Theory "list"

Signature

Definitions

SUM_ACC_DEF

|- (∀acc. SUM_ACC [] acc = acc) ∧
   ∀h t acc. SUM_ACC (h::t) acc = SUM_ACC t (h + acc)

REV_DEF

|- (∀acc. REV [] acc = acc) ∧ ∀h t acc. REV (h::t) acc = REV t (h::acc)

LEN_DEF

|- (∀n. LEN [] n = n) ∧ ∀h t n. LEN (h::t) n = LEN t (n + 1)

PAD_RIGHT

|- ∀c n s. PAD_RIGHT c n s = s ++ GENLIST (K c) (n − LENGTH s)

PAD_LEFT

|- ∀c n s. PAD_LEFT c n s = GENLIST (K c) (n − LENGTH s) ++ s

GENLIST_AUX

|- (∀f l. GENLIST_AUX f 0 l = l) ∧
   ∀f n l. GENLIST_AUX f (SUC n) l = GENLIST_AUX f n (f n::l)

GENLIST

|- (∀f. GENLIST f 0 = []) ∧ ∀f n. GENLIST f (SUC n) = SNOC (f n) (GENLIST f n)

SNOC

|- (∀x. SNOC x [] = [x]) ∧ ∀x x' l. SNOC x (x'::l) = x'::SNOC x l

isPREFIX

|- (∀l. [] ≼ l ⇔ T) ∧
   ∀h t l. h::t ≼ l ⇔ case l of [] => F | h'::t' => (h = h') ∧ t ≼ t'

SET_TO_LIST_primitive

|- SET_TO_LIST =
   WFREC (@R. WF R ∧ ∀s. FINITE s ∧ s ≠ ∅ ⇒ R (REST s) s)
     (λSET_TO_LIST s.
        I
          (if FINITE s then
             if s = ∅ then [] else CHOICE s::SET_TO_LIST (REST s)
           else ARB))

LRC_def

|- (∀R x y. LRC R [] x y ⇔ (x = y)) ∧
   ∀R h t x y. LRC R (h::t) x y ⇔ (x = h) ∧ ∃z. R x z ∧ LRC R t z y

ALL_DISTINCT

|- (ALL_DISTINCT [] ⇔ T) ∧
   ∀h t. ALL_DISTINCT (h::t) ⇔ ¬MEM h t ∧ ALL_DISTINCT t

DROP_def

|- (∀n. DROP n [] = []) ∧
   ∀n x xs. DROP n (x::xs) = if n = 0 then x::xs else DROP (n − 1) xs

TAKE_def

|- (∀n. TAKE n [] = []) ∧
   ∀n x xs. TAKE n (x::xs) = if n = 0 then [] else x::TAKE (n − 1) xs

FRONT_DEF

|- ∀h t. FRONT (h::t) = if t = [] then [] else h::FRONT t

LAST_DEF

|- ∀h t. LAST (h::t) = if t = [] then h else LAST t

REVERSE_DEF

|- (REVERSE [] = []) ∧ ∀h t. REVERSE (h::t) = REVERSE t ++ [h]

UNZIP

|- (UNZIP [] = ([],[])) ∧
   ∀x l. UNZIP (x::l) = (FST x::FST (UNZIP l),SND x::SND (UNZIP l))

ZIP

|- (ZIP ([],[]) = []) ∧
   ∀x1 l1 x2 l2. ZIP (x1::l1,x2::l2) = (x1,x2)::ZIP (l1,l2)

list_size_def

|- (∀f. list_size f [] = 0) ∧
   ∀f a0 a1. list_size f (a0::a1) = 1 + (f a0 + list_size f a1)

list_case_def

|- (∀v f. list_CASE [] v f = v) ∧ ∀a0 a1 v f. list_CASE (a0::a1) v f = f a0 a1

list_TY_DEF

|- ∃rep.
     TYPE_DEFINITION
       (λa0'.
          ∀'list' .
            (∀a0'.
               (a0' = ind_type$CONSTR 0 ARB (λn. ind_type$BOTTOM)) ∨
               (∃a0 a1.
                  (a0' =
                   (λa0 a1.
                      ind_type$CONSTR (SUC 0) a0
                        (ind_type$FCONS a1 (λn. ind_type$BOTTOM))) a0 a1) ∧
                  'list' a1) ⇒
               'list' a0') ⇒
            'list' a0') rep

NULL_DEF

|- (NULL [] ⇔ T) ∧ ∀h t. NULL (h::t) ⇔ F

HD

|- ∀h t. HD (h::t) = h

TL

|- ∀h t. TL (h::t) = t

SUM

|- (SUM [] = 0) ∧ ∀h t. SUM (h::t) = h + SUM t

APPEND

|- (∀l. [] ++ l = l) ∧ ∀l1 l2 h. h::l1 ++ l2 = h::(l1 ++ l2)

FLAT

|- (FLAT [] = []) ∧ ∀h t. FLAT (h::t) = h ++ FLAT t

LENGTH

|- (LENGTH [] = 0) ∧ ∀h t. LENGTH (h::t) = SUC (LENGTH t)

MAP

|- (∀f. MAP f [] = []) ∧ ∀f h t. MAP f (h::t) = f h::MAP f t

LIST_TO_SET_DEF

|- (∀x. LIST_TO_SET [] x ⇔ F) ∧
   ∀h t x. LIST_TO_SET (h::t) x ⇔ (x = h) ∨ LIST_TO_SET t x

FILTER

|- (∀P. FILTER P [] = []) ∧
   ∀P h t. FILTER P (h::t) = if P h then h::FILTER P t else FILTER P t

FOLDR

|- (∀f e. FOLDR f e [] = e) ∧ ∀f e x l. FOLDR f e (x::l) = f x (FOLDR f e l)

FOLDL

|- (∀f e. FOLDL f e [] = e) ∧ ∀f e x l. FOLDL f e (x::l) = FOLDL f (f e x) l

EVERY_DEF

|- (∀P. EVERY P [] ⇔ T) ∧ ∀P h t. EVERY P (h::t) ⇔ P h ∧ EVERY P t

EXISTS_DEF

|- (∀P. EXISTS P [] ⇔ F) ∧ ∀P h t. EXISTS P (h::t) ⇔ P h ∨ EXISTS P t

EL

|- (∀l. EL 0 l = HD l) ∧ ∀l n. EL (SUC n) l = EL n (TL l)

INDEX_FIND_def

|- (∀i P. INDEX_FIND i P [] = NONE) ∧
   ∀i P h t.
     INDEX_FIND i P (h::t) =
     if P h then SOME (i,h) else INDEX_FIND (SUC i) P t

FIND_def

|- ∀P. FIND P = OPTION_MAP SND o INDEX_FIND 0 P

INDEX_OF_def

|- ∀x. INDEX_OF x = OPTION_MAP FST o INDEX_FIND 0 ($= x)

LUPDATE_def

|- (∀e n. LUPDATE e n [] = []) ∧ (∀e x l. LUPDATE e 0 (x::l) = e::l) ∧
   ∀e n x l. LUPDATE e (SUC n) (x::l) = x::LUPDATE e n l

EVERYi_DEF

|- (∀P. EVERYi P [] ⇔ T) ∧
   ∀P h t. EVERYi P (h::t) ⇔ P 0 h ∧ EVERYi (P o SUC) t

splitAtPki_DEF

|- (∀P k. splitAtPki P k [] = k [] []) ∧
   ∀P k h t.
     splitAtPki P k (h::t) =
     if P 0 h then k [] (h::t) else splitAtPki (P o SUC) (λp s. k (h::p) s) t

LIST_BIND_DEF

|- ∀l f. LIST_BIND l f = FLAT (MAP f l)

LIST_IGNORE_BIND_DEF

|- ∀m1 m2. LIST_IGNORE_BIND m1 m2 = LIST_BIND m1 (K m2)

LIST_APPLY_DEF

|- ∀fs xs. fs <*> xs = LIST_BIND fs (combin$C MAP xs)

LIST_LIFT2_DEF

|- ∀f xs ys. LIST_LIFT2 f xs ys = MAP f xs <*> ys

LLEX_DEF

|- (∀R l2. LLEX R [] l2 ⇔ l2 ≠ []) ∧
   ∀R h1 t1 l2.
     LLEX R (h1::t1) l2 ⇔
     case l2 of
       [] => F
     | h2::t2 => if R h1 h2 then T else if h1 = h2 then LLEX R t1 t2 else F

nub_def

|- (nub [] = []) ∧ ∀x l. nub (x::l) = if MEM x l then nub l else x::nub l

dropWhile_def

|- (∀P. dropWhile P [] = []) ∧
   ∀P h t. dropWhile P (h::t) = if P h then dropWhile P t else h::t

Theorems

EXISTS_LIST

|- (∃l. P l) ⇔ P [] ∨ ∃h t. P (h::t)

SUM_SUM_ACC

|- ∀L. SUM L = SUM_ACC L 0

SUM_ACC_SUM_LEM

|- ∀L n. SUM_ACC L n = SUM L + n

REVERSE_REV

|- ∀L. REVERSE L = REV L []

