hcju/isabelleps huggingface dataset:
sm: shows "summable (\<lambda>n. - ((- 1) ^ n * 2 ^ (2 * n) / fact (2 * n)))"
derived_quot_of_subgroup_is_comm_group: fixes H :: "'a set" assumes "subgroup H G" shows "comm_group (G\<lparr>carrier := H\<rparr> Mod derived G H)"
minus_mult_commute: fixes a :: "'a"   and b :: "'a" shows "- a * b = a * - b"
valid_cfgD: fixes cfg :: "'s cfg" assumes "cfg \<in> valid_cfg" shows "cfg \<in> cfg_on (state cfg)"
circuit_in_dep_in: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> carrier"   and "circuit_in \<E> X" shows "\<not> indep_in \<E> X"
col_conga_col: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet A B C"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C D E F" shows "Tarski_neutral_dimensionless.Col Bet D E F"
diff_add: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" shows "b - a + a = b"
D: shows "D r \<tau> \<sigma> \<sigma>' \<tau>'"
dispath_instr_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and t1 :: "'b sparc_state"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> get_S (cpu_reg_val PSR s1) = 0 \<and> get_S (cpu_reg_val PSR s2) = 0 \<and> \<not> snd (dispatch_instruction instr s1) \<and> \<not> snd (dispatch_instruction instr s2) \<and> t1 = snd (fst (dispatch_instruction instr s1)) \<and> t2 = snd (fst (dispatch_instruction instr s2))" shows "low_equal t1 t2"
timpls_transformable_to_pred_if_equal_mod_timpls: fixes TI :: "('sets set \<times> 'sets set) list"   and t :: "(('fun, 'atom, 'sets) prot_fun, 'b) Term.term"   and s :: "(('fun, 'atom, 'sets) prot_fun, 'b) Term.term"   and A :: "'sets set set" assumes "equal_mod_timpls TI t s"   and "wf\<^sub>t\<^sub>r\<^sub>m t"   and "\<forall>f\<in>funs_term t. is_Abs f \<longrightarrow> the_Abs f \<in> A" shows "timpls_transformable_to_pred (A \<union> fst ` (set TI)\<^sup>+ \<union> snd ` (set TI)\<^sup>+) t s"
add_left_cancel: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> E"   and "y \<in> E"   and "z \<in> E" shows "(x + y = x + z) = (y = z)"
nth_tabulate: fixes k :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and x :: "nat" assumes "k < n" shows "tabulate f x n ! k = f (x + k)"
Nil: fixes less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ys :: "'d list" shows "ord.lexordp_eq less [] ys"
runit_coherence: fixes f :: "'a" assumes "ide f" shows "S.runit (S.UP f) = S.UP \<r>[f] \<cdot>\<^sub>S S.cmp\<^sub>U\<^sub>P (f, src f) \<cdot>\<^sub>S (S.UP f \<star>\<^sub>S UP.unit (src f))"
Suc_diff_Suc: fixes n :: "nat"   and m :: "nat" assumes "(n::nat) < (m::nat)" shows "Suc ((m::nat) - Suc (n::nat)) = m - n"
measurable_eq_pair: shows "(\<lambda>(y, x). x = y) \<in> COCOUNT \<Otimes>\<^sub>M POW \<rightarrow>\<^sub>M BOOL"
lexord_append_left_rightI: fixes a :: "'d"   and b :: "'d"   and r :: "('d \<times> 'd) set"   and u :: "'d list"   and x :: "'d list"   and y :: "'d list" assumes "(a, b) \<in> r" shows "(u @ a # x, u @ b # y) \<in> lexord r"
ratio_def: fixes za :: "complex_homo"   and zb :: "complex_homo"   and zc :: "complex_homo" shows "ratio za zb zc = (za -\<^sub>h zb) :\<^sub>h (za -\<^sub>h zc)"
perm_1: shows "\<forall>i<length xs. xs ! i = ys ! f i"
lemma3': fixes a :: "letter"   and vs :: "letter list list"   and ws :: "letter list list"   and as :: "letter list" assumes "T a vs ws"   and "L as vs" shows "L (a # as) ws"
conj_simps_5: fixes P :: "fm" shows "Ferrack.conj P P = P"
set_bd_1: fixes x :: "('l, 'k, 'm) F3" shows "|F3set1 x| \<le>o bd_F3"
Atm: fixes atm :: "'atom"   and s :: "'state" shows "(Atm atm, s) \<rightarrow>t aval atm s"
