url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
|---|---|---|---|---|---|---|---|---|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/DropRight.lean | List.rtake_rtake | [19, 1] | [20, 95] | rw [rtake_eq_reverse_take_reverse, ← take_reverse, take_take, rtake_eq_reverse_take_reverse] | α : Type u_1
l✝ l' l₀ l₁ l₂ : List α
a b : α
m✝ n✝ n m : ℕ
l : List α
⊢ (l.rtake m).rtake n = l.rtake (min n m) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/DropRight.lean | List.take_prefix_take | [37, 1] | [38, 91] | rw [prefix_iff_eq_take, length_take, take_take, min_right_comm, min_eq_left h, take_min] | α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
h : m ≤ n
⊢ take m l <+: take n l | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/DropRight.lean | List.IsPrefix.take | [48, 1] | [49, 44] | rw [hl.eq_take] | α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
hl : l <+: l'
h : l.length ≤ n
⊢ l <+: take n l' | α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
hl : l <+: l'
h : l.length ≤ n
⊢ take l.length l' <+: take n l' |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/DropRight.lean | List.IsPrefix.take | [48, 1] | [49, 44] | exact take_prefix_take h | α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
hl : l <+: l'
h : l.length ≤ n
⊢ take l.length l' <+: take n l' | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/DropRight.lean | List.IsSuffix.rtake | [51, 1] | [52, 47] | rw [hl.eq_rtake] | α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
hl : l <:+ l'
h : l.length ≤ n
⊢ l <:+ l'.rtake n | α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
hl : l <:+ l'
h : l.length ≤ n
⊢ l'.rtake l.length <:+ l'.rtake n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/DropRight.lean | List.IsSuffix.rtake | [51, 1] | [52, 47] | exact rtake_suffix_rtake h | α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
hl : l <:+ l'
h : l.length ≤ n
⊢ l'.rtake l.length <:+ l'.rtake n | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/DropRight.lean | List.Sublist.exists_intermediate | [71, 1] | [84, 23] | induction' l₀ with a l₀ ih generalizing n l₂ | α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
hl : l₀ <+ l₂
h₀ : l₀.length ≤ n
h₂ : n ≤ l₂.length
⊢ ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n | case nil
α : Type u_1
l l' l₁ : List α
a b : α
m : ℕ
l₂ : List α
n : ℕ
hl : [] <+ l₂
h₀ : [].length ≤ n
h₂ : n ≤ l₂.length
⊢ ∃ l₁, [] <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n
case cons
α : Type u_1
l l' l₁ : List α
a✝ b : α
m : ℕ
a : α
l₀ : List α
ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n
l₂ : List α
n : ℕ
hl : a :: l₀ <+ l₂
h₀ : (a :: l₀).length ≤ n
h₂ : n ≤ l₂.length
⊢ ∃ l₁, a :: l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/DropRight.lean | List.Sublist.exists_intermediate | [71, 1] | [84, 23] | cases n | case cons
α : Type u_1
l l' l₁ : List α
a✝ b : α
m : ℕ
a : α
l₀ : List α
ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n
l₂ : List α
n : ℕ
hl : a :: l₀ <+ l₂
h₀ : (a :: l₀).length ≤ n
h₂ : n ≤ l₂.length
⊢ ∃ l₁, a :: l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n | case cons.zero
α : Type u_1
l l' l₁ : List α
a✝ b : α
m : ℕ
a : α
l₀ : List α
ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n
l₂ : List α
hl : a :: l₀ <+ l₂
h₀ : (a :: l₀).length ≤ 0
h₂ : 0 ≤ l₂.length
⊢ ∃ l₁, a :: l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = 0
case cons.succ
α : Type u_1
l l' l₁ : List α
a✝ b : α
m : ℕ
a : α
l₀ : List α
ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n
l₂ : List α
hl : a :: l₀ <+ l₂
n✝ : ℕ
h₀ : (a :: l₀).length ≤ n✝ + 1
h₂ : n✝ + 1 ≤ l₂.length
⊢ ∃ l₁, a :: l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n✝ + 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/DropRight.lean | List.Sublist.exists_intermediate | [71, 1] | [84, 23] | cases' l₂ with b l₂ | case cons.succ
α : Type u_1
l l' l₁ : List α
a✝ b : α
m : ℕ
a : α
l₀ : List α
ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n
l₂ : List α
hl : a :: l₀ <+ l₂
n✝ : ℕ
h₀ : (a :: l₀).length ≤ n✝ + 1
h₂ : n✝ + 1 ≤ l₂.length
⊢ ∃ l₁, a :: l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n✝ + 1 | case cons.succ.nil
α : Type u_1
l l' l₁ : List α
a✝ b : α
m : ℕ
a : α
l₀ : List α
ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n
n✝ : ℕ
h₀ : (a :: l₀).length ≤ n✝ + 1
hl : a :: l₀ <+ []
h₂ : n✝ + 1 ≤ [].length
⊢ ∃ l₁, a :: l₀ <+ l₁ ∧ l₁ <+ [] ∧ l₁.length = n✝ + 1
case cons.succ.cons
α : Type u_1
l l' l₁ : List α
a✝ b✝ : α
m : ℕ
a : α
l₀ : List α
ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n
n✝ : ℕ
h₀ : (a :: l₀).length ≤ n✝ + 1
b : α
l₂ : List α
hl : a :: l₀ <+ b :: l₂
h₂ : n✝ + 1 ≤ (b :: l₂).length
⊢ ∃ l₁, a :: l₀ <+ l₁ ∧ l₁ <+ b :: l₂ ∧ l₁.length = n✝ + 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/DropRight.lean | List.Sublist.exists_intermediate | [71, 1] | [84, 23] | obtain ⟨l₁, h₀₁, h₁₂, rfl⟩ := ih sorry (Nat.succ_le_succ_iff.1 h₀) (Nat.le_of_succ_le h₂) | case cons.succ.cons
α : Type u_1
l l' l₁ : List α
a✝ b✝ : α
m : ℕ
a : α
l₀ : List α
ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n
n✝ : ℕ
h₀ : (a :: l₀).length ≤ n✝ + 1
b : α
l₂ : List α
hl : a :: l₀ <+ b :: l₂
h₂ : n✝ + 1 ≤ (b :: l₂).length
⊢ ∃ l₁, a :: l₀ <+ l₁ ∧ l₁ <+ b :: l₂ ∧ l₁.length = n✝ + 1 | case cons.succ.cons.intro.intro.intro
α : Type u_1
