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 β a \ a = β₯ | rw [symmDiff, sup_idem, sdiff_self] | [] | rw | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a : Ξ±
β’ a β a = β₯ |
a \ a = β₯ | rw [symmDiff, sup_idem, sdiff_self] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a : Ξ±
rw : a \ a β a \ a = β₯
β’ a β a = β₯ |
β₯ = β₯ | rw [symmDiff, sup_idem, sdiff_self] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a : Ξ±
rw : a \ a β a \ a = β₯
rwβ : a \ a = β₯
β’ a β a = β₯ |
β₯ = β₯ | rw [symmDiff, sup_idem, sdiff_self] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a : Ξ±
rw : a \ a β a \ a = β₯
rwβ : a \ a = β₯
rwβ : β₯ = β₯
β’ a β a = β₯ |
a \ β₯ β β₯ \ a = a | rw [symmDiff, sdiff_bot, bot_sdiff, sup_bot_eq] | [] | rw | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a : Ξ±
β’ a β β₯ = a |
a β β₯ \ a = a | rw [symmDiff, sdiff_bot, bot_sdiff, sup_bot_eq] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a : Ξ±
rw : a \ β₯ β β₯ \ a = a
β’ a β β₯ = a |
a β β₯ = a | rw [symmDiff, sdiff_bot, bot_sdiff, sup_bot_eq] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a : Ξ±
rw : a \ β₯ β β₯ \ a = a
rwβ : a β β₯ \ a = a
β’ a β β₯ = a |
a = a | rw [symmDiff, sdiff_bot, bot_sdiff, sup_bot_eq] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a : Ξ±
rw : a \ β₯ β β₯ \ a = a
rwβ : a β β₯ \ a = a
rwβ : a β β₯ = a
β’ a β β₯ = a |
a = a | rw [symmDiff, sdiff_bot, bot_sdiff, sup_bot_eq] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a : Ξ±
rw : a \ β₯ β β₯ \ a = a
rwβ : a β β₯ \ a = a
rwβ : a β β₯ = a
rwβ : a = a
β’ a β β₯ = a |
a β β₯ = a | rw [symmDiff_comm, symmDiff_bot] | [] | rw | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a : Ξ±
β’ β₯ β a = a |
a = a | rw [symmDiff_comm, symmDiff_bot] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a : Ξ±
rw : a β β₯ = a
β’ β₯ β a = a |
a = a | rw [symmDiff_comm, symmDiff_bot] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a : Ξ±
rw : a β β₯ = a
rwβ : a = a
β’ β₯ β a = a |
a \ b β b \ a = β₯ β a = b | simp_rw [symmDiff, sup_eq_bot_iff, sdiff_eq_bot_iff, le_antisymm_iff] | [] | simp_rw | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
β’ a β b = β₯ β a = b |
a \ b = β₯ β§ b \ a = β₯ β a = b | simp_rw [symmDiff, sup_eq_bot_iff, sdiff_eq_bot_iff, le_antisymm_iff] | [] | simp_rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
simp_rw : a \ b β b \ a = β₯ β a = b
β’ a β b = β₯ β a = b |
a β€ b β§ b β€ a β a = b | simp_rw [symmDiff, sup_eq_bot_iff, sdiff_eq_bot_iff, le_antisymm_iff] | [] | simp_rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
simp_rw : a \ b β b \ a = β₯ β a = b
simp_rwβ : a \ b = β₯ β§ b \ a = β₯ β a = b
β’ a β b = β₯ β a = b |
a \ b β b \ a = b \ a | rw [symmDiff, sdiff_eq_bot_iff.2 h, bot_sup_eq] | [] | rw | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
h : a β€ b
β’ a β b = b \ a |
β₯ β b \ a = b \ a | rw [symmDiff, sdiff_eq_bot_iff.2 h, bot_sup_eq] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
h : a β€ b
rw : a \ b β b \ a = b \ a
