hcju/isabelleps huggingface dataset:
remdups_on_subset_input: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" shows "set (remdups_on f xs) \<subseteq> set xs"
prj_chine_2: shows "t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1"
gcd_0_right: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "gcd a zero = normalize a"
flc: shows "round_num_denom_cost (dmu_ij_state state i (j - 1)) (d_state state j) = (fc, c1)"
is_cnf_encode_interfering_operator_exclusion: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "is_cnf (encode_interfering_operator_exclusion \<Pi> t)"
foldr_fold: fixes xs :: "'g list"   and f :: "'g \<Rightarrow> 'h \<Rightarrow> 'h" assumes "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set xs\<rbrakk> \<Longrightarrow> f y \<circ> f x = f x \<circ> f y" shows "foldr f xs = fold f xs"
mkM_kripke: fixes T :: "'s Trace set" shows "kripke (SPR.mkM T)"
prv_imp_monoR: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "prv (imp \<psi> \<chi>)" shows "prv (imp (imp \<chi> \<phi>) (imp \<psi> \<phi>))"
valid_path_shiftpath: fixes g :: "real \<Rightarrow> 'a"   and a :: "real" assumes "valid_path g"   and "pathfinish g = pathstart g"   and "a \<in> {0..1}" shows "valid_path (shiftpath a g)"
inner_Basis_scale2: fixes b1 :: "'a"   and v :: "'a"   and c :: "real" assumes "b1 \<in> S" shows "local.inner_Basis v (c *\<^sub>R b1) = c * local.inner_Basis v b1"
c: fixes m :: "nat"   and a :: "nat" assumes "\<not> m \<le> a" shows "a < m"
triangle_in_hom_3: shows "\<guillemotleft>\<l>[f] \<cdot> (\<xi> \<cdot> (f \<star> local.inv \<eta> \<star> g) \<cdot> (local.inv \<xi> \<star> f \<star> g) \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>) \<cdot> \<r>\<^sup>-\<^sup>1[f] : f \<Rightarrow> f\<guillemotright>"
fno: fixes x :: "'a" assumes "x \<in> span ((+) (- a) ` S)" shows "norm (f x) = norm x"
simps_4: fixes b :: "bexp"   and c1 :: "'a acom"   and c2 :: "'a acom"   and P :: "'a" shows "strip (IF b THEN c1 ELSE c2 {P}) = IF b THEN strip c1 ELSE strip c2"
finite_rhs_def2: fixes ES :: "('b \<times> 'c set) set" shows "finite_rhs ES = (\<forall>X rhs. (X, rhs) \<in> ES \<longrightarrow> finite rhs)"
less_le: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.order less_eq less" shows "less x y = (less_eq x y \<and> x \<noteq> y)"
simps_1: fixes c :: "'b" shows "polybound0 (c)\<^sub>p = True"
le_iff_sup: fixes sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.semilattice_sup sup less_eq less" shows "less_eq x y = (sup x y = y)"
refine: fixes A :: "('d \<times> ('e \<times> 'f) set) set"   and li :: "ereal"   and l :: "ereal"   and ui :: "ereal"   and u :: "ereal"   and Xi :: "'d list"   and X :: "('e \<times> 'f) set" assumes "single_valued A"   and "(li, l) \<in> ereal_rel"   and "(ui, u) \<in> ereal_rel"   and "(Xi, X) \<in> clw_rel A" shows "(nres_of (local.scaleRe_ivl_coll_impl li ui Xi), scaleRe_ivl_coll_spec l u X) \<in> \<langle>clw_rel (\<langle>A\<rangle>scaleR2_rel)\<rangle>nres_rel"
iso2Inv: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b" assumes "Category C"   and "ciso\<^bsub>C\<^esub> f" shows "cinv\<^bsub>C\<^esub> f Cinv\<^bsub>C\<^esub> f"
simps_2: fixes x21 :: "'b"   and x22 :: "'b"   and y21 :: "'b"   and y22 :: "'b" shows "(Edge x21 x22 = Edge y21 y22) = (x21 = y21 \<and> x22 = y22)"
