-(*
- ||M|| This file is part of HELM, an Hypertextual, Electronic
- ||A|| Library of Mathematics, developed at the Computer Science
- ||T|| Department of the University of Bologna, Italy.
- ||I||
- ||T||
- ||A|| This file is distributed under the terms of the
- \ / GNU General Public License Version 2
- \ /
- V_______________________________________________________________ *)
-
-include "arithmetics/nat.ma".
-include "basics/list.ma".
-
-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;
- eqb_true: ∀x,y. (eqb x y \ 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="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 (x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 y)
-}.
-
-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.
-
-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.
-
-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 · b" non associative with precedence 60 for @{ 'pc $a $b}.
-interpretation "cat" 'pc a b = (c ? 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 "ϵ" non associative with precedence 90 for @{ 'epsilon }.
-interpretation "epsilon" 'epsilon = (e ?).
-
-notation "∅" non associative with precedence 90 for @{ 'empty }.
-interpretation "empty" 'empty = (z ?).
-
-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 ≝
-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 ].
-
-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 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}.
-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.
-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).
-
-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).
-
-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 ≝
-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}.
-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}.
-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].
-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.
-
-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 ≝
- 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}.
-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}.
-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 ≝
-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).
-interpretation "in_pl mem" 'mem w l = (in_pl ? l w).
-
-ndefinition epsilon ≝ λS,b.if b then { ([ ] : word S) } else {}.
-
-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).
-
-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.
-
-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.
-
-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).
-
-ndefinition lc ≝ λS:Alpha.λbcast:∀S:Alpha.∀E:pitem S.pre S.λa,b:pre S.
- match a with [ mk_pair e1 b1 ⇒
- match b1 with
- [ false ⇒ 〈e1 · \fst b, \snd b〉
- | true ⇒ 〈e1 · \fst (bcast ? (\fst b)),\snd b || \snd (bcast ? (\fst b))〉]].
-
-notation < "a ⊙ b" left associative with precedence 60 for @{'lc $op $a $b}.
-interpretation "lc" 'lc op a b = (lc ? op a b).
-notation > "a ⊙ b" left associative with precedence 60 for @{'lc eclose $a $b}.
-
-ndefinition lk ≝ λS:Alpha.λbcast:∀S:Alpha.∀E:pitem S.pre S.λa:pre S.
- match a with [ mk_pair e1 b1 ⇒
- match b1 with
- [ false ⇒ 〈e1^*, false〉
- | true ⇒ 〈(\fst (bcast ? e1))^*, true〉]].
-
-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 > "•" non associative with precedence 60 for @{eclose ?}.
-nlet rec eclose (S: Alpha) (E: pitem S) on E : pre S ≝
- match E with
- [ pz ⇒ 〈 ∅, false 〉
- | pe ⇒ 〈 ϵ, true 〉
- | ps x ⇒ 〈 `.x, false 〉
- | pp x ⇒ 〈 `.x, false 〉
- | po E1 E2 ⇒ •E1 ⊕ •E2
- | pc E1 E2 ⇒ •E1 ⊙ 〈 E2, false 〉
- | pk E ⇒ 〈(\fst (•E))^*,true〉].
-notation < "• x" non associative with precedence 60 for @{'eclose $x}.
-interpretation "eclose" 'eclose x = (eclose ? x).
-notation > "• x" non associative with precedence 60 for @{'eclose $x}.
-
-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/; *; //; *;
-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 …); //;
-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.
-
-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; @;
-##[ *; #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;
-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_star : ∀S.∀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 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 sub0 : ∀S.∀a:word S → Prop. a - {} = a.
-#S a; napply extP; #w; nnormalize; @; /3/; *; //; nqed.
-
-nlemma subK : ∀S.∀a:word S → Prop. a - a = {}.
-#S a; napply extP; #w; nnormalize; @; *; /2/; nqed.
-
-nlemma subW : ∀S.∀a,b:word S → Prop.∀w.(a - b) w → a w.
-#S a b w; nnormalize; *; //; nqed.
-
-nlemma erase_bull : ∀S.∀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〉.
-#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.
-
-(* 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.
-#S e1 e2 th1; ncases e1; #e1' b1'; ncases b1';
-##[ nwhd in ⊢ (??(??%)?); 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…); //;##]
-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; @[]; @; //]
- *; #w1; *; #w2; *; *; #defw sube; *; #lw; *; #flx cj;
- @ (w1 :: lw); nrewrite < defw; nrewrite < flx; @; //;
- @; //; 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: @[]; @; //]
- @; //; @; //; @; *;##]##]##]
-nqed.
-
-(* theorem 16: 1 *)
-alias symbol "pc" (instance 13) = "cat lang".
-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|.
-#S e; nelim e; //;
- ##[ #a; napply extP; #w; nnormalize; @; *; /3/ by or_introl, or_intror;
- ##| #a; napply extP; #w; nnormalize; @; *; /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 …); //;
- ##| #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 …); //.
- ##| #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'^* ) ∪ {[ ]});
- 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.
