statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
sdiff_sdiff_left : a \ b \ c = a \ (b ⊔ c) | sdiff_sdiff _ _ _ | lemma | sdiff_sdiff_left | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sdiff_sdiff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_right_comm (a b c : α) : a \ b \ c = a \ c \ b | by simp_rw [sdiff_sdiff, sup_comm] | lemma | sdiff_right_comm | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sdiff_sdiff",
"sup_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_sdiff_comm : a \ b \ c = a \ c \ b | sdiff_right_comm _ _ _ | lemma | sdiff_sdiff_comm | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sdiff_right_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_idem : a \ b \ b = a \ b | by rw [sdiff_sdiff_left, sup_idem] | lemma | sdiff_idem | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sdiff_sdiff_left",
"sup_idem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_sdiff_self : a \ b \ a = ⊥ | by rw [sdiff_sdiff_comm, sdiff_self, bot_sdiff] | lemma | sdiff_sdiff_self | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"bot_sdiff",
"sdiff_sdiff_comm",
"sdiff_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_sdiff_distrib (a b c : α) : (a ⊔ b) \ c = a \ c ⊔ b \ c | eq_of_forall_ge_iff $ λ d, by simp_rw [sdiff_le_iff, sup_le_iff, sdiff_le_iff] | lemma | sup_sdiff_distrib | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"eq_of_forall_ge_iff",
"sdiff_le_iff",
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_inf_distrib (a b c : α) : a \ (b ⊓ c) = a \ b ⊔ a \ c | eq_of_forall_ge_iff $ λ d, by { rw [sup_le_iff, sdiff_le_comm, le_inf_iff], simp_rw sdiff_le_comm } | lemma | sdiff_inf_distrib | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"eq_of_forall_ge_iff",
"le_inf_iff",
"sdiff_le_comm",
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_sdiff : (a ⊔ b) \ c = a \ c ⊔ b \ c | sup_sdiff_distrib _ _ _ | lemma | sup_sdiff | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sup_sdiff_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_sdiff_right_self : (a ⊔ b) \ b = a \ b | by rw [sup_sdiff, sdiff_self, sup_bot_eq] | lemma | sup_sdiff_right_self | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sdiff_self",
"sup_bot_eq",
"sup_sdiff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_sdiff_left_self : (a ⊔ b) \ a = b \ a | by rw [sup_comm, sup_sdiff_right_self] | lemma | sup_sdiff_left_self | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sup_comm",
"sup_sdiff_right_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_le_sdiff_right (h : a ≤ b) : a \ c ≤ b \ c | sdiff_le_iff.2 $ h.trans $ le_sup_sdiff | lemma | sdiff_le_sdiff_right | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"le_sup_sdiff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_le_sdiff_left (h : a ≤ b) : c \ b ≤ c \ a | sdiff_le_iff.2 $ le_sup_sdiff.trans $ sup_le_sup_right h _ | lemma | sdiff_le_sdiff_left | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sup_le_sup_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_le_sdiff (hab : a ≤ b) (hcd : c ≤ d) : a \ d ≤ b \ c | (sdiff_le_sdiff_right hab).trans $ sdiff_le_sdiff_left hcd | lemma | sdiff_le_sdiff | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sdiff_le_sdiff_left",
"sdiff_le_sdiff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_inf : a \ (b ⊓ c) = a \ b ⊔ a \ c | sdiff_inf_distrib _ _ _ | lemma | sdiff_inf | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sdiff_inf_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_inf_self_left (a b : α) : a \ (a ⊓ b) = a \ b | by rw [sdiff_inf, sdiff_self, bot_sup_eq] | lemma | sdiff_inf_self_left | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"bot_sup_eq",
"sdiff_inf",
"sdiff_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_inf_self_right (a b : α) : b \ (a ⊓ b) = b \ a | by rw [sdiff_inf, sdiff_self, sup_bot_eq] | lemma | sdiff_inf_self_right | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sdiff_inf",
