hcju/isabelleps huggingface dataset:
simps: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and a1 :: "'c rule list"   and a2 :: "state" shows "\<Gamma>,\<gamma>,p\<turnstile> a1 \<Rightarrow>\<^sub>r a2 = (a1 = [] \<and> a2 = Undecided \<or> (\<exists>m rs. a1 = Rule m Accept # rs \<and> a2 = Decision FinalAllow \<and> matches \<gamma> m p) \<or> (\<exists>m rs. a1 = Rule m Drop # rs \<and> a2 = Decision FinalDeny \<and> matches \<gamma> m p) \<or> (\<exists>m rs. a1 = Rule m Reject # rs \<and> a2 = Decision FinalDeny \<and> matches \<gamma> m p) \<or> (\<exists>m rs. a1 = Rule m Return # rs \<and> a2 = Undecided \<and> matches \<gamma> m p) \<or> (\<exists>rs t m. a1 = Rule m Log # rs \<and> a2 = t \<and> \<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>r t) \<or> (\<exists>rs t m. a1 = Rule m Empty # rs \<and> a2 = t \<and> \<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>r t) \<or> (\<exists>m rs t a. a1 = Rule m a # rs \<and> a2 = t \<and> \<not> matches \<gamma> m p \<and> \<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>r t) \<or> (\<exists>m chain rs X rrs. a1 = Rule m (Call chain) # rrs \<and> a2 = Decision X \<and> matches \<gamma> m p \<and> \<Gamma> chain = Some rs \<and> \<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>r Decision X) \<or> (\<exists>chain rs rrs t m. a1 = Rule m (Call chain) # rrs \<and> a2 = t \<and> \<Gamma> chain = Some rs \<and> \<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>r Undecided \<and> \<Gamma>,\<gamma>,p\<turnstile> rrs \<Rightarrow>\<^sub>r t))"
weak_put_eq: fixes x :: "'d \<Longrightarrow> 'e"   and \<sigma> :: "'e"   and k :: "'d"   and u :: "'d"   and \<rho> :: "'e"   and v :: "'d" assumes "mwb_lens x"   and "\<sigma> \<in> \<S>\<^bsub>x\<^esub>"   and "get\<^bsub>x\<^esub> \<sigma> = k"   and "put\<^bsub>x\<^esub> \<sigma> u = put\<^bsub>x\<^esub> \<rho> v" shows "put\<^bsub>x\<^esub> \<rho> k = \<sigma>"
integral_norm_bound_integral_component: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'c"   and k :: "'c" assumes "f integrable_on S"   and "g integrable_on S"   and "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> g x \<bullet> k" shows "norm (integral S f) \<le> integral S g \<bullet> k"
neutral: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "\<forall>x\<in>A. g x = (1::'c)" shows "prod g A = (1::'c)"
monad_state_writerT: fixes return :: "'a \<times> 'b list \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b list \<Rightarrow> 'c) \<Rightarrow> 'c"   and get :: "('d \<Rightarrow> 'c) \<Rightarrow> 'c"   and put :: "'d \<Rightarrow> 'c \<Rightarrow> 'c" assumes "monad_state return bind get put" shows "monad_state (return_writer return) (bind_writer return bind) (get_writer get) (put_writer put)"
isomorphic_symmetric: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "VV.isomorphic f g" shows "VV.isomorphic g f"
Rep_quot_inverse: fixes x :: "'d quot" shows "Abs_quot (Rep_quot x) = x"
real_non_neg_vec_def: fixes v :: "(complex, 'b) Finite_Cartesian_Product.vec" shows "real_non_neg_vec v \<equiv> \<forall>a\<in>vec_elements_h v. a \<in> \<real> \<and> 0 \<le> Re a"
the_pure_pure: fixes R :: "('e \<times> 'f) set" shows "the_pure (pure R) = R"
