Split (1)

train · 189k rows

text stringlengths 0 3.34M
#include <boost/type_erasure/relaxed.hpp>
module ArithExample import Fix import ALaCarte %access public export -- DSL for arithmetic expressions data Val k = V Int data Add k = A k k data Mul k = M k k Expr : Type Expr = Fix (Sig [Val, Add, Mul]) Functor Val where map func (V x) = V x Functor Add where map func (A x y) = A (func x) (func y) Functor Mul where map func (M x y) = M (func x) (func y) -- Smart constructors val : {auto prf : Elem Val fs} -> Int -> Fix (Sig fs) val x = inject (V x) (+) : {auto prf : Elem Add fs} -> Fix (Sig fs) -> Fix (Sig fs) -> Fix (Sig fs) x + y = inject (A x y) () : {auto prf : Elem Mul fs} -> Fix (Sig fs) -> Fix (Sig fs) -> Fix (Sig fs) x y = inject (M x y) -- Evaluation Alg Val Int where alg (V x) = x Alg Add Int where alg (A x y) = x + y Alg Mul Int where alg (M x y) = x * y calc : Expr -> Int calc = eval -- Pretty Printing Alg Val String where alg (V x) = show x Alg Add String where alg (A x y) = "(" ++ x ++ " + " ++ y ++ ")" Alg Mul String where alg (M x y) = "(" ++ x ++ " * " ++ y ++ ")" pretty : Expr -> String pretty = eval -- Examples ex1 : Expr ex1 = (val 1 + val 2) * val 3 runEx : Expr -> IO () runEx e = do putStrLn (pretty e) putStrLn (show (calc e))
! MIT License ! ! Copyright (c) 2020 SHEMAT-Suite ! ! Permission is hereby granted, free of charge, to any person obtaining a copy ! of this software and associated documentation files (the "Software"), to deal ! in the Software without restriction, including without limitation the rights ! to use, copy, modify, merge, publish, distribute, sublicense, and/or sell ! copies of the Software, and to permit persons to whom the Software is ! furnished to do so, subject to the following conditions: ! ! The above copyright notice and this permission notice shall be included in all ! copies or substantial portions of the Software. ! ! THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR ! IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, ! FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE ! AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER ! LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, ! OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE ! SOFTWARE. !> @brief first initialisation for each realisation/sample/gradient/ensemble (original state variables) !> @param[in] ismpl local sample index SUBROUTINE single_init(ismpl) use arrays use mod_genrl use mod_conc use mod_time IMPLICIT NONE integer :: ismpl integer :: i, j, k INTEGER i_max, ilevel INTRINSIC max ! Here the parallelisation can be improved (second-inner level) ! ! But beware the MASTER (inside) and BARRIER (outside) construct at the end ! ! save "OPTI" state (before) ilevel = cgen_opti CALL dcopy(i0j0k0,headold(1,ilevel,idx_master),1,headold(1,ilevel,ismpl),1) CALL dcopy(i0j0k0,tempold(1,ilevel,idx_master),1,tempold(1,ilevel,ismpl),1) CALL dcopy(i0j0k0ntrans,concold(1,1,ilevel,idx_master),1,concold(1,1,ilevel,ismpl),1) CALL dcopy(i0j0*k0,presold(1,ilevel,idx_master),1,presold(1,ilevel,ismpl),1) ! !$OMP master i_max = max(maxunits,bc_maxunits) DO j = 1, nprop DO i = 1, i_max propunitold(i,j) = propunit(i,j,idx_master) END DO END DO DO k = 1, nbctp DO j = 1, 3 DO i = 1, ngsmax bcperiodold(i,j,k) = bcperiod(i,j,k,idx_master) END DO END DO END DO !$OMP end master ! RETURN END
context("tee pipe") test_that("Tee pipe related functionality works.", { dim_message <- function(data.) message(sprintf("Data has dimension %d x %d", NROW(data.), NCOL(data.))) expect_that(iris %T>% dim_message, shows_message(dim_message(iris))) expect_that(iris %T>% dim_message, is_identical_to(iris)) })
[GOAL] R : Type u_1 inst✝ : CommRing R n : ℕ ζ : R i : ℕ hi : i ∈ Nat.divisors n h : IsRoot (cyclotomic i R) ζ ⊢ ζ ^ n = 1 [PROOFSTEP] rcases n.eq_zero_or_pos with (rfl \| hn) [GOAL] case inl R : Type u_1 inst✝ : CommRing R ζ : R i : ℕ h : IsRoot (cyclotomic i R) ζ hi : i ∈ Nat.divisors 0 ⊢ ζ ^ 0 = 1 [PROOFSTEP] exact pow_zero _ [GOAL] case inr R : Type u_1 inst✝ : CommRing R n : ℕ ζ : R i : ℕ hi : i ∈ Nat.divisors n h : IsRoot (cyclotomic i R) ζ hn : n > 0 ⊢ ζ ^ n = 1 [PROOFSTEP] have := congr_arg (eval ζ) (prod_cyclotomic_eq_X_pow_sub_one hn R).symm [GOAL] case inr R : Type u_1 inst✝ : CommRing R n : ℕ ζ : R i : ℕ hi : i ∈ Nat.divisors n h : IsRoot (cyclotomic i R) ζ hn : n > 0 this : eval ζ (X ^ n - 1) = eval ζ (∏ i in Nat.divisors n, cyclotomic i R) ⊢ ζ ^ n = 1 [PROOFSTEP] rw [eval_sub, eval_pow, eval_X, eval_one] at this [GOAL] case inr R : Type u_1 inst✝ : CommRing R n : ℕ ζ : R i : ℕ hi : i ∈ Nat.divisors n h : IsRoot (cyclotomic i R) ζ hn : n > 0 this : ζ ^ n - 1 = eval ζ (∏ i in Nat.divisors n, cyclotomic i R) ⊢ ζ ^ n = 1 [PROOFSTEP] convert eq_add_of_sub_eq' this [GOAL] case h.e'_3 R : Type u_1 inst✝ : CommRing R n : ℕ ζ : R i : ℕ hi : i ∈ Nat.divisors n h : IsRoot (cyclotomic i R) ζ hn : n > 0 this : ζ ^ n - 1 = eval ζ (∏ i in Nat.divisors n, cyclotomic i R) ⊢ 1 = 1 + eval ζ (∏ i in Nat.divisors n, cyclotomic i R) [PROOFSTEP] convert (add_zero (M := R) _).symm [GOAL] case h.e'_3.h.e'_6 R : Type u_1 inst✝ : CommRing R n : ℕ ζ : R i : ℕ hi : i ∈ Nat.divisors n h : IsRoot (cyclotomic i R) ζ hn : n > 0 this : ζ ^ n - 1 = eval ζ (∏ i in Nat.divisors n, cyclotomic i R) ⊢ eval ζ (∏ i in Nat.divisors n, cyclotomic i R) = 0 [PROOFSTEP] apply eval_eq_zero_of_dvd_of_eval_eq_zero _ h [GOAL] R : Type u_1 inst✝ : CommRing R n : ℕ ζ : R i : ℕ hi : i ∈ Nat.divisors n h : IsRoot (cyclotomic i R) ζ hn : n > 0 this : ζ ^ n - 1 = eval ζ (∏ i in Nat.divisors n, cyclotomic i R) ⊢ cyclotomic i R ∣ ∏ i in Nat.divisors n, cyclotomic i R [PROOFSTEP] exact Finset.dvd_prod_of_mem _ hi [GOAL] R✝ : Type u_1 inst✝³ : CommRing R✝ n : ℕ inst✝² : IsDomain R✝ h : 0 < n R : Type u_2 inst✝¹ : CommRing R inst✝ : IsDomain R ζ : R ⊢ ζ ^ n = 1 ↔ ∃ i, i ∈ Nat.divisors n ∧ IsRoot (cyclotomic i R) ζ [PROOFSTEP] rw [← mem_nthRoots h, nthRoots, mem_roots <\| X_pow_sub_C_ne_zero h _, C_1, ← prod_cyclotomic_eq_X_pow_sub_one h, isRoot_prod] [GOAL] R : Type u_1 inst✝¹ : CommRing R n : ℕ inst✝ : IsDomain R hpos : 0 < n μ : R h : IsPrimitiveRoot μ n ⊢ IsRoot (cyclotomic n R) μ [PROOFSTEP] rw [← mem_roots (cyclotomic_ne_zero n R), cyclotomic_eq_prod_X_sub_primitiveRoots h, roots_prod_X_sub_C, ← Finset.mem_def] [GOAL] R : Type u_1 inst✝¹ : CommRing R n : ℕ inst✝ : IsDomain R hpos : 0 < n μ : R h : IsPrimitiveRoot μ n ⊢ μ ∈ primitiveRoots n R [PROOFSTEP] rwa [← mem_primitiveRoots hpos] at h [GOAL] R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n ⊢ IsRoot (cyclotomic n K) μ ↔ IsPrimitiveRoot μ n [PROOFSTEP] have hnpos : 0 < n := (NeZero.of_neZero_natCast K).out.bot_lt [GOAL] R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n ⊢ IsRoot (cyclotomic n K) μ ↔ IsPrimitiveRoot μ n [PROOFSTEP] refine' ⟨fun hμ => _, IsPrimitiveRoot.isRoot_cyclotomic hnpos⟩ [GOAL] R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμ : IsRoot (cyclotomic n K) μ ⊢ IsPrimitiveRoot μ n [PROOFSTEP] have hμn : μ ^ n = 1 := by rw [isRoot_of_unity_iff hnpos _] exact ⟨n, n.mem_divisors_self hnpos.ne', hμ⟩ [GOAL] R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμ : IsRoot (cyclotomic n K) μ ⊢ μ ^ n = 1 [PROOFSTEP] rw [isRoot_of_unity_iff hnpos _] [GOAL] R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμ : IsRoot (cyclotomic n K) μ ⊢ ∃ i, i ∈ Nat.divisors n ∧ IsRoot (cyclotomic i K) μ [PROOFSTEP] exact ⟨n, n.mem_divisors_self hnpos.ne', hμ⟩ [GOAL] R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμ : IsRoot (cyclotomic n K) μ hμn : μ ^ n = 1 ⊢ IsPrimitiveRoot μ n [PROOFSTEP] by_contra hnμ [GOAL] R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμ : IsRoot (cyclotomic n K) μ hμn : μ ^ n = 1 hnμ : ¬IsPrimitiveRoot μ n ⊢ False [PROOFSTEP] have ho : 0 < orderOf μ := by apply orderOf_pos' rw [isOfFinOrder_iff_pow_eq_one] exact ⟨n, hnpos, hμn⟩ [GOAL] R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμ : IsRoot (cyclotomic n K) μ hμn : μ ^ n = 1 hnμ : ¬IsPrimitiveRoot μ n ⊢ 0 < orderOf μ [PROOFSTEP] apply orderOf_pos' [GOAL] case h R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμ : IsRoot (cyclotomic n K) μ hμn : μ ^ n = 1 hnμ : ¬IsPrimitiveRoot μ n ⊢ IsOfFinOrder μ [PROOFSTEP] rw [isOfFinOrder_iff_pow_eq_one] [GOAL] case h R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμ : IsRoot (cyclotomic n K) μ hμn : μ ^ n = 1 hnμ : ¬IsPrimitiveRoot μ n ⊢ ∃ n, 0 < n ∧ μ ^ n = 1 [PROOFSTEP] exact ⟨n, hnpos, hμn⟩ [GOAL] R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμ : IsRoot (cyclotomic n K) μ hμn : μ ^ n = 1 hnμ : ¬IsPrimitiveRoot μ n ho : 0 < orderOf μ ⊢ False [PROOFSTEP] have := pow_orderOf_eq_one μ [GOAL] R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμ : IsRoot (cyclotomic n K) μ hμn : μ ^ n = 1 hnμ : ¬IsPrimitiveRoot μ n ho : 0 < orderOf μ this : μ ^ orderOf μ = 1 ⊢ False [PROOFSTEP] rw [isRoot_of_unity_iff ho] at this [GOAL] R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμ : IsRoot (cyclotomic n K) μ hμn : μ ^ n = 1 hnμ : ¬IsPrimitiveRoot μ n ho : 0 < orderOf μ this : ∃ i, i ∈ Nat.divisors (orderOf μ) ∧ IsRoot (cyclotomic i K) μ ⊢ False [PROOFSTEP] obtain ⟨i, hio, hiμ⟩ := this [GOAL] case intro.intro R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμ : IsRoot (cyclotomic n K) μ hμn : μ ^ n = 1 hnμ : ¬IsPrimitiveRoot μ n ho : 0 < orderOf μ i : ℕ hio : i ∈ Nat.divisors (orderOf μ) hiμ : IsRoot (cyclotomic i K) μ ⊢ False [PROOFSTEP] replace hio := Nat.dvd_of_mem_divisors hio [GOAL] case intro.intro R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμ : IsRoot (cyclotomic n K) μ hμn : μ ^ n = 1 hnμ : ¬IsPrimitiveRoot μ n ho : 0 < orderOf μ i : ℕ hiμ : IsRoot (cyclotomic i K) μ hio : i ∣ orderOf μ ⊢ False [PROOFSTEP] rw [IsPrimitiveRoot.not_iff] at hnμ [GOAL] case intro.intro R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμ : IsRoot (cyclotomic n K) μ hμn : μ ^ n = 1 hnμ : orderOf μ ≠ n ho : 0 < orderOf μ i : ℕ hiμ : IsRoot (cyclotomic i K) μ hio : i ∣ orderOf μ ⊢ False [PROOFSTEP] rw [← orderOf_dvd_iff_pow_eq_one] at hμn [GOAL] case intro.intro R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμ : IsRoot (cyclotomic n K) μ hμn : orderOf μ ∣ n hnμ : orderOf μ ≠ n ho : 0 < orderOf μ i : ℕ hiμ : IsRoot (cyclotomic i K) μ hio : i ∣ orderOf μ ⊢ False [PROOFSTEP] have key : i < n := (Nat.le_of_dvd ho hio).trans_lt ((Nat.le_of_dvd hnpos hμn).lt_of_ne hnμ) [GOAL] case intro.intro R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμ : IsRoot (cyclotomic n K) μ hμn : orderOf μ ∣ n hnμ : orderOf μ ≠ n ho : 0 < orderOf μ i : ℕ hiμ : IsRoot (cyclotomic i K) μ hio : i ∣ orderOf μ key : i < n ⊢ False [PROOFSTEP] have key' : i ∣ n := hio.trans hμn [GOAL] case intro.intro R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμ : IsRoot (cyclotomic n K) μ hμn : orderOf μ ∣ n hnμ : orderOf μ ≠ n ho : 0 < orderOf μ i : ℕ hiμ : IsRoot (cyclotomic i K) μ hio : i ∣ orderOf μ key : i < n key' : i ∣ n ⊢ False [PROOFSTEP] rw [← Polynomial.dvd_iff_isRoot] at hμ hiμ [GOAL] case intro.intro R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμ : X - ↑C μ ∣ cyclotomic n K hμn : orderOf μ ∣ n hnμ : orderOf μ ≠ n ho : 0 < orderOf μ i : ℕ hiμ : X - ↑C μ ∣ cyclotomic i K hio : i ∣ orderOf μ key : i < n key' : i ∣ n ⊢ False [PROOFSTEP] have hni : { i, n } ⊆ n.divisors := by simpa [Finset.insert_subset_iff, key'] using hnpos.ne' [GOAL] R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμ : X - ↑C μ ∣ cyclotomic n K hμn : orderOf μ ∣ n hnμ : orderOf μ ≠ n ho : 0 < orderOf μ i : ℕ hiμ : X - ↑C μ ∣ cyclotomic i K hio : i ∣ orderOf μ key : i < n key' : i ∣ n ⊢ {i, n} ⊆ Nat.divisors n [PROOFSTEP] simpa [Finset.insert_subset_iff, key'] using hnpos.ne' [GOAL] case intro.intro R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμ : X - ↑C μ ∣ cyclotomic n K hμn : orderOf μ ∣ n hnμ : orderOf μ ≠ n ho : 0 < orderOf μ i : ℕ hiμ : X - ↑C μ ∣ cyclotomic i K hio : i ∣ orderOf μ key : i < n key' : i ∣ n hni : {i, n} ⊆ Nat.divisors n ⊢ False [PROOFSTEP] obtain ⟨k, hk⟩ := hiμ [GOAL] case intro.intro.intro R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμ : X - ↑C μ ∣ cyclotomic n K hμn : orderOf μ ∣ n hnμ : orderOf μ ≠ n ho : 0 < orderOf μ i : ℕ hio : i ∣ orderOf μ key : i < n key' : i ∣ n hni : {i, n} ⊆ Nat.divisors n k : K[X] hk : cyclotomic i K = (X - ↑C μ) * k ⊢ False [PROOFSTEP] obtain ⟨j, hj⟩ := hμ [GOAL] case intro.intro.intro.intro R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμn : orderOf μ ∣ n hnμ : orderOf μ ≠ n ho : 0 < orderOf μ i : ℕ hio : i ∣ orderOf μ key : i < n key' : i ∣ n hni : {i, n} ⊆ Nat.divisors n k : K[X] hk : cyclotomic i K = (X - ↑C μ) * k j : K[X] hj : cyclotomic n K = (X - ↑C μ) * j ⊢ False [PROOFSTEP] have := prod_cyclotomic_eq_X_pow_sub_one hnpos K [GOAL] case intro.intro.intro.intro R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμn : orderOf μ ∣ n hnμ : orderOf μ ≠ n ho : 0 < orderOf μ i : ℕ hio : i ∣ orderOf μ key : i < n key' : i ∣ n hni : {i, n} ⊆ Nat.divisors n k : K[X] hk : cyclotomic i K = (X - ↑C μ) * k j : K[X] hj : cyclotomic n K = (X - ↑C μ) * j this : ∏ i in Nat.divisors n, cyclotomic i K = X ^ n - 1 ⊢ False [PROOFSTEP] rw [← Finset.prod_sdiff hni, Finset.prod_pair key.ne, hk, hj] at this [GOAL] case intro.intro.intro.intro R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμn : orderOf μ ∣ n hnμ : orderOf μ ≠ n ho : 0 < orderOf μ i : ℕ hio : i ∣ orderOf μ key : i < n key' : i ∣ n hni : {i, n} ⊆ Nat.divisors n k : K[X] hk : cyclotomic i K = (X - ↑C μ) * k j : K[X] hj : cyclotomic n K = (X - ↑C μ) * j this : (∏ x in Nat.divisors n \ {i, n}, cyclotomic x K) * ((X - ↑C μ) * k * ((X - ↑C μ) * j)) = X ^ n - 1 ⊢ False [PROOFSTEP] have hn := (X_pow_sub_one_separable_iff.mpr <\| NeZero.natCast_ne n K).squarefree [GOAL] case intro.intro.intro.intro R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμn : orderOf μ ∣ n hnμ : orderOf μ ≠ n ho : 0 < orderOf μ i : ℕ hio : i ∣ orderOf μ key : i < n key' : i ∣ n hni : {i, n} ⊆ Nat.divisors n k : K[X] hk : cyclotomic i K = (X - ↑C μ) * k j : K[X] hj : cyclotomic n K = (X - ↑C μ) * j this : (∏ x in Nat.divisors n \ {i, n}, cyclotomic x K) * ((X - ↑C μ) * k * ((X - ↑C μ) * j)) = X ^ n - 1 hn : Squarefree (X ^ n - 1) ⊢ False [PROOFSTEP] rw [← this, Squarefree] at hn [GOAL] case intro.intro.intro.intro R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμn : orderOf μ ∣ n hnμ : orderOf μ ≠ n ho : 0 < orderOf μ i : ℕ hio : i ∣ orderOf μ key : i < n key' : i ∣ n hni : {i, n} ⊆ Nat.divisors n k : K[X] hk : cyclotomic i K = (X - ↑C μ) * k j : K[X] hj : cyclotomic n K = (X - ↑C μ) * j this : (∏ x in Nat.divisors n \ {i, n}, cyclotomic x K) * ((X - ↑C μ) * k * ((X - ↑C μ) * j)) = X ^ n - 1 hn : ∀ (x : K[X]), x * x ∣ (∏ x in Nat.divisors n \ {i, n}, cyclotomic x K) * ((X - ↑C μ) * k * ((X - ↑C μ) * j)) → IsUnit x ⊢ False [PROOFSTEP] contrapose! hn [GOAL] case intro.intro.intro.intro R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμn : orderOf μ ∣ n hnμ : orderOf μ ≠ n ho : 0 < orderOf μ i : ℕ hio : i ∣ orderOf μ key : i < n key' : i ∣ n hni : {i, n} ⊆ Nat.divisors n k : K[X] hk : cyclotomic i K = (X - ↑C μ) * k j : K[X] hj : cyclotomic n K = (X - ↑C μ) * j this : (∏ x in Nat.divisors n \ {i, n}, cyclotomic x K) * ((X - ↑C μ) * k * ((X - ↑C μ) * j)) = X ^ n - 1 hn : ¬False ⊢ ∃ x, x * x ∣ (∏ x in Nat.divisors n \ {i, n}, cyclotomic x K) * ((X - ↑C μ) * k * ((X - ↑C μ) * j)) ∧ ¬IsUnit x [PROOFSTEP] refine' ⟨X - C μ, ⟨(∏ x in n.divisors \ { i, n }, cyclotomic x K) * k * j, by ring⟩, _⟩ [GOAL] R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμn : orderOf μ ∣ n hnμ : orderOf μ ≠ n ho : 0 < orderOf μ i : ℕ hio : i ∣ orderOf μ key : i < n key' : i ∣ n hni : {i, n} ⊆ Nat.divisors n k : K[X] hk : cyclotomic i K = (X - ↑C μ) * k j : K[X] hj : cyclotomic n K = (X - ↑C μ) * j this : (∏ x in Nat.divisors n \ {i, n}, cyclotomic x K) * ((X - ↑C μ) * k * ((X - ↑C μ) * j)) = X ^ n - 1 hn : ¬False ⊢ (∏ x in Nat.divisors n \ {i, n}, cyclotomic x K) * ((X - ↑C μ) * k * ((X - ↑C μ) * j)) = (X - ↑C μ) * (X - ↑C μ) * ((∏ x in Nat.divisors n \ {i, n}, cyclotomic x K) * k * j) [PROOFSTEP] ring [GOAL] case intro.intro.intro.intro R : Type u_1 inst✝³ : CommRing R n✝ : ℕ inst✝² : IsDomain R n : ℕ K : Type u_2 inst✝¹ : Field K μ : K inst✝ : NeZero ↑n hnpos : 0 < n hμn : orderOf μ ∣ n hnμ : orderOf μ ≠ n ho : 0 < orderOf μ i : ℕ hio : i ∣ orderOf μ key : i < n key' : i ∣ n hni : {i, n} ⊆ Nat.divisors n k : K[X] hk : cyclotomic i K = (X - ↑C μ) * k j : K[X] hj : cyclotomic n K = (X - ↑C μ) * j this : (∏ x in Nat.divisors n \ {i, n}, cyclotomic x K) * ((X - ↑C μ) * k * ((X - ↑C μ) * j)) = X ^ n - 1 hn : ¬False ⊢ ¬IsUnit (X - ↑C μ) [PROOFSTEP] simp [Polynomial.isUnit_iff_degree_eq_zero] [GOAL] R : Type u_1 inst✝² : CommRing R n : ℕ inst✝¹ : IsDomain R inst✝ : NeZero ↑n μ : R ⊢ IsRoot (cyclotomic n R) μ ↔ IsPrimitiveRoot μ n [PROOFSTEP] have hf : Function.Injective _ := IsFractionRing.injective R (FractionRing R) [GOAL] R : Type u_1 inst✝² : CommRing R n : ℕ inst✝¹ : IsDomain R inst✝ : NeZero ↑n μ : R hf : Function.Injective ↑(algebraMap R (FractionRing R)) ⊢ IsRoot (cyclotomic n R) μ ↔ IsPrimitiveRoot μ n [PROOFSTEP] haveI : NeZero (n : FractionRing R) := NeZero.nat_of_injective hf [GOAL] R : Type u_1 inst✝² : CommRing R n : ℕ inst✝¹ : IsDomain R inst✝ : NeZero ↑n μ : R hf : Function.Injective ↑(algebraMap R (FractionRing R)) this : NeZero ↑n ⊢ IsRoot (cyclotomic n R) μ ↔ IsPrimitiveRoot μ n [PROOFSTEP] rw [← isRoot_map_iff hf, ← IsPrimitiveRoot.map_iff_of_injective hf, map_cyclotomic, ← isRoot_cyclotomic_iff'] [GOAL] R : Type u_1 inst✝² : CommRing R n : ℕ inst✝¹ : IsDomain R inst✝ : NeZero ↑n ⊢ Multiset.Nodup (roots (cyclotomic n R)) [PROOFSTEP] obtain h \| ⟨ζ, hζ⟩ := (cyclotomic n R).roots.empty_or_exists_mem [GOAL] case inl R : Type u_1 inst✝² : CommRing R n : ℕ inst✝¹ : IsDomain R inst✝ : NeZero ↑n h : roots (cyclotomic n R) = 0 ⊢ Multiset.Nodup (roots (cyclotomic n R)) [PROOFSTEP] exact h.symm ▸ Multiset.nodup_zero [GOAL] case inr.intro R : Type u_1 inst✝² : CommRing R n : ℕ inst✝¹ : IsDomain R inst✝ : NeZero ↑n ζ : R hζ : ζ ∈ roots (cyclotomic n R) ⊢ Multiset.Nodup (roots (cyclotomic n R)) [PROOFSTEP] rw [mem_roots <\| cyclotomic_ne_zero n R, isRoot_cyclotomic_iff] at hζ [GOAL] case inr.intro R : Type u_1 inst✝² : CommRing R n : ℕ inst✝¹ : IsDomain R inst✝ : NeZero ↑n ζ : R hζ : IsPrimitiveRoot ζ n ⊢ Multiset.Nodup (roots (cyclotomic n R)) [PROOFSTEP] refine' Multiset.nodup_of_le (roots.le_of_dvd (X_pow_sub_C_ne_zero (NeZero.pos_of_neZero_natCast R) 1) <\| cyclotomic.dvd_X_pow_sub_one n R) hζ.nthRoots_nodup [GOAL] R : Type u_1 inst✝² : CommRing R n : ℕ inst✝¹ : IsDomain R inst✝ : NeZero ↑n ⊢ { val := roots (cyclotomic n R), nodup := (_ : Multiset.Nodup (roots (cyclotomic n R))) } = primitiveRoots n R [PROOFSTEP] ext a [GOAL] case a R : Type u_1 inst✝² : CommRing R n : ℕ inst✝¹ : IsDomain R inst✝ : NeZero ↑n a : R ⊢ a ∈ { val := roots (cyclotomic n R), nodup := (_ : Multiset.Nodup (roots (cyclotomic n R))) } ↔ a ∈ primitiveRoots n R [PROOFSTEP] simp only [mem_primitiveRoots, NeZero.pos_of_neZero_natCast R] [GOAL] case a R : Type u_1 inst✝² : CommRing R n : ℕ inst✝¹ : IsDomain R inst✝ : NeZero ↑n a : R ⊢ a ∈ { val := roots (cyclotomic n R), nodup := (_ : Multiset.Nodup (roots (cyclotomic n R))) } ↔ IsPrimitiveRoot a n [PROOFSTEP] convert isRoot_cyclotomic_iff (n := n) (μ := a) [GOAL] case h.e'_1.a R : Type u_1 inst✝² : CommRing R n : ℕ inst✝¹ : IsDomain R inst✝ : NeZero ↑n a : R ⊢ a ∈ { val := roots (cyclotomic n R), nodup := (_ : Multiset.Nodup (roots (cyclotomic n R))) } ↔ IsRoot (cyclotomic n R) a [PROOFSTEP] simp [cyclotomic_ne_zero n R] [GOAL] R : Type u_1 inst✝² : CommRing R n : ℕ inst✝¹ : IsDomain R inst✝ : NeZero ↑n ⊢ roots (cyclotomic n R) = (primitiveRoots n R).val [PROOFSTEP] rw [← cyclotomic.roots_to_finset_eq_primitiveRoots] [GOAL] R : Type u_1 inst✝¹ : CommRing R n : ℕ inst✝ : CharZero R ⊢ Function.Injective fun n => cyclotomic n R [PROOFSTEP] intro n m hnm [GOAL] R : Type u_1 inst✝¹ : CommRing R n✝ : ℕ inst✝ : CharZero R n m : ℕ hnm : (fun n => cyclotomic n R) n = (fun n => cyclotomic n R) m ⊢ n = m [PROOFSTEP] simp only at hnm [GOAL] R : Type u_1 inst✝¹ : CommRing R n✝ : ℕ inst✝ : CharZero R n m : ℕ hnm : cyclotomic n R = cyclotomic m R ⊢ n = m [PROOFSTEP] rcases eq_or_ne n 0 with (rfl \| hzero) [GOAL] case inl R : Type u_1 inst✝¹ : CommRing R n : ℕ inst✝ : CharZero R m : ℕ hnm : cyclotomic 0 R = cyclotomic m R ⊢ 0 = m [PROOFSTEP] rw [cyclotomic_zero] at hnm [GOAL] case inl R : Type u_1 inst✝¹ : CommRing R n : ℕ inst✝ : CharZero R m : ℕ hnm : 1 = cyclotomic m R ⊢ 0 = m [PROOFSTEP] replace hnm := congr_arg natDegree hnm [GOAL] case inl R : Type u_1 inst✝¹ : CommRing R n : ℕ inst✝ : CharZero R m : ℕ hnm : natDegree 1 = natDegree (cyclotomic m R) ⊢ 0 = m [PROOFSTEP] rw [natDegree_one, natDegree_cyclotomic] at hnm [GOAL] case inl R : Type u_1 inst✝¹ : CommRing R n : ℕ inst✝ : CharZero R m : ℕ hnm : 0 = Nat.totient m ⊢ 0 = m [PROOFSTEP] by_contra h [GOAL] case inl R : Type u_1 inst✝¹ : CommRing R n : ℕ inst✝ : CharZero R m : ℕ hnm : 0 = Nat.totient m h : ¬0 = m ⊢ False [PROOFSTEP] exact (Nat.totient_pos (zero_lt_iff.2 (Ne.symm h))).ne hnm [GOAL] case inr R : Type u_1 inst✝¹ : CommRing R n✝ : ℕ inst✝ : CharZero R n m : ℕ hnm : cyclotomic n R = cyclotomic m R hzero : n ≠ 0 ⊢ n = m [PROOFSTEP] haveI := NeZero.mk hzero [GOAL] case inr R : Type u_1 inst✝¹ : CommRing R n✝ : ℕ inst✝ : CharZero R n m : ℕ hnm : cyclotomic n R = cyclotomic m R hzero : n ≠ 0 this : NeZero n ⊢ n = m [PROOFSTEP] rw [← map_cyclotomic_int _ R, ← map_cyclotomic_int _ R] at hnm [GOAL] case inr R : Type u_1 inst✝¹ : CommRing R n✝ : ℕ inst✝ : CharZero R n m : ℕ hnm : map (Int.castRingHom R) (cyclotomic n ℤ) = map (Int.castRingHom R) (cyclotomic m ℤ) hzero : n ≠ 0 this : NeZero n ⊢ n = m [PROOFSTEP] replace hnm := map_injective (Int.castRingHom R) Int.cast_injective hnm [GOAL] case inr R : Type u_1 inst✝¹ : CommRing R n✝ : ℕ inst✝ : CharZero R n m : ℕ hzero : n ≠ 0 this : NeZero n hnm : cyclotomic n ℤ = cyclotomic m ℤ ⊢ n = m [PROOFSTEP] replace hnm := congr_arg (map (Int.castRingHom ℂ)) hnm [GOAL] case inr R : Type u_1 inst✝¹ : CommRing R n✝ : ℕ inst✝ : CharZero R n m : ℕ hzero : n ≠ 0 this : NeZero n hnm : map (Int.castRingHom ℂ) (cyclotomic n ℤ) = map (Int.castRingHom ℂ) (cyclotomic m ℤ) ⊢ n = m [PROOFSTEP] rw [map_cyclotomic_int, map_cyclotomic_int] at hnm [GOAL] case inr R : Type u_1 inst✝¹ : CommRing R n✝ : ℕ inst✝ : CharZero R n m : ℕ hzero : n ≠ 0 this : NeZero n hnm : cyclotomic n ℂ = cyclotomic m ℂ ⊢ n = m [PROOFSTEP] have hprim := Complex.isPrimitiveRoot_exp _ hzero [GOAL] case inr R : Type u_1 inst✝¹ : CommRing R n✝ : ℕ inst✝ : CharZero R n m : ℕ hzero : n ≠ 0 this : NeZero n hnm : cyclotomic n ℂ = cyclotomic m ℂ hprim : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) n ⊢ n = m [PROOFSTEP] have hroot := isRoot_cyclotomic_iff (R := ℂ).2 hprim [GOAL] case inr R : Type u_1 inst✝¹ : CommRing R n✝ : ℕ inst✝ : CharZero R n m : ℕ hzero : n ≠ 0 this : NeZero n hnm : cyclotomic n ℂ = cyclotomic m ℂ hprim : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) n hroot : IsRoot (cyclotomic n ℂ) (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) ⊢ n = m [PROOFSTEP] rw [hnm] at hroot [GOAL] case inr R : Type u_1 inst✝¹ : CommRing R n✝ : ℕ inst✝ : CharZero R n m : ℕ hzero : n ≠ 0 this : NeZero n hnm : cyclotomic n ℂ = cyclotomic m ℂ hprim : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) n hroot : IsRoot (cyclotomic m ℂ) (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) ⊢ n = m [PROOFSTEP] haveI hmzero : NeZero m := ⟨fun h => by simp [h] at hroot ⟩ [GOAL] R : Type u_1 inst✝¹ : CommRing R n✝ : ℕ inst✝ : CharZero R n m : ℕ hzero : n ≠ 0 this : NeZero n hnm : cyclotomic n ℂ = cyclotomic m ℂ hprim : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) n hroot : IsRoot (cyclotomic m ℂ) (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) h : m = 0 ⊢ False [PROOFSTEP] simp [h] at hroot [GOAL] case inr R : Type u_1 inst✝¹ : CommRing R n✝ : ℕ inst✝ : CharZero R n m : ℕ hzero : n ≠ 0 this : NeZero n hnm : cyclotomic n ℂ = cyclotomic m ℂ hprim : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) n hroot : IsRoot (cyclotomic m ℂ) (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) hmzero : NeZero m ⊢ n = m [PROOFSTEP] rw [isRoot_cyclotomic_iff (R := ℂ)] at hroot [GOAL] case inr R : Type u_1 inst✝¹ : CommRing R n✝ : ℕ inst✝ : CharZero R n m : ℕ hzero : n ≠ 0 this : NeZero n hnm : cyclotomic n ℂ = cyclotomic m ℂ hprim : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) n hroot : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) m hmzero : NeZero m ⊢ n = m [PROOFSTEP] replace hprim := hprim.eq_orderOf [GOAL] case inr R : Type u_1 inst✝¹ : CommRing R n✝ : ℕ inst✝ : CharZero R n m : ℕ hzero : n ≠ 0 this : NeZero n hnm : cyclotomic n ℂ = cyclotomic m ℂ hroot : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) m hmzero : NeZero m hprim : n = orderOf (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) ⊢ n = m [PROOFSTEP] rwa [← IsPrimitiveRoot.eq_orderOf hroot] at hprim [GOAL] R : Type u_1 inst✝² : CommRing R n✝ n : ℕ K : Type u_2 inst✝¹ : Field K μ : K h : IsPrimitiveRoot μ n hpos : 0 < n inst✝ : CharZero K ⊢ minpoly ℤ μ ∣ cyclotomic n ℤ [PROOFSTEP] apply minpoly.isIntegrallyClosed_dvd (h.isIntegral hpos) [GOAL] R : Type u_1 inst✝² : CommRing R n✝ n : ℕ K : Type u_2 inst✝¹ : Field K μ : K h : IsPrimitiveRoot μ n hpos : 0 < n inst✝ : CharZero K ⊢ ↑(aeval μ) (cyclotomic n ℤ) = 0 [PROOFSTEP] simpa [aeval_def, eval₂_eq_eval_map, IsRoot.def] using h.isRoot_cyclotomic hpos [GOAL] R✝ : Type u_1 inst✝⁵ : CommRing R✝ n✝ : ℕ K : Type u_2 inst✝⁴ : Field K R : Type u_3 inst✝³ : CommRing R inst✝² : IsDomain R μ : R n : ℕ inst✝¹ : Algebra K R hμ : IsPrimitiveRoot μ n h : Irreducible (cyclotomic n K) inst✝ : NeZero ↑n ⊢ cyclotomic n K = minpoly K μ [PROOFSTEP] haveI := NeZero.of_noZeroSMulDivisors K R n [GOAL] R✝ : Type u_1 inst✝⁵ : CommRing R✝ n✝ : ℕ K : Type u_2 inst✝⁴ : Field K R : Type u_3 inst✝³ : CommRing R inst✝² : IsDomain R μ : R n : ℕ inst✝¹ : Algebra K R hμ : IsPrimitiveRoot μ n h : Irreducible (cyclotomic n K) inst✝ : NeZero ↑n this : NeZero ↑n ⊢ cyclotomic n K = minpoly K μ [PROOFSTEP] refine' minpoly.eq_of_irreducible_of_monic h _ (cyclotomic.monic n K) [GOAL] R✝ : Type u_1 inst✝⁵ : CommRing R✝ n✝ : ℕ K : Type u_2 inst✝⁴ : Field K R : Type u_3 inst✝³ : CommRing R inst✝² : IsDomain R μ : R n : ℕ inst✝¹ : Algebra K R hμ : IsPrimitiveRoot μ n h : Irreducible (cyclotomic n K) inst✝ : NeZero ↑n this : NeZero ↑n ⊢ ↑(aeval μ) (cyclotomic n K) = 0 [PROOFSTEP] rwa [aeval_def, eval₂_eq_eval_map, map_cyclotomic, ← IsRoot.def, isRoot_cyclotomic_iff] [GOAL] R : Type u_1 inst✝² : CommRing R n✝ n : ℕ K : Type u_2 inst✝¹ : Field K μ : K h : IsPrimitiveRoot μ n hpos : 0 < n inst✝ : CharZero K ⊢ cyclotomic n ℤ = minpoly ℤ μ [PROOFSTEP] refine' eq_of_monic_of_dvd_of_natDegree_le (minpoly.monic (IsPrimitiveRoot.isIntegral h hpos)) (cyclotomic.monic n ℤ) (h.minpoly_dvd_cyclotomic hpos) _ [GOAL] R : Type u_1 inst✝² : CommRing R n✝ n : ℕ K : Type u_2 inst✝¹ : Field K μ : K h : IsPrimitiveRoot μ n hpos : 0 < n inst✝ : CharZero K ⊢ natDegree (cyclotomic n ℤ) ≤ natDegree (minpoly ℤ μ) [PROOFSTEP] simpa [natDegree_cyclotomic n ℤ] using totient_le_degree_minpoly h [GOAL] R : Type u_1 inst✝² : CommRing R n✝ n : ℕ K : Type u_2 inst✝¹ : Field K μ : K h : IsPrimitiveRoot μ n hpos : 0 < n inst✝ : CharZero K ⊢ cyclotomic n ℚ = minpoly ℚ μ [PROOFSTEP] rw [← map_cyclotomic_int, cyclotomic_eq_minpoly h hpos] [GOAL] R : Type u_1 inst✝² : CommRing R n✝ n : ℕ K : Type u_2 inst✝¹ : Field K μ : K h : IsPrimitiveRoot μ n hpos : 0 < n inst✝ : CharZero K ⊢ map (Int.castRingHom ℚ) (minpoly ℤ μ) = minpoly ℚ μ [PROOFSTEP] exact (minpoly.isIntegrallyClosed_eq_field_fractions' _ (IsPrimitiveRoot.isIntegral h hpos)).symm [GOAL] R : Type u_1 inst✝ : CommRing R n✝ n : ℕ hpos : 0 < n ⊢ Irreducible (cyclotomic n ℤ) [PROOFSTEP] rw [cyclotomic_eq_minpoly (isPrimitiveRoot_exp n hpos.ne') hpos] [GOAL] R : Type u_1 inst✝ : CommRing R n✝ n : ℕ hpos : 0 < n ⊢ Irreducible (minpoly ℤ (exp (2 * ↑Real.pi * I / ↑n))) [PROOFSTEP] apply minpoly.irreducible [GOAL] case hx R : Type u_1 inst✝ : CommRing R n✝ n : ℕ hpos : 0 < n ⊢ IsIntegral ℤ (exp (2 * ↑Real.pi * I / ↑n)) [PROOFSTEP] exact (isPrimitiveRoot_exp n hpos.ne').isIntegral hpos [GOAL] R : Type u_1 inst✝ : CommRing R n✝ n : ℕ hpos : 0 < n ⊢ Irreducible (cyclotomic n ℚ) [PROOFSTEP] rw [← map_cyclotomic_int] [GOAL] R : Type u_1 inst✝ : CommRing R n✝ n : ℕ hpos : 0 < n ⊢ Irreducible (map (Int.castRingHom ℚ) (cyclotomic n ℤ)) [PROOFSTEP] exact (IsPrimitive.irreducible_iff_irreducible_map_fraction_map (cyclotomic.isPrimitive n ℤ)).1 (cyclotomic.irreducible hpos) [GOAL] R : Type u_1 inst✝ : CommRing R n✝ n m : ℕ h : n ≠ m ⊢ IsCoprime (cyclotomic n ℚ) (cyclotomic m ℚ) [PROOFSTEP] rcases n.eq_zero_or_pos with (rfl \| hnzero) [GOAL] case inl R : Type u_1 inst✝ : CommRing R n m : ℕ h : 0 ≠ m ⊢ IsCoprime (cyclotomic 0 ℚ) (cyclotomic m ℚ) [PROOFSTEP] exact isCoprime_one_left [GOAL] case inr R : Type u_1 inst✝ : CommRing R n✝ n m : ℕ h : n ≠ m hnzero : n > 0 ⊢ IsCoprime (cyclotomic n ℚ) (cyclotomic m ℚ) [PROOFSTEP] rcases m.eq_zero_or_pos with (rfl \| hmzero) [GOAL] case inr.inl R : Type u_1 inst✝ : CommRing R n✝ n : ℕ hnzero : n > 0 h : n ≠ 0 ⊢ IsCoprime (cyclotomic n ℚ) (cyclotomic 0 ℚ) [PROOFSTEP] exact isCoprime_one_right [GOAL] case inr.inr R : Type u_1 inst✝ : CommRing R n✝ n m : ℕ h : n ≠ m hnzero : n > 0 hmzero : m > 0 ⊢ IsCoprime (cyclotomic n ℚ) (cyclotomic m ℚ) [PROOFSTEP] rw [Irreducible.coprime_iff_not_dvd <\| cyclotomic.irreducible_rat <\| hnzero] [GOAL] case inr.inr R : Type u_1 inst✝ : CommRing R n✝ n m : ℕ h : n ≠ m hnzero : n > 0 hmzero : m > 0 ⊢ ¬cyclotomic n ℚ ∣ cyclotomic m ℚ [PROOFSTEP] exact fun hdiv => h <\| cyclotomic_injective <\| eq_of_monic_of_associated (cyclotomic.monic n ℚ) (cyclotomic.monic m ℚ) <\| Irreducible.associated_of_dvd (cyclotomic.irreducible_rat hnzero) (cyclotomic.irreducible_rat hmzero) hdiv
lemma contour_integrable_reversepath: "valid_path g \<Longrightarrow> f contour_integrable_on g \<Longrightarrow> f contour_integrable_on (reversepath g)"
{-# OPTIONS --without-K --safe #-} open import Categories.Category -- Biproducts, a-la Karvonen. -- -- This definition has advantages over more traditional ones, -- namely that that we don't require either enrichment in CMon/Ab, or Zero Objects. -- -- See https://arxiv.org/abs/1801.06488 module Categories.Object.Biproduct {o ℓ e} (𝒞 : Category o ℓ e) where open import Level open import Categories.Object.Coproduct 𝒞 open import Categories.Object.Product 𝒞 open import Categories.Morphism 𝒞 open Category 𝒞 open HomReasoning private variable A B C D : Obj f g h : A ⇒ B record IsBiproduct {A B A⊕B : Obj} (π₁ : A⊕B ⇒ A) (π₂ : A⊕B ⇒ B) (i₁ : A ⇒ A⊕B) (i₂ : B ⇒ A⊕B) : Set (o ⊔ ℓ ⊔ e) where field isCoproduct : IsCoproduct i₁ i₂ isProduct : IsProduct π₁ π₂ π₁∘i₁≈id : π₁ ∘ i₁ ≈ id π₂∘i₂≈id : π₂ ∘ i₂ ≈ id permute : i₁ ∘ π₁ ∘ i₂ ∘ π₂ ≈ i₂ ∘ π₂ ∘ i₁ ∘ π₁ open IsCoproduct isCoproduct public renaming (unique to []-unique) open IsProduct isProduct public renaming (unique to ⟨⟩-unique) record Biproduct (A B : Obj) : Set (o ⊔ ℓ ⊔ e) where field A⊕B : Obj π₁ : A⊕B ⇒ A π₂ : A⊕B ⇒ B i₁ : A ⇒ A⊕B i₂ : B ⇒ A⊕B isBiproduct : IsBiproduct π₁ π₂ i₁ i₂ open IsBiproduct isBiproduct public IsBiproduct⇒Biproduct : {π₁ : C ⇒ A} {π₂ : C ⇒ B} {i₁ : A ⇒ C} {i₂ : B ⇒ C} → IsBiproduct π₁ π₂ i₁ i₂ → Biproduct A B IsBiproduct⇒Biproduct isBiproduct = record { isBiproduct = isBiproduct } Biproduct⇒IsBiproduct : (b : Biproduct A B) → IsBiproduct (Biproduct.π₁ b) (Biproduct.π₂ b) (Biproduct.i₁ b) (Biproduct.i₂ b) Biproduct⇒IsBiproduct biproduct = Biproduct.isBiproduct biproduct Biproduct⇒Product : Biproduct A B → Product A B Biproduct⇒Product b = record { ⟨_,_⟩ = ⟨_,_⟩ ; project₁ = project₁ ; project₂ = project₂ ; unique = ⟨⟩-unique } where open Biproduct b Biproduct⇒Coproduct : Biproduct A B → Coproduct A B Biproduct⇒Coproduct b = record { [_,_] = [_,_] ; inject₁ = inject₁ ; inject₂ = inject₂ ; unique = []-unique } where open Biproduct b
[STATEMENT] lemma supp_nat: fixes n::"nat" shows "(supp n) = {}" [PROOF STATE] proof (prove) goal (1 subgoal): 1. supp n = {} [PROOF STEP] by (simp add: supp_def perm_nat_def)
/** * APER_CHECK - create a FITS image showing the locations of * apertures. / #include "aper_check.h" #include <stdio.h> #include <stdlib.h> #include <gsl/gsl_vector.h> #include "spce_output.h" void mark_trace_in_aXe_mask(ap_pixel ap_p, aXe_mask mask) { gsl_vector_int trace_inds; int i, ind; trace_inds = (gsl_vector_int ) get_trace_inds (ap_p); for (i = 0; i < (int)trace_inds->size; i++) { ind = gsl_vector_int_get (trace_inds, i); fprintf(stderr,"%d %d\n",ap_p[ind].p_x,ap_p[ind].p_y); } gsl_vector_int_free(trace_inds); } /* Function to create and return a gsl array of the same size as the grism part of n observation structure. @param ob a pointer to an observation structure @return a pointer to a newly allocated aXe_mask structure / aXe_mask aXe_mask_init (observation * ob) { aXe_mask mask; long i,j; double v; / Allocated space for the mask array / mask = (aXe_mask ) malloc (sizeof (aXe_mask)); if (!mask) { aXe_message (aXe_M_FATAL, __FILE__, __LINE__, "Out of memory."); } /* Create the GSL image array / mask->img = gsl_matrix_alloc (ob->grism->size1, ob->grism->size2); // fprintf(stderr,"%ld %ld\n",ob->grism->size1,ob->grism->size2); / Initialise the mask array / for (i=0;i<(int)ob->grism->size1;i++) { for (j=0;j<(int)ob->grism->size2;j++) { v = gsl_matrix_get(ob->grism,i,j); gsl_matrix_set(mask->img, i,j, v); } } //gsl_matrix_set_all (mask->img, 1.0); return mask; } /* * This function sets all the pixels listed in an ap_pixel list * in an aXe_mask to 0.0 @param a pointer to a ap_pixel array @param mask an existing aXe_mask structure / void add_ap_p_to_aXe_mask (ap_pixel ap_p, aXe_mask * mask) { ap_pixel cur_p; if (ap_p==NULL) return; for (cur_p = ap_p; cur_p->p_x != -1; cur_p++) { / Loop over pixels which are valid */ if (!isnan (cur_p->count)) { // fprintf(stderr,"%d %d\n",cur_p->p_x, cur_p->p_y); gsl_matrix_set (mask->img, cur_p->p_x, cur_p->p_y, 0.0); } } }
function [EC,ec,degij] = edge_nei_overlap_bd(CIJ) % EDGE_NEI_OVERLAP_BD Overlap amongst neighbors of two adjacent nodes % % [EC,ec,degij] = edge_nei_bd(CIJ); % % This function determines the neighbors of two nodes that are linked by % an edge, and then computes their overlap. Connection matrix must be % binary and directed. Entries of 'EC' that are 'inf' indicate that no % edge is present. Entries of 'EC' that are 0 denote "local bridges", % i.e. edges that link completely non-overlapping neighborhoods. Low % values of EC indicate edges that are "weak ties". % % If CIJ is weighted, the weights are ignored. Neighbors of a node can be % linked by incoming, outgoing, or reciprocal connections. % % Inputs: CIJ, directed (binary/weighted) connection matrix % % Outputs: EC, edge neighborhood overlap matrix % ec, edge neighborhood overlap per edge, in vector format % degij, degrees of node pairs connected by each edge % % Reference: % % Easley and Kleinberg (2010) Networks, Crowds, and Markets. % Cambridge University Press, Chapter 3 % % Olaf Sporns, Indiana University, 2012 [ik,jk,ck] = find(CIJ); lel = length(ck); N = size(CIJ,1); [~,~,deg] = degrees_dir(CIJ); ec = zeros(1,lel); degij = zeros(2,lel); for e=1:lel neiik = setdiff(union(find(CIJ(ik(e),:)),find(CIJ(:,ik(e))')),[ik(e) jk(e)]); neijk = setdiff(union(find(CIJ(jk(e),:)),find(CIJ(:,jk(e))')),[ik(e) jk(e)]); ec(e) = length(intersect(neiik,neijk))/length(union(neiik,neijk)); degij(:,e) = [deg(ik(e)) deg(jk(e))]; end; ff = find(CIJ); EC = 1./zeros(N); EC(ff) = ec; %#ok<FNDSB>
! For testing of Fortran 90 modules ! David Engster <[email protected]> module testmodule implicit none type :: myModType integer :: one,two real :: three end type myType ! interface for external sub interface subroutine externalSub(arg1,arg2,arg3) integer :: arg1 type(myModType) :: arg2 real :: arg3 end subroutine externalSub end interface ! interface for polymorphism interface externalPolySub subroutine polySub_1(arg1,arg2) integer :: arg1,arg2 end subroutine polySub_1 subroutine polySub_2(arg1,arg2) real :: arg1,arg2 end subroutine polySub_2 end interface polySub contains double precision function myModFunc(arg1,arg2) double precision :: arg1,arg2 myModFunc = arg1arg2 end function myModFunc subroutine myModSub(arg1,arg2) integer,intent(in) :: arg1 integer,intent(out) :: arg2 arg2=2arg1 end subroutine myModSub end module testmodule
If $C$ is a convex set and $x, y \in C$ with $x < y$, then the interval $[x, y]$ is contained in $C$.
York achieved three successive wins for the first time in 2013 – 14 after beating Northampton 2 – 0 away , with Bowman and Fletcher scoring in three @-@ second half minutes . Defender John McCombe was signed on a two @-@ and @-@ a @-@ half @-@ year contract following his release from Mansfield , before Clay and Jamal Fyfield left York by mutual consent . Pope returned to York on loan from Charlton for the remainder of the season . York 's run of wins ended with a 0 – 0 draw at home to Bristol Rovers , before their first defeat of the year came after losing 2 – 0 away to Hartlepool . Preston winger Will Hayhurst , a Republic of Ireland under @-@ 21 international , was signed on a one @-@ month loan . York fell to a successive defeat for the first time since September 2013 after being beaten 2 – 0 at home by Chesterfield . Shortly after the match , Smith left the club by mutual consent to pursue first @-@ team football .
(* Copyright (C) 2017 M.A.L. Marques This Source Code Form is subject to the terms of the Mozilla Public License, v. 2.0. If a copy of the MPL was not distributed with this file, You can obtain one at http://mozilla.org/MPL/2.0/. ) ( type: mgga_exc ) b97mv_par_n := 6: b97mv_gamma_x := 0.004: b97mv_par_x := [ [ 0.85, 0, 0], [ 1.007, 0, 1], [ 0.259, 1, 0], [ 0, 0, 0], [ 0, 0, 0], [ 0, 0, 0] ]: b97mv_gamma_ss := 0.2: b97mv_par_ss := [ [ 0.443, 0, 0], [ -1.437, 0, 4], [ -4.535, 1, 0], [ -3.39, 2, 0], [ 4.278, 4, 3], [ 0, 0, 0] ]: b97mv_gamma_os := 0.006: b97mv_par_os := [ [ 1.000, 0, 0], [ 1.358, 1, 0], [ 2.924, 2, 0], [ -8.812, 2, 1], [ -1.39, 6, 0], [ 9.142, 6, 1] ]: $include "lda_x_erf.mpl" $include "b97mv.mpl" wb97mv_f := (rs, z, xs0, xs1, ts0, ts1) -> my_piecewise3(screen_dens_zeta(rs, z), 0, (1 + z)/2 lda_x_erf_spin(rs(2/(1 + z))^(1/3), 1) b97mv_g(b97mv_gamma_x, b97mv_wx_ss, b97mv_par_x, b97mv_par_n, xs0, ts0, 0)) + my_piecewise3(screen_dens_zeta(rs, -z), 0, (1 - z)/2 * lda_x_erf_spin(rs(2/(1 - z))^(1/3), 1) b97mv_g(b97mv_gamma_x, b97mv_wx_ss, b97mv_par_x, b97mv_par_n, xs1, ts1, 0)): f := (rs, z, xt, xs0, xs1, us0, us1, ts0, ts1) -> wb97mv_f(rs, z, xs0, xs1, ts0, ts1) + b97mv_f(rs, z, xs0, xs1, ts0, ts1):
[STATEMENT] lemma real_interval_plus: "real_interval (a + b) = real_interval a + real_interval b" [PROOF STATE] proof (prove) goal (1 subgoal): 1. real_interval (a + b) = real_interval a + real_interval b [PROOF STEP] by (simp add: interval_eqI)
Slice the meat into stip strips. Slice the garlic and the rawits fine. Heat in a wok the oil and fry the meat for 2 minutes. Add the garlic and the rawits and fry it for 1 minute more. Add the palmsugar and the rice and fry it for 2-3 minutes. Mix in the kecap, the fishsauce and the basil leaves and fry it for 2 minutes more, then serve.
require(plotrix) #start h2 54:48 0 #start h3 54:49 1 #A: start h1 54:50 2 #start h6 54:51 3 #B: start h5 54:52 4 #C: start vib 54:59 11 #D: complete vib 55:31 43 #E: start rotation 55:38 50 #F: complete rotation 56:16 88 #stop xterms 56:30 102 # Load data t09_l2_addresses <- read.table("data/t09_analysis_l2_addresses_r1",header=TRUE,sep=",") t09_l3_addresses <- read.table("data/t09_analysis_l3_addresses_r1",header=TRUE,sep=",") t09_h2latency <-read.table("data/t09_analysis_latency_r1",header=TRUE,sep=",") t09_h2latency_p95 <- read.table("data/t09_analysis_latency_p95_r1",header=TRUE,sep=",") t09_h1bandwidth <- read.table("data/t09_analysis_h1bandwidth_r1",header=TRUE,sep=",") t09_h5bandwidth <- read.table("data/t09_analysis_h5bandwidth_r1",header=TRUE,sep=",") phvals1=c(2, 11, 43, 50, 88) phlabels1=c("A", "C", "D", "E", "F") phvals2=c(4) phlabels2=c("B") plot_colors <- c("black","red") # ------ L2 Addresses ------ # Start pdf driver pdf(file="plots/t09_l2_addresses_r1.pdf", height=4, width=8) # Trim excess margin space par(mar=c(4.2, 3.9, 0.2, 0.5)) # Get range max_y <- max(t09_l2_addresses) # Plot plot(t09_l2_addresses, type="p", pch=".", col=plot_colors[1], ann=FALSE, yaxt="n", mgp=c(3, 0.5, 0)) maclabels <- c("00:","20:","40:","60:","80:","A0:","C0:","E0:") macvals <- c(0, max_y/8, 2max_y/8, 3max_y/8, 4max_y/8, 5max_y/8, 6max_y/8, 7max_y/8) axis(2,at=macvals,labels=maclabels,las=2) axis(1,at=phvals1,labels=phlabels1,las=1,mgp=c(3, 1.5, 0)) axis(1,at=phvals2,labels=phlabels2,las=1,mgp=c(3, 1.5, 0)) title(xlab="time (s)") title(ylab="MAC address") # Flush output to PDF dev.off() # ------ L3 Addresses ------ # Start pdf driver pdf(file="plots/t09_l3_addresses_r1.pdf", height=4, width=8) # Trim excess margin space par(mar=c(4.2, 3.9, 0.2, 0.5)) # Get range max_y <- max(t09_l3_addresses) # Plot plot(t09_l3_addresses, type="p", pch=".", col=plot_colors[1], ann=FALSE, yaxt="n", mgp=c(3, 0.5, 0)) iplabels <- c("0.","32.","64.","96.","128.","160.","192.","224.") ipvals <- c(0, max_y/8, 2max_y/8, 3max_y/8, 4max_y/8, 5max_y/8, 6max_y/8, 7max_y/8) axis(2,at=ipvals,labels=iplabels,las=2, cex.axis=0.5) axis(1,at=phvals1,labels=phlabels1,las=1,mgp=c(3, 1.5, 0)) axis(1,at=phvals2,labels=phlabels2,las=1,mgp=c(3, 1.5, 0)) title(xlab="time (s)") title(ylab="IP address") # Flush output to PDF dev.off() # ------ X axis limits for bandwidth plots ------ max_y_h1 <- max(t09_h1bandwidth[[2]]) max_y_h5 <- max(t09_h5bandwidth[[2]]) max_y <- max(max_y_h1,max_y_h5) min_y_h1 <- min(t09_h1bandwidth[[2]]) min_y_h5 <- min(t09_h5bandwidth[[2]]) min_y <- min(min_y_h1,min_y_h5) max_x <- max(t09_h1bandwidth[[1]]) # ------ Bandwidth ------ # Start pdf driver pdf(file="plots/t09_bandwidth_r1.pdf", height=4, width=8) # Trim excess margin space par(mar=c(4.2, 3.9, 0.2, 0.5)) # Plot #plot(t05_data[,4], t05_data[,8], type="n", xlim=c(min_x, max_x), ann=FALSE) plot(t09_h1bandwidth, type="n", col=plot_colors[1], ann=FALSE, mgp=c(3, 0.5, 0), ylim=c(min_y, max_y), xlim=c(0, max_x)) lines(t09_h1bandwidth, type="b", pch=20) points(t09_h5bandwidth, type="n", col=plot_colors[2], ann=FALSE, mgp=c(3, 0.5, 0), ylim=c(min_y, max_y), xlim=c(0, max_x)) lines(t09_h5bandwidth, type="b", pch=18, col=plot_colors[2]) axis(1,at=phvals1,labels=phlabels1,las=1,mgp=c(3, 1.5, 0)) axis(1,at=phvals2,labels=phlabels2,las=1,mgp=c(3, 1.5, 0)) title(xlab="time (s)") title(ylab="Mbps") # Create legend legend("topright", c("slice 1: h1 <-> h3","slice 2: h5 <-> h6"), col=plot_colors, pch=c(20,18), cex=0.8, box.lwd = 0, box.col = "white", bg = "white", inset=c(0.01, 0.01)) # Flush output to PDF dev.off() # ------ H2 Latency ------ # Start pdf driver pdf(file="plots/t09_h2latency_r1.pdf", height=4, width=8) # Trim excess margin space par(mar=c(4.2, 3.9, 0.2, 0.5)) plot(t09_h2latency, type="p", pch=".", col=plot_colors[1], ann=FALSE, mgp=c(3, 0.5, 0), xlab="") points(t09_h2latency_p95, type="p", pch=".", col=plot_colors[2]) axis(1,at=phvals1,labels=phlabels1,las=1,mgp=c(3, 1.5, 0)) axis(1,at=phvals2,labels=phlabels2,las=1,mgp=c(3, 1.5, 0)) title(xlab="time (s)") title(ylab="latency (ms)") # Create legend legend("topright", legend=c(expression('individual latency, p'[i]), expression('95th percentile, (p'[i-100]',...,p'[i]']')), col=plot_colors, pch=16, cex=0.8, box.lwd = 0, box.col = "white", bg = "white", inset=c(0.01, 0.01)) # Flush output to PDF dev.off()
library(dplyr) library(rvest) mufon <- rvest::read_html( 'https://mufoncms.com/last_20_report_public.html' ) %>% rvest::html_element("table") %>% rvest::html_table() %>% dplyr::bind_rows() # rename the columns names(mufon) <- paste(mufon[1, ], sep = "") mufon <- mufon[-1,] save(mufon, file = paste0("data_raw/data_", make.names(Sys.time()), ".Rda"))
In 1989 , the church , its 1912 rectory , and the rectory 's garage were listed in the National Register of Historic Places , as the work of noted Nebraska architect Jacob M. Nachtigall . A pupil of Thomas Rogers Kimball , Nachtigall designed a number of Catholic churches and other buildings in the state , several of which are also listed in the National Register .
module C where open import B C : Set C = B
#include <stdio.h> #include <stdlib.h> #include <omp.h> #include <sys/time.h> #include <gsl/gsl_cblas.h> #define N 5000; void mm_bruteforce_ijk(double a, double b, double c, int I, int K, int J) { for(int i = 0; i < I; i++) { for(int j = 0; j < J; j++) { for(int k = 0; k < K; k++) { c[i J + j] += a[i * K + k] * b[k * J + j]; } } } } void mm_bruteforce_ikj(double a, double b, double c, int I, int K, int J) { for(int i = 0; i < I; i++) { for(int k = 0; k < K; k++) { double dv = a[i K + k]; for(int j = 0; j < J; j++) { c[i * J + j] += dv * b[k * J + j]; } } } } void mm_omp(double a, double b, double c, int I, int K, int J) { #pragma omp parallel for for(int i = 0; i < I; i++) { for(int k = 0; k < K; k++) { register double dv = a[i K + k]; for(int j = 0; j < J; j++) { c[i * J + j] += dv * b[k * J + j]; } } } } void mm_cblas_dgemm(double a, double b, double c, int p, int q, int r) { int l = p; int m = q; int n = r; int lda = m; int ldb = n; int ldc = n; double alpha = 1.0; double beta = 0.0; cblas_dgemm(CblasRowMajor, CblasNoTrans, CblasNoTrans, l, n, m, alpha, a, lda, b, ldb, beta, c, ldc); } void init_arange(double mat, int a, int b) { for(int i = 0; i < (ab); i++) { mat[i] = i + 1; } } void init_value(double mat, int a, int b, double value) { for(int i = 0; i < (ab); i++) { mat[i] = value; } } void init_zero(double mat, int a, int b) { init_value(mat, a, b, 0); } void init_one(double mat, int a, int b) { init_value(mat, a, b, 1); } void timer_start(struct timeval pstv) { gettimeofday(pstv, NULL); } void timer_end(struct timeval petv) { gettimeofday(petv, NULL); } void timer_print(struct timeval pstv, struct timeval petv) { time_t sec; suseconds_t usec; sec = petv->tv_sec - pstv->tv_sec; usec = petv->tv_usec - pstv->tv_usec; if(usec < 0) { sec--; usec += 1000000; } printf("elapsed time : %ld.%ld\n", sec, usec); } typedef void (fptr_mm)(double a, double b, double c, int I, int K, int J); void check_etime_mm(double a, double b, double c, int I, int K, int J, fptr_mm mm) { struct timeval stv; struct timeval etv; init_zero(c, I, J); timer_start(&stv); mm(a, b, c, I, K, J); timer_end(&etv); timer_print(&stv, &etv); } int main(int argc, char argv[]) { int I, K, J; I = K = J = N; double a = (double)malloc(sizeof(double) I * K); double b = (double)malloc(sizeof(double) * K * J); double c = (double)malloc(sizeof(double) * I * J); init_arange(a, I, K); init_arange(b, K, J); check_etime_mm(a, b, c, I, K, J, mm_bruteforce_ijk); check_etime_mm(a, b, c, I, K, J, mm_bruteforce_ikj); check_etime_mm(a, b, c, I, K, J, mm_omp); check_etime_mm(a, b, c, I, K, J, mm_cblas_dgemm); return 0; }
Formal statement is: lemma filterlim_tendsto_pos_mult_at_top: assumes f: "(f \<longlongrightarrow> c) F" and c: "0 < c" and g: "LIM x F. g x :> at_top" shows "LIM x F. (f x * g x :: real) :> at_top" Informal statement is: If $f$ tends to a positive number $c$ and $g$ tends to infinity, then $f \cdot g$ tends to infinity.
( Weak Head Normalisation ) Require Import core core_axioms fintype fintype_axioms. Import ScopedNotations. From Chapter9 Require Export preservation. Definition E_ {m} (L: tm m -> Prop) (s : tm m) : Prop := exists v, star step s v /\ L v. Fixpoint L {m} (A : ty): tm m -> Prop := match A with \| Base => fun s => exists v, star step s v /\ value v \| Fun A1 A2 => fun e => match e with \| (lam B s) => forall k (xi: fin m -> fin k) v, L A1 v -> E_ (L A2) (subst_tm (scons v (xi >> var_tm)) s) \| _ => False end end. Lemma L_ren {m n} s A (xi: fin m -> fin n) : L A s -> L A (ren_tm xi s). Proof. revert m n s xi. induction A; eauto; intros m n s xi. - intros (?&?&?). exists (x⟨xi⟩). split. + substify. eauto using mstep_inst. + destruct x; eauto. - intros. destruct s; try contradiction. intros k zeta v H''. cbn in H. specialize (H _ (xi >> zeta) _ H''). destruct H as (?&?&?). exists x. split; eauto. asimpl. eauto. Qed. Definition G {m k} (Gamma : ctx m) : (fin m -> tm k) -> Prop := fun σ => forall (x : fin m), L (Gamma x) (σ x) . Definition has_ty_sem {m} (Gamma : ctx m) (s: tm m) (A: ty) : Prop := forall k (sigma: fin m -> tm k), G Gamma sigma -> E_ (L A) (subst_tm sigma s). Lemma val_inclusion {m} A (e: tm m) : L A e -> E_ (L A) e. Proof. intros. unfold E_. exists e. split; eauto. Qed. Lemma wn_fundamental_lam {m} Gamma (s: tm m) A: has_type Gamma s A -> has_ty_sem Gamma s A. Proof. intros C. unfold has_ty_sem, E_. induction C; subst; intros. - eauto. - apply val_inclusion. asimpl. intros m. intros. assert (G (S1 .: Gamma) (v .: (sigma >> ren_tm xi))). { unfold G. intros [\|]; cbn; eauto. - cbn in . specialize (H f). now apply L_ren. } (* TODO here we don't apply ext_tm because the goal is not an equality. How do we solve this? 1. waybe write another tactic that does something like assert (H: s[sigma] = ?x) by now asimpl. rewrite H which would take the instantiation as an argument (or somehow infer it maybe with `match Goal with _ \|- context[ subst_tm ... ]) 2. don't use simple apply in asimpl' but rather rewrite so that the normal asimpl tactic works here. But then we have the problem that ext_tm is much too general since we can rewrite every substitution with it. Maybe setoid rewriting can do it. We should somehow say that we can only rewrite with ext_tm if the substitution contains something that would be rewritten by renComp' (i.e. the one with funext) To do that we could form a predicate on subsitutions P : (nat -> tm) -> Prop with constructors for the atomic cases for renComp', renRen', etc. (so <xi> >> [sigma] / [sigma] >> [sigma], etc.) and other congruence constructors. Then declare a Proper instance to rewrite with ext_tm if the substitution fulfills the predicate. This might work but to build the proof of the predicate we probably need typeclasses which will probably lead to pain. ) ( assert (M[(@var_tm (S k) var_zero) .: sigma >> (ren_tm shift)][v .: xi >> (@var_tm m)] = M[_]) by (now asimpl). ) asimpl. ( rewrite renComp'_tm. ) specialize (IHC _ _ H1). destruct (IHC) as (v'&?&?). exists v'; split; eauto. revert H2. unfold funcomp. setoid_rewrite rinstInst_tm'. eauto using mstep_inst. - destruct (IHC1 _ _ H) as (v1&?&?). destruct (IHC2 _ _ H) as (v2&?&?). destruct v1; try contradiction. cbn in H1. destruct (H1 _ id _ H3) as (v&?&?). exists v; split; eauto. asimpl. enough (star step (app (lam t v1) v2) v). + eapply star_trans. eapply mstep_app; eauto. assumption. + eapply star_trans. eright. econstructor; eauto. constructor. * now asimpl in H4. Qed.
! ! ------------------------------------------------------------- ! C L E 0 S M ! ------------------------------------------------------------- ! ! * ! THIS PACKAGE DETERMINES THE VALUES OF FOLLOWING * ! --- --- * ! I Q S C I * ! CLEBSCH - GORDAN COEFFICIENT: I I * ! I QM 0 CM I * ! --- --- * ! * ! Written by G. Gaigalas, * ! Vilnius, Lithuania December 1993 * ! SUBROUTINE CLE0SM(Q, QM, S, C, CM, A) !----------------------------------------------- ! M o d u l e s !----------------------------------------------- USE vast_kind_param, ONLY: DOUBLE USE CONSTS_C !...Translated by Pacific-Sierra Research 77to90 4.3E 07:48:27 11/16/01 !...Switches: !----------------------------------------------- ! I n t e r f a c e B l o c k s !----------------------------------------------- USE ittk_I USE c1e0sm_I IMPLICIT NONE !----------------------------------------------- ! D u m m y A r g u m e n t s !----------------------------------------------- REAL(DOUBLE) :: Q REAL(DOUBLE) :: QM REAL(DOUBLE) , INTENT(IN) :: S REAL(DOUBLE) :: C REAL(DOUBLE) :: CM REAL(DOUBLE) :: A !----------------------------------------------- ! L o c a l V a r i a b l e s !----------------------------------------------- INTEGER :: IIQ, IIC, IIS !----------------------------------------------- A = ZERO IIQ = TWOQ + TENTH IIC = TWOC + TENTH IIS = TWO*S + TENTH IF (ITTK(IIQ,IIC,IIS) == 0) RETURN IF (S >= EPS) THEN CALL C1E0SM (Q, QM, C, CM, A) RETURN ENDIF IF (Q + TENTH < ABS(QM)) RETURN IF (C + TENTH < ABS(CM)) RETURN IF (ABS(Q - C) > EPS) RETURN IF (ABS(QM - CM) > EPS) RETURN A = ONE RETURN END SUBROUTINE CLE0SM
Set Implicit Arguments. Require Export Omega. Require Export ArithRing. Require Export Coq.Numbers.Natural.Peano.NPeano. (* [Nat] is a sub-module of [NPeano], which seems to contain many things. E.g. it defines [Nat.div], [Nat.pow], [Nat.log2]. E.g. it defines [Nat.max], which is the same as [max]. E.g. it has many properties of [max], see [Coq.Structures.GenericMinMax]. Unfortunately [Nat.le] is NOT the same as [le], which is [Peano.le]. For this reason, we do NOT import [Nat]. ) Notation log2 := Nat.log2. Require Import TLC.LibTactics. Require Import LibFunOrd. ( ---------------------------------------------------------------------------- ) ( A few simplification lemmas. ) Lemma plus_lt_plus: forall a x y, x < y -> x + a < y + a. Proof using. intros. omega. Qed. Lemma plus_le_plus: forall a x y, x <= y -> x + a <= y + a. Proof using. intros. omega. Qed. ( ---------------------------------------------------------------------------- ) ( [a <= b] is equivalent to [b = a + n] for some unknown [n]. ) Lemma leq_to_eq_plus: forall a b, a <= b -> exists n, b = a + n. Proof. intros. exists (b - a). omega. Qed. ( ---------------------------------------------------------------------------- ) ( Make [omega] a hint. ) Hint Extern 1 => omega : omega. ( ---------------------------------------------------------------------------- ) ( This lemma allows simplifying a [max] expression, by cases. ) Lemma max_case: forall m1 m2, m2 <= m1 /\ max m1 m2 = m1 \/ m1 <= m2 /\ max m1 m2 = m2. Proof using. intros. destruct (@le_gt_dec m1 m2); eauto using max_r, max_l with omega. Qed. ( This tactic looks for a [max] expression in the hypotheses or in the goal and applies the above lemma. ) Ltac max_case_m_n_as m n h := let i := fresh in destruct (max_case m n) as [ [ h i ] \| [ h i ] ]; rewrite i in ; clear i. Ltac max_case_as h := match goal with \| \|- context[max ?m ?n] => max_case_m_n_as m n h \| foo: context[max ?m ?n] \|- _ => max_case_m_n_as m n h end. Ltac max_case := let h := fresh in max_case_as h. (* [max m1 m2] is an upper bound for [m1] and [m2]. This can be sufficient to reason about [max] without introducing a case split. ) Lemma max_ub: forall m1 m2, m1 <= max m1 m2 /\ m2 <= max m1 m2. Proof using. intros. eauto using Nat.le_max_l, Nat.le_max_r. Qed. Ltac max_ub_m_n_as m n h1 h2 := destruct (max_ub m n) as [ h1 h2 ]; generalize dependent (max m n); intros. Ltac max_ub_as h1 h2 := match goal with \| \|- context[max ?m ?n] => max_ub_m_n_as m n h1 h2 \| foo: context[max ?m ?n] \|- _ => max_ub_m_n_as m n h1 h2 end. Ltac max_ub := let h1 := fresh in let h2 := fresh in max_ub_as h1 h2. ( ---------------------------------------------------------------------------- ) ( Properties of multiplication. ) Lemma mult_positive: forall m n, 0 < m -> 0 < n -> 0 < m n. Proof using. intros. destruct (eq_nat_dec (m * n) 0). forwards [ ? \| ? ]: mult_is_O. eauto. omega. omega. generalize dependent (m * n). intros. omega. Qed. Hint Resolve mult_positive : positive. Lemma mult_magnifies_left: forall m n, 0 < n -> m <= n * m. Proof using. intros. destruct n; [ omega \| simpl ]. generalize (n * m); intro. omega. Qed. Lemma mult_magnifies_right: forall m n, 0 < n -> m <= m * n. Proof using. intros. rewrite mult_comm. eauto using mult_magnifies_left. Qed. Lemma mult_magnifies_right_strict: forall m n, 0 < m -> 1 < n -> m < m * n. Proof using. intros. do 2 (destruct n; [ omega \| ]). rewrite mult_comm. simpl. generalize (n * m). intros. omega. Qed. (* ---------------------------------------------------------------------------- ) ( Properties of division. ) ( It is strange that the Coq standard library offers [divmod_spec], but lacks its corollary [div_spec]. ) Lemma div_spec: forall n k, 0 < k -> exists r, k (n / k) + r = n /\ 0 <= r < k. Proof using. intros. unfold Nat.div. destruct k; [ false; omega \| simpl ]. forwards: Nat.divmod_spec n k 0 k. eauto. destruct (Nat.divmod n k 0 k) as [ q r ]. unpack. simpl. exists (k - r). omega. Qed. (* Avoid undesired simplifications. ) ( TEMPORARY [plus] should be opaque too? ) Global Opaque mult Nat.div max. ( A tactic to reason about [n/2] in terms of its specification. ) Ltac div2 := match goal with \|- context[?n/2] => let h := fresh in forwards h: div_spec n 2; [ omega \| gen h; generalize (n/2); intros; unpack ] end. ( [./2] is monotonic. ) Lemma div2_monotonic: forall m n, m <= n -> m / 2 <= n / 2. Proof using. intros. repeat div2. omega. Qed. Lemma div2_step: forall n, (n + 2) / 2 = n/2 + 1. Proof using. intros. repeat div2. omega. Qed. Lemma div2_monotonic_strict: forall m n, m + 2 <= n -> m / 2 < n / 2. Proof using. intros. cut (m/2 + 1 <= n/2). omega. rewrite <- div2_step. eauto using div2_monotonic. Qed. Lemma mult_div_2: forall n, 2 (n / 2) <= n. Proof using. intros. div2. omega. Qed. Lemma div_mult_2: forall n, (2 * n) / 2 = n. Proof using. intros. div2. omega. Qed. (* A collection of lemmas about division by two and ordering. ) Lemma prove_div2_le: forall m n, m <= 2 n + 1 -> (* tight ) m / 2 <= n. Proof using. intros. div2. omega. Qed. Lemma use_div2_plus1_le: forall m n, (n + 1) / 2 <= m -> ( tight ) n <= 2 m. Proof using. intros m n. div2. omega. Qed. Lemma use_div2_le: forall m n, n / 2 <= m -> (* tight ) n <= 2 m + 1. Proof using. intros m n. div2. omega. Qed. Lemma prove_le_div2: forall m n, 2 * m <= n -> (* tight ) m <= n / 2. Proof using. intros. div2. omega. Qed. Lemma use_le_div2: forall m n, n <= m / 2 -> ( tight ) 2 n <= m. Proof using. intros m n. div2. omega. Qed. Lemma prove_div2_lt: forall m n, m < 2 * n -> (* tight ) m / 2 < n. Proof using. intros. div2. omega. Qed. Lemma use_div2_lt: forall m n, m / 2 < n -> ( tight ) m < 2 n. Proof using. intros m n. div2. omega. Qed. Lemma prove_lt_div2: forall m n, 2 * m < n - 1 -> (* tight ) m < n / 2. Proof using. intros. div2. omega. Qed. Lemma prove_lt_div2_zero: forall n, 1 < n -> ( tight ) 0 < n / 2. Proof using. intros. div2. omega. Qed. Lemma use_lt_div2: forall m n, m < (n + 1) / 2 -> ( tight ) 2 m < n. Proof using. intros m n. div2. omega. Qed. Hint Resolve prove_lt_div2_zero : positive. Hint Resolve prove_div2_le use_div2_plus1_le use_div2_le prove_le_div2 use_le_div2 prove_div2_lt use_div2_lt prove_lt_div2 use_lt_div2 : div2. Goal forall n, n <= 2 * (n / 2) + 1. Proof using. eauto with div2. Qed. (* ---------------------------------------------------------------------------- ) ( The [pow] function. ) Lemma power_positive: forall k n, 0 < n -> 0 < n^k. Proof using. induction k; simpl; intros. omega. eauto using mult_positive. Qed. Hint Resolve power_positive : positive. Lemma power_plus: forall k1 k2 n, n^(k1 + k2) = n^k1 n^k2. Proof using. induction k1; simpl; intros. omega. rewrite IHk1. ring. Qed. Lemma power_of_zero: forall k, 0 < k -> 0^k = 0. Proof using. induction k; simpl; intros; omega. Qed. Lemma power_monotonic_in_k: (* We must assume [n > 0], because [0^0] is 1, yet [0^1] is 0. ) forall n, 0 < n -> monotonic le le (fun k => n^k). Proof using. intros. intros k1 k2 ?. assert (f: k2 = k1 + (k2 - k1)). omega. rewrite f. rewrite power_plus. eapply mult_magnifies_right. eapply power_positive. assumption. Qed. Lemma power_strictly_monotonic_in_k: forall n, 1 < n -> monotonic lt lt (fun k => n^k). Proof using. intros. intros k1 k2 ?. assert (f: k2 = k1 + S (k2 - k1 - 1)). omega. rewrite f. rewrite power_plus. eapply mult_magnifies_right_strict. { eauto with positive omega. } { simpl. eapply lt_le_trans with (m := n). omega. eapply mult_magnifies_right. eauto with positive omega. } Qed. Lemma power_strictly_monotonic_in_n: forall k, 0 < k -> monotonic lt lt (fun n => n^k). Proof using. ( We first prove that this holds when [n1] is nonzero, and reformulate the hypothesis [k > 0] so that it is amenable to induction. ) assert (f: forall k n1 n2, 0 < n1 < n2 -> n1^(S k) < n2^(S k) ). { induction k; simpl; intros. omega. eapply lt_le_trans; [ eapply mult_lt_compat_r \| eapply mult_le_compat_l ]. ( wow ) omega. eapply power_positive with (k := S k). omega. forwards: IHk. eauto. simpl in . omega. } (* There remains to treat separately the case where [n1] is 0. ) intros. intros n1 n2 ?. destruct (le_gt_dec n1 0). { assert (n1 = 0). omega. subst. rewrite power_of_zero by assumption. eapply power_positive. omega. } { destruct k; [ omega \| ]. eapply f. omega. } ( ouf ) Qed. Hint Resolve power_monotonic_in_k power_strictly_monotonic_in_k power_strictly_monotonic_in_n : monotonic typeclass_instances. Lemma power_monotonic_in_n: forall k, monotonic le le (fun n => n^k). Proof using. intros. destruct (eq_nat_dec k 0). { subst. simpl. repeat intro. omega. } { eauto with monotonic omega. } Qed. Hint Resolve power_monotonic_in_n : monotonic typeclass_instances. ( TEMPORARY maybe explicitly use [inverse_monotonic] in lemmas below ) Lemma power_inverse_monotonic_in_k: forall n, 1 < n -> forall k1 k2, n^k1 <= n^k2 -> k1 <= k2. Proof using. intros. eapply monotonic_lt_lt_implies_inverse_monotonic_le_le with (f := fun k => n^k); eauto using power_strictly_monotonic_in_k. Qed. Lemma power_strictly_inverse_monotonic_in_k: forall n k1 k2, 0 < n -> n^k1 < n^k2 -> k1 < k2. Proof using. intros. eapply monotonic_le_le_implies_inverse_monotonic_lt_lt with (f := fun k => n^k); eauto using power_monotonic_in_k. Qed. Lemma power_strictly_inverse_monotonic_in_k_variant: forall n k1 k2, 0 < n -> n^k1 < n^(1 + k2) -> k1 <= k2. Proof using. intros. cut (k1 < 1 + k2). { omega. } eauto using power_strictly_inverse_monotonic_in_k. Qed. Lemma power_strictly_inverse_monotonic_in_k_frame: forall n k1 k2, 1 < n -> n^k2 <= n^k1 < n^(1 + k2) -> k1 = k2. Proof using. intros. assert (k2 <= k1). { eapply power_inverse_monotonic_in_k with (n := n); eauto with omega. } assert (k1 < 1 + k2). { eapply power_strictly_inverse_monotonic_in_k with (n := n); eauto with omega. } omega. Qed. Lemma power_inverse_monotonic_in_n: forall k, 0 < k -> forall n1 n2, n1^k <= n2^k -> n1 <= n2. Proof using. intros. eapply monotonic_lt_lt_implies_inverse_monotonic_le_le with (f := fun n => n^k); eauto using power_strictly_monotonic_in_n. Qed. ( A lower bound on [2^n]. ) Lemma n_lt_power: forall n, n < 2^n. Proof using. induction n; simpl; omega. Qed. ( ---------------------------------------------------------------------------- ) ( Base 2 logarithm. ) ( The Coq standard library gives us the following: Lemma log2_spec : forall n, 0<n -> 2^(log2 n) <= n < 2^(S (log2 n)). ) Ltac log2_spec := match goal with \|- context[log2 ?n] => let h := fresh in forwards h: Nat.log2_spec n; [ eauto with positive omega \| gen h; generalize (log2 n); intros; unpack ] end. ( The above specification is functional, i.e., it defines [log2 n] in a unique manner. ) Lemma log2_uniqueness_half: forall n k1 k2, 2^k1 <= n < 2^(1 + k1) -> 2^k2 <= n < 2^(1 + k2) -> k1 <= k2. Proof using. simpl. intros. unpack. eapply power_strictly_inverse_monotonic_in_k_variant with (n := 2). omega. eauto using le_lt_trans. Qed. Lemma log2_uniqueness: forall n k1 k2, 2^k1 <= n < 2^(1 + k1) -> 2^k2 <= n < 2^(1 + k2) -> k1 = k2. Proof using. intros. forwards: log2_uniqueness_half n k1 k2; eauto. forwards: log2_uniqueness_half n k2 k1; eauto. omega. Qed. ( When applied to a power of two, [log2] yields the exponent. ) Lemma log2_pow: forall k, log2 (2^k) = k. Proof using. intros k. log2_spec. symmetry. eapply power_strictly_inverse_monotonic_in_k_frame with (n := 2). omega. eauto. Qed. ( This is just a repetition of one half of [log2_spec]. ) Lemma pow_log2: forall n, 0 < n -> 2 ^ (log2 n) <= n. Proof using. intros. eapply Nat.log2_spec. eauto. Qed. Lemma pow_succ_log2: forall n, n < 2^(1 + log2 n). Proof using. intros. destruct (eq_nat_dec n 0). { subst. simpl. omega. } { eapply Nat.log2_spec. omega. } Qed. ( The inductive step of many arguments that involve divide-and-conquer. ) Lemma log2_step: forall n, 2 <= n -> 1 + log2 (n/2) = log2 n. Proof using. intros. repeat log2_spec. eapply log2_uniqueness; [ \| eauto ]. simpl. eauto with div2. Qed. ( [log2] is monotonic. ) Lemma log2_monotonic: monotonic le le log2. Proof using. intros m n ?. ( A special case for [m = 0]. ) destruct (eq_nat_dec m 0). { subst. unfold log2. simpl. omega. } ( Case [m > 0]. ) do 2 log2_spec. eapply power_strictly_inverse_monotonic_in_k_variant with (n := 2); simpl; omega. Qed. Hint Resolve log2_monotonic : monotonic typeclass_instances. ( A collection of lemmas involving [log2] and ordering. ) Lemma prove_le_log2: forall k n, 2^k <= n -> k <= log2 n. Proof using. intros. forwards: log2_monotonic. eauto. rewrite log2_pow in . assumption. Qed. Lemma prove_log2_le: forall k n, n <= 2^k -> log2 n <= k. Proof using. intros. forwards: log2_monotonic. eauto. rewrite log2_pow in . assumption. Qed. Lemma prove_log2_lt: forall k n, 0 < n -> n < 2^k -> log2 n < k. Proof using. intros. eapply monotonic_le_le_implies_inverse_monotonic_lt_lt with (f := fun n => 2^n). { eauto with monotonic. } eauto using le_lt_trans, pow_log2. Qed. Hint Resolve prove_le_log2 prove_log2_le prove_log2_lt : log2. ( An upper bound on [log2 n]. ) Lemma log2_lt_n: forall n, 0 < n -> log2 n < n. Proof using. eauto using prove_log2_lt, n_lt_power. Qed. ( ---------------------------------------------------------------------------- ) ( The existence of the function [log2] means that the sequence [n], [n/2], [n/4], etc. tends towards zero. We can exploit this by giving the following induction principle. ) Lemma div2_induction: forall (P : nat -> Prop), P 0 -> (forall n, P (n/2) -> P n) -> forall n, P n. Proof using. introv hbase hstep. assert (f: forall k n, log2 n < k -> P n). { induction k; intros. false. omega. ( Special cases for [n = 0] and [n = 1]. ) destruct (eq_nat_dec n 0); [ subst n \| ]. { eauto. } destruct (eq_nat_dec n 1); [ subst n \| ]. { eauto. } ( In the general case, [2 <= n], we use [log2_step]. ) eapply hstep. eapply IHk. cut (1 + log2 (n / 2) <= k). { omega. } rewrite log2_step by omega. omega. } intros. eapply f with (k := log2 n + 1). omega. Qed. ( The following variant of the above induction principle allows establishing a property [P] that mentions both [log2 n] and [n]. ) Lemma log2_induction: forall (P : nat -> nat -> Prop), P 0 0 -> P 0 1 -> (forall k n, 2 <= n -> P k (n/2) -> P (1+k) n) -> forall n, P (log2 n) n. Proof using. introv h00 h01 hkn. ( Maybe one could give a direct proof; anyway, we choose to give a proof based on the previous induction principle. ) assert (forall n, 1 <= n -> P (log2 n) n). { eapply (@div2_induction (fun n => 1 <= n -> P (log2 n) n)). ( The base case cannot arise. ) { intro. false. omega. } ( Step. ) { intros n IH ?. ( Special case for [n = 1]. ) destruct (eq_nat_dec n 1); [ subst n; exact h01 \| ]. ( In the general case, [2 <= n], we use [log2_step]. ) assert (2 <= n). { omega. } rewrite <- log2_step by assumption. eauto with div2. } } intro n. ( Special case for [n = 0]. ) destruct (eq_nat_dec n 0); [ subst n; exact h00 \| ]. ( General case. *) eauto with omega. Qed.
