Relation closures #

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This file defines the reflexive, transitive, and reflexive transitive closures of relations. It also proves some basic results on definitions in core, such as eqv_gen.

Note that this is about unbundled relations, that is terms of types of the form α → β → Prop. For the bundled version, see rel.

Definitions #

relation.refl_gen: Reflexive closure. refl_gen r relates everything r related, plus for all a it relates a with itself. So refl_gen r a b ↔ r a b ∨ a = b.
relation.trans_gen: Transitive closure. trans_gen r relates everything r related transitively. So trans_gen r a b ↔ ∃ x₀ ... xₙ, r a x₀ ∧ r x₀ x₁ ∧ ... ∧ r xₙ b.
relation.refl_trans_gen: Reflexive transitive closure. refl_trans_gen r relates everything r related transitively, plus for all a it relates a with itself. So refl_trans_gen r a b ↔ (∃ x₀ ... xₙ, r a x₀ ∧ r x₀ x₁ ∧ ... ∧ r xₙ b) ∨ a = b. It is the same as the reflexive closure of the transitive closure, or the transitive closure of the reflexive closure. In terms of rewriting systems, this means that a can be rewritten to b in a number of rewrites.
relation.comp: Relation composition. We provide notation ∘r. For r : α → β → Prop and s : β → γ → Prop, r ∘r srelates a : α and c : γ iff there exists b : β that's related to both.
relation.map: Image of a relation under a pair of maps. For r : α → β → Prop, f : α → γ, g : β → δ, map r f g is the relation γ → δ → Prop relating f a and g b for all a, b related by r.
relation.join: Join of a relation. For r : α → α → Prop, join r a b ↔ ∃ c, r a c ∧ r b c. In terms of rewriting systems, this means that a and b can be rewritten to the same term.

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theorem is_refl.reflexive {α : Type u_1} {r : α → α → Prop} [is_refl α r] :

reflexive r

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theorem reflexive.rel_of_ne_imp {α : Type u_1} {r : α → α → Prop} (h : reflexive r) {x y : α} (hr : x ≠ y → r x y) :

r x y

To show a reflexive relation r : α → α → Prop holds over x y : α, it suffices to show it holds when x ≠ y.

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theorem reflexive.ne_imp_iff {α : Type u_1} {r : α → α → Prop} (h : reflexive r) {x y : α} :

x ≠ y → r x y ↔ r x y

If a reflexive relation r : α → α → Prop holds over x y : α, then it holds whether or not x ≠ y.

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theorem reflexive_ne_imp_iff {α : Type u_1} {r : α → α → Prop} [is_refl α r] {x y : α} :

x ≠ y → r x y ↔ r x y

If a reflexive relation r : α → α → Prop holds over x y : α, then it holds whether or not x ≠ y. Unlike reflexive.ne_imp_iff, this uses [is_refl α r].

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@[protected]

theorem symmetric.iff {α : Type u_1} {r : α → α → Prop} (H : symmetric r) (x y : α) :

r x y ↔ r y x

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theorem symmetric.flip_eq {α : Type u_1} {r : α → α → Prop} (h : symmetric r) :

flip r = r

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theorem symmetric.swap_eq {α : Type u_1} {r : α → α → Prop} :

symmetric r → function.swap r = r

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theorem flip_eq_iff {α : Type u_1} {r : α → α → Prop} :

flip r = r ↔ symmetric r

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theorem swap_eq_iff {α : Type u_1} {r : α → α → Prop} :

function.swap r = r ↔ symmetric r

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theorem reflexive.comap {α : Type u_1} {β : Type u_2} {r : β → β → Prop} (h : reflexive r) (f : α → β) :

reflexive (r on f)

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theorem symmetric.comap {α : Type u_1} {β : Type u_2} {r : β → β → Prop} (h : symmetric r) (f : α → β) :

symmetric (r on f)

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theorem transitive.comap {α : Type u_1} {β : Type u_2} {r : β → β → Prop} (h : transitive r) (f : α → β) :

transitive (r on f)

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theorem equivalence.comap {α : Type u_1} {β : Type u_2} {r : β → β → Prop} (h : equivalence r) (f : α → β) :

equivalence (r on f)

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def relation.comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} (r : α → β → Prop) (p : β → γ → Prop) (a : α) (c : γ) :

Prop

The composition of two relations, yielding a new relation. The result relates a term of α and a term of γ if there is an intermediate term of β related to both.

