(* *)
(**************************************************************************)
-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.
+*)
-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 ⇒ ? | _ ⇒ ? ]
-| cons x xs ⇒ match l2 with [ nil ⇒ ? | cons y ys ⇒ x = y ∧ eq_list ? xs ys]].
-##[ napply True|napply False|napply False]nqed.
-
-ndefinition LIST : setoid → setoid.
-#S; @(list S); @(eq_list S); ncases admit; nqed.
-
-unification hint 0 ≔ S : setoid;
- 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 S) P1 P2 P3),
- T ≟ carr S
-(*-----------------------------------------------------------------------*) ⊢
- carr X ≡ list T.
-
-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.
ndefinition associative ≝ λA:setoid.λf:A → A → A.∀x,y,z.f x (f y z) = f (f x y) z.
-ninductive one : Type[0] ≝ unit : one.
-
-ndefinition force ≝
- λS:Type[2].λs:S.λT:Type[2].λt:T.λlock:one.
- match lock return λ_.Type[2] with [ unit ⇒ T].
-
-nlet rec lift (S:Type[2]) (s:S) (T:Type[2]) (t:T) (lock:one) on lock : force S s T t lock ≝
- match lock return λlock.force S s T t lock with [ unit ⇒ t ].
-
-ncoercion lift1 : ∀S:Type[1].∀s:S.∀T:Type[1].∀t:T.∀lock:one. force S s T t lock ≝ lift
- on s : ? to force ?????.
-
-ncoercion lift2 : ∀S:Type[2].∀s:S.∀T:Type[2].∀t:T.∀lock:one. force S s T t lock ≝ lift
- on s : ? to force ?????.
-
-unification hint 0 ≔ R : setoid;
- TR ≟ setoid, MR ≟ (carr R), lock ≟ unit
-(* ---------------------------------------- *) ⊢
- setoid ≡ force ?(*Type[0]*) MR TR R lock.
+ntheorem associative_append: ∀A:setoid.associative (list A) (append A).
+#A;#x;#y;#z;nelim x[ napply (refl ???) |#a;#x1;#H;nnormalize;/2/]nqed.
-unification hint 0 ≔ R : setoid1;
- TR ≟ setoid1, MR ≟ (carr1 R), lock ≟ unit
-(* ---------------------------------------- *) ⊢
- setoid1 ≡ force ? MR TR R lock.
+(* end move to list *)
-ntheorem associative_append: ∀A:setoid.associative (list A) (append A).
-#A;#x;#y;#z;nelim x[//|#a;#x1;#H;nnormalize;/2/]nqed.
+(* 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.
-notation < "a = b" non associative with precedence 45 for @{ 'eqpp $a $b}.
-interpretation "bool eq" 'eqpp 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.
-
nrecord Alpha : Type[1] ≝ {
acarr :> setoid;
eqb: acarr → acarr → bool;
eqb_true: ∀x,y. (eqb x y = true) = (x = y)
}.
-notation "a == b" non associative with precedence 45 for @{ 'eqb $a $b }.
-interpretation "eqb" 'eqb a b = (eqb ? a b).
+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.
-naxiom eq_re : ∀S:Alpha.re S → re S → CProp[0].
-ndefinition RE : Alpha → setoid.
-#A; @(re A); @(eq_re A); ncases admit. 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;
- P1 ≟ refl ? (eq0 (RE A)),
- P2 ≟ sym ? (eq0 (RE A)),
- P3 ≟ trans ? (eq0 (RE A)),
- X ≟ mk_setoid (re A) (mk_equivalence_relation ? (eq_re A) P1 P2 P3),
- T ≟ acarr A
-(*-----------------------------------------------------------------------*) ⊢
- carr X ≡ (re T).
-
notation < "a \sup ⋇" non associative with precedence 90 for @{ 'pk $a}.
-notation > "a ^ *" 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 "ϵ" non associative with precedence 90 for @{ 'epsilon }.
interpretation "epsilon" 'epsilon = (e ?).
-notation "0" non associative with precedence 90 for @{ 'empty }.
-interpretation "empty" 'empty = (z ?).
+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) }.
