From 3819ff5482f28f3bb9be822513c7bd73c47a46e0 Mon Sep 17 00:00:00 2001 From: Cosimo Oliboni Date: Tue, 4 Aug 2009 22:12:22 +0000 Subject: [PATCH] freescale porting, work in progress --- .../ng_assembly/common/list_utility.ma | 350 ++++++++++++++++++ .../ng_assembly/common/list_utility_lemmas.ma | 291 +++++++++++++++ .../matita/contribs/ng_assembly/common/nat.ma | 14 + .../contribs/ng_assembly/common/string.ma | 52 +++ .../ng_assembly/common/string_lemmas.ma | 101 +++++ .../matita/contribs/ng_assembly/depends | 6 +- 6 files changed, 813 insertions(+), 1 deletion(-) create mode 100755 helm/software/matita/contribs/ng_assembly/common/list_utility.ma create mode 100755 helm/software/matita/contribs/ng_assembly/common/list_utility_lemmas.ma create mode 100644 helm/software/matita/contribs/ng_assembly/common/string.ma create mode 100755 helm/software/matita/contribs/ng_assembly/common/string_lemmas.ma diff --git a/helm/software/matita/contribs/ng_assembly/common/list_utility.ma b/helm/software/matita/contribs/ng_assembly/common/list_utility.ma new file mode 100755 index 000000000..3ffd1d3c5 --- /dev/null +++ b/helm/software/matita/contribs/ng_assembly/common/list_utility.ma @@ -0,0 +1,350 @@ +(**************************************************************************) +(* ___ *) +(* ||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 *) +(* *) +(**************************************************************************) + +(* ********************************************************************** *) +(* Progetto FreeScale *) +(* *) +(* Sviluppato da: Cosimo Oliboni, oliboni@cs.unibo.it *) +(* Cosimo Oliboni, oliboni@cs.unibo.it *) +(* *) +(* ********************************************************************** *) + +include "common/list.ma". +include "common/option.ma". +include "common/nat_lemmas.ma". + +(* *************** *) +(* LISTE GENERICHE *) +(* *************** *) + +(* listlen *) +nlet rec len_list (T:Type) (l:list T) on l ≝ + match l with [ nil ⇒ O | cons _ t ⇒ S (len_list T t) ]. + +(* vuota? *) +ndefinition is_empty_list ≝ +λT:Type.λl:list T.match l with [ nil ⇒ True | cons _ _ ⇒ False ]. + +ndefinition isb_empty_list ≝ +λT:Type.λl:list T.match l with [ nil ⇒ true | cons _ _ ⇒ false ]. + +ndefinition isnot_empty_list ≝ +λT:Type.λl:list T.match l with [ nil ⇒ False | cons _ _ ⇒ True ]. + +ndefinition isnotb_empty_list ≝ +λT:Type.λl:list T.match l with [ nil ⇒ false | cons _ _ ⇒ true ]. + +(* reverse *) +nlet rec reverse_list (T:Type) (l:list T) on l ≝ + match l with + [ nil ⇒ nil T + | cons h t ⇒ (reverse_list T t)@[h] + ]. + +(* getFirst *) +ndefinition get_first_list ≝ +λT:Type.λl:list T.match l with + [ nil ⇒ None ? + | cons h _ ⇒ Some ? h ]. + +(* getLast *) +ndefinition get_last_list ≝ +λT:Type.λl:list T.match reverse_list T l with + [ nil ⇒ None ? + | cons h _ ⇒ Some ? h ]. + +(* cutFirst *) +ndefinition cut_first_list ≝ +λT:Type.λl:list T.match l with + [ nil ⇒ nil T + | cons _ t ⇒ t ]. + +(* cutLast *) +ndefinition cut_last_list ≝ +λT:Type.λl:list T.match reverse_list T l with + [ nil ⇒ nil T + | cons _ t ⇒ reverse_list T t ]. + +(* apply f *) +nlet rec apply_f_list (T1,T2:Type) (l:list T1) (f:T1 → T2) on l ≝ +match l with + [ nil ⇒ nil T2 + | cons h t ⇒ cons T2 (f h) (apply_f_list T1 T2 t f) ]. + +(* fold right *) +nlet rec fold_right_list (T1,T2:Type) (f:T1 → T2 → T2) (acc:T2) (l:list T1) on l ≝ + match l with + [ nil ⇒ acc + | cons h t ⇒ f h (fold_right_list T1 T2 f acc t) + ]. + +(* double fold right *) +nlemma fold_right_list2_aux1 : +∀T.∀h,t.len_list T [] = len_list T (h::t) → False. + #T; #h; #t; + nnormalize; + #H; + napply (nat_destruct_0_S ? H). +nqed. + +nlemma fold_right_list2_aux2 : +∀T.