--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* $Id: CSetoidFun.v,v 1.12 2004/09/22 11:06:10 loeb Exp $ *)
+
+set "baseuri" "cic:/matita/algebra/CoRN/SetoidFun".
+include "algebra/CoRN/Setoids.ma".
+
+
+definition ap_fun : \forall A,B : CSetoid. \forall f,g : CSetoid_fun A B. Prop \def
+\lambda A,B : CSetoid. \lambda f,g : CSetoid_fun A B.
+ \exists a:A. (csf_fun A B f) a \neq (csf_fun A B g) a.
+(* Definition ap_fun (A B : CSetoid) (f g : CSetoid_fun A B) :=
+ {a : A | f a[#]g a}. *)
+
+definition eq_fun : \forall A,B : CSetoid. \forall f,g : CSetoid_fun A B. Prop \def
+\lambda A,B : CSetoid. \lambda f,g : CSetoid_fun A B.
+ \forall a:A. (csf_fun A B f) a = (csf_fun A B g) a.
+(* Definition eq_fun (A B : CSetoid) (f g : CSetoid_fun A B) :=
+ forall a : A, f a[=]g a. *)
+
+lemma irrefl_apfun : \forall A,B : CSetoid.
+ irreflexive (CSetoid_fun A B) (ap_fun A B).
+intros.
+unfold irreflexive. intro f.
+unfold ap_fun.
+unfold.
+intro.
+elim H.
+cut (csf_fun A B f a = csf_fun A B f a)
+[
+ apply eq_imp_not_ap.
+ apply A.
+ assumption.
+ assumption.
+ auto.
+ auto
+|
+ apply eq_reflexive_unfolded
+]
+qed.
+
+(*
+Lemma irrefl_apfun : forall A B : CSetoid, irreflexive (ap_fun A B).
+intros A B.
+unfold irreflexive in |- *.
+intros f.
+unfold ap_fun in |- *.
+red in |- *.
+intro H.
+elim H.
+intros a H0.
+set (H1 := ap_irreflexive_unfolded B (f a)) in *.
+intuition.
+Qed.
+*)
+
+
+(*
+lemma cotrans_apfun : \forall A,B : CSetoid. cotransitive (CSetoid_fun A B) (ap_fun A B).
+intros (A B).
+unfold.
+unfold ap_fun.
+intros (f g H h).
+decompose.
+apply ap_cotransitive.
+apply (ap_imp_neq B (csf_fun A B x a) (csf_fun A B y a) H1).
+(* letin foo \def (\exist a:A.csf_fun A B x a\neq csf_fun A B z a). *)
+
+
+apply (ap_cotransitive B ? ? H1 ?).
+
+split.
+left.
+elim H.
+clear H.
+intros a H.
+set (H1 := ap_cotransitive B (f a) (g a) H (h a)) in *.
+elim H1.
+clear H1.
+intros H1.
+left.
+exists a.
+exact H1.
+
+clear H1.
+intro H1.
+right.
+exists a.
+exact H1.
+Qed.
+*)
+
+(*
+Lemma cotrans_apfun : forall A B : CSetoid, cotransitive (ap_fun A B).
+intros A B.
+unfold cotransitive in |- *.
+unfold ap_fun in |- *.
+intros f g H h.
+elim H.
+clear H.
+intros a H.
+set (H1 := ap_cotransitive B (f a) (g a) H (h a)) in *.
+elim H1.
+clear H1.
+intros H1.
+left.
+exists a.
+exact H1.
+
+clear H1.
+intro H1.
+right.
+exists a.
+exact H1.
+Qed.
+*)
+
+lemma ta_apfun : \forall A, B : CSetoid. tight_apart (CSetoid_fun A B) (eq_fun A B) (ap_fun A B).
+unfold tight_apart.
+intros (A B f g).
+unfold ap_fun.
+unfold eq_fun.
+split
+[ intros. elim H. unfold in H.
+generalize in match H. reduce. simplify .unfold.
+
+ |
+]
+intros.
+red in H.
+apply not_ap_imp_eq.
+red in |- *.
+intros H0.
+apply H.
+exists a.
+exact H0.
+intros H.
+red in |- *.
+
+intro H1.
+elim H1.
+intros a X.
+set (H2 := eq_imp_not_ap B (f a) (g a) (H a) X) in *.
+exact H2.
+Qed.
+
+(*
+Lemma ta_apfun : forall A B : CSetoid, tight_apart (eq_fun A B) (ap_fun A B).
+unfold tight_apart in |- *.
+unfold ap_fun in |- *.
+unfold eq_fun in |- *.
+intros A B f g.
+split.
+intros H a.
+red in H.
+apply not_ap_imp_eq.
+red in |- *.
+intros H0.
+apply H.
+exists a.
+exact H0.
+intros H.
+red in |- *.
+
+intro H1.
+elim H1.
+intros a X.
+set (H2 := eq_imp_not_ap B (f a) (g a) (H a) X) in *.
+exact H2.
+Qed.
+*)
+
+Lemma sym_apfun : forall A B : CSetoid, Csymmetric (ap_fun A B).
+unfold Csymmetric in |- *.
+unfold ap_fun in |- *.
+intros A B f g H.