LENGTH_LEN

|- ∀L. LENGTH L = LEN L 0

REV_REVERSE_LEM

|- ∀L1 L2. REV L1 L2 = REVERSE L1 ++ L2

LEN_LENGTH_LEM

|- ∀L n. LEN L n = LENGTH L + n

INFINITE_LIST_UNIV

|- INFINITE 𝕌(:α list)

MAP_ZIP_SAME

|- ∀ls f. MAP f (ZIP (ls,ls)) = MAP (λx. f (x,x)) ls

FOLDL_ZIP_SAME

|- ∀ls f e. FOLDL f e (ZIP (ls,ls)) = FOLDL (λx y. f x (y,y)) e ls

FOLDL_UNION_BIGUNION_paired

|- ∀f ls s.
     FOLDL (λs (x,y). s ∪ f x y) s ls =
     s ∪ BIGUNION (IMAGE (UNCURRY f) (LIST_TO_SET ls))

FOLDL_UNION_BIGUNION

|- ∀f ls s.
     FOLDL (λs x. s ∪ f x) s ls = s ∪ BIGUNION (IMAGE f (LIST_TO_SET ls))

REVERSE_GENLIST

|- REVERSE (GENLIST f n) = GENLIST (λm. f (PRE n − m)) n

EL_REVERSE

|- ∀n l. n < LENGTH l ⇒ (EL n (REVERSE l) = EL (PRE (LENGTH l − n)) l)

SUM_IMAGE_eq_SUM_MAP_SET_TO_LIST

|- FINITE s ⇒ (∑ f s = SUM (MAP f (SET_TO_LIST s)))

SUM_MAP_FOLDL

|- ∀ls. SUM (MAP f ls) = FOLDL (λa e. a + f e) 0 ls

SUM_APPEND

|- ∀l1 l2. SUM (l1 ++ l2) = SUM l1 + SUM l2

SUM_SNOC

|- ∀x l. SUM (SNOC x l) = SUM l + x

FOLDL_SNOC

|- ∀f e x l. FOLDL f e (SNOC x l) = f (FOLDL f e l) x

ALL_DISTINCT_GENLIST

|- ALL_DISTINCT (GENLIST f n) ⇔
   ∀m1 m2. m1 < n ∧ m2 < n ∧ (f m1 = f m2) ⇒ (m1 = m2)

ALL_DISTINCT_SNOC

|- ∀x l. ALL_DISTINCT (SNOC x l) ⇔ ¬MEM x l ∧ ALL_DISTINCT l

MEM_GENLIST

|- MEM x (GENLIST f n) ⇔ ∃m. m < n ∧ (x = f m)

GENLIST_NUMERALS

|- (GENLIST f 0 = []) ∧ (GENLIST f (NUMERAL n) = GENLIST_AUX f (NUMERAL n) [])

GENLIST_GENLIST_AUX

|- ∀n. GENLIST f n = GENLIST_AUX f n []

NULL_GENLIST

|- ∀n f. NULL (GENLIST f n) ⇔ (n = 0)

GENLIST_CONS

|- GENLIST f (SUC n) = f 0::GENLIST (f o SUC) n

ZIP_GENLIST

|- ∀l f n.
     (LENGTH l = n) ⇒ (ZIP (l,GENLIST f n) = GENLIST (λx. (EL x l,f x)) n)

TL_GENLIST

|- ∀f n. TL (GENLIST f (SUC n)) = GENLIST (f o SUC) n

EXISTS_GENLIST

|- ∀n. EXISTS P (GENLIST f n) ⇔ ∃i. i < n ∧ P (f i)

EVERY_GENLIST

|- ∀n. EVERY P (GENLIST f n) ⇔ ∀i. i < n ⇒ P (f i)

GENLIST_APPEND

|- ∀f a b. GENLIST f (a + b) = GENLIST f b ++ GENLIST (λt. f (t + b)) a

GENLIST_FUN_EQ

|- ∀n f g. (GENLIST f n = GENLIST g n) ⇔ ∀x. x < n ⇒ (f x = g x)

HD_GENLIST_COR

|- ∀n f. 0 < n ⇒ (HD (GENLIST f n) = f 0)

HD_GENLIST

|- HD (GENLIST f (SUC n)) = f 0

EL_GENLIST

|- ∀f n x. x < n ⇒ (EL x (GENLIST f n) = f x)

MAP_GENLIST

|- ∀f g n. MAP f (GENLIST g n) = GENLIST (f o g) n

GENLIST_AUX_compute

|- (∀f l. GENLIST_AUX f 0 l = l) ∧
   (∀f n l.
      GENLIST_AUX f (NUMERAL (BIT1 n)) l =
      GENLIST_AUX f (NUMERAL (BIT1 n) − 1) (f (NUMERAL (BIT1 n) − 1)::l)) ∧
   ∀f n l.
     GENLIST_AUX f (NUMERAL (BIT2 n)) l =
     GENLIST_AUX f (NUMERAL (BIT1 n)) (f (NUMERAL (BIT1 n))::l)

LENGTH_GENLIST

|- ∀f n. LENGTH (GENLIST f n) = n

SNOC_CASES

|- ∀ll. (ll = []) ∨ ∃x l. ll = SNOC x l

SNOC_INDUCT

|- ∀P. P [] ∧ (∀l. P l ⇒ ∀x. P (SNOC x l)) ⇒ ∀l. P l

SNOC_Axiom

|- ∀e f. ∃fn. (fn [] = e) ∧ ∀x l. fn (SNOC x l) = f x l (fn l)

REVERSE_SNOC

|- ∀x l. REVERSE (SNOC x l) = x::REVERSE l

REVERSE_SNOC_DEF

|- (REVERSE [] = []) ∧ ∀x l. REVERSE (x::l) = SNOC x (REVERSE l)

SNOC_11

|- ∀x y a b. (SNOC x y = SNOC a b) ⇔ (x = a) ∧ (y = b)

MEM_SNOC

|- ∀y x l. MEM y (SNOC x l) ⇔ (y = x) ∨ MEM y l

EXISTS_SNOC

|- ∀P x l. EXISTS P (SNOC x l) ⇔ P x ∨ EXISTS P l

EVERY_SNOC

|- ∀P x l. EVERY P (SNOC x l) ⇔ EVERY P l ∧ P x

APPEND_SNOC

|- ∀l1 x l2. l1 ++ SNOC x l2 = SNOC x (l1 ++ l2)

EL_LENGTH_SNOC

|- ∀l x. EL (LENGTH l) (SNOC x l) = x

EL_SNOC

|- ∀n l. n < LENGTH l ⇒ ∀x. EL n (SNOC x l) = EL n l

MAP_SNOC

|- ∀f x l. MAP f (SNOC x l) = SNOC (f x) (MAP f l)

LIST_TO_SET_SNOC

|- LIST_TO_SET (SNOC x ls) = x INSERT LIST_TO_SET ls

SNOC_APPEND

|- ∀x l. SNOC x l = l ++ [x]

FRONT_SNOC

|- ∀x l. FRONT (SNOC x l) = l

LAST_SNOC

|- ∀x l. LAST (SNOC x l) = x

LENGTH_SNOC

|- ∀x l. LENGTH (SNOC x l) = SUC (LENGTH l)

isPREFIX_THM

|- ([] ≼ l ⇔ T) ∧ (h::t ≼ [] ⇔ F) ∧ (h1::t1 ≼ h2::t2 ⇔ (h1 = h2) ∧ t1 ≼ t2)

ITSET_eq_FOLDL_SET_TO_LIST

|- ∀s. FINITE s ⇒ ∀f a. ITSET f s a = FOLDL (combin$C f) a (SET_TO_LIST s)

ALL_DISTINCT_SET_TO_LIST

|- ∀s. FINITE s ⇒ ALL_DISTINCT (SET_TO_LIST s)

SET_TO_LIST_SING

|- SET_TO_LIST {x} = [x]

MEM_SET_TO_LIST

|- ∀s. FINITE s ⇒ ∀x. MEM x (SET_TO_LIST s) ⇔ x ∈ s

SET_TO_LIST_IN_MEM

|- ∀s. FINITE s ⇒ ∀x. x ∈ s ⇔ MEM x (SET_TO_LIST s)

SET_TO_LIST_CARD

|- ∀s. FINITE s ⇒ (LENGTH (SET_TO_LIST s) = CARD s)

SET_TO_LIST_INV

|- ∀s. FINITE s ⇒ (LIST_TO_SET (SET_TO_LIST s) = s)

SET_TO_LIST_EMPTY

|- SET_TO_LIST ∅ = []

SET_TO_LIST_IND

|- ∀P. (∀s. (FINITE s ∧ s ≠ ∅ ⇒ P (REST s)) ⇒ P s) ⇒ ∀v. P v

SET_TO_LIST_THM

|- FINITE s ⇒
   (SET_TO_LIST s = if s = ∅ then [] else CHOICE s::SET_TO_LIST (REST s))

LIST_TO_SET_FILTER

|- LIST_TO_SET (FILTER P l) = {x | P x} ∩ LIST_TO_SET l

LIST_TO_SET_MAP

|- ∀f l. LIST_TO_SET (MAP f l) = IMAGE f (LIST_TO_SET l)

LIST_TO_SET_THM

|- (LIST_TO_SET [] = ∅) ∧ (LIST_TO_SET (h::t) = h INSERT LIST_TO_SET t)

