hcju/isabelleps huggingface dataset:
ge_gt_tpoly_trans: 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> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and r :: "('a pvar, hmultiset) tpoly"   and q :: "('a pvar, hmultiset) tpoly"   and p :: "('a pvar, hmultiset) tpoly" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "tkbo_coefs.ge_tpoly ground_heads_var \<delta> arity_sym wt_sym r q"   and "tkbo_coefs.gt_tpoly ground_heads_var \<delta> arity_sym wt_sym q p" shows "tkbo_coefs.gt_tpoly ground_heads_var \<delta> arity_sym wt_sym r p"
lt_A: fixes p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "p \<in> phull (set fs)"   and "p \<noteq> 0" obtains g :: "'t \<Rightarrow>\<^sub>0 'b" where "g \<in> A" and "g \<noteq> 0" and "lt g = lt p"
assms_1: fixes q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_red B q" shows "is_red A q"
simps: fixes \<phi> :: "'c stream \<Rightarrow> bool"   and \<psi> :: "'c stream \<Rightarrow> bool"   and a :: "'c stream" shows "(\<phi> until \<psi>) a = ((\<exists>xs. a = xs \<and> \<psi> xs) \<or> (\<exists>xs. a = xs \<and> \<phi> xs \<and> (\<phi> until \<psi>) (stl xs)))"
snoc_eq_iff_butlast: fixes xs :: "'g list"   and x :: "'g"   and ys :: "'g list" shows "(xs @ [x] = ys) = (ys \<noteq> [] \<and> butlast ys = xs \<and> last ys = x)"
m1_inv2r_servI: fixes x :: "'a m1r_state_scheme" assumes "\<And>A B Rb Kab nlb. \<lbrakk>B \<notin> bad; runs x Rb = Some (Resp, [A, B], aKey Kab # nlb)\<rbrakk> \<Longrightarrow> \<exists>Rs Na. Kab = sesK (Rs $ sk) \<and> runs x Rs = Some (Serv, [A, B], [aNon Na])" shows "x \<in> m1_inv2r_serv"
simps_1: fixes "end" :: "char list"   and t :: "char"   and ts :: "char list" shows "scan_upto end (t # ts) = (if map snd (zip end (t # ts)) = end then Inr (end, drop (length end) (t # ts)) else scan_upto end ts \<bind> (\<lambda>(res, ts'). Inr (t # res, ts')))"
assms: fixes G :: "\<Pi>\<^sub>1"   and \<phi> :: "\<nu> \<Rightarrow> \<kappa> \<Rightarrow> \<o>" shows "IsProperInX (\<lambda>x. \<lparr>G,\<^bold>\<iota>y. \<phi> y x\<rparr>)"
Re_power_real: fixes x :: "complex"   and n :: "nat" assumes "is_real x" shows "Re (x ^ n) = Re x ^ n"
finite_ri_commit: fixes runz :: "fid_t \<Rightarrow> (role_t \<times> agent list \<times> atom list) option"   and A :: "agent"   and B :: "agent"   and Kab :: "key"   and Ts :: "nat"   and Ta :: "nat" assumes "finite (dom runz)" shows "finite (ir_running runz A B Kab Ts Ta)"
sort_conv_fold: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and xs :: "'k list" assumes "class.linorder less_eq less" shows "linorder.sort_key less_eq (\<lambda>x. x) xs = fold (linorder.insort_key less_eq (\<lambda>x. x)) xs []"
one_before_two_aux: fixes j :: "state stream" assumes "\<exists>p r i. j = nxt (make_full_observation drinks (Some 1) r p) i" shows "alw (\<lambda>x. nxt (state_eq (Some 2)) x \<longrightarrow> state_eq (Some 1) x) j"
hyps: shows "\<not> is_final (steps0 (1, l, r) tcopy_loop stp)"
zmset_of_le: fixes M :: "'b multiset"   and N :: "'b multiset" shows "(zmset_of M \<le> zmset_of N) = (M \<le> N)"
locally_compact_closedin: fixes S :: "'a set"   and t :: "'a set" assumes "closedin (top_of_set S) t"   and "locally compact S" shows "locally compact t"
sorted_spvec_addmult_spvec_helper2: fixes y :: "'a"   and arr :: "(nat \<times> 'a) list"   and aa :: "nat"   and ba :: "'a"   and brr :: "(nat \<times> 'a) list"   and a :: "nat"   and b :: "'a" assumes "sorted_spvec (addmult_spvec y arr ((aa, ba) # brr))"   and "a < aa"   and "sorted_spvec ((a, b) # arr)"   and "sorted_spvec ((aa, ba) # brr)" shows "sorted_spvec ((a, b) # addmult_spvec y arr ((aa, ba) # brr))"
