hcju/isabelleps huggingface dataset:
gt_sameI: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> nat"   and s :: "('b, 'a) tm"   and t :: "('b, 'a) tm" assumes "kbo_std ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym"   and "vars_mset s \<subseteq># vars_mset t"   and "Lambda_Free_KBO_Std.kbo_std.wt ground_heads_var \<delta> arity_sym wt_sym t = Lambda_Free_KBO_Std.kbo_std.wt ground_heads_var \<delta> arity_sym wt_sym s"   and "head t = head s"   and "\<forall>f\<in>ground_heads.ground_heads ground_heads_var (head t). extf f (Lambda_Free_KBO_Std.kbo_std.gt ground_heads_var gt_sym \<delta> extf arity_sym wt_sym) (args t) (args s)" shows "Lambda_Free_KBO_Std.kbo_std.gt_same ground_heads_var gt_sym \<delta> extf arity_sym wt_sym t s"
induct: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and out :: "'c"   and x :: "('a, 'b, 'c, 'd) converter"   and P :: "('a, 'b, 'c, 'd) converter \<Rightarrow> bool" assumes "outsp_converter \<I> \<I>' out x"   and "\<And>conv a. \<lbrakk>out \<in> outs_gpv \<I>' (run_converter conv a); a \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> P conv"   and "\<And>b conv' conv a. \<lbrakk>(b, conv') \<in> results_gpv \<I>' (run_converter conv a); outsp_converter \<I> \<I>' out conv'; P conv'; a \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> P conv" shows "P x"
id: shows "(x < 0 \<or> n < 0) = False"
dual_comp: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "dual (times x y) = times (dual x) (dual y)"
pmf_expectation_eq_infsetsum: fixes p :: "'b pmf"   and f :: "'b \<Rightarrow> real" shows "measure_pmf.expectation p f = (\<Sum>\<^sub>ax. pmf p x * f x)"
t2: fixes X :: "real set" assumes "bdd_above (sqrt ` X)" shows "bdd_above X"
differentiable_on_def: fixes f :: "'a \<Rightarrow> 'b"   and s :: "'a set" shows "(f differentiable_on s) = (\<forall>x\<in>s. f differentiable at x within s)"
fls_X_pow_nonzero: fixes n :: "nat" shows "fls_X ^ n \<noteq> 0"
thetaFDW0_def: shows "thetaFDW0 \<equiv> {(c1, c2). discr0 c1 \<and> discr0 c2}"
distinct_list_subset_nths: fixes ss :: "'b list"   and ls :: "'b list" assumes "distinct ss"   and "set ss \<subseteq> set ls" obtains ids :: "nat list" where "distinct ids" and "set ids \<subseteq> {..<length ls}" and "ss = map ((!) ls) ids"
int_g: fixes f :: "'s stream \<Rightarrow> ennreal"   and s :: "'s" assumes "f \<in> borel_measurable (stream_space (count_space UNIV))" shows "\<integral>\<^sup>+ \<omega>. g s (f \<omega>) \<partial>T s = g s (integral\<^sup>N (T s) f)"
surj_vimage_empty: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'a set" assumes "surj f" shows "(f -` A = {}) = (A = {})"
assms_3: fixes i :: "'a" assumes "i \<le> least A"   and "i \<le> least B" shows "(i \<in> A) = (i \<in> B)"
pr: shows "F (i - 2) = Polynomial.smult (inverse (f (i - 1)) ^ (Suc (length (coeffs (F (i - 2)))) - length (coeffs (F (i - 1))))) (Q i) * F (i - 1) + Polynomial.smult (inverse (f (i - 1)) ^ (Suc (length (coeffs (F (i - 2)))) - length (coeffs (F (i - 1))))) (Polynomial.smult (\<beta> i) (F i))"
long_pow_exp: fixes r :: "'b list"   and m :: "nat" assumes "r \<noteq> \<epsilon>" shows "m \<le> \<^bold>|r \<^sup>@ m\<^bold>|"