β’ a β b = b \ a |
b \ a = b \ a | rw [symmDiff, sdiff_eq_bot_iff.2 h, bot_sup_eq] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
h : a β€ b
rw : a \ b β b \ a = b \ a
rwβ : β₯ β b \ a = b \ a
β’ a β b = b \ a |
b \ a = b \ a | rw [symmDiff, sdiff_eq_bot_iff.2 h, bot_sup_eq] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
h : a β€ b
rw : a \ b β b \ a = b \ a
rwβ : β₯ β b \ a = b \ a
rwβ : b \ a = b \ a
β’ a β b = b \ a |
a \ b β b \ a = a \ b | rw [symmDiff, sdiff_eq_bot_iff.2 h, sup_bot_eq] | [] | rw | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
h : b β€ a
β’ a β b = a \ b |
a \ b β β₯ = a \ b | rw [symmDiff, sdiff_eq_bot_iff.2 h, sup_bot_eq] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
h : b β€ a
rw : a \ b β b \ a = a \ b
β’ a β b = a \ b |
a \ b = a \ b | rw [symmDiff, sdiff_eq_bot_iff.2 h, sup_bot_eq] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
h : b β€ a
rw : a \ b β b \ a = a \ b
rwβ : a \ b β β₯ = a \ b
β’ a β b = a \ b |
a \ b = a \ b | rw [symmDiff, sdiff_eq_bot_iff.2 h, sup_bot_eq] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
h : b β€ a
rw : a \ b β b \ a = a \ b
rwβ : a \ b β β₯ = a \ b
rwβ : a \ b = a \ b
β’ a β b = a \ b |
a \ b β b \ a β€ c β a β€ b β c β§ b β€ a β c | simp_rw [symmDiff, sup_le_iff, sdiff_le_iff] | [] | simp_rw | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b c : Ξ±
β’ a β b β€ c β a β€ b β c β§ b β€ a β c |
a \ b β€ c β§ b \ a β€ c β a β€ b β c β§ b β€ a β c | simp_rw [symmDiff, sup_le_iff, sdiff_le_iff] | [] | simp_rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b c : Ξ±
simp_rw : a \ b β b \ a β€ c β a β€ b β c β§ b β€ a β c
β’ a β b β€ c β a β€ b β c β§ b β€ a β c |
a \ b β b \ a = a β b | rw [symmDiff, h.sdiff_eq_left, h.sdiff_eq_right] | [] | rw | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
h : Disjoint a b
β’ a β b = a β b |
a β b \ a = a β b | rw [symmDiff, h.sdiff_eq_left, h.sdiff_eq_right] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
h : Disjoint a b
rw : a \ b β b \ a = a β b
β’ a β b = a β b |
a β b = a β b | rw [symmDiff, h.sdiff_eq_left, h.sdiff_eq_right] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
h : Disjoint a b
rw : a \ b β b \ a = a β b
rwβ : a β b \ a = a β b
β’ a β b = a β b |
a β b = a β b | rw [symmDiff, h.sdiff_eq_left, h.sdiff_eq_right] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
h : Disjoint a b
rw : a \ b β b \ a = a β b
rwβ : a β b \ a = a β b
rwβ : a β b = a β b
β’ a β b = a β b |
(a \ b β b \ a) \ c = a \ (b β c) β b \ (a β c) | rw [symmDiff, sup_sdiff_distrib, sdiff_sdiff_left, sdiff_sdiff_left] | [] | rw | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b c : Ξ±
β’ a β b \ c = a \ (b β c) β b \ (a β c) |
(a \ b) \ c β (b \ a) \ c = a \ (b β c) β b \ (a β c) | rw [symmDiff, sup_sdiff_distrib, sdiff_sdiff_left, sdiff_sdiff_left] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b c : Ξ±
rw : (a \ b β b \ a) \ c = a \ (b β c) β b \ (a β c)
β’ a β b \ c = a \ (b β c) β b \ (a β c) |