l l' l₁✝ : List α
a✝ b✝ : α
m : ℕ
a : α
l₀ : List α
ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n
b : α
l₂ : List α
hl : a :: l₀ <+ b :: l₂
l₁ : List α
h₀₁ : l₀ <+ l₁
h₁₂ : l₁ <+ b :: l₂
h₀ : (a :: l₀).length ≤ l₁.length + 1
h₂ : l₁.length + 1 ≤ (b :: l₂).length
⊢ ∃ l₁_1, a :: l₀ <+ l₁_1 ∧ l₁_1 <+ b :: l₂ ∧ l₁_1.length = l₁.length + 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/DropRight.lean | List.Sublist.exists_intermediate | [71, 1] | [84, 23] | rw [cons_sublist_cons_iff'] at hl | case cons.succ.cons.intro.intro.intro
α : Type u_1
l l' l₁✝ : List α
a✝ b✝ : α
m : ℕ
a : α
l₀ : List α
ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n
b : α
l₂ : List α
hl : a :: l₀ <+ b :: l₂
l₁ : List α
h₀₁ : l₀ <+ l₁
h₁₂ : l₁ <+ b :: l₂
h₀ : (a :: l₀).length ≤ l₁.length + 1
h₂ : l₁.length + 1 ≤ (b :: l₂).length
⊢ ∃ l₁_1, a :: l₀ <+ l₁_1 ∧ l₁_1 <+ b :: l₂ ∧ l₁_1.length = l₁.length + 1 | case cons.succ.cons.intro.intro.intro
α : Type u_1
l l' l₁✝ : List α
a✝ b✝ : α
m : ℕ
a : α
l₀ : List α
ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n
b : α
l₂ : List α
hl : a :: l₀ <+ l₂ ∨ a = b ∧ l₀ <+ l₂
l₁ : List α
h₀₁ : l₀ <+ l₁
h₁₂ : l₁ <+ b :: l₂
h₀ : (a :: l₀).length ≤ l₁.length + 1
h₂ : l₁.length + 1 ≤ (b :: l₂).length
⊢ ∃ l₁_1, a :: l₀ <+ l₁_1 ∧ l₁_1 <+ b :: l₂ ∧ l₁_1.length = l₁.length + 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/DropRight.lean | List.Sublist.exists_intermediate | [71, 1] | [84, 23] | obtain hl | ⟨rfl, hl⟩ := hl | case cons.succ.cons.intro.intro.intro
α : Type u_1
l l' l₁✝ : List α
a✝ b✝ : α
m : ℕ
a : α
l₀ : List α
ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n
b : α
l₂ : List α
hl : a :: l₀ <+ l₂ ∨ a = b ∧ l₀ <+ l₂
l₁ : List α
h₀₁ : l₀ <+ l₁
h₁₂ : l₁ <+ b :: l₂
h₀ : (a :: l₀).length ≤ l₁.length + 1
h₂ : l₁.length + 1 ≤ (b :: l₂).length
⊢ ∃ l₁_1, a :: l₀ <+ l₁_1 ∧ l₁_1 <+ b :: l₂ ∧ l₁_1.length = l₁.length + 1 | case cons.succ.cons.intro.intro.intro.inl
α : Type u_1
l l' l₁✝ : List α
a✝ b✝ : α
m : ℕ
a : α
l₀ : List α
ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n
b : α
l₂ l₁ : List α
h₀₁ : l₀ <+ l₁
h₁₂ : l₁ <+ b :: l₂
h₀ : (a :: l₀).length ≤ l₁.length + 1
h₂ : l₁.length + 1 ≤ (b :: l₂).length
hl : a :: l₀ <+ l₂
⊢ ∃ l₁_1, a :: l₀ <+ l₁_1 ∧ l₁_1 <+ b :: l₂ ∧ l₁_1.length = l₁.length + 1
case cons.succ.cons.intro.intro.intro.inr.intro
α : Type u_1
l l' l₁✝ : List α
a✝ b : α
m : ℕ
a : α
l₀ : List α
ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n
l₂ l₁ : List α
h₀₁ : l₀ <+ l₁
h₀ : (a :: l₀).length ≤ l₁.length + 1
hl : l₀ <+ l₂
h₁₂ : l₁ <+ a :: l₂
h₂ : l₁.length + 1 ≤ (a :: l₂).length
⊢ ∃ l₁_1, a :: l₀ <+ l₁_1 ∧ l₁_1 <+ a :: l₂ ∧ l₁_1.length = l₁.length + 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/DropRight.lean | List.Sublist.exists_intermediate | [71, 1] | [84, 23] | exact (exists_sublist_length_eq h₂).imp (fun l₁ h ↦ ⟨nil_sublist _, h⟩) | case nil
α : Type u_1
l l' l₁ : List α
a b : α
m : ℕ
l₂ : List α
n : ℕ
hl : [] <+ l₂
h₀ : [].length ≤ n
h₂ : n ≤ l₂.length
⊢ ∃ l₁, [] <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/DropRight.lean | List.Sublist.exists_intermediate | [71, 1] | [84, 23] | cases h₀ | case cons.zero
α : Type u_1
l l' l₁ : List α
a✝ b : α
m : ℕ
a : α
l₀ : List α
ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n
l₂ : List α
hl : a :: l₀ <+ l₂
h₀ : (a :: l₀).length ≤ 0
h₂ : 0 ≤ l₂.length
⊢ ∃ l₁, a :: l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/DropRight.lean | List.Sublist.exists_intermediate | [71, 1] | [84, 23] | cases h₂ | case cons.succ.nil
α : Type u_1
l l' l₁ : List α
a✝ b : α
m : ℕ
a : α
l₀ : List α
ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n
n✝ : ℕ
h₀ : (a :: l₀).length ≤ n✝ + 1
hl : a :: l₀ <+ []
h₂ : n✝ + 1 ≤ [].length
⊢ ∃ l₁, a :: l₀ <+ l₁ ∧ l₁ <+ [] ∧ l₁.length = n✝ + 1 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/DropRight.lean | List.Subperm.exists_intermediate | [86, 1] | [90, 45] | obtain ⟨l₀', hl₀, hl'⟩ := hl | α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
hl : l₀ <+~ l₂
h₀ : l₀.length ≤ n
h₂ : n ≤ l₂.length
⊢ ∃ l₁, l₀ <+~ l₁ ∧ l₁ <+~ l₂ ∧ l₁.length = n | case intro.intro
α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
h₀ : l₀.length ≤ n
h₂ : n ≤ l₂.length
l₀' : List α
hl₀ : l₀' ~ l₀
hl' : l₀' <+ l₂
⊢ ∃ l₁, l₀ <+~ l₁ ∧ l₁ <+~ l₂ ∧ l₁.length = n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/DropRight.lean | List.Subperm.exists_intermediate | [86, 1] | [90, 45] | obtain ⟨l₁, h₀₁, h₁₂, hn⟩ := hl'.exists_intermediate (hl₀.length_eq.trans_le h₀) h₂ | case intro.intro
α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
h₀ : l₀.length ≤ n
h₂ : n ≤ l₂.length
l₀' : List α
hl₀ : l₀' ~ l₀
hl' : l₀' <+ l₂
⊢ ∃ l₁, l₀ <+~ l₁ ∧ l₁ <+~ l₂ ∧ l₁.length = n | case intro.intro.intro.intro.intro
α : Type u_1
l l' l₀ l₁✝ l₂ : List α
a b : α
m n : ℕ
h₀ : l₀.length ≤ n
h₂ : n ≤ l₂.length
l₀' : List α
hl₀ : l₀' ~ l₀
hl' : l₀' <+ l₂
l₁ : List α
h₀₁ : l₀' <+ l₁
h₁₂ : l₁ <+ l₂
hn : l₁.length = n
⊢ ∃ l₁, l₀ <+~ l₁ ∧ l₁ <+~ l₂ ∧ l₁.length = n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/DropRight.lean | List.Subperm.exists_intermediate | [86, 1] | [90, 45] | exact ⟨l₁, ⟨_, hl₀, h₀₁⟩, h₁₂.subperm, hn⟩ | case intro.intro.intro.intro.intro
α : Type u_1
l l' l₀ l₁✝ l₂ : List α
a b : α
m n : ℕ
h₀ : l₀.length ≤ n
h₂ : n ≤ l₂.length
l₀' : List α
hl₀ : l₀' ~ l₀
hl' : l₀' <+ l₂
l₁ : List α
h₀₁ : l₀' <+ l₁
