hcju/isabelleps huggingface dataset:
P: fixes i :: "nat" shows "P i \<in> Pi\<^sub>M {0..<i} M \<rightarrow>\<^sub>M subprob_algebra (M i)"
equiintegrable_minus: fixes F :: "('a \<Rightarrow> 'b) set"   and I :: "'a set" assumes "F equiintegrable_on I" shows "(\<Union>f\<in>F. {\<lambda>x. - f x}) equiintegrable_on I"
card_proots_unbounded_line: fixes st :: "complex"   and tt :: "complex"   and p :: "complex poly" assumes "st \<noteq> tt" shows "card (proots_within p (unbounded_line st tt)) = (let pc = p \<circ>\<^sub>p [:st, tt - st:]; pR = map_poly Re pc; pI = map_poly Im pc; g = gcd pR pI in nat (changes_R_smods g (pderiv g)))"
assms: shows "exp (\<i> * complex_of_real x) = z"
is_poincare_line_inverse_point: fixes H :: "circline"   and u :: "complex_homo" assumes "is_poincare_line H"   and "u \<in> circline_set H" shows "inversion u \<in> circline_set H"
intros_2: fixes \<Gamma> :: "binding list"   and X :: "tyvrs"   and T :: "ty" assumes "\<turnstile> \<Gamma> ok"   and "X \<sharp> ty_dom \<Gamma>"   and "T closed_in \<Gamma>" shows "\<turnstile> (TVarB X T # \<Gamma>) ok"
SPN_is_functor: shows "functor (\<cdot>) (\<bullet>) SPN"
tranclD2: fixes x :: "'a"   and y :: "'a"   and R :: "('a \<times> 'a) set" assumes "(x, y) \<in> R\<^sup>+" shows "\<exists>z. (x, z) \<in> R\<^sup>* \<and> (z, y) \<in> R"
order_refl: fixes x :: "'a" assumes "x \<in> U" shows "x \<le>\<^sub>o\<^sub>w x"
conf_widen: fixes P :: "'a prog"   and h :: "heap"   and v :: "val"   and T :: "ty"   and T' :: "ty" assumes "P,h \<turnstile> v :\<le> T"   and "P \<turnstile> T \<le> T'" shows "P,h \<turnstile> v :\<le> T'"
keys_impl_of: fixes t :: "('a, 'b) RBT.rbt" shows "RBT_Impl.keys (impl_of t) = keys t"
the_elem_set: fixes x :: "'b" shows "the_elem (set [x]) = x"
kuratowski_contr: fixes K :: "'a pair_pre_digraph" assumes "subgraph (with_proj K) (with_proj G)"   and "pair_pseudo_graph K"   and "K\<^bsub>3,3\<^esub> (with_proj (gen_contr_graph (with_proj K) (verts3 (with_proj K)))) \<or> K\<^bsub>5\<^esub> (with_proj (gen_contr_graph (with_proj K) (verts3 (with_proj K))))" shows "\<not> kuratowski_planar (with_proj G)"
to_tree_order_subset: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and nodes :: "(_) object_ptr list"   and node :: "(_) object_ptr"   and nodes' :: "(_) object_ptr list" assumes "Shadow_DOM.heap_is_wellformed h"   and "h \<turnstile> Shadow_DOM.to_tree_order ptr \<rightarrow>\<^sub>r nodes"   and "node \<in> set nodes"   and "h \<turnstile> Shadow_DOM.to_tree_order node \<rightarrow>\<^sub>r nodes'"   and "ShadowRootClass.known_ptrs h"   and "ShadowRootClass.type_wf h" shows "set nodes' \<subseteq> set nodes"
prod_pOp_mem: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme"   and X :: "'d \<Rightarrow> 'e"   and Y :: "'d \<Rightarrow> 'e" assumes "\<forall>k\<in>I. aGroup (A k)"   and "X \<in> carr_prodag I A"   and "Y \<in> carr_prodag I A" shows "prod_pOp I A X Y \<in> carr_prodag I A"
of_class_code1: fixes thy :: "exetheory"   and ty :: "typ"   and c :: "String.literal" assumes "exe_wf_theory' thy" shows "(has_sort (osig (sig (translate_theory thy))) ty (insert c full_sort) \<and> typ_ok (translate_theory thy) ty) = (exehas_sort (exesorts (exesig thy)) ty (insert c full_sort) \<and> exetyp_ok thy ty)"
