hcju/isabelleps huggingface dataset:
import_rel1_def: fixes R :: "('f \<times> 'e) set \<Rightarrow> ('h \<times> 'g) set" shows "import_rel1 R \<equiv> \<lambda>A c ci. \<up> (is_pure A \<and> (ci, c) \<in> \<langle>the_pure A\<rangle>R)"
finite_Diff2: fixes B :: "'b set"   and A :: "'b set" assumes "finite B" shows "finite (A - B) = finite A"
step_2: shows "h \<turnstile> get_component child \<rightarrow>\<^sub>r c"
VsetI: fixes b :: "V"   and i :: "V" assumes "rank b \<sqsubset> i" shows "b \<in>\<^sub>\<circ> Vset i"
Pi_cancel_merge_4: fixes J :: "'d set"   and I :: "'d set"   and x :: "'d \<Rightarrow> 'e"   and y :: "'d \<Rightarrow> 'e"   and B :: "'d \<Rightarrow> 'e set" assumes "J \<inter> I = {}" shows "(merge I J (x, y) \<in> Pi J B) = (y \<in> Pi J B)"
intro: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and C :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e" assumes "functor A B F"   and "functor B C G" shows "composite_functor A B C F G"
vfsequence_vsingleton: fixes a :: "V" shows "vfsequence (set {\<langle>[]\<^sub>\<circ>, a\<rangle>})"
finite_mult_not_eq_zero_prodI: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "finite {a. f a \<noteq> (0::'c)}"   and "finite {b. g b \<noteq> (0::'c)}" shows "finite {(a, b). f a * g b \<noteq> (0::'c)}"
SFAssInit\<^sub>1: fixes P :: "nat exp prog"   and e\<^sub>2 :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and v :: "val"   and h\<^sub>1 :: "heap"   and l\<^sub>1 :: "val list"   and sh\<^sub>1 :: "sheap"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and v' :: "val"   and h' :: "heap"   and l' :: "val list"   and sh' :: "sheap"   and sfs :: "sfields"   and i :: "init_state"   and sfs' :: "sfields"   and sh'' :: "sheap" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,(h\<^sub>1, l\<^sub>1, sh\<^sub>1)\<rangle>"   and "P \<turnstile> C has F,Static:t in D"   and "\<nexists>sfs. sh\<^sub>1 D = \<lfloor>(sfs, Done)\<rfloor>"   and "P \<turnstile>\<^sub>1 \<langle>INIT D ([D],False) \<leftarrow> unit,(h\<^sub>1, l\<^sub>1, sh\<^sub>1)\<rangle> \<Rightarrow> \<langle>Val v',(h', l', sh')\<rangle>"   and "sh' D = \<lfloor>(sfs, i)\<rfloor>"   and "sfs' = sfs(F \<mapsto> v)"   and "sh'' = sh'(D \<mapsto> (sfs', i))" shows "P \<turnstile>\<^sub>1 \<langle>C\<bullet>\<^sub>sF{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>unit,(h', l', sh'')\<rangle>"
runit_simps_1: fixes f :: "'a" assumes "C.ide f" shows "C.arr \<r>\<^sub>C[f]"
length_def: fixes h :: "heap"   and a :: "'a array" shows "Array.length h a = length (Array.get h a)"
surjective: fixes r :: "('a, 'b, 'c, 'd) configuration_scheme" shows "r = \<lparr>states = states r, msgs = msgs r, \<dots> = more r\<rparr>"
both_above_arg1: fixes t :: "'a ttree"   and t' :: "'a ttree" shows "t \<sqsubseteq> t \<otimes>\<otimes> t'"
var: fixes \<Gamma> :: "(name, value) fmap"   and name :: "name"   and val :: "value" assumes "fmlookup \<Gamma> name = Some val" shows "\<Gamma> \<turnstile>\<^sub>v Svar name \<down> val"
inner_node_standard_control_dependence_predecessor: fixes n :: "'b"   and as :: "'a list"   and as' :: "'a list" assumes "inner_node n"   and "Entry -as\<rightarrow>* n"   and "n -as'\<rightarrow>* Exit" obtains n' :: "'b" where "n' controls\<^sub>s n"
inj_on_cong: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "\<And>a. a \<in> A \<Longrightarrow> f a = g a" shows "inj_on f A = inj_on g A"
inj_monoid_mult_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "inj_monoid_mult_hom hom \<equiv> monoid_mult_hom hom \<and> inj_semigroup_mult_hom hom"
map_simp_ide: fixes a :: "'c" assumes "C.ide a" shows "Adj.\<epsilon>FoF\<eta>.map a = (Adj.\<epsilon> \<circ> F) a \<cdot>\<^sub>D (F \<circ> Adj.\<eta>) a"
