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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 "formal_topology/o-algebra.ma".
+include "formal_topology/notation.ma".
+
+record Obasic_pair: Type[2] ≝ { 
+   Oconcr: OA; Oform: OA; Orel: arrows2 ? Oconcr Oform
+}.
+
+(* FIX *)
+interpretation "o-basic pair relation indexed" 'Vdash2 x y c = (Orel c x y).
+interpretation "o-basic pair relation (non applied)" 'Vdash c = (Orel c).
+
+record Orelation_pair (BP1,BP2: Obasic_pair): Type[2] ≝ { 
+   Oconcr_rel: (Oconcr BP1) ⇒_\o2 (Oconcr BP2); Oform_rel: (Oform BP1) ⇒_\o2 (Oform BP2);
+   Ocommute: ⊩ ∘ Oconcr_rel =_2 Oform_rel ∘ ⊩
+}.
+ 
+(* FIX *)
+interpretation "o-concrete relation" 'concr_rel r = (Oconcr_rel ?? r). 
+interpretation "o-formal relation" 'form_rel r = (Oform_rel ?? r). 
+
+definition Orelation_pair_equality:
+ ∀o1,o2. equivalence_relation2 (Orelation_pair o1 o2).
+ intros;
+ constructor 1;
+  [ apply (λr,r'. ⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
+  | simplify;
+    intros;
+    apply refl2;
+  | simplify;
+    intros 2;
+    apply sym2;
+  | simplify;
+    intros 3;
+    apply trans2;
+  ]      
+qed.
+
+(* qui setoid1 e' giusto: ma non lo e'!!! *)
+definition Orelation_pair_setoid: Obasic_pair → Obasic_pair → setoid2.
+ intros;
+ constructor 1;
+  [ apply (Orelation_pair o o1)
+  | apply Orelation_pair_equality
+  ]
+qed.
+
+definition Orelation_pair_of_Orelation_pair_setoid: 
+  ∀P,Q. Orelation_pair_setoid P Q → Orelation_pair P Q ≝ λP,Q,x.x.
+coercion Orelation_pair_of_Orelation_pair_setoid.
+
+lemma eq_to_eq': ∀o1,o2.∀r,r': Orelation_pair_setoid o1 o2. r =_2 r' → r \sub\f ∘ ⊩ =_2 r'\sub\f ∘ ⊩.
+ intros 5 (o1 o2 r r' H); change in H with (⊩ ∘ r\sub\c = ⊩ ∘ r'\sub\c);
+ apply (.= ((Ocommute ?? r) ^ -1));
+ apply (.= H);
+ apply (.= (Ocommute ?? r'));
+ apply refl2;
+qed.
+
+
+definition Oid_relation_pair: ∀o:Obasic_pair. Orelation_pair o o.
+ intro;
+ constructor 1;
+  [1,2: apply id2;
+  | lapply (id_neutral_right2 ? (Oconcr o) ? (⊩)) as H;
+    lapply (id_neutral_left2 ?? (Oform o) (⊩)) as H1;
+    apply (.= H);
+    apply (H1^-1);]
+qed.
+
+lemma Orelation_pair_composition:
+ ∀o1,o2,o3:Obasic_pair.
+ Orelation_pair_setoid o1 o2 → Orelation_pair_setoid o2 o3→Orelation_pair_setoid o1 o3.
+intros 3 (o1 o2 o3);
+   intros (r r1);
+    constructor 1;
+     [ apply (r1 \sub\c ∘ r \sub\c) 
+     | apply (r1 \sub\f ∘ r \sub\f)
+     | lapply (Ocommute ?? r) as H;
+       lapply (Ocommute ?? r1) as H1;
+       apply rule (.= ASSOC);
+       apply (.= #‡H1);
+       apply rule (.= ASSOC ^ -1);
+       apply (.= H‡#);
+       apply rule ASSOC]
+qed.
+
+
+lemma Orelation_pair_composition_is_morphism:
+  ∀o1,o2,o3:Obasic_pair.
+  Πa,a':Orelation_pair_setoid o1 o2.Πb,b':Orelation_pair_setoid o2 o3.
