(* *)
(**************************************************************************)
-include "datatypes/pairs.ma".
-include "datatypes/bool.ma".
+include "datatypes/pairs-setoids.ma".
+include "datatypes/bool-setoids.ma".
+include "datatypes/list-setoids.ma".
include "sets/sets.ma".
+(*
ninductive Admit : CProp[0] ≝ .
naxiom admit : Admit.
+*)
-(* single = is for the abstract equality of setoids, == is for concrete
- equalities (that may be lifted to the setoid level when needed *)
-notation < "hvbox(a break mpadded width -50% (=)= b)" non associative with precedence 45 for @{ 'eq_low $a $b }.
-notation > "a == b" non associative with precedence 45 for @{ 'eq_low $a $b }.
-
-
-(* XXX move to lists.ma *)
-ninductive list (A:Type[0]) : Type[0] ≝
- | nil: list A
- | cons: A -> list A -> list A.
-
-nlet rec eq_list (A : setoid) (l1, l2 : list A) on l1 : CProp[0] ≝
-match l1 with
-[ nil ⇒ match l2 return λ_.CProp[0] with [ nil ⇒ True | _ ⇒ False ]
-| cons x xs ⇒ match l2 with [ nil ⇒ False | cons y ys ⇒ x = y ∧ eq_list ? xs ys]].
-
-interpretation "eq_list" 'eq_low a b = (eq_list ? a b).
-
-ndefinition LIST : setoid → setoid.
-#S; @(list S); @(eq_list S);
-##[ #l; nelim l; //; #; @; //;
-##| #l1; nelim l1; ##[ #y; ncases y; //] #x xs H y; ncases y; ##[*] #y ys; *; #; @; /2/;
-##| #l1; nelim l1; ##[ #l2 l3; ncases l2; ncases l3; /3/; #z zs y ys; *]
- #x xs H l2 l3; ncases l2; ncases l3; /2/; #z zs y yz; *; #H1 H2; *; #H3 H4; @; /3/;##]
-nqed.
-
-alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
-unification hint 0 ≔ S : setoid;
- T ≟ carr S,
- P1 ≟ refl ? (eq0 (LIST S)),
- P2 ≟ sym ? (eq0 (LIST S)),
- P3 ≟ trans ? (eq0 (LIST S)),
- X ≟ mk_setoid (list S) (mk_equivalence_relation ? (eq_list T) P1 P2 P3)
-(*-----------------------------------------------------------------------*) ⊢
- carr X ≡ list T.
-
-unification hint 0 ≔ S:setoid,a,b:list S;
- R ≟ eq0 (LIST S),
- L ≟ (list S)
-(* -------------------------------------------- *) ⊢
- eq_list S a b ≡ eq_rel L R a b.
-
-notation "hvbox(hd break :: tl)"
- right associative with precedence 47
- for @{'cons $hd $tl}.
-
-notation "[ list0 x sep ; ]"
- non associative with precedence 90
- for ${fold right @'nil rec acc @{'cons $x $acc}}.
+(* XXX move somewere else *)
+ndefinition if': ∀A,B:CPROP. A = B → A → B.
+#A B; *; /2/. nqed.
-notation "hvbox(l1 break @ l2)"
- right associative with precedence 47
- for @{'append $l1 $l2 }.
+ncoercion if : ∀A,B:CPROP. ∀p:A = B. A → B ≝ if' on _p : eq_rel1 ? (eq1 CPROP) ?? to ∀_:?.?.
-interpretation "nil" 'nil = (nil ?).
-interpretation "cons" 'cons hd tl = (cons ? hd tl).
+ndefinition ifs': ∀S.∀A,B:Ω^S. A = B → ∀x. x ∈ A → x ∈ B.
+#S A B; *; /2/. nqed.
-nlet rec append A (l1: list A) l2 on l1 ≝
- match l1 with
- [ nil ⇒ l2
- | cons hd tl ⇒ hd :: append A tl l2 ].
+ncoercion ifs : ∀S.∀A,B:Ω^S. ∀p:A = B.∀x. x ∈ A → x ∈ B ≝ ifs' on _p : eq_rel1 ? (eq1 (powerclass_setoid ?))?? to ∀_:?.?.
-interpretation "append" 'append l1 l2 = (append ? l1 l2).
