hcju/isabelleps huggingface dataset:
fmeasurableD2: fixes A :: "'b set"   and M :: "'b measure" assumes "A \<in> fmeasurable M" shows "emeasure M A \<noteq> top"
set_zero_plus: fixes C :: "'c set" shows "(0::'c) +o C = C"
sl_def: 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 mxs :: "nat"   and mxl :: "nat" shows "JVM_SemiType.sl P mxs mxl \<equiv> Err.sl (Opt.esl (Product.esl (stk_esl P mxs) (Err.esl (loc_sl P mxl))))"
iMOD_1: fixes r :: "nat" shows "[ r, mod Suc 0 ] = [r\<dots>]"
dagger_of_prod: fixes M :: "complex Matrix.mat"   and N :: "complex Matrix.mat" assumes "dim_col M = dim_row N" shows "M * N\<^sup>\<dagger> = N\<^sup>\<dagger> * M\<^sup>\<dagger>"
rangeIt_def: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'b" shows "rangeIt x f = range (\<lambda>n. (f ^^ n) x)"
index_mult_mat_vec: fixes i :: "nat"   and A :: "'a mat"   and v :: "'a vec" assumes "i < dim_row A" shows "(A *\<^sub>v v) $ i = row A i \<bullet> v"
simps_1: fixes u :: "'b local.term" shows "\<^bold>\<I> \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u = u"
IH_2: assumes "receive\<langle>t\<rangle> \<in> set A" shows "receive\<langle>t \<cdot> \<theta>\<rangle> \<in> set (A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>)"
sup_idem: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.semilattice_sup sup less_eq less" shows "sup x x = x"
length_cons_def: fixes s :: "'s" shows "local.length_cons s = length (unstream g s)"
reachable1_reachable: fixes G :: "('c, 'd) pre_digraph"   and v :: "'c"   and w :: "'c" assumes "wf_digraph G"   and "v \<rightarrow>\<^sup>+\<^bsub>G\<^esub> w" shows "v \<rightarrow>\<^sup>*\<^bsub>G\<^esub> w"
simps_3: fixes \<A> :: "'b \<Rightarrow> bool"   and F :: "'b formula" shows "\<A> \<Turnstile> \<^bold>\<not> F = (\<not> \<A> \<Turnstile> F)"
lead_coeff_of_nat: fixes n :: "nat" shows "lead_coeff (of_nat n) = of_nat n"
comp_in_homE: fixes g :: "'c"   and f :: "'c"   and a :: "'c"   and c :: "'c" assumes "\<guillemotleft>g \<cdot>\<^sub>C f : a \<Rightarrow>\<^sub>C c\<guillemotright>" obtains b :: "'c" where "\<guillemotleft>f : a \<Rightarrow>\<^sub>C b\<guillemotright>" and "\<guillemotleft>g : b \<Rightarrow>\<^sub>C c\<guillemotright>"
simps_128: fixes f1 :: "'d"   and f2 :: "('e \<Rightarrow> 'e) \<Rightarrow> 'd"   and f3 :: "('e \<times> 'e) set \<Rightarrow> 'd"   and f4 :: "('e, 'f, 'g) com \<Rightarrow> ('e, 'f, 'g) com \<Rightarrow> 'd"   and f5 :: "'e set \<Rightarrow> ('e, 'f, 'g) com \<Rightarrow> ('e, 'f, 'g) com \<Rightarrow> 'd"   and f6 :: "'e set \<Rightarrow> ('e, 'f, 'g) com \<Rightarrow> 'd"   and f7 :: "'f \<Rightarrow> 'd"   and f8 :: "('e \<Rightarrow> ('e, 'f, 'g) com) \<Rightarrow> 'd"   and f9 :: "'g \<Rightarrow> 'e set \<Rightarrow> ('e, 'f, 'g) com \<Rightarrow> 'd"   and f10 :: "'d"   and f11 :: "('e, 'f, 'g) com \<Rightarrow> ('e, 'f, 'g) com \<Rightarrow> 'd"   and x91 :: "'g"   and x92 :: "'e set"   and x93 :: "('e, 'f, 'g) com" shows "(case Guard x91 x92 x93 of Skip \<Rightarrow> f1 | Basic x \<Rightarrow> f2 x | Spec x \<Rightarrow> f3 x | Seq x xa \<Rightarrow> f4 x xa | Cond x xa xb \<Rightarrow> f5 x xa xb | While x xa \<Rightarrow> f6 x xa | Call x \<Rightarrow> f7 x | DynCom x \<Rightarrow> f8 x | Guard x xa xb \<Rightarrow> f9 x xa xb | Throw \<Rightarrow> f10 | Catch x xa \<Rightarrow> f11 x xa) = f9 x91 x92 x93"
