definition o_basic_pair_of_basic_pair: basic_pair → Obasic_pair.
intro b;
constructor 1;
- [ apply (map_objs2 ?? SUBSETS' (concr b));
- | apply (map_objs2 ?? SUBSETS' (form b));
- | apply (map_arrows2 ?? SUBSETS' (concr b) (form b) (rel b)); ]
+ [ apply (map_objs2 ?? POW (concr b));
+ | apply (map_objs2 ?? POW (form b));
+ | apply (map_arrows2 ?? POW (concr b) (form b) (rel b)); ]
qed.
definition o_relation_pair_of_relation_pair:
Orelation_pair (o_basic_pair_of_basic_pair BP1) (o_basic_pair_of_basic_pair BP2).
intros;
constructor 1;
- [ apply (map_arrows2 ?? SUBSETS' (concr BP1) (concr BP2) (r \sub \c));
- | apply (map_arrows2 ?? SUBSETS' (form BP1) (form BP2) (r \sub \f));
- | apply (.= (respects_comp2 ?? SUBSETS' (concr BP1) (concr BP2) (form BP2) r\sub\c (⊩\sub BP2) )^-1);
- cut (⊩ \sub BP2∘r \sub \c = r\sub\f ∘ ⊩ \sub BP1) as H;
+ [ apply (map_arrows2 ?? POW (concr BP1) (concr BP2) (r \sub \c));
+ | apply (map_arrows2 ?? POW (form BP1) (form BP2) (r \sub \f));
+ | apply (.= (respects_comp2 ?? POW (concr BP1) (concr BP2) (form BP2) r\sub\c (⊩\sub BP2) )^-1);
+ cut ( ⊩ \sub BP2 ∘ r \sub \c =_12 r\sub\f ∘ ⊩ \sub BP1) as H;
[ apply (.= †H);
- apply (respects_comp2 ?? SUBSETS' (concr BP1) (form BP1) (form BP2) (⊩\sub BP1) r\sub\f);
+ apply (respects_comp2 ?? POW (concr BP1) (form BP1) (form BP2) (⊩\sub BP1) r\sub\f);
| apply commute;]]
qed.
-definition BP_to_OBP: carr3 (arrows3 CAT2 (category2_of_category1 BP) OBP).
- constructor 1;
- [ apply o_basic_pair_of_basic_pair;
- | intros; constructor 1;
- [ apply (o_relation_pair_of_relation_pair S T);
- | intros (a b Eab); split; unfold o_relation_pair_of_relation_pair; simplify;
+lemma o_relation_pair_of_relation_pair_is_morphism :
+ ∀S,T:category2_of_category1 BP.
+ ∀a,b:arrows2 (category2_of_category1 BP) S T.a=b →
+ (eq2 (arrows2 OBP (o_basic_pair_of_basic_pair S) (o_basic_pair_of_basic_pair T)))
+ (o_relation_pair_of_relation_pair S T a) (o_relation_pair_of_relation_pair S T b).
+intros 2 (S T);
+ intros (a b Eab); split; unfold o_relation_pair_of_relation_pair; simplify;
unfold o_basic_pair_of_basic_pair; simplify;
[ change in match or_f_minus_star_ with (λq,w,x.fun12 ?? (or_f_minus_star q w) x);
| change in match or_f_minus_ with (λq,w,x.fun12 ?? (or_f_minus q w) x);
| change in match or_f_star_ with (λq,w,x.fun12 ?? (or_f_star q w) x);]
simplify;
apply (prop12);
- apply (.= (respects_comp2 ?? SUBSETS' (concr S) (concr T) (form T) (a\sub\c) (⊩\sub T))^-1);
+ apply (.= (respects_comp2 ?? POW (concr S) (concr T) (form T) (a\sub\c) (⊩\sub T))^-1);
apply sym2;
- apply (.= (respects_comp2 ?? SUBSETS' (concr S) (concr T) (form T) (b\sub\c) (⊩\sub T))^-1);
+ apply (.= (respects_comp2 ?? POW (concr S) (concr T) (form T) (b\sub\c) (⊩\sub T))^-1);
apply sym2;
apply prop12;
apply Eab;
- ]
- | simplify; intros; whd; split;
+qed.
+
+lemma o_relation_pair_of_relation_pair_morphism :
+ ∀S,T:category2_of_category1 BP.
+ unary_morphism2 (arrows2 (category2_of_category1 BP) S T)
+ (arrows2 OBP (o_basic_pair_of_basic_pair S) (o_basic_pair_of_basic_pair T)).
+intros (S T);
+ constructor 1;
+ [ apply (o_relation_pair_of_relation_pair S T);
+ | apply (o_relation_pair_of_relation_pair_is_morphism S T)]
+qed.
+
+lemma o_relation_pair_of_relation_pair_morphism_respects_id:
+ ∀o:category2_of_category1 BP.
+ o_relation_pair_of_relation_pair_morphism o o (id2 (category2_of_category1 BP) o)
+ = id2 OBP (o_basic_pair_of_basic_pair o).
