hcju/isabelleps huggingface dataset:
UNpart1_cases: fixes II :: "nat set"   and cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('d, 'e, 'f) cmd list"   and P :: "nat \<Rightarrow> nat set set" assumes "II \<in> UNpart1 cl dl P" obtains n :: "nat" and I :: "nat set" where "n \<in> theNFTBoth cl dl" and "I \<in> P n" and "II = local.shift cl n I"
induct: fixes P :: "'c list \<Rightarrow> 'c list \<Rightarrow> bool"   and a0 :: "'c list"   and a1 :: "'c list" assumes "\<And>x xs y ys. \<lbrakk>x = y \<Longrightarrow> P xs ys; \<lbrakk>x \<noteq> y; x < y\<rbrakk> \<Longrightarrow> P xs (y # ys); \<lbrakk>x \<noteq> y; \<not> x < y\<rbrakk> \<Longrightarrow> P (x # xs) ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)"   and "\<And>ys. P [] ys"   and "\<And>v va. P (v # va) []" shows "P a0 a1"
ShiftD: fixes kl :: "'d list"   and Kl :: "'d list set"   and k :: "'d" assumes "kl \<in> BNF_Greatest_Fixpoint.Shift Kl k" shows "k # kl \<in> Kl"
simps_3: fixes C' :: "char list"   and v :: "val" shows "not_init C' (Val v) = True"
det_one: fixes n :: "nat" shows "det (1\<^sub>m n) = (1::'c)"
distinct_cols_id: fixes d :: "nat" shows "distinct (cols (1\<^sub>m d))"
const_ghom: fixes F :: "('c, 'd) Group_scheme"   and G :: "('e, 'f) Group_scheme" assumes "Group F"   and "Group G" shows "1\<^bsub>F,G\<^esub> \<in> gHom F G"
cdf_lim_infty: shows "(\<lambda>i. cdf M (real i)) \<longlonglongrightarrow> Sigma_Algebra.measure M (space M)"
id: shows "four_block_mat (mat_of_cols nc [v]) (0\<^sub>m nc 0) (0\<^sub>m 0 1) (0\<^sub>m 0 0) = mat_of_cols nc [v]"
isometry_not_elliptic_has_repelling_fixed_point: fixes f :: "'b \<Rightarrow> 'b" assumes "isometry f"   and "\<not> elliptic_isometry f" shows "\<exists>xi\<in>Gromov_boundary. Gromov_extension f xi = xi \<and> 0 \<le> additive_strength f xi"
WHILEIT_to_monadic: fixes I :: "'a \<Rightarrow> bool"   and b :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a nres"   and s :: "'a" shows "WHILE\<^sub>T\<^bsup>I\<^esup> b f s = monadic_WHILEIT I (\<lambda>s. RETURN (b s)) f s"
index: shows "int (size (filter_mset ((\<subseteq>) x) \<B>)) = \<Lambda>"
inf_pp_commute: fixes x :: "'a"   and y :: "'a" shows "- - (- - x \<sqinter> y) = - - x \<sqinter> - - y"
simps: fixes a1 :: "Player"   and a2 :: "'a llist"   and a3 :: "'a \<Rightarrow> 'a" shows "path_conforms_with_strategy a1 a2 a3 = ((\<exists>p \<sigma>. a1 = p \<and> a2 = LNil \<and> a3 = \<sigma>) \<or> (\<exists>p v \<sigma>. a1 = p \<and> a2 = LCons v LNil \<and> a3 = \<sigma>) \<or> (\<exists>v p w \<sigma> Ps. a1 = p \<and> a2 = LCons v (LCons w Ps) \<and> a3 = \<sigma> \<and> v \<in> VV p \<and> w = \<sigma> v \<and> path_conforms_with_strategy p (LCons w Ps) \<sigma>) \<or> (\<exists>v p Ps \<sigma>. a1 = p \<and> a2 = LCons v Ps \<and> a3 = \<sigma> \<and> v \<notin> VV p \<and> path_conforms_with_strategy p Ps \<sigma>))"
eval_s: shows "eval (r_smn n m) (p # cs) \<down>= encode (Cn n (r_universal (n + length cs)) (r_constn (n - 1) p # map (r_constn (n - 1)) cs @ map (recf.Id n) [0..<n]))"
map_poly_pCons_hom: fixes a :: "'a mod_ring"   and p :: "'a mod_ring poly" shows "map_poly rebase (pCons a p) = pCons (@a) (map_poly rebase p)"
substNext_code_def: fixes g :: "'g"   and "next" :: "'node \<times> 'val" shows "substNext_code g next \<equiv> \<lambda>v. if v = snd next then substitution_code g next else v"