+elim H.
+clear H.
+intros a H.
+exists a.
+apply ap_symmetric_unfolded.
+exact H.
+Qed.
+
+Definition FS_is_CSetoid (A B : CSetoid) :=
+ Build_is_CSetoid (CSetoid_fun A B) (eq_fun A B) (ap_fun A B)
+ (irrefl_apfun A B) (sym_apfun A B) (cotrans_apfun A B) (ta_apfun A B).
+
+Definition FS_as_CSetoid (A B : CSetoid) :=
+ Build_CSetoid (CSetoid_fun A B) (eq_fun A B) (ap_fun A B)
+ (FS_is_CSetoid A B).
+
+(** **Nullary and n-ary operations
+*)
+
+Definition is_nullary_operation (S:CSetoid) (s:S):Prop := True.
+
+Fixpoint n_ary_operation (n:nat)(V:CSetoid){struct n}:CSetoid:=
+match n with
+|0 => V
+|(S m)=> (FS_as_CSetoid V (n_ary_operation m V))
+end.
+
+Section unary_function_composition.
+
+(** ** Composition of Setoid functions
+
+Let [S1], [S2] and [S3] be setoids, [f] a
+setoid function from [S1] to [S2], and [g] from [S2]
+to [S3] in the following definition of composition. *)
+
+Variables S1 S2 S3 : CSetoid.
+Variable f : CSetoid_fun S1 S2.
+Variable g : CSetoid_fun S2 S3.
+
+Definition compose_CSetoid_fun : CSetoid_fun S1 S3.
+apply (Build_CSetoid_fun _ _ (fun x : S1 => g (f x))).
+(* str_ext *)
+unfold fun_strext in |- *; intros x y H.
+apply (csf_strext _ _ f). apply (csf_strext _ _ g). assumption.
+Defined.
+
+End unary_function_composition.
+
+(** ***Composition as operation
+*)
+Definition comp (A B C : CSetoid) :
+ FS_as_CSetoid A B -> FS_as_CSetoid B C -> FS_as_CSetoid A C.
+intros A B C f g.
+set (H := compose_CSetoid_fun A B C f g) in *.
+exact H.
+Defined.
+
+Definition comp_as_bin_op (A:CSetoid) : CSetoid_bin_op (FS_as_CSetoid A A).
+intro A.
+unfold CSetoid_bin_op in |- *.
+eapply Build_CSetoid_bin_fun with (comp A A A).
+unfold bin_fun_strext in |- *.
+unfold comp in |- *.
+intros f1 f2 g1 g2.
+simpl in |- *.
+unfold ap_fun in |- *.
+unfold compose_CSetoid_fun in |- *.
+simpl in |- *.
+elim f1.
+unfold fun_strext in |- *.
+clear f1.
+intros f1 Hf1.
+elim f2.
+unfold fun_strext in |- *.
+clear f2.
+intros f2 Hf2.
+elim g1.
+unfold fun_strext in |- *.
+clear g1.
+intros g1 Hg1.
+elim g2.
+unfold fun_strext in |- *.
+clear g2.
+intros g2 Hg2.
+simpl in |- *.
+intro H.
+elim H.
+clear H.
+intros a H.
+set (H0 := ap_cotransitive A (g1 (f1 a)) (g2 (f2 a)) H (g2 (f1 a))) in *.
+elim H0.
+clear H0.
+intro H0.
+right.
+exists (f1 a).
+exact H0.
+
+clear H0.
+intro H0.
+left.
+exists a.
+apply Hg2.
+exact H0.
+Defined.
+
+Lemma assoc_comp : forall A : CSetoid, associative (comp_as_bin_op A).
+unfold associative in |- *.
+unfold comp_as_bin_op in |- *.
+intros A f g h.
+simpl in |- *.
+unfold eq_fun in |- *.
+simpl in |- *.
+intuition.
+Qed.
+
+Section unary_and_binary_function_composition.
+
+Definition compose_CSetoid_bin_un_fun (A B C : CSetoid)
+ (f : CSetoid_bin_fun B B C) (g : CSetoid_fun A B) : CSetoid_bin_fun A A C.
+intros A B C f g.
+apply (Build_CSetoid_bin_fun A A C (fun a0 a1 : A => f (g a0) (g a1))).
+intros x1 x2 y1 y2 H0.
+assert (H10:= csbf_strext B B C f).
+red in H10.
+assert (H40 := csf_strext A B g).
+red in H40.
+elim (H10 (g x1) (g x2) (g y1) (g y2) H0); [left | right]; auto.
+Defined.
+
+Definition compose_CSetoid_bin_fun A B C (f g : CSetoid_fun A B)
+ (h : CSetoid_bin_fun B B C) : CSetoid_fun A C.
+intros A B C f g h.
+apply (Build_CSetoid_fun A C (fun a : A => h (f a) (g a))).
+intros x y H.
+elim (csbf_strext _ _ _ _ _ _ _ _ H); apply csf_strext.
+Defined.
+
+Definition compose_CSetoid_un_bin_fun A B C (f : CSetoid_bin_fun B B C)
+ (g : CSetoid_fun C A) : CSetoid_bin_fun B B A.
+intros A0 B0 C f g.
