hcju/isabelleps huggingface dataset:
mertens_second_theorem_1: fixes x :: "real" assumes "2 \<le> x" shows "\<bar>prime_sum_upto (\<lambda>p. 1 / real p) x - ln (ln x) - meissel_mertens\<bar> \<le> 4 / ln x"
is_root_del_min: fixes h :: "'b tree" assumes "is_root h" shows "is_root (del_min h)"
filter2_sound: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and as :: "'c list"   and bs :: "'d list" shows "\<forall>i<length (fst (filter2 P as bs)). P (fst (filter2 P as bs) ! i) (snd (filter2 P as bs) ! i)"
h: fixes z :: "complex" assumes "z \<in> cball 0 1" shows "2 * f z - 1 = cos (complex_of_real pi * h z)"
Sup_fun_def: fixes A :: "('b \<Rightarrow> 'c) set" shows "Sup A = (\<lambda>x. SUP f\<in>A. f x)"
simps: fixes P :: "'a prog"   and a1 :: "String.literal"   and a2 :: "((String.literal \<times> String.literal) \<times> ty \<times> fmod) list" shows "(P \<turnstile> a1 has_fields a2) = ((\<exists>C D fs ms FDTs FDTs'. a1 = C \<and> a2 = FDTs' \<and> class P C = \<lfloor>(D, fs, ms)\<rfloor> \<and> C \<noteq> Object \<and> P \<turnstile> D has_fields FDTs \<and> FDTs' = map (\<lambda>(F, Tm). ((F, C), Tm)) fs @ FDTs) \<or> (\<exists>D fs ms FDTs. a1 = Object \<and> a2 = FDTs \<and> class P Object = \<lfloor>(D, fs, ms)\<rfloor> \<and> FDTs = map (\<lambda>(F, T). ((F, Object), T)) fs))"
sorted_filter: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq (map f xs)" shows "linorder.sorted less_eq (map f (filter P xs))"
fst_dominant_term_ms_aux_MSLCons: fixes x :: "'a \<times> real"   and xs :: "('a \<times> real) msllist" shows "fst (dominant_term_ms_aux (MSLCons x xs)) = fst (dominant_term (fst x))"
sum_prime_inverses_fraction_coprime: fixes A :: "'a set"   and f :: "'a \<Rightarrow> nat" assumes "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> prime (f x)"   and "inj_on f A" shows "coprime (\<Sum>x\<in>A. prod f (A - {x})) (prod f A)"
enc_and_TRelimpl_indRelRTPO_reflect_divergence: fixes TRel :: "('procT \<times> 'procT) set" assumes "enc_reflects_divergence"   and "rel_reflects_divergence TRel Target" shows "rel_reflects_divergence (indRelRTPO TRel) (STCal Source Target)"
densV: fixes x :: "'a" assumes "x \<in> G" shows "upper_asymptotic_density (V x) \<le> 2 / d * real_cond_exp M Invariants F2 x"
comm_monoid_mult_ow: shows "class.comm_monoid_mult = comm_monoid_mult_ow UNIV"
ring_iso_memE_3: fixes h :: "'c \<Rightarrow> 'd"   and R :: "('c, 'e) ring_scheme"   and S :: "('d, 'f) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "h \<in> ring_iso R S"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "h (x \<oplus>\<^bsub>R\<^esub> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
leq_trans: fixes x :: "'c llist"   and y :: "'c llist"   and z :: "'c llist" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> z" shows "x \<sqsubseteq> z"
get_shadow_root_reads: fixes element_ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" shows "reads (get_shadow_root_locs element_ptr) (get_shadow_root element_ptr) h h'"
f1'_le_f2': fixes f1 :: "real"   and f2 :: "real"   and f1' :: "real"   and f2' :: "real"   and e :: "real"   and c :: "real" assumes "Table0 f1 f2 f1' f2' e c" shows "f1' \<le> f2'"
bij_betw_subset: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and A' :: "'c set"   and B :: "'b set"   and B' :: "'c set" assumes "bij_betw f A A'"   and "B \<subseteq> A"   and "f ` B = B'" shows "bij_betw f B B'"
