hcju/isabelleps huggingface dataset:
space_PiM_empty_iff: fixes I :: "'a set"   and M :: "'a \<Rightarrow> 'b measure" shows "(space (Pi\<^sub>M I M) = {}) = (\<exists>i\<in>I. space (M i) = {})"
bit_double_iff: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'a"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "bit (times (numeral.numeral one plus (num.Bit0 num.One)) a) n = (bit a (n - 1) \<and> n \<noteq> 0 \<and> power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n \<noteq> zero)"
publicKey_image_eq: fixes b :: "keymode"   and x :: "agent"   and c :: "keymode"   and AA :: "agent set" shows "(publicKey b x \<in> publicKey c ` AA) = (b = c \<and> x \<in> AA)"
iterate_rule_insert_P: fixes m :: "'s"   and I :: "('k \<times> 'v) set \<Rightarrow> 'a \<Rightarrow> bool"   and \<sigma>0 :: "'a"   and f :: "'k \<times> 'v \<Rightarrow> 'a \<Rightarrow> 'a"   and P :: "'a \<Rightarrow> bool" assumes "invar m"   and "I {} \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>(k, v) \<in> map_to_set (\<alpha> m) - it; it \<subseteq> map_to_set (\<alpha> m); I it \<sigma>\<rbrakk> \<Longrightarrow> I (insert (k, v) it) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I (map_to_set (\<alpha> m)) \<sigma> \<Longrightarrow> P \<sigma>" shows "P (iterate m f \<sigma>0)"
upt_eq_Nil_conv: fixes i :: "nat"   and j :: "nat" shows "([i..<j] = []) = (j = 0 \<or> j \<le> i)"
comp_assoc_assoc'_2: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "\<a>\<^sub>D\<^sup>-\<^sup>1[f, g, h] \<cdot>\<^sub>D \<a>\<^sub>D[f, g, h] = (f \<star>\<^sub>D g) \<star>\<^sub>D h"
that_2: fixes f :: "'b \<Rightarrow>f 'a"   and a :: "'b"   and b :: "'a" assumes "x = f(a $:= b)" shows "thesis"
ext_inject: fixes red :: "('d \<times> nat) rgraph"   and black :: "('d, 'e, 'f) lts"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and init_conf :: "('e, 'f) conf"   and confs :: "'d \<times> nat \<Rightarrow> ('e, 'f) conf"   and marked :: "'d \<times> nat \<Rightarrow> bool"   and strengthenings :: "'d \<times> nat \<Rightarrow> ('e \<Rightarrow> 'f) \<Rightarrow> bool"   and more :: "'g"   and red' :: "('d \<times> nat) rgraph"   and black' :: "('d, 'e, 'f) lts"   and subs' :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and init_conf' :: "('e, 'f) conf"   and confs' :: "'d \<times> nat \<Rightarrow> ('e, 'f) conf"   and marked' :: "'d \<times> nat \<Rightarrow> bool"   and strengthenings' :: "'d \<times> nat \<Rightarrow> ('e \<Rightarrow> 'f) \<Rightarrow> bool"   and more' :: "'g" shows "(\<lparr>red = red, black = black, subs = subs, init_conf = init_conf, confs = confs, marked = marked, strengthenings = strengthenings, \<dots> = more\<rparr> = \<lparr>red = red', black = black', subs = subs', init_conf = init_conf', confs = confs', marked = marked', strengthenings = strengthenings', \<dots> = more'\<rparr>) = (red = red' \<and> black = black' \<and> subs = subs' \<and> init_conf = init_conf' \<and> confs = confs' \<and> marked = marked' \<and> strengthenings = strengthenings' \<and> more = more')"
assms_3: shows "\<guillemotleft>\<eta> : src\<^sub>C f \<Rightarrow>\<^sub>C g \<star>\<^sub>C f\<guillemotright>"
