hcju/isabelleps huggingface dataset:
orthogonal_cut': fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a antiparallel_edges.vertex \<times> 'a antiparallel_edges.vertex \<Rightarrow> ennreal"   and S :: "'a antiparallel_edges.vertex set" assumes "antiparallel_edges \<Delta>"   and "orthogonal (antiparallel_edges.\<Delta>'' \<Delta>) f S"   and "flow (antiparallel_edges.\<Delta>'' \<Delta>) f" shows "orthogonal \<Delta> (\<lambda>(x, y). f (antiparallel_edges.Edge x y, antiparallel_edges.Vertex y)) (antiparallel_edges.Vertex -` S)"
cinfinite_csum: fixes r1 :: "('a \<times> 'a) set"   and r2 :: "('b \<times> 'b) set" assumes "cinfinite r1 \<or> cinfinite r2" shows "cinfinite (r1 +c r2)"
rel_F_mono: fixes L1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and L1' :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and L2 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and L2' :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and L3 :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and L3' :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Co1 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Co1' :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Co2 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Co2' :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Co3 :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and Co3' :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and Contra1' :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and Contra1 :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and Contra2' :: "'ab \<Rightarrow> 'ac \<Rightarrow> bool"   and Contra2 :: "'ab \<Rightarrow> 'ac \<Rightarrow> bool"   and Contra3' :: "'ad \<Rightarrow> 'ae \<Rightarrow> bool"   and Contra3 :: "'ad \<Rightarrow> 'ae \<Rightarrow> bool" assumes "L1 \<le> L1'"   and "L2 \<le> L2'"   and "L3 \<le> L3'"   and "Co1 \<le> Co1'"   and "Co2 \<le> Co2'"   and "Co3 \<le> Co3'"   and "Contra1' \<le> Contra1"   and "Contra2' \<le> Contra2"   and "Contra3' \<le> Contra3" shows "rel_F L1 L2 L3 Co1 Co2 Co3 Contra1 Contra2 Contra3 \<le> rel_F L1' L2' L3' Co1' Co2' Co3' Contra1' Contra2' Contra3'"
disjI1_PL: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and A :: "'b set"   and q :: "'b" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL"   and "p \<in> Cn A" shows "disj_PL p q \<in> Cn A"
prv_prv_imp_trans: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "bprv (imp \<phi> \<chi>)"   and "bprv (imp \<chi> \<psi>)" shows "bprv (imp \<phi> \<psi>)"
lower_split_interval2: fixes X :: "'a interval"   and m :: "'a" shows "lower (snd (split_interval X m)) = min (upper X) m"
unique_connecting_path_tl: fixes G :: "('a, 'b) Graph_scheme"   and v :: "'a"   and u :: "'a"   and w :: "'a" assumes "Tree G"   and "v \<in> V\<^bsub>G\<^esub>"   and "u \<in> set (w \<leadsto>\<^bsub>G\<^esub> v)"   and "u \<rightarrow>\<^bsub>G\<^esub> w" shows "tl (w \<leadsto>\<^bsub>G\<^esub> v) = u \<leadsto>\<^bsub>G\<^esub> v"
simps_1: fixes f :: "'c \<Rightarrow> 'a + 'b" shows "mapM f [] = Inr []"
substAll_wf: fixes g :: "'g" assumes "i.redundant g" shows "card (dom (i.phis' g)) < card (dom ((phis(g := p)) g))"