pth_4_1: fixes x :: "'b" shows "0 *\<^sub>R x \<equiv> 0::'b"
degree_of_nat: fixes n :: "nat" shows "degree (of_nat n) = 0"
Gets_Server_message_form: fixes B :: "agent"   and M :: "nat"   and X :: "msg"   and Nb :: "nat"   and K :: "nat"   and evs :: "event list" assumes "Gets B \<lbrace>Nonce M, Crypt (shrK B) \<lbrace>X, Nonce Nb, Key K\<rbrace>\<rbrace> \<in> set evs"   and "evs \<in> orb" shows "K \<notin> range shrK \<and> (\<exists>A Na. X = Crypt (shrK A) \<lbrace>Nonce Na, Key K\<rbrace>) \<or> X \<in> analz (knows Spy evs)"
module_prod_def: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f" shows "module_prod s1 s2 \<equiv> module_pair s1 s2"
pr_confirm_says: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and n :: "nat"   and run :: "nat"   and X :: "agent"   and K :: "rat" assumes "(evs, S, A, U) \<in> protocol"   and "Says n run 5 X Spy (Crypt (sesK K) (Number 0)) \<in> set evs" shows "Says n run 5 Spy (Card n) (Crypt (sesK K) (Passwd n)) \<in> set evs"
B: shows "ExtMapK (S (User m, n, run)) = Some b"
assms_2: shows "\<rho>\<^sub>1 \<equiv> run\<^sub>t \<delta>\<^sub>1 q\<^sub>1 w"
mor_Rep_IF: shows "mor UNIV UNIV ctor1 ctor2 (car_init1 undefined) (car_init2 undefined) (str_init1 undefined) (str_init2 undefined) Rep_IF1 Rep_IF2"
disc_41: fixes x5 :: "'a MFOTL.formula" shows "formula.is_Exists (formula.Exists x5)"
leg0_uniquely_isomorphic_2: shows "\<exists>!\<theta>. \<guillemotleft>\<theta> : s\<^sub>0 \<star> \<mu>.chine \<Rightarrow> r\<^sub>0\<guillemotright>"
diff_union_swap: fixes a :: "'b"   and b :: "'b"   and M :: "'b multiset" assumes "a \<noteq> b" shows "b, M - (a, {#}) = (b, M) - (a, {#})"
Domainp_HMA_M: shows "Domainp Mod_Type_Connect.HMA_M = (\<lambda>A. A \<in> carrier_mat CARD('f) CARD('e))"
zero_rel_carrier: fixes U :: "'a set" assumes "U \<in> I" shows "\<lfloor>U, \<zero>\<^bsub>U\<^esub>\<rfloor> \<in> carrier_direct_lim"
entails_def: fixes uu :: "'b itself"   and C :: "'c set"   and T :: "('c Standard_Constant allegorical_term \<times> 'c Standard_Constant allegorical_term) set"   and S :: "'c Standard_Constant allegorical_term \<times> 'c Standard_Constant allegorical_term" shows "entails uu C T S \<equiv> \<forall>G. model C G T \<longrightarrow> G \<Turnstile> S"
h2b: shows "\<forall>q. List.member (qs1 @ qs2) q \<longrightarrow> coprime p q"
weakSimulation_def: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi" shows "P \<leadsto>\<^sup>^<Rel> Q \<equiv> \<forall>Rs. Q \<longmapsto> Rs \<longrightarrow> weakSimAct P Rs P Rel"
len_not_eq_0: fixes len_of :: "'k itself \<Rightarrow> nat" assumes "class.len len_of" shows "len_of TYPE('k) \<noteq> 0"
sorted_wrt_map_rev_linord: fixes l :: "('a \<times> 'd) list" shows "sorted_wrt (\<lambda>x y. fst y \<le> fst x) l = sorted (rev (map fst l))"
p_basic: fixes m :: "nat" assumes "m \<noteq> 0" shows "par m < m"
cdep_slice1: fixes n'' :: "'node SDG_node"   and n' :: "'node SDG_node"   and n :: "'node SDG_node" assumes "n'' s\<longrightarrow>\<^bsub>cd\<^esub> n'"   and "n' \<in> sum_SDG_slice1 n" shows "n'' \<in> sum_SDG_slice1 n"
smult_add_right: fixes a :: "'a"   and p :: "'a poly"   and q :: "'a poly" shows "smult a (p + q) = smult a p + smult a q"
