(* *)
(**************************************************************************)
+include "logic/pts.ma".
include "arithmetics/nat.ma".
ninductive list (A:Type[0]) : Type[0] ≝
interpretation "nil" 'nil = (nil ?).
interpretation "cons" 'cons hd tl = (cons ? hd tl).
-ntheorem nil_cons:
- ∀A:Type[0].∀l:list A.∀a:A. a::l ≠ [].
-#A;#l;#a; @; #H; ndestruct;
-nqed.
+nlet rec append A (l1: list A) l2 on l1 ≝
+ match l1 with
+ [ nil ⇒ l2
+ | cons hd tl ⇒ hd :: append A tl l2 ].
+
+interpretation "append" 'append l1 l2 = (append ? l1 l2).
nlet rec id_list A (l: list A) on l ≝
match l with
[ nil ⇒ []
| cons hd tl ⇒ hd :: id_list A tl ].
-nlet rec append A (l1: list A) l2 on l1 ≝
- match l1 with
- [ nil ⇒ l2
- | cons hd tl ⇒ hd :: append A tl l2 ].
ndefinition tail ≝ λA:Type[0].λl:list A.
match l with
[ nil ⇒ []
| cons hd tl ⇒ tl].
-interpretation "append" 'append l1 l2 = (append ? l1 l2).
+nlet rec flatten S (l : list (list S)) on l : list S ≝
+match l with [ nil ⇒ [ ] | cons w tl ⇒ w @ flatten ? tl ].
ntheorem append_nil: ∀A:Type.∀l:list A.l @ [] = l.
#A;#l;nelim l;//;
-#a;#l1;#IH;nnormalize;//;
+#a;#l1;#IH;nnormalize;
+nrewrite > IH;//;
nqed.
ntheorem associative_append: ∀A:Type[0].associative (list A) (append A).
*)
-nlet rec nth A l d n on n ≝
- match n with
- [ O ⇒ match l with
- [ nil ⇒ d
- | cons (x : A) _ ⇒ x ]
- | S n' ⇒ nth A (tail ? l) d n'].
+nlet rec nth A l d n on l ≝
+ match l with
+ [ nil ⇒ d
+ | cons (x:A) tl ⇒ match n with
+ [ O ⇒ x
+ | S n' ⇒ nth A tl d n' ] ].
nlet rec map A B f l on l ≝
match l with [ nil ⇒ nil B | cons (x:A) tl ⇒ f x :: map A B f tl ].
interpretation "list not member" 'notmem x l = (Not (in_list ? x l)).
interpretation "list inclusion" 'subseteq l1 l2 = (incl ? l1 l2).
-naxiom not_in_list_nil : \forall A,x.\lnot in_list A x [].
-(*intros.unfold.intro.inversion H
- [intros;lapply (sym_eq ? ? ? H2);destruct Hletin
- |intros;destruct H4]
-qed.*)
+nlemma not_in_list_nil : \forall A,x.\lnot in_list A x [].
+#A x;@;#H1;ninversion H1;
+##[#a0 al0 H2 H3;ndestruct (H3);
+##|#a0 a1 al0 H2 H3 H4 H5;ndestruct (H5)
+##]
+nqed.
-naxiom in_list_cons_case : \forall A,x,a,l.in_list A x (a::l) \to
+nlemma in_list_cons_case : \forall A,x,a,l.in_list A x (a::l) \to
x = a \lor in_list A x l.
-(*intros;inversion H;intros
- [destruct H2;left;reflexivity
- |destruct H4;right;assumption]
-qed.*)
-
-naxiom in_list_tail : \forall l,x,y.
- in_list nat x (y::l) \to x \neq y \to in_list nat x l.
-(*intros 4;elim (in_list_cons_case ? ? ? ? H)
- [elim (H2 H1)
- |assumption]
-qed.*)
-
-naxiom in_list_singleton_to_eq : \forall A,x,y.in_list A x [y] \to x = y.