ccExp_App: fixes e :: "exp"   and a :: "Arity"   and x :: "Vars.var" shows "\<G>\<^bsub>inc\<cdot>a\<^esub> e \<squnion> {x} G\<times> insert x (fv e) \<sqsubseteq> \<G>\<^bsub>a\<^esub> (App e x)"
llist_eq_lcons: fixes a :: "'b llist"   and b :: "'b llist" assumes "a \<noteq> LNil"   and "b \<noteq> LNil"   and "lhd a = lhd b"   and "ltl a = ltl b" shows "a = b"
Re_sin: fixes z :: "complex" shows "Re (sin z) = sin (Re z) * (exp (Im z) + exp (- Im z)) / 2"
MkIde_Dom: fixes a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "arr a" shows "MkIde (Dom a) = local.dom a"
assms: fixes i :: "'a" assumes "i \<in> I" shows "(0::'b) \<le> p i"
Px_nn: fixes x :: "'b" assumes "x \<in> space MX" shows "0 \<le> Px x"
True_steps_concD: fixes p :: "bool list"   and q :: "bool list"   and L :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and R :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and w :: "'b list" assumes "(True # p, q) \<in> NAe.steps (RegExp2NAe.conc L R) w" shows "(\<exists>r. (p, r) \<in> NAe.steps L w \<and> q = True # r) \<or> (\<exists>u v. w = u @ v \<and> (\<exists>r. (p, r) \<in> NAe.steps L u \<and> fin L r \<and> (\<exists>s. (start R, s) \<in> NAe.steps R v \<and> q = False # s)))"
intro: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and F :: "'ag \<Rightarrow> 'ah"   and G :: "'ag \<Rightarrow> 'ah"   and H :: "'ag \<Rightarrow> 'ah"   and \<sigma> :: "'ag \<Rightarrow> 'ah"   and \<tau> :: "'ag \<Rightarrow> 'ah" assumes "category A"   and "category B"   and "functor A B F"   and "functor A B G"   and "functor A B H"   and "natural_transformation A B F G \<sigma>"   and "natural_transformation A B G H \<tau>" shows "vertical_composite A B F G H \<sigma> \<tau>"
le_cases: fixes x :: "'a"   and y :: "'a" obtains "x \<sqsubseteq> y"   | "y \<sqsubseteq> x"
dim_resize_vec: fixes n :: "nat"   and v :: "'c vec" shows "dim_vec (resize_vec n v) = n"
mod_mult_self1: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus a (times c b)) b = modulo a b"
that: fixes f' :: "'a fls" assumes "f = fpxs_compose_power (fpxs_of_fls f') (1 / rat_of_nat d)" shows "thesis"
word_and_1_shiftl: fixes x :: "'b word"   and n :: "nat" shows "x AND (1 << n) = (if x !! n then 1 << n else 0)"
hyps_3: shows "finite {M. set_mset M \<subseteq> A \<and> size M = Suc k}"
subseteq_mset_size_eql: fixes X :: "'b multiset"   and Y :: "'b multiset" assumes "X \<subseteq># Y"   and "size Y = size X" shows "X = Y"
hyp: shows "\<forall>l\<le>k. unowned_shared (snd (snd (c (n + l)))) (fst (c (n + l)))"
set_of_apprs_ex_Cons: fixes xrs :: "real list"   and xs :: "'b list"   and b :: "'b" assumes "xrs \<in> set_of_appr xs" shows "\<exists>r. r # xrs \<in> set_of_appr (b # xs)"
it_b_b': fixes a :: "'b" assumes "a \<in> S_a" shows "it_b a = it_b' a"
inject_5: fixes x61 :: "label"   and x62 :: "expr"   and x63 :: "stmt"   and y61 :: "label"   and y62 :: "expr"   and y63 :: "stmt" shows "(x61\<bullet> While(x62) x63 = y61\<bullet> While(y62) y63) = (x61 = y61 \<and> x62 = y62 \<and> x63 = y63)"
fim: shows "f ` (sphere (0::'a) 1 \<inter> S) = sphere (0::'a) 1 \<inter> T"
less_trans: fixes x :: "'a Word.word"   and y :: "'a Word.word"   and z :: "'a Word.word" assumes "x <s y"   and "y <s z" shows "x <s z"
lt_mult_scalar: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0"   and "q \<noteq> 0" shows "lt (p \<odot> q) = punit.lt p \<oplus> lt q"
bdd_below_Icc: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.preorder less_eq less" shows "preorder.bdd_below less_eq (ord.atLeastAtMost less_eq a b)"
