--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti. C.Sacerdoti Coen. *)
+(* ||A|| E.Tassi. S.Zacchiroli *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+(* Code ported from the Coq theorem prover by Claudio Sacerdoti Coen *)
+(* Original author: Claudio Sacerdoti Coen. for the Coq system *)
+
+set "baseuri" "cic:/matita/technicalities/setoids".
+
+include "datatypes/constructors.ma".
+include "logic/coimplication.ma".
+include "logic/connectives2.ma".
+
+(* DEFINITIONS OF Relation_Class AND n-ARY Morphism_Theory *)
+
+(* X will be used to distinguish covariant arguments whose type is an *)
+(* Asymmetric* relation from contravariant arguments of the same type *)
+inductive X_Relation_Class (X: Type) : Type ≝
+ SymmetricReflexive :
+ ∀A,Aeq. symmetric A Aeq → reflexive ? Aeq → X_Relation_Class X
+ | AsymmetricReflexive : X → ∀A,Aeq. reflexive A Aeq → X_Relation_Class X
+ | SymmetricAreflexive : ∀A,Aeq. symmetric A Aeq → X_Relation_Class X
+ | AsymmetricAreflexive : X → ∀A.∀Aeq : relation A. X_Relation_Class X
+ | Leibniz : Type → X_Relation_Class X.
+
+inductive variance : Set ≝
+ Covariant : variance
+ | Contravariant : variance.
+
+definition Argument_Class ≝ X_Relation_Class variance.
+definition Relation_Class ≝ X_Relation_Class unit.
+
+inductive Reflexive_Relation_Class : Type :=
+ RSymmetric :
+ ∀A,Aeq. symmetric A Aeq → reflexive ? Aeq → Reflexive_Relation_Class
+ | RAsymmetric :
+ ∀A,Aeq. reflexive A Aeq → Reflexive_Relation_Class
+ | RLeibniz : Type → Reflexive_Relation_Class.
+
+inductive Areflexive_Relation_Class : Type :=
+ | ASymmetric : ∀A,Aeq. symmetric A Aeq → Areflexive_Relation_Class
+ | AAsymmetric : ∀A.∀Aeq : relation A. Areflexive_Relation_Class.
+
+definition relation_class_of_argument_class : Argument_Class → Relation_Class.
+ intros (a); cases a;
+ [ apply (SymmetricReflexive ? ? ? H H1)
+ | apply (AsymmetricReflexive ? something ? ? H)
+ | apply (SymmetricAreflexive ? ? ? H)
+ | apply (AsymmetricAreflexive ? something ? r)
+ | apply (Leibniz ? T)
+ ]
+qed.
+
+definition carrier_of_relation_class : ∀X. X_Relation_Class X → Type.
+ intros (X x); cases x (A o o o o A o o A o o o A o A); exact A.
+qed.
+
+definition relation_of_relation_class:
+ ∀X,R. carrier_of_relation_class X R → carrier_of_relation_class X R → Prop.
+intros 2; cases R; simplify; [1,2,3,4: assumption | apply (eq T) ]
+qed.
+
+lemma about_carrier_of_relation_class_and_relation_class_of_argument_class :
+ ∀R.
+ carrier_of_relation_class ? (relation_class_of_argument_class R) =
+ carrier_of_relation_class ? R.
+intro; cases R; reflexivity.
+qed.
+
+inductive nelistT (A : Type) : Type :=
+ singl : A → nelistT A
+ | cons : A → nelistT A → nelistT A.
+
+definition Arguments := nelistT Argument_Class.
+
+definition function_type_of_morphism_signature :
+ Arguments → Relation_Class → Type.
+ intros (In Out); elim In;
+ [ exact (carrier_of_relation_class ? t → carrier_of_relation_class ? Out)
+ | exact (carrier_of_relation_class ? t → T)
+ ]
+qed.
+
+definition make_compatibility_goal_aux:
+ ∀In,Out.∀f,g:function_type_of_morphism_signature In Out.Prop.
+ intros 2;
+ elim In (a); simplify in f f1;
+ generalize in match f1; clear f1;
+ generalize in match f; clear f;
+ [ elim a; simplify in f f1;
+ [ exact (∀x1,x2. r x1 x2 → relation_of_relation_class ? Out (f x1) (f1 x2))
+ | cases t;
+ [ exact (∀x1,x2. r x1 x2 → relation_of_relation_class ? Out (f x1) (f1 x2))
+ | exact (∀x1,x2. r x2 x1 → relation_of_relation_class ? Out (f x1) (f1 x2))
+ ]
+ | exact (∀x1,x2. r x1 x2 → relation_of_relation_class ? Out (f x1) (f1 x2))
+ | cases t;
+ [ exact (∀x1,x2. r x1 x2 → relation_of_relation_class ? Out (f x1) (f1 x2))
+ | exact (∀x1,x2. r x2 x1 → relation_of_relation_class ? Out (f x1) (f1 x2))
+ ]
+ | exact (∀x. relation_of_relation_class ? Out (f x) (f1 x))
+ ]
+ | change with
+ ((carrier_of_relation_class ? t → function_type_of_morphism_signature n Out) →
+ (carrier_of_relation_class ? t → function_type_of_morphism_signature n Out) →
+ Prop).
