hcju/isabelleps huggingface dataset:
sign_extended_def: fixes n :: "nat"   and w :: "'b word" shows "sign_extended n w \<equiv> \<forall>i>n. i < size w \<longrightarrow> w !! i = w !! n"
image_fst_zip: fixes ys :: "'b list"   and xs :: "'c list" assumes "length ys = length xs" shows "fst ` set (zip ys xs) = set ys"
last_in_set: fixes as :: "'k list" assumes "as \<noteq> []" shows "last as \<in> set as"
assms: fixes T :: "('a \<times> real) set"   and f :: "'a \<times> real \<Rightarrow> 'a"   and U :: "('a \<times> real) set" assumes "continuous_on T f"   and "f ` T \<subseteq> S"   and "closedin (top_of_set U) T" shows "\<exists>V g. T \<subseteq> V \<and> openin (top_of_set U) V \<and> continuous_on V g \<and> g ` V \<subseteq> S \<and> (\<forall>x\<in>T. g x = f x)"
disjnt_Union2: fixes B :: "'b set"   and \<A> :: "'b set set" shows "disjnt B (\<Union> \<A>) = (\<forall>A\<in>\<A>. disjnt B A)"
le_imp_less_or_eq: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.order less_eq less"   and "less_eq x y" shows "less x y \<or> x = y"
lookup_lift_poly_fun_2: fixes f :: "('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "lookup (lift_poly_fun_2 f p q) v = (lookup (f (proj_poly (component_of_term v) p) (proj_poly (component_of_term v) q)) (pp_of_term v) when component_of_term v \<in> keys (vectorize_poly p) \<union> keys (vectorize_poly q))"
theorem1: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "MinkowskiBetweenness \<E> \<P> betw"   and "betw a b c" shows "betw c b a \<and> \<not> betw b c a \<and> \<not> betw c a b"
lower_min_interval: fixes x :: "'a interval"   and y :: "'a interval" shows "lower (min_interval x y) = min (lower x) (lower y)"
tdghm_NTDGCod: shows "\<NN>\<lparr>NTDGCod\<rparr> = \<BB>"
flift1_Def: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "flift1 f\<cdot>(Def x) = f x"
topspace_map_topology: fixes f :: "'c \<Rightarrow> 'b"   and T :: "'c topology" shows "topspace (map_topology f T) = f ` topspace T"
digraph_iso_def: fixes G :: "('c, 'd) pre_digraph"   and H :: "('e, 'f) pre_digraph" shows "digraph_iso G H \<equiv> \<exists>f. pre_digraph.digraph_isomorphism G f \<and> H = app_iso f G"
prems: fixes m :: "nat" assumes "m < Suc n" shows "lnth xs m \<in> A"
g12D: shows "\<forall>x\<in>{0..1} - S. g1 +++ g2 differentiable at x"
S5safe: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> $S5 rmhist p \<and> ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<longrightarrow> [UNext memCh mm (resbar rmhist) p]_(rtrner memCh!p, resbar rmhist!p)"
is_path_undir_sym: fixes G :: "('c, 'd) graph"   and v :: "'c"   and p :: "('c \<times> 'd \<times> 'c) list"   and v' :: "'c" assumes "valid_graph G"   and "is_path_undir G v p v'" shows "is_path_undir G v' (rev (map (\<lambda>(u, w, u'). (u', w, u)) p)) v"
composite_with_leg0_is_map: shows "is_left_adjoint (s\<^sub>0 \<star> w)"
q_def: shows "q \<equiv> Poly (map (\<lambda>k. \<Sum>i\<le>min N k. c (N - i) * f (k - i)) [0..<N + m])"
int_2: fixes m :: "'b"   and n :: "'b" assumes "{m, n} \<subseteq> N" shows "B2 m \<inter> B2 n = {}"