plus_ms_aux_MSLNil_1: fixes F :: "('a \<times> real) msllist" shows "plus_ms_aux F MSLNil = F"
ext_inject: fixes CatDom :: "('d, 'f, 'h) Category_scheme"   and CatCod :: "('e, 'g, 'i) Category_scheme"   and MapM :: "'f \<Rightarrow> 'g"   and more :: "'j"   and CatDom' :: "('d, 'f, 'h) Category_scheme"   and CatCod' :: "('e, 'g, 'i) Category_scheme"   and MapM' :: "'f \<Rightarrow> 'g"   and more' :: "'j" shows "(\<lparr>CatDom = CatDom, CatCod = CatCod, MapM = MapM, \<dots> = more\<rparr> = \<lparr>CatDom = CatDom', CatCod = CatCod', MapM = MapM', \<dots> = more'\<rparr>) = (CatDom = CatDom' \<and> CatCod = CatCod' \<and> MapM = MapM' \<and> more = more')"
CCs_def: fixes E' :: "'edge set" shows "CCs E' = V // connected E'"
conditionalize: fixes A :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "finite A" shows "sum g A = (\<Sum>a. if a \<in> A then g a else (0::'b))"
map_convs_2: fixes a :: "'a \<Rightarrow> 'b option"   and b :: "'a \<Rightarrow> 'b option" shows "a + b = a ++ b"
p_cases: fixes m :: "nat" assumes "m \<noteq> 0" shows "m = s1 (par m) \<or> m = s2 (par m)"
det_echelon_form_of_column_k_det: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and det_P :: "'b"   and B :: "(('b, 'c) vec, 'c) vec"   and A :: "(('b, 'd) vec, 'd) vec"   and i :: "nat"   and k :: "nat" assumes "is_bezout_ext bezout"   and "det_P * det B = det A" shows "fst (echelon_form_of_column_k_det bezout (det_P, A, i) k) * det B = det (fst (snd (echelon_form_of_column_k_det bezout (det_P, A, i) k)))"
inorder_delete: fixes m :: "nat"   and t :: "('b \<times> nat) tree"   and x :: "'b" assumes "HBT m"   and "Sorted_Less.sorted (Tree2.inorder t)" shows "Tree2.inorder (HBT.delete m x t) = del_list x (Tree2.inorder t)"
mmupd_notin_upd: fixes k :: "'a"   and K :: "'a set"   and m :: "'a \<Rightarrow> 'b option"   and v :: "'b" assumes "k \<notin> K" shows "map_mmupd m K v k = m k"
H1H: shows "poly_mod.Mp p H1 = poly_mod.Mp p H"
top_clocked_unclocked_equiv: fixes env :: "v sem_env"   and s :: "'b state"   and tp :: "top0"   and s' :: "'b state"   and r :: "(v sem_env, v) result" shows "evaluate_top False env s tp (s', r) = (\<exists>c. evaluate_top True env (update_clock (\<lambda>_. c) s) tp (update_clock (\<lambda>_. 0) s', r) \<and> r \<noteq> Rerr (Rabort Rtimeout_error) \<and> clock s = clock s')"
q': fixes x :: "rat"   and y :: "rat" assumes "(x, y) \<in> set rxs_ys" shows "poly (of_int_poly q) x = y"
im: fixes x :: "real" assumes "0 \<le> x" shows "f x \<in> X"
assms: fixes x :: "'a" assumes "x = v" shows "f x \<le> ESPEC \<Phi> \<Psi>"
graph: fixes \<alpha> :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'b set" assumes "map \<alpha> S T" shows "\<alpha> \<in> S \<rightarrow>\<^sub>E T"
d: fixes modulus :: "'b poly" shows "power_poly_f_mod modulus = (\<lambda>a n. a ^ n mod modulus)"
assms_1: fixes a :: "'c"   and b :: "'d" assumes "p = Inl (a, b)" shows "thesis"
equal: fixes equal :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.equal equal" shows "equal = (=)"
consistent_one_step_axioms: shows "consistent_one_step (t 0) t1 x incr p B r L"
conjugate_def: fixes f :: "'a \<Rightarrow> 'a" shows "conjugate f \<equiv> THE g. conjugation_pair g f"
