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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|---|---|---|---|---|---|---|---|---|---|---|
rel_hom {α β : Type*} (r : α → α → Prop) (s : β → β → Prop) | (to_fun : α → β)
(map_rel' : ∀ {a b}, r a b → s (to_fun a) (to_fun b)) | structure | rel_hom | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | A relation homomorphism with respect to a given pair of relations `r` and `s`
is a function `f : α → β` such that `r a b → s (f a) (f b)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel_hom_class (F : Type*) {α β : out_param $ Type*}
(r : out_param $ α → α → Prop) (s : out_param $ β → β → Prop)
extends fun_like F α (λ _, β) | (map_rel : ∀ (f : F) {a b}, r a b → s (f a) (f b)) | class | rel_hom_class | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"fun_like"
] | `rel_hom_class F r s` asserts that `F` is a type of functions such that all `f : F`
satisfy `r a b → s (f a) (f b)`.
The relations `r` and `s` are `out_param`s since figuring them out from a goal is a higher-order
matching problem that Lean usually can't do unaided. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_irrefl [rel_hom_class F r s] (f : F) : ∀ [is_irrefl β s], is_irrefl α r | | ⟨H⟩ := ⟨λ a h, H _ (map_rel f h)⟩ | theorem | rel_hom_class.is_irrefl | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"rel_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_asymm [rel_hom_class F r s] (f : F) : ∀ [is_asymm β s], is_asymm α r | | ⟨H⟩ := ⟨λ a b h₁ h₂, H _ _ (map_rel f h₁) (map_rel f h₂)⟩ | theorem | rel_hom_class.is_asymm | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"rel_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
acc [rel_hom_class F r s] (f : F) (a : α) : acc s (f a) → acc r a | begin
generalize h : f a = b, intro ac,
induction ac with _ H IH generalizing a, subst h,
exact ⟨_, λ a' h, IH (f a') (map_rel f h) _ rfl⟩
end | theorem | rel_hom_class.acc | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"rel_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
well_founded [rel_hom_class F r s] (f : F) :
∀ (h : well_founded s), well_founded r | | ⟨H⟩ := ⟨λ a, rel_hom_class.acc f _ (H _)⟩ | theorem | rel_hom_class.well_founded | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"rel_hom_class",
"rel_hom_class.acc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_rel (f : r →r s) {a b} : r a b → s (f a) (f b) | f.map_rel' | theorem | rel_hom.map_rel | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_mk (f : α → β) (o) :
(@rel_hom.mk _ _ r s f o : α → β) = f | rfl | theorem | rel_hom.coe_fn_mk | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_to_fun (f : r →r s) : (f.to_fun : α → β) = f | rfl | theorem | rel_hom.coe_fn_to_fun | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_injective : @function.injective (r →r s) (α → β) coe_fn | fun_like.coe_injective | theorem | rel_hom.coe_fn_injective | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"fun_like.coe_injective"
] | The map `coe_fn : (r →r s) → (α → β)` is injective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext ⦃f g : r →r s⦄ (h : ∀ x, f x = g x) : f = g | fun_like.ext f g h | theorem | rel_hom.ext | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {f g : r →r s} : f = g ↔ ∀ x, f x = g x | fun_like.ext_iff | theorem | rel_hom.ext_iff | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"fun_like.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id (r : α → α → Prop) : r →r r | ⟨λ x, x, λ a b x, x⟩ | def | rel_hom.id | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | Identity map is a relation homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp (g : s →r t) (f : r →r s) : r →r t | ⟨λ x, g (f x), λ a b h, g.2 (f.2 h)⟩ | def | rel_hom.comp | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | Composition of two relation homomorphisms is a relation homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