+apply Build_CSetoid_bin_fun with (fun x y : B0 => g (f x y)).
+intros x1 x2 y1 y2.
+case f.
+simpl in |- *.
+unfold bin_fun_strext in |- *.
+case g.
+simpl in |- *.
+unfold fun_strext in |- *.
+intros gu gstrext fu fstrext H.
+apply fstrext.
+apply gstrext.
+exact H.
+Defined.
+
+End unary_and_binary_function_composition.
+
+(** ***Projections
+*)
+
+Section function_projection.
+
+Lemma proj_bin_fun : forall A B C (f : CSetoid_bin_fun A B C) a, fun_strext (f a).
+intros A B C f a.
+red in |- *.
+elim f.
+intro fo.
+simpl.
+intros csbf_strext0 x y H.
+elim (csbf_strext0 _ _ _ _ H); intro H0.
+ elim (ap_irreflexive_unfolded _ _ H0).
+exact H0.
+Qed.
+
+Definition projected_bin_fun A B C (f : CSetoid_bin_fun A B C) (a : A) :=
+ Build_CSetoid_fun B C (f a) (proj_bin_fun A B C f a).
+
+End function_projection.
+
+Section BinProj.
+
+Variable S : CSetoid.
+
+Definition binproj1 (x y:S) := x.
+
+Lemma binproj1_strext : bin_fun_strext _ _ _ binproj1.
+red in |- *; auto.
+Qed.
+
+Definition cs_binproj1 : CSetoid_bin_op S.
+red in |- *; apply Build_CSetoid_bin_op with binproj1.
+apply binproj1_strext.
+Defined.
+
+End BinProj.
+
+(** **Combining operations
+%\begin{convention}% Let [S1], [S2] and [S3] be setoids.
+%\end{convention}%
+*)
+
+Section CombiningOperations.
+Variables S1 S2 S3 : CSetoid.
+
+(**
+In the following definition, we assume [f] is a setoid function from
+[S1] to [S2], and [op] is an unary operation on [S2].
+Then [opOnFun] is the composition [op] after [f].
+*)
+Section CombiningUnaryOperations.
+Variable f : CSetoid_fun S1 S2.
+Variable op : CSetoid_un_op S2.
+
+Definition opOnFun : CSetoid_fun S1 S2.
+apply (Build_CSetoid_fun S1 S2 (fun x : S1 => op (f x))).
+(* str_ext *)
+unfold fun_strext in |- *; intros x y H.
+apply (csf_strext _ _ f x y).
+apply (csf_strext _ _ op _ _ H).
+Defined.
+
+End CombiningUnaryOperations.
+
+End CombiningOperations.
+
+Section p66E2b4.
+
+(** **The Free Setoid
+%\begin{convention}% Let [A:CSetoid].
+%\end{convention}%
+*)
+Variable A:CSetoid.
+
+Definition Astar := (list A).
+
+Definition empty_word := (nil A).
+
+Definition appA:= (app A).
+
+Fixpoint eq_fm (m:Astar)(k:Astar){struct m}:Prop:=
+match m with
+|nil => match k with
+ |nil => True
+ |cons a l => False
+ end
+|cons b n => match k with
+ |nil => False
+ |cons a l => b[=]a /\ (eq_fm n l)
+ end
+end.
+
+Fixpoint ap_fm (m:Astar)(k:Astar){struct m}: CProp :=
+match m with
+|nil => match k with
+ |nil => CFalse
+ |cons a l => CTrue
+ end
+|cons b n => match k with
+ |nil => CTrue
+ |cons a l => b[#]a or (ap_fm n l)
+ end
+end.
+
+Lemma ap_fm_irreflexive: (irreflexive ap_fm).
+unfold irreflexive.
+intro x.
+induction x.
+simpl.
+red.
+intuition.
+
+simpl.
+red.
+intro H.
+apply IHx.
+elim H.
+clear H.
+generalize (ap_irreflexive A a).
+unfold Not.
+intuition.
+
+intuition.
+Qed.
+
+
+Lemma ap_fm_symmetric: Csymmetric ap_fm.
+unfold Csymmetric.
+intros x.
+induction x.
+intro y.
+case y.
+simpl.
+intuition.
+
+simpl.
+intuition.
+simpl.
+intro y.
+case y.
+simpl.
+intuition.
+
+simpl.
+intros c l H0.
+elim H0.
+generalize (ap_symmetric A a c).
+intuition.
+clear H0.
+intro H0.
+right.
+apply IHx.
+exact H0.
+Qed.
+
+Lemma ap_fm_cotransitive : (cotransitive ap_fm).
+unfold cotransitive.
+intro x.
+induction x.
+simpl.
+intro y.
+case y.
+intuition.
+
+intros c l H z.
+case z.
+simpl.
+intuition.
+
+intuition.
+
+simpl.
+intro y.
+case y.
+intros H z.
+case z.
+intuition.
+
+simpl.
+intuition.
+
+intros c l H z.
+case z.
+intuition.
+
+simpl.
+intros c0 l0.
+elim H.
+clear H.
+intro H.
+generalize (ap_cotransitive A a c H c0).
+intuition.
+
+clear H.
+intro H.
+generalize (IHx l H l0).
