hcju/isabelleps huggingface dataset:
cases: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and a1 :: "char list"   and a2 :: "char list \<Rightarrow> ((ty list \<times> ty \<times> 'b) \<times> char list) option" assumes "P \<turnstile> a1 sees_methods a2" obtains D :: "char list" and fs :: "(char list \<times> ty) list" and ms :: "(char list \<times> ty list \<times> ty \<times> 'b) list" and Mm :: "char list \<Rightarrow> ((ty list \<times> ty \<times> 'b) \<times> char list) option" where "a1 = Object" and "a2 = Mm__" and "class P Object = \<lfloor>(D__, fs__, ms__)\<rfloor>" and "Mm__ = map_option (\<lambda>m. (m, Object)) \<circ> map_of ms__"   | C :: "char list" and D :: "char list" and fs :: "(char list \<times> ty) list" and ms :: "(char list \<times> ty list \<times> ty \<times> 'b) list" and Mm :: "char list \<Rightarrow> ((ty list \<times> ty \<times> 'b) \<times> char list) option" and Mm' :: "char list \<Rightarrow> ((ty list \<times> ty \<times> 'b) \<times> char list) option" where "a1 = C" and "a2 = Mm'" and "class P C = \<lfloor>(D, fs, ms)\<rfloor>" and "C \<noteq> Object" and "P \<turnstile> D sees_methods Mm" and "Mm' = Mm ++ (map_option (\<lambda>m. (m, C)) \<circ> map_of ms)"
simps: fixes a1 :: "'a list"   and a2 :: "'a list" shows "initial_segment a1 a2 = (\<exists>xs ys. a1 = xs \<and> a2 = xs @ ys)"
ind_def: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" shows "ind P n = (if 0 < n \<and> P n then 1::'b else (0::'b))"
summable: shows "(\<lambda>(x, y). f x y) abs_summable_on Sigma A B"
strict_sorted_append_iff: fixes xs :: "'a list"   and ys :: "'a list" shows "strict_sorted (xs @ ys) = (xs < ys \<and> strict_sorted xs \<and> strict_sorted ys)"
p'_4: fixes x :: "'n"   and K :: "'n set" assumes "(x, K) \<in> p" shows "\<exists>a b. K = cbox a b"
sumIdRight: fixes Rel :: "(ccs \<times> ccs) set"   and P :: "ccs" assumes "Id \<subseteq> Rel" shows "P \<leadsto>[Rel] P \<oplus> \<zero>"
assms_2: assumes "\<forall>i\<in>I. \<sigma>' i = \<sigma> i"   and "\<forall>j. j \<notin> I \<longrightarrow> P (\<sigma> j) (\<sigma>' j)" shows "R \<sigma> \<sigma>'"
iMOD_eq_conv: fixes m :: "nat"   and r :: "nat"   and r' :: "nat"   and m' :: "nat" assumes "0 < m" shows "([ r, mod m ] = [ r', mod m' ]) = (r = r' \<and> m = m')"
sub_of_denot_equivts: fixes E :: "char list \<Rightarrow> hterm"   and ts :: "fterm list" shows "eval\<^sub>t\<^sub>s E HFun (ts \<cdot>\<^sub>t\<^sub>s sub_of_denot E) = eval\<^sub>t\<^sub>s E HFun ts"
countably_compact_imp_compact: fixes U :: "'b set"   and B :: "'b set set" assumes "countably_compact U"   and "countable B"   and "\<forall>b\<in>B. open b"   and "\<And>T x. \<lbrakk>open T; x \<in> T; x \<in> U\<rbrakk> \<Longrightarrow> \<exists>b\<in>B. x \<in> b \<and> b \<inter> U \<subseteq> T" shows "compact U"
Process_eq_spec_optimized: fixes P :: "'a process"   and Q :: "'a process" shows "(P = Q) = (D P = D Q \<and> (D P = D Q \<longrightarrow> F P = F Q))"
img_inc: fixes m :: "nat" shows "enc ` cw_len_concat -` {m} \<subseteq> {bl. length bl = m}"
statesDecls_process_names: fixes v :: "procVarDecl"   and prog :: "'a program_scheme"   and p :: "'b pState_scheme" assumes "v \<in> statesDecls (states prog !! pState.idx p)" shows "procVarDeclName v \<in> process_names (states prog !! pState.idx p) (processes prog !! pState.idx p)"
