hcju/isabelleps huggingface dataset:
simps_3: fixes S :: "nat \<Rightarrow> typ"   and sch1 :: "type_scheme"   and sch2 :: "type_scheme" shows "bound_typ_inst S (sch1 =-> sch2) = bound_typ_inst S sch1 -> bound_typ_inst S sch2"
algebraI: fixes R :: "('a, 'b) ring_scheme"   and M :: "('a, 'c, 'd) module_scheme" assumes "cring R"   and "cring M"   and "\<And>a x. \<lbrakk>a \<in> carrier R; x \<in> carrier M\<rbrakk> \<Longrightarrow> a \<odot>\<^bsub>M\<^esub> x \<in> carrier M"   and "\<And>a b x. \<lbrakk>a \<in> carrier R; b \<in> carrier R; x \<in> carrier M\<rbrakk> \<Longrightarrow> (a \<oplus>\<^bsub>R\<^esub> b) \<odot>\<^bsub>M\<^esub> x = a \<odot>\<^bsub>M\<^esub> x \<oplus>\<^bsub>M\<^esub> b \<odot>\<^bsub>M\<^esub> x"   and "\<And>a x y. \<lbrakk>a \<in> carrier R; x \<in> carrier M; y \<in> carrier M\<rbrakk> \<Longrightarrow> a \<odot>\<^bsub>M\<^esub> (x \<oplus>\<^bsub>M\<^esub> y) = a \<odot>\<^bsub>M\<^esub> x \<oplus>\<^bsub>M\<^esub> a \<odot>\<^bsub>M\<^esub> y"   and "\<And>a b x. \<lbrakk>a \<in> carrier R; b \<in> carrier R; x \<in> carrier M\<rbrakk> \<Longrightarrow> a \<otimes>\<^bsub>R\<^esub> b \<odot>\<^bsub>M\<^esub> x = a \<odot>\<^bsub>M\<^esub> (b \<odot>\<^bsub>M\<^esub> x)"   and "\<And>x. x \<in> carrier M \<Longrightarrow> \<one>\<^bsub>R\<^esub> \<odot>\<^bsub>M\<^esub> x = x"   and "\<And>a x y. \<lbrakk>a \<in> carrier R; x \<in> carrier M; y \<in> carrier M\<rbrakk> \<Longrightarrow> a \<odot>\<^bsub>M\<^esub> x \<otimes>\<^bsub>M\<^esub> y = a \<odot>\<^bsub>M\<^esub> (x \<otimes>\<^bsub>M\<^esub> y)" shows "algebra R M"
reachable_awalk: fixes u :: "'a"   and v :: "'a" shows "(u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> {v. u \<rightarrow>\<^sup>* v}\<^esub> v) = (\<exists>p. c.awalk u p v)"
Spy_see_shrK: fixes evs :: "event list"   and A :: "agent" assumes "evs \<in> bankerberos" shows "(Key (shrK A) \<in> parts (knows Spy evs)) = (A \<in> bad)"
is_tangle_diagramness: fixes x :: "wall"   and y :: "wall" shows "is_tangle_diagram x \<and> is_tangle_diagram y \<longrightarrow> is_tangle_diagram (x \<otimes> y)"
set_intros_2: fixes y :: "'d"   and x32 :: "'d ifex"   and x31 :: "'d"   and x33 :: "'d ifex" assumes "y \<in> set_ifex x32" shows "y \<in> set_ifex (IF x31 x32 x33)"
cases: fixes x :: "'c option \<times> 'd option" obtains xa :: "'c" and y :: "'d" where "x = (Some xa, Some y)"   | uu_ :: "'c option" where "x = (uu_, None)"   | uv_ :: "'d option" where "x = (None, uv_)"
simplicial_imp_singular_simplex: fixes p :: "nat"   and S :: "('c \<Rightarrow> real) set"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "simplicial_simplex p S f" shows "singular_simplex p (subtopology (powertop_real UNIV) S) f"
wp_def: fixes x :: "'a"   and p :: "'a" shows "wp x p = ad (x \<cdot> ad p)"
