hcju/isabelleps huggingface dataset:
assms_2: fixes v :: "'a" assumes "v \<in> S" shows "\<exists>\<sigma>. strategy p \<sigma> \<and> strategy_attracts_via p \<sigma> v S W"
b: fixes b :: "'a set" assumes "b \<in># remove1_mset bl \<B>"   and "ps \<subseteq> b - bl" shows "ps \<subseteq> b"
simps_2: fixes f :: "'a fps"   and a :: "'a"   and v :: "nat" shows "fps_right_inverse_constructor f a (Suc v) = - a * (\<Sum>i = 1..Suc v. fps_nth f i * fps_right_inverse_constructor f a (Suc v - i))"
prob_eq_1: fixes A :: "'b set"   and M :: "'b pmf" assumes "A \<in> measure_pmf.events M" shows "(measure_pmf.prob M A = 1) = (AE x in measure_pmf M. x \<in> A)"
K_p_eq: fixes x :: "'s"   and ya :: "'s"   and m :: "nat" shows "K2.p (Some x) (Some ya) m = p x ya m"
refBy_order: fixes Q :: "'a upred"   and P :: "'a upred" shows "(P \<sqsubseteq> Q) = `Q \<Rightarrow> P`"
finite_cone_mono: fixes Ws :: "'a vec set"   and Vs :: "'a vec set" assumes "finite Ws"   and "Ws \<subseteq> carrier_vec n"   and "Vs \<subseteq> Ws" shows "finite_cone Vs \<subseteq> finite_cone Ws"
reach_def: fixes \<Sigma> :: "'c set"   and \<delta> :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and q\<^sub>0 :: "'b" shows "reach \<Sigma> \<delta> q\<^sub>0 = {run \<delta> q\<^sub>0 w n |w n. range w \<subseteq> \<Sigma>}"
coeff_poly_cutoff: fixes n :: "nat"   and p :: "'a poly"   and k :: "nat" shows "coeff (poly_cutoff n p) k = (if k < n then coeff p k else (0::'a))"
returns_error_def: fixes h :: "'d"   and p :: "('d, 'e, 'f) prog"   and e :: "'e" shows "h \<turnstile> p \<rightarrow>\<^sub>e e = (case h \<turnstile> p of Inl e' \<Rightarrow> e = e' | Inr x \<Rightarrow> False)"
reduce_basis_def: fixes \<alpha> :: "rat"   and fs :: "int vec list" shows "LLL_Impl.reduce_basis \<alpha> fs = (case fs of [] \<Rightarrow> fs | f # x \<Rightarrow> fs_state (basis_reduction \<alpha> (dim_vec f) fs))"
lasso_rel_ext_def: fixes Re :: "('ah \<times> 'aj) set"   and R :: "('ag \<times> 'ai) set" shows "\<langle>Re, R\<rangle>lasso_rel_ext = {(\<lparr>lasso_reach = r', lasso_va = va', lasso_cysfx = cysfx', \<dots> = m'\<rparr>, \<lparr>lasso_reach = r, lasso_va = va, lasso_cysfx = cysfx, \<dots> = m\<rparr>) |r' r va' va cysfx' cysfx m' m. (r', r) \<in> \<langle>R\<rangle>list_rel \<and> (va', va) \<in> R \<and> (cysfx', cysfx) \<in> \<langle>R\<rangle>list_rel \<and> (m', m) \<in> Re}"
bitnot_one: shows "(\<not>\<^sub>b \<one>) = \<zero>"
flip_altsublist_adjacent_sum_list_ball: fixes S :: "'b set" assumes "PreCoxeterSystem S" shows "\<forall>ss\<in>lists S. \<forall>ts. PreCoxeterSystem.flip_altsublist_adjacent ss ts \<longrightarrow> sum_list ts = sum_list ss"
simps_2: fixes m :: "'a simple_match"   and rs :: "'a simple_rule list"   and p :: "('a, 'b) simple_packet_scheme" shows "simple_fw (SimpleRule m Accept # rs) p = (if simple_matches m p then Decision FinalAllow else simple_fw rs p)"
expand: fixes prefix_match :: "'b prefix_match"   and prefix_match' :: "'b prefix_match" assumes "pfxm_prefix prefix_match = pfxm_prefix prefix_match' \<and> pfxm_length prefix_match = pfxm_length prefix_match'" shows "prefix_match = prefix_match'"
steps_cases: fixes x :: "nat"   and y :: "nat"   and rel :: "(nat \<times> nat) set"   and n :: "nat" assumes "(x, y) \<in> steps rel (Suc n)" obtains "(x, y) \<in> steps rel n"   | "(x, n) \<in> steps rel n" and "(n, y) \<in> steps rel n"