"sdiff_self",
"sup_bot_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint.sdiff_eq_left (h : disjoint a b) : a \ b = a | by { conv_rhs { rw ←@sdiff_bot _ _ a }, rw [←h.eq_bot, sdiff_inf_self_left] } | lemma | disjoint.sdiff_eq_left | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"disjoint",
"sdiff_bot",
"sdiff_inf_self_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint.sdiff_eq_right (h : disjoint a b) : b \ a = b | h.symm.sdiff_eq_left | lemma | disjoint.sdiff_eq_right | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint.sup_sdiff_cancel_left (h : disjoint a b) : (a ⊔ b) \ a = b | by rw [sup_sdiff, sdiff_self, bot_sup_eq, h.sdiff_eq_right] | lemma | disjoint.sup_sdiff_cancel_left | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"bot_sup_eq",
"disjoint",
"sdiff_self",
"sup_sdiff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint.sup_sdiff_cancel_right (h : disjoint a b) : (a ⊔ b) \ b = a | by rw [sup_sdiff, sdiff_self, sup_bot_eq, h.sdiff_eq_left] | lemma | disjoint.sup_sdiff_cancel_right | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"disjoint",
"sdiff_self",
"sup_bot_eq",
"sup_sdiff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint.le_sdiff_of_le_left (hac : disjoint a c) (hab : a ≤ b) : a ≤ b \ c | hac.sdiff_eq_left.ge.trans $ sdiff_le_sdiff_right hab | lemma | disjoint.le_sdiff_of_le_left | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"disjoint",
"sdiff_le_sdiff_right"
] | See `le_sdiff` for a stronger version in generalised Boolean algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sdiff_sdiff_le : a \ (a \ b) ≤ b | sdiff_le_iff.2 le_sdiff_sup | lemma | sdiff_sdiff_le | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"le_sdiff_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_triangle (a b c : α) : a \ c ≤ a \ b ⊔ b \ c | by { rw [sdiff_le_iff, sup_left_comm, ←sdiff_le_iff], exact sdiff_sdiff_le.trans le_sup_sdiff } | lemma | sdiff_triangle | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"le_sup_sdiff",
"sdiff_le_iff",
"sup_left_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_sup_sdiff_cancel (hba : b ≤ a) (hcb : c ≤ b) : a \ b ⊔ b \ c = a \ c | (sdiff_triangle _ _ _).antisymm' $ sup_le (sdiff_le_sdiff_left hcb) (sdiff_le_sdiff_right hba) | lemma | sdiff_sup_sdiff_cancel | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"antisymm'",
"sdiff_le_sdiff_left",
"sdiff_le_sdiff_right",
"sdiff_triangle",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_le_sdiff_of_sup_le_sup_left (h : c ⊔ a ≤ c ⊔ b) : a \ c ≤ b \ c | by { rw [←sup_sdiff_left_self, ←@sup_sdiff_left_self _ _ _ b], exact sdiff_le_sdiff_right h } | lemma | sdiff_le_sdiff_of_sup_le_sup_left | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sdiff_le_sdiff_right",
"sup_sdiff_left_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_le_sdiff_of_sup_le_sup_right (h : a ⊔ c ≤ b ⊔ c) : a \ c ≤ b \ c | by { rw [←sup_sdiff_right_self, ←@sup_sdiff_right_self _ _ b], exact sdiff_le_sdiff_right h } | lemma | sdiff_le_sdiff_of_sup_le_sup_right | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sdiff_le_sdiff_right",
"sup_sdiff_right_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_sdiff_sup_left : a \ c ⊓ (a ⊔ b) = a \ c | inf_of_le_left $ sdiff_le.trans le_sup_left | lemma | inf_sdiff_sup_left | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"le_sup_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_sdiff_sup_right : a \ c ⊓ (b ⊔ a) = a \ c | inf_of_le_left $ sdiff_le.trans le_sup_right | lemma | inf_sdiff_sup_right | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"le_sup_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generalized_coheyting_algebra.to_distrib_lattice : distrib_lattice α | { le_sup_inf := λ a b c, by simp_rw [←sdiff_le_iff, le_inf_iff, sdiff_le_iff, ←le_inf_iff],