h₁₂ : l₁ <+ l₂
hn : l₁.length = n
⊢ ∃ l₁, l₀ <+~ l₁ ∧ l₁ <+~ l₂ ∧ l₁.length = n | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/DropRight.lean | List.exists_subperm_length_eq | [92, 1] | [93, 65] | simpa using nil_subperm.exists_intermediate (Nat.zero_le _) hn | α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
hn : n ≤ l.length
⊢ ∃ l', l' <+~ l ∧ l'.length = n | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_single | [25, 1] | [26, 81] | simp [discConv] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w : R →₀ S
P : R[X]
r : R
s : S
⊢ discConv (fun₀ | r => s) P = s • P.comp (X + C r) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_zero | [28, 1] | [28, 82] | simp [discConv] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P : R[X]
w : R →₀ S
⊢ discConv w 0 = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_add | [30, 1] | [31, 75] | simp [discConv] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
w : R →₀ S
P Q : R[X]
⊢ discConv w (P + Q) = discConv w P + discConv w Q | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_sub | [33, 1] | [34, 85] | simp [discConv, smul_sub] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
w : R →₀ S
P Q : R[X]
⊢ discConv w (P - Q) = discConv w P - discConv w Q | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_neg | [36, 1] | [37, 18] | simp [discConv] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
w : R →₀ S
P : R[X]
⊢ discConv w (-P) = -discConv w P | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_sum | [39, 1] | [41, 64] | simp [discConv, smul_sum, Finsupp.sum] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
w : R →₀ S
s : Finset ι
P : ι → R[X]
⊢ discConv w (∑ i ∈ s, P i) = ∑ i ∈ s, discConv w (P i) | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
w : R →₀ S
s : Finset ι
P : ι → R[X]
⊢ ∑ x ∈ w.support, ∑ x_1 ∈ s, w x • (P x_1).comp (X + C x) = ∑ x ∈ s, ∑ x_1 ∈ w.support, w x_1 • (P x).comp (X + C x_1) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_sum | [39, 1] | [41, 64] | exact Finset.sum_comm | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
w : R →₀ S
s : Finset ι
P : ι → R[X]
⊢ ∑ x ∈ w.support, ∑ x_1 ∈ s, w x • (P x_1).comp (X + C x) = ∑ x ∈ s, ∑ x_1 ∈ w.support, w x_1 • (P x).comp (X + C x_1) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_zero_weight | [43, 1] | [43, 98] | simp [discConv] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w : R →₀ S
P✝ P : R[X]
⊢ discConv 0 P = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_add_weight | [45, 1] | [47, 56] | simp | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
v w : R →₀ S
P : R[X]
⊢ ∀ (a : R), 0 • P.comp (X + C a) = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_add_weight | [45, 1] | [47, 56] | simp [add_smul] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
v w : R →₀ S
P : R[X]
⊢ ∀ (a : R) (b₁ b₂ : S), (b₁ + b₂) • P.comp (X + C a) = b₁ • P.comp (X + C a) + b₂ • P.comp (X + C a) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_sub_weight | [49, 1] | [50, 99] | simp [sub_smul] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
v w : R →₀ S
P : R[X]
⊢ ∀ (a : R) (b₁ b₂ : S), (b₁ - b₂) • P.comp (X + C a) = b₁ • P.comp (X + C a) - b₂ • P.comp (X + C a) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_neg_weight | [52, 1] | [53, 63] | simp | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
w : R →₀ S
P : R[X]
⊢ ∀ (a : R), 0 • P.comp (X + C a) = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_neg_weight | [52, 1] | [53, 63] | simp [discConv] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
w : R →₀ S
P : R[X]
⊢ (w.sum fun a b => -b • P.comp (X + C a)) = -discConv w P | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_sum_weight | [55, 1] | [57, 56] | simp | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
s : Finset ι
w : ι → R →₀ S
P : R[X]
⊢ ∀ (i : R), 0 • P.comp (X + C i) = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_sum_weight | [55, 1] | [57, 56] | simp [add_smul] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
s : Finset ι
w : ι → R →₀ S
P : R[X]
⊢ ∀ (i : R) (x y : S), (x + y) • P.comp (X + C i) = x • P.comp (X + C i) + y • P.comp (X + C i) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.eval_discConv | [59, 1] | [61, 47] | simp [discConv, Finsupp.sum,eval_finset_sum] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
w : R →₀ S
P : R[X]
x : R
⊢ eval x (discConv w P) = w.sum fun r s => s • eval (x + r) P | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.eval_discForwardDiff | [63, 1] | [65, 72] | simp_rw [discForwardDiff, discConv_sub_weight, discConv_single] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w : R →₀ S
P✝ P : R[X]
x : R
⊢ eval x P.discForwardDiff = eval (x + 1) P - eval x P | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w : R →₀ S
P✝ P : R[X]
x : R
⊢ eval x (1 • P.comp (X + C 1) - 1 • P.comp (X + C 0)) = eval (x + 1) P - eval x P |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.eval_discForwardDiff | [63, 1] | [65, 72] | simp | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w : R →₀ S
P✝ P : R[X]
x : R
⊢ eval x (1 • P.comp (X + C 1) - 1 • P.comp (X + C 0)) = eval (x + 1) P - eval x P | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.eval_discBackwardDiff | [67, 1] | [69, 90] | simp_rw [discBackwardDiff, discConv_sub_weight, discConv_single] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w : R →₀ S
