(* *)
(**************************************************************************)
-include "logic/cprop.ma".
+(******************* SETS OVER TYPES *****************)
+
+include "logic/connectives.ma".
-nrecord powerset (A: setoid) : Type[1] ≝ { mem_op: unary_morphism1 A CPROP }.
+nrecord powerclass (A: Type[0]) : Type[1] ≝ { mem: A → CProp[0] }.
-interpretation "powerset" 'powerset A = (powerset A).
+interpretation "mem" 'mem a S = (mem ? S a).
+interpretation "powerclass" 'powerset A = (powerclass A).
+interpretation "subset construction" 'subset \eta.x = (mk_powerclass ? x).
-interpretation "subset construction" 'subset \eta.x =
- (mk_powerset ? (mk_unary_morphism1 ? CPROP x ?)).
+ndefinition subseteq ≝ λA.λU,V.∀a:A. a ∈ U → a ∈ V.
+interpretation "subseteq" 'subseteq U V = (subseteq ? U V).
-interpretation "mem" 'mem a S = (mem_op ? S a).
+ndefinition overlaps ≝ λA.λU,V.∃x:A.x ∈ U ∧ x ∈ V.
+interpretation "overlaps" 'overlaps U V = (overlaps ? U V).
-ndefinition subseteq ≝ λA:setoid.λU,V.∀a:A. a ∈ U → a ∈ V.
+ndefinition intersect ≝ λA.λU,V:Ω \sup A.{ x | x ∈ U ∧ x ∈ V }.
+interpretation "intersect" 'intersects U V = (intersect ? U V).
-interpretation "subseteq" 'subseteq U V = (subseteq ? U V).
+ndefinition union ≝ λA.λU,V:Ω \sup A.{ x | x ∈ U ∨ x ∈ V }.
+interpretation "union" 'union U V = (union ? U V).
-ntheorem subseteq_refl: ∀A.∀S:Ω \sup A.S ⊆ S.
- #A; #S; #x; #H; nassumption;
+ndefinition big_union ≝ λA.λT:Type[0].λf:T → Ω \sup A.{ x | ∃i. x ∈ f i }.
+
+ndefinition big_intersection ≝ λA.λT:Type[0].λf:T → Ω \sup A.{ x | ∀i. 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 ?). *)
+
+nlemma subseteq_refl: ∀A.∀S: Ω \sup A. S ⊆ S.
+ #A; #S; #x; #H; nassumption.
nqed.
-ntheorem subseteq_trans: ∀A.∀S1,S2,S3: Ω \sup A. S1 ⊆ S2 → S2 ⊆ S3 → S1 ⊆ S3.
- #A; #S1; #S2; #S3; #H12; #H23; #x; #H;
- napply H23; napply H12; nassumption;
+nlemma subseteq_trans: ∀A.∀S,T,U: Ω \sup A. S ⊆ T → T ⊆ U → S ⊆ U.
+ #A; #S; #T; #U; #H1; #H2; #x; #P; napply H2; napply H1; nassumption.
nqed.
-ndefinition powerset_setoid1: setoid → setoid1.
- #S; napply mk_setoid1
- [ napply (Ω \sup S)
- | napply mk_equivalence_relation1
- [ #A; #B; napply (∀x. iff (x ∈ A) (x ∈ B))
- | nwhd; #x; #x0; napply mk_iff; #H; nassumption
- | nwhd; #x; #y; #H; #A; napply mk_iff; #K
- [ napply (fi ?? (H ?)) | napply (if ?? (H ?)) ]
- nassumption
- | nwhd; #A; #B; #C; #H1; #H2; #H3; napply mk_iff; #H4
- [ napply (if ?? (H2 ?)); napply (if ?? (H1 ?)); nassumption
- | napply (fi ?? (H1 ?)); napply (fi ?? (H2 ?)); nassumption]##]
+include "properties/relations1.ma".
+
+ndefinition seteq: ∀A. equivalence_relation1 (Ω \sup A).
+ #A; napply mk_equivalence_relation1
+ [ 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;
+ ##[##2,5: nassumption |##1,4: ##skip |##*: nassumption]##]
+nqed.
+
+include "sets/setoids1.ma".
