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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|---|---|---|---|---|---|---|---|---|---|---|
cast_refl {α : Type u} {r : α → α → Prop}
(h₁ : α = α := rfl) (h₂ : r == r := heq.rfl) : rel_iso.cast h₁ h₂ = rel_iso.refl r | rfl | theorem | rel_iso.cast_refl | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"rel_iso.cast",
"rel_iso.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_trans {α β γ : Type u}
{r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} (h₁ : α = β) (h₁' : β = γ)
(h₂ : r == s) (h₂' : s == t): (rel_iso.cast h₁ h₂).trans (rel_iso.cast h₁' h₂') =
rel_iso.cast (h₁.trans h₁') (h₂.trans h₂') | ext $ λ x, by { subst h₁, refl } | theorem | rel_iso.cast_trans | 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 | |
swap (f : r ≃r s) : (swap r) ≃r (swap s) | ⟨f.to_equiv, λ _ _, f.map_rel_iff⟩ | def | rel_iso.swap | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | a relation isomorphism is also a relation isomorphism between dual relations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_fn_symm_mk (f o) : ((@rel_iso.mk _ _ r s f o).symm : β → α) = f.symm | rfl | theorem | rel_iso.coe_fn_symm_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 | |
apply_symm_apply (e : r ≃r s) (x : β) : e (e.symm x) = x | e.to_equiv.apply_symm_apply x | theorem | rel_iso.apply_symm_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 | |
symm_apply_apply (e : r ≃r s) (x : α) : e.symm (e x) = x | e.to_equiv.symm_apply_apply x | theorem | rel_iso.symm_apply_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 | |
rel_symm_apply (e : r ≃r s) {x y} : r x (e.symm y) ↔ s (e x) y | by rw [← e.map_rel_iff, e.apply_symm_apply] | theorem | rel_iso.rel_symm_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 | |
symm_apply_rel (e : r ≃r s) {x y} : r (e.symm x) y ↔ s x (e y) | by rw [← e.map_rel_iff, e.apply_symm_apply] | theorem | rel_iso.symm_apply_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 | |
bijective (e : r ≃r s) : bijective e | e.to_equiv.bijective | lemma | rel_iso.bijective | 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 (e : r ≃r s) : injective e | e.to_equiv.injective | lemma | rel_iso.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 | |
surjective (e : r ≃r s) : surjective e | e.to_equiv.surjective | lemma | rel_iso.surjective | 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 | |
eq_iff_eq (f : r ≃r s) {a b} : f a = f b ↔ a = b | f.injective.eq_iff | lemma | rel_iso.eq_iff_eq | 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 | |
preimage (f : α ≃ β) (s : β → β → Prop) : f ⁻¹'o s ≃r s | ⟨f, λ a b, iff.rfl⟩ | def | rel_iso.preimage | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | Any equivalence lifts to a relation isomorphism between `s` and its preimage. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_well_order.preimage {α : Type u} (r : α → α → Prop) [is_well_order α r] (f : β ≃ α) :
is_well_order β (f ⁻¹'o r) | @rel_embedding.is_well_order _ _ (f ⁻¹'o r) r (rel_iso.preimage f r) _ | instance | rel_iso.is_well_order.preimage | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"is_well_order",
"rel_embedding.is_well_order",
"rel_iso.preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_well_order.ulift {α : Type u} (r : α → α → Prop) [is_well_order α r] :
is_well_order (ulift α) (ulift.down ⁻¹'o r) | is_well_order.preimage r equiv.ulift | instance | rel_iso.is_well_order.ulift | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"equiv.ulift",
