hcju/isabelleps huggingface dataset:
subset_closed_ok_extend_imp_ok_project: fixes F :: "'a program"   and G :: "'c program"   and C :: "'c set" assumes "extend h F ok G"   and "subset_closed (AllowedActs F)" shows "F ok project h C G"
augment: fixes X :: "'a set"   and Y :: "'a set" assumes "indep X"   and "indep Y"   and "card Y < card X" shows "\<exists>x\<in>X - Y. indep (insert x Y)"
is_less: shows "\<forall>(a, c)\<in>set C. c \<in> cs \<and> is_le (a *R c) \<and> a \<noteq> 0"
sum_cong_aux: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "sum f A = sum g A"
X_def: fixes n :: "nat" shows "X n = (\<Inter>i\<in>{i. J i \<subseteq> {0..<n}}. PF.emb {0..<n} (J i) (X' i)) \<inter> space (Pi\<^sub>M {0..<n} M)"
S2: assumes "sf y \<le> sf x" shows "y \<le> x"
formulaFalseInEntailedValuationIsUnsatisfiable: fixes formula :: "Literal list list"   and valuation :: "Literal list" assumes "formulaFalse formula valuation"   and "formulaEntailsValuation formula valuation" shows "\<not> satisfiable formula"
alphaAbs_preserves_qGoodAbs2: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs"   and A' :: "('d, 'e, 'varSort, 'var, 'f) qAbs" assumes "A $= A'"   and "qGoodAbs A'" shows "qGoodAbs A"
simps_2: fixes b :: "nat"   and uu :: "event_data" shows "Formula.fvi_trm b (trm.Const uu) = {}"
set_lebesgue_integral_eq_integral_1: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "f absolutely_integrable_on S" shows "f integrable_on S"
ts_i: shows "ts ! i = (p, is, \<theta> |` (dom \<theta> - read_tmps sb), x, \<D>, \<O>, \<R>)"
CH_irc_idem: fixes A :: "'a set" shows "ManiPref'.CH (ManiPref'.CH A) = ManiPref'.CH A"
real_lim: fixes f :: "'a \<Rightarrow> complex"   and l :: "complex"   and F :: "'a filter"   and P :: "'a \<Rightarrow> bool" assumes "(f \<longlongrightarrow> l) F"   and "F \<noteq> bot"   and "eventually P F"   and "\<And>a. P a \<Longrightarrow> f a \<in> \<real>" shows "l \<in> \<real>"
nontrans1: fixes u :: "'a"   and r :: "'a" assumes "u \<parallel> r" shows "\<nexists>t. u \<parallel> t \<and> t \<parallel> r"
subst: fixes t :: "'a"   and env :: "(name, 'a) fmap" assumes "P t"   and "fmpred (\<lambda>_. P) env" shows "P (subst t env)"
line_through_K2_intersect_S: fixes p :: "proj2"   and l :: "proj2_line" assumes "p \<in> hyp2"   and "proj2_incident p l" shows "\<exists>r. r \<in> S \<and> proj2_incident r l"
intro: fixes \<Sigma> :: "'b set"   and \<delta> :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'b" assumes "finite \<Sigma>"   and "finite (reach \<Sigma> \<delta> q\<^sub>0)"   and "range w \<subseteq> \<Sigma>" shows "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"
properI: fixes S :: "'c set" assumes "\<And>x r. compact (cball x r \<inter> S)" shows "proper S"
comp_arr_dom: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a" assumes "H.VV.arr f"   and "H.VV.dom f = a" shows "H.VV.comp f a = f"
invP: shows "A \<TTurnstile> (I \<rightarrow>) P"
index_order2: fixes f :: "nat \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and z :: "'a"   and X :: "'a set"   and a :: "nat"   and b :: "nat"   and c :: "nat" assumes "[f[x .. y ..  z]X]"   and "f a = x"   and "f b = y"   and "f c = z"   and "finite X \<longrightarrow> a < card X"   and "finite X \<longrightarrow> b < card X"   and "finite X \<longrightarrow> c < card X" shows "a < b \<and> b < c \<or> c < b \<and> b < a"
