} (hf : times_cont_diff_at ℝ n f x) : times_cont_diff_at ℝ n (λ x, real.sin (f x)) x := real.times_cont_diff_sin.times_cont_diff_at.comp x hf lemma times_cont_diff_on.sin {n} (hf : times_cont_diff_on ℝ n f s) : times_cont_diff_on ℝ n (λ x, real.sin (f x)) s := real.times_cont_diff_sin.comp_times_cont_diff_on hf lemma times_cont_diff_within_at.sin {n} (hf : times_cont_diff_within_at ℝ n f s x) : times_cont_diff_within_at ℝ n (λ x, real.sin (f x)) s x := real.times_cont_diff_sin.times_cont_diff_at.comp_times_cont_diff_within_at x hf /-! #### `real.cosh` -/ lemma has_strict_fderiv_at.cosh (hf : has_strict_fderiv_at f f' x) : has_strict_fderiv_at (λ x, real.cosh (f x)) (real.sinh (f x) • f') x := (real.has_strict_deriv_at_cosh (f x)).comp_has_strict_fderiv_at x hf lemma has_fderiv_at.cosh (hf : has_fderiv_at f f' x) : has_fderiv_at (λ x, real.cosh (f x)) (real.sinh (f x) • f') x := (real.has_deriv_at_cosh (f x)).comp_has_fderiv_at x hf lemma has_fderiv_within_at.cosh (hf : has_fderiv_within_at f f' s x) : has_fderiv_within_at (λ x, real.cosh (f x)) (real.sinh (f x) • f') s x := (real.has_deriv_at_cosh (f x)).comp_has_fderiv_within_at x hf lemma differentiable_within_at.cosh (hf : differentiable_within_at ℝ f s x) : differentiable_within_at ℝ (λ x, real.cosh (f x)) s x := hf.has_fderiv_within_at.cosh.differentiable_within_at @[simp] lemma differentiable_at.cosh (hc : differentiable_at ℝ f x) : differentiable_at ℝ (λx, real.cosh (f x)) x := hc.has_fderiv_at.cosh.differentiable_at lemma differentiable_on.cosh (hc : differentiable_on ℝ f s) : differentiable_on ℝ (λx, real.cosh (f x)) s := λx h, (hc x h).cosh @[simp] lemma differentiable.cosh (hc : differentiable ℝ f) : differentiable ℝ (λx, real.cosh (f x)) := λx, (hc x).cosh lemma fderiv_within_cosh (hf : differentiable_within_at ℝ f s x) (hxs : unique_diff_within_at ℝ s x) : fderiv_within ℝ (λx, real.cosh (f x)) s x = real.sinh (f x) • (fderiv_within ℝ f s x) := hf.has_fderiv_within_at.cosh.fderiv_within hxs @[simp] lemma fderiv_cosh (hc : differentiable_at ℝ f x) : fderiv ℝ (λx, real.cosh (f x)) x = real.sinh (f x) • (fderiv ℝ f x) := hc.has_fderiv_at.cosh.fderiv lemma times_cont_diff.cosh {n} (h : times_cont_diff ℝ n f) : times_cont_diff ℝ n (λ x, real.cosh (f x)) := real.times_cont_diff_cosh.comp h lemma times_cont_diff_at.cosh {n} (hf : times_cont_diff_at ℝ n f x) : times_cont_diff_at ℝ n (λ x, real.cosh (f x)) x := real.times_cont_diff_cosh.times_cont_diff_at.comp x hf lemma times_cont_diff_on.cosh {n} (hf : times_cont_diff_on ℝ n f s) : times_cont_diff_on ℝ n (λ x, real.cosh (f x)) s := real.times_cont_diff_cosh.comp_times_cont_diff_on hf lemma times_cont_diff_within_at.cosh {n} (hf : times_cont_diff_within_at ℝ n f s x) : times_cont_diff_within_at ℝ n (λ x, real.cosh (f x)) s x := real.times_cont_diff_cosh.times_cont_diff_at.comp_times_cont_diff_within_at x hf /-! #### `real.sinh` -/ lemma has_strict_fderiv_at.sinh (hf : has_strict_fderiv_at f f' x) : has_strict_fderiv_at (λ x, real.sinh (f x)) (real.cosh (f x) • f') x := (real.has_strict_deriv_at_sinh (f x)).comp_has_strict_fderiv_at x hf lemma has_fderiv_at.sinh (hf : has_fderiv_at f f' x) : has_fderiv_at (λ x, real.sinh (f x)) (real.cosh (f x) • f') x := (real.has_deriv_at_sinh (f x)).comp_has_fderiv_at x hf lemma has_fderiv_within_at.sinh (hf : has_fderiv_within_at f f' s x) : has_fderiv_within_at (λ x, real.sinh (f x)) (real.cosh (f x) • f') s x := (real.has_deriv_at_sinh (f x)).comp_has_fderiv_within_at x hf lemma differentiable_within_at.sinh (hf : differentiable_within_at ℝ f s x) : differentiable_within_at ℝ (λ x, real.sinh (f x)) s x := hf.has_fderiv_within_at.sinh.differentiable_within_at @[simp] lemma differentiable_at.sinh (hc : differentiable_at ℝ f x) : differentiable_at ℝ (λx, real.sinh (f x)) x := hc.has_fderiv_at.sinh.differentiable_at lemma differentiable_on.sinh (hc : differentiable_on ℝ f s) : differentiable_on ℝ (λx, real.sinh (f x)) s := λx h, (hc x h).sinh @[simp] lemma differentiable.sinh (hc : differentiable ℝ f) : differentiable ℝ (λx, real.sinh (f x)) := λx, (hc x).sinh lemma fderiv_within_sinh (hf : differentiable_within_at ℝ f s x) (hxs : unique_diff_within_at ℝ s x) : fderiv_within ℝ (λx, real.sinh (f x)) s x = real.cosh (f x) • (fderiv_within ℝ f s x) := hf.has_fderiv_within_at.sinh.fderiv_within hxs @[simp] lemma fderiv_sinh (hc : differentiable_at ℝ f x) : fderiv ℝ (λx, real.sinh (f x)) x = real.cosh (f x) • (fderiv ℝ f x) := hc.has_fderiv_at.sinh.fderiv lemma times_cont_diff.sinh {n} (h : times_cont_diff ℝ n f) : times_cont_diff ℝ n (λ x, real.sinh (f x)) := real.times_cont_diff_sinh.comp h lemma times_cont_diff_at.sinh {n} (hf : times_cont_diff_at ℝ n f x) : times_cont_diff_at ℝ n (λ x, real.sinh (f x)) x := real.times_cont_diff_sinh.times_cont_diff_at.comp x hf lemma times_cont_diff_on.sinh {n} (hf : times_cont_diff_on ℝ n f s) : times_cont_diff_on ℝ n (λ x, real.sinh (f x)) s := real.times_cont_diff_sinh.comp_times_cont_diff_on hf lemma times_cont_diff_within_at.sinh {n} (hf : times_cont_diff_within_at ℝ n f s x) : times_cont_diff_within_at ℝ n (λ x, real.sinh (f x)) s x := real.times_cont_diff_sinh.times_cont_diff_at.comp_times_cont_diff_within_at x hf end namespace real lemma exists_cos_eq_zero : 0 ∈ cos '' Icc (1:ℝ) 2 := intermediate_value_Icc' (by norm_num) continuous_on_cos ⟨le_of_lt cos_two_neg, le_of_lt cos_one_pos⟩ /-- The number π = 3.14159265... Defined here using choice as twice a zero of cos in [1,2], from which one can derive all its properties. For explicit bounds on π, see `data.real.pi`. -/ protected noncomputable def pi : ℝ := 2 classical.some exists_cos_eq_zero localized "notation `π` := real.pi" in real @[simp] lemma cos_pi_div_two : cos (π / 2) = 0 := by rw [real.pi, mul_div_cancel_left _ (@two_ne_zero' ℝ _ _ _)]; exact (classical.some_spec exists_cos_eq_zero).2 lemma one_le_pi_div_two : (1 : ℝ) ≤ π / 2 := by rw [real.pi, mul_div_cancel_left _ (@two_ne_zero' ℝ _ _ _)]; exact (classical.some_spec exists_cos_eq_zero).1.1 lemma pi_div_two_le_two : π / 2 ≤ 2 := by rw [real.pi, mul_div_cancel_left _ (@two_ne_zero' ℝ _ _ _)]; exact (classical.some_spec exists_cos_eq_zero).1.2 lemma two_le_pi : (2 : ℝ) ≤ π := (div_le_div_right (show (0 : ℝ) < 2, by norm_num)).1 (by rw div_self (@two_ne_zero' ℝ _ _ _); exact one_le_pi_div_two) lemma pi_le_four : π ≤ 4 := (div_le_div_right (show (0 : ℝ) < 2, by norm_num)).1 (calc π / 2 ≤ 2 : pi_div_two_le_two ... = 4 / 2 : by norm_num) lemma pi_pos : 0 < π := lt_of_lt_of_le (by norm_num) two_le_pi lemma pi_ne_zero : π ≠ 0 := ne_of_gt pi_pos lemma pi_div_two_pos : 0 < π / 2 := half_pos pi_pos lemma two_pi_pos : 0 < 2 * π := by linarith [pi_pos] end real namespace nnreal open real open_locale real nnreal /-- `π` considered as a nonnegative real. -/ noncomputable def pi : ℝ≥0 := ⟨π, real.pi_pos.le⟩ @[simp] lemma coe_real_pi : (pi : ℝ) = π := rfl lemma pi_pos : 0 < pi := by exact_mod_cast real.pi_pos lemma pi_ne_zero : pi ≠ 0 := pi_pos.ne' end nnreal namespace real open_locale real @[simp] lemma sin_pi : sin π = 0 := by rw [← mul_div_cancel_left π (@two_ne_zero ℝ _ _), two_mul, add_div, sin_add, cos_pi_div_two]; simp @[simp] lemma cos_pi : cos π = -1 := by rw [← mul_div_cancel_left π (@two_ne_zero ℝ _ _), mul_div_assoc, cos_two_mul, cos_pi_div_two]; simp [bit0, pow_add] @[simp] lemma sin_two_pi : sin (2 * π) = 0 := by simp [two_mul, sin_add] @[simp] lemma cos_two_pi : cos (2 * π) = 1 := by simp [two_mul, cos_add] lemma sin_nat_mul_pi (n : ℕ) : sin (n * π) = 0 := by induction n; simp [add_mul, sin_add, ] lemma sin_int_mul_pi (n : ℤ) : sin (n π) = 0 := by cases n; simp [add_mul, sin_add, , sin_nat_mul_pi] lemma cos_nat_mul_two_pi (n : ℕ) : cos (n (2 * π)) = 1 := by induction n; simp [, mul_add, cos_add, add_mul, cos_two_pi, sin_two_pi] lemma cos_int_mul_two_pi (n : ℤ) : cos (n (2 * π)) = 1 := by cases n; simp only [cos_nat_mul_two_pi, int.of_nat_eq_coe, int.neg_succ_of_nat_coe, int.cast_coe_nat, int.cast_neg, ← neg_mul_eq_neg_mul, cos_neg] lemma sin_add_pi (x : ℝ) : sin (x + π) = -sin x := by simp [sin_add] lemma sin_add_int_mul_two_pi (x : ℝ) (n : ℤ) : sin (x + n * (2 * π)) = sin x := begin rw [sin_add, cos_int_mul_two_pi, ← mul_assoc], rw_mod_cast sin_int_mul_pi (n2), simp, end lemma sin_sub_int_mul_two_pi (x : ℝ) (n : ℤ) : sin (x - n (2 * π)) = sin x := by simpa using sin_add_int_mul_two_pi x (-n) lemma sin_add_nat_mul_two_pi (x : ℝ) (n : ℕ) : sin (x + n * (2 * π)) = sin x := by convert sin_add_int_mul_two_pi x n lemma sin_sub_nat_mul_two_pi (x : ℝ) (n : ℕ) : sin (x - n * (2 * π)) = sin x := by convert sin_sub_int_mul_two_pi x n lemma sin_add_two_pi (x : ℝ) : sin (x + 2 * π) = sin x := by simp [sin_add] lemma sin_sub_two_pi (x : ℝ) : sin (x - 2 * π) = sin x := by simp [sin_sub] lemma cos_add_int_mul_two_pi (x : ℝ) (n : ℤ) : cos (x + n * (2 * π)) = cos x := begin rw [cos_add, cos_int_mul_two_pi, ← mul_assoc], rw_mod_cast sin_int_mul_pi (n2), simp, end lemma cos_sub_int_mul_two_pi (x : ℝ) (n : ℤ) : cos (x - n (2 * π)) = cos x := by simpa using cos_add_int_mul_two_pi x (-n) lemma cos_add_nat_mul_two_pi (x : ℝ) (n : ℕ) : cos (x + n * (2 * π)) = cos x := by convert cos_add_int_mul_two_pi x n lemma cos_sub_nat_mul_two_pi (x : ℝ) (n : ℕ) : cos (x - n * (2 * π)) = cos x := by convert cos_sub_int_mul_two_pi x n lemma cos_int_mul_two_pi_add_pi (n : ℤ) : cos (n * (2 * π) + π) = -1 := by simp [add_comm, cos_add_int_mul_two_pi] lemma cos_int_mul_two_pi_sub_pi (n : ℤ) : cos (n * (2 * π) - π) = -1 := by simp [sub_eq_neg_add, cos_add_int_mul_two_pi] lemma cos_nat_mul_two_pi_add_pi (n : ℕ) : cos (n * (2 * π) + π) = -1 := by convert cos_int_mul_two_pi_add_pi n lemma cos_nat_mul_two_pi_sub_pi (n : ℕ) : cos (n * (2 * π) - π) = -1 := by convert cos_int_mul_two_pi_sub_pi n lemma cos_add_two_pi (x : ℝ) : cos (x + 2 * π) = cos x := by simp [cos_add] lemma cos_sub_two_pi (x : ℝ) : cos (x - 2 * π) = cos x := by simp [cos_sub] lemma sin_pi_sub (x : ℝ) : sin (π - x) = sin x := by simp [sub_eq_add_neg, sin_add] lemma cos_add_pi (x : ℝ) : cos (x + π) = -cos x := by simp [cos_add] lemma cos_sub_pi (x : ℝ) : cos (x - π) = -cos x := by simp [cos_sub] lemma cos_pi_sub (x : ℝ) : cos (π - x) = -cos x := by simp [cos_sub] lemma sin_pos_of_pos_of_lt_pi {x : ℝ} (h0x : 0 < x) (hxp : x < π) : 0 < sin x := if hx2 : x ≤ 2 then sin_pos_of_pos_of_le_two h0x hx2 else have (2 : ℝ) + 2 = 4, from rfl, have π - x ≤ 2, from sub_le_iff_le_add.2 (le_trans pi_le_four (this ▸ add_le_add_left (le_of_not_ge hx2) _)), sin_pi_sub x ▸ sin_pos_of_pos_of_le_two (sub_pos.2 hxp) this lemma sin_pos_of_mem_Ioo {x : ℝ} (hx : x ∈ Ioo 0 π) : 0 < sin x := sin_pos_of_pos_of_lt_pi hx.1 hx.2 lemma sin_nonneg_of_mem_Icc {x : ℝ} (hx : x ∈ Icc 0 π) : 0 ≤ sin x := begin rw ← closure_Ioo pi_pos at hx, exact closure_lt_subset_le continuous_const continuous_sin (closure_mono (λ y, sin_pos_of_mem_Ioo) hx) end lemma sin_nonneg_of_nonneg_of_le_pi {x : ℝ} (h0x : 0 ≤ x) (hxp : x ≤ π) : 0 ≤ sin x := sin_nonneg_of_mem_Icc ⟨h0x, hxp⟩ lemma sin_neg_of_neg_of_neg_pi_lt {x : ℝ} (hx0 : x < 0) (hpx : -π < x) : sin x < 0 := neg_pos.1 $ sin_neg x ▸ sin_pos_of_pos_of_lt_pi (neg_pos.2 hx0) (neg_lt.1 hpx) lemma sin_nonpos_of_nonnpos_of_neg_pi_le {x : ℝ} (hx0 : x ≤ 0) (hpx : -π ≤ x) : sin x ≤ 0 := neg_nonneg.1 $ sin_neg x ▸ sin_nonneg_of_nonneg_of_le_pi (neg_nonneg.2 hx0) (neg_le.1 hpx) @[simp] lemma sin_pi_div_two : sin (π / 2) = 1 := have sin (π / 2) = 1 ∨ sin (π / 2) = -1 := by simpa [sq, mul_self_eq_one_iff] using sin_sq_add_cos_sq (π / 2), this.resolve_right (λ h, (show ¬(0 : ℝ) < -1, by norm_num) $ h ▸ sin_pos_of_pos_of_lt_pi pi_div_two_pos (half_lt_self pi_pos)) lemma sin_add_pi_div_two (x : ℝ) : sin (x + π / 2) = cos x := by simp [sin_add] lemma sin_sub_pi_div_two (x : ℝ) : sin (x - π / 2) = -cos x := by simp [sub_eq_add_neg, sin_add] lemma sin_pi_div_two_sub (x : ℝ) : sin (π / 2 - x) = cos x := by simp [sub_eq_add_neg, sin_add] lemma cos_add_pi_div_two (x : ℝ) : cos (x + π / 2) = -sin x := by simp [cos_add] lemma cos_sub_pi_div_two (x : ℝ) : cos (x - π / 2) = sin x := by simp [sub_eq_add_neg, cos_add] lemma cos_pi_div_two_sub (x : ℝ) : cos (π / 2 - x) = sin x := by rw [← cos_neg, neg_sub, cos_sub_pi_div_two] lemma cos_pos_of_mem_Ioo {x : ℝ} (hx : x ∈ Ioo (-(π / 2)) (π / 2)) : 0 < cos x := sin_add_pi_div_two x ▸ sin_pos_of_mem_Ioo ⟨by linarith [hx.1], by linarith [hx.2]⟩ lemma cos_nonneg_of_mem_Icc {x : ℝ} (hx : x ∈ Icc (-(π / 2)) (π / 2)) : 0 ≤ cos x := sin_add_pi_div_two x ▸ sin_nonneg_of_mem_Icc ⟨by linarith [hx.1], by linarith [hx.2]⟩ lemma cos_nonneg_of_neg_pi_div_two_le_of_le {x : ℝ} (hl : -(π / 2) ≤ x) (hu : x ≤ π / 2) : 0 ≤ cos x := cos_nonneg_of_mem_Icc ⟨hl, hu⟩ lemma cos_neg_of_pi_div_two_lt_of_lt {x : ℝ} (hx₁ : π / 2 < x) (hx₂ : x < π + π / 2) : cos x < 0 := neg_pos.1 $ cos_pi_sub x ▸ cos_pos_of_mem_Ioo ⟨by linarith, by linarith⟩ lemma cos_nonpos_of_pi_div_two_le_of_le {x : ℝ} (hx₁ : π / 2 ≤ x) (hx₂ : x ≤ π + π / 2) : cos x ≤ 0 := neg_nonneg.1 $ cos_pi_sub x ▸ cos_nonneg_of_mem_Icc ⟨by linarith, by linarith⟩ lemma sin_eq_sqrt_one_sub_cos_sq {x : ℝ} (hl : 0 ≤ x) (hu : x ≤ π) : sin x = sqrt (1 - cos x ^ 2) := by rw [← abs_sin_eq_sqrt_one_sub_cos_sq, abs_of_nonneg (sin_nonneg_of_nonneg_of_le_pi hl hu)] lemma cos_eq_sqrt_one_sub_sin_sq {x : ℝ} (hl : -(π / 2) ≤ x) (hu : x ≤ π / 2) : cos x = sqrt (1 - sin x ^ 2) := by rw [← abs_cos_eq_sqrt_one_sub_sin_sq, abs_of_nonneg (cos_nonneg_of_mem_Icc ⟨hl, hu⟩)] lemma sin_eq_zero_iff_of_lt_of_lt {x : ℝ} (hx₁ : -π < x) (hx₂ : x < π) : sin x = 0 ↔ x = 0 := ⟨λ h, le_antisymm (le_of_not_gt (λ h0, lt_irrefl (0 : ℝ) $ calc 0 < sin x : sin_pos_of_pos_of_lt_pi h0 hx₂ ... = 0 : h)) (le_of_not_gt (λ h0, lt_irrefl (0 : ℝ) $ calc 0 = sin x : h.symm ... < 0 : sin_neg_of_neg_of_neg_pi_lt h0 hx₁)), λ h, by simp [h]⟩ lemma sin_eq_zero_iff {x : ℝ} : sin x = 0 ↔ ∃ n : ℤ, (n : ℝ) * π = x := ⟨λ h, ⟨⌊x / π⌋, le_antisymm (sub_nonneg.1 (sub_floor_div_mul_nonneg _ pi_pos)) (sub_nonpos.1 $ le_of_not_gt $ λ h₃, (sin_pos_of_pos_of_lt_pi h₃ (sub_floor_div_mul_lt _ pi_pos)).ne (by simp [sub_eq_add_neg, sin_add, h, sin_int_mul_pi]))⟩, λ ⟨n, hn⟩, hn ▸ sin_int_mul_pi _⟩ lemma sin_ne_zero_iff {x : ℝ} : sin x ≠ 0 ↔ ∀ n : ℤ, (n : ℝ) * π ≠ x := by rw [← not_exists, not_iff_not, sin_eq_zero_iff] lemma sin_eq_zero_iff_cos_eq {x : ℝ} : sin x = 0 ↔ cos x = 1 ∨ cos x = -1 := by rw [← mul_self_eq_one_iff, ← sin_sq_add_cos_sq x, sq, sq, ← sub_eq_iff_eq_add, sub_self]; exact ⟨λ h, by rw [h, mul_zero], eq_zero_of_mul_self_eq_zero ∘ eq.symm⟩ lemma cos_eq_one_iff (x : ℝ) : cos x = 1 ↔ ∃ n : ℤ, (n : ℝ) * (2 * π) = x := ⟨λ h, let ⟨n, hn⟩ := sin_eq_zero_iff.1 (sin_eq_zero_iff_cos_eq.2 (or.inl h)) in ⟨n / 2, (int.mod_two_eq_zero_or_one n).elim (λ hn0, by rwa [← mul_assoc, ← @int.cast_two ℝ, ← int.cast_mul, int.div_mul_cancel ((int.dvd_iff_mod_eq_zero _ _).2 hn0)]) (λ hn1, by rw [← int.mod_add_div n 2, hn1, int.cast_add, int.cast_one, add_mul, one_mul, add_comm, mul_comm (2 : ℤ), int.cast_mul, mul_assoc, int.cast_two] at hn; rw [← hn, cos_int_mul_two_pi_add_pi] at h; exact absurd h (by norm_num))⟩, λ ⟨n, hn⟩, hn ▸ cos_int_mul_two_pi _⟩ lemma cos_eq_one_iff_of_lt_of_lt {x : ℝ} (hx₁ : -(2 * π) < x) (hx₂ : x < 2 * π) : cos x = 1 ↔ x = 0 := ⟨λ h, begin rcases (cos_eq_one_iff _).1 h with ⟨n, rfl⟩, rw [mul_lt_iff_lt_one_left two_pi_pos] at hx₂, rw [neg_lt, neg_mul_eq_neg_mul, mul_lt_iff_lt_one_left two_pi_pos] at hx₁, norm_cast at hx₁ hx₂, obtain rfl : n = 0 := le_antisymm (by linarith) (by linarith), simp end, λ h, by simp [h]⟩ lemma cos_lt_cos_of_nonneg_of_le_pi_div_two {x y : ℝ} (hx₁ : 0 ≤ x) (hy₂ : y ≤ π / 2) (hxy : x < y) : cos y < cos x := begin rw [← sub_lt_zero, cos_sub_cos], have : 0 < sin ((y + x) / 2), { refine sin_pos_of_pos_of_lt_pi _ _; linarith }, have : 0 < sin ((y - x) / 2), { refine sin_pos_of_pos_of_lt_pi _ _; linarith }, nlinarith, end lemma cos_lt_cos_of_nonneg_of_le_pi {x y : ℝ} (hx₁ : 0 ≤ x) (hy₂ : y ≤ π) (hxy : x < y) : cos y < cos x := match (le_total x (π / 2) : x ≤ π / 2 ∨ π / 2 ≤ x), le_total y (π / 2) with \| or.inl hx, or.inl hy := cos_lt_cos_of_nonneg_of_le_pi_div_two hx₁ hy hxy \| or.inl hx, or.inr hy := (lt_or_eq_of_le hx).elim (λ hx, calc cos y ≤ 0 : cos_nonpos_of_pi_div_two_le_of_le hy (by linarith [pi_pos]) ... < cos x : cos_pos_of_mem_Ioo ⟨by linarith, hx⟩) (λ hx, calc cos y < 0 : cos_neg_of_pi_div_two_lt_of_lt (by linarith) (by linarith [pi_pos]) ... = cos x : by rw [hx, cos_pi_div_two]) \| or.inr hx, or.inl hy := by linarith \| or.inr hx, or.inr hy := neg_lt_neg_iff.1 (by rw [← cos_pi_sub, ← cos_pi_sub]; apply cos_lt_cos_of_nonneg_of_le_pi_div_two; linarith) end lemma strict_mono_decr_on_cos : strict_mono_decr_on cos (Icc 0 π) := λ x hx y hy hxy, cos_lt_cos_of_nonneg_of_le_pi hx.1 hy.2 hxy lemma cos_le_cos_of_nonneg_of_le_pi {x y : ℝ} (hx₁ : 0 ≤ x) (hy₂ : y ≤ π) (hxy : x ≤ y) : cos y ≤ cos x := (strict_mono_decr_on_cos.le_iff_le ⟨hx₁.trans hxy, hy₂⟩ ⟨hx₁, hxy.trans hy₂⟩).2 hxy lemma sin_lt_sin_of_lt_of_le_pi_div_two {x y : ℝ} (hx₁ : -(π / 2) ≤ x) (hy₂ : y ≤ π / 2) (hxy : x < y) : sin x < sin y := by rw [← cos_sub_pi_div_two, ← cos_sub_pi_div_two, ← cos_neg (x - _), ← cos_neg (y - _)]; apply cos_lt_cos_of_nonneg_of_le_pi; linarith lemma strict_mono_incr_on_sin : strict_mono_incr_on sin (Icc (-(π / 2)) (π / 2)) := λ x hx y hy hxy, sin_lt_sin_of_lt_of_le_pi_div_two hx.1 hy.2 hxy lemma sin_le_sin_of_le_of_le_pi_div_two {x y : ℝ} (hx₁ : -(π / 2) ≤ x) (hy₂ : y ≤ π / 2) (hxy : x ≤ y) : sin x ≤ sin y := (strict_mono_incr_on_sin.le_iff_le ⟨hx₁, hxy.trans hy₂⟩ ⟨hx₁.trans hxy, hy₂⟩).2 hxy lemma inj_on_sin : inj_on sin (Icc (-(π / 2)) (π / 2)) := strict_mono_incr_on_sin.inj_on lemma inj_on_cos : inj_on cos (Icc 0 π) := strict_mono_decr_on_cos.inj_on lemma surj_on_sin : surj_on sin (Icc (-(π / 2)) (π / 2)) (Icc (-1) 1) := by simpa only [sin_neg, sin_pi_div_two] using intermediate_value_Icc (neg_le_self pi_div_two_pos.le) continuous_sin.continuous_on lemma surj_on_cos : surj_on cos (Icc 0 π) (Icc (-1) 1) := by simpa only [cos_zero, cos_pi] using intermediate_value_Icc' pi_pos.le continuous_cos.continuous_on lemma sin_mem_Icc (x : ℝ) : sin x ∈ Icc (-1 : ℝ) 1 := ⟨neg_one_le_sin x, sin_le_one x⟩ lemma cos_mem_Icc (x : ℝ) : cos x ∈ Icc (-1 : ℝ) 1 := ⟨neg_one_le_cos x, cos_le_one x⟩ lemma maps_to_sin (s : set ℝ) : maps_to sin s (Icc (-1 : ℝ) 1) := λ x _, sin_mem_Icc x lemma maps_to_cos (s : set ℝ) : maps_to cos s (Icc (-1 : ℝ) 1) := λ x _, cos_mem_Icc x lemma bij_on_sin : bij_on sin (Icc (-(π / 2)) (π / 2)) (Icc (-1) 1) := ⟨maps_to_sin _, inj_on_sin, surj_on_sin⟩ lemma bij_on_cos : bij_on cos (Icc 0 π) (Icc (-1) 1) := ⟨maps_to_cos _, inj_on_cos, surj_on_cos⟩ @[simp] lemma range_cos : range cos = (Icc (-1) 1 : set ℝ) := subset.antisymm (range_subset_iff.2 cos_mem_Icc) surj_on_cos.subset_range @[simp] lemma range_sin : range sin = (Icc (-1) 1 : set ℝ) := subset.antisymm (range_subset_iff.2 sin_mem_Icc) surj_on_sin.subset_range lemma range_cos_infinite : (range real.cos).infinite := by { rw real.range_cos, exact Icc.infinite (by norm_num) } lemma range_sin_infinite : (range real.sin).infinite := by { rw real.range_sin, exact Icc.infinite (by norm_num) } lemma sin_lt {x : ℝ} (h : 0 < x) : sin x < x := begin cases le_or_gt x 1 with h' h', { have hx : abs x = x := abs_of_nonneg (le_of_lt h), have : abs x ≤ 1, rwa [hx], have := sin_bound this, rw [abs_le] at this, have := this.2, rw [sub_le_iff_le_add', hx] at this, apply lt_of_le_of_lt this, rw [sub_add], apply lt_of_lt_of_le _ (le_of_eq (sub_zero x)), apply sub_lt_sub_left, rw [sub_pos, div_eq_mul_inv (x ^ 3)], apply mul_lt_mul', { rw [pow_succ x 3], refine le_trans _ (le_of_eq (one_mul _)), rw mul_le_mul_right, exact h', apply pow_pos h }, norm_num, norm_num, apply pow_pos h }, exact lt_of_le_of_lt (sin_le_one x) h' end /- note 1: this inequality is not tight, the tighter inequality is sin x > x - x ^ 3 / 6. note 2: this is also true for x > 1, but it's nontrivial for x just above 1. -/ lemma sin_gt_sub_cube {x : ℝ} (h : 0 < x) (h' : x ≤ 1) : x - x ^ 3 / 4 < sin x := begin have hx : abs x = x := abs_of_nonneg (le_of_lt h), have : abs x ≤ 1, rwa [hx], have := sin_bound this, rw [abs_le] at this, have := this.1, rw [le_sub_iff_add_le, hx] at this, refine lt_of_lt_of_le _ this, rw [add_comm, sub_add, sub_neg_eq_add], apply sub_lt_sub_left, apply add_lt_of_lt_sub_left, rw (show x ^ 3 / 4 - x ^ 3 / 6 = x ^ 3 * 12⁻¹, by simp [div_eq_mul_inv, ← mul_sub]; norm_num), apply mul_lt_mul', { rw [pow_succ x 3], refine le_trans _ (le_of_eq (one_mul _)), rw mul_le_mul_right, exact h', apply pow_pos h }, norm_num, norm_num, apply pow_pos h end section cos_div_sq variable (x : ℝ) /-- the series `sqrt_two_add_series x n` is `sqrt(2 + sqrt(2 + ... ))` with `n` square roots, starting with `x`. We define it here because `cos (pi / 2 ^ (n+1)) = sqrt_two_add_series 0 n / 2` -/ @[simp, pp_nodot] noncomputable def sqrt_two_add_series (x : ℝ) : ℕ → ℝ \| 0 := x \| (n+1) := sqrt (2 + sqrt_two_add_series n) lemma sqrt_two_add_series_zero : sqrt_two_add_series x 0 = x := by simp lemma sqrt_two_add_series_one : sqrt_two_add_series 0 1 = sqrt 2 := by simp lemma sqrt_two_add_series_two : sqrt_two_add_series 0 2 = sqrt (2 + sqrt 2) := by simp lemma sqrt_two_add_series_zero_nonneg : ∀(n : ℕ), 0 ≤ sqrt_two_add_series 0 n \| 0 := le_refl 0 \| (n+1) := sqrt_nonneg _ lemma sqrt_two_add_series_nonneg {x : ℝ} (h : 0 ≤ x) : ∀(n : ℕ), 0 ≤ sqrt_two_add_series x n \| 0 := h \| (n+1) := sqrt_nonneg _ lemma sqrt_two_add_series_lt_two : ∀(n : ℕ), sqrt_two_add_series 0 n < 2 \| 0 := by norm_num \| (n+1) := begin refine lt_of_lt_of_le _ (le_of_eq $ sqrt_sq $ le_of_lt zero_lt_two), rw [sqrt_two_add_series, sqrt_lt, ← lt_sub_iff_add_lt'], { refine (sqrt_two_add_series_lt_two n).trans_le _, norm_num }, { exact add_nonneg zero_le_two (sqrt_two_add_series_zero_nonneg n) } end lemma sqrt_two_add_series_succ (x : ℝ) : ∀(n : ℕ), sqrt_two_add_series x (n+1) = sqrt_two_add_series (sqrt (2 + x)) n \| 0 := rfl \| (n+1) := by rw [sqrt_two_add_series, sqrt_two_add_series_succ, sqrt_two_add_series] lemma sqrt_two_add_series_monotone_left {x y : ℝ} (h : x ≤ y) : ∀(n : ℕ), sqrt_two_add_series x n ≤ sqrt_two_add_series y n \| 0 := h \| (n+1) := begin rw [sqrt_two_add_series, sqrt_two_add_series], exact sqrt_le_sqrt (add_le_add_left (sqrt_two_add_series_monotone_left _) _) end @[simp] lemma cos_pi_over_two_pow : ∀(n : ℕ), cos (π / 2 ^ (n+1)) = sqrt_two_add_series 0 n / 2 \| 0 := by simp \| (n+1) := begin have : (2 : ℝ) ≠ 0 := two_ne_zero, symmetry, rw [div_eq_iff_mul_eq this], symmetry, rw [sqrt_two_add_series, sqrt_eq_iff_sq_eq, mul_pow, cos_sq, ←mul_div_assoc, nat.add_succ, pow_succ, mul_div_mul_left _ _ this, cos_pi_over_two_pow, add_mul], congr, { norm_num }, rw [mul_comm, sq, mul_assoc, ←mul_div_assoc, mul_div_cancel_left, ←mul_div_assoc, mul_div_cancel_left]; try { exact this }, apply add_nonneg, norm_num, apply sqrt_two_add_series_zero_nonneg, norm_num, apply le_of_lt, apply cos_pos_of_mem_Ioo ⟨_, _⟩, { transitivity (0 : ℝ), rw neg_lt_zero, apply pi_div_two_pos, apply div_pos pi_pos, apply pow_pos, norm_num }, apply div_lt_div' (le_refl π) _ pi_pos _, refine lt_of_le_of_lt (le_of_eq (pow_one _).symm) _, apply pow_lt_pow, norm_num, apply nat.succ_lt_succ, apply nat.succ_pos, all_goals {norm_num} end lemma sin_sq_pi_over_two_pow (n : ℕ) : sin (π / 2 ^ (n+1)) ^ 2 = 1 - (sqrt_two_add_series 0 n / 2) ^ 2 := by rw [sin_sq, cos_pi_over_two_pow] lemma sin_sq_pi_over_two_pow_succ (n : ℕ) : sin (π / 2 ^ (n+2)) ^ 2 = 1 / 2 - sqrt_two_add_series 0 n / 4 := begin rw [sin_sq_pi_over_two_pow, sqrt_two_add_series, div_pow, sq_sqrt, add_div, ←sub_sub], congr, norm_num, norm_num, apply add_nonneg, norm_num, apply sqrt_two_add_series_zero_nonneg, end @[simp] lemma sin_pi_over_two_pow_succ (n : ℕ) : sin (π / 2 ^ (n+2)) = sqrt (2 - sqrt_two_add_series 0 n) / 2 := begin symmetry, rw [div_eq_iff_mul_eq], symmetry, rw [sqrt_eq_iff_sq_eq, mul_pow, sin_sq_pi_over_two_pow_succ, sub_mul], { congr, norm_num, rw [mul_comm], convert mul_div_cancel' _ _, norm_num, norm_num }, { rw [sub_nonneg], apply le_of_lt, apply sqrt_two_add_series_lt_two }, apply le_of_lt, apply mul_pos, apply sin_pos_of_pos_of_lt_pi, { apply div_pos pi_pos, apply pow_pos, norm_num }, refine lt_of_lt_of_le _ (le_of_eq (div_one _)), rw [div_lt_div_left], refine lt_of_le_of_lt (le_of_eq (pow_zero 2).symm) _, apply pow_lt_pow, norm_num, apply nat.succ_pos, apply pi_pos, apply pow_pos, all_goals {norm_num} end @[simp] lemma cos_pi_div_four : cos (π / 4) = sqrt 2 / 2 := by { transitivity cos (π / 2 ^ 2), congr, norm_num, simp } @[simp] lemma sin_pi_div_four : sin (π / 4) = sqrt 2 / 2 := by { transitivity sin (π / 2 ^ 2), congr, norm_num, simp } @[simp] lemma cos_pi_div_eight : cos (π / 8) = sqrt (2 + sqrt 2) / 2 := by { transitivity cos (π / 2 ^ 3), congr, norm_num, simp } @[simp] lemma sin_pi_div_eight : sin (π / 8) = sqrt (2 - sqrt 2) / 2 := by { transitivity sin (π / 2 ^ 3), congr, norm_num, simp } @[simp] lemma cos_pi_div_sixteen : cos (π / 16) = sqrt (2 + sqrt (2 + sqrt 2)) / 2 := by { transitivity cos (π / 2 ^ 4), congr, norm_num, simp } @[simp] lemma sin_pi_div_sixteen : sin (π / 16) = sqrt (2 - sqrt (2 + sqrt 2)) / 2 := by { transitivity sin (π / 2 ^ 4), congr, norm_num, simp } @[simp] lemma cos_pi_div_thirty_two : cos (π / 32) = sqrt (2 + sqrt (2 + sqrt (2 + sqrt 2))) / 2 := by { transitivity cos (π / 2 ^ 5), congr, norm_num, simp } @[simp] lemma sin_pi_div_thirty_two : sin (π / 32) = sqrt (2 - sqrt (2 + sqrt (2 + sqrt 2))) / 2 := by { transitivity sin (π / 2 ^ 5), congr, norm_num, simp } -- This section is also a convenient location for other explicit values of `sin` and `cos`. /-- The cosine of `π / 3` is `1 / 2`. -/ @[simp] lemma cos_pi_div_three : cos (π / 3) = 1 / 2 := begin have h₁ : (2 * cos (π / 3) - 1) ^ 2 * (2 * cos (π / 3) + 2) = 0, { have : cos (3 * (π / 3)) = cos π := by { congr' 1, ring }, linarith [cos_pi, cos_three_mul (π / 3)] }, cases mul_eq_zero.mp h₁ with h h, { linarith [pow_eq_zero h] }, { have : cos π < cos (π / 3), { refine cos_lt_cos_of_nonneg_of_le_pi _ rfl.ge _; linarith [pi_pos] }, linarith [cos_pi] } end /-- The square of the cosine of `π / 6` is `3 / 4` (this is sometimes more convenient than the result for cosine itself). -/ lemma sq_cos_pi_div_six : cos (π / 6) ^ 2 = 3 / 4 := begin have h1 : cos (π / 6) ^ 2 = 1 / 2 + 1 / 2 / 2, { convert cos_sq (π / 6), have h2 : 2 * (π / 6) = π / 3 := by cancel_denoms, rw [h2, cos_pi_div_three] }, rw ← sub_eq_zero at h1 ⊢, convert h1 using 1, ring end /-- The cosine of `π / 6` is `√3 / 2`. -/ @[simp] lemma cos_pi_div_six : cos (π / 6) = (sqrt 3) / 2 := begin suffices : sqrt 3 = cos (π / 6) * 2, { field_simp [(by norm_num : 0 ≠ 2)], exact this.symm }, rw sqrt_eq_iff_sq_eq, { have h1 := (mul_right_inj' (by norm_num : (4:ℝ) ≠ 0)).mpr sq_cos_pi_div_six, rw ← sub_eq_zero at h1 ⊢, convert h1 using 1, ring }, { norm_num }, { have : 0 < cos (π / 6) := by { apply cos_pos_of_mem_Ioo; split; linarith [pi_pos] }, linarith }, end /-- The sine of `π / 6` is `1 / 2`. -/ @[simp] lemma sin_pi_div_six : sin (π / 6) = 1 / 2 := begin rw [← cos_pi_div_two_sub, ← cos_pi_div_three], congr, ring end /-- The square of the sine of `π / 3` is `3 / 4` (this is sometimes more convenient than the result for cosine itself). -/ lemma sq_sin_pi_div_three : sin (π / 3) ^ 2 = 3 / 4 := begin rw [← cos_pi_div_two_sub, ← sq_cos_pi_div_six], congr, ring end /-- The sine of `π / 3` is `√3 / 2`. -/ @[simp] lemma sin_pi_div_three : sin (π / 3) = (sqrt 3) / 2 := begin rw [← cos_pi_div_two_sub, ← cos_pi_div_six], congr, ring end end cos_div_sq /-- The type of angles -/ def angle : Type := quotient_add_group.quotient (add_subgroup.gmultiples (2 * π)) namespace angle instance angle.add_comm_group : add_comm_group angle := quotient_add_group.add_comm_group _ instance : inhabited angle := ⟨0⟩ instance angle.has_coe : has_coe ℝ angle := ⟨quotient.mk'⟩ @[simp] lemma coe_zero : ↑(0 : ℝ) = (0 : angle) := rfl @[simp] lemma coe_add (x y : ℝ) : ↑(x + y : ℝ) = (↑x + ↑y : angle) := rfl @[simp] lemma coe_neg (x : ℝ) : ↑(-x : ℝ) = -(↑x : angle) := rfl @[simp] lemma coe_sub (x y : ℝ) : ↑(x - y : ℝ) = (↑x - ↑y : angle) := by rw [sub_eq_add_neg, sub_eq_add_neg, coe_add, coe_neg] @[simp, norm_cast] lemma coe_nat_mul_eq_nsmul (x : ℝ) (n : ℕ) : ↑((n : ℝ) * x) = n • (↑x : angle) := by simpa using add_monoid_hom.map_nsmul ⟨coe, coe_zero, coe_add⟩ _ _ @[simp, norm_cast] lemma coe_int_mul_eq_gsmul (x : ℝ) (n : ℤ) : ↑((n : ℝ) * x : ℝ) = n • (↑x : angle) := by simpa using add_monoid_hom.map_gsmul ⟨coe, coe_zero, coe_add⟩ _ _ @[simp] lemma coe_two_pi : ↑(2 * π : ℝ) = (0 : angle) := quotient.sound' ⟨-1, show (-1 : ℤ) • (2 * π) = _, by rw [neg_one_gsmul, add_zero]⟩ lemma angle_eq_iff_two_pi_dvd_sub {ψ θ : ℝ} : (θ : angle) = ψ ↔ ∃ k : ℤ, θ - ψ = 2 * π * k := by simp only [quotient_add_group.eq, add_subgroup.gmultiples_eq_closure, add_subgroup.mem_closure_singleton, gsmul_eq_mul', (sub_eq_neg_add _ _).symm, eq_comm] theorem cos_eq_iff_eq_or_eq_neg {θ ψ : ℝ} : cos θ = cos ψ ↔ (θ : angle) = ψ ∨ (θ : angle) = -ψ := begin split, { intro Hcos, rw [← sub_eq_zero, cos_sub_cos, mul_eq_zero, mul_eq_zero, neg_eq_zero, eq_false_intro two_ne_zero, false_or, sin_eq_zero_iff, sin_eq_zero_iff] at Hcos, rcases Hcos with ⟨n, hn⟩ \| ⟨n, hn⟩, { right, rw [eq_div_iff_mul_eq (@two_ne_zero ℝ _ _), ← sub_eq_iff_eq_add] at hn, rw [← hn, coe_sub, eq_neg_iff_add_eq_zero, sub_add_cancel, mul_assoc, coe_int_mul_eq_gsmul, mul_comm, coe_two_pi, gsmul_zero] }, { left, rw [eq_div_iff_mul_eq (@two_ne_zero ℝ _ _), eq_sub_iff_add_eq] at hn, rw [← hn, coe_add, mul_assoc, coe_int_mul_eq_gsmul, mul_comm, coe_two_pi, gsmul_zero, zero_add] }, apply_instance, }, { rw [angle_eq_iff_two_pi_dvd_sub, ← coe_neg, angle_eq_iff_two_pi_dvd_sub], rintro (⟨k, H⟩ \| ⟨k, H⟩), rw [← sub_eq_zero, cos_sub_cos, H, mul_assoc 2 π k, mul_div_cancel_left _ (@two_ne_zero ℝ _ _), mul_comm π _, sin_int_mul_pi, mul_zero], rw [← sub_eq_zero, cos_sub_cos, ← sub_neg_eq_add, H, mul_assoc 2 π k, mul_div_cancel_left _ (@two_ne_zero ℝ _ _), mul_comm π _, sin_int_mul_pi, mul_zero, zero_mul] } end theorem sin_eq_iff_eq_or_add_eq_pi {θ ψ : ℝ} : sin θ = sin ψ ↔ (θ : angle) = ψ ∨ (θ : angle) + ψ = π := begin split, { intro Hsin, rw [← cos_pi_div_two_sub, ← cos_pi_div_two_sub] at Hsin, cases cos_eq_iff_eq_or_eq_neg.mp Hsin with h h, { left, rw [coe_sub, coe_sub] at h, exact sub_right_inj.1 h }, right, rw [coe_sub, coe_sub, eq_neg_iff_add_eq_zero, add_sub, sub_add_eq_add_sub, ← coe_add, add_halves, sub_sub, sub_eq_zero] at h, exact h.symm }, { rw [angle_eq_iff_two_pi_dvd_sub, ←eq_sub_iff_add_eq, ←coe_sub, angle_eq_iff_two_pi_dvd_sub], rintro (⟨k, H⟩ \| ⟨k, H⟩), rw [← sub_eq_zero, sin_sub_sin, H, mul_assoc 2 π k, mul_div_cancel_left _ (@two_ne_zero ℝ _ _), mul_comm π _, sin_int_mul_pi, mul_zero, zero_mul], have H' : θ + ψ = (2 * k) * π + π := by rwa [←sub_add, sub_add_eq_add_sub, sub_eq_iff_eq_add, mul_assoc, mul_comm π _, ←mul_assoc] at H, rw [← sub_eq_zero, sin_sub_sin, H', add_div, mul_assoc 2 _ π, mul_div_cancel_left _ (@two_ne_zero ℝ _ _), cos_add_pi_div_two, sin_int_mul_pi, neg_zero, mul_zero] } end theorem cos_sin_inj {θ ψ : ℝ} (Hcos : cos θ = cos ψ) (Hsin : sin θ = sin ψ) : (θ : angle) = ψ := begin cases cos_eq_iff_eq_or_eq_neg.mp Hcos with hc hc, { exact hc }, cases sin_eq_iff_eq_or_add_eq_pi.mp Hsin with hs hs, { exact hs }, rw [eq_neg_iff_add_eq_zero, hs] at hc, cases quotient.exact' hc with n hn, change n • _ = _ at hn, rw [← neg_one_mul, add_zero, ← sub_eq_zero, gsmul_eq_mul, ← mul_assoc, ← sub_mul, mul_eq_zero, eq_false_intro (ne_of_gt pi_pos), or_false, sub_neg_eq_add, ← int.cast_zero, ← int.cast_one, ← int.cast_bit0, ← int.cast_mul, ← int.cast_add, int.cast_inj] at hn, have : (n * 2 + 1) % (2:ℤ) = 0 % (2:ℤ) := congr_arg (%(2:ℤ)) hn, rw [add_comm, int.add_mul_mod_self] at this, exact absurd this one_ne_zero end end angle /-- `real.sin` as an `order_iso` between `[-(π / 2), π / 2]` and `[-1, 1]`. -/ def sin_order_iso : Icc (-(π / 2)) (π / 2) ≃o Icc (-1:ℝ) 1 := (strict_mono_incr_on_sin.order_iso _ _).trans $ order_iso.set_congr _ _ bij_on_sin.image_eq @[simp] lemma coe_sin_order_iso_apply (x : Icc (-(π / 2)) (π / 2)) : (sin_order_iso x : ℝ) = sin x := rfl lemma sin_order_iso_apply (x : Icc (-(π / 2)) (π / 2)) : sin_order_iso x = ⟨sin x, sin_mem_Icc x⟩ := rfl /-- Inverse of the `sin` function, returns values in the range `-π / 2 ≤ arcsin x ≤ π / 2`. It defaults to `-π / 2` on `(-∞, -1)` and to `π / 2` to `(1, ∞)`. -/ @[pp_nodot] noncomputable def arcsin : ℝ → ℝ := coe ∘ Icc_extend (neg_le_self zero_le_one) sin_order_iso.symm lemma arcsin_mem_Icc (x : ℝ) : arcsin x ∈ Icc (-(π / 2)) (π / 2) := subtype.coe_prop _ @[simp] lemma range_arcsin : range arcsin = Icc (-(π / 2)) (π / 2) := by { rw [arcsin, range_comp coe], simp [Icc] } lemma arcsin_le_pi_div_two (x : ℝ) : arcsin x ≤ π / 2 := (arcsin_mem_Icc x).2 lemma neg_pi_div_two_le_arcsin (x : ℝ) : -(π / 2) ≤ arcsin x := (arcsin_mem_Icc x).1 lemma arcsin_proj_Icc (x : ℝ) : arcsin (proj_Icc (-1) 1 (neg_le_self $ @zero_le_one ℝ _) x) = arcsin x := by rw [arcsin, function.comp_app, Icc_extend_coe, function.comp_app, Icc_extend] lemma sin_arcsin' {x : ℝ} (hx : x ∈ Icc (-1 : ℝ) 1) : sin (arcsin x) = x := by simpa [arcsin, Icc_extend_of_mem _ _ hx, -order_iso.apply_symm_apply] using subtype.ext_iff.1 (sin_order_iso.apply_symm_apply ⟨x, hx⟩) lemma sin_arcsin {x : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) : sin (arcsin x) = x := sin_arcsin' ⟨hx₁, hx₂⟩ lemma arcsin_sin' {x : ℝ} (hx : x ∈ Icc (-(π / 2)) (π / 2)) : arcsin (sin x) = x := inj_on_sin (arcsin_mem_Icc _) hx $ by rw [sin_arcsin (neg_one_le_sin _) (sin_le_one _)] lemma arcsin_sin {x : ℝ} (hx₁ : -(π / 2) ≤ x) (hx₂ : x ≤ π / 2) : arcsin (sin x) = x := arcsin_sin' ⟨hx₁, hx₂⟩ lemma strict_mono_incr_on_arcsin : strict_mono_incr_on arcsin (Icc (-1) 1) := (subtype.strict_mono_coe _).comp_strict_mono_incr_on $ sin_order_iso.symm.strict_mono.strict_mono_incr_on_Icc_extend _ lemma monotone_arcsin : monotone arcsin := (subtype.mono_coe _).comp $ sin_order_iso.symm.monotone.Icc_extend _ lemma inj_on_arcsin : inj_on arcsin (Icc (-1) 1) := strict_mono_incr_on_arcsin.inj_on lemma arcsin_inj {x y : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) (hy₁ : -1 ≤ y) (hy₂ : y ≤ 1) : arcsin x = arcsin y ↔ x = y := inj_on_arcsin.eq_iff ⟨hx₁, hx₂⟩ ⟨hy₁, hy₂⟩ @[continuity] lemma continuous_arcsin : continuous arcsin := continuous_subtype_coe.comp sin_order_iso.symm.continuous.Icc_extend lemma continuous_at_arcsin {x : ℝ} : continuous_at arcsin x := continuous_arcsin.continuous_at lemma arcsin_eq_of_sin_eq {x y : ℝ} (h₁ : sin x = y) (h₂ : x ∈ Icc (-(π / 2)) (π / 2)) : arcsin y = x := begin subst y, exact inj_on_sin (arcsin_mem_Icc _) h₂ (sin_arcsin' (sin_mem_Icc x)) end @[simp] lemma arcsin_zero : arcsin 0 = 0 := arcsin_eq_of_sin_eq sin_zero ⟨neg_nonpos.2 pi_div_two_pos.le, pi_div_two_pos.le⟩ @[simp] lemma arcsin_one : arcsin 1 = π / 2 := arcsin_eq_of_sin_eq sin_pi_div_two $ right_mem_Icc.2 (neg_le_self pi_div_two_pos.le) lemma arcsin_of_one_le {x : ℝ} (hx : 1 ≤ x) : arcsin x = π / 2 := by rw [← arcsin_proj_Icc, proj_Icc_of_right_le _ hx, subtype.coe_mk, arcsin_one] lemma arcsin_neg_one : arcsin (-1) = -(π / 2) := arcsin_eq_of_sin_eq (by rw [sin_neg, sin_pi_div_two]) $ left_mem_Icc.2 (neg_le_self pi_div_two_pos.le) lemma arcsin_of_le_neg_one {x : ℝ} (hx : x ≤ -1) : arcsin x = -(π / 2) := by rw [← arcsin_proj_Icc, proj_Icc_of_le_left _ hx, subtype.coe_mk, arcsin_neg_one] @[simp] lemma arcsin_neg (x : ℝ) : arcsin (-x) = -arcsin x := begin cases le_total x (-1) with hx₁ hx₁, { rw [arcsin_of_le_neg_one hx₁, neg_neg, arcsin_of_one_le (le_neg.2 hx₁)] }, cases le_total 1 x with hx₂ hx₂, { rw [arcsin_of_one_le hx₂, arcsin_of_le_neg_one (neg_le_neg hx₂)] }, refine arcsin_eq_of_sin_eq _ _, { rw [sin_neg, sin_arcsin hx₁ hx₂] }, { exact ⟨neg_le_neg (arcsin_le_pi_div_two _), neg_le.2 (neg_pi_div_two_le_arcsin _)⟩ } end lemma arcsin_le_iff_le_sin {x y : ℝ} (hx : x ∈ Icc (-1 : ℝ) 1) (hy : y ∈ Icc (-(π / 2)) (π / 2)) : arcsin x ≤ y ↔ x ≤ sin y := by rw [← arcsin_sin' hy, strict_mono_incr_on_arcsin.le_iff_le hx (sin_mem_Icc _), arcsin_sin' hy] lemma arcsin_le_iff_le_sin' {x y : ℝ} (hy : y ∈ Ico (-(π / 2)) (π / 2)) : arcsin x ≤ y ↔ x ≤ sin y := begin cases le_total x (-1) with hx₁ hx₁, { simp [arcsin_of_le_neg_one hx₁, hy.1, hx₁.trans (neg_one_le_sin _)] }, cases lt_or_le 1 x with hx₂ hx₂, { simp [arcsin_of_one_le hx₂.le, hy.2.not_le, (sin_le_one y).trans_lt hx₂] }, exact arcsin_le_iff_le_sin ⟨hx₁, hx₂⟩ (mem_Icc_of_Ico hy) end lemma le_arcsin_iff_sin_le {x y : ℝ} (hx : x ∈ Icc (-(π / 2)) (π / 2)) (hy : y ∈ Icc (-1 : ℝ) 1) : x ≤ arcsin y ↔ sin x ≤ y := by rw [← neg_le_neg_iff, ← arcsin_neg, arcsin_le_iff_le_sin ⟨neg_le_neg hy.2, neg_le.2 hy.1⟩ ⟨neg_le_neg hx.2, neg_le.2 hx.1⟩, sin_neg, neg_le_neg_iff] lemma le_arcsin_iff_sin_le' {x y : ℝ} (hx : x ∈ Ioc (-(π / 2)) (π / 2)) : x ≤ arcsin y ↔ sin x ≤ y := by rw [← neg_le_neg_iff, ← arcsin_neg, arcsin_le_iff_le_sin' ⟨neg_le_neg hx.2, neg_lt.2 hx.1⟩, sin_neg, neg_le_neg_iff] lemma arcsin_lt_iff_lt_sin {x y : ℝ} (hx : x ∈ Icc (-1 : ℝ) 1) (hy : y ∈ Icc (-(π / 2)) (π / 2)) : arcsin x < y ↔ x < sin y := not_le.symm.trans $ (not_congr $ le_arcsin_iff_sin_le hy hx).trans not_le lemma arcsin_lt_iff_lt_sin' {x y : ℝ} (hy : y ∈ Ioc (-(π / 2)) (π / 2)) : arcsin x < y ↔ x < sin y := not_le.symm.trans $ (not_congr $ le_arcsin_iff_sin_le' hy).trans not_le lemma lt_arcsin_iff_sin_lt {x y : ℝ} (hx : x ∈ Icc (-(π / 2)) (π / 2)) (hy : y ∈ Icc (-1 : ℝ) 1) : x < arcsin y ↔ sin x < y := not_le.symm.trans $ (not_congr $ arcsin_le_iff_le_sin hy hx).trans not_le lemma lt_arcsin_iff_sin_lt' {x y : ℝ} (hx : x ∈ Ico (-(π / 2)) (π / 2)) : x < arcsin y ↔ sin x < y := not_le.symm.trans $ (not_congr $ arcsin_le_iff_le_sin' hx).trans not_le lemma arcsin_eq_iff_eq_sin {x y : ℝ} (hy : y ∈ Ioo (-(π / 2)) (π / 2)) : arcsin x = y ↔ x = sin y := by simp only [le_antisymm_iff, arcsin_le_iff_le_sin' (mem_Ico_of_Ioo hy), le_arcsin_iff_sin_le' (mem_Ioc_of_Ioo hy)] @[simp] lemma arcsin_nonneg {x : ℝ} : 0 ≤ arcsin x ↔ 0 ≤ x := (le_arcsin_iff_sin_le' ⟨neg_lt_zero.2 pi_div_two_pos, pi_div_two_pos.le⟩).trans $ by rw [sin_zero] @[simp] lemma arcsin_nonpos {x : ℝ} : arcsin x ≤ 0 ↔ x ≤ 0 := neg_nonneg.symm.trans $ arcsin_neg x ▸ arcsin_nonneg.trans neg_nonneg @[simp] lemma arcsin_eq_zero_iff {x : ℝ} : arcsin x = 0 ↔ x = 0 := by simp [le_antisymm_iff] @[simp] lemma zero_eq_arcsin_iff {x} : 0 = arcsin x ↔ x = 0 := eq_comm.trans arcsin_eq_zero_iff @[simp] lemma arcsin_pos {x : ℝ} : 0 < arcsin x ↔ 0 < x := lt_iff_lt_of_le_iff_le arcsin_nonpos @[simp] lemma arcsin_lt_zero {x : ℝ} : arcsin x < 0 ↔ x < 0 := lt_iff_lt_of_le_iff_le arcsin_nonneg @[simp] lemma arcsin_lt_pi_div_two {x : ℝ} : arcsin x < π / 2 ↔ x < 1 := (arcsin_lt_iff_lt_sin' (right_mem_Ioc.2 $ neg_lt_self pi_div_two_pos)).trans $ by rw sin_pi_div_two @[simp] lemma neg_pi_div_two_lt_arcsin {x : ℝ} : -(π / 2) < arcsin x ↔ -1 < x := (lt_arcsin_iff_sin_lt' $ left_mem_Ico.2 $ neg_lt_self pi_div_two_pos).trans $ by rw [sin_neg, sin_pi_div_two] @[simp] lemma arcsin_eq_pi_div_two {x : ℝ} : arcsin x = π / 2 ↔ 1 ≤ x := ⟨λ h, not_lt.1 $ λ h', (arcsin_lt_pi_div_two.2 h').ne h, arcsin_of_one_le⟩ @[simp] lemma pi_div_two_eq_arcsin {x} : π / 2 = arcsin x ↔ 1 ≤ x := eq_comm.trans arcsin_eq_pi_div_two @[simp] lemma pi_div_two_le_arcsin {x} : π / 2 ≤ arcsin x ↔ 1 ≤ x := (arcsin_le_pi_div_two x).le_iff_eq.trans pi_div_two_eq_arcsin @[simp] lemma arcsin_eq_neg_pi_div_two {x : ℝ} : arcsin x = -(π / 2) ↔ x ≤ -1 := ⟨λ h, not_lt.1 $ λ h', (neg_pi_div_two_lt_arcsin.2 h').ne' h, arcsin_of_le_neg_one⟩ @[simp] lemma neg_pi_div_two_eq_arcsin {x} : -(π / 2) = arcsin x ↔ x ≤ -1 := eq_comm.trans arcsin_eq_neg_pi_div_two @[simp] lemma arcsin_le_neg_pi_div_two {x} : arcsin x ≤ -(π / 2) ↔ x ≤ -1 := (neg_pi_div_two_le_arcsin x).le_iff_eq.trans arcsin_eq_neg_pi_div_two lemma maps_to_sin_Ioo : maps_to sin (Ioo (-(π / 2)) (π / 2)) (Ioo (-1) 1) := λ x h, by rwa [mem_Ioo, ← arcsin_lt_pi_div_two, ← neg_pi_div_two_lt_arcsin, arcsin_sin h.1.le h.2.le] /-- `real.sin` as a `local_homeomorph` between `(-π / 2, π / 2)` and `(-1, 1)`. -/ @[simp] def sin_local_homeomorph : local_homeomorph ℝ ℝ := { to_fun := sin, inv_fun := arcsin, source := Ioo (-(π / 2)) (π / 2), target := Ioo (-1) 1, map_source' := maps_to_sin_Ioo, map_target' := λ y hy, ⟨neg_pi_div_two_lt_arcsin.2 hy.1, arcsin_lt_pi_div_two.2 hy.2⟩, left_inv' := λ x hx, arcsin_sin hx.1.le hx.2.le, right_inv' := λ y hy, sin_arcsin hy.1.le hy.2.le, open_source := is_open_Ioo, open_target := is_open_Ioo, continuous_to_fun := continuous_sin.continuous_on, continuous_inv_fun := continuous_arcsin.continuous_on } lemma cos_arcsin_nonneg (x : ℝ) : 0 ≤ cos (arcsin x) := cos_nonneg_of_mem_Icc ⟨neg_pi_div_two_le_arcsin _, arcsin_le_pi_div_two _⟩ lemma cos_arcsin {x : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) : cos (arcsin x) = sqrt (1 - x ^ 2) := have sin (arcsin x) ^ 2 + cos (arcsin x) ^ 2 = 1 := sin_sq_add_cos_sq (arcsin x), begin rw [← eq_sub_iff_add_eq', ← sqrt_inj (sq_nonneg _) (sub_nonneg.2 (sin_sq_le_one (arcsin x))), sq, sqrt_mul_self (cos_arcsin_nonneg _)] at this, rw [this, sin_arcsin hx₁ hx₂], end lemma deriv_arcsin_aux {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) : has_strict_deriv_at arcsin (1 / sqrt (1 - x ^ 2)) x ∧ times_cont_diff_at ℝ ⊤ arcsin x := begin cases h₁.lt_or_lt with h₁ h₁, { have : 1 - x ^ 2 < 0, by nlinarith [h₁], rw [sqrt_eq_zero'.2 this.le, div_zero], have : arcsin =ᶠ[𝓝 x] λ _, -(π / 2) := (gt_mem_nhds h₁).mono (λ y hy, arcsin_of_le_neg_one hy.le), exact ⟨(has_strict_deriv_at_const _ _).congr_of_eventually_eq this.symm, times_cont_diff_at_const.congr_of_eventually_eq this⟩ }, cases h₂.lt_or_lt with h₂ h₂, { have : 0 < sqrt (1 - x ^ 2) := sqrt_pos.2 (by nlinarith [h₁, h₂]), simp only [← cos_arcsin h₁.le h₂.le, one_div] at this ⊢, exact ⟨sin_local_homeomorph.has_strict_deriv_at_symm ⟨h₁, h₂⟩ this.ne' (has_strict_deriv_at_sin _), sin_local_homeomorph.times_cont_diff_at_symm_deriv this.ne' ⟨h₁, h₂⟩ (has_deriv_at_sin _) times_cont_diff_sin.times_cont_diff_at⟩ }, { have : 1 - x ^ 2 < 0, by nlinarith [h₂], rw [sqrt_eq_zero'.2 this.le, div_zero], have : arcsin =ᶠ[𝓝 x] λ _, π / 2 := (lt_mem_nhds h₂).mono (λ y hy, arcsin_of_one_le hy.le), exact ⟨(has_strict_deriv_at_const _ _).congr_of_eventually_eq this.symm, times_cont_diff_at_const.congr_of_eventually_eq this⟩ } end lemma has_strict_deriv_at_arcsin {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) : has_strict_deriv_at arcsin (1 / sqrt (1 - x ^ 2)) x := (deriv_arcsin_aux h₁ h₂).1 lemma has_deriv_at_arcsin {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) : has_deriv_at arcsin (1 / sqrt (1 - x ^ 2)) x := (has_strict_deriv_at_arcsin h₁ h₂).has_deriv_at lemma times_cont_diff_at_arcsin {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) {n : with_top ℕ} : times_cont_diff_at ℝ n arcsin x := (deriv_arcsin_aux h₁ h₂).2.of_le le_top lemma has_deriv_within_at_arcsin_Ici {x : ℝ} (h : x ≠ -1) : has_deriv_within_at arcsin (1 / sqrt (1 - x ^ 2)) (Ici x) x := begin rcases em (x = 1) with (rfl\|h'), { convert (has_deriv_within_at_const _ _ (π / 2)).congr _ _; simp [arcsin_of_one_le] { contextual := tt } }, { exact (has_deriv_at_arcsin h h').has_deriv_within_at } end lemma has_deriv_within_at_arcsin_Iic {x : ℝ} (h : x ≠ 1) : has_deriv_within_at arcsin (1 / sqrt (1 - x ^ 2)) (Iic x) x := begin rcases em (x = -1) with (rfl\|h'), { convert (has_deriv_within_at_const _ _ (-(π / 2))).congr _ _; simp [arcsin_of_le_neg_one] { contextual := tt } }, { exact (has_deriv_at_arcsin h' h).has_deriv_within_at } end lemma differentiable_within_at_arcsin_Ici {x : ℝ} : differentiable_within_at ℝ arcsin (Ici x) x ↔ x ≠ -1 := begin refine ⟨_, λ h, (has_deriv_within_at_arcsin_Ici h).differentiable_within_at⟩, rintro h rfl, have : sin ∘ arcsin =ᶠ[𝓝[Ici (-1:ℝ)] (-1)] id, { filter_upwards [Icc_mem_nhds_within_Ici ⟨le_rfl, neg_lt_self (@zero_lt_one ℝ _ _)⟩], exact λ x, sin_arcsin' }, have := h.has_deriv_within_at.sin.congr_of_eventually_eq this.symm (by simp), simpa using (unique_diff_on_Ici _ _ left_mem_Ici).eq_deriv _ this (has_deriv_within_at_id _ _) end lemma differentiable_within_at_arcsin_Iic {x : ℝ} : differentiable_within_at ℝ arcsin (Iic x) x ↔ x ≠ 1 := begin refine ⟨λ h, _, λ h, (has_deriv_within_at_arcsin_Iic h).differentiable_within_at⟩, rw [← neg_neg x, ← image_neg_Ici] at h, have := (h.comp (-x) differentiable_within_at_id.neg (maps_to_image _ _)).neg, simpa [(∘), differentiable_within_at_arcsin_Ici] using this end lemma differentiable_at_arcsin {x : ℝ} : differentiable_at ℝ arcsin x ↔ x ≠ -1 ∧ x ≠ 1 := ⟨λ h, ⟨differentiable_within_at_arcsin_Ici.1 h.differentiable_within_at, differentiable_within_at_arcsin_Iic.1 h.differentiable_within_at⟩, λ h, (has_deriv_at_arcsin h.1 h.2).differentiable_at⟩ @[simp] lemma deriv_arcsin : deriv arcsin = λ x, 1 / sqrt (1 - x ^ 2) := begin funext x, by_cases h : x ≠ -1 ∧ x ≠ 1, { exact (has_deriv_at_arcsin h.1 h.2).deriv }, { rw [deriv_zero_of_not_differentiable_at (mt differentiable_at_arcsin.1 h)], simp only [not_and_distrib, ne.def, not_not] at h, rcases h with (rfl\|rfl); simp } end lemma differentiable_on_arcsin : differentiable_on ℝ arcsin {-1, 1}ᶜ := λ x hx, (differentiable_at_arcsin.2 ⟨λ h, hx (or.inl h), λ h, hx (or.inr h)⟩).differentiable_within_at lemma times_cont_diff_on_arcsin {n : with_top ℕ} : times_cont_diff_on ℝ n arcsin {-1, 1}ᶜ := λ x hx, (times_cont_diff_at_arcsin (mt or.inl hx) (mt or.inr hx)).times_cont_diff_within_at lemma times_cont_diff_at_arcsin_iff {x : ℝ} {n : with_top ℕ} : times_cont_diff_at ℝ n arcsin x ↔ n = 0 ∨ (x ≠ -1 ∧ x ≠ 1) := ⟨λ h, or_iff_not_imp_left.2 $ λ hn, differentiable_at_arcsin.1 $ h.differentiable_at $ with_top.one_le_iff_pos.2 (pos_iff_ne_zero.2 hn), λ h, h.elim (λ hn, hn.symm ▸ (times_cont_diff_zero.2 continuous_arcsin).times_cont_diff_at) $ λ hx, times_cont_diff_at_arcsin hx.1 hx.2⟩ lemma measurable_arcsin : measurable arcsin := continuous_arcsin.measurable /-- Inverse of the `cos` function, returns values in the range `0 ≤ arccos x` and `arccos x ≤ π`. If the argument is not between `-1` and `1` it defaults to `π / 2` -/ @[pp_nodot] noncomputable def arccos (x : ℝ) : ℝ := π / 2 - arcsin x lemma arccos_eq_pi_div_two_sub_arcsin (x : ℝ) : arccos x = π / 2 - arcsin x := rfl lemma arcsin_eq_pi_div_two_sub_arccos (x : ℝ) : arcsin x = π / 2 - arccos x := by simp [arccos] lemma arccos_le_pi (x : ℝ) : arccos x ≤ π := by unfold arccos; linarith [neg_pi_div_two_le_arcsin x] lemma arccos_nonneg (x : ℝ) : 0 ≤ arccos x := by unfold arccos; linarith [arcsin_le_pi_div_two x] lemma cos_arccos {x : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) : cos (arccos x) = x := by rw [arccos, cos_pi_div_two_sub, sin_arcsin hx₁ hx₂] lemma arccos_cos {x : ℝ} (hx₁ : 0 ≤ x) (hx₂ : x ≤ π) : arccos (cos x) = x := by rw [arccos, ← sin_pi_div_two_sub, arcsin_sin]; simp [sub_eq_add_neg]; linarith lemma strict_mono_decr_on_arccos : strict_mono_decr_on arccos (Icc (-1) 1) := λ x hx y hy h, sub_lt_sub_left (strict_mono_incr_on_arcsin hx hy h) _ lemma arccos_inj_on : inj_on arccos (Icc (-1) 1) := strict_mono_decr_on_arccos.inj_on lemma arccos_inj {x y : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) (hy₁ : -1 ≤ y) (hy₂ : y ≤ 1) : arccos x = arccos y ↔ x = y := arccos_inj_on.eq_iff ⟨hx₁, hx₂⟩ ⟨hy₁, hy₂⟩ @[simp] lemma arccos_zero : arccos 0 = π / 2 := by simp [arccos] @[simp] lemma arccos_one : arccos 1 = 0 := by simp [arccos] @[simp] lemma arccos_neg_one : arccos (-1) = π := by simp [arccos, add_halves] @[simp] lemma arccos_eq_zero {x} : arccos x = 0 ↔ 1 ≤ x := by simp [arccos, sub_eq_zero] @[simp] lemma arccos_eq_pi_div_two {x} : arccos x = π / 2 ↔ x = 0 := by simp [arccos, sub_eq_iff_eq_add] @[simp] lemma arccos_eq_pi {x} : arccos x = π ↔ x ≤ -1 := by rw [arccos, sub_eq_iff_eq_add, ← sub_eq_iff_eq_add', div_two_sub_self, neg_pi_div_two_eq_arcsin] lemma arccos_neg (x : ℝ) : arccos (-x) = π - arccos x := by rw [← add_halves π, arccos, arcsin_neg, arccos, add_sub_assoc, sub_sub_self, sub_neg_eq_add] lemma sin_arccos {x : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) : sin (arccos x) = sqrt (1 - x ^ 2) := by rw [arccos_eq_pi_div_two_sub_arcsin, sin_pi_div_two_sub, cos_arcsin hx₁ hx₂] @[continuity] lemma continuous_arccos : continuous arccos := continuous_const.sub continuous_arcsin lemma has_strict_deriv_at_arccos {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) : has_strict_deriv_at arccos (-(1 / sqrt (1 - x ^ 2))) x := (has_strict_deriv_at_arcsin h₁ h₂).const_sub (π / 2) lemma has_deriv_at_arccos {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) : has_deriv_at arccos (-(1 / sqrt (1 - x ^ 2))) x := (has_deriv_at_arcsin h₁ h₂).const_sub (π / 2) lemma times_cont_diff_at_arccos {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) {n : with_top ℕ} : times_cont_diff_at ℝ n arccos x := times_cont_diff_at_const.sub (times_cont_diff_at_arcsin h₁ h₂) lemma has_deriv_within_at_arccos_Ici {x : ℝ} (h : x ≠ -1) : has_deriv_within_at arccos (-(1 / sqrt (1 - x ^ 2))) (Ici x) x := (has_deriv_within_at_arcsin_Ici h).const_sub _ lemma has_deriv_within_at_arccos_Iic {x : ℝ} (h : x ≠ 1) : has_deriv_within_at arccos (-(1 / sqrt (1 - x ^ 2))) (Iic x) x := (has_deriv_within_at_arcsin_Iic h).const_sub _ lemma differentiable_within_at_arccos_Ici {x : ℝ} : differentiable_within_at ℝ arccos (Ici x) x ↔ x ≠ -1 := (differentiable_within_at_const_sub_iff _).trans differentiable_within_at_arcsin_Ici lemma differentiable_within_at_arccos_Iic {x : ℝ} : differentiable_within_at ℝ arccos (Iic x) x ↔ x ≠ 1 := (differentiable_within_at_const_sub_iff _).trans differentiable_within_at_arcsin_Iic lemma differentiable_at_arccos {x : ℝ} : differentiable_at ℝ arccos x ↔ x ≠ -1 ∧ x ≠ 1 := (differentiable_at_const_sub_iff _).trans differentiable_at_arcsin @[simp] lemma deriv_arccos : deriv arccos = λ x, -(1 / sqrt (1 - x ^ 2)) := funext $ λ x, (deriv_const_sub _).trans $ by simp only [deriv_arcsin] lemma differentiable_on_arccos : differentiable_on ℝ arccos {-1, 1}ᶜ := differentiable_on_arcsin.const_sub _ lemma times_cont_diff_on_arccos {n : with_top ℕ} : times_cont_diff_on ℝ n arccos {-1, 1}ᶜ := times_cont_diff_on_const.sub times_cont_diff_on_arcsin lemma times_cont_diff_at_arccos_iff {x : ℝ} {n : with_top ℕ} : times_cont_diff_at ℝ n arccos x ↔ n = 0 ∨ (x ≠ -1 ∧ x ≠ 1) := by refine iff.trans ⟨λ h, _, λ h, _⟩ times_cont_diff_at_arcsin_iff; simpa [arccos] using (@times_cont_diff_at_const _ _ _ _ _ _ _ _ _ _ (π / 2)).sub h lemma measurable_arccos : measurable arccos := continuous_arccos.measurable @[simp] lemma tan_pi_div_four : tan (π / 4) = 1 := begin rw [tan_eq_sin_div_cos, cos_pi_div_four, sin_pi_div_four], have h : (sqrt 2) / 2 > 0 := by cancel_denoms, exact div_self (ne_of_gt h), end @[simp] lemma tan_pi_div_two : tan (π / 2) = 0 := by simp [tan_eq_sin_div_cos] lemma tan_pos_of_pos_of_lt_pi_div_two {x : ℝ} (h0x : 0 < x) (hxp : x < π / 2) : 0 < tan x := by rw tan_eq_sin_div_cos; exact div_pos (sin_pos_of_pos_of_lt_pi h0x (by linarith)) (cos_pos_of_mem_Ioo ⟨by linarith, hxp⟩) lemma tan_nonneg_of_nonneg_of_le_pi_div_two {x : ℝ} (h0x : 0 ≤ x) (hxp : x ≤ π / 2) : 0 ≤ tan x := match lt_or_eq_of_le h0x, lt_or_eq_of_le hxp with \| or.inl hx0, or.inl hxp := le_of_lt (tan_pos_of_pos_of_lt_pi_div_two hx0 hxp) \| or.inl hx0, or.inr hxp := by simp [hxp, tan_eq_sin_div_cos] \| or.inr hx0, _ := by simp [hx0.symm] end lemma tan_neg_of_neg_of_pi_div_two_lt {x : ℝ} (hx0 : x < 0) (hpx : -(π / 2) < x) : tan x < 0 := neg_pos.1 (tan_neg x ▸ tan_pos_of_pos_of_lt_pi_div_two (by linarith) (by linarith [pi_pos])) lemma tan_nonpos_of_nonpos_of_neg_pi_div_two_le {x : ℝ} (hx0 : x ≤ 0) (hpx : -(π / 2) ≤ x) : tan x ≤ 0 := neg_nonneg.1 (tan_neg x ▸ tan_nonneg_of_nonneg_of_le_pi_div_two (by linarith) (by linarith)) lemma tan_lt_tan_of_nonneg_of_lt_pi_div_two {x y : ℝ} (hx₁ : 0 ≤ x) (hy₂ : y < π / 2) (hxy : x < y) : tan x < tan y := begin rw [tan_eq_sin_div_cos, tan_eq_sin_div_cos], exact div_lt_div (sin_lt_sin_of_lt_of_le_pi_div_two (by linarith) (le_of_lt hy₂) hxy) (cos_le_cos_of_nonneg_of_le_pi hx₁ (by linarith) (le_of_lt hxy)) (sin_nonneg_of_nonneg_of_le_pi (by linarith) (by linarith)) (cos_pos_of_mem_Ioo ⟨by linarith, hy₂⟩) end lemma tan_lt_tan_of_lt_of_lt_pi_div_two {x y : ℝ} (hx₁ : -(π / 2) < x) (hy₂ : y < π / 2) (hxy : x < y) : tan x < tan y := match le_total x 0, le_total y 0 with \| or.inl hx0, or.inl hy0 := neg_lt_neg_iff.1 $ by rw [← tan_neg, ← tan_neg]; exact tan_lt_tan_of_nonneg_of_lt_pi_div_two (neg_nonneg.2 hy0) (neg_lt.2 hx₁) (neg_lt_neg hxy) \| or.inl hx0, or.inr hy0 := (lt_or_eq_of_le hy0).elim (λ hy0, calc tan x ≤ 0 : tan_nonpos_of_nonpos_of_neg_pi_div_two_le hx0 (le_of_lt hx₁) ... < tan y : tan_pos_of_pos_of_lt_pi_div_two hy0 hy₂) (λ hy0, by rw [← hy0, tan_zero]; exact tan_neg_of_neg_of_pi_div_two_lt (hy0.symm ▸ hxy) hx₁) \| or.inr hx0, or.inl hy0 := by linarith \| or.inr hx0, or.inr hy0 := tan_lt_tan_of_nonneg_of_lt_pi_div_two hx0 hy₂ hxy end lemma strict_mono_incr_on_tan : strict_mono_incr_on tan (Ioo (-(π / 2)) (π / 2)) := λ x hx y hy, tan_lt_tan_of_lt_of_lt_pi_div_two hx.1 hy.2 lemma inj_on_tan : inj_on tan (Ioo (-(π / 2)) (π / 2)) := strict_mono_incr_on_tan.inj_on lemma tan_inj_of_lt_of_lt_pi_div_two {x y : ℝ} (hx₁ : -(π / 2) < x) (hx₂ : x < π / 2) (hy₁ : -(π / 2) < y) (hy₂ : y < π / 2) (hxy : tan x = tan y) : x = y := inj_on_tan ⟨hx₁, hx₂⟩ ⟨hy₁, hy₂⟩ hxy end real namespace complex open_locale real /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then real.arcsin (x.im / x.abs) else if 0 ≤ x.im then real.arcsin ((-x).im / x.abs) + π else real.arcsin ((-x).im / x.abs) - π lemma measurable_arg : measurable arg := have A : measurable (λ x : ℂ, real.arcsin (x.im / x.abs)), from real.measurable_arcsin.comp (measurable_im.div measurable_norm), have B : measurable (λ x : ℂ, real.arcsin ((-x).im / x.abs)), from real.measurable_arcsin.comp ((measurable_im.comp measurable_neg).div measurable_norm), measurable.ite (is_closed_le continuous_const continuous_re).measurable_set A $ measurable.ite (is_closed_le continuous_const continuous_im).measurable_set (B.add_const _) (B.sub_const _) lemma arg_le_pi (x : ℂ) : arg x ≤ π := if hx₁ : 0 ≤ x.re then by rw [arg, if_pos hx₁]; exact le_trans (real.arcsin_le_pi_div_two _) (le_of_lt (half_lt_self real.pi_pos)) else if hx₂ : 0 ≤ x.im then by rw [arg, if_neg hx₁, if_pos