"is_well_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_surjective (f : r ↪r s) (H : surjective f) : r ≃r s | ⟨equiv.of_bijective f ⟨f.injective, H⟩, λ a b, f.map_rel_iff⟩ | def | rel_iso.of_surjective | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [] | A surjective relation embedding is a relation isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_lex_congr {α₁ α₂ β₁ β₂ r₁ r₂ s₁ s₂}
(e₁ : @rel_iso α₁ β₁ r₁ s₁) (e₂ : @rel_iso α₂ β₂ r₂ s₂) :
sum.lex r₁ r₂ ≃r sum.lex s₁ s₂ | ⟨equiv.sum_congr e₁.to_equiv e₂.to_equiv, λ a b,
by cases e₁ with f hf; cases e₂ with g hg;
cases a; cases b; simp [hf, hg]⟩ | def | rel_iso.sum_lex_congr | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"rel_iso",
"sum.lex"
] | Given relation isomorphisms `r₁ ≃r s₁` and `r₂ ≃r s₂`, construct a relation isomorphism for the
lexicographic orders on the sum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_lex_congr {α₁ α₂ β₁ β₂ r₁ r₂ s₁ s₂}
(e₁ : @rel_iso α₁ β₁ r₁ s₁) (e₂ : @rel_iso α₂ β₂ r₂ s₂) :
prod.lex r₁ r₂ ≃r prod.lex s₁ s₂ | ⟨equiv.prod_congr e₁.to_equiv e₂.to_equiv,
λ a b, by simp [prod.lex_def, e₁.map_rel_iff, e₂.map_rel_iff]⟩ | def | rel_iso.prod_lex_congr | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"prod.lex_def",
"rel_iso"
] | Given relation isomorphisms `r₁ ≃r s₁` and `r₂ ≃r s₂`, construct a relation isomorphism for the
lexicographic orders on the product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel_iso_of_is_empty (r : α → α → Prop) (s : β → β → Prop) [is_empty α] [is_empty β] : r ≃r s | ⟨equiv.equiv_of_is_empty α β, is_empty_elim⟩ | def | rel_iso.rel_iso_of_is_empty | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"is_empty"
] | Two relations on empty types are isomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel_iso_of_unique_of_irrefl (r : α → α → Prop) (s : β → β → Prop)
[is_irrefl α r] [is_irrefl β s] [unique α] [unique β] : r ≃r s | ⟨equiv.equiv_of_unique α β,
λ x y, by simp [not_rel_of_subsingleton r, not_rel_of_subsingleton s]⟩ | def | rel_iso.rel_iso_of_unique_of_irrefl | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"not_rel_of_subsingleton",
"unique"
] | Two irreflexive relations on a unique type are isomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel_iso_of_unique_of_refl (r : α → α → Prop) (s : β → β → Prop)
[is_refl α r] [is_refl β s] [unique α] [unique β] : r ≃r s | ⟨equiv.equiv_of_unique α β,
λ x y, by simp [rel_of_subsingleton r, rel_of_subsingleton s]⟩ | def | rel_iso.rel_iso_of_unique_of_refl | order.rel_iso | src/order/rel_iso/basic.lean | [
"data.fun_like.basic",
"logic.embedding.basic",
"order.rel_classes"
] | [
"rel_of_subsingleton",
"unique"
] | Two reflexive relations on a unique type are isomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_one : ⇑(1 : r ≃r r) = id | rfl | lemma | rel_iso.coe_one | order.rel_iso | src/order/rel_iso/group.lean | [
"algebra.group.defs",
"order.rel_iso.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (e₁ e₂ : r ≃r r) : ⇑(e₁ * e₂) = e₁ ∘ e₂ | rfl | lemma | rel_iso.coe_mul | order.rel_iso | src/order/rel_iso/group.lean | [
"algebra.group.defs",
"order.rel_iso.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_apply (e₁ e₂ : r ≃r r) (x : α) : (e₁ * e₂) x = e₁ (e₂ x) | rfl | lemma | rel_iso.mul_apply | order.rel_iso | src/order/rel_iso/group.lean | [
"algebra.group.defs",
"order.rel_iso.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_apply_self (e : r ≃r r) (x) : e⁻¹ (e x) = x | e.symm_apply_apply x | lemma | rel_iso.inv_apply_self | order.rel_iso | src/order/rel_iso/group.lean | [
"algebra.group.defs",