decseq_Suc_iff: fixes f :: "nat \<Rightarrow> 'b" shows "decseq f = (\<forall>n. f (Suc n) \<le> f n)"
tconf_def: fixes P :: "'b prog"   and h :: "'heap"   and t :: "'thread_id" shows "P,h \<turnstile> t \<surd>t \<equiv> \<exists>C. typeof_addr h (thread_id2addr t) = \<lfloor>Class_type C\<rfloor> \<and> P \<turnstile> C \<preceq>\<^sup>* Thread"
cobounded1: fixes a :: "'c"   and b :: "'c" shows "a \<le> sup a b"
right_connected: fixes \<mu> :: "'b"   and \<mu>' :: "'b"   and \<nu> :: "'b" assumes "D.seq \<mu> \<mu>'" shows "(\<nu> \<star>\<^sub>D \<mu> \<noteq> D.null) = (\<nu> \<star>\<^sub>D \<mu>' \<noteq> D.null)"
simps: fixes i :: "float interval"   and p :: "float poly"   and e :: "float interval" shows "tm_inc_err i (TaylorModel p e) = TaylorModel p (e + i)"
geodesic_segment_endpoints_3: fixes G :: "'b set"   and x :: "'b"   and y :: "'b" assumes "geodesic_segment_between G x y" shows "G \<noteq> {}"
nretj: shows "path \<sigma> j \<noteq> local.return"
infinite_remove: fixes S :: "'d set"   and a :: "'d" assumes "infinite S" shows "infinite (S - {a})"
tanh_real_nonpos_iff: fixes x :: "real" shows "(tanh x \<le> 0) = (x \<le> 0)"
conforms_to_another_strategy: fixes \<sigma>' :: "'a \<Rightarrow> 'a" assumes "path_conforms_with_strategy p P \<sigma>'" shows "vmc_path G P v p \<sigma>'"
LE: fixes l :: "'b"   and r :: "'b" assumes "l \<in> S_Min_Lt"   and "r \<in> S_Max_Le" shows "l < r"
det_zero_imp_zero_row: fixes A :: "'b mat"   and n :: "nat"   and b :: "'c" assumes "A \<in> carrier_mat n n"   and "det A = (0::'b)" shows "\<exists>P. P \<in> Units (ring_mat TYPE('b) n b) \<and> row (P * A) (n - 1) = 0\<^sub>v n \<and> 0 < n \<and> row_echelon_form (P * A)"
pos_mult_pos_ge: fixes x :: "int"   and n :: "int" assumes "0 < x"   and "0 \<le> n" shows "n * 1 \<le> n * x"
TailIsPrefixList: fixes tail :: "'a list"   and l1 :: "'a list" assumes "tail \<noteq> []" shows "prefixList l1 (l1 @ tail)"
wa_precise_refine_def: fixes WAC :: "'a while_algo"   and WAA :: "'b while_algo"   and \<alpha> :: "'a \<Rightarrow> 'b" shows "wa_precise_refine WAC WAA \<alpha> \<equiv> wa_refine WAC WAA \<alpha> \<and> wa_precise_refine_axioms WAC WAA \<alpha>"
integral_le_const: fixes M :: "'d pmf"   and f :: "'d \<Rightarrow> real"   and c :: "real" assumes "integrable (measure_pmf M) f"   and "AE x in measure_pmf M. f x \<le> c" shows "measure_pmf.expectation M f \<le> c"
u0: fixes u :: "'a" assumes "u \<in> cball x e" shows "(u, 0::'c) \<in> H ` ball (x, y) r"
commute: fixes a :: "'f"   and b :: "'f" shows "a + b = b + a"
deg_comp_pos: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and u :: "'a"   and v :: "'a" assumes "cmp u v = Lt"   and "fst (rep_nat_term u) = fst (rep_nat_term v)" shows "deg_comp cmp u v = Lt"
is_UP_monomI': fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier (UP R)"   and "f = monom P (f (deg R f)) (deg R f)" shows "is_UP_monom f"