hx₂, ← le_sub_iff_add_le, sub_self, real.arcsin_nonpos, neg_im, neg_div, neg_nonpos]; exact div_nonneg hx₂ (abs_nonneg _) else by rw [arg, if_neg hx₁, if_neg hx₂]; exact sub_le_iff_le_add.2 (le_trans (real.arcsin_le_pi_div_two _) (by linarith [real.pi_pos])) lemma neg_pi_lt_arg (x : ℂ) : -π < arg x := if hx₁ : 0 ≤ x.re then by rw [arg, if_pos hx₁]; exact lt_of_lt_of_le (neg_lt_neg (half_lt_self real.pi_pos)) (real.neg_pi_div_two_le_arcsin _) else have hx : x ≠ 0, from λ h, by simpa [h, lt_irrefl] using hx₁, if hx₂ : 0 ≤ x.im then by rw [arg, if_neg hx₁, if_pos hx₂, ← sub_lt_iff_lt_add]; exact (lt_of_lt_of_le (by linarith [real.pi_pos]) (real.neg_pi_div_two_le_arcsin _)) else by rw [arg, if_neg hx₁, if_neg hx₂, lt_sub_iff_add_lt, neg_add_self, real.arcsin_pos, neg_im]; exact div_pos (neg_pos.2 (lt_of_not_ge hx₂)) (abs_pos.2 hx) lemma arg_eq_arg_neg_add_pi_of_im_nonneg_of_re_neg {x : ℂ} (hxr : x.re < 0) (hxi : 0 ≤ x.im) : arg x = arg (-x) + π := have 0 ≤ (-x).re, from le_of_lt $ by simpa [neg_pos], by rw [arg, arg, if_neg (not_le.2 hxr), if_pos this, if_pos hxi, abs_neg] lemma arg_eq_arg_neg_sub_pi_of_im_neg_of_re_neg {x : ℂ} (hxr : x.re < 0) (hxi : x.im < 0) : arg x = arg (-x) - π := have 0 ≤ (-x).re, from le_of_lt $ by simpa [neg_pos], by rw [arg, arg, if_neg (not_le.2 hxr), if_neg (not_le.2 hxi), if_pos this, abs_neg] @[simp] lemma arg_zero : arg 0 = 0 := by simp [arg, le_refl] @[simp] lemma arg_one : arg 1 = 0 := by simp [arg, zero_le_one] @[simp] lemma arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (@zero_lt_one ℝ _ _)] @[simp] lemma arg_I : arg I = π / 2 := by simp [arg, le_refl] @[simp] lemma arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] lemma sin_arg (x : ℂ) : real.sin (arg x) = x.im / x.abs := by unfold arg; split_ifs; simp [sub_eq_add_neg, arg, real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, real.sin_add, neg_div, real.arcsin_neg, real.sin_neg] private lemma cos_arg_of_re_nonneg {x : ℂ} (hx : x ≠ 0) (hxr : 0 ≤ x.re) : real.cos (arg x) = x.re / x.abs := have 0 ≤ 1 - (x.im / abs x) ^ 2, from sub_nonneg.2 $ by rw [sq, ← _root_.abs_mul_self, _root_.abs_mul, ← sq]; exact pow_le_one _ (_root_.abs_nonneg _) (abs_im_div_abs_le_one _), by rw [eq_div_iff_mul_eq (mt abs_eq_zero.1 hx), ← real.mul_self_sqrt (abs_nonneg x), arg, if_pos hxr, real.cos_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, ← real.sqrt_mul (abs_nonneg _), ← real.sqrt_mul this, sub_mul, div_pow, ← sq, div_mul_cancel _ (pow_ne_zero 2 (mt abs_eq_zero.1 hx)), one_mul, sq, mul_self_abs, norm_sq_apply, sq, add_sub_cancel, real.sqrt_mul_self hxr] lemma cos_arg {x : ℂ} (hx : x ≠ 0) : real.cos (arg x) = x.re / x.abs := if hxr : 0 ≤ x.re then cos_arg_of_re_nonneg hx hxr else have 0 ≤ (-x).re, from le_of_lt $ by simpa [neg_pos] using hxr, if hxi : 0 ≤ x.im then have 0 ≤ (-x).re, from le_of_lt $ by simpa [neg_pos] using hxr, by rw [arg_eq_arg_neg_add_pi_of_im_nonneg_of_re_neg (not_le.1 hxr) hxi, real.cos_add_pi, cos_arg_of_re_nonneg (neg_ne_zero.2 hx) this]; simp [neg_div] else by rw [arg_eq_arg_neg_sub_pi_of_im_neg_of_re_neg (not_le.1 hxr) (not_le.1 hxi)]; simp [sub_eq_add_neg, real.cos_add, neg_div, cos_arg_of_re_nonneg (neg_ne_zero.2 hx) this] lemma tan_arg {x : ℂ} : real.tan (arg x) = x.im / x.re := begin by_cases h : x = 0, { simp only [h, zero_div, complex.zero_im, complex.arg_zero, real.tan_zero, complex.zero_re] }, rw [real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (mt abs_eq_zero.1 h)] end lemma arg_cos_add_sin_mul_I {x : ℝ} (hx₁ : -π < x) (hx₂ : x ≤ π) : arg (cos x + sin x * I) = x := if hx₃ : -(π / 2) ≤ x ∧ x ≤ π / 2 then have hx₄ : 0 ≤ (cos x + sin x * I).re, by simp; exact real.cos_nonneg_of_mem_Icc hx₃, by rw [arg, if_pos hx₄]; simp [abs_cos_add_sin_mul_I, sin_of_real_re, real.arcsin_sin hx₃.1 hx₃.2] else if hx₄ : x < -(π / 2) then have hx₅ : ¬0 ≤ (cos x + sin x * I).re := suffices ¬ 0 ≤ real.cos x, by simpa, not_le.2 $ by rw ← real.cos_neg; apply real.cos_neg_of_pi_div_two_lt_of_lt; linarith, have hx₆ : ¬0 ≤ (cos ↑x + sin ↑x * I).im := suffices real.sin x < 0, by simpa, by apply real.sin_neg_of_neg_of_neg_pi_lt; linarith, suffices -π + -real.arcsin (real.sin x) = x, by rw [arg, if_neg hx₅, if_neg hx₆]; simpa [sub_eq_add_neg, add_comm, abs_cos_add_sin_mul_I, sin_of_real_re], by rw [← real.arcsin_neg, ← real.sin_add_pi, real.arcsin_sin]; try {simp [add_left_comm]}; linarith else have hx₅ : π / 2 < x, by cases not_and_distrib.1 hx₃; linarith, have hx₆ : ¬0 ≤ (cos x + sin x * I).re := suffices ¬0 ≤ real.cos x, by simpa, not_le.2 $ by apply real.cos_neg_of_pi_div_two_lt_of_lt; linarith, have hx₇ : 0 ≤ (cos x + sin x * I).im := suffices 0 ≤ real.sin x, by simpa, by apply real.sin_nonneg_of_nonneg_of_le_pi; linarith, suffices π - real.arcsin (real.sin x) = x, by rw [arg, if_neg hx₆, if_pos hx₇]; simpa [sub_eq_add_neg, add_comm, abs_cos_add_sin_mul_I, sin_of_real_re], by rw [← real.sin_pi_sub, real.arcsin_sin]; simp [sub_eq_add_neg]; linarith lemma arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := have hax : abs x ≠ 0, from (mt abs_eq_zero.1 hx), have hay : abs y ≠ 0, from (mt abs_eq_zero.1 hy), ⟨λ h, begin have hcos := congr_arg real.cos h, rw [cos_arg hx, cos_arg hy, div_eq_div_iff hax hay] at hcos, have hsin := congr_arg real.sin h, rw [sin_arg, sin_arg, div_eq_div_iff hax hay] at hsin, apply complex.ext, { rw [mul_re, ← of_real_div, of_real_re, of_real_im, zero_mul, sub_zero, mul_comm, ← mul_div_assoc, hcos, mul_div_cancel _ hax] }, { rw [mul_im, ← of_real_div, of_real_re, of_real_im, zero_mul, add_zero, mul_comm, ← mul_div_assoc, hsin, mul_div_cancel _ hax] } end, λ h, have hre : abs (y / x) * x.re = y.re, by rw ← of_real_div at h; simpa [-of_real_div, -is_R_or_C.of_real_div] using congr_arg re h, have hre' : abs (x / y) * y.re = x.re, by rw [← hre, abs_div, abs_div, ← mul_assoc, div_mul_div, mul_comm (abs _), div_self (mul_ne_zero hay hax), one_mul], have him : abs (y / x) * x.im = y.im, by rw ← of_real_div at h; simpa [-of_real_div, -is_R_or_C.of_real_div] using congr_arg im h, have him' : abs (x / y) * y.im = x.im, by rw [← him, abs_div, abs_div, ← mul_assoc, div_mul_div, mul_comm (abs _), div_self (mul_ne_zero hay hax), one_mul], have hxya : x.im / abs x = y.im / abs y, by rw [← him, abs_div, mul_comm, ← mul_div_comm, mul_div_cancel_left _ hay], have hnxya : (-x).im / abs x = (-y).im / abs y, by rw [neg_im, neg_im, neg_div, neg_div, hxya], if hxr : 0 ≤ x.re then have hyr : 0 ≤ y.re, from hre ▸ mul_nonneg (abs_nonneg _) hxr, by simp [arg, ] at else have hyr : ¬ 0 ≤ y.re, from λ hyr, hxr $ hre' ▸ mul_nonneg (abs_nonneg _) hyr, if hxi : 0 ≤ x.im then have hyi : 0 ≤ y.im, from him ▸ mul_nonneg (abs_nonneg _) hxi, by simp [arg, ] at else have hyi : ¬ 0 ≤ y.im, from λ hyi, hxi $ him' ▸ mul_nonneg (abs_nonneg _) hyi, by simp [arg, ] at ⟩ lemma arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := if hx : x = 0 then by simp [hx] else (arg_eq_arg_iff (mul_ne_zero (of_real_ne_zero.2 (ne_of_lt hr).symm) hx) hx).2 $ by rw [abs_mul, abs_of_nonneg (le_of_lt hr), ← mul_assoc, of_real_mul, mul_comm (r : ℂ), ← div_div_eq_div_mul, div_mul_cancel _ (of_real_ne_zero.2 (ne_of_lt hr).symm), div_self (of_real_ne_zero.2 (mt abs_eq_zero.1 hx)), one_mul] lemma ext_abs_arg {x y : ℂ} (h₁ : x.abs = y.abs) (h₂ : x.arg = y.arg) : x = y := if hy : y = 0 then by simp * at * else have hx : x ≠ 0, from λ hx, by simp [, eq_comm] at , by rwa [arg_eq_arg_iff hx hy, h₁, div_self (of_real_ne_zero.2 (mt abs_eq_zero.1 hy)), one_mul] at h₂ lemma arg_of_real_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] lemma arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := begin by_cases h₀ : z = 0, { simp [h₀, lt_irrefl, real.pi_ne_zero.symm] }, have h₀' : (abs z : ℂ) ≠ 0, by simpa, rw [← arg_neg_one, arg_eq_arg_iff h₀ (neg_ne_zero.2 one_ne_zero), abs_neg, abs_one, of_real_one, one_div, ← div_eq_inv_mul, div_eq_iff_mul_eq h₀', neg_one_mul, ext_iff, neg_im, of_real_im, neg_zero, @eq_comm _ z.im, and.congr_left_iff], rcases z with ⟨x, y⟩, simp only, rintro rfl, simp only [← of_real_def, of_real_eq_zero] at , simp [← ne.le_iff_lt h₀, @neg_eq_iff_neg_eq _ _ _ x, @eq_comm _ (-x)] end lemma arg_of_real_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ /-- Inverse of the `exp` function. Returns values such that `(log x).im > - π` and `(log x).im ≤ π`. `log 0 = 0`-/ @[pp_nodot] noncomputable def log (x : ℂ) : ℂ := x.abs.log + arg x I lemma measurable_log : measurable log := (measurable_of_real.comp $ real.measurable_log.comp measurable_norm).add $ (measurable_of_real.comp measurable_arg).mul_const I lemma log_re (x : ℂ) : x.log.re = x.abs.log := by simp [log] lemma log_im (x : ℂ) : x.log.im = x.arg := by simp [log] lemma neg_pi_lt_log_im (x : ℂ) : -π < (log x).im := by simp only [log_im, neg_pi_lt_arg] lemma log_im_le_pi (x : ℂ) : (log x).im ≤ π := by simp only [log_im, arg_le_pi] lemma exp_log {x : ℂ} (hx : x ≠ 0) : exp (log x) = x := by rw [log, exp_add_mul_I, ← of_real_sin, sin_arg, ← of_real_cos, cos_arg hx, ← of_real_exp, real.exp_log (abs_pos.2 hx), mul_add, of_real_div, of_real_div, mul_div_cancel' _ (of_real_ne_zero.2 (mt abs_eq_zero.1 hx)), ← mul_assoc, mul_div_cancel' _ (of_real_ne_zero.2 (mt abs_eq_zero.1 hx)), re_add_im] lemma range_exp : range exp = {x \| x ≠ 0} := set.ext $ λ x, ⟨by { rintro ⟨x, rfl⟩, exact exp_ne_zero x }, λ hx, ⟨log x, exp_log hx⟩⟩ lemma exp_inj_of_neg_pi_lt_of_le_pi {x y : ℂ} (hx₁ : -π < x.im) (hx₂ : x.im ≤ π) (hy₁ : - π < y.im) (hy₂ : y.im ≤ π) (hxy : exp x = exp y) : x = y := by rw [exp_eq_exp_re_mul_sin_add_cos, exp_eq_exp_re_mul_sin_add_cos y] at hxy; exact complex.ext (real.exp_injective $ by simpa [abs_mul, abs_cos_add_sin_mul_I] using congr_arg complex.abs hxy) (by simpa [(of_real_exp _).symm, - of_real_exp, arg_real_mul _ (real.exp_pos _), arg_cos_add_sin_mul_I hx₁ hx₂, arg_cos_add_sin_mul_I hy₁ hy₂] using congr_arg arg hxy) lemma log_exp {x : ℂ} (hx₁ : -π < x.im) (hx₂: x.im ≤ π) : log (exp x) = x := exp_inj_of_neg_pi_lt_of_le_pi (by rw log_im; exact neg_pi_lt_arg _) (by rw log_im; exact arg_le_pi _) hx₁ hx₂ (by rw [exp_log (exp_ne_zero _)]) lemma of_real_log {x : ℝ} (hx : 0 ≤ x) : (x.log : ℂ) = log x := complex.ext (by rw [log_re, of_real_re, abs_of_nonneg hx]) (by rw [of_real_im, log_im, arg_of_real_of_nonneg hx]) lemma log_of_real_re (x : ℝ) : (log (x : ℂ)).re = real.log x := by simp [log_re] @[simp] lemma log_zero : log 0 = 0 := by simp [log] @[simp] lemma log_one : log 1 = 0 := by simp [log] lemma log_neg_one : log (-1) = π * I := by simp [log] lemma log_I : log I = π / 2 * I := by simp [log] lemma log_neg_I : log (-I) = -(π / 2) * I := by simp [log] lemma exists_pow_nat_eq (x : ℂ) {n : ℕ} (hn : 0 < n) : ∃ z, z ^ n = x := begin by_cases hx : x = 0, { use 0, simp only [hx, zero_pow_eq_zero, hn] }, { use exp (log x / n), rw [← exp_nat_mul, mul_div_cancel', exp_log hx], exact_mod_cast (pos_iff_ne_zero.mp hn) } end lemma exists_eq_mul_self (x : ℂ) : ∃ z, x = z * z := begin obtain ⟨z, rfl⟩ := exists_pow_nat_eq x zero_lt_two, exact ⟨z, sq z⟩ end lemma two_pi_I_ne_zero : (2 * π * I : ℂ) ≠ 0 := by norm_num [real.pi_ne_zero, I_ne_zero] lemma exp_eq_one_iff {x : ℂ} : exp x = 1 ↔ ∃ n : ℤ, x = n * ((2 * π) * I) := have real.exp (x.re) * real.cos (x.im) = 1 → real.cos x.im ≠ -1, from λ h₁ h₂, begin rw [h₂, mul_neg_eq_neg_mul_symm, mul_one, neg_eq_iff_neg_eq] at h₁, have := real.exp_pos x.re, rw ← h₁ at this, exact absurd this (by norm_num) end, calc exp x = 1 ↔ (exp x).re = 1 ∧ (exp x).im = 0 : by simp [complex.ext_iff] ... ↔ real.cos x.im = 1 ∧ real.sin x.im = 0 ∧ x.re = 0 : begin rw exp_eq_exp_re_mul_sin_add_cos, simp [complex.ext_iff, cos_of_real_re, sin_of_real_re, exp_of_real_re, real.exp_ne_zero], split; finish [real.sin_eq_zero_iff_cos_eq] end ... ↔ (∃ n : ℤ, ↑n * (2 * π) = x.im) ∧ (∃ n : ℤ, ↑n * π = x.im) ∧ x.re = 0 : by rw [real.sin_eq_zero_iff, real.cos_eq_one_iff] ... ↔ ∃ n : ℤ, x = n * ((2 * π) * I) : ⟨λ ⟨⟨n, hn⟩, ⟨m, hm⟩, h⟩, ⟨n, by simp [complex.ext_iff, hn.symm, h]⟩, λ ⟨n, hn⟩, ⟨⟨n, by simp [hn]⟩, ⟨2 * n, by simp [hn, mul_comm, mul_assoc, mul_left_comm]⟩, by simp [hn]⟩⟩ lemma exp_eq_exp_iff_exp_sub_eq_one {x y : ℂ} : exp x = exp y ↔ exp (x - y) = 1 := by rw [exp_sub, div_eq_one_iff_eq (exp_ne_zero _)] lemma exp_eq_exp_iff_exists_int {x y : ℂ} : exp x = exp y ↔ ∃ n : ℤ, x = y + n * ((2 * π) * I) := by simp only [exp_eq_exp_iff_exp_sub_eq_one, exp_eq_one_iff, sub_eq_iff_eq_add'] /-- `complex.exp` as a `local_homeomorph` with `source = {z \| -π < im z < π}` and `target = {z \| 0 < re z} ∪ {z \| im z ≠ 0}`. This definition is used to prove that `complex.log` is complex differentiable at all points but the negative real semi-axis. -/ def exp_local_homeomorph : local_homeomorph ℂ ℂ := local_homeomorph.of_continuous_open { to_fun := exp, inv_fun := log, source := {z : ℂ \| z.im ∈ Ioo (- π) π}, target := {z : ℂ \| 0 < z.re} ∪ {z : ℂ \| z.im ≠ 0}, map_source' := begin rintro ⟨x, y⟩ ⟨h₁ : -π < y, h₂ : y < π⟩, refine (not_or_of_imp $ λ hz, _).symm, obtain rfl : y = 0, { rw exp_im at hz, simpa [(real.exp_pos _).ne', real.sin_eq_zero_iff_of_lt_of_lt h₁ h₂] using hz }, rw [mem_set_of_eq, ← of_real_def, exp_of_real_re], exact real.exp_pos x end, map_target' := λ z h, suffices 0 ≤ z.re ∨ z.im ≠ 0, by simpa [log_im, neg_pi_lt_arg, (arg_le_pi _).lt_iff_ne, arg_eq_pi_iff, not_and_distrib], h.imp (λ h, le_of_lt h) id, left_inv' := λ x hx, log_exp hx.1 (le_of_lt hx.2), right_inv' := λ x hx, exp_log $ by { rintro rfl, simpa [lt_irrefl] using hx } } continuous_exp.continuous_on is_open_map_exp (is_open_Ioo.preimage continuous_im) lemma has_strict_deriv_at_log {x : ℂ} (h : 0 < x.re ∨ x.im ≠ 0) : has_strict_deriv_at log x⁻¹ x := have h0 : x ≠ 0, by { rintro rfl, simpa [lt_irrefl] using h }, exp_local_homeomorph.has_strict_deriv_at_symm h h0 $ by simpa [exp_log h0] using has_strict_deriv_at_exp (log x) lemma times_cont_diff_at_log {x : ℂ} (h : 0 < x.re ∨ x.im ≠ 0) {n : with_top ℕ} : times_cont_diff_at ℂ n log x := exp_local_homeomorph.times_cont_diff_at_symm_deriv (exp_ne_zero $ log x) h (has_deriv_at_exp _) times_cont_diff_exp.times_cont_diff_at @[simp] lemma cos_pi_div_two : cos (π / 2) = 0 := calc cos (π / 2) = real.cos (π / 2) : by rw [of_real_cos]; simp ... = 0 : by simp @[simp] lemma sin_pi_div_two : sin (π / 2) = 1 := calc sin (π / 2) = real.sin (π / 2) : by rw [of_real_sin]; simp ... = 1 : by simp @[simp] lemma sin_pi : sin π = 0 := by rw [← of_real_sin, real.sin_pi]; simp @[simp] lemma cos_pi : cos π = -1 := by rw [← of_real_cos, real.cos_pi]; simp @[simp] lemma sin_two_pi : sin (2 * π) = 0 := by simp [two_mul, sin_add] @[simp] lemma cos_two_pi : cos (2 * π) = 1 := by simp [two_mul, cos_add] lemma sin_add_pi (x : ℂ) : sin (x + π) = -sin x := by simp [sin_add] lemma sin_add_two_pi (x : ℂ) : sin (x + 2 * π) = sin x := by simp [sin_add] lemma cos_add_two_pi (x : ℂ) : cos (x + 2 * π) = cos x := by simp [cos_add] lemma sin_pi_sub (x : ℂ) : sin (π - x) = sin x := by simp [sub_eq_add_neg, sin_add] lemma cos_add_pi (x : ℂ) : cos (x + π) = -cos x := by simp [cos_add] lemma cos_pi_sub (x : ℂ) : cos (π - x) = -cos x := by simp [sub_eq_add_neg, cos_add] lemma sin_add_pi_div_two (x : ℂ) : sin (x + π / 2) = cos x := by simp [sin_add] lemma sin_sub_pi_div_two (x : ℂ) : sin (x - π / 2) = -cos x := by simp [sub_eq_add_neg, sin_add] lemma sin_pi_div_two_sub (x : ℂ) : sin (π / 2 - x) = cos x := by simp [sub_eq_add_neg, sin_add] lemma cos_add_pi_div_two (x : ℂ) : cos (x + π / 2) = -sin x := by simp [cos_add] lemma cos_sub_pi_div_two (x : ℂ) : cos (x - π / 2) = sin x := by simp [sub_eq_add_neg, cos_add] lemma cos_pi_div_two_sub (x : ℂ) : cos (π / 2 - x) = sin x := by rw [← cos_neg, neg_sub, cos_sub_pi_div_two] lemma sin_nat_mul_pi (n : ℕ) : sin (n * π) = 0 := by induction n; simp [add_mul, sin_add, ] lemma sin_int_mul_pi (n : ℤ) : sin (n π) = 0 := by cases n; simp [add_mul, sin_add, , sin_nat_mul_pi] lemma cos_nat_mul_two_pi (n : ℕ) : cos (n (2 * π)) = 1 := by induction n; simp [, mul_add, cos_add, add_mul, cos_two_pi, sin_two_pi] lemma cos_int_mul_two_pi (n : ℤ) : cos (n (2 * π)) = 1 := by cases n; simp only [cos_nat_mul_two_pi, int.of_nat_eq_coe, int.neg_succ_of_nat_coe, int.cast_coe_nat, int.cast_neg, (neg_mul_eq_neg_mul _ _).symm, cos_neg] lemma cos_int_mul_two_pi_add_pi (n : ℤ) : cos (n * (2 * π) + π) = -1 := by simp [cos_add, sin_add, cos_int_mul_two_pi] lemma exp_pi_mul_I : exp (π * I) = -1 := by rw exp_mul_I; simp theorem cos_eq_zero_iff {θ : ℂ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1) * π / 2 := begin have h : (exp (θ * I) + exp (-θ * I)) / 2 = 0 ↔ exp (2 * θ * I) = -1, { rw [@div_eq_iff _ _ (exp (θ * I) + exp (-θ * I)) 2 0 two_ne_zero', zero_mul, add_eq_zero_iff_eq_neg, neg_eq_neg_one_mul, ← div_eq_iff (exp_ne_zero _), ← exp_sub], field_simp only, congr' 3, ring }, rw [cos, h, ← exp_pi_mul_I, exp_eq_exp_iff_exists_int, mul_right_comm], refine exists_congr (λ x, _), refine (iff_of_eq $ congr_arg _ _).trans (mul_right_inj' $ mul_ne_zero two_ne_zero' I_ne_zero), ring, end theorem cos_ne_zero_iff {θ : ℂ} : cos θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ (2 * k + 1) * π / 2 := by rw [← not_exists, not_iff_not, cos_eq_zero_iff] theorem sin_eq_zero_iff {θ : ℂ} : sin θ = 0 ↔ ∃ k : ℤ, θ = k * π := begin rw [← complex.cos_sub_pi_div_two, cos_eq_zero_iff], split, { rintros ⟨k, hk⟩, use k + 1, field_simp [eq_add_of_sub_eq hk], ring }, { rintros ⟨k, rfl⟩, use k - 1, field_simp, ring } end theorem sin_ne_zero_iff {θ : ℂ} : sin θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ k * π := by rw [← not_exists, not_iff_not, sin_eq_zero_iff] lemma sin_eq_zero_iff_cos_eq {z : ℂ} : sin z = 0 ↔ cos z = 1 ∨ cos z = -1 := by rw [← mul_self_eq_one_iff, ← sin_sq_add_cos_sq, sq, sq, ← sub_eq_iff_eq_add, sub_self]; exact ⟨λ h, by rw [h, mul_zero], eq_zero_of_mul_self_eq_zero ∘ eq.symm⟩ lemma tan_eq_zero_iff {θ : ℂ} : tan θ = 0 ↔ ∃ k : ℤ, θ = k * π / 2 := begin have h := (sin_two_mul θ).symm, rw mul_assoc at h, rw [tan, div_eq_zero_iff, ← mul_eq_zero, ← zero_mul ((1/2):ℂ), mul_one_div, cancel_factors.cancel_factors_eq_div h two_ne_zero', mul_comm], simpa only [zero_div, zero_mul, ne.def, not_false_iff] with field_simps using sin_eq_zero_iff, end lemma tan_ne_zero_iff {θ : ℂ} : tan θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ k * π / 2 := by rw [← not_exists, not_iff_not, tan_eq_zero_iff] lemma tan_int_mul_pi_div_two (n : ℤ) : tan (n * π/2) = 0 := tan_eq_zero_iff.mpr (by use n) lemma tan_int_mul_pi (n : ℤ) : tan (n * π) = 0 := by simp [tan, add_mul, sin_add, sin_int_mul_pi] lemma cos_eq_cos_iff {x y : ℂ} : cos x = cos y ↔ ∃ k : ℤ, y = 2 * k * π + x ∨ y = 2 * k * π - x := calc cos x = cos y ↔ cos x - cos y = 0 : sub_eq_zero.symm ... ↔ -2 * sin((x + y)/2) * sin((x - y)/2) = 0 : by rw cos_sub_cos ... ↔ sin((x + y)/2) = 0 ∨ sin((x - y)/2) = 0 : by simp [(by norm_num : (2:ℂ) ≠ 0)] ... ↔ sin((x - y)/2) = 0 ∨ sin((x + y)/2) = 0 : or.comm ... ↔ (∃ k : ℤ, y = 2 * k * π + x) ∨ (∃ k :ℤ, y = 2 * k * π - x) : begin apply or_congr; field_simp [sin_eq_zero_iff, (by norm_num : -(2:ℂ) ≠ 0), eq_sub_iff_add_eq', sub_eq_iff_eq_add, mul_comm (2:ℂ), mul_right_comm _ (2:ℂ)], split; { rintros ⟨k, rfl⟩, use -k, simp, }, end ... ↔ ∃ k : ℤ, y = 2 * k * π + x ∨ y = 2 * k * π - x : exists_or_distrib.symm lemma sin_eq_sin_iff {x y : ℂ} : sin x = sin y ↔ ∃ k : ℤ, y = 2 * k * π + x ∨ y = (2 * k + 1) * π - x := begin simp only [← complex.cos_sub_pi_div_two, cos_eq_cos_iff, sub_eq_iff_eq_add], refine exists_congr (λ k, or_congr _ _); refine eq.congr rfl _; field_simp; ring end lemma tan_add {x y : ℂ} (h : ((∀ k : ℤ, x ≠ (2 * k + 1) * π / 2) ∧ ∀ l : ℤ, y ≠ (2 * l + 1) * π / 2) ∨ ((∃ k : ℤ, x = (2 * k + 1) * π / 2) ∧ ∃ l : ℤ, y = (2 * l + 1) * π / 2)) : tan (x + y) = (tan x + tan y) / (1 - tan x * tan y) := begin rcases h with ⟨h1, h2⟩ \| ⟨⟨k, rfl⟩, ⟨l, rfl⟩⟩, { rw [tan, sin_add, cos_add, ← div_div_div_cancel_right (sin x * cos y + cos x * sin y) (mul_ne_zero (cos_ne_zero_iff.mpr h1) (cos_ne_zero_iff.mpr h2)), add_div, sub_div], simp only [←div_mul_div, ←tan, mul_one, one_mul, div_self (cos_ne_zero_iff.mpr h1), div_self (cos_ne_zero_iff.mpr h2)] }, { obtain ⟨t, hx, hy, hxy⟩ := ⟨tan_int_mul_pi_div_two, t (2k+1), t (2l+1), t (2k+1+(2l+1))⟩, simp only [int.cast_add, int.cast_bit0, int.cast_mul, int.cast_one, hx, hy] at hx hy hxy, rw [hx, hy, add_zero, zero_div, mul_div_assoc, mul_div_assoc, ← add_mul (2(k:ℂ)+1) (2l+1) (π/2), ← mul_div_assoc, hxy] }, end lemma tan_add' {x y : ℂ} (h : ((∀ k : ℤ, x ≠ (2 * k + 1) * π / 2) ∧ ∀ l : ℤ, y ≠ (2 * l + 1) * π / 2)) : tan (x + y) = (tan x + tan y) / (1 - tan x * tan y) := tan_add (or.inl h) lemma tan_two_mul {z : ℂ} : tan (2 * z) = 2 * tan z / (1 - tan z ^ 2) := begin by_cases h : ∀ k : ℤ, z ≠ (2 * k + 1) * π / 2, { rw [two_mul, two_mul, sq, tan_add (or.inl ⟨h, h⟩)] }, { rw not_forall_not at h, rw [two_mul, two_mul, sq, tan_add (or.inr ⟨h, h⟩)] }, end lemma tan_add_mul_I {x y : ℂ} (h : ((∀ k : ℤ, x ≠ (2 * k + 1) * π / 2) ∧ ∀ l : ℤ, y * I ≠ (2 * l + 1) * π / 2) ∨ ((∃ k : ℤ, x = (2 * k + 1) * π / 2) ∧ ∃ l : ℤ, y * I = (2 * l + 1) * π / 2)) : tan (x + yI) = (tan x + tanh y I) / (1 - tan x * tanh y * I) := by rw [tan_add h, tan_mul_I, mul_assoc] lemma tan_eq {z : ℂ} (h : ((∀ k : ℤ, (z.re:ℂ) ≠ (2 * k + 1) * π / 2) ∧ ∀ l : ℤ, (z.im:ℂ) * I ≠ (2 * l + 1) * π / 2) ∨ ((∃ k : ℤ, (z.re:ℂ) = (2 * k + 1) * π / 2) ∧ ∃ l : ℤ, (z.im:ℂ) * I = (2 * l + 1) * π / 2)) : tan z = (tan z.re + tanh z.im * I) / (1 - tan z.re * tanh z.im * I) := by convert tan_add_mul_I h; exact (re_add_im z).symm lemma has_strict_deriv_at_tan {x : ℂ} (h : cos x ≠ 0) : has_strict_deriv_at tan (1 / (cos x)^2) x := begin convert (has_strict_deriv_at_sin x).div (has_strict_deriv_at_cos x) h, rw ← sin_sq_add_cos_sq x, ring, end lemma has_deriv_at_tan {x : ℂ} (h : cos x ≠ 0) : has_deriv_at tan (1 / (cos x)^2) x := (has_strict_deriv_at_tan h).has_deriv_at lemma tendsto_abs_tan_of_cos_eq_zero {x : ℂ} (hx : cos x = 0) : tendsto (λ x, abs (tan x)) (𝓝[{x}ᶜ] x) at_top := begin simp only [tan_eq_sin_div_cos, ← norm_eq_abs, normed_field.norm_div], have A : sin x ≠ 0 := λ h, by simpa [, sq] using sin_sq_add_cos_sq x, have B : tendsto cos (𝓝[{x}ᶜ] (x)) (𝓝[{0}ᶜ] 0), { refine tendsto_inf.2 ⟨tendsto.mono_left _ inf_le_left, tendsto_principal.2 _⟩, exacts [continuous_cos.tendsto' x 0 hx, hx ▸ (has_deriv_at_cos _).eventually_ne (neg_ne_zero.2 A)] }, exact continuous_sin.continuous_within_at.norm.mul_at_top (norm_pos_iff.2 A) (tendsto_norm_nhds_within_zero.comp B).inv_tendsto_zero, end lemma tendsto_abs_tan_at_top (k : ℤ) : tendsto (λ x, abs (tan x)) (𝓝[{(2 k + 1) * π / 2}ᶜ] ((2 * k + 1) * π / 2)) at_top := tendsto_abs_tan_of_cos_eq_zero $ cos_eq_zero_iff.2 ⟨k, rfl⟩ @[simp] lemma continuous_at_tan {x : ℂ} : continuous_at tan x ↔ cos x ≠ 0 := begin refine ⟨λ hc h₀, _, λ h, (has_deriv_at_tan h).continuous_at⟩, exact not_tendsto_nhds_of_tendsto_at_top (tendsto_abs_tan_of_cos_eq_zero h₀) _ (hc.norm.tendsto.mono_left inf_le_left) end @[simp] lemma differentiable_at_tan {x : ℂ} : differentiable_at ℂ tan x ↔ cos x ≠ 0:= ⟨λ h, continuous_at_tan.1 h.continuous_at, λ h, (has_deriv_at_tan h).differentiable_at⟩ @[simp] lemma deriv_tan (x : ℂ) : deriv tan x = 1 / (cos x)^2 := if h : cos x = 0 then have ¬differentiable_at ℂ tan x := mt differentiable_at_tan.1 (not_not.2 h), by simp [deriv_zero_of_not_differentiable_at this, h, sq] else (has_deriv_at_tan h).deriv lemma continuous_on_tan : continuous_on tan {x \| cos x ≠ 0} := continuous_on_sin.div continuous_on_cos $ λ x, id @[continuity] lemma continuous_tan : continuous (λ x : {x \| cos x ≠ 0}, tan x) := continuous_on_iff_continuous_restrict.1 continuous_on_tan @[simp] lemma times_cont_diff_at_tan {x : ℂ} {n : with_top ℕ} : times_cont_diff_at ℂ n tan x ↔ cos x ≠ 0 := ⟨λ h, continuous_at_tan.1 h.continuous_at, times_cont_diff_sin.times_cont_diff_at.div times_cont_diff_cos.times_cont_diff_at⟩ lemma cos_eq_iff_quadratic {z w : ℂ} : cos z = w ↔ (exp (z * I)) ^ 2 - 2 * w * exp (z * I) + 1 = 0 := begin rw ← sub_eq_zero, field_simp [cos, exp_neg, exp_ne_zero], refine eq.congr _ rfl, ring end lemma cos_surjective : function.surjective cos := begin intro x, obtain ⟨w, w₀, hw⟩ : ∃ w ≠ 0, 1 * w * w + (-2 * x) * w + 1 = 0, { rcases exists_quadratic_eq_zero one_ne_zero (exists_eq_mul_self _) with ⟨w, hw⟩, refine ⟨w, _, hw⟩, rintro rfl, simpa only [zero_add, one_ne_zero, mul_zero] using hw }, refine ⟨log w / I, cos_eq_iff_quadratic.2 _⟩, rw [div_mul_cancel _ I_ne_zero, exp_log w₀], convert hw, ring end @[simp] lemma range_cos : range cos = set.univ := cos_surjective.range_eq lemma sin_surjective : function.surjective sin := begin intro x, rcases cos_surjective x with ⟨z, rfl⟩, exact ⟨z + π / 2, sin_add_pi_div_two z⟩ end @[simp] lemma range_sin : range sin = set.univ := sin_surjective.range_eq end complex section log_deriv open complex variables {α : Type} lemma measurable.carg [measurable_space α] {f : α → ℂ} (h : measurable f) : measurable (λ x, arg (f x)) := measurable_arg.comp h lemma measurable.clog [measurable_space α] {f : α → ℂ} (h : measurable f) : measurable (λ x, log (f x)) := measurable_log.comp h lemma filter.tendsto.clog {l : filter α} {f : α → ℂ} {x : ℂ} (h : tendsto f l (𝓝 x)) (hx : 0 < x.re ∨ x.im ≠ 0) : tendsto (λ t, log (f t)) l (𝓝 $ log x) := (has_strict_deriv_at_log hx).continuous_at.tendsto.comp h variables [topological_space α] lemma continuous_at.clog {f : α → ℂ} {x : α} (h₁ : continuous_at f x) (h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) : continuous_at (λ t, log (f t)) x := h₁.clog h₂ lemma continuous_within_at.clog {f : α → ℂ} {s : set α} {x : α} (h₁ : continuous_within_at f s x) (h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) : continuous_within_at (λ t, log (f t)) s x := h₁.clog h₂ lemma continuous_on.clog {f : α → ℂ} {s : set α} (h₁ : continuous_on f s) (h₂ : ∀ x ∈ s, 0 < (f x).re ∨ (f x).im ≠ 0) : continuous_on (λ t, log (f t)) s := λ x hx, (h₁ x hx).clog (h₂ x hx) lemma continuous.clog {f : α → ℂ} (h₁ : continuous f) (h₂ : ∀ x, 0 < (f x).re ∨ (f x).im ≠ 0) : continuous (λ t, log (f t)) := continuous_iff_continuous_at.2 $ λ x, h₁.continuous_at.clog (h₂ x) variables {E : Type} [normed_group E] [normed_space ℂ E] lemma has_strict_fderiv_at.clog {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} (h₁ : has_strict_fderiv_at f f' x) (h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) : has_strict_fderiv_at (λ t, log (f t)) ((f x)⁻¹ • f') x := (has_strict_deriv_at_log h₂).comp_has_strict_fderiv_at x h₁ lemma has_strict_deriv_at.clog {f : ℂ → ℂ} {f' x : ℂ} (h₁ : has_strict_deriv_at f f' x) (h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) : has_strict_deriv_at (λ t, log (f t)) (f' / f x) x := by { rw div_eq_inv_mul, exact (has_strict_deriv_at_log h₂).comp x h₁ } lemma has_fderiv_at.clog {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} (h₁ : has_fderiv_at f f' x) (h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) : has_fderiv_at (λ t, log (f t)) ((f x)⁻¹ • f') x := (has_strict_deriv_at_log h₂).has_deriv_at.comp_has_fderiv_at x h₁ lemma has_deriv_at.clog {f : ℂ → ℂ} {f' x : ℂ} (h₁ : has_deriv_at f f' x) (h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) : has_deriv_at (λ t, log (f t)) (f' / f x) x := by { rw div_eq_inv_mul, exact (has_strict_deriv_at_log h₂).has_deriv_at.comp x h₁ } lemma differentiable_at.clog {f : E → ℂ} {x : E} (h₁ : differentiable_at ℂ f x) (h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) : differentiable_at ℂ (λ t, log (f t)) x := (h₁.has_fderiv_at.clog h₂).differentiable_at lemma has_fderiv_within_at.clog {f : E → ℂ} {f' : E →L[ℂ] ℂ} {s : set E} {x : E} (h₁ : has_fderiv_within_at f f' s x) (h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) : has_fderiv_within_at (λ t, log (f t)) ((f x)⁻¹ • f') s x := (has_strict_deriv_at_log h₂).has_deriv_at.comp_has_fderiv_within_at x h₁ lemma has_deriv_within_at.clog {f : ℂ → ℂ} {f' x : ℂ} {s : set ℂ} (h₁ : has_deriv_within_at f f' s x) (h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) : has_deriv_within_at (λ t, log (f t)) (f' / f x) s x := by { rw div_eq_inv_mul, exact (has_strict_deriv_at_log h₂).has_deriv_at.comp_has_deriv_within_at x h₁ } lemma differentiable_within_at.clog {f : E → ℂ} {s : set E} {x : E} (h₁ : differentiable_within_at ℂ f s x) (h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) : differentiable_within_at ℂ (λ t, log (f t)) s x := (h₁.has_fderiv_within_at.clog h₂).differentiable_within_at lemma differentiable_on.clog {f : E → ℂ} {s : set E} (h₁ : differentiable_on ℂ f s) (h₂ : ∀ x ∈ s, 0 < (f x).re ∨ (f x).im ≠ 0) : differentiable_on ℂ (λ t, log (f t)) s := λ x hx, (h₁ x hx).clog (h₂ x hx) lemma differentiable.clog {f : E → ℂ} (h₁ : differentiable ℂ f) (h₂ : ∀ x, 0 < (f x).re ∨ (f x).im ≠ 0) : differentiable ℂ (λ t, log (f t)) := λ x, (h₁ x).clog (h₂ x) end log_deriv namespace polynomial.chebyshev open polynomial complex /-- The `n`-th Chebyshev polynomial of the first kind evaluates on `cos θ` to the value `cos (n * θ)`. -/ lemma T_complex_cos (θ : ℂ) : ∀ n, (T ℂ n).eval (cos θ) = cos (n * θ) \| 0 := by simp only [T_zero, eval_one, nat.cast_zero, zero_mul, cos_zero] \| 1 := by simp only [eval_X, one_mul, T_one, nat.cast_one] \| (n + 2) := begin simp only [eval_X, eval_one, T_add_two, eval_sub, eval_bit0, nat.cast_succ, eval_mul], rw [T_complex_cos (n + 1), T_complex_cos n], have aux : sin θ * sin θ = 1 - cos θ * cos θ, { rw ← sin_sq_add_cos_sq θ, ring, }, simp only [nat.cast_add, nat.cast_one, add_mul, cos_add, one_mul, sin_add, mul_assoc, aux], ring, end /-- `cos (n * θ)` is equal to the `n`-th Chebyshev polynomial of the first kind evaluated on `cos θ`. -/ lemma cos_nat_mul (n : ℕ) (θ : ℂ) : cos (n * θ) = (T ℂ n).eval (cos θ) := (T_complex_cos θ n).symm /-- The `n`-th Chebyshev polynomial of the second kind evaluates on `cos θ` to the value `sin ((n+1) * θ) / sin θ`. -/ lemma U_complex_cos (θ : ℂ) (n : ℕ) : (U ℂ n).eval (cos θ) * sin θ = sin ((n+1) * θ) := begin induction n with d hd, { simp only [U_zero, nat.cast_zero, eval_one, mul_one, zero_add, one_mul] }, { rw U_eq_X_mul_U_add_T, simp only [eval_add, eval_mul, eval_X, T_complex_cos, add_mul, mul_assoc, hd, one_mul], conv_rhs { rw [sin_add, mul_comm] }, push_cast, simp only [add_mul, one_mul] } end /-- `sin ((n + 1) * θ)` is equal to `sin θ` multiplied with the `n`-th Chebyshev polynomial of the second kind evaluated on `cos θ`. -/ lemma sin_nat_succ_mul (n : ℕ) (θ : ℂ) : sin ((n + 1) * θ) = (U ℂ n).eval (cos θ) * sin θ := (U_complex_cos θ n).symm end polynomial.chebyshev namespace real open_locale real lemma tan_add {x y : ℝ} (h : ((∀ k : ℤ, x ≠ (2 * k + 1) * π / 2) ∧ ∀ l : ℤ, y ≠ (2 * l + 1) * π / 2) ∨ ((∃ k : ℤ, x = (2 * k + 1) * π / 2) ∧ ∃ l : ℤ, y = (2 * l + 1) * π / 2)) : tan (x + y) = (tan x + tan y) / (1 - tan x * tan y) := by simpa only [← complex.of_real_inj, complex.of_real_sub, complex.of_real_add, complex.of_real_div, complex.of_real_mul, complex.of_real_tan] using @complex.tan_add (x:ℂ) (y:ℂ) (by convert h; norm_cast) lemma tan_add' {x y : ℝ} (h : ((∀ k : ℤ, x ≠ (2 * k + 1) * π / 2) ∧ ∀ l : ℤ, y ≠ (2 * l + 1) * π / 2)) : tan (x + y) = (tan x + tan y) / (1 - tan x * tan y) := tan_add (or.inl h) lemma tan_two_mul {x:ℝ} : tan (2 * x) = 2 * tan x / (1 - tan x ^ 2) := by simpa only [← complex.of_real_inj, complex.of_real_sub, complex.of_real_div, complex.of_real_pow, complex.of_real_mul, complex.of_real_tan, complex.of_real_bit0, complex.of_real_one] using complex.tan_two_mul theorem cos_eq_zero_iff {θ : ℝ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1) * π / 2 := by exact_mod_cast @complex.cos_eq_zero_iff θ theorem cos_ne_zero_iff {θ : ℝ} : cos θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ (2 * k + 1) * π / 2 := by rw [← not_exists, not_iff_not, cos_eq_zero_iff] lemma tan_ne_zero_iff {θ : ℝ} : tan θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ k * π / 2 := by rw [← complex.of_real_ne_zero, complex.of_real_tan, complex.tan_ne_zero_iff]; norm_cast lemma tan_eq_zero_iff {θ : ℝ} : tan θ = 0 ↔ ∃ k : ℤ, θ = k * π / 2 := by rw [← not_iff_not, not_exists, ← ne, tan_ne_zero_iff] lemma tan_int_mul_pi_div_two (n : ℤ) : tan (n * π/2) = 0 := tan_eq_zero_iff.mpr (by use n) lemma tan_int_mul_pi (n : ℤ) : tan (n * π) = 0 := by rw tan_eq_zero_iff; use (2n); field_simp [mul_comm ((n:ℝ)(π:ℝ)) 2, ← mul_assoc] lemma cos_eq_cos_iff {x y : ℝ} : cos x = cos y ↔ ∃ k : ℤ, y = 2 * k * π + x ∨ y = 2 * k * π - x := by exact_mod_cast @complex.cos_eq_cos_iff x y lemma sin_eq_sin_iff {x y : ℝ} : sin x = sin y ↔ ∃ k : ℤ, y = 2 * k * π + x ∨ y = (2 * k + 1) * π - x := by exact_mod_cast @complex.sin_eq_sin_iff x y lemma has_strict_deriv_at_tan {x : ℝ} (h : cos x ≠ 0) : has_strict_deriv_at tan (1 / (cos x)^2) x := by exact_mod_cast (complex.has_strict_deriv_at_tan (by exact_mod_cast h)).real_of_complex lemma has_deriv_at_tan {x : ℝ} (h : cos x ≠ 0) : has_deriv_at tan (1 / (cos x)^2) x := by exact_mod_cast (complex.has_deriv_at_tan (by exact_mod_cast h)).real_of_complex lemma tendsto_abs_tan_of_cos_eq_zero {x : ℝ} (hx : cos x = 0) : tendsto (λ x, abs (tan x)) (𝓝[{x}ᶜ] x) at_top := begin have hx : complex.cos x = 0, by exact_mod_cast hx, simp only [← complex.abs_of_real, complex.of_real_tan], refine (complex.tendsto_abs_tan_of_cos_eq_zero hx).comp _, refine tendsto.inf complex.continuous_of_real.continuous_at _, exact tendsto_principal_principal.2 (λ y, mt complex.of_real_inj.1) end lemma tendsto_abs_tan_at_top (k : ℤ) : tendsto (λ x, abs (tan x)) (𝓝[{(2 * k + 1) * π / 2}ᶜ] ((2 * k + 1) * π / 2)) at_top := tendsto_abs_tan_of_cos_eq_zero $ cos_eq_zero_iff.2 ⟨k, rfl⟩ lemma continuous_at_tan {x : ℝ} : continuous_at tan x ↔ cos x ≠ 0 := begin refine ⟨λ hc h₀, _, λ h, (has_deriv_at_tan h).continuous_at⟩, exact not_tendsto_nhds_of_tendsto_at_top (tendsto_abs_tan_of_cos_eq_zero h₀) _ (hc.norm.tendsto.mono_left inf_le_left) end lemma differentiable_at_tan {x : ℝ} : differentiable_at ℝ tan x ↔ cos x ≠ 0 := ⟨λ h, continuous_at_tan.1 h.continuous_at, λ h, (has_deriv_at_tan h).differentiable_at⟩ @[simp] lemma deriv_tan (x : ℝ) : deriv tan x = 1 / (cos x)^2 := if h : cos x = 0 then have ¬differentiable_at ℝ tan x := mt differentiable_at_tan.1 (not_not.2 h), by simp [deriv_zero_of_not_differentiable_at this, h, sq] else (has_deriv_at_tan h).deriv @[simp] lemma times_cont_diff_at_tan {n x} : times_cont_diff_at ℝ n tan x ↔ cos x ≠ 0 := ⟨λ h, continuous_at_tan.1 h.continuous_at, λ h, (complex.times_cont_diff_at_tan.2 $ by exact_mod_cast h).real_of_complex⟩ lemma continuous_on_tan : continuous_on tan {x \| cos x ≠ 0} := λ x hx, (continuous_at_tan.2 hx).continuous_within_at @[continuity] lemma continuous_tan : continuous (λ x : {x \| cos x ≠ 0}, tan x) := continuous_on_iff_continuous_restrict.1 continuous_on_tan lemma has_deriv_at_tan_of_mem_Ioo {x : ℝ} (h : x ∈ Ioo (-(π/2):ℝ) (π/2)) : has_deriv_at tan (1 / (cos x)^2) x := has_deriv_at_tan (cos_pos_of_mem_Ioo h).ne' lemma differentiable_at_tan_of_mem_Ioo {x : ℝ} (h : x ∈ Ioo (-(π/2):ℝ) (π/2)) : differentiable_at ℝ tan x := (has_deriv_at_tan_of_mem_Ioo h).differentiable_at lemma continuous_on_tan_Ioo : continuous_on tan (Ioo (-(π/2)) (π/2)) := λ x hx, (differentiable_at_tan_of_mem_Ioo hx).continuous_at.continuous_within_at lemma tendsto_sin_pi_div_two : tendsto sin (𝓝[Iio (π/2)] (π/2)) (𝓝 1) := by { convert continuous_sin.continuous_within_at, simp } lemma tendsto_cos_pi_div_two : tendsto cos (𝓝[Iio (π/2)] (π/2)) (𝓝[Ioi 0] 0) := begin apply tendsto_nhds_within_of_tendsto_nhds_of_eventually_within, { convert continuous_cos.continuous_within_at, simp }, { filter_upwards [Ioo_mem_nhds_within_Iio (right_mem_Ioc.mpr (norm_num.lt_neg_pos _ _ pi_div_two_pos pi_div_two_pos))] λ x hx, cos_pos_of_mem_Ioo hx }, end lemma tendsto_tan_pi_div_two : tendsto tan (𝓝[Iio (π/2)] (π/2)) at_top := begin convert tendsto_cos_pi_div_two.inv_tendsto_zero.at_top_mul zero_lt_one tendsto_sin_pi_div_two, simp only [pi.inv_apply, ← div_eq_inv_mul, ← tan_eq_sin_div_cos] end lemma tendsto_sin_neg_pi_div_two : tendsto sin (𝓝[Ioi (-(π/2))] (-(π/2))) (𝓝 (-1)) := by { convert continuous_sin.continuous_within_at, simp } lemma tendsto_cos_neg_pi_div_two : tendsto cos (𝓝[Ioi (-(π/2))] (-(π/2))) (𝓝[Ioi 0] 0) := begin apply tendsto_nhds_within_of_tendsto_nhds_of_eventually_within, { convert continuous_cos.continuous_within_at, simp }, { filter_upwards [Ioo_mem_nhds_within_Ioi (left_mem_Ico.mpr (norm_num.lt_neg_pos _ _ pi_div_two_pos pi_div_two_pos))] λ x hx, cos_pos_of_mem_Ioo hx }, end lemma tendsto_tan_neg_pi_div_two : tendsto tan (𝓝[Ioi (-(π/2))] (-(π/2))) at_bot := begin convert tendsto_cos_neg_pi_div_two.inv_tendsto_zero.at_top_mul_neg (by norm_num) tendsto_sin_neg_pi_div_two, simp only [pi.inv_apply, ← div_eq_inv_mul, ← tan_eq_sin_div_cos] end lemma surj_on_tan : surj_on tan (Ioo (-(π / 2)) (π / 2)) univ := have _ := neg_lt_self pi_div_two_pos, continuous_on_tan_Ioo.surj_on_of_tendsto (nonempty_Ioo.2 this) (by simp [tendsto_tan_neg_pi_div_two, this]) (by simp [tendsto_tan_pi_div_two, this]) lemma tan_surjective : function.surjective tan := λ x, surj_on_tan.subset_range trivial lemma image_tan_Ioo : tan '' (Ioo (-(π / 2)) (π / 2)) = univ := univ_subset_iff.1 surj_on_tan /-- `real.tan` as an `order_iso` between `(-(π / 2), π / 2)` and `ℝ`. -/ def tan_order_iso : Ioo (-(π / 2)) (π / 2) ≃o ℝ := (strict_mono_incr_on_tan.order_iso _ _).trans $ (order_iso.set_congr _ _ image_tan_Ioo).trans order_iso.set.univ /-- Inverse of the `tan` function, returns values in the range `-π / 2 < arctan x` and `arctan x < π / 2` -/ @[pp_nodot] noncomputable def arctan (x : ℝ) : ℝ := tan_order_iso.symm x @[simp] lemma tan_arctan (x : ℝ) : tan (arctan x) = x := tan_order_iso.apply_symm_apply x lemma arctan_mem_Ioo (x : ℝ) : arctan x ∈ Ioo (-(π / 2)) (π / 2) := subtype.coe_prop _ lemma arctan_tan {x : ℝ} (hx₁ : -(π / 2) < x) (hx₂ : x < π / 2) : arctan (tan x) = x := subtype.ext_iff.1 $ tan_order_iso.symm_apply_apply ⟨x, hx₁, hx₂⟩ lemma cos_arctan_pos (x : ℝ) : 0 < cos (arctan x) := cos_pos_of_mem_Ioo $ arctan_mem_Ioo x lemma cos_sq_arctan (x : ℝ) : cos (arctan x) ^ 2 = 1 / (1 + x ^ 2) := by rw [one_div, ← inv_one_add_tan_sq (cos_arctan_pos x).ne', tan_arctan] lemma sin_arctan (x : ℝ) : sin (arctan x) = x / sqrt (1 + x ^ 2) := by rw [← tan_div_sqrt_one_add_tan_sq (cos_arctan_pos x), tan_arctan] lemma cos_arctan (x : ℝ) : cos (arctan x) = 1 / sqrt (1 + x ^ 2) := by rw [one_div, ← inv_sqrt_one_add_tan_sq (cos_arctan_pos x), tan_arctan] lemma arctan_lt_pi_div_two (x : ℝ) : arctan x < π / 2 := (arctan_mem_Ioo x).2 lemma neg_pi_div_two_lt_arctan (x : ℝ) : -(π / 2) < arctan x := (arctan_mem_Ioo x).1 lemma arctan_eq_arcsin (x : ℝ) : arctan x = arcsin (x / sqrt (1 + x ^ 2)) := eq.symm $ arcsin_eq_of_sin_eq (sin_arctan x) (mem_Icc_of_Ioo $ arctan_mem_Ioo x) lemma arcsin_eq_arctan {x : ℝ} (h : x ∈ Ioo (-(1:ℝ)) 1) : arcsin x = arctan (x / sqrt (1 - x ^ 2)) := begin rw [arctan_eq_arcsin, div_pow, sq_sqrt, one_add_div, div_div_eq_div_mul, ← sqrt_mul, mul_div_cancel', sub_add_cancel, sqrt_one, div_one]; nlinarith [h.1, h.2], end @[simp] lemma arctan_zero : arctan 0 = 0 := by simp [arctan_eq_arcsin] lemma arctan_eq_of_tan_eq {x y : ℝ} (h : tan x = y) (hx : x ∈ Ioo (-(π / 2)) (π / 2)) : arctan y = x := inj_on_tan (arctan_mem_Ioo _) hx (by rw [tan_arctan, h]) @[simp] lemma arctan_one : arctan 1 = π / 4 := arctan_eq_of_tan_eq tan_pi_div_four $ by split; linarith [pi_pos] @[simp] lemma arctan_neg (x : ℝ) : arctan (-x) = - arctan x := by simp [arctan_eq_arcsin, neg_div] @[continuity] lemma continuous_arctan : continuous arctan := continuous_subtype_coe.comp tan_order_iso.to_homeomorph.continuous_inv_fun lemma continuous_at_arctan {x : ℝ} : continuous_at arctan x := continuous_arctan.continuous_at /-- `real.tan` as a `local_homeomorph` between `(-(π / 2), π / 2)` and the whole line. -/ def tan_local_homeomorph : local_homeomorph ℝ ℝ := { to_fun := tan, inv_fun := arctan, source := Ioo (-(π / 2)) (π / 2), target := univ, map_source' := maps_to_univ _ _, map_target' := λ y hy, arctan_mem_Ioo y, left_inv' := λ x hx, arctan_tan hx.1 hx.2, right_inv' := λ y hy, tan_arctan y, open_source := is_open_Ioo, open_target := is_open_univ, continuous_to_fun := continuous_on_tan_Ioo, continuous_inv_fun := continuous_arctan.continuous_on } @[simp] lemma coe_tan_local_homeomorph : ⇑tan_local_homeomorph = tan := rfl @[simp] lemma coe_tan_local_homeomorph_symm : ⇑tan_local_homeomorph.symm = arctan := rfl lemma has_strict_deriv_at_arctan (x : ℝ) : has_strict_deriv_at arctan (1 / (1 + x^2)) x := have A : cos (arctan x) ≠ 0 := (cos_arctan_pos x).ne', by simpa [cos_sq_arctan] using tan_local_homeomorph.has_strict_deriv_at_symm trivial (by simpa) (has_strict_deriv_at_tan A) lemma has_deriv_at_arctan (x : ℝ) : has_deriv_at arctan (1 / (1 + x^2)) x := (has_strict_deriv_at_arctan x).has_deriv_at lemma differentiable_at_arctan (x : ℝ) : differentiable_at ℝ arctan x := (has_deriv_at_arctan x).differentiable_at lemma differentiable_arctan : differentiable ℝ arctan := differentiable_at_arctan @[simp] lemma deriv_arctan : deriv arctan = (λ x, 1 / (1 + x^2)) := funext $ λ x, (has_deriv_at_arctan x).deriv lemma times_cont_diff_arctan {n : with_top ℕ} : times_cont_diff ℝ n arctan := times_cont_diff_iff_times_cont_diff_at.2 $ λ x, have cos (arctan x) ≠ 0 := (cos_arctan_pos x).ne', tan_local_homeomorph.times_cont_diff_at_symm_deriv (by simpa) trivial (has_deriv_at_tan this) (times_cont_diff_at_tan.2 this) lemma measurable_arctan : measurable arctan := continuous_arctan.measurable end real section /-! ### Lemmas for derivatives of the composition of `real.arctan` with a differentiable function In this section we register lemmas for the derivatives of the composition of `real.arctan` with a differentiable function, for standalone use and use with `simp`. -/ open real lemma measurable.arctan {α : Type} [measurable_space α] {f : α → ℝ} (hf : measurable f) : measurable (λ x, arctan (f x)) := measurable_arctan.comp hf section deriv variables {f : ℝ → ℝ} {f' x : ℝ} {s : set ℝ} lemma has_strict_deriv_at.arctan (hf : has_strict_deriv_at f f' x) : has_strict_deriv_at (λ x, arctan (f x)) ((1 / (1 + (f x)^2)) f') x := (real.has_strict_deriv_at_arctan (f x)).comp x hf lemma has_deriv_at.arctan (hf : has_deriv_at f f' x) : has_deriv_at (λ x, arctan (f x)) ((1 / (1 + (f x)^2)) * f') x := (real.has_deriv_at_arctan (f x)).comp x hf lemma has_deriv_within_at.arctan (hf : has_deriv_within_at f f' s x) : has_deriv_within_at (λ x, arctan (f x)) ((1 / (1 + (f x)^2)) * f') s x := (real.has_deriv_at_arctan (f x)).comp_has_deriv_within_at x hf lemma deriv_within_arctan (hf : differentiable_within_at ℝ f s x) (hxs : unique_diff_within_at ℝ s x) : deriv_within (λ x, arctan (f x)) s x = (1 / (1 + (f x)^2)) * (deriv_within f s x) := hf.has_deriv_within_at.arctan.deriv_within hxs @[simp] lemma deriv_arctan (hc : differentiable_at ℝ f x) : deriv (λ x, arctan (f x)) x = (1 / (1 + (f x)^2)) * (deriv f x) := hc.has_deriv_at.arctan.deriv end deriv section fderiv variables {E : Type*} [normed_group E] [normed_space ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} {s : set E} {n : with_top ℕ} lemma has_strict_fderiv_at.arctan (hf : has_strict_fderiv_at f f' x) : has_strict_fderiv_at (λ x, arctan (f x)) ((1 / (1 + (f x)^2)) • f') x := (has_strict_deriv_at_arctan (f x)).comp_has_strict_fderiv_at x hf lemma has_fderiv_at.arctan (hf : has_fderiv_at f f' x) : has_fderiv_at (λ x, arctan (f x)) ((1 / (1 + (f x)^2)) • f') x := (has_deriv_at_arctan (f x)).comp_has_fderiv_at x hf lemma has_fderiv_within_at.arctan (hf : has_fderiv_within_at f f' s x) : has_fderiv_within_at (λ x, arctan (f x)) ((1 / (1 + (f x)^2)) • f') s x := (has_deriv_at_arctan (f x)).comp_has_fderiv_within_at x hf lemma fderiv_within_arctan (hf : differentiable_within_at ℝ f s x) (hxs : unique_diff_within_at ℝ s x) : fderiv_within ℝ (λ x, arctan (f x)) s x = (1 / (1 + (f x)^2)) • (fderiv_within ℝ f s x) := hf.has_fderiv_within_at.arctan.fderiv_within hxs @[simp] lemma fderiv_arctan (hc : differentiable_at ℝ f x) : fderiv ℝ (λ x, arctan (f x)) x = (1 / (1 + (f x)^2)) • (fderiv ℝ f x) := hc.has_fderiv_at.arctan.fderiv lemma differentiable_within_at.arctan (hf : differentiable_within_at ℝ f s x) : differentiable_within_at ℝ (λ x, real.arctan (f x)) s x := hf.has_fderiv_within_at.arctan.differentiable_within_at @[simp] lemma differentiable_at.arctan (hc : differentiable_at ℝ f x) : differentiable_at ℝ (λ x, arctan (f x)) x := hc.has_fderiv_at.arctan.differentiable_at lemma differentiable_on.arctan (hc : differentiable_on ℝ f s) : differentiable_on ℝ (λ x, arctan (f x)) s := λ x h, (hc x h).arctan @[simp] lemma differentiable.arctan (hc : differentiable ℝ f) : differentiable ℝ (λ x, arctan (f x)) := λ x, (hc x).arctan lemma times_cont_diff_at.arctan (h : times_cont_diff_at ℝ n f x) : times_cont_diff_at ℝ n (λ x, arctan (f x)) x := times_cont_diff_arctan.times_cont_diff_at.comp x h lemma times_cont_diff.arctan (h : times_cont_diff ℝ n f) : times_cont_diff ℝ n (λ x, arctan (f x)) := times_cont_diff_arctan.comp h lemma times_cont_diff_within_at.arctan (h : times_cont_diff_within_at ℝ n f s x) : times_cont_diff_within_at ℝ n (λ x, arctan (f x)) s x := times_cont_diff_arctan.comp_times_cont_diff_within_at h lemma times_cont_diff_on.arctan (h : times_cont_diff_on ℝ n f s) : times_cont_diff_on ℝ n (λ x, arctan (f x)) s := times_cont_diff_arctan.comp_times_cont_diff_on h end fderiv end
theory "Differential_Axioms" imports Ordinary_Differential_Equations.ODE_Analysis "Ids" "Lib" "Syntax" "Denotational_Semantics" "Frechet_Correctness" "Axioms" "Coincidence" begin context ids begin section \<open>Differential Axioms\<close> text \<open>Differential axioms fall into two categories: Axioms for computing the derivatives of terms and axioms for proving properties of ODEs. The derivative axioms are all corollaries of the frechet correctness theorem. The ODE axioms are more involved, often requiring extensive use of the ODE libraries.\<close> subsection \<open>Derivative Axioms\<close> definition diff_const_axiom :: "('sf, 'sc, 'sz) formula" where [axiom_defs]:"diff_const_axiom \<equiv> Equals (Differential ($f fid1 empty)) (Const 0)" definition diff_var_axiom :: "('sf, 'sc, 'sz) formula" where [axiom_defs]:"diff_var_axiom \<equiv> Equals (Differential (Var vid1)) (DiffVar vid1)" definition state_fun ::"'sf \<Rightarrow> ('sf, 'sz) trm" where [axiom_defs]:"state_fun f = ($f f (\<lambda>i. Var i))" definition diff_plus_axiom :: "('sf, 'sc, 'sz) formula" where [axiom_defs]:"diff_plus_axiom \<equiv> Equals (Differential (Plus (state_fun fid1) (state_fun fid2))) (Plus (Differential (state_fun fid1)) (Differential (state_fun fid2)))" definition diff_times_axiom :: "('sf, 'sc, 'sz) formula" where [axiom_defs]:"diff_times_axiom \<equiv> Equals (Differential (Times (state_fun fid1) (state_fun fid2))) (Plus (Times (Differential (state_fun fid1)) (state_fun fid2)) (Times (state_fun fid1) (Differential (state_fun fid2))))" \<comment> \<open>\<open>[y=g(x)][y'=1](f(g(x))' = f(y)')\<close>\<close> definition diff_chain_axiom::"('sf, 'sc, 'sz) formula" where [axiom_defs]:"diff_chain_axiom \<equiv> [[Assign vid2 (f1 fid2 vid1)]]([[DiffAssign vid2 (Const 1)]] (Equals (Differential ($f fid1 (singleton (f1 fid2 vid1)))) (Times (Differential (f1 fid1 vid2)) (Differential (f1 fid2 vid1)))))" subsection \<open>ODE Axioms\<close> definition DWaxiom :: "('sf, 'sc, 'sz) formula" where [axiom_defs]:"DWaxiom = ([[EvolveODE (OVar vid1) (Predicational pid1)]](Predicational pid1))" definition DWaxiom' :: "('sf, 'sc, 'sz) formula" where [axiom_defs]:"DWaxiom' = ([[EvolveODE (OSing vid1 (Function fid1 (singleton (Var vid1)))) (Prop vid2 (singleton (Var vid1)))]](Prop vid2 (singleton (Var vid1))))" definition DCaxiom :: "('sf, 'sc, 'sz) formula" where [axiom_defs]:"DCaxiom = ( ([[EvolveODE (OVar vid1) (Predicational pid1)]]Predicational pid3) \<rightarrow> (([[EvolveODE (OVar vid1) (Predicational pid1)]](Predicational pid2)) \<leftrightarrow> ([[EvolveODE (OVar vid1) (And (Predicational pid1) (Predicational pid3))]]Predicational pid2)))" definition DEaxiom :: "('sf, 'sc, 'sz) formula" where [axiom_defs]:"DEaxiom = (([[EvolveODE (OSing vid1 (f1 fid1 vid1)) (p1 vid2 vid1)]] (P pid1)) \<leftrightarrow> ([[EvolveODE (OSing vid1 (f1 fid1 vid1)) (p1 vid2 vid1)]] [[DiffAssign vid1 (f1 fid1 vid1)]]P pid1))" definition DSaxiom :: "('sf, 'sc, 'sz) formula" where [axiom_defs]:"DSaxiom = (([[EvolveODE (OSing vid1 (f0 fid1)) (p1 vid2 vid1)]]p1 vid3 vid1) \<leftrightarrow> (Forall vid2 (Implies (Geq (Var vid2) (Const 0)) (Implies (Forall vid3 (Implies (And (Geq (Var vid3) (Const 0)) (Geq (Var vid2) (Var vid3))) (Prop vid2 (singleton (Plus (Var vid1) (Times (f0 fid1) (Var vid3))))))) ([[Assign vid1 (Plus (Var vid1) (Times (f0 fid1) (Var vid2)))]]p1 vid3 vid1)))))" \<comment> \<open>\<open>(Q \<rightarrow> [c&Q](f(x)' \<ge> g(x)'))\<close>\<close> \<comment> \<open>\<open>\<rightarrow>\<close>\<close> \<comment> \<open>\<open>([c&Q](f(x) \<ge> g(x))) --> (Q \<rightarrow> (f(x) \<ge> g(x))\<close>\<close> definition DIGeqaxiom :: "('sf, 'sc, 'sz) formula" where [axiom_defs]:"DIGeqaxiom = Implies (Implies (Prop vid1 empty) ([[EvolveODE (OVar vid1) (Prop vid1 empty)]](Geq (Differential (f1 fid1 vid1)) (Differential (f1 fid2 vid1))))) (Implies (Implies(Prop vid1 empty) (Geq (f1 fid1 vid1) (f1 fid2 vid1))) ([[EvolveODE (OVar vid1) (Prop vid1 empty)]](Geq (f1 fid1 vid1) (f1 fid2 vid1))))" \<comment> \<open>\<open>g(x) > h(x) \<rightarrow> [x'=f(x), c & p(x)](g(x)' \<ge> h(x)') \<rightarrow> [x'=f(x), c & p(x)]g(x) > h(x)\<close>\<close> \<comment> \<open>\<open>(Q \<rightarrow> [c&Q](f(x)' \<ge> g(x)'))\<close>\<close> \<comment> \<open>\<open>\<rightarrow>\<close>\<close> \<comment> \<open>\<open>([c&Q](f(x) > g(x))) <-> (Q \<rightarrow> (f(x) > g(x))\<close>\<close> definition DIGraxiom :: "('sf, 'sc, 'sz) formula" where [axiom_defs]:"DIGraxiom = Implies (Implies (Prop vid1 empty) ([[EvolveODE (OVar vid1) (Prop vid1 empty)]](Geq (Differential (f1 fid1 vid1)) (Differential (f1 fid2 vid1))))) (Implies (Implies(Prop vid1 empty) (Greater (f1 fid1 vid1) (f1 fid2 vid1))) ([[EvolveODE (OVar vid1) (Prop vid1 empty)]](Greater (f1 fid1 vid1) (f1 fid2 vid1))))" \<comment> \<open>\<open>[{1' = 1(1) & 1(1)}]2(1) <->\<close>\<close> \<comment> \<open>\<open>\<exists>2. [{1'=1(1), 2' = 2(1)2 + 3(1) & 1(1)}]2(1))\<close>\<close> definition DGaxiom :: "('sf, 'sc, 'sz) formula" where [axiom_defs]:"DGaxiom = (([[EvolveODE (OSing vid1 (f1 fid1 vid1)) (p1 vid1 vid1)]]p1 vid2 vid1) \<leftrightarrow> (Exists vid2 ([[EvolveODE (OProd (OSing vid1 (f1 fid1 vid1)) (OSing vid2 (Plus (Times (f1 fid2 vid1) (Var vid2)) (f1 fid3 vid1)))) (p1 vid1 vid1)]] p1 vid2 vid1)))" subsection \<open>Proofs for Derivative Axioms\<close> lemma constant_deriv_inner: assumes interp:"\<forall>x i. (Functions I i has_derivative FunctionFrechet I i x) (at x)" shows "FunctionFrechet I id1 (vec_lambda (\<lambda>i. sterm_sem I (empty i) (fst \<nu>))) (vec_lambda(\<lambda>i. frechet I (empty i) (fst \<nu>) (snd \<nu>)))= 0" proof - have empty_zero:"(vec_lambda(\<lambda>i. frechet I (empty i) (fst \<nu>) (snd \<nu>))) = 0" using local.empty_def Cart_lambda_cong frechet.simps(5) zero_vec_def apply auto apply(rule vec_extensionality) using local.empty_def Cart_lambda_cong frechet.simps(5) zero_vec_def by (simp add: local.empty_def) let ?x = "(vec_lambda (\<lambda>i. sterm_sem I (empty i) (fst \<nu>)))" from interp have has_deriv:"(Functions I id1 has_derivative FunctionFrechet I id1 ?x) (at ?x)" by auto then have f_linear:"linear (FunctionFrechet I id1 ?x)" using Deriv.has_derivative_linear by auto then show ?thesis using empty_zero f_linear linear_0 by (auto) qed lemma constant_deriv_zero:"is_interp I \<Longrightarrow> directional_derivative I ($f id1 empty) \<nu> = 0" apply(simp only: is_interp_def directional_derivative_def frechet.simps frechet_correctness) apply(rule constant_deriv_inner) apply(auto) done theorem diff_const_axiom_valid: "valid diff_const_axiom" apply(simp only: valid_def diff_const_axiom_def equals_sem) apply(rule allI \| rule impI)+ apply(simp only: dterm_sem.simps constant_deriv_zero sterm_sem.simps) done theorem diff_var_axiom_valid: "valid diff_var_axiom" apply(auto simp add: diff_var_axiom_def valid_def directional_derivative_def) by (metis inner_prod_eq) theorem diff_plus_axiom_valid: "valid diff_plus_axiom" apply(auto simp add: diff_plus_axiom_def valid_def) subgoal for I a b using frechet_correctness[of I "(Plus (state_fun fid1) (state_fun fid2))" b] unfolding state_fun_def apply (auto intro: dfree.intros) unfolding directional_derivative_def by auto done theorem diff_times_axiom_valid: "valid diff_times_axiom" apply(auto simp add: diff_times_axiom_def valid_def) subgoal for I a b using frechet_correctness[of I "(Times (state_fun fid1) (state_fun fid2))" b] unfolding state_fun_def apply (auto intro: dfree.intros) unfolding directional_derivative_def by auto done subsection \<open>Proofs for ODE Axioms\<close> lemma DW_valid:"valid DWaxiom" apply(unfold DWaxiom_def valid_def Let_def impl_sem ) apply(safe) apply(auto simp only: fml_sem.simps prog_sem.simps box_sem) subgoal for I aa ba ab bb sol t using mk_v_agree[of I "(OVar vid1)" "(ab,bb)" "sol t"] Vagree_univ[of "aa" "ba" "sol t" "ODEs I vid1 (sol t)"] solves_ode_domainD by (fastforce) done lemma DE_lemma: fixes ab bb::"'sz simple_state" and sol::"real \<Rightarrow> 'sz simple_state" and I::"('sf, 'sc, 'sz) interp" shows "repd (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)) vid1 (dterm_sem I (f1 fid1 vid1) (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t))) = mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)" proof have set_eq:" {Inl vid1, Inr vid1} = {Inr vid1, Inl vid1}" by auto have agree:"Vagree (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)) (mk_xode I (OSing vid1 (f1 fid1 vid1)) (sol t)) {Inl vid1, Inr vid1}" using mk_v_agree[of I "(OSing vid1 (f1 fid1 vid1))" "(ab, bb)" "(sol t)"] unfolding semBV.simps using set_eq by auto have fact:"dterm_sem I (f1 fid1 vid1) (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)) = snd (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)) $ vid1" using agree unfolding Vagree_def dterm_sem.simps f1_def mk_xode.simps proof - assume alls:"(\<forall>i. Inl i \<in> {Inl vid1, Inr vid1} \<longrightarrow> fst (mk_v I (OSing vid1 ($f fid1 (singleton (trm.Var vid1)))) (ab, bb) (sol t)) $ i = fst (sol t, ODE_sem I (OSing vid1 ($f fid1 (singleton (trm.Var vid1)))) (sol t)) $ i) \<and> (\<forall>i. Inr i \<in> {Inl vid1, Inr vid1} \<longrightarrow> snd (mk_v I (OSing vid1 ($f fid1 (singleton (trm.Var vid1)))) (ab, bb) (sol t)) $ i = snd (sol t, ODE_sem I (OSing vid1 ($f fid1 (singleton (trm.Var vid1)))) (sol t)) $ i)" hence atVid'':"snd (mk_v I (OSing vid1 ($f fid1 (singleton (trm.Var vid1)))) (ab, bb) (sol t)) $ vid1 = sterm_sem I ($f fid1 (singleton (trm.Var vid1))) (sol t)" by auto have argsEq:"(\<chi> i. dterm_sem I (singleton (trm.Var vid1) i) (mk_v I (OSing vid1 ($f fid1 (singleton (trm.Var vid1)))) (ab, bb) (sol t))) = (\<chi> i. sterm_sem I (singleton (trm.Var vid1) i) (sol t))" using alls f1_def by auto thus "Functions I fid1 (\<chi> i. dterm_sem I (singleton (trm.Var vid1) i) (mk_v I (OSing vid1 ($f fid1 (singleton (trm.Var vid1)))) (ab, bb) (sol t))) = snd (mk_v I (OSing vid1 ($f fid1 (singleton (trm.Var vid1)))) (ab, bb) (sol t)) $ vid1" by (simp only: atVid'' ODE_sem.simps sterm_sem.simps dterm_sem.simps) qed have eqSnd:"(\<chi> y. if vid1 = y then snd (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)) $ vid1 else snd (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)) $ y) = snd (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t))" by (simp add: vec_extensionality) have truth:"repd (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)) vid1 (dterm_sem I (f1 fid1 vid1) (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t))) = mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)" using fact by (auto simp only: eqSnd repd.simps fact prod.collapse split: if_split) thus "fst (repd (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)) vid1 (dterm_sem I (f1 fid1 vid1) (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)))) = fst (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t))" "snd (repd (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)) vid1 (dterm_sem I (f1 fid1 vid1) (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)))) = snd (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)) " by auto qed lemma DE_valid:"valid DEaxiom" proof - have dsafe:"dsafe ($f fid1 (singleton (trm.Var vid1)))" unfolding singleton_def by(auto intro: dsafe.intros) have osafe:"osafe(OSing vid1 (f1 fid1 vid1))" unfolding f1_def empty_def singleton_def using dsafe osafe.intros dsafe.intros by (simp add: osafe_Sing dfree_Const) have fsafe:"fsafe (p1 vid2 vid1)" unfolding p1_def singleton_def using hpsafe_fsafe.intros(10) using dsafe dsafe_Fun_simps image_iff by (simp add: dfree_Const) show "valid DEaxiom" apply(auto simp only: DEaxiom_def valid_def Let_def iff_sem impl_sem) apply(auto simp only: fml_sem.simps prog_sem.simps mem_Collect_eq box_sem) proof - fix I::"('sf,'sc,'sz) interp" and aa ba ab bb sol and t::real and ac bc assume "is_interp I" assume allw:"\<forall>\<omega>. (\<exists>\<nu> sol t. ((ab, bb), \<omega>) = (\<nu>, mk_v I (OSing vid1 (f1 fid1 vid1)) \<nu> (sol t)) \<and> 0 \<le> t \<and> (sol solves_ode (\<lambda>_. ODE_sem I (OSing vid1 (f1 fid1 vid1)))) {0..t} {x. mk_v I (OSing vid1 (f1 fid1 vid1)) \<nu> x \<in> fml_sem I (p1 vid2 vid1)} \<and> (sol 0) = (fst \<nu>) ) \<longrightarrow> \<omega> \<in> fml_sem I (P pid1)" assume t:"0 \<le> t" assume aaba:"(aa, ba) = mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)" assume solve:" (sol solves_ode (\<lambda>_. ODE_sem I (OSing vid1 (f1 fid1 vid1)))) {0..t} {x. mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) x \<in> fml_sem I (p1 vid2 vid1)}" assume sol0:" (sol 0) = (fst (ab, bb)) " assume rep:" (ac, bc) = repd (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)) vid1 (dterm_sem I (f1 fid1 vid1) (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)))" have aaba_sem:"(aa,ba) \<in> fml_sem I (P pid1)" using allw t aaba solve sol0 rep by blast have truth:"repd (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)) vid1 (dterm_sem I (f1 fid1 vid1) (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t))) = mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)" using DE_lemma by auto show " repd (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)) vid1 (dterm_sem I (f1 fid1 vid1) (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t))) \<in> fml_sem I (P pid1)" using aaba aaba_sem truth by (auto) next fix I::"('sf,'sc,'sz) interp" and aa ba ab bb sol and t::real assume "is_interp I" assume all:"\<forall>\<omega>. (\<exists>\<nu> sol t. ((ab, bb), \<omega>) = (\<nu>, mk_v I (OSing vid1 (f1 fid1 vid1)) \<nu> (sol t)) \<and> 0 \<le> t \<and> (sol solves_ode (\<lambda>_. ODE_sem I (OSing vid1 (f1 fid1 vid1)))) {0..t} {x. mk_v I (OSing vid1 (f1 fid1 vid1)) \<nu> x \<in> fml_sem I (p1 vid2 vid1)} \<and> (sol 0) = (fst \<nu>) ) \<longrightarrow> (\<forall>\<omega>'. \<omega>' = repd \<omega> vid1 (dterm_sem I (f1 fid1 vid1) \<omega>) \<longrightarrow> \<omega>' \<in> fml_sem I (P pid1))" hence justW:"(\<exists>\<nu> sol t. ((ab, bb), (aa, ba)) = (\<nu>, mk_v I (OSing vid1 (f1 fid1 vid1)) \<nu> (sol t)) \<and> 0 \<le> t \<and> (sol solves_ode (\<lambda>_. ODE_sem I (OSing vid1 (f1 fid1 vid1)))) {0..t} {x. mk_v I (OSing vid1 (f1 fid1 vid1)) \<nu> x \<in> fml_sem I (p1 vid2 vid1)} \<and> (sol 0) = (fst \<nu>)) \<longrightarrow> (\<forall>\<omega>'. \<omega>' = repd (aa, ba) vid1 (dterm_sem I (f1 fid1 vid1) (aa, ba)) \<longrightarrow> \<omega>' \<in> fml_sem I (P pid1))" by (rule allE) assume t:"0 \<le> t" assume aaba:"(aa, ba) = mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)" assume sol:"(sol solves_ode (\<lambda>_. ODE_sem I (OSing vid1 (f1 fid1 vid1)))) {0..t} {x. mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) x \<in> fml_sem I (p1 vid2 vid1)}" assume sol0:" (sol 0) = (fst (ab, bb))" have "repd (aa, ba) vid1 (dterm_sem I (f1 fid1 vid1) (aa, ba)) \<in> fml_sem I (P pid1)" using justW t aaba sol sol0 by auto hence foo:"repd (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)) vid1 (dterm_sem I (f1 fid1 vid1) (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t))) \<in> fml_sem I (P pid1)" using aaba by auto hence "repd (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)) vid1 (dterm_sem I (f1 fid1 vid1) (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t))) = mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)" using DE_lemma by auto thus "mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t) \<in> fml_sem I (P pid1)" using foo by auto qed qed lemma ODE_zero:"\<And>i. Inl i \<notin> BVO ODE \<Longrightarrow> Inr i \<notin> BVO ODE \<Longrightarrow> ODE_sem I ODE \<nu> $ i= 0" by(induction ODE, auto) lemma DE_sys_valid: assumes disj:"{Inl vid1, Inr vid1} \<inter> BVO ODE = {}" shows "valid (([[EvolveODE (OProd (OSing vid1 (f1 fid1 vid1)) ODE) (p1 vid2 vid1)]] (P pid1)) \<leftrightarrow> ([[EvolveODE ((OProd (OSing vid1 (f1 fid1 vid1))ODE)) (p1 vid2 vid1)]] [[DiffAssign vid1 (f1 fid1 vid1)]]P pid1))" proof - have dsafe:"dsafe ($f fid1 (singleton (trm.Var vid1)))" unfolding singleton_def by(auto intro: dsafe.intros) have osafe:"osafe(OSing vid1 (f1 fid1 vid1))" unfolding f1_def empty_def singleton_def using dsafe osafe.intros dsafe.intros by (simp add: osafe_Sing dfree_Const) have fsafe:"fsafe (p1 vid2 vid1)" unfolding p1_def singleton_def using hpsafe_fsafe.intros(10) using dsafe dsafe_Fun_simps image_iff by (simp add: dfree_Const) show "valid (([[EvolveODE (OProd (OSing vid1 (f1 fid1 vid1)) ODE) (p1 vid2 vid1)]] (P pid1)) \<leftrightarrow> ([[EvolveODE ((OProd (OSing vid1 (f1 fid1 vid1)) ODE)) (p1 vid2 vid1)]] [[DiffAssign vid1 (f1 fid1 vid1)]]P pid1))" apply(auto simp only: DEaxiom_def valid_def Let_def iff_sem impl_sem) apply(auto simp only: fml_sem.simps prog_sem.simps mem_Collect_eq box_sem f1_def p1_def P_def expand_singleton) proof - fix I ::"('sf,'sc,'sz) interp" and aa ba ab bb sol and t::real and ac bc assume good:"is_interp I" assume bigAll:" \<forall>\<omega>. (\<exists>\<nu> sol t. ((ab, bb), \<omega>) = (\<nu>, mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) ODE) \<nu> (sol t)) \<and> 0 \<le> t \<and> (sol solves_ode (\<lambda>_. ODE_sem I (OProd(OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) ODE ))) {0..t} {x. Predicates I vid2 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) \<nu> x))} \<and> sol 0 = fst \<nu>) \<longrightarrow> \<omega> \<in> fml_sem I (Pc pid1)" let ?my\<omega> = "mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) (ab,bb) (sol t)" assume t:"0 \<le> t" assume aaba:"(aa, ba) = mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) (ab, bb) (sol t)" assume sol:"(sol solves_ode (\<lambda>_. ODE_sem I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE))) {0..t} {x. Predicates I vid2 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) (ab, bb) x))}" assume sol0:"sol 0 = fst (ab, bb)" assume acbc:"(ac, bc) = repd (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) (ab, bb) (sol t)) vid1 (dterm_sem I ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) (ab, bb) (sol t)))" have bigEx:"(\<exists>\<nu> sol t. ((ab, bb), ?my\<omega>) = (\<nu>, mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) \<nu> (sol t)) \<and> 0 \<le> t \<and> (sol solves_ode (\<lambda>_. ODE_sem I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE))) {0..t} {x. Predicates I vid2 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) \<nu> x))} \<and> sol 0 = fst \<nu>)" apply(rule exI[where x="(ab, bb)"]) apply(rule exI[where x="sol"]) apply(rule exI[where x="t"]) apply(rule conjI) apply(rule refl) apply(rule conjI) apply(rule t) apply(rule conjI) using sol apply blast by (rule sol0) have bigRes:"?my\<omega> \<in> fml_sem I (Pc pid1)" using bigAll bigEx by blast have notin1:"Inl vid1 \<notin> BVO ODE" using disj by auto have notin2:"Inr vid1 \<notin> BVO ODE" using disj by auto have ODE_sem:"ODE_sem I ODE (sol t) $ vid1 = 0" using ODE_zero notin1 notin2 by blast have vec_eq:"(\<chi> i. sterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (sol t)) = (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) (ab, bb) (sol t)))" apply(rule vec_extensionality) apply simp using mk_v_agree[of I "(OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE)" "(ab, bb)" "(sol t)"] by(simp add: Vagree_def) have sem_eq:"(?my\<omega> \<in> fml_sem I (Pc pid1)) = ((repd (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) (ab, bb) (sol t)) vid1 (dterm_sem I ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) (ab, bb) (sol t)))) \<in> fml_sem I (Pc pid1))" apply(rule coincidence_formula) subgoal by simp subgoal by (rule Iagree_refl) using mk_v_agree[of "I" "(OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE)" "(ab, bb)" "(sol t)"] unfolding Vagree_def apply simp apply(erule conjE)+ apply(erule allE[where x="vid1"])+ apply(simp add: ODE_sem) using vec_eq by simp show "repd (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) (ab, bb) (sol t)) vid1 (dterm_sem I ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) (ab, bb) (sol t))) \<in> fml_sem I (Pc pid1)" using bigRes sem_eq by blast next fix I::"('sf,'sc,'sz)interp" and aa ba ab bb sol and t::real assume good_interp:"is_interp I" assume all:"\<forall>\<omega>. (\<exists>\<nu> sol t. ((ab, bb), \<omega>) = (\<nu>, mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) \<nu> (sol t)) \<and> 0 \<le> t \<and> (sol solves_ode (\<lambda>_. ODE_sem I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE))) {0..t} {x. Predicates I vid2 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) \<nu> x))} \<and> sol 0 = fst \<nu>) \<longrightarrow> (\<forall>\<omega>'. \<omega>' = repd \<omega> vid1 (dterm_sem I ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) \<omega>) \<longrightarrow> \<omega>' \<in> fml_sem I (Pc pid1))" let ?my\<omega> = "mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) (ab, bb) (sol t)" assume t:"0 \<le> t" assume aaba:"(aa, ba) = mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) (ab, bb) (sol t)" assume sol:" (sol solves_ode (\<lambda>_. ODE_sem I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE))) {0..t} {x. Predicates I vid2 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) (ab, bb) x))}" assume sol0:"sol 0 = fst (ab, bb)" have bigEx:"(\<exists>\<nu> sol t. ((ab, bb), ?my\<omega>) = (\<nu>, mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) \<nu> (sol t)) \<and> 0 \<le> t \<and> (sol solves_ode (\<lambda>_. ODE_sem I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE))) {0..t} {x. Predicates I vid2 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) \<nu> x))} \<and> sol 0 = fst \<nu>)" apply(rule exI[where x="(ab, bb)"]) apply(rule exI[where x=sol]) apply(rule exI[where x=t]) apply(rule conjI) apply(rule refl) apply(rule conjI) apply(rule t) apply(rule conjI) using sol sol0 by(blast)+ have rep_sem_eq:"repd (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) (ab, bb) (sol t)) vid1 (dterm_sem I ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) (ab, bb) (sol t))) \<in> fml_sem I (Pc pid1) = (repd ?my\<omega> vid1 (dterm_sem I ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) ?my\<omega>) \<in> fml_sem I (Pc pid1))" apply(rule coincidence_formula) subgoal by simp subgoal by (rule Iagree_refl) by(simp add: Vagree_def) have notin1:"Inl vid1 \<notin> BVO ODE" using disj by auto have notin2:"Inr vid1 \<notin> BVO ODE" using disj by auto have ODE_sem:"ODE_sem I ODE (sol t) $ vid1 = 0" using ODE_zero notin1 notin2 by blast have vec_eq:" (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) (ab, bb) (sol t))) = (\<chi> i. sterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (sol t))" apply(rule vec_extensionality) using mk_v_agree[of I "(OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE)" "(ab, bb)" "(sol t)"] by (simp add: Vagree_def) have sem_eq: "(repd ?my\<omega> vid1 (dterm_sem I ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) ?my\<omega>) \<in> fml_sem I (Pc pid1)) = (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) (ab, bb) (sol t) \<in> fml_sem I (Pc pid1)) " apply(rule coincidence_formula) subgoal by simp subgoal by (rule Iagree_refl) using mk_v_agree[of I "(OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE)" "(ab, bb)" "(sol t)"] unfolding Vagree_def apply simp apply(erule conjE)+ apply(erule allE[where x=vid1])+ by (simp add: ODE_sem vec_eq) have some_sem:"repd (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) (ab, bb) (sol t)) vid1 (dterm_sem I ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) (ab, bb) (sol t))) \<in> fml_sem I (Pc pid1)" using rep_sem_eq using all bigEx by blast have bigImp:"(\<forall>\<omega>'. \<omega>' = repd ?my\<omega> vid1 (dterm_sem I ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) ?my\<omega>) \<longrightarrow> \<omega>' \<in> fml_sem I (Pc pid1))" apply(rule allI) apply(rule impI) apply auto using some_sem by auto have fml_sem:"repd ?my\<omega> vid1 (dterm_sem I ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) ?my\<omega>) \<in> fml_sem I (Pc pid1)" using sem_eq bigImp by blast show "mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))ODE) (ab, bb) (sol t) \<in> fml_sem I (Pc pid1)" using fml_sem sem_eq by blast qed qed lemma DC_valid:"valid DCaxiom" proof (auto simp only: fml_sem.simps prog_sem.simps DCaxiom_def valid_def iff_sem impl_sem box_sem, auto) fix I::"('sf,'sc,'sz) interp" and aa ba bb sol t assume "is_interp I" and all3:"\<forall>a b. (\<exists>sola. sol 0 = sola 0 \<and> (\<exists>t. (a, b) = mk_v I (OVar vid1) (sola 0, bb) (sola t) \<and> 0 \<le> t \<and> (sola solves_ode (\<lambda>a. ODEs I vid1)) {0..t} {x. mk_v I (OVar vid1) (sola 0, bb) x \<in> Contexts I pid1 UNIV})) \<longrightarrow> (a, b) \<in> Contexts I pid3 UNIV" and all2:"\<forall>a b. (\<exists>sola. sol 0 = sola 0 \<and> (\<exists>t. (a, b) = mk_v I (OVar vid1) (sola 0, bb) (sola t) \<and> 0 \<le> t \<and> (sola solves_ode (\<lambda>a. ODEs I vid1)) {0..t} {x. mk_v I (OVar vid1) (sola 0, bb) x \<in> Contexts I pid1 UNIV})) \<longrightarrow> (a, b) \<in> Contexts I pid2 UNIV" and t:"0 \<le> t" and aaba:"(aa, ba) = mk_v I (OVar vid1) (sol 0, bb) (sol t)" and sol:"(sol solves_ode (\<lambda>a. ODEs I vid1)) {0..t} {x. mk_v I (OVar vid1) (sol 0, bb) x \<in> Contexts I pid1 UNIV \<and> mk_v I (OVar vid1) (sol 0, bb) x \<in> Contexts I pid3 UNIV}" from sol have sol1:"(sol solves_ode (\<lambda>a. ODEs I vid1)) {0..t} {x. mk_v I (OVar vid1) (sol 0, bb) x \<in> Contexts I pid1 UNIV}" by (metis (mono_tags, lifting) Collect_mono solves_ode_supset_range) from all2 have all2':"\<And>v. (\<exists>sola. sol 0 = sola 0 \<and> (\<exists>t. v = mk_v I (OVar vid1) (sola 0, bb) (sola t) \<and> 0 \<le> t \<and> (sola solves_ode (\<lambda>a. ODEs I vid1)) {0..t} {x. mk_v I (OVar vid1) (sola 0, bb) x \<in> Contexts I pid1 UNIV})) \<Longrightarrow> v \<in> Contexts I pid2 UNIV" by auto show "mk_v I (OVar vid1) (sol 0, bb) (sol t) \<in> Contexts I pid2 UNIV" apply(rule all2'[of "mk_v I (OVar vid1) (sol 0, bb) (sol t)"]) apply(rule exI[where x=sol]) apply(rule conjI) subgoal by (rule refl) subgoal using t sol1 by auto done next fix I::"('sf,'sc,'sz) interp" and aa ba bb sol t assume "is_interp I" and all3:"\<forall>a b. (\<exists>sola. sol 0 = sola 0 \<and> (\<exists>t. (a, b) = mk_v I (OVar vid1) (sola 0, bb) (sola t) \<and> 0 \<le> t \<and> (sola solves_ode (\<lambda>a. ODEs I vid1)) {0..t} {x. mk_v I (OVar vid1) (sola 0, bb) x \<in> Contexts I pid1 UNIV})) \<longrightarrow> (a, b) \<in> Contexts I pid3 UNIV" and all2:"\<forall>a b. (\<exists>sola. sol 0 = sola 0 \<and> (\<exists>t. (a, b) = mk_v I (OVar vid1) (sola 0, bb) (sola t) \<and> 0 \<le> t \<and> (sola solves_ode (\<lambda>a. ODEs I vid1)) {0..t} {x. mk_v I (OVar vid1) (sola 0, bb) x \<in> Contexts I pid1 UNIV \<and> mk_v I (OVar vid1) (sola 0, bb) x \<in> Contexts I pid3 UNIV})) \<longrightarrow> (a, b) \<in> Contexts I pid2 UNIV" and t:"0 \<le> t" and aaba:"(aa, ba) = mk_v I (OVar vid1) (sol 0, bb) (sol t)" and sol:"(sol solves_ode (\<lambda>a. ODEs I vid1)) {0..t} {x. mk_v I (OVar vid1) (sol 0, bb) x \<in> Contexts I pid1 UNIV}" from all2 have all2':"\<And>v. (\<exists>sola. sol 0 = sola 0 \<and> (\<exists>t. v = mk_v I (OVar vid1) (sola 0, bb) (sola t) \<and> 0 \<le> t \<and> (sola solves_ode (\<lambda>a. ODEs I vid1)) {0..t} {x. mk_v I (OVar vid1) (sola 0, bb) x \<in> Contexts I pid1 UNIV \<and> mk_v I (OVar vid1) (sola 0, bb) x \<in> Contexts I pid3 UNIV})) \<Longrightarrow> v \<in> Contexts I pid2 UNIV" by auto from all3 have all3':"\<And>v. (\<exists>sola. sol 0 = sola 0 \<and> (\<exists>t. v = mk_v I (OVar vid1) (sola 0, bb) (sola t) \<and> 0 \<le> t \<and> (sola solves_ode (\<lambda>a. ODEs I vid1)) {0..t} {x. mk_v I (OVar vid1) (sola 0, bb) x \<in> Contexts I pid1 UNIV})) \<Longrightarrow> v \<in> Contexts I pid3 UNIV" by auto have inp1:"\<And>s. 0 \<le> s \<Longrightarrow> s \<le> t \<Longrightarrow> mk_v I (OVar vid1) (sol 0, bb) (sol s) \<in> Contexts I pid1 UNIV" using sol solves_odeD atLeastAtMost_iff by blast have inp3:"\<And>s. 0 \<le> s \<Longrightarrow> s \<le> t \<Longrightarrow> mk_v I (OVar vid1) (sol 0, bb) (sol s) \<in> Contexts I pid3 UNIV" apply(rule all3') subgoal for s apply(rule exI [where x=sol]) apply(rule conjI) subgoal by (rule refl) apply(rule exI [where x=s]) apply(rule conjI) subgoal by (rule refl) apply(rule conjI) subgoal by assumption subgoal using sol by (meson atLeastatMost_subset_iff order_refl solves_ode_subset) done done have inp13:"\<And>s. 0 \<le> s \<Longrightarrow> s \<le> t \<Longrightarrow> mk_v I (OVar vid1) (sol 0, bb) (sol s) \<in> Contexts I pid1 UNIV \<and> mk_v I (OVar vid1) (sol 0, bb) (sol s) \<in> Contexts I pid3 UNIV" using inp1 inp3 by auto have sol13:"(sol solves_ode (\<lambda>a. ODEs I vid1)) {0..t} {x. mk_v I (OVar vid1) (sol 0, bb) x \<in> Contexts I pid1 UNIV \<and> mk_v I (OVar vid1) (sol 0, bb) x \<in> Contexts I pid3 UNIV}" apply(rule solves_odeI) subgoal using sol by (rule solves_odeD) subgoal for s using inp13[of s] by auto done show "mk_v I (OVar vid1) (sol 0, bb) (sol t) \<in> Contexts I pid2 UNIV" using t sol13 all2'[of "mk_v I (OVar vid1) (sol 0, bb) (sol t)"] by auto qed lemma DS_valid:"valid DSaxiom" proof - have dsafe:"dsafe($f fid1 (\<lambda>i. Const 0))" using dsafe_Const by auto have osafe:"osafe(OSing vid1 (f0 fid1))" unfolding f0_def empty_def using dsafe osafe.intros by (simp add: osafe_Sing dfree_Const) have fsafe:"fsafe(p1 vid2 vid1)" unfolding p1_def apply(rule fsafe_Prop) using singleton.simps dsafe_Const by (auto intro: dfree.intros) show "valid DSaxiom" apply(auto simp only: DSaxiom_def valid_def Let_def iff_sem impl_sem box_sem) apply(auto simp only: fml_sem.simps prog_sem.simps mem_Collect_eq iff_sem impl_sem box_sem forall_sem) proof - fix I::"('sf,'sc,'sz) interp" and a b r aa ba assume good_interp:"is_interp I" assume allW:"\<forall>\<omega>. (\<exists>\<nu> sol t. ((a, b), \<omega>) = (\<nu>, mk_v I (OSing vid1 (f0 fid1)) \<nu> (sol t)) \<and> 0 \<le> t \<and> (sol solves_ode (\<lambda>_. ODE_sem I (OSing vid1 (f0 fid1)))) {0..t} {x. mk_v I (OSing vid1 (f0 fid1)) \<nu> x \<in> fml_sem I (p1 vid2 vid1)} \<and> (sol 0) = (fst \<nu>)) \<longrightarrow> \<omega> \<in> fml_sem I (p1 vid3 vid1)" assume "dterm_sem I (Const 0) (repv (a, b) vid2 r) \<le> dterm_sem I (trm.Var vid2) (repv (a, b) vid2 r)" hence leq:"0 \<le> r" by (auto) assume "\<forall>ra. repv (repv (a, b) vid2 r) vid3 ra \<in> {v. dterm_sem I (Const 0) v \<le> dterm_sem I (trm.Var vid3) v} \<inter> {v. dterm_sem I (trm.Var vid3) v \<le> dterm_sem I (trm.Var vid2) v} \<longrightarrow> Predicates I vid2 (\<chi> i. dterm_sem I (singleton (Plus (trm.Var vid1) (Times (f0 fid1) (trm.Var vid3))) i) (repv (repv (a, b) vid2 r) vid3 ra))" hence constraint:"\<forall>ra. (0 \<le> ra \<and> ra \<le> r) \<longrightarrow> (repv (repv (a, b) vid2 r) vid3 ra) \<in> fml_sem I (Prop vid2 (singleton (Plus (Var vid1) (Times (f0 fid1) (Var vid3)))))" using leq by auto assume aaba:" (aa, ba) = repv (repv (a, b) vid2 r) vid1 (dterm_sem I (Plus (trm.Var vid1) (Times (f0 fid1) (trm.Var vid2))) (repv (a, b) vid2 r))" let ?abba = "repv (repd (a, b) vid1 (Functions I fid1 (\<chi> i. 0))) vid1 (dterm_sem I (Plus (trm.Var vid1) (Times (f0 fid1) (trm.Var vid2))) (repv (a, b) vid2 r))" from allW have thisW:"(\<exists>\<nu> sol t. ((a, b), ?abba) = (\<nu>, mk_v I (OSing vid1 (f0 fid1)) \<nu> (sol t)) \<and> 0 \<le> t \<and> (sol solves_ode (\<lambda>_. ODE_sem I (OSing vid1 (f0 fid1)))) {0..t} {x. mk_v I (OSing vid1 (f0 fid1)) \<nu> x \<in> fml_sem I (p1 vid2 vid1)} \<and> (sol 0) = (fst \<nu>)) \<longrightarrow> ?abba \<in> fml_sem I (p1 vid3 vid1)" by blast let ?c = "Functions I fid1 (\<chi> _. 0)" let ?sol = "(\<lambda>t. \<chi> i. if i = vid1 then (a $ i) + ?c * t else (a $ i))" have agrees:"Vagree (mk_v I (OSing vid1 (f0 fid1)) (a, b) (?sol r)) (a, b) (- semBV I (OSing vid1 (f0 fid1))) \<and> Vagree (mk_v I (OSing vid1 (f0 fid1)) (a, b) (?sol r)) (mk_xode I (OSing vid1 (f0 fid1)) (?sol r)) (semBV I (OSing vid1 (f0 fid1)))" using mk_v_agree[of "I" "(OSing vid1 (f0 fid1))" "(a,b)" "(?sol r)"] by auto have prereq1a:"fst ?abba = fst (mk_v I (OSing vid1 (f0 fid1)) (a,b) (?sol r))" using agrees aaba apply (auto simp add: aaba Vagree_def) apply (rule vec_extensionality) subgoal for i apply (cases "i = vid1") using vne12 agrees Vagree_def apply (auto simp add: aaba f0_def empty_def) done apply (rule vec_extensionality) subgoal for i apply (cases "i = vid1") apply(auto simp add: f0_def empty_def) done done have prereq1b:"snd (?abba) = snd (mk_v I (OSing vid1 (f0 fid1)) (a,b) (?sol r))" using agrees aaba apply (auto simp add: aaba Vagree_def) apply (rule vec_extensionality) subgoal for i apply (cases "i = vid1") using vne12 agrees Vagree_def apply (auto simp add: aaba f0_def empty_def ) done done have "?abba = mk_v I (OSing vid1 (f0 fid1)) (a,b) (?sol r)" using prod_eq_iff prereq1a prereq1b by blast hence req1:"((a, b), ?abba) = ((a, b), mk_v I (OSing vid1 (f0 fid1)) (a,b) (?sol r))" by auto have "sterm_sem I ($f fid1 (\<lambda>i. Const 0)) b = Functions I fid1 (\<chi> i. 0)" by auto hence vec_simp:"(\<lambda>a b. \<chi> i. if i = vid1 then sterm_sem I ($f fid1 (\<lambda>i. Const 0)) b else 0) = (\<lambda>a b. \<chi> i. if i = vid1 then Functions I fid1 (\<chi> i. 0) else 0)" by (auto simp add: vec_eq_iff cong: if_cong) have sub: "{0..r} \<subseteq> UNIV" by auto have sub2:"{x. mk_v I (OSing vid1 (f0 fid1)) (a,b) x \<in> fml_sem I (p1 vid2 vid1)} \<subseteq> UNIV" by auto have req3:"(?sol solves_ode (\<lambda>_. ODE_sem I (OSing vid1 (f0 fid1)))) {0..r} {x. mk_v I (OSing vid1 (f0 fid1)) (a,b) x \<in> fml_sem I (p1 vid2 vid1)}" apply(auto simp add: f0_def empty_def vec_simp) apply(rule solves_odeI) apply(auto simp only: has_vderiv_on_def has_vector_derivative_def box_sem) apply (rule has_derivative_vec[THEN has_derivative_eq_rhs]) defer apply (rule ext) apply (subst scaleR_vec_def) apply (rule refl) apply (auto intro!: derivative_eq_intros) \<comment> \<open>Domain constraint satisfied\<close> using constraint apply (auto) subgoal for t apply(erule allE[where x="t"]) apply(auto simp add: p1_def) proof - have eq:"(\<chi> i. dterm_sem I (if i = vid1 then Plus (trm.Var vid1) (Times (f0 fid1) (trm.Var vid3)) else Const 0) (\<chi> y. if vid3 = y then t else fst (\<chi> y. if vid2 = y then r else fst (a, b) $ y, b) $ y, b)) = (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OSing vid1 ($f fid1 (\<lambda>i. Const 0))) (a, b) (\<chi> i. if i = vid1 then a $ i + Functions I fid1 (\<chi> _. 0) * t else a $ i)))" using vne12 vne13 mk_v_agree[of "I" "(OSing vid1 ($f fid1 (\<lambda>i. Const 0)))" "(a, b)" "(\<chi> i. if i = vid1 then a $ i + Functions I fid1 (\<chi> _. 0) * t else a $ i)"] by (auto simp add: vec_eq_iff f0_def empty_def Vagree_def) show "0 \<le> t \<Longrightarrow> t \<le> r \<Longrightarrow> Predicates I vid2 (\<chi> i. dterm_sem I (if i = vid1 then Plus (trm.Var vid1) (Times (f0 fid1) (trm.Var vid3)) else Const 0) (\<chi> y. if vid3 = y then t else fst (\<chi> y. if vid2 = y then r else fst (a, b) $ y, b) $ y, b)) \<Longrightarrow> Predicates I vid2 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OSing vid1 ($f fid1 (\<lambda>i. Const 0))) (a, b) (\<chi> i. if i = vid1 then a $ i + Functions I fid1 (\<chi> _. 0) * t else a $ i)))" using eq by auto qed done have req4':"?sol 0 = fst (a,b)" by (auto simp: vec_eq_iff) then have req4: " (?sol 0) = (fst (a,b))" using VSagree_refl[of a] req4' unfolding VSagree_def by auto have inPred:"?abba \<in> fml_sem I (p1 vid3 vid1)" using req1 leq req3 req4 thisW by fastforce have sem_eq:"?abba \<in> fml_sem I (p1 vid3 vid1) \<longleftrightarrow> (aa,ba) \<in> fml_sem I (p1 vid3 vid1)" apply (rule coincidence_formula) apply (auto simp add: aaba Vagree_def p1_def f0_def empty_def) subgoal using Iagree_refl by auto done from inPred sem_eq have inPred':"(aa,ba) \<in> fml_sem I (p1 vid3 vid1)" by auto \<comment> \<open>thus by lemma 6 consequence for formulas\<close> show "repv (repv (a, b) vid2 r) vid1 (dterm_sem I (Plus (trm.Var vid1) (Times (f0 fid1) (trm.Var vid2))) (repv (a, b) vid2 r)) \<in> fml_sem I (p1 vid3 vid1)" using aaba inPred' by (auto) next fix I::"('sf,'sc,'sz) interp" and aa ba ab bb sol and t:: real assume good_interp:"is_interp I" assume all:" \<forall>r. dterm_sem I (Const 0) (repv (ab, bb) vid2 r) \<le> dterm_sem I (trm.Var vid2) (repv (ab, bb) vid2 r) \<longrightarrow> (\<forall>ra. repv (repv (ab, bb) vid2 r) vid3 ra \<in> {v. dterm_sem I (Const 0) v \<le> dterm_sem I (trm.Var vid3) v} \<inter> {v. dterm_sem I (trm.Var vid3) v \<le> dterm_sem I (trm.Var vid2) v} \<longrightarrow> Predicates I vid2 (\<chi> i. dterm_sem I (singleton (Plus (trm.Var vid1) (Times (f0 fid1) (trm.Var vid3))) i) (repv (repv (ab, bb) vid2 r) vid3 ra))) \<longrightarrow> (\<forall>\<omega>. \<omega> = repv (repv (ab, bb) vid2 r) vid1 (dterm_sem I (Plus (trm.Var vid1) (Times (f0 fid1) (trm.Var vid2))) (repv (ab, bb) vid2 r)) \<longrightarrow> \<omega> \<in> fml_sem I (p1 vid3 vid1))" assume t:"0 \<le> t" assume aaba:"(aa, ba) = mk_v I (OSing vid1 (f0 fid1)) (ab, bb) (sol t)" assume sol:"(sol solves_ode (\<lambda>_. ODE_sem I (OSing vid1 (f0 fid1)))) {0..t} {x. mk_v I (OSing vid1 (f0 fid1)) (ab, bb) x \<in> fml_sem I (p1 vid2 vid1)}" hence constraint:"\<And>s. s \<in> {0 .. t} \<Longrightarrow> sol s \<in> {x. mk_v I (OSing vid1 (f0 fid1)) (ab, bb) x \<in> fml_sem I (p1 vid2 vid1)}" using solves_ode_domainD by fastforce \<comment> \<open>\<open>sol 0 = fst (ab, bb)\<close>\<close> assume sol0:" (sol 0) = (fst (ab, bb)) " have impl:"dterm_sem I (Const 0) (repv (ab, bb) vid2 t) \<le> dterm_sem I (trm.Var vid2) (repv (ab, bb) vid2 t) \<longrightarrow> (\<forall>ra. repv (repv (ab, bb) vid2 t) vid3 ra \<in> {v. dterm_sem I (Const 0) v \<le> dterm_sem I (trm.Var vid3) v} \<inter> {v. dterm_sem I (trm.Var vid3) v \<le> dterm_sem I (trm.Var vid2) v} \<longrightarrow> Predicates I vid2 (\<chi> i. dterm_sem I (singleton (Plus (trm.Var vid1) (Times (f0 fid1) (trm.Var vid3))) i) (repv (repv (ab, bb) vid2 t) vid3 ra))) \<longrightarrow> (\<forall>\<omega>. \<omega> = repv (repv (ab, bb) vid2 t) vid1 (dterm_sem I (Plus (trm.Var vid1) (Times (f0 fid1) (trm.Var vid2))) (repv (ab, bb) vid2 t)) \<longrightarrow> \<omega> \<in> fml_sem I (p1 vid3 vid1))" using all by auto interpret ll:ll_on_open_it UNIV "(\<lambda>_. ODE_sem I (OSing vid1 (f0 fid1)))" "UNIV" 0 apply(standard) apply(auto) unfolding local_lipschitz_def f0_def empty_def sterm_sem.simps using gt_ex lipschitz_on_constant by blast have eq_UNIV:"ll.existence_ivl 0 (sol 0) = UNIV" apply(rule ll.existence_ivl_eq_domain) apply(auto) subgoal for tm tM t apply(unfold f0_def empty_def sterm_sem.simps) by(metis add.right_neutral mult_zero_left order_refl) done \<comment> \<open>Combine with \<open>flow_usolves_ode\<close> and \<open>equals_flowI\<close> to get uniqueness of solution\<close> let ?f = "(\<lambda>_. ODE_sem I (OSing vid1 (f0 fid1)))" have sol_UNIV: "\<And>t x. (ll.flow 0 x usolves_ode ?f from 0) (ll.existence_ivl 0 x) UNIV" using ll.flow_usolves_ode by auto from sol have sol': "(sol solves_ode (\<lambda>_. ODE_sem I (OSing vid1 (f0 fid1)))) {0..t} UNIV" apply (rule solves_ode_supset_range) by auto from sol' have sol'':"\<And>s. s \<ge> 0 \<Longrightarrow> s \<le> t \<Longrightarrow> (sol solves_ode (\<lambda>_. ODE_sem I (OSing vid1 (f0 fid1)))) {0..s} UNIV" by (simp add: solves_ode_subset) have sol0_eq:"sol 0 = ll.flow 0 (sol 0) 0" using ll.general.flow_initial_time_if by auto have isFlow:"\<And>s. s \<ge> 0 \<Longrightarrow> s \<le> t \<Longrightarrow> sol s = ll.flow 0 (sol 0) s" apply(rule ll.equals_flowI) apply(auto) subgoal using eq_UNIV by auto subgoal using sol'' closed_segment_eq_real_ivl t by (auto simp add: solves_ode_singleton) subgoal using eq_UNIV sol sol0_eq by auto done let ?c = "Functions I fid1 (\<chi> _. 0)" let ?sol = "(\<lambda>t. \<chi> i. if i = vid1 then (ab $ i) + ?c * t else (ab $ i))" have vec_simp:"(\<lambda>a b. \<chi> i. if i = vid1 then sterm_sem I ($f fid1 (\<lambda>i. Const 0)) b else 0) = (\<lambda>a b. \<chi> i. if i = vid1 then Functions I fid1 (\<chi> i. 0) else 0)" by (auto simp add: vec_eq_iff cong: if_cong) have exp_sol:"(?sol solves_ode (\<lambda>_. ODE_sem I (OSing vid1 (f0 fid1)))) {0..t} UNIV" apply(auto simp add: f0_def empty_def vec_simp) apply(rule solves_odeI) apply(auto simp only: has_vderiv_on_def has_vector_derivative_def box_sem) apply (rule has_derivative_vec[THEN has_derivative_eq_rhs]) defer apply (rule ext) apply (subst scaleR_vec_def) apply (rule refl) apply (auto intro!: derivative_eq_intros) done from exp_sol have exp_sol':"\<And>s. s \<ge> 0 \<Longrightarrow> s \<le> t \<Longrightarrow> (?sol solves_ode (\<lambda>_. ODE_sem I (OSing vid1 (f0 fid1)))) {0..s} UNIV" by (simp add: solves_ode_subset) have exp_sol0_eq:"?sol 0 = ll.flow 0 (?sol 0) 0" using ll.general.flow_initial_time_if by auto have more_eq:"(\<chi> i. if i = vid1 then ab $ i + Functions I fid1 (\<chi> _. 0) * 0 else ab $ i) = sol 0" using sol0 apply auto apply(rule vec_extensionality) by(auto) have exp_isFlow:"\<And>s. s \<ge> 0 \<Longrightarrow> s \<le> t \<Longrightarrow> ?sol s = ll.flow 0 (sol 0) s" apply(rule ll.equals_flowI) apply(auto) subgoal using eq_UNIV by auto defer subgoal for s using eq_UNIV apply auto subgoal using exp_sol exp_sol0_eq more_eq apply(auto) done done using exp_sol' closed_segment_eq_real_ivl t apply(auto) by (simp add: solves_ode_singleton) have sol_eq_exp:"\<And>s. s \<ge> 0 \<Longrightarrow> s \<le> t \<Longrightarrow> ?sol s = sol s" unfolding exp_isFlow isFlow by auto then have sol_eq_exp_t:"?sol t = sol t" using t by auto then have sol_eq_exp_t':"sol t $ vid1 = ?sol t $ vid1" by auto then have useful:"?sol t $ vid1 = ab $ vid1 + Functions I fid1 (\<chi> i. 0) * t" by auto from sol_eq_exp_t' useful have useful':"sol t $ vid1 = ab $ vid1 + Functions I fid1 (\<chi> i. 0) * t" by auto have sol_int:"((ll.flow 0 (sol 0)) usolves_ode ?f from 0) {0..t} {x. mk_v I (OSing vid1 (f0 fid1)) (ab, bb) x \<in> fml_sem I (p1 vid2 vid1)}" apply (rule usolves_ode_subset_range[of "(ll.flow 0 (sol 0))" "?f" "0" "{0..t}" "UNIV" "{x. mk_v I (OSing vid1 (f0 fid1)) (ab, bb) x \<in> fml_sem I (p1 vid2 vid1)}"]) subgoal using eq_UNIV sol_UNIV[of "(sol 0)"] apply (auto) apply (rule usolves_ode_subset) using t by(auto) apply(auto) using sol apply(auto dest!: solves_ode_domainD) subgoal for xa using isFlow[of xa] by(auto) done have thing:"\<And>s. 0 \<le> s \<Longrightarrow> s \<le> t \<Longrightarrow> fst (mk_v I (OSing vid1 ($f fid1 (\<lambda>i. Const 0))) (ab, bb) (?sol s)) $ vid1 = ab $ vid1 + Functions I fid1 (\<chi> i. 0) * s" subgoal for s using mk_v_agree[of I "(OSing vid1 ($f fid1 (\<lambda>i. Const 0)))" "(ab, bb)" "(?sol s)"] apply auto unfolding Vagree_def by auto done have thing':"\<And>s. 0 \<le> s \<Longrightarrow> s \<le> t \<Longrightarrow> fst (mk_v I (OSing vid1 ($f fid1 (\<lambda>i. Const 0))) (ab, bb) (sol s)) $ vid1 = ab $ vid1 + Functions I fid1 (\<chi> i. 0) * s" subgoal for s using thing[of s] sol_eq_exp[of s] by auto done have another_eq:"\<And>i s. 0 \<le> s \<Longrightarrow> s \<le> t \<Longrightarrow> dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OSing vid1 (f0 fid1)) (ab, bb) (sol s)) = dterm_sem I (if i = vid1 then Plus (trm.Var vid1) (Times (f0 fid1) (trm.Var vid3)) else Const 0) (\<chi> y. if vid3 = y then s else fst (\<chi> y. if vid2 = y then s else fst (ab, bb) $ y, bb) $ y, bb)" using mk_v_agree[of "I" "(OSing vid1 (f0 fid1))" "(ab, bb)" "(sol s)"] vne12 vne23 vne13 apply(auto simp add: f0_def p1_def empty_def) unfolding Vagree_def apply(simp add: f0_def empty_def) subgoal for s using thing' by auto done have allRa':"(\<forall>ra. repv (repv (ab, bb) vid2 t) vid3 ra \<in> {v. dterm_sem I (Const 0) v \<le> dterm_sem I (trm.Var vid3) v} \<inter> {v. dterm_sem I (trm.Var vid3) v \<le> dterm_sem I (trm.Var vid2) v} \<longrightarrow> Predicates I vid2 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OSing vid1 (f0 fid1)) (ab, bb) (sol ra))))" apply(rule allI) subgoal for ra using mk_v_agree[of "I" "(OSing vid1 (f0 fid1))" "(ab, bb)" "(sol ra)"] vne23 constraint[of ra] apply(auto simp add: Vagree_def p1_def) done done have anotherFact:"\<And>ra. 0 \<le> ra \<Longrightarrow> ra \<le> t \<Longrightarrow> (\<chi> i. if i = vid1 then ab $ i + Functions I fid1 (\<chi> _. 0) * ra else ab $ i) $ vid1 = ab $ vid1 + dterm_sem I (f0 fid1) (\<chi> y. if vid3 = y then ra else fst (\<chi> y. if vid2 = y then t else fst (ab, bb) $ y, bb) $ y, bb) * ra " subgoal for ra apply simp apply(rule disjI2) by (auto simp add: f0_def empty_def) done have thing':"\<And>ra i. 0 \<le> ra \<Longrightarrow> ra \<le> t \<Longrightarrow> dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OSing vid1 ($f fid1 (\<lambda>i. Const 0))) (ab, bb) (sol ra)) = dterm_sem I (if i = vid1 then Plus (trm.Var vid1) (Times (f0 fid1) (trm.Var vid3)) else Const 0) (\<chi> y. if vid3 = y then ra else fst (\<chi> y. if vid2 = y then t else fst (ab, bb) $ y, bb) $ y, bb) " subgoal for ra i using vne12 vne13 mk_v_agree[of I "OSing vid1 ($f fid1 (\<lambda>i. Const 0))" "(ab,bb)" "(sol ra)"] apply (auto) unfolding Vagree_def apply(safe) apply(erule allE[where x="vid1"])+ using sol_eq_exp[of ra] anotherFact[of ra] by auto done have allRa:"(\<forall>ra. repv (repv (ab, bb) vid2 t) vid3 ra \<in> {v. dterm_sem I (Const 0) v \<le> dterm_sem I (trm.Var vid3) v} \<inter> {v. dterm_sem I (trm.Var vid3) v \<le> dterm_sem I (trm.Var vid2) v} \<longrightarrow> Predicates I vid2 (\<chi> i. dterm_sem I (singleton (Plus (trm.Var vid1) (Times (f0 fid1) (trm.Var vid3))) i) (repv (repv (ab, bb) vid2 t) vid3 ra)))" apply(rule allI) subgoal for ra using mk_v_agree[of "I" "(OSing vid1 (f0 fid1))" "(ab, bb)" "(sol ra)"] vne23 constraint[of ra] apply(auto simp add: Vagree_def p1_def) using sol_eq_exp[of ra] apply (auto simp add: f0_def empty_def Vagree_def vec_eq_iff) using thing' by auto done have fml3:"\<And>ra. 0 \<le> ra \<Longrightarrow> ra \<le> t \<Longrightarrow> (\<forall>\<omega>. \<omega> = repv (repv (ab, bb) vid2 t) vid1 (dterm_sem I (Plus (trm.Var vid1) (Times (f0 fid1) (trm.Var vid2))) (repv (ab, bb) vid2 t)) \<longrightarrow> \<omega> \<in> fml_sem I (p1 vid3 vid1))" using impl allRa by auto have someEq:"(\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (\<chi> y. if vid1 = y then (if vid2 = vid1 then t else fst (ab, bb) $ vid1) + Functions I fid1 (\<chi> i. 0) * t else fst (\<chi> y. if vid2 = y then t else fst (ab, bb) $ y, bb) $ y, bb)) = (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OSing vid1 ($f fid1 (\<lambda>i. Const 0))) (ab, bb) (sol t)))" apply(rule vec_extensionality) using vne12 sol_eq_exp t thing by auto show "mk_v I (OSing vid1 (f0 fid1)) (ab, bb) (sol t) \<in> fml_sem I (p1 vid3 vid1)" using mk_v_agree[of I "OSing vid1 (f0 fid1)" "(ab, bb)" "sol t"] fml3[of t] unfolding f0_def p1_def empty_def Vagree_def using someEq by(auto simp add: sol_eq_exp_t' t vec_extensionality vne12) qed qed lemma MVT0_within: fixes f ::"real \<Rightarrow> real" and f'::"real \<Rightarrow> real \<Rightarrow> real" and s t :: real assumes f':"\<And>x. x \<in> {0..t} \<Longrightarrow> (f has_derivative (f' x)) (at x within {0..t})" assumes geq':"\<And>x. x \<in> {0..t} \<Longrightarrow> f' x s \<ge> 0" assumes int_s:"s > 0 \<and> s \<le> t" assumes t: "0 < t" shows "f s \<ge> f 0" proof - have "f 0 + 0 \<le> f s" apply (rule Lib.MVT_ivl'[OF f', of 0 s 0]) subgoal for x by assumption subgoal for x using geq' by auto using t int_s t apply auto subgoal for x by (metis int_s mult.commute mult.right_neutral order.trans real_mult_le_cancel_iff2) done then show "?thesis" by auto qed lemma MVT': fixes f g ::"real \<Rightarrow> real" fixes f' g'::"real \<Rightarrow> real \<Rightarrow> real" fixes s t ::real assumes f':"\<And>s. s \<in> {0..t} \<Longrightarrow> (f has_derivative (f' s)) (at s within {0..t})" assumes g':"\<And>s. s \<in> {0..t} \<Longrightarrow> (g has_derivative (g' s)) (at s within {0..t})" assumes geq':"\<And>x. x \<in> {0..t} \<Longrightarrow> f' x s \<ge> g' x s" assumes geq0:"f 0 \<ge> g 0" assumes int_s:"s > 0 \<and> s \<le> t" assumes t:"t > 0" shows "f s \<ge> g s" proof - let ?h = "(\<lambda>x. f x - g x)" let ?h' = "(\<lambda>s x. f' s x - g' s x)" have "?h s \<ge> ?h 0" apply(rule MVT0_within[of t ?h "?h'" s]) subgoal for s using f'[of s] g'[of s] by auto subgoal for sa using geq'[of sa] by auto subgoal using int_s by auto subgoal using t by auto done then show "?thesis" using geq0 by auto qed lemma MVT'_gr: fixes f g ::"real \<Rightarrow> real" fixes f' g'::"real \<Rightarrow> real \<Rightarrow> real" fixes s t ::real assumes f':"\<And>s. s \<in> {0..t} \<Longrightarrow> (f has_derivative (f' s)) (at s within {0..t})" assumes g':"\<And>s. s \<in> {0..t} \<Longrightarrow> (g has_derivative (g' s)) (at s within {0..t})" assumes geq':"\<And>x. x \<in> {0..t} \<Longrightarrow> f' x s \<ge> g' x s" assumes geq0:"f 0 > g 0" assumes int_s:"s > 0 \<and> s \<le> t" assumes t:"t > 0" shows "f s > g s" proof - let ?h = "(\<lambda>x. f x - g x)" let ?h' = "(\<lambda>s x. f' s x - g' s x)" have "?h s \<ge> ?h 0" apply(rule MVT0_within[of t ?h "?h'" s]) subgoal for s using f'[of s] g'[of s] by auto subgoal for sa using geq'[of sa] by auto subgoal using int_s by auto subgoal using t by auto done then show "?thesis" using geq0 by auto qed lemma frech_linear: fixes x \<theta> \<nu> \<nu>' I assumes good_interp:"is_interp I" assumes free:"dfree \<theta>" shows "x * frechet I \<theta> \<nu> \<nu>' = frechet I \<theta> \<nu> (x \<^sub>R \<nu>')" using frechet_linear[OF good_interp free] by (simp add: linear_simps) lemma rift_in_space_time: fixes sol I ODE \<psi> \<theta> t s b assumes good_interp:"is_interp I" assumes free:"dfree \<theta>" assumes osafe:"osafe ODE" assumes sol:"(sol solves_ode (\<lambda>_ \<nu>'. ODE_sem I ODE \<nu>')) {0..t} {x. mk_v I ODE (sol 0, b) x \<in> fml_sem I \<psi>}" assumes FVT:"FVT \<theta> \<subseteq> semBV I ODE" assumes ivl:"s \<in> {0..t}" shows "((\<lambda>t. sterm_sem I \<theta> (fst (mk_v I ODE (sol 0, b) (sol t)))) \<comment> \<open>This is Frechet derivative, so equivalent to:\<close> \<comment> \<open>\<open>has_real_derivative frechet I \<theta> (fst((mk_v I ODE (sol 0, b) (sol s)))) (snd (mk_v I ODE (sol 0, b) (sol s))))) (at s within {0..t})\<close>\<close> has_derivative (\<lambda>t'. t' frechet I \<theta> (fst((mk_v I ODE (sol 0, b) (sol s)))) (snd (mk_v I ODE (sol 0, b) (sol s))))) (at s within {0..t})" proof - let ?\<phi> = "(\<lambda>t. (mk_v I ODE (sol 0, b) (sol t)))" let ?\<phi>s = "(\<lambda>t. fst (?\<phi> t))" have sol_deriv:"\<And>s. s \<in> {0..t} \<Longrightarrow> (sol has_derivative (\<lambda>xa. xa \<^sub>R ODE_sem I ODE (sol s))) (at s within {0..t})" using sol apply simp apply (drule solves_odeD(1)) unfolding has_vderiv_on_def has_vector_derivative_def by auto have sol_dom:"\<And>s. s\<in> {0..t} \<Longrightarrow> ?\<phi> s \<in> fml_sem I \<psi>" using sol apply simp apply (drule solves_odeD(2)) by auto let ?h = "(\<lambda>t. sterm_sem I \<theta> (?\<phi>s t))" let ?g = "(\<lambda>\<nu>. sterm_sem I \<theta> \<nu>)" let ?f = "?\<phi>s" let ?f' = "(\<lambda>t'. t' \<^sub>R (\<chi> i. if i \<in> ODE_vars I ODE then ODE_sem I ODE (sol s) $ i else 0))" let ?g' = "(frechet I \<theta> (?\<phi>s s))" have heq:"?h = ?g \<circ> ?f" by (auto) have fact1:"\<And>i. i \<in> ODE_vars I ODE \<Longrightarrow> (\<lambda>t. ?\<phi>s(t) $ i) = (\<lambda>t. sol t $ i)" subgoal for i apply(rule ext) subgoal for t using mk_v_agree[of I ODE "(sol 0, b)" "sol t"] unfolding Vagree_def by auto done done have fact2:"\<And>i. i \<in> ODE_vars I ODE \<Longrightarrow> (\<lambda>t. if i \<in> ODE_vars I ODE then ODE_sem I ODE (sol t) $ i else 0) = (\<lambda>t. ODE_sem I ODE (sol t) $ i)" subgoal for i apply(rule ext) subgoal for t using mk_v_agree[of I ODE "(sol 0, b)" "sol t"] unfolding Vagree_def by auto done done have fact3:"\<And>i. i \<in> (-ODE_vars I ODE) \<Longrightarrow> (\<lambda>t. ?\<phi>s(t) $ i) = (\<lambda>t. sol 0 $ i)" subgoal for i apply(rule ext) subgoal for t using mk_v_agree[of I ODE "(sol 0, b)" "sol t"] unfolding Vagree_def by auto done done have fact4:"\<And>i. i \<in> (-ODE_vars I ODE) \<Longrightarrow> (\<lambda>t. if i \<in> ODE_vars I ODE then ODE_sem I ODE (sol t) $ i else 0) = (\<lambda>t. 0)" subgoal for i apply(rule ext) subgoal for t using mk_v_agree[of I ODE "(sol 0, b)" "sol t"] unfolding Vagree_def by auto done done have some_eq:"(\<lambda>v'. \<chi> i. v' \<^sub>R ODE_sem I ODE (sol s) $ i) = (\<lambda>v'. v' \<^sub>R ODE_sem I ODE (sol s))" apply(rule ext) apply(rule vec_extensionality) by auto have some_sol:"(sol has_derivative (\<lambda>v'. v' \<^sub>R ODE_sem I ODE (sol s))) (at s within {0..t})" using sol ivl unfolding solves_ode_def has_vderiv_on_def has_vector_derivative_def by auto have some_eta:"(\<lambda>t. \<chi> i. sol t $ i) = sol" by (rule ext, rule vec_extensionality, auto) have ode_deriv:"\<And>i. i \<in> ODE_vars I ODE \<Longrightarrow> ((\<lambda>t. sol t $ i) has_derivative (\<lambda> v'. v' \<^sub>R ODE_sem I ODE (sol s) $ i)) (at s within {0..t})" subgoal for i apply(rule has_derivative_proj) using some_eq some_sol some_eta by auto done have eta:"(\<lambda>t. (\<chi> i. ?f t $ i)) = ?f" by(rule ext, rule vec_extensionality, auto) have eta_esque:"(\<lambda>t'. \<chi> i. t' * (if i \<in> ODE_vars I ODE then ODE_sem I ODE (sol s) $ i else 0)) = (\<lambda>t'. t' \<^sub>R (\<chi> i. if i \<in> ODE_vars I ODE then ODE_sem I ODE (sol s) $ i else 0))" apply(rule ext \| rule vec_extensionality)+ subgoal for t' i by auto done have "((\<lambda>t. (\<chi> i. ?f t $ i)) has_derivative (\<lambda>t'. (\<chi> i. ?f' t' $ i))) (at s within {0..t})" apply (rule has_derivative_vec) subgoal for i apply(cases "i \<in> ODE_vars I ODE") subgoal using fact1[of i] fact2[of i] ode_deriv[of i] by auto subgoal using fact3[of i] fact4[of i] by auto done done then have fderiv:"(?f has_derivative ?f') (at s within {0..t})" using eta eta_esque by auto have gderiv:"(?g has_derivative ?g') (at (?f s) within ?f ` {0..t})" using has_derivative_at_withinI using frechet_correctness free good_interp by blast have chain:"((?g \<circ> ?f) has_derivative (?g' \<circ> ?f')) (at s within {0..t})" using fderiv gderiv diff_chain_within by blast let ?co\<nu>1 = "(fst (mk_v I ODE (sol 0, b) (sol s)), ODE_sem I ODE (fst (mk_v I ODE (sol 0, b) (sol s))))" let ?co\<nu>2 = "(fst (mk_v I ODE (sol 0, b) (sol s)), snd (mk_v I ODE (sol 0, b) (sol s)))" have sub_cont:"\<And>a .a \<notin> ODE_vars I ODE \<Longrightarrow> Inl a \<in> FVT \<theta> \<Longrightarrow> False" using FVT by auto have sub_cont2:"\<And>a .a \<notin> ODE_vars I ODE \<Longrightarrow> Inr a \<in> FVT \<theta> \<Longrightarrow> False" using FVT by auto have "Vagree (mk_v I ODE (sol 0, b) (sol s)) (sol s, b) (Inl ` ODE_vars I ODE)" using mk_v_agree[of I ODE "(sol 0, b)" "sol s"] unfolding Vagree_def by auto let ?co'\<nu>1 = "(\<lambda>x. (fst (mk_v I ODE (sol 0, b) (sol s)), x \<^sub>R (\<chi> i. if i \<in> ODE_vars I ODE then ODE_sem I ODE (sol s) $ i else 0)))" let ?co'\<nu>2 = "(\<lambda>x. (fst (mk_v I ODE (sol 0, b) (sol s)), x \<^sub>R snd (mk_v I ODE (sol 0, b) (sol s))))" have co_agree_sem:"\<And>s. Vagree (?co'\<nu>1 s) (?co'\<nu>2 s) (semBV I ODE)" subgoal for sa using mk_v_agree[of I ODE "(sol 0, b)" "sol s"] unfolding Vagree_def by auto done have co_agree_help:"\<And>s. Vagree (?co'\<nu>1 s) (?co'\<nu>2 s) (FVT \<theta>)" using agree_sub[OF FVT co_agree_sem] by auto have co_agree':"\<And>s. Vagree (?co'\<nu>1 s) (?co'\<nu>2 s) (FVDiff \<theta>)" subgoal for s using mk_v_agree[of I ODE "(sol 0, b)" "sol s"] unfolding Vagree_def apply auto subgoal for i x apply(cases x) subgoal for a apply(cases "a \<in> ODE_vars I ODE") by (simp \| metis (no_types, lifting) FVT ODE_vars_lr Vagree_def mk_v_agree mk_xode.elims subsetD snd_conv)+ subgoal for a apply(cases "a \<in> ODE_vars I ODE") by (simp \| metis (no_types, lifting) FVT Vagree_def mk_v_agree mk_xode.elims subsetD snd_conv)+ done subgoal for i x apply(cases x) subgoal for a apply(cases "a \<in> ODE_vars I ODE") using FVT ODE_vars_lr Vagree_def mk_v_agree mk_xode.elims subsetD snd_conv by auto subgoal for a apply(cases "a \<in> ODE_vars I ODE") apply(erule allE[where x=i])+ using FVT ODE_vars_lr Vagree_def mk_v_agree mk_xode.elims subsetD snd_conv by auto done done done have heq'':"(?g' \<circ> ?f') = (\<lambda>t'. t' \<^sub>R frechet I \<theta> (?\<phi>s s) (snd (?\<phi> s)))" using mk_v_agree[of I ODE "(sol 0, b)" "sol s"] unfolding comp_def apply auto apply(rule ext \| rule vec_extensionality)+ subgoal for x using frech_linear[of I \<theta> x "(fst (mk_v I ODE (sol 0, b) (sol s)))" "(snd (mk_v I ODE (sol 0, b) (sol s)))", OF good_interp free] using coincidence_frechet[OF free, of "(?co'\<nu>1 x)" "(?co'\<nu>2 x)", OF co_agree'[of x], of I] by auto done have "((?g \<circ> ?f) has_derivative (?g' \<circ> ?f')) (at s within {0..t})" using chain by auto then have "((?g \<circ> ?f) has_derivative (\<lambda>t'. t' * frechet I \<theta> (?\<phi>s s) (snd (?\<phi> s)))) (at s within {0..t})" using heq'' by auto then have result:"((\<lambda>t. sterm_sem I \<theta> (?\<phi>s t)) has_derivative (\<lambda>t. t * frechet I \<theta> (?\<phi>s s) (snd (?