vector_uminus_component: fixes x :: "('b, 'c) vec"   and i :: "'c" shows "(- x) $ i = - x $ i"
prev_eq_None_iff: fixes j :: "nat"   and i :: "nat" shows "(prev j i = None) = (j \<le> Suc 0 \<and> i = 0)"
IH: fixes y :: "nat" assumes "y < k"   and "3 \<le> y" shows "ord (2 ^ y) x = 2 ^ (y - 2)"
assms_2: shows "\<lbrace>P\<rbrace> c \<lbrace>Q\<rbrace>"
of_int_add: fixes i :: "int"   and j :: "int" shows "\<guillemotleft>i + j\<guillemotright> = \<guillemotleft>i\<guillemotright> \<oplus> \<guillemotleft>j\<guillemotright>"
Compl_Diff_eq: fixes A :: "'e set"   and B :: "'e set" shows "- (A - B) = - A \<union> B"
mult_smult_assoc_mat: fixes A :: "'a mat"   and nr :: "nat"   and n :: "nat"   and B :: "'a mat"   and nc :: "nat"   and k :: "'a" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc" shows "k \<cdot>\<^sub>m A * B = k \<cdot>\<^sub>m (A * B)"
extremum_unique: fixes a :: "'a upred" shows "(a \<sqsubseteq> true) = (a = true)"
le_less: fixes x :: "'c"   and y :: "'c" shows "(x \<le> y) = (x < y \<or> x = y)"
less_add_same_cancel1: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less a (plus a b) = less zero b"
fresh_atm: fixes a :: "NominalSequents.var"   and b :: "NominalSequents.var" shows "a \<sharp> b = (a \<noteq> b)"
zero_less_iff_neq_zero: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and n :: "'g" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less zero n = (n \<noteq> zero)"
Init_rhs_def: fixes CS :: "'b list set set"   and X :: "'b list set" shows "Init_rhs CS X \<equiv> if [] \<in> X then {Lam One} \<union> {Trn Y (Atom c) |Y c. Y \<in> CS \<and> Y \<Turnstile>c\<Rightarrow>X} else {Trn Y (Atom c) |Y c. Y \<in> CS \<and> Y \<Turnstile>c\<Rightarrow>X}"
adapt_stoch_proc_def: fixes F :: "'b \<Rightarrow> 'c measure"   and X :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and N :: "'d measure" shows "adapt_stoch_proc F X N = (\<forall>t. X t \<in> F t \<rightarrow>\<^sub>M N)"
T_Int: fixes a :: "(int, 'n) vec"   and b :: "(int, 'n) vec" assumes "a \<noteq> b" shows "T a \<inter> T b \<in> null_sets lebesgue"
cases: fixes a1 :: "nat"   and a2 :: "hmultiset" assumes "local.aligned\<^sub>e a1 a2" obtains M :: "hmultiset" and e :: "nat" where "a1 = e" and "a2 = M" and "\<forall>m\<in>#hmsetmset M. local.encode 0 e \<le> m"
lim_def: fixes X :: "nat \<Rightarrow> 'a" shows "lim X = (THE L. X \<longlonglongrightarrow> L)"
sorted_distinct_set_unique: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs"   and "distinct xs"   and "linorder.sorted less_eq ys"   and "distinct ys"   and "set xs = set ys" shows "xs = ys"
ab_2: shows "0 \<le> b \<and> b \<le> pi \<and> cos b = y"
agree: shows "matcher_agree_on_exact_matches \<gamma> \<beta>"
DBDeq_enabled: shows "\<turnstile> Enabled (<DBDeq>_(inp, mid, out, q1, q2)) = (q2 \<noteq> #[])"
not_outside_1: fixes i :: "'agent"   and x :: "'alt"   and y :: "'alt" assumes "R i x y" shows "i \<in> agents"
deficient_number_def: fixes n :: "nat" shows "deficient_number n \<equiv> aliquot_sum n < n"
