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tactic-haveDraft11

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a \ a βŠ” a \ a = βŠ₯
rw [symmDiff, sup_idem, sdiff_self]
[]
rw
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a : Ξ± ⊒ a βˆ† a = βŠ₯
a \ a = βŠ₯
rw [symmDiff, sup_idem, sdiff_self]
[]
rw₁
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a : Ξ± rw : a \ a βŠ” a \ a = βŠ₯ ⊒ a βˆ† a = βŠ₯
βŠ₯ = βŠ₯
rw [symmDiff, sup_idem, sdiff_self]
[]
rwβ‚‚
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a : Ξ± rw : a \ a βŠ” a \ a = βŠ₯ rw₁ : a \ a = βŠ₯ ⊒ a βˆ† a = βŠ₯
βŠ₯ = βŠ₯
rw [symmDiff, sup_idem, sdiff_self]
[]
rw₃
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a : Ξ± rw : a \ a βŠ” a \ a = βŠ₯ rw₁ : a \ a = βŠ₯ rwβ‚‚ : βŠ₯ = βŠ₯ ⊒ a βˆ† a = βŠ₯
a \ βŠ₯ βŠ” βŠ₯ \ a = a
rw [symmDiff, sdiff_bot, bot_sdiff, sup_bot_eq]
[]
rw
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a : Ξ± ⊒ a βˆ† βŠ₯ = a
a βŠ” βŠ₯ \ a = a
rw [symmDiff, sdiff_bot, bot_sdiff, sup_bot_eq]
[]
rw₁
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a : Ξ± rw : a \ βŠ₯ βŠ” βŠ₯ \ a = a ⊒ a βˆ† βŠ₯ = a
a βŠ” βŠ₯ = a
rw [symmDiff, sdiff_bot, bot_sdiff, sup_bot_eq]
[]
rwβ‚‚
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a : Ξ± rw : a \ βŠ₯ βŠ” βŠ₯ \ a = a rw₁ : a βŠ” βŠ₯ \ a = a ⊒ a βˆ† βŠ₯ = a
a = a
rw [symmDiff, sdiff_bot, bot_sdiff, sup_bot_eq]
[]
rw₃
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a : Ξ± rw : a \ βŠ₯ βŠ” βŠ₯ \ a = a rw₁ : a βŠ” βŠ₯ \ a = a rwβ‚‚ : a βŠ” βŠ₯ = a ⊒ a βˆ† βŠ₯ = a
a = a
rw [symmDiff, sdiff_bot, bot_sdiff, sup_bot_eq]
[]
rwβ‚„
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a : Ξ± rw : a \ βŠ₯ βŠ” βŠ₯ \ a = a rw₁ : a βŠ” βŠ₯ \ a = a rwβ‚‚ : a βŠ” βŠ₯ = a rw₃ : a = a ⊒ a βˆ† βŠ₯ = a
a βˆ† βŠ₯ = a
rw [symmDiff_comm, symmDiff_bot]
[]
rw
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a : Ξ± ⊒ βŠ₯ βˆ† a = a
a = a
rw [symmDiff_comm, symmDiff_bot]
[]
rw₁
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a : Ξ± rw : a βˆ† βŠ₯ = a ⊒ βŠ₯ βˆ† a = a
a = a
rw [symmDiff_comm, symmDiff_bot]
[]
rwβ‚‚
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a : Ξ± rw : a βˆ† βŠ₯ = a rw₁ : a = a ⊒ βŠ₯ βˆ† a = a
a \ b βŠ” b \ a = βŠ₯ ↔ a = b
simp_rw [symmDiff, sup_eq_bot_iff, sdiff_eq_bot_iff, le_antisymm_iff]
[]
simp_rw
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± ⊒ a βˆ† b = βŠ₯ ↔ a = b
a \ b = βŠ₯ ∧ b \ a = βŠ₯ ↔ a = b
simp_rw [symmDiff, sup_eq_bot_iff, sdiff_eq_bot_iff, le_antisymm_iff]
[]
simp_rw₁
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± simp_rw : a \ b βŠ” b \ a = βŠ₯ ↔ a = b ⊒ a βˆ† b = βŠ₯ ↔ a = b
a ≀ b ∧ b ≀ a ↔ a = b
simp_rw [symmDiff, sup_eq_bot_iff, sdiff_eq_bot_iff, le_antisymm_iff]
[]
simp_rwβ‚‚
