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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* $Id$ *)
(***********************************************************************
The "abstract" type index is defined to represent variables.
index : Set
index_eq : index -> bool
index_eq_prop: (n,m:index)(index_eq n m)=true -> n=m
index_lt : index -> bool
varmap : Type -> Type.
varmap_find : (A:Type)A -> index -> (varmap A) -> A.
The first arg. of varmap_find is the default value to take
if the object is not found in the varmap.
index_lt defines a total well-founded order, but we don't prove that.
***********************************************************************)
Set Implicit Arguments.
Section variables_map.
Variable A : Type.
Inductive varmap : Type :=
| Empty_vm : varmap
| Node_vm : A -> varmap -> varmap -> varmap.
Inductive index : Set :=
| Left_idx : index -> index
| Right_idx : index -> index
| End_idx : index.
Fixpoint varmap_find (default_value:A) (i:index) (v:varmap) {struct v} : A :=
match i, v with
| End_idx, Node_vm x _ _ => x
| Right_idx i1, Node_vm x v1 v2 => varmap_find default_value i1 v2
| Left_idx i1, Node_vm x v1 v2 => varmap_find default_value i1 v1
| _, _ => default_value
end.
Fixpoint index_eq (n m:index) {struct m} : bool :=
match n, m with
| End_idx, End_idx => true
| Left_idx n', Left_idx m' => index_eq n' m'
| Right_idx n', Right_idx m' => index_eq n' m'
| _, _ => false
end.
Fixpoint index_lt (n m:index) {struct m} : bool :=
match n, m with
| End_idx, Left_idx _ => true
| End_idx, Right_idx _ => true
| Left_idx n', Right_idx m' => true
| Right_idx n', Right_idx m' => index_lt n' m'
| Left_idx n', Left_idx m' => index_lt n' m'
| _, _ => false
end.
Lemma index_eq_prop : forall n m:index, index_eq n m = true -> n = m.
simple induction n; simple induction m; simpl in |- *; intros.
rewrite (H i0 H1); reflexivity.
discriminate.
discriminate.
discriminate.
rewrite (H i0 H1); reflexivity.
discriminate.
discriminate.
discriminate.
reflexivity.
Qed.
End variables_map.
Unset Implicit Arguments.
|
