hcju/isabelleps huggingface dataset:
ide_char: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and f :: "'b" assumes "set_category S \<AA>"   and "partial_magma.arr S f" shows "partial_magma.ide S f = (set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.dom S f) = set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.cod S f) \<and> set_category_given_img.Fun S (set_category.some_img S \<AA>) f = (\<lambda>x\<in>set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.dom S f). x))"
is_Zp_cauchyI': fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "\<forall>n. \<exists>k. \<forall>m. k \<le> int m \<longrightarrow> eint (int n) \<le> val_Zp (s (Suc m) \<ominus> s m)" shows "is_Zp_cauchy s"
map_of_rev_distinct: fixes m :: "('c \<times> 'd) list" assumes "distinct (map fst m)" shows "map_of (rev m) = map_of m"
rel_mono_strong: fixes R1 :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R3 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and x :: "('b, 'e, 'g) generat"   and y :: "('d, 'f, 'h) generat"   and R1a :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and R2a :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R3a :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "rel_generat R1 R2 R3 x y"   and "\<And>z1 y1. \<lbrakk>z1 \<in> generat_pures x; y1 \<in> generat_pures y; R1 z1 y1\<rbrakk> \<Longrightarrow> R1a z1 y1"   and "\<And>z2 y2. \<lbrakk>z2 \<in> generat_outs x; y2 \<in> generat_outs y; R2 z2 y2\<rbrakk> \<Longrightarrow> R2a z2 y2"   and "\<And>z3 y3. \<lbrakk>z3 \<in> generat_conts x; y3 \<in> generat_conts y; R3 z3 y3\<rbrakk> \<Longrightarrow> R3a z3 y3" shows "rel_generat R1a R2a R3a x y"
b: fixes x :: "'a"   and y :: "'a" shows "f x y = f y x"
le_Suc_eq: fixes m :: "nat"   and n :: "nat" shows "((m::nat) \<le> Suc (n::nat)) = (m \<le> n \<or> m = Suc n)"
satisfies_properties_sys2: shows "satisfies_properties_R p qs2 (get_subsets_R (calculate_data_R p qs2)) (get_signs_R (calculate_data_R p qs2)) (get_matrix_R (calculate_data_R p qs2))"
vector_space_row_space: fixes A :: "'a mat"   and nr :: "nat"   and n :: "nat" assumes "A \<in> carrier_mat nr n" shows "vectorspace class_ring (module_vec TYPE('a) n\<lparr>carrier := vec_space.row_space n A\<rparr>)"
sorted_wrt_true: fixes xs :: "'g list" shows "sorted_wrt (\<lambda>_ _. True) xs"
int_poly_rat_no_root_ge: fixes p :: "real poly"   and b :: "int"   and a :: "int" assumes "\<And>n. coeff p n \<in> \<int>"   and "0 < b"   and "poly p (real_of_int a / real_of_int b) \<noteq> 0" shows "1 / real_of_int b ^ degree p \<le> \<bar>poly p (real_of_int a / real_of_int b)\<bar>"
aSum_zero: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier A" shows "aSum A n \<zero> = \<zero>"
assms: shows "div_mod_int_poly p q = Some (r, m)"
Fresh_apply': fixes a :: "'a"   and h :: "'a \<Rightarrow> 'b" assumes "atom a \<sharp> h"   and "atom a \<sharp> h a" shows "Fresh h = h a"
arr_Rel_incl_RelI: fixes A :: "V"   and B :: "V" assumes "A \<in>\<^sub>\<circ> Vset \<alpha>"   and "B \<in>\<^sub>\<circ> Vset \<alpha>"   and "A \<subseteq>\<^sub>\<circ> B" shows "arr_Rel \<alpha> (incl_Rel A B)"