\<beta>: shows "\<beta> \<in> elts \<omega>1"
carrier_conv_generator: shows "carrier G = ([^]) \<^bold>g ` {..<order G}"
awalk_verts_conv': fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "cas u p v" shows "awalk_verts u p = (if p = [] then [u] else tail G (hd p) # map (head G) p)"
D: shows "\<forall>j k. k < length ns \<longrightarrow> Suc i \<le> j \<longrightarrow> j < k \<longrightarrow> 0 < offs_num (length ns) xs index key mi ma j \<longrightarrow> 0 < offs_num (length ns) xs index key mi ma k \<longrightarrow> ns ! j + offs_num (length ns) xs index key mi ma j \<le> ns ! k"
scale_left_distrib: fixes x :: "'vs"   and a :: "'f"   and b :: "'f" assumes "x \<in> U\<^sub>V\<^sub>S" shows "(a + b) *s\<^sub>o\<^sub>w x = a *s\<^sub>o\<^sub>w x +\<^sub>V\<^sub>S b *s\<^sub>o\<^sub>w x"
assms_8: shows "X = {(t, genericJoin J (newQuery J Q_J_pos (I, t)) (newQuery J Q_J_neg (I, t))) |t. t \<in> R_I}"
finfun_dom_const: fixes c :: "'b" shows "finfun_dom (finfun_const c) = finfun_const (finite UNIV \<and> c \<noteq> undefined)"
inner_Basis_SUP_left: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and Basis :: "'c set"   and abs :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and i :: "'c"   and f :: "'d \<Rightarrow> 'c"   and X :: "'d set" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less"   and "i \<in> Basis" shows "inner (Sup (f ` X)) i = (SUP x\<in>X. inner (f x) i)"
\<psi>: shows "wf_mformula \<sigma> j n R \<psi> \<psi>''"
quot_of_fract_quot_to_fract': fixes x :: "'b \<times> 'b" assumes "x \<in> normalized_fracts" shows "quot_of_fract (quot_to_fract x) = x"
ground_aux_simps_1: fixes S :: "atom set" shows "ground_aux Zero S = True"
encL: fixes Enc :: "'a \<Rightarrow> 'b"   and S :: "'a"   and SRel :: "('a \<times> 'a) set"   and TRel :: "('b \<times> 'b) set" shows "(TargetTerm (Enc S), SourceTerm S) \<in> encoding.indRelSTEQ Enc SRel TRel"
ennreal_between: fixes e :: "ennreal"   and x :: "ennreal" assumes "0 < e"   and "0 < x"   and "x < top" shows "x - e < x"
div65536: fixes a :: "nat"   and b :: "nat"   and c :: "nat"   and d :: "nat" assumes "a < 256"   and "b < 256"   and "c < 256"   and "d < 256" shows "nat ((int d + int (256 * c) + int (65536 * b) + int (16777216 * a)) div 65536 mod 256) = b"
comp_lunit_lunit'_1: fixes a :: "'a" assumes "ide a" shows "MC.lunit a \<cdot> MC.lunit' a = a"
nabla_unfold_eq: fixes x :: "'a" shows " |x\<rangle> \<nabla>x = \<nabla>x"
finter_absorb: fixes A :: "'a fset" shows "A |\<inter>| A = A"
inverse_map_def: fixes f :: "'e \<Rightarrow> 'd"   and S :: "'e set"   and T :: "'d set" shows "inverse_map f S T \<equiv> restrict (inv_into S f) T"
not_union_or: fixes x :: "'a"   and A :: "'a set"   and B :: "'a set" shows "(x \<notin> A \<union> B) = (x \<notin> A \<and> x \<notin> B)"
Ring_CongruenceD: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> Ring_Congruence" shows "a - b \<in> I"
synth_insert: fixes X :: "msg"   and H :: "msg set" shows "insert X (synth H) \<subseteq> synth (insert X H)"
vbrelation_VLambda: fixes A :: "V"   and f :: "V \<Rightarrow> V" shows "vbrelation (VLambda A f)"
nat_of_real_def: shows "nat_of_real = inv CZH_Sets_ZQR.real"