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± simp_rw : a \ b βŠ” b \ a = βŠ₯ ↔ a = b simp_rw₁ : a \ b = βŠ₯ ∧ b \ a = βŠ₯ ↔ a = b ⊒ a βˆ† b = βŠ₯ ↔ a = b
a \ b βŠ” b \ a = b \ a
rw [symmDiff, sdiff_eq_bot_iff.2 h, bot_sup_eq]
[]
rw
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± h : a ≀ b ⊒ a βˆ† b = b \ a
βŠ₯ βŠ” b \ a = b \ a
rw [symmDiff, sdiff_eq_bot_iff.2 h, bot_sup_eq]
[]
rw₁
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± h : a ≀ b rw : a \ b βŠ” b \ a = b \ a ⊒ a βˆ† b = b \ a
b \ a = b \ a
rw [symmDiff, sdiff_eq_bot_iff.2 h, bot_sup_eq]
[]
rwβ‚‚
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± h : a ≀ b rw : a \ b βŠ” b \ a = b \ a rw₁ : βŠ₯ βŠ” b \ a = b \ a ⊒ a βˆ† b = b \ a
b \ a = b \ a
rw [symmDiff, sdiff_eq_bot_iff.2 h, bot_sup_eq]
[]
rw₃
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± h : a ≀ b rw : a \ b βŠ” b \ a = b \ a rw₁ : βŠ₯ βŠ” b \ a = b \ a rwβ‚‚ : b \ a = b \ a ⊒ a βˆ† b = b \ a
a \ b βŠ” b \ a = a \ b
rw [symmDiff, sdiff_eq_bot_iff.2 h, sup_bot_eq]
[]
rw
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± h : b ≀ a ⊒ a βˆ† b = a \ b
a \ b βŠ” βŠ₯ = a \ b
rw [symmDiff, sdiff_eq_bot_iff.2 h, sup_bot_eq]
[]
rw₁
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± h : b ≀ a rw : a \ b βŠ” b \ a = a \ b ⊒ a βˆ† b = a \ b
a \ b = a \ b
rw [symmDiff, sdiff_eq_bot_iff.2 h, sup_bot_eq]
[]
rwβ‚‚
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± h : b ≀ a rw : a \ b βŠ” b \ a = a \ b rw₁ : a \ b βŠ” βŠ₯ = a \ b ⊒ a βˆ† b = a \ b
a \ b = a \ b
rw [symmDiff, sdiff_eq_bot_iff.2 h, sup_bot_eq]
[]
rw₃
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± h : b ≀ a rw : a \ b βŠ” b \ a = a \ b rw₁ : a \ b βŠ” βŠ₯ = a \ b rwβ‚‚ : a \ b = a \ b ⊒ a βˆ† b = a \ b
a \ b βŠ” b \ a ≀ c ↔ a ≀ b βŠ” c ∧ b ≀ a βŠ” c
simp_rw [symmDiff, sup_le_iff, sdiff_le_iff]
[]
simp_rw
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b c : Ξ± ⊒ a βˆ† b ≀ c ↔ a ≀ b βŠ” c ∧ b ≀ a βŠ” c
a \ b ≀ c ∧ b \ a ≀ c ↔ a ≀ b βŠ” c ∧ b ≀ a βŠ” c
simp_rw [symmDiff, sup_le_iff, sdiff_le_iff]
[]
simp_rw₁
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b c : Ξ± simp_rw : a \ b βŠ” b \ a ≀ c ↔ a ≀ b βŠ” c ∧ b ≀ a βŠ” c ⊒ a βˆ† b ≀ c ↔ a ≀ b βŠ” c ∧ b ≀ a βŠ” c
a \ b βŠ” b \ a = a βŠ” b
rw [symmDiff, h.sdiff_eq_left, h.sdiff_eq_right]
[]
rw
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± h : Disjoint a b ⊒ a βˆ† b = a βŠ” b
a βŠ” b \ a = a βŠ” b
rw [symmDiff, h.sdiff_eq_left, h.sdiff_eq_right]
[]
rw₁
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± h : Disjoint a b rw : a \ b βŠ” b \ a = a βŠ” b ⊒ a βˆ† b = a βŠ” b
a βŠ” b = a βŠ” b
rw [symmDiff, h.sdiff_eq_left, h.sdiff_eq_right]
[]
rwβ‚‚
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± h : Disjoint a b rw : a \ b βŠ” b \ a = a βŠ” b rw₁ : a βŠ” b \ a = a βŠ” b ⊒ a βˆ† b = a βŠ” b
