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".
ncoercion Full_set: ∀A. ext_powerclass A ≝ Full_set on A: setoid to ext_powerclass ?.
ndefinition ext_seteq: ∀A. equivalence_relation1 (𝛀^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))]
+ #A; @ [ napply (λS,S'. S = S') ] /2/.
nqed.
ndefinition ext_powerclass_setoid: setoid → setoid1.
- #A; @
- [ napply (ext_powerclass A)
- | napply (ext_seteq A) ]
+ #A; @ (ext_seteq A).
nqed.
unification hint 0 ≔ A;
(* ----------------------------------------------------- *) ⊢
carr1 R ≡ ext_powerclass 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_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 (. (ext_prop … Ha^-1)); nassumption;
- ##| napply Hb2; napply (. (ext_prop … Ha)); nassumption;
- ##]
- ##]
+ ∀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.
unification hint 0 ≔ A:setoid, x, S;
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))),
+ 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 (setoid1_of_setoid A) XX CPROP TT x S
+ fun11 (setoid1_of_setoid A)
+ (unary_morphism1_setoid1 XX CPROP) TT x S
≡ mem A SS x.
-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
- [ napply (subseteq_trans … a)
- [ nassumption | napply (subseteq_trans … b); nassumption ]
- ##| napply (subseteq_trans … a')
- [ nassumption | napply (subseteq_trans … b'); nassumption ] ##]
+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'
(* ------------------------------------------*) ⊢
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
- | napply Hb1; nassumption
- | napply Ha2; nassumption
- | napply Hb2; nassumption]
+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".
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: ext_powerclass A.λT: ext_powerclass B.λf:unary_morphism A B.
∀y. y ∈ T → ∃x. x ∈ S ∧ f x = y.
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.
}.
nlemma subseteq_intersection_l: ∀A.∀U,V,W:Ω^A.U ⊆ W ∨ V ⊆ W → U ∩ V ⊆ W.
-#A; #U; #V; #W; *; #H; #x; *; #xU; #xV; napply H; nassumption;
+#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; *; #Hx; ##[ napply H; ##| napply H1; ##] nassumption;
-nqed.
+#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.
-#A; #U; #V; #W; #H1; #H2; #x; #Hx; @; ##[ napply H1; ##| napply H2; ##] nassumption;
-nqed.
+/3/. nqed.
(*
nrecord isomorphism (A, B : setoid) (S: qpowerclass A) (T: qpowerclass B) : CProp[0] ≝
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