swap (f : r →r s) : swap r →r swap s | ⟨f, λ a b, f.map_rel⟩ | def | rel_hom.swap | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | A relation homomorphism is also a relation homomorphism between dual relations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preimage (f : α → β) (s : β → β → Prop) : f ⁻¹'o s →r s | ⟨f, λ a b, id⟩ | def | rel_hom.preimage | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | A function is a relation homomorphism from the preimage relation of `s` to `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
injective_of_increasing (r : α → α → Prop) (s : β → β → Prop) [is_trichotomous α r]
[is_irrefl β s] (f : α → β) (hf : ∀ {x y}, r x y → s (f x) (f y)) : injective f | begin
intros x y hxy,
rcases trichotomous_of r x y with h | h | h,
have := hf h, rw hxy at this, exfalso, exact irrefl_of s (f y) this,
exact h,
have := hf h, rw hxy at this, exfalso, exact irrefl_of s (f y) this
end | lemma | injective_of_increasing | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | An increasing function is injective | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel_hom.injective_of_increasing [is_trichotomous α r]
[is_irrefl β s] (f : r →r s) : injective f | injective_of_increasing r s f (λ x y, f.map_rel) | lemma | rel_hom.injective_of_increasing | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"injective_of_increasing"
] | An increasing function is injective | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
surjective.well_founded_iff {f : α → β} (hf : surjective f)
(o : ∀ {a b}, r a b ↔ s (f a) (f b)) : well_founded r ↔ well_founded s | iff.intro (begin
refine rel_hom_class.well_founded (rel_hom.mk _ _ : s →r r),
{ exact classical.some hf.has_right_inverse },
intros a b h, apply o.2, convert h,
iterate 2 { apply classical.some_spec hf.has_right_inverse },
end) (rel_hom_class.well_founded (⟨f, λ _ _, o.1⟩ : r →r s)) | theorem | surjective.well_founded_iff | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"rel_hom_class.well_founded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel_embedding {α β : Type*} (r : α → α → Prop) (s : β → β → Prop) extends α ↪ β | (map_rel_iff' : ∀ {a b}, s (to_embedding a) (to_embedding b) ↔ r a b) | structure | rel_embedding | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | A relation embedding with respect to a given pair of relations `r` and `s`
is an embedding `f : α ↪ β` such that `r a b ↔ s (f a) (f b)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subtype.rel_embedding {X : Type*} (r : X → X → Prop) (p : X → Prop) :
((subtype.val : subtype p → X) ⁻¹'o r) ↪r r | ⟨embedding.subtype p, λ x y, iff.rfl⟩ | definition | subtype.rel_embedding | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | The induced relation on a subtype is an embedding under the natural inclusion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preimage_equivalence {α β} (f : α → β) {s : β → β → Prop}
(hs : equivalence s) : equivalence (f ⁻¹'o s) | ⟨λ a, hs.1 _, λ a b h, hs.2.1 h, λ a b c h₁ h₂, hs.2.2 h₁ h₂⟩ | theorem | preimage_equivalence | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_rel_hom (f : r ↪r s) : (r →r s) | { to_fun := f.to_embedding.to_fun,
map_rel' := λ x y, (map_rel_iff' f).mpr } | def | rel_embedding.to_rel_hom | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | A relation embedding is also a relation homomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simps.apply (h : r ↪r s) : α → β | h
initialize_simps_projections rel_embedding (to_embedding_to_fun → apply, -to_embedding) | def | rel_embedding.simps.apply | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"rel_embedding"
] | See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_rel_hom_eq_coe (f : r ↪r s) : f.to_rel_hom = f | rfl | lemma | rel_embedding.to_rel_hom_eq_coe | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coe_fn (f : r ↪r s) : ((f : r →r s) : α → β) = f | rfl | lemma | rel_embedding.coe_coe_fn | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective (f : r ↪r s) : injective f | f.inj' | theorem | rel_embedding.injective | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inj (f : r ↪r s) {a b} : f a = f b ↔ a = b | f.injective.eq_iff | theorem | rel_embedding.inj | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_rel_iff (f : r ↪r s) {a b} : s (f a) (f b) ↔ r a b | f.map_rel_iff' | theorem | rel_embedding.map_rel_iff | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_mk (f : α ↪ β) (o) :
