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
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
of_dual_himp (a b : αᵒᵈ) : of_dual (a ⇨ b) = of_dual b \ of_dual a | rfl | lemma | of_dual_himp | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_dual_hnot (a : α) : to_dual ￢a = (to_dual a)ᶜ | rfl | lemma | to_dual_hnot | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_dual_sdiff (a b : α) : to_dual (a \ b) = to_dual b ⇨ to_dual a | rfl | lemma | to_dual_sdiff | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.coheyting_algebra [coheyting_algebra β] : coheyting_algebra (α × β) | { sdiff_le_iff := λ a b c, and_congr sdiff_le_iff sdiff_le_iff,
top_sdiff := λ a, prod.ext (top_sdiff' a.1) (top_sdiff' a.2),
..prod.lattice α β, ..prod.bounded_order α β, ..prod.has_sdiff, ..prod.has_hnot } | instance | prod.coheyting_algebra | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"coheyting_algebra",
"prod.ext",
"sdiff_le_iff",
"top_sdiff",
"top_sdiff'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.coheyting_algebra {α : ι → Type*} [Π i, coheyting_algebra (α i)] :
coheyting_algebra (Π i, α i) | by { pi_instance, exact λ a b c, forall_congr (λ i, sdiff_le_iff) } | instance | pi.coheyting_algebra | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"coheyting_algebra",
"sdiff_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_le_hnot : aᶜ ≤ ￢a | (disjoint_compl_left : disjoint _ a).le_of_codisjoint codisjoint_hnot_right | lemma | compl_le_hnot | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"codisjoint_hnot_right",
"disjoint",
"disjoint_compl_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Prop.heyting_algebra : heyting_algebra Prop | { himp := (→),
le_himp_iff := λ p q r, and_imp.symm,
himp_bot := λ p, rfl,
..Prop.has_compl, ..Prop.distrib_lattice, ..Prop.bounded_order } | instance | Prop.heyting_algebra | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"Prop.bounded_order",
"Prop.distrib_lattice",
"Prop.has_compl",
"heyting_algebra",
"himp_bot",
"le_himp_iff"
] | Propositions form a Heyting algebra with implication as Heyting implication and negation as
complement. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
himp_iff_imp (p q : Prop) : p ⇨ q ↔ p → q | iff.rfl | lemma | himp_iff_imp | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_iff_not (p : Prop) : pᶜ ↔ ¬ p | iff.rfl | lemma | compl_iff_not | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_order.to_biheyting_algebra [linear_order α] [bounded_order α] : biheyting_algebra α | { himp := λ a b, if a ≤ b then ⊤ else b,
compl := λ a, if a = ⊥ then ⊤ else ⊥,
le_himp_iff := λ a b c, begin
change _ ≤ ite _ _ _ ↔ _,
split_ifs,
{ exact iff_of_true le_top (inf_le_of_right_le h) },
{ rw [inf_le_iff, or_iff_left h] }
end,
himp_bot := λ a, if_congr le_bot_iff rfl rfl,
sdiff := ... | def | linear_order.to_biheyting_algebra | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"biheyting_algebra",
"bot_le",
"bounded_order",
"himp_bot",
"iff_of_true",
"inf_le_iff",
"inf_le_of_right_le",
"le_bot_iff",
"le_himp_iff",
"le_sup_iff",
"le_sup_of_le_left",
"le_top",
"linear_order.to_lattice",
"or_iff_left",
"or_iff_right",
"sdiff_le_iff",
"top_le_iff",
"top_sdif... | A bounded linear order is a bi-Heyting algebra by setting
* `a ⇨ b = ⊤` if `a ≤ b` and `a ⇨ b = b` otherwise.