simps_2: fixes s :: "val option"   and r :: "proc \<times> val"   and rs :: "(proc \<times> val) list" shows "s \<star> Cons r rs = (s \<star> rs) \<bullet> r"
to: shows "h \<turnstile> to_tree_order (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr) \<rightarrow>\<^sub>r to"
subst_apps: fixes \<rho> :: "'d \<Rightarrow> ('c, 'd) tm"   and s :: "('c, 'd) tm"   and ts :: "('c, 'd) tm list" shows "subst \<rho> (apps s ts) = apps (subst \<rho> s) (map (subst \<rho>) ts)"
star_n_export1: fixes x :: "'a"   and y :: "'a" shows "(!x \<cdot> y)\<^sup>\<star> \<cdot> !x \<le> !x \<cdot> y\<^sup>\<star>"
sum_incoming_alt: fixes g :: "nat \<times> nat \<Rightarrow> 'capacity" assumes "\<forall>e. (0::'capacity) \<le> g e \<and> g e \<le> c e" shows "\<forall>v\<in>V. sum g (incoming v) = (\<Sum>u\<in>V. g (u, v))"
endpoint_in_S_incident: fixes a :: "proj2"   and b :: "proj2"   and l :: "proj2_line" assumes "a \<noteq> b"   and "a \<in> hyp2 \<union> S"   and "b \<in> hyp2 \<union> S"   and "proj2_incident a l"   and "proj2_incident b l" shows "proj2_incident (endpoint_in_S a b) l"
distinct_3: fixes x1 :: "real" shows "ereal x1 \<noteq> MInfty"
R_abs_R1: shows "R \<circ> R1 = R"
fst_not_one_snd: fixes a :: "('e, 'f) OrdSum" assumes "fst (Rep_OrdSum a) \<noteq> (1::'e)" shows "snd (Rep_OrdSum a) = (1::'f)"
vnz: shows "lincomb l2 W \<noteq> 0\<^sub>v n"
matrix_vector_mul_linear_gen: fixes A :: "(('a, 'b) vec, 'c) vec" shows "linear (*s) (*s) ((*v) A)"
overlap_monotonicity: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "PM"   and "P x y"   and "O z x" shows "O z y"
ActualI: fixes \<phi> :: "\<o>"   and v :: "i" assumes "[\<phi> in dw]" shows "[\<^bold>\<A>\<phi> in v]"
simps_1: fixes w :: "nat \<Rightarrow> 'b" shows "w \<Turnstile>\<^sub>p false\<^sub>p = False"
leftmost_skip_prefix: fixes i :: "nat"   and u :: "symbol list"   and v :: "symbol list" assumes "leftmost i (u @ v)"   and "length u \<le> i" shows "leftmost (i - length u) v"
rts2a: shows "\<forall>p q t i. correct p t \<and> correct q t \<and> te q i + \<beta> \<le> t \<longrightarrow> te p i \<le> t"
bvars\<^sub>s\<^sub>s\<^sub>t_NegChecks: fixes A :: "('f, 'e) stateful_strand_step list" shows "bvars\<^sub>s\<^sub>s\<^sub>t A = bvars\<^sub>s\<^sub>s\<^sub>t (filter is_NegChecks A)"
simps: fixes z :: "'e list"   and W :: "'e list set" shows "mcp' z W = The (mcp z W)"
square_final_mat: shows "square_mat (get_matrix_R (reduce_system_R p (qs, M_mat_R signs subsets, subsets, signs)))"
seqE: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "H.VV.seq g f" obtains "H.VV.arr f" and "H.VV.arr g" and "H.VV.dom g = H.VV.cod f"
poly_subst_extend_indets: fixes f :: "'d option \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "poly_subst f (extend_indets p) = poly_subst (f \<circ> Some) p"
basis_for_obtain_unique_scalars: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and vs :: "'b list"   and v :: "'b" assumes "VectorSpace smult V"   and "set vs \<subseteq> V \<and> V = R_scalar_mult.RSpan UNIV smult vs \<and> R_scalar_mult.R_lin_independent UNIV smult vs"   and "v \<in> V" shows "\<exists>!as. length as = length vs \<and> v = scalar_mult.lincomb smult as vs"
