hcju/isabelleps huggingface dataset:
rel_flip: fixes R :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a pmf"   and b :: "'b pmf" shows "rel_pmf R\<inverse>\<inverse> a b = rel_pmf R b a"
is_natural_1: fixes f :: "'a" assumes "A.arr f" shows "G f \<cdot>\<^sub>B G (A.dom f) = G f"
ring_hom_ring_axioms: shows "ring_hom_ring R S h"
inorder_delete: fixes t :: "'b tree"   and x :: "'b" assumes "Sorted_Less.sorted (inorder t)" shows "inorder (delete x t) = del_list x (inorder t)"
polyadd_eq_const_degree: fixes p :: "poly"   and n0 :: "nat"   and q :: "poly"   and n1 :: "nat"   and c :: "int \<times> int" assumes "isnpolyh p n0"   and "isnpolyh q n1"   and "p +\<^sub>p q = C c" shows "Reflected_Multivariate_Polynomial.degree p = Reflected_Multivariate_Polynomial.degree q"
lossless_def: fixes \<A> :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool spmf" shows "ddh.lossless \<A> = (\<forall>\<alpha> \<beta> \<gamma>. lossless_spmf (\<A> \<alpha> \<beta> \<gamma>))"
prop_unfold_equiv_helper_1: fixes X :: "'b ltln set" assumes "finite (abs_ltln\<^sub>P ` X)" shows "finite (abs_ltln\<^sub>Q ` X)"
min_set_product_set: fixes A :: "'b fset set set" shows "\<Otimes>\<^sub>m A = min_set (\<Otimes> A)"
safe_cosafe: fixes r :: "'b regex"   and safe :: "safety \<Rightarrow> 'b \<Rightarrow> bool"   and fv :: "'b \<Rightarrow> 'c set"   and m :: "modality" assumes "\<And>x. \<lbrakk>x \<in> atms r; safe Strict x\<rbrakk> \<Longrightarrow> safe Lax x"   and "Regex.safe_regex fv safe m Strict r" shows "Regex.safe_regex fv safe m Lax r"
combine_in_tree: fixes l :: "('a, 'b) rbt"   and r :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b" assumes "inv2 l"   and "inv2 r"   and "bheight l = bheight r"   and "inv1 l"   and "inv1 r" shows "((k, v) \<in> set (entries (combine l r))) = ((k, v) \<in> set (entries l) \<or> (k, v) \<in> set (entries r))"
g: fixes x :: "'a" shows "monotone ord (ord_spmf (=)) (\<lambda>y. g y x)"
alpha_set: fixes bs :: "atom set"   and x :: "'a"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> atom set"   and p :: "perm"   and cs :: "atom set"   and y :: "'a" shows "(bs, x) \<approx>set R f p (cs, y) = (f x - bs = f y - cs \<and> (f x - bs) \<sharp>* p \<and> R (p \<bullet> x) y \<and> p \<bullet> bs = cs)"
Hausdorff_imp_t1_space: fixes X :: "'b topology" assumes "Hausdorff_space X" shows "t1_space X"
assms_4: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "P f []"
rep_eq: fixes x :: "('c, 'd) lifted_distrib_lattice_top"   and xa :: "('c, 'd) lifted_distrib_lattice_top" shows "Rep_lifted_distrib_lattice_top (x \<squnion> xa) = (\<lambda>f. Rep_lifted_distrib_lattice_top x f \<squnion> Rep_lifted_distrib_lattice_top xa f)"
ap_spec_add_pairs: fixes np :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and icrit :: "nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> bool"   and ncrit :: "nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> bool \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> bool"   and ocrit :: "nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> bool"   and comb :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" assumes "np_spec np"   and "icrit_spec icrit"   and "ncrit_spec ncrit"   and "ocrit_spec ocrit"   and "\<And>xs ys. set (comb xs ys) = set xs \<union> set ys" shows "ap_spec (add_pairs np icrit ncrit ocrit comb)"
assms_1: shows "i < card {i. i < dim_row A \<and> i \<in> I}"