cols_conjugate: fixes m :: "'b mat" shows "cols (conjugate m) = map conjugate (cols m)"
line_integral_sum_basis_2: fixes basis :: "'a set"   and F :: "'a \<Rightarrow> 'a"   and \<gamma> :: "real \<Rightarrow> 'a" assumes "finite basis"   and "\<forall>b\<in>basis. line_integral_exists F {b} \<gamma>" shows "line_integral_exists F basis \<gamma>"
vacant_upto_def: fixes a :: "'c zmultiset"   and t :: "'c" shows "vacant_upto a t = (\<forall>s\<le>t. zcount a s = 0)"
ipv6_unparsed_compressed_to_preferred_identity1: fixes ipv6compressed :: "ipv6addr_syntax_compressed"   and ipv6prferred :: "ipv6addr_syntax" shows "(ipv6_unparsed_compressed_to_preferred (ipv6addr_syntax_compressed_to_list ipv6compressed) = Some ipv6prferred) = (ipv6addr_c2p ipv6compressed = ipv6prferred)"
rel_pred2_neg_distr_cond_def: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_pred2_neg_distr_cond A B = (rel_pred (rel_pred (A OO B)) \<le> rel_pred (rel_pred A) OO rel_pred (rel_pred B))"
assms: fixes x :: "'a"   and y :: "'a" shows "P x y \<longrightarrow> Q x y"
X_\<mu>: shows "X \<subseteq> subformulas\<^sub>\<mu> \<phi>"
trace_and_start_determines_end: fixes c :: "('p, 's, 'm) configuration"   and t :: "('p, 's, 'm) event list"   and c' :: "('p, 's, 'm) configuration"   and d' :: "('p, 's, 'm) configuration" assumes "trace c t c'"   and "trace c t d'" shows "c' = d'"
keys_hom_componentD_1: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and n :: "nat" assumes "t \<in> keys (hom_component p n)" shows "t \<in> keys p"
random_perm_def: shows "random_perm \<equiv> \<lambda>xs. uniform_measure (count_space (permutations_of_set (set xs))) (permutations_of_set (set xs))"
compact_attains_sup: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set" assumes "class.linorder_topology less_eq less open"   and "topological_space.compact open S"   and "S \<noteq> {}" shows "\<exists>s\<in>S. \<forall>t\<in>S. less_eq t s"
permBottom: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and p :: "(name \<times> name) list" assumes "assertionAux SCompose SImp SBottom SChanEq" shows "p \<bullet> SBottom = SBottom"
padic_is_abelian_group: fixes p :: "int" assumes "prime p" shows "abelian_group (padic_int p)"
path_defs_4: fixes g :: "real \<Rightarrow> 'a" shows "path_image g = g ` {0..1}"
hermitean_mat_cnj: fixes H :: "complex \<times> complex \<times> complex \<times> complex" shows "hermitean H = hermitean (mat_cnj H)"
lim_stream_def: fixes x :: "real \<times> 'a" shows "K.lim_stream x = distr (K.lim_sequence x) (stream_space S) to_stream"
get_exists_matching_src_ips_executable: fixes iface :: "iface"   and m :: "'b common_primitive match_expr" shows "wordinterval_to_set (get_exists_matching_src_ips_executable iface m) = get_exists_matching_src_ips iface m"
trm_simps_2: fixes a :: "'a" assumes "ide a" shows "local.dom \<t>[a] = a"
closed_subset: fixes S :: "'a set"   and T :: "'a set" assumes "S \<subseteq> T"   and "closed S" shows "closedin (top_of_set T) S"
soldAllocationRestriction: fixes a :: "(integer \<times> 'c set) set"   and \<Omega> :: "'c set"   and N :: "integer set"   and n :: "integer" assumes "a \<in> soldAllocations N \<Omega>" shows "Outside' {n} a \<in> soldAllocations (N - {n}) \<Omega>"