P✝ P : R[X]
x : R
⊢ eval x P.discBackwardDiff = eval x P - eval (x - 1) P | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w : R →₀ S
P✝ P : R[X]
x : R
⊢ eval x (1 • P.comp (X + C 0) - 1 • P.comp (X + C (-1))) = eval x P - eval (x - 1) P |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.eval_discBackwardDiff | [67, 1] | [69, 90] | simp [sub_eq_add_neg] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w : R →₀ S
P✝ P : R[X]
x : R
⊢ eval x (1 • P.comp (X + C 0) - 1 • P.comp (X + C (-1))) = eval x P - eval (x - 1) P | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_discConv | [71, 1] | [76, 8] | simp [discConv] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
v w : AddMonoidAlgebra S R
P : R[X]
⊢ discConv v (discConv w P) = discConv (v * w) P | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
v w : AddMonoidAlgebra S R
P : R[X]
⊢ (Finsupp.sum v fun r s => s • (Finsupp.sum w fun r s => s • P.comp (X + C r)).comp (X + C r)) =
Finsupp.sum (v * w) fun r s => s • P.comp (X + C r) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_discConv | [71, 1] | [76, 8] | sorry | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
v w : AddMonoidAlgebra S R
P : R[X]
⊢ (Finsupp.sum v fun r s => s • (Finsupp.sum w fun r s => s • P.comp (X + C r)).comp (X + C r)) =
Finsupp.sum (v * w) fun r s => s • P.comp (X + C r) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.coeff_discConv_natDegree | [78, 1] | [80, 8] | sorry | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
w : R →₀ S
P : R[X]
⊢ (discConv w P).coeff P.natDegree = (w.sum fun x => id) • P.leadingCoeff | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discForwardDiff_aux | [84, 1] | [86, 62] | classical simp [Finsupp.sum_sub_index, -Finsupp.single_neg] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra S R
w : R →₀ S
P : R[X]
inst✝ : Nontrivial S
⊢ (((fun₀ | 1 => 1) - fun₀ | 0 => 1).sum fun x => id) = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discForwardDiff_aux | [84, 1] | [86, 62] | simp [Finsupp.sum_sub_index, -Finsupp.single_neg] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra S R
w : R →₀ S
P : R[X]
inst✝ : Nontrivial S
⊢ (((fun₀ | 1 => 1) - fun₀ | 0 => 1).sum fun x => id) = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discBackwardDiff_aux | [88, 1] | [90, 62] | classical simp [Finsupp.sum_sub_index, -Finsupp.single_neg] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra S R
w : R →₀ S
P : R[X]
inst✝ : Nontrivial S
⊢ (((fun₀ | 0 => 1) - fun₀ | -1 => 1).sum fun x => id) = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discBackwardDiff_aux | [88, 1] | [90, 62] | simp [Finsupp.sum_sub_index, -Finsupp.single_neg] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra S R
w : R →₀ S
P : R[X]
inst✝ : Nontrivial S
⊢ (((fun₀ | 0 => 1) - fun₀ | -1 => 1).sum fun x => id) = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.natDegree_discConv_le | [94, 1] | [99, 24] | refine' (natDegree_sum_le _ _).trans $ Finset.sup_le fun r _ ↦ _ | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
⊢ (discConv w P).natDegree ≤ P.natDegree | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
r : R
x✝ : r ∈ w.support
⊢ (natDegree ∘ fun a => (fun r s => s • P.comp (X + C r)) a (w a)) r ≤ P.natDegree |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.natDegree_discConv_le | [94, 1] | [99, 24] | dsimp | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
r : R
x✝ : r ∈ w.support
⊢ (natDegree ∘ fun a => (fun r s => s • P.comp (X + C r)) a (w a)) r ≤ P.natDegree | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
r : R
x✝ : r ∈ w.support
⊢ (w r • P.comp (X + C r)).natDegree ≤ P.natDegree |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.natDegree_discConv_le | [94, 1] | [99, 24] | simp_rw [algebra_compatible_smul R (w r)] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
r : R
x✝ : r ∈ w.support
⊢ (w r • P.comp (X + C r)).natDegree ≤ P.natDegree | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
r : R
x✝ : r ∈ w.support
⊢ ((algebraMap S R) (w r) • P.comp (X + C r)).natDegree ≤ P.natDegree |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.natDegree_discConv_le | [94, 1] | [99, 24] | refine (natDegree_smul_le _ _).trans ?_ | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
r : R
x✝ : r ∈ w.support
⊢ ((algebraMap S R) (w r) • P.comp (X + C r)).natDegree ≤ P.natDegree | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
r : R
x✝ : r ∈ w.support
⊢ (P.comp (X + C r)).natDegree ≤ P.natDegree |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.natDegree_discConv_le | [94, 1] | [99, 24] | simp [natDegree_comp] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
r : R
x✝ : r ∈ w.support
⊢ (P.comp (X + C r)).natDegree ≤ P.natDegree | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.degree_discConv_le | [101, 1] | [105, 30] | obtain rfl | hP := eq_or_ne P 0 | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
⊢ (discConv w P).degree ≤ P.degree | case inl
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
⊢ (discConv w 0).degree ≤ degree 0
case inr
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hP : P ≠ 0
