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) ... |
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 ⊔ ... |
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) ... |
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