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 *)
interpretation "empty" 'empty_r = (z ?).
notation > "'lang' S" non associative with precedence 90 for @{ Ω^(list $S) }.
-notation > "'Elang' S" non associative with precedence 90 for @{ 𝛀^(list $S) }.
+notation > "'Elang' S" non associative with precedence 90 for @{ 𝛀^(LIST $S) }.
(* setoid support for re *)
(*-----------------------------------------------------------------------*) ⊢
carr X ≡ re T.
-unification hint 0 ≔ A:Alpha,a,b:re A;
+unification hint 0 ≔ A:Alpha, a,b:re (carr (acarr A));
R ≟ eq0 (RE A),
- L ≟ re A
+ L ≟ re (carr (acarr A))
(* -------------------------------------------- *) ⊢
eq_re A a b ≡ eq_rel L R a b.
-
+
+(* XXX This seems to be a pattern for equations in setoid(0) *)
+unification hint 0 ≔ AA;
+ A ≟ carr (acarr AA),
+ R ≟ setoid1_of_setoid (RE AA)
+(*-----------------------------------------------*) ⊢
+ re A ≡ carr1 R.
+
+alias symbol "hint_decl" (instance 1) = "hint_decl_CProp2".
+unification hint 0 ≔ S : Alpha, x,y: re (carr (acarr S));
+ SS ≟ RE S,
+ TT ≟ setoid1_of_setoid SS,
+ T ≟ carr1 TT
+(*-----------------------------------------*) ⊢
+ eq_re S x y ≡ eq_rel1 T (eq1 TT) x y.
+
+(* contructors are morphisms *)
nlemma c_is_morph : ∀A:Alpha.(re A) ⇒_0 (re A) ⇒_0 (re A).
-#A; napply (mk_binary_morphism … (λs1,s2:re A. s1 · s2));
-#a; nelim a;
-##[##1,2: #a' b b'; ncases a'; nnormalize; /2/ by conj;
-##|#x a' b b'; ncases a'; /2/ by conj;
-##|##4,5: #r1 r2 IH1 IH2 a'; ncases a'; nnormalize; /2/ by conj;
-##|#r IH a'; ncases a'; nnormalize; /2/ by conj; ##]
-nqed.
+#A; napply (mk_binary_morphism … (λs1,s2:re A. s1 · s2)); #a; nelim a; /2/ by conj; nqed.
(* XXX This is the good format for hints about morphisms, fix the others *)
alias symbol "hint_decl" (instance 1) = "hint_decl_Type0".
-unification hint 0 ≔ S:Alpha, A,B:re S;
- MM ≟ mk_unary_morphism ??
- (λA:re S.mk_unary_morphism ?? (λB.A · B) (prop1 ?? (c_is_morph S A)))
- (prop1 ?? (c_is_morph S)),
+unification hint 0 ≔ S:Alpha, A,B:re (carr (acarr S));
+ SS ≟ carr (acarr S),
+ MM ≟ mk_unary_morphism ?? (λA.
+ mk_unary_morphism ??
+ (λB.A · B) (prop1 ?? (fun1 ?? (c_is_morph S) A)))
+ (prop1 ?? (c_is_morph S)),
T ≟ RE S
(*--------------------------------------------------------------------------*) ⊢
- fun1 T T (fun1 T (unary_morph_setoid T T) MM A) B ≡ A · B.
+ fun1 T T (fun1 T (unary_morph_setoid T T) MM A) B ≡ c SS A B.
nlemma o_is_morph : ∀A:Alpha.(re A) ⇒_0 (re A) ⇒_0 (re A).
-#A; napply (mk_binary_morphism … (λs1,s2:re A. s1 + s2));
-#a; nelim a;
-##[##1,2: #a' b b'; ncases a'; nnormalize; /2/ by conj;
-##|#x a' b b'; ncases a'; /2/ by conj;
-##|##4,5: #r1 r2 IH1 IH2 a'; ncases a'; nnormalize; /2/ by conj;
-##|#r IH a'; ncases a'; nnormalize; /2/ by conj; ##]
-nqed.