+intuition.
+Qed.
+
+Lemma ap_fm_tight : (tight_apart eq_fm ap_fm).
+unfold tight_apart.
+intros x.
+induction x.
+simpl.
+intro y.
+case y.
+red.
+unfold Not.
+intuition.
+
+intuition.
+
+intro y.
+simpl.
+case y.
+intuition.
+
+intros c l.
+generalize (IHx l).
+red.
+intro H0.
+elim H0.
+intros H1 H2.
+split.
+intro H3.
+split.
+red in H3.
+generalize (ap_tight A a c).
+intuition.
+
+apply H1.
+intro H4.
+apply H3.
+right.
+exact H4.
+
+intro H3.
+elim H3.
+clear H3.
+intros H3 H4.
+intro H5.
+elim H5.
+generalize (ap_tight A a c).
+intuition.
+
+apply H2.
+exact H4.
+Qed.
+
+Definition free_csetoid_is_CSetoid:(is_CSetoid Astar eq_fm ap_fm):=
+ (Build_is_CSetoid Astar eq_fm ap_fm ap_fm_irreflexive ap_fm_symmetric
+ ap_fm_cotransitive ap_fm_tight).
+
+Definition free_csetoid_as_csetoid:CSetoid:=
+(Build_CSetoid Astar eq_fm ap_fm free_csetoid_is_CSetoid).
+
+Lemma app_strext:
+ (bin_fun_strext free_csetoid_as_csetoid free_csetoid_as_csetoid
+ free_csetoid_as_csetoid appA).
+unfold bin_fun_strext.
+intros x1.
+induction x1.
+simpl.
+intro x2.
+case x2.
+simpl.
+intuition.
+
+intuition.
+
+intros x2 y1 y2.
+simpl.
+case x2.
+case y2.
+simpl.
+intuition.
+
+simpl.
+intuition.
+
+case y2.
+simpl.
+simpl in IHx1.
+intros c l H.
+elim H.
+intuition.
+
+clear H.
+generalize (IHx1 l y1 (nil A)).
+intuition.
+
+simpl.
+intros c l c0 l0.
+intro H.
+elim H.
+intuition.
+
+generalize (IHx1 l0 y1 (cons c l)).
+intuition.
+Qed.
+
+Definition app_as_csb_fun:
+(CSetoid_bin_fun free_csetoid_as_csetoid free_csetoid_as_csetoid
+ free_csetoid_as_csetoid):=
+ (Build_CSetoid_bin_fun free_csetoid_as_csetoid free_csetoid_as_csetoid
+ free_csetoid_as_csetoid appA app_strext).
+
+Lemma eq_fm_reflexive: forall (x:Astar), (eq_fm x x).
+intro x.
+induction x.
+simpl.
+intuition.
+
+simpl.
+intuition.
+Qed.
+
+End p66E2b4.
+
+(** **Partial Functions
+
+In this section we define a concept of partial function for an
+arbitrary setoid. Essentially, a partial function is what would be
+expected---a predicate on the setoid in question and a total function
+from the set of points satisfying that predicate to the setoid. There
+is one important limitations to this approach: first, the record we
+obtain has type [Type], meaning that we can't use, for instance,
+elimination of existential quantifiers.
+
+Furthermore, for reasons we will explain ahead, partial functions will
+not be defined via the [CSetoid_fun] record, but the whole structure
+will be incorporated in a new record.
+
+Finally, notice that to be completely general the domains of the
+functions have to be characterized by a [CProp]-valued predicate;
+otherwise, the use you can make of a function will be %\emph{%#<i>#a
+priori#</i>#%}% restricted at the moment it is defined.
+
+Before we state our definitions we need to do some work on domains.
+*)
+
+Section SubSets_of_G.
+
+(** ***Subsets of Setoids
+
+Subsets of a setoid will be identified with predicates from the
+carrier set of the setoid into [CProp]. At this stage, we do not make
+any assumptions about these predicates.
+
+We will begin by defining elementary operations on predicates, along
+with their basic properties. In particular, we will work with well
+defined predicates, so we will prove that these operations preserve
+welldefinedness.
+
+%\begin{convention}% Let [S:CSetoid] and [P,Q:S->CProp].
+%\end{convention}%
+*)
+
+Variable S : CSetoid.
+
+Section Conjunction.
+
+Variables P Q : S -> CProp.
+
+Definition conjP (x : S) : CProp := P x and Q x.
+
+Lemma prj1 : forall x : S, conjP x -> P x.
+intros x H; inversion_clear H; assumption.
+Qed.
+
+Lemma prj2 : forall x : S, conjP x -> Q x.
+intros x H; inversion_clear H; assumption.
+Qed.
+
+Lemma conj_wd : pred_wd _ P -> pred_wd _ Q -> pred_wd _ conjP.
+intros H H0.
+red in |- *; intros x y H1 H2.
+inversion_clear H1; split.
+
+apply H with x; assumption.
+
+apply H0 with x; assumption.
+Qed.
+
+End Conjunction.
+
+Section Disjunction.
+
+Variables P Q : S -> CProp.
+
+(**
+Although at this stage we never use it, for completeness's sake we also treat disjunction (corresponding to union of subsets).