inverse_minus_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "inverse (uminus a) = uminus (inverse a)"
resPres: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and x :: "name" assumes "P \<leadsto>\<^sup>^<Rel> Q"   and "\<And>P Q x. (P, Q) \<in> Rel \<Longrightarrow> (<\<nu>x>P, <\<nu>x>Q) \<in> Rel'"   and "Rel \<subseteq> Rel'"   and "eqvt Rel"   and "eqvt Rel'" shows "<\<nu>x>P \<leadsto>\<^sup>^<Rel'> <\<nu>x>Q"
univ_poly_is_principal: fixes K :: "'a set" assumes "subfield K R" shows "principal_domain (K [X])"
assms: shows "(r, t) \<in> unsorted_poly_rel O mset_poly_rel"
col_line_ext_1: fixes A :: "Points"   and B :: "Points"   and C :: "Points" assumes "col A B C"   and "A \<noteq> C" shows "B \<in> line_ext (line A C)"
OrdInd: fixes j :: "name"   and i :: "name"   and A :: "fm"   and H :: "fm set" assumes "atom j \<sharp> (i, A)"   and "H \<turnstile> OrdP (Var i)"   and "H \<turnstile> SyntaxN.All i (OrdP (Var i) IMP All2 j (Var i) (A(i::=Var j)) IMP A)" shows "H \<turnstile> A"
ncoplanar_perm_1: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "\<not> Coplanar A B C D" shows "\<not> Coplanar A B D C"
d1_eq: fixes x :: "'a" shows "x = - - x \<sqinter> x"
init: shows "[pns [\<mapsto>] vs] \<subseteq>\<^sub>m [pns [\<mapsto>] vs @ rest]"
others_same: shows "\<forall>j<length ts. j \<noteq> i \<longrightarrow> u_ts ! j = ts ! j"
sorted_distinct_imp_sorted_wrt: fixes xs :: "'a list" assumes "sorted xs"   and "distinct xs" shows "sorted_wrt (<) xs"
inject_3: fixes x3 :: "nat"   and y3 :: "nat" shows "(Regions.intv.Greater x3 = Regions.intv.Greater y3) = (x3 = y3)"
\<tau>Execsr_refl: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1 list"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" shows "\<tau>Exec_movesr ci P t e h s s"
similar_def: fixes M1 :: "moebius"   and M2 :: "moebius" shows "similar M1 M2 = (\<exists>I. moebius_mb I M1 = M2)"
Basic: fixes P :: "'d set"   and f :: "'d \<Rightarrow> 'd"   and Q :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and A :: "'d set" assumes "P \<subseteq> {s. f s \<in> Q}" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Basic f Q,A"
sign_simps_18: fixes aa :: "'b"   and ba :: "'b"   and ca :: "'b" shows "aa * (ba + ca) = aa * ba + aa * ca"
closure_halfspace_lt: fixes a :: "'a"   and b :: "real" assumes "a \<noteq> (0::'a)" shows "closure {x. a \<bullet> x < b} = {x. a \<bullet> x \<le> b}"
refl_trans_closure_of_indRelR: shows "indRelRPO = indRelR\<^sup>*"
fmfilter_upd: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'d"   and m :: "('c, 'd) fmap" shows "fmfilter P (fmupd x y m) = (if P x then fmupd x y (fmfilter P m) else fmfilter P m)"
valid_index_split: fixes "is" :: "nat list"   and ds1 :: "nat list"   and ds2 :: "nat list" assumes "is \<lhd> ds1 @ ds2" obtains is1 :: "nat list" and is2 :: "nat list" where "is1 @ is2 = is" and "is1 \<lhd> ds1" and "is2 \<lhd> ds2"
sup_closed: fixes A :: "'a set" assumes "A \<subseteq> carrier \<X>" shows "\<Squnion>\<^bsub>\<X>\<^esub>A \<in> carrier \<X>"
not_inf_Fault: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and m :: "'f" shows "\<not> AlternativeSmallStep.inf \<Gamma> cs css (Fault m)"
X\<^sub>0_def: shows "X\<^sub>0 \<equiv> {x \<in> X. isIntv (I x)}"