* `a \ b = ⊥` if `a ≤ b` and `a \ b = a` otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.injective.generalized_heyting_algebra [has_sup α] [has_inf α] [has_top α]
[has_himp α] [generalized_heyting_algebra β] (f : α → β) (hf : injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b)
(map_top : f ⊤ = ⊤) (map_himp : ∀ a b, f (a ⇨ b) = f a ⇨ f b) :
general... | { le_top := λ a, by { change f _ ≤ _, rw map_top, exact le_top },
le_himp_iff := λ a b c, by { change f _ ≤ _ ↔ f _ ≤ _, erw [map_himp, map_inf, le_himp_iff] },
..hf.lattice f map_sup map_inf, ..‹has_top α›, ..‹has_himp α› } | def | function.injective.generalized_heyting_algebra | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"generalized_heyting_algebra",
"has_himp",
"has_inf",
"has_sup",
"has_top",
"le_himp_iff",
"le_top"
] | Pullback a `generalized_heyting_algebra` along an injection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.injective.generalized_coheyting_algebra [has_sup α] [has_inf α] [has_bot α]
[has_sdiff α] [generalized_coheyting_algebra β] (f : α → β) (hf : injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b)
(map_bot : f ⊥ = ⊥) (map_sdiff : ∀ a b, f (a \ b) = f a \ f b) :
g... | { bot_le := λ a, by { change f _ ≤ _, rw map_bot, exact bot_le },
sdiff_le_iff := λ a b c, by { change f _ ≤ _ ↔ f _ ≤ _, erw [map_sdiff, map_sup, sdiff_le_iff] },
..hf.lattice f map_sup map_inf, ..‹has_bot α›, ..‹has_sdiff α› } | def | function.injective.generalized_coheyting_algebra | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"bot_le",
"generalized_coheyting_algebra",
"has_bot",
"has_inf",
"has_sup",
"sdiff_le_iff"
] | Pullback a `generalized_coheyting_algebra` along an injection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.injective.heyting_algebra [has_sup α] [has_inf α] [has_top α] [has_bot α]
[has_compl α] [has_himp α] [heyting_algebra β] (f : α → β) (hf : injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b)
(map_top : f ⊤ = ⊤) (map_bot : f ⊥ = ⊥) (map_compl : ∀ a, f aᶜ = (f a)ᶜ... | { bot_le := λ a, by { change f _ ≤ _, rw map_bot, exact bot_le },
himp_bot := λ a, hf $ by erw [map_himp, map_compl, map_bot, himp_bot],
..hf.generalized_heyting_algebra f map_sup map_inf map_top map_himp,
..‹has_bot α›, ..‹has_compl α› } | def | function.injective.heyting_algebra | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"bot_le",
"has_bot",
"has_compl",
"has_himp",
"has_inf",
"has_sup",
"has_top",
"heyting_algebra",
"himp_bot",
"map_compl"
] | Pullback a `heyting_algebra` along an injection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.injective.coheyting_algebra [has_sup α] [has_inf α] [has_top α] [has_bot α]
[has_hnot α] [has_sdiff α] [coheyting_algebra β] (f : α → β) (hf : injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b)
(map_top : f ⊤ = ⊤) (map_bot : f ⊥ = ⊥) (map_hnot : ∀ a, f ￢a = ￢f ... | { le_top := λ a, by { change f _ ≤ _, rw map_top, exact le_top },
top_sdiff := λ a, hf $ by erw [map_sdiff, map_hnot, map_top, top_sdiff'],
..hf.generalized_coheyting_algebra f map_sup map_inf map_bot map_sdiff,
..‹has_top α›, ..‹has_hnot α› } | def | function.injective.coheyting_algebra | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"coheyting_algebra",
"has_bot",
"has_hnot",
"has_inf",
"has_sup",
"has_top",
"le_top",
"map_hnot",
"top_sdiff",
"top_sdiff'"
] | Pullback a `coheyting_algebra` along an injection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.injective.biheyting_algebra [has_sup α] [has_inf α] [has_top α] [has_bot α]
[has_compl α] [has_hnot α] [has_himp α] [has_sdiff α] [biheyting_algebra β] (f : α → β)
(hf : injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b)
(map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) (map_top : f ⊤ = ⊤) (map_bot : f ⊥ = ⊥)
... | { ..hf.heyting_algebra f map_sup map_inf map_top map_bot map_compl map_himp,
..hf.coheyting_algebra f map_sup map_inf map_top map_bot map_hnot map_sdiff } | def | function.injective.biheyting_algebra | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"biheyting_algebra",
"has_bot",
"has_compl",
"has_himp",
"has_hnot",
"has_inf",
"has_sup",
"has_top",
"map_compl",
"map_hnot"
] | Pullback a `biheyting_algebra` along an injection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