only_one_odd_div_power2: fixes n :: "nat" assumes "n \<noteq> 0"   and "\<And>x. \<lbrakk>x dvd n; odd x\<rbrakk> \<Longrightarrow> x = 1" shows "\<exists>k. n = 2 ^ k"
sigma_sets_singletons: fixes S :: "'b set" assumes "countable S" shows "sigma_sets S ((\<lambda>s. {s}) ` S) = Pow S"
cr_mod_ring_def: shows "cr_mod_ring \<equiv> \<lambda>x y. x = Rep_mod_ring y"
set_integral_cont_up: fixes A :: "nat \<Rightarrow> 'a set"   and M :: "'a measure"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>i. A i \<in> sets M"   and "incseq A"   and "set_integrable M (\<Union> (range A)) f" shows "(\<lambda>i. set_lebesgue_integral M (A i) f) \<longlonglongrightarrow> set_lebesgue_integral M (\<Union> (range A)) f"
that: fixes vl' :: "'value list" assumes "consume trn vl vl'" shows "thesis"
add_im_sheaf_def: shows "idim.add_im_sheaf \<equiv> \<lambda>V. +\<^bsub>identity S \<^sup>\<inverse> S V\<^esub>"
inf_sup_distrib2: fixes y :: "'c multiset"   and z :: "'c multiset"   and x :: "'c multiset" shows "y \<union># z \<inter># x = y \<inter># x \<union># (z \<inter># x)"
size_eq_Suc_imp_eq_union: fixes M :: "'a multiset"   and n :: "nat" assumes "size M = Suc n" shows "\<exists>a N. M = add_mset a N"
cstrong_operator_topology_continuous_evaluation: fixes x :: "'a" shows "continuous_map cstrong_operator_topology euclidean (\<lambda>f. cblinfun_apply f x)"
elims: fixes x :: "bexp"   and y :: "bexp" assumes "not x = y" obtains "x = Bc True" and "y = Bc False"   | "x = Bc False" and "y = Bc True"   | v :: "bexp" where "x = bexp.Not v__" and "y = bexp.Not (bexp.Not v__)"   | v :: "bexp" and va :: "bexp" where "x = And va__ vaa__" and "y = bexp.Not (And va__ vaa__)"   | v :: "aexp" and va :: "aexp" where "x = Less v va" and "y = bexp.Not (Less v va)"
fags_def: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e" shows "fags f z i a = \<lparr>carrier = fags_carr f z i a, pop = fags_bpp f z i a, mop = fags_ipp f z i a, zero = z\<rparr>"
all_eql: fixes A :: "'p"   and B :: "'p"   and l1 :: "'p \<Rightarrow> 'p \<Rightarrow> bool"   and l2 :: "'p \<Rightarrow> 'p \<Rightarrow> bool" assumes "TarskiLen A B l1"   and "TarskiLen A B l2" shows "l1 = l2"
map_cong0: fixes x :: "'e list"   and f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f" assumes "\<And>z. z \<in> set x \<Longrightarrow> f z = g z" shows "map f x = map g x"
gen_cong_minimal: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and R' :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "R \<le> R'"   and "local.cong R'" shows "gen_cong R \<le> R'"
prv_imp_neg_rev: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "prv (imp \<phi> \<chi>)" shows "prv (imp (neg \<chi>) (neg \<phi>))"
both_below_singles1: fixes t :: "'b ttree"   and S :: "'b set"   and t' :: "'b ttree" assumes "t \<sqsubseteq> singles S"   and "carrier t' \<inter> S = {}" shows "t \<otimes>\<otimes> t' \<sqsubseteq> singles S"
int_interval_eq_image: fixes m :: "nat"   and n :: "nat" shows "{int m..int n} = int ` {m..n}"
less_eq_list_trans: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "local.less_eq_list xs ys"   and "local.less_eq_list ys zs" shows "local.less_eq_list xs zs"
face_cycle_pred_def: shows "H.face_cycle_pred \<equiv> rev_H \<circ> edge_pred HM"