cycle_def: fixes xs :: "'a list" shows "Graph_T1.cycle xs \<equiv> Graph_T1.path xs \<and> 2 < length xs \<and> last xs \<rightarrow>\<^bsub>T1\<^esub> hd xs"
ign_res_eq_def2: fixes P :: "'b assn"   and w :: "vals"   and Y :: "vals"   and s :: "state"   and Z :: "'b" shows "(P\<down>=w) Y s Z = ((\<exists>Y. P Y s Z) \<and> Y = w)"
moebius_cmat_eq_def: fixes A :: "complex \<times> complex \<times> complex \<times> complex"   and B :: "complex \<times> complex \<times> complex \<times> complex" shows "moebius_cmat_eq A B = (\<exists>k. k \<noteq> 0 \<and> B = k *\<^sub>s\<^sub>m A)"
vars_formula_def: fixes formula :: "Literal list list" shows "vars formula \<equiv> varsFormula formula"
count_mset_set_1: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "x \<in> A" shows "count (mset_set A) x = 1"
intro: fixes X :: "'a set"   and is_open :: "'a set \<Rightarrow> bool"   and U :: "'a set"   and I :: "real set"   and C :: "real \<Rightarrow> 'a set" assumes "open_cover_of_subset X is_open U I C"   and "open_cover_of_open_subset_axioms is_open U" shows "open_cover_of_open_subset X is_open U I C"
nud: fixes p :: "('a, 'd) simple_packet_scheme" shows "simple_fw fw p \<noteq> Undecided"
one_matrix_def: fixes n :: "nat" shows "one_matrix n = Abs_matrix (\<lambda>j i. if j = i \<and> j < n then 1::'b else (0::'b))"
constant_degree: fixes p :: "'b poly" shows "constant (poly p) = (degree p = 0)"
map_mset_unsorted_term_poly_list_rel: fixes s :: "(char list list \<times> int) list" assumes "\<And>a. a \<in> mononoms s \<Longrightarrow> distinct a"   and "\<forall>x\<in>mononoms s. distinct x"   and "\<forall>xs\<in>mononoms s. sorted_wrt (rel2p (Id \<union> var_order_rel)) xs" shows "(s, map (\<lambda>(a, y). (mset a, y)) s) \<in> \<langle>term_poly_list_rel \<times>\<^sub>r int_rel\<rangle>list_rel"
dang1: fixes x :: "'b"   and y :: "'b" shows "\<nu> (\<nu> x \<sqinter> y) \<le> \<nu> x"
right_add_zero_mat: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "A + 0\<^sub>m nr nc = A"
disj_move_sparse_row_vector_twice: fixes j :: "int"   and u :: "int"   and a :: "(nat \<times> 'a) list"   and i :: "int"   and b :: "(nat \<times> 'a) list"   and v :: "int" assumes "j \<noteq> u" shows "disj_matrices (move_matrix (sparse_row_vector a) j i) (move_matrix (sparse_row_vector b) u v)"
Sum_any_closed: fixes f :: "'c \<Rightarrow> 'a" assumes "\<And>x. f x \<in> A" shows "Sum_any f \<in> A"
hom_power: fixes hom :: "'a \<Rightarrow> 'b"   and x :: "'a"   and n :: "nat" assumes "monoid_mult_hom hom" shows "hom (x ^ n) = hom x ^ n"
remdups_adj_Cons: fixes x :: "'g"   and xs :: "'g list" shows "remdups_adj (x # xs) = (case remdups_adj xs of [] \<Rightarrow> [x] | y # xs \<Rightarrow> if x = y then y # xs else x # y # xs)"
class_add_sees_method_rev: fixes C :: "char list"   and cdec :: "jvm_method class"   and P :: "jvm_prog"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "jvm_method"   and D :: "char list" assumes "class_add P (C, cdec) \<turnstile> C\<^sub>0 sees M\<^sub>0, b :  Ts\<rightarrow>T = m in D"   and "(C\<^sub>0, C) \<notin> (subcls1 P)\<^sup>*" shows "P \<turnstile> C\<^sub>0 sees M\<^sub>0, b :  Ts\<rightarrow>T = m in D"
