hcju/isabelleps huggingface dataset:
irreducible\<^sub>d_dvd_smult: fixes p :: "'c poly"   and q :: "'c poly" assumes "0 < degree p"   and "irreducible\<^sub>d q"   and "p dvd q" shows "\<exists>c. c \<noteq> (0::'c) \<and> q = smult c p"
ln_float_interval_eq_Some_conv: fixes p :: "nat"   and x :: "float interval"   and y :: "float interval" shows "(ln_float_interval p x = Some y) = (lb_ln p (lower x) = Some (lower y) \<and> ub_ln p (upper x) = Some (upper y))"
inbetween_inset: fixes x :: "'a"   and xs :: "'a list"   and a :: "'a"   and b :: "'a" assumes "x \<in> set (between xs a b)" shows "x \<in> set xs"
chinese_remainder_nat: fixes A :: "'a set"   and m :: "'a \<Rightarrow> nat"   and u :: "'a \<Rightarrow> nat" assumes "finite A"   and "\<forall>i\<in>A. \<forall>j\<in>A. i \<noteq> j \<longrightarrow> coprime (m i) (m j)" shows "\<exists>x. \<forall>i\<in>A. [x = u i] (mod m i)"
n_ivl_mono: fixes x :: "ivl"   and y :: "ivl" assumes "x \<sqsubseteq> y" shows "n_ivl x \<le> n_ivl y"
prems_3: fixes f :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" assumes "f \<in> insert f__ F" shows "poly_deg f \<le> poly_deg f0"
pathin_subtopology: fixes X :: "'b topology"   and S :: "'b set"   and g :: "real \<Rightarrow> 'b" shows "pathin (subtopology X S) g = (pathin X g \<and> (\<forall>x\<in>{0..1}. g x \<in> S))"
finite_minimalE: fixes A :: "'c set"   and rel :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "finite A"   and "A \<noteq> {}"   and "irreflp rel"   and "transp rel" obtains a :: "'c" where "a \<in> A" and "\<And>b. rel b a \<Longrightarrow> b \<notin> A"
injectionsFromEmptyIsEmpty: fixes Y :: "'b list" shows "injectionsAlg [] Y = [{}]"
dist_triangle_half_r: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and y :: "'a"   and x1 :: "'a"   and e :: "real"   and x2 :: "'a" assumes "class.metric_space dist uniformity open"   and "dist y x1 < e / 2"   and "dist y x2 < e / 2" shows "dist x1 x2 < e"
conc_Un_distrib_2: fixes A :: "'b list set"   and B :: "'b list set"   and C :: "'b list set" shows "(A \<union> B) @@ C = A @@ C \<union> B @@ C"
inject_1: fixes x1 :: "exp0"   and y1 :: "exp0" shows "(Exp x1 = Exp y1) = (x1 = y1)"
assms_7: shows "line A B' = line A' B \<or> line B C' = line B' C \<or> line A C' = line A' C"
fold_fun_left_comm: fixes A :: "'d set"   and x :: "'d"   and z :: "'e" assumes "finite A" shows "f x (Finite_Set.fold f z A) = Finite_Set.fold f (f x z) A"
connected_components_of_eq_singleton: fixes X :: "'a topology"   and S :: "'a set" shows "(connected_components_of X = {S}) = (connected_space X \<and> topspace X \<noteq> {} \<and> S = topspace X)"
op_dg_components_3: fixes \<CC> :: "V" shows "op_dg \<CC>\<lparr>Dom\<rparr> = \<CC>\<lparr>Cod\<rparr>"
toList_ODList: fixes xs :: "'c list" shows "toList (ODList xs) = msort xs"
simps: fixes ires :: "cell list"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wcode_stop ires rs (l, r) = (\<exists>ln rn. l = Bk # ires \<and> r = Bk # Oc # Bk \<up> ln @ Bk # Bk # Oc \<up> Suc rs @ Bk \<up> rn)"
