hcju/isabelleps huggingface dataset:
select_convs_1: fixes igWls :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and igWlsAbs :: "'f \<times> 'b \<Rightarrow> 'i \<Rightarrow> bool"   and igVar :: "'f \<Rightarrow> 'h \<Rightarrow> 'c"   and igAbs :: "'f \<Rightarrow> 'h \<Rightarrow> 'c \<Rightarrow> 'i"   and igOp :: "'g \<Rightarrow> ('d \<Rightarrow> 'c option) \<Rightarrow> ('e \<Rightarrow> 'i option) \<Rightarrow> 'c"   and igFresh :: "'f \<Rightarrow> 'h \<Rightarrow> 'c \<Rightarrow> bool"   and igFreshAbs :: "'f \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> bool"   and igSwap :: "'f \<Rightarrow> 'h \<Rightarrow> 'h \<Rightarrow> 'c \<Rightarrow> 'c"   and igSwapAbs :: "'f \<Rightarrow> 'h \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> 'i"   and igSubst :: "'f \<Rightarrow> 'c \<Rightarrow> 'h \<Rightarrow> 'c \<Rightarrow> 'c"   and igSubstAbs :: "'f \<Rightarrow> 'c \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> 'i"   and more :: "'j" shows "igWls \<lparr>igWls = igWls, igWlsAbs = igWlsAbs, igVar = igVar, igAbs = igAbs, igOp = igOp, igFresh = igFresh, igFreshAbs = igFreshAbs, igSwap = igSwap, igSwapAbs = igSwapAbs, igSubst = igSubst, igSubstAbs = igSubstAbs, \<dots> = more\<rparr> = igWls"
simplex_furthest_lt: fixes S :: "'b set"   and a :: "'b" assumes "finite S" shows "\<forall>x\<in>convex hull S. x \<notin> S \<longrightarrow> (\<exists>y\<in>convex hull S. norm (x - a) < norm (y - a))"
rel_compp: fixes R1 :: "'d \<Rightarrow> 'j \<Rightarrow> bool"   and S1 :: "'j \<Rightarrow> 'g \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'k \<Rightarrow> bool"   and S2 :: "'k \<Rightarrow> 'h \<Rightarrow> bool"   and R3 :: "'f \<Rightarrow> 'l \<Rightarrow> bool"   and S3 :: "'l \<Rightarrow> 'i \<Rightarrow> bool" shows "F2rel (R1 OO S1) (R2 OO S2) (R3 OO S3) = F2rel R1 R2 R3 OO F2rel S1 S2 S3"
elims: fixes x :: "perm"   and xa :: "('a, 'b) S_action"   and y :: "('a, 'b) S_action" assumes "x \<bullet> xa = y" obtains p :: "perm" and \<alpha> :: "'a" where "x = pa__" and "xa = S_action.Act \<alpha>" and "y = S_action.Act (pa__ \<bullet> \<alpha>)"   | p :: "perm" and \<phi> :: "'b" where "x = p" and "xa = S_action.Pred \<phi>" and "y = S_action.Pred (p \<bullet> \<phi>)"
X: shows "x \<in> X \<and> X \<in> P"
\<W>_paths_le: fixes k :: "nat"   and P :: "(symbol \<times> character list) list set" shows "\<W> (paths_le k P) k = \<W> P k"
simps_1: fixes s :: "('c, 'd) fmap" shows "exec_plan s [] = s"
id_lens_compat: fixes X :: "'d \<Longrightarrow> 'e" assumes "vwb_lens X" shows "1\<^sub>L ##\<^sub>L X"
hyps: assumes "index < length L" shows "LeftDerivation \<alpha> (take (ladder_n L' index) D') (ladder_\<gamma> \<alpha> D' L' index) \<and> ladder_\<alpha> (\<delta> @ \<alpha>) D L index = \<delta> @ ladder_\<alpha> \<alpha> D' L' index \<and> ladder_\<gamma> (\<delta> @ \<alpha>) D L index = \<delta> @ ladder_\<gamma> \<alpha> D' L' index"
exec_instr_Load: fixes n :: "nat"   and P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" shows "exec_instr (Load n) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh = (None, h, (loc ! n # stk, loc, C\<^sub>0, M\<^sub>0, Suc pc, ics) # frs, sh)"
mod7_def: shows "mod7 = (\<lambda>x. x mod 7)"