assms_1: shows "y \<notin>\<^sub>\<circ> \<R>\<^sub>\<circ> xs"
dmf0: shows "- f 0 \<in> carrier_mat d d"
PfP_sf: fixes y :: "tm" shows "Sigma_fm (PfP y)"
div_comp_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and rs1 :: "word5"   and rd :: "word5"   and operand2 :: "word32"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (div_comp instr rs1 rd operand2 s1)) \<and> t2 = snd (fst (div_comp instr rs1 rd operand2 s2))" shows "low_equal t1 t2"
cbox: shows "cbox (0, 0) (1, 1) = {0..1} \<times> {0..1}"
4: shows "p \<le> p \<sqinter> - (1::'a) \<squnion> (1::'a)"
dilating_fun_def: fixes f :: "nat \<Rightarrow> nat"   and r :: "'b run" shows "dilating_fun f r \<equiv> strict_mono f \<and> f 0 = 0 \<and> (\<forall>n. n \<le> f n \<and> ((\<nexists>n\<^sub>0. f n\<^sub>0 = n) \<longrightarrow> (\<forall>c. \<not> hamlet (Rep_run r n c))) \<and> ((\<nexists>n\<^sub>0. f n\<^sub>0 = Suc n) \<longrightarrow> (\<forall>c. time (Rep_run r (Suc n) c) = time (Rep_run r n c))))"
group_left_neutral: fixes a :: "'d" shows "(0::'d) + a = a"
obs_singleton_element: fixes ms :: "'node list"   and ns :: "'node list"   and S :: "'node SDG_node set" assumes "ms \<in> obs ns \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "Ball (set (tl ns)) return_node" shows "obs ns \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub> = {ms}"
leftprefixlang: shows "\<L>\<^sub>P = {uu_. \<exists>u v. uu_ = u \<and> is_word u \<and> is_leftderivation (u @ v)}"
infsetsum_cong: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e"   and B :: "'d set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x"   and "A = B" shows "infsetsum f A = infsetsum g B"
VarNonOccFormP_graph_aux_def: shows "VarNonOccFormP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>VarNonOccFormP_aux v x. x1 = (v, x) \<and> x2 = FormP x AND SubstFormP v Zero x x)"
f_rw: shows "f = (fst f, fst (snd f), snd (snd f))"
vsv_vlrestriction: fixes A :: "V" shows "vsv (r\<inverse>\<^sub>\<circ> \<restriction>\<^sup>l\<^sub>\<circ> A)"
frechet_derivative_at': fixes f :: "'a \<Rightarrow> 'b"   and s :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "(f has_derivative s) (at x)" shows "frechet_derivative f (at x) = s"
grid_replace_dim: fixes d :: "nat"   and b' :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list"   and p :: "(nat \<times> int) list"   and ds :: "nat set"   and p' :: "(nat \<times> int) list" assumes "d < length b'"   and "d < length b"   and "p \<in> grid b ds"   and "p' \<in> grid b' ds" shows "p[d := p' ! d] \<in> grid (b[d := b' ! d]) ds"
bounded_linear_left: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and b :: "'b" assumes "bounded_bilinear prod" shows "bounded_linear (\<lambda>a. prod a b)"
assms_3: shows "nsqn rt1 dip < nsqn rt2 dip"
f4: shows "4 \<le> rk {A, A', C, a, c, Q}"
case_2: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "char list \<Rightarrow> 'a" shows "(case Boolean of Void \<Rightarrow> f1 | Boolean \<Rightarrow> f2 | Integer \<Rightarrow> f3 | NT \<Rightarrow> f4 | Class x \<Rightarrow> f5 x) = f2"
