hcju/isabelleps huggingface dataset:
inject: fixes \<Delta> :: "('a, 'b) network_scheme"   and x2 :: "'c"   and y2 :: "'c" assumes "flow_attainability \<Delta>" shows "(Inner x2 = Inner y2) = (x2 = y2)"
assms_1: shows "xs\<^sub>1 = xs\<^sub>2"
whiletrue: fixes b :: "'exp"   and m :: "'id \<Rightarrow> 'val"   and c :: "('exp, 'id) MWLfCom" assumes "BMap (E b m) = True" shows "SRabbr (while b do c od) m [c;while b do c od] m"
refl: fixes a :: "nat list" shows "a \<le>\<^sub>v a"
NatMultiples_Product: fixes k :: "nat"   and F :: "nat set"   and m :: "nat" assumes "k \<in> F"   and "m \<in> NatMultiples F" shows "k * m \<in> NatMultiples F"
ChiPlus_SA_OneAncestor: fixes S :: "'d"   and T :: "'d"   and A :: "('d, 'e, 'f) hierauto"   and U :: "'d"   and SA :: "('d, 'e, 'f) seqauto" assumes "(S, T) \<in> ChiPlus A"   and "U \<in> States SA"   and "SA \<in> SAs A"   and "T \<in> States SA" shows "(S, U) \<in> ChiPlus A"
pfa_init_rule: fixes n :: "nat"   and N :: "nat"   and x :: "'b" assumes "n \<le> N" shows "<emp> pfa_init N x n <is_pfa N (replicate n x)>"
rel_funI: fixes A :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and B :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and f :: "'r \<Rightarrow> 't"   and g :: "'s \<Rightarrow> 'u" assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)" shows "rel_fun A B f g"
range_construct_eq_span: fixes B :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "vs1.independent B" shows "range (construct B f) = vs2.span (f ` B)"
subseq_singleton_left: fixes x :: "'a"   and ys :: "'a list" shows "subseq [x] ys = (x \<in> set ys)"
set_image: fixes R :: "('c \<times> 'd) list"   and xs :: "'c list" assumes "distinct R"   and "distinct xs"   and "sorted R"   and "sorted xs" shows "set (List_local.image R xs) = set R `` set xs"
width_upper_bound_V: assumes "V \<noteq> {}" shows "width < card V"
content_iff: fixes x :: "'b"   and p :: "'b poly" shows "(x dvd content p) = (\<forall>c\<in>set (coeffs p). x dvd c)"
map_FGco_def: fixes l1 :: "'p \<Rightarrow> 'ac"   and co1 :: "'q \<Rightarrow> 'ad"   and co2 :: "'s \<Rightarrow> 'af"   and co3 :: "'t \<Rightarrow> 'ag"   and co4 :: "'u \<Rightarrow> 'ah"   and co5 :: "'r \<Rightarrow> 'ae"   and co6 :: "'y \<Rightarrow> 'ak"   and contra1 :: "'ai \<Rightarrow> 'v"   and contra2 :: "'aj \<Rightarrow> 'w"   and contra3 :: "'al \<Rightarrow> 'z"   and contra4 :: "'am \<Rightarrow> 'aa" shows "map_FGco l1 co1 co2 co3 co4 co5 co6 contra1 contra2 contra3 contra4 = map_F l1 co1 co5 (map_G co1 co2 co3 co4 contra1 contra2) co3 co6 contra1 contra3 contra4"
less_eq_option_Some: fixes x :: "'b"   and y :: "'b" shows "(Some x \<le> Some y) = (x \<le> y)"
assms_7: shows "sing_reg (t \<oplus> lt f) (punit.lt (rep_list f) \<oplus> lt p)"
compactin_subspace: fixes X :: "'c topology"   and S :: "'c set" shows "compactin X S = (S \<subseteq> topspace X \<and> compact_space (subtopology X S))"
isVar_imp_wls: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "isVar s X" shows "wls s X"