intros: fixes F :: "nat \<Rightarrow> 'b set" assumes "\<And>n. open (F n)" shows "gdelta (\<Inter> (range F))"
SUP_apply: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'a"   and A :: "'c set"   and x :: "'b" shows "(\<Squnion> \<P> f A) x = (\<Squnion>y\<in>A. f y x)"
lossless_sub_gpvsD: fixes \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv"   and gpv' :: "('f, 'd, 'e) gpv" assumes "lossless_gpv \<I> gpv"   and "gpv' \<in> sub_gpvs \<I> gpv" shows "lossless_gpv \<I> gpv'"
unitor_coincidence_2: fixes a :: "'a" assumes "B.obj a" shows "B.runit a = \<i>\<^sub>B[a]"
cases: fixes x :: "bool \<times> 'a \<times> 'a" obtains b :: "bool" and a1 :: "'a" and a2 :: "'a" where "x = (b, a1, a2)"
p_prod_distl: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<parallel> (y + z) = x \<parallel> y + x \<parallel> z"
t_4: assumes "f \<noteq> g" shows "is_Abs f \<and> is_Abs g \<and> (the_Abs f, the_Abs g) \<in> (set TI)\<^sup>+"
listrel1I1: fixes x :: "'g"   and y :: "'g"   and r :: "('g \<times> 'g) set"   and xs :: "'g list" assumes "(x, y) \<in> r" shows "(x # xs, y # xs) \<in> listrel1 r"
notin_paq: fixes x :: "complex" assumes "dist a x < kde" shows "x \<notin> path_image (linepath (a + complex_of_real kde) (a + complex_of_real e) +++ p +++ linepath (a - complex_of_real d) (a - complex_of_real kde))"
i_aggregate_take_nth: fixes n :: "nat"   and m :: "nat"   and k :: "nat"   and f :: "nat \<Rightarrow> 'b"   and ag :: "'b list \<Rightarrow> 'b" assumes "n < m div k" shows "f_aggregate (f \<Down> m) k ag ! n = i_aggregate f k ag n"
lc_in_carrier_1: shows "combine Cs Us \<in> carrier R"
col_transpose: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d" shows "col (Square_Matrix.transpose M) i = Square_Matrix.row M i"
subspace_mul: fixes S :: "'a set"   and x :: "'a"   and c :: "real" assumes "subspace S"   and "x \<in> S" shows "c *\<^sub>R x \<in> S"
balanced_sym: fixes k :: "'b"   and v :: "real"   and l :: "'b kdt"   and r :: "'b kdt"   and k' :: "'b"   and v' :: "real" assumes "balanced (Node k v l r)" shows "balanced (Node k' v' r l)"
approximation_preproc_push_neg_3: fixes a :: "real"   and b :: "real" shows "(a \<noteq> b) = (b < a \<or> a < b)"
seqI': fixes f :: "'s"   and a :: "'s"   and b :: "'s"   and g :: "'s"   and c :: "'s" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow> c\<guillemotright>" shows "S.seq g f"
subst_is_fresh_1: fixes y :: "Vars.var"   and z :: "Vars.var"   and e :: "exp" assumes "atom y \<sharp> z" shows "atom y \<sharp> e[y::=z]"
finite_has_maximal2: fixes A :: "'t set"   and a :: "'t" assumes "finite A"   and "a \<in> A" shows "\<exists>m\<in>A. a \<preceq>\<^sub>t m \<and> (\<forall>b\<in>A. m \<preceq>\<^sub>t b \<longrightarrow> m = b)"
size_list_def: shows "size_list \<equiv> \<lambda>f. rec_list 0 (\<lambda>x xa xb. f x + xb + Suc 0)"
earley_recognised_def: shows "earley_recognised = Bex \<II> is_finished"
pmaxsimpchain_def: fixes X :: "'b set set"   and xs :: "'b set list" assumes "SimplicialComplex X" shows "SimplicialComplex.pmaxsimpchain X xs \<equiv> (\<forall>x\<in>set xs. SimplicialComplex.maxsimp X x) \<and> padjacentchain xs"