+(* XXX move to list-setoids-theory.ma *)
ntheorem append_nil: ∀A:setoid.∀l:list A.l @ [] = l.
#A;#l;nelim l;//; #a;#l1;#IH;nnormalize;/2/;nqed.
ntheorem associative_append: ∀A:setoid.associative (list A) (append A).
#A;#x;#y;#z;nelim x[ napply (refl ???) |#a;#x1;#H;nnormalize;/2/]nqed.
-nlet rec flatten S (l : list (list S)) on l : list S ≝
-match l with [ nil ⇒ [ ] | cons w tl ⇒ w @ flatten ? tl ].
-
(* end move to list *)
+
+(* XXX to undestand what I want inside Alpha
+ the eqb part should be split away, but when available it should be
+ possible to obtain a leibnitz equality on lemmas proved on setoids
+*)
interpretation "iff" 'iff a b = (iff a b).
-ninductive eq (A:Type[0]) (x:A) : A → CProp[0] ≝ refl: eq A x x.
+ninductive eq (A:Type[0]) (x:A) : A → CProp[0] ≝ erefl: eq A x x.
nlemma eq_rect_Type0_r':
- ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → Type[0]. P a (refl A a) → P x p.
+ ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → Type[0]. P a (erefl A a) → P x p.
#A; #a; #x; #p; ncases p; #P; #H; nassumption.
nqed.
nlemma eq_rect_Type0_r:
- ∀A.∀a.∀P: ∀x:A. eq ? x a → Type[0]. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
+ ∀A.∀a.∀P: ∀x:A. eq ? x a → Type[0]. P a (erefl A a) → ∀x.∀p:eq ? x a.P x p.
#A; #a; #P; #p; #x0; #p0; napply (eq_rect_Type0_r' ??? p0); nassumption.
nqed.
nlemma eq_rect_CProp0_r':
- ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → CProp[0]. P a (refl A a) → P x p.
+ ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → CProp[0]. P a (erefl A a) → P x p.
#A; #a; #x; #p; ncases p; #P; #H; nassumption.
nqed.
nlemma eq_rect_CProp0_r:
- ∀A.∀a.∀P: ∀x:A. eq ? x a → CProp[0]. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
+ ∀A.∀a.∀P: ∀x:A. eq ? x a → CProp[0]. P a (erefl A a) → ∀x.∀p:eq ? x a.P x p.
#A; #a; #P; #p; #x0; #p0; napply (eq_rect_CProp0_r' ??? p0); nassumption.
nqed.
-(* XXX move to bool *)
-interpretation "bool eq" 'eq_low a b = (eq bool a b).
-
-ndefinition BOOL : setoid.
-@bool; @(eq bool); ncases admit.nqed.
-
-alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
-alias id "refl" = "cic:/matita/ng/properties/relations/refl.fix(0,1,3)".
-unification hint 0 ≔ ;
- P1 ≟ refl ? (eq0 BOOL),
- P2 ≟ sym ? (eq0 BOOL),
- P3 ≟ trans ? (eq0 BOOL),
- X ≟ mk_setoid bool (mk_equivalence_relation ? (eq bool) P1 P2 P3)
-(*-----------------------------------------------------------------------*) ⊢
- carr X ≡ bool.
-
-unification hint 0 ≔ a,b;
- R ≟ eq0 BOOL,
- L ≟ bool
-(* -------------------------------------------- *) ⊢
- eq bool a b ≡ eq_rel L R a b.
-
nrecord Alpha : Type[1] ≝ {
acarr :> setoid;
eqb: acarr → acarr → bool;
}.
interpretation "eqb" 'eq_low a b = (eqb ? a b).
+(* end alpha *)
+(* re *)
ninductive re (S: Type[0]) : Type[0] ≝
z: re S
| e: re S
| o: re S → re S → re S
| k: re S → re S.
+notation < "a \sup ⋇" non associative with precedence 90 for @{ 'pk $a}.
+notation > "a ^ *" non associative with precedence 75 for @{ 'pk $a}.
+interpretation "star" 'pk a = (k ? a).
+interpretation "or" 'plus a b = (o ? a b).
+
+notation "a · b" non associative with precedence 60 for @{ 'pc $a $b}.
+interpretation "cat" 'pc a b = (c ? a b).
+
+(* to get rid of \middot *)
+ncoercion c : ∀S.∀p:re S. re S → re S ≝ c on _p : re ? to ∀_:?.?.