top_eq : (⊤ : punit) = star | rfl | lemma | punit.top_eq | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_eq : (⊥ : punit) = star | rfl | lemma | punit.bot_eq | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_eq : a ⊔ b = star | rfl | lemma | punit.sup_eq | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_eq : a ⊓ b = star | rfl | lemma | punit.inf_eq | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_eq : aᶜ = star | rfl | lemma | punit.compl_eq | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_eq : a \ b = star | rfl | lemma | punit.sdiff_eq | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sdiff_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hnot_eq : ￢a = star | rfl | lemma | punit.hnot_eq | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
himp_eq : a ⇨ b = star | rfl | lemma | punit.himp_eq | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"himp_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
boundary (a : α) : α | a ⊓ ￢a | def | coheyting.boundary | order.heyting | src/order/heyting/boundary.lean | [
"order.boolean_algebra"
] | [] | The boundary of an element of a co-Heyting algebra is the intersection of its Heyting negation
with itself. Note that this is always `⊥` for a boolean algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_hnot_self (a : α) : a ⊓ ￢a = ∂ a | rfl | lemma | coheyting.inf_hnot_self | order.heyting | src/order/heyting/boundary.lean | [
"order.boolean_algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
boundary_le : ∂ a ≤ a | inf_le_left | lemma | coheyting.boundary_le | order.heyting | src/order/heyting/boundary.lean | [
"order.boolean_algebra"
] | [
"inf_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
boundary_le_hnot : ∂ a ≤ ￢a | inf_le_right | lemma | coheyting.boundary_le_hnot | order.heyting | src/order/heyting/boundary.lean | [
"order.boolean_algebra"
] | [
"inf_le_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
boundary_bot : ∂ (⊥ : α) = ⊥ | bot_inf_eq | lemma | coheyting.boundary_bot | order.heyting | src/order/heyting/boundary.lean | [
"order.boolean_algebra"
] | [
"bot_inf_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
boundary_top : ∂ (⊤ : α) = ⊥ | by rw [boundary, hnot_top, inf_bot_eq] | lemma | coheyting.boundary_top | order.heyting | src/order/heyting/boundary.lean | [
"order.boolean_algebra"
] | [
"hnot_top",
"inf_bot_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
boundary_hnot_le (a : α) : ∂ ￢a ≤ ∂ a | inf_comm.trans_le $ inf_le_inf_right _ hnot_hnot_le | lemma | coheyting.boundary_hnot_le | order.heyting | src/order/heyting/boundary.lean | [
"order.boolean_algebra"
] | [
"hnot_hnot_le",
"inf_le_inf_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
boundary_hnot_hnot (a : α) : ∂ ￢￢a = ∂ ￢a | by simp_rw [boundary, hnot_hnot_hnot, inf_comm] | lemma | coheyting.boundary_hnot_hnot | order.heyting | src/order/heyting/boundary.lean | [
"order.boolean_algebra"
] | [
"hnot_hnot_hnot",
"inf_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hnot_boundary (a : α) : ￢ ∂ a = ⊤ | by rw [boundary, hnot_inf_distrib, sup_hnot_self] | lemma | coheyting.hnot_boundary | order.heyting | src/order/heyting/boundary.lean | [
"order.boolean_algebra"
] | [
"hnot_inf_distrib",
"sup_hnot_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
boundary_inf (a b : α) : ∂ (a ⊓ b) = ∂ a ⊓ b ⊔ a ⊓ ∂ b | by { unfold boundary, rw [hnot_inf_distrib, inf_sup_left, inf_right_comm, ←inf_assoc] } | lemma | coheyting.boundary_inf | order.heyting | src/order/heyting/boundary.lean | [
"order.boolean_algebra"
] | [
"hnot_inf_distrib",
"inf_right_comm",
"inf_sup_left"
] | **Leibniz rule** for the co-Heyting boundary. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
boundary_inf_le : ∂ (a ⊓ b) ≤ ∂ a ⊔ ∂ b | (boundary_inf _ _).trans_le $ sup_le_sup inf_le_left inf_le_right | lemma | coheyting.boundary_inf_le | order.heyting | src/order/heyting/boundary.lean | [
"order.boolean_algebra"
] | [
"inf_le_left",
"inf_le_right",
"sup_le_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
boundary_sup_le : ∂ (a ⊔ b) ≤ ∂ a ⊔ ∂ b | begin
rw [boundary, inf_sup_right],
exact sup_le_sup (inf_le_inf_left _ $ hnot_anti le_sup_left)
(inf_le_inf_left _ $ hnot_anti le_sup_right),
end | lemma | coheyting.boundary_sup_le | order.heyting | src/order/heyting/boundary.lean | [
"order.boolean_algebra"
] | [
"hnot_anti",
"inf_le_inf_left",
"inf_sup_right",
"le_sup_left",
"le_sup_right",
"sup_le_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
boundary_le_boundary_sup_sup_boundary_inf_left : ∂ a ≤ ∂ (a ⊔ b) ⊔ ∂ (a ⊓ b) | begin