totalizer_in_R2: fixes d :: "nat" shows "totalizer d \<in> \<R>\<^sup>2"
axioms_1: fixes X :: "'b set set" assumes "ChamberComplex X" shows "SimplicialComplex X"
compute_pdevs_domain: fixes xs :: "(nat, 'd) slist" shows "pdevs_domain (Pdevs xs) = set (map fst (list_of_slist xs))"
power_eq_prime_powerD: fixes p :: "'a"   and n :: "nat"   and x :: "'a"   and k :: "nat" assumes "prime p"   and "0 < n"   and "x ^ n = p ^ k" shows "\<exists>i. normalize x = normalize (p ^ i)"
rel_cases: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b sequent"   and b :: "'c sequent" assumes "rel_sequent R a b" obtains x1a :: "'b form multiset" and x2a :: "'b form multiset" and y1a :: "'c form multiset" and y2a :: "'c form multiset" where "a = ( x1a \<Rightarrow>* x2a)" and "b = ( y1a \<Rightarrow>* y2a)" and "rel_mset (rel_form R) x1a y1a" and "rel_mset (rel_form R) x2a y2a"
right_translate_sum: fixes G :: "'a set"   and g :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "Group G"   and "g \<in> G" shows "sum f G = (\<Sum>h\<in>G. f (h + g))"
A6': fixes a :: "'a"   and b :: "'a" assumes "B a b a" shows "a = b"
elem_mpj: fixes m :: "'a"   and H :: "'a set" assumes "m \<in> carrier M"   and "submodule R M H" shows "mpj M H m = m \<uplus>\<^bsub>M\<^esub> H"
hyperDiff1_def: fixes A :: "char list set option"   and a :: "char list" shows "A \<ominus> a \<equiv> case A of None \<Rightarrow> None | \<lfloor>A\<rfloor> \<Rightarrow> \<lfloor>A - {a}\<rfloor>"
inv': fixes x :: "'b" assumes "x \<in> space M" shows "f (f' x) = x"
widen_Array_Class: fixes G :: "prog"   and S :: "ty"   and C :: "qtname" assumes "G\<turnstile>S.[]\<preceq>Class C" shows "C = Object"
gamma_Step_subcomm: fixes num' :: "int \<Rightarrow> 'a"   and plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and \<gamma> :: "'a \<Rightarrow> int set"   and f1 :: "char list \<Rightarrow> aexp \<Rightarrow> (char list \<Rightarrow> int) set \<Rightarrow> (char list \<Rightarrow> int) set"   and f2 :: "char list \<Rightarrow> aexp \<Rightarrow> 'a st option \<Rightarrow> 'a st option"   and g1 :: "bexp \<Rightarrow> (char list \<Rightarrow> int) set \<Rightarrow> (char list \<Rightarrow> int) set"   and g2 :: "bexp \<Rightarrow> 'a st option \<Rightarrow> 'a st option"   and S :: "'a st option"   and C :: "'a st option acom" assumes "Gamma_semilattice num' plus' \<gamma>"   and "\<And>x e S. f1 x e (\<gamma>_option (\<gamma>_st \<gamma>) S) \<subseteq> \<gamma>_option (\<gamma>_st \<gamma>) (f2 x e S)"   and "\<And>b S. g1 b (\<gamma>_option (\<gamma>_st \<gamma>) S) \<subseteq> \<gamma>_option (\<gamma>_st \<gamma>) (g2 b S)" shows "Step f1 g1 (\<gamma>_option (\<gamma>_st \<gamma>) S) (ACom.map_acom (\<gamma>_option (\<gamma>_st \<gamma>)) C) \<le> ACom.map_acom (\<gamma>_option (\<gamma>_st \<gamma>)) (Step f2 g2 S C)"
dbl_dec_def: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_dec minus plus one x = minus (plus x x) one"
vertex_web_of_network: fixes \<Delta> :: "('a, 'b) network_scheme"   and x :: "'a"   and y :: "'a" shows "vertex (web_of_network \<Delta>) (x, y) = (edge \<Delta> x y \<and> (\<exists>z. edge \<Delta> y z \<or> edge \<Delta> z x))"
