X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fformal_topology%2Foverlap%2Fo-algebra.ma;h=a84fc24770ac859b40040b56a3ecdaf826d5c8ca;hb=bd9161789ad35aae35b66e1f9bba660d8fde3c61;hp=cb211341427ed8cc934f3e390d76b611af124f80;hpb=6dac171410066642dd976b5cb88f83da9dfa70d8;p=helm.git 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 cb2113414..a84fc2477 100644 --- a/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma +++ b/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma @@ -12,10 +12,10 @@ (* *) (**************************************************************************) -include "datatypes/categories.ma". +include "categories.ma". include "logic/cprop_connectives.ma". -inductive bool : Type := true : bool | false : bool. +inductive bool : Type0 := true : bool | false : bool. lemma BOOL : objs1 SET. constructor 1; [apply bool] constructor 1; @@ -27,15 +27,17 @@ constructor 1; [apply bool] constructor 1; qed. lemma IF_THEN_ELSE_p : - ∀S:setoid.∀a,b:S.∀x,y:BOOL.x = y → + ∀S:setoid1.∀a,b:S.∀x,y:BOOL.x = y → (λm.match m with [ true ⇒ a | false ⇒ b ]) x = (λm.match m with [ true ⇒ a | false ⇒ b ]) y. whd in ⊢ (?→?→?→%→?); -intros; cases x in H; cases y; simplify; intros; try apply refl; whd in H; cases H; -qed. +intros; cases x in e; cases y; simplify; intros; try apply refl1; whd in e; cases e; +qed. interpretation "unary morphism comprehension with no proof" 'comprehension T P = (mk_unary_morphism T _ P _). +interpretation "unary morphism1 comprehension with no proof" 'comprehension T P = + (mk_unary_morphism1 T _ P _). notation > "hvbox({ ident i ∈ s | term 19 p | by })" with precedence 90 for @{ 'comprehension_by $s (λ${ident i}. $p) $by}. @@ -44,17 +46,27 @@ for @{ 'comprehension_by $s (λ${ident i}:$_. $p) $by}. interpretation "unary morphism comprehension with proof" 'comprehension_by s \eta.f p = (mk_unary_morphism s _ f p). +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 *) -(* Far salire SET usando setoidi1 *) -record OAlgebra : Type := { - oa_P :> SET; +(*alias symbol "comprehension_by" = "unary morphism comprehension with proof".*) +record OAlgebra : Type2 := { + oa_P :> SET1; oa_leq : binary_morphism1 oa_P oa_P CPROP; (* CPROP is setoid1, CPROP importante che sia small *) oa_overlap: binary_morphism1 oa_P oa_P CPROP; - oa_meet: ∀I:SET.unary_morphism (arrows1 SET I oa_P) oa_P; - oa_join: ∀I:SET.unary_morphism (arrows1 SET I oa_P) oa_P; + oa_meet: ∀I:SET.unary_morphism2 (arrows2 SET1 I oa_P) oa_P; + oa_join: ∀I:SET.unary_morphism2 (arrows2 SET1 I oa_P) oa_P; oa_one: oa_P; oa_zero: oa_P; oa_leq_refl: ∀a:oa_P. oa_leq a a; @@ -66,13 +78,13 @@ record OAlgebra : Type := { oa_join_sup: ∀I.∀p_i.∀p:oa_P.oa_leq (oa_join I p_i) p → ∀i:I.oa_leq (p_i i) p; oa_zero_bot: ∀p:oa_P.oa_leq oa_zero p; oa_one_top: ∀p:oa_P.oa_leq p oa_one; - (* preservers!! (typo) *) - oa_overlap_preservers_meet_: - ∀p,q.oa_overlap p q → oa_overlap p + oa_overlap_preserves_meet_: + ∀p,q:oa_P.oa_overlap p q → oa_overlap p (oa_meet ? { x ∈ BOOL | match x with [ true ⇒ p | false ⇒ q ] | IF_THEN_ELSE_p oa_P p q }); (* ⇔ deve essere =, l'esiste debole *) oa_join_split: - ∀I:SET.