+
+notation < "a" non associative with precedence 90 for @{ 'ps $a}.
+notation > "` term 90 a" non associative with precedence 90 for @{ 'ps $a}.
+interpretation "atom" 'ps a = (s ? a).
+
+notation "ϵ" non associative with precedence 90 for @{ 'epsilon }.
+interpretation "epsilon" 'epsilon = (e ?).
+
+notation "0" non associative with precedence 90 for @{ 'empty_r }.
+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) }.
+
+(* setoid support for re *)
+
nlet rec eq_re (S:Alpha) (a,b : re S) on a : CProp[0] ≝
match a with
[ z ⇒ match b with [ z ⇒ True | _ ⇒ False]
#r2 r3; /3/; ##]##]
nqed.
-alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
-alias id "carr" = "cic:/matita/ng/sets/setoids/carr.fix(0,0,1)".
unification hint 0 ≔ A : Alpha;
S ≟ acarr A,
T ≟ carr S,
(*-----------------------------------------------------------------------*) ⊢
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.
-notation < "a \sup ⋇" non associative with precedence 90 for @{ 'pk $a}.
-notation > "a ^ *" non associative with precedence 75 for @{ 'pk $a}.
-interpretation "star" 'pk a = (k ? a).
-interpretation "or" 'plus a b = (o ? a b).
-
-notation "a · b" non associative with precedence 60 for @{ 'pc $a $b}.
-interpretation "cat" 'pc a b = (c ? a b).
-
-(* to get rid of \middot *)
-ncoercion c : ∀S.∀p:re S. re S → re S ≝ c on _p : re ? to ∀_:?.?.
-
-notation < "a" non associative with precedence 90 for @{ 'ps $a}.
-notation > "` term 90 a" non associative with precedence 90 for @{ 'ps $a}.
-interpretation "atom" 'ps a = (s ? a).
-
-notation "ϵ" non associative with precedence 90 for @{ 'epsilon }.
-interpretation "epsilon" 'epsilon = (e ?).
+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; /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 (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 ≡ 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; /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 MM A ≡ k SS A.
+
+nlemma s_is_morph : ∀A:Alpha.A ⇒_0 (re A).
+#A; @(λs1:A. s ? s1 ); #x y E; //; nqed.
-notation "0" non associative with precedence 90 for @{ 'empty_r }.
-interpretation "empty" 'empty_r = (z ?).
+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.
-notation > "'lang' S" non associative with precedence 90 for @{ Ω^(list $S) }.
-notation > "'Elang' S" non associative with precedence 90 for @{ 𝛀^(list $S) }.
+(* end setoids support for re *)
nlet rec conjunct S (l : list (list S)) (L : lang S) on l: CProp[0] ≝
match l with [ nil ⇒ True | cons w tl ⇒ w ∈ L ∧ conjunct ? tl L ].
-(*
-ndefinition sing_lang : ∀A:setoid.∀x:A.Ω^A ≝ λS.λx.{ w | x = w }.
-interpretation "sing lang" 'singl x = (sing_lang ? x).
-*)
-
interpretation "subset construction with type" 'comprehension t \eta.x =
(mk_powerclass t x).
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).
-notation "a || b" left associative with precedence 30 for @{'orb $a $b}.
-ndefinition orb ≝ λa,b:bool. match a with [ true ⇒ true | _ ⇒ b ].
-interpretation "orb" 'orb a b = (orb a b).
+(* 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
| po: pitem S → pitem S → pitem S
| pk: pitem S → pitem S.
+interpretation "pstar" 'pk a = (pk ? a).
+interpretation "por" 'plus a b = (po ? a b).
+interpretation "pcat" 'pc a b = (pc ? a b).
+notation < ".a" non associative with precedence 90 for @{ 'pp $a}.
+notation > "`. term 90 a" non associative with precedence 90 for @{ 'pp $a}.
+interpretation "ppatom" 'pp a = (pp ? a).
+(* to get rid of \middot *)
+ncoercion pc : ∀S.∀p:pitem S. pitem S → pitem S ≝ pc on _p : pitem ? to ∀_:?.?.
+interpretation "patom" 'ps a = (ps ? a).
+interpretation "pepsilon" 'epsilon = (pe ?).