-
-unification hint 0 ≔ S:Alpha, A,B:re S;
- MM ≟ mk_unary_morphism ??
- (λA:re S.mk_unary_morphism ?? (λB.A + B) (prop1 ?? (o_is_morph S A)))
- (prop1 ?? (o_is_morph S)),
+#A; napply (mk_binary_morphism … (λs1,s2:re A. s1 + s2)); #a; nelim a; /2/ by conj; nqed.
+
+unification hint 0 ≔ S:Alpha, A,B:re (carr (acarr S));
+ SS ≟ carr (acarr S),
+ MM ≟ mk_unary_morphism ?? (λA.
+ mk_unary_morphism ??
+ (λB.A + B) (prop1 ?? (fun1 ?? (o_is_morph S) A)))
+ (prop1 ?? (o_is_morph S)),
+ T ≟ RE S
+(*--------------------------------------------------------------------------*) ⊢
+ fun1 T T (fun1 T (unary_morph_setoid T T) MM A) B ≡ o SS A B.
+
+nlemma k_is_morph : ∀A:Alpha.(re A) ⇒_0 (re A).
+#A; @(λs1:re A. s1^* ); #a; nelim a; /2/ by conj; nqed.
+
+unification hint 0 ≔ S:Alpha, A:re (carr (acarr S));
+ SS ≟ carr (acarr S),
+ MM ≟ mk_unary_morphism ?? (λB.B^* ) (prop1 ?? (k_is_morph S)),
T ≟ RE S
(*--------------------------------------------------------------------------*) ⊢
- fun1 T T (fun1 T (unary_morph_setoid T T) MM A) B ≡ A + B.
+ fun1 T T MM A ≡ k SS A.
+
+nlemma s_is_morph : ∀A:Alpha.A ⇒_0 (re A).
+#A; @(λs1:A. s ? s1 ); #x y E; //; nqed.
+
+unification hint 0 ≔ S:Alpha, a: carr (acarr S);
+ SS ≟ carr (acarr S),
+ MM ≟ mk_unary_morphism ?? (λb.s ? b ) (prop1 ?? (s_is_morph S)),
+ T ≟ RE S, T1 ≟ acarr S
+(*--------------------------------------------------------------------------*) ⊢
+ fun1 T1 T MM a ≡ s SS a.
(* end setoids support for re *)
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,
+ BB ≟ ext_carr AA B,
+ CC ≟ ext_carr AA C,
+ R ≟ mk_ext_powerclass AA
+ (cat A (ext_carr AA B) (ext_carr AA C))
+ (ext_prop AA (cat_is_ext A B C))
+(*----------------------------------------------------------*) ⊢
+ ext_carr AA R ≡ cat A BB CC.
+
+unification hint 0 ≔ S:setoid, A,B:lang (carr S);
+ T ≟ powerclass_setoid (list (carr S)),
+ MM ≟ mk_unary_morphism1 T (unary_morphism1_setoid1 T T)
+ (λA:lang (carr S).
+ mk_unary_morphism1 T T
+ (λB:lang (carr S).cat S A B)
+ (prop11 T T (fun11 ?? (cat_is_morph S) A)))
+ (prop11 T (unary_morphism1_setoid1 T T) (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 AAS,
+ R ≟ mk_unary_morphism1 T (unary_morphism1_setoid1 T T) (λX:Elang AA.
+ mk_unary_morphism1 T T (λY:Elang AA.
+ mk_ext_powerclass AAS
+ (cat AA (ext_carr ? X) (ext_carr ? Y))
+ (ext_prop AAS (cat_is_ext AA X Y)))
+ (prop11 T T (fun11 ?? (cat_is_ext_morph AA) X)))
+ (prop11 T (unary_morphism1_setoid1 T T) (cat_is_ext_morph AA)),
+ BB ≟ ext_carr ? B,
+ CC ≟ ext_carr ? C
+(*------------------------------------------------------*) ⊢
+ ext_carr AAS (fun11 T T (fun11 T (unary_morphism1_setoid1 T T) 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).
