hcju/isabelleps huggingface dataset:
red_lA_incr: fixes e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "P \<turnstile> \<langle>e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>e',(h', l', sh'),b'\<rangle>" shows "\<lfloor>dom l\<rfloor> \<squnion> \<A> e \<sqsubseteq> \<lfloor>dom l'\<rfloor> \<squnion> \<A> e'"
that: fixes n :: "'n"   and interchange_A :: "(('a, 'b) vec, 'n) vec"   and A' :: "(('a, 'b) vec, 'n) vec" assumes "(LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n) = n"   and "A $ n $ j \<noteq> (0::'a)"   and "i \<le> n"   and "interchange_A = interchange_rows A i n"   and "A' = mult_row interchange_A i ((1::'a) / interchange_A $ i $ j)"   and "Gauss_Jordan_in_ij A i j = (\<chi>s. if s = i then A' $ s else row_add A' s i (- interchange_A $ s $ j) $ s)" shows "thesis"
hta_union_def: fixes H1 :: "('a, 'c) hashedTa"   and H2 :: "('b, 'c) hashedTa" shows "hta_union H1 H2 \<equiv> init_hta (hs.union (hh_set_xy.g_image USW1 (hta_Qi H1)) (hh_set_xy.g_image USW2 (hta_Qi H2))) (ls.union_dj (ll_set_xy.g_image (remap_rule USW1) (hta_\<delta> H1)) (ll_set_xy.g_image (remap_rule USW2) (hta_\<delta> H2)))"
axb: shows "a \<le> x \<and> x \<le> b \<and> P x"
dagger_of_prod: fixes M :: "complex mat"   and N :: "complex mat" assumes "dim_col M = dim_row N" shows "M * N\<^sup>\<dagger> = N\<^sup>\<dagger> * M\<^sup>\<dagger>"
mHom_test: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g" assumes "R module N"   and "f \<in> carrier M \<rightarrow> carrier N \<and> f \<in> extensional (carrier M) \<and> (\<forall>m\<in>carrier M. \<forall>n\<in>carrier M. f (m \<plusminus> n) = f m \<plusminus>\<^bsub>N\<^esub> f n) \<and> (\<forall>a\<in>carrier R. \<forall>m\<in>carrier M. f (a \<cdot>\<^sub>s m) = a \<cdot>\<^sub>s\<^bsub>N\<^esub> f m)" shows "f \<in> mHom R M N"
nonneg: fixes x :: "nat \<Rightarrow> 'a" assumes "x \<in> space (Pi\<^sub>M {0..<n + m} M)" shows "0 \<le> f x"
assms_3: shows "S' = S + (\<Sum>n\<in>A. f n - g n)"
0: shows "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2 \<and> w\<^sub>2 \<preceq>\<^sub>F w\<^sub>1"
T_eq_T': fixes K :: "'c \<Rightarrow> 'c pmf"   and s :: "'c" shows "MC_syntax.T K s = MC_syntax.T' K (K s)"
Fvars_neg: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla" shows "Fvars (neg \<phi>) = Fvars \<phi>"
borel_measurableI: fixes f :: "'b \<Rightarrow> 'a"   and M :: "'b measure" assumes "\<And>S. open S \<Longrightarrow> f -` S \<inter> space M \<in> sets M" shows "f \<in> borel_measurable M"
Min_insert: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "A \<noteq> {}" shows "Min (insert x A) = min x (Min A)"
is_aligned_add_conv: fixes w :: "'b word"   and n :: "nat"   and off :: "'b word" assumes "is_aligned w n"   and "off < 2 ^ n" shows "to_bl (w + off) = take (LENGTH('b) - n) (to_bl w) @ drop (LENGTH('b) - n) (to_bl off)"
assms_3: shows "rk {A', B', P, Q} = 4"
Auxk_def: shows "Auxk \<equiv> \<lambda>s. k s < length (M s) \<and> (M s ! k s \<noteq> Black \<or> \<not> BtoW (E s ! ind s, M s) \<or> obc s \<subset> Blacks (M s) \<or> \<not> z s \<and> ind s = R \<and> snd (E s ! R) = T \<and> (\<exists>r>ind s. r < length (E s) \<and> BtoW (E s ! r, M s)))"
