hcju/isabelleps huggingface dataset:
lexordp_eq_trans: fixes xs :: "'t list"   and ys :: "'t list"   and zs :: "'t list" assumes "ord_term_lin.lexordp_eq xs ys"   and "ord_term_lin.lexordp_eq ys zs" shows "ord_term_lin.lexordp_eq xs zs"
finite_incidence_system_def: fixes point_set :: "'c set"   and block_collection :: "'c set multiset" shows "finite_incidence_system point_set block_collection \<equiv> incidence_system point_set block_collection \<and> finite_incidence_system_axioms point_set"
col_one: fixes j :: "nat"   and n :: "nat" assumes "j < n" shows "col (1\<^sub>m n) j = unit_vec n j"
type_conf_Boolean: fixes 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 E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr" shows "(P,E,h \<turnstile> e :\<^bsub>NT\<^esub> Boolean) = (P,E,h \<turnstile> e : Boolean)"
fg_sub: shows "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S"
decr_grading_term_le: fixes d :: "'a \<Rightarrow> nat"   and n :: "nat"   and v :: "'t" assumes "hom_grading d" shows "d (pp_of_term (decr_grading_term d n v)) \<le> n"
d: shows "CARD('pq) = CARD('p) * CARD('q)"
sff_4: assumes "p = 0" shows "c = 0 \<and> qis = []"
span_mono: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B" shows "More_Modules.ideal A \<subseteq> More_Modules.ideal B"
Ops: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and Ops :: "('b \<Rightarrow> 'b) set"   and enc :: "('b \<Rightarrow> 'b) \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b"   and t :: "'b" assumes "TermEncode var trm fmla Var FvarsT substT Fvars subst num Ops enc"   and "f \<in> Ops"   and "t \<in> trm" shows "f t \<in> trm"
target_alt_def: fixes r :: "(('e \<times> 'f) \<times> 'd) list"   and p :: "'d" shows "target r p = last (p # states r p)"
assms_1: shows "e \<le> v * - v\<^sup>T"
finite_ordLess_var: fixes S :: "'d set" shows "(|S| <o |UNIV| \<or> finite S) = (|S| <o |UNIV|)"
mod_eq_0_iff_dvd: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "(modulo a b = zero) = dvd.dvd times b a"
inner_prod_expand: fixes a :: "complex Matrix.vec"   and b :: "complex Matrix.vec"   and c :: "complex Matrix.vec"   and d :: "complex Matrix.vec" assumes "dim_vec a = dim_vec b"   and "dim_vec a = dim_vec c"   and "dim_vec a = dim_vec d" shows "\<langle>a + b|c + d\<rangle> = \<langle>a|c\<rangle> + \<langle>a|d\<rangle> + \<langle>b|c\<rangle> + \<langle>b|d\<rangle>"
not_in_segment2: fixes y :: "'c"   and f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "y \<notin> segment f x y"
1: shows "finite (\<Union>x\<in>L ` V. matching_i x V E M L)"
simps_6: fixes ve :: "16 word"   and m :: "of_match_field set" shows "prerequisites (VlanPriority ve) m = (\<exists>id. let v = VlanId id in v \<in> m \<and> prerequisites v m)"
simps_10: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> 'a"   and f3 :: "'d \<Rightarrow> 'a"   and x1 :: "'b" shows "(case In1 x1 of In1 x \<Rightarrow> f1 x | In2 x \<Rightarrow> f2 x | In3 x \<Rightarrow> f3 x) = f1 x1"
rel_assoc: fixes u :: "'a"   and v :: "'a"   and x :: "'a"   and w :: "'a" shows "(\<exists>y. \<rho> y u v \<and> \<rho> x y w) = (\<exists>z. \<rho> z v w \<and> \<rho> x u z)"