a \ (b β c) β (b \ a) \ c = a \ (b β c) β b \ (a β c) | rw [symmDiff, sup_sdiff_distrib, sdiff_sdiff_left, sdiff_sdiff_left] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b c : Ξ±
rw : (a \ b β b \ a) \ c = a \ (b β c) β b \ (a β c)
rwβ : (a \ b) \ c β (b \ a) \ c = a \ (b β c) β b \ (a β c)
β’ a β b \ c = a \ (b β c) β b \ (a β c) |
a \ (b β c) β b \ (a β c) = a \ (b β c) β b \ (a β c) | rw [symmDiff, sup_sdiff_distrib, sdiff_sdiff_left, sdiff_sdiff_left] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b c : Ξ±
rw : (a \ b β b \ a) \ c = a \ (b β c) β b \ (a β c)
rwβ : (a \ b) \ c β (b \ a) \ c = a \ (b β c) β b \ (a β c)
rwβ : a \ (b β c) β (b \ a) \ c = a \ (b β c) β b \ (a β c)
β’ a β b \ c = a \ (b β c) β b \ (a β c) |
a \ (b β c) β b \ (a β c) = a \ (b β c) β b \ (a β c) | rw [symmDiff, sup_sdiff_distrib, sdiff_sdiff_left, sdiff_sdiff_left] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b c : Ξ±
rw : (a \ b β b \ a) \ c = a \ (b β c) β b \ (a β c)
rwβ : (a \ b) \ c β (b \ a) \ c = a \ (b β c) β b \ (a β c)
rwβ : a \ (b β c) β (b \ a) \ c = a \ (b β c) β b \ (a β c)
rwβ : a \ (b β c) β b \ (a β c) = a \ (b β c) β b \ (a β c)
β’ a β b \ c = a \ (b β c) ... |
a \ (b β a β b) β b \ (a β a β b) = a β b | rw [symmDiff_sdiff] | [] | rw | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
β’ a β b \ (a β b) = a β b |
a \ (b \ a) β b \ a = a β b | rw [symmDiff, sdiff_idem] | [] | rw | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
β’ a β (b \ a) = a β b |
b β (a \ b) = a β b | rw [symmDiff_comm, symmDiff_sdiff_eq_sup, sup_comm] | [] | rw | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
β’ (a \ b) β b = a β b |
b β a = a β b | rw [symmDiff_comm, symmDiff_sdiff_eq_sup, sup_comm] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
rw : b β (a \ b) = a β b
β’ (a \ b) β b = a β b |
a β b = a β b | rw [symmDiff_comm, symmDiff_sdiff_eq_sup, sup_comm] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
rw : b β (a \ b) = a β b
rwβ : b β a = a β b
β’ (a \ b) β b = a β b |
a β b = a β b | rw [symmDiff_comm, symmDiff_sdiff_eq_sup, sup_comm] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
rw : b β (a \ b) = a β b
rwβ : b β a = a β b
rwβ : a β b = a β b
β’ (a \ b) β b = a β b |
a β b β€ a β b β a β b | refine le_antisymm (sup_le symmDiff_le_sup inf_le_sup) ?_ | [] | refine | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
β’ a β b β a β b = a β b |
a β b β€ (a \ b β b \ a β a) β (a \ b β b \ a β b) | rw [sup_inf_left, symmDiff] | [] | rw | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
β’ a β b β€ a β b β a β b |
a β€ a \ b β b \ a β b | refine sup_le (le_inf le_sup_right ?_) (le_inf ?_ le_sup_right) | [] | refine_1 | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
β’ a β b β€ (a \ b β b \ a β a) β (a \ b β b \ a β b) |
b β€ a \ b β b \ a β a | refine sup_le (le_inf le_sup_right ?_) (le_inf ?_ le_sup_right) | [] | refine_2 | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
refine_1 : a β€ a \ b β b \ a β b
β’ a β b β€ (a \ b β b \ a β a) β (a \ b β b \ a β b) |