disc_rfr_proc_Suc_div: fixes r :: "real"   and n :: "nat"   and w :: "'b" assumes "- 1 < r" shows "disc_rfr_proc r (Suc n) w / disc_rfr_proc r n w = 1 + r"
simps_2: fixes opt :: "atom fm \<Rightarrow> atom fm"   and step :: "nat \<Rightarrow> (atom list \<times> atom fm list \<times> nat) list \<Rightarrow> atom fm"   and \<phi>\<^sub>1 :: "atom fm"   and \<phi>\<^sub>2 :: "atom fm" shows "QE_dnf' opt step (Or \<phi>\<^sub>1 \<phi>\<^sub>2) = or (QE_dnf' opt step \<phi>\<^sub>1) (QE_dnf' opt step \<phi>\<^sub>2)"
simps: fixes s :: "name"   and x :: "tm"   and y :: "tm" assumes "atom s \<sharp> (x, y)" shows "WRP x y = SyntaxN.Ex s (SeqWRP (Var s) x y)"
CX5h_bilateral_substitutes: fixes h :: "H2" shows "BSI.bilateral_substitutes_on UNIV (CX5h h)"
relevant_vars_Entry: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and backward_slice :: "'b set \<Rightarrow> 'b set"   and Exit :: "'b"   and H :: "'d set"   and L :: "'d set"   and High :: "'b"   and Low :: "'b"   and V :: "'d"   and S :: "'b set" assumes "NonInterferenceIntraGraph sourcenode targetnode kind valid_edge Entry Def Use state_val backward_slice Exit H L High Low"   and "V \<in> BackwardSlice.relevant_vars sourcenode targetnode valid_edge Def Use backward_slice S Entry"   and "High \<notin> backward_slice S" shows "V \<in> L"
obs_consistent_med01iai: shows "obs_consistent R0sk1iai med0sk1iai a0i skl1"
RR: fixes i :: "nat" shows "(if i = k then coeff P p i else \<zero>) \<in> carrier R"
sum_gp_strict: fixes x :: "'a"   and n :: "nat" shows "sum ((^) x) {..<n} = (if x = (1::'a) then of_nat n else ((1::'a) - x ^ n) / ((1::'a) - x))"
sin_npi: fixes n :: "nat" shows "sin (real n * pi) = 0"
assms: shows "(case (a, b) of (x, y) \<Rightarrow> (cx x y, cy x y)) = 0"
nn_integral_measure_pmf: fixes p :: "'a pmf"   and f :: "'a \<Rightarrow> ennreal" shows "integral\<^sup>N (measure_pmf p) f = (\<Sum>\<^sup>+ x. ennreal (pmf p x) * f x)"
linear_orderable_3_implies_5: fixes x :: "'a" assumes "linear_orderable_3 x" shows "transitive x \<and> linear_orderable_6 x"
face_of_imp_eq_affine_Int: fixes S :: "'b set"   and T :: "'b set" assumes "convex S"   and "T face_of S" shows "T = affine hull T \<inter> S"
fut_invariant_def: shows "fut_invariant = (\<lambda>(\<phi>, \<Phi>). local.wf idx \<phi> \<and> (\<forall>\<phi>\<in>set \<Phi>. local.wf idx \<phi>) \<and> (\<exists>k. \<phi> = fut_derivs k \<psi> \<and> \<Phi> = map (\<lambda>i. fut_derivs i \<psi>) (rev [0..<k])))"
set_oo_map_alt: fixes f :: "'a \<Rightarrow> 'b" shows "(set \<circ>\<circ> map) f = (\<lambda>l. f ` set l)"
that_1: assumes "LENGTH('a) = 1"   and "a = 0"   and "sint a = 0" shows "P"
simps: fixes xs :: "'a list"   and ys :: "'a list" shows "balance (xs, ys) = (if length xs \<le> length ys then (xs, ys) else ([], ys @ rev xs))"
cotrace_all_have_snapshotted_and_no_markers: fixes t :: "('a, 'b, 'c) event llist" assumes "cotrace init t" shows "\<exists>i. enat i \<le> llength t \<and> (\<forall>p. ps (coS t i) p \<noteq> None) \<and> (\<forall>c. Marker \<notin> set (msgs (coS t i) c))"
non_empty_cycle_root_move_root_forward: fixes r :: "'a"   and x :: "'a"   and q :: "'a" assumes "non_empty_cycle_root r x"   and "point q"   and "r \<le> x\<^sup>\<star> ; q" shows "non_empty_cycle_root q x"
