hcju/isabelleps huggingface dataset:
freshAction_5: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and c :: "name" assumes "P \<longmapsto> a<\<nu>x> \<prec> P'"   and "c \<sharp> P" shows "c \<noteq> a"
Ks_4: shows "set Ks \<noteq> {\<zero>\<^bsub>S\<^esub>}"
sup_least: fixes y :: "'c multiset"   and x :: "'c multiset"   and z :: "'c multiset" assumes "y \<subseteq># x"   and "z \<subseteq># x" shows "y \<union># z \<subseteq># x"
dvd_mult: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.comm_monoid_mult times one"   and "dvd.dvd times a c" shows "dvd.dvd times a (times b c)"
seqSubstTau: fixes P :: "pi"   and \<sigma> :: "(name \<times> name) list" shows "\<tau>.P[<\<sigma>>] = \<tau>.(P[<\<sigma>>])"
0: fixes t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" assumes "t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t set M" shows "\<exists>\<delta>. wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> t \<in> set M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta>"
reduce_system_matrix_signs_helper_aux: fixes S :: "nat list"   and signs :: "rat list list"   and p :: "real poly"   and subsets :: "nat list list" assumes "\<forall>x. List.member S x \<longrightarrow> x < length signs"   and "p \<noteq> 0" shows "alt_matrix_A (take_indices signs S) subsets = take_cols_from_matrix (alt_matrix_A signs subsets) S"
af_homomorphism: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "pure f \<diamondop> pure x = pure (f x)"
intros_8: fixes vi :: "v list"   and j :: "nat"   and s :: "s"   and v :: "v"   and vs :: "v list"   and i :: "nat" assumes "length vi = j" shows "\<lparr>s;vi @ [v] @ vs;[$Get_local j]\<rparr> \<leadsto>_ i \<lparr>s;vi @ [v] @ vs;[$C v]\<rparr>"
assms_4: shows "\<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>u\<^bold>\<rfloor>"
zero_less_HNatInfinite: fixes x :: "nat star" assumes "x \<in> HNatInfinite" shows "0 < x"
map_simp_2: shows "local.map J.One = C.cod f0"
map_replicate: fixes f :: "'h \<Rightarrow> 'g"   and n :: "nat"   and x :: "'h" shows "map f (replicate n x) = replicate n (f x)"
uint64_div_def: fixes x :: "uint64"   and y :: "uint64" shows "uint64_div x y = fst (uint64_divmod x y)"
map_of_delete_iff: fixes x :: "'b"   and \<Gamma> :: "('b \<times> 'a) list"   and xa :: "'b"   and e :: "'a" shows "(map_of (delete x \<Gamma>) xa = Some e) = (map_of \<Gamma> xa = Some e \<and> xa \<noteq> x)"
PartDom_PartNum_distinct: fixes i :: "nat"   and d :: "'b dataspace"   and j :: "nat"   and p :: "'b" assumes "i < PartNum d"   and "j < PartNum d"   and "i \<noteq> j"   and "p \<in> d !D! i" shows "p \<notin> d !D! j"
mult_cancel2: fixes m :: "nat"   and k :: "nat"   and n :: "nat" shows "(m * k = n * k) = (m = n \<or> k = 0)"
c_ent_e: shows "CC \<TTurnstile>e EE"
solution_pell_cnj: fixes z :: "'a \<times> 'a" assumes "solution z" shows "solution (pell_cnj z)"
integral_indicator_finite_real: fixes A :: "'b set"   and M :: "'b measure"   and f :: "'b \<Rightarrow> real" assumes "finite A"   and "\<And>a. a \<in> A \<Longrightarrow> {a} \<in> sets M"   and "\<And>a. a \<in> A \<Longrightarrow> emeasure M {a} < \<infinity>" shows "LINT x|M. f x * indicat_real A x = (\<Sum>a\<in>A. f a * Sigma_Algebra.measure M {a})"
DomainE: fixes a :: "'a"   and r :: "('a \<times> 'b) set" assumes "a \<in> Domain r" obtains b :: "'b" where "(a, b) \<in> r"
nextnat_le_seqsim: fixes n :: "nat"   and s :: "nat \<Rightarrow> 'b" assumes "n < nextnat s" shows "s \<approx> s |\<^sub>s n"
dvd_euclidean_size_eq_imp_dvd: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> (0::'a)"   and "euclidean_size a = euclidean_size b"   and "b dvd a" shows "a dvd b"