lunit_eqI: fixes f :: "'a"   and a :: "'a"   and \<mu> :: "'a" assumes "Left.in_hom f a (trg f)"   and "Left.in_hom \<mu> (Left.L f) f"   and "Left.L \<mu> = Left.comp (\<i>[trg f] \<star> f) (local.inv \<a>[trg f, trg f, f])" shows "\<mu> = Left.lunit f"
C: shows "\<exists>z'. z' \<in> ipurge_ref I D (D y) zs Z"
convex_same_rel_interior_closure: fixes S :: "'b set"   and T :: "'b set" assumes "convex S"   and "convex T" shows "(rel_interior S = rel_interior T) = (closure S = closure T)"
Object_subcls_conv: fixes P :: "'a prog"   and C :: "String.literal" shows "P \<turnstile> Object \<preceq>\<^sup>* C = (C = Object)"
isPrefixPrefixToLevel_aux: fixes t :: "('a \<times> bool) list"   and l :: "nat"   and i :: "nat" shows "\<exists>s. prefixToLevel_aux t l i @ s = t"
in_apartments_1: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and B :: "'b set set"   and B' :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and E :: "'b set"   and z :: "'b set" assumes "ChamberComplexApartmentSystemTriangle X \<A> A B B' C D E z" shows "C \<in> A \<inter> B"
match_raw_bool: fixes \<beta> :: "'b \<Rightarrow> 'c \<Rightarrow> ternaryvalue"   and \<alpha> :: "action \<Rightarrow> 'c \<Rightarrow> bool"   and expr :: "'b"   and a :: "action"   and p :: "'c" shows "matches (\<beta>, \<alpha>) (Match expr) a p = (case ternary_to_bool (\<beta> expr p) of None \<Rightarrow> \<alpha> a p | Some r \<Rightarrow> r)"
node_histories_def: fixes history :: "nat \<Rightarrow> 'c list" shows "node_histories history \<equiv> \<forall>i. distinct (history i)"
step: fixes s :: "'a"   and a :: "'b"   and s' :: "'a" assumes "s \<in> reachable A I"   and "P s"   and "(s, a, s') \<in> automaton.trans A"   and "I a" shows "P s'"
LTS_is_inf_run_def: fixes \<Delta> :: "('d \<times> 'e \<times> 'd) set"   and w :: "nat \<Rightarrow> 'e"   and r :: "nat \<Rightarrow> 'd" shows "LTS_is_inf_run \<Delta> w r = (\<forall>i. (r i, w i, r (Suc i)) \<in> \<Delta>)"
derived_point_subset_orig: fixes ps :: "'a set" assumes "ps \<subseteq> bl" shows "ps \<subset> \<V>"
split_pairs: fixes A :: "'c"   and B :: "'d"   and X :: "'c \<times> 'd" shows "((A, B) = X) = (fst X = A \<and> snd X = B)"
strict_sorted_list_of_set: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and A :: "'g set" assumes "class.linorder less_eq less" shows "linorder.strict_sorted less (linorder.sorted_list_of_set less_eq A)"
Restrp_transitive: shows "transitive UNIV ((\<sqsubset>) \<restriction> A)"
casi: shows "\<forall>i<n. CAs ! i = Cs ! i + poss (AAs ! i)"
Cv_3: shows "<\<bottom>, outM x> \<in># msgs cv"
is_faithful: fixes f :: "'b"   and f' :: "'b" assumes "F.B.par f f'"   and "G f = G f'" shows "f = f'"
limit_foTn_over_n': fixes f :: "'a \<Rightarrow> real" assumes "integrable M f" shows "AE x in M. (\<lambda>n. f ((T ^^ n) x) / real n) \<longlonglongrightarrow> 0"
replete_set_category_axioms: shows "replete_set_category (\<cdot>)"
taylor_deg_1_eval': fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and fa :: "'a"   and f'a :: "'a" assumes "f \<in> carrier P"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "c = to_fun (shift 2 (T\<^bsub>a\<^esub> f)) b"   and "fa = to_fun f a"   and "f'a = deriv f a" shows "to_fun f (a \<oplus> b) = fa \<oplus> f'a \<otimes> b \<oplus> c \<otimes> b [^] 2"