simps_6: fixes wf_atom :: "nat \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "nat"   and r :: "'c rexp"   and s :: "'c rexp" shows "alphabet.wf wf_atom n (Times r s) = (alphabet.wf wf_atom n r \<and> alphabet.wf wf_atom n s)"
n_is_len_\<gamma>\<delta>_C\<V>i: shows "n = length ((\<gamma> @ \<delta>) \<upharpoonleft> C\<^bsub>\<V>i\<^esub>)"
intros_1: fixes c :: "('d, 'e) conf" shows "se_star c [] c"
RUN_all_tickfree_traces1: fixes A :: "'b set" shows "T (RUN A) = {s. set s \<subseteq> ev ` A}"
isOp_stOf_Op: fixes delta :: "'opSym"   and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option" assumes "wlsInp delta inp"   and "wlsBinp delta binp" shows "isOp (stOf delta) (Op delta inp binp)"
takeWhile_cong: fixes l :: "'d list"   and k :: "'d list"   and P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "l = k"   and "\<And>x. x \<in> set l \<Longrightarrow> P x = Q x" shows "takeWhile P l = takeWhile Q k"
length_list_of_vec: fixes v :: "'c vec" shows "length (list_of_vec v) = dim_vec v"
vmc_path_ltl: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c" assumes "vmc_path_no_deadend G P v0 p \<sigma>" shows "vmc_path G (ltl P) (vmc_path_no_deadend.w0 P) p \<sigma>"
invertible: fixes G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and u :: "'c" assumes "Group_Theory.group G composition unit"   and "u \<in> G" shows "monoid.invertible G composition unit u"
bound_invD_2: fixes outside :: "bool"   and upw :: "bool"   and i :: "nat"   and fs :: "int vec list" assumes "LLL_bound_invariant outside upw i fs" shows "f_bound outside i fs"
intro: fixes e :: "'a"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and F :: "'b \<Rightarrow> 'a"   and a :: "'b" assumes "e \<in> set_category_data.set S (set_category.some_img S (cardSuc (cmax |UNIV| natLeq))) (F a)" shows "yoneda_lemma_fixed_e_axioms S F a e"
short_vector_def: shows "local.short_vector = hd local.reduce_basis"
mask_eqs_6: fixes b :: "'b word"   and n :: "nat"   and a :: "'b word" shows "(b AND mask n) * a AND mask n = b * a AND mask n"
urel32_plus: fixes x :: "uint32"   and y :: "int"   and x' :: "uint32"   and y' :: "int" assumes "urel32 x y"   and "urel32 x' y'" shows "urel32 (plus_p32 pp x x') (plus_p p y y')"
assms_3: shows "proots_count (map_poly complex_of_real P) circle_01_diam = 0"
nat_mod_eq': fixes a :: "nat"   and n :: "nat" assumes "a < n" shows "a mod n = a"
post_map_acom: fixes f :: "'d \<Rightarrow> 'c"   and c :: "'d acom" shows "post (ACom.map_acom f c) = f (post c)"
distinct_33: fixes x31 :: "nat"   and x32 :: "'e"   and x7 :: "'e" shows "unicast x31 x32 \<noteq> receive x7"
min_for_in: fixes x :: "'a"   and S :: "'a set"   and y :: "'a" assumes "x \<in> S"   and "y \<in> S" shows "min_for x y \<in> S"
word_mult_def: fixes a :: "'b word"   and b :: "'b word" shows "a * b = word_of_int (uint a * uint b)"
eventually_strongly_fair: fixes enabled :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and taken :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<diamond>strongly_fair enabled taken = strongly_fair enabled taken"
bhv2cnf_ge_lActive: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and n :: "nat" shows "dynamic_component.lActive active c t \<le> dynamic_component.bhv2cnf tCMP active c t n"
cnf_semantics_def: fixes \<A> :: "'a \<Rightarrow> bool"   and S :: "'a literal set set" shows "cnf_semantics \<A> S \<equiv> \<forall>C\<in>S. clause_semantics \<A> C"
backflows_un: fixes A :: "('a \<times> 'a) set"   and B :: "('a \<times> 'a) set" shows "backflows (A \<union> B) = backflows A \<union> backflows B"