finite_integrable_vp: fixes pf :: "'c \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and Mkt :: "('a, 'c) discrete_market"   and N :: "'a measure" assumes "\<forall>n. \<forall>asset\<in>support_set pf. finite (prices Mkt asset n ` space M)"   and "\<forall>n. \<forall>asset\<in>support_set pf. finite (pf asset n ` space M)"   and "prob_space N"   and "filt_equiv F M N"   and "trading_strategy pf"   and "\<forall>n. \<forall>asset\<in>support_set pf. random_variable borel (prices Mkt asset n)" shows "\<forall>n. \<forall>asset\<in>support_set pf. integrable N (\<lambda>w. prices Mkt asset n w * pf asset (Suc n) w)"
pr_rec: fixes g :: "nat \<Rightarrow> nat"   and h :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat" assumes "g \<in> PrimRec1"   and "h \<in> PrimRec3" shows "PrimRecOp g h \<in> PrimRec2"
assms: shows "\<forall>F\<in>S. F \<subseteq> X"
mapping_comparator: assumes "ID ccompare \<noteq> None" shows "comparator ccomp"
uminus_finite_2_def: shows "uminus = (\<lambda>x. x)"
nonempty_subgraphs_def: fixes G :: "nat set \<times> nat set set" shows "nonempty_subgraphs G = {G'. uwellformed G' \<and> subgraph G' G \<and> nonempty_graph G'}"
dvd_mult_cancel1: fixes m :: "nat"   and n :: "nat" assumes "0 < m" shows "(m * n dvd m) = (n = 1)"
ord_spmf_eqD_measure_spmf: fixes p :: "'a spmf"   and q :: "'a spmf" assumes "ord_spmf (=) p q" shows "measure_spmf p \<le> measure_spmf q"
intros_4: fixes P :: "('a \<times> real) msllist \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and ys :: "('a \<times> real) msllist"   and g :: "real \<Rightarrow> real" assumes "ms_closure P xs f basis"   and "ms_closure P ys g basis" shows "ms_closure P (plus_ms_aux xs ys) (\<lambda>x. f x + g x) basis"
may_lock_iff: fixes l :: "('b \<times> nat) option"   and t :: "'b" shows "may_lock l t = (case l of None \<Rightarrow> True | \<lfloor>(t', n)\<rfloor> \<Rightarrow> t = t')"
rel_coinduct: fixes P :: "('n, 'o, 'p, 'q, 'r, 's) U \<Rightarrow> ('t, 'o, 'p, 'q, 'r, 's) U \<Rightarrow> bool"   and x :: "('n, 'o, 'p, 'q, 'r, 's) U"   and y :: "('t, 'o, 'p, 'q, 'r, 's) U"   and R1 :: "'n \<Rightarrow> 't \<Rightarrow> bool" assumes "P x y"   and "\<And>U U'. P U U' \<Longrightarrow> rell_G P R1 (D_U U) (D_U U')" shows "rell_U R1 x y"
comp_cod_arr: fixes f :: "'a arrow_of_spans_data"   and b :: "'a arrow_of_spans_data" assumes "V.arr f"   and "V.cod f = b" shows "b \<bullet> f = f"
veval_welldefined: fixes \<Gamma> :: "(name, value) fmap"   and t :: "sterm"   and v :: "value" assumes "rs, \<Gamma> \<turnstile>\<^sub>v t \<down> v"   and "fmpred (\<lambda>_. vwelldefined) \<Gamma>"   and "consts t |\<subseteq>| all_consts" shows "vwelldefined v"
mem_val_w32_10_unchanged: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and a :: "word36"   and addr :: "word36"   and val :: "word8" assumes "mem_equal s1 s2 a" shows "mem_val_w32 10 a (s1\<lparr>mem := (mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))\<rparr>) = mem_val_w32 10 a (s2\<lparr>mem := (mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))\<rparr>)"
tag_name_eq: fixes ptr' :: "(_) element_ptr"   and tag :: "char list" shows "h \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r tag = h' \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r tag"
f0: shows "gate 3 (Quantum.Id 2 \<Otimes> Z)"
