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-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-include "subsets.ma".
-
-record binary_relation (A,B: SET) : Type1 ≝
- { satisfy:> binary_morphism1 A B CPROP }.
-
-notation < "hvbox (x \nbsp \natur term 90 r \nbsp y)"  with precedence 45 for @{'satisfy $r $x $y}.
-notation > "hvbox (x \natur term 90 r y)"  with precedence 45 for @{'satisfy $r $x $y}.
-interpretation "relation applied" 'satisfy r x y = (fun21 ??? (satisfy ?? r) x y).
-
-definition binary_relation_setoid: SET → SET → setoid1.
- intros (A B);
- constructor 1;
-  [ apply (binary_relation A B)
-  | constructor 1;
-     [ apply (λA,B.λr,r': binary_relation A B. ∀x,y. r x y ↔ r' x y)
-     | simplify; intros 3; split; intro; assumption
-     | simplify; intros 5; split; intro;
-       [ apply (fi ?? (f ??)) | apply (if ?? (f ??))] assumption
-     | simplify;  intros 7; split; intro;
-        [ apply (if ?? (f1 ??)) | apply (fi ?? (f ??)) ]
-        [ apply (if ?? (f ??)) | apply (fi ?? (f1 ??)) ]
-       assumption]]
-qed.
-
-definition binary_relation_of_binary_relation_setoid : 
-  ∀A,B.binary_relation_setoid A B → binary_relation A B ≝ λA,B,c.c.
-coercion binary_relation_of_binary_relation_setoid.
-
-definition composition:
- ∀A,B,C.
-  (binary_relation_setoid A B) × (binary_relation_setoid B C) ⇒_1 (binary_relation_setoid A C).
- intros;
- constructor 1;
-  [ intros (R12 R23);
-    constructor 1;
-    constructor 1;
-     [ apply (λs1:A.λs3:C.∃s2:B. s1 ♮R12 s2 ∧ s2 ♮R23 s3);
-     | intros;
-       split; intro; cases e2 (w H3); clear e2; exists; [1,3: apply w ]
-        [ apply (. (e^-1‡#)‡(#‡e1^-1)); assumption
-        | apply (. (e‡#)‡(#‡e1)); assumption]]
-  | intros 8; split; intro H2; simplify in H2 ⊢ %;
-    cases H2 (w H3); clear H2; exists [1,3: apply w] cases H3 (H2 H4); clear H3;
-    [ lapply (if ?? (e x w) H2) | lapply (fi ?? (e x w) H2) ]
-    [ lapply (if ?? (e1 w y) H4)| lapply (fi ?? (e1 w y) H4) ]
-    exists; try assumption;
-    split; assumption]
-qed.
-
-definition REL: category1.
- constructor 1;
-  [ apply setoid
-  | intros (T T1); apply (binary_relation_setoid T T1)
-  | intros; constructor 1;
-    constructor 1; unfold setoid1_of_setoid; simplify;
-     [ (* changes required to avoid universe inconsistency *)
-       change with (carr o → carr o → CProp); intros; apply (eq ? c c1)
-     | intros; split; intro; change in a a' b b' with (carr o);
-       change in e with (eq ? a a'); change in e1 with (eq ? b b');
-        [ apply (.= (e ^ -1));
-          apply (.= e2);
-          apply e1
-        | apply (.= e);
-          apply (.= e2);
-          apply (e1 ^ -1)]]
-  | apply composition
-  | intros 9;
-    split; intro;
-    cases f (w H); clear f; cases H; clear H;
-    [cases f (w1 H); clear f | cases f1 (w1 H); clear f1]
-    cases H; clear H;
-    exists; try assumption;
-    split; try assumption;
-    exists; try assumption;
-    split; assumption
-  |6,7: intros 5; unfold composition; simplify; split; intro;
-        unfold setoid1_of_setoid in x y; simplify in x y;
-        [1,3: cases e (w H1); clear e; cases H1; clear H1; unfold;
-          [ apply (. (e : eq1 ? x w)‡#); assumption
-          | apply (. #‡(e : eq1 ? w y)^-1); assumption]
-        |2,4: exists; try assumption; split;
-          (* change required to avoid universe inconsistency *)
-          change in x with (carr o1); change in y with (carr o2);
-          first [apply refl | assumption]]]
-qed.
-
-definition setoid_of_REL : objs1 REL → setoid ≝ λx.x.
-coercion setoid_of_REL.
-
-definition binary_relation_setoid_of_arrow1_REL : 
-  ∀P,Q. arrows1 REL P Q → binary_relation_setoid P Q ≝ λP,Q,x.x.
-coercion binary_relation_setoid_of_arrow1_REL.
-
-
-notation > "B ⇒_\r1 C" right associative with precedence 72 for @{'arrows1_REL $B $C}.
-notation "B ⇒\sub (\r 1) C" right associative with precedence 72 for @{'arrows1_REL $B $C}.
-interpretation "'arrows1_SET" 'arrows1_REL A B = (arrows1 REL A B).
-
-
-definition full_subset: ∀s:REL. Ω^s.
