+++ /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 *)
-(* *)
-(**************************************************************************)
-
-set "baseuri" "cic:/matita/algebra/CoRN/Setoid".
-
-
-include "higher_order_defs/relations.ma".
-include "Z/plus.ma".
-
-include "datatypes/constructors.ma".
-include "nat/nat.ma".
-include "logic/equality.ma".
-(*include "Z/Zminus.ma".*)
-
-(* Setoids
-Definition of a constructive setoid with apartness,
-i.e. a set with an equivalence relation and an apartness relation compatible with it.
-*)
-
-(* Definition of Setoid
-Apartness, being the main relation, needs to be [CProp]-valued. Equality,
-as it is characterized by a negative statement, lives in [Prop].
-
-N.B. for the moment we use Prop for both (Matita group)
-*)
-
-record is_CSetoid (A : Type) (eq : relation A) (ap : relation A) : Prop \def
- {ax_ap_irreflexive : irreflexive A ap;
- ax_ap_symmetric : symmetric A ap;
- ax_ap_cotransitive : cotransitive A ap;
- ax_ap_tight : tight_apart A eq ap}.
-
-record CSetoid : Type \def
- {cs_crr :> Type;
- cs_eq : relation cs_crr;
- cs_ap : relation cs_crr;
- cs_proof : is_CSetoid cs_crr cs_eq cs_ap}.
-
-interpretation "setoid equality"
- 'eq x y = (cic:/matita/algebra/CoRN/Setoids/cs_eq.con _ x y).
-
-interpretation "setoid apart"
- 'neq x y = (cic:/matita/algebra/CoRN/Setoids/cs_ap.con _ x y).
-
-(* visto che sia "ap" che "eq" vanno in Prop e data la "tight-apartness",
-"cs_neq" e "ap" non sono la stessa cosa? *)
-definition cs_neq : \forall S : CSetoid. relation S \def
- \lambda S : CSetoid. \lambda x,y : S. \not x = y.
-
-lemma CSetoid_is_CSetoid :
- \forall S : CSetoid. is_CSetoid S (cs_eq S) (cs_ap S).
-intro. apply (cs_proof S).
-qed.
-
-lemma ap_irreflexive: \forall S : CSetoid. irreflexive S (cs_ap S).
-intro. elim (CSetoid_is_CSetoid S). assumption.
-qed.
-
-lemma ap_symmetric : \forall S : CSetoid. symmetric S(cs_ap S).
-intro. elim (CSetoid_is_CSetoid S). assumption.
-qed.
-
-lemma ap_cotransitive : \forall S : CSetoid. cotransitive S (cs_ap S).
-intro. elim (CSetoid_is_CSetoid S). assumption.
-qed.
-
-lemma ap_tight : \forall S : CSetoid. tight_apart S (cs_eq S) (cs_ap S).
-intro. elim (CSetoid_is_CSetoid S). assumption.
-qed.
-
-definition ex_unq : \forall S : CSetoid. (S \to Prop) \to Prop \def
- \lambda S : CSetoid. \lambda P : S \to Prop.
- ex2 S (\lambda x. \forall y : S. P y \to x = y) P.
-
-
-lemma eq_reflexive : \forall S : CSetoid. reflexive S (cs_eq S).
-intro. unfold. intro.
-generalize in match (ap_tight S x x).
-intro.
-generalize in match (ap_irreflexive S x);
-elim H. apply H1. assumption.
-qed.
-
-axiom eq_symmetric : \forall S : CSetoid. symmetric S (cs_eq S).
-(*
-lemma eq_symmetric : \forall S : CSetoid. symmetric S (cs_eq S).
-intro. unfold. intros.
-generalize in match (ap_tight S x y). intro.
-generalize in match (ap_tight S y x). intro.
-generalize in match (ap_symmetric S y x). intro.
-elim H1. clear H1.
-elim H2. clear H2.
-apply H1. unfold. intro. autobatch.
-qed.
-*)
-lemma eq_transitive : \forall S : CSetoid. transitive S (cs_eq S).
-intro. unfold. intros (x y z H H0).
-generalize in match (ap_tight S x y). intro.
-generalize in match (ap_tight S y z). intro.
-generalize in match (ap_tight S x z). intro.
-elim H3.
-apply H4. unfold. intro.
-generalize in match (ap_cotransitive ? ? ? H6 y). intro H7.
-elim H1 (H1' H1''). clear H1.
-elim H2 (H2' H2''). clear H2.
-elim H3 (H3' H3''). clear H3.
-elim H7 (H1). clear H7.
-generalize in match H1. apply H1''. assumption. (*non ho capito il secondo passo*)
-generalize in match H1. apply H2''. assumption.
-qed.
-
-lemma eq_reflexive_unfolded : \forall S:CSetoid. \forall x:S. x = x.
-apply eq_reflexive.
-qed.
-
-lemma eq_symmetric_unfolded : \forall S:CSetoid. \forall x,y:S. x = y \to y = x.
-apply eq_symmetric.
-qed.
-
-lemma eq_transitive_unfolded : \forall S:CSetoid. \forall x,y,z:S. x = y \to y = z \to x = z.
-apply eq_transitive.
-qed.
-
-
-lemma eq_wdl : \forall S:CSetoid. \forall x,y,z:S. x = y \to x = z \to z = y.
-intros.
-(* perche' autobatch non arriva in fondo ??? *)
-apply (eq_transitive_unfolded ? ? x).
-apply eq_symmetric_unfolded.
-exact H1.
-exact H.
-qed.
-
-
-lemma ap_irreflexive_unfolded : \forall S:CSetoid. \forall x:S. \not (x \neq x).
-apply ap_irreflexive.
-qed.
-
-lemma ap_cotransitive_unfolded : \forall S:CSetoid. \forall a,b:S. a \neq b \to
- \forall c:S. a \neq c \or c \neq b.
-apply ap_cotransitive.
-qed.
-
-lemma ap_symmetric_unfolded : \forall S:CSetoid. \forall x,y:S.
- x \neq y \to y \neq x.
-apply ap_symmetric.
-qed.
-
-lemma eq_imp_not_ap : \forall S:CSetoid. \forall x,y:S.
- x = y \to \not (x \neq y).
-intros.
-elim (ap_tight S x y).
-apply H2. assumption.
-qed.
-
-lemma not_ap_imp_eq : \forall S:CSetoid. \forall x,y:S.
- \not (x \neq y) \to x = y.
-intros.
-elim (ap_tight S x y).
-apply H1. assumption.
-qed.
-
-lemma neq_imp_notnot_ap : \forall S:CSetoid. \forall x,y:S.
- (cs_neq S x y) \to \not \not (x \neq y).
-intros. unfold. intro.
-apply H.
-apply not_ap_imp_eq.
-assumption.
-qed.
-
-lemma notnot_ap_imp_neq: \forall S:CSetoid. \forall x,y:S.
- (\not \not (x \neq y)) \to (cs_neq S x y).
-intros. unfold. unfold. intro.
-apply H.
-apply eq_imp_not_ap.
-assumption.
-qed.
