+include "tutorial/chapter3.ma".
-include "arithmetics/nat.ma".
-include "basics/list.ma".
+(* As a simple application of lists, let us now consider strings of characters
+over a given alphabet Alpha. We shall assume to have a decidable equality between
+characters, that is a (computable) function eqb associating a boolean value true
+or false to each pair of characters; eqb is correct, in the sense that (eqb x y)
+if and only if (x = y). The type Alpha of alphabets is hence defined by the
+following record *)
-interpretation "iff" 'iff a b = (iff a b).
+interpretation "iff" 'iff a b = (iff a b).
record Alpha : Type[1] ≝ { carr :> Type[0];
eqb: carr → carr → \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6;
notation "a == b" non associative with precedence 45 for @{ 'eqb $a $b }.
interpretation "eqb" 'eqb a b = (eqb ? a b).
-definition word ≝ λS:\ 5a href="cic:/matita/tutorial/re/Alpha.ind(1,0,0)"\ 6Alpha\ 5/a\ 6.\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 S.
+definition word ≝ λS: \ 5a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"\ 6Alpha\ 5/a\ 6.\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 S.
-inductive re (S: \ 5a href="cic:/matita/tutorial/re/Alpha.ind(1,0,0)"\ 6Alpha\ 5/a\ 6) : Type[0] ≝
- z: re S
- | e: re S
- | s: S → re S
- | c: re S → re S → re S
- | o: re S → re S → re S
- | k: re S → re S.
+inductive re (S: \ 5a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"\ 6Alpha\ 5/a\ 6) : Type[0] ≝
+ zero: re S
+ | epsilon: re S
+ | char: S → re S
+ | concat: re S → re S → re S
+ | or: re S → re S → re S
+ | star: re S → re S.
-notation < "a \sup ⋇" non associative with precedence 90 for @{ 'pk $a}.
-notation > "a ^ *" non associative with precedence 90 for @{ 'pk $a}.
-interpretation "star" 'pk a = (k ? a).
-interpretation "or" 'plus a b = (o ? a b).
+(* notation < "a \sup ⋇" non associative with precedence 90 for @{ 'pk $a}. *)
+notation "a ^ *" non associative with precedence 90 for @{ 'kstar $a}.
+interpretation "star" 'kstar a = (star ? a).
+interpretation "or" 'plus a b = (or ? a b).
-notation "a · b" non associative with precedence 60 for @{ 'pc $a $b}.
-interpretation "cat" 'pc a b = (c ? a b).
+notation "a · b" non associative with precedence 60 for @{ 'concat $a $b}.
+interpretation "cat" 'concat a b = (concat ? a b).
(* to get rid of \middot
coercion c : ∀S:Alpha.∀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 < "a" non associative with precedence 90 for @{ 'ps $a}. *)
+notation "` term 90 a" non associative with precedence 90 for @{ 'atom $a}.
+interpretation "atom" 'atom a = (char ? a).
notation "ϵ" non associative with precedence 90 for @{ 'epsilon }.
-interpretation "epsilon" 'epsilon = (e ?).
+interpretation "epsilon" 'epsilon = (epsilon ?).
-notation "∅" non associative with precedence 90 for @{ 'empty }.
-interpretation "empty" 'empty = (z ?).
+notation "∅" (* slash emptyv *) non associative with precedence 90 for @{ 'empty }.
+interpretation "empty" 'empty = (zero ?).
-let rec flatten (S : \ 5a href="cic:/matita/tutorial/re/Alpha.ind(1,0,0)"\ 6Alpha\ 5/a\ 6) (l : \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/re/word.def(3)"\ 6word\ 5/a\ 6 S)) on l : \ 5a href="cic:/matita/tutorial/re/word.def(3)"\ 6word\ 5/a\ 6 S ≝
+let rec flatten (S : \ 5a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"\ 6Alpha\ 5/a\ 6) (l : \ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter4/word.def(3)"\ 6word\ 5/a\ 6 S)) on l : \ 5a href="cic:/matita/tutorial/chapter4/word.def(3)"\ 6word\ 5/a\ 6 S ≝
match l with [ nil ⇒ \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 ] | cons w tl ⇒ w \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 flatten ? tl ].
