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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|---|---|---|---|---|---|---|---|---|---|---|
coe_comp (f : biheyting_hom β γ) (g : biheyting_hom α β) : ⇑(f.comp g) = f ∘ g | rfl | lemma | biheyting_hom.coe_comp | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"biheyting_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : biheyting_hom β γ) (g : biheyting_hom α β) (a : α) :
f.comp g a = f (g a) | rfl | lemma | biheyting_hom.comp_apply | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"biheyting_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : biheyting_hom γ δ) (g : biheyting_hom β γ) (h : biheyting_hom α β) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | biheyting_hom.comp_assoc | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"biheyting_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : biheyting_hom α β) : f.comp (biheyting_hom.id α) = f | ext $ λ a, rfl | lemma | biheyting_hom.comp_id | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"biheyting_hom",
"biheyting_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : biheyting_hom α β) : (biheyting_hom.id β).comp f = f | ext $ λ a, rfl | lemma | biheyting_hom.id_comp | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"biheyting_hom",
"biheyting_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left (hg : injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨λ h, biheyting_hom.ext $ λ a, hg $ by rw [←comp_apply, h, comp_apply], congr_arg _⟩ | lemma | biheyting_hom.cancel_left | order.heyting | src/order/heyting/hom.lean | [
"order.hom.lattice"
] | [
"biheyting_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_regular (a : α) : Prop | aᶜᶜ = a | def | heyting.is_regular | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [
"is_regular"
] | An element of an Heyting algebra is regular if its double complement is itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_regular.eq : is_regular a → aᶜᶜ = a | id | lemma | heyting.is_regular.eq | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [
"is_regular"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_regular.decidable_pred [decidable_eq α] : @decidable_pred α is_regular | λ _, ‹decidable_eq α› _ _ | instance | heyting.is_regular.decidable_pred | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [
"is_regular"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_regular_bot : is_regular (⊥ : α) | by rw [is_regular, compl_bot, compl_top] | lemma | heyting.is_regular_bot | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [
"compl_bot",
"compl_top",
"is_regular"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_regular_top : is_regular (⊤ : α) | by rw [is_regular, compl_top, compl_bot] | lemma | heyting.is_regular_top | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [
"compl_bot",
"compl_top",
"is_regular"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_regular.inf (ha : is_regular a) (hb : is_regular b) : is_regular (a ⊓ b) | by rw [is_regular, compl_compl_inf_distrib, ha.eq, hb.eq] | lemma | heyting.is_regular.inf | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [
"compl_compl_inf_distrib",
"is_regular"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_regular.himp (ha : is_regular a) (hb : is_regular b) : is_regular (a ⇨ b) | by rw [is_regular, compl_compl_himp_distrib, ha.eq, hb.eq] | lemma | heyting.is_regular.himp | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [
"compl_compl_himp_distrib",
"is_regular"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_regular_compl (a : α) : is_regular aᶜ | compl_compl_compl _ | lemma | heyting.is_regular_compl | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [
"compl_compl_compl",
"is_regular"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_regular.disjoint_compl_left_iff (ha : is_regular a) : disjoint aᶜ b ↔ b ≤ a | by rw [←le_compl_iff_disjoint_left, ha.eq] | lemma | heyting.is_regular.disjoint_compl_left_iff | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [
"disjoint",
"is_regular"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_regular.disjoint_compl_right_iff (hb : is_regular b) : disjoint a bᶜ ↔ a ≤ b | by rw [←le_compl_iff_disjoint_right, hb.eq] | lemma | heyting.is_regular.disjoint_compl_right_iff | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [
"disjoint",
"is_regular"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.boolean_algebra.of_regular (h : ∀ a : α, is_regular (a ⊔ aᶜ)) : boolean_algebra α | have ∀ a : α, is_compl a aᶜ := λ a, ⟨disjoint_compl_right, codisjoint_iff.2 $
by erw [←(h a).eq, compl_sup, inf_compl_eq_bot, compl_bot]⟩,
{ himp_eq := λ a b, eq_of_forall_le_iff $ λ c,
le_himp_iff.trans ((this _).le_sup_right_iff_inf_left_le).symm,