-
-(* theorem 16: 3 *)
-nlemma odot_dot:
- ∀S.∀e1,e2: pre S. 𝐋\p (e1 ⊙ e2) = 𝐋\p e1 · 𝐋 |\fst e2| ∪ 𝐋\p e2.
-#S e1 e2; napply odot_dot_aux; napply (bull_cup S (\fst e2)); nqed.
-
-nlemma dot_star_epsilon : ∀S.∀e:re S.𝐋 e · 𝐋 e^* ∪ {[]} = 𝐋 e^*.
-#S e; napply extP; #w; nnormalize; @;
-##[ *; ##[##2: #H; nrewrite < H; @[]; /3/] *; #w1; *; #w2;
- *; *; #defw Hw1; *; #wl; *; #defw2 Hwl; @(w1 :: wl);
- nrewrite < defw; nrewrite < defw2; @; //; @;//;
-##| *; #wl; *; #defw Hwl; ncases wl in defw Hwl; ##[#defw; #; @2; nrewrite < defw; //]
- #x xs defw; *; #Hx Hxs; @; @x; @(flatten ? xs); nrewrite < defw;
- @; /2/; @xs; /2/;##]
- nqed.
-
-nlemma nil_star : ∀S.∀e:re S. [ ] ∈ e^*.
-#S e; @[]; /2/; nqed.
-
-nlemma cupID : ∀S.∀l:word S → Prop.l ∪ l = l.
-#S l; napply extP; #w; @; ##[*]//; #; @; //; nqed.
-
-nlemma cup_star_nil : ∀S.∀l:word S → Prop. l^* ∪ {[]} = l^*.
-#S a; napply extP; #w; @; ##[*; //; #H; nrewrite < H; @[]; @; //] #;@; //;nqed.
-
-nlemma rcanc_sing : ∀S.∀A,C:word S → Prop.∀b:word S .
- ¬ (A b) → A ∪ { (b) } = C → A = C - { (b) }.
-#S A C b nbA defC; nrewrite < defC; napply extP; #w; @;
-##[ #Aw; /3/| *; *; //; #H nH; ncases nH; #abs; nlapply (abs H); *]
-nqed.
-
-(* theorem 16: 4 *)
-nlemma star_dot: ∀S.∀e:pre S. 𝐋\p (e^⊛) = 𝐋\p e · (𝐋 |\fst e|)^*.
-#S p; ncases p; #e b; ncases b;
-##[ nchange in match (〈e,true〉^⊛) with 〈?,?〉;
- nletin e' ≝ (\fst (•e)); nletin b' ≝ (\snd (•e));
- nchange in ⊢ (??%?) with (?∪?);
- nchange in ⊢ (??(??%?)?) with (𝐋\p e' · 𝐋 |e'|^* );
- nrewrite > (?: 𝐋\p e' = 𝐋\p e ∪ (𝐋 |e| - ϵ b' )); ##[##2:
- nlapply (bull_cup ? e); #bc;
- nchange in match (𝐋\p (•e)) in bc with (?∪?);
- nchange in match b' in bc with b';
- ncases b' in bc; ##[##2: nrewrite > (cup0…); nrewrite > (sub0…); //]
- nrewrite > (cup_sub…); ##[napply rcanc_sing] //;##]
- nrewrite > (cup_dotD…); nrewrite > (cupA…);nrewrite > (erase_bull…);
- nrewrite > (sub_dot_star…);
- nchange in match (𝐋\p 〈?,?〉) with (?∪?);
- nrewrite > (cup_dotD…); nrewrite > (epsilon_dot…); //;
-##| nwhd in match (〈e,false〉^⊛); nchange in match (𝐋\p 〈?,?〉) with (?∪?);
- nrewrite > (cup0…);
- nchange in ⊢ (??%?) with (𝐋\p e · 𝐋 |e|^* );
- nrewrite < (cup0 ? (𝐋\p e)); //;##]
-nqed.
-
-nlet rec pre_of_re (S : Alpha) (e : re S) on e : pitem S ≝
- match e with
- [ z ⇒ pz ?
- | e ⇒ pe ?
- | s x ⇒ ps ? x
- | c e1 e2 ⇒ pc ? (pre_of_re ? e1) (pre_of_re ? e2)
- | o e1 e2 ⇒ po ? (pre_of_re ? e1) (pre_of_re ? e2)
- | k e1 ⇒ pk ? (pre_of_re ? e1)].
-
-nlemma notFalse : ¬False. @; //; nqed.
-
-nlemma dot0 : ∀S.∀A:word S → Prop. {} · A = {}.
-#S A; nnormalize; napply extP; #w; @; ##[##2: *]
-*; #w1; *; #w2; *; *; //; nqed.
-
-nlemma Lp_pre_of_re : ∀S.∀e:re S. 𝐋\p (pre_of_re ? e) = {}.