index_mat_addrow_5: fixes ad :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mul :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and k :: "nat"   and l :: "nat"   and A :: "'c mat" shows "dim_col (mat_addrow_gen ad mul a k l A) = dim_col A"
LEAST_FP_def: fixes L :: "('a, 'b) gorder_scheme"   and f :: "'a \<Rightarrow> 'a" shows "LFP\<^bsub>L\<^esub> f = \<Sqinter>\<^bsub>L\<^esub>{u \<in> carrier L. f u \<sqsubseteq>\<^bsub>L\<^esub> u}"
simps_15: fixes x21 :: "'b"   and x22 :: "'b tree"   and x23 :: "'b tree"   and x24 :: "nat" shows "set_of (MKT x21 x22 x23 x24) = Set.insert x21 (set_of x22 \<union> set_of x23)"
i: shows "i < IArray.length (matrix_to_iarray (Gram_Schmidt_column_k_efficient A k))"
sq_norm_vec_eq_0: fixes v :: "'b vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "(\<parallel>v\<parallel>\<^sup>2 = (0::'b)) = (v = 0\<^sub>v n)"
gen_bg_impl_rel_eext_def: fixes Rm :: "('i \<times> 'k) set"   and Rf :: "('h \<times> 'j set) set" shows "\<langle>Rm, Rf\<rangle>gen_bg_impl_rel_eext = {(\<lparr>bgi_F = Fi, \<dots> = mi\<rparr>, \<lparr>bg_F = F, \<dots> = m\<rparr>) |Fi mi F m. (Fi, F) \<in> Rf \<and> (mi, m) \<in> Rm}"
MapsToMorDomCod_1: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e"   and g :: "'e" assumes "Category C"   and "f \<approx>>\<^bsub>C\<^esub> g" shows "f ;;\<^bsub>C\<^esub> g \<in> mor\<^bsub>C\<^esub>"
ui_re_is_be: fixes prb :: "('d, 'e, 'f) pre_RedBlack"   and re :: "('d \<times> nat) edge" assumes "RedBlack prb"   and "re \<in> edges (red prb)" shows "ui_edge re \<in> edges (black prb)"
cases: fixes x :: "(('a list \<times> 'a list) option + unit) \<times> ('a list \<times> 'a list) insec_query" obtains uu_ :: "('a list \<times> 'a list) insec_query" where "x = (Inr (), uu_)"   | a' :: "'a list" and m' :: "'a list" where "x = (Inl None, Edit (a'__, m'__))"   | a :: "'a list" and m :: "'a list" and a' :: "'a list" and m' :: "'a list" where "x = (Inl (Some (a__, m__)), Edit (a', m'))"   | "x = (Inl None, Forward)"   | uv_ :: "'a list \<times> 'a list" where "x = (Inl (Some uv_), Forward)"   | "x = (Inl None, Drop)"   | uw_ :: "'a list \<times> 'a list" where "x = (Inl (Some uw_), Drop)"   | a :: "'a list" and m :: "'a list" where "x = (Inl (Some (a, m)), Look)"   | "x = (Inl None, Look)"
Well_R: shows "\<forall>r\<in>R. Well_order r"
llcp_same_conv_length: fixes xs :: "'a llist" shows "llcp xs xs = llength xs"
assms: shows "range_vars \<delta> \<subseteq> fv\<^sub>s\<^sub>t S"
mapi_branch_def: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> ('b, 'c) fm \<Rightarrow> ('b, 'c) fm"   and branch :: "(('b, 'c) fm list \<times> 'c) list" shows "mapi_branch f branch \<equiv> mapi (\<lambda>v. mapi_block (f v)) branch"
prems_2: shows "(aa_, a'a_) \<in> Rv"
holfg: shows "(\<lambda>z. f z + \<i> * g z) holomorphic_on S"
in_poly_roots_iff: fixes p :: "'b poly"   and x :: "'b" assumes "p \<noteq> 0" shows "(x \<in># poly_roots p) = (poly p x = (0::'b))"
aux: fixes f :: "'c \<Rightarrow>\<^sub>0 'd" shows "(lookup f = (\<lambda>_. 0::'d)) = (f = 0)"
\<Gamma>'_def: shows "\<Gamma>' = \<Gamma>\<lparr>weight := \<lambda>x. if x \<in> A \<Gamma> then weight \<Gamma> x - d_OUT g x else weight \<Gamma> x\<rparr>"
commit_correct: shows "OR_\<Sigma>_commit.abstract_com.correct"
