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author | Stephane Glondu <steph@glondu.net> | 2010-07-21 09:46:51 +0200 |
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committer | Stephane Glondu <steph@glondu.net> | 2010-07-21 09:46:51 +0200 |
commit | 5b7eafd0f00a16d78f99a27f5c7d5a0de77dc7e6 (patch) | |
tree | 631ad791a7685edafeb1fb2e8faeedc8379318ae /theories/Structures/DecidableTypeEx.v | |
parent | da178a880e3ace820b41d38b191d3785b82991f5 (diff) |
Imported Upstream snapshot 8.3~beta0+13298
Diffstat (limited to 'theories/Structures/DecidableTypeEx.v')
-rw-r--r-- | theories/Structures/DecidableTypeEx.v | 96 |
1 files changed, 96 insertions, 0 deletions
diff --git a/theories/Structures/DecidableTypeEx.v b/theories/Structures/DecidableTypeEx.v new file mode 100644 index 00000000..4407ead4 --- /dev/null +++ b/theories/Structures/DecidableTypeEx.v @@ -0,0 +1,96 @@ +(***********************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) +(* \VV/ *************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(***********************************************************************) + +(* $Id$ *) + +Require Import DecidableType OrderedType OrderedTypeEx. +Set Implicit Arguments. +Unset Strict Implicit. + +(** NB: This file is here only for compatibility with earlier version of + [FSets] and [FMap]. Please use [Structures/Equalities.v] directly now. *) + +(** * Examples of Decidable Type structures. *) + +(** A particular case of [DecidableType] where + the equality is the usual one of Coq. *) + +Module Type UsualDecidableType := Equalities.UsualDecidableTypeOrig. + +(** a [UsualDecidableType] is in particular an [DecidableType]. *) + +Module UDT_to_DT (U:UsualDecidableType) <: DecidableType := U. + +(** an shortcut for easily building a UsualDecidableType *) + +Module Type MiniDecidableType := Equalities.MiniDecidableType. + +Module Make_UDT (M:MiniDecidableType) <: UsualDecidableType + := Equalities.Make_UDT M. + +(** An OrderedType can now directly be seen as a DecidableType *) + +Module OT_as_DT (O:OrderedType) <: DecidableType := O. + +(** (Usual) Decidable Type for [nat], [positive], [N], [Z] *) + +Module Nat_as_DT <: UsualDecidableType := Nat_as_OT. +Module Positive_as_DT <: UsualDecidableType := Positive_as_OT. +Module N_as_DT <: UsualDecidableType := N_as_OT. +Module Z_as_DT <: UsualDecidableType := Z_as_OT. + +(** From two decidable types, we can build a new DecidableType + over their cartesian product. *) + +Module PairDecidableType(D1 D2:DecidableType) <: DecidableType. + + Definition t := prod D1.t D2.t. + + Definition eq x y := D1.eq (fst x) (fst y) /\ D2.eq (snd x) (snd y). + + Lemma eq_refl : forall x : t, eq x x. + Proof. + intros (x1,x2); red; simpl; auto. + Qed. + + Lemma eq_sym : forall x y : t, eq x y -> eq y x. + Proof. + intros (x1,x2) (y1,y2); unfold eq; simpl; intuition. + Qed. + + Lemma eq_trans : forall x y z : t, eq x y -> eq y z -> eq x z. + Proof. + intros (x1,x2) (y1,y2) (z1,z2); unfold eq; simpl; intuition eauto. + Qed. + + Definition eq_dec : forall x y, { eq x y }+{ ~eq x y }. + Proof. + intros (x1,x2) (y1,y2); unfold eq; simpl. + destruct (D1.eq_dec x1 y1); destruct (D2.eq_dec x2 y2); intuition. + Defined. + +End PairDecidableType. + +(** Similarly for pairs of UsualDecidableType *) + +Module PairUsualDecidableType(D1 D2:UsualDecidableType) <: UsualDecidableType. + Definition t := prod D1.t D2.t. + Definition eq := @eq t. + Definition eq_refl := @refl_equal t. + Definition eq_sym := @sym_eq t. + Definition eq_trans := @trans_eq t. + Definition eq_dec : forall x y, { eq x y }+{ ~eq x y }. + Proof. + intros (x1,x2) (y1,y2); + destruct (D1.eq_dec x1 y1); destruct (D2.eq_dec x2 y2); + unfold eq, D1.eq, D2.eq in *; simpl; + (left; f_equal; auto; fail) || + (right; intro H; injection H; auto). + Defined. + +End PairUsualDecidableType. |