include "logic/cprop.ma".
-nrecord qpowerclass (A: setoid) : Type[1] ≝
- { pc:> Ω^A; (* qui pc viene dichiarato con un target preciso...
+nrecord ext_powerclass (A: setoid) : Type[1] ≝
+ { ext_carr:> Ω^A; (* qui pc viene dichiarato con un target preciso...
forse lo si vorrebbe dichiarato con un target più lasco
ma la sintassi :> non lo supporta *)
- mem_ok': ∀x,x':A. x=x' → (x ∈ pc) = (x' ∈ pc)
+ ext_prop: ∀x,x':A. x=x' → (x ∈ ext_carr) = (x' ∈ ext_carr)
}.
notation > "𝛀 ^ term 90 A" non associative with precedence 70
-for @{ 'qpowerclass $A }.
+for @{ 'ext_powerclass $A }.
notation "Ω term 90 A \atop ≈" non associative with precedence 70
-for @{ 'qpowerclass $A }.
+for @{ 'ext_powerclass $A }.
-interpretation "qpowerclass" 'qpowerclass a = (qpowerclass a).
+interpretation "extensional powerclass" 'ext_powerclass a = (ext_powerclass a).
ndefinition Full_set: ∀A. 𝛀^A.
#A; @[ napply A | #x; #x'; #H; napply refl1]
nqed.
-ncoercion Full_set: ∀A. qpowerclass A ≝ Full_set on A: setoid to qpowerclass ?.
+ncoercion Full_set: ∀A. ext_powerclass A ≝ Full_set on A: setoid to ext_powerclass ?.
-ndefinition qseteq: ∀A. equivalence_relation1 (𝛀^A).
+ndefinition ext_seteq: ∀A. equivalence_relation1 (𝛀^A).
#A; @
[ napply (λS,S'. S = S')
| #S; napply (refl1 ? (seteq A))
| #S; #T; #U; napply (trans1 ? (seteq A))]
nqed.
-ndefinition qpowerclass_setoid: setoid → setoid1.
+ndefinition ext_powerclass_setoid: setoid → setoid1.
#A; @
- [ napply (qpowerclass A)
- | napply (qseteq A) ]
+ [ napply (ext_powerclass A)
+ | napply (ext_seteq A) ]
nqed.
-unification hint 0 ≔ A ⊢
- carr1 (mk_setoid1 (𝛀^A) (eq1 (qpowerclass_setoid A)))
-≡ qpowerclass A.
+unification hint 0 ≔ A;
+ R ≟ (mk_setoid1 (𝛀^A) (eq1 (ext_powerclass_setoid A)))
+ (* ----------------------------------------------------- *) ⊢
+ carr1 R ≡ ext_powerclass A.
-ncoercion pc' : ∀A.∀x:qpowerclass_setoid A. Ω^A ≝ pc
-on _x : (carr1 (qpowerclass_setoid ?)) to (Ω^?).
+interpretation "prop21 mem" 'prop2 l r = (prop21 (setoid1_of_setoid ?) (ext_powerclass_setoid ?) ? ? ???? l r).
+
+(*
+ncoercion ext_carr' : ∀A.∀x:ext_powerclass_setoid A. Ω^A ≝ ext_carr
+on _x : (carr1 (ext_powerclass_setoid ?)) to (Ω^?).
+*)
-nlemma mem_ok: ∀A. binary_morphism1 (setoid1_of_setoid A) (qpowerclass_setoid A) CPROP.
+nlemma mem_ext_powerclass_setoid_is_morph:
+ ∀A. binary_morphism1 (setoid1_of_setoid A) (ext_powerclass_setoid A) CPROP.
#A; @
[ napply (λx,S. x ∈ S)
| #a; #a'; #b; #b'; #Ha; *; #Hb1; #Hb2; @; #H;
- ##[ napply Hb1; napply (. (mem_ok' …)); ##[##3: napply H| napply Ha^-1;##]
- ##| napply Hb2; napply (. (mem_ok' …)); ##[##3: napply H| napply Ha;##]
+ ##[ napply Hb1; napply (. (ext_prop … Ha^-1)); nassumption;
+ ##| napply Hb2; napply (. (ext_prop … Ha)); nassumption;
##]
##]
nqed.
unification hint 0 ≔ A:setoid, x, S;
- SS ≟ (pc ? S),
- TT ≟ (mk_binary_morphism1 ???
- (λx:setoid1_of_setoid ?.λS:qpowerclass_setoid ?. x ∈ S)
- (prop21 ??? (mem_ok A)))
-
+ SS ≟ (ext_carr ? S),
+ TT ≟ (mk_binary_morphism1 ???
+ (λx:setoid1_of_setoid ?.λS:ext_powerclass_setoid ?. x ∈ S)
+ (prop21 ??? (mem_ext_powerclass_setoid_is_morph A))),
+ XX ≟ (ext_powerclass_setoid A)
(*-------------------------------------*) ⊢
- fun21 ? ? ? TT x S
- ≡ mem A SS x.
