+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 "o-algebra.ma".
-include "notation.ma".
-
-record Obasic_pair: Type2 ≝ {
- 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): Type2 ≝ {
- 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.
-
-(*
-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)).
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