hcju/isabelleps huggingface dataset:
process_mru_Max: fixes sa :: "'a v_state_scheme"   and p :: "process"   and r :: "nat"   and v :: "val" assumes "sa \<in> Vinv1"   and "process_mru (votes sa) p = Some (r, v)" shows "votes sa r p = Some v \<and> (\<forall>r'>r. votes sa r' p = None)"
sas_plus_problem_to_strips_problem_def: fixes \<Psi> :: "('c, 'd) sas_plus_problem" shows "\<phi> \<Psi>  \<equiv> let vs = concat (map (\<lambda>v. map (\<lambda>as. as) (possible_assignments_for \<Psi> v)) (\<Psi>\<^sub>\<V>\<^sub>+)); ops = map (sasp_op_to_strips \<Psi>) (\<Psi>\<^sub>\<O>\<^sub>+); I = \<phi>\<^sub>S \<Psi> \<Psi>\<^sub>I\<^sub>+ in Let (\<phi>\<^sub>S \<Psi> \<Psi>\<^sub>G\<^sub>+) (problem_for vs ops I)"
assms_1: fixes r :: "char list"   and s :: "char list" shows "a r @ s = a (r @ s)"
simps_2: fixes p :: "int"   and s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and k :: "nat" assumes "padic_integers p" shows "res_seq s (Suc k) = const_res_subseq (Suc k) (res_seq s k)"
healthy_nearly_healthy: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "healthy t" shows "nearly_healthy t"
TimesL_def: fixes r :: "'b regex"   and S :: "'b regex set" shows "TimesL r S = Times r ` S"
submonoid_def: fixes N :: "'c set"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" shows "submonoid N M composition unit \<equiv> Group_Theory.monoid M composition unit \<and> submonoid_axioms N M composition unit"
Abs_bset_inverse: fixes y :: "'a set" assumes "y \<in> {A. |A| <o natLeq +c |UNIV|}" shows "set_bset (Abs_bset y) = y"
lm008: fixes n :: "'a"   and N :: "'a set"   and G :: "'b" assumes "n \<in> N" shows "{(G, n)} \<in> injections {G} N"
simps_2: fixes n :: "nat" shows "Suc n\<^sub>\<nat> = ZFC_in_HOL.succ (n\<^sub>\<nat>)"
size1_size: fixes t :: "'b tree" shows "size1 t = size t + 1"
properfactor_trans2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "properfactor (add_monoid R) a b"   and "b divides\<^bsub>add_monoid R\<^esub> c"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "properfactor (add_monoid R) a c"
l_find_slot\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   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 get_mode :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, shadow_root_mode) prog"   and get_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, char list option) prog"   and get_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   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 first_in_tree_order :: "(_) object_ptr \<Rightarrow> ((_) object_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog) \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and find_slot :: "bool \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and assigned_slot :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<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" shows "l_find_slot\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_parent get_parent_locs get_shadow_root get_shadow_root_locs get_mode get_mode_locs get_attribute get_attribute_locs get_tag_name get_tag_name_locs first_in_tree_order find_slot assigned_slot type_wf known_ptr known_ptrs get_child_nodes