bdt_Some_Leaf_var_le_1: fixes p :: "ref"   and low :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref"   and t :: "dag"   and var :: "ref \<Rightarrow> nat"   and x :: "bdt" assumes "Dag p low high t"   and "bdt t var = Some x"   and "isLeaf_pt p low high" shows "var p \<le> 1"
smult_diff_left: fixes a :: "'c"   and b :: "'c"   and p :: "'c poly" shows "Polynomial.smult (a - b) p = Polynomial.smult a p - Polynomial.smult b p"
ring_simprules_1: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y \<in> carrier R"
rtranclp_induct4: fixes r :: "'a \<times> 'b \<times> 'c \<times> 'd \<Rightarrow> 'a \<times> 'b \<times> 'c \<times> 'd \<Rightarrow> bool"   and ax :: "'a"   and ay :: "'b"   and az :: "'c"   and aw :: "'d"   and bx :: "'a"   and "by" :: "'b"   and bz :: "'c"   and bw :: "'d"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "r\<^sup>*\<^sup>* (ax, ay, az, aw) (bx, by, bz, bw)"   and "P ax ay az aw"   and "\<And>a aa ab b ac ad ae ba. \<lbrakk>r\<^sup>*\<^sup>* (ax, ay, az, aw) (a, aa, ab, b); r (a, aa, ab, b) (ac, ad, ae, ba); P a aa ab b\<rbrakk> \<Longrightarrow> P ac ad ae ba" shows "P bx by bz bw"
mgu_wf_trm: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and s :: "('a, 'c) Term.term"   and t :: "('a, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('a, 'c) Term.term"   and v :: "'c" assumes "intruder_model arity Ana"   and "mgu s t = Some \<sigma>"   and "intruder_model.wf\<^sub>t\<^sub>r\<^sub>m arity s"   and "intruder_model.wf\<^sub>t\<^sub>r\<^sub>m arity t" shows "intruder_model.wf\<^sub>t\<^sub>r\<^sub>m arity (\<sigma> v)"
assms_4: fixes x :: "variable \<times> typ" assumes "x \<in> \<Union> (fv ` snd ` set insts)" shows "x \<notin> fst ` set insts"
irreducible_hom_imp_irreducible: fixes hom :: "'a \<Rightarrow> 'b"   and a :: "'a" assumes "unit_preserving_hom hom"   and "irreducible (hom a)" shows "irreducible a"
cong_add_lcancel: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and uminus :: "'d \<Rightarrow> 'd"   and division_segment :: "'d \<Rightarrow> 'd"   and a :: "'d"   and x :: "'d"   and y :: "'d"   and n :: "'d" assumes "class.unique_euclidean_ring divide plus minus zero times one modulo euclidean_size uminus division_segment" shows "unique_euclidean_semiring.cong modulo (plus a x) (plus a y) n = unique_euclidean_semiring.cong modulo x y n"
crit_pair_def: fixes p :: "'t \<Rightarrow>\<^sub>0 'd"   and q :: "'t \<Rightarrow>\<^sub>0 'd" shows "crit_pair p q = (if component_of_term (lt p) = component_of_term (lt q) then (monom_mult ((1::'d) / lc p) (lcs (lp p) (lp q) - lp p) (tail p), monom_mult ((1::'d) / lc q) (lcs (lp p) (lp q) - lp q) (tail q)) else (0, 0))"
ge: fixes x :: "real" assumes "0 < x"   and "x < 1" shows "e \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
vertical_composite_axioms: shows "vertical_composite A B F H K \<rho>\<sigma>.map \<tau>"
is_aligned_neg_mask: fixes m :: "nat"   and n :: "nat"   and x :: "'b word" assumes "m \<le> n" shows "is_aligned (x AND NOT (mask n)) m"
injective_sup: fixes t :: "'a"   and e :: "'a" assumes "injective t"   and "coreflexive (e * t\<^sup>T)"   and "injective e" shows "injective (t \<squnion> e)"
partial_order_on_well_order_on: fixes r :: "('d \<times> 'd) set"   and A :: "'d set" assumes "finite r"   and "partial_order_on A r" shows "wf (r - Id)"