..‹generalized_coheyting_algebra α› } | instance | generalized_coheyting_algebra.to_distrib_lattice | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"distrib_lattice",
"le_inf_iff",
"le_sup_inf",
"sdiff_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.generalized_coheyting_algebra [generalized_coheyting_algebra β] :
generalized_coheyting_algebra (α × β) | { sdiff_le_iff := λ a b c, and_congr sdiff_le_iff sdiff_le_iff,
..prod.lattice α β, ..prod.order_bot α β, ..prod.has_sdiff, ..prod.has_hnot } | instance | prod.generalized_coheyting_algebra | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"generalized_coheyting_algebra",
"sdiff_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.generalized_coheyting_algebra {α : ι → Type*}
[Π i, generalized_coheyting_algebra (α i)] : generalized_coheyting_algebra (Π i, α i) | by { pi_instance, exact λ a b c, forall_congr (λ i, sdiff_le_iff) } | instance | pi.generalized_coheyting_algebra | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"generalized_coheyting_algebra",
"sdiff_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
himp_bot (a : α) : a ⇨ ⊥ = aᶜ | heyting_algebra.himp_bot _ | lemma | himp_bot | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_himp (a : α) : ⊥ ⇨ a = ⊤ | himp_eq_top_iff.2 bot_le | lemma | bot_himp | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"bot_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_sup_distrib (a b : α) : (a ⊔ b)ᶜ = aᶜ ⊓ bᶜ | by simp_rw [←himp_bot, sup_himp_distrib] | lemma | compl_sup_distrib | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sup_himp_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_sup : (a ⊔ b)ᶜ = aᶜ ⊓ bᶜ | compl_sup_distrib _ _ | lemma | compl_sup | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"compl_sup_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_le_himp : aᶜ ≤ a ⇨ b | (himp_bot _).ge.trans $ himp_le_himp_left bot_le | lemma | compl_le_himp | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"bot_le",
"himp_bot",
"himp_le_himp_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_sup_le_himp : aᶜ ⊔ b ≤ a ⇨ b | sup_le compl_le_himp le_himp | lemma | compl_sup_le_himp | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"compl_le_himp",
"le_himp",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_compl_le_himp : b ⊔ aᶜ ≤ a ⇨ b | sup_le le_himp compl_le_himp | lemma | sup_compl_le_himp | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"compl_le_himp",
"le_himp",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
himp_compl (a : α) : a ⇨ aᶜ = aᶜ | by rw [←himp_bot, himp_himp, inf_idem] | lemma | himp_compl | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"himp_himp",
"inf_idem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
himp_compl_comm (a b : α) : a ⇨ bᶜ = b ⇨ aᶜ | by simp_rw [←himp_bot, himp_left_comm] | lemma | himp_compl_comm | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"himp_left_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_compl_iff_disjoint_right : a ≤ bᶜ ↔ disjoint a b | by rw [←himp_bot, le_himp_iff, disjoint_iff_inf_le] | lemma | le_compl_iff_disjoint_right | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"disjoint",
"disjoint_iff_inf_le",
"le_himp_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_compl_iff_disjoint_left : a ≤ bᶜ ↔ disjoint b a | le_compl_iff_disjoint_right.trans disjoint.comm | lemma | le_compl_iff_disjoint_left | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"disjoint",
"disjoint.comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_compl_comm : a ≤ bᶜ ↔ b ≤ aᶜ | by rw [le_compl_iff_disjoint_right, le_compl_iff_disjoint_left] | lemma | le_compl_comm | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"le_compl_iff_disjoint_left",