∀h,t.len_list T (h::t) = len_list T [] → False. + #T; #h; #t; + nnormalize; + #H; + napply (nat_destruct_S_0 ? H). +nqed. + +nlemma fold_right_list2_aux3 : +∀T.∀h,h',t,t'.len_list T (h::t) = len_list T (h'::t') → len_list T t = len_list T t'. + #T; #h; #h'; #t; #t'; + nelim t; + nelim t'; + ##[ ##1: nnormalize; #H; napply refl_eq + ##| ##2: #a; #l'; #H; #H1; + nchange in H1:(%) with ((S O) = (S (S (len_list T l')))); + nelim (nat_destruct_0_S ? (nat_destruct_S_S … H1)) + ##| ##3: #a; #l'; #H; #H1; + nchange in H1:(%) with ((S (S (len_list T l'))) = (S O)); + nelim (nat_destruct_S_0 ? (nat_destruct_S_S … H1)) + ##| ##4: #a; #l; #H; #a1; #l1; #H1; #H2; + nchange in H2:(%) with ((S (S (len_list T l1))) = (S (S (len_list T l)))); + nchange with ((S (len_list T l1)) = (S (len_list T l))); + nrewrite > (nat_destruct_S_S … H2); + napply refl_eq + ##] +nqed. + +nlet rec fold_right_list2 (T1,T2:Type) (f:T1 → T1 → T2 → T2) (acc:T2) (l1:list T1) on l1 ≝ + match l1 + return λl1.Πl2.len_list T1 l1 = len_list T1 l2 → T2 + with + [ nil ⇒ λl2.match l2 return λl2.len_list T1 [] = len_list T1 l2 → T2 with + [ nil ⇒ λp:len_list T1 [] = len_list T1 [].acc + | cons h t ⇒ λp:len_list T1 [] = len_list T1 (h::t). + False_rect_Type0 ? (fold_right_list2_aux1 T1 h t p) + ] + | cons h t ⇒ λl2.match l2 return λl2.len_list T1 (h::t) = len_list T1 l2 → T2 with + [ nil ⇒ λp:len_list T1 (h::t) = len_list T1 []. + False_rect_Type0 ? (fold_right_list2_aux2 T1 h t p) + | cons h' t' ⇒ λp:len_list T1 (h::t) = len_list T1 (h'::t'). + f h h' (fold_right_list2 T1 T2 f acc t t' (fold_right_list2_aux3 T1 h h' t t' p)) + ] + ]. + +nlet rec bfold_right_list2 (T1:Type) (f:T1 → T1 → bool) (l1,l2:list T1) on l1 ≝ + match l1 with + [ nil ⇒ match l2 with + [ nil ⇒ true | cons h t ⇒ false ] + | cons h t ⇒ match l2 with + [ nil ⇒ false | cons h' t' ⇒ (f h h') ⊗ (bfold_right_list2 T1 f t t') + ] + ]. + +(* nth elem *) +nlet rec nth_list (T:Type) (l:list T) (n:nat) on l ≝ + match l with + [ nil ⇒ None ? + | cons h t ⇒ match n with + [ O ⇒ Some ? h | S n' ⇒ nth_list T t n' ] + ]. + +(* abs elem *) +ndefinition abs_list_aux1 : ∀T:Type.∀n.((len_list T []) > n) = true → False. + #T; nnormalize; #n; #H; napply (bool_destruct … H). nqed. + +ndefinition abs_list_aux2 : ∀T:Type.∀h:T.∀t:list T.∀n.((len_list T (h::t)) > (S n) = true) → ((len_list T t) > n) = true. + #T; #h; #t; #n; nnormalize; #H; napply H. nqed. + +nlet rec abs_list (T:Type) (l:list T) on l ≝ + match l + return λl.Πn.(((len_list T l) > n) = true) → T + with + [ nil ⇒ λn.λp:(((len_list T []) > n) = true).False_rect_Type0 ? (abs_list_aux1 T n p) + | cons h t ⇒ λn. + match n with + [ O ⇒ λp:(((len_list T (h::t)) > O) = true).h + | S n' ⇒ λp:(((len_list T (h::t)) > (S n')) = true). + abs_list T t n' (abs_list_aux2 T h t n' p) + ] + ]. + +(* esempio: abs_list ? [ 1; 2; 3; 4 ] 2 (refl_eq …) = 0. *) + +(* ************** *) +(* NON-EMPTY LIST *) +(* ************** *) + +(* listlen *) +nlet rec len_neList (T:Type) (nl:ne_list T) on nl ≝ + match nl with [ ne_nil _ ⇒ 1 | ne_cons _ t ⇒ S (len_neList T t) ]. + +(* reverse *) +nlet rec reverse_neList (T:Type) (nl:ne_list T) on nl ≝ + match nl with + [ ne_nil h ⇒ ne_nil T h + | ne_cons h t ⇒ (reverse_neList T t)&(ne_nil T h) + ]. + +(* getFirst *) +ndefinition get_first_neList ≝ +λT:Type.λnl:ne_list T.match nl with + [ ne_nil h ⇒ h + | ne_cons h _ ⇒ h ]. + +(* getLast *) +ndefinition get_last_neList ≝ +λT:Type.λnl:ne_list T.match reverse_neList T nl with + [ ne_nil h ⇒ h + | ne_cons h _ ⇒ h ]. + +(* cutFirst *) +ndefinition cut_first_neList ≝ +λT:Type.