# Integração numérica Também conhecido na literatura como quadratura. Como calcular $$ \int_a^b f(\xi)\:d\xi $$ para uma função qualquer? * Na mão: difícil mesmo nos casos mais simples * No computador com cálculo simbólico: útil algumas vezes mas pouco eficiente * Métodos numéricos - na prática o que é usado. <!-- TEASER_END --> # Usar a aproximação! É bom lembrar que chegamos em várias maneiras para aproximar uma função: $$ y = f(x) \approx y^\delta = \sum_{i=1}^n a_i\phi_i(x) $$ Uma primeira possibilidade é integrar termo a termo: $$ \int_a^bf(\xi)\:d\xi \approx \sum_{i=1}^n a_i\int_a^b\phi_i(\xi)\:d\xi $$ Como $\phi_i(x)$ foram escolhidos levando em conta simplicidade, esta abordagem pode ser utilizada. # Regra do ponto médio: Dados os pontos $x_1, x_2, \ldots, x_n$, usando interpoladores constantes ```julia using PyPlot using CurveFit using BenchmarkTools ``` ```julia x = 0:0.1:1 xx = 0:0.002:1 f(x) = sin(πx) + 0.5 + cos(x) y = f.(x); yy = f.(xx); ``` ```julia plot(xx, yy, "b-") plot(x, y, "rs") nx = length(x) lty = "k-" for i in 1:nx-1 xm = 0.5 (x[i] + x[i+1]) ym = f(xm) plot([x[i], x[i], x[i+1], x[i+1]], [0, ym, ym, 0], lty ) plot([xm, xm], [0, ym], "k:") end ``` Analisando a figura acima, pode-se facilmante ver que a integral vale (para pontos igualmente espaçados): $$ \int_a^b f(\xi)\:d\xi \approx \frac{b-a}{n-1} \sum_{i=1}^{n-1}\left[ f\left(x_{i+\frac{1}{2}}\right) \right] $$ ```julia function midrule(a, b, f, n) dx = (b-a) / n x = range(a, b, length=n+1) s = 0.0 for i in 1:n s += f(x[i]+dx/2) end return dx * s end ``` ```julia ## Análise de convergência: nn = 2:1000; a = 0.0; b = 1.0; Ie = 2/π + 0.5 + sin(1); Inn = midrule.(a, b, f, nn); err = abs.(Inn .- Ie); loglog(nn, err, "bo"); xlabel("n"); ylabel("ε") nn1 = 10:300; ee1 = 5 .* nn1.^(-2) plot(nn1, ee1, "r--"); text(50, 0.005, L"$\mathcal{O}(n^{-2})$") ``` Assim, a estimativa do erro vale: $$ \varepsilon = \mathcal{O}\left(n^{-2}\right) $$ # Regra do trapézio Agora, ao invés de usar interpolação consante entre os pontos, pode-se ligar os pontos por retas. ```julia function trapezoidal(a, b, f, n) x = range(a, b, length=n+1) s = f(x[1])/2 + f(x[end])/2 for i in 2:n s += f(x[i]) end return (b-a) * s/n end ``` ```julia nn2 = nn1 Inn2 = trapezoidal.(a, b, f, nn2); err2 = abs.(Inn2 .- Ie); ``` ```julia loglog(nn2, err2, "bo"); xlabel("n"); ylabel("ε") mm = 10:300; ee1 = 5 .* mm.^(-2) plot(mm, ee1, "r--") text(50, 0.005, L"$\mathcal{O}(n^{-2})$") ``` ### Estimativa de erro da regra do trapézio: O erro ao integrar uma seção usando a regra do trapézio vale: $$ \epsilon_i = R_i(h) - \int_{i-1}^i f(x)\:dx = f''(\xi) \frac{h^3}{12} $$ Erro total: $$ \epsilon = \sum_{i=1}^n \epsilon_i \approx f''(\xi)\frac{h^2(b-a)}{12} \sim h^2 $$ # Regra de Simpson Usa três pontos para aproximar uma parábola: $$ \int_{i-1}^{i+1} f(x)\:dx = S_i(h) = \frac{h}{3}\left[f(x_{i-1}) + 4f(x_i) + f(x_{i+1})\right] $$ Assim, no domínio inteiro, $$ \int_a^b f(x)\:dx \approx \frac{h}{3}\left[f(x_1) + 4f(x_2) + 2f(x_3) + 4f(x_4) + \ldots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)\right] $$ ### Estimativa do erro: $$ \int_{i-1}^{i+1} f(x)\:dx = \frac{h}{3}\left[f(x_{i-1}) + 4f(x_i) + f(x_{i+1})\right] - \frac{h^5}{90}f^{(4)}(\xi) $$ $$ \epsilon_i = \mathcal{O}(h^5) \quad\Longrightarrow\quad \epsilon = \mathcal{O}(h^4) $$ ```julia # Detalhe n tem que ser divisível por 2. function simpsonsrule(a, b, f, n) x = range(a, b, length=n+1) s = f(x[1]) + f(x[end]) w, w2 = 4,2 for i = 2:n s += wf(x[i]) w, w2 = w2, w end return s (b-a)/(3n) end ``` ```julia nn3 = 2:2:1000 Inn3 = simpsonsrule.(a, b, f, nn3); err3 = abs.(Inn3 .- Ie); ``` ```julia loglog(nn3, err3, "bo") grid(ls=":") xlabel("n") ylabel("ε") nn1 = 10:300 ee1 = 5 .* nn1.^(-4) plot(nn1, ee1, "r--") text(50, 0.000002, L"$\mathcal{O}(n^{-4})$") ``` # Generalização do processo Usando as idéias acima com interpoladores polinomiais de ordem crescente pode-se obter regras de ordem superior para a integração. A idéia é usar interpoladores de Lagrange para calcular as integrais.Com isso se chega às formulas de Newton-Cotes (<https://pt.wikipedia.org/wiki/F%C3%B3rmulas_de_Newton-Cotes>) Cuidado com isso!!! Lembre-se das oscilações dos polinômios de Lagrange. # Integração de Romberg Um processo parecido com a extrapolação de Richardson (veremos isso mais tarde). Suponhamos que queremos estimar $$ \int_a^b f(\xi) \: d\xi $$ Uma primeira aproximação é usar a regra do trapézio nas extremidades. O intervalo pode ser dividido por dois sequencialmente, obtendo-se: \begin{align} R_{1,1} &= \frac{h_1}{2}\left[f(a) + f(b)\right] \\ R_{2,1} &= \frac{h_2}{2}\left[f(a) + f(b) + 2f(a+h_2)\right] = \frac{1}{2}\left[ R_{1,1} + h_1 f(a+h_2)\right]\\ \vdots & \\ R_{k,1} &= \frac{1}{2}\left\{ R_{k-1,1} + h_{k-1}\sum_{i=1}^{2^k-2}f\left[a + \left(2i-1\right)h_k\right]\right\} \end{align} onde $$h_k = \frac{b-a}{2^{k-1}}$$ O erro da integral vale (difícil de demonstrar mas possível): $$ \int_a^b f(x)\:dx - R_{k,1} = \sum_{i=1}^\infty K_i h_k^{2i} = K_1 h_k^2 + \sum_{i=2}^\infty K_i h_k^{2i} $$ por outro lado, lembrando que $$\epsilon_{k+1} = \frac{\epsilon_{k}}{4}$$ chega-se a: $$ \int_a^b f(x)\:dx - R_{k+1,1} = \sum_{i=1}^\infty K_i h_{k+1}^{2i} = \sum_{i=1}^\infty \frac{K_i h_{k}^{2i}}{2^{2i}} = \frac{K_1 h_k^2}{4} + \sum_{i=2}^\infty \frac{K_i h_k^{2i}}{4^i} $$ substraindo a equação anterior de 4 vezes a equação acima, chega-se à seguinte relação: $$ R_{k,2} = R_{k,1} + \frac{R_{k,1} - R_{k-1,1}}{3} $$ Repetindo este processo: $$ R_{k,j} = R_{k,j-1} + \frac{R_{k,j-1} - R_{k-1,j-1}}{4^{j-1}-1} $$ ```julia function romberg1(a, b, f, n) R = zeros(n, n) h = Float64(b-a) R[1,1] = h/2 * (f(a) + f(b)) for i in 2:n s = 0.0 for k in 1:(2^(i-2)) s += f(a + (k-0.5)h) end R[i,1] = 0.5 (R[i-1,1] + h * s) for k in 2:i R[i,k] = R[i,k-1] + (R[i,k-1] - R[i-1,k-1]) / (4^(k-1) - 1) end h = h/2 end return R end ``` ```julia romberg1(0, π, sin, 6) ``` ```julia function romberg(a, b, f, n) R1 = zeros(n) R2 = zeros(n) h = Float64(b-a) R1[1] = h/2 * (f(a) + f(b)) for i in 2:n s = 0.0 for k in 1:(2^(i-2)) s += f(a + (k-0.5)h) end R2[1] = 0.5 (R1[1] + h * s) for k in 2:i R2[k] = R2[k-1] + (R2[k-1] - R1[k-1]) / (4^(k-1) - 1) end for k in 1:i R1[k] = R2[k] end h = h/2 end return R2[n] end ``` ```julia function countromberg(n) #R1 = zeros(n) #R2 = zeros(n) #h = Float64(b-a) #R1[1] = h/2 * (f(a) - f(b)) countfun = 2 countfloat = 2 for i in 2:n for k in 1:(2^(i-2)) countfun += 1 countfloat += 4 end for k in 2:i countfloat += 5 end countfloat += 1 end return countfun, countfloat end ``` ```julia nnx = 1:20 nny = countromberg.(nnx) nfun = [x[1] for x in nny] nfloat = [x[2] for x in nny]; ``` ```julia semilogy(nnx, nfun) grid() xlabel("Número de iterações") ylabel("Número de chamadas de função") ``` ```julia nn4 = nnx Inn4 = romberg.(a, b, f, nn4); err4 = abs.(Inn4 .- Ie); ``` ```julia rom = loglog(nfun, err4, "bo", label="Romberg") sim = plot(nn3, err3, "r-", label="Regra de Simpson") mid = plot(nn, err, "g-", label="Regra do ponto médio") grid() xlabel("Número de Chamadas de função") ylabel("ε") legend() ``` ```julia ## Benchmark para comparar tempo de execução: I1a = midrule(a, b, f, 400) I2a = simpsonsrule(a, b, f, 22) I3a = romberg(a, b, f, 4) abs(I1a - Ie), abs(I2a - Ie), abs(I3a - Ie) ``` ```julia @benchmark midrule(a, b, f, 400) ``` ```julia @benchmark simpsonsrule(a, b, f, 22) ``` ```julia @benchmark romberg(a, b, f, 4) ``` ```julia I1b = midrule(a, b, f, 800000) I2b = simpsonsrule(a, b, f, 1000) I3b = romberg(a, b, f, 6) abs(I1b - Ie), abs(I2b - Ie), abs(I3b - Ie) ``` ```julia @benchmark midrule(a, b, f, 800000) ``` ```julia @benchmark simpsonsrule(a, b, f, 1000) ``` ```julia @benchmark romberg(a, b, f, 6) ``` # Problemas ## Problema 1 Faça uma função para calcular a integral usando a regra do ponto médio com pontos cujo espaçamento não é igual ```julia ``` ## Problema 2 Faça uma função para calcular a integral usando a regra do trapézio com pontos cujo espaçamento não é igual ```julia ``` ## Problema 3 Calcule usando os diversos métodos apresentados a seguinte integral: $$ \int_{-1}^1 \left(1- \| x \|\right) \: dx $$ ```julia ``` ## Problema 4 A regra do ponto médio apresenta o mesmo erro assintótico que a regra do trapézio. Vamos explorar um pouco isso: 1. Você consegue criar um método com erro $\mathcal{O}(h)$? 2. Se você criou o método acima do jeito mais simples você consegue criar um outro método bem semelhante com o mesmo tipo de erro. 3. Qual a relação entre estes dois métodos e a regra ponto médio ```julia ```
x2 <- seq(0,2pi,len=100) y2 <- sin(x2) y4 <- cos(x2) plot(y2) plot(y4) par(mfrow = c(1,2)) plot(y2,y4) polygon(y2,y4,col='lightgreen') plot(y2,y4, asp=1, type='n') polygon(y2,y4,col='lightgreen') #install.packages("GISTools", depend = T) library(GISTools) data(georgia) class(georgia) georgia@data appling <- georgia.polys[[1]] View(georgia@data) plot.new() plot(appling, asp=1, type='n', xlab="Easting", ylab="Northing") polygon(appling, density=10, angle=145) View(appling[[1]]) colours() polygon(appling, col="blue") polygon(appling, col=rgb(0,0.5,0.7,0.4)) # set the plot extent plot(appling, asp=1, type='n', xlab="Easting", ylab="Northing") # plot the points points(x = runif(500,126,132)10000, y = runif(500,103,108)10000, pch=16, col='red') # plot the polygon with a transparency factor polygon(appling, col=rgb(0,0.5,0.7,0.4)) plot(c(-1.5,1.5),c(-1.5,1.5),asp=1, type='n') # plot the green/blue rectangle rect(-0.5,-0.5,0.5,0.5, border=NA, col=rgb(0,0.5,0.5,0.7)) # then the second one rect(0,0,1,1, col=rgb(1,0.5,0.5,0.7)) data(meuse.grid) mat = SpatialPixelsDataFrame(points = meuse.grid[c("x", "y")], data = meuse.grid) image(mat, "dist") library(RColorBrewer) greenpal <- brewer.pal(7,'Greens') image(mat, "dist", col=greenpal) library(ggplot2) theme_bw() theme_dark() library(gridExtra) appling appling <- data.frame(appling) colnames(appling) <- c("X", "Y") #ggplot2 p1 <- qplot(X, Y, data = appling, geom = "polygon", asp = 1, colour = I("black"), fill=I(rgb(0,0.5,0.7,0.4))) + theme(axis.text=element_text(size=12), axis.title=element_text(size=20)) # create a data.frame to hold the points p1 df <- data.frame(x = runif(500,126,132)10000, y = runif(500,103,108)10000) p2 <- ggplot(appling, aes(x = X, y= Y)) + geom_polygon(fill = I(rgb(0,0.5,0.7,0.4))) + geom_point(data = df, aes(x, y),col=I('red')) + coord_fixed() + theme(axis.text=element_text(size=12), axis.title=element_text(size=20)) p2 grid.arrange(p1, p2, ncol = 2) #is.element("sf", installed.packages()) #install.packages("sf", dep = TRUE) #help(sf) #??sf #library(GISTools) data(newhaven) ls() class(breach) class(appling) class(Rooms) class(roads) head(data.frame(blocks)) head(blocks@data) plot(blocks) par(mar = c(0,0,0,0)) plot(roads, col="red") plot(blocks, add = T) ### ------------------------- library(sf) vignette(package = "sf") vignette("sf1", package = "sf") # load the georgia data data(georgia) # conversion to sf georgia_sf = st_as_sf(georgia) class(georgia_sf) View(georgia_sf) View(georgia@data) # all attributes plot(georgia_sf) # selected attribute plot(georgia_sf[, 6]) # selected attributes plot(georgia_sf[,c(4,5)]) georgia_sf[,c(4,5)] georgia_sf ## sp SpatialPolygonDataFrame object head(data.frame(georgia)) ## sf polygon object head(data.frame(georgia_sf)) #######3 --BLOCK --> Spatial frame roads_sf <- st_as_sf(roads) class(roads_sf) r2 <- as(roads_sf, "Spatial") class(r2) ############## library(rgdal) writeOGR(obj=georgia, dsn="./nam", layer="abc", driver="ESRI Shapefile", overwrite_layer=T) new.georgia <- readOGR("test/georgia.shp") class(new.georgia) writeOGR(new.georgia, dsn = "./test", layer = "georgia", driver="ESRI Shapefile", overwrite_layer = T) #st_read() #st_write() g2 <- st_read("test/georgia.shp") st_write(g2, "test/georgia.shp", delete_layer = T) st_write(g2, dsn = "test/georgia.shp", layer = "georgia.shp", driver = "ESRI Shapefile", delete_layer = T) #mapping rm(list=ls()) data(georgia) georgia_sf <- st_as_sf(georgia) ##### PLOTING USING TMAP ---_ggpolot2 library(tmap) qtm(georgia, fill = "red", style = "natural") qtm(georgia_sf, fill="MedInc", text="Name", text.size=1, format="World_wide", style="classic", text.root=5, fill.title="Median Income") # sf --> # PostGIS --> st_area g <- st_union(georgia_sf) ## using tmap libary ggplot tm_shape(georgia_sf) + tm_fill("tomato") tm_shape(georgia_sf) + tm_fill("tomato") + tm_borders(lty = "dashed", col = "gold") tm_shape(georgia_sf) + tm_fill("tomato") + tm_borders(lty = "dashed", col = "gold") + tm_style("natural", bg.color = "grey90") tm_shape(georgia_sf) + tm_fill("tomato") + tm_borders(lty = "dashed", col = "gold") + tm_style("natural", bg.color = "grey90") + # now add the outline tm_shape(g) + tm_borders(lwd = 2) tm_shape(georgia_sf) + tm_fill("tomato") + tm_borders(lty = "dashed", col = "gold") + tm_style("natural", bg.color = "grey90") + # now add the outline tm_shape(g) + tm_borders(lwd = 2) + tm_layout(title = "The State of Georgia", title.size = 1, title.position = c(0.55, "top")) # 1st plot of georgia t1 <- tm_shape(georgia_sf) + tm_fill("coral") + tm_borders() + tm_layout(bg.color = "grey85") # 2nd plot of georgia2 t2 <- tm_shape(georgia2) + tm_fill("orange") + tm_borders() + # the asp paramter controls aspect # this is makes the 2nd plot align tm_layout(asp = 0.86,bg.color = "grey95") library(grid) # open a new plot page grid.newpage() # set up the layout pushViewport(viewport(layout=grid.layout(1,2))) # plot using the print command print(t1, vp=viewport(layout.pos.col = 1, height = 5)) print(t2, vp=viewport(layout.pos.col = 2, height = 5)) ################## data.frame(georgia_sf)[,13] tm_shape(georgia_sf) + tm_fill("white") + tm_borders()+ tm_text("Name", size = 0.3)+ tm_layout(frame = FALSE) ######## ---- index <- c(81, 82, 83, 150, 62, 53, 21, 16, 124, 121, 17) georgia_sf.sub <- georgia_sf[index,] tm_shape(georgia_sf.sub) + tm_fill("gold1") + tm_borders("grey") + tm_text("Name", size = 1) + # add the outline tm_shape(g) + tm_borders(lwd = 2) + # specify some layout parameters tm_layout(frame = FALSE, title = "A subset of Georgia", title.size = 1.5, title.position = c(0., "bottom")) # the 1st layer tm_shape(georgia_sf) + tm_fill("white") + tm_borders("grey", lwd = 0.5) + # the 2nd layer tm_shape(g) + tm_borders(lwd = 2) + # the 3rd layer tm_shape(georgia_sf.sub) + tm_fill("lightblue") + tm_borders() + # specify some layout parameters tm_layout(frame = T, title = "Georgia with a subset of counties", title.size = 1, title.position = c(0.02, "bottom")) ##### plolty --> interactive plot with ggplot2 # library(OpenStreetMap) # # define upper left, lower right corners # georgia.sub <- georgia[index,] # # # ul <- as.vector(cbind(bbox(georgia.sub)[2,2], # bbox(georgia.sub)[1,1])) # lr <- as.vector(cbind(bbox(georgia.sub)[2,1], # bbox(georgia.sub)[1,2])) # # download the map tile # MyMap <- openmap(ul,lr) # # now plot the layer and the backdrop # par(mar = c(0,0,0,0)) # plot(MyMap, removeMargin=FALSE) # plot(spTransform(georgia.sub, osm()), add = TRUE, lwd = 2) # load the package library(RgoogleMaps) # convert the subset shp <- SpatialPolygons2PolySet(georgia.sub) # determine the extent of the subset bb <- qbbox(lat = shp[,"Y"], lon = shp[,"X"]) # download map data and store it MyMap <- GetMap.bbox(bb$lonR, bb$latR, destfile = "DC.jpg") # now plot the layer and the backdrop par(mar = c(0,0,0,0)) PlotPolysOnStaticMap(MyMap, shp, lwd=2, col = rgb(0.25,0.25,0.25,0.025), add = F) #### tmap_mode('view') tm_shape(georgia_sf.sub) + tm_polygons(col = "#C6DBEF80" ) tmap_mode("plot") # load package and data library(GISTools) data(newhaven) proj4string(roads) <- proj4string(blocks) # plot spatial data tm_shape(blocks) + tm_borders() + tm_shape(roads) + tm_lines(col = "red") + # embellish the map tm_scale_bar(width = 0.22) + tm_compass(position = c(0.8, 0.07)) + tm_layout(frame = F, title = "New Haven, CT", title.size = 1.5, title.position = c(0.55, "top"), legend.outside = T) # sf --> # load package and data pts_sf <- st_centroid(georgia_sf) pts_sf plot(pts_sf) png(filename = "centroid.png", w = 5, h = 7, units = "in", res = 150) tm_shape(georgia_sf) + tm_fill("olivedrab4") + tm_borders("grey", lwd = 1) + # the points layer tm_shape(pts_sf) + tm_bubbles("PctBlack", title.size = "% Black", col = "gold") # clear workspace rm(list = ls()) # load & list the data data(newhaven) ls() # convert to sf blocks_sf <- st_as_sf(blocks) breach_sf <- st_as_sf(breach) tracts_sf <- st_as_sf(tracts) # have a look at the attributes and object class summary(blocks_sf) class(blocks_sf) summary(breach_sf) class(breach_sf) summary(tracts_sf) class(tracts_sf) data.frame(blocks_sf) head(data.frame(blocks_sf)) colnames(data.frame(blocks_sf)) names(blocks_sf) colnames(blocks@data) head(blocks@data) data.frame(blocks_sf$P_VACANT) blocks$P_VACANT attach(data.frame(blocks_sf)) #attach() function in R Language is used to access the variables present in the data framework without calling the data frame. hist(blocks$P_VACANT) detach(data.frame(blocks_sf)) breach.dens = st_as_sf(kde.points(breach,lims=tracts)) #kde is to create kernel density surface summary(breach.dens) breach.dens plot(breach.dens) blocks_sf$RandVar <- rnorm(nrow(blocks_sf)) tmap_mode('plot') tm_shape(blocks_sf) + tm_polygons("P_OWNEROCC") ################### tm_shape(blocks_sf) + tm_polygons("P_OWNEROCC", breaks=seq(0, 100, by=25)) tm_shape(blocks_sf) + tm_polygons("P_OWNEROCC", breaks=c(10, 40, 60, 90)) tm_shape(blocks_sf) + tm_polygons("P_OWNEROCC", title = "Owner Occ") + tm_layout(legend.title.size = 1, legend.text.size = 1, legend.position = c(0.1, 0.1)) display.brewer.all() brewer.pal(5,'Blues') tm_shape(blocks_sf) + tm_polygons("P_OWNEROCC", title = "Owner Occ", palette = "Reds") + tm_layout(legend.title.size = 1) tm_shape(blocks_sf) + tm_fill("P_OWNEROCC", title = "Owner Occ", palette = "Blues") + tm_layout(legend.title.size = 1) # with equal intervals: the tmap default p1 <- tm_shape(blocks_sf) + tm_polygons("P_OWNEROCC", title = "Owner Occ", palette = "Blues") + tm_layout(legend.title.size = 0.7) # with style = kmeans p2 <- tm_shape(blocks_sf) + tm_polygons("P_OWNEROCC", title = "Owner Occ", palette = "Oranges", style = "kmeans") + tm_layout(legend.title.size = 0.7) # with quantiles p3 <- tm_shape(blocks_sf) + tm_polygons("P_OWNEROCC", title = "Owner Occ", palette = "Greens", breaks = c(0, round(quantileCuts(blocks$P_OWNEROCC, 6), 1))) + tm_layout(legend.title.size = 0.7) p3 library(grid) grid.newpage() pushViewport(viewport(layout=grid.layout(1,3))) print(p1, vp=viewport(layout.pos.col = 1, height = 5)) print(p2, vp=viewport(layout.pos.col = 2, height = 5)) print(p3, vp=viewport(layout.pos.col = 3, height = 5)) tm_shape(blocks_sf) + tm_polygons("P_OWNEROCC", title = "Owner Occ", palette = "-GnBu", breaks = c(0, round(quantileCuts(blocks$P_OWNEROCC, 6), 1)), legend.hist = T)+ tm_scale_bar(width = 0.22) + tm_compass(position = c(0.8, 0.07)) + tm_layout(frame = F, title = "New Haven", title.size = 2, title.position = c(0.55, "top"), legend.hist.size = 0.5) # add a projection to tracts data and convert tracts data to sf proj4string(tracts) <- proj4string(blocks) tracts_sf <- st_as_sf(tracts) class(tracts_sf) tracts_sf <- st_transform(tracts_sf, "+proj=longlat +ellps=WGS84") # plot tm_shape(blocks_sf) + tm_fill(col="POP1990", convert2density=TRUE, style="kmeans", title=expression("Population (per " km^2 * ")"), legend.hist=F, id="name") + tm_borders("grey25", alpha=.5) + # add tracts context tm_shape(tracts_sf) + tm_borders("grey40", lwd=2) # add an area in km^2 to blocks blocks_sf$area = st_area(blocks_sf) / (1000*1000) # calculate population density manually summary(blocks_sf$POP1990/blocks_sf$are) tm_shape(blocks_sf) + tm_fill(c("P_RENTROCC", "P_BLACK")) + tm_borders() + tm_layout(legend.format = list(digits = 0), legend.position = c("left", "bottom"), legend.text.size = 0.5, legend.title.size = 0.8) ### Mapping Point and Atttribute tm_shape(blocks_sf) + tm_polygons("white") + tm_shape(breach_sf) + tm_dots(size = 0.5, shape = 19, col = "red", alpha = 1) tm_shape(breach_sf) + tm_dots(size = 0.5, shape = 19, col = "red", alpha = 0.5) # load the data data(quakes) # look at the first 6 records head(quakes) # define the coordinates coords.tmp <- cbind(quakes$long, quakes$lat) # create the SpatialPointsDataFrame quakes.sp <- SpatialPointsDataFrame(coords.tmp, data = data.frame(quakes), proj4string = CRS("+proj=longlat ")) # convert to sf quakes_sf <- st_as_sf(quakes.sp) tm_shape(quakes_sf) + tm_dots(size = 0.5, alpha = 0.3) library(grid) # by size p1 <- tm_shape(quakes_sf) + tm_bubbles("depth", scale = 1, shape = 19, alpha = 0.3, title.size="Quake Depths") # by colour p2 <- tm_shape(quakes_sf) + tm_dots("depth", shape = 19, alpha = 0.5, size = 0.6, palette = "PuBuGn", title="Quake Depths") # Multiple plots using the grid package grid.newpage() # set up the layout pushViewport(viewport(layout=grid.layout(1,2))) # plot using he print command print(p1, vp=viewport(layout.pos.col = 1, height = 5)) print(p2, vp=viewport(layout.pos.col = 2, height = 5)) # create the index index <- quakes_sf$mag > 5.5 summary(index) # select the subset assign to tmp tmp <- quakes_sf[index,] # plot the subset tm_shape(tmp) + tm_dots(col = brewer.pal(5, "Reds")[4], shape = 19, alpha = 0.5, size = 1) + tm_layout(title="Quakes > 5.5", title.position = c("centre", "top")) library(RgoogleMaps) # define Lat and Lon Lat <- as.vector(quakes$lat) Long <- as.vector(quakes$long) # get the map tiles # you will need to be online MyMap <- MapBackground(lat=Lat, lon=Long, zoom = 10, maptype = "satellite") # define a size vector tmp <- 1+(quakes$mag - min(quakes$mag))/max(quakes$mag) PlotOnStaticMap(MyMap,Lat,Long,cex=tmp,pch=1,col='#FB6A4A30') ### Mapping line and attribute data(newhaven) proj4string(roads) <- proj4string(blocks) xmin <- bbox(roads)[1,1] ymin <- bbox(roads)[2,1] xmax <- xmin + diff(bbox(roads)[1,]) / 2 ymax <- ymin + diff(bbox(roads)[2,]) / 2 xx = as.vector(c(xmin, xmin, xmax, xmax, xmin)) yy = as.vector(c(ymin, ymax, ymax, ymin, ymin)) crds <- cbind(xx,yy) Pl <- Polygon(crds) ID <- "clip" Pls <- Polygons(list(Pl), ID=ID) Pls SPls <- SpatialPolygons(list(Pls)) df <- data.frame(value=1, row.names=ID) clip.bb <- SpatialPolygonsDataFrame(SPls, df) proj4string(clip.bb) <- proj4string(blocks) # 3. convert to sf # convert the data to sf clip_sf <- st_as_sf(clip.bb) roads_sf <- st_as_sf(roads) # 4. clip out the roads and the data frame roads_tmp <- st_intersection(st_cast(clip_sf), roads_sf) tm_shape(clip_sf) + tm_polygons("white") + tm_shape(roads_sf) + tm_lines(size = 0.5, shape = 19, col = "red", alpha = 1)+ tm_shape(roads_tmp)+ tm_lines(size = 0.5, shape = 19, col = "red", alpha = 1)
The mainstay of Ali 's act was " water spouting " . After swallowing large amounts of water , 60 to 100 glasses at a time , he spouted the water in a continuous stream for a sustained period of time , sometimes approaching one minute . Another common trick was to swallow 30 to 50 unshelled hazelnuts ( although one of his posters advertised 40 pecans ) , followed by another nut of a different variety , such as an almond . Ali then brought them up one by one with the odd @-@ nut @-@ out produced at a mark called out by the audience . In another trick , Ali swallowed three to six handkerchiefs of different hues and then produced them in a color order requested by audience members .
module Url import Control.Monad.Either import TyTTP.URL main : IO () main = eitherT putStrLn pure $ do Left EmptyString <- pure $ Simple.parse "" \| _ => throwError "empty url is not matched" Left EmptyString <- pure $ Simple.parse " \t\r\n" \| _ => throwError "blank url is not matched" Left MissingAuthorityOrPath <- pure $ Simple.parse "http:" \| _ => throwError "http scheme not recognized" Right (MkURL (Just HTTP) Nothing "something" "") <- pure $ Simple.parse "http:something" \| _ => throwError "http scheme not recognized" Right (MkURL (Just HTTPS) Nothing "som" "") <- pure $ Simple.parse " \thttps:som" \| _ => throwError "https scheme with whitespace not recognized" Right (MkURL Nothing (Just "something") "/" "") <- pure $ Simple.parse "//something" \| _ => throwError "simple authority not recognized" Right (MkURL (Just HTTP) (Just "user:passwd@something") "/path/like" "?query=string") <- pure $ Simple.parse "http://user:passwd@something/path/like?query=string" \| _ => throwError "simple authority not recognized" pure ()
#Load positions pos <- read.table("optimaltransport.ascii",header=F) p = nrow(pos) par(mfrow=c(1,1)) plot(pos) set.seed(3453443) #Calculate pairwise distances between cities d = as.matrix((dist(pos,diag=TRUE,upper=TRUE))) #Convert to vector in order to access many components at a time d = as.vector(d) #Random order of visits as initialization #We implement the fact that the distribution must start and finish at home town theta = c(1, sample(2:p,p-1), 1) #Convert sequential pairs into index of the d-vector ind = (theta[-p]-1)p+theta[-1] #Calculate total distance of order V = sum(d[ind]) Vopt = V Vseq = V #Make table of neighbor search #first two columns are indices to be swapped #We exclude the hometown from it, starting from 2 instead of 1 num = (p-1)(p-2)/2 searchtab = matrix(0,nrow=num,ncol=2) ind = 1 for(i1 in 2:(p-1)) for(i2 in (i1+1):p) { searchtab[ind,1:2] = c(i1,i2) ind = ind+1 } searchtab #Perform neighbor search, changing best two components more = TRUE tabu = NULL H = NULL tau = 5 #while(more) for(it in 1:10000) { V2opt = V+1000 #Just to get some initial value to beat i1opt = NA for(i in 1:num) { if(is.na(pmatch(i,H))) { #Find indices to swap i1 = searchtab[i,1] i2 = searchtab[i,2] #Swap components, put into theta2 theta2 = theta theta2[i1] = theta[i2] theta2[i2] = theta[i1] #Calculate value for new configuration ind2 = (theta2[-p]-1)*p+theta2[-1] V2 = sum(d[ind2]) #If best so far, store it if(V2<V2opt) { V2opt = V2 iopt = i i1opt = i1 i2opt = i2 } } } #Change to best configuration found theta2 = theta theta2[i1opt] = theta[i2opt] theta2[i2opt] = theta[i1opt] theta = theta2 V = V2opt Vseq = c(Vseq,V) #Include the swap in TABU table H = c(H,iopt) #If table is too large, remove first element (oldest swap) if(length(H)>tau) H = H[-1] #Check if better than best so far if(V < Vopt) { theta.opt = theta Vopt = V } } #par(mfrow=c(2,1),mar=c(0.5,1,0.5,1)) plot(pos) lines(pos[theta.opt,1],pos[theta.opt,2]) points(pos[1,1],pos[1,2], type = 'o', col = 'red') dev.copy2pdf(file="../doc/travel_sale_opt1.pdf") plot.ts(Vseq) show(min(Vseq))
%% subImage % Below is a demonstration of the features of the \|subImage\| function %% clear; close all; clc; %% Syntax % \|[M,linearInd]=subImage(M,nSub);\| %% Description % UNDOCUMENTED %% Examples % %% % % <<gibbVerySmall.gif>> % % _GIBBON_ % <www.gibboncode.org> % % _Kevin Mattheus Moerman_, <[email protected]> %% % _GIBBON footer text_ % % License: <https://github.com/gibbonCode/GIBBON/blob/master/LICENSE> % % GIBBON: The Geometry and Image-based Bioengineering add-On. A toolbox for % image segmentation, image-based modeling, meshing, and finite element % analysis. % % Copyright (C) 2006-2022 Kevin Mattheus Moerman and the GIBBON contributors % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see <http://www.gnu.org/licenses/>.
# Kinematika togega telesa Prof. dr. Janko Slavič, Univerza v Ljubljani, Fakulteta za strojništvo ## Vrtilna količina - vektorski izračun ```python import sympy as sym import numpy as np import matplotlib.pyplot as plt %matplotlib inline sym.init_printing() ``` Če uporabite Google Colab, potem odkomentirajte (odstranite #) spodnje vrstice in jih sprožite: ```python #def custom_latex_printer(exp,options): # from google.colab.output._publish import javascript # url = "https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.3/latest.js?config=default" # javascript(url=url) # return sym.printing.latex(exp,options) #sym.init_printing(use_latex="mathjax",latex_printer=custom_latex_printer) ``` Definirajmo simbole: ```python ωx, ωy, ωz, m = sym.symbols("omega_x, omega_y, omega_z m") x = sym.Function('x')(m) y = sym.Function('y')(m) z = sym.Function('z')(m) ``` in vektorje: ```python r = sym.Matrix([[x], [y], [z]]) ω = sym.Matrix([[ωx], [ωy], [ωz]]) ``` ```python r ``` ```python ω ``` Želimo izračunati: $$\boldsymbol{L}=\int_m \boldsymbol{r}\times\boldsymbol{\omega}\times\boldsymbol{r}\,\,\textrm{d}m$$ Izračunajmo podintegralsko funkcijo ```python B = r.cross(ω.cross(r)) ``` ```python B ``` Ker je kotna hitrost $\boldsymbol{\omega}$ konstantna za vse masne delce, jo lahko izpostavimo. Poglejmo si najprej za prvo vrstico $B$: ```python B[0] ``` ```python B[0].expand().coeff(ωx) ``` ```python B[0].expand().coeff(ωy) ``` ```python B[0].expand().coeff(ωz) ``` In sedaj posplošimo: ```python L = sym.Matrix([[wsym.integrate(b.expand().coeff(w),m) for w in ω] for b in B]) L ``` Iz zapisa je jasno, da je smiselno zapisati masni vztrajnostni tenzor (pri togem telesu je konstanta) kot: ## Masni vztrajnostni moment trikotne plošče (V mapi `\ostalo\` se nahaja SolidWorks datoteka obravnavane plošče; preverite MVM v modelu!) Simboli ```python a, b, h, x, y, ρ, m = sym.symbols('\ a, b, h, x, y, rho, m') ``` Podatki ```python podatki = {a: 1, b:2, h: 0.01, ρ: 2700} ``` Funkcija stranica: ```python fx = a - a/by ``` Priprava numeričnih podatkov: ```python fxp = sym.lambdify(y, fx.subs(podatki), 'numpy') y_n = np.linspace(0, podatki[b], 10) x_n = fxp(y_n) y_n = np.append(y_n, 0.) # zaradi prikaza dodamo 0 na koncu x_n = np.append(x_n, 0.) # zaradi prikaza dodamo 0 na koncu ``` Prikažimo trikotnik ```python ax = plt.subplot(111) plt.fill(x_n[::-1], y_n[::-1], alpha=0.3, closed=True) ax.axis('equal') plt.grid(True) plt.xlim(-.1, podatki[a]+0.1) plt.ylim(-.1, podatki[b]+0.1) plt.show() ``` Masni vztrajnostni momenti (integriramo: $y^2 \, f(x)\,\rho\,h$) ```python J_xx = sym.integrate(y*2 fx * ρ * h, (y, 0, b)) J_xx ``` Vpeljemo maso $m$: ```python J_xx = J_xx.subs(hrab/2, m) J_xx ``` Na podlagi rezultata $J_{xx}$, definiramo še $J_{yy}$ in nato $J_{zz}$: ```python J_yy = J_xx.subs(b,a) J_zz = J_xx + J_yy ``` Izvedemo še integral za deviacijske MVM, najprej $J_{xy}=\int_0^{b} \left(\int_0^{f(x)} x\,y\,h\,\rho\, \textrm{d}x\right) \textrm{d}y$: ```python J_xy = sym.integrate( sym.integrate(xy * h * r, (x, 0, fx)), (y, 0, b)).subs(hrab/2, m) J_xy ``` Ker je telo simetrično glede na ravnini $xz$ in $yz$, sta naslednja MVM enaka nič: ```python J_xz = 0 J_yz = 0 ``` Matrika MVM torej je: ```python J = sym.Matrix([[+J_xx, -J_xy, -J_xz], [-J_xy, +J_yy, -J_yz], [-J_xz, -J_yz, +J_zz]]) ``` ```python J ``` Numerične vrednosti: ```python Jnum = np.array(J.subs(m, hrab/2).evalf(subs=podatki), dtype=float) Jnum ``` array([[18. , -4.5, 0. ], [-4.5, 4.5, 0. ], [ 0. , 0. , 22.5]]) Sedaj lahko izračunamo glavne MVM: ```python eig = J.eigenvects() eig ``` Ali v numerični obliki: ```python eig = np.linalg.eig(Jnum) eig ``` (array([19.36249037, 3.13750963, 22.5 ]), array([[ 0.95709203, 0.28978415, 0. ], [-0.28978415, 0.95709203, 0. ], [ 0. , 0. , 1. ]])) Glavni MVM v numerični obliki so: ```python eig[0] ``` array([19.36249037, 3.13750963, 22.5 ]) In pripadajoči lastni vektorji (stolpci): ```python eig[1] ``` array([[ 0.95709203, 0.28978415, 0. ], [-0.28978415, 0.95709203, 0. ], [ 0. , 0. , 1. ]]) ```python ax = plt.subplot(111) plt.fill(x_n[::-1], y_n[::-1], alpha=0.3, closed=True) ax.axis('equal') plt.grid(True) plt.xlim(-1.1, podatki[a]+1.1) plt.ylim(-1.1, podatki[b]+1.1) ax.quiver(0,0,eig[1][0,0], eig[1][0,1],angles='xy',scale_units='xy',scale=1)#prvi vektor ax.quiver(0,0,eig[1][1,0], eig[1][1,1],angles='xy',scale_units='xy',scale=1)#drugi vektor plt.show() ``` ```python eig[0] ``` array([19.36249037, 3.13750963, 22.5 ]) ```python eig[1][1] ``` array([-0.28978415, 0.95709203, 0. ]) ## Zvrnitev osi - izpeljava Izpeljava zvrnitve osi; definirajmo najprej simbole: ```python Lx, Ly, Lz, x, y, z = sym.symbols("lambda_x, lambda_y, lambda_z, x, y, z") ``` Vektor lege $\mathbf{r}$ in vektor zvrnjene osi: $\mathbf{\lambda}$: ```python r = sym.Matrix([[x], [y], [z]]) L = sym.Matrix([[Lx], [Ly], [Lz]]) ``` ```python r ``` ```python L ``` Podintegralska funkcija je: ```python tmp1 = r.cross(L).dot(r.cross(L)) tmp1 ``` ```python tmp2 = tmp1.expand() tmp2 ``` Opzimo, da konstantne člene $\lambda$ lahko izpostavimo in ostanejo členi, ki definirajo MVM: ```python tmp2.coeff(-2LxLy) ``` ```python ```
import algebra.category.FinVect import algebra.group.units import linear_algebra.dual import data.matrix.notation import linear_algebra.matrix.determinant import group_theory.perm.fin import tactic.norm_swap import Tangle variables (K: Type) [field K] @[simp] def K_2: Module K := Module.of K (fin 2 → K) def FinVect_K_2: FinVect K := ⟨ K_2 K, by change finite_dimensional K (fin 2 → K); apply_instance, ⟩ @[simp] def FinVect_dual (V: FinVect K): FinVect K := ⟨ Module.of K (module.dual K V), by change finite_dimensional K (module.dual K V); apply_instance ⟩ @[simp] def FinVect_tensor (X Y: FinVect K): FinVect K := ⟨ Module.of K (tensor_product K X Y), by change finite_dimensional K (tensor_product K X Y); apply_instance ⟩ namespace Tangle @[simp] def toFinVect: Tangle → FinVect K \| id := ⟨Module.of K K, finite_dimensional.finite_dimensional_self K⟩ \| (of tt) := FinVect_K_2 K \| (of ff) := FinVect_dual K (FinVect_K_2 K) \| (tensor x y) := FinVect_tensor K x.toFinVect y.toFinVect @[simp] def rotate_to_dual (a: Tangle): a.rotate.toFinVect K = FinVect_dual K (a.toFinVect K) := begin dsimp [Tangle.rotate], sorry end end Tangle def functor_map (q: units K): Π {X Y: Tangle}, (X ⟶ᵐ Y) → (X.toFinVect K ⟶ Y.toFinVect K) \| _ _ (𝟙 a) := linear_map.id \| _ _ (f ≫ g) := functor_map g ∘ₗ functor_map f \| _ _ (f ⊗ᵐ g) := tensor_product.map (functor_map f) (functor_map g) \| _ _ (α a b c) := begin have f := tensor_product.assoc K (a.toFinVect K) (b.toFinVect K) (c.toFinVect K), exact Module.of_hom f.to_linear_map, end \| _ _ (α⁻¹ a b c) := begin have f := tensor_product.assoc K (a.toFinVect K) (b.toFinVect K) (c.toFinVect K), exact Module.of_hom f.symm.to_linear_map, end \| _ _ (ℓ a) := begin have f := tensor_product.lid K (a.toFinVect K), exact Module.of_hom f.to_linear_map, end \| _ _ (ℓ⁻¹ a) := begin have f := tensor_product.lid K (a.toFinVect K), exact Module.of_hom f.symm.to_linear_map, end \| _ _ (ρ a) := begin have f := tensor_product.rid K (a.toFinVect K), exact Module.of_hom f.to_linear_map, end \| _ _ (ρ⁻¹ a) := begin have f := tensor_product.rid K (a.toFinVect K), exact Module.of_hom f.symm.to_linear_map, end \| _ _ (hom.evaluation_1 tt) := begin have f := module.dual.eval K ((Tangle.of tt).toFinVect K), have g := tensor_product.uncurry K _ _ _ f, simp, dsimp [Tangle.rotate_to_dual K (Tangle.of tt)], exact Module.of_hom g, end \| _ _ (hom.evaluation_1 ff) := begin have f := module.dual.eval K ((Tangle.of ff).toFinVect K), have g := tensor_product.uncurry K _ _ _ f, rw Tangle.toFinVect at g, simp, dsimp [Tangle.rotate_to_dual K (Tangle.of ff)], have h := Module.of_hom g, exact h, end \| _ _ (hom.coevaluation_1 a) := begin have f := coevaluation K (a.toFinVect K), have g := (tensor_product.comm K _ _).to_linear_map ∘ₗ f, simp, dsimp [a.rotate_to_dual K], exact Module.of_hom g, end \| _ _ hom.braiding_dd_hom := begin have mat: matrix (fin 4) (fin 4) K := ![ ![q^(1/2), 0, 0, 0], ![0, 0, q, 0], ![0, q, q^(1/2)-q^(3/2), 0], ![0, 0, 0, q^(1/2)] ], have X := (↓ ⊗ᵗ ↓).toFinVect K, have b: basis _ _ _ := sorry, have f := matrix.to_lin b b mat, exact f, end \| _ _ hom.braiding_dd_inv := sorry def functor_tangle: Tangle ⥤ FinVect K := { obj := Tangle.toFinVect K, map := begin rintro X Y f, end --by rintro X Y ⟨f⟩; exact functor_map K f, } namespace test open_locale matrix def f: fin 3 → rat := λ i, i + 2 -- (2, 3, 4) @[simp] def iota (n: ℕ): fin n → ℤ := λ i, i /- iota 3 = (0, 1, 2) iota 4 = (0, 1, 2, 3) -/ example: iota 3 = ![0, 1, 2] := begin ext i, apply @fin.cons_induction i ![0, 1, 2], simp, cases i, induction i_val, simp, end example (i: fin 3): ![0, 1, 2] i = i := begin cases i, induction i_val, simp, have h' : i_val_n < 3 := by calc i_val_n < i_val_n.succ : sorry ... < 3 : i_property, have h := matrix.cons_val_succ _ _ ⟨i_val_n, h'⟩, end @[simp] def mat_id (n: nat): matrix (fin n) (fin n) rat := λ i j, if (i = j) then 1 else 0 example: mat_id 2 = ![![1, 0], ![0, 1]] := begin funext, simp, end example: ![![1, 0], ![0, 1]] ⬝ ![![1, 0], ![0, 1]] = ![![1, 0], ![0, 1]] := begin simp, end example (n: nat): mat_id n ⬝ mat_id n = mat_id n := begin funext, rw matrix.mul, dsimp [matrix.dot_product], simp, end end test
[GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G ε : ι → ℝ C : ℝ hε : ∀ (i : ι), 0 < ε i c : ι → 𝕜 hc : ∀ (i : ι), 1 < ‖c i‖ hf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m : (i : ι) → E i ⊢ ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ [PROOFSTEP] rcases em (∃ i, m i = 0) with (⟨i, hi⟩ \| hm) <;> [skip; push_neg at hm ] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G ε : ι → ℝ C : ℝ hε : ∀ (i : ι), 0 < ε i c : ι → 𝕜 hc : ∀ (i : ι), 1 < ‖c i‖ hf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m : (i : ι) → E i ⊢ ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ [PROOFSTEP] rcases em (∃ i, m i = 0) with (⟨i, hi⟩ \| hm) [GOAL] case inl.intro 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G ε : ι → ℝ C : ℝ hε : ∀ (i : ι), 0 < ε i c : ι → 𝕜 hc : ∀ (i : ι), 1 < ‖c i‖ hf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m : (i : ι) → E i i : ι hi : m i = 0 ⊢ ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ [PROOFSTEP] skip [GOAL] case inr 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G ε : ι → ℝ C : ℝ hε : ∀ (i : ι), 0 < ε i c : ι → 𝕜 hc : ∀ (i : ι), 1 < ‖c i‖ hf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m : (i : ι) → E i hm : ¬∃ i, m i = 0 ⊢ ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ [PROOFSTEP] push_neg at hm [GOAL] case inl.intro 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G ε : ι → ℝ C : ℝ hε : ∀ (i : ι), 0 < ε i c : ι → 𝕜 hc : ∀ (i : ι), 1 < ‖c i‖ hf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m : (i : ι) → E i i : ι hi : m i = 0 ⊢ ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ [PROOFSTEP] simp [f.map_coord_zero i hi, prod_eq_zero (mem_univ i), hi] [GOAL] case inr 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G ε : ι → ℝ C : ℝ hε : ∀ (i : ι), 0 < ε i c : ι → 𝕜 hc : ∀ (i : ι), 1 < ‖c i‖ hf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m : (i : ι) → E i hm : ∀ (i : ι), m i ≠ 0 ⊢ ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ [PROOFSTEP] choose δ hδ0 hδm_lt hle_δm _ using fun i => rescale_to_shell (hc i) (hε i) (hm i) [GOAL] case inr 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G ε : ι → ℝ C : ℝ hε : ∀ (i : ι), 0 < ε i c : ι → 𝕜 hc : ∀ (i : ι), 1 < ‖c i‖ hf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m : (i : ι) → E i hm : ∀ (i : ι), m i ≠ 0 δ : ι → 𝕜 hδ0 : ∀ (i : ι), δ i ≠ 0 hδm_lt : ∀ (i : ι), ‖δ i • m i‖ < ε i hle_δm : ∀ (i : ι), ε i / ‖c i‖ ≤ ‖δ i • m i‖ ✝ : ∀ (i : ι), ‖δ i‖⁻¹ ≤ (ε i)⁻¹ * ‖c i‖ * ‖m i‖ ⊢ ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ [PROOFSTEP] have hδ0 : 0 < ∏ i, ‖δ i‖ := prod_pos fun i _ => norm_pos_iff.2 (hδ0 i) [GOAL] case inr 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G ε : ι → ℝ C : ℝ hε : ∀ (i : ι), 0 < ε i c : ι → 𝕜 hc : ∀ (i : ι), 1 < ‖c i‖ hf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m : (i : ι) → E i hm : ∀ (i : ι), m i ≠ 0 δ : ι → 𝕜 hδ0✝ : ∀ (i : ι), δ i ≠ 0 hδm_lt : ∀ (i : ι), ‖δ i • m i‖ < ε i hle_δm : ∀ (i : ι), ε i / ‖c i‖ ≤ ‖δ i • m i‖ ✝ : ∀ (i : ι), ‖δ i‖⁻¹ ≤ (ε i)⁻¹ * ‖c i‖ * ‖m i‖ hδ0 : 0 < ∏ i : ι, ‖δ i‖ ⊢ ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ [PROOFSTEP] simpa [map_smul_univ, norm_smul, prod_mul_distrib, mul_left_comm C, mul_le_mul_left hδ0] using hf (fun i => δ i • m i) hle_δm hδm_lt [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G hf : Continuous ↑f ⊢ ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ [PROOFSTEP] cases isEmpty_or_nonempty ι [GOAL] case inl 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G hf : Continuous ↑f h✝ : IsEmpty ι ⊢ ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ [PROOFSTEP] refine' ⟨‖f 0‖ + 1, add_pos_of_nonneg_of_pos (norm_nonneg _) zero_lt_one, fun m => _⟩ [GOAL] case inl 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G hf : Continuous ↑f h✝ : IsEmpty ι m : (i : ι) → E i ⊢ ‖↑f m‖ ≤ (‖↑f 0‖ + 1) * ∏ i : ι, ‖m i‖ [PROOFSTEP] obtain rfl : m = 0 [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G hf : Continuous ↑f h✝ : IsEmpty ι m : (i : ι) → E i ⊢ m = 0 case inl 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G hf : Continuous ↑f h✝ : IsEmpty ι ⊢ ‖↑f 0‖ ≤ (‖↑f 0‖ + 1) * ∏ i : ι, ‖OfNat.ofNat 0 i‖ [PROOFSTEP] exact funext (IsEmpty.elim ‹_›) [GOAL] case inl 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G hf : Continuous ↑f h✝ : IsEmpty ι ⊢ ‖↑f 0‖ ≤ (‖↑f 0‖ + 1) * ∏ i : ι, ‖OfNat.ofNat 0 i‖ [PROOFSTEP] simp [univ_eq_empty, zero_le_one] [GOAL] case inr 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G hf : Continuous ↑f h✝ : Nonempty ι ⊢ ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ [PROOFSTEP] obtain ⟨ε : ℝ, ε0 : 0 < ε, hε : ∀ m : ∀ i, E i, ‖m - 0‖ < ε → ‖f m - f 0‖ < 1⟩ := NormedAddCommGroup.tendsto_nhds_nhds.1 (hf.tendsto 0) 1 zero_lt_one [GOAL] case inr.intro.intro 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G hf : Continuous ↑f h✝ : Nonempty ι ε : ℝ ε0 : 0 < ε hε : ∀ (m : (i : ι) → E i), ‖m - 0‖ < ε → ‖↑f m - ↑f 0‖ < 1 ⊢ ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ [PROOFSTEP] simp only [sub_zero, f.map_zero] at hε [GOAL] case inr.intro.intro 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G hf : Continuous ↑f h✝ : Nonempty ι ε : ℝ ε0 : 0 < ε hε : ∀ (m : (i : ι) → E i), ‖m‖ < ε → ‖↑f m‖ < 1 ⊢ ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ [PROOFSTEP] rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩ [GOAL] case inr.intro.intro.intro 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G hf : Continuous ↑f h✝ : Nonempty ι ε : ℝ ε0 : 0 < ε hε : ∀ (m : (i : ι) → E i), ‖m‖ < ε → ‖↑f m‖ < 1 c : 𝕜 hc : 1 < ‖c‖ ⊢ ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ [PROOFSTEP] have : 0 < (‖c‖ / ε) ^ Fintype.card ι := pow_pos (div_pos (zero_lt_one.trans hc) ε0) _ [GOAL] case inr.intro.intro.intro 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G hf : Continuous ↑f h✝ : Nonempty ι ε : ℝ ε0 : 0 < ε hε : ∀ (m : (i : ι) → E i), ‖m‖ < ε → ‖↑f m‖ < 1 c : 𝕜 hc : 1 < ‖c‖ this : 0 < (‖c‖ / ε) ^ Fintype.card ι ⊢ ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ [PROOFSTEP] refine' ⟨_, this, _⟩ [GOAL] case inr.intro.intro.intro 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G hf : Continuous ↑f h✝ : Nonempty ι ε : ℝ ε0 : 0 < ε hε : ∀ (m : (i : ι) → E i), ‖m‖ < ε → ‖↑f m‖ < 1 c : 𝕜 hc : 1 < ‖c‖ this : 0 < (‖c‖ / ε) ^ Fintype.card ι ⊢ ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ (‖c‖ / ε) ^ Fintype.card ι * ∏ i : ι, ‖m i‖ [PROOFSTEP] refine' f.bound_of_shell (fun _ => ε0) (fun _ => hc) fun m hcm hm => _ [GOAL] case inr.intro.intro.intro 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G hf : Continuous ↑f h✝ : Nonempty ι ε : ℝ ε0 : 0 < ε hε : ∀ (m : (i : ι) → E i), ‖m‖ < ε → ‖↑f m‖ < 1 c : 𝕜 hc : 1 < ‖c‖ this : 0 < (‖c‖ / ε) ^ Fintype.card ι m : (i : ι) → E i hcm : ∀ (i : ι), ε / ‖c‖ ≤ ‖m i‖ hm : ∀ (i : ι), ‖m i‖ < ε ⊢ ‖↑f m‖ ≤ (‖c‖ / ε) ^ Fintype.card ι * ∏ i : ι, ‖m i‖ [PROOFSTEP] refine' (hε m ((pi_norm_lt_iff ε0).2 hm)).le.trans _ [GOAL] case inr.intro.intro.intro 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G hf : Continuous ↑f h✝ : Nonempty ι ε : ℝ ε0 : 0 < ε hε : ∀ (m : (i : ι) → E i), ‖m‖ < ε → ‖↑f m‖ < 1 c : 𝕜 hc : 1 < ‖c‖ this : 0 < (‖c‖ / ε) ^ Fintype.card ι m : (i : ι) → E i hcm : ∀ (i : ι), ε / ‖c‖ ≤ ‖m i‖ hm : ∀ (i : ι), ‖m i‖ < ε ⊢ 1 ≤ (‖c‖ / ε) ^ Fintype.card ι * ∏ i : ι, ‖m i‖ [PROOFSTEP] rw [← div_le_iff' this, one_div, ← inv_pow, inv_div, Fintype.card, ← prod_const] [GOAL] case inr.intro.intro.intro 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G hf : Continuous ↑f h✝ : Nonempty ι ε : ℝ ε0 : 0 < ε hε : ∀ (m : (i : ι) → E i), ‖m‖ < ε → ‖↑f m‖ < 1 c : 𝕜 hc : 1 < ‖c‖ this : 0 < (‖c‖ / ε) ^ Fintype.card ι m : (i : ι) → E i hcm : ∀ (i : ι), ε / ‖c‖ ≤ ‖m i‖ hm : ∀ (i : ι), ‖m i‖ < ε ⊢ ∏ _x : ι, ε / ‖c‖ ≤ ∏ i : ι, ‖m i‖ [PROOFSTEP] exact prod_le_prod (fun _ _ => div_nonneg ε0.le (norm_nonneg _)) fun i _ => hcm i [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i ⊢ ‖↑f m₁ - ↑f m₂‖ ≤ C * ∑ i : ι, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ [PROOFSTEP] have A : ∀ s : Finset ι, ‖f m₁ - f (s.piecewise m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ := by intro s induction' s using Finset.induction with i s his Hrec · simp have I : ‖f (s.piecewise m₂ m₁) - f ((insert i s).piecewise m₂ m₁)‖ ≤ C * ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ := by have A : (insert i s).piecewise m₂ m₁ = Function.update (s.piecewise m₂ m₁) i (m₂ i) := s.piecewise_insert _ _ _ have B : s.piecewise m₂ m₁ = Function.update (s.piecewise m₂ m₁) i (m₁ i) := by ext j by_cases h : j = i · rw [h] simp [his] · simp [h] rw [B, A, ← f.map_sub] apply le_trans (H _) gcongr with j · exact fun j _ => norm_nonneg _ by_cases h : j = i · rw [h] simp · by_cases h' : j ∈ s <;> simp [h', h, le_refl] calc ‖f m₁ - f ((insert i s).piecewise m₂ m₁)‖ ≤ ‖f m₁ - f (s.piecewise m₂ m₁)‖ + ‖f (s.piecewise m₂ m₁) - f ((insert i s).piecewise m₂ m₁)‖ := by rw [← dist_eq_norm, ← dist_eq_norm, ← dist_eq_norm] exact dist_triangle _ _ _ _ ≤ (C * ∑ i in s, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) + C * ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ := (add_le_add Hrec I) _ = C * ∑ i in insert i s, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ := by simp [his, add_comm, left_distrib] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i ⊢ ∀ (s : Finset ι), ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ [PROOFSTEP] intro s [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i s : Finset ι ⊢ ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ [PROOFSTEP] induction' s using Finset.induction with i s his Hrec [GOAL] case empty 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i ⊢ ‖↑f m₁ - ↑f (piecewise ∅ m₂ m₁)‖ ≤ C * ∑ i in ∅, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ [PROOFSTEP] simp [GOAL] case insert 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι s : Finset ι his : ¬i ∈ s Hrec : ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ ⊢ ‖↑f m₁ - ↑f (piecewise (insert i s) m₂ m₁)‖ ≤ C * ∑ i in insert i s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ [PROOFSTEP] have I : ‖f (s.piecewise m₂ m₁) - f ((insert i s).piecewise m₂ m₁)‖ ≤ C * ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ := by have A : (insert i s).piecewise m₂ m₁ = Function.update (s.piecewise m₂ m₁) i (m₂ i) := s.piecewise_insert _ _ _ have B : s.piecewise m₂ m₁ = Function.update (s.piecewise m₂ m₁) i (m₁ i) := by ext j by_cases h : j = i · rw [h] simp [his] · simp [h] rw [B, A, ← f.map_sub] apply le_trans (H _) gcongr with j · exact fun j _ => norm_nonneg _ by_cases h : j = i · rw [h] simp · by_cases h' : j ∈ s <;> simp [h', h, le_refl] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι s : Finset ι his : ¬i ∈ s Hrec : ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ ⊢ ‖↑f (piecewise s m₂ m₁) - ↑f (piecewise (insert i s) m₂ m₁)‖ ≤ C * ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ [PROOFSTEP] have A : (insert i s).piecewise m₂ m₁ = Function.update (s.piecewise m₂ m₁) i (m₂ i) := s.piecewise_insert _ _ _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι s : Finset ι his : ¬i ∈ s Hrec : ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ A : piecewise (insert i s) m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₂ i) ⊢ ‖↑f (piecewise s m₂ m₁) - ↑f (piecewise (insert i s) m₂ m₁)‖ ≤ C * ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ [PROOFSTEP] have B : s.piecewise m₂ m₁ = Function.update (s.piecewise m₂ m₁) i (m₁ i) := by ext j by_cases h : j = i · rw [h] simp [his] · simp [h] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι s : Finset ι his : ¬i ∈ s Hrec : ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ A : piecewise (insert i s) m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₂ i) ⊢ piecewise s m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₁ i) [PROOFSTEP] ext j [GOAL] case h 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι s : Finset ι his : ¬i ∈ s Hrec : ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ A : piecewise (insert i s) m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₂ i) j : ι ⊢ piecewise s m₂ m₁ j = Function.update (piecewise s m₂ m₁) i (m₁ i) j [PROOFSTEP] by_cases h : j = i [GOAL] case pos 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι s : Finset ι his : ¬i ∈ s Hrec : ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ A : piecewise (insert i s) m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₂ i) j : ι h : j = i ⊢ piecewise s m₂ m₁ j = Function.update (piecewise s m₂ m₁) i (m₁ i) j [PROOFSTEP] rw [h] [GOAL] case pos 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι s : Finset ι his : ¬i ∈ s Hrec : ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ A : piecewise (insert i s) m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₂ i) j : ι h : j = i ⊢ piecewise s m₂ m₁ i = Function.update (piecewise s m₂ m₁) i (m₁ i) i [PROOFSTEP] simp [his] [GOAL] case neg 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι s : Finset ι his : ¬i ∈ s Hrec : ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ A : piecewise (insert i s) m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₂ i) j : ι h : ¬j = i ⊢ piecewise s m₂ m₁ j = Function.update (piecewise s m₂ m₁) i (m₁ i) j [PROOFSTEP] simp [h] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι s : Finset ι his : ¬i ∈ s Hrec : ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ A : piecewise (insert i s) m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₂ i) B : piecewise s m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₁ i) ⊢ ‖↑f (piecewise s m₂ m₁) - ↑f (piecewise (insert i s) m₂ m₁)‖ ≤ C * ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ [PROOFSTEP] rw [B, A, ← f.map_sub] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι s : Finset ι his : ¬i ∈ s Hrec : ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ A : piecewise (insert i s) m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₂ i) B : piecewise s m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₁ i) ⊢ ‖↑f (Function.update (piecewise s m₂ m₁) i (m₁ i - m₂ i))‖ ≤ C * ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ [PROOFSTEP] apply le_trans (H _) [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι s : Finset ι his : ¬i ∈ s Hrec : ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ A : piecewise (insert i s) m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₂ i) B : piecewise s m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₁ i) ⊢ C * ∏ i_1 : ι, ‖Function.update (piecewise s m₂ m₁) i (m₁ i - m₂ i) i_1‖ ≤ C * ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ [PROOFSTEP] gcongr with j [GOAL] case h.h0 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι s : Finset ι his : ¬i ∈ s Hrec : ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ A : piecewise (insert i s) m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₂ i) B : piecewise s m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₁ i) ⊢ ∀ (i_1 : ι), i_1 ∈ univ → 0 ≤ ‖Function.update (piecewise s m₂ m₁) i (m₁ i - m₂ i) i_1‖ [PROOFSTEP] exact fun j _ => norm_nonneg _ [GOAL] case h.h1 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι s : Finset ι his : ¬i ∈ s Hrec : ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ A : piecewise (insert i s) m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₂ i) B : piecewise s m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₁ i) j : ι a✝ : j ∈ univ ⊢ ‖Function.update (piecewise s m₂ m₁) i (m₁ i - m₂ i) j‖ ≤ if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ [PROOFSTEP] by_cases h : j = i [GOAL] case pos 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι s : Finset ι his : ¬i ∈ s Hrec : ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ A : piecewise (insert i s) m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₂ i) B : piecewise s m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₁ i) j : ι a✝ : j ∈ univ h : j = i ⊢ ‖Function.update (piecewise s m₂ m₁) i (m₁ i - m₂ i) j‖ ≤ if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ [PROOFSTEP] rw [h] [GOAL] case pos 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι s : Finset ι his : ¬i ∈ s Hrec : ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ A : piecewise (insert i s) m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₂ i) B : piecewise s m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₁ i) j : ι a✝ : j ∈ univ h : j = i ⊢ ‖Function.update (piecewise s m₂ m₁) i (m₁ i - m₂ i) i‖ ≤ if i = i then ‖m₁ i - m₂ i‖ else max ‖m₁ i‖ ‖m₂ i‖ [PROOFSTEP] simp [GOAL] case neg 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι s : Finset ι his : ¬i ∈ s Hrec : ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ A : piecewise (insert i s) m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₂ i) B : piecewise s m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₁ i) j : ι a✝ : j ∈ univ h : ¬j = i ⊢ ‖Function.update (piecewise s m₂ m₁) i (m₁ i - m₂ i) j‖ ≤ if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ [PROOFSTEP] by_cases h' : j ∈ s [GOAL] case pos 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι s : Finset ι his : ¬i ∈ s Hrec : ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ A : piecewise (insert i s) m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₂ i) B : piecewise s m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₁ i) j : ι a✝ : j ∈ univ h : ¬j = i h' : j ∈ s ⊢ ‖Function.update (piecewise s m₂ m₁) i (m₁ i - m₂ i) j‖ ≤ if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ [PROOFSTEP] simp [h', h, le_refl] [GOAL] case neg 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι s : Finset ι his : ¬i ∈ s Hrec : ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ A : piecewise (insert i s) m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₂ i) B : piecewise s m₂ m₁ = Function.update (piecewise s m₂ m₁) i (m₁ i) j : ι a✝ : j ∈ univ h : ¬j = i h' : ¬j ∈ s ⊢ ‖Function.update (piecewise s m₂ m₁) i (m₁ i - m₂ i) j‖ ≤ if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ [PROOFSTEP] simp [h', h, le_refl] [GOAL] case insert 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι s : Finset ι his : ¬i ∈ s Hrec : ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ I : ‖↑f (piecewise s m₂ m₁) - ↑f (piecewise (insert i s) m₂ m₁)‖ ≤ C * ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ ⊢ ‖↑f m₁ - ↑f (piecewise (insert i s) m₂ m₁)‖ ≤ C * ∑ i in insert i s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ [PROOFSTEP] calc ‖f m₁ - f ((insert i s).piecewise m₂ m₁)‖ ≤ ‖f m₁ - f (s.piecewise m₂ m₁)‖ + ‖f (s.piecewise m₂ m₁) - f ((insert i s).piecewise m₂ m₁)‖ := by rw [← dist_eq_norm, ← dist_eq_norm, ← dist_eq_norm] exact dist_triangle _ _ _ _ ≤ (C * ∑ i in s, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) + C * ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ := (add_le_add Hrec I) _ = C * ∑ i in insert i s, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ := by simp [his, add_comm, left_distrib] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι s : Finset ι his : ¬i ∈ s Hrec : ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ I : ‖↑f (piecewise s m₂ m₁) - ↑f (piecewise (insert i s) m₂ m₁)‖ ≤ C * ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ ⊢ ‖↑f m₁ - ↑f (piecewise (insert i s) m₂ m₁)‖ ≤ ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ + ‖↑f (piecewise s m₂ m₁) - ↑f (piecewise (insert i s) m₂ m₁)‖ [PROOFSTEP] rw [← dist_eq_norm, ← dist_eq_norm, ← dist_eq_norm] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι s : Finset ι his : ¬i ∈ s Hrec : ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ I : ‖↑f (piecewise s m₂ m₁) - ↑f (piecewise (insert i s) m₂ m₁)‖ ≤ C * ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ ⊢ dist (↑f m₁) (↑f (piecewise (insert i s) m₂ m₁)) ≤ dist (↑f m₁) (↑f (piecewise s m₂ m₁)) + dist (↑f (piecewise s m₂ m₁)) (↑f (piecewise (insert i s) m₂ m₁)) [PROOFSTEP] exact dist_triangle _ _ _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι s : Finset ι his : ¬i ∈ s Hrec : ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ I : ‖↑f (piecewise s m₂ m₁) - ↑f (piecewise (insert i s) m₂ m₁)‖ ≤ C * ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ ⊢ ((C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) + C * ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) = C * ∑ i in insert i s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ [PROOFSTEP] simp [his, add_comm, left_distrib] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i A : ∀ (s : Finset ι), ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ ⊢ ‖↑f m₁ - ↑f m₂‖ ≤ C * ∑ i : ι, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ [PROOFSTEP] convert A univ [GOAL] case h.e'_3.h.e'_3.h.e'_6.h.e'_6 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G inst✝ : DecidableEq ι C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i A : ∀ (s : Finset ι), ‖↑f m₁ - ↑f (piecewise s m₂ m₁)‖ ≤ C * ∑ i in s, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ ⊢ m₂ = piecewise univ m₂ m₁ [PROOFSTEP] simp [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i ⊢ ‖↑f m₁ - ↑f m₂‖ ≤ C * ↑(Fintype.card ι) * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) * ‖m₁ - m₂‖ [PROOFSTEP] classical have A : ∀ i : ι, ∏ j, (if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤ ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) := by intro i calc ∏ j, (if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤ ∏ j : ι, Function.update (fun _ => max ‖m₁‖ ‖m₂‖) i ‖m₁ - m₂‖ j := by apply prod_le_prod · intro j _ by_cases h : j = i <;> simp [h, norm_nonneg] · intro j _ by_cases h : j = i · rw [h] simp exact norm_le_pi_norm (m₁ - m₂) i · simp [h, -le_max_iff, -max_le_iff, max_le_max, norm_le_pi_norm (_ : ∀ i, E i)] _ = ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) := by rw [prod_update_of_mem (Finset.mem_univ _)] simp [card_univ_diff] calc ‖f m₁ - f m₂‖ ≤ C * ∑ i, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ := f.norm_image_sub_le_of_bound' hC H m₁ m₂ _ ≤ C * ∑ i, ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) := by gcongr; apply A _ = C * Fintype.card ι * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) * ‖m₁ - m₂‖ := by rw [sum_const, card_univ, nsmul_eq_mul] ring [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i ⊢ ‖↑f m₁ - ↑f m₂‖ ≤ C * ↑(Fintype.card ι) * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) * ‖m₁ - m₂‖ [PROOFSTEP] have A : ∀ i : ι, ∏ j, (if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤ ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) := by intro i calc ∏ j, (if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤ ∏ j : ι, Function.update (fun _ => max ‖m₁‖ ‖m₂‖) i ‖m₁ - m₂‖ j := by apply prod_le_prod · intro j _ by_cases h : j = i <;> simp [h, norm_nonneg] · intro j _ by_cases h : j = i · rw [h] simp exact norm_le_pi_norm (m₁ - m₂) i · simp [h, -le_max_iff, -max_le_iff, max_le_max, norm_le_pi_norm (_ : ∀ i, E i)] _ = ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) := by rw [prod_update_of_mem (Finset.mem_univ _)] simp [card_univ_diff] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i ⊢ ∀ (i : ι), (∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤ ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) [PROOFSTEP] intro i [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι ⊢ (∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤ ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) [PROOFSTEP] calc ∏ j, (if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤ ∏ j : ι, Function.update (fun _ => max ‖m₁‖ ‖m₂‖) i ‖m₁ - m₂‖ j := by apply prod_le_prod · intro j _ by_cases h : j = i <;> simp [h, norm_nonneg] · intro j _ by_cases h : j = i · rw [h] simp exact norm_le_pi_norm (m₁ - m₂) i · simp [h, -le_max_iff, -max_le_iff, max_le_max, norm_le_pi_norm (_ : ∀ i, E i)] _ = ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) := by rw [prod_update_of_mem (Finset.mem_univ _)] simp [card_univ_diff] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι ⊢ (∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤ ∏ j : ι, Function.update (fun x => max ‖m₁‖ ‖m₂‖) i ‖m₁ - m₂‖ j [PROOFSTEP] apply prod_le_prod [GOAL] case h0 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι ⊢ ∀ (i_1 : ι), i_1 ∈ univ → 0 ≤ if i_1 = i then ‖m₁ i - m₂ i‖ else max ‖m₁ i_1‖ ‖m₂ i_1‖ [PROOFSTEP] intro j _ [GOAL] case h0 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i j : ι a✝ : j ∈ univ ⊢ 0 ≤ if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ [PROOFSTEP] by_cases h : j = i [GOAL] case pos 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i j : ι a✝ : j ∈ univ h : j = i ⊢ 0 ≤ if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ [PROOFSTEP] simp [h, norm_nonneg] [GOAL] case neg 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i j : ι a✝ : j ∈ univ h : ¬j = i ⊢ 0 ≤ if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ [PROOFSTEP] simp [h, norm_nonneg] [GOAL] case h1 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι ⊢ ∀ (i_1 : ι), i_1 ∈ univ → (if i_1 = i then ‖m₁ i - m₂ i‖ else max ‖m₁ i_1‖ ‖m₂ i_1‖) ≤ Function.update (fun x => max ‖m₁‖ ‖m₂‖) i ‖m₁ - m₂‖ i_1 [PROOFSTEP] intro j _ [GOAL] case h1 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i j : ι a✝ : j ∈ univ ⊢ (if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤ Function.update (fun x => max ‖m₁‖ ‖m₂‖) i ‖m₁ - m₂‖ j [PROOFSTEP] by_cases h : j = i [GOAL] case pos 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i j : ι a✝ : j ∈ univ h : j = i ⊢ (if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤ Function.update (fun x => max ‖m₁‖ ‖m₂‖) i ‖m₁ - m₂‖ j [PROOFSTEP] rw [h] [GOAL] case pos 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i j : ι a✝ : j ∈ univ h : j = i ⊢ (if i = i then ‖m₁ i - m₂ i‖ else max ‖m₁ i‖ ‖m₂ i‖) ≤ Function.update (fun x => max ‖m₁‖ ‖m₂‖) i ‖m₁ - m₂‖ i [PROOFSTEP] simp [GOAL] case pos 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i j : ι a✝ : j ∈ univ h : j = i ⊢ ‖m₁ i - m₂ i‖ ≤ ‖m₁ - m₂‖ [PROOFSTEP] exact norm_le_pi_norm (m₁ - m₂) i [GOAL] case neg 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i j : ι a✝ : j ∈ univ h : ¬j = i ⊢ (if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤ Function.update (fun x => max ‖m₁‖ ‖m₂‖) i ‖m₁ - m₂‖ j [PROOFSTEP] simp [h, -le_max_iff, -max_le_iff, max_le_max, norm_le_pi_norm (_ : ∀ i, E i)] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι ⊢ ∏ j : ι, Function.update (fun x => max ‖m₁‖ ‖m₂‖) i ‖m₁ - m₂‖ j = ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) [PROOFSTEP] rw [prod_update_of_mem (Finset.mem_univ _)] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i i : ι ⊢ ‖m₁ - m₂‖ * ∏ x in univ \ {i}, max ‖m₁‖ ‖m₂‖ = ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) [PROOFSTEP] simp [card_univ_diff] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i A : ∀ (i : ι), (∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤ ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) ⊢ ‖↑f m₁ - ↑f m₂‖ ≤ C * ↑(Fintype.card ι) * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) * ‖m₁ - m₂‖ [PROOFSTEP] calc ‖f m₁ - f m₂‖ ≤ C * ∑ i, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ := f.norm_image_sub_le_of_bound' hC H m₁ m₂ _ ≤ C * ∑ i, ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) := by gcongr; apply A _ = C * Fintype.card ι * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) * ‖m₁ - m₂‖ := by rw [sum_const, card_univ, nsmul_eq_mul] ring [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i A : ∀ (i : ι), (∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤ ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) ⊢ (C * ∑ i : ι, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤ C * ∑ i : ?m.189282, ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) [PROOFSTEP] gcongr [GOAL] case h.h 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i A : ∀ (i : ι), (∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤ ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) i✝ : ι a✝ : i✝ ∈ univ ⊢ (∏ j : ι, if j = i✝ then ‖m₁ i✝ - m₂ i✝‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤ ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) [PROOFSTEP] apply A [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i A : ∀ (i : ι), (∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤ ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) ⊢ C * ∑ i : ι, ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) = C * ↑(Fintype.card ι) * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) * ‖m₁ - m₂‖ [PROOFSTEP] rw [sum_const, card_univ, nsmul_eq_mul] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ hC : 0 ≤ C H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ m₁ m₂ : (i : ι) → E i A : ∀ (i : ι), (∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤ ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) ⊢ C * (↑(Fintype.card ι) * (‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1))) = C * ↑(Fintype.card ι) * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) * ‖m₁ - m₂‖ [PROOFSTEP] ring [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ ⊢ Continuous ↑f [PROOFSTEP] let D := max C 1 [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ D : ℝ := max C 1 ⊢ Continuous ↑f [PROOFSTEP] have D_pos : 0 ≤ D := le_trans zero_le_one (le_max_right _ _) [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ D : ℝ := max C 1 D_pos : 0 ≤ D ⊢ Continuous ↑f [PROOFSTEP] replace H : ∀ m, ‖f m‖ ≤ D * ∏ i, ‖m i‖ [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ D : ℝ := max C 1 D_pos : 0 ≤ D ⊢ ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ D * ∏ i : ι, ‖m i‖ [PROOFSTEP] intro m [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ D : ℝ := max C 1 D_pos : 0 ≤ D m : (i : ι) → E i ⊢ ‖↑f m‖ ≤ D * ∏ i : ι, ‖m i‖ [PROOFSTEP] apply le_trans (H m) (mul_le_mul_of_nonneg_right (le_max_left _ _) _) [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖ D : ℝ := max C 1 D_pos : 0 ≤ D m : (i : ι) → E i ⊢ 0 ≤ ∏ i : ι, ‖m i‖ [PROOFSTEP] exact prod_nonneg fun (i : ι) _ => norm_nonneg (m i) [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ D : ℝ := max C 1 D_pos : 0 ≤ D H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ D * ∏ i : ι, ‖m i‖ ⊢ Continuous ↑f [PROOFSTEP] refine' continuous_iff_continuousAt.2 fun m => _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ D : ℝ := max C 1 D_pos : 0 ≤ D H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ D * ∏ i : ι, ‖m i‖ m : (i : ι) → E i ⊢ ContinuousAt (↑f) m [PROOFSTEP] refine' continuousAt_of_locally_lipschitz zero_lt_one (D * Fintype.card ι * (‖m‖ + 1) ^ (Fintype.card ι - 1)) fun m' h' => _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ D : ℝ := max C 1 D_pos : 0 ≤ D H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ D * ∏ i : ι, ‖m i‖ m m' : (i : ι) → E i h' : dist m' m < 1 ⊢ dist (↑f m') (↑f m) ≤ D * ↑(Fintype.card ι) * (‖m‖ + 1) ^ (Fintype.card ι - 1) * dist m' m [PROOFSTEP] rw [dist_eq_norm, dist_eq_norm] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ D : ℝ := max C 1 D_pos : 0 ≤ D H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ D * ∏ i : ι, ‖m i‖ m m' : (i : ι) → E i h' : dist m' m < 1 ⊢ ‖↑f m' - ↑f m‖ ≤ D * ↑(Fintype.card ι) * (‖m‖ + 1) ^ (Fintype.card ι - 1) * ‖m' - m‖ [PROOFSTEP] have : max ‖m'‖ ‖m‖ ≤ ‖m‖ + 1 := by simp [zero_le_one, norm_le_of_mem_closedBall (le_of_lt h')] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ D : ℝ := max C 1 D_pos : 0 ≤ D H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ D * ∏ i : ι, ‖m i‖ m m' : (i : ι) → E i h' : dist m' m < 1 ⊢ max ‖m'‖ ‖m‖ ≤ ‖m‖ + 1 [PROOFSTEP] simp [zero_le_one, norm_le_of_mem_closedBall (le_of_lt h')] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ D : ℝ := max C 1 D_pos : 0 ≤ D H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ D * ∏ i : ι, ‖m i‖ m m' : (i : ι) → E i h' : dist m' m < 1 this : max ‖m'‖ ‖m‖ ≤ ‖m‖ + 1 ⊢ ‖↑f m' - ↑f m‖ ≤ D * ↑(Fintype.card ι) * (‖m‖ + 1) ^ (Fintype.card ι - 1) * ‖m' - m‖ [PROOFSTEP] calc ‖f m' - f m‖ ≤ D * Fintype.card ι * max ‖m'‖ ‖m‖ ^ (Fintype.card ι - 1) * ‖m' - m‖ := f.norm_image_sub_le_of_bound D_pos H m' m _ ≤ D * Fintype.card ι * (‖m‖ + 1) ^ (Fintype.card ι - 1) * ‖m' - m‖ := by gcongr [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E G C : ℝ D : ℝ := max C 1 D_pos : 0 ≤ D H : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ D * ∏ i : ι, ‖m i‖ m m' : (i : ι) → E i h' : dist m' m < 1 this : max ‖m'‖ ‖m‖ ≤ ‖m‖ + 1 ⊢ D * ↑(Fintype.card ι) * max ‖m'‖ ‖m‖ ^ (Fintype.card ι - 1) * ‖m' - m‖ ≤ D * ↑(Fintype.card ι) * (‖m‖ + 1) ^ (Fintype.card ι - 1) * ‖m' - m‖ [PROOFSTEP] gcongr [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n✝ : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n✝) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n✝)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n✝)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f✝ : MultilinearMap 𝕜 E G k n : ℕ f : MultilinearMap 𝕜 (fun x => G) G' s : Finset (Fin n) hk : card s = k z : G C : ℝ H : ∀ (m : Fin n → G), ‖↑f m‖ ≤ C * ∏ i : Fin n, ‖m i‖ v : Fin k → G ⊢ ‖↑(restr f s hk z) v‖ ≤ C * ‖z‖ ^ (n - k) * ∏ i : Fin k, ‖v i‖ [PROOFSTEP] rw [mul_right_comm, mul_assoc] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n✝ : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n✝) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n✝)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n✝)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f✝ : MultilinearMap 𝕜 E G k n : ℕ f : MultilinearMap 𝕜 (fun x => G) G' s : Finset (Fin n) hk : card s = k z : G C : ℝ H : ∀ (m : Fin n → G), ‖↑f m‖ ≤ C * ∏ i : Fin n, ‖m i‖ v : Fin k → G ⊢ ‖↑(restr f s hk z) v‖ ≤ C * ((∏ i : Fin k, ‖v i‖) * ‖z‖ ^ (n - k)) [PROOFSTEP] convert H _ using 2 [GOAL] case h.e'_4.h.e'_6 𝕜 : Type u ι : Type v ι' : Type v' n✝ : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n✝) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n✝)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n✝)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f✝ : MultilinearMap 𝕜 E G k n : ℕ f : MultilinearMap 𝕜 (fun x => G) G' s : Finset (Fin n) hk : card s = k z : G C : ℝ H : ∀ (m : Fin n → G), ‖↑f m‖ ≤ C * ∏ i : Fin n, ‖m i‖ v : Fin k → G ⊢ (∏ i : Fin k, ‖v i‖) * ‖z‖ ^ (n - k) = ∏ i : Fin n, ‖if h : i ∈ s then v (↑(OrderIso.symm (orderIsoOfFin s hk)) { val := i, property := h }) else z‖ [PROOFSTEP] simp only [apply_dite norm, Fintype.prod_dite, prod_const ‖z‖, Finset.card_univ, Fintype.card_of_subtype sᶜ fun _ => mem_compl, card_compl, Fintype.card_fin, hk, mk_coe, ← (s.orderIsoOfFin hk).symm.bijective.prod_comp fun x => ‖v x‖] [GOAL] case h.e'_4.h.e'_6 𝕜 : Type u ι : Type v ι' : Type v' n✝ : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n✝) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n✝)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n✝)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f✝ : MultilinearMap 𝕜 E G k n : ℕ f : MultilinearMap 𝕜 (fun x => G) G' s : Finset (Fin n) hk : card s = k z : G C : ℝ H : ∀ (m : Fin n → G), ‖↑f m‖ ≤ C * ∏ i : Fin n, ‖m i‖ v : Fin k → G ⊢ (∏ i : { x // x ∈ s }, ‖v (↑(OrderIso.symm (orderIsoOfFin s hk)) i)‖) * ‖z‖ ^ (n - k) = (∏ a : { a // a ∈ s }, ‖v (↑(OrderIso.symm (orderIsoOfFin s hk)) { val := ↑a, property := (_ : ↑a ∈ s) })‖) * ‖z‖ ^ (n - k) [PROOFSTEP] convert rfl [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i ⊢ ‖↑f m‖ ≤ ‖f‖ * ∏ i : ι, ‖m i‖ [PROOFSTEP] have A : 0 ≤ ∏ i, ‖m i‖ := prod_nonneg fun j _ => norm_nonneg _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i A : 0 ≤ ∏ i : ι, ‖m i‖ ⊢ ‖↑f m‖ ≤ ‖f‖ * ∏ i : ι, ‖m i‖ [PROOFSTEP] cases' A.eq_or_lt with h hlt [GOAL] case inl 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i A : 0 ≤ ∏ i : ι, ‖m i‖ h : 0 = ∏ i : ι, ‖m i‖ ⊢ ‖↑f m‖ ≤ ‖f‖ * ∏ i : ι, ‖m i‖ [PROOFSTEP] rcases prod_eq_zero_iff.1 h.symm with ⟨i, _, hi⟩ [GOAL] case inl.intro.intro 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i A : 0 ≤ ∏ i : ι, ‖m i‖ h : 0 = ∏ i : ι, ‖m i‖ i : ι left✝ : i ∈ univ hi : ‖m i‖ = 0 ⊢ ‖↑f m‖ ≤ ‖f‖ * ∏ i : ι, ‖m i‖ [PROOFSTEP] rw [norm_eq_zero] at hi [GOAL] case inl.intro.intro 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i A : 0 ≤ ∏ i : ι, ‖m i‖ h : 0 = ∏ i : ι, ‖m i‖ i : ι left✝ : i ∈ univ hi : m i = 0 ⊢ ‖↑f m‖ ≤ ‖f‖ * ∏ i : ι, ‖m i‖ [PROOFSTEP] have : f m = 0 := f.map_coord_zero i hi [GOAL] case inl.intro.intro 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i A : 0 ≤ ∏ i : ι, ‖m i‖ h : 0 = ∏ i : ι, ‖m i‖ i : ι left✝ : i ∈ univ hi : m i = 0 this : ↑f m = 0 ⊢ ‖↑f m‖ ≤ ‖f‖ * ∏ i : ι, ‖m i‖ [PROOFSTEP] rw [this, norm_zero] [GOAL] case inl.intro.intro 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i A : 0 ≤ ∏ i : ι, ‖m i‖ h : 0 = ∏ i : ι, ‖m i‖ i : ι left✝ : i ∈ univ hi : m i = 0 this : ↑f m = 0 ⊢ 0 ≤ ‖f‖ * ∏ i : ι, ‖m i‖ [PROOFSTEP] exact mul_nonneg (op_norm_nonneg f) A [GOAL] case inr 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i A : 0 ≤ ∏ i : ι, ‖m i‖ hlt : 0 < ∏ i : ι, ‖m i‖ ⊢ ‖↑f m‖ ≤ ‖f‖ * ∏ i : ι, ‖m i‖ [PROOFSTEP] rw [← div_le_iff hlt] [GOAL] case inr 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i A : 0 ≤ ∏ i : ι, ‖m i‖ hlt : 0 < ∏ i : ι, ‖m i‖ ⊢ ‖↑f m‖ / ∏ i : ι, ‖m i‖ ≤ ‖f‖ [PROOFSTEP] apply le_csInf bounds_nonempty [GOAL] case inr 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i A : 0 ≤ ∏ i : ι, ‖m i‖ hlt : 0 < ∏ i : ι, ‖m i‖ ⊢ ∀ (b : ℝ), b ∈ {c \| 0 ≤ c ∧ ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ c * ∏ i : ι, ‖m i‖} → ‖↑f m‖ / ∏ i : ι, ‖m i‖ ≤ b [PROOFSTEP] rintro c ⟨_, hc⟩ [GOAL] case inr.intro 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c✝ : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i A : 0 ≤ ∏ i : ι, ‖m i‖ hlt : 0 < ∏ i : ι, ‖m i‖ c : ℝ left✝ : 0 ≤ c hc : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ c * ∏ i : ι, ‖m i‖ ⊢ ‖↑f m‖ / ∏ i : ι, ‖m i‖ ≤ c [PROOFSTEP] rw [div_le_iff hlt] [GOAL] case inr.intro 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c✝ : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i A : 0 ≤ ∏ i : ι, ‖m i‖ hlt : 0 < ∏ i : ι, ‖m i‖ c : ℝ left✝ : 0 ≤ c hc : ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ c * ∏ i : ι, ‖m i‖ ⊢ ‖↑f m‖ ≤ c * ∏ i : ι, ‖m i‖ [PROOFSTEP] apply hc [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i h : ‖m‖ ≤ 1 ⊢ ‖f‖ * ∏ i : ι, 1 = ‖f‖ [PROOFSTEP] simp [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m x : (i : ι) → E i ⊢ ‖↑(f + g) x‖ ≤ (‖f‖ + ‖g‖) * ∏ i : ι, ‖x i‖ [PROOFSTEP] rw [add_mul] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m x : (i : ι) → E i ⊢ ‖↑(f + g) x‖ ≤ ‖f‖ * ∏ i : ι, ‖x i‖ + ‖g‖ * ∏ i : ι, ‖x i‖ [PROOFSTEP] exact norm_add_le_of_le (le_op_norm _ _) (le_op_norm _ _) [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m✝ m : (i : ι) → E i ⊢ ‖↑0 m‖ ≤ 0 * ∏ i : ι, ‖m i‖ [PROOFSTEP] simp [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i h : ‖f‖ = 0 ⊢ f = 0 [PROOFSTEP] ext m [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m✝ : (i : ι) → E i h : ‖f‖ = 0 m : (i : ι) → E i ⊢ ↑f m = ↑0 m [PROOFSTEP] simpa [h] using f.le_op_norm m [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i ⊢ f = 0 → ‖f‖ = 0 [PROOFSTEP] rintro rfl [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i ⊢ ‖0‖ = 0 [PROOFSTEP] exact op_norm_zero [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' c✝ : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝² : NormedField 𝕜' inst✝¹ : NormedSpace 𝕜' G inst✝ : SMulCommClass 𝕜 𝕜' G c : 𝕜' ⊢ ∀ (m : (i : ι) → E i), ‖↑(c • f) m‖ ≤ ‖c‖ * ‖f‖ * ∏ i : ι, ‖m i‖ [PROOFSTEP] intro m [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' c✝ : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m✝ : (i : ι) → E i 𝕜' : Type u_1 inst✝² : NormedField 𝕜' inst✝¹ : NormedSpace 𝕜' G inst✝ : SMulCommClass 𝕜 𝕜' G c : 𝕜' m : (i : ι) → E i ⊢ ‖↑(c • f) m‖ ≤ ‖c‖ * ‖f‖ * ∏ i : ι, ‖m i‖ [PROOFSTEP] erw [norm_smul, mul_assoc] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' c✝ : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m✝ : (i : ι) → E i 𝕜' : Type u_1 inst✝² : NormedField 𝕜' inst✝¹ : NormedSpace 𝕜' G inst✝ : SMulCommClass 𝕜 𝕜' G c : 𝕜' m : (i : ι) → E i ⊢ ‖c‖ * ‖↑f.toMultilinearMap m‖ ≤ ‖c‖ * (‖f‖ * ∏ i : ι, ‖m i‖) [PROOFSTEP] exact mul_le_mul_of_nonneg_left (le_op_norm _ _) (norm_nonneg _) [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝² : NormedField 𝕜' inst✝¹ : NormedSpace 𝕜' G inst✝ : SMulCommClass 𝕜 𝕜' G ⊢ ‖-f‖ = ‖f‖ [PROOFSTEP] rw [norm_def] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝² : NormedField 𝕜' inst✝¹ : NormedSpace 𝕜' G inst✝ : SMulCommClass 𝕜 𝕜' G ⊢ sInf {c \| 0 ≤ c ∧ ∀ (m : (i : ι) → E i), ‖↑(-f) m‖ ≤ c * ∏ i : ι, ‖m i‖} = ‖f‖ [PROOFSTEP] apply congr_arg [GOAL] case h 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝² : NormedField 𝕜' inst✝¹ : NormedSpace 𝕜' G inst✝ : SMulCommClass 𝕜 𝕜' G ⊢ {c \| 0 ≤ c ∧ ∀ (m : (i : ι) → E i), ‖↑(-f) m‖ ≤ c * ∏ i : ι, ‖m i‖} = {c \| 0 ≤ c ∧ ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ c * ∏ i : ι, ‖m i‖} [PROOFSTEP] ext [GOAL] case h.h 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝² : NormedField 𝕜' inst✝¹ : NormedSpace 𝕜' G inst✝ : SMulCommClass 𝕜 𝕜' G x✝ : ℝ ⊢ x✝ ∈ {c \| 0 ≤ c ∧ ∀ (m : (i : ι) → E i), ‖↑(-f) m‖ ≤ c * ∏ i : ι, ‖m i‖} ↔ x✝ ∈ {c \| 0 ≤ c ∧ ∀ (m : (i : ι) → E i), ‖↑f m‖ ≤ c * ∏ i : ι, ‖m i‖} [PROOFSTEP] simp [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝² : NormedField 𝕜' inst✝¹ : NormedSpace 𝕜' G inst✝ : SMulCommClass 𝕜 𝕜' G b : ℝ hm : ∀ (i : ι), ‖m i‖ ≤ b ⊢ ‖↑f m‖ ≤ ‖f‖ * b ^ Fintype.card ι [PROOFSTEP] simpa only [prod_const] using f.le_op_norm_mul_prod_of_le m hm [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei✝ : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁹ : Fintype ι inst✝¹⁸ : Fintype ι' inst✝¹⁷ : NontriviallyNormedField 𝕜 inst✝¹⁶ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁵ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹² : (i : ι') → NormedAddCommGroup (E' i) inst✝¹¹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝¹⁰ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei✝ i) inst✝⁹ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei✝ i) inst✝⁸ : NormedAddCommGroup G inst✝⁷ : NormedSpace 𝕜 G inst✝⁶ : NormedAddCommGroup G' inst✝⁵ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m✝ : (i : ι) → E i 𝕜' : Type u_1 inst✝⁴ : NormedField 𝕜' inst✝³ : NormedSpace 𝕜' G inst✝² : SMulCommClass 𝕜 𝕜' G Ei : Fin n → Type u_2 inst✝¹ : (i : Fin n) → NormedAddCommGroup (Ei i) inst✝ : (i : Fin n) → NormedSpace 𝕜 (Ei i) f : ContinuousMultilinearMap 𝕜 Ei G m : (i : Fin n) → Ei i b : ℝ hm : ‖m‖ ≤ b ⊢ ‖↑f m‖ ≤ ‖f‖ * b ^ n [PROOFSTEP] simpa only [Fintype.card_fin] using f.le_op_norm_mul_pow_card_of_le m fun i => (norm_le_pi_norm m i).trans hm [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝² : NormedField 𝕜' inst✝¹ : NormedSpace 𝕜' G inst✝ : SMulCommClass 𝕜 𝕜' G ⊢ ↑‖↑f m‖₊ ≤ ↑(‖f‖₊ * ∏ i : ι, ‖m i‖₊) [PROOFSTEP] push_cast [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝² : NormedField 𝕜' inst✝¹ : NormedSpace 𝕜' G inst✝ : SMulCommClass 𝕜 𝕜' G ⊢ ‖↑f m‖ ≤ ‖f‖ * ∏ x : ι, ‖m x‖ [PROOFSTEP] exact f.le_op_norm m [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' c : 𝕜 f✝ g✝ : ContinuousMultilinearMap 𝕜 E G m✝ : (i : ι) → E i 𝕜' : Type u_1 inst✝² : NormedField 𝕜' inst✝¹ : NormedSpace 𝕜' G inst✝ : SMulCommClass 𝕜 𝕜' G f : ContinuousMultilinearMap 𝕜 E G g : ContinuousMultilinearMap 𝕜 E G' m : (i : ι) → E i ⊢ ‖↑(prod f g) m‖ ≤ max ‖f‖ ‖g‖ * ∏ i : ι, ‖m i‖ [PROOFSTEP] have H : 0 ≤ ∏ i, ‖m i‖ := prod_nonneg fun _ _ => norm_nonneg _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' c : 𝕜 f✝ g✝ : ContinuousMultilinearMap 𝕜 E G m✝ : (i : ι) → E i 𝕜' : Type u_1 inst✝² : NormedField 𝕜' inst✝¹ : NormedSpace 𝕜' G inst✝ : SMulCommClass 𝕜 𝕜' G f : ContinuousMultilinearMap 𝕜 E G g : ContinuousMultilinearMap 𝕜 E G' m : (i : ι) → E i H : 0 ≤ ∏ i : ι, ‖m i‖ ⊢ ‖↑(prod f g) m‖ ≤ max ‖f‖ ‖g‖ * ∏ i : ι, ‖m i‖ [PROOFSTEP] simpa only [prod_apply, Prod.norm_def, max_mul_of_nonneg, H] using max_le_max (f.le_op_norm m) (g.le_op_norm m) [GOAL] 𝕜 : Type u ι : Type v ι'✝ : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E'✝ : ι'✝ → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝²⁰ : Fintype ι inst✝¹⁹ : Fintype ι'✝ inst✝¹⁸ : NontriviallyNormedField 