simps_1: fixes s :: "nat \<Rightarrow> 'd" shows "inf_truncate s 0 = [s 0]"
oexp_Well_order: fixes r :: "('d \<times> 'd) set"   and s :: "('e \<times> 'e) set" assumes "Well_order r"   and "Well_order s" shows "Well_order (r ^o s)"
FGModuleHomSetI: fixes G :: "'a set"   and fgsmult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and fgsmult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd"   and W :: "'d set" assumes "FGModuleHom G fgsmult V fgsmult' T"   and "T ` V \<subseteq> W" shows "T \<in> FGModuleHomSet G fgsmult V fgsmult' W"
zgt_0_zge_1: fixes z :: "int" assumes "0 < z" shows "1 \<le> z"
has_integral_neg: fixes f :: "'c \<Rightarrow> 'd"   and k :: "'d"   and S :: "'c set" assumes "(f has_integral k) S" shows "((\<lambda>x. - f x) has_integral - k) S"
con1: fixes env :: "v sem_env"   and cn :: "(char list, char list) id0 option"   and es :: "exp0 list"   and vs :: "v list"   and v :: "v"   and s1 :: "'a state"   and s2 :: "'a state" assumes "do_con_check (c env) cn (length es)"   and "build_conv (c env) cn (rev vs) = Some v"   and "Big_Step_Unclocked_Single.evaluate_list (Big_Step_Unclocked_Single.evaluate env) s1 (rev es) (s2, Rval vs)" shows "Big_Step_Unclocked_Single.evaluate env s1 (Con cn es) (s2, Rval v)"
hyps: shows "\<Psi> \<rhd> M\<langle>N\<rangle>.Q \<longmapsto> \<tau> \<prec> Q'"
assms_2: shows "f \<longlonglongrightarrow> l"
carrier_nhd: fixes x :: "nat" assumes "x \<in> carrier" shows "carrier \<in> nhds x"
remove: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<in> A" shows "sorted_list_of_set A = insort x (sorted_list_of_set (A - {x}))"
elims: fixes x :: "('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr"   and y :: "('d, 'e, 'f) expr" assumes "x\<^sub>e = y" obtains \<sigma> :: "'e \<Rightarrow> ('d, 'e, 'f) val option" and \<tau> :: "'e \<Rightarrow> ('d, 'e, 'f) val option" and e :: "('d, 'e, 'f) expr" where "x = (\<sigma>, \<tau>, e)" and "y = e"
le_funD: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "f \<le> g" shows "f x \<le> g x"
paint_bd: fixes t :: "('a, 'b) rbt" assumes "bd_inv t"   and "t \<noteq> Leaf"   and "cl t = B" shows "black_depth (paint R t) = black_depth t - 1"
pm: fixes i :: "'a"   and x :: "'c \<times> 'd \<Rightarrow>\<^sub>L 'c" assumes "i \<in> I"   and "x \<in> X0 i" shows "case x of (x, d) \<Rightarrow> returns_to p x \<and> fst ` X0 i \<subseteq> S \<and> return_time p differentiable at x within S \<and> (\<exists>D. (poincare_map p has_derivative blinfun_apply D) (at x within S) \<and> (poincare_map p x, D o\<^sub>L d) \<in> X1 i) \<and> (\<forall>t\<in>{0<..<return_time p x}. flow0 x t \<in> CX i)"
repeat_stabilize_induct: fixes P :: "'c \<Rightarrow> bool"   and m :: "'c"   and f :: "'c \<Rightarrow> 'c"   and n :: "nat" assumes "P m"   and "\<And>m. P m \<Longrightarrow> P (f m)" shows "P (repeat_stabilize n f m)"
exec_step_input_StepCD: fixes P :: "jvm_prog"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and C' :: "char list"   and Cs :: "char list list" assumes "exec_step_input P C M pc ics = StepC C' Cs" shows "ics = Calling C' Cs"
zero_scaleRat: fixes v :: "'a" shows "0 *R v = (0::'a)"
hyps_3: shows "l \<in> Wellfounded.acc (listrel1 r)"
map_gpv'_def: shows "map_gpv' \<equiv> \<lambda>uu uua uub uuc. corec_gpv (\<lambda>((f, g), h, gpv). map_spmf (map_generat f g ((\<circ>) (\<lambda>uu. Inr ((f, g), h, uu)))) (map_spmf (map_generat id id (h ---> id)) (the_gpv gpv))) ((uu, uua), uub, uuc)"
graph: shows "\<alpha> \<in> S \<rightarrow>\<^sub>E T"
