hcju/isabelleps huggingface dataset:
axioms_2: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and type_wf :: "(_) 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 get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and create_element :: "(_) document_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog" assumes "l_create_element_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr known_ptrs type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel set_tag_name set_tag_name_locs set_disconnected_nodes set_disconnected_nodes_locs create_element" shows "l_new_element_get_disconnected_nodes get_disconnected_nodes_locs"
ZOretr_def: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set" shows "ZOretr theta \<equiv> {(c, d). matchC_LC theta c d}"
basis_in_indep_in_card: fixes \<E> :: "'a set"   and X :: "'a set"   and B :: "'a set" assumes "\<E> \<subseteq> carrier"   and "indep_in \<E> X"   and "basis_in \<E> B" shows "card X \<le> card B"
gen_model_world_refl: fixes w :: "'e"   and M :: "('f, 'g, 'e) KripkeStructure" assumes "w \<in> worlds M" shows "w \<in> worlds (gen_model M w)"
vdisjnt_singleton1: fixes a :: "V"   and A :: "V" shows "(set {a} \<inter>\<^sub>\<circ> A = []\<^sub>\<circ>) = (a \<notin>\<^sub>\<circ> A)"
refl: fixes lc :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)" shows "\<langle>fst (fst lc), snd (fst lc), snd lc\<rangle>\<^sub>A \<in> abs.loc_reach lc"
\<tau>Execsr_induct: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and es :: "'addr expr1 list"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and stk'' :: "'addr val list"   and loc'' :: "'addr val list"   and pc'' :: "nat"   and xcp'' :: "'addr option"   and Q :: "'addr val list \<Rightarrow> 'addr val list \<Rightarrow> nat \<Rightarrow> 'addr option \<Rightarrow> bool" assumes "\<tau>Exec_movesr ci P t es h (stk, loc, pc, xcp) (stk'', loc'', pc'', xcp'')"   and "Q stk loc pc xcp"   and "\<And>stk' loc' pc' xcp' stk'' loc'' pc'' xcp''. \<lbrakk>\<tau>Exec_movesr ci P t es h (stk, loc, pc, xcp) (stk', loc', pc', xcp'); \<tau>exec_moves ci P t es h (stk', loc', pc', xcp') (stk'', loc'', pc'', xcp''); Q stk' loc' pc' xcp'\<rbrakk> \<Longrightarrow> Q stk'' loc'' pc'' xcp''" shows "Q stk'' loc'' pc'' xcp''"
card_ran_le_dom: fixes m :: "'b \<Rightarrow> 'c option" assumes "finite (dom m)" shows "card (ran m) \<le> card (dom m)"
X_2: shows "X \<longlonglongrightarrow> a"
assms_2: fixes j :: "'j" assumes "J.arr j" shows "\<guillemotleft>F j : a \<rightarrow> D j\<guillemotright>"
le_cases3: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat"   and z :: "'x \<Rightarrow>\<^sub>0 nat" obtains "x \<preceq> y" and "y \<preceq> z"   | "y \<preceq> x" and "x \<preceq> z"   | "x \<preceq> z" and "z \<preceq> y"   | "z \<preceq> y" and "y \<preceq> x"   | "y \<preceq> z" and "z \<preceq> x"   | "z \<preceq> x" and "x \<preceq> y"
vors1_3: shows "incrIndexList es |es| |vertices f|"
word_and_full_mask_simp: fixes x :: "'a word" shows "x AND mask LENGTH('a) = x"
get_CWP_def: fixes psr :: "word32" shows "get_CWP psr \<equiv> ucast (psr AND 31)"
X1D: fixes a :: "'a"   and b :: "'c \<Rightarrow>\<^sub>L 'a" assumes "(a, b) \<in> X1'" shows "\<exists>c. (a, c) \<in> X1"
prems_2: shows "Suc n < mod_type_class.to_nat a"