LIST_TO_SET_REVERSE

|- ∀ls. LIST_TO_SET (REVERSE ls) = LIST_TO_SET ls

ALL_DISTINCT_CARD_LIST_TO_SET

|- ∀ls. ALL_DISTINCT ls ⇒ (CARD (LIST_TO_SET ls) = LENGTH ls)

CARD_LIST_TO_SET

|- CARD (LIST_TO_SET ls) ≤ LENGTH ls

INJ_MAP_EQ

|- ∀f l1 l2.
     INJ f (LIST_TO_SET l1 ∪ LIST_TO_SET l2) 𝕌(:β) ∧ (MAP f l1 = MAP f l2) ⇒
     (l1 = l2)

SUM_MAP_MEM_bound

|- ∀f x ls. MEM x ls ⇒ f x ≤ SUM (MAP f ls)

SUM_IMAGE_LIST_TO_SET_upper_bound

|- ∀ls. ∑ f (LIST_TO_SET ls) ≤ SUM (MAP f ls)

FINITE_LIST_TO_SET

|- ∀l. FINITE (LIST_TO_SET l)

LIST_TO_SET_EQ_EMPTY

|- ((LIST_TO_SET l = ∅) ⇔ (l = [])) ∧ ((∅ = LIST_TO_SET l) ⇔ (l = []))

UNION_APPEND

|- ∀l1 l2. LIST_TO_SET l1 ∪ LIST_TO_SET l2 = LIST_TO_SET (l1 ++ l2)

LIST_TO_SET_APPEND

|- ∀l1 l2. LIST_TO_SET (l1 ++ l2) = LIST_TO_SET l1 ∪ LIST_TO_SET l2

LRC_MEM_right

|- LRC R (h::t) x y ∧ MEM e t ⇒ ∃z p. R z e ∧ LRC R p x z

LRC_MEM

|- LRC R ls x y ∧ MEM e ls ⇒ ∃z t. R e z ∧ LRC R t z y

NRC_LRC

|- NRC R n x y ⇔ ∃ls. LRC R ls x y ∧ (LENGTH ls = n)

ALL_DISTINCT_REVERSE

|- ∀l. ALL_DISTINCT (REVERSE l) ⇔ ALL_DISTINCT l

ALL_DISTINCT_ZIP_SWAP

|- ∀l1 l2.
     ALL_DISTINCT (ZIP (l1,l2)) ∧ (LENGTH l1 = LENGTH l2) ⇒
     ALL_DISTINCT (ZIP (l2,l1))

ALL_DISTINCT_ZIP

|- ∀l1 l2.
     ALL_DISTINCT l1 ∧ (LENGTH l1 = LENGTH l2) ⇒ ALL_DISTINCT (ZIP (l1,l2))

ALL_DISTINCT_SING

|- ∀x. ALL_DISTINCT [x]

ALL_DISTINCT_APPEND

|- ∀l1 l2.
     ALL_DISTINCT (l1 ++ l2) ⇔
     ALL_DISTINCT l1 ∧ ALL_DISTINCT l2 ∧ ∀e. MEM e l1 ⇒ ¬MEM e l2

ALL_DISTINCT_EL_IMP

|- ∀l n1 n2.
     ALL_DISTINCT l ∧ n1 < LENGTH l ∧ n2 < LENGTH l ⇒
     ((EL n1 l = EL n2 l) ⇔ (n1 = n2))

EL_ALL_DISTINCT_EL_EQ

|- ∀l.
     ALL_DISTINCT l ⇔
     ∀n1 n2. n1 < LENGTH l ∧ n2 < LENGTH l ⇒ ((EL n1 l = EL n2 l) ⇔ (n1 = n2))

ALL_DISTINCT_MAP

|- ∀f ls. ALL_DISTINCT (MAP f ls) ⇒ ALL_DISTINCT ls

FILTER_ALL_DISTINCT

|- ∀P l. ALL_DISTINCT l ⇒ ALL_DISTINCT (FILTER P l)

ALL_DISTINCT_FILTER

|- ∀l. ALL_DISTINCT l ⇔ ∀x. MEM x l ⇒ (FILTER ($= x) l = [x])

EVERY2_mono

|- (∀x y. R1 x y ⇒ R2 x y) ⇒ LIST_REL R1 l1 l2 ⇒ LIST_REL R2 l1 l2

EVERY2_LENGTH

|- ∀P l1 l2. LIST_REL P l1 l2 ⇒ (LENGTH l1 = LENGTH l2)

EVERY2_EVERY

|- ∀l1 l2 f.
     LIST_REL f l1 l2 ⇔
     (LENGTH l1 = LENGTH l2) ∧ EVERY (UNCURRY f) (ZIP (l1,l2))

MAP_EQ_EVERY2

|- ∀f1 f2 l1 l2.
     (MAP f1 l1 = MAP f2 l2) ⇔
     (LENGTH l1 = LENGTH l2) ∧ LIST_REL (λx y. f1 x = f2 y) l1 l2

EVERY2_cong

|- ∀l1 l1' l2 l2' P P'.
     (l1 = l1') ∧ (l2 = l2') ∧
     (∀x y. MEM x l1' ∧ MEM y l2' ⇒ (P x y ⇔ P' x y)) ⇒
     (LIST_REL P l1 l2 ⇔ LIST_REL P' l1' l2')

FOLDL2_FOLDL

|- ∀l1 l2.
     (LENGTH l1 = LENGTH l2) ⇒
     ∀f a. FOLDL2 f a l1 l2 = FOLDL (λa. UNCURRY (f a)) a (ZIP (l1,l2))

FOLDL2_cong

|- ∀l1 l1' l2 l2' a a' f f'.
     (l1 = l1') ∧ (l2 = l2') ∧ (a = a') ∧
     (∀z b c. MEM b l1' ∧ MEM c l2' ⇒ (f z b c = f' z b c)) ⇒
     (FOLDL2 f a l1 l2 = FOLDL2 f' a' l1' l2')

FOLDL2_def

|- (∀f cs c bs b a. FOLDL2 f a (b::bs) (c::cs) = FOLDL2 f (f a b c) bs cs) ∧
   (∀f cs a. FOLDL2 f a [] cs = a) ∧ ∀v7 v6 f a. FOLDL2 f a (v6::v7) [] = a

FOLDL2_ind

|- ∀P.
     (∀f a b bs c cs. P f (f a b c) bs cs ⇒ P f a (b::bs) (c::cs)) ∧
     (∀f a cs. P f a [] cs) ∧ (∀f a v6 v7. P f a (v6::v7) []) ⇒
     ∀v v1 v2 v3. P v v1 v2 v3

DROP_NIL

|- ∀ls n. (DROP n ls = []) ⇔ n ≥ LENGTH ls

MEM_DROP

|- ∀x ls n.
     MEM x (DROP n ls) ⇔
     n < LENGTH ls ∧ (x = EL n ls) ∨ MEM x (DROP (SUC n) ls)

LENGTH_DROP

|- ∀n l. LENGTH (DROP n l) = LENGTH l − n

TAKE_DROP

|- ∀n l. TAKE n l ++ DROP n l = l

DROP_0

|- DROP 0 l = l

TAKE_APPEND2

|- ∀n. LENGTH l1 < n ⇒ (TAKE n (l1 ++ l2) = l1 ++ TAKE (n − LENGTH l1) l2)

TAKE_APPEND1

|- ∀n. n ≤ LENGTH l1 ⇒ (TAKE n (l1 ++ l2) = TAKE n l1)

LENGTH_TAKE

|- ∀n l. n ≤ LENGTH l ⇒ (LENGTH (TAKE n l) = n)

TAKE_LENGTH_ID

|- ∀l. TAKE (LENGTH l) l = l

TAKE_0

|- TAKE 0 l = []

LAST_APPEND_CONS

|- ∀h l1 l2. LAST (l1 ++ h::l2) = LAST (h::l2)

LAST_CONS_cond

|- LAST (h::t) = if t = [] then h else LAST t

APPEND_FRONT_LAST

|- ∀l. l ≠ [] ⇒ (FRONT l ++ [LAST l] = l)

FRONT_CONS_EQ_NIL

|- (∀x xs. (FRONT (x::xs) = []) ⇔ (xs = [])) ∧
   (∀x xs. ([] = FRONT (x::xs)) ⇔ (xs = [])) ∧
   ∀x xs. NULL (FRONT (x::xs)) ⇔ NULL xs

LENGTH_FRONT_CONS

|- ∀x xs. LENGTH (FRONT (x::xs)) = LENGTH xs

FRONT_CONS

|- (∀x. FRONT [x] = []) ∧ ∀x y z. FRONT (x::y::z) = x::FRONT (y::z)

LAST_EL

|- ∀ls. ls ≠ [] ⇒ (LAST ls = EL (PRE (LENGTH ls)) ls)

LAST_CONS

|- (∀x. LAST [x] = x) ∧ ∀x y z. LAST (x::y::z) = LAST (y::z)

FILTER_REVERSE

|- ∀l P. FILTER P (REVERSE l) = REVERSE (FILTER P l)

REVERSE_EQ_SING

|- (REVERSE l = [e]) ⇔ (l = [e])