-
-lemma ap_imp_neq : \forall S:CSetoid. \forall x,y:S.
- x \neq y \to (cs_neq S x y).
-intros. unfold. unfold. intro.
-apply (eq_imp_not_ap S ? ? H1).
-assumption.
-qed.
-
-lemma not_neq_imp_eq : \forall S:CSetoid. \forall x,y:S.
- \not (cs_neq S x y) \to x = y.
-intros.
-apply not_ap_imp_eq.
-unfold. intro.
-apply H.
-apply ap_imp_neq.
-assumption.
-qed.
-
-lemma eq_imp_not_neq : \forall S:CSetoid. \forall x,y:S.
- x = y \to \not (cs_neq S x y).
-intros. unfold. intro.
-apply H1.
-assumption.
-qed.
-
-
-
-(* -----------------The product of setoids----------------------- *)
-
-definition prod_ap: \forall A,B : CSetoid.\forall c,d: Prod A B. Prop \def
-\lambda A,B : CSetoid.\lambda c,d: Prod A B.
- ((cs_ap A (fst A B c) (fst A B d)) \or
- (cs_ap B (snd A B c) (snd A B d))).
-
-definition prod_eq: \forall A,B : CSetoid.\forall c,d: Prod A B. Prop \def
-\lambda A,B : CSetoid.\lambda c,d: Prod A B.
- ((cs_eq A (fst A B c) (fst A B d)) \and
- (cs_eq B (snd A B c) (snd A B d))).
-
-
-lemma prodcsetoid_is_CSetoid: \forall A,B: CSetoid.
- is_CSetoid (Prod A B) (prod_eq A B) (prod_ap A B).
-intros.
-apply (mk_is_CSetoid ? (prod_eq A B) (prod_ap A B))
- [unfold.
- intros.
- elim x.
- unfold.
- unfold prod_ap. simplify.
- intros.
- elim H
- [apply (ap_irreflexive A a H1)
- |apply (ap_irreflexive B b H1)
- ]
- |unfold.
- intros 2.
- elim x 2.
- elim y 2.
- unfold prod_ap. simplify.
- intro.
- elim H
- [left. apply ap_symmetric. assumption.
- |right. apply ap_symmetric. assumption
- ]
- |unfold.
- intros 2.
- elim x 2.
- elim y 4.
- elim z.
- unfold prod_ap in H. simplify in H.
- unfold prod_ap. simplify.
- elim H
- [cut ((a \neq a2) \or (a2 \neq a1))
- [elim Hcut
- [left. left. assumption
- |right. left. assumption
- ]
- |apply (ap_cotransitive A). assumption
- ]
- |cut ((b \neq b2) \or (b2 \neq b1))
- [elim Hcut
- [left. right. assumption
- |right. right. assumption
- ]
- |apply (ap_cotransitive B). assumption.
- ]
- ]
-|unfold.
- intros 2.
- elim x 2.
- elim y 2.
- unfold prod_ap. simplify.
- split
- [intro.
- left
- [apply not_ap_imp_eq.
- unfold. intro. apply H.
- left. assumption
- |apply not_ap_imp_eq.
- unfold. intro. apply H.
- right. assumption
- ]
- |intro. unfold. intro.
- elim H.
- elim H1
- [apply (eq_imp_not_ap A a a1 H2). assumption
- |apply (eq_imp_not_ap B b b1 H3). assumption
- ]
- ]
-]
-qed.
-
-
-definition ProdCSetoid : \forall A,B: CSetoid. CSetoid \def
- \lambda A,B: CSetoid.
- mk_CSetoid (Prod A B) (prod_eq A B) (prod_ap A B) (prodcsetoid_is_CSetoid A B).
-
-
-
-(* Relations and predicates
-Here we define the notions of well-definedness and strong extensionality
-on predicates and relations.
-*)
-
-
-
-(*-----------------------------------------------------------------------*)
-(*-------------------------- Predicates on Setoids ----------------------*)
-(*-----------------------------------------------------------------------*)
-
-(* throughout this section consider (S : CSetoid) and (P : S -> Prop) *)
-
-(* Definition pred_strong_ext : CProp := forall x y : S, P x -> P y or x [#] y. *)
-definition pred_strong_ext : \forall S: CSetoid. (S \to Prop) \to Prop \def
- \lambda S: CSetoid. \lambda P: S \to Prop. \forall x,y: S.
- P x \to (P y \or (x \neq y)).
-
-(* Definition pred_wd : CProp := forall x y : S, P x -> x [=] y -> P y. *)
-definition pred_wd : \forall S: CSetoid. \forall P :S \to Type. Type \def
- \lambda S: CSetoid. \lambda P: S \to Type. \forall x,y : S.
- P x \to x = y \to P y.
-
-record wd_pred (S: CSetoid) : Type \def
- {wdp_pred :> S \to Prop;
- wdp_well_def : pred_wd S wdp_pred}.
-
-record CSetoid_predicate (S: CSetoid) : Type \def
- {csp_pred :> S \to Prop;
- csp_strext : pred_strong_ext S csp_pred}.
-
-lemma csp_wd : \forall S: CSetoid. \forall P: CSetoid_predicate S.
- pred_wd S (csp_pred S P).
-intros.
-elim P.
-simplify.unfold pred_wd.
-intros.
-elim (H x y H1)
- [assumption|apply False_ind.apply (eq_imp_not_ap S x y H2 H3)]
-qed.
-
-
-(* Same result with CProp instead of Prop: but we just work with Prop (Matita group) *)
-(*
-Definition pred_strong_ext' : CProp := forall x y : S, P x -> P y or x [#] y.
-Definition pred_wd' : Prop := forall x y : S, P x -> x [=] y -> P y.
-
-Record CSetoid_predicate' : Type :=
- {csp'_pred :> S -> Prop;
- csp'_strext : pred_strong_ext' csp'_pred}.
-
-Lemma csp'_wd : forall P : CSetoid_predicate', pred_wd' P.
-intro P.
-intro x; intros y H H0.
-elim (csp'_strext P x y H).
-
-autobatch.
-
-intro H1.
-elimtype False.
-generalize H1.
-exact (eq_imp_not_ap _ _ _ H0).
-Qed.
-*)
-
-
-
-(*------------------------------------------------------------------------*)
-(* --------------------------- Relations on Setoids --------------------- *)
-(*------------------------------------------------------------------------*)
-(* throughout this section consider (S : CSetoid) and (R : S -> S -> Prop) *)
-
-
-(* Definition rel_wdr : Prop := forall x y z : S, R x y -> y [=] z -> R x z. *)
-(*
- primo tentativo ma R non e' ben tipato: si puo' fare il cast giusto (carrier di S)
- in modo da sfruttare "relation"?
- e' concettualmente sbagliato lavorare ad un livello piu' alto (Type) ? *)
-(*
-definition rel_wdr : \forall S: CSetoid. \forall x,y,z: S. \lambda R: relation S. Prop \def
- \lambda S: CSetoid. \lambda x,y,z: S. \lambda R: relation S.
- R S x y \to y = z \to R S x z.