mod_field_poly_i_def: fixes ops :: "'b arith_ops_record"   and cf :: "'b list"   and cg :: "'b list" shows "mod_field_poly_i ops cf cg = (if cg = [] then cf else let ilc = arith_ops_record.inverse ops (last cg); ch = map (arith_ops_record.times ops ilc) cg; r = mod_poly_one_main_i ops (rev cf) (rev ch) (1 + length cf - length cg) in poly_of_list_i ops (rev r))"
mod_0_imp_div_mult_left_eq: fixes n :: "nat"   and m :: "nat"   and k :: "nat" assumes "n mod m = 0" shows "k * n div m = k * (n div m)"
VectorSpace_def: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set" shows "VectorSpace smult V \<equiv> Module smult V"
comp_min_basis_subset: fixes xs :: "('a \<Rightarrow>\<^sub>0 'c) list" shows "set (punit.comp_min_basis xs) \<subseteq> set xs"
FinalDeny_approximating_in_doubt_deny: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and \<beta> :: "'c \<Rightarrow> 'd \<Rightarrow> ternaryvalue"   and rs :: "'c rule list"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and p :: "'d" assumes "matcher_agree_on_exact_matches \<gamma> \<beta>"   and "good_ruleset rs"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> Decision FinalDeny" shows "(\<beta>, in_doubt_deny),p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalDeny"
rem_initial_append1: fixes b :: "bit"   and xs :: "bit list"   and ys :: "bit list" assumes "rem_initial b xs \<noteq> []" shows "rem_initial b (xs @ ys) = rem_initial b xs @ ys"
strict_dom_trans': fixes n :: "'node"   and m :: "'node"   and g :: "'g"   and m' :: "'node" assumes "n \<noteq> m \<and> old.dominates g n m"   and "m \<noteq> m' \<and> old.dominates g m m'" shows "n \<noteq> m' \<and> old.dominates g n m'"
w_prop: shows "\<forall>y<w. b * y < y\<^sup>2"
only_one_L_per_bin: fixes U :: "'b set"   and w :: "'b \<Rightarrow> real"   and c :: "nat"   and S :: "'b set"   and L :: "'b set"   and P :: "'b set set"   and B :: "'b set" assumes "BinPacking U w c S L"   and "BinPacking.bp U w c P"   and "B \<in> P" shows "\<forall>x\<in>B. \<forall>y\<in>B. x \<noteq> y \<longrightarrow> x \<notin> L \<or> y \<notin> L"
no_collision_reactI: fixes S :: "real set" assumes "\<And>t. t \<in> S \<Longrightarrow> u t \<noteq> other.s t" shows "no_collision_react S"
dynmethd_C_C: fixes G :: "prog"   and C :: "qtname"   and sig :: "sig" assumes "is_class G C"   and "ws_prog G" shows "dynmethd G C C sig = methd G C sig"
cop2_ts__tsp: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and X :: "'p"   and D :: "'p"   and E :: "'p"   and Y :: "'p" assumes "\<not> Coplanar A B C X"   and "Coplanar A B C D"   and "Coplanar A B C E"   and "D E TS X Y" shows "A B CTSP X Y"
scale_right_distrib: fixes a :: "'a"   and x :: "'a"   and y :: "'a" shows "a * (x + y) = a * x + a * y"
inject_1: fixes x1 :: "agent"   and y1 :: "agent" shows "(cAgent x1 = cAgent y1) = (x1 = y1)"
2: shows "snd ` set (pairs (add_pairs_single_sorted (rel data)) False hs) \<subseteq> set hs \<times> (set gs \<union> set bs \<union> set hs)"
SC_refl: fixes a :: "'p"   and b :: "'p" shows "a b \<equiv> a b"
insert: fixes g :: "'c \<Rightarrow> 'b"   and x :: "'c"   and A :: "'c multiset" shows "\<Sum>\<^sub># (image_mset g (x, A)) = g x + \<Sum>\<^sub># (image_mset g A)"
lr_of_tran_s1_valid: fixes rt :: "('b, 'c) routing_rule_scheme list" assumes "valid_prefixes rt" shows "gsfw_valid (lr_of_tran_s1 rt)"