-let rec conjunct (S : \ 5a href="cic:/matita/tutorial/re/Alpha.ind(1,0,0)"\ 6Alpha\ 5/a\ 6) (l : \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/re/word.def(3)"\ 6word\ 5/a\ 6 S)) (r : \ 5a href="cic:/matita/tutorial/re/word.def(3)"\ 6word\ 5/a\ 6 S → Prop) on l: Prop ≝
-match l with [ nil ⇒ \ 5a href="cic:/matita/basics/logic/True.ind(1,0,0)"\ 6True\ 5/a\ 6 | cons w tl ⇒ r w \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 conjunct ? tl r ].
+let rec conjunct (S : \ 5a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"\ 6Alpha\ 5/a\ 6) (l : \ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter4/word.def(3)"\ 6word\ 5/a\ 6 S)) (L :\ 5a href="cic:/matita/tutorial/chapter4/word.def(3)"\ 6word\ 5/a\ 6 S → Prop) on l: Prop ≝
+match l with [ nil ⇒ \ 5a href="cic:/matita/basics/logic/True.ind(1,0,0)"\ 6True\ 5/a\ 6 | cons w tl ⇒ L w \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 conjunct ? tl L ].
-definition empty_lang ≝ λS.λw:\ 5a href="cic:/matita/tutorial/re/word.def(3)"\ 6word\ 5/a\ 6 S.\ 5a href="cic:/matita/basics/logic/False.ind(1,0,0)"\ 6False\ 5/a\ 6.
-notation "{}" non associative with precedence 90 for @{'empty_lang}.
-interpretation "empty lang" 'empty_lang = (empty_lang ?).
+definition empty_lang ≝ λS.λw:\ 5a href="cic:/matita/tutorial/chapter4/word.def(3)"\ 6word\ 5/a\ 6 S.\ 5a href="cic:/matita/basics/logic/False.ind(1,0,0)"\ 6False\ 5/a\ 6.
+(* notation "{}" non associative with precedence 90 for @{'empty_lang}. *)
+interpretation "empty lang" 'empty = (empty_lang ?).
-definition sing_lang ≝ λS.λx,w:\ 5a href="cic:/matita/tutorial/re/word.def(3)"\ 6word\ 5/a\ 6 S.x\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6w.
-notation "{x}" non associative with precedence 90 for @{'sing_lang $x}.
+definition sing_lang ≝ λS.λx,w:\ 5a href="cic:/matita/tutorial/chapter4/word.def(3)"\ 6word\ 5/a\ 6 S.x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 w.
+notation "{: x }" non associative with precedence 90 for @{'sing_lang $x}.
interpretation "sing lang" 'sing_lang x = (sing_lang ? x).
-definition union : ∀S,l1,l2,w.Prop ≝ λS.λl1,l2.λw:\ 5a href="cic:/matita/tutorial/re/word.def(3)"\ 6word\ 5/a\ 6 S.l1 w \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 l2 w.
+definition union : ∀S,L1,L2,w.Prop ≝ λS,L1,L2.λw: \ 5a href="cic:/matita/tutorial/chapter4/word.def(3)"\ 6word\ 5/a\ 6 S.L1 w \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 L2 w.
interpretation "union lang" 'union a b = (union ? a b).
definition cat : ∀S,l1,l2,w.Prop ≝
- λS.λl1,l2.λw:\ 5a href="cic:/matita/tutorial/re/word.def(3)"\ 6word\ 5/a\ 6 S.\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6w1,w2.w1 \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 w2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 w \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 l1 w1 \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 l2 w2.
-interpretation "cat lang" 'pc a b = (cat ? a b).
+ λS.λl1,l2.λw:\ 5a href="cic:/matita/tutorial/chapter4/word.def(3)"\ 6word\ 5/a\ 6 S.\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6w1,w2.w1 \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 w2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 w \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 l1 w1 \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 l2 w2.
+interpretation "cat lang" 'concat a b = (cat ? a b).
-definition star ≝ λS.λl.λw:\ 5a href="cic:/matita/tutorial/re/word.def(3)"\ 6word\ 5/a\ 6 S.\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6lw.\ 5a href="cic:/matita/tutorial/re/flatten.fix(0,1,4)"\ 6flatten\ 5/a\ 6 ? lw \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 w \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/re/conjunct.fix(0,1,4)"\ 6conjunct\ 5/a\ 6 ? lw l.