a β€ a \ b β b β b \ a | rw [sup_right_comm] | [] | refine_1 | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
β’ a β€ a \ b β b \ a β b |
b β€ a \ b β (b \ a β a) | rw [sup_assoc] | [] | refine_2 | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
β’ b β€ a \ b β b \ a β a |
a β b β a β b = a β b | rw [sup_comm, symmDiff_sup_inf] | [] | rw | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
β’ a β b β a β b = a β b |
a β b = a β b | rw [sup_comm, symmDiff_sup_inf] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
rw : a β b β a β b = a β b
β’ a β b β a β b = a β b |
a β b = a β b | rw [sup_comm, symmDiff_sup_inf] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
rw : a β b β a β b = a β b
rwβ : a β b = a β b
β’ a β b β a β b = a β b |
(a β b \ (a β b)) β (a β b) = a β b | rw [β symmDiff_sdiff_inf a, sdiff_symmDiff_eq_sup, symmDiff_sup_inf] | [] | rw | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
β’ a β b β (a β b) = a β b |
a β b β a β b = a β b | rw [β symmDiff_sdiff_inf a, sdiff_symmDiff_eq_sup, symmDiff_sup_inf] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
rw : (a β b \ (a β b)) β (a β b) = a β b
β’ a β b β (a β b) = a β b |
a β b = a β b | rw [β symmDiff_sdiff_inf a, sdiff_symmDiff_eq_sup, symmDiff_sup_inf] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
rw : (a β b \ (a β b)) β (a β b) = a β b
rwβ : a β b β a β b = a β b
β’ a β b β (a β b) = a β b |
a β b = a β b | rw [β symmDiff_sdiff_inf a, sdiff_symmDiff_eq_sup, symmDiff_sup_inf] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
rw : (a β b \ (a β b)) β (a β b) = a β b
rwβ : a β b β a β b = a β b
rwβ : a β b = a β b
β’ a β b β (a β b) = a β b |
a β b β (a β b) = a β b | rw [symmDiff_comm, symmDiff_symmDiff_inf] | [] | rw | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
β’ (a β b) β (a β b) = a β b |
a β b = a β b | rw [symmDiff_comm, symmDiff_symmDiff_inf] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
rw : a β b β (a β b) = a β b
β’ (a β b) β (a β b) = a β b |
a β b = a β b | rw [symmDiff_comm, symmDiff_symmDiff_inf] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
rw : a β b β (a β b) = a β b
rwβ : a β b = a β b
β’ (a β b) β (a β b) = a β b |
a \ b β b \ c β (c \ b β b \ a) = a β b β b β c | refine (sup_le_sup (sdiff_triangle a b c) <| sdiff_triangle _ b _).trans_eq ?_ | [] | refine | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b c : Ξ±
β’ a β c β€ a β b β b β c |
a \ b β b \ c β (b \ a β c \ b) = a β b β b β c | rw [sup_comm (c \ b), sup_sup_sup_comm, symmDiff, symmDiff] | [] | rw | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b c : Ξ±
β’ a \ b β b \ c β (c \ b β b \ a) = a β b β b β c |
a \ b β b \ a β (b \ c β c \ b) = a β b β b β c | rw [sup_comm (c \ b), sup_sup_sup_comm, symmDiff, symmDiff] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b c : Ξ±
rw : a \ b β b \ c β (b \ a β c \ b) = a β b β b β c
β’ a \ b β b \ c β (c \ b β b \ a) = a β b β b β c |