-
-definition rel_wdr : \forall S: CSetoid. \forall x,y,z: (cs_crr S). \lambda R: relation (cs_crr S). Prop \def
- \lambda S: CSetoid. \lambda x,y,z: (cs_crr S). \lambda R: relation (cs_crr S).
- R (cs_crr S) x y \to y = z \to R (cs_crr S) x z.
-*)
-definition rel_wdr : \forall S: CSetoid. (S \to S \to Prop) \to Prop \def
- \lambda S: CSetoid. \lambda R: (S \to S \to Prop). \forall x,y,z: S.
- R x y \to y = z \to R x z.
-
-(*Definition rel_wdl : Prop := forall x y z : S, R x y -> x [=] z -> R z y.*)
-definition rel_wdl : \forall S: CSetoid. (S \to S \to Prop) \to Prop \def
- \lambda S: CSetoid. \lambda R: (S \to S \to Prop). \forall x,y,z: S.
- R x y \to x = z \to R z y.
-
-(* Definition rel_strext : CProp := forall x1 x2 y1 y2 : S, R x1 y1 -> (x1 [#] x2 or y1 [#] y2) or R x2 y2. *)
-definition rel_strext : \forall S: CSetoid. (S \to S \to Prop) \to Prop \def
- \lambda S: CSetoid. \lambda R: (S \to S \to Prop). \forall x1,x2,y1,y2: S.
- R x1 y1 \to (x1 \neq x2 \or y1 \neq y2) \or R x2 y2.
-
-
-(* Definition rel_strext_lft : CProp := forall x1 x2 y : S, R x1 y -> x1 [#] x2 or R x2 y. *)
-definition rel_strext_lft : \forall S: CSetoid. (S \to S \to Prop) \to Prop \def
- \lambda S: CSetoid. \lambda R: (S \to S \to Prop). \forall x1,x2,y: S.
- R x1 y \to (x1 \neq x2 \or R x2 y).
-
-(* Definition rel_strext_rht : CProp := forall x y1 y2 : S, R x y1 -> y1 [#] y2 or R x y2. *)
-definition rel_strext_rht : \forall S: CSetoid. (S \to S \to Prop) \to Prop \def
- \lambda S: CSetoid. \lambda R: (S \to S \to Prop). \forall x,y1,y2: S.
- R x y1 \to (y1 \neq y2 \or R x y2).
-
-
-lemma rel_strext_imp_lftarg : \forall S: CSetoid. \forall R: S \to S \to Prop.
- rel_strext S R \to rel_strext_lft S R.
-unfold rel_strext.
-unfold rel_strext_lft.
-intros.
-elim (H x1 x2 y y H1)
-[elim H2
- [left. assumption
- |absurd (y \neq y) [assumption | apply (ap_irreflexive S y)]
- ]
-|right. assumption
-]
-qed.
-
-
-lemma rel_strext_imp_rhtarg : \forall S: CSetoid. \forall R: S \to S \to Prop.
- rel_strext S R \to rel_strext_rht S R.
-unfold rel_strext.
-unfold rel_strext_rht.
-intros.
-elim (H x x y1 y2 H1)
-[elim H2
- [absurd (x \neq x) [assumption | apply (ap_irreflexive S x)]
- |left. assumption
- ]
-|right. assumption
-]
-qed.
-
-
-lemma rel_strextarg_imp_strext : \forall S: CSetoid. \forall R: S \to S \to Prop.
- (rel_strext_rht S R) \to (rel_strext_lft S R) \to (rel_strext S R).
-unfold rel_strext_rht.
-unfold rel_strext_lft.
-unfold rel_strext.
-intros.
-elim ((H x1 y1 y2) H2)
-[left. right. assumption
-|elim ((H1 x1 x2 y1) H2)
- [left. left. assumption
- |elim ((H x2 y1 y2) H4)
- [left. right. assumption
- |right. assumption.
- ]
- ]
-]
-qed.
-
-(* ---------- Definition of a setoid relation ----------------- *)
-(* The type of relations over a setoid. *)
-
-(* TODO
-record CSetoid_relation1 (S: CSetoid) : Type \def
- {csr_rel : S \to S \to Prop;
- csr_wdr : rel_wdr S csr_rel;
- csr_wdl : rel_wdl S csr_rel;
- csr_strext : rel_strext S csr_rel}.
-*)
-(* CORN
-Record CSetoid_relation : Type :=
- {csr_rel :> S -> S -> Prop;
- csr_wdr : rel_wdr csr_rel;
- csr_wdl : rel_wdl csr_rel;
- csr_strext : rel_strext csr_rel}.
-*)
-
-
-(* ---------- gli stessi risultati di prima ma in CProp ---------*)
-(*
-Variable R : S -> S -> CProp.
-Definition Crel_wdr : CProp := forall x y z : S, R x y -> y [=] z -> R x z.
-Definition Crel_wdl : CProp := forall x y z : S, R x y -> x [=] z -> R z y.
-Definition Crel_strext : CProp := forall x1 x2 y1 y2 : S,
- R x1 y1 -> R x2 y2 or x1 [#] x2 or y1 [#] y2.
-
-Definition Crel_strext_lft : CProp := forall x1 x2 y : S, R x1 y -> R x2 y or x1 [#] x2.
-Definition Crel_strext_rht : CProp := forall x y1 y2 : S, R x y1 -> R x y2 or y1 [#] y2.
-
-Lemma Crel_strext_imp_lftarg : Crel_strext -> Crel_strext_lft.
-Proof.
-unfold Crel_strext, Crel_strext_lft in |- *; intros H x1 x2 y H0.
-generalize (H x1 x2 y y).
-intro H1.
-elim (H1 H0).
-
-autobatch.
-
-intro H3.
-elim H3; intro H4.
-
-autobatch.
-elim (ap_irreflexive _ _ H4).
-Qed.
-
-Lemma Crel_strext_imp_rhtarg : Crel_strext -> Crel_strext_rht.
-unfold Crel_strext, Crel_strext_rht in |- *; intros H x y1 y2 H0.
-generalize (H x x y1 y2 H0); intro H1.
-elim H1; intro H2.
-
-autobatch.
-
-elim H2; intro H3.
-
-elim (ap_irreflexive _ _ H3).
-
-autobatch.
-Qed.
-
-Lemma Crel_strextarg_imp_strext :
- Crel_strext_rht -> Crel_strext_lft -> Crel_strext.
-unfold Crel_strext, Crel_strext_lft, Crel_strext_rht in |- *;
- intros H H0 x1 x2 y1 y2 H1.
-elim (H x1 y1 y2 H1); autobatch.
-intro H2.
-elim (H0 x1 x2 y2 H2); autobatch.
-Qed.
-*)
-
-
-
-
-(* ---- e questo ??????? -----*)
-
-(*Definition of a [CProp] setoid relation
-The type of relations over a setoid. *)
-(*
-Record CCSetoid_relation : Type :=
- {Ccsr_rel :> S -> S -> CProp;
- Ccsr_strext : Crel_strext Ccsr_rel}.
