From 1c406089d385be2d444308a783bc051bd28be463 Mon Sep 17 00:00:00 2001 From: Claudio Sacerdoti Coen Date: Sat, 3 Jan 2009 16:16:09 +0000 Subject: [PATCH] More reorganization. --- .../formal_topology/overlap/categories.ma | 17 +++-- .../formal_topology/overlap/o-algebra.ma | 64 ++++++++++--------- .../formal_topology/overlap/o-saturations.ma | 14 ++-- 3 files changed, 53 insertions(+), 42 deletions(-) diff --git a/helm/software/matita/contribs/formal_topology/overlap/categories.ma b/helm/software/matita/contribs/formal_topology/overlap/categories.ma index ea246ef6c..f09e0ee6c 100644 --- a/helm/software/matita/contribs/formal_topology/overlap/categories.ma +++ b/helm/software/matita/contribs/formal_topology/overlap/categories.ma @@ -181,6 +181,7 @@ interpretation "prop11" 'prop1 c = (prop11 _____ c). interpretation "prop12" 'prop1 c = (prop12 _____ c). interpretation "prop2" 'prop2 l r = (prop2 ________ l r). interpretation "prop21" 'prop2 l r = (prop21 ________ l r). +interpretation "prop22" 'prop2 l r = (prop22 ________ l r). interpretation "refl" 'refl = (refl ___). interpretation "refl1" 'refl = (refl1 ___). interpretation "refl2" 'refl = (refl2 ___). @@ -288,8 +289,8 @@ notation "'ASSOC'" with precedence 90 for @{'assoc}. notation "'ASSOC1'" with precedence 90 for @{'assoc1}. notation "'ASSOC2'" with precedence 90 for @{'assoc2}. -interpretation "category1 composition" 'compose x y = (fun22 ___ (comp2 ____) y x). -interpretation "category1 assoc" 'assoc1 = (comp_assoc2 ________). +interpretation "category2 composition" 'compose x y = (fun22 ___ (comp2 ____) y x). +interpretation "category2 assoc" 'assoc1 = (comp_assoc2 ________). interpretation "category1 composition" 'compose x y = (fun21 ___ (comp1 ____) y x). interpretation "category1 assoc" 'assoc1 = (comp_assoc1 ________). interpretation "category composition" 'compose x y = (fun2 ___ (comp ____) y x). @@ -387,10 +388,18 @@ definition prop11_SET1 : intros; apply (prop11 A B w a b e); qed. -definition hint: Type_OF_category2 SET1 → Type1. +definition setoid2_OF_category2: Type_OF_category2 SET1 → setoid2. + intro; apply (setoid2_of_setoid1 t); qed. +coercion setoid2_OF_category2. + +definition objs2_OF_category1: Type_OF_category1 SET → objs2 SET1. + intro; apply (setoid1_of_setoid t); qed. +coercion objs2_OF_category1. + +definition Type1_OF_SET1: Type_OF_category2 SET1 → Type1. intro; whd in t; apply (carr1 t); qed. -coercion hint. +coercion Type1_OF_SET1. interpretation "SET dagger" 'prop1 h = (prop11_SET1 _ _ _ _ _ h). interpretation "unary morphism1" 'Imply a b = (arrows2 SET1 a b). diff --git a/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma b/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma index ce9583da3..029e93258 100644 --- a/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma +++ b/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma @@ -49,14 +49,6 @@ interpretation "unary morphism comprehension with proof" 'comprehension_by s \et interpretation "unary morphism1 comprehension with proof" 'comprehension_by s \eta.f p = (mk_unary_morphism1 s _ f p). -definition hint: Type_OF_category2 SET1 → setoid2. - intro; apply (setoid2_of_setoid1 t); qed. -coercion hint. - -definition hint2: Type_OF_category1 SET → objs2 SET1. - intro; apply (setoid1_of_setoid t); qed. -coercion hint2. - (* per il set-indexing vedere capitolo BPTools (foundational tools), Sect. 0.3.4 complete lattices, Definizione 0.9 *) (* USARE L'ESISTENZIALE DEBOLE *) @@ -147,7 +139,11 @@ for @{ 'oa_meet_bin $a $b }. interpretation "o-algebra binary meet" 'oa_meet_bin a b = (fun21 ___ (binary_meet _) a b). +coercion Type1_OF_OAlgebra nocomposites. + lemma oa_overlap_preservers_meet: ∀O:OAlgebra.∀p,q:O.p >< q → p >< (p ∧ q). +(* next change to avoid universe inconsistency *) +change in ⊢ (?