get_child_nodes_locs to_tree_order \<equiv> ((l_get_parent type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs \<and> l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs) \<and> l_get_mode type_wf get_mode get_mode_locs \<and> l_get_attribute type_wf get_attribute get_attribute_locs) \<and> (l_get_tag_name type_wf get_tag_name get_tag_name_locs \<and> l_to_tree_order to_tree_order) \<and> l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M to_tree_order first_in_tree_order \<and> l_find_slot\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_parent get_shadow_root get_mode get_attribute get_tag_name first_in_tree_order find_slot assigned_slot"
null_sets_outer_le: fixes S :: "'c set"   and M :: "'c measure" shows "(S \<in> null_sets (completion M)) = (\<forall>e>0. \<exists>T\<in>fmeasurable (completion M). S \<subseteq> T \<and> Sigma_Algebra.measure (completion M) T \<le> e)"
is_faithful: fixes f :: "'a"   and f' :: "'a" assumes "par f f'"   and "cod \<iota> \<otimes> f = cod \<iota> \<otimes> f'" shows "f = f'"
insert_rank_invar: fixes q :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and e :: "'a"   and a :: "'b" assumes "SkewBinomialHeapStruc.rank_skew_invar q" shows "SkewBinomialHeapStruc.rank_skew_invar (SkewBinomialHeapStruc.insert e a q)"
subspace_scale: fixes S :: "'b set"   and x :: "'b"   and c :: "'a" assumes "subspace S"   and "x \<in> S" shows "c *s x \<in> S"
summable_norm_cancel: fixes f :: "nat \<Rightarrow> 'b" assumes "summable (\<lambda>n. norm (f n))" shows "summable f"
x_1: shows "(x, \<tau>') \<notin> FV \<Gamma>"
ws_interface_induct: fixes G :: "prog"   and I :: "qtname"   and P :: "qtname \<Rightarrow> bool" assumes "is_iface G I"   and "ws_prog G"   and "\<And>I i. \<lbrakk>iface G I = Some i; \<forall>J\<in>set (isuperIfs i). G\<turnstile>I\<prec>I1J \<and> P J \<and> is_iface G J\<rbrakk> \<Longrightarrow> P I" shows "P I"
more_resources_suffice: fixes f :: "'b \<rightarrow> 'a"   and r :: "'b"   and r' :: "'b" assumes "f\<cdot>r \<noteq> \<bottom>"   and "r \<sqsubseteq> r'" shows "f\<cdot>r' \<noteq> \<bottom>"
dg_prod_components_4: fixes I :: "V"   and \<AA> :: "V \<Rightarrow> V" shows "dg_prod I \<AA>\<lparr>Cod\<rparr> = (\<lambda>f\<in>\<^sub>\<circ>\<Prod>\<^sub>\<circ>i\<in>\<^sub>\<circ>I. \<AA> i\<lparr>Arr\<rparr>. \<lambda>i\<in>\<^sub>\<circ>I. \<AA> i\<lparr>Cod\<rparr>\<lparr>f\<lparr>i\<rparr>\<rparr>)"
coalg_F1set2: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('f, 'd, 'e) F2"   and a :: "'d" assumes "coalg B1 B2 s1 s2"   and "a \<in> B1" shows "F1set2 (s1 a) \<subseteq> B1"
assms: shows "[last p1, hd p2] \<notin> {[(fst e, w), (w, snd e)], [(snd e, w), (w, fst e)]}"
cases: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "integrable M f" obtains x :: "'d" where "has_bochner_integral M f x"
bindET_throwET: fixes e :: "'g"   and k :: "'h \<rightarrow> 'e\<cdot>('f, 'g) errorT" shows "bindET\<cdot>(throwET\<cdot>e)\<cdot>k = throwET\<cdot>e"
le_max_iff_disj: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and z :: "'k"   and x :: "'k"   and y :: "'k" assumes "class.linorder less_eq less" shows "less_eq z (ord.max less_eq x y) = (less_eq z x \<or> less_eq z y)"
