(* *)
(**************************************************************************)
-include "datatypes/pairs.ma".
-include "datatypes/bool.ma".
-include "logic/cprop.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;
- T ≟ carr S,
- P1 ≟ refl ? (eq (LIST S)),
- P2 ≟ sym ? (eq (LIST S)),
- P3 ≟ trans ? (eq (LIST S)),
- X ≟ mk_setoid (list S) (mk_equivalence_relation ? (eq_list S) P1 P2 P3)
-(*-----------------------------------------------------------------------*) ⊢
- 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}}.
-
-notation "hvbox(l1 break @ l2)"
- right associative with precedence 47
- for @{'append $l1 $l2 }.
-
-interpretation "nil" 'nil = (nil ?).
-interpretation "cons" 'cons hd tl = (cons ? hd tl).
+(* XXX move somewere else *)
+ndefinition if': ∀A,B:CPROP. A = B → A → B.
+#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 if : ∀A,B:CPROP. ∀p:A = B. A → B ≝ if' on _p : eq_rel1 ???? 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.
+ntheorem associative_append: ∀A:setoid.associative (list A) (append A).
+#A;#x;#y;#z;nelim x[ napply (refl ???) |#a;#x1;#H;nnormalize;/2/]nqed.
-ndefinition force ≝
- λS:Type[1].λs:S.λT:Type[1].λt:T.λlock:one.
- match lock return λ_.Type[1] with [ unit ⇒ T].
+(* end move to list *)
-nlet rec lift (S:Type[1]) (s:S) (T:Type[1]) (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 lift : ∀S:Type[1].∀s:S.∀T:Type[1].∀t:T.∀lock:one. force S s T t lock ≝ lift
- on s : ? to force ?????.
+(* 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).
-unification hint 0 ≔ R : setoid;
- TR ≟ setoid, MR ≟ (carr R), lock ≟ unit
-(* ---------------------------------------- *) ⊢
- setoid ≡ force ?(*Type[0]*) MR TR R lock.
+ninductive eq (A:Type[0]) (x:A) : A → CProp[0] ≝ erefl: eq A x x.
-ntheorem associative_append: ∀A:setoid.associative (list A) (append A).
-#A;#x;#y;#z;nelim x[//|#a;#x1;#H;nnormalize;/2/]nqed.
+nlemma eq_rect_Type0_r':
+ ∀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.
-interpretation "iff" 'iff a b = (iff a b).
+nlemma eq_rect_Type0_r:
+ ∀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.
-naxiom eq_bool : bool → bool → CProp[0].
-ndefinition BOOL : setoid.
-@bool; @eq_bool; ncases admit.nqed.
+nlemma eq_rect_CProp0_r':
+ ∀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.
-unification hint 0 ≔ ;
- P1 ≟ refl ? (eq BOOL),
- P2 ≟ sym ? (eq BOOL),
- P3 ≟ trans ? (eq BOOL),
- X ≟ mk_setoid bool (mk_equivalence_relation ? eq_bool P1 P2 P3)
-(*-----------------------------------------------------------------------*) ⊢
- carr X ≡ bool.
+nlemma eq_rect_CProp0_r:
+ ∀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.
nrecord Alpha : Type[1] ≝ {
acarr :> setoid;
- eqb: acarr → acarr → bool (*;
- eqb_true: ∀x,y. (eqb x y = true) = (x = y)*)
+ 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;
- T ≟ acarr A,
- P1 ≟ refl ? (eq (RE A)),
- P2 ≟ sym ? (eq (RE A)),
- P3 ≟ trans ? (eq (RE A)),
- X ≟ mk_setoid (re T) (mk_equivalence_relation ? (eq_re A) P1 P2 P3)
-(*-----------------------------------------------------------------------*) ⊢
- 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 "∅" 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) }.
+
+(* 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).
-nrecord Setl (A : Type[0]) : Type[1] ≝ { in_set : A → CProp[0] }.
-ndefinition Lang ≝ λA.Setl (list A).
+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.
-interpretation "in Setl" 'mem x l = (in_set ? l x).
-interpretation "compr Lang" 'comprehension t f = (mk_Setl t f).