+ fun21 (setoid1_of_setoid A) XX CPROP TT x S
+ ≡ mem A SS x.
-nlemma subseteq_ok: ∀A. binary_morphism1 (qpowerclass_setoid A) (qpowerclass_setoid A) CPROP.
+nlemma subseteq_is_morph: ∀A. binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) CPROP.
#A; @
[ napply (λS,S'. S ⊆ S')
| #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; @; #H
(*-----------------------------------------------------------------*) ⊢
eq_rel ? (eq A) a a' ≡ eq_rel1 ? (eq1 (setoid1_of_setoid A)) a a'.
-nlemma intersect_ok: ∀A. 𝛀^A → 𝛀^A → 𝛀^A.
+nlemma intersect_is_ext: ∀A. 𝛀^A → 𝛀^A → 𝛀^A.
#A; #S; #S'; @ (S ∩ S');
- #a; #a'; #Ha; @; *; #H1; #H2; @
+ #a; #a'; #Ha; @; *; #H1; #H2; @
[##1,2: napply (. Ha^-1‡#); nassumption;
##|##3,4: napply (. Ha‡#); nassumption]
nqed.
alias symbol "hint_decl" = "hint_decl_Type1".
-unification hint 1 ≔
- A : setoid, B,C : qpowerclass A ⊢
- pc A (mk_qpowerclass ? (B ∩ C) (mem_ok' ? (intersect_ok ? B C)))
- ≡ intersect ? (pc ? B) (pc ? C).
-
-nlemma intersect_ok': ∀A. binary_morphism1 (powerclass_setoid A) (powerclass_setoid A) (powerclass_setoid A).
+unification hint 0 ≔
+ A : setoid, B,C : ext_powerclass A;
+ R ≟ (mk_ext_powerclass ? (B ∩ C) (ext_prop ? (intersect_is_ext ? B C)))
+
+ (* ------------------------------------------*) ⊢
+ ext_carr A R ≡ intersect ? (ext_carr ? B) (ext_carr ? C).
+
+nlemma intersect_is_morph:
+ ∀A. binary_morphism1 (powerclass_setoid A) (powerclass_setoid A) (powerclass_setoid A).
#A; @ (λS,S'. S ∩ S');
#a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; @; #x; nnormalize; *; #Ka; #Kb; @
[ napply Ha1; nassumption
alias symbol "hint_decl" = "hint_decl_Type1".
unification hint 0 ≔
- A : Type[0], B,C : powerclass A ⊢
- fun21 …
- (mk_binary_morphism1 …
+ A : Type[0], B,C : Ω^A;
+ R ≟ (mk_binary_morphism1 …
(λS,S'.S ∩ S')
- (prop21 … (intersect_ok' A))) B C
- ≡ intersect ? B C.
-
-ndefinition prop21_mem :
- ∀A,C.∀f:binary_morphism1 (setoid1_of_setoid A) (qpowerclass_setoid A) C.
- ∀a,a':setoid1_of_setoid A.
- ∀b,b':qpowerclass_setoid A.a = a' → b = b' → f a b = f a' b'.
-#A; #C; #f; #a; #a'; #b; #b'; #H1; #H2; napply prop21; nassumption;
-nqed.
-
-interpretation "prop21 mem" 'prop2 l r = (prop21_mem ??????? l r).
+ (prop21 … (intersect_is_morph A)))
+ ⊢
+ fun21 (powerclass_setoid A) (powerclass_setoid A) (powerclass_setoid A) R B C
+ ≡ intersect ? B C.
-nlemma intersect_ok'':
- ∀A. binary_morphism1 (qpowerclass_setoid A) (qpowerclass_setoid A) (qpowerclass_setoid A).
- #A; @ (intersect_ok A); nlapply (prop21 … (intersect_ok' A)); #H;
- #a; #a'; #b; #b'; #H1; #H2; napply H; nassumption;
+interpretation "prop21 ext" 'prop2 l r = (prop21 (ext_powerclass_setoid ?) (ext_powerclass_setoid ?) ? ? ???? l r).
+
+nlemma intersect_is_ext_morph:
+ ∀A. binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) (ext_powerclass_setoid A).
+ #A; @ (intersect_is_ext …); nlapply (prop21 … (intersect_is_morph A));
+#H; #a; #a'; #b; #b'; #H1; #H2; napply H; nassumption;
nqed.
unification hint 1 ≔
- A:?, B,C : 𝛀^A ⊢
- fun21 …
- (mk_binary_morphism1 …
- (λS,S':qpowerclass_setoid A.S ∩ S')
- (prop21 … (intersect_ok'' A))) B C
- ≡ intersect ? B C.