restrict_extensional: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" shows "restrict f A \<in> extensional A"
ord_of_nat_\<omega>: fixes n :: "nat" shows "ord_of_nat n \<in> elts \<omega>"
ss_merge: shows "ss = merges f (step (hd b) (a ! hd b)) a"
sorted_iff_nth_mono: fixes xs :: "'a list" shows "linorder.sorted (\<lambda>x y. y \<le> x) xs = (\<forall>i j. i \<le> j \<longrightarrow> j < length xs \<longrightarrow> xs ! j \<le> xs ! i)"
distinct_11: fixes x21 :: "nat"   and x22 :: "nat"   and x23 :: "nat"   and x24 :: "nat"   and x25 :: "nat"   and x41 :: "nat"   and x42 :: "nat" shows "Rrep x21 x22 x23 x24 x25 \<noteq> Newpkt x41 x42"
vD_iD_gives_kD_1: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "ip \<in> vD rt" shows "ip \<in> kD rt"
allow_deny_dom: fixes a :: "(int \<times> int \<times> protocol) set set"   and b :: "(int \<times> int \<times> protocol) set set"   and po :: "protocol \<times> int" shows "dom (Cp (AllowPortFromTo a b po)) \<subseteq> dom (Cp (DenyAllFromTo a b))"
same: shows "inv_map \<pi> x = inv_map \<pi> y"
refine: fixes R_Iec :: "('b \<times> unit) set" shows "(impl_blue_dfs_ahs, blue_dfs) \<in> bg_impl_rel_ext R_Iec Id \<rightarrow> \<langle>\<langle>\<langle>Id\<rangle>list_rel \<times>\<^sub>r \<langle>Id\<rangle>list_rel\<rangle>option_rel\<rangle>nres_rel"
adv: fixes \<A> :: "nat \<Rightarrow> ('g + 'h, 'i + 'j) distinguisher" assumes "\<And>\<eta>. \<I>_real2 \<eta> \<oplus>\<^sub>\<I> \<I>_common2 \<eta> \<turnstile>g \<A> \<eta> \<surd>"   and "\<And>\<eta>. interaction_any_bounded_by (\<A> \<eta>) (bound2 \<eta>)"   and "\<And>\<eta>. lossless2 \<Longrightarrow> plossless_gpv (\<I>_real2 \<eta> \<oplus>\<^sub>\<I> \<I>_common2 \<eta>) (\<A> \<eta>)" shows "Negligible.negligible (\<lambda>\<eta>. advantage (\<A> \<eta>) (sim2 \<eta> |\<^sub>= 1\<^sub>C \<rhd> ideal2 \<eta>) (real2 \<eta>))"
arctan_upper_42_def: shows "arctan_upper_42 \<equiv> \<lambda>x. (15159 * x ^ 6 + 147455 * x ^ 4 + 345345 * x\<^sup>2 + 225225) * x / (35 * (35 * x ^ 8 + 1260 * x ^ 6 + 6930 * x ^ 4 + 12012 * x\<^sup>2 + 6435))"
landau_product_preprocess_7: fixes m :: "num" shows "num.Bit0 m + num.One = num.Bit1 m"
holomorphic_on_sin: fixes S :: "complex set" shows "sin holomorphic_on S"
is_coext_def: fixes x' :: "'c"   and f :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and g :: "'c" shows "x.is_coext x' f g \<equiv> \<guillemotleft>g : x' \<rightarrow>\<^sub>C a\<guillemotright> \<and> f = x \<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] local.map g"
r: shows "reduced_row_echelon_form_upt_k A (Suc k)"
arr_UP: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "S.arr (S.UP \<mu>)"
T_chop_le: fixes d :: "nat"   and xs :: "'a list" shows "T_chop d xs \<le> 5 * length xs + 1"
until_def: fixes \<gamma>' :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma> :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool" shows "\<gamma>' \<UU>\<^sub>c \<gamma> \<equiv> \<lambda>t n. \<exists>n''\<ge>n. \<gamma> t n'' \<and> (\<forall>n'\<ge>n. n' < n'' \<longrightarrow> \<gamma>' t n')"