sinterleave_snth_1: fixes n :: "nat"   and s1 :: "'d stream"   and s2 :: "'d stream" assumes "even n" shows "sinterleave s1 s2 !! n = s1 !! (n div 2)"
iprod_left_diff_distrib: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" shows "\<langle>xs - ys,zs\<rangle> = \<langle>xs,zs\<rangle> - \<langle>ys,zs\<rangle>"
Not\<^sub>\<alpha>_def: shows "FL_Formula.Not\<^sub>\<alpha> \<equiv> (FL_Formula.rep_Tree\<^sub>\<alpha> ---> FL_Formula.abs_Tree\<^sub>\<alpha>) FL_Formula.Tree.tNot"
coalg_F2in: fixes B1 :: "'d set"   and A :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('f, 'd, 'e) F2"   and x :: "'e" assumes "coalg B1 A s1 s2"   and "x \<in> A" shows "s2 x \<in> F2in UNIV B1 A"
bv_length: fixes xs :: "bool list"   and ys :: "bool list" assumes "xs \<preceq>\<^sub>b ys" shows "length xs = length ys"
prems_18: shows "getSATFlag state' = TRUE \<longrightarrow> satisfiable F0'"
assms_2: shows "\<forall>t\<in>subst_range \<delta>. fv t = {} \<or> (\<exists>x. t = Var x)"
of_rat_of_nat_eq: fixes n :: "nat" shows "of_rat (rat_of_nat n) = of_nat n"
push_down_state_rank_token_run: fixes q :: "'a ltl_prop_equiv_quotient"   and n :: "nat"   and i :: "nat" assumes "state_rank q n = Some i" shows "\<exists>x. token_run x n = q \<and> x \<le> n"
completion: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and C :: "'c set"   and B :: "'c set"   and B' :: "'c set" assumes "F \<in> A \<longmapsto> A' \<union> C"   and "F \<in> A' co A' \<union> C"   and "F \<in> B \<longmapsto> B' \<union> C"   and "F \<in> B' co B' \<union> C" shows "F \<in> A \<inter> B \<longmapsto> A' \<inter> B' \<union> C"
Step_AxSubst: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and i :: "nat"   and a :: "axiom"   and \<sigma> :: "('sf, 'sc, 'sz) subst" shows "step_result (SG, C) (i, AxSubst a \<sigma>) = (closeI SG i, C)"
Int: fixes a :: "'a set"   and b :: "'a set" assumes "a \<in> M"   and "b \<in> M" shows "a \<inter> b \<in> M"
ideD_2: fixes a :: "(bool, 'arr list) concrete_category.arr" assumes "ide a" shows "local.dom a = a"
sg4: shows "if activation2 t = [] then nStore j t = recv t else nStore j t = []"
cring_simprules_14: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> y = y \<otimes>\<^bsub>R\<^esub> x"
intro: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set" assumes "Markov_Decision_Process K"   and "Markov_Decision_Process_Invariant_axioms K S" shows "Markov_Decision_Process_Invariant K S"
wax_def: fixes \<sigma> :: "'c" shows "wax \<sigma> \<equiv> [Pos (Pr (Guard \<sigma>) [Fn (Wit \<sigma>) []])]"
subst_list_def: fixes \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and ys :: "(('a, 'b) Term.term \<times> ('a, 'b) Term.term) list" shows "subst_list \<sigma> ys = map (\<lambda>p. (fst p \<cdot> \<sigma>, snd p \<cdot> \<sigma>)) ys"
accept_def: fixes G :: "('a, 'b, 'c) sa_rec_scheme"   and w :: "nat \<Rightarrow> 'b" assumes "sa G" shows "sa.accept G w \<equiv> \<exists>r. graph_defs.is_run G r \<and> w = sa_L G \<circ> r"
