Structure listTheory
signature listTheory =
sig
type thm = Thm.thm
(* Definitions *)
val ALL_DISTINCT : thm
val APPEND : thm
val DROP_def : thm
val EL : thm
val EVERY2_curried_def : thm
val EVERY2_tupled_primitive_def : thm
val EVERY_DEF : thm
val EXISTS_DEF : thm
val FILTER : thm
val FLAT : thm
val FOLDL : thm
val FOLDL2_curried_def : thm
val FOLDL2_tupled_primitive_def : thm
val FOLDR : thm
val FRONT_DEF : thm
val GENLIST : thm
val GENLIST_AUX : thm
val HD : thm
val LAST_DEF : thm
val LENGTH : thm
val LEN_DEF : thm
val LIST_TO_SET : thm
val LRC_def : thm
val LUPDATE_DEF : thm
val MAP : thm
val MAP2_curried_def : thm
val MAP2_tupled_primitive_def : thm
val MEM : thm
val NULL_DEF : thm
val PAD_LEFT : thm
val PAD_RIGHT : thm
val REVERSE_DEF : thm
val REV_DEF : thm
val SET_TO_LIST_primitive : thm
val SNOC : thm
val SUM : thm
val TAKE_def : thm
val TL : thm
val UNZIP : thm
val ZIP : thm
val isPREFIX : thm
val listRel : thm
val list_TY_DEF : thm
val list_case_def : thm
val list_size_def : thm
(* Theorems *)
val ALL_DISTINCT_APPEND : thm
val ALL_DISTINCT_CARD_LIST_TO_SET : thm
val ALL_DISTINCT_EL_IMP : thm
val ALL_DISTINCT_FILTER : thm
val ALL_DISTINCT_GENLIST : thm
val ALL_DISTINCT_MAP : thm
val ALL_DISTINCT_REVERSE : thm
val ALL_DISTINCT_SET_TO_LIST : thm
val ALL_DISTINCT_SING : thm
val ALL_DISTINCT_SNOC : thm
val ALL_DISTINCT_ZIP : thm
val ALL_DISTINCT_ZIP_SWAP : thm
val APPEND_11 : thm
val APPEND_11_LENGTH : thm
val APPEND_ASSOC : thm
val APPEND_EQ_APPEND : thm
val APPEND_EQ_APPEND_MID : thm
val APPEND_EQ_CONS : thm
val APPEND_EQ_SELF : thm
val APPEND_EQ_SING : thm
val APPEND_FRONT_LAST : thm
val APPEND_LENGTH_EQ : thm
val APPEND_NIL : thm
val APPEND_SNOC : thm
val APPEND_eq_NIL : thm
val CARD_LIST_TO_SET : thm
val CONS : thm
val CONS_11 : thm
val CONS_ACYCLIC : thm
val DROP_0 : thm
val DROP_NIL : thm
val DROP_compute : thm
val EL_ALL_DISTINCT_EL_EQ : thm
val EL_GENLIST : thm
val EL_LENGTH_SNOC : thm
val EL_LUPDATE : thm
val EL_MAP : thm
val EL_REVERSE : thm
val EL_SNOC : thm
val EL_ZIP : thm
val EL_compute : thm
val EL_restricted : thm
val EL_simp : thm
val EL_simp_restricted : thm
val EQ_LIST : thm
val EVERY2_EVERY : thm
val EVERY2_LENGTH : thm
val EVERY2_cong : thm
val EVERY2_def : thm
val EVERY2_ind : thm
val EVERY2_mono : thm
val EVERY_APPEND : thm
val EVERY_CONG : thm
val EVERY_CONJ : thm
val EVERY_EL : thm
val EVERY_FILTER : thm
val EVERY_FILTER_IMP : thm
val EVERY_GENLIST : thm
val EVERY_MAP : thm
val EVERY_MEM : thm
val EVERY_MONOTONIC : thm
val EVERY_NOT_EXISTS : thm
val EVERY_SIMP : thm
val EVERY_SNOC : thm
val EXISTS_APPEND : thm
val EXISTS_CONG : thm
val EXISTS_GENLIST : thm
val EXISTS_LIST : thm
val EXISTS_MAP : thm
val EXISTS_MEM : thm
val EXISTS_NOT_EVERY : thm
val EXISTS_SIMP : thm
val EXISTS_SNOC : thm
val FILTER_ALL_DISTINCT : thm
val FILTER_APPEND_DISTRIB : thm
val FILTER_COND_REWRITE : thm
val FILTER_EQ_APPEND : thm
val FILTER_EQ_CONS : thm
val FILTER_EQ_ID : thm
val FILTER_EQ_NIL : thm
val FILTER_NEQ_ID : thm
val FILTER_NEQ_NIL : thm
val FILTER_REVERSE : thm
val FINITE_LIST_TO_SET : thm
val FOLDL2_FOLDL : thm
val FOLDL2_cong : thm
val FOLDL2_def : thm
val FOLDL2_ind : thm
val FOLDL_CONG : thm
val FOLDL_EQ_FOLDR : thm
val FOLDL_SNOC : thm
val FOLDL_UNION_BIGUNION : thm
val FOLDL_UNION_BIGUNION_paired : thm
val FOLDL_ZIP_SAME : thm
val FOLDR_CONG : thm
val FOLDR_CONS : thm
val FORALL_LIST : thm
val FRONT_CONS : thm
val FRONT_CONS_EQ_NIL : thm
val FRONT_SNOC : thm
val GENLIST_APPEND : thm
val GENLIST_AUX_compute : thm
val GENLIST_CONS : thm
val GENLIST_FUN_EQ : thm
val GENLIST_GENLIST_AUX : thm
val GENLIST_NUMERALS : thm
val HD_GENLIST : thm
val HD_GENLIST_COR : thm
val INFINITE_LIST_UNIV : thm
val INJ_MAP_EQ : thm
val IN_LIST_TO_SET : thm
val ITSET_eq_FOLDL_SET_TO_LIST : thm
val LAST_APPEND_CONS : thm
val LAST_CONS : thm
val LAST_CONS_cond : thm
val LAST_EL : thm
val LAST_SNOC : thm
val LAST_compute : thm
val LENGTH_APPEND : thm
val LENGTH_CONS : thm
val LENGTH_DROP : thm
val LENGTH_EQ_CONS : thm
val LENGTH_EQ_NIL : thm
val LENGTH_EQ_NUM : thm
val LENGTH_EQ_NUM_compute : thm
val LENGTH_EQ_SUM : thm
val LENGTH_FRONT_CONS : thm
val LENGTH_GENLIST : thm
val LENGTH_LEN : thm
val LENGTH_LUPDATE : thm
val LENGTH_MAP : thm
val LENGTH_NIL : thm
val LENGTH_NIL_SYM : thm
val LENGTH_REVERSE : thm
val LENGTH_SNOC : thm
val LENGTH_TAKE : thm
val LENGTH_TL : thm
val LENGTH_UNZIP : thm
val LENGTH_ZIP : thm
val LEN_LENGTH_LEM : thm
val LIST_EQ : thm
val LIST_EQ_REWRITE : thm
val LIST_NOT_EQ : thm
val LIST_REL_CONJ : thm
val LIST_REL_CONS1 : thm
val LIST_REL_CONS2 : thm
val LIST_REL_EL_EQN : thm
val LIST_REL_EVERY_ZIP : thm
val LIST_REL_MAP1 : thm
val LIST_REL_MAP2 : thm
val LIST_REL_NIL : thm
val LIST_REL_cases : thm
val LIST_REL_def : thm
val LIST_REL_ind : thm
val LIST_REL_mono : thm
val LIST_REL_rules : thm
val LIST_REL_strongind : thm
val LIST_TO_SET_APPEND : thm
val LIST_TO_SET_EQ_EMPTY : thm
val LIST_TO_SET_FILTER : thm
val LIST_TO_SET_MAP : thm
val LIST_TO_SET_REVERSE : thm
val LIST_TO_SET_SNOC : thm
val LIST_TO_SET_THM : thm
val LRC_MEM : thm
val LRC_MEM_right : thm
val LUPDATE_LENGTH : thm
val LUPDATE_SNOC : thm
val MAP2 : thm