⊢ (discConv w P).degree ≤ P.degree |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.degree_discConv_le | [101, 1] | [105, 30] | refine degree_le_natDegree.trans $ (Nat.cast_le.2 natDegree_discConv_le).trans ?_ | case inr
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hP : P ≠ 0
⊢ (discConv w P).degree ≤ P.degree | case inr
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hP : P ≠ 0
⊢ ↑P.natDegree ≤ P.degree |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.degree_discConv_le | [101, 1] | [105, 30] | rw [degree_eq_natDegree hP] | case inr
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hP : P ≠ 0
⊢ ↑P.natDegree ≤ P.degree | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.degree_discConv_le | [101, 1] | [105, 30] | simp | case inl
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
⊢ (discConv w 0).degree ≤ degree 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.degree_discConv_lt | [108, 1] | [115, 8] | have := coeff_discConv_natDegree w P | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hw : (w.sum fun x => id) = 0
hP : P ≠ 0
⊢ (discConv w P).degree < P.degree | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hw : (w.sum fun x => id) = 0
hP : P ≠ 0
this : (discConv w P).coeff P.natDegree = (w.sum fun x => id) • P.leadingCoeff
⊢ (discConv w P).degree < P.degree |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.degree_discConv_lt | [108, 1] | [115, 8] | rw [hw, zero_smul] at this | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hw : (w.sum fun x => id) = 0
hP : P ≠ 0
this : (discConv w P).coeff P.natDegree = (w.sum fun x => id) • P.leadingCoeff
⊢ (discConv w P).degree < P.degree | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hw : (w.sum fun x => id) = 0
hP : P ≠ 0
this : (discConv w P).coeff P.natDegree = 0
⊢ (discConv w P).degree < P.degree |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.degree_discConv_lt | [108, 1] | [115, 8] | refine' (degree_sum_le _ _).trans_lt $ (Finset.sup_lt_iff _).2 _ | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hw : (w.sum fun x => id) = 0
hP : P ≠ 0
this : (discConv w P).coeff P.natDegree = 0
⊢ (discConv w P).degree < P.degree | case refine'_1
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hw : (w.sum fun x => id) = 0
hP : P ≠ 0
this : (discConv w P).coeff P.natDegree = 0
⊢ ⊥ < P.degree
case refine'_2
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hw : (w.sum fun x => id) = 0
hP : P ≠ 0
this : (discConv w P).coeff P.natDegree = 0
⊢ ∀ b ∈ w.support, ((fun r s => s • P.comp (X + C r)) b (w b)).degree < P.degree |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.degree_discConv_lt | [108, 1] | [115, 8] | sorry | case refine'_1
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hw : (w.sum fun x => id) = 0
hP : P ≠ 0
this : (discConv w P).coeff P.natDegree = 0
⊢ ⊥ < P.degree
case refine'_2
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hw : (w.sum fun x => id) = 0
hP : P ≠ 0
this : (discConv w P).coeff P.natDegree = 0
⊢ ∀ b ∈ w.support, ((fun r s => s • P.comp (X + C r)) b (w b)).degree < P.degree | case refine'_2
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hw : (w.sum fun x => id) = 0
hP : P ≠ 0
this : (discConv w P).coeff P.natDegree = 0
⊢ ∀ b ∈ w.support, ((fun r s => s • P.comp (X + C r)) b (w b)).degree < P.degree |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.degree_discConv_lt | [108, 1] | [115, 8] | sorry | case refine'_2
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hw : (w.sum fun x => id) = 0
hP : P ≠ 0
this : (discConv w P).coeff P.natDegree = 0
⊢ ∀ b ∈ w.support, ((fun r s => s • P.comp (X + C r)) b (w b)).degree < P.degree | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/VanDenBergKesten.lean | Finset.mem_certificator | [45, 1] | [50, 37] | rw [certificator, mem_filter, and_iff_right_of_imp] | α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a : α
⊢ a ∈ s □ t ↔ ∃ x y, IsCompl x y ∧ (∀ ⦃b : α⦄, a ⊓ x = b ⊓ x → b ∈ s) ∧ ∀ ⦃b : α⦄, a ⊓ y = b ⊓ y → b ∈ t | α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a : α
⊢ (∃ x y, IsCompl x y ∧ (∀ ⦃b : α⦄, a ⊓ x = b ⊓ x → b ∈ s) ∧ ∀ ⦃b : α⦄, a ⊓ y = b ⊓ y → b ∈ t) → a ∈ s ∩ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/VanDenBergKesten.lean | Finset.mem_certificator | [45, 1] | [50, 37] | rintro ⟨u, v, _, hu, hv⟩ | α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a : α
⊢ (∃ x y, IsCompl x y ∧ (∀ ⦃b : α⦄, a ⊓ x = b ⊓ x → b ∈ s) ∧ ∀ ⦃b : α⦄, a ⊓ y = b ⊓ y → b ∈ t) → a ∈ s ∩ t | case intro.intro.intro.intro
α : Type u_1
inst✝ : BooleanAlgebra α
s t u✝ : Finset α
a u v : α
left✝ : IsCompl u v
hu : ∀ ⦃b : α⦄, a ⊓ u = b ⊓ u → b ∈ s
hv : ∀ ⦃b : α⦄, a ⊓ v = b ⊓ v → b ∈ t
⊢ a ∈ s ∩ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/VanDenBergKesten.lean | Finset.mem_certificator | [45, 1] | [50, 37] | exact mem_inter.2 ⟨hu rfl, hv rfl⟩ | case intro.intro.intro.intro
α : Type u_1
inst✝ : BooleanAlgebra α
s t u✝ : Finset α
a u v : α
left✝ : IsCompl u v
hu : ∀ ⦃b : α⦄, a ⊓ u = b ⊓ u → b ∈ s
hv : ∀ ⦃b : α⦄, a ⊓ v = b ⊓ v → b ∈ t
⊢ a ∈ s ∩ t | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/VanDenBergKesten.lean | Finset.certificator_subset_disjSups | [54, 1] | [59, 57] | simp_rw [subset_iff, mem_certificator, mem_disjSups] | α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a : α
⊢ s □ t ⊆ s ○ t | α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a : α
⊢ ∀ ⦃x : α⦄,