nstd_lemma: fixes c :: "nat" shows "rec_exec rec_NSTD [c] = nstd c"
add_pred_def: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and P :: "'d set"   and \<S> :: "'a set \<times> 'a set"   and e :: "'d" assumes "sifum_lang_no_dma eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars" shows "sifum_lang_no_dma.add_pred bexp_vars P \<S> e \<equiv> if \<forall>x\<in>bexp_vars e. sifum_lang_no_dma.stable \<S> x then P \<union> {e} else P"
Two_map: shows "Two \<in> set (map (h1 k 0) js)"
prod_eqI: fixes p :: "'d \<times> 'e"   and q :: "'d \<times> 'e" assumes "fst p = fst q"   and "snd p = snd q" shows "p = q"
writes_def: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem" shows "IFC_def.writes prob = ifc_problem.writes prob"
normalize_mult_unit_factor: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "times (normalize a) (unit_factor a) = a"
poincare_map_plane_has_derivative: fixes S :: "'a set"   and i :: "'a"   and c :: "real"   and x :: "'a" assumes "returns_to {x \<in> S. x \<bullet> i = c} x"   and "closed S"   and "f (poincare_map {x \<in> S. x \<bullet> i = c} x) \<bullet> i \<noteq> 0"   and "\<forall>\<^sub>F x in at (poincare_map {x \<in> S. x \<bullet> i = c} x). x \<bullet> i = c \<longrightarrow> x \<in> S"   and "x \<notin> S \<or> x \<bullet> i \<noteq> c" shows "(poincare_map {x \<in> S. x \<bullet> i = c} has_derivative blinfun_apply (Dpoincare_map' i c S x)) (at x)"
progress_MatchF_le: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and I :: "\<I>"   and r :: "Formula.formula Regex.regex"   and j :: "nat" shows "Monitor.progress \<sigma> P (formula.MatchF I r) j \<le> progress_regex \<sigma> P r j"
iT_Plus_def: fixes I :: "nat set"   and k :: "nat" shows "I \<oplus> k \<equiv> (\<lambda>n. n + k) ` I"
chaum_ped_asymp_axioms: shows "chaum_ped_asymp \<G>"
nonpos_Ints_subset_nonpos_Reals: shows "\<int>\<^sub>\<le>\<^sub>0 \<subseteq> \<real>\<^sub>\<le>\<^sub>0"
nz: fixes t :: "real" assumes "t \<in> existence_ivl0 x"   and "0 \<le> t" shows "0 < max (- V' (flow0 x t) (f (flow0 x t))) ((V (flow0 x t))\<^sup>2)"
diff_fun_c_diff: fixes b :: "'a \<Rightarrow> real"   and x :: "real" assumes "b \<in> diff_fun_space" shows "(\<lambda>x. b (c x)) differentiable at x"
ns: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in># X"   and "y \<in># Y"   and "z \<in># Z"   and "(x, y) \<in> ns"   and "(y, z) \<in> s" shows "(x, z) \<in> s"
groupGSi: shows "Group.group (G\<lparr>carrier := \<GG> ! (i + 1)\<rparr>)"
subtermeq_vars_subset: fixes M :: "('a, 'b) Term.term"   and N :: "('a, 'b) Term.term" assumes "M \<sqsubseteq> N" shows "fv M \<subseteq> fv N"
left_distrib: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<cdot> (y \<squnion> z) = x \<cdot> y \<squnion> x \<cdot> z"
igFreshCls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'e, 'f, 'g, 'h) model_scheme" shows "igFreshCls MOD \<equiv> igFreshIGVar MOD \<and> igFreshIGAbs1 MOD \<and> igFreshIGAbs2 MOD \<and> igFreshIGOp MOD"
content_std_simplex: shows "content (convex hull insert (0::'a) Basis) = 1 / fact DIM('a)"
tensor_one_from_lookup: shows "1 = tensor_from_lookup [] (\<lambda>_. 1::'a)"