star_false_2: fixes P :: "assn" shows "P * false = false"
odd_v: shows "v \<in> odd_nodes_set (del_unEdge v w v' G)"
pick_asTerm: fixes qX :: "('c, 'd, 'varSort, 'var, 'e) qTerm" assumes "qGood qX" shows "pick (asTerm qX) #= qX"
pentagon: fixes f :: "'b"   and g :: "'b"   and h :: "'b"   and k :: "'b" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "D.ide k"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h"   and "src\<^sub>D h = trg\<^sub>D k" shows "(f \<star>\<^sub>D \<a>\<^sub>D[g, h, k]) \<cdot>\<^sub>D \<a>\<^sub>D[f, g \<star>\<^sub>D h, k] \<cdot>\<^sub>D (\<a>\<^sub>D[f, g, h] \<star>\<^sub>D k) = \<a>\<^sub>D[f, g, h \<star>\<^sub>D k] \<cdot>\<^sub>D \<a>\<^sub>D[f \<star>\<^sub>D g, h, k]"
cot_of_real: fixes x :: "real" shows "of_real (cot x) = cot (of_real x)"
T_eq: fixes cfg :: "'s cfg" shows "T cfg = measure_pmf (K_cfg cfg) \<bind> (\<lambda>cfg'. distr (T cfg') St ((##) (state cfg')))"
atLeastatMost_empty_iff: fixes a :: "'e word"   and b :: "'e word" shows "(signed.atLeastAtMost a b = {}) = (\<not> a \<le>s b)"
negateI: fixes P :: "'a \<Rightarrow> bool"   and s :: "'a" assumes "\<not> P s" shows "\<N> P s"
reach_redef: shows "reach (set \<Sigma>) \<delta> q\<^sub>0 = {foldl \<delta> q\<^sub>0 w |w. set w \<subseteq> set \<Sigma>}"
subst_term_preserves_term_ok: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and x :: "variable"   and \<tau> :: "typ"   and y :: "variable"   and A :: "Core.term" shows "term_ok \<Theta> (subst_term [((x, \<tau>), Fv y \<tau>)] A) = term_ok \<Theta> A"
sublist_cons_exists: fixes h :: "'a"   and l1 :: "'a list"   and l2 :: "'a list" shows "subseq (h # l1) l2 = (\<exists>l2a l2b. l2 = l2a @ [h] @ l2b \<and> \<not> ListMem h l2a \<and> subseq l1 l2b)"
norm_Pair: fixes a :: "'d"   and b :: "'e" shows "norm (a, b) = sqrt ((norm a)\<^sup>2 + (norm b)\<^sup>2)"
Abs_poly_mapping_cases: fixes x :: "'b \<Rightarrow>\<^sub>0 'c" obtains y :: "'b \<Rightarrow> 'c" where "x = Abs_poly_mapping y" and "y \<in> {f. finite {x. f x \<noteq> (0::'c)}}"
singleton_distinct_path: fixes G :: "('a, 'b) graph"   and v :: "'a"   and w :: "'b"   and v' :: "'a" assumes "valid_unMultigraph G"   and "(v, w, v') \<in> edges G" shows "valid_unMultigraph.is_trail G v [(v, w, v')] v'"
rep_eq: fixes x :: "('a, 'b) ffun" shows "fdom x = pdom (pfun_of x)"
vdomain_vrange_is_vempty: fixes r :: "V" assumes "vsv r" shows "(\<D>\<^sub>\<circ> r = []\<^sub>\<circ>) = (\<R>\<^sub>\<circ> r = []\<^sub>\<circ>)"
wf_firstRecursiveCall: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and n :: "nat"   and V :: "nat set"   and Qp :: "(nat set \<times> 'b option list set) set"   and Qn :: "(nat set \<times> 'b option list set) set"   and I :: "nat set"   and J :: "nat set"   and Q_I_pos :: "(nat set \<times> 'b option list set) set"   and Q_I_neg :: "(nat set \<times> 'b option list set) set" assumes "getIJ getIJ"   and "rwf_query n V Qp Qn"   and "2 \<le> card V"   and "(I, J) = getIJ Qp Qn V"   and "Q_I_pos = New_max.projectQuery I (New_max.filterQuery I Qp)"   and "Q_I_neg = New_max.filterQueryNeg I Qn" shows "rwf_query n I Q_I_pos Q_I_neg"