binop_slp_of_fa_induction_step: fixes fa1 :: "floatarith"   and xs :: "real list"   and fa2 :: "floatarith"   and Binop :: "floatarith \<Rightarrow> floatarith \<Rightarrow> floatarith"   and M :: "(floatarith, nat) mapping"   and slp :: "floatarith list"   and M' :: "(floatarith, nat) mapping"   and slp' :: "floatarith list"   and binop :: "real \<Rightarrow> real \<Rightarrow> real" assumes "\<And>M slp M' slp'. \<lbrakk>slp_of_fa fa1 M slp = (M', slp'); \<And>f. f \<in> Mapping.keys M \<Longrightarrow> subterms f \<subseteq> Mapping.keys M; \<And>f. f \<in> Mapping.keys M \<Longrightarrow> the (Mapping.lookup M f) < length slp; \<And>f. f \<in> Mapping.keys M \<Longrightarrow> interpret_slp slp xs ! slp_index_lookup slp M f = interpret_floatarith f xs\<rbrakk> \<Longrightarrow> subterms fa1 \<subseteq> Mapping.keys M' \<and> Mapping.keys M \<subseteq> Mapping.keys M' \<and> (\<forall>f\<in>Mapping.keys M'. subterms f \<subseteq> Mapping.keys M' \<and> the (Mapping.lookup M' f) < length slp' \<and> interpret_slp slp' xs ! slp_index_lookup slp' M' f = interpret_floatarith f xs)"   and "\<And>M slp M' slp'. \<lbrakk>slp_of_fa fa2 M slp = (M', slp'); \<And>f. f \<in> Mapping.keys M \<Longrightarrow> subterms f \<subseteq> Mapping.keys M; \<And>f. f \<in> Mapping.keys M \<Longrightarrow> the (Mapping.lookup M f) < length slp; \<And>f. f \<in> Mapping.keys M \<Longrightarrow> interpret_slp slp xs ! slp_index_lookup slp M f = interpret_floatarith f xs\<rbrakk> \<Longrightarrow> subterms fa2 \<subseteq> Mapping.keys M' \<and> Mapping.keys M \<subseteq> Mapping.keys M' \<and> (\<forall>f\<in>Mapping.keys M'. subterms f \<subseteq> Mapping.keys M' \<and> the (Mapping.lookup M' f) < length slp' \<and> interpret_slp slp' xs ! slp_index_lookup slp' M' f = interpret_floatarith f xs)"   and "(case slp_of_fa fa1 M slp of (M1, slp1) \<Rightarrow> case slp_of_fa fa2 M1 slp1 of (x, xa) \<Rightarrow> slp_of_fa_bin Binop fa1 fa2 M slp x xa) = (M', slp')"   and "\<And>f. f \<in> Mapping.keys M \<Longrightarrow> subterms f \<subseteq> Mapping.keys M"   and "\<And>f. f \<in> Mapping.keys M \<Longrightarrow> the (Mapping.lookup M f) < length slp"   and "\<And>f. f \<in> Mapping.keys M \<Longrightarrow> interpret_slp slp xs ! slp_index_lookup slp M f = interpret_floatarith f xs"   and "\<And>a b. subterms (Binop a b) = insert (Binop a b) (subterms a \<union> subterms b)"   and "\<And>a b xs. interpret_floatarith (Binop a b) xs = binop (interpret_floatarith a xs) (interpret_floatarith b xs)" shows "insert (Binop fa1 fa2) (subterms fa1 \<union> subterms fa2) \<subseteq> Mapping.keys M' \<and> Mapping.keys M \<subseteq> Mapping.keys M' \<and> (\<forall>f\<in>Mapping.keys M'. subterms f \<subseteq> Mapping.keys M' \<and> the (Mapping.lookup M' f) < length slp' \<and> interpret_slp slp' xs ! slp_index_lookup slp' M' f = interpret_floatarith f xs)"
sat_dual_def: fixes A :: "'a mat"   and c :: "'a vec" shows "sat_dual A c = {y. [y\<^sub>v*A]=c \<and> (\<forall>i<dim_vec y. (0::'a) \<le> y $ i)}"
