ndefinition full_set: ∀A. Ω^A ≝ λA.{ x | True }.
nlemma subseteq_refl: ∀A.∀S: Ω^A. S ⊆ S.
- #A; #S; #x; #H; nassumption.
-nqed.
+//.nqed.
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.
+/3/.nqed.
include "properties/relations1.ma".
ndefinition seteq: ∀A. equivalence_relation1 (Ω^A).
#A; @
[ napply (λS,S'. S ⊆ S' ∧ S' ⊆ S)
- | #S; @; napply subseteq_refl
- | #S; #S'; *; #H1; #H2; @; nassumption
- | #S; #T; #U; *; #H1; #H2; *; #H3; #H4; @; napply subseteq_trans;
- ##[##2,5: nassumption |##1,4: ##skip |##*: nassumption]##]
+ | /2/
+ | #S; #S'; *; /2/
+ | #S; #T; #U; *; #H1; #H2; *; /3/]
nqed.
include "sets/setoids1.ma".
ncoercion full_set : ∀A:Type[0]. Ω^A ≝ full_set on A: Type[0] to (Ω^?).
ndefinition powerclass_setoid: Type[0] → setoid1.
- #A; @[ napply (Ω^A)| napply seteq ]
+ #A; @(Ω^A);//.
nqed.
include "hints_declaration.ma".
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 @{ 'ext_powerclass $A }.
+
+notation "Ω term 90 A \atop ≈" non associative with precedence 70
+for @{ 'ext_powerclass $A }.
-ndefinition Full_set: ∀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 (qpowerclass A).
- #A; @
- [ napply (λS,S'. S = S')
- | #S; napply (refl1 ? (seteq A))
- | #S; #S'; napply (sym1 ? (seteq A))
- | #S; #T; #U; napply (trans1 ? (seteq A))]
+ndefinition ext_seteq: ∀A. equivalence_relation1 (𝛀^A).
+ #A; @ [ napply (λS,S'. S = S') ] /2/.
nqed.
-ndefinition qpowerclass_setoid: setoid → setoid1.
- #A; @
- [ napply (qpowerclass A)
- | napply (qseteq A) ]
+ndefinition ext_powerclass_setoid: setoid → setoid1.
+ #A; @ (ext_seteq A).
nqed.
+
+unification hint 0 ≔ A;
+ R ≟ (mk_setoid1 (𝛀^A) (eq1 (ext_powerclass_setoid A)))
+ (* ----------------------------------------------------- *) ⊢
+ carr1 R ≡ ext_powerclass A.
-unification hint 0 ≔ A ⊢
- carr1 (qpowerclass_setoid A) ≡ qpowerclass A.
-
-(*CSC: non va!
-unification hint 0 ≔ A ⊢
- carr1 (mk_setoid1 (qpowerclass A) (eq1 (qpowerclass_setoid A))) ≡ qpowerclass A.*)
+(*
+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.
- #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;##]
- ##]
- ##]
+nlemma mem_ext_powerclass_setoid_is_morph:
+ ∀A. unary_morphism1 (setoid1_of_setoid A) (unary_morphism1_setoid1 (ext_powerclass_setoid A) CPROP).
+ #A; napply (mk_binary_morphism1 … (λx:setoid1_of_setoid A.λS: 𝛀^A. x ∈ S));
+ #a; #a'; #b; #b'; #Ha; *; #Hb1; #Hb2; @; #H
+ [ napply (. (ext_prop … Ha^-1)) | napply (. (ext_prop … Ha)) ] /2/.
nqed.
-(*CSC: bug qui se metto x o S al posto di ?
-nlemma foo: True.
-nletin xxx ≝ (λA:setoid.λx,S. let SS ≝ pc ? S in
- fun21 ??? (mk_binary_morphism1 ??? (λx.λS. ? ∈ ?) (prop21 ??? (mem_ok A))) x S);
-*)
unification hint 0 ≔ A:setoid, x, S;
- SS ≟ (pc ? S)
+ SS ≟ (ext_carr ? S),
+ TT ≟ (mk_unary_morphism1 …
+ (λx:setoid1_of_setoid ?.
+ mk_unary_morphism1 …
+ (λS:ext_powerclass_setoid ?. x ∈ S)
+ (prop11 … (mem_ext_powerclass_setoid_is_morph A x)))
+ (prop11 … (mem_ext_powerclass_setoid_is_morph A))),
+ XX ≟ (ext_powerclass_setoid A)
(*-------------------------------------*) ⊢
- fun21 ??? (mk_binary_morphism1 ??? (λx,S. x ∈ S) (prop21 ??? (mem_ok A))) x S ≡ mem A SS x.
-
-nlemma subseteq_ok: ∀A. binary_morphism1 (qpowerclass_setoid A) (qpowerclass_setoid A) CPROP.