simps: fixes s :: "'s" shows "local.terminates_within s = (case g s of Done \<Rightarrow> Some 0 | Skip s' \<Rightarrow> map_option (\<lambda>n. n + 1) (local.terminates_within s') | Yield a s' \<Rightarrow> map_option (\<lambda>n. n + 1) (local.terminates_within s'))"
cclfp_def: fixes f :: "'b \<Rightarrow> 'b" shows "cclfp f = (SUP i. (f ^^ i) bot)"
proots_upper_card_def: fixes p :: "complex poly" shows "proots_upper_card p = card (proots_within p {x. 0 < Im x})"
smallo_powr_iff: fixes p :: "real"   and f :: "'b \<Rightarrow> real"   and F :: "'b filter"   and g :: "'b \<Rightarrow> real" assumes "0 < p"   and "\<forall>\<^sub>F x in F. 0 \<le> f x"   and "\<forall>\<^sub>F x in F. 0 \<le> g x" shows "((\<lambda>x. f x powr p) \<in> o[F](\<lambda>x. g x powr p)) = (f \<in> o[F](g))"
rHom_mem: fixes f :: "'d \<Rightarrow> 'e"   and A :: "('d, 'f) Ring_scheme"   and R :: "('e, 'g) Ring_scheme"   and a :: "'d" assumes "f \<in> rHom A R"   and "a \<in> carrier A" shows "f a \<in> carrier R"
f_aggregate_take: fixes n :: "nat"   and xs :: "'b list"   and k :: "nat"   and ag :: "'b list \<Rightarrow> 'b" shows "f_aggregate (xs \<down> n) k ag = f_aggregate xs k ag \<down> (n div k)"
stieltjes_gamma_real_limit_form: fixes n :: "nat" shows "(\<lambda>m. (\<Sum>k = 1..m. ln (real k) ^ n / real k) - ln (real m) ^ (n + 1) / real (n + 1)) \<longlonglongrightarrow> stieltjes_gamma n"
IH: fixes x :: "pat"   and y :: "value"   and env :: "(name, value) fmap" assumes "name = name' \<and> length ps = length vs"   and "x \<in> set ps"   and "y \<in> set vs"   and "vmatch x y = Some env"   and "pred y" shows "fmpred (\<lambda>_. pred) env"
deg_monom: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and n :: "nat" assumes "UP_ring R"   and "a \<noteq> \<zero>\<^bsub>R\<^esub>"   and "a \<in> carrier R" shows "deg R (monom (UP R) a n) = n"
keys_m1_m2: shows "keys (monomial (lc p) (lt p) + monomial (tc q) (tt q)) = {lt p, tt q}"
height_le_size_tree: fixes t :: "'b tree" shows "height t \<le> size t"
s: shows "conv_abscissa f < ereal s1 \<and> s1 < s2 \<and> s2 < s \<bullet> (1::'a)"
SeqAuto_def: fixes S :: "'d set"   and I :: "'d"   and L :: "(('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) set"   and D :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set" shows "SeqAuto S I L D = (I \<in> S \<and> S \<noteq> {} \<and> finite S \<and> finite D \<and> (\<forall>(s, l, t)\<in>D. s \<in> S \<and> t \<in> S \<and> l \<in> L))"
components_are_iso: fixes a :: "'c \<times> 'c" assumes "C.VV.ide a" shows "C.iso (fst a \<star>\<^sub>C snd a)"
sandwich_type: fixes A :: "'o"   and B :: "'o"   and a :: "'a" assumes "A \<in> Ob"   and "B \<in> Ob"   and "a \<in> F \<^bsub>\<o>\<^esub> A" shows "\<sigma>(A,a) B \<in> Hom\<^bsub>Set\<^esub> (Hom A B) F \<^bsub>\<o>\<^esub> B"
eval: shows "(cms\<^sub>1, mem\<^sub>1) \<rightarrow> (cms\<^sub>1', mem\<^sub>1')"
nth_map_subst: fixes i :: "nat"   and T :: "('c, 'd) Term.term list"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term" assumes "i < length T" shows "(T \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<theta>) ! i = T ! i \<cdot> \<theta>"
r2f_inj_iff: fixes f :: "'a \<Rightarrow> 'b set"   and g :: "'a \<Rightarrow> 'b set" shows "(\<R> f = \<R> g) = (f = g)"
