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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 *)
(************************************************************************)
(*i $Id$: i*)
Set Implicit Arguments.
(* DEFINITIONS OF Relation_Class AND n-ARY Morphism_Theory *)
Definition is_reflexive (A: Type) (Aeq: A -> A -> Prop) : Prop :=
forall x:A, Aeq x x.
Definition is_symmetric (A: Type) (Aeq: A -> A -> Prop) : Prop :=
forall (x y:A), Aeq x y -> Aeq y x.
Inductive Relation_Class : Type :=
Reflexive : forall A Aeq, (@is_reflexive A Aeq) -> Relation_Class
| Leibniz : Type -> Relation_Class.
Implicit Type Hole Out: Relation_Class.
Definition carrier_of_relation_class : Relation_Class -> Type.
intro; case X; intros.
exact A.
exact T.
Defined.
Inductive nelistT (A : Type) : Type :=
singl : A -> (nelistT A)
| cons : A -> (nelistT A) -> (nelistT A).
Implicit Type In: (nelistT Relation_Class).
Definition function_type_of_morphism_signature :
(nelistT Relation_Class) -> Relation_Class -> Type.
intros In Out.
induction In.
exact (carrier_of_relation_class a -> carrier_of_relation_class Out).
exact (carrier_of_relation_class a -> IHIn).
Defined.
Definition make_compatibility_goal_aux:
forall In Out
(f g: function_type_of_morphism_signature In Out), Prop.
intros; induction In; simpl in f, g.
induction a; destruct Out; simpl in f, g.
exact (forall (x1 x2: A), (Aeq x1 x2) -> (Aeq0 (f x1) (g x2))).
exact (forall (x1 x2: A), (Aeq x1 x2) -> f x1 = g x2).
exact (forall (x: T), (Aeq (f x) (g x))).
exact (forall (x: T), f x = g x).
induction a; simpl in f, g.
exact (forall (x1 x2: A), (Aeq x1 x2) -> IHIn (f x1) (g x2)).
exact (forall (x: T), IHIn (f x) (g x)).
Defined.
Definition make_compatibility_goal :=
(fun In Out f => make_compatibility_goal_aux In Out f f).
Record Morphism_Theory In Out : Type :=
{Function : function_type_of_morphism_signature In Out;
Compat : make_compatibility_goal In Out Function}.
Definition list_of_Leibniz_of_list_of_types:
nelistT Type -> nelistT Relation_Class.
induction 1.
exact (singl (Leibniz a)).
exact (cons (Leibniz a) IHX).
Defined.
(* every function is a morphism from Leibniz+ to Leibniz *)
Definition morphism_theory_of_function :
forall (In: nelistT Type) (Out: Type),
let In' := list_of_Leibniz_of_list_of_types In in
let Out' := Leibniz Out in
function_type_of_morphism_signature In' Out' ->
Morphism_Theory In' Out'.
intros.
exists X.
induction In; unfold make_compatibility_goal; simpl.
reflexivity.
intro; apply (IHIn (X x)).
Defined.
(* THE Prop RELATION CLASS *)
Add Relation Prop iff reflexivity proved by iff_refl symmetry proved by iff_sym.
Definition Prop_Relation_Class : Relation_Class.
eapply (@Reflexive _ iff).
exact iff_refl.
Defined.
(* every predicate is morphism from Leibniz+ to Prop_Relation_Class *)
Definition morphism_theory_of_predicate :
forall (In: nelistT Type),
let In' := list_of_Leibniz_of_list_of_types In in
function_type_of_morphism_signature In' Prop_Relation_Class ->
Morphism_Theory In' Prop_Relation_Class.
intros.
exists X.
induction In; unfold make_compatibility_goal; simpl.
intro; apply iff_refl.
intro; apply (IHIn (X x)).
Defined.
