qed.
interpretation "unary morphism comprehension with no proof" 'comprehension T P =
- (mk_unary_morphism T _ P _).
+ (mk_unary_morphism T ? P ?).
interpretation "unary morphism1 comprehension with no proof" 'comprehension T P =
- (mk_unary_morphism1 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}.
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).
+ (mk_unary_morphism s ? f p).
interpretation "unary morphism1 comprehension with proof" 'comprehension_by s \eta.f p =
- (mk_unary_morphism1 s _ f p).
+ (mk_unary_morphism1 s ? f p).
(* per il set-indexing vedere capitolo BPTools (foundational tools), Sect. 0.3.4 complete
lattices, Definizione 0.9 *)
(* USARE L'ESISTENZIALE DEBOLE *)
-(*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_leq : binary_morphism1 oa_P oa_P CPROP;
oa_overlap: binary_morphism1 oa_P oa_P CPROP;
- 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_meet: ∀I:SET.unary_morphism2 (I ⇒ oa_P) oa_P;
+ oa_join: ∀I:SET.unary_morphism2 (I ⇒ oa_P) oa_P;
oa_one: oa_P;
oa_zero: oa_P;
oa_leq_refl: ∀a:oa_P. oa_leq a a;
oa_leq_antisym: ∀a,b:oa_P.oa_leq a b → oa_leq b a → a = b;
oa_leq_trans: ∀a,b,c:oa_P.oa_leq a b → oa_leq b c → oa_leq a c;
oa_overlap_sym: ∀a,b:oa_P.oa_overlap a b → oa_overlap b a;
- (* Errore: = in oa_meet_inf e oa_join_sup *)
- oa_meet_inf: ∀I.∀p_i.∀p:oa_P.oa_leq p (oa_meet I p_i) → ∀i:I.oa_leq p (p_i i);
- 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_meet_inf:
+ ∀I:SET.∀p_i:I ⇒ oa_P.∀p:oa_P.
+ oa_leq p (oa_meet I p_i) = (∀i:I.oa_leq p (p_i i));
+ oa_join_sup: ∀I:SET.∀p_i:I ⇒ oa_P.∀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;
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:arrows2 SET1 I oa_P.
- oa_overlap p (oa_join I q) ⇔ ∃i:carr I.oa_overlap p (q i);
+ ∀I:SET.∀p.∀q: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
∀p,q.(∀r.oa_overlap p r → oa_overlap q r) → oa_leq p q
}.
-interpretation "o-algebra leq" 'leq a b = (fun21 ___ (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 = (fun21 ___ (oa_overlap _) a b).
+interpretation "o-algebra overlap" 'overlap a b = (fun21 ??? (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 $p }.
notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∧) \below (ident i ∈ I) break term 90 p)"
non associative with precedence 50 for @{ 'oa_meet_mk (λ${ident i}:$I.$p) }.
-(*
-notation < "hovbox(a ∧ b)" left associative with precedence 35
-for @{ 'oa_meet_mk (λ${ident i}:$_.match $i with [ true ⇒ $a | false ⇒ $b ]) }.
-*)
notation > "hovbox(∧ f)" non associative with precedence 60
for @{ 'oa_meet $f }.
-(*
-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 =
- (fun12 __ (oa_meet __) f).
+ (fun12 ?? (oa_meet ??) f).
interpretation "o-algebra meet with explicit function" 'oa_meet_mk f =
- (fun12 __ (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.
+notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∨) \below (\emsp) \nbsp term 90 p)"
+non associative with precedence 50 for @{ 'oa_join $p }.
+notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∨) \below (ident i ∈ I) break term 90 p)"
+non associative with precedence 50 for @{ 'oa_join_mk (λ${ident i}:$I.$p) }.
-definition hint4: ∀A. setoid2_OF_OAlgebra A → hint3 A.
- intros; apply t;
-qed.
-coercion hint4.
+notation > "hovbox(∨ f)" non associative with precedence 60
+for @{ 'oa_join $f }.
+interpretation "o-algebra join" 'oa_join f =
+ (fun12 ?? (oa_join ??) f).
+interpretation "o-algebra join with explicit function" 'oa_join_mk f =
+ (fun12 ?? (oa_join ??) (mk_unary_morphism ?? f ?)).
definition binary_meet : ∀O:OAlgebra. binary_morphism1 O O O.
intros; split;
qed.
interpretation "o-algebra binary meet" 'and a b =
- (fun21 ___ (binary_meet _) a b).
+ (fun21 ??? (binary_meet ?) a b).