+interpretation "pempty" 'empty_r = (pz ?).
+
+(* setoids for pitem *)
nlet rec eq_pitem (S : Alpha) (p1, p2 : pitem S) on p1 : CProp[0] ≝
match p1 with
[ pz ⇒ match p2 with [ pz ⇒ True | _ ⇒ False]
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.
-(* XXX move to pair.ma *)
-nlet rec eq_pair (A, B : setoid) (a : A × B) (b : A × B) on a : CProp[0] ≝
- match a with [ mk_pair a1 a2 ⇒
- match b with [ mk_pair b1 b2 ⇒ a1 = b1 ∧ a2 = b2 ]].
-
-interpretation "eq_pair" 'eq_low a b = (eq_pair ?? a b).
-
-nlemma PAIR : ∀A,B:setoid. setoid.
-#A B; @(A × B); @(eq_pair …);
-##[ #ab; ncases ab; #a b; @; napply #;
-##| #ab cd; ncases ab; ncases cd; #a1 a2 b1 b2; *; #E1 E2;
- @; napply (?^-1); //;
-##| #a b c; ncases a; ncases b; ncases c; #c1 c2 b1 b2 a1 a2;
- *; #E1 E2; *; #E3 E4; @; ##[ napply (.= E1); //] napply (.= E2); //.##]
-nqed.
-
-alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
-unification hint 0 ≔ AA, BB;
- A ≟ carr AA, B ≟ carr BB,
- P1 ≟ refl ? (eq0 (PAIR AA BB)),
- P2 ≟ sym ? (eq0 (PAIR AA BB)),
- P3 ≟ trans ? (eq0 (PAIR AA BB)),
- R ≟ mk_setoid (A × B) (mk_equivalence_relation ? (eq_pair …) P1 P2 P3)
-(*---------------------------------------------------------------------------*)⊢
- carr R ≡ A × B.
-
-unification hint 0 ≔ S1,S2,a,b;
- R ≟ PAIR S1 S2,
- L ≟ (pair S1 S2)
-(* -------------------------------------------- *) ⊢
- eq_pair S1 S2 a b ≡ eq_rel L (eq0 R) a b.
+(* end setoids for pitem *)
-(* end move to pair *)
-
ndefinition pre ≝ λS.pitem S × bool.
notation "\fst term 90 x" non associative with precedence 90 for @{'fst $x}.
notation > "\snd term 90 x" non associative with precedence 90 for @{'snd $x}.
interpretation "snd" 'snd x = (snd ? ? x).
-interpretation "pstar" 'pk a = (pk ? a).
-interpretation "por" 'plus a b = (po ? a b).
-interpretation "pcat" 'pc a b = (pc ? a b).
-notation < ".a" non associative with precedence 90 for @{ 'pp $a}.
-notation > "`. term 90 a" non associative with precedence 90 for @{ 'pp $a}.
-interpretation "ppatom" 'pp a = (pp ? a).
-(* to get rid of \middot *)
-ncoercion pc : ∀S.∀p:pitem S. pitem S → pitem S ≝ pc on _p : pitem ? to ∀_:?.?.
-interpretation "patom" 'ps a = (ps ? a).
-interpretation "pepsilon" 'epsilon = (pe ?).
-interpretation "pempty" 'empty_r = (pz ?).
-
notation > "|term 19 e|" non associative with precedence 70 for @{forget ? $e}.
nlet rec forget (S: Alpha) (l : pitem S) on l: re S ≝
match l with
| pc E1 E2 ⇒ (|E1| · |E2|)
| po E1 E2 ⇒ (|E1| + |E2|)
| pk E ⇒ |E|^* ].
+
notation < "|term 19 e|" non associative with precedence 70 for @{'forget $e}.
interpretation "forget" 'forget a = (forget ? a).
notation "𝐋\sub(\p) term 70 E" non associative with precedence 75 for @{'L_pi $E}.
interpretation "in_pl" 'L_pi E = (L_pi ? E).
-(* The caml, as some patches for it *)
-ncoercion setoid1_of_setoid : ∀s:setoid. setoid1 ≝ setoid1_of_setoid on _s: setoid to setoid1.
-
-alias symbol "hint_decl" (instance 1) = "hint_decl_CProp2".