\<phi> s)))) (at s within {0..t})" using heq by auto then show "?thesis" by auto qed lemma dterm_sterm_dfree: "dfree \<theta> \<Longrightarrow> (\<And>\<nu> \<nu>'. sterm_sem I \<theta> \<nu> = dterm_sem I \<theta> (\<nu>, \<nu>'))" by(induction rule: dfree.induct, auto) \<comment> \<open>\<open>g(x)\<ge> h(x) \<rightarrow> [x'=f(x), c & p(x)](g(x)' \<ge> h(x)') \<rightarrow> [x'=f(x), c]g(x) \<ge> h(x)\<close>\<close> lemma DG_valid:"valid DGaxiom" proof - have osafe:"osafe (OSing vid1 (f1 fid1 vid1))" by(auto simp add: osafe_Sing dfree_Fun dfree_Const f1_def expand_singleton) have fsafe:"fsafe (p1 vid1 vid1)" by(auto simp add: p1_def dfree_Const) have osafe2:"osafe (OProd (OSing vid1 (f1 fid1 vid1)) (OSing vid2 (Plus (Times (f1 fid2 vid1) (trm.Var vid2)) (f1 fid3 vid1))))" by(auto simp add: f1_def expand_singleton osafe.intros dfree.intros vne12) note sem = ode_alt_sem[OF osafe fsafe] note sem2 = ode_alt_sem[OF osafe2 fsafe] have p2safe:"fsafe (p1 vid2 vid1)" by(auto simp add: p1_def dfree_Const) show "valid DGaxiom" apply(auto simp del: prog_sem.simps(8) simp add: DGaxiom_def valid_def sem sem2) apply(rule exI[where x=0], auto simp add: f1_def p1_def expand_singleton) subgoal for I a b aa ba sol t proof - assume good_interp:"is_interp I" assume " \<forall>aa ba. (\<exists>sol t. (aa, ba) = mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (a, b) (sol t) \<and> 0 \<le> t \<and> (sol solves_ode (\<lambda>a b. \<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) b else 0)) {0..t} {x. Predicates I vid1 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (a, b) x))} \<and> VSagree (sol 0) a {uu. uu = vid1 \<or> Inl uu \<in> Inl ` {x. \<exists>xa. Inl x \<in> FVT (if xa = vid1 then trm.Var vid1 else Const 0)} \<or> (\<exists>x. Inl uu \<in> FVT (if x = vid1 then trm.Var vid1 else Const 0))}) \<longrightarrow> Predicates I vid2 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (aa, ba))" then have bigAll:" \<And>aa ba. (\<exists>sol t. (aa, ba) = mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (a, b) (sol t) \<and> 0 \<le> t \<and> (sol solves_ode (\<lambda>a b. \<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) b else 0)) {0..t} {x. Predicates I vid1 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (a, b) x))} \<and> VSagree (sol 0) a {uu. uu = vid1 \<or> (\<exists>x. Inl uu \<in> FVT (if x = vid1 then trm.Var vid1 else Const 0))}) \<longrightarrow> Predicates I vid2 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (aa, ba))" by (auto) assume aaba:"(aa, ba) = mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))) (\<chi> y. if vid2 = y then 0 else fst (a, b) $ y, b) (sol t)" assume t:"0 \<le> t" assume sol:" (sol solves_ode (\<lambda>a b. (\<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) b else 0) + (\<chi> i. if i = vid2 then sterm_sem I (Plus (Times (f1 fid2 vid1) (trm.Var vid2)) (f1 fid3 vid1)) b else 0))) {0..t} {x. Predicates I vid1 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))) (\<chi> y. if vid2 = y then 0 else fst (a, b) $ y, b) x))}" assume VSag:"VSagree (sol 0) (\<chi> y. if vid2 = y then 0 else fst (a, b) $ y) {x. x = vid2 \<or> x = vid1 \<or> x = vid2 \<or> x = vid1 \<or> Inl x \<in> Inl ` {x. x = vid2 \<or> x = vid1} \<or> x = vid1}" let ?sol = "(\<lambda>t. \<chi> i. if i = vid1 then sol t $ vid1 else 0)" let ?aaba' = "mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (a, b) (?sol t)" from bigAll[of "fst ?aaba'" "snd ?aaba'"] have bigEx:"(\<exists>sol t. ?aaba' = mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (a, b) (sol t) \<and> 0 \<le> t \<and> (sol solves_ode (\<lambda>a b. \<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) b else 0)) {0..t} {x. Predicates I vid1 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (a, b) x))} \<and> VSagree (sol 0) a {uu. uu = vid1 \<or> (\<exists>x. Inl uu \<in> FVT (if x = vid1 then trm.Var vid1 else Const 0))}) \<longrightarrow> Predicates I vid2 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (?aaba'))" by simp have pre1:"?aaba' = mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (a, b) (?sol t)" by (rule refl) have agreeL:"\<And>s. fst (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))) (\<chi> y. if vid2 = y then 0 else fst (a, b) $ y, b) (sol s)) $ vid1 = sol s $ vid1" subgoal for s using mk_v_agree[of I "(OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))))" "(\<chi> y. if vid2 = y then 0 else fst (a, b) $ y, b)" "(sol s)"] unfolding Vagree_def by auto done have agreeR:"\<And>s. fst (mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (a, b) (\<chi> i. if i = vid1 then sol s $ vid1 else 0)) $ vid1 = sol s $ vid1" subgoal for s using mk_v_agree[of "I" "(OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))" "(a, b)" "(\<chi> i. if i = vid1 then sol s $ vid1 else 0)"] unfolding Vagree_def by auto done have FV:"(FVF (p1 vid1 vid1)) = {Inl vid1}" unfolding p1_def expand_singleton apply auto subgoal for x xa apply(cases "xa = vid1") by auto done have agree:"\<And>s. Vagree (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))) (\<chi> y. if vid2 = y then 0 else fst (a, b) $ y, b) (sol s)) (mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (a, b) (\<chi> i. if i = vid1 then sol s $ vid1 else 0)) (FVF (p1 vid1 vid1))" using agreeR agreeL unfolding Vagree_def FV by auto note con_sem_eq = coincidence_formula[OF fsafe Iagree_refl agree] have constraint:"\<And>s. 0 \<le> s \<and> s \<le> t \<Longrightarrow> Predicates I vid1 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (a, b) (\<chi> i. if i = vid1 then sol s $ vid1 else 0)))" using sol apply simp apply(drule solves_odeD(2)) apply auto[1] subgoal for s using con_sem_eq by (auto simp add: p1_def expand_singleton) done have eta:"sol = (\<lambda>t. \<chi> i. sol t $ i)" by (rule ext, rule vec_extensionality, simp) have yet_another_eq:"\<And>x. (\<lambda>xa. xa \<^sub>R ((\<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) (sol x) else 0) + (\<chi> i. if i = vid2 then sterm_sem I (Plus (Times (f1 fid2 vid1) (trm.Var vid2)) (f1 fid3 vid1)) (sol x) else 0))) = (\<lambda>xa. (\<chi> i. (xa \<^sub>R ((\<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) (sol x) else 0) + (\<chi> i. if i = vid2 then sterm_sem I (Plus (Times (f1 fid2 vid1) (trm.Var vid2)) (f1 fid3 vid1)) (sol x) else 0))) $ i))" subgoal for x by (rule ext, rule vec_extensionality, simp) done have sol_deriv:"\<And>x. x \<in>{0..t} \<Longrightarrow> (sol has_derivative (\<lambda>xa. xa \<^sub>R ((\<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) (sol x) else 0) + (\<chi> i. if i = vid2 then sterm_sem I (Plus (Times (f1 fid2 vid1) (trm.Var vid2)) (f1 fid3 vid1)) (sol x) else 0)))) (at x within {0..t})" using sol apply simp apply(drule solves_odeD(1)) unfolding has_vderiv_on_def has_vector_derivative_def by auto then have sol_deriv:"\<And>x. x \<in> {0..t} \<Longrightarrow> ((\<lambda>t. \<chi> i. sol t $ i) has_derivative (\<lambda>xa. (\<chi> i. (xa \<^sub>R ((\<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) (sol x) else 0) + (\<chi> i. if i = vid2 then sterm_sem I (Plus (Times (f1 fid2 vid1) (trm.Var vid2)) (f1 fid3 vid1)) (sol x) else 0))) $ i))) (at x within {0..t})" using yet_another_eq eta by auto have sol_deriv1: "\<And>x. x \<in> {0..t} \<Longrightarrow> ((\<lambda>t. sol t $ vid1) has_derivative (\<lambda>xa. (xa \<^sub>R ((\<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) (sol x) else 0) + (\<chi> i. if i = vid2 then sterm_sem I (Plus (Times (f1 fid2 vid1) (trm.Var vid2)) (f1 fid3 vid1)) (sol x) else 0)) $ vid1))) (at x within {0..t})" subgoal for s ( I heard higher-order unification is hard.) apply(rule has_derivative_proj[of "(\<lambda> i t. sol t $ i)" "(\<lambda>j xa. (xa \<^sub>R ((\<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) (sol s) else 0) + (\<chi> i. if i = vid2 then sterm_sem I (Plus (Times (f1 fid2 vid1) (trm.Var vid2)) (f1 fid3 vid1)) (sol s) else 0)) $ j))" "at s within {0..t}""vid1"]) using sol_deriv[of s] by auto done have hmm:"\<And>s. (\<chi> i. sterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (sol s)) = (\<chi> i. sterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (\<chi> i. if i = vid1 then sol s $ vid1 else 0))" by(rule vec_extensionality, auto) have aha:"\<And>s. (\<lambda>xa. xa * sterm_sem I (f1 fid1 vid1) (sol s)) = (\<lambda>xa. xa * sterm_sem I (f1 fid1 vid1) (\<chi> i. if i = vid1 then sol s $ vid1 else 0))" subgoal for s apply(rule ext) subgoal for xa using hmm by (auto simp add: f1_def) done done let ?sol' = "(\<lambda>s. (\<lambda>xa. \<chi> i. if i = vid1 then xa * sterm_sem I (f1 fid1 vid1) (\<chi> i. if i = vid1 then sol s $ vid1 else 0) else 0))" let ?project_me_plz = "(\<lambda>t. (\<chi> i. if i = vid1 then ?sol t $ vid1 else 0))" have sol_deriv_eq:"\<And>s. s \<in>{0..t} \<Longrightarrow> ((\<lambda>t. (\<chi> i. if i = vid1 then ?sol t $ vid1 else 0)) has_derivative ?sol' s) (at s within {0..t})" subgoal for s apply(rule has_derivative_vec) subgoal for i apply (cases "i = vid1", cases "i = vid2", auto) using vne12 apply simp using sol_deriv1[of s] using aha by auto done done have yup:"(\<lambda>t. (\<chi> i. if i = vid1 then ?sol t $ vid1 else 0) $ vid1) = (\<lambda>t. sol t $ vid1)" by(rule ext, auto) have maybe:"\<And>s. (\<lambda>xa. xa * sterm_sem I (f1 fid1 vid1) (\<chi> i. if i = vid1 then sol s $ vid1 else 0)) = (\<lambda>xa. (\<chi> i. if i = vid1 then xa * sterm_sem I (f1 fid1 vid1) (\<chi> i. if i = vid1 then sol s $ vid1 else 0) else 0) $ vid1) " by(rule ext, auto) have almost:"(\<lambda>x. if vid1 = vid1 then (\<chi> i. if i = vid1 then sol x $ vid1 else 0) $ vid1 else 0) = (\<lambda>x. (\<chi> i. if i = vid1 then sol x $ vid1 else 0) $ vid1)" by(rule ext, auto) have almost':"\<And>s. (\<lambda>h. if vid1 = vid1 then h * sterm_sem I (f1 fid1 vid1) (\<chi> i. if i = vid1 then sol s $ vid1 else 0) else 0) = (\<lambda>h. h * sterm_sem I (f1 fid1 vid1) (\<chi> i. if i = vid1 then sol s $ vid1 else 0))" by(rule ext, auto) have deriv':" \<And>x. x \<in> {0..t} \<Longrightarrow> ((\<lambda>t. \<chi> i. if i = vid1 then sol t $ vid1 else 0) has_derivative (\<lambda>xa. (\<chi> i. xa \<^sub>R (if i = vid1 then sterm_sem I (f1 fid1 vid1) (\<chi> i. if i = vid1 then sol x $ vid1 else 0) else 0)))) (at x within {0..t})" subgoal for s apply(rule has_derivative_vec) subgoal for i apply(cases "i = vid1") prefer 2 subgoal by auto apply auto using has_derivative_proj[OF sol_deriv_eq[of s], of vid1] using yup maybe[of s] almost almost'[of s] by fastforce done done have derEq:"\<And>s. (\<lambda>xa. (\<chi> i. xa \<^sub>R (if i = vid1 then sterm_sem I (f1 fid1 vid1) (\<chi> i. if i = vid1 then sol s $ vid1 else 0) else 0))) = (\<lambda>xa. xa \<^sub>R (\<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) (\<chi> i. if i = vid1 then sol s $ vid1 else 0) else 0))" subgoal for s apply (rule ext, rule vec_extensionality) by auto done have "\<And>x. x \<in> {0..t} \<Longrightarrow> ((\<lambda>t. \<chi> i. if i = vid1 then sol t $ vid1 else 0) has_derivative (\<lambda>xa. xa \<^sub>R (\<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) (\<chi> i. if i = vid1 then sol x $ vid1 else 0) else 0))) (at x within {0..t})" subgoal for s using deriv'[of s] derEq[of s] by auto done then have deriv:"((\<lambda>t. \<chi> i. if i = vid1 then sol t $ vid1 else 0) has_vderiv_on (\<lambda>t. \<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) (\<chi> i. if i = vid1 then sol t $ vid1 else 0) else 0)) {0..t}" unfolding has_vderiv_on_def has_vector_derivative_def by auto have pre2:"(?sol solves_ode (\<lambda>a b. \<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) b else 0)) {0..t} {x. Predicates I vid1 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (a, b) x))}" apply(rule solves_odeI) subgoal by (rule deriv) subgoal for s using constraint by auto done have pre3:"VSagree (?sol 0) a {u. u = vid1 \<or> (\<exists>x. Inl u \<in> FVT (if x = vid1 then trm.Var vid1 else Const 0))}" using vne12 VSag unfolding VSagree_def by simp have bigPre:"(\<exists>sol t. ?aaba' = mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then Var vid1 else Const 0))) (a, b) (sol t) \<and> 0 \<le> t \<and> (sol solves_ode (\<lambda>a b. \<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) b else 0)) {0..t} {x. Predicates I vid1 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then Var vid1 else Const 0))) (a, b) x))} \<and> VSagree (sol 0) a {u. u = vid1 \<or> (\<exists>x. Inl u \<in> FVT (if x = vid1 then Var vid1 else Const 0))})" apply(rule exI[where x="?sol"]) apply(rule exI[where x=t]) apply(rule conjI) apply(rule pre1) apply(rule conjI) apply(rule t) apply(rule conjI) apply(rule pre2) by(rule pre3) have pred2:"Predicates I vid2 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) ?aaba')" using bigEx bigPre by auto then have pred2':"?aaba' \<in> fml_sem I (p1 vid2 vid1)" unfolding p1_def expand_singleton by auto let ?res_state = "(mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))) (\<chi> y. if vid2 = y then 0 else fst (a, b) $ y, b) (sol t))" have aabaX:"(fst ?aaba') $ vid1 = sol t $ vid1" using aaba mk_v_agree[of "I" "(OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))" "(a, b)" "(?sol t)"] proof - assume " Vagree (mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (a, b) (\<chi> i. if i = vid1 then sol t $ vid1 else 0)) (a, b) (- semBV I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))) \<and> Vagree (mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (a, b) (\<chi> i. if i = vid1 then sol t $ vid1 else 0)) (mk_xode I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (\<chi> i. if i = vid1 then sol t $ vid1 else 0)) (semBV I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))" then have ag:" Vagree (mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (a, b) (?sol t)) (mk_xode I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (?sol t)) (semBV I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))" by auto have sembv:"(semBV I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))) = {Inl vid1, Inr vid1}" by auto have sub:"{Inl vid1} \<subseteq> {Inl vid1, Inr vid1}" by auto have ag':"Vagree (mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (a, b) (?sol t)) (mk_xode I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (?sol t)) {Inl vid1}" using ag agree_sub[OF sub] sembv by auto then have eq1:"fst (mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (a, b) (?sol t)) $ vid1 = fst (mk_xode I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (?sol t)) $ vid1" unfolding Vagree_def by auto moreover have "... = sol t $ vid1" by auto ultimately show ?thesis by auto qed have res_stateX:"(fst ?res_state) $ vid1 = sol t $ vid1" using mk_v_agree[of I "(OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))))" "(\<chi> y. if vid2 = y then 0 else fst (a, b) $ y, b)" "(sol t)"] proof - assume "Vagree (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))) (\<chi> y. if vid2 = y then 0 else fst (a, b) $ y, b) (sol t)) (\<chi> y. if vid2 = y then 0 else fst (a, b) $ y, b) (- semBV I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))))) \<and> Vagree (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))) (\<chi> y. if vid2 = y then 0 else fst (a, b) $ y, b) (sol t)) (mk_xode I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))) (sol t)) (semBV I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))))" then have ag:" Vagree (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))) (\<chi> y. if vid2 = y then 0 else fst (a, b) $ y, b) (sol t)) (mk_xode I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))) (sol t)) (semBV I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))))" by auto have sembv:"(semBV I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))))) = {Inl vid1, Inr vid1, Inl vid2, Inr vid2}" by auto have sub:"{Inl vid1} \<subseteq> {Inl vid1, Inr vid1, Inl vid2, Inr vid2}" by auto have ag':"Vagree (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))) (\<chi> y. if vid2 = y then 0 else fst (a, b) $ y, b) (sol t)) (mk_xode I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))) (sol t)) {Inl vid1}" using ag sembv agree_sub[OF sub] by auto then have "fst ?res_state $ vid1 = fst ((mk_xode I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))) (sol t))) $ vid1" unfolding Vagree_def by blast moreover have "... = sol t $ vid1" by auto ultimately show "?thesis" by linarith qed have agree:"Vagree ?aaba' (?res_state) (FVF (p1 vid2 vid1))" unfolding p1_def Vagree_def using aabaX res_stateX by auto have fml_sem_eq:"(?res_state \<in> fml_sem I (p1 vid2 vid1)) = (?aaba' \<in> fml_sem I (p1 vid2 vid1))" using coincidence_formula[OF p2safe Iagree_refl agree, of I] by auto then show "Predicates I vid2 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))) (\<chi> y. if vid2 = y then 0 else fst (a, b) $ y, b) (sol t)))" using pred2 unfolding p1_def expand_singleton by auto qed subgoal for I a b r aa ba sol t proof - assume good_interp:"is_interp I" assume bigAll:" \<forall>aa ba. (\<exists>sol t. (aa, ba) = mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))) (\<chi> y. if vid2 = y then r else fst (a, b) $ y, b) (sol t) \<and> 0 \<le> t \<and> (sol solves_ode (\<lambda>a b. (\<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) b else 0) + (\<chi> i. if i = vid2 then sterm_sem I (Plus (Times (f1 fid2 vid1) (trm.Var vid2)) (f1 fid3 vid1)) b else 0))) {0..t} {x. Predicates I vid1 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))) (\<chi> y. if vid2 = y then r else fst (a, b) $ y, b) x))} \<and> VSagree (sol 0) (\<chi> y. if vid2 = y then r else fst (a, b) $ y) {uu. uu = vid2 \<or> uu = vid1 \<or> uu = vid2 \<or> uu = vid1 \<or> Inl uu \<in> Inl ` ({x. \<exists>xa. Inl x \<in> FVT (if xa = vid1 then trm.Var vid1 else Const 0)} \<union> {x. x = vid2 \<or> (\<exists>xa. Inl x \<in> FVT (if xa = vid1 then trm.Var vid1 else Const 0))}) \<or> (\<exists>x. Inl uu \<in> FVT (if x = vid1 then trm.Var vid1 else Const 0))}) \<longrightarrow> Predicates I vid2 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (aa, ba))" assume aaba:"(aa, ba) = mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (a, b) (sol t)" assume t:"0 \<le> t" assume sol:"(sol solves_ode (\<lambda>a b. \<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) b else 0)) {0..t} {x. Predicates I vid1 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (a, b) x))}" assume VSA:"VSagree (sol 0) a {uu. uu = vid1 \<or> Inl uu \<in> Inl ` {x. \<exists>xa. Inl x \<in> FVT (if xa = vid1 then trm.Var vid1 else Const 0)} \<or> (\<exists>x. Inl uu \<in> FVT (if x = vid1 then trm.Var vid1 else Const 0))}" let ?xode = "(\<lambda>a b. \<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) b else 0)" let ?xconstraint = UNIV let ?ivl = "ll_on_open.existence_ivl {0 .. t} ?xode ?xconstraint 0 (sol 0)" have freef1:"dfree ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))" by(auto simp add: dfree_Fun dfree_Const) have simple_term_inverse':"\<And>\<theta>. dfree \<theta> \<Longrightarrow> raw_term (simple_term \<theta>) = \<theta>" using simple_term_inverse by auto have old_lipschitz:"local_lipschitz (UNIV::real set) UNIV (\<lambda>a b. \<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) b else 0)" apply(rule c1_implies_local_lipschitz[where f'="(\<lambda> (t,b). blinfun_vec(\<lambda> i. if i = vid1 then blin_frechet (good_interp I) (simple_term (Function fid1 (\<lambda> i. if i = vid1 then Var vid1 else Const 0))) b else Blinfun(\<lambda> _. 0)))"]) apply auto subgoal for x apply(rule has_derivative_vec) subgoal for i apply(auto simp add: bounded_linear_Blinfun_apply good_interp_inverse good_interp) apply(auto simp add: simple_term_inverse'[OF freef1]) apply(cases "i = vid1") apply(auto simp add: f1_def expand_singleton) proof - let ?h = "(\<lambda>b. Functions I fid1 (\<chi> i. sterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) b))" let ?h' = "(\<lambda>b'. FunctionFrechet I fid1 (\<chi> i. sterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) x) (\<chi> i. frechet I (if i = vid1 then trm.Var vid1 else Const 0) x b'))" let ?f = "(\<lambda> b. (\<chi> i. sterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) b))" let ?f' = "(\<lambda> b'. (\<chi> i. frechet I (if i = vid1 then trm.Var vid1 else Const 0) x b'))" let ?g = "Functions I fid1" let ?g'= "FunctionFrechet I fid1 (?f x)" have heq:"?h = ?g \<circ> ?f" by(rule ext, auto) have heq':"?h' = ?g' \<circ> ?f'" by(rule ext, auto) have fderiv:"(?f has_derivative ?f') (at x)" apply(rule has_derivative_vec) by (auto simp add: svar_deriv axis_def) have gderiv:"(?g has_derivative ?g') (at (?f x))" using good_interp unfolding is_interp_def by blast have gfderiv: "((?g \<circ> ?f) has_derivative(?g' \<circ> ?f')) (at x)" using fderiv gderiv diff_chain_at by blast have boring_eq:"(\<lambda>b. Functions I fid1 (\<chi> i. sterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) b)) = sterm_sem I ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))" by(rule ext, auto) have "(?h has_derivative ?h') (at x)" using gfderiv heq heq' by auto then show "(sterm_sem I ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) has_derivative (\<lambda>v'. (THE f'. \<forall>x. (Functions I fid1 has_derivative f' x) (at x)) (\<chi> i. sterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) x) (\<chi> i. frechet I (if i = vid1 then trm.Var vid1 else Const 0) x v'))) (at x)" using boring_eq by auto qed done proof - have the_thing:"continuous_on (UNIV::('sz Rvec set)) (\<lambda>b. blinfun_vec (\<lambda>i. if i = vid1 then blin_frechet (good_interp I) (simple_term ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) b else Blinfun (\<lambda>_. 0)))" apply(rule continuous_blinfun_vec') subgoal for i apply(cases "i = vid1") apply(auto) using frechet_continuous[OF good_interp freef1] by (auto simp add: continuous_on_const) done have another_cont:"continuous_on (UNIV) (\<lambda>x. blinfun_vec (\<lambda>i. if i = vid1 then blin_frechet (good_interp I) (simple_term ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (snd x) else Blinfun (\<lambda>_. 0)))" apply(rule continuous_on_compose2[of UNIV "(\<lambda>b. blinfun_vec (\<lambda>i. if i = vid1 then blin_frechet (good_interp I) (simple_term ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) b else Blinfun (\<lambda>_. 0)))"]) apply(rule the_thing) by (auto intro!: continuous_intros) have ext:"(\<lambda>x. case x of (t, b) \<Rightarrow> blinfun_vec (\<lambda>i. if i = vid1 then blin_frechet (good_interp I) (simple_term ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) b else Blinfun (\<lambda>_. 0))) =(\<lambda>x. blinfun_vec (\<lambda>i. if i = vid1 then blin_frechet (good_interp I) (simple_term ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (snd x) else Blinfun (\<lambda>_. 0))) " apply(rule ext, auto) by (metis snd_conv) then show "continuous_on (UNIV) (\<lambda>x. case x of (t, b) \<Rightarrow> blinfun_vec (\<lambda>i. if i = vid1 then blin_frechet (good_interp I) (simple_term ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) b else Blinfun (\<lambda>_. 0)))" using another_cont by (simp add: another_cont local.ext) qed have old_continuous:" \<And>x. x \<in> UNIV \<Longrightarrow> continuous_on UNIV (\<lambda>t. \<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) x else 0)" by(rule continuous_on_const) interpret ll_old: ll_on_open_it "UNIV" ?xode ?xconstraint 0 apply(standard) subgoal by auto prefer 3 subgoal by auto prefer 3 subgoal by auto apply(rule old_lipschitz) by (rule old_continuous) let ?ivl = "(ll_old.existence_ivl 0 (sol 0))" let ?flow = "ll_old.flow 0 (sol 0)" have tclosed:"{0..t} = {0--t}" using t real_Icc_closed_segment by auto have "(sol solves_ode (\<lambda>a b. \<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) b else 0)) {0..t} UNIV" apply(rule solves_ode_supset_range) apply(rule sol) by auto then have sol':"(sol solves_ode (\<lambda>a b. \<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) b else 0)) {0--t} UNIV" using tclosed by auto have sub:"{0--t} \<subseteq> ll_old.existence_ivl 0 (sol 0)" apply(rule ll_old.closed_segment_subset_existence_ivl) apply(rule ll_old.existence_ivl_maximal_segment) apply(rule sol') apply(rule refl) by auto have usol_old:"(?flow usolves_ode ?xode from 0) ?ivl UNIV" by(rule ll_old.flow_usolves_ode, auto) have sol_old:"(ll_old.flow 0 (sol 0) solves_ode ?xode) ?ivl UNIV" by(rule ll_old.flow_solves_ode, auto) have another_sub:"\<And>s. s \<in> {0..t} \<Longrightarrow> {s--0} \<subseteq> {0..t}" unfolding closed_segment_def apply auto by (metis diff_0_right diff_left_mono mult.commute mult_left_le order.trans) have sol_eq_flow:"\<And>s. s \<in> {0..t} \<Longrightarrow> sol s = ?flow s" using usol_old apply simp apply(drule usolves_odeD(4)) (* 7 subgoals) apply auto subgoal for s x proof - assume xs0:"x \<in> {s--0}" assume s0:"0 \<le> s" and st: "s \<le> t" have "{s--0} \<subseteq> {0..t}" using another_sub[of s] s0 st by auto then have "x \<in> {0..t}" using xs0 by auto then have "x \<in> {0--t}" using tclosed by auto then show "x \<in> ll_old.existence_ivl 0 (sol 0)" using sub by auto qed apply(rule solves_ode_subset) using sol' apply auto[1] subgoal for s proof - assume s0:"0 \<le> s" and st:"s \<le> t" show "{s--0} \<subseteq> {0--t}" using tclosed unfolding closed_segment using s0 st using another_sub intervalE by blast qed done have sol_deriv_orig:"\<And>s. s\<in>?ivl \<Longrightarrow> (?flow has_derivative (\<lambda>xa. xa \<^sub>R (\<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) (?flow s) else 0))) (at s within ?ivl)" using sol_old apply simp apply(drule solves_odeD(1)) by (auto simp add: has_vderiv_on_def has_vector_derivative_def) have sol_eta:"(\<lambda>t. \<chi> i. ?flow t $ i) = ?flow" by(rule ext, rule vec_extensionality, auto) have sol_deriv_eq1:"\<And>s i. (\<lambda>xa. xa \<^sub>R (\<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) (?flow s) else 0)) = (\<lambda>xa. \<chi> i. xa (if i = vid1 then sterm_sem I (f1 fid1 vid1) (?flow s) else 0))" by(rule ext, rule vec_extensionality, auto) have sol_deriv_proj:"\<And>s i. s\<in>?ivl \<Longrightarrow> ((\<lambda>t. ?flow t $ i) has_derivative (\<lambda>xa. (xa \<^sub>R (\<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) (?flow s) else 0)) $ i)) (at s within ?ivl)" subgoal for s i apply(rule has_derivative_proj[of "(\<lambda> i t. ?flow t $ i)" "(\<lambda> i t'. (t' \<^sub>R (\<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) (?flow s) else 0)) $ i)" "(at s within ?ivl)" "i"]) using sol_deriv_orig[of s] sol_eta sol_deriv_eq1 by auto done have sol_deriv_eq2:"\<And>s i. (\<lambda>xa. xa * (if i = vid1 then sterm_sem I (f1 fid1 vid1) (?flow s) else 0)) = (\<lambda>xa. (xa \<^sub>R (\<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) (?flow s) else 0)) $ i)" by(rule ext, auto) have sol_deriv_proj':"\<And>s i. s\<in>?ivl \<Longrightarrow> ((\<lambda>t. ?flow t $ i) has_derivative (\<lambda>xa. xa (if i = vid1 then sterm_sem I (f1 fid1 vid1) (?flow s) else 0))) (at s within ?ivl)" subgoal for s i using sol_deriv_proj[of s i] sol_deriv_eq2[of i s] by metis done have sol_deriv_proj_vid1:"\<And>s. s\<in>?ivl \<Longrightarrow> ((\<lambda>t. ?flow t $ vid1) has_derivative (\<lambda>xa. xa * (sterm_sem I (f1 fid1 vid1) (?flow s)))) (at s within ?ivl)" subgoal for s using sol_deriv_proj'[of s vid1] by auto done have deriv1_args:"\<And>s. s \<in> ?ivl \<Longrightarrow> ((\<lambda> t. (\<chi> i. sterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (?flow t))) has_derivative ((\<lambda> t'. \<chi> i . t' * (if i = vid1 then sterm_sem I (f1 fid1 vid1) (?flow s) else 0)))) (at s within ?ivl)" apply(rule has_derivative_vec) by (auto simp add: sol_deriv_proj_vid1) have con_fid:"\<And>fid. continuous_on ?ivl (\<lambda>x. sterm_sem I (f1 fid vid1) (?flow x))" subgoal for fid apply(rule has_derivative_continuous_on[of "?ivl" "(\<lambda>x. sterm_sem I (f1 fid vid1) (?flow x))" "(\<lambda>t t'. FunctionFrechet I fid (\<chi> i. sterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (?flow t)) (\<chi> i . t' * (if i = vid1 then sterm_sem I (f1 fid1 vid1) (?flow t) else 0)))"]) proof - fix s assume ivl:"s \<in> ?ivl" let ?h = "(\<lambda>x. sterm_sem I (f1 fid vid1) (?flow x))" let ?g = "Functions I fid" let ?f = "(\<lambda>x. (\<chi> i. sterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (?flow x)))" let ?h' = "(\<lambda>t'. FunctionFrechet I fid (\<chi> i. sterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (?flow s)) (\<chi> i. t' * (if i = vid1 then sterm_sem I (f1 fid1 vid1) (?flow s) else 0)))" let ?g' = "FunctionFrechet I fid (?f s)" let ?f' = "(\<lambda> t'. \<chi> i . t' * (if i = vid1 then sterm_sem I (f1 fid1 vid1) (?flow s) else 0))" have heq:"?h = ?g \<circ> ?f" unfolding comp_def f1_def expand_singleton by auto have heq':"?h' = ?g' \<circ> ?f'" unfolding comp_def by auto have fderiv:"(?f has_derivative ?f') (at s within ?ivl)" using deriv1_args[OF ivl] by auto have gderiv:"(?g has_derivative ?g') (at (?f s) within (?f ` ?ivl))" using good_interp unfolding is_interp_def using has_derivative_within_subset by blast have gfderiv:"((?g \<circ> ?f) has_derivative (?g' \<circ> ?f')) (at s within ?ivl)" using fderiv gderiv diff_chain_within by blast show "((\<lambda>x. sterm_sem I (f1 fid vid1) (?flow x)) has_derivative (\<lambda>t'. FunctionFrechet I fid (\<chi> i. sterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (?flow s)) (\<chi> i. t' * (if i = vid1 then sterm_sem I (f1 fid1 vid1) (?flow s) else 0)))) (at s within ?ivl)" using heq heq' gfderiv by auto qed done have con:"\<And>x. continuous_on (?ivl) (\<lambda>t. x * sterm_sem I (f1 fid2 vid1) (?flow t) + sterm_sem I (f1 fid3 vid1) (?flow t))" apply(rule continuous_on_add) apply(rule continuous_on_mult_left) apply(rule con_fid[of fid2]) by(rule con_fid[of fid3]) let ?axis = "(\<lambda> i. Blinfun(axis i))" have bounded_linear_deriv:"\<And>t. bounded_linear (\<lambda>y' . y' \<^sub>R sterm_sem I (f1 fid2 vid1) (ll_old.flow 0 (sol 0) t))" using bounded_linear_scaleR_left by blast have ll:"local_lipschitz (ll_old.existence_ivl 0 (sol 0)) UNIV (\<lambda>t y. y sterm_sem I (f1 fid2 vid1) (?flow t) + sterm_sem I (f1 fid3 vid1) (?flow t))" apply(rule c1_implies_local_lipschitz[where f'="(\<lambda> (t,y). Blinfun(\<lambda>y' . y' \<^sub>R sterm_sem I (f1 fid2 vid1) (ll_old.flow 0 (sol 0) t)))"]) apply auto subgoal for t x apply(rule has_derivative_add_const) proof - have deriv:"((\<lambda>x. x sterm_sem I (f1 fid2 vid1) (ll_old.flow 0 (sol 0) t)) has_derivative (\<lambda>x. x * sterm_sem I (f1 fid2 vid1) (ll_old.flow 0 (sol 0) t))) (at x)" by(auto intro: derivative_eq_intros) have eq:"(\<lambda>x. x * sterm_sem I (f1 fid2 vid1) (ll_old.flow 0 (sol 0) t)) = blinfun_apply (Blinfun (\<lambda>y'. y' * sterm_sem I (f1 fid2 vid1) (ll_old.flow 0 (sol 0) t)))" apply(rule ext) using bounded_linear_deriv[of t] by (auto simp add: bounded_linear_Blinfun_apply) show "((\<lambda>x. x * sterm_sem I (f1 fid2 vid1) (ll_old.flow 0 (sol 0) t)) has_derivative blinfun_apply (Blinfun (\<lambda>y'. y' * sterm_sem I (f1 fid2 vid1) (ll_old.flow 0 (sol 0) t)))) (at x)" using deriv eq by auto qed apply(auto intro: continuous_intros simp add: split_beta') proof - have bounded_linear:"\<And>x. bounded_linear (\<lambda>y'. y' * sterm_sem I (f1 fid2 vid1) x)" by (simp add: bounded_linear_mult_left) have eq:"(\<lambda>x. Blinfun (\<lambda>y'. y' * sterm_sem I (f1 fid2 vid1) x)) = (\<lambda>x. (sterm_sem I (f1 fid2 vid1) x) \<^sub>R id_blinfun)" apply(rule ext, rule blinfun_eqI) subgoal for x i using bounded_linear[of x] apply(auto simp add: bounded_linear_Blinfun_apply) by (simp add: blinfun.scaleR_left) done have conFlow:"continuous_on (ll_old.existence_ivl 0 (sol 0)) (ll_old.flow 0 (sol 0))" using ll_old.general.flow_continuous_on by blast have conF':"continuous_on (ll_old.flow 0 (sol 0) ` ll_old.existence_ivl 0 (sol 0)) (\<lambda>x. (sterm_sem I (f1 fid2 vid1) x) \<^sub>R id_blinfun)" apply(rule continuous_on_scaleR) apply(auto intro: continuous_intros) apply(rule sterm_continuous') apply(rule good_interp) by(auto simp add: f1_def intro: dfree.intros) have conF:"continuous_on (ll_old.flow 0 (sol 0) ` ll_old.existence_ivl 0 (sol 0)) (\<lambda>x. Blinfun (\<lambda>y'. y' * sterm_sem I (f1 fid2 vid1) x))" apply(rule continuous_on_compose2[of "UNIV" "(\<lambda>x. Blinfun (\<lambda>y'. y' * x))" "(ll_old.flow 0 (sol 0) ` ll_old.existence_ivl 0 (sol 0))" "sterm_sem I (f1 fid2 vid1)"]) subgoal by (metis blinfun_mult_left.abs_eq bounded_linear_blinfun_mult_left continuous_on_eq linear_continuous_on) apply(rule sterm_continuous') apply(rule good_interp) by(auto simp add: f1_def intro: dfree.intros) show "continuous_on (ll_old.existence_ivl 0 (sol 0) \<times> UNIV) (\<lambda>x. Blinfun (\<lambda>y'. y' * sterm_sem I (f1 fid2 vid1) (ll_old.flow 0 (sol 0) (fst x))))" apply(rule continuous_on_compose2[of "ll_old.existence_ivl 0 (sol 0)" "(\<lambda>x. Blinfun (\<lambda>y'. y' * sterm_sem I (f1 fid2 vid1) (ll_old.flow 0 (sol 0) x)))" "(ll_old.existence_ivl 0 (sol 0) \<times> UNIV)" "fst"]) apply(rule continuous_on_compose2[of "(ll_old.flow 0 (sol 0) ` ll_old.existence_ivl 0 (sol 0))" "(\<lambda>x. Blinfun (\<lambda>y'. y' * sterm_sem I (f1 fid2 vid1) x))" "(ll_old.existence_ivl 0 (sol 0))" "(ll_old.flow 0 (sol 0))"]) using conF conFlow by (auto intro!: continuous_intros) qed let ?ivl = "ll_old.existence_ivl 0 (sol 0)" \<comment> \<open>Construct solution to ODE for \<open>y'\<close> here:\<close> let ?yode = "(\<lambda>t y. y * sterm_sem I (f1 fid2 vid1) (?flow t) + sterm_sem I (f1 fid3 vid1) (?flow t))" let ?ysol0 = r interpret ll_new: ll_on_open_it "?ivl" "?yode" "UNIV" 0 apply(standard) apply(auto) apply(rule ll) by(rule con) have sol_new:"(ll_new.flow 0 r solves_ode ?yode) (ll_new.existence_ivl 0 r) UNIV" by(rule ll_new.flow_solves_ode, auto) have more_lipschitz:"\<And>tm tM. tm \<in> ll_old.existence_ivl 0 (sol 0) \<Longrightarrow> tM \<in> ll_old.existence_ivl 0 (sol 0) \<Longrightarrow> \<exists>M L. \<forall>t\<in>{tm..tM}. \<forall>x. \<bar>x * sterm_sem I (f1 fid2 vid1) (?flow t) + sterm_sem I (f1 fid3 vid1) (?flow t)\<bar> \<le> M + L * \<bar>x\<bar>" proof - fix tm tM assume tm:"tm \<in> ll_old.existence_ivl 0 (sol 0)" assume tM:"tM \<in> ll_old.existence_ivl 0 (sol 0)" let ?f2 = "(\<lambda>t. sterm_sem I (f1 fid2 vid1) (ll_old.flow 0 (sol 0) t))" let ?f3 = "(\<lambda>t. sterm_sem I (f1 fid3 vid1) (ll_old.flow 0 (sol 0) t))" let ?boundLP = "(\<lambda>L t . (tm \<le> t \<and> t \<le> tM \<longrightarrow> \<bar>?f2 t\<bar> \<le> L))" let ?boundL = "(SOME L. (\<forall>t. ?boundLP L t))" have compactT:"compact {tm..tM}" by auto have sub:"{tm..tM} \<subseteq> ll_old.existence_ivl 0 (sol 0)" by (metis atLeastatMost_empty_iff empty_subsetI ll_old.general.segment_subset_existence_ivl real_Icc_closed_segment tM tm) let ?f2abs = "(\<lambda>x. abs(?f2 x))" have neg_compact:"\<And>S::real set. compact S \<Longrightarrow> compact ((\<lambda>x. -x) ` S)" by(rule compact_continuous_image, auto intro: continuous_intros) have compactf2:"compact (?f2 ` {tm..tM})" apply(rule compact_continuous_image) apply(rule continuous_on_compose2[of UNIV "sterm_sem I (f1 fid2 vid1)" "{tm..tM}" "ll_old.flow 0 (sol 0)"]) apply(rule sterm_continuous) apply(rule good_interp) subgoal by (auto intro: dfree.intros simp add: f1_def) apply(rule continuous_on_subset) prefer 2 apply (rule sub) subgoal using ll_old.general.flow_continuous_on by blast by auto then have boundedf2:"bounded (?f2 ` {tm..tM})" using compact_imp_bounded by auto then have boundedf2neg:"bounded ((\<lambda>x. -x) ` ?f2 ` {tm..tM})" using compact_imp_bounded neg_compact by auto then have bdd_above_f2neg:"bdd_above ((\<lambda>x. -x) ` ?f2 ` {tm..tM})" by (rule bounded_imp_bdd_above) then have bdd_above_f2:"bdd_above ( ?f2 ` {tm..tM})" using bounded_imp_bdd_above boundedf2 by auto have bdd_above_f2_abs:"bdd_above (abs ` ?f2 ` {tm..tM})" using bdd_above_f2neg bdd_above_f2 unfolding bdd_above_def apply auto subgoal for M1 M2 apply(rule exI[where x="max M1 M2"]) by fastforce done then have theBound:"\<exists>L. (\<forall>t. ?boundLP L t)" unfolding bdd_above_def norm_conv_dist by (auto simp add: Ball_def Bex_def norm_conv_dist image_iff norm_bcontfun_def dist_blinfun_def) then have boundLP:"\<forall>t. ?boundLP (?boundL) t" using someI[of "(\<lambda> L. \<forall>t. ?boundLP L t)"] by blast let ?boundMP = "(\<lambda>M t. (tm \<le> t \<and> t \<le> tM \<longrightarrow> \<bar>?f3 t\<bar> \<le> M))" let ?boundM = "(SOME M. (\<forall>t. ?boundMP M t))" have compactf3:"compact (?f3 ` {tm..tM})" apply(rule compact_continuous_image) apply(rule continuous_on_compose2[of UNIV "sterm_sem I (f1 fid3 vid1)" "{tm..tM}" "ll_old.flow 0 (sol 0)"]) apply(rule sterm_continuous) apply(rule good_interp) subgoal by (auto intro: dfree.intros simp add: f1_def) apply(rule continuous_on_subset) prefer 2 apply (rule sub) subgoal using ll_old.general.flow_continuous_on by blast by auto then have boundedf3:"bounded (?f3 ` {tm..tM})" using compact_imp_bounded by auto then have boundedf3neg:"bounded ((\<lambda>x. -x) ` ?f3 ` {tm..tM})" using compact_imp_bounded neg_compact by auto then have bdd_above_f3neg:"bdd_above ((\<lambda>x. -x) ` ?f3 ` {tm..tM})" by (rule bounded_imp_bdd_above) then have bdd_above_f3:"bdd_above ( ?f3 ` {tm..tM})" using bounded_imp_bdd_above boundedf3 by auto have bdd_above_f3_abs:"bdd_above (abs ` ?f3 ` {tm..tM})" using bdd_above_f3neg bdd_above_f3 unfolding bdd_above_def apply auto subgoal for M1 M2 apply(rule exI[where x="max M1 M2"]) by fastforce done then have theBound:"\<exists>L. (\<forall>t. ?boundMP L t)" unfolding bdd_above_def norm_conv_dist by (auto simp add: Ball_def Bex_def norm_conv_dist image_iff norm_bcontfun_def dist_blinfun_def) then have boundMP:"\<forall>t. ?boundMP (?boundM) t" using someI[of "(\<lambda> M. \<forall>t. ?boundMP M t)"] by blast show "\<exists>M L. \<forall>t\<in>{tm..tM}. \<forall>x. \<bar>x * ?f2 t + ?f3 t\<bar> \<le> M + L * \<bar>x\<bar>" apply(rule exI[where x="?boundM"]) apply(rule exI[where x="?boundL"]) apply auto proof - fix t and x :: real assume ttm:"tm \<le> t" assume ttM:"t \<le> tM" from ttm ttM have ttmM:"tm \<le> t \<and> t \<le> tM" by auto have leqf3:"\<bar>?f3 t\<bar> \<le> ?boundM" using boundMP ttmM by auto have leqf2:"\<bar>?f2 t\<bar> \<le> ?boundL" using boundLP ttmM by auto have gr0:" \<bar>x\<bar> \<ge> 0" by auto have leqf2x:"\<bar>?f2 t\<bar> * \<bar>x\<bar> \<le> ?boundL * \<bar>x\<bar>" using gr0 leqf2 by (metis (no_types, lifting) real_scaleR_def scaleR_right_mono) have "\<bar>x * ?f2 t + ?f3 t\<bar> \<le> \<bar>x\<bar> * \<bar>?f2 t\<bar> + \<bar>?f3 t\<bar>" proof - have f1: "\<And>r ra. \<bar>r::real\<bar> * \<bar>ra\<bar> = \<bar>r * ra\<bar>" by (metis norm_scaleR real_norm_def real_scaleR_def) have "\<And>r ra. \<bar>(r::real) + ra\<bar> \<le> \<bar>r\<bar> + \<bar>ra\<bar>" by (metis norm_triangle_ineq real_norm_def) then show ?thesis using f1 by presburger qed moreover have "... = \<bar>?f3 t\<bar> + \<bar>?f2 t\<bar> * \<bar>x\<bar>" by auto moreover have "... \<le> ?boundM + \<bar>?f2 t\<bar> * \<bar>x\<bar>" using leqf3 by linarith moreover have "... \<le> ?boundM + ?boundL * \<bar>x\<bar>" using leqf2x by linarith ultimately show "\<bar>x * ?f2 t + ?f3 t\<bar> \<le> ?boundM + ?boundL * \<bar>x\<bar>" by linarith qed qed have ivls_eq:"(ll_new.existence_ivl 0 r) = (ll_old.existence_ivl 0 (sol 0))" apply(rule ll_new.existence_ivl_eq_domain) apply auto apply (rule more_lipschitz) by auto have sub':"{0--t} \<subseteq> ll_new.existence_ivl 0 r" using sub ivls_eq by auto have sol_new':"(ll_new.flow 0 r solves_ode ?yode) {0--t} UNIV" by(rule solves_ode_subset, rule sol_new, rule sub') let ?soly = "ll_new.flow 0 r" let ?sol' = "(\<lambda>t. \<chi> i. if i = vid2 then ?soly t else sol t $ i)" let ?aaba' = "mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))) (\<chi> y. if vid2 = y then r else fst (a, b) $ y, b) (?sol' t)" have duh:"(fst ?aaba', snd ?aaba') = ?aaba'" by auto note bigEx = spec[OF spec[OF bigAll, where x="fst ?aaba'"], where x="snd ?aaba'"] have sol_deriv:"\<And>s. s \<in> {0..t} \<Longrightarrow> (sol has_derivative (\<lambda>xa. xa \<^sub>R (\<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) (sol s) else 0))) (at s within {0..t})" using sol apply simp by(drule solves_odeD(1), auto simp add: has_vderiv_on_def has_vector_derivative_def) have silly_eq1:"(\<lambda>t. \<chi> i. sol t $ i) = sol" by(rule ext, rule vec_extensionality, auto) have silly_eq2:"\<And>s. (\<lambda>xa. \<chi> i. (xa \<^sub>R (\<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) (sol s) else 0)) $ i) = (\<lambda>xa. xa \<^sub>R (\<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) (sol s) else 0))" by(rule ext, rule vec_extensionality, auto) have sol_proj_deriv:"\<And>s i. s \<in> {0..t} \<Longrightarrow> ((\<lambda> t. sol t $ i) has_derivative (\<lambda>xa. (xa \<^sub>R (\<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) (sol s) else 0)) $ i)) (at s within {0..t})" subgoal for s i apply(rule has_derivative_proj) using sol_deriv[of s] silly_eq1 silly_eq2[of s] by auto done have sol_proj_deriv_vid1:"\<And>s. s \<in> {0..t} \<Longrightarrow> ((\<lambda> t. sol t $ vid1) has_derivative (\<lambda>xa. xa * sterm_sem I (f1 fid1 vid1) (sol s))) (at s within {0..t})" subgoal for s using sol_proj_deriv[of s vid1] by auto done have sol_proj_deriv_other:"\<And>s i. s \<in> {0..t} \<Longrightarrow> i \<noteq> vid1 \<Longrightarrow> ((\<lambda> t. sol t $ i) has_derivative (\<lambda>xa. 0)) (at s within {0..t})" subgoal for s i using sol_proj_deriv[of s i] by auto done have fact:"\<And>x. x \<in>{0..t} \<Longrightarrow> (ll_new.flow 0 r has_derivative (\<lambda>xa. xa \<^sub>R (ll_new.flow 0 r x sterm_sem I (f1 fid2 vid1) (ll_old.flow 0 (sol 0) x) + sterm_sem I (f1 fid3 vid1) (ll_old.flow 0 (sol 0) x)))) (at x within {0 .. t})" using sol_new' apply simp apply(drule solves_odeD(1)) using tclosed unfolding has_vderiv_on_def has_vector_derivative_def by auto have new_sol_deriv:"\<And>s. s \<in> {0..t} \<Longrightarrow> (ll_new.flow 0 r has_derivative (\<lambda>xa. xa \<^sub>R (ll_new.flow 0 r s sterm_sem I (f1 fid2 vid1) (sol s) + sterm_sem I (f1 fid3 vid1) (sol s)))) (at s within {0.. t})" subgoal for s using fact[of s] tclosed sol_eq_flow[of s] by auto done have sterm_agree:"\<And>s. Vagree (\<chi> i. if i = vid2 then ll_new.flow 0 r s else sol s $ i, undefined) (sol s, undefined) {Inl vid1}" subgoal for s unfolding Vagree_def using vne12 by auto done have FVF:"(FVT (f1 fid2 vid1)) = {Inl vid1}" unfolding f1_def expand_singleton apply auto subgoal for x xa by (cases "xa = vid1", auto) done have FVF2:"(FVT (f1 fid3 vid1)) = {Inl vid1}" unfolding f1_def expand_singleton apply auto subgoal for x xa by (cases "xa = vid1", auto) done have sterm_agree_FVF:"\<And>s. Vagree (\<chi> i. if i = vid2 then ll_new.flow 0 r s else sol s $ i, undefined) (sol s, undefined) (FVT (f1 fid2 vid1))" using sterm_agree FVF by auto have sterm_agree_FVF2:"\<And>s. Vagree (\<chi> i. if i = vid2 then ll_new.flow 0 r s else sol s $ i, undefined) (sol s, undefined) (FVT (f1 fid3 vid1))" using sterm_agree FVF2 by auto have y_component_sem_eq2:"\<And>s. sterm_sem I (f1 fid2 vid1) (\<chi> i. if i = vid2 then ll_new.flow 0 r s else sol s $ i) = sterm_sem I (f1 fid2 vid1) (sol s)" using coincidence_sterm[OF sterm_agree_FVF, of I] by auto have y_component_sem_eq3:"\<And>s. sterm_sem I (f1 fid3 vid1) (\<chi> i. if i = vid2 then ll_new.flow 0 r s else sol s $ i) = sterm_sem I (f1 fid3 vid1) (sol s)" using coincidence_sterm[OF sterm_agree_FVF2, of I] by auto have y_component_ode_eq:"\<And>s. s \<in> {0..t} \<Longrightarrow> (\<lambda>xa. xa * (ll_new.flow 0 r s * sterm_sem I (f1 fid2 vid1) (sol s) + sterm_sem I (f1 fid3 vid1) (sol s))) = (\<lambda>xa. xa * (sterm_sem I (f1 fid2 vid1) (\<chi> i. if i = vid2 then ll_new.flow 0 r s else sol s $ i) * ll_new.flow 0 r s + sterm_sem I (f1 fid3 vid1) (\<chi> i. if i = vid2 then ll_new.flow 0 r s else sol s $ i)))" subgoal for s apply(rule ext) subgoal for xa using y_component_sem_eq2 y_component_sem_eq3 by auto done done have agree_vid1:"\<And>s. Vagree (sol s, undefined) (\<chi> i. if i = vid2 then ll_new.flow 0 r s else sol s $ i, undefined) {Inl vid1}" unfolding Vagree_def using vne12 by auto have FVT_vid1:"FVT(f1 fid1 vid1) = {Inl vid1}" apply(auto simp add: f1_def) subgoal for x xa apply(cases "xa = vid1") by auto done have agree_vid1_FVT:"\<And>s. Vagree (sol s, undefined) (\<chi> i. if i = vid2 then ll_new.flow 0 r s else sol s $ i, undefined) (FVT (f1 fid1 vid1))" using FVT_vid1 agree_vid1 by auto have sterm_eq_vid1:"\<And>s. sterm_sem I (f1 fid1 vid1) (sol s) = sterm_sem I (f1 fid1 vid1) (\<chi> i. if i = vid2 then ll_new.flow 0 r s else sol s $ i)" subgoal for s using coincidence_sterm[OF agree_vid1_FVT[of s], of I] by auto done have vid1_deriv_eq:"\<And>s. (\<lambda>xa. xa * sterm_sem I (f1 fid1 vid1) (sol s)) = (\<lambda>xa. xa * sterm_sem I (f1 fid1 vid1) (\<chi> i. if i = vid2 then ll_new.flow 0 r s else sol s $ i))" subgoal for s apply(rule ext) subgoal for x' using sterm_eq_vid1[of s] by auto done done have inner_deriv:"\<And>s. s \<in> {0..t} \<Longrightarrow> ((\<lambda>t. \<chi> i. if i = vid2 then ll_new.flow 0 r t else sol t $ i) has_derivative (\<lambda>xa. (\<chi> i. xa * (if i = vid1 then sterm_sem I (f1 fid1 vid1) (\<chi> i. if i = vid2 then ll_new.flow 0 r s else sol s $ i) else if i = vid2 then sterm_sem I (Plus (Times (f1 fid2 vid1) (trm.Var vid2)) (f1 fid3 vid1)) (\<chi> i. if i = vid2 then ll_new.flow 0 r s else sol s $ i) else 0)))) (at s within {0..t})" subgoal for s apply(rule has_derivative_vec) subgoal for i apply(cases "i = vid2") subgoal using vne12 using new_sol_deriv[of s] using y_component_ode_eq by auto subgoal apply(cases "i = vid1") using sol_proj_deriv_vid1[of s] vid1_deriv_eq[of s] sol_proj_deriv_other[of s i] by auto done done done have deriv_eta:"\<And>s. (\<lambda>xa. xa \<^sub>R ((\<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) (\<chi> i. if i = vid2 then ll_new.flow 0 r s else sol s $ i) else 0) + (\<chi> i. if i = vid2 then sterm_sem I (Plus (Times (f1 fid2 vid1) (trm.Var vid2)) (f1 fid3 vid1)) (\<chi> i. if i = vid2 then ll_new.flow 0 r s else sol s $ i) else 0))) = (\<lambda>xa. (\<chi> i. xa (if i = vid1 then sterm_sem I (f1 fid1 vid1) (\<chi> i. if i = vid2 then ll_new.flow 0 r s else sol s $ i) else if i = vid2 then sterm_sem I (Plus (Times (f1 fid2 vid1) (trm.Var vid2)) (f1 fid3 vid1)) (\<chi> i. if i = vid2 then ll_new.flow 0 r s else sol s $ i) else 0))) " subgoal for s apply(rule ext) apply(rule vec_extensionality) using vne12 by auto done have sol'_deriv:"\<And>s. s \<in> {0..t} \<Longrightarrow> ((\<lambda>t. \<chi> i. if i = vid2 then ll_new.flow 0 r t else sol t $ i) has_derivative (\<lambda>xa. xa *\<^sub>R ((\<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) (\<chi> i. if i = vid2 then ll_new.flow 0 r s else sol s $ i) else 0) + (\<chi> i. if i = vid2 then sterm_sem I (Plus (Times (f1 fid2 vid1) (trm.Var vid2)) (f1 fid3 vid1)) (\<chi> i. if i = vid2 then ll_new.flow 0 r s else sol s $ i) else 0)))) (at s within {0..t})" subgoal for s using inner_deriv[of s] deriv_eta[of s] by auto done have FVT:"\<And>i. FVT (if i = vid1 then trm.Var vid1 else Const 0) \<subseteq> {Inl vid1}" by auto have agree:"\<And>s. Vagree (mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (a, b) (sol s)) (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))) (\<chi> y. if vid2 = y then r else fst (a, b) $ y, b) (\<chi> i. if i = vid2 then ll_new.flow 0 r s else sol s $ i)) {Inl vid1}" subgoal for s using mk_v_agree [of "I" "(OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))" "(a, b)" "(sol s)"] using mk_v_agree [of I "(OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))))" "(\<chi> y. if vid2 = y then r else fst (a, b) $ y, b)" "(\<chi> i. if i = vid2 then ll_new.flow 0 r s else sol s $ i)"] unfolding Vagree_def using vne12 by simp done have agree':"\<And>s i. Vagree (mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (a, b) (sol s)) (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))) (\<chi> y. if vid2 = y then r else fst (a, b) $ y, b) (\<chi> i. if i = vid2 then ll_new.flow 0 r s else sol s $ i)) (FVT (if i = vid1 then trm.Var vid1 else Const 0))" subgoal for s i using agree_sub[OF FVT[of i] agree[of s]] by auto done have safe:"\<And>i. dsafe (if i = vid1 then trm.Var vid1 else Const 0)" subgoal for i apply(cases "i = vid1", auto) done done have dterm_sem_eq:"\<And>s i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (a, b) (sol s)) = dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))) (\<chi> y. if vid2 = y then r else fst (a, b) $ y, b) (\<chi> i. if i = vid2 then ll_new.flow 0 r s else sol s $ i))" subgoal for s i using coincidence_dterm[OF safe[of i] agree'[of s i], of I] by auto done have dterm_vec_eq:"\<And>s. (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (a, b) (sol s))) = (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))) (\<chi> y. if vid2 = y then r else fst (a, b) $ y, b) (\<chi> i. if i = vid2 then ll_new.flow 0 r s else sol s $ i)))" subgoal for s apply(rule vec_extensionality) subgoal for i using dterm_sem_eq[of i s] by auto done done have pred_same:"\<And>s. s \<in> {0..t} \<Longrightarrow> Predicates I vid1 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (a, b) (sol s))) \<Longrightarrow> Predicates I vid1 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))) (\<chi> y. if vid2 = y then r else fst (a, b) $ y, b) (\<chi> i. if i = vid2 then ll_new.flow 0 r s else sol s $ i)))" subgoal for s using dterm_vec_eq[of s] by auto done have sol'_domain:"\<And>s. 0 \<le> s \<Longrightarrow> s \<le> t \<Longrightarrow> Predicates I vid1 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))) (\<chi> y. if vid2 = y then r else fst (a, b) $ y, b) (\<chi> i. if i = vid2 then ll_new.flow 0 r s else sol s $ i)))" subgoal for s using sol apply simp apply(drule solves_odeD(2)) using pred_same[of s] by auto done have sol':"(?sol' solves_ode (\<lambda>a b. (\<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) b else 0) + (\<chi> i. if i = vid2 then sterm_sem I (Plus (Times (f1 fid2 vid1) (trm.Var vid2)) (f1 fid3 vid1)) b else 0))) {0..t} {x. Predicates I vid1 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))) (\<chi> y. if vid2 = y then r else fst (a, b) $ y, b) x))}" apply(rule solves_odeI) subgoal unfolding has_vderiv_on_def has_vector_derivative_def using sol'_deriv by auto by(auto, rule sol'_domain, auto) have set_eq:"{y. y = vid2 \<or> y = vid1 \<or> y = vid2 \<or> y = vid1 \<or> (\<exists>x. Inl y \<in> FVT (if x = vid1 then trm.Var vid1 else Const 0))} = {vid1, vid2}" by auto have "VSagree (?sol' 0) (\<chi> y. if vid2 = y then r else fst (a, b) $ y) {vid1, vid2}" using VSA unfolding VSagree_def by simp then have VSA':" VSagree (?sol' 0) (\<chi> y. if vid2 = y then r else fst (a, b) $ y) {y. y = vid2 \<or> y = vid1 \<or> y = vid2 \<or> y = vid1 \<or> (\<exists>x. Inl y \<in> FVT (if x = vid1 then trm.Var vid1 else Const 0))} " by (auto simp add: set_eq) have bigPre:"(\<exists>sol t. (fst ?aaba', snd ?aaba') = mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))) ((\<chi> y. if vid2 = y then r else fst (a,b) $ y), b) (sol t) \<and> 0 \<le> t \<and> (sol solves_ode (\<lambda>a b. (\<chi> i. if i = vid1 then sterm_sem I (f1 fid1 vid1) b else 0) + (\<chi> i. if i = vid2 then sterm_sem I (Plus (Times (f1 fid2 vid1) (trm.Var vid2)) (f1 fid3 vid1)) b else 0))) {0..t} {x. Predicates I vid1 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))))) ((\<chi> y. if vid2 = y then r else (fst (a,b)) $ y), b) x))} \<and> VSagree (sol 0) (\<chi> y. if vid2 = y then r else fst (a,b) $ y) {uu. uu = vid2 \<or> uu = vid1 \<or> uu = vid2 \<or> uu = vid1 \<or> Inl uu \<in> Inl ` ({x. \<exists>xa. Inl x \<in> FVT (if xa = vid1 then trm.Var vid1 else Const 0)} \<union> {x. x = vid2 \<or> (\<exists>xa. Inl x \<in> FVT (if xa = vid1 then trm.Var vid1 else Const 0))}) \<or> (\<exists>x. Inl uu \<in> FVT (if x = vid1 then trm.Var vid1 else Const 0))})" apply(rule exI[where x="?sol'"]) apply(rule exI[where x=t]) apply(rule conjI) subgoal by simp apply(rule conjI) subgoal by (rule t) apply(rule conjI) apply(rule sol') using VSA' unfolding VSagree_def by auto have pred_sem:"Predicates I vid2 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) ?aaba')" using mp[OF bigEx bigPre] by auto let ?other_state = "(mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (a, b) (sol t))" have agree:"Vagree (?aaba') (?other_state) {Inl vid1} " using mk_v_agree [of "I" "(OProd (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (OSing vid2 (Plus (Times ($f fid2 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)) (trm.Var vid2)) ($f fid3 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))))" "(\<chi> y. if vid2 = y then r else fst (a, b) $ y, b)" "(?sol' t)"] using mk_v_agree [of "I" "(OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0)))" "(a, b)" "(sol t)"] unfolding Vagree_def using vne12 by simp have sub:"\<And>i. FVT (if i = vid1 then trm.Var vid1 else Const 0) \<subseteq> {Inl vid1}" by auto have agree':"\<And>i. Vagree (?aaba') (?other_state) (FVT (if i = vid1 then trm.Var vid1 else Const 0)) " subgoal for i using agree_sub[OF sub[of i] agree] by auto done have silly_safe:"\<And>i. dsafe (if i = vid1 then trm.Var vid1 else Const 0)" subgoal for i apply(cases "i = vid1") by (auto simp add: dsafe_Var dsafe_Const) done have dsem_eq:"(\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) ?aaba') = (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) ?other_state)" apply(rule vec_extensionality) subgoal for i using coincidence_dterm[OF silly_safe[of i] agree'[of i], of I] by auto done show "Predicates I vid2 (\<chi> i. dterm_sem I (if i = vid1 then trm.Var vid1 else Const 0) (mk_v I (OSing vid1 ($f fid1 (\<lambda>i. if i = vid1 then trm.Var vid1 else Const 0))) (a, b) (sol t)))" using pred_sem dsem_eq by auto qed done qed end end