subst_ext_iff: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and \<sigma> :: "'c"   and \<tau> :: "'c" assumes "substitution subst_atm id_subst comp_subst renamings_apart atm_of_atms" shows "(\<sigma> = \<tau>) = (\<forall>A. subst_atm A \<sigma> = subst_atm A \<tau>)"
lossless2: fixes s2 :: "'a"   and x :: "'c" assumes "bad2 s2" shows "lossless_spmf (oracle2 s2 x)"
execute_lookup: fixes r :: "'a ref"   and h :: "heap" shows "execute (!r) h = Some (Ref.get h r, h)"
sc_bounded_by: fixes P :: "'c \<Rightarrow> real"   and c :: "real" assumes "sound P"   and "0 \<le> c" shows "bounded_by (c * bound_of P) (\<lambda>x. c * P x)"
action_proj_in_prob_proj: fixes a :: "('a, 'b) fmap \<times> ('a, 'c) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'c) fmap) set"   and vs :: "'a set" assumes "a \<in> PROB" shows "action_proj a vs \<in> prob_proj PROB vs"
dom_if: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" shows "dom (\<lambda>x. if P x then f x else g x) = dom f \<inter> {x. P x} \<union> dom g \<inter> {x. \<not> P x}"
hoare_def: fixes p :: "'b Assertion"   and S :: "'b"   and q :: "'b Assertion" shows "hoare p S q = (p \<le> wp S q)"
eSubst_simp2: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and ys :: "'varSort"   and Y :: "'c"   and y :: "'var"   and eX :: "'c withERR" assumes "\<not> igWls MOD (asSort ys) Y" shows "eSubst MOD ys (OK Y) y eX = ERR"
no_cs_change_if_no_event: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> c'"   and "isRecv ev \<longrightarrow> getId ev \<noteq> i"   and "regular_event ev" shows "cs c i = cs c' i"
walk_decomp_1: fixes G :: "('c, 'd) Graph_scheme"   and xs :: "'c list"   and ys :: "'c list" assumes "Graph G"   and "Graph.walk G (xs @ ys)" shows "Graph.walk G xs"
UN_upt: fixes f :: "'d \<Rightarrow> 'c set"   and xs :: "'d list" shows "(\<Union>i\<in>{0..<length xs}. f (xs ! i)) = \<Union> (f ` set xs)"
size_4: fixes x21 :: "'b"   and x22 :: "'b list" shows "length (x21 # x22) = length x22 + Suc 0"
pathCard_def: fixes p :: "Vars.var list"   and x :: "Vars.var" shows "pathCard p x = (if no_call_in_path x p then none else if one_call_in_path x p then once else many)"
inf_top_left: fixes x :: "('a \<times> 'a set) set" shows "Multirelations.U \<inter> x = x"
plus_adds_left: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a + b adds c" shows "a adds c"
assms_5: shows "Q_I_neg = filterQueryNeg I Qn"
card_1_singleton_conv: fixes A :: "'a set" shows "(card A = Suc 0) = (\<exists>a. A = {a})"
lambda_p_q_p_eq_q: fixes v :: "i"   and p :: "\<o>"   and q :: "\<o>" shows "[\<^bold>\<lambda>\<^sup>0 p \<^bold>= \<^bold>\<lambda>\<^sup>0 q \<^bold>\<equiv> p \<^bold>= q in v]"
sublist_transfer: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "bi_unique A" shows "(list_all2 A ===> list_all2 A ===> (=)) sublist sublist"
matrix_vector_mult_in_columnspace_gen: fixes A :: "(('b, 'd) vec, 'c) vec"   and x :: "('b, 'd) vec" shows "A *v x \<in> vec.span (columns A)"
fine_rsum_const: fixes \<delta> :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and D :: "(real \<times> real \<times> real) list"   and c :: "real" assumes "fine \<delta> (a, b) D" shows "rsum D (\<lambda>x. c) = c * (b - a)"
assoc_end_min_def: shows "assoc_end_min \<equiv> fst \<circ> snd"
unitaryI2: fixes P :: "'c \<Rightarrow> real" assumes "nneg P"   and "bounded_by 1 P" shows "unitary P"
name_id: fixes a :: "name"   and x :: "'b" shows "[(a, a)] \<bullet> x = x"
monoidal_category_axioms: shows "monoidal_category (\<cdot>\<^sub>D) T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D"