app_def: fixes i :: "instr"   and P :: "'a prog"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and pc :: "nat"   and mpc :: "nat"   and xt :: "ex_table"   and t :: "(ty list \<times> ty err list) option" shows "app i P mxs T\<^sub>r pc mpc xt t = (case t of None \<Rightarrow> True | \<lfloor>\<tau>\<rfloor> \<Rightarrow> app\<^sub>i (i, P, pc, mxs, T\<^sub>r, \<tau>) \<and> xcpt_app i P pc mxs xt \<tau> \<and> (\<forall>(pc', \<tau>')\<in>set (eff i P pc xt t). pc' < mpc))"
assms_5: fixes n :: "nat" shows "(deriv ^^ n) f z = (deriv ^^ n) g z"
arg_div_real_negative: fixes k :: "real"   and z :: "complex" assumes "k < 0" shows "arg (z / cor k) = arg (- z)"
rel_fun_conversep: fixes A :: "'f \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'g \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_fun A\<inverse>\<inverse> B\<inverse>\<inverse> = (rel_fun A B)\<inverse>\<inverse>"
elims_2: fixes x :: "('a \<times> ('b \<times> 'c list) list) list" assumes "exe_ars_conds' x" obtains arss :: "('a \<times> ('b \<times> 'c list) list) list" where "x = arss" and "alist_conds arss \<and> (\<forall>ars\<in>snd ` set arss. alist_conds ars)"
UINF_as_Sup: fixes \<P> :: "('d, 'e) uexpr set" shows "(\<Sqinter> P \<in> \<P> \<bullet> P) = \<Sqinter>\<P>"
loop_super_orientable_def: fixes x :: "'a" shows "orientable_1 x \<equiv> \<exists>y. loop_super_orientation x y"
context_decompose: fixes s :: "('g, 'h, 'i, 'j) com list \<times> 'h option \<times> 'j"   and \<alpha> :: "('g, 'i) seq_label"   and s' :: "('g, 'h, 'i, 'j) com list \<times> 'h option \<times> 'j" shows "s \<rightarrow>\<^bsub>\<alpha>\<^esub> s' = (\<exists>(c, ictxt, fctxt)\<in>decomposeLS s. cPGM s = ictxt c # tl (cPGM s) \<and> (c # fctxt c @ tl (cPGM s), cTKN s, cLST s) \<rightarrow>\<^bsub>\<alpha>\<^esub> s' \<and> (\<forall>l\<in>atC c. cTKN s' = Some l))"
angle_def: fixes A :: "state \<times> state \<Rightarrow> bool"   and v :: "state \<Rightarrow> 'b" shows "<A>_v \<equiv> A \<and> \<not> unchanged v"
bind_return_distr: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and N :: "'d measure" assumes "space M \<noteq> {}"   and "f \<in> M \<rightarrow>\<^sub>M N" shows "M \<bind> return N \<circ> f = distr M N f"
sup_ge2: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y \<le> y"
del_left_braun: fixes t :: "'b tree"   and x :: "'b"   and t' :: "'b tree" assumes "del_left t = (x, t')"   and "braun t"   and "t \<noteq> \<langle>\<rangle>" shows "braun t'"
simps_1: fixes x :: "Preliminaries.var" shows "tNN (Var x) = (if unprot (TE.tpOfV x) \<or> protFw (TE.tpOfV x) then Var x else Fn (Tag (TE.tpOfV x)) [Var x])"
lemma_conf: fixes A :: "agent"   and B :: "agent"   and evs :: "event list"   and Tk :: "nat"   and K :: "nat" assumes "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> bankerb_gets"   and "Says Server A (Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, Crypt (shrK B) \<lbrace>Number Tk, Agent A, Key K\<rbrace>\<rbrace>) \<in> set evs"   and "Key K \<in> analz (knows Spy evs)" shows "Notes Spy \<lbrace>Number Tk, Key K\<rbrace> \<in> set evs"
drop_last_in: fixes n :: "nat"   and ls :: "'c list" assumes "n < |ls|" shows "last ls \<in> set (drop n ls)"
configuration_Min_in: fixes q :: "'b"   and n :: "nat" assumes "configuration q n \<noteq> {}" shows "Min (configuration q n) \<in> configuration q n"
lens_override_overshadow_right: fixes X :: "'a \<Longrightarrow> 'b"   and S\<^sub>1 :: "'b"   and S\<^sub>2 :: "'b"   and S\<^sub>3 :: "'b" assumes "mwb_lens X" shows "S\<^sub>1 \<triangleleft>\<^bsub>X\<^esub> (S\<^sub>2 \<triangleleft>\<^bsub>X\<^esub> S\<^sub>3) = S\<^sub>1 \<triangleleft>\<^bsub>X\<^esub> S\<^sub>3"