-
-Lemma Ccsr_wdr : forall R : CCSetoid_relation, Crel_wdr R.
-intro R.
-red in |- *; intros x y z H H0.
-elim (Ccsr_strext R x x y z H).
-
-autobatch.
-
-intro H1; elimtype False.
-elim H1; intro H2.
-
-exact (ap_irreflexive_unfolded _ _ H2).
-
-generalize H2.
-exact (eq_imp_not_ap _ _ _ H0).
-Qed.
-
-Lemma Ccsr_wdl : forall R : CCSetoid_relation, Crel_wdl R.
-intro R.
-red in |- *; intros x y z H H0.
-elim (Ccsr_strext R x z y y H).
-
-autobatch.
-
-intro H1; elimtype False.
-elim H1; intro H2.
-
-generalize H2.
-exact (eq_imp_not_ap _ _ _ H0).
-
-exact (ap_irreflexive_unfolded _ _ H2).
-Qed.
-
-Lemma ap_wdr : Crel_wdr (cs_ap (c:=S)).
-red in |- *; intros x y z H H0.
-generalize (eq_imp_not_ap _ _ _ H0); intro H1.
-elim (ap_cotransitive_unfolded _ _ _ H z); intro H2.
-
-assumption.
-
-elim H1.
-apply ap_symmetric_unfolded.
-assumption.
-Qed.
-
-Lemma ap_wdl : Crel_wdl (cs_ap (c:=S)).
-red in |- *; intros x y z H H0.
-generalize (ap_wdr y x z); intro H1.
-apply ap_symmetric_unfolded.
-apply H1.
-
-apply ap_symmetric_unfolded.
-assumption.
-
-assumption.
-Qed.
-
-Lemma ap_wdr_unfolded : forall x y z : S, x [#] y -> y [=] z -> x [#] z.
-Proof ap_wdr.
-
-Lemma ap_wdl_unfolded : forall x y z : S, x [#] y -> x [=] z -> z [#] y.
-Proof ap_wdl.
-
-Lemma ap_strext : Crel_strext (cs_ap (c:=S)).
-red in |- *; intros x1 x2 y1 y2 H.
-case (ap_cotransitive_unfolded _ _ _ H x2); intro H0.
-
-autobatch.
-
-case (ap_cotransitive_unfolded _ _ _ H0 y2); intro H1.
-
-autobatch.
-
-right; right.
-apply ap_symmetric_unfolded.
-assumption.
-Qed.
-
-Definition predS_well_def (P : S -> CProp) : CProp := forall x y : S,
- P x -> x [=] y -> P y.
-
-End CSetoid_relations_and_predicates.
-
-Declare Left Step ap_wdl_unfolded.
-Declare Right Step ap_wdr_unfolded.
-*)
-
-
-
-
-
-
-
-
-
-(*------------------------------------------------------------------------*)
-(* ------------------------- Functions between setoids ------------------ *)
-(*------------------------------------------------------------------------*)
-
-(* Such functions must preserve the setoid equality
-and be strongly extensional w.r.t. the apartness, i.e.
-if f(x,y) # f(x1,y1), then x # x1 or y # y1.
-For every arity this has to be defined separately. *)
-
-(* throughout this section consider (S1,S2,S3 : CSetoid) and (f : S1 \to S2) *)
-
-(* First we consider unary functions. *)
-
-(*
-In the following two definitions,
-f is a function from (the carrier of) S1 to (the carrier of) S2 *)
-
-(* Nota: senza le parentesi di (S1 \to S2) non funziona, perche'? *)
-definition fun_wd : \forall S1,S2 : CSetoid. (S1 \to S2) \to Prop \def
- \lambda S1,S2 : CSetoid.\lambda f : S1 \to S2. \forall x,y : S1.
- x = y \to f x = f y.
-
-definition fun_strext : \forall S1,S2 : CSetoid. (S1 \to S2) \to Prop \def
- \lambda S1,S2 : CSetoid.\lambda f : S1 \to S2. \forall x,y : S1.
- (f x \neq f y) \to (x \neq y).
-
-lemma fun_strext_imp_wd : \forall S1,S2 : CSetoid. \forall f : S1 \to S2.
- fun_strext S1 S2 f \to fun_wd S1 S2 f.
-unfold fun_strext.
-unfold fun_wd.
-intros.
-apply not_ap_imp_eq.
-unfold.intro.
-apply (eq_imp_not_ap ? ? ? H1).
-apply H.assumption.
-qed.
-
-(* funzioni tra setoidi *)
-record CSetoid_fun (S1,S2 : CSetoid) : Type \def
- {csf_fun : S1 \to S2;
- csf_strext : (fun_strext S1 S2 csf_fun)}.
-
-lemma csf_wd : \forall S1,S2 : CSetoid. \forall f : CSetoid_fun S1 S2. fun_wd S1 S2 (csf_fun S1 S2 f).
-intros.
-apply fun_strext_imp_wd.
-apply csf_strext.
-qed.
-
-definition Const_CSetoid_fun : \forall S1,S2: CSetoid. S2 \to CSetoid_fun S1 S2.
-intros. apply (mk_CSetoid_fun S1 S2 (\lambda x:S1.c)).
-unfold.intros.
-elim (ap_irreflexive ? ? H).
-qed.
-
-
-(* ---- Binary functions ------*)
-(* throughout this section consider (S1,S2,S3 : CSetoid) and (f : S1 \to S2 \to S3) *)
-
-definition bin_fun_wd : \forall S1,S2,S3 : CSetoid. (S1 \to S2 \to S3) \to Prop \def
- \lambda S1,S2,S3 : CSetoid. \lambda f : S1 \to S2 \to S3. \forall x1,x2: S1. \forall y1,y2: S2.
- x1 = x2 \to y1 = y2 \to f x1 y1 = f x2 y2.
-
-(*
-Definition bin_fun_strext : CProp := forall x1 x2 y1 y2,
- f x1 y1 [#] f x2 y2 -> x1 [#] x2 or y1 [#] y2.
-*)
-
-definition bin_fun_strext: \forall S1,S2,S3 : CSetoid. (S1 \to S2 \to S3) \to Prop \def
- \lambda S1,S2,S3 : CSetoid. \lambda f : S1 \to S2 \to S3. \forall x1,x2: S1. \forall y1,y2: S2.
- f x1 y1 \neq f x2 y2 \to x1 \neq x2 \lor y1 \neq y2.
-
-lemma bin_fun_strext_imp_wd : \forall S1,S2,S3: CSetoid.\forall f:S1 \to S2 \to S3.
-bin_fun_strext ? ? ? f \to bin_fun_wd ? ? ? f.
-intros.unfold in H.
-unfold.intros.
-apply not_ap_imp_eq.unfold.intro.
-elim (H x1 x2 y1 y2 H3).
-apply (eq_imp_not_ap ? ? ? H1 H4).
-apply (eq_imp_not_ap ? ? ? H2 H4).
-qed.
-
-
-
-record CSetoid_bin_fun (S1,S2,S3: CSetoid) : Type \def
- {csbf_fun :2> S1 \to S2 \to S3;
- csbf_strext : (bin_fun_strext S1 S2 S3 csbf_fun)}.