cos_aform_err_def: fixes p :: "nat"   and X :: "(real \<times> real pdevs) \<times> real" shows "cos_aform_err p X = range_reduce_aform_err p X \<bind> (\<lambda>X. let ivl = ivl_of_aform_err p X; l = lower ivl; u = upper ivl; L = Num l; U = Num u in if 0 \<le> l \<and> u \<le> lb_pi p then min_range_antimono p Cos (\<lambda>x. Minus (Sin x)) l u X else if ub_pi p \<le> l \<and> u \<le> 2 * lb_pi p then min_range_mono p Cos (\<lambda>x. Minus (Sin x)) l u X else Some (ivl_err (real_interval (cos_float_interval p ivl))))"
dom: fixes i :: "'b" assumes "i \<in> S" shows "\<psi> (local.dom (\<phi> i)) \<in> C'.domains i"
normalizePrQ_def: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" shows "normalizePrQ p = removeAllDuplicates (insertDenies (separate (qsort (removeShadowRules2 (remdups (rm_MT_rules Cp (insertDeny (removeShadowRules1 (policy2list p)))))) (Nets_List p))))"
generate_field_min_subfield1: fixes H :: "'a set"   and E :: "'a set" assumes "H \<subseteq> carrier R"   and "subfield E R"   and "H \<subseteq> E" shows "generate_field R H \<subseteq> E"
that: fixes u :: "nat list"   and v :: "nat list" assumes "u @ v \<le>\<^sub>v x @ y"   and "(u, v) \<in> Minimal_Solutions a b" shows "thesis"
int_rw2: shows "sum ((^) (int (x ^ a))) {..<q} = int (sum ((^) (x ^ a)) {..<q})"
D_base_finite: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and i :: "nat"   and j :: "nat"   and k :: "nat" shows "finite {linordered_monoid.len m i j xs |xs. set xs \<subseteq> {0..k} \<and> distinct xs}"
vimage_Union: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'b set set" shows "f -` \<Union> A = \<Union> ((-`) f ` A)"
awalk_verts_append_cas: fixes u :: "'a"   and p :: "'b list"   and q :: "'b list"   and v :: "'a" assumes "cas u (p @ q) v" shows "awalk_verts u (p @ q) = awalk_verts u p @ tl (awalk_verts (awlast u p) q)"
map_of_def: fixes heap :: "heap"   and k :: "'a" shows "state_dp_consistency.map_of heap k = fst (run_state (lookup' k) heap)"
left_adjoint_preserved_by_iso: fixes f :: "'a"   and \<phi> :: "'a"   and f' :: "'a" assumes "B.is_left_adjoint f"   and "\<guillemotleft>\<phi> : f \<Rightarrow>\<^sub>B f'\<guillemotright>"   and "B.iso \<phi>" shows "B.is_left_adjoint f'"
walk_first_edge': fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and xs :: "'c list" assumes "Graph G"   and "Graph.walk G (v # xs)"   and "xs \<noteq> []" shows "v \<rightarrow>\<^bsub>G\<^esub> hd xs"
mcont_contD: fixes luba :: "'b set \<Rightarrow> 'b"   and orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and lubb :: "'c set \<Rightarrow> 'c"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and Y :: "'b set" assumes "mcont luba orda lubb ordb f"   and "Complete_Partial_Order.chain orda Y"   and "Y \<noteq> {}" shows "f (luba Y) = lubb (f ` Y)"
cinfinite_csum: fixes r1 :: "('c \<times> 'c) set"   and r2 :: "('d \<times> 'd) set" assumes "cinfinite r1 \<or> cinfinite r2" shows "cinfinite (r1 +c r2)"
coinduct: fixes f :: "'f \<Rightarrow> 'f"   and X :: "'f" assumes "mono f"   and "X \<le> f (sup X (gfp f))" shows "X \<le> gfp f"
contg: shows "continuous_on (flow0 x ` (existence_ivl0 x \<inter> {0..})) g"
steps_append: fixes A :: "'c \<times> ('d option \<Rightarrow> 'c \<Rightarrow> 'c set) \<times> ('c \<Rightarrow> bool)"   and v :: "'d list"   and w :: "'d list" shows "NAe.steps A (v @ w) = NAe.steps A v O NAe.steps A w"
Nonce_def: shows "Nonce \<equiv> (id ---> abs_msg) NONCE"
brent_def: fixes f :: "'b \<Rightarrow> 'b"   and x0 :: "'b" shows "fx0.brent f x0 \<equiv> fx0.find_lambda f x0 ;; fx0.find_mu f x0"