REVERSE_EQ_NIL

|- (REVERSE l = []) ⇔ (l = [])

LENGTH_REVERSE

|- ∀l. LENGTH (REVERSE l) = LENGTH l

MEM_REVERSE

|- ∀l x. MEM x (REVERSE l) ⇔ MEM x l

REVERSE_11

|- ∀l1 l2. (REVERSE l1 = REVERSE l2) ⇔ (l1 = l2)

REVERSE_REVERSE

|- ∀l. REVERSE (REVERSE l) = l

REVERSE_APPEND

|- ∀l1 l2. REVERSE (l1 ++ l2) = REVERSE l2 ++ REVERSE l1

LIST_REL_EVERY_ZIP

|- ∀R l1 l2.
     LIST_REL R l1 l2 ⇔
     (LENGTH l1 = LENGTH l2) ∧ EVERY (UNCURRY R) (ZIP (l1,l2))

SUM_MAP_PLUS_ZIP

|- ∀ls1 ls2.
     (LENGTH ls1 = LENGTH ls2) ∧ (∀x y. f (x,y) = g x + h y) ⇒
     (SUM (MAP f (ZIP (ls1,ls2))) = SUM (MAP g ls1) + SUM (MAP h ls2))

MEM_EL

|- ∀l x. MEM x l ⇔ ∃n. n < LENGTH l ∧ (x = EL n l)

MAP_ZIP

|- (LENGTH l1 = LENGTH l2) ⇒
   (MAP FST (ZIP (l1,l2)) = l1) ∧ (MAP SND (ZIP (l1,l2)) = l2) ∧
   (MAP (f o FST) (ZIP (l1,l2)) = MAP f l1) ∧
   (MAP (g o SND) (ZIP (l1,l2)) = MAP g l2)

MAP2_MAP

|- ∀l1 l2.
     (LENGTH l1 = LENGTH l2) ⇒
     ∀f. MAP2 f l1 l2 = MAP (UNCURRY f) (ZIP (l1,l2))

MAP2_ZIP

|- ∀l1 l2.
     (LENGTH l1 = LENGTH l2) ⇒
     ∀f. MAP2 f l1 l2 = MAP (UNCURRY f) (ZIP (l1,l2))

EL_ZIP

|- ∀l1 l2 n.
     (LENGTH l1 = LENGTH l2) ∧ n < LENGTH l1 ⇒
     (EL n (ZIP (l1,l2)) = (EL n l1,EL n l2))

MEM_ZIP

|- ∀l1 l2 p.
     (LENGTH l1 = LENGTH l2) ⇒
     (MEM p (ZIP (l1,l2)) ⇔ ∃n. n < LENGTH l1 ∧ (p = (EL n l1,EL n l2)))

ZIP_MAP

|- ∀l1 l2 f1 f2.
     (LENGTH l1 = LENGTH l2) ⇒
     (ZIP (MAP f1 l1,l2) = MAP (λp. (f1 (FST p),SND p)) (ZIP (l1,l2))) ∧
     (ZIP (l1,MAP f2 l2) = MAP (λp. (FST p,f2 (SND p))) (ZIP (l1,l2)))

UNZIP_ZIP

|- ∀l1 l2. (LENGTH l1 = LENGTH l2) ⇒ (UNZIP (ZIP (l1,l2)) = (l1,l2))

ZIP_UNZIP

|- ∀l. ZIP (UNZIP l) = l

LENGTH_UNZIP

|- ∀pl.
     (LENGTH (FST (UNZIP pl)) = LENGTH pl) ∧
     (LENGTH (SND (UNZIP pl)) = LENGTH pl)

LENGTH_ZIP

|- ∀l1 l2.
     (LENGTH l1 = LENGTH l2) ⇒
     (LENGTH (ZIP (l1,l2)) = LENGTH l1) ∧ (LENGTH (ZIP (l1,l2)) = LENGTH l2)

UNZIP_MAP

|- ∀L. UNZIP L = (MAP FST L,MAP SND L)

UNZIP_THM

|- (UNZIP [] = ([],[])) ∧
   (UNZIP ((x,y)::t) = (let (L1,L2) = UNZIP t in (x::L1,y::L2)))

EVERY_MONOTONIC

|- ∀P Q. (∀x. P x ⇒ Q x) ⇒ ∀l. EVERY P l ⇒ EVERY Q l

EVERY_CONG

|- ∀l1 l2 P P'.
     (l1 = l2) ∧ (∀x. MEM x l2 ⇒ (P x ⇔ P' x)) ⇒ (EVERY P l1 ⇔ EVERY P' l2)

EXISTS_CONG

|- ∀l1 l2 P P'.
     (l1 = l2) ∧ (∀x. MEM x l2 ⇒ (P x ⇔ P' x)) ⇒ (EXISTS P l1 ⇔ EXISTS P' l2)

MAP2_CONG

|- ∀l1 l1' l2 l2' f f'.
     (l1 = l1') ∧ (l2 = l2') ∧
     (∀x y. MEM x l1' ∧ MEM y l2' ⇒ (f x y = f' x y)) ⇒
     (MAP2 f l1 l2 = MAP2 f' l1' l2')

MAP_CONG

|- ∀l1 l2 f f'.
     (l1 = l2) ∧ (∀x. MEM x l2 ⇒ (f x = f' x)) ⇒ (MAP f l1 = MAP f' l2)

FOLDL_CONG

|- ∀l l' b b' f f'.
     (l = l') ∧ (b = b') ∧ (∀x a. MEM x l' ⇒ (f a x = f' a x)) ⇒
     (FOLDL f b l = FOLDL f' b' l')

FOLDR_CONG

|- ∀l l' b b' f f'.
     (l = l') ∧ (b = b') ∧ (∀x a. MEM x l' ⇒ (f x a = f' x a)) ⇒
     (FOLDR f b l = FOLDR f' b' l')

list_size_cong

|- ∀M N f f'.
     (M = N) ∧ (∀x. MEM x N ⇒ (f x = f' x)) ⇒ (list_size f M = list_size f' N)

LIST_REL_LENGTH

|- ∀x y. LIST_REL R x y ⇒ (LENGTH x = LENGTH y)

LIST_REL_MAP2

|- LIST_REL (λa b. R a b) l1 (MAP f l2) ⇔ LIST_REL (λa b. R a (f b)) l1 l2

LIST_REL_MAP1

|- LIST_REL R (MAP f l1) l2 ⇔ LIST_REL (R o f) l1 l2

LIST_REL_CONJ

|- LIST_REL (λa b. P a b ∧ Q a b) l1 l2 ⇔
   LIST_REL (λa b. P a b) l1 l2 ∧ LIST_REL (λa b. Q a b) l1 l2

LIST_REL_CONS2

|- LIST_REL R xs (h::t) ⇔ ∃h' t'. (xs = h'::t') ∧ R h' h ∧ LIST_REL R t' t

LIST_REL_CONS1

|- LIST_REL R (h::t) xs ⇔ ∃h' t'. (xs = h'::t') ∧ R h h' ∧ LIST_REL R t t'

LIST_REL_NIL

|- (LIST_REL R [] x ⇔ (x = [])) ∧ (LIST_REL R [] y ⇔ (y = []))

LIST_REL_mono

|- (∀x y. R1 x y ⇒ R2 x y) ⇒ LIST_REL R1 l1 l2 ⇒ LIST_REL R2 l1 l2

LIST_REL_def

|- (LIST_REL R [] [] ⇔ T) ∧ (LIST_REL R (a::as) [] ⇔ F) ∧
   (LIST_REL R [] (b::bs) ⇔ F) ∧
   (LIST_REL R (a::as) (b::bs) ⇔ R a b ∧ LIST_REL R as bs)

WF_LIST_PRED

|- WF (λL1 L2. ∃h. L2 = h::L1)

NULL_FILTER

|- ∀P ls. NULL (FILTER P ls) ⇔ ∀x. MEM x ls ⇒ ¬P x

SUM_eq_0

|- ∀ls. (SUM ls = 0) ⇔ ∀x. MEM x ls ⇒ (x = 0)

EL_simp_restricted

|- (EL (NUMERAL (BIT1 n)) (l::ls) = EL (PRE (NUMERAL (BIT1 n))) ls) ∧
   (EL (NUMERAL (BIT2 n)) (l::ls) = EL (NUMERAL (BIT1 n)) ls)

EL_restricted

|- (EL 0 = HD) ∧ (EL (SUC n) (l::ls) = EL n ls)

EL_simp

|- (EL (NUMERAL (BIT1 n)) l = EL (PRE (NUMERAL (BIT1 n))) (TL l)) ∧
   (EL (NUMERAL (BIT2 n)) l = EL (NUMERAL (BIT1 n)) (TL l))

EL_compute

|- ∀n. EL n l = if n = 0 then HD l else EL (PRE n) (TL l)

NOT_NULL_MEM

|- ∀l. ¬NULL l ⇔ ∃e. MEM e l

FILTER_COND_REWRITE

|- (FILTER P [] = []) ∧ (∀h. P h ⇒ (FILTER P (h::l) = h::FILTER P l)) ∧
   ∀h. ¬P h ⇒ (FILTER P (h::l) = FILTER P l)

EVERY_FILTER_IMP

|- ∀P1 P2 l. EVERY P1 l ⇒ EVERY P1 (FILTER P2 l)