+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]
+ | e ⇒ match b with [ e ⇒ True | _ ⇒ False]
+ | s x ⇒ match b with [ s y ⇒ x = y | _ ⇒ False]
+ | c r1 r2 ⇒ match b with [ c s1 s2 ⇒ eq_re ? r1 s1 ∧ eq_re ? r2 s2 | _ ⇒ False]
+ | o r1 r2 ⇒ match b with [ o s1 s2 ⇒ eq_re ? r1 s1 ∧ eq_re ? r2 s2 | _ ⇒ False]
+ | k r1 ⇒ match b with [ k r2 ⇒ eq_re ? r1 r2 | _ ⇒ False]].
+
+interpretation "eq_re" 'eq_low a b = (eq_re ? a b).
-nlet rec flatten S (l : list (list S)) on l : list S ≝
-match l with [ nil ⇒ [ ] | cons w tl ⇒ w @ flatten ? tl ].
+ndefinition RE : Alpha → setoid.
+#A; @(re A); @(eq_re A);
+##[ #p; nelim p; /2/;
+##| #p1; nelim p1; ##[##1,2: #p2; ncases p2; /2/;
+ ##|##2,3: #x p2; ncases p2; /2/;
+ ##|##4,5: #e1 e2 H1 H2 p2; ncases p2; /3/; #e3 e4; *; #; @; /2/;
+ ##|#r H p2; ncases p2; /2/;##]
+##| #p1; nelim p1;
+ ##[ ##1,2: #y z; ncases y; ncases z; //; nnormalize; #; ncases (?:False); //;
+ ##| ##3: #a; #y z; ncases y; ncases z; /2/; nnormalize; #; ncases (?:False); //;
+ ##| ##4,5: #r1 r2 H1 H2 y z; ncases y; ncases z; //; nnormalize;
+ ##[##1,3,4,5,6,8: #; ncases (?:False); //;##]
+ #r1 r2 r3 r4; nnormalize; *; #H1 H2; *; #H3 H4; /3/;
+ ##| #r H y z; ncases y; ncases z; //; nnormalize; ##[##1,2,3: #; ncases (?:False); //]
+ #r2 r3; /3/; ##]##]
+nqed.
-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 ].
+unification hint 0 ≔ A : Alpha;
+ S ≟ acarr A,
+ T ≟ carr S,
+ P1 ≟ refl ? (eq0 (RE A)),
+ P2 ≟ sym ? (eq0 (RE A)),
+ P3 ≟ trans ? (eq0 (RE A)),
+ X ≟ mk_setoid (re T) (mk_equivalence_relation ? (eq_re A) P1 P2 P3)
+(*-----------------------------------------------------------------------*) ⊢
+ carr X ≡ re T.
+unification hint 0 ≔ A:Alpha, a,b:re (carr (acarr A));
+ R ≟ eq0 (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; /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.
-ndefinition empty_set : ∀A.Ω^A ≝ λA.{ w | False }.
-notation "∅" non associative with precedence 90 for @{'emptyset}.
-interpretation "empty set" 'emptyset = (empty_set ?).
+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 "{}" non associative with precedence 90 for @{'empty_lang}.
-interpretation "empty lang" 'empty_lang = (empty_lang ?).
-*)
+(* end setoids support for re *)
-ndefinition sing_lang : ∀A:setoid.∀x:A.Ω^A ≝ λS.λx.{ w | x = w }.
-interpretation "sing lang" 'singl x = (sing_lang ? x).
+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 ].
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.
-ndefinition pre ≝ λS.pitem S × bool.
-
-notation "\fst term 90 x" non associative with precedence 90 for @{'fst $x}.
-interpretation "fst" 'fst x = (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).
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 = (pz ?).
+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]
+ | pe ⇒ match p2 with [ pe ⇒ True | _ ⇒ False]
+ | ps x ⇒ match p2 with [ ps y ⇒ x = y | _ ⇒ False]
+ | pp x ⇒ match p2 with [ pp y ⇒ x = y | _ ⇒ False]
+ | pc a1 a2 ⇒ match p2 with [ pc b1 b2 ⇒ eq_pitem ? a1 b1 ∧ eq_pitem ? a2 b2| _ ⇒ False]
+ | po a1 a2 ⇒ match p2 with [ po b1 b2 ⇒ eq_pitem ? a1 b1 ∧ eq_pitem ? a2 b2| _ ⇒ False]
+ | pk a ⇒ match p2 with [ pk b ⇒ eq_pitem ? a b | _ ⇒ False]].