val MAP2_CONG : thm
val MAP2_MAP : thm
val MAP2_ZIP : thm
val MAP2_def : thm
val MAP2_ind : thm
val MAP_APPEND : thm
val MAP_CONG : thm
val MAP_EQ_EVERY2 : thm
val MAP_EQ_NIL : thm
val MAP_EQ_f : thm
val MAP_GENLIST : thm
val MAP_ID : thm
val MAP_MAP_o : thm
val MAP_SNOC : thm
val MAP_TL : thm
val MAP_ZIP : thm
val MAP_ZIP_SAME : thm
val MAP_o : thm
val MEM_APPEND : thm
val MEM_DROP : thm
val MEM_EL : thm
val MEM_FILTER : thm
val MEM_FLAT : thm
val MEM_GENLIST : thm
val MEM_MAP : thm
val MEM_REVERSE : thm
val MEM_SET_TO_LIST : thm
val MEM_SNOC : thm
val MEM_SPLIT : thm
val MEM_SPLIT_APPEND_first : thm
val MEM_SPLIT_APPEND_last : thm
val MEM_ZIP : thm
val MONO_EVERY : thm
val MONO_EXISTS : thm
val MONO_listRel : thm
val NOT_CONS_NIL : thm
val NOT_EQ_LIST : thm
val NOT_EVERY : thm
val NOT_EXISTS : thm
val NOT_NIL_CONS : thm
val NOT_NULL_MEM : thm
val NRC_LRC : thm
val NULL : thm
val NULL_EQ : thm
val NULL_GENLIST : thm
val NULL_LENGTH : thm
val REVERSE_APPEND : thm
val REVERSE_EQ_NIL : thm
val REVERSE_EQ_SING : thm
val REVERSE_GENLIST : thm
val REVERSE_REV : thm
val REVERSE_REVERSE : thm
val REVERSE_SNOC : thm
val REVERSE_SNOC_DEF : thm
val REV_REVERSE_LEM : thm
val SET_TO_LIST_CARD : thm
val SET_TO_LIST_EMPTY : thm
val SET_TO_LIST_IND : thm
val SET_TO_LIST_INV : thm
val SET_TO_LIST_IN_MEM : thm
val SET_TO_LIST_SING : thm
val SET_TO_LIST_THM : thm
val SNOC_11 : thm
val SNOC_APPEND : thm
val SNOC_Axiom : thm
val SNOC_CASES : thm
val SNOC_INDUCT : thm
val SUM_APPEND : thm
val SUM_IMAGE_LIST_TO_SET_upper_bound : thm
val SUM_IMAGE_eq_SUM_MAP_SET_TO_LIST : thm
val SUM_MAP_FOLDL : thm
val SUM_MAP_MEM_bound : thm
val SUM_MAP_PLUS_ZIP : thm
val SUM_SNOC : thm
val TAKE_0 : thm
val TAKE_APPEND1 : thm
val TAKE_APPEND2 : thm
val TAKE_DROP : thm
val TAKE_LENGTH_ID : thm
val TAKE_compute : thm
val TL_GENLIST : thm
val UNION_APPEND : thm
val UNZIP_MAP : thm
val UNZIP_THM : thm
val UNZIP_ZIP : thm
val WF_LIST_PRED : thm
val ZIP_GENLIST : thm
val ZIP_MAP : thm
val ZIP_UNZIP : thm
val datatype_list : thm
val isPREFIX_THM : thm
val listRel_CONS : thm
val listRel_LENGTH : thm
val listRel_NIL : thm
val listRel_cases : thm
val listRel_ind : thm
val listRel_rules : thm
val listRel_strong_ind : thm
val listRel_strongind : thm
val list_11 : thm
val list_Axiom : thm
val list_Axiom_old : thm
val list_CASES : thm
val list_INDUCT : thm
val list_case_compute : thm
val list_case_cong : thm
val list_distinct : thm
val list_induction : thm
val list_nchotomy : thm
val list_size_cong : thm
val list_grammars : type_grammar.grammar * term_grammar.grammar
(*
[ind_type] Parent theory of "list"
[operator] Parent theory of "list"
[pred_set] Parent theory of "list"
[ALL_DISTINCT] Definition
|- (ALL_DISTINCT [] ⇔ T) ∧
∀h t. ALL_DISTINCT (h::t) ⇔ ¬MEM h t ∧ ALL_DISTINCT t
[APPEND] Definition
|- (∀l. [] ++ l = l) ∧ ∀l1 l2 h. h::l1 ++ l2 = h::(l1 ++ l2)
[DROP_def] Definition
|- (∀n. DROP n [] = []) ∧
∀n x xs. DROP n (x::xs) = if n = 0 then x::xs else DROP (n − 1) xs
[EL] Definition
|- (∀l. EL 0 l = HD l) ∧ ∀l n. EL (SUC n) l = EL n (TL l)
[EVERY2_curried_def] Definition
|- ∀x x1 x2. EVERY2 x x1 x2 ⇔ EVERY2_tupled (x,x1,x2)
[EVERY2_tupled_primitive_def] Definition
|- EVERY2_tupled =
WFREC (@R. WF R ∧ ∀b a bs as P. R (P,as,bs) (P,a::as,b::bs))
(λEVERY2_tupled a'.
case a' of
(P,[],v3) => I (([] = []) ∧ (v3 = []))
| (P,a::as,[]) => I ((a::as = []) ∧ ([] = []))
| (P,a::as,b::bs) => I (P a b ∧ EVERY2_tupled (P,as,bs)))
[EVERY_DEF] Definition
|- (∀P. EVERY P [] ⇔ T) ∧ ∀P h t. EVERY P (h::t) ⇔ P h ∧ EVERY P t
[EXISTS_DEF] Definition
|- (∀P. EXISTS P [] ⇔ F) ∧ ∀P h t. EXISTS P (h::t) ⇔ P h ∨ EXISTS P t
[FILTER] Definition
|- (∀P. FILTER P [] = []) ∧
∀P h t.
FILTER P (h::t) = if P h then h::FILTER P t else FILTER P t
[FLAT] Definition
|- (FLAT [] = []) ∧ ∀h t. FLAT (h::t) = h ++ FLAT t
[FOLDL] Definition
|- (∀f e. FOLDL f e [] = e) ∧
∀f e x l. FOLDL f e (x::l) = FOLDL f (f e x) l
[FOLDL2_curried_def] Definition
|- ∀x x1 x2 x3. FOLDL2 x x1 x2 x3 = FOLDL2_tupled (x,x1,x2,x3)
[FOLDL2_tupled_primitive_def] Definition
|- FOLDL2_tupled =
WFREC
(@R.
WF R ∧ ∀cs bs c b a f. R (f,f a b c,bs,cs) (f,a,b::bs,c::cs))
(λFOLDL2_tupled a'.
case a' of
(f,a,[],v5) => I a
| (f,a,b::bs,[]) => I a
| (f,a,b::bs,c::cs) => I (FOLDL2_tupled (f,f a b c,bs,cs)))
[FOLDR] Definition
|- (∀f e. FOLDR f e [] = e) ∧
∀f e x l. FOLDR f e (x::l) = f x (FOLDR f e l)
[FRONT_DEF] Definition
|- ∀h t. FRONT (h::t) = if t = [] then [] else h::FRONT t
[GENLIST] Definition
|- (∀f. GENLIST f 0 = []) ∧
∀f n. GENLIST f (SUC n) = SNOC (f n) (GENLIST f n)
[GENLIST_AUX] Definition
|- (∀f l. GENLIST_AUX f 0 l = l) ∧
∀f n l. GENLIST_AUX f (SUC n) l = GENLIST_AUX f n (f n::l)
[HD] Definition
|- ∀h t. HD (h::t) = h
[LAST_DEF] Definition
|- ∀h t. LAST (h::t) = if t = [] then h else LAST t
[LENGTH] Definition
|- (LENGTH [] = 0) ∧ ∀h t. LENGTH (h::t) = SUC (LENGTH t)
[LEN_DEF] Definition
|- (∀n. LEN [] n = n) ∧ ∀h t n. LEN (h::t) n = LEN t (n + 1)
[LIST_TO_SET] Definition
|- set = combin$C MEM
[LRC_def] Definition
|- (∀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
[LUPDATE_DEF] Definition
|- (∀n x. LUPDATE n x [] = []) ∧
∀n x y ys.