-#S e; nelim e; ##[##1,2,3: //]
-##[ #e1 e2 H1 H2; nchange in match (𝐋\p (pre_of_re S (e1 e2))) with (?∪?);
- nrewrite > H1; nrewrite > H2; nrewrite > (dot0…); nrewrite > (cupID…);//
-##| #e1 e2 H1 H2; nchange in match (𝐋\p (pre_of_re S (e1+e2))) with (?∪?);
- nrewrite > H1; nrewrite > H2; nrewrite > (cupID…); //
-##| #e1 H1; nchange in match (𝐋\p (pre_of_re S (e1^* ))) with (𝐋\p (pre_of_re ??) · ?);
- nrewrite > H1; napply dot0; ##]
-nqed.
-
-nlemma erase_pre_of_reK : ∀S.∀e. 𝐋 |pre_of_re S e| = 𝐋 e.
-#S A; nelim A; //;
-##[ #e1 e2 H1 H2; nchange in match (𝐋 (e1 · e2)) with (𝐋 e1·?);
- nrewrite < H1; nrewrite < H2; //
-##| #e1 e2 H1 H2; nchange in match (𝐋 (e1 + e2)) with (𝐋 e1 ∪ ?);
- nrewrite < H1; nrewrite < H2; //
-##| #e1 H1; nchange in match (𝐋 (e1^* )) with ((𝐋 e1)^* );
- nrewrite < H1; //]
-nqed.
-
-(* corollary 17 *)
-nlemma L_Lp_bull : ∀S.∀e:re S.𝐋 e = 𝐋\p (•pre_of_re ? e).
-#S e; nrewrite > (bull_cup…); nrewrite > (Lp_pre_of_re…);
-nrewrite > (cupC…); nrewrite > (cup0…); nrewrite > (erase_pre_of_reK…); //;
-nqed.
-
-nlemma Pext : ∀S.∀f,g:word S → Prop. f = g → ∀w.f w → g w.
-#S f g H; nrewrite > H; //; nqed.
+include "basics/types.ma".
+
+(* Most of the types we have seen so far are enumerated types, composed by a
+finite set of alternatives, and records, composed by tuples of heteregoneous
+elements. A more interesting case of type definition is when some of the rules
+defining its elements are recursive, i.e. they allow the formation of more
+elements of the type in terms of the already defined ones. The most typical case
+is provided by the natural numbers, that can be defined as the smallest set
+generated by a constant 0 and a successor function from natural numbers to natural
+numbers *)
+
+inductive nat : Type[0] ≝
+| O :nat
+| S: nat →nat.
+
+(* The two terms O and S are called constructors: they define the signature of the
+type, whose objects are the elements freely generated by means of them. So,
+examples of natural numbers are O, S O, S (S O), S (S (S O)) and so on.
+
+The language of Matita allows the definition of well founded recursive functions
+over inductive types; in order to guarantee termination of recursion you are only
+allowed to make recursive calls on structurally smaller arguments than the ones
+you received in input. Most mathematical functions can be naturally defined in this
+way. For instance, the sum of two natural numbers can be defined as follows *)
+
+let rec add n m ≝
+match n with
+[ O ⇒ m
+| S a ⇒ \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (add a m)
+].
+
+(* It is worth to observe that the previous algorithm works by recursion over the
+first argument. This means that, for instance, (add O x) will reduce to x, as
+expected, but (add x O) is stuck.
+How can we prove that, for a generic x, (add x O) = x? The mathematical tool to do
+it is called induction. The induction principle states that, given a property P(n)
+over natural numbers, if we prove P(0) and prove that, for any m, P(m) implies P(S m),
+than we can conclude P(n) for any n.
+
+The elim tactic, allow you to apply induction in a very simple way. If your goal is
+P(n), the invocation of
+ elim n
+will break down your task to prove the two subgoals P(0) and ∀m.P(m) → P(S m).
+
+Let us apply it to our case *)
+
+lemma add_0: ∀a. \ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 a \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 a.
+#a elim a
+
+(* If you stop the computation here, you will see on the right the two subgoals
+ - add O O = O
+ - ∀x. add x 0 = x → add (S x) O = S x
+After normalization, both goals are trivial.
+*)
+
+normalize // qed.
+
+(* In a similar way, it is convenient to state a lemma about the behaviour of
+add when the second argument is not zero. *)
+
+lemma add_S : ∀a,b. \ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 a (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 b) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 a b).
+
+(* In the same way as before, we proceed by induction over a. *)
+
+#a #b elim a normalize //
+qed.
+
+(* We are now in the position to prove the commutativity of the sum *)
+
+theorem add_comm : ∀a,b. \ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 a b \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 b a.
+#a elim a normalize
+
+(* We have two sub goals:
+ G1: ∀b. b = add b O
+ G2: ∀x.(∀b. add x b = add b x) → ∀b. S (add x b) = add b (S x).
+G1 is just our lemma add_O. For G2, we start introducing x and the induction
+hypothesis IH; then, the goal is proved by rewriting using add_S and IH.
+For Matita, the task is trivial and we can simply close the goal with // *)
+
+// qed.
+
+(* COERCIONS *)
+
+inductive bool : Type[0] ≝
+| tt : bool
+| ff : bool.
+
+definition nat_of_bool ≝ λb. match b with
+[ tt ⇒ \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6
+| ff ⇒ \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6
+].
