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Diffstat (limited to 'theories/Logic/DecidableTypeEx.v')
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diff --git a/theories/Logic/DecidableTypeEx.v b/theories/Logic/DecidableTypeEx.v deleted file mode 100644 index 9c59c519..00000000 --- a/theories/Logic/DecidableTypeEx.v +++ /dev/null @@ -1,109 +0,0 @@ -(***********************************************************************) -(* 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: DecidableTypeEx.v 11699 2008-12-18 11:49:08Z letouzey $ *) - -Require Import DecidableType OrderedType OrderedTypeEx. -Set Implicit Arguments. -Unset Strict Implicit. - -(** * Examples of Decidable Type structures. *) - -(** A particular case of [DecidableType] where - the equality is the usual one of Coq. *) - -Module Type UsualDecidableType. - Parameter Inline t : Type. - Definition eq := @eq t. - Definition eq_refl := @refl_equal t. - Definition eq_sym := @sym_eq t. - Definition eq_trans := @trans_eq t. - Parameter eq_dec : forall x y, { eq x y }+{~eq x y }. -End UsualDecidableType. - -(** 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. - Parameter Inline t : Type. - Parameter eq_dec : forall x y:t, { x=y }+{ x<>y }. -End MiniDecidableType. - -Module Make_UDT (M:MiniDecidableType) <: UsualDecidableType. - Definition t:=M.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 := M.eq_dec. -End Make_UDT. - -(** 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. |