a βŠ” b = a βŠ” b
rw [symmDiff, h.sdiff_eq_left, h.sdiff_eq_right]
[]
rw₃
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± h : Disjoint a b rw : a \ b βŠ” b \ a = a βŠ” b rw₁ : a βŠ” b \ a = a βŠ” b rwβ‚‚ : a βŠ” b = a βŠ” b ⊒ a βˆ† b = a βŠ” b
(a \ b βŠ” b \ a) \ c = a \ (b βŠ” c) βŠ” b \ (a βŠ” c)
rw [symmDiff, sup_sdiff_distrib, sdiff_sdiff_left, sdiff_sdiff_left]
[]
rw
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b c : Ξ± ⊒ a βˆ† b \ c = a \ (b βŠ” c) βŠ” b \ (a βŠ” c)
(a \ b) \ c βŠ” (b \ a) \ c = a \ (b βŠ” c) βŠ” b \ (a βŠ” c)
rw [symmDiff, sup_sdiff_distrib, sdiff_sdiff_left, sdiff_sdiff_left]
[]
rw₁
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b c : Ξ± rw : (a \ b βŠ” b \ a) \ c = a \ (b βŠ” c) βŠ” b \ (a βŠ” c) ⊒ a βˆ† b \ c = a \ (b βŠ” c) βŠ” b \ (a βŠ” c)
a \ (b βŠ” c) βŠ” (b \ a) \ c = a \ (b βŠ” c) βŠ” b \ (a βŠ” c)
rw [symmDiff, sup_sdiff_distrib, sdiff_sdiff_left, sdiff_sdiff_left]
[]
rwβ‚‚
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b c : Ξ± rw : (a \ b βŠ” b \ a) \ c = a \ (b βŠ” c) βŠ” b \ (a βŠ” c) rw₁ : (a \ b) \ c βŠ” (b \ a) \ c = a \ (b βŠ” c) βŠ” b \ (a βŠ” c) ⊒ a βˆ† b \ c = a \ (b βŠ” c) βŠ” b \ (a βŠ” c)
a \ (b βŠ” c) βŠ” b \ (a βŠ” c) = a \ (b βŠ” c) βŠ” b \ (a βŠ” c)
rw [symmDiff, sup_sdiff_distrib, sdiff_sdiff_left, sdiff_sdiff_left]
[]
rw₃
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b c : Ξ± rw : (a \ b βŠ” b \ a) \ c = a \ (b βŠ” c) βŠ” b \ (a βŠ” c) rw₁ : (a \ b) \ c βŠ” (b \ a) \ c = a \ (b βŠ” c) βŠ” b \ (a βŠ” c) rwβ‚‚ : a \ (b βŠ” c) βŠ” (b \ a) \ c = a \ (b βŠ” c) βŠ” b \ (a βŠ” c) ⊒ a βˆ† b \ c = a \ (b βŠ” c) βŠ” b \ (a βŠ” c)
a \ (b βŠ” c) βŠ” b \ (a βŠ” c) = a \ (b βŠ” c) βŠ” b \ (a βŠ” c)
rw [symmDiff, sup_sdiff_distrib, sdiff_sdiff_left, sdiff_sdiff_left]
[]
rwβ‚„
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b c : Ξ± rw : (a \ b βŠ” b \ a) \ c = a \ (b βŠ” c) βŠ” b \ (a βŠ” c) rw₁ : (a \ b) \ c βŠ” (b \ a) \ c = a \ (b βŠ” c) βŠ” b \ (a βŠ” c) rwβ‚‚ : a \ (b βŠ” c) βŠ” (b \ a) \ c = a \ (b βŠ” c) βŠ” b \ (a βŠ” c) rw₃ : a \ (b βŠ” c) βŠ” b \ (a βŠ” c) = a \ (b βŠ” c) βŠ” b \ (a βŠ” c) ⊒ a βˆ† b \ c = a \ (b βŠ” c) βŠ” b \ (a βŠ” c)
a \ (b βŠ” a βŠ“ b) βŠ” b \ (a βŠ” a βŠ“ b) = a βˆ† b
rw [symmDiff_sdiff]
[]
rw
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± ⊒ a βˆ† b \ (a βŠ“ b) = a βˆ† b
a \ (b \ a) βŠ” b \ a = a βŠ” b
rw [symmDiff, sdiff_idem]
[]
rw
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± ⊒ a βˆ† (b \ a) = a βŠ” b
b βˆ† (a \ b) = a βŠ” b
rw [symmDiff_comm, symmDiff_sdiff_eq_sup, sup_comm]
[]
rw
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± ⊒ (a \ b) βˆ† b = a βŠ” b
b βŠ” a = a βŠ” b
rw [symmDiff_comm, symmDiff_sdiff_eq_sup, sup_comm]
[]
rw₁
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± rw : b βˆ† (a \ b) = a βŠ” b ⊒ (a \ b) βˆ† b = a βŠ” b