Equations

relation.comp r p a c = ∃ (b : β), r a b ∧ p b c

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theorem relation.comp_eq {α : Type u_1} {β : Type u_2} {r : α → β → Prop} :

relation.comp r eq = r

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theorem relation.eq_comp {α : Type u_1} {β : Type u_2} {r : α → β → Prop} :

relation.comp eq r = r

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theorem relation.iff_comp {α : Type u_1} {r : Prop → α → Prop} :

relation.comp iff r = r

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theorem relation.comp_iff {α : Type u_1} {r : α → Prop → Prop} :

relation.comp r iff = r

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theorem relation.comp_assoc {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {r : α → β → Prop} {p : β → γ → Prop} {q : γ → δ → Prop} :

relation.comp (relation.comp r p) q = relation.comp r (relation.comp p q)

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theorem relation.flip_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} {r : α → β → Prop} {p : β → γ → Prop} :

flip (relation.comp r p) = relation.comp (flip p) (flip r)

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def relation.fibration {α : Type u_1} {β : Type u_2} (rα : α → α → Prop) (rβ : β → β → Prop) (f : α → β) :

Prop

A function f : α → β is a fibration between the relation rα and rβ if for all a : α and b : β, whenever b : β and f a are related by rβ, b is the image of some a' : α under f, and a' and a are related by rα.

Equations

relation.fibration rα rβ f = ∀ ⦃a : α⦄ ⦃b : β⦄, rβ b (f a) → (∃ (a' : α), rα a' a ∧ f a' = b)

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theorem acc.of_fibration {α : Type u_1} {β : Type u_2} {rα : α → α → Prop} {rβ : β → β → Prop} (f : α → β) (fib : relation.fibration rα rβ f) {a : α} (ha : acc rα a) :

acc rβ (f a)

If f : α → β is a fibration between relations rα and rβ, and a : α is accessible under rα, then f a is accessible under rβ.

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theorem acc.of_downward_closed {α : Type u_1} {β : Type u_2} {rβ : β → β → Prop} (f : α → β) (dc : ∀ {a : α} {b : β}, rβ b (f a) → (∃ (c : α), f c = b)) (a : α) (ha : acc (inv_image rβ f) a) :

acc rβ (f a)

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def relation.map {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (r : α → β → Prop) (f : α → γ) (g : β → δ) :

γ → δ → Prop

The map of a relation r through a pair of functions pushes the relation to the codomains of the functions. The resulting relation is defined by having pairs of terms related if they have preimages related by r.

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relation.map r f g = λ (c : γ) (d : δ), ∃ (a : α) (b : β), r a b ∧ f a = c ∧ g b = d

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inductive relation.refl_trans_gen {α : Type u_1} (r : α → α → Prop) (a : α) :

α → Prop

refl : ∀ {α : Type u_1} {r : α → α → Prop} {a : α}, relation.refl_trans_gen r a a
tail : ∀ {α : Type u_1} {r : α → α → Prop} {a b c : α}, relation.refl_trans_gen r a b → r b c → relation.refl_trans_gen r a c

refl_trans_gen r: reflexive transitive closure of r

Instances for relation.refl_trans_gen

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theorem relation.refl_trans_gen.cases_tail_iff {α : Type u_1} (r : α → α → Prop) (a ᾰ : α) :

relation.refl_trans_gen r a ᾰ ↔ ᾰ = a ∨ ∃ {b : α}, relation.refl_trans_gen r a b ∧ r b ᾰ

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inductive relation.refl_gen {α : Type u_1} (r : α → α → Prop) (a : α) :