hyps: fixes P :: "complex mat" assumes "is_quantum_predicate P"   and "k < n" shows "\<turnstile>\<^sub>p {adjoint (exexH_k k) * P * exexH_k k} hadamard_n (Suc k) {P}"
segment_intersection_singleton: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and xl :: "real" assumes "fst a \<noteq> fst b"   and "closed_segment a b \<inter> {p. fst p = xl} \<noteq> {}" obtains p1 :: "real \<times> real" where "closed_segment a b \<inter> {p. fst p = xl} = {p1}"
exhaust: fixes y :: "'a tm" obtains x1 :: "'a" and x2 :: "nat" where "y = TM x1 x2"
assms_1: shows "\<forall>x\<in>X. P x"
fst_apfst: fixes f :: "'c \<Rightarrow> 'a"   and x :: "'c \<times> 'b" shows "fst (apfst f x) = f (fst x)"
lf: shows "\<forall>n s. numbound0 s \<longrightarrow> isrlfm (f n s)"
Deliver_added_files: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and i :: "'a"   and e :: "'b" assumes "xs prefix of j"   and "i \<in> set (added_files xs e)" shows "Deliver (i, Append i e) \<in> set xs \<or> (\<exists>mo. Deliver (i, Store e mo i) \<in> set xs)"
assms: shows "\<tau> \<Turnstile> H .allInstances()->includes\<^sub>S\<^sub>e\<^sub>t(x)"
neg_less_0_iff_less: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less (uminus a) zero = less zero a"
assms: shows "\<forall>c. c \<in> set w \<longrightarrow> c = a"
distinct_18: fixes x51 :: "pdf_type"   and x52 :: "pdf_type" shows "PRODUCT x51 x52 \<noteq> INTEG"
univ_def: fixes f :: "'b \<Rightarrow> 'a"   and X :: "'b set" shows "univ f X \<equiv> f (SOME x. x \<in> X)"
less_eq_char_def: shows "less_eq_char = (\<le>)"
assms_1: shows "ik\<^sub>e\<^sub>s\<^sub>t A \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile> t \<cdot> \<I>"
mpj_surjec: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and H :: "'e set" assumes "R module M"   and "submodule R M H" shows "surjec\<^bsub>M,M /\<^sub>m H\<^esub> mpj M H"
induced_inj_on: shows "inj_on induced kernel.Partition = (Ker = Ker)"
opp_half_apartmentI: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b set"   and C :: "'b set" assumes "ChamberComplexFolding X f"   and "x \<in> X"   and "C \<in> ChamberComplex.chamber_system X - f \<turnstile> ChamberComplex.chamber_system X"   and "order.greater_eq C x" shows "x \<in> ChamberComplexFolding.opp_half_apartment X f"
f4: shows "Deliver (i, Store e2 mo1 i) \<in> set (history k)"
a: fixes n :: "'b" shows "esqrt (extended_Gromov_distance (u n) x) = \<infinity>"
coprime_self: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one" shows "algebraic_semidom.coprime times one a a = dvd.dvd times a one"
imp_cnj_compat_eqvL: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<chi> \<in> fmla" shows "prv (imp (eqv \<phi>1 \<phi>2) (eqv (cnj \<phi>1 \<chi>) (cnj \<phi>2 \<chi>)))"
cont_Seq_finished: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and c1 :: "('c, 'd, 'a) cmd"   and s :: "'b"   and i :: "nat"   and c2 :: "('c, 'd, 'a) cmd" assumes "PL cval"   and "finished (PL.cont aval tval cval c1 s i)" shows "PL.cont aval tval cval (c1 ;; c2) s i = c2"
field_poly_irreducible_imp_prime: fixes p :: "'b poly" assumes "irreducible p" shows "prime_elem p"
nn_set_integral_add: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure"   and g :: "'b \<Rightarrow> ennreal"   and A :: "'b set" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "A \<in> sets M" shows "\<integral>\<^sup>+x\<in>A. (f x + g x)\<partial>M = set_nn_integral M A f + set_nn_integral M A g"
UP_subring_deriv_closed: fixes S :: "'a set"   and g :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "subring S R"   and "g \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))"   and "a \<in> S" shows "deriv g a \<in> S"