(@rel_embedding.mk _ _ r s f o : α → β) = f | rfl | theorem | rel_embedding.coe_fn_mk | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_to_embedding (f : r ↪r s) : (f.to_embedding : α → β) = f | rfl | theorem | rel_embedding.coe_fn_to_embedding | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_injective : @function.injective (r ↪r s) (α → β) coe_fn | fun_like.coe_injective | theorem | rel_embedding.coe_fn_injective | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"fun_like.coe_injective"
] | The map `coe_fn : (r ↪r s) → (α → β)` is injective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext ⦃f g : r ↪r s⦄ (h : ∀ x, f x = g x) : f = g | fun_like.ext _ _ h | theorem | rel_embedding.ext | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {f g : r ↪r s} : f = g ↔ ∀ x, f x = g x | fun_like.ext_iff | theorem | rel_embedding.ext_iff | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"fun_like.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl (r : α → α → Prop) : r ↪r r | ⟨embedding.refl _, λ a b, iff.rfl⟩ | def | rel_embedding.refl | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | Identity map is a relation embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans (f : r ↪r s) (g : s ↪r t) : r ↪r t | ⟨f.1.trans g.1, λ a b, by simp [f.map_rel_iff, g.map_rel_iff]⟩ | def | rel_embedding.trans | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | Composition of two relation embeddings is a relation embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_apply (f : r ↪r s) (g : s ↪r t) (a : α) : (f.trans g) a = g (f a) | rfl | theorem | rel_embedding.trans_apply | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_trans (f : r ↪r s) (g : s ↪r t) : ⇑(f.trans g) = g ∘ f | rfl | theorem | rel_embedding.coe_trans | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
swap (f : r ↪r s) : swap r ↪r swap s | ⟨f.to_embedding, λ a b, f.map_rel_iff⟩ | def | rel_embedding.swap | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | A relation embedding is also a relation embedding between dual relations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preimage (f : α ↪ β) (s : β → β → Prop) : f ⁻¹'o s ↪r s | ⟨f, λ a b, iff.rfl⟩ | def | rel_embedding.preimage | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | If `f` is injective, then it is a relation embedding from the
preimage relation of `s` to `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_preimage (f : r ↪r s) : r = f ⁻¹'o s | by { ext a b, exact f.map_rel_iff.symm } | theorem | rel_embedding.eq_preimage | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_irrefl (f : r ↪r s) [is_irrefl β s] : is_irrefl α r | ⟨λ a, mt f.map_rel_iff.2 (irrefl (f a))⟩ | theorem | rel_embedding.is_irrefl | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_refl (f : r ↪r s) [is_refl β s] : is_refl α r | ⟨λ a, f.map_rel_iff.1 $ refl _⟩ | theorem | rel_embedding.is_refl | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_symm (f : r ↪r s) [is_symm β s] : is_symm α r | ⟨λ a b, imp_imp_imp f.map_rel_iff.2 f.map_rel_iff.1 symm⟩ | theorem | rel_embedding.is_symm | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"imp_imp_imp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_asymm (f : r ↪r s) [is_asymm β s] : is_asymm α r | ⟨λ a b h₁ h₂, asymm (f.map_rel_iff.2 h₁) (f.map_rel_iff.2 h₂)⟩ | theorem | rel_embedding.is_asymm | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_antisymm : ∀ (f : r ↪r s) [is_antisymm β s], is_antisymm α r | | ⟨f, o⟩ ⟨H⟩ := ⟨λ a b h₁ h₂, f.inj' (H _ _ (o.2 h₁) (o.2 h₂))⟩ | theorem | rel_embedding.is_antisymm | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_trans : ∀ (f : r ↪r s) [is_trans β s], is_trans α r | | ⟨f, o⟩ ⟨H⟩ := ⟨λ a b c h₁ h₂, o.1 (H _ _ _ (o.2 h₁) (o.2 h₂))⟩ | theorem | rel_embedding.is_trans | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_total : ∀ (f : r ↪r s) [is_total β s], is_total α r | | ⟨f, o⟩ ⟨H⟩ := ⟨λ a b, (or_congr o o).1 (H _ _)⟩ | theorem | rel_embedding.is_total | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preorder : ∀ (f : r ↪r