overlap_eq: fixes x :: "i"   and y :: "i" assumes "PM" shows "O x y = (\<exists>z. P z x \<and> P z y)"
WTrt_hext_shext_mono: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and T :: "ty"   and h' :: "heap"   and sh' :: "sheap" assumes "P,E,h,sh \<turnstile> e : T"   and "h \<unlhd> h'"   and "sh \<unlhd>\<^sub>s sh'"   and "\<not> sub_RI e" shows "P,E,h',sh' \<turnstile> e : T"
eventually_True: fixes F :: "'b filter" shows "\<forall>\<^sub>F x in F. True"
partial_span': fixes i :: "nat" assumes "i \<le> length (map of_int_hom.vec_hom fs)" shows "gs.span (gs1.gso ` {0..<i}) = gs.span ((!) (map of_int_hom.vec_hom fs) ` {0..<i})"
all_not_in_conv: fixes A :: "'k set" shows "(\<forall>x. x \<notin> A) = (A = {})"
valuate_add_lemma: fixes f1 :: "'a \<Rightarrow> rat"   and f2 :: "'a \<Rightarrow> rat"   and v :: "'a \<Rightarrow> 'b" assumes "finite {v. f1 v \<noteq> 0}"   and "finite {v. f2 v \<noteq> 0}" shows "(\<Sum>x\<in>{v. f1 v + f2 v \<noteq> 0}. (f1 x + f2 x) *R v x) = (\<Sum>x\<in>{v. f1 v \<noteq> 0}. f1 x *R v x) + (\<Sum>x\<in>{v. f2 v \<noteq> 0}. f2 x *R v x)"
a_known_ptr_def: fixes ptr :: "(_) object_ptr" shows "DocumentClass.known_ptr ptr = (CharacterDataClass.known_ptr ptr \<or> is_document_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)"
domain_eq: shows "Domainp pcr_bool2 = (\<lambda>x. x)"
simps: fixes a1 :: "'d set"   and a2 :: "'d \<Rightarrow> nat"   and a3 :: "'d \<Rightarrow> intv"   and a4 :: "('d \<times> 'd) set" shows "valid_region a1 a2 a3 a4 = (\<exists>X\<^sub>0 X I r k. a1 = X \<and> a2 = k \<and> a3 = I \<and> a4 = r \<and> X\<^sub>0 = {x \<in> X. \<exists>d. I x = Intv d} \<and> refl_on X\<^sub>0 r \<and> trans r \<and> total_on X\<^sub>0 r \<and> (\<forall>x\<in>X. valid_intv (k x) (I x)))"
prob0_deutsch_algo_const: assumes "is_const" shows "prob0 2 deutsch_algo 0 = 1"
lm161: fixes a :: "real"   and b :: "real"   and c :: "real" shows "a * b - a * c = a * (b - c)"
det_perm_cols: fixes p :: "'c \<Rightarrow> 'c"   and A :: "('d, 'c) sq_matrix" assumes "p permutes UNIV" shows "Square_Matrix.det (perm_cols A p) = of_int (sign p) * Square_Matrix.det A"
heap_is_wellformed_children_disc_nodes: fixes h :: "(_) heap"   and node_ptr :: "(_) node_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "node_ptr |\<in>| node_ptr_kinds h"   and "\<not> (\<exists>parent\<in>fset (object_ptr_kinds h). node_ptr \<in> set |h \<turnstile> Shadow_DOM.get_child_nodes parent|\<^sub>r)" shows "\<exists>document_ptr\<in>fset (document_ptr_kinds h). node_ptr \<in> set |h \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r"
add_edge_wf: fixes G :: "'c graph"   and v :: "'c"   and v' :: "'c" assumes "wf_graph G" shows "wf_graph (add_edge v v' G)"
TESL_interp_remdups_absorb: fixes \<Phi> :: "'b TESL_atomic list" shows "\<lbrakk>\<lbrakk> \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = \<lbrakk>\<lbrakk> remdups \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