collapse_2: fixes prot_fun :: "('e, 'f, 'g) prot_fun" assumes "is_Set prot_fun" shows "Set (the_Set prot_fun) = prot_fun"
trace_eq: shows "A \<turnstile>\<^sub>R res1 \<approx> res2"
free_const_distinct: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and name\<^sub>1 :: "name"   and name\<^sub>2 :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "free name\<^sub>1 \<noteq> const name\<^sub>2"
simps_1: shows "PX3d D1 = linord_of_list [X3.Xd1, X3.Xd1']"
col_transpose: fixes i :: "nat"   and A :: "'b mat" assumes "i < dim_row A" shows "col A\<^sup>T i = Matrix.row A i"
termi_while: fixes b :: "state \<Rightarrow> bool"   and c :: "com"   and f :: "nat \<Rightarrow> state"   and k :: "nat" assumes "WHILE b DO c \<down> f k"   and "\<forall>i. f i -c\<rightarrow> f (Suc i)" shows "\<exists>i. \<not> b (f i)"
R10_ad_2: shows "pmf (sds R10) d = 1 / 2"
fresh_frun_def: shows "fresh_frun \<equiv> frun_fresh"
cop_stable_no_blocking_on: fixes ds :: "'b set" shows "ManiPref.stable_no_blocking_on ds (ManiPref.cop ds)"
assms_5: shows "[g ^ (p - 1) \<noteq> 1] (mod p\<^sup>2)"
normalize_trick: fixes p :: "'a list list" shows "p = replicate (length p - length (UP.normalize p)) \<zero>\<^bsub>K [X]\<^esub> @ UP.normalize p"
simps_8: fixes f :: "'c \<Rightarrow> 'a"   and p :: "('c, 'b) form" shows "psubst f (Forall p) = Forall (psubst f p)"
a: shows "endo a \<and> E.Obj (Dom a) \<and> Map a = EVAL (Dom a)"
deduct_synth_subterm: fixes M :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term"   and s :: "('fun, 'var) Term.term" assumes "M \<turnstile>\<^sub>c t"   and "s \<sqsubseteq> t"   and "\<forall>m\<in>M. \<forall>s\<in>subterms m. M \<turnstile>\<^sub>c s" shows "M \<turnstile>\<^sub>c s"
preserves_reflects_arr: fixes f :: "'a arrow_of_spans_data" shows "arr (L f) = arr f"
asym_iff: fixes R :: "('b \<times> 'b) set" shows "asym R = (\<forall>x y. (x, y) \<in> R \<longrightarrow> (y, x) \<notin> R)"
strict_ext_def: fixes base_r :: "('a \<times> 'a) set"   and r :: "('a \<times> 'a) set" shows "strict_ext base_r r \<equiv> strict_partial_order r \<and> base_r \<subseteq> r"
OrdP_Mem_E: fixes z :: "name"   and k :: "tm"   and l :: "tm"   and H :: "fm set"   and A :: "fm" assumes "atom z \<sharp> (k, l)"   and "insert (OrdP k) H \<turnstile> l IN k"   and "insert (l SUBS k) (insert (All2 z l (Var z SUBS l)) H) \<turnstile> A" shows "insert (OrdP k) H \<turnstile> A"
hyps_2: fixes T' :: "ty" assumes "P,E \<turnstile> e :: T'" shows "T = T'"
right_inverse: fixes a :: "'d" shows "a + - a = (0::'d)"
distinct_43: fixes x21 :: "char list"   and x22 :: "expr"   and x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list" shows "Cast x21 x22 \<noteq> Call x101 x102 x103 x104"
assms_4: shows "\<p>\<^sub>0[f, g] \<cdot> h = \<p>\<^sub>0[f, g] \<cdot> h'"
greaterThanAtMost_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and l :: "'c"   and u :: "'c" shows "ord.greaterThanAtMost less_eq less l u \<equiv> ord.greaterThan less l \<inter> ord.atMost less_eq u"
coboundedI1: fixes c :: "'a multiset"   and a :: "'a multiset"   and b :: "'a multiset" assumes "c \<subseteq># a" shows "c \<subseteq># a \<union># b"
indRelSTEQ_impl_SRel_and_TRel_weakly_preserve_barbs_2: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_weakly_preserves_barbs (indRelSTEQ SRel TRel) (STCalWB SWB TWB)" shows "rel_weakly_preserves_barbs TRel TWB"