+
+(* coercion nat_of_bool. ?? *)
-(* corollary 18 *)
-ntheorem bull_true_epsilon : ∀S.∀e:pitem S. \snd (•e) = true ↔ [ ] ∈ |e|.
-#S e; @;
-##[ #defsnde; nlapply (bull_cup ? e); nchange in match (𝐋\p (•e)) with (?∪?);
- nrewrite > defsnde; #H;
- nlapply (Pext ??? H [ ] ?); ##[ @2; //] *; //;
-
-STOP
-
-notation > "\move term 90 x term 90 E"
-non associative with precedence 60 for @{move ? $x $E}.
-nlet rec move (S: Alpha) (x:S) (E: pitem S) on E : pre S ≝
- match E with
- [ pz ⇒ 〈 ∅, false 〉
- | pe ⇒ 〈 ϵ, false 〉
- | ps y ⇒ 〈 `y, false 〉
- | pp y ⇒ 〈 `y, x == y 〉
- | po e1 e2 ⇒ \move x e1 ⊕ \move x e2
- | pc e1 e2 ⇒ \move x e1 ⊙ \move x e2
- | pk e ⇒ (\move x e)^⊛ ].
-notation < "\move\shy x\shy E" non associative with precedence 60 for @{'move $x $E}.
-notation > "\move term 90 x term 90 E" non associative with precedence 60 for @{'move $x $E}.
-interpretation "move" 'move x E = (move ? x E).
-
-ndefinition rmove ≝ λS:Alpha.λx:S.λe:pre S. \move x (\fst e).
-interpretation "rmove" 'move x E = (rmove ? x E).
-
-nlemma XXz : ∀S:Alpha.∀w:word S. w ∈ ∅ → False.
-#S w abs; ninversion abs; #; ndestruct;
-nqed.
-
-
-nlemma XXe : ∀S:Alpha.∀w:word S. w .∈ ϵ → False.
-#S w abs; ninversion abs; #; ndestruct;
-nqed.
-
-nlemma XXze : ∀S:Alpha.∀w:word S. w .∈ (∅ · ϵ) → False.
-#S w abs; ninversion abs; #; ndestruct; /2/ by XXz,XXe;
-nqed.
-
-
-naxiom in_move_cat:
- ∀S.∀w:word S.∀x.∀E1,E2:pitem S. w .∈ \move x (E1 · E2) →
- (∃w1.∃w2. w = w1@w2 ∧ w1 .∈ \move x E1 ∧ w2 ∈ .|E2|) ∨ w .∈ \move x E2.
-#S w x e1 e2 H; nchange in H with (w .∈ \move x e1 ⊙ \move x e2);
-ncases e1 in H; ncases e2;
-##[##1: *; ##[*; nnormalize; #; ndestruct]
- #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct]
- nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz,XXze;
-##|##2: *; ##[*; nnormalize; #; ndestruct]
- #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct]
- nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz,XXze;
-##| #r; *; ##[ *; nnormalize; #; ndestruct]
- #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct]
- ##[##2: nnormalize; #; ndestruct; @2; @2; //.##]
- nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz;
-##| #y; *; ##[ *; nnormalize; #defw defx; ndestruct; @2; @1; /2/ by conj;##]
- #H; ninversion H; nnormalize; #; ndestruct;
- ##[ncases (?:False); /2/ by XXz] /3/ by or_intror;
-##| #r1 r2; *; ##[ *; #defw]
- ...
-nqed.
-
-ntheorem move_ok:
- ∀S:Alpha.∀E:pre S.∀a,w.w .∈ \move a E ↔ (a :: w) .∈ E.
-#S E; ncases E; #r b; nelim r;
-##[##1,2: #a w; @;
- ##[##1,3: nnormalize; *; ##[##1,3: *; #; ndestruct; ##| #abs; ncases (XXz … abs); ##]
- #H; ninversion H; #; ndestruct;
- ##|##*:nnormalize; *; ##[##1,3: *; #; ndestruct; ##| #H1; ncases (XXz … H1); ##]
- #H; ninversion H; #; ndestruct;##]
-##|#a c w; @; nnormalize; ##[*; ##[*; #; ndestruct; ##] #abs; ninversion abs; #; ndestruct;##]
- *; ##[##2: #abs; ninversion abs; #; ndestruct; ##] *; #; ndestruct;
-##|#a c w; @; nnormalize;
- ##[ *; ##[ *; #defw; nrewrite > defw; #ca; @2; nrewrite > (eqb_t … ca); @; ##]
- #H; ninversion H; #; ndestruct;
- ##| *; ##[ *; #; ndestruct; ##] #H; ninversion H; ##[##2,3,4,5,6: #; ndestruct]
- #d defw defa; ndestruct; @1; @; //; nrewrite > (eqb_true S d d); //. ##]
-##|#r1 r2 H1 H2 a w; @;
- ##[ #H; ncases (in_move_cat … H);
- ##[ *; #w1; *; #w2; *; *; #defw w1m w2m;
- ncases (H1 a w1); #H1w1; #_; nlapply (H1w1 w1m); #good;
- nrewrite > defw; @2; @2 (a::w1); //; ncases good; ##[ *; #; ndestruct] //.