s) [is_preorder β s], is_preorder α r | | f H := by exactI {..f.is_refl, ..f.is_trans} | theorem | rel_embedding.is_preorder | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_partial_order : ∀ (f : r ↪r s) [is_partial_order β s], is_partial_order α r | | f H := by exactI {..f.is_preorder, ..f.is_antisymm} | theorem | rel_embedding.is_partial_order | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_linear_order : ∀ (f : r ↪r s) [is_linear_order β s], is_linear_order α r | | f H := by exactI {..f.is_partial_order, ..f.is_total} | theorem | rel_embedding.is_linear_order | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_strict_order : ∀ (f : r ↪r s) [is_strict_order β s], is_strict_order α r | | f H := by exactI {..f.is_irrefl, ..f.is_trans} | theorem | rel_embedding.is_strict_order | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_trichotomous : ∀ (f : r ↪r s) [is_trichotomous β s], is_trichotomous α r | | ⟨f, o⟩ ⟨H⟩ := ⟨λ a b, (or_congr o (or_congr f.inj'.eq_iff o)).1 (H _ _)⟩ | theorem | rel_embedding.is_trichotomous | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_strict_total_order :
∀ (f : r ↪r s) [is_strict_total_order β s], is_strict_total_order α r | | f H := by exactI {..f.is_trichotomous, ..f.is_strict_order} | theorem | rel_embedding.is_strict_total_order | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
acc (f : r ↪r s) (a : α) : acc s (f a) → acc r a | begin
generalize h : f a = b, intro ac,
induction ac with _ H IH generalizing a, subst h,
exact ⟨_, λ a' h, IH (f a') (f.map_rel_iff.2 h) _ rfl⟩
end | theorem | rel_embedding.acc | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
well_founded : ∀ (f : r ↪r s) (h : well_founded s), well_founded r | | f ⟨H⟩ := ⟨λ a, f.acc _ (H _)⟩ | theorem | rel_embedding.well_founded | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_well_founded (f : r ↪r s) [is_well_founded β s] : is_well_founded α r | ⟨f.well_founded is_well_founded.wf⟩ | theorem | rel_embedding.is_well_founded | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"is_well_founded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_well_order : ∀ (f : r ↪r s) [is_well_order β s], is_well_order α r | | f H := by exactI {wf := f.well_founded H.wf, ..f.is_strict_total_order} | theorem | rel_embedding.is_well_order | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"is_well_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype.well_founded_lt [has_lt α] [well_founded_lt α] (p : α → Prop) :
well_founded_lt (subtype p) | (subtype.rel_embedding (<) p).is_well_founded | instance | subtype.well_founded_lt | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"is_well_founded",
"subtype.rel_embedding",
"well_founded_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype.well_founded_gt [has_lt α] [well_founded_gt α] (p : α → Prop) :
well_founded_gt (subtype p) | (subtype.rel_embedding (>) p).is_well_founded | instance | subtype.well_founded_gt | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"is_well_founded",
"subtype.rel_embedding",
"well_founded_gt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient.mk_rel_hom [setoid α] {r : α → α → Prop} (H) :
r →r quotient.lift₂ r H | ⟨@quotient.mk α _, λ _ _, id⟩ | def | quotient.mk_rel_hom | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | `quotient.mk` as a relation homomorphism between the relation and the lift of a relation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient.out_rel_embedding [setoid α] {r : α → α → Prop} (H) :
quotient.lift₂ r H ↪r r | ⟨embedding.quotient_out α, begin
refine λ x y, quotient.induction_on₂ x y (λ a b, _),
apply iff_iff_eq.2 (H _ _ _ _ _ _);
apply quotient.mk_out
end⟩ | def | quotient.out_rel_embedding | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"quotient.mk_out"
] | `quotient.out` as a relation embedding between the lift of a relation and the relation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient.out'_rel_embedding {s : setoid α} {r : α → α → Prop} (H) :
(λ a b, quotient.lift_on₂' a b r H) ↪r r | { to_fun := quotient.out',
..quotient.out_rel_embedding _ } | def | quotient.out'_rel_embedding | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"quotient.lift_on₂'",