llcp_LNil1: fixes ys :: "'c llist" shows "llcp LNil ys = 0"
o_x_axis_def: shows "o_x_axis \<equiv> abs_ocircline x_axis_clmat"
not_iMOD_iIN_subset: fixes m' :: "nat"   and r' :: "nat"   and n :: "nat"   and d :: "nat" assumes "0 < m'" shows "\<not> [ r', mod m' ] \<subseteq> [n\<dots>,d]"
path_nonempt: fixes SES :: "('c, 'd) SES_rec"   and s1 :: "'c"   and e :: "'d"   and t :: "'d list" shows "path SES s1 (e # t) = (if \<exists>s2. s1 e\<longrightarrow>\<^bsub>SES\<^esub> s2 then path SES (the (T\<^bsub>SES\<^esub> s1 e)) t else None)"
erase_r_bal: fixes l :: "'a tree"   and r :: "'a tree"   and n :: "'a" assumes "hinv l"   and "hinv r"   and "height (erase r) = height (erase l) + 2" shows "erase (r_bal n l r) = r_bal\<^sub>0 n (erase l) (erase r)"
genPrefix_take_append: fixes r :: "('a \<times> 'a) set"   and xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "refl r"   and "(xs, ys) \<in> genPrefix r" shows "(xs @ zs, take (length xs) ys @ zs) \<in> genPrefix r"
independent_extend_basis: fixes B :: "'b set" assumes "independent B" shows "independent (extend_basis B)"
real_linorder_cases: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_real_line plus zero minus less_eq less times"   and "less a zero \<or> a = zero \<or> less zero a"   and "less b zero \<or> b = zero \<or> less zero b" obtains "less a b"   | "a = b"   | "less b a"
len: shows "length (sha1 M) + sLen + 16 \<le> length (nat_to_bv (p * q)) - Suc 0"
mat_of_rows_carrier_2: fixes n :: "nat"   and vs :: "'c vec list" shows "dim_row (mat_of_rows n vs) = length vs"
assms: shows "\<forall>A\<in>carrier_mat 1 n. is_SNF A (Smith_1xn A)"
sum_Inf: fixes f :: "'a \<Rightarrow> ereal"   and A :: "'a set" shows "(\<bar>sum f A\<bar> = \<infinity>) = (finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>))"
pure_the_pure: fixes R :: "'b \<Rightarrow> 'd \<Rightarrow> assn" assumes "is_pure R" shows "pure (the_pure R) = R"
in_minimize_wrtD: fixes Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and x :: "'a"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>x y. Q x y \<Longrightarrow> \<not> Q y x"   and "sorted_wrt Q xs"   and "x \<in> set (minimize_wrt P xs)"   and "\<And>x y. \<not> P x y \<Longrightarrow> Q x y"   and "\<And>x. P x x" shows "x \<in> set xs \<and> (\<forall>y\<in>set xs. P y x)"
lc_Call: fixes t :: "sterm"   and a :: "sterm"   and l :: "Label" assumes "lc t"   and "lc a" shows "lc (Call t l a)"
A2: fixes l' :: "real"   and r' :: "real"   and lr :: "real"   and r :: "real" assumes "1 \<le> l'"   and "1 \<le> r'"   and "1 \<le> lr"   and "1 \<le> r" shows "\<alpha> * (l' + r') * (lr + r) powr \<beta> * (lr + r' + r) powr \<beta> \<le> (l' + r') powr \<beta> * (l' + lr + r') powr \<beta> * (l' + lr + r' + r)"
assms_2: shows "A, n \<turnstile> ((\<^bold>\<not> q) # (\<^bold>\<not> p) # ps, a) # branch"
powser_insidea: fixes f :: "nat \<Rightarrow> 'a"   and x :: "'a"   and z :: "'a" assumes "summable (\<lambda>n. f n * x ^ n)"   and "norm z < norm x" shows "summable (\<lambda>n. norm (f n * z ^ n))"
rangedom: fixes x :: "'a" shows "r (d x) = d x"