+ elim t; simplify in f f1;
+ [1,3: exact (∀x1,x2. r x1 x2 → R (f x1) (f1 x2))
+ |2,4: cases t1;
+ [1,3: exact (∀x1,x2. r x1 x2 → R (f x1) (f1 x2))
+ |2,4: exact (∀x1,x2. r x2 x1 → R (f x1) (f1 x2))
+ ]
+ | exact (∀x. R (f x) (f1 x))
+ ]
+ ]
+qed.
+
+definition make_compatibility_goal :=
+ λIn,Out,f. make_compatibility_goal_aux In Out f f.
+
+record Morphism_Theory (In: Arguments) (Out: Relation_Class) : 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 → Arguments.
+ intro;
+ elim n;
+ [ apply (singl ? (Leibniz ? t))
+ | apply (cons ? (Leibniz ? t) a)
+ ]
+qed.
+
+(* every function is a morphism from Leibniz+ to Leibniz *)
+definition morphism_theory_of_function :
+ ∀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;
+ apply (mk_Morphism_Theory ? ? f);
+ unfold In' in f ⊢ %; clear In';
+ unfold Out' in f ⊢ %; clear Out';
+ generalize in match f; clear f;
+ elim In;
+ [ unfold make_compatibility_goal;
+ whd; intros;
+ reflexivity
+ | simplify;
+ intro;
+ unfold In' in f;
+ unfold Out' in f;
+ exact (H (f1 x))
+ ]
+qed.
+
+(* THE iff RELATION CLASS *)
+
+definition Iff_Relation_Class : Relation_Class.
+ apply (SymmetricReflexive unit ? iff);
+ [ exact symmetric_iff
+ | exact reflexive_iff
+ ]
+qed.
+
+(* THE impl RELATION CLASS *)
+
+definition impl \def \lambda A,B:Prop. A → B.
+
+theorem impl_refl: reflexive ? impl.
+ unfold reflexive;
+ intros;
+ unfold impl;
+ intro;
+ assumption.
+qed.
+
+definition Impl_Relation_Class : Relation_Class.
+ unfold Relation_Class;
+ apply (AsymmetricReflexive unit something ? impl);
+ exact impl_refl.
+qed.
+
+(* UTILITY FUNCTIONS TO PROVE THAT EVERY TRANSITIVE RELATION IS A MORPHISM *)
+
+definition equality_morphism_of_symmetric_areflexive_transitive_relation:
+ ∀A: Type.∀Aeq: relation A.∀sym: symmetric ? Aeq.∀trans: transitive ? Aeq.
+ let ASetoidClass := SymmetricAreflexive ? ? ? sym in
+ (Morphism_Theory (cons ? ASetoidClass (singl ? ASetoidClass))
+ Iff_Relation_Class).
+ intros;
+ apply mk_Morphism_Theory;
+ [ exact Aeq
+ | unfold make_compatibility_goal;
+ simplify; unfold ASetoidClass; simplify;
+ intros;
+ split;
+ unfold transitive in H;
+ unfold symmetric in sym;
+ intro;
+ [ apply (H x2 x1 x3 ? ?);
+ [apply (sym x1 x2 ?).
+ apply (H1).
+ |apply (H x1 x x3 ? ?);
+ [apply (H3).
+ |apply (H2).
+ ]
+ ]
+ | apply (H x1 x3 x ? ?);
+ [apply (H x1 x2 x3 ? ?);
+ [apply (H1).
+ |apply (H3).
+ ]
+ |apply (sym x x3 ?).
+ apply (H2).
+ ]
+ ]
+ ].
+qed.
+
+definition equality_morphism_of_symmetric_reflexive_transitive_relation:
+ ∀A: Type.∀Aeq: relation A.∀refl: reflexive ? Aeq.∀sym: symmetric ? Aeq.
+ ∀trans: transitive ? Aeq.
+ let ASetoidClass := SymmetricReflexive ? ? ? sym refl in
+ (Morphism_Theory (cons ? ASetoidClass (singl ? ASetoidClass)) Iff_Relation_Class).
+ intros;
+ apply mk_Morphism_Theory;
+ normalize;
+ [ exact Aeq
+ | intros;
+ split;
+ intro;
+ unfold transitive in H;
+ unfold symmetric in sym;
+ [ apply (H x2 x1 x3 ? ?);
+ [apply (sym x1 x2 ?).
+ apply (H1).
+ |apply (H x1 x x3 ? ?);
+ [apply (H3).
+ |apply (H2).
+ ]
+ ]
+ | apply (H x1 x2 x ? ?);
+ [apply (H1).
+ |apply (H x2 x3 x ? ?);
+ [apply (H3).
+ |apply (sym x x3 ?).
+ apply (H x x3 x3 ? ?);
+ [apply (H2).
+ |apply (refl x3).
+ ]
+ ]
+ ]
+ ]
+ ]
+qed.
+
+definition equality_morphism_of_asymmetric_areflexive_transitive_relation:
+ ∀A: Type.∀Aeq: relation A.∀trans: transitive ? Aeq.
+ let ASetoidClass1 := AsymmetricAreflexive ? Contravariant ? Aeq in
+ let ASetoidClass2 := AsymmetricAreflexive ? Covariant ? Aeq in
+ (Morphism_Theory (cons ? ASetoidClass1 (singl ? ASetoidClass2)) Impl_Relation_Class).
+ intros;
+ apply mk_Morphism_Theory;
+ [ simplify;
+ apply Aeq
+ | simplify; unfold ASetoidClass1; simplify; unfold ASetoidClass2; simplify;
+ intros;
+ whd;
+ intros;
+ apply (H x2 x1 x3 ? ?);
+ [apply (H1).