+   a=a' →b=b' →
+      Orelation_pair_composition o1 o2 o3 a b
+      = Orelation_pair_composition o1 o2 o3 a' b'.
+intros;
+    change with (⊩ ∘ (b\sub\c ∘ a\sub\c) = ⊩ ∘ (b'\sub\c ∘ a'\sub\c));  
+    change in e with (⊩ ∘ a \sub\c = ⊩ ∘ a' \sub\c);
+    change in e1 with (⊩ ∘ b \sub\c = ⊩ ∘ b' \sub\c);
+    apply rule (.= ASSOC);
+    apply (.= #‡e1);
+    apply (.= #‡(Ocommute ?? b'));
+    apply rule (.= ASSOC^-1);
+    apply (.= e‡#);
+    apply rule (.= ASSOC);
+    apply (.= #‡(Ocommute ?? b')^-1);
+    apply rule (ASSOC^-1);
+qed.
+
+definition Orelation_pair_composition_morphism:
+ ∀o1,o2,o3. binary_morphism2 (Orelation_pair_setoid o1 o2) (Orelation_pair_setoid o2 o3) (Orelation_pair_setoid o1 o3).
+intros; constructor 1;
+[ apply Orelation_pair_composition;
+| apply Orelation_pair_composition_is_morphism;]
+qed.
+
+lemma Orelation_pair_composition_morphism_assoc:
+∀o1,o2,o3,o4:Obasic_pair
+   .Πa12:Orelation_pair_setoid o1 o2
+    .Πa23:Orelation_pair_setoid o2 o3
+     .Πa34:Orelation_pair_setoid o3 o4
+      .Orelation_pair_composition_morphism o1 o3 o4
+       (Orelation_pair_composition_morphism o1 o2 o3 a12 a23) a34
+       =Orelation_pair_composition_morphism o1 o2 o4 a12
+        (Orelation_pair_composition_morphism o2 o3 o4 a23 a34).  
+   intros;
+    change with (⊩ ∘ (a34\sub\c ∘ (a23\sub\c ∘ a12\sub\c)) =
+                 ⊩ ∘ ((a34\sub\c ∘ a23\sub\c) ∘ a12\sub\c));
+    apply rule (ASSOC‡#);
+qed.
+
+lemma Orelation_pair_composition_morphism_respects_id:
+Πo1:Obasic_pair
+.Πo2:Obasic_pair
+ .Πa:Orelation_pair_setoid o1 o2
+  .Orelation_pair_composition_morphism o1 o1 o2 (Oid_relation_pair o1) a=a.
+   intros;
+    change with (⊩ ∘ (a\sub\c ∘ (Oid_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
+    apply ((id_neutral_right2 ????)‡#);
+qed.
+
+lemma Orelation_pair_composition_morphism_respects_id_r:
+Πo1:Obasic_pair
+.Πo2:Obasic_pair
+ .Πa:Orelation_pair_setoid o1 o2
+  .Orelation_pair_composition_morphism o1 o2 o2 a (Oid_relation_pair o2)=a.
+intros;
+    change with (⊩ ∘ ((Oid_relation_pair o2)\sub\c ∘ a\sub\c) = ⊩ ∘ a\sub\c);
+    apply ((id_neutral_left2 ????)‡#);
+qed.
+
+definition OBP: category2.
+ constructor 1;
+  [ apply Obasic_pair
+  | apply Orelation_pair_setoid
+  | apply Oid_relation_pair
+  | apply Orelation_pair_composition_morphism
+  | apply Orelation_pair_composition_morphism_assoc;
+  | apply Orelation_pair_composition_morphism_respects_id;
+  | apply Orelation_pair_composition_morphism_respects_id_r;]
+qed.
+
+definition Obasic_pair_of_objs2_OBP: objs2 OBP → Obasic_pair ≝ λx.x.
+coercion Obasic_pair_of_objs2_OBP.
+
+definition Orelation_pair_setoid_of_arrows2_OBP: 
+  ∀P,Q.arrows2 OBP P Q → Orelation_pair_setoid P Q ≝ λP,Q,c.c.
+coercion Orelation_pair_setoid_of_arrows2_OBP.