+
+ndefinition powerclass_setoid: Type[0] → setoid1.
+ #A; napply mk_setoid1
+ [ napply (Ω \sup A)
+ | napply seteq ]
+nqed.
+
+unification hint 0 (∀A. (λx,y.True) (carr1 (powerclass_setoid A)) (Ω \sup A)).
+
+(************ SETS OVER SETOIDS ********************)
+
+include "logic/cprop.ma".
+
+nrecord qpowerclass (A: setoid) : Type[1] ≝
+ { pc:> Ω \sup A;
+ mem_ok': ∀x,x':A. x=x' → (x ∈ pc) = (x' ∈ pc)
+ }.
+
+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')
+ | #S; napply (refl1 ? (seteq A))
+ | #S; #S'; napply (sym1 ? (seteq A))
+ | #S; #T; #U; napply (trans1 ? (seteq A))]
nqed.
-unification hint 0 (∀A.(λx,y.True) (Ω \sup A) (carr1 (powerset_setoid1 A))).
+ndefinition qpowerclass_setoid: setoid → setoid1.
+ #A; napply mk_setoid1
+ [ napply (qpowerclass A)
+ | napply (qseteq A) ]
+nqed.
-ndefinition mem: ∀A:setoid. binary_morphism1 A (powerset_setoid1 A) CPROP.
+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 ?).
+
+nlemma mem_ok: ∀A. binary_morphism1 (setoid1_of_setoid A) (qpowerclass_setoid A) CPROP.
#A; napply mk_binary_morphism1
- [ napply (λa.λA.a ∈ A)
- | #a; #a'; #B; #B'; #Ha; #HB; napply mk_iff; #H
- [ napply (. (†Ha^-1)); (* CSC: notation for ∈ not working *)
- napply (if ?? (HB ?)); nassumption
- | napply (. (†Ha)); napply (fi ?? (HB ?)); nassumption]##]
+ [ 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) (mem_op A x S) (fun21 ??? (mem A) S x)).
+unification hint 0 (∀A,x,S. (λx,y.True) (fun21 ??? (mem_ok A) x S) (mem A S 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 ] ##]
+nqed.
-ndefinition overlaps ≝ λA.λU,V:Ω \sup A.∃x:A.x ∈ U ∧ x ∈ V.
+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
+ [ 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
+ | napply (. ((#‡Ha)‡(#‡Hb))); nassumption ]##]
+nqed.
-interpretation "overlaps" 'overlaps U V = (overlaps ? U V).
+unification hint 0 (∀A.∀U,V.(λx,y.True) (fun21 ??? (intersect_ok A) U V) (intersect A U V)).
-ndefinition intersects ≝ λA.λU,V:Ω \sup A. {x | x ∈ U ∧ x ∈ V }.
- #a; #a'; #H; napply mk_iff; *; #H1; #H2
- [ napply (. ((H^-1‡#)‡(H^-1‡#))); nwhd; napply conj; nassumption
- | napply (. ((H‡#)‡(H‡#))); nwhd; napply conj; nassumption]
+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.
nqed.
-
-(*interpretation "intersects" 'intersects U V = (intersects ? U V).*)
(*
-ndefinition union ≝ λA.λU,V:Ω \sup A. {x | x ∈ U ∨ x ∈ V }.
+(* 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 ≝
+ 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 *)
+ }.
+
+ndefinition quotient: ∀A. compatible_equivalence_relation A → setoid.
+ #A; #R; napply mk_setoid
+ [ napply A
+ | napply R]
+nqed.
-interpretation "union" 'union U V = (union ? U V).
+(******************* 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
+ [ 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 ]
+nqed.
-ndefinition singleton ≝ λA:setoid.λa:A.{b | a=b}.
+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) ]
+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
+ [ 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).
+ #A; #B; #f; #x; napply refl;
+nqed.
-interpretation "singleton" 'singl a = (singleton ? a).*)
\ No newline at end of file
+ndefinition surjective ≝ λA,B.λf:unary_morphism A B. ∀y.∃x. f x = y.
+
+ndefinition injective ≝ λA,B.λf:unary_morphism A B. ∀x,x'. 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.
+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.
+nqed.
projects / helm.git / blobdiff