bisim1'_refl: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and n :: "nat"   and xs :: "'addr val list" shows "P,e,n,h \<turnstile>' (e, xs) \<leftrightarrow> ([], xs, 0, None)"
diff_is_0_eq': fixes m :: "nat"   and n :: "nat" assumes "order.greater_eq n m" shows "m - n = 0"
negligible_Un_eq: fixes s :: "'a set"   and t :: "'a set" shows "negligible (s \<union> t) = (negligible s \<and> negligible t)"
normalize_prime: fixes p :: "'a" assumes "prime p" shows "normalize p = p"
dickson_grading_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and d :: "'a \<Rightarrow> nat" shows "plus.dickson_grading plus d = ((\<forall>s t. d (plus s t) = max (d s) (d t)) \<and> (\<forall>n. almost_full_on (plus.adds plus) {x. d x \<le> n}))"
eq\<rho>I: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and \<rho>1 :: "Vars.var \<Rightarrow> Value"   and \<rho>2 :: "Vars.var \<Rightarrow> Value" assumes "\<And>x a. ae x = up\<cdot>a \<Longrightarrow> eq a (\<rho>1 x) (\<rho>2 x)" shows "eq\<rho> ae \<rho>1 \<rho>2"
Lbot: shows "([],  \<LM> ff  \<RM> \<Rightarrow>* \<Empt>) \<in> Ax"
Ana_subterm: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and t :: "('a, 'b) Term.term"   and K :: "('a, 'b) Term.term list"   and T :: "('a, 'b) Term.term list" assumes "intruder_model arity Ana"   and "Ana t = (K, T)" shows "set T \<subset> subterms t"
cases: fixes a1 :: "char list \<Rightarrow> ty"   and a2 :: "com" assumes "a1 \<turnstile> a2" obtains \<Gamma> :: "char list \<Rightarrow> ty" where "a1 = \<Gamma>'__" and "a2 = SKIP"   | \<Gamma> :: "char list \<Rightarrow> ty" and a :: "aexp" and x :: "char list" where "a1 = \<Gamma>''__" and "a2 = x ::= a" and "\<Gamma>''__ \<turnstile> a : \<Gamma>''__ x"   | \<Gamma> :: "char list \<Rightarrow> ty" and c1 :: "com" and c2 :: "com" where "a1 = \<Gamma>'''__" and "a2 = c1__;; c2__" and "\<Gamma>'''__ \<turnstile> c1__" and "\<Gamma>'''__ \<turnstile> c2__"   | \<Gamma> :: "char list \<Rightarrow> ty" and b :: "bexp" and c1 :: "com" and c2 :: "com" where "a1 = \<Gamma>''''__" and "a2 = IF b__ THEN c1 ELSE c2" and "\<Gamma>''''__ \<turnstile> b__" and "\<Gamma>''''__ \<turnstile> c1" and "\<Gamma>''''__ \<turnstile> c2"   | \<Gamma> :: "char list \<Rightarrow> ty" and b :: "bexp" and c :: "com" where "a1 = \<Gamma>" and "a2 = WHILE b DO c" and "\<Gamma> \<turnstile> b" and "\<Gamma> \<turnstile> c"
poly_prod_list: fixes ps :: "'b poly list"   and x :: "'b" shows "poly (prod_list ps) x = (\<Prod>p\<leftarrow>ps. poly p x)"
authK_not_AKcryptSK: fixes A :: "agent"   and authK :: "nat"   and tk :: "msg"   and evs :: "event list"   and K :: "nat" assumes "Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, tk\<rbrace> \<in> parts (knows Spy evs)"   and "evs \<in> kerbIV_gets" shows "\<not> AKcryptSK K authK evs"
psubst_Var_simp1: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm set option"   and xs :: "'varSort"   and x :: "'var" assumes "goodEnv rho"   and "rho xs x = None" shows "Var xs x #[rho] = Var xs x"
divide_inverse': fixes a :: "'a"   and b :: "'a" shows "a / b = inverse b * a"
prefix_suffix: fixes x :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "x = prefix n x \<frown> x [n..]"