λnl:ne_list T.match nl with + [ ne_nil h ⇒ ne_nil T h + | ne_cons _ t ⇒ t ]. + +(* cutLast *) +ndefinition cut_last_neList ≝ +λT:Type.λnl:ne_list T.match reverse_neList T nl with + [ ne_nil h ⇒ ne_nil T h + | ne_cons _ t ⇒ reverse_neList T t ]. + +(* apply f *) +nlet rec apply_f_neList (T1,T2:Type) (nl:ne_list T1) (f:T1 → T2) on nl ≝ +match nl with + [ ne_nil h ⇒ ne_nil T2 (f h) + | ne_cons h t ⇒ ne_cons T2 (f h) (apply_f_neList T1 T2 t f) ]. + +(* fold right *) +nlet rec fold_right_neList (T1,T2:Type) (f:T1 → T2 → T2) (acc:T2) (nl:ne_list T1) on nl ≝ + match nl with + [ ne_nil h ⇒ f h acc + | ne_cons h t ⇒ f h (fold_right_neList T1 T2 f acc t) + ]. + +(* double fold right *) +nlemma fold_right_neList2_aux1 : +∀T.∀h,h',t'.len_neList T «£h» = len_neList T (h'§§t') → False. + #T; #h; #h'; #t'; + nnormalize; + ncases t'; + nnormalize; + ##[ ##1: #x; #H; nelim (nat_destruct_0_S ? (nat_destruct_S_S … H)) + ##| ##2: #x; #l; #H; nelim (nat_destruct_0_S ? (nat_destruct_S_S … H)) + ##] +nqed. + +nlemma fold_right_neList2_aux2 : +∀T.∀h,h',t.len_neList T (h§§t) = len_neList T «£h'» → False. + #T; #h; #h'; #t; + nnormalize; + ncases t; + nnormalize; + ##[ ##1: #x; #H; nelim (nat_destruct_S_0 ? (nat_destruct_S_S … H)) + ##| ##2: #x; #l; #H; nelim (nat_destruct_S_0 ? (nat_destruct_S_S … H)) + ##] +nqed. + +nlemma fold_right_neList2_aux3 : +∀T.∀h,h',t,t'.len_neList T (h§§t) = len_neList T (h'§§t') → len_neList T t = len_neList T t'. + #T; #h; #h'; #t; #t'; + nelim t; + nelim t'; + ##[ ##1: nnormalize; #x; #y; #H; napply refl_eq + ##| ##2: #a; #l'; #H; #x; #H1; + nchange in H1:(%) with ((S (len_neList T «£x»)) = (S (len_neList T (a§§l')))); + nrewrite > (nat_destruct_S_S … H1); + napply refl_eq + ##| ##3: #x; #a; #l'; #H; #H1; + nchange in H1:(%) with ((S (len_neList T (a§§l')))= (S (len_neList T «£x»))); + nrewrite > (nat_destruct_S_S … H1); + napply refl_eq + ##| ##4: #a; #l; #H; #a1; #l1; #H1; #H2; + nchange in H2:(%) with ((S (len_neList T (a1§§l1))) = (S (len_neList T (a§§l)))); + nrewrite > (nat_destruct_S_S … H2); + napply refl_eq + ##] +nqed. + +nlet rec fold_right_neList2 (T1,T2:Type) (f:T1 → T1 → T2 → T2) (acc:T2) (l1:ne_list T1) on l1 ≝ + match l1 + return λl1.Πl2.len_neList T1 l1 = len_neList T1 l2 → T2 + with + [ ne_nil h ⇒ λl2.match l2 return λl2.len_neList T1 «£h» = len_neList T1 l2 → T2 with + [ ne_nil h' ⇒ λp:len_neList T1 «£h» = len_neList T1 «£h'». + f h h' acc + | ne_cons h' t' ⇒ λp:len_neList T1 «£h» = len_neList T1 (h'§§t'). + False_rect_Type0 ? (fold_right_neList2_aux1 T1 h h' t' p) + ] + | ne_cons h t ⇒ λl2.match l2 return λl2.len_neList T1 (h§§t) = len_neList T1 l2 → T2 with + [ ne_nil h' ⇒ λp:len_neList T1 (h§§t) = len_neList T1 «£h'». + False_rect_Type0 ? (fold_right_neList2_aux2 T1 h h' t p) + | ne_cons h' t' ⇒ λp:len_neList T1 (h§§t) = len_neList T1 (h'§§t'). + f h h' (fold_right_neList2 T1 T2 f acc t t' (fold_right_neList2_aux3 T1 h h' t t' p)) + ] + ]. + +nlet rec bfold_right_neList2 (T1:Type) (f:T1 → T1 → bool) (l1,l2:ne_list T1) on l1 ≝ + match l1 with + [ ne_nil h ⇒ match l2 with + [ ne_nil h' ⇒ f h h' | ne_cons h' t' ⇒ false ] + | ne_cons h t ⇒ match l2 with + [ ne_nil h' ⇒ false | ne_cons h' t' ⇒ (f h h') ⊗ (bfold_right_neList2 T1 f t t') + ] + ]. + +(* nth elem *) +nlet rec nth_neList (T:Type) (nl:ne_list T) (n:nat) on nl ≝ + match nl with + [ ne_nil h ⇒ match n with + [ O ⇒ Some ? h | S _ ⇒ None ? ] + | ne_cons h t ⇒ match n with + [ O ⇒ Some ? h | S n' ⇒ nth_neList T t n' ] + ]. + +(* abs elem *) +ndefinition abs_neList_aux1 : ∀T:Type.∀h:T.∀n.((len_neList T («£h»)) > (S n)) = true → False. + #T; #h; #n; nnormalize; #H; napply (bool_destruct … H). nqed. + +ndefinition abs_neList_aux2 : ∀T:Type.∀h:T.∀t:ne_list T.