EVERY_FILTER

|- ∀P1 P2 l. EVERY P1 (FILTER P2 l) ⇔ EVERY (λx. P2 x ⇒ P1 x) l

FILTER_EQ_APPEND

|- ∀P l l1 l2.
     (FILTER P l = l1 ++ l2) ⇔
     ∃l3 l4. (l = l3 ++ l4) ∧ (FILTER P l3 = l1) ∧ (FILTER P l4 = l2)

MEM

|- (∀x. MEM x [] ⇔ F) ∧ ∀x h t. MEM x (h::t) ⇔ (x = h) ∨ MEM x t

FILTER_APPEND_DISTRIB

|- ∀P L M. FILTER P (L ++ M) = FILTER P L ++ FILTER P M

FILTER_EQ_CONS

|- ∀P l h lr.
     (FILTER P l = h::lr) ⇔
     ∃l1 l2.
       (l = l1 ++ [h] ++ l2) ∧ (FILTER P l1 = []) ∧ (FILTER P l2 = lr) ∧ P h

FILTER_NEQ_ID

|- ∀P l. FILTER P l ≠ l ⇔ ∃x. MEM x l ∧ ¬P x

FILTER_EQ_ID

|- ∀P l. (FILTER P l = l) ⇔ EVERY P l

FILTER_NEQ_NIL

|- ∀P l. FILTER P l ≠ [] ⇔ ∃x. MEM x l ∧ P x

FILTER_EQ_NIL

|- ∀P l. (FILTER P l = []) ⇔ EVERY (λx. ¬P x) l

LENGTH_TL

|- ∀l. 0 < LENGTH l ⇒ (LENGTH (TL l) = LENGTH l − 1)

FOLDR_CONS

|- ∀f ls a. FOLDR (λx y. f x::y) a ls = MAP f ls ++ a

FOLDL_EQ_FOLDR

|- ∀f l e. ASSOC f ∧ COMM f ⇒ (FOLDL f e l = FOLDR f e l)

LIST_EQ

|- ∀l1 l2.
     (LENGTH l1 = LENGTH l2) ∧ (∀x. x < LENGTH l1 ⇒ (EL x l1 = EL x l2)) ⇒
     (l1 = l2)

LIST_EQ_REWRITE

|- ∀l1 l2.
     (l1 = l2) ⇔
     (LENGTH l1 = LENGTH l2) ∧ ∀x. x < LENGTH l1 ⇒ (EL x l1 = EL x l2)

MEM_SPLIT

|- ∀x l. MEM x l ⇔ ∃l1 l2. l = l1 ++ x::l2

APPEND_EQ_SELF

|- (∀l1 l2. (l1 ++ l2 = l1) ⇔ (l2 = [])) ∧
   (∀l1 l2. (l1 ++ l2 = l2) ⇔ (l1 = [])) ∧
   (∀l1 l2. (l1 = l1 ++ l2) ⇔ (l2 = [])) ∧ ∀l1 l2. (l2 = l1 ++ l2) ⇔ (l1 = [])

APPEND_11_LENGTH

|- (∀l1 l2 l1' l2'.
      (LENGTH l1 = LENGTH l1') ⇒
      ((l1 ++ l2 = l1' ++ l2') ⇔ (l1 = l1') ∧ (l2 = l2'))) ∧
   ∀l1 l2 l1' l2'.
     (LENGTH l2 = LENGTH l2') ⇒
     ((l1 ++ l2 = l1' ++ l2') ⇔ (l1 = l1') ∧ (l2 = l2'))

APPEND_LENGTH_EQ

|- ∀l1 l1'.
     (LENGTH l1 = LENGTH l1') ⇒
     ∀l2 l2'.
       (LENGTH l2 = LENGTH l2') ⇒
       ((l1 ++ l2 = l1' ++ l2') ⇔ (l1 = l1') ∧ (l2 = l2'))

APPEND_11

|- (∀l1 l2 l3. (l1 ++ l2 = l1 ++ l3) ⇔ (l2 = l3)) ∧
   ∀l1 l2 l3. (l2 ++ l1 = l3 ++ l1) ⇔ (l2 = l3)

APPEND_EQ_SING

|- (l1 ++ l2 = [e]) ⇔ (l1 = [e]) ∧ (l2 = []) ∨ (l1 = []) ∧ (l2 = [e])

MAP_EQ_APPEND

|- (MAP f l = l1 ++ l2) ⇔
   ∃l10 l20. (l = l10 ++ l20) ∧ (l1 = MAP f l10) ∧ (l2 = MAP f l20)

APPEND_eq_NIL

|- (∀l1 l2. ([] = l1 ++ l2) ⇔ (l1 = []) ∧ (l2 = [])) ∧
   ∀l1 l2. (l1 ++ l2 = []) ⇔ (l1 = []) ∧ (l2 = [])

CONS_ACYCLIC

|- ∀l x. l ≠ x::l ∧ x::l ≠ l

LENGTH_EQ_NIL

|- ∀P. (∀l. (LENGTH l = 0) ⇒ P l) ⇔ P []

LENGTH_EQ_NUM_compute

|- (∀l. (LENGTH l = 0) ⇔ (l = [])) ∧
   (∀l n.
      (LENGTH l = NUMERAL (BIT1 n)) ⇔
      ∃h l'. (LENGTH l' = NUMERAL (BIT1 n) − 1) ∧ (l = h::l')) ∧
   (∀l n.
      (LENGTH l = NUMERAL (BIT2 n)) ⇔
      ∃h l'. (LENGTH l' = NUMERAL (BIT1 n)) ∧ (l = h::l')) ∧
   ∀l n1 n2.
     (LENGTH l = n1 + n2) ⇔
     ∃l1 l2. (LENGTH l1 = n1) ∧ (LENGTH l2 = n2) ∧ (l = l1 ++ l2)

LENGTH_EQ_NUM

|- (∀l. (LENGTH l = 0) ⇔ (l = [])) ∧
   (∀l n. (LENGTH l = SUC n) ⇔ ∃h l'. (LENGTH l' = n) ∧ (l = h::l')) ∧
   ∀l n1 n2.
     (LENGTH l = n1 + n2) ⇔
     ∃l1 l2. (LENGTH l1 = n1) ∧ (LENGTH l2 = n2) ∧ (l = l1 ++ l2)

LENGTH_EQ_SUM

|- ∀l n1 n2.
     (LENGTH l = n1 + n2) ⇔
     ∃l1 l2. (LENGTH l1 = n1) ∧ (LENGTH l2 = n2) ∧ (l = l1 ++ l2)

LENGTH_EQ_CONS

|- ∀P n.
     (∀l. (LENGTH l = SUC n) ⇒ P l) ⇔
     ∀l. (LENGTH l = n) ⇒ (λl. ∀x. P (x::l)) l

LENGTH_CONS

|- ∀l n. (LENGTH l = SUC n) ⇔ ∃h l'. (LENGTH l' = n) ∧ (l = h::l')

NULL_LENGTH

|- ∀l. NULL l ⇔ (LENGTH l = 0)

NULL_EQ

|- ∀l. NULL l ⇔ (l = [])

LENGTH_NIL_SYM

|- (0 = LENGTH l) ⇔ (l = [])

LENGTH_NIL

|- ∀l. (LENGTH l = 0) ⇔ (l = [])

MEM_MAP

|- ∀l f x. MEM x (MAP f l) ⇔ ∃y. (x = f y) ∧ MEM y l

NOT_EXISTS

|- ∀P l. ¬EXISTS P l ⇔ EVERY ($~ o P) l

NOT_EVERY

|- ∀P l. ¬EVERY P l ⇔ EXISTS ($~ o P) l

EXISTS_APPEND

|- ∀P l1 l2. EXISTS P (l1 ++ l2) ⇔ EXISTS P l1 ∨ EXISTS P l2

EVERY_APPEND

|- ∀P l1 l2. EVERY P (l1 ++ l2) ⇔ EVERY P l1 ∧ EVERY P l2

FLAT_APPEND

|- ∀l1 l2. FLAT (l1 ++ l2) = FLAT l1 ++ FLAT l2

MEM_FLAT

|- ∀x L. MEM x (FLAT L) ⇔ ∃l. MEM l L ∧ MEM x l

MEM_FILTER

|- ∀P L x. MEM x (FILTER P L) ⇔ P x ∧ MEM x L

MEM_APPEND

|- ∀e l1 l2. MEM e (l1 ++ l2) ⇔ MEM e l1 ∨ MEM e l2

EXISTS_NOT_EVERY

|- ∀P l. EXISTS P l ⇔ ¬EVERY (λx. ¬P x) l

EVERY_NOT_EXISTS

|- ∀P l. EVERY P l ⇔ ¬EXISTS (λx. ¬P x) l

MONO_EXISTS

|- (∀x. P x ⇒ Q x) ⇒ EXISTS P l ⇒ EXISTS Q l

EXISTS_SIMP

|- ∀c l. EXISTS (λx. c) l ⇔ l ≠ [] ∧ c

EXISTS_MAP

|- ∀P f l. EXISTS P (MAP f l) ⇔ EXISTS (λx. P (f x)) l

EXISTS_MEM

|- ∀P l. EXISTS P l ⇔ ∃e. MEM e l ∧ P e

MONO_EVERY

|- (∀x. P x ⇒ Q x) ⇒ EVERY P l ⇒ EVERY Q l

EVERY_SIMP

|- ∀c l. EVERY (λx. c) l ⇔ (l = []) ∨ c

EVERY_MAP

|- ∀P f l. EVERY P (MAP f l) ⇔ EVERY (λx. P (f x)) l

EVERY_MEM

|- ∀P l. EVERY P l ⇔ ∀e. MEM e l ⇒ P e

EVERY_CONJ

|- ∀P Q l. EVERY (λx. P x ∧ Q x) l ⇔ EVERY P l ∧ EVERY Q l

EVERY_EL

|- ∀l P. EVERY P l ⇔ ∀n. n < LENGTH l ⇒ P (EL n l)