inf_right_dist_sum: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and conv :: "'c \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'c"   and x :: "'c" assumes "class.stone_relation_algebra inf less_eq less sup bot top uminus one times conv" shows "comm_monoid_add.sum sup bot (\<lambda>k. inf (f k) x) {k. True} = inf (comm_monoid_add.sum sup bot f {k. True}) x"
Skip: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and s :: "'e"   and n :: "nat" shows "\<Gamma>\<turnstile> \<langle>Skip,Normal s\<rangle> =n\<Rightarrow> Normal s"
pp_of_term_def: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.pp_of_term pair_of_term v = fst (pair_of_term v)"
homotopic_paths_rinv: fixes p :: "real \<Rightarrow> 'a"   and s :: "'a set" assumes "path p"   and "path_image p \<subseteq> s" shows "homotopic_paths s (p +++ reversepath p) (linepath (pathstart p) (pathstart p))"
IH: shows "lift_streams (rev xs) = smap rev (lift_streams xs)"
less_restrictive_complete: fixes atom_ordering :: "('a \<times> 'a) set"   and R1 :: "'a Literal set \<Rightarrow> 'a Literal set \<Rightarrow> 'a Literal set \<Rightarrow> bool"   and R2 :: "'a Literal set \<Rightarrow> 'a Literal set \<Rightarrow> 'a Literal set \<Rightarrow> bool" assumes "propositional_atoms atom_ordering"   and "propositional_atoms.less_restrictive R1 R2"   and "propositional_atoms.Complete R2" shows "propositional_atoms.Complete R1"
s\<^sub>2: shows "s\<^sub>2 = s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>(l \<mapsto> v), \<E> [VE (CV Unit)]))"
nres_more_simps_6: fixes x :: "'b"   and y :: "'b" shows "(RETURN x = RETURN y) = (x = y)"
dvd_int_poly_non_0_def: fixes q :: "int poly"   and p :: "int poly" shows "dvd_int_poly_non_0 q p = (div_int_poly p q \<noteq> None)"
m_lcomm: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a"   and z :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P"   and "z \<in> carrier P" shows "x \<oplus>\<^bsub>P\<^esub> (y \<oplus>\<^bsub>P\<^esub> z) = y \<oplus>\<^bsub>P\<^esub> (x \<oplus>\<^bsub>P\<^esub> z)"
eq: fixes n :: "nat" shows "(1 / root DIM('a) (real (Suc n))) ^ DIM('a) = 1 / real (Suc n)"
\<beta>s_rev: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (subst_bv (Fv x \<tau>') s) (Abs \<tau>' s $ Fv x \<tau>')"
domI: fixes m :: "'b \<Rightarrow> 'a option"   and a :: "'b"   and b :: "'a" assumes "m a = \<lfloor>b\<rfloor>" shows "a \<in> dom m"
coprime_iff_invertible_nat: fixes a :: "nat"   and m :: "nat" shows "coprime a m = (\<exists>x. [a * x = Suc 0] (mod m))"
formulaTrueCommutativity: fixes f1 :: "Literal list list"   and f2 :: "Literal list list"   and valuation :: "Literal list" shows "formulaTrue (f1 @ f2) valuation = formulaTrue (f2 @ f1) valuation"
is_aligned_add_not_aligned: fixes p :: "'b word"   and n :: "nat"   and q :: "'b word" assumes "is_aligned p n"   and "\<not> is_aligned q n" shows "\<not> is_aligned (p + q) n"
mdinit_def: shows "mdinit \<equiv> FOREACH V (\<lambda>v wl. RETURN (wl(v \<mapsto> infty.Infty))) Map.empty \<bind> (\<lambda>wl. RETURN (wl(v0 \<mapsto> Num (0::'W)), [v0 \<mapsto> ([], 0::'W)]))"
integrable_indicator_iff: fixes M :: "'c measure"   and A :: "'c set" shows "integrable M (indicat_real A) = (A \<inter> space M \<in> sets M \<and> emeasure M (A \<inter> space M) < \<infinity>)"