simps_3: fixes B :: "'a set"   and C :: "'a set"   and Cs :: "'a set list" shows "wall_crossings (B # C # Cs) = THE_default {} (\<lambda>H. H \<in> (\<Union>(f, g)\<in>foldpairs. {{f \<turnstile> \<C>, g \<turnstile> \<C>}}) \<and> separated_by H B C) # wall_crossings (C # Cs)"
Cs_def: shows "Cs \<equiv> ARG_MIN length Cs. gallery (C # Cs @ [D])"
preserves_reflects_arr: fixes f :: "'b" shows "arr (\<rr>'.map f) = arr f"
affine_hull_convex_hull: fixes S :: "'c set" shows "affine hull (convex hull S) = affine hull S"
hyps_10: fixes Vs :: "char list list"   and ls :: "val list" assumes "fv body \<subseteq> set Vs"   and "l\<^sub>2' \<subseteq>\<^sub>m [Vs [\<mapsto>] ls]"   and "length Vs + max_vars body \<le> length ls" shows "\<exists>ls'. compP\<^sub>1 P \<turnstile>\<^sub>1 \<langle>compE\<^sub>1 Vs body,(h\<^sub>2, ls)\<rangle> \<Rightarrow> \<langle>fin\<^sub>1 b',(h\<^sub>3, ls')\<rangle> \<and> l\<^sub>3 \<subseteq>\<^sub>m [Vs [\<mapsto>] ls']"
lower: fixes t :: "real" assumes "t \<in> S"   and "t0 < t" shows "v' t < f t (v t)"
Nsetn_sub_mem1: fixes n :: "nat" shows "\<forall>j. j \<in> {j. j \<le> n} \<longrightarrow> j \<in> {j. j \<le> Suc n}"
ab_left_minus: fixes a :: "'b" shows "- a + a = (0::'b)"
set_iterator_genord_foldli_correct: fixes xs :: "'d list"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "distinct xs"   and "sorted_wrt R xs" shows "set_iterator_genord (foldli xs) (set xs) R"
comp_simp: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "VoV.seq g f" shows "VoV.comp g f = VxV.comp g f"
greatest_postfixpoint_same: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "is_greatest_postfixpoint f x" shows "x = p\<nu> f"
num_params_polymul: fixes p1 :: "'a poly"   and p2 :: "'a poly" shows "num_params (p1 *\<^sub>p p2) \<le> max (num_params p1) (num_params p2)"
hmlsl_def: fixes sensors :: "cars \<Rightarrow> traffic \<Rightarrow> cars \<Rightarrow> real" shows "hmlsl sensors \<equiv> \<forall>e ts c. 0 < sensors e ts c"
simps_1: fixes \<nu> :: "'b"   and u :: "'b E.term" shows "\<^bold>\<langle>\<nu>\<^bold>\<rangle>\<^sub>0 \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u = u"
ts_bool_True: fixes x :: "nat \<Rightarrow> bool list"   and i :: "nat" assumes "ts x"   and "x i \<noteq> [False]" shows "x i = [True]"
order_finite_chain: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b"   and X :: "'b set"   and i :: "nat"   and j :: "nat"   and l :: "nat" assumes "MinkowskiBetweenness \<E> \<P> betw"   and "MinkowskiBetweenness.long_ch_by_ord betw f X"   and "finite X"   and "0 \<le> i \<and> i < j \<and> j < l \<and> l < card X" shows "betw (f i) (f j) (f l)"
fds_abs_converges_remainder: fixes f :: "'b fds"   and s :: "'b"   and m :: "nat" assumes "fds_abs_converges f s" shows "fds_abs_converges (fds_remainder m f) s"
subst_atyping: fixes E :: "char list \<Rightarrow> Poly_Types.ty"   and a :: "aexp"   and t :: "Poly_Types.ty"   and S :: "nat \<Rightarrow> Poly_Types.ty" assumes "E \<turnstile>p a : t" shows "tsubst S \<circ> E \<turnstile>p a : tsubst S t"
in_hom_char: fixes f :: "(bool, 'arr list) concrete_category.arr"   and a :: "(bool, 'arr list) concrete_category.arr"   and b :: "(bool, 'arr list) concrete_category.arr" shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright> = (arr f \<and> ide a \<and> ide b \<and> Dom f = Dom a \<and> Cod f = Dom b)"
