hcju/isabelleps huggingface dataset:
CACH_INVS: fixes C :: "Class"   and m :: "Method"   and l :: "int"   and D :: "Class"   and m' :: "Method"   and par :: "Var list"   and code :: "(int \<times> Instr) list"   and l0 :: "int"   and Mspec :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and Minv :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and Anno :: "(int \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and k :: "nat"   and Anno2 :: "(int \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and nk :: "nat"   and n :: "nat"   and G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list" assumes "ins_is C m l (invokeS D m')"   and "mbody_is D m' (par, code, l0)"   and "MST\<down>(C, m) = Some (Mspec, Minv, Anno)"   and "Anno\<down>l = None"   and "MST\<down>(D, m') = Some (mkSPEC (Cachera k) Anno2)"   and "nk = n + k"   and "derivAssum G C m (l + 1) (Cachera n)" shows "deriv G C m l (Cachera nk)"
shift1_induct: fixes d :: "nat"   and args :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list \<times> ((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set"   and P :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list \<times> ((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set \<Rightarrow> ((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list \<times> ((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set \<Rightarrow> bool" assumes "shift1_inv d args"   and "\<And>qs B. \<lbrakk>shift1_inv d (qs, B); card B \<le> 1\<rbrakk> \<Longrightarrow> P (qs, B) (qs, B)"   and "\<And>qs B h U x. \<lbrakk>shift1_inv d (qs, B); 1 < card B; (h, U) = (SOME b. b \<in> B \<and> card (snd b) = Suc m); x = (SOME y. y \<in> U); finite U; x \<in> U; card (U - {x}) = m; P (shift_list (h, U) x qs, B - {(h, U)}) (shift1 (shift_list (h, U) x qs, B - {(h, U)}))\<rbrakk> \<Longrightarrow> P (qs, B) (shift1 (shift_list (h, U) x qs, B - {(h, U)}))" shows "P args (shift1 args)"
span_swap: fixes E :: "'c set"   and u :: "'c"   and v :: "'c" assumes "finite E"   and "E \<subseteq> carrier V"   and "u \<in> carrier V"   and "u \<notin> span E"   and "v \<in> carrier V"   and "u \<in> span (insert v E)" shows "span (insert u E) \<subseteq> span (insert v E)"
int_of_fbint: fixes x :: "fbint" shows "int_of_fbint x \<in> UNIV"
splitAt_1: fixes ram :: "'a" shows "splitAt ram [] = ([], [])"
ex_inj: assumes "class.countable TYPE('d)" shows "\<exists>to_nat. inj to_nat"
formulaIsDecreasedByLearn_1: fixes stateA :: "State"   and stateB :: "State" assumes "appliedLearn stateA stateB" shows "getM stateA = getM stateB"
igSubstIGVar2_def: fixes MOD :: "('b, 'c, 'varSort, 'sort, 'd, 'e, 'f, 'g, 'h) model_scheme" shows "igSubstIGVar2 MOD \<equiv> \<forall>ys y Y. igWls MOD (asSort ys) Y \<longrightarrow> igSubst MOD ys Y y (igVar MOD ys y) = Y"
pseudo_mod_monic_2: fixes g :: "'b poly"   and f :: "'b poly" assumes "monic g" shows "pseudo_mod f g = 0 \<or> degree (pseudo_mod f g) < degree g"
Gauss_Jordan_PA_def: fixes A :: "(('b, 'd) vec, 'c) vec" shows "Gauss_Jordan_PA A = Gauss_Jordan_upt_k_PA A (ncols A - 1)"
mcont_const: fixes Sup :: "'a set \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and luba :: "'b set \<Rightarrow> 'b"   and orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "'a" assumes "class.ccpo Sup less_eq less" shows "mcont luba orda Sup less_eq (\<lambda>x. c)"
g_def: shows "g = (\<lambda>i. r * cos i + Re z - Re z0)"
L1: shows "c_graph (chf A) = {c_pair x (chf A x) |x. x \<in> UNIV}"
Inf_trans_def: fixes S :: "(('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real) set" shows "Inf_trans S = (\<lambda>P. Inf_exp {t P |t. t \<in> S})"
