+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 "freescale/theory.ma".
-include "freescale/option.ma".
-
-(* ************** *)
-(* Non-Empty List *)
-(* ************** *)
-
-(* lista non vuota *)
-ninductive ne_list (A:Type) : Type ≝
- | ne_nil: A → ne_list A
- | ne_cons: A → ne_list A → ne_list A.
-
-(* append *)
-nlet rec ne_append (A:Type) (l1,l2:ne_list A) on l1 ≝
- match l1 with
- [ ne_nil hd ⇒ ne_cons A hd l2
- | ne_cons hd tl ⇒ ne_cons A hd (ne_append A tl l2) ].
-
-notation "hvbox(hd break §§ tl)"
- right associative with precedence 46
- for @{'ne_cons $hd $tl}.
-
-notation "« list0 x sep ; break £ y break »"
- non associative with precedence 90
- for ${fold right @{'ne_nil $y } rec acc @{'ne_cons $x $acc}}.
-
-notation "hvbox(l1 break & l2)"
- right associative with precedence 47
- for @{'ne_append $l1 $l2 }.
-
-interpretation "ne_nil" 'ne_nil hd = (ne_nil ? hd).
-interpretation "ne_cons" 'ne_cons hd tl = (ne_cons ? hd tl).
-interpretation "ne_append" 'ne_append l1 l2 = (ne_append ? l1 l2).
-
-(* ************ *)
-(* List Utility *)
-(* ************ *)
-
-(* listlen *)
-nlet rec len_list (T:Type) (l:list T) on l ≝
- match l with [ nil ⇒ O | cons _ t ⇒ S (len_list T t) ].
-
-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) ].
-
-(* 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 ].
-
-(* 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) ].
-
-nlet rec list_to_neList_aux (T:Type) (l:list T) on l : option (ne_list T) ≝
- match l with
- [ nil ⇒ None (ne_list T)
- | cons h t ⇒ match list_to_neList_aux T t with
- [ None ⇒ Some (ne_list T) «£h»
- | Some t' ⇒ Some (ne_list T) («£h»&t') ]].
-
-ndefinition list_to_neList ≝
-λT:Type.λl:list T.
- match l
- return λl:list T.isnot_empty_list T l → ne_list T
- with
- [ nil ⇒ λp:isnot_empty_list T (nil T).False_rect_Type0 ? p
- | cons h t ⇒ λp:isnot_empty_list T (cons T h t).
- match list_to_neList_aux T t with
- [ None ⇒ «£h»
- | Some t' ⇒ «£h»&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' ]
- ].
-
-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' ]
- ].
-
-nlet rec abs_nth_neList (T:Type) (nl:ne_list T) (n:nat) on nl ≝
- match nl with
- [ ne_nil h ⇒ h
- | ne_cons h t ⇒ match n with
- [ O ⇒ h | S n' ⇒ abs_nth_neList T t n' ]
- ].
-
-(* 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]
- ].
-
-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)
- ].
-
-(* getLast *)
-ndefinition get_last_list ≝
-λT:Type.λl:list T.match reverse_list T l with
- [ nil ⇒ None ?
- | cons h _ ⇒ Some ? h ].
-
-ndefinition get_last_neList ≝
-λT:Type.λnl:ne_list T.match reverse_neList T nl with
- [ ne_nil h ⇒ h
- | ne_cons h _ ⇒ h ].
-
-(* 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 ].
-
-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 ].
-
-(* getFirst *)
-ndefinition get_first_list ≝
-λT:Type.λl:list T.match l with
- [ nil ⇒ None ?
- | cons h _ ⇒ Some ? h ].
-
-ndefinition get_first_neList ≝
-λT:Type.λnl:ne_list T.match nl with
- [ ne_nil h ⇒ h
- | ne_cons h _ ⇒ h ].
-
-(* cutFirst *)
-ndefinition cut_first_list ≝
-λT:Type.λl:list T.match l with
- [ nil ⇒ nil T
- | cons _ t ⇒ t ].
-
-ndefinition cut_first_neList ≝
-λT:Type.λnl:ne_list T.match nl with
- [ ne_nil h ⇒ ne_nil T h
- | ne_cons _ 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) ].
-
-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_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)
- ].
-
-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_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')
- ]
- ].
-
-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')
- ]
- ].
-
-(* ******** *)
-(* naturali *)
-(* ******** *)
-
-ndefinition isZero ≝ λn:nat.match n with [ O ⇒ True | S _ ⇒ False ].
-
-ndefinition isZerob ≝ λn:nat.match n with [ O ⇒ true | S _ ⇒ false ].
-
-ndefinition ltb ≝ λn1,n2:nat.(le_nat n1 n2) ⊗ (⊖ (eq_nat n1 n2)).
-
-ndefinition geb ≝ λn1,n2:nat.(⊖ (le_nat n1 n2)) ⊕ (eq_nat n1 n2).
-
-ndefinition gtb ≝ λn1,n2:nat.⊖ (le_nat n1 n2).
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