lsc_at_ereal: fixes x0 :: "'b"   and f :: "'b \<Rightarrow> ereal" shows "lsc_at x0 f = (\<forall>C<f x0. \<exists>T. open T \<and> x0 \<in> T \<and> (\<forall>y\<in>T. C < f y))"
Q_def: shows "Q \<equiv> intruder_synth' public arity {}"
ev_def: shows "ev \<equiv> \<lambda>\<phi>. lfp (\<lambda>p x. (\<exists>xs. x = xs \<and> \<phi> xs) \<or> (\<exists>xs. x = xs \<and> p (stl xs)))"
rank_monotonic: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat"   and i :: "nat"   and m :: "nat"   and j :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "semi_mojmir_def.rank \<Sigma> \<delta> q\<^sub>0 w x n = Some i"   and "semi_mojmir_def.rank \<Sigma> \<delta> q\<^sub>0 w x (n + m) = Some j" shows "j \<le> i"
o_assoc: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and h :: "'a \<Rightarrow> 'd" shows "f \<circ> (g \<circ> h) = f \<circ> g \<circ> h"
P1: shows "length xs = length (map ((!) xs) [0..<length xs])"
is_valid_problem_sas_plus_dom_sas_plus_problem_range_of: fixes \<Psi> :: "('c, 'd) sas_plus_problem" assumes "is_valid_problem_sas_plus \<Psi>" shows "\<forall>v\<in>set (\<Psi>\<^sub>\<V>\<^sub>+). v \<in> dom (range_of \<Psi>)"
Sup_real_def: fixes X :: "real set" shows "Sup X = (LEAST z. \<forall>x\<in>X. x \<le> z)"
borel_measurable_continuous_countable_exceptions: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "countable X"   and "continuous_on (- X) f" shows "f \<in> borel_measurable borel"
lift_PDG_scd: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and Exit :: "'g"   and nx :: "'g"   and H :: "'i set"   and L :: "'i set" assumes "PDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit (Postdomination.standard_control_dependence sourcenode targetnode valid_edge Exit)"   and "Postdomination sourcenode targetnode kind valid_edge Entry Exit"   and "CFGExit.inner_node sourcenode targetnode valid_edge Entry Exit nx" shows "PDG src trg knd (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry (lift_Def Def Entry Exit H L) (lift_Use Use Entry Exit H L) state_val NewExit (Postdomination.standard_control_dependence src trg (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewExit)"
n_dvd_one: fixes n :: "nat" assumes "is_unit n" shows "n = 1"
e_def: fixes p :: "PrIds" shows "e p = caller memCh!p"
power2_minus: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.ring_1 one times plus zero minus uminus" shows "power.power one times (uminus a) 2 = power.power one times a 2"
nop_n: shows "n \<in>\<^sub>\<circ> \<omega>"
map_option_case: fixes f :: "'f \<Rightarrow> 'e"   and y :: "'f option" shows "map_option f y = (case y of None \<Rightarrow> None | Some x \<Rightarrow> Some (f x))"
a2_eq: fixes x :: "'a"   and y :: "'a" shows "ad (x \<cdot> ans_d y) = ad (x \<cdot> y)"
sorted_lin_data_in': shows "sorted (map fst (linearize data_in'))"
intros: fixes \<delta> :: "('a, 'b) ta_rule set"   and rcm :: "('a, 'b) ta_rule \<Rightarrow> nat option" assumes "\<And>r. r \<in> \<delta> \<Longrightarrow> rcm r = Some (card (set (rhsq r)))" shows "(br_iq \<delta>, br_iq \<delta>, rcm) \<in> br'_initial \<delta>"
m_inv_closed: fixes x :: "'a" assumes "x \<in> B" shows "inv x \<in> B"
pred_inject_1: fixes P :: "'a \<Rightarrow> bool" shows "pred_option P None"
convex_simplex: fixes n :: "int"   and S :: "'b set" assumes "n simplex S" shows "convex S"
horner_sum_foldr: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'c"   and a :: "'c"   and xs :: "'d list" assumes "class.comm_semiring_0 plus zero times" shows "comm_semiring_0.horner_sum plus zero times f a xs = foldr (\<lambda>x b. plus (f x) (times a b)) xs zero"