bounded_coeffs_imp_abs_conv_abscissa_le: fixes f :: "'b fds"   and s :: "'b"   and c :: "ereal" assumes "Bseq (\<lambda>n. fds_nth f n * nat_power n s)"   and "ereal (1 - s \<bullet> (1::'b)) \<le> c" shows "abs_conv_abscissa f \<le> c"
scope_find: fixes v :: "'d"   and ps :: "'d \<times> 'b"   and v' :: "'d"   and pth :: "(('d \<times> 'c) \<times> 'd \<times> 'b) list" assumes "v \<in> scope ps"   and "terminal_vertex v'"   and "path v v' pth" shows "ps \<in> snd ` set pth"
set_bind_spmf: fixes M :: "'e spmf"   and f :: "'e \<Rightarrow> 'd spmf" shows "set_spmf (M \<bind> f) = set_spmf M \<bind> set_spmf \<circ> f"
inv_inv: fixes x :: "'a \<Rightarrow> complex" assumes "x \<in> carrier (Characters G)" shows "inv\<^bsub>Characters G\<^esub> (inv\<^bsub>Characters G\<^esub> x) = x"
intros: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and dma :: "'b \<Rightarrow> Sec"   and \<R> :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and lc\<^sub>1 :: "('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)"   and lc\<^sub>2 :: "('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)" assumes "sifum_security stop eval some_val some_val'"   and "sifum_security.strong_low_bisim_mm dma eval \<R>"   and "(lc\<^sub>1, lc\<^sub>2) \<in> \<R>" shows "(lc\<^sub>1, lc\<^sub>2) \<in> sifum_security.mm_equiv dma eval"
assms: shows "separateAtoms L = (eq, les, leq, neq)"
AbGroup: fixes R :: "'a set" assumes "Ring1 R" shows "AbGroup R"
reach_a': shows "tail G a' \<rightarrow>\<^sup>*\<^bsub>H\<^esub> head G a'"
hom_component_plus: fixes p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and q :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and n :: "nat" shows "hom_component (p + q) n = hom_component p n + hom_component q n"
singleton_quotient: fixes x :: "'c"   and r :: "('c \<times> 'c) set" shows "{x} // r = {r `` {x}}"
gaugeI: fixes \<gamma> :: "'b \<Rightarrow> 'b set" assumes "\<And>x. x \<in> \<gamma> x"   and "\<And>x. open (\<gamma> x)" shows "gauge \<gamma>"
zmset_subseteq_add_iff2: fixes i :: "nat"   and j :: "nat"   and u :: "'b zmultiset"   and m :: "'b zmultiset"   and n :: "'b zmultiset" assumes "i \<le> j" shows "(repeat_zmset i u + m \<subseteq>#\<^sub>z repeat_zmset j u + n) = (m \<subseteq>#\<^sub>z repeat_zmset (j - i) u + n)"
assms_1: shows "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>m \<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<CC>"
subspace_add: fixes S :: "'b set"   and x :: "'b"   and y :: "'b" assumes "csubspace S"   and "x \<in> S"   and "y \<in> S" shows "x + y \<in> S"
add_le_cancel_right: fixes a :: "'c multiset"   and c :: "'c multiset"   and b :: "'c multiset" shows "(a + c \<subseteq># b + c) = (a \<subseteq># b)"
zag_def: shows "zag = {(x, y) |x y. x * y = 0 \<and> (y = - 1 \<or> y = 0 \<or> y = 1)}"
eval_Runit: fixes t :: "'c local.term" assumes "Arr t" shows "\<lbrace>\<^bold>\<r>\<^bold>[t\<^bold>]\<rbrace> = \<r>[\<lbrace>Cod t\<rbrace>] \<cdot> (\<lbrace>t\<rbrace> \<star> src \<lbrace>t\<rbrace>)"
Pd_shuffle_to_top_simps_1: fixes d :: "'b"   and ds' :: "'b set"   and Y :: "'a set" shows "Pd_shuffle_to_top (insert d ds') Y = (Pd_shuffle_to_top ds' Y)(d := shuffle_to_top Y d)"