Union_disjoint: fixes C :: "'c set set"   and g :: "'c \<Rightarrow> 'a" assumes "\<forall>A\<in>C. finite A"   and "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {}" shows "F g (\<Union> C) = (F \<circ> F) g C"
only_vertical_division: shows "only_vertical_division (boundary C) two_chain_typeI"
is_target_def: fixes step :: "nat \<Rightarrow> 'c \<Rightarrow> (nat \<times> 'c) list"   and \<tau>s :: "'c list"   and pc' :: "nat" shows "is_target step \<tau>s pc' = (\<exists>pc s'. pc' \<noteq> pc + 1 \<and> pc < length \<tau>s \<and> (pc', s') \<in> set (step pc (\<tau>s ! pc)))"
small_step_terminated: fixes s :: "('g, 'h, 'i, 'j) com list \<times> 'h option \<times> 'j"   and x :: "('g, 'i) seq_label"   and s' :: "('g, 'h, 'i, 'j) com list \<times> 'h option \<times> 'j" assumes "s \<rightarrow>\<^bsub>x\<^esub> s'"   and "atCs (cPGM s) = {}" shows "atCs (cPGM s') = {}"
summandTransition_1: fixes P :: "pi"   and P' :: "pi" assumes "hnf P" shows "P \<longmapsto> \<tau> \<prec> P' = (\<tau>.P' \<in> summands P)"
vsubstEnv_preserves_good: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and xs :: "'varSort"   and x1 :: "'var"   and x :: "'var" assumes "goodEnv rho" shows "goodEnv (rho &[x1 // x]_xs)"
continuous_on_arccos': shows "continuous_on {- 1..1} arccos"
curry_le: fixes f :: "'a \<times> 'b \<Rightarrow> 'c"   and g :: "'a \<times> 'b \<Rightarrow> 'c" shows "(curry f \<le> curry g) = (f \<le> g)"
character_axioms: shows "character (Characters G) (principal_char (Characters G))"
unique_factorization_m_smult: fixes f :: "int poly"   and c :: "int"   and fs :: "int poly multiset"   and di :: "int"   and d :: "int" assumes "unique_factorization_m f (c, fs)"   and "M (di * d) = 1" shows "unique_factorization_m (Polynomial.smult d f) (c * d, fs)"
simps_9: fixes v :: "nat" shows "\<top>\<^sub>1 +\<^sub>1 [v]\<^sub>1 = \<top>\<^sub>1"
ret_obs_all_obs: fixes \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat"   and i :: "nat"   and l :: "nat" assumes "is_path \<pi>"   and "is_kth_obs \<pi> k i"   and "\<pi> i = local.return"   and "k < l" obtains j :: "nat" where "is_kth_obs \<pi> l j"
n_sorted: shows "\<not> Sorted_Less.sorted (rev (p # rev [0..<length (g_V G)]))"
equalityD2: fixes A :: "'a set"   and B :: "'a set" assumes "A = B" shows "B \<subseteq> A"
ord_le_eq_trans: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" assumes "a \<subseteq># b"   and "b = c" shows "a \<subseteq># c"
ternary_to_bool_SomeE: fixes t :: "ternaryvalue"   and X :: "bool" assumes "ternary_to_bool t = Some X" obtains "t = TernaryTrue" and "X = True"   | "t = TernaryFalse" and "X = False"
that: fixes f :: "V \<Rightarrow> V" assumes "f \<in> elts A \<rightarrow> elts (vcard A)"   and "inj_on f (elts A)" shows "thesis"