bv_to_int_upper_range: fixes w :: "bit list" shows "bv_to_int w < 2 ^ (length w - 1)"
contractible_space_def: fixes X :: "'b topology" shows "contractible_space X \<equiv> \<exists>a. homotopic_with (\<lambda>x. True) X X id (\<lambda>x. a)"
msed_map_invR: fixes f :: "'b \<Rightarrow> 'a"   and M :: "'b multiset"   and b :: "'a"   and N :: "'a multiset" assumes "image_mset f M = b, N" shows "\<exists>M1 a. M = a, M1 \<and> f a = b \<and> image_mset f M1 = N"
Act_Basic_7: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<A>\<phi> \<^bold>\<equiv> \<^bold>\<box>\<^bold>\<A>\<phi> in v]"
map_le_map_add: fixes f :: "'b \<Rightarrow> 'c option"   and g :: "'b \<Rightarrow> 'c option" shows "f \<subseteq>\<^sub>m f \<Oplus> g"
distinct_109: fixes x51 :: "expr"   and x52 :: "bop"   and x53 :: "expr"   and x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list" shows "x51 \<guillemotleft>x52\<guillemotright> x53 \<noteq> Call x101 x102 x103 x104"
matrix_to_iarray_option_inverse_matrix: fixes A :: "(('b, 'c) vec, 'c) vec" shows "matrix_to_iarray_option (inverse_matrix A) = inverse_matrix_iarray (matrix_to_iarray A)"
comp_arr_mkPoint: fixes f :: "'s"   and t :: "'s" assumes "S.arr f"   and "t \<in> S.Dom f" shows "f \<cdot>\<^sub>S S.mkPoint (S.dom f) t = S.mkPoint (S.cod f) (S.Fun f t)"
hs_pending_locs_def: shows "hs_pending_locs = prefixed ''hs_'' - {hs_load_pending, local.hs_pending}"
real_of_float_inject: fixes x :: "Float.float"   and y :: "Float.float" shows "(real_of_float x = real_of_float y) = (x = y)"
rep_eq: fixes x :: "'b trace" shows "\<Gamma> x = (\<lambda>i. fst (Rep_trace x !! i))"
Suc_less_pred_conv: fixes m :: "nat"   and n :: "nat" shows "(Suc m < n) = (m < n - Suc 0)"
merge_coeffs_is_normalize_poly_p: fixes xs :: "(char list list \<times> int) list"   and ys :: "(char list multiset \<times> int) multiset" assumes "(xs, ys) \<in> sorted_repeat_poly_list_rel (rel2p (Id \<union> term_order_rel))" shows "\<exists>r. (merge_coeffs xs, r) \<in> sorted_poly_rel \<and> normalize_poly_p\<^sup>*\<^sup>* ys r"
Disjoint_E: fixes H :: "fm set"   and t :: "tm"   and u :: "tm"   and a :: "tm"   and A :: "fm" assumes "H \<turnstile> Disjoint t u"   and "H \<turnstile> a IN t"   and "H \<turnstile> a IN u" shows "H \<turnstile> A"
prod_mset_multiplicity: fixes times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and M :: "'k multiset" assumes "class.comm_monoid_mult times one" shows "comm_monoid_mult.prod_mset times one M = comm_monoid_mult.prod times one (\<lambda>x. power.power one times x (count M x)) (set_mset M)"
simps: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" shows "gt_chop a1 a2 = (\<exists>t s. a1 = t \<and> a2 = s \<and> is_App t \<and> chop t \<ge>\<^sub>t s)"
mset_subset_eq_add_right: fixes B :: "'c multiset"   and A :: "'c multiset" shows "B \<subseteq># A + B"
inf_llist_def: fixes f :: "nat \<Rightarrow> 'b" shows "inf_llist f = lmap f (iterates Suc 0)"
some: shows "compress_normalize_primitive (disc, sel) C f m = Some m'"
Pareto_iff: fixes x :: "'b pmf"   and y :: "'b pmf" shows "Pareto (SD \<circ> R) x y = (\<forall>i\<in>agents. (SD \<circ> R) i x y)"