+(* hints for star *)
+nlemma star_is_morph : ∀A:setoid. (lang A) ⇒_1 (lang A).
+#X; @(λA:lang X.A^* ); #a1 a2 E; @; #x; *; #wl; *; #defx Px; @wl; @; //;
+nelim wl in Px; //; #s tl IH; *; #a1s a1tl; /4/; nqed.
+
+nlemma star_is_ext: ∀A:setoid. (Elang A) → (Elang A).
+ #S A; @ ((ext_carr … A) ^* ); #w1 w2 E; @; *; #wl; *; #defw1 Pwl;
+ @wl; @; //; napply (.=_0 defw1); /2/; nqed.
+
+alias symbol "hint_decl" = "hint_decl_Type1".
+unification hint 0 ≔ A : setoid, B : Elang A;
+ AA ≟ LIST A,
+ BB ≟ ext_carr AA B,
+ R ≟ mk_ext_powerclass ?
+ ((ext_carr ? B)^* ) (ext_prop ? (star_is_ext ? B))
+(*--------------------------------------------------------------------*) ⊢
+ ext_carr AA R ≡ star A BB.
+
+unification hint 0 ≔ S:setoid, A:lang (carr S);
+ T ≟ powerclass_setoid (list (carr S)),
+ MM ≟ mk_unary_morphism1 T T
+ (λB:lang (carr S).star S B) (prop11 T T (star_is_morph S))
+(*--------------------------------------------------------------------------*) ⊢
+ fun11 T T MM A ≡ star S A.
+
+nlemma star_is_ext_morph:∀A:setoid.(Elang A) ⇒_1 (Elang A).
+#A; @(star_is_ext …);
+#x1 x2 Ex; napply (prop11 … (star_is_morph A)); nassumption.
+nqed.
+
+unification hint 1 ≔ AA : setoid, B : Elang AA;
+ AAS ≟ LIST AA,
+ T ≟ ext_powerclass_setoid AAS,
+ R ≟ mk_unary_morphism1 T T
+ (λS:Elang AA.
+ mk_ext_powerclass AAS (star AA (ext_carr ? S))
+ (ext_prop AAS (star_is_ext AA S)))
+ (prop11 T T (star_is_ext_morph AA)),
+ BB ≟ ext_carr ? B
+(*------------------------------------------------------*) ⊢
+ ext_carr AAS (fun11 T T R B) ≡ star AA BB.
+
+(* end hints for star *)
+
notation > "𝐋 term 70 E" non associative with precedence 75 for @{L_re ? $E}.
nlet rec L_re (S : Alpha) (r : re S) on r : lang S ≝
match r with
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 (carr (acarr S));
+ SS ≟ LIST (acarr 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 (carr (acarr A));
+ T ≟ setoid1_of_setoid (RE A),
+ T2 ≟ powerclass_setoid (list (carr (acarr A))),
+ MM ≟ mk_unary_morphism1 ??
+ (λa:carr1 (setoid1_of_setoid (RE A)).𝐋 a) (prop11 ?? (L_re_is_morph A))
+(*--------------------------------------------------------------------------*) ⊢
+ fun11 T T2 MM a ≡ L_re A 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;
+ncut (𝐋 b = 𝐋 a); ##[ napply (.=_1 (┼_1 E^-1)); // ] #EL;
+@; #x H; nchange in H ⊢ % with (x ∈ 𝐋 ?);
+##[ napply (. (# ╪_1 ?)); ##[##3: napply H |##2: ##skip ] napply EL;
+##| napply (. (# ╪_1 ?)); ##[##3: napply H |##2: ##skip ] napply (EL^-1)]
+nqed.