ncg_sym: fixes x :: "'a"   and y :: "'a" assumes "ncg x y" shows "ncg y x"
index_extract_matrix: fixes i :: "nat"   and m :: "nat"   and j :: "nat"   and n :: "nat"   and a :: "nat \<Rightarrow> 'a" assumes "i < m"   and "j < n" shows "extract_matrix a m n $$ (i, j) = a (i * n + j)"
dg_Hom_vifunion_in_Vset: fixes A :: "V"   and B :: "V" assumes "A \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"   and "B \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"   and "A \<in>\<^sub>\<circ> Vset \<alpha>"   and "B \<in>\<^sub>\<circ> Vset \<alpha>" shows "\<Union>\<^sub>\<circ> ((\<lambda>a\<in>\<^sub>\<circ>A. \<Union>\<^sub>\<circ> (VLambda B (Hom \<CC> a) `\<^sub>\<circ> B)) `\<^sub>\<circ> A) \<in>\<^sub>\<circ> Vset \<alpha>"
eq0: shows "fst ` set (fst aux) - {0} = fst ` set (fst aux)"
distinct_1: fixes name1 :: "name"   and name2 :: "name"   and pi :: "pi"   and freeRes' :: "freeRes"   and pi' :: "pi" shows "name1<\<nu>name2> \<prec> pi \<noteq> freeRes' \<prec> pi'"
intro: fixes h' :: "'b \<Rightarrow> 'b"   and b :: "'b" assumes "h' b = b" shows "recursive h' b"
parts_crypt: fixes K :: "key"   and X :: "msg"   and H :: "msg set" assumes "Crypt K X \<in> parts H" shows "X \<in> parts H"
norm_ket_k_lt_K: fixes k :: "nat" assumes "k < K" shows "inner_prod (ket_k k) (ket_k k) = 1"
collapse_1: fixes sum :: "'b + 'c" assumes "isl sum" shows "Inl (projl sum) = sum"
mult_inv_character: fixes k :: "nat" shows "dc2c \<chi>\<^sub>0 k * inv_character (dc2c \<chi>\<^sub>0) k = principal_char G k"
mult_left_sub_dist_sup_left: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<le> x \<sqinter> (y \<squnion> z)"
hyps: shows "map (map f) (subseqs xs) = subseqs (map f xs)"
simps_6: fixes bs :: "real list"   and a :: "MIR.num"   and b :: "MIR.num" shows "Inum bs (Sub a b) = Inum bs a - Inum bs b"
assms: shows "\<forall>i\<le>n. f i \<le> 1"
dvd_trans: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a dvd b"   and "b dvd c" shows "a dvd c"
set_zip_leftI: fixes xs :: "'a list"   and ys :: "'b list"   and y :: "'b" assumes "length xs = length ys"   and "y \<in> set ys" obtains x :: "'a" where "(x, y) \<in> set (zip xs ys)"
A_def: shows "A = (\<lambda>i. {x \<in> space M. \<epsilon> < \<bar>X i x - Y x\<bar>})"
g: shows "continuous_map (subtopology X (X closure_of {x. \<not> P x})) Y g"
listsetD_1: fixes ys :: "'c list"   and xs :: "'c set list" assumes "ys \<in> listset xs" shows "length ys = length xs"
create_opt_sol_sat_eq_mat: fixes c :: "rat vec"   and A :: "rat mat"   and b :: "rat vec"   and X :: "(nat, rat) mapping"   and x :: "rat vec"   and y :: "rat vec" assumes "dim_vec c = dim_row A\<^sup>T"   and "dim_vec b = dim_col A\<^sup>T"   and "create_optimal_solutions A b c = Inr X"   and "(x, y) = split_n_m_x (dim_vec c) (dim_vec b) X" shows "[y\<^sub>v*A]=c"
minus_set_fold: fixes A :: "'a set"   and xs :: "'a list" shows "A - set xs = fold Set.remove xs A"
min_lvar_not_in_bounds_Some: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat" shows "min_lvar_not_in_bounds s = Some x\<^sub>i \<longrightarrow> \<not> in_bounds x\<^sub>i \<langle>\<V> s\<rangle> (\<B> s)"