+ simplify; intros; whd; split;
[ change in match or_f_minus_star_ with (λq,w,x.fun12 ?? (or_f_minus_star q w) x);
| change in match or_f_minus_ with (λq,w,x.fun12 ?? (or_f_minus q w) x);
| change in match or_f_ with (λq,w,x.fun12 ?? (or_f q w) x);
simplify;
apply prop12;
apply prop22;[2,4,6,8: apply rule #;]
- apply (respects_id2 ?? SUBSETS' (concr o));
- | simplify; intros; whd; split;
+ apply (respects_id2 ?? POW (concr o));
+qed.
+
+lemma o_relation_pair_of_relation_pair_morphism_respects_comp:
+ ∀o1,o2,o3:category2_of_category1 BP.
+ ∀f1:arrows2 (category2_of_category1 BP) o1 o2.
+ ∀f2:arrows2 (category2_of_category1 BP) o2 o3.
+ (eq2 (arrows2 OBP (o_basic_pair_of_basic_pair o1) (o_basic_pair_of_basic_pair o3)))
+ (o_relation_pair_of_relation_pair_morphism o1 o3 (f2 ∘ f1))
+ (comp2 OBP ???
+ (o_relation_pair_of_relation_pair_morphism o1 o2 f1)
+ (o_relation_pair_of_relation_pair_morphism o2 o3 f2)).
+ simplify; intros; whd; split;
[ change in match or_f_minus_star_ with (λq,w,x.fun12 ?? (or_f_minus_star q w) x);
| change in match or_f_minus_ with (λq,w,x.fun12 ?? (or_f_minus q w) x);
| change in match or_f_ with (λq,w,x.fun12 ?? (or_f q w) x);
simplify;
apply prop12;
apply prop22;[2,4,6,8: apply rule #;]
- apply (respects_comp2 ?? SUBSETS' (concr o1) (concr o2) (concr o3) f1\sub\c f2\sub\c);]
+ apply (respects_comp2 ?? POW (concr o1) (concr o2) (concr o3) f1\sub\c f2\sub\c);
qed.
+definition BP_to_OBP: carr3 (arrows3 CAT2 (category2_of_category1 BP) OBP).
+ constructor 1;
+ [ apply o_basic_pair_of_basic_pair;
+ | intros; apply o_relation_pair_of_relation_pair_morphism;
+ | apply o_relation_pair_of_relation_pair_morphism_respects_id;
+ | apply o_relation_pair_of_relation_pair_morphism_respects_comp;]
+qed.
-(*
theorem BP_to_OBP_faithful:
∀S,T.∀f,g:arrows2 (category2_of_category1 BP) S T.
map_arrows2 ?? BP_to_OBP ?? f = map_arrows2 ?? BP_to_OBP ?? g → f=g.
- intros; unfold BP_to_OBP in e; simplify in e; cases e;
- unfold orelation_of_relation in e3; simplify in e3; clear e e1 e2 e4;
- intros 2; change in match or_f_ in e3 with (λq,w,x.fun12 ?? (or_f q w) x);
- simplify in e3; STOP lapply (e3 (singleton ? x)); cases Hletin;
- split; intro; [ lapply (s y); | lapply (s1 y); ]
- [2,4: exists; [1,3:apply x] split; [1,3: assumption |*: change with (x=x); apply rule #]
- |*: cases Hletin1; cases x1; change in f3 with (eq1 ? x w); apply (. f3‡#); assumption;]
+ intros; change with ( (⊩) ∘ f \sub \c = (⊩) ∘ g \sub \c);
+ apply (POW_faithful);
+ apply (.= respects_comp2 ?? POW (concr S) (concr T) (form T) f \sub \c (⊩ \sub T));
+ apply sym2;
+ apply (.= respects_comp2 ?? POW (concr S) (concr T) (form T) g \sub \c (⊩ \sub T));
+ apply sym2;
+ apply e;
qed.
-*)
-(*
-theorem SUBSETS_full: ∀S,T.∀f. exT22 ? (λg. map_arrows2 ?? BP_to_OBP S T g = f).
- intros; exists;
-
-*)
\ No newline at end of file
+theorem BP_to_OBP_full:
+ ∀S,T.∀f. exT22 ? (λg. map_arrows2 ?? BP_to_OBP S T g = f).
+ intros;
+ cases (POW_full (concr S) (concr T) (Oconcr_rel ?? f)) (gc Hgc);
+ cases (POW_full (form S) (form T) (Oform_rel ?? f)) (gf Hgf);
+ exists[
+ constructor 1; [apply gc|apply gf]
+ apply (POW_faithful);
+ apply (let xxxx ≝POW in .= respects_comp2 ?? POW (concr S) (concr T) (form T) gc (rel T));
+ apply rule (.= Hgc‡#);
+ apply (.= Ocommute ?? f);
+ apply (.= #‡Hgf^-1);
+ apply (let xxxx ≝POW in (respects_comp2 ?? POW (concr S) (form S) (form T) (rel S) gf)^-1)]
+ split;
+ [ change in match or_f_minus_star_ with (λq,w,x.fun12 ?? (or_f_minus_star q w) x);
+ | change in match or_f_minus_ with (λq,w,x.fun12 ?? (or_f_minus q w) x);
+ | change in match or_f_ with (λq,w,x.fun12 ?? (or_f q w) x);
+ | change in match or_f_star_ with (λq,w,x.fun12 ?? (or_f_star q w) x);]
+ simplify; apply (†(Hgc‡#));
+qed.
projects / helm.git / blobdiff