three_fixed_points: fixes z1 :: "complex_homo"   and z2 :: "complex_homo"   and z3 :: "complex_homo"   and M :: "moebius" assumes "z1 \<noteq> z2"   and "z1 \<noteq> z3"   and "z2 \<noteq> z3"   and "moebius_pt M z1 = z1"   and "moebius_pt M z2 = z2"   and "moebius_pt M z3 = z3" shows "M = id_moebius"
assms_2: shows "x mod 8 \<in> {3, 5}"
p: fixes N :: "'a stream measure"   and M :: "'a stream measure"   and y :: "'a" assumes "R N M"   and "y \<in> set_pmf (p N M)" shows "R (n N M y) (m N M y)"
v: shows "(1, \<lambda>y. (d / 2, 0)) \<noteq> (1, \<lambda>x. ((x - 1 / 2) * d, - (d * sqrt (1 / 4 - (x - 1 / 2) * (x - 1 / 2)))))"
check_plan'_def: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and \<pi>s :: "char list list" shows "check_plan' problem \<pi>s = (case simulate_plan' problem \<pi>s (initial_state' problem) of None \<Rightarrow> False | Some s' \<Rightarrow> match_pres (ast_problem.astG problem) s')"
merges_incr_lemma: fixes n :: "nat"   and ps :: "(nat \<times> 'a) list" shows "\<forall>xs. xs \<in> list n A \<longrightarrow> (\<forall>(p, x)\<in>set ps. p < length xs \<and> x \<in> A) \<longrightarrow> xs [\<sqsubseteq>\<^bsub>r\<^esub>] merges f ps xs"
word_upto_Nil: fixes y :: "'b word"   and x :: "'b word" assumes "y < x" shows "[x .e. y] = []"
comp_ide_arr: fixes b :: "'d"   and f :: "'d" assumes "D.ide b"   and "D.seq b f" shows "b \<cdot>\<^sub>D f = f"
fls_regpart_one: shows "fls_regpart 1 = 1"
nth_normalize: fixes p :: "(real, 2) vec"   and i :: "2" shows "\<bar>normalize p $ i\<bar> \<le> 1"
iT_Div_assoc: fixes I :: "nat set"   and a :: "nat"   and b :: "nat" shows "I \<oslash> a \<oslash> b = I \<oslash> a * b"
poly_mult_r_distr: fixes K :: "'a set"   and p1 :: "'a list"   and p2 :: "'a list"   and p3 :: "'a list" assumes "subring K R"   and "polynomial K p1"   and "polynomial K p2"   and "polynomial K p3" shows "poly_mult p1 (poly_add p2 p3) = poly_add (poly_mult p1 p2) (poly_mult p1 p3)"
euler_char_def: shows "G'.euler_char \<equiv> int (card (verts G)) - int (card (arcs G) div 2) + int (card G'.face_cycle_sets)"
OclInt8_def: shows "\<eight> = (\<lambda>_. \<lfloor>\<lfloor>8\<rfloor>\<rfloor>)"
to_nat_on_infinite: fixes S :: "'b set" assumes "countable S"   and "infinite S" shows "bij_betw (to_nat_on S) S UNIV"
emeasure_C: fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> 'b) \<Rightarrow> 'b measure"   and M :: "nat \<Rightarrow> 'b measure"   and m :: "nat"   and m' :: "nat"   and A :: "(nat \<Rightarrow> 'b) set"   and n :: "nat"   and x :: "nat \<Rightarrow> 'b" assumes "Ionescu_Tulcea P M"   and "m \<le> m'"   and "A \<in> sets (Pi\<^sub>M {0..<n + m} M)"   and "x \<in> space (Pi\<^sub>M {0..<n} M)" shows "emeasure (Ionescu_Tulcea.C P M n m' x) (prod_emb {0..<n + m'} M {0..<n + m} A) = emeasure (Ionescu_Tulcea.C P M n m x) A"
is_serial_sol_then_valid_plan: fixes ops :: "(nat, nat) sas_plus_operator list" assumes "SAS_Plus_Semantics.is_serial_solution_for_problem abs_prob ops" shows "valid_plan (decode_abs_plan ops)"
lincomb_list_Nil: fixes c :: "nat \<Rightarrow> 'a" shows "m.lincomb_list c [] = 0\<^sub>v m"