+
+prefer coercion Type1_OF_OAlgebra.
+
+definition binary_join : ∀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; lapply (prop12 ? O (oa_join 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.
-coercion Type1_OF_OAlgebra nocomposites.
+interpretation "o-algebra binary join" 'or a b =
+ (fun21 ??? (binary_join ?) a b).
lemma oa_overlap_preservers_meet: ∀O:OAlgebra.∀p,q:O.p >< q → p >< (p ∧ q).
(* next change to avoid universe inconsistency *)
for @{ 'oa_join (mk_unary_morphism BOOL ? (λx__:bool.match x__ with [ true ⇒ $a | false ⇒ $b ]) (IF_THEN_ELSE_p ? $a $b)) }.
interpretation "o-algebra join" 'oa_join f =
- (fun12 __ (oa_join __) f).
+ (fun12 ?? (oa_join ??) f).
interpretation "o-algebra join with explicit function" 'oa_join_mk f =
- (fun12 __ (oa_join __) (mk_unary_morphism _ _ f _)).
-
-definition hint5: OAlgebra → objs2 SET1.
- intro; apply (oa_P o);
-qed.
-coercion hint5.
+ (fun12 ?? (oa_join ??) (mk_unary_morphism ?? f ?)).
record ORelation (P,Q : OAlgebra) : Type2 ≝ {
- or_f_ : P ⇒ Q;
- or_f_minus_star_ : P ⇒ Q;
- or_f_star_ : Q ⇒ P;
- or_f_minus_ : Q ⇒ P;
+ or_f_ : carr2 (P ⇒ Q);
+ or_f_minus_star_ : carr2(P ⇒ Q);
+ or_f_star_ : carr2(Q ⇒ P);
+ or_f_minus_ : carr2(Q ⇒ P);
or_prop1_ : ∀p,q. (or_f_ p ≤ q) = (p ≤ or_f_star_ q);
or_prop2_ : ∀p,q. (or_f_minus_ p ≤ q) = (p ≤ or_f_minus_star_ q);
or_prop3_ : ∀p,q. (or_f_ p >< q) = (p >< or_f_minus_ q)
| constructor 1;
(* tenere solo una uguaglianza e usare la proposizione 9.9 per
le altre (unicita' degli aggiunti e del simmetrico) *)
- [ apply (λp,q. And42 (eq2 ? (or_f_minus_star_ ?? p) (or_f_minus_star_ ?? q))
- (eq2 ? (or_f_minus_ ?? p) (or_f_minus_ ?? q))
- (eq2 ? (or_f_ ?? p) (or_f_ ?? q))
- (eq2 ? (or_f_star_ ?? p) (or_f_star_ ?? q)));
+ [ apply (λp,q. And42
+ (or_f_minus_star_ ?? p = or_f_minus_star_ ?? q)
+ (or_f_minus_ ?? p = or_f_minus_ ?? q)
+ (or_f_ ?? p = or_f_ ?? q)
+ (or_f_star_ ?? p = or_f_star_ ?? q));
| whd; simplify; intros; repeat split; intros; apply refl2;
| whd; simplify; intros; cases a; clear a; split;
intro a; apply sym1; generalize in match a;assumption;
| apply (.= (e3 a)); apply e7;]]]
qed.
+definition ORelation_of_ORelation_setoid :
+ ∀P,Q.ORelation_setoid P Q → ORelation P Q ≝ λP,Q,x.x.
+coercion ORelation_of_ORelation_setoid.
+
definition or_f_minus_star:
∀P,Q:OAlgebra.unary_morphism2 (ORelation_setoid P Q) (P ⇒ Q).
intros; constructor 1;
| intros; cases e; assumption]
qed.
-coercion or_f.
-
definition or_f_minus: ∀P,Q:OAlgebra.unary_morphism2 (ORelation_setoid P Q) (Q ⇒ P).
intros; constructor 1;
[ apply or_f_minus_;
| intros; cases e; assumption]
qed.
-lemma arrows1_OF_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → (P ⇒ Q).
-intros; apply (or_f ?? t);
-qed.
-
-coercion arrows1_OF_ORelation_setoid.
-
-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. ORelation_setoid B C → B → C.
-intros; apply ((fun11 ?? t) t1);
-qed.
-
-coercion uncurry_arrows 1.
-
-lemma hint6: ∀P,Q. Type_OF_setoid2 (hint5 P ⇒ hint5 Q) → unary_morphism1 P Q.
- intros; apply t;
+lemma arrows1_of_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → (P ⇒ Q).
