type
stringlengths 1
129k
| tactic
stringlengths 1
8.85k
| removals
listlengths 0
38
| name
stringlengths 1
85
| kind
stringclasses 3
values | goal
stringlengths 7
91.3k
|
|---|---|---|---|---|---|
a \ a ⊔ a \ a = ⊥
|
rw [symmDiff, sup_idem, sdiff_self]
|
[] |
rw
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a : α
⊢ a ∆ a = ⊥
|
a \ a = ⊥
|
rw [symmDiff, sup_idem, sdiff_self]
|
[] |
rw₁
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a : α
rw : a \ a ⊔ a \ a = ⊥
⊢ a ∆ a = ⊥
|
⊥ = ⊥
|
rw [symmDiff, sup_idem, sdiff_self]
|
[] |
rw₂
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a : α
rw : a \ a ⊔ a \ a = ⊥
rw₁ : a \ a = ⊥
⊢ a ∆ a = ⊥
|
⊥ = ⊥
|
rw [symmDiff, sup_idem, sdiff_self]
|
[] |
rw₃
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a : α
rw : a \ a ⊔ a \ a = ⊥
rw₁ : a \ a = ⊥
rw₂ : ⊥ = ⊥
⊢ a ∆ a = ⊥
|
a \ ⊥ ⊔ ⊥ \ a = a
|
rw [symmDiff, sdiff_bot, bot_sdiff, sup_bot_eq]
|
[] |
rw
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a : α
⊢ a ∆ ⊥ = a
|
a ⊔ ⊥ \ a = a
|
rw [symmDiff, sdiff_bot, bot_sdiff, sup_bot_eq]
|
[] |
rw₁
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a : α
rw : a \ ⊥ ⊔ ⊥ \ a = a
⊢ a ∆ ⊥ = a
|
a ⊔ ⊥ = a
|
rw [symmDiff, sdiff_bot, bot_sdiff, sup_bot_eq]
|
[] |
rw₂
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a : α
rw : a \ ⊥ ⊔ ⊥ \ a = a
rw₁ : a ⊔ ⊥ \ a = a
⊢ a ∆ ⊥ = a
|
a = a
|
rw [symmDiff, sdiff_bot, bot_sdiff, sup_bot_eq]
|
[] |
rw₃
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a : α
rw : a \ ⊥ ⊔ ⊥ \ a = a
rw₁ : a ⊔ ⊥ \ a = a
rw₂ : a ⊔ ⊥ = a
⊢ a ∆ ⊥ = a
|
a = a
|
rw [symmDiff, sdiff_bot, bot_sdiff, sup_bot_eq]
|
[] |
rw₄
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a : α
rw : a \ ⊥ ⊔ ⊥ \ a = a
rw₁ : a ⊔ ⊥ \ a = a
rw₂ : a ⊔ ⊥ = a
rw₃ : a = a
⊢ a ∆ ⊥ = a
|
a ∆ ⊥ = a
|
rw [symmDiff_comm, symmDiff_bot]
|
[] |
rw
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a : α
⊢ ⊥ ∆ a = a
|
a = a
|
rw [symmDiff_comm, symmDiff_bot]
|
[] |
rw₁
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a : α
rw : a ∆ ⊥ = a
⊢ ⊥ ∆ a = a
|
a = a
|
rw [symmDiff_comm, symmDiff_bot]
|
[] |
rw₂
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a : α
rw : a ∆ ⊥ = a
rw₁ : a = a
⊢ ⊥ ∆ a = a
|
a \ b ⊔ b \ a = ⊥ ↔ a = b
|
simp_rw [symmDiff, sup_eq_bot_iff, sdiff_eq_bot_iff, le_antisymm_iff]
|
[] |
simp_rw
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
⊢ a ∆ b = ⊥ ↔ a = b
|
a \ b = ⊥ ∧ b \ a = ⊥ ↔ a = b
|
simp_rw [symmDiff, sup_eq_bot_iff, sdiff_eq_bot_iff, le_antisymm_iff]
|
[] |
simp_rw₁
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
simp_rw : a \ b ⊔ b \ a = ⊥ ↔ a = b
⊢ a ∆ b = ⊥ ↔ a = b
|
a ≤ b ∧ b ≤ a ↔ a = b
|
simp_rw [symmDiff, sup_eq_bot_iff, sdiff_eq_bot_iff, le_antisymm_iff]
|
[] |
simp_rw₂
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
simp_rw : a \ b ⊔ b \ a = ⊥ ↔ a = b
simp_rw₁ : a \ b = ⊥ ∧ b \ a = ⊥ ↔ a = b