5: shows "run ((v\<^sub>1 @ l) @- w\<^sub>2 @- u) q"
fSup_unfold: fixes f :: "nat \<Rightarrow> 'c \<times> 'd" shows "sup_prod (f 0) (Sup_prod (range (\<lambda>n. f (Suc n)))) = Sup_prod (range f)"
indep_vars_def: fixes i :: "'b"   and M' :: "'d \<Rightarrow> 'e measure"   and X :: "'d \<Rightarrow> ennreal \<Rightarrow> 'e"   and I :: "'d set" shows "Y.indep_vars i M' X I = ((\<forall>i\<in>I. Y.random_variable i (M' i) (X i)) \<and> Y.indep_sets i (\<lambda>i. sigma_sets (space (distr M borel (Y i))) {X i -` A \<inter> space (distr M borel (Y i)) |A. A \<in> sets (M' i)}) I)"
heq: fixes y :: "'c" assumes "y \<in> U" shows "h (0, y) = p (f y)"
fresh_star_list_nil_2: fixes Xs :: "name set" shows "Xs \<sharp>* []"
log_spied: fixes s :: "(agent \<times> msg) set"   and X :: "msg" assumes "s\<^sub>0 \<Turnstile> s"   and "Log X \<in> parts (used s)" shows "Log X \<in> spied s"
incomplete_imp_incomp_block: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int"   and bl :: "'b set" assumes "incomplete_design point_set block_collection u_block_size"   and "bl \<in># block_collection" shows "card bl < card point_set \<and> bl \<in># block_collection"
resize_vec_def: fixes n :: "nat"   and v :: "'c vec" shows "resize_vec n v = vec n (($) v)"
rep_eq: fixes x :: "'d \<Rightarrow>\<^sub>0 'e"   and xa :: "'d \<Rightarrow>\<^sub>0 'e" shows "lookup (x + xa) = (\<lambda>k. lookup x k + lookup xa k)"
\<alpha>_def: fixes SBIP :: "'a list \<times> nat list \<times> ('a \<Rightarrow> node_state option) \<times> (nat \<times> 'a set) list" shows "GS.\<alpha> SBIP \<equiv> (GS.p_\<alpha> SBIP, GS.D_\<alpha> SBIP, GS.pE_\<alpha> SBIP)"
size_int: fixes l :: "int"   and r :: "int" shows "int_index_def.size (Bound l r) = nat (r - l)"
eq_\<I>_comp_cong: fixes \<I>1 :: "('y, 'z) \<I>"   and \<I>2 :: "('aa, 'ab) \<I>"   and conv1 :: "('y, 'z, 'aa, 'ab) converter"   and conv1' :: "('y, 'z, 'aa, 'ab) converter"   and \<I>3 :: "('ac, 'ad) \<I>"   and conv2 :: "('aa, 'ab, 'ac, 'ad) converter"   and conv2' :: "('aa, 'ab, 'ac, 'ad) converter" assumes "\<I>1,\<I>2 \<turnstile>\<^sub>C conv1 \<sim> conv1'"   and "\<I>2,\<I>3 \<turnstile>\<^sub>C conv2 \<sim> conv2'" shows "\<I>1,\<I>3 \<turnstile>\<^sub>C conv1 \<odot> conv2 \<sim> conv1' \<odot> conv2'"
fls_divide_X_intpow: fixes f :: "'a fls"   and i :: "int" shows "f / fls_X_intpow i = fls_shift i f"
rel_pmf_measureI: fixes p :: "'c spmf"   and q :: "'d spmf"   and R :: "'c option \<Rightarrow> 'd option \<Rightarrow> bool" assumes "\<And>A. measure_pmf.prob p A \<le> measure_pmf.prob q {y. \<exists>x\<in>A. R x y}" shows "rel_pmf R p q"
funs_term_subterms_eq_1: fixes t :: "('a, 'b) Term.term" shows "\<Union> (funs_term ` subterms t) = funs_term t"
simps: fixes x :: "real"   and y :: "real" shows "Rmax x + Rmax y = Rmax (max x y)"
join_below_iff: fixes x :: "'b"   and y :: "'b"   and a :: "'b" assumes "compatible x y" shows "(x \<squnion> y \<sqsubseteq> a) = (x \<sqsubseteq> a \<and> y \<sqsubseteq> a)"
inside_eq_outside: fixes S :: "'a set" shows "(inside S = outside S) = (S = UNIV)"
assms_2: shows "n \<in>\<^sub>\<circ> Vset \<alpha>"
ListSum_compl1: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list"   and f :: "'b \<Rightarrow> nat" shows "ListSum (filter (\<lambda>x. \<not> P x) xs) f + ListSum (filter P xs) f = ListSum xs f"