+*)
+
+Definition disj (x : S) : CProp := P x or Q x.
+
+Lemma inj1 : forall x : S, P x -> disj x.
+intros; left; assumption.
+Qed.
+
+Lemma inj2 : forall x : S, Q x -> disj x.
+intros; right; assumption.
+Qed.
+
+Lemma disj_wd : pred_wd _ P -> pred_wd _ Q -> pred_wd _ disj.
+intros H H0.
+red in |- *; intros x y H1 H2.
+inversion_clear H1.
+
+left; apply H with x; assumption.
+
+right; apply H0 with x; assumption.
+Qed.
+
+End Disjunction.
+
+Section Extension.
+
+(**
+The next definition is a bit tricky, and is useful for choosing among the elements that satisfy a predicate [P] those that also satisfy [R] in the case where [R] is only defined for elements satisfying [P]---consider [R] to be a condition on the image of an object via a function with domain [P]. We chose to call this operation [extension].
+*)
+
+Variable P : S -> CProp.
+Variable R : forall x : S, P x -> CProp.
+
+Definition extend (x : S) : CProp := P x and (forall H : P x, R x H).
+
+Lemma ext1 : forall x : S, extend x -> P x.
+intros x H; inversion_clear H; assumption.
+Qed.
+
+Lemma ext2_a : forall x : S, extend x -> {H : P x | R x H}.
+intros x H; inversion_clear H.
+exists X; auto.
+Qed.
+
+Lemma ext2 : forall (x : S) (Hx : extend x), R x (ProjT1 (ext2_a x Hx)).
+intros; apply projT2.
+Qed.
+
+Lemma extension_wd : pred_wd _ P ->
+ (forall (x y : S) Hx Hy, x [=] y -> R x Hx -> R y Hy) -> pred_wd _ extend.
+intros H H0.
+red in |- *; intros x y H1 H2.
+elim H1; intros H3 H4; split.
+
+apply H with x; assumption.
+
+intro H5; apply H0 with x H3; [ apply H2 | apply H4 ].
+Qed.
+
+End Extension.
+
+End SubSets_of_G.
+
+Notation Conj := (conjP _).
+Implicit Arguments disj [S].
+
+Implicit Arguments extend [S].
+Implicit Arguments ext1 [S P R x].
+Implicit Arguments ext2 [S P R x].
+
+(** ***Operations
+
+We are now ready to define the concept of partial function between arbitrary setoids.
+*)
+
+Record BinPartFunct (S1 S2 : CSetoid) : Type :=
+ {bpfdom : S1 -> CProp;
+ bdom_wd : pred_wd S1 bpfdom;
+ bpfpfun :> forall x : S1, bpfdom x -> S2;
+ bpfstrx : forall x y Hx Hy, bpfpfun x Hx [#] bpfpfun y Hy -> x [#] y}.
+
+
+Notation BDom := (bpfdom _ _).
+Implicit Arguments bpfpfun [S1 S2].
+
+(**
+The next lemma states that every partial function is well defined.
+*)
+
+Lemma bpfwdef : forall S1 S2 (F : BinPartFunct S1 S2) x y Hx Hy,
+ x [=] y -> F x Hx [=] F y Hy.
+intros.
+apply not_ap_imp_eq; intro H0.
+generalize (bpfstrx _ _ _ _ _ _ _ H0).
+exact (eq_imp_not_ap _ _ _ H).
+Qed.
+
+(** Similar for automorphisms. *)
+
+Record PartFunct (S : CSetoid) : Type :=
+ {pfdom : S -> CProp;
+ dom_wd : pred_wd S pfdom;
+ pfpfun :> forall x : S, pfdom x -> S;
+ pfstrx : forall x y Hx Hy, pfpfun x Hx [#] pfpfun y Hy -> x [#] y}.
+
+Notation Dom := (pfdom _).
+Notation Part := (pfpfun _).
+Implicit Arguments pfpfun [S].
+
+(**
+The next lemma states that every partial function is well defined.
+*)
+
+Lemma pfwdef : forall S (F : PartFunct S) x y Hx Hy, x [=] y -> F x Hx [=] F y Hy.
+intros.
+apply not_ap_imp_eq; intro H0.
+generalize (pfstrx _ _ _ _ _ _ H0).
+exact (eq_imp_not_ap _ _ _ H).
+Qed.
+
+(**
+A few characteristics of this definition should be explained:
+ - The domain of the partial function is characterized by a predicate
+that is required to be well defined but not strongly extensional. The
+motivation for this choice comes from two facts: first, one very
+important subset of real numbers is the compact interval
+[[a,b]]---characterized by the predicate [ fun x : IR => a [<=] x /\ x
+[<=] b], which is not strongly extensional; on the other hand, if we
+can apply a function to an element [s] of a setoid [S] it seems
+reasonable (and at some point we do have to do it) to apply that same
+function to any element [s'] which is equal to [s] from the point of
+view of the setoid equality.
+ - The last two conditions state that [pfpfun] is really a subsetoid
+function. The reason why we do not write it that way is the
+following: when applying a partial function [f] to an element [s] of
+[S] we also need a proof object [H]; with this definition the object
+we get is [f(s,H)], where the proof is kept separate from the object.
