Hint Resolve refl_equal. Require Import Relations.
Require Import Confluence.

Set Implicit Arguments.
Unset Strict Implicit.

Theorem Rstar_reflexive :
 ∀ (U : Type) (R : Relation U), Reflexive (Rstar R).
Proof.
auto.
Qed.

Theorem Rplus_contains_R :
 ∀ (U : Type) (R : Relation U), Contains (Rplus R) R.
Proof.
auto.
Qed.

Theorem Rstar_contains_R :
 ∀ (U : Type) (R : Relation U), Contains (Rstar R) R.
Proof.
intros U R; red in |- *; intros x y H'; apply Rstar_n with y; auto.
Qed.

Theorem Rstar_contains_Rplus :
 ∀ (U : Type) (R : Relation U), Contains (Rstar R) (Rplus R).
Proof.
intros U R; red in |- ×.
intros x y H'; elim H'.
generalize Rstar_contains_R; intro T; red in T; auto.
intros x0 y0 z H'0 H'1 H'2; apply Rstar_n with y0; auto.
Qed.

Theorem Rstar_transitive :
 ∀ (U : Type) (R : Relation U), Transitive (Rstar R).
Proof.
intros U R; red in |- ×.
intros x y z H'; elim H'; auto.
intros x0 y0 z0 H'0 H'1 H'2 H'3; apply Rstar_n with y0; auto.
Qed.

Theorem Rstar_cases :
 ∀ (U : Type) (R : Relation U) (x y : U),
 Rstar R x y → x = y ∨ (∃ u : U, R x u ∧ Rstar R u y).
Proof.
intros U R x y H'; elim H'; auto.
intros x0 y0 z H'0 H'1 H'2; right; ∃ y0; auto.
Qed.

Theorem Rstar_equiv_Rstar1 :
 ∀ (U : Type) (R : Relation U), Same_relation (Rstar R) (Rstar1 R).
Proof.
generalize Rstar_contains_R; intro T; red in T.
intros U R; unfold Same_relation, Contains in |- ×.
split; intros x y H'; elim H'; auto.
generalize Rstar_transitive; intro T1; red in T1.
intros x0 y0 z H'0 H'1 H'2 H'3; apply T1 with y0; auto.
intros x0 y0 z H'0 H'1 H'2; apply Rstar1_n with y0; auto.
Qed.

Theorem Rsym_imp_Rstarsym :
 ∀ (U : Type) (R : Relation U), Symmetric R → Symmetric (Rstar R).
Proof.
intros U R H'; red in |- ×.
intros x y H'0; elim H'0; auto.
intros x0 y0 z H'1 H'2 H'3.
generalize Rstar_transitive; intro T; red in T.
apply T with y0; auto.
apply Rstar_n with x0; auto.
Qed.

Theorem Sstar_contains_Rstar :
 ∀ (U : Type) (R S : Relation U),
 Contains (Rstar S) R → Contains (Rstar S) (Rstar R).
Proof.
unfold Contains in |- ×.
intros U R S H' x y H'0; elim H'0; auto.
generalize Rstar_transitive; intro T; red in T.
intros x0 y0 z H'1 H'2 H'3; apply T with y0; auto.
Qed.

Theorem star_monotone :
 ∀ (U : Type) (R S : Relation U),
 Contains S R → Contains (Rstar S) (Rstar R).
Proof.
intros U R S H'.
apply Sstar_contains_Rstar.
generalize (Rstar_contains_R (R:=S)); auto.
Qed.

Theorem RstarRplus_RRstar :
 ∀ (U : Type) (R : Relation U) (x y z : U),
 Rstar R x y → Rplus R y z → ∃ u : U, R x u ∧ Rstar R u z.
Proof.
generalize Rstar_contains_Rplus; intro T; red in T.
generalize Rstar_transitive; intro T1; red in T1.
intros U R x y z H'; elim H'.
intros x0 H'0; elim H'0.
intros; ∃ y0; auto.
intros; ∃ y0; auto.
intros; ∃ y0; auto.
split; [ try assumption | idtac ].
apply T1 with z0; auto.
Qed.

Theorem Rstar_imp_coherent :
 ∀ (U : Type) (R : Relation U) (x y : U), Rstar R x y → coherent R x y.
Proof.
intros U R x y H'; red in |- ×.
∃ y; auto.
Qed.
Hint Resolve Rstar_imp_coherent.

Theorem coherent_symmetric :
 ∀ (U : Type) (R : Relation U), Symmetric (coherent R).
Proof.
unfold coherent at 1 in |- ×.
intros U R; red in |- ×.
intros x y h; elim h; intros z h0; elim h0; intros H' H'0; clear h h0.
∃ z; auto.
Qed.