Smith_normal_form_of_JNF_def: fixes A :: "'a mat"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "Smith_normal_form_of_JNF A bezout = (let (P'', D, Q'') = diagonalize_JNF A bezout; (P', S, Q') = diagonal_to_Smith_PQ_JNF D bezout in (P' * P'', S, Q'' * Q'))"
prod_defs_1: fixes f :: "nat \<Rightarrow> 'e"   and M :: "nat"   and p :: "'e" shows "raw_has_prod f M p \<equiv> (\<lambda>n. \<Prod>i\<le>n. f (i + M)) \<longlonglongrightarrow> p \<and> p \<noteq> (0::'e)"
assms_2: shows "\<forall>i. 0 \<le> c i"
step': fixes y :: "'a"   and x :: "'b" assumes "y \<in> U (F (C (inf (U (C (lfp (\<lambda>f. U (F (C f)))))) (\<lambda>x. {y. P x y})))) x" shows "P x y"
monom_mult_sum_left: fixes f :: "'e \<Rightarrow> 'd"   and C :: "'e set"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'd" shows "monom_mult (sum f C) t p = (\<Sum>c\<in>C. monom_mult (f c) t p)"
dvd: shows "even ((\<v> - 1) * (\<Lambda> * (\<Lambda> - 1)))"
Suc_le_D: fixes n :: "nat"   and m' :: "nat" assumes "Suc n \<le> m'" shows "\<exists>m. m' = Suc m"
gatherDomain_def: fixes p :: "('c \<Rightarrow> 'd option) list" shows "gatherDomain p = gatherDomain_aux (butlast p)"
valid_return_list_def: fixes cs :: "'edge list"   and n :: "'node" shows "valid_return_list cs n \<equiv> \<forall>cs' c cs''. cs = cs' @ c # cs'' \<longrightarrow> valid_edge c \<and> (\<exists>Q p f. kind c = Q\<hookleftarrow>\<^bsub>p\<^esub>f \<and> p = get_proc (case cs' of [] \<Rightarrow> n | a # list \<Rightarrow> last (targetnodes cs')))"
owhile_inv_def: fixes C :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'c \<Rightarrow> 'd option"   and x :: "'d"   and I :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and M :: "'d \<Rightarrow> 'c \<Rightarrow> nat" shows "owhile_inv C B x I M = owhile C B x"
nn_integral_indicator_singleton: fixes y :: "'b"   and M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" assumes "{y} \<in> sets M" shows "\<integral>\<^sup>+ x. f x * indicator {y} x \<partial>M = f y * emeasure M {y}"
mutual_exclusion': fixes s :: "nat \<times> nat \<times> (nat \<Rightarrow> thread)"   and t :: "nat"   and t' :: "nat"   and tk :: "nat"   and tk' :: "nat" assumes "A.reachable s"   and "t < N"   and "t' < N"   and "t \<noteq> t'"   and "locks_ticket (tts s t) tk"   and "locks_ticket (tts s t') tk'" shows "False"
Rep_SkewBinomialHeap: fixes x :: "('a, 'b) SkewBinomialHeap" shows "Rep_SkewBinomialHeap x \<in> {q. BsSkewBinomialHeapStruc.bs_invar q}"
ass_2: shows "(k, \<gamma>) \<in> set (a # l)"
Statuses_def: shows "Statuses = fst \<circ> Rep_kripke"
dvd_minus_iff: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "dvd.dvd times x (uminus y) = dvd.dvd times x y"
assms_3: shows "\<D>\<^sub>\<circ> xs = vcard xs"
Trg_Dom: fixes t :: "'a local.term" assumes "Arr t" shows "Trg (Dom t) = Trg t"
h2: shows "h \<turnstile> forall_M (\<lambda>(slot, assigned_nodes). Heap_Error_Monad.bind (Heap_Error_Monad.bind (get_child_nodes (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r slot)) (forall_M remove)) (\<lambda>_. forall_M (append_child (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r slot)) assigned_nodes)) tups \<rightarrow>\<^sub>h h2"