- apply (λs.{x | True});
- intros; simplify; split; intro; assumption.
-qed.
-
-coercion full_subset.
-
-definition comprehension: ∀b:REL. (b ⇒_1. CPROP) → Ω^b.
- apply (λb:REL. λP: b ⇒_1 CPROP. {x | P x});
- intros; simplify;
- apply (.= †e); apply refl1.
-qed.
-
-interpretation "subset comprehension" 'comprehension s p =
- (comprehension s (mk_unary_morphism1 ?? p ?)).
-
-definition ext: ∀X,S:REL. (X ⇒_\r1 S) × S ⇒_1 (Ω^X).
- intros (X S); constructor 1; 
-  [ apply (λr:X ⇒_\r1 S.λf:S.{x ∈ X | x ♮r f}); intros; simplify; apply (.= (e‡#)); apply refl1
-  | intros; simplify; split; intros; simplify;
-     [ change with (∀x. x ♮a b → x ♮a' b'); intros;
-       apply (. (#‡e1^-1)); whd in e; apply (if ?? (e ??)); assumption
-     | change with (∀x. x ♮a' b' → x ♮a b); intros;
-       apply (. (#‡e1)); whd in e; apply (fi ?? (e ??));assumption]]
-qed.
-
-(*
-definition extS: ∀X,S:REL. ∀r: arrows1 ? X S. Ω \sup S ⇒ Ω \sup X.
- (* ∃ is not yet a morphism apply (λX,S,r,F.{x ∈ X | ∃a. a ∈ F ∧ x ♮r a});*)
- intros (X S r); constructor 1;
-  [ intro F; constructor 1; constructor 1;
-    [ apply (λx. x ∈ X ∧ ∃a:S. a ∈ F ∧ x ♮r a);
-    | intros; split; intro; cases f (H1 H2); clear f; split;
-       [ apply (. (H‡#)); assumption
-       |3: apply (. (H\sup -1‡#)); assumption
-       |2,4: cases H2 (w H3); exists; [1,3: apply w]
-         [ apply (. (#‡(H‡#))); assumption
-         | apply (. (#‡(H \sup -1‡#))); assumption]]]
-  | intros; split; simplify; intros; cases f; cases H1; split;
-     [1,3: assumption
-     |2,4: exists; [1,3: apply w]
-      [ apply (. (#‡H)‡#); assumption
-      | apply (. (#‡H\sup -1)‡#); assumption]]]
-qed.
-
-lemma equalset_extS_id_X_X: ∀o:REL.∀X.extS ?? (id1 ? o) X = X.
- intros;
- unfold extS; simplify;
- split; simplify;
-  [ intros 2; change with (a ∈ X);
-    cases f; clear f;
-    cases H; clear H;
-    cases x; clear x;
-    change in f2 with (eq1 ? a w);
-    apply (. (f2\sup -1‡#));
-    assumption
-  | intros 2; change in f with (a ∈ X);
-    split;
-     [ whd; exact I 
-     | exists; [ apply a ]
-       split;
-        [ assumption
-        | change with (a = a); apply refl]]]
-qed.
-
-lemma extS_com: ∀o1,o2,o3,c1,c2,S. extS o1 o3 (c2 ∘ c1) S = extS o1 o2 c1 (extS o2 o3 c2 S).
- intros; unfold extS; simplify; split; intros; simplify; intros;
-  [ cases f (H1 H2); cases H2 (w H3); clear f H2; split; [assumption]
-    cases H3 (H4 H5); cases H5 (w1 H6); clear H3 H5; cases H6 (H7 H8); clear H6;
-    exists; [apply w1] split [2: assumption] constructor 1; [assumption]
-    exists; [apply w] split; assumption
-  | cases f (H1 H2); cases H2 (w H3); clear f H2; split; [assumption]
-    cases H3 (H4 H5); cases H4 (w1 H6); clear H3 H4; cases H6 (w2 H7); clear H6;
-    cases H7; clear H7; exists; [apply w2] split; [assumption] exists [apply w] split;
-    assumption]
-qed.
-*)
-
-(* the same as ⋄ for a basic pair *)
-definition image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^U ⇒_1 Ω^V.
- intros; constructor 1;
-  [ apply (λr:U ⇒_\r1 V.λS: Ω \sup U. {y | ∃x:U. x ♮r y ∧ x ∈ S });
-    intros; simplify; split; intro; cases e1; exists [1,3: apply w]
-     [ apply (. (#‡e^-1)‡#); assumption
-     | apply (. (#‡e)‡#); assumption]
-  | intros; split; simplify; intros; cases e2; exists [1,3: apply w]
-     [ apply (. #‡(#‡e1^-1)); cases x; split; try assumption;
-       apply (if ?? (e ??)); assumption
-     | apply (. #‡(#‡e1)); cases x; split; try assumption;
-       apply (if ?? (e ^ -1 ??)); assumption]]
-qed.
-
-(* the same as □ for a basic pair *)
-definition minus_star_image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^U ⇒_1 Ω^V.