fmeasurable_UN_bound: fixes I :: "'a set"   and A :: "'a \<Rightarrow> 'b set"   and M :: "'b measure"   and B :: "real" assumes "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> fmeasurable M"   and "\<And>I'. \<lbrakk>I' \<subseteq> I; finite I'\<rbrakk> \<Longrightarrow> Sigma_Algebra.measure M (\<Union> (A ` I')) \<le> B" shows "\<Union> (A ` I) \<in> fmeasurable M"
is_lasso_pre_prpl_of_lasso: fixes ma :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b"   and m :: "'a"   and L :: "('Q \<times> nat) lasso" shows "graph.is_lasso_prpl_pre (b_graph.to_gbg_ext (degeneralize_ext ma) m) (prpl_of_lasso L) = graph.is_lasso_pre (b_graph.to_gbg_ext (degeneralize_ext ma) m) L"
finite_entropy_nn: fixes S :: "'c measure"   and X :: "'a \<Rightarrow> 'c"   and Px :: "'c \<Rightarrow> real"   and x :: "'c" assumes "finite_entropy S X Px"   and "x \<in> space S" shows "0 \<le> Px x"
cl9_d: fixes x :: "'a" shows "d (x \<sqinter> 1\<^sub>\<sigma>) = x \<sqinter> 1\<^sub>\<sigma>"
homI: fixes CC :: "('b, 'c, 'd) category_scheme"   and f :: "'c"   and A :: "'b"   and B :: "'b" assumes "category CC"   and "f \<in> Ar\<^bsub>CC\<^esub>"   and "Dom\<^bsub>CC\<^esub> f = A"   and "Cod\<^bsub>CC\<^esub> f = B" shows "f \<in> Hom\<^bsub>CC\<^esub> A B"
simply_connected_sphere: fixes a :: "'a"   and r :: "real" assumes "3 \<le> DIM('a)" shows "simply_connected (sphere a r)"
sorted_Cons_iff: fixes x :: "'b"   and xs :: "'b list" shows "sorted (x # xs) = ((\<forall>y\<in>set xs. x < y) \<and> sorted xs)"
not_ex: fixes P :: "'d \<Rightarrow> bool" shows "(\<nexists>x. P x) = (\<forall>x. \<not> P x)"
assms_2: shows "(of_int_poly a1, of_int_poly a2) = (div_rat_poly_step (of_int_poly q) \<circ> rat_of_int) b2 (of_int_poly c1, of_int_poly c2)"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "monoid_mult_hom_1 hom" shows "comm_monoid_mult_hom_1 hom"
hyps_2: fixes xs :: "['a]" assumes "distinct xs"   and "\<forall>a\<in>set ys. \<forall>x\<in>set xs. eq\<cdot>a\<cdot>x \<noteq> \<bottom>" shows "set (xs \\ ys) = set xs - set ys"
atLeast1_atMost_eq: fixes g :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "prod g {Suc 0..n} = (\<Prod>k<n. g (Suc k))"
rTau: fixes Q' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> Q \<longmapsto> \<tau> \<prec> Q'"   and "yvec \<sharp>* Q'" shows "\<exists>P'. \<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P' \<and> (\<Psi>, P', Q') \<in> Rel"
recurring_condition: fixes P :: "nat \<times> 'a \<Rightarrow> bool"   and p :: "nat \<times> 'a"   and A :: "('b, 'a) nba"   and w :: "'b stream"   and V :: "(nat \<times> 'a) set" assumes "P p"   and "\<And>p. P p \<Longrightarrow> \<exists>r. r \<noteq> [] \<and> gpath A w V r p \<and> P (gtarget r p)" obtains r :: "(nat \<times> 'a) stream" where "grun A w V r p" and "infs P (p ## gtrace r p)"
mod0_uint8_def: fixes x :: "uint8" shows "mod0_uint8 x = undefined (mod) x 0"
cong_solve_coprime_int: fixes a :: "int"   and n :: "int" assumes "coprime a n" shows "\<exists>x. [a * x = 1] (mod n)"
in\<alpha>_def: shows "in\<alpha> = fst\<^sub>L"
distr_pair_swap: shows "S \<Otimes>\<^sub>M T = distr (T \<Otimes>\<^sub>M S) (S \<Otimes>\<^sub>M T) (\<lambda>(x, y). (y, x))"