lex_ord_pair_valE: fixes x :: "'o"   and xs :: "('a \<times> 'd) list"   and ys :: "('a \<times> 'e) list"   and f :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> order option"   and aux :: "order option" assumes "oalist_inv_raw x xs"   and "oalist_inv_raw x ys"   and "lex_ord_pair x f xs ys = aux"   and "aux \<noteq> Some Eq" obtains k :: "'a" where "k \<in> fst ` set xs \<union> fst ` set ys" and "aux = f k (lookup_pair x xs k) (lookup_pair x ys k)" and "\<And>k'. \<lbrakk>k' \<in> fst ` set xs \<union> fst ` set ys; lt x k' k\<rbrakk> \<Longrightarrow> f k' (lookup_pair x xs k') (lookup_pair x ys k') = Some Eq"
len_hchop_left_right_border: fixes v :: "view"   and ts :: "traffic"   and c :: "cars"   and v1 :: "view"   and v2 :: "view" shows "len v ts c = ext v \<and> v=v1\<parallel>v2 \<longrightarrow> right (len v1 ts c) = right (ext v1)"
North_R3_def: shows "North_R3 = (0, 0, 1)"
is_sur_def: fixes x :: "'a" shows "is_sur x \<equiv> 1' \<le> x\<^sup>\<smile> ; x"
induct: fixes P :: "ty \<Rightarrow> bool"   and ty :: "ty" assumes "\<And>list. P (PR list)"   and "\<And>ty. P ty \<Longrightarrow> P (NOT ty)"   and "\<And>ty1 ty2. \<lbrakk>P ty1; P ty2\<rbrakk> \<Longrightarrow> P (ty1 AND ty2)"   and "\<And>ty1 ty2. \<lbrakk>P ty1; P ty2\<rbrakk> \<Longrightarrow> P (ty1 OR ty2)"   and "\<And>ty1 ty2. \<lbrakk>P ty1; P ty2\<rbrakk> \<Longrightarrow> P (ty1 IMP ty2)" shows "P ty"
is_interval_compact: fixes S :: "'b set" shows "(is_interval S \<and> compact S) = (\<exists>a b. S = cbox a b)"
cSup_cInf: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "S \<noteq> {}"   and "preorder.bdd_above less_eq S" shows "Sup S = Inf {x. \<forall>s\<in>S. less_eq s x}"
cert_valid_lemma: fixes i :: "nat"   and N :: "nat"   and C :: "agent"   and cardSK :: "nat"   and N1 :: "msg"   and evs :: "event list" assumes "Crypt (priSK (CA i)) \<lbrace>Hash \<lbrace>Nonce N, Pan (pan C)\<rbrace>, Key cardSK, N1\<rbrace> \<in> parts (knows Spy evs)"   and "CA i \<notin> bad"   and "evs \<in> set_cr" shows "\<exists>KC2 X Y. Says (CA i) C (Crypt KC2 \<lbrace>X, certC (pan C) cardSK N onlySig (priSK (CA i)), Y\<rbrace>) \<in> set evs"
is_sig_GB_upt_le: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and u :: "'t"   and v :: "'t" assumes "is_sig_GB_upt d G u"   and "v \<preceq>\<^sub>t u" shows "is_sig_GB_upt d G v"
combine_non_Leaves: fixes t :: "'b tree"   and u :: "'b tree" assumes "t \<noteq> Leaf"   and "u \<noteq> Leaf" shows "combine t u = Node (l t) (r u) (val t + val u) (discharge t) u"
Says_Server_message_form: fixes A :: "agent"   and K' :: "nat"   and N :: "msg"   and B :: "agent"   and K :: "nat"   and X :: "msg"   and evs :: "event list" assumes "Says Server A (Crypt K' \<lbrace>N, Agent B, Key K, X\<rbrace>) \<in> set evs"   and "evs \<in> ns_shared" shows "K \<notin> range shrK \<and> X = Crypt (shrK B) \<lbrace>Key K, Agent A\<rbrace> \<and> K' = shrK A"
ChiPlus_ChiRel_Ex: fixes S :: "'d"   and T :: "'d"   and A :: "('d, 'e, 'f) hierauto" assumes "(S, T) \<in> ChiPlus A"   and "(S, T) \<notin> ChiRel A" shows "\<exists>U. (S, U) \<in> ChiPlus A \<and> (U, T) \<in> ChiRel A"
cbelow_on_symmetric: fixes A :: "'b set"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and z :: "'b"   and rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "cbelow_on A ord z rel a b"   and "symp rel" shows "cbelow_on A ord z rel b a"