+Using subsetoid notation, we would get $f(\langle
+s,H\rangle)$#f(⟨s,H⟩)#; from this we need to apply two
+projections to get either the original object or the proof, and we
+need to apply an extra constructor to get $f(\langle
+s,H\rangle)$#f(⟨s,H⟩)# from [s] and [H]. This amounts
+to spending more resources when actually working with these objects.
+ - This record has type [Type], which is very unfortunate, because it
+means in particular that we cannot use the well behaved set
+existential quantification over partial functions; however, later on
+we will manage to avoid this problem in a way that also justifies that
+we don't really need to use that kind of quantification. Another
+approach to this definition that completely avoid this complication
+would be to make [PartFunct] a dependent type, receiving the predicate
+as an argument. This does work in that it allows us to give
+[PartFunct] type [Set] and do some useful stuff with it; however, we
+are not able to define something as simple as an operator that gets a
+function and returns its domain (because of the restrictions in the
+type elimination rules). This sounds very unnatural, and soon gets us
+into strange problems that yield very unlikely definitions, which is
+why we chose to altogether do away with this approach.
+
+%\begin{convention}% All partial functions will henceforth be denoted by capital letters.
+%\end{convention}%
+
+We now present some methods for defining partial functions.
+*)
+
+Hint Resolve CI: core.
+
+Section CSetoid_Ops.
+
+Variable S : CSetoid.
+
+(**
+To begin with, we want to be able to ``see'' each total function as a partial function.
+*)
+
+Definition total_eq_part : CSetoid_un_op S -> PartFunct S.
+intros f.
+apply
+ Build_PartFunct with (fun x : S => CTrue) (fun (x : S) (H : CTrue) => f x).
+red in |- *; intros; auto.
+intros x y Hx Hy H.
+exact (csf_strext_unfolded _ _ f _ _ H).
+Defined.
+
+Section Part_Function_Const.
+
+(**
+In any setoid we can also define constant functions (one for each element of the setoid) and an identity function:
+
+%\begin{convention}% Let [c:S].
+%\end{convention}%
+*)
+
+Variable c : S.
+
+Definition Fconst := total_eq_part (Const_CSetoid_fun _ _ c).
+
+End Part_Function_Const.
+
+Section Part_Function_Id.
+
+Definition Fid := total_eq_part (id_un_op S).
+
+End Part_Function_Id.
+
+(**
+(These happen to be always total functions, but that is more or less obvious, as we have no information on the setoid; however, we will be able to define partial functions just applying other operators to these ones.)
+
+If we have two setoid functions [F] and [G] we can always compose them. The domain of our new function will be the set of points [s] in the domain of [F] for which [F(s)] is in the domain of [G]#. #%\footnote{%Notice that the use of extension here is essential.%}.% The inversion in the order of the variables is done to maintain uniformity with the usual mathematical notation.
+
+%\begin{convention}% Let [G,F:(PartFunct S)] and denote by [Q] and [P], respectively, the predicates characterizing their domains.
+%\end{convention}%
+*)
+
+Section Part_Function_Composition.
+
+Variables G F : PartFunct S.
+
+(* begin hide *)
+Let P := Dom F.
+Let Q := Dom G.
+(* end hide *)
+Let R x := {Hx : P x | Q (F x Hx)}.
+
+Lemma part_function_comp_strext : forall x y (Hx : R x) (Hy : R y),
+ G (F x (ProjT1 Hx)) (ProjT2 Hx) [#] G (F y (ProjT1 Hy)) (ProjT2 Hy) -> x [#] y.
+intros x y Hx Hy H.
+exact (pfstrx _ _ _ _ _ _ (pfstrx _ _ _ _ _ _ H)).
+Qed.
+
+Lemma part_function_comp_dom_wd : pred_wd S R.
+red in |- *; intros x y H H0.
+unfold R in |- *; inversion_clear H.
+exists (dom_wd _ F x y x0 H0).
+apply (dom_wd _ G) with (F x x0).
+assumption.
+apply pfwdef; assumption.
+Qed.
+
+Definition Fcomp := Build_PartFunct _ R part_function_comp_dom_wd
+ (fun x (Hx : R x) => G (F x (ProjT1 Hx)) (ProjT2 Hx))
+ part_function_comp_strext.
+
+End Part_Function_Composition.
+
+End CSetoid_Ops.
+
+(**
+%\begin{convention}% Let [F:(BinPartFunct S1 S2)] and [G:(PartFunct S2 S3)], and denote by [Q] and [P], respectively, the predicates characterizing their domains.
+%\end{convention}%
+*)
+
+Section BinPart_Function_Composition.
+
+Variables S1 S2 S3 : CSetoid.
+
+Variable G : BinPartFunct S2 S3.
+Variable F : BinPartFunct S1 S2.
+
+(* begin hide *)
+Let P := BDom F.
+Let Q := BDom G.
+(* end hide *)
+Let R x := {Hx : P x | Q (F x Hx)}.
+
+Lemma bin_part_function_comp_strext : forall x y (Hx : R x) (Hy : R y),
+ G (F x (ProjT1 Hx)) (ProjT2 Hx) [#] G (F y (ProjT1 Hy)) (ProjT2 Hy) -> x [#] y.