simps_1: fixes carrier :: "'c set"   and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and zero :: "'c"   and add :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and more :: "'d" shows "\<zero>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, zero = zero, add = add, \<dots> = more\<rparr>\<^esub> = zero"
lcoset_rel_sym: fixes G :: "'a set" assumes "Group G" shows "sym (lcoset_rel G)"
composite_is_tabulation: shows "tabulation (\<cdot>) (\<star>) \<a> \<i> src trg ((f \<star> g) \<star> h) THfgh.\<rho>\<sigma>.tab (tab\<^sub>0 h \<star> THfgh.p\<^sub>0) (tab\<^sub>1 (f \<star> g) \<star> THfgh.p\<^sub>1)"
i_def: shows "i \<equiv> set (\<Union> (elts ` rank ` A))"
AbstrLevels_S3_A21: fixes i :: "AbstrLevelsID" assumes "sS3 \<in> AbstrLevel i" shows "sA21 \<notin> AbstrLevel i"
indicator_scaleR_eq_if: fixes S :: "'c set"   and x :: "'c"   and f :: "'c \<Rightarrow> 'b" shows "indicat_real S x *\<^sub>R f x = (if x \<in> S then f x else (0::'b))"
T2: shows "\<lfloor>(\<lambda>w. \<P> (\<lambda>x. \<forall>xa. \<P> xa w \<longrightarrow> xa x) w)\<rfloor>"
Input: fixes a :: "name"   and x :: "name"   and P :: "pi"   and u :: "name" shows "a<x>.P \<Longrightarrow>\<^sup>^a<u> \<prec> P[x::=u]"
keys_filter: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "('b, 'c) mapping" shows "keys (Mapping.filter P m) \<subseteq> keys m"
span_tree_component: fixes t :: "'a"   and g :: "'a"   and r :: "'a" assumes "spanning_tree t g r" shows "component g r = component t r"
class_of_0_in: fixes U :: "'a set" assumes "U \<in> I" shows "\<zero>\<^bsub>U\<^esub> \<in> \<FF> U"
ereal_sqrt2: shows "ereal (sqrt 2) = esqrt 2"
assms_4: shows "g <# H1 #> inv g = g <# H2 #> inv g"
thetaParWL2_Wretr: shows "thetaParWL2 \<subseteq> Wretr (thetaParWL2 \<union> Wbis)"
idn_nmaps: fixes \<Gamma> :: "(name \<times> ty) list"   and a :: "coname"   and x :: "name"   and b :: "coname"   and M :: "trm" assumes "idn \<Gamma> a nmaps x to Some (b, M)" shows "M = Ax x b"
Int_Un_distrib: fixes A :: "'e set"   and B :: "'e set"   and C :: "'e set" shows "A \<inter> (B \<union> C) = A \<inter> B \<union> A \<inter> C"
g: shows "poly_of_vec g = q' * u + smult k r'"
tendsto_cong_limit: fixes f :: "'c \<Rightarrow> 'd"   and l :: "'d"   and F :: "'c filter"   and k :: "'d" assumes "(f \<longlongrightarrow> l) F"   and "k = l" shows "(f \<longlongrightarrow> k) F"
inv_push_forward_inverse: fixes X :: "('c \<Rightarrow> real) \<Rightarrow> real"   and p :: "'c" assumes "X \<in> dest.tangent_space p"   and "p \<in> dest.carrier" shows "push_forward (inv.push_forward X) = X"
lemSomeFunc: fixes P :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'c"   and f :: "'c \<Rightarrow> 'b" assumes "\<exists>y. P x y"   and "f = someFunc P" shows "P x (f x)"
bounded_by_const: fixes c :: "real"   and b :: "real" assumes "c \<le> b" shows "bounded_by b (\<lambda>x. c)"
P: shows "rel_fun R (rel_fun P (=)) (\<lambda>f x. f x = A) (\<lambda>V y. y \<in> V)"
simps_4: fixes f1 :: "'c"   and f2 :: "'c" shows "(case False of True \<Rightarrow> f1 | False \<Rightarrow> f2) = f2"