strand_sem_append_1: fixes M :: "('fun, 'var) Term.term set"   and S :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S' :: "('fun, 'var) strand_step list" assumes "\<lbrakk>M; S\<rbrakk>\<^sub>c \<theta>"   and "\<lbrakk>M \<union> (ik\<^sub>s\<^sub>t S \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>); S'\<rbrakk>\<^sub>c \<theta>" shows "\<lbrakk>M; S @ S'\<rbrakk>\<^sub>c \<theta>"
distinct_4: fixes x1 :: "nat" shows "(_Exit_) \<noteq> (_ x1 _)"
nonneg_mat_power: fixes A :: "'a mat"   and n :: "nat"   and k :: "nat" assumes "A \<in> carrier_mat n n"   and "nonneg_mat A" shows "nonneg_mat (A ^\<^sub>m k)"
eq_0_height: fixes kdt :: "'b kdt" shows "(0 = height kdt) = (\<exists>p. kdt = Leaf p)"
intro: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<iota> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c"   and u :: "'c"   and w :: "'c"   and \<theta> :: "'c"   and w' :: "'c"   and \<theta>' :: "'c"   and \<beta> :: "'c" assumes "uw\<theta>w'\<theta>' V H \<a> \<iota> src trg r \<rho> f g u w \<theta> w' \<theta>'"   and "uw\<theta>w'\<theta>'\<beta>_axioms V H \<a> r \<rho> f g u w \<theta> w' \<theta>' \<beta>" shows "uw\<theta>w'\<theta>'\<beta> V H \<a> \<iota> src trg r \<rho> f g u w \<theta> w' \<theta>' \<beta>"
s_cond_def_2: fixes t :: "real" assumes "0 \<le> t"   and "t \<le> t_stop" shows "s t = p t"
assms: fixes m :: "'a"   and n :: "'a" assumes "m |\<in>| M"   and "n |\<in>| M" shows "P m n"
arcs'_decomp: fixes xs :: "nat list"   and i :: "nat"   and j :: "nat" assumes "1 < length xs"   and "(i, j) \<in> arcs' xs" shows "\<exists>zs ys. xs = zs @ i # j # ys"
simps: fixes cs :: "('a \<times> constraint) list" shows "init_simplex cs = (let tons_cs = to_ns cs in Simplex_State (map snd tons_cs, case preprocess tons_cs of (t, as, trans_v, ui) \<Rightarrow> ((create_map as, trans_v, remdups ui), init_state t)))"
trans: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "R (h x) (h y)"   and "R (h y) (h z)" shows "R (h x) (h z)"
rep_number_unfold_set: fixes x :: "'a" assumes "x \<in> \<V>" shows "int (size (filter_mset ((\<in>) x) \<B>)) = \<Lambda> * (\<v> - 1) div (\<k> - 1)"
list_match_cong: fixes l :: "'a list"   and l' :: "'b list"   and A :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and A' :: "'a \<Rightarrow> 'b \<Rightarrow> assn" assumes "\<And>x x'. \<lbrakk>x \<in> set l; x' \<in> set l'\<rbrakk> \<Longrightarrow> hn_ctxt A x x' \<Longrightarrow>\<^sub>t hn_ctxt A' x x'" shows "hn_ctxt (list_assn A) l l' \<Longrightarrow>\<^sub>t hn_ctxt (list_assn A') l l'"
sigma_finite_embed_measure: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b" assumes "sigma_finite_measure M"   and "inj f" shows "sigma_finite_measure (embed_measure M f)"
linear_injective_left_inverse: fixes s1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and s2 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c" assumes "vector_space_pair s1 s2"   and "linear s1 s2 f"   and "inj f" shows "\<exists>g. linear s2 s1 g \<and> g \<circ> f = id"