⊢ a ∆ b = ⊥ ↔ a = b
|
a \ b ⊔ b \ a = b \ a
|
rw [symmDiff, sdiff_eq_bot_iff.2 h, bot_sup_eq]
|
[] |
rw
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
h : a ≤ b
⊢ a ∆ b = b \ a
|
⊥ ⊔ b \ a = b \ a
|
rw [symmDiff, sdiff_eq_bot_iff.2 h, bot_sup_eq]
|
[] |
rw₁
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
h : a ≤ b
rw : a \ b ⊔ b \ a = b \ a
⊢ a ∆ b = b \ a
|
b \ a = b \ a
|
rw [symmDiff, sdiff_eq_bot_iff.2 h, bot_sup_eq]
|
[] |
rw₂
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
h : a ≤ b
rw : a \ b ⊔ b \ a = b \ a
rw₁ : ⊥ ⊔ b \ a = b \ a
⊢ a ∆ b = b \ a
|
b \ a = b \ a
|
rw [symmDiff, sdiff_eq_bot_iff.2 h, bot_sup_eq]
|
[] |
rw₃
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
h : a ≤ b
rw : a \ b ⊔ b \ a = b \ a
rw₁ : ⊥ ⊔ b \ a = b \ a
rw₂ : b \ a = b \ a
⊢ a ∆ b = b \ a
|
a \ b ⊔ b \ a = a \ b
|
rw [symmDiff, sdiff_eq_bot_iff.2 h, sup_bot_eq]
|
[] |
rw
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
h : b ≤ a
⊢ a ∆ b = a \ b
|
a \ b ⊔ ⊥ = a \ b
|
rw [symmDiff, sdiff_eq_bot_iff.2 h, sup_bot_eq]
|
[] |
rw₁
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
h : b ≤ a
rw : a \ b ⊔ b \ a = a \ b
⊢ a ∆ b = a \ b
|
a \ b = a \ b
|
rw [symmDiff, sdiff_eq_bot_iff.2 h, sup_bot_eq]
|
[] |
rw₂
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
h : b ≤ a
rw : a \ b ⊔ b \ a = a \ b
rw₁ : a \ b ⊔ ⊥ = a \ b
⊢ a ∆ b = a \ b
|
a \ b = a \ b
|
rw [symmDiff, sdiff_eq_bot_iff.2 h, sup_bot_eq]
|
[] |
rw₃
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
h : b ≤ a
rw : a \ b ⊔ b \ a = a \ b
rw₁ : a \ b ⊔ ⊥ = a \ b
rw₂ : a \ b = a \ b
⊢ a ∆ b = a \ b
|
a \ b ⊔ b \ a ≤ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c
|
simp_rw [symmDiff, sup_le_iff, sdiff_le_iff]
|
[] |
simp_rw
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c : α
⊢ a ∆ b ≤ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c
|
a \ b ≤ c ∧ b \ a ≤ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c
|
simp_rw [symmDiff, sup_le_iff, sdiff_le_iff]
|
[] |
simp_rw₁
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c : α
simp_rw : a \ b ⊔ b \ a ≤ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c
⊢ a ∆ b ≤ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c
|
a \ b ⊔ b \ a = a ⊔ b
|
rw [symmDiff, h.sdiff_eq_left, h.sdiff_eq_right]
|
[] |
rw
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
h : Disjoint a b
⊢ a ∆ b = a ⊔ b
|
a ⊔ b \ a = a ⊔ b
|
rw [symmDiff, h.sdiff_eq_left, h.sdiff_eq_right]
|
[] |
rw₁
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
h : Disjoint a b
rw : a \ b ⊔ b \ a = a ⊔ b
⊢ a ∆ b = a ⊔ b
|
a ⊔ b = a ⊔ b
|
rw [symmDiff, h.sdiff_eq_left, h.sdiff_eq_right]
|
[] |
rw₂
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
h : Disjoint a b
rw : a \ b ⊔ b \ a = a ⊔ b
rw₁ : a ⊔ b \ a = a ⊔ b
⊢ a ∆ b = a ⊔ b
|
a ⊔ b = a ⊔ b
|