+
+notation > "B ⇒_\obp2 C" right associative with precedence 72 for @{'arrows2_OBP $B $C}.
+notation "B ⇒\sub (\obp 2) C" right associative with precedence 72 for @{'arrows2_OBP $B $C}.
+interpretation "'arrows2_OBP" 'arrows2_OBP A B = (arrows2 OBP A B).
+
+(*
+definition BPext: ∀o: BP. form o ⇒ Ω \sup (concr o).
+ intros; constructor 1;
+  [ apply (ext ? ? (rel o));
+  | intros;
+    apply (.= #‡H);
+    apply refl1]
+qed.
+
+definition BPextS: ∀o: BP. Ω \sup (form o) ⇒ Ω \sup (concr o) ≝
+ λo.extS ?? (rel o).
+*)
+
+(*
+definition fintersects: ∀o: BP. binary_morphism1 (form o) (form o) (Ω \sup (form o)).
+ intros (o); constructor 1;
+  [ apply (λa,b: form o.{c | BPext o c ⊆ BPext o a ∩ BPext o b });
+    intros; simplify; apply (.= (†H)‡#); apply refl1
+  | intros; split; simplify; intros;
+     [ apply (. #‡((†H)‡(†H1))); assumption
+     | apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
+qed.
+
+interpretation "fintersects" 'fintersects U V = (fun1 ??? (fintersects ?) U V).
+
+definition fintersectsS:
+ ∀o:BP. binary_morphism1 (Ω \sup (form o)) (Ω \sup (form o)) (Ω \sup (form o)).
+ intros (o); constructor 1;
+  [ apply (λo: basic_pair.λa,b: Ω \sup (form o).{c | BPext o c ⊆ BPextS o a ∩ BPextS o b });
+    intros; simplify; apply (.= (†H)‡#); apply refl1
+  | intros; split; simplify; intros;
+     [ apply (. #‡((†H)‡(†H1))); assumption
+     | apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
+qed.
+
+interpretation "fintersectsS" 'fintersects U V = (fun1 ??? (fintersectsS ?) U V).
+*)
+
+(*
+definition relS: ∀o: BP. binary_morphism1 (concr o) (Ω \sup (form o)) CPROP.
+ intros (o); constructor 1;
+  [ apply (λx:concr o.λS: Ω \sup (form o).∃y: form o.y ∈ S ∧ x ⊩ y);
+  | intros; split; intros; cases H2; exists [1,3: apply w]
+     [ apply (. (#‡H1)‡(H‡#)); assumption
+     | apply (. (#‡H1 \sup -1)‡(H \sup -1‡#)); assumption]]
+qed.
+
+interpretation "basic pair relation for subsets" 'Vdash2 x y = (fun1 (concr ?) ?? (relS ?) x y).
+interpretation "basic pair relation for subsets (non applied)" 'Vdash = (fun1 ??? (relS ?)).
+*)
+
+notation "□ \sub b" non associative with precedence 90 for @{'box $b}.
+notation > "□⎽term 90 b" non associative with precedence 90 for @{'box $b}.
+interpretation "Universal image ⊩⎻*" 'box x = (fun12 ? ? (or_f_minus_star ? ?) (Orel x)).
+ 
+notation "◊ \sub b" non associative with precedence 90 for @{'diamond $b}.
+notation > "◊⎽term 90 b" non associative with precedence 90 for @{'diamond $b}.
+interpretation "Existential image ⊩" 'diamond x = (fun12 ? ? (or_f ? ?) (Orel x)).
+
+notation "'Rest' \sub b" non associative with precedence 90 for @{'rest $b}.
+notation > "'Rest'⎽term 90 b" non associative with precedence 90 for @{'rest $b}.
+interpretation "Universal pre-image ⊩*" 'rest x = (fun12 ? ? (or_f_star ? ?) (Orel x)).
+
+notation "'Ext' \sub b" non associative with precedence 90 for @{'ext $b}.
+notation > "'Ext'⎽term 90 b" non associative with precedence 90 for @{'ext $b}.
+interpretation "Existential pre-image ⊩⎻" 'ext x = (fun12 ? ? (or_f_minus ? ?) (Orel x)).