simp only [boundary, sup_inf_left, sup_inf_right, sup_right_idem, le_inf_iff, sup_assoc,
@sup_comm _ _ _ a],
refine ⟨⟨⟨_, _⟩, _⟩, ⟨_, _⟩, _⟩;
try { exact le_sup_of_le_left inf_le_left };
refine inf_le_of_right_le _,
{ rw [hnot_le_iff_codisjoint_right, codisjoint_left_comm],
exact codisjoint_... | lemma | coheyting.boundary_le_boundary_sup_sup_boundary_inf_left | order.heyting | src/order/heyting/boundary.lean | [
"order.boolean_algebra"
] | [
"codisjoint_hnot_left",
"codisjoint_left_comm",
"hnot_anti",
"hnot_le_iff_codisjoint_right",
"inf_le_left",
"inf_le_of_right_le",
"le_inf_iff",
"le_sup_of_le_left",
"le_sup_of_le_right",
"sup_assoc",
"sup_comm",
"sup_inf_left",
"sup_inf_right",
"sup_right_idem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
boundary_le_boundary_sup_sup_boundary_inf_right : ∂ b ≤ ∂ (a ⊔ b) ⊔ ∂ (a ⊓ b) | by { rw [@sup_comm _ _ a, inf_comm], exact boundary_le_boundary_sup_sup_boundary_inf_left } | lemma | coheyting.boundary_le_boundary_sup_sup_boundary_inf_right | order.heyting | src/order/heyting/boundary.lean | [
"order.boolean_algebra"
] | [
"inf_comm",
"sup_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
boundary_sup_sup_boundary_inf (a b : α) : ∂ (a ⊔ b) ⊔ ∂ (a ⊓ b) = ∂ a ⊔ ∂ b | le_antisymm (sup_le boundary_sup_le boundary_inf_le) $ sup_le
boundary_le_boundary_sup_sup_boundary_inf_left boundary_le_boundary_sup_sup_boundary_inf_right | lemma | coheyting.boundary_sup_sup_boundary_inf | order.heyting | src/order/heyting/boundary.lean | [
"order.boolean_algebra"
] | [
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
boundary_idem (a : α) : ∂ ∂ a = ∂ a | by rw [boundary, hnot_boundary, inf_top_eq] | lemma | coheyting.boundary_idem | order.heyting | src/order/heyting/boundary.lean | [
"order.boolean_algebra"
] | [
"inf_top_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hnot_hnot_sup_boundary (a : α) : ￢￢a ⊔ ∂ a = a | by { rw [boundary, sup_inf_left, hnot_sup_self, inf_top_eq, sup_eq_right], exact hnot_hnot_le } | lemma | coheyting.hnot_hnot_sup_boundary | order.heyting | src/order/heyting/boundary.lean | [
"order.boolean_algebra"
] | [
"hnot_hnot_le",
"hnot_sup_self",
"inf_top_eq",
"sup_eq_right",
"sup_inf_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hnot_eq_top_iff_exists_boundary : ￢a = ⊤ ↔ ∃ b, ∂ b = a | ⟨λ h, ⟨a, by rw [boundary, h, inf_top_eq]⟩, by { rintro ⟨b, rfl⟩, exact hnot_boundary _ }⟩ | lemma | coheyting.hnot_eq_top_iff_exists_boundary | order.heyting | src/order/heyting/boundary.lean | [
"order.boolean_algebra"
] | [
"inf_top_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coheyting.boundary_eq_bot (a : α) : ∂ a = ⊥ | inf_compl_eq_bot | lemma | coheyting.boundary_eq_bot | order.heyting | src/order/heyting/boundary.lean | [
"order.boolean_algebra"
] | [
"inf_compl_eq_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
heyting_hom (α β : Type*) [heyting_algebra α] [heyting_algebra β]
extends lattice_hom α β | (map_bot' : to_fun ⊥ = ⊥)
(map_himp' : ∀ a b, to_fun (a ⇨ b) = to_fun a ⇨ to_fun b) | structure | heyting_hom | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"heyting_algebra",
"lattice_hom"
] | The type of Heyting homomorphisms from `α` to `β`. Bounded lattice homomorphisms that preserve
Heyting implication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coheyting_hom (α β : Type*) [coheyting_algebra α] [coheyting_algebra β]
extends lattice_hom α β | (map_top' : to_fun ⊤ = ⊤)
(map_sdiff' : ∀ a b, to_fun (a \ b) = to_fun a \ to_fun b) | structure | coheyting_hom | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"coheyting_algebra",
"lattice_hom",
"map_sdiff'"
] | The type of co-Heyting homomorphisms from `α` to `β`. Bounded lattice homomorphisms that
preserve difference. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biheyting_hom (α β : Type*) [biheyting_algebra α] [biheyting_algebra β]
extends lattice_hom α β | (map_himp' : ∀ a b, to_fun (a ⇨ b) = to_fun a ⇨ to_fun b)
(map_sdiff' : ∀ a b, to_fun (a \ b) = to_fun a \ to_fun b) | structure | biheyting_hom | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"biheyting_algebra",
"lattice_hom",
"map_sdiff'"
] | The type of bi-Heyting homomorphisms from `α` to `β`. Bounded lattice homomorphisms that
preserve Heyting implication and difference. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
heyting_hom_class (F : Type*) (α β : out_param $ Type*) [heyting_algebra α]