-interpretation "star lang" 'pk l = (star ? l).
+definition star_lang ≝ λS.λl.λw:\ 5a href="cic:/matita/tutorial/chapter4/word.def(3)"\ 6word\ 5/a\ 6 S.\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6lw. \ 5a href="cic:/matita/tutorial/chapter4/flatten.fix(0,1,4)"\ 6flatten\ 5/a\ 6 ? lw \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 w \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter4/conjunct.fix(0,1,4)"\ 6conjunct\ 5/a\ 6 ? lw l.
+interpretation "star lang" 'kstar l = (star_lang ? l).
-notation > "𝐋 term 70 E" non associative with precedence 75 for @{in_l ? $E}.
+(* notation "ℓ term 70 E" non associative with precedence 75 for @{in_l ? $E}. *)
-let rec in_l (S : Alpha) (r : re S) on r : word S → Prop ≝
+let rec in_l (S : \ 5a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"\ 6Alpha\ 5/a\ 6) (r : \ 5a href="cic:/matita/tutorial/chapter4/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S) on r : \ 5a href="cic:/matita/tutorial/chapter4/word.def(3)"\ 6word\ 5/a\ 6 S → Prop ≝
match r with
-[ 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 70 E" non associative with precedence 75 for @{'in_l $E}.
+ [ zero ⇒ \ 5a title="empty lang" href="cic:/fakeuri.def(1)"\ 6∅\ 5/a\ 6
+ | epsilon ⇒ \ 5a title="sing lang" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6: \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6] }
+ | char x ⇒ \ 5a title="sing lang" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6: x\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6] }
+ | concat r1 r2 ⇒ in_l ? r1 \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 in_l ? r2
+ | or r1 r2 ⇒ in_l ? r1 \ 5a title="union lang" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 in_l ? r2
+ | star r1 ⇒ (in_l ? r1)\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*
+ ].
+
+notation "ℓ term 70 E" non associative with precedence 75 for @{'in_l $E}.
interpretation "in_l" 'in_l E = (in_l ? E).
interpretation "in_l mem" 'mem w l = (in_l ? l w).
-notation "a || b" left associative with precedence 30 for @{'orb $a $b}.
+notation "a ∨ b" left associative with precedence 30 for @{'orb $a $b}.
interpretation "orb" 'orb a b = (orb a b).
-ndefinition if_then_else ≝ λT:Type[0].λe,t,f.match e return λ_.T with [ true ⇒ t | false ⇒ f].
+(* ndefinition if_then_else ≝ λT:Type[0].λe,t,f.match e return λ_.T with [ true ⇒ t | false ⇒ f].
notation > "'if' term 19 e 'then' term 19 t 'else' term 19 f" non associative with precedence 19 for @{ 'if_then_else $e $t $f }.
notation < "'if' \nbsp term 19 e \nbsp 'then' \nbsp term 19 t \nbsp 'else' \nbsp term 90 f \nbsp" non associative with precedence 19 for @{ 'if_then_else $e $t $f }.
-interpretation "if_then_else" 'if_then_else e t f = (if_then_else ? e t f).
-
-ninductive pitem (S: Alpha) : Type[0] ≝
- pz: pitem S
- | pe: pitem S
- | ps: S → pitem S
- | pp: S → pitem S
- | pc: pitem S → pitem S → pitem S
- | po: pitem S → pitem S → pitem S
- | pk: pitem S → pitem S.
+interpretation "if_then_else" 'if_then_else e t f = (if_then_else ? e t f). *)
+
+inductive pitem (S: \ 5a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"\ 6Alpha\ 5/a\ 6) : Type[0] ≝
+ | pzero: pitem S
+ | pepsilon: pitem S
+ | pchar: S → pitem S
+ | ppoint: S → pitem S
+ | pconcat: pitem S → pitem S → pitem S
+ | por: pitem S → pitem S → pitem S
+ | pstar: pitem S → pitem S.
-ndefinition pre ≝ λS.pitem S × bool.
-
-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 = (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 ≝
+definition pre ≝ λS.\ 5a href="cic:/matita/tutorial/chapter4/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S \ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)" title="null"\ 6bool\ 5/a\ 6.