"quotient.out'",
"quotient.out_rel_embedding"
] | `quotient.out'` as a relation embedding between the lift of a relation and the relation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
acc_lift₂_iff [setoid α] {r : α → α → Prop} {H} {a} :
acc (quotient.lift₂ r H) ⟦a⟧ ↔ acc r a | begin
split,
{ exact rel_hom_class.acc (quotient.mk_rel_hom H) a, },
{ intro ac,
induction ac with _ H IH, dsimp at IH,
refine ⟨_, λ q h, _⟩,
obtain ⟨a', rfl⟩ := q.exists_rep,
exact IH a' h, },
end | theorem | acc_lift₂_iff | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"quotient.mk_rel_hom",
"rel_hom_class.acc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
acc_lift_on₂'_iff {s : setoid α} {r : α → α → Prop} {H} {a} :
acc (λ x y, quotient.lift_on₂' x y r H) (quotient.mk' a : quotient s) ↔ acc r a | acc_lift₂_iff | theorem | acc_lift_on₂'_iff | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"acc_lift₂_iff",
"quotient.lift_on₂'",
"quotient.mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
well_founded_lift₂_iff [setoid α] {r : α → α → Prop} {H} :
well_founded (quotient.lift₂ r H) ↔ well_founded r | begin
split,
{ exact rel_hom_class.well_founded (quotient.mk_rel_hom H), },
{ refine λ wf, ⟨λ q, _⟩,
obtain ⟨a, rfl⟩ := q.exists_rep,
exact acc_lift₂_iff.2 (wf.apply a), },
end | theorem | well_founded_lift₂_iff | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"quotient.mk_rel_hom",
"rel_hom_class.well_founded"
] | A relation is well founded iff its lift to a quotient is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
well_founded_lift_on₂'_iff {s : setoid α} {r : α → α → Prop} {H} :
well_founded (λ x y : quotient s, quotient.lift_on₂' x y r H) ↔ well_founded r | well_founded_lift₂_iff | theorem | well_founded_lift_on₂'_iff | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"quotient.lift_on₂'",
"well_founded_lift₂_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_map_rel_iff (f : α → β) [is_antisymm α r] [is_refl β s]
(hf : ∀ a b, s (f a) (f b) ↔ r a b) : r ↪r s | { to_fun := f,
inj' := λ x y h, antisymm ((hf _ _).1 (h ▸ refl _)) ((hf _ _).1 (h ▸ refl _)),
map_rel_iff' := hf } | def | rel_embedding.of_map_rel_iff | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | To define an relation embedding from an antisymmetric relation `r` to a reflexive relation `s` it
suffices to give a function together with a proof that it satisfies `s (f a) (f b) ↔ r a b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_map_rel_iff_coe (f : α → β) [is_antisymm α r] [is_refl β s]
(hf : ∀ a b, s (f a) (f b) ↔ r a b) :
⇑(of_map_rel_iff f hf : r ↪r s) = f | rfl | lemma | rel_embedding.of_map_rel_iff_coe | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_monotone [is_trichotomous α r] [is_asymm β s] (f : α → β)
(H : ∀ a b, r a b → s (f a) (f b)) : r ↪r s | begin
haveI := @is_asymm.is_irrefl β s _,
refine ⟨⟨f, λ a b e, _⟩, λ a b, ⟨λ h, _, H _ _⟩⟩,
{ refine ((@trichotomous _ r _ a b).resolve_left _).resolve_right _;
exact λ h, @irrefl _ s _ _ (by simpa [e] using H _ _ h) },
{ refine (@trichotomous _ r _ a b).resolve_right (or.rec (λ e, _) (λ h', _)),
{ subs... | def | rel_embedding.of_monotone | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"is_asymm.is_irrefl"
] | It suffices to prove `f` is monotone between strict relations
to show it is a relation embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_monotone_coe [is_trichotomous α r] [is_asymm β s] (f : α → β) (H) :
(@of_monotone _ _ r s _ _ f H : α → β) = f | rfl | theorem | rel_embedding.of_monotone_coe | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_is_empty (r : α → α → Prop) (s : β → β → Prop) [is_empty α] : r ↪r s | ⟨embedding.of_is_empty, is_empty_elim⟩ | def | rel_embedding.of_is_empty | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"is_empty"
] | A relation embedding from an empty type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_lift_rel_inl (r : α → α → Prop) (s : β → β → Prop) : r ↪r sum.lift_rel r s | { to_fun := sum.inl,
inj' := sum.inl_injective,
map_rel_iff' := λ a b, sum.lift_rel_inl_inl } | def | rel_embedding.sum_lift_rel_inl | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"sum.inl_injective",