snoc_1: fixes v\<^sub>1 :: "'item list"   and v\<^sub>2 :: "'item list" assumes "u @ v\<^sub>1 =\<^sub>F u @ v\<^sub>2" shows "v\<^sub>1 =\<^sub>F v\<^sub>2"
integral_one_over_log_squared_bigo: shows "(\<lambda>x. integral {2..x} (\<lambda>t. 1 / (ln t)\<^sup>2)) \<in> O(\<lambda>x. x / (ln x)\<^sup>2)"
euler_def: fixes f :: "real \<Rightarrow> 'b \<Rightarrow> 'b" shows "euler f = grid_function (discrete_evolution (euler_increment f))"
ft_invar_def: fixes t :: "('a, 'b) FingerTreeStruc" shows "FingerTreeStruc.ft_invar t \<equiv> FingerTreeStruc.is_leveln_ftree 0 t \<and> FingerTreeStruc.is_measured_ftree t"
Melem_subst_atm_mset: fixes A :: "'a"   and AA :: "'a multiset"   and \<sigma> :: "'s" shows "(A \<in># AA \<cdot>am \<sigma>) = (\<exists>B. B \<in># AA \<and> A = B \<cdot>a \<sigma>)"
bind_eq_Nil_iff: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'a list" shows "(List.bind xs f = []) = (\<forall>x\<in>set xs. f x = [])"
ends_del_vert: fixes u :: "'a" shows "arc_to_ends (H.del_vert u) = arc_to_ends (del_vert u)"
simps: fixes a :: "'b" shows "algebraic_int a = (\<exists>p x. a = x \<and> lead_coeff p = (1::'b) \<and> (\<forall>i. coeff p i \<in> \<int>) \<and> poly p x = (0::'b))"
schroeder_3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x * y \<le> z) = (x\<^sup>T * - z \<le> - y)"
eq_infE: fixes w\<^sub>1 :: "'item stream"   and w\<^sub>2 :: "'item stream" assumes "w\<^sub>1 =\<^sub>I w\<^sub>2" obtains "w\<^sub>1 \<preceq>\<^sub>I w\<^sub>2" and "w\<^sub>2 \<preceq>\<^sub>I w\<^sub>1"
R2_imp_P2: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat option" assumes "f \<in> \<R>\<^sup>2" shows "f \<in> \<P>\<^sup>2"
Gromov_product_at_quasi_isometry_2: fixes lambda :: "real"   and C :: "real"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "lambda C -quasi_isometry f" shows "Gromov_product_at (f x) (f y) (f z) \<le> lambda * Gromov_product_at x y z + 187 * lambda\<^sup>2 * (C + deltaG TYPE('c) + deltaG TYPE('d))"
correct_closed: shows "\<forall>p s t. s \<le> t \<and> correct p t \<longrightarrow> correct p s"
ra_ia_rb_ib_rc_ic_def_3: shows "rb = Re b"
acyclic_imp_goal: assumes "acyclic step_rel" shows "goal"
distinct_5: fixes x1 :: "'d" shows "Normal x1 \<noteq> Stuck"
antisym: fixes x :: "'a"   and y :: "'a" assumes "less_eq0 x y"   and "less_eq0 y x" shows "x = y"
assms_1: shows "(\<gamma> \<and>\<^sup>c \<gamma>') t n"
PP_inject: fixes x :: "'a \<Rightarrow>\<^sub>0 'b"   and y :: "'a \<Rightarrow>\<^sub>0 'b" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "(PP x = PP y) = (x = y)"
P: fixes f :: "'b \<Rightarrow> 'c" assumes "continuous_on t f"   and "f ` t \<subseteq> U"   and "Q f" shows "P (f \<circ> h)"
fract_poly_add: fixes p :: "'a poly"   and q :: "'a poly" shows "map_poly to_fract (p + q) = map_poly to_fract p + map_poly to_fract q"
mat_def: shows "Matrix.mat \<equiv> map_fun id (map_fun id (map_fun id Abs_mat)) (\<lambda>nr nc f. (nr, nc, mk_mat nr nc f))"
mono_derived: fixes K :: "'a set"   and H :: "'a set" assumes "K \<subseteq> H" shows "derived G K \<subseteq> derived G H"
x0: shows "0 < real_of_1 (p, l, r)"