- ##|
- ...
-##|
-##|
-##]
-nqed.
-
-
-notation > "x ↦* E" non associative with precedence 60 for @{move_star ? $x $E}.
-nlet rec move_star (S : decidable) w E on w : bool × (pre S) ≝
- match w with
- [ nil ⇒ E
- | cons x w' ⇒ w' ↦* (x ↦ \snd E)].
-
-ndefinition in_moves ≝ λS:decidable.λw.λE:bool × (pre S). \fst(w ↦* E).
-
-ncoinductive equiv (S:decidable) : bool × (pre S) → bool × (pre S) → Prop ≝
- mk_equiv:
- ∀E1,E2: bool × (pre S).
- \fst E1 = \fst E2 →
- (∀x. equiv S (x ↦ \snd E1) (x ↦ \snd E2)) →
- equiv S E1 E2.
-
-ndefinition NAT: decidable.
- @ nat eqb; /2/.
-nqed.
-
-include "hints_declaration.ma".
-
-alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
-unification hint 0 ≔ ; X ≟ NAT ⊢ carr X ≡ nat.
-
-ninductive unit: Type[0] ≝ I: unit.
-
-nlet corec foo_nop (b: bool):
- equiv ?
- 〈 b, pc ? (ps ? 0) (pk ? (pc ? (ps ? 1) (ps ? 0))) 〉
- 〈 b, pc ? (pk ? (pc ? (ps ? 0) (ps ? 1))) (ps ? 0) 〉 ≝ ?.
- @; //; #x; ncases x
- [ nnormalize in ⊢ (??%%); napply (foo_nop false)
- | #y; ncases y
- [ nnormalize in ⊢ (??%%); napply (foo_nop false)
- | #w; nnormalize in ⊢ (??%%); napply (foo_nop false) ]##]
-nqed.
-
-(*
-nlet corec foo (a: unit):
- equiv NAT
- (eclose NAT (pc ? (ps ? 0) (pk ? (pc ? (ps ? 1) (ps ? 0)))))
- (eclose NAT (pc ? (pk ? (pc ? (ps ? 0) (ps ? 1))) (ps ? 0)))
-≝ ?.
- @;
- ##[ nnormalize; //
- ##| #x; ncases x
- [ nnormalize in ⊢ (??%%);
- nnormalize in foo: (? → ??%%);
- @; //; #y; ncases y
- [ nnormalize in ⊢ (??%%); napply foo_nop
- | #y; ncases y
- [ nnormalize in ⊢ (??%%);
-
- ##| #z; nnormalize in ⊢ (??%%); napply foo_nop ]##]
- ##| #y; nnormalize in ⊢ (??%%); napply foo_nop
- ##]
-nqed.
+(* Let us now define the following function: *)
+
+definition twice ≝ λn.\ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 n n.
+
+(* We are interested to prove that for any natural number n there exists a natural
+number m that is the integer half of n. This will give us the opportunity to
+introduce new connectives and quantifiers and, later on, to make some interesting
+consideration on proofs and computations. *)
+
+theorem ex_half: ∀n.\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6m. n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 m \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 m).
+#n elim n normalize
+
+(* We proceed by induction on n, that breaks down to the following goals:
+ G1: ∃m.O = add O O ∨ O = S (add m m)
+ G2: ∀x.(∃m. x = add m m ∨ x = S (add m m))→ ∃m. S x = add m m ∨ S x = S (add m m)
+The only way we have to prove an existential goal is by exhibiting the witness,
+that in the case of first goal is O. We do it by apply the term called ex_intro
+instantiated by the witness. Then, it is clear that we must follow the left branch
+of the disjunction. One way to do it is by applying the term or_introl, that is
+the first constructor of the disjunction. However, remembering the names of
+constructors can be annyoing: we can invoke the application of the n-th
+constructor of an inductive type (inferred by the current goal) by typing %n. At
+this point we are left with the subgoal O = add O O, that is closed by
+computation. It is worth to observe that invoking automation at depth /3/ would
+also automatically close G1.
+*)
+ [@(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6) %1 //
+
+(* The case of G2 is more complex. We should start introducing x and the
+inductive hypothesis
+ IH: ∃m. x = add m m ∨ x = S (add m m)
+At this point we should assume the existence of m enjoying the inductive
+hypothesis. To eliminate the existential from the context we can just use the
+case tactic. This situation where we introduce something into the context and
+immediately eliminate it by case analysis is so frequent that Matita provides a
+convenient shorthand: you can just type a single "*".
+The star symbol should be reminiscent of an explosion: the idea is that you have
+a structured hypothesis, and you ask to explode it into its constituents. In the
+case of the existential, it allows to pass from a goal of the shape
+ (∃x.P x) → Q
+to a goal of the shape
+ ∀x.P x → Q
+*)
+ |#x *
+(* At this point we are left with a new goal with the following shape
+ G3: ∀m. x = add m m ∨ x = S (add m m) → ....