path_from_to_def: fixes G :: "('e, 'f) Graph_scheme"   and v :: "'e"   and xs :: "'e list"   and w :: "'e" shows "v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w \<equiv> (Graph.walk G xs \<and> distinct xs) \<and> xs \<noteq> [] \<and> hd xs = v \<and> last xs = w"
wf: fixes ext :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "ext_wf ext"   and "wfP (\<lambda>x y. gt y x)" shows "wfP (\<lambda>xs ys. ext gt ys xs)"
dual_SUP: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "\<Squnion>(f ` A) ^ o = (\<Sqinter>a\<in>A. f a ^ o)"
T_A'_eq_lem: shows "sum t_A' {0..<length qs} = int (T (s_A' 0) (drop 0 qs) (drop 0 acts))"
mreq_end3: fixes C :: "'b \<Rightarrow> 'c \<Rightarrow> 'd option"   and x :: "'c"   and p :: "'b list"   and b :: "'b" assumes "applied_rule_rev C x p \<noteq> \<bottom>" shows "applied_rule_rev C x (b # p) = applied_rule_rev C x p"
eq_fold: fixes g :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "prod g A = Finite_Set.fold ((*) \<circ> g) (1::'a) A"
join_left: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "dlat le" shows "le x (dlat.join le x y)"
correctCompositionKS_negP: fixes PQ :: "specID"   and P :: "specID"   and Q :: "specID"   and ks :: "KS" assumes "subcomponents PQ = {P, Q}"   and "correctCompositionKS PQ"   and "ks \<notin> specKeysSecrets PQ" shows "ks \<notin> specKeysSecrets P"
mk_or_semantics: fixes w :: "nat \<Rightarrow> 'a set"   and x :: "'a ltln"   and y :: "'a ltln" shows "w \<Turnstile>\<^sub>n mk_or x y = w \<Turnstile>\<^sub>n x or\<^sub>n y"
multiple_1_same: shows "des.multiple_blocks 1 = multiple_blocks n"
coinduct_strong: fixes R :: "'e llist \<Rightarrow> 'e llist \<Rightarrow> bool"   and llist :: "'e llist"   and llist' :: "'e llist" assumes "R llist llist'"   and "\<And>llist llist'. R llist llist' \<Longrightarrow> lnull llist = lnull llist' \<and> (\<not> lnull llist \<longrightarrow> \<not> lnull llist' \<longrightarrow> lhd llist = lhd llist' \<and> (R (ltl llist) (ltl llist') \<or> ltl llist = ltl llist'))" shows "llist = llist'"
card_UN_disjoint: fixes I :: "'d set"   and A :: "'d \<Rightarrow> 'e set" assumes "finite I"   and "\<forall>i\<in>I. finite (A i)"   and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}" shows "card (\<Union> (A ` I)) = (\<Sum>i\<in>I. card (A i))"
someI_ex: fixes P :: "'e \<Rightarrow> bool" assumes "\<exists>x. P x" shows "P (SOME x. P x)"
keys_hom_component: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and n :: "nat" shows "keys (hom_component p n) = {t \<in> keys p. deg_pm t = n}"
kat_while_def: fixes b :: "'a"   and x :: "'a" shows "while b do x = (\<tt>\<tt> b \<cdot> x)\<^sup>\<star> \<cdot> !b"
arrI: fixes f :: "'a" assumes "\<guillemotleft>f : src e \<rightarrow> src (trg f)\<guillemotright>" shows "hom'.arr f"
simps: fixes n :: "'c" shows "get_arctic_delta_num (Num_arc_delta n) = n"
graph_Entry_axioms: shows "graph_Entry \<alpha>e2 \<alpha>n2 invar2 inEdges2 Entry2"