MAP_TL

|- ∀l f. ¬NULL l ⇒ (MAP f (TL l) = TL (MAP f l))

EL_MAP

|- ∀n l. n < LENGTH l ⇒ ∀f. EL n (MAP f l) = f (EL n l)

MAP_MAP_o

|- ∀f g l. MAP f (MAP g l) = MAP (f o g) l

MAP_o

|- ∀f g. MAP (f o g) = MAP f o MAP g

MAP_EQ_f

|- ∀f1 f2 l. (MAP f1 l = MAP f2 l) ⇔ ∀e. MEM e l ⇒ (f1 e = f2 e)

MAP_EQ_SING

|- (MAP f l = [x]) ⇔ ∃x0. (l = [x0]) ∧ (x = f x0)

MAP_EQ_CONS

|- (MAP f l = h::t) ⇔ ∃x0 t0. (l = x0::t0) ∧ (h = f x0) ∧ (t = MAP f t0)

MAP_EQ_NIL

|- ∀l f. ((MAP f l = []) ⇔ (l = [])) ∧ (([] = MAP f l) ⇔ (l = []))

LENGTH_MAP

|- ∀l f. LENGTH (MAP f l) = LENGTH l

MAP_ID

|- (MAP (λx. x) l = l) ∧ (MAP I l = l)

MAP_APPEND

|- ∀f l1 l2. MAP f (l1 ++ l2) = MAP f l1 ++ MAP f l2

LENGTH_APPEND

|- ∀l1 l2. LENGTH (l1 ++ l2) = LENGTH l1 + LENGTH l2

APPEND_ASSOC

|- ∀l1 l2 l3. l1 ++ (l2 ++ l3) = l1 ++ l2 ++ l3

APPEND_NIL

|- ∀l. l ++ [] = l

CONS

|- ∀l. ¬NULL l ⇒ (HD l::TL l = l)

EQ_LIST

|- ∀h1 h2. (h1 = h2) ⇒ ∀l1 l2. (l1 = l2) ⇒ (h1::l1 = h2::l2)

NOT_EQ_LIST

|- ∀h1 h2. h1 ≠ h2 ⇒ ∀l1 l2. h1::l1 ≠ h2::l2

LIST_NOT_EQ

|- ∀l1 l2. l1 ≠ l2 ⇒ ∀h1 h2. h1::l1 ≠ h2::l2

NOT_CONS_NIL

|- ∀a1 a0. a0::a1 ≠ []

NOT_NIL_CONS

|- ∀a1 a0. [] ≠ a0::a1

CONS_11

|- ∀a0 a1 a0' a1'. (a0::a1 = a0'::a1') ⇔ (a0 = a0') ∧ (a1 = a1')

list_case_compute

|- ∀l. list_CASE l b f = if NULL l then b else f (HD l) (TL l)

list_nchotomy

|- ∀l. (l = []) ∨ ∃h t. l = h::t

list_case_cong

|- ∀M M' v f.
     (M = M') ∧ ((M' = []) ⇒ (v = v')) ∧
     (∀a0 a1. (M' = a0::a1) ⇒ (f a0 a1 = f' a0 a1)) ⇒
     (list_CASE M v f = list_CASE M' v' f')

list_distinct

|- ∀a1 a0. [] ≠ a0::a1

list_11

|- ∀a0 a1 a0' a1'. (a0::a1 = a0'::a1') ⇔ (a0 = a0') ∧ (a1 = a1')

datatype_list

|- DATATYPE (list [] CONS)

list_Axiom_old

|- ∀x f. ∃!fn1. (fn1 [] = x) ∧ ∀h t. fn1 (h::t) = f (fn1 t) h t

LIST_TO_SET

|- (LIST_TO_SET [] = ∅) ∧ (LIST_TO_SET (h::t) = h INSERT LIST_TO_SET t)

IN_LIST_TO_SET

|- T

MAP2_ind

|- ∀P.
     (∀f h1 t1 h2 t2. P f t1 t2 ⇒ P f (h1::t1) (h2::t2)) ∧ (∀f y. P f [] y) ∧
     (∀f v4 v5. P f (v4::v5) []) ⇒
     ∀v v1 v2. P v v1 v2

MAP2_def

|- (∀t2 t1 h2 h1 f. MAP2 f (h1::t1) (h2::t2) = f h1 h2::MAP2 f t1 t2) ∧
   (∀y f. MAP2 f [] y = []) ∧ ∀v5 v4 f. MAP2 f (v4::v5) [] = []

MAP2

|- (∀f. MAP2 f [] [] = []) ∧
   ∀f h1 t1 h2 t2. MAP2 f (h1::t1) (h2::t2) = f h1 h2::MAP2 f t1 t2

NULL

|- NULL [] ∧ ∀h t. ¬NULL (h::t)

list_INDUCT

|- ∀P. P [] ∧ (∀t. P t ⇒ ∀h. P (h::t)) ⇒ ∀l. P l

list_Axiom

|- ∀f0 f1. ∃fn. (fn [] = f0) ∧ ∀a0 a1. fn (a0::a1) = f1 a0 a1 (fn a1)

list_induction

|- ∀P. P [] ∧ (∀t. P t ⇒ ∀h. P (h::t)) ⇒ ∀l. P l

LIST_REL_EL_EQN

|- ∀R l1 l2.
     LIST_REL R l1 l2 ⇔
     (LENGTH l1 = LENGTH l2) ∧ ∀n. n < LENGTH l1 ⇒ R (EL n l1) (EL n l2)

LIST_REL_cases

|- ∀R a0 a1.
     LIST_REL R a0 a1 ⇔
     (a0 = []) ∧ (a1 = []) ∨
     ∃h1 h2 t1 t2. (a0 = h1::t1) ∧ (a1 = h2::t2) ∧ R h1 h2 ∧ LIST_REL R t1 t2

LIST_REL_strongind

|- ∀R LIST_REL'.
     LIST_REL' [] [] ∧
     (∀h1 h2 t1 t2.
        R h1 h2 ∧ LIST_REL R t1 t2 ∧ LIST_REL' t1 t2 ⇒
        LIST_REL' (h1::t1) (h2::t2)) ⇒
     ∀a0 a1. LIST_REL R a0 a1 ⇒ LIST_REL' a0 a1

LIST_REL_ind

|- ∀R LIST_REL'.
     LIST_REL' [] [] ∧
     (∀h1 h2 t1 t2. R h1 h2 ∧ LIST_REL' t1 t2 ⇒ LIST_REL' (h1::t1) (h2::t2)) ⇒
     ∀a0 a1. LIST_REL R a0 a1 ⇒ LIST_REL' a0 a1

LIST_REL_rules

|- ∀R.
     LIST_REL R [] [] ∧
     ∀h1 h2 t1 t2. R h1 h2 ∧ LIST_REL R t1 t2 ⇒ LIST_REL R (h1::t1) (h2::t2)

list_CASES

|- ∀l. (l = []) ∨ ∃h t. l = h::t

FORALL_LIST

|- (∀l. P l) ⇔ P [] ∧ ∀h t. P (h::t)

MEM_SPLIT_APPEND_first

|- MEM e l ⇔ ∃pfx sfx. (l = pfx ++ [e] ++ sfx) ∧ ¬MEM e pfx

MEM_SPLIT_APPEND_last

|- MEM e l ⇔ ∃pfx sfx. (l = pfx ++ [e] ++ sfx) ∧ ¬MEM e sfx

APPEND_EQ_APPEND

|- (l1 ++ l2 = m1 ++ m2) ⇔
   (∃l. (l1 = m1 ++ l) ∧ (m2 = l ++ l2)) ∨ ∃l. (m1 = l1 ++ l) ∧ (l2 = l ++ m2)

APPEND_EQ_CONS

|- (l1 ++ l2 = h::t) ⇔
   (l1 = []) ∧ (l2 = h::t) ∨ ∃lt. (l1 = h::lt) ∧ (t = lt ++ l2)

APPEND_EQ_APPEND_MID

|- (l1 ++ [e] ++ l2 = m1 ++ m2) ⇔
   (∃l. (m1 = l1 ++ [e] ++ l) ∧ (l2 = l ++ m2)) ∨
   ∃l. (l1 = m1 ++ l) ∧ (m2 = l ++ [e] ++ l2)

LUPDATE_SEM

|- (∀e n l. LENGTH (LUPDATE e n l) = LENGTH l) ∧
   ∀e n l p.
     p < LENGTH l ⇒ (EL p (LUPDATE e n l) = if p = n then e else EL p l)