+
+interpretation "eq_pitem" 'eq_low a b = (eq_pitem ? a b).
+
+nlemma PITEM : ∀S:Alpha.setoid.
+#S; @(pitem S); @(eq_pitem …);
+##[ #p; nelim p; //; nnormalize; #; @; //;
+##| #p; nelim p; ##[##1,2: #y; ncases y; //; ##|##3,4: #x y; ncases y; //; #; napply (?^-1); nassumption;
+ ##|##5,6: #r1 r2 H1 H2 p2; ncases p2; //; #s1 s2; nnormalize; *; #; @; /2/;
+ ##| #r H y; ncases y; //; nnormalize; /2/;##]
+##| #x; nelim x;
+ ##[ ##1,2: #y z; ncases y; ncases z; //; nnormalize; #; ncases (?:False); //;
+ ##| ##3,4: #a; #y z; ncases y; ncases z; /2/; nnormalize; #; ncases (?:False); //;
+ ##| ##5,6: #r1 r2 H1 H2 y z; ncases y; ncases z; //; nnormalize;
+ ##[##1,2,5,6,7,8,4,10: #; ncases (?:False); //;##]
+ #r1 r2 r3 r4; nnormalize; *; #H1 H2; *; #H3 H4; /3/;
+ ##| #r H y z; ncases y; ncases z; //; nnormalize; ##[##1,2,3,4: #; ncases (?:False); //]
+ #r2 r3; /3/; ##]##]
+nqed.
+
+alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
+unification hint 0 ≔ SS:Alpha;
+ S ≟ acarr SS,
+ A ≟ carr S,
+ P1 ≟ refl ? (eq0 (PITEM SS)),
+ P2 ≟ sym ? (eq0 (PITEM SS)),
+ P3 ≟ trans ? (eq0 (PITEM SS)),
+ 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 (carr (acarr S));
+ R ≟ PITEM S, L ≟ pitem (carr (acarr S))
+(* -------------------------------------------- *) ⊢
+ eq_pitem S a b ≡ eq_rel L (eq0 R) a b.
+
+(* end setoids for pitem *)
+
+ndefinition pre ≝ λS.pitem S × bool.
+
+notation "\fst term 90 x" non associative with precedence 90 for @{'fst $x}.
+interpretation "fst" 'fst x = (fst ? ? x).
+notation > "\snd term 90 x" non associative with precedence 90 for @{'snd $x}.
+interpretation "snd" 'snd x = (snd ? ? x).
notation > "|term 19 e|" non associative with precedence 70 for @{forget ? $e}.
nlet rec forget (S: Alpha) (l : pitem S) on l: re S ≝
| 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).
-unification hint 0 ≔ S,a,b;
- R ≟ LIST S
-(* -------------------------------------------- *) ⊢
- eq_list S a b ≡ eq_rel (list S) (eq0 R) a b.
-
-notation > "B ⇒_0 C" right associative with precedence 72 for @{'umorph0 $B $C}.
-notation > "B ⇒_1 C" right associative with precedence 72 for @{'umorph1 $B $C}.
-notation "B ⇒\sub 0 C" right associative with precedence 72 for @{'umorph0 $B $C}.
-notation "B ⇒\sub 1 C" right associative with precedence 72 for @{'umorph1 $B $C}.
-
-interpretation "unary morphism 0" 'umorph0 A B = (unary_morphism A B).
-interpretation "unary morphism 1" 'umorph1 A B = (unary_morphism1 A B).
-
-ncoercion setoid1_of_setoid : ∀s:setoid. setoid1 ≝ setoid1_of_setoid on _s: setoid to setoid1.
-
-nlemma exists_is_morph: (* BUG *) ∀S,T:setoid.∀P: S ⇒_1 (T ⇒_1 (CProp[0]:?)).
- ∀y,z:S.y =_0 z → (Ex T (P y)) = (Ex T (P z)).
-#S T P y z E; @;
-##[ *; #x Px; @x; alias symbol "refl" (instance 4) = "refl".
- alias symbol "prop2" (instance 2) = "prop21".
- napply (. E^-1‡#); napply Px;
-##| *; #x Px; @x; napply (. E‡#); napply Px;##]
-nqed.