+ |apply (H x1 x x3 ? ?);
+ [apply (H3).
+ |apply (H2).
+ ]
+ ]
+ ].
+qed.
+
+definition equality_morphism_of_asymmetric_reflexive_transitive_relation:
+ ∀A: Type.∀Aeq: relation A.∀refl: reflexive ? Aeq.∀trans: transitive ? Aeq.
+ let ASetoidClass1 := AsymmetricReflexive ? Contravariant ? ? refl in
+ let ASetoidClass2 := AsymmetricReflexive ? Covariant ? ? refl in
+ (Morphism_Theory (cons ? ASetoidClass1 (singl ? ASetoidClass2)) Impl_Relation_Class).
+ intros;
+ apply mk_Morphism_Theory;
+ [ simplify;
+ apply Aeq
+ | simplify; unfold ASetoidClass1; simplify; unfold ASetoidClass2; simplify;
+ intros;
+ whd;
+ intro;
+ apply (H x2 x1 x3 ? ?);
+ [apply (H1).
+ |apply (H x1 x x3 ? ?);
+ [apply (H3).
+ |apply (H2).
+ ]
+ ]
+ ].
+qed.
+
+(* iff AS A RELATION *)
+
+(*DA PORTARE:Add Relation Prop iff
+ reflexivity proved by iff_refl
+ symmetry proved by iff_sym
+ transitivity proved by iff_trans
+ as iff_relation.*)
+
+(* every predicate is morphism from Leibniz+ to Iff_Relation_Class *)
+definition morphism_theory_of_predicate :
+ ∀(In: nelistT Type).
+ let In' := list_of_Leibniz_of_list_of_types In in
+ function_type_of_morphism_signature In' Iff_Relation_Class →
+ Morphism_Theory In' Iff_Relation_Class.
+ intros;
+ apply mk_Morphism_Theory;
+ [ apply f
+ | generalize in match f; clear f;
+ unfold In'; clear In';
+ elim In;
+ [ normalize;
+ intro;
+ apply iff_refl
+ | simplify;
+ intro x;
+ apply (H (f1 x))
+ ]
+ ].
+qed.
+
+(* impl AS A RELATION *)
+
+theorem impl_trans: transitive ? impl.
+ whd;
+ unfold impl;
+ intros;
+ apply (H1 ?).apply (H ?).apply (H2).
+ autobatch.
+qed.
+
+(*DA PORTARE: Add Relation Prop impl
+ reflexivity proved by impl_refl
+ transitivity proved by impl_trans
+ as impl_relation.*)
+
+(* THE CIC PART OF THE REFLEXIVE TACTIC (SETOID REWRITE) *)
+
+inductive rewrite_direction : Type :=
+ Left2Right: rewrite_direction
+ | Right2Left: rewrite_direction.
+
+definition variance_of_argument_class : Argument_Class → option variance.
+ intros;
+ elim a;
+ [ apply None
+ | apply (Some ? t)
+ | apply None
+ | apply (Some ? t)
+ | apply None
+ ]
+qed.
+
+definition opposite_direction :=
+ λdir.
+ match dir with
+ [ Left2Right ⇒ Right2Left
+ | Right2Left ⇒ Left2Right
+ ].
+
+lemma opposite_direction_idempotent:
+ ∀dir. opposite_direction (opposite_direction dir) = dir.
+ intros;
+ elim dir;
+ reflexivity.
+qed.
+
+inductive check_if_variance_is_respected :
+ option variance → rewrite_direction → rewrite_direction → Prop
+:=
+ MSNone : ∀dir,dir'. check_if_variance_is_respected (None ?) dir dir'
+ | MSCovariant : ∀dir. check_if_variance_is_respected (Some ? Covariant) dir dir
+ | MSContravariant :
+ ∀dir.
+ check_if_variance_is_respected (Some ? Contravariant) dir (opposite_direction dir).
+
+definition relation_class_of_reflexive_relation_class:
+ Reflexive_Relation_Class → Relation_Class.
+ intro;
+ elim r;
+ [ apply (SymmetricReflexive ? ? ? H H1)
+ | apply (AsymmetricReflexive ? something ? ? H)
+ | apply (Leibniz ? T)
+ ]
+qed.
+
+definition relation_class_of_areflexive_relation_class:
+ Areflexive_Relation_Class → Relation_Class.
+ intro;
+ elim a;
+ [ apply (SymmetricAreflexive ? ? ? H)
+ | apply (AsymmetricAreflexive ? something ? r)
+ ]
+qed.
+
+definition carrier_of_reflexive_relation_class :=
+ λR.carrier_of_relation_class ? (relation_class_of_reflexive_relation_class R).
+
+definition carrier_of_areflexive_relation_class :=
+ λR.carrier_of_relation_class ? (relation_class_of_areflexive_relation_class R).
+
+definition relation_of_areflexive_relation_class :=
+ λR.relation_of_relation_class ? (relation_class_of_areflexive_relation_class R).
+
+inductive Morphism_Context (Hole: Relation_Class) (dir:rewrite_direction) : Relation_Class → rewrite_direction → Type :=
+ App :
+ ∀In,Out,dir'.
+ Morphism_Theory In Out → Morphism_Context_List Hole dir dir' In →
+ Morphism_Context Hole dir Out dir'
+ | ToReplace : Morphism_Context Hole dir Hole dir
+ | ToKeep :
+ ∀S,dir'.