+
+unification hint 1 ≔ AA : Alpha, a: re (carr (acarr AA));
+ T ≟ RE AA, T1 ≟ LIST (acarr AA), T2 ≟ setoid1_of_setoid T,
+ TT ≟ ext_powerclass_setoid T1,
+ R ≟ mk_unary_morphism1 T2 TT
+ (λa:carr1 (setoid1_of_setoid T).
+ mk_ext_powerclass T1 (𝐋 a) (ext_prop T1 (L_re_is_ext AA a)))
+ (prop11 T2 TT (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 (carr 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)
+unification hint 0 ≔ S:Alpha,a,b:pitem (carr (acarr S));
+ R ≟ PITEM S, L ≟ pitem (carr (acarr 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.
-unification hint 0 ≔ S : Alpha,e : pitem S;
- SS ≟ (list S),
- X ≟ (mk_ext_powerclass SS (𝐋\p e) (ext_prop SS (L_pi_ext S e)))
+unification hint 0 ≔ S : Alpha,e : pitem (carr (acarr S));
+ SS ≟ LIST (acarr S),
+ X ≟ mk_ext_powerclass SS (𝐋\p e) (ext_prop SS (L_pi_ext S e))
(*-----------------------------------------------------------------*)⊢
ext_carr SS X ≡ 𝐋\p e.
notation < "ϵ b" non associative with precedence 90 for @{'app_epsilon $b}.
interpretation "epsilon lang" 'app_epsilon b = (epsilon ? b).
+(* hints for epsilon *)
+nlemma epsilon_is_morph : ∀A:Alpha. (setoid1_of_setoid bool) ⇒_1 (lang A).
+#X; @; ##[#b; napply(ϵ b)] #a1 a2; ncases a1; ncases a2; //; *; nqed.
+
+nlemma epsilon_is_ext: ∀A:Alpha. (setoid1_of_setoid bool) → (Elang A).
+ #S b; @(ϵ b); #w1 w2 E; ncases b; @; ##[##3,4:*]
+nchange in match (w1 ∈ ϵ true) with ([] =_0 w1);
+nchange in match (w2 ∈ ϵ true) with ([] =_0 w2); #H; napply (.= H); /2/;
+nqed.
+
+alias symbol "hint_decl" = "hint_decl_Type1".
+unification hint 0 ≔ A : Alpha, B : bool;
+ AA ≟ LIST (acarr A),
+ R ≟ mk_ext_powerclass ?
+ (ϵ B) (ext_prop ? (epsilon_is_ext ? B))
+(*--------------------------------------------------------------------*) ⊢
+ ext_carr AA R ≡ epsilon A B.
+
+unification hint 0 ≔ S:Alpha, A:bool;
+ B ≟ setoid1_of_setoid BOOL,
+ T ≟ powerclass_setoid (list (carr (acarr S))),
+ MM ≟ mk_unary_morphism1 B T
+ (λB.ϵ B) (prop11 B T (epsilon_is_morph S))
+(*--------------------------------------------------------------------------*) ⊢
+ fun11 B T MM A ≡ epsilon S A.
+
+nlemma epsilon_is_ext_morph:∀A:Alpha. (setoid1_of_setoid bool) ⇒_1 (Elang A).
+#A; @(epsilon_is_ext …);
+#x1 x2 Ex; napply (prop11 … (epsilon_is_morph A)); nassumption.
+nqed.
+
+unification hint 1 ≔ AA : Alpha, B : bool;
+ AAS ≟ LIST (acarr AA),
+ BB ≟ setoid1_of_setoid BOOL,
+ T ≟ ext_powerclass_setoid AAS,
+ R ≟ mk_unary_morphism1 BB T
+ (λS.
+ mk_ext_powerclass AAS (epsilon AA S)
+ (ext_prop AAS (epsilon_is_ext AA S)))
+ (prop11 BB T (epsilon_is_ext_morph AA))
+(*------------------------------------------------------*) ⊢
+ ext_carr AAS (fun11 BB T R B) ≡ epsilon AA B.