-
-ndefinition ex_morph : ∀S:setoid. (S ⇒_1 CPROP) ⇒_1 CPROP.
-#S; @; ##[ #P; napply (Ex ? P); ##| #P1 P2 E; @;
-*; #x; #H; @ x; nlapply (E x x ?); //; *; /2/;
-nqed.
-
-nlemma d : ∀S:Alpha.
- ∀ee: (setoid1_of_setoid (list S)) ⇒_1 (setoid1_of_setoid (list S)) ⇒_1 CPROP.
- ∀x,y:list S.x = y →
- let form ≝ comp1_unary_morphisms ??? (ex_morph (list S)) ee in
- form x = form y.
- #S ee x y E;
- nletin F ≝ (comp1_unary_morphisms ??? (ex_morph (list S)) ee);
-
- nnormalize;
- nlapply (exists_is_morph (list S) (list S) ? ?? E);
-
- nchange with (F x = F y);
- nchange in E with (eq_rel1 ? (eq1 (setoid1_of_setoid (LIST S))) x y);
- napply (.= † E);
- napply #.
+(* 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;
+ 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 (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.
-E : w = w2
+(* end set support for 𝐋\p *)
-
- Σ(λx.(#‡E)‡#) : ∃x.x = w ∧ m → ∃x.x = w2 ∧ m
- λx.(#‡E)‡# : ∀x.x = w ∧ m → x = w2 ∧ m
-
-
-w;
-F ≟ ex_moprh ∘ G
-g ≟ fun11 G
-------------------------------
-ex (λx.g w x) ==?== fun11 F w
-
-x ⊢ fun11 ?h ≟ λw. ?g w x
-?G ≟ morphism_leibniz (T → S) ∘ ?h
-------------------------------
-(λw. (λx:T.?g w x)) ==?== fun11 ?G
-
-...
-x ⊢ fun11 ?h ==?== λw. eq x w ∧ m [w]
-(λw,x.eq x w ∧ m[w]) ==?== fun11 ?G
-ex (λx.?g w x) ==?== ex (λx.x = w ∧ m[w])
-
-ndefinition ex_morph : ∀S:setoid. (S ⇒_1 CPROP) ⇒_1 CPROP.
-#S; @; ##[ #P; napply (Ex ? P); ##| ncases admit. ##] nqed.
+ndefinition epsilon ≝
+ λS:Alpha.λb.match b return λ_.lang S with [ true ⇒ { [ ] } | _ ⇒ ∅ ].
-ndefinition ex_morph1 : ∀S:setoid. (S ⇒_1 CPROP) ⇒_1 CPROP.
-#S; @; ##[ #P; napply (Ex ? (λx.P); ##| ncases admit. ##] nqed.
+interpretation "epsilon" 'epsilon = (epsilon ?).
+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 d : ∀S:Alpha.
- ∀ee: (setoid1_of_setoid (list S)) ⇒_1 (setoid1_of_setoid (list S)) ⇒_1 CPROP.
- ∀x,y:list S.x = y →
- let form ≝ comp1_unary_morphisms ??? (ex_morph (list S)) ee in
- form x = form y.
- #S ee x y E;
- nletin F ≝ (comp1_unary_morphisms ??? (ex_morph (list S)) ee);
-
- nnormalize;
-
- nchange with (F x = F y);
- nchange in E with (eq_rel1 ? (eq1 (setoid1_of_setoid (LIST S))) x y);
- napply (.= † E);
- napply #.
+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.
-
-nlemma d : ∀S:Alpha.∀ee: (setoid1_of_setoid (list S)) ⇒_1 (setoid1_of_setoid (list S)) ⇒_1 CPROP.∀x,y:list S.x = y →
- let form ≝ comp1_unary_morphisms ??? (ex_morph (list S)) ee in
- form x = form y.
- #S ee x y E;
- nletin F ≝ (comp1_unary_morphisms ??? (ex_morph (list S)) ee);
-
- nnormalize;
-
- nchange with (F x = F y);
- nchange in E with (eq_rel1 ? (eq1 (setoid1_of_setoid (LIST S))) x y);
- napply (.= † E);
- napply #.
-nqed.
-
-
-nlemma d : ∀S:Alpha.(setoid1_of_setoid (list S)) ⇒_1 CPROP.