a βŠ” b = a βŠ” b
rw [symmDiff_comm, symmDiff_sdiff_eq_sup, sup_comm]
[]
rwβ‚‚
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± rw : b βˆ† (a \ b) = a βŠ” b rw₁ : b βŠ” a = a βŠ” b ⊒ (a \ b) βˆ† b = a βŠ” b
a βŠ” b = a βŠ” b
rw [symmDiff_comm, symmDiff_sdiff_eq_sup, sup_comm]
[]
rw₃
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± rw : b βˆ† (a \ b) = a βŠ” b rw₁ : b βŠ” a = a βŠ” b rwβ‚‚ : a βŠ” b = a βŠ” b ⊒ (a \ b) βˆ† b = a βŠ” b
a βŠ” b ≀ a βˆ† b βŠ” a βŠ“ b
refine le_antisymm (sup_le symmDiff_le_sup inf_le_sup) ?_
[]
refine
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± ⊒ a βˆ† b βŠ” a βŠ“ b = a βŠ” b
a βŠ” b ≀ (a \ b βŠ” b \ a βŠ” a) βŠ“ (a \ b βŠ” b \ a βŠ” b)
rw [sup_inf_left, symmDiff]
[]
rw
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± ⊒ a βŠ” b ≀ a βˆ† b βŠ” a βŠ“ b
a ≀ a \ b βŠ” b \ a βŠ” b
refine sup_le (le_inf le_sup_right ?_) (le_inf ?_ le_sup_right)
[]
refine_1
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± ⊒ a βŠ” b ≀ (a \ b βŠ” b \ a βŠ” a) βŠ“ (a \ b βŠ” b \ a βŠ” b)
b ≀ a \ b βŠ” b \ a βŠ” a
refine sup_le (le_inf le_sup_right ?_) (le_inf ?_ le_sup_right)
[]
refine_2
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± refine_1 : a ≀ a \ b βŠ” b \ a βŠ” b ⊒ a βŠ” b ≀ (a \ b βŠ” b \ a βŠ” a) βŠ“ (a \ b βŠ” b \ a βŠ” b)
a ≀ a \ b βŠ” b βŠ” b \ a
rw [sup_right_comm]
[]
refine_1
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± ⊒ a ≀ a \ b βŠ” b \ a βŠ” b
b ≀ a \ b βŠ” (b \ a βŠ” a)
rw [sup_assoc]
[]
refine_2
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± ⊒ b ≀ a \ b βŠ” b \ a βŠ” a
a βˆ† b βŠ” a βŠ“ b = a βŠ” b
rw [sup_comm, symmDiff_sup_inf]
[]
rw
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± ⊒ a βŠ“ b βŠ” a βˆ† b = a βŠ” b
a βŠ” b = a βŠ” b
rw [sup_comm, symmDiff_sup_inf]
[]
rw₁
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± rw : a βˆ† b βŠ” a βŠ“ b = a βŠ” b ⊒ a βŠ“ b βŠ” a βˆ† b = a βŠ” b
a βŠ” b = a βŠ” b
rw [sup_comm, symmDiff_sup_inf]
[]
rwβ‚‚
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± rw : a βˆ† b βŠ” a βŠ“ b = a βŠ” b rw₁ : a βŠ” b = a βŠ” b ⊒ a βŠ“ b βŠ” a βˆ† b = a βŠ” b
(a βˆ† b \ (a βŠ“ b)) βˆ† (a βŠ“ b) = a βŠ” b
rw [← symmDiff_sdiff_inf a, sdiff_symmDiff_eq_sup, symmDiff_sup_inf]
[]
rw
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± ⊒ a βˆ† b βˆ† (a βŠ“ b) = a βŠ” b
a βˆ† b βŠ” a βŠ“ b = a βŠ” b
rw [← symmDiff_sdiff_inf a, sdiff_symmDiff_eq_sup, symmDiff_sup_inf]
[]
rw₁
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± rw : (a βˆ† b \ (a βŠ“ b)) βˆ† (a βŠ“ b) = a βŠ” b ⊒ a βˆ† b βˆ† (a βŠ“ b) = a βŠ” b
a βŠ” b = a βŠ” b
rw [← symmDiff_sdiff_inf a, sdiff_symmDiff_eq_sup, symmDiff_sup_inf]
[]
rwβ‚‚