inf_compl_le_bot := λ a, (this _).1.le_bot,
top_le_sup_compl... | def | boolean_algebra.of_regular | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [
"boolean_algebra",
"compl_bot",
"compl_sup",
"eq_of_forall_le_iff",
"generalized_heyting_algebra.to_distrib_lattice",
"himp_eq",
"inf_compl_eq_bot",
"is_compl",
"is_regular"
] | A Heyting algebra with regular excluded middle is a boolean algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
regular : Type* | {a : α // is_regular a} | def | heyting.regular | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [
"is_regular"
] | The boolean algebra of Heyting regular elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_injective : injective (coe : regular α → α) | subtype.coe_injective | lemma | heyting.regular.coe_injective | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inj {a b : regular α} : (a : α) = b ↔ a = b | subtype.coe_inj | lemma | heyting.regular.coe_inj | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [
"subtype.coe_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_top : ((⊤ : regular α) : α) = ⊤ | rfl | lemma | heyting.regular.coe_top | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bot : ((⊥ : regular α) : α) = ⊥ | rfl | lemma | heyting.regular.coe_bot | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inf (a b : regular α) : (↑(a ⊓ b) : α) = a ⊓ b | rfl | lemma | heyting.regular.coe_inf | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_himp (a b : regular α) : (↑(a ⇨ b) : α) = a ⇨ b | rfl | lemma | heyting.regular.coe_himp | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_compl (a : regular α) : (↑(aᶜ) : α) = aᶜ | rfl | lemma | heyting.regular.coe_compl | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_le_coe {a b : regular α} : (a : α) ≤ b ↔ a ≤ b | iff.rfl | lemma | heyting.regular.coe_le_coe | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_lt_coe {a b : regular α} : (a : α) < b ↔ a < b | iff.rfl | lemma | heyting.regular.coe_lt_coe | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_regular : α →o regular α | ⟨λ a, ⟨aᶜᶜ, is_regular_compl _⟩, λ a b h, coe_le_coe.1 $ compl_le_compl $ compl_le_compl h⟩ | def | heyting.regular.to_regular | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [
"compl_le_compl"
] | **Regularization** of `a`. The smallest regular element greater than `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_regular (a : α) : (to_regular a : α) = aᶜᶜ | rfl | lemma | heyting.regular.coe_to_regular | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_regular_coe (a : regular α) : to_regular (a : α) = a | coe_injective a.2 | lemma | heyting.regular.to_regular_coe | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gi : galois_insertion to_regular (coe : regular α → α) | { choice := λ a ha, ⟨a, ha.antisymm le_compl_compl⟩,
gc := λ a b, coe_le_coe.symm.trans $
⟨le_compl_compl.trans, λ h, (compl_anti $ compl_anti h).trans_eq b.2⟩,
le_l_u := λ _, le_compl_compl,
choice_eq := λ a ha, coe_injective $ le_compl_compl.antisymm ha } | def | heyting.regular.gi | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [
"compl_anti",
"galois_insertion",
"le_compl_compl"
] | The Galois insertion between `regular.to_regular` and `coe`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_sup (a b : regular α) : (↑(a ⊔ b) : α) = (a ⊔ b)ᶜᶜ | rfl | lemma | heyting.regular.coe_sup | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sdiff (a b : regular α) : (↑(a \ b) : α) = a ⊓ bᶜ | rfl | lemma | heyting.regular.coe_sdiff | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_regular_of_boolean : ∀ a : α, is_regular a | compl_compl | lemma | heyting.is_regular_of_boolean | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [
"compl_compl",
"is_regular"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_regular_of_decidable (p : Prop) [decidable p] : is_regular p | propext $ decidable.not_not_iff _ | lemma | heyting.is_regular_of_decidable | order.heyting | src/order/heyting/regular.lean | [
"order.galois_connection"
] | [
"is_regular"
] | A decidable proposition is intuitionistically Heyting-regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_hom (α β : Type*) [preorder α] [preorder β] | (to_fun : α → β)
(monotone' : monotone to_fun) | structure | order_hom | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"monotone"
] | Bundled monotone (aka, increasing) function | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_embedding (α β : Type*) [has_le α] [has_le β] | @rel_embedding α β (≤) (≤) | abbreviation | order_embedding | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"rel_embedding"
] | An order embedding is an embedding `f : α ↪ β` such that `a ≤ b ↔ (f a) ≤ (f b)`.