(* THE CIC PART OF THE REFLEXIVE TACTIC (SETOID REWRITE) *)
Inductive Morphism_Context Hole : Relation_Class -> Type :=
App : forall In Out,
Morphism_Theory In Out -> Morphism_Context_List Hole In ->
Morphism_Context Hole Out
| Toreplace : Morphism_Context Hole Hole
| Tokeep :
forall (S: Relation_Class),
carrier_of_relation_class S -> Morphism_Context Hole S
| Imp :
Morphism_Context Hole Prop_Relation_Class ->
Morphism_Context Hole Prop_Relation_Class ->
Morphism_Context Hole Prop_Relation_Class
with Morphism_Context_List Hole: nelistT Relation_Class -> Type :=
fcl_singl :
forall (S: Relation_Class), Morphism_Context Hole S ->
Morphism_Context_List Hole (singl S)
| fcl_cons :
forall (S: Relation_Class) (L: nelistT Relation_Class),
Morphism_Context Hole S -> Morphism_Context_List Hole L ->
Morphism_Context_List Hole (cons S L).
Scheme Morphism_Context_rect2 := Induction for Morphism_Context Sort Type
with Morphism_Context_List_rect2 := Induction for Morphism_Context_List Sort Type.
Inductive prodT (A B: Type) : Type :=
pairT : A -> B -> prodT A B.
Definition product_of_relation_class_list : nelistT Relation_Class -> Type.
induction 1.
exact (carrier_of_relation_class a).
exact (prodT (carrier_of_relation_class a) IHX).
Defined.
Definition relation_of_relation_class:
forall Out,
carrier_of_relation_class Out -> carrier_of_relation_class Out -> Prop.
destruct Out.
exact Aeq.
exact (@eq T).
Defined.
Definition relation_of_product_of_relation_class_list:
forall In,
product_of_relation_class_list In -> product_of_relation_class_list In -> Prop.
induction In.
exact (relation_of_relation_class a).
simpl; intros.
destruct X; destruct X0.
exact (relation_of_relation_class a c c0 /\ IHIn p p0).
Defined.
Definition apply_morphism:
forall In Out (m: function_type_of_morphism_signature In Out)
(args: product_of_relation_class_list In), carrier_of_relation_class Out.
intros.
induction In.
exact (m args).
simpl in m, args.
destruct args.
exact (IHIn (m c) p).
Defined.
Theorem apply_morphism_compatibility:
forall In Out (m1 m2: function_type_of_morphism_signature In Out)
(args1 args2: product_of_relation_class_list In),
make_compatibility_goal_aux _ _ m1 m2 ->
relation_of_product_of_relation_class_list _ args1 args2 ->
relation_of_relation_class _
(apply_morphism _ _ m1 args1)
(apply_morphism _ _ m2 args2).
intros.
induction In.
simpl; simpl in m1, m2, args1, args2, H0.
destruct a; destruct Out.
apply H; exact H0.
simpl; apply H; exact H0.
simpl; rewrite H0; apply H.
simpl; rewrite H0; apply H.
simpl in args1, args2, H0.
destruct args1; destruct args2; simpl.
destruct H0.
simpl in H.
destruct a; simpl in H.
apply IHIn.
apply H; exact H0.
exact H1.
rewrite H0; apply IHIn.
apply H.
exact H1.
Qed.
Definition interp :
forall Hole Out, carrier_of_relation_class Hole ->
Morphism_Context Hole Out -> carrier_of_relation_class Out.
intros Hole Out H t.
elim t using
(@Morphism_Context_rect2 Hole (fun S _ => carrier_of_relation_class S)
(fun L fcl => product_of_relation_class_list L));
intros.
exact (apply_morphism _ _ (Function m) X).
exact H.
exact c.
exact (X -> X0).
exact X.
split; [ exact X | exact X0 ].
Defined.
(*CSC: interp and interp_relation_class_list should be mutually defined, since
the proof term of each one contains the proof term of the other one. However
I cannot do that interactively (I should write the Fix by hand) *)
Definition interp_relation_class_list :
forall Hole (L: nelistT Relation_Class), carrier_of_relation_class Hole ->
Morphism_Context_List Hole L -> product_of_relation_class_list L.
intros Hole L H t.
elim t using
(@Morphism_Context_List_rect2 Hole (fun S _ => carrier_of_relation_class S)
(fun L fcl => product_of_relation_class_list L));
intros.
exact (apply_morphism _ _ (Function m) X).
exact H.
exact c.
exact (X -> X0).
exact X.
split; [ exact X | exact X0 ].