+intros x y Hx Hy H.
+exact (bpfstrx _ _ _ _ _ _ _ (bpfstrx _ _ _ _ _ _ _ H)).
+Qed.
+
+Lemma bin_part_function_comp_dom_wd : pred_wd S1 R.
+red in |- *; intros x y H H0.
+unfold R in |- *; inversion_clear H.
+exists (bdom_wd _ _ F x y x0 H0).
+apply (bdom_wd _ _ G) with (F x x0).
+assumption.
+apply bpfwdef; assumption.
+Qed.
+
+Definition BinFcomp := Build_BinPartFunct _ _ R bin_part_function_comp_dom_wd
+ (fun x (Hx : R x) => G (F x (ProjT1 Hx)) (ProjT2 Hx))
+ bin_part_function_comp_strext.
+
+End BinPart_Function_Composition.
+
+(* Different tokens for compatibility with coqdoc *)
+
+Implicit Arguments Fconst [S].
+Notation "[-C-] x" := (Fconst x) (at level 2, right associativity).
+
+Notation FId := (Fid _).
+
+Implicit Arguments Fcomp [S].
+Infix "[o]" := Fcomp (at level 65, no associativity).
+
+Hint Resolve pfwdef bpfwdef: algebra.
+
+Section bijections.
+(** **Bijections *)
+
+Definition injective A B (f : CSetoid_fun A B) := (forall a0 a1 : A,
+ a0 [#] a1 -> f a0 [#] f a1):CProp.
+
+Definition injective_weak A B (f : CSetoid_fun A B) := forall a0 a1 : A,
+ f a0 [=] f a1 -> a0 [=] a1.
+
+Definition surjective A B (f : CSetoid_fun A B) := (forall b : B, {a : A | f a [=] b}):CProp.
+
+Implicit Arguments injective [A B].
+Implicit Arguments injective_weak [A B].
+Implicit Arguments surjective [A B].
+
+Lemma injective_imp_injective_weak : forall A B (f : CSetoid_fun A B),
+ injective f -> injective_weak f.
+intros A B f.
+unfold injective in |- *.
+intro H.
+unfold injective_weak in |- *.
+intros a0 a1 H0.
+apply not_ap_imp_eq.
+red in |- *.
+intro H1.
+set (H2 := H a0 a1 H1) in *.
+set (H3 := ap_imp_neq B (f a0) (f a1) H2) in *.
+set (H4 := eq_imp_not_neq B (f a0) (f a1) H0) in *.
+apply H4.
+exact H3.
+Qed.
+
+Definition bijective A B (f:CSetoid_fun A B) := injective f and surjective f.
+
+Implicit Arguments bijective [A B].
+
+Lemma id_is_bij : forall A, bijective (id_un_op A).
+intro A.
+split.
+ red; simpl; auto.
+intro b; exists b; apply eq_reflexive.
+Qed.
+
+Lemma comp_resp_bij : forall A B C f g, bijective f -> bijective g ->
+ bijective (compose_CSetoid_fun A B C f g).
+intros A B C f g.
+intros H0 H1.
+elim H0; clear H0; intros H00 H01.
+elim H1; clear H1; intros H10 H11.
+split.
+ intros a0 a1; simpl; intro.
+ apply H10; apply H00; auto.
+intro c; simpl.
+elim (H11 c); intros b H20.
+elim (H01 b); intros a H30.
+exists a.
+Step_final (g b).
+Qed.
+
+Lemma inv : forall A B (f:CSetoid_fun A B),
+ bijective f -> forall b : B, {a : A | f a [=] b}.
+unfold bijective in |- *.
+unfold surjective in |- *.
+intuition.
+Qed.
+
+Implicit Arguments inv [A B].
+
+Definition invfun A B (f : CSetoid_fun A B) (H : bijective f) : B -> A.
+intros A B f H H0.
+elim (inv f H H0); intros a H2.
+apply a.
+Defined.
+
+Implicit Arguments invfun [A B].
+
+Lemma inv1 : forall A B (f : CSetoid_fun A B) (H : bijective f) (b : B),
+ f (invfun f H b) [=] b.
+intros A B f H b.
+unfold invfun in |- *; case inv.
+simpl; auto.
+Qed.
+
+Lemma inv2 : forall A B (f : CSetoid_fun A B) (H : bijective f) (a : A),
+ invfun f H (f a) [=] a.
+intros.
+unfold invfun in |- *; case inv; simpl.
+elim H; clear H; intros H0 H1.
+intro x.
+apply injective_imp_injective_weak.
+auto.
+Qed.
+
+Lemma inv_strext : forall A B (f : CSetoid_fun A B) (H : bijective f),
+ fun_strext (invfun f H).
+intros A B f H x y.
+intro H1.
+elim H; intros H00 H01.
+elim (H01 x); intros a0 H2.
+elim (H01 y); intros a1 H3.
+astepl (f a0).
+astepr (f a1).
+apply H00.
+astepl (invfun f H x).
+astepr (invfun f H y).
+exact H1.
+astepl (invfun f H (f a1)).
+apply inv2.
+apply injective_imp_injective_weak with (f := f); auto.
+astepl (f a1).