assms_2: shows "\<forall>l. \<forall>m\<in>M l. homogeneous\<^sub>l\<^sub>s\<^sub>t m \<A> Sec"
f_Exec_Stream_Acc_LocalState_length: fixes k :: "nat"   and localState :: "'e \<Rightarrow> 'd"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and xs :: "'f message_af list"   and c :: "'e" assumes "0 < k" shows "length (f_Exec_Comp_Stream_Acc_LocalState k localState trans_fun xs c) = length xs"
iMOD_E: fixes x :: "nat"   and r :: "nat"   and m :: "nat" assumes "x \<in> [ r, mod m ]" obtains "x mod m = r mod m" and "r \<le> x"
unfold_tree_unique: fixes f :: "'b \<Rightarrow> 'c tree"   and ROOT :: "'b \<Rightarrow> 'c"   and LEFT :: "'b \<Rightarrow> 'b"   and RIGHT :: "'b \<Rightarrow> 'b"   and s :: "'b" assumes "\<And>s. root (f s) = ROOT s"   and "\<And>s. left (f s) = f (LEFT s)"   and "\<And>s. right (f s) = f (RIGHT s)" shows "f s = unfold_tree ROOT LEFT RIGHT s"
fbox_temp_dyn: fixes a :: "real"   and U :: "(real, 4) vec \<Rightarrow> real set"   and L :: "real"   and G :: "(real, 4) vec \<Rightarrow> bool"   and Q :: "(real, 4) vec \<Rightarrow> bool" assumes "0 < a"   and "\<And>s. s \<in> UNIV \<Longrightarrow> 0 \<in> U s \<and> is_interval (U s) \<and> U s \<subseteq> UNIV" shows "wp x\<acute>=\<lambda>t. temp_vec_field a L & G on U UNIV @ 0 \<lceil>Q\<rceil> = \<lceil>\<lambda>s. s \<in> UNIV \<longrightarrow> (\<forall>t\<in>U s. (\<forall>\<tau>\<in>down (U s) t. G (temp_flow a L \<tau> s)) \<longrightarrow> Q (temp_flow a L t s))\<rceil>"
negligibleD: fixes f :: "nat \<Rightarrow> real"   and c :: "real" assumes "negligible f"   and "0 < c" shows "f \<in> o(\<lambda>x. inverse (real x powr c))"
simps_1: fixes cs :: "'edge list" shows "upd_cs cs [] = cs"
map_dual_func1: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" shows "\<partial>\<^sub>F (f \<circ> g) = \<partial>\<^sub>F f \<circ> \<partial>\<^sub>F g"
int_xor_lt0: fixes x :: "int"   and y :: "int" shows "(x XOR y < 0) = ((x < 0) \<noteq> (y < 0))"
l_coset_eq_rcong: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and a :: "'c" assumes "subgroup H G"   and "Group.group G"   and "a \<in> carrier G" shows "a <#\<^bsub>G\<^esub> H = rcong\<^bsub>G\<^esub> H `` {a}"
mu_small: shows "\<mu>_small_row i fs (Suc j)"
mono_mult2_s: fixes ns :: "('a \<times> 'a) set"   and ns' :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set"   and s' :: "('a \<times> 'a) set" assumes "ns \<subseteq> ns'"   and "s \<subseteq> s'" shows "mult2_s ns s \<subseteq> mult2_s ns' s'"
sort_oalist_ko_ntm_def: shows "sort_oalist_ko_ntm \<equiv> oalist_raw.sort_oalist_raw key_order_of_nat_term_order_inv"
empty_interior_convex_hull: fixes S :: "'b set" assumes "finite S"   and "card S \<le> DIM('b)" shows "interior (convex hull S) = {}"
PostConjI: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and R :: "'e set"   and B :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c R,B" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c (Q \<inter> R),(A \<inter> B)"
assms_3: shows "set (xs @ [(oid, oper)]) = set (ys @ [(oid, oper)] @ zs)"
