statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
with_bot_map (f : α →o β) : with_bot α →o with_bot β | ⟨with_bot.map f, f.mono.with_bot_map⟩ | def | order_hom.with_bot_map | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"with_bot"
] | Lift an order homomorphism `f : α →o β` to an order homomorphism `with_bot α →o with_bot β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_top_map (f : α →o β) : with_top α →o with_top β | ⟨with_top.map f, f.mono.with_top_map⟩ | def | order_hom.with_top_map | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"with_top"
] | Lift an order homomorphism `f : α →o β` to an order homomorphism `with_top α →o with_top β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel_embedding.order_embedding_of_lt_embedding [partial_order α] [partial_order β]
(f : ((<) : α → α → Prop) ↪r ((<) : β → β → Prop)) :
α ↪o β | { map_rel_iff' := by { intros, simp [le_iff_lt_or_eq,f.map_rel_iff, f.injective.eq_iff] }, .. f } | def | rel_embedding.order_embedding_of_lt_embedding | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | Embeddings of partial orders that preserve `<` also preserve `≤`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel_embedding.order_embedding_of_lt_embedding_apply [partial_order α] [partial_order β]
{f : ((<) : α → α → Prop) ↪r ((<) : β → β → Prop)} {x : α} :
rel_embedding.order_embedding_of_lt_embedding f x = f x | rfl | lemma | rel_embedding.order_embedding_of_lt_embedding_apply | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"rel_embedding.order_embedding_of_lt_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_embedding : ((<) : α → α → Prop) ↪r ((<) : β → β → Prop) | { map_rel_iff' := by intros; simp [lt_iff_le_not_le, f.map_rel_iff], .. f } | def | order_embedding.lt_embedding | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | `<` is preserved by order embeddings of preorders. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lt_embedding_apply (x : α) : f.lt_embedding x = f x | rfl | lemma | order_embedding.lt_embedding_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 | |
le_iff_le {a b} : (f a) ≤ (f b) ↔ a ≤ b | f.map_rel_iff | theorem | order_embedding.le_iff_le | 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 | |
lt_iff_lt {a b} : f a < f b ↔ a < b | f.lt_embedding.map_rel_iff | theorem | order_embedding.lt_iff_lt | 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 | |
eq_iff_eq {a b} : f a = f b ↔ a = b | f.injective.eq_iff | lemma | order_embedding.eq_iff_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 | |
monotone : monotone f | order_hom_class.monotone f | theorem | order_embedding.monotone | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"monotone",
"order_hom_class.monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono : strict_mono f | λ x y, f.lt_iff_lt.2 | theorem | order_embedding.strict_mono | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
acc (a : α) : acc (<) (f a) → acc (<) a | f.lt_embedding.acc a | theorem | order_embedding.acc | 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 | |
well_founded :
well_founded ((<) : β → β → Prop) → well_founded ((<) : α → α → Prop) | f.lt_embedding.well_founded | theorem | order_embedding.well_founded | 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 | |
is_well_order [is_well_order β (<)] : is_well_order α (<) | f.lt_embedding.is_well_order | theorem | order_embedding.is_well_order | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"is_well_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual : αᵒᵈ ↪o βᵒᵈ | ⟨f.to_embedding, λ a b, f.map_rel_iff⟩ | def | order_embedding.dual | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | An order embedding is also an order embedding between dual orders. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_bot_map (f : α ↪o β) : with_bot α ↪o with_bot β | { to_fun := with_bot.map f,
map_rel_iff' := with_bot.map_le_iff f (λ a b, f.map_rel_iff),
.. f.to_embedding.option_map } | def | order_embedding.with_bot_map | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"with_bot",
"with_bot.map",
"with_bot.map_le_iff"
] | A version of `with_bot.map` for order embeddings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_top_map (f : α ↪o β) : with_top α ↪o with_top β | { to_fun := with_top.map f,