"order.rel_iso.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_inv_self (e : r ≃r r) (x) : e (e⁻¹ x) = x | e.apply_symm_apply x | lemma | rel_iso.apply_inv_self | order.rel_iso | src/order/rel_iso/group.lean | [
"algebra.group.defs",
"order.rel_iso.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_inf [semilattice_inf α] [linear_order β]
[rel_hom_class F ((<) : β → β → Prop) ((<) : α → α → Prop)]
(a : F) (m n : β) : a (m ⊓ n) = a m ⊓ a n | (strict_mono.monotone $ λ x y, map_rel a).map_inf m n | lemma | rel_hom_class.map_inf | order.rel_iso | src/order/rel_iso/set.lean | [
"order.rel_iso.basic",
"logic.embedding.set"
] | [
"rel_hom_class",
"semilattice_inf",
"strict_mono.monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sup [semilattice_sup α] [linear_order β]
[rel_hom_class F ((>) : β → β → Prop) ((>) : α → α → Prop)]
(a : F) (m n : β) : a (m ⊔ n) = a m ⊔ a n | @map_inf αᵒᵈ βᵒᵈ _ _ _ _ _ _ _ | lemma | rel_hom_class.map_sup | order.rel_iso | src/order/rel_iso/set.lean | [
"order.rel_iso.basic",
"logic.embedding.set"
] | [
"rel_hom_class",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_eq (e : r ≃r s) : set.range e = set.univ | e.surjective.range_eq | lemma | rel_iso.range_eq | order.rel_iso | src/order/rel_iso/set.lean | [
"order.rel_iso.basic",
"logic.embedding.set"
] | [
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subrel (r : α → α → Prop) (p : set α) : p → p → Prop | (coe : p → α) ⁻¹'o r | def | subrel | order.rel_iso | src/order/rel_iso/set.lean | [
"order.rel_iso.basic",
"logic.embedding.set"
] | [] | `subrel r p` is the inherited relation on a subset. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subrel_val (r : α → α → Prop) (p : set α)
{a b} : subrel r p a b ↔ r a.1 b.1 | iff.rfl | theorem | subrel_val | order.rel_iso | src/order/rel_iso/set.lean | [
"order.rel_iso.basic",
"logic.embedding.set"
] | [
"subrel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel_embedding (r : α → α → Prop) (p : set α) :
subrel r p ↪r r | ⟨embedding.subtype _, λ a b, iff.rfl⟩ | def | subrel.rel_embedding | order.rel_iso | src/order/rel_iso/set.lean | [
"order.rel_iso.basic",
"logic.embedding.set"
] | [
"rel_embedding",
"subrel"
] | The relation embedding from the inherited relation on a subset. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel_embedding_apply (r : α → α → Prop) (p a) :
subrel.rel_embedding r p a = a.1 | rfl | theorem | subrel.rel_embedding_apply | order.rel_iso | src/order/rel_iso/set.lean | [
"order.rel_iso.basic",
"logic.embedding.set"
] | [
"subrel.rel_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel_embedding.cod_restrict (p : set β) (f : r ↪r s) (H : ∀ a, f a ∈ p) : r ↪r subrel s p | ⟨f.to_embedding.cod_restrict p H, λ _ _, f.map_rel_iff'⟩ | def | rel_embedding.cod_restrict | order.rel_iso | src/order/rel_iso/set.lean | [
"order.rel_iso.basic",
"logic.embedding.set"
] | [
"subrel"
] | Restrict the codomain of a relation embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel_embedding.cod_restrict_apply (p) (f : r ↪r s) (H a) :
rel_embedding.cod_restrict p f H a = ⟨f a, H a⟩ | rfl | theorem | rel_embedding.cod_restrict_apply | order.rel_iso | src/order/rel_iso/set.lean | [
"order.rel_iso.basic",
"logic.embedding.set"
] | [
"rel_embedding.cod_restrict"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_order (α : Type*) [preorder α] | (succ : α → α)
(le_succ : ∀ a, a ≤ succ a)
(max_of_succ_le {a} : succ a ≤ a → is_max a)
(succ_le_of_lt {a b} : a < b → succ a ≤ b)
(le_of_lt_succ {a b} : a < succ b → a ≤ b) | class | succ_order | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"is_max"
] | Order equipped with a sensible successor function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pred_order (α : Type*) [preorder α] | (pred : α → α)