nsum_suc: fixes R :: "('d, 'e) aGroup_scheme"   and f :: "nat \<Rightarrow> 'd"   and n :: "nat" shows "\<Sigma>\<^sub>e R f Suc n = \<Sigma>\<^sub>e R f n \<plusminus>\<^bsub>R\<^esub> f (Suc n)"
val_simps_2: fixes m :: "'c tm"   and f :: "'c \<Rightarrow> 'b tm" shows "Time_Monad.val (m \<bind> f) = (let x = Time_Monad.val m in Time_Monad.val (f x))"
has_density_equal_on_space: fixes M :: "'a measure"   and N :: "'a measure"   and f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal" assumes "has_density M N f"   and "\<And>x. x \<in> space N \<Longrightarrow> f x = g x" shows "has_density M N g"
simps_3: fixes v :: "val" shows "fv (Val v) = {}"
diff_fun_const: fixes c :: "'d" shows "diff_fun k charts (\<lambda>x. c)"
gtr2gtrp: fixes s :: "'f list"   and c :: "'f list multiset"   and w :: "('g, 'h) label list list"   and c' :: "'f list multiset"   and fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme" assumes "(add_mset s c, w, c') \<in> trcl (ntr fg)" obtains s' :: "'f list" and ce' :: "'f list multiset" and ww :: "('g, 'h) label list el_step list" where "c' = add_mset s' ce'" and "w = map le_rem_s ww" and "((s, c), ww, s', ce') \<in> trcl (ntrp fg)"
fls_regpart_subdegree_conv: fixes f :: "'a fls" assumes "0 \<le> fls_subdegree f" shows "subdegree (fls_regpart f) = nat (fls_subdegree f)"
arrI: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>" shows "Adj.C.arr f"
holomorphic_zeta: fixes A :: "complex set" assumes "1 \<notin> A" shows "zeta holomorphic_on A"
elims: fixes x :: "w_node"   and xa :: "nat"   and y :: "w_node" assumes "x \<oplus> xa = y" obtains l :: "nat" and i :: "nat" where "x = (_ l _)" and "xa = i__" and "y = (_ l + i__ _)"   | i :: "nat" where "x = (_Entry_)" and "xa = ia__" and "y = (_Entry_)"   | i :: "nat" where "x = (_Exit_)" and "xa = i" and "y = (_Exit_)"
C: shows "ipurge_tr_aux I D (the ` U) (map (inv p) xs) = map (inv p) (ipurge_tr_aux (con_comp_pol I) (con_comp_map D E p q) U xs)"
simps_1: fixes f :: "'fsym" shows "TE_resOf (Oldf f) = resOf f"
lprefix_ltlI: fixes xs :: "'b llist"   and ys :: "'b llist" assumes "lprefix xs ys" shows "lprefix (ltl xs) (ltl ys)"
ptensor_vec_def: fixes v1 :: "'b vec"   and v2 :: "'b vec" shows "ptensor_vec v1 v2 = partial_state.tensor_vec dims0 vars1' v1 v2"
fold_insert: fixes A :: "'b fset set set"   and x :: "'b fset set"   and z :: "'b fset set" assumes "finite A"   and "x \<notin> A" shows "Finite_Set.fold (\<otimes>) z (insert x A) = x \<otimes> Finite_Set.fold (\<otimes>) z A"
inner_node_Entry_edge: fixes n :: "'b" assumes "inner_node n" obtains a :: "'a" where "valid_edge a" and "intra_kind (kind a)" and "inner_node (targetnode a)" and "sourcenode a = Entry"
distinct_7: fixes x1 :: "agent"   and x5 :: "msg" shows "msg.Agent x1 \<noteq> Hash x5"
