hcju/isabelleps huggingface dataset:
finite_field_i_sound: fixes f' :: "'i list"   and f :: "int poly"   and c' :: "'i"   and fs' :: "'i list list"   and fs :: "int poly list"   and c :: "int" assumes "f' = of_int_poly_i ff_ops (Mp f)"   and "finite_field_factorization_i p ff_ops f' = (c', fs')"   and "square_free_m f"   and "fs = map (to_int_poly_i ff_ops) fs'"   and "c = arith_ops_record.to_int ff_ops c'" shows "unique_factorization_m f (c, mset fs) \<and> c \<in> {0..<p} \<and> (\<forall>fi\<in>set fs. set (coeffs fi) \<subseteq> {0..<p})"
nat_pow_nonzero: fixes x :: "nat \<Rightarrow> int"   and n :: "nat" assumes "x \<in> nonzero Zp" shows "x [^] n \<in> nonzero Zp"
lift_cancels_to: fixes x :: "(bool \<times> 'd) list"   and y :: "(bool \<times> 'd) list"   and gens :: "'d set"   and f :: "'d \<Rightarrow> 'a" assumes "cancels_to x y"   and "x \<in> lists (UNIV \<times> gens)"   and "f \<in> gens \<rightarrow> carrier G" shows "lift f x = lift f y"
residue_neg: fixes s :: "complex set"   and z :: "complex"   and f :: "complex \<Rightarrow> complex" assumes "open s"   and "z \<in> s"   and "f holomorphic_on s - {z}" shows "residue (\<lambda>z. - f z) z = - residue f z"
divide_poly: fixes g :: "'a poly"   and f :: "'a poly" assumes "g \<noteq> 0" shows "f * g div g = f"
dg_Set_Cod_app: fixes \<alpha> :: "V"   and x :: "V" assumes "arr_Set \<alpha> x" shows "dg_Set \<alpha>\<lparr>Cod\<rparr>\<lparr>x\<rparr> = x\<lparr>ArrCod\<rparr>"
slice_kind_Pred_obs_nearer_not_SOME: fixes a :: "'edge"   and S :: "'node SDG_node set"   and Q :: "('var \<Rightarrow> 'val option) \<Rightarrow> bool"   and m :: "'node"   and x :: "nat" assumes "sourcenode a \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "kind a = (Q)\<^sub>\<surd>"   and "m \<in> obs_intra (sourcenode a) \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "distance (targetnode a) m x"   and "distance (sourcenode a) m (x + 1)"   and "targetnode a \<noteq> (SOME nx'. \<exists>a'. sourcenode a = sourcenode a' \<and> distance (targetnode a') m x \<and> valid_edge a' \<and> intra_kind (kind a') \<and> targetnode a' = nx')" shows "slice_kind S a = (\<lambda>s. False)\<^sub>\<surd>"
Extend_def: fixes S :: "(nat, 'a) form set"   and C :: "(nat, 'a) form set set"   and f :: "nat \<Rightarrow> (nat, 'a) form" shows "Extend S C f = \<Union> (range (extend S C f))"
absurd: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf"   and A1 :: "'a"   and A2 :: "'a"   and A3 :: "'a"   and A4 :: "'a"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "sds_impossibility_4_4 agents alts sds A1 A2 A3 A4 a b c d" shows "False"
coeff_poly_lift: fixes p :: "'a poly"   and i :: "nat" shows "coeff (poly_lift p) i = [:coeff p i:]"
2: shows "if a \<in> externals (ioa.asig A) then tr = [a] else tr = []"
mult_commute: fixes z1 :: "complex_homo"   and z2 :: "complex_homo" shows "z1 *\<^sub>h z2 = z2 *\<^sub>h z1"
Bex_cartesian_product: fixes A :: "'b set"   and B :: "'c set"   and P :: "'b \<times> 'c \<Rightarrow> bool" shows "\<exists>xy\<in>A \<times> B. P xy \<equiv> \<exists>x\<in>A. \<exists>y\<in>B. P (x, y)"
chainD: fixes ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and x :: "'b"   and y :: "'b" assumes "Complete_Partial_Order.chain ord S"   and "x \<in> S"   and "y \<in> S" shows "ord x y \<or> ord y x"
rel_gpv''_GPV: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and C :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and p :: "('d, 'f, ('d, 'f, 'h) rpv) generat spmf"   and q :: "('e, 'g, ('e, 'g, 'i) rpv) generat spmf" shows "rel_gpv'' A C R (GPV p) (GPV q) = rel_spmf (rel_generat A C (rel_fun R (rel_gpv'' A C R))) p q"