trans: fixes P :: "conc_interface"   and Q :: "conc_interface"   and R :: "conc_interface" assumes "P \<simeq> Q"   and "Q \<simeq> R" shows "P \<simeq> R"
simps_2: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and as :: "(String.literal \<times> value list) list" shows "observe_execution e s r ((l, i) # as) = (let viable = possible_steps e s r l i in if viable = {||} then [] else let (s', t) = SOME x. x |\<in>| viable in evaluate_outputs t i r # observe_execution e s' (evaluate_updates t i r) as)"
skip: fixes k :: "nat" assumes "k < N" shows "replicate N SKIP ! k = SKIP"
strict_implies_not_eq: fixes b :: "'a Word.word"   and a :: "'a Word.word" assumes "b <s a" shows "a \<noteq> b"
intros_1: 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 V :: "'d"   and n' :: "'b" assumes "DynPDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit dyn_control_dependence"   and "V \<in> Use n'" shows "(V, [], []) \<in> DynPDG.dependent_live_vars sourcenode targetnode valid_edge Def Use dyn_control_dependence n'"
ord_p_tail: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0"   and "lt p = lt q"   and "p \<prec>\<^sub>p q" shows "tail p \<prec>\<^sub>p tail q"
cases: fixes h :: "real \<Rightarrow> real"   and F :: "real filter"   and a1 :: "real \<Rightarrow> real"   and a2 :: "real msstream" assumes "asymp_powser h F a1 a2" obtains f :: "real \<Rightarrow> real" and cs :: "real msstream" and f' :: "real \<Rightarrow> real" and c :: "real" where "a1 = f'" and "a2 = MSSCons c cs" and "asymp_powser h F f cs" and "f' \<in> O[F](\<lambda>_. 1)" and "\<forall>\<^sub>F x in F. f' x = c + f x * h x"
cod_MkArr: fixes A :: "bool"   and B :: "bool"   and F :: "'arr list" assumes "arr (MkArr A B F)" shows "cod (MkArr A B F) = MkIde B"
cone_axioms: shows "D.cone (G (J_C.MkIde D)) (\<chi>.\<Phi>.Ya.Cop_S.Map (Adj.\<epsilon> (J_C.MkIde D)))"
inner_prod_def: fixes u :: "complex Matrix.vec"   and v :: "complex Matrix.vec" shows "\<langle>u|v\<rangle> \<equiv> \<Sum>i = 0..<dim_vec v. cnj (u $ i) * v $ i"
plus_mono: fixes weights :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and s1 :: "'b"   and s2 :: "'b"   and s3 :: "'b"   and s4 :: "'b" assumes "graph weights"   and "s1 \<le> s2"   and "s3 \<le> s4" shows "s1 + s3 \<le> s2 + s4"
siso0_transC: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "('b, 'c) com"   and s :: "'a"   and c' :: "('b, 'c) com"   and s' :: "'a" assumes "PL_Indis indis"   and "PL_Indis.siso0 tval aval indis c"   and "PL.mustT tval aval c s"   and "PL.transC tval aval (c, s) (c', s')" shows "PL_Indis.siso0 tval aval indis c'"
lub_mono: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'b"   and Y :: "nat \<Rightarrow> 'b" assumes "class.cpo below"   and "po.chain below X"   and "po.chain below Y"   and "\<And>i. below (X i) (Y i)" shows "below (po.lub below (range X)) (po.lub below (range Y))"
CFGExit_wf_axioms: shows "CFGExit_wf src trg knd (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry (lift_get_proc get_proc Main) (lift_get_return_edges get_return_edges valid_edge sourcenode targetnode kind) procs Main NewExit (lift_Def Def Entry Exit H L) (lift_Use Use Entry Exit H L) (lift_ParamDefs ParamDefs) (lift_ParamUses ParamUses)"