elims_3: fixes x :: "'a formula" assumes "\<not> is_disj x" obtains F :: "'a formula" and G :: "'a formula" where "x = F \<^bold>\<or> G" and "\<not> (is_lit_plus F \<and> is_disj G)"   | v :: "'a" where "x = Atom v__" and "\<not> is_lit_plus (Atom v__)"   | "x = \<bottom>" and "\<not> is_lit_plus \<bottom>"   | v :: "'a formula" where "x = \<^bold>\<not> va__" and "\<not> is_lit_plus (\<^bold>\<not> va__)"   | v :: "'a formula" and va :: "'a formula" where "x = vb__ \<^bold>\<and> vaa__" and "\<not> is_lit_plus (vb__ \<^bold>\<and> vaa__)"   | v :: "'a formula" and va :: "'a formula" where "x = v \<^bold>\<rightarrow> va" and "\<not> is_lit_plus (v \<^bold>\<rightarrow> va)"
obtain_set_qFresh_card_of: fixes V :: "'d set"   and XS :: "('f, 'g, 'e, 'd, 'h) qTerm set"   and xs :: "'e" assumes "FixVars TYPE('d) TYPE('e)"   and "|V| <o |UNIV|"   and "|XS| <o |UNIV|"   and "\<forall>X\<in>XS. FixVars.qGood X" shows "\<exists>W. infinite W \<and> W \<inter> V = {} \<and> (\<forall>x\<in>W. \<forall>X\<in>XS. qAFresh xs x X \<and> qFresh xs x X)"
not_thesis: shows "(sint a sdiv sint b < 2 ^ (size a - 1)) \<noteq> (a \<noteq> - (2 ^ (size a - 1)) \<or> b \<noteq> - 1)"
product_cone_axioms: shows "product_cone J.comp (\<cdot>) (\<lambda>i. if i \<in> Collect J0.arr then D0 i else if i \<in> Collect J1.arr then D1 i else null) (local.dom p0) (\<lambda>i. if i \<in> Collect J0.arr then \<pi>0 i \<cdot> p0 else if i \<in> Collect J1.arr then \<pi>1 i \<cdot> p1 else null)"
sup2CI: fixes B :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b"   and A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<not> B x y \<Longrightarrow> A x y" shows "(A \<squnion> B) x y"
elims_3: fixes x :: "('g, 'h, 'i, 'sort, 'j, 'k, 'l) semDom_scheme"   and xa :: "'varSort \<times> 'sort"   and xb :: "('varSort, 'k) sAbs" assumes "\<not> sWlsAbs x xa xb" obtains SEM :: "('g, 'h, 'i, 'sort, 'j, 'k, 'l) semDom_scheme" and xs :: "'varSort" and s :: "'sort" and xs' :: "'varSort" and sF :: "'k \<Rightarrow> 'k" where "x = SEM" and "xa = (xs, s)" and "xb = sAbs xs' sF" and "\<not> (isInBar (xs, s) \<and> xs = xs' \<and> (\<forall>sX. if sWls SEM (asSort xs) sX then sWls SEM s (sF sX) else sF sX = sDummy SEM s))"
telescope_summable: fixes f :: "nat \<Rightarrow> 'b"   and c :: "'b" assumes "f \<longlonglongrightarrow> c" shows "summable (\<lambda>n. f (Suc n) - f n)"
simps_1: fixes i :: "nat"   and lev :: "nat"   and args :: "Core.term list" shows "subst_bvs1' (Bv i) lev args = (if i < lev then Bv i else if i - lev < length args then args ! (i - lev) else Bv (i - length args))"
if_eqvt: fixes pi :: "('d \<times> 'd) list"   and b :: "bool"   and c1 :: "'c"   and c2 :: "'c" shows "pi \<bullet> (if b then c1 else c2) = (if pi \<bullet> b then pi \<bullet> c1 else pi \<bullet> c2)"
compl_inf: fixes x :: "'a"   and y :: "'a" shows "- (x \<cdot> y) = - x + - y"
summable_zero_power: shows "summable ((^) (0::'a))"
geq: fixes w :: "complex" assumes "w \<in> ball \<xi> r1" shows "f w = (w - \<xi>) ^ n * g w"