fold_def: fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> 'f \<Rightarrow> 'f"   and t :: "('g, 'h) rbt" shows "RBT_Impl.fold f t = fold (\<lambda>(x, y). f x y) (RBT_Impl.entries t)"
assms: shows "edge (bipartite_web_of \<Gamma>) uv uv'"
polynomial_function_diff: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "polynomial_function f"   and "polynomial_function g" shows "polynomial_function (\<lambda>x. f x - g x)"
error_free_throw: fixes s :: "state"   and x :: "val" assumes "error_free s" shows "error_free (abupd (throw x) s)"
check_ipresIGVar: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" shows "ipresIGVar check (errMOD MOD) MOD"
gcd_left_idem: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "gcd a (gcd a b) = gcd a b"
conj_disj_distrib: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<^bold>\<squnion> y \<^bold>\<sqinter> z = (x \<^bold>\<squnion> y) \<^bold>\<sqinter> (x \<^bold>\<squnion> z)"
f2r_epi: shows "\<forall>f g. f \<circ> \<R> = g \<circ> \<R> \<longrightarrow> f = g"
prv_imp_eqvER: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and \<phi>1 :: "'c"   and \<phi>2 :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "prv (Syntax_with_Connectives.eqv cnj imp \<phi>1 \<phi>2)" shows "prv (imp \<phi>2 \<phi>1)"
simps: fixes a1 :: "'b"   and a2 :: "'c"   and r :: "('c \<times> 'b) set" shows "((a1, a2) \<in> r\<inverse>) = (\<exists>a b. a1 = b \<and> a2 = a \<and> (a, b) \<in> r)"
monom_mult_one_left: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "monom_mult (1::'c) (0::'a) p = p"
nat_list_0: fixes x :: "nat"   and xs :: "nat list" assumes "x \<in> set xs"   and "0 \<notin> set xs" shows "0 < x"
eq_fin_concat_end: fixes w\<^sub>1 :: "'item list"   and w :: "'item list"   and w\<^sub>2 :: "'item list" shows "(w\<^sub>1 @ w =\<^sub>F w\<^sub>2 @ w) = (w\<^sub>1 =\<^sub>F w\<^sub>2)"
facs_2: shows "u \<le>f s \<^sup>@ l"
bernoulli'_odd_eq_0: fixes n :: "nat" assumes "n \<noteq> 1"   and "odd n" shows "bernoulli' n = 0"
bexpE: fixes A :: "('a \<times> 'b) set"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<exists>(x, y)\<in>A. P x y" obtains x :: "'a" and y :: "'b" where "(x, y) \<in> A" and "P x y"
path_connected_arc_complement: fixes \<gamma> :: "real \<Rightarrow> 'b" assumes "arc \<gamma>"   and "2 \<le> DIM('b)" shows "path_connected (- path_image \<gamma>)"
discI_4: fixes "term" :: "Core.term"   and x41 :: "typ"   and x42 :: "Core.term" assumes "term = Abs x41 x42" shows "is_Abs term"
map_of_eq_None_iff: fixes xys :: "('f \<times> 'e) list"   and x :: "'f" shows "(map_of xys x = None) = (x \<notin> fst ` set xys)"
cons_perm_eq: fixes z :: "'b"   and xs :: "'b list"   and ys :: "'b list" shows "(z # xs <~~> z # ys) = (xs <~~> ys)"
components_f4_red_aux_subset: fixes bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "component_of_term ` Keys (set (f4_red_aux bs ps)) \<subseteq> component_of_term ` Keys (args_to_set ([], bs, ps))"
low_eq_def: 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 mem\<^sub>1 :: "'b \<Rightarrow> 'c"   and mem\<^sub>2 :: "'b \<Rightarrow> 'c" assumes "sifum_security stop eval some_val some_val'" shows "sifum_security.low_eq dma mem\<^sub>1 mem\<^sub>2 \<equiv> \<forall>x. dma x = Low \<longrightarrow> mem\<^sub>1 x = mem\<^sub>2 x"
edge_space_def: fixes p :: "real" shows "edge_space p \<equiv> 0 \<le> p \<and> p \<le> 1"
full_True: shows "intTF = (\<lambda>\<sigma> D. True)"