-(*intros;elim (in_list_cons_case ? ? ? ? H)
- [assumption
- |elim (not_in_list_nil ? ? H1)]
-qed.*)
-
-naxiom in_list_to_in_list_append_l: \forall A.\forall x:A.
+#A a0 a1 al0 H1;ninversion H1
+##[#a2 al1 H2 H3;ndestruct (H3);@;@
+##|#a2 a3 al1 H2 H3 H4 H5;ndestruct (H5);@2;//
+##]
+nqed.
+
+nlemma in_list_tail : \forall A,l,x,y.
+ in_list A x (y::l) \to x \neq y \to in_list A x l.
+#A;#l;#x;#y;#H;#Hneq;
+ninversion H;
+##[#x1;#l1;#Hx;#Hl;ndestruct;nelim Hneq;#Hfalse;
+ nelim (Hfalse ?);@;
+##|#x1;#y1;#l1;#H1;#_;#Hx;#Heq;ndestruct;//;
+##]
+nqed.
+
+nlemma in_list_singleton_to_eq : \forall A,x,y.in_list A x [y] \to x = y.
+#A a0 a1 H1;ncases (in_list_cons_case ???? H1)
+##[//
+##|#H2;napply False_ind;ncases (not_in_list_nil ? a0);#H3;/2/
+##]
+nqed.
+
+nlemma in_list_to_in_list_append_l: \forall A.\forall x:A.
\forall l1,l2.in_list ? x l1 \to in_list ? x (l1@l2).
-(*intros.
-elim H;simplify
- [apply in_list_head
- |apply in_list_cons;assumption
- ]
-qed.*)
-
-naxiom in_list_to_in_list_append_r: \forall A.\forall x:A.
+#A a0 al0 al1 H1;nelim H1
+##[#a1 al2;@;
+##|#a1 a2 al2 H2 H3;@2;//
+##]
+nqed.
+
+nlemma in_list_to_in_list_append_r: \forall A.\forall x:A.
\forall l1,l2. in_list ? x l2 \to in_list ? x (l1@l2).
-(*intros 3.
-elim l1;simplify
- [assumption
- |apply in_list_cons;apply H;assumption
- ]
-qed.*)
-
-naxiom in_list_append_to_or_in_list: \forall A:Type.\forall x:A.
+#A a0 al0 al1 H1;nelim al0
+##[napply H1
+##|#a1 al2 IH;@2;napply IH
+##]
+nqed.
+
+nlemma in_list_append_to_or_in_list: \forall A:Type.\forall x:A.
\forall l,l1. in_list ? x (l@l1) \to in_list ? x l \lor in_list ? x l1.
-(*intros 3.
-elim l
- [right.apply H
- |simplify in H1.inversion H1;intros; destruct;
- [left.apply in_list_head
- | elim (H l2)
- [left.apply in_list_cons. assumption
- |right.assumption
- |assumption
- ]
- ]
- ]
-qed.*)
+#A a0 al0;nelim al0
+##[#al1 H1;@2;napply H1
+##|#a1 al1 IH al2 H1;nnormalize in H1;
+ ncases (in_list_cons_case ???? H1);#H2
+ ##[@;nrewrite > H2;@
+ ##|ncases (IH … H2);#H3
+ ##[@;@2;//
+ ##|@2;//
+ ##]
+ ##]
+##]
+nqed.
nlet rec mem (A:Type) (eq: A → A → bool) x (l: list A) on l ≝
match l with
]
].
-naxiom mem_true_to_in_list :
+nlemma mem_true_to_in_list :
\forall A,equ.
(\forall x,y.equ x y = true \to x = y) \to
\forall x,l.mem A equ x l = true \to in_list A x l.