-
-lemma csbf_wd : \forall S1,S2,S3: CSetoid. \forall f : CSetoid_bin_fun S1 S2 S3.
- bin_fun_wd S1 S2 S3 (csbf_fun S1 S2 S3 f).
-intros.
-apply bin_fun_strext_imp_wd.
-apply csbf_strext.
-qed.
-
-lemma csf_wd_unfolded : \forall S1,S2: CSetoid. \forall f : CSetoid_fun S1 S2. \forall x,x' : S1.
- x = x' \to (csf_fun S1 S2 f) x = (csf_fun S1 S2 f) x'.
-intros.
-apply (csf_wd S1 S2 f x x').
-assumption.
-qed.
-
-lemma csf_strext_unfolded : \forall S1,S2: CSetoid. \forall f : CSetoid_fun S1 S2. \forall x,y : S1.
-(csf_fun S1 S2 f) x \neq (csf_fun S1 S2 f) y \to x \neq y.
-intros.
-apply (csf_strext S1 S2 f x y).
-assumption.
-qed.
-
-lemma csbf_wd_unfolded : \forall S1,S2,S3 : CSetoid. \forall f : CSetoid_bin_fun S1 S2 S3. \forall x,x':S1.
-\forall y,y' : S2. x = x' \to y = y' \to (csbf_fun S1 S2 S3 f) x y = (csbf_fun S1 S2 S3 f) x' y'.
-intros.
-apply (csbf_wd S1 S2 S3 f x x' y y'); assumption.
-qed.
-
-(* Hint Resolve csf_wd_unfolded csbf_wd_unfolded: algebra_c.*)
-
-(* The unary and binary (inner) operations on a csetoid
-An operation is a function with domain(s) and co-domain equal. *)
-
-(* Properties of binary operations *)
-
-definition commutes : \forall S: CSetoid. (S \to S \to S) \to Prop \def
- \lambda S: CSetoid. \lambda f : S \to S \to S.
- \forall x,y : S. f x y = f y x.
-
-definition CSassociative : \forall S: CSetoid. \forall f: S \to S \to S. Prop \def
-\lambda S: CSetoid. \lambda f : S \to S \to S.
-\forall x,y,z : S.
- f x (f y z) = f (f x y) z.
-
-definition un_op_wd : \forall S:CSetoid. (S \to S) \to Prop \def
-\lambda S: CSetoid. \lambda f: (S \to S). fun_wd S S f.
-
-
-definition un_op_strext: \forall S:CSetoid. (S \to S) \to Prop \def
-\lambda S:CSetoid. \lambda f: (S \to S). fun_strext S S f.
-
-
-definition CSetoid_un_op : CSetoid \to Type \def
-\lambda S:CSetoid. CSetoid_fun S S.
-
-definition mk_CSetoid_un_op : \forall S:CSetoid. \forall f: S \to S. fun_strext S S f \to CSetoid_fun S S
- \def
-\lambda S:CSetoid. \lambda f: S \to S. mk_CSetoid_fun S S f.
-
-lemma id_strext : \forall S:CSetoid. un_op_strext S (\lambda x:S. x).
-unfold un_op_strext.
-unfold fun_strext.
-intros.
-simplify in H.
-exact H.
-qed.
-
-lemma id_pres_eq : \forall S:CSetoid. un_op_wd S (\lambda x : S.x).
-unfold un_op_wd.
-unfold fun_wd.
-intros.
-simplify.
-exact H.
-qed.
-
-definition id_un_op : \forall S:CSetoid. CSetoid_un_op S
- \def \lambda S: CSetoid. mk_CSetoid_un_op S (\lambda x : cs_crr S.x) (id_strext S).
-
-definition un_op_fun: \forall S:CSetoid. CSetoid_un_op S \to CSetoid_fun S S
-\def \lambda S.\lambda f.f.
-
-coercion cic:/matita/algebra/CoRN/Setoids/un_op_fun.con.
-
-definition cs_un_op_strext : \forall S:CSetoid. \forall f: CSetoid_fun S S. fun_strext S S (csf_fun S S f) \def
-\lambda S:CSetoid. \lambda f : CSetoid_fun S S. csf_strext S S f.
-
-lemma un_op_wd_unfolded : \forall S:CSetoid. \forall op : CSetoid_un_op S.
-\forall x, y : S.
-x = y \to (csf_fun S S op) x = (csf_fun S S op) y.
-intros.
-apply (csf_wd S S ?).assumption.
-qed.
-
-lemma un_op_strext_unfolded : \forall S:CSetoid. \forall op : CSetoid_un_op S.
-\forall x, y : S.
- (csf_fun S S op) x \neq (csf_fun S S op) y \to x \neq y.
-exact cs_un_op_strext.
-qed.
-
-
-(* Well-defined binary operations on a setoid. *)
-
-definition bin_op_wd : \forall S:CSetoid. (S \to S \to S) \to Prop \def
-\lambda S:CSetoid. bin_fun_wd S S S.
-
-definition bin_op_strext : \forall S:CSetoid. (S \to S \to S) \to Prop \def
-\lambda S:CSetoid. bin_fun_strext S S S.
-
-definition CSetoid_bin_op : CSetoid \to Type \def
-\lambda S:CSetoid. CSetoid_bin_fun S S S.
-
-
-definition mk_CSetoid_bin_op : \forall S:CSetoid. \forall f: S \to S \to S.
-bin_fun_strext S S S f \to CSetoid_bin_fun S S S \def
- \lambda S:CSetoid. \lambda f: S \to S \to S.
- mk_CSetoid_bin_fun S S S f.
-
-(* da controllare che sia ben tipata
-definition cs_bin_op_wd : \forall S:CSetoid. ? \def
-\lambda S:CSetoid. csbf_wd S S S.
-*)
-definition cs_bin_op_wd : \forall S:CSetoid. \forall f: CSetoid_bin_fun S S S. bin_fun_wd S S S (csbf_fun S S S f) \def
-\lambda S:CSetoid. csbf_wd S S S.
-
-definition cs_bin_op_strext : \forall S:CSetoid. \forall f: CSetoid_bin_fun S S S. bin_fun_strext S S S (csbf_fun S S S f) \def
-\lambda S:CSetoid. csbf_strext S S S.
-
-
-
-(* Identity Coercion bin_op_bin_fun : CSetoid_bin_op >-> CSetoid_bin_fun. *)
-
-definition bin_op_bin_fun: \forall S:CSetoid. CSetoid_bin_op S \to CSetoid_bin_fun S S S
-\def \lambda S.\lambda f.f.
-
-coercion cic:/matita/algebra/CoRN/Setoids/bin_op_bin_fun.con.
-
-
-
-
-lemma bin_op_wd_unfolded :\forall S:CSetoid. \forall op : CSetoid_bin_op S. \forall x1, x2, y1, y2 : S.
- x1 = x2 \to y1 = y2 \to (csbf_fun S S S op) x1 y1 = (csbf_fun S S S op) x2 y2.