replacefacesAt_nth2: fixes k :: "nat"   and F :: "face list list"   and oldf :: "face"   and newfs :: "face list" assumes "k < |F|" shows "replacefacesAt [k] oldf newfs F ! k = replace oldf newfs (F ! k)"
hyps: assumes "p \<noteq> 0" shows "proots_count p B = proots_count (fcompose p q1 q2) A"
poincare_line_type: fixes z :: "complex_homo"   and w :: "complex_homo" assumes "z \<noteq> w" shows "circline_type (poincare_line z w) = - 1"
Ch_iia: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and h :: "'f" assumes "ContractsWithSubstitutes Xd Xh Pd Ch" shows "iia (Ch h)"
ascii_of_def: fixes c :: "char" shows "ascii_of c = Char (digit0 c) (digit1 c) (digit2 c) (digit3 c) (digit4 c) (digit5 c) (digit6 c) False"
obs0: shows "{t\<^sub>0--t} \<subseteq> U s"
jordan_block_pow_carrier: fixes n :: "nat"   and a :: "'b"   and r :: "nat" shows "jordan_block n a ^\<^sub>m r \<in> carrier_mat n n"
psi_preserve_order: fixes l :: "'a sort_key extended"   and u :: "'a sort_key extended"   and i :: "'a"   and l' :: "'a sort_key extended"   and u' :: "'a sort_key extended"   and i' :: "'a" assumes "elem \<lbrakk>\<Psi> (l, u) i\<rbrakk> (l', u')"   and "elem \<lbrakk>\<Psi> (l', u') i'\<rbrakk> (l, u)" shows "preserve_order i i' \<lbrakk>\<Psi> (l, u) i\<rbrakk> \<lbrakk>\<Psi> (l', u') i'\<rbrakk>"
bij_inv: fixes f :: "'b \<Rightarrow> 'b" assumes "bijection f" shows "bij (inv f)"
bounded_clinear_compose: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'a" assumes "bounded_clinear f"   and "bounded_clinear g" shows "bounded_clinear (\<lambda>x. f (g x))"
simps_4: fixes \<pi> :: "'b prm"   and x :: "'b"   and T :: "PreSimplyTyped.type"   and A :: "'b ptrm" shows "\<pi> \<bullet> PFn x T A = PFn (\<pi> $ x) T (\<pi> \<bullet> A)"
simps_6: fixes t :: "nat"   and e :: "expr"   and s :: "stmt" shows "valid_sops_stmt t (While e s) = (valid_sops_expr t e \<and> valid_sops_stmt t s)"
Lcm_0_iff_nat: fixes A :: "nat set" assumes "finite A" shows "(Lcm A = 0) = (0 \<in> A)"
image_ring_def: fixes f :: "'d \<Rightarrow> 'c"   and R :: "('d, 'e) ring_scheme" shows "image_ring f R \<equiv> monoid.extend (image_group f R) \<lparr>zero = f \<zero>\<^bsub>R\<^esub>, add = \<lambda>a b. f (inv_into (carrier R) f a \<oplus>\<^bsub>R\<^esub> inv_into (carrier R) f b)\<rparr>"
intros: fixes W :: "'a list"   and \<Sigma>i :: "'a set" assumes "distinct W"   and "set W = \<Sigma>i" shows "(\<Sigma>i, W) \<in> dfs_initial \<Sigma>i"
add_independentS_doubleI: fixes B :: "'b set"   and A :: "'b set" assumes "\<And>b a. \<lbrakk>b \<in> B; a \<in> A; a + b = (0::'b)\<rbrakk> \<Longrightarrow> a = (0::'b)" shows "add_independentS [A, B]"
check_s_def: fixes s :: "('i, 'a) state" shows "check_s s = (let s' = check s in if \<U> s' then Inl (the (\<U>\<^sub>c s')) else Inr s')"
vcomp_transformation_cone: fixes J :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and C :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and D :: "'ag \<Rightarrow> 'ah"   and a :: "'ah"   and \<chi> :: "'ag \<Rightarrow> 'ah"   and D' :: "'ag \<Rightarrow> 'ah"   and \<tau> :: "'ag \<Rightarrow> 'ah" assumes "Limit.cone J C D a \<chi>"   and "natural_transformation J C D D' \<tau>" shows "Limit.cone J C D' a (vertical_composite.map J C \<chi> \<tau>)"