ennreal_suminf_bound_add: fixes f :: "nat \<Rightarrow> ennreal"   and y :: "ennreal"   and x :: "ennreal" assumes "\<And>N. sum f {..<N} + y \<le> x" shows "suminf f + y \<le> x"
replacefacesAt2_in: fixes i :: "nat"   and "is" :: "nat list"   and Fss :: "face list list"   and olfF :: "face"   and newFs :: "face list" assumes "i \<in> set is"   and "distinct is"   and "i < |Fss|" shows "replacefacesAt2 is olfF newFs Fss ! i = replace olfF newFs (Fss ! i)"
unique: fixes A :: "'a"   and B :: "'b"   and B' :: "'b" assumes "(A, B) \<in> F"   and "(A, B') \<in> F" shows "B = B'"
analz_Un_implConfidSet_2_aux_3: fixes H :: "msg set"   and Ag :: "(agent \<times> agent) set"   and G :: "msg set" assumes "Enc_keys_clean H"   and "Ag \<inter> broken (parts H \<inter> range LtK) = {}" shows "analz (implSecureSet_aux Ag G \<union> H) \<subseteq> synth (analz H) \<union> - payload"
l3_inv8_derived: shows "l3_inv2 \<inter> l3_inv3 \<inter> l3_inv4 \<subseteq> l3_inv8"
intro: fixes F :: "('b, 'd, 'a, 'e, 'c, 'f, 'g) Functor_scheme" assumes "\<And>f g. f \<approx>>\<^bsub>CatDom F\<^esub> g \<Longrightarrow> F ## f ;;\<^bsub>CatDom F\<^esub> g = F ## f ;;\<^bsub>CatCod F\<^esub> (F ## g)"   and "\<And>X. X \<in> obj\<^bsub>CatDom F\<^esub> \<Longrightarrow> \<exists>Y\<in>obj\<^bsub>CatCod F\<^esub>. F ## id\<^bsub>CatDom F\<^esub> X = id\<^bsub>CatCod F\<^esub> Y"   and "Category (CatDom F)"   and "Category (CatCod F)" shows "PreFunctor F"
assms_1: shows "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"
finite_entropy_def: fixes S :: "'c measure"   and X :: "'a \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> real" shows "finite_entropy S X f = (distributed M S X (\<lambda>x. ennreal (f x)) \<and> integrable S (\<lambda>x. f x * log b (f x)) \<and> (\<forall>x\<in>space S. 0 \<le> f x))"
mk_until_atoms: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "atoms_ltln (mk_until \<phi> \<psi>) \<subseteq> atoms_ltln \<phi> \<union> atoms_ltln \<psi>"
density_unique2: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal"   and f' :: "'b \<Rightarrow> ennreal" assumes "sigma_finite_measure M"   and "f \<in> borel_measurable M"   and "f' \<in> borel_measurable M"   and "\<And>A. A \<in> sets M \<Longrightarrow> set_nn_integral M A f = set_nn_integral M A f'" shows "AE x in M. f x = f' x"
Cons_1: fixes s :: "'a \<Rightarrow> bool option" assumes "\<forall>op\<in>set ops. v \<notin> set (add_effects_of op) \<and> v \<notin> set (delete_effects_of op)" shows "foldl (++) s (map (map_of \<circ> effect_to_assignments) ops) v = s v"
bind_lunit: fixes x :: "'e"   and f :: "'e \<Rightarrow> 'd option" shows "Option.bind (\<lfloor>x\<rfloor>) f = f x"
sup_commute: fixes x :: "'k multiset"   and y :: "'k multiset" shows "x \<union># y = y \<union># x"
vertices_makeFaceFinal: fixes f :: "face"   and g :: "graph" shows "vertices (makeFaceFinal f g) = vertices g"
prime_power: fixes f :: "nat \<Rightarrow> 'b"   and f_prime_power :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and f_prime :: "nat \<Rightarrow> 'b"   and p :: "nat"   and k :: "nat" assumes "multiplicative_function' f f_prime_power f_prime"   and "prime p"   and "0 < k" shows "f (p ^ k) = f_prime_power p k"
induce_reachable_is_in_sccs: fixes u :: "'a" assumes "u \<in> verts G" shows "G \<restriction> {v. u \<rightarrow>\<^sup>* v} \<in> sccs"