finite_dom_fmap: fixes F :: "'c \<Rightarrow> 'd option" shows "finite (dom F)"
nat_neq_4k3: fixes m :: "nat"   and k :: "nat"   and n :: "nat" shows "of_nat m \<noteq> ((4::'b) * of_nat k + (3::'b)) / ((2::'b) * (2::'b) ^ n)"
lnull_llist_of: fixes xs :: "'b list" shows "lnull (llist_of xs) = (xs = [])"
defNode_var_disjoint: fixes p :: "'val"   and g :: "'g"   and q :: "'val" assumes "p \<in> allVars g"   and "q \<in> allVars g"   and "p \<noteq> q"   and "defNode g p = defNode g q" shows "var g p \<noteq> var g q"
homeomorphic_map_connectedness: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and U :: "'c set" assumes "homeomorphic_map X Y f"   and "U \<subseteq> topspace X" shows "connectedin Y (f ` U) = connectedin X U"
RBTi2_axioms: shows "RBTi2 TYPE('a) bal_i e"
qSwapAll_compose: fixes zs' :: "'c"   and x' :: "'d"   and y' :: "'d"   and zs :: "'c"   and x :: "'d"   and y :: "'d"   and X :: "('a, 'b, 'c, 'd, 'e) qTerm"   and A :: "('a, 'b, 'c, 'd, 'e) qAbs" shows "X #[[x \<and> y]]_zs #[[x' \<and> y']]_zs' = X #[[x' \<and> y']]_zs' #[[x @zs[x' \<and> y']_zs' \<and> y @zs[x' \<and> y']_zs']]_zs \<and> A $[[x \<and> y]]_zs $[[x' \<and> y']]_zs' = A $[[x' \<and> y']]_zs' $[[x @zs[x' \<and> y']_zs' \<and> y @zs[x' \<and> y']_zs']]_zs"
A2: shows "A\<^sub>2 \<in> carrier_mat nr\<^sub>2 n"
of_nat_nat: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and z :: "int" assumes "class.ring_1 one times plus zero minus uminus"   and "0 \<le> z" shows "semiring_1.of_nat one plus zero (nat z) = ring_1.of_int one plus zero minus z"
hd_Basis_list: shows "hd Basis_list \<in> Basis"
eff_def: fixes r :: "'rule"   and s :: "'state"   and sl :: "'state fset" shows "eff r s sl \<equiv> eff' r s = Some sl"
const_list_comb_free: fixes name :: "name"   and args :: "'c list" shows "\<not> is_free (name $$ args)"
in_streams: fixes s :: "'b stream"   and S :: "'b set" assumes "stl s \<in> streams S"   and "shd s \<in> S" shows "s \<in> streams S"
bind_pure_I: fixes f :: "('l, 'm, 'n) prog"   and h :: "'l"   and g :: "'n \<Rightarrow> ('l, 'm, 'o) prog" assumes "pure f h"   and "\<And>x. h \<turnstile> f \<rightarrow>\<^sub>r x \<Longrightarrow> pure (g x) h" shows "pure (Heap_Error_Monad.bind f g) h"
Domain_iff: fixes a :: "'h"   and r :: "('h \<times> 'i) set" shows "(a \<in> Domain r) = (\<exists>y. (a, y) \<in> r)"
collapse_7: fixes stateful_strand_step :: "('c, 'd) stateful_strand_step" assumes "is_NegChecks stateful_strand_step" shows "\<forall>bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p stateful_strand_step\<langle>\<or>\<noteq>: the_eqs stateful_strand_step \<or>\<notin>: the_ins stateful_strand_step\<rangle> = stateful_strand_step"
assms_5: shows "(P\<^sub>1 \<union> wrap B\<^sub>1) \<inter> (P\<^sub>2 \<union> wrap B\<^sub>2) = {}"
inv_into_into: fixes x :: "'ag"   and f :: "'ah \<Rightarrow> 'ag"   and A :: "'ah set" assumes "x \<in> f ` A" shows "inv_into A f x \<in> A"
fps_ext: fixes p :: "'a fps"   and q :: "'a fps" assumes "\<And>n. fps_nth p n = fps_nth q n" shows "p = q"
even: shows "even (a\<^sup>2 + 3 * b\<^sup>2)"
assms: shows "subseq l1 l2 \<and> list_all P l2"