wtl_complete: fixes A :: "'a set"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "'a"   and B :: "'a"   and step :: "nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> 'a) list"   and \<tau>s :: "'a list"   and s :: "'a"   and ins :: "'b list" assumes "lbvc A r f T B step \<tau>s"   and "wt_step r T step \<tau>s"   and "s \<sqsubseteq>\<^bsub>r\<^esub> \<tau>s ! 0"   and "s \<in> A"   and "s \<noteq> T"   and "length ins = length \<tau>s" shows "wtl_inst_list ins (make_cert step \<tau>s B) f r T B step 0 s \<noteq> T"
select_convs_5: fixes iiface :: "iface"   and oiface :: "iface"   and src :: "'a word \<times> nat"   and dst :: "'a word \<times> nat"   and proto :: "protocol"   and sports :: "16 word \<times> 16 word"   and dports :: "16 word \<times> 16 word"   and more :: "'b" shows "proto \<lparr>iiface = iiface, oiface = oiface, src = src, dst = dst, proto = proto, sports = sports, dports = dports, \<dots> = more\<rparr> = proto"
order_def: fixes a :: "'c"   and p :: "'c poly" shows "order a p = (LEAST n. \<not> [:- a, 1::'c:] ^ Suc n dvd p)"
part_antisymmetry: fixes x :: "i"   and y :: "i" assumes "P x y"   and "P y x" shows "x = y"
subst_apply_pairs_pset_subst: fixes F :: "(('d, 'f) Term.term \<times> ('d, 'f) Term.term) list"   and \<theta> :: "'f \<Rightarrow> ('d, 'e) Term.term" shows "set (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<theta>) = set F \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<theta>"
prefix_imp_pfixLe: fixes xs :: "nat list"   and ys :: "nat list" assumes "xs \<le> ys" shows "xs pfixLe ys"
dvd_lcmI1: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a b" shows "dvd.dvd times a (lcm b c)"
sdrop_snth: fixes n :: "nat"   and s :: "'b stream"   and m :: "nat" shows "sdrop n s !! m = s !! (n + m)"
i_2: shows "d / real n \<le> \<bar>(f z - z) \<bullet> i\<bar>"
step: fixes x :: "'a"   and y :: "'b" assumes "orda x (ccpo.fixp luba orda f)"   and "ordb y (ccpo.fixp lubb ordb g)"   and "P x y" shows "P (f x) (g y)"
cA: fixes c :: "'a" assumes "extreme_bound B (\<sqsubseteq>) B.C c" shows "c \<in> B"
Inf_V_def: fixes X :: "V set" shows "\<Sqinter> X \<equiv> if X = {} \<Rightarrow> 0 | otherwise \<Rightarrow> set (\<Inter> (elts ` X))"
lift_closed: fixes f :: "'d \<Rightarrow> 'a"   and gens :: "'d set"   and x :: "(bool \<times> 'd) list" assumes "f \<in> gens \<rightarrow> carrier G"   and "x \<in> lists (UNIV \<times> gens)" shows "lift f x \<in> carrier G"
that: fixes xs :: "'a list" assumes "xs' = filter P xs"   and "distinct xs"   and "set xs = S" shows "thesis"
16: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<squnion> z \<sqinter> y = (x \<squnion> z) \<sqinter> y"
const_exists_def: fixes c :: "'b" shows "const_exists c \<equiv> transl_rule (\<top> \<sqsubseteq> A_Cmp (A_Cmp \<top> (A_Lbl (S_Const c))) \<top>)"
subch: fixes R :: "('a \<times> 'a) set set" assumes "R \<in> Chains I" shows "chain\<^sub>\<subseteq> R"
tappend_TCons: fixes a :: "'a"   and tr :: "('a, 'c) tllist"   and f :: "'c \<Rightarrow> ('a, 'b) tllist" shows "tappend (TCons a tr) f = TCons a (tappend tr f)"
blinfun_of_matrix_apply: fixes a :: "'d \<Rightarrow> 'e \<Rightarrow> real"   and x :: "'e" shows "blinfun_apply (blinfun_of_matrix a) x = (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. (x \<bullet> j * a i j) *\<^sub>R i)"
fds_totient: shows "fds (\<lambda>n. of_nat (totient n)) = fds of_nat * fds moebius_mu"
assms_2: shows "conv_abscissa f < ereal (g s \<bullet> (1::'a))"
ringhom1: fixes A :: "('d, 'e) Ring_scheme"   and R :: "('f, 'g) Ring_scheme"   and x :: "'d"   and y :: "'d"   and f :: "'d \<Rightarrow> 'f" assumes "Ring A"   and "Ring R"   and "x \<in> carrier A"   and "y \<in> carrier A"   and "f \<in> rHom A R" shows "f (x \<plusminus>\<^bsub>A\<^esub> y) = f x \<plusminus>\<^bsub>R\<^esub> f y"
assms_3: fixes zs :: "'varSort"   and z :: "'var" shows "freshEnv zs z rho \<or> fresh zs z X"
predfl: shows "pred_cof L' ((f(l \<mapsto> t)) l) (f' l) l"
mod_lemma: fixes x :: "nat"   and y :: "nat" shows "rec_exec rec_mod [x, y] = x mod y"
that: shows "Guard f b c \<subseteq>\<^sub>g d"
set_integral_space: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "integrable M f" shows "set_lebesgue_integral M (space M) f = integral\<^sup>L M f"
lens_prod_def: fixes X :: "'d \<Longrightarrow> 'f"   and Y :: "'e \<Longrightarrow> 'g" shows "X \<times>\<^sub>L Y = \<lparr>lens_get = map_prod get\<^bsub>X\<^esub> get\<^bsub>Y\<^esub>, lens_put = \<lambda>(u, v) (x, y). (put\<^bsub>X\<^esub> u x, put\<^bsub>Y\<^esub> v y)\<rparr>"
un: shows "Y -` (A \<inter> range Y) = \<Union> (range prY)"
assms: shows "N2 t1 a t2 \<in> B (Suc n)"
2: shows "fmlookup (action_needed_asses a s) v = fmlookup s v"
isolate_islimpt_iff: fixes x :: "'c"   and S :: "'c set" shows "x isolate S = (\<not> x islimpt S \<and> x \<in> S)"
Rep_assn_inverse: fixes x :: "assn" shows "Abs_assn (Rep_assn x) = x"
Pbij_inverseI: fixes l1 :: "Location"   and l2 :: "Location"   and \<beta> :: "(Location \<times> Location) set" assumes "(l1, l2) \<in> \<beta>" shows "(l2, l1) \<in> Pbij_inverse \<beta>"
assms_1: fixes n :: "nat" shows "u n \<in> space\<^sub>N N"
vec0: fixes nr :: "nat"   and ze :: "'c" shows "vec nr (vec0I ze nr)"
dX_def: fixes X :: "'x set"   and d :: "'d" shows "dX X d = {x \<in> X. Xd x = d}"
volume_positive: fixes M :: "'b set set"   and f :: "'b set \<Rightarrow> ennreal"   and a :: "'b set" assumes "volume M f"   and "a \<in> M" shows "0 \<le> f a"
rw: shows "P' * (P * A * Q) * Q' = A"
intpow_pos: fixes n :: "int"   and x :: "'b" assumes "0 \<le> n" shows "x ^\<^sub>Z n = x ^ nat n"
blinfun_eqI: fixes x :: "'c \<Rightarrow>\<^sub>L 'd"   and y :: "'c \<Rightarrow>\<^sub>L 'd" assumes "\<And>i. blinfun_apply x i = blinfun_apply y i" shows "x = y"
Arg2pi_Ln: fixes z :: "complex" assumes "0 < Arg2pi z" shows "Arg2pi z = Im (Ln (- z)) + pi"
loose_linkable_bounded: assumes "loose (quotient_web \<Gamma> f)" shows "linkable (quotient_web \<Gamma> f)"
complete_ht_delete: fixes l :: "('a \<times> 'b) list list"   and ht :: "('a, 'b) hashtable"   and k :: "'a" shows "<is_hashtable l ht> hm_delete k ht <is_hashtable (l[bounded_hashcode_nat (length l) k := fst (ls_delete k (l ! bounded_hashcode_nat (length l) k))])>"
"0_2": shows "list_all is_Assignment (unlabel (transaction_selects T))"
positive_outer_measure: fixes f :: "'a set \<Rightarrow> ennreal" assumes "positive M f" shows "positive (Pow \<Omega>) (outer_measure M f)"
dom_eqI: fixes a :: "'b"   and f :: "'b" assumes "B.ide a"   and "B.seq f a" shows "B.dom f = a"
add_max: fixes ns :: "nat list" assumes "ns \<noteq> []" shows "Max (set ns) \<le> foldl (+) 0 ns"