+
+interpretation "pstar" 'kstar a = (pstar ? a).
+interpretation "por" 'plus a b = (por ? a b).
+interpretation "pcat" 'concat a b = (pconcat ? a b).
+
+notation "• a" non associative with precedence 90 for @{ 'ppoint $a}.
+(* notation > "`. term 90 a" non associative with precedence 90 for @{ 'pp $a}. *)
+
+interpretation "ppatom" 'ppoint a = (ppoint ? a).
+(* to get rid of \middot
+ncoercion pc : ∀S.∀p:pitem S. pitem S → pitem S ≝ pc on _p : pitem ? to ∀_:?.?. *)
+interpretation "patom" 'pchar a = (pchar ? a).
+interpretation "pepsilon" 'epsilon = (pepsilon ?).
+interpretation "pempty" 'empty = (pzero ?).
+
+notation "\boxv term 19 e \boxv" (* slash boxv *) non associative with precedence 70 for @{forget ? $e}.
+
+let rec forget (S: \ 5a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"\ 6Alpha\ 5/a\ 6) (l : \ 5a href="cic:/matita/tutorial/chapter4/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S) on l: \ 5a href="cic:/matita/tutorial/chapter4/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S ≝
match l with
- [ pz ⇒ ∅
- | 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}.
+ [ pzero ⇒ \ 5a title="empty" href="cic:/fakeuri.def(1)"\ 6∅\ 5/a\ 6
+ | pepsilon ⇒ \ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6
+ | pchar x ⇒ \ 5a href="cic:/matita/tutorial/chapter4/re.con(0,3,1)"\ 6char\ 5/a\ 6 ? x
+ | ppoint x ⇒ \ 5a href="cic:/matita/tutorial/chapter4/re.con(0,3,1)"\ 6char\ 5/a\ 6 ? x
+ | pconcat e1 e2 ⇒ │e1│ \ 5a title="cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 │e2│
+ | por e1 e2 ⇒ │e1│ \ 5a title="or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 │e2│
+ | pstar e ⇒ │e│\ 5a title="star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*
+ ].
+
+notation "│ term 19 e │" non associative with precedence 70 for @{'forget $e}.
interpretation "forget" 'forget a = (forget ? a).
-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}.
+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 > "𝐋\p\ term 70 E" non associative with precedence 75 for @{in_pl ? $E}.
-nlet rec in_pl (S : Alpha) (r : pitem S) on r : word S → Prop ≝
+notation "ℓ term 70 E" non associative with precedence 75 for @{in_pl ? $E}.
+
+let rec in_pl (S : \ 5a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"\ 6Alpha\ 5/a\ 6) (r : \ 5a href="cic:/matita/tutorial/chapter4/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S) on r : \ 5a href="cic:/matita/tutorial/chapter4/word.def(3)"\ 6word\ 5/a\ 6 S → Prop ≝
match r with
-[ pz ⇒ {}
-| pe ⇒ {}
-| ps _ ⇒ {}
-| pp x ⇒ { [x] }
-| 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 70 E" non associative with precedence 75 for @{'in_pl $E}.
-notation "𝐋\sub(\p) term 70 E" non associative with precedence 75 for @{'in_pl $E}.
-interpretation "in_pl" 'in_pl E = (in_pl ? E).
+ [ pzero ⇒ \ 5a title="empty lang" href="cic:/fakeuri.def(1)"\ 6∅\ 5/a\ 6
+ | pepsilon ⇒ \ 5a title="empty lang" href="cic:/fakeuri.def(1)"\ 6∅\ 5/a\ 6
+ | pchar _ ⇒ \ 5a title="empty lang" href="cic:/fakeuri.def(1)"\ 6∅\ 5/a\ 6
+ | ppoint x ⇒ \ 5a title="sing lang" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6: x\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6] }
+ | pconcat pe1 pe2 ⇒ in_pl ? pe1 \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6ℓ\ 5/a\ 6 │pe2│ \ 5a title="union lang" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 in_pl ? pe2
+ | por pe1 pe2 ⇒ in_pl ? pe1 \ 5a title="union lang" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 in_pl ? pe2
+ | pstar pe ⇒ in_pl ? pe \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6ℓ\ 5/a\ 6 │pe│\ 5a title="star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*
+ ].