clause_saturated_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and S :: "'b eclause set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.clause_saturated trm_ord sel pos_ord filter_trms S = (\<forall>C P \<sigma> C' D \<theta> \<eta>. basic_superposition.derivable trm_ord sel pos_ord filter_trms C P S \<sigma> Ground C' \<longrightarrow> ground_clause (cl_ecl C) \<longrightarrow> basic_superposition.derivable trm_ord sel pos_ord filter_trms D P S \<theta> FirstOrder C' \<longrightarrow> trms_subsumes D C \<eta> \<longrightarrow> \<sigma> \<doteq> \<theta> \<lozenge> \<eta> \<longrightarrow> basic_superposition.redundant_clause trm_ord (subst_ecl D \<eta>) S \<sigma> C')"
assms: shows "\<forall>y\<in>S. f x \<le> f y"
subgroup_self: fixes G :: "('a, 'b) monoid_scheme" assumes "Group.group G" shows "subgroup (carrier G) G"
verit_sko_ex': fixes P :: "'e \<Rightarrow> bool"   and A :: "bool" assumes "P (SOME x. P x) = A" shows "(\<exists>x. P x) = A"
padr_unpadr: fixes v :: "'b vec"   and m :: "nat"   and U :: "'b vec set" assumes "v \<in> (\<lambda>v. v @\<^sub>v 0\<^sub>v m) ` U" shows "vardim.unpadr m v @\<^sub>v 0\<^sub>v m = v"
fun_type_param_ex: fixes f :: "'fun"   and T :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and g :: "'fun"   and S :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" assumes "\<Gamma> (Fun f T) = Fun g (map \<Gamma> S)"   and "t \<in> set S" shows "\<exists>s\<in>set T. \<Gamma> s = \<Gamma> t"
msb_shift: fixes w :: "'b word" shows "msb w = (w >> LENGTH('b) - 1 \<noteq> 0)"
mono_empty: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.order less_eq less" shows "order.mono_set less_eq {}"
asset_seskey_secret: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and SK :: "nat option \<times> nat set \<times> nat set" assumes "s\<^sub>0 \<Turnstile> s"   and "n \<notin> bad_shakey \<inter> (bad_pwd \<union> bad_prikey)"   and "(Asset n, Crypt (SesK SK) (Num 0)) \<in> s" shows "SesKey SK \<notin> spied s"
residualEq: fixes \<alpha> :: "'a action"   and P :: "('a, 'b, 'c) psi"   and \<beta> :: "'a action"   and Q :: "('a, 'b, 'c) psi" assumes "\<alpha> \<prec> P = \<beta> \<prec> Q"   and "bn \<alpha> \<sharp>* bn \<beta>"   and "distinct (bn \<alpha>)"   and "distinct (bn \<beta>)"   and "bn \<alpha> \<sharp>* (\<alpha> \<prec> P)"   and "bn \<beta> \<sharp>* (\<beta> \<prec> Q)" obtains p :: "(name \<times> name) list" where "set p \<subseteq> set (bn \<alpha>) \<times> set (bn (p \<bullet> \<alpha>))" and "distinctPerm p" and "\<beta> = p \<bullet> \<alpha>" and "Q = p \<bullet> P" and "bn \<alpha> \<sharp>* \<beta>" and "bn \<alpha> \<sharp>* Q" and "bn (p \<bullet> \<alpha>) \<sharp>* \<alpha>" and "bn (p \<bullet> \<alpha>) \<sharp>* P"
set_tag_name_get_child_nodes: fixes ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) object_ptr" shows "\<forall>w\<in>set_tag_name_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>Shadow_DOM.get_child_nodes_locs ptr'. r h h')"
len_gt_eq: shows "dim_col (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets))) \<le> length (rows_to_keep (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets))))"