α → Prop

refl : ∀ {α : Type u_1} {r : α → α → Prop} {a : α}, relation.refl_gen r a a
single : ∀ {α : Type u_1} {r : α → α → Prop} {a b : α}, r a b → relation.refl_gen r a b

refl_gen r: reflexive closure of r

Instances for relation.refl_gen

relation.refl_gen.is_refl

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theorem relation.refl_gen_iff {α : Type u_1} (r : α → α → Prop) (a ᾰ : α) :

relation.refl_gen r a ᾰ ↔ ᾰ = a ∨ r a ᾰ

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inductive relation.trans_gen {α : Type u_1} (r : α → α → Prop) (a : α) :

α → Prop

single : ∀ {α : Type u_1} {r : α → α → Prop} {a b : α}, r a b → relation.trans_gen r a b
tail : ∀ {α : Type u_1} {r : α → α → Prop} {a b c : α}, relation.trans_gen r a b → r b c → relation.trans_gen r a c

trans_gen r: transitive closure of r

Instances for relation.trans_gen

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theorem relation.trans_gen_iff {α : Type u_1} (r : α → α → Prop) (a ᾰ : α) :

relation.trans_gen r a ᾰ ↔ r a ᾰ ∨ ∃ {b : α}, relation.trans_gen r a b ∧ r b ᾰ

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theorem relation.refl_gen.to_refl_trans_gen {α : Type u_1} {r : α → α → Prop} {a b : α} :

relation.refl_gen r a b → relation.refl_trans_gen r a b

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theorem relation.refl_gen.mono {α : Type u_1} {r p : α → α → Prop} (hp : ∀ (a b : α), r a b → p a b) {a b : α} :

relation.refl_gen r a b → relation.refl_gen p a b

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def relation.refl_gen.is_refl {α : Type u_1} {r : α → α → Prop} :

is_refl α (relation.refl_gen r)

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@[trans]

theorem relation.refl_trans_gen.trans {α : Type u_1} {r : α → α → Prop} {a b c : α} (hab : relation.refl_trans_gen r a b) (hbc : relation.refl_trans_gen r b c) :

relation.refl_trans_gen r a c

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theorem relation.refl_trans_gen.single {α : Type u_1} {r : α → α → Prop} {a b : α} (hab : r a b) :

relation.refl_trans_gen r a b

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theorem relation.refl_trans_gen.head {α : Type u_1} {r : α → α → Prop} {a b c : α} (hab : r a b) (hbc : relation.refl_trans_gen r b c) :

relation.refl_trans_gen r a c

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theorem relation.refl_trans_gen.symmetric {α : Type u_1} {r : α → α → Prop} (h : symmetric r) :

symmetric (relation.refl_trans_gen r)

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theorem relation.refl_trans_gen.cases_tail {α : Type u_1} {r : α → α → Prop} {a b : α} :

relation.refl_trans_gen r a b → (b = a ∨ ∃ (c : α), relation.refl_trans_gen r a c ∧ r c b)

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theorem relation.refl_trans_gen.head_induction_on {α : Type u_1} {r : α → α → Prop} {b : α} {P : Π (a : α), relation.refl_trans_gen r a b → Prop} {a : α} (h : relation.refl_trans_gen r a b) (refl : P b relation.refl_trans_gen.refl) (head : ∀ {a c : α} (h' : r a c) (h : relation.refl_trans_gen r c b), P c h → P a _) :

P a h

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theorem relation.refl_trans_gen.trans_induction_on {α : Type u_1} {r : α → α → Prop} {P : Π {a b : α}, relation.refl_trans_gen r a b → Prop} {a b : α} (h : relation.refl_trans_gen r a b) (ih₁ : ∀ (a : α), P relation.refl_trans_gen.refl) (ih₂ : ∀ {a b : α} (h : r a b), P _) (ih₃ : ∀ {a b c : α} (h₁ : relation.refl_trans_gen r a b) (h₂ : relation.refl_trans_gen r b c), P h₁ → P h₂ → P _) :

P h

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theorem relation.refl_trans_gen.cases_head {α : Type u_1} {r : α → α → Prop} {a b : α} (h : relation.refl_trans_gen r a b) :

a = b ∨ ∃ (c : α), r a c ∧ relation.refl_trans_gen r c b

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theorem relation.refl_trans_gen.cases_head_iff {α : Type u_1} {r : α → α → Prop} {a b : α} :

relation.refl_trans_gen r a b ↔ a = b ∨ ∃ (c : α), r a c ∧ relation.refl_trans_gen r c b