component_leaks_stateful_def: fixes n :: "'a"   and \<A> :: "('a strand_label \<times> ('fun, 'var) stateful_strand_step) list"   and Sec :: "('fun, 'var) Term.term set" shows "component_leaks_stateful n \<A> Sec \<equiv> \<exists>\<A>' \<I>\<^sub>\<tau>. (subst_domain \<I>\<^sub>\<tau> = \<V> \<and> ground (subst_range \<I>\<^sub>\<tau>)) \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<I>\<^sub>\<tau> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<I>\<^sub>\<tau>) \<and> prefix \<A>' \<A> \<and> (\<exists>t\<in>Sec - declassified\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A>' \<I>\<^sub>\<tau>. \<lbrakk>{}; {}; proj_unl n \<A>' @ [send\<langle>t\<rangle>]\<rbrakk>\<^sub>s \<I>\<^sub>\<tau>)"
assms_1: fixes v :: "i"   and x :: "\<nu>" shows "[\<lparr>A!,x\<^sup>P\<rparr> \<^bold>\<equiv> \<^bold>\<not>\<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr> in v]"
disj_2: shows "h\<^sub>1 \<bottom> h"
init_config_def: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c :: "('d, 'e, 'f) configuration_scheme" assumes "dataflow_topology summary results_in" shows "init_config c \<equiv> \<forall>loc. c_imp c loc = {#}\<^sub>z \<and> c_work c loc = zmset_of (mset_set (set_antichain (frontier (c_pts c loc))))"
to_fun_monic_monom_diff: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c"   and n :: "nat" assumes "UP_cring R"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "\<exists>c. c \<in> carrier R \<and> UP_cring.to_fun R (monom (UP R) \<one>\<^bsub>R\<^esub> n) a \<ominus>\<^bsub>R\<^esub> UP_cring.to_fun R (monom (UP R) \<one>\<^bsub>R\<^esub> n) b = (a \<ominus>\<^bsub>R\<^esub> b) \<otimes>\<^bsub>R\<^esub> c"
acc_sim2: fixes r :: "nat \<Rightarrow> 'v" assumes "G'.is_acc r" shows "is_acc (fi \<circ> r)"
ir_pre: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and P' :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "ir_hoare P c c' Q"   and "\<And>s s'. P s s' \<Longrightarrow> P' s s'" shows "ir_hoare P' c c' Q"
raise_iff: fixes P :: "assn"   and s :: "String.literal"   and Q :: "'d \<Rightarrow> assn" shows "<P> raise s <Q> = (P = false)"
zs1: shows "remdups_adj (xs1 @ xs @ xs2) = zs1 @ remdups_adj (xs @ xs2)"
k: shows "k \<noteq> 0 \<and> H2 = cor k *\<^sub>s\<^sub>m H1"
sig_trd_spp_aux_def: shows "sig_trd_spp_aux = tailrec.fun (\<lambda>x. fst x = 0) snd sig_trd_spp_body"
BW_meas: fixes n :: "nat" shows "BW n \<in> sets M"
simps_1: fixes w :: "nat"   and b :: "'a"   and a :: "'a" shows "depth (Leaf w b) a = 0"
cycle_freeD: fixes g :: "'b ugraph"   and u :: "'b"   and p :: "('b \<times> 'b) list" assumes "cycle_free g"   and "path g u p u"   and "p \<noteq> []"   and "simple p" shows "False"
le_cINF_iff: fixes A :: "'k set"   and f :: "'k \<Rightarrow> 'l multiset"   and u :: "'l multiset" assumes "A \<noteq> {}"   and "subset_mset.bdd_below (f ` A)" shows "(u \<subseteq># Inf (f ` A)) = (\<forall>x\<in>A. u \<subseteq># f x)"
divide_divide_eq_left: 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 a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide (divide a b) c = divide a (times b c)"
vimage_singleton_eq: fixes a :: "'d"   and f :: "'d \<Rightarrow> 'e"   and b :: "'e" shows "(a \<in> f -` {b}) = (f a = b)"
fdemodalisation2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(antidomain_semiringl.fdia a (\<cdot>) x y \<le> a (a z)) = (x \<cdot> a (a y) \<le> a (a z) \<cdot> x)"