"le_compl_iff_disjoint_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_compl_left : disjoint aᶜ a | disjoint_iff_inf_le.mpr $ le_himp_iff.1 (himp_bot _).ge | lemma | disjoint_compl_left | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"disjoint",
"himp_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_compl_right : disjoint a aᶜ | disjoint_compl_left.symm | lemma | disjoint_compl_right | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_le.le.disjoint_compl_left (h : b ≤ a) : disjoint aᶜ b | disjoint_compl_left.mono_right h | lemma | has_le.le.disjoint_compl_left | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_le.le.disjoint_compl_right (h : a ≤ b) : disjoint a bᶜ | disjoint_compl_right.mono_left h | lemma | has_le.le.disjoint_compl_right | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compl.compl_eq (h : is_compl a b) : aᶜ = b | h.1.le_compl_left.antisymm' $ disjoint.le_of_codisjoint disjoint_compl_left h.2 | lemma | is_compl.compl_eq | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"disjoint.le_of_codisjoint",
"disjoint_compl_left",
"is_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compl.eq_compl (h : is_compl a b) : a = bᶜ | h.1.le_compl_right.antisymm $ disjoint.le_of_codisjoint disjoint_compl_left h.2.symm | lemma | is_compl.eq_compl | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"disjoint.le_of_codisjoint",
"disjoint_compl_left",
"is_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_unique (h₀ : a ⊓ b = ⊥) (h₁ : a ⊔ b = ⊤) : aᶜ = b | (is_compl.of_eq h₀ h₁).compl_eq | lemma | compl_unique | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"is_compl.of_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_compl_self (a : α) : a ⊓ aᶜ = ⊥ | disjoint_compl_right.eq_bot | lemma | inf_compl_self | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_inf_self (a : α) : aᶜ ⊓ a = ⊥ | disjoint_compl_left.eq_bot | lemma | compl_inf_self | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_compl_eq_bot : a ⊓ aᶜ = ⊥ | inf_compl_self _ | lemma | inf_compl_eq_bot | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"inf_compl_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_inf_eq_bot : aᶜ ⊓ a = ⊥ | compl_inf_self _ | lemma | compl_inf_eq_bot | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"compl_inf_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_top : (⊤ : α)ᶜ = ⊥ | eq_of_forall_le_iff $ λ a, by rw [le_compl_iff_disjoint_right, disjoint_top, le_bot_iff] | lemma | compl_top | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"disjoint_top",
"eq_of_forall_le_iff",
"le_bot_iff",
"le_compl_iff_disjoint_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_bot : (⊥ : α)ᶜ = ⊤ | by rw [←himp_bot, himp_self] | lemma | compl_bot | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"himp_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_compl_compl : a ≤ aᶜᶜ | disjoint_compl_right.le_compl_right | lemma | le_compl_compl | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_anti : antitone (compl : α → α) | λ a b h, le_compl_comm.1 $ h.trans le_compl_compl | lemma | compl_anti | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"antitone",
"le_compl_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_le_compl (h : a ≤ b) : bᶜ ≤ aᶜ | compl_anti h | lemma | compl_le_compl | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"compl_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_compl_compl (a : α) : aᶜᶜᶜ = aᶜ | (compl_anti le_compl_compl).antisymm le_compl_compl | lemma | compl_compl_compl | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"compl_anti",
"le_compl_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_compl_compl_left_iff : disjoint aᶜᶜ b ↔ disjoint a b | by simp_rw [←le_compl_iff_disjoint_left, compl_compl_compl] | lemma | disjoint_compl_compl_left_iff | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"compl_compl_compl",