eval_red_Hcomp: fixes a :: "'c local.term"   and b :: "'c local.term" assumes "Ide a"   and "Ide b" shows "\<lbrace>(a \<^bold>\<star> b)\<^bold>\<down>\<rbrace> = \<lbrace>\<^bold>\<lfloor>a\<^bold>\<rfloor> \<^bold>\<Down> \<^bold>\<lfloor>b\<^bold>\<rfloor>\<rbrace> \<cdot> (\<lbrace>a\<^bold>\<down>\<rbrace> \<star> \<lbrace>b\<^bold>\<down>\<rbrace>)"
termlemma2: fixes i :: "'b ifex"   and t :: "'b ifex"   and e :: "'b ifex"   and xa :: "'b"   and val :: "bool" assumes "lowest_tops [i, t, e] = Some xa" shows "size (restrict_top i xa val) + size (restrict_top t xa val) + size (restrict_top e xa val) < size i + size t + size e"
normalize_replace_primitive_matchexpr_preserves_normalized_not_has_disc: fixes m :: "'b match_expr"   and disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and C :: "'c \<Rightarrow> 'b"   and disc2 :: "'b \<Rightarrow> bool"   and replace_f :: "'c negation_type \<Rightarrow> 'b match_expr"   and m' :: "'b match_expr" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "\<not> has_disc disc2 m"   and "\<And>a. \<not> has_disc disc2 (replace_f a)"   and "m' \<in> set (normalize_match (replace_primitive_matchexpr (disc, sel) replace_f m))" shows "\<not> has_disc disc2 m'"
rel_compp_Grp: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_pmf R = (BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf fst))\<inverse>\<inverse> OO BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf snd)"
comp_in_homI: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and a :: "'b"   and b :: "'b"   and g :: "'b"   and c :: "'b" assumes "category C"   and "partial_magma.in_hom C f a b"   and "partial_magma.in_hom C g b c" shows "partial_magma.in_hom C (C g f) a c"
is_bcv_kiljvm: shows "is_bcv r Err step (length is) A (kiljvm P mxs mxl T\<^sub>r is xt)"
Exe1E: fixes v :: "i"   and F :: "\<Pi>\<^sub>1"   and x :: "\<kappa>" assumes "[\<lparr>F,x\<rparr> in v]" shows "\<exists>r o\<^sub>1. Some r = Semantics.d\<^sub>1 F \<and> Some o\<^sub>1 = Semantics.d\<^sub>\<kappa> x \<and> o\<^sub>1 \<in> Semantics.ex1 r v"
aff_dim_eq_0: fixes S :: "'b set" shows "(aff_dim S = 0) = (\<exists>a. S = {a})"
to_nat_0: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "mod_type.to_nat Rep zero = 0"
list_all2_triple: shows "list_all2 (\<lambda>x y. triple_eq_pair x y (\<lambda>tp'. filter_a1_map pos tp' a1_map) (\<lambda>ts' tp'. filter_a2_map I ts' tp' a2_map)) (drop (length done) auxlist) (zip (linearize tss) [tp - len..<tp])"
disc_25: fixes x51 :: "nat"   and x52 :: "'a"   and x53 :: "'a" shows "isRecvMarker (RecvMarker x51 x52 x53)"
cases: fixes a :: "'a list" assumes "local.sorted a" obtains "a = []"   | xs :: "'a list" and x :: "'a" where "a = x # xs" and "\<forall>y\<in>set xs. x \<prec> y" and "local.sorted xs"
meet_leftcomm: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "bounded_lattice L"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<sqinter>\<^bsub>L\<^esub> (y \<sqinter>\<^bsub>L\<^esub> z) = y \<sqinter>\<^bsub>L\<^esub> (x \<sqinter>\<^bsub>L\<^esub> z)"
runErrorT_unitET: fixes x :: "'e" shows "runErrorT\<cdot>(unitET\<cdot>x) = return\<cdot>(Ok\<cdot>x)"
hom_from_subgroup_generated: fixes G :: "('e, 'f) monoid_scheme"   and h :: "'e \<Rightarrow> 'g"   and H :: "('g, 'h) monoid_scheme"   and A :: "'e set" assumes "Group.group G"   and "h \<in> hom G H" shows "h \<in> hom (subgroup_generated G A) H"
assms_3: shows "kKS key \<notin> LocalSecrets PQ"