reds_preserves_lconf: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and es :: "expr list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and es' :: "expr list"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option"   and Ts :: "ty list" assumes "wwf_prog P"   and "P,E \<turnstile> \<langle>es,(h, l)\<rangle> [\<rightarrow>] \<langle>es',(h', l')\<rangle>"   and "P,E,h \<turnstile> es [:] Ts"   and "P,h \<turnstile> l (:\<le>)\<^sub>w E"   and "P \<turnstile> E \<surd>" shows "P,h' \<turnstile> l' (:\<le>)\<^sub>w E"
Complete_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and k :: "nat"   and I :: "item set" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "Complete k I = I \<union> {uu_. \<exists>x y. uu_ = inc_item x k \<and> x \<in> bin I (item_origin y) \<and> y \<in> bin I k \<and> is_complete y \<and> next_symbol x = Some (item_nonterminal y)}"
char_one: shows "principal_char (G\<lparr>carrier := {\<one>}\<rparr>) \<one>\<^bsub>G\<lparr>carrier := {\<one>}\<rparr>\<^esub> = 1"
exec_moves_def: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and es :: "'addr expr1 list" shows "exec_moves ci P t es \<equiv> exec_meth ci (compP2 P) (compEs2 es) (compxEs2 es 0 0) t"
I: shows "strip_all_vars (mk_all_list vs t) = rev (map snd vs)"
assms_4: shows "qf_1 = (\<lambda>x y. x\<^sup>2 + y\<^sup>2 + b1 / A1 * x + b2 / A1 * y + c / A1)"
\<Sigma>_protocols_to_commitments_def: fixes init :: "'b \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'e) spmf"   and response :: "'d \<Rightarrow> 'c \<Rightarrow> 'f \<Rightarrow> 'g spmf"   and check :: "'b \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> bool"   and Rel :: "('b \<times> 'c) set"   and S_raw :: "'b \<Rightarrow> 'f \<Rightarrow> ('e \<times> 'g) spmf"   and \<A>ss :: "'b \<Rightarrow> 'e \<times> 'f \<times> 'g \<Rightarrow> 'e \<times> 'f \<times> 'g \<Rightarrow> 'c spmf"   and challenge_space :: "'f set"   and valid_pub :: "'b set"   and G :: "('b \<times> 'c) spmf" shows "\<Sigma>_protocols_to_commitments init response check Rel S_raw \<A>ss challenge_space valid_pub G \<equiv> \<Sigma>_protocols_base Rel valid_pub \<and> \<Sigma>_protocols_to_commitments_axioms init response check Rel S_raw \<A>ss challenge_space valid_pub G"
P_subset_A: shows "P \<subseteq> A\<^sup>\<spadesuit>"
assms_4: shows "support_set pf \<subseteq> stocks Mkt"
lex_pp_plus_monotone: fixes s :: "('a, 'b) pp"   and t :: "('a, 'b) pp"   and u :: "('a, 'b) pp" assumes "lex_pp s t" shows "lex_pp (s + u) (t + u)"
nop: shows "\<lparr>[$Nop]\<rparr> \<leadsto> \<lparr>[]\<rparr>"
sat_push_nonsat_potential: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and e :: "nat \<times> nat" assumes "Height_Bounded_Labeling c s t f l"   and "Network.sat_push_precond c f l e" shows "Network.nonsat_potential c (Network.push_effect c f e) l \<le> Network.nonsat_potential c f l + 2 * card (Graph.V c)"
HEndPhase0_HInv1: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HInv1 s"   and "HEndPhase0 s s' p" shows "HInv1 s'"