invar_def: shows "invar \<equiv> \<lambda>v0 D0 (p, D, pE). invar_loc G v0 D0 p D pE"
simps_5: shows "J.BT \<cdot>\<^sub>J J.BB = J.BT"
exp_eq_0_imp_not_bit: 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 n :: "nat"   and a :: "'b" assumes "class.semiring_bits plus minus zero times one divide modulo bit"   and "power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n = zero" shows "\<not> bit a n"
assms_3: shows "ModalSequents.mset (extend S c) = \<LM> B  \<RM>"
eventually_subst: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and F :: "'b filter" assumes "\<forall>\<^sub>F n in F. P n = Q n" shows "eventually P F = eventually Q F"
convex_imp_Borsukian: fixes S :: "'a set" assumes "convex S" shows "Borsukian S"
Int_Un_eq_4: fixes T :: "'e set"   and S :: "'e set" shows "T \<union> S \<inter> T = T"
dividend_less_div_times: fixes n :: "nat"   and m :: "nat" assumes "0 < n" shows "m < n + m div n * n"
is_ubD: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c set"   and u :: "'c"   and x :: "'c" assumes "class.po below"   and "po.is_ub below S u"   and "x \<in> S" shows "below x u"
measure_le_0_iff: fixes M :: "'b measure"   and X :: "'b set" shows "(Sigma_Algebra.measure M X \<le> 0) = (Sigma_Algebra.measure M X = 0)"
Compr_image_eq: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and P :: "'a \<Rightarrow> bool" shows "{x \<in> f ` A. P x} = f ` {x \<in> A. P (f x)}"
assms: shows "exec_step_input P C M pc ics = StepI i"
successively_cong: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ys :: "'a list" assumes "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set xs\<rbrakk> \<Longrightarrow> P x y = Q x y"   and "xs = ys" shows "successively P xs = successively Q ys"
pmdl_closed_red: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and A :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "pmdl B \<subseteq> pmdl A"   and "p \<in> pmdl A"   and "red B p q" shows "q \<in> pmdl A"
assms_1: shows "CFG_SSA_Transformed \<alpha>e \<alpha>n invar inEdges Entry oldDefs oldUses defs (\<lambda>g. lookup_multimap (uses g)) (\<lambda>g. Mapping.lookup (phis g)) var"
prv_cnjR_fls: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla" shows "prv (eqv (cnj fls \<phi>) fls)"
index_zero_vec_1: fixes i :: "nat"   and n :: "nat" assumes "i < n" shows "0\<^sub>v n $v i = (0::'a)"
simps_15: fixes s :: "'a"   and t :: "'b"   and va :: "('a, 'b) rbt"   and vb :: "'a"   and vc :: "'b"   and vd :: "('a, 'b) rbt" shows "balance Empty s t (Branch B va vb vc vd) = Branch B Empty s t (Branch B va vb vc vd)"
pf_notcong1: fixes x :: "nat" assumes "x \<in># prime_factorization (p - 1)" shows "[a ^ ((p - 1) div x) \<noteq> 1] (mod p)"
assms_1: shows "SubRel.ces v1 es v2 subs1"
monotoneD: fixes orda :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and ordb :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'h"   and x :: "'g"   and y :: "'g" assumes "monotone orda ordb f"   and "orda x y" shows "ordb (f x) (f y)"
ls_insls_map_of: fixes ld :: "('a \<times> 'b) list list"   and xs :: "('a \<times> 'b) list" assumes "ht_hash ld"   and "ht_distinct ld"   and "1 < length ld"   and "distinct (map fst xs)" shows "map_of (concat (ls_insls xs ld)) = map_of (concat ld) ++ map_of xs"
simps_2: fixes x :: "'a"   and xs :: "'a list" shows "index (x # xs) 0 = Some x"
rA': shows "local.sub2 d l (r, A) = (r', A')"
xor_def2: fixes x :: "'b"   and y :: "'b" shows "x XOR y = (x OR y) AND (NOT x OR NOT y)"
ipresWlsAll_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('e, 'f, 'varSort, 'sort, 'g, 'h, 'c, 'd, 'i) Iteration.model_scheme" shows "ipresWlsAll h hA MOD \<equiv> ipresWls h MOD \<and> ipresWlsAbs hA MOD"