eval_fm_Induction_axP: fixes e :: "(name, hf) finfun"   and p :: "tm" shows "eval_fm e (Induction_axP p) = (\<lbrakk>p\<rbrakk>e \<in> Induction_ax)"
compl_code_1: fixes xs :: "('b, unit) RBT.rbt" shows "- Set xs = Coset xs"
groups_size: shows "1 < card {G \<times> I |G. G \<in> \<G>}"
payload_rep_cpayload: fixes x :: "msg" shows "(Re x \<in> cpayload) = (x \<in> payload)"
edges_are_in_V_1: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and w :: "'c" assumes "Graph G"   and "v \<rightarrow>\<^bsub>G\<^esub> w" shows "v \<in> V\<^bsub>G\<^esub>"
similar_sq_mtx_matrix: fixes A :: "'a sq_mtx"   and B :: "'a sq_mtx" shows "(A \<sim> B) = similar_matrix (to_vec A) (to_vec B)"
fineI: fixes s :: "('a \<times> 'b set) set"   and d :: "'a \<Rightarrow> 'b set" assumes "\<And>x k. (x, k) \<in> s \<Longrightarrow> k \<subseteq> d x" shows "d fine s"
lossless_restrict_gpvD: fixes \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv" assumes "lossless_gpv \<I> (restrict_gpv \<I> gpv)"   and "\<I> \<turnstile>g gpv \<surd>" shows "lossless_gpv \<I> gpv"
rel_OO: shows "rel_U L1 (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2') x y"
meth: shows "P \<turnstile> C sees M: Ts\<rightarrow>T = (mxs, mxl\<^sub>0, ins, xt) in C"
len_v_borders: shows "left (len v ts c) = right (len v ts c)"
BetaNegImpl: fixes A :: "('a, 'b) form"   and G :: "('a, 'b) form list"   and B :: "('a, 'b) form" assumes "\<turnstile> A # G"   and "\<turnstile> Neg B # G" shows "\<turnstile> Neg (Impl A B) # G"
true_cls_mset_singleton: fixes I :: "'b set"   and C :: "'b literal multiset" shows "(I \<Turnstile>m {#C#}) = (I \<Turnstile> C)"
deformation_retraction_imp_isomorphic_relative_homology_groups: fixes X :: "'a topology"   and Y :: "'b topology"   and r :: "'a \<Rightarrow> 'b"   and s :: "'b \<Rightarrow> 'a"   and U :: "'a set"   and V :: "'b set"   and p :: "int" assumes "retraction_maps X Y r s"   and "r ` U \<subseteq> V"   and "s ` V \<subseteq> U"   and "homotopic_with (\<lambda>h. h ` U \<subseteq> U) X X (s \<circ> r) id" shows "relative_homology_group p X U \<cong> relative_homology_group p Y V"
Rep_uexpr_inverse: fixes x :: "('e, 'f) uexpr" shows "mk\<^sub>e \<lbrakk>x\<rbrakk>\<^sub>e = x"
PO_a0n_inv1_niagree_trans: shows "{a0n_inv1_niagree} TS.trans a0n {> a0n_inv1_niagree}"
valid_eqvt: fixes P :: "'state"   and x :: "('b, 'pred, 'act) formula"   and p :: "perm" assumes "P \<Turnstile> x" shows "p \<bullet> P \<Turnstile> p \<bullet> x"
assms: fixes bl :: "'a set" assumes "bl \<in> bs" shows "p \<in> bl"
bin_rsplit_len: fixes n :: "nat"   and nw :: "nat"   and w :: "int" assumes "n \<noteq> 0" shows "length (bin_rsplit n (nw, w)) = (nw + n - 1) div n"
S4: shows "A = {z. univ_for_pr (c_fst z) = c_snd z}"
prefix_suffix: fixes x :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "x = prefix n x \<frown> suffix n x"
word_sless_alt: fixes a :: "'c word"   and b :: "'c word" shows "(a <s b) = (sint a < sint b)"
lselect_restrict_indices: fixes w :: "'b llist"   and s :: "nat set" shows "lselect {i \<in> s. enat i < llength w} w = lselect s w"
fls_inverse_shift_nonzero: fixes f :: "'b fls"   and m :: "int" assumes "f \<noteq> 0" shows "inverse (fls_shift m f) = fls_shift (- m) (inverse f)"