r_right_minus_eq: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "(a \<ominus> b = \<zero>) = (a = b)"
cases: fixes a1 :: "'vtx"   and a2 :: "'vtx"   and a3 :: "('vtx \<times> 'lbl \<times> 'vtx) list" assumes "path a1 a2 a3" obtains l1 :: "'vtx" and l2 :: "'vtx" where "a1 = l1a__" and "a2 = l2a__" and "a3 = []" and "l1a__ = l2a__"   | l1 :: "'vtx" and l2 :: "'vtx" and xs :: "('vtx \<times> 'lbl \<times> 'vtx) list" and lbl :: "'lbl" and l3 :: "'vtx" where "a1 = l1" and "a2 = l3" and "a3 = xs @ [(l2, lbl, l3)]" and "path l1 l2 xs" and "lbl \<in>\<^sub>A weights l2 l3"
8: shows "\<forall>x y z. enum_lex_less_eq x y \<and> enum_lex_less_eq y z \<longrightarrow> enum_lex_less_eq x z"
mod_mult_eq_imp_mod_eq: fixes a :: "nat"   and m :: "nat"   and k :: "nat"   and b :: "nat" assumes "a mod (m * k) = b mod (m * k)" shows "a mod m = b mod m"
eWlsAbs_invert: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and us :: "'varSort"   and s :: "'sort"   and eA :: "'d withERR" assumes "eWlsAbs MOD (us, s) eA" shows "\<exists>A. eA = OK A \<and> igWlsAbs MOD (us, s) A"
simps: fixes a1 :: "init_state"   and a2 :: "init_state" shows "(a1 \<le>\<^sub>i a2) = ((\<exists>i. a1 = Prepared \<and> a2 = i) \<or> a1 = Processing \<and> a2 = Done \<or> a1 = Processing \<and> a2 = Error \<or> (\<exists>i. a1 = i \<and> a2 = i))"
minus_uexpr_def: fixes u :: "('a, 'b) uexpr"   and v :: "('a, 'b) uexpr" shows "u - v = bop (-) u v"
sff_1: shows "p = Polynomial.smult c (\<Prod>(a, i)\<in>set qis. a ^ Suc i)"
inverse_zero: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "inverse zero = zero"
induced_map_conservative: fixes A :: "'a set" assumes "A \<in> sets (restrict_space M A)" shows "conservative (restrict_space (restrict_space M A) A) (A.induced_map A)"
gdd_index_zero_iff: fixes ps :: "'a set" assumes "ps \<subseteq> \<V>"   and "card ps = 2" shows "(\<B> index ps = 0) = (size (filter_mset ((\<subseteq>) ps) (mset_set \<G>)) = 1)"
coset_join1: fixes H :: "'a set"   and x :: "'a" assumes "H #> x = H"   and "x \<in> carrier G"   and "subgroup H G" shows "x \<in> H"
strict_implies_not_eq: fixes a :: "'c multiset"   and b :: "'c multiset" assumes "a \<subset># b" shows "a \<noteq> b"
partJoin_def: fixes P :: "'a set set"   and Q :: "'a set set" shows "partJoin P Q \<equiv> partGen (P \<union> Q)"
hyps_1: shows "class P C_ = \<lfloor>(D_, fs_, ms_)\<rfloor>"
domo_widen_subset: fixes S1 :: "'a Abs_State.st option"   and S2 :: "'a Abs_State.st option" shows "domo (S1 \<nabla> S2) \<subseteq> domo S1 \<union> domo S2"
semiring_norm_163: fixes a :: "'a" shows "(0::'a) + a = a"
monom_zero: fixes R :: "('a, 'b) ring_scheme"   and n :: "nat" assumes "UP_ring R" shows "monom (UP R) \<zero>\<^bsub>R\<^esub> n = \<zero>\<^bsub>UP R\<^esub>"
Aexp_subst_restr: fixes x :: "Vars.var"   and S :: "Vars.var set"   and y :: "Vars.var"   and e :: "exp"   and a :: "Arity" assumes "x \<notin> S"   and "y \<notin> S" shows "\<A>\<^bsub>a\<^esub> e[x::=y] f|` S = \<A>\<^bsub>a\<^esub> e f|` S"
lcoset_inj_on: fixes a :: "'b" shows "inj ((+o) a)"