This definition is an abbreviation of `rel_embedding (≤) (≤)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_iso (α β : Type*) [has_le α] [has_le β] | @rel_iso α β (≤) (≤) | abbreviation | order_iso | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"rel_iso"
] | An order isomorphism is an equivalence such that `a ≤ b ↔ (f a) ≤ (f b)`.
This definition is an abbreviation of `rel_iso (≤) (≤)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_hom_class (F : Type*) (α β : out_param Type*) [has_le α] [has_le β] | rel_hom_class F ((≤) : α → α → Prop) ((≤) : β → β → Prop) | abbreviation | order_hom_class | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"rel_hom_class"
] | `order_hom_class F α b` asserts that `F` is a type of `≤`-preserving morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_iso_class (F : Type*) (α β : out_param Type*) [has_le α] [has_le β]
extends equiv_like F α β | (map_le_map_iff (f : F) {a b : α} : f a ≤ f b ↔ a ≤ b) | class | order_iso_class | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"equiv_like"
] | `order_iso_class F α β` states that `F` is a type of order isomorphisms.
You should extend this class when you extend `order_iso`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_iso_class.to_order_hom_class [has_le α] [has_le β] [order_iso_class F α β] :
order_hom_class F α β | { map_rel := λ f a b, (map_le_map_iff f).2, ..equiv_like.to_embedding_like } | instance | order_iso_class.to_order_hom_class | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"equiv_like.to_embedding_like",
"order_hom_class",
"order_iso_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone (f : F) : monotone (f : α → β) | λ _ _, map_rel f | lemma | order_hom_class.monotone | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono (f : F) : monotone (f : α → β) | λ _ _, map_rel f | lemma | order_hom_class.mono | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_inv_le_iff (f : F) {a : α} {b : β} : equiv_like.inv f b ≤ a ↔ b ≤ f a | by { convert (map_le_map_iff _).symm, exact (equiv_like.right_inv _ _).symm } | lemma | map_inv_le_iff | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_map_inv_iff (f : F) {a : α} {b : β} : a ≤ equiv_like.inv f b ↔ f a ≤ b | by { convert (map_le_map_iff _).symm, exact (equiv_like.right_inv _ _).symm } | lemma | le_map_inv_iff | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_lt_map_iff (f : F) {a b : α} : f a < f b ↔ a < b | lt_iff_lt_of_le_iff_le' (map_le_map_iff f) (map_le_map_iff f) | lemma | map_lt_map_iff | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"lt_iff_lt_of_le_iff_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_inv_lt_iff (f : F) {a : α} {b : β} : equiv_like.inv f b < a ↔ b < f a | by { convert (map_lt_map_iff _).symm, exact (equiv_like.right_inv _ _).symm } | lemma | map_inv_lt_iff | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"map_lt_map_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_map_inv_iff (f : F) {a : α} {b : β} : a < equiv_like.inv f b ↔ f a < b | by { convert (map_lt_map_iff _).symm, exact (equiv_like.right_inv _ _).symm } | lemma | lt_map_inv_iff | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"map_lt_map_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone (f : α →o β) : monotone f | f.monotone' | lemma | order_hom.monotone | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono (f : α →o β) : monotone f | f.monotone | lemma | order_hom.mono | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe {f : α →o β} : f.to_fun = f | rfl | lemma | order_hom.to_fun_eq_coe | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fun_mk {f : α → β} (hf : _root_.monotone f) : (mk f hf : α → β) = f | rfl | lemma | order_hom.coe_fun_mk | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext (f g : α →o β) (h : (f : α → β) = g) : f = g | fun_like.coe_injective h | lemma | order_hom.ext | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"fun_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_eq (f : α →o β) : coe f = f | by ext ; refl | lemma | order_hom.coe_eq | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : α →o β) (f' : α → β) (h : f' = f) : α →o β | ⟨f', h.symm.subst f.monotone'⟩ | def | order_hom.copy | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | Copy of an `order_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 : α →o β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | order_hom.coe_copy | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : α →o β) (f' : α → β) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | order_hom.copy_eq | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : α →o α | ⟨id, monotone_id⟩ | def | order_hom.id | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | The identity function as bundled monotone function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_def {f g : α →o β} : f ≤ g ↔ ∀ x, f x ≤ g x | iff.rfl | lemma | order_hom.le_def | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_le_coe {f g : α →o β} : (f : α → β) ≤ g ↔ f ≤ g | iff.rfl | lemma | order_hom.coe_le_coe | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_le_mk {f g : α → β} {hf hg} : mk f hf ≤ mk g hg ↔ f ≤ g | iff.rfl | lemma | order_hom.mk_le_mk | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_mono {f g : α →o β} {x y : α} (h₁ : f ≤ g) (h₂ : x ≤ y) :
f x ≤ g y | (h₁ x).trans $ g.mono h₂ | lemma | order_hom.apply_mono | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
curry : (α × β →o γ) ≃o (α →o β →o γ) | { to_fun := λ f, ⟨λ x, ⟨function.curry f x, λ y₁ y₂ h, f.mono ⟨le_rfl, h⟩⟩,
λ x₁ x₂ h y, f.mono ⟨h, le_rfl⟩⟩,
inv_fun := λ f, ⟨function.uncurry (λ x, f x), λ x y h, (f.mono h.1 x.2).trans $ (f y.1).mono h.2⟩,
left_inv := λ f, by { ext ⟨x, y⟩, refl },
right_inv := λ f, by { ext x y, refl },
map_rel_iff' := λ... | def | order_hom.curry | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"inv_fun"
] | Curry/uncurry as an order isomorphism between `α × β →o γ` and `α →o β →o γ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
curry_apply (f : α × β →o γ) (x : α) (y : β) : curry f x y = f (x, y) | rfl | lemma | order_hom.curry_apply | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"curry_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
curry_symm_apply (f : α →o β →o γ) (x : α × β) : curry.symm f x = f x.1 x.2 | rfl | lemma | order_hom.curry_symm_apply | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (g : β →o γ) (f : α →o β) : α →o γ | ⟨g ∘ f, g.mono.comp f.mono⟩ | def | order_hom.comp | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | The composition of two bundled monotone functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_mono ⦃g₁ g₂ : β →o γ⦄ (hg : g₁ ≤ g₂) ⦃f₁ f₂ : α →o β⦄ (hf : f₁ ≤ f₂) :
g₁.comp f₁ ≤ g₂.comp f₂ | λ x, (hg _).trans (g₂.mono $ hf _) | lemma | order_hom.comp_mono | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compₘ : (β →o γ) →o (α →o β) →o α →o γ | curry ⟨λ f : (β →o γ) × (α →o β), f.1.comp f.2, λ f₁ f₂ h, comp_mono h.1 h.2⟩ | def | order_hom.compₘ | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | The composition of two bundled monotone functions, a fully bundled version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_id (f : α →o β) : comp f id = f | by { ext, refl } | lemma | order_hom.comp_id | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : α →o β) : comp id f = f | by { ext, refl } | lemma | order_hom.id_comp | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const (α : Type*) [preorder α] {β : Type*} [preorder β] : β →o α →o β | { to_fun := λ b, ⟨function.const α b, λ _ _ _, le_rfl⟩,
monotone' := λ b₁ b₂ h x, h } | def | order_hom.const | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | Constant function bundled as a `order_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const_comp (f : α →o β) (c : γ) : (const β c).comp f = const α c | rfl | lemma | order_hom.const_comp | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_const (γ : Type*) [preorder γ] (f : α →o β) (c : α) :