(∃ x_1 y, IsCompl x_1 y ∧ (∀ ⦃b : α⦄, x ⊓ x_1 = b ⊓ x_1 → b ∈ s) ∧ ∀ ⦃b : α⦄, x ⊓ y = b ⊓ y → b ∈ t) →
∃ a ∈ s, ∃ b ∈ t, Disjoint a b ∧ a ⊔ b = x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/VanDenBergKesten.lean | Finset.certificator_subset_disjSups | [54, 1] | [59, 57] | rintro x ⟨u, v, huv, hu, hv⟩ | α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a : α
⊢ ∀ ⦃x : α⦄,
(∃ x_1 y, IsCompl x_1 y ∧ (∀ ⦃b : α⦄, x ⊓ x_1 = b ⊓ x_1 → b ∈ s) ∧ ∀ ⦃b : α⦄, x ⊓ y = b ⊓ y → b ∈ t) →
∃ a ∈ s, ∃ b ∈ t, Disjoint a b ∧ a ⊔ b = x | case intro.intro.intro.intro
α : Type u_1
inst✝ : BooleanAlgebra α
s t u✝ : Finset α
a x u v : α
huv : IsCompl u v
hu : ∀ ⦃b : α⦄, x ⊓ u = b ⊓ u → b ∈ s
hv : ∀ ⦃b : α⦄, x ⊓ v = b ⊓ v → b ∈ t
⊢ ∃ a ∈ s, ∃ b ∈ t, Disjoint a b ∧ a ⊔ b = x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/VanDenBergKesten.lean | Finset.certificator_subset_disjSups | [54, 1] | [59, 57] | refine' ⟨x ⊓ u, hu (inf_right_idem _ _).symm, x ⊓ v, hv (inf_right_idem _ _).symm,
huv.disjoint.mono inf_le_right inf_le_right, _⟩ | case intro.intro.intro.intro
α : Type u_1
inst✝ : BooleanAlgebra α
s t u✝ : Finset α
a x u v : α
huv : IsCompl u v
hu : ∀ ⦃b : α⦄, x ⊓ u = b ⊓ u → b ∈ s
hv : ∀ ⦃b : α⦄, x ⊓ v = b ⊓ v → b ∈ t
⊢ ∃ a ∈ s, ∃ b ∈ t, Disjoint a b ∧ a ⊔ b = x | case intro.intro.intro.intro
α : Type u_1
inst✝ : BooleanAlgebra α
s t u✝ : Finset α
a x u v : α
huv : IsCompl u v
hu : ∀ ⦃b : α⦄, x ⊓ u = b ⊓ u → b ∈ s
hv : ∀ ⦃b : α⦄, x ⊓ v = b ⊓ v → b ∈ t
⊢ x ⊓ u ⊔ x ⊓ v = x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/VanDenBergKesten.lean | Finset.certificator_subset_disjSups | [54, 1] | [59, 57] | rw [← inf_sup_left, huv.codisjoint.eq_top, inf_top_eq] | case intro.intro.intro.intro
α : Type u_1
inst✝ : BooleanAlgebra α
s t u✝ : Finset α
a x u v : α
huv : IsCompl u v
hu : ∀ ⦃b : α⦄, x ⊓ u = b ⊓ u → b ∈ s
hv : ∀ ⦃b : α⦄, x ⊓ v = b ⊓ v → b ∈ t
⊢ x ⊓ u ⊔ x ⊓ v = x | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/VanDenBergKesten.lean | Finset.certificator_comm | [63, 1] | [64, 75] | ext s | α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a : α
⊢ s □ t = t □ s | case a
α : Type u_1
inst✝ : BooleanAlgebra α
s✝ t u : Finset α
a s : α
⊢ s ∈ s✝ □ t ↔ s ∈ t □ s✝ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/VanDenBergKesten.lean | Finset.certificator_comm | [63, 1] | [64, 75] | rw [mem_certificator, exists_comm] | case a
α : Type u_1
inst✝ : BooleanAlgebra α
s✝ t u : Finset α
a s : α
⊢ s ∈ s✝ □ t ↔ s ∈ t □ s✝ | case a
α : Type u_1
inst✝ : BooleanAlgebra α
s✝ t u : Finset α
a s : α
⊢ (∃ b a, IsCompl a b ∧ (∀ ⦃b : α⦄, s ⊓ a = b ⊓ a → b ∈ s✝) ∧ ∀ ⦃b_1 : α⦄, s ⊓ b = b_1 ⊓ b → b_1 ∈ t) ↔ s ∈ t □ s✝ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/VanDenBergKesten.lean | Finset.certificator_comm | [63, 1] | [64, 75] | simp [isCompl_comm, and_comm] | case a
α : Type u_1
inst✝ : BooleanAlgebra α
s✝ t u : Finset α
a s : α
⊢ (∃ b a, IsCompl a b ∧ (∀ ⦃b : α⦄, s ⊓ a = b ⊓ a → b ∈ s✝) ∧ ∀ ⦃b_1 : α⦄, s ⊓ b = b_1 ⊓ b → b_1 ∈ t) ↔ s ∈ t □ s✝ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/VanDenBergKesten.lean | Finset.IsUpperSet.certificator_eq_inter | [66, 1] | [72, 59] | refine'
certificator_subset_inter.antisymm fun a ha ↦ mem_certificator.2 ⟨a, aᶜ, isCompl_compl, _⟩ | α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a : α
hs : IsUpperSet ↑s
ht : IsLowerSet ↑t
⊢ s □ t = s ∩ t | α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a✝ : α
hs : IsUpperSet ↑s
ht : IsLowerSet ↑t
a : α
ha : a ∈ s ∩ t
⊢ (∀ ⦃b : α⦄, a ⊓ a = b ⊓ a → b ∈ s) ∧ ∀ ⦃b : α⦄, a ⊓ aᶜ = b ⊓ aᶜ → b ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/VanDenBergKesten.lean | Finset.IsUpperSet.certificator_eq_inter | [66, 1] | [72, 59] | rw [mem_inter] at ha | α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a✝ : α
hs : IsUpperSet ↑s
ht : IsLowerSet ↑t
a : α
ha : a ∈ s ∩ t
⊢ (∀ ⦃b : α⦄, a ⊓ a = b ⊓ a → b ∈ s) ∧ ∀ ⦃b : α⦄, a ⊓ aᶜ = b ⊓ aᶜ → b ∈ t | α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a✝ : α
hs : IsUpperSet ↑s
ht : IsLowerSet ↑t
a : α
ha : a ∈ s ∧ a ∈ t
⊢ (∀ ⦃b : α⦄, a ⊓ a = b ⊓ a → b ∈ s) ∧ ∀ ⦃b : α⦄, a ⊓ aᶜ = b ⊓ aᶜ → b ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/VanDenBergKesten.lean | Finset.IsUpperSet.certificator_eq_inter | [66, 1] | [72, 59] | simp only [@eq_comm _ ⊥, ← sdiff_eq, inf_idem, right_eq_inf, _root_.sdiff_self, sdiff_eq_bot_iff] | α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a✝ : α
hs : IsUpperSet ↑s
ht : IsLowerSet ↑t
a : α
ha : a ∈ s ∧ a ∈ t
⊢ (∀ ⦃b : α⦄, a ⊓ a = b ⊓ a → b ∈ s) ∧ ∀ ⦃b : α⦄, a ⊓ aᶜ = b ⊓ aᶜ → b ∈ t | α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a✝ : α
hs : IsUpperSet ↑s
ht : IsLowerSet ↑t
a : α
ha : a ∈ s ∧ a ∈ t
⊢ (∀ ⦃b : α⦄, a ≤ b → b ∈ s) ∧ ∀ ⦃b : α⦄, b ≤ a → b ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/VanDenBergKesten.lean | Finset.IsUpperSet.certificator_eq_inter | [66, 1] | [72, 59] | exact ⟨fun b hab ↦ hs hab ha.1, fun b hab ↦ ht hab ha.2⟩ | α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a✝ : α
hs : IsUpperSet ↑s
ht : IsLowerSet ↑t
a : α
ha : a ∈ s ∧ a ∈ t
⊢ (∀ ⦃b : α⦄, a ≤ b → b ∈ s) ∧ ∀ ⦃b : α⦄, b ≤ a → b ∈ t | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/VanDenBergKesten.lean | Finset.IsLowerSet.certificator_eq_inter | [74, 1] | [80, 59] | refine' certificator_subset_inter.antisymm fun a ha ↦
mem_certificator.2 ⟨aᶜ, a, isCompl_compl.symm, _⟩ | α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a : α