that: fixes R :: "complex \<Rightarrow> complex"   and C :: "real" assumes "\<forall>s. s \<notin> \<real>\<^sub>\<le>\<^sub>0 \<longrightarrow> ln_Gamma s = (s - 1 / 2) * Ln s - s + complex_of_real (ln (2 * pi) / 2) + (\<Sum>k = 1..<m. complex_of_real (bernoulli (k + 1)) / (of_nat (k * (k + 1)) * s ^ k)) - R s"   and "\<forall>s. s \<noteq> 0 \<and> \<bar>arg s\<bar> \<le> \<alpha> \<longrightarrow> cmod (R s) \<le> C / cmod s ^ m" shows "thesis"
coeff_length: fixes p :: "'a list"   and i :: "nat" assumes "length p \<le> i" shows "local.coeff p i = \<zero>"
map_3: fixes f :: "'b \<Rightarrow> 'a"   and x3 :: "'b" shows "map\<^sub>f\<^sub>m f (Atom x3) = Atom (f x3)"
subset_proj_absorb_1_a: fixes vs1 :: "'d set"   and vs2 :: "'d set"   and f :: "('d, 'e) fmap" assumes "vs1 \<subseteq> vs2" shows "fmrestrict_set vs1 (fmrestrict_set vs2 f) = fmrestrict_set vs1 f"
T_vrestr_same_measure_2: fixes A :: "'a set"   and n :: "nat" assumes "A \<in> sets M" shows "Sigma_Algebra.measure M ((T ^^ n) --` A) = Sigma_Algebra.measure M A"
univ_poly_subfield_of_consts: fixes K :: "'a set" assumes "subfield K (L\<lparr>carrier := A\<rparr>)" shows "subfield (Hom.poly_of_const ` K) (K [X]\<^bsub>L\<lparr>carrier := A\<rparr>\<^esub>)"
sel_1: fixes x1 :: "'a"   and x2 :: "'b"   and x3 :: "'c list" shows "state.fun_env (State x1 x2 x3) = x1"
dvd_minus_mod: 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 b :: "'a"   and a :: "'a" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "dvd.dvd times b (minus a (modulo a b))"
seqSubstMatch: fixes a :: "name"   and b :: "name"   and P :: "pi"   and \<sigma> :: "(name \<times> name) list" shows "[a\<frown>b]P[<\<sigma>>] = [seq_subst_name a \<sigma>\<frown>seq_subst_name b \<sigma>](P[<\<sigma>>])"
seqI': fixes f :: "'c"   and a :: "'c"   and b :: "'c"   and g :: "'c"   and c :: "'c" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>C b\<guillemotright>"   and "\<guillemotleft>g : b \<Rightarrow>\<^sub>C c\<guillemotright>" shows "C.seq g f"
App_Betas: fixes s :: "Lambda_Z.term"   and t :: "Lambda_Z.term"   and u :: "Lambda_Z.term"   and v :: "Lambda_Z.term" assumes "s \<rightarrow>\<^sub>\<beta>\<^sup>* t"   and "u \<rightarrow>\<^sub>\<beta>\<^sup>* v" shows "App s u \<rightarrow>\<^sub>\<beta>\<^sup>* App t v"
p_inf_pp_pp: fixes x :: "'a"   and y :: "'a" shows "- (- - x \<sqinter> - - y) = - (x \<sqinter> y)"
Inf_dual_var: fixes X :: "'a set" shows "\<partial> (\<Sqinter> X) = \<Squnion> (\<partial> ` X)"
fun_def: shows "fun \<equiv> map_fun rep_st id fun_rep"
fun_of_rel_def: fixes R :: "('d \<times> 'c) set"   and x :: "'d" shows "fun_of_rel R x \<equiv> SOME y. (x, y) \<in> R"
DisjI1: fixes A :: "fm"   and B :: "fm" shows "A IMP A OR B \<in> boolean_axioms"
AboveS_Field: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" shows "AboveS r A \<subseteq> Field r"
assms_3: shows "token_run x (Suc n) = token_run y (Suc n)"
neq: shows "fst (splitAt ram1 vs) \<noteq> []"
sub_id_iff: shows "(A \<inter> B \<subseteq> {\<one>}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<otimes> y = \<one> \<longrightarrow> x = \<one> \<and> y = \<one>)"