factors_equal: fixes a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" assumes "a = b"   and "c = d" shows "a * c = b * d"
simps_5: fixes \<psi>\<^sub>1 :: "'b ltln"   and \<psi>\<^sub>2 :: "'b ltln"   and Y :: "'b ltln set" shows "(\<psi>\<^sub>1 M\<^sub>n \<psi>\<^sub>2)[Y]\<^sub>\<mu> = \<psi>\<^sub>1[Y]\<^sub>\<mu> M\<^sub>n \<psi>\<^sub>2[Y]\<^sub>\<mu>"
part: shows "pdevs_val e (pdevs_of_list (x # xs)) = pdevs_val f (pdevs_of_list (filter (coll 0 x) (x # xs))) + pdevs_val g (pdevs_of_list (filter (Not \<circ> coll 0 x) (x # xs)))"
path_components_of_eq_singleton: fixes X :: "'b topology"   and S :: "'b set" shows "(path_components_of X = {S}) = (path_connected_space X \<and> topspace X \<noteq> {} \<and> S = topspace X)"
position_def: fixes A :: "'b llist set"   and N :: "enat"   and e :: "'b list" assumes "GSgame A N" shows "GSgame.position N e \<equiv> enat (length e) \<le> 2 * N"
valid_path_same_level_path_Append: fixes as :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list"   and as' :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list" assumes "CFGExit.valid_path as"   and "CFGExit.same_level_path as'" shows "CFGExit.valid_path (as @ as')"
f_deriv: fixes x :: "real" assumes "x \<in>\<^sub>r compute_bound_tm prec I a tg"   and "c (compute_bound_tm prec I a tg)" shows "isDERIV 0 f [x]"
add_implies_diff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and c :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.cancel_comm_monoid_add plus minus zero"   and "plus c b = a" shows "c = minus a b"
fully_abstract_wrt_target_relation: fixes RelT :: "('procT \<times> 'procT) set" shows "\<exists>RelS. \<forall>S1 S2. ((S1, S2) \<in> RelS) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> RelT)"
asc_main_theorem: fixes M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'f) FSM"   and m :: "nat"   and FAIL :: "'f \<times> 'f"   and PM :: "('d, 'e, 'f \<times> 'f) FSM"   and V :: "'d list set"   and \<Omega> :: "('d, 'e) ATC set"   and i :: "nat" assumes "OFSM M1"   and "OFSM M2"   and "asc_fault_domain M2 M1 m"   and "test_tools M2 M1 FAIL PM V \<Omega>"   and "final_iteration M2 M1 \<Omega> V m i" shows "M1 \<preceq> M2 = M1 \<preceq>\<lbrakk>(TS M2 M1 \<Omega> V m i).\<Omega>\<rbrakk> M2"
lsorted_LCons: fixes x :: "'a"   and xs :: "'a llist" shows "lsorted (LCons x xs) = (lsorted xs \<and> (\<forall>y\<in>lset xs. x \<le> y))"
domintros_9: fixes v :: "op0"   and va :: "exp0 list" shows "is_value_dom (App v va)"
substE_length: fixes \<Gamma> :: "binding list"   and k :: "nat"   and U :: "POPLmark.type" shows "\<parallel>\<Gamma>[k \<mapsto>\<^sub>\<tau> U]\<^sub>e\<parallel> = \<parallel>\<Gamma>\<parallel>"
sum_by_parts_ivt: fixes X :: "'b set"   and G :: "'c set"   and g :: "'b \<Rightarrow> 'c"   and c :: "'b \<Rightarrow> real" assumes "finite X"   and "convex G"   and "\<And>i. i \<in> X \<Longrightarrow> g i \<in> G"   and "\<And>i. i \<in> X \<Longrightarrow> 0 \<le> c i" obtains y :: "'c" where "y \<in> G" and "(\<Sum>x\<in>X. c x *\<^sub>R g x) = sum c X *\<^sub>R y"   | "G = {}"
