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-rw-r--r--theories/Logic/DecidableType.v173
-rw-r--r--theories/Logic/DecidableTypeEx.v109
2 files changed, 0 insertions, 282 deletions
diff --git a/theories/Logic/DecidableType.v b/theories/Logic/DecidableType.v
deleted file mode 100644
index 625f776bfa..0000000000
--- a/theories/Logic/DecidableType.v
+++ /dev/null
@@ -1,173 +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 *)
-(***********************************************************************)
-
-(* $Id$ *)
-
-Require Export SetoidList.
-Set Implicit Arguments.
-Unset Strict Implicit.
-
-(** * Types with Equalities, and nothing more (for subtyping purpose) *)
-
-Module Type EqualityType.
-
- Parameter Inline t : Type.
-
- Parameter Inline eq : t -> t -> Prop.
-
- Axiom eq_refl : forall x : t, eq x x.
- Axiom eq_sym : forall x y : t, eq x y -> eq y x.
- Axiom eq_trans : forall x y z : t, eq x y -> eq y z -> eq x z.
-
- Hint Immediate eq_sym.
- Hint Resolve eq_refl eq_trans.
-
-End EqualityType.
-
-(** * Types with decidable Equalities (but no ordering) *)
-
-Module Type DecidableType.
-
- Parameter Inline t : Type.
-
- Parameter Inline eq : t -> t -> Prop.
-
- Axiom eq_refl : forall x : t, eq x x.
- Axiom eq_sym : forall x y : t, eq x y -> eq y x.
- Axiom eq_trans : forall x y z : t, eq x y -> eq y z -> eq x z.
-
- Parameter eq_dec : forall x y : t, { eq x y } + { ~ eq x y }.
-
- Hint Immediate eq_sym.
- Hint Resolve eq_refl eq_trans.
-
-End DecidableType.
-
-(** * Additional notions about keys and datas used in FMap *)
-
-Module KeyDecidableType(D:DecidableType).
- Import D.
-
- Section Elt.
- Variable elt : Type.
- Notation key:=t.
-
- Definition eqk (p p':key*elt) := eq (fst p) (fst p').
- Definition eqke (p p':key*elt) :=
- eq (fst p) (fst p') /\ (snd p) = (snd p').
-
- Hint Unfold eqk eqke.
- Hint Extern 2 (eqke ?a ?b) => split.
-
- (* eqke is stricter than eqk *)
-
- Lemma eqke_eqk : forall x x', eqke x x' -> eqk x x'.
- Proof.
- unfold eqk, eqke; intuition.
- Qed.
-
- (* eqk, eqke are equalities *)
-
- Lemma eqk_refl : forall e, eqk e e.
- Proof. auto. Qed.
-
- Lemma eqke_refl : forall e, eqke e e.
- Proof. auto. Qed.
-
- Lemma eqk_sym : forall e e', eqk e e' -> eqk e' e.
- Proof. auto. Qed.
-
- Lemma eqke_sym : forall e e', eqke e e' -> eqke e' e.
- Proof. unfold eqke; intuition. Qed.
-
- Lemma eqk_trans : forall e e' e'', eqk e e' -> eqk e' e'' -> eqk e e''.
- Proof. eauto. Qed.
-
- Lemma eqke_trans : forall e e' e'', eqke e e' -> eqke e' e'' -> eqke e e''.
- Proof.
- unfold eqke; intuition; [ eauto | congruence ].
- Qed.
-
- Hint Resolve eqk_trans eqke_trans eqk_refl eqke_refl.
- Hint Immediate eqk_sym eqke_sym.
-
- Lemma InA_eqke_eqk :
- forall x m, InA eqke x m -> InA eqk x m.
- Proof.
- unfold eqke; induction 1; intuition.
- Qed.
- Hint Resolve InA_eqke_eqk.
-
- Lemma InA_eqk : forall p q m, eqk p q -> InA eqk p m -> InA eqk q m.
- Proof.
- intros; apply InA_eqA with p; auto; apply eqk_trans; auto.
- Qed.
-
- Definition MapsTo (k:key)(e:elt):= InA eqke (k,e).
- Definition In k m := exists e:elt, MapsTo k e m.
-
- Hint Unfold MapsTo In.
-
- (* An alternative formulation for [In k l] is [exists e, InA eqk (k,e) l] *)
-
- Lemma In_alt : forall k l, In k l <-> exists e, InA eqk (k,e) l.
- Proof.
- firstorder.
- exists x; auto.
- induction H.
- destruct y.
- exists e; auto.
- destruct IHInA as [e H0].
- exists e; auto.
- Qed.
-
- Lemma MapsTo_eq : forall l x y e, eq x y -> MapsTo x e l -> MapsTo y e l.
- Proof.
- intros; unfold MapsTo in *; apply InA_eqA with (x,e); eauto.
- Qed.
-
- Lemma In_eq : forall l x y, eq x y -> In x l -> In y l.
- Proof.
- destruct 2 as (e,E); exists e; eapply MapsTo_eq; eauto.
- Qed.
-
- Lemma In_inv : forall k k' e l, In k ((k',e) :: l) -> eq k k' \/ In k l.
- Proof.
- inversion 1.
- inversion_clear H0; eauto.
- destruct H1; simpl in *; intuition.
- Qed.
-
- Lemma In_inv_2 : forall k k' e e' l,
- InA eqk (k, e) ((k', e') :: l) -> ~ eq k k' -> InA eqk (k, e) l.
- Proof.
- inversion_clear 1; compute in H0; intuition.
- Qed.
-
- Lemma In_inv_3 : forall x x' l,
- InA eqke x (x' :: l) -> ~ eqk x x' -> InA eqke x l.
- Proof.
- inversion_clear 1; compute in H0; intuition.
- Qed.
-
- End Elt.
-
- Hint Unfold eqk eqke.
- Hint Extern 2 (eqke ?a ?b) => split.
- Hint Resolve eqk_trans eqke_trans eqk_refl eqke_refl.
- Hint Immediate eqk_sym eqke_sym.
- Hint Resolve InA_eqke_eqk.
- Hint Unfold MapsTo In.
- Hint Resolve In_inv_2 In_inv_3.
-
-End KeyDecidableType.
-
-
-
-
-
diff --git a/theories/Logic/DecidableTypeEx.v b/theories/Logic/DecidableTypeEx.v
deleted file mode 100644
index 022102f70d..0000000000
--- 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 *)
-(***********************************************************************)
-
-(* $Id$ *)
-
-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.
generated by cgit v1.2.3 (git 2.25.1) at 2026-04-23 20:44:25 +0000