a \ b β b \ a β (b \ c β c \ b) = a \ b β b \ a β b β c | rw [sup_comm (c \ b), sup_sup_sup_comm, symmDiff, symmDiff] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b c : Ξ±
rw : a \ b β b \ c β (b \ a β c \ b) = a β b β b β c
rwβ : a \ b β b \ a β (b \ c β c \ b) = a β b β b β c
β’ a \ b β b \ c β (c \ b β b \ a) = a β b β b β c |
a \ b β b \ a β (b \ c β c \ b) = a \ b β b \ a β (b \ c β c \ b) | rw [sup_comm (c \ b), sup_sup_sup_comm, symmDiff, symmDiff] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b c : Ξ±
rw : a \ b β b \ c β (b \ a β c \ b) = a β b β b β c
rwβ : a \ b β b \ a β (b \ c β c \ b) = a β b β b β c
rwβ : a \ b β b \ a β (b \ c β c \ b) = a \ b β b \ a β b β c
β’ a \ b β b \ c β (c \ b β b \ a) = a β b β b β c |
a \ b β b \ a β (b \ c β c \ b) = a \ b β b \ a β (b \ c β c \ b) | rw [sup_comm (c \ b), sup_sup_sup_comm, symmDiff, symmDiff] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b c : Ξ±
rw : a \ b β b \ c β (b \ a β c \ b) = a β b β b β c
rwβ : a \ b β b \ a β (b \ c β c \ b) = a β b β b β c
rwβ : a \ b β b \ a β (b \ c β c \ b) = a \ b β b \ a β b β c
rwβ : a \ b β b \ a β (b \ c β c \ b) = a \ b β b \ a β (b \ c β c \ b)
β’ a \ b β b \ c β ... |
a = a β β₯ | convert symmDiff_triangle a b β₯ | [] | h | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
β’ a β€ a β b β b |
b = b β β₯ | convert symmDiff_triangle a b β₯ | [] | hβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
h : a = a β β₯
β’ a β€ a β b β b |
a = a | rw [symmDiff_bot] | [] | h | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
β’ a = a β β₯ |
a = a | rw [symmDiff_bot] | [] | hβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
h : a = a
β’ a = a β β₯ |
b = b | rw [symmDiff_bot] | [] | h | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
β’ b = b β β₯ |
b = b | rw [symmDiff_bot] | [] | hβ | goal | Ξ± : Type u_2
instβ : GeneralizedCoheytingAlgebra Ξ±
a b : Ξ±
h : b = b
β’ b = b β β₯ |
(a β¨ a) β (a β¨ a) = β€ | rw [bihimp, inf_idem, himp_self] | [] | rw | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a : Ξ±
β’ a β a = β€ |
a β¨ a = β€ | rw [bihimp, inf_idem, himp_self] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a : Ξ±
rw : (a β¨ a) β (a β¨ a) = β€
β’ a β a = β€ |
β€ = β€ | rw [bihimp, inf_idem, himp_self] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a : Ξ±
rw : (a β¨ a) β (a β¨ a) = β€
rwβ : a β¨ a = β€
β’ a β a = β€ |
β€ = β€ | rw [bihimp, inf_idem, himp_self] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a : Ξ±
rw : (a β¨ a) β (a β¨ a) = β€
rwβ : a β¨ a = β€
rwβ : β€ = β€
β’ a β a = β€ |
(β€ β¨ a) β (a β¨ β€) = a | rw [bihimp, himp_top, top_himp, inf_top_eq] | [] | rw | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a : Ξ±
β’ a β β€ = a |
(β€ β¨ a) β β€ = a | rw [bihimp, himp_top, top_himp, inf_top_eq] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a : Ξ±
rw : (β€ β¨ a) β (a β¨ β€) = a
β’ a β β€ = a |
a β β€ = a | rw [bihimp, himp_top, top_himp, inf_top_eq] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a : Ξ±
rw : (β€ β¨ a) β (a β¨ β€) = a
rwβ : (β€ β¨ a) β β€ = a