+ carrier_of_reflexive_relation_class S →
+ Morphism_Context Hole dir (relation_class_of_reflexive_relation_class S) dir'
+ | ProperElementToKeep :
+ ∀S,dir'.∀x: carrier_of_areflexive_relation_class S.
+ relation_of_areflexive_relation_class S x x →
+ Morphism_Context Hole dir (relation_class_of_areflexive_relation_class S) dir'
+with Morphism_Context_List :
+ rewrite_direction → Arguments → Type
+:=
+ fcl_singl :
+ ∀S,dir',dir''.
+ check_if_variance_is_respected (variance_of_argument_class S) dir' dir'' →
+ Morphism_Context Hole dir (relation_class_of_argument_class S) dir' →
+ Morphism_Context_List Hole dir dir'' (singl ? S)
+ | fcl_cons :
+ ∀S,L,dir',dir''.
+ check_if_variance_is_respected (variance_of_argument_class S) dir' dir'' →
+ Morphism_Context Hole dir (relation_class_of_argument_class S) dir' →
+ Morphism_Context_List Hole dir dir'' L →
+ Morphism_Context_List Hole dir dir'' (cons ? S L).
+
+lemma Morphism_Context_rect2:
+ ∀Hole,dir.
+ ∀P:
+ ∀r:Relation_Class.∀r0:rewrite_direction.Morphism_Context Hole dir r r0 → Type.
+ ∀P0:
+ ∀r:rewrite_direction.∀a:Arguments.Morphism_Context_List Hole dir r a → Type.
+ (∀In,Out,dir'.
+ ∀m:Morphism_Theory In Out.∀m0:Morphism_Context_List Hole dir dir' In.
+ P0 dir' In m0 → P Out dir' (App Hole ? ? ? ? m m0)) →
+ P Hole dir (ToReplace Hole dir) →
+ (∀S:Reflexive_Relation_Class.∀dir'.∀c:carrier_of_reflexive_relation_class S.
+ P (relation_class_of_reflexive_relation_class S) dir'
+ (ToKeep Hole dir S dir' c)) →
+ (∀S:Areflexive_Relation_Class.∀dir'.
+ ∀x:carrier_of_areflexive_relation_class S.
+ ∀r:relation_of_areflexive_relation_class S x x.
+ P (relation_class_of_areflexive_relation_class S) dir'
+ (ProperElementToKeep Hole dir S dir' x r)) →
+ (∀S:Argument_Class.∀dir',dir''.
+ ∀c:check_if_variance_is_respected (variance_of_argument_class S) dir' dir''.
+ ∀m:Morphism_Context Hole dir (relation_class_of_argument_class S) dir'.
+ P (relation_class_of_argument_class S) dir' m ->
+ P0 dir'' (singl ? S) (fcl_singl ? ? S ? ? c m)) →
+ (∀S:Argument_Class.∀L:Arguments.∀dir',dir''.
+ ∀c:check_if_variance_is_respected (variance_of_argument_class S) dir' dir''.
+ ∀m:Morphism_Context Hole dir (relation_class_of_argument_class S) dir'.
+ P (relation_class_of_argument_class S) dir' m →
+ ∀m0:Morphism_Context_List Hole dir dir'' L.
+ P0 dir'' L m0 → P0 dir'' (cons ? S L) (fcl_cons ? ? S ? ? ? c m m0)) →
+ ∀r:Relation_Class.∀r0:rewrite_direction.∀m:Morphism_Context Hole dir r r0.
+ P r r0 m
+≝
+ λHole,dir,P,P0,f,f0,f1,f2,f3,f4.
+ let rec
+ F (rc:Relation_Class) (r0:rewrite_direction)
+ (m:Morphism_Context Hole dir rc r0) on m : P rc r0 m
+ ≝
+ match m return λrc.λr0.λm0.P rc r0 m0 with
+ [ App In Out dir' m0 m1 ⇒ f In Out dir' m0 m1 (F0 dir' In m1)
+ | ToReplace ⇒ f0
+ | ToKeep S dir' c ⇒ f1 S dir' c
+ | ProperElementToKeep S dir' x r1 ⇒ f2 S dir' x r1
+ ]
+ and
+ F0 (r:rewrite_direction) (a:Arguments)
+ (m:Morphism_Context_List Hole dir r a) on m : P0 r a m
+ ≝
+ match m return λr.λa.λm0.P0 r a m0 with
+ [ fcl_singl S dir' dir'' c m0 ⇒
+ f3 S dir' dir'' c m0 (F (relation_class_of_argument_class S) dir' m0)
+ | fcl_cons S L dir' dir'' c m0 m1 ⇒
+ f4 S L dir' dir'' c m0 (F (relation_class_of_argument_class S) dir' m0)
+ m1 (F0 dir'' L m1)
+ ]
+in F.
+
+lemma Morphism_Context_List_rect2:
+ ∀Hole,dir.
+ ∀P:
+ ∀r:Relation_Class.∀r0:rewrite_direction.Morphism_Context Hole dir r r0 → Type.
+ ∀P0:
+ ∀r:rewrite_direction.∀a:Arguments.Morphism_Context_List Hole dir r a → Type.
+ (∀In,Out,dir'.
+ ∀m:Morphism_Theory In Out.∀m0:Morphism_Context_List Hole dir dir' In.
+ P0 dir' In m0 → P Out dir' (App Hole ? ? ? ? m m0)) →
+ P Hole dir (ToReplace Hole dir) →
+ (∀S:Reflexive_Relation_Class.∀dir'.∀c:carrier_of_reflexive_relation_class S.