+intros; apply (or_f ?? c);
qed.
-coercion hint6.
+coercion arrows1_of_ORelation_setoid.
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}.
notation "r \sup ⎻" non associative with precedence 90 for @{'OR_f_minus $r}.
notation > "r⎻" non associative with precedence 90 for @{'OR_f_minus $r}.
-interpretation "o-relation f⎻*" 'OR_f_minus_star r = (fun12 __ (or_f_minus_star _ _) r).
-interpretation "o-relation f⎻" 'OR_f_minus r = (fun12 __ (or_f_minus _ _) r).
-interpretation "o-relation f*" 'OR_f_star r = (fun12 __ (or_f_star _ _) r).
+interpretation "o-relation f⎻*" 'OR_f_minus_star r = (fun12 ?? (or_f_minus_star ? ?) r).
+interpretation "o-relation f⎻" 'OR_f_minus r = (fun12 ?? (or_f_minus ? ?) r).
+interpretation "o-relation f*" 'OR_f_star r = (fun12 ?? (or_f_star ? ?) r).
definition or_prop1 : ∀P,Q:OAlgebra.∀F:ORelation_setoid P Q.∀p,q.
(F p ≤ q) = (p ≤ F* q).
apply or_prop3;
]
| intros; split; simplify;
- [1,3: unfold umorphism_setoid_OF_ORelation_setoid; unfold arrows1_OF_ORelation_setoid; apply ((†e)‡(†e1));
+ [3: unfold arrows1_of_ORelation_setoid; apply ((†e)‡(†e1));
+ |1: apply ((†e)‡(†e1));
|2,4: apply ((†e1)‡(†e));]]
qed.
| intros; split; unfold ORelation_composition; simplify; apply id_neutral_right2;]
qed.
-lemma setoid1_of_OA: OA → setoid1.
- intro; apply (oa_P t);
+definition OAlgebra_of_objs2_OA: objs2 OA → OAlgebra ≝ λx.x.
+coercion OAlgebra_of_objs2_OA.
+
+definition ORelation_setoid_of_arrows2_OA:
+ ∀P,Q. arrows2 OA P Q → ORelation_setoid P Q ≝ λP,Q,c.c.
+coercion ORelation_setoid_of_arrows2_OA.
+
+prefer coercion Type_OF_objs2.
+
+(* alias symbol "eq" = "setoid1 eq". *)
+
+(* qui la notazione non va *)
+lemma leq_to_eq_join: ∀S:OA.∀p,q:S. p ≤ q → q = (binary_join ? p q).
+ intros;
+ apply oa_leq_antisym;
+ [ apply oa_density; intros;
+ apply oa_overlap_sym;
+ unfold binary_join; simplify;
+ apply (. (oa_join_split : ?));
+ exists; [ apply false ]
+ apply oa_overlap_sym;
+ assumption
+ | unfold binary_join; simplify;
+ apply (. (oa_join_sup : ?)); intro;
+ cases i; whd in ⊢ (? ? ? ? ? % ?);
+ [ assumption | apply oa_leq_refl ]]
+qed.
+
+lemma overlap_monotone_left: ∀S:OA.∀p,q,r:S. p ≤ q → p >< r → q >< r.
+ intros;
+ apply (. (leq_to_eq_join : ?)‡#);
+ [ apply f;
+ | skip
+ | apply oa_overlap_sym;
+ unfold binary_join; simplify;
+ apply (. (oa_join_split : ?));
+ exists [ apply true ]
+ apply oa_overlap_sym;
+ assumption; ]
+qed.
+
+(* Part of proposition 9.9 *)
+lemma f_minus_image_monotone: ∀S,T.∀R:arrows2 OA S T.∀p,q. p ≤ q → R⎻ p ≤ R⎻ q.
+ intros;
+ apply (. (or_prop2 : ?));
+ apply oa_leq_trans; [2: apply f; | skip | apply (. (or_prop2 : ?)^ -1); apply oa_leq_refl;]
+qed.
+
+(* Part of proposition 9.9 *)
+lemma f_minus_star_image_monotone: ∀S,T.∀R:arrows2 OA S T.∀p,q. p ≤ q → R⎻* p ≤ R⎻* q.
+ intros;
+ apply (. (or_prop2 : ?)^ -1);
+ apply oa_leq_trans; [3: apply f; | skip | apply (. (or_prop2 : ?)); apply oa_leq_refl;]
+qed.
+
+(* Part of proposition 9.9 *)
+lemma f_image_monotone: ∀S,T.∀R:arrows2 OA S T.∀p,q. p ≤ q → R p ≤ R q.