minimal_sequence_to_failure_extending_implies_Rep_Cov: fixes V :: "'d list set"   and M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'f) FSM"   and vs :: "('d \<times> 'e) list"   and xs :: "('d \<times> 'e) list"   and FAIL :: "'f \<times> 'f"   and PM :: "('d, 'e, 'f \<times> 'f) FSM"   and \<Omega> :: "('d, 'e) ATC set"   and V'' :: "('d \<times> 'e) list set"   and xsR :: "('d \<times> 'e) list" assumes "minimal_sequence_to_failure_extending V M1 M2 vs xs"   and "OFSM M1"   and "OFSM M2"   and "test_tools M2 M1 FAIL PM V \<Omega>"   and "V'' \<in> N (vs @ xsR) M1 V"   and "prefix xsR xs" shows "\<not> Rep_Cov M2 M1 V'' vs xsR"
eventually_filtermap: fixes P :: "'b \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'b"   and F :: "'d filter" shows "eventually P (filtermap f F) = (\<forall>\<^sub>F x in F. P (f x))"
divisor_sigma_prime_power_nat: fixes p :: "nat"   and x :: "nat"   and k :: "nat" assumes "prime p" shows "divisor_sigma x (p ^ k) = (if x = 0 then Suc k else (p ^ (x * Suc k) - 1) div (p ^ x - 1))"
fun_relD: fixes f :: "'a \<Rightarrow> 'c"   and f' :: "'d \<Rightarrow> 'e"   and A :: "('a \<times> 'd) set"   and B :: "('c \<times> 'e) set"   and x :: "'a"   and x' :: "'d" assumes "(f, f') \<in> A \<rightarrow> B"   and "(x, x') \<in> A" shows "(f x, f' x') \<in> B"
iMin_insert: fixes I :: "'a set"   and n :: "'a" assumes "I \<noteq> {}" shows "iMin (insert n I) = min n (iMin I)"
surj_iff: fixes f :: "'b \<Rightarrow> 'a" shows "surj f = ((f \<circ>\<circ>\<circ> inv_into) UNIV f = id)"
inside_Int_outside: fixes S :: "'a set" shows "inside S \<inter> outside S = {}"
sub: fixes i :: "name"   and t :: "tm"   and u :: "tm"   and x :: "tm" assumes "atom i \<sharp> tms" shows "(P t u)(i::=x) = P (subst i x t) (subst i x u)"
connected_component_sym_eq: fixes S :: "'b set"   and x :: "'b"   and y :: "'b" shows "connected_component S x y = connected_component S y x"
1: shows "\<forall>l s. wpw b c (Suc (l (new Q))) Q l s \<longrightarrow> (\<exists>t. (c, s) \<Rightarrow> t \<and> wpw b c (l (new Q)) Q l t)"
Or\<^sub>n_semantics: fixes \<Phi> :: "'b ltln set"   and w :: "nat \<Rightarrow> 'b set" assumes "finite \<Phi>" shows "w \<Turnstile>\<^sub>n Or\<^sub>n \<Phi> = (\<exists>\<phi>\<in>\<Phi>. w \<Turnstile>\<^sub>n \<phi>)"
order_trans: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and y :: "'k"   and z :: "'k" assumes "class.preorder less_eq less"   and "less_eq x y"   and "less_eq y z" shows "less_eq x z"
a_ptr_disconnected_node_rel_finite: fixes h :: "(_) heap" shows "finite (local.a_ptr_disconnected_node_rel h)"
subst_term_single_no_change: fixes x :: "variable"   and \<tau> :: "typ"   and B :: "Core.term"   and t :: "Core.term" assumes "(x, \<tau>) \<notin> fv B" shows "subst_term [((x, \<tau>), t)] B = B"
in_homI: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "D.arr f"   and "D.dom f = a"   and "D.cod f = b" shows "\<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright>"