- intros; constructor 1;
-  [ apply (λr:U ⇒_\r1 V.λS: Ω \sup U. {y | ∀x:U. x ♮r y → x ∈ S});
-    intros; simplify; split; intros; apply f;
-     [ apply (. #‡e); assumption
-     | apply (. #‡e ^ -1); assumption]
-  | intros; split; simplify; intros; [ apply (. #‡e1^ -1); | apply (. #‡e1 )]
-    apply f; [ apply (if ?? (e ^ -1 ??)); | apply (if ?? (e ??)) ] assumption]
-qed.
-
-(* the same as Rest for a basic pair *)
-definition star_image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^V ⇒_1 Ω^U.
- intros; constructor 1;
-  [ apply (λr:U ⇒_\r1 V.λS: Ω \sup V. {x | ∀y:V. x ♮r y → y ∈ S});
-    intros; simplify; split; intros; apply f;
-     [ apply (. e ‡#); assumption
-     | apply (. e^ -1‡#); assumption]
-  | intros; split; simplify; intros; [ apply (. #‡e1 ^ -1); | apply (. #‡e1)]
-    apply f; [ apply (if ?? (e ^ -1 ??)); | apply (if ?? (e ??)) ] assumption]
-qed.
-
-(* the same as Ext for a basic pair *)
-definition minus_image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^V ⇒_1 Ω^U.
- intros; constructor 1;
-  [ apply (λr:U ⇒_\r1 V.λS: Ω \sup V. {x | (*∃x:U. x ♮r y ∧ x ∈ S*)
-      exT ? (λy:V.x ♮r y ∧ y ∈ S) });
-    intros; simplify; split; intro; cases e1; exists [1,3: apply w]
-     [ apply (. (e ^ -1‡#)‡#); assumption
-     | apply (. (e‡#)‡#); assumption]
-  | intros; split; simplify; intros; cases e2; exists [1,3: apply w]
-     [ apply (. #‡(#‡e1 ^ -1)); cases x; split; try assumption;
-       apply (if ?? (e ??)); assumption
-     | apply (. #‡(#‡e1)); cases x; split; try assumption;
-       apply (if ?? (e ^ -1 ??)); assumption]]
-qed.
-
-(* minus_image is the same as ext *)
-
-theorem image_id: ∀o,U. image o o (id1 REL o) U = U.
- intros; unfold image; simplify; split; simplify; intros;
-  [ change with (a ∈ U);
-    cases e; cases x; change in f with (eq1 ? w a); apply (. f^-1‡#); assumption
-  | change in f with (a ∈ U);
-    exists; [apply a] split; [ change with (a = a); apply refl1 | assumption]]
-qed.
-
-theorem minus_star_image_id: ∀o,U. minus_star_image o o (id1 REL o) U = U.
- intros; unfold minus_star_image; simplify; split; simplify; intros;
-  [ change with (a ∈ U); apply f; change with (a=a); apply refl1
-  | change in f1 with (eq1 ? x a); apply (. f1‡#); apply f]
-qed.
-
-alias symbol "compose" (instance 2) = "category1 composition".
-theorem image_comp: ∀A,B,C,r,s,X. image A C (r ∘ s) X = image B C r (image A B s X).
- intros; unfold image; simplify; split; simplify; intros; cases e; clear e; cases x;
- clear x; [ cases f; clear f; | cases f1; clear f1 ]
- exists; try assumption; cases x; clear x; split; try assumption;
- exists; try assumption; split; assumption.
-qed.
-
-theorem minus_star_image_comp:
- ∀A,B,C,r,s,X.
-  minus_star_image A C (r ∘ s) X = minus_star_image B C r (minus_star_image A B s X).
- intros; unfold minus_star_image; simplify; split; simplify; intros; whd; intros;
-  [ apply f; exists; try assumption; split; assumption
-  | change with (x ∈ X); cases f1; cases x1; apply f; assumption]
-qed.
-
-(*
-(*CSC: unused! *)
-theorem ext_comp:
- ∀o1,o2,o3: REL.
-  ∀a: arrows1 ? o1 o2.
-   ∀b: arrows1 ? o2 o3.
-    ∀x. ext ?? (b∘a) x = extS ?? a (ext ?? b x).
- intros;
- unfold ext; unfold extS; simplify; split; intro; simplify; intros;
- cases f; clear f; split; try assumption;
-  [ cases f2; clear f2; cases x1; clear x1; exists; [apply w] split;
-     [1: split] assumption;
-  | cases H; clear H; cases x1; clear x1; exists [apply w]; split;
-     [2: cases f] assumption]
-qed.
-
-theorem extS_singleton:
- ∀o1,o2.∀a:arrows1 ? o1 o2.∀x.extS o1 o2 a (singleton o2 x) = ext o1 o2 a x.
- intros; unfold extS; unfold ext; unfold singleton; simplify;
- split; intros 2; simplify; cases f; split; try assumption;
-  [ cases H; cases x1; change in f2 with (eq1 ? x w); apply (. #‡f2 \sup -1);
-    assumption
-  | exists; try assumption; split; try assumption; change with (x = x); apply refl]
-qed.
-*)