not_less: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat" shows "(\<not> lex_pm_strict x y) = lex_pm y x"
a: shows "rep_list (monomial (1::'b) (term_of_pair (0::'a, j))) = fs ! j"
b3_def: shows "b3 = ball x e \<inter> {y. (y - a) \<bullet> rot (a - b) = 0}"
tiny_dg_Cod_in_Vset: shows "\<CC>\<lparr>Cod\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
map_of_eq_empty_iff: fixes xys :: "('c \<times> 'd) list" shows "(map_of xys = Map.empty) = (xys = [])"
map_FGf_def: fixes l1 :: "'w \<Rightarrow> 'ag"   and l2 :: "'x \<Rightarrow> 'ah"   and co1 :: "'z \<Rightarrow> 'ai"   and co2 :: "'aa \<Rightarrow> 'aj"   and contra1 :: "'ak \<Rightarrow> 'ac"   and contra2 :: "'al \<Rightarrow> 'ad" shows "map_FGf l1 l2 co1 co2 contra1 contra2 = map_F l1 l2 id co1 co2 id contra1 contra2 id"
x_gt_inf: fixes x :: "ant" assumes "\<infinity> \<le> x" shows "x = \<infinity>"
multi_member_last: fixes x :: "'b" shows "x \<in># {#x#}"
simps_1: fixes \<gamma> :: "'c \<Rightarrow> 'b set" shows "\<gamma>_option \<gamma> None = {}"
example: fixes n :: "nat" shows "\<turnstile>\<^sub>2 {\<lambda>s. enat (2 + 3 * n) + \<up> (s ''x'' = int n)} ''y'' ::= N 0;; wsum {\<lambda>s. 0}"
pdevs_apply_project_coord_pdevs: fixes sctn :: "'a sctn"   and i :: "'a"   and cxs :: "'a pdevs"   and x :: "nat" assumes "normal sctn \<in> Basis \<or> - normal sctn \<in> Basis"   and "i \<in> Basis" shows "pdevs_apply (project_coord_pdevs sctn cxs) x \<bullet> i = (if i = \<bar>normal sctn\<bar> then 0 else pdevs_apply cxs x \<bullet> i)"
reduced_word_for_eq_length: fixes A :: "'a set"   and a :: "'a"   and as :: "'a list"   and bs :: "'a list" assumes "reduced_word_for A a as"   and "reduced_word_for A a bs" shows "length as = length bs"
subset_box_2: fixes c :: "'b"   and d :: "'b"   and a :: "'b"   and b :: "'b" shows "(cbox c d \<subseteq> box a b) = ((\<forall>i\<in>Basis. c \<bullet> i \<le> d \<bullet> i) \<longrightarrow> (\<forall>i\<in>Basis. a \<bullet> i < c \<bullet> i \<and> d \<bullet> i < b \<bullet> i))"
span_superset: fixes Sa :: "('a \<Rightarrow> real) set" assumes "Sa \<subseteq> diff_fun_space" shows "Sa \<subseteq> diff_fun_space.m1.span Sa"
propex\<^sub>1: fixes F :: "\<Pi>\<^sub>1" shows "\<exists>r. Some r = d\<^sub>1 F"
tso_no_pending_marksD: fixes s :: "'d process_name \<Rightarrow> ('a, 'd, 'b, 'c) Model.local_state"   and p :: "'d process_name"   and r :: "'c" assumes "filter is_mw_Mark (sys_mem_store_buffers p s) = []" shows "sys_load p (mr_Mark r) (s sys) = mv_Mark (map_option obj_mark (sys_heap s r))"
i_eq_HComplex_0_1: shows "iii = HComplex 0 1"
isomorphic_def: fixes a :: "'c"   and a' :: "'c" shows "G.B.isomorphic a a' = (\<exists>f. \<guillemotleft>f : a \<rightarrow> a'\<guillemotright> \<and> G.B.iso f)"
subsFAll_def: fixes atms :: "(signs \<times> predicate \<times> vbl list) list"   and nAs :: "(nat \<times> formula) list"   and n :: "nat"   and z :: "signs"   and A :: "formula"   and frees :: "vbl set" shows "subsFAll atms nAs n z A frees = (case z of Pos \<Rightarrow> {let v = freshVar frees in (atms, (0, instanceF v A) # nAs)} | Neg \<Rightarrow> {(atms, (0, instanceF (X n) A) # nAs @ [(Suc n, FAll Neg A)])})"
