(* per il set-indexing vedere capitolo BPTools (foundational tools), Sect. 0.3.4 complete
lattices, Definizione 0.9 *)
(* USARE L'ESISTENZIALE DEBOLE *)
+
+definition if_then_else ≝ λT:Type.λe,t,f.match e return λ_.T with [ true ⇒ t | false ⇒ f].
+notation > "'If' term 19 e 'then' term 19 t 'else' term 90 f" non associative with precedence 19 for @{ 'if_then_else $e $t $f }.
+notation < "'If' \nbsp term 19 e \nbsp 'then' \nbsp term 19 t \nbsp 'else' \nbsp term 90 f \nbsp" non associative with precedence 19 for @{ 'if_then_else $e $t $f }.
+interpretation "Formula if_then_else" 'if_then_else e t f = (if_then_else ? e t f).
+
+notation > "hvbox(a break ≤ b)" non associative with precedence 45 for @{oa_leq $a $b}.
+notation > "a >< b" non associative with precedence 45 for @{oa_overlap $a $b}.
+notation > "⋁ p" non associative with precedence 45 for @{oa_join ? $p}.
+notation > "⋀ p" non associative with precedence 45 for @{oa_meet ? $p}.
+notation > "𝟙" non associative with precedence 90 for @{oa_one}.
+notation > "𝟘" non associative with precedence 90 for @{oa_zero}.
record OAlgebra : Type2 := {
oa_P :> SET1;
- oa_leq : binary_morphism1 oa_P oa_P CPROP;
- oa_overlap: binary_morphism1 oa_P oa_P CPROP;
- oa_meet: ∀I:SET.unary_morphism2 (I ⇒ oa_P) oa_P;
- oa_join: ∀I:SET.unary_morphism2 (I ⇒ oa_P) oa_P;
+ oa_leq : oa_P × oa_P ⇒_1 CPROP;
+ oa_overlap: oa_P × oa_P ⇒_1 CPROP;
+ oa_meet: ∀I:SET.(I ⇒_2 oa_P) ⇒_2. oa_P;
+ oa_join: ∀I:SET.(I ⇒_2 oa_P) ⇒_2. 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;
- 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 });
- oa_join_split:
- ∀I:SET.∀p.∀q:I ⇒ oa_P.
- oa_overlap p (oa_join I q) = (∃i:I.oa_overlap p (q i));
+ oa_leq_refl: ∀a:oa_P. a ≤ a;
+ oa_leq_antisym: ∀a,b:oa_P.a ≤ b → b ≤ a → a = b;
+ oa_leq_trans: ∀a,b,c:oa_P.a ≤ b → b ≤ c → a ≤ c;
+ oa_overlap_sym: ∀a,b:oa_P.a >< b → b >< a;
+ oa_meet_inf: ∀I:SET.∀p_i:I ⇒_2 oa_P.∀p:oa_P.p ≤ (⋀ p_i) = (∀i:I.p ≤ (p_i i));
+ oa_join_sup: ∀I:SET.∀p_i:I ⇒_2 oa_P.∀p:oa_P.(⋁ p_i) ≤ p = (∀i:I.p_i i ≤ p);
+ oa_zero_bot: ∀p:oa_P.𝟘 ≤ p;
+ oa_one_top: ∀p:oa_P.p ≤ 𝟙;
+ oa_overlap_preserves_meet_: ∀p,q:oa_P.p >< q →
+ p >< (⋀ { x ∈ BOOL | If x then p else q(*match x with [ true ⇒ p | false ⇒ q ]*) | IF_THEN_ELSE_p oa_P p q });
+ oa_join_split: ∀I:SET.∀p.∀q:I ⇒_2 oa_P.p >< (⋁ q) = (∃i:I.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
oa_enum : ums oa_base oa_P;
oa_density: ∀p,q.(∀i.oa_overlap p (oa_enum i) → oa_overlap q (oa_enum i)) → oa_leq p q
*)
- oa_density:
- ∀p,q.(∀r.oa_overlap p r → oa_overlap q r) → oa_leq p q
+ oa_density: ∀p,q.(∀r.p >< r → q >< r) → p ≤ q
}.