LUPDATE n x (y::ys) =
if n = 0 then x::ys else y::LUPDATE (PRE n) x ys
[MAP] Definition
|- (∀f. MAP f [] = []) ∧ ∀f h t. MAP f (h::t) = f h::MAP f t
[MAP2_curried_def] Definition
|- ∀x x1 x2. MAP2 x x1 x2 = MAP2_tupled (x,x1,x2)
[MAP2_tupled_primitive_def] Definition
|- MAP2_tupled =
WFREC (@R. WF R ∧ ∀h2 h1 t2 t1 f. R (f,t1,t2) (f,h1::t1,h2::t2))
(λMAP2_tupled a.
case a of
(f,[],v3) => I []
| (f,h1::t1,[]) => I []
| (f,h1::t1,h2::t2) => I (f h1 h2::MAP2_tupled (f,t1,t2)))
[MEM] Definition
|- (∀x. MEM x [] ⇔ F) ∧ ∀x h t. MEM x (h::t) ⇔ (x = h) ∨ MEM x t
[NULL_DEF] Definition
|- (NULL [] ⇔ T) ∧ ∀h t. NULL (h::t) ⇔ F
[PAD_LEFT] Definition
|- ∀c n s. PAD_LEFT c n s = GENLIST (K c) (n − LENGTH s) ++ s
[PAD_RIGHT] Definition
|- ∀c n s. PAD_RIGHT c n s = s ++ GENLIST (K c) (n − LENGTH s)
[REVERSE_DEF] Definition
|- (REVERSE [] = []) ∧ ∀h t. REVERSE (h::t) = REVERSE t ++ [h]
[REV_DEF] Definition
|- (∀acc. REV [] acc = acc) ∧
∀h t acc. REV (h::t) acc = REV t (h::acc)
[SET_TO_LIST_primitive] Definition
|- 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))
[SNOC] Definition
|- (∀x. SNOC x [] = [x]) ∧ ∀x x' l. SNOC x (x'::l) = x'::SNOC x l
[SUM] Definition
|- (SUM [] = 0) ∧ ∀h t. SUM (h::t) = h + SUM t
[TAKE_def] Definition
|- (∀n. TAKE n [] = []) ∧
∀n x xs. TAKE n (x::xs) = if n = 0 then [] else x::TAKE (n − 1) xs
[TL] Definition
|- ∀h t. TL (h::t) = t
[UNZIP] Definition
|- (UNZIP [] = ([],[])) ∧
∀x l. UNZIP (x::l) = (FST x::FST (UNZIP l),SND x::SND (UNZIP l))
[ZIP] Definition
|- (ZIP ([],[]) = []) ∧
∀x1 l1 x2 l2. ZIP (x1::l1,x2::l2) = (x1,x2)::ZIP (l1,l2)
[isPREFIX] Definition
|- (∀l. [] ≼ l ⇔ T) ∧
∀h t l. h::t ≼ l ⇔ case l of [] => F | h'::t' => (h = h') ∧ t ≼ t'
[listRel] Definition
|- listRel =
(λR a0 a1.
∀listRel'.
(∀a0 a1.
(a0 = []) ∧ (a1 = []) ∨
(∃h1 h2 t1 t2.
(a0 = h1::t1) ∧ (a1 = h2::t2) ∧ R h1 h2 ∧
listRel' t1 t2) ⇒
listRel' a0 a1) ⇒
listRel' a0 a1)
[list_TY_DEF] Definition
|- ∃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
[list_case_def] Definition
|- (∀v f. list_case v f [] = v) ∧
∀v f a0 a1. list_case v f (a0::a1) = f a0 a1
[list_size_def] Definition
|- (∀f. list_size f [] = 0) ∧
∀f a0 a1. list_size f (a0::a1) = 1 + (f a0 + list_size f a1)
[ALL_DISTINCT_APPEND] Theorem
|- ∀l1 l2.
ALL_DISTINCT (l1 ++ l2) ⇔
ALL_DISTINCT l1 ∧ ALL_DISTINCT l2 ∧ ∀e. MEM e l1 ⇒ ¬MEM e l2
[ALL_DISTINCT_CARD_LIST_TO_SET] Theorem
|- ∀ls. ALL_DISTINCT ls ⇒ (CARD (set ls) = LENGTH ls)
[ALL_DISTINCT_EL_IMP] Theorem
|- ∀l n1 n2.
ALL_DISTINCT l ∧ n1 < LENGTH l ∧ n2 < LENGTH l ⇒
((EL n1 l = EL n2 l) ⇔ (n1 = n2))
[ALL_DISTINCT_FILTER] Theorem
|- ∀l. ALL_DISTINCT l ⇔ ∀x. MEM x l ⇒ (FILTER ($= x) l = [x])
[ALL_DISTINCT_GENLIST] Theorem
|- ALL_DISTINCT (GENLIST f n) ⇔
∀m1 m2. m1 < n ∧ m2 < n ∧ (f m1 = f m2) ⇒ (m1 = m2)
[ALL_DISTINCT_MAP] Theorem
|- ∀f ls. ALL_DISTINCT (MAP f ls) ⇒ ALL_DISTINCT ls
[ALL_DISTINCT_REVERSE] Theorem
|- ∀l. ALL_DISTINCT (REVERSE l) ⇔ ALL_DISTINCT l
[ALL_DISTINCT_SET_TO_LIST] Theorem
|- ∀s. FINITE s ⇒ ALL_DISTINCT (SET_TO_LIST s)
[ALL_DISTINCT_SING] Theorem
|- ∀x. ALL_DISTINCT [x]
[ALL_DISTINCT_SNOC] Theorem
|- ∀x l. ALL_DISTINCT (SNOC x l) ⇔ ¬MEM x l ∧ ALL_DISTINCT l
[ALL_DISTINCT_ZIP] Theorem
|- ∀l1 l2.
ALL_DISTINCT l1 ∧ (LENGTH l1 = LENGTH l2) ⇒
ALL_DISTINCT (ZIP (l1,l2))
[ALL_DISTINCT_ZIP_SWAP] Theorem
|- ∀l1 l2.
ALL_DISTINCT (ZIP (l1,l2)) ∧ (LENGTH l1 = LENGTH l2) ⇒
ALL_DISTINCT (ZIP (l2,l1))
[APPEND_11] Theorem
|- (∀l1 l2 l3. (l1 ++ l2 = l1 ++ l3) ⇔ (l2 = l3)) ∧
∀l1 l2 l3. (l2 ++ l1 = l3 ++ l1) ⇔ (l2 = l3)
[APPEND_11_LENGTH] Theorem
|- (∀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_ASSOC] Theorem
|- ∀l1 l2 l3. l1 ++ (l2 ++ l3) = l1 ++ l2 ++ l3
[APPEND_EQ_APPEND] Theorem
|- (l1 ++ l2 = m1 ++ m2) ⇔
(∃l. (l1 = m1 ++ l) ∧ (m2 = l ++ l2)) ∨
∃l. (m1 = l1 ++ l) ∧ (l2 = l ++ m2)
[APPEND_EQ_APPEND_MID] Theorem
|- (l1 ++ [e] ++ l2 = m1 ++ m2) ⇔
(∃l. (m1 = l1 ++ [e] ++ l) ∧ (l2 = l ++ m2)) ∨
∃l. (l1 = m1 ++ l) ∧ (m2 = l ++ [e] ++ l2)
[APPEND_EQ_CONS] Theorem
|- (l1 ++ l2 = h::t) ⇔
(l1 = []) ∧ (l2 = h::t) ∨ ∃lt. (l1 = h::lt) ∧ (t = lt ++ l2)
[APPEND_EQ_SELF] Theorem
|- (∀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_EQ_SING] Theorem
|- (l1 ++ l2 = [e]) ⇔ (l1 = [e]) ∧ (l2 = []) ∨ (l1 = []) ∧ (l2 = [e])
[APPEND_FRONT_LAST] Theorem
|- ∀l. l ≠ [] ⇒ (FRONT l ++ [LAST l] = l)
[APPEND_LENGTH_EQ] Theorem
|- ∀l1 l1'.