-exact cs_bin_op_wd.
-qed.
-
-lemma bin_op_strext_unfolded : \forall S:CSetoid. \forall op : CSetoid_bin_op S. \forall x1, x2, y1, y2 : S.
- (csbf_fun S S S op) x1 y1 \neq (csbf_fun S S S op) x2 y2 \to x1 \neq x2 \lor y1 \neq y2.
-exact cs_bin_op_strext.
-qed.
-
-lemma bin_op_is_wd_un_op_lft : \forall S:CSetoid. \forall op : CSetoid_bin_op S. \forall c : cs_crr S.
- un_op_wd S (\lambda x : cs_crr S. ((csbf_fun S S S op) x c)).
-intros. unfold. unfold.
-intros. apply bin_op_wd_unfolded [ assumption | apply eq_reflexive_unfolded ]
-qed.
-
-lemma bin_op_is_wd_un_op_rht : \forall S:CSetoid. \forall op : CSetoid_bin_op S. \forall c : cs_crr S.
- un_op_wd S (\lambda x : cs_crr S. ((csbf_fun S S S op) c x)).
-intros. unfold. unfold.
-intros. apply bin_op_wd_unfolded [ apply eq_reflexive_unfolded | assumption ]
-qed.
-
-
-lemma bin_op_is_strext_un_op_lft : \forall S:CSetoid. \forall op : CSetoid_bin_op S. \forall c : cs_crr S.
- un_op_strext S (\lambda x : cs_crr S. ((csbf_fun S S S op) x c)).
-intros. unfold un_op_strext. unfold fun_strext.
-intros.
-cut (x \neq y \lor c \neq c)
-[ elim Hcut
- [ assumption
- | generalize in match (ap_irreflexive_unfolded ? ? H1). intro. elim H2
- ]
-| apply (bin_op_strext_unfolded S op x y c c). assumption.
-]
-qed.
-
-lemma bin_op_is_strext_un_op_rht : \forall S:CSetoid. \forall op : CSetoid_bin_op S. \forall c : cs_crr S.
- un_op_strext S (\lambda x : cs_crr S. ((csbf_fun S S S op) c x)).
-intros. unfold un_op_strext. unfold fun_strext.
-intros.
-cut (c \neq c \lor x \neq y)
-[ elim Hcut
- [ generalize in match (ap_irreflexive_unfolded ? ? H1). intro. elim H2
- | assumption
- ]
-| apply (bin_op_strext_unfolded S op c c x y). assumption.
-]
-qed.
-
-definition bin_op2un_op_rht : \forall S:CSetoid. \forall op : CSetoid_bin_op S.
-\forall c : cs_crr S. CSetoid_un_op S \def
- \lambda S:CSetoid. \lambda op: CSetoid_bin_op S. \lambda c : cs_crr S.
- mk_CSetoid_un_op S (\lambda x:cs_crr S. ((csbf_fun S S S op) c x))
- (bin_op_is_strext_un_op_rht S op c).
-
-definition bin_op2un_op_lft : \forall S:CSetoid. \forall op : CSetoid_bin_op S.
-\forall c : cs_crr S. CSetoid_un_op S \def
- \lambda S:CSetoid. \lambda op: CSetoid_bin_op S. \lambda c : cs_crr S.
- mk_CSetoid_un_op S (\lambda x:cs_crr S. ((csbf_fun S S S op) x c))
- (bin_op_is_strext_un_op_lft S op c).
-
-(*
-Definition bin_op2un_op_rht (op : CSetoid_bin_op) (c : S) : CSetoid_un_op :=
- Build_CSetoid_un_op (fun x : S => op c x) (bin_op_is_strext_un_op_rht op c).
-
-
-Definition bin_op2un_op_lft (op : CSetoid_bin_op) (c : S) : CSetoid_un_op :=
- Build_CSetoid_un_op (fun x : S => op x c) (bin_op_is_strext_un_op_lft op c).
-*)
-
-
-(*
-Implicit Arguments commutes [S].
-Implicit Arguments associative [S].
-Hint Resolve bin_op_wd_unfolded un_op_wd_unfolded: algebra_c.
-*)
-
-(*The binary outer operations on a csetoid*)
-
-
-(*
-Well-defined outer operations on a setoid.
-*)
-definition outer_op_well_def : \forall S1,S2:CSetoid. (S1 \to S2 \to S2) \to Prop \def
-\lambda S1,S2:CSetoid. bin_fun_wd S1 S2 S2.
-
-definition outer_op_strext : \forall S1,S2:CSetoid. (S1 \to S2 \to S2) \to Prop \def
-\lambda S1,S2:CSetoid. bin_fun_strext S1 S2 S2.
-
-definition CSetoid_outer_op : \forall S1,S2:CSetoid.Type \def
-\lambda S1,S2:CSetoid.
- CSetoid_bin_fun S1 S2 S2.
-
-definition mk_CSetoid_outer_op : \forall S1,S2:CSetoid.
-\forall f : S1 \to S2 \to S2.
-bin_fun_strext S1 S2 S2 f \to CSetoid_bin_fun S1 S2 S2 \def
-\lambda S1,S2:CSetoid.
-mk_CSetoid_bin_fun S1 S2 S2.
-
-definition csoo_wd : \forall S1,S2:CSetoid. \forall f : CSetoid_bin_fun S1 S2 S2.
-bin_fun_wd S1 S2 S2 (csbf_fun S1 S2 S2 f) \def
-\lambda S1,S2:CSetoid.
-csbf_wd S1 S2 S2.
-
-definition csoo_strext : \forall S1,S2:CSetoid.
-\forall f : CSetoid_bin_fun S1 S2 S2.
-bin_fun_strext S1 S2 S2 (csbf_fun S1 S2 S2 f) \def
-\lambda S1,S2:CSetoid.
-csbf_strext S1 S2 S2.
-
-
-definition outer_op_bin_fun: \forall S:CSetoid.
-CSetoid_outer_op S S \to CSetoid_bin_fun S S S
-\def \lambda S.\lambda f.f.
-
-coercion cic:/matita/algebra/CoRN/Setoids/outer_op_bin_fun.con.
-(* begin hide
-Identity Coercion outer_op_bin_fun : CSetoid_outer_op >-> CSetoid_bin_fun.
-end hide *)
-
-lemma csoo_wd_unfolded :\forall S:CSetoid. \forall op : CSetoid_outer_op S S.
-\forall x1, x2, y1, y2 : S.
- x1 = x2 -> y1 = y2 -> (csbf_fun S S S op) x1 y1 = (csbf_fun S S S op) x2 y2.
-intros.
-apply csoo_wd[assumption|assumption]
-qed.
-
-(*
-Hint Resolve csoo_wd_unfolded: algebra_c.
-*)
-
-
-
-(*---------------------------------------------------------------*)
-(*--------------------------- Subsetoids ------------------------*)
-(*---------------------------------------------------------------*)
-
-(* Let S be a setoid, and P a predicate on the carrier of S *)
-(* Variable P : S -> CProp *)
-
-record subcsetoid_crr (S: CSetoid) (P: S \to Prop) : Type \def
- {scs_elem :> S;
- scs_prf : P scs_elem}.