cases: fixes r :: "('a, 'b, 'c, 'd, 'e) SPRstate" obtains sprFst :: "('a, 'b, 'c, 'd, 'e) BEState" and sprLst :: "('a, 'b, 'c, 'd, 'e) BEState" and sprCRel :: "(('a, 'b, 'c, 'd, 'e) BEState \<times> ('a, 'b, 'c, 'd, 'e) BEState) set" where "r = \<lparr>sprFst = sprFst, sprLst = sprLst, sprCRel = sprCRel\<rparr>"
induced_modulus_def: fixes d :: "nat" shows "\<Phi>.induced_modulus d = (d dvd d__ \<and> (\<forall>a. coprime a d__ \<and> [a = 1] (mod d) \<longrightarrow> \<Phi> a = 1))"
rep_eq: fixes x :: "'a" shows "(*\<^sub>V) (cblinfun_scaleC_left x) = (\<lambda>y. y *\<^sub>C x)"
ub_rangeI: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'b \<Rightarrow> 'a"   and x :: "'a" assumes "class.po below"   and "\<And>i. below (S i) x" shows "po.is_ub below (range S) x"
topspace_discrete_topology: fixes U :: "'a set" shows "topspace (discrete_topology U) = U"
val_nonzero_noninf: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "x \<noteq> \<zero>" shows "v x \<noteq> \<infinity>"
homeomorphism_def: fixes s :: "'c set"   and t :: "'d set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" shows "homeomorphism s t f g = ((\<forall>x\<in>s. g (f x) = x) \<and> f ` s = t \<and> continuous_on s f \<and> (\<forall>y\<in>t. f (g y) = y) \<and> g ` t = s \<and> continuous_on t g)"
keys_tail_less_lt: fixes v :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "v \<in> keys (punit.tail p)" shows "v \<prec> punit.lt p"
real_sqrt_ge_abs1: fixes x :: "real"   and y :: "real" shows "\<bar>x\<bar> \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
last_stays_if_no_recv_marker_and_no_send: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and j :: "nat"   and cid :: "nat"   and p :: "'a"   and q :: "'a" assumes "trace init t final"   and "i < j"   and "j \<le> length t"   and "last (msgs (S t i) cid) = Marker"   and "Marker \<in> set (msgs (S t i) cid)"   and "Marker \<in> set (msgs (S t j) cid)"   and "\<forall>k. i \<le> k \<and> k < j \<longrightarrow> (\<nexists>u u' m. t ! k = Send cid p q u u' m)"   and "channel cid = Some (p, q)" shows "last (msgs (S t j) cid) = Marker"
euclidean_eq_iff: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and x :: "'b"   and y :: "'b" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "(x = y) = (\<forall>b\<in>Basis. inner x b = inner y b)"
andfold_MatchExp_normalized_normalized_n_primitive_single: fixes disc :: "'c \<Rightarrow> bool"   and C :: "'b \<Rightarrow> 'c"   and s :: "'c match_expr"   and xs :: "'b list"   and sel :: "'c \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> bool" assumes "\<forall>a. \<not> disc (C a)"   and "s \<in> set (normalize_match (andfold_MatchExp (map (Match \<circ> C) xs)))" shows "normalized_n_primitive (disc, sel) f s"