assoc: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" shows "a \<union># b \<union># c = a \<union># (b \<union># c)"
cltn2_assoc: fixes A :: "cltn2"   and B :: "cltn2"   and C :: "cltn2" shows "cltn2_compose (cltn2_compose A B) C = cltn2_compose A (cltn2_compose B C)"
addr_loc_type_def: fixes P :: "'b prog" shows "addr_loc_type P = h.addr_loc_type P undefined"
polyneg: fixes bs :: "'a list"   and p :: "poly" shows "\<lparr>~\<^sub>p p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = - \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
div_positive: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'b \<times> 'b"   and divmod_step :: "num \<Rightarrow> 'b \<times> 'b \<Rightarrow> 'b \<times> 'b"   and b :: "'b"   and a :: "'b" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step"   and "less zero b"   and "less_eq b a" shows "less zero (divide a b)"
s_def: shows "s \<equiv> \<lambda>x xa xb. incr' x xa xb - incr x xa xb"
Rep_mini_alg_unique: fixes x :: "mini_alg_unique" shows "Rep_mini_alg_unique x \<in> {r. ma_coeff r = 0 \<and> ma_base r = 0 \<or> ma_coeff r \<noteq> 0 \<and> prime_product (ma_base r)}"
StepConf_def: fixes A :: "('a, 'b, 'c) hierauto"   and C :: "'a set"   and TS :: "('a \<times> (('a, 'b) expr \<times> ('c data \<Rightarrow> bool) \<times> 'b set \<times> 'c pupdate) \<times> 'a) set" shows "StepConf A C TS \<equiv> C - ChiStar A `` Source TS \<union> Target TS \<union> ChiRel A `` Target TS \<inter> HAInitStates A \<union> (Restr (ChiRel A) (HAInitStates A))\<^sup>+ `` (ChiRel A `` Target TS \<inter> HAInitStates A)"
f_newind: fixes anything :: "nat list" shows "{\<lambda>nl. nl = xs @ 0 \<up> (fft - far) @ anything} fap {\<lambda>nl. nl = xs @ rec_exec f xs # 0 \<up> (fft - Suc far) @ anything}"
div_positive: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and divmod :: "Num.num \<Rightarrow> Num.num \<Rightarrow> 'a \<times> 'a"   and divmod_step :: "Num.num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"   and b :: "'a"   and a :: "'a" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step"   and "less zero b"   and "less_eq b a" shows "less zero (divide a b)"
fact_code: fixes n :: "nat" shows "fact n = of_nat (fold_atLeastAtMost_nat (*) 2 n 1)"
Vd_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and src :: "nat"   and d :: "nat" shows "pre_bfs_invar.Vd c src d \<equiv> {v. Graph.connected c src v \<and> Graph.min_dist c src v = d}"
good_item_simps_39: fixes inp :: "'a \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and binpa :: "'b \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qAbs set option"   and xsa :: "'varSort"   and xa :: "'var"   and delta :: "'c" assumes "goodInp inp"   and "goodBinp binpa" shows "fresh xsa xa (Op delta inp binpa) = (freshInp xsa xa inp \<and> freshBinp xsa xa binpa)"
not_eq_extremum: fixes a :: "'a" shows "(a \<noteq> \<bottom>) = (\<bottom> < a)"
finsum_over_singleton: fixes R :: "('a, 'b) ring_scheme"   and M :: "('a, 'c, 'd) module_scheme"   and f :: "'e \<Rightarrow> 'c"   and x :: "'e" assumes "Module.module R M"   and "f x \<in> carrier M" shows "finsum M f {x} = f x"
Inf_greatest: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and A :: "'d set"   and z :: "'d" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq z x" shows "less_eq z (Inf A)"
square_part_def: fixes n :: "nat" shows "square_part n = (if n = 0 then 0 else \<Prod>p\<in>prime_factors n. p ^ (multiplicity p n div 2))"
Pair_Exp: fixes X :: "msg"   and Y :: "msg"   and X' :: "msg"   and Y' :: "msg" shows "\<langle>X, Y\<rangle> \<noteq> Exp X' Y'"