eventually_at_top_linorder: fixes P :: "'c \<Rightarrow> bool" shows "eventually P at_top = (\<exists>N. \<forall>n\<ge>N. P n)"
correct_state_hext_mono: fixes \<Phi> :: "String.literal \<Rightarrow> String.literal \<Rightarrow> (ty list \<times> ty err list) option list"   and t :: "'thread_id"   and xcp :: "'addr option"   and h :: "'heap"   and frs :: "'addr frame list"   and h' :: "'heap" assumes "\<Phi> \<turnstile> t:(xcp, h, frs) \<surd>"   and "h \<unlhd> h'"   and "hconf h'" shows "\<Phi> \<turnstile> t:(xcp, h', frs) \<surd>"
Nlt_def: shows "(<\<^sub>N) = (\<lambda>a b. 0>\<^sub>N (a -\<^sub>N b))"
Max_wrt_among_of_weak_ranking_Cons1: fixes A :: "'alt set" assumes "x \<inter> A = {}" shows "Max_wrt_among (of_weak_ranking (x # xs)) A = Max_wrt_among (of_weak_ranking xs) A"
subgraph_separation_extend: fixes v :: "'a"   and S :: "'a set" assumes "v \<noteq> v0"   and "v \<noteq> v1"   and "v \<in> V"   and "Separation (remove_vertex v) v0 v1 S" shows "Separation G v0 v1 (insert v S)"
conj_cong: fixes P :: "bool"   and P' :: "bool"   and Q :: "bool"   and Q' :: "bool" assumes "P = P'"   and "P' \<Longrightarrow> Q = Q'" shows "(P \<and> Q) = (P' \<and> Q')"
perm2: fixes p :: "perm" assumes "supp p \<sharp>* c" shows "p \<bullet> f (bs \<inter> supp y) y c = f (p \<bullet> (bs \<inter> supp y)) (p \<bullet> y) c"
assms_10: shows "\<forall>\<^sub>F n in sequentially. g n = - sum (residue (\<lambda>z. f z / z ^ Suc n)) S"
runiq_def: fixes R :: "('c \<times> 'd) set" shows "runiq R = (\<forall>X. trivial X \<longrightarrow> trivial (R `` X))"
real_of_int_floor_ge_diff_one: fixes r :: "real" shows "r - 1 \<le> real_of_int \<lfloor>r\<rfloor>"
\<beta>': fixes p :: "fm"   and a :: "real"   and bs :: "real list"   and d :: "int" assumes "iszlfm p (a # bs)"   and "d_\<beta> p 1"   and "d_\<delta> p d"   and "0 < d" shows "\<forall>x. \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>set (\<beta> p). Ifm ((Inum (a # bs) b + real_of_int j) # bs) p) \<longrightarrow> Ifm (real_of_int x # bs) p \<longrightarrow> Ifm (real_of_int (x - d) # bs) p"
rel_optionTE: fixes M :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and m :: "('i, 'g) optionT"   and m' :: "('j, 'h) optionT" assumes "rel_optionT M m m'" obtains x :: "'g" and y :: "'h" where "m = OptionT x" and "m' = OptionT y" and "M x y"
k1_def: shows "k1 \<equiv> Suc p - nat (bitlen \<bar>m1\<bar>)"
r_pdec12_def: shows "r_pdec12 \<equiv> Cn 1 r_pdec1 [r_pdec2]"
array_rules_2: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c"   and y :: "'b"   and z :: "'c" shows "(f(x := y)) z = (if z = x then y else f z)"
lcm_abs_mult_right_int: fixes x :: "int"   and a :: "int"   and y :: "int" shows "lcm x (\<bar>a\<bar> * y) = lcm x (a * y)"
hom_zero: fixes a :: "'a" shows "poly 0 a = (0::'a)"
limit_subset: fixes f :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "limit f \<subseteq> f ` {n..}"
assms_1: shows "analytically_valid (cubeImage diamond_cube) (\<lambda>x. F x \<bullet> i) j"
pt1: fixes x :: "'a" assumes "pt TYPE('a) TYPE('b)" shows "[] \<bullet> x = x"