+
+(* end hints for epsilon *)
+
ndefinition L_pr ≝ λS : Alpha.λp:pre S. 𝐋\p\ (\fst p) ∪ ϵ (\snd p).
interpretation "L_pr" 'L_pi E = (L_pr ? E).
(* 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#);
nlemma subW : ∀S.∀a,b:Ω^S.∀w.w ∈ (a - b) → w ∈ a.
#S a b w; nnormalize; *; //; nqed.
+alias symbol "eclose" (instance 3) = "eclose".
nlemma erase_bull : ∀S:Alpha.∀a:pitem S. |\fst (•a)| = |a|.
#S a; nelim a; // by {};
##[ #e1 e2 IH1 IH2;
- napply (?^-1);
+ napply (?^-1);
napply (.=_0 (IH1^-1)╪_0 (IH2^-1));
nchange in match (•(e1 · ?)) with (?⊙?);
ncases (•e1); #e3 b; ncases b; ##[ nnormalize; ncases (•e2); /3/ by refl, conj]
napply Hw2;
nqed.
+
+
(* 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 ⊢ (???(??%?)); //;
-##| ncases e2; #e2' b2'; nchange in match (〈e1',false〉⊙?) with 〈?,?〉;
- nchange in match (𝐋\p ?) with (?∪?);
- nchange in match (𝐋\p (e1'·?)) with (?∪?);
- nchange in match (𝐋\p 〈e1',?〉) with (?∪?);
- nrewrite > (cup0…);
- nrewrite > (cupA…); //;##]
+ 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 …));
+ napply (.=_1 (((# ╪_1 (┼_1 (erase_bull S e2')) )╪_1 #)╪_1 #));
+ //;
+##| ncases e2; #e2' b2'; nchange in match (𝐋\p ?) with (?∪?∪?);
+ napply (.=_1 (cupA…));
+ napply (?^-1); nchange in match (𝐋\p 〈?,false〉) with (?∪?);
+ napply (.=_1 ((cup0…)╪_1#)╪_1#);
+ //]
nqed.
+
+
nlemma sub_dot_star :
- ∀S.∀X:word S → Prop.∀b. (X - ϵ b) · X^* ∪ {[]} = X^*.
-#S X b; napply extP; #w; @;
-##[ *; ##[##2: nnormalize; #defw; nrewrite < defw; @[]; @; //]
+ ∀S:Alpha.∀X:Elang S.∀b. (X - ϵ b) · (ext_carr … X)^* ∪ {[]} = (ext_carr … X)^*.
+#S X b; napply ext_set; #w; @;
+##[ *; ##[##2: #defw; @[]; @; //]
*; #w1; *; #w2; *; *; #defw sube; *; #lw; *; #flx cj;
- @ (w1 :: lw); nrewrite < defw; nrewrite < flx; @; //;
+ @ (w1 :: lw); @; ##[ napply (.=_0 # ╪_0 flx); napply (?^-1); //]
@; //; napply (subW … sube);
-##| *; #wl; *; #defw Pwl; nrewrite < defw; nelim wl in Pwl; ##[ #_; @2; //]
- #w' wl' IH; *; #Pw' IHp; nlapply (IH IHp); *;
- ##[ *; #w1; *; #w2; *; *; #defwl' H1 H2;
- @; ncases b in H1; #H1;
- ##[##2: nrewrite > (sub0…); @w'; @(w1@w2);
- nrewrite > (associative_append ? w' w1 w2);
- nrewrite > defwl'; @; ##[@;//] @(wl'); @; //;
- ##| ncases w' in Pw';
- ##[ #ne; @w1; @w2; nrewrite > defwl'; @; //; @; //;
- ##| #x xs Px; @(x::xs); @(w1@w2);
- nrewrite > (defwl'); @; ##[@; //; @; //; @; nnormalize; #; ndestruct]