monomials_Const: fixes x :: "'a" shows "monomials (Const x) = (if x = (0::'a) then {} else {0})"
hom_0_iff: fixes x :: "'a"   and d :: "nat" shows "(Polynomial.monom x d = 0) = (x = (0::'a))"
ide_trg: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "B.ide (trg \<mu>)"
typ_of_betaply: fixes t :: "Core.term"   and uty :: "typ"   and tty :: "typ"   and u :: "Core.term" assumes "typ_of t = Some (uty \<rightarrow> tty)"   and "typ_of u = Some uty" shows "typ_of (t \<bullet> u) = Some tty"
disc_1: fixes x11 :: "'d"   and x12 :: "'e"   and x13 :: "'e" shows "isTrans (Trans x11 x12 x13)"
SN_imp_SN_trancl: fixes R :: "('b \<times> 'b) set" assumes "SN R" shows "SN (R\<^sup>+)"
pairs_inf_closed: fixes phi :: "'c \<Rightarrow> 'd Filters.filter"   and x :: "'c"   and y :: "'d Filters.filter"   and z :: "'c"   and w :: "'d Filters.filter" assumes "triple phi"   and "(x, y) \<in> triple.pairs phi"   and "(z, w) \<in> triple.pairs phi" shows "triple.pairs_inf (x, y) (z, w) \<in> triple.pairs phi"
rtrancl3p_cases_1: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "rtrancl3p r x [] y" obtains "y = x"
Fs: fixes F :: "int poly" assumes "F \<in> set Fs" shows "Mp F = F \<and> monic F"
axioms_3: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "digraph \<alpha> \<AA>"
ring_record_simps_12: fixes carrier :: "'c set"   and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and zero :: "'c"   and add :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and more :: "'d" shows "(\<oplus>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, zero = zero, add = add, \<dots> = more\<rparr>\<^esub>) = add"
mat_mult_limit: fixes B :: "complex mat"   and m :: "nat"   and X :: "nat \<Rightarrow> complex mat"   and A :: "complex mat" assumes "B \<in> carrier_mat m m"   and "limit_mat X A m" shows "limit_mat (mat_mult_seq B X) (B * A) m"
Ide_Cod: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Arr V src trg t" shows "bicategorical_language.Ide V src trg (bicategorical_language.Cod V src trg t)"
arr_Rel_ArrCod_in_Vset: shows "T\<lparr>ArrCod\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
co_c1: shows "connected (path_image c1 - {pathstart c1, pathfinish c1})"
put_cont_def: fixes s :: "'s"   and m :: "('b, 'm) contT" shows "local.put_cont s m = ContT (\<lambda>k. put s (run_cont m k))"
cartesian_product: fixes g :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and B :: "'d set"   and A :: "'c set" shows "(\<Sum>x\<in>A. sum (g x) B) = (\<Sum>(x, y)\<in>A \<times> B. g x y)"
chosen_right_adjoints_axioms: shows "chosen_right_adjoints (\<cdot>) (\<star>) \<a> \<i> src trg"
NSLIM_Im: fixes f :: "'a \<Rightarrow> complex"   and a :: "'a"   and L :: "complex" assumes "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L" shows "(\<lambda>x. Im (f x)) \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S Im L"
vfinite_vimage: fixes A :: "V" assumes "vfinite A" shows "vfinite (f `\<^sub>\<circ> A)"
intros_9: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" shows "(P \<oplus> Q) \<oplus> R \<equiv>\<^sub>e P \<oplus> Q \<oplus> R"
hom_char: fixes f :: "'c"   and b :: "'c"   and a :: "'c" shows "Cop.in_hom f b a = \<guillemotleft>f : a \<rightarrow> b\<guillemotright>"
rel_refl: fixes Ra :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'f \<rightharpoonup>\<^sub>f 'e" assumes "\<And>x. Ra x x" shows "fmrel Ra x x"
assms: shows "low_equal s1 s2 \<and> t1 = snd (fst (save_restore_instr instr s1)) \<and> t2 = snd (fst (save_restore_instr instr s2))"
no_incoming_s: shows "\<forall>u. (u, s) \<notin> E"