ch: shows "x \<in> convex hull set (map snd (half_segments_of_aform X))"
measurable_enabled: fixes s :: "'s" shows "Measurable.pred (stream_space (count_space UNIV)) (enabled s)"
vec_of_basis_enum_zero: shows "vec_of_basis_enum (0::'b) = 0\<^sub>v (length canonical_basis)"
psiFreshVec_8: fixes xvec :: "name list" shows "xvec \<sharp>* \<zero>"
fds_converges_shift: fixes c :: "'a"   and f :: "'a fds"   and z :: "'a" shows "fds_converges (fds_shift c f) z = fds_converges f (z - c)"
cr_word_def: shows "cr_word \<equiv> \<lambda>x. (=) (Word.Word x)"
homologous_rel_restrict: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set" shows "homologous_rel p X (topspace X \<inter> S) = homologous_rel p X S"
f11: shows "\<not> 4294967296 \<le> 0"
projection_eqI: fixes M :: "'b set"   and h :: "'b"   and x :: "'b" assumes "closed_csubspace M"   and "h - x \<in> orthogonal_complement M"   and "x \<in> M" shows "projection M h = x"
atLeast_subset_iff: fixes x :: "'e"   and y :: "'e" shows "({x..} \<subseteq> {y..}) = (y \<le> x)"
idem: fixes a :: "'a::semilattice_sup" shows "(a::'a::semilattice_sup) \<squnion> a = a"
red_union: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "red (F \<union> G) p q = (red F p q \<or> red G p q)"
eqpoll_refl: fixes A :: "'a set" shows "A \<approx> A"
fstI: fixes x :: "'d \<times> 'e"   and y :: "'d"   and z :: "'e" assumes "x = (y, z)" shows "fst x = y"
extend_2: shows "set w\<^sub>2 \<subseteq> invisible"
canonical_subs_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "nat"   and I :: "nat set"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b" assumes "class.linordered_ab_monoid_add plus zero less_eq less" shows "linordered_ab_monoid_add.canonical_subs plus less_eq n I m = (\<forall>i j k. i \<le> n \<and> k \<le> n \<and> j \<in> I \<longrightarrow> less_eq (m i k) (plus (m i j) (m j k)))"
pref_prod_pref: fixes u :: "'b list"   and z :: "'b list"   and w :: "'b list" assumes "u \<le>p z \<cdot> w"   and "u \<le>p w" shows "u \<le>p z \<cdot> u"
oalist_inv_alt: fixes xs :: "('a \<times> 'c) list"   and ko :: "'o" shows "oalist_inv (xs, ko) = oalist_inv_raw ko xs"
prems: shows "\<forall>b\<in>set (a # A) \<union> B. case b of (j, p) \<Rightarrow> \<forall>a\<in>set (a # A) \<union> B. case a of (k, q) \<Rightarrow> p = q \<longrightarrow> j = k"
assms: fixes a :: "'b"   and r :: "'a" shows "f (h a * r) = f (g a * r)"
order_iff: fixes b :: "'d"   and a :: "'d" shows "(b \<le> a) = (a = sup a b)"
list_set_cons_cases: fixes w :: "'d list"   and a :: "'d"   and S :: "'d list set" assumes "w \<in> a \<cdot> S" obtains w' :: "'d list" where "w = a # w'" and "w' \<in> S"
kat_3_equiv_opp: fixes z :: "'a"   and y :: "'a"   and x :: "'a" shows "(ad z \<cdot> y \<cdot> d x = (zero_class.zero::'a)) = (y \<cdot> d x = d z \<cdot> y \<cdot> d x)"
prop_in_f_3_a: fixes v :: "i" shows "[\<^bold>\<not>Indiscriminate E! in v]"
norm_vec_def: fixes x :: "('d, 'e) vec" shows "\<parallel>x\<parallel> = L2_set (\<lambda>i. \<parallel>x $ i\<parallel>) UNIV"
edges_of_walk_middle: fixes G :: "('a, 'b) Graph_scheme"   and v :: "'a"   and w :: "'a"   and xs :: "'a list"   and xs' :: "'a list" assumes "Digraph G" shows "(v, w) \<in> Digraph.edges_of_walk (xs @ v # w # xs')"