assms_1: shows "\<Gamma>(chain \<mapsto> rs),\<gamma>,p\<turnstile> \<langle>[Rule m a], Undecided\<rangle> \<Rightarrow> Undecided"
eq_imp_diff_eq: fixes m :: "nat"   and n :: "nat"   and k :: "nat" assumes "m = n" shows "m - k = n - k"
simps_9: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" shows "\<lbrace>\<^bold>\<a>\<^bold>[t, u, v\<^bold>]\<rbrace> = \<a>[\<lbrace>t\<rbrace>, \<lbrace>u\<rbrace>, \<lbrace>v\<rbrace>]"
Fvars_fmla_simps_5: shows "Fvars fls = {}"
OrI1: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula" assumes "\<Gamma> \<turnstile> F" shows "\<Gamma> \<turnstile> F \<^bold>\<or> G"
trace_parallel_plan_strips_head_is_initial_state: fixes I :: "'a \<Rightarrow> bool option"   and \<pi> :: "'a strips_operator list list" shows "trace_parallel_plan_strips I \<pi> ! 0 = I"
map_disjI: fixes h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option" assumes "dom h\<^sub>0 \<inter> dom h\<^sub>1 = {}" shows "h\<^sub>0 \<bottom> h\<^sub>1"
is_full_functor: shows "full_functor (\<cdot>) Cop_S.comp local.map"
IH: shows "card {s. fmdom' s = F} = 2 ^ card F"
order_iff_strict: fixes b :: "'b list"   and a :: "'b list" shows "prefix b a = (strict_prefix b a \<or> a = b)"
red_def: shows "red \<equiv> {(xa, x). redp xa x}"
mod_star_trueI: fixes P :: "assn"   and h :: "heap \<times> nat set" assumes "h \<Turnstile> P" shows "h \<Turnstile> P * true"
map_of_map_keys: fixes xs :: "'c list"   and m :: "'c \<Rightarrow> 'd option" assumes "set xs = dom m" shows "map_of (map (\<lambda>k. (k, the (m k))) xs) = m"
RPD_fv_regex_le: fixes ms :: "mregex"   and mr :: "mregex"   and \<phi>s :: "Formula.formula list" assumes "ms \<in> RPD mr" shows "fv_regex (from_mregex ms \<phi>s) \<subseteq> fv_regex (from_mregex mr \<phi>s)"
ring_hom_memE_4: fixes h :: "'c \<Rightarrow> 'd"   and R :: "('c, 'e) ring_scheme"   and S :: "('d, 'f) ring_scheme" assumes "h \<in> ring_hom R S" shows "h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>"
axioms_1: fixes M :: "'c measure" assumes "prob_space M" shows "finite_measure M"
\<P>_induct: fixes p :: "(symbol \<times> character list) list"   and k :: "nat"   and u :: "nat"   and P :: "(symbol \<times> character list) list \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" assumes "p \<in> \<P> k u"   and "\<And>p k u. \<lbrakk>\<And>p' k' u'. \<lbrakk>p' \<in> \<P> k' u'; indexlt k' u' k u\<rbrakk> \<Longrightarrow> P p' k' u'; p \<in> \<P> k u\<rbrakk> \<Longrightarrow> P p k u" shows "P p k u"
encR: fixes Enc :: "'a \<Rightarrow> 'b"   and S :: "'a" shows "encoding.indRelRp Enc (SourceTerm S) (TargetTerm (Enc S))"
shared_eq: shows "\<forall>a\<in>all_acquired sb. \<S>' a = \<S> a"
x_def: shows "x = r\<lparr>i\<rparr>"
compact_rel_boundary: fixes S :: "'c set" assumes "compact S" shows "compact (S - rel_interior S)"
vars_mset_subst_db: fixes lam :: "'b"   and db :: "nat \<Rightarrow> 'b"   and i :: "nat"   and x :: "'a"   and s :: "('b, 'a) tm" shows "vars_mset (lambda_encoding.subst_db lam db i x s) = {#y \<in># vars_mset s. y \<noteq> x#}"
empty_sem_env_def: fixes C_info :: "(name, dt_def) fmap" shows "Compiler.empty_sem_env C_info = make_sem_env nsEmpty (constructors.as_static_cenv C_info)"