sq_mtx_one_idty_1: fixes A :: "'a sq_mtx" shows "1 * A = A"
wlconf_empty_vals: fixes G :: "prog"   and s :: "st"   and ts :: "'a \<Rightarrow> ty option" shows "G,s\<turnstile>Map.empty[\<sim>\<Colon>\<preceq>]ts"
f': fixes x :: "'a" assumes "x \<in> topspace X" shows "f' x = f x"
faces: fixes x :: "'a set"   and y :: "'a set" assumes "x \<in> X"   and "order.greater_eq x y" shows "y \<in> X"
perp_in_col: fixes X :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "X PerpAt A B C D " shows "Col A B X \<and> Col C D X"
hisPrev_nex_less: fixes n' :: "nat"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid" assumes "\<exists>n''<n'. \<exists>nid'. (n'', nid') \<in> his t n nid" shows "\<not> (\<exists>x\<in>his t n nid. fst x < n' \<and> hisPred t n nid n' < fst x)"
\<xi>: shows "[\<lparr>A!,b\<^sup>P\<rparr> \<^bold>& (\<^bold>\<lambda>z. \<lparr>R,a\<^sup>P,z\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>z. \<lparr>R,b\<^sup>P,z\<^sup>P\<rparr>) \<^bold>& \<^bold>\<not>\<lbrace>b\<^sup>P,\<^bold>\<lambda>z. \<lparr>R,a\<^sup>P,z\<^sup>P\<rparr>\<rbrace> in v]"
inject_1: fixes x1 :: "char list \<times> Unary_PCF.type"   and y1 :: "char list \<times> Unary_PCF.type" shows "(\<langle>x1\<rangle> = \<langle>y1\<rangle>) = (x1 = y1)"
ex_n_lists_def: fixes P :: "'a list \<Rightarrow> bool"   and n :: "nat" shows "ex_n_lists P n = (\<exists>xs\<in>set (List.n_lists n enum). P xs)"
cardinal_idem: fixes a :: "V" shows "vcard (vcard a) = vcard a"
Sources_2DSources: fixes i :: "AbstrLevelsID"   and S :: "CSet"   and C1 :: "CSet"   and C2 :: "CSet" assumes "DSources i S = {C1, C2}" shows "Sources i S = {C1, C2} \<union> Sources i C1 \<union> Sources i C2"
alw_ev_Ord: obtains ks :: "'ord stream" where "alw (\<lambda>ks. (shd (stl ks), shd ks) \<in> Ord) ks" and "alw (ev (\<lambda>ks. shd (stl ks) \<noteq> shd ks)) ks"
Un_insert_right: fixes A :: "'b set"   and a :: "'b"   and B :: "'b set" shows "A \<union> (a \<triangleright> B) = a \<triangleright> A \<union> B"
remdups_rev_code_removeAll: fixes ps :: "'a list"   and p :: "'a"   and rsa :: "'a list" shows "remdups_rev_code ps (removeAll p rsa) = removeAll p (remdups_rev_code ps rsa)"
simps_2: fixes M :: "('e, 'f, 'g) KripkeStructure"   and w :: "'g"   and \<phi> :: "('e, 'f) Kform" shows "M, w \<Turnstile> Knot \<phi> = (\<not> M, w \<Turnstile> \<phi>)"
large_m_factor: fixes u :: "int poly"   and vs :: "'a poly list"   and v :: "int poly" assumes "large_m u vs"   and "v dvd u" shows "large_m v vs"
gcd_eq_gcd_factorial: fixes a :: "'a"   and b :: "'a" shows "gcd a b = gcd_factorial a b"
compare_Eq_is_eq: fixes compare :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a" assumes "class.compare compare" shows "(compare x y = Eq) = (x = y)"
wary_num_args_le_arity_head: fixes s :: "('s, 'v) tm" assumes "wary s" shows "enat (num_args s) \<le> arity_hd (head s)"
ufa_\<alpha>_def: fixes l :: "nat list" shows "ufa_\<alpha> l = {(x, y). x < length l \<and> y < length l \<and> rep_of l x = rep_of l y}"
MOST_nat_le: fixes P :: "nat \<Rightarrow> bool" shows "(\<forall>\<^sub>\<infinity>n. P n) = (\<exists>m. \<forall>n\<ge>m. P n)"
ucast_nat_def: fixes x :: "'c word" shows "word_of_nat (unat x) = UCAST('c \<rightarrow> 'b) x"
elims: fixes x :: "rat mat"   and xa :: "rat list list"   and xb :: "(nat list \<times> nat list) list"   and xc :: "rat vec"   and y :: "rat mat \<times> (nat list \<times> nat list) list \<times> rat list list" assumes "reduction_step_R x xa xb xc = y" obtains A :: "rat mat" and signs :: "rat list list" and subsets :: "(nat list \<times> nat list) list" and lhs_vec :: "rat vec" where "x = A" and "xa = signs" and "xb = subsets" and "xc = lhs_vec" and "y = (let reduce_cols_A = reduce_mat_cols A lhs_vec; rows_keep = rows_to_keep reduce_cols_A in (take_rows_from_matrix reduce_cols_A rows_keep, take_indices subsets rows_keep, take_indices signs (find_nonzeros_from_input_vec lhs_vec)))"