- #A; @
- [ napply (λS,S'. S ⊆ S')
- | #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; @; #H
- [ napply (subseteq_trans … a)
- [ nassumption | napply (subseteq_trans … b); nassumption ]
- ##| napply (subseteq_trans … a')
- [ nassumption | napply (subseteq_trans … b'); nassumption ] ##]
+ fun11 (setoid1_of_setoid A)
+ (unary_morphism1_setoid1 XX CPROP) TT x S
+ ≡ mem A SS x.
+
+nlemma subseteq_is_morph: ∀A. unary_morphism1 (ext_powerclass_setoid A)
+ (unary_morphism1_setoid1 (ext_powerclass_setoid A) CPROP).
+ #A; napply (mk_binary_morphism1 … (λS,S':𝛀^A. S ⊆ S'));
+ #a; #a'; #b; #b'; *; #H1; #H2; *; /4/.
nqed.
unification hint 0 ≔ A,a,a'
(*-----------------------------------------------------------------*) ⊢
eq_rel ? (eq A) a a' ≡ eq_rel1 ? (eq1 (setoid1_of_setoid A)) a a'.
-nlemma intersect_ok: ∀A. binary_morphism1 (qpowerclass_setoid A) (qpowerclass_setoid A) (qpowerclass_setoid A).
+nlemma intersect_is_ext: ∀A. 𝛀^A → 𝛀^A → 𝛀^A.
+ #A; #S; #S'; @ (S ∩ S');
+ #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 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. unary_morphism1 (powerclass_setoid A) (unary_morphism1_setoid1 (powerclass_setoid A) (powerclass_setoid A)).
+ #A; napply (mk_binary_morphism1 … (λS,S'. S ∩ S'));
+ #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; @; #x; nnormalize; *;/3/.
+nqed.
+
+alias symbol "hint_decl" = "hint_decl_Type1".
+unification hint 0 ≔
+ A : Type[0], B,C : Ω^A;
+ R ≟ (mk_binary_morphism1 …
+ (λS,S'.S ∩ S')
+ (prop21 … (intersect_is_morph A)))
+ ⊢
+ fun21 (powerclass_setoid A) (powerclass_setoid A) (powerclass_setoid A) R B C
+ ≡ intersect ? B C.
+
+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: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: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 mem".
+ alias symbol "invert" = "setoid1 symmetry".
+ napply (. K^-1‡H);
+ nassumption;
+nqed.
+
+
+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 ⊢
- pc A (intersect_ok A B C) ≡ intersect ? (pc ? B) (pc ? C).
+ 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')))
+ (prop21 … (intersect_ok A)))
+ B
+ C)
+ ≡ intersect ? (pc ? B) (pc ? C).
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.
+*)
ndefinition image: ∀A,B. (carr A → carr B) → Ω^A → Ω^B ≝
λA,B:setoid.λf:carr A → carr B.λSa:Ω^A.
ndefinition eqrel_of_morphism:
∀A,B. unary_morphism A B → compatible_equivalence_relation A.
#A; #B; #f; @
- [ @
- [ napply (λx,y. f x = f y)
- | #x; napply refl | #x; #y; napply sym | #x; #y; #z; napply trans]
+ [ @ [ napply (λx,y. f x = f y) ] /2/;
##| #x; #x'; #H; nwhd; alias symbol "prop1" = "prop1".
-napply (.= (†H)); napply refl ]
+napply (.= (†H)); // ]
nqed.
ndefinition canonical_proj: ∀A,R. unary_morphism A (quotient A R).
#A; #R; @
- [ 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.
- #A; #B; #f; @
- [ napply f | #a; #a'; #H; nassumption]
+ #A; #B; #f; @ [ napply f ] //.
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).
- #A; #B; #f; #x; napply refl;
-nqed.
+//. nqed.
+alias symbol "eq" = "setoid eq".
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: unary_morphism A B.
surjective … (Full_set ?) (Full_set ?) (canonical_proj ? (eqrel_of_morphism … f)).
- #A; #B; #f; nwhd; #y; #Hy; @ y; @ I ; napply refl;
- (* bug, prova @ I refl *)
-nqed.
+/3/. nqed.
nlemma first_omomorphism_theorem_functions3:
∀A,B.∀f: unary_morphism A B.
#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;
f_inj: injective … S iso_f
}.
+nlemma subseteq_intersection_l: ∀A.∀U,V,W:Ω^A.U ⊆ W ∨ V ⊆ W → U ∩ V ⊆ W.
+#A; #U; #V; #W; *; #H; #x; *; /2/.
+nqed.
+
+nlemma subseteq_union_l: ∀A.∀U,V,W:Ω^A.U ⊆ W → V ⊆ W → U ∪ V ⊆ W.
+#A; #U; #V; #W; #H; #H1; #x; *; /2/.
+nqed.
+
+nlemma subseteq_intersection_r: ∀A.∀U,V,W:Ω^A.W ⊆ U → W ⊆ V → W ⊆ U ∩ V.
+/3/. nqed.
+
(*
nrecord isomorphism (A, B : setoid) (S: qpowerclass A) (T: qpowerclass B) : CProp[0] ≝
{ iso_f:> unary_morphism A B;
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