[heyting_algebra β] extends lattice_hom_class F α β | (map_bot (f : F) : f ⊥ = ⊥)
(map_himp (f : F) : ∀ a b, f (a ⇨ b) = f a ⇨ f b) | class | heyting_hom_class | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"heyting_algebra",
"lattice_hom_class"
] | `heyting_hom_class F α β` states that `F` is a type of Heyting homomorphisms.
You should extend this class when you extend `heyting_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coheyting_hom_class (F : Type*) (α β : out_param $ Type*) [coheyting_algebra α]
[coheyting_algebra β] extends lattice_hom_class F α β | (map_top (f : F) : f ⊤ = ⊤)
(map_sdiff (f : F) : ∀ a b, f (a \ b) = f a \ f b) | class | coheyting_hom_class | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"coheyting_algebra",
"lattice_hom_class"
] | `coheyting_hom_class F α β` states that `F` is a type of co-Heyting homomorphisms.
You should extend this class when you extend `coheyting_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biheyting_hom_class (F : Type*) (α β : out_param $ Type*) [biheyting_algebra α]
[biheyting_algebra β] extends lattice_hom_class F α β | (map_himp (f : F) : ∀ a b, f (a ⇨ b) = f a ⇨ f b)
(map_sdiff (f : F) : ∀ a b, f (a \ b) = f a \ f b) | class | biheyting_hom_class | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"biheyting_algebra",
"lattice_hom_class"
] | `biheyting_hom_class F α β` states that `F` is a type of bi-Heyting homomorphisms.
You should extend this class when you extend `biheyting_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
heyting_hom_class.to_bounded_lattice_hom_class [heyting_algebra α] [heyting_algebra β]
[heyting_hom_class F α β] : bounded_lattice_hom_class F α β | { map_top := λ f, by rw [←@himp_self α _ ⊥, ←himp_self, map_himp],
..‹heyting_hom_class F α β› } | instance | heyting_hom_class.to_bounded_lattice_hom_class | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"bounded_lattice_hom_class",
"heyting_algebra",
"heyting_hom_class",
"himp_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coheyting_hom_class.to_bounded_lattice_hom_class [coheyting_algebra α]
[coheyting_algebra β] [coheyting_hom_class F α β] : bounded_lattice_hom_class F α β | { map_bot := λ f, by rw [←@sdiff_self α _ ⊤, ←sdiff_self, map_sdiff],
..‹coheyting_hom_class F α β› } | instance | coheyting_hom_class.to_bounded_lattice_hom_class | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"bounded_lattice_hom_class",
"coheyting_algebra",
"coheyting_hom_class",
"sdiff_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biheyting_hom_class.to_heyting_hom_class [biheyting_algebra α] [biheyting_algebra β]
[biheyting_hom_class F α β] :
heyting_hom_class F α β | { map_bot := λ f, by rw [←@sdiff_self α _ ⊤, ←sdiff_self, biheyting_hom_class.map_sdiff],
..‹biheyting_hom_class F α β› } | instance | biheyting_hom_class.to_heyting_hom_class | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"biheyting_algebra",
"biheyting_hom_class",
"heyting_hom_class",
"sdiff_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biheyting_hom_class.to_coheyting_hom_class [biheyting_algebra α] [biheyting_algebra β]
[biheyting_hom_class F α β] :
coheyting_hom_class F α β | { map_top := λ f, by rw [←@himp_self α _ ⊥, ←himp_self, map_himp],
..‹biheyting_hom_class F α β› } | instance | biheyting_hom_class.to_coheyting_hom_class | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"biheyting_algebra",
"biheyting_hom_class",
"coheyting_hom_class",
"himp_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso_class.to_heyting_hom_class [heyting_algebra α] [heyting_algebra β]
[order_iso_class F α β] :
heyting_hom_class F α β | { map_himp := λ f a b, eq_of_forall_le_iff $ λ c,
by { simp only [←map_inv_le_iff, le_himp_iff], rw ←order_iso_class.map_le_map_iff f, simp },
..order_iso_class.to_bounded_lattice_hom_class } | instance | order_iso_class.to_heyting_hom_class | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"eq_of_forall_le_iff",
"heyting_algebra",
"heyting_hom_class",
"le_himp_iff",
"order_iso_class",
"order_iso_class.to_bounded_lattice_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso_class.to_coheyting_hom_class [coheyting_algebra α] [coheyting_algebra β]
[order_iso_class F α β] :
coheyting_hom_class F α β | { map_sdiff := λ f a b, eq_of_forall_ge_iff $ λ c,
by { simp only [←le_map_inv_iff, sdiff_le_iff], rw ←order_iso_class.map_le_map_iff f, simp },