(LENGTH l1 = LENGTH l1') ⇒
∀l2 l2'.
(LENGTH l2 = LENGTH l2') ⇒
((l1 ++ l2 = l1' ++ l2') ⇔ (l1 = l1') ∧ (l2 = l2'))
[APPEND_NIL] Theorem
|- ∀l. l ++ [] = l
[APPEND_SNOC] Theorem
|- ∀l1 x l2. l1 ++ SNOC x l2 = SNOC x (l1 ++ l2)
[APPEND_eq_NIL] Theorem
|- (∀l1 l2. ([] = l1 ++ l2) ⇔ (l1 = []) ∧ (l2 = [])) ∧
∀l1 l2. (l1 ++ l2 = []) ⇔ (l1 = []) ∧ (l2 = [])
[CARD_LIST_TO_SET] Theorem
|- CARD (set ls) ≤ LENGTH ls
[CONS] Theorem
|- ∀l. ¬NULL l ⇒ (HD l::TL l = l)
[CONS_11] Theorem
|- ∀a0 a1 a0' a1'. (a0::a1 = a0'::a1') ⇔ (a0 = a0') ∧ (a1 = a1')
[CONS_ACYCLIC] Theorem
|- ∀l x. l ≠ x::l ∧ x::l ≠ l
[DROP_0] Theorem
|- DROP 0 l = l
[DROP_NIL] Theorem
|- ∀ls n. (DROP n ls = []) ⇔ n ≥ LENGTH ls
[DROP_compute] Theorem
|- (∀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
[EL_ALL_DISTINCT_EL_EQ] Theorem
|- ∀l.
ALL_DISTINCT l ⇔
∀n1 n2.
n1 < LENGTH l ∧ n2 < LENGTH l ⇒
((EL n1 l = EL n2 l) ⇔ (n1 = n2))
[EL_GENLIST] Theorem
|- ∀f n x. x < n ⇒ (EL x (GENLIST f n) = f x)
[EL_LENGTH_SNOC] Theorem
|- ∀l x. EL (LENGTH l) (SNOC x l) = x
[EL_LUPDATE] Theorem
|- ∀ys x i k.
EL i (LUPDATE k x ys) =
if (i = k) ∧ k < LENGTH ys then x else EL i ys
[EL_MAP] Theorem
|- ∀n l. n < LENGTH l ⇒ ∀f. EL n (MAP f l) = f (EL n l)
[EL_REVERSE] Theorem
|- ∀n l.
n < LENGTH l ⇒ (EL n (REVERSE l) = EL (PRE (LENGTH l − n)) l)
[EL_SNOC] Theorem
|- ∀n l. n < LENGTH l ⇒ ∀x. EL n (SNOC x l) = EL n l
[EL_ZIP] Theorem
|- ∀l1 l2 n.
(LENGTH l1 = LENGTH l2) ∧ n < LENGTH l1 ⇒
(EL n (ZIP (l1,l2)) = (EL n l1,EL n l2))
[EL_compute] Theorem
|- ∀n. EL n l = if n = 0 then HD l else EL (PRE n) (TL l)
[EL_restricted] Theorem
|- (EL 0 = HD) ∧ (EL (SUC n) (l::ls) = EL n ls)
[EL_simp] Theorem
|- (EL (NUMERAL (BIT1 n)) l = EL (PRE (NUMERAL (BIT1 n))) (TL l)) ∧
(EL (NUMERAL (BIT2 n)) l = EL (NUMERAL (BIT1 n)) (TL l))
[EL_simp_restricted] Theorem
|- (EL (NUMERAL (BIT1 n)) (l::ls) = EL (PRE (NUMERAL (BIT1 n))) ls) ∧
(EL (NUMERAL (BIT2 n)) (l::ls) = EL (NUMERAL (BIT1 n)) ls)
[EQ_LIST] Theorem
|- ∀h1 h2. (h1 = h2) ⇒ ∀l1 l2. (l1 = l2) ⇒ (h1::l1 = h2::l2)
[EVERY2_EVERY] Theorem
|- ∀l1 l2 f.
EVERY2 f l1 l2 ⇔
(LENGTH l1 = LENGTH l2) ∧ EVERY (UNCURRY f) (ZIP (l1,l2))
[EVERY2_LENGTH] Theorem
|- ∀P l1 l2. EVERY2 P l1 l2 ⇒ (LENGTH l1 = LENGTH l2)
[EVERY2_cong] Theorem
|- ∀l1 l1' l2 l2' P P'.
(l1 = l1') ∧ (l2 = l2') ∧
(∀x y. MEM x l1' ∧ MEM y l2' ⇒ (P x y ⇔ P' x y)) ⇒
(EVERY2 P l1 l2 ⇔ EVERY2 P' l1' l2')
[EVERY2_def] Theorem
|- (∀bs b as a P.
EVERY2 P (a::as) (b::bs) ⇔ P a b ∧ EVERY2 P as bs) ∧
(∀bs P. EVERY2 P [] bs ⇔ ([] = []) ∧ (bs = [])) ∧
∀v5 v4 P. EVERY2 P (v4::v5) [] ⇔ (v4::v5 = []) ∧ ([] = [])
[EVERY2_ind] Theorem
|- ∀P'.
(∀P a as b bs. P' P as bs ⇒ P' P (a::as) (b::bs)) ∧
(∀P bs. P' P [] bs) ∧ (∀P v4 v5. P' P (v4::v5) []) ⇒
∀v v1 v2. P' v v1 v2
[EVERY2_mono] Theorem
|- (∀x y. P x y ⇒ Q x y) ⇒ EVERY2 P l1 l2 ⇒ EVERY2 Q l1 l2
[EVERY_APPEND] Theorem
|- ∀P l1 l2. EVERY P (l1 ++ l2) ⇔ EVERY P l1 ∧ EVERY P l2
[EVERY_CONG] Theorem
|- ∀l1 l2 P P'.
(l1 = l2) ∧ (∀x. MEM x l2 ⇒ (P x ⇔ P' x)) ⇒
(EVERY P l1 ⇔ EVERY P' l2)
[EVERY_CONJ] Theorem
|- ∀P Q l. EVERY (λx. P x ∧ Q x) l ⇔ EVERY P l ∧ EVERY Q l
[EVERY_EL] Theorem
|- ∀l P. EVERY P l ⇔ ∀n. n < LENGTH l ⇒ P (EL n l)
[EVERY_FILTER] Theorem
|- ∀P1 P2 l. EVERY P1 (FILTER P2 l) ⇔ EVERY (λx. P2 x ⇒ P1 x) l
[EVERY_FILTER_IMP] Theorem
|- ∀P1 P2 l. EVERY P1 l ⇒ EVERY P1 (FILTER P2 l)
[EVERY_GENLIST] Theorem
|- ∀n. EVERY P (GENLIST f n) ⇔ ∀i. i < n ⇒ P (f i)
[EVERY_MAP] Theorem
|- ∀P f l. EVERY P (MAP f l) ⇔ EVERY (λx. P (f x)) l
[EVERY_MEM] Theorem
|- ∀P l. EVERY P l ⇔ ∀e. MEM e l ⇒ P e
[EVERY_MONOTONIC] Theorem
|- ∀P Q. (∀x. P x ⇒ Q x) ⇒ ∀l. EVERY P l ⇒ EVERY Q l
[EVERY_NOT_EXISTS] Theorem
|- ∀P l. EVERY P l ⇔ ¬EXISTS (λx. ¬P x) l
[EVERY_SIMP] Theorem
|- ∀c l. EVERY (λx. c) l ⇔ (l = []) ∨ c
[EVERY_SNOC] Theorem
|- ∀P x l. EVERY P (SNOC x l) ⇔ EVERY P l ∧ P x
[EXISTS_APPEND] Theorem
|- ∀P l1 l2. EXISTS P (l1 ++ l2) ⇔ EXISTS P l1 ∨ EXISTS P l2
[EXISTS_CONG] Theorem
|- ∀l1 l2 P P'.