+
+interpretation "in_pl" 'in_l E = (in_pl ? E).
interpretation "in_pl mem" 'mem w l = (in_pl ? l w).
-ndefinition epsilon ≝ λS,b.if b then { ([ ] : word S) } else {}.
+definition eps: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"\ 6Alpha\ 5/a\ 6.\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)" title="null"\ 6bool\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter4/word.def(3)"\ 6word\ 5/a\ 6 S → Prop
+ ≝ λS,b. \ 5a href="cic:/matita/basics/bool/if_then_else.def(1)"\ 6if_then_else\ 5/a\ 6 ? b \ 5a title="sing lang" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6: \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6] } \ 5a title="empty lang" href="cic:/fakeuri.def(1)"\ 6∅\ 5/a\ 6.
-interpretation "epsilon" 'epsilon = (epsilon ?).
-notation < "ϵ b" non associative with precedence 90 for @{'app_epsilon $b}.
-interpretation "epsilon lang" 'app_epsilon b = (epsilon ? b).
+(* interpretation "epsilon" 'epsilon = (epsilon ?). *)
+notation "ϵ _ b" non associative with precedence 90 for @{'app_epsilon $b}.
+interpretation "epsilon lang" 'app_epsilon b = (eps ? b).
-ndefinition in_prl ≝ λS : Alpha.λp:pre S. 𝐋\p (\fst p) ∪ ϵ (\snd p).
+definition in_prl ≝ λS : \ 5a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"\ 6Alpha\ 5/a\ 6.λp:\ 5a href="cic:/matita/tutorial/chapter4/pre.def(1)"\ 6pre\ 5/a\ 6 S. \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6ℓ\ 5/a\ 6 (\ 5a title="fst" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p) \ 5a title="union lang" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="epsilon lang" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6_(\ 5a title="snd" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p).
interpretation "in_prl mem" 'mem w l = (in_prl ? l w).
-interpretation "in_prl" 'in_pl E = (in_prl ? 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 {};//;
-nqed.
+interpretation "in_prl" 'in_l E = (in_prl ? E).
+
+lemma not_epsilon_lp :∀S.∀pi:\ 5a href="cic:/matita/tutorial/chapter4/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S. \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6 ((\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6ℓ\ 5/a\ 6 pi) \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]).
+#S #pi (elim pi) normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Not.con(0,1,1)"\ 6nmk\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+ [#pi1 #pi2 #H1 #H2 % * /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ * #w1 * #w2 * * #appnil
+ cases (\ 5a href="cic:/matita/tutorial/chapter3/nil_to_nil.def(5)"\ 6nil_to_nil\ 5/a\ 6 … appnil) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+ |#p11 #p12 #H1 #H2 % * /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+ |#pi #H % * #w1 * #w2 * * #appnil (cases (\ 5a href="cic:/matita/tutorial/chapter3/nil_to_nil.def(5)"\ 6nil_to_nil\ 5/a\ 6 … appnil)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+ ]
+qed.
-nlemma not_epsilon_lp : ∀S:Alpha.∀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/;##]
-nqed.
+lemma if_true_epsilon: ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter4/pre.def(1)"\ 6pre\ 5/a\ 6 S. \ 5a title="snd" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 e \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → ((\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6ℓ\ 5/a\ 6 e) \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]).
+#S #e #H %2 >H // qed.
+
+lemma if_epsilon_true : ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter4/pre.def(1)"\ 6pre\ 5/a\ 6 S. \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 ] \ 5a title="in_prl mem" href="cic:/fakeuri.def(1)"\ 6∈\ 5/a\ 6 e → \ 5a title="snd" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 e \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
+#S * #pi #b * [normalize #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /2/] cases b normalize // @False_ind
+qed.
+
+definition lor ≝ λS:Alpha.λa,b:pre S.〈\fst a + \fst b,\snd a ∨ \snd b〉.
-ndefinition lo ≝ λS:Alpha.λa,b:pre S.〈\fst a + \fst b,\snd a || \snd b〉.
notation "a ⊕ b" left associative with precedence 60 for @{'oplus $a $b}.
interpretation "oplus" 'oplus a b = (lo ? a b).
projects / helm.git / blobdiff