+ P (relation_class_of_reflexive_relation_class S) dir'
+ (ToKeep Hole dir S dir' c)) →
+ (∀S:Areflexive_Relation_Class.∀dir'.
+ ∀x:carrier_of_areflexive_relation_class S.
+ ∀r:relation_of_areflexive_relation_class S x x.
+ P (relation_class_of_areflexive_relation_class S) dir'
+ (ProperElementToKeep Hole dir S dir' x r)) →
+ (∀S:Argument_Class.∀dir',dir''.
+ ∀c:check_if_variance_is_respected (variance_of_argument_class S) dir' dir''.
+ ∀m:Morphism_Context Hole dir (relation_class_of_argument_class S) dir'.
+ P (relation_class_of_argument_class S) dir' m ->
+ P0 dir'' (singl ? S) (fcl_singl ? ? S ? ? c m)) →
+ (∀S:Argument_Class.∀L:Arguments.∀dir',dir''.
+ ∀c:check_if_variance_is_respected (variance_of_argument_class S) dir' dir''.
+ ∀m:Morphism_Context Hole dir (relation_class_of_argument_class S) dir'.
+ P (relation_class_of_argument_class S) dir' m →
+ ∀m0:Morphism_Context_List Hole dir dir'' L.
+ P0 dir'' L m0 → P0 dir'' (cons ? S L) (fcl_cons ? ? S ? ? ? c m m0)) →
+ ∀r:rewrite_direction.∀a:Arguments.∀m:Morphism_Context_List Hole dir r a.
+ P0 r a m
+≝
+ λHole,dir,P,P0,f,f0,f1,f2,f3,f4.
+ let rec
+ F (rc:Relation_Class) (r0:rewrite_direction)
+ (m:Morphism_Context Hole dir rc r0) on m : P rc r0 m
+ ≝
+ match m return λrc.λr0.λm0.P rc r0 m0 with
+ [ App In Out dir' m0 m1 ⇒ f In Out dir' m0 m1 (F0 dir' In m1)
+ | ToReplace ⇒ f0
+ | ToKeep S dir' c ⇒ f1 S dir' c
+ | ProperElementToKeep S dir' x r1 ⇒ f2 S dir' x r1
+ ]
+ and
+ F0 (r:rewrite_direction) (a:Arguments)
+ (m:Morphism_Context_List Hole dir r a) on m : P0 r a m
+ ≝
+ match m return λr.λa.λm0.P0 r a m0 with
+ [ fcl_singl S dir' dir'' c m0 ⇒
+ f3 S dir' dir'' c m0 (F (relation_class_of_argument_class S) dir' m0)
+ | fcl_cons S L dir' dir'' c m0 m1 ⇒
+ f4 S L dir' dir'' c m0 (F (relation_class_of_argument_class S) dir' m0)
+ m1 (F0 dir'' L m1)
+ ]
+in F0.
+
+definition product_of_arguments : Arguments → Type.
+ intro;
+ elim a;
+ [ apply (carrier_of_relation_class ? t)
+ | apply (Prod (carrier_of_relation_class ? t) T)
+ ]
+qed.
+
+definition get_rewrite_direction: rewrite_direction → Argument_Class → rewrite_direction.
+ intros (dir R);
+ cases (variance_of_argument_class R) (a);
+ [ exact dir
+ | cases a;
+ [ exact dir (* covariant *)
+ | exact (opposite_direction dir) (* contravariant *)
+ ]
+ ]
+qed.
+
+definition directed_relation_of_relation_class:
+ ∀dir:rewrite_direction.∀R: Relation_Class.
+ carrier_of_relation_class ? R → carrier_of_relation_class ? R → Prop.
+ intros;
+ cases r;
+ [ exact (relation_of_relation_class ? ? c c1)
+ | apply (relation_of_relation_class ? ? c1 c)
+ ]
+qed.
+
+definition directed_relation_of_argument_class:
+ ∀dir:rewrite_direction.∀R: Argument_Class.
+ carrier_of_relation_class ? R → carrier_of_relation_class ? R → Prop.
+ intros (dir R c c1);
+ rewrite < (about_carrier_of_relation_class_and_relation_class_of_argument_class R) in c c1;
+ exact (directed_relation_of_relation_class dir (relation_class_of_argument_class R) c c1).
+qed.
+
+
+definition relation_of_product_of_arguments:
+ ∀dir:rewrite_direction.∀In.
+ product_of_arguments In → product_of_arguments In → Prop.
+ intros 2;
+ elim In 0;
+ [ simplify;
+ intro;
+ exact (directed_relation_of_argument_class (get_rewrite_direction r t) t)
+ | intros;
+ change in p with (Prod (carrier_of_relation_class variance t) (product_of_arguments n));
+ change in p1 with (Prod (carrier_of_relation_class variance t) (product_of_arguments n));
+ cases p (c p2);
+ cases p1 (c1 p3);
+ apply And;
+ [ exact
+ (directed_relation_of_argument_class (get_rewrite_direction r t) t c c1)
+ | exact (R p2 p3)
+ ]
+ ]
+qed.
+
+definition apply_morphism:
+ ∀In,Out.∀m: function_type_of_morphism_signature In Out.
+ ∀args: product_of_arguments In. carrier_of_relation_class ? Out.