rw [symmDiff, h.sdiff_eq_left, h.sdiff_eq_right]
|
[] |
rw₃
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
h : Disjoint a b
rw : a \ b ⊔ b \ a = a ⊔ b
rw₁ : a ⊔ b \ a = a ⊔ b
rw₂ : a ⊔ b = a ⊔ b
⊢ a ∆ b = a ⊔ b
|
(a \ b ⊔ b \ a) \ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c)
|
rw [symmDiff, sup_sdiff_distrib, sdiff_sdiff_left, sdiff_sdiff_left]
|
[] |
rw
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c : α
⊢ a ∆ b \ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c)
|
(a \ b) \ c ⊔ (b \ a) \ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c)
|
rw [symmDiff, sup_sdiff_distrib, sdiff_sdiff_left, sdiff_sdiff_left]
|
[] |
rw₁
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c : α
rw : (a \ b ⊔ b \ a) \ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c)
⊢ a ∆ b \ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c)
|
a \ (b ⊔ c) ⊔ (b \ a) \ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c)
|
rw [symmDiff, sup_sdiff_distrib, sdiff_sdiff_left, sdiff_sdiff_left]
|
[] |
rw₂
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c : α
rw : (a \ b ⊔ b \ a) \ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c)
rw₁ : (a \ b) \ c ⊔ (b \ a) \ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c)
⊢ a ∆ b \ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c)
|
a \ (b ⊔ c) ⊔ b \ (a ⊔ c) = a \ (b ⊔ c) ⊔ b \ (a ⊔ c)
|
rw [symmDiff, sup_sdiff_distrib, sdiff_sdiff_left, sdiff_sdiff_left]
|
[] |
rw₃
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c : α
rw : (a \ b ⊔ b \ a) \ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c)
rw₁ : (a \ b) \ c ⊔ (b \ a) \ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c)
rw₂ : a \ (b ⊔ c) ⊔ (b \ a) \ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c)
⊢ a ∆ b \ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c)
|
a \ (b ⊔ c) ⊔ b \ (a ⊔ c) = a \ (b ⊔ c) ⊔ b \ (a ⊔ c)
|
rw [symmDiff, sup_sdiff_distrib, sdiff_sdiff_left, sdiff_sdiff_left]
|
[] |
rw₄
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c : α
rw : (a \ b ⊔ b \ a) \ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c)
rw₁ : (a \ b) \ c ⊔ (b \ a) \ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c)
rw₂ : a \ (b ⊔ c) ⊔ (b \ a) \ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c)
rw₃ : a \ (b ⊔ c) ⊔ b \ (a ⊔ c) = a \ (b ⊔ c) ⊔ b \ (a ⊔ c)
⊢ a ∆ b \ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c)
|
a \ (b ⊔ a ⊓ b) ⊔ b \ (a ⊔ a ⊓ b) = a ∆ b
|
rw [symmDiff_sdiff]
|
[] |
rw
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
⊢ a ∆ b \ (a ⊓ b) = a ∆ b
|
a \ (b \ a) ⊔ b \ a = a ⊔ b
|
rw [symmDiff, sdiff_idem]
|
[] |
rw
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
⊢ a ∆ (b \ a) = a ⊔ b
|
b ∆ (a \ b) = a ⊔ b
|
rw [symmDiff_comm, symmDiff_sdiff_eq_sup, sup_comm]
|
[] |
rw
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
⊢ (a \ b) ∆ b = a ⊔ b
