--- /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 "formal_topology/basic_pairs.ma".
+include "formal_topology/o-basic_pairs.ma".
+include "formal_topology/relations_to_o-algebra.ma".
+
+definition o_basic_pair_of_basic_pair: basic_pair → Obasic_pair.
+ intro b;
+ constructor 1;
+ [ apply (POW (concr b));
+ | apply (POW (form b));
+ | apply (POW⎽⇒ ?); apply (rel b); ]
+qed.
+
+definition o_relation_pair_of_relation_pair:
+ ∀BP1,BP2. relation_pair BP1 BP2 →
+ Orelation_pair (o_basic_pair_of_basic_pair BP1) (o_basic_pair_of_basic_pair BP2).
+ intros;
+ constructor 1;
+ [ unfold o_basic_pair_of_basic_pair; simplify; apply (POW⎽⇒ ?); apply (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 ?? POW (concr BP1) (form BP1) (form BP2) (⊩\sub BP1) r\sub\f);
+ | apply commute;]]
+qed.
+
+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_ 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 (prop12);
+ apply (.= (respects_comp2 ?? POW (concr S) (concr T) (form T) (a\sub\c) (⊩\sub T))^-1);
+ apply sym2;
+ apply (.= (respects_comp2 ?? POW (concr S) (concr T) (form T) (b\sub\c) (⊩\sub T))^-1);
+ apply sym2;
+ apply prop12;
+ apply Eab;
+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);
+ | change in match or_f_star_ with (λq,w,x.fun12 ?? (or_f_star q w) x);]
+ simplify;
+ apply prop12;
+ apply prop22;[2,4,6,8: apply rule #;]
+ 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);
+ | change in match or_f_star_ with (λq,w,x.fun12 ?? (or_f_star q w) x);]
+ simplify;
+ apply prop12;
+ apply prop22;[2,4,6,8: apply rule #;]
+ 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: faithful2 ?? BP_to_OBP.
+ intros 5 (S T); 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 BP_to_OBP_full: full2 ?? BP_to_OBP.
+ intros 3 (S T);
+ 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