abs_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b set \<Rightarrow> 'd"   and Rep :: "'d \<Rightarrow> 'b set"   and x :: "'b" assumes "quot_type R Abs Rep" shows "quot_type.abs R Abs x = Abs (Collect (R x))"
transversal_segment_exists: fixes x :: "'a" assumes "x \<in> X"   and "f x \<noteq> (0::'a)" obtains a :: "'a" and b :: "'a" where "x \<in> {a<--<b}" and "transversal_segment a b"
c_def: shows "c = complex_of_real (2 * pi) * \<i>"
step_Normal_elim_cases_3: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and u :: "('e, 'd, 'f) com list \<times> (('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (c # cs, css, Stuck) \<rightarrow> u" obtains "u = (cs, css, Stuck)"
BotL_canonical: fixes \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" shows "\<bottom>, \<Gamma> \<Rightarrow> \<Delta>"
irreducible_content: fixes p :: "'a poly" assumes "irreducible p" shows "degree p = 0 \<or> primitive p"
continuous_map_closedin_preimage_eq: fixes X :: "'d topology"   and Y :: "'e topology"   and f :: "'d \<Rightarrow> 'e" shows "continuous_map X Y f = (f ` topspace X \<subseteq> topspace Y \<and> (\<forall>U. closedin Y U \<longrightarrow> closedin X (topspace X \<inter> f -` U)))"
EX: shows "\<exists>\<phi>. \<guillemotleft>\<phi> : a \<rightarrow> a'\<guillemotright> \<and> C.iso \<phi> \<and> local.map \<phi> = Cop_S.MkArr (\<Phi>'.Y a) (\<Phi>'.Y a') \<Phi>\<Psi>'.map"
psubst_Var_simp1: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and xs :: "'varSort"   and x :: "'var" assumes "goodEnv rho"   and "rho xs x = None" shows "Var xs x #[rho] = Var xs x"
bounded_countable_web_def: fixes \<Gamma> :: "('a, 'b) web_scheme" shows "bounded_countable_web \<Gamma> \<equiv> countable_web \<Gamma> \<and> bounded_countable_web_axioms \<Gamma>"
extreme_unique: fixes X :: "'a set"   and x :: "'a"   and y :: "'a" assumes "X \<subseteq> A"   and "extreme X (\<sqsubseteq>) x" shows "extreme X (\<sqsubseteq>) y = (x = y)"
mpyf: shows "AEeq M (cls_val_process Mkt pf matur) pyf"
COPY_def: fixes x :: "'j" shows "COPY x \<equiv> x"
assms: shows "LD_1' r \<beta> \<alpha> \<sigma>1 \<sigma>2 \<sigma>3"
mu_fusion_1: fixes l :: "'a \<Rightarrow> 'a"   and u :: "'a \<Rightarrow> 'a"   and h :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "galois l u"   and "isotone h"   and "has_least_prefixpoint g"   and "has_least_fixpoint h"   and "l (g (u (\<mu> h))) \<le> h (l (u (\<mu> h)))" shows "l (p\<mu> g) \<le> \<mu> h"
poly_cond2I: fixes p :: "int poly" assumes "poly_cond p"   and "1 < degree p" shows "poly_cond2 p"
\<B>\<I>_set_unsat: fixes I :: "'b list"   and s :: "('b, 'a) state" shows "\<B>\<I> (set_unsat I s) = \<B>\<I> s"
subclsR_def: shows "subclsR \<equiv> \<lambda>P. {(xa, x). subclsRp P xa x}"
antisym_conv1: fixes x :: "'t \<Rightarrow>\<^sub>0 'c"   and y :: "'t \<Rightarrow>\<^sub>0 'c" assumes "\<not> ord_strict_p x y" shows "ord_p x y = (x = y)"
vread: shows "v \<in> local.reads (path \<sigma> k)"
flowD_finite: fixes g :: "'v \<times> 'v \<Rightarrow> ennreal"   and e :: "'v \<times> 'v" assumes "flow (residual_network f) g" shows "g e \<noteq> \<top>"
case_rews_3: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b" shows "(case EQ of LT \<Rightarrow> f1 | EQ \<Rightarrow> f2 | GT \<Rightarrow> f3) = f2"
insert_subset_eq_iff: fixes a :: "'a"   and A :: "'a multiset"   and B :: "'a multiset" shows "(add_mset a A \<subseteq># B) = (a \<in># B \<and> A \<subseteq># B - {#a#})"
UP_simps_4: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "local.dom (UP \<mu>) = UP (B.dom \<mu>)"
cut_less_mono: fixes t :: "'b"   and t' :: "'b"   and I :: "'b set" assumes "t \<le> t'" shows "I \<down>< t \<subseteq> I \<down>< t'"