theI: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b" assumes "P a"   and "\<And>x. P x \<Longrightarrow> x = a" shows "P (THE x. P x)"
emeasure_eq_1_AE: fixes M :: "'c measure"   and S :: "'c set" assumes "prob_space M"   and "S \<in> sets M"   and "AE x in M. x \<in> S" shows "emeasure M S = 1"
strict_implies_not_eq: fixes a :: "'k multiset"   and b :: "'k multiset" assumes "a \<subset># b" shows "a \<noteq> b"
3: shows "- v * - v\<^sup>T \<sqinter> top * v\<^sup>T = bot"
get_def: fixes h :: "heap" shows "Ref.get h = from_nat \<circ> refs h TYPEREP('b) \<circ> addr_of_ref"
simps_5: fixes t :: "Core.term"   and u :: "Core.term"   and \<rho> :: "variable \<Rightarrow> String.literal set \<Rightarrow> typ" shows "tsubst (t $ u) \<rho> = tsubst t \<rho> $ tsubst u \<rho>"
d2: shows "degree_m [:k:] = degree_m (u * h)"
split_sel: fixes P :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd \<Rightarrow> 'a"   and prod :: "'b \<times> 'd" shows "P (case prod of (x, xa) \<Rightarrow> f x xa) = (prod = (fst prod, snd prod) \<longrightarrow> P (f (fst prod) (snd prod)))"
multiplicity_zero: fixes p :: "'a" shows "multiplicity p (0::'a) = 0"
radical_sqrt_linear_equation: fixes a :: "real"   and b :: "real"   and x :: "real" assumes "a \<in> radical_sqrt"   and "b \<in> radical_sqrt"   and "\<not> (a = 0 \<and> b = 0)"   and "a * x + b = 0" shows "x \<in> radical_sqrt"
properfactor_trans1: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "Group.monoid G"   and "a divides\<^bsub>G\<^esub> b"   and "properfactor G b c"   and "a \<in> carrier G"   and "c \<in> carrier G" shows "properfactor G a c"
trivial_implies_finite: fixes X :: "'a set" assumes "trivial X" shows "finite X"
mat_res_pointwise: fixes x :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'b"   and j :: "'b" shows "(x\<^sup>\<dagger> \<otimes> (x \<otimes> y)\<^sup>c) i j \<le> (y\<^sup>c) i j"
surjective_map_axioms: shows "surjective_map additive.kernel.Class R additive.kernel.Partition"
abs_eq: fixes xa :: "'a"   and x :: "'b" shows "monomial x xa = Abs_poly_mapping (\<lambda>k'. x when xa = k')"
nodes_inv: fixes ts :: "('c, 'd) Node list"   and n :: "nat" assumes "\<forall>x\<in>set ts. is_leveln_node n x \<and> is_measured_node x"   and "2 \<le> length ts" shows "\<forall>x\<in>set (nodes ts). is_leveln_node (Suc n) x \<and> is_measured_node x"
unary_wt_sym_0_gt\<^sub>h: fixes f :: "'s"   and g :: "'s" assumes "arity_sym\<^sub>h f = 1"   and "wt_sym f = 0" shows "f >\<^sub>s g \<or> g = f"
Init: fixes s :: "'b"   and F :: "'b program" assumes "s \<in> Init F" shows "s \<in> reachable F"
bags_edges: fixes v :: "'a"   and w :: "'a" assumes "v \<rightarrow> w" shows "\<exists>t\<in>V\<^bsub>T\<^esub>. v \<in> bag t \<and> w \<in> bag t"
closed_Int_compact: fixes s :: "'a set"   and t :: "'a set" assumes "closed s"   and "compact t" shows "compact (s \<inter> t)"
Max_wrt_among_total_preorder: fixes A :: "'a set" shows "Max_wrt_among le A = {x \<in> carrier \<inter> A. \<forall>y\<in>carrier \<inter> A. le y x}"