β’ a β β€ = a |
a = a | rw [bihimp, himp_top, top_himp, inf_top_eq] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a : Ξ±
rw : (β€ β¨ a) β (a β¨ β€) = a
rwβ : (β€ β¨ a) β β€ = a
rwβ : a β β€ = a
β’ a β β€ = a |
a = a | rw [bihimp, himp_top, top_himp, inf_top_eq] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a : Ξ±
rw : (β€ β¨ a) β (a β¨ β€) = a
rwβ : (β€ β¨ a) β β€ = a
rwβ : a β β€ = a
rwβ : a = a
β’ a β β€ = a |
a β β€ = a | rw [bihimp_comm, bihimp_top] | [] | rw | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a : Ξ±
β’ β€ β a = a |
a = a | rw [bihimp_comm, bihimp_top] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a : Ξ±
rw : a β β€ = a
β’ β€ β a = a |
a = a | rw [bihimp_comm, bihimp_top] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a : Ξ±
rw : a β β€ = a
rwβ : a = a
β’ β€ β a = a |
(b β¨ a) β (a β¨ b) = b β¨ a | rw [bihimp, himp_eq_top_iff.2 h, inf_top_eq] | [] | rw | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a b : Ξ±
h : a β€ b
β’ a β b = b β¨ a |
(b β¨ a) β β€ = b β¨ a | rw [bihimp, himp_eq_top_iff.2 h, inf_top_eq] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a b : Ξ±
h : a β€ b
rw : (b β¨ a) β (a β¨ b) = b β¨ a
β’ a β b = b β¨ a |
b β¨ a = b β¨ a | rw [bihimp, himp_eq_top_iff.2 h, inf_top_eq] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a b : Ξ±
h : a β€ b
rw : (b β¨ a) β (a β¨ b) = b β¨ a
rwβ : (b β¨ a) β β€ = b β¨ a
β’ a β b = b β¨ a |
b β¨ a = b β¨ a | rw [bihimp, himp_eq_top_iff.2 h, inf_top_eq] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a b : Ξ±
h : a β€ b
rw : (b β¨ a) β (a β¨ b) = b β¨ a
rwβ : (b β¨ a) β β€ = b β¨ a
rwβ : b β¨ a = b β¨ a
β’ a β b = b β¨ a |
(b β¨ a) β (a β¨ b) = a β¨ b | rw [bihimp, himp_eq_top_iff.2 h, top_inf_eq] | [] | rw | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a b : Ξ±
h : b β€ a
β’ a β b = a β¨ b |
β€ β (a β¨ b) = a β¨ b | rw [bihimp, himp_eq_top_iff.2 h, top_inf_eq] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a b : Ξ±
h : b β€ a
rw : (b β¨ a) β (a β¨ b) = a β¨ b
β’ a β b = a β¨ b |
a β¨ b = a β¨ b | rw [bihimp, himp_eq_top_iff.2 h, top_inf_eq] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a b : Ξ±
h : b β€ a
rw : (b β¨ a) β (a β¨ b) = a β¨ b
rwβ : β€ β (a β¨ b) = a β¨ b
β’ a β b = a β¨ b |
a β¨ b = a β¨ b | rw [bihimp, himp_eq_top_iff.2 h, top_inf_eq] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a b : Ξ±
h : b β€ a
rw : (b β¨ a) β (a β¨ b) = a β¨ b
rwβ : β€ β (a β¨ b) = a β¨ b
rwβ : a β¨ b = a β¨ b
β’ a β b = a β¨ b |
a β€ (c β¨ b) β (b β¨ c) β a β b β€ c β§ a β c β€ b | simp_rw [bihimp, le_inf_iff, le_himp_iff, and_comm] | [] | simp_rw | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a b c : Ξ±
β’ a β€ b β c β a β b β€ c β§ a β c β€ b |
a β€ c β¨ b β§ a β€ b β¨ c β a β b β€ c β§ a β c β€ b | simp_rw [bihimp, le_inf_iff, le_himp_iff, and_comm] | [] | simp_rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a b c : Ξ±
simp_rw : a β€ (c β¨ b) β (b β¨ c) β a β b β€ c β§ a β c β€ b
β’ a β€ b β c β a β b β€ c β§ a β c β€ b |