checker_r_def: fixes s\<^sub>e :: "real"   and v\<^sub>e :: "real"   and a\<^sub>e :: "real"   and s\<^sub>o :: "real"   and v\<^sub>o :: "real"   and a\<^sub>o :: "real"   and \<delta> :: "real" shows "checker_r s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o \<delta> \<equiv> let distance = s\<^sub>o - s\<^sub>e; precond = check_precond_r s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o \<delta>; vo_star = v\<^sub>o + a\<^sub>o * \<delta>; t_stop_o_star = - vo_star / a\<^sub>o; t_stop_e = - v\<^sub>e / a\<^sub>e; t_stop_o = - v\<^sub>o / a\<^sub>o; safe_dist0 = safe_distance_1r a\<^sub>e v\<^sub>e \<delta>; safe_dist1 = safe_distance_2r a\<^sub>e v\<^sub>e a\<^sub>o v\<^sub>o \<delta>; safe_dist2 = safe_distance_4r a\<^sub>e v\<^sub>e a\<^sub>o v\<^sub>o \<delta>; safe_dist3 = safe_distance_3r a\<^sub>e v\<^sub>e a\<^sub>o v\<^sub>o \<delta> in if \<not> precond then False else if safe_dist0 < distance then True else if \<delta> \<le> t_stop_o \<and> safe_dist3 < distance then True else if \<delta> \<le> t_stop_o \<and> a\<^sub>e < a\<^sub>o \<and> vo_star < v\<^sub>e \<and> t_stop_e < t_stop_o_star then safe_dist2 < distance else safe_dist1 < distance"
H: shows "\<turnstile> fps [\<rhd>] (l, t) \<Colon> fs \<Rightarrow> ts"
listToFuns_ext: fixes xs :: "'b list"   and g :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c list" assumes "set xs = UNIV"   and "distinct xs" shows "(g \<in> set (listToFuns f xs)) = (\<forall>x. g x \<in> set (f x))"
empty_no_progress: fixes es :: "e list"   and s :: "s"   and vs :: "v list"   and i :: "nat"   and s' :: "s"   and vs' :: "v list"   and es' :: "e list" assumes "es = []" shows "\<not> \<lparr>s;vs;es\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"
cInf_cSup: fixes S :: "'a multiset set" assumes "S \<noteq> {}"   and "subset_mset.bdd_below S" shows "\<Sqinter> S = \<Squnion> {x. \<forall>s\<in>S. x \<subseteq># s}"
group_l_invI: assumes "\<And>x. x \<in> carrier G \<Longrightarrow> \<exists>y\<in>carrier G. y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>" shows "Group.group G"
insert_UNIV: fixes x :: "'a" shows "insert x UNIV = UNIV"
Functor_def: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme" shows "Functor F \<equiv> FunctorM F \<and> FunctorExt F"
abs_eq: fixes x :: "int" shows "uint64_of_int x = Abs_uint64 (word_of_int x)"
C: shows "set_mset K \<subseteq> r \<down>m J"
continuous_pow_A_1: fixes R :: "(real, 'n) Finite_Cartesian_Product.vec set" shows "continuous_on R pow_A_1"
s_Unauth: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and m :: "mode"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, e\<ggreater> m\<rightarrow> \<lless>\<pi>', e'\<ggreater>" shows "\<lless>\<pi>, Unauth e\<ggreater> m\<rightarrow> \<lless>\<pi>', Unauth e'\<ggreater>"
find_closest_dist: fixes p :: "int \<times> int"   and ps :: "(int \<times> int) list"   and \<delta> :: "real" assumes "sorted_snd (p # ps)"   and "\<exists>q\<in>set ps. dist (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case q of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) < \<delta>" shows "\<forall>q\<in>set ps. dist (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case find_closest p \<delta> ps of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) \<le> dist (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case q of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))"