map_comp: fixes g :: "'c \<Rightarrow> 'b"   and f :: "'d \<Rightarrow> 'c"   and v :: "'d pmf" shows "map_pmf g (map_pmf f v) = map_pmf (g \<circ> f) v"
subspace_dim_equal: fixes S :: "('b, 'c) vec set"   and T :: "('b, 'c) vec set" assumes "vec.subspace S"   and "vec.subspace T"   and "S \<subseteq> T"   and "vec.dim T \<le> vec.dim S" shows "S = T"
that: fixes out :: "'b"   and c :: "'c" assumes "generat = IO out c" shows "thesis"
simps_2: fixes x :: "'a"   and xs :: "'a list"   and ys :: "'b list" shows "List.product (x # xs) ys = map (Pair x) ys @ List.product xs ys"
less_top: fixes a :: "'b" shows "(a \<noteq> top) = (a < top)"
basic_cqe_conv1_4: shows "(\<exists>x. poly 0 x = (0::'a)) = True"
diff_delta_arctan_upper_14_def: shows "diff_delta_arctan_upper_14 \<equiv> \<lambda>x. - 4 / ((1 + 3 * x\<^sup>2)\<^sup>2 * (1 + x\<^sup>2))"
may_lock_iff: fixes l :: "('c \<times> nat) option"   and t :: "'c" shows "may_lock l t = (case l of None \<Rightarrow> True | \<lfloor>(t', n)\<rfloor> \<Rightarrow> t = t')"
tuple_in_Some_None: fixes as :: "'a option list"   and t :: "nat"   and X :: "'a option list set" assumes "Mapping.lookup tuple_in as = Some t"   and "as \<in> X"   and "(t, X) \<in> set discard" shows "Mapping.lookup tuple_in' as = None"
bin_nth_eq_iff: fixes x :: "int"   and y :: "int" shows "(bit x = bit y) = (x = y)"
F: fixes a :: "'a"   and b :: "'b" assumes "a \<in> basisA"   and "b \<in> basisB" shows "\<exists>F\<in>{butterfly a b |a b. a \<in> basisA \<and> b \<in> basisB}. \<forall>b'\<in>basisB. F *\<^sub>V b' = (if b' = b then a else (0::'a))"
starP2_def: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "*p2* P = (\<lambda>x y. unstar (star_of P \<star> x \<star> y))"
equal_fin_string_def: fixes X :: "fin_string"   and Y :: "fin_string" shows "equal_class.equal X Y = (X \<le> Y \<and> Y \<le> X)"
characteristic_formula_is_characteristic': fixes satisfies :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and transition :: "'b \<Rightarrow> ('d, 'b) residual \<Rightarrow> bool"   and Q :: "'b"   and P :: "'b" assumes "indexed_nominal_ts TYPE('a) satisfies transition" shows "nominal_ts.valid satisfies transition Q (indexed_nominal_ts.characteristic_formula satisfies transition P) = indexed_nominal_ts.logically_equivalent TYPE('a) satisfies transition P Q"
fmlookup_ran'_iff: fixes y :: "'c"   and m :: "('d, 'c) fmap" shows "(y \<in> fmran' m) = (\<exists>x. fmlookup m x = Some y)"
axioms_1: fixes ext :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "ext_compat_cons ext" shows "ext ext"
ssorted_monoI: fixes s :: "'b stream" assumes "\<forall>i j. i \<le> j \<longrightarrow> s !! i \<le> s !! j" shows "ssorted s"
L8: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'b list"   and ys :: "'b list" assumes "associative f"   and "xs \<noteq> []"   and "ys \<noteq> []" shows "foldl1 f (xs @ ys) = f (foldl1 f xs) (foldl1 f ys)"
simps: fixes a1 :: "('test, 'atom) com"   and a2 :: "'state" shows "mayDiverge a1 a2 = (\<exists>c s c' s'. a1 = c \<and> a2 = s \<and> (c, s) \<rightarrow>c (c', s') \<and> mayDiverge c' s')"
induct: fixes P :: "'c list list \<Rightarrow> bool"   and a0 :: "'c list list" assumes "P []"   and "\<And>xs. P [xs]"   and "\<And>xs ys zss. P zss \<Longrightarrow> P (xs # ys # zss)" shows "P a0"
that: fixes p :: "'state"   and v :: "'action stream" assumes "w = smap int (p ## trace v p)"   and "init p"   and "run v p" shows "thesis"