\subsection{Neural Network Design and Training} \label{subsec:nntraining} %\begin{wrapfigure}{r}{0.5\textwidth} % \centering % \includegraphics[width=0.35\textwidth]{img/nnlayout} % \caption{Network Layers} % \label{fig:network-layers} %\end{wrapfigure} The network was implemented in PyTorch \cite{Paszke:2019aa} as well as Tensorflow \cite{MartinAbadi:2015aa}. The backend was later exclusively switched to PyTorch (which is also the most common deep learning framework in Science) due to its better support of quantization. The layers of the neural network are depicted in Figure~\ref{fig:eggnet}. The network was trained on the \emph{MNIST} dataset which consists of 60.000 images of handwritten digits. Those where split up in 50.000 images used for training and 10.000 used for evaluation. The network maps mathematically an $28 \times 28$ input image $x \in \mathbb{R}^{28,28}$ to an output of vector of probabilities $y \in \mathbb{R}^{10}$ where each value corresponds how likely the input image belongs to that class. The layers of the network are listed in Table~\ref{tab:eggnet-layers}. \begin{table}[h] \centering \begin{tabular}{clcl} \toprule Layer & Type & Data Size (output) & Purpose \\ \midrule 1 & Conv01 & $[28,28,16]$ & Extract features ($[3,3]$ kernels) \\ 2 & ReLU & $[28,28,16]$ & Introduce nonlinearity \\ 3 & Pool01 & $[14,14,16]$ & Introduce nonlinearity, reduce dimensions \\ 4 & Conv02 & $[14,14,32]$ & Extract features ($[3,3]$ kernels) \\ 5 & ReLU & $[14,14,32]$ & Introduce nonlinearity \\ 6 & Pool02 & $[7,7,32]$ & Introduce nonlinearity, reduce dimensions \\ 7 & Dense01 & $[32]$ & Combine features \\ 8 & ReLU & $[32]$ & Introduce nonlinearity \\ 9 & Dense02 & $[10]$ & Combine features \\ 10 & Softmax & $[10]$ & Normalize output \\ \bottomrule \end{tabular} \caption{Eggnet Layers} \label{tab:eggnet-layers} \end{table} Further for training of the network the \emph{ADAM} optimization algorithm \cite{Kingma:2014aa} was used to minimize the cross-entropy-loss function which is defined as \begin{equation} J = - y \log(h) + (1-y) \log(1-h) \end{equation} This loss function is standard for classification problems where each sample belongs to exactly one class. The ADAM algorithm can be adjusted by parameters, which control the speed of convergence. The recommended values, listed in Table~\ref{tab:train-params}, by \cite{Kingma:2014aa} were used. \begin{table}[ht] \centering \begin{tabular}{cc} \toprule Parameter & Value \\ \midrule $\alpha$ & $0.001$ \\ $\beta_1$ & $0.9$ \\ $\beta_2$ & $0.999$ \\ \bottomrule \end{tabular} \caption{Network Training Parameters} \label{tab:train-params} \end{table} A useful guide for implementing convolutions can be found in \cite{dumoulin2016guide}. The training of the network yielded very high accuracy rates that are typical for the MNIST dataset, which is an easy challenge for machine learning. Even though the network performance could be improved, e.g. by hyperparameter tuning the results were acceptable for our case. The progress of the training in terms of accuracy and loss can be seen in Figure~\ref{fig:network-train-acc} respectively in Figure~\ref{fig:network-train-loss}. The final output of the network over the training is evaluated in Figure~\ref{fig:network-test-cm} for real values and in Figure~\ref{fig:network-test-qcm} for fake quantized values. Both, PyTorch and Tensorflow/Keras are Python libraries but most \gls{acr:NN} operations are not implemented in Python directly because of performance reasons. This makes debugging more difficult which is why we reimplemented the operations we needed in Python. This was done using NumPy and SWIG, which will be discussed more extensively in Section~\ref{sec:sw-python}, and enabled us to verify the correct ordering of parameters, weights and activations. \begin{figure}[hbtp] \centering \begin{subfigure}[t]{0.5\textwidth} \includegraphics[width=0.8\textwidth]{../../net/images/training_loss} \caption{Training Loss} \label{fig:network-train-loss} \end{subfigure}% ~ \begin{subfigure}[t]{0.5\textwidth} \includegraphics[width=0.8\textwidth]{../../net/images/training_accuracy} \caption{Training Accuracy} \label{fig:network-train-acc} \end{subfigure} \caption[Network loss and accuracy over the training iterations]{Network loss and accuracy over the training iterations. The blue lines show spikes which occur because of the randomly selected mini batches. The orange line shows the smoothed version over 50 periods} \label{fig:network-training-graphs} \end{figure} \begin{figure}[hbtp] \centering \begin{subfigure}[t]{0.5\textwidth} \includegraphics[width=0.9\textwidth]{../../net/images/cm} \caption{Floating Point} \label{fig:network-test-cm} \end{subfigure}% ~ \begin{subfigure}[t]{0.5\textwidth} \includegraphics[width=0.9\textwidth]{../../net/images/qcm} \caption{Quantized Values} \label{fig:network-test-qcm} \end{subfigure} \caption{Confusion matrix for the floating point and quantized version of the network.} \label{fig:network-confusion-matrix} \end{figure} \subsection{Quantization} \label{sec:nn-quant} The network is trained and created using \SI{32}{\bit} floating point values in Python. Directly porting this all the weights and biases to the FPGA is due to the limited amount of available resources not feasible. The goal is therefore to reduce the amount of required hardware cells by switching from floating point arithmetic to the less expensive integer arithmetic. Then a floating point value $v$ can be approximately represented as \begin{equation} v \approx Q \cdot 2 ^{-m} \end{equation} where $Q$ and $m$ are integers. In our case all input values of the first layer are guaranteed to lie in the interval $[0,1]$ and all layer weights are known from training. It is therefore possible to precompute the expected range where the output values will be. Depending on this range it is then possible to select a suitable bit width for both $Q$ and $m$. This is a cost-accuracy trade-off where higher bit widths would improve accuracy as well as increase the amount of hardware resources needed. In \cite{Wu:2018aa} different strategies of choosing bit widths for $Q$ and $m$ are compared and they observed three main configurations, which are (from simple to advanced): \begin{enumerate} \item Use a $(Q,m)$ configuration for the whole network \item Use a $(Q,m)$ configuration for each layer \item Use a $(Q,m)$ configuration for each output channel \end{enumerate} In the third configuration the authors could reduce the bit widths the most without sacrificing accuracy this increases the complexity in transferring the weights from layer to layer because the additional shift operations are necessary in order to adjust for the different values of $m$. In \cite{Wu:2018aa} the authors also deduced from their experiments that the accuracy of the weights can be reduced the most, followed by the activations. By analysing the weights of our network (see Figure~\ref{fig:network-weight-distributions}) a per channel quantization is not necessary, because all weights in a Convolutional Layer are equally distributed among the output channels. Another important property that can be noted is the that the weights do have zero mean and most of the values lie very close to zero. Because of the usage of ReLU layer the situation is different for the activations where unsigned integers can be used, the distributions are shown in Figure~\ref{fig:network-activations-distributions}. Using the distribution histograms we then derived the necessary bitwidths for $Q$ and $m$. In our experiments we were able to reduce them to \SI{8}{\bit}, if we used a single configuration for the whole network and also reducing them down to \SI{4}{\bit} if the bitwidth configuration is selected for each layer independently with an accuracy drop from around \SI{98.35}{\percent} to \SI{97.37}{\percent}. The strategy to the select the values for $(Q,m)$ was \begin{enumerate} \item Find the value range of the weights and output activations of each layer \item Select suitable $(Q,m)$ values that most activations fall in that range \item Calculate the bit widths and exponents of the multiplication operation \item Add $\lceil \log_2(n) \rceil$ extra bits to account for the accumulation of $n$ values \item Compare the accumulated exponents and with the exponents of the successive layers input exponents. The difference is the amount of shift required \end{enumerate} It is noteworthy that the values for $m$ do not need to be stored in the final network, because those are only used to determine the amount of shifts between the layers. Also the values need to be clipped to their maximum and minimum values. The complete configuration of the network is summarized in Table~\ref{tab:quantization-linear-params}. Ad 4 and 5: The transition from a layer to the next often changes the exponent $m$ and the available bitwidth. To account for this the values need to accordingly shifted. Also the decreased bitwidth needs clipping to maximum available values for the target bitwidth. This directly alters the behaviour of the network which should be accounted for during training, which is done via a saturated version of ReLU, defined as: \begin{equation} \text{ReLU}_{\text{sat}}: ~ f(x;p) = \begin{cases} 0 \quad \text{if} \quad x < 0 \\ p \quad \text{if} \quad x > p \\ x \quad \text{else} \end{cases} \end{equation} For our network only linear quantization has been used but also non-linear quantization, e.g. in a $\log_2$ way which is proposed in \cite{Lee:2017aa}. Experiments showed that using this technique even further down to \SI{3}{\bit} weights in our case. Another optimization technique that could be explored is the systematically removing of weights (connections) of the network and reduce the amount of operations needed to be performed, a process referred to as ''pruning'' \cite{Zhu:2017aa}. This was not explicitly performed but is implicitly done by low bit quantization. \begin{table}[hbt] \centering \begin{tabular}{lcccc} \toprule Network Part & $\|Q\|$ & $m$ & $\pm$ & $v$ (real value range) \\ \midrule Input & 8 & 8 & $+$ & $[0,1] $ \\ L1: Weights & 4 & 2 & $\pm$ & $[-2,2] $ \\ L1: Intermediates & 12 & 10 & $\pm$ & $[-2,2] $ \\ L1: Accumulated & 16 & 10 & $\pm$ & \\ \midrule L1 $\to$ L2 & \multicolumn{4}{c}{Rshift by $10-2$ and clip values in range $[0,15]$} \\ \midrule L2: Input & 4 & 2 & $+$ & $[-2,2] $ \\ L2: Weights & 4 & 5 & $\pm$ & $[-0.5,0.5] $ \\ L2: Intermediates & 8 & 7 & $\pm$ & $[-1,1] $ \\ L2: Accumulated & 16 & 7 & $\pm$ & \\ \midrule L2 $\to$ L3 & \multicolumn{4}{c}{Rshift by $7-0$ and clip values in range $[0,15]$} \\ \midrule L3: Input & 4 & 0 & $+$ & $[0,15] $ \\ L3: Weights & 4 & 5 & $\pm$ & $[-0.5,0.5] $ \\ L3: Intermediates & 8 & 5 & $\pm$ & $[-7.5,7.5] $ \\ L3: Accumulated & 19 & 5 & $\pm$ & \\ \midrule L3 $\to$ L4 & \multicolumn{4}{c}{Rshift by $5-0$ and clip values in range $[0,15]$} \\ \midrule L4: Input & 4 & 0 & $+$ & $[0,15] $ \\ L4: Weights & 4 & 5 & $\pm$ & $[-0.5,0.5] $ \\ L4: Intermediates & 8 & 5 & $\pm$ & $[-7.5,7.5] $ \\ L4: Accumulated & 14 & 5 & $\pm$ & $[0,1] $ \\ \bottomrule \end{tabular} \caption[Quantization parameters for the \SI{4}{\bit} network]{Quantization parameters for the \SI{4}{\bit} network. The intermediate terms are the values after the multiplication operation and the accumulated term denotes values after summing up of weighted inputs including bias in a channel.} \label{tab:quantization-linear-params} \end{table} %% WEIGHT DISTTRIBUTIONS \begin{figure}[htbp] \centering \begin{subfigure}[t]{0.5\textwidth} \centering \includegraphics[height=1.6in]{../../net/images/hist_cn1_k} \caption{Convolutional Layer 1} \end{subfigure}% ~ \begin{subfigure}[t]{0.5\textwidth} \centering \includegraphics[height=1.6in]{../../net/images/hist_cn2_k} \caption{Convolutional Layer 2} \end{subfigure}% \\ \begin{subfigure}[t]{0.5\textwidth} \centering \includegraphics[height=1.6in]{../../net/images/hist_fc1_w} \caption{Fully Connected Layer 1} \end{subfigure}% ~ \begin{subfigure}[t]{0.5\textwidth} \centering \includegraphics[height=1.6in]{../../net/images/hist_fc2_w} \caption{Fully Connected Layer 2} \end{subfigure} \caption[Distribution of the network weights for the different layers]{Distribution of the network weights for the different layers. It can be seen that the weights are distributed close to zero.} \label{fig:network-weight-distributions} \end{figure} %% ACTIVATIONS DISTTRIBUTIONS \begin{figure}[htbp] \centering \begin{subfigure}[t]{0.5\textwidth} \centering \includegraphics[height=1.6in]{../../net/images/hist_ao1} \caption{Convolutional Layer 1} \end{subfigure}% ~ \begin{subfigure}[t]{0.5\textwidth} \centering \includegraphics[height=1.6in]{../../net/images/hist_ao2} \caption{Convolutional Layer 2} \end{subfigure}% \\ \begin{subfigure}[t]{0.5\textwidth} \centering \includegraphics[height=1.6in]{../../net/images/hist_ao3} \caption{Fully Connected Layer 1} \end{subfigure}% ~ \begin{subfigure}[t]{0.5\textwidth} \centering \includegraphics[height=1.6in]{../../net/images/hist_ao4} \caption{Fully Connected Layer 2} \end{subfigure} \caption[Distribution of the activations for a randomly selected batch of the input data]{Distribution of the activations for a randomly selected batch of the input data. The occurring minimum and maximum values were used to define the quantization bit width.} \label{fig:network-activations-distributions} \end{figure}
(\| ###################### Coq inference behavior ###################### :Link: https://stackoverflow.com/q/36996020 \|) (\| Question ******* I'm trying to write the following Agda snippet in Coq. .. code-block:: agda open import Data.Fin using (Fin; suc; zero) open import Data.Nat using (ℕ; suc; zero) thin : {n : ℕ} -> Fin (suc n) -> Fin n -> Fin (suc n) thin zero y = suc y thin (suc x) zero = zero thin (suc x) (suc y) = suc (thin x y) I thought this could be straightforwardly translated to Coq as: \|) Inductive Fin : nat -> Type := \| fz {n : nat} : Fin (S n) \| fs {n : nat} : Fin n -> Fin (S n). Fail Fixpoint thin {n : nat} (x : Fin (S n)) (y : Fin n) : Fin (S n) := match x, y with \| fz, y' => fs y' \| fs x', fz => fz \| fs x', fs y' => fs (thin x' y') end. ( .fails ) (\| I believe Coq should be able to figure out the implicit parameter ``n`` so I have no idea what is going on. I think I am unaware of a difference between the type systems of Agda and Coq since the former typechecks fine. \|) (\| Answer ****** When pattern matching with dependent types, Coq does not usually consider some essential relations between the variables in the context and the variables introduced in the branches. The easiest solution is to define the function in proof mode, at least to understand what is going on. This gives: \|) Fixpoint thin {n : nat} (x : Fin (S n)) (y : Fin n) : Fin (S n). Proof. remember (S n) as n1. ( trick to keep the information when destructing ) destruct x; apply eq_add_S in Heqn1; subst n0. - apply fs. assumption. - destruct y. + apply fz. + apply fs. apply thin; assumption. Defined. ( to create a transparent constant, as given by a classic Fixpoint ) (\| You can then print the value and read the lambda term to understand how to define it directly. This could give: \|) Print thin. ( .unfold .messages ) (\| The ``return`` clauses of the pattern matchings is used to solve the problem presented above: they connect the variables introduced in the branches and the ones in the context. This is discussed more deeply here: http://adam.chlipala.net/cpdt/html/MoreDep.html. Also note that this particular inductive type was discussed a few weeks ago on the coq-club mailing list. See https://sympa.inria.fr/sympa/arc/coq-club/2016-03/msg00206.html. ---- Q: This helps a lot, thanks! I should have finished Chlipala's book before actually trying to do stuff in Coq. One more question I have: is it considered bad practice to define functions in proof mode? A: I don't really know. Usually, I do that to check the lambda term and write it myself. Anyway, I try to use mostly low-level tactics to create a readable lambda term. Tactics like ``inversion`` create A: Tactics like ``inversion`` create non-natural terms and should be avoided. Maybe other users have stronger opinions on this particular subject. \|*)
% !TEX root = ../root.tex \section{Problem formulation}\label{sec:problem} The aim of the project is to design and develop a UAVCAN-compatible indoor positioning system for aerial unmanned veicles (UAVs) based on the \emph{decaWave} devices. Using these radio modules, an algorithm shall be devised and implemented in order to estimate the UAV position in a 2D coordinate system. The resulting information is to be broadcasted through the UAVCAN bus --- a message definition shall be chosen amongst the pool of standard definitions, or designed from scratch. The performance of the system ought to be in line with decaWave's claims in term of maximum accuracy and range, and the device has to be suitable for battery operation. Moreover, the system shall comply with the UAVCAN specifications and support its major standard functionalities. The project is carried out in collaboration with \emph{UAVComponents ApS}, which will lend its expertise in hardware design and provide the necessary development tools.
-- Reported by nils.anders.danielsson, Feb 17, 2015 -- See also Issue 292 , Issue 1406 , and Issue 1427. -- The code below is accepted by Agda 2.4.2.2, but not by the current -- maintenance or master branches. data Box (A : Set) : Set where [_] : A → Box A data _≡_ (A : Set) : Set → Set₁ where refl : A ≡ A data _≅_ {A : Set₁} (x : A) : {B : Set₁} → B → Set₂ where refl : x ≅ x -- C could be a typed DSEL. data C : Set → Set₁ where c₁ c₂ : (A : Set) → C (Box A) -- If A is considered forced, the code no longer type-checks. -- D could be some kind of semantics for C. data D : {A : Set} → C A → Set₂ where d₁ : (A : Set) → D (c₁ A) d₂ : (A : Set) → D (c₂ A) module Doesn't-work where -- Let's try to write an eliminator for the part of the semantics -- that concerns c₁ programs. The basic approach doesn't work: D-elim-c₁ : (P : {A : Set} → D (c₁ A) → Set₂) → ((A : Set) → P (d₁ A)) → {A : Set} (x : D (c₁ A)) → P x D-elim-c₁ P p (d₁ A) = p A -- The following trick also fails (but for some reason the absurd -- case is accepted): -- Jesper 2015-12-18 update: this is no longer accepted by the new unifier. --D-elim-c₁-helper : -- (P : {A B : Set} {c : C A} → -- D c → A ≡ Box B → c ≅ c₁ B → Set₂) → -- ((A : Set) → P (d₁ A) refl refl) → -- {A B : Set} {c : C A} -- (x : D c) (eq₂ : c ≅ c₁ B) (eq₁ : A ≡ Box B) → P x eq₁ eq₂ --D-elim-c₁-helper P p (d₂ A) () _ --D-elim-c₁-helper P p (d₁ A) refl refl = p A module Works where -- I can define the eliminators by first defining and proving no -- confusion (following McBride, Goguen and McKinna). However, this -- requires a fair amount of work, and easy dependent pattern -- matching is arguably one of the defining features of Agda. -- -- A quote from "A Few Constructions on Constructors": "The Epigram -- language and system [25, 23] takes these constructions for -- granted. We see no reason why the users of other systems should -- work harder than we do." data ⊥ : Set₁ where No-confusion : ∀ {A B} → C A → C B → Set₁ No-confusion (c₁ A) (c₁ B) = A ≡ B No-confusion (c₂ A) (c₂ B) = A ≡ B No-confusion _ _ = ⊥ no-confusion : ∀ {A B} (x : C A) (y : C B) → A ≡ B → x ≅ y → No-confusion x y no-confusion (c₁ A) .(c₁ A) refl refl = refl no-confusion (c₂ A) .(c₂ A) refl refl = refl D-elim-c₁-helper : (P : {A B : Set} {c : C A} → D c → A ≡ Box B → c ≅ c₁ B → Set₂) → ((A : Set) → P (d₁ A) refl refl) → {A B : Set} {c : C A} (x : D c) (eq₂ : c ≅ c₁ B) (eq₁ : A ≡ Box B) → P x eq₁ eq₂ D-elim-c₁-helper P p (d₁ A) eq₂ eq₁ with no-confusion _ _ eq₁ eq₂ D-elim-c₁-helper P p (d₁ B) refl refl \| refl = p B D-elim-c₁-helper P p (d₂ A) eq₂ eq₁ with no-confusion _ _ eq₁ eq₂ D-elim-c₁-helper P p (d₂ A) eq₂ eq₁ \| () cast : {A B : Set} {x : C A} {y : C B} → A ≡ B → x ≅ y → D x → D y cast refl refl x = x D-elim-c₁ : (P : {A : Set} → D (c₁ A) → Set₂) → ((A : Set) → P (d₁ A)) → {A : Set} (x : D (c₁ A)) → P x D-elim-c₁ P p x = D-elim-c₁-helper (λ x eq₁ eq₂ → P (cast eq₁ eq₂ x)) p x refl refl -- should type-check
/- Copyright (c) 2023 Huub Vromen. All rights reserved. Author: Huub Vromen -/ import data.set.basic /-- First-order semantics for Aristotle's assertoric syllogisms A first-order logic semantics is a variant of a set-theoretic semantics. See, for instance Malink (2013, ch. 3). Terms are interpreted as non-empty subsets of some set of individuals. -/ variable {α : Type} variable {x : α} variables {A B C : α → Prop} /-- semantics of the `a` relation -/ def universal_affirmative (A: α → Prop) (B: α → Prop) : Prop := ∀x, B x → A x infixr ` a ` : 80 := universal_affirmative /-- semantics of the `e` relation -/ def universal_negative (A: α → Prop) (B: α → Prop) : Prop := ∀x, B x → ¬ A x infixr ` e ` : 80 := universal_negative /-- semantics of the `i` relation -/ def particular_affirmative (A: α → Prop) (B: α → Prop) : Prop := ∃x, A x ∧ B x -- existential import needs to be stipulated infixr ` i ` : 80 := particular_affirmative /-- semantics of the `o` relation -/ def particular_negative (A: α → Prop) (B: α → Prop) : Prop := ∃x, B x ∧ ¬ A x infixr ` o ` : 80 := particular_negative /-- semantics of contradictory: contradictory is defined as negation -/ def c (p : Prop) : Prop := ¬ p /-- We prove the soundness of the axiom system DR -/ lemma Barbara₁ : A a B → B a C → A a C := begin intros h1 h2, rw universal_affirmative, { intros p h3, have h4 : B p := by exact h2 p h3, exact h1 p h4 }, end lemma Celarent₁ : A e B → B a C → A e C := begin intros h1 h2 p h3, have h4 : B p := by exact h2 p h3, exact h1 p h4 end lemma e_conv : A e B → B e A := begin intros h1 b h2, by_contra, show false, from (h1 b h) h2, end lemma a_conv (hex: ∃x, B x) : A a B → B i A := begin intro h1, rw universal_affirmative at h1, rw particular_affirmative, cases hex with p hp, apply exists.intro p (and.intro hp (h1 p hp)) end lemma contr {p r : Prop} : (c r → c p) → p → r := begin intros h1, contrapose, assumption end /-- We can also prove the contradictories axioms -/ lemma contr_a : c (A a B) = A o B := by simp [c, particular_negative, universal_affirmative] lemma contr_e : c (A e B) = A i B := begin simp [c, particular_affirmative, universal_negative], finish end lemma contr_i : c (A i B) = A e B := begin simp [c, particular_affirmative, universal_negative], finish end lemma contr_o : c (A o B) = A a B := by simp [c, particular_negative, universal_affirmative] /-- it is, of course, also possible to prove the redundant axioms -/ lemma Darii₁ : A a B → B i C → A i C := begin intros h1 h2, cases h2 with p h, apply exists.intro p, exact and.intro (h1 p h.1) h.2 end lemma Ferio₁ : A e B → B i C → A o C := begin intros h1 h2, cases h2 with p h, --rw universal_denial at h1, have h3 : ¬ A p := by exact h1 p h.1, --rw particular_denial, apply exists.intro p (and.intro h.2 h3) end lemma i_conv : A i B → B i A := begin intros h1, cases h1 with p h2, cases h2 with q r, apply exists.intro p (and.intro r q) end #lint
module Nat.Sum where open import Data.Nat open import Relation.Binary open import Relation.Binary.PropositionalEquality open DecTotalOrder decTotalOrder hiding (refl) +id : (n : ℕ) → n + zero ≡ n +id zero = refl +id (suc n) = cong suc (+id n) +assoc : (m n : ℕ) → m + suc n ≡ suc (m + n) +assoc zero n = refl +assoc (suc m) n = cong suc (+assoc m n)
lemma limitin_subsequence: "\<lbrakk>strict_mono r; limitin X f l sequentially\<rbrakk> \<Longrightarrow> limitin X (f \<circ> r) l sequentially"
If $c$ is a component of $s$, then $s - c$ is the union of all components of $s$ other than $c$.
Formal statement is: lemma Lim_transform_away_within: fixes a b :: "'a::t1_space" assumes "a \<noteq> b" and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x" and "(f \<longlongrightarrow> l) (at a within S)" shows "(g \<longlongrightarrow> l) (at a within S)" Informal statement is: If $f$ and $g$ are functions defined on a set $S$ and $f$ converges to $l$ at $a$ within $S$, then $g$ converges to $l$ at $a$ within $S$ if $f$ and $g$ agree on $S$ except possibly at $a$ and $b$.
Formal statement is: proposition maximum_modulus_frontier: assumes holf: "f holomorphic_on (interior S)" and contf: "continuous_on (closure S) f" and bos: "bounded S" and leB: "\<And>z. z \<in> frontier S \<Longrightarrow> norm(f z) \<le> B" and "\<xi> \<in> S" shows "norm(f \<xi>) \<le> B" Informal statement is: If $f$ is holomorphic on the interior of a bounded set $S$ and continuous on the closure of $S$, and if $f$ is bounded on the boundary of $S$, then $f$ is bounded on $S$.
module Tables using LinearAlgebra, DataValueInterfaces, DataAPI, TableTraits, IteratorInterfaceExtensions export rowtable, columntable if !hasmethod(getproperty, Tuple{Tuple, Int}) Base.getproperty(t::Tuple, i::Int) = t[i] end "Abstract row type with a simple required interface: row values are accessible via `getproperty(row, field)`; for example, a NamedTuple like `nt = (a=1, b=2, c=3)` can access its value for `a` like `nt.a` which turns into a call to the function `getproperty(nt, :a)`" abstract type Row end """ The Tables.jl package provides simple, yet powerful interface functions for working with all kinds tabular data through predictable access patterns. ```julia Tables.rows(table) => Rows Tables.columns(table) => Columns ``` Where `Rows` and `Columns` are the duals of each other: * `Rows` is an iterator of property-accessible objects (any type that supports `propertynames(row)` and `getproperty(row, nm::Symbol`) * `Columns` is a property-accessible object of iterators (i.e. each column is an iterator) In addition to these `Rows` and `Columns` objects, it's useful to be able to query properties of these objects: * `Tables.schema(x::Union{Rows, Columns}) => Union{Tables.Schema, Nothing}`: returns a `Tables.Schema` object, or `nothing` if the table's schema is unknown * For the `Tables.Schema` object: * column names can be accessed as a tuple of Symbols like `sch.names` * column types can be accessed as a tuple of types like `sch.types` * See `?Tables.Schema` for more details on this type A big part of the power in these simple interface functions is that each (`Tables.rows` & `Tables.columns`) is defined for any table type, even if the table type only explicitly implements one interface function or the other. This is accomplished by providing performant, generic fallback definitions in Tables.jl itself (though obviously nothing prevents a table type from implementing each interface function directly). With these simple definitions, powerful workflows are enabled: * A package providing data cleansing, manipulation, visualization, or analysis can automatically handle any number of decoupled input table types * A tabular file format can have automatic integration with in-memory structures and translation to other file formats So how does one go about satisfying the Tables.jl interface functions? It mainly depends on what you've already defined and the natural access patterns of your table: First: * `Tables.istable(::Type{<:MyTable}) = true`: this provides an explicit affirmation that your type implements the Tables interface To support `Rows`: * Define `Tables.rowaccess(::Type{<:MyTable}) = true`: this signals to other types that `MyTable` supports valid `Row`-iteration * Define `Tables.rows(x::MyTable)`: return a `Row`-iterator object (perhaps the table itself if already defined) * Define `Tables.schema(Tables.rows(x::MyTable))` to either return a `Tables.Schema` object, or `nothing` if the schema is unknown or non-inferrable for some reason To support `Columns`: * Define `Tables.columnaccess(::Type{<:MyTable}) = true`: this signals to other types that `MyTable` supports returning a valid `Columns` object * Define `Tables.columns(x::MyTable)`: return a `Columns`, property-accessible object (perhaps the table itself if it naturally supports property-access to columns) * Define `Tables.schema(Tables.columns(x::MyTable))` to either return a `Tables.Schema` object, or `nothing` if the schema is unknown or non-inferrable for some reason The final question is how `MyTable` can be a "sink" for any other table type. The answer is quite simple: use the interface functions! * Define a function or constructor that takes, at a minimum, a single, untyped argument and then calls `Tables.rows` or `Tables.columns` on that argument to construct an instance of `MyTable` For example, if `MyTable` is a row-oriented format, I might define my "sink" function like: ```julia function MyTable(x) Tables.istable(x) \|\| throw(ArgumentError("MyTable requires a table input")) rows = Tables.rows(x) sch = Tables.schema(rows) names = sch.names types = sch.types # custom constructor that creates an "empty" MyTable according to given column names & types # note that the "unknown" schema case should be considered, i.e. when `sch.types => nothing` mytbl = MyTable(names, types) for row in rows # a convenience function provided in Tables.jl for "unrolling" access to each column/property of a `Row` # it works by applying a provided function to each value; see `?Tables.eachcolumn` for more details Tables.eachcolumn(sch, row) do val, col, name push!(mytbl[col], val) end end return mytbl end ``` Alternatively, if `MyTable` is column-oriented, perhaps my definition would be more like: ```julia function MyTable(x) Tables.istable(x) \|\| throw(ArgumentError("MyTable requires a table input")) cols = Tables.columns(x) # here we use Tables.eachcolumn to iterate over each column in a `Columns` object return MyTable(collect(propertynames(cols)), [collect(col) for col in Tables.eachcolumn(cols)]) end ``` Obviously every table type is different, but via a combination of `Tables.rows` and `Tables.columns` each table type should be able to construct an instance of itself. """ abstract type Table end # default definitions istable(x::T) where {T} = istable(T) \|\| TableTraits.isiterabletable(x) === true istable(::Type{T}) where {T} = false rowaccess(x::T) where {T} = rowaccess(T) rowaccess(::Type{T}) where {T} = false columnaccess(x::T) where {T} = columnaccess(T) columnaccess(::Type{T}) where {T} = false schema(x) = nothing materializer(x) = columntable # Schema implementation """ Tables.Schema(names, types) Create a `Tables.Schema` object that holds the column names and types for a tabular data object. `Tables.Schema` is dual-purposed: provide an easy interface for users to query these properties, as well as provide a convenient "structural" type for code generation. To get a table's schema, one can call `Tables.schema(tbl)`, but also note that a table may return `nothing`, indicating that it's column names and/or column types are unknown (usually not inferrable). This is similar to the `Base.EltypeUnknown()` trait for iterators when `Base.IteratorEltype` is called. Users should account for the `Tables.schema(tbl) => nothing` case by using the properties of the results of `Tables.rows(x)` and `Tables.columns(x)` directly. To access the names, one can simply call `sch.names` to return the tuple of Symbols. To access column types, one can similarly call `sch.types`, which will return a tuple of types (like `(Int64, Float64, String)`). The actual type definition is ```julia struct Schema{names, types} end ``` Where `names` is a tuple of Symbols, and `types` is a tuple _type_ of types (like `Tuple{Int64, Float64, String}`). Encoding the names & types as type parameters allows convenient use of the type in generated functions and other optimization use-cases. """ struct Schema{names, types} end Schema(names::Tuple{Vararg{Symbol}}, types::Type{T}) where {T <: Tuple} = Schema{names, T}() Schema(::Type{NamedTuple{names, types}}) where {names, types} = Schema{names, types}() # pass through Ints to allow Tuples to act as rows sym(x) = Symbol(x) sym(x::Int) = x Schema(names, ::Nothing) = Schema{Tuple(Base.map(sym, names)), nothing}() Schema(names, types) = Schema{Tuple(Base.map(sym, names)), Tuple{types...}}() function Base.show(io::IO, sch::Schema{names, types}) where {names, types} println(io, "Tables.Schema:") Base.print_matrix(io, hcat(collect(names), types === nothing ? fill(nothing, length(names)) : collect(fieldtype(types, i) for i = 1:fieldcount(types)))) end function Base.getproperty(sch::Schema{names, types}, field::Symbol) where {names, types} if field === :names return names elseif field === :types return types === nothing ? nothing : Tuple(fieldtype(types, i) for i = 1:fieldcount(types)) else throw(ArgumentError("unsupported property for Tables.Schema")) end end Base.propertynames(sch::Schema) = (:names, :types) # helper functions include("utils.jl") # reference implementations: Vector of NamedTuples and NamedTuple of Vectors include("namedtuples.jl") # generic fallback definitions include("fallbacks.jl") # allow any valid iterator to be a table include("tofromdatavalues.jl") # simple table operations on table inputs include("operations.jl") # matrix integration include("matrix.jl") "Return the column index (1-based) of a `colname` in a table with a known schema; returns 0 if `colname` doesn't exist in table" columnindex(table, colname) = columnindex(schema(table).names, colname) "Return the column type of a `colname` in a table with a known schema; returns Union{} if `colname` doesn't exist in table" columntype(table, colname) = columntype(schema(table), colname) end # module
(** * System Fsub subtyping without arrow types ) Require Export unscoped. Reserved Notation " Γ \|-' s <: t" (at level 68, s, t at next level). (* ** Syntax of types ) Inductive type : Type := \| var_type : fin -> type \| top : type \| all : type -> type -> type. Fixpoint ren_type (ξ : fin -> fin) (t : type) : type := match t return type with \| var_type n => var_type (ξ n) \| top => top \| all t0 t1 => all (ren_type ξ t0) (ren_type (up_ren ξ) t1) end. Fixpoint subst_type (θ : fin -> type) (t : type) : type := match t return type with \| var_type n => θ n \| top => top \| all t0 t1 => all (subst_type θ t0) (subst_type (var_type 0 .: θ >> ren_type ↑) t1) end. (* ** Subtyping relation ) Inductive sub' (Γ : list type) : type -> type -> Prop := \| Refl τ : Γ \|-' τ <: τ \| Trans σ τ υ : Γ \|-' σ <: τ -> Γ \|-' τ <: υ -> Γ \|-' σ <: υ \| Top τ : Γ \|-' τ <: top \| Var n : Γ \|-' var_type n <: nth_default (var_type n) Γ n \| All σ1 σ2 τ1 τ2 : Γ \|-' τ1 <: σ1 -> map (ren_type ↑) (τ1 :: Γ) \|-' σ2 <: τ2 -> Γ \|-' (all σ1 σ2) <: (all τ1 τ2) where "Γ \|-' σ <: τ" := (sub' Γ σ τ). (* ** Subtyping problem ) Definition Fsub'_SUBTYPE : (list type (type * type)) -> Prop := fun ctt => let (Γ, tt) := ctt in let (σ, τ) := tt in Γ \|-' σ <: τ.
module SemVar.Parser import SemVar.Lexer import SemVar.Data import Text.Parser import Text.Token import Data.List import public SemVar.Tokens %default total release : Grammar _ SemVarToken True String release = do match Hyphen match Text metadata : Grammar _ SemVarToken True String metadata = do match Plus match Text dotOrDefault : Grammar _ SemVarToken False Int dotOrDefault = option 0 ( do match Dot match Number ) version : Grammar _ SemVarToken True Version version = do major <- match Number minor <- dotOrDefault patch <- dotOrDefault release <- optional release metadata <- optional metadata pure (MkVersion major minor patch release metadata) tilde : Grammar _ SemVarToken True Requirement tilde = do match Tilde v <- version pure $ AND (GTE v) (LT $ nextMinor v) pin : Grammar _ SemVarToken True Requirement pin = do match Caret v <- version pure $ case v of MkVersion 0 0 patch Nothing Nothing => EQ v _ => AND (GTE v) (LT $ nextMajor v) exact : Grammar _ SemVarToken True Requirement exact = do ignore $ optional (match CmpEQ) v <- version pure $ EQ v gt : Grammar _ SemVarToken True Requirement gt = do ignore $ optional (match CmpGT) v <- version pure $ GT v lt : Grammar _ SemVarToken True Requirement lt = do ignore $ optional (match CmpLT) v <- version pure $ LT v gte : Grammar _ SemVarToken True Requirement gte = do ignore $ optional (match CmpGTE) v <- version pure $ GTE v lte : Grammar _ SemVarToken True Requirement lte = do ignore $ optional (match CmpLTE) v <- version pure $ LTE v range : Grammar _ SemVarToken True Requirement range = do v0 <- version ignore $ optional (match Whitespace) match Hyphen ignore $ optional (match Whitespace) v1 <- version pure $ AND (GTE v0) (LTE v1) simpleRequirement : Grammar _ SemVarToken True Requirement simpleRequirement = ( range <\|> tilde <\|> pin <\|> exact <\|> gte <\|> gt <\|> lte <\|> lt ) conj : Grammar _ SemVarToken True Requirement conj = do v0 <- simpleRequirement match Whitespace v1 <- simpleRequirement pure $ AND v0 v1 disjuction : Grammar _ SemVarToken True Requirement disjuction = do v0 <- simpleRequirement ignore $ optional (match Whitespace) ignore $ match Pipe ignore $ optional (match Whitespace) v1 <- simpleRequirement pure $ OR v0 v1 requirement : Grammar _ SemVarToken True Requirement requirement = ( conj <\|> disjuction <\|> simpleRequirement ) export parseVersionToks : List (WithBounds SemVarToken) -> Maybe Version parseVersionToks toks = case parse version toks of Right (j, []) => Just j _ => Nothing export parseRequirementToks : List (WithBounds SemVarToken) -> Maybe Requirement parseRequirementToks toks = case parse requirement toks of Right (j, []) => Just j _ => Nothing
State Before: α : Type ?u.8872 β : Type u_2 ι : Type ?u.8878 E : Type u_1 F : Type ?u.8884 𝕜 : Type ?u.8887 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ ⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ‖f x - y₀‖₊ State After: α : Type ?u.8872 β : Type u_2 ι : Type ?u.8878 E : Type u_1 F : Type ?u.8884 𝕜 : Type ?u.8887 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ this : edist (↑(approxOn f hf s y₀ h₀ n) x) (f x) ≤ edist y₀ (f x) ⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ‖f x - y₀‖₊ Tactic: have := edist_approxOn_le hf h₀ x n State Before: α : Type ?u.8872 β : Type u_2 ι : Type ?u.8878 E : Type u_1 F : Type ?u.8884 𝕜 : Type ?u.8887 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ this : edist (↑(approxOn f hf s y₀ h₀ n) x) (f x) ≤ edist y₀ (f x) ⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ‖f x - y₀‖₊ State After: α : Type ?u.8872 β : Type u_2 ι : Type ?u.8878 E : Type u_1 F : Type ?u.8884 𝕜 : Type ?u.8887 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ this : edist (↑(approxOn f hf s y₀ h₀ n) x) (f x) ≤ edist (f x) y₀ ⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ‖f x - y₀‖₊ Tactic: rw [edist_comm y₀] at this State Before: α : Type ?u.8872 β : Type u_2 ι : Type ?u.8878 E : Type u_1 F : Type ?u.8884 𝕜 : Type ?u.8887 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ this : edist (↑(approxOn f hf s y₀ h₀ n) x) (f x) ≤ edist (f x) y₀ ⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ‖f x - y₀‖₊ State After: α : Type ?u.8872 β : Type u_2 ι : Type ?u.8878 E : Type u_1 F : Type ?u.8884 𝕜 : Type ?u.8887 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ this : ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ↑‖f x - y₀‖₊ ⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ‖f x - y₀‖₊ Tactic: simp only [edist_nndist, nndist_eq_nnnorm] at this State Before: α : Type ?u.8872 β : Type u_2 ι : Type ?u.8878 E : Type u_1 F : Type ?u.8884 𝕜 : Type ?u.8887 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ this : ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ↑‖f x - y₀‖₊ ⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ‖f x - y₀‖₊ State After: no goals Tactic: exact_mod_cast this
```python %matplotlib inline ``` # Bayesian optimization with `skopt` Gilles Louppe, Manoj Kumar July 2016. Reformatted by Holger Nahrstaedt 2020 .. currentmodule:: skopt ## Problem statement We are interested in solving \begin{align}x^* = arg \min_x f(x)\end{align} under the constraints that - $f$ is a black box for which no closed form is known (nor its gradients); - $f$ is expensive to evaluate; - and evaluations of $y = f(x)$ may be noisy. Disclaimer. If you do not have these constraints, then there is certainly a better optimization algorithm than Bayesian optimization. This example uses :class:`plots.plot_gaussian_process` which is available since version 0.8. ## Bayesian optimization loop For $t=1:T$: 1. Given observations $(x_i, y_i=f(x_i))$ for $i=1:t$, build a probabilistic model for the objective $f$. Integrate out all possible true functions, using Gaussian process regression. 2. optimize a cheap acquisition/utility function $u$ based on the posterior distribution for sampling the next point. $x_{t+1} = arg \min_x u(x)$ Exploit uncertainty to balance exploration against exploitation. 3. Sample the next observation $y_{t+1}$ at $x_{t+1}$. ## Acquisition functions Acquisition functions $u(x)$ specify which sample $x$: should be tried next: - Expected improvement (default): $-EI(x) = -\mathbb{E} [f(x) - f(x_t^+)]$ - Lower confidence bound: $LCB(x) = \mu_{GP}(x) + \kappa \sigma_{GP}(x)$ - Probability of improvement: $-PI(x) = -P(f(x) \geq f(x_t^+) + \kappa)$ where $x_t^+$ is the best point observed so far. In most cases, acquisition functions provide knobs (e.g., $\kappa$) for controlling the exploration-exploitation trade-off. - Search in regions where $\mu_{GP}(x)$ is high (exploitation) - Probe regions where uncertainty $\sigma_{GP}(x)$ is high (exploration) ```python print(__doc__) import numpy as np np.random.seed(237) import matplotlib.pyplot as plt from skopt.plots import plot_gaussian_process ``` ## Toy example Let assume the following noisy function $f$: ```python noise_level = 0.1 def f(x, noise_level=noise_level): return np.sin(5 * x[0]) * (1 - np.tanh(x[0] ** 2))\ + np.random.randn() * noise_level ``` Note. In `skopt`, functions $f$ are assumed to take as input a 1D vector $x$: represented as an array-like and to return a scalar $f(x)$:. ```python # Plot f(x) + contours x = np.linspace(-2, 2, 400).reshape(-1, 1) fx = [f(x_i, noise_level=0.0) for x_i in x] plt.plot(x, fx, "r--", label="True (unknown)") plt.fill(np.concatenate([x, x[::-1]]), np.concatenate(([fx_i - 1.9600 * noise_level for fx_i in fx], [fx_i + 1.9600 * noise_level for fx_i in fx[::-1]])), alpha=.2, fc="r", ec="None") plt.legend() plt.grid() plt.show() ``` Bayesian optimization based on gaussian process regression is implemented in :class:`gp_minimize` and can be carried out as follows: ```python from skopt import gp_minimize res = gp_minimize(f, # the function to minimize [(-2.0, 2.0)], # the bounds on each dimension of x acq_func="EI", # the acquisition function n_calls=15, # the number of evaluations of f n_random_starts=5, # the number of random initialization points noise=0.1*2, # the noise level (optional) random_state=1234) # the random seed ``` Accordingly, the approximated minimum is found to be: ```python "x^=%.4f, f(x^)=%.4f" % (res.x[0], res.fun) ``` For further inspection of the results, attributes of the `res` named tuple provide the following information: - `x` [float]: location of the minimum. - `fun` [float]: function value at the minimum. - `models`: surrogate models used for each iteration. - `x_iters` [array]: location of function evaluation for each iteration. - `func_vals` [array]: function value for each iteration. - `space` [Space]: the optimization space. - `specs` [dict]: parameters passed to the function. ```python print(res) ``` Together these attributes can be used to visually inspect the results of the minimization, such as the convergence trace or the acquisition function at the last iteration: ```python from skopt.plots import plot_convergence plot_convergence(res); ``` Let us now visually examine 1. The approximation of the fit gp model to the original function. 2. The acquisition values that determine the next point to be queried. ```python plt.rcParams["figure.figsize"] = (8, 14) def f_wo_noise(x): return f(x, noise_level=0) ``` Plot the 5 iterations following the 5 random points ```python for n_iter in range(5): # Plot true function. plt.subplot(5, 2, 2n_iter+1) if n_iter == 0: show_legend = True else: show_legend = False ax = plot_gaussian_process(res, n_calls=n_iter, objective=f_wo_noise, noise_level=noise_level, show_legend=show_legend, show_title=False, show_next_point=False, show_acq_func=False) ax.set_ylabel("") ax.set_xlabel("") # Plot EI(x) plt.subplot(5, 2, 2*n_iter+2) ax = plot_gaussian_process(res, n_calls=n_iter, show_legend=show_legend, show_title=False, show_mu=False, show_acq_func=True, show_observations=False, show_next_point=True) ax.set_ylabel("") ax.set_xlabel("") plt.show() ``` The first column shows the following: 1. The true function. 2. The approximation to the original function by the gaussian process model 3. How sure the GP is about the function. The second column shows the acquisition function values after every surrogate model is fit. It is possible that we do not choose the global minimum but a local minimum depending on the minimizer used to minimize the acquisition function. At the points closer to the points previously evaluated at, the variance dips to zero. Finally, as we increase the number of points, the GP model approaches the actual function. The final few points are clustered around the minimum because the GP does not gain anything more by further exploration: ```python plt.rcParams["figure.figsize"] = (6, 4) # Plot f(x) + contours _ = plot_gaussian_process(res, objective=f_wo_noise, noise_level=noise_level) plt.show() ```
source("pattern_to_number.r") computing_frequencies <- function(text, k) { frequency_array <- array(0, 4^k) for(i in 0:(nchar(text) - k)) { pattern <- substr(text, i + 1, i + k) j <- pattern_to_number(pattern) frequency_array[j + 1] <- frequency_array[j + 1] + 1 } return(frequency_array) } # message("Text") # text <- scan("stdin", what="character", nlines=1) # message("k") # k <- scan("stdin", nlines=1) # # cat(computing_frequencies(text, k))
C @(#)sys100.f 20.3 2/13/96 subroutine sys100 C include 'ipfinc/parametr.inc' include 'ipfinc/alpha.inc' include 'ipfinc/blank.inc' include 'ipfinc/bus.inc' include 'ipfinc/dc2t.inc' include 'ipfinc/dcmt.inc' include 'ipfinc/ikk.inc' include 'ipfinc/merge.inc' include 'ipfinc/mrgsys.inc' include 'ipfinc/prt.inc' include 'ipfinc/qksrt.inc' include 'ipfinc/red2.inc' dimension mtrx(MAXBUS) character ktrpos1,own13,own23 integer find_bus external find_bus, kpface, spface logical found ksy = ksy +1 C Establish connection matrix: add 2-terminal dc to y-matrix do i = 1,kdtot k1 = dc2t(1,i) k2 = dc2t(3,i) kt = inp2opt(k1) mt = inp2opt(k2) C Add branch (KT,MT) to y-matrix do isw = 1, 2 ln = km(kt) - 1 ls = kmlen(kt) do l = 1, ls if (ikmu(l+ln) .eq. mt) go to 102 enddo c c Appending an entity to km() requires relocating branch c list to end of km(). c ln = km(kt) - 1 ls = kmlen(kt) do l = 1, ls ikmu(l+yptr) = ikmu(l+ln) enddo ikmu(ls+yptr+1) = mt yptr = yptr + ls + 1 km(kt) = yptr kmlen(kt) = ls + 1 102 continue kt = inp2opt(k2) ! swap kt and mt mt = inp2opt(k1) enddo enddo C Add N-terminal d-c to y-matrix do i=1,mtdcln k1=dcmtln(1,i) k2=dcmtln(2,i) kt = inp2opt(k1) mt = inp2opt(k2) do isw = 1, 2 ln = km(kt) - 1 ls = kmlen(kt) do l = 1, ls if (ikmu(l+ln) .eq. mt) go to 104 enddo c c Appending an entity to km() requires relocating branch c list to end of km(). c ln = km(kt) - 1 ls = kmlen(kt) do l = 1, ls ikmu(l+yptr) = ikmu(l+ln) enddo ikmu(ls+yptr+1) = mt km(kt) = yptr + 1 kmlen(kt) = ls + 1 yptr = yptr + ls + 1 104 continue kt = inp2opt(k2) ! swap kt and mt mt = inp2opt(k1) enddo enddo C c IKK(1,) = 0 - bus is in eliminated subsystem. c 1 - bus is in retained subsystem. c c IKK(2,) = 0 - bus is not specifically classified. c 1 - bus is specifically classified. C c IKK(3,) = isystm - subsystem number C c IKKIND(1,) = is - starting KOLUM() index c IKKIND(2,) = il - KOLUM length C itot=1 do kt=1,ntot k=inp2opt(kt) jtot=0 do l = km(k), km(k)-1+kmlen(k) mt = ikmu(l) kolum(itot+jtot)=opt2inp(mt) jtot=jtot+1 enddo ikk(1,kt)=1 ikk(2,kt)=0 ikk(3,kt)=0 ikk(4,kt)=0 ikkind(1,kt)=itot ikkind(2,kt)=jtot itot=itot+jtot enddo if (itot .gt. MAXYE) then write (errbuf(1),330) itot, MAXYE 330 format ('Total branches in system (including transposes)', 1 'is ',i5,'. Limit is ',i5,'.') call prterx ('F',1) kerrsw = kerrsw + 1 endif C Identify merge subsystem ntotx=max0(nbsys1,nbsys2) do 370 i=1,ntotx kt = find_bus(mrgbus(i),mrgbas(i)) if (kt .le. 0) then if (mrgbus(i) .ne. srtlst) then write (errbuf(1),350) mrgbus(i),mrgbas(i) 350 format ('Merged subsystem bus (',a8,f6.1, & ') is not in system. Bus ignored. ') call prterx ('W',1) endif else ikk(1,kt)=0 endif 370 continue C C Identify all interface branches C nsave=0 do 380 i=1,ntot if (ikk(1,i).eq.1) nsave=nsave+1 380 continue if (nsave .eq. 0) then write (errbuf(1),390) 390 format ('0 Caution - Merge and Base system are identical. ', & 'Interface ignored. ') call prterx ('W',1) go to 580 endif nsyst=0 itface=0 na=0 nl=0 do while (nl .le. nsave) c c Search for kernel node KT to prime topological emanation c kt = 1 found = .false. do while (kt .le. ntot .and. .not. found) if (ikk(1,kt) .eq. 1 .and. ikk(3,kt) .eq. 0) then found = .true. else kt = kt + 1 endif enddo if (.not. found) then write (errbuf(1), 400) 400 format ('Base merge failed to establish interface ') call prterx ('F',1) kerrsw = kerrsw + 1 go to 900 endif c c Add node KT as first node in subsystem NSYST c na=na+1 mtrx(na)=kt nl=na nsyst=nsyst+1 ikk(3,kt)=nsyst do while (nl .le. na) kt = mtrx(nl) i5 = ikkind(1,kt) i6 = i5+ikkind(2,kt)-1 do 470 l=i5,i6 mt=kolum(l) if (ikk(1,mt) .eq. 0) then ikk(2,kt)=1 ikk(2,mt)=1 do 430 i=1,itface if (face(1,i) .eq. kt .and. & face(2,i) .eq. mt) go to 470 430 continue itface=itface+1 if (itface .ge. 399) then write (errbuf(1),450) bus(kt), base(kt), bus(mt), & base(mt) 450 format (' More that 400 merge interface branches', & ' branch ignored :(',a8,f6.1, 2x,a8,f6.1,')') call prterx ('W',1) itface = 399 kerrsw = kerrsw + 1 else face(1,itface)=kt face(2,itface)=mt facec(itface)='0' // owner(kt) // owner(mt) face(4,itface)=nsyst itface=itface+1 face(1,itface)=mt face(2,itface)=kt facec (itface)= '1' // owner(mt) // owner(kt) face(4,itface)=nsyst endif else if (ikk(3,mt) .eq. 0) then c c Add node MT as subsequent node in subsystem NSYST c na=na+1 mtrx(na)=mt ikk(3,mt)=nsyst endif 470 continue nl=nl+1 enddo enddo if (nl .lt. nsave) then write (errbuf(1), 480) 480 format ('Base merge failed with an incomplete enclosure ') call prterx ('F',1) kerrsw = kerrsw + 1 go to 900 endif C END PROCEDURE 490 continue C SUMMARIZE "FACE" ARRAY BY SUBSYSTEMS if (itface .eq. 0) go to 580 if (itface .gt. 1) then key=1 call qiksrt(1,itface,kpface,spface) endif nf=1 do 570 is=1,nsyst write (outbuf,520) is 520 format ('0 Summary of Interface Branches for Subsystem ',i2) call prtout(1) write (outbuf,530) 530 format ('0 Base System Merge System Owner1 Owner2 ') call prtout(1) do 550 i=nf,itface if (face(4,i).ne.is) go to 560 kt=face(1,i) mt=face(2,i) ktrpos=facec(i)(1:1) own1=facec(i)(2:4) own2=facec(i)(5:7) if (ktrpos .ne. '1') then write (outbuf,540) bus(kt), base(kt), bus(mt), base(mt), & own1,own2 540 format (2x,a8,f6.1,2x,a8,f6.1,3x,a3,5x,a3) call prtout(1) endif 550 continue i=itface+1 560 nf=i 570 continue 580 continue 900 continue return end
Defender Economy Safety Glasses provide medium impact protection, anti-scratch lenses and 99.9% UV protection. They are an economical choice that offers a mix of protection and comfort. Meets the Australian Standard AS/NZS 1337.1:2010 ensuring the best optical protection in most work environments.
State Before: 𝕜 : Type u_1 inst✝⁸ : NontriviallyNormedField 𝕜 E : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace 𝕜 E F : Type u_3 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace 𝕜 F G : Type ?u.1390092 inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G G' : Type ?u.1390187 inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f f₀ f₁ g : E → F f' f₀' f₁' g' e : E →L[𝕜] F x : E s t : Set E L L₁ L₂ : Filter E c : F hxs : UniqueDiffWithinAt 𝕜 s x ⊢ fderivWithin 𝕜 (fun x => c) s x = 0 State After: 𝕜 : Type u_1 inst✝⁸ : NontriviallyNormedField 𝕜 E : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace 𝕜 E F : Type u_3 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace 𝕜 F G : Type ?u.1390092 inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G G' : Type ?u.1390187 inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f f₀ f₁ g : E → F f' f₀' f₁' g' e : E →L[𝕜] F x : E s t : Set E L L₁ L₂ : Filter E c : F hxs : UniqueDiffWithinAt 𝕜 s x ⊢ fderiv 𝕜 (fun x => c) x = 0 Tactic: rw [DifferentiableAt.fderivWithin (differentiableAt_const _) hxs] State Before: 𝕜 : Type u_1 inst✝⁸ : NontriviallyNormedField 𝕜 E : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace 𝕜 E F : Type u_3 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace 𝕜 F G : Type ?u.1390092 inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G G' : Type ?u.1390187 inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f f₀ f₁ g : E → F f' f₀' f₁' g' e : E →L[𝕜] F x : E s t : Set E L L₁ L₂ : Filter E c : F hxs : UniqueDiffWithinAt 𝕜 s x ⊢ fderiv 𝕜 (fun x => c) x = 0 State After: no goals Tactic: exact fderiv_const_apply _
using IntervalCensored using Test @testset "IntervalCensored.jl" begin # Write your tests here. end
import numpy as np from sklearn_import.fixes import _parse_version np_version = _parse_version(np.__version__) def sparse_min_max(X, axis): return (X.min(axis=axis).toarray().ravel(), X.max(axis=axis).toarray().ravel())
= = = = Trip to Europe = = = =
create_output_files <- function(JOBID, max_date) { three_panel <- readRDS(paste0("usa/results/", "three-panel-data-",JOBID,".RDS")) infectious <- readRDS(paste0("usa/results/", "infectious-individuals-out-",JOBID,".RDS")) model_output <- merge(three_panel, infectious, by = c('state','date')) write.csv(model_output, paste0("usa/results/", "model-output-",max_date,".csv"), row.names = FALSE) death_scenario <- readRDS(paste0('usa/results/deaths-scenario-out-',JOBID,'.RDS')) mobzero <- death_scenario[death_scenario$key=='Constant mobility',] mob20 <- death_scenario[death_scenario$key=='Increased mobility 20%',] mob40 <- death_scenario[death_scenario$key=='Increased mobility 40%',] mobzero <- mobzero[,c("state", "date","estimated_deaths","deaths_min","deaths_max")] mob20 <- mob20[,c("state", "date","estimated_deaths","deaths_min","deaths_max")] mob40 <- mob40[,c("state", "date","estimated_deaths","deaths_min","deaths_max")] colnames(mobzero) <- c("state", "date","constant_mobility_estimated_deaths_mean","constant_mobility_estimated_deaths_lower_CI_95","constant_mobility_estimated_deaths_higher_CI_95") colnames(mob20) <- c("state", "date","mobility_increase_20_estimated_deaths_mean","mobility_increase_20_estimated_deaths_lower_CI_95","mobility_increase_20_estimated_deaths_higher_CI_95") colnames(mob40) <- c("state", "date","mobility_increase_40__estimated_deaths_mean","mobility_increase_40__estimated_deaths_lower_CI_95","mobility_increase_40_estimated_deaths_higher_CI_95") deaths_scenarios <- merge(mobzero, mob20,by = c('state', 'date')) deaths_scenarios <- merge(deaths_scenarios, mob40,by = c('state', 'date')) deaths_scenarios <- deaths_scenarios[deaths_scenarios$date > max_date,] write.csv(deaths_scenarios, file = paste0('usa/results/deaths-scenarios-',max_date,'.csv'), row.names = FALSE) cases_scenario <- readRDS(paste0('usa/results/cases-scenario-out-',JOBID,'.RDS')) mobzero <- cases_scenario[cases_scenario$key=='Constant mobility',] mob20 <- cases_scenario[cases_scenario$key=='Increased mobility 20%',] mob40 <- cases_scenario[cases_scenario$key=='Increased mobility 40%',] mobzero <- mobzero[,c("state", "date","predicted_cases","cases_min","cases_max")] mob20 <- mob20[,c("state", "date","predicted_cases","cases_min","cases_max")] mob40 <- mob40[,c("state", "date","predicted_cases","cases_min","cases_max")] colnames(mobzero) <- c("state", "date","constant_mobility_estimated_infections_mean","constant_mobility_estimated_infections_lower_CI_95","constant_mobility_estimated_infections_higher_CI_95") colnames(mob20) <- c("state", "date","mobility_increase_20_estimated_infections_mean","mobility_increase_20_estimated_infections_lower_CI_95","mobility_increase_20_estimated_infections_higher_CI_95") colnames(mob40) <- c("state", "date","mobility_increase_40__estimated_infections_mean","mobility_increase_40__estimated_infections_lower_CI_95","mobility_increase_40_estimated_infections_higher_CI_95") cases_scenarios <- merge(mobzero, mob20,by = c('state', 'date')) cases_scenarios <- merge(cases_scenarios, mob40,by = c('state', 'date')) cases_scenarios <- cases_scenarios[cases_scenarios$date > max_date,] write.csv(cases_scenarios, file = paste0('usa/results/infections-scenarios-',max_date,'.csv'), row.names = FALSE) rt_scenario <- readRDS(paste0('usa/results/rt-scenario-out-',JOBID,'.RDS')) mobzero <- rt_scenario[rt_scenario$key=='Constant mobility',] mob20 <- rt_scenario[rt_scenario$key=='Increased mobility 20%',] mob40 <- rt_scenario[rt_scenario$key=='Increased mobility 40%',] mobzero <- mobzero[,c("state", "date","rt","rt_min","rt_max")] mob20 <- mob20[,c("state", "date","rt","rt_min","rt_max")] mob40 <- mob40[,c("state", "date","rt","rt_min","rt_max")] colnames(mobzero) <- c("state", "date","constant_mobility_mean_time_varying_reproduction_number_R(t)","constant_mobility_time_varying_reproduction_number_R(t)_lower_CI_95","time_varying_reproduction_number_R(t)_Higher_CI_95") colnames(mob20) <- c("state", "date","mobility_increase_20_mean_time_varying_reproduction_number_R(t)","mobility_increase_20_time_varying_reproduction_number_R(t)_lower_CI_95","mobility_increase_20_time_varying_reproduction_number_R(t)_Higher_CI_95") colnames(mob40) <- c("state", "date","mobility_increase_20_mean_time_varying_reproduction_number_R(t)","mobility_increase_40_time_varying_reproduction_number_R(t)_lower_CI_95","mobility_increase_40_time_varying_reproduction_number_R(t)_Higher_CI_95") rt_scenarios <- merge(mobzero, mob20,by = c('state', 'date')) rt_scenarios <- merge(rt_scenarios, mob40,by = c('state', 'date')) rt_scenarios <- rt_scenarios[rt_scenarios$date > max_date,] write.csv(rt_scenarios, file = paste0('usa/results/time-varying-reproduction-number-scenarios-',max_date,'.csv'), row.names = FALSE) }
[STATEMENT] lemma OclIsKindOf\<^sub>O\<^sub>c\<^sub>l\<^sub>A\<^sub>n\<^sub>y_Person_strict2[simp] : "(null::Person) .oclIsKindOf(OclAny) = true" [PROOF STATE] proof (prove) goal (1 subgoal): 1. null.oclIsKindOf(OclAny) = true [PROOF STEP] by(rule ext, simp add: null_fun_def null_option_def bot_option_def OclIsKindOf\<^sub>O\<^sub>c\<^sub>l\<^sub>A\<^sub>n\<^sub>y_Person)
-- Andreas, 2013-11-07 -- Instance candidates are now considered module judgemental equality. module Issue899 where postulate A B : Set f : {{ x : A }} → B instance a : A instance a' : A a' = a test : B test = f {- The previous code fails with the following message: Resolve implicit argument _x_257 : A. Candidates: [a : A, a : A] There are indeed two values in scope of type A (a and a'), but given that they are definitionally equal, Agda should not complain about it but just pick any one of them. -}
Require Import Raft. Require Import RaftRefinementInterface. Section VotedForMoreUpToDate. Context {orig_base_params : BaseParams}. Context {one_node_params : OneNodeParams orig_base_params}. Context {raft_params : RaftParams orig_base_params}. Definition votedFor_moreUpToDate (net : network) : Prop := forall t h h', currentTerm (snd (nwState net h)) = t -> type (snd (nwState net h)) = Candidate -> votedFor (snd (nwState net h')) = Some h -> currentTerm (snd (nwState net h')) = t -> exists vl, moreUpToDate (maxTerm (log (snd (nwState net h)))) (maxIndex (log (snd (nwState net h)))) (maxTerm vl) (maxIndex vl) = true /\ In (t, h, vl) (votesWithLog (fst (nwState net h'))). Class votedFor_moreUpToDate_interface : Prop := { votedFor_moreUpToDate_invariant : forall net, refined_raft_intermediate_reachable net -> votedFor_moreUpToDate net }. End VotedForMoreUpToDate.