- #S; napply (comp1_unary_morphisms ??? (ex_morph (list S)) ?);
- napply (eq1).
-
-
-
-(*
-ndefinition comp_ex : ∀X,S,T,K.∀P:X ⇒_1 (S ⇒_1 T).∀Pc : (S ⇒_1 T) ⇒_1 K. X ⇒_1 K.
- *)
-
-ndefinition L_pi_ext : ∀S:Alpha.∀r:pitem S.Elang S.
-#S r; @(𝐋\p r); #w1 w2 E; nelim r; /2/;
-##[ #x; @; #H; ##[ nchange in H ⊢ % with ([?]=?); napply ((.= H) E)]
- nchange in H ⊢ % with ([?]=?); napply ((.= H) E^-1);
-##| #e1 e2 H1 H2;
- nchange in match (w1 ∈ 𝐋\p (?·?)) with ((∃_.?)∨?);
- napply (.= (#‡H2));
- napply (.= (Eexists ?? ? w1 w2 E)‡#);
+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.
-
- nchange in match (w2 ∈ 𝐋\p (?·?)) with (?∨?);
- napply (.=
-
-
- //; napply (trans ?? ??? H E);
- napply (trans (list S) (eq0 (LIST S)) ? [?] ?(*w1 [x] w2*) H E);
- nlapply (trans (list S) (eq0 (LIST S))).
-
-napply (.= H); nnormalize; nlapply (trans ? [x] w1 w2 E H); napply (.= ?) [napply (w1 = [x])] ##[##2: napply #; napply sym1; napply refl1 ]
+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.
-ndefinition epsilon ≝
- λS:Alpha.λb.match b return λ_.lang S with [ true ⇒ { [ ] } | _ ⇒ ∅ ].
+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.
-interpretation "epsilon" 'epsilon = (epsilon ?).
-notation < "ϵ b" non associative with precedence 90 for @{'app_epsilon $b}.
-interpretation "epsilon lang" 'app_epsilon b = (epsilon ? 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 = [ ].
+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 {}]
*; #w1; *; #w2; *; *;
-##[ #defw1 H1 foo; napply H; napply (. #‡#); (append_eq_nil … defw1)^-1‡#);
-
- nrewrite > (append_eq_nil ? … w1 w2 …); /3/ by {};//;
+##[ #defw1 H1 foo; napply H;
+ napply (. (append_eq_nil ? ?? defw1)^-1╪_1#);
+ nassumption;
+##| #defw1 H1 foo; napply H;
+ napply (. (append_eq_nil ? ?? defw1)^-1╪_1#);
+ nassumption;
+##]
nqed.
-nlemma not_epsilon_lp : ∀S.∀e:pitem S. ¬ (𝐋\p e [ ]).
-#S e; nelim e; nnormalize; /2/ by nmk;
-##[ #; @; #; ndestruct;
-##| #r1 r2 n1 n2; @; *; /2/; *; #w1; *; #w2; *; *; #H;
- nrewrite > (append_eq_nil …H…); /2/;
-##| #r1 r2 n1 n2; @; *; /2/;
-##| #r n; @; *; #w1; *; #w2; *; *; #H;
- nrewrite > (append_eq_nil …H…); /2/;##]
+nlemma not_epsilon_lp : ∀S:Alpha.∀e:pitem S. ¬ ([ ] ∈ (𝐋\p e)).
+#S e; nelim e; ##[##1,2,3,4: nnormalize;/2/]
+##[ #p1 p2 np1 np2; *; ##[##2: napply np2] *; #w1; *; #w2; *; *; #abs;
+ nlapply (append_eq_nil ??? abs); # defw1; #; napply np1;
+ napply (. defw1^-1╪_1#);
+ nassumption;
+##| #p1 p2 np1 np2; *; nchange with (¬?); //;
+##| #r n; *; #w1; *; #w2; *; *; #abs; #; napply n;
+ nlapply (append_eq_nil ??? abs); # defw1; #;
+ napply (. defw1^-1╪_1#);
+ nassumption;##]
nqed.
ndefinition lo ≝ λS:Alpha.λa,b:pre S.〈\fst a + \fst b,\snd a || \snd b〉.
notation < "a \sup ⊛" non associative with precedence 90 for @{'lk $op $a}.
interpretation "lk" 'lk op a = (lk ? op a).