maxsimpchain_rev: fixes xs :: "'a set list" assumes "maxsimpchain xs" shows "maxsimpchain (rev xs)"
arrE: fixes f :: "'a" assumes "L.arr f" shows "left g f"
etas: fixes t :: "dB" assumes "\<forall>i\<in>frees t. n \<le> i" shows "(Abs ^^ n) (var_dist vs' ((Abs ^^ n) (var_dist vs (liftn n t 0)))) \<leftrightarrow> strip_context n t 0"
replicate_def: shows "replicate = (\<Lambda> n x. take\<cdot>n\<cdot>(repeat\<cdot>x))"
SC_counterexample': fixes C :: "'b list"   and D :: "'b list"   and \<Gamma> :: "'b formula list"   and \<Delta> :: "'b formula list" assumes "(C, D) \<in> sc \<Gamma> [] \<Delta> []" shows "(\<lambda>k. k \<in> set C) \<not>\<Turnstile> mset \<Gamma> \<Rightarrow> mset \<Delta>"
assms_4: shows "A B C Cong3 A' B' C2"
assms_2: shows "eval c t t' n (\<diamond>\<^sub>b\<gamma>)"
ccompare: fixes ccompare :: "('c \<Rightarrow> 'c \<Rightarrow> order) option"   and comp :: "'c \<Rightarrow> 'c \<Rightarrow> order" assumes "class.ccompare ccompare"   and "ccompare = Some comp" shows "comparator comp"
induct: fixes P :: "('a \<times> 'a \<times> 'a poly) list \<Rightarrow> bool"   and a0 :: "('a \<times> 'a \<times> 'a poly) list" assumes "\<And>xi xj p_ij xsi xsj p_sisj rest. P ((xsi, xsj, p_sisj) # rest) \<Longrightarrow> P ((xi, xj, p_ij) # (xsi, xsj, p_sisj) # rest)"   and "\<And>uu_. P [uu_]"   and "P []" shows "P a0"
scaleC: fixes r :: "complex"   and b :: "'d" shows "g (r *\<^sub>C b) = r *\<^sub>C g b"
aa1: fixes f :: "nat \<Rightarrow> 'o"   and i :: "nat" shows "A f i \<le> U (f i) (state f i)"
ord_eq_le_subst: fixes a :: "'a"   and f :: "'b \<Rightarrow> 'a"   and b :: "'b"   and c :: "'b" assumes "a = f b"   and "b \<le> c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "a \<le> f c"
det_echelon_form_of_det: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and A :: "(('b, 'c) vec, 'c) vec" assumes "is_bezout_ext bezout" shows "fst (echelon_form_of_det A bezout) * det A = det (snd (echelon_form_of_det A bezout))"
assms_2: shows "h = [] \<or> (\<exists>hs. h = [x, x] @ hs)"
tt_zero: shows "punit.tt 0 = punit.min_term"
simps_2: fixes v :: "'a" shows "Some v \<triangle> None = None"
emb_u_def: shows "EMB('e\<^sub>\<bottom>) = u_emb oo liftemb"
Id_on_def: fixes A :: "'b set" shows "Id_on A = (\<Union>x\<in>A. {(x, x)})"
cosh_real_nonzero: fixes x :: "real" shows "cosh x \<noteq> 0"
deleteMin_invar: fixes bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list" assumes "BinomialHeapStruc.invar bq"   and "bq \<noteq> []" shows "BinomialHeapStruc.invar (BinomialHeapStruc.deleteMin bq)"
Inf_atLeast: fixes x :: "'a" shows "\<Sqinter>{x..} = x"
beta_reduction_left_appE: fixes A :: "'a trm"   and B :: "'a trm"   and M' :: "'a trm" assumes "App A B \<rightarrow>\<beta> M'" shows "(\<exists>x T X. A = Fn x T X \<and> M' = X[x ::= B]) \<or> (\<exists>A'. A \<rightarrow>\<beta> A' \<and> M' = App A' B) \<or> (\<exists>B'. B \<rightarrow>\<beta> B' \<and> M' = App A B')"
simps_1: fixes x :: "'b" shows "lookup_pair_tc [] x = (0::'a)"
le_imp_max_y_ge: fixes u :: "nat list"   and x :: "nat list"   and j :: "nat" assumes "u \<le>\<^sub>v x"   and "length x \<le> m" shows "max_y x j \<le> max_y u j"
