ndefinition if': ∀A,B:CPROP. A = B → A → B.
#A B; *; /2/. nqed.
-ncoercion if : ∀A,B:CPROP. ∀p:A = B. A → B ≝ if' on _p : eq_rel1 ???? to ∀_:?.?.
+ncoercion if : ∀A,B:CPROP. ∀p:A = B. A → B ≝ if' on _p : eq_rel1 ? (eq1 CPROP) ?? to ∀_:?.?.
+
+ndefinition ifs': ∀S.∀A,B:Ω^S. A = B → ∀x. x ∈ A → x ∈ B.
+#S A B; *; /2/. nqed.
+
+ncoercion ifs : ∀S.∀A,B:Ω^S. ∀p:A = B.∀x. x ∈ A → x ∈ B ≝ ifs' on _p : eq_rel1 ? (eq1 (powerclass_setoid ?))?? to ∀_:?.?.
(* XXX move to list-setoids-theory.ma *)
λS.λl1,l2.{ w ∈ list S | ∃w1,w2.w =_0 w1 @ w2 ∧ w1 ∈ l1 ∧ w2 ∈ l2}.
interpretation "cat lang" 'pc a b = (cat ? a b).
+(* hints for cat *)
+nlemma cat_is_morph : ∀A:setoid. (lang A) ⇒_1 (lang A) ⇒_1 (lang A).
+#X; napply (mk_binary_morphism1 … (λA,B:lang X.A · B));
+#A1 A2 B1 B2 EA EB; napply ext_set; #x;
+ncut (∀y,x:list X.(x ∈ B1) =_1 (x ∈ B2)); ##[
+ #_; #y; ncases EA; ncases EB; #h1 h2 h3 h4; @; ##[ napply h1 | napply h2] ##] #YY;
+ncut (∀x,y:list X.(x ∈ A1) =_1 (x ∈ A2)); ##[
+ #y; #y; ncases EA; ncases EB; #h1 h2 h3 h4; @; ##[ napply h3 | napply h4] ##] #XX;
+napply (.=_1 (∑w1, w2. XX w1 w2/ E ; (# ╪_1 E) ╪_1 #));
+napply (.=_1 (∑w1, w2. YY w1 w2/ E ; # ╪_1 E)); //;
+nqed.
+
+nlemma cat_is_ext: ∀A:setoid. (Elang A) → (Elang A) → (Elang A).
+ #S A B; @ (ext_carr … A · ext_carr … B); (* XXX coercion ext_carr che non funge *)
+#x y Exy;
+ncut (∀w1,w2.(x == w1@w2) = (y == w1@w2)); ##[
+ #w1 w2; @; #H; ##[ napply (.= Exy^-1) | napply (.= Exy)] // ]
+#E; @; #H;
+##[ napply (. (∑w1,w2. (E w1 w2)^-1 / E ; (E ╪_1 #) ╪_1 #)); napply H;
+##| napply (. (∑w1,w2. E w1 w2 / E ; (E ╪_1 #) ╪_1 #)); napply H ]
+nqed.
+
+alias symbol "hint_decl" = "hint_decl_Type1".
+unification hint 0 ≔ A : setoid, B,C : Elang A;
+ AA ≟ LIST A,
+ R ≟ mk_ext_powerclass ?
+ (ext_carr ? B · ext_carr ? C) (ext_prop ? (cat_is_ext ? B C))
+(*--------------------------------------------------------------------*) ⊢
+ ext_carr AA R ≡ cat A (ext_carr AA B) (ext_carr AA C).
+
+unification hint 0 ≔ S:setoid, A,B:lang S;
+ T ≟ powerclass_setoid (list S),
+ TT ≟ unary_morphism1_setoid1 T T,
+ MM ≟ mk_unary_morphism1 T TT
+ (λA:lang S.
+ mk_unary_morphism1 T T
+ (λB:lang S.A · B) (prop11 T T (cat_is_morph S A)))
+ (prop11 T TT (cat_is_morph S))
+(*--------------------------------------------------------------------------*) ⊢
+ fun11 T T (fun11 T (unary_morphism1_setoid1 T T) MM A) B ≡ cat S A B.
+
+nlemma cat_is_ext_morph:∀A:setoid.(Elang A) ⇒_1 (Elang A) ⇒_1 (Elang A).