ide_def: fixes a :: "'a \<times> 'a" shows "VV.ide a \<equiv> VV.comp a a \<noteq> VV.null \<and> (\<forall>f. (VV.comp f a \<noteq> VV.null \<longrightarrow> VV.comp f a = f) \<and> (VV.comp a f \<noteq> VV.null \<longrightarrow> VV.comp a f = f))"
u_def: shows "u \<equiv> \<lambda>x. snd (G (x, 0::'c))"
fps_compose_0: fixes a :: "'a fps" shows "0 oo a = 0"
fps_mult_nth_conv_inside_subdegrees: fixes f :: "'a fps"   and g :: "'a fps"   and n :: "nat" shows "fps_nth (f * g) n = (\<Sum>i = subdegree f..n - subdegree g. fps_nth f i * fps_nth g (n - i))"
Path_upd: fixes u :: "'b"   and as :: "'b list"   and f :: "'b \<Rightarrow> 'b ref"   and v :: "'b ref"   and x :: "'b ref"   and y :: "'b ref" assumes "u \<notin> set as" shows "Path (f(u := v)) x as y = Path f x as y"
has_as_product_def: fixes J :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'c \<Rightarrow> 'b"   and a :: "'b" shows "J.has_as_product J D a \<equiv> \<exists>\<pi>. product_cone J J.comp D a \<pi>"
vars_def: fixes p :: "'a mpoly" shows "vars p = \<Union> (keys ` keys (mapping_of p))"
coset_mult_assoc: fixes G :: "('e, 'f) monoid_scheme"   and M :: "'e set"   and g :: "'e"   and h :: "'e" assumes "Group.group G"   and "M \<subseteq> carrier G"   and "g \<in> carrier G"   and "h \<in> carrier G" shows "M #>\<^bsub>G\<^esub> g #>\<^bsub>G\<^esub> h = M #>\<^bsub>G\<^esub> g \<otimes>\<^bsub>G\<^esub> h"
tauPres: fixes P :: "pi"   and Q :: "pi" assumes "P \<approx> Q" shows "\<tau>.P \<approx> \<tau>.Q"
assms_2: shows "u \<in> [M]\<^bsub>v,n\<^esub>"
d: shows "Y \<in> obj\<^bsub>C\<^esub>"
convex_Inter: fixes f :: "'c set set" assumes "\<And>s. s \<in> f \<Longrightarrow> convex s" shows "convex (\<Inter> f)"
connected_contains_Ioo: fixes A :: "'b set"   and a :: "'b"   and b :: "'b" assumes "connected A"   and "a \<in> A"   and "b \<in> A" shows "{a<..<b} \<subseteq> A"
const_def: fixes x :: "'e" shows "pure x = (\<lambda>_. x)"
is_extensional: fixes f :: "'a" assumes "\<not> B.arr f" shows "src\<^sub>B f = B.null"
lborel_integrable_real_affine_iff: fixes c :: "real"   and f :: "real \<Rightarrow> 'a"   and t :: "real" assumes "c \<noteq> 0" shows "integrable lborel (\<lambda>x. f (t + c * x)) = integrable lborel f"
fromArr_def: fixes x :: "bool" shows "fromArr x \<equiv> if x \<in> {False, True} then MkArr False True [x] else null"
floor_diff_one: fixes x :: "'a" shows "\<lfloor>x - (1::'a)\<rfloor> = \<lfloor>x\<rfloor> - 1"
eq_lift_def: shows "eq \<equiv> \<Lambda> (Def x) (Def y). Def (x = y)"
scaleR: fixes i :: "'a"   and r :: "real"   and b :: "'b" shows "f i (r *\<^sub>R b) = r *\<^sub>R f i b"
seq_filter_def: fixes xs :: "'b list"   and A :: "'b set" shows "xs \<restriction>\<^sub>l A = filter (\<lambda>x. x \<in> A) xs"
A: shows "xps @ yps @ [yp] = c_tr step out s\<^sub>0 (map fst (xps @ yps) @ [fst yp])"
create_gba_from_nodes__invar: fixes \<phi> :: "'b ltlr"   and qs :: "'b node set" assumes "create_gba_from_nodes_precond \<phi> qs" shows "gba (create_gba_from_nodes \<phi> qs)"
subcategory_axioms: shows "subcategory C (\<lambda>f. C.arr f \<and> Ide (C.dom f) \<and> Ide (C.cod f))"
cinner_sup_norm: fixes \<psi> :: "'a" shows "norm \<psi> = (SUP \<phi>. cmod (\<phi> \<bullet>\<^sub>C \<psi>) / norm \<phi>)"
a1: shows "low_equal s1 s2 \<and> t1 = out_reg_mod w curr_win rd s1 \<and> t2 = out_reg_mod w curr_win rd s2"