∀n.((len_neList T (h§§t)) > (S n) = true) → ((len_neList T t) > n) = true. + #T; #h; #t; #n; nnormalize; #H; napply H. nqed. + +naxiom daemon : False. + +nlet rec abs_neList (T:Type) (l:ne_list T) on l ≝ + match l + return λl.Πn.(((len_neList T l) > n) = true) → T + with + [ ne_nil h ⇒ λn. + match n + return λn.(((len_neList T (ne_nil T h)) > n) = true) → T + with + [ O ⇒ λp:(((len_neList T (ne_nil T h)) > O) = true).h + | S n' ⇒ λp:(((len_neList T (ne_nil T h)) > (S n')) = true). + False_rect_Type0 ? (abs_neList_aux1 T h n' p) + ] + | ne_cons h t ⇒ λn. + match n with + [ O ⇒ λp:(((len_neList T (ne_cons T h t)) > O) = true).h + | S n' ⇒ λp:(((len_neList T (ne_cons T h t)) > (S n')) = true). + abs_neList T t n' (abs_neList_aux2 T h t n' p) + ] + ]. + +(* esempio: abs_neList ? « 1; 2; 3 £ 4 » 0 (refl_eq …) = 0. *) + +(* conversione *) +nlet rec neList_to_list (T:Type) (nl:ne_list T) on nl : list T ≝ + match nl with [ ne_nil h ⇒ [h] | ne_cons h t ⇒ [h]@(neList_to_list T t) ]. diff --git a/helm/software/matita/contribs/ng_assembly/common/list_utility_lemmas.ma b/helm/software/matita/contribs/ng_assembly/common/list_utility_lemmas.ma new file mode 100755 index 000000000..5ae7fce81 --- /dev/null +++ b/helm/software/matita/contribs/ng_assembly/common/list_utility_lemmas.ma @@ -0,0 +1,291 @@ +(**************************************************************************) +(* ___ *) +(* ||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 *) +(* *) +(**************************************************************************) + +(* ********************************************************************** *) +(* Progetto FreeScale *) +(* *) +(* Sviluppato da: Cosimo Oliboni, oliboni@cs.unibo.it *) +(* Cosimo Oliboni, oliboni@cs.unibo.it *) +(* *) +(* ********************************************************************** *) + +include "common/list_utility.ma". +include "common/list_lemmas.ma". + +(* *************** *) +(* LISTE GENERICHE *) +(* *************** *) + +nlemma symmetric_lenlist : ∀T.∀l1,l2:list T.len_list T l1 = len_list T l2 → len_list T l2 = len_list T l1. + #T; #l1; + nelim l1; + ##[ ##1: #l2; ncases l2; nnormalize; + ##[ ##1: #H; napply refl_eq + ##| ##2: #h; #t; #H; nelim (nat_destruct_0_S ? H) + ##] + ##| ##2: #h; #l2; ncases l2; nnormalize; + ##[ ##1: #H; #l; #H1; nrewrite < H1; napply refl_eq + ##| ##2: #h; #l; #H; #l3; #H1; nrewrite < H1; napply refl_eq + ##] + ##] +nqed. + +nlemma symmetric_foldrightlist2_aux + : ∀T1,T2:Type.∀f:T1 → T1 → T2 → T2.∀acc:T2.∀l1,l2:list T1. + ∀H1:len_list T1 l1 = len_list T1 l2.∀H2:len_list T1 l2 = len_list T1 l1. + (∀x,y,z.f x y z = f y x z) → + fold_right_list2 T1 T2 f acc l1 l2 H1 = fold_right_list2 T1 T2 f acc l2 l1 H2. + #T1; #T2; #f; #acc; #l1; + nelim l1; + ##[ ##1: #l2; ncases l2; + ##[ ##1: nnormalize; #H1; #H2; #H3; napply refl_eq + ##| ##2: #h; #l; #H1; #H2; #H3; + nchange in H1:(%) with (O = (S (len_list ? l))); + nelim (nat_destruct_0_S ? H1) + ##] + ##| ##2: #h3; #l3; #H; #l2; ncases l2; + ##[ ##1: #H1; #H2; #H3; nchange in H1:(%) with ((S (len_list ? l3)) = O); + nelim (nat_destruct_S_0 ? H1) + ##| ##2: #h4; #l4; #H1; #H2; #H3; + nchange in H1:(%) with ((S (len_list ? l3)) = (S (len_list ? l4))); + nchange in H2:(%) with ((S (len_list ? l4)) = (S (len_list ? l3))); + nchange with ((f h3 h4 (fold_right_list2 T1 T2 f acc l3 l4 (fold_right_list2_aux3 T1 h3 h4 l3 l4 ?))) = + (f h4 h3 (fold_right_list2 T1 T2 f acc l4 l3 (fold_right_list2_aux3 T1 h4 h3 l4 l3 ?)))); + nrewrite < (H l4 (fold_right_list2_aux3 T1 h3 h4 l3 l4 H1) (fold_right_list2_aux3 T1 h4 h3 l4 l3 H2) H3); + nrewrite > (H3 h3 h4 (fold_right_list2 T1 T2 f acc l3 l4 ?)); + napply refl_eq + ##] + ##] +nqed. + +nlemma symmetric_foldrightlist2 + : ∀T1,T2:Type.