exhaust: fixes y :: "val" obtains "y = Unit"   | "y = Null"   | x3 :: "bool" where "y = Bool x3"   | x4 :: "int" where "y = Intg x4"   | x5 :: "nat \<times> char list list" where "y = Ref x5"
standard_sound: fixes P :: "'b \<Rightarrow> bool" shows "sound \<guillemotleft> P \<guillemotright>"
PiE_empty_domain: fixes T :: "'d \<Rightarrow> 'e set" shows "Pi\<^sub>E {} T = {\<lambda>x. undefined}"
dirichlet_prod'_inversion: fixes a :: "nat \<Rightarrow> 'a"   and ainv :: "nat \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a"   and f :: "real \<Rightarrow> 'a" assumes "dirichlet_prod a ainv = (\<lambda>n. if n = 1 then 1::'a else (0::'a))" shows "(\<forall>x\<ge>1. g x = dirichlet_prod' a f x) = (\<forall>x\<ge>1. f x = dirichlet_prod' ainv g x)"
col2_orth_at__orth_at: fixes U :: "'p"   and V :: "'p"   and P :: "'p"   and Q :: "'p"   and X :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p" assumes "U \<noteq> V"   and "Col P Q U"   and "Col P Q V"   and "X OrthAt A B C P Q" shows "X OrthAt A B C U V"
of_complex_cvec_def: fixes z :: "complex" shows "of_complex_cvec z = (z, 1)"
coeff_monom: fixes a :: "'a"   and m :: "nat"   and n :: "nat" shows "poly.coeff (Polynomial.monom a m) n = (if m = n then a else (0::'a))"
strict_iff_order: fixes a :: "'b list"   and b :: "'b list" shows "strict_prefix a b = (prefix a b \<and> a \<noteq> b)"
l_delete_shadow_root_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_delete_shadow_root_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs \<equiv> l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs"
wp_choice: fixes P :: "('b, 'c) urel"   and Q :: "('b, 'c) urel"   and R :: "'c upred" shows "P \<sqinter> Q wp R = (P wp R \<and> Q wp R)"
validI: fixes \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'e) com option"   and c :: "('d, 'f, 'e) com"   and P :: "'d set"   and F :: "'e set"   and Q :: "'d set"   and A :: "'d set" assumes "\<And>s t. \<lbrakk>\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow> t; s \<in> P; t \<notin> Fault ` F\<rbrakk> \<Longrightarrow> t \<in> Normal ` Q \<union> Abrupt ` A" shows "\<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A"
gfinprod_def: fixes G :: "('d, 'e) monoid_scheme"   and f :: "'f \<Rightarrow> 'd"   and A :: "'f set" shows "gfinprod G f A = (if finite {x \<in> A. f x \<noteq> \<one>\<^bsub>G\<^esub>} then finprod G f {x \<in> A. f x \<noteq> \<one>\<^bsub>G\<^esub>} else \<one>\<^bsub>G\<^esub>)"
norm_cblinfun_id: shows "norm id_cblinfun = 1"
trancl_Image_in_Range: fixes R :: "('b \<times> 'b) set"   and V :: "'b set" shows "R\<^sup>+ `` V \<subseteq> Range R"
decompose_com_ictxt: fixes c' :: "('g, 'h, 'i, 'j) com"   and ictxt :: "('g, 'h, 'i, 'j) com \<Rightarrow> ('g, 'h, 'i, 'j) com"   and fctxt :: "('g, 'h, 'i, 'j) com \<Rightarrow> ('g, 'h, 'i, 'j) com list"   and c :: "('g, 'h, 'i, 'j) com" assumes "(c', ictxt, fctxt) \<in> decompose_com c" shows "ictxt c' = c"
to_nat_mono': fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "less_eq a b" shows "mod_type.to_nat Rep a \<le> mod_type.to_nat Rep b"
b: shows "abc_steps_l (0, lm) (A [+] B) n = abc_steps_l (0, lm) A n"