f.comp (const γ c) = const γ (f c) | rfl | lemma | order_hom.comp_const | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod (f : α →o β) (g : α →o γ) : α →o (β × γ) | ⟨λ x, (f x, g x), λ x y h, ⟨f.mono h, g.mono h⟩⟩ | def | order_hom.prod | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | Given two bundled monotone maps `f`, `g`, `f.prod g` is the map `x ↦ (f x, g x)` bundled as a
`order_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_mono {f₁ f₂ : α →o β} (hf : f₁ ≤ f₂) {g₁ g₂ : α →o γ} (hg : g₁ ≤ g₂) :
f₁.prod g₁ ≤ f₂.prod g₂ | λ x, prod.le_def.2 ⟨hf _, hg _⟩ | lemma | order_hom.prod_mono | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_prod_comp_same (f₁ f₂ : β →o γ) (g : α →o β) :
(f₁.comp g).prod (f₂.comp g) = (f₁.prod f₂).comp g | rfl | lemma | order_hom.comp_prod_comp_same | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prodₘ : (α →o β) →o (α →o γ) →o α →o β × γ | curry ⟨λ f : (α →o β) × (α →o γ), f.1.prod f.2, λ f₁ f₂ h, prod_mono h.1 h.2⟩ | def | order_hom.prodₘ | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | Given two bundled monotone maps `f`, `g`, `f.prod g` is the map `x ↦ (f x, g x)` bundled as a
`order_hom`. This is a fully bundled version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diag : α →o α × α | id.prod id | def | order_hom.diag | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | Diagonal embedding of `α` into `α × α` as a `order_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
on_diag (f : α →o α →o β) : α →o β | (curry.symm f).comp diag | def | order_hom.on_diag | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | Restriction of `f : α →o α →o β` to the diagonal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fst : α × β →o α | ⟨prod.fst, λ x y h, h.1⟩ | def | order_hom.fst | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | `prod.fst` as a `order_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd : α × β →o β | ⟨prod.snd, λ x y h, h.2⟩ | def | order_hom.snd | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | `prod.snd` as a `order_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fst_prod_snd : (fst : α × β →o α).prod snd = id | by { ext ⟨x, y⟩ : 2, refl } | lemma | order_hom.fst_prod_snd | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_comp_prod (f : α →o β) (g : α →o γ) : fst.comp (f.prod g) = f | ext _ _ rfl | lemma | order_hom.fst_comp_prod | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_comp_prod (f : α →o β) (g : α →o γ) : snd.comp (f.prod g) = g | ext _ _ rfl | lemma | order_hom.snd_comp_prod | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_iso : (α →o β × γ) ≃o (α →o β) × (α →o γ) | { to_fun := λ f, (fst.comp f, snd.comp f),
inv_fun := λ f, f.1.prod f.2,
left_inv := λ f, by ext; refl,
right_inv := λ f, by ext; refl,
map_rel_iff' := λ f g, forall_and_distrib.symm } | def | order_hom.prod_iso | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"inv_fun"
] | Order isomorphism between the space of monotone maps to `β × γ` and the product of the spaces
of monotone maps to `β` and `γ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_map (f : α →o β) (g : γ →o δ) : α × γ →o β × δ | ⟨prod.map f g, λ x y h, ⟨f.mono h.1, g.mono h.2⟩⟩ | def | order_hom.prod_map | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"prod_map"
] | `prod.map` of two `order_hom`s as a `order_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.pi.eval_order_hom (i : ι) : (Π j, π j) →o π i | ⟨function.eval i, function.monotone_eval i⟩ | def | pi.eval_order_hom | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"function.monotone_eval"
] | Evaluation of an unbundled function at a point (`function.eval`) as a `order_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_fn_hom : (α →o β) →o (α → β) | { to_fun := λ f, f,