sets_sup_measure': fixes B :: "'b measure"   and A :: "'b measure" assumes "sets B = sets A" shows "sets (sup_measure' A B) = sets A"
is_closest_exists: fixes s :: "('a, 'b) IEEE.float set"   and v :: "('a, 'b) IEEE.float \<Rightarrow> real"   and x :: "real" assumes "finite s"   and "s \<noteq> {}" shows "\<exists>a. is_closest v s x a"
class_add_sees_methods_rev_Obj: fixes C :: "char list"   and cdec :: "jvm_method class"   and P :: "jvm_prog"   and Mm :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> jvm_method) \<times> char list) option" assumes "class_add P (C, cdec) \<turnstile> Object sees_methods Mm"   and "C \<noteq> Object" shows "P \<turnstile> Object sees_methods Mm"
Un_insert_right: fixes A :: "'a set"   and a :: "'a"   and B :: "'a set" shows "A \<union> insert a B = insert a (A \<union> B)"
maddux_3_5: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> x = x \<squnion> - (y \<squnion> - y)"
learn_bc_prenum_eventually_zero: fixes s :: "nat \<Rightarrow> nat option"   and b :: "nat" assumes "r1_bc s" shows "\<exists>k. \<phi> (the (s (e_append_zeros b k))) (e_length b + k) \<down>= 0"
conjuncts_in_consistency: fixes C :: "'b fm set set"   and S :: "'b fm set"   and p :: "'b fm"   and q :: "'b fm" assumes "consistency C"   and "subset_closed C"   and "S \<union> {p \<^bold>\<and> q} \<in> C" shows "S \<union> {p, q} \<in> C"
par_comp_prot_def: fixes \<P> :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list set"   and Sec :: "('fun, 'var) Term.term set" shows "par_comp_prot \<P> Sec \<equiv> (\<forall>l1 l2. l1 \<noteq> l2 \<longrightarrow> GSMP_disjoint (\<Union>\<A>\<in>\<P>. trms\<^sub>l\<^sub>s\<^sub>t (proj l1 \<A>)) (\<Union>\<A>\<in>\<P>. trms\<^sub>l\<^sub>s\<^sub>t (proj l2 \<A>)) Sec) \<and> ground Sec \<and> (\<forall>s\<in>Sec. \<forall>s'\<in>subterms s. {} \<turnstile>\<^sub>c s' \<or> s' \<in> Sec) \<and> typing_cond_prot \<P>"
permutation_lemma: fixes S :: "'e set"   and p :: "'e \<Rightarrow> 'e" assumes "finite S"   and "bij p"   and "\<forall>x. x \<notin> S \<longrightarrow> p x = x" shows "permutation p"
cont2mono: fixes f :: "'d \<Rightarrow> 'e" assumes "cont f" shows "monofun f"
\<Delta>_simps_1: shows "seq (\<pi> \<star> s\<^sub>0) \<sigma>"
accessible_imp_states: fixes q :: "hf" assumes "q \<in> accessible" shows "q \<in> dfa.states M"
INV_rule_from_inv_rule: fixes T :: "('e, 'f) TS_scheme"   and I :: "'e set" assumes "init T \<subseteq> I"   and "{I \<inter> reach T} TS.trans T {> I}" shows "reach T \<subseteq> I"
integrable_mult_right: fixes c :: "'d"   and M :: "'e measure"   and f :: "'e \<Rightarrow> 'd" assumes "c \<noteq> (0::'d) \<Longrightarrow> integrable M f" shows "integrable M (\<lambda>x. c * f x)"
prime_factors_prod: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "finite A"   and "(0::'c) \<notin> f ` A" shows "prime_factors (prod f A) = \<Union> ((prime_factors \<circ> f) ` A)"
obs_consistent_med1a1: shows "obs_consistent R1a1 med1a1 m1a m1"
get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def: fixes node_ptr :: "(_) node_ptr"   and h :: "(_) heap" shows "get node_ptr h = Option.bind (get (cast node_ptr) h) cast"
is_processT9: fixes P :: "'a process" shows "\<forall>s. s @ [tick] \<in> D P \<longrightarrow> s \<in> D P"