"sum.lift_rel",
"sum.lift_rel_inl_inl"
] | `sum.inl` as a relation embedding into `sum.lift_rel r s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_lift_rel_inr (r : α → α → Prop) (s : β → β → Prop) : s ↪r sum.lift_rel r s | { to_fun := sum.inr,
inj' := sum.inr_injective,
map_rel_iff' := λ a b, sum.lift_rel_inr_inr } | def | rel_embedding.sum_lift_rel_inr | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"sum.inr_injective",
"sum.lift_rel",
"sum.lift_rel_inr_inr"
] | `sum.inr` as a relation embedding into `sum.lift_rel r s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_lift_rel_map (f : r ↪r s) (g : t ↪r u) : sum.lift_rel r t ↪r sum.lift_rel s u | { to_fun := sum.map f g,
inj' := f.injective.sum_map g.injective,
map_rel_iff' := by { rintro (a | b) (c | d); simp [f.map_rel_iff, g.map_rel_iff] } } | def | rel_embedding.sum_lift_rel_map | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"sum.lift_rel",
"sum.map"
] | `sum.map` as a relation embedding between `sum.lift_rel` relations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_lex_inl (r : α → α → Prop) (s : β → β → Prop) : r ↪r sum.lex r s | { to_fun := sum.inl,
inj' := sum.inl_injective,
map_rel_iff' := λ a b, sum.lex_inl_inl } | def | rel_embedding.sum_lex_inl | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"sum.inl_injective",
"sum.lex",
"sum.lex_inl_inl"
] | `sum.inl` as a relation embedding into `sum.lex r s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_lex_inr (r : α → α → Prop) (s : β → β → Prop) : s ↪r sum.lex r s | { to_fun := sum.inr,
inj' := sum.inr_injective,
map_rel_iff' := λ a b, sum.lex_inr_inr } | def | rel_embedding.sum_lex_inr | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"sum.inr_injective",
"sum.lex",
"sum.lex_inr_inr"
] | `sum.inr` as a relation embedding into `sum.lex r s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_lex_map (f : r ↪r s) (g : t ↪r u) : sum.lex r t ↪r sum.lex s u | { to_fun := sum.map f g,
inj' := f.injective.sum_map g.injective,
map_rel_iff' := by { rintro (a | b) (c | d); simp [f.map_rel_iff, g.map_rel_iff] } } | def | rel_embedding.sum_lex_map | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"sum.lex",
"sum.map"
] | `sum.map` as a relation embedding between `sum.lex` relations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_lex_mk_left (s : β → β → Prop) {a : α} (h : ¬ r a a) : s ↪r prod.lex r s | { to_fun := prod.mk a,
inj' := prod.mk.inj_left a,
map_rel_iff' := λ b₁ b₂, by simp [prod.lex_def, h] } | def | rel_embedding.prod_lex_mk_left | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"prod.lex_def",
"prod.mk.inj_left"
] | `λ b, prod.mk a b` as a relation embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_lex_mk_right (r : α → α → Prop) {b : β} (h : ¬ s b b) : r ↪r prod.lex r s | { to_fun := λ a, (a, b),
inj' := prod.mk.inj_right b,
map_rel_iff' := λ a₁ a₂, by simp [prod.lex_def, h] } | def | rel_embedding.prod_lex_mk_right | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"prod.lex_def",
"prod.mk.inj_right"
] | `λ a, prod.mk a b` as a relation embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_lex_map (f : r ↪r s) (g : t ↪r u) : prod.lex r t ↪r prod.lex s u | { to_fun := prod.map f g,
inj' := f.injective.prod_map g.injective,
map_rel_iff' := λ a b, by simp [prod.lex_def, f.map_rel_iff, g.map_rel_iff] } | def | rel_embedding.prod_lex_map | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"prod.lex_def"
] | `prod.map` as a relation embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel_iso {α β : Type*} (r : α → α → Prop) (s : β → β → Prop) extends α ≃ β | (map_rel_iff' : ∀ {a b}, s (to_equiv a) (to_equiv b) ↔ r a b) | structure | rel_iso | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | A relation isomorphism is an equivalence that is also a relation embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_rel_embedding (f : r ≃r s) : r ↪r s | ⟨f.to_equiv.to_embedding, λ _ _, f.map_rel_iff'⟩ | def | rel_iso.to_rel_embedding | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | Convert an `rel_iso` to an `rel_embedding`. This function is also available as a coercion