(pred_le : ∀ a, pred a ≤ a)
(min_of_le_pred {a} : a ≤ pred a → is_min a)
(le_pred_of_lt {a b} : a < b → a ≤ pred b)
(le_of_pred_lt {a b} : pred a < b → a ≤ b) | class | pred_order | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"is_min"
] | Order equipped with a sensible predecessor function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
succ_order.of_succ_le_iff_of_le_lt_succ (succ : α → α)
(hsucc_le_iff : ∀ {a b}, succ a ≤ b ↔ a < b) (hle_of_lt_succ : ∀ {a b}, a < succ b → a ≤ b) :
succ_order α | { succ := succ,
le_succ := λ a, (hsucc_le_iff.1 le_rfl).le,
max_of_succ_le := λ a ha, (lt_irrefl a $ hsucc_le_iff.1 ha).elim,
succ_le_of_lt := λ a b, hsucc_le_iff.2,
le_of_lt_succ := λ a b, hle_of_lt_succ } | def | succ_order.of_succ_le_iff_of_le_lt_succ | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"le_rfl",
"succ_order"
] | A constructor for `succ_order α` usable when `α` has no maximal element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pred_order.of_le_pred_iff_of_pred_le_pred (pred : α → α)
(hle_pred_iff : ∀ {a b}, a ≤ pred b ↔ a < b) (hle_of_pred_lt : ∀ {a b}, pred a < b → a ≤ b) :
pred_order α | { pred := pred,
pred_le := λ a, (hle_pred_iff.1 le_rfl).le,
min_of_le_pred := λ a ha, (lt_irrefl a $ hle_pred_iff.1 ha).elim,
le_pred_of_lt := λ a b, hle_pred_iff.2,
le_of_pred_lt := λ a b, hle_of_pred_lt } | def | pred_order.of_le_pred_iff_of_pred_le_pred | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"le_rfl",
"pred_order"
] | A constructor for `pred_order α` usable when `α` has no minimal element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
succ_order.of_core (succ : α → α) (hn : ∀ {a}, ¬ is_max a → ∀ b, a < b ↔ succ a ≤ b)
(hm : ∀ a, is_max a → succ a = a) : succ_order α | { succ := succ,
succ_le_of_lt := λ a b, classical.by_cases (λ h hab, (hm a h).symm ▸ hab.le) (λ h, (hn h b).mp),
le_succ := λ a, classical.by_cases (λ h, (hm a h).symm.le)
(λ h, le_of_lt $ by simpa using (hn h a).not),
le_of_lt_succ := λ a b hab, clas... | def | succ_order.of_core | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"is_max",
"succ_order"
] | A constructor for `succ_order α` for `α` a linear order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pred_order.of_core {α} [linear_order α] (pred : α → α)
(hn : ∀ {a}, ¬ is_min a → ∀ b, b ≤ pred a ↔ b < a) (hm : ∀ a, is_min a → pred a = a) :
pred_order α | { pred := pred,
le_pred_of_lt := λ a b, classical.by_cases (λ h hab, (hm b h).symm ▸ hab.le) (λ h, (hn h a).mpr),
pred_le := λ a, classical.by_cases (λ h, (hm a h).le)
(λ h, le_of_lt $ by simpa using (hn h a).not),
le_of_pred_lt := λ a b hab, classica... | def | pred_order.of_core | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"is_min",
"pred_order"
] | A constructor for `pred_order α` for `α` a linear order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
succ_order.of_succ_le_iff (succ : α → α) (hsucc_le_iff : ∀ {a b}, succ a ≤ b ↔ a < b) :
succ_order α | { succ := succ,
le_succ := λ a, (hsucc_le_iff.1 le_rfl).le,
max_of_succ_le := λ a ha, (lt_irrefl a $ hsucc_le_iff.1 ha).elim,
succ_le_of_lt := λ a b, hsucc_le_iff.2,
le_of_lt_succ := λ a b h, le_of_not_lt ((not_congr hsucc_le_iff).1 h.not_le) } | def | succ_order.of_succ_le_iff | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"le_rfl",
"succ_order"
] | A constructor for `succ_order α` usable when `α` is a linear order with no maximal element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pred_order.of_le_pred_iff (pred : α → α) (hle_pred_iff : ∀ {a b}, a ≤ pred b ↔ a < b) :