simps_3: fixes disc :: "'b \<Rightarrow> bool"   and m :: "'b match_expr" shows "has_disc disc (MatchNot m) = has_disc disc m"
invertible_imp_bijective: fixes scaleB :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c" assumes "invertible_lf scaleB scaleC f" shows "bij f"
face_cycle_succ_def: fixes M :: "'c pre_map" shows "pre_digraph_map.face_cycle_succ M \<equiv> edge_succ M \<circ> edge_rev M"
normalize_mod_ring_def: fixes x :: "'b mod_ring" shows "normalize x = (if x = 0 then 0 else 1)"
set_orbit_list: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'b" assumes "x \<in> orbit f x" shows "set (orbit_list f x) = orbit f x"
sees_wf_mdecl: fixes wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and C :: "char list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'b"   and D :: "char list" assumes "wf_prog wf_md P"   and "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D" shows "wf_mdecl wf_md P D (M, b, Ts, T, m)"
assms_2: shows "bob \<le> vimage2p hb hb (=)"
hom_nat_pow: fixes x :: "int \<times> int"   and n :: "nat" assumes "x \<in> carrier R" shows "\<phi> (x [^]\<^bsub>R\<^esub> n) = \<phi> x [^]\<^bsub>S\<^esub> n"
3: shows "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
help1: fixes xys :: "('a \<times> 'b) list"   and P :: "'a \<times> 'b \<Rightarrow> bool" assumes "distinct (map fst xys)" shows "map_of (filter P xys) = map_of xys |` {x. \<exists>y. (x, y) \<in> set xys \<and> P (x, y)}"
ccpo1: shows "class.ccpo luba orda (mk_less orda)"
d_OUT_current_of_bipartite: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "('a + 'a) \<times> ('a + 'a) \<Rightarrow> ennreal"   and x :: "'a" assumes "web \<Gamma>"   and "current (bipartite_web_of \<Gamma>) f" shows "d_OUT (web.current_of_bipartite \<Gamma> f) x = d_OUT f (Inl x) - f (Inl x, Inr x)"
halfspace_gt_in_halfspace: fixes i :: "'c"   and A :: "'c set"   and a :: "real" assumes "i \<in> A" shows "{x. a < x \<bullet> i} \<in> sigma_sets UNIV ((\<lambda>(a, i). {x. x \<bullet> i < a}) ` (UNIV \<times> A))"
distinct_1: fixes x1 :: "bool"   and x2 :: "nat" shows "Det x1 \<noteq> Indet x2"
sin_minus_converges: fixes x :: "'a" shows "(\<lambda>n. - (sin_coeff n *\<^sub>R (- x) ^ n)) sums sin x"
assms_2: shows "diff_invariant I\<^sub>2 f U S t\<^sub>0 G"
scope_find: fixes v :: "'vertex"   and ps :: "'vertex \<times> ('form, 'var) antecedent"   and v' :: "'vertex"   and pth :: "(('vertex \<times> ('form, 'var) out_port) \<times> 'vertex \<times> ('form, 'var) antecedent) list" assumes "v \<in> scope ps"   and "terminal_vertex v'"   and "path v v' pth" shows "ps \<in> snd ` set pth"
diamond_top_left_edge_def: shows "diamond_top_left_edge = (- 1, \<lambda>x. (diamond_x (1 / 2 * x), diamond_x (1 / 2 * x) + d / 2))"
prod_n_ideal_contTr0: fixes n :: "nat"   and J :: "nat \<Rightarrow> 'a set" shows "(\<forall>l\<le>n. ideal R (J l)) \<longrightarrow> i\<Pi>\<^bsub>R,n\<^esub> J \<subseteq> \<Inter> {X. \<exists>k\<le>n. X = J k}"
norm_nth_le: fixes i :: "'c"   and x :: "'c" assumes "i \<in> Basis" shows "norm (x \<bullet> i) \<le> norm x"
elims_3: fixes x :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "\<not> SkewBinomialHeapStruc.rank_invar x" obtains t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" and t' :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" where "x = t # t' # bq" and "\<not> (SkewBinomialHeapStruc.rank t < SkewBinomialHeapStruc.rank t' \<and> SkewBinomialHeapStruc.rank_invar (t' # bq))"