+ intros;
+ apply (. (or_prop1 : ?));
+ apply oa_leq_trans; [2: apply f; | skip | apply (. (or_prop1 : ?)^ -1); apply oa_leq_refl;]
+qed.
+
+(* Part of proposition 9.9 *)
+lemma f_star_image_monotone: ∀S,T.∀R:arrows2 OA S T.∀p,q. p ≤ q → R* p ≤ R* q.
+ intros;
+ apply (. (or_prop1 : ?)^ -1);
+ apply oa_leq_trans; [3: apply f; | skip | apply (. (or_prop1 : ?)); apply oa_leq_refl;]
+qed.
+
+lemma lemma_10_2_a: ∀S,T.∀R:arrows2 OA S T.∀p. p ≤ R⎻* (R⎻ p).
+ intros;
+ apply (. (or_prop2 : ?)^-1);
+ apply oa_leq_refl.
+qed.
+
+lemma lemma_10_2_b: ∀S,T.∀R:arrows2 OA S T.∀p. R⎻ (R⎻* p) ≤ p.
+ intros;
+ apply (. (or_prop2 : ?));
+ apply oa_leq_refl.
+qed.
+
+lemma lemma_10_2_c: ∀S,T.∀R:arrows2 OA S T.∀p. p ≤ R* (R p).
+ intros;
+ apply (. (or_prop1 : ?)^-1);
+ apply oa_leq_refl.
+qed.
+
+lemma lemma_10_2_d: ∀S,T.∀R:arrows2 OA S T.∀p. R (R* p) ≤ p.
+ intros;
+ apply (. (or_prop1 : ?));
+ apply oa_leq_refl.
qed.
-coercion setoid1_of_OA.
-lemma SET1_of_OA: OA → SET1.
- intro; whd; apply (setoid1_of_OA t);
+lemma lemma_10_3_a: ∀S,T.∀R:arrows2 OA S T.∀p. R⎻ (R⎻* (R⎻ p)) = R⎻ p.
+ intros; apply oa_leq_antisym;
+ [ apply lemma_10_2_b;
+ | apply f_minus_image_monotone;
+ apply lemma_10_2_a; ]
qed.
-coercion SET1_of_OA.
-lemma objs2_SET1_OF_OA: OA → objs2 SET1.
- intro; whd; apply (setoid1_of_OA t);
+lemma lemma_10_3_b: ∀S,T.∀R:arrows2 OA S T.∀p. R* (R (R* p)) = R* p.
+ intros; apply oa_leq_antisym;
+ [ apply f_star_image_monotone;
+ apply (lemma_10_2_d ?? R p);
+ | apply lemma_10_2_c; ]
qed.
-coercion objs2_SET1_OF_OA.
-lemma Type_OF_category2_OF_SET1_OF_OA: OA → Type_OF_category2 SET1.
- intro; apply (oa_P t);
+lemma lemma_10_3_c: ∀S,T.∀R:arrows2 OA S T.∀p. R (R* (R p)) = R p.
+ intros; apply oa_leq_antisym;
+ [ apply lemma_10_2_d;
+ | apply f_image_monotone;
+ apply (lemma_10_2_c ?? R p); ]
qed.
-coercion Type_OF_category2_OF_SET1_OF_OA.
+
+lemma lemma_10_3_d: ∀S,T.∀R:arrows2 OA S T.∀p. R⎻* (R⎻ (R⎻* p)) = R⎻* p.
+ intros; apply oa_leq_antisym;
+ [ apply f_minus_star_image_monotone;
+ apply (lemma_10_2_b ?? R p);
+ | apply lemma_10_2_a; ]
+qed.
+
+lemma lemma_10_4_a: ∀S,T.∀R:arrows2 OA S T.∀p. R⎻* (R⎻ (R⎻* (R⎻ p))) = R⎻* (R⎻ p).
+ intros; apply (†(lemma_10_3_a ?? R p));
+qed.
+
+lemma lemma_10_4_b: ∀S,T.∀R:arrows2 OA S T.∀p. R (R* (R (R* p))) = R (R* p).
+intros; unfold in ⊢ (? ? ? % %); apply (†(lemma_10_3_b ?? R p));
+qed.
+
+lemma oa_overlap_sym': ∀o:OA.∀U,V:o. (U >< V) = (V >< U).
+ intros; split; intro; apply oa_overlap_sym; assumption.
+qed.
\ No newline at end of file
projects / helm.git / blobdiff