.. f.dual.with_bot_map.dual } | def | order_embedding.with_top_map | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"with_top",
"with_top.map"
] | A version of `with_top.map` for order embeddings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_map_le_iff {α β} [partial_order α] [preorder β] (f : α → β)
(hf : ∀ a b, f a ≤ f b ↔ a ≤ b) : α ↪o β | rel_embedding.of_map_rel_iff f hf | def | order_embedding.of_map_le_iff | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"rel_embedding.of_map_rel_iff"
] | To define an order embedding from a partial order to a preorder it suffices to give a function
together with a proof that it satisfies `f a ≤ f b ↔ a ≤ b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of_map_le_iff {α β} [partial_order α] [preorder β] {f : α → β} (h) :
⇑(of_map_le_iff f h) = f | rfl | lemma | order_embedding.coe_of_map_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 | |
of_strict_mono {α β} [linear_order α] [preorder β] (f : α → β)
(h : strict_mono f) : α ↪o β | of_map_le_iff f (λ _ _, h.le_iff_le) | def | order_embedding.of_strict_mono | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"strict_mono"
] | A strictly monotone map from a linear order is an order embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of_strict_mono {α β} [linear_order α] [preorder β] {f : α → β}
(h : strict_mono f) : ⇑(of_strict_mono f h) = f | rfl | lemma | order_embedding.coe_of_strict_mono | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype (p : α → Prop) : subtype p ↪o α | ⟨function.embedding.subtype p, λ x y, iff.rfl⟩ | def | order_embedding.subtype | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | Embedding of a subtype into the ambient type as an `order_embedding`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_order_hom {X Y : Type*} [preorder X] [preorder Y] (f : X ↪o Y) : X →o Y | { to_fun := f,
monotone' := f.monotone } | def | order_embedding.to_order_hom | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | Convert an `order_embedding` to a `order_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_order_hom : α →o β | { to_fun := f,
monotone' := strict_mono.monotone (λ x y, f.map_rel), } | def | rel_hom.to_order_hom | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"strict_mono.monotone"
] | A bundled expression of the fact that a map between partial orders that is strictly monotone
is weakly monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel_embedding.to_order_hom_injective (f : ((<) : α → α → Prop) ↪r ((<) : β → β → Prop)) :
function.injective (f : ((<) : α → α → Prop) →r ((<) : β → β → Prop)).to_order_hom | λ _ _ h, f.injective h | lemma | rel_embedding.to_order_hom_injective | 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 | |
to_fun_eq_coe {f : α ≃o β} : f.to_fun = f | rfl | lemma | order_iso.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 | |
ext {f g : α ≃o β} (h : (f : α → β) = g) : f = g | fun_like.coe_injective h | lemma | order_iso.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 | |
to_order_embedding (e : α ≃o β) : α ↪o β | e.to_rel_embedding | def | order_iso.to_order_embedding | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | Reinterpret an order isomorphism as an order embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_order_embedding (e : α ≃o β) :
⇑(e.to_order_embedding) = e | rfl | lemma | order_iso.coe_to_order_embedding | 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 | |
bijective (e : α ≃o β) : function.bijective e | e.to_equiv.bijective | lemma | order_iso.bijective | 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 | |
injective (e : α ≃o β) : function.injective e | e.to_equiv.injective | lemma | order_iso.injective | 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 | |
surjective (e : α ≃o β) : function.surjective e | e.to_equiv.surjective | lemma | order_iso.surjective | 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_eq_iff_eq (e : α ≃o β) {x y : α} : e x = e y ↔ x = y | e.to_equiv.apply_eq_iff_eq | lemma | order_iso.apply_eq_iff_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 | |
refl (α : Type*) [has_le α] : α ≃o α | rel_iso.refl (≤) | def | order_iso.refl | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"rel_iso.refl"