-unification hint 0 ≔ S : setoid, x,y;
- SS ≟ LIST S,
- TT ≟ setoid1_of_setoid SS
-(*-----------------------------------------*) ⊢
- eq_list S x y ≡ eq_rel1 ? (eq1 TT) x y.
-
-unification hint 0 ≔ SS : setoid;
- S ≟ carr SS,
- TT ≟ setoid1_of_setoid (LIST SS)
-(*-----------------------------------------------------------------*) ⊢
- list S ≡ carr1 TT.
-
-(* XXX Ex setoid support *)
-nlemma Sig: ∀S,T:setoid.∀P: S → (T → CPROP).
- ∀y,z:T.y = z → (∀x.y=z → P x y = P x z) → (Ex S (λx.P x y)) =_1 (Ex S (λx.P x z)).
-#S T P y z Q E; @; *; #x Px; @x; nlapply (E x Q); *; /2/; nqed.
-
-notation "∑" non associative with precedence 90 for @{Sig ?????}.
-
-nlemma test : ∀S:setoid. ∀ee: S ⇒_1 S ⇒_1 CPROP.
- ∀x,y:setoid1_of_setoid S.x =_1 y → (Ex S (λw.ee x w ∧ True)) =_1 (Ex S (λw.ee y w ∧ True)).
-#S m x y E;
-napply (.=_1 (∑ E (λw,H.(H ╪_1 #)╪_1 #))).
-napply #.
-nqed.
-
-nlemma test2 : ∀S:setoid. ∀ee: S ⇒_1 S ⇒_1 CPROP.
- ∀x,y:setoid1_of_setoid S.x =_1 y →
- (True ∧ (Ex S (λw.ee x w ∧ True))) =_1 (True ∧ (Ex S (λw.ee y w ∧ True))).
-#S m x y E;
-napply (.=_1 #╪_1(∑ E (λw,H.(H ╪_1 #) ╪_1 #))).
-napply #.
-nqed.
-
-nlemma ex_setoid : ∀S:setoid.(S ⇒_1 CPROP) → setoid.
-#T P; @ (Ex T (λx:T.P x)); @;
-##[ #H1 H2; napply True |##*: //; ##]
-nqed.
-
-unification hint 0 ≔ T,P ; S ≟ (ex_setoid T P) ⊢
- Ex T (λx:T.P x) ≡ carr S.
-
-nlemma test3 : ∀S:setoid. ∀ee: S ⇒_1 S ⇒_1 CPROP.
- ∀x,y:setoid1_of_setoid S.x =_1 y →
- ((Ex S (λw.ee x w ∧ True) ∨ True)) =_1 ((Ex S (λw.ee y w ∧ True) ∨ True)).
-#S m x y E;
-napply (.=_1 (∑ E (λw,H.(H ╪_1 #) ╪_1 #)) ╪_1 #).
-napply #.
-nqed.
-(* Ex setoid support end *)
-
+(* set support for 𝐋\p *)
ndefinition L_pi_ext : ∀S:Alpha.∀r:pitem S.Elang S.
#S r; @(𝐋\p r); #w1 w2 E; nelim r;
##[ ##1,2: /2/;
##| #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 ((∃_.?)∨?);
- napply (.= (#‡H2));
- napply (.=_1 (∑ E (λx1,H1.∑ E (λx2,H2.?)))╪_1 #); ##[
- ncut ((w1 = (x1@x2)) = (w2 = (x1@x2)));##[
- @; #X; ##[ napply ((.= H1^-1) X) | napply ((.= H2) X) ] ##] #X;
- napply ( (X‡#)‡#); ##]
- napply #;
##| #e1 e2 H1 H2;
- nnormalize in ⊢ (???%%);
- napply (H1‡H2);
-##| #e H; nnormalize in ⊢ (???%%);
- napply (.=_1 (∑ E (λx1,H1.∑ E (λx2,H2.?)))); ##[
- ncut ((w1 = (x1@x2)) = (w2 = (x1@x2)));##[
- @; #X; ##[ napply ((.= H1^-1) X) | napply ((.= H2) X) ] ##] #X;
- napply ((X‡#)‡#); ##]
- napply #;##]
+ 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);
+##| #e H;
+ 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#);
+##]
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.
+
+(* end set support for 𝐋\p *)
ndefinition epsilon ≝
λS:Alpha.λb.match b return λ_.lang S with [ true ⇒ { [ ] } | _ ⇒ ∅ ].