pred_order α | { pred := pred,
pred_le := λ a, (hle_pred_iff.1 le_rfl).le,
min_of_le_pred := λ a ha, (lt_irrefl a $ hle_pred_iff.1 ha).elim,
le_pred_of_lt := λ a b, hle_pred_iff.2,
le_of_pred_lt := λ a b h, le_of_not_lt ((not_congr hle_pred_iff).1 h.not_le) } | def | pred_order.of_le_pred_iff | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"le_rfl",
"pred_order"
] | A constructor for `pred_order α` usable when `α` is a linear order with no minimal element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
succ : α → α | succ_order.succ | def | order.succ | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [] | The successor of an element. If `a` is not maximal, then `succ a` is the least element greater
than `a`. If `a` is maximal, then `succ a = a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_succ : ∀ a : α, a ≤ succ a | succ_order.le_succ | lemma | order.le_succ | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_of_succ_le {a : α} : succ a ≤ a → is_max a | succ_order.max_of_succ_le | lemma | order.max_of_succ_le | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_le_of_lt {a b : α} : a < b → succ a ≤ b | succ_order.succ_le_of_lt | lemma | order.succ_le_of_lt | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_lt_succ {a b : α} : a < succ b → a ≤ b | succ_order.le_of_lt_succ | lemma | order.le_of_lt_succ | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_le_iff_is_max : succ a ≤ a ↔ is_max a | ⟨max_of_succ_le, λ h, h $ le_succ _⟩ | lemma | order.succ_le_iff_is_max | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_succ_iff_not_is_max : a < succ a ↔ ¬ is_max a | ⟨not_is_max_of_lt, λ ha, (le_succ a).lt_of_not_le $ λ h, ha $ max_of_succ_le h⟩ | lemma | order.lt_succ_iff_not_is_max | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"is_max",
"lt_of_not_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wcovby_succ (a : α) : a ⩿ succ a | ⟨le_succ a, λ b hb, (succ_le_of_lt hb).not_lt⟩ | lemma | order.wcovby_succ | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covby_succ_of_not_is_max (h : ¬ is_max a) : a ⋖ succ a | (wcovby_succ a).covby_of_lt $ lt_succ_of_not_is_max h | lemma | order.covby_succ_of_not_is_max | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_succ_iff_of_not_is_max (ha : ¬ is_max a) : b < succ a ↔ b ≤ a | ⟨le_of_lt_succ, λ h, h.trans_lt $ lt_succ_of_not_is_max ha⟩ | lemma | order.lt_succ_iff_of_not_is_max | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_le_iff_of_not_is_max (ha : ¬ is_max a) : succ a ≤ b ↔ a < b | ⟨(lt_succ_of_not_is_max ha).trans_le, succ_le_of_lt⟩ | lemma | order.succ_le_iff_of_not_is_max | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_lt_succ_iff_of_not_is_max (ha : ¬ is_max a) (hb : ¬ is_max b) :
succ a < succ b ↔ a < b | by rw [lt_succ_iff_of_not_is_max hb, succ_le_iff_of_not_is_max ha] | lemma | order.succ_lt_succ_iff_of_not_is_max | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_le_succ_iff_of_not_is_max (ha : ¬ is_max a) (hb : ¬ is_max b) :
succ a ≤ succ b ↔ a ≤ b | by rw [succ_le_iff_of_not_is_max ha, lt_succ_iff_of_not_is_max hb] | lemma | order.succ_le_succ_iff_of_not_is_max | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_le_succ (h : a ≤ b) : succ a ≤ succ b | begin
by_cases hb : is_max b,
{ by_cases hba : b ≤ a,
{ exact (hb $ hba.trans $ le_succ _).trans (le_succ _) },
{ exact succ_le_of_lt ((h.lt_of_not_le hba).trans_le $ le_succ b) } },
{ rwa [succ_le_iff_of_not_is_max (λ ha, hb $ ha.mono h), lt_succ_iff_of_not_is_max hb] }