GSMP_disjointI: fixes A' :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and B' :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and A :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and B :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and Sec :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set" assumes "list_all (wf\<^sub>t\<^sub>r\<^sub>m' arity) A'"   and "list_all (wf\<^sub>t\<^sub>r\<^sub>m' arity) B'"   and "has_all_wt_instances_of \<Gamma> (set A') (set A)"   and "has_all_wt_instances_of \<Gamma> (set B') (set (B \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t var_rename (max_var_set (fv\<^sub>s\<^sub>e\<^sub>t (set A)))))"   and "finite_SMP_representation arity Ana \<Gamma> A"   and "finite_SMP_representation arity Ana \<Gamma> (B \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t var_rename (max_var_set (fv\<^sub>s\<^sub>e\<^sub>t (set A))))"   and "\<forall>t\<in>set A. \<forall>s\<in>set (B \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t var_rename (max_var_set (fv\<^sub>s\<^sub>e\<^sub>t (set A)))). \<Gamma> t = \<Gamma> s \<and> mgu t s \<noteq> None \<longrightarrow> intruder_synth' public arity {} t \<and> intruder_synth' public arity {} s \<or> (\<exists>u\<in>Sec. is_wt_instance_of_cond \<Gamma> t u) \<and> (\<exists>u\<in>Sec. is_wt_instance_of_cond \<Gamma> s u)"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s Sec" shows "GSMP_disjoint (set A') (set B') ({t \<cdot> \<delta> |t \<delta>. t \<in> Sec \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> fv (t \<cdot> \<delta>) = {}} - {m. {} \<turnstile>\<^sub>c m})"
simps_1: fixes v :: "nat \<Rightarrow> 'a"   and l :: "linear_poly"   and r :: "'a" shows "v \<Turnstile>\<^sub>n\<^sub>s LEQ_ns l r = (l \<lbrace> v \<rbrace> \<le> r)"
one_ratfps_def: shows "1 \<equiv> Abs_ratfps 1"
assms_1: shows "diagram J (\<cdot>\<^sub>C) E"
ltake_eSuc_LCons: fixes n :: "enat"   and x :: "'a"   and xs :: "'a llist" shows "ltake (eSuc n) (LCons x xs) = LCons x (ltake n xs)"
perm_append_Cons: fixes a :: "'a"   and xs :: "'a list"   and ys :: "'a list" shows "a # xs @ ys <~~> xs @ a # ys"
Abs_mod_ring_inject: fixes x :: "int"   and y :: "int" assumes "x \<in> {0..<int CARD('b)}"   and "y \<in> {0..<int CARD('b)}" shows "(Abs_mod_ring x = Abs_mod_ring y) = (x = y)"
restrict_space_def: fixes M :: "'b measure"   and \<Omega> :: "'b set" shows "restrict_space M \<Omega> = measure_of (\<Omega> \<inter> space M) (sets.restricted_space M \<Omega>) (emeasure M)"
revis_subexpansion: fixes K :: "'a set"   and A :: "'a set"   and \<psi> :: "'a"   and \<phi> :: "'a" assumes "K = Cn A"   and ".\<not> \<psi> \<notin> K \<^bold>* \<phi>" shows "(K \<^bold>* \<phi>) \<oplus> \<psi> \<subseteq> K \<^bold>* (\<phi> .\<and>. \<psi>)"
complete_c_K33: shows "K\<^bsub>3,3\<^esub> (with_proj c_K33)"
fun_ord_def: fixes ord :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'e"   and g :: "'g \<Rightarrow> 'f" shows "fun_ord ord f g = (\<forall>x. ord (f x) (g x))"
"case": fixes f :: "(('e \<Rightarrow> 'f) \<Rightarrow> 'f) \<Rightarrow> 'd"   and x :: "('e \<Rightarrow> 'f) \<Rightarrow> 'f" shows "(case ContT x of ContT x \<Rightarrow> f x) = f x"
d_swap_unchanged: fixes F1 :: "int vec list"   and i :: "nat"   and k :: "nat"   and F2 :: "int vec list" assumes "length F1 = m"   and "i \<noteq> 0"   and "i < m"   and "k \<noteq> i"   and "k \<le> m"   and "F2 = F1[i := F1 ! (i - 1), i - 1 := F1 ! i]" shows "d F1 k = d F2 k"
RSpanD_lincomb_arb_len_coeffs: fixes ms :: "'m list" assumes "set ms \<subseteq> M" shows "RSpan ms = {rs \<bullet>\<cdot> ms |rs. set rs \<subseteq> R}"
mkarr_Diagonalize_rep: fixes f :: "'c local.term set" assumes "\<F>C.arr f"   and "Diag (\<F>C.DOM f)"   and "Diag (\<F>C.COD f)" shows "\<F>C.mkarr \<^bold>\<lfloor>\<F>C.rep f\<^bold>\<rfloor> = f"