SATAxiomTerminal: fixes gamma :: "(signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list" assumes "SATAxiom (sequent gamma)" shows "terminal subs gamma"
rtranclp_tranclp_tranclp: fixes r :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g"   and z :: "'g" assumes "r\<^sup>*\<^sup>* x y"   and "r\<^sup>+\<^sup>+ y z" shows "r\<^sup>+\<^sup>+ x z"
measure_times: fixes A :: "'i \<Rightarrow> 'a set" assumes "\<And>ia. ia \<in> {i} \<Longrightarrow> A ia \<in> sets (M ia)" shows "emeasure (Pi\<^sub>M {i} M) (Pi\<^sub>E {i} A) = (\<Prod>i\<in>{i}. emeasure (M i) (A i))"
eval: shows "prg Env\<turnstile>s0 \<midarrow>v=\<succ>vf\<rightarrow> s1"
any_node_non_sdom0: fixes k :: "nat" shows "non_strict_dominate k 0"
a1: shows "s' = snd (fst (add_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
ring_simprules_20: fixes R :: "('e, 'f) ring_scheme"   and x :: "'e" assumes "ring R"   and "x \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> (\<ominus>\<^bsub>R\<^esub> x) = x"
cSup_insert_If: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and X :: "'a set"   and a :: "'a" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "preorder.bdd_above less_eq X" shows "Sup (insert a X) = (if X = {} then a else sup a (Sup X))"
loopstep: fixes Z :: "(variable \<Rightarrow> real) set" assumes "Z \<subseteq> game_sem I \<alpha>** X"   and "P Z" shows "P (X \<union> game_sem I \<alpha> Z)"
dg_is_arrD_2: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<BB>\<^esub> b" shows "a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
term_ok: shows "term_ok \<Theta> (mk_eq f g)"
case_2: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c llist \<Rightarrow> 'b"   and x21 :: "'c"   and x22 :: "'c llist" shows "(case x21 ## x22 of LNil \<Rightarrow> f1 | x ## xa \<Rightarrow> f2 x xa) = f2 x21 x22"
sDef: shows "Sources i C = DSources i C \<union> \<Union> (Sources i ` DSources i C)"
get_IFS_get_ACS_select_simps_4: fixes M :: "('b SecurityInvariant \<times> 'b SecurityInvariant_configured) list" assumes "\<forall>(m_impl, m_spec)\<in>set M. SecurityInvariant_complies_formal_def m_impl m_spec" shows "\<forall>(m_impl, m_spec)\<in>set (zip (TopoS_Composition_Theory_impl.get_ACS (get_impl M)) (TopoS_Composition_Theory.get_ACS (get_spec M))). SecurityInvariant_complies_formal_def m_impl m_spec"
in_hom_def: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and a :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and b :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>] b\<guillemotright> \<equiv> a \<in> A_B.domains f \<and> b \<in> A_B.codomains f"
tauFreshChain: fixes xvec :: "name list"   and P :: "('a, 'b, 'c) psi" shows "xvec \<sharp>* (\<tau>.P) = xvec \<sharp>* P"
disjE_PL: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b"   and A :: "'b set"   and r :: "'b" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL"   and "disj_PL p q \<in> Cn A"   and "imp_PL p r \<in> Cn A"   and "imp_PL q r \<in> Cn A" shows "r \<in> Cn A"
max_flow_min_cut': assumes "\<exists>f S. web_flow (web_of_network \<Delta>) f \<and> separating (web_of_network \<Delta>) S \<and> orthogonal_current (web_of_network \<Delta>) f S"   and "\<And>x. \<not> edge \<Delta> x (source \<Delta>)"   and "\<And>y. \<not> edge \<Delta> (sink \<Delta>) y"   and "\<And>x y. edge \<Delta> x y \<Longrightarrow> (\<exists>z. edge \<Delta> y z) \<or> (\<exists>z. edge \<Delta> z x)"   and "\<not> edge \<Delta> (source \<Delta>) (sink \<Delta>)"   and "\<And>x. \<not> edge \<Delta> x x"   and "\<And>e. e \<in> \<^bold>E \<Longrightarrow> 0 < capacity \<Delta> e" shows "\<exists>f S. flow \<Delta> f \<and> MFMC_Network.cut \<Delta> S \<and> orthogonal \<Delta> f S"
