Local Set Implicit Arguments.
Local Unset Strict Implicit.
(* *Some basic things concerning lists *)
(* * Definitions *)
Local Definition symbol := nat.
Local Definition string := (string nat).
Local Definition card : Type := (string * string).
Local Definition stack := list card.
Local Definition SRS := SRS nat.
Implicit Types a b : symbol.
Implicit Types x y z : string.
Implicit Types d e : card.
Implicit Types A R P : stack.
Coercion sing (n : nat) := [n].
(* ** String Rewriting *)
Scheme rewt_ind' := Induction for rewt Sort Prop.
Instance PreOrder_rewt R : PreOrder (rewt R).
Instance Proper_rewt R : Proper (rewt R ==> rewt R ==> rewt R) (@app symbol).

Lemma do_rew (R : SRS) x1 x2 x y u v :
In (u, v) R ->
x1 = x ++ u ++ y ->
x2 = x ++ v ++ y ->
rew R x1 x2.