ring_record_simps_11: fixes carrier :: "'a set"   and mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and zero :: "'a"   and add :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and more :: "'b" shows "\<zero>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, zero = zero, add = add, \<dots> = more\<rparr>\<^esub> = zero"
simps: fixes I :: "'e set"   and v :: "nat \<Rightarrow> 'f"   and as :: "('e \<times> 'f atom) set" shows "(I, v) \<Turnstile>\<^sub>i\<^sub>a\<^sub>s as = v \<Turnstile>\<^sub>a\<^sub>s (snd ` (as \<inter> I \<times> UNIV))"
plus_subset1: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> o[F](g)" shows "O[F](g) \<subseteq> O[F](\<lambda>x. f x + g x)"
rsa_inv: fixes d :: "nat"   and e :: "nat"   and P :: "nat"   and Q :: "nat"   and x :: "nat" assumes "d = nat (fst (bezw e ((P - 1) * (Q - 1))) mod int ((P - 1) * (Q - 1)))"   and "coprime e ((P - 1) * (Q - 1))"   and "prime P"   and "prime Q"   and "P \<noteq> Q"   and "1 < e"   and "1 < d" shows "(x ^ e mod (P * Q)) ^ d mod (P * Q) = x mod (P * Q)"
sup_bot_left: fixes x :: "'a" shows "(0::'a) + x = x"
FSwSbImorph_termMOD: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e, 'f) model_scheme" shows "FSwSbImorph h hA termMOD MOD = termFSwSbImorph h hA MOD"
le: shows "(x' \<le> x) = (int_of_uint64 x' \<le> int_of_uint64 x)"
imp_finite: assumes "\<forall>j\<in>{0..<m}. finite {n. P j n}" shows "finite {n. \<exists>j\<in>{0..<m}. P j n}"
is_leftderivation_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and u :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "CFG.is_leftderivation \<NN> \<TT> \<RR> \<SS> u = CFG.leftderives \<NN> \<TT> \<RR> [\<SS>] u"
length_filter_conv_card: fixes p :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "length (filter p xs) = card {i. i < length xs \<and> p (xs ! i)}"
FlexRay_def: fixes n :: "nat"   and nReturn :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nC :: "nat \<Rightarrow> Config"   and nStore :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "nat \<Rightarrow> nat \<Rightarrow> nat list" shows "FlexRay n nReturn nC nStore nGet \<equiv> CorrectSheaf n \<and> ((\<forall>i<n. msg 1 (nReturn i)) \<and> DisjointSchedules n nC \<and> IdenticCycleLength n nC \<longrightarrow> FrameTransmission n nStore nReturn nGet nC \<and> (\<forall>i<n. msg 1 (nGet i) \<and> msg 1 (nStore i)))"
differentiable_mult: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and s :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "f differentiable at x within s"   and "g differentiable at x within s" shows "(\<lambda>x. f x * g x) differentiable at x within s"
assms_3: shows "\<exists>v\<in>wfrestrictedvars\<^sub>s\<^sub>t S. t \<cdot> \<I> = \<I> v"
ab: shows "a \<squnion>\<^bsub>f\<^esub> b \<in> A"
rel_G_mono': fixes L1 :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and L2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and Co1 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Co2 :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and Contra1 :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and Contra2 :: "'m \<Rightarrow> 'n \<Rightarrow> bool"   and x :: "('b, 'e, 'g, 'i, 'k, 'm, 'o) G"   and y :: "('d, 'f, 'h, 'j, 'l, 'n, 'o) G"   and L1' :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and L2' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and Co1' :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Co2' :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and Contra1' :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and Contra2' :: "'m \<Rightarrow> 'n \<Rightarrow> bool" assumes "rel_G L1 L2 Co1 Co2 Contra1 Contra2 x y"   and "L1 \<le> L1'"   and "L2 \<le> L2'"   and "Co1 \<le> Co1'"   and "Co2 \<le> Co2'"   and "Contra1' \<le> Contra1"   and "Contra2' \<le> Contra2" shows "rel_G L1' L2' Co1' Co2' Contra1' Contra2' x y"