set_restriction_fun_def: fixes P :: "'b \<Rightarrow> bool" shows "set_restriction_fun P \<equiv> \<lambda>A. {x \<in> A. P x}"
Nset_pre: fixes x :: "nat"   and n :: "nat" assumes "x \<in> {i. i \<le> Suc n}"   and "x \<noteq> Suc n" shows "x \<in> {i. i \<le> n}"
const_def: shows "rs.const = (\<lambda>x. if x \<in> Field s then zero r else undefined)"
rtrancl_eq_or_trancl: fixes x :: "'a"   and y :: "'a"   and R :: "('a \<times> 'a) set" shows "((x, y) \<in> R\<^sup>*) = (x = y \<or> x \<noteq> y \<and> (x, y) \<in> R\<^sup>+)"
assms_2: shows "\<rho>' \<equiv> run\<^sub>t (\<delta>\<^sub>m k) q\<^sub>0 w"
share_all_until_volatile_write_flush_commute: fixes sb :: "'b memref list"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and \<S> :: "nat \<Rightarrow> bool option"   and i :: "nat"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and sb' :: "'b memref list"   and R :: "nat set"   and L :: "nat set"   and A :: "nat set"   and p' :: "'b"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and \<D>' :: "bool"   and \<O>' :: "nat set"   and \<R>' :: "nat \<Rightarrow> bool option"   and W :: "nat set" assumes "takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb = []"   and "ownership_distinct ts"   and "sharing_consis \<S> ts"   and "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "\<forall>i p is \<O> \<R> \<D> \<theta> sb. i < length ts \<longrightarrow> ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> (all_shared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<union> all_unshared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<union> all_acquired (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)) \<inter> (all_shared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb') \<union> all_unshared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb') \<union> all_acquired (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb')) = {}"   and "\<forall>j p is \<O> \<R> \<D> \<theta> sb. j < length ts \<longrightarrow> i \<noteq> j \<longrightarrow> ts ! j = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> (all_shared sb \<union> all_unshared sb \<union> all_acquired sb) \<inter> (R \<union> L \<union> A) = {}" shows "share_all_until_volatile_write (ts[i := (p', is', \<theta>', sb', \<D>', \<O>', \<R>')]) (\<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L) = share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb') (share_all_until_volatile_write ts \<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L)"
gcd_coprime: fixes a :: "'b"   and b :: "'b"   and a' :: "'b"   and b' :: "'b" assumes "gcd a b \<noteq> (0::'b)"   and "a = a' * gcd a b"   and "b = b' * gcd a b" shows "coprime a' b'"
GlobalIncomingInfo_def: fixes c :: "('d, 'c) configuration"   and k :: "nat"   and p :: "'d"   and q :: "'d" shows "GlobalIncomingInfo c k p q = (\<Sum>p'\<in>UNIV. IncomingInfo c (if p' = p then k else 0) p' q)"
extensionalityI: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and g :: "'d \<Rightarrow> 'e" assumes "f \<in> extensional A"   and "g \<in> extensional A"   and "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "f = g"
conv_dist_sup: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y)\<^sup>T = x\<^sup>T \<squnion> y\<^sup>T"