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± rw : (a βˆ† b \ (a βŠ“ b)) βˆ† (a βŠ“ b) = a βŠ” b rw₁ : a βˆ† b βŠ” a βŠ“ b = a βŠ” b ⊒ a βˆ† b βˆ† (a βŠ“ b) = a βŠ” b
a βŠ” b = a βŠ” b
rw [← symmDiff_sdiff_inf a, sdiff_symmDiff_eq_sup, symmDiff_sup_inf]
[]
rw₃
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± rw : (a βˆ† b \ (a βŠ“ b)) βˆ† (a βŠ“ b) = a βŠ” b rw₁ : a βˆ† b βŠ” a βŠ“ b = a βŠ” b rwβ‚‚ : a βŠ” b = a βŠ” b ⊒ a βˆ† b βˆ† (a βŠ“ b) = a βŠ” b
a βˆ† b βˆ† (a βŠ“ b) = a βŠ” b
rw [symmDiff_comm, symmDiff_symmDiff_inf]
[]
rw
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± ⊒ (a βŠ“ b) βˆ† (a βˆ† b) = a βŠ” b
a βŠ” b = a βŠ” b
rw [symmDiff_comm, symmDiff_symmDiff_inf]
[]
rw₁
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± rw : a βˆ† b βˆ† (a βŠ“ b) = a βŠ” b ⊒ (a βŠ“ b) βˆ† (a βˆ† b) = a βŠ” b
a βŠ” b = a βŠ” b
rw [symmDiff_comm, symmDiff_symmDiff_inf]
[]
rwβ‚‚
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± rw : a βˆ† b βˆ† (a βŠ“ b) = a βŠ” b rw₁ : a βŠ” b = a βŠ” b ⊒ (a βŠ“ b) βˆ† (a βˆ† b) = a βŠ” b
a \ b βŠ” b \ c βŠ” (c \ b βŠ” b \ a) = a βˆ† b βŠ” b βˆ† c
refine (sup_le_sup (sdiff_triangle a b c) <| sdiff_triangle _ b _).trans_eq ?_
[]
refine
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b c : Ξ± ⊒ a βˆ† c ≀ a βˆ† b βŠ” b βˆ† c
a \ b βŠ” b \ c βŠ” (b \ a βŠ” c \ b) = a βˆ† b βŠ” b βˆ† c
rw [sup_comm (c \ b), sup_sup_sup_comm, symmDiff, symmDiff]
[]
rw
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b c : Ξ± ⊒ a \ b βŠ” b \ c βŠ” (c \ b βŠ” b \ a) = a βˆ† b βŠ” b βˆ† c
a \ b βŠ” b \ a βŠ” (b \ c βŠ” c \ b) = a βˆ† b βŠ” b βˆ† c
rw [sup_comm (c \ b), sup_sup_sup_comm, symmDiff, symmDiff]
[]
rw₁
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b c : Ξ± rw : a \ b βŠ” b \ c βŠ” (b \ a βŠ” c \ b) = a βˆ† b βŠ” b βˆ† c ⊒ a \ b βŠ” b \ c βŠ” (c \ b βŠ” b \ a) = a βˆ† b βŠ” b βˆ† c
a \ b βŠ” b \ a βŠ” (b \ c βŠ” c \ b) = a \ b βŠ” b \ a βŠ” b βˆ† c
rw [sup_comm (c \ b), sup_sup_sup_comm, symmDiff, symmDiff]
[]
rwβ‚‚
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b c : Ξ± rw : a \ b βŠ” b \ c βŠ” (b \ a βŠ” c \ b) = a βˆ† b βŠ” b βˆ† c rw₁ : a \ b βŠ” b \ a βŠ” (b \ c βŠ” c \ b) = a βˆ† b βŠ” b βˆ† c ⊒ a \ b βŠ” b \ c βŠ” (c \ b βŠ” b \ a) = a βˆ† b βŠ” b βˆ† c
a \ b βŠ” b \ a βŠ” (b \ c βŠ” c \ b) = a \ b βŠ” b \ a βŠ” (b \ c βŠ” c \ b)
rw [sup_comm (c \ b), sup_sup_sup_comm, symmDiff, symmDiff]
[]
rw₃
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b c : Ξ± rw : a \ b βŠ” b \ c βŠ” (b \ a βŠ” c \ b) = a βˆ† b βŠ” b βˆ† c rw₁ : a \ b βŠ” b \ a βŠ” (b \ c βŠ” c \ b) = a βˆ† b βŠ” b βˆ† c rwβ‚‚ : a \ b βŠ” b \ a βŠ” (b \ c βŠ” c \ b) = a \ b βŠ” b \ a βŠ” b βˆ† c ⊒ a \ b βŠ” b \ c βŠ” (c \ b βŠ” b \ a) = a βˆ† b βŠ” b βˆ† c
a \ b βŠ” b \ a βŠ” (b \ c βŠ” c \ b) = a \ b βŠ” b \ a βŠ” (b \ c βŠ” c \ b)