..order_iso_class.to_bounded_lattice_hom_class } | instance | order_iso_class.to_coheyting_hom_class | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"coheyting_algebra",
"coheyting_hom_class",
"eq_of_forall_ge_iff",
"order_iso_class",
"order_iso_class.to_bounded_lattice_hom_class",
"sdiff_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso_class.to_biheyting_hom_class [biheyting_algebra α] [biheyting_algebra β]
[order_iso_class F α β] :
biheyting_hom_class F α β | { map_himp := λ f a b, eq_of_forall_le_iff $ λ c,
by { simp only [←map_inv_le_iff, le_himp_iff], rw ←order_iso_class.map_le_map_iff f, simp },
map_sdiff := λ f a b, eq_of_forall_ge_iff $ λ c,
by { simp only [←le_map_inv_iff, sdiff_le_iff], rw ←order_iso_class.map_le_map_iff f, simp },
..order_iso_class.to_l... | instance | order_iso_class.to_biheyting_hom_class | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"biheyting_algebra",
"biheyting_hom_class",
"eq_of_forall_ge_iff",
"eq_of_forall_le_iff",
"le_himp_iff",
"order_iso_class",
"order_iso_class.to_lattice_hom_class",
"sdiff_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_lattice_hom_class.to_biheyting_hom_class [boolean_algebra α] [boolean_algebra β]
[bounded_lattice_hom_class F α β] :
biheyting_hom_class F α β | { map_himp := λ f a b, by rw [himp_eq, himp_eq, map_sup, (is_compl_compl.map _).compl_eq],
map_sdiff := λ f a b, by rw [sdiff_eq, sdiff_eq, map_inf, (is_compl_compl.map _).compl_eq],
..‹bounded_lattice_hom_class F α β› } | def | bounded_lattice_hom_class.to_biheyting_hom_class | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"biheyting_hom_class",
"boolean_algebra",
"bounded_lattice_hom_class",
"himp_eq",
"sdiff_eq"
] | This can't be an instance because of typeclass loops. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_compl (a : α) : f aᶜ = (f a)ᶜ | by rw [←himp_bot, ←himp_bot, map_himp, map_bot] | lemma | map_compl | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_bihimp (a b : α) : f (a ⇔ b) = f a ⇔ f b | by simp_rw [bihimp, map_inf, map_himp] | lemma | map_bihimp | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"bihimp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_hnot (a : α) : f ￢a = ￢f a | by rw [←top_sdiff', ←top_sdiff', map_sdiff, map_top] | lemma | map_hnot | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_symm_diff (a b : α) : f (a ∆ b) = f a ∆ f b | by simp_rw [symm_diff, map_sup, map_sdiff] | lemma | map_symm_diff | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"symm_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe {f : heyting_hom α β} : f.to_fun = (f : α → β) | rfl | lemma | heyting_hom.to_fun_eq_coe | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"heyting_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : heyting_hom α β} (h : ∀ a, f a = g a) : f = g | fun_like.ext f g h | lemma | heyting_hom.ext | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"fun_like.ext",
"heyting_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : heyting_hom α β) (f' : α → β) (h : f' = f) : heyting_hom α β | { to_fun := f',
map_sup' := by simpa only [h] using map_sup f,
map_inf' := by simpa only [h] using map_inf f,
map_bot' := by simpa only [h] using map_bot f,
map_himp' := by simpa only [h] using map_himp f } | def | heyting_hom.copy | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"heyting_hom"
] | Copy of a `heyting_hom` with a new `to_fun` equal to the old one. Useful to fix definitional
equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (f : heyting_hom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | heyting_hom.coe_copy | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"heyting_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : heyting_hom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | heyting_hom.copy_eq | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"fun_like.ext'",
"heyting_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : heyting_hom α α | { to_lattice_hom := lattice_hom.id _,
map_himp' := λ a b, rfl,
..bot_hom.id _ } | def | heyting_hom.id | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"bot_hom.id",
"heyting_hom",
"lattice_hom.id"
] | `id` as a `heyting_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(heyting_hom.id α) = id | rfl | lemma | heyting_hom.coe_id | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"heyting_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (a : α) : heyting_hom.id α a = a | rfl | lemma | heyting_hom.id_apply | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"heyting_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : heyting_hom β γ) (g : heyting_hom α β) : heyting_hom α γ | { to_fun := f ∘ g,