Fun_comp: fixes g :: "'s"   and f :: "'s" assumes "seq g f" shows "Fun (g \<cdot> f) = restrict (Fun g \<circ> Fun f) (Dom f)"
measurable_inequality_set_1: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "{x \<in> space M. f x \<le> g x} \<in> sets M"
schroeder_3_p: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x * y \<le> - z) = (x[[z]] \<le> - y)"
inject: fixes x1 :: "'a"   and x2 :: "'b"   and y1 :: "'a"   and y2 :: "'b" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
weakSync: fixes P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs"   and Q :: "ccs"   and Q' :: "ccs" assumes "P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P'"   and "Q \<Longrightarrow>\<^sup>^coAction \<alpha> \<prec> Q'"   and "\<alpha> \<noteq> \<tau>" shows "P \<parallel> Q \<Longrightarrow>\<^sup>^\<tau> \<prec> P' \<parallel> Q'"
adj_antivertex: fixes x :: "'b set"   and y :: "'b set" assumes "x \<sim> y"   and "x \<noteq> y" shows "\<exists>!v. v \<in> x - y"
rt_fresh_as_nsqnr: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "dip \<in> kD rt1"   and "dip \<in> kD rt2"   and "rt1 \<approx>\<^bsub>dip\<^esub> rt2" shows "nsqn\<^sub>r (the (rt2 dip)) = nsqn\<^sub>r (the (rt1 dip))"
cases: fixes x :: "('f \<Rightarrow> 'g) \<times> ('h \<Rightarrow> 'g) \<times> ('f \<Rightarrow> 'e) \<times> ('h \<Rightarrow> 'e) \<times> 'e" obtains xa :: "'e" and p :: "'f \<Rightarrow> 'e" and D :: "'f \<Rightarrow> 'g" and E :: "'h \<Rightarrow> 'g" and q :: "'h \<Rightarrow> 'e" where "xa__ \<in> range pa__" and "x = (Da__, Ea__, pa__, qa__, xa__)"   | xa :: "'e" and p :: "'f \<Rightarrow> 'e" and q :: "'h \<Rightarrow> 'e" and D :: "'f \<Rightarrow> 'g" and E :: "'h \<Rightarrow> 'g" where "xaa__ \<notin> range pb__" and "xaa__ \<in> range qb__" and "x = (Db__, Eb__, pb__, qb__, xaa__)"   | xa :: "'e" and p :: "'f \<Rightarrow> 'e" and q :: "'h \<Rightarrow> 'e" and D :: "'f \<Rightarrow> 'g" and E :: "'h \<Rightarrow> 'g" where "xa \<notin> range p" and "xa \<notin> range q" and "x = (D, E, p, q, xa)"
assms: shows "M \<le>\<^sub>n SPEC (\<lambda>c. (c, a) \<in> R)"
traces_def: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "traces ind \<equiv> \<forall>a b. ind a b \<longrightarrow> ind b a"
Zero_not_eq_fromArr: shows "Zero \<notin> fromArr ` {False, True}"
pw_rwof_in_bounded_WHILE: fixes m0 :: "'c nres"   and cond :: "'c \<Rightarrow> bool"   and step :: "'c \<Rightarrow> 'c nres"   and x :: "'c" assumes "rwof m0 cond step x" shows "\<exists>n. inres (bounded_WHILE n cond step m0) x"
count: fixes x :: "'b multiset" shows "count x \<in> multiset"
l_set_attribute_get_attribute_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, char list option) prog"   and get_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> char list option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_attribute_get_attribute type_wf get_attribute get_attribute_locs set_attribute set_attribute_locs \<equiv> l_get_attribute type_wf get_attribute get_attribute_locs \<and> l_set_attribute type_wf set_attribute set_attribute_locs \<and> l_set_attribute_get_attribute_axioms get_attribute set_attribute"
eq_\<I>_exec_gpv: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "('c, 'd) \<I>"   and gpv :: "('a, 'c, 'd) gpv"   and gpv' :: "('b, 'c, 'd) gpv"   and s :: "('c, 'd) resource" assumes "eq_\<I>_gpv A \<I> gpv gpv'"   and "\<I> \<turnstile>res s \<surd>" shows "rel_spmf (rel_prod A (eq_onp (WT_resource \<I>))) (exec_gpv run_resource gpv s) (exec_gpv run_resource gpv' s)"