occuring_gens_in_element: fixes x :: "(bool \<times> 'c) list"   and gens :: "'c set" assumes "x \<in> carrier \<F>\<^bsub>gens\<^esub>" shows "x \<in> lists (UNIV \<times> gens)"
production_subseteq_INTERP: fixes C :: "'a literal multiset" shows "production C \<subseteq> INTERP"
Seed_holds_facesAt_distinct: fixes p :: "nat" shows "facesAt_distinct Seed\<^bsub>p\<^esub>"
lin_tss_not_Nil: shows "linearize tss \<noteq> []"
simps_6: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'d \<Rightarrow> 'b"   and x2 :: "'d" shows "(case Inr x2 of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = f2 x2"
TESL_interp_idem: fixes \<Phi> :: "'b TESL_atomic list" shows "\<lbrakk>\<lbrakk> \<Phi> @ \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = \<lbrakk>\<lbrakk> \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
t1: shows "(1 - t1) * (a \<bullet> c) + t1 * (b \<bullet> c) = d"
l: fixes v :: "'b" shows "lookup (monomial c (0::'b) * r) v = c * lookup r v"
comp_closed_set_def: fixes X :: "'c set" shows "comp_closed_set X = (\<forall>x\<in>X. \<forall>y\<in>X. x \<cdot> y \<in> X)"
simps_4: fixes ob :: "'c set"   and ar :: "'b set"   and dom :: "'b \<Rightarrow> 'c"   and cod :: "'b \<Rightarrow> 'c"   and id :: "'c \<Rightarrow> 'b"   and comp :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and more :: "'d" shows "cod \<lparr>ob = ob, ar = ar, dom = dom, cod = cod, id = id, comp = comp, \<dots> = more\<rparr> = cod"
A: fixes u :: "nat \<Rightarrow> 'a"   and z :: "'a" assumes "abs_Gromov_completion u = abs_Gromov_completion (\<lambda>n. z)"   and "Gromov_completion_rel u u" shows "u = (\<lambda>n. z)"
mat_leqb_eqc_split_simplex_correct2: fixes c :: "rat vec"   and A :: "rat mat"   and b :: "rat vec"   and X :: "(nat, rat) mapping"   and x :: "rat vec"   and y :: "rat vec" assumes "dim_vec c = dim_row A\<^sup>T"   and "dim_vec b = dim_col A\<^sup>T"   and "simplex (mat_leqb_eqc A b c) = Inr X"   and "(x, y) = split_n_m_x (dim_row A\<^sup>T) (dim_col A\<^sup>T) X" shows "[y\<^sub>v*A]=c"
exec2: fixes q :: "nat" assumes "q \<notin> set (map fst (step (hd b) (a ! hd b)))"   and "q < length a" shows "ss ! q = a ! q"
simps_6: fixes \<psi>\<^sub>1 :: "'b ltln"   and \<psi>\<^sub>2 :: "'b ltln"   and X :: "'b ltln set" shows "(\<psi>\<^sub>1 U\<^sub>n \<psi>\<^sub>2)[X]\<^sub>\<nu> = (if \<psi>\<^sub>1 U\<^sub>n \<psi>\<^sub>2 \<in> X then \<psi>\<^sub>1[X]\<^sub>\<nu> W\<^sub>n \<psi>\<^sub>2[X]\<^sub>\<nu> else false\<^sub>n)"
conv_radius_geI_ex': fixes R :: "ereal"   and f :: "nat \<Rightarrow> 'b" assumes "\<And>r. \<lbrakk>0 < r; ereal r < R\<rbrakk> \<Longrightarrow> summable (\<lambda>n. f n * of_real r ^ n)" shows "R \<le> conv_radius f"
a_notin: shows "a \<notin> outstanding_refs is_Write\<^sub>s\<^sub>b (r # sb)"
frefI: fixes P :: "'d \<Rightarrow> bool"   and R :: "('c \<times> 'd) set"   and f :: "'c \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f"   and S :: "('e \<times> 'f) set" assumes "\<And>x y. \<lbrakk>P y; (x, y) \<in> R\<rbrakk> \<Longrightarrow> (f x, g y) \<in> S" shows "(f, g) \<in> [P]\<^sub>f R \<rightarrow> S"
