Library Relation

Copyright (c) 2018 by Karl Crary
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The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
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Require Export Relation_Definitions.

Require Import Tact.

reflexive, transitive closure
Inductive star {T : Type} (R : T -> T -> Prop) : T -> T -> Prop :=
| star_refl {x}
    : star R x x

| star_step {x y z}
    : R x y
      -> star R y z
      -> star R x z.

Inductive starr {T : Type} (R : T -> T -> Prop) : T -> T -> Prop :=
| starr_refl {x}
    : starr R x x

| starr_step {x y z}
    : starr R x y
      -> R y z
      -> starr R x z.

Lemma star_trans :
  forall (T : Type) (R : T -> T -> Prop) x y z,
    star R x y
    -> star R y z
    -> star R x z.

Lemma star_transitive :
  forall (T : Type) (R : T -> T -> Prop),
    transitive T (star R).

Lemma star_one :
  forall (T : Type) (R : T -> T -> Prop) x y,
    R x y -> star R x y.

Lemma star_stepr :
  forall (T : Type) (R : T -> T -> Prop) x y z,
    star R x y
    -> R y z
    -> star R x z.

Lemma star_mono :
  forall (T : Type) (R R' : T -> T -> Prop),
    (forall x y, R x y -> R' x y)
    -> forall x y, star R x y -> star R' x y.
refl
Lemma star_map :
  forall (S T : Type) (R : S -> S -> Prop) (R' : T -> T -> Prop) (f : S -> T),
    (forall x y, R x y -> R' (f x) (f y))
    -> forall x y, star R x y -> star R' (f x) (f y).
refl
Lemma star_starr :
  forall (T : Type) (R : T -> T -> Prop) x y,
    star R x y
    -> starr R x y.
refl
Lemma starr_star :
  forall (T : Type) (R : T -> T -> Prop) x y,
    starr R x y
    -> star R x y.
refl
Definition compose {T : Type} (R R' : T -> T -> Prop) (x x' : T) : Prop :=
  exists x'', R x x'' /\ R' x'' x'.

transitive closure
Definition plus {T : Type} (R : T -> T -> Prop) : T -> T -> Prop :=
  compose R (star R).

Definition plusr {T : Type} (R : T -> T -> Prop) : T -> T -> Prop :=
  compose (star R) R.

Inductive plusi {T : Type} (R : T -> T -> Prop) : T -> T -> Prop :=
| plusi_one {x y}
    : R x y
      -> plusi R x y

| plusi_step {x y z}
    : R x y
      -> plusi R y z
      -> plusi R x z.

Inductive plusri {T : Type} (R : T -> T -> Prop) : T -> T -> Prop :=
| plusri_one {x y}
    : R x y
      -> plusri R x y

| plusri_step {x y z}
    : plusri R x y
      -> R y z
      -> plusri R x z.

Lemma plus_star :
  forall (T : Type) (R : T -> T -> Prop) x y,
    plus R x y -> star R x y.

Lemma star_plus :
  forall (T : Type) (R : T -> T -> Prop) x y,
    star R x y -> x = y \/ plus R x y.

Lemma star_neq_plus :
  forall (T : Type) (R : T -> T -> Prop) x y,
    star R x y -> x <> y -> plus R x y.

Lemma plus_one :
  forall (T : Type) (R : T -> T -> Prop) x y,
    R x y -> plus R x y.

Lemma plusr_plus :
  forall (T : Type) (R : T -> T -> Prop) x y,
    plusr R x y -> plus R x y.

Lemma plus_plusr :
  forall (T : Type) (R : T -> T -> Prop) x y,
    plus R x y -> plusr R x y.
refl
Lemma plus_trans :
  forall (T : Type) (R : T -> T -> Prop) x y z,
    plus R x y -> plus R y z -> plus R x z.

Lemma plus_transitive :
  forall (T : Type) (R : T -> T -> Prop),
    transitive T (plus R).

Lemma plus_star_trans :
  forall (T : Type) (R : T -> T -> Prop) x y z,
    plus R x y -> star R y z -> plus R x z.

Lemma star_plus_trans :
  forall (T : Type) (R : T -> T -> Prop) x y z,
    star R x y -> plus R y z -> plus R x z.

Lemma plus_plusi :
  forall (T : Type) (R : T -> T -> Prop) x y,
    plus R x y -> plusi R x y.

Lemma plusi_plus :
  forall (T : Type) (R : T -> T -> Prop) x y,
    plusi R x y -> plus R x y.

Lemma plus_plusri :
  forall (T : Type) (R : T -> T -> Prop) x y,
    plus R x y -> plusri R x y.

Lemma plusri_plus :
  forall (T : Type) (R : T -> T -> Prop) x y,
    plusri R x y -> plus R x y.
one
Lemma plus_step :
  forall (T : Type) (R : T -> T -> Prop) x y z,
    R x y -> plus R y z -> plus R x z.

Lemma plus_mono :
  forall (T : Type) (R R' : T -> T -> Prop),
    (forall x y, R x y -> R' x y)
    -> forall x y, plus R x y -> plus R' x y.

Lemma star_map' :
  forall (T : Type) (R : T -> T -> Prop) f,
    (forall x y, R x y -> R (f x) (f y))
    -> forall x y, star R x y -> star R (f x) (f y).
refl
Lemma plus_map' :
  forall (T : Type) (R : T -> T -> Prop) f,
    (forall x y, R x y -> R (f x) (f y))
    -> forall x y, plus R x y -> plus R (f x) (f y).

Lemma star_mono_map :
  forall (T : Type) (R R' : T -> T -> Prop) f,
    (forall x y, R x y -> R' (f x) (f y))
    -> forall x y, star R x y -> star R' (f x) (f y).
refl
Lemma plus_mono_map :
  forall (T : Type) (R R' : T -> T -> Prop) f,
    (forall x y, R x y -> R' (f x) (f y))
    -> forall x y, plus R x y -> plus R' (f x) (f y).

Lemma plus_idem :
  forall (T : Type) (R : T -> T -> Prop) x y,
    plus R x y <-> plus (plus R) x y.

Lemma plus_of_transitive :
  forall (T : Type) (R : T -> T -> Prop),
    transitive T R
    -> forall x y, plus R x y -> R x y.

Lemma plus_well_founded :
  forall (T : Type) (R : T -> T -> Prop),
    well_founded R
    -> well_founded (plus R).

Lemma plus_ind :
  forall (T : Type) (R P : T -> T -> Prop),
    (forall x y z, R x y -> (y = z \/ (plus R y z /\ P y z)) -> P x z)
    -> forall x y, plus R x y -> P x y.
one
Lemma plus_ind_r :
  forall (T : Type) (R P : T -> T -> Prop),
    (forall x y z, (x = y \/ (plus R x y /\ P x y)) -> R y z -> P x z)
    -> forall x y, plus R x y -> P x y.
one
Lemma well_founded_impl_irrefl :
  forall (T : Type) (P : T -> T -> Prop),
    well_founded P
    -> forall x, P x x -> False.