hs : IsLowerSet ↑s
ht : IsUpperSet ↑t
⊢ s □ t = s ∩ t | α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a✝ : α
hs : IsLowerSet ↑s
ht : IsUpperSet ↑t
a : α
ha : a ∈ s ∩ t
⊢ (∀ ⦃b : α⦄, a ⊓ aᶜ = b ⊓ aᶜ → b ∈ s) ∧ ∀ ⦃b : α⦄, a ⊓ a = b ⊓ a → b ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/VanDenBergKesten.lean | Finset.IsLowerSet.certificator_eq_inter | [74, 1] | [80, 59] | rw [mem_inter] at ha | α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a✝ : α
hs : IsLowerSet ↑s
ht : IsUpperSet ↑t
a : α
ha : a ∈ s ∩ t
⊢ (∀ ⦃b : α⦄, a ⊓ aᶜ = b ⊓ aᶜ → b ∈ s) ∧ ∀ ⦃b : α⦄, a ⊓ a = b ⊓ a → b ∈ t | α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a✝ : α
hs : IsLowerSet ↑s
ht : IsUpperSet ↑t
a : α
ha : a ∈ s ∧ a ∈ t
⊢ (∀ ⦃b : α⦄, a ⊓ aᶜ = b ⊓ aᶜ → b ∈ s) ∧ ∀ ⦃b : α⦄, a ⊓ a = b ⊓ a → b ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/VanDenBergKesten.lean | Finset.IsLowerSet.certificator_eq_inter | [74, 1] | [80, 59] | simp only [@eq_comm _ ⊥, ← sdiff_eq, inf_idem, right_eq_inf, _root_.sdiff_self, sdiff_eq_bot_iff] | α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a✝ : α
hs : IsLowerSet ↑s
ht : IsUpperSet ↑t
a : α
ha : a ∈ s ∧ a ∈ t
⊢ (∀ ⦃b : α⦄, a ⊓ aᶜ = b ⊓ aᶜ → b ∈ s) ∧ ∀ ⦃b : α⦄, a ⊓ a = b ⊓ a → b ∈ t | α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a✝ : α
hs : IsLowerSet ↑s
ht : IsUpperSet ↑t
a : α
ha : a ∈ s ∧ a ∈ t
⊢ (∀ ⦃b : α⦄, b ≤ a → b ∈ s) ∧ ∀ ⦃b : α⦄, a ≤ b → b ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/VanDenBergKesten.lean | Finset.IsLowerSet.certificator_eq_inter | [74, 1] | [80, 59] | exact ⟨fun b hab ↦ hs hab ha.1, fun b hab ↦ ht hab ha.2⟩ | α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a✝ : α
hs : IsLowerSet ↑s
ht : IsUpperSet ↑t
a : α
ha : a ∈ s ∧ a ∈ t
⊢ (∀ ⦃b : α⦄, b ≤ a → b ∈ s) ∧ ∀ ⦃b : α⦄, a ≤ b → b ∈ t | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/VanDenBergKesten.lean | Finset.IsUpperSet.certificator_eq_disjSups | [82, 1] | [88, 75] | refine' certificator_subset_disjSups.antisymm fun a ha ↦ mem_certificator.2 _ | α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a : α
hs : IsUpperSet ↑s
ht : IsUpperSet ↑t
⊢ s □ t = s ○ t | α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a✝ : α
hs : IsUpperSet ↑s
ht : IsUpperSet ↑t
a : α
ha : a ∈ s ○ t
⊢ ∃ x y, IsCompl x y ∧ (∀ ⦃b : α⦄, a ⊓ x = b ⊓ x → b ∈ s) ∧ ∀ ⦃b : α⦄, a ⊓ y = b ⊓ y → b ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/VanDenBergKesten.lean | Finset.IsUpperSet.certificator_eq_disjSups | [82, 1] | [88, 75] | obtain ⟨x, hx, y, hy, hxy, rfl⟩ := mem_disjSups.1 ha | α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a✝ : α
hs : IsUpperSet ↑s
ht : IsUpperSet ↑t
a : α
ha : a ∈ s ○ t
⊢ ∃ x y, IsCompl x y ∧ (∀ ⦃b : α⦄, a ⊓ x = b ⊓ x → b ∈ s) ∧ ∀ ⦃b : α⦄, a ⊓ y = b ⊓ y → b ∈ t | case intro.intro.intro.intro.intro
α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a : α
hs : IsUpperSet ↑s
ht : IsUpperSet ↑t
x : α
hx : x ∈ s
y : α
hy : y ∈ t
hxy : Disjoint x y
ha : x ⊔ y ∈ s ○ t
⊢ ∃ x_1 y_1, IsCompl x_1 y_1 ∧ (∀ ⦃b : α⦄, (x ⊔ y) ⊓ x_1 = b ⊓ x_1 → b ∈ s) ∧ ∀ ⦃b : α⦄, (x ⊔ y) ⊓ y_1 = b ⊓ y_1 → b ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/VanDenBergKesten.lean | Finset.IsUpperSet.certificator_eq_disjSups | [82, 1] | [88, 75] | refine' ⟨x, xᶜ, isCompl_compl, _⟩ | case intro.intro.intro.intro.intro
α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a : α
hs : IsUpperSet ↑s
ht : IsUpperSet ↑t
x : α
hx : x ∈ s
y : α
hy : y ∈ t
hxy : Disjoint x y
ha : x ⊔ y ∈ s ○ t
⊢ ∃ x_1 y_1, IsCompl x_1 y_1 ∧ (∀ ⦃b : α⦄, (x ⊔ y) ⊓ x_1 = b ⊓ x_1 → b ∈ s) ∧ ∀ ⦃b : α⦄, (x ⊔ y) ⊓ y_1 = b ⊓ y_1 → b ∈ t | case intro.intro.intro.intro.intro
α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a : α
hs : IsUpperSet ↑s
ht : IsUpperSet ↑t
x : α
hx : x ∈ s
y : α
hy : y ∈ t
hxy : Disjoint x y
ha : x ⊔ y ∈ s ○ t
⊢ (∀ ⦃b : α⦄, (x ⊔ y) ⊓ x = b ⊓ x → b ∈ s) ∧ ∀ ⦃b : α⦄, (x ⊔ y) ⊓ xᶜ = b ⊓ xᶜ → b ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/VanDenBergKesten.lean | Finset.IsUpperSet.certificator_eq_disjSups | [82, 1] | [88, 75] | simp only [inf_of_le_right, le_sup_left, right_eq_inf, ← sdiff_eq, hxy.sup_sdiff_cancel_left] | case intro.intro.intro.intro.intro
α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a : α
hs : IsUpperSet ↑s
ht : IsUpperSet ↑t
x : α
hx : x ∈ s
y : α
hy : y ∈ t
hxy : Disjoint x y
ha : x ⊔ y ∈ s ○ t
⊢ (∀ ⦃b : α⦄, (x ⊔ y) ⊓ x = b ⊓ x → b ∈ s) ∧ ∀ ⦃b : α⦄, (x ⊔ y) ⊓ xᶜ = b ⊓ xᶜ → b ∈ t | case intro.intro.intro.intro.intro
α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a : α
hs : IsUpperSet ↑s
ht : IsUpperSet ↑t
x : α
hx : x ∈ s
y : α
hy : y ∈ t
hxy : Disjoint x y
ha : x ⊔ y ∈ s ○ t
⊢ (∀ ⦃b : α⦄, x ≤ b → b ∈ s) ∧ ∀ ⦃b : α⦄, y = b \ x → b ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/VanDenBergKesten.lean | Finset.IsUpperSet.certificator_eq_disjSups | [82, 1] | [88, 75] | exact ⟨fun b hab ↦ hs hab hx, fun b hab ↦ ht (hab.trans_le sdiff_le) hy⟩ | case intro.intro.intro.intro.intro
α : Type u_1
inst✝ : BooleanAlgebra α
s t u : Finset α
a : α
hs : IsUpperSet ↑s
ht : IsUpperSet ↑t
x : α
hx : x ∈ s
y : α
hy : y ∈ t
hxy : Disjoint x y
ha : x ⊔ y ∈ s ○ t
⊢ (∀ ⦃b : α⦄, x ≤ b → b ∈ s) ∧ ∀ ⦃b : α⦄, y = b \ x → b ∈ t | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | SBtw.sbtw.symm | [27, 1] | [28, 95] | simpa [dist_comm, add_comm] using d | V : Type u_1