rtrancl_refl: fixes a :: "'h"   and r :: "('h \<times> 'h) set" shows "(a, a) \<in> r\<^sup>*"
sum_every_even_summand_is_zero: fixes n :: "nat" assumes "1 \<le> n" shows "\<forall>f. (\<forall>i. i < 2 ^ (n + 1) \<and> even i \<longrightarrow> f i = 0) \<longrightarrow> sum f {0..<2 ^ (n + 1)} = (\<Sum>k = 0..<2 ^ n. f (2 * k + 1))"
str_del_point_blocks_def: fixes p :: "'a" shows "des.str_del_point_blocks p \<equiv> filter_mset ((\<notin>) p) (str_del_point_blocks x)"
lcos_module_rev: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and x' :: "'c" assumes "subgroup H G"   and "Group.group G"   and "x \<in> carrier G"   and "x' \<in> carrier G"   and "inv\<^bsub>G\<^esub> x \<otimes>\<^bsub>G\<^esub> x' \<in> H" shows "x' \<in> x <#\<^bsub>G\<^esub> H"
resultant_impl_generic_def: fixes div_exp :: "'a \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and f :: "'a poly"   and g :: "'a poly" shows "resultant_impl_generic div_exp f g = (if length (coeffs g) \<le> length (coeffs f) then resultant_impl_main div_exp f g else let res = resultant_impl_main div_exp g f in if even (degree f) \<or> even (degree g) then res else - res)"
rel_generat_mono: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and C :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and x :: "('d, 'f, 'h) generat"   and y :: "('e, 'g, 'i) generat"   and A' :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B' :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and C' :: "'h \<Rightarrow> 'i \<Rightarrow> bool" assumes "rel_generat A B C x y"   and "\<And>x y. A x y \<Longrightarrow> A' x y"   and "\<And>x y. B x y \<Longrightarrow> B' x y"   and "\<And>x y. C x y \<Longrightarrow> C' x y" shows "rel_generat A' B' C' x y"
tendsto_const: fixes k :: "'a"   and F :: "'c filter" shows "((\<lambda>x. k) \<longlongrightarrow> k) F"
dim_eq_0: fixes S :: "('b, 'c) vec set" shows "(vec.dim S = 0) = (S \<subseteq> {0})"
A_objects_unique: fixes v :: "i"   and \<phi> :: "\<Pi>\<^sub>1 \<Rightarrow> \<o>" shows "[\<^bold>\<exists>!x. \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall>F. \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<phi> F) in v]"
setwise_def: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and X :: "'b set"   and Y :: "'c set" shows "setwise r X Y \<equiv> \<forall>x\<in>X. \<forall>y\<in>Y. r x y"
exhaust: fixes y :: "unop" obtains "y = UPlus"   | "y = UMinus"   | "y = UBitNot"   | "y = UNot"
invertible_row_add: fixes a :: "'b"   and b :: "'b"   and q :: "'c" assumes "a \<noteq> b" shows "invertible (row_add (Finite_Cartesian_Product.mat (1::'c)) a b q)"
acute_one_side_aux0: fixes A :: "'p"   and C :: "'p"   and P :: "'p"   and B :: "'p" assumes "Col A C P"   and "Acute A C P"   and "C A Perp B C" shows "C B OS A P"
prems_4: fixes p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c"   and q :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c" assumes "processed (p, q) (gs @ bs) ps"   and "(p, q) \<notin>\<^sub>p D"   and "fst p \<noteq> 0"   and "fst q \<noteq> 0" shows "crit_pair_cbelow_on d m (fst ` (set gs \<union> set bs)) (fst p) (fst q)"
preal_nonempty: fixes A :: "rat set" assumes "Dedekind_Real.cut A" shows "\<exists>x\<in>A. 0 < x"