|
b ⊔ a = a ⊔ b
|
rw [symmDiff_comm, symmDiff_sdiff_eq_sup, sup_comm]
|
[] |
rw₁
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
rw : b ∆ (a \ b) = a ⊔ b
⊢ (a \ b) ∆ b = a ⊔ b
|
a ⊔ b = a ⊔ b
|
rw [symmDiff_comm, symmDiff_sdiff_eq_sup, sup_comm]
|
[] |
rw₂
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
rw : b ∆ (a \ b) = a ⊔ b
rw₁ : b ⊔ a = a ⊔ b
⊢ (a \ b) ∆ b = a ⊔ b
|
a ⊔ b = a ⊔ b
|
rw [symmDiff_comm, symmDiff_sdiff_eq_sup, sup_comm]
|
[] |
rw₃
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
rw : b ∆ (a \ b) = a ⊔ b
rw₁ : b ⊔ a = a ⊔ b
rw₂ : a ⊔ b = a ⊔ b
⊢ (a \ b) ∆ b = a ⊔ b
|
a ⊔ b ≤ a ∆ b ⊔ a ⊓ b
|
refine le_antisymm (sup_le symmDiff_le_sup inf_le_sup) ?_
|
[] |
refine
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
⊢ a ∆ b ⊔ a ⊓ b = a ⊔ b
|
a ⊔ b ≤ (a \ b ⊔ b \ a ⊔ a) ⊓ (a \ b ⊔ b \ a ⊔ b)
|
rw [sup_inf_left, symmDiff]
|
[] |
rw
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
⊢ a ⊔ b ≤ a ∆ b ⊔ a ⊓ b
|
a ≤ a \ b ⊔ b \ a ⊔ b
|
refine sup_le (le_inf le_sup_right ?_) (le_inf ?_ le_sup_right)
|
[] |
refine_1
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
⊢ a ⊔ b ≤ (a \ b ⊔ b \ a ⊔ a) ⊓ (a \ b ⊔ b \ a ⊔ b)
|
b ≤ a \ b ⊔ b \ a ⊔ a
|
refine sup_le (le_inf le_sup_right ?_) (le_inf ?_ le_sup_right)
|
[] |
refine_2
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
refine_1 : a ≤ a \ b ⊔ b \ a ⊔ b
⊢ a ⊔ b ≤ (a \ b ⊔ b \ a ⊔ a) ⊓ (a \ b ⊔ b \ a ⊔ b)
|
a ≤ a \ b ⊔ b ⊔ b \ a
|
rw [sup_right_comm]
|
[] |
refine_1
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
⊢ a ≤ a \ b ⊔ b \ a ⊔ b
|
b ≤ a \ b ⊔ (b \ a ⊔ a)
|
rw [sup_assoc]
|
[] |
refine_2
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
⊢ b ≤ a \ b ⊔ b \ a ⊔ a
|
a ∆ b ⊔ a ⊓ b = a ⊔ b
|
rw [sup_comm, symmDiff_sup_inf]
|
[] |
rw
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
⊢ a ⊓ b ⊔ a ∆ b = a ⊔ b
|
a ⊔ b = a ⊔ b
|
rw [sup_comm, symmDiff_sup_inf]
|
[] |
rw₁
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
rw : a ∆ b ⊔ a ⊓ b = a ⊔ b
⊢ a ⊓ b ⊔ a ∆ b = a ⊔ b
|
a ⊔ b = a ⊔ b
|
rw [sup_comm, symmDiff_sup_inf]
|
[] |
rw₂
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
rw : a ∆ b ⊔ a ⊓ b = a ⊔ b
rw₁ : a ⊔ b = a ⊔ b
⊢ a ⊓ b ⊔ a ∆ b = a ⊔ b
|
(a ∆ b \ (a ⊓ b)) ∆ (a ⊓ b) = a ⊔ b
|
rw [← symmDiff_sdiff_inf a, sdiff_symmDiff_eq_sup, symmDiff_sup_inf]
|
[] |
rw
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
⊢ a ∆ b ∆ (a ⊓ b) = a ⊔ b
|
a ∆ b ⊔ a ⊓ b = a ⊔ b
|
rw [← symmDiff_sdiff_inf a, sdiff_symmDiff_eq_sup, symmDiff_sup_inf]
|
[] |
rw₁
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
rw : (a ∆ b \ (a ⊓ b)) ∆ (a ⊓ b) = a ⊔ b
⊢ a ∆ b ∆ (a ⊓ b) = a ⊔ b
|
a ⊔ b = a ⊔ b
|
rw [← symmDiff_sdiff_inf a, sdiff_symmDiff_eq_sup, symmDiff_sup_inf]
|
[] |
rw₂
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
rw : (a ∆ b \ (a ⊓ b)) ∆ (a ⊓ b) = a ⊔ b