complete_refl_on: fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and x :: "'c"   and y :: "'c" assumes "complete A r"   and "refl_on A r"   and "x \<in> A"   and "y \<in> A" shows "x \<^bsub>r\<^esub>\<preceq> y \<or> y \<^bsub>r\<^esub>\<preceq> x"
trancl_rtrancl_trancl: fixes a :: "'c"   and b :: "'c"   and r :: "('c \<times> 'c) set"   and c :: "'c" assumes "(a, b) \<in> r\<^sup>+"   and "(b, c) \<in> r\<^sup>*" shows "(a, c) \<in> r\<^sup>+"
homeomorphic_ball01_UNIV: shows "ball (0::'b) 1 homeomorphic UNIV"
quasi_inverses_def: fixes f :: "'b"   and g :: "'b" shows "D.quasi_inverses f g \<equiv> \<exists>\<phi> \<psi>. D.internal_equivalence f g \<phi> \<psi>"
f_shrink_last_nth: fixes n :: "nat"   and xs :: "'d list"   and k :: "nat" assumes "n < length xs div k" shows "xs \<div>\<^sub>l k ! n = xs ! (n * k + k - Suc 0)"
is_sum4sq_nat_def: fixes n :: "nat" shows "is_sum4sq_nat n \<equiv> \<exists>a b c d. n = sum4sq_nat a b c d"
vec_of_basis_enum_times: fixes \<psi> :: "'b"   and \<phi> :: "'b" shows "vec_of_basis_enum (\<psi> * \<phi>) = vec_of_list [vec_of_basis_enum \<psi> $ 0 * vec_of_basis_enum \<phi> $ 0]"
scale_right_distrib_on: fixes x :: "'b"   and y :: "'b"   and a :: "'a" assumes "x \<in> U\<^sub>V\<^sub>S"   and "y \<in> U\<^sub>V\<^sub>S" shows "a *s (x + y) = a *s x + a *s y"
assms_1: shows "\<forall>t\<in>M. \<exists>c. t = Fun c []"
rbl_word_plus': fixes x :: "'a word"   and y :: "'a word" shows "to_bl (x + y) = rev (rbl_plus False (rev (to_bl x)) (rev (to_bl y)))"
94: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<squnion> - z \<sqinter> x) = y \<squnion> x"
simps_1: fixes a :: "'a" shows "n1 (L2 a) = N2 N0 a N0"
unat_shiftl_less_t2n: fixes x :: "'b word"   and m :: "nat"   and n :: "nat" assumes "unat x < 2 ^ (m - n)"   and "m < LENGTH('b)" shows "unat (x << n) < 2 ^ m"
2: fixes t :: "'a \<Rightarrow>\<^sub>0 nat" assumes "t \<noteq> 0" shows "0 \<prec> t"
consistent_signs_prop_R: fixes p :: "real poly"   and x :: "real"   and sign :: "rat list"   and qs :: "real poly list" assumes "p \<noteq> 0"   and "x \<in> {x. poly p x = 0}"   and "sign = consistent_sign_vec qs x" shows "list_all (\<lambda>s. s = 1 \<or> s = 0 \<or> s = - 1) sign"
e: fixes i :: "nat" shows "e i \<in> {- 1..1}"
that: fixes y :: "'a formula"   and z :: "'a formula"   and b :: "'b"   and c :: "'b" assumes "x = ifte (var i) y z"   and "a = ifte (f i) b c"   and "hom_graph f S y b"   and "hom_graph f S z c" shows "thesis"
measurable_prod_emb: fixes J :: "'d set"   and L :: "'d set"   and X :: "('d \<Rightarrow> 'e) set"   and M :: "'d \<Rightarrow> 'e measure" assumes "J \<subseteq> L"   and "X \<in> sets (Pi\<^sub>M J M)" shows "prod_emb L M J X \<in> sets (Pi\<^sub>M L M)"
component_of_wg: fixes F :: "'b program"   and H :: "'b program"   and X :: "'b program set" assumes "F component_of H" shows "(H \<in> wg F X) = (H \<in> X)"
ag_mOp_closed: fixes x :: "'a" assumes "x \<in> carrier M" shows "-\<^sub>a x \<in> carrier M"
vimage_def: fixes f :: "'b \<Rightarrow> 'c"   and B :: "'c set" shows "f -` B \<equiv> {x. f x \<in> B}"
assms_1: shows "f \<in> (\<Pi>\<^sub>E i\<in>I. carrier (G i))"