simps_3: fixes R :: "('f, 'g, 'h) Resumption.resumption set" assumes "\<forall>r\<in>R. Resumption.resumption.is_Done r" shows "Resumption.resumption.result (resumption_lub R) = flat_lub None (Resumption.resumption.result ` R)"
q: fixes Y :: "'a multiset"   and y :: "'a" assumes "(Y, X) \<in> (mult (r\<^sup>+))\<^sup>*"   and "y \<in># Y" shows "y \<in> ((r\<^sup>+)\<^sup>*)\<inverse> `` set_mset X"
monotone_results: shows "monotone resumption_ord (\<subseteq>) results"
infimum_pair: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a \<in> infimum {x. x = b \<or> x = c}) = (a = b \<sqinter> c)"
span_finite: fixes S :: "'b set" assumes "finite S" shows "lf.vs1.span S = range (\<lambda>u. \<Sum>v\<in>S. scaleB (u v) v)"
pmod_def: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list"   and q :: "'c list" assumes "ring R" shows "ring.pmod R p q = (if q = [] then p else snd (ring.long_division R p q))"
distinct_member_remove1: fixes equal :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and x :: "'c" assumes "equal equal"   and "equal_base.list_distinct equal xs" shows "equal_base.list_member equal (equal_base.list_remove1 equal x xs) = (equal_base.list_member equal xs)(x := False)"
simps_1: fixes s :: "'a \<Rightarrow> 'b option" shows "trace_parallel_plan_sas_plus s [] = [s]"
mcalc_iff: fixes \<sigma> :: "'tp"   and c :: "('c, 'd) lit list" shows "(\<sigma> \<turnstile> c) = (infTp \<sigma> \<or> (\<forall>x\<in>nvC c. GE.tpOfV x \<noteq> \<sigma>))"
l1_defs_2: shows "l1_init \<equiv> {\<lparr>ik = {}, secret = {}, progress = Map.empty, signals = \<lambda>x. 0\<rparr>}"
af_subsequence_U: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set"   and n :: "nat" shows "af (\<phi> U\<^sub>n \<psi>) (prefix (Suc n) w) = af \<psi> (prefix (Suc n) w) or\<^sub>n (af \<phi> (prefix (Suc n) w) and\<^sub>n af (\<phi> U\<^sub>n \<psi>) (w [1 \<rightarrow> Suc n]))"
uint_of_bl_is_bl_to_bin: fixes l :: "bool list" assumes "length l \<le> LENGTH('b)" shows "uint (of_bl l) = bl_to_bin l"
set_filter: fixes P :: "'e \<Rightarrow> bool"   and xs :: "'e list" shows "set (filter P xs) = {x \<in> set xs. P x}"
simps: fixes a1 :: "'a option"   and a2 :: "('a \<times> 'a option) list" shows "insert_seq a1 a2 = ((\<exists>start oid. a1 = start \<and> a2 = [(oid, start)]) \<or> (\<exists>start list prev ref oid. a1 = start \<and> a2 = list @ [(prev, ref), (oid, Some prev)] \<and> insert_seq start (list @ [(prev, ref)])))"
hyps_3: fixes I :: "'a set" assumes "i \<notin> J"   and "\<forall>a\<in>J. i dvd a"   and "i \<in> I" shows "ideal_generated I = ideal_generated (I - J)"
rejects_subset: fixes \<F> :: "nat set set"   and S :: "nat set"   and M :: "nat set"   and N :: "nat set" assumes "rejects \<F> S M"   and "N \<subseteq> M" shows "rejects \<F> S N"
toSink: fixes L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Sink :: "'a set"   and s :: "'a" assumes "Shallow_Idle L AP' S s0 \<delta> AP f Sink"   and "s \<in> S" shows "Shallow_Idle.toSink \<delta> Sink s \<in> \<delta> s \<inter> Sink"
fun_ordD: fixes ord :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and x :: "'e" assumes "fun_ord ord f g" shows "ord (f x) (g x)"
span_subspace_induct: fixes x :: "'d"   and S :: "'d set"   and P :: "'d set" assumes "x \<in> cspan S"   and "csubspace P"   and "\<And>x. x \<in> S \<Longrightarrow> x \<in> P" shows "x \<in> P"