-(* intros 5.elim l
- [simplify in H1;destruct H1
- |simplify in H2;apply (bool_elim ? (equ x a))
- [intro;rewrite > (H ? ? H3);apply in_list_head
- |intro;rewrite > H3 in H2;simplify in H2;
- apply in_list_cons;apply H1;assumption]]
-qed.*)
-
-naxiom in_list_to_mem_true :
+#A equ H1 a0 al0;nelim al0
+##[nnormalize;#H2;ndestruct (H2)
+##|#a1 al1 IH H2;nwhd in H2:(??%?);
+ nlapply (refl ? (equ a0 a1));ncases (equ a0 a1) in ⊢ (???% → %);#H3
+ ##[nrewrite > (H1 … H3);@
+ ##|@2;napply IH;nrewrite > H3 in H2;nnormalize;//;
+ ##]
+##]
+nqed.
+
+nlemma in_list_to_mem_true :
\forall A,equ.
(\forall x.equ x x = true) \to
\forall x,l.in_list A x l \to mem A equ x l = true.
-(*intros 5.elim l
- [elim (not_in_list_nil ? ? H1)
- |elim H2
- [simplify;rewrite > H;reflexivity
- |simplify;rewrite > H4;apply (bool_elim ? (equ a1 a2));intro;reflexivity]].
-qed.*)
-
-naxiom in_list_filter_to_p_true : \forall A,l,x,p.
+#A equ H1 a0 al0;nelim al0
+##[#H2;napply False_ind;ncases (not_in_list_nil ? a0);/2/
+##|#a1 al1 IH H2;nelim H2
+ ##[nnormalize;#a2 al2;nrewrite > (H1 …);@
+ ##|#a2 a3 al2 H3 H4;nnormalize;ncases (equ a2 a3);nnormalize;//;
+ ##]
+##]
+nqed.
+
+nlemma in_list_filter_to_p_true : \forall A,l,x,p.
in_list A x (filter A l p) \to p x = true.
-(* intros 4;elim l
- [simplify in H;elim (not_in_list_nil ? ? H)
- |simplify in H1;apply (bool_elim ? (p a));intro;rewrite > H2 in H1;
- simplify in H1
- [elim (in_list_cons_case ? ? ? ? H1)
- [rewrite > H3;assumption
- |apply (H H3)]
- |apply (H H1)]]
-qed.*)
-
-naxiom in_list_filter : \forall A,l,p,x.in_list A x (filter A l p) \to in_list A x l.
-(*intros 4;elim l
- [simplify in H;elim (not_in_list_nil ? ? H)
- |simplify in H1;apply (bool_elim ? (p a));intro;rewrite > H2 in H1;
- simplify in H1
- [elim (in_list_cons_case ? ? ? ? H1)
- [rewrite > H3;apply in_list_head
- |apply in_list_cons;apply H;assumption]
- |apply in_list_cons;apply H;assumption]]
-qed.*)
-
-naxiom in_list_filter_r : \forall A,l,p,x.
+#A al0 a0 p;nelim al0
+##[nnormalize;#H1;napply False_ind;ncases (not_in_list_nil ? a0);/2/
+##|#a1 al1 IH H1;nnormalize in H1;nlapply (refl ? (p a1));
+ ngeneralize in match H1;ncases (p a1) in ⊢ (???% -> ???% → %);
+ ##[#H2 H3;ncases (in_list_cons_case ???? H2);#H4
+ ##[nrewrite > H4;//
+ ##|napply (IH H4);
+ ##]
+ ##|#H2 H3;napply (IH H2);
+ ##]
+##]
+nqed.
+
+nlemma in_list_filter : \forall A,l,p,x.in_list A x (filter A l p) \to in_list A x l.
+#A al0 p a0;nelim al0
+##[nnormalize;//;
+##|#a1 al1 IH H1;nnormalize in H1;
+ nlapply (refl ? (p a1));ncases (p a1) in ⊢ (???% → %);#H2
+ ##[nrewrite > H2 in H1;#H1;ncases (in_list_cons_case ???? H1);#H3
+ ##[nrewrite > H3;@
+ ##|@2;napply IH;napply H3
+ ##]
+ ##|@2;napply IH;nrewrite > H2 in H1;#H1;napply H1;
+ ##]
+##]
+nqed.