that: fixes p1 :: "enat list"   and x :: "'v"   and y :: "'v"   and p2 :: "enat list" assumes "p = p1 @ w (x, y) # p2"   and "x \<in> S"   and "y \<notin> S"   and "path u p1 x"   and "path y p2 v" shows "thesis"
assms: shows "is_pappus1 A B C A' B' C' P Q R"
entailsD: fixes P :: "assn"   and Q :: "assn"   and h :: "heap \<times> nat set" assumes "P \<Longrightarrow>\<^sub>A Q"   and "h \<Turnstile> P" shows "h \<Turnstile> Q"
scaleR_left_mono_neg: fixes b :: "'a"   and a :: "'a"   and c :: "real" assumes "b \<le> a"   and "c \<le> 0" shows "c *\<^sub>R a \<le> c *\<^sub>R b"
prems_3: shows "sharing_consistent \<S> \<O> (x # sb)"
nat_to_bv_helper_legacy_induct: fixes P :: "nat \<Rightarrow> bool"   and x :: "nat" assumes "\<And>n. n \<noteq> 0 \<longrightarrow> P (n div 2) \<Longrightarrow> P n" shows "P x"
ctr_transfer_1: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "list_all2 R [] []"
vseq_implies_hpar_1: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "B.seq \<nu> \<mu>" shows "src\<^sub>B \<nu> = src\<^sub>B \<mu>"
distinct_take: fixes xs :: "'b list"   and i :: "nat" assumes "distinct xs" shows "distinct (take i xs)"
distinct_adj_altdef: fixes xs :: "'g list" shows "distinct_adj xs = (remdups_adj xs = xs)"
restrictto_extends: fixes X :: "(variable \<Rightarrow> real) set"   and V :: "variable set" shows "X \<subseteq> restrictto X V"
lucas_lehmer_hom_in_carrier: fixes x :: "int \<times> int" shows "lucas_lehmer_hom m x \<in> {..<m} \<times> {..<m}"
finite_UB: fixes xs :: "real list"   and f :: "atom fm" shows "finite (UB f xs)"
deg_comp_monotone: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and u :: "'a"   and v :: "'a"   and t :: "'a" assumes "cmp u v = Lt \<Longrightarrow> cmp (splus t u) (splus t v) = Lt"   and "deg_comp cmp u v = Lt" shows "deg_comp cmp (splus t u) (splus t v) = Lt"
list_eq_iff: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs = ys) = (length xs = length ys \<and> (\<forall>i<length xs. xs ! i = ys ! i))"
l3_step3_def: fixes Ra :: "fid_t"   and A :: "agent"   and B :: "agent"   and gny :: "msg" shows "l3_step3 Ra A B gny \<equiv> {(s, s'). guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> progress s Ra = Some {xnx, xgnx} \<and> guessed_frame Ra xgny = Some gny \<and> guessed_frame Ra xsk = Some (Exp gny (NonceF (Ra $ nx))) \<and> implAuth B A \<langle>Number 0, gny, Exp Gen (NonceF (Ra $ nx))\<rangle> \<in> ik s \<and> s' = s\<lparr>progress := progress s(Ra \<mapsto> {xnx, xgnx, xgny, xsk, xEnd}), ik := {implAuth A B \<langle>Number 1, Exp Gen (NonceF (Ra $ nx)), gny\<rangle>} \<union> ik s, secret := {x. x = Exp gny (NonceF (Ra $ nx)) \<and> Ra = test} \<union> secret s, signalsInit := if can_signal s A B then addSignal (signalsInit s) (Commit A B (Exp gny (NonceF (Ra $ nx)))) else signalsInit s, signalsResp := if can_signal s A B then addSignal (signalsResp s) (Running A B (Exp gny (NonceF (Ra $ nx)))) else signalsResp s\<rparr>}"
QR_lemma_04: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "card (QR.BuC p q) = card ({(int p - 1) div 2<..int p - 1} \<times> {0<..(int q - 1) div 2})"
a_p_subset: shows "\<forall>p<length a. set (a ! p) \<subseteq> set nodes"
assms: shows "{(\<Psi>, P, P) |\<Psi> P. True} \<subseteq> Rel"
