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.λT:Type[0].λf:T → Ω \sup A.{ x | ∃i. 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.λT:Type[0].λf:T → Ω \sup A.{ x | ∀i. 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 }.
-(* bug dichiarazione coercion qui *)
-(* 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 }.
-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).
- #A; napply mk_equivalence_relation1
+ndefinition seteq: ∀A. equivalence_relation1 (Ω^A).
+ #A; @
[ napply (λS,S'. S ⊆ S' ∧ S' ⊆ S)
- | #S; napply conj; napply subseteq_refl
- | #S; #S'; *; #H1; #H2; napply conj; nassumption
- | #S; #T; #U; *; #H1; #H2; *; #H3; #H4; napply conj; napply subseteq_trans;
+ | #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]##]
-nqed.
+nqed.
include "sets/setoids1.ma".
+(* this has to be declared here, so that it is combined with carr *)
+ncoercion full_set : ∀A:Type[0]. Ω^A ≝ full_set on A: Type[0] to (Ω^?).
+
ndefinition powerclass_setoid: Type[0] → setoid1.
- #A; napply mk_setoid1
- [ napply (Ω \sup A)
- | napply seteq ]
+ #A; @[ napply (Ω^A)| napply seteq ]
nqed.
-unification hint 0 (∀A. (λx,y.True) (carr1 (powerclass_setoid A)) (Ω \sup A)).
+include "hints_declaration.ma".
+
+alias symbol "hint_decl" = "hint_decl_Type2".
+unification hint 0 ≔ A ⊢ carr1 (mk_setoid1 (Ω^A) (eq1 (powerclass_setoid A))) ≡ Ω^A.
(************ SETS OVER SETOIDS ********************)
include "logic/cprop.ma".
nrecord qpowerclass (A: setoid) : Type[1] ≝
- { pc:> Ω \sup A;
+ { pc:> 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)
}.
+ndefinition Full_set: ∀A. qpowerclass A.
+ #A; @[ napply A | #x; #x'; #H; napply refl1]
+nqed.
+ncoercion Full_set: ∀A. qpowerclass A ≝ Full_set on A: setoid to qpowerclass ?.
+
ndefinition qseteq: ∀A. equivalence_relation1 (qpowerclass A).
- #A; napply mk_equivalence_relation1
- [ napply (λS,S':qpowerclass A. eq_rel1 ? (eq1 (powerclass_setoid A)) S S')
+ #A; @
+ [ napply (λS,S'. S = S')
| #S; napply (refl1 ? (seteq A))
| #S; #S'; napply (sym1 ? (seteq A))
| #S; #T; #U; napply (trans1 ? (seteq A))]
nqed.
ndefinition qpowerclass_setoid: setoid → setoid1.
- #A; napply mk_setoid1
+ #A; @
[ napply (qpowerclass A)
| napply (qseteq A) ]
nqed.
-unification hint 0 (∀A. (λx,y.True) (carr1 (qpowerclass_setoid A)) (qpowerclass A)).
-ncoercion qpowerclass_hint: ∀A: setoid. ∀S: qpowerclass_setoid A. Ω \sup A ≝ λA.λS.S
- on _S: (carr1 (qpowerclass_setoid ?)) to (Ω \sup ?).
+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.*)
nlemma mem_ok: ∀A. binary_morphism1 (setoid1_of_setoid A) (qpowerclass_setoid A) CPROP.
- #A; napply mk_binary_morphism1
- [ napply (λx.λS: qpowerclass_setoid A. x ∈ S) (* CSC: ??? *)
- | #a; #a'; #b; #b'; #Ha; #Hb; (* CSC: qui *; non funziona *)
- nwhd; nwhd in ⊢ (? (? % ??) (? % ??)); napply mk_iff; #H
- [ ncases Hb; #Hb1; #_; napply Hb1; napply (. (mem_ok' …))
- [ nassumption | napply Ha^-1 | ##skip ]
- ##| ncases Hb; #_; #Hb2; napply Hb2; napply (. (mem_ok' …))
- [ nassumption | napply Ha | ##skip ]##]
-nqed.
-
-unification hint 0 (∀A,x,S. (λx,y.True) (fun21 ??? (mem_ok A) x S) (mem A S x)).
-
+ #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;##]
+ ##]
+ ##]
+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)
+ (*-------------------------------------*) ⊢
+ 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 mk_binary_morphism1
- [ napply (λS,S': qpowerclass_setoid ?. S ⊆ S')
- | #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; napply mk_iff; #H
- [ napply (subseteq_trans … a' a) (* anche qui, perche' serve a'? *)
- [ nassumption | napply (subseteq_trans … a b); nassumption ]
- ##| napply (subseteq_trans … a a') (* anche qui, perche' serve a'? *)
- [ nassumption | napply (subseteq_trans … a' b'); nassumption ] ##]
+ #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 ] ##]
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).