-notation > "a^⊛" non associative with precedence 90 for @{'lk eclose $a}.
+notation > "a ^ ⊛" non associative with precedence 75 for @{'lk eclose $a}.
notation > "•" non associative with precedence 60 for @{eclose ?}.
nlet rec eclose (S: Alpha) (E: pitem S) on E : pre S ≝
match E with
- [ pz ⇒ 〈 ∅, false 〉
+ [ pz ⇒ 〈 0, false 〉
| pe ⇒ 〈 ϵ, true 〉
| ps x ⇒ 〈 `.x, false 〉
| pp x ⇒ 〈 `.x, false 〉
ndefinition reclose ≝ λS:Alpha.λp:pre S.let p' ≝ •\fst p in 〈\fst p',\snd p || \snd p'〉.
interpretation "reclose" 'eclose x = (reclose ? x).
-ndefinition eq_f1 ≝ λS.λa,b:word S → Prop.∀w.a w ↔ b w.
-notation > "A =1 B" non associative with precedence 45 for @{'eq_f1 $A $B}.
-notation "A =\sub 1 B" non associative with precedence 45 for @{'eq_f1 $A $B}.
-interpretation "eq f1" 'eq_f1 a b = (eq_f1 ? a b).
-
-naxiom extP : ∀S.∀p,q:word S → Prop.(p =1 q) → p = q.
-
nlemma epsilon_or : ∀S:Alpha.∀b1,b2. ϵ(b1 || b2) = ϵ b1 ∪ ϵ b2. ##[##2: napply S]
-#S b1 b2; ncases b1; ncases b2; napply extP; #w; nnormalize; @; /2/; *; //; *;
+#S b1 b2; ncases b1; ncases b2;
+nchange in match (true || true) with true;
+nchange in match (true || false) with true;
+nchange in match (ϵ true) with {[]};
+nchange in match (ϵ false) with ∅;
+##[##1,4: napply ((cupID…)^-1);
+##| napply ((cup0 ? {[]})^-1);
+##| napply (.= (cup0 ? {[]})^-1); napply cupC; ##]
nqed.
-nlemma cupA : ∀S.∀a,b,c:word S → Prop.a ∪ b ∪ c = a ∪ (b ∪ c).
-#S a b c; napply extP; #w; nnormalize; @; *; /3/; *; /3/; nqed.
-
-nlemma cupC : ∀S. ∀a,b:word S → Prop.a ∪ b = b ∪ a.
-#S a b; napply extP; #w; @; *; nnormalize; /2/; 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;
-nwhd in ⊢ (??(??%)?);
-nchange in ⊢(??%?) with (𝐋\p (e1 + e2) ∪ ϵ (b1 || b2));
-nchange in ⊢(??(??%?)?) with (𝐋\p (e1) ∪ 𝐋\p (e2));
-nrewrite > (epsilon_or S …); nrewrite > (cupA S (𝐋\p e1) …);
-nrewrite > (cupC ? (ϵ b1) …); nrewrite < (cupA S (𝐋\p e2) …);
-nrewrite > (cupC ? ? (ϵ b1) …); nrewrite < (cupA …); //;
+napply (.=_1 #╪_1 (epsilon_or ???));
+napply (.=_1 (cupA…)^-1);
+napply (.=_1 (cupA…)╪_1#);
+napply (.=_1 (#╪_1(cupC…))╪_1#);
+napply (.=_1 (cupA…)^-1╪_1#);
+napply (.=_1 (cupA…));
+//;
nqed.
-nlemma odotEt :
- ∀S.∀e1,e2:pitem S.∀b2. 〈e1,true〉 ⊙ 〈e2,b2〉 = 〈e1 · \fst (•e2),b2 || \snd (•e2)〉.
-#S e1 e2 b2; nnormalize; ncases (•e2); //; nqed.
-nlemma LcatE : ∀S.∀e1,e2:pitem S.𝐋\p (e1 · e2) = 𝐋\p e1 · 𝐋 .|e2| ∪ 𝐋\p e2. //; nqed.
+(* 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; @; /3/ by refl, conj, I; nqed.
-nlemma cup_dotD : ∀S.∀p,q,r:word S → Prop.(p ∪ q) · r = (p · r) ∪ (q · r).