is_ub_lub: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "f n << lub f"
rat_LP_def: fixes A :: "rat mat"   and b :: "rat vec"   and c :: "rat vec"   and m :: "nat"   and n :: "nat" shows "rat_LP A b c m n \<equiv> abstract_LP A b c m n"
up_below: fixes x :: "'b"   and y :: "'b" shows "(up\<cdot>x \<sqsubseteq> up\<cdot>y) = (x \<sqsubseteq> y)"
eucl_ivals_4: fixes a :: "'a" shows "{a..} \<in> sets borel"
set_rel_empty_iff_1: fixes y :: "'b set"   and A :: "('a \<times> 'b) set" shows "(({}, y) \<in> \<langle>A\<rangle>set_rel) = (y = {})"
assms: shows "set {a} \<in>\<^sub>\<circ> Vset \<alpha>"
non_neutral': fixes g :: "'c \<Rightarrow> 'b"   and I :: "'c set" shows "sum' g {x \<in> I. g x \<noteq> (0::'b)} = sum' g I"
P_LCons': 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 "P = LCons v0 (LCons (vmc_path_no_deadend.w0 P) (ltl (ltl P)))"
e_c_a_b: fixes c :: "'a"   and a :: "'a"   and b :: "'a" shows "e_aux c a b = e_aux a b c"
induct: fixes P :: "complex \<Rightarrow> nat \<Rightarrow> (complex \<times> nat) list \<Rightarrow> bool"   and a0 :: "complex"   and a1 :: "nat"   and a2 :: "(complex \<times> nat) list" assumes "\<And>x i y j yjs. \<lbrakk>\<lbrakk>x = y; Suc j \<noteq> i; \<not> i < Suc j\<rbrakk> \<Longrightarrow> P x (i - Suc j) yjs; x \<noteq> y \<Longrightarrow> P x i yjs\<rbrakk> \<Longrightarrow> P x i ((y, j) # yjs)"   and "\<And>uu_ uv_. P uu_ uv_ []" shows "P a0 a1 a2"
path_is_path: fixes \<sigma> :: "'var \<Rightarrow> 'val" shows "is_path (path \<sigma>)"
det_mult_row: fixes A :: "(('b, 'c) vec, 'c) vec"   and a :: "'c"   and k :: "'b" shows "det (mult_row A a k) = k * det A"
inject_2: fixes x2 :: "'d"   and y2 :: "'d" shows "(Abrupt x2 = Abrupt y2) = (x2 = y2)"
degen_visit_acc: fixes m :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b"   and q :: "'Q"   and n :: "nat"   and p :: "('Q \<times> nat) list"   and q' :: "'Q"   and n' :: "nat" assumes "path (g_E (degeneralize_ext m)) (q, n) p (q', n')"   and "n \<noteq> n'" shows "\<exists>qa. (qa, n) \<in> set p \<and> n \<in> acc qa"
map_closed: fixes a :: "'c set \<times> 'd" assumes "a \<in> Sigma {U. is_open\<^sub>Y U \<and> f x \<in> U} \<O>\<^sub>Y" shows "eqY.Class a \<in> eqY.Partition"
ex_le_of_int: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.archimedean_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "\<exists>z. less_eq x (ring_1.of_int one plus zero minus z)"
c_def: shows "c = (\<lambda>i. if i \<le> N then c_aux i else 0)"
braun_indices_if_braun: fixes t :: "'a tree" assumes "braun t" shows "braun_indices t = {1..size t}"
that: fixes tx :: "'a \<Rightarrow>\<^sub>0 nat"   and e :: "nat" assumes "tx \<in> .[X]"   and "t = monomial e x + tx" shows "thesis"
reachable_def: shows "local.reachable \<equiv> {v. (src, v) \<in> E\<^sup>*}"
atLeast_Suc_atMost: fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'b" assumes "m \<le> n" shows "prod g {m..n} = g m * prod g {Suc m..n}"