convergent_powser_arctan: shows "convergent_powser (msllist_of_msstream arctan_series_stream)"
select_convs_2: fixes carrier :: "'b set"   and mult :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and more :: "'c" shows "\<one>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, \<dots> = more\<rparr>\<^esub> = one"
left_unique_rel_set_lemma: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and Y :: "'e set"   and X :: "'f set" assumes "left_unique R"   and "rel_set R Y X" obtains f :: "'f \<Rightarrow> 'e" where "Y = f ` X" and "\<forall>x\<in>X. R (f x) x"
WHILEIT_def: shows "cWHILEIT \<equiv> \<lambda>I b f. REC\<^sub>T (WHILEI_body cbind creturn I b f)"
uncountable_infinite: fixes A :: "'g set" assumes "uncountable A" shows "infinite A"
simps_12: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and y21 :: "'b"   and y22 :: "'b list" shows "\<not> list_all2 R [] (y21 # y22)"
exp_induct: fixes P :: "exp \<Rightarrow> bool"   and exp :: "exp" assumes "\<And>var. P (Var var)"   and "\<And>exp var. P exp \<Longrightarrow> P (App exp var)"   and "\<And>\<Gamma> exp. \<lbrakk>\<And>x. x \<in> domA \<Gamma> \<Longrightarrow> P (the (map_of \<Gamma> x)); P exp\<rbrakk> \<Longrightarrow> P (Terms.Let \<Gamma> exp)"   and "\<And>var exp. P exp \<Longrightarrow> P (Lam [var]. exp)"   and "\<And>b. P (Bool b)"   and "\<And>scrut e1 e2. \<lbrakk>P scrut; P e1; P e2\<rbrakk> \<Longrightarrow> P (scrut ? e1 : e2)" shows "P exp"
simps_5: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'c"   and x2 :: "'d" shows "(case Some x2 of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f2 x2"
R'_def: shows "R' \<equiv> (\<lambda>c. c inp) ` Resumption.resumption.resume ` (R \<inter> {r. \<not> Resumption.resumption.is_Done r})"
assms_2: fixes s :: "'a" assumes "s \<in> basis" shows "F *\<^sub>V s = (0::'b)"
IH: assumes "\<And>a a'. \<lbrakk>a \<in> F; a' \<in> F; a \<noteq> a'\<rbrakk> \<Longrightarrow> \<langle>f a, f a'\<rangle> = 0" shows "(norm (sum f F))\<^sup>2 = (\<Sum>a\<in>F. (norm (f a))\<^sup>2)"
dX_def: fixes X :: "'a set"   and d :: "'b" shows "dX X d = {x \<in> X. Xd x = d}"
sub_args: fixes s :: "('c, 'd) tm"   and t :: "('c, 'd) tm" assumes "s \<in> set (args t)" shows "sub s t"
projr_def: fixes sum :: "'h + 'g" shows "projr sum = (case sum of Inr x2 \<Rightarrow> x2)"
summatory_divisor_eq: fixes m :: "nat" shows "sum_upto (\<lambda>n. real (divisor_count n)) (real m) = real (card {(n, d). n \<in> {0<..m} \<and> d dvd n})"
extend_7: shows "b =\<^sub>F b\<^sub>1 @ b\<^sub>2"
path_connectedin_def: fixes X :: "'b topology"   and S :: "'b set" shows "path_connectedin X S \<equiv> S \<subseteq> topspace X \<and> path_connected_space (subtopology X S)"
mono_extend_get_offending_flows: fixes M :: "'b SecurityInvariant_configured list"   and V :: "'b set"   and E :: "('b \<times> 'b) set"   and E' :: "('b \<times> 'b) set"   and F' :: "('b \<times> 'b) set" assumes "valid_reqs M"   and "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "E' \<subseteq> E"   and "F' \<in> get_offending_flows M \<lparr>nodes = V, edges = E'\<rparr>" shows "\<exists>F\<in>get_offending_flows M \<lparr>nodes = V, edges = E\<rparr>. F' \<subseteq> F"
