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/meta/metatheory/CoqUniquenessTacEx.v

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Possible License(s): BSD-3-Clause
 1(* This file is distributed under the terms of the MIT License, also
 2   known as the X11 Licence.  A copy of this license is in the README
 3   file that accompanied the original distribution of this file.
 4
 5   Based on code written by:
 6     Brian Aydemir *)
 7
 8Require Import Coq.Arith.Peano_dec.
 9Require Import Coq.Lists.SetoidList.
10Require Import Coq.omega.Omega.
11
12Require Import CoqUniquenessTac.
13
14
15(* *********************************************************************** *)
16(** * Examples *)
17
18(** The examples go through more smoothly if we declare [eq_nat_dec]
19    as a hint. *)
20
21Hint Resolve eq_nat_dec : eq_dec.
22
23
24(* *********************************************************************** *)
25(** ** Predicates on natural numbers *)
26
27Scheme le_ind' := Induction for le Sort Prop.
28
29Lemma le_unique : forall (x y : nat) (p q: x <= y), p = q.
30Proof.
31  induction p using le_ind'; uniqueness 1.
32  assert False by omega; intuition.
33  assert False by omega; intuition.
34Qed.
35
36
37(* ********************************************************************** *)
38(** ** Predicates on lists *)
39
40(** Uniqueness of proofs for predicates on lists often comes up when
41    discussing extensional equality on finite sets, as implemented by
42    the FSets library. *)
43
44Section Uniqueness_Of_SetoidList_Proofs.
45
46  Variable A : Type.
47  Variable R : A -> A -> Prop.
48
49  Hypothesis R_unique : forall (x y : A) (p q : R x y), p = q.
50  Hypothesis list_eq_dec : forall (xs ys : list A), {xs = ys} + {xs <> ys}.
51
52  Scheme lelistA_ind' := Induction for lelistA Sort Prop.
53  Scheme sort_ind'    := Induction for sort Sort Prop.
54  Scheme eqlistA_ind' := Induction for eqlistA Sort Prop.
55
56  Theorem lelistA_unique :
57    forall (x : A) (xs : list A) (p q : lelistA R x xs), p = q.
58  Proof. induction p using lelistA_ind'; uniqueness 1. Qed.
59
60  Theorem sort_unique :
61    forall (xs : list A) (p q : sort R xs), p = q.
62  Proof. induction p using sort_ind'; uniqueness 1. apply lelistA_unique. Qed.
63
64  Theorem eqlistA_unique :
65    forall (xs ys : list A) (p q : eqlistA R xs ys), p = q.
66  Proof. induction p using eqlistA_ind'; uniqueness 2. Qed.
67
68End Uniqueness_Of_SetoidList_Proofs.
69
70
71(* *********************************************************************** *)
72(** ** Vectors *)
73
74(** [uniqueness] can show that the only vector of length zero is the
75    empty vector.  This shows that the tactic is not restricted to
76    working only on [Prop]s. *)
77
78Inductive vector (A : Type) : nat -> Type :=
79  | vnil : vector A 0
80  | vcons : forall (n : nat) (a : A), vector A n -> vector A (S n).
81
82Theorem vector_O_eq : forall (A : Type) (v : vector A 0),
83  v = vnil _.
84Proof. intros. uniqueness 1. Qed.