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theorem relation.refl_trans_gen.total_of_right_unique {α : Type u_1} {r : α → α → Prop} {a b c : α} (U : relator.right_unique r) (ab : relation.refl_trans_gen r a b) (ac : relation.refl_trans_gen r a c) :

relation.refl_trans_gen r b c ∨ relation.refl_trans_gen r c b

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theorem relation.trans_gen.to_refl {α : Type u_1} {r : α → α → Prop} {a b : α} (h : relation.trans_gen r a b) :

relation.refl_trans_gen r a b

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@[trans]

theorem relation.trans_gen.trans_left {α : Type u_1} {r : α → α → Prop} {a b c : α} (hab : relation.trans_gen r a b) (hbc : relation.refl_trans_gen r b c) :

relation.trans_gen r a c

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@[trans]

theorem relation.trans_gen.trans {α : Type u_1} {r : α → α → Prop} {a b c : α} (hab : relation.trans_gen r a b) (hbc : relation.trans_gen r b c) :

relation.trans_gen r a c

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theorem relation.trans_gen.head' {α : Type u_1} {r : α → α → Prop} {a b c : α} (hab : r a b) (hbc : relation.refl_trans_gen r b c) :

relation.trans_gen r a c

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theorem relation.trans_gen.tail' {α : Type u_1} {r : α → α → Prop} {a b c : α} (hab : relation.refl_trans_gen r a b) (hbc : r b c) :

relation.trans_gen r a c

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theorem relation.trans_gen.head {α : Type u_1} {r : α → α → Prop} {a b c : α} (hab : r a b) (hbc : relation.trans_gen r b c) :

relation.trans_gen r a c

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theorem relation.trans_gen.head_induction_on {α : Type u_1} {r : α → α → Prop} {b : α} {P : Π (a : α), relation.trans_gen r a b → Prop} {a : α} (h : relation.trans_gen r a b) (base : ∀ {a : α} (h : r a b), P a _) (ih : ∀ {a c : α} (h' : r a c) (h : relation.trans_gen r c b), P c h → P a _) :

P a h

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theorem relation.trans_gen.trans_induction_on {α : Type u_1} {r : α → α → Prop} {P : Π {a b : α}, relation.trans_gen r a b → Prop} {a b : α} (h : relation.trans_gen r a b) (base : ∀ {a b : α} (h : r a b), P _) (ih : ∀ {a b c : α} (h₁ : relation.trans_gen r a b) (h₂ : relation.trans_gen r b c), P h₁ → P h₂ → P _) :

P h

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@[trans]

theorem relation.trans_gen.trans_right {α : Type u_1} {r : α → α → Prop} {a b c : α} (hab : relation.refl_trans_gen r a b) (hbc : relation.trans_gen r b c) :

relation.trans_gen r a c

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theorem relation.trans_gen.tail'_iff {α : Type u_1} {r : α → α → Prop} {a c : α} :

relation.trans_gen r a c ↔ ∃ (b : α), relation.refl_trans_gen r a b ∧ r b c

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theorem relation.trans_gen.head'_iff {α : Type u_1} {r : α → α → Prop} {a c : α} :

relation.trans_gen r a c ↔ ∃ (b : α), r a b ∧ relation.refl_trans_gen r b c

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theorem acc.trans_gen {α : Type u_1} {r : α → α → Prop} {a : α} (h : acc r a) :

acc (relation.trans_gen r) a

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theorem acc_trans_gen_iff {α : Type u_1} {r : α → α → Prop} {a : α} :

acc (relation.trans_gen r) a ↔ acc r a

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theorem well_founded.trans_gen {α : Type u_1} {r : α → α → Prop} (h : well_founded r) :

well_founded (relation.trans_gen r)

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theorem relation.trans_gen_eq_self {α : Type u_1} {r : α → α → Prop} (trans : transitive r) :

relation.trans_gen r = r

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theorem relation.transitive_trans_gen {α : Type u_1} {r : α → α → Prop} :

transitive (relation.trans_gen r)