get_ancestors_si_also_host: fixes h :: "(_) heap"   and some_ptr :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list"   and shadow_root :: "(_) shadow_root_ptr"   and host :: "(_) element_ptr" assumes "heap_is_wellformed h"   and "h \<turnstile> get_ancestors_si some_ptr \<rightarrow>\<^sub>r ancestors"   and "cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^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 shadow_root \<in> set ancestors"   and "h \<turnstile> get_host shadow_root \<rightarrow>\<^sub>r host"   and "type_wf h"   and "known_ptrs h" shows "cast\<^sub>e\<^sub>l\<^sub>e\<^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 host \<in> set ancestors"
rep_eq: fixes x :: "i"   and xa :: "\<o>" shows "(x \<Turnstile> xa) = eval\<o> xa dj x"
fmranI: fixes m :: "('d, 'c) fmap"   and x :: "'d"   and y :: "'c" assumes "fmlookup m x = Some y" shows "y |\<in>| fmran m"
absorb2: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'c"   and a :: "'c" assumes "semilattice_order f less_eq less"   and "less_eq b a" shows "f a b = b"
t2_mem: shows "flow0 x t2 \<in> {a<--<b}"
3: shows "vectorspace.dim class_ring (vs (local.span (set (cols A)))) = card (set (cols A))"
step: fixes r :: "nat" assumes "P r"   and "CHOnextConfig A r (rho r) (HOs r) (coords (Suc r)) (rho (Suc r))" shows "P (Suc r)"
simps_1: fixes Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "vc SKIP Q = True"
not_distinct_conv_prefix: fixes as :: "'d list" shows "(\<not> distinct as) = (\<exists>xs y ys. y \<in> set xs \<and> distinct xs \<and> as = xs @ y # ys)"
some_geodesic_in_subset: fixes x :: "'b"   and S :: "'b set"   and y :: "'b" assumes "x \<in> S"   and "y \<in> S" shows "{x--S--y} \<subseteq> S"
effect_tapI: fixes h' :: "heap"   and h :: "heap"   and r :: "'b"   and f :: "heap \<Rightarrow> 'b" assumes "h' = h"   and "r = f h" shows "effect (tap f) h h' r"
le_iff_sup: fixes x :: "'b multiset"   and y :: "'b multiset" shows "(x \<subseteq># y) = (x \<union># y = y)"
map_vector_def: fixes f :: "'f \<Rightarrow> 'd"   and v :: "('f, 'e) Finite_Cartesian_Product.vec" shows "map_vector f v \<equiv> \<chi>i. f (v $h i)"
D: shows "(Asset n, \<lbrace>Num 2, PubKey B\<rbrace>) \<in> s"
assms_2: shows "Z \<subseteq>\<^sub>\<circ> Y"
relpow_Suc_E2': fixes x :: "'b"   and z :: "'b"   and n :: "nat"   and A :: "('b \<times> 'b) set" assumes "(x, z) \<in> A ^^ Suc n" obtains y :: "'b" where "(x, y) \<in> A" and "(y, z) \<in> A\<^sup>*"
fmrestrict_fset_twice: fixes S :: "'c fset"   and T :: "'c fset"   and m :: "('c, 'd) fmap" shows "fmrestrict_fset S (fmrestrict_fset T m) = fmrestrict_fset (S |\<inter>| T) m"
assms_2: shows "x1 0 \<in> carrier_mat d d"
compute_SCC_code_def: fixes node_eq_impl :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and node_hash_impl :: "nat \<Rightarrow> 'c \<Rightarrow> nat"   and node_def_hash_size :: "nat"   and G_impl :: "('c \<Rightarrow> bool, 'c \<Rightarrow> 'c list, 'c list, 'd) gen_g_impl_scheme" shows "compute_SCC_code node_eq_impl node_hash_impl node_def_hash_size G_impl \<equiv> stat_start_nres \<bind> (\<lambda>x. let y' = ([], Impl_Array_Hash_Map.ahm_empty node_def_hash_size) in LIST_FOREACH' (RETURN (id (gi_V0 G_impl))) (\<lambda>xa. True) (\<lambda>xa (xb, xc). if \<not> (case Impl_Array_Hash_Map.ahm_lookup node_eq_impl node_hash_impl xa xc of None \<Rightarrow> False | Some i \<Rightarrow> if 0 \<le> i then False else True) then let