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).
nlemma append_eq_nil : ∀S:setoid.∀w1,w2:list S. [ ] = w1 @ w2 → w1 = [ ].
#S w1; ncases w1; //. nqed.
-(* lemma 12 *)
+(* lemma 12 *) (* XXX: a case where Leibnitz equality could be exploited for H *)
nlemma epsilon_in_true : ∀S:Alpha.∀e:pre S. [ ] ∈ 𝐋\p e = (\snd e = true).
-#S r; ncases r; #e b; @; ##[##2: #H; nrewrite > H; @2; /2/; ##] ncases b;//;
-*; ##[##2:*] nelim e;
+#S r; ncases r; #e b; @; ##[##2: #H; ncases b in H; ##[##2:*] #; @2; /2/; ##]
+ncases b; //; *; ##[##2:*] nelim e;
##[ ##1,2: *; ##| #c; *; ##| #c; *| ##7: #p H;
##| #r1 r2 H G; *; ##[##2: nassumption; ##]
##| #r1 r2 H1 H2; *; /2/ by {}]
##| napply (.= (cup0 ? {[]})^-1); napply cupC; ##]
nqed.
-(* XXX move somewere else *)
-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 ∀_:?.?.
-
(* 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;
-nwhd in ⊢ (???(??%)?);
-nchange in ⊢(???%?) with (𝐋\p (e1 + e2) ∪ ϵ (b1 || b2));
-nchange in ⊢(???(??%?)?) with (𝐋\p (e1) ∪ 𝐋\p (e2));
napply (.=_1 #╪_1 (epsilon_or ???));
napply (.=_1 (cupA…)^-1);
napply (.=_1 (cupA…)╪_1#);
(* XXX problem: auto does not find # (refl) when it has a concrete == *)
nlemma odotEt : ∀S:Alpha.∀e1,e2:pitem S.∀b2:bool.
〈e1,true〉 ⊙ 〈e2,b2〉 = 〈e1 · \fst (•e2),b2 || \snd (•e2)〉.
-#S e1 e2 b2; ncases b2; nnormalize; @; //; @; napply refl; nqed.
+#S e1 e2 b2; ncases b2; @; /3/ by refl, conj, I; nqed.
+(*
nlemma LcatE : ∀S:Alpha.∀e1,e2:pitem S.
𝐋\p (e1 · e2) = 𝐋\p e1 · 𝐋 |e2| ∪ 𝐋\p e2. //; nqed.
+*)
nlemma cup_dotD : ∀S:Alpha.∀p,q,r:lang S.(p ∪ q) · r = (p · r) ∪ (q · r).
#S p q r; napply ext_set; #w; nnormalize; @;
nlemma mem_single : ∀S:setoid.∀a,b:S. a ∈ {(b)} → a = b.
#S a b; nnormalize; /2/; nqed.
-notation < "[\setoid\emsp\of\emsp term 19 x]" non associative with precedence 90 for @{'mk_setoid $x}.
-interpretation "mk_setoid" 'mk_setoid x = (mk_setoid x ?).
-
nlemma cup_sub: ∀S.∀A,B:𝛀^S.∀x. ¬ (x ∈ A) → A ∪ (B - {(x)}) = (A ∪ B) - {(x)}.
#S A B x H; napply ext_set; #w; @;
##[ *; ##[ #wa; @; ##[@;//] #H2; napply H; napply (. (mem_single ??? H2)^-1╪_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; nchange in ⊢ (???%) with (|e1| · |e2|);
-
-finqui: manca · morfismo, oppure un lemma che dice che == è come Leibnitz.
-
- nrewrite < IH1; nrewrite < IH2;
- nchange in ⊢ (??(??%)?) with (\fst (•e1 ⊙ 〈e2,false〉));
- ncases (•e1); #e3 b; ncases b; nnormalize;
- ##[ ncases (•e2); //; ##| nrewrite > IH2; //]
-##| #e1 e2 IH1 IH2; nchange in ⊢ (???%) with (.|e1| + .|e2|);
- nrewrite < IH2; nrewrite < IH1;
- nchange in ⊢ (??(??%)?) with (\fst (•e1 ⊕ •e2));
+##[ #e1 e2 IH1 IH2;
+ 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 (.=_0 #╪_0 (IH2)); //;
+##| #e1 e2 IH1 IH2; napply (?^-1);
+ napply (.=_0 (IH1^-1)╪_0(IH2^-1));
+ nchange in match (•(e1+?)) with (?⊕?);
ncases (•e1); ncases (•e2); //]
-##| #e IH; nchange in ⊢ (???%) with (.|e|^* ); nrewrite < IH;
- nchange in ⊢ (??(??%)?) with (\fst (•e))^*; //; ##]
nqed.