[STATEMENT] lemma infs_suffix: "infs A w \<longleftrightarrow> (\<forall> u v. w = u @- v \<longrightarrow> sset v \<inter> A \<noteq> {})" [PROOF STATE] proof (prove) goal (1 subgoal): 1. infs A w = (\<forall>u v. w = u @- v \<longrightarrow> sset v \<inter> A \<noteq> {}) [PROOF STEP] using alwD alw_iff_sdrop alw_shift ev_HLD_sset stake_sdrop [PROOF STATE] proof (prove) using this: alw ?P ?x \<Longrightarrow> ?P ?x alw ?P ?\<omega> = (\<forall>m. ?P (sdrop m ?\<omega>)) alw ?\<phi> (?xl @- ?xs) \<Longrightarrow> alw ?\<phi> ?xs ev (HLD ?A) ?w = (sset ?w \<inter> ?A \<noteq> {}) stake ?n ?s @- sdrop ?n ?s = ?s goal (1 subgoal): 1. infs A w = (\<forall>u v. w = u @- v \<longrightarrow> sset v \<inter> A \<noteq> {}) [PROOF STEP] by metis
[GOAL] d : ℕ h : d ≠ 0 w : Nat.coprime (Int.natAbs 0) d ⊢ mk' 0 d = 0 [PROOFSTEP] congr [GOAL] q : ℚ ⊢ q.num = 0 ↔ q = 0 [PROOFSTEP] induction q [GOAL] case mk' num✝ : ℤ den✝ : ℕ den_nz✝ : den✝ ≠ 0 reduced✝ : Nat.coprime (Int.natAbs num✝) den✝ ⊢ (mk' num✝ den✝).num = 0 ↔ mk' num✝ den✝ = 0 [PROOFSTEP] constructor [GOAL] case mk'.mp num✝ : ℤ den✝ : ℕ den_nz✝ : den✝ ≠ 0 reduced✝ : Nat.coprime (Int.natAbs num✝) den✝ ⊢ (mk' num✝ den✝).num = 0 → mk' num✝ den✝ = 0 [PROOFSTEP] rintro rfl [GOAL] case mk'.mp den✝ : ℕ den_nz✝ : den✝ ≠ 0 reduced✝ : Nat.coprime (Int.natAbs 0) den✝ ⊢ mk' 0 den✝ = 0 [PROOFSTEP] exact zero_mk _ _ _ [GOAL] case mk'.mpr num✝ : ℤ den✝ : ℕ den_nz✝ : den✝ ≠ 0 reduced✝ : Nat.coprime (Int.natAbs num✝) den✝ ⊢ mk' num✝ den✝ = 0 → (mk' num✝ den✝).num = 0 [PROOFSTEP] exact congr_arg num [GOAL] a b : ℤ b0 : b ≠ 0 ⊢ a /. b = 0 ↔ a = 0 [PROOFSTEP] rw [← zero_divInt b, divInt_eq_iff b0 b0, zero_mul, mul_eq_zero, or_iff_left b0] [GOAL] C : ℚ → Sort u n : ℤ d : ℕ h : d ≠ 0 c : Nat.coprime (Int.natAbs n) d H : (n : ℤ) → (d : ℕ) → 0 < d → Nat.coprime (Int.natAbs n) d → C (n /. ↑d) ⊢ C (mk' n d) [PROOFSTEP] rw [num_den'] [GOAL] C : ℚ → Sort u n : ℤ d : ℕ h : d ≠ 0 c : Nat.coprime (Int.natAbs n) d H : (n : ℤ) → (d : ℕ) → 0 < d → Nat.coprime (Int.natAbs n) d → C (n /. ↑d) ⊢ C (n /. ↑d) [PROOFSTEP] exact H n d (Nat.pos_of_ne_zero h) c [GOAL] f : ℚ → ℚ → ℚ f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ fv : ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : Nat.coprime (Int.natAbs n₁) d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0} {c₂ : Nat.coprime (Int.natAbs n₂) d₂}, f (mk' n₁ d₁) (mk' n₂ d₂) = f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂ f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0 a b c d : ℤ b0 : b ≠ 0 d0 : d ≠ 0 H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂ ⊢ f (a /. b) (c /. d) = f₁ a b c d /. f₂ a b c d [PROOFSTEP] generalize ha : a /. b = x [GOAL] f : ℚ → ℚ → ℚ f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ fv : ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : Nat.coprime (Int.natAbs n₁) d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0} {c₂ : Nat.coprime (Int.natAbs n₂) d₂}, f (mk' n₁ d₁) (mk' n₂ d₂) = f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂ f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0 a b c d : ℤ b0 : b ≠ 0 d0 : d ≠ 0 H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂ x : ℚ ha : a /. b = x ⊢ f x (c /. d) = f₁ a b c d /. f₂ a b c d [PROOFSTEP] cases' x with n₁ d₁ h₁ c₁ [GOAL] case mk' f : ℚ → ℚ → ℚ f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ fv : ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : Nat.coprime (Int.natAbs n₁) d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0} {c₂ : Nat.coprime (Int.natAbs n₂) d₂}, f (mk' n₁ d₁) (mk' n₂ d₂) = f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂ f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0 a b c d : ℤ b0 : b ≠ 0 d0 : d ≠ 0 H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂ n₁ : ℤ d₁ : ℕ h₁ : d₁ ≠ 0 c₁ : Nat.coprime (Int.natAbs n₁) d₁ ha : a /. b = mk' n₁ d₁ ⊢ f (mk' n₁ d₁) (c /. d) = f₁ a b c d /. f₂ a b c d [PROOFSTEP] rw [num_den'] at ha [GOAL] case mk' f : ℚ → ℚ → ℚ f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ fv : ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : Nat.coprime (Int.natAbs n₁) d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0} {c₂ : Nat.coprime (Int.natAbs n₂) d₂}, f (mk' n₁ d₁) (mk' n₂ d₂) = f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂ f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0 a b c d : ℤ b0 : b ≠ 0 d0 : d ≠ 0 H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂ n₁ : ℤ d₁ : ℕ h₁ : d₁ ≠ 0 c₁ : Nat.coprime (Int.natAbs n₁) d₁ ha : a /. b = n₁ /. ↑d₁ ⊢ f (mk' n₁ d₁) (c /. d) = f₁ a b c d /. f₂ a b c d [PROOFSTEP] generalize hc : c /. d = x [GOAL] case mk' f : ℚ → ℚ → ℚ f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ fv : ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : Nat.coprime (Int.natAbs n₁) d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0} {c₂ : Nat.coprime (Int.natAbs n₂) d₂}, f (mk' n₁ d₁) (mk' n₂ d₂) = f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂ f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0 a b c d : ℤ b0 : b ≠ 0 d0 : d ≠ 0 H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂ n₁ : ℤ d₁ : ℕ h₁ : d₁ ≠ 0 c₁ : Nat.coprime (Int.natAbs n₁) d₁ ha : a /. b = n₁ /. ↑d₁ x : ℚ hc : c /. d = x ⊢ f (mk' n₁ d₁) x = f₁ a b c d /. f₂ a b c d [PROOFSTEP] cases' x with n₂ d₂ h₂ c₂ [GOAL] case mk'.mk' f : ℚ → ℚ → ℚ f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ fv : ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : Nat.coprime (Int.natAbs n₁) d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0} {c₂ : Nat.coprime (Int.natAbs n₂) d₂}, f (mk' n₁ d₁) (mk' n₂ d₂) = f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂ f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0 a b c d : ℤ b0 : b ≠ 0 d0 : d ≠ 0 H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂ n₁ : ℤ d₁ : ℕ h₁ : d₁ ≠ 0 c₁ : Nat.coprime (Int.natAbs n₁) d₁ ha : a /. b = n₁ /. ↑d₁ n₂ : ℤ d₂ : ℕ h₂ : d₂ ≠ 0 c₂ : Nat.coprime (Int.natAbs n₂) d₂ hc : c /. d = mk' n₂ d₂ ⊢ f (mk' n₁ d₁) (mk' n₂ d₂) = f₁ a b c d /. f₂ a b c d [PROOFSTEP] rw [num_den'] at hc [GOAL] case mk'.mk' f : ℚ → ℚ → ℚ f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ fv : ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : Nat.coprime (Int.natAbs n₁) d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0} {c₂ : Nat.coprime (Int.natAbs n₂) d₂}, f (mk' n₁ d₁) (mk' n₂ d₂) = f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂ f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0 a b c d : ℤ b0 : b ≠ 0 d0 : d ≠ 0 H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂ n₁ : ℤ d₁ : ℕ h₁ : d₁ ≠ 0 c₁ : Nat.coprime (Int.natAbs n₁) d₁ ha : a /. b = n₁ /. ↑d₁ n₂ : ℤ d₂ : ℕ h₂ : d₂ ≠ 0 c₂ : Nat.coprime (Int.natAbs n₂) d₂ hc : c /. d = n₂ /. ↑d₂ ⊢ f (mk' n₁ d₁) (mk' n₂ d₂) = f₁ a b c d /. f₂ a b c d [PROOFSTEP] rw [fv] [GOAL] case mk'.mk' f : ℚ → ℚ → ℚ f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ fv : ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : Nat.coprime (Int.natAbs n₁) d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0} {c₂ : Nat.coprime (Int.natAbs n₂) d₂}, f (mk' n₁ d₁) (mk' n₂ d₂) = f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂ f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0 a b c d : ℤ b0 : b ≠ 0 d0 : d ≠ 0 H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂ n₁ : ℤ d₁ : ℕ h₁ : d₁ ≠ 0 c₁ : Nat.coprime (Int.natAbs n₁) d₁ ha : a /. b = n₁ /. ↑d₁ n₂ : ℤ d₂ : ℕ h₂ : d₂ ≠ 0 c₂ : Nat.coprime (Int.natAbs n₂) d₂ hc : c /. d = n₂ /. ↑d₂ ⊢ f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂ = f₁ a b c d /. f₂ a b c d [PROOFSTEP] have d₁0 := ne_of_gt (Int.ofNat_lt.2 $ Nat.pos_of_ne_zero h₁) [GOAL] case mk'.mk' f : ℚ → ℚ → ℚ f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ fv : ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : Nat.coprime (Int.natAbs n₁) d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0} {c₂ : Nat.coprime (Int.natAbs n₂) d₂}, f (mk' n₁ d₁) (mk' n₂ d₂) = f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂ f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0 a b c d : ℤ b0 : b ≠ 0 d0 : d ≠ 0 H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂ n₁ : ℤ d₁ : ℕ h₁ : d₁ ≠ 0 c₁ : Nat.coprime (Int.natAbs n₁) d₁ ha : a /. b = n₁ /. ↑d₁ n₂ : ℤ d₂ : ℕ h₂ : d₂ ≠ 0 c₂ : Nat.coprime (Int.natAbs n₂) d₂ hc : c /. d = n₂ /. ↑d₂ d₁0 : ↑d₁ ≠ ↑0 ⊢ f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂ = f₁ a b c d /. f₂ a b c d [PROOFSTEP] have d₂0 := ne_of_gt (Int.ofNat_lt.2 $ Nat.pos_of_ne_zero h₂) [GOAL] case mk'.mk' f : ℚ → ℚ → ℚ f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ fv : ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : Nat.coprime (Int.natAbs n₁) d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0} {c₂ : Nat.coprime (Int.natAbs n₂) d₂}, f (mk' n₁ d₁) (mk' n₂ d₂) = f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂ f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0 a b c d : ℤ b0 : b ≠ 0 d0 : d ≠ 0 H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂ n₁ : ℤ d₁ : ℕ h₁ : d₁ ≠ 0 c₁ : Nat.coprime (Int.natAbs n₁) d₁ ha : a /. b = n₁ /. ↑d₁ n₂ : ℤ d₂ : ℕ h₂ : d₂ ≠ 0 c₂ : Nat.coprime (Int.natAbs n₂) d₂ hc : c /. d = n₂ /. ↑d₂ d₁0 : ↑d₁ ≠ ↑0 d₂0 : ↑d₂ ≠ ↑0 ⊢ f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂ = f₁ a b c d /. f₂ a b c d [PROOFSTEP] exact (divInt_eq_iff (f0 d₁0 d₂0) (f0 b0 d0)).2 (H ((divInt_eq_iff b0 d₁0).1 ha) ((divInt_eq_iff d0 d₂0).1 hc)) [GOAL] a b c : ℚ n : ℤ d : ℕ h : d ≠ 0 ⊢ n /. ↑d + 0 = n /. ↑d [PROOFSTEP] rw [← zero_divInt d, add_def'', zero_mul, add_zero, divInt_mul_right] [GOAL] a b c : ℚ n : ℤ d : ℕ h : d ≠ 0 ⊢ ↑d ≠ 0 [PROOFSTEP] simp [h] [GOAL] case b0 a b c : ℚ n : ℤ d : ℕ h : d ≠ 0 ⊢ ↑d ≠ 0 [PROOFSTEP] simp [h] [GOAL] case d0 a b c : ℚ n : ℤ d : ℕ h : d ≠ 0 ⊢ ↑d ≠ 0 [PROOFSTEP] simp [h] [GOAL] a b c : ℚ n : ℤ d : ℕ h : d ≠ 0 ⊢ 0 + n /. ↑d = n /. ↑d [PROOFSTEP] rw [← zero_divInt d, add_def'', zero_mul, zero_add, divInt_mul_right] [GOAL] a b c : ℚ n : ℤ d : ℕ h : d ≠ 0 ⊢ ↑d ≠ 0 [PROOFSTEP] simp [h] [GOAL] case b0 a b c : ℚ n : ℤ d : ℕ h : d ≠ 0 ⊢ ↑d ≠ 0 [PROOFSTEP] simp [h] [GOAL] case d0 a b c : ℚ n : ℤ d : ℕ h : d ≠ 0 ⊢ ↑d ≠ 0 [PROOFSTEP] simp [h] [GOAL] a b c : ℚ n₁ : ℤ d₁ : ℕ h₁ : d₁ ≠ 0 n₂ : ℤ d₂ : ℕ h₂ : d₂ ≠ 0 ⊢ n₁ /. ↑d₁ + n₂ /. ↑d₂ = n₂ /. ↑d₂ + n₁ /. ↑d₁ [PROOFSTEP] simp [h₁, h₂, add_comm, mul_comm] [GOAL] a b c : ℚ n₁ : ℤ d₁ : ℕ h₁ : d₁ ≠ 0 n₂ : ℤ d₂ : ℕ h₂ : d₂ ≠ 0 n₃ : ℤ d₃ : ℕ h₃ : d₃ ≠ 0 ⊢ n₁ /. ↑d₁ + n₂ /. ↑d₂ + n₃ /. ↑d₃ = n₁ /. ↑d₁ + (n₂ /. ↑d₂ + n₃ /. ↑d₃) [PROOFSTEP] simp [h₁, h₂, h₃] [GOAL] a b c : ℚ n₁ : ℤ d₁ : ℕ h₁ : d₁ ≠ 0 n₂ : ℤ d₂ : ℕ h₂ : d₂ ≠ 0 n₃ : ℤ d₃ : ℕ h₃ : d₃ ≠ 0 ⊢ ((n₁ * ↑d₂ + n₂ * ↑d₁) * ↑d₃ + n₃ * (↑d₁ * ↑d₂)) /. (↑d₁ * ↑d₂ * ↑d₃) = (n₁ * (↑d₂ * ↑d₃) + (n₂ * ↑d₃ + n₃ * ↑d₂) * ↑d₁) /. (↑d₁ * (↑d₂ * ↑d₃)) [PROOFSTEP] rw [mul_assoc, add_mul, add_mul, mul_assoc, add_assoc] [GOAL] a b c : ℚ n₁ : ℤ d₁ : ℕ h₁ : d₁ ≠ 0 n₂ : ℤ d₂ : ℕ h₂ : d₂ ≠ 0 n₃ : ℤ d₃ : ℕ h₃ : d₃ ≠ 0 ⊢ (n₁ * (↑d₂ * ↑d₃) + (n₂ * ↑d₁ * ↑d₃ + n₃ * (↑d₁ * ↑d₂))) /. (↑d₁ * (↑d₂ * ↑d₃)) = (n₁ * (↑d₂ * ↑d₃) + (n₂ * ↑d₃ * ↑d₁ + n₃ * ↑d₂ * ↑d₁)) /. (↑d₁ * (↑d₂ * ↑d₃)) [PROOFSTEP] congr 2 [GOAL] case e_a.e_a a b c : ℚ n₁ : ℤ d₁ : ℕ h₁ : d₁ ≠ 0 n₂ : ℤ d₂ : ℕ h₂ : d₂ ≠ 0 n₃ : ℤ d₃ : ℕ h₃ : d₃ ≠ 0 ⊢ n₂ * ↑d₁ * ↑d₃ + n₃ * (↑d₁ * ↑d₂) = n₂ * ↑d₃ * ↑d₁ + n₃ * ↑d₂ * ↑d₁ [PROOFSTEP] ac_rfl [GOAL] a b c : ℚ n : ℤ d : ℕ h : d ≠ 0 ⊢ -(n /. ↑d) + n /. ↑d = 0 [PROOFSTEP] simp [h, mkRat_add_mkRat] [GOAL] a b c : ℚ ⊢ 0 /. 1 = 0 [PROOFSTEP] rw [divInt] [GOAL] a b c : ℚ ⊢ (match 0, 1 with \| n, Int.ofNat d => inline (mkRat n d) \| n, Int.negSucc d => normalize (-n) (Nat.succ d)) = 0 [PROOFSTEP] simp [GOAL] a b c : ℚ ⊢ 1 /. 1 = 1 [PROOFSTEP] rw [divInt] [GOAL] a b c : ℚ ⊢ (match 1, 1 with \| n, Int.ofNat d => inline (mkRat n d) \| n, Int.negSucc d => normalize (-n) (Nat.succ d)) = 1 [PROOFSTEP] simp [GOAL] a b c : ℚ ⊢ -1 /. 1 = -1 [PROOFSTEP] rw [divInt] [GOAL] a b c : ℚ ⊢ (match -1, 1 with \| n, Int.ofNat d => inline (mkRat n d) \| n, Int.negSucc d => normalize (-n) (Nat.succ d)) = -1 [PROOFSTEP] simp [GOAL] a b c : ℚ n : ℤ ⊢ n /. 1 = ↑n [PROOFSTEP] rw [divInt] [GOAL] a b c : ℚ n : ℤ ⊢ (match n, 1 with \| n, Int.ofNat d => inline (mkRat n d) \| n, Int.negSucc d => normalize (-n) (Nat.succ d)) = ↑n [PROOFSTEP] simp [mkRat, normalize] [GOAL] a b c : ℚ n : ℤ ⊢ mk' n 1 = ↑n [PROOFSTEP] rfl [GOAL] a b c : ℚ n : ℤ ⊢ mkRat n 1 = ↑n [PROOFSTEP] simp [Rat.mkRat_eq, Rat.divInt_one] [GOAL] a b c : ℚ n₁ : ℤ d₁ : ℕ h₁ : d₁ ≠ 0 n₂ : ℤ d₂ : ℕ h₂ : d₂ ≠ 0 n₃ : ℤ d₃ : ℕ h₃ : d₃ ≠ 0 ⊢ n₁ /. ↑d₁ * (n₂ /. ↑d₂) * (n₃ /. ↑d₃) = n₁ /. ↑d₁ * (n₂ /. ↑d₂ * (n₃ /. ↑d₃)) [PROOFSTEP] simp [h₁, h₂, h₃, mul_ne_zero, mul_comm, mul_assoc, mul_left_comm] [GOAL] a b c : ℚ n₁ : ℤ d₁ : ℕ h₁ : d₁ ≠ 0 n₂ : ℤ d₂ : ℕ h₂ : d₂ ≠ 0 n₃ : ℤ d₃ : ℕ h₃ : d₃ ≠ 0 ⊢ (n₁ /. ↑d₁ + n₂ /. ↑d₂) * (n₃ /. ↑d₃) = n₁ /. ↑d₁ * (n₃ /. ↑d₃) + n₂ /. ↑d₂ * (n₃ /. ↑d₃) [PROOFSTEP] simp [h₁, h₂, h₃, mul_ne_zero] [GOAL] a b c : ℚ n₁ : ℤ d₁ : ℕ h₁ : d₁ ≠ 0 n₂ : ℤ d₂ : ℕ h₂ : d₂ ≠ 0 n₃ : ℤ d₃ : ℕ h₃ : d₃ ≠ 0 ⊢ (n₁ * ↑d₂ + n₂ * ↑d₁) * n₃ /. (↑d₁ * ↑d₂ * ↑d₃) = (n₁ * n₃ * (↑d₂ * ↑d₃) + n₂ * n₃ * (↑d₁ * ↑d₃)) /. (↑d₁ * ↑d₃ * (↑d₂ * ↑d₃)) [PROOFSTEP] rw [← divInt_mul_right (Int.coe_nat_ne_zero.2 h₃), add_mul, add_mul] [GOAL] a b c : ℚ n₁ : ℤ d₁ : ℕ h₁ : d₁ ≠ 0 n₂ : ℤ d₂ : ℕ h₂ : d₂ ≠ 0 n₃ : ℤ d₃ : ℕ h₃ : d₃ ≠ 0 ⊢ (n₁ * ↑d₂ * n₃ * ↑d₃ + n₂ * ↑d₁ * n₃ * ↑d₃) /. (↑d₁ * ↑d₂ * ↑d₃ * ↑d₃) = (n₁ * n₃ * (↑d₂ * ↑d₃) + n₂ * n₃ * (↑d₁ * ↑d₃)) /. (↑d₁ * ↑d₃ * (↑d₂ * ↑d₃)) [PROOFSTEP] ac_rfl [GOAL] a b c : ℚ ⊢ a * (b + c) = a * b + a * c [PROOFSTEP] rw [Rat.mul_comm, Rat.add_mul, Rat.mul_comm, Rat.mul_comm c a] [GOAL] a b c : ℚ ⊢ 0 ≠ 1 [PROOFSTEP] rw [ne_comm, ← divInt_one_one, divInt_ne_zero one_ne_zero] [GOAL] a b c : ℚ ⊢ 1 ≠ 0 [PROOFSTEP] exact one_ne_zero [GOAL] a b c : ℚ n : ℤ d : ℕ h : d ≠ 0 a0 : n /. ↑d ≠ 0 ⊢ n /. ↑d * (n /. ↑d)⁻¹ = 1 [PROOFSTEP] have n0 : n ≠ 0 := mt (by rintro rfl; simp) a0 [GOAL] a b c : ℚ n : ℤ d : ℕ h : d ≠ 0 a0 : n /. ↑d ≠ 0 ⊢ n = 0 → n /. ↑d = 0 [PROOFSTEP] rintro rfl [GOAL] a b c : ℚ d : ℕ h : d ≠ 0 a0 : 0 /. ↑d ≠ 0 ⊢ 0 /. ↑d = 0 [PROOFSTEP] simp [GOAL] a b c : ℚ n : ℤ d : ℕ h : d ≠ 0 a0 : n /. ↑d ≠ 0 n0 : n ≠ 0 ⊢ n /. ↑d * (n /. ↑d)⁻¹ = 1 [PROOFSTEP] simpa [h, n0, mul_comm] using @divInt_mul_right 1 1 (n * d) (by simp [h, n0]) [GOAL] a b c : ℚ n : ℤ d : ℕ h : d ≠ 0 a0 : n /. ↑d ≠ 0 n0 : n ≠ 0 ⊢ n * ↑d ≠ 0 [PROOFSTEP] simp [h, n0] [GOAL] a b c : ℚ n : ℕ ⊢ NatCast.natCast (n + 1) = NatCast.natCast n + 1 [PROOFSTEP] simp only [coe_int_eq_divInt, add_def'' one_ne_zero one_ne_zero, ← divInt_one_one, Nat.cast_add, Nat.cast_one, mul_one] [GOAL] a b c : ℚ ⊢ 0⁻¹ = 0 [PROOFSTEP] change Rat.inv 0 = 0 [GOAL] a b c : ℚ ⊢ Rat.inv 0 = 0 [PROOFSTEP] rw [Rat.inv_def] [GOAL] a b c : ℚ ⊢ ↑0.den /. 0.num = 0 [PROOFSTEP] rfl [GOAL] a b c : ℚ ⊢ Nontrivial ℚ [PROOFSTEP] infer_instance [GOAL] a b c : ℚ ⊢ CommSemiring ℚ [PROOFSTEP] infer_instance [GOAL] a b c : ℚ ⊢ Semiring ℚ [PROOFSTEP] infer_instance [GOAL] a b c : ℚ ⊢ AddCommGroup ℚ [PROOFSTEP] infer_instance [GOAL] a b c : ℚ ⊢ AddGroup ℚ [PROOFSTEP] infer_instance [GOAL] a b c : ℚ ⊢ AddCommMonoid ℚ [PROOFSTEP] infer_instance [GOAL] a b c : ℚ ⊢ AddMonoid ℚ [PROOFSTEP] infer_instance [GOAL] a b c : ℚ ⊢ AddLeftCancelSemigroup ℚ [PROOFSTEP] infer_instance [GOAL] a b c : ℚ ⊢ AddRightCancelSemigroup ℚ [PROOFSTEP] infer_instance [GOAL] a b c : ℚ ⊢ AddCommSemigroup ℚ [PROOFSTEP] infer_instance [GOAL] a b c : ℚ ⊢ AddSemigroup ℚ [PROOFSTEP] infer_instance [GOAL] a b c : ℚ ⊢ CommMonoid ℚ [PROOFSTEP] infer_instance [GOAL] a b c : ℚ ⊢ Monoid ℚ [PROOFSTEP] infer_instance [GOAL] a b c : ℚ ⊢ CommSemigroup ℚ [PROOFSTEP] infer_instance [GOAL] a b c : ℚ ⊢ Semigroup ℚ [PROOFSTEP] infer_instance [GOAL] a b c p q : ℚ ⊢ p = q ↔ p.num * ↑q.den = q.num * ↑p.den [PROOFSTEP] conv => lhs rw [← @num_den p, ← @num_den q] [GOAL] a b c p q : ℚ \| p = q ↔ p.num * ↑q.den = q.num * ↑p.den [PROOFSTEP] lhs rw [← @num_den p, ← @num_den q] [GOAL] a b c p q : ℚ \| p = q ↔ p.num * ↑q.den = q.num * ↑p.den [PROOFSTEP] lhs rw [← @num_den p, ← @num_den q] [GOAL] a b c p q : ℚ \| p = q ↔ p.num * ↑q.den = q.num * ↑p.den [PROOFSTEP] lhs [GOAL] a b c p q : ℚ \| p = q [PROOFSTEP] rw [← @num_den p, ← @num_den q] [GOAL] a b c p q : ℚ ⊢ p.num /. ↑p.den = q.num /. ↑q.den ↔ p.num * ↑q.den = q.num * ↑p.den [PROOFSTEP] apply Rat.divInt_eq_iff [GOAL] case z₁ a b c p q : ℚ ⊢ ↑p.den ≠ 0 [PROOFSTEP] rw [← Nat.cast_zero, Ne, Int.ofNat_inj] [GOAL] case z₁ a b c p q : ℚ ⊢ ¬p.den = 0 [PROOFSTEP] apply den_nz [GOAL] case z₂ a b c p q : ℚ ⊢ ↑q.den ≠ 0 [PROOFSTEP] rw [← Nat.cast_zero, Ne, Int.ofNat_inj] [GOAL] case z₂ a b c p q : ℚ ⊢ ¬q.den = 0 [PROOFSTEP] apply den_nz [GOAL] a b c q : ℚ hq : q.num = 0 ⊢ q = 0 [PROOFSTEP] have : q = q.num /. q.den := num_den.symm [GOAL] a b c q : ℚ hq : q.num = 0 this : q = q.num /. ↑q.den ⊢ q = 0 [PROOFSTEP] simpa [hq] using this [GOAL] a b c q : ℚ x✝ : q = 0 ⊢ q.num = 0 [PROOFSTEP] simp [] [GOAL] a b c q : ℚ n d : ℤ hq : q ≠ 0 hqnd : q = n /. d this : n = 0 ⊢ q = 0 [PROOFSTEP] simpa [this] using hqnd [GOAL] a b c q : ℚ n d : ℤ hq : q ≠ 0 hqnd : q = n /. d this : d = 0 ⊢ q = 0 [PROOFSTEP] simpa [this] using hqnd [GOAL] a b c q r : ℚ ⊢ q r = q.num * r.num /. ↑(q.den * r.den) [PROOFSTEP] have hq' : (↑q.den : ℤ) ≠ 0 := by have := den_nz q; simpa [GOAL] a b c q r : ℚ ⊢ ↑q.den ≠ 0 [PROOFSTEP] have := den_nz q [GOAL] a b c q r : ℚ this : q.den ≠ 0 ⊢ ↑q.den ≠ 0 [PROOFSTEP] simpa [GOAL] a b c q r : ℚ hq' : ↑q.den ≠ 0 ⊢ q * r = q.num * r.num /. ↑(q.den * r.den) [PROOFSTEP] have hr' : (↑r.den : ℤ) ≠ 0 := by have := den_nz r; simpa [GOAL] a b c q r : ℚ hq' : ↑q.den ≠ 0 ⊢ ↑r.den ≠ 0 [PROOFSTEP] have := den_nz r [GOAL] a b c q r : ℚ hq' : ↑q.den ≠ 0 this : r.den ≠ 0 ⊢ ↑r.den ≠ 0 [PROOFSTEP] simpa [GOAL] a b c q r : ℚ hq' : ↑q.den ≠ 0 hr' : ↑r.den ≠ 0 ⊢ q * r = q.num * r.num /. ↑(q.den * r.den) [PROOFSTEP] suffices q.num /. ↑q.den * (r.num /. ↑r.den) = q.num * r.num /. ↑(q.den * r.den) by simpa [num_den] using this [GOAL] a b c q r : ℚ hq' : ↑q.den ≠ 0 hr' : ↑r.den ≠ 0 this : q.num /. ↑q.den * (r.num /. ↑r.den) = q.num * r.num /. ↑(q.den * r.den) ⊢ q * r = q.num * r.num /. ↑(q.den * r.den) [PROOFSTEP] simpa [num_den] using this [GOAL] a b c q r : ℚ hq' : ↑q.den ≠ 0 hr' : ↑r.den ≠ 0 ⊢ q.num /. ↑q.den * (r.num /. ↑r.den) = q.num * r.num /. ↑(q.den * r.den) [PROOFSTEP] simp [mul_def' hq' hr'] [GOAL] a b c q r : ℚ hr : r.num = 0 ⊢ q / r = q.num * ↑r.den /. (↑q.den * r.num) [PROOFSTEP] have hr' : r = 0 := zero_of_num_zero hr [GOAL] a b c q r : ℚ hr : r.num = 0 hr' : r = 0 ⊢ q / r = q.num * ↑r.den /. (↑q.den * r.num) [PROOFSTEP] simp [] [GOAL] a b c q r : ℚ hr : ¬r.num = 0 ⊢ q r⁻¹ = q.num /. ↑q.den * (r.num /. ↑r.den)⁻¹ [PROOFSTEP] simp [num_den] [GOAL] a b c q r : ℚ hr : ¬r.num = 0 ⊢ q.num /. ↑q.den * (r.num /. ↑r.den)⁻¹ = q.num /. ↑q.den * (↑r.den /. r.num) [PROOFSTEP] rw [inv_def'] [GOAL] a b c q r : ℚ hr : ¬r.num = 0 ⊢ ↑q.den ≠ 0 [PROOFSTEP] simpa using den_nz q [GOAL] a✝ b✝ c✝ : ℚ a b c : ℤ h : c = 0 ⊢ (a + b) /. c = a /. c + b /. c [PROOFSTEP] simp [h] [GOAL] a✝ b✝ c✝ : ℚ a b c : ℤ h : ¬c = 0 ⊢ (a + b) /. c = a /. c + b /. c [PROOFSTEP] rw [add_def'' h h, divInt_eq_iff h (mul_ne_zero h h)] [GOAL] a✝ b✝ c✝ : ℚ a b c : ℤ h : ¬c = 0 ⊢ (a + b) * (c * c) = (a * c + b * c) * c [PROOFSTEP] simp [add_mul, mul_assoc] [GOAL] a b c : ℚ n d : ℤ ⊢ n /. d = ↑n / ↑d [PROOFSTEP] by_cases d0 : d = 0 [GOAL] case pos a b c : ℚ n d : ℤ d0 : d = 0 ⊢ n /. d = ↑n / ↑d [PROOFSTEP] simp [d0, div_zero] [GOAL] case neg a b c : ℚ n d : ℤ d0 : ¬d = 0 ⊢ n /. d = ↑n / ↑d [PROOFSTEP] simp [division_def, coe_int_eq_divInt, mul_def' one_ne_zero d0] [GOAL] a b c : ℚ x : ℤ hx : x ≠ 0 n d : ℤ ⊢ n /. x * (x /. d) = n /. d [PROOFSTEP] by_cases hd : d = 0 [GOAL] case pos a b c : ℚ x : ℤ hx : x ≠ 0 n d : ℤ hd : d = 0 ⊢ n /. x * (x /. d) = n /. d [PROOFSTEP] rw [hd] [GOAL] case pos a b c : ℚ x : ℤ hx : x ≠ 0 n d : ℤ hd : d = 0 ⊢ n /. x * (x /. 0) = n /. 0 [PROOFSTEP] simp [GOAL] case neg a b c : ℚ x : ℤ hx : x ≠ 0 n d : ℤ hd : ¬d = 0 ⊢ n /. x * (x /. d) = n /. d [PROOFSTEP] rw [mul_def' hx hd, mul_comm x, divInt_mul_right hx] [GOAL] a b c : ℚ x : ℤ hx : x ≠ 0 n d : ℤ ⊢ n /. x / (d /. x) = n /. d [PROOFSTEP] rw [div_eq_mul_inv, inv_def', divInt_mul_divInt_cancel hx] [GOAL] a b c : ℚ x : ℤ hx : x ≠ 0 n d : ℤ ⊢ x /. n / (x /. d) = d /. n [PROOFSTEP] rw [div_eq_mul_inv, inv_def', mul_comm, divInt_mul_divInt_cancel hx] [GOAL] a b c : ℚ n d : ℤ ⊢ ↑n / ↑d = n /. d [PROOFSTEP] repeat' rw [coe_int_eq_divInt] [GOAL] a b c : ℚ n d : ℤ ⊢ ↑n / ↑d = n /. d [PROOFSTEP] rw [coe_int_eq_divInt] [GOAL] a b c : ℚ n d : ℤ ⊢ n /. 1 / ↑d = n /. d [PROOFSTEP] rw [coe_int_eq_divInt] [GOAL] a b c : ℚ n d : ℤ ⊢ n /. 1 / (d /. 1) = n /. d [PROOFSTEP] rw [coe_int_eq_divInt] [GOAL] a b c : ℚ n d : ℤ ⊢ n /. 1 / (d /. 1) = n /. d [PROOFSTEP] exact divInt_div_divInt_cancel_left one_ne_zero n d [GOAL] a b c r : ℚ ⊢ ↑r.num / ↑r.den = r [PROOFSTEP] rw [← Int.cast_ofNat] [GOAL] a b c r : ℚ ⊢ ↑r.num / ↑↑r.den = r [PROOFSTEP] erw [← divInt_eq_div, num_den] [GOAL] a b c q : ℚ hq : q.den = 1 ⊢ ↑q.num = q [PROOFSTEP] conv_rhs => rw [← @num_den q, hq] [GOAL] a b c q : ℚ hq : q.den = 1 \| q [PROOFSTEP] rw [← @num_den q, hq] [GOAL] a b c q : ℚ hq : q.den = 1 \| q [PROOFSTEP] rw [← @num_den q, hq] [GOAL] a b c q : ℚ hq : q.den = 1 \| q [PROOFSTEP] rw [← @num_den q, hq] [GOAL] a b c q : ℚ hq : q.den = 1 ⊢ ↑q.num = q.num /. ↑1 [PROOFSTEP] rw [coe_int_eq_divInt] [GOAL] a b c q : ℚ hq : q.den = 1 ⊢ q.num /. 1 = q.num /. ↑1 [PROOFSTEP] rfl [GOAL] a b c : ℚ n : ℕ ⊢ ↑n = ↑n /. 1 [PROOFSTEP] rw [← Int.cast_ofNat, coe_int_eq_divInt] [GOAL] a b c : ℚ n : ℕ ⊢ (↑n).num = ↑n [PROOFSTEP] rw [← Int.cast_ofNat, coe_int_num] [GOAL] a b c : ℚ n : ℕ ⊢ (↑n).den = 1 [PROOFSTEP] rw [← Int.cast_ofNat, coe_int_den] [GOAL] a b c : ℚ n : ℤ d : ℕ ⊢ mkRat n d = ↑n / ↑d [PROOFSTEP] simp [mkRat] [GOAL] a b c : ℚ n : ℤ d : ℕ ⊢ (if h : d = 0 then 0 else normalize n d) = ↑n / ↑d [PROOFSTEP] by_cases d = 0 [GOAL] a b c : ℚ n : ℤ d : ℕ ⊢ (if h : d = 0 then 0 else normalize n d) = ↑n / ↑d [PROOFSTEP] by_cases d = 0 [GOAL] case pos a b c : ℚ n : ℤ d : ℕ h : d = 0 ⊢ (if h : d = 0 then 0 else normalize n d) = ↑n / ↑d [PROOFSTEP] simp [h] [GOAL] case neg a b c : ℚ n : ℤ d : ℕ h : ¬d = 0 ⊢ (if h : d = 0 then 0 else normalize n d) = ↑n / ↑d [PROOFSTEP] simp [h, HDiv.hDiv, Rat.div, Div.div] [GOAL] case neg a b c : ℚ n : ℤ d : ℕ h : ¬d = 0 ⊢ normalize n d = ↑n * Rat.inv ↑d [PROOFSTEP] unfold Rat.inv [GOAL] case neg a b c : ℚ n : ℤ d : ℕ h : ¬d = 0 ⊢ normalize n d = ↑n * if h : (↑d).num < 0 then mk' (-↑(↑d).den) (Int.natAbs (↑d).num) else if h : (↑d).num > 0 then mk' (↑(↑d).den) (Int.natAbs (↑d).num) else ↑d [PROOFSTEP] have h₁ : 0 < d := Nat.pos_iff_ne_zero.2 h [GOAL] case neg a b c : ℚ n : ℤ d : ℕ h : ¬d = 0 h₁ : 0 < d ⊢ normalize n d = ↑n * if h : (↑d).num < 0 then mk' (-↑(↑d).den) (Int.natAbs (↑d).num) else if h : (↑d).num > 0 then mk' (↑(↑d).den) (Int.natAbs (↑d).num) else ↑d [PROOFSTEP] have h₂ : ¬(d : ℤ) < 0 := by simp [GOAL] a b c : ℚ n : ℤ d : ℕ h : ¬d = 0 h₁ : 0 < d ⊢ ¬↑d < 0 [PROOFSTEP] simp [GOAL] case neg a b c : ℚ n : ℤ d : ℕ h : ¬d = 0 h₁ : 0 < d h₂ : ¬↑d < 0 ⊢ normalize n d = ↑n * if h : (↑d).num < 0 then mk' (-↑(↑d).den) (Int.natAbs (↑d).num) else if h : (↑d).num > 0 then mk' (↑(↑d).den) (Int.natAbs (↑d).num) else ↑d [PROOFSTEP] simp [h, h₁, h₂, ← Rat.normalize_eq_mk', Rat.normalize_eq_mkRat, ← mkRat_one, Rat.mkRat_mul_mkRat]
[STATEMENT] lemma addfunsetD_supp : "f \<in> addfunset A M \<Longrightarrow> supp f \<subseteq> A" [PROOF STATE] proof (prove) goal (1 subgoal): 1. f \<in> addfunset A M \<Longrightarrow> supp f \<subseteq> A [PROOF STEP] unfolding addfunset_def [PROOF STATE] proof (prove) goal (1 subgoal): 1. f \<in> {f. supp f \<subseteq> A \<and> range f \<subseteq> M \<and> (\<forall>x\<in>A. \<forall>y\<in>A. f (x + y) = f x + f y)} \<Longrightarrow> supp f \<subseteq> A [PROOF STEP] by fast
#ifndef SB_HPP_DEF__ #define SB_HPP_DEF__ #include<rexio/tk/toolkit.h++> #include <string> #include <map> #include <boost/function.hpp> //! displays basic information //! about... anything class StatusBar: public Scr::Tk::Widget { private: //! message static std::string Status; public: typedef std::map<Scr::Uint,boost::function<void ()> > KeyMap; void OnRedraw(Scr::Screen &screen)throw(); //!status is global inline static void SetStatus(std::string s) { Status.assign(s); } void OnKeyDown(Scr::Key key)throw(); }; #endif
module Specdris.ConsoleColor %access export %default total \|\|\| Console colors public export data Color = Red \| Green \| Yellow \| Blue \| Magenta \| Cyan \| White Eq Color where (==) Red Red = True (==) Green Green = True (==) Yellow Yellow = True (==) Blue Blue = True (==) Magenta Magenta = True (==) Cyan Cyan = True (==) White White = True (==) _ _ = False Show Color where show Red = "red" show Green = "green" show Yellow = "yellow" show Blue = "blue" show Magenta = "magenta" show Cyan = "cyan" show White = "white" private colorToCode : Color -> Nat colorToCode Red = 1 colorToCode Green = 2 colorToCode Yellow = 3 colorToCode Blue = 4 colorToCode Magenta = 5 colorToCode Cyan = 6 colorToCode White = 7 private colorToAnsi : Color -> String colorToAnsi code = "\ESC[3" ++ show (colorToCode code) ++ "m" colorise : Color -> String -> String colorise color str = colorToAnsi color ++ str ++ "\ESC[0m" indent : (level : Nat) -> String indent level = foldr (\el, acc => el ++ acc) "" (replicate level " ") format : String -> Color -> (level : Nat) -> String format str color level = colorise color $ indent level ++ str
{-# OPTIONS --safe #-} module Cubical.Algebra.OrderedCommMonoid.Properties where open import Cubical.Foundations.Prelude open import Cubical.Foundations.Structure open import Cubical.Foundations.HLevels open import Cubical.Foundations.SIP using (TypeWithStr) open import Cubical.Data.Sigma open import Cubical.Algebra.CommMonoid open import Cubical.Algebra.OrderedCommMonoid.Base open import Cubical.Relation.Binary.Poset private variable ℓ ℓ' ℓ'' : Level module _ (M : OrderedCommMonoid ℓ ℓ') (P : ⟨ M ⟩ → hProp ℓ'') where open OrderedCommMonoidStr (snd M) module _ (·Closed : (x y : ⟨ M ⟩) → ⟨ P x ⟩ → ⟨ P y ⟩ → ⟨ P (x · y) ⟩) (εContained : ⟨ P ε ⟩) where private subtype = Σ[ x ∈ ⟨ M ⟩ ] ⟨ P x ⟩ submonoid = makeSubCommMonoid (OrderedCommMonoid→CommMonoid M) P ·Closed εContained _≤ₛ_ : (x y : ⟨ submonoid ⟩) → Type ℓ' x ≤ₛ y = (fst x) ≤ (fst y) pres≤ : (x y : ⟨ submonoid ⟩) (x≤y : x ≤ₛ y) → (fst x) ≤ (fst y) pres≤ x y x≤y = x≤y makeOrderedSubmonoid : OrderedCommMonoid _ ℓ' fst makeOrderedSubmonoid = subtype OrderedCommMonoidStr._≤_ (snd makeOrderedSubmonoid) = _≤ₛ_ OrderedCommMonoidStr._·_ (snd makeOrderedSubmonoid) = CommMonoidStr._·_ (snd submonoid) OrderedCommMonoidStr.ε (snd makeOrderedSubmonoid) = CommMonoidStr.ε (snd submonoid) OrderedCommMonoidStr.isOrderedCommMonoid (snd makeOrderedSubmonoid) = IsOrderedCommMonoidFromIsCommMonoid (CommMonoidStr.isCommMonoid (snd submonoid)) (λ x y → is-prop-valued (fst x) (fst y)) (λ x → is-refl (fst x)) (λ x y z → is-trans (fst x) (fst y) (fst z)) (λ x y x≤y y≤x → Σ≡Prop (λ x → snd (P x)) (is-antisym (fst x) (fst y) (pres≤ x y x≤y) (pres≤ y x y≤x))) (λ x y z x≤y → MonotoneR (pres≤ x y x≤y)) λ x y z x≤y → MonotoneL (pres≤ x y x≤y)
import tactic basic namespace complex /-! # `ext` : A mathematical triviality -/ /- Two complex numbers with the same and imaginary parts are equal. This is an "extensionality lemma", i.e. a lemma of the form "if two things are made from the same pieces, they are equal". This is not hard to prove, but we want to give the result a name so we can tag it with the `ext` attribute, meaning that the `ext` tactic will know it. To add to the confusion, let's call the theorem `ext` :-) -/ /-- If two complex numbers z and w have equal real and imaginary parts, they are equal -/ @[ext] theorem ext {z w : ℂ} (hre : re(z) = re(w)) (him : im(z) = im(w)) : z = w := begin cases z with zr zi, cases w with ww wi, simp at , /- goal now a logic puzzle hre : zr = ww, him : zi = wi ⊢ zr = ww ∧ zi = wi -/ cc, end /-! # Theorem: The complex numbers are a commutative ring -/ -- Proof: we've defined all the structure, and every axiom can be checked by reducing it -- to checking real and imaginary parts with `ext`, expanding everything out with `simp` -- and then using the fact that the real numbers are a commutative ring. instance : comm_ring ℂ := begin -- first the data refine_struct { zero := (0 : ℂ), add := (+), neg := has_neg.neg, one := 1, mul := (), ..}, -- now the axioms -- of which there seem to be 11 -- introduce the variables all_goals {intros}, -- we now have to prove an equality between two complex numbers. -- It suffices to check on real and imaginary parts all_goals {ext}, -- the simplifier can simplify stuff like re(a+0) all_goals {simp}, -- all the goals now are identities between real numbers, -- and the reals are already known to be a ring all_goals {ring}, end -- That is the end of the proof that the complexes form a ring. We built -- a basic API which was honed towards the general idea that to prove -- certain statements about the complex numbers, for example distributivity, -- we could just check on real and imaginary parts. We trained the -- simplifier to expand out things like re(zw) in terms -- of re(z), im(z), re(w), im(w). /-! # Optional section for mathematicians : more basic infrastructure, and term mode -/ /-! ## `ext` revisited Recall extensionality: `theorem ext {z w : ℂ} (hre : re(z) = re(w)) (him : im(z) = im(w)) : z = w := ...` Here is another tactic mode proof of extensionality. Note that we have moved the hypotheses to the other side of the colon; this does not change the theorem. This proof shows the power of the `rintros` tactic. -/ theorem ext' : ∀ z w : ℂ, z.re = w.re → z.im = w.im → z = w := begin rintros ⟨zr, zi⟩ ⟨_, _⟩ ⟨rfl⟩ ⟨rfl⟩, refl, end /- Explanation: `rintros` does `cases` as many times as you like using this cool `⟨ ⟩` syntax for the case splits. Note that if you say that a proof of `a = b` is `rfl` then Lean will define a to be b, or b to be a, and not even introduce new notation for it. -/ -- Here is the same proof in term mode. theorem ext'' : ∀ {z w : ℂ}, z.re = w.re → z.im = w.im → z = w \| ⟨zr, zi⟩ ⟨_, _⟩ rfl rfl := rfl /-! ## `eta` -/ /- We prove the mathematically obvious statement that the complex number whose real part is re(z) and whose imaginary part is im(z) is of course equal to z. -/ /-- ⟨z.re, z.im⟩ is equal to z -/ @[simp] theorem eta : ∀ z : ℂ, complex.mk z.re z.im = z := begin intro z, cases z with x y, /- goal now looks complicated, and contains terms which look like {re := a, im := b}.re which obviously simplify to a. The `dsimp` tactic will do some tidying up for us, although it is not logically necessary. `dsimp` does definitional simplification. -/ dsimp, -- Now we see the goal can be solved by reflexivity refl, end /- The proof was "unfold everything, and it's true by definition". This proof does not teach a mathematician anything, so we may as well write it in term mode. Many tactics have term mode equivalent. The equation compiler does the `intro` and `cases` steps, and `dsimp` was unnecessary -- the two sides of the equation were definitionally equal. -/ theorem eta' : ∀ z : ℂ, complex.mk z.re z.im = z \| ⟨x, y⟩ := rfl /-! ## ext_iff -/ /- Note that `ext` is an implication -- if re(z)=re(w) and im(z)=im(w) then z=w. The below variant `ext_iff` is the two-way implication: two complex numbers are equal if and only if they have the same real and imaginary part. Let's first see a tactic mode proof. See how the `ext` tactic is used? After it is applied, we have two goals, both of which are hypotheses. The semicolon means "apply the next tactic to all the goals produced by this one" -/ theorem ext_iff {z w : ℂ} : z = w ↔ z.re = w.re ∧ z.im = w.im := begin split, { intro H, simp [H]}, { rintro ⟨hre, him⟩, ext; assumption, } end -- Again this is easy to write in term mode, and no mathematician -- wants to read the proof anyway. theorem ext_iff' {z w : ℂ} : z = w ↔ z.re = w.re ∧ z.im = w.im := ⟨λ H, by simp [H], and.rec ext⟩ end complex /-! # some last comments on the `simp` tactic Some equalities, even if obvious, had to be given names, because we want `simp` to be able to use them. In short, the `simp` tactic tries to solve goals of the form A = B, when `refl` doesn't work (i.e. the goals are not definitionally equal) but when any mathematician would be able to simplify A and B via "obvious" steps such as `0 + x = x` or `⟨z.re, z.im⟩ = z`. These things are sometimes not true by definition, but they should be tagged as being well-known ways to simplify an equality. When building our API for the complex numbers, if we prove a theorem of the form `A = B` where `B` is a bit simpler than `A`, we should probably tag it with the `@[simp]` attribute, so `simp` can use it. Note: `simp` does not* prove "all simple things". It proves equalities. It proves `A = B` when, and only when, it can do it by applying its "simplification rules", where a simplification rule is simply a proof of a theorem of the form `A = B` and `B` is simpler than `A`. -/ ------------------ VERY INTERESTING BITS ------------------------- /-- The canonical map from ℝ to ℂ. -/ def of_real (r : ℝ) : ℂ := ⟨r, 0⟩ /- We make this map into a coercion, which means that if `(r : ℝ)` is a real number, then `(r : ℂ)` or `(↑r : ℂ)` will indicate the corresponding complex number with no imaginary part. This is the notation we shall use in our `simp` lemmas. -/ /-- The coercion from ℝ to ℂ sending `r` to the complex number `⟨r, 0⟩` -/ instance : has_coe ℝ ℂ := ⟨of_real⟩
– Daily Express – Mention for the IW Walking Festival. Further coverage was received following a media visit by leading Dutch garden writers, Marieke van Gessel and Modeste Herweg. Please click here to view this exposure. It’s been used by HDC Media: http://www.hc-hdc.nl/overhdcmedia/ several editions, circulation 255.000, read by 830.000 people and is valued at approx. £12,000. Our PR media value (ad value) coverage for 2014 now stands as £2.3million. This figure currently excludes local, on-line and any TV coverage. You will doubtless have seen the fantastic images as some of the world’s best ice climbers took on the challenge of ascending the sheer cliffs of chalk at The Needles as the first ever Red Bull White Cliff competition came to the Island on Sunday 19th October. The film of this spectacular climb is on YouTube, or you can see it by clicking here. Katie Jones of VIOW who helped to set up the event together with The Needles Park, The National Trust and the ferry operators is hopeful of the event returning to the Island to become a real tourist attraction in 2015. And for those who missed the Isle of Wight episode of the new BBC1 television series “Holiday of my Lifetime” featuring Len Goodman and Esther Rantzen, more details about the programme which show Len taking Esther back to the Island to relive her holiday as a 6-year old in 1946 can be found by clicking here. In the film they can be seen in and around Bembridge, watching the daily demonstration of the donkeys at Carisbrooke Castle and visiting The Needles Park where Len surprises Esther with a chairlift ride with stunning views of the Island’s coastline. Robin Hill’s new event attraction “Electric Woods” also features in the programme as well as Isle of Wight Adventure Activities. Along the way Esther reveals that her journalistic curiosity may have started during her time on the Isle of Wight. We were first approached by the programme makers back in June when they requested our assistance in finding suitable film locations and we were delighted to be able to provide support and act as a liaison between the television crew and relevant tourism providers.
section \<open>Map Interface\<close> theory IICF_Map imports "../../Sepref" begin subsection \<open>Parametricity for Maps\<close> definition [to_relAPP]: "map_rel K V \<equiv> (K \<rightarrow> \<langle>V\<rangle>option_rel) \<inter> { (mi,m). dom mi \<subseteq> Domain K \<and> dom m \<subseteq> Range K }" (* definition [to_relAPP]: "map_rel K V \<equiv> (K \<rightarrow> \<langle>V\<rangle>option_rel) \<inter> { (mi,m). dom mi \<subseteq> Domain K \<and> dom m \<subseteq> Range K \<and> ran mi \<subseteq> Domain V \<and> ran m \<subseteq> Range V }" *) lemma bi_total_map_rel_eq: "\<lbrakk>IS_RIGHT_TOTAL K; IS_LEFT_TOTAL K\<rbrakk> \<Longrightarrow> \<langle>K,V\<rangle>map_rel = K \<rightarrow> \<langle>V\<rangle>option_rel" unfolding map_rel_def IS_RIGHT_TOTAL_def IS_LEFT_TOTAL_def by (auto dest: fun_relD) lemma map_rel_empty1_simp[simp]: "(Map.empty,m)\<in>\<langle>K,V\<rangle>map_rel \<longleftrightarrow> m=Map.empty" apply (auto simp: map_rel_def) by (meson RangeE domIff option_rel_simp(1) subsetCE tagged_fun_relD_none) lemma map_rel_empty2_simp[simp]: "(m,Map.empty)\<in>\<langle>K,V\<rangle>map_rel \<longleftrightarrow> m=Map.empty" apply (auto simp: map_rel_def) by (meson Domain.cases domIff fun_relD2 option_rel_simp(2) subset_eq) lemma map_rel_obtain1: assumes 1: "(m,n)\<in>\<langle>K,V\<rangle>map_rel" assumes 2: "n l = Some w" obtains k v where "m k = Some v" "(k,l)\<in>K" "(v,w)\<in>V" using 1 unfolding map_rel_def proof clarsimp assume R: "(m, n) \<in> K \<rightarrow> \<langle>V\<rangle>option_rel" assume "dom n \<subseteq> Range K" with 2 obtain k where "(k,l)\<in>K" by auto moreover from fun_relD[OF R this] have "(m k, n l) \<in> \<langle>V\<rangle>option_rel" . with 2 obtain v where "m k = Some v" "(v,w)\<in>V" by (cases "m k"; auto) ultimately show thesis by - (rule that) qed lemma map_rel_obtain2: assumes 1: "(m,n)\<in>\<langle>K,V\<rangle>map_rel" assumes 2: "m k = Some v" obtains l w where "n l = Some w" "(k,l)\<in>K" "(v,w)\<in>V" using 1 unfolding map_rel_def proof clarsimp assume R: "(m, n) \<in> K \<rightarrow> \<langle>V\<rangle>option_rel" assume "dom m \<subseteq> Domain K" with 2 obtain l where "(k,l)\<in>K" by auto moreover from fun_relD[OF R this] have "(m k, n l) \<in> \<langle>V\<rangle>option_rel" . with 2 obtain w where "n l = Some w" "(v,w)\<in>V" by (cases "n l"; auto) ultimately show thesis by - (rule that) qed subsection \<open>Interface Type\<close> sepref_decl_intf ('k,'v) i_map is "'k \<rightharpoonup> 'v" subsection \<open>Operations\<close> sepref_decl_op map_empty: "Map.empty" :: "\<langle>K,V\<rangle>map_rel" . sepref_decl_op map_is_empty: "(=) Map.empty" :: "\<langle>K,V\<rangle>map_rel \<rightarrow> bool_rel" apply (rule fref_ncI) apply parametricity apply (rule fun_relI; auto) done sepref_decl_op map_update: "\<lambda>k v m. m(k\<mapsto>v)" :: "K \<rightarrow> V \<rightarrow> \<langle>K,V\<rangle>map_rel \<rightarrow> \<langle>K,V\<rangle>map_rel" where "single_valued K" "single_valued (K\<inverse>)" apply (rule fref_ncI) apply parametricity unfolding map_rel_def apply (intro fun_relI) apply (elim IntE; rule IntI) apply (intro fun_relI) apply parametricity apply (simp add: pres_eq_iff_svb) apply auto done sepref_decl_op map_delete: "\<lambda>k m. fun_upd m k None" :: "K \<rightarrow> \<langle>K,V\<rangle>map_rel \<rightarrow> \<langle>K,V\<rangle>map_rel" where "single_valued K" "single_valued (K\<inverse>)" apply (rule fref_ncI) apply parametricity unfolding map_rel_def apply (intro fun_relI) apply (elim IntE; rule IntI) apply (intro fun_relI) apply parametricity apply (simp add: pres_eq_iff_svb) apply auto done sepref_decl_op map_lookup: "\<lambda>k (m::'k\<rightharpoonup>'v). m k" :: "K \<rightarrow> \<langle>K,V\<rangle>map_rel \<rightarrow> \<langle>V\<rangle>option_rel" apply (rule fref_ncI) apply parametricity unfolding map_rel_def apply (intro fun_relI) apply (elim IntE) apply parametricity done lemma in_dom_alt: "k\<in>dom m \<longleftrightarrow> \<not>is_None (m k)" by (auto split: option.split) sepref_decl_op map_contains_key: "\<lambda>k m. k\<in>dom m" :: "K \<rightarrow> \<langle>K,V\<rangle>map_rel \<rightarrow> bool_rel" unfolding in_dom_alt apply (rule fref_ncI) apply parametricity unfolding map_rel_def apply (elim IntE) apply parametricity done subsection \<open>Patterns\<close> lemma pat_map_empty[pat_rules]: "\<lambda>\<^sub>2_. None \<equiv> op_map_empty" by simp lemma pat_map_is_empty[pat_rules]: "(=) $m$(\<lambda>\<^sub>2_. None) \<equiv> op_map_is_empty$m" "(=) $(\<lambda>\<^sub>2_. None)$m \<equiv> op_map_is_empty$m" "(=) $(dom$m)${} \<equiv> op_map_is_empty$m" "(=) ${}$(dom$m) \<equiv> op_map_is_empty$m" unfolding atomize_eq by (auto dest: sym) lemma pat_map_update[pat_rules]: "fun_upd$m$k$(Some$v) \<equiv> op_map_update$'k$'v$'m" by simp lemma pat_map_lookup[pat_rules]: "m$k \<equiv> op_map_lookup$'k$'m" by simp lemma op_map_delete_pat[pat_rules]: "(\|`) $ m $ (uminus $ (insert $ k $ {})) \<equiv> op_map_delete$'k$'m" "fun_upd$m$k$None \<equiv> op_map_delete$'k$'m" by (simp_all add: map_upd_eq_restrict) lemma op_map_contains_key[pat_rules]: "(\<in>) $ k $ (dom$m) \<equiv> op_map_contains_key$'k$'m" "Not$((=) $(m$k)$None) \<equiv> op_map_contains_key$'k$'m" by (auto intro!: eq_reflection) subsection \<open>Parametricity\<close> locale map_custom_empty = fixes op_custom_empty :: "'k\<rightharpoonup>'v" assumes op_custom_empty_def: "op_custom_empty = op_map_empty" begin sepref_register op_custom_empty :: "('kx,'vx) i_map" lemma fold_custom_empty: "Map.empty = op_custom_empty" "op_map_empty = op_custom_empty" "mop_map_empty = RETURN op_custom_empty" unfolding op_custom_empty_def by simp_all end end
If $a > b$, then eventually $x \in (b, a)$ as $x$ approaches $a$ from the left.