Agent_Number: fixes X :: "agent"   and X' :: "nat" shows "Messages.Agent X \<noteq> Number X'"
div0: shows "\<forall>a\<in>set as. divisor a \<noteq> 0"
le_antisym: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c" assumes "partial_order L"   and "x \<sqsubseteq>\<^bsub>L\<^esub> y"   and "y \<sqsubseteq>\<^bsub>L\<^esub> x"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "x = y"
ancestors: shows "h \<turnstile> get_ancestors_si ptr \<rightarrow>\<^sub>r ancestors"
lisetD_1: fixes i :: "nat"   and A :: "'a set"   and w :: "'a llist" assumes "i \<in> liset A w" shows "enat i < llength w"
that: fixes c :: "'a" assumes "(sum_upto f \<longlongrightarrow> c) at_top" shows "thesis"
integrable_on_lebesgue_on: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "integrable (lebesgue_on S) f"   and "S \<in> sets lebesgue" shows "f integrable_on S"
le_supI: fixes a :: "'e multiset"   and x :: "'e multiset"   and b :: "'e multiset" assumes "a \<subseteq># x"   and "b \<subseteq># x" shows "a \<union># b \<subseteq># x"
spanning_subset_independent: fixes B :: "'b set"   and A :: "'b set" assumes "B \<subseteq> A"   and "independent A"   and "A \<subseteq> span B" shows "A = B"
e2rv_def: fixes e :: "nat" shows "e2rv e \<equiv> pdec2 e"
glb_in_lattice: fixes cl :: "'b potype"   and S :: "'b set" assumes "CL cl"   and "S \<subseteq> pset cl" shows "glb S cl \<in> pset cl"
BotL: fixes \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset" assumes "\<bottom> \<in># \<Gamma>" shows "\<Gamma> \<Rightarrow> \<Delta>"
sort_eqv_normalize_sort: fixes s :: "String.literal set" assumes "finite s"   and "sort_ex cs s" shows "sort_eqv cs s (normalize_sort cs s)"
nb\<^sub>1: shows "cnf f = {{literal_formula_to_literal f}}"
simps_9: fixes e :: "nat \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a list \<Rightarrow> bool"   and p :: "('b, 'c) form" shows "eval e f g (Exists p) = (\<exists>z. eval (e\<langle>0:z\<rangle>) f g p)"
map_pmf_of_set_bij_betw: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "bij_betw f A B"   and "A \<noteq> {}"   and "finite A" shows "map_pmf f (pmf_of_set A) = pmf_of_set B"
InvariantGetReasonIsReasonQSubset: fixes Q :: "Literal set"   and Q' :: "Literal set"   and GetReason :: "Literal \<Rightarrow> nat option"   and F :: "Literal list list"   and M :: "(Literal \<times> bool) list" assumes "Q \<subseteq> Q'"   and "InvariantGetReasonIsReason GetReason F M Q'" shows "InvariantGetReasonIsReason GetReason F M Q"
smc_SemiCAT_components_2: fixes \<alpha> :: "V" shows "smc_SemiCAT \<alpha>\<lparr>Arr\<rparr> = all_smcfs \<alpha>"
add2_Some_Some: fixes x :: "'a"   and y :: "'a" shows "Some x ++ Some y = x \<oplus> y"
smallo_const_inverse: fixes f :: "'b \<Rightarrow> real"   and F :: "'b filter"   and c :: "real" assumes "filterlim f at_top F"   and "F \<noteq> bot" shows "((\<lambda>_. c) \<in> o[F](\<lambda>x. inverse (f x))) = (c = 0)"
HistoryLemma: shows "\<turnstile> Init (\<forall>p. ImpInit p) \<and> \<box>(\<forall>p. ImpNext p) \<longrightarrow> (\<exists>\<exists> rmhist. Init (\<forall>p. HInit rmhist p) \<and> \<box>(\<forall>p. [HNext rmhist p]_(c p, r p, m p, rmhist!p)))"