+We should introduce m, the hypothesis H: x = add m m ∨ x = S (add m m), and
+then reason by cases on this hypothesis. It is the same situation as before:
+we explode the disjunctive hypothesis into its possible consituents. In the case
+of a disjunction, the * tactic allows to pass from a goal of the form
+ A∨B → Q
+to two subgoals of the form
+ A → Q and B → Q
+*)
+ #m * #eqx
+(* In the first subgoal, we are under the assumption that x = add m m. The half
+of (S x) is hence m, and we have to prove the right branch of the disjunction.
+In the second subgoal, we are under the assumption that x = S (add m m). The halh
+of (S x) is hence (S m), and have to follow the left branch of the disjunction.
+*)
+ [@(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … m) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ | @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m)) normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+ ]
+qed.
+
+(* Instead of proving the existence of a number corresponding to the half of n,
+we could be interested in computing it. The best way to do it is to define this
+division operation together with the remainder, that in our case is just a
+boolean value: tt if the input term is even, and ff if the input term is odd.
+Since we must return a pair, we could use a suitably defined record type, or
+simply a product type nat × bool, defined in the basic library. The product type
+is just a sort of general purpose record, with standard fields fst and snd, called
+projections.
+A pair of values n and m is written (pair … m n) or \langle n,m \rangle - visually
+rendered as 〈n,m〉
+
+We first write down the function, and then discuss it.*)
+
+let rec div2 n ≝
+match n with
+[ O ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6,\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6〉
+| S a ⇒ \ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6
+ let p ≝ (div2 a) in
+ match (\ 5a href="cic:/matita/basics/types/snd.def(1)"\ 6snd\ 5/a\ 6 … p) with
+ [ tt ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (\ 5a href="cic:/matita/basics/types/fst.def(1)"\ 6fst\ 5/a\ 6 … p),\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6〉
+ | ff ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/types/fst.def(1)"\ 6fst\ 5/a\ 6 … p, \ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,1,0)"\ 6tt\ 5/a\ 6〉
+ ]
+].
+
+(* The function is computed by recursion over the input n. If n is 0, then the
+quotient is 0 and the remainder is tt. If n = S a, we start computing the half
+of a, say 〈q,b〉. Then we have two cases according to the possible values of b:
+if b is tt, then we must return 〈q,ff〉, while if b = ff then we must return
+〈S q,tt〉.
+
+It is important to point out the deep, substantial analogy between the algorithm
+for computing div2 and the the proof of ex_half. In particular ex_half returns a
+proof of the kind ∃n.A(n)∨B(n): the really informative content in it is the
+witness n and a boolean indicating which one between the two conditions A(n) and
+B(n) is met. This is precisely the quotient-remainder pair returned by div2.
+In both cases we proceed by recurrence (respectively, induction or recursion) over
+the input argument n. In case n = 0, we conclude the proof in ex_half by providing
+the witness O and a proof of A(O); this corresponds to returning the pair 〈O,ff〉 in
+div2. Similarly, in the inductive case n = S a, we must exploit the inductive
+hypothesis for a (i.e. the result of the recursive call), distinguishing two subcases
+according to the the two possibilites A(a) or B(a) (i.e. the two possibile values of
+the remainder for a). The reader is strongly invited to check all remaining details.
+
+Let us now prove that our div2 function has the expected behaviour.
+*)
+
+lemma surjective_pairing: ∀A,B.∀p:A\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6B. p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/types/fst.def(1)"\ 6fst\ 5/a\ 6 … p,\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 … p〉.
+#A #B * // qed.
+
+lemma div2SO: ∀n,q. \ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6q,\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6〉 → \ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6q,\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,1,0)"\ 6tt\ 5/a\ 6〉.
+#n #q #H normalize >H normalize // qed.
+
+lemma div2S1: ∀n,q. \ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6q,\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,1,0)"\ 6tt\ 5/a\ 6〉 → \ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 q,\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6〉.
+#n #q #H normalize >H normalize // qed.
+
+lemma div2_ok: ∀n,q,r. \ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6q,r〉 → n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 q) (\ 5a href="cic:/matita/tutorial/chapter2/nat_of_bool.def(1)"\ 6nat_of_bool\ 5/a\ 6 r).
+#n elim n
+ [#q #r normalize #H destruct //
+ |#a #Hind #q #r
+ cut (\ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/types/fst.def(1)"\ 6fst\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 a), \ 5a href="cic:/matita/basics/types/snd.def(1)"\ 6snd\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 a)〉) [//]
+ cases (\ 5a href="cic:/matita/basics/types/snd.def(1)"\ 6snd\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 a))
+ [#H >(\ 5a href="cic:/matita/tutorial/chapter2/div2S1.def(3)"\ 6div2S1\ 5/a\ 6 … H) #H1 destruct @\ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6 \ 5span style="text-decoration: underline;"\ 6>\ 5/span\ 6\ 5a href="cic:/matita/tutorial/chapter2/add_S.def(2)"\ 6add_S\ 5/a\ 6 whd in ⊢ (???%); <\ 5a href="cic:/matita/tutorial/chapter2/add_S.def(2)"\ 6add_S\ 5/a\ 6 @(Hind … H)
+ |#H >(\ 5a href="cic:/matita/tutorial/chapter2/div2SO.def(3)"\ 6div2SO\ 5/a\ 6 … H) #H1 destruct >\ 5a href="cic:/matita/tutorial/chapter2/add_S.def(2)"\ 6add_S\ 5/a\ 6 @\ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6 @(Hind … H)
+ ]
+qed.