-#S p q r; napply extP; #w; nnormalize; @;
+(*
+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; @;
##[ *; #x; *; #y; *; *; #defw; *; /7/ by or_introl, or_intror, ex_intro, conj;
##| *; *; #x; *; #y; *; *; /7/ by or_introl, or_intror, ex_intro, conj; ##]
nqed.
-nlemma cup0 :∀S.∀p:word S → Prop.p ∪ {} = p.
-#S p; napply extP; #w; nnormalize; @; /2/; *; //; *; nqed.
-nlemma erase_dot : ∀S.∀e1,e2:pitem S.𝐋 .|e1 · e2| = 𝐋 .|e1| · 𝐋 .|e2|.
-#S e1 e2; napply extP; nnormalize; #w; @; *; #w1; *; #w2; *; *; /7/ by ex_intro, conj;
+nlemma erase_dot : ∀S:Alpha.∀e1,e2:pitem S.𝐋 |e1 · e2| = 𝐋 |e1| · 𝐋 |e2|.
+#S e1 e2; napply ext_set; nnormalize; #w; @; *; #w1; *; #w2; *; *; /7/ by ex_intro, conj;
nqed.
-nlemma erase_plus : ∀S.∀e1,e2:pitem S.𝐋 .|e1 + e2| = 𝐋 .|e1| ∪ 𝐋 .|e2|.
-#S e1 e2; napply extP; nnormalize; #w; @; *; /4/ by or_introl, or_intror; nqed.
+nlemma erase_plus : ∀S:Alpha.∀e1,e2:pitem S.𝐋 |e1 + e2| = 𝐋 |e1| ∪ 𝐋 |e2|.
+#S e1 e2; napply ext_set; nnormalize; #w; @; *; /4/ by or_introl, or_intror; nqed.
-nlemma erase_star : ∀S.∀e1:pitem S.𝐋 .|e1|^* = 𝐋 .|e1^*|. //; nqed.
+nlemma erase_star : ∀S:Alpha.∀e1:pitem S.𝐋 |e1|^* = 𝐋 |e1^*|. //; nqed.
-ndefinition substract := λS.λp,q:word S → Prop.λw.p w ∧ ¬ q w.
-interpretation "substract" 'minus a b = (substract ? a b).
+nlemma mem_single : ∀S:setoid.∀a,b:S. a ∈ {(b)} → a = b.
+#S a b; nnormalize; /2/; nqed.
-nlemma cup_sub: ∀S.∀a,b:word S → Prop. ¬ (a []) → a ∪ (b - {[]}) = (a ∪ b) - {[]}.
-#S a b c; napply extP; #w; nnormalize; @; *; /4/; *; /4/; nqed.
+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#); //]
+ *; #wb nwn; @; ##[@2;//] //;
+##| *; *; ##[ #wa nwn; @; //] #wb nwn; @2; @; //;##]
+nqed.
-nlemma sub0 : ∀S.∀a:word S → Prop. a - {} = a.
-#S a; napply extP; #w; nnormalize; @; /3/; *; //; nqed.
+nlemma sub0 : ∀S.∀a:Ω^S. a - ∅ = a.
+#S a; napply ext_set; #w; nnormalize; @; /3/; *; //; nqed.
-nlemma subK : ∀S.∀a:word S → Prop. a - a = {}.
-#S a; napply extP; #w; nnormalize; @; *; /2/; nqed.
+nlemma subK : ∀S.∀a:Ω^S. a - a = ∅.
+#S a; napply ext_set; #w; nnormalize; @; *; /2/; nqed.
-nlemma subW : ∀S.∀a,b:word S → Prop.∀w.(a - b) w → a w.
+nlemma subW : ∀S.∀a,b:Ω^S.∀w.w ∈ (a - b) → w ∈ a.
#S a b w; nnormalize; *; //; nqed.
-nlemma erase_bull : ∀S.∀a:pitem S. .|\fst (•a)| = .|a|.
+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|);
- 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));
- ncases (•e1); ncases (•e2); //;
-##| #e IH; nchange in ⊢ (???%) with (.|e|^* ); nrewrite < IH;
- nchange in ⊢ (??(??%)?) with (\fst (•e))^*; //; ##]
+##[ #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); //]
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:
projects / helm.git / blobdiff