action_order_read_before_write: fixes E :: "('thread_id \<times> ('addr, 'thread_id) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat"   and w :: "nat"   and r :: "nat"   and adal :: "'addr \<times> addr_loc" assumes "E \<in> \<E>"   and "P \<turnstile> (E, ws) \<surd>"   and "E \<turnstile> w \<le>a r"   and "r \<in> read_actions E"   and "w \<in> write_actions E"   and "adal \<in> action_loc P E r"   and "adal \<in> action_loc P E w"   and "\<And>a. \<lbrakk>a < r; a \<in> read_actions E\<rbrakk> \<Longrightarrow> P,E \<turnstile> a \<leadsto>mrw ws a" shows "w < r"
simps_2: fixes \<Sigma> :: "nat \<Rightarrow> 'b set"   and n :: "nat"   and w :: "'b"   and ws :: "'b list" shows "alphabet.wf_word \<Sigma> n (w # ws) = (w \<in> \<Sigma> n \<and> alphabet.wf_word \<Sigma> n ws)"
locally_compact_closedin_open: fixes S :: "'a set" assumes "locally compact S" obtains T :: "'a set" where "open T" and "closedin (top_of_set T) S"
assms_3: fixes a :: "'a \<times> 'b"   and b :: "'a \<times> 'b" shows "comp (fst (f a)) (fst (f b)) = comp (fst a) (fst b)"
comparator_option_simps_2: fixes comp\<^sub>'\<^sub>a :: "'c \<Rightarrow> 'c \<Rightarrow> order"   and y :: "'c" shows "comparator_option comp\<^sub>'\<^sub>a None (Some y) = Lt"
extend_restrict_indets_eq_dehomogenize: fixes p :: "('c option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "extend_indets (restrict_indets p) = dehomogenize None p"
ball_subset_ball_iff: fixes a :: "'b"   and r :: "real"   and a' :: "'b"   and r' :: "real" shows "(ball a r \<subseteq> ball a' r') = (dist a a' + r \<le> r' \<or> r \<le> 0)"
someI: fixes P :: "'d \<Rightarrow> bool"   and x :: "'d" assumes "P x" shows "P (Eps P)"
rtrancl_imp_relpow: fixes p :: "'a \<times> 'a"   and R :: "('a \<times> 'a) set" assumes "p \<in> R\<^sup>*" shows "\<exists>n. p \<in> R ^^ n"
Sup_distl: fixes x :: "'c \<times> 'd"   and Y :: "('c \<times> 'd) set" shows "times_prod' x (Sup_prod Y) = Sup_prod {times_prod' x y |y. y \<in> Y}"
succ_ran_fin: fixes succi :: "'c \<Rightarrow> 'c list"   and E :: "('d \<times> 'd) set"   and Rv :: "('c \<times> 'd) set"   and v :: "'d" assumes "(succi, E) \<in> \<langle>Rv\<rangle>succg_rel"   and "v \<in> Range Rv" shows "finite (E `` {v})"
a: shows "\<exists>a. atom a \<sharp> (h, h a)"
erfc_unroll: fixes x :: "real"   and n :: "nat" assumes "0 < x" shows "erfc x = exp (- x\<^sup>2) / sqrt pi * (\<Sum>i<n. (- 1) ^ i * fact (2 * i) / (4 ^ i * fact i) / x ^ (2 * i + 1)) + erf_remainder n x"
cl_inv_ins: fixes t :: "('a, 'b) rbt"   and x :: "'a"   and v :: "'b" assumes "cl_inv t" shows "cl_inv' (ins x v t)"
Merge_insert_alt: fixes A :: "'a\<^sub>m set"   and x :: "'a\<^sub>m" assumes "finite A" shows "Merge (insert x A) = Finite_Set.fold (\<lambda>x yo. yo \<bind> m x) (Some x) A"
dominatesI: fixes a :: "'b"   and u :: "'a"   and v :: "'a" assumes "arc_to_ends G a = (u, v)"   and "a \<in> arcs G" shows "u \<rightarrow>\<^bsub>G\<^esub> v"
cfi_rel_alt: shows "cfi_rel = {(cf, f). cf = cf_of f \<and> NFlow c s t f}"
norm_eff_def: fixes i :: "'c instr"   and P :: "'d prog"   and pc :: "nat"   and \<tau> :: "ty list \<times> ty err list" shows "norm_eff i P pc \<tau> \<equiv> map (\<lambda>pc'. (pc', \<lfloor>eff\<^sub>i (i, P, \<tau>)\<rfloor>)) (succs i \<tau> pc)"
is_Throws_conv: fixes es :: "('e, 'f, 'g) exp list" shows "is_Throws es = (\<exists>vs a es'. es = map Val vs @ Throw a # es')"
enum_ex_finite_1_def: fixes P :: "finite_1 \<Rightarrow> bool" shows "enum_ex P = P a\<^sub>1"
