(* *)
(**************************************************************************)
-include "logic/equality.ma".
+(******************* SETS OVER TYPES *****************)
-nrecord powerset (A: Type) : Type[1] ≝ { mem: A → CProp }.
-(* This is a projection! *)
-ndefinition mem ≝ λA.λr:powerset A.match r with [mk_powerset f ⇒ f].
+include "logic/connectives.ma".
-interpretation "powerset" 'powerset A = (powerset A).
-
-interpretation "subset construction" 'subset \eta.x = (mk_powerset ? x).
+nrecord powerclass (A: Type[0]) : Type[1] ≝ { mem: A → CProp[0] }.
interpretation "mem" 'mem a S = (mem ? S a).
+interpretation "powerclass" 'powerset A = (powerclass A).
+interpretation "subset construction" 'subset \eta.x = (mk_powerclass ? x).
ndefinition subseteq ≝ λA.λU,V.∀a:A. a ∈ U → a ∈ V.
-
interpretation "subseteq" 'subseteq U V = (subseteq ? U V).
-ntheorem subseteq_refl: ∀A.∀S:Ω \sup A.S ⊆ S.
- #A; #S; #x; #H; nassumption;
+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^A.{ x | x ∈ U ∧ x ∈ V }.
+interpretation "intersect" 'intersects U V = (intersect ? U 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^A.λf:A → Ω^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^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^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^A. S ⊆ T → T ⊆ U → S ⊆ U.
+ #A; #S; #T; #U; #H1; #H2; #x; #P; napply H2; napply H1; nassumption.
nqed.
-ndefinition overlaps ≝ λA.λU,V:Ω \sup A.∃x:A.x ∈ U ∧ x ∈ V.
+include "properties/relations1.ma".
-interpretation "overlaps" 'overlaps U V = (overlaps ? U V).
+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
+ | #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.
-ndefinition intersects ≝ λA.λU,V:Ω \sup A. {x | x ∈ U ∧ x ∈ V }.
+include "sets/setoids1.ma".
-interpretation "intersects" 'intersects U V = (intersects ? U V).
+ndefinition powerclass_setoid: Type[0] → setoid1.
+ #A; napply mk_setoid1
+ [ napply (Ω^A)
+ | napply seteq ]
+nqed.
-ndefinition union ≝ λA.λU,V:Ω \sup A. {x | x ∈ U ∨ x ∈ V }.
+include "hints_declaration.ma".
-interpretation "union" 'union U V = (union ? U V).
+alias symbol "hint_decl" = "hint_decl_Type2".
+unification hint 0 ≔ A ⊢ carr1 (powerclass_setoid A) ≡ Ω^A.
+
+(************ SETS OVER SETOIDS ********************)
+
+include "logic/cprop.ma".
+
+nrecord qpowerclass (A: setoid) : Type[1] ≝
+ { 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 mk_qpowerclass
+ [ napply (full_set A)
+ | #x; #x'; #H; napply refl1; ##]
+nqed.
+
+ndefinition qseteq: ∀A. equivalence_relation1 (qpowerclass A).
+ #A; napply mk_equivalence_relation1
+ [ 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
+ [ napply (qpowerclass A)
+ | napply (qseteq A) ]
+nqed.
+
+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
+ [ napply (λx,S. x ∈ S)
+ | #a; #a'; #b; #b'; #Ha; *; #Hb1; #Hb2; napply mk_iff; #H;
+ ##[ napply Hb1; napply (. (mem_ok' …)); ##[##3: napply H| napply Ha^-1;##]
+ ##| napply Hb2; napply (. (mem_ok' …)); ##[##3: napply H| napply Ha;##]
+ ##]
+ ##]
+nqed.
+
+unification hint 0 ≔
+ A : setoid, x, S ⊢ (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'. S ⊆ S')
+ | #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; napply mk_iff; #H
+ [ napply (subseteq_trans … a)
+ [ nassumption | napply (subseteq_trans … b); nassumption ]
+ ##| napply (subseteq_trans … a')
+ [ nassumption | napply (subseteq_trans … b'); nassumption ] ##]
+nqed.
-ndefinition singleton ≝ λA.λa:A.{b | a=b}.
+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 |##2,5: nassumption |##*: ##skip]
+ ##|##3,4: napply (. (mem_ok' …)) [##1,3,4,6: 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.
+
+(* 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).
+
+(* 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.
+
+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' → 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.
+ #A; #R; napply mk_setoid
+ [ napply A
+ | napply 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
+ [ 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 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.
+
+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.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 … (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; nwhd; #x; #x'; #Hx; #Hx'; #K; nassumption.
+nqed.
-interpretation "singleton" 'singl a = (singleton ? a).
+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
+ }.
projects / helm.git / blobdiff