deleteMin_correct_1: fixes q :: "('a, 'b) BinomialHeapStruc.BinomialTree list" assumes "BinomialHeapStruc.invar q"   and "q \<noteq> []" shows "BinomialHeapStruc.invar (BinomialHeapStruc.deleteMin q)"
hom_mult_eq_zero: fixes x :: "int"   and y :: "int" assumes "x * y = 0" shows "of_int x * of_int y = (0::'b)"
scaleR_times: fixes u :: "num"   and w :: "num"   and a :: "'b" shows "numeral u *\<^sub>R (numeral w * a) = (numeral u * numeral w) *\<^sub>R a"
WTrt_elim_cases_8: fixes 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"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and F :: "char list"   and D :: "char list"   and v :: "char list exp"   and T :: "ty" assumes "P,E,h \<turnstile> e\<bullet>F{D} := v : T" obtains C :: "char list" and T :: "ty" and T\<^sub>2 :: "ty" where "Ta__ = Void" and "P,E,h \<turnstile> e : Class C" and "P \<turnstile> C has F:T in D" and "P,E,h \<turnstile> v : T\<^sub>2__" and "P \<turnstile> T\<^sub>2__ \<le> T"   | T\<^sub>2 :: "ty" where "Ta__ = Void" and "P,E,h \<turnstile> e : NT" and "P,E,h \<turnstile> v : T\<^sub>2"
tl_app: assumes "status \<noteq> Idle" shows "\<forall>l. tl front @ l = tl (front @ l)"
1: fixes x :: "'a prefix_match option"   and y :: "'a wordinterval"   and x2 :: "'a prefix_match" assumes "\<not> wordinterval_empty rs"   and "(x, y) = wordinterval_CIDR_split1 rs"   and "x = Some x2" shows "distinct (wordinterval_CIDR_split_prefixmatch y)"
IH: assumes "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets M" shows "Sigma_Algebra.measure M (\<Union> (F ` I)) \<le> (\<Sum>i\<in>I. Sigma_Algebra.measure M (F i))"
set_child_nodes_get_child_nodes_different_pointers: fixes ptr :: "(_) object_ptr"   and ptr' :: "(_) object_ptr"   and w :: "((_) heap, exception, unit) prog"   and h :: "(_) heap"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool" assumes "ptr \<noteq> ptr'"   and "w \<in> set_child_nodes_locs ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'"   and "r \<in> get_child_nodes_locs ptr'" shows "r h h'"
exists_s_factor_dvd_h_s: fixes P :: "'a mod_ring poly set"   and f :: "'a mod_ring poly"   and fi :: "'a mod_ring poly"   and h :: "'a mod_ring poly" assumes "finite P"   and "f = \<Prod>P"   and "P \<subseteq> {q. irreducible q \<and> monic q}"   and "fi \<in> P"   and "h \<in> {v. [v ^ CARD('a) = v] (mod f)}" shows "\<exists>s. fi dvd h - [:s:]"
S_3: shows "\<exists>\<^sub>F t in at_top. infdist (flow0 x t) (\<omega>_limit_set x) \<notin> S"
DBM_set_diag: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'b DBMEntry" assumes "[M]\<^bsub>v,n\<^esub> \<noteq> {}" shows "[M]\<^bsub>v,n\<^esub> = [(\<lambda>i j. if i = j then Le (0::'b) else M i j)]\<^bsub>v,n\<^esub>"
filter_equals_takeWhile_sorted_rev: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'g"   and xs :: "'h list"   and t :: "'g" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq (rev (map f xs))" shows "filter (\<lambda>x. less t (f x)) xs = takeWhile (\<lambda>x. less t (f x)) xs"
cs_path_shift_set_cd: fixes \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat"   and n :: "nat" assumes "is_path \<pi>" shows "cs\<^bsup>\<pi> \<guillemotleft> k\<^esup> n = map \<pi> (sorted_list_of_set {i. k + n cd\<^bsup>\<pi>\<^esup>\<rightarrow> i \<and> k \<le> i}) @ [\<pi> (k + n)]"
assms: fixes t :: "'a rexp" shows "P t \<equiv> \<exists>t'. t \<sim> ro t +\<^sub>r t' \<and> ro t' = 0\<^sub>r"
R1_of_list: fixes x :: "'b list" shows "list_R1_to_list (list_to_list_R1 x) = x"