simps_1: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "PAC_Checker_Init.merge f (x # xs) (y # ys) = (if f x y then x # PAC_Checker_Init.merge f xs (y # ys) else y # PAC_Checker_Init.merge f (x # xs) ys)"
assms_1: shows "((\<sigma>, s), a, \<sigma>', s') \<in> oparp_sos i (oseqp_sos \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V i) (seqp_sos \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G)"
simps_2: fixes s :: "'b set"   and t :: "'b set"   and ts :: "'b set list" shows "difference_list_opt s (t # ts) = (t - s) # difference_list_opt (s - t) ts"
less_minus_one_simps_4: 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 sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "\<not> less one (uminus one)"
trans: fixes r :: "'h"   and s :: "'h"   and t :: "'h" assumes "r = s"   and "s = t" shows "r = t"
return_bind: fixes x :: "'e"   and f :: "'e \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'c) set \<times> bool" shows "return x >>= f = f x"
Cons_genPrefix_Cons: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list"   and r :: "('a \<times> 'a) set" shows "((x # xs, y # ys) \<in> genPrefix r) = ((x, y) \<in> r \<and> (xs, ys) \<in> genPrefix r)"
append_same_eq: fixes ys :: "'d list"   and xs :: "'d list"   and zs :: "'d list" shows "(ys @ xs = zs @ xs) = (ys = zs)"
alt_group_is_unsolvable: fixes n :: "nat" assumes "5 \<le> n" shows "\<not> solvable (alt_group n)"
applied_rule_rev_def: fixes C :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and a :: "'b"   and x :: "'a list" shows "applied_rule_rev C a x = applied_rule C a (rev x)"
exhaust: fixes y :: "'b lift" obtains "y = \<bottom>"   | a :: "'b" where "y = Def a"
conth0: shows "path (\<lambda>u. h (u, 0))"
LIMSEQ_I: fixes X :: "nat \<Rightarrow> 'a"   and L :: "'a" assumes "\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r" shows "X \<longlonglongrightarrow> L"
degree_divisor2: fixes f :: "'b mod_ring poly"   and d :: "nat"   and c :: "nat" assumes "irreducible f"   and "degree f = d"   and "1 \<le> c"   and "c < d" shows "\<not> f dvd monom 1 1 ^ CARD('b) ^ c - monom 1 1"
If: fixes G :: "'b triple set"   and P :: "'b assn"   and b :: "state \<Rightarrow> bool"   and c :: "com"   and Q :: "'b assn"   and d :: "com" assumes "G|-{P &> b}. c .{Q}"   and "G|-{P &> Not \<circ> b}. d .{Q}" shows "G|-{P}. IF b THEN c ELSE d .{Q}"
degeneralize_language: fixes A :: "('a, 'b) ngba" shows "NBA.language (degeneralize A) = NGBA.language A"
tensor_preserves_iso: fixes f :: "'d"   and g :: "'d" assumes "D.iso f"   and "D.iso g" shows "D.iso (f \<otimes>\<^sub>D g)"
nonzero_imp_inverse_nonzero: fixes a :: "'a" assumes "a \<noteq> (0::'a)" shows "inverse a \<noteq> (0::'a)"
lookup_default_def: fixes d :: "'a"   and m :: "('b, 'a) mapping"   and k :: "'b" shows "Mapping.lookup_default d m k = (case Mapping.lookup m k of None \<Rightarrow> d | Some v \<Rightarrow> v)"
range_bintrunc: fixes n :: "nat" shows "range (take_bit n) = {i. 0 \<le> i \<and> i < 2 ^ n}"
two_powr_int_float: fixes i :: "int" shows "2 powr real_of_int i \<in> float"
Int_space_eq1: fixes x :: "'a set" assumes "x \<in> sigma_sets \<Omega> E" shows "\<Omega> \<inter> x = x"