image_fold_insert: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "finite A" shows "f ` A = Finite_Set.fold (\<lambda>k. insert (f k)) {} A"
le_eqvt: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "perm"   and x :: "'a"   and y :: "'a" assumes "class.le_eqvt less_eq less" shows "p \<bullet> (x \<le> y) = (p \<bullet> x \<le> p \<bullet> y)"
that: fixes e :: "real" assumes "0 < e"   and "ball x e \<subseteq> S" shows "thesis"
zeros_is_alpha: fixes v :: "nat"   and n :: "nat" shows "list_all (is_alph v) (zeros n v)"
inner_image_cbox: assumes "1 \<in> Basis"   and "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i" shows "(\<lambda>x. x \<bullet> 1) ` cbox a b = {a \<bullet> 1..b \<bullet> 1}"
rep_eq: fixes x :: "real" shows "(*\<^sub>v) (blinfun_scaleR_right x) = (*\<^sub>R) x"
read_only_unacquired_share: fixes \<O> :: "nat set"   and S :: "nat \<Rightarrow> bool option"   and sb :: "'b memref list"   and a :: "nat" assumes "\<O> \<inter> read_only S = {}"   and "weak_sharing_consistent \<O> sb"   and "a \<in> read_only S"   and "a \<notin> all_acquired sb" shows "a \<in> read_only (share sb S)"
abs_summable_on_empty: fixes f :: "'a \<Rightarrow> 'b" shows "f abs_summable_on {}"
abs_eq: fixes x :: "'d" shows "blinfun_scaleR_left x = Blinfun (\<lambda>y. y *\<^sub>R x)"
part_circlepath_mirror: fixes a' :: "real"   and a :: "real"   and k :: "int"   and b' :: "real"   and b :: "real"   and c' :: "complex"   and c :: "complex"   and r :: "real" assumes "a' = a + pi + 2 * pi * real_of_int k"   and "b' = b + pi + 2 * pi * real_of_int k"   and "c' = - c" shows "- part_circlepath c r a b = part_circlepath c' r a' b'"
cases: fixes a1 :: "'V"   and a2 :: "'V set \<times> ('V \<Rightarrow> ('V \<times> 'W \<times> 'V) list option)"   and a3 :: "'V set \<times> ('V \<Rightarrow> ('V \<times> 'W \<times> 'V) list option)" assumes "update_spec a1 a2 a3" obtains res' :: "'V \<Rightarrow> ('V \<times> 'W \<times> 'V) list option" and res :: "'V \<Rightarrow> ('V \<times> 'W \<times> 'V) list option" and v :: "'V" and wl :: "'V set" where "a1 = v" and "a2 = (wl, res)" and "a3 = (wl, res')" and "\<forall>v'\<in>V. res' v' \<in> least_map path_weight' ({res v'} \<union> {Some (p @ [(v, w, v')]) |p w. res v = Some p \<and> (v, w, v') \<in> E})"
comp_termFSbImorph: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'g"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'h"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'g, 'h, 'i) model_scheme"   and h' :: "'g \<Rightarrow> 'j"   and hA' :: "'h \<Rightarrow> 'k"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'j, 'k, 'l) model_scheme" assumes "termFSbImorph h hA MOD"   and "FSbImorph h' hA' MOD MOD'" shows "termFSbImorph (h' \<circ> h) (hA' \<circ> hA) MOD'"
tBD_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and grouping :: "int"   and index :: "int"   and sizes :: "int set" shows "tBD point_set block_collection grouping index sizes \<equiv> t_wise_balance point_set block_collection grouping index \<and> K_block_design point_set block_collection sizes \<and> tBD_axioms grouping sizes"