in_image_insert_iff: fixes B :: "'a set set"   and x :: "'a"   and A :: "'a set" assumes "\<And>C. C \<in> B \<Longrightarrow> x \<notin> C" shows "(A \<in> insert x ` B) = (x \<in> A \<and> A - {x} \<in> B)"
class_from_belongs_stalk: fixes s :: "('a set set \<times> ('a set \<Rightarrow> ('a \<times> 'a) set)) set" assumes "s \<in> st.carrier_stalk" obtains U :: "'a set set" and s' :: "'a set \<Rightarrow> ('a \<times> 'a) set" where "is_zariski_open U" and "\<pp> \<in> U" and "s' \<in> \<O> U" and "s = st.class_of U s'"
assms_2: assumes "\<paragraph>''else''"   and "\<not> bval b s" shows "wlp \<pi> c\<^sub>2 Q s"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_append: fixes A :: "('e strand_label \<times> ('f, 'g) stateful_strand_step) list"   and B :: "('e strand_label \<times> ('f, 'g) stateful_strand_step) list" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (A @ B) = dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A @ dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t B"
consistent: fixes a :: "'a"   and b :: "'a" assumes "a \<in> F"   and "(a, b) \<in> r" shows "b \<in> F"
integrable_const: fixes a :: "'b" shows "integrable M (\<lambda>x. a)"
l1_trans_def: shows "l1_trans \<equiv> \<Union>m Ra Rb A B x. (l1_step1 Ra A B \<union> l1_step2 Rb A B x \<union> l1_step3 Ra A B x \<union> l1_step4 Rb A B x \<union> l1_learn m)\<^sup>="
carrier_subgroup_generated_subset: fixes G :: "('a, 'b) monoid_scheme"   and A :: "'a set" assumes "Group.group G" shows "carrier (subgroup_generated G A) \<subseteq> carrier G"
summable_zero: shows "summable (\<lambda>n. 0::'b)"
mult_fract: fixes a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" shows "Fraction_Field.Fract a b * Fraction_Field.Fract c d = Fraction_Field.Fract (a * c) (b * d)"
list_all3_list_all2_conv: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list" shows "list_all3 R xs xs ys = list_all2 (\<lambda>x. R x x) xs ys"
eq1: shows "f \<equiv> C1 (ccpo.fixp (fun_lub luba) (fun_ord orda) (\<lambda>f. U1 (F (C1 f))))"
integrable_cmul: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and c :: "real" assumes "f integrable_on S" shows "(\<lambda>x. c *\<^sub>R f x) integrable_on S"
splits_at_append: fixes u :: "symbol list"   and i :: "nat"   and u1 :: "symbol list"   and N :: "symbol"   and u2 :: "symbol list"   and v :: "symbol list" assumes "splits_at u i u1 N u2" shows "splits_at (u @ v) i u1 N (u2 @ v)"
i_to_ns_sat: fixes I :: "'i set"   and v' :: "(nat, 'a) mapping"   and cs :: "('i \<times> constraint) list" assumes "(I, \<langle>v'\<rangle>) \<Turnstile>\<^sub>i\<^sub>n\<^sub>s\<^sub>s  set (to_ns cs)" shows "(I, \<langle>from_ns v' (map snd (to_ns cs))\<rangle>) \<Turnstile>\<^sub>i\<^sub>c\<^sub>s set cs"
le_infE: fixes x :: "'a multiset"   and a :: "'a multiset"   and b :: "'a multiset" assumes "x \<subseteq># a \<inter># b" obtains "x \<subseteq># a" and "x \<subseteq># b"
is_rbt_rbt_sorted: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "('a, 'b) RBT_Impl.rbt" assumes "ord.is_rbt less t" shows "ord.rbt_sorted less t"
orbit_equiv: fixes b :: "'a"   and f :: "'a perm"   and a :: "'a" assumes "b \<in> orbit f a" shows "orbit f b = orbit f a"