seq_if_composable: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "\<nu> \<star>\<^sub>B \<mu> \<noteq> B.null" shows "src\<^sub>B \<nu> = trg\<^sub>B \<mu>"
ht_hash_ht_distinct_map_of: fixes l :: "('a \<times> 'b) list list"   and k :: "'a" assumes "ht_hash l"   and "ht_distinct l"   and "1 < length l" shows "map_of (concat l) k = map_of (l ! bounded_hashcode_nat (length l) k) k"
fundchamber_WS_image_adjacent: fixes w :: "'a permutation"   and s :: "'a permutation" assumes "w \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>"   and "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})" shows "w `\<rightarrow> C0 \<sim> (w + s) `\<rightarrow> C0"
infinite_cyclic_subgroup_order: fixes x :: "'a" assumes "x \<in> carrier G" shows "infinite (carrier (subgroup_generated G {x})) = (ord x = 0)"
i\<^sub>\<MM>_def: shows "\<MM>.smallest_accepting_rank\<^sub>R = Some i\<^sub>\<MM>"
del_undirected: fixes v :: "'a"   and e :: "'b"   and v' :: "'a"   and g :: "('a, 'b, 'c) graph_scheme" shows "del_unEdge v e v' g = delete_edge v' e v (delete_edge v e v' g)"
qGood_qGoodInp: fixes delta :: "'e"   and qinp :: "'c \<Rightarrow> ('c, 'd, 'varSort, 'var, 'e) qTerm option"   and qbinp :: "'d \<Rightarrow> ('c, 'd, 'varSort, 'var, 'e) qAbs option" shows "qGood (qOp delta qinp qbinp) = (qGoodInp qinp \<and> qGoodBinp qbinp)"
add_mset_eq_single: fixes b :: "'f"   and M :: "'f multiset"   and a :: "'f" shows "(add_mset b M = {#a#}) = (b = a \<and> M = {#})"
atMost_Suc: fixes g :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "sum g {..Suc n} = sum g {..n} + g (Suc n)"
GF_advice_nested_prop_atoms\<^sub>\<nu>: fixes \<phi> :: "'b ltln"   and X :: "'b ltln set" shows "nested_prop_atoms (\<phi>[X]\<^sub>\<nu>) \<subseteq> nested_prop_atoms\<^sub>\<nu> \<phi> X"
surjective: fixes r :: "('b, 'c) potype_scheme" shows "r = \<lparr>pset = pset r, order = order r, \<dots> = more r\<rparr>"
count_image_mset_ge_count: fixes A :: "'c multiset"   and b :: "'c"   and f :: "'c \<Rightarrow> 'd" shows "count A b \<le> count (image_mset f A) (f b)"
ZObisT_prefix: shows "ZObisT \<subseteq> ZOretrT ZObisT"
to_stream_snth: fixes w :: "nat \<Rightarrow> 'b"   and k :: "nat" shows "to_stream w !! k = w k"
simps_1: fixes C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom" shows "varacom (C\<^sub>1;; C\<^sub>2) = varacom C\<^sub>1 \<union> varacom C\<^sub>2"
set_mrbst_inter_diff: fixes t' :: "'a tree"   and b :: "bool"   and t1 :: "'a tree"   and t2 :: "'a tree" assumes "t' \<in> set_pmf (mrbst_inter_diff b t1 t2)"   and "bst t1"   and "bst t2" shows "set_tree t' = (if b then (\<inter>) else (-)) (set_tree t1) (set_tree t2)"
diff_sup_eq_inf: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "minus a (sup b c) = plus a (inf (uminus b) (uminus c))"
VarTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and v :: "String.literal"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E Var v : \<tau>" obtains "fmlookup \<Gamma> v = Some \<tau>"
proper_map: fixes S :: "'a set"   and K :: "'a set"   and T :: "'b set"   and f :: "'a \<Rightarrow> 'b" assumes "closedin (top_of_set S) K"   and "\<And>U. \<lbrakk>U \<subseteq> T; compact U\<rbrakk> \<Longrightarrow> compact (S \<inter> f -` U)"   and "f ` S \<subseteq> T" shows "closedin (top_of_set T) (f ` K)"