subset_entailed: fixes Bot :: "'b set"   and entails :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and N2 :: "'b set"   and N1 :: "'b set" assumes "consequence_relation Bot entails"   and "N2 \<subseteq> N1" shows "entails N1 N2"
pseudofunctor_axioms: shows "pseudofunctor (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F"
3: shows "pd_kleene_allegory_class.acyclic (p \<sqinter> - (1::'a))"
expand: fixes literal :: "'a literal"   and literal' :: "'a literal" assumes "is_pos literal = is_pos literal'"   and "\<lbrakk>is_pos literal; is_pos literal'\<rbrakk> \<Longrightarrow> atm_of literal = atm_of literal'"   and "\<lbrakk>is_neg literal; is_neg literal'\<rbrakk> \<Longrightarrow> atm_of literal = atm_of literal'" shows "literal = literal'"
LIMSEQ_const_iff: fixes k :: "'b"   and l :: "'b" shows "(\<lambda>n. k) \<longlonglongrightarrow> l = (k = l)"
vbrelation_vinsertI: fixes a :: "V"   and b :: "V" shows "vbrelation (vinsert \<langle>a, b\<rangle> r)"
norm_in_ARR: fixes f :: "'c local.term set"   and t :: "'c local.term" assumes "ARR f"   and "t \<in> f" shows "\<^bold>\<parallel>t\<^bold>\<parallel> \<in> f"
phi_i_total: fixes x :: "nat" shows "\<phi> i x \<down>"
in_steps_epsclosure: fixes p :: "'b"   and q :: "'b"   and A :: "'b \<times> ('c option \<Rightarrow> 'b \<Rightarrow> 'b set) \<times> ('b \<Rightarrow> bool)"   and r :: "'b"   and w :: "'c list" assumes "(p, q) \<in> (eps A)\<^sup>*"   and "(q, r) \<in> NAe.steps A w" shows "(p, r) \<in> NAe.steps A w"
assms: shows "timpls_transformable_to' TI (Fun f T) t"
generate_lists_def: fixes n :: "nat"   and xs :: "'c list" shows "generate_lists n xs \<equiv> concat_lists (map (\<lambda>_. xs) [0..<n])"
Abs_formula_inverse: fixes t\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha>" assumes "hereditarily_fs t\<^sub>\<alpha>" shows "Rep_formula (Abs_formula t\<^sub>\<alpha>) = t\<^sub>\<alpha>"
simps_1: fixes B :: "'c mat"   and i :: "nat" shows "eliminate_entries_rec B i [] = B"
butterfly_0_left: fixes a :: "'a" shows "butterfly (0::'b) a = 0"
continuous_on_open_vimage: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "open s" shows "continuous_on s f = (\<forall>B. open B \<longrightarrow> open (f -` B \<inter> s))"
bisim_def: shows "Strong_Late_Bisim.bisim \<equiv> {(xa, x). Strong_Late_Bisim.bisimp xa x}"
finite_SigmaI: fixes A :: "'b set"   and B :: "'b \<Rightarrow> 'c set" assumes "finite A"   and "\<And>a. a \<in> A \<Longrightarrow> finite (B a)" shows "finite (Sigma A B)"
lex_ext_def: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> bool \<times> bool"   and n :: "nat"   and ss :: "'c list"   and ts :: "'c list" shows "lex_ext f n ss ts = (let lts = length ts in if length ss = lts \<or> lts \<le> n then lex_ext_unbounded f ss ts else (False, False))"
is_const_simps_2: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and t :: "'b"   and u :: "'b" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "\<not> pre_term.is_const unconst (app t u)"
partial_equiv_funD: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "f \<sim> g"   and "x \<in> domain"   and "y \<in> domain"   and "x \<sim> y" shows "f x \<sim> g y"