module MyPackage using LinearAlgebra: norm include("myfile.jl") end
/- Copyright (c) 2018 Simon Hudon. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Simon Hudon -/ import control.functor import data.sum.basic /-! # Functors with two arguments This file defines bifunctors. A bifunctor is a function `F : Type* → Type* → Type` along with a bimap which turns `F α β` into `F α' β'` given two functions `α → α'` and `β → β'`. It further respects the identity: `bimap id id = id` * composes in the obvious way: `(bimap f' g') ∘ (bimap f g) = bimap (f' ∘ f) (g' ∘ g)` ## Main declarations * `bifunctor`: A typeclass for the bare bimap of a bifunctor. * `is_lawful_bifunctor`: A typeclass asserting this bimap respects the bifunctor laws. -/ universes u₀ u₁ u₂ v₀ v₁ v₂ open function /-- Lawless bifunctor. This typeclass only holds the data for the bimap. -/ class bifunctor (F : Type u₀ → Type u₁ → Type u₂) := (bimap : Π {α α' β β'}, (α → α') → (β → β') → F α β → F α' β') export bifunctor ( bimap ) /-- Bifunctor. This typeclass asserts that a lawless `bifunctor` is lawful. -/ class is_lawful_bifunctor (F : Type u₀ → Type u₁ → Type u₂) [bifunctor F] := (id_bimap : Π {α β} (x : F α β), bimap id id x = x) (bimap_bimap : Π {α₀ α₁ α₂ β₀ β₁ β₂} (f : α₀ → α₁) (f' : α₁ → α₂) (g : β₀ → β₁) (g' : β₁ → β₂) (x : F α₀ β₀), bimap f' g' (bimap f g x) = bimap (f' ∘ f) (g' ∘ g) x) export is_lawful_bifunctor (id_bimap bimap_bimap) attribute [higher_order bimap_id_id] id_bimap attribute [higher_order bimap_comp_bimap] bimap_bimap export is_lawful_bifunctor (bimap_id_id bimap_comp_bimap) variables {F : Type u₀ → Type u₁ → Type u₂} [bifunctor F] namespace bifunctor /-- Left map of a bifunctor. -/ @[reducible] def fst {α α' β} (f : α → α') : F α β → F α' β := bimap f id /-- Right map of a bifunctor. -/ @[reducible] def snd {α β β'} (f : β → β') : F α β → F α β' := bimap id f variable [is_lawful_bifunctor F] @[higher_order fst_id] lemma id_fst : Π {α β} (x : F α β), fst id x = x := @id_bimap _ _ _ @[higher_order snd_id] lemma id_snd : Π {α β} (x : F α β), snd id x = x := @id_bimap _ _ _ @[higher_order fst_comp_fst] @[higher_order fst_comp_snd] lemma fst_snd {α₀ α₁ β₀ β₁} (f : α₀ → α₁) (f' : β₀ → β₁) (x : F α₀ β₀) : fst f (snd f' x) = bimap f f' x := by simp [fst,bimap_bimap] @[higher_order snd_comp_fst] lemma snd_fst {α₀ α₁ β₀ β₁} (f : α₀ → α₁) (f' : β₀ → β₁) (x : F α₀ β₀) : snd f' (fst f x) = bimap f f' x := by simp [snd,bimap_bimap] @[higher_order snd_comp_snd] lemma comp_snd {α β₀ β₁ β₂} (g : β₀ → β₁) (g' : β₁ → β₂) (x : F α β₀) : snd g' (snd g x) = snd (g' ∘ g) x := by simp [snd,bimap_bimap] attribute [functor_norm] bimap_bimap comp_snd comp_fst snd_comp_snd snd_comp_fst fst_comp_snd fst_comp_fst bimap_comp_bimap bimap_id_id fst_id snd_id end bifunctor open functor instance : bifunctor prod := { bimap := @prod.map } instance : is_lawful_bifunctor prod := by refine { .. }; intros; cases x; refl instance bifunctor.const : bifunctor const := { bimap := (λ α α' β β f _, f) } instance is_lawful_bifunctor.const : is_lawful_bifunctor const := by refine { .. }; intros; refl instance bifunctor.flip : bifunctor (flip F) := { bimap := (λ α α' β β' f f' x, (bimap f' f x : F β' α')) } instance is_lawful_bifunctor.flip [is_lawful_bifunctor F] : is_lawful_bifunctor (flip F) := by refine { .. }; intros; simp [bimap] with functor_norm instance : bifunctor sum := { bimap := @sum.map } instance : is_lawful_bifunctor sum := by refine { .. }; intros; cases x; refl open bifunctor functor @[priority 10] instance bifunctor.functor {α} : functor (F α) := { map := λ _ _, snd } @[priority 10] instance bifunctor.is_lawful_functor [is_lawful_bifunctor F] {α} : is_lawful_functor (F α) := by refine {..}; intros; simp [functor.map] with functor_norm section bicompl variables (G : Type* → Type u₀) (H : Type* → Type u₁) [functor G] [functor H] instance : bifunctor (bicompl F G H) := { bimap := λ α α' β β' f f' x, (bimap (map f) (map f') x : F (G α') (H β')) } instance [is_lawful_functor G] [is_lawful_functor H] [is_lawful_bifunctor F] : is_lawful_bifunctor (bicompl F G H) := by constructor; intros; simp [bimap,map_id,map_comp_map] with functor_norm end bicompl section bicompr variables (G : Type u₂ → Type*) [functor G] instance : bifunctor (bicompr G F) := { bimap := λ α α' β β' f f' x, (map (bimap f f') x : G (F α' β')) } instance [is_lawful_functor G] [is_lawful_bifunctor F] : is_lawful_bifunctor (bicompr G F) := by constructor; intros; simp [bimap] with functor_norm end bicompr
module Js.React.DOM import Js import Js.Object import Js.DOM.Element import Js.React.Element %default total export %inline render : (mount : Js.DOM.Element.Element) -> (element : Js.React.Element.Element) -> JS_IO () render mount = js "ReactDOM.render(%1, %0)" (Ptr -> Ptr -> JS_IO ()) (ptr mount) . ptr
State Before: C : Type u inst✝¹ : Category C inst✝ : NonPreadditiveAbelian C X Y Z : C f g : X ⟶ Y h : Y ⟶ Z ⊢ (f + g) ≫ h = f ≫ h + g ≫ h State After: no goals Tactic: rw [add_def, sub_comp, neg_def, sub_comp, zero_comp, add_def, neg_def]
# Imports and setup: ```python # import required packages import os import csv from sympy import symbols, init_printing import numpy as np import matplotlib %matplotlib inline import seaborn as sb from matplotlib import pyplot as plt import sympy import itertools import scipy import datetime import matplotlib.dates as mdates from pygom import DeterministicOde, Transition, SimulateOde, TransitionType, SquareLoss from scipy.optimize import minimize import pickle as pk import jsonpickle as jpk from cycler import cycler from pandas.plotting import register_matplotlib_converters register_matplotlib_converters() import pwlf import pprint ppr = pprint.PrettyPrinter() ``` /Users/n/.pyenv/versions/3.7.2/lib/python3.7/site-packages/statsmodels/tools/_testing.py:19: FutureWarning: pandas.util.testing is deprecated. Use the functions in the public API at pandas.testing instead. import pandas.util.testing as tm ```python savefigs = False # whether to save specific figures for paper to .../figures directory ``` ```python # Jupyter Specifics from IPython.display import display, HTML from ipywidgets.widgets import interact, interactive, IntSlider, FloatSlider, Layout, ToggleButton, ToggleButtons, fixed display(HTML("<style>.container { width:100% !important; }</style>")) style = {'description_width': '100px'} slider_layout = Layout(width='99%') ``` <style>.container { width:100% !important; }</style> # Model Definitions ```python #from models import * from model_fits import * ``` loading data.py... getting JHU data... number of countries listed 274 done with JHU data. Got 274 countries. getting owid data... data for population changes only slowly if at all in OWID database done with owid data. Got 212 countries WHO acute file found dictionary acute_who ICU file found dictionary icus_2012 --------------------------------- Done with data. --------------------------------- done with data.py. making the models... SEI3R SC3EI3R SC3UEI3R done with the models. ```python print(acute_dict) # WHO 2014 acute beds per 100000 edited by JSMcC to fill in missing data for 5 countries from previous years ``` {'Armenia': 341.07, 'Austria': 579.96, 'Azerbaijan': 354.18, 'Belgium': 567.27, 'Bulgaria': 595.04, 'Belarus': 866.42, 'Cyprus': 341.32, 'Czechia': 425.03, 'Germany': 620.83, 'Denmark': 261.67, 'Spain': 238.54, 'Estonia': 366.51, 'Finland': 307.55, 'France': 428.04, 'United Kingdom': 227.79, 'Georgia': 210.02, 'Greece': 346.25, 'Hungary': 428.87, 'Croatia': 398.84, 'Iceland': 267.32, 'Ireland': 240.44, 'Israel': 240.6, 'Italy': 274.57, 'Kazakhstan': 418.76, 'Kyrgyzstan': 353.86, 'Latvia': 339.21, 'Lithuania': 634.39, 'Luxembourg': 418.99, 'Monaco': 1119.0, 'Macedonia': 301.66, 'Moldova': 454.62, 'Malta': 323.9, 'Montenegro': 391.18, 'Netherlands': 332.38, 'Norway': 342.52, 'Poland': 487.68, 'Portugal': 325.13, 'Romania': 402.61, 'Russia': 640.69, 'Serbia': 456.25, 'Slovakia': 491.29, 'Slovenia': 423.03, 'Sweden': 234.5, 'Switzerland': 375.08, 'Tajikistan': 425.14, 'Turkey': 258.8, 'Turkmenistan': 306.4, 'Ukraine': 613.31, 'Uzbekistan': 333.87, 'ARM': 341.07, 'AUT': 579.96, 'AZE': 354.18, 'BEL': 567.27, 'BGR': 595.04, 'BLR': 866.42, 'CYP': 341.32, 'CZE': 425.03, 'DEU': 620.83, 'DNK': 261.67, 'ESP': 238.54, 'EST': 366.51, 'FIN': 307.55, 'FRA': 428.04, 'GBR': 227.79, 'GEO': 210.02, 'GRC': 346.25, 'HUN': 428.87, 'HRV': 398.84, 'ISL': 267.32, 'IRL': 240.44, 'ISR': 240.6, 'ITA': 274.57, 'KAZ': 418.76, 'KGZ': 353.86, 'LVA': 339.21, 'LTU': 634.39, 'LUX': 418.99, 'MCO': 1119.0, 'MKD': 301.66, 'MDA': 454.62, 'MLT': 323.9, 'MNE': 391.18, 'NLD': 332.38, 'NOR': 342.52, 'POL': 487.68, 'PRT': 325.13, 'ROU': 402.61, 'RUS': 640.69, 'SRB': 456.25, 'SVK': 491.29, 'SVN': 423.03, 'SWE': 234.5, 'CHE': 375.08, 'TJK': 425.14, 'TUR': 258.8, 'TKM': 306.4, 'UKR': 613.31, 'UZB': 333.87} ```python print(icu_dict) # Intensive Care Med (2012) 38:1647–1653 DOI 10.1007/s00134-012-2627-8 IMCUs and ICUs per 100000 in 2012 ``` {'Andorra': 7.1, 'Austria': 21.8, 'Belgium': 15.9, 'Bulgaria': 12.2, 'Croatia': 14.7, 'Cyprus': 11.4, 'Czechia': 11.6, 'Denmark': 6.7, 'Estonia': 14.6, 'Finland': 6.1, 'France': 11.6, 'Germany': 29.2, 'Greece': 6.0, 'Hungary': 13.8, 'Iceland': 9.1, 'Ireland': 6.5, 'Italy': 12.5, 'Latvia': 9.7, 'Lithuania': 15.5, 'Luxembourg': 24.8, 'Netherlands': 6.4, 'Norway': 8.0, 'Poland': 6.9, 'Portugal': 4.2, 'Romania': 21.4, 'Slovakia': 9.2, 'Slovenia': 6.4, 'Spain': 9.7, 'Sweden': 5.8, 'Switzerland': 11.0, 'United Kingdom': 6.6} ```python (len(icu_dict),len(acute_dict)) ``` (31, 98) # Test ModelFit ```python testfit = ModelFit('SC3EI3R',country='Germany',run_id="_tst") ``` ============= SC3EI3R_Germany_tst warning: changing run_id from SC3EI3R_Germany_tst to SC3EI3R_Germany_tst loaded params from /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC3EI3R_Germany_tst.pk : ```python testfit.dumpparams() ``` dumped params to /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC3EI3R_Germany_tst.pk ```python testfit.stopdate ``` '09/27/20' ```python testfit.prparams() ``` params: {'N': 1, 'alpha': 0.2, 'beta_1': 0.36, 'beta_2': 0.0, 'beta_3': 0.0, 'c_0': 0.1, 'c_1': 0.012, 'c_2': 4881.0, 'gamma_1': 0.06999999999999999, 'gamma_2': 0.16666666666666666, 'gamma_3': 0.10000000000000003, 'mu': 0.09999999999999998, 'p_1': 0.030000000000000013, 'p_2': 0.08333333333333334} sbparams: {'CFR': 0.05, 'DurHosp': 4, 'DurMildInf': 10, 'Exposure': 0.4, 'FracCritical': 0.10000000000000003, 'FracMild': 0.7, 'FracSevere': 0.2, 'I0': 2.2387211385683378e-08, 'ICUFrac': 0.001, 'IncubPeriod': 5, 'TimeICUDeath': 5} pfbarams: {'FracConfirmedDet': 1.0, 'FracDeathsDet': 1.0, 'FracRecoveredDet': 1.0} cbparams: {'CautionFactor': 0.1, 'CautionICUFrac': 0.1, 'CautionRetention': 66.66666666666667, 'EconomicCostOfCaution': 0.5, 'EconomicRetention': 66.66666666666667, 'EconomicStriction': 66.66666666666667, 'EconomyRelaxation': 66.66666666666667} dbparams: {'country': 'Germany', 'data_src': 'owid', 'run_name': 'SC3EI3R'} initial_values: (array([9.999997e-01, 0.000000e+00, 3.000000e-07, 0.000000e+00, 0.000000e+00, 0.000000e+00, 0.000000e+00, 0.000000e+00, 0.000000e+00, 0.000000e+00]), 0) ```python # this cell for debugging methods, as needed. # copy method here, uncomment appropriate setattr() call below, debug, copy method back to model_fits.py #MyClass = ModelFit('SC3EI3R',run_id='tst').__class__ #setattr(MyClass,'__init__',__init__) #setattr(MyClass,'dumpparams',dumpparams) #setattr(MyClass,'__init__',__init__) #setattr(MyClass,'loadparams',loadparams) #setattr(MyClass,'prparams',prparams) ``` ```python testfit.loadparams() ``` loaded params from /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC3EI3R_Germany_tst.pk : True ```python testfit2 = ModelFit('SC3UEI3R',model=None,country='Italy',run_id='testfit2',datatypes='all',data_src='owid',startdate='2/15/20',stopdate='8/31/20',simdays=300) ``` ============= testfit2 warning: changing run_id from testfit2 to testfit2 no file available with this run_id: /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/testfit2.pk Problem loading paramfile for testfit2 ... using default set of parameters for model type SC3UEI3R stopdate 8/31/20 ```python print(testfit2.startdate,'to',testfit2.stopdate) print(len(testfit2.tsim),len(testfit2.tdata)) ``` 02/15/20 to 08/31/20 300 199 ## Debugging of plotdata Following cells are set up to debug methods of the ModelFit class without restarting kernel and reloading model_fits.py. Edit functions here, then when debugged, copy them into the class definition of model_fits.py. ```python # simple plot of data : finessing dates issue def plotdata(self,dtypes=['confirmed','deaths']): if type(dtypes)==str: dtypes = [dtypes] xx = np.array(range(len(self.tdata)-1)) print(len(xx)) print([(x,len(self.data[x])) for x in dtypes]) for dt in dtypes: try: yy = self.data[dt] except: print("data type '"+dt+"' not found.") try: plt.plot(xx,yy) except: print("couldn't plot xx,yy",xx,yy) plt.show() ``` ## Plotting simulation and data with solveplot method ```python testfit.solveplot(species=['confirmed','deaths'],mag={'deaths':10},summing='daily',averaging='weekly',axes=None, scale='linear',plottitle= '',label='',newplot = True, gbrcolors=False, figsize = (12,12), outfile = 'tst.pdf') ``` ```python testfit.prparams() ``` params: {'N': 1, 'alpha': 0.2, 'beta_1': 0.36, 'beta_2': 0.0, 'beta_3': 0.0, 'c_0': 0.1, 'c_1': 0.012, 'c_2': 4881.0, 'gamma_1': 0.06999999999999999, 'gamma_2': 0.16666666666666666, 'gamma_3': 0.10000000000000003, 'mu': 0.09999999999999998, 'p_1': 0.030000000000000013, 'p_2': 0.08333333333333334} sbparams: {'CFR': 0.05, 'DurHosp': 4, 'DurMildInf': 10, 'Exposure': 0.4, 'FracCritical': 0.10000000000000003, 'FracMild': 0.7, 'FracSevere': 0.2, 'I0': 2.2387211385683378e-08, 'ICUFrac': 0.001, 'IncubPeriod': 5, 'TimeICUDeath': 5} pfbarams: {'FracConfirmedDet': 1.0, 'FracDeathsDet': 1.0, 'FracRecoveredDet': 1.0} cbparams: {'CautionFactor': 0.1, 'CautionICUFrac': 0.1, 'CautionRetention': 66.66666666666667, 'EconomicCostOfCaution': 0.5, 'EconomicRetention': 66.66666666666667, 'EconomicStriction': 66.66666666666667, 'EconomyRelaxation': 66.66666666666667} dbparams: {'country': 'Germany', 'data_src': 'owid', 'run_name': 'SC3EI3R'} initial_values: (array([9.999997e-01, 0.000000e+00, 3.000000e-07, 0.000000e+00, 0.000000e+00, 0.000000e+00, 0.000000e+00, 0.000000e+00, 0.000000e+00, 0.000000e+00]), 0) ```python testfit.prparams() ``` params: {'N': 1, 'alpha': 0.2, 'beta_1': 0.36, 'beta_2': 0.0, 'beta_3': 0.0, 'c_0': 0.1, 'c_1': 0.012, 'c_2': 4881.0, 'gamma_1': 0.06999999999999999, 'gamma_2': 0.16666666666666666, 'gamma_3': 0.10000000000000003, 'mu': 0.09999999999999998, 'p_1': 0.030000000000000013, 'p_2': 0.08333333333333334} sbparams: {'CFR': 0.05, 'DurHosp': 4, 'DurMildInf': 10, 'Exposure': 0.4, 'FracCritical': 0.10000000000000003, 'FracMild': 0.7, 'FracSevere': 0.2, 'I0': 2.2387211385683378e-08, 'ICUFrac': 0.001, 'IncubPeriod': 5, 'TimeICUDeath': 5} pfbarams: {'FracConfirmedDet': 1.0, 'FracDeathsDet': 1.0, 'FracRecoveredDet': 1.0} cbparams: {'CautionFactor': 0.1, 'CautionICUFrac': 0.1, 'CautionRetention': 66.66666666666667, 'EconomicCostOfCaution': 0.5, 'EconomicRetention': 66.66666666666667, 'EconomicStriction': 66.66666666666667, 'EconomyRelaxation': 66.66666666666667} dbparams: {'country': 'Germany', 'data_src': 'owid', 'run_name': 'SC3EI3R'} initial_values: (array([9.999997e-01, 0.000000e+00, 3.000000e-07, 0.000000e+00, 0.000000e+00, 0.000000e+00, 0.000000e+00, 0.000000e+00, 0.000000e+00, 0.000000e+00]), 0) # Fitting simulation and data with sliders ## Germany ```python testfit = ModelFit('SC3EI3R',country='Germany',run_id='SC3EI3R_Germany_tst',simdays=200) ``` ============= SC3EI3R_Germany_tst warning: changing run_id from SC3EI3R_Germany_tst to SC3EI3R_Germany_tst loaded params from /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC3EI3R_Germany_tst.pk : ```python def slidefitplot(modelfit,beta_1,mu,c_0,c_1,c_2,logI_0): params={ 'beta_1':beta_1, 'mu':mu, 'c_0':c_0, 'c_1':c_1, 'c_2':c_2} for pp in params: # careful to keep intact rest of params that don't have sliders modelfit.params[pp] = params[pp] modelfit.model.parameters = modelfit.params I0 = 10*logI_0 x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,0.] modelfit.model.initial_values = (x0,0) modelfit.sbparams['I0'] = I0 modelfit.solveplot(species=['confirmed','deaths'],mag={'deaths':10},summing='daily',averaging='weekly',axes=None, scale='linear',plottitle= '',label='',newplot = True, gbrcolors=False, figsize = (15,10),outfile = testfit.run_id+'.pdf') ``` ```python params=testfit.params logI_0 = np.log10(testfit.sbparams['I0']) w =interactive(slidefitplot,modelfit=fixed(testfit), beta_1=FloatSlider(min=0,max=1,step=0.01,value=params['beta_1'],description='beta_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), mu=FloatSlider(min=0,max=0.2,step=0.002,value=params['mu'],description='mu', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_2=FloatSlider(min=0,max=20000,step=1,value=params['c_2'],description='c_2', style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'), logI_0=FloatSlider(min=-10,max=0,step=0.01,value=logI_0,description='log I_0', style=style,layout=slider_layout,continuous_update=False) ) display(w) ``` interactive(children=(FloatSlider(value=0.36, continuous_update=False, description='beta_1', layout=Layout(wid… ```python testfit.prparams(testfit.run_id+'.params') ``` ```python testfit.prparams() ``` params: {'N': 1, 'alpha': 0.2, 'beta_1': 0.36, 'beta_2': 0.0, 'beta_3': 0.0, 'c_0': 0.1, 'c_1': 0.012, 'c_2': 4881.0, 'gamma_1': 0.06999999999999999, 'gamma_2': 0.16666666666666666, 'gamma_3': 0.10000000000000003, 'mu': 0.09999999999999998, 'p_1': 0.030000000000000013, 'p_2': 0.08333333333333334} sbparams: {'CFR': 0.05, 'DurHosp': 4, 'DurMildInf': 10, 'Exposure': 0.4, 'FracCritical': 0.10000000000000003, 'FracMild': 0.7, 'FracSevere': 0.2, 'I0': 2.3442288153199228e-08, 'ICUFrac': 0.001, 'IncubPeriod': 5, 'TimeICUDeath': 5} pfbarams: {'FracConfirmedDet': 1.0, 'FracDeathsDet': 1.0, 'FracRecoveredDet': 1.0} cbparams: {'CautionFactor': 0.1, 'CautionICUFrac': 0.1, 'CautionRetention': 66.66666666666667, 'EconomicCostOfCaution': 0.5, 'EconomicRetention': 66.66666666666667, 'EconomicStriction': 66.66666666666667, 'EconomyRelaxation': 66.66666666666667} dbparams: {'country': 'Germany', 'data_src': 'owid', 'run_name': 'SC3EI3R'} initial_values: (array([9.999997e-01, 0.000000e+00, 3.000000e-07, 0.000000e+00, 0.000000e+00, 0.000000e+00, 0.000000e+00, 0.000000e+00, 0.000000e+00, 0.000000e+00]), 0) ```python testfit.plotdata() # I left this cell in so you can see how one can use the mdates together with my data structures to add real dates to the plots # Have to replace xx with the dates in the solveplot method in that case def truncx(xx,daystart,daystop): """truncate array xx to run from daystart to daystop do this before trying to extend the arrays if required""" daymin = max(daystart,0) daymax = min(daystop,(xx[-1]-xx[0]).days) return xx[daymin:daymax+1] def truncy(xx,yy,daystart,daystop): """truncate arrays xx and yy to run from daystart to daystop do this before trying to extend the arrays if required""" daymin = max(daystart,0) daymax = min(daystop,(xx[-1]-xx[0]).days) return yy[daymin:daymax+1] def plotCountry_(country_s, datatype='confirmed', dataaccum='cumulative', fittype=None, ax=None, ax2=False, symbol='o--', step=None, firstdate=None, lastdate=None, intdates=False, linecolor=None, maxyval=None, minconfirmed=0,nsegments=3,database='jhu'): """ plots selected data for a list of countries or single country datatypes allowed are 'confirmed','deaths','recovered' dataaccum specifies either 'cumulative' or 'daily' or averaged over 7 days 'cum_av_weekly' or 'daily_av_weekly' fittypes allowed are currently None, 'piecewise-linear' ax graphical axes to use for plot: default None -> new axes ax2 true if second axes as twin axes for overlay plotting symbol to use for plotting step whether to use step plotting instead of points: default None -> points firstdate to plot (maybe before first date in data - pad with 0) lastdate to plot (maybe after last date in data - pad with 0) intdates : whether to plot dates as integers for compatibility (default as dates) linecolor is default color to use for a single trace, instead of listed set) """ global covid_ts, covid_ts_owid import math import warnings # extract list of countries in [(country,region),...] format from first parameter countries = [] if isinstance(country_s,list): for country in country_s: if isinstance(country,str) and database == 'jhu': country = (country,'') countries.append(country) elif isinstance(country_s,str): if database == 'jhu': countries = [( country_s,'')] else: countries = [country_s] else: # single ('country','reg') entry countries = [country_s] # get data with datatype and extend dates to padd desired interval specified by firstdate,lastdate if database == 'jhu': popkeyed = covid_ts[datatype] dates = popkeyed['dates'] fmt = '%m/%d/%y' elif database == 'owid': popkeyed = covid_owid_ts[datatype] dates = popkeyed['dates'] fmt = '%m/%d/%y' # fmt = '%Y-%m-%d' the owid date format was converted to the jhu date format in get_data_owid xxd = [datetime.datetime.strptime(dd,fmt) for dd in dates ] if firstdate: firstdate_d = datetime.datetime.strptime(firstdate,fmt) else: firstdate_d = datetime.datetime.strptime(dates[0],fmt) if lastdate: lastdate_d = datetime.datetime.strptime(lastdate,fmt) else: lastdate_d = datetime.datetime.strptime(dates[-1],fmt) daystart = (firstdate_d-xxd[0]).days daystop = (lastdate_d-xxd[0]).days xx = [0.](daystop-daystart+1) xx = truncx(xxd,daystart,daystop) # truncates list to chosen date interval !!!!!!!!!!!!!!!!!!!!!!!!! # print('1 len xx',len(xx)) if daystart <0: xx0 = [xx[0]+datetime.timedelta(days=i) for i in range(daystart,0)] yy0 = [0.](-daystart) else: xx0 = [] yy0 = [] if daystop > (xxd[-1]-xxd[0]).days: xx1 = [xxd[-1]+datetime.timedelta(days=i) for i in range(daystop-(xxd[-1]-xxd[0]).days)] yy1 = [' '](daystop-(xxd[-1]-xxd[0]).days) else: xx1 = [] yy1 = [] xx = xx0 + xx + xx1 # print('2 len xx',len(xx)) #print('len xx1 yy1',len(xx1),len(yy1)) # print('len xx',len(xx)) if fittype == 'piecewise-linear': xxi = [Float((x-xx[0]).days) for x in xx ] # print(xxi) # print('len xxi',len(xxi)) # locator = mdates.MonthLocator() locator = mdates.AutoDateLocator(minticks=5, maxticks=13) formatter= mdates.ConciseDateFormatter(locator) if not ax: fig,ax = plt.subplots(1,1,figsize=(9,6)) ax2 = ax elif ax2: ax2 = ax.twinx() else: ax2 = ax colors = ['k', 'b', 'c', 'm', 'y', 'g', 'olive', 'chocolate'] i = 0 j = 0 for country in countries: try: yyd = popkeyed[country] if np.max(yyd) >= minconfirmed: j = j+1 else: i = i + 1 continue except: print('country not found',country) i = i + 1 continue yy = truncy(xxd,yyd,daystart,daystop) # print(country,'1 len yy yyd',len(yy),len(yyd)) yyf = [Float(y) for y in yy] yy = yy0 + yyf + yy1 # print(country,'2 len yy',len(yy)) # ymax=np.max(np.array(yy)) yyf = [Float(y) for y in yy] if dataaccum == 'daily': yy = [0.]len(yy) yy[0] = yyf[0] for k in range(1,len(yy)): yy[k] = yyf[k]-yyf[k-1] elif dataaccum == 'cum_av_weekly': yy = [0.]len(yy) moving_av = 0. for k in range(len(yy)): if k-7 >= 0: moving_av = moving_av - yyf[k-7] moving_av = moving_av + yyf[k] yy[k] = moving_av/min(7.0,float(k+1)) elif dataaccum == 'daily_av_weekly': yy = [0.]len(yyf) yy[0] = yyf[0] for k in range(1,len(yy)): yy[k] = yyf[k]-yyf[k-1] yyf = [y for y in yy] yy = [0.]len(yy) moving_av = 0. for k in range(len(yy)): if k-7 >= 0: moving_av = moving_av - yyf[k-7] moving_av = moving_av + yyf[k] yy[k] = moving_av/min(7.0,float(k+1)) if intdates: xx = range(len(xx)) if step: ax2.step(xx,yy,label = country[0]) else: # print(ax,ax2) # ax2.set_ylim(ymax,0) if linecolor: color = linecolor else: color = colors[i] ax2.plot(xx, yy, symbol, markersize=3, color = color, alpha=0.8, label = country[0]) if maxyval: ax.set_ylim(0,maxyval) if maxyval: ax2.set_ylim(0,maxyval) plt.title(country[0]+'-'+country[1]) # +' '+datatype) if fittype == 'piecewise-linear': warnings.filterwarnings("ignore", message="Warning: zero length interval encountered in pwlf.py calc_slopes") # initialize piecewise linear fit with your x and y data # yyf = [Float(y) for y in yy] yyf = [Float(y) if not math.isnan(y) else 0.0 for y in yy] # print(yyf) my_pwlf = pwlf.PiecewiseLinFit(xxi, yyf) # fit the data for three line segments res = my_pwlf.fit(nsegments) ppp = my_pwlf.p_values(method='non-linear', step_size=1e-4) se = my_pwlf.se # standard errors parameters = np.concatenate((my_pwlf.beta, my_pwlf.fit_breaks[1:-1])) header = ['Parameter type', 'Parameter value ', 'Standard error ', 't ', 'P > np.abs(t) (p-value)'] print(header, sep=' \| ') fltfmt = "{:12.3f}".format values = np.zeros((parameters.size, 5), dtype=np.object_) values[:, 1] = [fltfmt(np.around(term, decimals=3)) for term in parameters] values[:, 2] = [fltfmt(np.around(term, decimals=3)) for term in se] values[:, 3] = [fltfmt(np.around(term, decimals=3)) for term in parameters/se] values[:, 4] = [fltfmt(np.around(term, decimals=3)) for term in ppp] for iii, row in enumerate(values): if iii < my_pwlf.beta.size: row[0] = 'Slope ' print(row, sep=' \| ') else: row[0] = 'Breakpoint' print(row, sep=' \| ') print("") # predict for the determined points xHat = np.linspace(min(xxi), max(xxi), num=len(xx)) # print(len(xHat),len(xxi)) yHat = my_pwlf.predict(xHat) ax2.plot(xx, yHat, color = colors[i], alpha=0.5, label = country[0]+' fit') i = i+1 if j==0: ax.axis("off") else: if j > 1: plt.legend(loc="upper left") plt.title('countries '+datatype+dataaccum) if not intdates: ax2.xaxis.set_major_formatter(formatter) ax2.xaxis.set_major_locator(locator) for tick in ax2.get_xticklabels(): tick.set_rotation(40) ``` ## Spain ```python testfit = ModelFit('SC3EI3R',country='Spain',run_id='_tst',stopdate='09/10/20') ``` ============= SC3EI3R_Spain_tst warning: changing run_id from SC3EI3R_Spain_tst to SC3EI3R_Spain_tst loaded params from /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC3EI3R_Spain_tst.pk : stopdate 09/10/20 ```python print([x for x in dir(testfit) if 'date' in x]) testfit.stopdate ``` ['dates', 'startdate', 'stopdate'] '09/10/20' ```python [pp for pp in testfit.params] ``` ['beta_1', 'beta_2', 'beta_3', 'alpha', 'gamma_1', 'gamma_2', 'gamma_3', 'p_1', 'p_2', 'mu', 'c_0', 'c_1', 'c_2', 'N'] ```python def slidefitplot(modelfit,beta_1,alpha,gamma_1,gamma_2,gamma_3,mu,c_0,c_1,c_2,logI_0): params={ 'beta_1':beta_1, 'alpha':alpha,'gamma_1':gamma_1,'gamma_2':gamma_2,'gamma_3':gamma_3, 'mu':mu, 'c_0':c_0, 'c_1':c_1, 'c_2':c_2} for pp in params: # careful to keep intact rest of params that don't have sliders modelfit.params[pp] = params[pp] modelfit.model.parameters = modelfit.params I0 = 10logI_0 x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,0.] modelfit.model.initial_values = (x0,0) modelfit.sbparams['I0'] = I0 modelfit.solveplot(species=['confirmed','deaths'],mag={'deaths':10},summing='daily',averaging='weekly',axes=None, scale='linear',plottitle= '',label='',newplot = True, gbrcolors=False, figsize = (15,10),outfile = testfit.run_id+'.pdf') ``` ```python params=testfit.params logI_0 = np.log10(testfit.sbparams['I0']) w =interactive(slidefitplot,modelfit=fixed(testfit), beta_1=FloatSlider(min=0,max=1,step=0.01,value=params['beta_1'],description='beta_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), alpha=FloatSlider(min=0,max=1,step=0.01,value=params['alpha'],description='alpha', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), gamma_1=FloatSlider(min=0,max=0.5,step=0.002,value=params['gamma_1'],description='gamma_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), gamma_2=FloatSlider(min=0,max=0.5,step=0.002,value=params['gamma_2'],description='gamma_2', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), gamma_3=FloatSlider(min=0,max=0.5,step=0.002,value=params['gamma_3'],description='gamma_3', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), mu=FloatSlider(min=0,max=0.2,step=0.002,value=params['mu'],description='mu', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_2=FloatSlider(min=0,max=20000,step=1,value=params['c_2'],description='c_2', style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'), logI_0=FloatSlider(min=-10,max=0,step=0.01,value=logI_0,description='log I_0', style=style,layout=slider_layout,continuous_update=False) ) display(w) ``` interactive(children=(FloatSlider(value=0.4, continuous_update=False, description='beta_1', layout=Layout(widt… ```python testfit.prparams(testfit.run_id+'.params') ``` ```python testfit.prparams() ``` params: {'N': 1, 'alpha': 0.2, 'beta_1': 0.4, 'beta_2': 0.0, 'beta_3': 0.0, 'c_0': 0.12, 'c_1': 0.04, 'c_2': 2575.0, 'gamma_1': 0.06999999999999999, 'gamma_2': 0.16666666666666666, 'gamma_3': 0.10000000000000003, 'mu': 0.042, 'p_1': 0.030000000000000013, 'p_2': 0.08333333333333334} sbparams: {'CFR': 0.05, 'DurHosp': 4, 'DurMildInf': 10, 'Exposure': 0.4, 'FracCritical': 0.10000000000000003, 'FracMild': 0.7, 'FracSevere': 0.2, 'I0': 1.5848931924611143e-08, 'ICUFrac': 0.001, 'IncubPeriod': 5, 'TimeICUDeath': 5} pfbarams: {'FracConfirmedDet': 1.0, 'FracDeathsDet': 1.0, 'FracRecoveredDet': 1.0} cbparams: {'CautionFactor': 0.1, 'CautionICUFrac': 0.1, 'CautionRetention': 66.66666666666667, 'EconomicCostOfCaution': 0.5, 'EconomicRetention': 66.66666666666667, 'EconomicStriction': 66.66666666666667, 'EconomyRelaxation': 66.66666666666667} dbparams: {'country': 'Germany', 'data_src': 'owid', 'run_name': 'SC3EI3R'} initial_values: (array([9.999997e-01, 0.000000e+00, 3.000000e-07, 0.000000e+00, 0.000000e+00, 0.000000e+00, 0.000000e+00, 0.000000e+00, 0.000000e+00, 0.000000e+00]), 0) ## Italy ```python testfit = ModelFit('SC3EI3R',country='Italy',run_id='_tst',stopdate='09/10/20') ``` ============= SC3EI3R_Italy_tst warning: changing run_id from SC3EI3R_Italy_tst to SC3EI3R_Italy_tst no file available with this run_id /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC3EI3R_Italy_tst.pk Problem loading paramfile for _tst ... using default set of parameters for model type SC3EI3R stopdate 09/10/20 ```python print([x for x in dir(testfit) if 'date' in x]) testfit.stopdate ``` ['dates', 'startdate', 'stopdate'] '09/10/20' ```python def slidefitplot(modelfit,beta_1,alpha,gamma_1,gamma_2,gamma_3,mu,c_0,c_1,c_2,logI_0): params={ 'beta_1':beta_1, 'alpha':alpha,'gamma_1':gamma_1,'gamma_2':gamma_2,'gamma_3':gamma_3, 'mu':mu, 'c_0':c_0, 'c_1':c_1, 'c_2':c_2} for pp in params: # careful to keep intact rest of params that don't have sliders modelfit.params[pp] = params[pp] modelfit.model.parameters = modelfit.params I0 = 10logI_0 x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,0.] modelfit.model.initial_values = (x0,0) modelfit.sbparams['I0'] = I0 modelfit.solveplot(species=['confirmed','deaths'],mag={'deaths':10},summing='daily',averaging='weekly',axes=None, scale='linear',plottitle= '',label='',newplot = True, gbrcolors=False, figsize = (15,10),outfile = testfit.run_id+'.pdf') ``` ```python params=testfit.params logI_0 = np.log10(testfit.sbparams['I0']) w =interactive(slidefitplot,modelfit=fixed(testfit), beta_1=FloatSlider(min=0,max=1,step=0.01,value=params['beta_1'],description='beta_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), alpha=FloatSlider(min=0,max=1,step=0.01,value=params['alpha'],description='alpha', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), gamma_1=FloatSlider(min=0,max=0.5,step=0.002,value=params['gamma_1'],description='gamma_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), gamma_2=FloatSlider(min=0,max=0.5,step=0.002,value=params['gamma_2'],description='gamma_2', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), gamma_3=FloatSlider(min=0,max=0.5,step=0.002,value=params['gamma_3'],description='gamma_3', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), mu=FloatSlider(min=0,max=0.2,step=0.002,value=params['mu'],description='mu', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_2=FloatSlider(min=0,max=20000,step=1,value=params['c_2'],description='c_2', style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'), logI_0=FloatSlider(min=-10,max=0,step=0.01,value=logI_0,description='log I_0', style=style,layout=slider_layout,continuous_update=False) ) display(w) ``` interactive(children=(FloatSlider(value=0.4, continuous_update=False, description='beta_1', layout=Layout(widt… ```python from shutil import copyfile ``` ```python testfit.prparams(testfit.run_id+'.params') paramfile = './params/'+testfit.run_id+'.pk' copyfile(paramfile,testfit.run_id+'.pk') ``` 'SC3EI3R_Italy_tst.pk' ## Brazil ```python testfit = ModelFit('SC3EI3R',country='Brazil',run_id='_tst',stopdate='09/10/20') ``` ============= SC3EI3R_Brazil_tst warning: changing run_id from SC3EI3R_Brazil_tst to SC3EI3R_Brazil_tst no file available with this run_id /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC3EI3R_Brazil_tst.pk Problem loading paramfile for _tst ... using default set of parameters for model type SC3EI3R stopdate 09/10/20 ```python print([x for x in dir(testfit) if 'date' in x]) testfit.stopdate ``` ['dates', 'startdate', 'stopdate'] '09/10/20' ```python def slidefitplot(modelfit,beta_1,alpha,gamma_1,gamma_2,gamma_3,mu,c_0,c_1,c_2,logI_0): params={ 'beta_1':beta_1, 'alpha':alpha,'gamma_1':gamma_1,'gamma_2':gamma_2,'gamma_3':gamma_3, 'mu':mu, 'c_0':c_0, 'c_1':c_1, 'c_2':c_2} for pp in params: # careful to keep intact rest of params that don't have sliders modelfit.params[pp] = params[pp] modelfit.model.parameters = modelfit.params I0 = 10logI_0 x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,0.] modelfit.model.initial_values = (x0,0) modelfit.sbparams['I0'] = I0 modelfit.solveplot(species=['confirmed','deaths'],mag={'deaths':10},summing='daily',averaging='weekly',axes=None, scale='linear',plottitle= '',label='',newplot = True, gbrcolors=False, figsize = (15,10),outfile = testfit.run_id+'.pdf') ``` ```python params=testfit.params logI_0 = np.log10(testfit.sbparams['I0']) w =interactive(slidefitplot,modelfit=fixed(testfit), beta_1=FloatSlider(min=0,max=1,step=0.01,value=params['beta_1'],description='beta_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), alpha=FloatSlider(min=0,max=1,step=0.01,value=params['alpha'],description='alpha', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), gamma_1=FloatSlider(min=0,max=0.5,step=0.002,value=params['gamma_1'],description='gamma_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), gamma_2=FloatSlider(min=0,max=0.5,step=0.002,value=params['gamma_2'],description='gamma_2', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), gamma_3=FloatSlider(min=0,max=0.5,step=0.002,value=params['gamma_3'],description='gamma_3', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), mu=FloatSlider(min=0,max=0.2,step=0.002,value=params['mu'],description='mu', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_2=FloatSlider(min=0,max=20000,step=1,value=params['c_2'],description='c_2', style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'), logI_0=FloatSlider(min=-10,max=0,step=0.01,value=logI_0,description='log I_0', style=style,layout=slider_layout,continuous_update=False) ) display(w) ``` interactive(children=(FloatSlider(value=0.4, continuous_update=False, description='beta_1', layout=Layout(widt… ```python from shutil import copyfile ``` ```python testfit.prparams(testfit.run_id+'.params') paramfile = './params/'+testfit.run_id+'.pk' copyfile(paramfile,testfit.run_id+'.pk') ``` 'SC3EI3R_Italy_tst.pk' ## Sweden ### SC3EI3R ```python testfit = ModelFit('SC3EI3R',country='Sweden',run_id='_tst',stopdate='09/10/20') ``` ============= SC3EI3R_Sweden_tst warning: changing run_id from SC3EI3R_Sweden_tst to SC3EI3R_Sweden_tst no file available with this run_id /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC3EI3R_Sweden_tst.pk Problem loading paramfile for _tst ... using default set of parameters for model type SC3EI3R stopdate 09/10/20 ```python print([x for x in dir(testfit) if 'date' in x]) testfit.stopdate ``` ['dates', 'startdate', 'stopdate'] '09/10/20' ```python def slidefitplot(modelfit,beta_1,alpha,gamma_1,gamma_2,gamma_3,mu,c_0,c_1,c_2,logI_0): params={ 'beta_1':beta_1, 'alpha':alpha,'gamma_1':gamma_1,'gamma_2':gamma_2,'gamma_3':gamma_3, 'mu':mu, 'c_0':c_0, 'c_1':c_1, 'c_2':c_2} for pp in params: # careful to keep intact rest of params that don't have sliders modelfit.params[pp] = params[pp] modelfit.model.parameters = modelfit.params I0 = 10logI_0 x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,0.] modelfit.model.initial_values = (x0,0) modelfit.sbparams['I0'] = I0 modelfit.solveplot(species=['confirmed','deaths'],mag={'deaths':10},summing='daily',averaging='weekly',axes=None, scale='linear',plottitle= '',label='',newplot = True, gbrcolors=False, figsize = (15,10),outfile = testfit.run_id+'.pdf') ``` ```python params=testfit.params logI_0 = np.log10(testfit.sbparams['I0']) w =interactive(slidefitplot,modelfit=fixed(testfit), beta_1=FloatSlider(min=0,max=1,step=0.01,value=params['beta_1'],description='beta_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), alpha=FloatSlider(min=0,max=1,step=0.01,value=params['alpha'],description='alpha', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), gamma_1=FloatSlider(min=0,max=0.5,step=0.002,value=params['gamma_1'],description='gamma_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), gamma_2=FloatSlider(min=0,max=0.5,step=0.002,value=params['gamma_2'],description='gamma_2', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), gamma_3=FloatSlider(min=0,max=0.5,step=0.002,value=params['gamma_3'],description='gamma_3', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), mu=FloatSlider(min=0,max=0.2,step=0.002,value=params['mu'],description='mu', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_2=FloatSlider(min=0,max=20000,step=1,value=params['c_2'],description='c_2', style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'), logI_0=FloatSlider(min=-10,max=0,step=0.01,value=logI_0,description='log I_0', style=style,layout=slider_layout,continuous_update=False) ) display(w) ``` interactive(children=(FloatSlider(value=0.4, continuous_update=False, description='beta_1', layout=Layout(widt… ```python from shutil import copyfile ``` ```python testfit.prparams(testfit.run_id+'.params') paramfile = './params/'+testfit.run_id+'.pk' copyfile(paramfile,testfit.run_id+'.pk') ``` 'SC3EI3R_Sweden_tst.pk' ### SC3UEI3R ```python testfit = ModelFit('SC3UEI3R',country='Sweden',run_id='_tst',stopdate='09/10/20') ``` ============= SC3UEI3R_Sweden_tst warning: changing run_id from SC3UEI3R_Sweden_tst to SC3UEI3R_Sweden_tst no file available with this run_id /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC3UEI3R_Sweden_tst.pk Problem loading paramfile for _tst ... using default set of parameters for model type SC3UEI3R stopdate 09/10/20 ```python print([x for x in dir(testfit) if 'date' in x]) testfit.stopdate ``` ['dates', 'startdate', 'stopdate'] '09/10/20' ```python def slidefitplot(modelfit,beta_1,alpha,gamma_1,gamma_2,gamma_3,mu,c_0,c_1,c_2,logI_0,k_u,k_1,k_w,kappa): params={ 'beta_1':beta_1, 'alpha':alpha,'gamma_1':gamma_1,'gamma_2':gamma_2,'gamma_3':gamma_3, 'mu':mu, 'c_0':c_0, 'c_1':c_1, 'c_2':c_2,'k_u':k_u, 'k_1':k_1, 'k_w':k_w, 'kappa':kappa} for pp in params: # careful to keep intact rest of params that don't have sliders modelfit.params[pp] = params[pp] modelfit.model.parameters = modelfit.params I0 = 10logI_0 x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,0.,0.0,0.0] modelfit.model.initial_values = (x0,0) modelfit.sbparams['I0'] = I0 modelfit.solveplot(species=['confirmed','deaths'],mag={'deaths':10},summing='daily',averaging='weekly',axes=None, scale='linear',plottitle= '',label='',newplot = True, gbrcolors=False, figsize = (15,10),outfile = testfit.run_id+'.pdf') ``` ```python params=testfit.params logI_0 = np.log10(testfit.sbparams['I0']) w =interactive(slidefitplot,modelfit=fixed(testfit), beta_1=FloatSlider(min=0,max=1,step=0.01,value=params['beta_1'],description='beta_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), alpha=FloatSlider(min=0,max=1,step=0.01,value=params['alpha'],description='alpha', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), gamma_1=FloatSlider(min=0,max=0.5,step=0.002,value=params['gamma_1'],description='gamma_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), gamma_2=FloatSlider(min=0,max=0.5,step=0.002,value=params['gamma_2'],description='gamma_2', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), gamma_3=FloatSlider(min=0,max=0.5,step=0.002,value=params['gamma_3'],description='gamma_3', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), mu=FloatSlider(min=0,max=0.2,step=0.002,value=params['mu'],description='mu', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_2=FloatSlider(min=0,max=20000,step=1,value=params['c_2'],description='c_2', style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'), logI_0=FloatSlider(min=-10,max=0,step=0.01,value=logI_0,description='log I_0', style=style,layout=slider_layout,continuous_update=False), k_u=FloatSlider(min=0,max=1,step=0.001,value=params['k_u'],description='k_u', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), k_1=FloatSlider(min=0,max=1,step=0.001,value=params['k_1'],description='k_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), k_w=FloatSlider(min=0,max=1,step=0.001,value=params['k_w'],description='k_w', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), kappa=FloatSlider(min=0,max=1,step=0.001,value=params['kappa'],description='kappa', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f') ) display(w) ``` interactive(children=(FloatSlider(value=0.4, continuous_update=False, description='beta_1', layout=Layout(widt… ```python from shutil import copyfile ``` ```python testfit.prparams(testfit.run_id+'.params') paramfile = './params/'+testfit.run_id+'.pk' copyfile(paramfile,testfit.run_id+'.pk') ``` 'SC3UEI3R_Sweden_tst.pk' # Data ## Oxford data ```python owid_file = '../../covid-policy-tracker/data/OxCGRT_latest.csv' oxdat = [] with open(owid_file,'r',newline='') as fp: myreader = csv.DictReader(fp,delimiter=',') # rows = list(itertools.islice(myreader,4)) for row in myreader: oxdat.append(row) ``` ```python oxdat[0].keys() ``` odict_keys(['CountryName', 'CountryCode', 'RegionName', 'RegionCode', 'Date', 'C1_School closing', 'C1_Flag', 'C2_Workplace closing', 'C2_Flag', 'C3_Cancel public events', 'C3_Flag', 'C4_Restrictions on gatherings', 'C4_Flag', 'C5_Close public transport', 'C5_Flag', 'C6_Stay at home requirements', 'C6_Flag', 'C7_Restrictions on internal movement', 'C7_Flag', 'C8_International travel controls', 'E1_Income support', 'E1_Flag', 'E2_Debt/contract relief', 'E3_Fiscal measures', 'E4_International support', 'H1_Public information campaigns', 'H1_Flag', 'H2_Testing policy', 'H3_Contact tracing', 'H4_Emergency investment in healthcare', 'H5_Investment in vaccines', 'M1_Wildcard', 'ConfirmedCases', 'ConfirmedDeaths', 'StringencyIndex', 'StringencyIndexForDisplay', 'StringencyLegacyIndex', 'StringencyLegacyIndexForDisplay', 'GovernmentResponseIndex', 'GovernmentResponseIndexForDisplay', 'ContainmentHealthIndex', 'ContainmentHealthIndexForDisplay', 'EconomicSupportIndex', 'EconomicSupportIndexForDisplay']) ```python countries = list(set([(dd['CountryName'],dd['CountryCode']) for dd in oxdat])) ``` ```python len(countries) ``` 185 ```python [countries[i] for i in range(10)] ``` [('Saudi Arabia', 'SAU'), ('South Korea', 'KOR'), ('Greece', 'GRC'), ('Fiji', 'FJI'), ('Montserrat', 'MSR'), ('Ecuador', 'ECU'), ('Rwanda', 'RWA'), ('Romania', 'ROU'), ('Turks and Caicos Islands', 'TCA'), ('Slovak Republic', 'SVK')] ```python dates = np.sort(list([dd['Date'] for dd in oxdat])) dates_uniq = np.unique(dates) dates_uniq ``` array(['20200101', '20200102', '20200103', '20200104', '20200105', '20200106', '20200107', '20200108', '20200109', '20200110', '20200111', '20200112', '20200113', '20200114', '20200115', '20200116', '20200117', '20200118', '20200119', '20200120', '20200121', '20200122', '20200123', '20200124', '20200125', '20200126', '20200127', '20200128', '20200129', '20200130', '20200131', '20200201', '20200202', '20200203', '20200204', '20200205', '20200206', '20200207', '20200208', '20200209', '20200210', '20200211', '20200212', '20200213', '20200214', '20200215', '20200216', '20200217', '20200218', '20200219', '20200220', '20200221', '20200222', '20200223', '20200224', '20200225', '20200226', '20200227', '20200228', '20200229', '20200301', '20200302', '20200303', '20200304', '20200305', '20200306', '20200307', '20200308', '20200309', '20200310', '20200311', '20200312', '20200313', '20200314', '20200315', '20200316', '20200317', '20200318', '20200319', '20200320', '20200321', '20200322', '20200323', '20200324', '20200325', '20200326', '20200327', '20200328', '20200329', '20200330', '20200331', '20200401', '20200402', '20200403', '20200404', '20200405', '20200406', '20200407', '20200408', '20200409', '20200410', '20200411', '20200412', '20200413', '20200414', '20200415', '20200416', '20200417', '20200418', '20200419', '20200420', '20200421', '20200422', '20200423', '20200424', '20200425', '20200426', '20200427', '20200428', '20200429', '20200430', '20200501', '20200502', '20200503', '20200504', '20200505', '20200506', '20200507', '20200508', '20200509', '20200510', '20200511', '20200512', '20200513', '20200514', '20200515', '20200516', '20200517', '20200518', '20200519', '20200520', '20200521', '20200522', '20200523', '20200524', '20200525', '20200526', '20200527', '20200528', '20200529', '20200530', '20200531', '20200601', '20200602', '20200603', '20200604', '20200605', '20200606', '20200607', '20200608', '20200609', '20200610', '20200611', '20200612', '20200613', '20200614', '20200615', '20200616', '20200617', '20200618', '20200619', '20200620', '20200621', '20200622', '20200623', '20200624', '20200625', '20200626', '20200627', '20200628', '20200629', '20200630', '20200701', '20200702', '20200703', '20200704', '20200705', '20200706', '20200707', '20200708', '20200709', '20200710', '20200711', '20200712', '20200713', '20200714', '20200715', '20200716', '20200717', '20200718', '20200719', '20200720', '20200721', '20200722', '20200723', '20200724', '20200725', '20200726', '20200727', '20200728', '20200729', '20200730', '20200731', '20200801', '20200802', '20200803', '20200804', '20200805', '20200806', '20200807', '20200808', '20200809', '20200810', '20200811', '20200812', '20200813', '20200814', '20200815', '20200816', '20200817', '20200818', '20200819', '20200820', '20200821', '20200822', '20200823', '20200824', '20200825', '20200826', '20200827', '20200828', '20200829', '20200830', '20200831'], dtype='<U8') ```python foo = [(dd['Date'],dd['C1_School closing'],dd['C1_Flag']) for dd in oxdat if dd['CountryCode']=='USA'] ``` ```python xx = foo[0:10] print(xx) xx=foo[-10:] print(xx) ``` [('20200101', '0.00', ''), ('20200102', '0.00', ''), ('20200103', '0.00', ''), ('20200104', '0.00', ''), ('20200105', '0.00', ''), ('20200106', '0.00', ''), ('20200107', '0.00', ''), ('20200108', '0.00', ''), ('20200109', '0.00', ''), ('20200110', '0.00', '')] [('20200822', '', ''), ('20200823', '', ''), ('20200824', '', ''), ('20200825', '', ''), ('20200826', '', ''), ('20200827', '', ''), ('20200828', '', ''), ('20200829', '', ''), ('20200830', '', ''), ('20200831', '', '')] ```python ccodes = [cc[1] for cc in countries] for cc in ccodes: foo = [(dd['CountryCode'],dd['RegionName']) for dd in oxdat if dd['CountryCode'] == cc] if len(set([x[1] for x in foo]))>1: print(cc) print( set([x[1] for x in foo])) ``` USA {'', 'New Hampshire', 'California', 'Maryland', 'Tennessee', 'Rhode Island', 'Georgia', 'Louisiana', 'Virgin Islands', 'Arkansas', 'Washington', 'Iowa', 'Hawaii', 'Virginia', 'Colorado', 'Vermont', 'North Carolina', 'Oklahoma', 'West Virginia', 'Indiana', 'Delaware', 'Alaska', 'Idaho', 'Nebraska', 'Wyoming', 'Illinois', 'Texas', 'Missouri', 'New Jersey', 'Minnesota', 'South Dakota', 'Kentucky', 'New York', 'Ohio', 'North Dakota', 'Florida', 'Mississippi', 'Alabama', 'Montana', 'South Carolina', 'Wisconsin', 'Michigan', 'Washington DC', 'Oregon', 'Kansas', 'Connecticut', 'Maine', 'Arizona', 'New Mexico', 'Massachusetts', 'Nevada', 'Pennsylvania', 'Utah'} GBR {'', 'Wales', 'England', 'Northern Ireland', 'Scotland'} ```python [cc for cc in countries if cc[1]=='USA'] ``` [('United States', 'USA')] ```python foo = [(dd['Date'],dd['C1_School closing'],dd['C1_Flag']) for dd in oxdat if dd['CountryCode']=='USA'] fmt = '%Y%m%d' first = 31+22-1 xx = [datetime.datetime.strptime(dd[0],fmt) for dd in foo][first:] yy = [] for dd in foo[first:]: try: yy.append(float(dd[1])) except: yy.append(float('NaN')) locator = mdates.MonthLocator() formatter= mdates.ConciseDateFormatter(locator) fig,ax = plt.subplots(1,1) ax.step(xx,yy) ax.xaxis.set_major_formatter(formatter) ax.xaxis.set_major_locator(locator) for tick in ax.get_xticklabels(): tick.set_rotation(40) plt.title('USA') plt.show() ``` Note: no wildcard data: ```python foo = [(dd['Date'],dd['CountryCode'],dd['M1_Wildcard']) for dd in oxdat if dd['M1_Wildcard'] != ''] ``` ```python foo ``` [] Notes: first version of `datcodes` leaves out all flags (general vs. targeted implementation) * 2nd version leaves out all variables measured in dollars ```python datcodes = ['C1_School closing', 'C2_Workplace closing', 'C3_Cancel public events', 'C4_Restrictions on gatherings', 'C5_Close public transport', 'C6_Stay at home requirements', 'C7_Restrictions on internal movement', 'C8_International travel controls', 'E1_Income support', 'E2_Debt/contract relief', 'E3_Fiscal measures', 'E4_International support', 'H1_Public information campaigns', 'H2_Testing policy', 'H3_Contact tracing', 'H4_Emergency investment in healthcare', 'H5_Investment in vaccines'] datcodes = ['C1_School closing', 'C2_Workplace closing', 'C3_Cancel public events', 'C4_Restrictions on gatherings', 'C5_Close public transport', 'C6_Stay at home requirements', 'C7_Restrictions on internal movement', 'C8_International travel controls', 'E1_Income support', 'E2_Debt/contract relief', 'H1_Public information campaigns', 'H2_Testing policy', 'H3_Contact tracing'] ``` ```python countrycodes = list(set([dd['CountryCode'] for dd in oxdat if Float(dd['ConfirmedCases'])>10000])) len(countrycodes) ``` 90 ```python countriesox = list(set([dd['CountryName'] for dd in oxdat if Float(dd['ConfirmedCases'])>10000])) ``` ```python len(countriesox) ``` 90 ```python def Float(x): try: rtn = float(x) except: rtn = float('NaN') return rtn ``` ```python datcodes ``` ['C1_School closing', 'C2_Workplace closing', 'C3_Cancel public events', 'C4_Restrictions on gatherings', 'C5_Close public transport', 'C6_Stay at home requirements', 'C7_Restrictions on internal movement', 'C8_International travel controls', 'E1_Income support', 'E2_Debt/contract relief', 'H1_Public information campaigns', 'H2_Testing policy', 'H3_Contact tracing'] ```python def get_codedat(countrycode='USA'): codedat = [] foo = [[dd['Date']]+ [Float(dd[c]) for c in datcodes] for dd in oxdat if dd['CountryCode']==countrycode] return foo ``` ```python get_codedat() ``` [['20200101', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200102', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200103', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200104', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200105', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200106', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200107', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200108', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200109', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200110', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200111', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200112', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200113', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200114', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200115', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200116', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200117', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200118', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200119', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200120', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200121', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200122', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200123', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200124', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200125', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200126', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200127', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200128', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200129', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200130', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200131', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200201', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200202', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200203', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200204', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200205', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200206', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200207', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200208', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200209', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200210', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200211', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200212', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200213', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200214', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200215', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200216', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200217', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200218', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200219', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200220', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200221', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200222', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200223', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200224', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200225', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200226', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200227', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0], ['20200228', 0.0, 0.0, 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1.0, 0.0, 1.0, 3.0, 1.0, 2.0, 2.0, 1.0, 2.0], ['20200321', 3.0, 2.0, 2.0, 4.0, 1.0, 0.0, 1.0, 3.0, 1.0, 2.0, 2.0, 1.0, 2.0], ['20200322', 3.0, 3.0, 2.0, 4.0, 1.0, 2.0, 1.0, 3.0, 1.0, 2.0, 2.0, 1.0, 2.0], ['20200323', 3.0, 3.0, 2.0, 4.0, 1.0, 2.0, 1.0, 3.0, 1.0, 2.0, 2.0, 1.0, 2.0], ['20200324', 3.0, 3.0, 2.0, 4.0, 1.0, 2.0, 1.0, 3.0, 1.0, 2.0, 2.0, 1.0, 2.0], ['20200325', 3.0, 3.0, 2.0, 4.0, 1.0, 2.0, 1.0, 3.0, 1.0, 2.0, 2.0, 1.0, 2.0], ['20200326', 3.0, 3.0, 2.0, 4.0, 1.0, 2.0, 1.0, 3.0, 1.0, 2.0, 2.0, 1.0, 2.0], ['20200327', 3.0, 3.0, 2.0, 4.0, 1.0, 2.0, 1.0, 3.0, 2.0, 2.0, 2.0, 1.0, 2.0], ['20200328', 3.0, 3.0, 2.0, 4.0, 1.0, 2.0, 2.0, 3.0, 2.0, 2.0, 2.0, 1.0, 2.0], ['20200329', 3.0, 3.0, 2.0, 4.0, 1.0, 2.0, 2.0, 3.0, 2.0, 2.0, 2.0, 1.0, 2.0], ['20200330', 3.0, 3.0, 2.0, 4.0, 1.0, 2.0, 2.0, 3.0, 2.0, 2.0, 2.0, 1.0, 2.0], ['20200331', 3.0, 3.0, 2.0, 4.0, 1.0, 2.0, 2.0, 3.0, 2.0, 2.0, 2.0, 1.0, 2.0], ['20200401', 3.0, 3.0, 2.0, 4.0, 1.0, 2.0, 2.0, 3.0, 2.0, 2.0, 2.0, 1.0, 2.0], 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2.0, 1.0, 2.0, 2.0, 2.0], ['20200415', 3.0, 3.0, 2.0, 4.0, 2.0, 2.0, 2.0, 3.0, 2.0, 1.0, 2.0, 2.0, 2.0], ['20200416', 3.0, 3.0, 2.0, 4.0, 2.0, 2.0, 2.0, 3.0, 2.0, 1.0, 2.0, 2.0, 2.0], ['20200417', 3.0, 3.0, 2.0, 4.0, 2.0, 2.0, 2.0, 3.0, 2.0, 1.0, 2.0, 2.0, 2.0], ['20200418', 3.0, 3.0, 2.0, 4.0, 2.0, 2.0, 2.0, 3.0, 2.0, 1.0, 2.0, 2.0, 2.0], ['20200419', 3.0, 3.0, 2.0, 4.0, 2.0, 2.0, 2.0, 3.0, 2.0, 1.0, 2.0, 2.0, 2.0], ['20200420', 3.0, 3.0, 2.0, 4.0, 2.0, 2.0, 2.0, 3.0, 2.0, 1.0, 2.0, 2.0, 2.0], ['20200421', 3.0, 3.0, 2.0, 4.0, 2.0, 2.0, 2.0, 3.0, 2.0, 1.0, 2.0, 2.0, 2.0], ['20200422', 3.0, 3.0, 2.0, 4.0, 2.0, 2.0, 2.0, 3.0, 2.0, 1.0, 2.0, 2.0, 2.0], ['20200423', 3.0, 3.0, 2.0, 4.0, 2.0, 2.0, 2.0, 3.0, 2.0, 1.0, 2.0, 2.0, 2.0], ['20200424', 3.0, 2.0, 2.0, 3.0, 2.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 2.0, 2.0], ['20200425', 3.0, 2.0, 2.0, 3.0, 2.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 2.0, 2.0], ['20200426', 3.0, 2.0, 2.0, 3.0, 2.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 2.0, 2.0], ['20200427', 3.0, 2.0, 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['20200729', 3.0, 2.0, 1.0, 3.0, 0.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 3.0, 1.0], ['20200730', 3.0, 2.0, 1.0, 3.0, 0.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 3.0, 1.0], ['20200731', 3.0, 2.0, 1.0, 3.0, 0.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 3.0, 1.0], ['20200801', 3.0, 2.0, 1.0, 3.0, 0.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 3.0, 1.0], ['20200802', 3.0, 2.0, 1.0, 3.0, 0.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 3.0, 1.0], ['20200803', 3.0, 2.0, 1.0, 3.0, 0.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 3.0, 1.0], ['20200804', 3.0, 2.0, 1.0, 3.0, 0.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 3.0, 1.0], ['20200805', 3.0, 2.0, 1.0, 3.0, 0.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 3.0, 1.0], ['20200806', 3.0, 2.0, 1.0, 3.0, 0.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 3.0, 1.0], ['20200807', 3.0, 2.0, 1.0, 3.0, 0.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 3.0, 1.0], ['20200808', 3.0, 2.0, 1.0, 3.0, 0.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 3.0, 1.0], ['20200809', 3.0, 2.0, 1.0, 3.0, 0.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 3.0, 1.0], ['20200810', 3.0, 2.0, 1.0, 3.0, 0.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 3.0, 1.0], ['20200811', 3.0, 2.0, 1.0, 3.0, 0.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 3.0, 1.0], ['20200812', 3.0, 2.0, 1.0, 3.0, 0.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 3.0, 1.0], ['20200813', 3.0, 2.0, 1.0, 3.0, 0.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 3.0, 1.0], ['20200814', 3.0, 2.0, 1.0, 3.0, 0.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 3.0, 1.0], ['20200815', 3.0, 2.0, 1.0, 3.0, 0.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 3.0, 1.0], ['20200816', 3.0, 2.0, 1.0, 3.0, 0.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 3.0, 1.0], ['20200817', 3.0, 2.0, 1.0, 3.0, 0.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 3.0, 1.0], ['20200818', 3.0, 2.0, 1.0, 3.0, 0.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 3.0, 2.0], ['20200819', 3.0, 2.0, 1.0, 3.0, 0.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 3.0, 2.0], ['20200820', 3.0, 2.0, 1.0, 3.0, 0.