subtypings_index: fixes CT :: "nat \<Rightarrow> classDef option"   and Cs :: "nat list"   and Ds :: "nat list"   and i :: "nat" assumes "CT \<turnstile>+ Cs <: Ds"   and "i < length Cs" shows "CT \<turnstile> Cs ! i <: Ds ! i"
assms_6: shows "InvariantGetReasonIsReason (getReason state) (getF state) (getM state) (set (getQ state))"
not_undec: fixes p :: "(32, 'b) simple_packet_scheme" shows "simple_fw fw p \<noteq> Undecided"
gb_schema_aux_domI1: fixes data :: "nat \<times> nat \<times> 'd"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list" shows "gb_schema_aux_dom (data, bs, [])"
sup_mono: fixes a :: "'d multiset"   and c :: "'d multiset"   and b :: "'d multiset"   and d :: "'d multiset" assumes "a \<subseteq># c"   and "b \<subseteq># d" shows "a \<union># b \<subseteq># c \<union># d"
p: shows "p \<in> setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p a"
Abs_fresh_star_iff_1: fixes as :: "atom set"   and bs :: "atom set"   and x :: "'a" shows "as \<sharp>* ([bs]set. x) = (as - bs) \<sharp>* x"
inf_V_def: fixes x :: "V"   and y :: "V" shows "x \<inter>\<^sub>\<circ> y \<equiv> set (elts x \<inter> elts y)"
exec_assign: fixes \<sigma> :: "'a control_state_scheme"   and f :: "'a control_state_scheme \<Rightarrow> 'a control_state_scheme"   and M :: "'a control_state_scheme \<Rightarrow> (bool \<times> 'a control_state_scheme) option" assumes "exec_stop \<sigma>" shows "(\<sigma> \<Turnstile> _bind_SE _ (assign f) M) = (\<sigma> \<Turnstile> M)"
exec_instr_Throw: fixes 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 Throw P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh = (let xp' = if hd stk = Null then \<lfloor>addr_of_sys_xcpt NullPointer\<rfloor> else \<lfloor>the_Addr (hd stk)\<rfloor> in (xp', h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, ics) # frs, sh))"
simps_7: fixes c :: "CTypeId" shows "isprimitive (CClassT c) = False"
invariant3p_correct_jvm_state_mexecdT: fixes addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and empty_heap :: "'c"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'c \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_read :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> bool"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and hconf :: "'c \<Rightarrow> bool"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and \<Phi> :: "String.literal \<Rightarrow> String.literal \<Rightarrow> (ty list \<times> ty err list) option list"   and spurious_wakeups :: "bool" assumes "JVM_conf_read addr2thread_id thread_id2addr empty_heap allocate typeof_addr heap_read heap_write hconf P"   and "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P" shows "invariant3p (multithreaded_base.redT JVM_final (\<lambda>t ((xcp, frstls), h) ta ((xcp', frstls'), h'). JVM_heap_base.exec_1_d addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write P t (Normal (xcp, h, frstls)) ta (Normal (xcp', h', frstls'))) convert_RA) (JVM_heap_conf_base.correct_jvm_state thread_id2addr empty_heap allocate typeof_addr hconf P \<Phi>)"
game01'_def: shows "game01' = (\<lambda>decrypt \<A>. key_gen \<bind> (\<lambda>key. coin_spmf \<bind> (\<lambda>b. exec_gpv (\<dagger>(ind_cca'.oracle_encrypt key b) \<oplus>\<^sub>O decrypt key) \<A> (False, {}) \<bind> (\<lambda>(b', bad', L'). return_spmf (b = b', bad')))))"