map_sum_sel_2: fixes s :: "'d + 'e"   and f :: "'d \<Rightarrow> 'g"   and g :: "'e \<Rightarrow> 'f" assumes "\<not> isl s" shows "projr (map_sum f g s) = g (projr s)"
assms_2: shows "case_nat t2 \<Gamma> \<turnstile>\<^sub>c f : t3"
assms: fixes x :: "(real, 'n) vec" assumes "x \<in> Basis" shows "A *v x = B *v x"
inv_matching_def: fixes E :: "'b set set"   and C :: "'c set"   and F :: "'b set set"   and M :: "'b set set" assumes "Edges E" shows "Edges.inv_matching E C F M = (matching M \<and> M \<subseteq> E \<and> card C \<le> 2 * card M \<and> (\<forall>e\<in>M. \<forall>f\<in>F. e \<inter> f = {}))"
sat_constraint_valuation: fixes c :: "'a ns_constraint"   and v1 :: "nat \<Rightarrow> 'a"   and v2 :: "nat \<Rightarrow> 'a" assumes "\<forall>var\<in>vars (poly c). v1 var = v2 var" shows "v1 \<Turnstile>\<^sub>n\<^sub>s c = v2 \<Turnstile>\<^sub>n\<^sub>s c"
continuity_lem_lt0_expanded_var: fixes a :: "real"   and r :: "real"   and b :: "real"   and c :: "real" assumes "a * r\<^sup>2 + b * r + c < 0" shows "\<exists>y'>r. \<forall>x\<in>{r<..y'}. a * x\<^sup>2 + b * x + c \<le> 0"
indicator_scaleR_eq_if: fixes S :: "'d set"   and x :: "'d"   and f :: "'d \<Rightarrow> 'c" shows "indicat_real S x *\<^sub>R f x = (if x \<in> S then f x else (0::'c))"
maxne0_times_sum_list_gt_dotprod: fixes b :: "nat list"   and ys :: "nat list" assumes "length b = length ys" shows "b \<bullet> ys \<le> maxne0 ys b * sum_list ys"
sgn_blinfun_def: fixes x :: "'c \<Rightarrow>\<^sub>L 'd" shows "sgn x = x /\<^sub>R norm x"
OP_def: fixes x :: "'b" shows "Autoref_Tagging.OP x \<equiv> x"
mult_closed: fixes r :: "('b, 'a) aezfun"   and s :: "('b, 'a) aezfun" assumes "r \<in> FG"   and "s \<in> FG" shows "r * s \<in> FG"
g_rec_is_pr: shows "g_rec \<in> PrimRec2"
rep_eq: fixes x :: "uint64"   and xa :: "uint64" shows "Rep_uint64 (x * xa) = Rep_uint64 x * Rep_uint64 xa"
mV: shows "max_Var_floatariths ode_e \<le> DIM('a)"
mult_less_mono1: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i < j"   and "0 < k" shows "i * k < j * k"
le: shows "(x = 0) = (int_of_uint64 x = 0)"
closure_union: fixes A :: "'a list set"   and B :: "'a list set" assumes "regular A"   and "regular B" shows "regular (A \<union> B)"
order_pderiv: fixes p :: "'a poly"   and x :: "'a" assumes "p \<noteq> 0"   and "poly p x = (0::'a)" shows "order x p = Suc (order x (pderiv p))"
x_in_MPLS_charles_then: fixes x :: "nat"   and PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set" assumes "x \<in> MPLS_charles PROB" shows "\<exists>s as. s \<in> valid_states PROB \<and> as \<in> valid_plans PROB \<and> x = Inf (PLS_charles s as PROB)"
hypreal_of_real_not_eq_omega: fixes x :: "real" shows "hypreal_of_real x \<noteq> \<omega>"
assert_imp_def: fixes P :: "cell list \<times> cell list \<Rightarrow> bool"   and Q :: "cell list \<times> cell list \<Rightarrow> bool" shows "P \<mapsto> Q \<equiv> \<forall>l r. P (l, r) \<longrightarrow> Q (l, r)"
UniqueSucStates_Ancestor_StepConf: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and S :: "'d"   and SA :: "('d, 'e, 'f) seqauto"   and T :: "'d" assumes "TS \<in> HPT ST"   and "S \<in> HAStates (HA ST)"   and "SA \<in> the (CompFun (HA ST) S)"   and "T \<in> States SA"   and "T \<in> StepConf (HA ST) (Conf ST) TS" shows "S \<in> StepConf (HA ST) (Conf ST) TS"