+
+nlemma in_list_filter_r : \forall A,l,p,x.
in_list A x l \to p x = true \to in_list A x (filter A l p).
-(* intros 4;elim l
- [elim (not_in_list_nil ? ? H)
- |elim (in_list_cons_case ? ? ? ? H1)
- [rewrite < H3;simplify;rewrite > H2;simplify;apply in_list_head
- |simplify;apply (bool_elim ? (p a));intro;simplify;
- [apply in_list_cons;apply H;assumption
- |apply H;assumption]]]
-qed.*)
-
-naxiom incl_A_A: ∀T,A.incl T A A.
-(*intros.unfold incl.intros.assumption.
-qed.*)
-
-naxiom incl_append_l : ∀T,A,B.incl T A (A @ B).
-(*unfold incl; intros;autobatch.
-qed.*)
-
-naxiom incl_append_r : ∀T,A,B.incl T B (A @ B).
-(*unfold incl; intros;autobatch.
-qed.*)
-
-naxiom incl_cons : ∀T,A,B,x.incl T A B → incl T (x::A) (x::B).
-(*unfold incl; intros;elim (in_list_cons_case ? ? ? ? H1);autobatch.
-qed.*)
+#A al0 p a0;nelim al0
+##[#H1;napply False_ind;ncases (not_in_list_nil ? a0);/2/
+##|#a1 al1 IH H1 H2;ncases (in_list_cons_case ???? H1);#H3
+ ##[nnormalize;nrewrite < H3;nrewrite > H2;@
+ ##|nnormalize;ncases (p a1);nnormalize;
+ ##[@2;napply IH;//
+ ##|napply IH;//
+ ##]
+ ##]
+##]
+nqed.
+
+nlemma incl_A_A: ∀T,A.incl T A A.
+#A al0 a0 H1;//;
+nqed.
+nlemma incl_append_l : ∀T,A,B.incl T A (A @ B).
+#A al0 al1 a0 H1;/2/;
+nqed.
+
+nlemma incl_append_r : ∀T,A,B.incl T B (A @ B).
+#A al0 al1 a0 H1;/2/;
+nqed.
+
+nlemma incl_cons : ∀T,A,B,x.incl T A B → incl T (x::A) (x::B).
+#A al0 al1 a0 H1 a1 H2;ncases (in_list_cons_case ???? H2);/2/;
+#H3;@2;napply H1;//;
+nqed.
+
+nlet rec foldl (A,B:Type[0]) (f:A → B → A) (a:A) (l:list B) on l ≝
+ match l with
+ [ nil ⇒ a
+ | cons b bl ⇒ foldl A B f (f a b) bl ].
+
+nlet rec foldl2 (A,B,C:Type[0]) (f:A → B → C → A) (a:A) (bl:list B) (cl:list C) on bl ≝
+ match bl with
+ [ nil ⇒ a
+ | cons b0 bl0 ⇒ match cl with
+ [ nil ⇒ a
+ | cons c0 cl0 ⇒ foldl2 A B C f (f a b0 c0) bl0 cl0 ] ].
+
+nlet rec foldr2 (A,B : Type[0]) (X : Type[0]) (f: A → B → X → X) (x:X)
+ (al : list A) (bl : list B) on al : X ≝
+ match al with
+ [ nil ⇒ x
+ | cons a al1 ⇒ match bl with
+ [ nil ⇒ x
+ | cons b bl1 ⇒ f a b (foldr2 ??? f x al1 bl1) ] ].
+
+nlet rec rev (A:Type[0]) (l:list A) on l ≝
+ match l with
+ [ nil ⇒ nil A
+ | cons hd tl ⇒ (rev A tl)@[hd] ].