viterbi_final_max_prob: fixes s :: "'s"   and os :: "'t list"   and as :: "'s list"   and v :: "ennreal" assumes "viterbi_final s os = (as, v)"   and "s \<in> \<S>" shows "v = max_prob s os"
diagonal_to_Smith_PQ: fixes A :: "(('a, 'b) vec, 'c) vec"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and P :: "(('a, 'c) vec, 'c) vec"   and B :: "(('a, 'b) vec, 'c) vec"   and Q :: "(('a, 'b) vec, 'b) vec" assumes "isDiagonal A"   and "is_bezout_ext bezout"   and "(P, B, Q) = diagonal_to_Smith_PQ A bezout" shows "B = P ** A ** Q \<and> invertible P \<and> invertible Q \<and> B = diagonal_to_Smith A bezout"
subdiv_ends_G_rev_1: shows "tail I (rev_I uv) = v"
ccsqrt: fixes s :: "complex"   and z :: "complex" assumes "s * s = z" shows "s = ccsqrt z \<or> s = - ccsqrt z"
normal_iff_subgroup: fixes G :: "('b, 'c) monoid_scheme"   and N :: "'b set" assumes "comm_group G" shows "N \<lhd> G = subgroup N G"
A'_3: shows "n' < funpow_dist1 f x y"
norm1_nonzero: fixes v :: "('b, 'c) Finite_Cartesian_Product.vec" assumes "v \<noteq> 0" shows "0 < norm1 v"
ih2: assumes " s1 \<longrightarrow>\<^sub>1 s'" shows " s' \<longrightarrow>\<^sub>1 s2"
simps_2: fixes pu :: "'a \<Rightarrow> bool"   and ar :: "'a \<Rightarrow> nat"   and M :: "('a, 'b) Term.term set"   and f :: "'a"   and T :: "('a, 'b) Term.term list" shows "intruder_synth' pu ar M (Fun f T) = (Fun f T \<in> M \<or> pu f \<and> length T = ar f \<and> list_all (intruder_synth' pu ar M) T)"
assms: shows "\<exists>P'. P \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel"
memberid_fdecl_simp: fixes vn :: "vname"   and f :: "Decl.field" shows "memberid (fdecl (vn, f)) = fid vn"
that: fixes f :: "'a \<Rightarrow> real" assumes "f \<in> borel_measurable M"   and "\<And>x. 0 < f x"   and "\<And>x. f x \<le> 1"   and "integrable M f" shows "thesis"
T_deter_rand: fixes step :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'b"   and t :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> nat"   and s0 :: "'b"   and qs :: "'c list"   and A :: "('b \<Rightarrow> 'e) \<times> ('b \<times> 'e \<Rightarrow> 'c \<Rightarrow> 'd \<times> 'e)"   and x :: "'e" shows "real (On_Off.T step t s0 qs (On_Off.off2 step A (s0, x) qs)) = On_Off.T_on_rand' step t (On_Off.embed A) (return_pmf (s0, x)) qs"
le_fun_parametric: fixes A1 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and A3 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and A2 :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "(A1 ===> A3 ===> (=)) (\<le>) (\<le>)"   and "bi_total A2" shows "((A2 ===> A1) ===> (A2 ===> A3) ===> (=)) (\<le>) (\<le>)"
not_finite_existsD: fixes P :: "'c \<Rightarrow> bool" assumes "infinite {a. P a}" shows "\<exists>a. P a"
graph_Entry_def: fixes \<alpha>e :: "'f \<Rightarrow> ('g \<times> 'h \<times> 'g) set"   and \<alpha>n :: "'f \<Rightarrow> 'g list"   and invar :: "'f \<Rightarrow> bool"   and inEdges' :: "'f \<Rightarrow> 'g \<Rightarrow> ('g \<times> 'h) list"   and Entry :: "'f \<Rightarrow> 'g" shows "graph_Entry \<alpha>e \<alpha>n invar inEdges' Entry \<equiv> graph_path \<alpha>e \<alpha>n invar inEdges' \<and> graph_Entry_axioms \<alpha>n invar inEdges' Entry"
lepoll_times2: fixes B :: "'a set"   and A :: "'b set" assumes "B \<noteq> {}" shows "A \<lesssim> B \<times> A"