inner_vec_def: fixes x :: "('a, 'rows) vec"   and y :: "('a, 'rows) vec" shows "inner_rows x y = (\<Sum>i\<in>UNIV. x $ i * cnj_class.cnj (y $ i))"
simps_2: fixes x :: "udom"   and f :: "udom\<cdot>'a error" shows "plusU\<cdot>(Ok\<cdot>x)\<cdot>f = Ok\<cdot>x"
unit_wfactors_empty: fixes a :: "'a"   and fs :: "'a list" assumes "a \<in> Units G"   and "wfactors G fs a"   and "set fs \<subseteq> carrier G" shows "fs = []"
arcs_G: shows "arcs I = arcs H \<union> {uv, rev_I uv} - {uw, rev_H uw, vw, rev_H vw}"
mult_right_mono_neg: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and b :: "'d"   and a :: "'d"   and c :: "'d" assumes "class.ordered_ring plus minus zero less_eq less times uminus"   and "less_eq b a"   and "less_eq c zero" shows "less_eq (times a c) (times b c)"
assms: fixes r :: "real" shows "\<exists>N. \<forall>n\<ge>N. r < X n"
DomainPI: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a"   and b :: "'b" assumes "r a b" shows "Domainp r a"
Conj_def: fixes xset :: "('a, 'b, 'c) Formula.formula set['a]" shows "Formula.Conj xset = Formula.formula.Abs_formula (Formula.Conj\<^sub>\<alpha> (map_bset Formula.formula.Rep_formula xset))"
idEnv_def: shows "idEnv \<equiv> \<lambda>xs. Map.empty"
shift1_invD_1: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and B :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set" assumes "shift1_inv (qs, B)" shows "B = {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)}"
order_iff_strict: fixes b :: "'a"   and a :: "'a" shows "lesseq_b b a = (less_b b a \<or> a = b)"
finsertI2: fixes a :: "'m"   and B :: "'m fset"   and b :: "'m" assumes "a |\<in>| B" shows "a |\<in>| finsert b B"
zmod_zminus1_eq_if: fixes a :: "int"   and b :: "int" shows "- a mod b = (if a mod b = 0 then 0 else b - a mod b)"
quotient_of_int: fixes n :: "int" shows "quotient_of (rat_of_int n) = (n, 1)"
wprepare_goto_first_end_Bk_nonempty_fst: fixes m :: "nat"   and lm :: "nat list"   and b :: "cell list"   and list :: "cell list" assumes "wprepare_goto_first_end m lm (b, Bk # list)" shows "b \<noteq> []"
one_less_floor: fixes x :: "'b" shows "(1 < \<lfloor>x\<rfloor>) = ((2::'b) \<le> x)"
awalk_cyc_decompE: fixes G :: "('c, 'd) pre_digraph"   and p :: "'d list"   and q :: "'d list"   and r :: "'d list"   and s :: "'d list"   and u :: "'c"   and v :: "'c" assumes "wf_digraph G"   and "wf_digraph.awalk_cyc_decomp G p = (q, r, s)"   and "pre_digraph.awalk G u p v"   and "\<not> distinct (pre_digraph.awalk_verts G u p)" obtains "p = q @ r @ s" and "distinct (pre_digraph.awalk_verts G u q)" and "\<exists>w. pre_digraph.awalk G u q w \<and> pre_digraph.awalk G w r w \<and> pre_digraph.awalk G w s v" and "wf_digraph.closed_w G r"
inject_2: fixes x21 :: "'b"   and x22 :: "('b, 'c) Kruskal_Examples.term list"   and y21 :: "'b"   and y22 :: "('b, 'c) Kruskal_Examples.term list" shows "(Fun x21 x22 = Fun y21 y22) = (x21 = y21 \<and> x22 = y22)"
main_step_R: fixes qs :: "real poly list" shows "set (find_consistent_signs_R qs) = consistent_sign_vectors_R qs UNIV"
nu_star_nu: fixes x :: "'a" shows "\<nu> (\<nu> x\<^sup>\<star>) = \<nu> x\<^sup>\<star>"
assms: shows "P {x. y + 1 = 1 + x}"
subst_apply_fv_unfold': fixes t :: "('b, 'c) Term.term"   and \<delta> :: "'c \<Rightarrow> ('b, 'a) Term.term" shows "fv (t \<cdot> \<delta>) = (\<Union>v\<in>fv t. fv (\<delta> v))"
word_and_mask_le_2pm1: fixes w :: "'b word"   and n :: "nat" shows "w AND mask n \<le> 2 ^ n - 1"
unit_in_vhom: shows "\<guillemotleft>\<eta> : src f \<Rightarrow> g \<star> f\<guillemotright>"