map_bot' := by simp,
map_himp' := λ a b, by simp,
..f.to_lattice_hom.comp g.to_lattice_hom } | def | heyting_hom.comp | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"heyting_hom"
] | Composition of `heyting_hom`s as a `heyting_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : heyting_hom β γ) (g : heyting_hom α β) : ⇑(f.comp g) = f ∘ g | rfl | lemma | heyting_hom.coe_comp | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"heyting_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : heyting_hom β γ) (g : heyting_hom α β) (a : α) :
f.comp g a = f (g a) | rfl | lemma | heyting_hom.comp_apply | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"heyting_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : heyting_hom γ δ) (g : heyting_hom β γ) (h : heyting_hom α β) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | heyting_hom.comp_assoc | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"heyting_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : heyting_hom α β) : f.comp (heyting_hom.id α) = f | ext $ λ a, rfl | lemma | heyting_hom.comp_id | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"heyting_hom",
"heyting_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : heyting_hom α β) : (heyting_hom.id β).comp f = f | ext $ λ a, rfl | lemma | heyting_hom.id_comp | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"heyting_hom",
"heyting_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right (hf : surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨λ h, ext $ hf.forall.2 $ fun_like.ext_iff.1 h, congr_arg _⟩ | lemma | heyting_hom.cancel_right | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left (hg : injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨λ h, heyting_hom.ext $ λ a, hg $ by rw [←comp_apply, h, comp_apply], congr_arg _⟩ | lemma | heyting_hom.cancel_left | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"heyting_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe {f : coheyting_hom α β} : f.to_fun = (f : α → β) | rfl | lemma | coheyting_hom.to_fun_eq_coe | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"coheyting_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : coheyting_hom α β} (h : ∀ a, f a = g a) : f = g | fun_like.ext f g h | lemma | coheyting_hom.ext | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"coheyting_hom",
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : coheyting_hom α β) (f' : α → β) (h : f' = f) : coheyting_hom α β | { to_fun := f',
map_sup' := by simpa only [h] using map_sup f,
map_inf' := by simpa only [h] using map_inf f,
map_top' := by simpa only [h] using map_top f,
map_sdiff' := by simpa only [h] using map_sdiff f } | def | coheyting_hom.copy | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"coheyting_hom",
"map_sdiff'"
] | Copy of a `coheyting_hom` with a new `to_fun` equal to the old one. Useful to fix definitional
equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (f : coheyting_hom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | coheyting_hom.coe_copy | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"coheyting_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : coheyting_hom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | coheyting_hom.copy_eq | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"coheyting_hom",
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : coheyting_hom α α | { to_lattice_hom := lattice_hom.id _,
map_sdiff' := λ a b, rfl,
..top_hom.id _ } | def | coheyting_hom.id | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"coheyting_hom",
"lattice_hom.id",
"map_sdiff'",
"top_hom.id"
] | `id` as a `coheyting_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(coheyting_hom.id α) = id | rfl | lemma | coheyting_hom.coe_id | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"coheyting_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (a : α) : coheyting_hom.id α a = a | rfl | lemma | coheyting_hom.id_apply | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"coheyting_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : coheyting_hom β γ) (g : coheyting_hom α β) : coheyting_hom α γ | { to_fun := f ∘ g,