set_map_1: fixes f1 :: "'f \<Rightarrow> 'c"   and f2 :: "'g \<Rightarrow> 'd"   and f3 :: "'h \<Rightarrow> 'e"   and v :: "('f, 'g, 'h) F1" shows "F1set1 (F1map f1 f2 f3 v) = f1 ` F1set1 v"
closedin_retract: fixes S :: "'b set"   and T :: "'b set" assumes "S retract_of T" shows "closedin (top_of_set T) S"
cinner_sum_left: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<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 dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and y :: "'a" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner (comm_monoid_add.sum plus zero f A) y = (\<Sum>x\<in>A. cinner (f x) y)"
induct: fixes P :: "nat \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "'a list" assumes "\<And>k xs. \<lbrakk>\<And>x xa xb xc xd xe y xda ya xdb xdc. \<lbrakk>\<not> length xs \<le> 20; x = chop 5 xs; xa = map slow_median x; xb = ((length xs + 4) div 5 - 1) div 2; xc = mom_select xb xa; xd = partition3 xc xs; (xe, y) = xd; (xda, ya) = y; xdb = length xe; xdc = length xda; k < xdb\<rbrakk> \<Longrightarrow> P k xe; \<And>x xa xb xc xd xe y xda ya xdb xdc. \<lbrakk>\<not> length xs \<le> 20; x = chop 5 xs; xa = map slow_median x; xb = ((length xs + 4) div 5 - 1) div 2; xc = mom_select xb xa; xd = partition3 xc xs; (xe, y) = xd; (xda, ya) = y; xdb = length xe; xdc = length xda; \<not> k < xdb; \<not> k < xdb + xdc\<rbrakk> \<Longrightarrow> P (k - xdb - xdc) ya; \<And>x xa xb xc xd xe y xda ya xdb xdc. \<lbrakk>\<not> length xs \<le> 20; x = chop 5 xs; xa = map slow_median x; xb = ((length xs + 4) div 5 - 1) div 2; xc = mom_select xb xa; xd = partition3 xc xs; (xe, y) = xd; (xda, ya) = y; xdb = length xe; xdc = length xda\<rbrakk> \<Longrightarrow> P xb xa\<rbrakk> \<Longrightarrow> P k xs" shows "P a0 a1"
blocksFinal: fixes Vs :: "char list list"   and Ts :: "ty list"   and vs :: "val list"   and e :: "char list exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" assumes "length Vs = length Ts"   and "length vs = length Ts"   and "final e" shows "P \<turnstile> \<langle>blocks (Vs, Ts, vs, e),(h, l)\<rangle> \<rightarrow>* \<langle>e,(h, l)\<rangle>"
simps_2: fixes x :: "'c ltl"   and xs :: "'c ltl list" shows "And (x # xs) = foldl LTLAnd x xs"
G_def: fixes p :: "'c \<Rightarrow> 'b"   and I :: "'c set" shows "sum' p I \<equiv> if finite {x \<in> I. p x \<noteq> (0::'b)} then sum p {x \<in> I. p x \<noteq> (0::'b)} else (0::'b)"
addN_def: fixes g :: "'g"   and n :: "'node" shows "addN g n \<equiv> fold (\<lambda>v. Mapping.map_default v {} (insert n)) (sorted_list_of_set (uses g n))"
ports_to_set_wordinterval: fixes ps :: "(16 word \<times> 16 word) list" shows "ports_to_set ps = wordinterval_to_set (l2wi ps)"
map_dds_rel_cong: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and xb :: "('b, 'd) dds"   and yb :: "('c, 'e) dds"   and A' :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'f \<Rightarrow> 'b"   and y :: "'g \<Rightarrow> 'c"   and B' :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and xa :: "'d \<Rightarrow> 'h"   and ya :: "'e \<Rightarrow> 'i" assumes "rel_dds A B xb yb"   and "\<And>xa ya. A' xa ya \<Longrightarrow> A (x xa) (y ya)"   and "\<And>x y. B x y \<Longrightarrow> B' (xa x) (ya y)" shows "rel_dds A' B' (map_dds x xa xb) (map_dds y ya yb)"
moebius_preserve_circline_angle: fixes M :: "moebius"   and H1 :: "ocircline"   and H2 :: "ocircline" shows "cos_angle (moebius_ocircline M H1) (moebius_ocircline M H2) = cos_angle H1 H2"