+
+notation > "hvbox(a break \liff b)"
+ left associative with precedence 25
+for @{ 'iff $a $b }.
+
+notation "hvbox(a break \leftrightarrow b)"
+ left associative with precedence 25
+for @{ 'iff $a $b }.
+
+interpretation "logical iff" 'iff x y = (iff x y).
+
+ndefinition coincl : ∀A.list A → list A → Prop ≝ λA,l1,l2.∀x.x ∈ l1 ↔ x ∈ l2.
+
+notation > "hvbox(a break ≡ b)"
+ non associative with precedence 45
+for @{'equiv $a $b}.
+
+notation < "hvbox(term 46 a break ≡ term 46 b)"
+ non associative with precedence 45
+for @{'equiv $a $b}.
+
+interpretation "list coinclusion" 'equiv x y = (coincl ? x y).
+
+nlemma refl_coincl : ∀A.∀l:list A.l ≡ l.
+#;@;#;//;
+nqed.
+
+nlemma coincl_rev : ∀A.∀l:list A.l ≡ rev ? l.
+#A l x;@;nelim l
+##[##1,3:#H;napply False_ind;ncases (not_in_list_nil ? x);
+ #H1;napply (H1 H);
+##|#a l0 IH H;ncases (in_list_cons_case ???? H);#H1
+ ##[napply in_list_to_in_list_append_r;nrewrite > H1;@
+ ##|napply in_list_to_in_list_append_l;/2/
+ ##]
+##|#a l0 IH H;ncases (in_list_append_to_or_in_list ???? H);#H1
+ ##[/3/;
+ ##|nrewrite > (in_list_singleton_to_eq ??? H1);@
+ ##]
+##]
+nqed.
+
+nlemma not_in_list_nil_r : ∀A.∀l:list A.l = [] → ∀x.x ∉ l.
+#A l;nelim l
+##[#;napply not_in_list_nil
+##|#a l0 IH Hfalse;ndestruct (Hfalse)
+##]
+nqed.
+
+nlemma eq_filter_append :
+ ∀A,p,l1,l2.filter A (l1@l2) p = filter A l1 p@filter A l2 p.
+#A p l1 l2;nelim l1
+##[@
+##|#a0 l0 IH;nwhd in ⊢ (??%(??%?));ncases (p a0)
+ ##[nwhd in ⊢ (??%%);nrewrite < IH;@
+ ##|nwhd in ⊢ (??%(??%?));nrewrite < IH;@
+ ##]
+##]
+nqed.
+
+nlemma map_ind :
+ ∀A,B:Type[0].∀f:A→B.∀P:B → Prop.
+ ∀al.(∀a.a ∈ al → P (f a)) →
+ ∀b. b ∈ map ?? f al → P b.
+#A B f P al;nelim al
+##[#H1 b Hfalse;napply False_ind;
+ ncases (not_in_list_nil ? b);#H2;napply H2;napply Hfalse;
+##|#a1 al1 IH H1 b Hin;nwhd in Hin:(???%);ncases (in_list_cons_case ???? Hin);
+ ##[#e;nrewrite > e;napply H1;@
+ ##|#Hin1;napply IH;
+ ##[#a2 Hin2;napply H1;@2;//;
+ ##|//
+ ##]
+ ##]
+##]
+nqed.
+
+nlemma map_compose :
+ ∀A,B,C,f,g,l.map B C f (map A B g l) = map A C (λx.f (g x)) l.
+#A B C f g l;nelim l
+##[@
+##|#a0 al0 IH;nchange in ⊢ (??%%) with (cons ???);
+ napply eq_f2; //;
+##]
+nqed.
+
+nlemma incl_incl_to_incl_append :
+ ∀A.∀l1,l2,l1',l2':list A.l1 ⊆ l1' → l2 ⊆ l2' → l1@l2 ⊆ l1'@l2'.