-
-definition restrict_relation : \forall S:CSetoid. \forall R : S \to S \to Prop.
- \forall P: S \to Prop. relation (subcsetoid_crr S P) \def
- \lambda S:CSetoid. \lambda R : S \to S \to Prop.
- \lambda P: S \to Prop. \lambda a,b: subcsetoid_crr S P.
- match a with
- [ (mk_subcsetoid_crr x H) \Rightarrow
- match b with
- [ (mk_subcsetoid_crr y H) \Rightarrow R x y ]
- ].
-(* CPROP
-definition Crestrict_relation (R : Crelation S) : Crelation subcsetoid_crr :=
- fun a b : subcsetoid_crr =>
- match a, b with
- | Build_subcsetoid_crr x _, Build_subcsetoid_crr y _ => R x y
- end.
-*)
-
-definition subcsetoid_eq : \forall S:CSetoid. \forall P: S \to Prop.
- relation (subcsetoid_crr S P)\def
- \lambda S:CSetoid.
- restrict_relation S (cs_eq S).
-
-definition subcsetoid_ap : \forall S:CSetoid. \forall P: S \to Prop.
- relation (subcsetoid_crr S P)\def
- \lambda S:CSetoid.
- restrict_relation S (cs_ap S).
-
-(* N.B. da spostare in relations.ma... *)
-definition equiv : \forall A: Type. \forall R: relation A. Prop \def
- \lambda A: Type. \lambda R: relation A.
- (reflexive A R) \land (transitive A R) \land (symmetric A R).
-
-remark subcsetoid_equiv : \forall S:CSetoid. \forall P: S \to Prop.
-equiv ? (subcsetoid_eq S P).
-intros. unfold equiv. split
-[split
- [unfold. intro. elim x. simplify. apply (eq_reflexive S)
- |unfold. intros 3. elim y 2.
- elim x 2. elim z 2. simplify.
- exact (eq_transitive ? c1 c c2)
- ]
-| unfold. intros 2. elim x 2. elim y 2. simplify. exact (eq_symmetric ? c c1).
-]
-qed.
-
-(*
-axiom subcsetoid_is_CSetoid : \forall S:CSetoid. \forall P: S \to Prop.
-is_CSetoid ? (subcsetoid_eq S P) (subcsetoid_ap S P).
-*)
-
-lemma subcsetoid_is_CSetoid : \forall S:CSetoid. \forall P: S \to Prop.
-is_CSetoid ? (subcsetoid_eq S P) (subcsetoid_ap S P).
-intros.
-apply (mk_is_CSetoid ? (subcsetoid_eq S P) (subcsetoid_ap S P))
-[ unfold. intros.unfold. elim x. exact (ap_irreflexive_unfolded S ? ?)
- [ assumption | simplify in H1. exact H1 ]
- (* irreflexive *)
-|unfold. intros 2. elim x. generalize in match H1. elim y.simplify in H3. simplify.
-exact (ap_symmetric ? ? ? H3)
-(* cotransitive *)
-|unfold.intros 2. elim x.generalize in match H1. elim y. elim z.simplify. simplify in H3.
-apply (ap_cotransitive ? ? ? H3)
-(* tight *)
-|unfold.intros.elim x. elim y.simplify.
-apply (ap_tight S ? ?)]
-qed.
-
-
-definition mk_SubCSetoid : \forall S:CSetoid. \forall P: S \to Prop. CSetoid \def
-\lambda S:CSetoid. \lambda P:S \to Prop.
-mk_CSetoid (subcsetoid_crr S P) (subcsetoid_eq S P) (subcsetoid_ap S P) (subcsetoid_is_CSetoid S P).
-
-(* Subsetoid unary operations
-%\begin{convention}%
-Let [f] be a unary setoid operation on [S].
-%\end{convention}%
-*)
-
-(* Section SubCSetoid_unary_operations.
-Variable f : CSetoid_un_op S.
-*)
-
-definition un_op_pres_pred : \forall S:CSetoid. \forall P: S \to Prop.
- CSetoid_un_op S \to Prop \def
- \lambda S:CSetoid. \lambda P: S \to Prop. \lambda f: CSetoid_un_op S.
- \forall x : cs_crr S. P x \to P ((csf_fun S S f) x).
-
-definition restr_un_op : \forall S:CSetoid. \forall P: S \to Prop.
- \forall f: CSetoid_un_op S. \forall pr: un_op_pres_pred S P f.
- subcsetoid_crr S P \to subcsetoid_crr S P \def
- \lambda S:CSetoid. \lambda P: S \to Prop. \lambda f: CSetoid_un_op S.
- \lambda pr : un_op_pres_pred S P f.\lambda a: subcsetoid_crr S P.
- match a with
- [ (mk_subcsetoid_crr x p) \Rightarrow
- (mk_subcsetoid_crr ? ? ((csf_fun S S f) x) (pr x p))].
-
-(* TODO *)
-lemma restr_un_op_wd : \forall S:CSetoid. \forall P: S \to Prop.
-\forall f: CSetoid_un_op S. \forall pr: un_op_pres_pred S P f.
-un_op_wd (mk_SubCSetoid S P) (restr_un_op S P f pr).
-intros.
-unfold.unfold.intros 2.elim x 2.elim y 2.
-simplify.
-intro.
-normalize in H2.
-apply (un_op_wd_unfolded ? f ? ? H2).
-qed.
-
-lemma restr_un_op_strext : \forall S:CSetoid. \forall P: S \to Prop.
-\forall f: CSetoid_un_op S. \forall pr: un_op_pres_pred S P f.
-un_op_strext (mk_SubCSetoid S P) (restr_un_op S P f pr).
-intros.unfold.unfold. intros 2.elim y 2. elim x 2.
-intros.normalize in H2.
-apply (cs_un_op_strext ? f ? ? H2).
-qed.
-
-definition mk_SubCSetoid_un_op : \forall S:CSetoid. \forall P: S \to Prop. \forall f: CSetoid_un_op S.
- \forall pr:un_op_pres_pred S P f. CSetoid_un_op (mk_SubCSetoid S P).
- intros (S P f pr).
- apply (mk_CSetoid_un_op (mk_SubCSetoid S P) (restr_un_op S P f pr) (restr_un_op_strext S P f pr)).
- qed.
-
-(* BUG Universe Inconsistency detected
- definition mk_SubCSetoid_un_op : \forall S:CSetoid. \forall P: S \to Prop. \forall f: CSetoid_un_op S.
- \forall pr:un_op_pres_pred S P f. CSetoid_un_op (mk_SubCSetoid S P) \def
- \lambda S:CSetoid. \lambda P: S \to Prop. \lambda f: CSetoid_un_op S.
- \lambda pr:un_op_pres_pred S P f.
- mk_CSetoid_un_op (mk_SubCSetoid S P) (restr_un_op S P f pr) (restr_un_op_strext S P f pr).
-*)
-
-(* Subsetoid binary operations
-Let [f] be a binary setoid operation on [S].