eval_fds_log_deriv: fixes f :: "'a fds"   and s :: "'a" assumes "fds_nth f 1 \<noteq> (0::'a)"   and "abs_conv_abscissa f < ereal (s \<bullet> (1::'a))"   and "abs_conv_abscissa (fds_deriv f / f) < ereal (s \<bullet> (1::'a))"   and "eval_fds f s \<noteq> (0::'a)" shows "eval_fds (fds_deriv f / f) s = eval_fds (fds_deriv f) s / eval_fds f s"
chain_mono: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and C :: "'a set" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; P x y\<rbrakk> \<Longrightarrow> Q x y"   and "pred_on.chain A P C" shows "pred_on.chain A Q C"
permutation_bijective: fixes p :: "'a \<Rightarrow> 'a" assumes "permutation p" shows "bij p"
compute_monomial: fixes t :: "'a"   and c :: "'b" shows "monomial c t = (if c = (0::'b) then 0 else sparse\<^sub>0 [(t, c)])"
finite_UnI: fixes F :: "'g set"   and G :: "'g set" assumes "finite F"   and "finite G" shows "finite (F \<union> G)"
supteq_antisym: fixes s :: "('d, 'e) Term.term"   and t :: "('d, 'e) Term.term" assumes "s \<unrhd> t"   and "t \<unrhd> s" shows "s = t"
UP_root_div_def: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a" shows "UP_root_div f a = Cring_Poly.compose R (poly_shift (T\<^bsub>a\<^esub> f)) (X_poly_minus R a)"
take_nths_take_nths: fixes i :: "nat"   and m :: "nat"   and j :: "nat"   and n :: "nat"   and xs :: "'b list" shows "take_nths i m (take_nths j n xs) = take_nths (i * 2 ^ n + j) (m + n) xs"
simps_30: fixes v :: "'a rexp"   and va :: "'a rexp"   and vk :: "'a rexp"   and vl :: "'a rexp" shows "local.le_rexp (Times v va) (Plus vk vl) = False"
bij_betw_Diff: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and A' :: "'a set"   and B' :: "'b set" assumes "bij_betw f A B"   and "bij_betw f A' B'"   and "A' \<subseteq> A"   and "B' \<subseteq> B" shows "bij_betw f (A - A') (B - B')"
vs2: shows "vs = fst (splitAt ram1 vs) @ [ram1] @ r"
integrable_pbernpoly_1: fixes s :: "complex"   and N :: "nat" assumes "s \<notin> \<real>\<^sub>\<le>\<^sub>0"   and "0 < N" shows "(\<lambda>x. complex_of_real (- pbernpoly 1 x) / (complex_of_real x + s)) integrable_on {0..real N}"
leg1_composite_2: shows "HH\<mu>\<nu>H\<pi>\<rho>.dom.leg1 = \<mu>.leg1 \<cdot> \<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>1\<^sub>1"
fcd: fixes x :: "complex" assumes "x \<in> S - k" shows "f field_differentiable at x"
tauStepChainFreshChain: fixes P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and xvec :: "name list" assumes "\<Psi> \<rhd> P \<Longrightarrow>\<^sub>\<tau> P'"   and "xvec \<sharp>* P" shows "xvec \<sharp>* P'"
cring_sub_monom: fixes a :: "'a"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "a \<noteq> \<zero>"   and "f = monom P a n"   and "g \<in> carrier P"   and "a \<otimes> g (deg R g) [^] n \<noteq> \<zero>" shows "deg R (Cring_Poly.compose R f g) = n * deg R g"
gf: fixes x :: "'a" assumes "x \<in> span ((+) (- a) ` S)" shows "g (f x) = x"
trg_hcomp: fixes \<mu> :: "'b"   and \<nu> :: "'b" assumes "D.hseq \<mu> \<nu>" shows "trg\<^sub>D (\<mu> \<star>\<^sub>D \<nu>) = trg\<^sub>D \<mu>"
dims1_def: shows "ps2_P.dims1 = nths dims vars1"
tensor_mat_positive_le: fixes m1 :: "complex mat"   and dims :: "nat list"   and vars :: "nat set"   and m2 :: "complex mat"   and A :: "complex mat"   and B :: "complex mat" assumes "m1 \<in> carrier_mat (partial_state.d1 dims vars) (partial_state.d1 dims vars)"   and "m2 \<in> carrier_mat (partial_state.d2 dims vars) (partial_state.d2 dims vars)"   and "positive m1"   and "positive m2"   and "m1 \<le>\<^sub>L A"   and "m2 \<le>\<^sub>L B" shows "tensor_mat dims vars m1 m2 \<le>\<^sub>L tensor_mat dims vars A B"