rw₁ : a ∆ b ⊔ a ⊓ b = a ⊔ b
⊢ a ∆ b ∆ (a ⊓ b) = a ⊔ b
|
a ⊔ b = a ⊔ b
|
rw [← symmDiff_sdiff_inf a, sdiff_symmDiff_eq_sup, symmDiff_sup_inf]
|
[] |
rw₃
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
rw : (a ∆ b \ (a ⊓ b)) ∆ (a ⊓ b) = a ⊔ b
rw₁ : a ∆ b ⊔ a ⊓ b = a ⊔ b
rw₂ : a ⊔ b = a ⊔ b
⊢ a ∆ b ∆ (a ⊓ b) = a ⊔ b
|
a ∆ b ∆ (a ⊓ b) = a ⊔ b
|
rw [symmDiff_comm, symmDiff_symmDiff_inf]
|
[] |
rw
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
⊢ (a ⊓ b) ∆ (a ∆ b) = a ⊔ b
|
a ⊔ b = a ⊔ b
|
rw [symmDiff_comm, symmDiff_symmDiff_inf]
|
[] |
rw₁
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
rw : a ∆ b ∆ (a ⊓ b) = a ⊔ b
⊢ (a ⊓ b) ∆ (a ∆ b) = a ⊔ b
|
a ⊔ b = a ⊔ b
|
rw [symmDiff_comm, symmDiff_symmDiff_inf]
|
[] |
rw₂
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
rw : a ∆ b ∆ (a ⊓ b) = a ⊔ b
rw₁ : a ⊔ b = a ⊔ b
⊢ (a ⊓ b) ∆ (a ∆ b) = a ⊔ b
|
a \ b ⊔ b \ c ⊔ (c \ b ⊔ b \ a) = a ∆ b ⊔ b ∆ c
|
refine (sup_le_sup (sdiff_triangle a b c) <| sdiff_triangle _ b _).trans_eq ?_
|
[] |
refine
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c : α
⊢ a ∆ c ≤ a ∆ b ⊔ b ∆ c
|
a \ b ⊔ b \ c ⊔ (b \ a ⊔ c \ b) = a ∆ b ⊔ b ∆ c
|
rw [sup_comm (c \ b), sup_sup_sup_comm, symmDiff, symmDiff]
|
[] |
rw
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c : α
⊢ a \ b ⊔ b \ c ⊔ (c \ b ⊔ b \ a) = a ∆ b ⊔ b ∆ c
|
a \ b ⊔ b \ a ⊔ (b \ c ⊔ c \ b) = a ∆ b ⊔ b ∆ c
|
rw [sup_comm (c \ b), sup_sup_sup_comm, symmDiff, symmDiff]
|
[] |
rw₁
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c : α
rw : a \ b ⊔ b \ c ⊔ (b \ a ⊔ c \ b) = a ∆ b ⊔ b ∆ c
⊢ a \ b ⊔ b \ c ⊔ (c \ b ⊔ b \ a) = a ∆ b ⊔ b ∆ c
|
a \ b ⊔ b \ a ⊔ (b \ c ⊔ c \ b) = a \ b ⊔ b \ a ⊔ b ∆ c
|
rw [sup_comm (c \ b), sup_sup_sup_comm, symmDiff, symmDiff]
|
[] |
rw₂
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c : α
rw : a \ b ⊔ b \ c ⊔ (b \ a ⊔ c \ b) = a ∆ b ⊔ b ∆ c
rw₁ : a \ b ⊔ b \ a ⊔ (b \ c ⊔ c \ b) = a ∆ b ⊔ b ∆ c
⊢ a \ b ⊔ b \ c ⊔ (c \ b ⊔ b \ a) = a ∆ b ⊔ b ∆ c
|
a \ b ⊔ b \ a ⊔ (b \ c ⊔ c \ b) = a \ b ⊔ b \ a ⊔ (b \ c ⊔ c \ b)
|
rw [sup_comm (c \ b), sup_sup_sup_comm, symmDiff, symmDiff]
|
[] |
rw₃
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c : α
rw : a \ b ⊔ b \ c ⊔ (b \ a ⊔ c \ b) = a ∆ b ⊔ b ∆ c
rw₁ : a \ b ⊔ b \ a ⊔ (b \ c ⊔ c \ b) = a ∆ b ⊔ b ∆ c
rw₂ : a \ b ⊔ b \ a ⊔ (b \ c ⊔ c \ b) = a \ b ⊔ b \ a ⊔ b ∆ c
⊢ a \ b ⊔ b \ c ⊔ (c \ b ⊔ b \ a) = a ∆ b ⊔ b ∆ c
|
a \ b ⊔ b \ a ⊔ (b \ c ⊔ c \ b) = a \ b ⊔ b \ a ⊔ (b \ c ⊔ c \ b)
|
rw [sup_comm (c \ b), sup_sup_sup_comm, symmDiff, symmDiff]
|
[] |
rw₄
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c : α
rw : a \ b ⊔ b \ c ⊔ (b \ a ⊔ c \ b) = a ∆ b ⊔ b ∆ c
rw₁ : a \ b ⊔ b \ a ⊔ (b \ c ⊔ c \ b) = a ∆ b ⊔ b ∆ c
rw₂ : a \ b ⊔ b \ a ⊔ (b \ c ⊔ c \ b) = a \ b ⊔ b \ a ⊔ b ∆ c
rw₃ : a \ b ⊔ b \ a ⊔ (b \ c ⊔ c \ b) = a \ b ⊔ b \ a ⊔ (b \ c ⊔ c \ b)