𝕜 inst✝¹⁷ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁶ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁵ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁴ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹³ : (i : ι'✝) → NormedAddCommGroup (E'✝ i) inst✝¹² : (i : ι'✝) → NormedSpace 𝕜 (E'✝ i) inst✝¹¹ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝¹⁰ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁹ : NormedAddCommGroup G inst✝⁸ : NormedSpace 𝕜 G inst✝⁷ : NormedAddCommGroup G' inst✝⁶ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝⁵ : NormedField 𝕜' inst✝⁴ : NormedSpace 𝕜' G inst✝³ : SMulCommClass 𝕜 𝕜' G ι' : Type v' inst✝² : Fintype ι' E' : ι' → Type wE' inst✝¹ : (i' : ι') → NormedAddCommGroup (E' i') inst✝ : (i' : ι') → NormedSpace 𝕜 (E' i') f : (i' : ι') → ContinuousMultilinearMap 𝕜 E (E' i') ⊢ ‖pi f‖ = ‖f‖ [PROOFSTEP] apply le_antisymm [GOAL] case a 𝕜 : Type u ι : Type v ι'✝ : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E'✝ : ι'✝ → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝²⁰ : Fintype ι inst✝¹⁹ : Fintype ι'✝ inst✝¹⁸ : NontriviallyNormedField 𝕜 inst✝¹⁷ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁶ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁵ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁴ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹³ : (i : ι'✝) → NormedAddCommGroup (E'✝ i) inst✝¹² : (i : ι'✝) → NormedSpace 𝕜 (E'✝ i) inst✝¹¹ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝¹⁰ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁹ : NormedAddCommGroup G inst✝⁸ : NormedSpace 𝕜 G inst✝⁷ : NormedAddCommGroup G' inst✝⁶ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝⁵ : NormedField 𝕜' inst✝⁴ : NormedSpace 𝕜' G inst✝³ : SMulCommClass 𝕜 𝕜' G ι' : Type v' inst✝² : Fintype ι' E' : ι' → Type wE' inst✝¹ : (i' : ι') → NormedAddCommGroup (E' i') inst✝ : (i' : ι') → NormedSpace 𝕜 (E' i') f : (i' : ι') → ContinuousMultilinearMap 𝕜 E (E' i') ⊢ ‖pi f‖ ≤ ‖f‖ [PROOFSTEP] refine' op_norm_le_bound _ (norm_nonneg f) fun m => _ [GOAL] case a 𝕜 : Type u ι : Type v ι'✝ : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E'✝ : ι'✝ → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝²⁰ : Fintype ι inst✝¹⁹ : Fintype ι'✝ inst✝¹⁸ : NontriviallyNormedField 𝕜 inst✝¹⁷ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁶ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁵ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁴ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹³ : (i : ι'✝) → NormedAddCommGroup (E'✝ i) inst✝¹² : (i : ι'✝) → NormedSpace 𝕜 (E'✝ i) inst✝¹¹ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝¹⁰ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁹ : NormedAddCommGroup G inst✝⁸ : NormedSpace 𝕜 G inst✝⁷ : NormedAddCommGroup G' inst✝⁶ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m✝ : (i : ι) → E i 𝕜' : Type u_1 inst✝⁵ : NormedField 𝕜' inst✝⁴ : NormedSpace 𝕜' G inst✝³ : SMulCommClass 𝕜 𝕜' G ι' : Type v' inst✝² : Fintype ι' E' : ι' → Type wE' inst✝¹ : (i' : ι') → NormedAddCommGroup (E' i') inst✝ : (i' : ι') → NormedSpace 𝕜 (E' i') f : (i' : ι') → ContinuousMultilinearMap 𝕜 E (E' i') m : (i : ι) → E i ⊢ ‖↑(pi f) m‖ ≤ ‖f‖ * ∏ i : ι, ‖m i‖ [PROOFSTEP] dsimp [GOAL] case a 𝕜 : Type u ι : Type v ι'✝ : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E'✝ : ι'✝ → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝²⁰ : Fintype ι inst✝¹⁹ : Fintype ι'✝ inst✝¹⁸ : NontriviallyNormedField 𝕜 inst✝¹⁷ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁶ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁵ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁴ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹³ : (i : ι'✝) → NormedAddCommGroup (E'✝ i) inst✝¹² : (i : ι'✝) → NormedSpace 𝕜 (E'✝ i) inst✝¹¹ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝¹⁰ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁹ : NormedAddCommGroup G inst✝⁸ : NormedSpace 𝕜 G inst✝⁷ : NormedAddCommGroup G' inst✝⁶ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m✝ : (i : ι) → E i 𝕜' : Type u_1 inst✝⁵ : NormedField 𝕜' inst✝⁴ : NormedSpace 𝕜' G inst✝³ : SMulCommClass 𝕜 𝕜' G ι' : Type v' inst✝² : Fintype ι' E' : ι' → Type wE' inst✝¹ : (i' : ι') → NormedAddCommGroup (E' i') inst✝ : (i' : ι') → NormedSpace 𝕜 (E' i') f : (i' : ι') → ContinuousMultilinearMap 𝕜 E (E' i') m : (i : ι) → E i ⊢ ‖fun j => ↑(f j) m‖ ≤ ‖f‖ * ∏ i : ι, ‖m i‖ [PROOFSTEP] rw [pi_norm_le_iff_of_nonneg] [GOAL] case a 𝕜 : Type u ι : Type v ι'✝ : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E'✝ : ι'✝ → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝²⁰ : Fintype ι inst✝¹⁹ : Fintype ι'✝ inst✝¹⁸ : NontriviallyNormedField 𝕜 inst✝¹⁷ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁶ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁵ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁴ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹³ : (i : ι'✝) → NormedAddCommGroup (E'✝ i) inst✝¹² : (i : ι'✝) → NormedSpace 𝕜 (E'✝ i) inst✝¹¹ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝¹⁰ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁹ : NormedAddCommGroup G inst✝⁸ : NormedSpace 𝕜 G inst✝⁷ : NormedAddCommGroup G' inst✝⁶ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m✝ : (i : ι) → E i 𝕜' : Type u_1 inst✝⁵ : NormedField 𝕜' inst✝⁴ : NormedSpace 𝕜' G inst✝³ : SMulCommClass 𝕜 𝕜' G ι' : Type v' inst✝² : Fintype ι' E' : ι' → Type wE' inst✝¹ : (i' : ι') → NormedAddCommGroup (E' i') inst✝ : (i' : ι') → NormedSpace 𝕜 (E' i') f : (i' : ι') → ContinuousMultilinearMap 𝕜 E (E' i') m : (i : ι) → E i ⊢ ∀ (i : ι'), ‖↑(f i) m‖ ≤ ‖f‖ * ∏ i : ι, ‖m i‖ case a 𝕜 : Type u ι : Type v ι'✝ : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E'✝ : ι'✝ → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝²⁰ : Fintype ι inst✝¹⁹ : Fintype ι'✝ inst✝¹⁸ : NontriviallyNormedField 𝕜 inst✝¹⁷ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁶ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁵ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁴ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹³ : (i : ι'✝) → NormedAddCommGroup (E'✝ i) inst✝¹² : (i : ι'✝) → NormedSpace 𝕜 (E'✝ i) inst✝¹¹ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝¹⁰ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁹ : NormedAddCommGroup G inst✝⁸ : NormedSpace 𝕜 G inst✝⁷ : NormedAddCommGroup G' inst✝⁶ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m✝ : (i : ι) → E i 𝕜' : Type u_1 inst✝⁵ : NormedField 𝕜' inst✝⁴ : NormedSpace 𝕜' G inst✝³ : SMulCommClass 𝕜 𝕜' G ι' : Type v' inst✝² : Fintype ι' E' : ι' → Type wE' inst✝¹ : (i' : ι') → NormedAddCommGroup (E' i') inst✝ : (i' : ι') → NormedSpace 𝕜 (E' i') f : (i' : ι') → ContinuousMultilinearMap 𝕜 E (E' i') m : (i : ι) → E i ⊢ 0 ≤ ‖f‖ * ∏ i : ι, ‖m i‖ [PROOFSTEP] exacts [fun i => (f i).le_of_op_norm_le m (norm_le_pi_norm f i), mul_nonneg (norm_nonneg f) (prod_nonneg fun _ _ => norm_nonneg _)] [GOAL] case a 𝕜 : Type u ι : Type v ι'✝ : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E'✝ : ι'✝ → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝²⁰ : Fintype ι inst✝¹⁹ : Fintype ι'✝ inst✝¹⁸ : NontriviallyNormedField 𝕜 inst✝¹⁷ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁶ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁵ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁴ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹³ : (i : ι'✝) → NormedAddCommGroup (E'✝ i) inst✝¹² : (i : ι'✝) → NormedSpace 𝕜 (E'✝ i) inst✝¹¹ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝¹⁰ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁹ : NormedAddCommGroup G inst✝⁸ : NormedSpace 𝕜 G inst✝⁷ : NormedAddCommGroup G' inst✝⁶ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝⁵ : NormedField 𝕜' inst✝⁴ : NormedSpace 𝕜' G inst✝³ : SMulCommClass 𝕜 𝕜' G ι' : Type v' inst✝² : Fintype ι' E' : ι' → Type wE' inst✝¹ : (i' : ι') → NormedAddCommGroup (E' i') inst✝ : (i' : ι') → NormedSpace 𝕜 (E' i') f : (i' : ι') → ContinuousMultilinearMap 𝕜 E (E' i') ⊢ ‖f‖ ≤ ‖pi f‖ [PROOFSTEP] refine' (pi_norm_le_iff_of_nonneg (norm_nonneg _)).2 fun i => _ [GOAL] case a 𝕜 : Type u ι : Type v ι'✝ : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E'✝ : ι'✝ → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝²⁰ : Fintype ι inst✝¹⁹ : Fintype ι'✝ inst✝¹⁸ : NontriviallyNormedField 𝕜 inst✝¹⁷ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁶ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁵ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁴ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹³ : (i : ι'✝) → NormedAddCommGroup (E'✝ i) inst✝¹² : (i : ι'✝) → NormedSpace 𝕜 (E'✝ i) inst✝¹¹ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝¹⁰ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁹ : NormedAddCommGroup G inst✝⁸ : NormedSpace 𝕜 G inst✝⁷ : NormedAddCommGroup G' inst✝⁶ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝⁵ : NormedField 𝕜' inst✝⁴ : NormedSpace 𝕜' G inst✝³ : SMulCommClass 𝕜 𝕜' G ι' : Type v' inst✝² : Fintype ι' E' : ι' → Type wE' inst✝¹ : (i' : ι') → NormedAddCommGroup (E' i') inst✝ : (i' : ι') → NormedSpace 𝕜 (E' i') f : (i' : ι') → ContinuousMultilinearMap 𝕜 E (E' i') i : ι' ⊢ ‖f i‖ ≤ ‖pi f‖ [PROOFSTEP] refine' op_norm_le_bound _ (norm_nonneg _) fun m => _ [GOAL] case a 𝕜 : Type u ι : Type v ι'✝ : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E'✝ : ι'✝ → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝²⁰ : Fintype ι inst✝¹⁹ : Fintype ι'✝ inst✝¹⁸ : NontriviallyNormedField 𝕜 inst✝¹⁷ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁶ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁵ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁴ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹³ : (i : ι'✝) → NormedAddCommGroup (E'✝ i) inst✝¹² : (i : ι'✝) → NormedSpace 𝕜 (E'✝ i) inst✝¹¹ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝¹⁰ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁹ : NormedAddCommGroup G inst✝⁸ : NormedSpace 𝕜 G inst✝⁷ : NormedAddCommGroup G' inst✝⁶ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m✝ : (i : ι) → E i 𝕜' : Type u_1 inst✝⁵ : NormedField 𝕜' inst✝⁴ : NormedSpace 𝕜' G inst✝³ : SMulCommClass 𝕜 𝕜' G ι' : Type v' inst✝² : Fintype ι' E' : ι' → Type wE' inst✝¹ : (i' : ι') → NormedAddCommGroup (E' i') inst✝ : (i' : ι') → NormedSpace 𝕜 (E' i') f : (i' : ι') → ContinuousMultilinearMap 𝕜 E (E' i') i : ι' m : (i : ι) → E i ⊢ ‖↑(f i) m‖ ≤ ‖pi f‖ * ∏ i : ι, ‖m i‖ [PROOFSTEP] refine' le_trans _ ((pi f).le_op_norm m) [GOAL] case a 𝕜 : Type u ι : Type v ι'✝ : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E'✝ : ι'✝ → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝²⁰ : Fintype ι inst✝¹⁹ : Fintype ι'✝ inst✝¹⁸ : NontriviallyNormedField 𝕜 inst✝¹⁷ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁶ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁵ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁴ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹³ : (i : ι'✝) → NormedAddCommGroup (E'✝ i) inst✝¹² : (i : ι'✝) → NormedSpace 𝕜 (E'✝ i) inst✝¹¹ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝¹⁰ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁹ : NormedAddCommGroup G inst✝⁸ : NormedSpace 𝕜 G inst✝⁷ : NormedAddCommGroup G' inst✝⁶ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m✝ : (i : ι) → E i 𝕜' : Type u_1 inst✝⁵ : NormedField 𝕜' inst✝⁴ : NormedSpace 𝕜' G inst✝³ : SMulCommClass 𝕜 𝕜' G ι' : Type v' inst✝² : Fintype ι' E' : ι' → Type wE' inst✝¹ : (i' : ι') → NormedAddCommGroup (E' i') inst✝ : (i' : ι') → NormedSpace 𝕜 (E' i') f : (i' : ι') → ContinuousMultilinearMap 𝕜 E (E' i') i : ι' m : (i : ι) → E i ⊢ ‖↑(f i) m‖ ≤ ‖↑(pi f) m‖ [PROOFSTEP] convert norm_le_pi_norm (fun j => f j m) i [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁸ : Fintype ι inst✝¹⁷ : Fintype ι' inst✝¹⁶ : NontriviallyNormedField 𝕜 inst✝¹⁵ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁴ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹³ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹¹ : (i : ι') → NormedAddCommGroup (E' i) inst✝¹⁰ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁹ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁷ : NormedAddCommGroup G inst✝⁶ : NormedSpace 𝕜 G inst✝⁵ : NormedAddCommGroup G' inst✝⁴ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m✝ : (i : ι) → E i 𝕜' : Type u_1 inst✝³ : NormedField 𝕜' inst✝² : NormedSpace 𝕜' G inst✝¹ : SMulCommClass 𝕜 𝕜' G inst✝ : Subsingleton ι i' : ι m : ι → G ⊢ ‖↑(ofSubsingleton 𝕜 G i') m‖ ≤ 1 * ∏ i : ι, ‖m i‖ [PROOFSTEP] rw [Fintype.prod_subsingleton _ i', one_mul, ofSubsingleton_apply] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁹ : Fintype ι inst✝¹⁸ : Fintype ι' inst✝¹⁷ : NontriviallyNormedField 𝕜 inst✝¹⁶ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁵ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹² : (i : ι') → NormedAddCommGroup (E' i) inst✝¹¹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝¹⁰ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁹ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁸ : NormedAddCommGroup G inst✝⁷ : NormedSpace 𝕜 G inst✝⁶ : NormedAddCommGroup G' inst✝⁵ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝⁴ : NormedField 𝕜' inst✝³ : NormedSpace 𝕜' G inst✝² : SMulCommClass 𝕜 𝕜' G inst✝¹ : Subsingleton ι inst✝ : Nontrivial G i' : ι ⊢ ‖ofSubsingleton 𝕜 G i'‖ = 1 [PROOFSTEP] apply le_antisymm (norm_ofSubsingleton_le 𝕜 G i') [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁹ : Fintype ι inst✝¹⁸ : Fintype ι' inst✝¹⁷ : NontriviallyNormedField 𝕜 inst✝¹⁶ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁵ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹² : (i : ι') → NormedAddCommGroup (E' i) inst✝¹¹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝¹⁰ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁹ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁸ : NormedAddCommGroup G inst✝⁷ : NormedSpace 𝕜 G inst✝⁶ : NormedAddCommGroup G' inst✝⁵ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝⁴ : NormedField 𝕜' inst✝³ : NormedSpace 𝕜' G inst✝² : SMulCommClass 𝕜 𝕜' G inst✝¹ : Subsingleton ι inst✝ : Nontrivial G i' : ι ⊢ 1 ≤ ‖ofSubsingleton 𝕜 G i'‖ [PROOFSTEP] obtain ⟨g, hg⟩ := exists_ne (0 : G) [GOAL] case intro 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁹ : Fintype ι inst✝¹⁸ : Fintype ι' inst✝¹⁷ : NontriviallyNormedField 𝕜 inst✝¹⁶ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁵ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹² : (i : ι') → NormedAddCommGroup (E' i) inst✝¹¹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝¹⁰ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁹ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁸ : NormedAddCommGroup G inst✝⁷ : NormedSpace 𝕜 G inst✝⁶ : NormedAddCommGroup G' inst✝⁵ : NormedSpace 𝕜 G' c : 𝕜 f g✝ : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝⁴ : NormedField 𝕜' inst✝³ : NormedSpace 𝕜' G inst✝² : SMulCommClass 𝕜 𝕜' G inst✝¹ : Subsingleton ι inst✝ : Nontrivial G i' : ι g : G hg : g ≠ 0 ⊢ 1 ≤ ‖ofSubsingleton 𝕜 G i'‖ [PROOFSTEP] rw [← norm_ne_zero_iff] at hg [GOAL] case intro 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁹ : Fintype ι inst✝¹⁸ : Fintype ι' inst✝¹⁷ : NontriviallyNormedField 𝕜 inst✝¹⁶ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁵ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹² : (i : ι') → NormedAddCommGroup (E' i) inst✝¹¹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝¹⁰ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁹ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁸ : NormedAddCommGroup G inst✝⁷ : NormedSpace 𝕜 G inst✝⁶ : NormedAddCommGroup G' inst✝⁵ : NormedSpace 𝕜 G' c : 𝕜 f g✝ : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝⁴ : NormedField 𝕜' inst✝³ : NormedSpace 𝕜' G inst✝² : SMulCommClass 𝕜 𝕜' G inst✝¹ : Subsingleton ι inst✝ : Nontrivial G i' : ι g : G hg : ‖g‖ ≠ 0 ⊢ 1 ≤ ‖ofSubsingleton 𝕜 G i'‖ [PROOFSTEP] have := (ofSubsingleton 𝕜 G i').ratio_le_op_norm fun _ => g [GOAL] case intro 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁹ : Fintype ι inst✝¹⁸ : Fintype ι' inst✝¹⁷ : NontriviallyNormedField 𝕜 inst✝¹⁶ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁵ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹² : (i : ι') → NormedAddCommGroup (E' i) inst✝¹¹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝¹⁰ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁹ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁸ : NormedAddCommGroup G inst✝⁷ : NormedSpace 𝕜 G inst✝⁶ : NormedAddCommGroup G' inst✝⁵ : NormedSpace 𝕜 G' c : 𝕜 f g✝ : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝⁴ : NormedField 𝕜' inst✝³ : NormedSpace 𝕜' G inst✝² : SMulCommClass 𝕜 𝕜' G inst✝¹ : Subsingleton ι inst✝ : Nontrivial G i' : ι g : G hg : ‖g‖ ≠ 0 this : ‖↑(ofSubsingleton 𝕜 G i') fun x => g‖ / ∏ i : ι, ‖g‖ ≤ ‖ofSubsingleton 𝕜 G i'‖ ⊢ 1 ≤ ‖ofSubsingleton 𝕜 G i'‖ [PROOFSTEP] rwa [Fintype.prod_subsingleton _ i', ofSubsingleton_apply, div_self hg] at this [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁸ : Fintype ι inst✝¹⁷ : Fintype ι' inst✝¹⁶ : NontriviallyNormedField 𝕜 inst✝¹⁵ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁴ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹³ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹¹ : (i : ι') → NormedAddCommGroup (E' i) inst✝¹⁰ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁹ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁷ : NormedAddCommGroup G inst✝⁶ : NormedSpace 𝕜 G inst✝⁵ : NormedAddCommGroup G' inst✝⁴ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝³ : NormedField 𝕜' inst✝² : NormedSpace 𝕜' G inst✝¹ : SMulCommClass 𝕜 𝕜' G inst✝ : IsEmpty ι x : G ⊢ ‖constOfIsEmpty 𝕜 E x‖ = ‖x‖ [PROOFSTEP] apply le_antisymm [GOAL] case a 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁸ : Fintype ι inst✝¹⁷ : Fintype ι' inst✝¹⁶ : NontriviallyNormedField 𝕜 inst✝¹⁵ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁴ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹³ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹¹ : (i : ι') → NormedAddCommGroup (E' i) inst✝¹⁰ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁹ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁷ : NormedAddCommGroup G inst✝⁶ : NormedSpace 𝕜 G inst✝⁵ : NormedAddCommGroup G' inst✝⁴ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝³ : NormedField 𝕜' inst✝² : NormedSpace 𝕜' G inst✝¹ : SMulCommClass 𝕜 𝕜' G inst✝ : IsEmpty ι x : G ⊢ ‖constOfIsEmpty 𝕜 E x‖ ≤ ‖x‖ [PROOFSTEP] refine' op_norm_le_bound _ (norm_nonneg _) fun x => _ [GOAL] case a 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁸ : Fintype ι inst✝¹⁷ : Fintype ι' inst✝¹⁶ : NontriviallyNormedField 𝕜 inst✝¹⁵ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁴ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹³ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹¹ : (i : ι') → NormedAddCommGroup (E' i) inst✝¹⁰ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁹ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁷ : NormedAddCommGroup G inst✝⁶ : NormedSpace 𝕜 G inst✝⁵ : NormedAddCommGroup G' inst✝⁴ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝³ : NormedField 𝕜' inst✝² : NormedSpace 𝕜' G inst✝¹ : SMulCommClass 𝕜 𝕜' G inst✝ : IsEmpty ι x✝ : G x : (i : ι) → E i ⊢ ‖↑(constOfIsEmpty 𝕜 E x✝) x‖ ≤ ‖x✝‖ * ∏ i : ι, ‖x i‖ [PROOFSTEP] rw [Fintype.prod_empty, mul_one, constOfIsEmpty_apply] [GOAL] case a 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁸ : Fintype ι inst✝¹⁷ : Fintype ι' inst✝¹⁶ : NontriviallyNormedField 𝕜 inst✝¹⁵ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁴ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹³ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹¹ : (i : ι') → NormedAddCommGroup (E' i) inst✝¹⁰ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁹ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁷ : NormedAddCommGroup G inst✝⁶ : NormedSpace 𝕜 G inst✝⁵ : NormedAddCommGroup G' inst✝⁴ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝³ : NormedField 𝕜' inst✝² : NormedSpace 𝕜' G inst✝¹ : SMulCommClass 𝕜 𝕜' G inst✝ : IsEmpty ι x : G ⊢ ‖x‖ ≤ ‖constOfIsEmpty 𝕜 E x‖ [PROOFSTEP] simpa using (constOfIsEmpty 𝕜 E x).le_op_norm 0 [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝² : NormedField 𝕜' inst✝¹ : NormedSpace 𝕜' G inst✝ : SMulCommClass 𝕜 𝕜' G f : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G' ⊢ (fun f => (ContinuousLinearMap.compContinuousMultilinearMap (ContinuousLinearMap.fst 𝕜 G G') f, ContinuousLinearMap.compContinuousMultilinearMap (ContinuousLinearMap.snd 𝕜 G G') f)) (AddHom.toFun { toAddHom := { toFun := fun f => prod f.fst f.snd, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G'), (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) }, map_smul' := (_ : ∀ (c : 𝕜) (f : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G'), AddHom.toFun { toFun := fun f => prod f.fst f.snd, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G'), (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) } (c • f) = AddHom.toFun { toFun := fun f => prod f.fst f.snd, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G'), (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) } (c • f)) }.toAddHom f) = f [PROOFSTEP] ext [GOAL] case h₁.H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝² : NormedField 𝕜' inst✝¹ : NormedSpace 𝕜' G inst✝ : SMulCommClass 𝕜 𝕜' G f : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G' x✝ : (i : ι) → E i ⊢ ↑((fun f => (ContinuousLinearMap.compContinuousMultilinearMap (ContinuousLinearMap.fst 𝕜 G G') f, ContinuousLinearMap.compContinuousMultilinearMap (ContinuousLinearMap.snd 𝕜 G G') f)) (AddHom.toFun { toAddHom := { toFun := fun f => prod f.fst f.snd, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G'), (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) }, map_smul' := (_ : ∀ (c : 𝕜) (f : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G'), AddHom.toFun { toFun := fun f => prod f.fst f.snd, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G'), (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) } (c • f) = AddHom.toFun { toFun := fun f => prod f.fst f.snd, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G'), (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) } (c • f)) }.toAddHom f)).fst x✝ = ↑f.fst x✝ [PROOFSTEP] rfl [GOAL] case h₂.H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝² : NormedField 𝕜' inst✝¹ : NormedSpace 𝕜' G inst✝ : SMulCommClass 𝕜 𝕜' G f : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G' x✝ : (i : ι) → E i ⊢ ↑((fun f => (ContinuousLinearMap.compContinuousMultilinearMap (ContinuousLinearMap.fst 𝕜 G G') f, ContinuousLinearMap.compContinuousMultilinearMap (ContinuousLinearMap.snd 𝕜 G G') f)) (AddHom.toFun { toAddHom := { toFun := fun f => prod f.fst f.snd, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G'), (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) }, map_smul' := (_ : ∀ (c : 𝕜) (f : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G'), AddHom.toFun { toFun := fun f => prod f.fst f.snd, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G'), (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) } (c • f) = AddHom.toFun { toFun := fun f => prod f.fst f.snd, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G'), (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) } (c • f)) }.toAddHom f)).snd x✝ = ↑f.snd x✝ [PROOFSTEP] rfl [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝² : NormedField 𝕜' inst✝¹ : NormedSpace 𝕜' G inst✝ : SMulCommClass 𝕜 𝕜' G f : ContinuousMultilinearMap 𝕜 E (G × G') ⊢ AddHom.toFun { toAddHom := { toFun := fun f => prod f.fst f.snd, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G'), (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) }, map_smul' := (_ : ∀ (c : 𝕜) (f : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G'), AddHom.toFun { toFun := fun f => prod f.fst f.snd, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G'), (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) } (c • f) = AddHom.toFun { toFun := fun f => prod f.fst f.snd, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G'), (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) } (c • f)) }.toAddHom ((fun f => (ContinuousLinearMap.compContinuousMultilinearMap (ContinuousLinearMap.fst 𝕜 G G') f, ContinuousLinearMap.compContinuousMultilinearMap (ContinuousLinearMap.snd 𝕜 G G') f)) f) = f [PROOFSTEP] ext [GOAL] case H.h₁ 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝² : NormedField 𝕜' inst✝¹ : NormedSpace 𝕜' G inst✝ : SMulCommClass 𝕜 𝕜' G f : ContinuousMultilinearMap 𝕜 E (G × G') x✝ : (i : ι) → E i ⊢ (↑(AddHom.toFun { toAddHom := { toFun := fun f => prod f.fst f.snd, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G'), (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) }, map_smul' := (_ : ∀ (c : 𝕜) (f : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G'), AddHom.toFun { toFun := fun f => prod f.fst f.snd, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G'), (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) } (c • f) = AddHom.toFun { toFun := fun f => prod f.fst f.snd, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G'), (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) } (c • f)) }.toAddHom ((fun f => (ContinuousLinearMap.compContinuousMultilinearMap (ContinuousLinearMap.fst 𝕜 G G') f, ContinuousLinearMap.compContinuousMultilinearMap (ContinuousLinearMap.snd 𝕜 G G') f)) f)) x✝).fst = (↑f x✝).fst [PROOFSTEP] rfl [GOAL] case H.h₂ 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i 𝕜' : Type u_1 inst✝² : NormedField 𝕜' inst✝¹ : NormedSpace 𝕜' G inst✝ : SMulCommClass 𝕜 𝕜' G f : ContinuousMultilinearMap 𝕜 E (G × G') x✝ : (i : ι) → E i ⊢ (↑(AddHom.toFun { toAddHom := { toFun := fun f => prod f.fst f.snd, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G'), (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) }, map_smul' := (_ : ∀ (c : 𝕜) (f : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G'), AddHom.toFun { toFun := fun f => prod f.fst f.snd, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G'), (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) } (c • f) = AddHom.toFun { toFun := fun f => prod f.fst f.snd, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G'), (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) } (c • f)) }.toAddHom ((fun f => (ContinuousLinearMap.compContinuousMultilinearMap (ContinuousLinearMap.fst 𝕜 G G') f, ContinuousLinearMap.compContinuousMultilinearMap (ContinuousLinearMap.snd 𝕜 G G') f)) f)) x✝).snd = (↑f x✝).snd [PROOFSTEP] rfl [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i ⊢ Continuous fun p => ↑p.fst p.snd [PROOFSTEP] apply continuous_iff_continuousAt.2 fun p => ?_ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i p : ContinuousMultilinearMap 𝕜 E G × ((i : ι) → E i) ⊢ ContinuousAt (fun p => ↑p.fst p.snd) p [PROOFSTEP] apply continuousAt_of_locally_lipschitz zero_lt_one ((‖p‖ + 1) * Fintype.card ι * (‖p‖ + 1) ^ (Fintype.card ι - 1) + ∏ i, ‖p.2 i‖) fun q hq => ?_ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i p q : ContinuousMultilinearMap 𝕜 E G × ((i : ι) → E i) hq : dist q p < 1 ⊢ dist (↑q.fst q.snd) (↑p.fst p.snd) ≤ ((‖p‖ + 1) * ↑(Fintype.card ι) * (‖p‖ + 1) ^ (Fintype.card ι - 1) + ∏ i : ι, ‖Prod.snd p i‖) * dist q p [PROOFSTEP] have : 0 ≤ max ‖q.2‖ ‖p.2‖ := by simp [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i p q : ContinuousMultilinearMap 𝕜 E G × ((i : ι) → E i) hq : dist q p < 1 ⊢ 0 ≤ max ‖q.snd‖ ‖p.snd‖ [PROOFSTEP] simp [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i p q : ContinuousMultilinearMap 𝕜 E G × ((i : ι) → E i) hq : dist q p < 1 this : 0 ≤ max ‖q.snd‖ ‖p.snd‖ ⊢ dist (↑q.fst q.snd) (↑p.fst p.snd) ≤ ((‖p‖ + 1) * ↑(Fintype.card ι) * (‖p‖ + 1) ^ (Fintype.card ι - 1) + ∏ i : ι, ‖Prod.snd p i‖) * dist q p [PROOFSTEP] have : 0 ≤ ‖p‖ + 1 := zero_le_one.trans ((le_add_iff_nonneg_left 1).2 <\| norm_nonneg p) [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i p q : ContinuousMultilinearMap 𝕜 E G × ((i : ι) → E i) hq : dist q p < 1 this✝ : 0 ≤ max ‖q.snd‖ ‖p.snd‖ this : 0 ≤ ‖p‖ + 1 ⊢ dist (↑q.fst q.snd) (↑p.fst p.snd) ≤ ((‖p‖ + 1) * ↑(Fintype.card ι) * (‖p‖ + 1) ^ (Fintype.card ι - 1) + ∏ i : ι, ‖Prod.snd p i‖) * dist q p [PROOFSTEP] have A : ‖q‖ ≤ ‖p‖ + 1 := norm_le_of_mem_closedBall hq.le [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i p q : ContinuousMultilinearMap 𝕜 E G × ((i : ι) → E i) hq : dist q p < 1 this✝ : 0 ≤ max ‖q.snd‖ ‖p.snd‖ this : 0 ≤ ‖p‖ + 1 A : ‖q‖ ≤ ‖p‖ + 1 ⊢ dist (↑q.fst q.snd) (↑p.fst p.snd) ≤ ((‖p‖ + 1) * ↑(Fintype.card ι) * (‖p‖ + 1) ^ (Fintype.card ι - 1) + ∏ i : ι, ‖Prod.snd p i‖) * dist q p [PROOFSTEP] have : max ‖q.2‖ ‖p.2‖ ≤ ‖p‖ + 1 := (max_le_max (norm_snd_le q) (norm_snd_le p)).trans (by simp [A, zero_le_one]) [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i p q : ContinuousMultilinearMap 𝕜 E G × ((i : ι) → E i) hq : dist q p < 1 this✝ : 0 ≤ max ‖q.snd‖ ‖p.snd‖ this : 0 ≤ ‖p‖ + 1 A : ‖q‖ ≤ ‖p‖ + 1 ⊢ max ‖q‖ ‖p‖ ≤ ‖p‖ + 1 [PROOFSTEP] simp [A, zero_le_one] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i p q : ContinuousMultilinearMap 𝕜 E G × ((i : ι) → E i) hq : dist q p < 1 this✝¹ : 0 ≤ max ‖q.snd‖ ‖p.snd‖ this✝ : 0 ≤ ‖p‖ + 1 A : ‖q‖ ≤ ‖p‖ + 1 this : max ‖q.snd‖ ‖p.snd‖ ≤ ‖p‖ + 1 ⊢ dist (↑q.fst q.snd) (↑p.fst p.snd) ≤ ((‖p‖ + 1) * ↑(Fintype.card ι) * (‖p‖ + 1) ^ (Fintype.card ι - 1) + ∏ i : ι, ‖Prod.snd p i‖) * dist q p [PROOFSTEP] have : ∀ i : ι, i ∈ univ → 0 ≤ ‖p.2 i‖ := fun i _ => norm_nonneg _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i p q : ContinuousMultilinearMap 𝕜 E G × ((i : ι) → E i) hq : dist q p < 1 this✝² : 0 ≤ max ‖q.snd‖ ‖p.snd‖ this✝¹ : 0 ≤ ‖p‖ + 1 A : ‖q‖ ≤ ‖p‖ + 1 this✝ : max ‖q.snd‖ ‖p.snd‖ ≤ ‖p‖ + 1 this : ∀ (i : ι), i ∈ univ → 0 ≤ ‖Prod.snd p i‖ ⊢ dist (↑q.fst q.snd) (↑p.fst p.snd) ≤ ((‖p‖ + 1) * ↑(Fintype.card ι) * (‖p‖ + 1) ^ (Fintype.card ι - 1) + ∏ i : ι, ‖Prod.snd p i‖) * dist q p [PROOFSTEP] calc dist (q.1 q.2) (p.1 p.2) ≤ dist (q.1 q.2) (q.1 p.2) + dist (q.1 p.2) (p.1 p.2) := dist_triangle _ _ _ _ = ‖q.1 q.2 - q.1 p.2‖ + ‖q.1 p.2 - p.1 p.2‖ := by rw [dist_eq_norm, dist_eq_norm] _ ≤ ‖q.1‖ * Fintype.card ι * max ‖q.2‖ ‖p.2‖ ^ (Fintype.card ι - 1) * ‖q.2 - p.2‖ + ‖q.1 - p.1‖ * ∏ i, ‖p.2 i‖ := (add_le_add (norm_image_sub_le _ _ _) ((q.1 - p.1).le_op_norm p.2)) _ ≤ (‖p‖ + 1) * Fintype.card ι * (‖p‖ + 1) ^ (Fintype.card ι - 1) * ‖q - p‖ + ‖q - p‖ * ∏ i, ‖p.2 i‖ := by apply_rules [add_le_add, mul_le_mul, le_refl, le_trans (norm_fst_le q) A, Nat.cast_nonneg, mul_nonneg, pow_le_pow_of_le_left, pow_nonneg, norm_snd_le (q - p), norm_nonneg, norm_fst_le (q - p), prod_nonneg] _ = ((‖p‖ + 1) * Fintype.card ι * (‖p‖ + 1) ^ (Fintype.card ι - 1) + ∏ i, ‖p.2 i‖) * dist q p := by rw [dist_eq_norm] ring [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i p q : ContinuousMultilinearMap 𝕜 E G × ((i : ι) → E i) hq : dist q p < 1 this✝² : 0 ≤ max ‖q.snd‖ ‖p.snd‖ this✝¹ : 0 ≤ ‖p‖ + 1 A : ‖q‖ ≤ ‖p‖ + 1 this✝ : max ‖q.snd‖ ‖p.snd‖ ≤ ‖p‖ + 1 this : ∀ (i : ι), i ∈ univ → 0 ≤ ‖Prod.snd p i‖ ⊢ dist (↑q.fst q.snd) (↑q.fst p.snd) + dist (↑q.fst p.snd) (↑p.fst p.snd) = ‖↑q.fst q.snd - ↑q.fst p.snd‖ + ‖↑q.fst p.snd - ↑p.fst p.snd‖ [PROOFSTEP] rw [dist_eq_norm, dist_eq_norm] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i p q : ContinuousMultilinearMap 𝕜 E G × ((i : ι) → E i) hq : dist q p < 1 this✝² : 0 ≤ max ‖q.snd‖ ‖p.snd‖ this✝¹ : 0 ≤ ‖p‖ + 1 A : ‖q‖ ≤ ‖p‖ + 1 this✝ : max ‖q.snd‖ ‖p.snd‖ ≤ ‖p‖ + 1 this : ∀ (i : ι), i ∈ univ → 0 ≤ ‖Prod.snd p i‖ ⊢ ‖q.fst‖ * ↑(Fintype.card ι) * max ‖q.snd‖ ‖p.snd‖ ^ (Fintype.card ι - 1) * ‖q.snd - p.snd‖ + ‖q.fst - p.fst‖ * ∏ i : ι, ‖Prod.snd p i‖ ≤ (‖p‖ + 1) * ↑(Fintype.card ι) * (‖p‖ + 1) ^ (Fintype.card ι - 1) * ‖q - p‖ + ‖q - p‖ * ∏ i : ι, ‖Prod.snd p i‖ [PROOFSTEP] apply_rules [add_le_add, mul_le_mul, le_refl, le_trans (norm_fst_le q) A, Nat.cast_nonneg, mul_nonneg, pow_le_pow_of_le_left, pow_nonneg, norm_snd_le (q - p), norm_nonneg, norm_fst_le (q - p), prod_nonneg] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i p q : ContinuousMultilinearMap 𝕜 E G × ((i : ι) → E i) hq : dist q p < 1 this✝² : 0 ≤ max ‖q.snd‖ ‖p.snd‖ this✝¹ : 0 ≤ ‖p‖ + 1 A : ‖q‖ ≤ ‖p‖ + 1 this✝ : max ‖q.snd‖ ‖p.snd‖ ≤ ‖p‖ + 1 this : ∀ (i : ι), i ∈ univ → 0 ≤ ‖Prod.snd p i‖ ⊢ (‖p‖ + 1) * ↑(Fintype.card ι) * (‖p‖ + 1) ^ (Fintype.card ι - 1) * ‖q - p‖ + ‖q - p‖ * ∏ i : ι, ‖Prod.snd p i‖ = ((‖p‖ + 1) * ↑(Fintype.card ι) * (‖p‖ + 1) ^ (Fintype.card ι - 1) + ∏ i : ι, ‖Prod.snd p i‖) * dist q p [PROOFSTEP] rw [dist_eq_norm] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i p q : ContinuousMultilinearMap 𝕜 E G × ((i : ι) → E i) hq : dist q p < 1 this✝² : 0 ≤ max ‖q.snd‖ ‖p.snd‖ this✝¹ : 0 ≤ ‖p‖ + 1 A : ‖q‖ ≤ ‖p‖ + 1 this✝ : max ‖q.snd‖ ‖p.snd‖ ≤ ‖p‖ + 1 this : ∀ (i : ι), i ∈ univ → 0 ≤ ‖Prod.snd p i‖ ⊢ (‖p‖ + 1) * ↑(Fintype.card ι) * (‖p‖ + 1) ^ (Fintype.card ι - 1) * ‖q - p‖ + ‖q - p‖ * ∏ i : ι, ‖Prod.snd p i‖ = ((‖p‖ + 1) * ↑(Fintype.card ι) * (‖p‖ + 1) ^ (Fintype.card ι - 1) + ∏ i : ι, ‖Prod.snd p i‖) * ‖q - p‖ [PROOFSTEP] ring [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m✝ : (i : ι) → E i α : Type u_1 p : α → ContinuousMultilinearMap 𝕜 E G q : ContinuousMultilinearMap 𝕜 E G h : HasSum p q m : (i : ι) → E i ⊢ HasSum (fun a => ↑(p a) m) (↑q m) [PROOFSTEP] dsimp [HasSum] at h ⊢ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m✝ : (i : ι) → E i α : Type u_1 p : α → ContinuousMultilinearMap 𝕜 E G q : ContinuousMultilinearMap 𝕜 E G h : Filter.Tendsto (fun s => ∑ b in s, p b) Filter.atTop (nhds q) m : (i : ι) → E i ⊢ Filter.Tendsto (fun s => ∑ b in s, ↑(p b) m) Filter.atTop (nhds (↑q m)) [PROOFSTEP] convert ((continuous_eval_left m).tendsto _).comp h using 1 [GOAL] case h.e'_3 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m✝ : (i : ι) → E i α : Type u_1 p : α → ContinuousMultilinearMap 𝕜 E G q : ContinuousMultilinearMap 𝕜 E G h : Filter.Tendsto (fun s => ∑ b in s, p b) Filter.atTop (nhds q) m : (i : ι) → E i ⊢ (fun s => ∑ b in s, ↑(p b) m) = (fun p => ↑p m) ∘ fun s => ∑ b in s, p b [PROOFSTEP] ext s [GOAL] case h.e'_3.h 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m✝ : (i : ι) → E i α : Type u_1 p : α → ContinuousMultilinearMap 𝕜 E G q : ContinuousMultilinearMap 𝕜 E G h : Filter.Tendsto (fun s => ∑ b in s, p b) Filter.atTop (nhds q) m : (i : ι) → E i s : Finset α ⊢ ∑ b in s, ↑(p b) m = ((fun p => ↑p m) ∘ fun s => ∑ b in s, p b) s [PROOFSTEP] simp [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G ⊢ CompleteSpace (ContinuousMultilinearMap 𝕜 E G) [PROOFSTEP] have nonneg : ∀ v : ∀ i, E i, 0 ≤ ∏ i, ‖v i‖ := fun v => Finset.prod_nonneg fun i _ => norm_nonneg _ -- We show that every Cauchy sequence converges. [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ ⊢ CompleteSpace (ContinuousMultilinearMap 𝕜 E G) [PROOFSTEP] refine' Metric.complete_of_cauchySeq_tendsto fun f hf => _ -- We now expand out the definition of a Cauchy sequence, [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f ⊢ ∃ a, Tendsto f atTop (𝓝 a) [PROOFSTEP] rcases cauchySeq_iff_le_tendsto_0.1 hf with ⟨b, b0, b_bound, b_lim⟩ -- and establish that the evaluation at any point `v : Π i, E i` is Cauchy. [GOAL] case intro.intro.intro 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) ⊢ ∃ a, Tendsto f atTop (𝓝 a) [PROOFSTEP] have cau : ∀ v, CauchySeq fun n => f n v := by intro v apply cauchySeq_iff_le_tendsto_0.2 ⟨fun n => b n * ∏ i, ‖v i‖, _, _, _⟩ · intro exact mul_nonneg (b0 _) (nonneg v) · intro n m N hn hm rw [dist_eq_norm] apply le_trans ((f n - f m).le_op_norm v) _ exact mul_le_mul_of_nonneg_right (b_bound n m N hn hm) (nonneg v) · simpa using b_lim.mul tendsto_const_nhds [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) ⊢ ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v [PROOFSTEP] intro v [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) v : (i : ι) → E i ⊢ CauchySeq fun n => ↑(f n) v [PROOFSTEP] apply cauchySeq_iff_le_tendsto_0.2 ⟨fun n => b n * ∏ i, ‖v i‖, _, _, _⟩ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) v : (i : ι) → E i ⊢ ∀ (n : ℕ), 0 ≤ (fun n => b n * ∏ i : ι, ‖v i‖) n [PROOFSTEP] intro [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) v : (i : ι) → E i n✝ : ℕ ⊢ 0 ≤ (fun n => b n * ∏ i : ι, ‖v i‖) n✝ [PROOFSTEP] exact mul_nonneg (b0 _) (nonneg v) [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) v : (i : ι) → E i ⊢ ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (↑(f n) v) (↑(f m) v) ≤ (fun n => b n * ∏ i : ι, ‖v i‖) N [PROOFSTEP] intro n m N hn hm [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n✝ : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n✝) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n✝)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n✝)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m✝ : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) v : (i : ι) → E i n m N : ℕ hn : N ≤ n hm : N ≤ m ⊢ dist (↑(f n) v) (↑(f m) v) ≤ (fun n => b n * ∏ i : ι, ‖v i‖) N [PROOFSTEP] rw [dist_eq_norm] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n✝ : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n✝) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n✝)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n✝)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m✝ : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) v : (i : ι) → E i n m N : ℕ hn : N ≤ n hm : N ≤ m ⊢ ‖↑(f n) v - ↑(f m) v‖ ≤ (fun n => b n * ∏ i : ι, ‖v i‖) N [PROOFSTEP] apply le_trans ((f n - f m).le_op_norm v) _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n✝ : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n✝) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n✝)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n✝)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m✝ : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) v : (i : ι) → E i n m N : ℕ hn : N ≤ n hm : N ≤ m ⊢ ‖f n - f m‖ * ∏ i : ι, ‖v i‖ ≤ (fun n => b n * ∏ i : ι, ‖v i‖) N [PROOFSTEP] exact mul_le_mul_of_nonneg_right (b_bound n m N hn hm) (nonneg v) [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) v : (i : ι) → E i ⊢ Tendsto (fun n => b n * ∏ i : ι, ‖v i‖) atTop (𝓝 0) [PROOFSTEP] simpa using b_lim.mul tendsto_const_nhds [GOAL] case intro.intro.intro 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v ⊢ ∃ a, Tendsto f atTop (𝓝 a) [PROOFSTEP] choose F hF using fun v => cauchySeq_tendsto_of_complete (cau v) -- Next, we show that this `F` is multilinear, [GOAL] case intro.intro.intro 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) ⊢ ∃ a, Tendsto f atTop (𝓝 a) [PROOFSTEP] let Fmult : MultilinearMap 𝕜 E G := { toFun := F map_add' := fun v i x y => by skip have A := hF (Function.update v i (x + y)) have B := (hF (Function.update v i x)).add (hF (Function.update v i y)) simp at A B exact tendsto_nhds_unique A B map_smul' := fun v i c x => by skip have A := hF (Function.update v i (c • x)) have B := Filter.Tendsto.smul (@tendsto_const_nhds _ ℕ _ c _) (hF (Function.update v i x)) simp at A B exact tendsto_nhds_unique A B } -- and that `F` has norm at most `(b 0 + ‖f 0‖)`. [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedSpace 𝕜 G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝¹ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) inst✝ : DecidableEq ι v : (i : ι) → E i i : ι x y : E i ⊢ F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y) [PROOFSTEP] skip [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedSpace 𝕜 G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝¹ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) inst✝ : DecidableEq ι v : (i : ι) → E i i : ι x y : E i ⊢ F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y) [PROOFSTEP] have A := hF (Function.update v i (x + y)) [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedSpace 𝕜 G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝¹ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) inst✝ : DecidableEq ι v : (i : ι) → E i i : ι x y : E i A : Tendsto (fun n => ↑(f n) (Function.update v i (x + y))) atTop (𝓝 (F (Function.update v i (x + y)))) ⊢ F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y) [PROOFSTEP] have B := (hF (Function.update v i x)).add (hF (Function.update v i y)) [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedSpace 𝕜 G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝¹ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) inst✝ : DecidableEq ι v : (i : ι) → E i i : ι x y : E i A : Tendsto (fun n => ↑(f n) (Function.update v i (x + y))) atTop (𝓝 (F (Function.update v i (x + y)))) B : Tendsto (fun x_1 => ↑(f x_1) (Function.update v i x) + ↑(f x_1) (Function.update v i y)) atTop (𝓝 (F (Function.update v i x) + F (Function.update v i y))) ⊢ F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y) [PROOFSTEP] simp at A B [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedSpace 𝕜 G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝¹ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) inst✝ : DecidableEq ι v : (i : ι) → E i i : ι x y : E i B : Tendsto (fun x_1 => ↑(f x_1) (Function.update v i x) + ↑(f x_1) (Function.update v i y)) atTop (𝓝 (F (Function.update v i x) + F (Function.update v i y))) A : Tendsto (fun n => ↑(f n) (Function.update v i x) + ↑(f n) (Function.update v i y)) atTop (𝓝 (F (Function.update v i (x + y)))) ⊢ F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y) [PROOFSTEP] exact tendsto_nhds_unique A B [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedSpace 𝕜 G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace 𝕜 G' c✝ : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝¹ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) inst✝ : DecidableEq ι v : (i : ι) → E i i : ι c : 𝕜 x : E i ⊢ F (Function.update v i (c • x)) = c • F (Function.update v i x) [PROOFSTEP] skip [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedSpace 𝕜 G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace 𝕜 G' c✝ : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝¹ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) inst✝ : DecidableEq ι v : (i : ι) → E i i : ι c : 𝕜 x : E i ⊢ F (Function.update v i (c • x)) = c • F (Function.update v i x) [PROOFSTEP] have A := hF (Function.update v i (c • x)) [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedSpace 𝕜 G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace 𝕜 G' c✝ : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝¹ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) inst✝ : DecidableEq ι v : (i : ι) → E i i : ι c : 𝕜 x : E i A : Tendsto (fun n => ↑(f n) (Function.update v i (c • x))) atTop (𝓝 (F (Function.update v i (c • x)))) ⊢ F (Function.update v i (c • x)) = c • F (Function.update v i x) [PROOFSTEP] have B := Filter.Tendsto.smul (@tendsto_const_nhds _ ℕ _ c _) (hF (Function.update v i x)) [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedSpace 𝕜 G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace 𝕜 G' c✝ : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝¹ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) inst✝ : DecidableEq ι v : (i : ι) → E i i : ι c : 𝕜 x : E i A : Tendsto (fun n => ↑(f n) (Function.update v i (c • x))) atTop (𝓝 (F (Function.update v i (c • x)))) B : Tendsto (fun x_1 => c • ↑(f x_1) (Function.update v i x)) atTop (𝓝 (c • F (Function.update v i x))) ⊢ F (Function.update v i (c • x)) = c • F (Function.update v i x) [PROOFSTEP] simp at A B [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedSpace 𝕜 G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace 𝕜 G' c✝ : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝¹ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) inst✝ : DecidableEq ι v : (i : ι) → E i i : ι c : 𝕜 x : E i B : Tendsto (fun x_1 => c • ↑(f x_1) (Function.update v i x)) atTop (𝓝 (c • F (Function.update v i x))) A : Tendsto (fun n => c • ↑(f n) (Function.update v i x)) atTop (𝓝 (F (Function.update v i (c • x)))) ⊢ F (Function.update v i (c • x)) = c • F (Function.update v i x) [PROOFSTEP] exact tendsto_nhds_unique A B [GOAL] case intro.intro.intro 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) Fmult : MultilinearMap 𝕜 E G := { toFun := F, map_add' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (x y : E i), F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), F (Function.update v i (c • x)) = c • F (Function.update v i x)) } ⊢ ∃ a, Tendsto f atTop (𝓝 a) [PROOFSTEP] have Fnorm : ∀ v, ‖F v‖ ≤ (b 0 + ‖f 0‖) * ∏ i, ‖v i‖ := by intro v have A : ∀ n, ‖f n v‖ ≤ (b 0 + ‖f 0‖) * ∏ i, ‖v i‖ := by intro n apply le_trans ((f n).le_op_norm _) _ apply mul_le_mul_of_nonneg_right _ (nonneg v) calc ‖f n‖ = ‖f n - f 0 + f 0‖ := by congr 1 abel _ ≤ ‖f n - f 0‖ + ‖f 0‖ := (norm_add_le _ _) _ ≤ b 0 + ‖f 0‖ := by apply add_le_add_right simpa [dist_eq_norm] using b_bound n 0 0 (zero_le _) (zero_le _) exact le_of_tendsto (hF v).norm (eventually_of_forall A) -- Thus `F` is continuous, and we propose that as the limit point of our original Cauchy sequence. [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) Fmult : MultilinearMap 𝕜 E G := { toFun := F, map_add' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (x y : E i), F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), F (Function.update v i (c • x)) = c • F (Function.update v i x)) } ⊢ ∀ (v : (i : ι) → E i), ‖F v‖ ≤ (b 0 + ‖f 0‖) * ∏ i : ι, ‖v i‖ [PROOFSTEP] intro v [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) Fmult : MultilinearMap 𝕜 E G := { toFun := F, map_add' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (x y : E i), F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), F (Function.update v i (c • x)) = c • F (Function.update v i x)) } v : (i : ι) → E i ⊢ ‖F v‖ ≤ (b 0 + ‖f 0‖) * ∏ i : ι, ‖v i‖ [PROOFSTEP] have A : ∀ n, ‖f n v‖ ≤ (b 0 + ‖f 0‖) * ∏ i, ‖v i‖ := by intro n apply le_trans ((f n).le_op_norm _) _ apply mul_le_mul_of_nonneg_right _ (nonneg v) calc ‖f n‖ = ‖f n - f 0 + f 0‖ := by congr 1 abel _ ≤ ‖f n - f 0‖ + ‖f 0‖ := (norm_add_le _ _) _ ≤ b 0 + ‖f 0‖ := by apply add_le_add_right simpa [dist_eq_norm] using b_bound n 0 0 (zero_le _) (zero_le _) [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) Fmult : MultilinearMap 𝕜 E G := { toFun := F, map_add' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (x y : E i), F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), F (Function.update v i (c • x)) = c • F (Function.update v i x)) } v : (i : ι) → E i ⊢ ∀ (n : ℕ), ‖↑(f n) v‖ ≤ (b 0 + ‖f 0‖) * ∏ i : ι, ‖v i‖ [PROOFSTEP] intro n [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n✝ : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n✝) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n✝)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n✝)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) Fmult : MultilinearMap 𝕜 E G := { toFun := F, map_add' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (x y : E i), F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), F (Function.update v i (c • x)) = c • F (Function.update v i x)) } v : (i : ι) → E i n : ℕ ⊢ ‖↑(f n) v‖ ≤ (b 0 + ‖f 0‖) * ∏ i : ι, ‖v i‖ [PROOFSTEP] apply le_trans ((f n).le_op_norm _) _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n✝ : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n✝) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n✝)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n✝)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) Fmult : MultilinearMap 𝕜 E G := { toFun := F, map_add' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (x y : E i), F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), F (Function.update v i (c • x)) = c • F (Function.update v i x)) } v : (i : ι) → E i n : ℕ ⊢ ‖f n‖ * ∏ i : ι, ‖v i‖ ≤ (b 0 + ‖f 0‖) * ∏ i : ι, ‖v i‖ [PROOFSTEP] apply mul_le_mul_of_nonneg_right _ (nonneg v) [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n✝ : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n✝) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n✝)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n✝)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) Fmult : MultilinearMap 𝕜 E G := { toFun := F, map_add' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (x y : E i), F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), F (Function.update v i (c • x)) = c • F (Function.update v i x)) } v : (i : ι) → E i n : ℕ ⊢ ‖f n‖ ≤ b 0 + ‖f 0‖ [PROOFSTEP] calc ‖f n‖ = ‖f n - f 0 + f 0‖ := by congr 1 abel _ ≤ ‖f n - f 0‖ + ‖f 0‖ := (norm_add_le _ _) _ ≤ b 0 + ‖f 0‖ := by apply add_le_add_right simpa [dist_eq_norm] using b_bound n 0 0 (zero_le _) (zero_le _) [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n✝ : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n✝) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n✝)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n✝)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) Fmult : MultilinearMap 𝕜 E G := { toFun := F, map_add' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (x y : E i), F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), F (Function.update v i (c • x)) = c • F (Function.update v i x)) } v : (i : ι) → E i n : ℕ ⊢ ‖f n‖ = ‖f n - f 0 + f 0‖ [PROOFSTEP] congr 1 [GOAL] case e_a 𝕜 : Type u ι : Type v ι' : Type v' n✝ : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n✝) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n✝)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n✝)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) Fmult : MultilinearMap 𝕜 E G := { toFun := F, map_add' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (x y : E i), F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), F (Function.update v i (c • x)) = c • F (Function.update v i x)) } v : (i : ι) → E i n : ℕ ⊢ f n = f n - f 0 + f 0 [PROOFSTEP] abel [GOAL] case e_a 𝕜 : Type u ι : Type v ι' : Type v' n✝ : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n✝) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n✝)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n✝)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) Fmult : MultilinearMap 𝕜 E G := { toFun := F, map_add' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (x y : E i), F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), F (Function.update v i (c • x)) = c • F (Function.update v i x)) } v : (i : ι) → E i n : ℕ ⊢ f n = f n - f 0 + f 0 [PROOFSTEP] abel [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n✝ : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n✝) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n✝)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n✝)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) Fmult : MultilinearMap 𝕜 E G := { toFun := F, map_add' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (x y : E i), F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), F (Function.update v i (c • x)) = c • F (Function.update v i x)) } v : (i : ι) → E i n : ℕ ⊢ ‖f n - f 0‖ + ‖f 0‖ ≤ b 0 + ‖f 0‖ [PROOFSTEP] apply add_le_add_right [GOAL] case bc 𝕜 : Type u ι : Type v ι' : Type v' n✝ : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n✝) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n✝)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n✝)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) Fmult : MultilinearMap 𝕜 E G := { toFun := F, map_add' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (x y : E i), F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), F (Function.update v i (c • x)) = c • F (Function.update v i x)) } v : (i : ι) → E i n : ℕ ⊢ ‖f n - f 0‖ ≤ b 0 [PROOFSTEP] simpa [dist_eq_norm] using b_bound n 0 0 (zero_le _) (zero_le _) [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) Fmult : MultilinearMap 𝕜 E G := { toFun := F, map_add' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (x y : E i), F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), F (Function.update v i (c • x)) = c • F (Function.update v i x)) } v : (i : ι) → E i A : ∀ (n : ℕ), ‖↑(f n) v‖ ≤ (b 0 + ‖f 0‖) * ∏ i : ι, ‖v i‖ ⊢ ‖F v‖ ≤ (b 0 + ‖f 0‖) * ∏ i : ι, ‖v i‖ [PROOFSTEP] exact le_of_tendsto (hF v).norm (eventually_of_forall A) -- Thus `F` is continuous, and we propose that as the limit point of our original Cauchy sequence. [GOAL] case intro.intro.intro 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) Fmult : MultilinearMap 𝕜 E G := { toFun := F, map_add' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (x y : E i), F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), F (Function.update v i (c • x)) = c • F (Function.update v i x)) } Fnorm : ∀ (v : (i : ι) → E i), ‖F v‖ ≤ (b 0 + ‖f 0‖) * ∏ i : ι, ‖v i‖ ⊢ ∃ a, Tendsto f atTop (𝓝 a) [PROOFSTEP] let Fcont := Fmult.mkContinuous _ Fnorm [GOAL] case intro.intro.intro 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) Fmult : MultilinearMap 𝕜 E G := { toFun := F, map_add' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (x y : E i), F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), F (Function.update v i (c • x)) = c • F (Function.update v i x)) } Fnorm : ∀ (v : (i : ι) → E i), ‖F v‖ ≤ (b 0 + ‖f 0‖) * ∏ i : ι, ‖v i‖ Fcont : ContinuousMultilinearMap 𝕜 E G := MultilinearMap.mkContinuous Fmult (b 0 + ‖f 0‖) Fnorm ⊢ ∃ a, Tendsto f atTop (𝓝 a) [PROOFSTEP] use Fcont [GOAL] case h 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) Fmult : MultilinearMap 𝕜 E G := { toFun := F, map_add' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (x y : E i), F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), F (Function.update v i (c • x)) = c • F (Function.update v i x)) } Fnorm : ∀ (v : (i : ι) → E i), ‖F v‖ ≤ (b 0 + ‖f 0‖) * ∏ i : ι, ‖v i‖ Fcont : ContinuousMultilinearMap 𝕜 E G := MultilinearMap.mkContinuous Fmult (b 0 + ‖f 0‖) Fnorm ⊢ Tendsto f atTop (𝓝 Fcont) [PROOFSTEP] have : ∀ n, ‖f n - Fcont‖ ≤ b n := by intro n apply op_norm_le_bound _ (b0 n) fun v => ?_ have A : ∀ᶠ m in atTop, ‖(f n - f m) v‖ ≤ b n * ∏ i, ‖v i‖ := by refine' eventually_atTop.2 ⟨n, fun m hm => _⟩ apply le_trans ((f n - f m).le_op_norm _) _ exact mul_le_mul_of_nonneg_right (b_bound n m n le_rfl hm) (nonneg v) have B : Tendsto (fun m => ‖(f n - f m) v‖) atTop (𝓝 ‖(f n - Fcont) v‖) := Tendsto.norm (tendsto_const_nhds.sub (hF v)) exact le_of_tendsto B A [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) Fmult : MultilinearMap 𝕜 E G := { toFun := F, map_add' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (x y : E i), F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), F (Function.update v i (c • x)) = c • F (Function.update v i x)) } Fnorm : ∀ (v : (i : ι) → E i), ‖F v‖ ≤ (b 0 + ‖f 0‖) * ∏ i : ι, ‖v i‖ Fcont : ContinuousMultilinearMap 𝕜 E G := MultilinearMap.mkContinuous Fmult (b 0 + ‖f 0‖) Fnorm ⊢ ∀ (n : ℕ), ‖f n - Fcont‖ ≤ b n [PROOFSTEP] intro n [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n✝ : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n✝) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n✝)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n✝)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) Fmult : MultilinearMap 𝕜 E G := { toFun := F, map_add' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (x y : E i), F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), F (Function.update v i (c • x)) = c • F (Function.update v i x)) } Fnorm : ∀ (v : (i : ι) → E i), ‖F v‖ ≤ (b 0 + ‖f 0‖) * ∏ i : ι, ‖v i‖ Fcont : ContinuousMultilinearMap 𝕜 E G := MultilinearMap.mkContinuous Fmult (b 0 + ‖f 0‖) Fnorm n : ℕ ⊢ ‖f n - Fcont‖ ≤ b n [PROOFSTEP] apply op_norm_le_bound _ (b0 n) fun v => ?_ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n✝ : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n✝) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n✝)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n✝)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) Fmult : MultilinearMap 𝕜 E G := { toFun := F, map_add' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (x y : E i), F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), F (Function.update v i (c • x)) = c • F (Function.update v i x)) } Fnorm : ∀ (v : (i : ι) → E i), ‖F v‖ ≤ (b 0 + ‖f 0‖) * ∏ i : ι, ‖v i‖ Fcont : ContinuousMultilinearMap 𝕜 E G := MultilinearMap.mkContinuous Fmult (b 0 + ‖f 0‖) Fnorm n : ℕ v : (i : ι) → E i ⊢ ‖↑(f n - Fcont) v‖ ≤ b n * ∏ i : ι, ‖v i‖ [PROOFSTEP] have A : ∀ᶠ m in atTop, ‖(f n - f m) v‖ ≤ b n * ∏ i, ‖v i‖ := by refine' eventually_atTop.2 ⟨n, fun m hm => _⟩ apply le_trans ((f n - f m).le_op_norm _) _ exact mul_le_mul_of_nonneg_right (b_bound n m n le_rfl hm) (nonneg v) [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n✝ : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n✝) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n✝)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n✝)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) Fmult : MultilinearMap 𝕜 E G := { toFun := F, map_add' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (x y : E i), F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), F (Function.update v i (c • x)) = c • F (Function.update v i x)) } Fnorm : ∀ (v : (i : ι) → E i), ‖F v‖ ≤ (b 0 + ‖f 0‖) * ∏ i : ι, ‖v i‖ Fcont : ContinuousMultilinearMap 𝕜 E G := MultilinearMap.mkContinuous Fmult (b 0 + ‖f 0‖) Fnorm n : ℕ v : (i : ι) → E i ⊢ ∀ᶠ (m : ℕ) in atTop, ‖↑(f n - f m) v‖ ≤ b n * ∏ i : ι, ‖v i‖ [PROOFSTEP] refine' eventually_atTop.2 ⟨n, fun m hm => _⟩ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n✝ : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n✝) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n✝)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n✝)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m✝ : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) Fmult : MultilinearMap 𝕜 E G := { toFun := F, map_add' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (x y : E i), F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), F (Function.update v i (c • x)) = c • F (Function.update v i x)) } Fnorm : ∀ (v : (i : ι) → E i), ‖F v‖ ≤ (b 0 + ‖f 0‖) * ∏ i : ι, ‖v i‖ Fcont : ContinuousMultilinearMap 𝕜 E G := MultilinearMap.mkContinuous Fmult (b 0 + ‖f 0‖) Fnorm n : ℕ v : (i : ι) → E i m : ℕ hm : m ≥ n ⊢ ‖↑(f n - f m) v‖ ≤ b n * ∏ i : ι, ‖v i‖ [PROOFSTEP] apply le_trans ((f n - f m).le_op_norm _) _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n✝ : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n✝) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n✝)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n✝)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m✝ : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) Fmult : MultilinearMap 𝕜 E G := { toFun := F, map_add' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (x y : E i), F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), F (Function.update v i (c • x)) = c • F (Function.update v i x)) } Fnorm : ∀ (v : (i : ι) → E i), ‖F v‖ ≤ (b 0 + ‖f 0‖) * ∏ i : ι, ‖v i‖ Fcont : ContinuousMultilinearMap 𝕜 E G := MultilinearMap.mkContinuous Fmult (b 0 + ‖f 0‖) Fnorm n : ℕ v : (i : ι) → E i m : ℕ hm : m ≥ n ⊢ ‖f n - f m‖ * ∏ i : ι, ‖v i‖ ≤ b n * ∏ i : ι, ‖v i‖ [PROOFSTEP] exact mul_le_mul_of_nonneg_right (b_bound n m n le_rfl hm) (nonneg v) [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n✝ : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n✝) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n✝)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n✝)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) Fmult : MultilinearMap 𝕜 E G := { toFun := F, map_add' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (x y : E i), F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), F (Function.update v i (c • x)) = c • F (Function.update v i x)) } Fnorm : ∀ (v : (i : ι) → E i), ‖F v‖ ≤ (b 0 + ‖f 0‖) * ∏ i : ι, ‖v i‖ Fcont : ContinuousMultilinearMap 𝕜 E G := MultilinearMap.mkContinuous Fmult (b 0 + ‖f 0‖) Fnorm n : ℕ v : (i : ι) → E i A : ∀ᶠ (m : ℕ) in atTop, ‖↑(f n - f m) v‖ ≤ b n * ∏ i : ι, ‖v i‖ ⊢ ‖↑(f n - Fcont) v‖ ≤ b n * ∏ i : ι, ‖v i‖ [PROOFSTEP] have B : Tendsto (fun m => ‖(f n - f m) v‖) atTop (𝓝 ‖(f n - Fcont) v‖) := Tendsto.norm (tendsto_const_nhds.sub (hF v)) [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n✝ : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n✝) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n✝)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n✝)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) Fmult : MultilinearMap 𝕜 E G := { toFun := F, map_add' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (x y : E i), F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), F (Function.update v i (c • x)) = c • F (Function.update v i x)) } Fnorm : ∀ (v : (i : ι) → E i), ‖F v‖ ≤ (b 0 + ‖f 0‖) * ∏ i : ι, ‖v i‖ Fcont : ContinuousMultilinearMap 𝕜 E G := MultilinearMap.mkContinuous Fmult (b 0 + ‖f 0‖) Fnorm n : ℕ v : (i : ι) → E i A : ∀ᶠ (m : ℕ) in atTop, ‖↑(f n - f m) v‖ ≤ b n * ∏ i : ι, ‖v i‖ B : Tendsto (fun m => ‖↑(f n - f m) v‖) atTop (𝓝 ‖↑(f n - Fcont) v‖) ⊢ ‖↑(f n - Fcont) v‖ ≤ b n * ∏ i : ι, ‖v i‖ [PROOFSTEP] exact le_of_tendsto B A [GOAL] case h 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) Fmult : MultilinearMap 𝕜 E G := { toFun := F, map_add' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (x y : E i), F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), F (Function.update v i (c • x)) = c • F (Function.update v i x)) } Fnorm : ∀ (v : (i : ι) → E i), ‖F v‖ ≤ (b 0 + ‖f 0‖) * ∏ i : ι, ‖v i‖ Fcont : ContinuousMultilinearMap 𝕜 E G := MultilinearMap.mkContinuous Fmult (b 0 + ‖f 0‖) Fnorm this : ∀ (n : ℕ), ‖f n - Fcont‖ ≤ b n ⊢ Tendsto f atTop (𝓝 Fcont) [PROOFSTEP] erw [tendsto_iff_norm_tendsto_zero] [GOAL] case h 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' c : 𝕜 f✝ g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i inst✝ : CompleteSpace G nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖ f : ℕ → ContinuousMultilinearMap 𝕜 E G hf : CauchySeq f b : ℕ → ℝ b0 : ∀ (n : ℕ), 0 ≤ b n b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N b_lim : Tendsto b atTop (𝓝 0) cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => ↑(f n) v F : (v : (i : ι) → E i) → (fun x => G) v hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => ↑(f n) v) atTop (𝓝 (F v)) Fmult : MultilinearMap 𝕜 E G := { toFun := F, map_add' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (x y : E i), F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (v : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), F (Function.update v i (c • x)) = c • F (Function.update v i x)) } Fnorm : ∀ (v : (i : ι) → E i), ‖F v‖ ≤ (b 0 + ‖f 0‖) * ∏ i : ι, ‖v i‖ Fcont : ContinuousMultilinearMap 𝕜 E G := MultilinearMap.mkContinuous Fmult (b 0 + ‖f 0‖) Fnorm this : ∀ (n : ℕ), ‖f n - Fcont‖ ≤ b n ⊢ Tendsto (fun e => ‖f e - Fcont‖) atTop (𝓝 0) [PROOFSTEP] exact squeeze_zero (fun n => norm_nonneg _) this b_lim [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n✝ : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n✝) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n✝)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n✝)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' k n : ℕ f : ContinuousMultilinearMap 𝕜 (fun i => G) G' s : Finset (Fin n) hk : card s = k z : G ⊢ ‖restr f s hk z‖ ≤ ‖f‖ * ‖z‖ ^ (n - k) [PROOFSTEP] apply MultilinearMap.mkContinuous_norm_le [GOAL] case hC 𝕜 : Type u ι : Type v ι' : Type v' n✝ : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n✝) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n✝)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n✝)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' k n : ℕ f : ContinuousMultilinearMap 𝕜 (fun i => G) G' s : Finset (Fin n) hk : card s = k z : G ⊢ 0 ≤ ‖f‖ * ‖z‖ ^ (n - k) [PROOFSTEP] exact mul_nonneg (norm_nonneg _) (pow_nonneg (norm_nonneg _) _) [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' A : Type u_1 inst✝² : NormedCommRing A inst✝¹ : NormedAlgebra 𝕜 A inst✝ : Nonempty ι ⊢ ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ ≤ 1 [PROOFSTEP] have := fun f => @op_norm_le_bound 𝕜 ι (fun _ => A) A _ _ _ _ _ _ f _ zero_le_one [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' A : Type u_1 inst✝² : NormedCommRing A inst✝¹ : NormedAlgebra 𝕜 A inst✝ : Nonempty ι this : ∀ (f : ContinuousMultilinearMap 𝕜 (fun x => A) A), (∀ (m : ι → A), ‖↑f m‖ ≤ 1 * ∏ i : ι, ‖m i‖) → ‖f‖ ≤ 1 ⊢ ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ ≤ 1 [PROOFSTEP] refine' this _ _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' A : Type u_1 inst✝² : NormedCommRing A inst✝¹ : NormedAlgebra 𝕜 A inst✝ : Nonempty ι this : ∀ (f : ContinuousMultilinearMap 𝕜 (fun x => A) A), (∀ (m : ι → A), ‖↑f m‖ ≤ 1 * ∏ i : ι, ‖m i‖) → ‖f‖ ≤ 1 ⊢ ∀ (m : ι → A), ‖↑(ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A) m‖ ≤ 1 * ∏ i : ι, ‖m i‖ [PROOFSTEP] intro m [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' A : Type u_1 inst✝² : NormedCommRing A inst✝¹ : NormedAlgebra 𝕜 A inst✝ : Nonempty ι this : ∀ (f : ContinuousMultilinearMap 𝕜 (fun x => A) A), (∀ (m : ι → A), ‖↑f m‖ ≤ 1 * ∏ i : ι, ‖m i‖) → ‖f‖ ≤ 1 m : ι → A ⊢ ‖↑(ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A) m‖ ≤ 1 * ∏ i : ι, ‖m i‖ [PROOFSTEP] simp only [ContinuousMultilinearMap.mkPiAlgebra_apply, one_mul] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' A : Type u_1 inst✝² : NormedCommRing A inst✝¹ : NormedAlgebra 𝕜 A inst✝ : Nonempty ι this : ∀ (f : ContinuousMultilinearMap 𝕜 (fun x => A) A), (∀ (m : ι → A), ‖↑f m‖ ≤ 1 * ∏ i : ι, ‖m i‖) → ‖f‖ ≤ 1 m : ι → A ⊢ ‖∏ i : ι, m i‖ ≤ ∏ x : ι, ‖m x‖ [PROOFSTEP] exact norm_prod_le' _ univ_nonempty _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' A : Type u_1 inst✝² : NormedCommRing A inst✝¹ : NormedAlgebra 𝕜 A inst✝ : IsEmpty ι ⊢ ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ = ‖1‖ [PROOFSTEP] apply le_antisymm [GOAL] case a 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' A : Type u_1 inst✝² : NormedCommRing A inst✝¹ : NormedAlgebra 𝕜 A inst✝ : IsEmpty ι ⊢ ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ ≤ ‖1‖ [PROOFSTEP] have := fun f => @op_norm_le_bound 𝕜 ι (fun _ => A) A _ _ _ _ _ _ f _ (norm_nonneg (1 : A)) [GOAL] case a 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' A : Type u_1 inst✝² : NormedCommRing A inst✝¹ : NormedAlgebra 𝕜 A inst✝ : IsEmpty ι this : ∀ (f : ContinuousMultilinearMap 𝕜 (fun x => A) A), (∀ (m : ι → A), ‖↑f m‖ ≤ ‖1‖ * ∏ i : ι, ‖m i‖) → ‖f‖ ≤ ‖1‖ ⊢ ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ ≤ ‖1‖ [PROOFSTEP] refine' this _ _ [GOAL] case a 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' A : Type u_1 inst✝² : NormedCommRing A inst✝¹ : NormedAlgebra 𝕜 A inst✝ : IsEmpty ι this : ∀ (f : ContinuousMultilinearMap 𝕜 (fun x => A) A), (∀ (m : ι → A), ‖↑f m‖ ≤ ‖1‖ * ∏ i : ι, ‖m i‖) → ‖f‖ ≤ ‖1‖ ⊢ ∀ (m : ι → A), ‖↑(ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A) m‖ ≤ ‖1‖ * ∏ i : ι, ‖m i‖ [PROOFSTEP] simp -- Porting note: have to annotate types to get mvars to unify [GOAL] case a 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' A : Type u_1 inst✝² : NormedCommRing A inst✝¹ : NormedAlgebra 𝕜 A inst✝ : IsEmpty ι ⊢ ‖1‖ ≤ ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ [PROOFSTEP] convert ratio_le_op_norm (ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A) fun _ => (1 : A) [GOAL] case h.e'_3 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' A : Type u_1 inst✝² : NormedCommRing A inst✝¹ : NormedAlgebra 𝕜 A inst✝ : IsEmpty ι ⊢ ‖1‖ = ‖↑(ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A) fun x => 1‖ / ∏ i : ι, ‖1‖ [PROOFSTEP] simp [eq_empty_of_isEmpty (univ : Finset ι)] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' A : Type u_1 inst✝² : NormedCommRing A inst✝¹ : NormedAlgebra 𝕜 A inst✝ : NormOneClass A ⊢ ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ = 1 [PROOFSTEP] cases isEmpty_or_nonempty ι [GOAL] case inl 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' A : Type u_1 inst✝² : NormedCommRing A inst✝¹ : NormedAlgebra 𝕜 A inst✝ : NormOneClass A h✝ : IsEmpty ι ⊢ ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ = 1 [PROOFSTEP] simp [norm_mkPiAlgebra_of_empty] [GOAL] case inr 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' A : Type u_1 inst✝² : NormedCommRing A inst✝¹ : NormedAlgebra 𝕜 A inst✝ : NormOneClass A h✝ : Nonempty ι ⊢ ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ = 1 [PROOFSTEP] refine' le_antisymm norm_mkPiAlgebra_le _ [GOAL] case inr 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' A : Type u_1 inst✝² : NormedCommRing A inst✝¹ : NormedAlgebra 𝕜 A inst✝ : NormOneClass A h✝ : Nonempty ι ⊢ 1 ≤ ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ [PROOFSTEP] convert ratio_le_op_norm (ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A) fun _ => 1 [GOAL] case h.e'_3 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' A : Type u_1 inst✝² : NormedCommRing A inst✝¹ : NormedAlgebra 𝕜 A inst✝ : NormOneClass A h✝ : Nonempty ι ⊢ 1 = ‖↑(ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A) fun x => 1‖ / ∏ i : ι, ‖1‖ [PROOFSTEP] simp [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedSpace 𝕜 G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace 𝕜 G' A : Type u_1 inst✝¹ : NormedRing A inst✝ : NormedAlgebra 𝕜 A ⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 (Nat.succ n) A‖ ≤ 1 [PROOFSTEP] have := fun f => @op_norm_le_bound 𝕜 (Fin n.succ) (fun _ => A) A _ _ _ _ _ _ f _ zero_le_one [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedSpace 𝕜 G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace 𝕜 G' A : Type u_1 inst✝¹ : NormedRing A inst✝ : NormedAlgebra 𝕜 A this : ∀ (f : ContinuousMultilinearMap 𝕜 (fun x => A) A), (∀ (m : Fin (Nat.succ n) → A), ‖↑f m‖ ≤ 1 * ∏ i : Fin (Nat.succ n), ‖m i‖) → ‖f‖ ≤ 1 ⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 (Nat.succ n) A‖ ≤ 1 [PROOFSTEP] refine' this _ _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedSpace 𝕜 G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace 𝕜 G' A : Type u_1 inst✝¹ : NormedRing A inst✝ : NormedAlgebra 𝕜 A this : ∀ (f : ContinuousMultilinearMap 𝕜 (fun x => A) A), (∀ (m : Fin (Nat.succ n) → A), ‖↑f m‖ ≤ 1 * ∏ i : Fin (Nat.succ n), ‖m i‖) → ‖f‖ ≤ 1 ⊢ ∀ (m : Fin (Nat.succ n) → A), ‖↑(ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 (Nat.succ n) A) m‖ ≤ 1 * ∏ i : Fin (Nat.succ n), ‖m i‖ [PROOFSTEP] intro m [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedSpace 𝕜 G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace 𝕜 G' A : Type u_1 inst✝¹ : NormedRing A inst✝ : NormedAlgebra 𝕜 A this : ∀ (f : ContinuousMultilinearMap 𝕜 (fun x => A) A), (∀ (m : Fin (Nat.succ n) → A), ‖↑f m‖ ≤ 1 * ∏ i : Fin (Nat.succ n), ‖m i‖) → ‖f‖ ≤ 1 m : Fin (Nat.succ n) → A ⊢ ‖↑(ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 (Nat.succ n) A) m‖ ≤ 1 * ∏ i : Fin (Nat.succ n), ‖m i‖ [PROOFSTEP] simp only [ContinuousMultilinearMap.mkPiAlgebraFin_apply, one_mul, List.ofFn_eq_map, Fin.prod_univ_def, Multiset.coe_map, Multiset.coe_prod] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedSpace 𝕜 G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace 𝕜 G' A : Type u_1 inst✝¹ : NormedRing A inst✝ : NormedAlgebra 𝕜 A this : ∀ (f : ContinuousMultilinearMap 𝕜 (fun x => A) A), (∀ (m : Fin (Nat.succ n) → A), ‖↑f m‖ ≤ 1 * ∏ i : Fin (Nat.succ n), ‖m i‖) → ‖f‖ ≤ 1 m : Fin (Nat.succ n) → A ⊢ ‖List.prod (List.map m (List.finRange (Nat.succ n)))‖ ≤ List.prod (List.map (fun i => ‖m i‖) (List.finRange (Nat.succ n))) [PROOFSTEP] refine' (List.norm_prod_le' _).trans_eq _ [GOAL] case refine'_1 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedSpace 𝕜 G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace 𝕜 G' A : Type u_1 inst✝¹ : NormedRing A inst✝ : NormedAlgebra 𝕜 A this : ∀ (f : ContinuousMultilinearMap 𝕜 (fun x => A) A), (∀ (m : Fin (Nat.succ n) → A), ‖↑f m‖ ≤ 1 * ∏ i : Fin (Nat.succ n), ‖m i‖) → ‖f‖ ≤ 1 m : Fin (Nat.succ n) → A ⊢ List.map m (List.finRange (Nat.succ n)) ≠ [] [PROOFSTEP] rw [Ne.def, List.map_eq_nil, List.finRange_eq_nil] [GOAL] case refine'_1 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedSpace 𝕜 G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace 𝕜 G' A : Type u_1 inst✝¹ : NormedRing A inst✝ : NormedAlgebra 𝕜 A this : ∀ (f : ContinuousMultilinearMap 𝕜 (fun x => A) A), (∀ (m : Fin (Nat.succ n) → A), ‖↑f m‖ ≤ 1 * ∏ i : Fin (Nat.succ n), ‖m i‖) → ‖f‖ ≤ 1 m : Fin (Nat.succ n) → A ⊢ ¬Nat.succ n = 0 [PROOFSTEP] exact Nat.succ_ne_zero _ [GOAL] case refine'_2 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedSpace 𝕜 G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace 𝕜 G' A : Type u_1 inst✝¹ : NormedRing A inst✝ : NormedAlgebra 𝕜 A this : ∀ (f : ContinuousMultilinearMap 𝕜 (fun x => A) A), (∀ (m : Fin (Nat.succ n) → A), ‖↑f m‖ ≤ 1 * ∏ i : Fin (Nat.succ n), ‖m i‖) → ‖f‖ ≤ 1 m : Fin (Nat.succ n) → A ⊢ List.prod (List.map norm (List.map m (List.finRange (Nat.succ n)))) = List.prod (List.map (fun i => ‖m i‖) (List.finRange (Nat.succ n))) [PROOFSTEP] rw [List.map_map] [GOAL] case refine'_2 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedSpace 𝕜 G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace 𝕜 G' A : Type u_1 inst✝¹ : NormedRing A inst✝ : NormedAlgebra 𝕜 A this : ∀ (f : ContinuousMultilinearMap 𝕜 (fun x => A) A), (∀ (m : Fin (Nat.succ n) → A), ‖↑f m‖ ≤ 1 * ∏ i : Fin (Nat.succ n), ‖m i‖) → ‖f‖ ≤ 1 m : Fin (Nat.succ n) → A ⊢ List.prod (List.map (norm ∘ m) (List.finRange (Nat.succ n))) = List.prod (List.map (fun i => ‖m i‖) (List.finRange (Nat.succ n))) [PROOFSTEP] rfl [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedSpace 𝕜 G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace 𝕜 G' A : Type u_1 inst✝¹ : NormedRing A inst✝ : NormedAlgebra 𝕜 A hn : 0 < n ⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n A‖ ≤ 1 [PROOFSTEP] obtain ⟨n, rfl⟩ := Nat.exists_eq_succ_of_ne_zero hn.ne' [GOAL] case intro 𝕜 : Type u ι : Type v ι' : Type v' E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : NormedAddCommGroup G inst✝⁶ : NormedSpace 𝕜 G inst✝⁵ : NormedAddCommGroup G' inst✝⁴ : NormedSpace 𝕜 G' A : Type u_1 inst✝³ : NormedRing A inst✝² : NormedAlgebra 𝕜 A n : ℕ Ei : Fin (Nat.succ (Nat.succ n)) → Type wEi inst✝¹ : (i : Fin (Nat.succ (Nat.succ n))) → NormedAddCommGroup (Ei i) inst✝ : (i : Fin (Nat.succ (Nat.succ n))) → NormedSpace 𝕜 (Ei i) hn : 0 < Nat.succ n ⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 (Nat.succ n) A‖ ≤ 1 [PROOFSTEP] exact norm_mkPiAlgebraFin_succ_le [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedSpace 𝕜 G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace 𝕜 G' A : Type u_1 inst✝¹ : NormedRing A inst✝ : NormedAlgebra 𝕜 A ⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A‖ = ‖1‖ [PROOFSTEP] refine' le_antisymm _ _ [GOAL] case refine'_1 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedSpace 𝕜 G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace 𝕜 G' A : Type u_1 inst✝¹ : NormedRing A inst✝ : NormedAlgebra 𝕜 A ⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A‖ ≤ ‖1‖ [PROOFSTEP] have := fun f => @op_norm_le_bound 𝕜 (Fin 0) (fun _ => A) A _ _ _ _ _ _ f _ (norm_nonneg (1 : A)) [GOAL] case refine'_1 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedSpace 𝕜 G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace 𝕜 G' A : Type u_1 inst✝¹ : NormedRing A inst✝ : NormedAlgebra 𝕜 A this : ∀ (f : ContinuousMultilinearMap 𝕜 (fun x => A) A), (∀ (m : Fin 0 → A), ‖↑f m‖ ≤ ‖1‖ * ∏ i : Fin 0, ‖m i‖) → ‖f‖ ≤ ‖1‖ ⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A‖ ≤ ‖1‖ [PROOFSTEP] refine' this _ _ [GOAL] case refine'_1 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedSpace 𝕜 G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace 𝕜 G' A : Type u_1 inst✝¹ : NormedRing A inst✝ : NormedAlgebra 𝕜 A this : ∀ (f : ContinuousMultilinearMap 𝕜 (fun x => A) A), (∀ (m : Fin 0 → A), ‖↑f m‖ ≤ ‖1‖ * ∏ i : Fin 0, ‖m i‖) → ‖f‖ ≤ ‖1‖ ⊢ ∀ (m : Fin 0 → A), ‖↑(ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A) m‖ ≤ ‖1‖ * ∏ i : Fin 0, ‖m i‖ [PROOFSTEP] simp [GOAL] case refine'_2 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedSpace 𝕜 G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace 𝕜 G' A : Type u_1 inst✝¹ : NormedRing A inst✝ : NormedAlgebra 𝕜 A ⊢ ‖1‖ ≤ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A‖ [PROOFSTEP] convert ratio_le_op_norm (ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A) fun _ => (1 : A) [GOAL] case h.e'_3 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁶ : Fintype ι inst✝¹⁵ : Fintype ι' inst✝¹⁴ : NontriviallyNormedField 𝕜 inst✝¹³ : (i : ι) → NormedAddCommGroup (E i) inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedSpace 𝕜 G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace 𝕜 G' A : Type u_1 inst✝¹ : NormedRing A inst✝ : NormedAlgebra 𝕜 A ⊢ ‖1‖ = ‖↑(ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A) fun x => 1‖ / ∏ i : Fin 0, ‖1‖ [PROOFSTEP] simp [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedSpace 𝕜 G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace 𝕜 G' A : Type u_1 inst✝² : NormedRing A inst✝¹ : NormedAlgebra 𝕜 A inst✝ : NormOneClass A ⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n A‖ = 1 [PROOFSTEP] cases n [GOAL] case zero 𝕜 : Type u ι : Type v ι' : Type v' E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : NormedAddCommGroup G inst✝⁷ : NormedSpace 𝕜 G inst✝⁶ : NormedAddCommGroup G' inst✝⁵ : NormedSpace 𝕜 G' A : Type u_1 inst✝⁴ : NormedRing A inst✝³ : NormedAlgebra 𝕜 A inst✝² : NormOneClass A Ei : Fin (Nat.succ Nat.zero) → Type wEi inst✝¹ : (i : Fin (Nat.succ Nat.zero)) → NormedAddCommGroup (Ei i) inst✝ : (i : Fin (Nat.succ Nat.zero)) → NormedSpace 𝕜 (Ei i) ⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 Nat.zero A‖ = 1 [PROOFSTEP] rw [norm_mkPiAlgebraFin_zero] [GOAL] case zero 𝕜 : Type u ι : Type v ι' : Type v' E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : NormedAddCommGroup G inst✝⁷ : NormedSpace 𝕜 G inst✝⁶ : NormedAddCommGroup G' inst✝⁵ : NormedSpace 𝕜 G' A : Type u_1 inst✝⁴ : NormedRing A inst✝³ : NormedAlgebra 𝕜 A inst✝² : NormOneClass A Ei : Fin (Nat.succ Nat.zero) → Type wEi inst✝¹ : (i : Fin (Nat.succ Nat.zero)) → NormedAddCommGroup (Ei i) inst✝ : (i : Fin (Nat.succ Nat.zero)) → NormedSpace 𝕜 (Ei i) ⊢ ‖1‖ = 1 [PROOFSTEP] simp [GOAL] case succ 𝕜 : Type u ι : Type v ι' : Type v' E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : NormedAddCommGroup G inst✝⁷ : NormedSpace 𝕜 G inst✝⁶ : NormedAddCommGroup G' inst✝⁵ : NormedSpace 𝕜 G' A : Type u_1 inst✝⁴ : NormedRing A inst✝³ : NormedAlgebra 𝕜 A inst✝² : NormOneClass A n✝ : ℕ Ei : Fin (Nat.succ (Nat.succ n✝)) → Type wEi inst✝¹ : (i : Fin (Nat.succ (Nat.succ n✝))) → NormedAddCommGroup (Ei i) inst✝ : (i : Fin (Nat.succ (Nat.succ n✝))) → NormedSpace 𝕜 (Ei i) ⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 (Nat.succ n✝) A‖ = 1 [PROOFSTEP] refine' le_antisymm norm_mkPiAlgebraFin_succ_le _ [GOAL] case succ 𝕜 : Type u ι : Type v ι' : Type v' E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : NormedAddCommGroup G inst✝⁷ : NormedSpace 𝕜 G inst✝⁶ : NormedAddCommGroup G' inst✝⁵ : NormedSpace 𝕜 G' A : Type u_1 inst✝⁴ : NormedRing A inst✝³ : NormedAlgebra 𝕜 A inst✝² : NormOneClass A n✝ : ℕ Ei : Fin (Nat.succ (Nat.succ n✝)) → Type wEi inst✝¹ : (i : Fin (Nat.succ (Nat.succ n✝))) → NormedAddCommGroup (Ei i) inst✝ : (i : Fin (Nat.succ (Nat.succ n✝))) → NormedSpace 𝕜 (Ei i) ⊢ 1 ≤ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 (Nat.succ n✝) A‖ [PROOFSTEP] convert ratio_le_op_norm (ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 (Nat.succ _) A) fun _ => 1 [GOAL] case h.e'_3 𝕜 : Type u ι : Type v ι' : Type v' E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' G : Type wG G' : Type wG' inst✝¹⁷ : Fintype ι inst✝¹⁶ : Fintype ι' inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i) inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁸ : NormedAddCommGroup G inst✝⁷ : NormedSpace 𝕜 G inst✝⁶ : NormedAddCommGroup G' inst✝⁵ : NormedSpace 𝕜 G' A : Type u_1 inst✝⁴ : NormedRing A inst✝³ : NormedAlgebra 𝕜 A inst✝² : NormOneClass A n✝ : ℕ Ei : Fin (Nat.succ (Nat.succ n✝)) → Type wEi inst✝¹ : (i : Fin (Nat.succ (Nat.succ n✝))) → NormedAddCommGroup (Ei i) inst✝ : (i : Fin (Nat.succ (Nat.succ n✝))) → NormedSpace 𝕜 (Ei i) ⊢ 1 = ‖↑(ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 (Nat.succ n✝) A) fun x => 1‖ / ∏ i : Fin (Nat.succ n✝), ‖1‖ [PROOFSTEP] simp [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' z : G m : ι → 𝕜 ⊢ ‖↑(MultilinearMap.mkPiRing 𝕜 ι z) m‖ ≤ ‖z‖ * ∏ i : ι, ‖m i‖ [PROOFSTEP] simp only [MultilinearMap.mkPiRing_apply, norm_smul, norm_prod, mul_comm] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' z : G m : ι → 𝕜 ⊢ ‖z‖ * ∏ b : ι, ‖m b‖ ≤ ‖z‖ * ∏ i : ι, ‖m i‖ [PROOFSTEP] rfl [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' z : G ⊢ ‖z‖ ≤ ‖MultilinearMap.mkContinuous (MultilinearMap.mkPiRing 𝕜 ι z) ‖z‖ (_ : ∀ (m : ι → 𝕜), ‖↑(MultilinearMap.mkPiRing 𝕜 ι z) m‖ ≤ ‖z‖ * ∏ i : ι, ‖m i‖)‖ [PROOFSTEP] simpa using (ContinuousMultilinearMap.mkPiField 𝕜 ι z).le_op_norm fun _ => 1 [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' z₁ z₂ : G ⊢ ContinuousMultilinearMap.mkPiField 𝕜 ι z₁ = ContinuousMultilinearMap.mkPiField 𝕜 ι z₂ ↔ z₁ = z₂ [PROOFSTEP] rw [← toMultilinearMap_injective.eq_iff] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' z₁ z₂ : G ⊢ (ContinuousMultilinearMap.mkPiField 𝕜 ι z₁).toMultilinearMap = (ContinuousMultilinearMap.mkPiField 𝕜 ι z₂).toMultilinearMap ↔ z₁ = z₂ [PROOFSTEP] exact MultilinearMap.mkPiRing_eq_iff [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' ⊢ ContinuousMultilinearMap.mkPiField 𝕜 ι 0 = 0 [PROOFSTEP] ext [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' x✝ : ι → 𝕜 ⊢ ↑(ContinuousMultilinearMap.mkPiField 𝕜 ι 0) x✝ = ↑0 x✝ [PROOFSTEP] rw [mkPiField_apply, smul_zero, ContinuousMultilinearMap.zero_apply] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' z : G ⊢ ContinuousMultilinearMap.mkPiField 𝕜 ι z = 0 ↔ z = 0 [PROOFSTEP] rw [← mkPiField_zero, mkPiField_eq_iff] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' z z' : G ⊢ (fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) (z + z') = (fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) z + (fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) z' [PROOFSTEP] ext m [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' z z' : G m : ι → 𝕜 ⊢ ↑((fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) (z + z')) m = ↑((fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) z + (fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) z') m [PROOFSTEP] simp [smul_add] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 z : G ⊢ AddHom.toFun { toFun := fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z, map_add' := (_ : ∀ (z z' : G), (fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) (z + z') = (fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) z + (fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) z') } (c • z) = ↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z, map_add' := (_ : ∀ (z z' : G), (fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) (z + z') = (fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) z + (fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) z') } z [PROOFSTEP] ext m [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 z : G m : ι → 𝕜 ⊢ ↑(AddHom.toFun { toFun := fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z, map_add' := (_ : ∀ (z z' : G), (fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) (z + z') = (fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) z + (fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) z') } (c • z)) m = ↑(↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z, map_add' := (_ : ∀ (z z' : G), (fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) (z + z') = (fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) z + (fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) z') } z) m [PROOFSTEP] simp [smul_smul, mul_comm] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' z : G ⊢ (fun f => ↑f fun i => 1) (AddHom.toFun { toAddHom := { toFun := fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z, map_add' := (_ : ∀ (z z' : G), (fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) (z + z') = (fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) z + (fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) z') }, map_smul' := (_ : ∀ (c : 𝕜) (z : G), AddHom.toFun { toFun := fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z, map_add' := (_ : ∀ (z z' : G), (fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) (z + z') = (fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) z + (fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) z') } (c • z) = ↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z, map_add' := (_ : ∀ (z z' : G), (fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) (z + z') = (fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) z + (fun z => ContinuousMultilinearMap.mkPiField 𝕜 ι z) z') } z) }.toAddHom z) = z [PROOFSTEP] simp [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : G →L[𝕜] G' g₁ g₂ : ContinuousMultilinearMap 𝕜 E G ⊢ compContinuousMultilinearMap f (g₁ + g₂) = compContinuousMultilinearMap f g₁ + compContinuousMultilinearMap f g₂ [PROOFSTEP] ext1 [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : G →L[𝕜] G' g₁ g₂ : ContinuousMultilinearMap 𝕜 E G x✝ : (i : ι) → E i ⊢ ↑(compContinuousMultilinearMap f (g₁ + g₂)) x✝ = ↑(compContinuousMultilinearMap f g₁ + compContinuousMultilinearMap f g₂) x✝ [PROOFSTEP] apply f.map_add [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f : G →L[𝕜] G' g : ContinuousMultilinearMap 𝕜 E G ⊢ compContinuousMultilinearMap f (c • g) = c • compContinuousMultilinearMap f g [PROOFSTEP] ext1 [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f : G →L[𝕜] G' g : ContinuousMultilinearMap 𝕜 E G x✝ : (i : ι) → E i ⊢ ↑(compContinuousMultilinearMap f (c • g)) x✝ = ↑(c • compContinuousMultilinearMap f g) x✝ [PROOFSTEP] simp [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : G →L[𝕜] G' g : ContinuousMultilinearMap 𝕜 E G ⊢ ‖↑(↑(LinearMap.mk₂ 𝕜 compContinuousMultilinearMap (_ : ∀ (f₁ f₂ : G →L[𝕜] G') (g : ContinuousMultilinearMap 𝕜 E G), compContinuousMultilinearMap (f₁ + f₂) g = compContinuousMultilinearMap (f₁ + f₂) g) (_ : ∀ (c : 𝕜) (f : G →L[𝕜] G') (g : ContinuousMultilinearMap 𝕜 E G), compContinuousMultilinearMap (c • f) g = compContinuousMultilinearMap (c • f) g) (_ : ∀ (f : G →L[𝕜] G') (g₁ g₂ : ContinuousMultilinearMap 𝕜 E G), compContinuousMultilinearMap f (g₁ + g₂) = compContinuousMultilinearMap f g₁ + compContinuousMultilinearMap f g₂) (_ : ∀ (c : 𝕜) (f : G →L[𝕜] G') (g : ContinuousMultilinearMap 𝕜 E G), compContinuousMultilinearMap f (c • g) = c • compContinuousMultilinearMap f g)) f) g‖ ≤ 1 * ‖f‖ * ‖g‖ [PROOFSTEP] rw [one_mul] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : G →L[𝕜] G' g : ContinuousMultilinearMap 𝕜 E G ⊢ ‖↑(↑(LinearMap.mk₂ 𝕜 compContinuousMultilinearMap (_ : ∀ (f₁ f₂ : G →L[𝕜] G') (g : ContinuousMultilinearMap 𝕜 E G), compContinuousMultilinearMap (f₁ + f₂) g = compContinuousMultilinearMap (f₁ + f₂) g) (_ : ∀ (c : 𝕜) (f : G →L[𝕜] G') (g : ContinuousMultilinearMap 𝕜 E G), compContinuousMultilinearMap (c • f) g = compContinuousMultilinearMap (c • f) g) (_ : ∀ (f : G →L[𝕜] G') (g₁ g₂ : ContinuousMultilinearMap 𝕜 E G), compContinuousMultilinearMap f (g₁ + g₂) = compContinuousMultilinearMap f g₁ + compContinuousMultilinearMap f g₂) (_ : ∀ (c : 𝕜) (f : G →L[𝕜] G') (g : ContinuousMultilinearMap 𝕜 E G), compContinuousMultilinearMap f (c • g) = c • compContinuousMultilinearMap f g)) f) g‖ ≤ ‖f‖ * ‖g‖ [PROOFSTEP] exact f.norm_compContinuousMultilinearMap_le g [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' g : G ≃L[𝕜] G' src✝ : (fun x => ContinuousMultilinearMap 𝕜 E G →L[𝕜] ContinuousMultilinearMap 𝕜 E G') ↑g := ↑(compContinuousMultilinearMapL 𝕜 E G G') ↑g ⊢ Function.LeftInverse (↑(↑(compContinuousMultilinearMapL 𝕜 E G' G) ↑(ContinuousLinearEquiv.symm g))) src✝.toFun [PROOFSTEP] intro f [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' g : G ≃L[𝕜] G' src✝ : (fun x => ContinuousMultilinearMap 𝕜 E G →L[𝕜] ContinuousMultilinearMap 𝕜 E G') ↑g := ↑(compContinuousMultilinearMapL 𝕜 E G G') ↑g f : ContinuousMultilinearMap 𝕜 E G ⊢ ↑(↑(compContinuousMultilinearMapL 𝕜 E G' G) ↑(ContinuousLinearEquiv.symm g)) (AddHom.toFun src✝.toAddHom f) = f [PROOFSTEP] ext1 m [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' g : G ≃L[𝕜] G' src✝ : (fun x => ContinuousMultilinearMap 𝕜 E G →L[𝕜] ContinuousMultilinearMap 𝕜 E G') ↑g := ↑(compContinuousMultilinearMapL 𝕜 E G G') ↑g f : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i ⊢ ↑(↑(↑(compContinuousMultilinearMapL 𝕜 E G' G) ↑(ContinuousLinearEquiv.symm g)) (AddHom.toFun src✝.toAddHom f)) m = ↑f m [PROOFSTEP] simp [compContinuousMultilinearMapL] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' g : G ≃L[𝕜] G' src✝ : (fun x => ContinuousMultilinearMap 𝕜 E G →L[𝕜] ContinuousMultilinearMap 𝕜 E G') ↑g := ↑(compContinuousMultilinearMapL 𝕜 E G G') ↑g ⊢ Function.RightInverse (↑(↑(compContinuousMultilinearMapL 𝕜 E G' G) ↑(ContinuousLinearEquiv.symm g))) src✝.toFun [PROOFSTEP] intro f [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' g : G ≃L[𝕜] G' src✝ : (fun x => ContinuousMultilinearMap 𝕜 E G →L[𝕜] ContinuousMultilinearMap 𝕜 E G') ↑g := ↑(compContinuousMultilinearMapL 𝕜 E G G') ↑g f : ContinuousMultilinearMap 𝕜 E G' ⊢ AddHom.toFun src✝.toAddHom (↑(↑(compContinuousMultilinearMapL 𝕜 E G' G) ↑(ContinuousLinearEquiv.symm g)) f) = f [PROOFSTEP] ext1 m [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' g : G ≃L[𝕜] G' src✝ : (fun x => ContinuousMultilinearMap 𝕜 E G →L[𝕜] ContinuousMultilinearMap 𝕜 E G') ↑g := ↑(compContinuousMultilinearMapL 𝕜 E G G') ↑g f : ContinuousMultilinearMap 𝕜 E G' m : (i : ι) → E i ⊢ ↑(AddHom.toFun src✝.toAddHom (↑(↑(compContinuousMultilinearMapL 𝕜 E G' G) ↑(ContinuousLinearEquiv.symm g)) f)) m = ↑f m [PROOFSTEP] simp [compContinuousMultilinearMapL] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : G →L[𝕜] ContinuousMultilinearMap 𝕜 E G' m : (i : ι) → E i x y : G ⊢ (fun x => ↑(↑f x) m) (x + y) = (fun x => ↑(↑f x) m) x + (fun x => ↑(↑f x) m) y [PROOFSTEP] simp only [map_add, ContinuousMultilinearMap.add_apply] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : G →L[𝕜] ContinuousMultilinearMap 𝕜 E G' m : (i : ι) → E i c : 𝕜 x : G ⊢ AddHom.toFun { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) } (c • x) = ↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) } x [PROOFSTEP] simp only [ContinuousMultilinearMap.smul_apply, map_smul, RingHom.id_apply] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : G →L[𝕜] ContinuousMultilinearMap 𝕜 E G' m : (i : ι) → E i x : G ⊢ ‖↑{ toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } x‖ ≤ (‖f‖ * ∏ i : ι, ‖m i‖) * ‖x‖ [PROOFSTEP] rw [mul_right_comm] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : G →L[𝕜] ContinuousMultilinearMap 𝕜 E G' m : (i : ι) → E i x : G ⊢ ‖↑{ toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } x‖ ≤ ‖f‖ * ‖x‖ * ∏ i : ι, ‖m i‖ [PROOFSTEP] exact (f x).le_of_op_norm_le _ (f.le_op_norm x) [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : G →L[𝕜] ContinuousMultilinearMap 𝕜 E G' inst✝ : DecidableEq ι m : (i : ι) → E i i : ι x y : E i ⊢ (fun m => LinearMap.mkContinuous { toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } (‖f‖ * ∏ i : ι, ‖m i‖) (_ : ∀ (x : G), ‖↑{ toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } x‖ ≤ (‖f‖ * ∏ i : ι, ‖m i‖) * ‖x‖)) (Function.update m i (x + y)) = (fun m => LinearMap.mkContinuous { toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } (‖f‖ * ∏ i : ι, ‖m i‖) (_ : ∀ (x : G), ‖↑{ toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } x‖ ≤ (‖f‖ * ∏ i : ι, ‖m i‖) * ‖x‖)) (Function.update m i x) + (fun m => LinearMap.mkContinuous { toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } (‖f‖ * ∏ i : ι, ‖m i‖) (_ : ∀ (x : G), ‖↑{ toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } x‖ ≤ (‖f‖ * ∏ i : ι, ‖m i‖) * ‖x‖)) (Function.update m i y) [PROOFSTEP] ext1 [GOAL] case h 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : G →L[𝕜] ContinuousMultilinearMap 𝕜 E G' inst✝ : DecidableEq ι m : (i : ι) → E i i : ι x y : E i x✝ : G ⊢ ↑((fun m => LinearMap.mkContinuous { toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } (‖f‖ * ∏ i : ι, ‖m i‖) (_ : ∀ (x : G), ‖↑{ toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } x‖ ≤ (‖f‖ * ∏ i : ι, ‖m i‖) * ‖x‖)) (Function.update m i (x + y))) x✝ = ↑((fun m => LinearMap.mkContinuous { toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } (‖f‖ * ∏ i : ι, ‖m i‖) (_ : ∀ (x : G), ‖↑{ toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } x‖ ≤ (‖f‖ * ∏ i : ι, ‖m i‖) * ‖x‖)) (Function.update m i x) + (fun m => LinearMap.mkContinuous { toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } (‖f‖ * ∏ i : ι, ‖m i‖) (_ : ∀ (x : G), ‖↑{ toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } x‖ ≤ (‖f‖ * ∏ i : ι, ‖m i‖) * ‖x‖)) (Function.update m i y)) x✝ [PROOFSTEP] simp only [add_apply, ContinuousMultilinearMap.map_add, LinearMap.coe_mk, LinearMap.mkContinuous_apply, AddHom.coe_mk] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : G →L[𝕜] ContinuousMultilinearMap 𝕜 E G' inst✝ : DecidableEq ι m : (i : ι) → E i i : ι c : 𝕜 x : E i ⊢ (fun m => LinearMap.mkContinuous { toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } (‖f‖ * ∏ i : ι, ‖m i‖) (_ : ∀ (x : G), ‖↑{ toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } x‖ ≤ (‖f‖ * ∏ i : ι, ‖m i‖) * ‖x‖)) (Function.update m i (c • x)) = c • (fun m => LinearMap.mkContinuous { toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } (‖f‖ * ∏ i : ι, ‖m i‖) (_ : ∀ (x : G), ‖↑{ toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } x‖ ≤ (‖f‖ * ∏ i : ι, ‖m i‖) * ‖x‖)) (Function.update m i x) [PROOFSTEP] ext1 [GOAL] case h 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : G →L[𝕜] ContinuousMultilinearMap 𝕜 E G' inst✝ : DecidableEq ι m : (i : ι) → E i i : ι c : 𝕜 x : E i x✝ : G ⊢ ↑((fun m => LinearMap.mkContinuous { toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } (‖f‖ * ∏ i : ι, ‖m i‖) (_ : ∀ (x : G), ‖↑{ toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } x‖ ≤ (‖f‖ * ∏ i : ι, ‖m i‖) * ‖x‖)) (Function.update m i (c • x))) x✝ = ↑(c • (fun m => LinearMap.mkContinuous { toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } (‖f‖ * ∏ i : ι, ‖m i‖) (_ : ∀ (x : G), ‖↑{ toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } x‖ ≤ (‖f‖ * ∏ i : ι, ‖m i‖) * ‖x‖)) (Function.update m i x)) x✝ [PROOFSTEP] simp only [coe_smul', ContinuousMultilinearMap.map_smul, LinearMap.coe_mk, LinearMap.mkContinuous_apply, Pi.smul_apply, AddHom.coe_mk] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : G →L[𝕜] ContinuousMultilinearMap 𝕜 E G' m : (i : ι) → E i ⊢ ‖↑{ toFun := fun m => LinearMap.mkContinuous { toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } (‖f‖ * ∏ i : ι, ‖m i‖) (_ : ∀ (x : G), ‖↑{ toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } x‖ ≤ (‖f‖ * ∏ i : ι, ‖m i‖) * ‖x‖), map_add' := (_ : ∀ [inst : DecidableEq ι] (m : (i : ι) → E i) (i : ι) (x y : E i), (fun m => LinearMap.mkContinuous { toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } (‖f‖ * ∏ i : ι, ‖m i‖) (_ : ∀ (x : G), ‖↑{ toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } x‖ ≤ (‖f‖ * ∏ i : ι, ‖m i‖) * ‖x‖)) (Function.update m i (x + y)) = (fun m => LinearMap.mkContinuous { toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } (‖f‖ * ∏ i : ι, ‖m i‖) (_ : ∀ (x : G), ‖↑{ toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } x‖ ≤ (‖f‖ * ∏ i : ι, ‖m i‖) * ‖x‖)) (Function.update m i x) + (fun m => LinearMap.mkContinuous { toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } (‖f‖ * ∏ i : ι, ‖m i‖) (_ : ∀ (x : G), ‖↑{ toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } x‖ ≤ (‖f‖ * ∏ i : ι, ‖m i‖) * ‖x‖)) (Function.update m i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (m : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), (fun m => LinearMap.mkContinuous { toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } (‖f‖ * ∏ i : ι, ‖m i‖) (_ : ∀ (x : G), ‖↑{ toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } x‖ ≤ (‖f‖ * ∏ i : ι, ‖m i‖) * ‖x‖)) (Function.update m i (c • x)) = c • (fun m => LinearMap.mkContinuous { toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } (‖f‖ * ∏ i : ι, ‖m i‖) (_ : ∀ (x : G), ‖↑{ toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } x‖ ≤ (‖f‖ * ∏ i : ι, ‖m i‖) * ‖x‖)) (Function.update m i x)) } m‖ ≤ ‖f‖ * ∏ i : ι, ‖m i‖ [PROOFSTEP] dsimp only [MultilinearMap.coe_mk] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : G →L[𝕜] ContinuousMultilinearMap 𝕜 E G' m : (i : ι) → E i ⊢ ‖LinearMap.mkContinuous { toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } (‖f‖ * ∏ i : ι, ‖m i‖) (_ : ∀ (x : G), ‖↑{ toAddHom := { toFun := fun x => ↑(↑f x) m, map_add' := (_ : ∀ (x y : G), ↑(↑f (x + y)) m = ↑(↑f x) m + ↑(↑f y) m) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), ↑(↑f (c • x)) m = c • ↑(↑f x) m) } x‖ ≤ (‖f‖ * ∏ i : ι, ‖m i‖) * ‖x‖)‖ ≤ ‖f‖ * ∏ i : ι, ‖m i‖ [PROOFSTEP] refine LinearMap.mkContinuous_norm_le _ (mul_nonneg (norm_nonneg f) (prod_nonneg fun i _ => norm_nonneg (m i))) _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' g : G →ₗᵢ[𝕜] G' f : ContinuousMultilinearMap 𝕜 E G ⊢ ‖ContinuousLinearMap.compContinuousMultilinearMap (toContinuousLinearMap g) f‖ = ‖f‖ [PROOFSTEP] simp only [ContinuousLinearMap.compContinuousMultilinearMap_coe, LinearIsometry.coe_toContinuousLinearMap, LinearIsometry.norm_map, ContinuousMultilinearMap.norm_def, Function.comp_apply] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : G →ₗ[𝕜] MultilinearMap 𝕜 E G' C : ℝ H : ∀ (x : G) (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖ x y : G ⊢ (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) (x + y) = (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) x + (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) y [PROOFSTEP] ext1 [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : G →ₗ[𝕜] MultilinearMap 𝕜 E G' C : ℝ H : ∀ (x : G) (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖ x y : G x✝ : (i : ι) → E i ⊢ ↑((fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) (x + y)) x✝ = ↑((fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) x + (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) y) x✝ [PROOFSTEP] simp only [_root_.map_add] [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : G →ₗ[𝕜] MultilinearMap 𝕜 E G' C : ℝ H : ∀ (x : G) (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖ x y : G x✝ : (i : ι) → E i ⊢ ↑(mkContinuous (↑f x + ↑f y) (C * ‖x + y‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x + ↑f y) m‖ ≤ C * ‖x + y‖ * ∏ i : ι, ‖m i‖)) x✝ = ↑(mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖) + mkContinuous (↑f y) (C * ‖y‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f y) m‖ ≤ C * ‖y‖ * ∏ i : ι, ‖m i‖)) x✝ [PROOFSTEP] rfl [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : G →ₗ[𝕜] MultilinearMap 𝕜 E G' C : ℝ H : ∀ (x : G) (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖ c : 𝕜 x : G ⊢ AddHom.toFun { toFun := fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖), map_add' := (_ : ∀ (x y : G), (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) (x + y) = (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) x + (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) y) } (c • x) = ↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖), map_add' := (_ : ∀ (x y : G), (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) (x + y) = (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) x + (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) y) } x [PROOFSTEP] ext1 [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : G →ₗ[𝕜] MultilinearMap 𝕜 E G' C : ℝ H : ∀ (x : G) (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖ c : 𝕜 x : G x✝ : (i : ι) → E i ⊢ ↑(AddHom.toFun { toFun := fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖), map_add' := (_ : ∀ (x y : G), (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) (x + y) = (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) x + (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) y) } (c • x)) x✝ = ↑(↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖), map_add' := (_ : ∀ (x y : G), (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) (x + y) = (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) x + (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) y) } x) x✝ [PROOFSTEP] simp only [SMulHomClass.map_smul] [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : G →ₗ[𝕜] MultilinearMap 𝕜 E G' C : ℝ H : ∀ (x : G) (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖ c : 𝕜 x : G x✝ : (i : ι) → E i ⊢ ↑(mkContinuous (c • ↑f x) (C * ‖c • x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(c • ↑f x) m‖ ≤ C * ‖c • x‖ * ∏ i : ι, ‖m i‖)) x✝ = ↑(↑(RingHom.id 𝕜) c • mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) x✝ [PROOFSTEP] rfl [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : G →ₗ[𝕜] MultilinearMap 𝕜 E G' C : ℝ H : ∀ (x : G) (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖ x : G ⊢ ‖↑{ toAddHom := { toFun := fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖), map_add' := (_ : ∀ (x y : G), (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) (x + y) = (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) x + (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) y) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), AddHom.toFun { toFun := fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖), map_add' := (_ : ∀ (x y : G), (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) (x + y) = (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) x + (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) y) } (c • x) = ↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖), map_add' := (_ : ∀ (x y : G), (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) (x + y) = (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) x + (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) y) } x) } x‖ ≤ max C 0 * ‖x‖ [PROOFSTEP] rw [LinearMap.coe_mk, AddHom.coe_mk] -- Porting note: added [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : G →ₗ[𝕜] MultilinearMap 𝕜 E G' C : ℝ H : ∀ (x : G) (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖ x : G ⊢ ‖mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)‖ ≤ max C 0 * ‖x‖ [PROOFSTEP] exact ((f x).mkContinuous_norm_le' _).trans_eq <\| by rw [max_mul_of_nonneg _ _ (norm_nonneg x), zero_mul] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : G →ₗ[𝕜] MultilinearMap 𝕜 E G' C : ℝ H : ∀ (x : G) (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖ x : G ⊢ max (C * ‖x‖) 0 = max C 0 * ‖x‖ [PROOFSTEP] rw [max_mul_of_nonneg _ _ (norm_nonneg x), zero_mul] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : G →ₗ[𝕜] MultilinearMap 𝕜 E G' C : ℝ H : ∀ (x : G) (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖ ⊢ ‖mkContinuousLinear f C H‖ ≤ max C 0 [PROOFSTEP] dsimp only [mkContinuousLinear] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : G →ₗ[𝕜] MultilinearMap 𝕜 E G' C : ℝ H : ∀ (x : G) (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖ ⊢ ‖LinearMap.mkContinuous { toAddHom := { toFun := fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖), map_add' := (_ : ∀ (x y : G), (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) (x + y) = (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) x + (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) y) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), AddHom.toFun { toFun := fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖), map_add' := (_ : ∀ (x y : G), (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) (x + y) = (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) x + (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) y) } (c • x) = ↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖), map_add' := (_ : ∀ (x y : G), (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) (x + y) = (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) x + (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) y) } x) } (max C 0) (_ : ∀ (x : G), ‖↑{ toAddHom := { toFun := fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖), map_add' := (_ : ∀ (x y : G), (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) (x + y) = (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) x + (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) y) }, map_smul' := (_ : ∀ (c : 𝕜) (x : G), AddHom.toFun { toFun := fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖), map_add' := (_ : ∀ (x y : G), (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) (x + y) = (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) x + (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) y) } (c • x) = ↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖), map_add' := (_ : ∀ (x y : G), (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) (x + y) = (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) x + (fun x => mkContinuous (↑f x) (C * ‖x‖) (_ : ∀ (m : (i : ι) → E i), ‖↑(↑f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)) y) } x) } x‖ ≤ max C 0 * ‖x‖)‖ ≤ max C 0 [PROOFSTEP] exact LinearMap.mkContinuous_norm_le _ (le_max_right _ _) _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E (MultilinearMap 𝕜 E' G) C : ℝ H : ∀ (m₁ : (i : ι) → E i) (m₂ : (i : ι') → E' i), ‖↑(↑f m₁) m₂‖ ≤ (C * ∏ i : ι, ‖m₁ i‖) * ∏ i : ι', ‖m₂ i‖ inst✝ : DecidableEq ι m : (i : ι) → E i i : ι x y : E i ⊢ (fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖)) (Function.update m i (x + y)) = (fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖)) (Function.update m i x) + (fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖)) (Function.update m i y) [PROOFSTEP] ext1 [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E (MultilinearMap 𝕜 E' G) C : ℝ H : ∀ (m₁ : (i : ι) → E i) (m₂ : (i : ι') → E' i), ‖↑(↑f m₁) m₂‖ ≤ (C * ∏ i : ι, ‖m₁ i‖) * ∏ i : ι', ‖m₂ i‖ inst✝ : DecidableEq ι m : (i : ι) → E i i : ι x y : E i x✝ : (i : ι') → E' i ⊢ ↑((fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖)) (Function.update m i (x + y))) x✝ = ↑((fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖)) (Function.update m i x) + (fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖)) (Function.update m i y)) x✝ [PROOFSTEP] simp [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E (MultilinearMap 𝕜 E' G) C : ℝ H : ∀ (m₁ : (i : ι) → E i) (m₂ : (i : ι') → E' i), ‖↑(↑f m₁) m₂‖ ≤ (C * ∏ i : ι, ‖m₁ i‖) * ∏ i : ι', ‖m₂ i‖ inst✝ : DecidableEq ι m : (i : ι) → E i i : ι c : 𝕜 x : E i ⊢ (fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖)) (Function.update m i (c • x)) = c • (fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖)) (Function.update m i x) [PROOFSTEP] ext1 [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E (MultilinearMap 𝕜 E' G) C : ℝ H : ∀ (m₁ : (i : ι) → E i) (m₂ : (i : ι') → E' i), ‖↑(↑f m₁) m₂‖ ≤ (C * ∏ i : ι, ‖m₁ i‖) * ∏ i : ι', ‖m₂ i‖ inst✝ : DecidableEq ι m : (i : ι) → E i i : ι c : 𝕜 x : E i x✝ : (i : ι') → E' i ⊢ ↑((fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖)) (Function.update m i (c • x))) x✝ = ↑(c • (fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖)) (Function.update m i x)) x✝ [PROOFSTEP] simp [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E (MultilinearMap 𝕜 E' G) C : ℝ H : ∀ (m₁ : (i : ι) → E i) (m₂ : (i : ι') → E' i), ‖↑(↑f m₁) m₂‖ ≤ (C * ∏ i : ι, ‖m₁ i‖) * ∏ i : ι', ‖m₂ i‖ m : (i : ι) → E i ⊢ ‖↑{ toFun := fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖), map_add' := (_ : ∀ [inst : DecidableEq ι] (m : (i : ι) → E i) (i : ι) (x y : E i), (fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖)) (Function.update m i (x + y)) = (fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖)) (Function.update m i x) + (fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖)) (Function.update m i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (m : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), (fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖)) (Function.update m i (c • x)) = c • (fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖)) (Function.update m i x)) } m‖ ≤ max C 0 * ∏ i : ι, ‖m i‖ [PROOFSTEP] simp only [coe_mk] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E (MultilinearMap 𝕜 E' G) C : ℝ H : ∀ (m₁ : (i : ι) → E i) (m₂ : (i : ι') → E' i), ‖↑(↑f m₁) m₂‖ ≤ (C * ∏ i : ι, ‖m₁ i‖) * ∏ i : ι', ‖m₂ i‖ m : (i : ι) → E i ⊢ ‖mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖)‖ ≤ max C 0 * ∏ i : ι, ‖m i‖ [PROOFSTEP] refine ((f m).mkContinuous_norm_le' _).trans_eq ?_ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E (MultilinearMap 𝕜 E' G) C : ℝ H : ∀ (m₁ : (i : ι) → E i) (m₂ : (i : ι') → E' i), ‖↑(↑f m₁) m₂‖ ≤ (C * ∏ i : ι, ‖m₁ i‖) * ∏ i : ι', ‖m₂ i‖ m : (i : ι) → E i ⊢ max (C * ∏ i : ι, ‖m i‖) 0 = max C 0 * ∏ i : ι, ‖m i‖ [PROOFSTEP] rw [max_mul_of_nonneg, zero_mul] [GOAL] case hc 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E (MultilinearMap 𝕜 E' G) C : ℝ H : ∀ (m₁ : (i : ι) → E i) (m₂ : (i : ι') → E' i), ‖↑(↑f m₁) m₂‖ ≤ (C * ∏ i : ι, ‖m₁ i‖) * ∏ i : ι', ‖m₂ i‖ m : (i : ι) → E i ⊢ 0 ≤ ∏ i : ι, ‖m i‖ [PROOFSTEP] exact prod_nonneg fun _ _ => norm_nonneg _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E (MultilinearMap 𝕜 E' G) C : ℝ H : ∀ (m₁ : (i : ι) → E i) (m₂ : (i : ι') → E' i), ‖↑(↑f m₁) m₂‖ ≤ (C * ∏ i : ι, ‖m₁ i‖) * ∏ i : ι', ‖m₂ i‖ ⊢ ‖mkContinuousMultilinear f C H‖ ≤ max C 0 [PROOFSTEP] dsimp only [mkContinuousMultilinear] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : MultilinearMap 𝕜 E (MultilinearMap 𝕜 E' G) C : ℝ H : ∀ (m₁ : (i : ι) → E i) (m₂ : (i : ι') → E' i), ‖↑(↑f m₁) m₂‖ ≤ (C * ∏ i : ι, ‖m₁ i‖) * ∏ i : ι', ‖m₂ i‖ ⊢ ‖mkContinuous { toFun := fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖), map_add' := (_ : ∀ [inst : DecidableEq ι] (m : (i : ι) → E i) (i : ι) (x y : E i), (fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖)) (Function.update m i (x + y)) = (fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖)) (Function.update m i x) + (fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖)) (Function.update m i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (m : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), (fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖)) (Function.update m i (c • x)) = c • (fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖)) (Function.update m i x)) } (max C 0) (_ : ∀ (m : (i : ι) → E i), ‖↑{ toFun := fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖), map_add' := (_ : ∀ [inst : DecidableEq ι] (m : (i : ι) → E i) (i : ι) (x y : E i), (fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖)) (Function.update m i (x + y)) = (fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖)) (Function.update m i x) + (fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖)) (Function.update m i y)), map_smul' := (_ : ∀ [inst : DecidableEq ι] (m : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), (fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖)) (Function.update m i (c • x)) = c • (fun m => mkContinuous (↑f m) (C * ∏ i : ι, ‖m i‖) (_ : ∀ (m₂ : (i : ι') → E' i), ‖↑(↑f m) m₂‖ ≤ (C * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m₂ i‖)) (Function.update m i x)) } m‖ ≤ max C 0 * ∏ i : ι, ‖m i‖)‖ ≤ max C 0 [PROOFSTEP] exact mkContinuous_norm_le _ (le_max_right _ _) _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' g : ContinuousMultilinearMap 𝕜 E₁ G f : (i : ι) → E i →L[𝕜] E₁ i m : (i : ι) → E i ⊢ ‖g‖ * ∏ i : ι, ‖f i‖ * ‖m i‖ = (‖g‖ * ∏ i : ι, ‖f i‖) * ∏ i : ι, ‖m i‖ [PROOFSTEP] rw [prod_mul_distrib, mul_assoc] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' g : ContinuousMultilinearMap 𝕜 E₁ G f : (i : ι) → E i →ₗᵢ[𝕜] E₁ i ⊢ ‖compContinuousLinearMap g fun i => LinearIsometry.toContinuousLinearMap (f i)‖ ≤ ‖g‖ [PROOFSTEP] refine op_norm_le_bound _ (norm_nonneg _) fun m => ?_ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' g : ContinuousMultilinearMap 𝕜 E₁ G f : (i : ι) → E i →ₗᵢ[𝕜] E₁ i m : (i : ι) → E i ⊢ ‖↑(compContinuousLinearMap g fun i => LinearIsometry.toContinuousLinearMap (f i)) m‖ ≤ ‖g‖ * ∏ i : ι, ‖m i‖ [PROOFSTEP] apply (g.le_op_norm _).trans _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' g : ContinuousMultilinearMap 𝕜 E₁ G f : (i : ι) → E i →ₗᵢ[𝕜] E₁ i m : (i : ι) → E i ⊢ ‖g‖ * ∏ i : ι, ‖↑((fun i => ↑((fun i => LinearIsometry.toContinuousLinearMap (f i)) i)) i) (m i)‖ ≤ ‖g‖ * ∏ i : ι, ‖m i‖ [PROOFSTEP] simp only [ContinuousLinearMap.coe_coe, LinearIsometry.coe_toContinuousLinearMap, LinearIsometry.norm_map] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' g : ContinuousMultilinearMap 𝕜 E₁ G f : (i : ι) → E i →ₗᵢ[𝕜] E₁ i m : (i : ι) → E i ⊢ ‖g‖ * ∏ x : ι, ‖m x‖ ≤ ‖g‖ * ∏ i : ι, ‖m i‖ [PROOFSTEP] rfl [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' g : ContinuousMultilinearMap 𝕜 E₁ G f : (i : ι) → E i ≃ₗᵢ[𝕜] E₁ i ⊢ ‖compContinuousLinearMap g fun i => ↑(ContinuousLinearEquiv.mk (f i).toLinearEquiv)‖ = ‖g‖ [PROOFSTEP] apply le_antisymm (g.norm_compContinuous_linearIsometry_le fun i => (f i).toLinearIsometry) [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' g : ContinuousMultilinearMap 𝕜 E₁ G f : (i : ι) → E i ≃ₗᵢ[𝕜] E₁ i ⊢ ‖g‖ ≤ ‖compContinuousLinearMap g fun i => LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry (f i))‖ [PROOFSTEP] have : g = (g.compContinuousLinearMap fun i => (f i : E i →L[𝕜] E₁ i)).compContinuousLinearMap fun i => ((f i).symm : E₁ i →L[𝕜] E i) := by ext1 m simp only [compContinuousLinearMap_apply, LinearIsometryEquiv.coe_coe'', LinearIsometryEquiv.apply_symm_apply] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' g : ContinuousMultilinearMap 𝕜 E₁ G f : (i : ι) → E i ≃ₗᵢ[𝕜] E₁ i ⊢ g = compContinuousLinearMap (compContinuousLinearMap g fun i => ↑(ContinuousLinearEquiv.mk (f i).toLinearEquiv)) fun i => ↑(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm (f i)).toLinearEquiv) [PROOFSTEP] ext1 m [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' g : ContinuousMultilinearMap 𝕜 E₁ G f : (i : ι) → E i ≃ₗᵢ[𝕜] E₁ i m : (i : ι) → E₁ i ⊢ ↑g m = ↑(compContinuousLinearMap (compContinuousLinearMap g fun i => ↑(ContinuousLinearEquiv.mk (f i).toLinearEquiv)) fun i => ↑(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm (f i)).toLinearEquiv)) m [PROOFSTEP] simp only [compContinuousLinearMap_apply, LinearIsometryEquiv.coe_coe'', LinearIsometryEquiv.apply_symm_apply] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' g : ContinuousMultilinearMap 𝕜 E₁ G f : (i : ι) → E i ≃ₗᵢ[𝕜] E₁ i this : g = compContinuousLinearMap (compContinuousLinearMap g fun i => ↑(ContinuousLinearEquiv.mk (f i).toLinearEquiv)) fun i => ↑(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm (f i)).toLinearEquiv) ⊢ ‖g‖ ≤ ‖compContinuousLinearMap g fun i => LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry (f i))‖ [PROOFSTEP] conv_lhs => rw [this] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' g : ContinuousMultilinearMap 𝕜 E₁ G f : (i : ι) → E i ≃ₗᵢ[𝕜] E₁ i this : g = compContinuousLinearMap (compContinuousLinearMap g fun i => ↑(ContinuousLinearEquiv.mk (f i).toLinearEquiv)) fun i => ↑(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm (f i)).toLinearEquiv) \| ‖g‖ [PROOFSTEP] rw [this] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' g : ContinuousMultilinearMap 𝕜 E₁ G f : (i : ι) → E i ≃ₗᵢ[𝕜] E₁ i this : g = compContinuousLinearMap (compContinuousLinearMap g fun i => ↑(ContinuousLinearEquiv.mk (f i).toLinearEquiv)) fun i => ↑(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm (f i)).toLinearEquiv) \| ‖g‖ [PROOFSTEP] rw [this] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' g : ContinuousMultilinearMap 𝕜 E₁ G f : (i : ι) → E i ≃ₗᵢ[𝕜] E₁ i this : g = compContinuousLinearMap (compContinuousLinearMap g fun i => ↑(ContinuousLinearEquiv.mk (f i).toLinearEquiv)) fun i => ↑(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm (f i)).toLinearEquiv) \| ‖g‖ [PROOFSTEP] rw [this] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' g : ContinuousMultilinearMap 𝕜 E₁ G f : (i : ι) → E i ≃ₗᵢ[𝕜] E₁ i this : g = compContinuousLinearMap (compContinuousLinearMap g fun i => ↑(ContinuousLinearEquiv.mk (f i).toLinearEquiv)) fun i => ↑(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm (f i)).toLinearEquiv) ⊢ ‖compContinuousLinearMap (compContinuousLinearMap g fun i => ↑(ContinuousLinearEquiv.mk (f i).toLinearEquiv)) fun i => ↑(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm (f i)).toLinearEquiv)‖ ≤ ‖compContinuousLinearMap g fun i => LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry (f i))‖ [PROOFSTEP] apply (g.compContinuousLinearMap fun i => (f i : E i →L[𝕜] E₁ i)).norm_compContinuous_linearIsometry_le fun i => (f i).symm.toLinearIsometry [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : (i : ι) → E i ≃L[𝕜] E₁ i src✝ : ContinuousMultilinearMap 𝕜 (fun i => E₁ i) G →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E i) G := compContinuousLinearMapL fun i => ↑(f i) ⊢ Function.LeftInverse (↑(compContinuousLinearMapL fun i => ↑(ContinuousLinearEquiv.symm (f i)))) src✝.toFun [PROOFSTEP] intro g [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : (i : ι) → E i ≃L[𝕜] E₁ i src✝ : ContinuousMultilinearMap 𝕜 (fun i => E₁ i) G →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E i) G := compContinuousLinearMapL fun i => ↑(f i) g : ContinuousMultilinearMap 𝕜 E₁ G ⊢ ↑(compContinuousLinearMapL fun i => ↑(ContinuousLinearEquiv.symm (f i))) (AddHom.toFun src✝.toAddHom g) = g [PROOFSTEP] ext1 m [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : (i : ι) → E i ≃L[𝕜] E₁ i src✝ : ContinuousMultilinearMap 𝕜 (fun i => E₁ i) G →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E i) G := compContinuousLinearMapL fun i => ↑(f i) g : ContinuousMultilinearMap 𝕜 E₁ G m : (i : ι) → E₁ i ⊢ ↑(↑(compContinuousLinearMapL fun i => ↑(ContinuousLinearEquiv.symm (f i))) (AddHom.toFun src✝.toAddHom g)) m = ↑g m [PROOFSTEP] simp only [LinearMap.toFun_eq_coe, ContinuousLinearMap.coe_coe, compContinuousLinearMapL_apply, compContinuousLinearMap_apply, ContinuousLinearEquiv.coe_coe, ContinuousLinearEquiv.apply_symm_apply] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : (i : ι) → E i ≃L[𝕜] E₁ i src✝ : ContinuousMultilinearMap 𝕜 (fun i => E₁ i) G →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E i) G := compContinuousLinearMapL fun i => ↑(f i) ⊢ Function.RightInverse (↑(compContinuousLinearMapL fun i => ↑(ContinuousLinearEquiv.symm (f i)))) src✝.toFun [PROOFSTEP] intro g [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : (i : ι) → E i ≃L[𝕜] E₁ i src✝ : ContinuousMultilinearMap 𝕜 (fun i => E₁ i) G →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E i) G := compContinuousLinearMapL fun i => ↑(f i) g : ContinuousMultilinearMap 𝕜 E G ⊢ AddHom.toFun src✝.toAddHom (↑(compContinuousLinearMapL fun i => ↑(ContinuousLinearEquiv.symm (f i))) g) = g [PROOFSTEP] ext1 m [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : (i : ι) → E i ≃L[𝕜] E₁ i src✝ : ContinuousMultilinearMap 𝕜 (fun i => E₁ i) G →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E i) G := compContinuousLinearMapL fun i => ↑(f i) g : ContinuousMultilinearMap 𝕜 E G m : (i : ι) → E i ⊢ ↑(AddHom.toFun src✝.toAddHom (↑(compContinuousLinearMapL fun i => ↑(ContinuousLinearEquiv.symm (f i))) g)) m = ↑g m [PROOFSTEP] simp only [compContinuousLinearMapL_apply, LinearMap.toFun_eq_coe, ContinuousLinearMap.coe_coe, compContinuousLinearMap_apply, ContinuousLinearEquiv.coe_coe, ContinuousLinearEquiv.symm_apply_apply] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G m : (i : Fin (Nat.succ n)) → Ei i ⊢ ‖f‖ * ‖m 0‖ * ∏ i : Fin n, ‖tail m i‖ = ‖f‖ * (‖m 0‖ * ∏ i : Fin n, ‖tail m i‖) [PROOFSTEP] ring [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G m : (i : Fin (Nat.succ n)) → Ei i ⊢ ‖f‖ * (‖m 0‖ * ∏ i : Fin n, ‖tail m i‖) = ‖f‖ * ∏ i : Fin (Nat.succ n), ‖m i‖ [PROOFSTEP] rw [prod_univ_succ] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G m : (i : Fin (Nat.succ n)) → Ei i ⊢ ‖f‖ * (‖m 0‖ * ∏ i : Fin n, ‖tail m i‖) = ‖f‖ * (‖m 0‖ * ∏ i : Fin n, ‖m (succ i)‖) [PROOFSTEP] rfl [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G) m : (i : Fin (Nat.succ n)) → Ei i ⊢ ‖f‖ * ((∏ i : Fin n, ‖init m i‖) * ‖m (last n)‖) = ‖f‖ * ∏ i : Fin (Nat.succ n), ‖m i‖ [PROOFSTEP] rw [prod_univ_castSucc] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G) m : (i : Fin (Nat.succ n)) → Ei i ⊢ ‖f‖ * ((∏ i : Fin n, ‖init m i‖) * ‖m (last n)‖) = ‖f‖ * ((∏ i : Fin n, ‖m (castSucc i)‖) * ‖m (last n)‖) [PROOFSTEP] rfl [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 Ei G x : Ei 0 m : (i : Fin n) → Ei (succ i) ⊢ ‖f‖ * ∏ i : Fin (n + 1), ‖Fin.cons x m i‖ = ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖ [PROOFSTEP] rw [prod_univ_succ] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 Ei G x : Ei 0 m : (i : Fin n) → Ei (succ i) ⊢ ‖f‖ * (‖Fin.cons x m 0‖ * ∏ i : Fin n, ‖Fin.cons x m (succ i)‖) = ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖ [PROOFSTEP] simp [mul_assoc] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 Ei G m : (i : Fin n) → Ei (castSucc i) x : Ei (last n) ⊢ ‖f‖ * ∏ i : Fin (n + 1), ‖snoc m x i‖ = (‖f‖ * ∏ i : Fin n, ‖m i‖) * ‖x‖ [PROOFSTEP] rw [prod_univ_castSucc] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 Ei G m : (i : Fin n) → Ei (castSucc i) x : Ei (last n) ⊢ ‖f‖ * ((∏ i : Fin n, ‖snoc m x (castSucc i)‖) * ‖snoc m x (last n)‖) = (‖f‖ * ∏ i : Fin n, ‖m i‖) * ‖x‖ [PROOFSTEP] simp [mul_assoc] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G m : (i : Fin (Nat.succ n)) → Ei i ⊢ ‖↑(LinearMap.uncurryLeft (LinearMap.comp toMultilinearMapLinear ↑f)) m‖ ≤ ‖f‖ * ∏ i : Fin (Nat.succ n), ‖m i‖ [PROOFSTEP] exact ContinuousLinearMap.norm_map_tail_le f m [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 Ei G x y : Ei 0 ⊢ (fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) (x + y) = (fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) x + (fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) y [PROOFSTEP] ext m [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 Ei G x y : Ei 0 m : (i : Fin n) → Ei (succ i) ⊢ ↑((fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) (x + y)) m = ↑((fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) x + (fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) y) m [PROOFSTEP] exact f.cons_add m x y [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 Ei G c : 𝕜 x : Ei 0 ⊢ AddHom.toFun { toFun := fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖), map_add' := (_ : ∀ (x y : Ei 0), (fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) (x + y) = (fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) x + (fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) y) } (c • x) = ↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖), map_add' := (_ : ∀ (x y : Ei 0), (fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) (x + y) = (fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) x + (fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) y) } x [PROOFSTEP] ext m [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 Ei G c : 𝕜 x : Ei 0 m : (i : Fin n) → Ei (succ i) ⊢ ↑(AddHom.toFun { toFun := fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖), map_add' := (_ : ∀ (x y : Ei 0), (fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) (x + y) = (fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) x + (fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) y) } (c • x)) m = ↑(↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖), map_add' := (_ : ∀ (x y : Ei 0), (fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) (x + y) = (fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) x + (fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) y) } x) m [PROOFSTEP] exact f.cons_smul m c x [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 Ei G x : Ei 0 ⊢ ‖↑{ toAddHom := { toFun := fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖), map_add' := (_ : ∀ (x y : Ei 0), (fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) (x + y) = (fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) x + (fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) y) }, map_smul' := (_ : ∀ (c : 𝕜) (x : Ei 0), AddHom.toFun { toFun := fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖), map_add' := (_ : ∀ (x y : Ei 0), (fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) (x + y) = (fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) x + (fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) y) } (c • x) = ↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖), map_add' := (_ : ∀ (x y : Ei 0), (fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) (x + y) = (fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) x + (fun x => MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)) y) } x) } x‖ ≤ ‖f‖ * ‖x‖ [PROOFSTEP] rw [LinearMap.coe_mk, AddHom.coe_mk] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 Ei G x : Ei 0 ⊢ ‖MultilinearMap.mkContinuous (↑(MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖) (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖↑f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖)‖ ≤ ‖f‖ * ‖x‖ [PROOFSTEP] exact MultilinearMap.mkContinuous_norm_le _ (mul_nonneg (norm_nonneg _) (norm_nonneg _)) _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G ⊢ curryLeft (uncurryLeft f) = f [PROOFSTEP] ext m x [GOAL] case h.H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G m : Ei 0 x : (i : Fin n) → Ei (succ i) ⊢ ↑(↑(curryLeft (uncurryLeft f)) m) x = ↑(↑f m) x [PROOFSTEP] simp only [tail_cons, ContinuousLinearMap.uncurryLeft_apply, ContinuousMultilinearMap.curryLeft_apply] [GOAL] case h.H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G m : Ei 0 x : (i : Fin n) → Ei (succ i) ⊢ ↑(↑f (Fin.cons m x 0)) x = ↑(↑f m) x [PROOFSTEP] rw [cons_zero] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f₁ f₂ : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G ⊢ ContinuousLinearMap.uncurryLeft (f₁ + f₂) = ContinuousLinearMap.uncurryLeft f₁ + ContinuousLinearMap.uncurryLeft f₂ [PROOFSTEP] ext m [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f₁ f₂ : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G m : (i : Fin (Nat.succ n)) → Ei i ⊢ ↑(ContinuousLinearMap.uncurryLeft (f₁ + f₂)) m = ↑(ContinuousLinearMap.uncurryLeft f₁ + ContinuousLinearMap.uncurryLeft f₂) m [PROOFSTEP] rfl [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G ⊢ AddHom.toFun { toFun := ContinuousLinearMap.uncurryLeft, map_add' := (_ : ∀ (f₁ f₂ : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G), ContinuousLinearMap.uncurryLeft (f₁ + f₂) = ContinuousLinearMap.uncurryLeft f₁ + ContinuousLinearMap.uncurryLeft f₂) } (c • f) = ↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := ContinuousLinearMap.uncurryLeft, map_add' := (_ : ∀ (f₁ f₂ : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G), ContinuousLinearMap.uncurryLeft (f₁ + f₂) = ContinuousLinearMap.uncurryLeft f₁ + ContinuousLinearMap.uncurryLeft f₂) } f [PROOFSTEP] ext m [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G m : (i : Fin (Nat.succ n)) → Ei i ⊢ ↑(AddHom.toFun { toFun := ContinuousLinearMap.uncurryLeft, map_add' := (_ : ∀ (f₁ f₂ : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G), ContinuousLinearMap.uncurryLeft (f₁ + f₂) = ContinuousLinearMap.uncurryLeft f₁ + ContinuousLinearMap.uncurryLeft f₂) } (c • f)) m = ↑(↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := ContinuousLinearMap.uncurryLeft, map_add' := (_ : ∀ (f₁ f₂ : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G), ContinuousLinearMap.uncurryLeft (f₁ + f₂) = ContinuousLinearMap.uncurryLeft f₁ + ContinuousLinearMap.uncurryLeft f₂) } f) m [PROOFSTEP] rfl [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G ⊢ ‖↑{ toLinearMap := { toAddHom := { toFun := ContinuousLinearMap.uncurryLeft, map_add' := (_ : ∀ (f₁ f₂ : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G), ContinuousLinearMap.uncurryLeft (f₁ + f₂) = ContinuousLinearMap.uncurryLeft f₁ + ContinuousLinearMap.uncurryLeft f₂) }, map_smul' := (_ : ∀ (c : 𝕜) (f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G), AddHom.toFun { toFun := ContinuousLinearMap.uncurryLeft, map_add' := (_ : ∀ (f₁ f₂ : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G), ContinuousLinearMap.uncurryLeft (f₁ + f₂) = ContinuousLinearMap.uncurryLeft f₁ + ContinuousLinearMap.uncurryLeft f₂) } (c • f) = ↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := ContinuousLinearMap.uncurryLeft, map_add' := (_ : ∀ (f₁ f₂ : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G), ContinuousLinearMap.uncurryLeft (f₁ + f₂) = ContinuousLinearMap.uncurryLeft f₁ + ContinuousLinearMap.uncurryLeft f₂) } f) }, invFun := curryLeft, left_inv := (_ : ∀ (f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G), curryLeft (ContinuousLinearMap.uncurryLeft f) = f), right_inv := (_ : ∀ (f : ContinuousMultilinearMap 𝕜 Ei G), ContinuousLinearMap.uncurryLeft (curryLeft f) = f) } f‖ ≤ ‖f‖ [PROOFSTEP] simp only [LinearEquiv.coe_mk] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G ⊢ ‖ContinuousLinearMap.uncurryLeft f‖ ≤ ‖f‖ [PROOFSTEP] exact MultilinearMap.mkContinuous_norm_le _ (norm_nonneg f) _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 Ei G ⊢ ‖↑(LinearEquiv.symm { toLinearMap := { toAddHom := { toFun := ContinuousLinearMap.uncurryLeft, map_add' := (_ : ∀ (f₁ f₂ : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G), ContinuousLinearMap.uncurryLeft (f₁ + f₂) = ContinuousLinearMap.uncurryLeft f₁ + ContinuousLinearMap.uncurryLeft f₂) }, map_smul' := (_ : ∀ (c : 𝕜) (f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G), AddHom.toFun { toFun := ContinuousLinearMap.uncurryLeft, map_add' := (_ : ∀ (f₁ f₂ : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G), ContinuousLinearMap.uncurryLeft (f₁ + f₂) = ContinuousLinearMap.uncurryLeft f₁ + ContinuousLinearMap.uncurryLeft f₂) } (c • f) = ↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := ContinuousLinearMap.uncurryLeft, map_add' := (_ : ∀ (f₁ f₂ : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G), ContinuousLinearMap.uncurryLeft (f₁ + f₂) = ContinuousLinearMap.uncurryLeft f₁ + ContinuousLinearMap.uncurryLeft f₂) } f) }, invFun := curryLeft, left_inv := (_ : ∀ (f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei (succ i)) G), curryLeft (ContinuousLinearMap.uncurryLeft f) = f), right_inv := (_ : ∀ (f : ContinuousMultilinearMap 𝕜 Ei G), ContinuousLinearMap.uncurryLeft (curryLeft f) = f) }) f‖ ≤ ‖f‖ [PROOFSTEP] simp only [LinearEquiv.coe_symm_mk] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 Ei G ⊢ ‖curryLeft f‖ ≤ ‖f‖ [PROOFSTEP] exact LinearMap.mkContinuous_norm_le _ (norm_nonneg f) _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G) inst✝ : DecidableEq (Fin n) m : (i : Fin n) → Ei (castSucc i) i : Fin n x y : Ei (castSucc i) ⊢ (fun m => ↑(↑f m)) (update m i (x + y)) = (fun m => ↑(↑f m)) (update m i x) + (fun m => ↑(↑f m)) (update m i y) [PROOFSTEP] simp [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G) inst✝ : DecidableEq (Fin n) m : (i : Fin n) → Ei (castSucc i) i : Fin n c : 𝕜 x : Ei (castSucc i) ⊢ (fun m => ↑(↑f m)) (update m i (c • x)) = c • (fun m => ↑(↑f m)) (update m i x) [PROOFSTEP] simp [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 Ei G inst✝ : DecidableEq (Fin n) m : (i : Fin n) → Ei (castSucc i) i : Fin n x y : Ei (castSucc i) ⊢ (fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ i : Fin n, ‖m i‖) (_ : ∀ (x : Ei (last n)), ‖↑f (snoc m x)‖ ≤ (‖f‖ * ∏ i : Fin n, ‖m i‖) * ‖x‖)) (update m i (x + y)) = (fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ i : Fin n, ‖m i‖) (_ : ∀ (x : Ei (last n)), ‖↑f (snoc m x)‖ ≤ (‖f‖ * ∏ i : Fin n, ‖m i‖) * ‖x‖)) (update m i x) + (fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ i : Fin n, ‖m i‖) (_ : ∀ (x : Ei (last n)), ‖↑f (snoc m x)‖ ≤ (‖f‖ * ∏ i : Fin n, ‖m i‖) * ‖x‖)) (update m i y) [PROOFSTEP] simp [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 Ei G inst✝ : DecidableEq (Fin n) m : (i : Fin n) → Ei (castSucc i) i : Fin n x y : Ei (castSucc i) ⊢ LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) (update m i x) + ↑(MultilinearMap.curryRight f.toMultilinearMap) (update m i y)) (‖f‖ * ∏ x_1 : Fin n, ‖update m i (x + y) x_1‖) (_ : ∀ (x_1 : Ei (last n)), ‖↑(↑(MultilinearMap.curryRight f.toMultilinearMap) (update m i x) + ↑(MultilinearMap.curryRight f.toMultilinearMap) (update m i y)) x_1‖ ≤ (‖f‖ * ∏ x_2 : Fin n, ‖update m i (x + y) x_2‖) * ‖x_1‖) = LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) (update m i x)) (‖f‖ * ∏ i_1 : Fin n, ‖update m i x i_1‖) (_ : ∀ (x_1 : Ei (last n)), ‖↑f (snoc (update m i x) x_1)‖ ≤ (‖f‖ * ∏ i_1 : Fin n, ‖update m i x i_1‖) * ‖x_1‖) + LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) (update m i y)) (‖f‖ * ∏ i_1 : Fin n, ‖update m i y i_1‖) (_ : ∀ (x : Ei (last n)), ‖↑f (snoc (update m i y) x)‖ ≤ (‖f‖ * ∏ i_1 : Fin n, ‖update m i y i_1‖) * ‖x‖) [PROOFSTEP] rfl [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 Ei G inst✝ : DecidableEq (Fin n) m : (i : Fin n) → Ei (castSucc i) i : Fin n c : 𝕜 x : Ei (castSucc i) ⊢ (fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ i : Fin n, ‖m i‖) (_ : ∀ (x : Ei (last n)), ‖↑f (snoc m x)‖ ≤ (‖f‖ * ∏ i : Fin n, ‖m i‖) * ‖x‖)) (update m i (c • x)) = c • (fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ i : Fin n, ‖m i‖) (_ : ∀ (x : Ei (last n)), ‖↑f (snoc m x)‖ ≤ (‖f‖ * ∏ i : Fin n, ‖m i‖) * ‖x‖)) (update m i x) [PROOFSTEP] simp [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁵ : Fintype ι inst✝¹⁴ : Fintype ι' inst✝¹³ : NontriviallyNormedField 𝕜 inst✝¹² : (i : ι) → NormedAddCommGroup (E i) inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i) inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedSpace 𝕜 G inst✝² : NormedAddCommGroup G' inst✝¹ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 Ei G inst✝ : DecidableEq (Fin n) m : (i : Fin n) → Ei (castSucc i) i : Fin n c : 𝕜 x : Ei (castSucc i) ⊢ LinearMap.mkContinuous (c • ↑(MultilinearMap.curryRight f.toMultilinearMap) (update m i x)) (‖f‖ * ∏ x_1 : Fin n, ‖update m i (c • x) x_1‖) (_ : ∀ (x_1 : Ei (last n)), ‖↑(c • ↑(MultilinearMap.curryRight f.toMultilinearMap) (update m i x)) x_1‖ ≤ (‖f‖ * ∏ x_2 : Fin n, ‖update m i (c • x) x_2‖) * ‖x_1‖) = c • LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) (update m i x)) (‖f‖ * ∏ i_1 : Fin n, ‖update m i x i_1‖) (_ : ∀ (x_1 : Ei (last n)), ‖↑f (snoc (update m i x) x_1)‖ ≤ (‖f‖ * ∏ i_1 : Fin n, ‖update m i x i_1‖) * ‖x_1‖) [PROOFSTEP] rfl [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 Ei G f' : MultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G) := { toFun := fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ i : Fin n, ‖m i‖) (_ : ∀ (x : Ei (last n)), ‖↑f (snoc m x)‖ ≤ (‖f‖ * ∏ i : Fin n, ‖m i‖) * ‖x‖), map_add' := (_ : ∀ [inst : DecidableEq (Fin n)] (m : (i : Fin n) → Ei (castSucc i)) (i : Fin n) (x y : Ei (castSucc i)), (fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ i : Fin n, ‖m i‖) (_ : ∀ (x : Ei (last n)), ‖↑f (snoc m x)‖ ≤ (‖f‖ * ∏ i : Fin n, ‖m i‖) * ‖x‖)) (update m i (x + y)) = (fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ i : Fin n, ‖m i‖) (_ : ∀ (x : Ei (last n)), ‖↑f (snoc m x)‖ ≤ (‖f‖ * ∏ i : Fin n, ‖m i‖) * ‖x‖)) (update m i x) + (fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ i : Fin n, ‖m i‖) (_ : ∀ (x : Ei (last n)), ‖↑f (snoc m x)‖ ≤ (‖f‖ * ∏ i : Fin n, ‖m i‖) * ‖x‖)) (update m i y)), map_smul' := (_ : ∀ [inst : DecidableEq (Fin n)] (m : (i : Fin n) → Ei (castSucc i)) (i : Fin n) (c : 𝕜) (x : Ei (castSucc i)), (fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ i : Fin n, ‖m i‖) (_ : ∀ (x : Ei (last n)), ‖↑f (snoc m x)‖ ≤ (‖f‖ * ∏ i : Fin n, ‖m i‖) * ‖x‖)) (update m i (c • x)) = c • (fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ i : Fin n, ‖m i‖) (_ : ∀ (x : Ei (last n)), ‖↑f (snoc m x)‖ ≤ (‖f‖ * ∏ i : Fin n, ‖m i‖) * ‖x‖)) (update m i x)) } m : (i : Fin n) → Ei (castSucc i) ⊢ ‖↑f' m‖ ≤ ‖f‖ * ∏ i : Fin n, ‖m i‖ [PROOFSTEP] simp only [MultilinearMap.coe_mk] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 Ei G f' : MultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G) := { toFun := fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ i : Fin n, ‖m i‖) (_ : ∀ (x : Ei (last n)), ‖↑f (snoc m x)‖ ≤ (‖f‖ * ∏ i : Fin n, ‖m i‖) * ‖x‖), map_add' := (_ : ∀ [inst : DecidableEq (Fin n)] (m : (i : Fin n) → Ei (castSucc i)) (i : Fin n) (x y : Ei (castSucc i)), (fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ i : Fin n, ‖m i‖) (_ : ∀ (x : Ei (last n)), ‖↑f (snoc m x)‖ ≤ (‖f‖ * ∏ i : Fin n, ‖m i‖) * ‖x‖)) (update m i (x + y)) = (fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ i : Fin n, ‖m i‖) (_ : ∀ (x : Ei (last n)), ‖↑f (snoc m x)‖ ≤ (‖f‖ * ∏ i : Fin n, ‖m i‖) * ‖x‖)) (update m i x) + (fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ i : Fin n, ‖m i‖) (_ : ∀ (x : Ei (last n)), ‖↑f (snoc m x)‖ ≤ (‖f‖ * ∏ i : Fin n, ‖m i‖) * ‖x‖)) (update m i y)), map_smul' := (_ : ∀ [inst : DecidableEq (Fin n)] (m : (i : Fin n) → Ei (castSucc i)) (i : Fin n) (c : 𝕜) (x : Ei (castSucc i)), (fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ i : Fin n, ‖m i‖) (_ : ∀ (x : Ei (last n)), ‖↑f (snoc m x)‖ ≤ (‖f‖ * ∏ i : Fin n, ‖m i‖) * ‖x‖)) (update m i (c • x)) = c • (fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ i : Fin n, ‖m i‖) (_ : ∀ (x : Ei (last n)), ‖↑f (snoc m x)‖ ≤ (‖f‖ * ∏ i : Fin n, ‖m i‖) * ‖x‖)) (update m i x)) } m : (i : Fin n) → Ei (castSucc i) ⊢ ‖LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ i : Fin n, ‖m i‖) (_ : ∀ (x : Ei (last n)), ‖↑f (snoc m x)‖ ≤ (‖f‖ * ∏ i : Fin n, ‖m i‖) * ‖x‖)‖ ≤ ‖f‖ * ∏ i : Fin n, ‖m i‖ [PROOFSTEP] exact LinearMap.mkContinuous_norm_le _ (mul_nonneg (norm_nonneg _) (prod_nonneg fun _ _ => norm_nonneg _)) _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G) ⊢ curryRight (uncurryRight f) = f [PROOFSTEP] ext m x [GOAL] case H.h 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G) m : (i : Fin n) → Ei (castSucc i) x : Ei (last n) ⊢ ↑(↑(curryRight (uncurryRight f)) m) x = ↑(↑f m) x [PROOFSTEP] simp only [snoc_last, ContinuousMultilinearMap.curryRight_apply, ContinuousMultilinearMap.uncurryRight_apply] [GOAL] case H.h 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G) m : (i : Fin n) → Ei (castSucc i) x : Ei (last n) ⊢ ↑(↑f (init (snoc m x))) x = ↑(↑f m) x [PROOFSTEP] rw [init_snoc] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 Ei G ⊢ uncurryRight (curryRight f) = f [PROOFSTEP] ext m [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 Ei G m : (i : Fin (Nat.succ n)) → Ei i ⊢ ↑(uncurryRight (curryRight f)) m = ↑f m [PROOFSTEP] simp [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f₁ f₂ : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G) ⊢ uncurryRight (f₁ + f₂) = uncurryRight f₁ + uncurryRight f₂ [PROOFSTEP] ext m [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f₁ f₂ : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G) m : (i : Fin (Nat.succ n)) → Ei i ⊢ ↑(uncurryRight (f₁ + f₂)) m = ↑(uncurryRight f₁ + uncurryRight f₂) m [PROOFSTEP] rfl [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G) ⊢ AddHom.toFun { toFun := uncurryRight, map_add' := (_ : ∀ (f₁ f₂ : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)), uncurryRight (f₁ + f₂) = uncurryRight f₁ + uncurryRight f₂) } (c • f) = ↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := uncurryRight, map_add' := (_ : ∀ (f₁ f₂ : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)), uncurryRight (f₁ + f₂) = uncurryRight f₁ + uncurryRight f₂) } f [PROOFSTEP] ext m [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G) m : (i : Fin (Nat.succ n)) → Ei i ⊢ ↑(AddHom.toFun { toFun := uncurryRight, map_add' := (_ : ∀ (f₁ f₂ : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)), uncurryRight (f₁ + f₂) = uncurryRight f₁ + uncurryRight f₂) } (c • f)) m = ↑(↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := uncurryRight, map_add' := (_ : ∀ (f₁ f₂ : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)), uncurryRight (f₁ + f₂) = uncurryRight f₁ + uncurryRight f₂) } f) m [PROOFSTEP] rfl [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G) ⊢ ‖↑{ toLinearMap := { toAddHom := { toFun := uncurryRight, map_add' := (_ : ∀ (f₁ f₂ : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)), uncurryRight (f₁ + f₂) = uncurryRight f₁ + uncurryRight f₂) }, map_smul' := (_ : ∀ (c : 𝕜) (f : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)), AddHom.toFun { toFun := uncurryRight, map_add' := (_ : ∀ (f₁ f₂ : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)), uncurryRight (f₁ + f₂) = uncurryRight f₁ + uncurryRight f₂) } (c • f) = ↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := uncurryRight, map_add' := (_ : ∀ (f₁ f₂ : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)), uncurryRight (f₁ + f₂) = uncurryRight f₁ + uncurryRight f₂) } f) }, invFun := curryRight, left_inv := (_ : ∀ (f : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)), curryRight (uncurryRight f) = f), right_inv := (_ : ∀ (f : ContinuousMultilinearMap 𝕜 Ei G), uncurryRight (curryRight f) = f) } f‖ ≤ ‖f‖ [PROOFSTEP] simp only [uncurryRight, LinearEquiv.coe_mk] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G) ⊢ ‖MultilinearMap.mkContinuous (MultilinearMap.uncurryRight { toFun := fun m => ↑(↑f m), map_add' := (_ : ∀ [inst : DecidableEq (Fin n)] (m : (i : Fin n) → Ei (castSucc i)) (i : Fin n) (x y : Ei (castSucc i)), ↑(↑f (update m i (x + y))) = ↑(↑f (update m i x)) + ↑(↑f (update m i y))), map_smul' := (_ : ∀ [inst : DecidableEq (Fin n)] (m : (i : Fin n) → Ei (castSucc i)) (i : Fin n) (c : 𝕜) (x : Ei (castSucc i)), ↑(↑f (update m i (c • x))) = c • ↑(↑f (update m i x))) }) ‖f‖ (_ : ∀ (m : (i : Fin (Nat.succ n)) → Ei i), ‖↑(↑f (init m)) (m (last n))‖ ≤ ‖f‖ * ∏ i : Fin (Nat.succ n), ‖m i‖)‖ ≤ ‖f‖ [PROOFSTEP] exact MultilinearMap.mkContinuous_norm_le _ (norm_nonneg f) _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 Ei G ⊢ ‖↑(LinearEquiv.symm { toLinearMap := { toAddHom := { toFun := uncurryRight, map_add' := (_ : ∀ (f₁ f₂ : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)), uncurryRight (f₁ + f₂) = uncurryRight f₁ + uncurryRight f₂) }, map_smul' := (_ : ∀ (c : 𝕜) (f : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)), AddHom.toFun { toFun := uncurryRight, map_add' := (_ : ∀ (f₁ f₂ : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)), uncurryRight (f₁ + f₂) = uncurryRight f₁ + uncurryRight f₂) } (c • f) = ↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := uncurryRight, map_add' := (_ : ∀ (f₁ f₂ : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)), uncurryRight (f₁ + f₂) = uncurryRight f₁ + uncurryRight f₂) } f) }, invFun := curryRight, left_inv := (_ : ∀ (f : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)), curryRight (uncurryRight f) = f), right_inv := (_ : ∀ (f : ContinuousMultilinearMap 𝕜 Ei G), uncurryRight (curryRight f) = f) }) f‖ ≤ ‖f‖ [PROOFSTEP] simp only [curryRight, LinearEquiv.coe_symm_mk] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 Ei G ⊢ ‖MultilinearMap.mkContinuous { toFun := fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ x : Fin n, ‖m x‖) (_ : ∀ (x : Ei (last n)), ‖↑(↑(MultilinearMap.curryRight f.toMultilinearMap) m) x‖ ≤ (‖f‖ * ∏ x : Fin n, ‖m x‖) * ‖x‖), map_add' := (_ : ∀ [inst : DecidableEq (Fin n)] (m : (i : Fin n) → (fun i => Ei (castSucc i)) i) (i : Fin n) (x y : (fun i => Ei (castSucc i)) i), (fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ x : Fin n, ‖m x‖) (_ : ∀ (x : Ei (last n)), ‖↑(↑(MultilinearMap.curryRight f.toMultilinearMap) m) x‖ ≤ (‖f‖ * ∏ x : Fin n, ‖m x‖) * ‖x‖)) (update m i (x + y)) = (fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ x : Fin n, ‖m x‖) (_ : ∀ (x : Ei (last n)), ‖↑(↑(MultilinearMap.curryRight f.toMultilinearMap) m) x‖ ≤ (‖f‖ * ∏ x : Fin n, ‖m x‖) * ‖x‖)) (update m i x) + (fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ x : Fin n, ‖m x‖) (_ : ∀ (x : Ei (last n)), ‖↑(↑(MultilinearMap.curryRight f.toMultilinearMap) m) x‖ ≤ (‖f‖ * ∏ x : Fin n, ‖m x‖) * ‖x‖)) (update m i y)), map_smul' := (_ : ∀ [inst : DecidableEq (Fin n)] (m : (i : Fin n) → (fun i => Ei (castSucc i)) i) (i : Fin n) (c : 𝕜) (x : (fun i => Ei (castSucc i)) i), (fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ x : Fin n, ‖m x‖) (_ : ∀ (x : Ei (last n)), ‖↑(↑(MultilinearMap.curryRight f.toMultilinearMap) m) x‖ ≤ (‖f‖ * ∏ x : Fin n, ‖m x‖) * ‖x‖)) (update m i (c • x)) = c • (fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ x : Fin n, ‖m x‖) (_ : ∀ (x : Ei (last n)), ‖↑(↑(MultilinearMap.curryRight f.toMultilinearMap) m) x‖ ≤ (‖f‖ * ∏ x : Fin n, ‖m x‖) * ‖x‖)) (update m i x)) } ‖f‖ (_ : ∀ (m : (i : Fin n) → (fun i => Ei (castSucc i)) i), ‖↑{ toFun := fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ x : Fin n, ‖m x‖) (_ : ∀ (x : Ei (last n)), ‖↑(↑(MultilinearMap.curryRight f.toMultilinearMap) m) x‖ ≤ (‖f‖ * ∏ x : Fin n, ‖m x‖) * ‖x‖), map_add' := (_ : ∀ [inst : DecidableEq (Fin n)] (m : (i : Fin n) → (fun i => Ei (castSucc i)) i) (i : Fin n) (x y : (fun i => Ei (castSucc i)) i), (fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ x : Fin n, ‖m x‖) (_ : ∀ (x : Ei (last n)), ‖↑(↑(MultilinearMap.curryRight f.toMultilinearMap) m) x‖ ≤ (‖f‖ * ∏ x : Fin n, ‖m x‖) * ‖x‖)) (update m i (x + y)) = (fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ x : Fin n, ‖m x‖) (_ : ∀ (x : Ei (last n)), ‖↑(↑(MultilinearMap.curryRight f.toMultilinearMap) m) x‖ ≤ (‖f‖ * ∏ x : Fin n, ‖m x‖) * ‖x‖)) (update m i x) + (fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ x : Fin n, ‖m x‖) (_ : ∀ (x : Ei (last n)), ‖↑(↑(MultilinearMap.curryRight f.toMultilinearMap) m) x‖ ≤ (‖f‖ * ∏ x : Fin n, ‖m x‖) * ‖x‖)) (update m i y)), map_smul' := (_ : ∀ [inst : DecidableEq (Fin n)] (m : (i : Fin n) → (fun i => Ei (castSucc i)) i) (i : Fin n) (c : 𝕜) (x : (fun i => Ei (castSucc i)) i), (fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ x : Fin n, ‖m x‖) (_ : ∀ (x : Ei (last n)), ‖↑(↑(MultilinearMap.curryRight f.toMultilinearMap) m) x‖ ≤ (‖f‖ * ∏ x : Fin n, ‖m x‖) * ‖x‖)) (update m i (c • x)) = c • (fun m => LinearMap.mkContinuous (↑(MultilinearMap.curryRight f.toMultilinearMap) m) (‖f‖ * ∏ x : Fin n, ‖m x‖) (_ : ∀ (x : Ei (last n)), ‖↑(↑(MultilinearMap.curryRight f.toMultilinearMap) m) x‖ ≤ (‖f‖ * ∏ x : Fin n, ‖m x‖) * ‖x‖)) (update m i x)) } m‖ ≤ ‖f‖ * ∏ i : Fin n, ‖m i‖)‖ ≤ ‖f‖ [PROOFSTEP] exact MultilinearMap.mkContinuous_norm_le _ (norm_nonneg f) _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun i => G) G' x : Fin 0 → G ⊢ curry0 𝕜 G (↑f x) = f [PROOFSTEP] ext m [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun i => G) G' x m : Fin 0 → G ⊢ ↑(curry0 𝕜 G (↑f x)) m = ↑f m [PROOFSTEP] simp [(Subsingleton.elim _ _ : x = m)] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun i => G) G' ⊢ curry0 𝕜 G (uncurry0 f) = f [PROOFSTEP] simp [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun i => G) G' x : Fin 0 → G ⊢ ‖↑f x‖ = ‖f‖ [PROOFSTEP] obtain rfl : x = 0 := Subsingleton.elim _ _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun i => G) G' ⊢ ‖↑f 0‖ = ‖f‖ [PROOFSTEP] refine' le_antisymm (by simpa using f.le_op_norm 0) _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun i => G) G' ⊢ ‖↑f 0‖ ≤ ‖f‖ [PROOFSTEP] simpa using f.le_op_norm 0 [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun i => G) G' ⊢ ‖f‖ ≤ ‖↑f 0‖ [PROOFSTEP] have : ‖ContinuousMultilinearMap.curry0 𝕜 G f.uncurry0‖ ≤ ‖f.uncurry0‖ := ContinuousMultilinearMap.op_norm_le_bound _ (norm_nonneg _) fun m => by simp [-ContinuousMultilinearMap.apply_zero_curry0] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun i => G) G' m : Fin 0 → G ⊢ ‖↑(curry0 𝕜 G (uncurry0 f)) m‖ ≤ ‖uncurry0 f‖ * ∏ i : Fin 0, ‖m i‖ [PROOFSTEP] simp [-ContinuousMultilinearMap.apply_zero_curry0] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun i => G) G' this : ‖curry0 𝕜 G (uncurry0 f)‖ ≤ ‖uncurry0 f‖ ⊢ ‖f‖ ≤ ‖↑f 0‖ [PROOFSTEP] simpa [-Matrix.zero_empty] using this [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun i => G) G' ⊢ ‖uncurry0 f‖ = ‖f‖ [PROOFSTEP] simp [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' σ : ι ≃ ι' f : ContinuousMultilinearMap 𝕜 (fun x => G) G' ⊢ ‖domDomCongr σ f‖ = ‖f‖ [PROOFSTEP] simp only [norm_def, LinearEquiv.coe_mk, ← σ.prod_comp, (σ.arrowCongr (Equiv.refl G)).surjective.forall, domDomCongr_apply, Equiv.arrowCongr_apply, Equiv.coe_refl, comp.left_id, comp_apply, Equiv.symm_apply_apply, id] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun x => G) G' m : ι → G m' : ι' → G ⊢ ‖↑(↑(MultilinearMap.currySum f.toMultilinearMap) m) m'‖ ≤ (‖f‖ * ∏ i : ι, ‖m i‖) * ∏ i : ι', ‖m' i‖ [PROOFSTEP] simpa [Fintype.prod_sum_type, mul_assoc] using f.le_op_norm (Sum.elim m m') [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun x => G) (ContinuousMultilinearMap 𝕜 (fun x => G) G') m : ι ⊕ ι' → G ⊢ ‖↑(MultilinearMap.uncurrySum (LinearMap.compMultilinearMap toMultilinearMapLinear f.toMultilinearMap)) m‖ ≤ ‖f‖ * ∏ i : ι ⊕ ι', ‖m i‖ [PROOFSTEP] simpa [Fintype.prod_sum_type, mul_assoc] using (f (m ∘ Sum.inl)).le_of_op_norm_le (m ∘ Sum.inr) (f.le_op_norm _) [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f g : ContinuousMultilinearMap 𝕜 (fun x => G) G' ⊢ currySum (f + g) = currySum f + currySum g [PROOFSTEP] ext [GOAL] case H.H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f g : ContinuousMultilinearMap 𝕜 (fun x => G) G' x✝¹ : ι → G x✝ : ι' → G ⊢ ↑(↑(currySum (f + g)) x✝¹) x✝ = ↑(↑(currySum f + currySum g) x✝¹) x✝ [PROOFSTEP] rfl [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f : ContinuousMultilinearMap 𝕜 (fun x => G) G' ⊢ AddHom.toFun { toFun := currySum, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 (fun x => G) G'), currySum (f + g) = currySum f + currySum g) } (c • f) = ↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := currySum, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 (fun x => G) G'), currySum (f + g) = currySum f + currySum g) } f [PROOFSTEP] ext [GOAL] case H.H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' c : 𝕜 f : ContinuousMultilinearMap 𝕜 (fun x => G) G' x✝¹ : ι → G x✝ : ι' → G ⊢ ↑(↑(AddHom.toFun { toFun := currySum, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 (fun x => G) G'), currySum (f + g) = currySum f + currySum g) } (c • f)) x✝¹) x✝ = ↑(↑(↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := currySum, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 (fun x => G) G'), currySum (f + g) = currySum f + currySum g) } f) x✝¹) x✝ [PROOFSTEP] rfl [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun x => G) G' ⊢ uncurrySum (AddHom.toFun { toAddHom := { toFun := currySum, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 (fun x => G) G'), currySum (f + g) = currySum f + currySum g) }, map_smul' := (_ : ∀ (c : 𝕜) (f : ContinuousMultilinearMap 𝕜 (fun x => G) G'), AddHom.toFun { toFun := currySum, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 (fun x => G) G'), currySum (f + g) = currySum f + currySum g) } (c • f) = ↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := currySum, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 (fun x => G) G'), currySum (f + g) = currySum f + currySum g) } f) }.toAddHom f) = f [PROOFSTEP] ext m [GOAL] case H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun x => G) G' m : ι ⊕ ι' → G ⊢ ↑(uncurrySum (AddHom.toFun { toAddHom := { toFun := currySum, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 (fun x => G) G'), currySum (f + g) = currySum f + currySum g) }, map_smul' := (_ : ∀ (c : 𝕜) (f : ContinuousMultilinearMap 𝕜 (fun x => G) G'), AddHom.toFun { toFun := currySum, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 (fun x => G) G'), currySum (f + g) = currySum f + currySum g) } (c • f) = ↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := currySum, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 (fun x => G) G'), currySum (f + g) = currySum f + currySum g) } f) }.toAddHom f)) m = ↑f m [PROOFSTEP] exact congr_arg f (Sum.elim_comp_inl_inr m) [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun x => G) (ContinuousMultilinearMap 𝕜 (fun x => G) G') ⊢ AddHom.toFun { toAddHom := { toFun := currySum, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 (fun x => G) G'), currySum (f + g) = currySum f + currySum g) }, map_smul' := (_ : ∀ (c : 𝕜) (f : ContinuousMultilinearMap 𝕜 (fun x => G) G'), AddHom.toFun { toFun := currySum, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 (fun x => G) G'), currySum (f + g) = currySum f + currySum g) } (c • f) = ↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := currySum, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 (fun x => G) G'), currySum (f + g) = currySum f + currySum g) } f) }.toAddHom (uncurrySum f) = f [PROOFSTEP] ext m₁ m₂ [GOAL] case H.H 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun x => G) (ContinuousMultilinearMap 𝕜 (fun x => G) G') m₁ : ι → G m₂ : ι' → G ⊢ ↑(↑(AddHom.toFun { toAddHom := { toFun := currySum, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 (fun x => G) G'), currySum (f + g) = currySum f + currySum g) }, map_smul' := (_ : ∀ (c : 𝕜) (f : ContinuousMultilinearMap 𝕜 (fun x => G) G'), AddHom.toFun { toFun := currySum, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 (fun x => G) G'), currySum (f + g) = currySum f + currySum g) } (c • f) = ↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := currySum, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 (fun x => G) G'), currySum (f + g) = currySum f + currySum g) } f) }.toAddHom (uncurrySum f)) m₁) m₂ = ↑(↑f m₁) m₂ [PROOFSTEP] rfl [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun x => G) (ContinuousMultilinearMap 𝕜 (fun x => G) G') ⊢ ‖↑(LinearEquiv.symm { toLinearMap := { toAddHom := { toFun := currySum, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 (fun x => G) G'), currySum (f + g) = currySum f + currySum g) }, map_smul' := (_ : ∀ (c : 𝕜) (f : ContinuousMultilinearMap 𝕜 (fun x => G) G'), AddHom.toFun { toFun := currySum, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 (fun x => G) G'), currySum (f + g) = currySum f + currySum g) } (c • f) = ↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := currySum, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 (fun x => G) G'), currySum (f + g) = currySum f + currySum g) } f) }, invFun := uncurrySum, left_inv := (_ : ∀ (f : ContinuousMultilinearMap 𝕜 (fun x => G) G'), uncurrySum (AddHom.toFun { toAddHom := { toFun := currySum, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 (fun x => G) G'), currySum (f + g) = currySum f + currySum g) }, map_smul' := (_ : ∀ (c : 𝕜) (f : ContinuousMultilinearMap 𝕜 (fun x => G) G'), AddHom.toFun { toFun := currySum, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 (fun x => G) G'), currySum (f + g) = currySum f + currySum g) } (c • f) = ↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := currySum, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 (fun x => G) G'), currySum (f + g) = currySum f + currySum g) } f) }.toAddHom f) = f), right_inv := (_ : ∀ (f : ContinuousMultilinearMap 𝕜 (fun x => G) (ContinuousMultilinearMap 𝕜 (fun x => G) G')), AddHom.toFun { toAddHom := { toFun := currySum, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 (fun x => G) G'), currySum (f + g) = currySum f + currySum g) }, map_smul' := (_ : ∀ (c : 𝕜) (f : ContinuousMultilinearMap 𝕜 (fun x => G) G'), AddHom.toFun { toFun := currySum, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 (fun x => G) G'), currySum (f + g) = currySum f + currySum g) } (c • f) = ↑(RingHom.id 𝕜) c • AddHom.toFun { toFun := currySum, map_add' := (_ : ∀ (f g : ContinuousMultilinearMap 𝕜 (fun x => G) G'), currySum (f + g) = currySum f + currySum g) } f) }.toAddHom (uncurrySum f) = f) }) f‖ ≤ ‖f‖ [PROOFSTEP] simp only [LinearEquiv.coe_symm_mk] [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' f : ContinuousMultilinearMap 𝕜 (fun x => G) (ContinuousMultilinearMap 𝕜 (fun x => G) G') ⊢ ‖uncurrySum f‖ ≤ ‖f‖ [PROOFSTEP] exact MultilinearMap.mkContinuous_norm_le _ (norm_nonneg f) _ [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' k l : ℕ s : Finset (Fin n) hk : card s = k hl : card sᶜ = l f : ContinuousMultilinearMap 𝕜 (fun i => G) G' x y : G ⊢ (↑(↑(↑(curryFinFinset 𝕜 G G' hk hl) f) fun x_1 => x) fun x => y) = ↑f (piecewise s (fun x_1 => x) fun x => y) [PROOFSTEP] refine' (curryFinFinset_symm_apply_piecewise_const hk hl _ _ _).symm.trans _ -- `rw` fails [GOAL] 𝕜 : Type u ι : Type v ι' : Type v' n : ℕ E : ι → Type wE E₁ : ι → Type wE₁ E' : ι' → Type wE' Ei : Fin (Nat.succ n) → Type wEi G : Type wG G' : Type wG' inst✝¹⁴ : Fintype ι inst✝¹³ : Fintype ι' inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i) inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i) inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i) inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i) inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i) inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i) inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i) inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i) inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G inst✝¹ : NormedAddCommGroup G' inst✝ : NormedSpace 𝕜 G' k l : ℕ s : Finset (Fin n) hk : card s = k hl : card sᶜ = l f : ContinuousMultilinearMap 𝕜 (fun i => G) G' x y : G ⊢ ↑(↑(LinearIsometryEquiv.symm (curryFinFinset 𝕜 G G' hk hl)) (↑(curryFinFinset 𝕜 G G' hk hl) f)) (piecewise s (fun x_1 => x) fun x => y) = ↑f (piecewise s (fun x_1 => x) fun x => y) [PROOFSTEP] rw [LinearIsometryEquiv.symm_apply_apply]
I recently went on a trip with my youngest daughter to Atlanta, Georgia. I had not been to Georgia before. I used to live in Alabama and Florida (at different times), so I figured it would be mostly the same. What I didn't know was that Atlanta is Gotham City. You may not have known this, and I'm sorry if I'm ruining some fantasy-other-wordliness aspect of Gotham City, but....it's in Georgia. Our adventure began when we got the rental car and headed out in the safe hands of GPS (aka evil maniacal devil guide) to our hotel. I don't know when I decided Siri was a reliable source again. She has burned me way too many times in the past for me to have any sane reason to trust her, but here I was, across the country in the middle of a rainy night, trusting my life to the voice in my phone with a desire to kill me. How do I know she wants to kill me? Let's recap. I have gotten lost more times than not when using her. Usually it is because she decides to tell me to take U turns in the middle of freeways or busy roads. Other times it is because she leads me to my destination which actually turns out to be a lonely spot in the middle of nowhere, perfect for murder. This time, it was to the back side of a prison. Yes, just pull into the guarded driveway...don't mind the guns..."your destination is on the right"....sure it is. I didn't fall for it. I had a feeling that no matter what Siri said, I should probably head towards the taller, better lighted buildings in the distance. By some miracle, we ended up at our hotel. I prefer to believe it was devine guidance. On the way, we discovered we were in Gotham. It wasn't hard to tell, all we had to do was look up. The buildings were somewhat shrouded in fog. Did I mention it was raining? Constantly? The entire weekend? The fog swirled around the tops of the buildings. One building in particular had lights shining off the top, cutting through the fog, just waiting for the Bat signal to be blasted into the sky. It was Gotham. I'm sure of it. If that wasn't enough proof, the sirens that screamed all night were the clincher. I didn't actually SEE Batman, but I'm pretty sure I saw something dark and caped streak past my 9th floor hotel window in the middle of the night. The rest of our trip went fairly well. We went to a Art walk/block party, very much NOT like the art walks here in Arizona. I expected to see cool art that made my mind scream, "How did they think of that!?" like what happens at the art walks here. Instead, what my mind was screaming was more of a, "WHY did they think of that? What is wrong with them!?" My daughter assured me it was all about the experience, not the product. That had to be the only resoning behind the live music in the "Mammal Bar" which turned out to be a very warm, over crowded room (bar), with a group of adult size toddlers playing on xylophones, recorders, and bongos....with no apparent melody in common. My brain hurt and I wanted to curl up with something familiar, like a cactus. It was painful. All in all, I totally enjoyed my trip with my daughter. She is an awesome person. I'd want to be her friend if we weren't related. She's totally cool. She is also blessed/cursed with the same talent as I have. We both see celebrity faces in random people. On this trip we saw young DiCaprio, Ice T's uncle, A white Bruno Mars, and the perfect blend of Bradley Cooper and Will Arnett. I admit to following the latter around with my eyes for quite a while and even snapping a picture. Is that weird? It's not like I'm going to stalk him or google his picture online and find out his name and whether or not that girl with him was his girlfriend or send him bunnies or anything, it was purely for research. Have never asked Siri anything but have of course used a GPS well Tim has I have never even tried to punch in an address in one. They are great except when they get you lost or send you down a one way street the wrong way and that has happened a few time to me and Tim.