LT5: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> \<box>(F \<longrightarrow> \<diamond>G) \<longrightarrow> \<diamond>F \<longrightarrow> \<diamond>G"
slice1_call_slice1: fixes nx :: "'node SDG_node"   and n :: "'node SDG_node"   and p :: "'pname"   and n' :: "'node SDG_node" assumes "nx \<in> sum_SDG_slice1 n"   and "n s-p\<rightarrow>\<^bsub>call\<^esub> n'" shows "nx \<in> sum_SDG_slice1 n'"
pair_bidirected_digraph_subdivide: fixes e :: "'a \<times> 'a"   and w :: "'a" assumes "e \<in> parcs H"   and "w \<notin> pverts H" shows "pair_bidirected_digraph (subdivide H e w)"
k_def: shows "k = \<lfloor>x / (2 * pi)\<rfloor>"
T_size_tree_def: fixes t :: "'b tree" shows "T_size_tree t = time (size_tree_tm t)"
finite_enumerate_in_set: fixes S :: "'a set"   and n :: "nat" assumes "finite S"   and "n < card S" shows "enum S n \<in> S"
simps_2: fixes u :: "'a"   and e :: "'b"   and es :: "'b list" shows "awalk_verts u (e # es) = tail G e # awalk_verts (head G e) es"
assms_5: shows "(\<xi>, p) \<in> reachable A I'"
cond_spmf_spmf_of_set: fixes A :: "'a set"   and B :: "'a set" assumes "finite A" shows "cond_spmf (spmf_of_set A) B = spmf_of_set (A \<inter> B)"
list_decomp_1: fixes l :: "'b list" assumes "length l = 1" shows "\<exists>a. l = [a]"
fmadd_restrict_right_dom: fixes n :: "'c \<rightharpoonup>\<^sub>f 'd"   and m :: "'c \<rightharpoonup>\<^sub>f 'd" shows "fmrestrict_fset (fmdom n) (m ++\<^sub>f n) = n"
sup_impr_inf: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<squnion>1 b r\<rightarrow> c = (a r\<rightarrow> c) \<sqinter> (b r\<rightarrow> c)"
refl: fixes a :: "'f multiset" shows "a \<subseteq># a"
A_trusts_YM3: fixes A :: "agent"   and B :: "agent"   and K :: "nat"   and na :: "msg"   and evs :: "event list" assumes "Crypt (shrK A) \<lbrace>Agent B, Key K, na\<rbrace> \<in> parts (knows Spy evs)"   and "A \<notin> bad"   and "evs \<in> yahalom" shows "\<exists>nb. Says Server A \<lbrace>nb, Crypt (shrK A) \<lbrace>Agent B, Key K, na\<rbrace>, Crypt (shrK B) \<lbrace>Agent A, Agent B, Key K, nb\<rbrace>\<rbrace> \<in> set evs"
MarkGuardsI_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and f :: "'f" assumes "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/{}\<^esub> P c Q,A" shows "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/{}\<^esub> P mark_guards f c Q,A"
assm: shows "consec j k \<and> consec i (j \<squnion> k)"
residue_add_closed: fixes k :: "nat"   and x :: "int"   and y :: "int" shows "x \<oplus>\<^bsub>residue_ring (p ^ k)\<^esub> y \<in> carrier (residue_ring (p ^ k))"
lexordp_simps_1: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ys :: "'a list" shows "ord.lexordp less [] ys = (ys \<noteq> [])"
op_reachable_def: fixes E :: "('b \<times> 'b) set"   and u :: "'b"   and v :: "'b" shows "op_reachable E u v \<equiv> (u, v) \<in> E\<^sup>*"
freeword_funlift_0: fixes f :: "'c \<Rightarrow> 'b" shows "freeword_funlift f 0 = (0::'b)"
UN_iff: fixes b :: "'e"   and B :: "'f \<Rightarrow> 'e set"   and A :: "'f set" shows "(b \<in> \<Union> (B ` A)) = (\<exists>x\<in>A. b \<in> B x)"
less_list_def': fixes xs :: "'b list"   and ys :: "'b list" shows "(xs < ys) = strict_prefix xs ys"
