(**************************************************************************)
include "o-algebra.ma".
-include "datatypes/categories.ma".
-record basic_pair: Type ≝
+record basic_pair: Type2 ≝
{ concr: OA;
form: OA;
- rel: arrows1 ? concr form
+ rel: arrows2 ? concr form
}.
-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).
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);
+(*DIFFER*)
+
+alias symbol "eq" = "setoid2 eq".
+alias symbol "compose" = "category2 composition".
+record relation_pair (BP1,BP2: basic_pair): Type2 ≝
+ { concr_rel: arrows2 ? (concr BP1) (concr BP2);
+ form_rel: arrows2 ? (form BP1) (form BP2);
commute: ⊩ ∘ concr_rel = form_rel ∘ ⊩
}.
interpretation "formal relation" 'form_rel r = (form_rel __ r).
definition relation_pair_equality:
- ∀o1,o2. equivalence_relation1 (relation_pair o1 o2).
+ ∀o1,o2. equivalence_relation2 (relation_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 relation_pair_setoid: basic_pair → basic_pair → setoid2.
intros;
constructor 1;
[ apply (relation_pair b b1)
]
qed.
+definition relation_pair_of_relation_pair_setoid:
+ ∀P,Q. relation_pair_setoid P Q → relation_pair P Q ≝ λP,Q,x.x.
+coercion relation_pair_of_relation_pair_setoid.
+
lemma eq_to_eq': ∀o1,o2.∀r,r': relation_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 (.= H);
apply (.= (commute ?? r'));
- apply refl1;
+ apply refl2;
qed.
definition id_relation_pair: ∀o:basic_pair. relation_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 ? (concr o) ? (⊩)) as H;
+ lapply (id_neutral_left2 ?? (form o) (⊩)) as H1;
apply (.= H);
apply (H1 \sup -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).
+ ∀o1,o2,o3. binary_morphism2 (relation_pair_setoid o1 o2) (relation_pair_setoid o2 o3) (relation_pair_setoid o1 o3).
intros;
constructor 1;
[ intros (r r1);
| apply (r1 \sub\f ∘ r \sub\f)
| lapply (commute ?? r) as H;
lapply (commute ?? r1) as H1;
- apply (.= ASSOC1);
+ apply rule (.= ASSOC);
apply (.= #‡H1);
- apply (.= ASSOC1\sup -1);
+ apply rule (.= ASSOC ^ -1);
apply (.= H‡#);
- apply ASSOC1]
+ apply rule ASSOC]
| 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);
+ 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 (.= #‡(commute ?? b'));
- apply (.= ASSOC1 \sup -1);
- apply (.= H‡#);
- apply (.= ASSOC1);
+ apply rule (.= ASSOC \sup -1);
+ apply (.= e‡#);
+ apply rule (.= ASSOC);
apply (.= #‡(commute ?? b')\sup -1);
- apply (ASSOC1 \sup -1)]
+ apply rule (ASSOC \sup -1)]
qed.
-definition BP: category1.
+definition BP: category2.
constructor 1;
[ apply basic_pair
| apply relation_pair_setoid
| intros;
change with (⊩ ∘ (a34\sub\c ∘ (a23\sub\c ∘ a12\sub\c)) =
⊩ ∘ ((a34\sub\c ∘ a23\sub\c) ∘ a12\sub\c));
- apply (ASSOC1‡#);
+ apply rule (ASSOC‡#);
| intros;
change with (⊩ ∘ (a\sub\c ∘ (id_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
- apply ((id_neutral_right1 ????)‡#);
+ apply ((id_neutral_right2 ????)‡#);
| intros;
change with (⊩ ∘ ((id_relation_pair o2)\sub\c ∘ a\sub\c) = ⊩ ∘ a\sub\c);
- apply ((id_neutral_left1 ????)‡#);]
+ apply ((id_neutral_left2 ????)‡#);]
qed.
+definition basic_pair_of_objs2_BP: objs2 BP → basic_pair ≝ λx.x.
+coercion basic_pair_of_objs2_BP.
+
+definition relation_pair_setoid_of_arrows2_BP:
+ ∀P,Q.arrows2 BP P Q → relation_pair_setoid P Q ≝ λP,Q,c.c.
+coercion relation_pair_setoid_of_arrows2_BP.
(*
definition BPext: ∀o: BP. form o ⇒ Ω \sup (concr o).
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
+*)
+
+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 _ _) (rel 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 _ _) (rel 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 _ _) (rel 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 _ _) (rel x)).
projects / helm.git / blobdiff