[STATEMENT] lemma codegree_normalized_altY: assumes "y \<in> Y" and "y' \<in> Y" shows "codegree_normalized y y' X = card (neighbors_ss y X \<inter> neighbors_ss y' X) / card X" [PROOF STATE] proof (prove) goal (1 subgoal): 1. codegree_normalized y y' X = real (card (neighbors_ss y X \<inter> neighbors_ss y' X)) / real (card X) [PROOF STEP] proof - [PROOF STATE] proof (state) goal (1 subgoal): 1. codegree_normalized y y' X = real (card (neighbors_ss y X \<inter> neighbors_ss y' X)) / real (card X) [PROOF STEP] have "neighbors_ss y X \<inter> neighbors_ss y' X = neighborhood y \<inter> neighborhood y'" [PROOF STATE] proof (prove) goal (1 subgoal): 1. neighbors_ss y X \<inter> neighbors_ss y' X = neighborhood y \<inter> neighborhood y' [PROOF STEP] using neighbors_ss_eq_neighborhoodY assms [PROOF STATE] proof (prove) using this: ?v \<in> Y \<Longrightarrow> neighborhood ?v = neighbors_ss ?v X y \<in> Y y' \<in> Y goal (1 subgoal): 1. neighbors_ss y X \<inter> neighbors_ss y' X = neighborhood y \<inter> neighborhood y' [PROOF STEP] by auto [PROOF STATE] proof (state) this: neighbors_ss y X \<inter> neighbors_ss y' X = neighborhood y \<inter> neighborhood y' goal (1 subgoal): 1. codegree_normalized y y' X = real (card (neighbors_ss y X \<inter> neighbors_ss y' X)) / real (card X) [PROOF STEP] then [PROOF STATE] proof (chain) picking this: neighbors_ss y X \<inter> neighbors_ss y' X = neighborhood y \<inter> neighborhood y' [PROOF STEP] show ?thesis [PROOF STATE] proof (prove) using this: neighbors_ss y X \<inter> neighbors_ss y' X = neighborhood y \<inter> neighborhood y' goal (1 subgoal): 1. codegree_normalized y y' X = real (card (neighbors_ss y X \<inter> neighbors_ss y' X)) / real (card X) [PROOF STEP] unfolding codegree_normalized_def [PROOF STATE] proof (prove) using this: neighbors_ss y X \<inter> neighbors_ss y' X = neighborhood y \<inter> neighborhood y' goal (1 subgoal): 1. real (codegree y y') / real (card X) = real (card (neighbors_ss y X \<inter> neighbors_ss y' X)) / real (card X) [PROOF STEP] using codegree_def codegree_neighbors [PROOF STATE] proof (prove) using this: neighbors_ss y X \<inter> neighbors_ss y' X = neighborhood y \<inter> neighborhood y' codegree ?v ?u \<equiv> card {x \<in> V. vert_adj ?v x \<and> vert_adj ?u x} codegree ?v ?u = card (neighborhood ?v \<inter> neighborhood ?u) goal (1 subgoal): 1. real (codegree y y') / real (card X) = real (card (neighbors_ss y X \<inter> neighbors_ss y' X)) / real (card X) [PROOF STEP] by presburger [PROOF STATE] proof (state) this: codegree_normalized y y' X = real (card (neighbors_ss y X \<inter> neighbors_ss y' X)) / real (card X) goal: No subgoals! [PROOF STEP] qed
function [F_RF,F_BB]=OMP_Precoding() global Ns Nt Nrf H Codebook_v F_RF = []; [~,~,V] = svd(H); F_opt = V(:,1:Ns); Fres = F_opt; for i = 1:Nrf y = Codebook_v'Fres; k = find(diag(yy')==max(diag(yy'))); F_RF (:,i) = Codebook_v(:,k); F_BB = (F_RF' F_RF)^(-1) * F_RF' F_opt; Fres = (F_opt - F_RF F_BB)/norm(F_opt - F_RF * F_BB,'fro'); end F_BB = sqrt(Ns) * F_BB / norm(F_RF * F_BB,'fro');
Text provided under a Creative Commons Attribution license, CC-BY. Code under MIT license. (c)2014 Lorena A. Barba, Olivier Mesnard. Thanks: NSF for support via CAREER award #1149784. [@LorenaABarba](https://twitter.com/LorenaABarba) ##### Version 0.3 -- February 2015 # Vortex Let's recap. In the first three lessons of _AeroPython_, you computed: 1. [a source-sink pair](http://nbviewer.ipython.org/github/barbagroup/AeroPython/blob/master/lessons/01_Lesson01_sourceSink.ipynb)—you learned how to make a `streamplot()` and use `scatter()` to mark the location of the singular points. If you did your challenge task, you saw that equipotential lines are perpendicular to streamlines. 2. [a source-sink pair in a free stream](http://nbviewer.ipython.org/github/barbagroup/AeroPython/blob/master/lessons/02_Lesson02_sourceSinkFreestream.ipynb), creating a _Rankine oval_. For the first time, we see that we can model the flow around an object using potential flow solutions. You also learned to define custom functions, to make Python work for you and look even more like plain English. 3. [a doublet](http://nbviewer.ipython.org/github/barbagroup/AeroPython/blob/master/lessons/03_Lesson03_doublet.ipynb)—this one is wonderful, because the stream-line pattern gives the flow around a circular cylinder (in two dimensions). You encountered the _D'Alembert paradox_ and hopefully that got you thinking. (If not, go back to that lesson and _think_!) Did you also do the [assignment](http://nbviewer.ipython.org/github/barbagroup/AeroPython/blob/master/lessons/03_Lesson03_Assignment.ipynb)? It shows you how a distribution of sources can be used to model potential flow around an airfoil. This starts looking like aerodynamics! But what is the most important thing we want from applied aerodynamics? We want to make things fly, of course! And to fly, there must be a force of aerodynamic lift to counteract the weight of the object. In this section of the course, we learn about lift. First, we will compute the flow of a potential vortex. It turns out, vortex circulation and lift are intimately related. ## What's a vortex? This question is deeper than you might think! The simple answer is that a vortex is motion in circular streamlines. Imagine streamlines that are concentric circles about a given point—what's confusing is that this _does not mean_ that fluid elements are themselves rotating! In an irrotational vortex, the tangential velocity is constant along a (circular) streamline and inversely proportional to the radius, while the radial velocity is zero. In polar coordinates: \begin{equation} u_\theta\left(r,\theta\right) = \frac{\text{constant}}{r} \quad \text{,} \quad u_r\left(r,\theta\right) = 0 \end{equation} The vorticity is zero everywhere, except at the location of the point vortex, where the derivative of $u_\theta$ is infinite. We introduced the concept of circulation in the first lesson ([Source & Sink](http://nbviewer.ipython.org/urls/github.com/barbagroup/AeroPython/blob/master/lessons/01_Lesson01_sourceSink.ipynb)). Let's use that. Around any circular streamline enclosing the vortex, and using the sign convention that a negative vortex circulates anti-clockwise, we have: \begin{equation}\Gamma = -\oint \mathbf{v}\cdot d\vec{l} = -u_\theta 2 \pi r\end{equation} Thus, the constant in the expression for $u_\theta$ in $(1)$ is equal to $\Gamma/2\pi$, and we now write: \begin{equation}u_\theta\left(r,\theta\right) = \frac{\Gamma}{2\pi r}\end{equation} We can get the stream function by integrating the velocity components: \begin{equation}\psi\left(r,\theta\right) = \frac{\Gamma}{2\pi}\ln r\end{equation} In Cartesian coordinates, the stream function is \begin{equation}\psi\left(x,y\right) = \frac{\Gamma}{4\pi}\ln\left(x^2+y^2\right)\end{equation} while the velocity components would be: \begin{equation}u\left(x,y\right) = \frac{\Gamma}{2\pi}\frac{y}{x^2+y^2} \qquad v\left(x,y\right) = -\frac{\Gamma}{2\pi}\frac{x}{x^2+y^2}\end{equation} This vortex flow is irrotational everywhere, except at the vortex center, where it is infinite. The strenth of the point vortex is equal to the circulation $\Gamma$ around it. ## Let's compute a vortex The set-up is the same as before: we load our favorite libraries, and we create a grid of points to evaluate the velocity field. ```python import numpy import math from matplotlib import pyplot ``` ```python N = 50 # Number of points in each direction x_start, x_end = -2.0, 2.0 # x-direction boundaries y_start, y_end = -1.0, 1.0 # y-direction boundaries x = numpy.linspace(x_start, x_end, N) # computes a 1D-array for x y = numpy.linspace(y_start, y_end, N) # computes a 1D-array for y X, Y = numpy.meshgrid(x, y) # generates a mesh grid ``` Give your vortex a strength $\Gamma=5$ and place it at the center of your domain: ```python gamma = 5.0 # strength of the vortex x_vortex, y_vortex = 0.0, 0.0 # location of the vortex ``` We will define two functions, * `get_velocity_vortex()` and * `get_stream_function_vortex()`, to compute the velocity components and the stream function on our Cartesian grid, given the strength and the location of the vortex. Then, we will use our custom functions to evaluate everything on the grid points. Let's write those functions first. ```python def get_velocity_vortex(strength, xv, yv, X, Y): """Returns the velocity field generated by a vortex. Arguments --------- strength -- strength of the vortex. xv, yv -- coordinates of the vortex. X, Y -- mesh grid. """ u = + strength/(2math.pi)(Y-yv)/((X-xv)2+(Y-yv)2) v = - strength/(2math.pi)(X-xv)/((X-xv)2+(Y-yv)2) return u, v ``` ```python def get_stream_function_vortex(strength, xv, yv, X, Y): """Returns the stream-function generated by a vortex. Arguments --------- strength -- strength of the vortex. xv, yv -- coordinates of the vortex. X, Y -- mesh grid. """ psi = strength/(4math.pi)numpy.log((X-xv)2+(Y-yv)2) return psi ``` An now, call the functions with the vortex strength and position, plus the coordinates of the evaluation grid, to get the velocity and streamfunction of the vortex. ```python # computes the velocity field on the mesh grid u_vortex, v_vortex = get_velocity_vortex(gamma, x_vortex, y_vortex, X, Y) # computes the stream-function on the mesh grid psi_vortex = get_stream_function_vortex(gamma, x_vortex, y_vortex, X, Y) ``` We are now able to visualize the streamlines of a vortex, and they look like concentric circles around the vortex center, as expected. ```python # plots the streamlines %matplotlib inline size = 10 pyplot.figure(figsize=(size, (y_end-y_start)/(x_end-x_start)size)) pyplot.xlabel('x', fontsize=16) pyplot.ylabel('y', fontsize=16) pyplot.xlim(x_start, x_end) pyplot.ylim(y_start, y_end) pyplot.streamplot(X, Y, u_vortex, v_vortex, density=2, linewidth=1, arrowsize=1, arrowstyle='->') pyplot.scatter(x_vortex, y_vortex, color='#CD2305', s=80, marker='o'); ``` ## Vortex & Sink For fun, let's use our superposition powers. Add a vortex to a sink, using two new functions to compute the velocity components and the stream function of the sink, and adding to those of a vortex (remember that the sink can be easily replaced by a source by just changing the sign of the strength). ```python strength_sink = -1.0 # strength of the sink x_sink, y_sink = 0.0, 0.0 # location of the sink ``` ```python def get_velocity_sink(strength, xs, ys, X, Y): """Returns the velocity field generated by a sink. Arguments --------- strength -- strength of the sink. xs, ys -- coordinates of the sink. X, Y -- mesh grid. """ u = strength/(2math.pi)(X-xs)/((X-xs)2+(Y-ys)2) v = strength/(2math.pi)(Y-ys)/((X-xs)2+(Y-ys)2) return u, v ``` ```python def get_stream_function_sink(strength, xs, ys, X, Y): """Returns the stream-function generated by a sink. Arguments --------- strength -- strength of the sink. xs, ys -- coordinates of the sink. X, Y -- mesh grid. """ psi = strength/(2math.pi)numpy.arctan2((Y-ys), (X-xs)) return psi ``` ```python # computes the velocity field on the mesh grid u_sink, v_sink = get_velocity_sink(strength_sink, x_sink, y_sink, X, Y) # computes the stream-function on the mesh grid psi_sink = get_stream_function_sink(strength_sink, x_sink, y_sink, X, Y) ``` Now, let's visualize the streamlines of the vortex-sink, and admire our artistic creation: ```python # superposition of the sink and the vortex u = u_vortex + u_sink v = v_vortex + v_sink psi = psi_vortex + psi_sink # plots the streamlines size = 10 pyplot.figure(figsize=(size, (y_end-y_start)/(x_end-x_start)size)) pyplot.xlabel('x', fontsize=16) pyplot.ylabel('y', fontsize=16) pyplot.xlim(x_start, x_end) pyplot.ylim(y_start, y_end) pyplot.streamplot(X, Y, u, v, density=2, linewidth=1, arrowsize=1, arrowstyle='->') pyplot.scatter(x_vortex, y_vortex, color='#CD2305', s=80, marker='o'); ``` Very cool op-art. (And a good model for your typical bath-tub vortex.) But is all this useful? Yes! First, you will take your training wheels off and—on your own—compute the flow of an [infinite row of vortices](http://nbviewer.ipython.org/github/barbagroup/AeroPython/blob/master/lessons/05_Lesson05_InfiniteRowOfVortices.ipynb). Your mission, should you choose to accept it, is in the next notebook. After that, we will learn about the connection between a vortex, and the force of lift. This turns out to be very important in aerodynamics! ## What's this "irrotational" vortex thing? I know what you are thinking. What does it mean that the vortex is irrotational? Surely if there's a vortex, there's rotation! You are not crazy. It's natural to think this way, but the potential vortex is a flow where streamlines are circular, yet fluid elements do not rotate around themselves—they just go around the circular path. This classic video will help you understand ... just watch the 25 seconds of video after time 4m 25s, and see a "vorticity meter" go around a free vortex without rotating itself. ```python from IPython.display import YouTubeVideo from datetime import timedelta start=int(timedelta(hours=0, minutes=4, seconds=25).total_seconds()) YouTubeVideo("loCLkcYEWD4", start=start) ``` Remember: vorticity measures the local angular velocity of each fluid element. If the fluid elements go around a circular path, but do not spin themselves, there is no vorticity! This animation from [Wikipedia](http://en.wikipedia.org/wiki/Vortex#Irrotational_vortices) helps further illustrate what happens in an irrotational vortex: the orange markers with a line across them are going around in circles, but they are not themselves rotating (notice the white lines keep their orientation). ## Next ... The next IPython Notebook of the AeroPython series presents the exercise ["Infinite row of vortices,"](http://nbviewer.ipython.org/github/barbagroup/AeroPython/blob/master/lessons/05_Lesson05_InfiniteRowOfVortices.ipynb) as independent student work. --- Please ignore the cell below. It just loads our style for the notebook. ```python from IPython.core.display import HTML def css_styling(): styles = open('../styles/custom.css', 'r').read() return HTML(styles) css_styling() ``` <link href='http://fonts.googleapis.com/css?family=Fenix' rel='stylesheet' type='text/css'> <link href='http://fonts.googleapis.com/css?family=Alegreya+Sans:100,300,400,500,700,800,900,100italic,300italic,400italic,500italic,700italic,800italic,900italic' rel='stylesheet' type='text/css'> <link href='http://fonts.googleapis.com/css?family=Source+Code+Pro:300,400' rel='stylesheet' type='text/css'> <style> @font-face { font-family: "Computer Modern"; src: url('http://mirrors.ctan.org/fonts/cm-unicode/fonts/otf/cmunss.otf'); } div.cell{ width:800px; margin-left:16% !important; margin-right:auto; } h1 { font-family: 'Alegreya Sans', sans-serif; } h2 { font-family: 'Fenix', serif; } h3{ font-family: 'Fenix', serif; margin-top:12px; margin-bottom: 3px; } h4{ font-family: 'Fenix', serif; } h5 { font-family: 'Alegreya Sans', sans-serif; } div.text_cell_render{ font-family: 'Alegreya Sans',Computer Modern, "Helvetica Neue", Arial, Helvetica, Geneva, sans-serif; line-height: 135%; font-size: 120%; width:600px; margin-left:auto; margin-right:auto; } .CodeMirror{ font-family: "Source Code Pro"; font-size: 90%; } /* .prompt{ display: None; }*/ .text_cell_render h1 { font-weight: 200; font-size: 50pt; line-height: 100%; color:#CD2305; margin-bottom: 0.5em; margin-top: 0.5em; display: block; } .text_cell_render h5 { font-weight: 300; font-size: 16pt; color: #CD2305; font-style: italic; margin-bottom: .5em; margin-top: 0.5em; display: block; } .warning{ color: rgb( 240, 20, 20 ) } </style>
Formal statement is: lemma space_empty_eq_bot: "space a = {} \<longleftrightarrow> a = bot" Informal statement is: A measure space is empty if and only if it is the trivial measure space.
theorem caratheodory: "convex hull p = {x::'a::euclidean_space. \<exists>S. finite S \<and> S \<subseteq> p \<and> card S \<le> DIM('a) + 1 \<and> x \<in> convex hull S}"
[GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B hx : IsIntegral A x ⊢ Monic (minpoly A x) [PROOFSTEP] delta minpoly [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B hx : IsIntegral A x ⊢ Monic (if hx : IsIntegral A x then WellFounded.min (_ : WellFounded fun p q => degree p < degree q) (fun x_1 => Monic x_1 ∧ eval₂ (algebraMap A B) x x_1 = 0) hx else 0) [PROOFSTEP] rw [dif_pos hx] [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B hx : IsIntegral A x ⊢ Monic (WellFounded.min (_ : WellFounded fun p q => degree p < degree q) (fun x_1 => Monic x_1 ∧ eval₂ (algebraMap A B) x x_1 = 0) hx) [PROOFSTEP] exact (degree_lt_wf.min_mem _ hx).1 [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x✝ : B f : B →ₐ[A] B' hf : Injective ↑f x : B ⊢ minpoly A (↑f x) = minpoly A x [PROOFSTEP] refine' dif_ctx_congr (isIntegral_algHom_iff _ hf) (fun _ => _) fun _ => rfl [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x✝¹ : B f : B →ₐ[A] B' hf : Injective ↑f x : B x✝ : IsIntegral A x ⊢ WellFounded.min (_ : WellFounded fun p q => degree p < degree q) (fun x_1 => Monic x_1 ∧ eval₂ (algebraMap A ((fun x => B') x)) (↑f x) x_1 = 0) (_ : IsIntegral A (↑f x)) = WellFounded.min (_ : WellFounded fun p q => degree p < degree q) (fun x_1 => Monic x_1 ∧ eval₂ (algebraMap A B) x x_1 = 0) x✝ [PROOFSTEP] simp_rw [← Polynomial.aeval_def, aeval_algHom, AlgHom.comp_apply, _root_.map_eq_zero_iff f hf] [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B ⊢ ↑(Polynomial.aeval x) (minpoly A x) = 0 [PROOFSTEP] delta minpoly [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B ⊢ ↑(Polynomial.aeval x) (if hx : IsIntegral A x then WellFounded.min (_ : WellFounded fun p q => degree p < degree q) (fun x_1 => Monic x_1 ∧ eval₂ (algebraMap A B) x x_1 = 0) hx else 0) = 0 [PROOFSTEP] split_ifs with hx [GOAL] case pos A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B hx : IsIntegral A x ⊢ ↑(Polynomial.aeval x) (if hx : IsIntegral A x then WellFounded.min (_ : WellFounded fun p q => degree p < degree q) (fun x_1 => Monic x_1 ∧ eval₂ (algebraMap A B) x x_1 = 0) hx else 0) = 0 [PROOFSTEP] rw [dif_pos hx] [GOAL] case pos A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B hx : IsIntegral A x ⊢ ↑(Polynomial.aeval x) (WellFounded.min (_ : WellFounded fun p q => degree p < degree q) (fun x_1 => Monic x_1 ∧ eval₂ (algebraMap A B) x x_1 = 0) hx) = 0 [PROOFSTEP] exact (degree_lt_wf.min_mem _ hx).2 [GOAL] case neg A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B hx : ¬IsIntegral A x ⊢ ↑(Polynomial.aeval x) (if hx : IsIntegral A x then WellFounded.min (_ : WellFounded fun p q => degree p < degree q) (fun x_1 => Monic x_1 ∧ eval₂ (algebraMap A B) x x_1 = 0) hx else 0) = 0 [PROOFSTEP] rw [dif_neg hx] [GOAL] case neg A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B hx : ¬IsIntegral A x ⊢ ↑(Polynomial.aeval x) 0 = 0 [PROOFSTEP] exact aeval_zero _ [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁵ : CommRing A inst✝⁴ : Ring B inst✝³ : Ring B' inst✝² : Algebra A B inst✝¹ : Algebra A B' x : B inst✝ : Nontrivial B ⊢ minpoly A x ≠ 1 [PROOFSTEP] intro h [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁵ : CommRing A inst✝⁴ : Ring B inst✝³ : Ring B' inst✝² : Algebra A B inst✝¹ : Algebra A B' x : B inst✝ : Nontrivial B h : minpoly A x = 1 ⊢ False [PROOFSTEP] refine' (one_ne_zero : (1 : B) ≠ 0) _ [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁵ : CommRing A inst✝⁴ : Ring B inst✝³ : Ring B' inst✝² : Algebra A B inst✝¹ : Algebra A B' x : B inst✝ : Nontrivial B h : minpoly A x = 1 ⊢ 1 = 0 [PROOFSTEP] simpa using congr_arg (Polynomial.aeval x) h [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁷ : CommRing A inst✝⁶ : Ring B inst✝⁵ : Ring B' inst✝⁴ : Algebra A B inst✝³ : Algebra A B' x : B inst✝² : Nontrivial B R : Type u_4 inst✝¹ : Semiring R inst✝ : Nontrivial R f : A →+* R ⊢ map f (minpoly A x) ≠ 1 [PROOFSTEP] by_cases hx : IsIntegral A x [GOAL] case pos A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁷ : CommRing A inst✝⁶ : Ring B inst✝⁵ : Ring B' inst✝⁴ : Algebra A B inst✝³ : Algebra A B' x : B inst✝² : Nontrivial B R : Type u_4 inst✝¹ : Semiring R inst✝ : Nontrivial R f : A →+* R hx : IsIntegral A x ⊢ map f (minpoly A x) ≠ 1 [PROOFSTEP] exact mt ((monic hx).eq_one_of_map_eq_one f) (ne_one A x) [GOAL] case neg A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁷ : CommRing A inst✝⁶ : Ring B inst✝⁵ : Ring B' inst✝⁴ : Algebra A B inst✝³ : Algebra A B' x : B inst✝² : Nontrivial B R : Type u_4 inst✝¹ : Semiring R inst✝ : Nontrivial R f : A →+* R hx : ¬IsIntegral A x ⊢ map f (minpoly A x) ≠ 1 [PROOFSTEP] rw [eq_zero hx, Polynomial.map_zero] [GOAL] case neg A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁷ : CommRing A inst✝⁶ : Ring B inst✝⁵ : Ring B' inst✝⁴ : Algebra A B inst✝³ : Algebra A B' x : B inst✝² : Nontrivial B R : Type u_4 inst✝¹ : Semiring R inst✝ : Nontrivial R f : A →+* R hx : ¬IsIntegral A x ⊢ 0 ≠ 1 [PROOFSTEP] exact zero_ne_one [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁵ : CommRing A inst✝⁴ : Ring B inst✝³ : Ring B' inst✝² : Algebra A B inst✝¹ : Algebra A B' x : B inst✝ : Nontrivial B ⊢ ¬IsUnit (minpoly A x) [PROOFSTEP] haveI : Nontrivial A := (algebraMap A B).domain_nontrivial [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁵ : CommRing A inst✝⁴ : Ring B inst✝³ : Ring B' inst✝² : Algebra A B inst✝¹ : Algebra A B' x : B inst✝ : Nontrivial B this : Nontrivial A ⊢ ¬IsUnit (minpoly A x) [PROOFSTEP] by_cases hx : IsIntegral A x [GOAL] case pos A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁵ : CommRing A inst✝⁴ : Ring B inst✝³ : Ring B' inst✝² : Algebra A B inst✝¹ : Algebra A B' x : B inst✝ : Nontrivial B this : Nontrivial A hx : IsIntegral A x ⊢ ¬IsUnit (minpoly A x) [PROOFSTEP] exact mt (monic hx).eq_one_of_isUnit (ne_one A x) [GOAL] case neg A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁵ : CommRing A inst✝⁴ : Ring B inst✝³ : Ring B' inst✝² : Algebra A B inst✝¹ : Algebra A B' x : B inst✝ : Nontrivial B this : Nontrivial A hx : ¬IsIntegral A x ⊢ ¬IsUnit (minpoly A x) [PROOFSTEP] rw [eq_zero hx] [GOAL] case neg A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁵ : CommRing A inst✝⁴ : Ring B inst✝³ : Ring B' inst✝² : Algebra A B inst✝¹ : Algebra A B' x : B inst✝ : Nontrivial B this : Nontrivial A hx : ¬IsIntegral A x ⊢ ¬IsUnit 0 [PROOFSTEP] exact not_isUnit_zero [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B hx : degree (minpoly A x) = 1 ⊢ x ∈ RingHom.range (algebraMap A B) [PROOFSTEP] have h : IsIntegral A x := by by_contra h rw [eq_zero h, degree_zero, ← WithBot.coe_one] at hx exact ne_of_lt (show ⊥ < ↑1 from WithBot.bot_lt_coe 1) hx [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B hx : degree (minpoly A x) = 1 ⊢ IsIntegral A x [PROOFSTEP] by_contra h [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B hx : degree (minpoly A x) = 1 h : ¬IsIntegral A x ⊢ False [PROOFSTEP] rw [eq_zero h, degree_zero, ← WithBot.coe_one] at hx [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B hx : ⊥ = ↑1 h : ¬IsIntegral A x ⊢ False [PROOFSTEP] exact ne_of_lt (show ⊥ < ↑1 from WithBot.bot_lt_coe 1) hx [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B hx : degree (minpoly A x) = 1 h : IsIntegral A x ⊢ x ∈ RingHom.range (algebraMap A B) [PROOFSTEP] have key := minpoly.aeval A x [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B hx : degree (minpoly A x) = 1 h : IsIntegral A x key : ↑(Polynomial.aeval x) (minpoly A x) = 0 ⊢ x ∈ RingHom.range (algebraMap A B) [PROOFSTEP] rw [eq_X_add_C_of_degree_eq_one hx, (minpoly.monic h).leadingCoeff, C_1, one_mul, aeval_add, aeval_C, aeval_X, ← eq_neg_iff_add_eq_zero, ← RingHom.map_neg] at key [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B hx : degree (minpoly A x) = 1 h : IsIntegral A x key : x = ↑(algebraMap A B) (-coeff (minpoly A x) 0) ⊢ x ∈ RingHom.range (algebraMap A B) [PROOFSTEP] exact ⟨-(minpoly A x).coeff 0, key.symm⟩ [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B p : A[X] pmonic : Monic p hp : ↑(Polynomial.aeval x) p = 0 ⊢ degree (minpoly A x) ≤ degree p [PROOFSTEP] delta minpoly [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B p : A[X] pmonic : Monic p hp : ↑(Polynomial.aeval x) p = 0 ⊢ degree (if hx : IsIntegral A x then WellFounded.min (_ : WellFounded fun p q => degree p < degree q) (fun x_1 => Monic x_1 ∧ eval₂ (algebraMap A B) x x_1 = 0) hx else 0) ≤ degree p [PROOFSTEP] split_ifs with hx [GOAL] case pos A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B p : A[X] pmonic : Monic p hp : ↑(Polynomial.aeval x) p = 0 hx : IsIntegral A x ⊢ degree (WellFounded.min (_ : WellFounded fun p q => degree p < degree q) (fun x_1 => Monic x_1 ∧ eval₂ (algebraMap A B) x x_1 = 0) hx) ≤ degree p [PROOFSTEP] exact le_of_not_lt (degree_lt_wf.not_lt_min _ hx ⟨pmonic, hp⟩) [GOAL] case neg A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B p : A[X] pmonic : Monic p hp : ↑(Polynomial.aeval x) p = 0 hx : ¬IsIntegral A x ⊢ degree 0 ≤ degree p [PROOFSTEP] simp only [degree_zero, bot_le] [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B p : A[X] hm : Monic p hp : ↑(Polynomial.aeval x) p = 0 hl : ∀ (q : A[X]), degree q < degree p → q = 0 ∨ ↑(Polynomial.aeval x) q ≠ 0 ⊢ p = minpoly A x [PROOFSTEP] nontriviality A [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B p : A[X] hm : Monic p hp : ↑(Polynomial.aeval x) p = 0 hl : ∀ (q : A[X]), degree q < degree p → q = 0 ∨ ↑(Polynomial.aeval x) q ≠ 0 ✝ : Nontrivial A ⊢ p = minpoly A x [PROOFSTEP] have hx : IsIntegral A x := ⟨p, hm, hp⟩ [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B p : A[X] hm : Monic p hp : ↑(Polynomial.aeval x) p = 0 hl : ∀ (q : A[X]), degree q < degree p → q = 0 ∨ ↑(Polynomial.aeval x) q ≠ 0 ✝ : Nontrivial A hx : IsIntegral A x ⊢ p = minpoly A x [PROOFSTEP] obtain h \| h := hl _ ((minpoly A x).degree_modByMonic_lt hm) [GOAL] case inl A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B p : A[X] hm : Monic p hp : ↑(Polynomial.aeval x) p = 0 hl : ∀ (q : A[X]), degree q < degree p → q = 0 ∨ ↑(Polynomial.aeval x) q ≠ 0 ✝ : Nontrivial A hx : IsIntegral A x h : minpoly A x %ₘ p = 0 ⊢ p = minpoly A x case inr A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B p : A[X] hm : Monic p hp : ↑(Polynomial.aeval x) p = 0 hl : ∀ (q : A[X]), degree q < degree p → q = 0 ∨ ↑(Polynomial.aeval x) q ≠ 0 ✝ : Nontrivial A hx : IsIntegral A x h : ↑(Polynomial.aeval x) (minpoly A x %ₘ p) ≠ 0 ⊢ p = minpoly A x [PROOFSTEP] swap [GOAL] case inr A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B p : A[X] hm : Monic p hp : ↑(Polynomial.aeval x) p = 0 hl : ∀ (q : A[X]), degree q < degree p → q = 0 ∨ ↑(Polynomial.aeval x) q ≠ 0 ✝ : Nontrivial A hx : IsIntegral A x h : ↑(Polynomial.aeval x) (minpoly A x %ₘ p) ≠ 0 ⊢ p = minpoly A x [PROOFSTEP] exact (h <\| (aeval_modByMonic_eq_self_of_root hm hp).trans <\| aeval A x).elim [GOAL] case inl A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B p : A[X] hm : Monic p hp : ↑(Polynomial.aeval x) p = 0 hl : ∀ (q : A[X]), degree q < degree p → q = 0 ∨ ↑(Polynomial.aeval x) q ≠ 0 ✝ : Nontrivial A hx : IsIntegral A x h : minpoly A x %ₘ p = 0 ⊢ p = minpoly A x [PROOFSTEP] obtain ⟨r, hr⟩ := (dvd_iff_modByMonic_eq_zero hm).1 h [GOAL] case inl.intro A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B p : A[X] hm : Monic p hp : ↑(Polynomial.aeval x) p = 0 hl : ∀ (q : A[X]), degree q < degree p → q = 0 ∨ ↑(Polynomial.aeval x) q ≠ 0 ✝ : Nontrivial A hx : IsIntegral A x h : minpoly A x %ₘ p = 0 r : A[X] hr : minpoly A x = p * r ⊢ p = minpoly A x [PROOFSTEP] rw [hr] [GOAL] case inl.intro A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B p : A[X] hm : Monic p hp : ↑(Polynomial.aeval x) p = 0 hl : ∀ (q : A[X]), degree q < degree p → q = 0 ∨ ↑(Polynomial.aeval x) q ≠ 0 ✝ : Nontrivial A hx : IsIntegral A x h : minpoly A x %ₘ p = 0 r : A[X] hr : minpoly A x = p * r ⊢ p = p * r [PROOFSTEP] have hlead := congr_arg leadingCoeff hr [GOAL] case inl.intro A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B p : A[X] hm : Monic p hp : ↑(Polynomial.aeval x) p = 0 hl : ∀ (q : A[X]), degree q < degree p → q = 0 ∨ ↑(Polynomial.aeval x) q ≠ 0 ✝ : Nontrivial A hx : IsIntegral A x h : minpoly A x %ₘ p = 0 r : A[X] hr : minpoly A x = p * r hlead : leadingCoeff (minpoly A x) = leadingCoeff (p * r) ⊢ p = p * r [PROOFSTEP] rw [mul_comm, leadingCoeff_mul_monic hm, (monic hx).leadingCoeff] at hlead [GOAL] case inl.intro A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B p : A[X] hm : Monic p hp : ↑(Polynomial.aeval x) p = 0 hl : ∀ (q : A[X]), degree q < degree p → q = 0 ∨ ↑(Polynomial.aeval x) q ≠ 0 ✝ : Nontrivial A hx : IsIntegral A x h : minpoly A x %ₘ p = 0 r : A[X] hr : minpoly A x = p * r hlead : 1 = leadingCoeff r ⊢ p = p * r [PROOFSTEP] have : natDegree r ≤ 0 := by have hr0 : r ≠ 0 := by rintro rfl exact ne_zero hx (mul_zero p ▸ hr) apply_fun natDegree at hr rw [hm.natDegree_mul' hr0] at hr apply Nat.le_of_add_le_add_left rw [add_zero] exact hr.symm.trans_le (natDegree_le_natDegree <\| min A x hm hp) [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B p : A[X] hm : Monic p hp : ↑(Polynomial.aeval x) p = 0 hl : ∀ (q : A[X]), degree q < degree p → q = 0 ∨ ↑(Polynomial.aeval x) q ≠ 0 ✝ : Nontrivial A hx : IsIntegral A x h : minpoly A x %ₘ p = 0 r : A[X] hr : minpoly A x = p * r hlead : 1 = leadingCoeff r ⊢ natDegree r ≤ 0 [PROOFSTEP] have hr0 : r ≠ 0 := by rintro rfl exact ne_zero hx (mul_zero p ▸ hr) [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B p : A[X] hm : Monic p hp : ↑(Polynomial.aeval x) p = 0 hl : ∀ (q : A[X]), degree q < degree p → q = 0 ∨ ↑(Polynomial.aeval x) q ≠ 0 ✝ : Nontrivial A hx : IsIntegral A x h : minpoly A x %ₘ p = 0 r : A[X] hr : minpoly A x = p * r hlead : 1 = leadingCoeff r ⊢ r ≠ 0 [PROOFSTEP] rintro rfl [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B p : A[X] hm : Monic p hp : ↑(Polynomial.aeval x) p = 0 hl : ∀ (q : A[X]), degree q < degree p → q = 0 ∨ ↑(Polynomial.aeval x) q ≠ 0 ✝ : Nontrivial A hx : IsIntegral A x h : minpoly A x %ₘ p = 0 hr : minpoly A x = p * 0 hlead : 1 = leadingCoeff 0 ⊢ False [PROOFSTEP] exact ne_zero hx (mul_zero p ▸ hr) [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B p : A[X] hm : Monic p hp : ↑(Polynomial.aeval x) p = 0 hl : ∀ (q : A[X]), degree q < degree p → q = 0 ∨ ↑(Polynomial.aeval x) q ≠ 0 ✝ : Nontrivial A hx : IsIntegral A x h : minpoly A x %ₘ p = 0 r : A[X] hr : minpoly A x = p * r hlead : 1 = leadingCoeff r hr0 : r ≠ 0 ⊢ natDegree r ≤ 0 [PROOFSTEP] apply_fun natDegree at hr [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B p : A[X] hm : Monic p hp : ↑(Polynomial.aeval x) p = 0 hl : ∀ (q : A[X]), degree q < degree p → q = 0 ∨ ↑(Polynomial.aeval x) q ≠ 0 ✝ : Nontrivial A hx : IsIntegral A x h : minpoly A x %ₘ p = 0 r : A[X] hlead : 1 = leadingCoeff r hr0 : r ≠ 0 hr : natDegree (minpoly A x) = natDegree (p * r) ⊢ natDegree r ≤ 0 [PROOFSTEP] rw [hm.natDegree_mul' hr0] at hr [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B p : A[X] hm : Monic p hp : ↑(Polynomial.aeval x) p = 0 hl : ∀ (q : A[X]), degree q < degree p → q = 0 ∨ ↑(Polynomial.aeval x) q ≠ 0 ✝ : Nontrivial A hx : IsIntegral A x h : minpoly A x %ₘ p = 0 r : A[X] hlead : 1 = leadingCoeff r hr0 : r ≠ 0 hr : natDegree (minpoly A x) = natDegree p + natDegree r ⊢ natDegree r ≤ 0 [PROOFSTEP] apply Nat.le_of_add_le_add_left [GOAL] case h A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B p : A[X] hm : Monic p hp : ↑(Polynomial.aeval x) p = 0 hl : ∀ (q : A[X]), degree q < degree p → q = 0 ∨ ↑(Polynomial.aeval x) q ≠ 0 ✝ : Nontrivial A hx : IsIntegral A x h : minpoly A x %ₘ p = 0 r : A[X] hlead : 1 = leadingCoeff r hr0 : r ≠ 0 hr : natDegree (minpoly A x) = natDegree p + natDegree r ⊢ ?a + natDegree r ≤ ?a + 0 case a A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B p : A[X] hm : Monic p hp : ↑(Polynomial.aeval x) p = 0 hl : ∀ (q : A[X]), degree q < degree p → q = 0 ∨ ↑(Polynomial.aeval x) q ≠ 0 ✝ : Nontrivial A hx : IsIntegral A x h : minpoly A x %ₘ p = 0 r : A[X] hlead : 1 = leadingCoeff r hr0 : r ≠ 0 hr : natDegree (minpoly A x) = natDegree p + natDegree r ⊢ ℕ [PROOFSTEP] rw [add_zero] [GOAL] case h A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B p : A[X] hm : Monic p hp : ↑(Polynomial.aeval x) p = 0 hl : ∀ (q : A[X]), degree q < degree p → q = 0 ∨ ↑(Polynomial.aeval x) q ≠ 0 ✝ : Nontrivial A hx : IsIntegral A x h : minpoly A x %ₘ p = 0 r : A[X] hlead : 1 = leadingCoeff r hr0 : r ≠ 0 hr : natDegree (minpoly A x) = natDegree p + natDegree r ⊢ ?h + natDegree r ≤ ?h case h A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B p : A[X] hm : Monic p hp : ↑(Polynomial.aeval x) p = 0 hl : ∀ (q : A[X]), degree q < degree p → q = 0 ∨ ↑(Polynomial.aeval x) q ≠ 0 ✝ : Nontrivial A hx : IsIntegral A x h : minpoly A x %ₘ p = 0 r : A[X] hlead : 1 = leadingCoeff r hr0 : r ≠ 0 hr : natDegree (minpoly A x) = natDegree p + natDegree r ⊢ ℕ [PROOFSTEP] exact hr.symm.trans_le (natDegree_le_natDegree <\| min A x hm hp) [GOAL] case inl.intro A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Ring B' inst✝¹ : Algebra A B inst✝ : Algebra A B' x : B p : A[X] hm : Monic p hp : ↑(Polynomial.aeval x) p = 0 hl : ∀ (q : A[X]), degree q < degree p → q = 0 ∨ ↑(Polynomial.aeval x) q ≠ 0 ✝ : Nontrivial A hx : IsIntegral A x h : minpoly A x %ₘ p = 0 r : A[X] hr : minpoly A x = p * r hlead : 1 = leadingCoeff r this : natDegree r ≤ 0 ⊢ p = p * r [PROOFSTEP] rw [eq_C_of_natDegree_le_zero this, ← Nat.eq_zero_of_le_zero this, ← leadingCoeff, ← hlead, C_1, mul_one] [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁵ : CommRing A inst✝⁴ : Ring B inst✝³ : Ring B' inst✝² : Algebra A B inst✝¹ : Algebra A B' x : B inst✝ : Subsingleton B ⊢ minpoly A x = 1 [PROOFSTEP] nontriviality A [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁵ : CommRing A inst✝⁴ : Ring B inst✝³ : Ring B' inst✝² : Algebra A B inst✝¹ : Algebra A B' x : B inst✝ : Subsingleton B ✝ : Nontrivial A ⊢ minpoly A x = 1 [PROOFSTEP] have := minpoly.min A x monic_one (Subsingleton.elim _ _) [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁵ : CommRing A inst✝⁴ : Ring B inst✝³ : Ring B' inst✝² : Algebra A B inst✝¹ : Algebra A B' x : B inst✝ : Subsingleton B ✝ : Nontrivial A this : degree (minpoly A x) ≤ degree 1 ⊢ minpoly A x = 1 [PROOFSTEP] rw [degree_one] at this [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁵ : CommRing A inst✝⁴ : Ring B inst✝³ : Ring B' inst✝² : Algebra A B inst✝¹ : Algebra A B' x : B inst✝ : Subsingleton B ✝ : Nontrivial A this : degree (minpoly A x) ≤ 0 ⊢ minpoly A x = 1 [PROOFSTEP] cases' le_or_lt (minpoly A x).degree 0 with h h [GOAL] case inl A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁵ : CommRing A inst✝⁴ : Ring B inst✝³ : Ring B' inst✝² : Algebra A B inst✝¹ : Algebra A B' x : B inst✝ : Subsingleton B ✝ : Nontrivial A this : degree (minpoly A x) ≤ 0 h : degree (minpoly A x) ≤ 0 ⊢ minpoly A x = 1 [PROOFSTEP] rwa [(monic ⟨1, monic_one, by simp⟩ : (minpoly A x).Monic).degree_le_zero_iff_eq_one] at h [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁵ : CommRing A inst✝⁴ : Ring B inst✝³ : Ring B' inst✝² : Algebra A B inst✝¹ : Algebra A B' x : B inst✝ : Subsingleton B ✝ : Nontrivial A this : degree (minpoly A x) ≤ 0 h : degree (minpoly A x) ≤ 0 ⊢ eval₂ (algebraMap A B) x 1 = 0 [PROOFSTEP] simp [GOAL] case inr A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁵ : CommRing A inst✝⁴ : Ring B inst✝³ : Ring B' inst✝² : Algebra A B inst✝¹ : Algebra A B' x : B inst✝ : Subsingleton B ✝ : Nontrivial A this : degree (minpoly A x) ≤ 0 h : 0 < degree (minpoly A x) ⊢ minpoly A x = 1 [PROOFSTEP] exact (this.not_lt h).elim [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝³ : CommRing A inst✝² : Ring B inst✝¹ : Algebra A B x : B inst✝ : Nontrivial B hx : IsIntegral A x ⊢ 0 < natDegree (minpoly A x) [PROOFSTEP] rw [pos_iff_ne_zero] [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝³ : CommRing A inst✝² : Ring B inst✝¹ : Algebra A B x : B inst✝ : Nontrivial B hx : IsIntegral A x ⊢ natDegree (minpoly A x) ≠ 0 [PROOFSTEP] intro ndeg_eq_zero [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝³ : CommRing A inst✝² : Ring B inst✝¹ : Algebra A B x : B inst✝ : Nontrivial B hx : IsIntegral A x ndeg_eq_zero : natDegree (minpoly A x) = 0 ⊢ False [PROOFSTEP] have eq_one : minpoly A x = 1 := by rw [eq_C_of_natDegree_eq_zero ndeg_eq_zero] convert C_1 (R := A) simpa only [ndeg_eq_zero.symm] using (monic hx).leadingCoeff [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝³ : CommRing A inst✝² : Ring B inst✝¹ : Algebra A B x : B inst✝ : Nontrivial B hx : IsIntegral A x ndeg_eq_zero : natDegree (minpoly A x) = 0 ⊢ minpoly A x = 1 [PROOFSTEP] rw [eq_C_of_natDegree_eq_zero ndeg_eq_zero] [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝³ : CommRing A inst✝² : Ring B inst✝¹ : Algebra A B x : B inst✝ : Nontrivial B hx : IsIntegral A x ndeg_eq_zero : natDegree (minpoly A x) = 0 ⊢ ↑C (coeff (minpoly A x) 0) = 1 [PROOFSTEP] convert C_1 (R := A) [GOAL] case h.e'_2.h.e'_6 A : Type u_1 B : Type u_2 B' : Type u_3 inst✝³ : CommRing A inst✝² : Ring B inst✝¹ : Algebra A B x : B inst✝ : Nontrivial B hx : IsIntegral A x ndeg_eq_zero : natDegree (minpoly A x) = 0 ⊢ coeff (minpoly A x) 0 = 1 [PROOFSTEP] simpa only [ndeg_eq_zero.symm] using (monic hx).leadingCoeff [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝³ : CommRing A inst✝² : Ring B inst✝¹ : Algebra A B x : B inst✝ : Nontrivial B hx : IsIntegral A x ndeg_eq_zero : natDegree (minpoly A x) = 0 eq_one : minpoly A x = 1 ⊢ False [PROOFSTEP] simpa only [eq_one, AlgHom.map_one, one_ne_zero] using aeval A x [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝² : CommRing A inst✝¹ : Ring B inst✝ : Algebra A B x : B a : A hf : Injective ↑(algebraMap A B) ⊢ minpoly A (↑(algebraMap A B) a) = X - ↑C a [PROOFSTEP] nontriviality A [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝² : CommRing A inst✝¹ : Ring B inst✝ : Algebra A B x : B a : A hf : Injective ↑(algebraMap A B) ✝ : Nontrivial A ⊢ minpoly A (↑(algebraMap A B) a) = X - ↑C a [PROOFSTEP] refine' (unique' A _ (monic_X_sub_C a) _ _).symm [GOAL] case refine'_1 A : Type u_1 B : Type u_2 B' : Type u_3 inst✝² : CommRing A inst✝¹ : Ring B inst✝ : Algebra A B x : B a : A hf : Injective ↑(algebraMap A B) ✝ : Nontrivial A ⊢ ↑(Polynomial.aeval (↑(algebraMap A B) a)) (X - ↑C a) = 0 [PROOFSTEP] rw [map_sub, aeval_C, aeval_X, sub_self] [GOAL] case refine'_2 A : Type u_1 B : Type u_2 B' : Type u_3 inst✝² : CommRing A inst✝¹ : Ring B inst✝ : Algebra A B x : B a : A hf : Injective ↑(algebraMap A B) ✝ : Nontrivial A ⊢ ∀ (q : A[X]), degree q < degree (X - ↑C a) → q = 0 ∨ ↑(Polynomial.aeval (↑(algebraMap A B) a)) q ≠ 0 [PROOFSTEP] simp_rw [or_iff_not_imp_left] [GOAL] case refine'_2 A : Type u_1 B : Type u_2 B' : Type u_3 inst✝² : CommRing A inst✝¹ : Ring B inst✝ : Algebra A B x : B a : A hf : Injective ↑(algebraMap A B) ✝ : Nontrivial A ⊢ ∀ (q : A[X]), degree q < degree (X - ↑C a) → ¬q = 0 → ↑(Polynomial.aeval (↑(algebraMap A B) a)) q ≠ 0 [PROOFSTEP] intro q hl h0 [GOAL] case refine'_2 A : Type u_1 B : Type u_2 B' : Type u_3 inst✝² : CommRing A inst✝¹ : Ring B inst✝ : Algebra A B x : B a : A hf : Injective ↑(algebraMap A B) ✝ : Nontrivial A q : A[X] hl : degree q < degree (X - ↑C a) h0 : ¬q = 0 ⊢ ↑(Polynomial.aeval (↑(algebraMap A B) a)) q ≠ 0 [PROOFSTEP] rw [← natDegree_lt_natDegree_iff h0, natDegree_X_sub_C, Nat.lt_one_iff] at hl [GOAL] case refine'_2 A : Type u_1 B : Type u_2 B' : Type u_3 inst✝² : CommRing A inst✝¹ : Ring B inst✝ : Algebra A B x : B a : A hf : Injective ↑(algebraMap A B) ✝ : Nontrivial A q : A[X] hl : natDegree q = 0 h0 : ¬q = 0 ⊢ ↑(Polynomial.aeval (↑(algebraMap A B) a)) q ≠ 0 [PROOFSTEP] rw [eq_C_of_natDegree_eq_zero hl] at h0 ⊢ [GOAL] case refine'_2 A : Type u_1 B : Type u_2 B' : Type u_3 inst✝² : CommRing A inst✝¹ : Ring B inst✝ : Algebra A B x : B a : A hf : Injective ↑(algebraMap A B) ✝ : Nontrivial A q : A[X] hl : natDegree q = 0 h0 : ¬↑C (coeff q 0) = 0 ⊢ ↑(Polynomial.aeval (↑(algebraMap A B) a)) (↑C (coeff q 0)) ≠ 0 [PROOFSTEP] rwa [aeval_C, map_ne_zero_iff _ hf, ← C_ne_zero] [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝² : CommRing A inst✝¹ : Ring B inst✝ : Algebra A B x : B a : A[X] hx : IsIntegral A x hamonic : Monic a hdvd : DvdNotUnit a (minpoly A x) ⊢ ↑(Polynomial.aeval x) a ≠ 0 [PROOFSTEP] refine' fun ha => (min A x hamonic ha).not_lt (degree_lt_degree _) [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝² : CommRing A inst✝¹ : Ring B inst✝ : Algebra A B x : B a : A[X] hx : IsIntegral A x hamonic : Monic a hdvd : DvdNotUnit a (minpoly A x) ha : ↑(Polynomial.aeval x) a = 0 ⊢ natDegree a < natDegree (minpoly A x) [PROOFSTEP] obtain ⟨_, c, hu, he⟩ := hdvd [GOAL] case intro.intro.intro A : Type u_1 B : Type u_2 B' : Type u_3 inst✝² : CommRing A inst✝¹ : Ring B inst✝ : Algebra A B x : B a : A[X] hx : IsIntegral A x hamonic : Monic a ha : ↑(Polynomial.aeval x) a = 0 left✝ : a ≠ 0 c : A[X] hu : ¬IsUnit c he : minpoly A x = a * c ⊢ natDegree a < natDegree (minpoly A x) [PROOFSTEP] have hcm := hamonic.of_mul_monic_left (he.subst <\| monic hx) [GOAL] case intro.intro.intro A : Type u_1 B : Type u_2 B' : Type u_3 inst✝² : CommRing A inst✝¹ : Ring B inst✝ : Algebra A B x : B a : A[X] hx : IsIntegral A x hamonic : Monic a ha : ↑(Polynomial.aeval x) a = 0 left✝ : a ≠ 0 c : A[X] hu : ¬IsUnit c he : minpoly A x = a * c hcm : Monic c ⊢ natDegree a < natDegree (minpoly A x) [PROOFSTEP] rw [he, hamonic.natDegree_mul hcm] -- TODO: port Nat.lt_add_of_zero_lt_left from lean3 core [GOAL] case intro.intro.intro A : Type u_1 B : Type u_2 B' : Type u_3 inst✝² : CommRing A inst✝¹ : Ring B inst✝ : Algebra A B x : B a : A[X] hx : IsIntegral A x hamonic : Monic a ha : ↑(Polynomial.aeval x) a = 0 left✝ : a ≠ 0 c : A[X] hu : ¬IsUnit c he : minpoly A x = a * c hcm : Monic c ⊢ natDegree a < natDegree a + natDegree c [PROOFSTEP] apply lt_add_of_pos_right [GOAL] case intro.intro.intro.h A : Type u_1 B : Type u_2 B' : Type u_3 inst✝² : CommRing A inst✝¹ : Ring B inst✝ : Algebra A B x : B a : A[X] hx : IsIntegral A x hamonic : Monic a ha : ↑(Polynomial.aeval x) a = 0 left✝ : a ≠ 0 c : A[X] hu : ¬IsUnit c he : minpoly A x = a * c hcm : Monic c ⊢ 0 < natDegree c [PROOFSTEP] refine (lt_of_not_le fun h => hu ?_) [GOAL] case intro.intro.intro.h A : Type u_1 B : Type u_2 B' : Type u_3 inst✝² : CommRing A inst✝¹ : Ring B inst✝ : Algebra A B x : B a : A[X] hx : IsIntegral A x hamonic : Monic a ha : ↑(Polynomial.aeval x) a = 0 left✝ : a ≠ 0 c : A[X] hu : ¬IsUnit c he : minpoly A x = a * c hcm : Monic c h : natDegree c ≤ 0 ⊢ IsUnit c [PROOFSTEP] rw [eq_C_of_natDegree_le_zero h, ← Nat.eq_zero_of_le_zero h, ← leadingCoeff, hcm.leadingCoeff, C_1] [GOAL] case intro.intro.intro.h A : Type u_1 B : Type u_2 B' : Type u_3 inst✝² : CommRing A inst✝¹ : Ring B inst✝ : Algebra A B x : B a : A[X] hx : IsIntegral A x hamonic : Monic a ha : ↑(Polynomial.aeval x) a = 0 left✝ : a ≠ 0 c : A[X] hu : ¬IsUnit c he : minpoly A x = a * c hcm : Monic c h : natDegree c ≤ 0 ⊢ IsUnit 1 [PROOFSTEP] exact isUnit_one [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Algebra A B x : B inst✝¹ : IsDomain A inst✝ : IsDomain B hx : IsIntegral A x ⊢ Irreducible (minpoly A x) [PROOFSTEP] refine' (irreducible_of_monic (monic hx) <\| ne_one A x).2 fun f g hf hg he => _ [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Algebra A B x : B inst✝¹ : IsDomain A inst✝ : IsDomain B hx : IsIntegral A x f g : A[X] hf : Monic f hg : Monic g he : f * g = minpoly A x ⊢ f = 1 ∨ g = 1 [PROOFSTEP] rw [← hf.isUnit_iff, ← hg.isUnit_iff] [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Algebra A B x : B inst✝¹ : IsDomain A inst✝ : IsDomain B hx : IsIntegral A x f g : A[X] hf : Monic f hg : Monic g he : f * g = minpoly A x ⊢ IsUnit f ∨ IsUnit g [PROOFSTEP] by_contra' h [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Algebra A B x : B inst✝¹ : IsDomain A inst✝ : IsDomain B hx : IsIntegral A x f g : A[X] hf : Monic f hg : Monic g he : f * g = minpoly A x h : ¬IsUnit f ∧ ¬IsUnit g ⊢ False [PROOFSTEP] have heval := congr_arg (Polynomial.aeval x) he [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Algebra A B x : B inst✝¹ : IsDomain A inst✝ : IsDomain B hx : IsIntegral A x f g : A[X] hf : Monic f hg : Monic g he : f * g = minpoly A x h : ¬IsUnit f ∧ ¬IsUnit g heval : ↑(Polynomial.aeval x) (f * g) = ↑(Polynomial.aeval x) (minpoly A x) ⊢ False [PROOFSTEP] rw [aeval A x, aeval_mul, mul_eq_zero] at heval [GOAL] A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Algebra A B x : B inst✝¹ : IsDomain A inst✝ : IsDomain B hx : IsIntegral A x f g : A[X] hf : Monic f hg : Monic g he : f * g = minpoly A x h : ¬IsUnit f ∧ ¬IsUnit g heval : ↑(Polynomial.aeval x) f = 0 ∨ ↑(Polynomial.aeval x) g = 0 ⊢ False [PROOFSTEP] cases' heval with heval heval [GOAL] case inl A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Algebra A B x : B inst✝¹ : IsDomain A inst✝ : IsDomain B hx : IsIntegral A x f g : A[X] hf : Monic f hg : Monic g he : f * g = minpoly A x h : ¬IsUnit f ∧ ¬IsUnit g heval : ↑(Polynomial.aeval x) f = 0 ⊢ False [PROOFSTEP] exact aeval_ne_zero_of_dvdNotUnit_minpoly hx hf ⟨hf.ne_zero, g, h.2, he.symm⟩ heval [GOAL] case inr A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Algebra A B x : B inst✝¹ : IsDomain A inst✝ : IsDomain B hx : IsIntegral A x f g : A[X] hf : Monic f hg : Monic g he : f * g = minpoly A x h : ¬IsUnit f ∧ ¬IsUnit g heval : ↑(Polynomial.aeval x) g = 0 ⊢ False [PROOFSTEP] refine' aeval_ne_zero_of_dvdNotUnit_minpoly hx hg ⟨hg.ne_zero, f, h.1, _⟩ heval [GOAL] case inr A : Type u_1 B : Type u_2 B' : Type u_3 inst✝⁴ : CommRing A inst✝³ : Ring B inst✝² : Algebra A B x : B inst✝¹ : IsDomain A inst✝ : IsDomain B hx : IsIntegral A x f g : A[X] hf : Monic f hg : Monic g he : f * g = minpoly A x h : ¬IsUnit f ∧ ¬IsUnit g heval : ↑(Polynomial.aeval x) g = 0 ⊢ minpoly A x = g * f [PROOFSTEP] rw [mul_comm, he]
import data.nat.gcd import algebra.big_operators import data.zmod.basic def divides (d n: ℤ):= ∃l:ℤ, n = dl def is_congruent_mod_n (a b n: ℤ) := divides n (b-a) def relatively_prime(a b:ℤ) := ∀l:ℤ, ¬(divides l a) ∨ ¬(divides l b) theorem relatively_prime_sum_to_one (a b : ℤ) (h : relatively_prime a b) : ∃ m n: ℤ, m a + n * b = 1 := sorry --helper theorem that is not true :(-- -- the iff is not true, but I can't use it without it-- theorem divides_in_product (a b c: ℤ): ab = c ↔ divides a c:= sorry --Generalized Euclid Lemma -- theorem divides_product_coprime_and_not_coprime (a b n:ℤ) (c: divides n (ab)) (d: relatively_prime a n): divides n b:= begin -- cases breaks down existential qualifiers cases c with l k, cases relatively_prime_sum_to_one a n d with m h, cases h with q manb_eq_one, -- multiplies both sides by b have := congr_arg (has_mul.mul b) manb_eq_one, -- does not work if I covert to naturals rw [mul_one] at this, -- this : b * (m * a + q * n) = b -- order-changing rw [mul_add] at this, rw [mul_comm] at this, rw [mul_assoc] at this, rw k at this, rw [mul_comm] at this, rw [mul_assoc] at this, rw [add_comm] at this, rw [mul_comm] at this, rw [mul_assoc] at this, rw [mul_left_comm] at this, rw <- mul_add at this, rw divides_in_product n (qb+lm) b at this, -- can use "apply" instead and get rid of iff apply this, end theorem cancel_out_mod_n (a b c n: ℤ) (p: is_congruent_mod_n (ab) (ac) n) (q: relatively_prime a n) : is_congruent_mod_n b c n := begin cases p with j k, rw <-mul_sub at k, --does not work if I convert to naturals rw eq_comm at k, rw divides_in_product n j (a(c-b)) at k, -- cannot use "apply" and get rid of iff apply divides_product_coprime_and_not_coprime a (c-b) n, { apply k, }, { apply q, }, end -- residue class -- def reduced_residue_class (n : nat) : finset nat := (finset.range n).filter $ assume k, k.coprime n def prod_rsc (n) : nat := (reduced_residue_class n).prod id def Euler_phi (n) : nat := (reduced_residue_class n).sum $ λ k, 1 -- not scaled :( -- def scaled_residue_class (a n: nat): finset nat := (reduced_residue_class n).map $ λ k, ka -- def prod_scaled_rsc (a n: nat): nat := (scaled_residue_class a n).prod id --congruence of residue classes -- -- theorem congruent_residue_classes (a n: nat) (c:a.coprime n): -- is_congruent_mod_n (prod_rsc n)(prod_scaled_rsc a n) n:= sorry R ≃+* R -- mult by a is automorphism theorem mult_by_a_is_auto (a n : nat) (c:a.coprime n): ∃ inv : (reduced_residue_class n) → (reduced_residue_class n), (inv ∘ (λ k, a * k) = id) ∧ ((λ k, a * k) ∘ inv = id) := begin sorry, end --Zulip proof of Bézout lemma bezout (a : ℕ) (b : ℕ) (ha : a ≥ 1) (hb : b ≥ 1) (h : nat.gcd a b = 1) : ∃ x : ℕ, ∃ y : ℕ, ax + 1 = by := begin let k := max (int.nat_abs (nat.gcd_a a b)) (int.nat_abs (nat.gcd_b a b)) + 1, let x : ℤ := bk - nat.gcd_a a b, let y : ℤ := ak + nat.gcd_b a b, refine ⟨int.to_nat x, int.to_nat y, int.coe_nat_inj _⟩, suffices : (a * int.to_nat x + 1 : ℤ) = b * int.to_nat y, {simpa}, have k1 : 1 ≤ k := nat.le_add_left _ _, have ha' : (1:ℤ) ≤ a := int.coe_nat_le.2 ha, have hb' : (1:ℤ) ≤ b := int.coe_nat_le.2 hb, have x0 : 0 ≤ x, { refine sub_nonneg.2 _, have := mul_le_mul_of_nonneg_right hb' (int.coe_nat_nonneg k), rw one_mul at this, refine le_trans (le_trans int.le_nat_abs _) this, refine int.coe_nat_le.2 _, exact nat.le_succ_of_le (le_max_left _ _) }, have y0 : 0 ≤ y, { refine sub_le_iff_le_add.1 _, rw zero_sub, have := mul_le_mul_of_nonneg_right ha' (int.coe_nat_nonneg k), rw one_mul at this, refine le_trans (le_trans int.le_nat_abs _) this, rw [int.nat_abs_neg, int.coe_nat_le], exact nat.le_succ_of_le (le_max_right _ _) }, rw [int.to_nat_of_nonneg x0, int.to_nat_of_nonneg y0], have := nat.gcd_eq_gcd_ab a b, rw [h, int.coe_nat_one] at this, rw [this, mul_sub, ← add_assoc, sub_add_cancel, mul_left_comm, ← mul_add], end -- alternative definitions of residue classes -- -- lean definition for residue class -- example (n : ℕ+) : units (zmod n) ≃ {x : zmod n // nat.coprime x.val n} := zmod.units_equiv_coprime -- lean doesn't know if this is finite def rsc (n: nat) := {k: nat\| k<n ∧ k.coprime n} #print units
#ifndef VALUE_R #define VALUE_R #include <stdarg.h> struct Type { void * (* new) (va_list ap); double (* exec) (const void * tree); void (* delete) (void * tree); }; double exec (const void * tree); struct Bin { const void * type; void * left, * right; }; #define left(tree) (((struct Bin ) tree) -> left) #define right(tree) (((struct Bin ) tree) -> right) void * mkBin (va_list ap); void freeBin (void * tree); #endif
# Scenario E - Peakshape Variation (pVoigt model - results evaluation) This file is used to evaluate the inference results. In this scenario the peakshape in a spectrum with a fixed number of peaks is varied from Gaussian (n = 0.0) to Lorentzian (n = 1.0). All datasets contain 3 peaks and the noise level is kept constant at 1%. The model used in the inference of the parameters is formulated as follows: \begin{equation} \large y = f(x) = \sum\limits_{m=1}^M \big[A_m \cdot f_{pseudo-Voigt}(x)\big] + \epsilon \end{equation} where: \begin{equation} \large f_{pseudo-Voigt}(x) = \eta \cdot \frac{\sigma_m^2}{(x-\mu_m)^2 + \sigma_m^2} + (1 - \eta) \cdot e^{-\frac{(x-\mu_m)^2}{2\cdot\sigma_m^2}} \end{equation} ```python %matplotlib inline import numpy as np import pandas as pd import matplotlib.pyplot as plt import pymc3 as pm import arviz as az #az.style.use('arviz-darkgrid') print('Running on PyMC3 v{}'.format(pm.__version__)) ``` WARNING (theano.tensor.blas): Using NumPy C-API based implementation for BLAS functions. Running on PyMC3 v3.8 ## Load results summary ```python # load results from disk #fname = './scenario_peakshape_pvoigt.csv' fname = './scenario_peakshape_pvoigt_lrun.csv' df = pd.read_csv(fname) df.index += 1 df ``` <div> <style scoped> .dataframe tbody tr th:only-of-type { vertical-align: middle; } .dataframe tbody tr th { vertical-align: top; } .dataframe thead th { text-align: right; } </style> <table border="1" class="dataframe"> <thead> <tr style="text-align: right;"> <th></th> <th>r_hat</th> <th>mcse</th> <th>ess</th> <th>bfmi</th> <th>r2</th> <th>waic</th> <th>epsilon</th> <th>epsilon_real</th> <th>eta</th> <th>eta_real</th> <th>cat</th> </tr> </thead> <tbody> <tr> <th>1</th> <td>1.780909</td> <td>0.226636</td> <td>104.363636</td> <td>1.249257</td> <td>0.999856</td> <td>-4675.470979</td> <td>0.049972</td> <td>0.05</td> <td>0.005661</td> <td>0.0</td> <td>1</td> </tr> <tr> <th>2</th> <td>1.000000</td> <td>0.000000</td> <td>4383.363636</td> <td>1.029887</td> <td>0.999035</td> <td>-2331.742859</td> <td>0.109758</td> <td>0.05</td> <td>0.000332</td> <td>0.0</td> <td>1</td> </tr> <tr> <th>3</th> <td>1.070909</td> <td>0.049455</td> <td>684.363636</td> <td>1.014762</td> <td>0.999820</td> <td>-4207.406117</td> <td>0.059134</td> <td>0.05</td> <td>0.000384</td> <td>0.0</td> <td>1</td> </tr> <tr> <th>4</th> <td>1.005455</td> <td>0.003818</td> <td>1720.727273</td> <td>1.037991</td> <td>0.999818</td> <td>-4715.487126</td> <td>0.050034</td> <td>0.05</td> <td>0.008271</td> <td>0.0</td> <td>1</td> </tr> <tr> <th>5</th> <td>1.000000</td> <td>0.000727</td> <td>2124.454545</td> <td>1.044191</td> <td>0.999872</td> <td>-4726.544298</td> <td>0.049924</td> <td>0.05</td> <td>0.001852</td> <td>0.0</td> <td>1</td> </tr> <tr> <th>...</th> <td>...</td> <td>...</td> <td>...</td> <td>...</td> <td>...</td> <td>...</td> <td>...</td> <td>...</td> <td>...</td> <td>...</td> <td>...</td> </tr> <tr> <th>196</th> <td>1.000000</td> <td>0.000182</td> <td>2297.909091</td> <td>1.000964</td> <td>0.999846</td> <td>-4766.885947</td> <td>0.049235</td> <td>0.05</td> <td>0.998066</td> <td>1.0</td> <td>5</td> </tr> <tr> <th>197</th> <td>1.450909</td> <td>0.557818</td> <td>1605.181818</td> <td>1.008004</td> <td>0.999950</td> <td>-4836.525261</td> <td>0.048113</td> <td>0.05</td> <td>0.999041</td> <td>1.0</td> <td>5</td> </tr> <tr> <th>198</th> <td>1.000000</td> <td>0.000000</td> <td>4905.090909</td> <td>1.054635</td> <td>0.999424</td> <td>-2786.356407</td> <td>0.094399</td> <td>0.05</td> <td>0.977116</td> <td>1.0</td> <td>5</td> </tr> <tr> <th>199</th> <td>1.000000</td> <td>0.000000</td> <td>3922.636364</td> <td>0.997197</td> <td>0.999916</td> <td>-4743.426499</td> <td>0.049649</td> <td>0.05</td> <td>0.999503</td> <td>1.0</td> <td>5</td> </tr> <tr> <th>200</th> <td>1.000000</td> <td>0.000000</td> <td>3774.636364</td> <td>1.053122</td> <td>0.999882</td> <td>-4733.435338</td> <td>0.049827</td> <td>0.05</td> <td>0.999430</td> <td>1.0</td> <td>5</td> </tr> </tbody> </table> <p>200 rows × 11 columns</p> </div> ```python # number of succesfull convergences suc = df.loc[(df['r_hat'] <= 1.1) & (df['r2'] >= 0.99)] len(suc) ``` 189 ```python import seaborn as sns sns.set(style="ticks", rc={'figure.figsize':(8,6)}, font_scale=1.35) # color palette pal = sns.cubehelix_palette(8, rot=-.5, dark=.3) ax = sns.violinplot(x='eta_real', y='epsilon', data=suc, palette=pal, linewidth=2) \ .set_title("Peakshape eta (real) vs. Noise level (inferred)") plt.savefig('violinplot_noise.png', dpi=150) ``` ```python #ax = sns.boxplot(x='eta_real', y='epsilon', data=suc, palette=pal, linewidth=2, fliersize=2) \ # .set_title("Peakshape eta (real) vs. Noise level (inferred)") ax = sns.boxplot(x='eta_real', y='epsilon', data=suc, palette=sns.color_palette("Purples"), linewidth=1, fliersize=2.5) plt.savefig('boxplot_noise.png', dpi=150) ``` ```python ax = sns.violinplot(x='eta_real', y='eta', data=suc, palette=pal, linewidth=2) \ .set_title("Peakshape eta (real) vs. Peakshape eta (inferred)") plt.savefig('violinplot_peakshape.png', dpi=150) ``` ```python #ax = sns.boxplot(x='eta_real', y='eta', data=suc, palette=pal, linewidth=2, fliersize=2) \ # .set_title("Peakshape eta (real) vs. Peakshape eta (inferred)") #ax = sns.boxplot(x='eta_real', y='eta', data=suc, palette=pal, linewidth=2, fliersize=2) ax = sns.boxplot(x='eta_real', y='eta', data=suc, palette=sns.color_palette("Purples"), linewidth=1, fliersize=2.5) plt.savefig('boxplot_peakshape.png', dpi=150) ``` ```python # table values suc_000 = suc.loc[(suc['eta_real'] == 0.0)] print("N = {0}".format(len(suc_000))) suc_000['eta'].quantile([.25, .5, .75]) ``` N = 35 0.25 0.000404 0.50 0.000938 0.75 0.001384 Name: eta, dtype: float64 ```python suc_025 = suc.loc[(suc['eta_real'] == 0.25)] print("N = {0}".format(len(suc_025))) suc_025['eta'].quantile([.25, .5, .75]) ``` N = 39 0.25 0.227337 0.50 0.246641 0.75 0.250805 Name: eta, dtype: float64 ```python suc_050 = suc.loc[(suc['eta_real'] == 0.5)] print("N = {0}".format(len(suc_050))) suc_050['eta'].quantile([.25, .5, .75]) ``` N = 39 0.25 0.489433 0.50 0.499222 0.75 0.500169 Name: eta, dtype: float64 ```python suc_075 = suc.loc[(suc['eta_real'] == 0.75)] print("N = {0}".format(len(suc_075))) suc_075['eta'].quantile([.25, .5, .75]) ``` N = 38 0.25 0.733126 0.50 0.748641 0.75 0.750410 Name: eta, dtype: float64 ```python suc_100 = suc.loc[(suc['eta_real'] == 1.0)] print("N = {0}".format(len(suc_100))) suc_100['eta'].quantile([.25, .5, .75]) ``` N = 38 0.25 0.990277 0.50 0.997331 0.75 0.998399 Name: eta, dtype: float64 ```python ax = sns.scatterplot(x="eta", y="epsilon", data=suc, hue="eta_real", s=100, palette="jet", legend="full") plt.savefig('scatterplot_noise_peakshape.png', dpi=150) ``` ## Statistics per peak shape factor ```python for i in [1,2,3,4,5]: cat = df.loc[(df['cat'] == i)] print('cat: {0} r-hat: {1:.2f}'.format(i, cat['r_hat'].mean())) print('cat: {0} r2 : {1:.4f}'.format(i, cat['r2'].mean())) print('cat: {0} waic : {1:.2f}'.format(i, cat['waic'].mean())) print('cat: {0} mcse : {1:.4f}'.format(i, cat['mcse'].mean())) print('cat: {0} ess : {1:.1f}'.format(i, cat['ess'].mean())) print('cat: {0} bfmi : {1:.4f}'.format(i, cat['bfmi'].mean())) print('cat: {0} epsi : {1:.4f}'.format(i, cat['epsilon'].mean())) print('cat: {0} eta : {1:.4f}'.format(i, cat['eta'].mean())) print('\n') ``` cat: 1 r-hat: 1.08 cat: 1 r2 : 0.9997 cat: 1 waic : -4342.65 cat: 1 mcse : 0.0823 cat: 1 ess : 3298.4 cat: 1 bfmi : 1.0416 cat: 1 epsi : 0.0585 cat: 1 eta : 0.0014 cat: 2 r-hat: 1.01 cat: 2 r2 : 0.9995 cat: 2 waic : -3972.92 cat: 2 mcse : 0.0267 cat: 2 ess : 3879.6 cat: 2 bfmi : 1.0516 cat: 2 epsi : 0.0762 cat: 2 eta : 0.2313 cat: 3 r-hat: 1.01 cat: 3 r2 : 0.9995 cat: 3 waic : -4069.55 cat: 3 mcse : 0.0199 cat: 3 ess : 3676.4 cat: 3 bfmi : 1.0493 cat: 3 epsi : 0.0713 cat: 3 eta : 0.4816 cat: 4 r-hat: 1.00 cat: 4 r2 : 0.9988 cat: 4 waic : -3637.68 cat: 4 mcse : 0.0039 cat: 4 ess : 3647.7 cat: 4 bfmi : 1.0568 cat: 4 epsi : 0.0971 cat: 4 eta : 0.7067 cat: 5 r-hat: 1.02 cat: 5 r2 : 0.9996 cat: 5 waic : -4145.78 cat: 5 mcse : 0.0217 cat: 5 ess : 3287.5 cat: 5 bfmi : 1.0257 cat: 5 epsi : 0.0649 cat: 5 eta : 0.9747 ```python ```
module SymRCM # (C) 2020, Petr Krysl export symrcm using SparseArrays """ adjgraph(A) Compute the adjacency graph from a sparse matrix. The sparse matrix `A` is assumed to be symmetric. The results will be wrong if it isn't. - `sortbydeg`: Should the neighbor lists be sorted by column degree? The default is `true`, but often results of very similar quality are obtained when this is set to `false` and the lists are not sorted. The second option is much faster, as the sorting is expensive. """ function adjgraph(A::SparseMatrixCSC; sortbydeg = true) colptr = A.colptr rowval = A.rowval ncols = length(colptr)-1 neighbors = Vector{Vector{eltype(colptr)}}(undef, ncols) cdeg = diff(colptr) # the degree is colptr[j+1]-colptr[j] for j in 1:ncols cstart = colptr[j] jdeg = cdeg[j] neighbors[j] = [rowval[cstart+m-1] for m in 1:jdeg] end # All of these sorts can be done in parallel, they are totally independent. # The question is when to switch over to parallel execution so as to # amortize the cost of starting up threads. if sortbydeg for j in 1:ncols sort!(neighbors[j], by = j -> cdeg[j]) end end return neighbors end """ nodedegrees(adjgr::Vector{Vector{Int}}) Compute the degrees of the nodes in the adjacency graph. conn = [9 1 8 4; 1 3 2 8; 8 2 7 5; 2 6 7 7]; nfens = 9; adjgr = adjgraph(conn, nfens) nodedegrees(adjgr) julia> degrees = node_degrees(adjgr) 9-element Array{Int64,1}: 5 6 3 3 3 2 4 7 3 """ function nodedegrees(adjgr::Vector{Vector{T}}) where {T} degrees = fill(0, length(adjgr)) for k = 1:length(adjgr) degrees[k] = length(adjgr[k]) end return degrees end """ symrcm(adjgr::Vector{Vector{T}}, degrees::Vector{T}) where {T} Reverse Cuthill-McKee node-renumbering algorithm. """ function symrcm(adjgr::Vector{Vector{T}}, degrees::Vector{T}) where {T} @assert length(adjgr) == length(degrees) # Initialization n = length(adjgr) ndegperm = sortperm(degrees) # sorted nodal degrees inR = fill(false, n) # Is a node in the result list? inQ = fill(false, n) # Is a node in the queue? R = T[] sizehint!(R, n) Q = T[] # Node queue sizehint!(Q, n) while true P = zero(T) # Find the next node to start from while !isempty(ndegperm) i = popfirst!(ndegperm) if !inR[i] P = i break end end if P == zero(T) break # That was the last node end # Now we have a node to start from: put it into the result list push!(R, P); inR[P] = true empty!(Q) # empty the queue append!(Q, adjgr[P]); inQ[adjgr[P]] .= true # put adjacent nodes in queue while length(Q) >= 1 C = popfirst!(Q) # child to put into the result list inQ[C] = false # make note: it is not in the queue anymore if !inR[C] push!(R, C); inR[C] = true end for i in adjgr[C] # add all adjacent nodes into the queue if (!inR[i]) && (!inQ[i]) # contingent on not being in result/queue push!(Q, i); inQ[i] = true end end end end return reverse(R) # reverse the result list end """ symrcm(A::SparseMatrixCSC; sortbydeg = true) Reverse Cuthill-McKee node-renumbering algorithm. Compute the adjacency graph from a sparse matrix. The sparse matrix `A` is assumed to be symmetric. The results will be wrong if it isn't. - `sortbydeg`: Should the neighbor lists be sorted by column degree? The default is `true`, but often results of very similar quality are obtained when this is set to `false` and the lists are not sorted. The second option can be much faster, as the sorting is expensive when the neighbor lists are long. """ function symrcm(A::SparseMatrixCSC; sortbydeg = true) ag = adjgraph(A; sortbydeg = sortbydeg) nd = nodedegrees(ag) return symrcm(ag, nd) end end # module
{-# OPTIONS --without-K --safe #-} -- Multicategories but over an 'index' type, rather than forcing Fin n module Categories.Multi.Category.Indexed where open import Level open import Data.Fin.Base using (Fin) open import Data.Product using (Σ; uncurry; curry; _×_; _,_; proj₁; proj₂) open import Data.Product.Properties open import Data.Unit.Polymorphic using (⊤; tt) open import Data.Vec.Functional open import Function.Base using (const) renaming (_∘_ to _●_; id to id→) open import Function.Equality using (_⟨$⟩_) -- note how this is Function.Inverse instead of the one from Function. open import Function.Inverse as Inv renaming (id to id↔; _∘_ to trans) open import Relation.Binary.Core using (Rel) open import Relation.Binary.PropositionalEquality using (_≡_; refl) -- any point can be lifted to a function from ⊤ pointed : {s t : Level} {S : Set s} (x : S) → ⊤ {t} → S pointed x _ = x -- the standard library doesn't seem to have the 'right' version of these. ⊤×K↔K : {t k : Level} {K : Set k} → (⊤ {t} × K) ↔ K ⊤×K↔K = inverse proj₂ (tt ,_) (λ _ → refl) λ _ → refl K×⊤↔K : {t k : Level} {K : Set k} → (K × ⊤ {t}) ↔ K K×⊤↔K = inverse proj₁ (_, tt) (λ _ → refl) λ _ → refl ⊤×⊤↔⊤ : {t : Level} → (⊤ {t} × ⊤) ↔ ⊤ ⊤×⊤↔⊤ = inverse proj₁ (λ x → x , x) (λ _ → refl) λ _ → refl Σ-assoc : {a b c : Level} {I : Set a} {J : I → Set b} {K : Σ I J → Set c} → Σ (Σ I J) K ↔ Σ I (λ i → Σ (J i) (curry K i)) Σ-assoc {I = I} {J} {K} = inverse g f (λ _ → refl) λ _ → refl where f : Σ I (λ i → Σ (J i) (λ j → K (i , j))) → Σ (Σ I J) K f (i , j , k) = (i , j) , k g : Σ (Σ I J) K → Σ I (λ i → Σ (J i) (λ j → K (i , j))) g ((i , j) , k) = i , j , k -- the ι level is for the 'index' (and to not steal 'i') -- The important part is that in _∘_, there is no flattening of the -- index set. But also _≈[_]_ builds in an explicit equivalence -- that allows one to properly re-index things. The classical view -- of MultiCategory sweeps all of that under the rug, which gives -- Agda conniptions (and rightfully so). The advantage of doing it -- this way makes it clear that the 3 laws are based on the underlying -- 3 laws that hold for (dependent!) product. -- The upshot is that this version of MultiCategory makes no finiteness -- assumption whatsoever. The index sets involved could be huge, -- without any issues. -- Note that this still isn't Symmetric Multicategory. The renaming that -- happens on indices say nothing about the relation to the contents -- of the other Hom set. record MultiCategory {o ℓ e ı : Level} : Set (suc (o ⊔ ℓ ⊔ e ⊔ ı)) where infix 4 _≈[_]_ infixr 9 _∘_ field Obj : Set o Hom : {I : Set ı} → (I → Obj) → Obj → Set ℓ id : (o : Obj) → Hom {⊤} (pointed o) o _∘_ : {I : Set ı} {aₙ : I → Obj} {a : Obj} {J : I → Set ı} {v : (i : I) → J i → Obj} {b : I → Obj} → Hom {I} aₙ a → ((i : I) → Hom (v i) (b i)) → Hom {Σ I J} (uncurry v) a _≈[_]_ : {I J : Set ı} {aₙ : I → Obj} {a : Obj} → Hom {I} aₙ a → (σ : I ↔ J) → Hom {J} (aₙ ● ( Inverse.from σ ⟨$⟩_ )) a → Set e identityˡ : {K : Set ı} {aₖ : K → Obj} {a : Obj} {f : Hom aₖ a} → id a ∘ pointed f ≈[ ⊤×K↔K ] f identityʳ : {K : Set ı} {aₖ : K → Obj} {a : Obj} {f : Hom aₖ a} → f ∘ (id ● aₖ) ≈[ K×⊤↔K ] f identity² : {a : Obj} → id a ∘ pointed (id a) ≈[ ⊤×⊤↔⊤ ] id a assoc : -- the 3 index sets {I : Set ı} {J : I → Set ı} {K : Σ I J → Set ı} -- the 3 sets of (indexed) objects {vh : I → Obj} {bh : Obj} {vg : (i : I) → J i → Obj} {bg : I → Obj} {vf : (h : Σ I J) → K h → Obj} {bf : Σ I J → Obj} -- the 3 Homs {h : Hom vh bh} {g : (i : I) → Hom (vg i) (bg i)} {f : (k : Σ I J) → Hom (vf k) (bf k)} → -- and their relation under composition (h ∘ g) ∘ f ≈[ Σ-assoc ] h ∘ (λ i → g i ∘ curry f i) -- we also need that _≈[_]_ is, in an appropriate sense, an equivalence relation, which in this case -- means that _≈[ id↔ ]_ is. In other words, we don't care when transport is over 'something else'. refl≈ : {I : Set ı} {aₙ : I → Obj} {a : Obj} → {h : Hom {I} aₙ a} → h ≈[ id↔ ] h sym≈ : {I : Set ı} {aₙ : I → Obj} {a : Obj} → {g h : Hom {I} aₙ a} → g ≈[ id↔ ] h → h ≈[ id↔ ] g trans≈ : {I : Set ı} {aₙ : I → Obj} {a : Obj} → {f g h : Hom {I} aₙ a} → f ≈[ id↔ ] g → g ≈[ id↔ ] h → f ≈[ id↔ ] h -- we probably need ∘-resp-≈ too.