select_convs_2: fixes states :: "hf"   and init :: "hf"   and final :: "hf"   and "next" :: "hf \<Rightarrow> 'b \<Rightarrow> hf \<Rightarrow> bool"   and more :: "'c" shows "init \<lparr>states = states, init = init, final = final, next = next, \<dots> = more\<rparr> = init"
star_unfold_left: fixes A :: "'b list set" shows "A\<star> = A \<cdot> A\<star> \<union> {[]}"
simp_num_pair_l: fixes t :: "Ferrack.num"   and n :: "int"   and t' :: "Ferrack.num"   and n' :: "int" assumes "numbound0 t"   and "0 < n"   and "simp_num_pair (t, n) = (t', n')" shows "numbound0 t' \<and> 0 < n'"
assms_2: shows "H \<turnstile> NotInDom d r"
open_riemann_sphere_def: fixes U :: "riemann_sphere set" shows "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> U)"
assms_2: shows "\<parallel>tid\<parallel>\<^bsub>t n\<^esub>"
hash_ineq_card: fixes W :: "'a set"   and seed_gen :: "'b spmf"   and hash :: "'b \<Rightarrow> 'a \<Rightarrow> 'c" assumes "finite W" shows "spmf (PRF_UHF.hash.game_hash_set seed_gen hash W) True \<le> PRF_UHF.hash.\<epsilon>_uh seed_gen hash * real (card W) * real (card W)"
cinner_mult_right: fixes a :: "'c"   and m :: "complex"   and b :: "'c" shows "cinner a (of_complex m * b) = m * cinner a b"
help''': fixes ram :: "'c"   and xs :: "'c list"   and b :: "'c list" shows "fst (splitAtRec ram xs b) = xs @ fst (splitAtRec ram [] b)"
implements_represents: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'g"   and jviewInit :: "'b \<Rightarrow> 'g \<Rightarrow> 'f"   and jviewIncr :: "'b \<Rightarrow> 'g \<Rightarrow> 'f \<Rightarrow> 'f"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'h list"   and envTrans :: "'h \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and jp :: "'b \<Rightarrow> ('g, 'd, 'i) Protocol" assumes "IncrEnvironment jkbp jview envObs jviewInit jviewIncr"   and "IncrEnvironment.implements jkbp envInit envAction envTrans envVal jview envObs jp" shows "PreEnvironmentJView.represents jkbp envInit envAction envTrans envVal jview (IncrEnvironment.jpTraces envInit envAction envTrans envObs jp)"
simps_9: fixes f :: "'b \<Rightarrow> 'a"   and x21 :: "'b tree"   and x22 :: "'b"   and x23 :: "'b tree" shows "map_tree f \<langle>x21, x22, x23\<rangle> = \<langle>map_tree f x21, f x22, map_tree f x23\<rangle>"
assms_1: shows "kKS m \<notin> specKeysSecrets P"
funcset_mem: fixes f :: "'h \<Rightarrow> 'i"   and A :: "'h set"   and B :: "'i set"   and x :: "'h" assumes "f \<in> A \<rightarrow> B"   and "x \<in> A" shows "f x \<in> B"
bounded_linear_mult_right: fixes x :: "'c" shows "bounded_linear ((*) x)"
continuous_on_arcsin: fixes s :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "continuous_on s f"   and "\<forall>x\<in>s. - 1 \<le> f x \<and> f x \<le> 1" shows "continuous_on s (\<lambda>x. arcsin (f x))"
sorted_alist_product: fixes xs :: "('a \<times> 'f) list"   and ys :: "('b \<times> 'g) list"   and f :: "'a \<Rightarrow> 'f \<Rightarrow> 'b \<Rightarrow> 'g \<Rightarrow> 'h" assumes "linorder.sorted (\<sqsubseteq>\<^sub>a) (map fst xs)"   and "distinct (map fst xs)"   and "linorder.sorted (\<sqsubseteq>\<^sub>b) (map fst ys)" shows "linorder.sorted (less_eq_prod (\<sqsubseteq>\<^sub>a) (\<sqsubset>\<^sub>a) (\<sqsubseteq>\<^sub>b)) (map fst (alist_product f xs ys))"