+
+(* There is still another possibility, however, namely to mix the program and its
+specification into a single entity. The idea is to refine the output type of the
+div2 function: it should not be just a generic pair 〈q,r〉 of natural numbers but a
+specific pair satisfying the specification of the function. In other words, we need
+the possibility to define, for a type A and a property P over A, the subset type
+{a:A|P(a)} of all elements a of type A that satisfy the property P. Subset types
+are just a particular case of the so called dependent types, that is types that
+can depend over arguments (such as arrays of a specified length, taken as a
+parameter).These kind of types are quite unusual in traditional programming
+languages, and their study is one of the new frontiers of the current research on
+type systems.
+
+There is nothing special in a subset type {a:A|P(a)}: it is just a record composed
+by an element of a of type A and a proof of P(a). The crucial point is to have a
+language reach enough to comprise proofs among its expressions.
*)
-ndefinition test1 : pre ? ≝ ❨ `0 | `1 ❩^* `0.
-ndefinition test2 : pre ? ≝ ❨ (`0`1)^* `0 | (`0`1)^* `1 ❩.
-ndefinition test3 : pre ? ≝ (`0 (`0`1)^* `1)^*.
-
-
-nlemma foo: in_moves ? [0;0;1;0;1;1] (ɛ test3) = true.
- nnormalize in match test3;
- nnormalize;
-//;
-nqed.
-
-(**********************************************************)
-
-ninductive der (S: Type[0]) (a: S) : re S → re S → CProp[0] ≝
- der_z: der S a (z S) (z S)
- | der_e: der S a (e S) (z S)
- | der_s1: der S a (s S a) (e ?)
- | der_s2: ∀b. a ≠ b → der S a (s S b) (z S)
- | der_c1: ∀e1,e2,e1',e2'. in_l S [] e1 → der S a e1 e1' → der S a e2 e2' →
- der S a (c ? e1 e2) (o ? (c ? e1' e2) e2')
- | der_c2: ∀e1,e2,e1'. Not (in_l S [] e1) → der S a e1 e1' →
- der S a (c ? e1 e2) (c ? e1' e2)
- | der_o: ∀e1,e2,e1',e2'. der S a e1 e1' → der S a e2 e2' →
- der S a (o ? e1 e2) (o ? e1' e2').
-
-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; ncases p; #P; #H; nassumption.
-nqed.
-
-nlemma append1: ∀A.∀a:A.∀l. [a] @ l = a::l. //. nqed.
-
-naxiom in_l1: ∀S,r1,r2,w. in_l S [ ] r1 → in_l S w r2 → in_l S w (c S r1 r2).
-(* #S; #r1; #r2; #w; nelim r1
- [ #K; ninversion K
- | #H1; #H2; napply (in_c ? []); //
- | (* tutti casi assurdi *) *)
-
-ninductive in_l' (S: Type[0]) : word S → re S → CProp[0] ≝
- in_l_empty1: ∀E.in_l S [] E → in_l' S [] E
- | in_l_cons: ∀a,w,e,e'. in_l' S w e' → der S a e e' → in_l' S (a::w) e.
-
-ncoinductive eq_re (S: Type[0]) : re S → re S → CProp[0] ≝
- mk_eq_re: ∀E1,E2.
- (in_l S [] E1 → in_l S [] E2) →
- (in_l S [] E2 → in_l S [] E1) →
- (∀a,E1',E2'. der S a E1 E1' → der S a E2 E2' → eq_re S E1' E2') →
- eq_re S E1 E2.
-
-(* serve il lemma dopo? *)
-ntheorem eq_re_is_eq: ∀S.∀E1,E2. eq_re S E1 E2 → ∀w. in_l ? w E1 → in_l ? w E2.
- #S; #E1; #E2; #H1; #w; #H2; nelim H2 in E2 H1 ⊢ %
- [ #r; #K (* ok *)
- | #a; #w; #R1; #R2; #K1; #K2; #K3; #R3; #K4; @2 R2; //; ncases K4;
-
-(* IL VICEVERSA NON VALE *)
-naxiom in_l_to_in_l: ∀S,w,E. in_l' S w E → in_l S w E.
-(* #S; #w; #E; #H; nelim H
- [ //
- | #a; #w'; #r; #r'; #H1; (* e si cade qua sotto! *)
+record Sub (A:Type[0]) (P:A → Prop) : Type[0] ≝
+ {witness: A;
+ proof: P witness}.
+
+definition qr_spec ≝ λn.λp.∀q,r. p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6q,r〉 → n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 q) (\ 5a href="cic:/matita/tutorial/chapter2/nat_of_bool.def(1)"\ 6nat_of_bool\ 5/a\ 6 r).