end | lemma | order.succ_le_succ | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_mono : monotone (succ : α → α) | λ a b, succ_le_succ | lemma | order.succ_mono | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_succ_iterate (k : ℕ) (x : α) : x ≤ (succ^[k] x) | begin
conv_lhs { rw (by simp only [function.iterate_id, id.def] : x = (id^[k] x)) },
exact monotone.le_iterate_of_le succ_mono le_succ k x,
end | lemma | order.le_succ_iterate | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"function.iterate_id",
"monotone.le_iterate_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_iterate_succ_of_eq_of_lt {n m : ℕ}
(h_eq : (succ^[n] a) = (succ^[m] a)) (h_lt : n < m) :
is_max (succ^[n] a) | begin
refine max_of_succ_le (le_trans _ h_eq.symm.le),
have : succ (succ^[n] a) = (succ^[n + 1] a), by rw function.iterate_succ',
rw this,
have h_le : n + 1 ≤ m := nat.succ_le_of_lt h_lt,
exact monotone.monotone_iterate_of_le_map succ_mono (le_succ a) h_le,
end | lemma | order.is_max_iterate_succ_of_eq_of_lt | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"function.iterate_succ'",
"is_max",
"monotone.monotone_iterate_of_le_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_iterate_succ_of_eq_of_ne {n m : ℕ}
(h_eq : (succ^[n] a) = (succ^[m] a)) (h_ne : n ≠ m) :
is_max (succ^[n] a) | begin
cases le_total n m,
{ exact is_max_iterate_succ_of_eq_of_lt h_eq (lt_of_le_of_ne h h_ne), },
{ rw h_eq,
exact is_max_iterate_succ_of_eq_of_lt h_eq.symm (lt_of_le_of_ne h h_ne.symm), },
end | lemma | order.is_max_iterate_succ_of_eq_of_ne | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iio_succ_of_not_is_max (ha : ¬ is_max a) : Iio (succ a) = Iic a | set.ext $ λ x, lt_succ_iff_of_not_is_max ha | lemma | order.Iio_succ_of_not_is_max | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"is_max",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ici_succ_of_not_is_max (ha : ¬ is_max a) : Ici (succ a) = Ioi a | set.ext $ λ x, succ_le_iff_of_not_is_max ha | lemma | order.Ici_succ_of_not_is_max | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"is_max",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ico_succ_right_of_not_is_max (hb : ¬ is_max b) : Ico a (succ b) = Icc a b | by rw [←Ici_inter_Iio, Iio_succ_of_not_is_max hb, Ici_inter_Iic] | lemma | order.Ico_succ_right_of_not_is_max | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ioo_succ_right_of_not_is_max (hb : ¬ is_max b) : Ioo a (succ b) = Ioc a b | by rw [←Ioi_inter_Iio, Iio_succ_of_not_is_max hb, Ioi_inter_Iic] | lemma | order.Ioo_succ_right_of_not_is_max | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Icc_succ_left_of_not_is_max (ha : ¬ is_max a) : Icc (succ a) b = Ioc a b | by rw [←Ici_inter_Iic, Ici_succ_of_not_is_max ha, Ioi_inter_Iic] | lemma | order.Icc_succ_left_of_not_is_max | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ico_succ_left_of_not_is_max (ha : ¬ is_max a) : Ico (succ a) b = Ioo a b | by rw [←Ici_inter_Iio, Ici_succ_of_not_is_max ha, Ioi_inter_Iio] | lemma | order.Ico_succ_left_of_not_is_max | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_succ (a : α) : a < succ a | lt_succ_of_not_is_max $ not_is_max a | lemma | order.lt_succ | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"not_is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_succ_iff : a < succ b ↔ a ≤ b | lt_succ_iff_of_not_is_max $ not_is_max b | lemma | order.lt_succ_iff | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"not_is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_le_iff : succ a ≤ b ↔ a < b | succ_le_iff_of_not_is_max $ not_is_max a | lemma | order.succ_le_iff | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"not_is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_le_succ_iff : succ a ≤ succ b ↔ a ≤ b | by simp | lemma | order.succ_le_succ_iff | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_lt_succ_iff : succ a < succ b ↔ a < b | by simp | lemma | order.succ_lt_succ_iff | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_strict_mono : strict_mono (succ : α → α) | λ a b, succ_lt_succ | lemma | order.succ_strict_mono | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covby_succ (a : α) : a ⋖ succ a | covby_succ_of_not_is_max $ not_is_max a | lemma | order.covby_succ | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"not_is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iio_succ (a : α) : Iio (succ a) = Iic a | Iio_succ_of_not_is_max $ not_is_max _ | lemma | order.Iio_succ | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"not_is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ici_succ (a : α) : Ici (succ a) = Ioi a | Ici_succ_of_not_is_max $ not_is_max _ | lemma | order.Ici_succ | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"not_is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ico_succ_right (a b : α) : Ico a (succ b) = Icc a b | Ico_succ_right_of_not_is_max $ not_is_max _ | lemma | order.Ico_succ_right | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"not_is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ioo_succ_right (a b : α) : Ioo a (succ b) = Ioc a b | Ioo_succ_right_of_not_is_max $ not_is_max _ | lemma | order.Ioo_succ_right | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"not_is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Icc_succ_left (a b : α) : Icc (succ a) b = Ioc a b | Icc_succ_left_of_not_is_max $ not_is_max _ | lemma | order.Icc_succ_left | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"not_is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ico_succ_left (a b : α) : Ico (succ a) b = Ioo a b | Ico_succ_left_of_not_is_max $ not_is_max _ | lemma | order.Ico_succ_left | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"not_is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_eq_iff_is_max : succ a = a ↔ is_max a | ⟨λ h, max_of_succ_le h.le, λ h, h.eq_of_ge $ le_succ _⟩ | lemma | order.succ_eq_iff_is_max | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_eq_succ_iff_of_not_is_max (ha : ¬ is_max a) (hb : ¬ is_max b) :
succ a = succ b ↔ a = b | by rw [eq_iff_le_not_lt, eq_iff_le_not_lt,
succ_le_succ_iff_of_not_is_max ha hb, succ_lt_succ_iff_of_not_is_max ha hb] | lemma | order.succ_eq_succ_iff_of_not_is_max | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"eq_iff_le_not_lt",
"is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_le_succ_iff : a ≤ b ∧ b ≤ succ a ↔ b = a ∨ b = succ a | begin
refine ⟨λ h, or_iff_not_imp_left.2 $ λ hba : b ≠ a,
h.2.antisymm (succ_le_of_lt $ h.1.lt_of_ne $ hba.symm), _⟩,
rintro (rfl | rfl),
{ exact ⟨le_rfl, le_succ b⟩ },
{ exact ⟨le_succ a, le_rfl⟩ }
end | lemma | order.le_le_succ_iff | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.covby.succ_eq (h : a ⋖ b) : succ a = b | (succ_le_of_lt h.lt).eq_of_not_lt $ λ h', h.2 (lt_succ_of_not_is_max h.lt.not_is_max) h' | lemma | covby.succ_eq | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.wcovby.le_succ (h : a ⩿ b) : b ≤ succ a | begin
obtain h | rfl := h.covby_or_eq,
{ exact h.succ_eq.ge },
{ exact le_succ _ }
end | lemma | wcovby.le_succ | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_succ_iff_eq_or_le : a ≤ succ b ↔ a = succ b ∨ a ≤ b | begin
by_cases hb : is_max b,
{ rw [hb.succ_eq, or_iff_right_of_imp le_of_eq] },
{ rw [←lt_succ_iff_of_not_is_max hb, le_iff_eq_or_lt] }