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_compl_compl_right_iff : disjoint a bᶜᶜ ↔ disjoint a b | by simp_rw [←le_compl_iff_disjoint_right, compl_compl_compl] | lemma | disjoint_compl_compl_right_iff | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"compl_compl_compl",
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_sup_compl_le : aᶜ ⊔ bᶜ ≤ (a ⊓ b)ᶜ | sup_le (compl_anti inf_le_left) $ compl_anti inf_le_right | lemma | compl_sup_compl_le | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"compl_anti",
"inf_le_left",
"inf_le_right",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_compl_inf_distrib (a b : α) : (a ⊓ b)ᶜᶜ = aᶜᶜ ⊓ bᶜᶜ | begin
refine ((compl_anti compl_sup_compl_le).trans (compl_sup_distrib _ _).le).antisymm _,
rw [le_compl_iff_disjoint_right, disjoint_assoc, disjoint_compl_compl_left_iff,
disjoint_left_comm, disjoint_compl_compl_left_iff, ←disjoint_assoc, inf_comm],
exact disjoint_compl_right,
end | lemma | compl_compl_inf_distrib | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"compl_anti",
"compl_sup_compl_le",
"compl_sup_distrib",
"disjoint_assoc",
"disjoint_compl_compl_left_iff",
"disjoint_compl_right",
"disjoint_left_comm",
"inf_comm",
"le_compl_iff_disjoint_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_compl_himp_distrib (a b : α) : (a ⇨ b)ᶜᶜ = aᶜᶜ ⇨ bᶜᶜ | begin
refine le_antisymm _ _,
{ rw [le_himp_iff, ←compl_compl_inf_distrib],
exact compl_anti (compl_anti himp_inf_le) },
{ refine le_compl_comm.1 ((compl_anti compl_sup_le_himp).trans _),
rw [compl_sup_distrib, le_compl_iff_disjoint_right, disjoint_right_comm,
←le_compl_iff_disjoint_right],
exac... | lemma | compl_compl_himp_distrib | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"compl_anti",
"compl_sup_distrib",
"compl_sup_le_himp",
"disjoint_right_comm",
"himp_inf_le",
"inf_himp_le",
"le_compl_iff_disjoint_right",
"le_himp_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_dual_hnot (a : αᵒᵈ) : of_dual ￢a = (of_dual a)ᶜ | rfl | lemma | of_dual_hnot | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_dual_compl (a : α) : to_dual aᶜ = ￢to_dual a | rfl | lemma | to_dual_compl | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.heyting_algebra [heyting_algebra β] : heyting_algebra (α × β) | { himp_bot := λ a, prod.ext (himp_bot a.1) (himp_bot a.2),
..prod.generalized_heyting_algebra, ..prod.bounded_order α β, ..prod.has_compl } | instance | prod.heyting_algebra | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"heyting_algebra",
"himp_bot",
"prod.ext",
"prod.generalized_heyting_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.heyting_algebra {α : ι → Type*} [Π i, heyting_algebra (α i)] :
heyting_algebra (Π i, α i) | by { pi_instance, exact λ a b c, forall_congr (λ i, le_himp_iff) } | instance | pi.heyting_algebra | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"heyting_algebra",
"le_himp_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_sdiff' (a : α) : ⊤ \ a = ￢a | coheyting_algebra.top_sdiff _ | lemma | top_sdiff' | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_top (a : α) : a \ ⊤ = ⊥ | sdiff_eq_bot_iff.2 le_top | lemma | sdiff_top | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"le_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hnot_inf_distrib (a b : α) : ￢ (a ⊓ b) = ￢a ⊔ ￢b | by simp_rw [←top_sdiff', sdiff_inf_distrib] | lemma | hnot_inf_distrib | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sdiff_inf_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_le_hnot : a \ b ≤ ￢b | (sdiff_le_sdiff_right le_top).trans_eq $ top_sdiff' _ | lemma | sdiff_le_hnot | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"le_top",
"sdiff_le_sdiff_right",
"top_sdiff'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_le_inf_hnot : a \ b ≤ a ⊓ ￢b | le_inf sdiff_le sdiff_le_hnot | lemma | sdiff_le_inf_hnot | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"le_inf",