+#A; napply (mk_binary_morphism1 … (cat_is_ext …));
+#x1 x2 y1 y2 Ex Ey; napply (prop11 … (cat_is_morph A)); nassumption.
+nqed.
+
+unification hint 1 ≔ AA : setoid, B,C : Elang AA;
+ AAS ≟ LIST AA,
+ T ≟ ext_powerclass_setoid (list AA),
+ TT ≟ unary_morphism1_setoid1 T T,
+ R ≟ mk_unary_morphism1 ??
+ (λS:Elang AA.
+ mk_unary_morphism1 ??
+ (λS':Elang AA.
+ mk_ext_powerclass (list AA) (ext_carr ? S · ext_carr ? S')
+ (ext_prop (list AA) (cat_is_ext AA S S')))
+ (prop11 ?? (cat_is_ext_morph AA S)))
+ (prop11 ?? (cat_is_ext_morph AA)),
+ BB ≟ ext_carr ? B,
+ CC ≟ ext_carr ? C
+(*------------------------------------------------------*) ⊢
+ ext_carr AAS (fun11 T T (fun11 T TT R B) C) ≡ cat AA BB CC.
+
+(* end hints for cat *)
+
ndefinition star : ∀A:setoid.∀l:lang A.lang A ≝
λS.λl.{ w ∈ list S | ∃lw.flatten ? lw = w ∧ conjunct ? lw l}.
interpretation "star lang" 'pk l = (star ? l).
notation "𝐋 term 70 E" non associative with precedence 75 for @{'L_re $E}.
interpretation "in_l" 'L_re E = (L_re ? E).
+(* support for 𝐋 as an extensional set *)
+ndefinition L_re_is_ext : ∀S:Alpha.∀r:re S.Elang S.
+#S r; @(𝐋 r); #w1 w2 E; nelim r;
+##[ ##1,2: /2/; @; #defw1; napply (.=_0 (defw1 : [ ] = ?)); //; napply (?^-1); //;
+##| #x; @; #defw1; napply (.=_0 (defw1 : [x] = ?)); //; napply (?^-1); //;
+##| #e1 e2 H1 H2; (* not shure I shoud Inline *)
+ @; *; #s1; *; #s2; *; *; #defw1 s1L1 s2L2;
+ ##[ nlapply (trans … E^-1 defw1); #defw2;
+ ##| nlapply (trans … E defw1); #defw2; ##] @s1; @s2; /3/;
+##| #e1 e2 H1 H2; napply (H1‡H2); (* good! *)
+##| #e H; @; *; #l; *; #defw1 Pl; @l; @; //; napply (.=_1 defw1); /2/; ##]
+nqed.
+
+unification hint 0 ≔ S : Alpha,e : re S;
+ SS ≟ LIST S,
+ X ≟ mk_ext_powerclass SS (𝐋 e) (ext_prop SS (L_re_is_ext S e))
+(*-----------------------------------------------------------------*)⊢
+ ext_carr SS X ≡ L_re S e.
+
+nlemma L_re_is_morph:∀A:Alpha.(setoid1_of_setoid (re A)) ⇒_1 Ω^(list A).
+#A; @; ##[ napply (λr:re A.𝐋 r); ##] #r1; nelim r1;
+##[##1,2: #r2; ncases r2; //; ##[##1,6: *|##2,7,5,12,10: #a; *|##3,4,8,9: #a1 a2; *]
+##|#x r2; ncases r2; ##[##1,2: *|##4,5: #a1 a2; *|##6: #a; *] #y E; @; #z defz;
+ ncases z in defz; ##[##1,3: *] #zh ztl; ncases ztl; ##[##2,4: #d dl; *; #_; *]
+ *; #defx; #_; @; //; napply (?^-1); napply (.= defx^-1); //; napply (?^-1); //;
+##|#e1 e2 IH1 IH2 r2; ncases r2; ##[##1,2: *|##5: #a1 a2; *|##3,6: #a1; *]
+ #f1 f2; *; #E1 E2; nlapply (IH2 … E2); nlapply (IH1 … E1); #H1 H2;
+ nchange in match (𝐋 (e1 · e2)) with (?·?);
+ napply (.=_1 (H1 ╪_1 H2)); //;
+##|#e1 e2 IH1 IH2 r2; ncases r2; ##[##1,2: *|##4: #a1 a2; *|##3,6: #a1; *]
+ #f1 f2; *; #E1 E2; nlapply (IH2 … E2); nlapply (IH1 … E1); #H1 H2;
+ napply (.=_1 H1╪_1H2); //;
+##|#r IH r2; ncases r2; ##[##1,2: *|##4,5: #a1 a2; *|##3: #a1; *]
+ #e; #defe; nlapply (IH e defe); #H;
+ @; #x; *; #wl; *; #defx Px; @wl; @; //; nelim wl in Px; //; #l ls IH; *; #lr Pr;
+ ##[ nlapply (ifs' … H … lr) | nlapply (ifs' … H^-1 … lr) ] #le;
+ @; ##[##1,3: nassumption] /2/; ##]
+nqed.