(l1 = l2) ∧ (∀x. MEM x l2 ⇒ (P x ⇔ P' x)) ⇒
(EXISTS P l1 ⇔ EXISTS P' l2)
[EXISTS_GENLIST] Theorem
|- ∀n. EXISTS P (GENLIST f n) ⇔ ∃i. i < n ∧ P (f i)
[EXISTS_LIST] Theorem
|- (∃l. P l) ⇔ P [] ∨ ∃h t. P (h::t)
[EXISTS_MAP] Theorem
|- ∀P f l. EXISTS P (MAP f l) ⇔ EXISTS (λx. P (f x)) l
[EXISTS_MEM] Theorem
|- ∀P l. EXISTS P l ⇔ ∃e. MEM e l ∧ P e
[EXISTS_NOT_EVERY] Theorem
|- ∀P l. EXISTS P l ⇔ ¬EVERY (λx. ¬P x) l
[EXISTS_SIMP] Theorem
|- ∀c l. EXISTS (λx. c) l ⇔ l ≠ [] ∧ c
[EXISTS_SNOC] Theorem
|- ∀P x l. EXISTS P (SNOC x l) ⇔ P x ∨ EXISTS P l
[FILTER_ALL_DISTINCT] Theorem
|- ∀P l. ALL_DISTINCT l ⇒ ALL_DISTINCT (FILTER P l)
[FILTER_APPEND_DISTRIB] Theorem
|- ∀P L M. FILTER P (L ++ M) = FILTER P L ++ FILTER P M
[FILTER_COND_REWRITE] Theorem
|- (FILTER P [] = []) ∧
(∀h. P h ⇒ (FILTER P (h::l) = h::FILTER P l)) ∧
∀h. ¬P h ⇒ (FILTER P (h::l) = FILTER P l)
[FILTER_EQ_APPEND] Theorem
|- ∀P l l1 l2.
(FILTER P l = l1 ++ l2) ⇔
∃l3 l4. (l = l3 ++ l4) ∧ (FILTER P l3 = l1) ∧ (FILTER P l4 = l2)
[FILTER_EQ_CONS] Theorem
|- ∀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_EQ_ID] Theorem
|- ∀P l. (FILTER P l = l) ⇔ EVERY P l
[FILTER_EQ_NIL] Theorem
|- ∀P l. (FILTER P l = []) ⇔ EVERY (λx. ¬P x) l
[FILTER_NEQ_ID] Theorem
|- ∀P l. FILTER P l ≠ l ⇔ ∃x. MEM x l ∧ ¬P x
[FILTER_NEQ_NIL] Theorem
|- ∀P l. FILTER P l ≠ [] ⇔ ∃x. MEM x l ∧ P x
[FILTER_REVERSE] Theorem
|- ∀l P. FILTER P (REVERSE l) = REVERSE (FILTER P l)
[FINITE_LIST_TO_SET] Theorem
|- ∀l. FINITE (set l)
[FOLDL2_FOLDL] Theorem
|- ∀l1 l2.
(LENGTH l1 = LENGTH l2) ⇒
∀f a.
FOLDL2 f a l1 l2 = FOLDL (λa. UNCURRY (f a)) a (ZIP (l1,l2))
[FOLDL2_cong] Theorem
|- ∀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] Theorem
|- (∀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] Theorem
|- ∀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
[FOLDL_CONG] Theorem
|- ∀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')
[FOLDL_EQ_FOLDR] Theorem
|- ∀f l e. ASSOC f ∧ COMM f ⇒ (FOLDL f e l = FOLDR f e l)
[FOLDL_SNOC] Theorem
|- ∀f e x l. FOLDL f e (SNOC x l) = f (FOLDL f e l) x
[FOLDL_UNION_BIGUNION] Theorem
|- ∀f ls s.
FOLDL (λs x. s ∪ f x) s ls = s ∪ BIGUNION (IMAGE f (set ls))
[FOLDL_UNION_BIGUNION_paired] Theorem
|- ∀f ls s.
FOLDL (λs (x,y). s ∪ f x y) s ls =
s ∪ BIGUNION (IMAGE (UNCURRY f) (set ls))
[FOLDL_ZIP_SAME] Theorem
|- ∀ls f e. FOLDL f e (ZIP (ls,ls)) = FOLDL (λx y. f x (y,y)) e ls
[FOLDR_CONG] Theorem
|- ∀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')
[FOLDR_CONS] Theorem
|- ∀f ls a. FOLDR (λx y. f x::y) a ls = MAP f ls ++ a
[FORALL_LIST] Theorem
|- (∀l. P l) ⇔ P [] ∧ ∀h t. P (h::t)
[FRONT_CONS] Theorem
|- (∀x. FRONT [x] = []) ∧ ∀x y z. FRONT (x::y::z) = x::FRONT (y::z)
[FRONT_CONS_EQ_NIL] Theorem
|- (∀x xs. (FRONT (x::xs) = []) ⇔ (xs = [])) ∧
(∀x xs. ([] = FRONT (x::xs)) ⇔ (xs = [])) ∧
∀x xs. NULL (FRONT (x::xs)) ⇔ NULL xs
[FRONT_SNOC] Theorem
|- ∀x l. FRONT (SNOC x l) = l
[GENLIST_APPEND] Theorem
|- ∀f a b.
GENLIST f (a + b) = GENLIST f b ++ GENLIST (λt. f (t + b)) a
[GENLIST_AUX_compute] Theorem
|- (∀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)
[GENLIST_CONS] Theorem
|- GENLIST f (SUC n) = f 0::GENLIST (f o SUC) n
[GENLIST_FUN_EQ] Theorem
|- ∀n f g. (GENLIST f n = GENLIST g n) ⇔ ∀x. x < n ⇒ (f x = g x)
[GENLIST_GENLIST_AUX] Theorem
|- ∀n. GENLIST f n = GENLIST_AUX f n []
[GENLIST_NUMERALS] Theorem
|- (GENLIST f 0 = []) ∧
(GENLIST f (NUMERAL n) = GENLIST_AUX f (NUMERAL n) [])
[HD_GENLIST] Theorem
|- HD (GENLIST f (SUC n)) = f 0
[HD_GENLIST_COR] Theorem
|- ∀n f. 0 < n ⇒ (HD (GENLIST f n) = f 0)
[INFINITE_LIST_UNIV] Theorem
|- INFINITE pred_set$UNIV
[INJ_MAP_EQ] Theorem
|- ∀f l1 l2.
INJ f (set l1 ∪ set l2) pred_set$UNIV ∧ (MAP f l1 = MAP f l2) ⇒
(l1 = l2)
[IN_LIST_TO_SET] Theorem
|- x ∈ set l ⇔ MEM x l
[ITSET_eq_FOLDL_SET_TO_LIST] Theorem
|- ∀s.
FINITE s ⇒
∀f a. ITSET f s a = FOLDL (combin$C f) a (SET_TO_LIST s)
[LAST_APPEND_CONS] Theorem
|- ∀h l1 l2. LAST (l1 ++ h::l2) = LAST (h::l2)
[LAST_CONS] Theorem
|- (∀x. LAST [x] = x) ∧ ∀x y z. LAST (x::y::z) = LAST (y::z)
[LAST_CONS_cond] Theorem
|- LAST (h::t) = if t = [] then h else LAST t
[LAST_EL] Theorem
|- ∀ls. ls ≠ [] ⇒ (LAST ls = EL (PRE (LENGTH ls)) ls)
[LAST_SNOC] Theorem
|- ∀x l. LAST (SNOC x l) = x
[LAST_compute] Theorem
|- (∀x. LAST [x] = x) ∧ ∀h1 h2 t. LAST (h1::h2::t) = LAST (h2::t)
[LENGTH_APPEND] Theorem
|- ∀l1 l2. LENGTH (l1 ++ l2) = LENGTH l1 + LENGTH l2
[LENGTH_CONS] Theorem
|- ∀l n. (LENGTH l = SUC n) ⇔ ∃h l'. (LENGTH l' = n) ∧ (l = h::l')
[LENGTH_DROP] Theorem
|- ∀n l. LENGTH (DROP n l) = LENGTH l − n
[LENGTH_EQ_CONS] Theorem
|- ∀P n.