Jacob Morgan is a well-known voice in field of OpenWork thanks to a multimedia approach that includes two books, the Future of Work podcast, a regular column on Forbes and an engaging YouTube channel. We spoke with Jacob to get his read on the current state of the future of work. OpenWork: How did you come to be interested in the future of work? Jacob Morgan: Basically, I had bad jobs working for other people. My first job out of college, I went to work for a technology company in Los Angeles. When I interviewed for this job, they told me I was going to be doing all these great things and working on fun projects and traveling. Of course, I took the job. My commute was an hour and a half each way every day, and a couple months into the job, I was still doing data entry, cold calling, PowerPoint and just really boring, mundane type of work. Then one day, the CEO came into the office and handed me a $10 bill, and he said, “Go get me a cup of coffee.” That was one of two experiences I’ve had that led me to go off on my own. I really felt trapped, stuck, like a cog in the machine, so to speak. That was the last full-time job I had working for anybody else. I actually quit and went to that conference anyway, which is the best decision I’ve ever made. Those were two things in my life that helped me realized that there’s something very wrong with the way that we think about work — our expectations, our values, the way companies think about work. Ever since then, I’ve become very fascinated with what the future of work is going to look like. How do we prevent these types of experiences from happening? Why do they happen? That was about nine years ago, and I’ve been in the space ever since. OW: What do you think is the most pressing challenge for employers and workplaces to address right now regarding the evolution of work? JM: I think the biggest challenge is a mindset shift. The big assumption that organizations have always had is that people need to work there. They need money, they need jobs, they are working at your company because they need to. And because they need to work there, you can treat them however you want, and you can do whatever you want. Nobody cared about engagement, inspiration, creativity or health and wellbeing. All of this stuff is relatively new. What we’re realizing is that we now live in a very, very different world where you don’t need to work for a company anymore. In fact, not only do people have a lot of choices about where we can work — freelance, Uber, Airbnb, create stuff and sell it on Etsy, raise money on GoFundMe or Kickstarter — but also, technology has made it very, very competitive. If you’re working at an organization you don’t enjoy, very quickly you’ll find there are other opportunities out there. The biggest challenge companies have is: how do we change from creating an environment where people need to show up to creating an environment where people actually want to show up? That is where we’re seeing all these changes and shifts happening. OW: As a workplace futurist, what are the most interesting forecasts about workplace change or innovation you’ve seen come to fruition? JM: The biggest and best prediction that we’re starting to see is that all the assumptions we’ve always had about how work gets done are being challenged. Everything we had assumed to be true over the past hundred years since modern organizations were created, we’re starting to challenge and question. These are basic things like: does our office have to have walls? Do people need to show up to the office? Do managers have to have all the information and make all of the decisions on behalf of everyone else? Do we all have to use the same technologies and use the same tools? Do we need dress codes? All of these very basic things that have gone unquestioned, organizations are now starting to think about. I think that is the most amazing thing to see. OW: Are there any companies or industries that are, perhaps unexpectedly, great at addressing the evolving needs of the workplace? JM: I have seen organizations in pretty much every industry that are doing well and are forward-thinking, as well as those that aren’t. In our Future of Work Community, it’s interesting to see that we have a lot of organizations that are quite old and doing some very interesting things. And we also have a lot of financial institutions that are part of the Community that are thinking very differently about how work is getting done. I definitely don’t think the future of work is focused on technology companies. I think every organization around the world is aware that change is starting to happen, and they are making shifts towards acting on that change. I haven’t seen any really large industry disparities, and that’s what’s really cool about the future of work — it’s impacting pretty much every organization you can think of. OW: You mentioned your Future of Work Community. What were the key takeaways from the group’s Fall Future of Work Forum event held in New York earlier this month? JM: The Future of Work community includes companies from around the world that are all thinking differently about how the workplace is changing. Throughout the year, they have a digital community where they interact, they share what they’re working on, what their challenges are and what they’re struggling with. And these senior-level leaders provide feedback and help each other and support each other. There were several themes that I thought resonated at this month’s event. One was the idea of focusing on employee experiences [which] I define as a combination of three distinct environments: the physical environment, the cultural environment and the technological environment. For the first time, organizations are realizing how the combination of these three things really has amazing and positive impact for employees. Another theme is people analytics, which is huge. How do we take data and analytics from employees and start making sense of data to improve the workplace? That’s been a fascinating area. Another theme I noted is that HR and IT are building stronger relationships than ever before. HR is realizing they cannot propel organizations into the future of work without technology, and the technology professionals are realizing that they need to leverage HR to drive change. Jacob Morgan’s latest book is “The Future of Work: Attract New Talent, Build Better Leaders, and Create a Competitive Organization.” He can be reached at [email protected].
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 conj25synthconj1 : forall (lv0 : natural) (lv1 : natural) (lv2 : natural) (lv3 : natural), (@eq natural (plus (mult lv0 lv1) (plus Zero (plus lv1 lv2))) (plus lv3 lv2)). Admitted. QuickChick conj25synthconj1.
function p = getExpParams(predidx) switch predidx case 0 p.name = 'Pishchulin et., CVPR''13'; p.predFilename = ''; case 1 p.name = 'Pishchulin et., ICCV''13'; p.predFilename = './pred/pishchulin13iccv/pred_keypoints_lsp_oc'; p.colorIdxs = [1 1]; case 2 p.name = 'Tompson et al., NIPS''14'; p.predFilename = './pred/tompson14nips/pred_keypoints_lsp_pc'; p.colorIdxs = [2 1]; case 3 p.name = 'Chen&Yuille, NIPS''14'; p.predFilename = './pred/chen14nips/pred_keypoints_lsp_oc'; p.colorIdxs = [5 1]; case 4 p.name = 'Ramakrishna et al., ECCV''14'; p.predFilename = './pred/ramakrishna14eccv/pred_keypoints_lsp_oc'; p.colorIdxs = [4 1]; case 5 p.name = 'Ouyang et al., CVPR''14'; p.predFilename = './pred/ouyang14cvpr/pred_keypoints_lsp_oc'; p.colorIdxs = [7 1]; case 6 p.name = 'Pishchulin et., ICCV''13'; p.predFilename = './pred/pishchulin13iccv/pred_keypoints_lsp_pc'; p.colorIdxs = [1 1]; case 7 p.name = 'Pishchulin et., ICCV''13'; p.predFilename = './pred/pishchulin13iccv/pred_sticks_lsp_oc'; p.colorIdxs = [1 1]; case 8 p.name = 'Pishchulin et., ICCV''13'; p.predFilename = './pred/pishchulin13iccv/pred_sticks_lsp_pc'; p.colorIdxs = [1 1]; case 9 p.name = 'Tompson et al., NIPS''14'; p.predFilename = './pred/tompson14nips/pred_sticks_lsp_pc'; p.colorIdxs = [2 1]; case 10 p.name = 'Ramakrishna et al., ECCV''14'; p.predFilename = './pred/ramakrishna14eccv/pred_sticks_lsp_oc'; p.colorIdxs = [4 1]; case 11 p.name = 'Chen&Yuille, NIPS''14'; p.predFilename = './pred/chen14nips/pred_sticks_lsp_oc'; p.colorIdxs = [5 1]; case 12 p.name = 'Ouyang et al., CVPR''14'; p.predFilename = './pred/ouyang14cvpr/pred_sticks_lsp_oc'; p.colorIdxs = [7 1]; case 13 p.name = 'Pishchulin et al., CVPR''13'; p.predFilename = './pred/pishchulin13cvpr/pred_sticks_lsp_oc'; p.colorIdxs = [8 1]; case 14 p.name = 'Kiefel&Gehler, ECCV''14'; p.predFilename = './pred/kiefel14eccv/pred_sticks_lsp_oc'; p.colorIdxs = [8 2]; case 15 p.name = 'Kiefel&Gehler, ECCV''14'; p.predFilename = './pred/kiefel14eccv/pred_keypoints_lsp_oc'; p.colorIdxs = [8 3]; case 19 p.name = 'Bulat et al., ECCV''16'; p.predFilename = 'pred/1st/pred_keypoints_lsp_pc'; p.colorIdxs = [1 1]; case 20 p.name = 'Wei el al., CVPR''16'; p.predFilename = 'pred/2nd/pred_keypoints_lsp_pc'; p.colorIdxs = [2 1]; case 21 p.name = 'Insafutdinov et al., ECCV''16 .'; p.predFilename = 'pred/3rd/pred_keypoints_lsp_pc'; p.colorIdxs = [3 1]; case 22 p.name = 'Pishchulin et al., CVPR''16'; p.predFilename = 'pred/4th/pred_keypoints_lsp_pc'; p.colorIdxs = [4 1]; case 23 p.name = 'Lifshitz et al., ECCV''16'; p.predFilename = 'pred/5th/pred_keypoints_lsp_pc'; p.colorIdxs = [5 1]; case 24 p.name = 'Belagiannis et al., FG''17'; p.predFilename = 'pred/6th/pred_keypoints_lsp_pc'; p.colorIdxs = [6 1]; case 25 p.name = 'Yu et al., ECCV''16'; p.predFilename = 'pred/7th/pred_keypoints_lsp_pc'; p.colorIdxs = [7 1]; case 26 p.name = 'Rafi et al., BMVC''16'; p.predFilename = 'pred/8th/pred_keypoints_lsp_pc'; p.colorIdxs = [1 2]; case 27 p.name = 'Yang et al., CVPR''16'; p.predFilename = 'pred/9th/pred_keypoints_lsp_pc'; p.colorIdxs = [2 2]; case 28 p.name = 'Chen&Yuille, NIPS''14'; p.predFilename = 'pred/10th/pred_keypoints_lsp_pc'; p.colorIdxs = [3 2]; case 29 p.name = 'Fan et al., CVPR''15'; p.predFilename = 'pred/12th/pred_keypoints_lsp_pc'; p.colorIdxs = [4 2]; case 30 p.name = 'Tompson et al., NIPS''14'; p.predFilename = 'pred/13th/pred_keypoints_lsp_pc'; p.colorIdxs = [4 3]; case 31 p.name = 'Pishchulin et al., ICCV''13'; p.predFilename = 'pred/14th/pred_keypoints_lsp_pc'; p.colorIdxs = [4 4]; case 32 p.name = 'Wang&Li, CVPR''13'; p.predFilename = 'pred/15th/pred_keypoints_lsp_pc'; p.colorIdxs = [2 3]; case 33 % Add GNet. Guanghan Ning. Dec 2016 p.name = 'Ours'; p.predFilename = 'pred/ning17iccv/pred_keypoints_lsp'; p.colorIdxs = [8 1]; end p.colorName = getColor(p.colorIdxs); p.colorName = p.colorName ./ 255; end
State Before: l✝ : Type ?u.20447 m✝ : Type ?u.20450 n✝ : Type ?u.20453 o✝ : Type ?u.20456 p : Type ?u.20459 q : Type ?u.20462 m' : o✝ → Type ?u.20467 n' : o✝ → Type ?u.20472 p' : o✝ → Type ?u.20477 R : Type ?u.20480 S : Type ?u.20483 α✝ : Type ?u.20486 β : Type ?u.20489 l : Type u_1 m : Type u_2 n : Type u_3 o : Type u_4 α : Type u_5 A : Matrix n l α B : Matrix n m α C : Matrix o l α D : Matrix o m α ⊢ submatrix (fromBlocks A B C D) Sum.swap Sum.swap = fromBlocks D C B A State After: no goals Tactic: simp
function f = f_function ( x ) %****************************************************************************80 % %% F_FUNCTION evaluates the right hand side of the finite element system. % % Licensing: % % This code is distributed under the GNU LGPL license. % % Modified: % % 10 March 2011 % % Author: % % Jeff Borggaard, John Burkardt, Catalin Trenchea, Clayton Webster % % Parameters: % % Input, real X(), the evaluation points. % % Output, real F(), the function values. % f = - 15x.^4 + 3x.^2 - 6x; return end
In this notebook, you will implement the forward longitudinal vehicle model. The model accepts throttle inputs and steps through the longitudinal dynamic equations. Once implemented, you will be given a set of inputs that drives over a small road slope to test your model. The input to the model is a throttle percentage $x_\theta \in [0,1]$ which provides torque to the engine and subsequently accelerates the vehicle for forward motion. The dynamic equations consist of many stages to convert throttle inputs to wheel speed (engine -> torque converter -> transmission -> wheel). These stages are bundled together in a single inertia term $J_e$ which is used in the following combined engine dynamic equations. \begin{align} J_e \dot{\omega}_e &= T_e - (GR)(r_{eff} F_{load}) \\ m\ddot{x} &= F_x - F_{load} \end{align} Where $T_e$ is the engine torque, $GR$ is the gear ratio, $r_{eff}$ is the effective radius, $m$ is the vehicle mass, $x$ is the vehicle position, $F_x$ is the tire force, and $F_{load}$ is the total load force. The engine torque is computed from the throttle input and the engine angular velocity $\omega_e$ using a simplified quadratic model. \begin{align} T_e = x_{\theta}(a_0 + a_1 \omega_e + a_2 \omega_e^2) \end{align} The load forces consist of aerodynamic drag $F_{aero}$, rolling friction $R_x$, and gravitational force $F_g$ from an incline at angle $\alpha$. The aerodynamic drag is a quadratic model and the friction is a linear model. \begin{align} F_{load} &= F_{aero} + R_x + F_g \\ F_{aero} &= \frac{1}{2} C_a \rho A \dot{x}^2 = c_a \dot{x}^2\\ R_x &= N(\hat{c}_{r,0} + \hat{c}_{r,1}\|\dot{x}\| + \hat{c}_{r,2}\dot{x}^2) \approx c_{r,1} \dot{x}\\ F_g &= mg\sin{\alpha} \end{align} Note that the absolute value is ignored for friction since the model is used for only forward motion ($\dot{x} \ge 0$). The tire force is computed using the engine speed and wheel slip equations. \begin{align} \omega_w &= (GR)\omega_e \\ s &= \frac{\omega_w r_e - \dot{x}}{\dot{x}}\\ F_x &= \left\{\begin{array}{lr} cs, & \|s\| < 1\\ F_{max}, & \text{otherwise} \end{array}\right\} \end{align} Where $\omega_w$ is the wheel angular velocity and $s$ is the slip ratio. We setup the longitudinal model inside a Python class below. The vehicle begins with an initial velocity of 5 m/s and engine speed of 100 rad/s. All the relevant parameters are defined and like the bicycle model, a sampling time of 10ms is used for numerical integration. ```python import sys import math import numpy as np import matplotlib.pyplot as plt import matplotlib.image as mpimg class Vehicle(): def __init__(self): # ================================== # Parameters # ================================== #Throttle to engine torque self.a_0 = 400 self.a_1 = 0.1 self.a_2 = -0.0002 # Gear ratio, effective radius, mass + inertia self.GR = 0.35 self.r_e = 0.3 self.J_e = 10 # inertia self.m = 2000 self.g = 9.81 # Aerodynamic and friction coefficients self.c_a = 1.36 self.c_r1 = 0.01 # Tire force self.c = 10000 self.F_max = 10000 # State variables self.x = 0 # position self.v = 5 # velocity self.a = 0 # acceleration self.w_e = 100 # engine angular velocity self.w_e_dot = 0 # engine angular acceleration self.sample_time = 0.01 def reset(self): # reset state variables self.x = 0 self.v = 5 self.a = 0 self.w_e = 100 self.w_e_dot = 0 ``` Implement the combined engine dynamic equations along with the force equations in the cell below. The function $\textit{step}$ takes the throttle $x_\theta$ and incline angle $\alpha$ as inputs and performs numerical integration over one timestep to update the state variables. Hint: Integrate to find the current position, velocity, and engine speed first, then propagate those values into the set of equations. ```python class Vehicle(Vehicle): def step(self, throttle, alpha): # ================================== # Implement vehicle model here # Integrate to find the current position, velocity, and engine speed first, # then propagate those values into the set of equations. # x theta is throttle # alpha is incline angle in radian x_dot = self.v # velocity v_dot = self.a # acceleration w_e_dot = self.w_e_dot # engine angular acceleration self.x += x_dot * self.sample_time # position self.v += v_dot * self.sample_time # velocity self.w_e += w_e_dot * self.sample_time # engine speed omega_w = self.GR * self.w_e # Wheel speed = geer ratio * engine speed s = omega_w * self.r_e / self.v - 1 # Slip ratio formula F_x = self.c * s if abs(s) < 1 else self.F_max # tire force F_g = self.m * self.g * np.sin(alpha) # gravitational force R_x = self.c_r1 * x_dot # rolling friction F_aero = self.c_a * x_dot*2 # aerodynamic drag F_load = F_aero + R_x + F_g # total load force T_e = throttle (self.a_0 + self.a_1 * self.w_e + self.a_2 * self.w_e*2) # engine torque self.a = (F_x - F_load) / self.m # resulted acceleration self.w_e_dot = (T_e - self.GR self.r_e * F_load) / self.J_e # resulted angular acceleration # ================================== pass ``` Using the model, you can send constant throttle inputs to the vehicle in the cell below. You will observe that the velocity converges to a fixed value based on the throttle input due to the aerodynamic drag and tire force limit. A similar velocity profile can be seen by setting a negative incline angle $\alpha$. In this case, gravity accelerates the vehicle to a terminal velocity where it is balanced by the drag force. ```python sample_time = 0.01 time_end = 100 model = Vehicle() t_data = np.arange(0,time_end,sample_time) v_data = np.zeros_like(t_data) # throttle percentage between 0 and 1 throttle = 0.2 # incline angle (in radians) alpha = 0 for i in range(t_data.shape[0]): v_data[i] = model.v model.step(throttle, alpha) plt.plot(t_data, v_data) plt.show() ``` We will now drive the vehicle over a slope as shown in the diagram below. To climb the slope, a trapezoidal throttle input is provided for the next 20 seconds as shown in the figure below. The vehicle begins at 20% throttle and gradually increases to 50% throttle. This is maintained for 10 seconds as the vehicle climbs the steeper slope. Afterwards, the vehicle reduces the throttle to 0. In the cell below, implement the ramp angle profile $\alpha (x)$ and throttle profile $x_\theta (t)$ and step them through the vehicle dynamics. The vehicle position $x(t)$ is saved in the array $\textit{x_data}$. This will be used to grade your solution. ```python time_end = 20 t_data = np.arange(0,time_end,sample_time) x_data = np.zeros_like(t_data) # reset the states model.reset() # ================================== # Learner solution begins here # ================================== ramp_angle_data = np.zeros_like(t_data) throttle_data = np.zeros_like(t_data) n = t_data.shape[0] for i in range(n//4): throttle_data[i] = 0.2 + (0.3/(n//4)) * i for i in range(n//4, 3n//4): throttle_data[i] = 0.5 for i in range(3n//4, n): throttle_data[i] = 0.5 - (i-3n//4) (0.5/(n//4)) for i in range(n): x_data[i] = model.x # store current position v_data[i] = model.v # store current velocity if model.x < 60: ramp_angle_data[i] = math.atan2(3, 60) # return arc tangent in radian elif model.x < 150: ramp_angle_data[i] = math.atan2(12-3, 90) # return arc tangent in radian else: ramp_angle_data[i] = 0 model.step(throttle_data[i], ramp_angle_data[i]) # update dynamic model with new throttle and ramp angle # ================================== # Learner solution ends here # ================================== # Plot x vs t for visualization plt.plot(t_data, x_data) plt.show() ``` If you have implemented the vehicle model and inputs correctly, you should see that the vehicle crosses the ramp at ~15s where the throttle input begins to decrease. The cell below will save the time and vehicle inputs as text file named $\textit{xdata.txt}$. To locate the file, change the end of your web directory to $\textit{/notebooks/Course_1_Module_4/xdata.txt}$ Once you are there, you can download the file and submit to the Coursera grader to complete this assessment. ```python data = np.vstack([t_data, x_data]).T np.savetxt('xdata.txt', data, delimiter=', ') ``` Congratulations! You have now completed the assessment! Feel free to test the vehicle model with different inputs in the cell below, and see what trajectories they form. In the next module, you will see the longitudinal model being used for speed control. See you there! ```python sample_time = 0.01 time_end = 30 model.reset() t_data = np.arange(0,time_end,sample_time) x_data = np.zeros_like(t_data) # ================================== # Test various inputs here # ================================== for i in range(t_data.shape[0]): model.step(0,0) plt.axis('equal') plt.plot(t_data, x_data) plt.show() ```
If $S$ is a bounded set, then for all $x, y \in S$, we have $d(x, y) \leq \text{diameter}(S)$. Moreover, for all $d > 0$, if $d < \text{diameter}(S)$, then there exist $x, y \in S$ such that $d(x, y) > d$.
[STATEMENT] lemma innermost_intvl: "\<lbrakk> nqfree f; nolub f xs l x u; l < x; x < u; x \<notin> EQ f xs; DLO.I f (x#xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f (y#xs)" [PROOF STATE] proof (prove) goal (1 subgoal): 1. \<lbrakk>nqfree f; nolub f xs l x u; l < x; x < u; x \<notin> EQ f xs; DLO.I f (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f (y # xs) [PROOF STEP] proof(induct f) [PROOF STATE] proof (state) goal (7 subgoals): 1. \<lbrakk>nqfree TrueF; nolub TrueF xs l x u; l < x; x < u; x \<notin> EQ TrueF xs; DLO.I TrueF (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I TrueF (y # xs) 2. \<lbrakk>nqfree FalseF; nolub FalseF xs l x u; l < x; x < u; x \<notin> EQ FalseF xs; DLO.I FalseF (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I FalseF (y # xs) 3. \<And>xa. \<lbrakk>nqfree (Atom xa); nolub (Atom xa) xs l x u; l < x; x < u; x \<notin> EQ (Atom xa) xs; DLO.I (Atom xa) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (Atom xa) (y # xs) 4. \<And>f1 f2. \<lbrakk>\<lbrakk>nqfree f1; nolub f1 xs l x u; l < x; x < u; x \<notin> EQ f1 xs; DLO.I f1 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f1 (y # xs); \<lbrakk>nqfree f2; nolub f2 xs l x u; l < x; x < u; x \<notin> EQ f2 xs; DLO.I f2 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f2 (y # xs); nqfree (And f1 f2); nolub (And f1 f2) xs l x u; l < x; x < u; x \<notin> EQ (And f1 f2) xs; DLO.I (And f1 f2) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (And f1 f2) (y # xs) 5. \<And>f1 f2. \<lbrakk>\<lbrakk>nqfree f1; nolub f1 xs l x u; l < x; x < u; x \<notin> EQ f1 xs; DLO.I f1 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f1 (y # xs); \<lbrakk>nqfree f2; nolub f2 xs l x u; l < x; x < u; x \<notin> EQ f2 xs; DLO.I f2 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f2 (y # xs); nqfree (Or f1 f2); nolub (Or f1 f2) xs l x u; l < x; x < u; x \<notin> EQ (Or f1 f2) xs; DLO.I (Or f1 f2) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (Or f1 f2) (y # xs) 6. \<And>f. \<lbrakk>\<lbrakk>nqfree f; nolub f xs l x u; l < x; x < u; x \<notin> EQ f xs; DLO.I f (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f (y # xs); nqfree (Neg f); nolub (Neg f) xs l x u; l < x; x < u; x \<notin> EQ (Neg f) xs; DLO.I (Neg f) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (Neg f) (y # xs) 7. \<And>f. \<lbrakk>\<lbrakk>nqfree f; nolub f xs l x u; l < x; x < u; x \<notin> EQ f xs; DLO.I f (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f (y # xs); nqfree (ExQ f); nolub (ExQ f) xs l x u; l < x; x < u; x \<notin> EQ (ExQ f) xs; DLO.I (ExQ f) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (ExQ f) (y # xs) [PROOF STEP] case (Atom a) [PROOF STATE] proof (state) this: nqfree (Atom a) nolub (Atom a) xs l x u l < x x < u x \<notin> EQ (Atom a) xs DLO.I (Atom a) (x # xs) l < y y < u goal (7 subgoals): 1. \<lbrakk>nqfree TrueF; nolub TrueF xs l x u; l < x; x < u; x \<notin> EQ TrueF xs; DLO.I TrueF (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I TrueF (y # xs) 2. \<lbrakk>nqfree FalseF; nolub FalseF xs l x u; l < x; x < u; x \<notin> EQ FalseF xs; DLO.I FalseF (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I FalseF (y # xs) 3. \<And>xa. \<lbrakk>nqfree (Atom xa); nolub (Atom xa) xs l x u; l < x; x < u; x \<notin> EQ (Atom xa) xs; DLO.I (Atom xa) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (Atom xa) (y # xs) 4. \<And>f1 f2. \<lbrakk>\<lbrakk>nqfree f1; nolub f1 xs l x u; l < x; x < u; x \<notin> EQ f1 xs; DLO.I f1 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f1 (y # xs); \<lbrakk>nqfree f2; nolub f2 xs l x u; l < x; x < u; x \<notin> EQ f2 xs; DLO.I f2 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f2 (y # xs); nqfree (And f1 f2); nolub (And f1 f2) xs l x u; l < x; x < u; x \<notin> EQ (And f1 f2) xs; DLO.I (And f1 f2) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (And f1 f2) (y # xs) 5. \<And>f1 f2. \<lbrakk>\<lbrakk>nqfree f1; nolub f1 xs l x u; l < x; x < u; x \<notin> EQ f1 xs; DLO.I f1 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f1 (y # xs); \<lbrakk>nqfree f2; nolub f2 xs l x u; l < x; x < u; x \<notin> EQ f2 xs; DLO.I f2 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f2 (y # xs); nqfree (Or f1 f2); nolub (Or f1 f2) xs l x u; l < x; x < u; x \<notin> EQ (Or f1 f2) xs; DLO.I (Or f1 f2) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (Or f1 f2) (y # xs) 6. \<And>f. \<lbrakk>\<lbrakk>nqfree f; nolub f xs l x u; l < x; x < u; x \<notin> EQ f xs; DLO.I f (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f (y # xs); nqfree (Neg f); nolub (Neg f) xs l x u; l < x; x < u; x \<notin> EQ (Neg f) xs; DLO.I (Neg f) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (Neg f) (y # xs) 7. \<And>f. \<lbrakk>\<lbrakk>nqfree f; nolub f xs l x u; l < x; x < u; x \<notin> EQ f xs; DLO.I f (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f (y # xs); nqfree (ExQ f); nolub (ExQ f) xs l x u; l < x; x < u; x \<notin> EQ (ExQ f) xs; DLO.I (ExQ f) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (ExQ f) (y # xs) [PROOF STEP] show ?case [PROOF STATE] proof (prove) goal (1 subgoal): 1. DLO.I (Atom a) (y # xs) [PROOF STEP] proof (cases a) [PROOF STATE] proof (state) goal (2 subgoals): 1. \<And>x11 x12. a = Less x11 x12 \<Longrightarrow> DLO.I (Atom a) (y # xs) 2. \<And>x21 x22. a = Eq x21 x22 \<Longrightarrow> DLO.I (Atom a) (y # xs) [PROOF STEP] case (Less i j) [PROOF STATE] proof (state) this: a = Less i j goal (2 subgoals): 1. \<And>x11 x12. a = Less x11 x12 \<Longrightarrow> DLO.I (Atom a) (y # xs) 2. \<And>x21 x22. a = Eq x21 x22 \<Longrightarrow> DLO.I (Atom a) (y # xs) [PROOF STEP] then [PROOF STATE] proof (chain) picking this: a = Less i j [PROOF STEP] show ?thesis [PROOF STATE] proof (prove) using this: a = Less i j goal (1 subgoal): 1. DLO.I (Atom a) (y # xs) [PROOF STEP] using Atom [PROOF STATE] proof (prove) using this: a = Less i j nqfree (Atom a) nolub (Atom a) xs l x u l < x x < u x \<notin> EQ (Atom a) xs DLO.I (Atom a) (x # xs) l < y y < u goal (1 subgoal): 1. DLO.I (Atom a) (y # xs) [PROOF STEP] unfolding nolub_def [PROOF STATE] proof (prove) using this: a = Less i j nqfree (Atom a) (\<forall>y\<in>{l<..<x}. y \<notin> LB (Atom a) xs) \<and> (\<forall>y\<in>{x<..<u}. y \<notin> UB (Atom a) xs) l < x x < u x \<notin> EQ (Atom a) xs DLO.I (Atom a) (x # xs) l < y y < u goal (1 subgoal): 1. DLO.I (Atom a) (y # xs) [PROOF STEP] by (clarsimp simp: nth.simps Ball_def split:if_split_asm nat.splits) (metis not_le_imp_less order_antisym order_less_trans)+ [PROOF STATE] proof (state) this: DLO.I (Atom a) (y # xs) goal (1 subgoal): 1. \<And>x21 x22. a = Eq x21 x22 \<Longrightarrow> DLO.I (Atom a) (y # xs) [PROOF STEP] next [PROOF STATE] proof (state) goal (1 subgoal): 1. \<And>x21 x22. a = Eq x21 x22 \<Longrightarrow> DLO.I (Atom a) (y # xs) [PROOF STEP] case [simp]: (Eq i j) [PROOF STATE] proof (state) this: a = Eq i j goal (1 subgoal): 1. \<And>x21 x22. a = Eq x21 x22 \<Longrightarrow> DLO.I (Atom a) (y # xs) [PROOF STEP] show ?thesis [PROOF STATE] proof (prove) goal (1 subgoal): 1. DLO.I (Atom a) (y # xs) [PROOF STEP] proof (cases i) [PROOF STATE] proof (state) goal (2 subgoals): 1. i = 0 \<Longrightarrow> DLO.I (Atom a) (y # xs) 2. \<And>nat. i = Suc nat \<Longrightarrow> DLO.I (Atom a) (y # xs) [PROOF STEP] case [simp]: 0 [PROOF STATE] proof (state) this: i = 0 goal (2 subgoals): 1. i = 0 \<Longrightarrow> DLO.I (Atom a) (y # xs) 2. \<And>nat. i = Suc nat \<Longrightarrow> DLO.I (Atom a) (y # xs) [PROOF STEP] show ?thesis [PROOF STATE] proof (prove) goal (1 subgoal): 1. DLO.I (Atom a) (y # xs) [PROOF STEP] proof (cases j) [PROOF STATE] proof (state) goal (2 subgoals): 1. j = 0 \<Longrightarrow> DLO.I (Atom a) (y # xs) 2. \<And>nat. j = Suc nat \<Longrightarrow> DLO.I (Atom a) (y # xs) [PROOF STEP] case 0 [PROOF STATE] proof (state) this: j = 0 goal (2 subgoals): 1. j = 0 \<Longrightarrow> DLO.I (Atom a) (y # xs) 2. \<And>nat. j = Suc nat \<Longrightarrow> DLO.I (Atom a) (y # xs) [PROOF STEP] thus ?thesis [PROOF STATE] proof (prove) using this: j = 0 goal (1 subgoal): 1. DLO.I (Atom a) (y # xs) [PROOF STEP] using Atom [PROOF STATE] proof (prove) using this: j = 0 nqfree (Atom a) nolub (Atom a) xs l x u l < x x < u x \<notin> EQ (Atom a) xs DLO.I (Atom a) (x # xs) l < y y < u goal (1 subgoal): 1. DLO.I (Atom a) (y # xs) [PROOF STEP] by simp [PROOF STATE] proof (state) this: DLO.I (Atom a) (y # xs) goal (1 subgoal): 1. \<And>nat. j = Suc nat \<Longrightarrow> DLO.I (Atom a) (y # xs) [PROOF STEP] next [PROOF STATE] proof (state) goal (1 subgoal): 1. \<And>nat. j = Suc nat \<Longrightarrow> DLO.I (Atom a) (y # xs) [PROOF STEP] case Suc [PROOF STATE] proof (state) this: j = Suc nat_ goal (1 subgoal): 1. \<And>nat. j = Suc nat \<Longrightarrow> DLO.I (Atom a) (y # xs) [PROOF STEP] thus ?thesis [PROOF STATE] proof (prove) using this: j = Suc nat_ goal (1 subgoal): 1. DLO.I (Atom a) (y # xs) [PROOF STEP] using Atom [PROOF STATE] proof (prove) using this: j = Suc nat_ nqfree (Atom a) nolub (Atom a) xs l x u l < x x < u x \<notin> EQ (Atom a) xs DLO.I (Atom a) (x # xs) l < y y < u goal (1 subgoal): 1. DLO.I (Atom a) (y # xs) [PROOF STEP] by(simp add:EQ_def) [PROOF STATE] proof (state) this: DLO.I (Atom a) (y # xs) goal: No subgoals! [PROOF STEP] qed [PROOF STATE] proof (state) this: DLO.I (Atom a) (y # xs) goal (1 subgoal): 1. \<And>nat. i = Suc nat \<Longrightarrow> DLO.I (Atom a) (y # xs) [PROOF STEP] next [PROOF STATE] proof (state) goal (1 subgoal): 1. \<And>nat. i = Suc nat \<Longrightarrow> DLO.I (Atom a) (y # xs) [PROOF STEP] case [simp]: Suc [PROOF STATE] proof (state) this: i = Suc nat_ goal (1 subgoal): 1. \<And>nat. i = Suc nat \<Longrightarrow> DLO.I (Atom a) (y # xs) [PROOF STEP] show ?thesis [PROOF STATE] proof (prove) goal (1 subgoal): 1. DLO.I (Atom a) (y # xs) [PROOF STEP] proof (cases j) [PROOF STATE] proof (state) goal (2 subgoals): 1. j = 0 \<Longrightarrow> DLO.I (Atom a) (y # xs) 2. \<And>nat. j = Suc nat \<Longrightarrow> DLO.I (Atom a) (y # xs) [PROOF STEP] case 0 [PROOF STATE] proof (state) this: j = 0 goal (2 subgoals): 1. j = 0 \<Longrightarrow> DLO.I (Atom a) (y # xs) 2. \<And>nat. j = Suc nat \<Longrightarrow> DLO.I (Atom a) (y # xs) [PROOF STEP] thus ?thesis [PROOF STATE] proof (prove) using this: j = 0 goal (1 subgoal): 1. DLO.I (Atom a) (y # xs) [PROOF STEP] using Atom [PROOF STATE] proof (prove) using this: j = 0 nqfree (Atom a) nolub (Atom a) xs l x u l < x x < u x \<notin> EQ (Atom a) xs DLO.I (Atom a) (x # xs) l < y y < u goal (1 subgoal): 1. DLO.I (Atom a) (y # xs) [PROOF STEP] by(simp add:EQ_def) [PROOF STATE] proof (state) this: DLO.I (Atom a) (y # xs) goal (1 subgoal): 1. \<And>nat. j = Suc nat \<Longrightarrow> DLO.I (Atom a) (y # xs) [PROOF STEP] next [PROOF STATE] proof (state) goal (1 subgoal): 1. \<And>nat. j = Suc nat \<Longrightarrow> DLO.I (Atom a) (y # xs) [PROOF STEP] case Suc [PROOF STATE] proof (state) this: j = Suc nat_ goal (1 subgoal): 1. \<And>nat. j = Suc nat \<Longrightarrow> DLO.I (Atom a) (y # xs) [PROOF STEP] thus ?thesis [PROOF STATE] proof (prove) using this: j = Suc nat_ goal (1 subgoal): 1. DLO.I (Atom a) (y # xs) [PROOF STEP] using Atom [PROOF STATE] proof (prove) using this: j = Suc nat_ nqfree (Atom a) nolub (Atom a) xs l x u l < x x < u x \<notin> EQ (Atom a) xs DLO.I (Atom a) (x # xs) l < y y < u goal (1 subgoal): 1. DLO.I (Atom a) (y # xs) [PROOF STEP] by simp [PROOF STATE] proof (state) this: DLO.I (Atom a) (y # xs) goal: No subgoals! [PROOF STEP] qed [PROOF STATE] proof (state) this: DLO.I (Atom a) (y # xs) goal: No subgoals! [PROOF STEP] qed [PROOF STATE] proof (state) this: DLO.I (Atom a) (y # xs) goal: No subgoals! [PROOF STEP] qed [PROOF STATE] proof (state) this: DLO.I (Atom a) (y # xs) goal (6 subgoals): 1. \<lbrakk>nqfree TrueF; nolub TrueF xs l x u; l < x; x < u; x \<notin> EQ TrueF xs; DLO.I TrueF (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I TrueF (y # xs) 2. \<lbrakk>nqfree FalseF; nolub FalseF xs l x u; l < x; x < u; x \<notin> EQ FalseF xs; DLO.I FalseF (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I FalseF (y # xs) 3. \<And>f1 f2. \<lbrakk>\<lbrakk>nqfree f1; nolub f1 xs l x u; l < x; x < u; x \<notin> EQ f1 xs; DLO.I f1 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f1 (y # xs); \<lbrakk>nqfree f2; nolub f2 xs l x u; l < x; x < u; x \<notin> EQ f2 xs; DLO.I f2 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f2 (y # xs); nqfree (And f1 f2); nolub (And f1 f2) xs l x u; l < x; x < u; x \<notin> EQ (And f1 f2) xs; DLO.I (And f1 f2) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (And f1 f2) (y # xs) 4. \<And>f1 f2. \<lbrakk>\<lbrakk>nqfree f1; nolub f1 xs l x u; l < x; x < u; x \<notin> EQ f1 xs; DLO.I f1 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f1 (y # xs); \<lbrakk>nqfree f2; nolub f2 xs l x u; l < x; x < u; x \<notin> EQ f2 xs; DLO.I f2 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f2 (y # xs); nqfree (Or f1 f2); nolub (Or f1 f2) xs l x u; l < x; x < u; x \<notin> EQ (Or f1 f2) xs; DLO.I (Or f1 f2) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (Or f1 f2) (y # xs) 5. \<And>f. \<lbrakk>\<lbrakk>nqfree f; nolub f xs l x u; l < x; x < u; x \<notin> EQ f xs; DLO.I f (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f (y # xs); nqfree (Neg f); nolub (Neg f) xs l x u; l < x; x < u; x \<notin> EQ (Neg f) xs; DLO.I (Neg f) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (Neg f) (y # xs) 6. \<And>f. \<lbrakk>\<lbrakk>nqfree f; nolub f xs l x u; l < x; x < u; x \<notin> EQ f xs; DLO.I f (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f (y # xs); nqfree (ExQ f); nolub (ExQ f) xs l x u; l < x; x < u; x \<notin> EQ (ExQ f) xs; DLO.I (ExQ f) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (ExQ f) (y # xs) [PROOF STEP] next [PROOF STATE] proof (state) goal (6 subgoals): 1. \<lbrakk>nqfree TrueF; nolub TrueF xs l x u; l < x; x < u; x \<notin> EQ TrueF xs; DLO.I TrueF (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I TrueF (y # xs) 2. \<lbrakk>nqfree FalseF; nolub FalseF xs l x u; l < x; x < u; x \<notin> EQ FalseF xs; DLO.I FalseF (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I FalseF (y # xs) 3. \<And>f1 f2. \<lbrakk>\<lbrakk>nqfree f1; nolub f1 xs l x u; l < x; x < u; x \<notin> EQ f1 xs; DLO.I f1 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f1 (y # xs); \<lbrakk>nqfree f2; nolub f2 xs l x u; l < x; x < u; x \<notin> EQ f2 xs; DLO.I f2 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f2 (y # xs); nqfree (And f1 f2); nolub (And f1 f2) xs l x u; l < x; x < u; x \<notin> EQ (And f1 f2) xs; DLO.I (And f1 f2) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (And f1 f2) (y # xs) 4. \<And>f1 f2. \<lbrakk>\<lbrakk>nqfree f1; nolub f1 xs l x u; l < x; x < u; x \<notin> EQ f1 xs; DLO.I f1 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f1 (y # xs); \<lbrakk>nqfree f2; nolub f2 xs l x u; l < x; x < u; x \<notin> EQ f2 xs; DLO.I f2 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f2 (y # xs); nqfree (Or f1 f2); nolub (Or f1 f2) xs l x u; l < x; x < u; x \<notin> EQ (Or f1 f2) xs; DLO.I (Or f1 f2) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (Or f1 f2) (y # xs) 5. \<And>f. \<lbrakk>\<lbrakk>nqfree f; nolub f xs l x u; l < x; x < u; x \<notin> EQ f xs; DLO.I f (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f (y # xs); nqfree (Neg f); nolub (Neg f) xs l x u; l < x; x < u; x \<notin> EQ (Neg f) xs; DLO.I (Neg f) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (Neg f) (y # xs) 6. \<And>f. \<lbrakk>\<lbrakk>nqfree f; nolub f xs l x u; l < x; x < u; x \<notin> EQ f xs; DLO.I f (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f (y # xs); nqfree (ExQ f); nolub (ExQ f) xs l x u; l < x; x < u; x \<notin> EQ (ExQ f) xs; DLO.I (ExQ f) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (ExQ f) (y # xs) [PROOF STEP] case (And f1 f2) [PROOF STATE] proof (state) this: \<lbrakk>nqfree f1; nolub f1 xs l x u; l < x; x < u; x \<notin> EQ f1 xs; DLO.I f1 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f1 (y # xs) \<lbrakk>nqfree f2; nolub f2 xs l x u; l < x; x < u; x \<notin> EQ f2 xs; DLO.I f2 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f2 (y # xs) nqfree (And f1 f2) nolub (And f1 f2) xs l x u l < x x < u x \<notin> EQ (And f1 f2) xs DLO.I (And f1 f2) (x # xs) l < y y < u goal (6 subgoals): 1. \<lbrakk>nqfree TrueF; nolub TrueF xs l x u; l < x; x < u; x \<notin> EQ TrueF xs; DLO.I TrueF (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I TrueF (y # xs) 2. \<lbrakk>nqfree FalseF; nolub FalseF xs l x u; l < x; x < u; x \<notin> EQ FalseF xs; DLO.I FalseF (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I FalseF (y # xs) 3. \<And>f1 f2. \<lbrakk>\<lbrakk>nqfree f1; nolub f1 xs l x u; l < x; x < u; x \<notin> EQ f1 xs; DLO.I f1 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f1 (y # xs); \<lbrakk>nqfree f2; nolub f2 xs l x u; l < x; x < u; x \<notin> EQ f2 xs; DLO.I f2 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f2 (y # xs); nqfree (And f1 f2); nolub (And f1 f2) xs l x u; l < x; x < u; x \<notin> EQ (And f1 f2) xs; DLO.I (And f1 f2) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (And f1 f2) (y # xs) 4. \<And>f1 f2. \<lbrakk>\<lbrakk>nqfree f1; nolub f1 xs l x u; l < x; x < u; x \<notin> EQ f1 xs; DLO.I f1 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f1 (y # xs); \<lbrakk>nqfree f2; nolub f2 xs l x u; l < x; x < u; x \<notin> EQ f2 xs; DLO.I f2 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f2 (y # xs); nqfree (Or f1 f2); nolub (Or f1 f2) xs l x u; l < x; x < u; x \<notin> EQ (Or f1 f2) xs; DLO.I (Or f1 f2) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (Or f1 f2) (y # xs) 5. \<And>f. \<lbrakk>\<lbrakk>nqfree f; nolub f xs l x u; l < x; x < u; x \<notin> EQ f xs; DLO.I f (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f (y # xs); nqfree (Neg f); nolub (Neg f) xs l x u; l < x; x < u; x \<notin> EQ (Neg f) xs; DLO.I (Neg f) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (Neg f) (y # xs) 6. \<And>f. \<lbrakk>\<lbrakk>nqfree f; nolub f xs l x u; l < x; x < u; x \<notin> EQ f xs; DLO.I f (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f (y # xs); nqfree (ExQ f); nolub (ExQ f) xs l x u; l < x; x < u; x \<notin> EQ (ExQ f) xs; DLO.I (ExQ f) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (ExQ f) (y # xs) [PROOF STEP] thus ?case [PROOF STATE] proof (prove) using this: \<lbrakk>nqfree f1; nolub f1 xs l x u; l < x; x < u; x \<notin> EQ f1 xs; DLO.I f1 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f1 (y # xs) \<lbrakk>nqfree f2; nolub f2 xs l x u; l < x; x < u; x \<notin> EQ f2 xs; DLO.I f2 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f2 (y # xs) nqfree (And f1 f2) nolub (And f1 f2) xs l x u l < x x < u x \<notin> EQ (And f1 f2) xs DLO.I (And f1 f2) (x # xs) l < y y < u goal (1 subgoal): 1. DLO.I (And f1 f2) (y # xs) [PROOF STEP] by (fastforce) [PROOF STATE] proof (state) this: DLO.I (And f1 f2) (y # xs) goal (5 subgoals): 1. \<lbrakk>nqfree TrueF; nolub TrueF xs l x u; l < x; x < u; x \<notin> EQ TrueF xs; DLO.I TrueF (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I TrueF (y # xs) 2. \<lbrakk>nqfree FalseF; nolub FalseF xs l x u; l < x; x < u; x \<notin> EQ FalseF xs; DLO.I FalseF (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I FalseF (y # xs) 3. \<And>f1 f2. \<lbrakk>\<lbrakk>nqfree f1; nolub f1 xs l x u; l < x; x < u; x \<notin> EQ f1 xs; DLO.I f1 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f1 (y # xs); \<lbrakk>nqfree f2; nolub f2 xs l x u; l < x; x < u; x \<notin> EQ f2 xs; DLO.I f2 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f2 (y # xs); nqfree (Or f1 f2); nolub (Or f1 f2) xs l x u; l < x; x < u; x \<notin> EQ (Or f1 f2) xs; DLO.I (Or f1 f2) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (Or f1 f2) (y # xs) 4. \<And>f. \<lbrakk>\<lbrakk>nqfree f; nolub f xs l x u; l < x; x < u; x \<notin> EQ f xs; DLO.I f (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f (y # xs); nqfree (Neg f); nolub (Neg f) xs l x u; l < x; x < u; x \<notin> EQ (Neg f) xs; DLO.I (Neg f) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (Neg f) (y # xs) 5. \<And>f. \<lbrakk>\<lbrakk>nqfree f; nolub f xs l x u; l < x; x < u; x \<notin> EQ f xs; DLO.I f (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f (y # xs); nqfree (ExQ f); nolub (ExQ f) xs l x u; l < x; x < u; x \<notin> EQ (ExQ f) xs; DLO.I (ExQ f) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (ExQ f) (y # xs) [PROOF STEP] next [PROOF STATE] proof (state) goal (5 subgoals): 1. \<lbrakk>nqfree TrueF; nolub TrueF xs l x u; l < x; x < u; x \<notin> EQ TrueF xs; DLO.I TrueF (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I TrueF (y # xs) 2. \<lbrakk>nqfree FalseF; nolub FalseF xs l x u; l < x; x < u; x \<notin> EQ FalseF xs; DLO.I FalseF (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I FalseF (y # xs) 3. \<And>f1 f2. \<lbrakk>\<lbrakk>nqfree f1; nolub f1 xs l x u; l < x; x < u; x \<notin> EQ f1 xs; DLO.I f1 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f1 (y # xs); \<lbrakk>nqfree f2; nolub f2 xs l x u; l < x; x < u; x \<notin> EQ f2 xs; DLO.I f2 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f2 (y # xs); nqfree (Or f1 f2); nolub (Or f1 f2) xs l x u; l < x; x < u; x \<notin> EQ (Or f1 f2) xs; DLO.I (Or f1 f2) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (Or f1 f2) (y # xs) 4. \<And>f. \<lbrakk>\<lbrakk>nqfree f; nolub f xs l x u; l < x; x < u; x \<notin> EQ f xs; DLO.I f (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f (y # xs); nqfree (Neg f); nolub (Neg f) xs l x u; l < x; x < u; x \<notin> EQ (Neg f) xs; DLO.I (Neg f) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (Neg f) (y # xs) 5. \<And>f. \<lbrakk>\<lbrakk>nqfree f; nolub f xs l x u; l < x; x < u; x \<notin> EQ f xs; DLO.I f (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f (y # xs); nqfree (ExQ f); nolub (ExQ f) xs l x u; l < x; x < u; x \<notin> EQ (ExQ f) xs; DLO.I (ExQ f) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (ExQ f) (y # xs) [PROOF STEP] case (Or f1 f2) [PROOF STATE] proof (state) this: \<lbrakk>nqfree f1; nolub f1 xs l x u; l < x; x < u; x \<notin> EQ f1 xs; DLO.I f1 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f1 (y # xs) \<lbrakk>nqfree f2; nolub f2 xs l x u; l < x; x < u; x \<notin> EQ f2 xs; DLO.I f2 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f2 (y # xs) nqfree (Or f1 f2) nolub (Or f1 f2) xs l x u l < x x < u x \<notin> EQ (Or f1 f2) xs DLO.I (Or f1 f2) (x # xs) l < y y < u goal (5 subgoals): 1. \<lbrakk>nqfree TrueF; nolub TrueF xs l x u; l < x; x < u; x \<notin> EQ TrueF xs; DLO.I TrueF (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I TrueF (y # xs) 2. \<lbrakk>nqfree FalseF; nolub FalseF xs l x u; l < x; x < u; x \<notin> EQ FalseF xs; DLO.I FalseF (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I FalseF (y # xs) 3. \<And>f1 f2. \<lbrakk>\<lbrakk>nqfree f1; nolub f1 xs l x u; l < x; x < u; x \<notin> EQ f1 xs; DLO.I f1 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f1 (y # xs); \<lbrakk>nqfree f2; nolub f2 xs l x u; l < x; x < u; x \<notin> EQ f2 xs; DLO.I f2 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f2 (y # xs); nqfree (Or f1 f2); nolub (Or f1 f2) xs l x u; l < x; x < u; x \<notin> EQ (Or f1 f2) xs; DLO.I (Or f1 f2) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (Or f1 f2) (y # xs) 4. \<And>f. \<lbrakk>\<lbrakk>nqfree f; nolub f xs l x u; l < x; x < u; x \<notin> EQ f xs; DLO.I f (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f (y # xs); nqfree (Neg f); nolub (Neg f) xs l x u; l < x; x < u; x \<notin> EQ (Neg f) xs; DLO.I (Neg f) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (Neg f) (y # xs) 5. \<And>f. \<lbrakk>\<lbrakk>nqfree f; nolub f xs l x u; l < x; x < u; x \<notin> EQ f xs; DLO.I f (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f (y # xs); nqfree (ExQ f); nolub (ExQ f) xs l x u; l < x; x < u; x \<notin> EQ (ExQ f) xs; DLO.I (ExQ f) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (ExQ f) (y # xs) [PROOF STEP] thus ?case [PROOF STATE] proof (prove) using this: \<lbrakk>nqfree f1; nolub f1 xs l x u; l < x; x < u; x \<notin> EQ f1 xs; DLO.I f1 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f1 (y # xs) \<lbrakk>nqfree f2; nolub f2 xs l x u; l < x; x < u; x \<notin> EQ f2 xs; DLO.I f2 (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f2 (y # xs) nqfree (Or f1 f2) nolub (Or f1 f2) xs l x u l < x x < u x \<notin> EQ (Or f1 f2) xs DLO.I (Or f1 f2) (x # xs) l < y y < u goal (1 subgoal): 1. DLO.I (Or f1 f2) (y # xs) [PROOF STEP] by (fastforce) [PROOF STATE] proof (state) this: DLO.I (Or f1 f2) (y # xs) goal (4 subgoals): 1. \<lbrakk>nqfree TrueF; nolub TrueF xs l x u; l < x; x < u; x \<notin> EQ TrueF xs; DLO.I TrueF (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I TrueF (y # xs) 2. \<lbrakk>nqfree FalseF; nolub FalseF xs l x u; l < x; x < u; x \<notin> EQ FalseF xs; DLO.I FalseF (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I FalseF (y # xs) 3. \<And>f. \<lbrakk>\<lbrakk>nqfree f; nolub f xs l x u; l < x; x < u; x \<notin> EQ f xs; DLO.I f (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f (y # xs); nqfree (Neg f); nolub (Neg f) xs l x u; l < x; x < u; x \<notin> EQ (Neg f) xs; DLO.I (Neg f) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (Neg f) (y # xs) 4. \<And>f. \<lbrakk>\<lbrakk>nqfree f; nolub f xs l x u; l < x; x < u; x \<notin> EQ f xs; DLO.I f (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I f (y # xs); nqfree (ExQ f); nolub (ExQ f) xs l x u; l < x; x < u; x \<notin> EQ (ExQ f) xs; DLO.I (ExQ f) (x # xs); l < y; y < u\<rbrakk> \<Longrightarrow> DLO.I (ExQ f) (y # xs) [PROOF STEP] qed simp+
A common question from Sea Kayakers new to Clayoquot Sound revolves around the tidal currents. In many locations around British Columbia there are detailed current speed and time predictions published to assist mariners in route planning; no such predictions are published for Clayoquot Sound. Instead we rely on the Tofino Tidal Predictions published by the Canadian Hydrographic Service. When paddling in more inland waterways (such as the Gulf Islands, San Juan Islands, Desolation Sound, Broughton Archipelago) mariners need to pay very close attention to Current Speed / Turn times vs Tidal High/Low Times. In inland waterways the time difference between the tide high-or-low and current slack water can often have hours long differences. When the tides are rising, the current will move flood from the open ocean towards the inlets. When the tides are falling, the current will ebb from the inlets towards the open ocean. Examining the bathymetry lines (the shape of the sea floor) on a nautical chart, one can easily see the deep channels scoured by the currents. As in a river, the fastest current can be expected in the middle of those channels, while the most turbulent areas will be located along the edges. Tidal currents experienced in Clayoquot Sound will be directly proportional to the size of the tidal range experienced. For instance on a day with a 4 foot difference between and high and low tide one should expect much slower current speeds than a day where we experience a 10 foot range between high and low tides. For reference, the largest tidal ranges we experience in Tofino are about 13 feet (4 meters). Significant changes to barometric pressure (low-pressure from a storm, or high pressure during fair weather) can alter the tidal heights. A strong winter storm will lower the barometric pressure and result in larger tides. Conversely a strong ridge of high pressure will decrease the tidal range. A general rule of thumb; a change of 1 mb of pressure results in a 1 cm change in sea level. Tidal predictions are based on the average air pressure of 1013 mb. Therefore a strong high pressure system of 1033 will lower tidal heights by 20 cm. A low-pressure system of 980 will increase tidal heights by 33 cm. Pay attention to fast changes to weather systems. For the most part the currents manifest themselves as a gentle river moving up and down the passages. Eddy lines can be expected at headlands but once within most current channels the only reminder of the current will be the effect it has on your speed. Understanding how to ‘Ferry’ across the current is an important skill, both while crossing small passages in the inner islands, and larger crossings on the outer coast. On Long crossings, a strong flood or ebb tide can significantly alter your course. Please don’t mistake the lack of published current predictions as a sign the currents should be ignored. The currents can significantly influence the sea-state experienced in Clayoquot Sound. On calm days with a large tidal range, the currents themselves can create hazards. There are locations we can experience up to 5 knots of current which can generate a standing wave big enough to surf. Many kayakers have run into difficulty by not appreciating the speed of the currents within some channels. Many, but not all, of these locations are indicated on the Marine Chart by current directional arrows. On windy days, the current will play an important role. Within the inner waters, the current can drastically change the waves experienced. The waters directly in the Tofino Harbour are a great example. It is very common to experience a North Westerly wind in the afternoon during the summer months. During a rising tide with a flood current, the wind and current will be moving in the same direction; causing the waves to be elongated and less rough. However with a falling tide / ebb current the wind waves will be accentuated and much rougher conditions can be expected. Perhaps the largest consideration with currents in Clayoquot Sound is the influence it can have on the incoming open ocean swell. An outgoing ebb current, will shorten the wave length of incoming ocean swell making the waves much steeper and closer together. Couple this with wind waves, and hazardous conditions can occur on the outer coast. A few locations of note include Moser Pt. by Vargas Island, the Gerrard Group by Flores Island, and Sharp Pt by Hot Springs Cove. Regardless of where you paddle a healthy respect for the tides and currents will serve you well.
# defines the SimpleTropical module with the Tropical type module SimpleTropical import Base.isinf, Base.show, Base.+, Base., Base.inv, Base.== import Base.isequal, Base.^, Base.convert export Tropical, TropicalInf struct Tropical{T<:Real} <: Number val::T inf_flag::Bool function Tropical{T}(xx::Real, ii::Bool=false) where T TT = typeof(xx) if isinf(xx) \|\| ii return new(zero(TT),true) end return new(xx,false) end end Tropical(x::T) where T<:Real = Tropical{T}(x) function Tropical(x::T, i::Bool) where T<:Real if i return Tropical{T}(zero(T),true) end return Tropical(x) end """ `TropicalInf` is a constant that represents infinity in the tropical semiring. """ const TropicalInf = Tropical{Bool}(0,true) isinf(X::Tropical) = X.inf_flag Base.promote_rule(::Type{Tropical{T}}, ::Type{S}) where {T<:Real, S<:Real} = Tropical{promote_type(T, S)} Base.promote_rule(::Type{Tropical{T}}, ::Type{Tropical{S}}) where {T<:Real, S<:Real} = Tropical{promote_type(T, S)} convert(::Type{Tropical}, x::T) where {T<:Real} = Tropical{T}(x) convert(::Type{Tropical{T}}, x::S) where {T<:Real,S<:Tropical} = Tropical(convert(T, x.val), x.inf_flag) function show(io::IO, t::Tropical) if isinf(t) print(io,"Tropical($(Char(8734)))") # infinity character else print(io,"Tropical{$(typeof(t.val))}($(t.val))") end end function (+)(x::Tropical{T}, y::Tropical{T}) where {T} if isinf(x) if isinf(y) # when X,Y both are infinite return Tropical(zero(T),true) # create common infinite else return Tropical(y) end end if isinf(y) return Tropical(x) end return Tropical(min(x.val, y.val)) end (+)(x::Tropical{T}, y::Tropical{S}) where {T,S} = +(promote(x, y)...) (+)(x::Tropical{T}, y::Real) where T = +(promote(x, y)...) (+)(x::Real, y::Tropical{T}) where T = +(promote(x, y)...) function ()(x::Tropical{T}, y::Tropical{T}) where {T} if isinf(x) \|\| isinf(y) return Tropical(zero(T),true) end return Tropical(x.val + y.val) end ()(x::Tropical{T}, y::Tropical{S}) where {T,S} = (promote(x, y)...) ()(x::Tropical{T}, y::Real) where T = (promote(x, y)...) ()(x::Real, y::Tropical{T}) where T = (promote(x, y)...) function inv(X::Tropical) @assert !isinf(X) "TropicalInf is not invertible" return Tropical(-X.val) end function (^)(X::Tropical, p::Integer) if isinf(X) @assert p>0 "Cannot raise tropical infinity to a nonpositive power." return X end return Tropical(X.val * p) end function isequal(X::Tropical, Y::Tropical) if !isinf(X) && !isinf(Y) return isequal(X.val, Y.val) else return isinf(X) && isinf(Y) end end function ==(X::Tropical, Y::Tropical) if !isinf(X) && !isinf(Y) return X.val == Y.val else return isinf(X) && isinf(Y) end end end # end of module
-- Export only the experiments that are expected to compile (without -- any holes) {-# OPTIONS --cubical --no-import-sorts #-} module Cubical.Experiments.Everything where open import Cubical.Experiments.Brunerie public open import Cubical.Experiments.EscardoSIP public open import Cubical.Experiments.Generic public open import Cubical.Experiments.NatMinusTwo open import Cubical.Experiments.Problem open import Cubical.Experiments.FunExtFromUA public open import Cubical.Experiments.HoTT-UF open import Cubical.Experiments.Rng
Gioachino Rossini (born Pesaro, 29 February 1792; died Paris, 13 November 1868). Angelo Anelli (written for Luigi Mosca, performed at La Scala, Milan in 1808). First performance: Venice (Teatro San Benedetto), 22 May 1813. First UK performance: London (His Majesty’s Theatre, Haymarket), 26 January 1819. First performance in Scotland: Edinburgh (Theatre Royal), 8 May 1838. Scottish Opera premiere: Glasgow (Theatre Royal), 21 October 2009. Rossini came from a family of musicians, and his talent was recognised when still a child. His operas, both serious and comic, dominated the operatic world throughout his active career, which lasted twenty years from 1810. It is the eleventh of Rossini’s operas (he eventually composed around forty). The Italian Girl was an instant success in spite of being prepared at short notice and adapting an old text. In the early 19th century, Barbary pirates were still a serious problem for sea traders and for coastal towns in parts of Italy. The abduction and enslavement of Europeans therefore seems an unlikely subject for farcical humour, though Mozart showed the way with his Seraglio of 1782. Rossini has the chorus of eunuchs sung by tenors and basses. In addition, the idea of Italian patriotism was also problematic, with the north of Italy still to be under Austrian control for many years. Mustafà is tired of his wife and wants to find a new Italian bride. Lindoro, a young Italian, has been captured and is working as a slave for Mustafà. He has become something of a favourite. He laments his separation from his lover, Isabella. Mustafà suggests he should marry Elvira and take her to Italy. More European prisoners arrive, among them an Italian Girl. This is Isabella, who has set out to rescue Lindoro. She has with her Taddeo, who wants to marry her, but agrees to pose as her uncle meantime. Mustafà falls for Isabella at first sight and she flirts with him. When Lindoro, Elvira and Zulma come to say their farewells they meet Isabella. The first act ends in complete confusion, with one of Rossini’s craziest inventions. The second act continues the farcical activity as Isabella and Lindoro plot their escape, along with the other Italian captives. Elvira is re-united with Mustafà. Jennifer Larmore (Isabella), Raúl Giménez (Lindoro), John del Carlo (Mustafà). An excellent modern recording, stylishly conducted. Jennifer Larmore is one of the leading American mezzos singing the Rossini repertoire at the moment. Many of the best Rossinian tenors in recent years have come from Latin America, and Argentinian Raúl Giménez sails through the high-lying lines with ease. Lucia Valentini-Terrani (Isabella), Francisco Araiza (Lindoro), Wladimiro Ganzarolli (Mustafà). Lucia Valentini-Terrani was that rare thing, a real Italian who also had a fine sense of Rossinian style. She delights in the rapid-fire language. Her early death from leukaemia was a tragic loss. Araiza, a Mexican this time, has no problems with the high line, and gives no indication that his voice would develop (briefly) in a Wagnerian direction. Ganzarolli is a born comedian. This recording was an early attempt to play Rossini on original instruments and Ferro is generally a natural conductor. Marilyn Horne (Isabella), Ernesto Palacio (Lindoro), Samuel Ramey (Mustafa). Horne featured in many important revivals of Rossini’s serious operas, but was able to lighten her touch successfully when she sang the comedies, having particular success as Isabella. Palacio is noted now as the teacher of his fellow-Peruvian Juan Diego Flórez, but he was a notable performer in revivals of little-known Rossini and Handel works. Ramey’s long career covered nearly every possible style of singing except perhaps Wagner, and his technique sounds unusually effortless for a bass. Teresa Berganza (Isabella), Luigi Alva (Lindoro), Fernando Corena (Mustafà). This recording dates from a time when only a handful of Rossini’s operas were known, and it did much to restore the Italian Girl to the general repertoire. Varviso conducts delicately and the soloists were the leading performers of the day. Berganza always sounds lovely as well as characterful. Alva (another Peruvian) was the dominant Rossini tenor for twenty years, and is at his peak here. Corena also projects plenty of humour.
Formal statement is: lemma in_closure_iff_infdist_zero: assumes "A \<noteq> {}" shows "x \<in> closure A \<longleftrightarrow> infdist x A = 0" Informal statement is: A point $x$ is in the closure of a set $A$ if and only if the infimum of the distance from $x$ to $A$ is zero.