antiframe_hoare_r: fixes a :: "'a \<Longrightarrow> 'b"   and r :: "'b upred"   and q :: "'b upred"   and p :: "'b upred"   and P :: "'b hrel" assumes "vwb_lens a"   and "a \<natural> r"   and "a \<sharp> q"   and "\<lbrace>p\<rbrace> P \<lbrace>q\<rbrace>\<^sub>u" shows "\<lbrace>p \<and> r\<rbrace> {a}:\<lbrakk>P\<rbrakk> \<lbrace>q \<and> r\<rbrace>\<^sub>u"
return_time_returns: fixes P :: "'a set"   and x :: "'a" assumes "returns_to P x"   and "closed P" shows "flow0 x (return_time P x) \<in> P"
NAKTr3: fixes H :: "'a set" assumes "M fgover R"   and "H \<subseteq> carrier M"   and "finite H"   and "card H < (LEAST j. \<exists>L. finite_generator R M L \<and> j = card L)" shows "\<not> finite_generator R M H"
functor_axioms: shows "functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (A_B.Cod (fst Fg))"
elementLevelAppend: fixes a :: "'a"   and M :: "('a \<times> bool) list"   and M' :: "('a \<times> bool) list" assumes "a \<in> set (elements M)" shows "elementLevel a M = elementLevel a (M @ M')"
deg_eqI: fixes n :: "nat"   and p :: "nat \<Rightarrow> 'a" assumes "\<And>m. n < m \<Longrightarrow> coeff P p m = \<zero>"   and "\<And>n. n \<noteq> 0 \<Longrightarrow> coeff P p n \<noteq> \<zero>"   and "p \<in> carrier P" shows "deg R p = n"
sumhr_less_bounds_zero: fixes n :: "nat star"   and m :: "nat star"   and f :: "nat \<Rightarrow> real" assumes "n < m" shows "sumhr (m, n, f) = 0"
InitDone: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and P :: "(char list list \<times> char list exp) prog"   and C' :: "char list"   and Cs :: "char list list"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "sh C = \<lfloor>(sfs, Done)\<rfloor>"   and "P \<turnstile> \<langle>INIT C' (Cs,True) \<leftarrow> e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
3: shows "z\<^sup>\<star> * z\<^sup>+ * y * x\<^sup>\<star> \<le> z\<^sup>\<star> * (x\<^sup>\<star> \<squnion> y * x\<^sup>\<star>) * z\<^sup>\<star>"
Int_insert_right_if0: fixes a :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "a \<notin> A" shows "A \<inter> insert a B = A \<inter> B"
lin_indpt_lin_comb_list: fixes ss :: "'b vec list"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b"   and i :: "nat" assumes "distinct ss"   and "\<not> module.lin_dep class_ring (module_vec TYPE('b) n) (set ss)"   and "set ss \<subseteq> carrier_vec n"   and "module.lincomb_list (module_vec TYPE('b) n) f ss = 0\<^sub>v n"   and "i < length ss" shows "f i = (0::'b)"
hd_pair: fixes x :: "nat"   and V :: "nat set"   and y :: "nat"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal" assumes "x \<in> V"   and "y \<in> V"   and "x \<noteq> y" shows "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d <Var x ,  Var y> \<Rightarrow> marg_dens2 (V, V', \<Gamma>, \<delta>) x y"
collision_trim_subset: fixes S :: "real set"   and T :: "real set" assumes "collision_react S"   and "no_collision_react T"   and "T \<subseteq> S" shows "collision_react (S - T)"
map_eq_append_conv: fixes f :: "'d \<Rightarrow> 'c"   and ls :: "'d list"   and fl :: "'c list"   and fl' :: "'c list" shows "(map f ls = fl @ fl') = (\<exists>l l'. ls = l @ l' \<and> map f l = fl \<and> map f l' = fl')"