⊢ a \ b ⊔ b \ c ⊔ (c \ b ⊔ b \ a) = a ∆ b ⊔ b ∆ c
|
a = a ∆ ⊥
|
convert symmDiff_triangle a b ⊥
|
[] |
h
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
⊢ a ≤ a ∆ b ⊔ b
|
b = b ∆ ⊥
|
convert symmDiff_triangle a b ⊥
|
[] |
h₁
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
h : a = a ∆ ⊥
⊢ a ≤ a ∆ b ⊔ b
|
a = a
|
rw [symmDiff_bot]
|
[] |
h
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
⊢ a = a ∆ ⊥
|
a = a
|
rw [symmDiff_bot]
|
[] |
h₁
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
h : a = a
⊢ a = a ∆ ⊥
|
b = b
|
rw [symmDiff_bot]
|
[] |
h
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
⊢ b = b ∆ ⊥
|
b = b
|
rw [symmDiff_bot]
|
[] |
h₁
|
goal
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
h : b = b
⊢ b = b ∆ ⊥
|
(a ⇨ a) ⊓ (a ⇨ a) = ⊤
|
rw [bihimp, inf_idem, himp_self]
|
[] |
rw
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a : α
⊢ a ⇔ a = ⊤
|
a ⇨ a = ⊤
|
rw [bihimp, inf_idem, himp_self]
|
[] |
rw₁
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a : α
rw : (a ⇨ a) ⊓ (a ⇨ a) = ⊤
⊢ a ⇔ a = ⊤
|
⊤ = ⊤
|
rw [bihimp, inf_idem, himp_self]
|
[] |
rw₂
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a : α
rw : (a ⇨ a) ⊓ (a ⇨ a) = ⊤
rw₁ : a ⇨ a = ⊤
⊢ a ⇔ a = ⊤
|
⊤ = ⊤
|
rw [bihimp, inf_idem, himp_self]
|
[] |
rw₃
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a : α
rw : (a ⇨ a) ⊓ (a ⇨ a) = ⊤
rw₁ : a ⇨ a = ⊤
rw₂ : ⊤ = ⊤
⊢ a ⇔ a = ⊤
|
(⊤ ⇨ a) ⊓ (a ⇨ ⊤) = a
|
rw [bihimp, himp_top, top_himp, inf_top_eq]
|
[] |
rw
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a : α
⊢ a ⇔ ⊤ = a
|
(⊤ ⇨ a) ⊓ ⊤ = a
|
rw [bihimp, himp_top, top_himp, inf_top_eq]
|
[] |
rw₁
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a : α
rw : (⊤ ⇨ a) ⊓ (a ⇨ ⊤) = a
⊢ a ⇔ ⊤ = a
|
a ⊓ ⊤ = a
|
rw [bihimp, himp_top, top_himp, inf_top_eq]
|
[] |
rw₂
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a : α
rw : (⊤ ⇨ a) ⊓ (a ⇨ ⊤) = a
rw₁ : (⊤ ⇨ a) ⊓ ⊤ = a
⊢ a ⇔ ⊤ = a
|
a = a
|
rw [bihimp, himp_top, top_himp, inf_top_eq]
|
[] |
rw₃
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a : α
rw : (⊤ ⇨ a) ⊓ (a ⇨ ⊤) = a
rw₁ : (⊤ ⇨ a) ⊓ ⊤ = a
rw₂ : a ⊓ ⊤ = a
⊢ a ⇔ ⊤ = a
|
a = a
|
rw [bihimp, himp_top, top_himp, inf_top_eq]
|
[] |
rw₄
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a : α
rw : (⊤ ⇨ a) ⊓ (a ⇨ ⊤) = a
rw₁ : (⊤ ⇨ a) ⊓ ⊤ = a
rw₂ : a ⊓ ⊤ = a
rw₃ : a = a
⊢ a ⇔ ⊤ = a
|
a ⇔ ⊤ = a
|
rw [bihimp_comm, bihimp_top]
|
[] |
rw
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a : α
⊢ ⊤ ⇔ a = a
|
a = a
|
rw [bihimp_comm, bihimp_top]
|
[] |
rw₁
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a : α
rw : a ⇔ ⊤ = a
⊢ ⊤ ⇔ a = a
|
a = a
|
rw [bihimp_comm, bihimp_top]
|
[] |
rw₂
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a : α
rw : a ⇔ ⊤ = a
rw₁ : a = a
⊢ ⊤ ⇔ a = a
|
(b ⇨ a) ⊓ (a ⇨ b) = b ⇨ a
|
rw [bihimp, himp_eq_top_iff.2 h, inf_top_eq]
|
[] |
rw