+ intro;
+ elim In;
+ [ exact (f p)
+ | change in p with (Prod (carrier_of_relation_class variance t) (product_of_arguments n));
+ elim p;
+ change in f1 with (carrier_of_relation_class variance t → function_type_of_morphism_signature n Out);
+ exact (f ? (f1 t1) t2)
+ ]
+qed.
+
+theorem apply_morphism_compatibility_Right2Left:
+ ∀In,Out.∀m1,m2: function_type_of_morphism_signature In Out.
+ ∀args1,args2: product_of_arguments In.
+ make_compatibility_goal_aux ? ? m1 m2 →
+ relation_of_product_of_arguments Right2Left ? args1 args2 →
+ directed_relation_of_relation_class Right2Left ?
+ (apply_morphism ? ? m2 args1)
+ (apply_morphism ? ? m1 args2).
+ intro;
+ elim In;
+ [ simplify in m1 m2 args1 args2 ⊢ %;
+ change in H1 with
+ (directed_relation_of_argument_class
+ (get_rewrite_direction Right2Left t) t args1 args2);
+ generalize in match H1; clear H1;
+ generalize in match H; clear H;
+ generalize in match args2; clear args2;
+ generalize in match args1; clear args1;
+ generalize in match m2; clear m2;
+ generalize in match m1; clear m1;
+ elim t 0; simplify;
+ [ intros (T1 r Hs Hr m1 m2 args1 args2 H H1);
+ apply H;
+ exact H1
+ | intros 8 (v T1 r Hr m1 m2 args1 args2);
+ cases v;
+ simplify;
+ intros (H H1);
+ apply (H ? ? H1);
+ | intros;
+ apply H1;
+ exact H2
+ | intros 7 (v);
+ cases v; simplify;
+ intros (H H1);
+ apply H;
+ exact H1
+ | intros;
+ simplify in H1;
+ rewrite > H1;
+ apply H;
+ exact H1
+ ]
+ | change in m1 with
+ (carrier_of_relation_class variance t →
+ function_type_of_morphism_signature n Out);
+ change in m2 with
+ (carrier_of_relation_class variance t →
+ function_type_of_morphism_signature n Out);
+ change in args1 with
+ ((carrier_of_relation_class ? t) × (product_of_arguments n));
+ change in args2 with
+ ((carrier_of_relation_class ? t) × (product_of_arguments n));
+ generalize in match H2; clear H2;
+ elim args2 0; clear args2;
+ elim args1; clear args1;
+ simplify in H2;
+ change in H2:(? ? %) with
+ (relation_of_product_of_arguments Right2Left n t2 t4);
+ elim H2; clear H2;
+ change with
+ (relation_of_relation_class unit Out (apply_morphism n Out (m1 t3) t4)
+ (apply_morphism n Out (m2 t1) t2));
+ generalize in match H3; clear H3;
+ generalize in match t3; clear t3;
+ generalize in match t1; clear t1;
+ generalize in match H1; clear H1;
+ generalize in match m2; clear m2;
+ generalize in match m1; clear m1;
+ elim t 0;
+ [ intros (T1 r Hs Hr m1 m2 H1 t1 t3 H3);
+ simplify in H3;
+ change in H1 with
+ (∀x1,x2:T1.r x1 x2 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2));
+ | intro v;
+ elim v 0;
+ [ intros (T1 r Hr m1 m2 H1 t1 t3 H3);
+ simplify in H3;
+ change in H1 with
+ (∀x1,x2:T1.r x1 x2 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2));
+ | intros (T1 r Hr m1 m2 H1 t1 t3 H3);
+ simplify in H3;
+ change in H1 with
+ (∀x1,x2:T1.r x2 x1 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2));
+ ]
+ | intros (T1 r Hs m1 m2 H1 t1 t3 H3);
+ simplify in H3;
+ change in H1 with
+ (∀x1,x2:T1.r x1 x2 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2));
+ | intro v;
+ elim v 0;
+ [ intros (T1 r m1 m2 H1 t1 t3 H3);
+ simplify in H3;
+ change in H1 with
+ (∀x1,x2:T1.r x1 x2 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2));
+ | intros (T1 r m1 m2 H1 t1 t3 H3);
+ simplify in H3;
+ change in H1 with
+ (∀x1,x2:T1.r x2 x1 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2));
+ ]
+ | intros (T m1 m2 H1 t1 t3 H3);
+ simplify in H3;
+ change in H1 with
+ (∀x:T. make_compatibility_goal_aux n Out (m1 x) (m2 x));
+ rewrite > H3;
+ simplify in H;
+ apply H;
+ [ apply H1
+ | assumption
+ ]
+ ] ;
+ simplify in H;
+ apply H;
+ [1,3,5,7,9,11:
+ apply H1;
+ assumption
+ |2,4,6,8,10,12:
+ assumption
+ ]
+ ]
+qed.
+
+theorem apply_morphism_compatibility_Left2Right:
+ ∀In,Out.∀m1,m2: function_type_of_morphism_signature In Out.
+ ∀args1,args2: product_of_arguments In.
+ make_compatibility_goal_aux ? ? m1 m2 →
+ relation_of_product_of_arguments Left2Right ? args1 args2 →
+ directed_relation_of_relation_class Left2Right ?
+ (apply_morphism ? ? m1 args1)
+ (apply_morphism ? ? m2 args2).