monotone' := λ x y h, h } | def | order_hom.coe_fn_hom | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | The "forgetful functor" from `α →o β` to `α → β` that takes the underlying function,
is monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply (x : α) : (α →o β) →o β | (pi.eval_order_hom x).comp coe_fn_hom | def | order_hom.apply | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"pi.eval_order_hom"
] | Function application `λ f, f a` (for fixed `a`) is a monotone function from the
monotone function space `α →o β` to `β`. See also `pi.eval_order_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi (f : Π i, α →o π i) : α →o (Π i, π i) | ⟨λ x i, f i x, λ x y h i, (f i).mono h⟩ | def | order_hom.pi | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | Construct a bundled monotone map `α →o Π i, π i` from a family of monotone maps
`f i : α →o π i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_iso : (α →o Π i, π i) ≃o Π i, α →o π i | { to_fun := λ f i, (pi.eval_order_hom i).comp f,
inv_fun := pi,
left_inv := λ f, by { ext x i, refl },
right_inv := λ f, by { ext x i, refl },
map_rel_iff' := λ f g, forall_swap } | def | order_hom.pi_iso | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"forall_swap",
"inv_fun",
"pi.eval_order_hom"
] | Order isomorphism between bundled monotone maps `α →o Π i, π i` and families of bundled monotone
maps `Π i, α →o π i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subtype.val (p : α → Prop) : subtype p →o α | ⟨subtype.val, λ x y h, h⟩ | def | order_hom.subtype.val | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | `subtype.val` as a bundled monotone function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique [subsingleton α] : unique (α →o α) | { default := order_hom.id, uniq := λ a, ext _ _ (subsingleton.elim _ _) } | instance | order_hom.unique | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"order_hom.id",
"unique"
] | There is a unique monotone map from a subsingleton to itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_hom_eq_id [subsingleton α] (g : α →o α) : g = order_hom.id | subsingleton.elim _ _ | lemma | order_hom.order_hom_eq_id | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"order_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual : (α →o β) ≃ (αᵒᵈ →o βᵒᵈ) | { to_fun := λ f, ⟨order_dual.to_dual ∘ f ∘ order_dual.of_dual, f.mono.dual⟩,
inv_fun := λ f, ⟨order_dual.of_dual ∘ f ∘ order_dual.to_dual, f.mono.dual⟩,
left_inv := λ f, ext _ _ rfl,
right_inv := λ f, ext _ _ rfl } | def | order_hom.dual | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"inv_fun",
"order_dual.of_dual",
"order_dual.to_dual"
] | Reinterpret a bundled monotone function as a monotone function between dual orders. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dual_id : (order_hom.id : α →o α).dual = order_hom.id | rfl | lemma | order_hom.dual_id | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"order_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual_comp (g : β →o γ) (f : α →o β) : (g.comp f).dual = g.dual.comp f.dual | rfl | lemma | order_hom.dual_comp | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_dual_id : order_hom.dual.symm order_hom.id = (order_hom.id : α →o α) | rfl | lemma | order_hom.symm_dual_id | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"order_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_dual_comp (g : βᵒᵈ →o γᵒᵈ) (f : αᵒᵈ →o βᵒᵈ) :
order_hom.dual.symm (g.comp f) = (order_hom.dual.symm g).comp (order_hom.dual.symm f) | rfl | lemma | order_hom.symm_dual_comp | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual_iso (α β : Type*) [preorder α] [preorder β] : (α →o β) ≃o (αᵒᵈ →o βᵒᵈ)ᵒᵈ | { to_equiv := order_hom.dual.trans order_dual.to_dual,
map_rel_iff' := λ f g, iff.rfl } | def | order_hom.dual_iso | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"order_dual.to_dual"
] | `order_hom.dual` as an order isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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