EL_LUPDATE

|- ∀ys x i k.
     EL i (LUPDATE x k ys) = if (i = k) ∧ k < LENGTH ys then x else EL i ys

LENGTH_LUPDATE

|- ∀x n ys. LENGTH (LUPDATE x n ys) = LENGTH ys

LUPDATE_LENGTH

|- ∀xs x y ys. LUPDATE x (LENGTH xs) (xs ++ y::ys) = xs ++ x::ys

LUPDATE_SNOC

|- ∀ys k x y.
     LUPDATE x k (SNOC y ys) =
     if k = LENGTH ys then SNOC x ys else SNOC y (LUPDATE x k ys)

MEM_LUPDATE_E

|- ∀l x y i. MEM x (LUPDATE y i l) ⇒ (x = y) ∨ MEM x l

MEM_LUPDATE

|- ∀l x y i.
     MEM x (LUPDATE y i l) ⇔
     i < LENGTH l ∧ (x = y) ∨ ∃j. j < LENGTH l ∧ i ≠ j ∧ (EL j l = x)

LUPDATE_compute

|- (∀e n. LUPDATE e n [] = []) ∧ (∀e x l. LUPDATE e 0 (x::l) = e::l) ∧
   (∀e n x l.
      LUPDATE e (NUMERAL (BIT1 n)) (x::l) =
      x::LUPDATE e (NUMERAL (BIT1 n) − 1) l) ∧
   ∀e n x l.
     LUPDATE e (NUMERAL (BIT2 n)) (x::l) = x::LUPDATE e (NUMERAL (BIT1 n)) l

LUPDATE_MAP

|- ∀x n l f. MAP f (LUPDATE x n l) = LUPDATE (f x) n (MAP f l)

splitAtPki_APPEND

|- ∀l1 l2 P k.
     EVERYi (λi. $~ o P i) l1 ∧ (0 < LENGTH l2 ⇒ P (LENGTH l1) (HD l2)) ⇒
     (splitAtPki P k (l1 ++ l2) = k l1 l2)

splitAtPki_EQN

|- splitAtPki P k l =
   case OLEAST i. i < LENGTH l ∧ P i (EL i l) of
     NONE => k l []
   | SOME i => k (TAKE i l) (DROP i l)

TAKE_LENGTH_TOO_LONG

|- ∀l n. LENGTH l ≤ n ⇒ (TAKE n l = l)

DROP_LENGTH_TOO_LONG

|- ∀l n. LENGTH l ≤ n ⇒ (DROP n l = [])

TAKE_splitAtPki

|- TAKE n l = splitAtPki (K o $= n) K l

DROP_splitAtPki

|- DROP n l = splitAtPki (K o $= n) (K I) l

LIST_BIND_THM

|- (LIST_BIND [] f = []) ∧ (LIST_BIND (h::t) f = f h ++ LIST_BIND t f)

LIST_BIND_ID

|- (LIST_BIND l (λx. x) = FLAT l) ∧ (LIST_BIND l I = FLAT l)

LIST_BIND_APPEND

|- LIST_BIND (l1 ++ l2) f = LIST_BIND l1 f ++ LIST_BIND l2 f

LIST_BIND_MAP

|- LIST_BIND (MAP f l) g = LIST_BIND l (g o f)

MAP_LIST_BIND

|- MAP f (LIST_BIND l g) = LIST_BIND l (MAP f o g)

LIST_BIND_LIST_BIND

|- LIST_BIND (LIST_BIND l g) f = LIST_BIND l (combin$C LIST_BIND f o g)

SINGL_LIST_APPLY_L

|- LIST_BIND [x] f = f x

SINGL_LIST_APPLY_R

|- LIST_BIND l (λx. [x]) = l

SINGL_APPLY_MAP

|- [f] <*> l = MAP f l

SINGL_SINGL_APPLY

|- [f] <*> [x] = [f x]

SINGL_APPLY_PERMUTE

|- fs <*> [x] = [(λf. f x)] <*> fs

MAP_FLAT

|- MAP f (FLAT l) = FLAT (MAP (MAP f) l)

LIST_APPLY_o

|- [$o] <*> fs <*> gs <*> xs = fs <*> (gs <*> xs)

LLEX_THM

|- (¬LLEX R [] [] ∧ ¬LLEX R (h1::t1) []) ∧ LLEX R [] (h2::t2) ∧
   (LLEX R (h1::t1) (h2::t2) ⇔ R h1 h2 ∨ (h1 = h2) ∧ LLEX R t1 t2)

LLEX_MONO

|- (∀x y. R1 x y ⇒ R2 x y) ⇒ LLEX R1 x y ⇒ LLEX R2 x y

LLEX_CONG

|- ∀R l1 l2 R' l1' l2'.
     (l1 = l1') ∧ (l2 = l2') ∧
     (∀a b. MEM a l1' ∧ MEM b l2' ⇒ (R a b ⇔ R' a b)) ⇒
     (LLEX R l1 l2 ⇔ LLEX R' l1' l2')

LLEX_NIL2

|- ¬LLEX R l []

LLEX_transitive

|- transitive R ⇒ transitive (LLEX R)

LLEX_total

|- total (RC R) ⇒ total (RC (LLEX R))

LLEX_not_WF

|- (∃a b. R a b) ⇒ ¬WF (LLEX R)

nub_set

|- ∀l. LIST_TO_SET (nub l) = LIST_TO_SET l

all_distinct_nub

|- ∀l. ALL_DISTINCT (nub l)

nub_append

|- ∀l1 l2. nub (l1 ++ l2) = nub (FILTER (λx. ¬MEM x l2) l1) ++ nub l2

list_to_set_diff

|- ∀l1 l2.
     LIST_TO_SET l2 DIFF LIST_TO_SET l1 =
     LIST_TO_SET (FILTER (λx. ¬MEM x l1) l2)

length_nub_append

|- ∀l1 l2.
     LENGTH (nub (l1 ++ l2)) =
     LENGTH (nub l1) + LENGTH (nub (FILTER (λx. ¬MEM x l1) l2))

ALL_DISTINCT_DROP

|- ∀ls n. ALL_DISTINCT ls ⇒ ALL_DISTINCT (DROP n ls)

EXISTS_LIST_EQ_MAP

|- ∀ls f. EVERY (λx. ∃y. x = f y) ls ⇒ ∃l. ls = MAP f l

LIST_TO_SET_FLAT

|- ∀ls. LIST_TO_SET (FLAT ls) = BIGUNION (LIST_TO_SET (MAP LIST_TO_SET ls))

MEM_APPEND_lemma

|- ∀a b c d x.
     (a ++ [x] ++ b = c ++ [x] ++ d) ∧ ¬MEM x b ∧ ¬MEM x a ⇒ (a = c) ∧ (b = d)

EVERY2_REVERSE

|- ∀R l1 l2. LIST_REL R l1 l2 ⇒ LIST_REL R (REVERSE l1) (REVERSE l2)

SUM_MAP_PLUS

|- ∀f g ls. SUM (MAP (λx. f x + g x) ls) = SUM (MAP f ls) + SUM (MAP g ls)

TAKE_LENGTH_ID_rwt

|- ∀l m. (m = LENGTH l) ⇒ (TAKE m l = l)

ZIP_DROP

|- ∀a b n.
     n ≤ LENGTH a ∧ (LENGTH a = LENGTH b) ⇒
     (ZIP (DROP n a,DROP n b) = DROP n (ZIP (a,b)))

GENLIST_EL

|- ∀ls f n.
     (n = LENGTH ls) ∧ (∀i. i < n ⇒ (f i = EL i ls)) ⇒ (GENLIST f n = ls)

EVERY2_trans

|- (∀x y z. R x y ∧ R y z ⇒ R x z) ⇒
   ∀x y z. LIST_REL R x y ∧ LIST_REL R y z ⇒ LIST_REL R x z

EVERY2_sym

|- (∀x y. R1 x y ⇒ R2 y x) ⇒ ∀x y. LIST_REL R1 x y ⇒ LIST_REL R2 y x

EVERY2_LUPDATE_same

|- ∀P l1 l2 v1 v2 n.
     P v1 v2 ∧ LIST_REL P l1 l2 ⇒
     LIST_REL P (LUPDATE v1 n l1) (LUPDATE v2 n l2)

EVERY2_refl

|- (∀x. MEM x ls ⇒ R x x) ⇒ LIST_REL R ls ls

EVERY2_THM

|- (∀P ys. LIST_REL P [] ys ⇔ (ys = [])) ∧
   (∀P yys x xs.
      LIST_REL P (x::xs) yys ⇔
      ∃y ys. (yys = y::ys) ∧ P x y ∧ LIST_REL P xs ys) ∧
   (∀P xs. LIST_REL P xs [] ⇔ (xs = [])) ∧
   ∀P xxs y ys.
     LIST_REL P xxs (y::ys) ⇔ ∃x xs. (xxs = x::xs) ∧ P x y ∧ LIST_REL P xs ys