snoc_5: shows "pac p \<inter> set (w @ [b]) = {}"
GPicard3: fixes S :: "complex set"   and w :: "complex"   and Y :: "(complex \<Rightarrow> complex) set"   and X :: "(complex \<Rightarrow> complex) set"   and K :: "complex set" assumes "open S"   and "connected S"   and "w \<in> S"   and "Y \<subseteq> X"   and "\<And>h. h \<in> X \<Longrightarrow> h holomorphic_on S"   and "\<And>h z. \<lbrakk>h \<in> X; z \<in> S\<rbrakk> \<Longrightarrow> h z \<noteq> 0 \<and> h z \<noteq> 1"   and "\<And>h. h \<in> Y \<Longrightarrow> cmod (h w) \<le> 1"   and "compact K"   and "K \<subseteq> S" obtains B :: "real" where "\<And>h z. \<lbrakk>h \<in> Y; z \<in> K\<rbrakk> \<Longrightarrow> cmod (h z) \<le> B"
mod0_uint_def: fixes x :: "uint" shows "mod0_uint x = undefined (mod) x 0"
gauss_cnj_of_nat: fixes n1 :: "nat" shows "gauss_cnj (of_nat n1) = of_nat n1"
fst_set_add_indices: fixes ns :: "(('f \<Rightarrow>\<^sub>0 'g) \<times> 'h) list \<times> 'i"   and data' :: "nat \<times> 'i" shows "fst ` set (fst (add_indices ns data')) = fst ` set (fst ns)"
eqv_class_def: fixes a :: "'b" shows "\<lfloor>a\<rfloor> = Abs_quot {x. a \<sim> x}"
hg: shows "Cod\<^bsub>set_cat U\<^esub> h = Dom\<^bsub>set_cat U\<^esub> g"
mult1_mono: fixes A :: "'b set"   and R :: "('b \<times> 'b) set"   and S :: "('b \<times> 'b) set"   and M :: "'b multiset"   and N :: "'b multiset" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; (x, y) \<in> R\<rbrakk> \<Longrightarrow> (x, y) \<in> S"   and "M \<in> multisets A"   and "N \<in> multisets A"   and "(M, N) \<in> mult1 R" shows "(M, N) \<in> mult1 S"
ennreal_SUP_of_nat_eq_top: shows "\<Squnion> range of_nat = \<top>"
all_final_exceptD: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x1 \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and Ts :: "'b set"   and t :: "'b" assumes "r1.all_final_except s Ts"   and "r1.not_final_thread s t" shows "t \<in> Ts"
x: shows "x \<in> {f. \<forall>k\<in>UNIV. f (a [^] k) = b [^]\<^bsub>H\<^esub> k}"
that: fixes n :: "nat" assumes "take_cols A inds \<in> carrier_mat nr n" shows "thesis"
f': fixes T :: "real set"   and X :: "'b set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and f' :: "real \<times> 'b \<Rightarrow> real \<times> 'b \<Rightarrow> 'b"   and tx :: "real \<times> 'b" assumes "derivative_on_prod T X f f'"   and "tx \<in> T \<times> X" shows "((\<lambda>(t, x). f t x) has_derivative f' tx) (at tx within T \<times> X)"
rtranclp_rtrancl_eq: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" shows "r\<^sup>*\<^sup>* x y = ((x, y) \<in> {(x, y). r x y}\<^sup>*)"
underS_incr: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a" assumes "trans r"   and "antisym r"   and "(a, b) \<in> r" shows "underS r a \<subseteq> underS r b"
iptables_bigstepD_8: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state"   and X :: "final_decision" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>r, s\<rangle> \<Rightarrow> t"   and "s = Decision X" shows "t = Decision X"
components_are_equivalences: fixes a :: "'c" assumes "C.obj a" shows "D.equivalence_map (\<tau>\<^sub>0 a)"
simps: fixes a :: "('b \<times> 'b) set" shows "asym a = (\<exists>R. a = R \<and> (\<forall>x xa. (x, xa) \<in> R \<longrightarrow> (xa, x) \<notin> R))"
approx_slp_spec_def: fixes fas :: "floatarith list"   and l :: "nat"   and slp :: "floatarith list"   and env :: "real list set" shows "approx_slp_spec fas l slp env = ASSERT (slp_of_fas fas = slp \<and> length fas = l) \<bind> (\<lambda>_. SPEC (\<lambda>R. case R of None \<Rightarrow> True | Some R \<Rightarrow> \<exists>e. env_len env e \<and> env_len R l \<and> (\<forall>e\<in>env. interpret_floatariths fas e \<in> R)))"