(∀l. (LENGTH l = SUC n) ⇒ P l) ⇔
∀l. (LENGTH l = n) ⇒ (λl. ∀x. P (x::l)) l
[LENGTH_EQ_NIL] Theorem
|- ∀P. (∀l. (LENGTH l = 0) ⇒ P l) ⇔ P []
[LENGTH_EQ_NUM] Theorem
|- (∀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_NUM_compute] Theorem
|- (∀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_SUM] Theorem
|- ∀l n1 n2.
(LENGTH l = n1 + n2) ⇔
∃l1 l2. (LENGTH l1 = n1) ∧ (LENGTH l2 = n2) ∧ (l = l1 ++ l2)
[LENGTH_FRONT_CONS] Theorem
|- ∀x xs. LENGTH (FRONT (x::xs)) = LENGTH xs
[LENGTH_GENLIST] Theorem
|- ∀f n. LENGTH (GENLIST f n) = n
[LENGTH_LEN] Theorem
|- ∀L. LENGTH L = LEN L 0
[LENGTH_LUPDATE] Theorem
|- ∀ys n x. LENGTH (LUPDATE n x ys) = LENGTH ys
[LENGTH_MAP] Theorem
|- ∀l f. LENGTH (MAP f l) = LENGTH l
[LENGTH_NIL] Theorem
|- ∀l. (LENGTH l = 0) ⇔ (l = [])
[LENGTH_NIL_SYM] Theorem
|- (0 = LENGTH l) ⇔ (l = [])
[LENGTH_REVERSE] Theorem
|- ∀l. LENGTH (REVERSE l) = LENGTH l
[LENGTH_SNOC] Theorem
|- ∀x l. LENGTH (SNOC x l) = SUC (LENGTH l)
[LENGTH_TAKE] Theorem
|- ∀n l. n ≤ LENGTH l ⇒ (LENGTH (TAKE n l) = n)
[LENGTH_TL] Theorem
|- ∀l. 0 < LENGTH l ⇒ (LENGTH (TL l) = LENGTH l − 1)
[LENGTH_UNZIP] Theorem
|- ∀pl.
(LENGTH (FST (UNZIP pl)) = LENGTH pl) ∧
(LENGTH (SND (UNZIP pl)) = LENGTH pl)
[LENGTH_ZIP] Theorem
|- ∀l1 l2.
(LENGTH l1 = LENGTH l2) ⇒
(LENGTH (ZIP (l1,l2)) = LENGTH l1) ∧
(LENGTH (ZIP (l1,l2)) = LENGTH l2)
[LEN_LENGTH_LEM] Theorem
|- ∀L n. LEN L n = LENGTH L + n
[LIST_EQ] Theorem
|- ∀l1 l2.
(LENGTH l1 = LENGTH l2) ∧
(∀x. x < LENGTH l1 ⇒ (EL x l1 = EL x l2)) ⇒
(l1 = l2)
[LIST_EQ_REWRITE] Theorem
|- ∀l1 l2.
(l1 = l2) ⇔
(LENGTH l1 = LENGTH l2) ∧
∀x. x < LENGTH l1 ⇒ (EL x l1 = EL x l2)
[LIST_NOT_EQ] Theorem
|- ∀l1 l2. l1 ≠ l2 ⇒ ∀h1 h2. h1::l1 ≠ h2::l2
[LIST_REL_CONJ] Theorem
|- 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_CONS1] Theorem
|- LIST_REL R (h::t) xs ⇔
∃h' t'. (xs = h'::t') ∧ R h h' ∧ LIST_REL R t t'
[LIST_REL_CONS2] Theorem
|- LIST_REL R xs (h::t) ⇔
∃h' t'. (xs = h'::t') ∧ R h' h ∧ LIST_REL R t' t
[LIST_REL_EL_EQN] Theorem
|- ∀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_EVERY_ZIP] Theorem
|- ∀R l1 l2.
LIST_REL R l1 l2 ⇔
(LENGTH l1 = LENGTH l2) ∧ EVERY (UNCURRY R) (ZIP (l1,l2))
[LIST_REL_MAP1] Theorem
|- LIST_REL R (MAP f l1) l2 ⇔ LIST_REL (R o f) l1 l2
[LIST_REL_MAP2] Theorem
|- LIST_REL (λa b. R a b) l1 (MAP f l2) ⇔
LIST_REL (λa b. R a (f b)) l1 l2
[LIST_REL_NIL] Theorem
|- (LIST_REL R [] x ⇔ (x = [])) ∧ (LIST_REL R [] y ⇔ (y = []))
[LIST_REL_cases] Theorem
|- ∀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_def] Theorem
|- (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)
[LIST_REL_ind] Theorem
|- ∀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_mono] Theorem
|- (∀x y. R1 x y ⇒ R2 x y) ⇒ LIST_REL R1 l1 l2 ⇒ LIST_REL R2 l1 l2
[LIST_REL_rules] Theorem
|- ∀R.
LIST_REL R [] [] ∧
∀h1 h2 t1 t2.
R h1 h2 ∧ LIST_REL R t1 t2 ⇒ LIST_REL R (h1::t1) (h2::t2)
[LIST_REL_strongind] Theorem
|- ∀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_TO_SET_APPEND] Theorem
|- ∀l1 l2. set (l1 ++ l2) = set l1 ∪ set l2
[LIST_TO_SET_EQ_EMPTY] Theorem
|- ((set l = ∅) ⇔ (l = [])) ∧ ((∅ = set l) ⇔ (l = []))
[LIST_TO_SET_FILTER] Theorem
|- set (FILTER P l) = {x | P x} ∩ set l
[LIST_TO_SET_MAP] Theorem
|- ∀f l. set (MAP f l) = IMAGE f (set l)
[LIST_TO_SET_REVERSE] Theorem
|- ∀ls. set (REVERSE ls) = set ls
[LIST_TO_SET_SNOC] Theorem
|- set (SNOC x ls) = x INSERT set ls
[LIST_TO_SET_THM] Theorem
|- (set [] = ∅) ∧ (set (h::t) = h INSERT set t)
[LRC_MEM] Theorem
|- LRC R ls x y ∧ MEM e ls ⇒ ∃z t. R e z ∧ LRC R t z y
[LRC_MEM_right] Theorem
|- LRC R (h::t) x y ∧ MEM e t ⇒ ∃z p. R z e ∧ LRC R p x z
[LUPDATE_LENGTH] Theorem
|- ∀xs x y ys. LUPDATE (LENGTH xs) x (xs ++ y::ys) = xs ++ x::ys
[LUPDATE_SNOC] Theorem
|- ∀ys k x y.
LUPDATE k x (SNOC y ys) =
if k = LENGTH ys then SNOC x ys else SNOC y (LUPDATE k x ys)
[MAP2] Theorem
|- (∀f. MAP2 f [] [] = []) ∧
∀f h1 t1 h2 t2. MAP2 f (h1::t1) (h2::t2) = f h1 h2::MAP2 f t1 t2
[MAP2_CONG] Theorem
|- ∀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')
[MAP2_MAP] Theorem
|- ∀l1 l2.
(LENGTH l1 = LENGTH l2) ⇒
∀f. MAP2 f l1 l2 = MAP (UNCURRY f) (ZIP (l1,l2))
[MAP2_ZIP] Theorem
|- ∀l1 l2.
(LENGTH l1 = LENGTH l2) ⇒
∀f. MAP2 f l1 l2 = MAP (UNCURRY f) (ZIP (l1,l2))
[MAP2_def] Theorem
|- (∀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_ind] Theorem
|- ∀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
[MAP_APPEND] Theorem
|- ∀f l1 l2. MAP f (l1 ++ l2) = MAP f l1 ++ MAP f l2
[MAP_CONG] Theorem
|- ∀l1 l2 f f'.