that: fixes inside :: "'a set"   and outside :: "'a set" assumes "inside \<noteq> {}"   and "open inside"   and "connected inside"   and "outside \<noteq> {}"   and "open outside"   and "connected outside"   and "bounded inside"   and "\<not> bounded outside"   and "inside \<inter> outside = {}"   and "inside \<union> outside = - path_image c"   and "frontier inside = path_image c"   and "frontier outside = path_image c" shows "thesis"
idx_iteratei_def: fixes get :: "'d \<Rightarrow> nat \<Rightarrow> 'e"   and sz :: "'d \<Rightarrow> nat"   and l :: "'d"   and c :: "'c \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'c \<Rightarrow> 'c"   and \<sigma> :: "'c" shows "idx_iteratei get sz l c f \<sigma> \<equiv> idx_iteratei_aux get (sz l) (sz l) l c f \<sigma>"
left_impl_one: fixes a :: "'b" shows "a r\<rightarrow> a = (1::'b)"
empH_implies_dHindistinguishable_eq_dindistinguishable: fixes E :: "'c \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and DA :: "'d \<Rightarrow> 'f"   and d :: "'f"   and e :: "'c"   and e' :: "'c" shows "WHATWHERE_Secure_Programs.dH_indistinguishable E DA d {} e e' = WHATWHERE_Secure_Programs.d_indistinguishable E DA d e e'"
simps_1: fixes uu :: "String.literal"   and T :: "typ" shows "tvs (Ct uu T) = tvsT T"
abs_eq: fixes y :: "'b \<Rightarrow> 'c option" shows "Mapping.lookup (Mapping y) = y"
simps_6: fixes bs :: "'b list"   and a :: "'b poly"   and b :: "'b poly" shows "Ipoly bs (Mul a b) = Ipoly bs a * Ipoly bs b"
poly_mapping_adds_iff: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" shows "(s adds t) = (lookup s adds lookup t)"
refl: fixes t :: "('a, 'b) tm" shows "P t t"
ASSERT_eq: shows "aASSERT \<equiv> iASSERT areturn"
sup2: fixes B :: "'a"   and k :: "'a"   and b :: "'a"   and a :: "'a" assumes "B \<equiv> sup k b" shows "sup a B \<equiv> sup k (sup a b)"
lm085: fixes P :: "'c \<Rightarrow> bool"   and xx :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "P xx" shows "{(x, f x) |x. P x} ,, xx = f xx"
im_sheaf_is_sheaf: shows "sheaf_of_rings S' is_open' local.im_sheaf im_sheaf_morphisms b add_im_sheaf mult_im_sheaf zero_im_sheaf one_im_sheaf"
cmp_simps_2: fixes f :: "'c"   and g :: "'c" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "src\<^sub>D (cmp (f, g)) = src\<^sub>D (local.map g)"
AxParallelConesE: fixes outsideCone :: "'a Point \<Rightarrow> 'a Cone \<Rightarrow> bool"   and f :: "'a Point"   and cone :: "'a Cone"   and onCone :: "'a Point \<Rightarrow> 'a Cone \<Rightarrow> bool" shows "outsideCone f cone \<longrightarrow> (\<exists>e. onCone e cone \<and> e \<noteq> vertex cone \<and> inPlane f (tangentPlane e cone))"
uconnected_refl: fixes E :: "'b uprod set" shows "refl (uconnected E)"
LIMSEQ_inverse_real_of_nat_add_minus: fixes r :: "real" shows "(\<lambda>n. r + - inverse (real (Suc n))) \<longlonglongrightarrow> r"
reflection: shows "rel_reflects_binary_pred (indRelSTEQ SRel TRel) (\<lambda>b. hasBarb b (STCalWB SWB TWB))"
app_correct_2: fixes t1 :: "('a, 'b) FingerTreeStruc"   and t2 :: "('a, 'b) FingerTreeStruc" assumes "FingerTreeStruc.ft_invar t1"   and "FingerTreeStruc.ft_invar t2" shows "FingerTreeStruc.ft_invar (FingerTreeStruc.app t1 t2)"
is_processT5: fixes P :: "'c process" shows "\<forall>s X Y. (s, X) \<in> F P \<and> (\<forall>c. c \<in> Y \<longrightarrow> (s @ [c], {}) \<notin> F P) \<longrightarrow> (s, X \<union> Y) \<in> F P"