arr_char: fixes i :: "'b" shows "C'.arr i = (i \<in> S)"
ldropn_ltl: fixes n :: "nat"   and xs :: "'b llist" shows "ldropn n (ltl xs) = ldropn (Suc n) xs"
Nil_is_sublist: fixes ys :: "'a list"   and i :: "nat" shows "sublist_at [] ys i = (i \<le> length ys)"
igAbsRenSTR_imp_igAbsRen: fixes MOD :: "('e, 'f, 'varSort, 'sort, 'g, 'h, 'i, 'j, 'k) model_scheme" assumes "igAbsRenSTR MOD" shows "igAbsRen MOD"
closest_point_in_rel_frontier: fixes S :: "'a set"   and x :: "'a" assumes "closed S"   and "S \<noteq> {}"   and "x \<in> affine hull S - rel_interior S" shows "closest_point S x \<in> rel_frontier S"
when_simps_1: fixes zero :: "'a"   and a :: "'a" shows "zero.when zero a True = a"
prems: shows "v \<notin> ifex_var_set (IF v1 t e)"
id_constant: fixes i :: "nat" shows "ptoy i \<TTurnstile> onl \<Gamma>\<^sub>T\<^sub>O\<^sub>Y (\<lambda>(\<xi>, uu_). state.id \<xi> = i)"
subdegreeI: fixes f :: "'b fps"   and d :: "nat" assumes "f $ d \<noteq> (0::'b)"   and "\<And>i. i < d \<Longrightarrow> f $ i = (0::'b)" shows "subdegree f = d"
Max_insert: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and x :: "'a" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "linorder.Max less_eq (x \<triangleright> A) = ord.max less_eq x (linorder.Max less_eq A)"
finite_cone_empty: shows "finite_cone {} = {0\<^sub>v n}"
complete_development_triangle: fixes A :: "'a trm"   and D :: "'a trm"   and B :: "'a trm" assumes "A >>> D"   and "A >> B" shows "B >> D"
igFreshInp_termMOD: shows "igFreshInp termMOD = freshInp"
ang_circ_a_pTrue: fixes E :: "complex"   and \<mu>1 :: "complex"   and \<mu>2 :: "complex"   and p1 :: "bool"   and p2 :: "bool" assumes "E \<noteq> \<mu>1"   and "E \<noteq> \<mu>2" shows "ang_circ_a E \<mu>1 \<mu>2 p1 p2 = ang_circ_a' E \<mu>1 \<mu>2"
rep_eq: fixes x :: "'d" shows "set_bset (bsingleton x) = {x}"
H_cosets_aux: fixes x :: "nat" assumes "euler_witness (int x) p" shows "(H #>\<^bsub>G\<^esub> x) \<inter> H = {}"
ring_simprules_14: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "ring R" shows "x \<ominus>\<^bsub>R\<^esub> y = x \<oplus>\<^bsub>R\<^esub> \<ominus>\<^bsub>R\<^esub> y"
inner_dirsum_singleD: fixes M :: "'a set" shows "(\<Oplus>N\<leftarrow>[M]. N) = M"
SN_rel_mono': fixes R :: "('a \<times> 'a) set"   and R' :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set"   and S' :: "('a \<times> 'a) set" assumes "R \<subseteq> R'"   and "S \<subseteq> R' \<union> S'"   and "SN_rel R' S'" shows "SN_rel R S"
create_graph_finite: fixes \<phi> :: "'b ltlr" shows "create_graph \<phi> \<le> SPEC finite"
proj_nil_2: fixes n :: "'b" shows "proj_unl n [] = []"
pos_zcount_in_zmset: fixes M :: "'c zmultiset"   and x :: "'c" assumes "0 < zcount M x" shows "x \<in>#\<^sub>z M"
Dup: fixes A :: "'c set"   and n :: "nat"   and p :: "('d, 'c) fm"   and ps :: "('d, 'c) fm list"   and a :: "'c"   and branch :: "(('d, 'c) fm list \<times> 'c) list" assumes "A, n \<turnstile> (p # ps, a) # branch"   and "\<not> new p a ((ps, a) # branch)" shows "A, n \<turnstile> (ps, a) # branch"
comm_add_mat: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'c mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "A + B = B + A"