→%→%→?) with (Type1_OF_OAlgebra O); intros; lapply (oa_overlap_preserves_meet_ O p q f); lapply (prop21 O O CPROP (oa_overlap O) p p ? (p ∧ q) # ?); [3: apply (if ?? (Hletin1)); apply Hletin;|skip] apply refl1; @@ -239,28 +235,29 @@ qed. coercion arrows1_OF_ORelation_setoid. -lemma umorphism_OF_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → P ⇒ Q. +lemma umorphism_OF_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → unary_morphism1 P Q. intros; apply (or_f ?? t); qed. coercion umorphism_OF_ORelation_setoid. +lemma umorphism_setoid_OF_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → unary_morphism1_setoid1 P Q. +intros; apply (or_f ?? t); +qed. + +coercion umorphism_setoid_OF_ORelation_setoid. -lemma uncurry_arrows : ∀B,C. arrows1 SET B C → B → C. -intros; apply ((fun1 ?? t) t1); +lemma uncurry_arrows : ∀B,C. ORelation_setoid B C → B → C. +intros; apply ((fun11 ?? t) t1); qed. coercion uncurry_arrows 1. -lemma hint6 : ∀P,Q. arrows1 SET P Q → P ⇒ Q. intros; apply t;qed. +lemma hint6: ∀P,Q. Type_OF_setoid2 (hint5 P ⇒ hint5 Q) → unary_morphism1 P Q. + intros; apply t; +qed. coercion hint6. -(* -lemma hint2: OAlgebra → setoid. intros; apply (oa_P o). qed. -coercion hint2 nocomposites. -*) - - notation "r \sup *" non associative with precedence 90 for @{'OR_f_star $r}. notation > "r *" non associative with precedence 90 for @{'OR_f_star $r}. @@ -290,13 +287,13 @@ intros; apply (or_prop3_ ?? F p q); qed. definition ORelation_composition : ∀P,Q,R. - binary_morphism1 (ORelation_setoid P Q) (ORelation_setoid Q R) (ORelation_setoid P R). + binary_morphism2 (ORelation_setoid P Q) (ORelation_setoid Q R) (ORelation_setoid P R). intros; constructor 1; [ intros (F G); constructor 1; [ apply (G ∘ F); - | apply (G⎻* ∘ F⎻* ); + | apply rule (G⎻* ∘ F⎻* ); | apply (F* ∘ G* ); | apply (F⎻ ∘ G⎻); | intros; @@ -312,24 +309,29 @@ constructor 1; apply or_prop3; ] | intros; split; simplify; - [1,3: unfold arrows1_OF_ORelation_setoid; apply ((†H)‡(†H1)); - |2,4: apply ((†H1)‡(†H));]] + [1,3: unfold umorphism_setoid_OF_ORelation_setoid; unfold arrows1_OF_ORelation_setoid; apply ((†e)‡(†e1)); + |2,4: apply ((†e1)‡(†e));]] qed. -definition OA : category1. +definition OA : category2. split; [ apply (OAlgebra); | intros; apply (ORelation_setoid o o1); | intro O; split; - [1,2,3,4: apply id1; + [1,2,3,4: apply id2; |5,6,7:intros; apply refl1;] | apply ORelation_composition; | intros (P Q R S F G H); split; [ change with (H⎻* ∘ G⎻* ∘ F⎻* = H⎻* ∘ (G⎻* ∘ F⎻* )); - apply (comp_assoc1 ????? (F⎻* ) (G⎻* ) (H⎻* )); - | apply ((comp_assoc1 ????? (H⎻) (G⎻) (F⎻))^-1); - | apply ((comp_assoc1 ????? F G H)^-1); - | apply ((comp_assoc1 ????? H* G* F* ));] -| intros; split; unfold ORelation_composition; simplify; apply id_neutral_left1; -| intros; split; unfold ORelation_composition; simplify; apply id_neutral_right1;] -qed. \ No newline at end of file + apply (comp_assoc2 ????? (F⎻* ) (G⎻* ) (H⎻* )); + | apply ((comp_assoc2 ????? (H⎻) (G⎻) (F⎻))^-1); + | apply ((comp_assoc2 ????? F G H)^-1); + | apply ((comp_assoc2 ????? H* G* F* ));] +| intros; split; unfold ORelation_composition; simplify; apply id_neutral_left2; +| intros; split; unfold ORelation_composition; simplify; apply id_neutral_right2;] +qed. + +lemma setoid1_of_OA: OA → setoid1. + intro; apply (oa_P t); +qed. +coercion setoid1_of_OA. diff --git a/helm/software/matita/contribs/formal_topology/overlap/o-saturations.ma b/helm/software/matita/contribs/formal_topology/overlap/o-saturations.ma index 22c8fbdea..0511edd34 100644 --- a/helm/software/matita/contribs/formal_topology/overlap/o-saturations.ma +++ b/helm/software/matita/contribs/formal_topology/overlap/o-saturations.ma @@ -16,28 +16,28 @@ include "o-algebra.ma". alias symbol "eq" = "setoid1 eq". definition is_saturation ≝ - λC:OA.λA:unary_morphism (oa_P C) (oa_P C). + λC:OA.λA:unary_morphism1 C C. ∀U,V. (U ≤ A V) = (A U ≤ A V). definition is_reduction ≝ - λC:OA.λJ:unary_morphism (oa_P C) (oa_P C). + λC:OA.λJ:unary_morphism1 C C. ∀U,V. (J U ≤ V) = (J U ≤ J V). theorem saturation_expansive: ∀C,A. is_saturation C A → ∀U. U ≤ A U. - intros; apply (fi ?? (H ??)); apply (oa_leq_refl C). + intros; apply (fi ?? (i ??)); apply (oa_leq_refl C). qed. theorem saturation_monotone: ∀C,A. is_saturation C A → ∀U,V. U ≤ V → A U ≤ A V. - intros; apply (if ?? (H ??)); apply (oa_leq_trans C); + intros; apply (if ?? (i ??)); apply (oa_leq_trans C); [apply V|3: apply saturation_expansive ] assumption. qed. theorem saturation_idempotent: ∀C,A. is_saturation C A → ∀U. - eq (oa_P C) (A (A U)) (A U). + eq1 C (A (A U)) (A U). intros; apply (oa_leq_antisym C); - [ apply (if ?? (H (A U) U)); apply (oa_leq_refl C). + [ apply (if ?? (i (A U) U)); apply (oa_leq_refl C). | apply saturation_expansive; assumption] -qed. +qed. \ No newline at end of file -- 2.39.2