From 85521efd364ec494e4cc024bbf87182a312e1b7b Mon Sep 17 00:00:00 2001 From: Claudio Sacerdoti Coen Date: Tue, 4 May 2010 16:40:26 +0000 Subject: [PATCH] Regular expressions. --- helm/software/matita/nlibrary/re/re.ma | 178 +++++++++++++++++++++++++ 1 file changed, 178 insertions(+) create mode 100644 helm/software/matita/nlibrary/re/re.ma diff --git a/helm/software/matita/nlibrary/re/re.ma b/helm/software/matita/nlibrary/re/re.ma new file mode 100644 index 000000000..10f446bc5 --- /dev/null +++ b/helm/software/matita/nlibrary/re/re.ma @@ -0,0 +1,178 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "logic/connectives.ma". +(*include "logic/equality.ma".*) +include "datatypes/list.ma". +include "datatypes/bool.ma". +include "datatypes/pairs.ma". + +include "Plogic/equality.ma". + +ndefinition word ≝ λS:Type[0].list S. + +ninductive re (S: Type[0]) : 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. + +nlemma foo1: ∀S. ¬ (z S = e S). #S; @; #H; ndestruct. nqed. +nlemma foo2: ∀S,x. ¬ (z S = s S x). #S; #x; @; #H; ndestruct. nqed. +nlemma foo3: ∀S,x1,x2. ¬ (z S = c S x1 x2). #S; #x1; #x2; @; #H; ndestruct. nqed. +nlemma foo4: ∀S,x1,x2. ¬ (z S = o S x1 x2). #S; #x1; #x2; @; #H; ndestruct. nqed. +nlemma foo5: ∀S,x. ¬ (z S = k S x). #S; #x; @; #H; ndestruct. nqed. +nlemma foo6: ∀S,x. ¬ (e S = s S x). #S; #x; @; #H; ndestruct. nqed. +nlemma foo7: ∀S,x1,x2. ¬ (e S = c S x1 x2). #S; #x1; #x2; @; #H; ndestruct. nqed. +nlemma foo8: ∀S,x1,x2. ¬ (e S = o S x1 x2). #S; #x1; #x2; @; #H; ndestruct. nqed. +nlemma foo9: ∀S,x. ¬ (e S = k S x). #S; #x; @; #H; ndestruct. nqed. + +ninductive in_l (S: Type[0]): word S → re S → CProp[0] ≝ + in_e: in_l S [] (e ?) + | in_s: ∀x. in_l S [x] (s ? x) + | in_c: ∀w1,w2,e1,e2. in_l ? w1 e1 → in_l ? w2 e2 → in_l S (w1@w2) (c ? e1 e2) + | in_o1: ∀w,e1,e2. in_l ? w e1 → in_l S w (o ? e1 e2) + | in_o2: ∀w,e1,e2. in_l ? w e2 → in_l S w (o ? e1 e2) + | in_ke: ∀e. in_l S [] (k ? e) + | in_ki: ∀w1,w2,e. in_l ? w1 e → in_l ? w2 (k ? e) → in_l S (w1@w2) (k ? e). + +ninductive pre (S: Type[0]) : Type[0] ≝ + pp: pre S → pre S + | pz: pre S + | pe: pre S + | ps: S → pre S + | pc: pre S → pre S → pre S + | po: pre S → pre S → pre S + | pk: pre S → pre S. + +nlet rec forget (S: Type[0]) (l : pre S) on l: re S ≝ + match l with + [ pp E ⇒ forget S E + | pz ⇒ z S + | pe ⇒ e S + | ps x ⇒ s S x + | pc E1 E2 ⇒ c S (forget ? E1) (forget ? E2) + | po E1 E2 ⇒ o S (forget ? E1) (forget ? E2) + | pk E ⇒ k S (forget ? E) ]. + +ninductive in_pl (S: Type[0]): word S → pre S → CProp[0] ≝ + in_pp: ∀w,E. in_l S w (forget ? E) → in_pl S w (pp S E) + | in_pc1: ∀w1,w2,e1,e2. in_pl ? w1 e1 → in_l ? w2 (forget ? e2) → + in_pl S (w1@w2) (pc ? e1 e2) + | in_pc2: ∀w,e1,e2. in_pl ? w e2 → in_pl S w (pc ? e1 e2) + | in_po1: ∀w,e1,e2. in_pl ? w e1 → in_pl S w (po ? e1 e2) + | in_po2: ∀w,e1,e2. in_pl ? w e2 → in_pl S w (po ? e1 e2) + | in_pki: ∀w1,w2,e. in_pl ? w1 e → in_l ? w2 (k ? (forget ? e)) → + in_pl S (w1@w2) (pk ? e). + +nlet rec eclose (S: Type[0]) (b: bool) (E: pre S) on E ≝ + match E with + [ pp E' ⇒ eclose ? true E' + | pz ⇒ 〈 false, pz ? 〉 + | pe ⇒ 〈 b, pe ? 〉 + | ps x ⇒ 〈 false, ps ? x 〉 + | pc E1 E2 ⇒ + let E1' ≝ eclose ? b E1 in + let E1'' ≝ snd … E1' in + let E2' ≝ eclose ? (fst … E1') E2 in + 〈 fst … E2', pc ? E1'' (snd … E2') 〉 + | po E1 E2 ⇒ + let E1' ≝ eclose ? b E1 in + let E2' ≝ eclose ? b E2 in + 〈 fst … E1' ∨ fst … E2', po ? (snd … E1') (snd … E2') 〉 + | pk E ⇒ + 〈 true, pk ? (snd … (eclose S b E)) 〉 ]. + +(**********************************************************) + +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! *) + ] +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; + -- 2.39.2