] | Identity order isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_refl : ⇑(refl α) = id | rfl | lemma | order_iso.coe_refl | 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 | |
refl_apply (x : α) : refl α x = x | rfl | lemma | order_iso.refl_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 | |
refl_to_equiv : (refl α).to_equiv = equiv.refl α | rfl | lemma | order_iso.refl_to_equiv | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm (e : α ≃o β) : β ≃o α | e.symm | def | order_iso.symm | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | Inverse of an order isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_symm_apply (e : α ≃o β) (x : β) : e (e.symm x) = x | e.to_equiv.apply_symm_apply x | lemma | order_iso.apply_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 | |
symm_apply_apply (e : α ≃o β) (x : α) : e.symm (e x) = x | e.to_equiv.symm_apply_apply x | lemma | order_iso.symm_apply_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 | |
symm_refl (α : Type*) [has_le α] : (refl α).symm = refl α | rfl | lemma | order_iso.symm_refl | 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_eq_iff_eq_symm_apply (e : α ≃o β) (x : α) (y : β) : e x = y ↔ x = e.symm y | e.to_equiv.apply_eq_iff_eq_symm_apply | lemma | order_iso.apply_eq_iff_eq_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 | |
symm_apply_eq (e : α ≃o β) {x : α} {y : β} : e.symm y = x ↔ y = e x | e.to_equiv.symm_apply_eq | theorem | order_iso.symm_apply_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 | |
symm_symm (e : α ≃o β) : e.symm.symm = e | by { ext, refl } | lemma | order_iso.symm_symm | 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_injective : function.injective (symm : (α ≃o β) → (β ≃o α)) | λ e e' h, by rw [← e.symm_symm, h, e'.symm_symm] | lemma | order_iso.symm_injective | 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 | |
to_equiv_symm (e : α ≃o β) : e.to_equiv.symm = e.symm.to_equiv | rfl | lemma | order_iso.to_equiv_symm | 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 | |
trans (e : α ≃o β) (e' : β ≃o γ) : α ≃o γ | e.trans e' | def | order_iso.trans | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | Composition of two order isomorphisms is an order isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_trans (e : α ≃o β) (e' : β ≃o γ) : ⇑(e.trans e') = e' ∘ e | rfl | lemma | order_iso.coe_trans | 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 | |
trans_apply (e : α ≃o β) (e' : β ≃o γ) (x : α) : e.trans e' x = e' (e x) | rfl | lemma | order_iso.trans_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 | |
refl_trans (e : α ≃o β) : (refl α).trans e = e | by { ext x, refl } | lemma | order_iso.refl_trans | 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 | |
trans_refl (e : α ≃o β) : e.trans (refl β) = e | by { ext x, refl } | lemma | order_iso.trans_refl | 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_trans_apply (e₁ : α ≃o β) (e₂ : β ≃o γ) (c : γ) :
(e₁.trans e₂).symm c = e₁.symm (e₂.symm c) | rfl | lemma | order_iso.symm_trans_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 | |
symm_trans (e₁ : α ≃o β) (e₂ : β ≃o γ) : (e₁.trans e₂).symm = e₂.symm.trans e₁.symm | rfl | lemma | order_iso.symm_trans | 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_comm : (α × β) ≃o (β × α) | { to_equiv := equiv.prod_comm α β,
map_rel_iff' := λ a b, prod.swap_le_swap } | def | order_iso.prod_comm | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"equiv.prod_comm",
"prod.swap_le_swap"
] | `prod.swap` as an `order_iso`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_prod_comm : ⇑(prod_comm : (α × β) ≃o (β × α)) = prod.swap | rfl | lemma | order_iso.coe_prod_comm | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"prod.swap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_comm_symm : (prod_comm : (α × β) ≃o (β × α)).symm = prod_comm | rfl | lemma | order_iso.prod_comm_symm | 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_dual : α ≃o αᵒᵈᵒᵈ | refl α | def | order_iso.dual_dual | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | The order isomorphism between a type and its double dual. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_dual_dual : ⇑(dual_dual α) = to_dual ∘ to_dual | rfl | lemma | order_iso.coe_dual_dual | 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_dual_dual_symm : ⇑(dual_dual α).symm = of_dual ∘ of_dual | rfl | lemma | order_iso.coe_dual_dual_symm | 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_dual_apply (a : α) : dual_dual α a = to_dual (to_dual a) | rfl | lemma | order_iso.dual_dual_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 | |