in_results_gpv_sub_gvps: fixes x :: "'d"   and \<I> :: "('e, 'f) \<I>"   and gpv' :: "('d, 'e, 'f) gpv"   and gpv :: "('d, 'e, 'f) gpv" assumes "x \<in> results_gpv \<I> gpv'"   and "gpv' \<in> sub_gpvs \<I> gpv" shows "x \<in> results_gpv \<I> gpv"
lin_part_def: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c" assumes "UP_cring R" shows "UP_cring.lin_part R f = UP_cring.trms_of_deg_leq R 1 f"
nn_integral_pmf: fixes A :: "'c set"   and p :: "'c pmf" shows "\<integral>\<^sup>+ x. ennreal (pmf p x) \<partial>count_space A = emeasure (measure_pmf p) A"
graphE: fixes x :: "'a"   and y :: "real"   and F :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "(x, y) \<in> graph F f" obtains "x \<in> F" and "y = f x"
brn_ParT_simp: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and cl :: "('c, 'd, 'a) cmd list" assumes "PL cval" shows "PL.brn (ParT cl) = PL.brnL cl (length cl)"
modes_eq: shows "map snd cms\<^sub>1 = map snd cms\<^sub>2"
simps: fixes i :: "nat"   and j :: "nat" shows "fs.gs.\<mu> i j = (if j < i then map of_int_hom.vec_hom fs ! i \<bullet> fs.gs.gso j / \<parallel>fs.gs.gso j\<parallel>\<^sup>2 else if i = j then 1 else 0)"
nn_integral_indicator_neq_infty: fixes f :: "'a \<Rightarrow> 'c"   and A :: "'c set" assumes "f -` A \<in> sets M" shows "\<integral>\<^sup>+ x. indicator A (f x) \<partial>M \<noteq> \<infinity>"
trg_inv: fixes \<mu> :: "'d" assumes "D.iso \<mu>" shows "trg\<^sub>D (D.inv \<mu>) = trg\<^sub>D \<mu>"
th3_2: fixes a :: "(real, 'b) vec"   and b :: "(real, 'b) vec"   and c :: "(real, 'b) vec" assumes "B\<^sub>\<real> a b c" shows "B\<^sub>\<real> c b a"
card_partitions_k_parts: fixes n :: "nat"   and k :: "nat" shows "card {p. p partitions n \<and> sum p {..n} = k} = Partition n k"
fwi_len': fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat"   and i' :: "nat"   and n :: "nat"   and j' :: "nat"   and i :: "nat"   and j :: "nat" assumes "(0::'a) \<le> m k k"   and "i' \<le> n"   and "j' \<le> n"   and "k \<le> n"   and "i \<le> i'"   and "j \<le> j'" shows "\<exists>xs. set xs \<subseteq> {k} \<and> fwi m n k i' j' i j = len m i j xs"
measure_pmf_of_set: fixes S :: "'a set"   and A :: "'a set" assumes "S \<noteq> {}"   and "finite S" shows "measure_pmf.prob (pmf_of_set S) A = real (card (S \<inter> A)) / real (card S)"
FSwSbImorph_iff: fixes h :: "'e \<Rightarrow> 'f"   and hA :: "'g \<Rightarrow> 'h"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'i, 'e, 'g, 'j) model_scheme"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'i, 'f, 'h, 'k) model_scheme" shows "FSwSbImorph h hA MOD MOD' = (FSbImorph h hA MOD MOD' \<and> ipresIGSwapAll h hA MOD MOD')"
separatedin_Un_1: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set"   and U :: "'c set" shows "separatedin X S (T \<union> U) = (separatedin X S T \<and> separatedin X S U)"
next_imp: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "|~ \<circle>(F \<longrightarrow> G) = (\<circle>F \<longrightarrow> \<circle>G)"
L1_space: fixes M :: "'a measure" shows "lspace M 1 = {f. integrable M f}"
correctCompositionDiffLevelsS2: shows "correctCompositionDiffLevels sS2"
dn_km1: shows "rat_of_int (d fs (k - 1) * d fs (k - 1)) * \<parallel>fs.gs.gso (k - 1)\<parallel>\<^sup>2 = rat_of_int (d fs k) * rat_of_int (d fs (k - 1))"
mult_sheaf_spec_def: fixes U :: "('a \<times> 'a) set set set"   and s :: "('a \<times> 'a) set set \<Rightarrow> (('a \<times> 'a) set \<times> ('a \<times> 'a) set) set"   and s' :: "('a \<times> 'a) set set \<Rightarrow> (('a \<times> 'a) set \<times> ('a \<times> 'a) set) set" shows "local.local.mult_sheaf_spec U s s' \<equiv> \<lambda>\<pp>\<in>U. quotient_ring.mult_rel (local.carrier_quotient_ring\<setminus>\<pp>) local.carrier_quotient_ring local.add_rel local.mult_rel local.zero_rel (s \<pp>) (s' \<pp>)"