communicating_def: shows "K1.communicating = K1.acc \<inter> K1.acc\<inverse>"
put_nondet_def: fixes put :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and s :: "'f"   and m :: "('d, 'e) nondetT" shows "nondetM_base.put_nondet put s m = NondetT (put s (run_nondet m))"
preserves_ide: fixes a :: "'b" assumes "Adj.D.ide a" shows "Adj.C.ide (F a)"
ord0_Lim_le: fixes f :: "nat \<Rightarrow> ord0"   and g :: "nat \<Rightarrow> ord0" assumes "\<forall>n. f n \<le> g n" shows "ord0_Lim f \<le> ord0_Lim g"
CHAR_eq0_iff: shows "(CHAR = 0) = (\<forall>n>0. of_nat n \<noteq> (0::'a))"
single_eq_single: fixes a :: "'k"   and b :: "'k" shows "({#a#} = {#b#}) = (a = b)"
top: fixes c :: "alt"   and i :: "indi" assumes "b \<noteq> c" shows "c <\<^bsub>Top {a, b} (P i)\<^esub> b"
isLub_in_lattice: fixes cl :: "'b potype"   and S :: "'b set"   and L :: "'b" assumes "CL cl"   and "isLub S cl L" shows "L \<in> pset cl"
asymp_equiv_refl_ev: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "\<forall>\<^sub>F x in F. f x = g x" shows "f \<sim>[F] g"
simplify_and_membership: fixes S :: "'at Literal set set"   and T :: "'at Literal set set"   and C :: "'at Literal set" assumes "all_fulfill finite S"   and "T = simplify S"   and "C \<in> S" shows "redundant C T"
fact_gt_zero: fixes n :: "nat" shows "(0::'a) < fact n"
directional_derivative_add: fixes k :: "enat"   and a :: "'c"   and x :: "'c"   and y :: "'c" assumes "k \<noteq> 0" shows "directional_derivative k a (x + y) = directional_derivative k a x + directional_derivative k a y"
pred_count_space_const1: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and c :: "'d" assumes "f \<in> M \<rightarrow>\<^sub>M count_space UNIV" shows "pred M (\<lambda>x. f x = c)"
quot_atom_subset: fixes c :: "'a" shows "UNIV // \<approx>{[c]} \<subseteq> {{[]}, {[c]}, UNIV - {[], [c]}}"
ss: shows "set (s_TS init h \<sigma> (length (as @ [x] @ bs @ [x]) + i + 1)) = set init"
nat_pow_0: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a" shows "x [^]\<^bsub>G\<^esub> 0 = \<one>\<^bsub>G\<^esub>"
in_bd: fixes A1 :: "'d set"   and A2 :: "'e set"   and A3 :: "'f set" shows "|F2in A1 A2 A3| \<le>o ((|A1| +c |A2| +c |A3|) +c ctwo) ^c F2bd'"
max_topo_def: fixes M :: "'b SecurityInvariant_configured list"   and G :: "'b graph" shows "max_topo M G \<equiv> all_security_requirements_fulfilled M G \<and> (\<forall>(v1, v2)\<in>nodes G \<times> nodes G - edges G. \<not> all_security_requirements_fulfilled M (add_edge v1 v2 G))"
hom_0_iff: fixes x :: "'a" shows "(hom x = (0::'b)) = (x = (0::'a))"
f_def: shows "f = (\<lambda>k x. if x \<in> {c..real k} then exp (- a * x) else 0)"
cod_tensor: fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a"   and d :: "'a" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>g : c \<rightarrow> d\<guillemotright>" shows "cod (MC.tensor f g) = MC.tensor b d"
POWSER'_def: shows "POWSER' \<equiv> \<lambda>f z. \<Sum>n. diffs f n * z ^ n"
split: fixes P :: "'f \<Rightarrow> bool"   and f1 :: "'f"   and f2 :: "'g \<Rightarrow> 'f"   and option :: "'g option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = ((option = None \<longrightarrow> P f1) \<and> (\<forall>x2. option = Some x2 \<longrightarrow> P (f2 x2)))"
assms_1: shows "f \<in> S \<rightarrow>\<^sub>E T - {a}"