"sdiff_le",
"sdiff_le_hnot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coheyting_algebra.to_distrib_lattice : distrib_lattice α | { le_sup_inf := λ a b c, by simp_rw [←sdiff_le_iff, le_inf_iff, sdiff_le_iff, ←le_inf_iff],
..‹coheyting_algebra α› } | instance | coheyting_algebra.to_distrib_lattice | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"distrib_lattice",
"le_inf_iff",
"le_sup_inf",
"sdiff_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hnot_sdiff (a : α) : ￢a \ a = ￢a | by rw [←top_sdiff', sdiff_sdiff, sup_idem] | lemma | hnot_sdiff | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sdiff_sdiff",
"sup_idem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hnot_sdiff_comm (a b : α) : ￢a \ b = ￢b \ a | by simp_rw [←top_sdiff', sdiff_right_comm] | lemma | hnot_sdiff_comm | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sdiff_right_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hnot_le_iff_codisjoint_right : ￢a ≤ b ↔ codisjoint a b | by rw [←top_sdiff', sdiff_le_iff, codisjoint_iff_le_sup] | lemma | hnot_le_iff_codisjoint_right | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"codisjoint",
"codisjoint_iff_le_sup",
"sdiff_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hnot_le_iff_codisjoint_left : ￢a ≤ b ↔ codisjoint b a | hnot_le_iff_codisjoint_right.trans codisjoint.comm | lemma | hnot_le_iff_codisjoint_left | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"codisjoint",
"codisjoint.comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hnot_le_comm : ￢a ≤ b ↔ ￢b ≤ a | by rw [hnot_le_iff_codisjoint_right, hnot_le_iff_codisjoint_left] | lemma | hnot_le_comm | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"hnot_le_iff_codisjoint_left",
"hnot_le_iff_codisjoint_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
codisjoint_hnot_right : codisjoint a (￢a) | codisjoint_iff_le_sup.2 $ sdiff_le_iff.1 (top_sdiff' _).le | lemma | codisjoint_hnot_right | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"codisjoint",
"top_sdiff'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
codisjoint_hnot_left : codisjoint (￢a) a | codisjoint_hnot_right.symm | lemma | codisjoint_hnot_left | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"codisjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_le.le.codisjoint_hnot_left (h : a ≤ b) : codisjoint (￢a) b | codisjoint_hnot_left.mono_right h | lemma | has_le.le.codisjoint_hnot_left | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"codisjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_le.le.codisjoint_hnot_right (h : b ≤ a) : codisjoint a (￢b) | codisjoint_hnot_right.mono_left h | lemma | has_le.le.codisjoint_hnot_right | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"codisjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compl.hnot_eq (h : is_compl a b) : ￢a = b | h.2.hnot_le_right.antisymm $ disjoint.le_of_codisjoint h.1.symm codisjoint_hnot_right | lemma | is_compl.hnot_eq | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"codisjoint_hnot_right",
"disjoint.le_of_codisjoint",
"is_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compl.eq_hnot (h : is_compl a b) : a = ￢b | h.2.hnot_le_left.antisymm' $ disjoint.le_of_codisjoint h.1 codisjoint_hnot_right | lemma | is_compl.eq_hnot | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"codisjoint_hnot_right",
"disjoint.le_of_codisjoint",
"is_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_hnot_self (a : α) : a ⊔ ￢a = ⊤ | codisjoint.eq_top codisjoint_hnot_right | lemma | sup_hnot_self | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"codisjoint.eq_top",
"codisjoint_hnot_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hnot_sup_self (a : α) : ￢a ⊔ a = ⊤ | codisjoint.eq_top codisjoint_hnot_left | lemma | hnot_sup_self | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"codisjoint.eq_top",