+ intro;
+ elim In 0; simplify; intros;
+ [ change in H1 with
+ (directed_relation_of_argument_class
+ (get_rewrite_direction Left2Right t) t args1 args2);
+ generalize in match H1; clear H1;
+ generalize in match H; clear H;
+ generalize in match args2; clear args2;
+ generalize in match args1; clear args1;
+ generalize in match m2; clear m2;
+ generalize in match m1; clear m1;
+ elim t 0; simplify;
+ [ intros (T1 r Hs Hr m1 m2 args1 args2 H H1);
+ apply H;
+ exact H1
+ | intros 8 (v T1 r Hr m1 m2 args1 args2);
+ cases v;
+ intros (H H1);
+ simplify in H1;
+ apply H;
+ exact H1
+ | intros;
+ apply H1;
+ exact H2
+ | intros 7 (v);
+ cases v;
+ intros (H H1);
+ simplify in H1;
+ apply H;
+ exact H1
+ | intros;
+ simplify in H1;
+ rewrite > H1;
+ apply H;
+ exact H1
+ ]
+ | change in m1 with
+ (carrier_of_relation_class variance t →
+ function_type_of_morphism_signature n Out);
+ change in m2 with
+ (carrier_of_relation_class variance t →
+ function_type_of_morphism_signature n Out);
+ change in args1 with
+ ((carrier_of_relation_class ? t) × (product_of_arguments n));
+ change in args2 with
+ ((carrier_of_relation_class ? t) × (product_of_arguments n));
+ generalize in match H2; clear H2;
+ elim args2 0; clear args2;
+ elim args1; clear args1;
+ simplify in H2; change in H2:(? ? %) with
+ (relation_of_product_of_arguments Left2Right n t2 t4);
+ elim H2; clear H2;
+ change with
+ (relation_of_relation_class unit Out (apply_morphism n Out (m1 t1) t2)
+ (apply_morphism n Out (m2 t3) t4));
+ generalize in match H3; clear H3;
+ generalize in match t3; clear t3;
+ generalize in match t1; clear t1;
+ generalize in match H1; clear H1;
+ generalize in match m2; clear m2;
+ generalize in match m1; clear m1;
+ elim t 0;
+ [ intros (T1 r Hs Hr m1 m2 H1 t1 t3 H3);
+ change in H1 with
+ (∀x1,x2:T1.r x1 x2 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2));
+ | intro v;
+ elim v 0;
+ [ intros (T1 r Hr m1 m2 H1 t1 t3 H3);
+ simplify in H3;
+ change in H1 with
+ (∀x1,x2:T1.r x1 x2 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2));
+ | intros (T1 r Hr m1 m2 H1 t1 t3 H3);
+ simplify in H3;
+ change in H1 with
+ (∀x1,x2:T1.r x2 x1 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2));
+ ]
+ | intros (T1 r Hs m1 m2 H1 t1 t3 H3);
+ simplify in H3;
+ change in H1 with
+ (∀x1,x2:T1.r x1 x2 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2));
+ | intro v;
+ elim v 0;
+ [ intros (T1 r m1 m2 H1 t1 t3 H3);
+ simplify in H3;
+ change in H1 with
+ (∀x1,x2:T1.r x1 x2 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2));
+ | intros (T1 r m1 m2 H1 t1 t3 H3);
+ simplify in H3;
+ change in H1 with
+ (∀x1,x2:T1.r x2 x1 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2));
+ ]
+ | intros (T m1 m2 H1 t1 t3 H3);
+ simplify in H3;
+ change in H1 with
+ (∀x:T. make_compatibility_goal_aux n Out (m1 x) (m2 x));
+ rewrite > H3;
+ simplify in H;
+ apply H;
+ [ apply H1
+ | assumption
+ ]
+ ] ;
+ simplify in H;
+ apply H;
+ [1,3,5,7,9,11:
+ apply H1;
+ assumption
+ |2,4,6,8,10,12:
+ assumption
+ ]
+ ]
+qed.
+
+definition interp :
+ ∀Hole,dir,Out,dir'. carrier_of_relation_class ? Hole →
+ Morphism_Context Hole dir Out dir' → carrier_of_relation_class ? Out.
+ intros (Hole dir Out dir' H t).
+ apply
+ (Morphism_Context_rect2 Hole dir (λS,xx,yy. carrier_of_relation_class ? S)
+ (λxx,L,fcl.product_of_arguments L));
+ intros;
+ [8: apply t
+ |7: skip
+ | exact (apply_morphism ? ? (Function ? ? m) p)
+ | exact H
+ | exact c
+ | exact x
+ | simplify;
+ rewrite <
+ (about_carrier_of_relation_class_and_relation_class_of_argument_class S);
+ exact c1
+ | simplify;split;
+ [ rewrite <
+ (about_carrier_of_relation_class_and_relation_class_of_argument_class S);
+ exact c1
+ | exact p
+ ]
+ ]
+qed.
+
+
+(*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 :
+ ∀Hole,dir,dir'.∀L: Arguments. carrier_of_relation_class ? Hole →
+ Morphism_Context_List Hole dir dir' L → product_of_arguments L.
+ intros (Hole dir dir' L H t);
+ apply
+ (Morphism_Context_List_rect2 Hole dir (λS,xx,yy.carrier_of_relation_class ? S)
+ (λxx,L,fcl.product_of_arguments L));
+ intros;
+ [8: apply t
+ |7: skip
+ | exact (apply_morphism ? ? (Function ? ? m) p)
+ | exact H
+ | exact c
+ | exact x
+ | simplify;
+ rewrite <
+ (about_carrier_of_relation_class_and_relation_class_of_argument_class S);
+ exact c1
+ | simplify; split;
+ [ rewrite <
+ (about_carrier_of_relation_class_and_relation_class_of_argument_class S);
+ exact c1
+ | exact p
+ ]
+ ]
+qed.