inv_eq_imp_eq: fixes x :: "'a"   and y :: "'a" assumes "x \<in> Units (add_monoid R)"   and "y \<in> Units (add_monoid R)"   and "inv\<^bsub>add_monoid R\<^esub> x = inv\<^bsub>add_monoid R\<^esub> y" shows "x = y"
colours_sweep_loop_free_1: fixes r :: "'a"   and s :: "'b process_name \<Rightarrow> ('c, 'b, 'd, 'a) Model.local_state"   and r' :: "'a" shows "black r (s(sys := s sys\<lparr>heap := (sys_heap s)(r' := None)\<rparr>)) = (black r s \<and> r \<noteq> r')"
assms_1: shows "h \<turnstile> get_elements_by_class_name ptr name \<rightarrow>\<^sub>r results"
deriv: shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P p Q,A"
triple_implies: fixes x :: "'a"   and y :: "'a" shows "x \<leadsto> y \<leadsto> y \<leadsto> y = x \<leadsto> y"
gen_lossless_gpvI: fixes gpv :: "('d, 'e, 'f) gpv"   and \<I> :: "('e, 'f) \<I>"   and co :: "bool" assumes "lossless_spmf (the_gpv gpv)"   and "\<And>out c input. \<lbrakk>IO out c \<in> set_spmf (the_gpv gpv); input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> gen_lossless_gpv co \<I> (c input)" shows "gen_lossless_gpv co \<I> gpv"
outward: fixes H1 :: "circline"   and H2 :: "circline" assumes "is_poincare_line H1"   and "is_poincare_line H2"   and "intersects_x_axis_positive H1"   and "intersects_x_axis_positive H2"   and "outward H1 H2" shows "Re (to_complex (calc_x_axis_intersection H2)) \<le> Re (to_complex (calc_x_axis_intersection H1))"
exhaust_sel: fixes tree :: "'b tree" obtains "tree = Leaf"   | "tree = Node (l tree) (r tree) (val tree) (lchild tree) (rchild tree)"
nat_pow_distrib: fixes x :: "'a vec"   and y :: "'a vec"   and n :: "nat" assumes "x \<in> carrier_vec n"   and "y \<in> carrier_vec n" shows "[n] \<cdot>\<^bsub>V\<^esub> (x + y) = [n] \<cdot>\<^bsub>V\<^esub> x + [n] \<cdot>\<^bsub>V\<^esub> y"
Complexs_1: shows "(1::'b) \<in> \<complex>"
exhaust_disc: fixes event :: "('a, 'b, 'c) event" obtains "isTrans event"   | "isSend event"   | "isRecv event"   | "isSnapshot event"   | "isRecvMarker event"
l2: shows "(\<lambda>n. snd (f (r1 (r2 n)))) \<longlonglongrightarrow> l2"
add: fixes b1 :: "'m_1"   and b2 :: "'m_1" assumes "b1 \<in> U\<^sub>M\<^sub>_\<^sub>1"   and "b2 \<in> U\<^sub>M\<^sub>_\<^sub>1" shows "f (b1 +\<^sub>M\<^sub>_\<^sub>1 b2) = f b1 +\<^sub>M\<^sub>_\<^sub>2 f b2"
simps_6: fixes I :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and Is :: "char list set"   and S :: "(char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int"   and E :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and Es :: "char list set"   and SS :: "char list \<Rightarrow> char list set"   and b :: "bexp"   and C :: "acom"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "pre ({((I, Is), S, E, Es, SS)} WHILE b DO C) Q = I"
CastRedsThrow: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and a :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>throw a,s',b'\<rangle>" shows "P \<turnstile> \<langle>Cast C e,s,b\<rangle> \<rightarrow>* \<langle>throw a,s',b'\<rangle>"
rep_eq: fixes x :: "'a covector"   and xa :: "'a covector" shows "(x \<le> xa) = (Rep_covector x \<le> Rep_covector xa)"
eq1: shows "ord_term_lin.max_list (v # vs) = v"
sums: fixes x :: "real" shows "(\<lambda>n. P x n / (real (Suc n))\<^sup>2) sums (if x = 0 then K else g x / x\<^sup>2)"
test_meet_comm: fixes test :: "'c \<Rightarrow> 'd"   and not :: "'d \<Rightarrow> 'd"   and p :: "'c"   and q :: "'c" assumes "drat test not" shows "test p \<cdot> test q = test q \<cdot> test p"