∀p.∀q:arrows1 SET I oa_P.oa_overlap p (oa_join I q) ⇔ ∃i:I.oa_overlap p (q i); + ∀I:SET.∀p.∀q:arrows2 SET1 I oa_P. + oa_overlap p (oa_join I q) ⇔ ∃i:I.oa_overlap p (q i); (*oa_base : setoid; 1) enum non e' il nome giusto perche' non e' suriettiva 2) manca (vedere altro capitolo) la "suriettivita'" come immagine di insiemi di oa_base @@ -83,11 +95,11 @@ record OAlgebra : Type := { ∀p,q.(∀r.oa_overlap p r → oa_overlap q r) → oa_leq p q }. -interpretation "o-algebra leq" 'leq a b = (fun1 ___ (oa_leq _) a b). +interpretation "o-algebra leq" 'leq a b = (fun21 ___ (oa_leq _) a b). notation "hovbox(a mpadded width -150% (>)< b)" non associative with precedence 45 for @{ 'overlap $a $b}. -interpretation "o-algebra overlap" 'overlap a b = (fun1 ___ (oa_overlap _) a b). +interpretation "o-algebra overlap" 'overlap a b = (fun22 ___ (oa_overlap _) a b). notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∧) \below (\emsp) \nbsp term 90 p)" non associative with precedence 50 for @{ 'oa_meet $p }. @@ -105,22 +117,35 @@ notation > "hovbox(a ∧ b)" left associative with precedence 50 for @{ 'oa_meet (mk_unary_morphism BOOL ? (λx__:bool.match x__ with [ true ⇒ $a | false ⇒ $b ]) (IF_THEN_ELSE_p ? $a $b)) }. *) interpretation "o-algebra meet" 'oa_meet f = - (fun_1 __ (oa_meet __) f). + (fun12 __ (oa_meet __) f). interpretation "o-algebra meet with explicit function" 'oa_meet_mk f = - (fun_1 __ (oa_meet __) (mk_unary_morphism _ _ f _)). + (fun12 __ (oa_meet __) (mk_unary_morphism _ _ f _)). + +definition hint3: OAlgebra → setoid1. + intro; apply (oa_P o); +qed. +coercion hint3. + +definition hint4: ∀A. setoid2_OF_OAlgebra A → hint3 A. + intros; apply t; +qed. +coercion hint4. definition binary_meet : ∀O:OAlgebra. binary_morphism1 O O O. intros; split; [ intros (p q); apply (∧ { x ∈ BOOL | match x with [ true ⇒ p | false ⇒ q ] | IF_THEN_ELSE_p ? p q }); -| intros; apply (prop_1 ?? (oa_meet O BOOL)); intro x; simplify; - cases x; simplify; assumption;] +| intros; lapply (prop12 ? O (oa_meet O BOOL)); + [2: apply ({ x ∈ BOOL | match x with [ true ⇒ a | false ⇒ b ] | IF_THEN_ELSE_p ? a b }); + |3: apply ({ x ∈ BOOL | match x with [ true ⇒ a' | false ⇒ b' ] | IF_THEN_ELSE_p ? a' b' }); + | apply Hletin;] + intro x; simplify; cases x; simplify; assumption;] qed. notation "hovbox(a ∧ b)" left associative with precedence 35 for @{ 'oa_meet_bin $a $b }. interpretation "o-algebra binary meet" 'oa_meet_bin a b = - (fun1 ___ (binary_meet _) a b). + (fun21 ___ (binary_meet _) a b). lemma oa_overlap_preservers_meet: ∀O:OAlgebra.∀p,q:O.p >< q → p >< (p ∧ q). intros; lapply (oa_overlap_preservers_meet_ O p q f); @@ -302,4 +327,4 @@ split; | 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. +qed. \ No newline at end of file