map_top' := by simp,
map_sdiff' := λ a b, by simp,
..f.to_lattice_hom.comp g.to_lattice_hom } | def | coheyting_hom.comp | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"coheyting_hom",
"map_sdiff'"
] | Composition of `coheyting_hom`s as a `coheyting_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : coheyting_hom β γ) (g : coheyting_hom α β) : ⇑(f.comp g) = f ∘ g | rfl | lemma | coheyting_hom.coe_comp | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"coheyting_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : coheyting_hom β γ) (g : coheyting_hom α β) (a : α) :
f.comp g a = f (g a) | rfl | lemma | coheyting_hom.comp_apply | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"coheyting_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : coheyting_hom γ δ) (g : coheyting_hom β γ) (h : coheyting_hom α β) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | coheyting_hom.comp_assoc | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"coheyting_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : coheyting_hom α β) : f.comp (coheyting_hom.id α) = f | ext $ λ a, rfl | lemma | coheyting_hom.comp_id | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"coheyting_hom",
"coheyting_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : coheyting_hom α β) : (coheyting_hom.id β).comp f = f | ext $ λ a, rfl | lemma | coheyting_hom.id_comp | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"coheyting_hom",
"coheyting_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left (hg : injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨λ h, coheyting_hom.ext $ λ a, hg $ by rw [←comp_apply, h, comp_apply], congr_arg _⟩ | lemma | coheyting_hom.cancel_left | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"coheyting_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe {f : biheyting_hom α β} : f.to_fun = (f : α → β) | rfl | lemma | biheyting_hom.to_fun_eq_coe | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"biheyting_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : biheyting_hom α β} (h : ∀ a, f a = g a) : f = g | fun_like.ext f g h | lemma | biheyting_hom.ext | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"biheyting_hom",
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : biheyting_hom α β) (f' : α → β) (h : f' = f) : biheyting_hom α β | { to_fun := f',
map_sup' := by simpa only [h] using map_sup f,
map_inf' := by simpa only [h] using map_inf f,
map_himp' := by simpa only [h] using map_himp f,
map_sdiff' := by simpa only [h] using map_sdiff f } | def | biheyting_hom.copy | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"biheyting_hom",
"map_sdiff'"
] | Copy of a `biheyting_hom` with a new `to_fun` equal to the old one. Useful to fix definitional
equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (f : biheyting_hom α β) (f' : α → β) (h : f' = f) :
⇑(f.copy f' h) = f' | rfl | lemma | biheyting_hom.coe_copy | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"biheyting_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : biheyting_hom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | biheyting_hom.copy_eq | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"biheyting_hom",
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : biheyting_hom α α | { to_lattice_hom := lattice_hom.id _,
..heyting_hom.id _, ..coheyting_hom.id _ } | def | biheyting_hom.id | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"biheyting_hom",
"coheyting_hom.id",
"heyting_hom.id",
"lattice_hom.id"
] | `id` as a `biheyting_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(biheyting_hom.id α) = id | rfl | lemma | biheyting_hom.coe_id | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"biheyting_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (a : α) : biheyting_hom.id α a = a | rfl | lemma | biheyting_hom.id_apply | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"biheyting_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : biheyting_hom β γ) (g : biheyting_hom α β) : biheyting_hom α γ | { to_fun := f ∘ g,
map_himp' := λ a b, by simp,
map_sdiff' := λ a b, by simp,
..f.to_lattice_hom.comp g.to_lattice_hom } | def | biheyting_hom.comp | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"biheyting_hom",
"map_sdiff'"
] | Composition of `biheyting_hom`s as a `biheyting_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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