∀f:T1 → T1 → T2 → T2.∀acc:T2.∀l1,l2:list T1.∀H:len_list T1 l1 = len_list T1 l2. + (∀x,y,z.f x y z = f y x z) → + fold_right_list2 T1 T2 f acc l1 l2 H = fold_right_list2 T1 T2 f acc l2 l1 (symmetric_lenlist T1 l1 l2 H). + #T1; #T2; #f; #acc; #l1; #l2; #H; #H1; + nrewrite > (symmetric_foldrightlist2_aux T1 T2 f acc l1 l2 H (symmetric_lenlist T1 l1 l2 H) H1); + napply refl_eq. +nqed. + +nlemma symmetric_bfoldrightlist2 + : ∀T1:Type.∀f:T1 → T1 → bool.∀l1,l2:list T1. + (∀x,y.f x y = f y x) → + bfold_right_list2 T1 f l1 l2 = bfold_right_list2 T1 f l2 l1. + #T; #f; #l1; + nelim l1; + ##[ ##1: #l2; ncases l2; + ##[ ##1: #H; nnormalize; napply refl_eq + ##| ##2: #hh2; #ll2; #H; nnormalize; napply refl_eq + ##] + ##| ##2: #hh1; #ll1; #H; #l2; ncases l2; + ##[ ##1: #H1; nnormalize; napply refl_eq + ##| ##2: #hh2; #ll2; #H1; nnormalize; + nrewrite > (H ll2 H1); + nrewrite > (H1 hh1 hh2); + napply refl_eq + ##] + ##] +nqed. + +nlemma bfoldrightlist2_to_eq + : ∀T1:Type.∀f:T1 → T1 → bool.∀l1,l2:list T1. + (∀x,y.(f x y = true → x = y)) → + (bfold_right_list2 T1 f l1 l2 = true → l1 = l2). + #T; #f; #l1; + nelim l1; + ##[ ##1: #l2; ncases l2; + ##[ ##1: #H; #H1; napply refl_eq + ##| ##2: #hh2; #ll2; #H; nnormalize; #H1; napply (bool_destruct … H1) + ##] + ##| ##2: #hh1; #ll1; #H; #l2; ncases l2; + ##[ ##1: #H1; nnormalize; #H2; napply (bool_destruct … H2) + ##| ##2: #hh2; #ll2; #H1; #H2; + nchange in H2:(%) with (((f hh1 hh2)⊗(bfold_right_list2 T f ll1 ll2)) = true); + nrewrite > (H1 hh1 hh2 (andb_true_true_l … H2)); + nrewrite > (H ll2 H1 (andb_true_true_r … H2)); + napply refl_eq + ##] + ##] +nqed. + +nlemma eq_to_bfoldrightlist2 + : ∀T1:Type.∀f:T1 → T1 → bool.∀l1,l2:list T1. + (∀x,y.(x = y → f x y = true)) → + (l1 = l2 → bfold_right_list2 T1 f l1 l2 = true). + #T; #f; #l1; + nelim l1; + ##[ ##1: #l2; ncases l2; + ##[ ##1: #H; #H1; nnormalize; napply refl_eq + ##| ##2: #hh2; #ll2; #H; #H1; nelim (list_destruct_nil_cons ??? H1) + ##] + ##| ##2: #hh1; #ll1; #H; #l2; ncases l2; + ##[ ##1: #H1; #H2; nelim (list_destruct_cons_nil ??? H2) + ##| ##2: #hh2; #ll2; #H1; #H2; nnormalize; + nrewrite > (list_destruct_1 … H2); + nrewrite > (H1 hh2 hh2 (refl_eq …)); + nnormalize; + nrewrite > (H ll2 H1 (list_destruct_2 … H2)); + napply refl_eq + ##] + ##] +nqed. + +(* ************** *) +(* NON-EMPTY LIST *) +(* ************** *) + +nlemma symmetric_lennelist : ∀T.∀l1,l2:ne_list T.len_neList T l1 = len_neList T l2 → len_neList T l2 = len_neList T l1. + #T; #l1; + nelim l1; + ##[ ##1: #h; #l2; ncases l2; nnormalize; + ##[ ##1: #H; #H1; napply refl_eq + ##| ##2: #h; #t; #H; nrewrite > H; napply refl_eq + ##] + ##| ##2: #h; #l2; ncases l2; nnormalize; + ##[ ##1: #h1; #H; #l; #H1; nrewrite < H1; napply refl_eq + ##| ##2: #h; #l; #H; #l3; #H1; nrewrite < H1; napply refl_eq + ##] + ##] +nqed. + +nlemma symmetric_foldrightnelist2_aux + : ∀T1,T2:Type.∀f:T1 → T1 → T2 → T2.∀acc:T2.∀l1,l2:ne_list T1. + ∀H1:len_neList T1 l1 = len_neList T1 l2.∀H2:len_neList T1 l2 = len_neList T1 l1. + (∀x,y,z.f x y z = f y x z) → + fold_right_neList2 T1 T2 f acc l1 l2 H1 = fold_right_neList2 T1 T2 f acc l2 l1 H2. + #T1; #T2; #f; #acc; #l1; + nelim l1; + ##[ ##1: #h; #l2; ncases l2; + ##[ ##1: #h1; nnormalize; #H1; #H2; #H3; nrewrite > (H3 h h1 acc); napply refl_eq + ##| ##2: #h1; #l; ncases l; + ##[ ##1: #h3; #H1; #H2; #H3; + nchange in H1:(%) with ((S O) = (S (S O))); + nelim (nat_destruct_0_S ? (nat_destruct_S_S … H1)) + ##| ##2: #h3; #l3; #H1; #H2; #H3; + nchange in H1:(%) with ((S O) = (S (S (len_neList ? l3)))); + nelim (nat_destruct_0_S ? (nat_destruct_S_S … H1)) + ##] + ##] + ##| ##2: #h3; #l3; #H; #l2; ncases l2; + ##[ ##1: #h4; ncases l3; + ##[ ##1: #h5; #H1; #H2; #H3; + nchange in H1:(%) with ((S (S O)) = (S O)); + nelim (nat_destruct_S_0 ? (nat_destruct_S_S … H1)) + ##| ##2: #h5; #l5; #H1; #H2; #H3; + nchange in H1:(%) with ((S (S (len_neList ? l5))) = (S O)); + nelim (nat_destruct_S_0 ? (nat_destruct_S_S … H1)) + ##] + ##| ##2: #h4; #l4; #H1; #H2; #H3; + nchange in H1:(%) with ((S (len_neList ? l3)) = (S (len_neList ? l4))); + nchange in H2:(%) with ((S (len_neList ? l4)) = (S (len_neList ? l3))); + nchange with ((f h3 h4 (fold_right_neList2 T1 T2 f acc l3 l4 (fold_right_neList2_aux3 T1 h3 h4 l3 l4 ?))) = + (f h4 h3 (fold_right_neList2 T1 T2 f acc l4 l3 (fold_right_neList2_aux3 T1 h4 h3 l4 l3 ?)))); + nrewrite < (H l4 (fold_right_neList2_aux3 T1 h3 h4 l3 l4 H1) (fold_right_neList2_aux3 T1 h4 h3 l4 l3 H2) H3); + nrewrite > (H3 h3 h4 (fold_right_neList2 T1 T2 f acc l3 l4 ?)); + napply refl_eq + ##] + ##] +nqed. + +nlemma symmetric_foldrightnelist2 + : ∀T1,T2:Type.∀f:T1 → T1 → T2 → T2.∀acc:T2.∀l1,l2:ne_list T1.∀H:len_neList T1 l1 = len_neList T1 l2. + (∀x,y,z.f x y z = f y x z) → + fold_right_neList2 T1 T2 f acc l1 l2 H = fold_right_neList2 T1 T2 f acc l2 l1 (symmetric_lennelist T1 l1 l2 H). + #T1; #T2; #f; #acc; #l1; #l2; #H; #H1; + nrewrite > (symmetric_foldrightnelist2_aux T1 T2 f acc l1 l2 H (symmetric_lennelist T1 l1 l2 H) H1); + napply refl_eq. +nqed. + +nlemma symmetric_bfoldrightnelist2 + : ∀T1:Type.∀f:T1 → T1 → bool.∀l1,l2:ne_list T1. + (∀x,y.f x y = f y x) → + bfold_right_neList2 T1 f l1 l2 = bfold_right_neList2 T1 f l2 l1. + #T; #f; #l1; + nelim l1; + ##[ ##1: #hh1; #l2; ncases l2; + ##[ ##1: #hh2; #H; nnormalize; nrewrite > (H hh1 hh2); napply refl_eq + ##| ##2: #hh2; #ll2; #H; nnormalize; napply refl_eq + ##] + ##| ##2: #hh1; #ll1; #H; #l2; ncases l2; + ##[ ##1: #hh2; #H1; nnormalize; napply refl_eq + ##| ##2: #hh2; #ll2; #H1; nnormalize; + nrewrite > (H ll2 H1); + nrewrite > (H1 hh1 hh2); + napply refl_eq + ##] + ##] +nqed. + +nlemma bfoldrightnelist2_to_eq + : ∀T1:Type.∀f:T1 → T1 → bool.∀l1,l2:ne_list T1. + (∀x,y.(f x y = true → x = y)) → + (bfold_right_neList2 T1 f l1 l2 = true → l1 = l2). + #T; #f; #l1; + nelim l1; + ##[ ##1: #hh1; #l2; ncases l2; + ##[ ##1: #hh2; #H; #H1; nnormalize in H1:(%); nrewrite > (H hh1 hh2 H1); napply refl_eq + ##| ##2: #hh2; #ll2; #H; nnormalize; #H1; napply (bool_destruct … H1) + ##] + ##| ##2: #hh1; #ll1; #H; #l2; ncases l2; + ##[ ##1: #hh2; #H1; nnormalize; #H2; napply (bool_destruct … H2) + ##| ##2: #hh2; #ll2; #H1; #H2; + nchange in H2:(%) with (((f hh1 hh2)⊗(bfold_right_neList2 T f ll1 ll2)) = true); + nrewrite > (H1 hh1 hh2 (andb_true_true_l … H2)); + nrewrite > (H ll2 H1 (andb_true_true_r … H2)); + napply refl_eq + ##] + ##] +nqed. + +nlemma eq_to_bfoldrightnelist2 + : ∀T1:Type.∀f:T1 → T1 → bool.∀l1,l2:ne_list T1. + (∀x,y.(x = y → f x y = true)) → + (l1 = l2 → bfold_right_neList2 T1 f l1 l2 = true). + #T; #f; #l1; + nelim l1; + ##[ ##1: #hh1; #l2; ncases l2; + ##[ ##1: #hh2; #H; #H1; nnormalize; + nrewrite > (H hh1 hh2 (nelist_destruct_nil_nil … H1)); + napply refl_eq + ##| ##2: #hh2; #ll2; #H; #H1; nelim (nelist_destruct_nil_cons ???? H1) + ##] + ##| ##2: #hh1; #ll1; #H; #l2; ncases l2; + ##[ ##1: #hh2; #H1; #H2; nelim (nelist_destruct_cons_nil ???? H2) + ##| ##2: #hh2; #ll2; #H1; #H2; nnormalize; + nrewrite > (nelist_destruct_cons_cons_1 … H2); + nrewrite > (H1 hh2 hh2 (refl_eq …)); + nnormalize; + nrewrite > (H ll2 H1 (nelist_destruct_cons_cons_2 … H2)); + napply refl_eq + ##] + ##] +nqed. + +nlemma isbemptylist_to_isemptylist : ∀T,l.isb_empty_list T l = true → is_empty_list T l. + #T; #l; + ncases l; + nnormalize; + ##[ ##1: #H; napply I + ##| ##2: #x; #l; #H; napply (bool_destruct … H) + ##] +nqed. + +nlemma isnotbemptylist_to_isnotemptylist : ∀T,l.isnotb_empty_list T l = true → isnot_empty_list T l. + #T; #l; + ncases l; + nnormalize; + ##[ ##1: #H; napply (bool_destruct … H) + ##| ##2: #x; #l; #H; napply I + ##] +nqed. diff --git a/helm/software/matita/contribs/ng_assembly/common/nat.ma b/helm/software/matita/contribs/ng_assembly/common/nat.ma index 874ab7d8c..7e9ec23e4 100755 --- a/helm/software/matita/contribs/ng_assembly/common/nat.ma +++ b/helm/software/matita/contribs/ng_assembly/common/nat.ma @@ -49,6 +49,20 @@ nlet rec le_nat n m ≝ [ O ⇒ false | (S q) ⇒ le_nat p q ] ]. +interpretation "natural 'less or equal to'" 'leq x y = (le_nat x y). + +ndefinition lt_nat ≝ λn1,n2:nat.(le_nat n1 n2) ⊗ (⊖ (eq_nat n1 n2)). + +interpretation "natural 'less than'" 'lt x y = (lt_nat x y). + +ndefinition ge_nat ≝ λn1,n2:nat.(⊖ (le_nat n1 n2)) ⊕ (eq_nat n1 n2). + +interpretation "natural 'greater or equal to'" 'geq x y = (ge_nat x y). + +ndefinition gt_nat ≝ λn1,n2:nat.⊖ (le_nat n1 n2). + +interpretation "natural 'greater than'" 'gt x y = (gt_nat x y). + nlet rec plus (n1,n2:nat) on n1 ≝ match n1 with [ O ⇒ n2 diff --git a/helm/software/matita/contribs/ng_assembly/common/string.ma b/helm/software/matita/contribs/ng_assembly/common/string.ma new file mode 100644 index 000000000..8763ec8fc --- /dev/null +++ b/helm/software/matita/contribs/ng_assembly/common/string.ma @@ -0,0 +1,52 @@ +(**************************************************************************) +(* ___ *) +(* ||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 *) +(* *) +(**************************************************************************) + +(* ********************************************************************** *) +(* Progetto FreeScale *) +(* *) +(* Sviluppato da: Cosimo Oliboni, oliboni@cs.unibo.it *) +(* Cosimo Oliboni, oliboni@cs.unibo.it *) +(* *) +(* ********************************************************************** *) + +include "common/ascii.ma". +include "common/list_utility.ma". + +(* ************************ *) +(* MANIPOLAZIONE DI STRINGA *) +(* ************************ *) + +(* tipo pubblico *) +ndefinition aux_str_type ≝ list ascii. + +(* strcmp *) +ndefinition eq_str ≝ + bfold_right_list2 ascii (λx,y.eq_ascii x y). + +(* ************ *) +(* STRINGA + ID *) +(* ************ *) + +(* tipo pubblico *) +nrecord strId : Type ≝ + { + str_elem: aux_str_type; + id_elem: nat + }. + +(* confronto *) +ndefinition eq_strId ≝ +λsid,sid':strId. + (eq_str (str_elem sid) (str_elem sid'))⊗ + (eq_nat (id_elem sid) (id_elem sid')). diff --git a/helm/software/matita/contribs/ng_assembly/common/string_lemmas.ma b/helm/software/matita/contribs/ng_assembly/common/string_lemmas.ma new file mode 100755 index 000000000..67590ae1e --- /dev/null +++ b/helm/software/matita/contribs/ng_assembly/common/string_lemmas.ma @@ -0,0 +1,101 @@ +(**************************************************************************) +(* ___ *) +(* ||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 *) +(* *) +(**************************************************************************) + +(* ********************************************************************** *) +(* Progetto FreeScale *) +(* *) +(* Sviluppato da: Cosimo Oliboni, oliboni@cs.unibo.it *) +(* Cosimo Oliboni, oliboni@cs.unibo.it *) +(* *) +(* ********************************************************************** *) + +include "common/string.ma". +include "common/ascii_lemmas2.ma". +include "common/list_utility_lemmas.ma". + +(* ************************ *) +(* MANIPOLAZIONE DI STRINGA *) +(* ************************ *) + +nlemma symmetric_eqstr : symmetricT (list ascii) bool eq_str. + #s1; #s2; + napply (symmetric_bfoldrightlist2 ascii eq_ascii s1 s2 symmetric_eqascii). +nqed. + +nlemma eqstr_to_eq : ∀s,s'.eq_str s s' = true → s = s'. + #s1; #s2; + napply (bfoldrightlist2_to_eq ascii eq_ascii s1 s2 eqascii_to_eq). +nqed. + +nlemma eq_to_eqstr : ∀s,s'.s = s' → eq_str s s' = true. + #s1; #s2; + napply (eq_to_bfoldrightlist2 ascii eq_ascii s1 s2 eq_to_eqascii). +nqed. + +(* ************ *) +(* STRINGA + ID *) +(* ************ *) + +nlemma strid_destruct_1 : ∀x1,x2,y1,y2.mk_strId x1 y1 = mk_strId x2 y2 → x1 = x2. + #x1; #x2; #y1; #y2; #H; + nchange with (match mk_strId x2 y2 with [ mk_strId a _ ⇒ x1 = a ]); + nrewrite < H; + nnormalize; + napply refl_eq. +nqed. + +nlemma strid_destruct_2 : ∀x1,x2,y1,y2.mk_strId x1 y1 = mk_strId x2 y2 → y1 = y2. + #x1; #x2; #y1; #y2; #H; + nchange with (match mk_strId x2 y2 with [ mk_strId _ b ⇒ y1 = b ]); + nrewrite < H; + nnormalize; + napply refl_eq. +nqed. + +nlemma symmetric_eqstrid : symmetricT strId bool eq_strId. + #si1; #si2; + nchange with ( + ((eq_str (str_elem si1) (str_elem si2))⊗(eq_nat (id_elem si1) (id_elem si2))) = + ((eq_str (str_elem si2) (str_elem si1))⊗(eq_nat (id_elem si2) (id_elem si1)))); + nrewrite > (symmetric_eqstr (str_elem si1) (str_elem si2)); + nrewrite > (symmetric_eqnat (id_elem si1) (id_elem si2)); + napply refl_eq. +nqed. + +nlemma eqstrid_to_eq : ∀s,s'.eq_strId s s' = true → s = s'. + #si1; #si2; + nelim si1; + #l1; #n1; + nelim si2; + #l2; #n2; #H; + nchange in H:(%) with (((eq_str l1 l2)⊗(eq_nat n1 n2)) = true); + nrewrite > (eqstr_to_eq l1 l2 (andb_true_true_l … H)); + nrewrite > (eqnat_to_eq n1 n2 (andb_true_true_r … H)); + napply refl_eq. +nqed. + +nlemma eq_to_eqstrid : ∀s,s'.s = s' → eq_strId s s' = true. + #si1; #si2; + nelim si1; + #l1; #n1; + nelim si2; + #l2; #n2; #H; + nchange with (((eq_str l1 l2)⊗(eq_nat n1 n2)) = true); + nrewrite > (strid_destruct_1 … H); + nrewrite > (strid_destruct_2 … H); + nrewrite > (eq_to_eqstr l2 l2 (refl_eq …)); + nrewrite > (eq_to_eqnat n2 n2 (refl_eq …)); + nnormalize; + napply refl_eq. +nqed. diff --git a/helm/software/matita/contribs/ng_assembly/depends b/helm/software/matita/contribs/ng_assembly/depends index e9d1d4c63..adad69897 100644 --- a/helm/software/matita/contribs/ng_assembly/depends +++ b/helm/software/matita/contribs/ng_assembly/depends @@ -9,10 +9,12 @@ common/ascii_lemmas1.ma common/ascii.ma num/word16_lemmas.ma num/byte8_lemmas.ma num/word16.ma common/prod.ma num/bool.ma common/ascii_lemmas2.ma common/ascii_lemmas1.ma num/bool_lemmas.ma +common/list_utility_lemmas.ma common/list_lemmas.ma common/list_utility.ma +common/string_lemmas.ma common/ascii_lemmas2.ma common/list_utility_lemmas.ma common/string.ma num/bool_lemmas.ma num/bool.ma common/nat_lemmas.ma common/nat.ma num/bool_lemmas.ma -num/exadecim_lemmas.ma num/bool_lemmas.ma num/exadecim.ma common/option_lemmas.ma common/option.ma num/bool_lemmas.ma +num/exadecim_lemmas.ma num/bool_lemmas.ma num/exadecim.ma num/exadecim.ma common/prod.ma num/bool.ma num/oct.ma num/quatern.ma common/theory.ma num/quatern.ma num/bool.ma @@ -24,6 +26,8 @@ common/option.ma num/bool.ma common/prod_lemmas.ma common/prod.ma num/bool_lemmas.ma num/bool.ma common/theory.ma num/word16.ma num/byte8.ma +common/string.ma common/ascii.ma common/list_utility.ma common/ascii.ma num/bool.ma common/list.ma common/theory.ma +common/list_utility.ma common/list.ma common/nat_lemmas.ma common/option.ma num/bitrigesim_lemmas.ma num/bitrigesim.ma num/bool_lemmas.ma -- 2.39.2