simps_2: fixes y :: "bool"   and ys :: "bool list"   and x :: "bool list" shows "rbl_add (y # ys) x = (let ws = rbl_add ys (tl x) in (y \<noteq> hd x) # (if hd x \<and> y then rbl_succ ws else ws))"
RBT_HM_1: shows "m = RBT_HM m'"
strict_mono_enumerate: fixes S :: "nat set" assumes "infinite S" shows "strict_mono (enum S)"
lappend_lnull2: fixes ys :: "'a llist"   and xs :: "'a llist" assumes "lnull ys" shows "lappend xs ys = xs"
gen_dfs_post_def: fixes E :: "('b \<times> 'b) set"   and U :: "'c"   and S :: "'b set"   and V0 :: "'b set"   and u0 :: "'b"   and V :: "'b set"   and brk :: "bool" shows "gen_dfs_post E U S V0 u0 V brk \<equiv> (\<not> brk \<longrightarrow> E `` (V - S) \<subseteq> V) \<and> u0 \<in> V \<and> V0 \<subseteq> V \<and> V \<subseteq> V0 \<union> (E - UNIV \<times> S)\<^sup>* `` {u0}"
3: fixes x :: "real" shows "\<exists>d>0. \<forall>a b. a \<le> x \<and> x \<le> b \<and> b - a < d \<longrightarrow> P a b"
xs_decomp: shows "xs = xs_pre @ new_last # xs_post"
Quotient_uint: shows "Quotient (=) Abs_uint Rep_uint cr_uint"
iso_inv_iso: fixes f :: "'c" assumes "C.iso f" shows "C.iso (C.inv f)"
lexn_transI: fixes r :: "('g \<times> 'g) set"   and n :: "nat" assumes "trans r" shows "trans (lexn r n)"
AA: shows "swapcols J (Suc J) A \<in> carrier_mat n m"
rev: assumes "(v, u) \<in> E"   and "(u, v) \<notin> E" shows "thesis"
smap_lub: fixes A :: "'a set"   and z :: "'a" assumes "\<And>x. x \<in> A \<Longrightarrow> sleq x z" shows "sleq (smap A) z"
safe_length: fixes x :: "'c" assumes "x \<in> Csafe" shows "length ode_e = DIM('c)"
has_field_derivative_subset: fixes f :: "'b \<Rightarrow> 'b"   and y :: "'b"   and x :: "'b"   and s :: "'b set"   and t :: "'b set" assumes "(f has_field_derivative y) (at x within s)"   and "t \<subseteq> s" shows "(f has_field_derivative y) (at x within t)"
upd_rev_cs_Cons_Return: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and p :: "'pname"   and f :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option"   and cs :: "'edge list"   and as :: "'edge list" assumes "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f" shows "upd_rev_cs cs (a # as) = a # upd_rev_cs cs as"
downset_iso_iff: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "(prod.downset x \<subseteq> prod.downset y) = less_eq_prod x y"
bound: fixes x :: "nat"   and xs :: "nat list"   and s :: "nat" assumes "B < x" shows "cond (x # xs) s = False"
infsetsum'_converges_union_disjoint: fixes A :: "'c set"   and f :: "'d \<Rightarrow> 'e"   and B :: "'c \<Rightarrow> 'd set" assumes "finite A"   and "\<And>a. a \<in> A \<Longrightarrow> infsetsum'_converges f (B a)"   and "\<And>a a'. \<lbrakk>a \<in> A; a' \<in> A; a \<noteq> a'\<rbrakk> \<Longrightarrow> B a \<inter> B a' = {}" shows "infsetsum'_converges f (\<Union> (B ` A))"
indRelLTPO_impl_TRel_preserves_barbs: fixes TRel :: "('procT \<times> 'procT) set" assumes "rel_preserves_binary_pred (indRelLTPO TRel) (\<lambda>a. hasBarb a (STCalWB SWB TWB))" shows "rel_preserves_binary_pred TRel (\<lambda>a. hasBarb a TWB)"
norm_sum_f: fixes A :: "(nat \<times> nat) set" shows "norm (\<Sum>(i, j)\<in>A. norm (a i) * norm (b j)) = (\<Sum>(i, j)\<in>A. norm (a i) * norm (b j))"