"codisjoint_hnot_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hnot_bot : ￢(⊥ : α) = ⊤ | eq_of_forall_ge_iff $ λ a, by rw [hnot_le_iff_codisjoint_left, codisjoint_bot, top_le_iff] | lemma | hnot_bot | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"codisjoint_bot",
"eq_of_forall_ge_iff",
"hnot_le_iff_codisjoint_left",
"top_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hnot_top : ￢(⊤ : α) = ⊥ | by rw [←top_sdiff', sdiff_self] | lemma | hnot_top | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sdiff_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hnot_hnot_le : ￢￢a ≤ a | codisjoint_hnot_right.hnot_le_left | lemma | hnot_hnot_le | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hnot_anti : antitone (hnot : α → α) | λ a b h, hnot_le_comm.1 $ hnot_hnot_le.trans h | lemma | hnot_anti | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"antitone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hnot_le_hnot (h : a ≤ b) : ￢b ≤ ￢a | hnot_anti h | lemma | hnot_le_hnot | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"hnot_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hnot_hnot_hnot (a : α) : ￢￢￢a = ￢a | hnot_hnot_le.antisymm $ hnot_anti hnot_hnot_le | lemma | hnot_hnot_hnot | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"hnot_anti",
"hnot_hnot_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
codisjoint_hnot_hnot_left_iff : codisjoint (￢￢a) b ↔ codisjoint a b | by simp_rw [←hnot_le_iff_codisjoint_right, hnot_hnot_hnot] | lemma | codisjoint_hnot_hnot_left_iff | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"codisjoint",
"hnot_hnot_hnot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
codisjoint_hnot_hnot_right_iff : codisjoint a (￢￢b) ↔ codisjoint a b | by simp_rw [←hnot_le_iff_codisjoint_left, hnot_hnot_hnot] | lemma | codisjoint_hnot_hnot_right_iff | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"codisjoint",
"hnot_hnot_hnot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_hnot_inf_hnot : ￢ (a ⊔ b) ≤ ￢a ⊓ ￢b | le_inf (hnot_anti le_sup_left) $ hnot_anti le_sup_right | lemma | le_hnot_inf_hnot | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"hnot_anti",
"le_inf",
"le_sup_left",
"le_sup_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hnot_hnot_sup_distrib (a b : α) : ￢￢(a ⊔ b) = ￢￢a ⊔ ￢￢b | begin
refine ((hnot_inf_distrib _ _).ge.trans $ hnot_anti le_hnot_inf_hnot).antisymm' _,
rw [hnot_le_iff_codisjoint_left, codisjoint_assoc, codisjoint_hnot_hnot_left_iff,
codisjoint_left_comm, codisjoint_hnot_hnot_left_iff, ←codisjoint_assoc, sup_comm],
exact codisjoint_hnot_right,
end | lemma | hnot_hnot_sup_distrib | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"antisymm'",
"codisjoint_assoc",
"codisjoint_hnot_hnot_left_iff",
"codisjoint_hnot_right",
"codisjoint_left_comm",
"hnot_anti",
"hnot_inf_distrib",
"hnot_le_iff_codisjoint_left",
"le_hnot_inf_hnot",
"sup_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hnot_hnot_sdiff_distrib (a b : α) : ￢￢(a \ b) = ￢￢a \ ￢￢b | begin
refine le_antisymm _ _,
{ refine hnot_le_comm.1 ((hnot_anti sdiff_le_inf_hnot).trans' _),
rw [hnot_inf_distrib, hnot_le_iff_codisjoint_right, codisjoint_left_comm,
←hnot_le_iff_codisjoint_right],
exact le_sdiff_sup },
{ rw [sdiff_le_iff, ←hnot_hnot_sup_distrib],
exact hnot_anti (hnot_anti ... | lemma | hnot_hnot_sdiff_distrib | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"codisjoint_left_comm",
"hnot_anti",
"hnot_inf_distrib",
"hnot_le_iff_codisjoint_right",
"le_sdiff_sup",
"le_sup_sdiff",
"sdiff_le_iff",
"sdiff_le_inf_hnot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_dual_compl (a : αᵒᵈ) : of_dual aᶜ = ￢of_dual a | rfl | lemma | of_dual_compl | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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