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def relation.trans_gen.is_trans {α : Type u_1} {r : α → α → Prop} :

is_trans α (relation.trans_gen r)

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theorem relation.trans_gen_idem {α : Type u_1} {r : α → α → Prop} :

relation.trans_gen (relation.trans_gen r) = relation.trans_gen r

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theorem relation.trans_gen.lift {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {p : β → β → Prop} {a b : α} (f : α → β) (h : ∀ (a b : α), r a b → p (f a) (f b)) (hab : relation.trans_gen r a b) :

relation.trans_gen p (f a) (f b)

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theorem relation.trans_gen.lift' {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {p : β → β → Prop} {a b : α} (f : α → β) (h : ∀ (a b : α), r a b → relation.trans_gen p (f a) (f b)) (hab : relation.trans_gen r a b) :

relation.trans_gen p (f a) (f b)

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theorem relation.trans_gen.closed {α : Type u_1} {r : α → α → Prop} {a b : α} {p : α → α → Prop} :

(∀ (a b : α), r a b → relation.trans_gen p a b) → relation.trans_gen r a b → relation.trans_gen p a b

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theorem relation.trans_gen.mono {α : Type u_1} {r : α → α → Prop} {a b : α} {p : α → α → Prop} :

(∀ (a b : α), r a b → p a b) → relation.trans_gen r a b → relation.trans_gen p a b

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theorem relation.trans_gen.swap {α : Type u_1} {r : α → α → Prop} {a b : α} (h : relation.trans_gen r b a) :

relation.trans_gen (function.swap r) a b

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theorem relation.trans_gen_swap {α : Type u_1} {r : α → α → Prop} {a b : α} :

relation.trans_gen (function.swap r) a b ↔ relation.trans_gen r b a

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theorem relation.refl_trans_gen_iff_eq {α : Type u_1} {r : α → α → Prop} {a b : α} (h : ∀ (b : α), ¬r a b) :

relation.refl_trans_gen r a b ↔ b = a

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theorem relation.refl_trans_gen_iff_eq_or_trans_gen {α : Type u_1} {r : α → α → Prop} {a b : α} :

relation.refl_trans_gen r a b ↔ b = a ∨ relation.trans_gen r a b

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theorem relation.refl_trans_gen.lift {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {p : β → β → Prop} {a b : α} (f : α → β) (h : ∀ (a b : α), r a b → p (f a) (f b)) (hab : relation.refl_trans_gen r a b) :

relation.refl_trans_gen p (f a) (f b)

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theorem relation.refl_trans_gen.mono {α : Type u_1} {r : α → α → Prop} {a b : α} {p : α → α → Prop} :

(∀ (a b : α), r a b → p a b) → relation.refl_trans_gen r a b → relation.refl_trans_gen p a b

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theorem relation.refl_trans_gen_eq_self {α : Type u_1} {r : α → α → Prop} (refl : reflexive r) (trans : transitive r) :

relation.refl_trans_gen r = r

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theorem relation.reflexive_refl_trans_gen {α : Type u_1} {r : α → α → Prop} :

reflexive (relation.refl_trans_gen r)

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theorem relation.transitive_refl_trans_gen {α : Type u_1} {r : α → α → Prop} :

transitive (relation.refl_trans_gen r)

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def relation.refl_trans_gen.is_refl {α : Type u_1} {r : α → α → Prop} :

is_refl α (relation.refl_trans_gen r)

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@[protected, instance]

def relation.refl_trans_gen.is_trans {α : Type u_1} {r : α → α → Prop} :

is_trans α (relation.refl_trans_gen r)

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theorem relation.refl_trans_gen_idem {α : Type u_1} {r : α → α → Prop} :

relation.refl_trans_gen (relation.refl_trans_gen r) = relation.refl_trans_gen r

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theorem relation.refl_trans_gen.lift' {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {p : β → β → Prop} {a b : α} (f : α → β) (h : ∀ (a b : α), r a b → relation.refl_trans_gen p (f a) (f b)) (hab : relation.refl_trans_gen r a b) :

relation.refl_trans_gen p (f a) (f b)

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theorem relation.refl_trans_gen_closed {α : Type u_1} {r : α → α → Prop} {a b : α} {p : α → α → Prop} :