rw [sup_comm (c \ b), sup_sup_sup_comm, symmDiff, symmDiff]
[]
rwβ‚„
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b c : Ξ± rw : a \ b βŠ” b \ c βŠ” (b \ a βŠ” c \ b) = a βˆ† b βŠ” b βˆ† c rw₁ : a \ b βŠ” b \ a βŠ” (b \ c βŠ” c \ b) = a βˆ† b βŠ” b βˆ† c rwβ‚‚ : a \ b βŠ” b \ a βŠ” (b \ c βŠ” c \ b) = a \ b βŠ” b \ a βŠ” b βˆ† c rw₃ : a \ b βŠ” b \ a βŠ” (b \ c βŠ” c \ b) = a \ b βŠ” b \ a βŠ” (b \ c βŠ” c \ b) ⊒ a \ b βŠ” b \ c βŠ” (c \ b βŠ” b \ a) = a βˆ† b βŠ” b βˆ† c
a = a βˆ† βŠ₯
convert symmDiff_triangle a b βŠ₯
[]
h
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± ⊒ a ≀ a βˆ† b βŠ” b
b = b βˆ† βŠ₯
convert symmDiff_triangle a b βŠ₯
[]
h₁
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± h : a = a βˆ† βŠ₯ ⊒ a ≀ a βˆ† b βŠ” b
a = a
rw [symmDiff_bot]
[]
h
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± ⊒ a = a βˆ† βŠ₯
a = a
rw [symmDiff_bot]
[]
h₁
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± h : a = a ⊒ a = a βˆ† βŠ₯
b = b
rw [symmDiff_bot]
[]
h
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± ⊒ b = b βˆ† βŠ₯
b = b
rw [symmDiff_bot]
[]
h₁
goal
Ξ± : Type u_2 inst✝ : GeneralizedCoheytingAlgebra Ξ± a b : Ξ± h : b = b ⊒ b = b βˆ† βŠ₯
(a ⇨ a) βŠ“ (a ⇨ a) = ⊀
rw [bihimp, inf_idem, himp_self]
[]
rw
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a : Ξ± ⊒ a ⇔ a = ⊀
a ⇨ a = ⊀
rw [bihimp, inf_idem, himp_self]
[]
rw₁
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a : Ξ± rw : (a ⇨ a) βŠ“ (a ⇨ a) = ⊀ ⊒ a ⇔ a = ⊀
⊀ = ⊀
rw [bihimp, inf_idem, himp_self]
[]
rwβ‚‚
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a : Ξ± rw : (a ⇨ a) βŠ“ (a ⇨ a) = ⊀ rw₁ : a ⇨ a = ⊀ ⊒ a ⇔ a = ⊀
⊀ = ⊀
rw [bihimp, inf_idem, himp_self]
[]
rw₃
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a : Ξ± rw : (a ⇨ a) βŠ“ (a ⇨ a) = ⊀ rw₁ : a ⇨ a = ⊀ rwβ‚‚ : ⊀ = ⊀ ⊒ a ⇔ a = ⊀
(⊀ ⇨ a) βŠ“ (a ⇨ ⊀) = a
rw [bihimp, himp_top, top_himp, inf_top_eq]
[]
rw
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a : Ξ± ⊒ a ⇔ ⊀ = a
(⊀ ⇨ a) βŠ“ ⊀ = a
rw [bihimp, himp_top, top_himp, inf_top_eq]
[]
rw₁
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a : Ξ± rw : (⊀ ⇨ a) βŠ“ (a ⇨ ⊀) = a ⊒ a ⇔ ⊀ = a
a βŠ“ ⊀ = a
rw [bihimp, himp_top, top_himp, inf_top_eq]
[]
rwβ‚‚
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a : Ξ± rw : (⊀ ⇨ a) βŠ“ (a ⇨ ⊀) = a rw₁ : (⊀ ⇨ a) βŠ“ ⊀ = a ⊒ a ⇔ ⊀ = a
a = a
rw [bihimp, himp_top, top_himp, inf_top_eq]
[]
rw₃
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a : Ξ± rw : (⊀ ⇨ a) βŠ“ (a ⇨ ⊀) = a rw₁ : (⊀ ⇨ a) βŠ“ ⊀ = a rwβ‚‚ : a βŠ“ ⊀ = a ⊒ a ⇔ ⊀ = a
a = a
rw [bihimp, himp_top, top_himp, inf_top_eq]
[]
rwβ‚„
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a : Ξ± rw : (⊀ ⇨ a) βŠ“ (a ⇨ ⊀) = a rw₁ : (⊀ ⇨ a) βŠ“ ⊀ = a rwβ‚‚ : a βŠ“ ⊀ = a rw₃ : a = a ⊒ a ⇔ ⊀ = a
a ⇔ ⊀ = a
rw [bihimp_comm, bihimp_top]
[]
rw