+#A al0 al1 al2 al3 H1 H2 a0 H3;
+ncases (in_list_append_to_or_in_list ???? H3);#H4;
+##[napply in_list_to_in_list_append_l;napply H1;//
+##|napply in_list_to_in_list_append_r;napply H2;//
+##]
+nqed.
+
+nlemma eq_map_append :
+ ∀A,B,f,l1,l2.map A B f (l1@l2) = map A B f l1@map A B f l2.
+#A B f al1 al2;nelim al1
+##[@
+##|#a0 al3 IH;nnormalize;nrewrite > IH;@;
+##]
+nqed.
+
+nlemma not_in_list_to_mem_false :
+ ∀A,equ.
+ (∀x,y.equ x y = true → x = y) →
+ ∀x:A.∀l. x ∉ l → mem A equ x l = false.
+#A equ H1 a0 al0;nelim al0
+##[#_;@
+##|#a1 al1 IH H2;nwhd in ⊢ (??%?);
+ nlapply (refl ? (equ a0 a1));ncases (equ a0 a1) in ⊢ (???% → %);#H3;
+ ##[napply False_ind;ncases H2;#H4;napply H4;
+ nrewrite > (H1 … H3);@
+ ##|napply IH;@;#H4;ncases H2;#H5;napply H5;@2;//
+ ##]
+##]
+nqed.
+
+nlet rec list_forall (A:Type[0]) (l:list A) (p:A → bool) on l : bool ≝
+ match l with
+ [ nil ⇒ (true:bool)
+ | cons a al ⇒ p a ∧ list_forall A al p ].
+
+nlemma eq_map_f_g :
+ ∀A,B,f,g,xl.(∀x.x ∈ xl → f x = g x) → map A B f xl = map A B g xl.
+#A B f g xl;nelim xl
+##[#;@
+##|#a al IH H1;nwhd in ⊢ (??%%);napply eq_f2
+ ##[napply H1;@;
+ ##|napply IH;#x Hx;napply H1;@2;//
+ ##]
+##]
+nqed.
+
+nlemma x_in_map_to_eq :
+ ∀A,B,f,x,l.x ∈ map A B f l → ∃x'.x = f x' ∧ x' ∈ l.
+#A B f x l;nelim l
+##[#H;ncases (not_in_list_nil ? x);#H1;napply False_ind;napply (H1 H)
+##|#a l0 IH H;ncases (in_list_cons_case ???? H);#H1
+ ##[nrewrite > H1;@ a;@;@
+ ##|ncases (IH H1);#a0;*;#H2 H3;@a0;@
+ ##[// ##|@2;// ##]
+ ##]
+##]
+nqed.
+
+nlemma list_forall_false :
+ ∀A:Type[0].∀x,xl,p. p x = false → x ∈ xl → list_forall A xl p = false.
+#A x xl p H1;nelim xl
+##[#Hfalse;napply False_ind;ncases (not_in_list_nil ? x);#H2;napply (H2 Hfalse)
+##|#x0 xl0 IH H2;ncases (in_list_cons_case ???? H2);#H3
+ ##[nwhd in ⊢ (??%?);nrewrite < H3;nrewrite > H1;@
+ ##|nwhd in ⊢ (??%?);ncases (p x0)
+ ##[nrewrite > (IH H3);@
+ ##|@
+ ##]
+ ##]
+##]
+nqed.
+
+nlemma list_forall_true :
+ ∀A:Type[0].∀xl,p. (∀x.x ∈ xl → p x = true) → list_forall A xl p = true.
+#A xl p;nelim xl
+##[#;@
+##|#x0 xl0 IH H1;nwhd in ⊢ (??%?);nrewrite > (H1 …)
+ ##[napply IH;#x Hx;napply H1;@2;//
+ ##|@
+ ##]
+##]
+nqed.
projects / helm.git / blobdiff