intros: fixes C :: "'a sequent"   and F :: "'a"   and Fs :: "'a form list"   and Ps :: "'a sequent list" assumes "C = ( \<Empt> \<Rightarrow>* \<LM> Compound F Fs  \<RM>)" shows "rightPrincipal (Ps, C) (Compound F Fs)"
some_best_in_best: shows "local.some_best \<in> local.best"
harm_integral_0_1_fraction_2: fixes n :: "nat" shows "LBINT x=0..1. (1 - x ^ n) / (1 - x) = harm n"
dvd_add_times_triv_left_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 a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.comm_semiring_1_cancel plus minus zero times one" shows "dvd.dvd times a (plus (times c a) b) = dvd.dvd times a b"
part1: shows "preprocess_part_1 ncs = (t1, as, ui)"
bij: shows "bij_betw (ordermap H r) H (elts (\<alpha> ! i))"
set1_F_bound: fixes x :: "('a, 'j, 'k, 'c, 'd, 'e, 'f, 'g, 'h, 'i) F" shows "|set1_F x| \<le>o bd_F"
is_max: fixes \<aa> :: "'a set" assumes "lideal \<aa> R (+) (\<cdot>) \<zero> \<one>"   and "\<aa> \<noteq> R"   and "I \<subseteq> \<aa>" shows "I = \<aa>"
singleton_insert_inj_eq: fixes b :: "'b"   and a :: "'b"   and A :: "'b set" shows "({b} = insert a A) = (a = b \<and> order.greater_eq {b} A)"
approximation_miv_def: fixes n :: "nat"   and S :: "'a set" shows "approximation_miv n S = (independent_vertices E S \<and> (\<forall>S'. independent_vertices E S' \<longrightarrow> card S' \<le> card S * n))"
topspace_euclidean_subtopology: fixes S :: "'d set" shows "topspace (top_of_set S) = S"
reduce_basis_inv: fixes fs :: "int vec list" assumes "reduce_basis = fs" shows "LLL_invariant True m fs"
fst_sym_preproc_addnew_dgrad_set_le: fixes d :: "'a \<Rightarrow> nat"   and gs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and vs :: "'t list"   and fs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and v :: "'t" assumes "dickson_grading d" shows "dgrad_set_le d (pp_of_term ` set (fst (sym_preproc_addnew gs vs fs v))) (pp_of_term ` (Keys (set gs) \<union> insert v (set vs)))"
eventually_less_ceiling: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and F :: "'a filter" assumes "(f \<longlongrightarrow> l) F"   and "l \<notin> \<int>" shows "\<forall>\<^sub>F x in F. f x < of_int \<lceil>l\<rceil>"
wtl_sound_strong: fixes A :: "'a set"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "'a"   and B :: "'a"   and step :: "nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> 'a) list"   and c :: "'a list"   and ins :: "'b list"   and s\<^sub>0 :: "'a" assumes "lbvs A r f T B step c ins"   and "wtl_inst_list ins c f r T B step 0 s\<^sub>0 \<noteq> T"   and "s\<^sub>0 \<in> A"   and "0 < length ins" shows "\<exists>\<tau>s\<in>list (length ins) A. wt_step r T step \<tau>s \<and> s\<^sub>0 \<sqsubseteq>\<^bsub>r\<^esub> \<tau>s ! 0"
iso_iff_section_and_epi: fixes f :: "'d" shows "Dop.iso f = (Dop.section f \<and> Dop.epi f)"
simps_2: fixes x :: "'a"   and xs :: "'a list" shows "llist_of (x # xs) = LCons x (llist_of xs)"
supteq_supt_conv: fixes s :: "('c, 'd) Term.term"   and t :: "('c, 'd) Term.term" shows "s \<unrhd> t = (s \<rhd> t \<or> s = t)"
assms_1: shows "\<forall>G F nP i. wf_graph G \<and> F \<in> SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP \<and> i \<in> snd ` F \<longrightarrow> \<not> sinvar G (nP(i := otherbot))"
atLeast0_atMost_Suc: fixes g :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "prod g {0..Suc n} = prod g {0..n} * g (Suc n)"
intro: fixes S :: "'c set"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" assumes "subgroup S R addition zero"   and "submonoid S R multiplication unit" shows "subring S R addition multiplication zero unit"