end | lemma | order.le_succ_iff_eq_or_le | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"is_max",
"le_iff_eq_or_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_succ_iff_eq_or_lt_of_not_is_max (hb : ¬ is_max b) : a < succ b ↔ a = b ∨ a < b | (lt_succ_iff_of_not_is_max hb).trans le_iff_eq_or_lt | lemma | order.lt_succ_iff_eq_or_lt_of_not_is_max | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"is_max",
"le_iff_eq_or_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iic_succ (a : α) : Iic (succ a) = insert (succ a) (Iic a) | ext $ λ _, le_succ_iff_eq_or_le | lemma | order.Iic_succ | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Icc_succ_right (h : a ≤ succ b) : Icc a (succ b) = insert (succ b) (Icc a b) | by simp_rw [←Ici_inter_Iic, Iic_succ, inter_insert_of_mem (mem_Ici.2 h)] | lemma | order.Icc_succ_right | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ioc_succ_right (h : a < succ b) : Ioc a (succ b) = insert (succ b) (Ioc a b) | by simp_rw [←Ioi_inter_Iic, Iic_succ, inter_insert_of_mem (mem_Ioi.2 h)] | lemma | order.Ioc_succ_right | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iio_succ_eq_insert_of_not_is_max (h : ¬is_max a) : Iio (succ a) = insert a (Iio a) | ext $ λ _, lt_succ_iff_eq_or_lt_of_not_is_max h | lemma | order.Iio_succ_eq_insert_of_not_is_max | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ico_succ_right_eq_insert_of_not_is_max (h₁ : a ≤ b) (h₂ : ¬is_max b) :
Ico a (succ b) = insert b (Ico a b) | by simp_rw [←Iio_inter_Ici, Iio_succ_eq_insert_of_not_is_max h₂, insert_inter_of_mem (mem_Ici.2 h₁)] | lemma | order.Ico_succ_right_eq_insert_of_not_is_max | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ioo_succ_right_eq_insert_of_not_is_max (h₁ : a < b) (h₂ : ¬is_max b) :
Ioo a (succ b) = insert b (Ioo a b) | by simp_rw [←Iio_inter_Ioi, Iio_succ_eq_insert_of_not_is_max h₂, insert_inter_of_mem (mem_Ioi.2 h₁)] | lemma | order.Ioo_succ_right_eq_insert_of_not_is_max | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_eq_succ_iff : succ a = succ b ↔ a = b | succ_eq_succ_iff_of_not_is_max (not_is_max a) (not_is_max b) | lemma | order.succ_eq_succ_iff | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"not_is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_injective : injective (succ : α → α) | λ a b, succ_eq_succ_iff.1 | lemma | order.succ_injective | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_ne_succ_iff : succ a ≠ succ b ↔ a ≠ b | succ_injective.ne_iff | lemma | order.succ_ne_succ_iff | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_succ_iff_eq_or_lt : a < succ b ↔ a = b ∨ a < b | lt_succ_iff.trans le_iff_eq_or_lt | lemma | order.lt_succ_iff_eq_or_lt | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"le_iff_eq_or_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_eq_iff_covby : succ a = b ↔ a ⋖ b | ⟨by { rintro rfl, exact covby_succ _ }, covby.succ_eq⟩ | lemma | order.succ_eq_iff_covby | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iio_succ_eq_insert (a : α) : Iio (succ a) = insert a (Iio a) | Iio_succ_eq_insert_of_not_is_max $ not_is_max a | lemma | order.Iio_succ_eq_insert | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"not_is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ico_succ_right_eq_insert (h : a ≤ b) : Ico a (succ b) = insert b (Ico a b) | Ico_succ_right_eq_insert_of_not_is_max h $ not_is_max b | lemma | order.Ico_succ_right_eq_insert | order.succ_pred | src/order/succ_pred/basic.lean | [
"order.complete_lattice",
"order.cover",
"order.iterate"
] | [
"not_is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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