ndefinition overlaps ≝ λA.λU,V.∃x:A.x ∈ U ∧ x ∈ V.
interpretation "overlaps" 'overlaps U V = (overlaps ? U V).
-ndefinition intersect ≝ λA.λU,V:Ω \sup A.{ x | x ∈ U ∧ x ∈ V }.
+ndefinition intersect ≝ λA.λU,V:Ω^A.{ x | x ∈ U ∧ x ∈ V }.
interpretation "intersect" 'intersects U V = (intersect ? U V).
-ndefinition union ≝ λA.λU,V:Ω \sup A.{ x | x ∈ U ∨ x ∈ V }.
+ndefinition union ≝ λA.λU,V:Ω^A.{ x | x ∈ U ∨ x ∈ V }.
interpretation "union" 'union U V = (union ? U V).
-ndefinition big_union ≝ λA,B.λT:Ω \sup A.λf:A → Ω \sup B.{ x | ∃i. i ∈ T ∧ x ∈ f i }.
+ndefinition big_union ≝ λA,B.λT:Ω^A.λf:A → Ω^B.{ x | ∃i. i ∈ T ∧ x ∈ f i }.
-ndefinition big_intersection ≝ λA,B.λT:Ω \sup A.λf:A → Ω \sup B.{ x | ∀i. i ∈ T → x ∈ f i }.
+ndefinition big_intersection ≝ λA,B.λT:Ω^A.λf:A → Ω^B.{ x | ∀i. i ∈ T → x ∈ f i }.
-ndefinition full_set: ∀A. Ω \sup A ≝ λA.{ x | True }.
-ncoercion full_set : ∀A:Type[0]. Ω \sup A ≝ full_set on A: Type[0] to (Ω \sup ?).
+ndefinition full_set: ∀A. Ω^A ≝ λA.{ x | True }.
+ncoercion full_set : ∀A:Type[0]. Ω^A ≝ full_set on A: Type[0] to (Ω^?).
-nlemma subseteq_refl: ∀A.∀S: Ω \sup A. S ⊆ S.
+nlemma subseteq_refl: ∀A.∀S: Ω^A. S ⊆ S.
#A; #S; #x; #H; nassumption.
nqed.
-nlemma subseteq_trans: ∀A.∀S,T,U: Ω \sup A. S ⊆ T → T ⊆ U → S ⊆ U.
+nlemma subseteq_trans: ∀A.∀S,T,U: Ω^A. S ⊆ T → T ⊆ U → S ⊆ U.
#A; #S; #T; #U; #H1; #H2; #x; #P; napply H2; napply H1; nassumption.
nqed.
include "properties/relations1.ma".
-ndefinition seteq: ∀A. equivalence_relation1 (Ω \sup A).
+ndefinition seteq: ∀A. equivalence_relation1 (Ω^A).
#A; napply mk_equivalence_relation1
[ napply (λS,S'. S ⊆ S' ∧ S' ⊆ S)
| #S; napply conj; napply subseteq_refl
| napply (qseteq A) ]
nqed.
-unification hint 0 ≔ A : ? ⊢ carr1 (qpowerclass_setoid A) ≡ qpowerclass A.
+unification hint 0 ≔ A : ? ⊢
+ carr1 (qpowerclass_setoid A) ≡ qpowerclass A.
nlemma mem_ok: ∀A. binary_morphism1 (setoid1_of_setoid A) (qpowerclass_setoid A) CPROP.
#A; napply mk_binary_morphism1
alias symbol "hint_decl" = "hint_decl_Type1".
unification hint 0 ≔
A : setoid, B : qpowerclass A, C : qpowerclass A ⊢
- pc A (fun21 ??? (intersect_ok A) B C) ≡ intersect ? (pc ? B) (pc ? C).
+ pc A (intersect_ok A B C) ≡ intersect ? (pc ? B) (pc ? C).
-(* hints can pass under mem *)
-unification hint 0 (
- ∀A,B,x.
- let C ≝ B in
- (λa,b.Prop) (mem A B x) (mem A C x)).
+(* hints can pass under mem *) (* ??? XXX why is it needed? *)
+unification hint 0 ≔ A:?, B:?, x:?;
+ C ≟ B
+ (*---------------------*) ⊢
+ mem A B x ≡ mem A C x.
nlemma test: ∀A:setoid. ∀U,V:qpowerclass A. ∀x,x':setoid1_of_setoid A. x=x' → x ∈ U ∩ V → x' ∈ U ∩ V.
#A; #U; #V; #x; #x'; #H; #p; napply (. (H^-1‡#)); nassumption.
nqed.