n_ivl_narrow: fixes x :: "ivl"   and y :: "ivl" assumes "\<not> x \<sqsubseteq> x \<triangle> y" shows "n_ivl (x \<triangle> y) < n_ivl x"
in_set_real_times_interval: fixes x :: "real"   and X :: "float interval"   and y :: "real"   and Y :: "float interval" assumes "x \<in>\<^sub>r X"   and "y \<in>\<^sub>r Y" shows "x * y \<in>\<^sub>r X * Y"
S_list_not_min_gallery_double_split: fixes ss :: "'a permutation list" assumes "ss \<in> lists (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"   and "ss \<noteq> []"   and "\<not> min_gallery (map (\<lambda>w. w `\<rightarrow> C0) (sums ss))" shows "\<exists>f g as s bs t cs. (f, g) \<in> foldpairs \<and> sum_list as `\<rightarrow> C0 \<in> f \<turnstile> \<C> \<and> sum_list (as @ [s]) `\<rightarrow> C0 \<in> g \<turnstile> \<C> \<and> sum_list (as @ [s] @ bs) `\<rightarrow> C0 \<in> g \<turnstile> \<C> \<and> sum_list (as @ [s] @ bs @ [t]) `\<rightarrow> C0 \<in> f \<turnstile> \<C> \<and> ss = as @ [s] @ bs @ [t] @ cs"
assms_1: fixes i :: "'a" assumes "i \<in> I" shows "x i \<in> carrier (G i)"
Inf_fun_def: fixes A :: "('b \<Rightarrow> 'c) set" shows "\<Sqinter> A = (\<lambda>x. \<Sqinter>f\<in>A. f x)"
relpow_Suc_I: fixes x :: "'a"   and y :: "'a"   and n :: "nat"   and R :: "('a \<times> 'a) set"   and z :: "'a" assumes "(x, y) \<in> R ^^ n"   and "(y, z) \<in> R" shows "(x, z) \<in> R ^^ Suc n"
supp_zerofun_subset_any: fixes A :: "'a set" shows "supp 0 \<subseteq> A"
sup_top_left: fixes x :: "'b" shows "\<top> \<squnion> x = \<top>"
simps_2: fixes xs :: "char list list"   and n :: "int" shows "merge_coeffs [(xs, n)] = [(xs, n)]"
bisim1AAccFail: fixes P :: "'a prog"   and h :: "'heap"   and a :: "'addr expr1"   and i :: "'addr expr1"   and ad :: "'addr"   and xs :: "'addr val list"   and v :: "'addr val"   and v' :: "'addr val" shows "P,a\<lfloor>i\<rceil>,h \<turnstile> (Throw ad, xs) \<leftrightarrow> ([v, v'], xs, length (compE2 a) + length (compE2 i), \<lfloor>ad\<rfloor>)"
simps_2: fixes pred :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and trn :: "('a, 'b, 'c) trans"   and func :: "('a, 'b, 'c) trans \<Rightarrow> 'd"   and tr :: "('a, 'b, 'c) trans list" assumes "\<not> pred trn" shows "BD_Security.filtermap pred func (trn # tr) = BD_Security.filtermap pred func tr"
wf_pre_RedBlack_def: fixes prb :: "('d, 'e, 'f) pre_RedBlack" shows "wf_pre_RedBlack prb \<equiv> (wf_rgraph (red prb) \<and> wf_lts (black prb)) \<and> fst (root (red prb)) = init (black prb) \<and> (\<forall>e. e \<in> edges (red prb) \<longrightarrow> ui_edge e \<in> edges (black prb))"
Special_Solutions_in_Minimal_Solutions: fixes x :: "nat list"   and y :: "nat list" assumes "(x, y) \<in> Special_Solutions" shows "(x, y) \<in> local.Minimal_Solutions"
uc_par_meet: fixes x :: "('a \<times> 'a set) set"   and y :: "('a \<times> 'a set) set" shows "x \<parallel> Multirelations.U \<inter> (y \<parallel> Multirelations.U) = x \<parallel> Multirelations.U \<parallel> y \<parallel> Multirelations.U"
append_self_conv2: fixes xs :: "'a list"   and ys :: "'a list" shows "(xs @ ys = ys) = (xs = [])"