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b : α
h : a ≤ b
⊢ a ⇔ b = b ⇨ a
|
(b ⇨ a) ⊓ ⊤ = b ⇨ a
|
rw [bihimp, himp_eq_top_iff.2 h, inf_top_eq]
|
[] |
rw₁
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b : α
h : a ≤ b
rw : (b ⇨ a) ⊓ (a ⇨ b) = b ⇨ a
⊢ a ⇔ b = b ⇨ a
|
b ⇨ a = b ⇨ a
|
rw [bihimp, himp_eq_top_iff.2 h, inf_top_eq]
|
[] |
rw₂
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b : α
h : a ≤ b
rw : (b ⇨ a) ⊓ (a ⇨ b) = b ⇨ a
rw₁ : (b ⇨ a) ⊓ ⊤ = b ⇨ a
⊢ a ⇔ b = b ⇨ a
|
b ⇨ a = b ⇨ a
|
rw [bihimp, himp_eq_top_iff.2 h, inf_top_eq]
|
[] |
rw₃
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b : α
h : a ≤ b
rw : (b ⇨ a) ⊓ (a ⇨ b) = b ⇨ a
rw₁ : (b ⇨ a) ⊓ ⊤ = b ⇨ a
rw₂ : b ⇨ a = b ⇨ a
⊢ a ⇔ b = b ⇨ a
|
(b ⇨ a) ⊓ (a ⇨ b) = a ⇨ b
|
rw [bihimp, himp_eq_top_iff.2 h, top_inf_eq]
|
[] |
rw
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b : α
h : b ≤ a
⊢ a ⇔ b = a ⇨ b
|
⊤ ⊓ (a ⇨ b) = a ⇨ b
|
rw [bihimp, himp_eq_top_iff.2 h, top_inf_eq]
|
[] |
rw₁
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b : α
h : b ≤ a
rw : (b ⇨ a) ⊓ (a ⇨ b) = a ⇨ b
⊢ a ⇔ b = a ⇨ b
|
a ⇨ b = a ⇨ b
|
rw [bihimp, himp_eq_top_iff.2 h, top_inf_eq]
|
[] |
rw₂
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b : α
h : b ≤ a
rw : (b ⇨ a) ⊓ (a ⇨ b) = a ⇨ b
rw₁ : ⊤ ⊓ (a ⇨ b) = a ⇨ b
⊢ a ⇔ b = a ⇨ b
|
a ⇨ b = a ⇨ b
|
rw [bihimp, himp_eq_top_iff.2 h, top_inf_eq]
|
[] |
rw₃
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b : α
h : b ≤ a
rw : (b ⇨ a) ⊓ (a ⇨ b) = a ⇨ b
rw₁ : ⊤ ⊓ (a ⇨ b) = a ⇨ b
rw₂ : a ⇨ b = a ⇨ b
⊢ a ⇔ b = a ⇨ b
|
a ≤ (c ⇨ b) ⊓ (b ⇨ c) ↔ a ⊓ b ≤ c ∧ a ⊓ c ≤ b
|
simp_rw [bihimp, le_inf_iff, le_himp_iff, and_comm]
|
[] |
simp_rw
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b c : α
⊢ a ≤ b ⇔ c ↔ a ⊓ b ≤ c ∧ a ⊓ c ≤ b
|
a ≤ c ⇨ b ∧ a ≤ b ⇨ c ↔ a ⊓ b ≤ c ∧ a ⊓ c ≤ b
|
simp_rw [bihimp, le_inf_iff, le_himp_iff, and_comm]
|
[] |
simp_rw₁
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b c : α
simp_rw : a ≤ (c ⇨ b) ⊓ (b ⇨ c) ↔ a ⊓ b ≤ c ∧ a ⊓ c ≤ b
⊢ a ≤ b ⇔ c ↔ a ⊓ b ≤ c ∧ a ⊓ c ≤ b
|
a ⊓ c ≤ b ∧ a ⊓ b ≤ c ↔ a ⊓ b ≤ c ∧ a ⊓ c ≤ b
|
simp_rw [bihimp, le_inf_iff, le_himp_iff, and_comm]
|
[] |
simp_rw₂
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b c : α
simp_rw : a ≤ (c ⇨ b) ⊓ (b ⇨ c) ↔ a ⊓ b ≤ c ∧ a ⊓ c ≤ b
simp_rw₁ : a ≤ c ⇨ b ∧ a ≤ b ⇨ c ↔ a ⊓ b ≤ c ∧ a ⊓ c ≤ b
⊢ a ≤ b ⇔ c ↔ a ⊓ b ≤ c ∧ a ⊓ c ≤ b
|
(b ⇨ a) ⊓ (a ⇨ b) = a ⊓ b
|
rw [bihimp, h.himp_eq_left, h.himp_eq_right]
|
[] |
rw
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b : α
h : Codisjoint a b
⊢ a ⇔ b = a ⊓ b
|
(b ⇨ a) ⊓ b = a ⊓ b
|
rw [bihimp, h.himp_eq_left, h.himp_eq_right]
|
[] |
rw₁
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b : α
h : Codisjoint a b
rw : (b ⇨ a) ⊓ (a ⇨ b) = a ⊓ b
⊢ a ⇔ b = a ⊓ b
|
a ⊓ b = a ⊓ b
|
rw [bihimp, h.himp_eq_left, h.himp_eq_right]
|
[] |
rw₂
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b : α