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a : Ξ± ⊒ ⊀ ⇔ a = a
a = a
rw [bihimp_comm, bihimp_top]
[]
rw₁
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a : Ξ± rw : a ⇔ ⊀ = a ⊒ ⊀ ⇔ a = a
a = a
rw [bihimp_comm, bihimp_top]
[]
rwβ‚‚
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a : Ξ± rw : a ⇔ ⊀ = a rw₁ : a = a ⊒ ⊀ ⇔ a = a
(b ⇨ a) βŠ“ (a ⇨ b) = b ⇨ a
rw [bihimp, himp_eq_top_iff.2 h, inf_top_eq]
[]
rw
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a b : Ξ± h : a ≀ b ⊒ a ⇔ b = b ⇨ a
(b ⇨ a) βŠ“ ⊀ = b ⇨ a
rw [bihimp, himp_eq_top_iff.2 h, inf_top_eq]
[]
rw₁
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a b : Ξ± h : a ≀ b rw : (b ⇨ a) βŠ“ (a ⇨ b) = b ⇨ a ⊒ a ⇔ b = b ⇨ a
b ⇨ a = b ⇨ a
rw [bihimp, himp_eq_top_iff.2 h, inf_top_eq]
[]
rwβ‚‚
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a b : Ξ± h : a ≀ b rw : (b ⇨ a) βŠ“ (a ⇨ b) = b ⇨ a rw₁ : (b ⇨ a) βŠ“ ⊀ = b ⇨ a ⊒ a ⇔ b = b ⇨ a
b ⇨ a = b ⇨ a
rw [bihimp, himp_eq_top_iff.2 h, inf_top_eq]
[]
rw₃
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a b : Ξ± h : a ≀ b rw : (b ⇨ a) βŠ“ (a ⇨ b) = b ⇨ a rw₁ : (b ⇨ a) βŠ“ ⊀ = b ⇨ a rwβ‚‚ : b ⇨ a = b ⇨ a ⊒ a ⇔ b = b ⇨ a
(b ⇨ a) βŠ“ (a ⇨ b) = a ⇨ b
rw [bihimp, himp_eq_top_iff.2 h, top_inf_eq]
[]
rw
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a b : Ξ± h : b ≀ a ⊒ a ⇔ b = a ⇨ b
⊀ βŠ“ (a ⇨ b) = a ⇨ b
rw [bihimp, himp_eq_top_iff.2 h, top_inf_eq]
[]
rw₁
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a b : Ξ± h : b ≀ a rw : (b ⇨ a) βŠ“ (a ⇨ b) = a ⇨ b ⊒ a ⇔ b = a ⇨ b
a ⇨ b = a ⇨ b
rw [bihimp, himp_eq_top_iff.2 h, top_inf_eq]
[]
rwβ‚‚
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a b : Ξ± h : b ≀ a rw : (b ⇨ a) βŠ“ (a ⇨ b) = a ⇨ b rw₁ : ⊀ βŠ“ (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
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a b : Ξ± h : b ≀ a rw : (b ⇨ a) βŠ“ (a ⇨ b) = a ⇨ b rw₁ : ⊀ βŠ“ (a ⇨ b) = a ⇨ b rwβ‚‚ : a ⇨ b = a ⇨ b ⊒ a ⇔ b = a ⇨ b
a ≀ (c ⇨ b) βŠ“ (b ⇨ c) ↔ a βŠ“ b ≀ c ∧ a βŠ“ c ≀ b
simp_rw [bihimp, le_inf_iff, le_himp_iff, and_comm]
[]
simp_rw
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a b c : Ξ± ⊒ a ≀ b ⇔ c ↔ a βŠ“ b ≀ c ∧ a βŠ“ c ≀ b
a ≀ c ⇨ b ∧ a ≀ b ⇨ c ↔ a βŠ“ b ≀ c ∧ a βŠ“ c ≀ b
simp_rw [bihimp, le_inf_iff, le_himp_iff, and_comm]
[]
simp_rw₁
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a b c : Ξ± simp_rw : a ≀ (c ⇨ b) βŠ“ (b ⇨ c) ↔ a βŠ“ b ≀ c ∧ a βŠ“ c ≀ b ⊒ a ≀ b ⇔ c ↔ a βŠ“ b ≀ c ∧ a βŠ“ c ≀ b
a βŠ“ c ≀ b ∧ a βŠ“ b ≀ c ↔ a βŠ“ b ≀ c ∧ a βŠ“ c ≀ b
simp_rw [bihimp, le_inf_iff, le_himp_iff, and_comm]
[]
simp_rwβ‚‚
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a b c : Ξ± simp_rw : a ≀ (c ⇨ b) βŠ“ (b ⇨ c) ↔ a βŠ“ b ≀ c ∧ a βŠ“ c ≀ b simp_rw₁ : a ≀ c ⇨ b ∧ a ≀ b ⇨ c ↔ a βŠ“ b ≀ c ∧ a βŠ“ c ≀ b ⊒ a ≀ b ⇔ c ↔ a βŠ“ b ≀ c ∧ a βŠ“ c ≀ b