(l1 = l2) ∧ (∀x. MEM x l2 ⇒ (f x = f' x)) ⇒
(MAP f l1 = MAP f' l2)
[MAP_EQ_EVERY2] Theorem
|- ∀f1 f2 l1 l2.
(MAP f1 l1 = MAP f2 l2) ⇔
(LENGTH l1 = LENGTH l2) ∧ EVERY2 (λx y. f1 x = f2 y) l1 l2
[MAP_EQ_NIL] Theorem
|- ∀l f. ((MAP f l = []) ⇔ (l = [])) ∧ (([] = MAP f l) ⇔ (l = []))
[MAP_EQ_f] Theorem
|- ∀f1 f2 l. (MAP f1 l = MAP f2 l) ⇔ ∀e. MEM e l ⇒ (f1 e = f2 e)
[MAP_GENLIST] Theorem
|- ∀f g n. MAP f (GENLIST g n) = GENLIST (f o g) n
[MAP_ID] Theorem
|- (MAP (λx. x) l = l) ∧ (MAP I l = l)
[MAP_MAP_o] Theorem
|- ∀f g l. MAP f (MAP g l) = MAP (f o g) l
[MAP_SNOC] Theorem
|- ∀f x l. MAP f (SNOC x l) = SNOC (f x) (MAP f l)
[MAP_TL] Theorem
|- ∀l f. ¬NULL l ⇒ (MAP f (TL l) = TL (MAP f l))
[MAP_ZIP] Theorem
|- (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)
[MAP_ZIP_SAME] Theorem
|- ∀ls f. MAP f (ZIP (ls,ls)) = MAP (λx. f (x,x)) ls
[MAP_o] Theorem
|- ∀f g. MAP (f o g) = MAP f o MAP g
[MEM_APPEND] Theorem
|- ∀e l1 l2. MEM e (l1 ++ l2) ⇔ MEM e l1 ∨ MEM e l2
[MEM_DROP] Theorem
|- ∀x ls n.
MEM x (DROP n ls) ⇔
n < LENGTH ls ∧ (x = EL n ls) ∨ MEM x (DROP (SUC n) ls)
[MEM_EL] Theorem
|- ∀l x. MEM x l ⇔ ∃n. n < LENGTH l ∧ (x = EL n l)
[MEM_FILTER] Theorem
|- ∀P L x. MEM x (FILTER P L) ⇔ P x ∧ MEM x L
[MEM_FLAT] Theorem
|- ∀x L. MEM x (FLAT L) ⇔ ∃l. MEM l L ∧ MEM x l
[MEM_GENLIST] Theorem
|- MEM x (GENLIST f n) ⇔ ∃m. m < n ∧ (x = f m)
[MEM_MAP] Theorem
|- ∀l f x. MEM x (MAP f l) ⇔ ∃y. (x = f y) ∧ MEM y l
[MEM_REVERSE] Theorem
|- ∀l x. MEM x (REVERSE l) ⇔ MEM x l
[MEM_SET_TO_LIST] Theorem
|- ∀s. FINITE s ⇒ ∀x. MEM x (SET_TO_LIST s) ⇔ x ∈ s
[MEM_SNOC] Theorem
|- ∀y x l. MEM y (SNOC x l) ⇔ (y = x) ∨ MEM y l
[MEM_SPLIT] Theorem
|- ∀x l. MEM x l ⇔ ∃l1 l2. l = l1 ++ x::l2
[MEM_SPLIT_APPEND_first] Theorem
|- MEM e l ⇔ ∃pfx sfx. (l = pfx ++ [e] ++ sfx) ∧ ¬MEM e pfx
[MEM_SPLIT_APPEND_last] Theorem
|- MEM e l ⇔ ∃pfx sfx. (l = pfx ++ [e] ++ sfx) ∧ ¬MEM e sfx
[MEM_ZIP] Theorem
|- ∀l1 l2 p.
(LENGTH l1 = LENGTH l2) ⇒
(MEM p (ZIP (l1,l2)) ⇔
∃n. n < LENGTH l1 ∧ (p = (EL n l1,EL n l2)))
[MONO_EVERY] Theorem
|- (∀x. P x ⇒ Q x) ⇒ EVERY P l ⇒ EVERY Q l
[MONO_EXISTS] Theorem
|- (∀x. P x ⇒ Q x) ⇒ EXISTS P l ⇒ EXISTS Q l
[MONO_listRel] Theorem
|- (∀x y. R x y ⇒ R' x y) ⇒ listRel R x y ⇒ listRel R' x y
[NOT_CONS_NIL] Theorem
|- ∀a1 a0. a0::a1 ≠ []
[NOT_EQ_LIST] Theorem
|- ∀h1 h2. h1 ≠ h2 ⇒ ∀l1 l2. h1::l1 ≠ h2::l2
[NOT_EVERY] Theorem
|- ∀P l. ¬EVERY P l ⇔ EXISTS ($~ o P) l
[NOT_EXISTS] Theorem
|- ∀P l. ¬EXISTS P l ⇔ EVERY ($~ o P) l
[NOT_NIL_CONS] Theorem
|- ∀a1 a0. [] ≠ a0::a1
[NOT_NULL_MEM] Theorem
|- ∀l. ¬NULL l ⇔ ∃e. MEM e l
[NRC_LRC] Theorem
|- NRC R n x y ⇔ ∃ls. LRC R ls x y ∧ (LENGTH ls = n)
[NULL] Theorem
|- NULL [] ∧ ∀h t. ¬NULL (h::t)
[NULL_EQ] Theorem
|- ∀l. NULL l ⇔ (l = [])
[NULL_GENLIST] Theorem
|- ∀n f. NULL (GENLIST f n) ⇔ (n = 0)
[NULL_LENGTH] Theorem
|- ∀l. NULL l ⇔ (LENGTH l = 0)
[REVERSE_APPEND] Theorem
|- ∀l1 l2. REVERSE (l1 ++ l2) = REVERSE l2 ++ REVERSE l1
[REVERSE_EQ_NIL] Theorem
|- (REVERSE l = []) ⇔ (l = [])
[REVERSE_EQ_SING] Theorem
|- (REVERSE l = [e]) ⇔ (l = [e])
[REVERSE_GENLIST] Theorem
|- REVERSE (GENLIST f n) = GENLIST (λm. f (PRE n − m)) n
[REVERSE_REV] Theorem
|- ∀L. REVERSE L = REV L []
[REVERSE_REVERSE] Theorem
|- ∀l. REVERSE (REVERSE l) = l
[REVERSE_SNOC] Theorem
|- ∀x l. REVERSE (SNOC x l) = x::REVERSE l
[REVERSE_SNOC_DEF] Theorem
|- (REVERSE [] = []) ∧ ∀x l. REVERSE (x::l) = SNOC x (REVERSE l)
[REV_REVERSE_LEM] Theorem
|- ∀L1 L2. REV L1 L2 = REVERSE L1 ++ L2
[SET_TO_LIST_CARD] Theorem
|- ∀s. FINITE s ⇒ (LENGTH (SET_TO_LIST s) = CARD s)
[SET_TO_LIST_EMPTY] Theorem
|- SET_TO_LIST ∅ = []
[SET_TO_LIST_IND] Theorem
|- ∀P. (∀s. (FINITE s ∧ s ≠ ∅ ⇒ P (REST s)) ⇒ P s) ⇒ ∀v. P v
[SET_TO_LIST_INV] Theorem
|- ∀s. FINITE s ⇒ (set (SET_TO_LIST s) = s)
[SET_TO_LIST_IN_MEM] Theorem
|- ∀s. FINITE s ⇒ ∀x. x ∈ s ⇔ MEM x (SET_TO_LIST s)
[SET_TO_LIST_SING] Theorem
|- SET_TO_LIST {x} = [x]
[SET_TO_LIST_THM] Theorem
|- FINITE s ⇒
(SET_TO_LIST s =
if s = ∅ then [] else CHOICE s::SET_TO_LIST (REST s))
[SNOC_11] Theorem
|- ∀x y a b. (SNOC x y = SNOC a b) ⇔ (x = a) ∧ (y = b)
[SNOC_APPEND] Theorem
|- ∀x l. SNOC x l = l ++ [x]
[SNOC_Axiom] Theorem
|- ∀e f. ∃fn. (fn [] = e) ∧ ∀x l. fn (SNOC x l) = f x l (fn l)
[SNOC_CASES] Theorem
|- ∀ll. (ll = []) ∨ ∃x l. ll = SNOC x l
[SNOC_INDUCT] Theorem
|- ∀P. P [] ∧ (∀l. P l ⇒ ∀x. P (SNOC x l)) ⇒ ∀l. P l
[SUM_APPEND] Theorem
|- ∀l1 l2. SUM (l1 ++ l2) = SUM l1 + SUM l2
[SUM_IMAGE_LIST_TO_SET_upper_bound] Theorem
|- ∀ls. ∑ f (set ls) ≤ SUM (MAP f ls)
[SUM_IMAGE_eq_SUM_MAP_SET_TO_LIST] Theorem
|- FINITE s ⇒ (∑ f s = SUM (MAP f (SET_TO_LIST s)))
[SUM_MAP_FOLDL] Theorem
|- ∀ls. SUM (MAP f ls) = FOLDL (λa e. a + f e) 0 ls
[SUM_MAP_MEM_bound] Theorem
|- ∀f x ls. MEM x ls ⇒ f x ≤ SUM (MAP f ls)
[SUM_MAP_PLUS_ZIP] Theorem
|- ∀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))
[SUM_SNOC] Theorem
|- ∀x l. SUM (SNOC x l) = SUM l + x
[TAKE_0] Theorem
|- TAKE 0 l = []
[TAKE_APPEND1] Theorem
|- ∀n. n ≤ LENGTH l1 ⇒ (TAKE n (l1 ++ l2) = TAKE n l1)
[TAKE_APPEND2] Theorem
|- ∀n.
LENGTH l1 < n ⇒
(TAKE n (l1 ++ l2) = l1 ++ TAKE (n − LENGTH l1) l2)
[TAKE_DROP] Theorem
|- ∀n l. TAKE n l ++ DROP n l = l
[TAKE_LENGTH_ID] Theorem
|- ∀l. TAKE (LENGTH l) l = l
[TAKE_compute] Theorem
|- (∀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
[TL_GENLIST] Theorem
|- ∀f n. TL (GENLIST f (SUC n)) = GENLIST (f o SUC) n
[UNION_APPEND] Theorem
|- ∀l1 l2. set l1 ∪ set l2 = set (l1 ++ l2)
[UNZIP_MAP] Theorem
|- ∀L. UNZIP L = (MAP FST L,MAP SND L)
[UNZIP_THM] Theorem
|- (UNZIP [] = ([],[])) ∧
(UNZIP ((x,y)::t) = (let (L1,L2) = UNZIP t in (x::L1,y::L2)))
[UNZIP_ZIP] Theorem
|- ∀l1 l2. (LENGTH l1 = LENGTH l2) ⇒ (UNZIP (ZIP (l1,l2)) = (l1,l2))
[WF_LIST_PRED] Theorem
|- WF (λL1 L2. ∃h. L2 = h::L1)
[ZIP_GENLIST] Theorem
|- ∀l f n.
(LENGTH l = n) ⇒
(ZIP (l,GENLIST f n) = GENLIST (λx. (EL x l,f x)) n)
[ZIP_MAP] Theorem
|- ∀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)))
[ZIP_UNZIP] Theorem
|- ∀l. ZIP (UNZIP l) = l
[datatype_list] Theorem
|- DATATYPE (list [] CONS)
[isPREFIX_THM] Theorem
|- ([] ≼ l ⇔ T) ∧ (h::t ≼ [] ⇔ F) ∧
(h1::t1 ≼ h2::t2 ⇔ (h1 = h2) ∧ t1 ≼ t2)
[listRel_CONS] Theorem
|- (listRel R (h::t) y ⇔
∃h' t'. (y = h'::t') ∧ R h h' ∧ listRel R t t') ∧
(listRel R x (h'::t') ⇔
∃h t. (x = h::t) ∧ R h h' ∧ listRel R t t')
[listRel_LENGTH] Theorem
|- ∀x y. listRel R x y ⇒ (LENGTH x = LENGTH y)
[listRel_NIL] Theorem
|- (listRel R [] y ⇔ (y = [])) ∧ (listRel R x [] ⇔ (x = []))
[listRel_cases] Theorem
|- ∀R a0 a1.
listRel R a0 a1 ⇔
(a0 = []) ∧ (a1 = []) ∨
∃h1 h2 t1 t2.
(a0 = h1::t1) ∧ (a1 = h2::t2) ∧ R h1 h2 ∧ listRel R t1 t2
[listRel_ind] Theorem
|- ∀R listRel'.
listRel' [] [] ∧
(∀h1 h2 t1 t2.
R h1 h2 ∧ listRel' t1 t2 ⇒ listRel' (h1::t1) (h2::t2)) ⇒
∀a0 a1. listRel R a0 a1 ⇒ listRel' a0 a1
[listRel_rules] Theorem
|- ∀R.
listRel R [] [] ∧
∀h1 h2 t1 t2.
R h1 h2 ∧ listRel R t1 t2 ⇒ listRel R (h1::t1) (h2::t2)
[listRel_strong_ind] Theorem
|- ∀R listRel'.
listRel' [] [] ∧
(∀h1 h2 t1 t2.
R h1 h2 ∧ listRel R t1 t2 ∧ listRel' t1 t2 ⇒
listRel' (h1::t1) (h2::t2)) ⇒
∀a0 a1. listRel R a0 a1 ⇒ listRel' a0 a1
[listRel_strongind] Theorem
|- ∀R listRel'.
listRel' [] [] ∧
(∀h1 h2 t1 t2.
R h1 h2 ∧ listRel R t1 t2 ∧ listRel' t1 t2 ⇒
listRel' (h1::t1) (h2::t2)) ⇒
∀a0 a1. listRel R a0 a1 ⇒ listRel' a0 a1
[list_11] Theorem
|- ∀a0 a1 a0' a1'. (a0::a1 = a0'::a1') ⇔ (a0 = a0') ∧ (a1 = a1')
[list_Axiom] Theorem
|- ∀f0 f1. ∃fn. (fn [] = f0) ∧ ∀a0 a1. fn (a0::a1) = f1 a0 a1 (fn a1)
[list_Axiom_old] Theorem
|- ∀x f. ∃!fn1. (fn1 [] = x) ∧ ∀h t. fn1 (h::t) = f (fn1 t) h t
[list_CASES] Theorem
|- ∀l. (l = []) ∨ ∃t h. l = h::t
[list_INDUCT] Theorem
|- ∀P. P [] ∧ (∀t. P t ⇒ ∀h. P (h::t)) ⇒ ∀l. P l
[list_case_compute] Theorem
|- ∀l. list_case b f l = if NULL l then b else f (HD l) (TL l)
[list_case_cong] Theorem
|- ∀M M' v f.
(M = M') ∧ ((M' = []) ⇒ (v = v')) ∧
(∀a0 a1. (M' = a0::a1) ⇒ (f a0 a1 = f' a0 a1)) ⇒
(list_case v f M = list_case v' f' M')
[list_distinct] Theorem
|- ∀a1 a0. [] ≠ a0::a1
[list_induction] Theorem
|- ∀P. P [] ∧ (∀t. P t ⇒ ∀h. P (h::t)) ⇒ ∀l. P l
[list_nchotomy] Theorem
|- ∀l. (l = []) ∨ ∃t h. l = h::t
[list_size_cong] Theorem
|- ∀M N f f'.
(M = N) ∧ (∀x. MEM x N ⇒ (f x = f' x)) ⇒
(list_size f M = list_size f' N)
*)
end
HOL 4, Kananaskis-8