+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 A;
+ R ≟ eq0 (RE A),
+ L ≟ re A
+(* -------------------------------------------- *) ⊢
+ eq_re A a b ≡ eq_rel L R a b.
+
+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.
-nlet rec flatten S (l : list (list S)) on l : list S ≝
-match l with [ nil ⇒ [ ] | cons w tl ⇒ w @ flatten ? tl ].
+(* 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)),
+ T ≟ RE S
+(*--------------------------------------------------------------------------*) ⊢
+ fun1 T T (fun1 T (unary_morph_setoid T T) MM A) B ≡ 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.
-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 ≔ 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)),
+ T ≟ RE S
+(*--------------------------------------------------------------------------*) ⊢
+ fun1 T T (fun1 T (unary_morph_setoid T T) MM A) B ≡ A + B.
-ndefinition empty_lang ≝ λS.{ w ∈ list S | False }.
-notation "{}" non associative with precedence 90 for @{'empty_lang}.
-interpretation "empty lang" 'empty_lang = (empty_lang ?).
+(* end setoids support for re *)
-ndefinition sing_lang ≝ λS:Alpha.λx.{ w ∈ list S | 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 ].
-ndefinition union ≝ λS.λl1,l2.{ w ∈ list S | w ∈ l1 ∨ w ∈ l2}.
-interpretation "union lang" 'union a b = (union ? a b).
+interpretation "subset construction with type" 'comprehension t \eta.x =
+ (mk_powerclass t x).
-ndefinition cat ≝
- λS:Alpha.λl1,l2.{ w ∈ list S | ∃w1,w2.w1 @ w2 = w ∧ w1 ∈ l1 ∧ w2 ∈ l2}.
+ndefinition cat : ∀A:setoid.∀l1,l2:lang A.lang A ≝
+ 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).
-ndefinition star ≝
- λS:Alpha.λl.{ w ∈ list S | ∃lw.flatten ? lw = w ∧ conjunct ? lw l}.
+ndefinition star : ∀A:setoid.∀l:lang A.lang A ≝
+ λS.λl.{ w ∈ list S | ∃lw.flatten ? lw = w ∧ conjunct ? lw l}.
interpretation "star lang" 'pk l = (star ? l).
-notation > "𝐋 term 90 E" non associative with precedence 75 for @{L_re ? $E}.
-nlet rec L_re (S : Alpha) (r : re S) on r : Lang S ≝
+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
-[ z ⇒ {}
+[ z ⇒ ∅
| e ⇒ { [ ] }
| s x ⇒ { [x] }
| c r1 r2 ⇒ 𝐋 r1 · 𝐋 r2
| o r1 r2 ⇒ 𝐋 r1 ∪ 𝐋 r2
| k r1 ⇒ (𝐋 r1) ^*].
-notation "𝐋 term 90 E" non associative with precedence 75 for @{'L_re $E}.
+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).
-
ninductive pitem (S: Type[0]) : Type[0] ≝
pz: pitem S
| pe: 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 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)
+(* -------------------------------------------- *) ⊢
+ 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 ≝
match l with
- [ pz ⇒ ∅
+ [ pz ⇒ 0
| pe ⇒ ϵ
| ps x ⇒ `x
| pp x ⇒ `x
| 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}.
+
+notation < "|term 19 e|" non associative with precedence 70 for @{'forget $e}.
interpretation "forget" 'forget a = (forget ? a).
-notation > "𝐋\p\ term 90 E" non associative with precedence 90 for @{in_pl ? $E}.
-nlet rec in_pl (S : Alpha) (r : pitem S) on r : word S → Prop ≝
+notation > "𝐋\p\ term 70 E" non associative with precedence 75 for @{L_pi ? $E}.
+nlet rec L_pi (S : Alpha) (r : pitem S) on r : lang S ≝
match r with
-[ pz ⇒ {}
-| pe ⇒ {}
-| ps _ ⇒ {}
+[ pz ⇒ ∅
+| pe ⇒ ∅
+| ps _ ⇒ ∅
| pp x ⇒ { [x] }
-| pc r1 r2 ⇒ 𝐋\p\ r1 · 𝐋 .|r2| ∪ 𝐋\p\ r2
+| pc r1 r2 ⇒ 𝐋\p\ r1 · 𝐋 |r2| ∪ 𝐋\p\ r2
| po r1 r2 ⇒ 𝐋\p\ r1 ∪ 𝐋\p\ r2
-| pk r1 ⇒ 𝐋\p\ r1 · 𝐋 (.|r1|^* ) ].