times_def: fixes a :: "'b G"   and b :: "'b G" shows "a * b \<equiv> Abs_G (Rep_G a - u + Rep_G b \<squnion> (0::'b))"
Psi_diff_Del: fixes i :: "real"   and d :: "real"   and n :: "nat"   and x\<^sub>1 :: "real"   and x\<^sub>2 :: "real"   and b :: "bool" assumes "0 < i"   and "0 \<le> d"   and "n \<noteq> 0"   and "x\<^sub>1 \<le> x\<^sub>2" shows "\<Psi> b i d x\<^sub>1 x\<^sub>2 (n - Suc 0) - \<Psi> b i d x\<^sub>1 x\<^sub>2 n \<le> d"
subadditive: fixes M :: "'b measure"   and f :: "'b set \<Rightarrow> ennreal"   and A :: "'c \<Rightarrow> 'b set"   and S :: "'c set" assumes "positive (sets M) f"   and "Measure_Space.additive (sets M) f"   and "A ` S \<subseteq> sets M"   and "finite S" shows "f (\<Union> (A ` S)) \<le> (\<Sum>i\<in>S. f (A i))"
set_spmf_def: fixes p :: "'a spmf" shows "set_spmf p = set_pmf p \<bind> set_option"
punit_monom_mult_monomial: fixes c :: "'c"   and s :: "'b"   and v :: "'b"   and d :: "'c" shows "MPoly_Type_Class.punit.monom_mult c s (monomial d v) = monomial (c * d) (s + v)"
add: fixes f :: "'y \<Rightarrow> 'z"   and x :: "'y"   and y :: "'y" assumes "Modules.additive f" shows "f (x + y) = f x + f y"
assms_2: shows "X \<subseteq>\<^sub>\<circ> Y"
web_of_network_sel_3: fixes \<Delta> :: "('a, 'b) network_scheme" shows "A (web_of_network \<Delta>) = {(source \<Delta>, x) |x. edge \<Delta> (source \<Delta>) x}"
comp_assoc: fixes h :: "'c"   and g :: "'c"   and f :: "'c" shows "(h \<cdot> g) \<cdot> f = h \<cdot> g \<cdot> f"
INF_ennreal_add_const: fixes f :: "nat \<Rightarrow> ennreal"   and c :: "ennreal" shows "(INF i. f i + c) = Inf (range f) + c"
incl_def: fixes f :: "'s" shows "incl f = (arr f \<and> local.set (local.dom f) \<subseteq> local.set (cod f) \<and> f = mkArr (local.set (local.dom f)) (local.set (cod f)) (\<lambda>x. x))"
nn_integral_liminf: fixes u :: "nat \<Rightarrow> 'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "\<And>i. u i \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. liminf (\<lambda>n. u n x) \<partial>M \<le> liminf (\<lambda>n. integral\<^sup>N M (u n))"
cong_trans: fixes b :: "'a"   and c :: "'a"   and a :: "'a"   and d :: "'a" assumes "[b = c] (mod a)"   and "[c = d] (mod a)" shows "[b = d] (mod a)"
min_worst_chain_Suc: fixes gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and i :: "nat" assumes "wf {(x, y). gt y x}"   and "bad p x"   and "p (worst_chain p gt i) x" shows "\<not> gt (worst_chain p gt (Suc i)) x"
fst_lens_plus: fixes y :: "'d \<Longrightarrow> 'e"   and x :: "'f \<Longrightarrow> 'e" assumes "wb_lens y" shows "fst\<^sub>L ;\<^sub>L (x +\<^sub>L y) = x"
set_intros_7: fixes x2 :: "'b" shows "x2 \<in> set2_fm (Nom x2)"
fold_mset_empty: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and s :: "'c" shows "fold_mset f s {#} = s"
simps: fixes x :: "'b extT"   and y :: "'b extT" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
Chernoff_d: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "f_range_on A f" shows "Chernoff_on A f = (\<forall>B C. B \<subseteq> A \<and> C \<subseteq> A \<longrightarrow> f (B \<union> C) \<subseteq> f (f B \<union> f C))"
star_simulation: fixes s1 :: "'state1"   and s2 :: "'state2"   and s2' :: "'state2" assumes "match s1 s2"   and "step2 s2 s2'" shows "\<exists>s1'. step1\<^sup>+\<^sup>+ s1 s1' \<and> match s1' s2'"