assms_2: shows "r\<^sup>*\<^sup>* x y1"
mat_inverse_var_def: fixes A :: "'b mat" shows "mat_inverse_var A = (if dim_row A = dim_col A then let one = 1\<^sub>m (dim_row A) in case gauss_jordan A one of (B, C) \<Rightarrow> if mat_equal B one then Some C else None else None)"
lset_lzipD2: fixes x :: "'c"   and y :: "'d"   and xs :: "'c llist"   and ys :: "'d llist" assumes "(x, y) \<in> lset (lzip xs ys)" shows "y \<in> lset ys"
ordinal_utility_expected_value': fixes outcomes :: "'a set"   and \<R> :: "('a pmf \<times> 'a pmf) set" assumes "rational_preference (\<P> outcomes) \<R>"   and "independent_vnm (\<P> outcomes) \<R>"   and "finite outcomes"   and "outcomes \<noteq> {}"   and "continuous_vnm (\<P> outcomes) \<R>" shows "\<exists>u. ordinal_utility (\<P> outcomes) \<R> (\<lambda>x. measure_pmf.expectation x u)"
lim: shows "((\<lambda>x. f x - eval C x) \<longlongrightarrow> 0) at_top"
pareto_loser_in_alts: fixes x :: "'b" assumes "x \<in> pareto_losers R'" shows "x \<in> alts'"
reduction_exists: fixes polarity :: "sign"   and t :: "'a trm"   and s :: "'a trm"   and C :: "'a eclause"   and S :: "'a eclause set"   and \<sigma> :: "('a \<times> 'a trm) list"   and L1 :: "'a literal" assumes "polarity = neg \<or> \<not> equivalent_eq_exists t s (cl_ecl C) (int_clset S) \<sigma> L1 \<or> proper_subterm_red t S \<sigma>"   and "\<forall>x y. x \<in> vars_of_cl (cl_ecl C) \<longrightarrow> occurs_in y (Var x \<lhd> \<sigma>) \<longrightarrow> trm_rep y S = y"   and "eligible_literal L1 C \<sigma>"   and "trm_rep (t \<lhd> \<sigma>) S \<noteq> t \<lhd> \<sigma>"   and "L1 \<in> cl_ecl C"   and "orient_lit_inst L1 t s polarity \<sigma>"   and "\<forall>x\<in>S. finite (cl_ecl x)"   and "ground_clause (subst_cl (cl_ecl C) \<sigma>)"   and "fo_interpretation (same_values (\<lambda>t. trm_rep t S))"   and "C \<in> S"   and "Ball S well_constrained"   and "all_trms_irreducible (subst_set (trms_ecl C) \<sigma>) (\<lambda>t. trm_rep t S)"   and "\<not> validate_ground_clause (int_clset S) (subst_cl (cl_ecl C) \<sigma>)"   and "closed_under_renaming S" shows "\<exists>\<sigma>' u u' p v D L2. reduction L1 C \<sigma>' t s polarity L2 u u' p v D (same_values (\<lambda>t. trm_rep t S)) S \<sigma> \<and> variable_disjoint C D"
paths_belowI: fixes t :: "'b ttree"   and t' :: "'b ttree" assumes "\<And>x xs. x # xs \<in> paths t \<Longrightarrow> x # xs \<in> paths t'" shows "t \<sqsubseteq> t'"
rep_eq: shows "Rep_complex_homo_coords \<infinity>\<^sub>h\<^sub>c = \<infinity>\<^sub>v"
M_rel_i_def: fixes ops :: "'b arith_ops_record"   and m :: "int"   and f :: "'b"   and F :: "int" shows "arith_ops_record.M_rel_i ops m f F = (arith_ops_record.to_int ops f = poly_mod.M m F)"