identity_functor_axioms: shows "identity_functor (\<cdot>\<^sub>D)"
dg_dghm_id_is_dghm: shows "dghm_id \<AA> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<AA>"
id: shows "integer_hull (polyhedron A b) = convex_hull H + cone C"
map_semantics_ltlc: fixes f :: "'a \<Rightarrow> 'b option"   and \<phi> :: "'a ltlc"   and \<xi> :: "nat \<Rightarrow> 'a set" assumes "inj_on f (dom f)"   and "atoms_ltlc \<phi> \<subseteq> dom f" shows "\<xi> \<Turnstile>\<^sub>c \<phi> = (map_props f \<circ> \<xi>) \<Turnstile>\<^sub>c map_ltlc (the \<circ> f) \<phi>"
k_def: shows "k = \<lfloor>log 2 (real_of_int \<bar>ai\<bar>)\<rfloor>"
while_inv: shows "\<forall>no\<in>set prx. repa no \<noteq> Null \<and> (if (repa \<propto> low) no = (repa \<propto> high) no \<and> low no \<noteq> Null then repa no = (repa \<propto> low) no else repa no = hd (filter (\<lambda>sn. repNodes_eq sn no low high repa) prx) \<and> repa (repa no) = repa no \<and> (\<forall>no1\<in>set prx. ((repa \<propto> high) no1 = (repa \<propto> high) no \<and> (repa \<propto> low) no1 = (repa \<propto> low) no) = (repa no = repa no1)))"
not_0_coeffs_not_Nil: fixes p :: "'a poly" assumes "p \<noteq> 0" shows "coeffs p \<noteq> []"
vimage_Diff: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'b set"   and B :: "'b set" shows "f -` (A - B) = f -` A - f -` B"
similar'_admI: fixes f :: "'b \<Rightarrow> Value"   and g :: "'b \<Rightarrow> CValue"   and n :: "nat" assumes "cont f"   and "cont g" shows "adm (\<lambda>x. f x \<triangleleft>\<triangleright>\<^bsub>n\<^esub> g x)"
allVars_def: fixes u :: "'g \<Rightarrow> 'node \<Rightarrow> 'val set"   and p :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option"   and g :: "'g" shows "uninst.allVars u p g \<equiv> \<Union>n\<in>set (\<alpha>n g). uninst.allDefs p g n \<union> uninst.allUses u p g n"
prems: fixes i :: "'sf"   and f' :: "('sf + 'sz, 'sz) trm" assumes "SFunctions \<sigma> i = Some f'" shows "dfree f'"
filter_cong: fixes xs :: "'b list"   and ys :: "'b list"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "xs = ys"   and "\<And>x. x \<in>. ys \<Longrightarrow> P x = Q x" shows "filter P xs = filter Q ys"
assms_2: shows "\<forall>n. \<forall>asset\<in>support_set p. prices Mkt asset n \<in> borel_measurable (F n)"
x_def: shows "x \<equiv> real_of_1 (p1, l1, r1)"
chainE: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Y :: "nat \<Rightarrow> 'c"   and i :: "nat" assumes "class.po below"   and "po.chain below Y" shows "below (Y i) (Y (Suc i))"
normalize_def'_1: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list" assumes "ring R" shows "p = replicate (length p - length (ring.normalize R p)) \<zero>\<^bsub>R\<^esub> @ drop (length p - length (ring.normalize R p)) p"
m1_inv3r_initD: fixes x :: "'a m1_state_scheme"   and Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and Kab :: "key" assumes "runs x Rb = Some (Resp, [A, B], [aKey Kab, END])"   and "x \<in> m1_inv3r_init"   and "B \<notin> bad"   and "A \<notin> bad"   and "Kab \<notin> Domain (m1_state.leak x)" shows "\<exists>Ra nla. runs x Ra = Some (Init, [A, B], aKey Kab # aNon (Rb $ nb) # nla)"