foundation14: fixes \<tau> :: "'b state \<times> 'b state"   and A :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(\<tau> \<Turnstile> A \<triangleq> false) = (\<tau> \<Turnstile> not A)"
foldseq_tail: fixes M :: "nat"   and N :: "nat"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "nat \<Rightarrow> 'b" assumes "M \<le> N" shows "foldseq f S N = foldseq f (\<lambda>k. if k < M then S k else foldseq f (\<lambda>k. S (k + M)) (N - M)) M"
lang_dual_pnorm_dual: fixes \<Sigma> :: "'a list"   and n :: "nat"   and r :: "'a atom rexp_dual" assumes "wf_dual \<Sigma> n r" shows "lang_dual \<Sigma> n (pnorm_dual r) = lang_dual \<Sigma> n r"
invariant_iff_pos_and_neg_invariant: fixes M :: "'a set" shows "rev.invariant M = (rev.positively_invariant M \<and> rev.negatively_invariant M)"
comp_ide_arr: fixes b :: "'a"   and f :: "'a" assumes "ide b"   and "seq b f" shows "b \<cdot> f = f"
mcp'_intro: fixes z :: "'e list"   and W :: "'e list set"   and p :: "'e list" assumes "mcp z W p" shows "mcp' z W = p"
bit_and_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and n :: "nat" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "bit (and a b) n = (bit a n \<and> bit b n)"
shift_bounds_cl_nat_ivl: fixes g :: "nat \<Rightarrow> 'e"   and m :: "nat"   and k :: "nat"   and n :: "nat" shows "prod g {m + k..n + k} = (\<Prod>i = m..n. g (i + k))"
Borsukian_separation_compact: fixes S :: "complex set" assumes "compact S" shows "Borsukian S = connected (- S)"
ab_semigroup_mult_hom_axioms: shows "ab_semigroup_mult_hom of_real"
exhaust: fixes y :: "le_rel" obtains "y = Leq_Rel"   | "y = Lt_Rel"
pullback_commutes': fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b" assumes "elementary_category_with_pullbacks C prj0 prj1"   and "partial_magma.arr C f \<and> partial_magma.arr C g \<and> partial_magma.cod C f = partial_magma.cod C g" shows "C f (prj1 f g) = C g (prj0 f g)"
hyps_5: shows "S \<union> ik\<^sub>e\<^sub>s\<^sub>t B \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile>\<^sub>c Fun f X \<cdot> \<I>"
cases: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and Exit :: "'b"   and dyn_control_dependence :: "'b \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> bool"   and a1 :: "'d"   and a2 :: "'a list"   and a3 :: "'a list"   and n' :: "'b" assumes "DynPDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit dyn_control_dependence"   and "(a1, a2, a3) \<in> DynPDG.dependent_live_vars sourcenode targetnode valid_edge Def Use dyn_control_dependence n'" obtains V :: "'d" where "a1 = Va__" and "a2 = []" and "a3 = []" and "Va__ \<in> Use n'"   | V :: "'d" and a :: "'a" and as' :: "'a list" and n'' :: "'b" and as'' :: "'a list" where "a1 = Vb__" and "a2 = []" and "a3 = a__ # as'a__ @ as''" and "Vb__ \<in> Use (sourcenode a__)" and "DynPDG.DynPDG_edge sourcenode targetnode valid_edge Def Use dyn_control_dependence (sourcenode a__) None (a__ # as'a__) n''" and "DynPDG.DynPDG_path sourcenode targetnode valid_edge Def Use dyn_control_dependence n'' as'' n'"   | V :: "'d" and as' :: "'a list" and as :: "'a list" and V' :: "'d" and a :: "'a" where "a1 = V'" and "a2 = []" and "a3 = aa__ # asa__" and "(Vc__, as'b__, asa__) \<in> DynPDG.dependent_live_vars sourcenode targetnode valid_edge Def Use dyn_control_dependence n'" and "V' \<in> Use (sourcenode aa__)" and "n' = last (CFG.targetnodes targetnode (aa__ # asa__))" and "DynPDG.DynPDG_edge sourcenode targetnode valid_edge Def Use dyn_control_dependence (sourcenode aa__) (Some Vc__) (aa__ # as'b__) (last (CFG.targetnodes targetnode (aa__ # as'b__)))"   | V :: "'d" and as' :: "'a list" and as :: "'a list" and a :: "'a" where "a1 = V" and "a2 = a # as'" and "a3 = a # as" and "(V, as', as) \<in> DynPDG.dependent_live_vars sourcenode targetnode valid_edge Def Use dyn_control_dependence n'" and "n' = last (CFG.targetnodes targetnode (a # as))" and "\<not> DynPDG.DynPDG_edge sourcenode targetnode valid_edge Def Use dyn_control_dependence (sourcenode a) (Some V) (a # as') (last (CFG.targetnodes targetnode (a # as')))"
clo: fixes C :: "'a set" assumes "C \<in> \<T>" shows "closed C"
complement_part: fixes x :: "i"   and y :: "i" assumes "CMC"   and "\<not> O x y" shows "P x (\<midarrow> y)"
le_fun_def: fixes f :: "'g \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'h" shows "(f \<le> g) = (\<forall>x. f x \<le> g x)"
weak_consis_take: shows "weak_sharing_consistent \<O> (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)"
subgroup_axioms: shows "subgroup Normal G (\<cdot>) \<one>"
terminates_impl_no_infinite_trans_computation: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile>c \<down> s" shows "\<nexists>f. f 0 = (c, s) \<and> (\<forall>i. \<Gamma>\<turnstile> f i \<rightarrow>\<^sup>+ f (Suc i))"
exec_instr_prealloc_pres: fixes h :: "heap"   and i :: "instr"   and P :: "jvm_prog"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap"   and xp' :: "nat option"   and h' :: "heap"   and frs' :: "frame list"   and sh' :: "sheap" assumes "preallocated h"   and "exec_instr i P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh = (xp', h', frs', sh')" shows "preallocated h'"
type_less_x_Required_intro: fixes \<tau> :: "'a OCL_Types.type"   and \<rho> :: "'a basic_type"   and \<sigma> :: "'a basic_type" assumes "\<tau> = \<rho>[1]"   and "\<rho> < \<sigma>" shows "\<tau> < \<sigma>[1]"
division: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and g :: "'b set \<Rightarrow> 'a"   and d :: "'b set set"   and a :: "'b"   and b :: "'b" assumes "operative f z g"   and "d division_of cbox a b" shows "comm_monoid_set.F f z g d = g (cbox a b)"
assms_3: fixes A :: "'a" assumes "reflexive_rewrite b A"   and "reflexive_rewrite c A" shows "goal"
CastFail: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and a :: "nat"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and D :: "char list"   and fs :: "Objects.fields"   and C :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,(h, l, sh)\<rangle>"   and "h a = \<lfloor>(D, fs)\<rfloor>"   and "(D, C) \<notin> (subcls1 P)\<^sup>*" shows "P \<turnstile> \<langle>Cast C e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW ClassCast,(h, l, sh)\<rangle>"
complex_nonpos_Reals_iff: fixes z :: "complex" shows "(z \<in> \<real>\<^sub>\<le>\<^sub>0) = (Re z \<le> 0 \<and> Im z = 0)"