LIST_REL_trans

|- ∀l1 l2 l3.
     (∀n.
        n < LENGTH l1 ∧ R (EL n l1) (EL n l2) ∧ R (EL n l2) (EL n l3) ⇒
        R (EL n l1) (EL n l3)) ∧ LIST_REL R l1 l2 ∧ LIST_REL R l2 l3 ⇒
     LIST_REL R l1 l3

SWAP_REVERSE

|- ∀l1 l2. (l1 = REVERSE l2) ⇔ (l2 = REVERSE l1)

SWAP_REVERSE_SYM

|- ∀l1 l2. (REVERSE l1 = l2) ⇔ (l1 = REVERSE l2)

BIGUNION_IMAGE_set_SUBSET

|- BIGUNION (IMAGE f (LIST_TO_SET ls)) ⊆ s ⇔ ∀x. MEM x ls ⇒ f x ⊆ s

IMAGE_EL_count_LENGTH

|- ∀f ls.
     IMAGE (λn. f (EL n ls)) (count (LENGTH ls)) = IMAGE f (LIST_TO_SET ls)

GENLIST_EL_MAP

|- ∀f ls. GENLIST (λn. f (EL n ls)) (LENGTH ls) = MAP f ls

LENGTH_FILTER_LEQ_MONO

|- ∀P Q. (∀x. P x ⇒ Q x) ⇒ ∀ls. LENGTH (FILTER P ls) ≤ LENGTH (FILTER Q ls)

LIST_EQ_MAP_PAIR

|- ∀l1 l2. (MAP FST l1 = MAP FST l2) ∧ (MAP SND l1 = MAP SND l2) ⇒ (l1 = l2)

TAKE_SUM

|- ∀n m l. n + m ≤ LENGTH l ⇒ (TAKE (n + m) l = TAKE n l ++ TAKE m (DROP n l))

ALL_DISTINCT_FILTER_EL_IMP

|- ∀P l n1 n2.
     ALL_DISTINCT (FILTER P l) ∧ n1 < LENGTH l ∧ n2 < LENGTH l ∧ P (EL n1 l) ∧
     (EL n1 l = EL n2 l) ⇒
     (n1 = n2)

FLAT_EQ_NIL

|- ∀ls. (FLAT ls = []) ⇔ EVERY ($= []) ls

ALL_DISTINCT_MAP_INJ

|- ∀ls f.
     (∀x y. MEM x ls ∧ MEM y ls ∧ (f x = f y) ⇒ (x = y)) ∧ ALL_DISTINCT ls ⇒
     ALL_DISTINCT (MAP f ls)

LENGTH_o_REVERSE

|- (LENGTH o REVERSE = LENGTH) ∧ (LENGTH o REVERSE o f = LENGTH o f)

REVERSE_o_REVERSE

|- REVERSE o REVERSE o f = f

GENLIST_PLUS_APPEND

|- GENLIST ($+ a) n1 ++ GENLIST ($+ (n1 + a)) n2 = GENLIST ($+ a) (n1 + n2)

LIST_TO_SET_GENLIST

|- ∀f n. LIST_TO_SET (GENLIST f n) = IMAGE f (count n)

MEM_ZIP_MEM_MAP

|- (LENGTH (FST ps) = LENGTH (SND ps)) ∧ MEM p (ZIP ps) ⇒
   MEM (FST p) (FST ps) ∧ MEM (SND p) (SND ps)

DISJOINT_GENLIST_PLUS

|- DISJOINT x (LIST_TO_SET (GENLIST ($+ n) (a + b))) ⇒
   DISJOINT x (LIST_TO_SET (GENLIST ($+ n) a)) ∧
   DISJOINT x (LIST_TO_SET (GENLIST ($+ (n + a)) b))

EVERY2_MAP

|- (LIST_REL P (MAP f l1) l2 ⇔ LIST_REL (λx y. P (f x) y) l1 l2) ∧
   (LIST_REL Q l1 (MAP g l2) ⇔ LIST_REL (λx y. Q x (g y)) l1 l2)

exists_list_GENLIST

|- (∃ls. P ls) ⇔ ∃n f. P (GENLIST f n)

EVERY_MEM_MONO

|- ∀P Q l. (∀x. MEM x l ∧ P x ⇒ Q x) ∧ EVERY P l ⇒ EVERY Q l

EVERY2_MEM_MONO

|- ∀P Q l1 l2.
     (∀x. MEM x (ZIP (l1,l2)) ∧ UNCURRY P x ⇒ UNCURRY Q x) ∧
     LIST_REL P l1 l2 ⇒
     LIST_REL Q l1 l2

mem_exists_set

|- ∀x y l. MEM (x,y) l ⇒ ∃z. (x = FST z) ∧ MEM z l

every_zip_snd

|- ∀l1 l2 P.
     (LENGTH l1 = LENGTH l2) ⇒
     (EVERY (λx. P (SND x)) (ZIP (l1,l2)) ⇔ EVERY P l2)

every_zip_fst

|- ∀l1 l2 P.
     (LENGTH l1 = LENGTH l2) ⇒
     (EVERY (λx. P (FST x)) (ZIP (l1,l2)) ⇔ EVERY P l1)

el_append3

|- ∀l1 x l2. EL (LENGTH l1) (l1 ++ [x] ++ l2) = x

lupdate_append

|- ∀x n l1 l2. n < LENGTH l1 ⇒ (LUPDATE x n (l1 ++ l2) = LUPDATE x n l1 ++ l2)

lupdate_append2

|- ∀v l1 x l2 l3. LUPDATE v (LENGTH l1) (l1 ++ [x] ++ l2) = l1 ++ [v] ++ l2

LAST_REVERSE

|- ∀ls. ls ≠ [] ⇒ (LAST (REVERSE ls) = HD ls)

dropWhile_splitAtPki

|- ∀P. dropWhile P = splitAtPki (combin$C (K o $~ o P)) (K I)

dropWhile_eq_nil

|- ∀P ls. (dropWhile P ls = []) ⇔ EVERY P ls

MEM_dropWhile_IMP

|- ∀P ls x. MEM x (dropWhile P ls) ⇒ MEM x ls

HD_dropWhile

|- ∀P ls. EXISTS ($~ o P) ls ⇒ ¬P (HD (dropWhile P ls))

LENGTH_dropWhile_LESS_EQ

|- ∀P ls. LENGTH (dropWhile P ls) ≤ LENGTH ls

dropWhile_APPEND_EVERY

|- ∀P l1 l2. EVERY P l1 ⇒ (dropWhile P (l1 ++ l2) = dropWhile P l2)

dropWhile_APPEND_EXISTS

|- ∀P l1 l2.
     EXISTS ($~ o P) l1 ⇒ (dropWhile P (l1 ++ l2) = dropWhile P l1 ++ l2)

EL_LENGTH_dropWhile_REVERSE

|- ∀P ls k.
     LENGTH (dropWhile P (REVERSE ls)) ≤ k ∧ k < LENGTH ls ⇒ P (EL k ls)

LENGTH_TAKE_EQ

|- LENGTH (TAKE n xs) = if n ≤ LENGTH xs then n else LENGTH xs

IMP_EVERY_LUPDATE

|- ∀xs h i. P h ∧ EVERY P xs ⇒ EVERY P (LUPDATE h i xs)

MAP_APPEND_MAP_EQ

|- ∀xs ys.
     (MAP f1 xs ++ MAP g1 ys = MAP f2 xs ++ MAP g2 ys) ⇔
     (MAP f1 xs = MAP f2 xs) ∧ (MAP g1 ys = MAP g2 ys)

LUPDATE_SOME_MAP

|- ∀xs n f h.
     LUPDATE (SOME (f h)) n (MAP (OPTION_MAP f) xs) =
     MAP (OPTION_MAP f) (LUPDATE (SOME h) n xs)

ZIP_EQ_NIL

|- ∀l1 l2.
     (LENGTH l1 = LENGTH l2) ⇒ ((ZIP (l1,l2) = []) ⇔ (l1 = []) ∧ (l2 = []))

LUPDATE_SAME

|- ∀n ls. n < LENGTH ls ⇒ (LUPDATE (EL n ls) n ls = ls)

LAST_compute

|- (∀x. LAST [x] = x) ∧ ∀h1 h2 t. LAST (h1::h2::t) = LAST (h2::t)

TAKE_compute

|- (∀l. TAKE 0 l = []) ∧ (∀n. TAKE (NUMERAL (BIT1 n)) [] = []) ∧
   (∀n. TAKE (NUMERAL (BIT2 n)) [] = []) ∧
   (∀n h t.
      TAKE (NUMERAL (BIT1 n)) (h::t) = h::TAKE (NUMERAL (BIT1 n) − 1) t) ∧
   ∀n h t. TAKE (NUMERAL (BIT2 n)) (h::t) = h::TAKE (NUMERAL (BIT1 n)) t

DROP_compute

|- (∀l. DROP 0 l = l) ∧ (∀n. DROP (NUMERAL (BIT1 n)) [] = []) ∧
   (∀n. DROP (NUMERAL (BIT2 n)) [] = []) ∧
   (∀n h t. DROP (NUMERAL (BIT1 n)) (h::t) = DROP (NUMERAL (BIT1 n) − 1) t) ∧
   ∀n h t. DROP (NUMERAL (BIT2 n)) (h::t) = DROP (NUMERAL (BIT1 n)) t