dense_ge: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and z :: "'c"   and y :: "'c" assumes "class.dense_linorder less_eq less"   and "\<And>x. less z x \<Longrightarrow> less_eq y x" shows "less_eq y z"
IH: fixes c :: "'a" assumes "u \<turnstile>\<^bsub>v,n\<^esub> m"   and "v c \<le> n"   and "v d \<le> n"   and "\<forall>c\<in>set cs. v c \<le> n" shows "dbm_entry_val u (Some c) (Some d) (len m (v c) (v d) (map v cs))"
tot_rec_impl_comp: fixes A :: "nat set" assumes "total_recursive (chf A)" shows "computable A"
continuous_power: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and n :: "nat" assumes "continuous F f" shows "continuous F (\<lambda>x. f x ^ n)"
avg_aux_correct: fixes s :: "(nat, unit) HashMap.hashmap" assumes "my_invar s" shows "avg_aux s = (card (my_\<alpha> s), \<Sum> (my_\<alpha> s))"
assms_4: shows "\<forall>e\<in>pred c1. finite (Bexp.vars e)"
sum_mset_add_diff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and A :: "'b multiset" assumes "class.ring_1 one times plus zero minus uminus" shows "comm_monoid_add.sum_mset plus zero {#minus (f x) (g x). x \<in># A#} = minus (comm_monoid_add.sum_mset plus zero (image_mset f A)) (comm_monoid_add.sum_mset plus zero (image_mset g A))"
sp_through_de_decomp: fixes g :: "('c, 'd) rgraph_scheme"   and e :: "'c edge"   and v1 :: "'c"   and es :: "'c edge list"   and v2 :: "'c" assumes "out_edges g (tgt e) = {}"   and "Graph.subpath g v1 es v2"   and "e \<in> set es" shows "\<exists>es'. es = es' @ [e] \<and> e \<notin> set es'"
select_convs_2: fixes g_V :: "'c set"   and g_E :: "('c \<times> 'c) set"   and g_V0 :: "'c set"   and more :: "'d" shows "g_E \<lparr>g_V = g_V, g_E = g_E, g_V0 = g_V0, \<dots> = more\<rparr> = g_E"
i: shows "is_zero_row_upt_k i (mod_type_class.to_nat k) A"
absorb_iff1: fixes b :: "'c"   and a :: "'c" shows "(b \<le> a) = (sup a b = a)"
sum_outgoing_pointwise: fixes g :: "nat \<times> nat \<Rightarrow> 'b"   and u :: "nat" shows "sum g (cf.outgoing u) = (\<Sum>v\<in>cf.E `` {u}. g (u, v))"
left_imp_options: fixes opt :: "game"   and g :: "game" assumes "zin opt (left_options g)" shows "zin opt (options g)"
simps: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b" shows "not_final_thread s t = ((\<exists>x ln. thr s t = \<lfloor>(x, ln)\<rfloor> \<and> \<not> final x) \<or> (\<exists>x ln. thr s t = \<lfloor>(x, ln)\<rfloor> \<and> ln \<noteq> no_wait_locks) \<or> (\<exists>x w ln. thr s t = \<lfloor>(x, ln)\<rfloor> \<and> wset s t = \<lfloor>w\<rfloor>))"
that: fixes f :: "'t \<Rightarrow>\<^sub>0 'b" assumes "f \<in> F"   and "f \<noteq> 0"   and "lt f adds\<^sub>t v" shows "thesis"
assms: fixes T :: "'a set set" assumes "Ball T S" shows "S (\<Inter> T)"
p_rw: shows "power_polys mult_p u curr_p (Suc n) ! i = (curr_p # power_polys mult_p u (curr_p * mult_p mod u) n) ! i"
NF_I: fixes a :: "'d"   and A :: "('d \<times> 'd) set" assumes "\<And>b. (a, b) \<notin> A" shows "a \<in> NF A"
primes_theta_def: shows "primes_theta = prime_sum_upto (\<lambda>p. ln (real p))"
simps_4: fixes v :: "'b"   and va :: "'a FOL_Fitting.term list" shows "compl (Pred v va) = Neg (Pred v va)"
One_pdevs_raw_def: fixes i :: "nat" shows "One_pdevs_raw i = (if i < length Basis_list then Basis_list ! i else (0::'b))"
extremum_uniqueI: fixes a :: "'c list" assumes "a \<le> []" shows "a = []"
sum_clauses_2: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "finite S" shows "sum f (insert x S) = (if x \<in> S then sum f S else f x + sum f S)"
r: shows "r = real_of_int i / real_of_int (int n)"