product_part_in_factors: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "CM"   and "O x y"   and "P z (x \<otimes> y)" shows "P z x \<and> P z y"
isomorphicI: fixes f :: "'a \<times> 'a" assumes "B.VV.iso f" shows "B.VV.isomorphic (B.VV.dom f) (B.VV.cod f)"
intro: fixes h :: "'e \<Rightarrow> 'f"   and R :: "('e, 'g) ring_scheme"   and S :: "('f, 'h) ring_scheme" assumes "h \<in> ring_hom R S" shows "ring_hom_ring_axioms R S h"
Oadmit_Sing: fixes \<sigma> :: "('a, 'b, 'c) subst"   and \<theta> :: "('a, 'c) trm"   and U :: "('c + 'c) set"   and x :: "'c" assumes "TUadmit \<sigma> \<theta> U"   and "TadmitF \<sigma> \<theta>" shows "Oadmit \<sigma> (OSing x \<theta>) U"
doesnt_read_annos: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and x :: "'Var"   and annos :: "'Var ModeUpd list" assumes "doesnt_read_or_modify c x" shows "doesnt_read_or_modify (c \<otimes> annos) x"
frac_eqI: fixes s1 :: "'a"   and r :: "'a"   and s :: "'a"   and r' :: "'a"   and s' :: "'a" assumes "s1 \<in> R\<setminus>I"   and "(r, s) \<in> R \<times> (R\<setminus>I)"   and "(r', s') \<in> R \<times> (R\<setminus>I)"   and "s1 \<cdot> s' \<cdot> r = s1 \<cdot> s \<cdot> r'" shows "cq.frac r s = cq.frac r' s'"
IC_rate2: fixes i :: "nat" shows "rho_bound2 (\<lambda>p. IC p i)"
degree_monom_eq: fixes a :: "'a"   and n :: "nat" assumes "a \<noteq> (0::'a)" shows "degree (monom a n) = n"
ereal_add_cancel_left: fixes a :: "ereal"   and b :: "ereal"   and c :: "ereal" assumes "a \<noteq> - \<infinity>" shows "(a + b = a + c) = (a = \<infinity> \<or> b = c)"
div_eq_0_iff: fixes a :: "'b"   and b :: "'b" assumes "division_segment a = division_segment b" shows "(a div b = (0::'b)) = (euclidean_size a < euclidean_size b \<or> b = (0::'b))"
f_join_all_conv: fixes xs :: "'b list"   and I :: "nat set" shows "(xs \<Join>\<^sub> I = xs) = ({..<length xs} \<subseteq> I)"
silent_moves_preserves_stacks: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and m :: "'node"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and m' :: "'node"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and cs :: "'edge list"   and rs :: "'edge list" assumes "S,f \<turnstile> (m # ms,s) =as\<Rightarrow>\<^sub>\<tau> (m' # ms',s')"   and "valid_node m"   and "valid_call_list cs m"   and "\<forall>i<length rs. rs ! i \<in> get_return_edges (cs ! i)"   and "valid_return_list rs m"   and "length rs = length cs"   and "ms = targetnodes rs" obtains cs' :: "'edge list" and rs' :: "'edge list" where "valid_node m'" and "valid_call_list cs' m'" and "\<forall>i<length rs'. rs' ! i \<in> get_return_edges (cs' ! i)" and "valid_return_list rs' m'" and "length rs' = length cs'" and "ms' = targetnodes rs'" and "upd_cs cs as = cs'"
seq_char: fixes g :: "'a"   and f :: "'a" shows "Left_a.seq g f = (Left_a.arr f \<and> Left_a.arr g \<and> seq g f)"
ipc_respects_policy: fixes tid :: "thread_id_t"   and u :: "partition_id_t"   and s :: "state_t"   and dir :: "ipc_direction_t"   and stage :: "ipc_stage_t"   and partner :: "thread_id_t"   and pag :: "page_t"   and ipt :: "int_point_t"   and page :: "page_t" assumes "\<not> Policy.ifp (Step.partition tid) u"   and "atomic_step_invariant s"   and "atomic_step_precondition s tid (SK_IPC dir stage partner pag)"   and "ipt = SK_IPC dir stage partner page" shows "vpeq u s (atomic_step_ipc tid dir stage partner page s)"