subseq_append_le_same_iff: fixes xs :: "'c list"   and ys :: "'c list" shows "subseq (xs @ ys) ys = (xs = [])"
ta_hb_consistent_lappendD2: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) list"   and E' :: "('b \<times> ('c, 'b) obs_event action) llist"   and E'' :: "('b \<times> ('c, 'b) obs_event action) llist" assumes "ta_hb_consistent P E (lappend E' E'')"   and "lfinite E'" shows "ta_hb_consistent P (E @ list_of E') E''"
M_in_subprob: fixes M :: "'c measure" assumes "prob_space M" shows "M \<in> space (subprob_algebra M)"
c_def: shows "c = (\<lambda>j. real_of_int (b j) / real_of_int (a j))"
list_eq_def: fixes xs :: "'a list"   and ys :: "'a list" shows "list_eq xs ys = (set xs = set ys)"
ffun_minus_self: fixes f :: "('a, 'b) ffun" shows "f - f = {}\<^sub>f"
h4a: shows "\<forall>i. insertion (nth_default 0 (L[var := C])) (isolate_variable_sparse p var i) = insertion (nth_default 0 (L[var := (A + B * C) / D])) (isolate_variable_sparse p var i)"
keys_tabulate: fixes ks :: "'b list"   and f :: "'b \<Rightarrow> 'c" shows "keys (tabulate ks f) = set ks"
exec_meth_drop_xt: fixes ci :: "('addr, 'heap) check_instr"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'addr instr list"   and ins' :: "'addr instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and xt' :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'thread_id"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and n :: "nat" assumes "exec_meth ci P (ins @ ins') (xt @ shift (length ins) xt') t h (stk, loc, length ins + pc, xcp) ta h' (stk', loc', pc', xcp')"   and "pcs xt \<subseteq> {..<length ins}"   and "jump_ok ins' 0 n" shows "exec_meth ci P ins' xt' t h (stk, loc, pc, xcp) ta h' (stk', loc', pc' - length ins, xcp')"
exhaust_sel: fixes option :: "'b option" obtains "option = None"   | "option = \<lfloor>the option\<rfloor>"
col_mat_of_cols: fixes j :: "nat"   and vs :: "'b vec list"   and n :: "nat" assumes "j < length vs"   and "vs ! j \<in> carrier_vec n" shows "col (mat_of_cols n vs) j = vs ! j"
distr_cong: fixes M :: "'c measure"   and K :: "'c measure"   and N :: "'d measure"   and L :: "'d measure"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "M = K"   and "sets N = sets L"   and "\<And>x. x \<in> space M \<Longrightarrow> f x = g x" shows "distr M N f = distr K L g"
poly_rel_def: fixes ops :: "'a arith_ops_record"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a list"   and x' :: "'b poly" assumes "ring_ops ops R" shows "ring_ops.poly_rel R x x' = list_all2 R x (coeffs x')"
intro: fixes hcond :: "'a hrel \<Rightarrow> 'a hrel" assumes "complete_lattice (utp_order hcond)" shows "utp_theory_lattice_axioms hcond"
elims_3: fixes cenum_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and x :: "'a list \<Rightarrow> bool"   and xa :: "nat" assumes "\<not> all_n_lists cenum_all x xa" obtains P :: "'a list \<Rightarrow> bool" and n :: "nat" where "x = P" and "xa = n" and "\<not> (if n = 0 then P [] else cenum_all (\<lambda>x. all_n_lists cenum_all (\<lambda>xs. P (x # xs)) (n - 1)))"
prems_2: shows "fv (e1 \<guillemotleft>bop\<guillemotright> e2) \<subseteq> dom s"
typeof_empty_is_type: fixes v :: "val"   and T :: "ty"   and G :: "prog" assumes "typeof Map.empty v = Some T" shows "is_type G T"