two: shows "1 \<noteq> 0 \<cdot> 1\<^sup>\<star>"
IH: fixes s1 :: "('a, bool) fmap"   and s2 :: "('a, bool) fmap" assumes "s1 \<subseteq>\<^sub>f s2"   and "sat_precond_as s1 as" shows "sat_precond_as s2 as"
arrE: fixes f :: "'a" assumes "R.arr f" shows "right f f"
non_zero_neighbour_alt: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and z :: "complex"   and \<beta> :: "complex" assumes "f holomorphic_on S"   and "open S"   and "connected S"   and "z \<in> S"   and "\<beta> \<in> S"   and "f \<beta> \<noteq> 0" shows "\<forall>\<^sub>F w in at z. f w \<noteq> 0 \<and> w \<in> S"
shiftpath_def: fixes a :: "real"   and f :: "real \<Rightarrow> 'a" shows "shiftpath a f = (\<lambda>x. if a + x \<le> 1 then f (a + x) else f (a + x - 1))"
E_prop: shows "E = V \<times> V - (\<Union>m\<in>set M. \<Union> (c_offending_flows m (fully_connected G)))"
assms_3: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and c :: "'a"   and exp :: "exp" assumes "atom ` domA \<Gamma> \<sharp>* c"   and "\<And>c x e. (x, e) \<in> set \<Gamma> \<Longrightarrow> P c e"   and "\<And>c. P c exp" shows "P c (Let \<Gamma> exp)"
v_1: shows "v \<in> w0 `\<rightarrow> C0"
Abs_inject: fixes x :: "int"   and y :: "int" assumes "x \<in> uints LENGTH('b)"   and "y \<in> uints LENGTH('b)" shows "(word_of_int x = word_of_int y) = (x = y)"
euler_mascheroni_upper: fixes n :: "nat" shows "euler_mascheroni \<le> harm (Suc n) - ln (real (n + 2)) + 1 / real (2 * (n + 1))"
fi_scalar_prod_gso: fixes n :: "nat"   and fs :: "'a vec list"   and i :: "nat"   and j :: "nat" assumes "gram_schmidt_fs_lin_indpt n fs"   and "i < length fs"   and "j < length fs" shows "fs ! i \<bullet> gram_schmidt_fs.gso n fs j = gram_schmidt_fs.\<mu> n fs i j * \<parallel>gram_schmidt_fs.gso n fs j\<parallel>\<^sup>2"
assms_3: shows "\<forall>z\<in>set xs. r x z"
Pd_linear': fixes d :: "'d" shows "Linear_order (Pd d)"
Regsum: shows "(\<lambda>x. Re (g x)) abs_summable_on A"
leq_path_unfold: fixes Cs :: "char list list"   and Ds :: "char list list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list" shows "P,C \<turnstile> Cs \<sqsubseteq> Ds = (leq_path1p P C)\<^sup>*\<^sup>* Cs Ds"
assms: fixes k :: "'a" shows "f k (0::'b) (0::'c) = (0::'d)"
unwind_to_def: fixes istate :: "'a"   and step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and \<phi> :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and f :: "('a, 'b, 'c) trans \<Rightarrow> 'd"   and \<gamma> :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and g :: "('a, 'b, 'c) trans \<Rightarrow> 'e"   and T :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and \<Delta> :: "'a \<Rightarrow> 'd list \<Rightarrow> 'a \<Rightarrow> 'd list \<Rightarrow> bool"   and \<Delta>s :: "('a \<Rightarrow> 'd list \<Rightarrow> 'a \<Rightarrow> 'd list \<Rightarrow> bool) set" shows "BD_Security.unwind_to istate step \<phi> f \<gamma> g T \<Delta> \<Delta>s \<equiv> \<forall>s vl s1 vl1. BD_Security.reachNT istate step T s \<and> IO_Automaton.reach istate step s1 \<and> \<Delta> s vl s1 vl1 \<longrightarrow> vl \<noteq> [] \<and> BD_Security.exit step \<phi> f T s (hd vl) \<or> BD_Security.iaction step \<phi> f \<gamma> (disjAll \<Delta>s) s vl s1 vl1 \<or> (vl \<noteq> [] \<or> vl1 = []) \<and> BD_Security.reaction step \<phi> f \<gamma> g T (disjAll \<Delta>s) s vl s1 vl1"
inject_5: fixes x6 :: "'a OCL_Types.type"   and y6 :: "'a OCL_Types.type" shows "(OrderedSet x6 = OrderedSet y6) = (x6 = y6)"