(∀ (a b : α), r a b → relation.refl_trans_gen p a b) → relation.refl_trans_gen r a b → relation.refl_trans_gen p a b

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theorem relation.refl_trans_gen.swap {α : Type u_1} {r : α → α → Prop} {a b : α} (h : relation.refl_trans_gen r b a) :

relation.refl_trans_gen (function.swap r) a b

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theorem relation.refl_trans_gen_swap {α : Type u_1} {r : α → α → Prop} {a b : α} :

relation.refl_trans_gen (function.swap r) a b ↔ relation.refl_trans_gen r b a

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def relation.join {α : Type u_1} (r : α → α → Prop) :

α → α → Prop

The join of a relation on a single type is a new relation for which pairs of terms are related if there is a third term they are both related to. For example, if r is a relation representing rewrites in a term rewriting system, then confluence is the property that if a rewrites to both b and c, then join r relates b and c (see relation.church_rosser).

Equations

relation.join r = λ (a b : α), ∃ (c : α), r a c ∧ r b c

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theorem relation.church_rosser {α : Type u_1} {r : α → α → Prop} {a b c : α} (h : ∀ (a b c : α), r a b → r a c → (∃ (d : α), relation.refl_gen r b d ∧ relation.refl_trans_gen r c d)) (hab : relation.refl_trans_gen r a b) (hac : relation.refl_trans_gen r a c) :

relation.join (relation.refl_trans_gen r) b c

A sufficient condition for the Church-Rosser property.

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theorem relation.join_of_single {α : Type u_1} {r : α → α → Prop} {a b : α} (h : reflexive r) (hab : r a b) :

relation.join r a b

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theorem relation.symmetric_join {α : Type u_1} {r : α → α → Prop} :

symmetric (relation.join r)

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theorem relation.reflexive_join {α : Type u_1} {r : α → α → Prop} (h : reflexive r) :

reflexive (relation.join r)

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theorem relation.transitive_join {α : Type u_1} {r : α → α → Prop} (ht : transitive r) (h : ∀ (a b c : α), r a b → r a c → relation.join r b c) :

transitive (relation.join r)

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theorem relation.equivalence_join {α : Type u_1} {r : α → α → Prop} (hr : reflexive r) (ht : transitive r) (h : ∀ (a b c : α), r a b → r a c → relation.join r b c) :

equivalence (relation.join r)

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theorem relation.equivalence_join_refl_trans_gen {α : Type u_1} {r : α → α → Prop} (h : ∀ (a b c : α), r a b → r a c → (∃ (d : α), relation.refl_gen r b d ∧ relation.refl_trans_gen r c d)) :

equivalence (relation.join (relation.refl_trans_gen r))

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theorem relation.join_of_equivalence {α : Type u_1} {r : α → α → Prop} {a b : α} {r' : α → α → Prop} (hr : equivalence r) (h : ∀ (a b : α), r' a b → r a b) :

relation.join r' a b → r a b

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theorem relation.refl_trans_gen_of_transitive_reflexive {α : Type u_1} {r : α → α → Prop} {a b : α} {r' : α → α → Prop} (hr : reflexive r) (ht : transitive r) (h : ∀ (a b : α), r' a b → r a b) (h' : relation.refl_trans_gen r' a b) :

r a b

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theorem relation.refl_trans_gen_of_equivalence {α : Type u_1} {r : α → α → Prop} {a b : α} {r' : α → α → Prop} (hr : equivalence r) :

(∀ (a b : α), r' a b → r a b) → relation.refl_trans_gen r' a b → r a b

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theorem equivalence.eqv_gen_iff {α : Type u_1} {r : α → α → Prop} {a b : α} (h : equivalence r) :

eqv_gen r a b ↔ r a b

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theorem equivalence.eqv_gen_eq {α : Type u_1} {r : α → α → Prop} (h : equivalence r) :

eqv_gen r = r

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theorem eqv_gen.mono {α : Type u_1} {a b : α} {r p : α → α → Prop} (hrp : ∀ (a b : α), r a b → p a b) (h : eqv_gen r a b) :

eqv_gen p a b