Defined.
Theorem setoid_rewrite:
forall Hole Out (E1 E2: carrier_of_relation_class Hole)
(E: Morphism_Context Hole Out),
(relation_of_relation_class Hole E1 E2) ->
(relation_of_relation_class Out (interp E1 E) (interp E2 E)).
intros.
elim E using
(@Morphism_Context_rect2 Hole
(fun S E => relation_of_relation_class S (interp E1 E) (interp E2 E))
(fun L fcl =>
relation_of_product_of_relation_class_list _
(interp_relation_class_list E1 fcl)
(interp_relation_class_list E2 fcl)));
intros.
change (relation_of_relation_class Out0
(apply_morphism _ _ (Function m) (interp_relation_class_list E1 m0))
(apply_morphism _ _ (Function m) (interp_relation_class_list E2 m0))).
apply apply_morphism_compatibility.
exact (Compat m).
exact H0.
exact H.
unfold interp, Morphism_Context_rect2.
(*CSC: reflexivity used here*)
destruct S.
apply i.
simpl; reflexivity.
change
(relation_of_relation_class Prop_Relation_Class
(interp E1 m -> interp E1 m0) (interp E2 m -> interp E2 m0)).
simpl; simpl in H0, H1.
tauto.
exact H0.
change
(relation_of_relation_class _ (interp E1 m) (interp E2 m) /\
relation_of_product_of_relation_class_list _
(interp_relation_class_list E1 m0) (interp_relation_class_list E2 m0)).
split.
exact H0.
exact H1.
Qed.
(* BEGIN OF UTILITY/BACKWARD COMPATIBILITY PART *)
Record Setoid_Theory (A: Type) (Aeq: A -> A -> Prop) : Prop :=
{Seq_refl : forall x:A, Aeq x x;
Seq_sym : forall x y:A, Aeq x y -> Aeq y x;
Seq_trans : forall x y z:A, Aeq x y -> Aeq y z -> Aeq x z}.
Definition relation_class_of_setoid_theory:
forall (A: Type) (Aeq: A -> A -> Prop),
Setoid_Theory Aeq -> Relation_Class.
intros.
apply (@Reflexive _ Aeq).
exact (Seq_refl H).
Defined.
Definition equality_morphism_of_setoid_theory:
forall (A: Type) (Aeq: A -> A -> Prop) (ST: Setoid_Theory Aeq),
let ASetoidClass := relation_class_of_setoid_theory ST in
(Morphism_Theory (cons ASetoidClass (singl ASetoidClass))
Prop_Relation_Class).
intros.
exists Aeq.
pose (sym := Seq_sym ST); clearbody sym.
pose (trans := Seq_trans ST); clearbody trans.
(*CSC: symmetry and transitivity used here *)
unfold make_compatibility_goal; simpl; split; eauto.
Defined.
(* END OF UTILITY/BACKWARD COMPATIBILITY PART *)
(* A FEW EXAMPLES *)
Add Morphism iff : Iff_Morphism.
tauto.
Defined.
(* impl IS A MORPHISM *)
Definition impl (A B: Prop) := A -> B.
Add Morphism impl : Impl_Morphism.
unfold impl; tauto.
Defined.
(* and IS A MORPHISM *)
Add Morphism and : And_Morphism.
tauto.
Defined.
(* or IS A MORPHISM *)
Add Morphism or : Or_Morphism.
tauto.
Defined.
(* not IS A MORPHISM *)
Add Morphism not : Not_Morphism.
tauto.
Defined.
(* FOR BACKWARD COMPATIBILITY *)
Implicit Arguments Setoid_Theory [].
Implicit Arguments Seq_refl [].
Implicit Arguments Seq_sym [].
Implicit Arguments Seq_trans [].
|