is_maxsplitter_auto_split: fixes A :: "'d \<times> ('c \<Rightarrow> 'd \<Rightarrow> 'd) \<times> ('d \<Rightarrow> bool)" shows "is_maxsplitter (accepts A) (\<lambda>xs. auto_split A (start A) ([], xs) [] xs)"
interrupts_init_fin_lift_stae: fixes s :: "status"   and \<sigma> :: "('b, 'a) locks \<times> (('a \<Rightarrow> ('c \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('a \<Rightarrow> 'e wait_set_status option) \<times> 'a set" shows "interrupts (init_fin_lift_state s \<sigma>) = interrupts \<sigma>"
not_col_permutation_2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Col Bet A B C" shows "\<not> Tarski_neutral_dimensionless.Col Bet C A B"
closure_imp_adh: fixes A :: "'b set"   and T :: "'b set set"   and x :: "'b" assumes "A \<subseteq> carrier\<^bsub>T\<^esub>"   and "x \<in> closure\<^bsub>T\<^esub> A" shows "x adh\<^bsub>T\<^esub> A"
b_def: shows "b = (\<lambda>p. ind prime p * ln (real p) / real p)"
less_eq_classes1_def: shows "(\<le>) \<equiv> subclass1\<^sup>=\<^sup>="
dlist_eq_iff: fixes dxs :: "'a dlist"   and dys :: "'a dlist" shows "(dxs = dys) = (list_of_dlist dxs = list_of_dlist dys)"
euler_witness_exists_weak: fixes n :: "nat" assumes "odd n"   and "\<not> prime n"   and "2 < n" shows "\<exists>a. euler_witness a n \<and> coprime a (int n)"
F2: shows "2 * \<rho> * \<beta> + 2 * \<Lambda> = 2 * \<beta> * \<rho> + 2 * \<Lambda>"
real_divl: fixes prec :: "nat"   and x :: "real"   and y :: "real" shows "real_divl prec x y \<le> x / y"
cmis: shows "m\<inverse> O s \<subseteq> ov\<inverse> \<union> d \<union> f"
get_put: fixes x :: "'c \<Longrightarrow> 'd"   and \<sigma> :: "'d" assumes "wb_lens x" shows "put\<^bsub>x\<^esub> \<sigma> (get\<^bsub>x\<^esub> \<sigma>) = \<sigma>"
cs_order: fixes \<pi> :: "nat \<Rightarrow> 'n"   and \<pi>' :: "nat \<Rightarrow> 'n"   and i :: "nat"   and i' :: "nat"   and j :: "nat"   and j' :: "nat" assumes "is_path \<pi>"   and "is_path \<pi>'"   and "cs\<^bsup>\<pi>\<^esup> i = cs\<^bsup>\<pi>'\<^esup> i'"   and "cs\<^bsup>\<pi>\<^esup> j = cs\<^bsup>\<pi>'\<^esup> j'"   and "\<pi> i \<noteq> local.return"   and "i < j" shows "i' < j'"
StepRecSeq1: fixes prog :: "cmd"   and c :: "cmd"   and s :: "char list \<Rightarrow> val option"   and l :: "nat"   and c' :: "cmd"   and s' :: "char list \<Rightarrow> val option"   and l' :: "nat"   and c\<^sub>2 :: "cmd" assumes "prog \<turnstile> \<langle>c,s,l\<rangle> \<leadsto> \<langle>c',s',l'\<rangle>" shows "prog;; c\<^sub>2 \<turnstile> \<langle>c;; c\<^sub>2,s,l\<rangle> \<leadsto> \<langle>c';; c\<^sub>2,s',l'\<rangle>"
A: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "(1 / 2) ^ n \<le> (1 / 2) ^ m"
add: fixes A :: "nat \<Rightarrow> 'i"   and j :: "'i" assumes "A \<in> UNIV \<rightarrow> I"   and "j \<in> I"   and "disjoint_family (G \<circ> A)"   and "(\<Union>i. G (A i)) = G j" shows "(\<Sum>n. \<mu> (A n)) = \<mu> j"
bal_d_def: fixes n :: "nat"   and dl :: "nat" shows "bal_d n dl = (real dl < cd * (real n + 1))"