inf_sup_aci_7: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.lattice (inf::'b \<Rightarrow> 'b \<Rightarrow> 'b) (less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (less::'b \<Rightarrow> 'b \<Rightarrow> bool) (sup::'b \<Rightarrow> 'b \<Rightarrow> 'b)" shows "(sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (x::'b) (sup (y::'b) (z::'b)) = sup y (sup x z)"
set_nemp_concat_nemp: fixes ws :: "'b list list" assumes "ws \<noteq> \<epsilon>"   and "\<epsilon> \<notin> set ws" shows "concat ws \<noteq> \<epsilon>"
rBang: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rs :: "('a, 'b, 'c) residual"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and C :: "'d" assumes "\<Psi> \<rhd> P \<parallel> !P \<longmapsto> Rs"   and "guarded P"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<And>C. Prop C \<Psi> (P \<parallel> !P) Rs A\<^sub>P (\<Psi>\<^sub>P \<otimes> \<one>)"   and "\<Psi>\<^sub>P \<simeq> \<one>"   and "supp \<Psi>\<^sub>P = {}"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Rs"   and "A\<^sub>P \<sharp>* C" shows "Prop C \<Psi> (!P) Rs [] (\<one>)"
put_inj: fixes \<sigma> :: "'b" shows "inj (put \<sigma>)"
no_var_subterm: fixes t :: "('a, 'b) Term.term"   and v :: "'b" shows "\<not> t \<sqsubset> Var v"
floor_int: fixes f :: "float" obtains k :: "int" where "real_of_int k = real_of_float (floor_fl f)"
Icc_subset_Iic_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and l :: "'a"   and h :: "'a"   and h' :: "'a" assumes "class.preorder less_eq less" shows "(ord.atLeastAtMost less_eq l h \<subseteq> ord.atMost less_eq h') = (\<not> less_eq l h \<or> less_eq h h')"
find_None_iff: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "(find P xs = None) = (\<nexists>x. x \<in> set xs \<and> P x)"
Quotient3I: fixes Abs :: "'d \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'd"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>a. Abs (Rep a) = a"   and "\<And>a. R (Rep a) (Rep a)"   and "\<And>r s. R r s = (R r r \<and> R s s \<and> Abs r = Abs s)" shows "Quotient3 R Abs Rep"
finsum_cong: fixes A :: "'e set"   and B :: "'e set"   and f :: "'e \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'c" assumes "A = B"   and "(f \<in> B \<rightarrow> carrier V) = True"   and "\<And>i. i \<in> B =simp=> f i = g i" shows "finsum V f A = finsum V g B"
fin: shows "integral\<^sup>N (count_space (\<^bold>E `` X)) g \<noteq> \<top>"
generator_closed: fixes G :: "('a, 'b) cyclic_group_scheme" assumes "Cyclic_Group.cyclic_group G" shows "\<^bold>g\<^bsub>G\<^esub> \<in> carrier G"
usolves_ode_on_subset: fixes x :: "real \<Rightarrow> 'a"   and S :: "real set"   and Y :: "'a set" assumes "(x usolves_ode f from t) (cball t et) (cball (flow t0 x0 t) ex)"   and "x t = flow t0 x0 t"   and "t \<in> S"   and "is_interval S"   and "S \<subseteq> cball t et"   and "cball (flow t0 x0 t) ex \<subseteq> Y" shows "(x usolves_ode f from t) S Y"
Sigma: fixes A :: "'c set"   and B :: "'c \<Rightarrow> 'd set"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> 'f" assumes "finite A"   and "\<forall>x\<in>A. finite (B x)" shows "(\<Sum>x\<in>A. sum (g x) (B x)) = (\<Sum>(x, y)\<in>Sigma A B. g x y)"
card_Inv_tri_ineq: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "dist_perm xs ys"   and "dist_perm ys zs" shows "card (Inv xs zs) \<le> card (Inv xs ys) + card (Inv ys zs)"
finsert_inter_finsert: fixes a :: "'b"   and A :: "'b fset"   and B :: "'b fset" shows "finsert a A |\<inter>| finsert a B = finsert a (A |\<inter>| B)"
we_n_def: shows "we_n = ce_rel_to_set r"
entails_def: fixes \<phi> :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"   and \<psi> :: "('a \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<phi> \<Turnstile>\<^sub>B \<psi> \<equiv> \<forall>\<sigma>. \<phi> \<sigma> \<longrightarrow> \<psi> \<sigma>"