qs_def: shows "qs = [a - f * normal_form L q, f * normal_form L q]"
connected_Int_frontier: fixes s :: "'a set"   and t :: "'a set" assumes "connected s"   and "s \<inter> t \<noteq> {}"   and "s - t \<noteq> {}" shows "s \<inter> frontier t \<noteq> {}"
hds: shows "head_\<omega> n1 \<noteq> head_\<omega> n2 \<or> n1 = 0 \<and> n2 = 0"
Cons_eq_map_conv: fixes x :: "'g"   and xs :: "'g list"   and f :: "'h \<Rightarrow> 'g"   and ys :: "'h list" shows "(x # xs = map f ys) = (\<exists>z zs. ys = z # zs \<and> x = f z \<and> xs = map f zs)"
preorder_family_axioms: shows "preorder_family agents lotteries (SD \<circ> R)"
pw_ERETURN_1: fixes x :: "'d" shows "nofail (ERETURN x)"
Not\<^sub>\<alpha>: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c) Formula.Tree\<^sub>\<alpha>" assumes "Formula.hereditarily_fs t\<^sub>\<alpha>" shows "Formula.hereditarily_fs (Formula.Not\<^sub>\<alpha> t\<^sub>\<alpha>)"
arith_special_11: shows "- (1::'a) + (1::'a) = (0::'a)"
elims: fixes x :: "'b rexp"   and y :: "'b rexp set" assumes "toplevel_summands x = y" obtains r :: "'b rexp" and s :: "'b rexp" where "x = Plus r s" and "y = toplevel_summands r \<union> toplevel_summands s"   | "x = Zero" and "y = {Zero}"   | "x = One" and "y = {One}"   | v :: "'b" where "x = Atom v__" and "y = {Atom v__}"   | v :: "'b rexp" and va :: "'b rexp" where "x = Times va__ va" and "y = {Times va__ va}"   | v :: "'b rexp" where "x = Star v" and "y = {Star v}"
WT_map_converter_idD: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and f :: "'a \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'b" assumes "\<I>, \<I>' \<turnstile>\<^sub>C map_converter id id f g 1\<^sub>C \<surd>" shows "\<I> \<le> map_\<I> f g \<I>'"
subgraph: fixes a :: "'a"   and b :: "'a" assumes "E' a b" shows "a \<rightarrow> b"
comm_neutral: fixes a :: "'a" shows "a + (0::'a) = a"
aux_1: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes> y \<in> carrier R"
ih: fixes t :: "real" shows "flow0 x (t + T * real n) = flow0 x t"
hyps_2: shows "r[x::=s] \<degree>\<degree> ss \<rightarrow>\<^sub>s t"
L0: shows "Infer (zip (prems_of \<iota>F) Ll) (concl_of \<iota>F, L0) \<in> Inf_FL"
v11_app_vconverse_app: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vint_uminus" shows "-\<^sub>\<int> vint_uminus\<inverse>\<^sub>\<circ>\<lparr>a\<rparr> = a"
comp_cod_arr: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and b :: "'a" assumes "category C"   and "partial_magma.arr C f"   and "partial_magma.cod C f = b" shows "C b f = f"
cy1: shows "sum cy {0..<length Vs} = (1::'a)"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_tag_name set_tag_name_locs"   and "l_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"   and "l_set_tag_name_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_tag_name set_tag_name_locs known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M" shows "l_set_tag_name_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_tag_name set_tag_name_locs known_ptr known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"