slice_tom_preds_vp: fixes S :: "'node SDG_node set"   and m :: "'node"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and m' :: "'node"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and cs :: "'edge list"   and rs :: "'edge list" assumes "S,slice_kind S \<turnstile> (m # ms,s) =as\<Rightarrow>* (m' # ms',s')"   and "valid_node m"   and "valid_call_list cs m"   and "\<forall>i<length rs. rs ! i \<in> get_return_edges (cs ! i)"   and "valid_return_list rs m"   and "length rs = length cs"   and "ms = targetnodes rs"   and "\<forall>mx\<in>set ms. \<exists>mx'. call_of_return_node mx mx' \<and> mx' \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>" obtains as' :: "'edge list" and cs' :: "'edge list" and rs' :: "'edge list" where "preds (slice_kinds S as') s" and "slice_edges S cs as' = as" and "m -as'\<rightarrow>* m'" and "valid_path_aux cs as'" and "upd_cs cs as' = cs'" and "valid_node m'" and "valid_call_list cs' m'" and "\<forall>i<length rs'. rs' ! i \<in> get_return_edges (cs' ! i)" and "valid_return_list rs' m'" and "length rs' = length cs'" and "ms' = targetnodes rs'" and "transfers (slice_kinds S as') s \<noteq> []" and "transfers (slice_kinds S (slice_edges S cs as')) s = transfers (slice_kinds S as') s"
hseqI: fixes \<mu> :: "'b"   and a :: "'b"   and b :: "'b"   and \<nu> :: "'b"   and c :: "'b" assumes "\<guillemotleft>\<mu> : a \<rightarrow>\<^sub>D b\<guillemotright>"   and "\<guillemotleft>\<nu> : b \<rightarrow>\<^sub>D c\<guillemotright>" shows "D.hseq \<nu> \<mu>"
mem_set_of_appr_appendE: fixes zs :: "real list"   and XS :: "'b list"   and YS :: "'b list" assumes "zs \<in> set_of_appr (XS @ YS)" obtains xs :: "real list" and ys :: "real list" where "zs = xs @ ys" and "xs \<in> set_of_appr XS" and "ys \<in> set_of_appr YS"
simps_8: fixes \<sigma> :: "(char list \<times> 'b list) trace"   and \<phi> :: "'b MFOTL.formula"   and I :: "\<I>"   and \<psi> :: "'b MFOTL.formula"   and j :: "nat" shows "Monitor.progress \<sigma> (formula.Since \<phi> I \<psi>) j = min (Monitor.progress \<sigma> \<phi> j) (Monitor.progress \<sigma> \<psi> j)"
simps_2: fixes v :: "String.literal"   and va :: "typ"   and u :: "Core.term" shows "Ct v va \<bullet> u = Ct v va $ u"
integrable_EM_remainder'_int: fixes a :: "int"   and b :: "int"   and f :: "real \<Rightarrow> 'a"   and n :: "nat" assumes "continuous_on {real_of_int a..real_of_int b} f" shows "(\<lambda>t. pbernpoly n t *\<^sub>R f t) integrable_on {real_of_int a..real_of_int b}"
supporters_def: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and x :: "'alt"   and y :: "'alt" assumes "is_pref_profile R" shows "supporters R x y = {i \<in> agents. R i y x}"
well_order_inductSL: fixes P :: "'a \<Rightarrow> bool"   and i :: "'a" assumes "\<And>i. \<lbrakk>local.aboveS i \<noteq> {}; P i\<rbrakk> \<Longrightarrow> P (succ i)"   and "\<And>i. \<lbrakk>isLim i; \<And>j. j \<in> local.underS i \<Longrightarrow> P j\<rbrakk> \<Longrightarrow> P i" shows "P i"
insert_eq_iff: fixes a :: "'a"   and A :: "'a set"   and b :: "'a"   and B :: "'a set" assumes "a \<notin> A"   and "b \<notin> B" shows "(insert a A = insert b B) = (if a = b then A = B else \<exists>C. A = insert b C \<and> b \<notin> C \<and> B = insert a C \<and> a \<notin> C)"
inv_inv_next_tame: fixes p :: "nat" shows "invariant Invariants.inv next_tame\<^bsub>p\<^esub>"
dataspace_def: shows "dataspace = {L |L. DataSpace.DataSpace L}"