[STATEMENT] lemma int_of_integer_inv: "int_of_integer (integer_of_int x) = x" [PROOF STATE] proof (prove) goal (1 subgoal): 1. int_of_integer (integer_of_int x) = x [PROOF STEP] by simp
/* * MapTagTest.hpp * * Unit tests for maptag. Uses .maptag database in home folder. * * * Created on: 13 maj 2013 * Author: Hirad Asadi * E-mail: [email protected] * * Copyright (C) 2013 Hirad Asadi * * This program is free software: you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by * the Free Software Foundation, either version 3 of the License, or * (at your option) any later version. * * This program is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * GNU General Public License for more details. * * You should have received a copy of the GNU General Public License * along with this program. If not, see <http://www.gnu.org/licenses/>. / #ifndef MAPTAGTEST_HPP_ #define MAPTAGTEST_HPP_ #include <iostream> #include <fstream> #include <boost/algorithm/string.hpp> #include "../src/core/FlagHandler.hpp" #include "../src/include/Flags.hpp" #include "Helper.hpp" namespace maptagtest { // Global test file path static std::string testFilePath; /* * Init test method. / void init() { // Create a test file in home folder std::stringstream ss; ss << getHomeFolder() << PATH_SEPARATOR << "tag_test_file.test"; testFilePath = new std::string(ss.str()); std::ofstream outfile(testFilePath); outfile << "Tag test file!" << std::endl; outfile.close(); } /* * Finish test method. / void finish() { // Remove test file remove(testFilePath->c_str()); delete testFilePath; } / * Print headline. / static inline void printHeadline(const std::string& h) { std::cout << "\n*************************************" << std::endl; std::cout << boost::to_upper_copy(h) << std::endl; std::cout << "*************************************\n" << std::endl; } / * Run flag with input against db. * Return true on success, else false. / static inline bool runFlagWithInput(const std::string& input, const maptag::Flag& flag, const std::string& path = std::string(testFilePath), const std::vector<maptag::Flag>& extraFlags = std::vector<maptag::Flag>()) { maptag::FlagHandler handler; std::vector<std::string> inputVec; if (!input.empty()) inputVec.push_back(input); if (!path.empty()) inputVec.push_back(path); // String for extra flags std::stringstream ss; for (unsigned int i = 0; i < extraFlags.size(); i++) { ss << extraFlags[i] << " "; } // Print input if (!input.empty()) std::cout << "\n>> Running: \"" << inputVec[0] << " " << path << "\" (flag = " << flag << ", extra flags = {" << ss.str() << "})\n" << std::endl; else std::cout << "\n>> Running: \" " << path << "\" (flag = " << flag << ", extra flags = {" << ss.str() << "})\n" << std::endl; bool ret = true; // Process flag with input if (extraFlags.empty()) ret = handler.processInput(inputVec, flag); else // Extra flags added ret = handler.processInput(inputVec, flag, extraFlags); // Error occurred if (!ret && handler.getError().err != maptag::NO_ERROR) std::cerr << handler.getError().msg << ".\t[" << handler.getError().err << "]" << std::endl; return ret; } /* * Generates vector of chars / std::vector<char> genChars() { std::vector<char> vec; for (int i = 32; i < 127; i++) { vec.push_back(i); } return vec; } } / namespace maptagtest / #endif / MAPTAGTEST_HPP_ */
\chapter{Introduction} \textit{Rendering} is generally the process of generating a two-dimensional image, called \textit{render}, from the mathematical description of a three-dimensional scene. %To oversimplify it, a scene is made of light sources, geometries and a camera, all defined in mathematical, hence rather abstract, terms. Several rendering algorithms have been developed over the years and many of them have been designed to produce images as \textit{photorealistic} and lifelike as possible. The most recent ones of these kind are all based upon simulating how light interacts with the scene, closely imitating its natural behavior. A particularly successful rendering algorithm following this philosophy is \textit{Path Tracing} \cite{kajiya1986rendering} since its derivations and itself are widespread across animation, visual effects and video games. In simple words, path tracing is based upon the idea of shooting rays out of an artificial camera towards the scene and make each bounce around the scene until it reaches a light source; then, the luminous energy carried by the ray and its bounces, called \textit{path}, can be computed --- or \textit{traced} back. Without diving further into its details --- which are left for section \ref{background} ---, it is clear how complex path tracing can get. This makes it difficult to grasp by people approaching it and difficult to debug. We are presenting a tool capable of showing the user an overview of the inner workings of a path tracer by providing interactive visualizations of the very core of a path tracer: paths and how they interact with the scene. We strived to create something that does not tell but --- literally --- shows the swarm of paths shot by a path tracer to help understand what is actually going on during the rendering process. The idea of providing interactive renderings of the data generated by a path tracer --- or a ray tracer --- is not original. Let us mention a few that inspired our work: \begin{itemize} \item The \textit{Ray tracing visualization toolkit} (\textit{rtVTK}) by C. Gribble et Al. \cite{gribble2012ray} is a nice starting point but it focuses on ray tracers \cite{whitted1979improved, cook1984distributed} more than on path tracers. Furthermore, it is designed to render only a single ray tree at once, making it difficult to picture a global overview on the data that concurred in the generation of the final render; after all, a good mantra of presenting visual data is, quoting Ben Shneiderman, “Overview first, zoom and filter, then details-on-demand” \cite{shneiderman2003eyes}. rtVTK practically presents only the details. \item The work presented in \textit{A Framework for Visual Dynamic Analysis of Ray Tracing Algorithms} by H. Lesev and A. Penev \cite{lesev2014framework} has extensive filtering and data gathering features which inspired us, but it works only on ray tracers, as much as rtVTK. \item Another great inspiration for us was the framework presented in \textit{Applying Visual Analytics to Physically Based Rendering} by G. Simons et Al. \cite{simons2019applying}, especially the visualization parts. We however drift off it because, while they reduce the data they gather from a path tracer by selecting only about 1/6000th of the total paths, our goal is to keep the whole dataset and let the user choose what to visualize from all the samples. \item Another tightly related work is \textit{EMCA}, which stands for \textit{Explorer of Monte-Carlo based Algorithms}, by C. Kreisl \cite{EMCA@2019}. It focuses on path tracing and visualizations but its client-server architecture with per-pixel path analysis lacks of the global view we are trying to achieve. \end{itemize} \section{Background} \label{background} %To fully comprehend the topics and challenges presented in this work a background is needed. We will now provide a quick overview on path tracing to establish a common ground of terminology between us and the reader. This is not meant to be in any way an exhaustive explanation, please refer to the cited literature to have a better insight. \subsection{The rendering equation} \label{secrendeq} As we introduced before, \textit{photorealistic rendering} is the process that given a mathematical description of a three-dimensional scene outputs an as life-like as possible image, know as \textit{render}. The process simulates how light travels from light sources to an abstract camera sensor while interacting with the scene. An important equation that describes how light behaves is the \textit{Rendering Equation} introduced by J. Kajiya in 1986 \cite{kajiya1986rendering} here presented with different notations: \begin{equation} L_o(\vec{x},\vec{\omega}_o) = L_e(\vec{x},\vec{\omega}_o) + L_r(\vec{x},\vec{\omega}_o) \label{rendeqeasy} \end{equation} It says that, considering a surface, the amount of light that leaves the surface point $\vec{x}$ in direction $\vec{\omega}_o$, called \textit{outgoing radiance} ($L_o$), is determined by the emitted radiance ($L_e$), which is the one emanated by the surface itself, plus the reflected radiance ($L_r$) which is given by: \begin{equation} L_r(\vec{x},\vec{\omega}_o) = \int_{\mathcal{H}^2} f(\vec{\omega}_o, \vec{x}, \vec{\omega}_i) L_i(\vec{x},\vec{\omega}_i) d\vec{\omega}_i \label{reflectioneq} \end{equation} This means it consists in the sum of the radiance incoming ($L_i$) from all possible directions ($\vec{\omega}_i \in \mathcal{H}^2$) hitting point $\vec{x}$, weighted by the reflective and refractive properties of the material ($f(\cdots)$). Now, image sensors, as well as all animal visual organs, are made of several small photo sensitive units that measure the incoming radiance hitting their surfaces over a short period of time. By putting together the values read by the units an image is produced. To compute the incoming radiance on each of these photo sensitive units called \textit{pixels}, the incoming radiance must be integrated over the pixel surface and scaled by its area to make the result independent of the sensor's size: \begin{equation} L_{pixel} = \frac{1}{A_{pixel}} \int_{\vec{q}\in pixel} L_i(\vec{q}, \vec{\omega}_r)dA_q \label{pixelradiance} \end{equation} Where $\vec{\omega}_r$ depends on the camera mathematical model, that can range from the simplest pinhole camera to a complex simulation of real camera lenses groups. At this point is clear that rendering is all about computing the radiance hitting a sensor but the rendering equation presented above (eq. \ref{rendeqeasy}) describes only outgoing radiance. Fortunately, thanks to the fact that radiance is constant along a ray\footnote{Radiance is constant along a ray only in perfect vacuum. All non-volumetric path tracers --- which we exclusively focus on --- assume vacuum between surfaces.}, it is possible to write: \begin{equation} L_i(\vec{x},\vec{\omega}_i) = L_o(\vec{y},\vec{\omega}_o) \end{equation} Where $\vec{y}$ is the first point on a surface hit by the ray shot from $\vec{x}$ in direction $\vec{\omega}_i$ and $\vec{\omega}_o$ is the direction pointing from $\vec{y}$ to $\vec{x}$. Introducing the Raytracing $RT$ operator: \begin{equation} \vec{y} = RT(\vec{x},\vec{\omega}_i) \end{equation} And rewriting $\vec{\omega}_o$ in function of $\vec{\omega}_i$: \begin{equation} \vec{\omega}_o = -\vec{\omega}_i \end{equation} We have that: \begin{equation} L_i(\vec{x},\vec{\omega}_i) = L_o(RT(\vec{x},\vec{\omega}_i),-\vec{\omega}_i) \label{lilo} \end{equation} The first thing that can be taken from this is that to compute the incoming radiance on a sensor it is sufficient to shoot a ray and compute the radiance on the scene point hit by the ray outgoing in the inverse ray direction. Secondly, plugging everything that has been said until now produces a more complete rendering equation: \begin{equation} L_o(\vec{x},\vec{\omega}_o) = L_e(\vec{x},\vec{\omega}_o) + \int_{\mathcal{H}^2} f(\vec{\omega}_o, \vec{x}, \vec{\omega}_i) L_o(RT(\vec{x},\vec{\omega}_i),-\vec{\omega}_i) d\vec{\omega}_i \label{renderingeq} \end{equation} %Where: %\begin{description} % \item[$\vec{x}$] A point in space. % \item[$\vec{\omega}_o$] An outgoing direction from $\vec{x}$. % \item[$\vec{\omega}_i$] An incoming direction towards $\vec{x}$ % \item[$L(\vec{x},\vec{\omega}_o)$] Radiance outgoing in direction $\vec{\omega}_o$ from $\vec{x}$ % \item[$L_e(\vec{x},\vec{\omega}_o)$] Radiance emitted in direction $\vec{\omega}_o$ by $\vec{x}$. % \item[$\mathcal{H}^2$] All the possible directions. % \item[$L(\vec{x},\vec{\omega}_i)$] Incoming radiance from direction $\vec{\omega}_i$ on $\vec{x}$. % \item[$f(\vec{\omega}_o, \vec{x}, \vec{\omega}_i)$] The Bidirectional Scattering Distribution Function (BSDF) computed on $\vec{x}$ on directions $\vec{\omega}_o$ and $\vec{\omega}_i$. % \item[$\hat{n}_{\vec{x}}$] The surface normal on $\vec{x}$. %\end{description} \subsection{Monte Carlo integration} Given all the equations above, all it is needed to render a convincing image is solving them. The first problem that surfaces is that they contain integrals. Integrals are infinitesimal sums and as such they do not conform too well with the discrete nature of electronic calculators. In other words, integrals cannot be numerically solved on computers. Their solution, though, can be approximated. Without lingering on the reasons, a particularly suitable approximation method for our case is the Monte Carlo integration \cite{kalos2009monte}. It randomly picks $N$ samples ($x_i, i \in [0,N)$) from the integration interval ($\Omega$), evaluates the integrand ($f(x)$) with each sample and does an average of the evaluated values weighting them by the probability density function ($pdf(x)$) of picking the corresponding sample from the interval. This gives: \begin{equation} \int_{\Omega}f(x)dx \approx \frac{1}{N} \sum_{i=0}^{N} \frac{f(x_i)}{pdf(x_i)} \label{montecarlo} \end{equation} The value computed by Monte Carlo gets closer to the correct result increasing the number of random samples $N$. It has been demonstrated that if the samples are picked from the interval using a uniform distribution, the relative error is directly proportional to $\frac{1}{\sqrt{N}}$. From this it can be deducted that Monte Carlo will never output the correct solution but it will asymptotically tend to it as samples increase; that is why literature usually talk about \textit{convergence} and sometimes about \textit{convergence speed}, which is an informal value expressing how many samples are needed to approximate an acceptable result, where what is considered acceptable varies case by case. The $\frac{1}{\sqrt{N}}$ factor can be improved by a technique called \textit{importance sampling} that is based upon the idea of drawing samples from a distribution as close as possible to the integrand in order to reduce the variance. The ideal case would be using the integrand itself as distribution but often, due to its complexity, is impossible to draw samples directly from it. Alternative distributions that roughly approximate the integrand are used instead. This can even be brought a step forward by stochastically combining two approximate distributions together using the so-called \textit{multiple importance sampling} (MIS) \cite{veach1995optimally}. \subsection{Path tracing} Now that we have a tool to numerically solve integrals, we can apply it to the integrals presented in section \ref{secrendeq}. The total incoming radiance on a pixel (eq. \ref{pixelradiance}) can then be calculated by randomly picking sample points on its surface and then applying Monte Carlo (eq. \ref{montecarlo}): \begin{equation} L_{pixel} \approx \frac{1}{N A_{pixel}} \sum^{N}_{i = 0} \frac{L_i(\vec{q}_i, \vec{\omega}_{ri})}{p(\vec{q}_i)} \end{equation} The number of samples picked here $N$ is usually constant for each pixel. Due to its direct relation between final render quality --- more samples means a solution closer to the real one --- and render time --- each additional sample adds computational burden --- it is a rather important parameter in all path tracers and it is called \textit{samples per pixel} or \textit{spp} for short. Now, it is a matter of computing $L_i$ for each pixel sample. Recalling the logical results given by the conservation of radiance along rays presented in equation \ref{lilo} the problem shifts on how to compute the outgoing radiance of the point of the scene hit by the ray shot from the sample position ($\vec{q}_i$) in its associated direction ($\vec{\omega}_{ri}$). This value, as presented by the complete version of the rendering equation (eq. \ref{renderingeq}), depends on another integral having all possible directions as its integration interval. Furthermore, inside the integral there is again an outgoing radiance, but this time computed with other parameters ($L_o(RT(\vec{x},\vec{\omega}_i),-\vec{\omega}_i)$). By substitution, it begins to become clear that evaluating one outgoing radiance theoretically leads to solving a recursively infinite amount of integrals. This, through the power of Monte Carlo integration, becomes manageable and a path tracer computes outgoing radiances by sampling all the possible directions just once. The rendering equation can then be written as: \begin{equation} L_o(\vec{x},\vec{\omega}_o) \approx L_e(\vec{x},\vec{\omega}_o) + \frac{1}{pdf(\vec{\omega}_i)} f(\vec{\omega}_o, \vec{x}, \vec{\omega}_i) L_o(RT(\vec{x},\vec{\omega}_i),-\vec{\omega}_i) \end{equation} With this approximation the calculations consist in shooting a ray, find an intersection with the scene, compute the properties of the material on the intersection, pick a random new direction, shoot a ray in that direction from the intersection point and then keep doing the same recursively, forming a path. In practice, to not make a path bounce indefinitely, a maximum number of bounces, called \textit{maximum path length}, is arbitrarily decided before rendering, as much as the number of samples per pixel is. Path tracers that behave as described until now are informally called na\"ive path tracers. Other more complex tracers employ techniques to speed up the Monte Carlo convergence. As already mentioned, a common technique is importance sampling, as it can be applied to path tracing by sampling new bounces' directions from probability distributions that naturally carry more radiance. This can happen in two ways: either paths are forced to follow the properties of the material they are bouncing on, or they are directed towards a light source. A usually even more efficient approach is to combine these two sampling strategies through multiple importance sampling. % Using this approximation, estimating radiance for each pixel sample can be written in pseudo code as shown in listing \ref{ptalgo}. % It is clear how % \begin{Listing} % \begin{lstlisting} % function estimate_radiance(ray): % intersection := intersect scene with ray % if no intersection: % return 0 % x := intersection point % omega_o := ray direction % omega_i := random direction picked from distribution d % pdf_omega_i := pdf of the distribution d evaluated for omega_i % new_ray := ray originating from x and direction omega_i % incoming_radiance := estimate_radiance(new_ray) % material := material properties computed with omega_o, x, and omega_i % return (1 / pdf_omega_i) * material * incoming_radiance % primary_ray := ray shot from the camera % estimate_radiance(primary_ray) % \end{lstlisting} % \caption{Pixel sample radiance estimation.} % \label{ptalgo} % \end{Listing} %\subsection{OpenGL} %It means that the algorithms that generated these datasets have different ways of choosing the direction of each path bounce in order to maximize the radiance carried by each path, speeding up the convergence of the Monte Carlo integration and hence trying to lower the amount of noise given the same amount of pixel samples. This technique, used in every application of Monte Carlo methods, is called \textit{importance sampling} \cite{kalos2009monte}. Multiple importance sampling \cite{veach1995optimally}. \section{Motivation} \label{motivation} %As introduced above, path tracing is an algorithm that is not easy to picture. For someone who got to understand it, the rendering equation is more than enough to tell how a path tracer work. At least to get started, everything a person needs to know is in that equation: once combined with Monte Carlo integration, everyone is theoretically ready to implement a path tracer. As introduced above, path tracing is not easy to picture. As many research groups before us thought, having a tool showing what a tracer is doing by visualizing its interactions with the 3D scene would help the professional developers lost in the debugging cycle as well as the students trying to get a hang of path tracing's inner workings. The challenge this vision has to face is the sheer amount of these interactions; talking about path tracing, to have a decent render\footnote{We considered as a decent render a $512 \times 512$ pixels and 256 spp image which is a good halfway point between production quality and being way too far from convergence.}, more than two hundreds paths have to be shot for each pixel: to produce an image, more than fifty million paths have to be shot. Postponing data size considerations for later (sec. \ref{datatogather}), leaves us with potential datasets that without proper data reduction or filtering are of no use to anyone. Related works usually perform substantial reductions of the datasets \cite{simons2019applying,EMCA@2019}, but we believe that having all the paths available for visualization would improve the experience. With these premises, a more precise idea for a visualization tool came to us. The user should have the power to visually and interactively explore all the paths previously generated by a path tracer during rendering. The paths have to be immersed in an interactive rendering of the scene: their most important property is their geometry or, in other words, where they bounce inside the scene. In the beginning, out of curiosity and naivety, we tried visualizing all the paths of some datasets and even if those were rather small, we achieved compelling yet cluttered results (fig. \ref{visual_clutter}); this made visualizing the millions of paths of a proper dataset out of question from the earliest stages. To navigate this ocean of paths, the user should be provided with a global summary overview of the dataset in its entirety, and they should also able to filter them using spatial queries. For the global overview we envisioned a heatmap applied as a texture to the 3D view of the scene that shows areas where there is more path activity, while for the spatial queries we thought the user might want to select path bouncing on certain noteworthy surfaces, such as lights or mirrors. \begin{figure} \centering \centering \begin{subfigure}[t]{0.49\linewidth} \includegraphics[width=\textwidth]{chapters/chapter_intro/clutter_early1} \caption{$64 \times 64$, 1 spp} \end{subfigure} \begin{subfigure}[t]{0.49\linewidth} \includegraphics[width=\textwidth]{chapters/chapter_intro/clutter_early2} \caption{$64 \times 64$, 64 spp} \end{subfigure} \caption{Visual clutter in an early version of the tool where the scene rendering has not been implemented yet. Both datasets have been generated on the \textit{Cornell Box} scene.} \label{visual_clutter} \end{figure} Later in the development, it became clear that the possibility of comparing different datasets would come in handy to users. It could be useful in many cases such as comparing two progressive versions of the same in-development path tracer to understand what changed or, more in an educational context, such as showing a ground truth next to a purposely wrong dataset to understand why one is wrong on a deeper level. Of course, before even talking about visualizing data, data has to be gathered from a path tracer. We envisioned a software, parallel to the visualization client, able to plug to any path tracer and gather with little effort the required data during the rendering process. Then the idea of a gathering library with a very simple interface came about; a library that can be plugged into any path tracer the user is currently working on by just calling the few required functions. To summarize, our vision consisted in a tool divided into a visualization client and a data gathering library which can be used both for debugging and teaching purposes. The library should have an easy to interact with interface, while the client would let the user explore different datasets simultaneously and in their entirety using the filtering and visualization options provided to them. %select a portion of a surface of the 3D scene the tracer has been run upon and see the paths that bounce there with a bunch of useful data. To be able to do that the whole set of paths shoot by a tracer are needed: by the very stochastic nature of a path tracer, it is impossible to determine which paths will end up bouncing where without resolving them all first. That is why it has been decided it was essential to store data about each path during the rendering process. To make the tool usable in most possible use cases, it had to be able to plug into an existing path tracer and this lead to the conception of the tool as a two software pieces suite: a \textit{data gatherer library} called \texttt{gatherer} and a \textit{visualization client} called \texttt{gathererclient}. %Now this “useful data” was not extremely well-defined during those early stages, so most of the efforts have been directed to the very essential: rendering the requested paths keeping interactivity.
section \<open>Two-step Optimized MRU Model\<close> theory Two_Step_MRU imports "../MRU_Vote_Opt" "../Two_Steps" begin text \<open>To make the coming proofs of concrete algorithms easier, in this model we split the @{term opt_mru_round} into two steps\<close> context mono_quorum begin subsection \<open>Model definition\<close> (*****************************************************************************) definition opt_mru_step0 :: "round \<Rightarrow> process set \<Rightarrow> process set \<Rightarrow> val \<Rightarrow> (opt_mru_state \<times> opt_mru_state) set" where "opt_mru_step0 r Q S v = {(s, s'). ( guards ) r = next_round s \<and> two_step r = 0 \<and> (S = {} \<or> opt_mru_guard (mru_vote s) Q v) \<and> ( actions ) s' = s\<lparr> mru_vote := mru_vote s ++ const_map (two_phase r, v) S , next_round := Suc r \<rparr> }" definition r_votes :: "opt_mru_state \<Rightarrow> round \<Rightarrow> (process, val)map" where "r_votes s r \<equiv> \<lambda>p. if (\<exists>v. mru_vote s p = Some (two_phase r, v)) then map_option snd (mru_vote s p) else None" definition opt_mru_step1 :: "round \<Rightarrow> (process, val)map \<Rightarrow> (opt_mru_state \<times> opt_mru_state) set" where "opt_mru_step1 r dec_f = {(s, s'). ( guards ) r = next_round s \<and> two_step r = 1 \<and> d_guard dec_f (r_votes s r) \<and> ( actions ) s' = s\<lparr> next_round := Suc r , decisions := decisions s ++ dec_f \<rparr> }" lemmas ts_lv_evt_defs = opt_mru_step0_def opt_mru_step1_def opt_mru_guard_def definition ts_lv_trans :: "(opt_mru_state \<times> opt_mru_state) set" where "ts_lv_trans = (\<Union>r Q S v. opt_mru_step0 r Q S v) \<union> (\<Union>r dec_f. opt_mru_step1 r dec_f) \<union> Id" definition ts_lv_TS :: "opt_mru_state TS" where "ts_lv_TS = \<lparr> init = opt_mru_init, trans = ts_lv_trans \<rparr>" lemmas ts_lv_TS_defs = ts_lv_TS_def opt_mru_init_def ts_lv_trans_def subsection \<open>Refinement\<close> (***************************************************************************) definition basic_rel where "basic_rel \<equiv> {(sa, sc). decisions sc = decisions sa \<and> next_round sa = two_phase (next_round sc) }" definition two_step0_rel :: "(opt_mru_state \<times> opt_mru_state)set" where "two_step0_rel \<equiv> basic_rel \<inter> {(sa, sc). two_step (next_round sc) = 0 \<and> mru_vote sc = mru_vote sa }" definition two_step1_rel :: "(opt_mru_state \<times> opt_mru_state)set" where "two_step1_rel \<equiv> basic_rel \<inter> {(sa, sc). (\<exists>sc' r Q S v. (sa, sc') \<in> two_step0_rel \<and> (sc', sc) \<in> opt_mru_step0 r Q S v) }" definition ts_ref_rel where "ts_ref_rel = {(sa, sc). (two_step (next_round sc) = 0 \<longrightarrow> (sa, sc) \<in> two_step0_rel) \<and> (two_step (next_round sc) = 1 \<longrightarrow> (sa, sc) \<in> two_step1_rel) }" lemmas ts_ref_rel_defs = basic_rel_def ts_ref_rel_def two_step0_rel_def two_step1_rel_def lemma step0_ref: "{ts_ref_rel} Id, opt_mru_step0 r Q S v {> ts_ref_rel}" apply(auto simp only: PO_rhoare_defs) apply(auto simp add: ts_ref_rel_def opt_mru_step0_def two_step_phase_Suc two_step1_rel_def) apply(auto simp add: two_step0_rel_def basic_rel_def two_step_phase_Suc) done lemma step1_ref: "{ts_ref_rel \<inter> OMRU_inv1 \<times> UNIV} \<Union>r' Q S' v dec_f'. opt_mru_round r' Q S' v dec_f', opt_mru_step1 r dec_f {> ts_ref_rel}" proof(auto simp only: PO_rhoare_defs) fix sa sc1 sc2 assume ainv: "sa \<in> OMRU_inv1" and R: "(sa, sc1) \<in> ts_ref_rel" and step1: "(sc1, sc2) \<in> opt_mru_step1 r dec_f" then obtain sc0 r0 S0 Q v where R0: "(sa, sc0) \<in> two_step0_rel" and step0: "(sc0, sc1) \<in> opt_mru_step0 r0 Q S0 v" by(auto simp add: ts_ref_rel_def two_step1_rel_def opt_mru_step1_def) have R1: "(sa, sc1) \<in> two_step1_rel" and r1_step: "two_step (next_round sc1) = Suc 0" and r2_step: "two_step (next_round sc2) = 0" using R step1 by(auto simp add: ts_ref_rel_def opt_mru_step1_def two_step_phase_Suc) have r: "r = Suc r0" and r0: "r0 = next_round sc0" and r0_step: "two_step r0 = 0" and r1: "r = next_round sc1" using step0 step1 by(auto simp add: opt_mru_step0_def opt_mru_step1_def) have abs_round1: "next_round sa = two_phase r" using R1 r1 by(auto simp add: two_step1_rel_def basic_rel_def) have abs_round0: "next_round sa = two_phase r0" using R0 r0 r by(auto simp add: two_step0_rel_def basic_rel_def two_step_phase_Suc) have r_votes: "r_votes sc1 r = const_map v S0" proof(rule ext) fix p show "r_votes sc1 r p = const_map v S0 p" proof(cases "r_votes sc1 r p") case None thus ?thesis using step0 abs_round0 abs_round1 by(auto simp add: r_votes_def opt_mru_step0_def const_map_def restrict_map_def map_add_def) next case (Some w) hence in_S0: "mru_vote sc0 p \<noteq> mru_vote sc1 p" using R0 step0 ainv r1_step r abs_round0 by(auto simp add: r_votes_def ts_ref_rel_defs two_step_phase_Suc opt_mru_step0_def dest: OMRU_inv1D[where p=p]) hence "p \<in> S0" using step0 by(auto simp add: opt_mru_step0_def map_add_def const_map_is_Some split: option.split_asm) moreover have "w=v" using R0 step0 ainv r1_step r abs_round0 Some by(auto simp add: r_votes_def ts_ref_rel_defs const_map_is_Some two_step_phase_Suc opt_mru_step0_def dest: OMRU_inv1D[where p=p]) ultimately show ?thesis using Some by(auto simp add: const_map_def) qed qed define sa' where "sa' = sa\<lparr> mru_vote := mru_vote sa ++ const_map (two_phase r, v) S0 , next_round := Suc (two_phase r) , decisions := decisions sa ++ dec_f \<rparr>" have "(sa, sa') \<in> opt_mru_round (two_phase r) Q S0 v dec_f \<and> (sa', sc2) \<in> ts_ref_rel" proof show "(sa', sc2) \<in> ts_ref_rel" using r2_step R0 step0 step1 by(auto simp add: ts_ref_rel_def two_step0_rel_def basic_rel_def opt_mru_step0_def opt_mru_step1_def sa'_def two_step_phase_Suc) next show "(sa, sa') \<in> opt_mru_round (two_phase r) Q S0 v dec_f" using R0 r0_step r1_step step0 step1 r0 r1 r r_votes by(auto simp add: ts_ref_rel_defs opt_mru_round_def two_step_phase_Suc opt_mru_step0_def opt_mru_step1_def sa'_def) qed thus "\<exists>y. (sa, y) \<in> (\<Union>r' Q S' v D'. opt_mru_round r' Q S' v D') \<and> (y, sc2) \<in> ts_ref_rel" by blast qed lemma TwoStep_Coordinated_Refines: "PO_refines (ts_ref_rel \<inter> OMRU_inv1 \<times> UNIV) lv_TS ts_lv_TS" proof(rule refine_using_invariants) show "init ts_lv_TS \<subseteq> ts_ref_rel `` init lv_TS" by(auto simp add: ts_lv_TS_defs lv_TS_def ts_ref_rel_def two_step0_rel_def two_step1_rel_def basic_rel_def) next show "{ts_ref_rel \<inter> OMRU_inv1 \<times> UNIV} TS.trans lv_TS, TS.trans (ts_lv_TS) {> ts_ref_rel}" apply(simp add: lv_TS_defs ts_lv_TS_defs) apply(auto simp add: ts_lv_trans_def intro!: step0_ref step1_ref) done qed(auto intro!: OMRU_inv1_inductive) subsection \<open>Invariants\<close> (****************************************************************************) definition TS_OMRU_inv1 where "TS_OMRU_inv1 \<equiv> {s. (two_step (next_round s) = 0 \<longrightarrow> (\<forall>p \<Phi> v. mru_vote s p = Some (\<Phi>, v) \<longrightarrow> \<Phi> < two_phase (next_round s))) \<and> (two_step (next_round s) = 1 \<longrightarrow> (\<forall>p \<Phi> v. mru_vote s p = Some (\<Phi>, v) \<longrightarrow> \<Phi> \<le> two_phase (next_round s))) }" lemma two_step_neqD: "two_step x \<noteq> Suc 0 \<Longrightarrow> two_step x = 0" "0 < two_step x \<Longrightarrow> two_step x = Suc 0" by(auto simp add: two_step_def) lemma TS_OMRU_inv1_inductive: "init ts_lv_TS \<subseteq> TS_OMRU_inv1" "{TS_OMRU_inv1} trans ts_lv_TS {> TS_OMRU_inv1}" apply(fastforce simp add: PO_hoare_defs ts_lv_TS_def opt_mru_init_def TS_OMRU_inv1_def)[1] apply(auto simp add: PO_hoare_defs ts_lv_TS_def ts_lv_trans_def opt_mru_step0_def opt_mru_step1_def) apply(auto simp add: TS_OMRU_inv1_def two_step_phase_Suc const_map_is_Some const_map_is_None less_Suc_eq_le dest: less_imp_le) done end end
-- Copyright © 2019 François G. Dorais. All rights reserved. -- Released under Apache 2.0 license as described in the file LICENSE. import .basic import .homomorphism import .identity namespace universal variables {τ : Type} {σ : Type} (sig : signature τ σ) namespace algebra section pi variables {sig} {ι : Type} (alg : ι → algebra sig) definition pi : algebra sig := { sort := λ t, Π j, (alg j).sort t , func := λ f xs j, (alg j).func f (λ i, xs i j) } @[simp] theorem pi_func (f) (xs) (j) : (pi alg).func f xs j = (alg j).func f (λ i, xs i j) := rfl definition pi_proj (j) : homomorphism (pi alg) (alg j) := { map := λ _ x, x j , func := λ _ _, rfl } theorem pi_eval {dom} : ∀ {cod} (t : term sig dom cod) (xs) (j), (pi alg).eval t xs j = (alg j).eval t (λ i, xs i j) \| _ (term.proj _) _ _ := rfl \| _ (term.func f ts) xs j := have (λ i, (pi alg).eval (ts i) xs j) = (λ i, (alg j).eval (ts i) (λ i, xs i j)), from funext $ λ i, pi_eval (ts i) xs j, calc _ = (alg j).func f (λ i, (pi alg).eval (ts i) xs j) : rfl ... = (alg j).func f (λ i, (alg j).eval (ts i) (λ i, xs i j)) : by rw this ... = (alg j).eval (term.func f ts) (λ i, xs i j) : by reflexivity theorem pi_satisfies (e : identity sig) : (∀ i, (alg i).satisfies e) → (pi alg).satisfies e := begin intros H val, funext, rw pi_eval, rw pi_eval, apply H, end section universal_property variables {alg} {alg₀ : algebra sig} (h : Π j, homomorphism alg₀ (alg j)) definition pi_hom : homomorphism alg₀ (pi alg) := { map := λ t x j, (h j).map t x , func := λ _ _, funext $ λ j, by simp [(h j).func] } @[simp] theorem pi_hom_proj (j) : (pi_proj alg j).comp (pi_hom h) = h j := homomorphism.ext $ λ _ _, rfl theorem pi_univ (h₀ : homomorphism alg₀ (pi alg)) : (∀ j, (pi_proj alg j).comp h₀ = h j) → h₀ = pi_hom h := λ H, homomorphism.ext $ λ t x, funext $ λ j, begin transitivity ((pi_proj alg j).comp h₀).map t x, reflexivity, rw H, reflexivity, end end universal_property end pi section prod variables {sig} (alg₁ : algebra sig) (alg₂ : algebra sig) definition prod : algebra sig := { sort := λ t, alg₁.sort t × alg₂.sort t , func := λ f xs, (alg₁.func f (λ i, (xs i).fst), alg₂.func f (λ i, (xs i).snd)) } @[simp] theorem prod_func (f) (xs) : (prod alg₁ alg₂).func f xs = (alg₁.func f (λ i, (xs i).fst), alg₂.func f (λ i, (xs i).snd)) := rfl definition prod_fst : homomorphism (prod alg₁ alg₂) alg₁ := { map := λ _, prod.fst , func := λ _ _, rfl } definition prod_snd : homomorphism (prod alg₁ alg₂) alg₂ := { map := λ _, prod.snd , func := λ _ _, rfl } theorem prod_eval {dom} : ∀ {cod} (t : term sig dom cod) (xs), (prod alg₁ alg₂).eval t xs = (alg₁.eval t (λ i, (xs i).fst), alg₂.eval t (λ i, (xs i).snd)) \| _ (term.proj _) _ := prod.eq rfl rfl \| _ (term.func f ts) xs := have (λ i, (prod alg₁ alg₂).eval (ts i) xs) = (λ i, (alg₁.eval (ts i) (λ i, (xs i).fst), alg₂.eval (ts i) (λ i, (xs i).snd))), from funext $ λ i, prod_eval (ts i) xs, calc (prod alg₁ alg₂).eval (term.func f ts) xs = (prod alg₁ alg₂).func f (λ i, (prod alg₁ alg₂).eval (ts i) xs) : rfl ... = (prod alg₁ alg₂).func f (λ i, (alg₁.eval (ts i) (λ i, (xs i).fst), alg₂.eval (ts i) (λ i, (xs i).snd))) : by rw this ... = (alg₁.eval (term.func f ts) (λ i, (xs i).fst), alg₂.eval (term.func f ts) (λ i, (xs i).snd)) : by reflexivity theorem prod_eval_fst {dom} {cod} (t : term sig dom cod) (xs) : ((prod alg₁ alg₂).eval t xs).fst = alg₁.eval t (λ i, (xs i).fst) := by rw prod_eval theorem prod_eval_snd {dom} {cod} (t : term sig dom cod) (xs) : ((prod alg₁ alg₂).eval t xs).snd = alg₂.eval t (λ i, (xs i).snd) := by rw prod_eval theorem prod_satisfies (e : identity sig) : alg₁.satisfies e → alg₂.satisfies e → (prod alg₁ alg₂).satisfies e := begin intros H₁ H₂ val, rw prod_eval, rw prod_eval, apply prod.eq, apply H₁, apply H₂, end section universal_property variables {alg₁ alg₂} {alg : algebra sig} (h₁ : homomorphism alg alg₁) (h₂ : homomorphism alg alg₂) include h₁ h₂ definition prod_hom : homomorphism alg (prod alg₁ alg₂) := { map := λ t x, (h₁.map t x, h₂.map t x) , func := λ f xs, prod.eq (by simp [h₁.func f]) (by simp [h₂.func f]) } @[simp] theorem prod_hom_fst : (prod_fst alg₁ alg₂).comp (prod_hom h₁ h₂) = h₁ := homomorphism.ext $ λ _ _, rfl @[simp] theorem prod_hom_snd : (prod_snd alg₁ alg₂).comp (prod_hom h₁ h₂) = h₂ := homomorphism.ext $ λ _ _, rfl theorem prod_univ (h : homomorphism alg (prod alg₁ alg₂)) : (prod_fst alg₁ alg₂).comp h = h₁ → (prod_snd alg₁ alg₂).comp h = h₂ → h = prod_hom h₁ h₂ := λ H₁ H₂, homomorphism.ext $ λ t x, prod.eq (begin transitivity ((prod_fst alg₁ alg₂).comp h).map t x, reflexivity, rw H₁, reflexivity, end) (begin transitivity ((prod_snd alg₁ alg₂).comp h).map t x, reflexivity, rw H₂, reflexivity, end) end universal_property end prod end algebra end universal
# JAX implementation of temporal parallelisation of Kalman filter and smoother This notebook implements the algorithms in the paper by S. Särkkä and Á. García-Fernández by leveraging the [JAX](https://github.com/google/jax) framework. ## Author: Adrien Corenflos ## Reference: Simo Särkkä and Ángel F. García-Fernández (2021). Temporal Parallelization of Bayesian Smoothers. IEEE Transactions on Automatic Control, Volume: 66, Issue: 1, Pages 299-306. arXiv: https://arxiv.org/abs/1905.13002 DOI: https://doi.org/10.1109/TAC.2020.2976316 - If you are on Colab don't forget to switch the Runtime (in the tab with the same name) type to GPU. - If you are using a personal machine, to install JAX with GPU support please read JAX README.md [file](https://github.com/google/jax). ``` !pip install -U jax==0.1.77 jaxlib==0.1.55+cuda101 -f https://storage.googleapis.com/jax-releases/jax_releases.html ``` Looking in links: https://storage.googleapis.com/jax-releases/jax_releases.html Requirement already up-to-date: jax==0.1.77 in /usr/local/lib/python3.6/dist-packages (0.1.77) Collecting jaxlib==0.1.55+cuda101 Using cached https://storage.googleapis.com/jax-releases/cuda101/jaxlib-0.1.55%2Bcuda101-cp36-none-manylinux2010_x86_64.whl Requirement already satisfied, skipping upgrade: numpy>=1.12 in /usr/local/lib/python3.6/dist-packages (from jax==0.1.77) (1.19.4) Requirement already satisfied, skipping upgrade: opt-einsum in /usr/local/lib/python3.6/dist-packages (from jax==0.1.77) (3.3.0) Requirement already satisfied, skipping upgrade: absl-py in /usr/local/lib/python3.6/dist-packages (from jax==0.1.77) (0.10.0) Requirement already satisfied, skipping upgrade: scipy in /usr/local/lib/python3.6/dist-packages (from jaxlib==0.1.55+cuda101) (1.4.1) Requirement already satisfied, skipping upgrade: six in /usr/local/lib/python3.6/dist-packages (from absl-py->jax==0.1.77) (1.15.0) Installing collected packages: jaxlib Found existing installation: jaxlib 0.1.55 Uninstalling jaxlib-0.1.55: Successfully uninstalled jaxlib-0.1.55 Successfully installed jaxlib-0.1.55 The above is necessary at the moment due to a batching (vectorization) bug in LU factorisation that was only fixed on 16 Dec 2020. It will be removed when next release of JAX (>2.7.0) reaches Colab. There should not be any interface change otherwise. ### Imports ``` # Python specific imports that will make our job easier and our code prettier from collections import namedtuple from functools import partial import time from tqdm.auto import trange, tqdm # JAX specific imports that we will use to code the logic from jax import jit, vmap, make_jaxpr, device_put, devices from jax.config import config from jax.core import eval_jaxpr # this will serve as a proxy compiler as JAX doesn't have an AOT function from jax.lax import associative_scan, psum, scan import jax.numpy as jnp import jax.scipy as jsc # Auxiliary libraries that we will use to report results and create the data import math import matplotlib.pyplot as plt import numpy as np import scipy as sc ``` ``` config.update("jax_enable_x64", True) # We use this so that we have the same precision as the pure numpy implementation # This can be useful in particular for large observations (long running series) ``` ### Utilities We define some utilities that will make the code more readable ``` LOG2PI = math.log(2 * math.pi) ``` ### Model In the vein of Example 3.6 in the book Bayesian Filtering and Smoothing (S. Särkkä) we consider a simple car tracking model where the car is characterised by its coordinates and directional speeds $\mathbf{z} = [x, y, \dot{x}, \dot{y}]^\top$ for which only the positional components are observed with some uncertainty. Formally this corresponds to the following state-space model: \begin{equation} \begin{split} x_k &= F \, x_{k-1} + q_{k-1},\\ y_k &= H \, x_k + r_k, \end{split} \end{equation} where $q_k \sim \mathrm{N}(0,Q)$, $r_k \sim \mathrm{N}(0,R)$, and \begin{equation} F = \begin{pmatrix} 1 & 0 & \Delta t & 0 \\ 0 & 1 & 0 & \Delta t \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} ,\quad Q = q \, \begin{pmatrix} \frac{\Delta t^3}{3} & 0 & \frac{\Delta t^2}{2} & 0 \\ 0 & \frac{\Delta t^3}{3} & 0 & \frac{\Delta t^2}{2} \\ \frac{\Delta t^2}{2} & 0 & \Delta t & 0 \\ 0 & \frac{\Delta t^2}{2} & 0 & \Delta t \end{pmatrix}, \end{equation} along with \begin{equation} H = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix}, \qquad R = \begin{pmatrix} \sigma^2 & 0 \\ 0 & \sigma^2 \end{pmatrix}. \end{equation} which translates into the following code: ``` StateSpaceModel = namedtuple("StateSpaceModel", ["F", "H", "Q", "R", "m0", "P0", "xdim", "ydim"]) ``` ``` def make_car_tracking_model(q: float, dt: float, r: float, m0: np.ndarray, P0: np.ndarray): F = np.eye(4) + dt * np.eye(4, k=2) H = np.eye(2, 4) Q = np.kron(np.array([[dt3/3, dt2/2], [dt2/2, dt]]), np.eye(2)) R = r 2 * np.eye(2) return StateSpaceModel(F, H, q * Q, R, m0, P0, m0.shape[0], H.shape[0]) ``` ``` car_tracking_model = make_car_tracking_model(q=1., dt=0.1, r=0.5, m0=np.array([0., 0., 1., -1.]), P0=np.eye(4)) ``` So that we can now write some generic code to sample from a SSM: ``` def get_data(model: StateSpaceModel, T:float, seed:int=0): # We first generate the normals we will be using to simulate the SSM: rng = np.random.RandomState(seed) normals = rng.randn(1 + T, model.xdim + model.ydim) # Then we allocate the arrays where the simulated path and observations will # be stored: xs = np.empty((T, model.xdim)) ys = np.empty((T, model.ydim)) # So that we can now run the sampling routine: Q_chol = sc.linalg.cholesky(model.Q, lower=True) R_chol = sc.linalg.cholesky(model.R, lower=True) P0_chol = sc.linalg.cholesky(model.P0, lower=True) x = model.m0 + P0_chol @ normals[0, :model.xdim] for i, norm in enumerate(normals[1:]): x = model.F @ x + Q_chol @ norm[:model.xdim] y = model.H @ x + R_chol @ norm[model.xdim:] xs[i] = x ys[i] = y return xs, ys ``` ``` log10T = 4 ``` ``` true_xs, ys = get_data(car_tracking_model, 10 ** log10T, 0) ``` This corresponds to the following car positions in space (truncated at 100 timesteps): ``` fig, ax = plt.subplots(figsize=(7, 7)) ax.plot(true_xs[:100, 0], true_xs[:100, 1], label="True State", color="b") ax.scatter(ys[:100].T, label="Observations", color="r") _ = plt.legend() ``` ### Sequential Kalman filter and smoother We can now write the Kalman filter and smoother sequential routines in a JAX fashion. For more information on these algorithms, we refer the reader to the book: Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. For which an online version is also available on his [webpage](https://users.aalto.fi/~ssarkka/). #### Filter: ``` def mvn_logpdf(x, mean, cov): n = mean.shape[0] upper = jsc.linalg.cholesky(cov, lower=False) log_det = 2 jnp.sum(jnp.log(jnp.abs(jnp.diag(upper)))) diff = x - mean scaled_diff = jsc.linalg.solve_triangular(upper, diff.T, lower=False) distance = jnp.sum(scaled_diff * scaled_diff, 0) return -0.5 * (distance + n * LOG2PI + log_det) ``` ``` def kf(model, observations): def body(carry, y): m, P = carry m = model.F @ m P = model.F @ P @ model.F.T + model.Q obs_mean = model.H @ m S = model.H @ P @ model.H.T + model.R K = jsc.linalg.solve(S, model.H @ P, sym_pos=True).T # notice the jsc here m = m + K @ (y - model.H @ m) P = P - K @ S @ K.T return (m, P), (m, P) _, (fms, fPs) = scan(body, (model.m0, model.P0), observations) return fms, fPs ``` ``` fms, fPs = kf(car_tracking_model, ys[:100]) ``` #### Smoother: ``` def ks(model, ms, Ps): def body(carry, inp): m, P = inp sm, sP = carry pm = model.F @ m pP = model.F @ P @ model.F.T + model.Q C = jsc.linalg.solve(pP, model.F @ P, sym_pos=True).T # notice the jsc here sm = m + C @ (sm - pm) sP = P + C @ (sP - pP) @ C.T return (sm, sP), (sm, sP) _, (sms, sPs) = scan(body, (ms[-1], Ps[-1]), (ms[:-1], Ps[:-1]), reverse=True) sms = jnp.append(sms, jnp.expand_dims(ms[-1], 0), 0) sPs = jnp.append(sPs, jnp.expand_dims(Ps[-1], 0), 0) return sms, sPs ``` ``` sms, sPs = ks(car_tracking_model, fms, fPs) ``` So that we can now plot the result agains the true state and the observations: ``` fig, ax = plt.subplots(figsize=(7, 7)) ax.plot(true_xs[:100, 0], true_xs[:100, 1], label="True State", color="b") ax.plot(fms[:100, 0], fms[:100, 1], label="Filtered", color="g", linestyle="--") ax.plot(sms[:100, 0], sms[:100, 1], label="Smoothed", color="k", linestyle="--") ax.scatter(ys[:100].T, label="Observations", color="r") _ = plt.legend() ``` #### Filter-Smoother We can finally combine the filter and the smoother into a single filter-smoother routine: ``` def kfs(model, observations): return ks(model, kf(model, observations, False)) ``` ### Parallel Kalman filter and smoother Now that we have a baseline for the sequential model, we can go on and implement the parallel version of the routine, leveraging the `associative_scan` function already offered by JAX. To do this we will need to do a bit more work, but before this let us offer some intuition around what exactly is happening. #### Prefix-Sum / Associative Scan Consider the simple problem of computing the cumulative sum of a given array of floats: ``` array_of_floats = np.arange(2 4, dtype=np.float32) ``` A simple way to do this would be to iterate (scan) through the array from left to right and accumulate the result as we go: ``` def cumulative_sum(arr): # obviously numpy implements cumulative sums already, but this is for illustrative purposes. res = np.empty_like(arr) temp = 0. for i, val in enumerate(arr): temp += val res[i] = temp return res ``` ``` cumulative_sum(array_of_floats) ``` array([ 0., 1., 3., 6., 10., 15., 21., 28., 36., 45., 55., 66., 78., 91., 105., 120.], dtype=float32) However, the addition operation has the fundamental property to be associative: $(a+b) + c = a + (b + c)$, under which accumulative scans have been proven to be parallelizable (Blelloch, 1990, see the paper for a reference) up to a span complexity of $O(log(N))$ in the context where many processors are available. If we were to write the algorithm in pure python, it would look like the below: ``` def prefix_sum(arr): # This is for illustration purposes only, and for instance doesn't take into # account the case when the array size is not a pure power of 2 res = np.copy(arr) n = res.shape[0] log_n = np.log2(n).astype(int) # Up pass for d in range(log_n): # this loop can't be done in parallel so it defines the span complexity under # parallelization for i in range(0, n, 2 (d + 1)): # this should be done in parallel, therefore would not be taken # into account in the span complexity provided we have at least # n / 2^{d+1} cores on our GPU i1 = i + 2 d - 1 i2 = i + 2 (d + 1) - 1 res[i2] += res[i1] res[-1] = 0 # Down pass for d in range(log_n-1, -1, -1): # this loop can't be done in parallel so it defines the span complexity under # parallelization for i in range(0, n, 2 (d + 1)): # this should be done in parallel, therefore would not be taken # into account in the span complexity provided we have at least # n / 2^{d+1} cores on our GPU i1 = i + 2 d - 1 i2 = i + 2 ** (d + 1) - 1 res[i1], res[i2] = res[i2], res[i1] + res[i2] # Extra pass res += arr return res ``` ``` prefix_sum(array_of_floats) ``` array([ 0., 1., 3., 6., 10., 15., 21., 28., 36., 45., 55., 66., 78., 91., 105., 120.], dtype=float32) #### Filter: In the case of Kalman filters, one can think of the addition as being the bayesian map operator coming from the fact of "adding" one observation to the model: \begin{equation} p(x_k \mid y_{1:k-1}) = \int p(x_k \mid x_{k-1}) \, p(x_{k-1} \mid y_{1:k-1}) \, \mathrm{d}x_{k-1}, \end{equation} \begin{equation} p(x_k \mid y_{1:k}) = \frac{p(y_k \mid x_k) \, p(x_k \mid y_{1:k-1})} {\int p(y_k \mid x_k) \, p(x_k \mid y_{1:k-1}) \, \mathrm{d}x_k}. \end{equation} As it is, this operation on the random variables $x_k$'s is not associative as switching around the observations will result in a completely different result. However S. Särkkä and Á. García-Fernández showed that these could be rewritten in terms of the associative operator given by: \begin{align} \left(f_{i},g_{i}\right)\otimes\left(f_{j},g_{j}\right) & =\left(f_{ij},g_{ij}\right), \end{align} \begin{align} f_{ij}\left(x\mid z\right) & =\frac{\int g_{j}\left(y\right)f_{j}\left(x\mid y\right)f_{i}\left(y\mid z\right)\mathrm{d}y}{\int g_{j}\left(y\right)f_{i}\left(y\mid z\right)\mathrm{d}y}, \\ g_{ij}\left(z\right) & =g_{i}\left(z\right)\int g_{j}\left(y\right)f_{i}\left(y\mid z\right)\mathrm{d}y. \end{align} This could in turn be rewritten in terms of the linear equations of the state parameters, which leads to the below algorithm: ##### Initialisation of the associative elements: We write the function that will initialise them (in parallel) based on the data and the observations: ``` def first_filtering_element(model, y): S = model.H @ model.Q @ model.H.T + model.R CF, low = jsc.linalg.cho_factor(S) # note the jsc m1 = model.F @ model.m0 P1 = model.F @ model.P0 @ model.F.T + model.Q S1 = model.H @ P1 @ model.H.T + model.R K1 = jsc.linalg.solve(S1, model.H @ P1, sym_pos=True).T # note the jsc A = jnp.zeros_like(model.F) b = m1 + K1 @ (y - model.H @ m1) C = P1 - K1 @ S1 @ K1.T # note the jsc eta = model.F.T @ model.H.T @ jsc.linalg.cho_solve((CF, low), y) J = model.F.T @ model.H.T @ jsc.linalg.cho_solve((CF, low), model.H @ model.F) return A, b, C, J, eta def generic_filtering_element(model, y): S = model.H @ model.Q @ model.H.T + model.R CF, low = jsc.linalg.cho_factor(S) # note the jsc K = jsc.linalg.cho_solve((CF, low), model.H @ model.Q).T # note the jsc A = model.F - K @ model.H @ model.F b = K @ y C = model.Q - K @ model.H @ model.Q # note the jsc eta = model.F.T @ model.H.T @ jsc.linalg.cho_solve((CF, low), y) J = model.F.T @ model.H.T @ jsc.linalg.cho_solve((CF, low), model.H @ model.F) return A, b, C, J, eta ``` And we can combine them properly: ``` def make_associative_filtering_elements(model, observations): first_elems = first_filtering_element(model, observations[0]) generic_elems = vmap(lambda o: generic_filtering_element(model, o))(observations[1:]) return tuple(jnp.concatenate([jnp.expand_dims(first_e, 0), gen_es]) for first_e, gen_es in zip(first_elems, generic_elems)) ``` ##### Associative filtering operator We now define the associative filtering operator ``` @vmap def filtering_operator(elem1, elem2): # # note the jsc everywhere A1, b1, C1, J1, eta1 = elem1 A2, b2, C2, J2, eta2 = elem2 dim = A1.shape[0] I = jnp.eye(dim) # note the jnp I_C1J2 = I + C1 @ J2 temp = jsc.linalg.solve(I_C1J2.T, A2.T, sym_pos=False).T A = temp @ A1 b = temp @ (b1 + C1 @ eta2) + b2 C = temp @ C1 @ A2.T + C2 I_J2C1 = I + J2 @ C1 temp = jsc.linalg.solve(I_J2C1.T, A1, sym_pos=False).T eta = temp @ (eta2 - J2 @ b1) + eta1 J = temp @ J2 @ A1 + J1 return A, b, C, J, eta ``` ##### Parallel routine And we can now combine all these to compute the filtering result in parallel: ``` def pkf(model, observations): initial_elements = make_associative_filtering_elements(model, observations) final_elements = associative_scan(filtering_operator, initial_elements) return final_elements[1], final_elements[2] ``` That we can compare with the sequential one: ``` pfms, pfPs = pkf(car_tracking_model, ys[:100]) ``` and we get a maximum absolute difference between the two methods of ``` np.abs(pfms - fms).max() ``` 2.353672812205332e-14 #### Smoother: The parallel implementation of the smoother follows similar principles and we refer the reader to the paper for more details about the associative formulation. ##### Initialisation of the associative elements: We now write the function that will initialise them (in parallel) based on the data and the observations: ``` def last_smoothing_element(m, P): return jnp.zeros_like(P), m, P def generic_smoothing_element(model, m, P): Pp = model.F @ P @ model.F.T + model.Q E = jsc.linalg.solve(Pp, model.F @ P, sym_pos=True).T g = m - E @ model.F @ m L = P - E @ Pp @ E.T return E, g, L ``` And we can combine them properly: ``` def make_associative_smoothing_elements(model, filtering_means, filtering_covariances): last_elems = last_smoothing_element(filtering_means[-1], filtering_covariances[-1]) generic_elems = vmap(lambda m, P: generic_smoothing_element(model, m, P))(filtering_means[:-1], filtering_covariances[:-1]) return tuple(jnp.append(gen_es, jnp.expand_dims(last_e, 0), axis=0) for gen_es, last_e in zip(generic_elems, last_elems)) ``` ##### Associative smoothing operator We now define the associative smoothing operator ``` @vmap def smoothing_operator(elem1, elem2): E1, g1, L1 = elem1 E2, g2, L2 = elem2 E = E2 @ E1 g = E2 @ g1 + g2 L = E2 @ L1 @ E2.T + L2 return E, g, L ``` ##### Parallel routine And we can now combine all these to compute the smoothing result in parallel: ``` def pks(model, filtered_means, filtered_covariances): initial_elements = make_associative_smoothing_elements(model, filtered_means, filtered_covariances) final_elements = associative_scan(smoothing_operator, initial_elements, reverse=True) # note the vmap return final_elements[1], final_elements[2] ``` That we can compare with the sequential one: ``` psms, psPs = pks(car_tracking_model, pfms, pfPs) ``` #### Filter-Smoother: Finally we can combine all these functions into one filter-smoother routine. ``` def pkfs(model, observations): return pks(model, pkf(model, observations)) ``` ### Runtime comparison To compare the runtime of both methods we will compile the routines on GPU and CPU and modify the size of the observation input fed to the routines. It is noteworthy that the parallel version will appear to be much slower due to a slow compilation in JAX. This could be improved by using a different implementation of the associative scan or by fixing the number of levels the way it is done in TensorFlow Probability. ``` cpu_kfs = jit(kfs, backend="cpu") cpu_pkfs = jit(pkfs, backend="cpu") gpu_kfs = jit(kfs, backend="gpu") gpu_pkfs = jit(pkfs, backend="gpu") ``` ``` input_sizes = np.logspace(2, log10T, num=10, base=10).astype(int) ``` ``` input_sizes ``` array([ 100, 166, 278, 464, 774, 1291, 2154, 3593, 5994, 10000]) ``` cpu = devices("cpu")[0] gpu = devices("gpu")[0] ``` ``` def get_average_runtimes(func, n_iter, device): runtimes = np.empty(input_sizes.shape) device_model = StateSpaceModel((device_put(jnp.asarray(val), device) for val in car_tracking_model)) for i, input_size in tqdm(enumerate(input_sizes), total=runtimes.shape[0]): observations = device_put(jnp.asarray(ys[:input_size]), device) res = func(device_model, observations) # compilation run for e in res: _ = e.block_until_ready() tic = time.time() for _ in trange(n_iter, leave=False): res = func(device_model, observations) for e in res: _ = e.block_until_ready() runtimes[i] = (time.time() - tic) / n_iter return runtimes ``` ``` n_iter = 5 ``` ``` cpu_sequential_runtimes = get_average_runtimes(cpu_kfs, n_iter, cpu) cpu_parallel_runtimes = get_average_runtimes(cpu_pkfs, n_iter, cpu) gpu_sequential_runtimes = get_average_runtimes(gpu_kfs, n_iter, gpu) gpu_parallel_runtimes = get_average_runtimes(gpu_pkfs, n_iter, gpu) ``` HBox(children=(FloatProgress(value=0.0, max=10.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=10.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=10.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=10.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value=''))) And we can plot these: ``` fig, axes = plt.subplots(ncols=2, figsize=(15, 6), sharex=True, sharey=True) axes[0].loglog(input_sizes, cpu_sequential_runtimes, label="Sequential-CPU", linestyle="-.", linewidth=3) axes[0].loglog(input_sizes, cpu_parallel_runtimes, label="Parallel-CPU", linewidth=3) axes[0].legend() axes[1].loglog(input_sizes, gpu_sequential_runtimes, label="Sequential-GPU", linestyle="-.", linewidth=3) axes[1].loglog(input_sizes, gpu_parallel_runtimes, label="Parallel-GPU", linewidth=3) _ = axes[0].set_ylabel("Average run time (seconds)") for ax in axes: _ = ax.set_xlabel("Number of data points") _ = fig.suptitle("Runtime comparison on CPU and GPU", size=15) _ = axes[1].legend() ```
Require Import Crypto.Arithmetic.PrimeFieldTheorems. Require Import Crypto.Specific.solinas32_2e488m17_19limbs.Synthesis. (* TODO : change this to field once field isomorphism happens *) Definition carry : { carry : feBW_loose -> feBW_tight \| forall a, phiBW_tight (carry a) = (phiBW_loose a) }. Proof. Set Ltac Profiling. Time synthesize_carry (). Show Ltac Profile. Time Defined. Print Assumptions carry.
{-# OPTIONS --without-K --rewriting #-} module HoTT where open import lib.Basics public open import lib.Equivalence2 public open import lib.NConnected public open import lib.NType2 public open import lib.Relation2 public open import lib.Function2 public open import lib.cubical.Cubical public open import lib.types.Types public open import lib.groups.Groups public open import lib.groupoids.Groupoids public open import lib.modalities.Modalities public {- To use coinduction in the form of [∞], [♭] and [♯] you can do: open import HoTT open Coinduction You can also use coinductive records and copatterns instead, that’s prettier (see experimental/GlobularTypes.agda for an example) -} module Coinduction where open import lib.Coinduction public -- deprecated operators module _ where infix 15 _∎ _∎ = _=∎ conn-elim = conn-extend conn-elim-β = conn-extend-β conn-elim-general = conn-extend-general conn-intro = conn-in if_then_else_ : ∀ {i} {A : Type i} → Bool → A → A → A if true then t else e = t if false then t else e = e
The infnorm of a vector is zero if and only if the vector is zero.
<unk> @-@ FM , active rock
function [pb] = pbHuman(pres,iid,r) if nargin<3, r=[0.25 0.99]; end segs = readSegs(pres,iid); pb = zeros(size(segs{1})); for i = 1:numel(segs), bmap = seg2bmap(segs{i}); pb = pb + bmap; end pb = pb / numel(segs); pb = (pb~=0) .* (pb * (r(2)-r(1)) + r(1)); pb = min(pb,r(2));
[STATEMENT] lemma cinner_canonical_basis_eq_0: defines "basisA == set (canonical_basis::'a::onb_enum list)" and "basisB == set (canonical_basis::'b::onb_enum list)" assumes "\<And>u v. u\<in>basisA \<Longrightarrow> v\<in>basisB \<Longrightarrow> is_orthogonal v (F \<^sub>V u)" shows "F = 0" [PROOF STATE] proof (prove) goal (1 subgoal): 1. F = 0 [PROOF STEP] proof- [PROOF STATE] proof (state) goal (1 subgoal): 1. F = 0 [PROOF STEP] have "F \<^sub>V u = 0" if "u\<in>basisA" for u [PROOF STATE] proof (prove) goal (1 subgoal): 1. F \<^sub>V u = (0::'b) [PROOF STEP] proof- [PROOF STATE] proof (state) goal (1 subgoal): 1. F \<^sub>V u = (0::'b) [PROOF STEP] have "\<And>v. v\<in>basisB \<Longrightarrow> is_orthogonal v (F \<^sub>V u)" [PROOF STATE] proof (prove) goal (1 subgoal): 1. \<And>v. v \<in> basisB \<Longrightarrow> is_orthogonal v (F \<^sub>V u) [PROOF STEP] by (simp add: assms(3) that) [PROOF STATE] proof (state) this: ?v \<in> basisB \<Longrightarrow> is_orthogonal ?v (F \<^sub>V u) goal (1 subgoal): 1. F \<^sub>V u = (0::'b) [PROOF STEP] moreover [PROOF STATE] proof (state) this: ?v \<in> basisB \<Longrightarrow> is_orthogonal ?v (F \<^sub>V u) goal (1 subgoal): 1. F \<^sub>V u = (0::'b) [PROOF STEP] have "(\<And>v. v\<in>basisB \<Longrightarrow> is_orthogonal v x) \<Longrightarrow> x = 0" for x [PROOF STATE] proof (prove) goal (1 subgoal): 1. (\<And>v. v \<in> basisB \<Longrightarrow> is_orthogonal v x) \<Longrightarrow> x = (0::'b) [PROOF STEP] proof- [PROOF STATE] proof (state) goal (1 subgoal): 1. (\<And>v. v \<in> basisB \<Longrightarrow> is_orthogonal v x) \<Longrightarrow> x = (0::'b) [PROOF STEP] assume r1: "\<And>v. v\<in>basisB \<Longrightarrow> is_orthogonal v x" [PROOF STATE] proof (state) this: ?v \<in> basisB \<Longrightarrow> is_orthogonal ?v x goal (1 subgoal): 1. (\<And>v. v \<in> basisB \<Longrightarrow> is_orthogonal v x) \<Longrightarrow> x = (0::'b) [PROOF STEP] have "is_orthogonal v x" for v [PROOF STATE] proof (prove) goal (1 subgoal): 1. is_orthogonal v x [PROOF STEP] proof- [PROOF STATE] proof (state) goal (1 subgoal): 1. is_orthogonal v x [PROOF STEP] have "cspan basisB = UNIV" [PROOF STATE] proof (prove) goal (1 subgoal): 1. cspan basisB = UNIV [PROOF STEP] using basisB_def is_generator_set [PROOF STATE] proof (prove) using this: basisB \<equiv> set canonical_basis cspan (set canonical_basis) = UNIV goal (1 subgoal): 1. cspan basisB = UNIV [PROOF STEP] by auto [PROOF STATE] proof (state) this: cspan basisB = UNIV goal (1 subgoal): 1. is_orthogonal v x [PROOF STEP] hence "v \<in> cspan basisB" [PROOF STATE] proof (prove) using this: cspan basisB = UNIV goal (1 subgoal): 1. v \<in> cspan basisB [PROOF STEP] by (smt iso_tuple_UNIV_I) [PROOF STATE] proof (state) this: v \<in> cspan basisB goal (1 subgoal): 1. is_orthogonal v x [PROOF STEP] hence "\<exists>t s. v = (\<Sum>a\<in>t. s a \<^sub>C a) \<and> finite t \<and> t \<subseteq> basisB" [PROOF STATE] proof (prove) using this: v \<in> cspan basisB goal (1 subgoal): 1. \<exists>t s. v = (\<Sum>a\<in>t. s a \<^sub>C a) \<and> finite t \<and> t \<subseteq> basisB [PROOF STEP] using complex_vector.span_explicit [PROOF STATE] proof (prove) using this: v \<in> cspan basisB cspan ?b = {\<Sum>a\<in>t. r a \<^sub>C a \|t r. finite t \<and> t \<subseteq> ?b} goal (1 subgoal): 1. \<exists>t s. v = (\<Sum>a\<in>t. s a \<^sub>C a) \<and> finite t \<and> t \<subseteq> basisB [PROOF STEP] by (smt mem_Collect_eq) [PROOF STATE] proof (state) this: \<exists>t s. v = (\<Sum>a\<in>t. s a \<^sub>C a) \<and> finite t \<and> t \<subseteq> basisB goal (1 subgoal): 1. is_orthogonal v x [PROOF STEP] then [PROOF STATE] proof (chain) picking this: \<exists>t s. v = (\<Sum>a\<in>t. s a \<^sub>C a) \<and> finite t \<and> t \<subseteq> basisB [PROOF STEP] obtain t s where b1: "v = (\<Sum>a\<in>t. s a \<^sub>C a)" and b2: "finite t" and b3: "t \<subseteq> basisB" [PROOF STATE] proof (prove) using this: \<exists>t s. v = (\<Sum>a\<in>t. s a \<^sub>C a) \<and> finite t \<and> t \<subseteq> basisB goal (1 subgoal): 1. (\<And>s t. \<lbrakk>v = (\<Sum>a\<in>t. s a \<^sub>C a); finite t; t \<subseteq> basisB\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis [PROOF STEP] by blast [PROOF STATE] proof (state) this: v = (\<Sum>a\<in>t. s a \<^sub>C a) finite t t \<subseteq> basisB goal (1 subgoal): 1. is_orthogonal v x [PROOF STEP] have "v \<bullet>\<^sub>C x = (\<Sum>a\<in>t. s a \<^sub>C a) \<bullet>\<^sub>C x" [PROOF STATE] proof (prove) goal (1 subgoal): 1. v \<bullet>\<^sub>C x = (\<Sum>a\<in>t. s a \<^sub>C a) \<bullet>\<^sub>C x [PROOF STEP] by (simp add: b1) [PROOF STATE] proof (state) this: v \<bullet>\<^sub>C x = (\<Sum>a\<in>t. s a \<^sub>C a) \<bullet>\<^sub>C x goal (1 subgoal): 1. is_orthogonal v x [PROOF STEP] also [PROOF STATE] proof (state) this: v \<bullet>\<^sub>C x = (\<Sum>a\<in>t. s a \<^sub>C a) \<bullet>\<^sub>C x goal (1 subgoal): 1. is_orthogonal v x [PROOF STEP] have "\<dots> = (\<Sum>a\<in>t. (s a \<^sub>C a) \<bullet>\<^sub>C x)" [PROOF STATE] proof (prove) goal (1 subgoal): 1. (\<Sum>a\<in>t. s a \<^sub>C a) \<bullet>\<^sub>C x = (\<Sum>a\<in>t. s a \<^sub>C a \<bullet>\<^sub>C x) [PROOF STEP] using cinner_sum_left [PROOF STATE] proof (prove) using this: sum ?f ?A \<bullet>\<^sub>C ?y = (\<Sum>x\<in>?A. ?f x \<bullet>\<^sub>C ?y) goal (1 subgoal): 1. (\<Sum>a\<in>t. s a \<^sub>C a) \<bullet>\<^sub>C x = (\<Sum>a\<in>t. s a \<^sub>C a \<bullet>\<^sub>C x) [PROOF STEP] by blast [PROOF STATE] proof (state) this: (\<Sum>a\<in>t. s a \<^sub>C a) \<bullet>\<^sub>C x = (\<Sum>a\<in>t. s a \<^sub>C a \<bullet>\<^sub>C x) goal (1 subgoal): 1. is_orthogonal v x [PROOF STEP] also [PROOF STATE] proof (state) this: (\<Sum>a\<in>t. s a \<^sub>C a) \<bullet>\<^sub>C x = (\<Sum>a\<in>t. s a \<^sub>C a \<bullet>\<^sub>C x) goal (1 subgoal): 1. is_orthogonal v x [PROOF STEP] have "\<dots> = (\<Sum>a\<in>t. cnj (s a) (a \<bullet>\<^sub>C x))" [PROOF STATE] proof (prove) goal (1 subgoal): 1. (\<Sum>a\<in>t. s a \<^sub>C a \<bullet>\<^sub>C x) = (\<Sum>a\<in>t. cnj (s a) (a \<bullet>\<^sub>C x)) [PROOF STEP] by auto [PROOF STATE] proof (state) this: (\<Sum>a\<in>t. s a \<^sub>C a \<bullet>\<^sub>C x) = (\<Sum>a\<in>t. cnj (s a) (a \<bullet>\<^sub>C x)) goal (1 subgoal): 1. is_orthogonal v x [PROOF STEP] also [PROOF STATE] proof (state) this: (\<Sum>a\<in>t. s a \<^sub>C a \<bullet>\<^sub>C x) = (\<Sum>a\<in>t. cnj (s a) (a \<bullet>\<^sub>C x)) goal (1 subgoal): 1. is_orthogonal v x [PROOF STEP] have "\<dots> = 0" [PROOF STATE] proof (prove) goal (1 subgoal): 1. (\<Sum>a\<in>t. cnj (s a) * (a \<bullet>\<^sub>C x)) = 0 [PROOF STEP] using b3 r1 subsetD [PROOF STATE] proof (prove) using this: t \<subseteq> basisB ?v \<in> basisB \<Longrightarrow> is_orthogonal ?v x \<lbrakk>?A \<subseteq> ?B; ?c \<in> ?A\<rbrakk> \<Longrightarrow> ?c \<in> ?B goal (1 subgoal): 1. (\<Sum>a\<in>t. cnj (s a) * (a \<bullet>\<^sub>C x)) = 0 [PROOF STEP] by force [PROOF STATE] proof (state) this: (\<Sum>a\<in>t. cnj (s a) * (a \<bullet>\<^sub>C x)) = 0 goal (1 subgoal): 1. is_orthogonal v x [PROOF STEP] finally [PROOF STATE] proof (chain) picking this: is_orthogonal v x [PROOF STEP] show ?thesis [PROOF STATE] proof (prove) using this: is_orthogonal v x goal (1 subgoal): 1. is_orthogonal v x [PROOF STEP] by simp [PROOF STATE] proof (state) this: is_orthogonal v x goal: No subgoals! [PROOF STEP] qed [PROOF STATE] proof (state) this: is_orthogonal ?v x goal (1 subgoal): 1. (\<And>v. v \<in> basisB \<Longrightarrow> is_orthogonal v x) \<Longrightarrow> x = (0::'b) [PROOF STEP] thus ?thesis [PROOF STATE] proof (prove) using this: is_orthogonal ?v x goal (1 subgoal): 1. x = (0::'b) [PROOF STEP] by (simp add: \<open>\<And>v. (v \<bullet>\<^sub>C x) = 0\<close> cinner_extensionality) [PROOF STATE] proof (state) this: x = (0::'b) goal: No subgoals! [PROOF STEP] qed [PROOF STATE] proof (state) this: (\<And>v. v \<in> basisB \<Longrightarrow> is_orthogonal v ?x) \<Longrightarrow> ?x = (0::'b) goal (1 subgoal): 1. F \<^sub>V u = (0::'b) [PROOF STEP] ultimately [PROOF STATE] proof (chain) picking this: ?v \<in> basisB \<Longrightarrow> is_orthogonal ?v (F \<^sub>V u) (\<And>v. v \<in> basisB \<Longrightarrow> is_orthogonal v ?x) \<Longrightarrow> ?x = (0::'b) [PROOF STEP] show ?thesis [PROOF STATE] proof (prove) using this: ?v \<in> basisB \<Longrightarrow> is_orthogonal ?v (F \<^sub>V u) (\<And>v. v \<in> basisB \<Longrightarrow> is_orthogonal v ?x) \<Longrightarrow> ?x = (0::'b) goal (1 subgoal): 1. F \<^sub>V u = (0::'b) [PROOF STEP] by simp [PROOF STATE] proof (state) this: F \<^sub>V u = (0::'b) goal: No subgoals! [PROOF STEP] qed [PROOF STATE] proof (state) this: ?u \<in> basisA \<Longrightarrow> F \<^sub>V ?u = (0::'b) goal (1 subgoal): 1. F = 0 [PROOF STEP] thus ?thesis [PROOF STATE] proof (prove) using this: ?u \<in> basisA \<Longrightarrow> F \<^sub>V ?u = (0::'b) goal (1 subgoal): 1. F = 0 [PROOF STEP] using basisA_def cblinfun_eq_0_on_canonical_basis [PROOF STATE] proof (prove) using this: ?u \<in> basisA \<Longrightarrow> F \<^sub>V ?u = (0::'b) basisA \<equiv> set canonical_basis (\<And>u. u \<in> set canonical_basis \<Longrightarrow> ?f *\<^sub>V u = (0::?'b)) \<Longrightarrow> ?f = 0 goal (1 subgoal): 1. F = 0 [PROOF STEP] by auto [PROOF STATE] proof (state) this: F = 0 goal: No subgoals! [PROOF STEP] qed
function order = level_to_order_open ( dim_num, level ) %***************************************************************************80 % %% LEVEL_TO_ORDER converts a level to an order for open rules. % % Discussion: % % Sparse grids can naturally be nested. A natural scheme is to use % a series of one-dimensional rules arranged in a series of "levels" % whose order roughly doubles with each step. % % The arrangement described here works naturally for the Fejer Type 1, % Fejer Type 2, Newton Cotes Open, Newton Cotes Half Open, % and Gauss-Patterson rules. It also can be used, partially, to describe % the growth of Gauss-Legendre rules. % % The idea is that we start with LEVEL = 0, ORDER = 1 indicating the single % point at the center, and for all values afterwards, we use the relationship % % ORDER = 2(LEVEL+1) - 1. % % The following table shows how the growth will occur: % % Level Order % % 0 1 % 1 3 = 4 - 1 % 2 7 = 8 - 1 % 3 15 = 16 - 1 % 4 31 = 32 - 1 % 5 63 = 64 - 1 % % For the Fejer Type 1, Fejer Type 2, Newton Cotes Open, % Newton Cotes Open Half, and Gauss-Patterson rules, the point growth is % nested. If we have ORDER points on a particular LEVEL, the next level % includes all these old points, plus ORDER+1 new points, formed in the % gaps between successive pairs of old points plus an extra point at each % end. % % Level Order = New + Old % % 0 1 = 1 + 0 % 1 3 = 2 + 1 % 2 7 = 4 + 3 % 3 15 = 8 + 7 % 4 31 = 16 + 15 % 5 63 = 32 + 31 % % If we use a series of Gauss-Legendre rules, then there is almost no % nesting, except that the central point is shared. If we insist on % producing a comparable series of such points, then the "nesting" behavior % is as follows: % % Level Order = New + Old % % 0 1 = 1 + 0 % 1 3 = 2 + 1 % 2 7 = 6 + 1 % 3 15 = 14 + 1 % 4 31 = 30 + 1 % 5 63 = 62 + 1 % % Moreover, if we consider ALL the points used in such a set of "nested" % Gauss-Legendre rules, then we must sum the "NEW" column, and we see that % we get roughly twice as many points as for the truly nested rules. % % In this routine, we assume that a vector of levels is given, % and the corresponding orders are desired. % % Licensing: % % This code is distributed under the GNU LGPL license. % % Modified: % % 18 April 2007 % % Author: % % John Burkardt % % Reference: % % Fabio Nobile, Raul Tempone, Clayton Webster, % A Sparse Grid Stochastic Collocation Method for Partial Differential % Equations with Random Input Data, % SIAM Journal on Numerical Analysis, % Volume 46, Number 5, 2008, pages 2309-2345. % % Parameters: % % Input, integer DIM_NUM, the spatial dimension. % % Input, integer LEVEL(DIM_NUM), the nesting level. % % Output, integer ORDER(DIM_NUM,1), the order (number of points) of the rule. % order = zeros ( dim_num, 1 ); for dim = 1 : dim_num if ( level(dim) < 0 ) order(dim,1) = -1; elseif ( level(dim) == 0 ) order(dim,1) = 1; else order(dim,1) = 2^( level(dim) + 1 ) - 1; end end return end
/- Copyright (c) 2019 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau, David Kurniadi Angdinata, Devon Tuma, Riccardo Brasca -/ import data.polynomial.div import ring_theory.polynomial.basic import ring_theory.ideal.quotient_operations /-! # Quotients of polynomial rings -/ open_locale polynomial namespace polynomial variables {R : Type} [comm_ring R] /-- For a commutative ring $R$, evaluating a polynomial at an element $x \in R$ induces an isomorphism of $R$-algebras $R[X] / \langle X - x \rangle \cong R$. -/ noncomputable def quotient_span_X_sub_C_alg_equiv (x : R) : (R[X] ⧸ ideal.span ({X - C x} : set R[X])) ≃ₐ[R] R := (alg_equiv.restrict_scalars R $ ideal.quotient_equiv_alg_of_eq R (by exact ker_eval_ring_hom x : ring_hom.ker (aeval x).to_ring_hom = _)).symm.trans $ ideal.quotient_ker_alg_equiv_of_right_inverse $ λ _, eval_C @[simp] lemma quotient_span_X_sub_C_alg_equiv_mk (x : R) (p : R[X]) : quotient_span_X_sub_C_alg_equiv x (ideal.quotient.mk _ p) = p.eval x := rfl @[simp] lemma quotient_span_X_sub_C_alg_equiv_symm_apply (x : R) (y : R) : (quotient_span_X_sub_C_alg_equiv x).symm y = algebra_map R _ y := rfl end polynomial namespace ideal noncomputable theory open polynomial variables {R : Type} [comm_ring R] lemma quotient_map_C_eq_zero {I : ideal R} : ∀ a ∈ I, ((quotient.mk (map (C : R →+* R[X]) I : ideal R[X])).comp C) a = 0 := begin intros a ha, rw [ring_hom.comp_apply, quotient.eq_zero_iff_mem], exact mem_map_of_mem _ ha, end lemma eval₂_C_mk_eq_zero {I : ideal R} : ∀ f ∈ (map (C : R →+* R[X]) I : ideal R[X]), eval₂_ring_hom (C.comp (quotient.mk I)) X f = 0 := begin intros a ha, rw ← sum_monomial_eq a, dsimp, rw eval₂_sum, refine finset.sum_eq_zero (λ n hn, _), dsimp, rw eval₂_monomial (C.comp (quotient.mk I)) X, refine mul_eq_zero_of_left (polynomial.ext (λ m, _)) (X ^ n), erw coeff_C, by_cases h : m = 0, { simpa [h] using quotient.eq_zero_iff_mem.2 ((mem_map_C_iff.1 ha) n) }, { simp [h] } end /-- If `I` is an ideal of `R`, then the ring polynomials over the quotient ring `I.quotient` is isomorphic to the quotient of `R[X]` by the ideal `map C I`, where `map C I` contains exactly the polynomials whose coefficients all lie in `I` -/ def polynomial_quotient_equiv_quotient_polynomial (I : ideal R) : (R ⧸ I)[X] ≃+* R[X] ⧸ (map C I : ideal R[X]) := { to_fun := eval₂_ring_hom (quotient.lift I ((quotient.mk (map C I : ideal R[X])).comp C) quotient_map_C_eq_zero) ((quotient.mk (map C I : ideal R[X]) X)), inv_fun := quotient.lift (map C I : ideal R[X]) (eval₂_ring_hom (C.comp (quotient.mk I)) X) eval₂_C_mk_eq_zero, map_mul' := λ f g, by simp only [coe_eval₂_ring_hom, eval₂_mul], map_add' := λ f g, by simp only [eval₂_add, coe_eval₂_ring_hom], left_inv := begin intro f, apply polynomial.induction_on' f, { intros p q hp hq, simp only [coe_eval₂_ring_hom] at hp, simp only [coe_eval₂_ring_hom] at hq, simp only [coe_eval₂_ring_hom, hp, hq, ring_hom.map_add] }, { rintros n ⟨x⟩, simp only [← smul_X_eq_monomial, C_mul', quotient.lift_mk, submodule.quotient.quot_mk_eq_mk, quotient.mk_eq_mk, eval₂_X_pow, eval₂_smul, coe_eval₂_ring_hom, ring_hom.map_pow, eval₂_C, ring_hom.coe_comp, ring_hom.map_mul, eval₂_X] } end, right_inv := begin rintro ⟨f⟩, apply polynomial.induction_on' f, { simp_intros p q hp hq, rw [hp, hq] }, { intros n a, simp only [← smul_X_eq_monomial, ← C_mul' a (X ^ n), quotient.lift_mk, submodule.quotient.quot_mk_eq_mk, quotient.mk_eq_mk, eval₂_X_pow, eval₂_smul, coe_eval₂_ring_hom, ring_hom.map_pow, eval₂_C, ring_hom.coe_comp, ring_hom.map_mul, eval₂_X] }, end, } @[simp] lemma polynomial_quotient_equiv_quotient_polynomial_symm_mk (I : ideal R) (f : R[X]) : I.polynomial_quotient_equiv_quotient_polynomial.symm (quotient.mk _ f) = f.map (quotient.mk I) := by rw [polynomial_quotient_equiv_quotient_polynomial, ring_equiv.symm_mk, ring_equiv.coe_mk, ideal.quotient.lift_mk, coe_eval₂_ring_hom, eval₂_eq_eval_map, ←polynomial.map_map, ←eval₂_eq_eval_map, polynomial.eval₂_C_X] @[simp] lemma polynomial_quotient_equiv_quotient_polynomial_map_mk (I : ideal R) (f : R[X]) : I.polynomial_quotient_equiv_quotient_polynomial (f.map I^.quotient.mk) = quotient.mk _ f := begin apply (polynomial_quotient_equiv_quotient_polynomial I).symm.injective, rw [ring_equiv.symm_apply_apply, polynomial_quotient_equiv_quotient_polynomial_symm_mk], end /-- If `P` is a prime ideal of `R`, then `R[x]/(P)` is an integral domain. -/ lemma is_domain_map_C_quotient {P : ideal R} (H : is_prime P) : is_domain (R[X] ⧸ (map (C : R →+* R[X]) P : ideal R[X])) := ring_equiv.is_domain (polynomial (R ⧸ P)) (polynomial_quotient_equiv_quotient_polynomial P).symm /-- Given any ring `R` and an ideal `I` of `R[X]`, we get a map `R → R[x] → R[x]/I`. If we let `R` be the image of `R` in `R[x]/I` then we also have a map `R[x] → R'[x]`. In particular we can map `I` across this map, to get `I'` and a new map `R' → R'[x] → R'[x]/I`. This theorem shows `I'` will not contain any non-zero constant polynomials -/ lemma eq_zero_of_polynomial_mem_map_range (I : ideal R[X]) (x : ((quotient.mk I).comp C).range) (hx : C x ∈ (I.map (polynomial.map_ring_hom ((quotient.mk I).comp C).range_restrict))) : x = 0 := begin let i := ((quotient.mk I).comp C).range_restrict, have hi' : (polynomial.map_ring_hom i).ker ≤ I, { refine λ f hf, polynomial_mem_ideal_of_coeff_mem_ideal I f (λ n, _), rw [mem_comap, ← quotient.eq_zero_iff_mem, ← ring_hom.comp_apply], rw [ring_hom.mem_ker, coe_map_ring_hom] at hf, replace hf := congr_arg (λ (f : polynomial _), f.coeff n) hf, simp only [coeff_map, coeff_zero] at hf, rwa [subtype.ext_iff, ring_hom.coe_range_restrict] at hf }, obtain ⟨x, hx'⟩ := x, obtain ⟨y, rfl⟩ := (ring_hom.mem_range).1 hx', refine subtype.eq _, simp only [ring_hom.comp_apply, quotient.eq_zero_iff_mem, zero_mem_class.coe_zero, subtype.val_eq_coe], suffices : C (i y) ∈ (I.map (polynomial.map_ring_hom i)), { obtain ⟨f, hf⟩ := mem_image_of_mem_map_of_surjective (polynomial.map_ring_hom i) (polynomial.map_surjective _ (((quotient.mk I).comp C).range_restrict_surjective)) this, refine sub_add_cancel (C y) f ▸ I.add_mem (hi' _ : (C y - f) ∈ I) hf.1, rw [ring_hom.mem_ker, ring_hom.map_sub, hf.2, sub_eq_zero, coe_map_ring_hom, map_C] }, exact hx, end end ideal namespace mv_polynomial variables {R : Type} {σ : Type} [comm_ring R] {r : R} lemma quotient_map_C_eq_zero {I : ideal R} {i : R} (hi : i ∈ I) : (ideal.quotient.mk (ideal.map (C : R →+* mv_polynomial σ R) I : ideal (mv_polynomial σ R))).comp C i = 0 := begin simp only [function.comp_app, ring_hom.coe_comp, ideal.quotient.eq_zero_iff_mem], exact ideal.mem_map_of_mem _ hi end lemma eval₂_C_mk_eq_zero {I : ideal R} {a : mv_polynomial σ R} (ha : a ∈ (ideal.map (C : R →+* mv_polynomial σ R) I : ideal (mv_polynomial σ R))) : eval₂_hom (C.comp (ideal.quotient.mk I)) X a = 0 := begin rw as_sum a, rw [coe_eval₂_hom, eval₂_sum], refine finset.sum_eq_zero (λ n hn, _), simp only [eval₂_monomial, function.comp_app, ring_hom.coe_comp], refine mul_eq_zero_of_left _ _, suffices : coeff n a ∈ I, { rw [← @ideal.mk_ker R _ I, ring_hom.mem_ker] at this, simp only [this, C_0] }, exact mem_map_C_iff.1 ha n end /-- If `I` is an ideal of `R`, then the ring `mv_polynomial σ I.quotient` is isomorphic as an `R`-algebra to the quotient of `mv_polynomial σ R` by the ideal generated by `I`. -/ def quotient_equiv_quotient_mv_polynomial (I : ideal R) : mv_polynomial σ (R ⧸ I) ≃ₐ[R] mv_polynomial σ R ⧸ (ideal.map C I : ideal (mv_polynomial σ R)) := { to_fun := eval₂_hom (ideal.quotient.lift I ((ideal.quotient.mk (ideal.map C I : ideal (mv_polynomial σ R))).comp C) (λ i hi, quotient_map_C_eq_zero hi)) (λ i, ideal.quotient.mk (ideal.map C I : ideal (mv_polynomial σ R)) (X i)), inv_fun := ideal.quotient.lift (ideal.map C I : ideal (mv_polynomial σ R)) (eval₂_hom (C.comp (ideal.quotient.mk I)) X) (λ a ha, eval₂_C_mk_eq_zero ha), map_mul' := ring_hom.map_mul _, map_add' := ring_hom.map_add _, left_inv := begin intro f, apply induction_on f, { rintro ⟨r⟩, rw [coe_eval₂_hom, eval₂_C], simp only [submodule.quotient.quot_mk_eq_mk, ideal.quotient.lift_mk, mv_polynomial.eval₂_hom_C, function.comp_app, ideal.quotient.mk_eq_mk, mv_polynomial.C_inj, ring_hom.coe_comp], }, { simp_intros p q hp hq only [ring_hom.map_add, mv_polynomial.coe_eval₂_hom, coe_eval₂_hom, mv_polynomial.eval₂_add], rw [hp, hq] }, { simp_intros p i hp only [coe_eval₂_hom], simp only [hp, coe_eval₂_hom, ideal.quotient.lift_mk, eval₂_mul, ring_hom.map_mul, eval₂_X] } end, right_inv := begin rintro ⟨f⟩, apply induction_on f, { intros r, simp only [submodule.quotient.quot_mk_eq_mk, ideal.quotient.lift_mk, ideal.quotient.mk_eq_mk, ring_hom.coe_comp, eval₂_hom_C] }, { simp_intros p q hp hq only [submodule.quotient.quot_mk_eq_mk, eval₂_add, ring_hom.map_add, coe_eval₂_hom, ideal.quotient.lift_mk, ideal.quotient.mk_eq_mk], rw [hp, hq] }, { simp_intros p i hp only [submodule.quotient.quot_mk_eq_mk, coe_eval₂_hom, ideal.quotient.lift_mk, ideal.quotient.mk_eq_mk, eval₂_mul, ring_hom.map_mul, eval₂_X], simp only [hp] } end, commutes' := λ r, eval₂_hom_C _ _ (ideal.quotient.mk I r) } end mv_polynomial