+
+unification hint 0 ≔ A:Alpha, a:re A;
+ T ≟ setoid1_of_setoid (RE A),
+ T1 ≟ LIST A,
+ T2 ≟ powerclass_setoid T1,
+ MM ≟ mk_unary_morphism1 ??
+ (λa:setoid1_of_setoid (RE A).𝐋 a) (prop11 ?? (L_re_is_morph A))
+(*--------------------------------------------------------------------------*) ⊢
+ fun11 T T2 MM a ≡ 𝐋 a.
+
+nlemma L_re_is_ext_morph:∀A:Alpha.(setoid1_of_setoid (re A)) ⇒_1 𝛀^(list A).
+#A; @; ##[ #a; napply (L_re_is_ext ? a); ##] #a b E; @; #x H;
+##[ nchange with (x ∈ 𝐋 b); napply (. #╪_1?);
+ ##[ nchange with (𝐋 b = ?); napply (.= ┼_1 E^-1); napply #| ##skip]
+ nassumption;
+##| nchange with (x ∈ 𝐋 a); napply (. #╪_1?);
+ ##[ nchange with (𝐋 a = ?); napply (.= ┼_1 E); napply #| ##skip]
+ nassumption; ##]
+nqed.
+
+unification hint 1 ≔ AA : Alpha, a: re AA;
+ T ≟ RE AA, T1 ≟ LIST AA, TT ≟ ext_powerclass_setoid T1,
+ R ≟ mk_unary_morphism1 ??
+ (λa:setoid1_of_setoid T.
+ mk_ext_powerclass ? (𝐋 a) (ext_prop ? (L_re_is_ext AA a)))
+ (prop11 ?? (L_re_is_ext_morph AA))
+(*------------------------------------------------------*) ⊢
+ ext_carr T1 (fun11 (setoid1_of_setoid T) TT R a) ≡ L_re AA a.
+
+(* end support for 𝐋 as an extensional set *)
+
ninductive pitem (S: Type[0]) : Type[0] ≝
pz: pitem S
| pe: pitem S
P1 ≟ refl ? (eq0 (PITEM SS)),
P2 ≟ sym ? (eq0 (PITEM SS)),
P3 ≟ trans ? (eq0 (PITEM SS)),
- R ≟ mk_setoid (pitem S) (mk_equivalence_relation (pitem A) (eq_pitem SS) P1 P2 P3)
-(*---------------------------*)⊢
+ R ≟ mk_setoid (pitem S)
+ (mk_equivalence_relation (pitem A) (eq_pitem SS) P1 P2 P3)
+(*-----------------------------------------------------------------*)⊢
carr R ≡ pitem A.
unification hint 0 ≔ S:Alpha,a,b:pitem S;
- R ≟ PITEM S,
- L ≟ (pitem S)
+ R ≟ PITEM S, L ≟ (pitem S)
(* -------------------------------------------- *) ⊢
eq_pitem S a b ≡ eq_rel L (eq0 R) a b.