-nlemma eta_lp : ∀S.∀p:pre S.𝐋\p p = 𝐋\p 〈\fst p, \snd p〉.
+(*
+nlemma eta_lp : ∀S:Alpha.∀p:pre S. 𝐋\p p = 𝐋\p 〈\fst p, \snd p〉.
#S p; ncases p; //; nqed.
+*)
+
+(* XXX coercion ext_carr non applica *)
+nlemma epsilon_dot: ∀S:Alpha.∀p:Elang S. {[]} · (ext_carr ? p) = p.
+#S e; napply ext_set; #w; @; ##[##2: #Hw; @[]; @w; @; //; @; //; napply #; (* XXX auto *) ##]
+*; #w1; *; #w2; *; *; #defw defw1 Hw2;
+napply (. defw╪_1#);
+napply (. ((defw1 : [ ] = ?)^-1 ╪_0 #)╪_1#);
+napply Hw2;
+nqed.
+
-nlemma epsilon_dot: ∀S.∀p:word S → Prop. {[]} · p = p.
-#S e; napply extP; #w; nnormalize; @; ##[##2: #Hw; @[]; @w; /3/; ##]
-*; #w1; *; #w2; *; *; #defw defw1 Hw2; nrewrite < defw; nrewrite < defw1;
-napply Hw2; nqed.
(* theorem 16: 1 → 3 *)
-nlemma odot_dot_aux : ∀S.∀e1,e2: pre S.
- 𝐋\p (•(\fst e2)) = 𝐋\p (\fst e2) ∪ 𝐋 .|\fst e2| →
- 𝐋\p (e1 ⊙ e2) = 𝐋\p e1 · 𝐋 .|\fst e2| ∪ 𝐋\p e2.
+nlemma odot_dot_aux : ∀S:Alpha.∀e1,e2: pre S.
+ 𝐋\p (•(\fst e2)) = 𝐋\p (\fst e2) ∪ 𝐋 |\fst e2| →
+ 𝐋\p (e1 ⊙ e2) = 𝐋\p e1 · 𝐋 |\fst e2| ∪ 𝐋\p e2.
#S e1 e2 th1; ncases e1; #e1' b1'; ncases b1';
-##[ nwhd in ⊢ (??(??%)?); nletin e2' ≝ (\fst e2); nletin b2' ≝ (\snd e2);
+##[ nchange in match (〈?,true〉⊙?) with 〈?,?〉;
+ nletin e2' ≝ (\fst e2); nletin b2' ≝ (\snd e2);
nletin e2'' ≝ (\fst (•(\fst e2))); nletin b2'' ≝ (\snd (•(\fst e2)));
- nchange in ⊢ (??%?) with (?∪?);
- nchange in ⊢ (??(??%?)?) with (?∪?);
- nchange in match (𝐋\p 〈?,?〉) with (?∪?);
- nrewrite > (epsilon_or …); nrewrite > (cupC ? (ϵ ?)…);
- nrewrite > (cupA …);nrewrite < (cupA ?? (ϵ?)…);
- nrewrite > (?: 𝐋\p e2'' ∪ ϵ b2'' = 𝐋\p e2' ∪ 𝐋 .|e2'|); ##[##2:
- nchange with (𝐋\p 〈e2'',b2''〉 = 𝐋\p e2' ∪ 𝐋 .|e2'|);
- ngeneralize in match th1;
- nrewrite > (eta_lp…); #th1; nrewrite > th1; //;##]
- 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 (# ╪_1 (epsilon_or …))); (* XXX … is too slow if combined with .= *)
+ nchange in match b2'' with b2''; (* XXX some unfoldings happened *)
+ nchange in match b2' with b2';
+ 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'|); ##[
+ napply (?^-1); napply (.=_1 th1^-1); //;##] #E;
+ napply (.=_1 (# ╪_1 (E ╪_1 #)));
+ 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:
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