-notation > "𝐋\p term 90 E" non associative with precedence 90 for @{'in_pl $E}.
-notation "𝐋\sub(\p) term 90 E" non associative with precedence 90 for @{'in_pl $E}.
-interpretation "in_pl" 'in_pl E = (in_pl ? E).
-interpretation "in_pl mem" 'mem w l = (in_pl ? l w).
+| pk r1 ⇒ 𝐋\p\ r1 · 𝐋 (|r1|^* ) ].
+notation > "𝐋\p term 70 E" non associative with precedence 75 for @{'L_pi $E}.
+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).
+
+(* 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 ((∃_.?)∨?); good! *)
+ 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! *)
+##| #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)))
+(*-----------------------------------------------------------------*)⊢
+ ext_carr SS X ≡ 𝐋\p e.
-ndefinition epsilon ≝ λS,b.if b then { ([ ] : word S) } else {}.
+(* end set support for 𝐋\p *)
+
+ndefinition epsilon ≝
+ λS:Alpha.λb.match b return λ_.lang S with [ true ⇒ { [ ] } | _ ⇒ ∅ ].
interpretation "epsilon" 'epsilon = (epsilon ?).
notation < "ϵ b" non associative with precedence 90 for @{'app_epsilon $b}.
interpretation "epsilon lang" 'app_epsilon b = (epsilon ? b).
-ndefinition in_prl ≝ λS : Alpha.λp:pre S. 𝐋\p\ (\fst p) ∪ ϵ (\snd p).
+ndefinition L_pr ≝ λS : Alpha.λp:pre S. 𝐋\p\ (\fst p) ∪ ϵ (\snd p).
-interpretation "in_prl mem" 'mem w l = (in_prl ? l w).
-interpretation "in_prl" 'in_pl E = (in_prl ? E).
+interpretation "L_pr" 'L_pi E = (L_pr ? E).
-nlemma append_eq_nil : ∀S.∀w1,w2:word S. w1 @ w2 = [ ] → w1 = [ ].
-#S w1; nelim w1; //. #x tl IH w2; nnormalize; #abs; ndestruct; nqed.
-
-(* lemma 12 *)
-nlemma epsilon_in_true : ∀S.∀e:pre S. [ ] ∈ e ↔ \snd e = true.
-#S r; ncases r; #e b; @; ##[##2: #H; nrewrite > H; @2; /2/; ##] ncases b;//;
-nnormalize; *; ##[##2:*] nelim e;
-##[ ##1,2: *; ##| #c; *; ##| #c; nnormalize; #; ndestruct; ##| ##7: #p H;
-##| #r1 r2 H G; *; ##[##2: /3/ by or_intror]
-##| #r1 r2 H1 H2; *; /3/ by or_intror, or_introl; ##]
-*; #w1; *; #w2; *; *; #defw1; nrewrite > (append_eq_nil … w1 w2 …); /3/ by {};//;
+nlemma append_eq_nil : ∀S:setoid.∀w1,w2:list S. [ ] = w1 @ w2 → w1 = [ ].
+#S w1; ncases w1; //. nqed.
+
+(* 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; 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╪_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 …); //;
+#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)); *)
+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 LcatE : ∀S:Alpha.∀e1,e2:pitem S.
+ 𝐋\p (e1 · e2) = 𝐋\p e1 · 𝐋 |e2| ∪ 𝐋\p e2. //; 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 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|.
+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))^*; //; ##]
-nqed.
-
-nlemma eta_lp : ∀S.∀p:pre S.𝐋\p p = 𝐋\p 〈\fst p, \snd p〉.
+##[ #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:Alpha.∀p:pre S. 𝐋\p p = 𝐋\p 〈\fst p, \snd p〉.
#S p; ncases p; //; 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.
+(* 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.
(* 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; //;##]
+ 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'|); ##[
+ 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…);