-(*
-(* qui non funziona una cippa *)
-ndefinition image: ∀A,B. (carr A → carr B) → Ω \sup A → Ω \sup B ≝
- λA,B:setoid.λf:carr A → carr B.λSa:Ω \sup A.
- {y | ∃x. x ∈ Sa ∧ eq_rel (carr B) (eq B) ? ?(*(f x) y*)}.
- ##[##2: napply (f x); ##|##3: napply y]
- #a; #a'; #H; nwhd; nnormalize; (* per togliere setoid1_of_setoid *) napply (mk_iff ????);
- *; #x; #Hx; napply (ex_intro … x)
- [ napply (. (#‡(#‡#)));
+ndefinition image: ∀A,B. (carr A → carr B) → Ω^A → Ω^B ≝
+ λA,B:setoid.λf:carr A → carr B.λSa:Ω^A.
+ {y | ∃x. x ∈ Sa ∧ eq_rel (carr B) (eq B) (f x) y}.
-ndefinition counter_image: ∀A,B. (A → B) → Ω \sup B → Ω \sup A ≝
+ndefinition counter_image: ∀A,B. (carr A → carr B) → Ω^B → Ω^A ≝
λA,B,f,Sb. {x | ∃y. y ∈ Sb ∧ f x = y}.
-*)
(******************* compatible equivalence relations **********************)
nrecord compatible_equivalence_relation (A: setoid) : Type[1] ≝
{ rel:> equivalence_relation A;
- compatibility: ∀x,x':A. x=x' → eq_rel ? rel x x' (* coercion qui non va *)
+ compatibility: ∀x,x':A. x=x' → rel x x'
+ (* coercion qui non andava per via di un Failure invece di Uncertain
+ ritornato dall'unificazione per il problema:
+ ?[] A =?= ?[Γ]->?[Γ+1]
+ *)
}.
ndefinition quotient: ∀A. compatible_equivalence_relation A → setoid.
ndefinition canonical_proj: ∀A,R. unary_morphism A (quotient A R).
#A; #R; napply mk_unary_morphism
- [ napply (λx.x) | #a; #a'; #H; napply (compatibility ? R … H) ]
+ [ napply (λx.x) | #a; #a'; #H; napply (compatibility … R … H) ]
nqed.
ndefinition quotiented_mor:
∀A,B.∀f:unary_morphism A B.
- unary_morphism (quotient ? (eqrel_of_morphism ?? f)) B.
+ unary_morphism (quotient … (eqrel_of_morphism … f)) B.
#A; #B; #f; napply mk_unary_morphism
[ napply f | #a; #a'; #H; nassumption]
nqed.
nlemma first_omomorphism_theorem_functions1:
∀A,B.∀f: unary_morphism A B.
- ∀x. f x = quotiented_mor ??? (canonical_proj ? (eqrel_of_morphism ?? f) x).
+ ∀x. f x = quotiented_mor … (canonical_proj … (eqrel_of_morphism … f) x).
#A; #B; #f; #x; napply refl;
nqed.
∀x,x'. x ∈ S → x' ∈ S → f x = f x' → x = x'.
nlemma first_omomorphism_theorem_functions2:
- ∀A,B.∀f: unary_morphism A B. surjective ?? (Full_set ?) (Full_set ?) (canonical_proj ? (eqrel_of_morphism ?? f)).
+ ∀A,B.∀f: unary_morphism A B.
+ surjective … (Full_set ?) (Full_set ?) (canonical_proj ? (eqrel_of_morphism … f)).
#A; #B; #f; nwhd; #y; #Hy; napply (ex_intro … y); napply conj
[ napply I | napply refl]
nqed.
nlemma first_omomorphism_theorem_functions3:
- ∀A,B.∀f: unary_morphism A B. injective ?? (Full_set ?) (quotiented_mor ?? f).
+ ∀A,B.∀f: unary_morphism A B.
+ injective … (Full_set ?) (quotiented_mor … f).
#A; #B; #f; nwhd; #x; #x'; #Hx; #Hx'; #K; nassumption.
nqed.
nrecord isomorphism (A) (B) (S: qpowerclass A) (T: qpowerclass B) : CProp[0] ≝
{ iso_f:> unary_morphism A B;
f_closed: ∀x. x ∈ S → iso_f x ∈ T;
- f_sur: surjective ?? S T iso_f;
- f_inj: injective ?? S iso_f
+ f_sur: surjective … S T iso_f;
+ f_inj: injective … S iso_f
}.
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