ivl_disj_un_9: fixes l :: "'b"   and u :: "'b" assumes "l \<le> u" shows "{..l} \<union> {l<..u} = {..u}"
strict_implies_not_eq: fixes b :: "'g multiset"   and a :: "'g multiset" assumes "b \<subset># a" shows "a \<noteq> b"
map_sel: fixes f1 :: "'e \<Rightarrow> 'b"   and f2 :: "'f \<Rightarrow> 'c"   and a :: "('a, 'e, 'f, 'd) converter" shows "run_converter (map_converter' f1 f2 a) = map_gpv (map_prod f1 (map_converter' f1 f2)) f2 \<circ> run_converter a"
inf_sup_distrib1: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "class.distrib_lattice inf less_eq less sup" shows "inf x (sup y z) = sup (inf x y) (inf x z)"
face_of_convex_hull_insert: fixes S :: "'b set"   and a :: "'b"   and T :: "'b set" assumes "finite S"   and "a \<notin> affine hull S"   and "T face_of convex hull S" shows "T face_of convex hull insert a S"
factors_of_int_poly_def: fixes p :: "int poly" shows "factors_of_int_poly p = map (abs_int_poly \<circ> fst) (snd (factorize_int_poly p))"
analz_items_subset: fixes H :: "msg set" shows "analz H \<subseteq> items H"
twoCisTypeI_2: shows "\<forall>x\<in>cbox a b. g2 x \<le> g1 x"
for_induct: fixes i :: "nat"   and j :: "nat"   and I :: "nat \<Rightarrow> heap \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b Heap"   and h :: "heap" assumes "i \<le> j"   and "\<And>k h. \<lbrakk>I k h; i \<le> k; k < j\<rbrakk> \<Longrightarrow> success (f k) h"   and "I i h"   and "\<And>k h h' r. \<lbrakk>I k h; i \<le> k; k < j; effect (f k) h h' r\<rbrakk> \<Longrightarrow> I (Suc k) h'" shows "\<exists>h'. effect (for [i..<j] f) h h' () \<and> I j h'"
seteqI: fixes A :: "'a set"   and B :: "'a set" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<in> B"   and "\<And>x. x \<in> B \<Longrightarrow> x \<in> A" shows "A = B"
reflect_poly_dvd: fixes f :: "'a poly"   and g :: "'a poly" assumes "f dvd g" shows "reflect_poly f dvd reflect_poly g"
roots_of_3_1: fixes p :: "int poly" assumes "p \<noteq> 0" shows "real_of_3 ` set (roots_of_3 p) = {x. poly (real_of_int_poly p) x = 0}"
marked_deletions_def: fixes m :: "'e" shows "mut_m.marked_deletions m = (\<lambda>s. \<forall>x. x \<in> set (sys_mem_store_buffers (mutator m) s) \<longrightarrow> marked_deletion x s)"
left_neutral: fixes a :: "'b" shows "(1::'b) \<cdot> a = a"
word_rsplit_def: fixes w :: "'b word" shows "word_rsplit w = map word_of_int (bin_rsplit LENGTH('a) (LENGTH('b), uint w))"
h_inj: fixes x1 :: "'a"   and x2 :: "'a" assumes "x1 \<in> carrier R"   and "x2 \<in> carrier R"   and "h x1 = h x2" shows "x1 = x2"
check_ipresIGSubst: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" shows "ipresIGSubst check (errMOD MOD) MOD"
simps_1: fixes k :: "'b" shows "atoms_of_lit (k\<^sup>+) = k"
logic_actual_nec_3: fixes \<phi> :: "'c \<Rightarrow> \<o>" shows "[[\<^bold>\<A>(\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) \<^bold>\<equiv> (\<^bold>\<forall>\<alpha>. \<^bold>\<A>\<phi> \<alpha>)]]"
step_def: fixes s :: "'state_t"   and oa :: "'action_t option" shows "int.step s oa \<equiv> case oa of None \<Rightarrow> s | Some a \<Rightarrow> kstep s a"
not_lta_and_conga: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" shows "\<not> (A B C LtA D E F \<and> A B C CongA D E F)"
that: fixes X :: "V" assumes "X \<in>\<^sub>\<circ> vproduct I A"   and "x = X \<restriction>\<^sup>l\<^sub>\<circ> J" shows "thesis"