del_block_def: fixes bl :: "'a set"   and b :: "'a set" shows "add_block_sys.del_block bl b \<equiv> remove1_mset b (add_point_sys.add_block bl)"
hyps_1: shows "(P, P') \<in> {(P, P') |P P'. P \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P'}\<^sup>*"
invf_def: shows "invf = the_inv_into (\<Union> X) f"
abs_sin_x_le_abs_x: fixes x :: "real" shows "\<bar>sin x\<bar> \<le> \<bar>x\<bar>"
has_field_derivative_rGamma: fixes z :: "'a"   and A :: "'a set" shows "(rGamma has_field_derivative (if z \<in> \<int>\<^sub>\<le>\<^sub>0 then (- (1::'a)) ^ nat \<lfloor>norm z\<rfloor> * fact (nat \<lfloor>norm z\<rfloor>) else - rGamma z * Digamma z)) (at z within A)"
ndet_cont: fixes f :: "'a \<Rightarrow> 'b process"   and g :: "'a \<Rightarrow> 'b process" assumes "cont f"   and "cont g" shows "cont (\<lambda>x. f x |-| g x)"
p_2: shows "\<forall>m. MPoly_Type.coeff p m \<in> \<rat>"
hyps: assumes "0 < m"   and "m \<le> n" shows "m dvd fact n"
orderE: fixes b :: "'a"   and a :: "'a" assumes "b \<le> a" obtains "a = max a b"
ring_chain: fixes R :: "'a ring" assumes "R \<in> C" shows "ring R"
binomial_fact: fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "of_nat (n choose k) = fact n / (fact k * fact (n - k))"
assms_4: shows "k \<in> fst ` set (fst xs) \<union> fst ` set (fst ys)"
intros_6: fixes r :: "'b rexp" shows "\<turnstile> Stars [] : Star r"
one_dvd: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d" assumes "class.comm_monoid_mult times one" shows "dvd.dvd times one a"
padd_commut: fixes b :: "'a list"   and a :: "'a list" shows "b +++ a = a +++ b"
subdensity_real: fixes T :: "'d \<Rightarrow> 'e"   and P :: "'d measure"   and Q :: "'e measure"   and M :: "'f measure"   and X :: "'f \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> real"   and Y :: "'f \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> real" assumes "T \<in> P \<rightarrow>\<^sub>M Q"   and "distributed M P X (\<lambda>x. ennreal (f x))"   and "distributed M Q Y (\<lambda>x. ennreal (g x))"   and "Y = T \<circ> X"   and "AE x in P. 0 \<le> g (T x)"   and "AE x in P. 0 \<le> f x" shows "AE x in P. g (T x) = 0 \<longrightarrow> f x = 0"
x: shows "x \<in> {x \<in> carrier G. h x \<in> I}"
prems_7: fixes j :: "nat" assumes "prime j"   and "j < i" shows "multiplicity j factor_pr \<le> 1"
vsv_vrange_vsubset_vifunion_app: fixes r :: "V"   and I :: "V"   and A :: "V \<Rightarrow> V" assumes "vsv r"   and "\<D>\<^sub>\<circ> r = I"   and "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> r\<lparr>i\<rparr> \<in>\<^sub>\<circ> A i" shows "\<R>\<^sub>\<circ> r \<subseteq>\<^sub>\<circ> \<Union>\<^sub>\<circ> (VLambda I A `\<^sub>\<circ> I)"
split_list_first: fixes x :: "'g"   and xs :: "'g list" assumes "x \<in> set xs" shows "\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys"
assms: shows "s' = snd (fst (store_sub2 instr curr_win rd asi address s)) \<and> get_S (cpu_reg_val PSR s) = 0"
space_Sup_measure'2: fixes M :: "'a measure set" shows "space (local.Sup_measure' M) = \<Union> (space ` M)"