dual_dual_symm_apply (a : αᵒᵈᵒᵈ) : (dual_dual α).symm a = of_dual (of_dual a) | rfl | lemma | order_iso.dual_dual_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 | |
le_iff_le (e : α ≃o β) {x y : α} : e x ≤ e y ↔ x ≤ y | e.map_rel_iff | lemma | order_iso.le_iff_le | 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_symm_apply (e : α ≃o β) {x : α} {y : β} : x ≤ e.symm y ↔ e x ≤ y | e.rel_symm_apply | lemma | order_iso.le_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 | |
symm_apply_le (e : α ≃o β) {x : α} {y : β} : e.symm y ≤ x ↔ y ≤ e x | e.symm_apply_rel | lemma | order_iso.symm_apply_le | 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 | |
monotone (e : α ≃o β) : monotone e | e.to_order_embedding.monotone | lemma | order_iso.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 | |
strict_mono (e : α ≃o β) : strict_mono e | e.to_order_embedding.strict_mono | lemma | order_iso.strict_mono | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_iff_lt (e : α ≃o β) {x y : α} : e x < e y ↔ x < y | e.to_order_embedding.lt_iff_lt | lemma | order_iso.lt_iff_lt | 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 | |
to_rel_iso_lt (e : α ≃o β) : ((<) : α → α → Prop) ≃r ((<) : β → β → Prop) | ⟨e.to_equiv, λ x y, lt_iff_lt e⟩ | def | order_iso.to_rel_iso_lt | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | Converts an `order_iso` into a `rel_iso (<) (<)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_rel_iso_lt_apply (e : α ≃o β) (x : α) : e.to_rel_iso_lt x = e x | rfl | lemma | order_iso.to_rel_iso_lt_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 | |
to_rel_iso_lt_symm (e : α ≃o β) : e.to_rel_iso_lt.symm = e.symm.to_rel_iso_lt | rfl | lemma | order_iso.to_rel_iso_lt_symm | 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 | |
of_rel_iso_lt {α β} [partial_order α] [partial_order β]
(e : ((<) : α → α → Prop) ≃r ((<) : β → β → Prop)) : α ≃o β | ⟨e.to_equiv, λ x y, by simp [le_iff_eq_or_lt, e.map_rel_iff]⟩ | def | order_iso.of_rel_iso_lt | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"le_iff_eq_or_lt"
] | Converts a `rel_iso (<) (<)` into an `order_iso`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_rel_iso_lt_apply {α β} [partial_order α] [partial_order β]
(e : ((<) : α → α → Prop) ≃r ((<) : β → β → Prop)) (x : α) : of_rel_iso_lt e x = e x | rfl | lemma | order_iso.of_rel_iso_lt_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 | |
of_rel_iso_lt_symm {α β} [partial_order α] [partial_order β]
(e : ((<) : α → α → Prop) ≃r ((<) : β → β → Prop)) :
(of_rel_iso_lt e).symm = of_rel_iso_lt e.symm | rfl | lemma | order_iso.of_rel_iso_lt_symm | 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 | |
of_rel_iso_lt_to_rel_iso_lt {α β} [partial_order α] [partial_order β] (e : α ≃o β) :
of_rel_iso_lt (to_rel_iso_lt e) = e | by { ext, simp } | lemma | order_iso.of_rel_iso_lt_to_rel_iso_lt | 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 | |
to_rel_iso_lt_of_rel_iso_lt {α β} [partial_order α] [partial_order β]
(e : ((<) : α → α → Prop) ≃r ((<) : β → β → Prop)) : to_rel_iso_lt (of_rel_iso_lt e) = e | by { ext, simp } | lemma | order_iso.to_rel_iso_lt_of_rel_iso_lt | 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 | |
of_cmp_eq_cmp {α β} [linear_order α] [linear_order β] (f : α → β) (g : β → α)
(h : ∀ (a : α) (b : β), cmp a (g b) = cmp (f a) b) : α ≃o β | have gf : ∀ (a : α), a = g (f a) := by { intro, rw [←cmp_eq_eq_iff, h, cmp_self_eq_eq] },
{ to_fun := f,
inv_fun := g,
left_inv := λ a, (gf a).symm,
right_inv := by { intro, rw [←cmp_eq_eq_iff, ←h, cmp_self_eq_eq] },
map_rel_iff' := by { intros, apply le_iff_le_of_cmp_eq_cmp, convert (h _ _).symm, apply gf } } | def | order_iso.of_cmp_eq_cmp | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"cmp_self_eq_eq",
"inv_fun",
"le_iff_le_of_cmp_eq_cmp"
] | To show that `f : α → β`, `g : β → α` make up an order isomorphism of linear orders,