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 3.0, 2.0], ['20200821', nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan], ['20200822', nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan], ['20200823', nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan], ['20200824', nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan], ['20200825', nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan], ['20200826', nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan], ['20200827', nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan], ['20200828', nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan], ['20200829', nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan], ['20200830', nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan], ['20200831', nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan], ['20200101', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200102', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200103', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200104', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200105', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200106', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200107', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200108', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200109', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200110', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200111', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200112', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200113', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200114', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200115', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200116', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200117', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200118', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200119', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200120', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], ['20200121', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 1.0], ['20200122', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 1.0], ['20200123', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 1.0], ['20200124', 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 1.0], ...] ### Stringency ```python [x for x in oxdat[0]] ``` ['CountryName', 'CountryCode', 'RegionName', 'RegionCode', 'Date', 'C1_School closing', 'C1_Flag', 'C2_Workplace closing', 'C2_Flag', 'C3_Cancel public events', 'C3_Flag', 'C4_Restrictions on gatherings', 'C4_Flag', 'C5_Close public transport', 'C5_Flag', 'C6_Stay at home requirements', 'C6_Flag', 'C7_Restrictions on internal movement', 'C7_Flag', 'C8_International travel controls', 'E1_Income support', 'E1_Flag', 'E2_Debt/contract relief', 'E3_Fiscal measures', 'E4_International support', 'H1_Public information campaigns', 'H1_Flag', 'H2_Testing policy', 'H3_Contact tracing', 'H4_Emergency investment in healthcare', 'H5_Investment in vaccines', 'M1_Wildcard', 'ConfirmedCases', 'ConfirmedDeaths', 'StringencyIndex', 'StringencyIndexForDisplay', 'StringencyLegacyIndex', 'StringencyLegacyIndexForDisplay', 'GovernmentResponseIndex', 'GovernmentResponseIndexForDisplay', 'ContainmentHealthIndex', 'ContainmentHealthIndexForDisplay', 'EconomicSupportIndex', 'EconomicSupportIndexForDisplay'] ```python country='DEU' reg = '' # to get entire country for UK, USA foo = [(dd['Date'],dd['StringencyIndex'],dd['C1_Flag']) for dd in oxdat if dd['CountryCode']==country and dd['RegionName']==reg] fmt_ox = '%Y%m%d' first = 31+22-1 # to get to Feb 22 where data starts xx = [datetime.datetime.strptime(dd[0],fmt_ox) for dd in foo][first:] yy = [] for dd in foo[first:]: try: yy.append(float(dd[1])) except: yy.append(float('NaN')) locator = mdates.MonthLocator() formatter= mdates.ConciseDateFormatter(locator) fig,ax = plt.subplots(1,1) ax.step(xx,yy) ax.xaxis.set_major_formatter(formatter) ax.xaxis.set_major_locator(locator) for tick in ax.get_xticklabels(): tick.set_rotation(40) plt.title(country+' Stringency') plt.show() ``` ```python print("number of countries",len(countrycodes)) ``` number of countries 90 ```python def get_stringency(country,to_database='jhu'): global fmt_jhu,fmt_ox,fmt_owid,dates_jhu,dates_owid stri = [(dd['Date'],dd['StringencyIndex'],dd['C1_Flag']) for dd in oxdat if dd['CountryName']==country and dd['RegionName']==''] fmt_ox = '%Y%m%d' #first = 31+22-1 # to get to Feb 22 where data starts dates_t = [datetime.datetime.strptime(dd[0],fmt_ox) for dd in stri] if to_database=='jhu': fmt = fmt_jhu dates = dates_jhu elif to_database=='owid': fmt = fmt_jhu # already converted from fmt_owid dates = dates_owid print('first date',dates[0],'last date',dates[-1]) firstdate = datetime.datetime.strptime(dates[0],fmt) lastdate = datetime.datetime.strptime(dates[-1],fmt) ndates=len(dates) daystart = (dates_t[0]-firstdate).days #dates = [date.strftime(fmt_jhu) for date in dates_t] yy = np.zeros((lastdate-firstdate).days+1) for i,dd in enumerate(stri): if daystart+i < len(dates): try: yy[daystart+i]=float(dd[1]) except: yy[daystart+i]=float('NaN') return([dates,yy]) ``` ```python stri = get_stringency('Germany',to_database='jhu') plt.plot([datetime.datetime.strptime(date,fmt_jhu) for date in stri[0]],stri[1]); ``` ```python stri = get_stringency('Germany') datesf = [datetime.datetime.strptime(date,fmt_jhu) for date in stri[0]] plt.figure(figsize=(12,12)) plt.plot(datesf,np.array(stri[1])/100.,label='Germany'); stri = get_stringency('Spain') plt.plot(datesf,np.array(stri[1])/100.,label='Spain'); stri = get_stringency('Italy') plt.plot(datesf,np.array(stri[1])/100.,label='Italy'); stri = get_stringency('United States') plt.plot(datesf,np.array(stri[1])/100.,label='United States'); plt.legend() ``` ```python import matplotlib.dates as mdates ``` ```python max_rows=12 max_cols=8 idx = 0 fig, axes = plt.subplots(nrows=max_rows, ncols=max_cols, figsize=(20,40)) fig.autofmt_xdate() for idx, country in enumerate(countriesox): foo = get_stringency(country) row = idx // max_cols col = idx % max_cols axes[row, col].set_ylim((0,110)) axes[row, col].xaxis.set_major_locator(plt.MaxNLocator(3)) axes[row, col].fmt_xdata = mdates.DateFormatter('%Y-%m-%d') axes[row, col].plot(foo[0],foo[1]) axes[row, col].set_title(country) for idx in range(len(countriesox),max_rowsmax_cols): row = idx // max_cols col = idx % max_cols axes[row, col].axis("off") ``` ```python len(countriesox) ``` 90 ```python import pickle as pk ``` ```python dicstringency={} for cc in countriesox: foo = get_stringency(cc) dicstringency[cc] = [x[1] for x in foo] # skip date dates = [x[0] for x in foo] stringency={} stringency['dates'] = dates stringency['data'] = dicstringency pk.dump(stringency,open('stringency.pk','wb')) ``` first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 first date 1/22/20 last date 8/30/20 ## Plots of data for Cautionary Model comparison Comment out line 1110 in pwlf.py (in /⁨usr⁩/local⁩/lib⁩/⁨python3.7⁩/site-packages⁩/pwlf⁩ directory) print("Warning: zero length interval encountered in pwlf.py calc_slopes"). to remove repeated warnings, which don't seem to harm final result Warning: zero length interval encountered in pwlf.py calc_slopes ```python plotCountry_(['Italy','Spain','Germany','France','United Kingdom','Sweden','Turkey'], 'confirmed','cum_av_weekly',firstdate='02/15/20',lastdate='09/1/20',fittype='piecewise-linear',nsegments=4) if savefigs: plt.savefig("covid-19-caution/figures/fig1a.pdf",bbox_inches='tight') ``` ```python plotCountry_(['Italy','Spain','Germany','France','United Kingdom','Sweden','Turkey'], 'confirmed','daily_av_weekly',firstdate='02/15/20',lastdate='08/31/20',database='jhu') if savefigs: plt.savefig("covid-19-caution/figures/fig1b.pdf",bbox_inches='tight') ``` ```python plotCountry_(['Italy','Spain','Germany','France','United Kingdom','Sweden','Turkey'], 'confirmed','daily_av_weekly',firstdate='02/15/20',lastdate='08/31/20',database='owid') if savefigs: plt.savefig("covid-19-caution/figures/fig1b_owid.pdf",bbox_inches='tight') ``` # Parameter fitting ## Fitting via sliders ### SC3EIR Model ```python len(t) ``` ```python model = 'SC3EIR' params={'beta':0.25,'alpha':1./5.,'gamma':0.1,'mu':0.05,'c_0':0.3, 'c_1':1/14., 'c_2':2000} def slidefitplot(beta,alpha,gamma,mu,c_0,c_1,c_2,logI_0): params={ 'beta':beta, 'alpha':alpha, 'gamma':gamma, 'mu':mu, 'c_0':c_0, 'c_1':c_1, 'c_2':c_2} cmodels[model].parameters = params I0 = 10logI_0 x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.] cmodels[model].initial_values = (x0,t[0]) solveplot(smodels=[model],species=['confirmed','recovered','deaths'],tmax=len(t),summing='daily',fitdata=y_jhu[test_country],scale='linear',plottitle= '',label='confirmed',newplot = True) ``` ```python interact(slidefitplot, beta=FloatSlider(min=0,max=1,step=0.01,value=params['beta'],description='beta', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), alpha=FloatSlider(min=0,max=1,step=0.01,value=params['alpha'],description='alpha', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), gamma=FloatSlider(min=0,max=1,step=0.01,value=params['gamma'],description='gamma', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), mu=FloatSlider(min=0,max=0.2,step=0.002,value=params['mu'],description='mu', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_2=FloatSlider(min=0,max=5000,step=1,value=params['c_2'],description='c_2', style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'), logI_0=FloatSlider(min=-10,max=0,step=0.01,value=-6,description='log I_0', style=style,layout=slider_layout,continuous_update=False)) ``` ### SC3EI3R Model #### Germany ```python # assumed data starting on firstdate test_country='Germany' N = 80000000 firstdate = '01/25/20' lastdate = '01/08/20' xx,xxf,yy0 = get_country_data(test_country,'confirmed',firstdate=firstdate,lastdate=lastdate) xx,xxf,yy1 = get_country_data(test_country,'recovered',firstdate=firstdate,lastdate=lastdate) xx,xxf,yy2 = get_country_data(test_country,'deaths',firstdate=firstdate,lastdate=lastdate) print(xxf) y_jhu={} y_jhu[test_country] = np.array([[yy0[i],yy1[i],yy2[i]] for i in range(0,len(yy0))])/N # data = np.array([[xxf[i],yy0[i],yy1[i],yy2[i]] for i in range(len(yy))]) # print(data) lastday = len(y_jhu[test_country]) print('days 0 to',lastday,'data stored in y_jhu') ``` ```python len(t) ``` ```python (1.0/TimeICUDeath)(CFR/FracCritical) ``` ```python model = 'SC3EI3R' # Define parameters based on clinical observations Dr. Alison Exposure=0.4 # Rate coefficient for exposure per individual in contact per day IncubPeriod=5 #Incubation period, days DurMildInf=10 #Duration of mild infections, days : includes time for reg. of recovery FracMild=0.7 #Fraction of infections that are mild FracSevere=0.20 #Fraction of infections that are severe FracCritical=0.1 #Fraction of infections that are critical CFR=0.05 #Case fatality rate (fraction of infections resulting in death) TimeICUDeath=5 #Time from ICU admission to death, days DurHosp=4 #Duration of hospitalization, days : includes 4 day reg of recovery # Model fitting extension to allow for incomplete detection FracConfirmedDet=1.0 # Fraction of recovered individuals measured : plots made with this parameter NYI FracRecoveredDet=FracConfirmedDet # Fraction of recovered individuals measured FracDeathsDet=1.0 # Model extension by John McCaskill to include caution CautionFactor= 0.1 # Fractional reduction of exposure rate for cautioned individuals CautionRetention= 60. # Duration of cautionary state of susceptibles (2 weeks) CautionICUFrac= 0.1 # Fraction of ICUs occupied leading to transition to caution @ 1/day ICUFrac= 0.001 # Fraction of ICUs relative to population size N params = {'beta_1' : Exposure/sum(x0_SC3EI3R), 'beta_2' : 0.0, 'beta_3' : 0.0, 'alpha' : 1.0/IncubPeriod, 'gamma_1': (1.0/DurMildInf)FracMild, 'gamma_2': (1.0/DurHosp)-(1/DurHosp)(FracCritical/(FracCritical+FracSevere)), 'gamma_3': (1.0/TimeICUDeath)-(1/TimeICUDeath)(CFR/FracCritical), 'p_1' : (1.0/DurMildInf)-(1.0/DurMildInf)FracMild, 'p_2' : (1.0/DurHosp)(FracCritical/(FracCritical+FracSevere)), 'mu' : (1.0/TimeICUDeath)(CFR/FracCritical), 'c_0' : CautionFactor, 'c_1' : 1.0/CautionRetention, 'c_2' : 1.0/(sum(x0_SC3EI3R)ICUFracCautionICUFrac), 'N' : sum(x0_SC3EI3R)} print(params) SC3EI3R_model.parameters = params def slidefitplot(beta_1,mu,c_0,c_1,c_2,logI_0): params={ 'beta_1':beta_1, 'mu':mu, 'c_0':c_0, 'c_1':c_1, 'c_2':c_2} cmodels[model].parameters = params I0 = 10*logI_0 x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,0.] cmodels[model].initial_values = (x0,t[0]) weights=np.array([1.,1.,1.]) solveplot(smodels=[model],species=['confirmed','recovered','deaths_x10'],tmax=len(t),summing='daily',averaging='weekly',fitdata=y_jhu[test_country]weights,scale='linear',plottitle= '',label='confirmed',newplot = True, figsize = (15,15)) ``` ```python w =interactive(slidefitplot, beta_1=FloatSlider(min=0,max=1,step=0.01,value=params['beta_1'],description='beta_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), mu=FloatSlider(min=0,max=0.2,step=0.002,value=params['mu'],description='mu', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_2=FloatSlider(min=0,max=20000,step=1,value=params['c_2'],description='c_2', style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'), logI_0=FloatSlider(min=-10,max=0,step=0.01,value=-6,description='log I_0', style=style,layout=slider_layout,continuous_update=False) ) display(w) ``` ```python params=w.kwargs print(params) ``` ```python ``` #### Spain ```python # assumed data starting on firstdate test_country='Spain' N = 80000000 firstdate = '01/25/20' lastdate = '01/08/20' xx,xxf,yy0 = get_country_data(test_country,'confirmed',firstdate=firstdate,lastdate=lastdate) xx,xxf,yy1 = get_country_data(test_country,'recovered',firstdate=firstdate,lastdate=lastdate) xx,xxf,yy2 = get_country_data(test_country,'deaths',firstdate=firstdate,lastdate=lastdate) print(xxf) y_jhu={} y_jhu[test_country] = np.array([[yy0[i],yy1[i],yy2[i]] for i in range(0,len(yy0))])/N # data = np.array([[xxf[i],yy0[i],yy1[i],yy2[i]] for i in range(len(yy))]) # print(data) lastday = len(y_jhu[test_country]) print('days 0 to',lastday,'data stored in y_jhu') ``` ```python len(t) ``` ```python (1.0/TimeICUDeath)(CFR/FracCritical) ``` ```python model = 'SC3EI3R' # Define parameters based on clinical observations Dr. Alison Exposure=0.4 # Rate coefficient for exposure per individual in contact per day IncubPeriod=5 #Incubation period, days DurMildInf=10 #Duration of mild infections, days : includes time for reg. of recovery FracMild=0.7 #Fraction of infections that are mild FracSevere=0.20 #Fraction of infections that are severe FracCritical=0.1 #Fraction of infections that are critical CFR=0.05 #Case fatality rate (fraction of infections resulting in death) TimeICUDeath=5 #Time from ICU admission to death, days DurHosp=4 #Duration of hospitalization, days : includes 4 day reg of recovery # Model fitting extension to allow for incomplete detection FracConfirmedDet=0.5 # Fraction of confirmed individuals measured : plots made with this parameter NYI FracRecoveredDet=FracConfirmedDet # Fraction of recovered individuals measured FracDeathsDet=1.0 # Model extension by John McCaskill to include caution CautionFactor= 0.1 # Fractional reduction of exposure rate for cautioned individuals CautionRetention= 60. # Duration of cautionary state of susceptibles (2 weeks) CautionICUFrac= 0.1 # Fraction of ICUs occupied leading to transition to caution @ 1/day ICUFrac= 0.001 # Fraction of ICUs relative to population size N params = {'beta_1' : Exposure/sum(x0_SC3EI3R), 'beta_2' : 0.0, 'beta_3' : 0.0, 'alpha' : 1.0/IncubPeriod, 'gamma_1': (1.0/DurMildInf)FracMild, 'gamma_2': (1.0/DurHosp)-(1/DurHosp)(FracCritical/(FracCritical+FracSevere)), 'gamma_3': (1.0/TimeICUDeath)-(1/TimeICUDeath)(CFR/FracCritical), 'p_1' : (1.0/DurMildInf)-(1.0/DurMildInf)FracMild, 'p_2' : (1.0/DurHosp)(FracCritical/(FracCritical+FracSevere)), 'mu' : (1.0/TimeICUDeath)(CFR/FracCritical), 'c_0' : CautionFactor, 'c_1' : 1.0/CautionRetention, 'c_2' : 1.0/(sum(x0_SC3EI3R)ICUFracCautionICUFrac), 'N' : sum(x0_SC3EI3R)} print(params) SC3EI3R_model.parameters = params def slidefitplot(beta_1,mu,c_0,c_1,c_2,logI_0): params={ 'beta_1':beta_1, 'mu':mu, 'c_0':c_0, 'c_1':c_1, 'c_2':c_2} cmodels[model].parameters = params I0 = 10logI_0 x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,0.] cmodels[model].initial_values = (x0,t[0]) weights=np.array([1.,1.,1.]) solveplot(smodels=[model],species=['confirmed','recovered','deaths_x10'],tmax=len(t),summing='daily',averaging='weekly',fitdata=y_jhu[test_country]weights,scale='linear',plottitle= '',label='confirmed',newplot = True, figsize = (15,15)) ``` ```python w =interactive(slidefitplot, beta_1=FloatSlider(min=0,max=1,step=0.01,value=params['beta_1'],description='beta_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), mu=FloatSlider(min=0,max=0.2,step=0.002,value=params['mu'],description='mu', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_2=FloatSlider(min=0,max=20000,step=1,value=params['c_2'],description='c_2', style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'), logI_0=FloatSlider(min=-10,max=0,step=0.01,value=-6,description='log I_0', style=style,layout=slider_layout,continuous_update=False) ) display(w) ``` ```python params=w.kwargs print(params) ``` #### Italy ```python # assumed data starting on firstdate test_country='Italy' N = 66650000 firstdate = '01/25/20' lastdate = '01/08/20' xx,xxf,yy0 = get_country_data(test_country,'confirmed',firstdate=firstdate,lastdate=lastdate) xx,xxf,yy1 = get_country_data(test_country,'recovered',firstdate=firstdate,lastdate=lastdate) xx,xxf,yy2 = get_country_data(test_country,'deaths',firstdate=firstdate,lastdate=lastdate) print(xxf) y_jhu={} y_jhu[test_country] = np.array([[yy0[i],yy1[i],yy2[i]] for i in range(0,len(yy0))])/N # data = np.array([[xxf[i],yy0[i],yy1[i],yy2[i]] for i in range(len(yy))]) # print(data) lastday = len(y_jhu[test_country]) print('days 0 to',lastday,'data stored in y_jhu') ``` ```python model = 'SC3EI3R' # Define parameters based on clinical observations Dr. Alison Exposure=0.4 # Rate coefficient for exposure per individual in contact per day IncubPeriod=5 #Incubation period, days DurMildInf=8 #Duration of mild infections, days FracMild=0.65 #Fraction of infections that are mild FracSevere=0.20 #Fraction of infections that are severe FracCritical=0.15 #Fraction of infections that are critical CFR=0.1 #Case fatality rate (fraction of infections resulting in death) TimeICUDeath=4 #Time from ICU admission to death, days DurHosp=4 #Duration of hospitalization, days # Model fitting extension to allow for incomplete detection FracConfirmedDet=0.5 # Fraction of infected individuals confirmed FracRecoveredDet=FracConfirmedDet # Fraction of recovered individuals measured FracDeathsDet=1.0 # Model extension by John McCaskill to include caution CautionFactor= 0.1 # Fractional reduction of exposure rate for cautioned individuals CautionRetention= 60. # Duration of cautionary state of susceptibles (2 weeks) CautionICUFrac= 0.1 # Fraction of ICUs occupied leading to transition to caution @ 1/day ICUFrac= 0.001 # Fraction of ICUs relative to population size N params = {'beta_1' : Exposure/sum(x0_SC3EI3R), 'beta_2' : 0.0, 'beta_3' : 0.0, 'alpha' : 1.0/IncubPeriod, 'gamma_1': (1.0/DurMildInf)FracMild, 'gamma_2': (1.0/DurHosp)-(1/DurHosp)(FracCritical/(FracCritical+FracSevere)), 'gamma_3': (1.0/TimeICUDeath)-(1/TimeICUDeath)(CFR/FracCritical), 'p_1' : (1.0/DurMildInf)-(1.0/DurMildInf)FracMild, 'p_2' : (1.0/DurHosp)(FracCritical/(FracCritical+FracSevere)), 'mu' : (1.0/TimeICUDeath)(CFR/FracCritical), 'c_0' : CautionFactor, 'c_1' : 1.0/CautionRetention, 'c_2' : 1.0/(sum(x0_SC3EI3R)ICUFracCautionICUFrac), 'N' : sum(x0_SC3EI3R)} print(params) SC3EI3R_model.parameters = params def slidefitplot(beta_1,mu,c_0,c_1,c_2,logI_0): params={ 'beta_1':beta_1, 'mu':mu, 'c_0':c_0, 'c_1':c_1, 'c_2':c_2} cmodels[model].parameters = params I0 = 10*logI_0 x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,0.] cmodels[model].initial_values = (x0,t[0]) weights=np.array([1.,1.,1.]) solveplot(smodels=[model],species=['confirmed','recovered','deaths_x10'],tmax=len(t),summing='daily',averaging='weekly',fitdata=y_jhu[test_country]weights,scale='linear',plottitle= '',label='confirmed',newplot = True, figsize = (15,15)) ``` ```python w =interactive(slidefitplot, beta_1=FloatSlider(min=0,max=1,step=0.01,value=params['beta_1'],description='beta_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), mu=FloatSlider(min=0,max=0.2,step=0.002,value=params['mu'],description='mu', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_2=FloatSlider(min=0,max=20000,step=1,value=params['c_2'],description='c_2', style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'), logI_0=FloatSlider(min=-10,max=0,step=0.01,value=-6,description='log I_0', style=style,layout=slider_layout,continuous_update=False) ) display(w) ``` ```python params=w.kwargs print(params) ``` Note that we have used 50% detection of confirmed and recovered, 100% for deaths in manual fit. It appears that Italy's registration of recovery, although the right overall magnitude is markedly delayed - check reporting delays. Italy also had at least two successive regional infections, as seen in the dual peak confirmed data, so not easy to fit with one model. See below for simulation of second peak. ```python w =interactive(slidefitplot, beta_1=FloatSlider(min=0,max=1,step=0.01,value=params['beta_1'],description='beta_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), mu=FloatSlider(min=0,max=0.2,step=0.002,value=params['mu'],description='mu', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_2=FloatSlider(min=0,max=20000,step=1,value=params['c_2'],description='c_2', style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'), logI_0=FloatSlider(min=-10,max=0,step=0.01,value=-6,description='log I_0', style=style,layout=slider_layout,continuous_update=False) ) display(w) ``` #### Brazil ```python # assumed data starting on firstdate test_country='Brazil' N = 210000000 firstdate = '01/25/20' lastdate = '01/08/20' xx,xxf,yy0 = get_country_data(test_country,'confirmed',firstdate=firstdate,lastdate=lastdate) xx,xxf,yy1 = get_country_data(test_country,'recovered',firstdate=firstdate,lastdate=lastdate) xx,xxf,yy2 = get_country_data(test_country,'deaths',firstdate=firstdate,lastdate=lastdate) print(xxf) y_jhu={} y_jhu[test_country] = np.array([[yy0[i],yy1[i],yy2[i]] for i in range(0,len(yy0))])/N # data = np.array([[xxf[i],yy0[i],yy1[i],yy2[i]] for i in range(len(yy))]) # print(data) lastday = len(y_jhu[test_country]) print('days 0 to',lastday,'data stored in y_jhu') ``` ```python model = 'SC3EI3R' # Define parameters based on clinical observations Dr. Alison Exposure=0.4 # Rate coefficient for exposure per individual in contact per day IncubPeriod=5 #Incubation period, days DurMildInf=8 #Duration of mild infections, days FracMild=0.65 #Fraction of infections that are mild FracSevere=0.20 #Fraction of infections that are severe FracCritical=0.15 #Fraction of infections that are critical CFR=0.1 #Case fatality rate (fraction of infections resulting in death) TimeICUDeath=4 #Time from ICU admission to death, days DurHosp=8 #Duration of hospitalization, days # Model fitting extension to allow for incomplete detection FracConfirmedDet=0. # Fraction of recovered individuals measured FracRecoveredDet=FracConfirmedDet # Fraction of recovered individuals measured FracDeathsDet=1.0 # Model extension by John McCaskill to include caution CautionFactor= 0.1 # Fractional reduction of exposure rate for cautioned individuals CautionRetention= 60. # Duration of cautionary state of susceptibles (2 weeks) CautionICUFrac= 0.1 # Fraction of ICUs occupied leading to transition to caution @ 1/day ICUFrac= 0.001 # Fraction of ICUs relative to population size N params = {'beta_1' : Exposure/sum(x0_SC3EI3R), 'beta_2' : 0.0, 'beta_3' : 0.0, 'alpha' : 1.0/IncubPeriod, 'gamma_1': (1.0/DurMildInf)FracMild, 'gamma_2': (1.0/DurHosp)-(1/DurHosp)(FracCritical/(FracCritical+FracSevere)), 'gamma_3': (1.0/TimeICUDeath)-(1/TimeICUDeath)(CFR/FracCritical), 'p_1' : (1.0/DurMildInf)-(1.0/DurMildInf)FracMild, 'p_2' : (1.0/DurHosp)(FracCritical/(FracCritical+FracSevere)), 'mu' : (1.0/TimeICUDeath)(CFR/FracCritical), 'c_0' : CautionFactor, 'c_1' : 1.0/CautionRetention, 'c_2' : 1.0/(sum(x0_SC3EI3R)ICUFracCautionICUFrac), 'N' : sum(x0_SC3EI3R)} print(params) SC3EI3R_model.parameters = params def slidefitplot(beta_1,mu,c_0,c_1,c_2,logI_0): params={ 'beta_1':beta_1, 'mu':mu, 'c_0':c_0, 'c_1':c_1, 'c_2':c_2} cmodels[model].parameters = params I0 = 10*logI_0 x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,0.] cmodels[model].initial_values = (x0,t[0]) weights=np.array([1.,1.,1.]) solveplot(smodels=[model],species=['confirmed','recovered','deaths_x10'],tmax=len(t),summing='cumulative',averaging='weekly',fitdata=y_jhu[test_country]weights,scale='linear',plottitle= '',label='confirmed',newplot = True, figsize = (15,15)) ``` ```python w =interactive(slidefitplot, beta_1=FloatSlider(min=0,max=1,step=0.01,value=params['beta_1'],description='beta_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), mu=FloatSlider(min=0,max=0.2,step=0.002,value=params['mu'],description='mu', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_2=FloatSlider(min=0,max=20000,step=1,value=params['c_2'],description='c_2', style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'), logI_0=FloatSlider(min=-10,max=0,step=0.01,value=-6,description='log I_0', style=style,layout=slider_layout,continuous_update=False) ) display(w) ``` The Brazil data shows that death is not as delayed as assumed. The process of progression is perhaps less clearly documented. #### Russia ```python # assumed data starting on firstdate test_country='Iran' N = 144500000 firstdate = '01/25/20' lastdate = '01/08/20' xx,xxf,yy0 = get_country_data(test_country,'confirmed',firstdate=firstdate,lastdate=lastdate) xx,xxf,yy1 = get_country_data(test_country,'recovered',firstdate=firstdate,lastdate=lastdate) xx,xxf,yy2 = get_country_data(test_country,'deaths',firstdate=firstdate,lastdate=lastdate) print(xxf) y_jhu={} y_jhu[test_country] = np.array([[yy0[i],yy1[i],yy2[i]] for i in range(0,len(yy0))])/N # data = np.array([[xxf[i],yy0[i],yy1[i],yy2[i]] for i in range(len(yy))]) # print(data) lastday = len(y_jhu[test_country]) print('days 0 to',lastday,'data stored in y_jhu') ``` ```python model = 'SC3EI3R' # Define parameters based on clinical observations Dr. Alison Exposure=0.4 # Rate coefficient for exposure per individual in contact per day IncubPeriod=5 #Incubation period, days DurMildInf=8 #Duration of mild infections, days FracMild=0.65 #Fraction of infections that are mild FracSevere=0.20 #Fraction of infections that are severe FracCritical=0.15 #Fraction of infections that are critical CFR=0.1 #Case fatality rate (fraction of infections resulting in death) TimeICUDeath=4 #Time from ICU admission to death, days DurHosp=8 #Duration of hospitalization, days # Model fitting extension to allow for incomplete detection FracConfirmedDet=0. # Fraction of recovered individuals measured FracRecoveredDet=FracConfirmedDet # Fraction of recovered individuals measured FracDeathsDet=1.0 # Model extension by John McCaskill to include caution CautionFactor= 0.1 # Fractional reduction of exposure rate for cautioned individuals CautionRetention= 60. # Duration of cautionary state of susceptibles (2 weeks) CautionICUFrac= 0.1 # Fraction of ICUs occupied leading to transition to caution @ 1/day ICUFrac= 0.002 # Fraction of ICUs relative to population size N params = {'beta_1' : Exposure/sum(x0_SC3EI3R), 'beta_2' : 0.0, 'beta_3' : 0.0, 'alpha' : 1.0/IncubPeriod, 'gamma_1': (1.0/DurMildInf)FracMild, 'gamma_2': (1.0/DurHosp)-(1/DurHosp)(FracCritical/(FracCritical+FracSevere)), 'gamma_3': (1.0/TimeICUDeath)-(1/TimeICUDeath)(CFR/FracCritical), 'p_1' : (1.0/DurMildInf)-(1.0/DurMildInf)FracMild, 'p_2' : (1.0/DurHosp)(FracCritical/(FracCritical+FracSevere)), 'mu' : (1.0/TimeICUDeath)(CFR/FracCritical), 'c_0' : CautionFactor, 'c_1' : 1.0/CautionRetention, 'c_2' : 1.0/(sum(x0_SC3EI3R)ICUFracCautionICUFrac), 'N' : sum(x0_SC3EI3R)} print(params) SC3EI3R_model.parameters = params def slidefitplot(beta_1,mu,c_0,c_1,c_2,logI_0): params={ 'beta_1':beta_1, 'mu':mu, 'c_0':c_0, 'c_1':c_1, 'c_2':c_2} cmodels[model].parameters = params I0 = 10*logI_0 x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,0.] cmodels[model].initial_values = (x0,t[0]) weights=np.array([1.,1.,1.]) solveplot(smodels=[model],species=['confirmed','recovered','deaths_x10'],tmax=len(t),summing='cumulative',averaging='weekly',fitdata=y_jhu[test_country]weights,scale='linear',plottitle= '',label='confirmed',newplot = True, figsize = (15,15)) ``` ```python w =interactive(slidefitplot, beta_1=FloatSlider(min=0,max=1,step=0.01,value=params['beta_1'],description='beta_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), mu=FloatSlider(min=0,max=0.2,step=0.002,value=params['mu'],description='mu', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_2=FloatSlider(min=0,max=20000,step=1,value=params['c_2'],description='c_2', style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'), logI_0=FloatSlider(min=-10,max=0,step=0.01,value=-6,description='log I_0', style=style,layout=slider_layout,continuous_update=False) ) display(w) ``` ### SC3UEIR Model ```python # assumed data starting on firstdate test_country='US' N = 66650000 firstdate = '01/25/20' lastdate = '01/08/20' xx,xxf,yy0 = get_country_data(test_country,'confirmed',firstdate=firstdate,lastdate=lastdate) xx,xxf,yy1 = get_country_data(test_country,'recovered',firstdate=firstdate,lastdate=lastdate) xx,xxf,yy2 = get_country_data(test_country,'deaths',firstdate=firstdate,lastdate=lastdate) print(xxf) y_jhu={} y_jhu[test_country] = np.array([[yy0[i],yy1[i],yy2[i]] for i in range(0,len(yy0))])/N # data = np.array([[xxf[i],yy0[i],yy1[i],yy2[i]] for i in range(len(yy))]) # print(data) lastday = len(y_jhu[test_country]) print('days 0 to',lastday,'data stored in y_jhu') ``` ```python model = 'SC3UEIR' I_0 = 0.00003 x0_SC3UEIR = [1.0-I_0, 0.0, I_0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0] SC3UEIR_model.initial_values = (x0_SC3UEIR, t[0]) # Define parameters based on clinical observations Dr. Alison Exposure=0.4 # Rate coefficient for exposure per individual in contact per day IncubPeriod=5 #Incubation period, days DurMildInf=8 #Duration of mild infections, days FracMild=0.65 #Fraction of infections that are mild FracSevere=0.20 #Fraction of infections that are severe FracCritical=0.15 #Fraction of infections that are critical CFR=0.1 #Case fatality rate (fraction of infections resulting in death) TimeICUDeath=4 #Time from ICU admission to death, days DurHosp=8 #Duration of hospitalization, days # Model fitting extension to allow for incomplete detection FracConfirmedDet=0.5 # Fraction of recovered individuals measured FracRecoveredDet=FracConfirmedDet # Fraction of recovered individuals measured FracDeathsDet=1.0 # Model extension by John McCaskill to include caution CautionFactor= 0.1 # Fractional reduction of exposure rate for cautioned individuals CautionRetention= 60. # Duration of cautionary state of susceptibles (2 weeks) CautionICUFrac= 0.1 # Fraction of ICUs occupied leading to transition to caution @ 1/day ICUFrac= 0.001 # Fraction of ICUs relative to population size N # Model extension by John McCaskill to include economic influence on caution EconomicCostOfCaution= 0.5 # Fractional reduction of economic contribution for cautioned individuals p = [0,(1.0/DurMildInf)-(1.0/DurMildInf)FracMild, (1.0/DurHosp)(FracCritical/(FracCritical+FracSevere))] g = [0,(1.0/DurMildInf)FracMild, (1.0/DurHosp)-(1/DurHosp)(FracCritical/(FracCritical+FracSevere)), (1.0/TimeICUDeath)-(1/TimeICUDeath)(CFR/FracCritical)] u = (1.0/TimeICUDeath)(CFR/FracCritical) params = {'beta' : Exposure/sum(x0_SC3UEIR), 'alpha' : 1.0/IncubPeriod, 'gamma' : g[1]+g[2](p[1]/(g[2]+p[2]))+g[3](p[1]/(g[2]+p[2]))(p[2]/(g[3]+u)), 'mu' : u(p[1]/(g[2]+p[2])(p[2]/(g[3]+u))), 'c_0' : CautionFactor, 'c_1' : 1.0/CautionRetention, 'c_2' : 1.0/(sum(x0_SC3UEIR)ICUFracCautionICUFrac), 'N' : sum(x0_SC3UEIR), 'k_u' : 1.0/CautionRetention, 'k_1' : 1.0/CautionRetention, 'k_w' : 1.0/CautionRetention, 'kappa' : EconomicCostOfCaution} print(params) SC3UEIR_model.parameters = params.copy() # solution = SCIR_model.integrate(t[1::]) def slidefitplot(beta,alpha,gamma,mu,c_0,c_1,c_2,logI_0,k_u,k_1,k_w,kappa): params={ 'beta':beta, 'alpha':alpha, 'gamma':gamma, 'mu':mu, 'c_0':c_0, 'c_1':c_1, 'c_2':c_2, 'k_u':k_u, 'k_1':k_1, 'k_w':k_w, 'kappa':kappa} cmodels[model].parameters = params I0 = 10logI_0 x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,1.] cmodels[model].initial_values = (x0,t[0]) weights=np.array([1.,1.,1.]) solveplot(smodels=[model],species=['confirmed','recovered','deaths_x10'],tmax=len(t),summing='daily',averaging='weekly',fitdata=y_jhu[test_country]weights,scale='linear',plottitle= '',label='confirmed',newplot = True, figsize = (15,15)) ``` ```python w = interactive(slidefitplot, beta=FloatSlider(min=0,max=1,step=0.01,value=params['beta'],description='beta', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), alpha=FloatSlider(min=0,max=1,step=0.01,value=params['alpha'],description='alpha', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), gamma=FloatSlider(min=0,max=1,step=0.01,value=params['gamma'],description='gamma', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), mu=FloatSlider(min=0,max=0.2,step=0.002,value=params['mu'],description='mu', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_2=FloatSlider(min=0,max=5000,step=1,value=params['c_2'],description='c_2', style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'), logI_0=FloatSlider(min=-10,max=0,step=0.01,value=-6,description='log I_0', style=style,layout=slider_layout,continuous_update=False), k_u=FloatSlider(min=0,max=1,step=0.001,value=params['k_u'],description='k_u', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), k_1=FloatSlider(min=0,max=1,step=0.001,value=params['k_1'],description='k_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), k_w=FloatSlider(min=0,max=1,step=0.001,value=params['k_w'],description='k_w', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), kappa=FloatSlider(min=0,max=1,step=0.001,value=params['kappa'],description='kappa', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f')) display(w) ``` ```python params=w.kwargs # not a good fit yet, did better last week print(params) ``` ### SC3UEI3R Model #### USA ```python # assumed data starting on firstdate test_country='US' N = 66650000 firstdate = '01/25/20' lastdate = '01/08/20' xx,xxf,yy0 = get_country_data(test_country,'confirmed',firstdate=firstdate,lastdate=lastdate) xx,xxf,yy1 = get_country_data(test_country,'recovered',firstdate=firstdate,lastdate=lastdate) xx,xxf,yy2 = get_country_data(test_country,'deaths',firstdate=firstdate,lastdate=lastdate) print(xxf) y_jhu={} y_jhu[test_country] = np.array([[yy0[i],yy1[i],yy2[i]] for i in range(0,len(yy0))])/N # data = np.array([[xxf[i],yy0[i],yy1[i],yy2[i]] for i in range(len(yy))]) # print(data) lastday = len(y_jhu[test_country]) print('days 0 to',lastday,'data stored in y_jhu') ``` ```python model = 'SC3UEI3R' I_0 = 0.00003 x0_SC3UEI3R = [1.0-I_0, 0.0, I_0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0] SC3UEI3R_model.initial_values = (x0_SC3UEI3R, t[0]) # Define parameters based on clinical observations Dr. Alison Exposure=0.4 # Rate coefficient for exposure per individual in contact per day IncubPeriod=5 #Incubation period, days DurMildInf=8 #Duration of mild infections, days FracMild=0.65 #Fraction of infections that are mild FracSevere=0.20 #Fraction of infections that are severe FracCritical=0.15 #Fraction of infections that are critical CFR=0.1 #Case fatality rate (fraction of infections resulting in death) TimeICUDeath=4 #Time from ICU admission to death, days DurHosp=5 #Duration of hospitalization, days # Model fitting extension to allow for incomplete detection FracConfirmedDet=0.5 # Fraction of recovered individuals measured FracRecoveredDet=FracConfirmedDet # Fraction of recovered individuals measured FracDeathsDet=1.0 # Model extension by John McCaskill to include caution CautionFactor= 0.1 # Fractional reduction of exposure rate for cautioned individuals CautionRetention= 60. # Duration of cautionary state of susceptibles (2 weeks) CautionICUFrac= 0.1 # Fraction of ICUs occupied leading to transition to caution @ 1/day ICUFrac= 0.001 # Fraction of ICUs relative to population size N # Model extension by John McCaskill to include economic influence on caution EconomicCostOfCaution= 0.5 # Fractional reduction of economic contribution for cautioned individuals p = [0,(1.0/DurMildInf)-(1.0/DurMildInf)FracMild, (1.0/DurHosp)(FracCritical/(FracCritical+FracSevere))] g = [0,(1.0/DurMildInf)FracMild, (1.0/DurHosp)-(1/DurHosp)(FracCritical/(FracCritical+FracSevere)), (1.0/TimeICUDeath)-(1/TimeICUDeath)(CFR/FracCritical)] u = (1.0/TimeICUDeath)(CFR/FracCritical) params = {'beta_1' : Exposure/sum(x0_SC3UEI3R), 'beta_2' : 0.0, 'beta_3' : 0.0, 'alpha' : 1.0/IncubPeriod, 'gamma_1': (1.0/DurMildInf)FracMild, 'gamma_2': (1.0/DurHosp)-(1/DurHosp)(FracCritical/(FracCritical+FracSevere)), 'gamma_3': (1.0/TimeICUDeath)-(1/TimeICUDeath)(CFR/FracCritical), 'p_1' : (1.0/DurMildInf)-(1.0/DurMildInf)FracMild, 'p_2' : (1.0/DurHosp)(FracCritical/(FracCritical+FracSevere)), 'mu' : (1.0/TimeICUDeath)(CFR/FracCritical), 'c_0' : CautionFactor, 'c_1' : 1.0/CautionRetention, 'c_2' : 1.0/(ICUFracCautionICUFrac), 'k_u' : 1.0/5., 'k_1' : 1.0/90, 'k_w' : 1.0/90, 'kappa' : EconomicCostOfCaution, 'N' : sum(x0_SC3UEI3R)} print(params) SC3UEI3R_model.parameters = params.copy() ``` ```python def slidefitplot(beta_1,alpha,mu,c_0,c_1,c_2,logI_0,k_u,k_1,k_w,kappa): params={ 'beta_1':beta_1, 'alpha':alpha, 'mu':mu, 'c_0':c_0, 'c_1':c_1, 'c_2':c_2, 'k_u':k_u, 'k_1':k_1, 'k_w':k_w, 'kappa':kappa} cmodels[model].parameters = params I0 = 10logI_0 x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,0.,0.,1.] cmodels[model].initial_values = (x0,t[0]) weights=np.array([1.,1.,1.]) solveplot(smodels=[model],species=['confirmed','recovered','deaths_x10'],tmax=len(t),summing='daily',averaging='weekly',fitdata=y_jhu[test_country]weights,scale='linear',plottitle= '',label='confirmed',newplot = True, figsize = (15,15)) ``` ```python w=interactive(slidefitplot, beta_1=FloatSlider(min=0,max=1,step=0.01,value=params['beta_1'],description='beta_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), alpha=FloatSlider(min=0,max=1,step=0.01,value=params['alpha'],description='alpha', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), mu=FloatSlider(min=0,max=0.2,step=0.002,value=params['mu'],description='mu', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_2=FloatSlider(min=0,max=5000,step=1,value=params['c_2'],description='c_2', style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'), logI_0=FloatSlider(min=-10,max=0,step=0.01,value=-6,description='log I_0', style=style,layout=slider_layout,continuous_update=False), k_u=FloatSlider(min=0,max=1,step=0.001,value=params['k_u'],description='k_u', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), k_1=FloatSlider(min=0,max=1,step=0.001,value=params['k_1'],description='k_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), k_w=FloatSlider(min=0,max=1,step=0.001,value=params['k_w'],description='k_w', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), kappa=FloatSlider(min=0,max=1,step=0.001,value=params['kappa'],description='kappa', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f')) display(w) ``` ```python params=w.kwargs print(params) ``` #### Spain ```python # assumed data starting on firstdate test_country='Spain' N = 66650000 firstdate = '01/25/20' lastdate = '01/08/20' xx,xxf,yy0 = get_country_data(test_country,'confirmed',firstdate=firstdate,lastdate=lastdate) xx,xxf,yy1 = get_country_data(test_country,'recovered',firstdate=firstdate,lastdate=lastdate) xx,xxf,yy2 = get_country_data(test_country,'deaths',firstdate=firstdate,lastdate=lastdate) print(xxf) y_jhu={} y_jhu[test_country] = np.array([[yy0[i],yy1[i],yy2[i]] for i in range(0,len(yy0))])/N # data = np.array([[xxf[i],yy0[i],yy1[i],yy2[i]] for i in range(len(yy))]) # print(data) lastday = len(y_jhu[test_country]) print('days 0 to',lastday,'data stored in y_jhu') ``` ```python model = 'SC3UEI3R' I_0 = 0.00003 x0_SC3UEI3R = [1.0-I_0, 0.0, I_0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0] SC3UEI3R_model.initial_values = (x0_SC3UEI3R, t[0]) # Define parameters based on clinical observations Dr. Alison Exposure=0.4 # Rate coefficient for exposure per individual in contact per day IncubPeriod=5 #Incubation period, days DurMildInf=8 #Duration of mild infections, days FracMild=0.65 #Fraction of infections that are mild FracSevere=0.20 #Fraction of infections that are severe FracCritical=0.15 #Fraction of infections that are critical CFR=0.1 #Case fatality rate (fraction of infections resulting in death) TimeICUDeath=4 #Time from ICU admission to death, days DurHosp=5 #Duration of hospitalization, days # Model fitting extension to allow for incomplete detection FracConfirmedDet=0.5 # Fraction of recovered individuals measured FracRecoveredDet=FracConfirmedDet # Fraction of recovered individuals measured FracDeathsDet=1.0 # Model extension by John McCaskill to include caution CautionFactor= 0.1 # Fractional reduction of exposure rate for cautioned individuals CautionRetention= 60. # Duration of cautionary state of susceptibles (2 weeks) CautionICUFrac= 0.1 # Fraction of ICUs occupied leading to transition to caution @ 1/day ICUFrac= 0.001 # Fraction of ICUs relative to population size N # Model extension by John McCaskill to include economic influence on caution EconomicCostOfCaution= 0.5 # Fractional reduction of economic contribution for cautioned individuals p = [0,(1.0/DurMildInf)-(1.0/DurMildInf)FracMild, (1.0/DurHosp)(FracCritical/(FracCritical+FracSevere))] g = [0,(1.0/DurMildInf)FracMild, (1.0/DurHosp)-(1/DurHosp)(FracCritical/(FracCritical+FracSevere)), (1.0/TimeICUDeath)-(1/TimeICUDeath)(CFR/FracCritical)] u = (1.0/TimeICUDeath)(CFR/FracCritical) params = {'beta_1' : Exposure/sum(x0_SC3UEI3R), 'beta_2' : 0.0, 'beta_3' : 0.0, 'alpha' : 1.0/IncubPeriod, 'gamma_1': (1.0/DurMildInf)FracMild, 'gamma_2': (1.0/DurHosp)-(1/DurHosp)(FracCritical/(FracCritical+FracSevere)), 'gamma_3': (1.0/TimeICUDeath)-(1/TimeICUDeath)(CFR/FracCritical), 'p_1' : (1.0/DurMildInf)-(1.0/DurMildInf)FracMild, 'p_2' : (1.0/DurHosp)(FracCritical/(FracCritical+FracSevere)), 'mu' : (1.0/TimeICUDeath)(CFR/FracCritical), 'c_0' : CautionFactor, 'c_1' : 1.0/CautionRetention, 'c_2' : 1.0/(ICUFracCautionICUFrac), 'k_u' : 1.0/5., 'k_1' : 1.0/90, 'k_w' : 1.0/90, 'kappa' : EconomicCostOfCaution, 'N' : sum(x0_SC3UEI3R)} print(params) SC3UEI3R_model.parameters = params.copy() ``` ```python def slidefitplot(beta_1,alpha,mu,c_0,c_1,c_2,logI_0,k_u,k_1,k_w,kappa): params={ 'beta_1':beta_1, 'alpha':alpha, 'mu':mu, 'c_0':c_0, 'c_1':c_1, 'c_2':c_2, 'k_u':k_u, 'k_1':k_1, 'k_w':k_w, 'kappa':kappa} cmodels[model].parameters = params I0 = 10logI_0 x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,0.,0.,1.] cmodels[model].initial_values = (x0,t[0]) weights=np.array([1.,1.,1.]) solveplot(smodels=[model],species=['confirmed','recovered','deaths_x10'],tmax=len(t),summing='daily',averaging='weekly',fitdata=y_jhu[test_country]weights,scale='linear',plottitle= '',label='confirmed',newplot = True, figsize = (15,15)) ``` ```python w=interactive(slidefitplot, beta_1=FloatSlider(min=0,max=1,step=0.01,value=params['beta_1'],description='beta_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), alpha=FloatSlider(min=0,max=1,step=0.01,value=params['alpha'],description='alpha', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), mu=FloatSlider(min=0,max=0.2,step=0.002,value=params['mu'],description='mu', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), c_2=FloatSlider(min=0,max=5000,step=1,value=params['c_2'],description='c_2', style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'), logI_0=FloatSlider(min=-10,max=0,step=0.01,value=-6,description='log I_0', style=style,layout=slider_layout,continuous_update=False), k_u=FloatSlider(min=0,max=1,step=0.001,value=params['k_u'],description='k_u', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), k_1=FloatSlider(min=0,max=1,step=0.001,value=params['k_1'],description='k_1', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), k_w=FloatSlider(min=0,max=1,step=0.001,value=params['k_w'],description='k_w', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), kappa=FloatSlider(min=0,max=1,step=0.001,value=params['kappa'],description='kappa', style=style,layout=slider_layout,continuous_update=False,readout_format='.3f')) display(w) ``` ```python params=w.kwargs print(params) ``` ## Fit SC3EI3R parameters to jhu data based on square_loss ### Fit c_0 , c_1 and c_2 as well as initial value of I_1 ```python SC3EI3R_model.parameters ``` ```python SC3EI3R_model.parameters = {'beta_1': 0.4, 'beta_2': 0.0, 'beta_3': 0.0, 'alpha': 0.2, 'gamma_1': 0.06999999999999999, 'gamma_2': 0.16666666666666669, 'gamma_3': 0.1, 'p_1': 0.030000000000000013, 'p_2': 0.08333333333333333, 'mu': 0.1, 'c_0': 0.1, 'c_1': 0.016666666666666666, 'c_2': 10000.0, 'N': 1.0} ``` ```python # Initial guess of parameters and initial condition, and bounding constraints I0 =10*-7 x0_SC3EI3R = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,0.] SC3EI3R_model.parameters={'beta_1': 0.41, 'mu': 0.1, 'c_0': 0.1, 'c_1': 0.0166, 'c_2': 10000.0} cautionparams = list(params.values())[-4:-1] theta = [0.3,0.0167,10000.] # cautionparams boxBounds = [(0.3,0.5),(0.01,0.02),(6000.,12000.)] # set up optimization function with cost and sensitivity (Jacobian) objSC3EI3R = SquareLoss(theta=theta, ode=SC3EI3R_model, x0=x0_SC3EI3R, t0=t[0], t=t[1::], y=y_jhu[test_country][1::,1:], state_weight=[1.,10.],state_name=['R','D'], target_param=['c_0','c_1','c_2'],target_state=['I_1']) # perform optimization res = minimize(fun=objSC3EI3R.costIV, jac=objSC3EI3R.sensitivityIV, x0=theta+[I0], bounds=boxBounds+[(0.00000001,0.000001)], #method='BFGS', method='SLSQP', #options={'disp':True,'maxiter':1000,'eps':0.01,'gtol':0.01}) #options={'disp':True}) options={'disp':True,'maxiter':1000,'eps':0.0001,'ftol':0.000001}) print(res) ``` ```python # model with fitted parameters print(params) print(x0_SC3EI3R) params_fit = params.copy() params_fit['c_0'] = res.x[0] params_fit['c_1'] = res.x[1] params_fit['c_2'] = res.x[2] SC3EI3R_model.params = params_fit print(SC3EI3R_model.params) #ode_fit = common_models.SEI3R({'beta':res.x[0], 'gamma':res.x[1],'alpha':res.x[2]}) #x0_fit = [1-1.27e-6, 1.27e-6, 0] x0_fit = x0_SC3EI3R.copy() x0_fit[2] = res.x[3] t_fit = np.linspace(0, 150, 1000) #t_fit = t SC3EI3R_model.initial_values = (x0_fit, t_fit[0]) # %timeit sol_fit =SCEI3R_model.integrate(t_fit[1::]) # use magic %timeit to time # sol_fit =SCEI3R_model.integrate(t_fit[0::]) sol_fit = scipy.integrate.odeint(SC3EI3R_model.ode, x0_fit, t_fit[1::]) # plt.figure(figsize=(15,10)) #plt.plot(t[1::],y_jhu[test_country][1::,0], 'o',color='lightgreen') # infected observations plt.plot(t[1::],y_jhu[test_country][1::,1], 'bo') # recovered plt.plot(t[1::],10y_jhu[test_country][1::,2], 'ro') # died x10 plt.plot(t_fit[1::], sol_fit[::,SC3EI3R_model.recovered]) # R plt.plot(t_fit[1::], 10sol_fit[::,SC3EI3R_model.deaths]) # D x10 #plt.ylim([0,0.004]) #plt.show(()) #ode_fit.plot() peak_i = np.argmax(sol_fit[:,2]) print('Peak infection (days)', t_fit[peak_i]) ``` ## Testing fitting ### Generate test data based on SCEI3R simulation ```python # Add noise y = solution[:,2:7].copy() #print('len(y)',len(y),'t',len(t),t[0],t[1],'...',t[-1]) np.random.seed(seed=6) noise = np.random.normal(0,1.e-2,[len(t),5]) # ynoise = y (1+noise) ynoise = y (1.0 + noise) ynoise[ynoise<0] = 0 plt.figure(figsize=(15,10)) plt.plot(t,ynoise[:,0], 'go', label='I_1') plt.plot(t,ynoise[:,1], 'go', label='I_2') plt.plot(t,ynoise[:,2], 'go', label='I_3') plt.plot(t,ynoise[:,3], 'bo', label='R') plt.plot(t,ynoise[:,4], 'ro', label='D') plt.plot(t,y[:,0], 'g', label='I_1') plt.plot(t,y[:,1], 'g', label='I_2') plt.plot(t,y[:,2], 'g', label='I_3') plt.plot(t,y[:,3], 'b', label='R') plt.plot(t,y[:,4], 'r', label='D') plt.legend() plt.ylim(0,0.003) plt.show() ``` ```python # model with generating parameters print(params) params_fit = params.copy() print(params_fit['c_0'],params_fit['c_1']) SCEI3R_model.params = params_fit x0_fit = x0_SCEI3R.copy() print(x0_fit) #t_fit = numpy.linspace(0, 150, 1000) t_fit = t SCEI3R_model.initial_values = (x0_fit, t_fit[0]) # %timeit sol_fit =SCEI3R_model.integrate(t_fit[1::]) # use magic %timeit to time sol_fit = scipy.integrate.odeint(SCEI3R_model.ode, x0_fit, t_fit[1::]) # print(len(sol_fit[0])) # plt.figure(figsize=(15,10)) plt.plot(t,ynoise[:,0], 'go',label='I_1') # infected observations plt.plot(t,ynoise[:,1], 'go',label='I_2') # infected observations plt.plot(t,ynoise[:,2], 'go',label='I_3') # infected observations plt.plot(t,ynoise[:,3], 'bo',label='R') # recoverd plt.plot(t,ynoise[:,4], 'ro',label='D') # died plt.gca().set_prop_cycle(color=['grey','orange','green','green','green','blue','red', 'black']) plt.plot(t_fit[1::], sol_fit) plt.ylim([0,0.004]) plt.legend() #plt.show(()) #ode_fit.plot() peak_i = np.argmax(sol_fit[:,2]) print('Peak infection (days)', t_fit[peak_i]) ``` ```python params # use list(...) to convert to list ``` ### Fit parameters to randomized simulation data based on square_loss #### Fit c_0 and c_1 only ```python # Initial guess of parameters, and bounding constraints cautionparams = list(params.values())[-4:-2] theta = [value for value in cautionparams] theta = [0.21,0.08] boxBounds = [(0.2,0.4),(0.05,0.15)] objSCEI3R = SquareLoss(theta=theta, ode=SCEI3R_model, x0=x0_SCEI3R, t0=t[0], t=t[1::], y=ynoise[1::,:], state_weight=[1.,1.,1.,1.0,10.0],state_name=['I_1','I_2','I_3','R','D'], target_param=['c_0','c_1']) # perform optimization res = minimize(fun=objSCEI3R.cost, jac=objSCEI3R.sensitivity, x0=theta, #bounds=boxBounds, method='BFGS', options={'disp':True,'maxiter':1000,'eps':0.0001})# ,'ftol':0.01}) #not BFGS print(res) ``` #### Fit c_0 and c_1 as well as initial value of E ##### Fit c_0 and c_1 as well as initial value of E with 'SLSQP' does not work well note use of special methods IV for initial value fitting of target_state ```python # Initial guess of parameters and initial condition, and bounding constraints cautionparams = list(params.values())[-4:-2] theta = [0.25,0.08] boxBounds = [(0.2,0.4),(0.05,0.15)] objSCEI3R = SquareLoss(theta=theta, ode=SCEI3R_model, x0=x0_SCEI3R, t0=t[0], t=t[1::], y=ynoise[1::,:], state_weight=[1.,1.,1.,1.,1.],state_name=['I_1','I_2','I_3','R','D'], target_param=['c_0','c_1'],target_state=['E']) # perform optimization res = minimize(fun=objSCEI3R.costIV, jac=objSCEI3R.sensitivityIV, x0=theta+[0.00005], #bounds=boxBounds+[(0.0000001,0.001)], method='SLSQP', options={'disp':True,'maxiter':1000,'eps':0.01,'ftol':0.01}) print(res) ``` ##### Fit c_0 and c_1 as well as initial value of E with BFGS works well: no constraints and gtol not ftol ```python # Initial guess of parameters and initial condition, and bounding constraints cautionparams = list(params.values())[-4:-2] theta = [0.25,0.08] boxBounds = [(0.2,0.4),(0.05,0.15)] objSCEI3R = SquareLoss(theta=theta, ode=SCEI3R_model, x0=x0_SCEI3R, t0=t[0], t=t[1::], y=ynoise[1::,:], state_weight=[1.,1.,1.,1.,1.],state_name=['I_1','I_2','I_3','R','D'], target_param=['c_0','c_1'],target_state=['E']) # perform optimization res = minimize(fun=objSCEI3R.costIV, jac=objSCEI3R.sensitivityIV, x0=theta+[0.00005], #bounds=boxBounds+[(0.0000001,0.001)], method='BFGS', options={'disp':True,'maxiter':1000,'eps':0.01,'gtol':0.01}) print(res) ``` ```python # model with fitted parameters print(params) print(x0_SCEI3R) params_fit = params.copy() #params_fit['c_0'] = res.x[0] #params_fit['c_1'] = res.x[1] SCEI3R_model.params = params_fit print(SCEI3R_model.params) #ode_fit = common_models.SEI3R({'beta':res.x[0], 'gamma':res.x[1],'alpha':res.x[2]}) #x0_fit = [1-1.27e-6, 1.27e-6, 0] x0_fit = x0.copy() #x0_fit[2] = res.x[2] #t_fit = numpy.linspace(0, 150, 1000) t_fit = t SCEI3R_model.initial_values = (x0_fit, t_fit[0]) # %timeit sol_fit =SCEI3R_model.integrate(t_fit[1::]) # use magic %timeit to time # sol_fit =SCEI3R_model.integrate(t_fit[0::]) sol_fit = scipy.integrate.odeint(SCEI3R_model.ode, x0_fit, t_fit[1::]) # plt.figure(figsize=(15,10)) plt.plot(t,ynoise[:,0], 'go') # infected observations plt.plot(t,ynoise[:,1], 'go') # infected observations plt.plot(t,ynoise[:,2], 'go') # infected observations plt.plot(t,ynoise[:,3], 'bo') # recoverd plt.plot(t,ynoise[:,4], 'ro') # died plt.plot(t_fit[1::], sol_fit) plt.ylim([0,0.004]) #plt.show(()) #ode_fit.plot() peak_i = np.argmax(sol_fit[:,2]) print('Peak infection (days)', t_fit[peak_i]) ``` ##### Fit c_0 and c_1 as well as initial value of E using L-BFGS-B this method does not work well ```python # Initial guess of parameters and initial condition, and bounding constraints cautionparams = list(params.values())[-4:-2] theta = [0.25,0.08] boxBounds = [(0.2,0.4),(0.05,0.15)] objSCEI3R = SquareLoss(theta=theta, ode=SCEI3R_model, x0=x0_SCEI3R, t0=t[0], t=t[1::], y=ynoise[1::,:], state_weight=[1.,1.,1.,1.,1.],state_name=['I_1','I_2','I_3','R','D'], target_param=['c_0','c_1'],target_state=['E']) # perform optimization res = minimize(fun=objSCEI3R.costIV, jac=objSCEI3R.sensitivityIV, x0=theta+[0.00005], bounds=boxBounds+[(0.0000001,0.001)], method='L-BFGS-B', options={'disp':True,'maxiter':1000,'eps':0.0001,'ftol':0.001}) print(res) ``` ```python objSCEI3R.residual() ``` ##### Fit c_0 and c_1 as well as initial value of E with Nelder-Mead no use of Jacobian and no constraints ```python # Initial guess of parameters and initial condition, and bounding constraints cautionparams = list(params.values())[-4:-2] theta = [0.25,0.08] boxBounds = [(0.2,0.4),(0.05,0.15)] objSCEI3R = SquareLoss(theta=theta, ode=SCEI3R_model, x0=x0_SCEI3R, t0=t[0], t=t[1::], y=ynoise[1::,:], state_weight=[1.,1.,1.,1.,1.],state_name=['I_1','I_2','I_3','R','D'], target_param=['c_0','c_1'],target_state=['E']) # perform optimization res = minimize(fun=objSCEI3R.costIV, #jac=objSCEI3R.sensitivityIV, x0=theta+[0.00005], #bounds=boxBounds+[(0.0000001,0.001)], method='Nelder-Mead', options={'disp':True,'maxiter':1000}) #,'eps':0.0001,'ftol':0.01}) #not NM print(res) ``` ```python # model with fitted parameters print(params) print(x0_SCEI3R) params_fit = params.copy() #params_fit['c_0'] = res.x[0] #params_fit['c_1'] = res.x[1] SCEI3R_model.params = params_fit print(SCEI3R_model.params) #ode_fit = common_models.SEI3R({'beta':res.x[0], 'gamma':res.x[1],'alpha':res.x[2]}) #x0_fit = [1-1.27e-6, 1.27e-6, 0] x0_fit = x0_SCEI3R.copy() #x0_fit[2] = res.x[2] #t_fit = numpy.linspace(0, 150, 1000) t_fit = t SCEI3R_model.initial_values = (x0_fit, t_fit[0]) # %timeit sol_fit =SCEI3R_model.integrate(t_fit[1::]) # use magic %timeit to time # sol_fit =SCEI3R_model.integrate(t_fit[0::]) sol_fit = scipy.integrate.odeint(SCEI3R_model.ode, x0_fit, t_fit[1::]) # plt.figure(figsize=(15,10)) plt.plot(t,ynoise[:,0], 'go') # infected observations plt.plot(t,ynoise[:,1], 'go') # infected observations plt.plot(t,ynoise[:,2], 'go') # infected observations plt.plot(t,ynoise[:,3], 'bo') # recoverd plt.plot(t,ynoise[:,4], 'ro') # died plt.plot(t_fit[1::], sol_fit) plt.ylim([0,0.004]) #plt.show(()) #ode_fit.plot() peak_i = np.argmax(sol_fit[:,2]) print('Peak infection (days)', t_fit[peak_i]) ``` ## Fit SC2IR parameters to jhu data based on square_loss ```python params=SC2IR_model.parameters print(params) ``` ```python # Initial guess of parameters and initial condition, and bounding constraints theta = [0.4,0.11,0.007,0.33,0.228,275.] boxBounds = [(0.2,0.5),(0.05,0.15),(0.005,0.015),(0.25,0.55),(0.15,0.4),(5.,2000.)] # setup cost function and Jacobian with target parameters and initial states objSC2IR = SquareLoss(theta=theta, ode=SC2IR_model, x0=x0, t0=t[0], t=t[1::], y=y_jhu[test_country][1::,1:3], state_weight=[0.2,1.],state_name=['R','D'], target_param=['beta','gamma','mu','c_0','c_1','c_2'], target_state=['I']) # perform optimization res = minimize(fun=objSC2IR.costIV, jac=objSC2IR.sensitivityIV, x0=theta+[0.000000001], bounds=boxBounds+[(0.0000000001,0.000001)], # method='L-BFGS-B', # method='Nelder-Mead', #options={'disp':True,'maxiter':1000,'eps':0.01,'gtol':0.01}) options={'disp':True,'maxiter':1000,'eps':0.000001,'ftol':0.000000001}) print(res) ``` ```python # model with fitted parameters startparams = SC2IR_model.parameters.copy() # save starting parameters (not fit) print(params) print(x0) params_fit = params.copy() params_fit['beta'] = res.x[0] params_fit['gamma'] = res.x[1] params_fit['mu'] = res.x[2] params_fit['c_0'] = res.x[3] params_fit['c_1'] = res.x[4] params_fit['c_2'] = res.x[5] SC2IR_model.params = params_fit print(SC2IR_model.params) x0_fit = x0.copy() x0_fit[1] = res.x[6] t_fit = t SC2IR_model.initial_values = (x0_fit, t_fit[0]) sol_fit = scipy.integrate.odeint(SC2IR_model.ode, x0_fit, t_fit[1::]) # plt.figure(figsize=(15,10)) plt.semilogy() plt.ylim([0.000001,1]) plt.plot(t,y_jhu[test_country][:,1], 'bo',label='R') # recovered plt.semilogy() plt.ylim([0.000001,1]) plt.plot(t,y_jhu[test_country][:,2], 'ro',label='D') # died plt.semilogy() plt.ylim([0.000001,1]) plt.gca().set_prop_cycle(color=['grey','green','blue','red', 'black']) plt.plot(t_fit[1::], sol_fit) plt.ylim([0.000001,1]) plt.legend(('R','D','S','I','R','D','S_c','I_c')) plt.semilogy() #plt.show(()) #ode_fit.plot() peak_i = np.argmax(sol_fit[:,1]) print('Peak infection (days)', t_fit[peak_i]) SC2IR_model.parameters = startparams ``` ## Fit SC3EI3R parameters to jhu data based on square_loss ### Fit c_0 and c_1 only ```python # Initial guess of parameters, and bounding constraints cautionparams = list(params.values())[-4:-1] theta = [value for value in cautionparams] print(theta) theta = [0.3,0.08,2500.] boxBounds = [(0.2,0.8),(0.05,0.15),(100.,10000.)] objSC3EI3R = SquareLoss(theta=theta, ode=SC3EI3R_model, x0=x0_SC3EI3R, t0=t[0], t=t[1::], y=y_jhu[test_country][1::,1:3], state_weight=[1.,1.],state_name=['R','D'], target_param=['c_0','c_1','c_2']) # perform optimization res = minimize(fun=objSC3EI3R.cost, #jac=objSC3EI3R.sensitivity, x0=theta, #bounds=boxBounds, method='L-BFGS-B', # method='Nelder-Mead', options={'disp':True,'maxiter':1000,'eps':0.00001})# ,'ftol':0.01}) #not BFGS print(res) ``` ### Fit c_0 and c_1 as well as initial value of E #### Fit c_0 and c_1 as well as initial value of E with 'SLSQP' does not work well note use of special methods IV for initial value fitting of target_state ```python # Initial guess of parameters and initial condition, and bounding constraints cautionparams = list(params.values())[-4:-3] theta = [value for value in cautionparams] theta = [0.21,0.08,2500.] objSC3EI3R = SquareLoss(theta=theta, ode=SC3EI3R_model, x0=x0_SC3EI3R, t0=t[0], t=t[1::], y=y_jhu[test_country][1::,1:3], state_weight=[1.,1.],state_name=['R','D'], target_param=['c_0','c_1','c_2'],target_state=['I_1']) # perform optimization res = minimize(fun=objSC3EI3R.costIV, jac=objSC3EI3R.sensitivityIV, x0=theta+[0.00005], bounds=boxBounds+[(0.0000001,0.001)], # method='BFGS', method='L-BFGS-B', options={'disp':True,'maxiter':1000,'eps':0.01,'gtol':0.01}) print(res) ``` #### Fit c_0 and c_1 as well as initial value of E with BFGS works well: no constraints and gtol not ftol ```python # Initial guess of parameters and initial condition, and bounding constraints cautionparams = list(params.values())[-4:-2] theta = [0.25,0.08] boxBounds = [(0.2,0.4),(0.05,0.15)] objSCEI3R = SquareLoss(theta=theta, ode=SCEI3R_model, x0=x0_SC3EI3R, t0=t[0], t=t[1::], y=ynoise[1::,:], state_weight=[1.,1.,1.,1.,1.],state_name=['I_1','I_2','I_3','R','D'], target_param=['c_0','c_1'],target_state=['E']) # perform optimization res = minimize(fun=objSCEI3R.costIV, jac=objSCEI3R.sensitivityIV, x0=theta+[0.00005], #bounds=boxBounds+[(0.0000001,0.001)], method='BFGS', options={'disp':True,'maxiter':1000,'eps':0.01,'gtol':0.01}) print(res) ``` ```python # model with fitted parameters print(params) print(x0_SC3EI3R) params_fit = params.copy() #params_fit['c_0'] = res.x[0] #params_fit['c_1'] = res.x[1] SC3EI3R_model.params = params_fit print(SC3EI3R_model.params) #ode_fit = common_models.SEI3R({'beta':res.x[0], 'gamma':res.x[1],'alpha':res.x[2]}) #x0_fit = [1-1.27e-6, 1.27e-6, 0] x0_fit = x0_SC3EI3R.copy() #x0_fit[2] = res.x[2] #t_fit = numpy.linspace(0, 150, 1000) t_fit = t SC3EI3R_model.initial_values = (x0_fit, t_fit[0]) # %timeit sol_fit =SCEI3R_model.integrate(t_fit[1::]) # use magic %timeit to time # sol_fit =SCEI3R_model.integrate(t_fit[0::]) sol_fit = scipy.integrate.odeint(SC3EI3R_model.ode, x0_fit, t_fit[1::]) # plt.figure(figsize=(15,10)) plt.plot(t,ynoise[:,0], 'o',color='lightgreen') # infected observations plt.plot(t,ynoise[:,1], 'o',color='green') # infected observations plt.plot(t,ynoise[:,2], 'o',color='darkgreen') # infected observations plt.plot(t,ynoise[:,3], 'bo') # recoverd plt.plot(t,ynoise[:,4], 'ro') # died plt.plot(t_fit[1::], sol_fit) plt.ylim([0,0.004]) #plt.show(()) #ode_fit.plot() peak_i = np.argmax(sol_fit[:,2]) print('Peak infection (days)', t_fit[peak_i]) ``` #### Fit c_0 and c_1 as well as initial value of E using L-BFGS-B this method does not work well ```python # Initial guess of parameters and initial condition, and bounding constraints cautionparams = list(params.values())[-4:-2] theta = [0.25,0.08] boxBounds = [(0.2,0.4),(0.05,0.15)] objSCEI3R = SquareLoss(theta=theta, ode=SCEI3R_model, x0=x0_SC3EI3R, t0=t[0], t=t[1::], y=ynoise[1::,:], state_weight=[1.,1.,1.,1.,1.],state_name=['I_1','I_2','I_3','R','D'], target_param=['c_0','c_1'],target_state=['E']) # perform optimization res = minimize(fun=objSCEI3R.costIV, jac=objSCEI3R.sensitivityIV, x0=theta+[0.00005], bounds=boxBounds+[(0.0000001,0.001)], method='L-BFGS-B', options={'disp':True,'maxiter':1000,'eps':0.0001,'ftol':0.001}) print(res) ``` ```python objSCEI3R.residual() ``` #### Fit c_0 and c_1 as well as initial value of E with Nelder-Mead no use of Jacobian and no constraints ```python # Initial guess of parameters and initial condition, and bounding constraints cautionparams = list(params.values())[-4:-2] theta = [0.25,0.08] boxBounds = [(0.2,0.4),(0.05,0.15)] objSCEI3R = SquareLoss(theta=theta, ode=SCEI3R_model, x0=x0_SC3EI3R, t0=t[0], t=t[1::], y=ynoise[1::,:], state_weight=[1.,1.,1.,1.,1.],state_name=['I_1','I_2','I_3','R','D'], target_param=['c_0','c_1'],target_state=['E']) # perform optimization res = minimize(fun=objSCEI3R.costIV, #jac=objSCEI3R.sensitivityIV, x0=theta+[0.00005], #bounds=boxBounds+[(0.0000001,0.001)], method='Nelder-Mead', options={'disp':True,'maxiter':1000}) #,'eps':0.0001,'ftol':0.01}) #not NM print(res) ``` ```python # model with fitted parameters print(params) print(x0_SC3EI3R) params_fit = params.copy() params_fit['c_0'] = res.x[0] params_fit['c_1'] = res.x[1] SC3EI3R_model.params = params_fit print(SC3EI3R_model.params) x0_fit = x0_SC3EI3R.copy() #x0_fit[2] = res.x[2] #t_fit = numpy.linspace(0, 150, 1000) t_fit = t SC3EI3R_model.initial_values = (x0_fit, t_fit[0]) # %timeit sol_fit =SC3EI3R_model.integrate(t_fit[1::]) # use magic %timeit to time # sol_fit =SC3EI3R_model.integrate(t_fit[0::]) sol_fit = scipy.integrate.odeint(SC3EI3R_model.ode, x0_fit, t_fit[1::]) # plt.figure(figsize=(15,10)) plt.plot(t,y_jhu[:,0], 'bo') # recoverd plt.plot(t,y_jhu[:,1], 'ro') # died plt.plot(t_fit[1::], sol_fit) plt.ylim([0,0.004]) #plt.show(()) #ode_fit.plot() peak_i = np.argmax(sol_fit[:,2]) print('Peak infection (days)', t_fit[peak_i]) ``` ### Information on method options ```python scipy.optimize.show_options(solver='minimize', method='SLSQP', disp=True) print(' ') scipy.optimize.show_options(solver='minimize', method='L-BFGS-B', disp=True) ``` ## Plot using full control ```python def plotmodel(solns,t,scale='linear',species='no_susc',plottitle= '',label='', newplot = True,models=['SEI3R','SCEI3R','SC3EI3R']): """ plot solns over times t interpreted as models indicated in models parameter scale: alternative 'linear' or 'log' species alternatives 'all', 'confirmed', 'deaths', 'daily confirmed', 'daily deaths' plottitle : title for plot label : label for curve when called as part of multicurve plot newplot : whether to open new plot True/False models : list of models to include, default all three of those possible """ nmodels = len(models) if len(solns) != len(models): print("Error: number of models must match number of solutions") return None nm = 0 if newplot == True: plt.figure(figsize=(nmodels8,6)) for nm in range(nmodels): soln = solns[nm] if models[nm] == 'SEI3R': #SEI3R plt.subplot(1,nmodels,nm+1) if scale == 'log': #Plot on log scale plt.semilogy() plt.ylim([1,10000]) elif species != 'daily confirmed': # Plot on normal linear scale #plt.ylim([0,10000]) pass if species == 'no_susc': plt.plot(tvec,soln[:,1:5],label=label) plt.legend(("E","I1","I2","I3")) elif species == 'confirmed' or species == 'daily confirmed': suma = np.sum(soln[:,2:7],axis=1) # print('length=',len(suma)) if species == 'daily confirmed': sumd = np.zeros(len(suma)) for i in range(1,len(suma)): sumd[i] = suma[i]-suma[i-1] #plt.ylim([0,1000]) plt.plot(tvec,sumd,label=label) else: #plt.ylim([0,200000]) plt.plot(t,suma,label=label) elif species == 'all': plt.plot(tvec,soln,label=label) plt.legend(("S","E","I1","I2","I3","R","D")) plt.xlabel("Time (days)") plt.ylabel("Portion of population N") plt.title('SEI3R %s' % plottitle) elif models[nm] == 'SCEI3R': #SCEI3R #Plot plt.subplot(1,nmodels,nm+1) if scale == 'log': #Plot on log scale plt.semilogy() plt.ylim([1,10000]) elif species != 'daily confirmed': # Plot on normal linear scale #plt.ylim([0,10000]) pass if species == 'no_susc': plt.plot(t,soln[:,1:5],label=label) plt.legend(("E","I1","I2","I3")) elif species == 'confirmed' or species == 'daily confirmed': suma = np.sum(soln[:,2:7],axis=1) # print('length=',len(suma)) if species == 'daily confirmed': sumd = np.zeros(len(suma)) for i in range(1,len(suma)): sumd[i] = suma[i]-suma[i-1] #plt.ylim([0,1000]) plt.plot(t,sumd,label=label) else: #plt.ylim([0,200000]) plt.plot(t,suma,label=label) elif species == 'all': plt.plot(t,soln,label=label) plt.legend(("S","E","I1","I2","I3","R","D","Sc")) plt.xlabel("Time (days)") plt.ylabel("Portion of population N") plt.title('SCEI3R %s' % plottitle) elif models[nm] == 'SC3EI3R': #SC3EI3R plt.subplot(1,nmodels,nm+1) if scale == 'log': #Plot on log scale plt.semilogy() plt.ylim([1,10000]) elif species != 'daily confirmed': # Plot on normal linear scale #plt.ylim([0,10000]) pass if species == 'no_susc': plt.plot(t,sol[:,1:5]) plt.legend(("E","I1","I2","I3")) elif species == 'confirmed' or species == 'daily confirmed': suma = np.sum(soln[:,2:7],axis=1) + soln[:,9] if species == 'daily confirmed': sumd = np.zeros(len(suma)) for i in range(1,len(suma)): sumd[i] = suma[i]-suma[i-1] # plt.ylim([0,1000]) plt.plot(t,sumd,label=label) else: # plt.ylim([0,200000]) plt.plot(t,suma,label=label) elif species == 'all': plt.plot(t,soln,label=label) plt.legend(("S","E","I1","I2","I3","R","D","Sc","Ec","I1c")) plt.xlabel("Time (days)") plt.ylabel("Portion of population N") plt.title('SC3EI3R %s' % plottitle) return True ``` ```python plotmodel([sol_fit],t_fit[1:],scale='linear',species='no_susc',plottitle= 'test',label='', newplot = True,models=['SCEI3R']) ``` ```python ```

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