a β c β€ b β§ a β b β€ c β a β b β€ c β§ a β c β€ b | simp_rw [bihimp, le_inf_iff, le_himp_iff, and_comm] | [] | simp_rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a b c : Ξ±
simp_rw : a β€ (c β¨ b) β (b β¨ c) β a β b β€ c β§ a β c β€ b
simp_rwβ : a β€ c β¨ b β§ a β€ b β¨ c β a β b β€ c β§ a β c β€ b
β’ a β€ b β c β a β b β€ c β§ a β c β€ b |
(b β¨ a) β (a β¨ b) = a β b | rw [bihimp, h.himp_eq_left, h.himp_eq_right] | [] | rw | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a b : Ξ±
h : Codisjoint a b
β’ a β b = a β b |
(b β¨ a) β b = a β b | rw [bihimp, h.himp_eq_left, h.himp_eq_right] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a b : Ξ±
h : Codisjoint a b
rw : (b β¨ a) β (a β¨ b) = a β b
β’ a β b = a β b |
a β b = a β b | rw [bihimp, h.himp_eq_left, h.himp_eq_right] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a b : Ξ±
h : Codisjoint a b
rw : (b β¨ a) β (a β¨ b) = a β b
rwβ : (b β¨ a) β b = a β b
β’ a β b = a β b |
a β b = a β b | rw [bihimp, h.himp_eq_left, h.himp_eq_right] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a b : Ξ±
h : Codisjoint a b
rw : (b β¨ a) β (a β¨ b) = a β b
rwβ : (b β¨ a) β b = a β b
rwβ : a β b = a β b
β’ a β b = a β b |
a β¨ (c β¨ b) β (b β¨ c) = (a β c β¨ b) β (a β b β¨ c) | rw [bihimp, himp_inf_distrib, himp_himp, himp_himp] | [] | rw | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a b c : Ξ±
β’ a β¨ b β c = (a β c β¨ b) β (a β b β¨ c) |
(a β¨ c β¨ b) β (a β¨ b β¨ c) = (a β c β¨ b) β (a β b β¨ c) | rw [bihimp, himp_inf_distrib, himp_himp, himp_himp] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a b c : Ξ±
rw : a β¨ (c β¨ b) β (b β¨ c) = (a β c β¨ b) β (a β b β¨ c)
β’ a β¨ b β c = (a β c β¨ b) β (a β b β¨ c) |
(a β c β¨ b) β (a β¨ b β¨ c) = (a β c β¨ b) β (a β b β¨ c) | rw [bihimp, himp_inf_distrib, himp_himp, himp_himp] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a b c : Ξ±
rw : a β¨ (c β¨ b) β (b β¨ c) = (a β c β¨ b) β (a β b β¨ c)
rwβ : (a β¨ c β¨ b) β (a β¨ b β¨ c) = (a β c β¨ b) β (a β b β¨ c)
β’ a β¨ b β c = (a β c β¨ b) β (a β b β¨ c) |
(a β c β¨ b) β (a β b β¨ c) = (a β c β¨ b) β (a β b β¨ c) | rw [bihimp, himp_inf_distrib, himp_himp, himp_himp] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a b c : Ξ±
rw : a β¨ (c β¨ b) β (b β¨ c) = (a β c β¨ b) β (a β b β¨ c)
rwβ : (a β¨ c β¨ b) β (a β¨ b β¨ c) = (a β c β¨ b) β (a β b β¨ c)
rwβ : (a β c β¨ b) β (a β¨ b β¨ c) = (a β c β¨ b) β (a β b β¨ c)
β’ a β¨ b β c = (a β c β¨ b) β (a β b β¨ c) |
(a β c β¨ b) β (a β b β¨ c) = (a β c β¨ b) β (a β b β¨ c) | rw [bihimp, himp_inf_distrib, himp_himp, himp_himp] | [] | rwβ | goal | Ξ± : Type u_2
instβ : GeneralizedHeytingAlgebra Ξ±
a b c : Ξ±
rw : a β¨ (c β¨ b) β (b β¨ c) = (a β c β¨ b) β (a β b β¨ c)
rwβ : (a β¨ c β¨ b) β (a β¨ b β¨ c) = (a β c β¨ b) β (a β b β¨ c)
rwβ : (a β c β¨ b) β (a β¨ b β¨ c) = (a β c β¨ b) β (a β b β¨ c)
rwβ : (a β c β¨ b) β (a β b β¨ c) = (a β c β¨ b) β (a β b β¨ c)
β’ a β¨ b β c = (a β c β¨ b) ... |
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