timpls_transformable_to'_iff_in_timpl_closure: fixes TI :: "('c set \<times> 'c set) list"   and s :: "(('d, 'e, 'c) prot_fun, 'f) Term.term"   and t :: "(('d, 'e, 'c) prot_fun, 'f) Term.term" shows "timpls_transformable_to' TI s t = (t \<in> timpl_closure s (set TI))"
ACI_nullable: fixes r :: "'b rexp"   and s :: "'b rexp" assumes "ACI r s" shows "nullable r = nullable s"
d_above: fixes x :: "'a"   and s :: "real" assumes "x \<in> {a--b}"   and "0 < s"   and "s \<le> d" shows "0 < (flow0 x s - x) \<bullet> rot n"
le_less_trans: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'f"   and y :: "'f"   and z :: "'f" assumes "class.preorder less_eq less"   and "less_eq x y"   and "less y z" shows "less x z"
llast_LCons2: fixes x :: "'b"   and y :: "'b"   and xs :: "'b llist" shows "llast (LCons x (LCons y xs)) = llast (LCons y xs)"
ball_eq_ball_iff: fixes x :: "'b"   and d :: "real"   and y :: "'b"   and e :: "real" shows "(ball x d = ball y e) = (d \<le> 0 \<and> e \<le> 0 \<or> x = y \<and> d = e)"
assms_3: shows "t_packing_design V B (int k) t \<Lambda>\<^sub>t"
self_in_downset: fixes x :: "'b" shows "x \<in> \<up>x"
tendsto_within_open: fixes a :: "'c"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and l :: "'d" assumes "a \<in> S"   and "open S" shows "(f \<longlongrightarrow> l) (at a within S) = f \<midarrow>a\<rightarrow> l"
rc: fixes s :: "'a"   and t :: "'a"   and a :: "'b"   and u :: "'a"   and b :: "'b" assumes "(s, t) \<in> (E a)\<inverse>"   and "(s, u) \<in> R b" shows "\<exists>xs. local.rcliff r a b xs \<and> (t, u) \<in> local.lconv xs"
ordLeq_total: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "Well_order r"   and "Well_order r'" shows "r \<le>o r' \<or> r' \<le>o r"
iMOD_div: fixes k :: "nat"   and m :: "nat"   and r :: "nat" assumes "0 < k"   and "m mod k = 0" shows "[ r, mod m ] \<oslash> k = [ r div k, mod m div k ]"
lcs_comm_fun: fixes s :: "'a \<Rightarrow> 'b"   and t :: "'a \<Rightarrow> 'b" shows "lcs s t = lcs t s"
restrictD: fixes G :: "('a, 'b) labeled_graph" assumes "graph G" shows "edges G \<subseteq> {(l, v1, v2). v1 \<in> vertices G \<and> v2 \<in> vertices G}"
carrier_AnalBinds_below: fixes \<Delta> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and a :: "Arity"   and x :: "Vars.var" shows "carrier ((FBinds \<Delta>\<cdot>(Aheap \<Delta> e\<cdot>a)) x) \<subseteq> edom (ABinds \<Delta>\<cdot>(Aheap \<Delta> e\<cdot>a))"
is_rbt_rbt_sorted: fixes t :: "('a, 'c) rbt" assumes "is_rbt t" shows "rbt_sorted t"
f3: shows "\<forall>r ra. - ra * r = ra * - r"
Afix_def: fixes \<Gamma> :: "(Vars.var \<times> exp) list" shows "Afix \<Gamma> = (\<Lambda> ae. \<mu> x. ABinds \<Gamma>\<cdot>x \<squnion> ae)"
ctr_1: fixes n :: "enat"   and xs :: "'a llist" assumes "n = 0 \<or> lnull xs" shows "ltake n xs = LNil"
pr_auth_key_analz: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and n :: "nat" assumes "(evs, S, A, U) \<in> protocol" shows "Pri_AgrK (priAK n) \<notin> analz (A \<union> spies evs)"
INum_normNum_iff: fixes x :: "int \<times> int"   and y :: "int \<times> int" shows "(INum x = INum y) = (normNum x = normNum y)"