ExecElim1: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and os :: "Val list"   and S :: "(Var \<times> Val) list"   and H :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and k :: "nat"   and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and v :: "Val" assumes "(M, l, (os, S, H), k, h, v) \<in> Exec" shows "get_ins M l = Some vreturn \<and> (\<exists>ops. os = v # ops \<and> k = Suc 0 \<and> h = H) \<or> (\<exists>n m t ll. (M, l, (os, S, H), n, ll, t) \<in> Step \<and> (M, ll, t, m, h, v) \<in> Exec \<and> k = Suc (max n m))"
map_is_Nil_conv: fixes f :: "'f \<Rightarrow> 'e"   and xs :: "'f list" shows "(map f xs = []) = (xs = [])"
divisor_chain_condition_monoid_axioms: shows "divisor_chain_condition_monoid G"
A: fixes k :: "nat" shows "\<exists>N. \<forall>j\<ge>N. real (card (A k \<inter> {..<j})) < (l + (1 / 2) ^ k) * real j"
appGetStatic: fixes C :: "char list"   and F :: "char list"   and D :: "char list"   and P :: "'a prog"   and pc :: "nat"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and s :: "ty list \<times> ty err list" shows "app\<^sub>i (Getstatic C F D, P, pc, mxs, T\<^sub>r, s) = (\<exists>vT ST LT. s = (ST, LT) \<and> length ST < mxs \<and> P \<turnstile> C sees F,Static:vT in D)"
vector_space_pair_def: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f" shows "vector_space_pair s1 s2 \<equiv> vector_space s1 \<and> vector_space s2"
in_HomD_2: fixes F :: "'a set"   and a :: "'a"   and b :: "'a" assumes "F \<in> Hom a b" shows "B.is_iso_class F"
IH: fixes i :: "'c" shows "continuous_on UNIV (sterm_sem I (args i))"
simps_1: fixes t :: "'b" shows "lock_lock None t = \<lfloor>(t, 0)\<rfloor>"
comp_next: fixes ls :: "nat"   and key :: "nat" shows "g_comp ls key = ls \<or> c_tl (g_comp ls key) = ls"
Some: fixes ord :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and x :: "'k"   and y :: "'l" assumes "ord x y" shows "ord_option ord (Some x) (Some y)"
finfun_curry_def: shows "finfun_curry = finfun_rec (finfun_const \<circ> finfun_const) (\<lambda>(a, b) c f. finfun_update f a (finfun_update (finfun_apply f a) b c))"
ran_ran_restrict: fixes f :: "'d \<Rightarrow> 'c option"   and B :: "'c set" shows "ran (f\<upharpoonleft>\<^bsub>B\<^esub>) = ran f \<inter> B"
close_eq: shows "close [(close (q # \<Gamma>) (p \<leftrightarrow> q), \<Delta>), (close \<Gamma> (p \<leftrightarrow> q), p # \<Delta>)] (\<Gamma>, \<Delta>) = [(close (q # \<Gamma>) (p \<leftrightarrow> q), \<Delta>), (close \<Gamma> (p \<leftrightarrow> q), p # \<Delta>)]"
eqfelem_imp_iff: fixes x :: "'x"   and y :: "'x"   and A :: "'x fset" assumes "x = y" shows "(x |\<in>| A) = (y |\<in>| A)"
wset_final_okD: fixes ws :: "'a \<Rightarrow> 'b wait_set_status option"   and ts :: "'a \<Rightarrow> ('x \<times> 'c \<Rightarrow>f nat) option"   and t :: "'a"   and w :: "'b wait_set_status" assumes "wset_final_ok ws ts"   and "ws t = \<lfloor>w\<rfloor>" shows "\<exists>x ln. ts t = \<lfloor>(x, ln)\<rfloor> \<and> \<not> final x"
real_cond_exp_discount_stock: fixes N :: "bool stream measure"   and q :: "real"   and n :: "nat" assumes "N = bernoulli_stream q"   and "0 < q"   and "q < 1" shows "AEeq N (real_cond_exp N (G n) (discounted_value r (prices Mkt stk) (Suc n))) (discounted_value r (\<lambda>m w. (q * u + (1 - q) * d) * prices Mkt stk n w) (Suc n))"