simps: fixes d :: "d"   and ds :: "d list"   and ve :: "(nat \<times> char list) \<times> contour \<Rightarrow> d option"   and b :: "contour" shows "|(d, ds, ve, b)| = (the_elem |d|, map AbsCFCorrect.abs ds, |ve|, |b|)"
frag_induction: fixes c :: "'b \<Rightarrow>\<^sub>0 int"   and S :: "'b set"   and P :: "('b \<Rightarrow>\<^sub>0 int) \<Rightarrow> bool" assumes "keys c \<subseteq> S"   and "P 0"   and "\<And>x. x \<in> S \<Longrightarrow> P (frag_of x)"   and "\<And>a b. \<lbrakk>P a; P b\<rbrakk> \<Longrightarrow> P (a - b)" shows "P c"
in_mul_eqI: fixes M :: "'a multiset"   and I :: "'a multiset"   and K :: "'a multiset"   and N :: "'a multiset"   and J :: "'a multiset"   and r :: "('a \<times> 'a) set" assumes "M = I + K"   and "N = I + J"   and "set_mset K \<subseteq> r \<down>m J" shows "(M, N) \<in> mul_eq r"
real_hyp2_C_exp_2dist: fixes a :: "hyp2"   and b :: "hyp2"   and c :: "hyp2"   and d :: "hyp2" assumes "a b \<congruent>\<^sub>K c d" shows "exp_2dist (Rep_hyp2 a) (Rep_hyp2 b) = exp_2dist (Rep_hyp2 c) (Rep_hyp2 d)"
bintr_uint: fixes n :: "nat"   and w :: "'c word" assumes "LENGTH('c) \<le> n" shows "take_bit n (uint w) = uint w"
union_single_eq_diff: fixes x :: "'k"   and M :: "'k multiset"   and N :: "'k multiset" assumes "add_mset x M = N" shows "M = N - {#x#}"
new_character_data_is_l_new_character_data: shows "l_new_character_data ShadowRootClass.type_wf"
Equality_3: shows "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r (S @ \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S') \<theta>"
oreachable_weakenE: fixes s :: "'a \<times> 'b"   and A :: "('a \<times> 'b, 'c) automaton"   and PS :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and PU :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and QS :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and QU :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "s \<in> oreachable A PS PU"   and "\<And>s s' a. PS s s' a \<Longrightarrow> QS s s' a"   and "\<And>s s'. PU s s' \<Longrightarrow> QU s s'" shows "s \<in> oreachable A QS QU"
tmps_le: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "bool"   and \<O> :: "nat set \<times> (nat \<Rightarrow> bool option)" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)" shows "dom \<theta> \<union> load_tmps is = {i. i < t}"
inject: fixes x2 :: "'b set"   and y2 :: "'b set" shows "(RES x2 = RES y2) = (x2 = y2)"
vcons_in_vcpowerE2: fixes xs :: "V"   and A :: "V"   and n :: "V" assumes "xs \<in>\<^sub>\<circ> A ^\<^sub>\<times> n"   and "n \<in>\<^sub>\<circ> \<omega>"   and "[]\<^sub>\<circ> \<in>\<^sub>\<circ> n" obtains x :: "V" and xs' :: "V" where "xs = xs' #\<^sub>\<circ> x" and "xs' \<in>\<^sub>\<circ> A ^\<^sub>\<times> vcard xs'" and "x \<in>\<^sub>\<circ> A" and "n = vcard (xs' #\<^sub>\<circ> x)"
open_Times: fixes S :: "'a set"   and T :: "'b set" assumes "open S"   and "open T" shows "open (S \<times> T)"
cone_subset_PolysI: fixes h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and X :: "'x set"   and U :: "'x set" assumes "h \<in> P[X]"   and "h \<noteq> 0 \<Longrightarrow> U \<subseteq> X" shows "cone (h, U) \<subseteq> P[X]"
cl_op_prop_var: fixes x :: "'b"   and y :: "'b" shows "(cl_op (x \<squnion> cl_op y) = cl_op y) = (cl_op x \<le> cl_op y)"