apply_singleton_matrix: fixes f :: "'b \<Rightarrow> 'a"   and j :: "nat"   and i :: "nat"   and x :: "'b" assumes "f (0::'b) = (0::'a)" shows "apply_matrix f (singleton_matrix j i x) = singleton_matrix j i (f x)"
kind_Predicate_notin_slice_slice_kind_Predicate: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and a :: "'a"   and S :: "'b SDG_node set"   and Q :: "('c \<Rightarrow> 'd option) \<Rightarrow> bool" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "sourcenode a \<notin> SDG.SDG_to_CFG_set (SDG.HRB_slice sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S)"   and "valid_edge a"   and "kind a = (Q)\<^sub>\<surd>" obtains Q' :: "('c \<Rightarrow> 'd option) \<Rightarrow> bool" where "SDG.slice_kind sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S a = (Q')\<^sub>\<surd>" and "Q' = (\<lambda>s. False) \<or> Q' = (\<lambda>s. True)"
simps_2: fixes n :: "nat" shows "nth_item (Suc n) = (case sum_decode n of Inl i \<Rightarrow> case sum_decode i of Inl j \<Rightarrow> In0 (nth_item j) | Inr j \<Rightarrow> In1 (nth_item j) | Inr i \<Rightarrow> case sum_decode i of Inl j \<Rightarrow> Old_Datatype.Leaf (from_nat j) | Inr j \<Rightarrow> case prod_decode j of (a, b) \<Rightarrow> Scons (nth_item a) (nth_item b))"
assms: shows "S_sum_lists \<equiv> \<Union>ss\<in>lists (S \<union> uminus ` S). {sum_list ss}"
elims_2: fixes x :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xa :: "'c list" assumes "sorted_wrt x xa" obtains P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" where "x = Pa__" and "xa = []"   | P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" and xb :: "'c" and ys :: "'c list" where "x = P" and "xa = xb # ys" and "Ball (set ys) (P xb) \<and> sorted_wrt P ys"
ex_redchainTr2: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set" assumes "d_gchain G (Suc n) f" shows "D_gchain G 0 (constmap {0} {f (Suc n)})"
ltl_const_equiv_and_const_2: fixes \<phi>\<^sub>1 :: "'a ltln"   and \<phi>\<^sub>2 :: "'a ltln" shows "\<phi>\<^sub>1 and\<^sub>n \<phi>\<^sub>2 \<sim>\<^sub>C false\<^sub>n = (\<phi>\<^sub>1 \<sim>\<^sub>C false\<^sub>n \<or> \<phi>\<^sub>2 \<sim>\<^sub>C false\<^sub>n)"
subst_atm_comp_subst: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and A :: "'b"   and \<sigma> :: "'c"   and \<tau> :: "'c" assumes "substitution subst_atm id_subst comp_subst renamings_apart atm_of_atms" shows "subst_atm A (comp_subst \<sigma> \<tau>) = subst_atm (subst_atm A \<sigma>) \<tau>"
rep_eq: fixes x :: "'a fset set" shows "fset (Inf x) = (if finite (\<Inter> (fset ` x)) then \<Inter> (fset ` x) else {})"
plus_eq_MkI_conv: fixes l :: "Integer"   and n :: "Integer"   and m :: "int" shows "(l + n = MkI\<cdot>m) = (\<exists>l' n'. l = MkI\<cdot>l' \<and> n = MkI\<cdot>n' \<and> m = l' + n')"
assms_2: shows "f \<in> A \<rightarrow>\<^sub>E B"
convex_box_2: fixes a :: "'c"   and b :: "'c" shows "convex (box a b)"
C''_1: shows "cv \<turnstile> msg \<mapsto> c''"
distinct_49: fixes x5 :: "nat"   and x81 :: "nat"   and x82 :: "msg" shows "Key x5 \<noteq> Crypt x81 x82"
simps: fixes t :: "'b trm"   and s :: "'b trm" shows "vars_of_eq (Eq t s) = vars_of t \<union> vars_of s"