inj_idom_hom_axioms: shows "inj_idom_hom (\<lambda>x. fpxs_compose_power x (1 / rat_of_nat b))"
Ax: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set" assumes "(F::'b formula) \<in> (\<Gamma>::'b formula set)" shows "(\<Gamma>::'b formula set) \<turnstile>\<^sub>H (F::'b formula)"
assms: shows "low_equal s1 s2 \<and> t1 = snd (fst (set_annul True s1)) \<and> t2 = snd (fst (set_annul True s2))"
mset_subset_insertD: fixes x :: "'b"   and A :: "'b multiset"   and B :: "'b multiset" assumes "add_mset x A \<subset># B" shows "x \<in># B \<and> A \<subset># B"
remove_Union_finite: fixes X :: "'a set"   and A :: "'b set"   and B :: "'a \<Rightarrow> 'b set" assumes "finite X" shows "remove_Union A X B = Finite_Set.fold (\<lambda>x A. A - B x) A X"
simps_2: fixes x :: "('idx, 'pred, 'act) formula"   and n :: "nat" shows "tau_steps x (Suc n) = Act \<tau> (tau_steps x n)"
circ_circ_split: fixes x :: "'a" shows "x\<^sup>\<circ>\<^sup>\<circ> = L \<squnion> x\<^sup>\<circ>"
neqE: fixes x :: "'d"   and y :: "'d" assumes "x \<noteq> y" obtains "y < x"   | "x < y"
eq_fst_iff: fixes a :: "'r"   and p :: "'r \<times> 's" shows "(a = prod.fst p) = (\<exists>b. p = (a, b))"
assms_1: shows "\<forall>x\<in>fv\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t S). (\<exists>f. \<sigma> x = Fun f []) \<or> (\<exists>y. \<sigma> x = Var y)"
check_eqv_def: fixes n :: "nat"   and \<phi> :: "'b formula"   and \<psi> :: "'b formula" shows "slow.check_eqv n \<phi> \<psi> = (wf_formula n (FOr \<phi> \<psi>) \<and> slow.check_eqvRE enum_class.enum n (Plus (rexp_of'' n (norm \<phi>)) One) (Plus (rexp_of'' n (norm \<psi>)) One))"
ah_update_def: fixes h :: "'b \<Rightarrow> 'b set"   and F :: "'b set \<times> 'b set"   and M :: "'b set"   and m :: "'b" shows "ah_update h F M m \<equiv> if m \<in> fst F then fst F \<union> snd F \<union> M else h m"
star2: fixes x :: "'b \<times> 'c" shows "prod.qstar (sup_prod one_prod x) = prod.qstar x"
infinite_chain_arbitrary_esize: fixes esize :: "'b \<Rightarrow> enat"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and C :: "'b set"   and n :: "nat" assumes "class.esize_order esize less_eq less"   and "Complete_Partial_Order.chain less_eq C"   and "infinite C" obtains x :: "'b" where "x \<in> C" and "enat n \<le> esize x"
wf_termi: fixes b :: "state \<Rightarrow> bool"   and c :: "com" shows "wf {(t, s). WHILE b DO c \<down> s \<and> b s \<and> s -c\<rightarrow> t}"
order_cpoly_gcd_eq: fixes pR :: "real poly"   and pI :: "real poly"   and t :: "real" assumes "pR \<noteq> 0 \<or> pI \<noteq> 0" shows "order (complex_of_real t) (cpoly_of pR pI) = order t (gcd pR pI)"
frac_def: fixes a :: "int"   and b :: "int" shows "Nitpick.frac a b \<equiv> Nitpick.Abs_Frac (Nitpick.norm_frac a b)"
embed_comp_embedS: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd"   and r'' :: "('e \<times> 'e) set"   and f' :: "'d \<Rightarrow> 'e" assumes "Well_order r"   and "Well_order r'"   and "embed r r' f"   and "embedS r' r'' f'" shows "embedS r r'' (f' \<circ> f)"