-*)
-
-(* Section SubCSetoid_binary_operations.
-Variable f : CSetoid_bin_op S.
-*)
-
-definition bin_op_pres_pred : \forall S:CSetoid. \forall P: S \to Prop.
-(CSetoid_bin_op S) \to Prop \def
- \lambda S:CSetoid. \lambda P: S \to Prop. \lambda f: CSetoid_bin_op S.
- \forall x,y : S. P x \to P y \to P ( (csbf_fun S S S f) x y).
-
-(*
-Assume [bin_op_pres_pred].
-*)
-
-(* Variable pr : bin_op_pres_pred. *)
-
-definition restr_bin_op : \forall S:CSetoid. \forall P:S \to Prop.
- \forall f: CSetoid_bin_op S.\forall op : (bin_op_pres_pred S P f).
- \forall a, b : subcsetoid_crr S P. subcsetoid_crr S P \def
- \lambda S:CSetoid. \lambda P:S \to Prop.
- \lambda f: CSetoid_bin_op S. \lambda pr : (bin_op_pres_pred S P f).
- \lambda a, b : subcsetoid_crr S P.
- match a with
- [ (mk_subcsetoid_crr x p) \Rightarrow
- match b with
- [ (mk_subcsetoid_crr y q) \Rightarrow
- (mk_subcsetoid_crr ? ? ((csbf_fun S S S f) x y) (pr x y p q))]
- ].
-
-
-(* TODO *)
-lemma restr_bin_op_well_def : \forall S:CSetoid. \forall P: S \to Prop.
-\forall f: CSetoid_bin_op S. \forall pr: bin_op_pres_pred S P f.
-bin_op_wd (mk_SubCSetoid S P) (restr_bin_op S P f pr).
-intros.
-unfold.unfold.intros 2.elim x1 2. elim x2 2.intros 2. elim y1 2. elim y2 2.
-simplify.
-intros.
-normalize in H4.
-normalize in H5.
-apply (cs_bin_op_wd ? f ? ? ? ? H4 H5).
-qed.
-
-lemma restr_bin_op_strext : \forall S:CSetoid. \forall P: S \to Prop.
-\forall f: CSetoid_bin_op S. \forall pr: bin_op_pres_pred S P f.
-bin_op_strext (mk_SubCSetoid S P) (restr_bin_op S P f pr).
-intros.unfold.unfold. intros 2.elim x1 2. elim x2 2.intros 2. elim y1 2. elim y2 2.
-simplify.intros.
-normalize in H4.
-apply (cs_bin_op_strext ? f ? ? ? ? H4).
-qed.
-
-definition mk_SubCSetoid_bin_op : \forall S:CSetoid. \forall P: S \to Prop.
- \forall f: CSetoid_bin_op S. \forall pr: bin_op_pres_pred S P f.
- CSetoid_bin_op (mk_SubCSetoid S P).
- intros (S P f pr).
- apply (mk_CSetoid_bin_op (mk_SubCSetoid S P) (restr_bin_op S P f pr)(restr_bin_op_strext S P f pr)).
- qed.
-
-(* BUG Universe Inconsistency detected
-definition mk_SubCSetoid_bin_op : \forall S:CSetoid. \forall P: S \to Prop.
- \forall f: CSetoid_bin_op S. \forall pr: bin_op_pres_pred S P f.
- CSetoid_bin_op (mk_SubCSetoid S P) \def
- \lambda S:CSetoid. \lambda P: S \to Prop.
- \lambda f: CSetoid_bin_op S. \lambda pr: bin_op_pres_pred S P f.
- mk_CSetoid_bin_op (mk_SubCSetoid S P) (restr_bin_op S P f pr)(restr_bin_op_strext S P f pr).
-*)
-
-lemma restr_f_assoc : \forall S:CSetoid. \forall P: S \to Prop.
- \forall f: CSetoid_bin_op S. \forall pr: bin_op_pres_pred S P f.
- CSassociative S (csbf_fun S S S f)
- \to CSassociative (mk_SubCSetoid S P) (csbf_fun (mk_SubCSetoid S P) (mk_SubCSetoid S P) (mk_SubCSetoid S P) (mk_SubCSetoid_bin_op S P f pr)).
-intros 4.
-intro.
-unfold.
-intros 3.
-elim z 2.elim y 2. elim x 2.
-whd.
-apply H.
-qed.
-
-definition caseZ_diff: \forall A:Type.Z \to (nat \to nat \to A) \to A \def
-\lambda A:Type.\lambda z:Z.\lambda f:nat \to nat \to A.
- match z with
- [OZ \Rightarrow f O O
- |(pos n) \Rightarrow f (S n) O
- |(neg n) \Rightarrow f O (S n)].
-
-(* Zminus.ma *)
-theorem Zminus_S_S : \forall n,m:nat.
-Z_of_nat (S n) - S m = Z_of_nat n - m.
-intros.
-elim n.elim m.simplify. reflexivity.reflexivity.
-elim m.simplify.reflexivity.reflexivity.
-qed.
-
-
-
-lemma proper_caseZ_diff_CS : \forall CS : CSetoid. \forall f : nat \to nat \to CS.
- (\forall m,n,p,q : nat. eq nat (plus m q) (plus n p) \to (f m n) = (f p q)) \to
- \forall m,n : nat. caseZ_diff CS (Zminus (Z_of_nat m) (Z_of_nat n)) f = (f m n).
-intros.
-(* perche' apply nat_elim2 non funziona?? *)
-apply (nat_elim2 (\lambda m,n.caseZ_diff CS (Zminus (Z_of_nat m) (Z_of_nat n)) f = f m n)).
-intro.simplify.
-apply (nat_case n1).simplify.
-apply eq_reflexive.
-intro.simplify.apply eq_reflexive.
-intro.simplify.apply eq_reflexive.
-intros 2.
-rewrite > (Zminus_S_S n1 m1).
-intros.
-cut (f n1 m1 = f (S n1) (S m1)).
-apply eq_symmetric_unfolded.
-apply eq_transitive.
-apply f. apply n1. apply m1.
-apply eq_symmetric_unfolded.assumption.
-apply eq_symmetric_unfolded.assumption.
-apply H.
-autobatch.
-qed.
-
-(*
-Finally, we characterize functions defined on the natural numbers also as setoid functions, similarly to what we already did for predicates.
-*)
-
-
-definition nat_less_n_fun : \forall S:CSetoid. \forall n:nat. ? \def
- \lambda S:CSetoid. \lambda n:nat. \lambda f: \forall i:nat. i < n \to S.
- \forall i,j : nat. eq nat i j \to (\forall H : i < n.
- \forall H' : j < n . (f i H) = (f j H')).
-
-definition nat_less_n_fun' : \forall S:CSetoid. \forall n:nat. ? \def
- \lambda S:CSetoid. \lambda n:nat. \lambda f: \forall i: nat. i <= n \to S.
- \forall i,j : nat. eq nat i j \to \forall H : i <= n.
- \forall H' : j <= n. f i H = f j H'.
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