rep_eq: fixes x :: "'b"   and xa :: "'c"   and xb :: "'b \<Rightarrow>\<^sub>0 'c" shows "lookup (update x xa xb) = (lookup xb)(x := xa)"
has_vector_derivative_polynomial_function: fixes p :: "real \<Rightarrow> 'a" assumes "polynomial_function p" obtains p' :: "real \<Rightarrow> 'a" where "polynomial_function p'" and "\<And>x. (p has_vector_derivative p' x) (at x)"
prune_def: fixes A :: "('b, 'a) nba"   and w :: "'b stream"   and V :: "(nat \<times> 'a) set" shows "prune A w V \<equiv> {v \<in> V. Bex (greachable A w V v) (gaccepting A)}"
timpl_closure_set_lfp: fixes M :: "(('a, 'b, 'c) prot_fun, 'd) Term.term set"   and TI :: "('c set \<times> 'c set) set" shows "lfp (\<lambda>X. M \<union> (\<Union>x\<in>X. \<Union>(a, b)\<in>TI. set \<langle>a --\<guillemotright> b\<rangle>\<langle>x\<rangle>)) = timpl_closure_set M TI"
list_wset_rel_def: fixes R :: "('e \<times> 'f) set" shows "\<langle>R\<rangle>list_wset_rel = br set top O \<langle>R\<rangle>set_rel"
complete_development_fnE: fixes x :: "'a"   and T :: "PreSimplyTyped.type"   and A :: "'a trm"   and X :: "'a trm" assumes "Fn x T A >>> X" shows "\<exists>A'. A >>> A' \<and> X = Fn x T A'"
in_reg_mod_privilege: fixes s' :: "'b sparc_state"   and a :: "word32"   and w :: "'b word"   and r :: "word5"   and s :: "'b sparc_state" assumes "s' = in_reg_mod a w r s \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
mat_row_def: fixes k :: "'d"   and i :: "'e" shows "mat_row k i = (\<chi>j. if i = j then k else (0::'d))"
CFG_call_determ: 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 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 Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list"   and a' :: "'a"   and Q' :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r' :: "'e"   and p' :: "'f"   and fs' :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list"   and s :: "(('c \<Rightarrow> 'd option) \<times> 'e) list" assumes "CFG_wf sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Def Use ParamDefs ParamUses"   and "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "valid_edge a'"   and "kind a' = Q':r'\<hookrightarrow>\<^bsub>p'\<^esub>fs'"   and "sourcenode a = sourcenode a'"   and "CFG.pred (kind a) s"   and "CFG.pred (kind a') s" shows "a = a'"
closed_Int_compactin: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "closedin X S"   and "compactin X T" shows "compactin X (S \<inter> T)"
wf_bs_polyneg: fixes bs :: "'b list"   and p :: "'b poly" assumes "wf_bs bs p" shows "wf_bs bs (~\<^sub>p p)"
finite_Sup_less_iff: fixes X :: "'a set"   and a :: "'a" assumes "finite X"   and "X \<noteq> {}" shows "(\<Squnion> X < a) = (\<forall>x\<in>X. x < a)"
sum_of_cubes: fixes n :: "nat" shows "real (\<Sum>k\<le>n. k ^ 3) = ((real n)\<^sup>2 + real n)\<^sup>2 / 4"
vlrestriction_vsubset_vpairs: fixes r :: "V"   and A :: "V" shows "r \<restriction>\<^sup>l\<^sub>\<circ> A \<subseteq>\<^sub>\<circ> vpairs r"
order_pair_def: fixes S :: "('b \<times> 'b) set"   and NS :: "('b \<times> 'b) set" shows "order_pair S NS \<equiv> pre_order_pair S NS \<and> compat_pair S NS"
A0_1: shows "lower_asymptotic_density A0 = 1"