- @wl'; @; //; ##] ##]
- ##| #wlnil; nchange in match (flatten ? (w'::wl')) with (w' @ flatten ? wl');
- nrewrite < (wlnil); nrewrite > (append_nil…); ncases b;
- ##[ ncases w' in Pw'; /2/; #x xs Pxs; @; @(x::xs); @([]);
- nrewrite > (append_nil…); @; ##[ @; //;@; //; nnormalize; @; #; ndestruct]
- @[]; @; //;
- ##| @; @w'; @[]; nrewrite > (append_nil…); @; ##[##2: @[]; @; //]
- @; //; @; //; @; *;##]##]##]
+##| *; #wl; *; #defw Pwl; napply (. (defw^-1 ╪_1 #));
+ nelim wl in Pwl; /2/;
+ #s tl IH; *; #Xs Ptl; ncases s in Xs; ##[ #; napply IH; //] #x xs Xxxs;
+ @; @(x :: xs); @(flatten ? tl); @;
+ ##[ @; ##[ napply #] @; ##[nassumption] ncases b; *; ##]
+ nelim tl in Ptl; ##[ #; @[]; /2/] #w ws IH; *; #Xw Pws; @(w :: ws); @; ##[ napply #]
+ @; //;##]
nqed.
(* theorem 16: 1 *)
alias symbol "in_pl" (instance 23) = "in_pl".
alias symbol "in_pl" (instance 5) = "in_pl".
alias symbol "eclose" (instance 21) = "eclose".
-ntheorem bull_cup : ∀S:Alpha. ∀e:pitem S. 𝐋\p (•e) = 𝐋\p e ∪ 𝐋 .|e|.
+ntheorem bull_cup : ∀S:Alpha. ∀e:pitem S. 𝐋\p (•e) = 𝐋\p e ∪ 𝐋 |e|.
#S e; nelim e; //;
- ##[ #a; napply extP; #w; nnormalize; @; *; /3/ by or_introl, or_intror;
- ##| #a; napply extP; #w; nnormalize; @; *; /3/ by or_introl; *;
+ ##[ #a; napply ext_set; #w; @; *; /3/ by or_introl, or_intror;
+ ##| #a; napply ext_set; #w; @; *; /3/ by or_introl; *;
##| #e1 e2 IH1 IH2;
- nchange in ⊢ (??(??(%))?) with (•e1 ⊙ 〈e2,false〉);
- nrewrite > (odot_dot_aux S (•e1) 〈e2,false〉 IH2);
- nrewrite > (IH1 …); nrewrite > (cup_dotD …);
- nrewrite > (cupA …); nrewrite > (cupC ?? (𝐋\p ?) …);
- nchange in match (𝐋\p 〈?,?〉) with (𝐋\p e2 ∪ {}); nrewrite > (cup0 …);
- nrewrite < (erase_dot …); nrewrite < (cupA …); //;
+ nchange in match (•(e1·e2)) with (•e1 ⊙ 〈e2,false〉);
+ napply (.=_1 (odot_dot_aux ?? 〈e2,false〉 IH2));
+ napply (.=_1 (IH1 ╪_1 #) ╪_1 #);
+ napply (.=_1 (cup_dotD …) ╪_1 #);
+ napply (.=_1 (cupA …));
+ napply (.=_1 # ╪_1 ((erase_dot ???)^-1 ╪_1 (cup0 ??)));
+ napply (.=_1 # ╪_1 (cupC…));
+ napply (.=_1 (cupA …)^-1); //;
##| #e1 e2 IH1 IH2;
- nchange in match (•(?+?)) with (•e1 ⊕ •e2); nrewrite > (oplus_cup …);
- nrewrite > (IH1 …); nrewrite > (IH2 …); nrewrite > (cupA …);
- nrewrite > (cupC ? (𝐋\p e2)…);nrewrite < (cupA ??? (𝐋\p e2)…);
- nrewrite > (cupC ?? (𝐋\p e2)…); nrewrite < (cupA …);
- nrewrite < (erase_plus …); //.