spec_ops_split: fixes xs :: "('c \<times> 'd) list"   and deps :: "'d \<Rightarrow> 'c set"   and oid :: "'c" assumes "spec_ops xs deps"   and "oid \<notin> set (map fst xs)" shows "\<exists>pre suf. xs = pre @ suf \<and> (\<forall>i\<in>set (map fst pre). i < oid) \<and> (\<forall>i\<in>set (map fst suf). oid < i)"
domain_unique: fixes a :: "'b"   and f :: "'b"   and a' :: "'b" assumes "a \<in> B.domains f"   and "a' \<in> B.domains f" shows "a = a'"
fresh_star_set_eq: fixes xs :: "'a list"   and c :: "'b" shows "set xs \<sharp>* c = xs \<sharp>* c"
purge_append: fixes G :: "'U set"   and ucl1 :: "('U \<times> 'C) list"   and ucl2 :: "('U \<times> 'C) list" shows "purge G (ucl1 @ ucl2) = purge G ucl1 @ purge G ucl2"
seqsim_sym: fixes s :: "nat \<Rightarrow> 'b"   and t :: "nat \<Rightarrow> 'b" assumes "s \<approx> t" shows "t \<approx> s"
elem_eq_class: fixes S :: "('c, 'd) eq_object_scheme"   and x :: "'c"   and y :: "'c" assumes "equivalence S"   and "x \<in> carrier S"   and "y \<in> carrier S"   and "x .=\<^bsub>S\<^esub> y" shows "class_of\<^bsub>S\<^esub> x = class_of\<^bsub>S\<^esub> y"
ordermap_mono: fixes w :: "'a"   and x :: "'a"   and r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "(w, x) \<in> r"   and "wf r"   and "w \<in> A"   and "small A" shows "ordermap A r w \<in> elts (ordermap A r x)"
poly_condD_3: fixes p :: "int poly" assumes "poly_cond p" shows "root_free p"
drop_drop: fixes n :: "nat"   and m :: "nat"   and xs :: "'a list" shows "drop n (drop m xs) = drop (n + m) xs"
simps_161: fixes x1 :: "('b, 'a, 'c) val" shows "LID\<^sub>E (VE x1) = LID\<^sub>V x1"
atLeast_Suc_lessThan: fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "m < n" shows "prod g {m..<n} = g m * prod g {Suc m..<n}"
preserves_reflects_arr: fixes f :: "J.Arr" shows "C.arr (\<chi> f) = J.arr f"
simps_1: fixes i :: "nat" shows "evalDdb (DBVar i) = (\<Lambda> \<rho>. \<rho>\<cdot>i)"
lex_nlex_pdevs: fixes x :: "(real \<times> real) pdevs"   and i :: "nat" shows "Counterclockwise_2D_Arbitrary.lex (pdevs_apply (nlex_pdevs x) i) 0"
w_def: shows "w = (\<lambda>n. (1 - t) *\<^sub>R u n + t *\<^sub>R v n)"
ev_holds_sset: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a stream" shows "ev (holds P) xs = (\<exists>x\<in>i.R xs. P x)"
in_arcs_eq: fixes w :: "'a" shows "in_arcs G w = rev_G ` out_arcs G w"
nat_group: fixes g :: "nat \<Rightarrow> 'a"   and k :: "nat"   and n :: "nat" shows "(\<Sum>m<n. sum g {m * k..<m * k + k}) = sum g {..<n * k}"
assms_3: fixes x :: "'a"   and y :: "'a" assumes "\<not> P x y" shows "Q x y"
is_faithful: fixes f :: "'a"   and f' :: "'a" assumes "par f f'"   and "f \<otimes> \<I> = f' \<otimes> \<I>" shows "f = f'"
collapse: fixes stream :: "'b stream" shows "shd stream ## stl stream = stream"
body_cost_def: shows "body_cost = 2 + (8 + 2 * n + 2 * m) * m"
mset_isort: fixes xs :: "'c list" shows "mset (isort xs) = mset xs"
cosy_def: shows "cosy = (\<lambda>U\<in>rcosets H. U #> inv y)"
FOREACH_def: fixes S :: "'c set" shows "FOREACH S \<equiv> FOREACH\<^sub>C S (\<lambda>_. True)"
isin_simps_3: fixes xs :: "'b list"   and x :: "'b" shows "sorted_less (xs @ [x]) = (sorted_less xs \<and> (\<forall>y\<in>set xs. y < x))"
in_space: fixes j :: "'i" shows "space (M j) = \<Union> (F j)"