it suffices to prove `cmp a (g b) = cmp (f a) b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_hom_inv {F G : Type*} [order_hom_class F α β] [order_hom_class G β α]
(f : F) (g : G) (h₁ : (f : α →o β).comp (g : β →o α) = order_hom.id)
(h₂ : (g : β →o α).comp (f : α →o β) = order_hom.id) : α ≃o β | { to_fun := f,
inv_fun := g,
left_inv := fun_like.congr_fun h₂,
right_inv := fun_like.congr_fun h₁,
map_rel_iff' := λ a b, ⟨λ h, by { replace h := map_rel g h, rwa [equiv.coe_fn_mk,
(show g (f a) = (g : β →o α).comp (f : α →o β) a, from rfl),
(show g (f b) = (g : β →o α).comp (f : α →o β) b, from rfl), ... | def | order_iso.of_hom_inv | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"equiv.coe_fn_mk",
"fun_like.congr_fun",
"inv_fun",
"monotone",
"order_hom.id",
"order_hom_class"
] | To show that `f : α →o β` and `g : β →o α` make up an order isomorphism it is enough to show
that `g` is the inverse of `f` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fun_unique (α β : Type*) [unique α] [preorder β] :
(α → β) ≃o β | { to_equiv := equiv.fun_unique α β,
map_rel_iff' := λ f g, by simp [pi.le_def, unique.forall_iff] } | def | order_iso.fun_unique | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"equiv.fun_unique",
"pi.le_def",
"unique",
"unique.forall_iff"
] | Order isomorphism between `α → β` and `β`, where `α` has a unique element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fun_unique_symm_apply {α β : Type*} [unique α] [preorder β] :
((fun_unique α β).symm : β → α → β) = function.const α | rfl | lemma | order_iso.fun_unique_symm_apply | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_order_iso (e : α ≃ β) (h₁ : monotone e) (h₂ : monotone e.symm) :
α ≃o β | ⟨e, λ x y, ⟨λ h, by simpa only [e.symm_apply_apply] using h₂ h, λ h, h₁ h⟩⟩ | def | equiv.to_order_iso | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"monotone"
] | If `e` is an equivalence with monotone forward and inverse maps, then `e` is an
order isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_order_iso (e : α ≃ β) (h₁ : monotone e) (h₂ : monotone e.symm) :
⇑(e.to_order_iso h₁ h₂) = e | rfl | lemma | equiv.coe_to_order_iso | 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_order_iso_to_equiv (e : α ≃ β) (h₁ : monotone e) (h₂ : monotone e.symm) :
(e.to_order_iso h₁ h₂).to_equiv = e | rfl | lemma | equiv.to_order_iso_to_equiv | 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 | |
order_iso_of_right_inverse
(g : β → α) (hg : function.right_inverse g f) : α ≃o β | { to_fun := f,
inv_fun := g,
left_inv := λ x, h_mono.injective $ hg _,
right_inv := hg,
.. order_embedding.of_strict_mono f h_mono } | def | strict_mono.order_iso_of_right_inverse | 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_embedding.of_strict_mono"
] | A strictly monotone function with a right inverse is an order isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_iso.dual [has_le α] [has_le β] (f : α ≃o β) : αᵒᵈ ≃o βᵒᵈ | ⟨f.to_equiv, λ _ _, f.map_rel_iff⟩ | def | order_iso.dual | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [] | An order isomorphism is also an order isomorphism between dual orders. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_iso.map_bot' [has_le α] [partial_order β] (f : α ≃o β) {x : α} {y : β}
(hx : ∀ x', x ≤ x') (hy : ∀ y', y ≤ y') : f x = y | by { refine le_antisymm _ (hy _), rw [← f.apply_symm_apply y, f.map_rel_iff], apply hx } | lemma | order_iso.map_bot' | 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 | |
order_iso.map_bot [has_le α] [partial_order β] [order_bot α] [order_bot β] (f : α ≃o β) :
f ⊥ = ⊥ | f.map_bot' (λ _, bot_le) (λ _, bot_le) | lemma | order_iso.map_bot | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"bot_le",
"order_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso.map_top' [has_le α] [partial_order β] (f : α ≃o β) {x : α} {y : β}
(hx : ∀ x', x' ≤ x) (hy : ∀ y', y' ≤ y) : f x = y | f.dual.map_bot' hx hy | lemma | order_iso.map_top' | 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 | |
order_iso.map_top [has_le α] [partial_order β] [order_top α] [order_top β] (f : α ≃o β) :