(b ⇨ a) βŠ“ (a ⇨ b) = a βŠ“ b
rw [bihimp, h.himp_eq_left, h.himp_eq_right]
[]
rw
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a b : Ξ± h : Codisjoint a b ⊒ a ⇔ b = a βŠ“ b
(b ⇨ a) βŠ“ b = a βŠ“ b
rw [bihimp, h.himp_eq_left, h.himp_eq_right]
[]
rw₁
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a b : Ξ± h : Codisjoint a b rw : (b ⇨ a) βŠ“ (a ⇨ b) = a βŠ“ b ⊒ a ⇔ b = a βŠ“ b
a βŠ“ b = a βŠ“ b
rw [bihimp, h.himp_eq_left, h.himp_eq_right]
[]
rwβ‚‚
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a b : Ξ± h : Codisjoint a b rw : (b ⇨ a) βŠ“ (a ⇨ b) = a βŠ“ b rw₁ : (b ⇨ a) βŠ“ b = a βŠ“ b ⊒ a ⇔ b = a βŠ“ b
a βŠ“ b = a βŠ“ b
rw [bihimp, h.himp_eq_left, h.himp_eq_right]
[]
rw₃
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a b : Ξ± h : Codisjoint a b rw : (b ⇨ a) βŠ“ (a ⇨ b) = a βŠ“ b rw₁ : (b ⇨ a) βŠ“ b = a βŠ“ b rwβ‚‚ : a βŠ“ b = a βŠ“ b ⊒ a ⇔ b = a βŠ“ b
a ⇨ (c ⇨ b) βŠ“ (b ⇨ c) = (a βŠ“ c ⇨ b) βŠ“ (a βŠ“ b ⇨ c)
rw [bihimp, himp_inf_distrib, himp_himp, himp_himp]
[]
rw
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a b c : Ξ± ⊒ a ⇨ b ⇔ c = (a βŠ“ c ⇨ b) βŠ“ (a βŠ“ b ⇨ c)
(a ⇨ c ⇨ b) βŠ“ (a ⇨ b ⇨ c) = (a βŠ“ c ⇨ b) βŠ“ (a βŠ“ b ⇨ c)
rw [bihimp, himp_inf_distrib, himp_himp, himp_himp]
[]
rw₁
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a b c : Ξ± rw : a ⇨ (c ⇨ b) βŠ“ (b ⇨ c) = (a βŠ“ c ⇨ b) βŠ“ (a βŠ“ b ⇨ c) ⊒ a ⇨ b ⇔ c = (a βŠ“ c ⇨ b) βŠ“ (a βŠ“ b ⇨ c)
(a βŠ“ c ⇨ b) βŠ“ (a ⇨ b ⇨ c) = (a βŠ“ c ⇨ b) βŠ“ (a βŠ“ b ⇨ c)
rw [bihimp, himp_inf_distrib, himp_himp, himp_himp]
[]
rwβ‚‚
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a b c : Ξ± rw : a ⇨ (c ⇨ b) βŠ“ (b ⇨ c) = (a βŠ“ c ⇨ b) βŠ“ (a βŠ“ b ⇨ c) rw₁ : (a ⇨ c ⇨ b) βŠ“ (a ⇨ b ⇨ c) = (a βŠ“ c ⇨ b) βŠ“ (a βŠ“ b ⇨ c) ⊒ a ⇨ b ⇔ c = (a βŠ“ c ⇨ b) βŠ“ (a βŠ“ b ⇨ c)
(a βŠ“ c ⇨ b) βŠ“ (a βŠ“ b ⇨ c) = (a βŠ“ c ⇨ b) βŠ“ (a βŠ“ b ⇨ c)
rw [bihimp, himp_inf_distrib, himp_himp, himp_himp]
[]
rw₃
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a b c : Ξ± rw : a ⇨ (c ⇨ b) βŠ“ (b ⇨ c) = (a βŠ“ c ⇨ b) βŠ“ (a βŠ“ b ⇨ c) rw₁ : (a ⇨ c ⇨ b) βŠ“ (a ⇨ b ⇨ c) = (a βŠ“ c ⇨ b) βŠ“ (a βŠ“ b ⇨ c) rwβ‚‚ : (a βŠ“ c ⇨ b) βŠ“ (a ⇨ b ⇨ c) = (a βŠ“ c ⇨ b) βŠ“ (a βŠ“ b ⇨ c) ⊒ a ⇨ b ⇔ c = (a βŠ“ c ⇨ b) βŠ“ (a βŠ“ b ⇨ c)
(a βŠ“ c ⇨ b) βŠ“ (a βŠ“ b ⇨ c) = (a βŠ“ c ⇨ b) βŠ“ (a βŠ“ b ⇨ c)
rw [bihimp, himp_inf_distrib, himp_himp, himp_himp]
[]
rwβ‚„
goal
Ξ± : Type u_2 inst✝ : GeneralizedHeytingAlgebra Ξ± a b c : Ξ± rw : a ⇨ (c ⇨ b) βŠ“ (b ⇨ c) = (a βŠ“ c ⇨ b) βŠ“ (a βŠ“ b ⇨ c) rw₁ : (a ⇨ c ⇨ b) βŠ“ (a ⇨ b ⇨ c) = (a βŠ“ c ⇨ b) βŠ“ (a βŠ“ b ⇨ c) rwβ‚‚ : (a βŠ“ c ⇨ b) βŠ“ (a ⇨ b ⇨ c) = (a βŠ“ c ⇨ b) βŠ“ (a βŠ“ b ⇨ c) rw₃ : (a βŠ“ c ⇨ b) βŠ“ (a βŠ“ b ⇨ c) = (a βŠ“ c ⇨ b) βŠ“ (a βŠ“ b ⇨ c) ⊒ a ⇨ b ⇔ c = (a βŠ“ c ⇨ b) βŠ“ (a βŠ“ b ⇨ c)
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