INF_mono: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and B :: "'b set"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. less_eq (f n) (g m)" shows "less_eq (Inf (f ` A)) (Inf (g ` B))"
set_embed_pmf: fixes f :: "'c \<Rightarrow> real" assumes "\<And>x. 0 \<le> f x"   and "\<integral>\<^sup>+ x. ennreal (f x) \<partial>count_space UNIV = 1" shows "set_pmf (embed_pmf f) = {x. f x \<noteq> 0}"
translate_eoracle_def: fixes translator :: "'t \<Rightarrow> 'v \<Rightarrow> ('s list \<times> 't) spmf"   and eoracle :: "'t \<times> 'u \<Rightarrow> 'w \<Rightarrow> (('r \<times> 'v list) \<times> 't \<times> 'u) spmf"   and state :: "'t \<times> 'u"   and inp :: "'w" shows "translate_eoracle translator eoracle state inp \<equiv> eoracle state inp \<bind> (\<lambda>((out, e_in), s). let conc = \<lambda>(es, st) e. map_spmf (map_prod ((@) es) id) (translator st e) in foldl_spmf conc (return_spmf ([], fst s)) e_in \<bind> (\<lambda>(e_out, s_event). return_spmf ((out, e_out), s_event, snd s)))"
cases: fixes a1 :: "'a list"   and a2 :: "('b, 'c) sequent" assumes "(a1, a2) \<in> Ax" obtains i :: "nat" where "a1 = []" and "a2 = ( \<LM> At i  \<RM> \<Rightarrow>* \<LM> At i  \<RM>)"   | "a1 = []" and "a2 = ( \<LM> ff  \<RM> \<Rightarrow>* \<Empt>)"
zero_fps_def: shows "0 = Abs_fps (\<lambda>n. 0::'d)"
simps: fixes xs :: "('a \<times> 'b) list"   and ko :: "'c" shows "oalist_raw.tl_raw (xs, ko) = (tl xs, ko)"
finite_dimensional_vector_space_def: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and Basis :: "'e set" shows "finite_dimensional_vector_space scale Basis \<equiv> vector_space scale \<and> finite_dimensional_vector_space_axioms scale Basis"
continuous_on_Re: fixes s :: "'b set"   and g :: "'b \<Rightarrow> complex" assumes "continuous_on s g" shows "continuous_on s (\<lambda>x. Re (g x))"
a0n_inv1_niagreeI: fixes x :: "'b a0n_state" assumes "\<And>h d. \<lbrakk>set h \<subseteq> good; d \<notin> corrupted x; 0 < signals x (Commit h d)\<rbrakk> \<Longrightarrow> 0 < signals x (Running h d)" shows "x \<in> a0n_inv1_niagree"
assms_2: shows "\<forall>j<k. g j = h j"
decliface_simp: fixes I :: "qtname"   and m :: "'b decl_scheme" shows "decliface (I, m) = I"
spn_simps_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b" assumes "bicategory_of_spans V H \<a> \<i> src trg"   and "partial_magma.arr V \<mu>" shows "partial_magma.ide V (bicategory_of_spans.spn V H \<a> \<i> src trg \<mu>)"
subdegree_def: fixes f :: "'a fps" shows "subdegree f = (if f = 0 then 0 else LEAST n. f $ n \<noteq> (0::'a))"
except_uminus: fixes p :: "'c \<Rightarrow>\<^sub>0 'd"   and S :: "'c set" shows "except (- p) S = - except p S"
elims: fixes x :: "('c, 'd) lit"   and y :: "Preliminaries.var set" assumes "varsL x = y" obtains at :: "('c, 'd) atm" where "x = Pos at__" and "y = varsA at__"   | at :: "('c, 'd) atm" where "x = Neg at" and "y = varsA at"
mulex_on_add_mset_mono: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "'b"   and n :: "'b"   and A :: "'b set"   and M :: "'b multiset"   and N :: "'b multiset" assumes "P m n"   and "m \<in> A"   and "n \<in> A"   and "mulex_on P A M N" shows "mulex_on P A (add_mset m M) (add_mset n N)"
assms_2: shows "k *\<^sub>s\<^sub>m A = B"