+
+(* We can now construct a function from n to {p|qr_spec n p} by composing the objects
+we already have *)
+
+definition div2P: ∀n.\ 5a href="cic:/matita/tutorial/chapter2/Sub.ind(1,0,2)"\ 6 Sub\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6\ 5span style="text-decoration: underline;"\ 6\ 5a href="cic:/matita/tutorial/chapter2/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6\ 5/span\ 6) (\ 5a href="cic:/matita/tutorial/chapter2/qr_spec.def(3)"\ 6qr_spec\ 5/a\ 6 n) ≝ λn.
+ \ 5a href="cic:/matita/tutorial/chapter2/Sub.con(0,1,2)"\ 6mk_Sub\ 5/a\ 6 ?? (\ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 n) (\ 5a href="cic:/matita/tutorial/chapter2/div2_ok.def(4)"\ 6div2_ok\ 5/a\ 6 n).
+
+(* But we can also try do directly build such an object *)
+
+definition div2Pagain : ∀n.\ 5a href="cic:/matita/tutorial/chapter2/Sub.ind(1,0,2)"\ 6Sub\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6\ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6\ 5a href="cic:/matita/tutorial/chapter2/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6) (\ 5a href="cic:/matita/tutorial/chapter2/qr_spec.def(3)"\ 6qr_spec\ 5/a\ 6 n).
+#n elim n
+ [@(\ 5a href="cic:/matita/tutorial/chapter2/Sub.con(0,1,2)"\ 6mk_Sub\ 5/a\ 6 … \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6,\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6〉) normalize #q #r #H destruct //
+ |#a * #p #qrspec
+ cut (p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/types/fst.def(1)"\ 6fst\ 5/a\ 6 … p, \ 5a href="cic:/matita/basics/types/snd.def(1)"\ 6snd\ 5/a\ 6 … p〉) [//]
+ cases (\ 5a href="cic:/matita/basics/types/snd.def(1)"\ 6snd\ 5/a\ 6 … p)
+ [#H @(\ 5a href="cic:/matita/tutorial/chapter2/Sub.con(0,1,2)"\ 6mk_Sub\ 5/a\ 6 … \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (\ 5a href="cic:/matita/basics/types/fst.def(1)"\ 6fst\ 5/a\ 6 … p),\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6〉) whd #q #r #H1 destruct @\ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6 \ 5span style="text-decoration: underline;"\ 6>\ 5/span\ 6\ 5a href="cic:/matita/tutorial/chapter2/add_S.def(2)"\ 6add_S\ 5/a\ 6
+ whd in ⊢ (???%); <\ 5a href="cic:/matita/tutorial/chapter2/add_S.def(2)"\ 6add_S\ 5/a\ 6 @(qrspec … H)
+ |#H @(\ 5a href="cic:/matita/tutorial/chapter2/Sub.con(0,1,2)"\ 6mk_Sub\ 5/a\ 6 … \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/types/fst.def(1)"\ 6fst\ 5/a\ 6 … p,\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,1,0)"\ 6tt\ 5/a\ 6〉) whd #q #r #H1 destruct >\ 5a href="cic:/matita/tutorial/chapter2/add_S.def(2)"\ 6add_S\ 5/a\ 6 @\ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6 @(qrspec … H)
]
-nqed. *)
-
-ntheorem der1: ∀S,a,e,e',w. der S a e e' → in_l S w e' → in_l S (a::w) e.
- #S; #a; #E; #E'; #w; #H; nelim H
- [##1,2: #H1; ninversion H1
- [##1,8: #_; #K; (* non va ndestruct K; *) ncases (?:False); (* perche' due goal?*) /2/
- |##2,9: #X; #Y; #K; ncases (?:False); /2/
- |##3,10: #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/
- |##4,11: #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
- |##5,12: #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
- |##6,13: #x; #y; #K; ncases (?:False); /2/
- |##7,14: #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/]
-##| #H1; ninversion H1
- [ //
- | #X; #Y; #K; ncases (?:False); /2/
- | #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/
- | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
- | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
- | #x; #y; #K; ncases (?:False); /2/
- | #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/ ]
-##| #H1; #H2; #H3; ninversion H3
- [ #_; #K; ncases (?:False); /2/
- | #X; #Y; #K; ncases (?:False); /2/
- | #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/
- | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
- | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
- | #x; #y; #K; ncases (?:False); /2/
- | #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/ ]
-##| #r1; #r2; #r1'; #r2'; #H1; #H2; #H3; #H4; #H5; #H6;
\ No newline at end of file
+qed.
+
+example quotient7: \ 5a href="cic:/matita/tutorial/chapter2/witness.fix(0,2,1)"\ 6witness\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter2/div2Pagain.def(4)"\ 6div2Pagain\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6)))))))) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6)),\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,1,0)"\ 6tt\ 5/a\ 6〉.
+// qed.
+
+example quotient8: \ 5a href="cic:/matita/tutorial/chapter2/witness.fix(0,2,1)"\ 6witness\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter2/div2Pagain.def(4)"\ 6div2Pagain\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6))))))
+ \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6))), \ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6〉.
+// qed.
+\ 5pre\ 6\ 5pre\ 6 \ 5/pre\ 6\ 5/pre\ 6
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projects / helm.git / blobdiff