+ nchange in match (•(?+?)) with (•e1 ⊕ •e2);
+ napply (.=_1 (oplus_cup …));
+ napply (.=_1 IH1 ╪_1 IH2);
+ napply (.=_1 (cupA …));
+ napply (.=_1 # ╪_1 (# ╪_1 (cupC…)));
+ napply (.=_1 # ╪_1 (cupA ????)^-1);
+ napply (.=_1 # ╪_1 (cupC…));
+ napply (.=_1 (cupA ????)^-1);
+ napply (.=_1 # ╪_1 (erase_plus ???)^-1); //;
##| #e; nletin e' ≝ (\fst (•e)); nletin b' ≝ (\snd (•e)); #IH;
- nchange in match (𝐋\p ?) with (𝐋\p 〈e'^*,true〉);
+ nchange in match (𝐋\p ?) with (𝐋\p 〈e'^*,true〉);
nchange in match (𝐋\p ?) with (𝐋\p (e'^* ) ∪ {[ ]});
- nchange in ⊢ (??(??%?)?) with (𝐋\p e' · 𝐋 .|e'|^* );
- nrewrite > (erase_bull…e);
- nrewrite > (erase_star …);
- nrewrite > (?: 𝐋\p e' = 𝐋\p e ∪ (𝐋 .|e| - ϵ b')); ##[##2:
- nchange in IH : (??%?) with (𝐋\p e' ∪ ϵ b'); ncases b' in IH;
- ##[ #IH; nrewrite > (cup_sub…); //; nrewrite < IH;
- nrewrite < (cup_sub…); //; nrewrite > (subK…); nrewrite > (cup0…);//;
- ##| nrewrite > (sub0 …); #IH; nrewrite < IH; nrewrite > (cup0 …);//; ##]##]
- nrewrite > (cup_dotD…); nrewrite > (cupA…);
- nrewrite > (?: ((?·?)∪{[]} = 𝐋 .|e^*|)); //;
- nchange in match (𝐋 .|e^*|) with ((𝐋. |e|)^* ); napply sub_dot_star;##]
- nqed.
+ (* nwhd in match (𝐋\p e'^* ); (* XXX bug uncertain *) *)
+ nchange in ⊢ (???(??%?)?) with (𝐋\p e' · ?);
+ napply (.=_1 (# ╪_1 (┼_1 (┼_0 (erase_bull S e)))) ╪_1 #);
+ napply (.=_1 (# ╪_1 (erase_star …)) ╪_1 #);
+ ncut ( 𝐋\p e' = 𝐋\p e ∪ (𝐋 |e| - ϵ b')); ##[
+ nchange in IH : (???%?) with (𝐋\p e' ∪ ϵ b'); ncases b' in IH;
+ ##[ #IH; napply (?^-1); napply (.=_1 (cup_sub … (not_epsilon_lp…)));
+ napply (.=_1 (IH^-1 ╪_1 #));
+ alias symbol "invert" = "setoid1 symmetry".
+ (* XXX too slow if ambiguous, since it tries with a ? (takes 12s) then
+ tries with sym0 and fails immediately, then with sym1 that is OK *)
+ napply (.=_1 (cup_sub …(not_epsilon_lp …))^-1);
+ napply (.=_1 # ╪_1 (subK…)); napply (.=_1 (cup0…)); //;
+ ##| #IH; napply (?^-1); napply (.=_1 # ╪_1 (sub0 …));
+ napply (.=_1 IH^-1); napply (.=_1 (cup0 …)); //; ##]##] #EE;
+ napply (.=_1 (EE ╪_1 #) ╪_1 #);
+ napply (.=_1 (cup_dotD…) ╪_1 #);
+ napply (.=_1 (cupA…));
+ napply (.=_1 # ╪_1 (sub_dot_star…)); //; ##]
+nqed.
+
+STOP
(* theorem 16: 3 *)
nlemma odot_dot: