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 EVERY_DEF : thm
val EVERYi_def : thm
val EXISTS_DEF : thm
val FILTER : thm
val FIND_def : thm
val FLAT : thm
val FOLDL : thm
val FOLDR : thm
val FRONT_DEF : thm
val GENLIST : thm
val GENLIST_AUX : thm
val HD : thm
val INDEX_FIND_def : thm
val INDEX_OF_def : thm
val LAST_DEF : thm
val LENGTH : thm
val LEN_DEF : thm
val LIST_APPLY_def : thm
val LIST_BIND_def : thm
val LIST_GUARD_def : thm
val LIST_IGNORE_BIND_def : thm
val LIST_LIFT2_def : thm
val LIST_TO_SET_DEF : thm
val LLEX_def : thm
val LRC_def : thm
val LUPDATE_def : thm
val MAP : thm
val NULL_DEF : thm
val OPT_MMAP_def : thm
val PAD_LEFT : thm
val PAD_RIGHT : thm
val REVERSE_DEF : thm
val REV_DEF : thm
val SET_TO_LIST_primitive_def : thm
val SHORTLEX_def : thm
val SNOC : thm
val SUM : thm
val SUM_ACC_DEF : thm
val TAKE_def : thm
val TL_DEF : thm
val UNIQUE_DEF : thm
val UNZIP : thm
val ZIP_def : thm
val adjacent_def : thm
val dropWhile_def : thm
val isPREFIX : thm
val list_TY_DEF : thm
val list_case_def : thm
val list_size_def : thm
val nub_def : thm
val oEL_def : thm
val oHD_def : thm
val splitAtPki_def : thm
(* Theorems *)
val ALL_DISTINCT_APPEND : thm
val ALL_DISTINCT_CARD_LIST_TO_SET : thm
val ALL_DISTINCT_DROP : thm
val ALL_DISTINCT_EL_IMP : thm
val ALL_DISTINCT_FILTER : thm
val ALL_DISTINCT_FILTER_EL_IMP : thm
val ALL_DISTINCT_FLAT_REVERSE : thm
val ALL_DISTINCT_GENLIST : thm
val ALL_DISTINCT_INDEX_OF_EL : thm
val ALL_DISTINCT_MAP : thm
val ALL_DISTINCT_MAP_INJ : 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 BIGUNION_IMAGE_set_SUBSET : thm
val CARD_LIST_TO_SET : thm
val CARD_LIST_TO_SET_ALL_DISTINCT : thm
val CARD_LIST_TO_SET_EQN : thm
val CONS : thm
val CONS_11 : thm
val CONS_ACYCLIC : thm
val DISJOINT_GENLIST_PLUS : thm
val DROP_0 : thm
val DROP_EQ_NIL : thm
val DROP_GENLIST : thm
val DROP_LENGTH_TOO_LONG : thm
val DROP_TAKE : thm
val DROP_compute : thm
val DROP_cons : thm
val DROP_nil : thm
val DROP_splitAtPki : thm
val EL_ALL_DISTINCT_EL_EQ : thm
val EL_APPEND_EQN : thm
val EL_DROP : thm
val EL_GENLIST : thm
val EL_LENGTH_SNOC : thm
val EL_LENGTH_dropWhile_REVERSE : thm
val EL_LUPDATE : thm
val EL_MAP : thm
val EL_MAP2 : thm
val EL_REVERSE : thm
val EL_SNOC : thm
val EL_TAKE : 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_LUPDATE_same : thm
val EVERY2_MAP : thm
val EVERY2_MEM_MONO : thm
val EVERY2_REVERSE : thm
val EVERY2_THM : thm
val EVERY2_cong : thm
val EVERY2_mono : thm
val EVERY2_refl : thm
val EVERY2_sym : thm
val EVERY2_trans : 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_FLAT : thm
val EVERY_GENLIST : thm
val EVERY_MAP : thm
val EVERY_MEM : thm
val EVERY_MEM_MONO : 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_LIST_EQ_MAP : 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_F : thm
val FILTER_NEQ_ID : thm
val FILTER_NEQ_NIL : thm
val FILTER_REVERSE : thm
val FILTER_T : thm
val FINITE_LIST_TO_SET : thm
val FLAT_APPEND : thm
val FLAT_EQ_NIL : thm
val FLAT_compute : 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_CONG : thm
val GENLIST_CONS : thm
val GENLIST_EL : thm
val GENLIST_EL_MAP : thm
val GENLIST_FUN_EQ : thm
val GENLIST_GENLIST_AUX : thm
val GENLIST_ID : thm
val GENLIST_NUMERALS : thm
val GENLIST_PLUS_APPEND : thm
val HD_DROP : thm
val HD_GENLIST : thm
val HD_GENLIST_COR : thm
val HD_REVERSE : thm
val HD_dropWhile : thm
val IMAGE_EL_count_LENGTH : thm
val IMP_EVERY_LUPDATE : thm
val INDEX_FIND_add : thm
val INDEX_OF_eq_NONE : thm
val INDEX_OF_eq_SOME : thm
val INFINITE_LIST_UNIV : thm
val INJ_MAP_EQ : thm
val INJ_MAP_EQ_IFF : 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_MAP : thm
val LAST_REVERSE : 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_FILTER_LEQ_MONO : thm
val LENGTH_FRONT_CONS : thm
val LENGTH_GENLIST : thm
val LENGTH_LEN : thm
val LENGTH_LT_SHORTLEX : thm
val LENGTH_LUPDATE : thm
val LENGTH_MAP : thm
val LENGTH_MAP2 : thm
val LENGTH_NIL : thm
val LENGTH_NIL_SYM : thm
val LENGTH_REVERSE : thm
val LENGTH_SNOC : thm
val LENGTH_TAKE : thm
val LENGTH_TAKE_EQ : thm
val LENGTH_TL : thm
val LENGTH_UNZIP : thm
val LENGTH_ZIP : thm
val LENGTH_ZIP_MIN : thm
val LENGTH_dropWhile_LESS_EQ : thm
val LENGTH_o_REVERSE : thm
val LEN_LENGTH_LEM : thm
val LIST_APPLY_o : thm
val LIST_BIND_APPEND : thm
val LIST_BIND_ID : thm
val LIST_BIND_LIST_BIND : thm
val LIST_BIND_MAP : thm
val LIST_BIND_THM : thm
val LIST_EQ : thm
val LIST_EQ_MAP_PAIR : thm
val LIST_EQ_REWRITE : thm
val LIST_NOT_EQ : thm
val LIST_REL_APPEND : thm
val LIST_REL_APPEND_EQ : thm
val LIST_REL_APPEND_IMP : thm
val LIST_REL_APPEND_suff : 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_LENGTH : thm
val LIST_REL_MAP1 : thm
val LIST_REL_MAP2 : thm
val LIST_REL_MAP_inv_image : thm
val LIST_REL_MEM_IMP : thm
val LIST_REL_NIL : thm
val LIST_REL_O : thm
val LIST_REL_SNOC : thm
val LIST_REL_SPLIT1 : thm
val LIST_REL_SPLIT2 : thm
val LIST_REL_cases : thm
val LIST_REL_def : thm
val LIST_REL_eq : thm
val LIST_REL_ind : thm
val LIST_REL_mono : thm
val LIST_REL_rules : thm
val LIST_REL_strongind : thm
val LIST_REL_trans : thm
val LIST_TO_SET : thm
val LIST_TO_SET_APPEND : thm
val LIST_TO_SET_EQ_EMPTY : thm
val LIST_TO_SET_FILTER : thm
val LIST_TO_SET_FLAT : thm
val LIST_TO_SET_GENLIST : 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 LLEX_CONG : thm
val LLEX_EL_THM : thm
val LLEX_MONO : thm
val LLEX_NIL2 : thm
val LLEX_THM : thm
val LLEX_not_WF : thm
val LLEX_total : thm
val LLEX_transitive : thm
val LRC_MEM : thm
val LRC_MEM_right : thm
val LUPDATE_GENLIST : thm
val LUPDATE_LENGTH : thm
val LUPDATE_MAP : thm
val LUPDATE_NIL : thm
val LUPDATE_SAME : thm
val LUPDATE_SEM : thm
val LUPDATE_SNOC : thm
val LUPDATE_SOME_MAP : thm
val LUPDATE_compute : thm
val MAP2 : thm
val MAP2_APPEND : thm
val MAP2_CONG : thm
val MAP2_DEF : thm
val MAP2_MAP : thm
val MAP2_NIL : thm
val MAP2_ZIP : thm
val MAP2_ind : thm
val MAP_APPEND : thm
val MAP_APPEND_MAP_EQ : thm
val MAP_CONG : thm
val MAP_DROP : thm
val MAP_EQ_APPEND : thm
val MAP_EQ_CONS : thm
val MAP_EQ_EVERY2 : thm
val MAP_EQ_NIL : thm
val MAP_EQ_SING : thm
val MAP_EQ_f : thm
val MAP_FLAT : thm
val MAP_FRONT : thm
val MAP_GENLIST : thm
val MAP_ID : thm
val MAP_LIST_BIND : thm
val MAP_MAP_o : thm
val MAP_SNOC : thm
val MAP_TAKE : thm
val MAP_TL : thm
val MAP_ZIP : thm
val MAP_ZIP_SAME : thm
val MAP_o : thm
val MEM : thm
val MEM_APPEND : thm
val MEM_APPEND_lemma : thm
val MEM_DROP : thm
val MEM_EL : thm
val MEM_FILTER : thm
val MEM_FLAT : thm
val MEM_GENLIST : thm
val MEM_LUPDATE : thm
val MEM_LUPDATE_E : thm
val MEM_MAP : thm
val MEM_MAP_f : 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_TL : thm
val MEM_ZIP : thm
val MEM_ZIP_MEM_MAP : thm
val MEM_dropWhile_IMP : thm
val MEM_nub : thm
val MONO_EVERY : thm
val MONO_EXISTS : 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_NIL_EQ_LENGTH_NOT_0 : thm
val NOT_NULL_MEM : thm
val NRC_LRC : thm
val NULL : thm
val NULL_APPEND : thm
val NULL_EQ : thm
val NULL_FILTER : thm
val NULL_GENLIST : thm
val NULL_LENGTH : thm
val OPT_MMAP_cong : thm
val REVERSE_11 : 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 REVERSE_o_REVERSE : 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 SHORTLEX_LENGTH_LE : thm
val SHORTLEX_MONO : thm
val SHORTLEX_NIL2 : thm
val SHORTLEX_THM : thm
val SHORTLEX_total : thm
val SHORTLEX_transitive : thm
val SINGL_APPLY_MAP : thm
val SINGL_APPLY_PERMUTE : thm
val SINGL_LIST_APPLY_L : thm
val SINGL_LIST_APPLY_R : thm
val SINGL_SINGL_APPLY : thm
val SING_HD : thm
val SNOC_11 : thm
val SNOC_APPEND : thm
val SNOC_Axiom : thm
val SNOC_CASES : thm
val SNOC_INDUCT : thm
val SUM_ACC_SUM_LEM : 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 : thm
val SUM_MAP_PLUS_ZIP : thm
val SUM_SNOC : thm
val SUM_SUM_ACC : thm
val SUM_eq_0 : thm
val SWAP_REVERSE : thm
val SWAP_REVERSE_SYM : thm
val TAKE1 : thm
val TAKE1_DROP : thm
val TAKE_0 : thm
val TAKE_APPEND1 : thm
val TAKE_APPEND2 : thm
val TAKE_DROP : thm
val TAKE_DROP_SWAP : thm
val TAKE_EQ_NIL : thm
val TAKE_GENLIST : thm
val TAKE_LENGTH_ID : thm
val TAKE_LENGTH_ID_rwt : thm
val TAKE_LENGTH_ID_rwt2 : thm
val TAKE_LENGTH_TOO_LONG : thm
val TAKE_SUM : thm
val TAKE_TAKE_MIN : thm
val TAKE_compute : thm
val TAKE_cons : thm
val TAKE_nil : thm
val TAKE_splitAtPki : thm
val TL : thm
val TL_GENLIST : thm
val UNION_APPEND : thm
val UNIQUE_FILTER : thm
val UNIQUE_LENGTH_FILTER : thm
val UNZIP_MAP : thm
val UNZIP_THM : thm
val UNZIP_ZIP : thm
val WF_LIST_PRED : thm
val WF_SHORTLEX : thm
val WF_SHORTLEX_same_lengths : thm
val ZIP : thm
val ZIP_DROP : thm
val ZIP_EQ_NIL : thm
val ZIP_GENLIST : thm
val ZIP_MAP : thm
val ZIP_UNZIP : thm
val adjacent_EL : thm
val adjacent_cases : thm
val adjacent_iff : thm
val adjacent_ind : thm
val adjacent_rules : thm
val adjacent_strongind : thm
val adjacent_thm : thm
val all_distinct_nub : thm
val all_distinct_nub_id : thm
val bounded_length_FINITE : thm
val datatype_list : thm
val dropWhile_APPEND_EVERY : thm
val dropWhile_APPEND_EXISTS : thm
val dropWhile_eq_nil : thm
val dropWhile_splitAtPki : thm
val el_append3 : thm
val every_zip_fst : thm
val every_zip_snd : thm
val exists_list_GENLIST : thm
val isPREFIX_CONSR : thm
val isPREFIX_NILR : thm
val isPREFIX_THM : thm
val last_drop : thm
val lazy_list_case_compute : thm
val length_nub_append : thm
val list_11 : thm
val list_Axiom : thm
val list_Axiom_old : thm
val list_CASES : thm
val list_INDUCT : thm
val list_INDUCT0 : thm
val list_case_compute : thm
val list_case_cong : thm
val list_case_eq : thm
val list_distinct : thm
val list_induction : thm
val list_nchotomy : thm
val list_size_cong : thm
val list_to_set_diff : thm
val lupdate_append : thm
val lupdate_append2 : thm
val mem_exists_set : thm
val nub_EQ0 : thm
val nub_MAP_INJ : thm
val nub_NIL : thm
val nub_append : thm
val nub_set : thm
val oEL_DROP : thm
val oEL_EQ_EL : thm
val oEL_LUPDATE : thm
val oEL_TAKE_E : thm
val oEL_THM : thm
val oHD_thm : thm
val splitAtPki_APPEND : thm
val splitAtPki_EQN : thm
val splitAtPki_MAP : thm
val splitAtPki_RAND : thm
val splitAtPki_change_predicate : thm
val list_grammars : type_grammar.grammar * term_grammar.grammar
(*
[ind_type] 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)
[EVERY_DEF] Definition
⊢ (∀P. EVERY P [] ⇔ T) ∧ ∀P h t. EVERY P (h::t) ⇔ P h ∧ EVERY P t
[EVERYi_def] Definition
⊢ (∀P. EVERYi P [] ⇔ T) ∧
∀P h t. EVERYi P (h::t) ⇔ P 0 h ∧ EVERYi (P ∘ SUC) 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
[FIND_def] Definition
⊢ ∀P. FIND P = OPTION_MAP SND ∘ INDEX_FIND 0 P
[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
[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
[INDEX_FIND_def] Definition
⊢ (∀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
[INDEX_OF_def] Definition
⊢ ∀x. INDEX_OF x = OPTION_MAP FST ∘ INDEX_FIND 0 ($= x)
[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_APPLY_def] Definition
⊢ ∀fs xs. fs <*> xs = LIST_BIND fs (flip MAP xs)
[LIST_BIND_def] Definition
⊢ ∀l f. LIST_BIND l f = FLAT (MAP f l)
[LIST_GUARD_def] Definition
⊢ ∀b. LIST_GUARD b = if b then [()] else []
[LIST_IGNORE_BIND_def] Definition
⊢ ∀m1 m2. LIST_IGNORE_BIND m1 m2 = LIST_BIND m1 (K m2)
[LIST_LIFT2_def] Definition
⊢ ∀f xs ys. LIST_LIFT2 f xs ys = MAP f xs <*> ys
[LIST_TO_SET_DEF] Definition
⊢ (∀x. set [] x ⇔ F) ∧ ∀h t x. set (h::t) x ⇔ x = h ∨ set t x
[LLEX_def] Definition
⊢ (∀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
[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
⊢ (∀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
[MAP] Definition
⊢ (∀f. MAP f [] = []) ∧ ∀f h t. MAP f (h::t) = f h::MAP f t
[NULL_DEF] Definition
⊢ (NULL [] ⇔ T) ∧ ∀h t. NULL (h::t) ⇔ F
[OPT_MMAP_def] Definition
⊢ (∀f. OPT_MMAP f [] = SOME []) ∧
∀f h0 t0.
OPT_MMAP f (h0::t0) =
OPTION_BIND (f h0)
(λh. OPTION_BIND (OPT_MMAP f t0) (λt. SOME (h::t)))
[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_def] Definition
⊢ SET_TO_LIST =
WFREC (@R. WF R ∧ ∀s. FINITE s ∧ s ≠ ∅ ⇒ R (REST s) s)
(λSET_TO_LIST a.
I
(if FINITE a then
if a = ∅ then [] else CHOICE a::SET_TO_LIST (REST a)
else ARB))
[SHORTLEX_def] Definition
⊢ (∀R l2. SHORTLEX R [] l2 ⇔ l2 ≠ []) ∧
∀R h1 t1 l2.
SHORTLEX R (h1::t1) l2 ⇔
case l2 of
[] => F
| h2::t2 =>
if LENGTH t1 < LENGTH t2 then T
else if LENGTH t1 = LENGTH t2 then
if R h1 h2 then T
else if h1 = h2 then SHORTLEX R t1 t2
else F
else F
[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
[SUM_ACC_DEF] Definition
⊢ (∀acc. SUM_ACC [] acc = acc) ∧
∀h t acc. SUM_ACC (h::t) acc = SUM_ACC t (h + acc)
[TAKE_def] Definition
⊢ (∀n. TAKE n [] = []) ∧
∀n x xs. TAKE n (x::xs) = if n = 0 then [] else x::TAKE (n − 1) xs
[TL_DEF] Definition
⊢ TL [] = [] ∧ ∀h t. TL (h::t) = t
[UNIQUE_DEF] Definition
⊢ ∀e L.
UNIQUE e L ⇔ ∃L1 L2. L1 ⧺ [e] ⧺ L2 = L ∧ ¬MEM e L1 ∧ ¬MEM e L2
[UNZIP] Definition
⊢ UNZIP [] = ([],[]) ∧
∀x l. UNZIP (x::l) = (FST x::FST (UNZIP l),SND x::SND (UNZIP l))
[ZIP_def] Definition
⊢ (∀l2. ZIP ([],l2) = []) ∧ (∀l1. ZIP (l1,[]) = []) ∧
∀x1 l1 x2 l2. ZIP (x1::l1,x2::l2) = (x1,x2)::ZIP (l1,l2)
[adjacent_def] Definition
⊢ adjacent =
(λa0 a1 a2.
∀adjacent'.
(∀a0 a1 a2.
(∃t. a0 = a1::a2::t) ∨
(∃h t. a0 = h::t ∧ adjacent' t a1 a2) ⇒
adjacent' a0 a1 a2) ⇒
adjacent' a0 a1 a2)
[dropWhile_def] Definition
⊢ (∀P. dropWhile P [] = []) ∧
∀P h t. dropWhile P (h::t) = if P h then dropWhile P t else h::t
[isPREFIX] Definition
⊢ (∀l. [] ≼ l ⇔ T) ∧
∀h t l. h::t ≼ l ⇔ case l of [] => F | h'::t' => h = h' ∧ t ≼ t'
[list_TY_DEF] Definition
⊢ ∃rep.
TYPE_DEFINITION
(λa0'.
∀ $var$('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 ∧ $var$('list') a1) ⇒
$var$('list') a0') ⇒
$var$('list') a0') rep
[list_case_def] Definition
⊢ (∀v f. list_CASE [] v f = v) ∧
∀a0 a1 v f. list_CASE (a0::a1) v f = 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)
[nub_def] Definition
⊢ nub [] = [] ∧
∀x l. nub (x::l) = if MEM x l then nub l else x::nub l
[oEL_def] Definition
⊢ (∀n. oEL n [] = NONE) ∧
∀n x xs. oEL n (x::xs) = if n = 0 then SOME x else oEL (n − 1) xs
[oHD_def] Definition
⊢ ∀l. oHD l = case l of [] => NONE | h::v1 => SOME h
[splitAtPki_def] Definition
⊢ (∀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 ∘ SUC) (λp s. k (h::p) s) t
[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_DROP] Theorem
⊢ ∀ls n. ALL_DISTINCT ls ⇒ ALL_DISTINCT (DROP n 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_FILTER_EL_IMP] Theorem
⊢ ∀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
[ALL_DISTINCT_FLAT_REVERSE] Theorem
⊢ ∀xs. ALL_DISTINCT (FLAT (REVERSE xs)) ⇔ ALL_DISTINCT (FLAT xs)
[ALL_DISTINCT_GENLIST] Theorem
⊢ ALL_DISTINCT (GENLIST f n) ⇔
∀m1 m2. m1 < n ∧ m2 < n ∧ f m1 = f m2 ⇒ m1 = m2
[ALL_DISTINCT_INDEX_OF_EL] Theorem
⊢ ∀l n. ALL_DISTINCT l ∧ n < LENGTH l ⇒ INDEX_OF (EL n l) l = SOME n
[ALL_DISTINCT_MAP] Theorem
⊢ ∀f ls. ALL_DISTINCT (MAP f ls) ⇒ ALL_DISTINCT ls
[ALL_DISTINCT_MAP_INJ] Theorem
⊢ ∀ls f.
(∀x y. MEM x ls ∧ MEM y ls ∧ f x = f y ⇒ x = y) ∧ ALL_DISTINCT ls ⇒
ALL_DISTINCT (MAP f 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 = []
[BIGUNION_IMAGE_set_SUBSET] Theorem
⊢ BIGUNION (IMAGE f (set ls)) ⊆ s ⇔ ∀x. MEM x ls ⇒ f x ⊆ s
[CARD_LIST_TO_SET] Theorem
⊢ CARD (set ls) ≤ LENGTH ls
[CARD_LIST_TO_SET_ALL_DISTINCT] Theorem
⊢ ∀ls. CARD (set ls) = LENGTH ls ⇒ ALL_DISTINCT ls
[CARD_LIST_TO_SET_EQN] Theorem
⊢ CARD (set l) = LENGTH (nub l)
[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
[DISJOINT_GENLIST_PLUS] Theorem
⊢ DISJOINT x (set (GENLIST ($+ n) (a + b))) ⇒
DISJOINT x (set (GENLIST ($+ n) a)) ∧
DISJOINT x (set (GENLIST ($+ (n + a)) b))
[DROP_0] Theorem
⊢ DROP 0 l = l
[DROP_EQ_NIL] Theorem
⊢ ∀ls n. DROP n ls = [] ⇔ LENGTH ls ≤ n
[DROP_GENLIST] Theorem
⊢ DROP n (GENLIST f m) = GENLIST (f ∘ $+ n) (m − n)
[DROP_LENGTH_TOO_LONG] Theorem
⊢ ∀l n. LENGTH l ≤ n ⇒ DROP n l = []
[DROP_TAKE] Theorem
⊢ ∀xs n k. DROP n (TAKE k xs) = TAKE (k − n) (DROP n xs)
[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
[DROP_cons] Theorem
⊢ 0 < n ⇒ DROP n (x::xs) = DROP (n − 1) xs
[DROP_nil] Theorem
⊢ ∀n. DROP n [] = []
[DROP_splitAtPki] Theorem
⊢ DROP n l = splitAtPki (K ∘ $= n) (K I) l
[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_APPEND_EQN] Theorem
⊢ ∀l1 l2 n.
EL n (l1 ⧺ l2) =
if n < LENGTH l1 then EL n l1 else EL (n − LENGTH l1) l2
[EL_DROP] Theorem
⊢ ∀m n l. m + n < LENGTH l ⇒ EL m (DROP n l) = EL (m + n) l
[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_LENGTH_dropWhile_REVERSE] Theorem
⊢ ∀P ls k.
LENGTH (dropWhile P (REVERSE ls)) ≤ k ∧ k < LENGTH ls ⇒
P (EL k ls)
[EL_LUPDATE] Theorem
⊢ ∀ys x i k.
EL i (LUPDATE x k 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_MAP2] Theorem
⊢ ∀ts tt n.
n < MIN (LENGTH ts) (LENGTH tt) ⇒
EL n (MAP2 f ts tt) = f (EL n ts) (EL n tt)
[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_TAKE] Theorem
⊢ ∀n x l. x < n ⇒ EL x (TAKE n l) = EL x 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.
LIST_REL f l1 l2 ⇔
LENGTH l1 = LENGTH l2 ∧ EVERY (UNCURRY f) (ZIP (l1,l2))
[EVERY2_LENGTH] Theorem
⊢ ∀P l1 l2. LIST_REL P l1 l2 ⇒ LENGTH l1 = LENGTH l2
[EVERY2_LUPDATE_same] Theorem
⊢ ∀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_MAP] Theorem
⊢ (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)
[EVERY2_MEM_MONO] Theorem
⊢ ∀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
[EVERY2_REVERSE] Theorem
⊢ ∀R l1 l2. LIST_REL R l1 l2 ⇒ LIST_REL R (REVERSE l1) (REVERSE l2)
[EVERY2_THM] Theorem
⊢ (∀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
[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)) ⇒
(LIST_REL P l1 l2 ⇔ LIST_REL P' l1' l2')
[EVERY2_mono] Theorem
⊢ (∀x y. R1 x y ⇒ R2 x y) ⇒ LIST_REL R1 l1 l2 ⇒ LIST_REL R2 l1 l2
[EVERY2_refl] Theorem
⊢ (∀x. MEM x ls ⇒ R x x) ⇒ LIST_REL R ls ls
[EVERY2_sym] Theorem
⊢ (∀x y. R1 x y ⇒ R2 y x) ⇒ ∀x y. LIST_REL R1 x y ⇒ LIST_REL R2 y x
[EVERY2_trans] Theorem
⊢ (∀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
[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_FLAT] Theorem
⊢ EVERY P (FLAT ls) ⇔ EVERY (EVERY P) ls
[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_MEM_MONO] Theorem
⊢ ∀P Q l. (∀x. MEM x l ∧ P x ⇒ Q x) ∧ EVERY P l ⇒ EVERY Q l
[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_LIST_EQ_MAP] Theorem
⊢ ∀ls f. EVERY (λx. ∃y. x = f y) ls ⇒ ∃l. ls = MAP f l
[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_F] Theorem
⊢ ∀xs. FILTER (λx. F) xs = []
[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)
[FILTER_T] Theorem
⊢ ∀xs. FILTER (λx. T) xs = xs
[FINITE_LIST_TO_SET] Theorem
⊢ ∀l. FINITE (set l)
[FLAT_APPEND] Theorem
⊢ ∀l1 l2. FLAT (l1 ⧺ l2) = FLAT l1 ⧺ FLAT l2
[FLAT_EQ_NIL] Theorem
⊢ ∀ls. FLAT ls = [] ⇔ EVERY ($= []) ls
[FLAT_compute] Theorem
⊢ FLAT [] = [] ∧ FLAT ([]::t) = FLAT t ∧
FLAT ((h::t1)::t2) = h::FLAT (t1::t2)
[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_CONG] Theorem
⊢ (∀m. m < n ⇒ f1 m = f2 m) ⇒ GENLIST f1 n = GENLIST f2 n
[GENLIST_CONS] Theorem
⊢ GENLIST f (SUC n) = f 0::GENLIST (f ∘ SUC) n
[GENLIST_EL] Theorem
⊢ ∀ls f n.
n = LENGTH ls ∧ (∀i. i < n ⇒ f i = EL i ls) ⇒ GENLIST f n = ls
[GENLIST_EL_MAP] Theorem
⊢ ∀f ls. GENLIST (λn. f (EL n ls)) (LENGTH ls) = MAP f ls
[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_ID] Theorem
⊢ ∀x. GENLIST (λi. EL i x) (LENGTH x) = x
[GENLIST_NUMERALS] Theorem
⊢ GENLIST f 0 = [] ∧
GENLIST f (NUMERAL n) = GENLIST_AUX f (NUMERAL n) []
[GENLIST_PLUS_APPEND] Theorem
⊢ GENLIST ($+ a) n1 ⧺ GENLIST ($+ (n1 + a)) n2 =
GENLIST ($+ a) (n1 + n2)
[HD_DROP] Theorem
⊢ ∀n l. n < LENGTH l ⇒ HD (DROP n l) = EL n l
[HD_GENLIST] Theorem
⊢ HD (GENLIST f (SUC n)) = f 0
[HD_GENLIST_COR] Theorem
⊢ ∀n f. 0 < n ⇒ HD (GENLIST f n) = f 0
[HD_REVERSE] Theorem
⊢ ∀x. x ≠ [] ⇒ HD (REVERSE x) = LAST x
[HD_dropWhile] Theorem
⊢ ∀P ls. EXISTS ($¬ ∘ P) ls ⇒ ¬P (HD (dropWhile P ls))
[IMAGE_EL_count_LENGTH] Theorem
⊢ ∀f ls.
IMAGE (λn. f (EL n ls)) (count (LENGTH ls)) = IMAGE f (set ls)
[IMP_EVERY_LUPDATE] Theorem
⊢ ∀xs h i. P h ∧ EVERY P xs ⇒ EVERY P (LUPDATE h i xs)
[INDEX_FIND_add] Theorem
⊢ ∀ls n.
INDEX_FIND n P ls =
OPTION_MAP (λ(i,x). (i + n,x)) (INDEX_FIND 0 P ls)
[INDEX_OF_eq_NONE] Theorem
⊢ ∀x l. INDEX_OF x l = NONE ⇔ ¬MEM x l
[INDEX_OF_eq_SOME] Theorem
⊢ ∀x l i.
INDEX_OF x l = SOME i ⇔
i < LENGTH l ∧ EL i l = x ∧ ∀j. j < i ⇒ EL j l ≠ x
[INFINITE_LIST_UNIV] Theorem
⊢ INFINITE 𝕌(:α list)
[INJ_MAP_EQ] Theorem
⊢ ∀f l1 l2.
INJ f (set l1 ∪ set l2) 𝕌(:β) ∧ MAP f l1 = MAP f l2 ⇒ l1 = l2
[INJ_MAP_EQ_IFF] Theorem
⊢ ∀f l1 l2.
INJ f (set l1 ∪ set l2) 𝕌(:β) ⇒ (MAP f l1 = MAP f l2 ⇔ l1 = l2)
[ITSET_eq_FOLDL_SET_TO_LIST] Theorem
⊢ ∀s. FINITE s ⇒ ∀f a. ITSET f s a = FOLDL (flip 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_MAP] Theorem
⊢ ∀l f. l ≠ [] ⇒ LAST (MAP f l) = f (LAST l)
[LAST_REVERSE] Theorem
⊢ ∀ls. ls ≠ [] ⇒ LAST (REVERSE ls) = HD 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_FILTER_LEQ_MONO] Theorem
⊢ ∀P Q.
(∀x. P x ⇒ Q x) ⇒
∀ls. LENGTH (FILTER P ls) ≤ LENGTH (FILTER Q ls)
[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_LT_SHORTLEX] Theorem
⊢ ∀l1 l2. LENGTH l1 < LENGTH l2 ⇒ SHORTLEX R l1 l2
[LENGTH_LUPDATE] Theorem
⊢ ∀x n ys. LENGTH (LUPDATE x n ys) = LENGTH ys
[LENGTH_MAP] Theorem
⊢ ∀l f. LENGTH (MAP f l) = LENGTH l
[LENGTH_MAP2] Theorem
⊢ ∀xs ys. LENGTH (MAP2 f xs ys) = MIN (LENGTH xs) (LENGTH ys)
[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_TAKE_EQ] Theorem
⊢ LENGTH (TAKE n xs) = if n ≤ LENGTH xs then n else LENGTH xs
[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
[LENGTH_ZIP_MIN] Theorem
⊢ ∀xs ys. LENGTH (ZIP (xs,ys)) = MIN (LENGTH xs) (LENGTH ys)
[LENGTH_dropWhile_LESS_EQ] Theorem
⊢ ∀P ls. LENGTH (dropWhile P ls) ≤ LENGTH ls
[LENGTH_o_REVERSE] Theorem
⊢ LENGTH ∘ REVERSE = LENGTH ∧ LENGTH ∘ REVERSE ∘ f = LENGTH ∘ f
[LEN_LENGTH_LEM] Theorem
⊢ ∀L n. LEN L n = LENGTH L + n
[LIST_APPLY_o] Theorem
⊢ [$o] <*> fs <*> gs <*> xs = fs <*> (gs <*> xs)
[LIST_BIND_APPEND] Theorem
⊢ LIST_BIND (l1 ⧺ l2) f = LIST_BIND l1 f ⧺ LIST_BIND l2 f
[LIST_BIND_ID] Theorem
⊢ LIST_BIND l (λx. x) = FLAT l ∧ LIST_BIND l I = FLAT l
[LIST_BIND_LIST_BIND] Theorem
⊢ LIST_BIND (LIST_BIND l g) f = LIST_BIND l (flip LIST_BIND f ∘ g)
[LIST_BIND_MAP] Theorem
⊢ LIST_BIND (MAP f l) g = LIST_BIND l (g ∘ f)
[LIST_BIND_THM] Theorem
⊢ LIST_BIND [] f = [] ∧ LIST_BIND (h::t) f = f h ⧺ LIST_BIND t f
[LIST_EQ] Theorem
⊢ ∀l1 l2.
LENGTH l1 = LENGTH l2 ∧ (∀x. x < LENGTH l1 ⇒ EL x l1 = EL x l2) ⇒
l1 = l2
[LIST_EQ_MAP_PAIR] Theorem
⊢ ∀l1 l2. MAP FST l1 = MAP FST l2 ∧ MAP SND l1 = MAP SND 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_APPEND] Theorem
⊢ LIST_REL R l1 l2 ∧ LIST_REL R l3 l4 ⇔
LIST_REL R (l1 ⧺ l3) (l2 ⧺ l4) ∧ LENGTH l1 = LENGTH l2 ∧
LENGTH l3 = LENGTH l4
[LIST_REL_APPEND_EQ] Theorem
⊢ LENGTH x1 = LENGTH x2 ⇒
(LIST_REL R (x1 ⧺ y1) (x2 ⧺ y2) ⇔
LIST_REL R x1 x2 ∧ LIST_REL R y1 y2)
[LIST_REL_APPEND_IMP] Theorem
⊢ ∀xs ys xs1 ys1.
LIST_REL P (xs ⧺ xs1) (ys ⧺ ys1) ∧ LENGTH xs = LENGTH ys ⇒
LIST_REL P xs ys ∧ LIST_REL P xs1 ys1
[LIST_REL_APPEND_suff] Theorem
⊢ LIST_REL R l1 l2 ∧ LIST_REL R l3 l4 ⇒
LIST_REL R (l1 ⧺ l3) (l2 ⧺ l4)
[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_LENGTH] Theorem
⊢ ∀x y. LIST_REL R x y ⇒ LENGTH x = LENGTH y
[LIST_REL_MAP1] Theorem
⊢ LIST_REL R (MAP f l1) l2 ⇔ LIST_REL (R ∘ 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_MAP_inv_image] Theorem
⊢ LIST_REL R (MAP f l1) (MAP f l2) ⇔ LIST_REL (inv_image R f) l1 l2
[LIST_REL_MEM_IMP] Theorem
⊢ ∀xs ys P x. LIST_REL P xs ys ∧ MEM x xs ⇒ ∃y. MEM y ys ∧ P x y
[LIST_REL_NIL] Theorem
⊢ (LIST_REL R [] y ⇔ y = []) ∧ (LIST_REL R x [] ⇔ x = [])
[LIST_REL_O] Theorem
⊢ ∀R1 R2. LIST_REL (R1 ∘ᵣ R2) = LIST_REL R1 ∘ᵣ LIST_REL R2
[LIST_REL_SNOC] Theorem
⊢ (LIST_REL R (SNOC x xs) yys ⇔
∃y ys. yys = SNOC y ys ∧ LIST_REL R xs ys ∧ R x y) ∧
(LIST_REL R xxs (SNOC y ys) ⇔
∃x xs. xxs = SNOC x xs ∧ LIST_REL R xs ys ∧ R x y)
[LIST_REL_SPLIT1] Theorem
⊢ ∀xs1 zs.
LIST_REL P (xs1 ⧺ xs2) zs ⇔
∃ys1 ys2.
zs = ys1 ⧺ ys2 ∧ LIST_REL P xs1 ys1 ∧ LIST_REL P xs2 ys2
[LIST_REL_SPLIT2] Theorem
⊢ ∀xs1 zs.
LIST_REL P zs (xs1 ⧺ xs2) ⇔
∃ys1 ys2.
zs = ys1 ⧺ ys2 ∧ LIST_REL P ys1 xs1 ∧ LIST_REL P ys2 xs2
[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_eq] Theorem
⊢ LIST_REL $= = $=
[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_REL_trans] Theorem
⊢ ∀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
[LIST_TO_SET] Theorem
⊢ set [] = ∅ ∧ set (h::t) = h INSERT set t
[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_FLAT] Theorem
⊢ ∀ls. set (FLAT ls) = BIGUNION (set (MAP set ls))
[LIST_TO_SET_GENLIST] Theorem
⊢ ∀f n. set (GENLIST f n) = IMAGE f (count n)
[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
[LLEX_CONG] Theorem
⊢ ∀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_EL_THM] Theorem
⊢ ∀R l1 l2.
LLEX R l1 l2 ⇔
∃n. n ≤ LENGTH l1 ∧ n < LENGTH l2 ∧ TAKE n l1 = TAKE n l2 ∧
(n < LENGTH l1 ⇒ R (EL n l1) (EL n l2))
[LLEX_MONO] Theorem
⊢ (∀x y. R1 x y ⇒ R2 x y) ⇒ LLEX R1 x y ⇒ LLEX R2 x y
[LLEX_NIL2] Theorem
⊢ ¬LLEX R l []
[LLEX_THM] Theorem
⊢ (¬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_not_WF] Theorem
⊢ (∃a b. R a b) ⇒ ¬WF (LLEX R)
[LLEX_total] Theorem
⊢ total (RC R) ⇒ total (RC (LLEX R))
[LLEX_transitive] Theorem
⊢ transitive R ⇒ transitive (LLEX R)
[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_GENLIST] Theorem
⊢ ∀m n e f. LUPDATE e n (GENLIST f m) = GENLIST f⦇n ↦ e⦈ m
[LUPDATE_LENGTH] Theorem
⊢ ∀xs x y ys. LUPDATE x (LENGTH xs) (xs ⧺ y::ys) = xs ⧺ x::ys
[LUPDATE_MAP] Theorem
⊢ ∀x n l f. MAP f (LUPDATE x n l) = LUPDATE (f x) n (MAP f l)
[LUPDATE_NIL] Theorem
⊢ ∀xs n x. LUPDATE x n xs = [] ⇔ xs = []
[LUPDATE_SAME] Theorem
⊢ ∀n ls. n < LENGTH ls ⇒ LUPDATE (EL n ls) n ls = ls
[LUPDATE_SEM] Theorem
⊢ (∀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
[LUPDATE_SNOC] Theorem
⊢ ∀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)
[LUPDATE_SOME_MAP] Theorem
⊢ ∀xs n f h.
LUPDATE (SOME (f h)) n (MAP (OPTION_MAP f) xs) =
MAP (OPTION_MAP f) (LUPDATE (SOME h) n xs)
[LUPDATE_compute] Theorem
⊢ (∀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
[MAP2] Theorem
⊢ (∀f. MAP2 f [] [] = []) ∧
∀f h1 t1 h2 t2. MAP2 f (h1::t1) (h2::t2) = f h1 h2::MAP2 f t1 t2
[MAP2_APPEND] Theorem
⊢ ∀xs ys xs1 ys1 f.
LENGTH xs = LENGTH xs1 ⇒
MAP2 f (xs ⧺ ys) (xs1 ⧺ ys1) = MAP2 f xs xs1 ⧺ MAP2 f ys ys1
[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_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_MAP] Theorem
⊢ ∀l1 l2.
LENGTH l1 = LENGTH l2 ⇒
∀f. MAP2 f l1 l2 = MAP (UNCURRY f) (ZIP (l1,l2))
[MAP2_NIL] Theorem
⊢ MAP2 f x [] = []
[MAP2_ZIP] Theorem
⊢ ∀l1 l2.
LENGTH l1 = LENGTH l2 ⇒
∀f. MAP2 f l1 l2 = MAP (UNCURRY f) (ZIP (l1,l2))
[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_APPEND_MAP_EQ] Theorem
⊢ ∀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
[MAP_CONG] Theorem
⊢ ∀l1 l2 f f'.
l1 = l2 ∧ (∀x. MEM x l2 ⇒ f x = f' x) ⇒ MAP f l1 = MAP f' l2
[MAP_DROP] Theorem
⊢ ∀l i. MAP f (DROP i l) = DROP i (MAP f l)
[MAP_EQ_APPEND] Theorem
⊢ MAP f l = l1 ⧺ l2 ⇔
∃l10 l20. l = l10 ⧺ l20 ∧ l1 = MAP f l10 ∧ l2 = MAP f l20
[MAP_EQ_CONS] Theorem
⊢ MAP f l = h::t ⇔ ∃x0 t0. l = x0::t0 ∧ h = f x0 ∧ t = MAP f t0
[MAP_EQ_EVERY2] Theorem
⊢ ∀f1 f2 l1 l2.
MAP f1 l1 = MAP f2 l2 ⇔
LENGTH l1 = LENGTH l2 ∧ LIST_REL (λx y. f1 x = f2 y) l1 l2
[MAP_EQ_NIL] Theorem
⊢ ∀l f. (MAP f l = [] ⇔ l = []) ∧ ([] = MAP f l ⇔ l = [])
[MAP_EQ_SING] Theorem
⊢ MAP f l = [x] ⇔ ∃x0. l = [x0] ∧ x = f x0
[MAP_EQ_f] Theorem
⊢ ∀f1 f2 l. MAP f1 l = MAP f2 l ⇔ ∀e. MEM e l ⇒ f1 e = f2 e
[MAP_FLAT] Theorem
⊢ MAP f (FLAT l) = FLAT (MAP (MAP f) l)
[MAP_FRONT] Theorem
⊢ ∀ls. ls ≠ [] ⇒ MAP f (FRONT ls) = FRONT (MAP f ls)
[MAP_GENLIST] Theorem
⊢ ∀f g n. MAP f (GENLIST g n) = GENLIST (f ∘ g) n
[MAP_ID] Theorem
⊢ MAP (λx. x) l = l ∧ MAP I l = l
[MAP_LIST_BIND] Theorem
⊢ MAP f (LIST_BIND l g) = LIST_BIND l (MAP f ∘ g)
[MAP_MAP_o] Theorem
⊢ ∀f g l. MAP f (MAP g l) = MAP (f ∘ g) l
[MAP_SNOC] Theorem
⊢ ∀f x l. MAP f (SNOC x l) = SNOC (f x) (MAP f l)
[MAP_TAKE] Theorem
⊢ ∀f n l. MAP f (TAKE n l) = TAKE n (MAP f l)
[MAP_TL] Theorem
⊢ ∀l f. 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 ∘ FST) (ZIP (l1,l2)) = MAP f l1 ∧
MAP (g ∘ 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 ∘ g) = MAP f ∘ MAP g
[MEM] Theorem
⊢ (∀x. MEM x [] ⇔ F) ∧ ∀x h t. MEM x (h::t) ⇔ x = h ∨ MEM x t
[MEM_APPEND] Theorem
⊢ ∀e l1 l2. MEM e (l1 ⧺ l2) ⇔ MEM e l1 ∨ MEM e l2
[MEM_APPEND_lemma] Theorem
⊢ ∀a b c d x.
a ⧺ [x] ⧺ b = c ⧺ [x] ⧺ d ∧ ¬MEM x b ∧ ¬MEM x a ⇒ a = c ∧ b = d
[MEM_DROP] Theorem
⊢ ∀x ls n.
MEM x (DROP n ls) ⇔ ∃m. m + n < LENGTH ls ∧ x = EL (m + 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_LUPDATE] Theorem
⊢ ∀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
[MEM_LUPDATE_E] Theorem
⊢ ∀l x y i. MEM x (LUPDATE y i l) ⇒ x = y ∨ MEM x l
[MEM_MAP] Theorem
⊢ ∀l f x. MEM x (MAP f l) ⇔ ∃y. x = f y ∧ MEM y l
[MEM_MAP_f] Theorem
⊢ ∀f l a. MEM a l ⇒ MEM (f a) (MAP f 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_TL] Theorem
⊢ ∀l x. MEM x (TL l) ⇒ MEM x l
[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))
[MEM_ZIP_MEM_MAP] Theorem
⊢ LENGTH (FST ps) = LENGTH (SND ps) ∧ MEM p (ZIP ps) ⇒
MEM (FST p) (FST ps) ∧ MEM (SND p) (SND ps)
[MEM_dropWhile_IMP] Theorem
⊢ ∀P ls x. MEM x (dropWhile P ls) ⇒ MEM x ls
[MEM_nub] Theorem
⊢ MEM x (nub l) ⇔ MEM x l
[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
[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 ($¬ ∘ P) l
[NOT_EXISTS] Theorem
⊢ ∀P l. ¬EXISTS P l ⇔ EVERY ($¬ ∘ P) l
[NOT_NIL_CONS] Theorem
⊢ ∀a1 a0. [] ≠ a0::a1
[NOT_NIL_EQ_LENGTH_NOT_0] Theorem
⊢ x ≠ [] ⇔ 0 < LENGTH x
[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_APPEND] Theorem
⊢ NULL (l1 ⧺ l2) ⇔ NULL l1 ∧ NULL l2
[NULL_EQ] Theorem
⊢ ∀l. NULL l ⇔ l = []
[NULL_FILTER] Theorem
⊢ ∀P ls. NULL (FILTER P ls) ⇔ ∀x. MEM x ls ⇒ ¬P x
[NULL_GENLIST] Theorem
⊢ ∀n f. NULL (GENLIST f n) ⇔ n = 0
[NULL_LENGTH] Theorem
⊢ ∀l. NULL l ⇔ LENGTH l = 0
[OPT_MMAP_cong] Theorem
⊢ ∀f1 f2 x1 x2.
x1 = x2 ∧ (∀a. MEM a x2 ⇒ f1 a = f2 a) ⇒
OPT_MMAP f1 x1 = OPT_MMAP f2 x2
[REVERSE_11] Theorem
⊢ ∀l1 l2. REVERSE l1 = REVERSE l2 ⇔ l1 = l2
[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)
[REVERSE_o_REVERSE] Theorem
⊢ REVERSE ∘ REVERSE ∘ f = f
[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)
[SHORTLEX_LENGTH_LE] Theorem
⊢ ∀l1 l2. SHORTLEX R l1 l2 ⇒ LENGTH l1 ≤ LENGTH l2
[SHORTLEX_MONO] Theorem
⊢ (∀x y. R1 x y ⇒ R2 x y) ⇒ SHORTLEX R1 x y ⇒ SHORTLEX R2 x y
[SHORTLEX_NIL2] Theorem
⊢ ¬SHORTLEX R l []
[SHORTLEX_THM] Theorem
⊢ (¬SHORTLEX R [] [] ∧ ¬SHORTLEX R (h1::t1) []) ∧
SHORTLEX R [] (h2::t2) ∧
(SHORTLEX R (h1::t1) (h2::t2) ⇔
LENGTH t1 < LENGTH t2 ∨
LENGTH t1 = LENGTH t2 ∧ (R h1 h2 ∨ h1 = h2 ∧ SHORTLEX R t1 t2))
[SHORTLEX_total] Theorem
⊢ total (RC R) ⇒ total (RC (SHORTLEX R))
[SHORTLEX_transitive] Theorem
⊢ transitive R ⇒ transitive (SHORTLEX R)
[SINGL_APPLY_MAP] Theorem
⊢ [f] <*> l = MAP f l
[SINGL_APPLY_PERMUTE] Theorem
⊢ fs <*> [x] = [(λf. f x)] <*> fs
[SINGL_LIST_APPLY_L] Theorem
⊢ LIST_BIND [x] f = f x
[SINGL_LIST_APPLY_R] Theorem
⊢ LIST_BIND l (λx. [x]) = l
[SINGL_SINGL_APPLY] Theorem
⊢ [f] <*> [x] = [f x]
[SING_HD] Theorem
⊢ ([HD xs] = xs ⇔ LENGTH xs = 1) ∧ (xs = [HD xs] ⇔ LENGTH xs = 1)
[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_ACC_SUM_LEM] Theorem
⊢ ∀L n. SUM_ACC L n = SUM L + n
[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] Theorem
⊢ ∀f g ls.
SUM (MAP (λx. f x + g x) ls) = SUM (MAP f ls) + SUM (MAP g 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
[SUM_SUM_ACC] Theorem
⊢ ∀L. SUM L = SUM_ACC L 0
[SUM_eq_0] Theorem
⊢ ∀ls. SUM ls = 0 ⇔ ∀x. MEM x ls ⇒ x = 0
[SWAP_REVERSE] Theorem
⊢ ∀l1 l2. l1 = REVERSE l2 ⇔ l2 = REVERSE l1
[SWAP_REVERSE_SYM] Theorem
⊢ ∀l1 l2. REVERSE l1 = l2 ⇔ l1 = REVERSE l2
[TAKE1] Theorem
⊢ ∀l. l ≠ [] ⇒ TAKE 1 l = [EL 0 l]
[TAKE1_DROP] Theorem
⊢ ∀n l. n < LENGTH l ⇒ TAKE 1 (DROP n l) = [EL n l]
[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_DROP_SWAP] Theorem
⊢ ∀xs k n. TAKE k (DROP n xs) = DROP n (TAKE (k + n) xs)
[TAKE_EQ_NIL] Theorem
⊢ TAKE n l = [] ⇔ n = 0 ∨ l = []
[TAKE_GENLIST] Theorem
⊢ TAKE n (GENLIST f m) = GENLIST f (MIN n m)
[TAKE_LENGTH_ID] Theorem
⊢ ∀l. TAKE (LENGTH l) l = l
[TAKE_LENGTH_ID_rwt] Theorem
⊢ ∀l m. m = LENGTH l ⇒ TAKE m l = l
[TAKE_LENGTH_ID_rwt2] Theorem
⊢ ∀l m. TAKE m l = l ⇔ LENGTH l ≤ m
[TAKE_LENGTH_TOO_LONG] Theorem
⊢ ∀l n. LENGTH l ≤ n ⇒ TAKE n l = l
[TAKE_SUM] Theorem
⊢ ∀n m l. TAKE (n + m) l = TAKE n l ⧺ TAKE m (DROP n l)
[TAKE_TAKE_MIN] Theorem
⊢ ∀m n. TAKE n (TAKE m l) = TAKE (MIN n m) 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
[TAKE_cons] Theorem
⊢ 0 < n ⇒ TAKE n (x::xs) = x::TAKE (n − 1) xs
[TAKE_nil] Theorem
⊢ ∀n. TAKE n [] = []
[TAKE_splitAtPki] Theorem
⊢ TAKE n l = splitAtPki (K ∘ $= n) K l
[TL] Theorem
⊢ ∀h t. TL (h::t) = t
[TL_GENLIST] Theorem
⊢ ∀f n. TL (GENLIST f (SUC n)) = GENLIST (f ∘ SUC) n
[UNION_APPEND] Theorem
⊢ ∀l1 l2. set l1 ∪ set l2 = set (l1 ⧺ l2)
[UNIQUE_FILTER] Theorem
⊢ ∀e L. UNIQUE e L ⇔ FILTER ($= e) L = [e]
[UNIQUE_LENGTH_FILTER] Theorem
⊢ ∀e L. UNIQUE e L ⇔ LENGTH (FILTER ($= e) L) = 1
[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)
[WF_SHORTLEX] Theorem
⊢ WF R ⇒ WF (SHORTLEX R)
[WF_SHORTLEX_same_lengths] Theorem
⊢ WF R ⇒
∀l s.
(∀d. d ∈ s ⇒ LENGTH d = l) ∧ (∃a. a ∈ s) ⇒
∃b. b ∈ s ∧ ∀c. SHORTLEX R c b ⇒ c ∉ s
[ZIP] Theorem
⊢ ZIP ([],[]) = [] ∧
∀x1 l1 x2 l2. ZIP (x1::l1,x2::l2) = (x1,x2)::ZIP (l1,l2)
[ZIP_DROP] Theorem
⊢ ∀a b n.
n ≤ LENGTH a ∧ LENGTH a = LENGTH b ⇒
ZIP (DROP n a,DROP n b) = DROP n (ZIP (a,b))
[ZIP_EQ_NIL] Theorem
⊢ ∀l1 l2.
LENGTH l1 = LENGTH l2 ⇒ (ZIP (l1,l2) = [] ⇔ l1 = [] ∧ l2 = [])
[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
[adjacent_EL] Theorem
⊢ adjacent L a b ⇔
∃i. i + 1 < LENGTH L ∧ a = EL i L ∧ b = EL (i + 1) L
[adjacent_cases] Theorem
⊢ ∀a0 a1 a2.
adjacent a0 a1 a2 ⇔
(∃t. a0 = a1::a2::t) ∨ ∃h t. a0 = h::t ∧ adjacent t a1 a2
[adjacent_iff] Theorem
⊢ adjacent (h1::h2::t) a b ⇔ h1 = a ∧ h2 = b ∨ adjacent (h2::t) a b
[adjacent_ind] Theorem
⊢ ∀adjacent'.
(∀a b t. adjacent' (a::b::t) a b) ∧
(∀a b h t. adjacent' t a b ⇒ adjacent' (h::t) a b) ⇒
∀a0 a1 a2. adjacent a0 a1 a2 ⇒ adjacent' a0 a1 a2
[adjacent_rules] Theorem
⊢ (∀a b t. adjacent (a::b::t) a b) ∧
∀a b h t. adjacent t a b ⇒ adjacent (h::t) a b
[adjacent_strongind] Theorem
⊢ ∀adjacent'.
(∀a b t. adjacent' (a::b::t) a b) ∧
(∀a b h t.
adjacent t a b ∧ adjacent' t a b ⇒ adjacent' (h::t) a b) ⇒
∀a0 a1 a2. adjacent a0 a1 a2 ⇒ adjacent' a0 a1 a2
[adjacent_thm] Theorem
⊢ (adjacent [] a b ⇔ F) ∧ (adjacent [e] a b ⇔ F) ∧
(adjacent (a::b::t) a b ⇔ T)
[all_distinct_nub] Theorem
⊢ ∀l. ALL_DISTINCT (nub l)
[all_distinct_nub_id] Theorem
⊢ ∀l. ALL_DISTINCT l ⇒ nub l = l
[bounded_length_FINITE] Theorem
⊢ FINITE 𝕌(:α) ⇒ ∀m s. (∀x. x ∈ s ⇒ LENGTH x ≤ m) ⇒ FINITE s
[datatype_list] Theorem
⊢ DATATYPE (list [] CONS)
[dropWhile_APPEND_EVERY] Theorem
⊢ ∀P l1 l2. EVERY P l1 ⇒ dropWhile P (l1 ⧺ l2) = dropWhile P l2
[dropWhile_APPEND_EXISTS] Theorem
⊢ ∀P l1 l2.
EXISTS ($¬ ∘ P) l1 ⇒ dropWhile P (l1 ⧺ l2) = dropWhile P l1 ⧺ l2
[dropWhile_eq_nil] Theorem
⊢ ∀P ls. dropWhile P ls = [] ⇔ EVERY P ls
[dropWhile_splitAtPki] Theorem
⊢ ∀P. dropWhile P = splitAtPki (flip (K ∘ $¬ ∘ P)) (K I)
[el_append3] Theorem
⊢ ∀l1 x l2. EL (LENGTH l1) (l1 ⧺ [x] ⧺ l2) = x
[every_zip_fst] Theorem
⊢ ∀l1 l2 P.
LENGTH l1 = LENGTH l2 ⇒
(EVERY (λx. P (FST x)) (ZIP (l1,l2)) ⇔ EVERY P l1)
[every_zip_snd] Theorem
⊢ ∀l1 l2 P.
LENGTH l1 = LENGTH l2 ⇒
(EVERY (λx. P (SND x)) (ZIP (l1,l2)) ⇔ EVERY P l2)
[exists_list_GENLIST] Theorem
⊢ (∃ls. P ls) ⇔ ∃n f. P (GENLIST f n)
[isPREFIX_CONSR] Theorem
⊢ x ≼ y::ys ⇔ x = [] ∨ ∃xs. x = y::xs ∧ xs ≼ ys
[isPREFIX_NILR] Theorem
⊢ x ≼ [] ⇔ x = []
[isPREFIX_THM] Theorem
⊢ ([] ≼ l ⇔ T) ∧ (h::t ≼ [] ⇔ F) ∧
(h1::t1 ≼ h2::t2 ⇔ h1 = h2 ∧ t1 ≼ t2)
[last_drop] Theorem
⊢ ∀l n. n < LENGTH l ⇒ LAST (DROP n l) = LAST l
[lazy_list_case_compute] Theorem
⊢ list_CASE = (λl b f. if NULL l then b else f (HD l) (TL l))
[length_nub_append] Theorem
⊢ ∀l1 l2.
LENGTH (nub (l1 ⧺ l2)) =
LENGTH (nub l1) + LENGTH (nub (FILTER (λx. ¬MEM x l1) l2))
[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 = [] ∨ ∃h t. l = h::t
[list_INDUCT] Theorem
⊢ ∀P. P [] ∧ (∀t. P t ⇒ ∀h. P (h::t)) ⇒ ∀l. P l
[list_INDUCT0] Theorem
⊢ ∀P. P [] ∧ (∀l. P l ⇒ ∀a. P (a::l)) ⇒ ∀l. P l
[list_case_compute] Theorem
⊢ ∀l. list_CASE l b f = 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 M v f = list_CASE M' v' f'
[list_case_eq] Theorem
⊢ list_CASE x v f = v' ⇔
x = [] ∧ v = v' ∨ ∃a l. x = a::l ∧ f a l = v'
[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 = [] ∨ ∃h t. 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
[list_to_set_diff] Theorem
⊢ ∀l1 l2. set l2 DIFF set l1 = set (FILTER (λx. ¬MEM x l1) l2)
[lupdate_append] Theorem
⊢ ∀x n l1 l2.
n < LENGTH l1 ⇒ LUPDATE x n (l1 ⧺ l2) = LUPDATE x n l1 ⧺ l2
[lupdate_append2] Theorem
⊢ ∀v l1 x l2 l3.
LUPDATE v (LENGTH l1) (l1 ⧺ [x] ⧺ l2) = l1 ⧺ [v] ⧺ l2
[mem_exists_set] Theorem
⊢ ∀x y l. MEM (x,y) l ⇒ ∃z. x = FST z ∧ MEM z l
[nub_EQ0] Theorem
⊢ nub l = [] ⇔ l = []
[nub_MAP_INJ] Theorem
⊢ INJ f (set ls) 𝕌(:β) ⇒ nub (MAP f ls) = MAP f (nub ls)
[nub_NIL] Theorem
⊢ nub [] = []
[nub_append] Theorem
⊢ ∀l1 l2. nub (l1 ⧺ l2) = nub (FILTER (λx. ¬MEM x l2) l1) ⧺ nub l2
[nub_set] Theorem
⊢ ∀l. set (nub l) = set l
[oEL_DROP] Theorem
⊢ oEL n (DROP m xs) = oEL (m + n) xs
[oEL_EQ_EL] Theorem
⊢ ∀xs n y. oEL n xs = SOME y ⇔ n < LENGTH xs ∧ y = EL n xs
[oEL_LUPDATE] Theorem
⊢ ∀xs i n x.
oEL n (LUPDATE x i xs) =
if i ≠ n then oEL n xs
else if i < LENGTH xs then SOME x
else NONE
[oEL_TAKE_E] Theorem
⊢ oEL n (TAKE m xs) = SOME x ⇒ oEL n xs = SOME x
[oEL_THM] Theorem
⊢ ∀xs n. oEL n xs = if n < LENGTH xs then SOME (EL n xs) else NONE
[oHD_thm] Theorem
⊢ oHD [] = NONE ∧ oHD (h::t) = SOME h
[splitAtPki_APPEND] Theorem
⊢ ∀l1 l2 P k.
EVERYi (λi. $¬ ∘ P i) l1 ∧
(0 < LENGTH l2 ⇒ P (LENGTH l1) (HD l2)) ⇒
splitAtPki P k (l1 ⧺ l2) = k l1 l2
[splitAtPki_EQN] Theorem
⊢ splitAtPki P k l =
case OLEAST i. i < LENGTH l ∧ P i (EL i l) of
NONE => k l []
| SOME i => k (TAKE i l) (DROP i l)
[splitAtPki_MAP] Theorem
⊢ splitAtPki P k (MAP f l) =
splitAtPki (flip ($o ∘ P) f) (flip ($o ∘ k ∘ MAP f) (MAP f)) l
[splitAtPki_RAND] Theorem
⊢ f (splitAtPki P k l) = splitAtPki P ($o f ∘ k) l
[splitAtPki_change_predicate] Theorem
⊢ (∀i. i < LENGTH l ⇒ (P1 i (EL i l) ⇔ P2 i (EL i l))) ⇒
splitAtPki P1 k l = splitAtPki P2 k l
*)
end
HOL 4, Kananaskis-14