+astepl y.
+apply eq_symmetric.
+apply inv1.
+apply eq_symmetric.
+apply inv1.
+astepl (invfun f H (f a0)).
+apply inv2.
+
+apply injective_imp_injective_weak with (f := f); auto.
+astepl (f a0).
+astepl x.
+apply eq_symmetric; apply inv1.
+apply eq_symmetric; apply inv1.
+Qed.
+
+Definition Inv A B f (H : bijective f) :=
+ Build_CSetoid_fun B A (invfun f H) (inv_strext A B f H).
+
+Implicit Arguments Inv [A B].
+
+Definition Inv_bij : forall A B (f : CSetoid_fun A B) (H : bijective f),
+ bijective (Inv f H).
+intros A B f H.
+unfold bijective in |- *.
+split.
+unfold injective in |- *.
+unfold bijective in H.
+unfold surjective in H.
+elim H.
+intros H0 H1.
+intros b0 b1 H2.
+elim (H1 b0).
+intros a0 H3.
+elim (H1 b1).
+intros a1 H4.
+astepl (Inv f (CAnd_intro _ _ H0 H1) (f a0)).
+astepr (Inv f (CAnd_intro _ _ H0 H1) (f a1)).
+cut (fun_strext f).
+unfold fun_strext in |- *.
+intros H5.
+apply H5.
+astepl (f a0).
+astepr (f a1).
+astepl b0.
+astepr b1.
+exact H2.
+apply eq_symmetric.
+unfold Inv in |- *.
+simpl in |- *.
+apply inv1.
+apply eq_symmetric.
+unfold Inv in |- *.
+simpl in |- *.
+apply inv1.
+elim f.
+intuition.
+
+unfold surjective in |- *.
+intro a.
+exists (f a).
+unfold Inv in |- *.
+simpl in |- *.
+apply inv2.
+Qed.
+
+
+End bijections.
+Implicit Arguments bijective [A B].
+Implicit Arguments injective [A B].
+Implicit Arguments injective_weak [A B].
+Implicit Arguments surjective [A B].
+Implicit Arguments inv [A B].
+Implicit Arguments invfun [A B].
+Implicit Arguments Inv [A B].
+
+Implicit Arguments conj_wd [S P Q].
+
+Notation Prj1 := (prj1 _ _ _ _).
+Notation Prj2 := (prj2 _ _ _ _).
--- /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/Setoid/cs_eq.con _ x y).
+
+interpretation "setoid apart"
+ 'neq x y = (cic:/matita/algebra/CoRN/Setoid/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.
+
+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. auto.
+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' auto non arriva in fondo ??? *)
+apply (eq_transitive_unfolded ? ? x). auto.
+assumption.
+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: ProdT A B. Prop \def
+\lambda A,B : CSetoid.\lambda c,d: ProdT A B.
+ ((cs_ap A (fstT A B c) (fstT A B d)) \or
+ (cs_ap B (sndT A B c) (sndT A B d))).
+
+definition prod_eq: \forall A,B : CSetoid.\forall c,d: ProdT A B. Prop \def
+\lambda A,B : CSetoid.\lambda c,d: ProdT A B.
+ ((cs_eq A (fstT A B c) (fstT A B d)) \and
+ (cs_eq B (sndT A B c) (sndT A B d))).
+
+
+lemma prodcsetoid_is_CSetoid: \forall A,B: CSetoid.
+ is_CSetoid (ProdT 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 t H1)
+ |apply (ap_irreflexive B t1 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 ((t \neq t4) \or (t4 \neq t2))
+ [elim Hcut
+ [left. left. assumption
+ |right. left. assumption
+ ]
+ |apply (ap_cotransitive A). assumption
+ ]
+ |cut ((t1 \neq t5) \or (t5 \neq t3))
+ [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 t t2 H2). assumption
+ |apply (eq_imp_not_ap B t1 t3 H3). assumption
+ ]
+ ]
+]
+qed.
+
+
+definition ProdCSetoid : \forall A,B: CSetoid. CSetoid \def
+ \lambda A,B: CSetoid.
+ mk_CSetoid (ProdT 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. (S \to Prop) \to Prop \def
+ \lambda S: CSetoid. \lambda P: S \to Prop. \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).
+
+auto.
+
+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).
+
+auto.
+
+intro H3.
+elim H3; intro H4.
+
+auto.
+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.
+
+auto.
+
+elim H2; intro H3.
+
+elim (ap_irreflexive _ _ H3).
+
+auto.
+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); auto.
+intro H2.
+elim (H0 x1 x2 y2 H2); auto.
+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).
+
+auto.
+
+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).
+
+auto.
+
+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.
+
+auto.
+
+case (ap_cotransitive_unfolded _ _ _ H0 y2); intro H1.
+
+auto.
+
+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 : 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 associative : \forall S: CSetoid. (S \to S \to S) \to 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.assumption.
+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.assumption.
+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/Setoid/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/Setoid/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/Setoid/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.
+
+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 | apply 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.
+reduce 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.reduce 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) \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.
+reduce in H4.
+reduce 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.
+reduce 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) \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.
+ associative S (csbf_fun S S S f)
+ \to associative (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.
+auto.
+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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