osum_Total: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "Total r"   and "Total r'" shows "Total (r +o r')"
exhaust: fixes y :: "acc_modi" obtains "y = Private"   | "y = Package"   | "y = Protected"   | "y = Public"
sum_diff1: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and a :: "'b" assumes "finite A" shows "sum f (A - {a}) = (if a \<in> A then sum f A - f a else sum f A)"
igSubstIGVar2_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igSubstIGVar2 MOD \<equiv> \<forall>ys y Y. igWls MOD (asSort ys) Y \<longrightarrow> igSubst MOD ys Y y (igVar MOD ys y) = Y"
cl_subset: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and X :: "'b set" assumes "matroid carrier indep"   and "X \<subseteq> carrier" shows "X \<subseteq> matroid.cl carrier indep X"
lm050: fixes a :: "('c \<times> 'd set) set"   and G :: "'d set"   and N :: "'c set" assumes "a \<in> allAllocations N G" shows "Range a \<subseteq> Pow G"
hyp: shows "\<Sum>\<^sub># M = (\<Sum>i\<in>set_mset M. count M i * i)"
FL_equivalence_implies_bisimilarity: fixes F :: "'effect fs_set"   and P :: "'state"   and Q :: "'state" assumes "FL_logically_equivalent F P Q" shows "P \<sim>\<cdot>[F] Q"
vors_3: shows "between (vertices f) ram1 ram2 = []"
union_assoc: fixes M :: "'c multiset"   and N :: "'c multiset"   and K :: "'c multiset" shows "M + N + K = M + (N + K)"
S_imp_NS: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term" assumes "S s t" shows "NS s t"
add_less_cancel_right: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "less (plus a c) (plus b c) = less a b"
freshInputTransition: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi"   and c :: "name" assumes "P \<Longrightarrow>a<b> \<prec> P'"   and "c \<sharp> P"   and "c \<noteq> b" shows "c \<sharp> P'"
wd_loop: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool" assumes "well_def body" shows "well_def (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip)"
simps_1: fixes uu :: "('a \<times> 'a) set"   and uv :: "'b \<Rightarrow> 'a"   and U :: "'a set" shows "sinks_aux uu uv U [] = U"
comp_converter_map1_out: fixes f :: "'h \<Rightarrow> 'n"   and g :: "'o \<Rightarrow> 'i"   and conv :: "('n, 'o, 'l, 'm) converter"   and conv' :: "('l, 'm, 'j, 'k) converter" shows "map_converter f g id id conv \<odot> conv' = map_converter f g id id (conv \<odot> conv')"
pair_ord_alt_spp: fixes x :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat"   and y :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat" shows "pair_ord x y = pair_ord_spp (app_pair spp_of x) (app_pair spp_of y)"
eSuc_pred: fixes n :: "enat" assumes "0 < n" shows "eSuc (n - eSuc 0) = n"
Rep_status_select: fixes ST :: "('d, 'e, 'f) status" shows "(HA ST, Conf ST, Events ST, Value ST) \<in> status"
MTF_competitive': fixes as :: "(nat \<times> nat list) list"   and rs :: "'b list"   and s0 :: "'b list" assumes "MTF_Off as rs s0" shows "T_on MTF s0 rs \<le> 2 * T s0 rs as"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and find_slot :: "bool \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and assigned_nodes :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and flatten_dom :: "((_) heap, exception, unit) prog"   and remove :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and append_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog" assumes "l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs"   and "l_set_shadow_root type_wf set_shadow_root set_shadow_root_locs"   and "l_find_slot find_slot"   and "l_get_tag_name type_wf get_tag_name get_tag_name_locs"   and "l_get_root_node get_root_node get_parent"   and "l_get_host get_host get_host_locs"   and "l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs"   and "l_to_tree_order to_tree_order"   and "l_assigned_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms assigned_nodes flatten_dom get_child_nodes get_tag_name get_root_node get_host find_slot remove append_child remove_shadow_root" shows "l_assigned_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr assigned_nodes flatten_dom get_child_nodes get_child_nodes_locs get_tag_name get_tag_name_locs get_root_node get_host get_host_locs find_slot remove append_child remove_shadow_root type_wf get_shadow_root get_shadow_root_locs set_shadow_root set_shadow_root_locs get_parent to_tree_order"
seq_chop_2: fixes s :: "'a"   and ss :: "('b \<times> 'a) list"   and ts :: "('b \<times> 'a) list"   and ars :: "('a \<times> 'b \<times> 'a) set" assumes "(s, ss @ ts) \<in> seq ars" shows "(lst (s, ss), ts) \<in> seq ars"
simps_4: fixes ux :: "typ"   and body :: "Core.term" shows "fv (Abs ux body) = fv body"
Read\<^sub>s\<^sub>b: fixes m :: "nat \<Rightarrow> nat"   and volatile :: "bool"   and a :: "nat"   and t :: "nat"   and v :: "nat"   and rs :: "'b memref list"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option" shows "(m, Read\<^sub>s\<^sub>b volatile a t v # rs, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>f (m, rs, \<O>, \<R>, \<S>)"
implc_get_offending_flows_Un: fixes M :: "'a SecurityInvariant list"   and G :: "'a list_graph" shows "set ` set (implc_get_offending_flows M G) = (\<Union>m\<in>set M. set ` set (implc_offending_flows m G))"
well_order_on_def: fixes A :: "'d set"   and r :: "('d \<times> 'd) set" shows "well_order_on A r \<equiv> linear_order_on A r \<and> wf (r - Id)"
map_rev_iterator_linord_is_it: fixes m :: "('c \<Rightarrow> bool) \<Rightarrow> ('d \<times> 'e \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and it :: "'d \<Rightarrow> 'e option" assumes "map_iterator_rev_linord m it" shows "map_iterator m it"
t_invk: fixes CT :: "nat \<Rightarrow> classDef option"   and \<Gamma> :: "nat \<Rightarrow> nat option"   and e0 :: "exp"   and C0 :: "nat"   and m :: "nat"   and Ds :: "nat list"   and C :: "nat"   and es :: "exp list"   and Cs :: "nat list" assumes "CT;\<Gamma> \<turnstile> e0 : C0"   and "mtype(CT,m,C0) = Ds \<rightarrow> C"   and "CT;\<Gamma> \<turnstile>+ es : Cs"   and "CT \<turnstile>+ Cs <: Ds"   and "length es = length Ds" shows "CT;\<Gamma> \<turnstile> MethodInvk e0 m es : C"
\<pi>_2: shows "permute_list \<pi> xs = ys"
j: shows "f `` {v} \<subseteq> vertices (S j)"
append_one_prefix: fixes xs :: "'a list"   and ys :: "'a list" assumes "prefix xs ys"   and "length xs < length ys" shows "prefix (xs @ [ys ! length xs]) ys"
simple_term_inverse: fixes y :: "('d, 'e) trm" assumes "y \<in> {\<theta>. dfree \<theta>}" shows "raw_term (simple_term y) = y"
refl': fixes x :: "'b"   and y :: "'b"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "x = y" shows "r\<^sup>*\<^sup>* x y"