dsumag_mOp: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)" shows "mop (a\<Oplus>\<^bsub>I\<^esub> A) = prod_mOp I A"
is_processT5_S7: fixes t :: "'c event list"   and P :: "'c process"   and A :: "'c event set" assumes "t \<in> T P"   and "(t, A) \<notin> F P" shows "\<exists>x. x \<in> A \<and> t @ [x] \<in> T P"
linear_continuous_on: fixes f :: "'c \<Rightarrow> 'd"   and s :: "'c set" assumes "bounded_linear f" shows "continuous_on s f"
fs_def: shows "fs = complex_poly (min_int_poly (Re x)) (min_int_poly (Im x))"
surj_hom_induced_retraction_map: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and p :: "int" assumes "retraction_map X Y f" shows "carrier (homology_group p Y) = hom_induced p X {} Y {} f ` carrier (homology_group p X)"
marker_must_stay_if_no_snapshot: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and i :: "nat"   and p :: "'d"   and q :: "'d"   and cid :: "nat" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final"   and "ps (distributed_system.s channel trans send recv init t i) p \<noteq> None"   and "\<not> ps (distributed_system.s channel trans send recv init t i) q \<noteq> None"   and "channel cid = Some (p, q)" shows "Marker \<in> set (msgs (distributed_system.s channel trans send recv init t i) cid)"
Ln_exp: fixes z :: "complex" assumes "- pi < Im z"   and "Im z \<le> pi" shows "Ln (exp z) = z"
subst_compose_fv: fixes \<theta> :: "'c \<Rightarrow> ('b, 'a) Term.term"   and x :: "'c"   and \<sigma> :: "'a \<Rightarrow> ('b, 'd) Term.term" assumes "fv (\<theta> x) = {}" shows "fv ((\<theta> \<circ>\<^sub>s \<sigma>) x) = {}"
problem_plan_bound_s_LESS_EQ_problem_plan_bound_thm: fixes PROB :: "(('c, bool) fmap \<times> ('c, bool) fmap) set"   and s :: "('c, bool) fmap" assumes "finite PROB"   and "s \<in> valid_states PROB" shows "problem_plan_bound_s PROB s < problem_plan_bound PROB + 1"
word_le_minus_cancel: fixes y :: "'b word"   and x :: "'b word"   and z :: "'b word" assumes "y - x \<le> z - x"   and "x \<le> z" shows "y \<le> z"
butlast_idx': fixes xs :: "'b list"   and i :: "nat"   and y :: "'b" assumes "xs ! i = y"   and "i < length xs - 1"   and "1 < length xs" shows "y \<in> set (butlast xs)"
chamberD_simplex: fixes X :: "'b set set"   and x :: "'b set" assumes "ChamberComplex X"   and "SimplicialComplex.maxsimp X x" shows "x \<in> X"
pcr_stream_def: fixes T :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "pcr_stream T \<equiv> llist_all2 T OO cr_stream"
lselect_llength: fixes w :: "'b llist"   and s :: "nat set" shows "llength (lselect s w) = esize {i \<in> s. enat i < llength w}"
prems_12: shows "InvariantGetReasonIsReason (getReason state') (getF state') (getM state') (set (getQ state'))"
WTrt_binop_simps_17: fixes P :: "'b prog"   and T1 :: "ty"   and T2 :: "ty"   and T :: "ty" shows "(P \<turnstile> T1\<guillemotleft>ShiftRightSigned\<guillemotright>T2 : T) = (T1 = Integer \<and> T2 = Integer \<and> T = Integer)"
measurable_lift: fixes a :: "'b set set"   and f :: "'b \<Rightarrow> 'c"   and b :: "'c set set" assumes "sigma_algebra a"   and "f \<in> measurable a b" shows "f \<in> measurable a (sigma b)"