map_add_ac_3: fixes h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option" shows "h\<^sub>0 \<bottom> h\<^sub>1 = h\<^sub>1 \<bottom> h\<^sub>0"
st2d1: shows "prod_list st2.dims1 = 2 ^ Suc k"
ref_union_closed_def: fixes P :: "'c process" shows "ref_union_closed P \<equiv> \<forall>xs A. (\<exists>X. X \<in> A) \<longrightarrow> (\<forall>X\<in>A. (xs, X) \<in> failures P) \<longrightarrow> (xs, \<Union>X\<in>A. X) \<in> failures P"
order_axioms_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "class.order_axioms less_eq \<equiv> \<forall>x y. less_eq x y \<longrightarrow> less_eq y x \<longrightarrow> x = y"
flow_funcset_ex_ivl: fixes s :: "'a" assumes "s \<in> S" shows "flow t\<^sub>0 s \<in> ex_ivl s \<rightarrow> S"
bad_guards: fixes t1 :: "transition"   and r :: "nat \<Rightarrow>f value option"   and t2 :: "transition" assumes "\<exists>i. can_take_transition t1 i r \<and> \<not> can_take_transition t2 i r" shows "\<not> subsumes t2 r t1"
simps_15: fixes M :: "char list"   and n :: "nat"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and pc :: "nat"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and ST :: "ty list"   and LT :: "ty err list" shows "app\<^sub>i (Invoke M n, P, pc, mxs, T\<^sub>r, ST, LT) = (n < length ST \<and> (ST ! n \<noteq> NT \<longrightarrow> (\<exists>C D Ts T m. ST ! n = Class C \<and> P \<turnstile> C sees M: Ts\<rightarrow>T = m in D \<and> P \<turnstile> rev (take n ST) [\<le>] Ts)))"
V_def: fixes i :: "nat" shows "V i = inv_chart (\<psi> i) ` ball (0::'b) 2"
assum: shows "(a, b) \<in> set (smash_list_list l1 l2)"
at_map_inf_pres: fixes x :: "'a"   and y :: "'a" shows "at_map (x \<sqinter> y) = at_map x \<sqinter> at_map y"
region_V: fixes R :: "('c \<Rightarrow> real) set" assumes "R \<in> \<R>" shows "R \<subseteq> V"
relcomp_subset_left_imp_relcomp_trancl_subset_left: fixes R :: "('b \<times> 'c) set"   and S :: "('c \<times> 'c) set" assumes "R O S \<subseteq> R" shows "R O S\<^sup>* \<subseteq> R"
fine_def: fixes d :: "'b \<Rightarrow> 'c set"   and s :: "('b \<times> 'c set) set" shows "(d fine s) = (\<forall>(x, k)\<in>s. k \<subseteq> d x)"
post_meas1_bell01_fst: shows "post_meas1 2 |\<beta>\<^sub>0\<^sub>1\<rangle> 0 = |unit_vec 4 2\<rangle>"
content_eq_0: fixes a :: "'b"   and b :: "'b" shows "(content (box a b) = 0) = (\<exists>i\<in>Basis. b \<bullet> i \<le> a \<bullet> i)"
assms_2: shows "monoid_homomorphism \<theta> B multB oneB C multC oneC"
Form_imp_is_fm: fixes x :: "hf"   and e :: "(name, hf) finfun" assumes "Form x" obtains A :: "fm" where "x = \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e"
downclosed_set_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'b set" shows "ord.downclosed_set less_eq X = (X \<in> ord.downsets less_eq)"
unitary_gen_iff': fixes M :: "complex \<times> complex \<times> complex \<times> complex" shows "unitary_gen M = (\<exists>k. k \<noteq> 0 \<and> congruence M (1, 0, 0, 1) = k *\<^sub>s\<^sub>m (1, 0, 0, 1))"
expand: fixes lit :: "lit"   and lit' :: "lit" assumes "pos lit = pos lit' \<and> var lit = var lit'" shows "lit = lit'"
distinct_191: fixes x111 :: "char list"   and x112 :: "ty"   and x113 :: "expr"   and x121 :: "expr"   and x122 :: "expr" shows "{x111:x112; x113} \<noteq> x121;; x122"
inverse_zero_imp_zero: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "inverse a = zero" shows "a = zero"