f ⊤ = ⊤ | f.dual.map_bot | lemma | order_iso.map_top | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"order_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_embedding.map_inf_le [semilattice_inf α] [semilattice_inf β] (f : α ↪o β) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y | f.monotone.map_inf_le x y | lemma | order_embedding.map_inf_le | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"semilattice_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_embedding.le_map_sup [semilattice_sup α] [semilattice_sup β] (f : α ↪o β) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) | f.monotone.le_map_sup x y | lemma | order_embedding.le_map_sup | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso.map_inf [semilattice_inf α] [semilattice_inf β] (f : α ≃o β) (x y : α) :
f (x ⊓ y) = f x ⊓ f y | begin
refine (f.to_order_embedding.map_inf_le x y).antisymm _,
apply f.symm.le_iff_le.1,
simpa using f.symm.to_order_embedding.map_inf_le (f x) (f y),
end | lemma | order_iso.map_inf | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"semilattice_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso.map_sup [semilattice_sup α] [semilattice_sup β] (f : α ≃o β) (x y : α) :
f (x ⊔ y) = f x ⊔ f y | f.dual.map_inf x y | lemma | order_iso.map_sup | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint.map_order_iso [semilattice_inf α] [order_bot α] [semilattice_inf β] [order_bot β]
{a b : α} (f : α ≃o β) (ha : disjoint a b) : disjoint (f a) (f b) | by { rw [disjoint_iff_inf_le, ←f.map_inf, ←f.map_bot], exact f.monotone ha.le_bot } | lemma | disjoint.map_order_iso | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"disjoint",
"disjoint_iff_inf_le",
"order_bot",
"semilattice_inf"
] | Note that this goal could also be stated `(disjoint on f) a b` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
codisjoint.map_order_iso [semilattice_sup α] [order_top α] [semilattice_sup β] [order_top β]
{a b : α} (f : α ≃o β) (ha : codisjoint a b) : codisjoint (f a) (f b) | by { rw [codisjoint_iff_le_sup, ←f.map_sup, ←f.map_top], exact f.monotone ha.top_le } | lemma | codisjoint.map_order_iso | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"codisjoint",
"codisjoint_iff_le_sup",
"order_top",
"semilattice_sup"
] | Note that this goal could also be stated `(codisjoint on f) a b` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
disjoint_map_order_iso_iff [semilattice_inf α] [order_bot α] [semilattice_inf β]
[order_bot β] {a b : α} (f : α ≃o β) : disjoint (f a) (f b) ↔ disjoint a b | ⟨λ h, f.symm_apply_apply a ▸ f.symm_apply_apply b ▸ h.map_order_iso f.symm, λ h, h.map_order_iso f⟩ | lemma | disjoint_map_order_iso_iff | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"disjoint",
"order_bot",
"semilattice_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
codisjoint_map_order_iso_iff [semilattice_sup α] [order_top α] [semilattice_sup β]
[order_top β] {a b : α} (f : α ≃o β) : codisjoint (f a) (f b) ↔ codisjoint a b | ⟨λ h, f.symm_apply_apply a ▸ f.symm_apply_apply b ▸ h.map_order_iso f.symm, λ h, h.map_order_iso f⟩ | lemma | codisjoint_map_order_iso_iff | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"codisjoint",
"order_top",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_dual_top_equiv [has_le α] : with_bot αᵒᵈ ≃o (with_top α)ᵒᵈ | order_iso.refl _ | def | with_bot.to_dual_top_equiv | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"order_iso.refl",
"with_bot",
"with_top"
] | Taking the dual then adding `⊥` is the same as adding `⊤` then taking the dual.
This is the order iso form of `with_bot.of_dual`, as proven by `coe_to_dual_top_equiv_eq`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_dual_top_equiv_coe [has_le α] (a : α) :
with_bot.to_dual_top_equiv ↑(to_dual a) = to_dual (a : with_top α) | rfl | lemma | with_bot.to_dual_top_equiv_coe | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"with_bot.to_dual_top_equiv",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_dual_top_equiv_symm_coe [has_le α] (a : α) :
with_bot.to_dual_top_equiv.symm (to_dual (a : with_top α)) = ↑(to_dual a) | rfl | lemma | with_bot.to_dual_top_equiv_symm_coe | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_dual_top_equiv_bot [has_le α] :
with_bot.to_dual_top_equiv (⊥ : with_bot αᵒᵈ) = ⊥ | rfl | lemma | with_bot.to_dual_top_equiv_bot | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"with_bot",
"with_bot.to_dual_top_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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