sin_exp_eq': fixes z :: "complex" shows "sin z = \<i> * (exp (- (\<i> * z)) - exp (\<i> * z)) / 2"
sub_minus_def: fixes x :: "'b"   and y :: "'b" shows "- x - - y = - (- - x \<squnion> - y)"
cases: fixes a1 :: "'b DBMEntry"   and a2 :: "'b DBMEntry" assumes "a1 \<prec> a2" obtains uu_ :: "'b" where "a1 = Lt uu_" and "a2 = \<infinity>"   | uv_ :: "'b" where "a1 = Le uv_" and "a2 = \<infinity>"   | a :: "'b" and b :: "'b" where "a1 = Le a__" and "a2 = Le b__" and "a__ < b__"   | a :: "'b" and b :: "'b" where "a1 = Le aa__" and "a2 = Lt ba__" and "aa__ < ba__"   | a :: "'b" and b :: "'b" where "a1 = Lt ab__" and "a2 = Le bb__" and "ab__ \<le> bb__"   | a :: "'b" and b :: "'b" where "a1 = Lt a" and "a2 = Lt b" and "a < b"
card_2_elements: fixes E :: "'a set" assumes "card E = 2" shows "\<exists>a b. E = {a, b} \<and> a \<noteq> b"
median_cong: fixes xs :: "'b list"   and xs' :: "'b list" assumes "mset xs = mset xs'" shows "median xs = median xs'"
elims_3: fixes x :: "primfun"   and xa :: "primfun" assumes "\<not> x \<le> xa" obtains x :: "nat" and y :: "nat" where "xa__ = LnChain x" and "xa = LnChain y" and "\<not> x \<le> y"
check_def: fixes P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and \<sigma> :: "'addr option \<times> 'heap \<times> 'addr frame list" shows "check P \<sigma> \<equiv> let (xcpt, h, frs) = \<sigma> in case frs of [] \<Rightarrow> True | (stk, loc, C, M, pc) # frs' \<Rightarrow> P \<turnstile> C has M \<and> (let (C', Ts, T, meth) = method P C M; (mxs, mxl\<^sub>0, ins, xt) = the meth; i = ins ! pc in meth \<noteq> None \<and> pc < length ins \<and> length stk \<le> mxs \<and> (case xcpt of None \<Rightarrow> check_instr i P h stk loc C M pc frs' | \<lfloor>a\<rfloor> \<Rightarrow> check_xcpt P h (length stk) pc xt a))"
smult_smult: fixes a :: "'a"   and b :: "'a"   and p :: "'a poly" shows "smult a (smult b p) = smult (a * b) p"
simps: fixes C :: "String.literal" shows "the_Class (Class C) = C"
measure_Union_AE: fixes F :: "'c set set"   and M :: "'c measure" assumes "finite F"   and "\<And>S. S \<in> F \<Longrightarrow> S \<in> fmeasurable M"   and "pairwise (\<lambda>S T. AE x in M. x \<notin> S \<or> x \<notin> T) F" shows "Sigma_Algebra.measure M (\<Union> F) = sum (Sigma_Algebra.measure M) F"
factor_exists: fixes n :: "nat" assumes "Suc 0 < n" shows "\<exists>ps. all_prime ps \<and> (\<Prod>m\<in>#mset ps. m) = n"
rel_R_def: fixes P :: "('a \<times> 'a) set"   and Q :: "('a \<times> 'a) set" shows "rel_R P Q = \<Union> {X. rel_kat.H P X Q}"
sigkey_secret: fixes s :: "(agent \<times> msg) set" assumes "s\<^sub>0 \<Turnstile> s" shows "SigKey \<notin> spied s"
\<gamma>_uminus: fixes i :: "int"   and iv :: "ivl" assumes "i \<in> \<gamma>_ivl iv" shows "- i \<in> \<gamma>_ivl (- iv)"
partition_onE: fixes A :: "'b set"   and P :: "'b set set" assumes "partition_on A P" obtains "\<And>p. p \<in> P \<Longrightarrow> p \<noteq> {}" and "\<Union> P = A" and "\<And>p p'. \<lbrakk>p \<in> P; p' \<in> P; p \<noteq> p'\<rbrakk> \<Longrightarrow> p \<inter> p' = {}"