axioms_1: fixes wtFsym :: "'e \<Rightarrow> bool"   and wtPsym :: "'f \<Rightarrow> bool"   and arOf :: "'e \<Rightarrow> 'g list"   and resOf :: "'e \<Rightarrow> 'g"   and intT :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and intF :: "'e \<Rightarrow> 'h list \<Rightarrow> 'h"   and intP :: "'f \<Rightarrow> 'h list \<Rightarrow> bool" assumes "Struct wtFsym wtPsym arOf resOf intT intF intP" shows "Signature TYPE('g) wtFsym wtPsym"
mono_dp_2: fixes w :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "is_direct_product w x"   and "is_direct_product y z" shows "\<Phi> w x y z ; z = x"
set_plus_def: fixes A :: "'b set"   and B :: "'b set" shows "A + B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a + b}"
prob_k_indep: fixes vs :: "nat set" assumes "vs \<in> {vs. vs \<subseteq> S_verts \<and> card vs = k}" shows "prob {es \<in> space P. vs \<in> independent_sets (edge_ugraph es)} = (1 - p) ^ (k choose 2)"
reflect_poly_simps_5: fixes p :: "'a poly"   and q :: "'a poly" shows "reflect_poly (p * q) = reflect_poly p * reflect_poly q"
comm_ring_axioms: shows "comm_ring A addA multA zeroA oneA"
ivl_disj_un_singleton_5: fixes l :: "'a"   and u :: "'a" assumes "l \<le> u" shows "{l} \<union> {l<..u} = {l..u}"
ltake_plus_conv_lappend: fixes n :: "enat"   and m :: "enat"   and xs :: "'b llist" shows "ltake (n + m) xs = lappend (ltake n xs) (ltake m (ldrop n xs))"
array_relD: fixes l :: "'a list"   and l' :: "'b list"   and R :: "('a \<times> 'b) set" assumes "(Array l, Array l') \<in> \<langle>R\<rangle>array_rel" shows "(l, l') \<in> \<langle>R\<rangle>list_rel"
FG_advice_inter: fixes Y :: "'b ltln set"   and \<phi> :: "'b ltln"   and S :: "'b ltln set" assumes "Y \<inter> subformulas\<^sub>\<nu> \<phi> \<subseteq> S" shows "\<phi>[Y \<inter> S]\<^sub>\<mu> = \<phi>[Y]\<^sub>\<mu>"
if_C_def: fixes c :: "'b control_state_scheme \<Rightarrow> bool"   and E :: "'b control_state_scheme \<Rightarrow> ('c \<times> 'b control_state_scheme) option"   and F :: "'b control_state_scheme \<Rightarrow> ('c \<times> 'b control_state_scheme) option" shows "_if_SECLEAN c E F = (\<lambda>\<sigma>. if exec_stop \<sigma> then Some (undefined, \<sigma>) else if c \<sigma> then E \<sigma> else F \<sigma>)"
zero_neq_one: fixes one :: "'b"   and zero :: "'b" assumes "class.zero_neq_one one zero" shows "zero \<noteq> one"
acyclic_power_vanishes: fixes x :: "'a" assumes "is_acyclic x" shows "\<exists>n. power.power 1' (;) x (Suc n) = 0"
A'': shows "q_All allw = \<lbrakk>quot_dbfm A''\<rbrakk>e"
antisym: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubseteq>\<^sub>a y"   and "y \<sqsubseteq>\<^sub>a x" shows "x = y"
A: shows "\<forall>i. \<exists>j. Y i < Y j"
insert_iff: fixes a :: "'e"   and b :: "'e"   and A :: "'e set" shows "(a \<in> insert b A) = (a = b \<or> a \<in> A)"
pmf_rel_mono': fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c pmf"   and y :: "'d pmf"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "rel_pmf P x y"   and "P \<le> Q" shows "rel_pmf Q x y"