quorum_for_def': fixes Quorum :: "process set set"   and Q :: "process set"   and v :: "val"   and v_f :: "process \<Rightarrow> val option" assumes "quorum_process Quorum" shows "quorum_process.quorum_for Quorum Q v v_f \<equiv> Q \<in> Quorum \<and> v_f ` Q = {Some v}"
1: shows "\<i>[a] \<star> a = (a \<star> \<i>[a]) \<cdot> \<a>[a, a, a]"
iso_def: fixes f :: "'a" shows "B.iso f \<equiv> \<exists>g. B.inverse_arrows f g"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Arr :: "'b \<Rightarrow> bool" assumes "category C"   and "subcategory C Arr" shows "inclusion_functor C Arr"
fx: shows "\<forall>y\<in>F. m \<le> y"
vimage_fst: fixes A :: "'c set" shows "fst -` A = A \<times> UNIV"
bounded_iff: fixes a :: "'k multiset"   and b :: "'k multiset"   and c :: "'k multiset" shows "(a \<subseteq># b \<inter># c) = (a \<subseteq># b \<and> a \<subseteq># c)"
hnr_comp: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and R1p :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and R :: "'d \<Rightarrow> 'c \<Rightarrow> assn"   and R1' :: "('a \<times> 'e) set"   and R' :: "('d \<times> 'f) set"   and a1 :: "'e"   and c1 :: "'b" assumes "\<And>b1 c1. P b1 \<Longrightarrow> hn_refine (R1 b1 c1 * emp) (c c1) (R1p b1 c1 * emp) R (b b1)"   and "\<And>a1 b1. \<lbrakk>Q a1; (b1, a1) \<in> R1'\<rbrakk> \<Longrightarrow> (b b1, a a1) \<in> \<langle>R'\<rangle>nres_rel"   and "\<And>a1 b1. \<lbrakk>Q a1; (b1, a1) \<in> R1'\<rbrakk> \<Longrightarrow> P b1"   and "Q a1" shows "hn_refine (hr_comp R1 R1' a1 c1 * emp) (c c1) (hr_comp R1p R1' a1 c1 * emp) (hr_comp R R') (a a1)"
delta_predict_predict: fixes der :: "bool stream \<Rightarrow> real"   and matur :: "nat"   and N :: "bool stream measure"   and q :: "real" assumes "der \<in> borel_measurable (G matur)"   and "N = bernoulli_stream q"   and "0 < q"   and "q < 1" shows "borel_predict_stoch_proc G (delta_predict N der matur)"
match_wf_tuple: fixes f :: "nat \<Rightarrow> event_data option"   and ts :: "Formula.trm list"   and xs :: "event_data list"   and n :: "nat" assumes "Some f = match ts xs" shows "wf_tuple n (\<Union> (fv_trm ` set ts)) (tabulate f 0 n)"
embed_unique: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e"   and a :: "'d" assumes "Well_order r"   and "Well_order r'"   and "embed r r' f"   and "embed r r' g" shows "a \<in> Field r \<longrightarrow> f a = g a"
bij_betw_mult_left: fixes x :: "'a" assumes "x \<in> carrier G" shows "bij_betw ((\<otimes>) x) (carrier G) (carrier G)"
eq_snd_iff: fixes b :: "'c"   and p :: "'d \<times> 'c" shows "(b = snd p) = (\<exists>a. p = (a, b))"
connected_component_of_eq_overlap: fixes X :: "'b topology"   and x :: "'b"   and y :: "'b" shows "(connected_component_of_set X x = connected_component_of_set X y) = (x \<notin> topspace X \<and> y \<notin> topspace X \<or> connected_component_of_set X x \<inter> connected_component_of_set X y \<noteq> {})"
algebraic_int_cnj: fixes x :: "complex" assumes "algebraic_int x" shows "algebraic_int (cnj x)"
hf_of_sum_def: shows "hf_of \<equiv> case_sum (HF.Inl \<circ> hf_of) (HF.Inr \<circ> hf_of)"