summable1: shows "(\<lambda>(n, p). f n p) abs_summable_on (SIGMA n:UNIV. {p. prime p \<and> p \<le> n})"
local_time_unbounded1: fixes A :: "'a set"   and k :: "nat" assumes "A \<in> sets M"   and "emeasure M A < \<infinity>" shows "(\<lambda>n. emeasure M {x \<in> (T ^^ n) --` A. local_time A n x < k}) \<longlonglongrightarrow> 0"
FEx_downward: fixes g :: "form"   and n :: "nat" assumes "infinite (deriv s)"   and "init s" shows "\<forall>m. (Suc m, FEx g) \<in> set (snd (f n)) \<longrightarrow> (\<exists>n'. (m, FEx g) \<in> set (snd (f n')))"
error_free_field_access: fixes G :: "prog"   and accC :: "qtname"   and statC :: "qtname"   and fn :: "vname"   and statDeclC :: "qtname"   and f :: "Decl.field"   and L :: "lenv"   and e :: "expr"   and s0 :: "st"   and s1 :: "state"   and a :: "val"   and s2 :: "state"   and v :: "vvar"   and s2' :: "state" assumes "accfield G accC statC fn = Some (statDeclC, f)"   and "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>e\<Colon>-Class statC"   and "G\<turnstile>Norm s0 \<midarrow>Init statDeclC\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>e-\<succ>a\<rightarrow> s2"   and "s2\<Colon>\<preceq>(G, L)"   and "normal s2 \<Longrightarrow> G,snd s2\<turnstile>a\<Colon>\<preceq>Class statC"   and "(v, s2') = fvar statDeclC (is_static f) fn a s2"   and "wf_prog G" shows "check_field_access G accC statDeclC fn (is_static f) a s2' = s2'"
vconverseD_2: fixes r :: "V"   and s :: "V"   and A :: "V" assumes "\<langle>r, s\<rangle> \<in>\<^sub>\<circ> vconverse A" shows "s = r\<inverse>\<^sub>\<circ>"
encode_goal_state_def: fixes \<Pi> :: "'a strips_problem"   and t :: "nat" shows "(\<Phi>\<^sub>G \<Pi>) t \<equiv> let vs = \<Pi>\<^sub>\<V>; G = \<Pi>\<^sub>G in \<^bold>\<And>map (\<lambda>v. encode_state_variable t (index vs v) (G v) \<^bold>\<or> \<bottom>) (filter (\<lambda>v. G v \<noteq> None) vs)"
fsmultD: fixes a :: "'f"   and v :: "'v" shows "a \<sharp>\<cdot> v = a \<delta>\<delta> (0::'g) \<cdot> v"
match_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and h :: "'b"   and g :: "'b"   and f :: "'b" assumes "category C"   and "partial_magma.arr C (C h (C g f))"   and "partial_magma.arr C (C g f)" shows "partial_magma.arr C (C h g)"
Blinfun_cases: fixes x :: "'d \<Rightarrow>\<^sub>L 'e" obtains y :: "'d \<Rightarrow> 'e" where "x = Blinfun y" and "y \<in> {f. bounded_linear f}"
timedelayed_e1: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and \<delta>\<tau> :: "'c tag_const"   and K\<^sub>2 :: "clock"   and K\<^sub>3 :: "clock"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "\<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^sub>e K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi>"
A: shows "X = (\<Union>n\<le>length xs. \<Union>W\<in>Rx n. W) \<and> (\<forall>W\<in>Rx 0. xs \<notin> sentences P \<and> None \<notin> set xs \<and> (xs, W) \<in> failures P \<or> xs \<in> sentences P \<and> (\<exists>U V. (xs, U) \<in> failures P \<and> ([], V) \<in> failures Q \<and> W = insert None U \<inter> V)) \<and> (\<forall>n\<in>{0<..length xs}. \<forall>W\<in>Rx n. take (length xs - n) xs \<in> sentences P \<and> (drop (length xs - n) xs, W) \<in> failures Q) \<and> (\<exists>n\<in>{..length xs}. \<exists>W. W \<in> Rx n)"