-
+ A:setoid, B,C : 𝛀^A;
+ R ≟ (mk_binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) (ext_powerclass_setoid A)
+ (λS,S':carr1 (ext_powerclass_setoid A).
+ mk_ext_powerclass A (S∩S') (ext_prop A (intersect_is_ext ? S S')))
+ (prop21 … (intersect_is_ext_morph A))) ,
+ BB ≟ (ext_carr ? B),
+ CC ≟ (ext_carr ? C)
+ (* ------------------------------------------------------*) ⊢
+ ext_carr A
+ (fun21
+ (ext_powerclass_setoid A)
+ (ext_powerclass_setoid A)
+ (ext_powerclass_setoid A) R B C) ≡
+ intersect (carr A) BB CC.
+(*
+alias symbol "hint_decl" = "hint_decl_Type2".
+unification hint 0 ≔
+ A : setoid, B,C : 𝛀^A ;
+ CC ≟ (ext_carr ? C),
+ BB ≟ (ext_carr ? B),
+ C1 ≟ (carr1 (powerclass_setoid (carr A))),
+ C2 ≟ (carr1 (ext_powerclass_setoid A))
+ ⊢
+ eq_rel1 C1 (eq1 (powerclass_setoid (carr A))) BB CC ≡
+ eq_rel1 C2 (eq1 (ext_powerclass_setoid A)) B C.
+
+unification hint 0 ≔
+ A, B : CPROP ⊢ iff A B ≡ eq_rel1 ? (eq1 CPROP) A B.
-
-nlemma test: ∀U.∀A,B:qpowerclass U. A ∩ B = A →
+nlemma test: ∀U.∀A,B:𝛀^U. A ∩ B = A →
∀x,y. x=y → x ∈ A → y ∈ A ∩ B.
#U; #A; #B; #H; #x; #y; #K; #K2;
- alias symbol "prop2" = "prop21".
-alias symbol "invert" = "setoid1 symmetry".
-napply (. (K^-1 ‡ (prop21 ??? (intersect_ok'' U) ???? H^-1 #))); STOP
-
- napply (. K^-1‡(H^-1‡#)); nassumption;
+ alias symbol "prop2" = "prop21 mem".
+ alias symbol "invert" = "setoid1 symmetry".
+ napply (. K^-1‡H);
+ nassumption;
nqed.
-(*
-nlemma intersect_ok: ∀A. binary_morphism1 (qpowerclass_setoid A) (qpowerclass_setoid A) (qpowerclass_setoid A).
+
+nlemma intersect_ok: ∀A. binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) (ext_powerclass_setoid A).
#A; @
[ #S; #S'; @
[ napply (S ∩ S')
##|##3,4: napply (. Ha‡#); nassumption]##]
##| #a; #a'; #b; #b'; #Ha; #Hb; nwhd; @; #x; nwhd in ⊢ (% → %); #H
[ alias symbol "invert" = "setoid1 symmetry".
- napply (. ((#‡Ha^-1)‡(#‡Hb^-1))); nassumption
+ alias symbol "refl" = "refl".
+alias symbol "prop2" = "prop21".
+napply (. ((#‡Ha^-1)‡(#‡Hb^-1))); nassumption
| napply (. ((#‡Ha)‡(#‡Hb))); nassumption ]##]
nqed.
(* unfold if intersect, exposing fun21 *)
alias symbol "hint_decl" = "hint_decl_Type1".
unification hint 0 ≔
- A : setoid, B,C : qpowerclass A ⊢
+ A : setoid, B,C : ext_powerclass A ⊢
pc A (fun21 …
(mk_binary_morphism1 …
(λS,S':qpowerclass_setoid A.mk_qpowerclass ? (S ∩ S') (mem_ok' ? (intersect_ok ? S S')))
nqed.
ndefinition surjective ≝
- λA,B.λS: qpowerclass A.λT: qpowerclass B.λf:unary_morphism A B.
+ λA,B.λS: ext_powerclass A.λT: ext_powerclass B.λf:unary_morphism A B.
∀y. y ∈ T → ∃x. x ∈ S ∧ f x = y.
ndefinition injective ≝
- λA,B.λS: qpowerclass A.λf:unary_morphism A B.
+ λA,B.λS: ext_powerclass A.λf:unary_morphism A B.
∀x,x'. x ∈ S → x' ∈ S → f x = f x' → x = x'.
nlemma first_omomorphism_theorem_functions2:
#A; #B; #f; nwhd; #x; #x'; #Hx; #Hx'; #K; nassumption.
nqed.
-nrecord isomorphism (A, B : setoid) (S: qpowerclass A) (T: qpowerclass B) : Type[0] ≝
+nrecord isomorphism (A, B : setoid) (S: ext_powerclass A) (T: ext_powerclass B) : Type[0] ≝
{ iso_f:> unary_morphism A B;
f_closed: ∀x. x ∈ S → iso_f x ∈ T;
f_sur: surjective … S T iso_f;
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