measurable_stl: fixes M :: "'a measure" shows "stl \<in> stream_space M \<rightarrow>\<^sub>M stream_space M"
euler_trail_app_isoI: fixes u :: "'a"   and p :: "('a \<times> nat \<times> 'a) list"   and v :: "'a"   and hom :: "('a, 'a \<times> nat \<times> 'a, 'c, 'd) digraph_isomorphism" assumes "H.euler_trail u p v"   and "H.digraph_isomorphism hom" shows "pre_digraph.euler_trail (app_iso hom (app_iso iso_f G)) (iso_verts hom u) (map (iso_arcs hom) p) (iso_verts hom v)"
Upper_empty: fixes L :: "('a, 'b) gorder_scheme" shows "Upper L {} = carrier L"
intros_1: fixes G :: "prog"   and S :: "ty"   and T :: "ty" assumes "G\<turnstile>S\<preceq>T" shows "G\<turnstile>S\<preceq>? T"
unitl_ex: fixes x :: "'a" shows "\<exists>e\<in>\<xi>. \<rho> x e x"
onorm_le: fixes f :: "'b \<Rightarrow> 'c"   and b :: "real" assumes "\<And>x. \<parallel>f x\<parallel> \<le> b * (\<parallel>x\<parallel>)" shows "onorm f \<le> b"
assms_1: shows "\<exists>x. \<alpha>' - \<alpha> = 2 * real_of_int x * pi"
dghm_comp_def: fixes \<GG> :: "V"   and \<FF> :: "V" shows "\<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF> = [\<GG>\<lparr>ObjMap\<rparr> \<circ>\<^sub>\<circ> \<FF>\<lparr>ObjMap\<rparr>, \<GG>\<lparr>ArrMap\<rparr> \<circ>\<^sub>\<circ> \<FF>\<lparr>ArrMap\<rparr>, \<FF>\<lparr>HomDom\<rparr>, \<GG>\<lparr>HomCod\<rparr>]\<^sub>\<circ>"
cover_of_subset_def: fixes X :: "'a set"   and U :: "'a set"   and index :: "real set"   and cover :: "real \<Rightarrow> 'a set" shows "cover_of_subset X U index cover \<equiv> U \<subseteq> X \<and> (\<forall>i. i \<in> index \<longrightarrow> cover i \<subseteq> X) \<and> U \<subseteq> \<Union> (cover ` index)"
Ord_LeastI2_wellorder: fixes a :: "V"   and P :: "V \<Rightarrow> bool"   and Q :: "V \<Rightarrow> bool" assumes "Ord a"   and "P a"   and "\<And>a. \<lbrakk>P a; \<forall>b. Ord b \<and> P b \<longrightarrow> a \<le> b\<rbrakk> \<Longrightarrow> Q a" shows "Q (LEAST i. Ord i \<and> P i)"
explode_Repl_eq: fixes A :: "ZF"   and f :: "ZF \<Rightarrow> ZF" shows "explode (Repl A f) = f ` explode A"
P_no_deadend_v0: assumes "\<not> lnull (ltl P')" shows "\<not> deadend (P $ n)"
sep_h: fixes y :: "'v" assumes "reachable y"   and "y \<in> B \<Gamma>"   and "y \<in> TER\<^bsub>\<Gamma>\<lparr>weight := weight \<Gamma> - u\<rparr>\<^esub> f" shows "y \<in> TER h"
edge'_simps_3: fixes y :: "'v option ??.MFMC_Flow_Attainability.flow_attainability.node" shows "??.local.\<Psi>.edge' y ??.MFMC_Flow_Attainability.flow_attainability.node.SOURCE = False"
map: fixes f :: "'b \<Rightarrow> 'a"   and x :: "nat" shows "map_ref f (Ref x) = Ref x"
lens_plus_def: fixes X :: "'a \<Longrightarrow> 'c"   and Y :: "'b \<Longrightarrow> 'c" shows "X +\<^sub>L Y = \<lparr>lens_get = \<lambda>\<sigma>. (get\<^bsub>X\<^esub> \<sigma>, get\<^bsub>Y\<^esub> \<sigma>), lens_put = \<lambda>\<sigma> (u, v). put\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) u\<rparr>"
r_neg2: fixes G :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "abelian_group G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<oplus>\<^bsub>G\<^esub> (\<ominus>\<^bsub>G\<^esub> x \<oplus>\<^bsub>G\<^esub> y) = y"