allDefs_disjoint: fixes n :: "'node"   and g :: "'g"   and m :: "'node" assumes "n \<in> set (\<alpha>n g)"   and "m \<in> set (\<alpha>n g)"   and "n \<noteq> m" shows "allDefs g n \<inter> allDefs g m = {}"
bisim1Call1: fixes P :: "'a prog"   and h :: "'heap"   and obj :: "'addr expr1"   and obj' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and M :: "String.literal"   and ps :: "'addr expr1 list" assumes "P,obj,h \<turnstile> (obj', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,obj\<bullet>M(ps),h \<turnstile> (obj'\<bullet>M(ps), xs) \<leftrightarrow> (stk, loc, pc, xcp)"
class_add_sees_field: fixes P :: "jvm_prog"   and C\<^sub>0 :: "char list"   and F :: "char list"   and b :: "staticb"   and T :: "ty"   and D :: "char list"   and C :: "char list"   and cdec :: "jvm_method class" assumes "P \<turnstile> C\<^sub>0 sees F,b:T in D"   and "\<not> is_class P C" shows "class_add P (C, cdec) \<turnstile> C\<^sub>0 sees F,b:T in D"
r2: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "x = y"   and "\<not> prefix xs ys"   and "P xs ys" shows "P (x # xs) (y # ys)"
isFault_simps_1: fixes s :: "'d" shows "isFault (Normal s) = False"
affine_borel_measurable_vector: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and a :: "'d"   and b :: "real" assumes "f \<in> borel_measurable M" shows "(\<lambda>x. a + b *\<^sub>R f x) \<in> borel_measurable M"
saturated_def: fixes r :: "'rule" shows "saturated r \<equiv> alw (\<lambda>xs. holds (enabledAtStep r) xs \<longrightarrow> ev (holds (\<lambda>step. snd step = r)) xs)"
simps_4: fixes x21 :: "'f"   and x22 :: "bool"   and x11 :: "'f"   and x12 :: "'g"   and x13 :: "'h" shows "ThreadExists x21 x22 \<noteq> NewThread x11 x12 x13"
polypow_norm: fixes p :: "'b poly"   and k :: "nat" assumes "isnpoly p" shows "isnpoly (p ^\<^sub>p k)"
key_compare_cases: fixes y :: "'a \<Rightarrow> 'a \<Rightarrow> order" assumes "y \<in> {compare. comparator compare}" obtains x :: "'a key_order" where "y = key_compare x"
ide_in_hom: fixes a :: "'b" shows "J.ide a = \<guillemotleft>a : a \<rightarrow>\<^sub>J a\<guillemotright>"
UN_extend_simps_3: fixes A :: "'b set"   and B :: "'c \<Rightarrow> 'b set"   and C :: "'c set" shows "A \<union> \<Union> (B ` C) = (if C = {} then A else \<Union>x\<in>C. A \<union> B x)"
assms_2: fixes x :: "'a"   and a\<^sub>1 :: "'b"   and a\<^sub>2 :: "'b" assumes "x \<in> S"   and "f x a\<^sub>1 a\<^sub>2" shows "g a\<^sub>1 = g a\<^sub>2"
prems_4: shows "wellformed_policy3Pr (ys @ [y])"
subst_of_map_def: fixes d :: "'e \<Rightarrow> 'd"   and \<sigma> :: "'e \<Rightarrow> 'd option"   and x :: "'e" shows "subst_of_map d \<sigma> x = (case \<sigma> x of None \<Rightarrow> d x | Some t \<Rightarrow> t)"
Always_LeadsToD: fixes F :: "'c program"   and INV :: "'c set"   and A :: "'c set"   and A' :: "'c set" assumes "F \<in> Always INV"   and "F \<in> A \<longmapsto>w A'" shows "F \<in> A \<longmapsto>w INV \<inter> A'"
atypes_inhabited: fixes a :: "'atom" shows "\<exists>c. \<Gamma> (Fun c []) = Var a \<and> wf\<^sub>t\<^sub>r\<^sub>m (Fun c []) \<and> public c \<and> arity c = 0"