##| #x; @; *;
##| #x; @; #H; nchange in H with ([?] =_0 ?); ##[ napply ((.=_0 H) E); ##]
napply ((.=_0 H) E^-1);
-##| #e1 e2 H1 H2; (*
- nchange in match (w1 ∈ 𝐋\p (?·?)) with ((∃_.?)∨?);
- nchange in match (w2 ∈ 𝐋\p (?·?)) with ((∃_.?)∨?); good! *)
+##| #e1 e2 H1 H2;
napply (.= (#‡H2));
ncut (∀x1,x2. (w1 = (x1@x2)) = (w2 = (x1@x2)));##[
#x1 x2; @; #X; ##[ napply ((.= E^-1) X) | napply ((.= E) X) ] ##] #X;
napply ((∑w1,w2. X w1 w2 / H ; (H╪_1#)╪_1#) ╪_1 #);
-##| #e1 e2 H1 H2; napply (H1‡H2); (* good! *)
+##| #e1 e2 H1 H2; napply (H1‡H2);
##| #e H;
ncut (∀x1,x2.(w1 = (x1@x2)) = (w2 = (x1@x2)));##[
#x1 x2; @; #X; ##[ napply ((.= E^-1) X) | napply ((.= E) X) ] ##] #X;
- (* nnormalize in ⊢ (???%%); good! (but a bit too hard) *)
napply (∑w1,w2. X w1 w2 / H ; (H╪_1#)╪_1#);
##]
nqed.
(* theorem 16: 2 *)
nlemma oplus_cup : ∀S:Alpha.∀e1,e2:pre S.𝐋\p (e1 ⊕ e2) = 𝐋\p e1 ∪ 𝐋\p e2.
-#S r1; ncases r1; #e1 b1 r2; ncases r2; #e2 b2; (* oh my!
-nwhd in ⊢ (???(??%)?);
-nchange in ⊢(???%?) with (𝐋\p (e1 + e2) ∪ ϵ (b1 || b2));
-nchange in ⊢(???(??%?)?) with (𝐋\p (e1) ∪ 𝐋\p (e2)); *)
+#S r1; ncases r1; #e1 b1 r2; ncases r2; #e2 b2;
napply (.=_1 #╪_1 (epsilon_or ???));
napply (.=_1 (cupA…)^-1);
napply (.=_1 (cupA…)╪_1#);
napply Hw2;
nqed.
+(* XXX This seems to be a pattern for equations *)
+alias symbol "hint_decl" (instance 1) = "hint_decl_CProp2".
+unification hint 0 ≔ S : Alpha, x,y: re S;
+ SS ≟ RE S,
+ TT ≟ setoid1_of_setoid SS,
+ T ≟ carr1 TT
+(*-----------------------------------------*) ⊢
+ eq_re S x y ≡ eq_rel1 T (eq1 TT) x y.
+(* XXX the previous hint does not work *)
+
(* theorem 16: 1 → 3 *)
nlemma odot_dot_aux : ∀S:Alpha.∀e1,e2: pre S.
𝐋\p (•(\fst e2)) = 𝐋\p (\fst e2) ∪ 𝐋 |\fst e2| →
napply (.=_1 (# ╪_1 (cupC …))); napply (.=_1 (cupA …));
napply (.=_1 (# ╪_1 (cupA …)^-1)); (* XXX slow, but not because of disamb! *)
ncut (𝐋\p e2'' ∪ ϵ b2'' = 𝐋\p e2' ∪ 𝐋 |e2'|); ##[
- nchange with (𝐋\p 〈e2'',b2''〉 = 𝐋\p e2' ∪ 𝐋 |e2'|);
napply (?^-1); napply (.=_1 th1^-1); //;##] #E;
napply (.=_1 (# ╪_1 (E ╪_1 #)));
- STOP
-
- nrewrite > (eta_lp ? e2);
- nchange in match (𝐋\p 〈\fst e2,?〉) with (𝐋\p e2'∪ ϵ b2');
- nrewrite > (cup_dotD …); nrewrite > (epsilon_dot…);
- nrewrite > (cupC ? (𝐋\p e2')…); nrewrite > (cupA…);nrewrite > (cupA…);
- nrewrite < (erase_bull S e2') in ⊢ (???(??%?)); //;
+ napply (?^-1);
+ napply (.=_1 (cup_dotD …) ╪_1 #);
+ napply (.=_1 (# ╪_1 (epsilon_dot …)) ╪_1 #);
+ napply (?^-1);
+ napply (.=_1 # ╪_1 ((cupC …) ╪_1 #));
+ napply (.=_1 (cupA …)^-1);
+ napply (.=_1 (cupA …)^-1 ╪_1 #);
+ napply (.=_1 (cupA …));
+ nlapply (erase_bull S e2'); #XX;
+ napply (.=_1 (((# ╪_1 (┼_1 ?) )╪_1 #)╪_1 #)); ##[##2: napply XX; ##| ##skip]
+ //;
##| ncases e2; #e2' b2'; nchange in match (〈e1',false〉⊙?) with 〈?,?〉;
nchange in match (𝐋\p ?) with (?∪?);
nchange in match (𝐋\p (e1'·?)) with (?∪?);
(* hints for ∩ *)
nlemma intersect_is_ext: ∀A. 𝛀^A → 𝛀^A → 𝛀^A.