+notation "hvbox(a break ≤ b)" non associative with precedence 45 for @{ '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
interpretation "o-algebra meet" '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_morphism1 ?? f ?)).
notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∨) \below (\emsp) \nbsp term 90 p)"
non associative with precedence 50 for @{ 'oa_join $p }.
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.
+definition binary_meet : ∀O:OAlgebra. O × O ⇒_1 O.
intros; split;
[ intros (p q);
apply (∧ { x ∈ BOOL | match x with [ true ⇒ p | false ⇒ q ] | IF_THEN_ELSE_p ? p q });
prefer coercion Type1_OF_OAlgebra.
-definition binary_join : ∀O:OAlgebra. binary_morphism1 O O O.
+definition binary_join : ∀O:OAlgebra. O × O ⇒_1 O.
intros; split;
[ intros (p q);
apply (∨ { x ∈ BOOL | match x with [ true ⇒ p | false ⇒ q ] | IF_THEN_ELSE_p ? p q });
(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 *)
-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;
+intros; lapply (oa_overlap_preserves_meet_ O p q f) as H; clear f;
+(** screenshot "screenoa". *)
+assumption;
qed.
notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∨) \below (\emsp) \nbsp term 90 p)"
(fun12 ?? (oa_join ??) (mk_unary_morphism ?? f ?)).
record ORelation (P,Q : OAlgebra) : Type2 ≝ {
- 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_f_ : P ⇒_2 Q;
+ or_f_minus_star_ : P ⇒_2 Q;
+ or_f_star_ : Q ⇒_2 P;
+ or_f_minus_ : Q ⇒_2 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)
∀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).
+definition or_f_minus_star: ∀P,Q:OAlgebra.(ORelation_setoid P Q) ⇒_2 (P ⇒_2 Q).
intros; constructor 1;
[ apply or_f_minus_star_;
| intros; cases e; assumption]
qed.
-definition or_f: ∀P,Q:OAlgebra.unary_morphism2 (ORelation_setoid P Q) (P ⇒ Q).
+definition or_f: ∀P,Q:OAlgebra.(ORelation_setoid P Q) ⇒_2 (P ⇒_2 Q).
intros; constructor 1;
[ apply or_f_;
| intros; cases e; assumption]
qed.
-definition or_f_minus: ∀P,Q:OAlgebra.unary_morphism2 (ORelation_setoid P Q) (Q ⇒ P).
+definition or_f_minus: ∀P,Q:OAlgebra.(ORelation_setoid P Q) ⇒_2 (Q ⇒_2 P).
intros; constructor 1;
[ apply or_f_minus_;
| intros; cases e; assumption]
qed.
-definition or_f_star: ∀P,Q:OAlgebra.unary_morphism2 (ORelation_setoid P Q) (Q ⇒ P).
+definition or_f_star: ∀P,Q:OAlgebra.(ORelation_setoid P Q) ⇒_2 (Q ⇒_2 P).
intros; constructor 1;
[ apply or_f_star_;
| intros; cases e; assumption]
qed.
-lemma arrows1_of_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → (P ⇒ Q).
+lemma arrows1_of_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → (P ⇒_2 Q).
intros; apply (or_f ?? c);
qed.
coercion arrows1_of_ORelation_setoid.
qed.
definition ORelation_composition : ∀P,Q,R.
- binary_morphism2 (ORelation_setoid P Q) (ORelation_setoid Q R) (ORelation_setoid P R).
+ (ORelation_setoid P Q) × (ORelation_setoid Q R) ⇒_2 (ORelation_setoid P R).
intros;
constructor 1;
[ intros (F G);
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;
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
+qed.
projects / helm.git / blobdiff