similar_mat_wit_smult: fixes A :: "'b mat"   and B :: "'b mat"   and P :: "'b mat"   and Q :: "'b mat"   and k :: "'b" assumes "similar_mat_wit A B P Q" shows "similar_mat_wit (k \<cdot>\<^sub>m A) (k \<cdot>\<^sub>m B) P Q"
w_cmpser_ns: fixes r :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and i :: "nat" assumes "0 < r"   and "w_cmpser G r f"   and "i \<le> r - 1" shows "\<natural>f i \<triangleright> f (Suc i)"
has_vderiv_eq: fixes f :: "real \<Rightarrow> 'b"   and f' :: "real \<Rightarrow> 'b"   and S :: "real set"   and g :: "real \<Rightarrow> 'b"   and g' :: "real \<Rightarrow> 'b"   and T :: "real set" assumes " D f = f' on S"   and "\<And>x. x \<in> S \<Longrightarrow> f x = g x"   and "\<And>x. x \<in> S \<Longrightarrow> f' x = g' x"   and "S = T" shows " D g = g' on T"
z_in_s: fixes n :: "nat" shows "z n \<in> s"
simps_1: fixes x21 :: "'k"   and x22 :: "'k list"   and y21 :: "'k"   and y22 :: "'k list" shows "(x21 # x22 = y21 # y22) = (x21 = y21 \<and> x22 = y22)"
mod_minus_right: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "modulo a (uminus b) = uminus (modulo (uminus a) b)"
locateTI_unique: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat"   and i :: "nat"   and ii :: "nat" assumes "properL cl"   and "n < length cl"   and "i < brn (cl ! n)"   and "ii = brnL cl n + i" shows "n = locateT cl ii \<and> i = locateI cl ii"
wt_attack_if_tfr_attack_d: fixes \<A> :: "('fun, 'var) strand_step list"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "wf\<^sub>s\<^sub>t {} \<A>"   and "fv\<^sub>s\<^sub>t \<A> \<inter> bvars\<^sub>s\<^sub>t \<A> = {}"   and "tfr\<^sub>s\<^sub>t \<A>"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>t \<A>)"   and "Ana_invar_subst (ik\<^sub>s\<^sub>t \<A> \<union> assignment_rhs\<^sub>s\<^sub>t \<A>)"   and "subst_domain \<I> = \<V> \<and> ground (subst_range \<I>)"   and "\<I> \<Turnstile> \<langle>\<A>\<rangle>" shows "\<exists>\<I>\<^sub>\<tau>. (subst_domain \<I>\<^sub>\<tau> = \<V> \<and> ground (subst_range \<I>\<^sub>\<tau>)) \<and> \<I>\<^sub>\<tau> \<Turnstile> \<langle>\<A>\<rangle> \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<I>\<^sub>\<tau> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<I>\<^sub>\<tau>)"
surjective: fixes r :: "('e, 'f) monoid_scheme" shows "r = \<lparr>carrier = carrier r, monoid.mult = (\<otimes>\<^bsub>r\<^esub>), one = \<one>\<^bsub>r\<^esub>, \<dots> = monoid.more r\<rparr>"
collinear_triples: fixes a :: "'b"   and b :: "'b"   and S :: "'b set" assumes "a \<noteq> b" shows "collinear (insert a (insert b S)) = (\<forall>x\<in>S. collinear {a, b, x})"
var_dist_frees: fixes vs :: "nat list"   and t :: "dB" shows "frees (var_dist vs t) = frees t \<union> set vs"
proj2_rep_in: fixes x :: "proj2" shows "proj2_rep x \<in> Rep_proj2 x"
id: shows "(\<lambda>p a. if a \<in> A then b_to_a (p (a_to_b a)) else a) ` (\<lambda>p b. if b \<in> B then a_to_b (p (b_to_a b)) else b) ` {p. p permutes A} = {p. p permutes A}"
intro: fixes cl :: "'b potype" assumes "PO cl"   and "CL_axioms cl" shows "CL cl"
degree_map_poly_hom: fixes p :: "'b poly" shows "degree (map_poly to_fract p) = degree p"
supp_10: fixes x1 :: "pi" shows "supp (!x1) = supp x1"
uminus_add_closed: fixes G :: "'a set"   and g :: "'a"   and h :: "'a" assumes "Group G"   and "g \<in> G"   and "h \<in> G" shows "- g + h \<in> G"
supp_alpha_supp_rel: fixes t\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha>" shows "supp t\<^sub>\<alpha> = supp_rel (=\<^sub>\<alpha>) (rep_Tree\<^sub>\<alpha> t\<^sub>\<alpha>)"
sorted_msort_key: fixes xs :: "'a list" shows "sorted (map key (local.msort_key xs))"