#S A B; @ (A ∩ B); #x y Exy; @; *; #H1 H2; @;
-##[##1,2: napply (. Exy^-1â\80¡#); nassumption;
+##[##1,2: napply (. Exy^-1â\95ª_1#); nassumption;
##|##3,4: napply (. Exy‡#); nassumption]
nqed.
alias symbol "hint_decl" = "hint_decl_Type1".
unification hint 0 ≔ A : Type[0], B,C : Ω^A;
+ T ≟ powerclass_setoid A,
R ≟ mk_unary_morphism1 ??
(λS. mk_unary_morphism1 ?? (λS'.S ∩ S') (prop11 ?? (intersect_is_morph A S)))
(prop11 ?? (intersect_is_morph A))
(*------------------------------------------------------------------------*) ⊢
- fun11 ?? (fun11 ?? R B) C ≡ intersect A B C.
+ fun11 T T (fun11 T (unary_morphism1_setoid1 T T) R B) C ≡ intersect A B C.
interpretation "prop21 ext" 'prop2 l r =
(prop11 (ext_powerclass_setoid ?)
ext_carr A R ≡ union ? (ext_carr ? B) (ext_carr ? C).
unification hint 0 ≔ S:Type[0], A,B:Ω^S;
+ T ≟ powerclass_setoid S,
MM ≟ mk_unary_morphism1 ??
(λA.mk_unary_morphism1 ?? (λB.A ∪ B) (prop11 ?? (union_is_morph S A)))
(prop11 ?? (union_is_morph S))
(*--------------------------------------------------------------------------*) ⊢
- fun11 ?? (fun11 ?? MM A) B ≡ A ∪ B.
+ fun11 T T (fun11 T (unary_morphism1_setoid1 T T) MM A) B ≡ A ∪ B.
nlemma union_is_ext_morph:∀A.𝛀^A ⇒_1 𝛀^A ⇒_1 𝛀^A.
#A; napply (mk_binary_morphism1 … (union_is_ext …));
ext_carr A R ≡ substract ? (ext_carr ? B) (ext_carr ? C).
unification hint 0 ≔ S:Type[0], A,B:Ω^S;
+ T ≟ powerclass_setoid S,
MM ≟ mk_unary_morphism1 ??
(λA.mk_unary_morphism1 ?? (λB.A - B) (prop11 ?? (substract_is_morph S A)))
(prop11 ?? (substract_is_morph S))
(*--------------------------------------------------------------------------*) ⊢
- fun11 ?? (fun11 ?? MM A) B ≡ A - B.
+ fun11 T T (fun11 T (unary_morphism1_setoid1 T T) MM A) B ≡ A - B.
nlemma substract_is_ext_morph:∀A.𝛀^A ⇒_1 𝛀^A ⇒_1 𝛀^A.
#A; napply (mk_binary_morphism1 … (substract_is_ext …));
ext_carr A R ≡ singleton A a.
unification hint 0 ≔ A:setoid, a:A;
+ T ≟ setoid1_of_setoid A,
MM ≟ mk_unary_morphism1 ??
(λa:setoid1_of_setoid A.{(a)}) (prop11 ?? (single_is_morph A))
(*--------------------------------------------------------------------------*) ⊢
- fun11 ?? MM a ≡ {(a)}.
+ fun11 T (powerclass_setoid A) MM a ≡ {(a)}.
nlemma single_is_ext_morph:∀A:setoid.(setoid1_of_setoid A) ⇒_1 𝛀^A.
#A; @; ##[ #a; napply (single_is_ext ? a); ##] #a b E; @; #x; /3/; nqed.
projects / helm.git / commitdiff