inst✝ : MetricSpace V
u v w : V
huv : u ≠ v
huw : u ≠ w
hvw : v ≠ w
d : Dist.dist u v + Dist.dist v w = Dist.dist u w
⊢ Dist.dist w v + Dist.dist v u = Dist.dist w u | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | sbtw_iff_of_ne | [32, 1] | [34, 48] | simp [MetricSpace.sbtw_iff, h12, h13, h23] | V : Type u_1
inst✝ : MetricSpace V
u v w : V
h12 : u ≠ v
h13 : u ≠ w
h23 : v ≠ w
⊢ sbtw u v w ↔ dist u v + dist v w = dist u w | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | sbtw_mk | [36, 1] | [42, 14] | refine ⟨h12, ?_, h23, h.antisymm (dist_triangle _ _ _)⟩ | V : Type u_1
inst✝ : MetricSpace V
u v w : V
h12 : u ≠ v
h23 : v ≠ w
h : dist u v + dist v w ≤ dist u w
⊢ sbtw u v w | V : Type u_1
inst✝ : MetricSpace V
u v w : V
h12 : u ≠ v
h23 : v ≠ w
h : dist u v + dist v w ≤ dist u w
⊢ u ≠ w |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | sbtw_mk | [36, 1] | [42, 14] | rintro rfl | V : Type u_1
inst✝ : MetricSpace V
u v w : V
h12 : u ≠ v
h23 : v ≠ w
h : dist u v + dist v w ≤ dist u w
⊢ u ≠ w | V : Type u_1
inst✝ : MetricSpace V
u v : V
h12 : u ≠ v
h23 : v ≠ u
h : dist u v + dist v u ≤ dist u u
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | sbtw_mk | [36, 1] | [42, 14] | rw [dist_self] at h | V : Type u_1
inst✝ : MetricSpace V
u v : V
h12 : u ≠ v
h23 : v ≠ u
h : dist u v + dist v u ≤ dist u u
⊢ False | V : Type u_1
inst✝ : MetricSpace V
u v : V
h12 : u ≠ v
h23 : v ≠ u
h : dist u v + dist v u ≤ 0
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | sbtw_mk | [36, 1] | [42, 14] | replace h : dist v u ≤ 0 := by linarith [dist_comm v u] | V : Type u_1
inst✝ : MetricSpace V
u v : V
h12 : u ≠ v
h23 : v ≠ u
h : dist u v + dist v u ≤ 0
⊢ False | V : Type u_1
inst✝ : MetricSpace V
u v : V
h12 : u ≠ v
h23 : v ≠ u
h : dist v u ≤ 0
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | sbtw_mk | [36, 1] | [42, 14] | simp only [dist_le_zero] at h | V : Type u_1
inst✝ : MetricSpace V
u v : V
h12 : u ≠ v
h23 : v ≠ u
h : dist v u ≤ 0
⊢ False | V : Type u_1
inst✝ : MetricSpace V
u v : V
h12 : u ≠ v
h23 : v ≠ u
h : v = u
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | sbtw_mk | [36, 1] | [42, 14] | exact h23 h | V : Type u_1
inst✝ : MetricSpace V
u v : V
h12 : u ≠ v
h23 : v ≠ u
h : v = u
⊢ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | sbtw_mk | [36, 1] | [42, 14] | linarith [dist_comm v u] | V : Type u_1
inst✝ : MetricSpace V
u v : V
h12 : u ≠ v
h23 : v ≠ u
h : dist u v + dist v u ≤ 0
⊢ dist v u ≤ 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | SBtw.sbtw.right_cancel | [44, 1] | [45, 99] | linarith [h.dist, h'.dist, dist_triangle u w x, dist_triangle u v w] | V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w✝ u v w x : V
h : sbtw u v x
h' : sbtw v w x
⊢ Dist.dist u v + Dist.dist v w ≤ Dist.dist u w | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | SBtw.sbtw.asymm_right | [47, 1] | [51, 24] | have := h'.dist | V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w u v x : V
h : sbtw u v x
h' : sbtw v u x
⊢ False | V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w u v x : V
h : sbtw u v x
h' : sbtw v u x
this : Dist.dist v u + Dist.dist u x = Dist.dist v x
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | SBtw.sbtw.asymm_right | [47, 1] | [51, 24] | rw [dist_comm] at this | V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w u v x : V
h : sbtw u v x
h' : sbtw v u x
this : Dist.dist v u + Dist.dist u x = Dist.dist v x
⊢ False | V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w u v x : V
h : sbtw u v x
h' : sbtw v u x
this : Dist.dist u v + Dist.dist u x = Dist.dist v x
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | SBtw.sbtw.asymm_right | [47, 1] | [51, 24] | have : Dist.dist u v = 0 := by linarith [h.dist] | V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w u v x : V
h : sbtw u v x
h' : sbtw v u x
this : Dist.dist u v + Dist.dist u x = Dist.dist v x
⊢ False | V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w u v x : V
h : sbtw u v x
h' : sbtw v u x
this✝ : Dist.dist u v + Dist.dist u x = Dist.dist v x
this : Dist.dist u v = 0
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | SBtw.sbtw.asymm_right | [47, 1] | [51, 24] | simp [h.ne12] at this | V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w u v x : V
h : sbtw u v x
h' : sbtw v u x
this✝ : Dist.dist u v + Dist.dist u x = Dist.dist v x
this : Dist.dist u v = 0
⊢ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | SBtw.sbtw.asymm_right | [47, 1] | [51, 24] | linarith [h.dist] | V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w u v x : V
h : sbtw u v x
h' : sbtw v u x
this : Dist.dist u v + Dist.dist u x = Dist.dist v x
⊢ Dist.dist u v = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | SBtw.sbtw.trans_right' | [53, 1] | [55, 77] | rintro rfl | V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w✝ u v w x : V
h : sbtw u v x
h' : sbtw v w x
⊢ u ≠ w | V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w u v x : V
h : sbtw u v x
h' : sbtw v u x
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | SBtw.sbtw.trans_right' | [53, 1] | [55, 77] | exact h.asymm_right h' | V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w u v x : V
h : sbtw u v x
h' : sbtw v u x
⊢ False | no goals |
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