module TTImp.Elab.Record import Core.Context import Core.Context.Log import Core.Core import Core.Env import Core.Metadata import Core.Normalise import Core.Unify import Core.TT import Core.Value import TTImp.Elab.Check import TTImp.Elab.Delayed import TTImp.TTImp import Data.List import Data.Maybe %default covering getRecordType : Env Term vars -> NF vars -> Maybe Name getRecordType env (NTCon _ n _ _ _) = Just n getRecordType env _ = Nothing data Rec : Type where Field : Maybe Name -> -- implicit argument name, if any String -> RawImp -> Rec -- field name on left, value on right Constr : Maybe Name -> -- implicit argument name, if any Name -> List (String, Rec) -> Rec Show Rec where show (Field mn n ty) = "Field " ++ show mn ++ "; " ++ show n ++ " : " ++ show ty show (Constr mn n args) = "Constr " ++ show mn ++ " " ++ show n ++ " " ++ show args applyImp : RawImp -> List (Maybe Name, RawImp) -> RawImp applyImp f [] = f applyImp f ((Nothing, arg) :: xs) = applyImp (IApp (getFC f) f arg) xs applyImp f ((Just n, arg) :: xs) = applyImp (INamedApp (getFC f) f n arg) xs toLHS' : FC -> Rec -> (Maybe Name, RawImp) toLHS' loc (Field mn@(Just _) n _) = (mn, IAs loc EmptyFC UseRight (UN $ Basic n) (Implicit loc True)) toLHS' loc (Field mn n _) = (mn, IBindVar EmptyFC n) toLHS' loc (Constr mn con args) = let args' = map (toLHS' loc . snd) args in (mn, applyImp (IVar loc con) args') toLHS : FC -> Rec -> RawImp toLHS fc r = snd (toLHS' fc r) toRHS' : FC -> Rec -> (Maybe Name, RawImp) toRHS' loc (Field mn _ val) = (mn, val) toRHS' loc (Constr mn con args) = let args' = map (toRHS' loc . snd) args in (mn, applyImp (IVar loc con) args') toRHS : FC -> Rec -> RawImp toRHS fc r = snd (toRHS' fc r) findConName : Defs -> Name -> Core (Maybe Name) findConName defs tyn = case !(lookupDefExact tyn (gamma defs)) of Just (TCon _ _ _ _ _ _ [con] _) => pure (Just con) _ => pure Nothing findFields : {auto c : Ref Ctxt Defs} -> Defs -> Name -> Core (Maybe (List (String, Maybe Name, Maybe Name))) findFields defs con = case !(lookupTyExact con (gamma defs)) of Just t => pure (Just !(getExpNames !(nf defs [] t))) _ => pure Nothing where getExpNames : NF [] -> Core (List (String, Maybe Name, Maybe Name)) getExpNames (NBind fc x (Pi _ _ p ty) sc) = do rest <- getExpNames !(sc defs (toClosure defaultOpts [] (Erased fc False))) let imp = case p of Explicit => Nothing _ => Just x pure $ (nameRoot x, imp, getRecordType [] !(evalClosure defs ty)) :: rest getExpNames _ = pure [] genFieldName : {auto u : Ref UST UState} -> String -> Core String genFieldName root = do ust <- get UST put UST (record { nextName $= (+1) } ust) pure (root ++ show (nextName ust)) -- There's probably a generic version of this in the prelude isn't -- there? replace : String -> Rec -> List (String, Rec) -> List (String, Rec) replace k v [] = [] replace k v ((k', v') :: vs) = if k == k' then ((k, v) :: vs) else ((k', v') :: replace k v vs) findPath : {auto c : Ref Ctxt Defs} -> {auto u : Ref UST UState} -> FC -> List String -> List String -> Maybe Name -> (String -> RawImp) -> Rec -> Core Rec findPath loc [] full tyn val (Field mn lhs _) = pure (Field mn lhs (val lhs)) findPath loc [] full tyn val rec = throw (IncompatibleFieldUpdate loc full) findPath loc (p :: ps) full Nothing val (Field mn n v) = throw (NotRecordField loc p Nothing) findPath loc (p :: ps) full (Just tyn) val (Field mn n v) = do defs <- get Ctxt Just con <- findConName defs tyn \| Nothing => throw (NotRecordType loc tyn) Just fs <- findFields defs con \| Nothing => throw (NotRecordType loc tyn) args <- mkArgs fs let rec' = Constr mn con args findPath loc (p :: ps) full (Just tyn) val rec' where mkArgs : List (String, Maybe Name, Maybe Name) -> Core (List (String, Rec)) mkArgs [] = pure [] mkArgs ((p, imp, _) :: ps) = do fldn <- genFieldName p args' <- mkArgs ps -- If it's an implicit argument, leave it as _ by default let arg = maybe (IVar EmptyFC (UN $ Basic fldn)) (const (Implicit loc False)) imp pure ((p, Field imp fldn arg) :: args') findPath loc (p :: ps) full tyn val (Constr mn con args) = do let Just prec = lookup p args \| Nothing => throw (NotRecordField loc p tyn) defs <- get Ctxt Just fs <- findFields defs con \| Nothing => pure (Constr mn con args) let Just (imp, mfty) = lookup p fs \| Nothing => throw (NotRecordField loc p tyn) prec' <- findPath loc ps full mfty val prec pure (Constr mn con (replace p prec' args)) getSides : {auto c : Ref Ctxt Defs} -> {auto u : Ref UST UState} -> FC -> IFieldUpdate -> Name -> RawImp -> Rec -> Core Rec getSides loc (ISetField path val) tyn orig rec -- update 'rec' so that 'path' is accessible on the lhs and rhs, -- then set the path on the rhs to 'val' = findPath loc path path (Just tyn) (const val) rec getSides loc (ISetFieldApp path val) tyn orig rec = findPath loc path path (Just tyn) (\n => apply val [IVar EmptyFC (UN $ Basic n)]) rec getAllSides : {auto c : Ref Ctxt Defs} -> {auto u : Ref UST UState} -> FC -> List IFieldUpdate -> Name -> RawImp -> Rec -> Core Rec getAllSides loc [] tyn orig rec = pure rec getAllSides loc (u :: upds) tyn orig rec = getAllSides loc upds tyn orig !(getSides loc u tyn orig rec) -- Convert the collection of high level field accesses into a case expression -- which does the updates all in one go export recUpdate : {vars : _} -> {auto c : Ref Ctxt Defs} -> {auto u : Ref UST UState} -> {auto e : Ref EST (EState vars)} -> RigCount -> ElabInfo -> FC -> NestedNames vars -> Env Term vars -> List IFieldUpdate -> (rec : RawImp) -> (grecty : Glued vars) -> Core RawImp recUpdate rigc elabinfo iloc nest env flds rec grecty = do defs <- get Ctxt rectynf <- getNF grecty let Just rectyn = getRecordType env rectynf \| Nothing => throw (RecordTypeNeeded iloc env) fldn <- genFieldName "__fld" sides <- getAllSides iloc flds rectyn rec (Field Nothing fldn (IVar vloc (UN $ Basic fldn))) pure $ ICase vloc rec (Implicit vloc False) [mkClause sides] where vloc : FC vloc = virtualiseFC iloc mkClause : Rec -> ImpClause mkClause rec = PatClause vloc (toLHS vloc rec) (toRHS vloc rec) needType : Error -> Bool needType (RecordTypeNeeded _ _) = True needType (InType _ _ err) = needType err needType (InCon _ _ err) = needType err needType (InLHS _ _ err) = needType err needType (InRHS _ _ err) = needType err needType (WhenUnifying _ _ _ _ _ err) = needType err needType _ = False export checkUpdate : {vars : _} -> {auto c : Ref Ctxt Defs} -> {auto m : Ref MD Metadata} -> {auto u : Ref UST UState} -> {auto e : Ref EST (EState vars)} -> RigCount -> ElabInfo -> NestedNames vars -> Env Term vars -> FC -> List IFieldUpdate -> RawImp -> Maybe (Glued vars) -> Core (Term vars, Glued vars) checkUpdate rig elabinfo nest env fc upds rec expected = do recty <- case expected of Just ret => pure ret _ => do (_, ty) <- checkImp rig elabinfo nest env rec Nothing pure ty let solvemode = case elabMode elabinfo of InLHS c => inLHS _ => inTerm delayOnFailure fc rig env (Just recty) needType RecordUpdate $ \delayed => do solveConstraints solvemode Normal exp <- getTerm recty -- We can't just use the old NF on the second attempt, -- because we might know more now, so recalculate it let recty' = if delayed then gnf env exp else recty logGlueNF "elab.record" 5 (show delayed ++ " record type " ++ show rec) env recty' rcase <- recUpdate rig elabinfo fc nest env upds rec recty' log "elab.record" 5 $ "Record update: " ++ show rcase check rig elabinfo nest env rcase expected
#include <boost/format/feed_args.hpp>
State Before: ι : Type ?u.105808 α : Type ?u.105811 M : Type u_1 N : Type ?u.105817 P : Type ?u.105820 M₀ : Type ?u.105823 G : Type ?u.105826 R : Type ?u.105829 inst✝⁴ : Monoid M inst✝³ : Monoid N inst✝² : Monoid P l✝ l₁ l₂ : List M a : M inst✝¹ : Preorder M inst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1 l : List M hl₁ : ∀ (x : M), x ∈ l → 1 ≤ x ⊢ 1 ≤ prod l State After: case nil ι : Type ?u.105808 α : Type ?u.105811 M : Type u_1 N : Type ?u.105817 P : Type ?u.105820 M₀ : Type ?u.105823 G : Type ?u.105826 R : Type ?u.105829 inst✝⁴ : Monoid M inst✝³ : Monoid N inst✝² : Monoid P l✝ l₁ l₂ : List M a : M inst✝¹ : Preorder M inst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1 l : List M hl₁✝ : ∀ (x : M), x ∈ l → 1 ≤ x hl₁ : ∀ (x : M), x ∈ [] → 1 ≤ x ⊢ 1 ≤ prod [] case cons ι : Type ?u.105808 α : Type ?u.105811 M : Type u_1 N : Type ?u.105817 P : Type ?u.105820 M₀ : Type ?u.105823 G : Type ?u.105826 R : Type ?u.105829 inst✝⁴ : Monoid M inst✝³ : Monoid N inst✝² : Monoid P l✝ l₁ l₂ : List M a : M inst✝¹ : Preorder M inst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1 l : List M hl₁✝ : ∀ (x : M), x ∈ l → 1 ≤ x hd : M tl : List M ih : (∀ (x : M), x ∈ tl → 1 ≤ x) → 1 ≤ prod tl hl₁ : ∀ (x : M), x ∈ hd :: tl → 1 ≤ x ⊢ 1 ≤ prod (hd :: tl) Tactic: induction' l with hd tl ih State Before: case cons ι : Type ?u.105808 α : Type ?u.105811 M : Type u_1 N : Type ?u.105817 P : Type ?u.105820 M₀ : Type ?u.105823 G : Type ?u.105826 R : Type ?u.105829 inst✝⁴ : Monoid M inst✝³ : Monoid N inst✝² : Monoid P l✝ l₁ l₂ : List M a : M inst✝¹ : Preorder M inst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1 l : List M hl₁✝ : ∀ (x : M), x ∈ l → 1 ≤ x hd : M tl : List M ih : (∀ (x : M), x ∈ tl → 1 ≤ x) → 1 ≤ prod tl hl₁ : ∀ (x : M), x ∈ hd :: tl → 1 ≤ x ⊢ 1 ≤ prod (hd :: tl) State After: case cons ι : Type ?u.105808 α : Type ?u.105811 M : Type u_1 N : Type ?u.105817 P : Type ?u.105820 M₀ : Type ?u.105823 G : Type ?u.105826 R : Type ?u.105829 inst✝⁴ : Monoid M inst✝³ : Monoid N inst✝² : Monoid P l✝ l₁ l₂ : List M a : M inst✝¹ : Preorder M inst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1 l : List M hl₁✝ : ∀ (x : M), x ∈ l → 1 ≤ x hd : M tl : List M ih : (∀ (x : M), x ∈ tl → 1 ≤ x) → 1 ≤ prod tl hl₁ : ∀ (x : M), x ∈ hd :: tl → 1 ≤ x ⊢ 1 ≤ hd * prod tl Tactic: rw [prod_cons] State Before: case cons ι : Type ?u.105808 α : Type ?u.105811 M : Type u_1 N : Type ?u.105817 P : Type ?u.105820 M₀ : Type ?u.105823 G : Type ?u.105826 R : Type ?u.105829 inst✝⁴ : Monoid M inst✝³ : Monoid N inst✝² : Monoid P l✝ l₁ l₂ : List M a : M inst✝¹ : Preorder M inst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1 l : List M hl₁✝ : ∀ (x : M), x ∈ l → 1 ≤ x hd : M tl : List M ih : (∀ (x : M), x ∈ tl → 1 ≤ x) → 1 ≤ prod tl hl₁ : ∀ (x : M), x ∈ hd :: tl → 1 ≤ x ⊢ 1 ≤ hd * prod tl State After: no goals Tactic: exact one_le_mul (hl₁ hd (mem_cons_self hd tl)) (ih fun x h => hl₁ x (mem_cons_of_mem hd h)) State Before: case nil ι : Type ?u.105808 α : Type ?u.105811 M : Type u_1 N : Type ?u.105817 P : Type ?u.105820 M₀ : Type ?u.105823 G : Type ?u.105826 R : Type ?u.105829 inst✝⁴ : Monoid M inst✝³ : Monoid N inst✝² : Monoid P l✝ l₁ l₂ : List M a : M inst✝¹ : Preorder M inst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1 l : List M hl₁✝ : ∀ (x : M), x ∈ l → 1 ≤ x hl₁ : ∀ (x : M), x ∈ [] → 1 ≤ x ⊢ 1 ≤ prod [] State After: no goals Tactic: rfl
(* Title: HOL/Auth/n_flash_lemma_on_inv__141.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_flash Protocol Case Study} theory n_flash_lemma_on_inv__141 imports n_flash_base begin section{All lemmas on causal relation between inv__141 and some rule r*} lemma n_NI_Local_Get_Put_HeadVsinv__141: assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_Get_Put_Head N src)" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_Get_Put_Head N src" apply fastforce done from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done moreover { assume b1: "(src=p__Inv4)" have "?P3 s" apply (cut_tac a1 a2 b1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''ShWbMsg'') ''Cmd'')) (Const SHWB_FAck)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''Pending'')) (Const false))))" in exI, auto) done then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4)" have "?P2 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } ultimately show "invHoldForRule s f r (invariants N)" by satx qed lemma n_NI_Remote_Get_PutVsinv__141: assumes a1: "(\<exists> src dst. src\<le>N\<and>dst\<le>N\<and>src~=dst\<and>r=n_NI_Remote_Get_Put src dst)" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a1 obtain src dst where a1:"src\<le>N\<and>dst\<le>N\<and>src~=dst\<and>r=n_NI_Remote_Get_Put src dst" apply fastforce done from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "(src=p__Inv4\<and>dst~=p__Inv4)\<or>(src~=p__Inv4\<and>dst=p__Inv4)\<or>(src~=p__Inv4\<and>dst~=p__Inv4)" apply (cut_tac a1 a2, auto) done moreover { assume b1: "(src=p__Inv4\<and>dst~=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4\<and>dst=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4\<and>dst~=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } ultimately show "invHoldForRule s f r (invariants N)" by satx qed lemma n_NI_Local_GetX_PutX_1Vsinv__141: assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_1 N src)" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_1 N src" apply fastforce done from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done moreover { assume b1: "(src=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } ultimately show "invHoldForRule s f r (invariants N)" by satx qed lemma n_NI_Local_GetX_PutX_2Vsinv__141: assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_2 N src)" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_2 N src" apply fastforce done from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done moreover { assume b1: "(src=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } ultimately show "invHoldForRule s f r (invariants N)" by satx qed lemma n_NI_Local_GetX_PutX_3Vsinv__141: assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_3 N src)" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_3 N src" apply fastforce done from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done moreover { assume b1: "(src=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } ultimately show "invHoldForRule s f r (invariants N)" by satx qed lemma n_NI_Local_GetX_PutX_4Vsinv__141: assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_4 N src)" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_4 N src" apply fastforce done from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done moreover { assume b1: "(src=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } ultimately show "invHoldForRule s f r (invariants N)" by satx qed lemma n_NI_Local_GetX_PutX_5Vsinv__141: assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_5 N src)" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_5 N src" apply fastforce done from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done moreover { assume b1: "(src=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } ultimately show "invHoldForRule s f r (invariants N)" by satx qed lemma n_NI_Local_GetX_PutX_6Vsinv__141: assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_6 N src)" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_6 N src" apply fastforce done from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done moreover { assume b1: "(src=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } ultimately show "invHoldForRule s f r (invariants N)" by satx qed lemma n_NI_Local_GetX_PutX_7__part__0Vsinv__141: assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_7__part__0 N src)" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_7__part__0 N src" apply fastforce done from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done moreover { assume b1: "(src=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } ultimately show "invHoldForRule s f r (invariants N)" by satx qed lemma n_NI_Local_GetX_PutX_7__part__1Vsinv__141: assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_7__part__1 N src)" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_7__part__1 N src" apply fastforce done from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done moreover { assume b1: "(src=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } ultimately show "invHoldForRule s f r (invariants N)" by satx qed lemma n_NI_Local_GetX_PutX_7_NODE_Get__part__0Vsinv__141: assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_7_NODE_Get__part__0 N src)" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_7_NODE_Get__part__0 N src" apply fastforce done from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done moreover { assume b1: "(src=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } ultimately show "invHoldForRule s f r (invariants N)" by satx qed lemma n_NI_Local_GetX_PutX_7_NODE_Get__part__1Vsinv__141: assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_7_NODE_Get__part__1 N src)" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_7_NODE_Get__part__1 N src" apply fastforce done from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done moreover { assume b1: "(src=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } ultimately show "invHoldForRule s f r (invariants N)" by satx qed lemma n_NI_Local_GetX_PutX_8_HomeVsinv__141: assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_8_Home N src)" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_8_Home N src" apply fastforce done from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done moreover { assume b1: "(src=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } ultimately show "invHoldForRule s f r (invariants N)" by satx qed lemma n_NI_Local_GetX_PutX_8_Home_NODE_GetVsinv__141: assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_8_Home_NODE_Get N src)" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_8_Home_NODE_Get N src" apply fastforce done from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done moreover { assume b1: "(src=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } ultimately show "invHoldForRule s f r (invariants N)" by satx qed lemma n_NI_Local_GetX_PutX_8Vsinv__141: assumes a1: "(\<exists> src pp. src\<le>N\<and>pp\<le>N\<and>src~=pp\<and>r=n_NI_Local_GetX_PutX_8 N src pp)" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a1 obtain src pp where a1:"src\<le>N\<and>pp\<le>N\<and>src~=pp\<and>r=n_NI_Local_GetX_PutX_8 N src pp" apply fastforce done from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "(src=p__Inv4\<and>pp~=p__Inv4)\<or>(src~=p__Inv4\<and>pp=p__Inv4)\<or>(src~=p__Inv4\<and>pp~=p__Inv4)" apply (cut_tac a1 a2, auto) done moreover { assume b1: "(src=p__Inv4\<and>pp~=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4\<and>pp=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4\<and>pp~=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } ultimately show "invHoldForRule s f r (invariants N)" by satx qed lemma n_NI_Local_GetX_PutX_8_NODE_GetVsinv__141: assumes a1: "(\<exists> src pp. src\<le>N\<and>pp\<le>N\<and>src~=pp\<and>r=n_NI_Local_GetX_PutX_8_NODE_Get N src pp)" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a1 obtain src pp where a1:"src\<le>N\<and>pp\<le>N\<and>src~=pp\<and>r=n_NI_Local_GetX_PutX_8_NODE_Get N src pp" apply fastforce done from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "(src=p__Inv4\<and>pp~=p__Inv4)\<or>(src~=p__Inv4\<and>pp=p__Inv4)\<or>(src~=p__Inv4\<and>pp~=p__Inv4)" apply (cut_tac a1 a2, auto) done moreover { assume b1: "(src=p__Inv4\<and>pp~=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4\<and>pp=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4\<and>pp~=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } ultimately show "invHoldForRule s f r (invariants N)" by satx qed lemma n_NI_Local_GetX_PutX_9__part__0Vsinv__141: assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_9__part__0 N src)" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_9__part__0 N src" apply fastforce done from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done moreover { assume b1: "(src=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } ultimately show "invHoldForRule s f r (invariants N)" by satx qed lemma n_NI_Local_GetX_PutX_9__part__1Vsinv__141: assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_9__part__1 N src)" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_9__part__1 N src" apply fastforce done from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done moreover { assume b1: "(src=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } ultimately show "invHoldForRule s f r (invariants N)" by satx qed lemma n_NI_Local_GetX_PutX_10_HomeVsinv__141: assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_10_Home N src)" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_10_Home N src" apply fastforce done from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done moreover { assume b1: "(src=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } ultimately show "invHoldForRule s f r (invariants N)" by satx qed lemma n_NI_Local_GetX_PutX_10Vsinv__141: assumes a1: "(\<exists> src pp. src\<le>N\<and>pp\<le>N\<and>src~=pp\<and>r=n_NI_Local_GetX_PutX_10 N src pp)" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a1 obtain src pp where a1:"src\<le>N\<and>pp\<le>N\<and>src~=pp\<and>r=n_NI_Local_GetX_PutX_10 N src pp" apply fastforce done from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "(src=p__Inv4\<and>pp~=p__Inv4)\<or>(src~=p__Inv4\<and>pp=p__Inv4)\<or>(src~=p__Inv4\<and>pp~=p__Inv4)" apply (cut_tac a1 a2, auto) done moreover { assume b1: "(src=p__Inv4\<and>pp~=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4\<and>pp=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4\<and>pp~=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } ultimately show "invHoldForRule s f r (invariants N)" by satx qed lemma n_NI_Local_GetX_PutX_11Vsinv__141: assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_11 N src)" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_11 N src" apply fastforce done from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done moreover { assume b1: "(src=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4)" have "?P1 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } ultimately show "invHoldForRule s f r (invariants N)" by satx qed lemma n_NI_Remote_GetX_PutXVsinv__141: assumes a1: "(\<exists> src dst. src\<le>N\<and>dst\<le>N\<and>src~=dst\<and>r=n_NI_Remote_GetX_PutX src dst)" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a1 obtain src dst where a1:"src\<le>N\<and>dst\<le>N\<and>src~=dst\<and>r=n_NI_Remote_GetX_PutX src dst" apply fastforce done from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "(src=p__Inv4\<and>dst~=p__Inv4)\<or>(src~=p__Inv4\<and>dst=p__Inv4)\<or>(src~=p__Inv4\<and>dst~=p__Inv4)" apply (cut_tac a1 a2, auto) done moreover { assume b1: "(src=p__Inv4\<and>dst~=p__Inv4)" have "?P3 s" apply (cut_tac a1 a2 b1, simp, rule_tac x="(neg (andForm (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true)) (eqn (IVar (Field (Para (Field (Ident ''Sta'') ''Proc'') dst) ''CacheState'')) (Const CACHE_E))))" in exI, auto) done then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4\<and>dst=p__Inv4)" have "?P3 s" apply (cut_tac a1 a2 b1, simp, rule_tac x="(neg (andForm (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true)) (eqn (IVar (Field (Para (Field (Ident ''Sta'') ''Proc'') p__Inv4) ''CacheState'')) (Const CACHE_E))))" in exI, auto) done then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume b1: "(src~=p__Inv4\<and>dst~=p__Inv4)" have "?P3 s" apply (cut_tac a1 a2 b1, simp, rule_tac x="(neg (andForm (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true)) (eqn (IVar (Field (Para (Field (Ident ''Sta'') ''Proc'') dst) ''CacheState'')) (Const CACHE_E))))" in exI, auto) done then have "invHoldForRule s f r (invariants N)" by auto } ultimately show "invHoldForRule s f r (invariants N)" by satx qed lemma n_NI_ReplaceVsinv__141: assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Replace src)" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Replace src" apply fastforce done from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done moreover { assume b1: "(src=p__Inv4)" have "((formEval (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) s))\<or>((formEval (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) s))" by auto moreover { assume c1: "((formEval (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) s))" have "?P1 s" proof(cut_tac a1 a2 b1 c1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume c1: "((formEval (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) s))" have "?P2 s" proof(cut_tac a1 a2 b1 c1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } ultimately have "invHoldForRule s f r (invariants N)" by satx } moreover { assume b1: "(src~=p__Inv4)" have "?P2 s" proof(cut_tac a1 a2 b1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } ultimately show "invHoldForRule s f r (invariants N)" by satx qed lemma n_PI_Local_GetX_PutX_HeadVld__part__0Vsinv__141: assumes a1: "(r=n_PI_Local_GetX_PutX_HeadVld__part__0 N )" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "?P1 s" proof(cut_tac a1 a2 , auto) qed then show "invHoldForRule s f r (invariants N)" by auto qed lemma n_PI_Local_GetX_PutX_HeadVld__part__1Vsinv__141: assumes a1: "(r=n_PI_Local_GetX_PutX_HeadVld__part__1 N )" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "?P1 s" proof(cut_tac a1 a2 , auto) qed then show "invHoldForRule s f r (invariants N)" by auto qed lemma n_NI_FAckVsinv__141: assumes a1: "(r=n_NI_FAck )" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "?P1 s" proof(cut_tac a1 a2 , auto) qed then show "invHoldForRule s f r (invariants N)" by auto qed lemma n_NI_ShWbVsinv__141: assumes a1: "(r=n_NI_ShWb N )" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s") proof - from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__141 p__Inv4" apply fastforce done have "((formEval (andForm (eqn (Const (index p__Inv4)) (IVar (Field (Field (Ident ''Sta'') ''ShWbMsg'') ''Proc''))) (eqn (IVar (Field (Field (Ident ''Sta'') ''ShWbMsg'') ''HomeProc'')) (Const false))) s))\<or>((formEval (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true)) s))\<or>((formEval (andForm (neg (eqn (Const (index p__Inv4)) (IVar (Field (Field (Ident ''Sta'') ''ShWbMsg'') ''Proc'')))) (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''ShWbMsg'') ''HomeProc'')) (Const false))) (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true)))) s))" by auto moreover { assume c1: "((formEval (andForm (eqn (Const (index p__Inv4)) (IVar (Field (Field (Ident ''Sta'') ''ShWbMsg'') ''Proc''))) (eqn (IVar (Field (Field (Ident ''Sta'') ''ShWbMsg'') ''HomeProc'')) (Const false))) s))" have "?P1 s" proof(cut_tac a1 a2 c1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume c1: "((formEval (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true)) s))" have "?P1 s" proof(cut_tac a1 a2 c1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume c1: "((formEval (andForm (neg (eqn (Const (index p__Inv4)) (IVar (Field (Field (Ident ''Sta'') ''ShWbMsg'') ''Proc'')))) (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true)))) s))" have "?P1 s" proof(cut_tac a1 a2 c1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } moreover { assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''ShWbMsg'') ''HomeProc'')) (Const false))) (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true)))) s))" have "?P1 s" proof(cut_tac a1 a2 c1, auto) qed then have "invHoldForRule s f r (invariants N)" by auto } ultimately show "invHoldForRule s f r (invariants N)" by satx qed lemma n_NI_Local_Get_Get__part__1Vsinv__141: assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_Local_Get_Get__part__1 src" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_Remote_GetX_PutX_HomeVsinv__141: assumes a1: "\<exists> dst. dst\<le>N\<and>r=n_NI_Remote_GetX_PutX_Home dst" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_PI_Remote_GetVsinv__141: assumes a1: "\<exists> src. src\<le>N\<and>r=n_PI_Remote_Get src" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_PI_Local_GetX_PutX__part__0Vsinv__141: assumes a1: "r=n_PI_Local_GetX_PutX__part__0 " and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_WbVsinv__141: assumes a1: "r=n_NI_Wb " and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_StoreVsinv__141: assumes a1: "\<exists> src data. src\<le>N\<and>data\<le>N\<and>r=n_Store src data" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_Local_GetX_GetX__part__1Vsinv__141: assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_GetX__part__1 src" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_InvAck_3Vsinv__141: assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_InvAck_3 N src" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_InvAck_1Vsinv__141: assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_InvAck_1 N src" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_PI_Local_GetX_GetX__part__1Vsinv__141: assumes a1: "r=n_PI_Local_GetX_GetX__part__1 " and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_PI_Local_GetX_GetX__part__0Vsinv__141: assumes a1: "r=n_PI_Local_GetX_GetX__part__0 " and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_PI_Remote_ReplaceVsinv__141: assumes a1: "\<exists> src. src\<le>N\<and>r=n_PI_Remote_Replace src" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_Store_HomeVsinv__141: assumes a1: "\<exists> data. data\<le>N\<and>r=n_Store_Home data" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_PI_Local_ReplaceVsinv__141: assumes a1: "r=n_PI_Local_Replace " and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_Local_GetX_Nak__part__1Vsinv__141: assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_Nak__part__1 src" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_Local_Get_Nak__part__1Vsinv__141: assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_Local_Get_Nak__part__1 src" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_Local_Get_Get__part__0Vsinv__141: assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_Local_Get_Get__part__0 src" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_InvAck_existsVsinv__141: assumes a1: "\<exists> src pp. src\<le>N\<and>pp\<le>N\<and>src~=pp\<and>r=n_NI_InvAck_exists src pp" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_Local_GetX_Nak__part__2Vsinv__141: assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_Nak__part__2 src" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_PI_Remote_PutXVsinv__141: assumes a1: "\<exists> dst. dst\<le>N\<and>r=n_PI_Remote_PutX dst" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_Remote_Get_Put_HomeVsinv__141: assumes a1: "\<exists> dst. dst\<le>N\<and>r=n_NI_Remote_Get_Put_Home dst" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_InvVsinv__141: assumes a1: "\<exists> dst. dst\<le>N\<and>r=n_NI_Inv dst" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_PI_Local_PutXVsinv__141: assumes a1: "r=n_PI_Local_PutX " and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_Local_Get_Nak__part__2Vsinv__141: assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_Local_Get_Nak__part__2 src" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_Local_GetX_GetX__part__0Vsinv__141: assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_GetX__part__0 src" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_PI_Local_Get_PutVsinv__141: assumes a1: "r=n_PI_Local_Get_Put " and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_Remote_GetX_Nak_HomeVsinv__141: assumes a1: "\<exists> dst. dst\<le>N\<and>r=n_NI_Remote_GetX_Nak_Home dst" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_Local_PutXAcksDoneVsinv__141: assumes a1: "r=n_NI_Local_PutXAcksDone " and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_Remote_GetX_NakVsinv__141: assumes a1: "\<exists> src dst. src\<le>N\<and>dst\<le>N\<and>src~=dst\<and>r=n_NI_Remote_GetX_Nak src dst" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_NakVsinv__141: assumes a1: "\<exists> dst. dst\<le>N\<and>r=n_NI_Nak dst" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_PI_Remote_GetXVsinv__141: assumes a1: "\<exists> src. src\<le>N\<and>r=n_PI_Remote_GetX src" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_PI_Local_GetX_PutX__part__1Vsinv__141: assumes a1: "r=n_PI_Local_GetX_PutX__part__1 " and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_Remote_Get_Nak_HomeVsinv__141: assumes a1: "\<exists> dst. dst\<le>N\<and>r=n_NI_Remote_Get_Nak_Home dst" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_Remote_PutXVsinv__141: assumes a1: "\<exists> dst. dst\<le>N\<and>r=n_NI_Remote_PutX dst" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_Remote_PutVsinv__141: assumes a1: "\<exists> dst. dst\<le>N\<and>r=n_NI_Remote_Put dst" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_Local_Get_PutVsinv__141: assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_Local_Get_Put src" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_Local_GetX_Nak__part__0Vsinv__141: assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_Nak__part__0 src" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_InvAck_exists_HomeVsinv__141: assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_InvAck_exists_Home src" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_Replace_HomeVsinv__141: assumes a1: "r=n_NI_Replace_Home " and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_Local_PutVsinv__141: assumes a1: "r=n_NI_Local_Put " and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_Remote_Get_NakVsinv__141: assumes a1: "\<exists> src dst. src\<le>N\<and>dst\<le>N\<and>src~=dst\<and>r=n_NI_Remote_Get_Nak src dst" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_Nak_ClearVsinv__141: assumes a1: "r=n_NI_Nak_Clear " and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_Local_Get_Put_DirtyVsinv__141: assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_Local_Get_Put_Dirty src" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_Local_Get_Nak__part__0Vsinv__141: assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_Local_Get_Nak__part__0 src" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_PI_Local_Get_GetVsinv__141: assumes a1: "r=n_PI_Local_Get_Get " and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_Nak_HomeVsinv__141: assumes a1: "r=n_NI_Nak_Home " and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done lemma n_NI_InvAck_2Vsinv__141: assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_InvAck_2 N src" and a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__141 p__Inv4)" shows "invHoldForRule s f r (invariants N)" apply (rule noEffectOnRule, cut_tac a1 a2, auto) done end
#` A fast version of row scaling of sparse TF matrix by IDF vector, this function wiil be invoked by function tfidf #` @param tf (sparse matrix) A saparse matrix of term frequency matrix #` @param idf (vector) A vector of inverse document frequency #` @return (sparse matrix) A sparse matrix of TF-IDF fast_tfidf = function(tf, idf) { tf <- t(tf) tf@x <- tf@x * rep.int(idf, diff(tf@p)) tf <- t(tf) return(tf) }
If $\|x - y\| < \frac{\epsilon}{2}$ and $\|x' - y\| < \frac{\epsilon}{2}$, then $\|x - x'\| < \epsilon$.
% Compute the log intensity for the inverse link function g(f) = 1/(1+exp(-f)). % % The function is used in GLM likelihoods such as likPoisson, likGamma, likBeta % and likInvGauss. % % Copyright (c) by Hannes Nickisch, 2013-10-16. function varargout = glm_invlink_logit(f) varargout = cell(nargout, 1); % allocate the right number of output arguments [varargout{:}] = glm_invlink_logistic(f); if nargout>0 elg = exp(varargout{1}); varargout{1} = f - elg; if nargout>1 dlg = varargout{2}; varargout{2} = 1 - elg.dlg; if nargout>2 d2lg = varargout{3}; varargout{3} = -elg.(dlg.^2+d2lg); if nargout>3 d3lg = varargout{4}; varargout{4} = -elg.(dlg.^3+3d2lg.*dlg+d3lg); end end end end
The Nosler 7mm Rem Mag Brass is used to build the perfect load. Since hand loaders need the best, they use Nosler cartridge brass. Since it was introduced in 2005, the Nosler brass line is ready to load. Because it is made with the Nosler philosophy of uncompromising attention to detail, Nosler cartridge brass is created to exact dimensional standards and tolerances. Nosler uses quality materials for maximum accuracy and consistency potential. Because of this, you will notice brass with extending case life. Lastly, Nosler brass undergoes strict quality control. The same as all premium Nosler bullets have experienced for over 60 years.
theorem open_mapping_thm: assumes holf: "f holomorphic_on S" and S: "open S" and "connected S" and "open U" and "U \<subseteq> S" and fne: "\<not> f constant_on S" shows "open (f ` U)"
restoreParams = function (bugsResult, ragged = NULL, extraX = NULL) { thearray = bugsResult$sims.array parnames = dimnames(thearray)[[3]] vecPars = grep("\\[[[:digit:]]+\\]$", parnames, value = TRUE) matPars = grep("[[:digit:]+],[[:digit:]]+\\]$", parnames, value = TRUE) scPars = parnames[!parnames %in% c(vecPars, matPars)] scPars = scPars[grep("^beta", scPars, invert = TRUE)] result = list() if (length(matPars)) { precisionIndex = grep("^T", matPars) precisions = matPars[precisionIndex] matPars = matPars[-precisionIndex] result$precision = thearray[, , precisions] precisions = unique(gsub("\\[[[:digit:]]+,[[:digit:]]+\\]", "", precisions)) for (D in precisions) { result[[paste("var", D, sep = "")]] = cholInvArray(result$precision, D) } colno = substr(matPars, regexpr(",[[:digit:]]+\\]", matPars) + 1, 10000) colno = substr(colno, 1, nchar(colno) - 1) maxcol = max(as.integer(colno)) if (is.na(maxcol)) warning("can't find max col number") interceptcols = grep(paste(",", maxcol, "\\]", sep = ""), matPars) slopecols = matPars[-interceptcols] result$slopes = thearray[, , slopecols] matPars = which(parnames %in% matPars) parnames[matPars] = gsub(paste(",", maxcol, "\\]", sep = ""), "\\]", parnames[matPars]) dimnames(thearray)[[3]] = parnames vecPars = grep("\\[[[:digit:]]+\\]$", parnames, value = TRUE) } thephi = grep("^phi", scPars, value = TRUE) for (D in thephi) { thesd = gsub("^phi", "SD", D) if (thesd %in% scPars) { scPars = grep(D, scPars, invert = TRUE, value = TRUE) therange = gsub("^phi", "range", D) result[[therange]] = thearray[, , D]/thearray[, , thesd] } } for (D in scPars) result[[D]] = thearray[, , D] if (!length(scPars)) warning("no parameter names") fixedEffects = grep("^X", names(ragged), value = TRUE) betas = NULL for (D in fixedEffects) { tobind = thearray[, , grep(gsub("^X", "beta", D), dimnames(thearray)[[3]]), drop = F] if (!dim(tobind)[3]) warning("can't find fixed effect parameters for ", D) newnames = dimnames(ragged[[D]])[[2]] if (length(newnames) == (dim(tobind)[3])) dimnames(tobind)[[3]] = newnames betas = abind::abind(betas, tobind, along = 3) } result$betas = betas groups = unique(gsub("\\[[[:digit:]]+\\]$", "", vecPars)) thebetas = grep("^beta", groups) if (length(thebetas)) groups = groups[-thebetas] for (D in groups) { thisGroup = grep(paste("^", D, "\\[", sep = ""), vecPars, value = TRUE) result[[D]] = thearray[, , thisGroup] } theSpatialGroups = grep("Spatial$", groups) if (length(theSpatialGroups)) { notSpatial = groups[-theSpatialGroups] } else { notSpatial = groups } for (D in notSpatial) { result[[paste("Fitted", D, sep = "")]] = result[[D]] } if (is.null(ragged)) { return(result) } groups = paste("S", substr(groups, 2, nchar(groups)), sep = "") randomEffects = groups[groups %in% names(ragged)] randomEffects = names(sort(unlist(lapply(ragged[randomEffects], length)))) randomEffects = substr(randomEffects, 2, nchar(randomEffects)) if (!length(randomEffects)) { warning(paste(toString(groups), ":cannot find random effects")) return(result) } theMeanOld = array(result[["intercept"]], c(dim(result[["intercept"]]), 1)) Nchain = dim(theMeanOld)[2] torep = rep(1, length(ragged[[paste("S", randomEffects[1], sep = "")]]) - 1) betanames = dimnames(result$betas)[[3]] for (D in randomEffects) { theR = paste("R", D, sep = "") theS = ragged[[paste("S", D, sep = "")]] thenames = names(theS)[-length(theS)] if (length(thenames) != (dim(result[[theR]])[3])) warning(D, "different dimensions in bugsResult and ragged") dimnames(result[[theR]])[[3]] = thenames dimnames(result[[paste("Fitted", theR, sep = "")]])[[3]] = thenames DX = paste("X", D, sep = "") Dbeta = paste("beta", D, sep = "") themean = theMeanOld[, , torep] if (!is.null(ragged[[DX]])) { theX = t(ragged[[DX]]) if (Dbeta %in% betanames) { theseBetas = result$betas[, , Dbeta, drop = F] } else { if (all(rownames(theX) %in% dimnames(result$betas)[[3]])) { theseBetas = result$betas[, , rownames(theX), drop = FALSE] } else { warning("cannot find ", D, " , ", DX) } } for (Dchain in 1:Nchain) { themean[, Dchain, ] = themean[, Dchain, ] + abind::adrop(theseBetas[, Dchain, , drop = FALSE], drop = 2) %% theX } } theMeanOld = result[[theR]] torep = diff(theS) torep = rep(1:length(torep), torep) result[[theR]] = result[[theR]] - themean } spatialEffects = paste("R", randomEffects, "Spatial", sep = "") spatialEffects = spatialEffects[spatialEffects %in% names(result)] for (D in spatialEffects) { DsubR = gsub("Spatial$", "", D) Dsub = gsub("^R", "", DsubR) Dfitted = paste("Fitted", DsubR, sep = "") thenames = names(ragged[[paste("num", Dsub, sep = "")]]) if (is.null(thenames)) { thenames = paste("noname", 1:ragged[[paste("N", Dsub, "Spatial", sep = "")]], sep = "") thenames[ragged[[paste("Sspatial", Dsub, sep = "")]]] = names(ragged[[paste("Sspatial", Dsub, sep = "")]]) } theID = dimnames(result[[D]])[[3]] theID = gsub("[[:graph:]]+\\[", "", theID) theID = gsub("\\]$", "", theID) dimnames(result[[D]])[[3]] = thenames[as.integer(theID)] regionsNoV = thenames[!thenames %in% dimnames(result[[DsubR]])[[3]]] if (length(regionsNoV)) { dimNoV = c(dim(result$intercept), length(regionsNoV)) sdBig = array(result[[paste("SD", Dsub, sep = "")]], dimNoV) VfornoV = stats::rnorm(prod(dim(sdBig)), 0, sdBig) VfornoV = array(VfornoV, dimNoV) dimnames(VfornoV)[[3]] = regionsNoV DsubSpatial = paste(DsubR, "Spatial", sep = "") withSpatial = regionsNoV[regionsNoV %in% dimnames(result[[DsubSpatial]])[[3]]] VfornoV[, , withSpatial] = VfornoV[, , withSpatial] + result[[DsubSpatial]][, , withSpatial] result[[DsubR]] = abind::abind(result[[DsubR]], VfornoV, along = 3) fittedForNoV = VfornoV + array(result$intercept, dimNoV) if (!is.null(extraX)) { haveExtraX = rownames(extraX)[rownames(extraX) %in% VfornoV] theBeta = result[[paste("beta", Dsub, sep = "")]] haveBeta = colnames(extraX)[colnames(extraX) %in% rownames(theBeta)] if (length(haveExtraX) & length(haveBeta)) { fittedForNoV = fittedForNoV + extraX[haveExtraX, haveBeta] %% theBeta[haveBeta, ] } } result[[Dfitted]] = abind::abind(result[[Dfitted]], fittedForNoV, along = 3) result[[DsubR]] = result[[DsubR]][, , thenames] result[[Dfitted]] = result[[Dfitted]][, , thenames] } } #if (dim(result$betas)[3] == 1) { # result$betas = result$betas[, , 1] #} return(result) }
[STATEMENT] lemma deriv_n_hf_diffr [iff]: "(deriv^^k) (hf n) field_differentiable at x" [PROOF STATE] proof (prove) goal (1 subgoal): 1. (deriv ^^ k) (hf n) field_differentiable at x [PROOF STEP] unfolding field_differentiable_def hf_deriv_int_poly [PROOF STATE] proof (prove) goal (1 subgoal): 1. \<exists>f'. ((\<lambda>x. 1 / fact n * (\<Sum>i = 0..2 * n - k. real_of_int (int (\<Prod>{i<..i + k}) * cf n (i + k)) * x ^ i)) has_real_derivative f') (at x) [PROOF STEP] by (rule derivative_eq_intros exI \| force)+
\chapter{File and Directory Access} \label{filesys} The modules described in this chapter deal with disk files and directories. For example, there are modules for reading the properties of files, manipulating paths in a portable way, and creating temporary files. The full list of modules in this chapter is: \localmoduletable % XXX can this be included in the seealso environment? --amk Also see section \ref{bltin-file-objects} for a description of Python's built-in file objects. \begin{seealso} \seemodule{os}{Operating system interfaces, including functions to work with files at a lower level than the built-in file object.} \end{seealso}
theory Counterexample imports MainRLT begin lemma bij_upto_surjE: "bij_upto P Q f \<Longrightarrow> Q y y \<Longrightarrow> \<exists>x. P x x \<and> Q y (f x)" unfolding bij_upto_def by blast (* Next is a formalization of the paper's Example 6, showing a type definition of a type \<tau> that, not being wide, cannot support a suitable relativization infrastructure on the defined type. As explained in the paper, the root cause for this is that the defining predicate t has smaller cardinality than its relativized version t_rlt, even after quotienting t_rlt to the PER relational interpretation on the host type 'a \<sigma>. To exhibit this problem, we take 'a \<sigma> to be ind\<Rightarrow>'a and we prove that, for a large enough instance of 'a, namely ind\<Rightarrow>bool and a small enough PER R on 'a (one given by a small enough set A), we have the following: -- t is a singleton, whereas -- the set of RIN(ind\<Rightarrow>bool)-classes that satisfy t_rlt has at least two elements.) definition t :: "(ind \<Rightarrow> 'a) \<Rightarrow> bool" where "t \<equiv> \<lambda>f. (\<forall>x. \<exists>i. f i = x) \<or> (\<forall>i. f i = (SOME x. True))" typedef 'a \<tau> = "{f::ind \<Rightarrow>'a. t f}" unfolding t_def by auto ( The extension of the predicate t is a singleton for large enough types 'a, for example taking 'a to be ind\<Rightarrow>bool: ) lemma t_singl: fixes f::"ind\<Rightarrow>(ind\<Rightarrow>bool)" shows "t f \<longleftrightarrow> f = (\<lambda>i. (SOME x. True))" proof - { assume "\<forall>x. \<exists>i. f i = x" then have "range (Collect \<circ> f) = Pow UNIV" unfolding image_def by auto (metis Collect_mem_eq) then have False using Cantors_paradox by blast } thus ?thesis unfolding t_def by auto qed ( Hence the newly defined type is also a singleton for that instance: ) lemma \<tau>_singl: "(x :: (ind\<Rightarrow>bool) \<tau>) = Abs_\<tau> (\<lambda>i. SOME f. True)" by (cases x) (auto simp: t_singl) ( We use the RLCST command to perform the relativization of t: ) local_setup \<open>RLCST @{term t}\<close> lemma t_rlt_alt: "neper R \<Longrightarrow> t_rlt R f \<longleftrightarrow> (\<forall>x. R x x \<longrightarrow> (\<exists>i. R (f i) x) \<or> (\<forall>x. R (f x) (SOME a. R a a)))" unfolding t_rlt_def by (auto simp: Eps_rlt_True) ( Now we are ready to show that \<tau> cannot have a relational interpretation in such a way that a basic sanity property is satisfied: the relativized version of the typedef theorem holds. ) ( Indeed, let us assume that such a relational interpretation rel_\<tau> existed: ) consts rel_\<tau> :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<tau> \<Rightarrow> 'a \<tau> \<Rightarrow> bool" ( ... and satisfied the aforementioned property: ) axiomatization where bij_upto_\<tau>: "\<And>R. neper R \<Longrightarrow> \<exists>f. bij_upto (rel_\<tau> R) (restr (rel_fun (=) R) (t_rlt R)) (f R)" ( That would yield a contradiction (a proof of False). The proof encodes the aforementioned cardinality mismatch argument, after instantiating 'a to ind\<Rightarrow>bool *) lemma False proof - define T where "T \<equiv> (\<lambda>_ :: ind. True)" define F where "F \<equiv> (\<lambda>_ :: ind. False)" define A where "A \<equiv> {T,F}" define R where "R \<equiv> perOfSet A" let ?RLT = "restr (rel_fun (=) R) (t_rlt R)" have neper[simp]: "neper R" unfolding R_def A_def by (meson insertI1 neper_def perOfSet_def per_perOfSet) from bij_upto_\<tau>[OF neper] obtain f where bij_upto: "bij_upto (rel_\<tau> R) ?RLT (f R)" by blast note surjE = bij_upto_surjE[where Q = ?RLT, OF bij_upto] have [simp]: "(SOME a. a \<in> A) \<in> A" by (metis A_def insertI1 someI) have 0: "rel_fun (=) R u u' \<longleftrightarrow> (u = u' \<and> (\<forall> i::ind. u i \<in> A))" for u u' unfolding rel_fun_def R_def perOfSet_def by auto have 1: "?RLT u u' \<longleftrightarrow> u = u' \<and> (range u = A \<or> u = (\<lambda>i. SOME a. a \<in> A))" for u u' unfolding 0 unfolding t_rlt_alt[OF neper] unfolding R_def restr_def perOfSet_def image_def by fastforce have disj: "Eps (\<lambda>a. a \<in> A) = T \<or> Eps (\<lambda>a. a \<in> A) = F" unfolding A_def by (metis insert_iff singletonD some_in_eq) then have "?RLT (\<lambda>_. Eps (\<lambda>a. a \<in> A)) (\<lambda>_. Eps (\<lambda>a. a \<in> A))" unfolding restr_def t_rlt_alt[OF neper] unfolding R_def rel_fun_def perOfSet_def by auto from surjE[OF this] obtain x where "rel_\<tau> R x x" "f R x = (\<lambda>_. Eps (\<lambda>a. a \<in> A))" by (auto simp: 1) moreover let ?u = "\<lambda>z. if z = Zero_Rep then T else F" from Suc_Rep_not_Zero_Rep have "?RLT ?u ?u" unfolding t_rlt_alt[OF neper] unfolding restr_def R_def A_def rel_fun_def perOfSet_def by auto from surjE[OF this] obtain y where "rel_\<tau> R y y" "f R y = ?u" by (auto simp: 1) moreover from disj Suc_Rep_not_Zero_Rep have "?u \<noteq> (\<lambda>_. Eps (\<lambda>a. a \<in> A))" by (auto simp: fun_eq_iff T_def F_def) ultimately show False by (auto simp: eq_onp_def \<tau>_singl[of x] \<tau>_singl[of y]) qed end
The constant function $c$ is integrable on the interval $[a,b]$.
import data.equiv universes u v w variables {α : Type u} {β : Type v} {γ : Type w} [group α] [group β] [group γ] variables (α β) structure isomorphism extends equiv α β := (map_mul : is_group_hom to_fun) namespace isomorphism variables {α β} (f : isomorphism α β) (g : isomorphism β γ) {x y : α} def comp : isomorphism α γ := { equiv.trans f.to_equiv g.to_equiv with map_mul := λ x y, calc g.to_fun (f.to_fun (x * y)) = g.to_fun (f.to_fun x * f.to_fun y) : congr_arg g.to_fun $ @isomorphism.map_mul _ _ _ _ f x y ... = g.to_fun (f.to_fun x) * g.to_fun (f.to_fun y) : @isomorphism.map_mul _ _ _ _ g _ _ } def inv : isomorphism β α := { equiv.symm f.to_equiv with map_mul := λ x y, calc f.inv_fun (x * y) = f.inv_fun (f.to_fun (f.inv_fun x) * y) : congr_arg f.inv_fun $ congr_arg (λ z, z * y) $ eq.symm $ f.right_inv x ... = f.inv_fun (f.to_fun (f.inv_fun x) * f.to_fun (f.inv_fun y)) : congr_arg f.inv_fun $ congr_arg (λ z, f.to_fun (f.inv_fun x) * z) $ eq.symm $ f.right_inv y ... = f.inv_fun (f.to_fun (f.inv_fun x * f.inv_fun y)) : congr_arg f.inv_fun $ eq.symm $ @isomorphism.map_mul _ _ _ _ f _ _ ... = f.inv_fun x * f.inv_fun y : f.left_inv (f.inv_fun x * f.inv_fun y) } def id : isomorphism α α := { equiv.refl α with map_mul := λ x y, rfl } end isomorphism structure automorphism extends isomorphism α α theorem automorphism.ext (f g : automorphism α) (H : ∀ x, f.to_isomorphism.to_equiv.to_fun x = g.to_isomorphism.to_equiv.to_fun x) : f = g := by cases f; cases f; cases g; cases g; congr; apply equiv.ext; exact H namespace automorphism instance : has_coe (automorphism α) (α → α) := ⟨λ f, f.to_isomorphism.to_equiv⟩ theorem map_mul (f : automorphism α) (x y : α) : (f : α → α) (x * y) = (f : α → α) x * (f : α → α) y := @isomorphism.map_mul _ _ _ _ f.to_isomorphism _ _ variable {α} theorem is_group_hom (f : automorphism α) : is_group_hom (f : α → α) := map_mul α f variable α instance : group (automorphism α) := { mul := λ f g, automorphism.mk $ g.to_isomorphism.comp f.to_isomorphism, mul_assoc := λ f g h, by unfold has_mul.mul; congr, one := automorphism.mk $ isomorphism.id, one_mul := λ f, by apply automorphism.ext; intro x; refl, mul_one := λ f, by apply automorphism.ext; intro x; refl, inv := λ f, automorphism.mk $ f.to_isomorphism.inv, mul_left_inv := λ f, by apply automorphism.ext; intro x; exact ((f.to_isomorphism).to_equiv).left_inv x} @[simp] lemma mul_apply (f g : automorphism α) (x : α) : (((f * g) : automorphism α) : α → α) x = (f:α → α) ((g:α → α) x) := rfl end automorphism variables (α β) def group_direct_product : group (α × β) := { mul := λ x y, (x.fst * y.fst, x.snd * y.snd), mul_assoc := λ x y z, by simp [mul_assoc], one := (1, 1), one_mul := λ x, by simp, mul_one := λ x, by rw prod.ext; split; apply mul_one, inv := λ x, (x.fst⁻¹, x.snd⁻¹), mul_left_inv := λ x, by rw prod.ext; split; apply mul_left_inv } def group_semidirect_product (f : β → automorphism α) (H : is_group_hom f) : group (α × β) := { mul := λ x y, (x.fst * (f x.snd : α → α) y.fst, x.snd * y.snd), mul_assoc := λ x y z, by rw prod.ext; split; simp [H, H x.snd, automorphism.map_mul α (f x.snd), mul_assoc], one := (1, 1), one_mul := λ x, by rw prod.ext; split; simp [H.one]; refl, mul_one := λ x, by rw prod.ext; split; simp [(), semigroup.mul, monoid.mul, group.mul, (f x.snd).is_group_hom.one]; apply mul_one, inv := λ x, ((f x.snd⁻¹ : α → α) (x.fst⁻¹), x.snd⁻¹), mul_left_inv := λ x, calc ((f x.snd⁻¹ : α → α) (x.fst⁻¹) (f x.snd⁻¹ : α → α) x.fst, x.snd⁻¹ * x.snd) = (1, 1) : by rw ← (f x.snd⁻¹).is_group_hom; simp [(f x.snd⁻¹).is_group_hom.one] }
Sal Genito is in charge of Buildings and Grounds, a part of Operations & Maintenance, which includes the Grounds Division Grounds Division, Custodial Division and agIndustrial Services. One of the programs that Sal has started at UC Davis is processing of UC Davis Olive Oil.
Require Import SpecDeps. Require Import RData. Require Import EventReplay. Require Import MoverTypes. Require Import Constants. Require Import CommonLib. Require Import AbsAccessor.Spec. Require Import BaremoreSMC.Spec. Local Open Scope Z_scope. Section Spec. Definition granule_delegate_ops_spec (g: Pointer) (addr: Z64) (adt: RData) : option RData := match addr with \| VZ64 addr => let gidx := offset g in rely is_int64 addr; rely (peq (base g) ginfo_loc); rely (gidx =? __addr_to_gidx addr); rely is_gidx gidx; rely Z.land (r_scr_el3 (cpu_regs (priv adt))) SCR_WORLD_MASK =? SCR_REALM_WORLD; when adt == query_oracle adt; let gn := (gs (share adt)) @ gidx in rely prop_dec ((buffer (priv adt)) @ SLOT_DELEGATED = None); rely prop_dec (glock gn = Some CPU_ID); rely prop_dec (gtype gn = GRANULE_STATE_NS); rely prop_dec ((gpt_lk (share adt)) @ gidx = None); rely prop_dec ((gpt (share adt)) @ gidx = false); let e1 := EVT CPU_ID (ACQ_GPT gidx) in let e2 := EVT CPU_ID (REL_GPT gidx true) in let g' := gn {ginfo: (ginfo gn) {g_tag: GRANULE_STATE_DELEGATED}} {gnorm: zero_granule_data_normal} {grec: zero_granule_data_rec} in let regs' := (cpu_regs (priv adt)) {r_x0: 0} {r_x1: addr} {r_esr_el3: ESR_EC_SMC} in let e' := EVT CPU_ID (REL gidx g') in Some adt {log: e' :: e2 :: e1 :: (log adt)} {share: (share adt) {gs: (gs (share adt)) # gidx == (g' {gtype: GRANULE_STATE_DELEGATED} {glock: None})} {gpt: (gpt (share adt)) # gidx == true}} {priv: (priv adt) {cpu_regs: regs'}} end. End Spec.
module Minecraft.Core.Block.Break.Export import public Minecraft.Core.Block.Break %default total
module Heap %default total %access private public export interface Heap (h : Type -> Type) where empty : Ord a => h a isEmpty : Ord a => h a -> Bool insert : Ord a => a -> h a -> h a merge : Ord a => h a -> h a -> h a findMin : Ord a => h a -> a deleteMin : Ord a => h a -> h a
Formal statement is: lemma [field_simps]: "c \<noteq> 0 \<Longrightarrow> a = b /\<^sub>R c \<longleftrightarrow> c \<^sub>R a = b" "c \<noteq> 0 \<Longrightarrow> b /\<^sub>R c = a \<longleftrightarrow> b = c \<^sub>R a" "c \<noteq> 0 \<Longrightarrow> a + b /\<^sub>R c = (c \<^sub>R a + b) /\<^sub>R c" "c \<noteq> 0 \<Longrightarrow> a /\<^sub>R c + b = (a + c \<^sub>R b) /\<^sub>R c" "c \<noteq> 0 \<Longrightarrow> a - b /\<^sub>R c = (c \<^sub>R a - b) /\<^sub>R c" "c \<noteq> 0 \<Longrightarrow> a /\<^sub>R c - b = (a - c \<^sub>R b) /\<^sub>R c" "c \<noteq> 0 \<Longrightarrow> - (a /\<^sub>R c) + b = (- a + c \<^sub>R b) /\<^sub>R c" "c \<noteq> 0 \<Longrightarrow> - (a /\<^sub>R c) - b = (- a - c \<^sub>R b) /\<^sub>R c" for a b :: "'a :: real_vector" Informal statement is: If $c \neq 0$, then $a = b/c$ if and only if $c \cdot a = b$.
State Before: α : Type u_1 β : Type ?u.3590 inst✝ : LinearOrder α a b : α h✝ : a ≤ b x : α h : a < b ⊢ projIcc a b (_ : a ≤ b) x = { val := a, property := (_ : a ∈ Icc a b) } ↔ x ≤ a State After: α : Type u_1 β : Type ?u.3590 inst✝ : LinearOrder α a b : α h✝ : a ≤ b x : α h : a < b h' : projIcc a b (_ : a ≤ b) x = { val := a, property := (_ : a ∈ Icc a b) } ⊢ x ≤ a Tactic: refine' ⟨fun h' => _, projIcc_of_le_left _⟩ State Before: α : Type u_1 β : Type ?u.3590 inst✝ : LinearOrder α a b : α h✝ : a ≤ b x : α h : a < b h' : projIcc a b (_ : a ≤ b) x = { val := a, property := (_ : a ∈ Icc a b) } ⊢ x ≤ a State After: α : Type u_1 β : Type ?u.3590 inst✝ : LinearOrder α a b : α h✝ : a ≤ b x : α h : a < b h' : x ≤ a ⊢ x ≤ a Tactic: simp_rw [Subtype.ext_iff_val, projIcc, max_eq_left_iff, min_le_iff, h.not_le, false_or_iff] at h' State Before: α : Type u_1 β : Type ?u.3590 inst✝ : LinearOrder α a b : α h✝ : a ≤ b x : α h : a < b h' : x ≤ a ⊢ x ≤ a State After: no goals Tactic: exact h'
\columnbreak \section{Model Validation} % === \subsection{Stability-based Validation} \emph{Stability:}\enspace Solutions on two data sets drawn from the same source should be similar. %\emph{(In-)Stability:} Expected (dis-)agreement of solutions from two data sets from the same source. % === \subsection{Information-theoretic Validation} \subsubsection{Shannon's Channel Coding Thm.} \begin{itemize} \item \textbf{Channel:}\enspace $(\mathcal S, \{p(\cdot\mid s)\}_{s\in\mathcal S} )$,\enspace $\mathcal S$: alphabet \begin{itemize} \item $\epsilon$-noisy binary channel: $p(\hat s\mid s) = \big\{ \substack{ 1-\epsilon \:\:\text{if}\:\: \hat s=s \\ \epsilon \phantom{1-} \:\:\text{if}\:\: \hat s\neq s}$ \end{itemize} \vspace{-3pt} \item \textbf{Capacity:}\enspace \highlight{$\mathrm{cap} = \max_p I(S;\hat S)$} $\rightsquigarrow p_S(s)$ \item \textbf{$\bm{(M,n)}$-code:}\enspace is a pair $(Enc,Dec)$\hfill $\color{gray} \leftarrow\text{scr. p.87}$\\\quad where $M$: \#messages, $n$: code-length \begin{itemize} \item \textbf{Rate:}\enspace $r = \frac{\log_2 M}{n}$ $\Leftrightarrow$ $M=\floor{2^{nr}}$ \item \textbf{Commu. err.:}\enspace $p\ped{err} \coloneqq \max\limits_{i\leq M} \P{ Dec(\widehat{ Enc(i) }) \neq i }$ \end{itemize} \end{itemize} Goal / \textbf{Best code:}\enspace \highlight{$\lim\limits_{n\to\infty} \frac{\log M}{n}$\enspace s.t.\enspace $\lim\limits_{n\to\infty} p\ped{err} \to 0$} \emph{Asymptotic equiparition property (AEP):} \vspace{-2pt} \begin{itemize} \item $A_\epsilon^{(n)}$: Typical set of sequences $(s_1,\ldots,s_n) \in \mathcal S^n$\\\qquad $\abs{-\frac1n \log p_{S^n} (\bm s^n) - H[S]} < \epsilon$\hfill $\color{gray} \leftarrow \text{scr. p.89}$ \item $\P{(\bm S^n, \bm{\hat S}^n) \in A_\epsilon^{(n)}} \overset{n\to\infty}{\to} 1$\hfill $\color{gray} \leftarrow \text{scr. p.90}$ \item \highlight{$p\ped{err} \leq 2^{-n(\mathrm{cap} - 3\epsilon -r)}$} $\overset{n\to\infty}{\to} 0$\enspace if\enspace \highlight{$r<\mathrm{cap}$} \end{itemize} \subsubsection{Algorithm Validation} \textbf{Assumptions:} \begin{itemize} \item Exponential solution space, i.e. $\log\abs{\mathcal C} = \mathcal O(n)$ \item $\mathcal A$'s output is probabilistic, i.e. $p(\cdot \mid X')$ \end{itemize} \emph{Ideal variant:} \textbf{Messages:}\enspace $\mathcal M = \brace{ X'_1, \ldots, X'_m }$ \vspace{-2pt} \textbf{Code:}\enspace $X'_i \xrightarrow{Enc_{\mathcal A}} p(\cdot\mid X'_i) \xrightarrow{\mathcal{C_A}} p(\cdot\mid X''_i) \xrightarrow{Dec_{\mathcal A}} \hat X$ \iffalse \begin{itemize} \item $Enc_{\mathcal A}$:\enspace encodes $X'_i \in \mathcal M$ as $p(\cdot\mid X'_i)$. \item $Dec_{\mathcal A}$:\enspace selects $\hat X = \arg\max\limits_X \kappa(X''_i, X)$\\ whereby $\kappa(X'', X') \coloneqq \sum_c p(c\mid X'') p(c\mid X')$ \end{itemize} \fi \emph{Empirical variant:} \textbf{Messages:}\enspace $\mathcal M = \brace{ \tau_1, \ldots, \tau_m }$ drawn u.a.r. from $\mathbb T$ \begin{itemize} \item Require\enspace $\sum_\tau p(c\mid \tau\circ X') \approx \frac{\abs{\mathbb T}}{\abs{\mathcal C}} \pm \rho$\hfill $\color{gray} \leftarrow \text{scr. p.95}$ \end{itemize} \vspace{-2pt} \textbf{Code:}\enspace $\tau_i \xrightarrow{Enc} p(\cdot\mid \tau_i \circ X') \xrightarrow{\mathcal{C_A}} p(\cdot\mid \tau_i \circ X'') \xrightarrow{Dec} \hat\tau$ \begin{itemize} \item $Enc_{\mathcal A}$:\enspace encodes $\tau_i \in \mathcal M$ as $p(\cdot\mid \tau_i \circ X')$ \item $Dec_{\mathcal A}$:\enspace selects $\hat\tau = \arg\max\limits_{\color{red}\tau} \kappa(\tau_i \circ X'', {\color{red}\tau} \circ X')$\\ whereby \highlight{$\kappa(X'', X') \coloneqq \sum_c p(c\mid X'') p(c\mid X')$} \end{itemize} \emph{Asymptotic Equipartition Property (AEP):} \textit{AEP fulfilled}\enspace if\enspace \highlight{$\log \kappa(X',X'') \overset{n\to\infty}{\to} \mathcal E$}\\ \enspace whereby $\mathcal E \coloneqq \E[X',X'']{\log\kappa(X',X'')}$ \begin{itemize} \item $A^{(n)}_\epsilon$: set of $(\epsilon,n)$-typical pairs $X', X''$\\\qquad $\abs{ \log\kappa(X',X'') - \mathcal E } < \epsilon$ \item \highlight{$p\ped{err} \leq P_{(n)} \color{gray} \text{ c.f. scr. (6.19)}$} $\overset{n\to\infty}{\to} 0$ \enspace if\enspace \highlight{$\frac{\log m}{\log\abs{\mathcal C}} < I$}\\ where $I \coloneqq \frac{1}{\log\abs{\mathcal C}} \E[X',X'']{\log(\abs{\mathcal C} \kappa(X',X''))}$ \end{itemize} % === \subsection{Applications of PA} \emph{PA:}\enskip \textit{quantifies the amount of information that algorithms extract from phenomena.} $\to$ quantified by \textbf{capacity} (max. \# distinguishable messages that can be communicated) \emph{Temperature:}\enspace $T^\ast = \arg\max_T \kappa(X',X'')$ \emph{Cost functions:}\enspace Given $R_1(\cdot,\cdot),\ldots,R_s(\cdot,\cdot)$\\ $\max\limits_{\ell\leq s} \kappa_\ell(X',X'') = \max\limits_{\ell\leq s} \frac{1}{Z_{X'} Z_{X''}} \sum\limits_c \eu^{-\frac1T R_\ell(c,X')} \eu^{-\frac1T R_\ell(c,X'')}$ \emph{Algorithms:}\enspace %\textit{here for MST (min. span. tree)} Many MST (min. spanning tree) algo's are \textbf{contractive} ($\to$ sequence of candidate sol's). \\ \textbf{Approximation Set Coding (ASC):}\\ \quad $p\ap{ASC}(c\mid X') = \begin{cases} 1 / \abs{G_\gamma(X')} & \text{if}\enspace c\in G_\gamma(X') \\[-2pt] 0 & \text{otw.} \end{cases}$\\ \quad $G_\gamma(X') \coloneqq \brace{ c\in\mathcal C : R(c,X') - \min\limits_{c\in\mathcal C} R(c,X') \leq \gamma }$ \begin{enumerate} \item Run $\mathcal A$ to compute $G_t^{\mathcal A}(X')$ and $G_t^{\mathcal A}(X'')$,\enspace for all $t$ \vspace{2pt} \item $t^\ast = \arg\max\limits_t \kappa(X',X'') = \arg\max\limits_t \frac{\abs{G_t^{\mathcal A}(X') \:\cap\: G_t^{\mathcal A}(X'')}}{\abs{G_t^{\mathcal A}(X')} \:\cdot\: \abs{G_t^{\mathcal A}(X'')}}$ \item $c^\ast \xleftarrow{\$\:\text{sample}} \mathrm{Unif}\paren*{G_{t^\ast}^{\mathcal A}(X') \cap G_{t^\ast}^{\mathcal A}(X'')}$ \end{enumerate} % ===
Require Export List. Require Export Bool. Require Export Arith. Require Export Peano_dec. Require Export Coq.Arith.PeanoNat. Require Import CpdtTactics. Require Export Coq.Program.Wf. Require Export Coq.Program.Tactics. Require Export Coq.Logic.FunctionalExtensionality. Require Export Recdef. Require Import wyv_common. Set Implicit Arguments. (* Here we model the case where an unseparated type on the left hand side, and a pure material is on the the right hand side. Here we know that following the syntax and the subtype semantics of Decidable Wyvern, every subsequent subtype check will feature a Material on the right hand side. We also know that if the left hand side is a shape, the only way to proceed would require a shape refinement on the right hand side. As we have already noted, this is not possible. Thus we can easily demonstrate termination as without shapes, cycles are not possible on the right hand side, and the algorithm is thus bound by the finiteness of specified types. ) Inductive tytree : Type := \| t_top : tytree \| t_bot : tytree \| t_sel_upp : var -> label -> tytree -> tytree \| t_sel_low : var -> label -> tytree -> tytree \| t_sel_equ : var -> label -> tytree -> tytree \| t_sel_nom : var -> label -> tytree -> tytree \| t_sel_bnd : var -> label -> tytree \| t_rfn_top : tytree -> tytree (first tytree is actually a list of trees constructed by t_nil and t_con) \| t_rfn_sel : var -> label -> tytree -> tytree -> tytree (first tytree is actually a list of trees constructed by t_nil and t_con) \| t_sha_top : var -> label -> decls -> tytree \| t_sha_sel : var -> label -> decls -> tytree -> tytree \| t_all : tytree -> tytree -> tytree \| t_upp : label -> tytree -> tytree \| t_low : label -> tytree -> tytree \| t_equ : label -> tytree -> tytree \| t_nom : label -> tytree -> tytree \| t_nil : tytree \| t_con : tytree -> tytree -> tytree. Inductive Tree (A: Set): Set := \| Leaf: A -> Tree A \| Node: list (Tree A) -> Tree A. Fixpoint bind' {A B:Set} (k:A -> Tree B) (m:Tree A) : Tree B := match m with \| Leaf a => k a \| Node ts => Node (map (bind' k) ts) end. Definition bind (A B:Set) m (k:A -> Tree B) := bind' k m. Fixpoint shape_depth (T : tytree) : nat := match T with \| t_sel_upp x L T' => 1 + shape_depth T' \| t_sel_low x L T' => 1 + shape_depth T' \| t_sel_equ x L T' => 1 + shape_depth T' \| t_sel_nom x L T' => 1 + shape_depth T' \| t_rfn_top Ts => 1 + shape_depth Ts \| t_rfn_sel _ _ Ts T' => 1 + shape_depth Ts + shape_depth T' \| t_sha_sel _ _ _ T' => 1 + shape_depth T' \| t_all T1 T2 => 1 + shape_depth T1 + shape_depth T2 \| t_upp _ T' => 1 + shape_depth T' \| t_low _ T' => 1 + shape_depth T' \| t_equ _ T' => 1 + shape_depth T' \| t_nom _ T' => 1 + shape_depth T' \| t_con T Ts => 1 + shape_depth T + shape_depth Ts \| _ => 1 end. Definition shape_depth_p (P : tytree tytree) := let (T1, T2) := P in shape_depth T1 + shape_depth T2. Function zip (Ts1 Ts2 : list tytree) : option (list (tytree * tytree)) := match Ts1, Ts2 with \| _, nil => Some nil \| T1::Ts1', T2::Ts2' => match zip Ts1' Ts2' with \| None => None \| Some Ps => Some ((T1, T2)::Ps) end \| nil, _::_ => None end. Check fold_right andb true (map negb nil). Compute fold_right andb true (map negb nil). Fixpoint fold_mapb {A B : Type} (f : A * B -> bool) (Ps : list (A * B)) : bool := match Ps with \| nil => true \| P::Ps' => andb (f P) (fold_mapb f Ps') end. Definition zip_mapb {A B : Type} (f : A * B -> bool) (o : option (list (A * B))) : bool := match o with \| None => false \| Some Ps => fold_mapb f Ps end. (Definition map {A B} (xs : list A) (f : forall (x:A), In x xs -> B) : list B. Proof. induction xs. exact nil. refine (f a _ :: IHxs _). - left. reflexivity. - intros. eapply f. right. eassumption. Defined.) (Fixpoint map {A B} (xs : list A) : forall (f : forall (x:A), In x xs -> B), list B := match xs with \| nil => fun _ => nil \| x :: xs => fun f => f x (or_introl eq_refl) :: map xs (fun y h => f y (or_intror h)) end.) (forall P0 : tytree tytree, shape_depth P0 < shape_depth P -> bool) Check in_cons. Definition map_tree (P1 : tytree tytree) (f : forall P2 : tytree * tytree, shape_depth_p P2 < shape_depth_p P1 -> bool) (Ps : list (tytree * tytree)) (lt : forall P2 : tytree * tytree, In P2 Ps -> shape_depth_p P2 < shape_depth_p P1) : list bool. Proof. induction Ps. exact nil. assert (HIna : In a (a :: Ps)); [apply in_eq\|apply lt in HIna]. refine (f a HIna :: IHPs _). intros. apply lt, in_cons; auto. Defined. (Fixpoint map (P1 : tytree tytree) (f : forall P2 : tytree * tytree, shape_depth P2 < shape_depth P1 -> bool) (Ps : list (tytree * tytree)) (lt : forall P2 : tytree * tytree, In P2 Ps -> shape_depth P2 < shape_depth P1) : list bool := match Ps with \| nil => nil \| P::Ps' => (f P lt)::(map P1 f Ps' lt) end.*) Fixpoint eq_ty (t1 t2 : ty): bool := match t1, t2 with \| top, top => true \| bot, bot => true \| sel x1 L1 , sel x2 L2 => eq_var x1 x2 && eq_label L1 L2 \| t1 str ss1 rts, t2 str ss2 rts => eq_ty t1 t2 && eq_decls ss1 ss2 \| all t1 ∙ t1', all t2 ∙ t2' => eq_ty t1 t2 && eq_ty t1' t2' \| _, _ => false end with eq_decl (s1 s2 : decl) : bool := match s1, s2 with \| type L1 ⩽ t1, type L2 ⩽ t2 => eq_label L1 L2 && eq_ty t1 t2 \| type L1 ⩾ t1, type L2 ⩾ t2 => eq_label L1 L2 && eq_ty t1 t2 \| type L1 ≝ t1, type L2 ≝ t2 => eq_label L1 L2 && eq_ty t1 t2 \| type L1 ⪯ t1, type L2 ⪯ t2 => eq_label L1 L2 && eq_ty t1 t2 \| _, _ => false end with eq_decls (ss1 ss2 : decls) : bool := match ss1, ss2 with \| d_nil, d_nil => true \| d_con s1 ss1, d_con s2 ss2 => eq_decl s1 s2 && eq_decls ss1 ss2 \| _, _ => false end. Fixpoint eq_tree (T1 T2 : tytree) : bool := match T1, T2 with \| t_top, t_top => true \| t_bot, t_bot => true \| t_sel_upp x1 L1 T1', t_sel_upp x2 L2 T2' => (eq_var x1 x2) && (eq_label L1 L2) && (eq_tree T1' T2') \| t_sel_low x1 L1 T1', t_sel_low x2 L2 T2' => (eq_var x1 x2) && (eq_label L1 L2) && (eq_tree T1' T2') \| t_sel_equ x1 L1 T1', t_sel_equ x2 L2 T2' => (eq_var x1 x2) && (eq_label L1 L2) && (eq_tree T1' T2') \| t_sel_nom x1 L1 T1', t_sel_nom x2 L2 T2' => (eq_var x1 x2) && (eq_label L1 L2) && (eq_tree T1' T2') \| t_all Ta1 Tb1, t_all Ta2 Tb2 => (eq_tree Ta1 Ta2) && (eq_tree Tb1 Tb2) \| t_rfn_top Ts1, t_rfn_top Ts2 => (eq_tree Ts1 Ts2) \| t_rfn_sel x1 L1 Ts1 T1', t_rfn_sel x2 L2 Ts2 T2' => (eq_var x1 x2) && (eq_label L1 L2) && (eq_tree Ts1 Ts2) && (eq_tree T1' T2') \| t_sha_top x1 L1 ss1, t_sha_top x2 L2 ss2 => (eq_var x1 x2) && (eq_label L1 L2) && (eq_decls ss1 ss2) \| t_sha_sel x1 L1 ss1 T1', t_sha_sel x2 L2 ss2 T2' => (eq_var x1 x2) && (eq_label L1 L2) && (eq_decls ss1 ss2) && (eq_tree T1' T2') \| t_upp L1 T1', t_upp L2 T2' => (eq_label L1 L2) && (eq_tree T1' T2') \| t_low L1 T1', t_low L2 T2' => (eq_label L1 L2) && (eq_tree T1' T2') \| t_equ L1 T1', t_equ L2 T2' => (eq_label L1 L2) && (eq_tree T1' T2') \| t_nom L1 T1', t_nom L2 T2' => (eq_label L1 L2) && (eq_tree T1' T2') \| t_nil, t_nil => true \| t_con T1' Ts1, t_con T2' Ts2 => (eq_tree T1' T2') && (eq_tree Ts1 Ts2) \| _, _ => false end. Parameter eq_tree_refl : forall T, eq_tree T T = true. Parameter eqb_tree_eq : forall T1 T2, eq_tree T1 T2 = true -> T1 = T2. Parameter neqb_tree_neq : forall T1 T2, eq_tree T1 T2 = false -> T1 <> T2. Parameter eq_tree_dec : forall T1 T2, {eq_tree T1 T2 = true} + {eq_tree T1 T2 = false}. Program Fixpoint subtype (T1 T2 : tytree) {measure (shape_depth T1 + shape_depth T2)} : bool := match T1 with \| t_top => match T2 with \| t_top => true \| t_sel_low x L T2' => subtype T1 T2' \| t_sel_equ x L T2' => subtype T1 T2' \| _ => false end \| t_bot => match T2 with \| t_upp _ _ => false \| t_low _ _ => false \| t_equ _ _ => false \| t_nom _ _ => false \| t_nil => false \| t_con _ _ => false \| _ => true end \| t_sel_upp x1 L1 T1' => match T2 with \| t_top => true \| t_sel_low x2 L2 T2' => orb (andb (eq_var x1 x2) (eq_label L1 L2)) (orb (subtype T1' T2) (subtype T1 T2')) \| t_sel_equ x2 L2 T2' => orb (andb (eq_var x1 x2) (eq_label L1 L2)) (orb (subtype T1' T2) (subtype T1 T2')) \| t_sel_upp x2 L2 _ => orb (andb (eq_var x1 x2) (eq_label L1 L2)) (subtype T1' T2) \| t_sel_nom x2 L2 _ => orb (andb (eq_var x1 x2) (eq_label L1 L2)) (subtype T1' T2) \| t_sel_bnd x2 L2 => orb (andb (eq_var x1 x2) (eq_label L1 L2)) (subtype T1' T2) \| t_upp L2 T2' => false \| t_low L2 T2' => false \| t_equ L2 T2' => false \| t_nom L2 T2' => false \| t_nil => false \| t_con _ _ => false \| _ => subtype T1' T2 end \| t_sel_equ x1 L1 T1' => match T2 with \| t_top => true \| t_sel_low x2 L2 T2' => orb (andb (eq_var x1 x2) (eq_label L1 L2)) (orb (subtype T1' T2) (subtype T1 T2')) \| t_sel_equ x2 L2 T2' => orb (andb (eq_var x1 x2) (eq_label L1 L2)) (orb (subtype T1' T2) (subtype T1 T2')) \| t_sel_upp x2 L2 _ => orb (andb (eq_var x1 x2) (eq_label L1 L2)) (subtype T1' T2) \| t_sel_nom x2 L2 _ => orb (andb (eq_var x1 x2) (eq_label L1 L2)) (subtype T1' T2) \| t_sel_bnd x2 L2 => orb (andb (eq_var x1 x2) (eq_label L1 L2)) (subtype T1' T2) \| t_upp L2 T2' => false \| t_low L2 T2' => false \| t_equ L2 T2' => false \| t_nom L2 T2' => false \| t_nil => false \| t_con _ _ => false \| _ => subtype T1' T2 end \| t_sel_nom x1 L1 T1' => match T2 with \| t_top => true \| t_sel_low x2 L2 T2' => orb (andb (eq_var x1 x2) (eq_label L1 L2)) (orb (subtype T1' T2) (subtype T1 T2')) \| t_sel_equ x2 L2 T2' => orb (andb (eq_var x1 x2) (eq_label L1 L2)) (orb (subtype T1' T2) (subtype T1 T2')) \| t_sel_upp x2 L2 _ => orb (andb (eq_var x1 x2) (eq_label L1 L2)) (subtype T1' T2) \| t_sel_nom x2 L2 _ => orb (andb (eq_var x1 x2) (eq_label L1 L2)) (subtype T1' T2) \| t_sel_bnd x2 L2 => orb (andb (eq_var x1 x2) (eq_label L1 L2)) (subtype T1' T2) \| t_upp L2 T2' => false \| t_low L2 T2' => false \| t_equ L2 T2' => false \| t_nom L2 T2' => false \| t_nil => false \| t_con _ _ => false \| _ => subtype T1' T2 end \| t_sel_low x1 L1 T1' => match T2 with \| t_top => true \| t_sel_low x2 L2 T2' => orb (andb (eq_var x1 x2) (eq_label L1 L2)) (subtype T1 T2') \| t_sel_equ x2 L2 T2' => orb (andb (eq_var x1 x2) (eq_label L1 L2)) (subtype T1 T2') \| t_sel_upp x2 L2 _ => andb (eq_var x1 x2) (eq_label L1 L2) \| t_sel_nom x2 L2 _ => andb (eq_var x1 x2) (eq_label L1 L2) \| t_sel_bnd x2 L2 => andb (eq_var x1 x2) (eq_label L1 L2) \| _ => false end \| t_sel_bnd x1 L1 => match T2 with \| t_top => true \| t_sel_low x2 L2 T2' => orb (andb (eq_var x1 x2) (eq_label L1 L2)) (subtype T1 T2') \| t_sel_equ x2 L2 T2' => orb (andb (eq_var x1 x2) (eq_label L1 L2)) (subtype T1 T2') \| t_sel_upp x2 L2 _ => andb (eq_var x1 x2) (eq_label L1 L2) \| t_sel_nom x2 L2 _ => andb (eq_var x1 x2) (eq_label L1 L2) \| t_sel_bnd x2 L2 => andb (eq_var x1 x2) (eq_label L1 L2) \| _ => false end \| t_rfn_top Ts1 => match T2 with \| t_top => true \| t_sel_low x L T => subtype T1 T \| t_sel_equ x L T => subtype T1 T \| t_rfn_top Ts2 => subtype Ts1 Ts2 \| _ => false end \| t_rfn_sel x1 L1 Ts1 T' => match T2 with \| t_top => true \| t_sel_low x L T => orb (subtype T1 T) (subtype T' T2) \| t_sel_equ x L T => orb (subtype T1 T) (subtype T' T2) \| t_rfn_top _ => subtype T' T2 \| t_rfn_sel x2 L2 Ts2 _ => orb (andb ((andb (eq_var x1 x2) (eq_label L1 L2))) (subtype Ts1 Ts2)) (subtype T' T2) \| t_upp _ _ => false \| t_low _ _ => false \| t_equ _ _ => false \| t_nom _ _ => false \| t_nil => false \| t_con _ _ => false \| _ => subtype T' T2 end \| t_sha_top _ _ _ => match T2 with \| t_top => true \| t_sel_low x L T => subtype T1 T \| t_sel_equ x L T => subtype T1 T \| _ => false end \| t_sha_sel _ _ _ T' => match T2 with \| t_top => true \| t_sel_low x L T => orb (subtype T1 T) (subtype T' T2) \| t_sel_equ x L T => orb (subtype T1 T) (subtype T' T2) \| t_upp _ _ => false \| t_low _ _ => false \| t_equ _ _ => false \| t_nom _ _ => false \| t_nil => false \| t_con _ _ => false \| _ => subtype T' T2 end \| t_all Ta1 Tb1 => match T2 with \| t_top => true \| t_sel_low x L T => subtype T1 T \| t_sel_equ x L T => subtype T1 T \| t_all Ta2 Tb2 => andb (eq_tree Ta2 Ta1) (subtype Tb1 Tb2) \| _ => false end \| t_upp L1 T1' => match T2 with \| t_upp L2 T2' => andb (eq_label L1 L2) (subtype T1' T2') \| _ => false end \| t_low L1 T1' => match T2 with \| t_low L2 T2' => andb (eq_label L1 L2) (subtype T2' T1') \| _ => false end \| t_equ L1 T1' => match T2 with \| t_upp L2 T2' => andb (eq_label L1 L2) (subtype T1' T2') \| t_low L2 T2' => andb (eq_label L1 L2) (subtype T2' T1') \| t_equ L2 T2' => andb (eq_label L1 L2) (andb (subtype T1' T2') (subtype T2' T1')) \| _ => false end \| t_nom L1 T1' => match T2 with \| t_upp L2 T2' => andb (eq_label L1 L2) (subtype T1' T2') \| t_nom L2 T2' => (eq_label L1 L2) && eq_tree T1' T2' \| _ => false end \| t_nil => match T2 with \| t_nil => true \| _ => false end \| t_con T1' Ts1 => match T2 with \| t_nil => true \| t_con T2' Ts2 => andb (subtype T1' T2') (subtype Ts1 Ts2) \| _ => false end end. 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Import WfExtensionality. Lemma subtype_top : forall T, (forall L T', T <> t_upp L T') -> (forall L T', T <> t_low L T') -> (forall L T', T <> t_equ L T') -> (forall L T', T <> t_nom L T') -> (T <> t_nil) -> (forall T' Ts, T <> t_con T' Ts) -> subtype T t_top = true. Proof. intros. unfold subtype, subtype_func; simpl; rewrite fix_sub_eq_ext; simpl; fold subtype_func; auto. destruct T; simpl; auto. contradiction (H l T); auto. contradiction (H0 l T); auto. contradiction (H1 l T); auto. contradiction (H2 l T); auto. contradiction (H3); auto. contradiction (H4 T1 T2); auto. Qed. Lemma subtype_bot : forall T, (forall L T', T <> t_upp L T') -> (forall L T', T <> t_low L T') -> (forall L T', T <> t_equ L T') -> (forall L T', T <> t_nom L T') -> (T <> t_nil) -> (forall T' Ts, T <> t_con T' Ts) -> subtype t_bot T = true. Proof. intros. unfold subtype, subtype_func; simpl; rewrite fix_sub_eq_ext; simpl; fold subtype_func; auto. destruct T; simpl; auto. contradiction (H l T); auto. contradiction (H0 l T); auto. contradiction (H1 l T); auto. contradiction (H2 l T); auto. contradiction (H3); auto. contradiction (H4 T1 T2); auto. Qed.
function write_doc(docformat::Pandoc2HTML, doc, rendered, out_path) _, weave_source = splitdir(abspath(doc.source)) weave_version, weave_date = weave_info() # Header is inserted from displayed plots header_script = doc.header_script self_contained = (header_script ≠ "") ? [] : "--self-contained" if haskey(doc.header, "bibliography") filt = "--filter" citeproc = "pandoc-citeproc" else filt = [] citeproc = [] end out_path = get_out_path(doc, out_path, "html") cd_back = let d = pwd(); () -> cd(d); end cd(dirname(out_path)) try out = basename(out_path) highlight_stylesheet = get_highlight_stylesheet(MIME("text/html"), docformat.highlight_theme) cmd = `pandoc -f markdown+raw_html -s --mathjax="" $filt $citeproc $(docformat.pandoc_options) --template $(docformat.template_path) -H $(docformat.stylesheet_path) $(self_contained) -V highlight_stylesheet=$(highlight_stylesheet) -V weave_version=$(weave_version) -V weave_date=$(weave_date) -V weave_source=$(weave_source) -V headerscript=$(header_script) -o $(out)` proc = open(cmd, "r+") println(proc.in, rendered) close(proc.in) proc_output = read(proc.out, String) catch rethrow() # TODO: just show error content instead of rethrow the err finally cd_back() end return out_path end function write_doc(docformat::Pandoc2PDF, doc, rendered, out_path) if haskey(doc.header, "bibliography") filt = "--filter" citeproc = "pandoc-citeproc" else filt = [] citeproc = [] end out_path = get_out_path(doc, out_path, "pdf") cd_back = let d = pwd(); () -> cd(d); end cd(dirname(out_path)) try out = basename(out_path) cmd = `pandoc -f markdown+raw_tex -s --pdf-engine=xelatex --highlight-style=tango $filt $citeproc $(docformat.pandoc_options) --include-in-header=$(docformat.header_template) -V fontsize=12pt -o $(out)` proc = open(cmd, "r+") println(proc.in, rendered) close(proc.in) proc_output = read(proc.out, String) catch rethrow() finally cd_back() end return out_path end
Formal statement is: lemma sigma_sets_superset_generator: "A \<subseteq> sigma_sets X A" Informal statement is: The sigma-algebra generated by a set $A$ is a superset of $A$.
/********************************************************\ Original Author: Richard Bateman and Georg Fritzsche Created: December 3, 2009 License: Dual license model; choose one of two: New BSD License http://www.opensource.org/licenses/bsd-license.php - or - GNU Lesser General Public License, version 2.1 http://www.gnu.org/licenses/lgpl-2.1.html Copyright 2009 PacketPass Inc, Georg Fritzsche, Firebreath development team \*******************************************************/ #include "JSObject.h" #include <boost/assign.hpp> #include "DOM/Document.h" #include "variant_list.h" using boost::assign::list_of; #include "SimpleMathAPI.h" SimpleMathAPI::SimpleMathAPI(const FB::BrowserHostPtr& host) : m_host(host) { registerMethod("add", make_method(this, &SimpleMathAPI::add)); registerMethod("sub", make_method(this, &SimpleMathAPI::sub)); registerMethod("mult", make_method(this, &SimpleMathAPI::mult)); registerMethod("div", make_method(this, &SimpleMathAPI::div)); registerProperty("getArray", make_property(this, &SimpleMathAPI::getArray)); registerProperty("self", make_property(this, &SimpleMathAPI::get_self)); registerMethod("GetSelf", make_method(this, &SimpleMathAPI::GetSelf)); } SimpleMathAPI::~SimpleMathAPI() { } // add Method FB::VariantList SimpleMathAPI::getArray() { std::vector<std::string> ret; ret.push_back("aa"); ret.push_back("bb"); ret.push_back("cc"); ret.push_back("dd"); return FB::make_variant_list(ret); } long SimpleMathAPI::add(long a, long b) { return a+b; } long SimpleMathAPI::sub(long a, long b) { return a-b; } long SimpleMathAPI::mult(long a, long b) { return ab; } long SimpleMathAPI::div(long a, long b) { return a/b; } const FB::JSAPIPtr SimpleMathAPI::get_self() { return shared_from_this(); } const FB::JSAPIPtr SimpleMathAPI::GetSelf() { return shared_from_this(); }
: platform s" sflinux" ; include afkit/ans/ffl/sflinux/ffl.f \ FFL: DOM; FFL loads FPMATH include afkit/dep/allegro5/allegro-5.2.x.f library libX11.so function: XOpenDisplay ( zdisplayname - display ) function: XDefaultScreen ( &display - display ) function: XSync ( display discard - ) function: XMapWindow ( display window - ) function: XRaiseWindow ( display window - ) function: XSetInputFocus ( display window revert time - ) function: XGetInputFocus ( display &window &revert - ) include afkit/plat/sf.f

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