but often it is easier to write `f.to_rel_embedding` than to write explicitly `r` and `s`
in the target type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_equiv_injective : injective (to_equiv : (r ≃r s) → α ≃ β) | | ⟨e₁, o₁⟩ ⟨e₂, o₂⟩ h := by { congr, exact h } | theorem | rel_iso.to_equiv_injective | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_rel_embedding_eq_coe (f : r ≃r s) : f.to_rel_embedding = f | rfl | lemma | rel_iso.to_rel_embedding_eq_coe | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coe_fn (f : r ≃r s) : ((f : r ↪r s) : α → β) = f | rfl | lemma | rel_iso.coe_coe_fn | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_rel_iff (f : r ≃r s) {a b} : s (f a) (f b) ↔ r a b | f.map_rel_iff' | theorem | rel_iso.map_rel_iff | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_mk (f : α ≃ β) (o : ∀ ⦃a b⦄, s (f a) (f b) ↔ r a b) :
(rel_iso.mk f o : α → β) = f | rfl | theorem | rel_iso.coe_fn_mk | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_to_equiv (f : r ≃r s) : (f.to_equiv : α → β) = f | rfl | theorem | rel_iso.coe_fn_to_equiv | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_injective : @function.injective (r ≃r s) (α → β) coe_fn | fun_like.coe_injective | theorem | rel_iso.coe_fn_injective | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"fun_like.coe_injective"
] | The map `coe_fn : (r ≃r s) → (α → β)` is injective. Lean fails to parse
`function.injective (λ e : r ≃r s, (e : α → β))`, so we use a trick to say the same. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext ⦃f g : r ≃r s⦄ (h : ∀ x, f x = g x) : f = g | fun_like.ext f g h | theorem | rel_iso.ext | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {f g : r ≃r s} : f = g ↔ ∀ x, f x = g x | fun_like.ext_iff | theorem | rel_iso.ext_iff | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"fun_like.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm (f : r ≃r s) : s ≃r r | ⟨f.to_equiv.symm, λ a b, by erw [← f.map_rel_iff, f.1.apply_symm_apply, f.1.apply_symm_apply]⟩ | def | rel_iso.symm | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | Inverse map of a relation isomorphism is a relation isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simps.apply (h : r ≃r s) : α → β | h | def | rel_iso.simps.apply | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simps.symm_apply (h : r ≃r s) : β → α | h.symm
initialize_simps_projections rel_iso
(to_equiv_to_fun → apply, to_equiv_inv_fun → symm_apply, -to_equiv) | def | rel_iso.simps.symm_apply | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"rel_iso"
] | See Note [custom simps projection]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl (r : α → α → Prop) : r ≃r r | ⟨equiv.refl _, λ a b, iff.rfl⟩ | def | rel_iso.refl | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | Identity map is a relation isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans (f₁ : r ≃r s) (f₂ : s ≃r t) : r ≃r t | ⟨f₁.to_equiv.trans f₂.to_equiv, λ a b, f₂.map_rel_iff.trans f₁.map_rel_iff⟩ | def | rel_iso.trans | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | Composition of two relation isomorphisms is a relation isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
default_def (r : α → α → Prop) : default = rel_iso.refl r | rfl | lemma | rel_iso.default_def | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"rel_iso.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast {α β : Type u} {r : α → α → Prop} {s : β → β → Prop}
(h₁ : α = β) (h₂ : r == s) : r ≃r s | ⟨equiv.cast h₁, λ a b, by { subst h₁, rw eq_of_heq h₂, refl }⟩ | def | rel_iso.cast | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | A relation isomorphism between equal relations on equal types. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cast_symm {α β : Type u} {r : α → α → Prop} {s : β → β → Prop}
(h₁ : α = β) (h₂ : r == s) : (rel_iso.cast h₁ h₂).symm = rel_iso.cast h₁.symm h₂.symm | rfl | theorem | rel_iso.cast_symm | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"rel_iso.cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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