+
+(*
+Theorem setoid_rewrite:
+ ∀Hole dir Out dir' (E1 E2: carrier_of_relation_class Hole)
+ (E: Morphism_Context Hole dir Out dir').
+ (directed_relation_of_relation_class dir Hole E1 E2) →
+ (directed_relation_of_relation_class dir' Out (interp E1 E) (interp E2 E)).
+ intros.
+ elim E using
+ (@Morphism_Context_rect2 Hole dir
+ (fun S dir'' E => directed_relation_of_relation_class dir'' S (interp E1 E) (interp E2 E))
+ (fun dir'' L fcl =>
+ relation_of_product_of_arguments dir'' ?
+ (interp_relation_class_list E1 fcl)
+ (interp_relation_class_list E2 fcl))); intros.
+ change (directed_relation_of_relation_class dir'0 Out0
+ (apply_morphism ? ? (Function m) (interp_relation_class_list E1 m0))
+ (apply_morphism ? ? (Function m) (interp_relation_class_list E2 m0))).
+ destruct dir'0.
+ apply apply_morphism_compatibility_Left2Right.
+ exact (Compat m).
+ exact H0.
+ apply apply_morphism_compatibility_Right2Left.
+ exact (Compat m).
+ exact H0.
+
+ exact H.
+
+ unfold interp. Morphism_Context_rect2.
+ (*CSC: reflexivity used here*)
+ destruct S; destruct dir'0; simpl; (apply r || reflexivity).
+
+ destruct dir'0; exact r.
+
+ destruct S; unfold directed_relation_of_argument_class; simpl in H0 |- *;
+ unfold get_rewrite_direction; simpl.
+ destruct dir'0; destruct dir'';
+ (exact H0 ||
+ unfold directed_relation_of_argument_class; simpl; apply s; exact H0).
+ (* the following mess with generalize/clear/intros is to help Coq resolving *)
+ (* second order unification problems. *)
+ generalize m c H0; clear H0 m c; inversion c;
+ generalize m c; clear m c; rewrite <- H1; rewrite <- H2; intros;
+ (exact H3 || rewrite (opposite_direction_idempotent dir'0); apply H3).
+ destruct dir'0; destruct dir'';
+ (exact H0 ||
+ unfold directed_relation_of_argument_class; simpl; apply s; exact H0).
+(* the following mess with generalize/clear/intros is to help Coq resolving *)
+ (* second order unification problems. *)
+ generalize m c H0; clear H0 m c; inversion c;
+ generalize m c; clear m c; rewrite <- H1; rewrite <- H2; intros;
+ (exact H3 || rewrite (opposite_direction_idempotent dir'0); apply H3).
+ destruct dir'0; destruct dir''; (exact H0 || hnf; symmetry; exact H0).
+
+ change
+ (directed_relation_of_argument_class (get_rewrite_direction dir'' S) S
+ (eq_rect ? (fun T : Type => T) (interp E1 m) ?
+ (about_carrier_of_relation_class_and_relation_class_of_argument_class S))
+ (eq_rect ? (fun T : Type => T) (interp E2 m) ?
+ (about_carrier_of_relation_class_and_relation_class_of_argument_class S)) /\
+ relation_of_product_of_arguments dir'' ?
+ (interp_relation_class_list E1 m0) (interp_relation_class_list E2 m0)).
+ split.
+ clear m0 H1; destruct S; simpl in H0 |- *; unfold get_rewrite_direction; simpl.
+ destruct dir''; destruct dir'0; (exact H0 || hnf; apply s; exact H0).
+ inversion c.
+ rewrite <- H3; exact H0.
+ rewrite (opposite_direction_idempotent dir'0); exact H0.
+ destruct dir''; destruct dir'0; (exact H0 || hnf; apply s; exact H0).
+ inversion c.
+ rewrite <- H3; exact H0.
+ rewrite (opposite_direction_idempotent dir'0); exact H0.
+ destruct dir''; destruct dir'0; (exact H0 || hnf; symmetry; exact H0).
+ exact H1.
+Qed.
+
+(* A FEW EXAMPLES ON iff *)
+
+(* impl IS A MORPHISM *)
+
+Add Morphism impl with signature iff ==> iff ==> iff as Impl_Morphism.
+unfold impl; tautobatch.
+Qed.
+
+(* and IS A MORPHISM *)
+
+Add Morphism and with signature iff ==> iff ==> iff as And_Morphism.
+ tautobatch.
+Qed.
+
+(* or IS A MORPHISM *)
+
+Add Morphism or with signature iff ==> iff ==> iff as Or_Morphism.
+ tautobatch.
+Qed.
+
+(* not IS A MORPHISM *)
+
+Add Morphism not with signature iff ==> iff as Not_Morphism.
+ tautobatch.
+Qed.
+
+(* THE SAME EXAMPLES ON impl *)
+
+Add Morphism and with signature impl ++> impl ++> impl as And_Morphism2.
+ unfold impl; tautobatch.
+Qed.
+
+Add Morphism or with signature impl ++> impl ++> impl as Or_Morphism2.
+ unfold impl; tautobatch.
+Qed.
+
+Add Morphism not with signature impl -→ impl as Not_Morphism2.
+ unfold impl; tautobatch.
+Qed.
+
+*)
projects / helm.git / blobdiff