(**************************************************************************)
include "o-algebra.ma".
-include "datatypes/categories.ma".
+include "notation.ma".
-record basic_pair: Type ≝
- { concr: OA;
- form: OA;
- rel: arrows1 ? concr form
+record Obasic_pair: Type2 ≝
+ { Oconcr: OA;
+ Oform: OA;
+ Orel: arrows2 ? Oconcr Oform
}.
-notation > "x ⊩ y" with precedence 45 for @{'Vdash2 $x $y ?}.
-notation < "x (⊩ \below c) y" with precedence 45 for @{'Vdash2 $x $y $c}.
-notation < "⊩ \sub c" with precedence 60 for @{'Vdash $c}.
-notation > "⊩ " with precedence 60 for @{'Vdash ?}.
-
-interpretation "basic pair relation indexed" 'Vdash2 x y c = (rel c x y).
-interpretation "basic pair relation (non applied)" 'Vdash c = (rel c).
+(* 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).
alias symbol "eq" = "setoid1 eq".
alias symbol "compose" = "category1 composition".
-record relation_pair (BP1,BP2: basic_pair): Type ≝
- { concr_rel: arrows1 ? (concr BP1) (concr BP2);
- form_rel: arrows1 ? (form BP1) (form BP2);
- commute: ⊩ ∘ concr_rel = form_rel ∘ ⊩
- }.
+(*DIFFER*)
-notation "hvbox (r \sub \c)" with precedence 90 for @{'concr_rel $r}.
-notation "hvbox (r \sub \f)" with precedence 90 for @{'form_rel $r}.
-
-interpretation "concrete relation" 'concr_rel r = (concr_rel __ r).
-interpretation "formal relation" 'form_rel r = (form_rel __ r).
+alias symbol "eq" = "setoid2 eq".
+alias symbol "compose" = "category2 composition".
+record Orelation_pair (BP1,BP2: Obasic_pair): Type2 ≝
+ { Oconcr_rel: arrows2 ? (Oconcr BP1) (Oconcr BP2);
+ Oform_rel: arrows2 ? (Oform BP1) (Oform BP2);
+ Ocommute: ⊩ ∘ Oconcr_rel = Oform_rel ∘ ⊩
+ }.
+
+(* FIX *)
+interpretation "o-concrete relation" 'concr_rel r = (Oconcr_rel ?? r).
+interpretation "o-formal relation" 'form_rel r = (Oform_rel ?? r).
-definition relation_pair_equality:
- ∀o1,o2. equivalence_relation1 (relation_pair o1 o2).
+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 refl1;
+ apply refl2;
| simplify;
intros 2;
- apply sym1;
+ apply sym2;
| simplify;
intros 3;
- apply trans1;
+ apply trans2;
]
qed.
-definition relation_pair_setoid: basic_pair → basic_pair → setoid1.
+(* qui setoid1 e' giusto: ma non lo e'!!! *)
+definition Orelation_pair_setoid: Obasic_pair → Obasic_pair → setoid2.
intros;
constructor 1;
- [ apply (relation_pair b b1)
- | apply relation_pair_equality
+ [ apply (Orelation_pair o o1)
+ | apply Orelation_pair_equality
]
qed.
-lemma eq_to_eq': ∀o1,o2.∀r,r': relation_pair_setoid o1 o2. r=r' → r \sub\f ∘ ⊩ = r'\sub\f ∘ ⊩.
+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=r' → r \sub\f ∘ ⊩ = r'\sub\f ∘ ⊩.
intros 5 (o1 o2 r r' H); change in H with (⊩ ∘ r\sub\c = ⊩ ∘ r'\sub\c);
- apply (.= ((commute ?? r) \sup -1));
+ apply (.= ((Ocommute ?? r) ^ -1));
apply (.= H);
- apply (.= (commute ?? r'));
- apply refl1;
+ apply (.= (Ocommute ?? r'));
+ apply refl2;
qed.
-definition id_relation_pair: ∀o:basic_pair. relation_pair o o.
+definition Oid_relation_pair: ∀o:Obasic_pair. Orelation_pair o o.
intro;
constructor 1;
- [1,2: apply id1;
- | lapply (id_neutral_right1 ? (concr o) ? (⊩)) as H;
- lapply (id_neutral_left1 ?? (form o) (⊩)) as H1;
+ [1,2: apply id2;
+ | lapply (id_neutral_right2 ? (Oconcr o) ? (⊩)) as H;
+ lapply (id_neutral_left2 ?? (Oform o) (⊩)) as H1;
apply (.= H);
- apply (H1 \sup -1);]
+ apply (H1^-1);]
qed.
-definition relation_pair_composition:
- ∀o1,o2,o3. binary_morphism1 (relation_pair_setoid o1 o2) (relation_pair_setoid o2 o3) (relation_pair_setoid o1 o3).
- intros;
- constructor 1;
- [ intros (r r1);
+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 (commute ?? r) as H;
- lapply (commute ?? r1) as H1;
- apply (.= ASSOC1);
+ | lapply (Ocommute ?? r) as H;
+ lapply (Ocommute ?? r1) as H1;
+ apply rule (.= ASSOC);
apply (.= #‡H1);
- apply (.= ASSOC1\sup -1);
+ apply rule (.= ASSOC ^ -1);
apply (.= H‡#);
- apply ASSOC1]
- | intros;
+ 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 H with (⊩ ∘ a \sub\c = ⊩ ∘ a' \sub\c);
- change in H1 with (⊩ ∘ b \sub\c = ⊩ ∘ b' \sub\c);
- apply (.= ASSOC1);
- apply (.= #‡H1);
- apply (.= #‡(commute ?? b'));
- apply (.= ASSOC1 \sup -1);
- apply (.= H‡#);
- apply (.= ASSOC1);
- apply (.= #‡(commute ?? b')\sup -1);
- apply (ASSOC1 \sup -1)]
+ 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 BP: category1.
- constructor 1;
- [ apply basic_pair
- | apply relation_pair_setoid
- | apply id_relation_pair
- | apply relation_pair_composition
- | intros;
+
+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 (ASSOC1‡#);
- | intros;
- change with (⊩ ∘ (a\sub\c ∘ (id_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
- apply ((id_neutral_right1 ????)‡#);
- | intros;
- change with (⊩ ∘ ((id_relation_pair o2)\sub\c ∘ a\sub\c) = ⊩ ∘ a\sub\c);
- apply ((id_neutral_left1 ????)‡#);]
+ 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).
| apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
qed.
-interpretation "fintersects" 'fintersects U V = (fun1 ___ (fintersects _) U V).
+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)).
| apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
qed.
-interpretation "fintersectsS" 'fintersects U V = (fun1 ___ (fintersectsS _) U V).
+interpretation "fintersectsS" 'fintersects U V = (fun1 ??? (fintersectsS ?) U V).
*)
(*
| 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 _)).
-*)
\ No newline at end of file
+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)).
\ No newline at end of file
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