y'a = (xb, as_singleton xa, as_singleton 0, Impl_Array_Hash_Map.ahm_update node_eq_impl node_hash_impl xa (int 0) xc, if is_Nil (gi_E G_impl xa) then as_empty () else as_singleton (0, gi_E G_impl xa)) in WHILE\<^sub>T (\<lambda>(xe, xf). \<not> as_is_empty (case xf of (xg, xh, xi, xj) \<Rightarrow> xg)) (\<lambda>(xe, xf). select_edge_code node_eq_impl xf \<bind> (\<lambda>(xg, xh). case xg of None \<Rightarrow> last_seg_code xh \<bind> (\<lambda>xi. pop_code node_eq_impl node_hash_impl xh \<bind> (\<lambda>xj. let y'b = xi # xe in RETURN (y'b, xj))) | Some xi \<Rightarrow> if case Impl_Array_Hash_Map.ahm_lookup node_eq_impl node_hash_impl xi (case xh of (xj, xk, xl, xm) \<Rightarrow> xl) of None \<Rightarrow> False | Some i \<Rightarrow> if 0 \<le> i then True else False then collapse_code node_eq_impl node_hash_impl xi xh \<bind> (\<lambda>xj. RETURN (xe, xj)) else if \<not> (case Impl_Array_Hash_Map.ahm_lookup node_eq_impl node_hash_impl xi (case xh of (xj, xk, xl, xm) \<Rightarrow> xl) of None \<Rightarrow> False | Some i \<Rightarrow> if 0 \<le> i then False else True) then RETURN (xe, push_code node_eq_impl node_hash_impl G_impl xi xh) else RETURN (xe, xh))) y'a \<bind> (\<lambda>(xd, xe, xf, xg, xh). RETURN (xd, xg)) else RETURN (xb, xc)) y' \<bind> (\<lambda>(xa, xb). stat_stop_nres \<bind> (\<lambda>xc. RETURN xa)))"
variation_0_2: fixes x :: "real" shows "variation x 0 = 0"
assms_2: fixes x :: "real"   and y :: "real" assumes "x \<in> {a..<b}"   and "x < y" shows "\<exists>z\<in>{x<..y}. dist (f z) (f x) \<le> M * (z - x)"
sm: fixes x :: "'a" assumes "x \<in> cball \<xi> r" shows "(\<lambda>n. a n * (x - \<xi>) ^ n) sums f x"
emeasure_lborel_cbox_eq: fixes l :: "'c"   and u :: "'c" shows "emeasure lborel (cbox l u) = ennreal (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then prod ((\<bullet>) (u - l)) Basis else 0)"
trg_hcomp: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "C.hseq \<mu> \<nu>" shows "trg\<^sub>C (\<mu> \<star>\<^sub>C \<nu>) = trg\<^sub>C \<mu>"
distinct_disc_4: fixes status :: "status" assumes "is_found status" shows "\<not> is_failed status"
simps: fixes b :: "bool"   and r :: "(bool \<times> 'b) rexp" shows "L_a (b, r) = Lm (follow b r) \<union> (if final_a (b, r) then {[]} else {})"
foldl_strict_1: fixes f :: "'b \<rightarrow> 'c \<rightarrow> 'b"   and d :: "'b" shows "foldl\<cdot>f\<cdot>d\<cdot>\<bottom> = \<bottom>"
equivalence_in_bicategory_axioms: shows "internal_equivalence w' w \<psi> (local.inv \<phi>)"
Try1Red: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and e' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list"   and C :: "String.literal"   and V :: "nat"   and e2 :: "'addr expr1" assumes "uf,P,t \<turnstile>1 \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>try e catch(C V) e2,s\<rangle> -ta\<rightarrow> \<langle>try e' catch(C V) e2,s'\<rangle>"
seq_type: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>' :: "'Var \<Rightarrow> Sec option"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>'' :: "'Var \<Rightarrow> Sec option" assumes "\<turnstile> \<Gamma> {c\<^sub>1} \<Gamma>'"   and "\<turnstile> \<Gamma>' {c\<^sub>2} \<Gamma>''" shows "\<turnstile> \<Gamma> {c\<^sub>1 ;; c\<^sub>2} \<Gamma>''"