h : Codisjoint a b
rw : (b ⇨ a) ⊓ (a ⇨ b) = a ⊓ b
rw₁ : (b ⇨ a) ⊓ b = a ⊓ b
⊢ a ⇔ b = a ⊓ b
|
a ⊓ b = a ⊓ b
|
rw [bihimp, h.himp_eq_left, h.himp_eq_right]
|
[] |
rw₃
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b : α
h : Codisjoint a b
rw : (b ⇨ a) ⊓ (a ⇨ b) = a ⊓ b
rw₁ : (b ⇨ a) ⊓ b = a ⊓ b
rw₂ : a ⊓ b = a ⊓ b
⊢ a ⇔ b = a ⊓ b
|
a ⇨ (c ⇨ b) ⊓ (b ⇨ c) = (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c)
|
rw [bihimp, himp_inf_distrib, himp_himp, himp_himp]
|
[] |
rw
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b c : α
⊢ a ⇨ b ⇔ c = (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c)
|
(a ⇨ c ⇨ b) ⊓ (a ⇨ b ⇨ c) = (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c)
|
rw [bihimp, himp_inf_distrib, himp_himp, himp_himp]
|
[] |
rw₁
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b c : α
rw : a ⇨ (c ⇨ b) ⊓ (b ⇨ c) = (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c)
⊢ a ⇨ b ⇔ c = (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c)
|
(a ⊓ c ⇨ b) ⊓ (a ⇨ b ⇨ c) = (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c)
|
rw [bihimp, himp_inf_distrib, himp_himp, himp_himp]
|
[] |
rw₂
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b c : α
rw : a ⇨ (c ⇨ b) ⊓ (b ⇨ c) = (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c)
rw₁ : (a ⇨ c ⇨ b) ⊓ (a ⇨ b ⇨ c) = (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c)
⊢ a ⇨ b ⇔ c = (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c)
|
(a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c) = (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c)
|
rw [bihimp, himp_inf_distrib, himp_himp, himp_himp]
|
[] |
rw₃
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b c : α
rw : a ⇨ (c ⇨ b) ⊓ (b ⇨ c) = (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c)
rw₁ : (a ⇨ c ⇨ b) ⊓ (a ⇨ b ⇨ c) = (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c)
rw₂ : (a ⊓ c ⇨ b) ⊓ (a ⇨ b ⇨ c) = (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c)
⊢ a ⇨ b ⇔ c = (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c)
|
(a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c) = (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c)
|
rw [bihimp, himp_inf_distrib, himp_himp, himp_himp]
|
[] |
rw₄
|
goal
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b c : α
rw : a ⇨ (c ⇨ b) ⊓ (b ⇨ c) = (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c)
rw₁ : (a ⇨ c ⇨ b) ⊓ (a ⇨ b ⇨ c) = (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c)
rw₂ : (a ⊓ c ⇨ b) ⊓ (a ⇨ b ⇨ c) = (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c)
rw₃ : (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c) = (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c)
⊢ a ⇨ b ⇔ c = (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c)
|
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