- #A; napply mk_binary_morphism1
- [ #S; #S'; napply mk_qpowerclass
+ #A; @
+ [ #S; #S'; @
[ napply (S ∩ S')
- | #a; #a'; #Ha; nwhd in ⊢ (? ? ? % %); napply mk_iff; *; #H1; #H2; napply conj
- [##1,2: napply (. (mem_ok' …)^-1) [##3,6: nassumption |##1,4: nassumption |##*: ##skip]
- ##|##3,4: napply (. (mem_ok' …)) [##2,5: nassumption |##1,4: nassumption |##*: ##skip]##]##]
- ##| #a; #a'; #b; #b'; #Ha; #Hb; nwhd; napply conj; #x; nwhd in ⊢ (% → %); #H
- [ napply (. ((#‡Ha^-1)‡(#‡Hb^-1))); nassumption
+ | #a; #a'; #Ha;
+ nwhd in ⊢ (? ? ? % %); @; *; #H1; #H2; @
+ [##1,2: napply (. Ha^-1‡#); nassumption;
+ ##|##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
| napply (. ((#‡Ha)‡(#‡Hb))); nassumption ]##]
nqed.
-unification hint 0 (∀A.∀U,V.(λx,y.True) (fun21 ??? (intersect_ok A) U V) (intersect A U V)).
+(* 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).
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;
- (* CSC: senza la change non funziona! *)
- nchange with (x' ∈ (fun21 ??? (intersect_ok A) U V));
- napply (. (H^-1‡#)); nassumption.
+ #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 counter_image: ∀A,B. (A → B) → Ω \sup B → Ω \sup A ≝
+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. (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'
}.
ndefinition quotient: ∀A. compatible_equivalence_relation A → setoid.
- #A; #R; napply mk_setoid
- [ napply A
- | napply R]
+ #A; #R; @ A R;
nqed.
(******************* first omomorphism theorem for sets **********************)
ndefinition eqrel_of_morphism:
∀A,B. unary_morphism A B → compatible_equivalence_relation A.
- #A; #B; #f; napply mk_compatible_equivalence_relation
- [ napply mk_equivalence_relation
+ #A; #B; #f; @
+ [ @
[ napply (λx,y. f x = f y)
| #x; napply refl | #x; #y; napply sym | #x; #y; #z; napply trans]
-##| #x; #x'; #H; nwhd; napply (.= (†H)); napply refl ]
+##| #x; #x'; #H; nwhd; alias symbol "prop1" = "prop1".
+napply (.= (†H)); napply refl ]
nqed.
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) ]
+ #A; #R; @
+ [ 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 mk_unary_morphism
+ unary_morphism (quotient … (eqrel_of_morphism … f)) B.
+ #A; #B; #f; @
[ 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.
-ndefinition surjective ≝ λA,B.λf:unary_morphism A B. ∀y.∃x. f x = y.
+ndefinition surjective ≝
+ λA,B.λS: qpowerclass A.λT: qpowerclass B.λf:unary_morphism A B.
+ ∀y. y ∈ T → ∃x. x ∈ S ∧ f x = y.
-ndefinition injective ≝ λA,B.λf:unary_morphism A B. ∀x,x'. f x = f x' → x = x'.
+ndefinition injective ≝
+ λA,B.λS: qpowerclass 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 ?? (canonical_proj ? (eqrel_of_morphism ?? f)).
- #A; #B; #f; nwhd; #y; napply (ex_intro … y); napply refl.
+ ∀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.
nlemma first_omomorphism_theorem_functions3:
- ∀A,B.∀f: unary_morphism A B. injective ?? (quotiented_mor ?? f).
- #A; #B; #f; nwhd; #x; #x'; #H; nassumption.
+ ∀A,B.∀f: unary_morphism A B.
+ injective … (Full_set ?) (quotiented_mor … f).
+ #A; #B; #f; nwhd; #x; #x'; #Hx; #Hx'; #K; nassumption.
nqed.
-(************************** partitions ****************************)
-
-nrecord partition (A: Type[0]) : Type[1] ≝
- { index_set: setoid;
- class: index_set → Ω \sup A;
- disjoint: ∀i,j. ¬ (i = j) → ¬(class i ≬ class j);
- covers: big_union ?? class = full_set A
+nrecord isomorphism (A, B : setoid) (S: qpowerclass A) (T: qpowerclass 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
}.
(*
-nrecord has_card (A: Type[0]) (S: Ω \sup A) (n: nat) : Prop ≝
- { f: ∀m:nat. m < n → S;
- f_inj: injective ?? f;
- f_sur: surjective ?? f
+nrecord isomorphism (A, B : setoid) (S: qpowerclass A) (T: qpowerclass B) : CProp[0] ≝
+ { iso_f:> unary_morphism A B;
+ f_closed: ∀x. x ∈ pc A S → fun1 ?? iso_f x ∈ pc B T}.
+
+
+ncheck (λA:?.
+ λB:?.
+ λS:?.
+ λT:?.
+ λxxx:isomorphism A B S T.
+ match xxx
+ return λxxx:isomorphism A B S T.
+ ∀x: carr A.
+ ∀x_72: mem (carr A) (pc A S) x.
+ mem (carr B) (pc B T) (fun1 A B ((λ_.?) A B S T xxx) x)
+ with [ mk_isomorphism _ yyy ⇒ yyy ] ).
+
+ ;
}.
-
-nlemma subset_of_finite:
- ∀A. ∃n. has_card ? (full_subset A) n → ∀S. ∃m. has_card ? S m.
-nqed.
-
-nlemma partition_splits_card:
- ∀A. ∀P: partition A. ∀s: index_set P → nat.
- (∀i. has_card ? (class i) = s i) →
- has_card ? (full_subset A) (big_plus ? (λi. s i)).
-nqed.
-*)
\ No newline at end of file
+*)
projects / helm.git / blobdiff