simps: fixes a :: "'node SDG_node"   and S :: "'node SDG_node set" shows "(a \<in> combine_SDG_slices S) = ((\<exists>n. a = n \<and> n \<in> S) \<or> (\<exists>n' n'' p n. a = n \<and> n' \<in> S \<and> n'' s-p\<rightarrow>\<^bsub>ret\<^esub> CFG_node (parent_node n') \<and> n \<in> sum_SDG_slice2 n'))"
final_weighted_RP: fixes Q :: "('a literal multiset \<times> nat) multiset"   and n :: "nat"   and St :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat" shows "\<not> ({#}, {#}, Q, n) \<leadsto>\<^sub>w St"
proj2_incident_iff: fixes p :: "proj2"   and q :: "proj2"   and l :: "proj2_line"   and r :: "proj2" assumes "p \<noteq> q"   and "proj2_incident p l"   and "proj2_incident q l" shows "proj2_incident r l = (r = p \<or> (\<exists>k. r = proj2_abs (k *\<^sub>R proj2_rep p + proj2_rep q)))"
sup_commute: fixes x :: "'b"   and y :: "'b" shows "x \<sqinter> y = y \<sqinter> x"
d: shows "Max (fst ` nonzero_positions (Rep_matrix A)) < m"
divisionTr: fixes g :: "'a"   and f :: "'a" assumes "Corps S"   and "g \<in> carrier R"   and "0 < deg R S X g"   and "f \<in> carrier R" shows "\<exists>q\<in>carrier R. \<exists>r\<in>carrier R. f = q \<cdot>\<^sub>r g \<plusminus> r \<and> deg R S X r < deg R S X g"
separators_split: fixes l :: "('c \<times> 'b) list"   and a :: "'c"   and b :: "'b"   and r :: "('c \<times> 'b) list" shows "set (separators (l @ (a, b) # r)) = set (separators l) \<union> set (separators r) \<union> {b}"
zip_fun_simps_4: fixes h :: "'a \<Rightarrow> 'b \<times> 'c" shows "fst \<circ> h \<parallel> snd \<circ> h = h"
G5: shows "dom (\<Psi>\<^sub>G\<^sub>+) \<subseteq> set (\<Psi>\<^sub>\<V>\<^sub>+)"
finite_indexed_bound: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> \<exists>n. P x n" shows "\<exists>m. \<forall>x\<in>A. \<exists>k\<le>m. P x k"
inject_3: fixes x51 :: "'b ltln"   and x52 :: "'b ltln"   and y51 :: "'b ltln"   and y52 :: "'b ltln" shows "(x51 and\<^sub>n x52 = y51 and\<^sub>n y52) = (x51 = y51 \<and> x52 = y52)"
resp_analz_insert: fixes K :: "nat"   and RB :: "msg"   and evsa :: "event list"   and evs :: "event list" assumes "Key K \<in> analz (insert RB (knows Spy evsa))"   and "evsa \<in> recur"   and "RB \<in> responses evs" shows "Key K \<in> parts {RB} \<or> Key K \<in> analz (knows Spy evsa)"
rel_conversep: fixes R1 :: "'g \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'h \<Rightarrow> 'e \<Rightarrow> bool"   and R3 :: "'i \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_generat R1\<inverse>\<inverse> R2\<inverse>\<inverse> R3\<inverse>\<inverse> = (rel_generat R1 R2 R3)\<inverse>\<inverse>"
residual_eq1: fixes f :: "'a \<Rightarrow> 'a"   and y :: "'a" assumes "residuated f" shows "residual f y = \<Squnion>{x. f x \<le> y}"
reduced_row_echelon_form_def': fixes A :: "(('d, 'e) vec, 'f) vec" shows "reduced_row_echelon_form A = ((\<forall>i. is_zero_row i A \<longrightarrow> \<not> (\<exists>j>i. \<not> is_zero_row j A)) \<and> (\<forall>i. \<not> is_zero_row i A \<longrightarrow> A $ i $ (LEAST k. A $ i $ k \<noteq> (0::'d)) = (1::'d)) \<and> (\<forall>i. i < i + (1::'f) \<and> \<not> is_zero_row i A \<and> \<not> is_zero_row (i + (1::'f)) A \<longrightarrow> (LEAST k. A $ i $ k \<noteq> (0::'d)) < (LEAST k. A $ (i + (1::'f)) $ k \<noteq> (0::'d))) \<and> (\<forall>i. \<not> is_zero_row i A \<longrightarrow> (\<forall>j. i \<noteq> j \<longrightarrow> A $ j $ (LEAST k. A $ i $ k \<noteq> (0::'d)) = (0::'d))))"
finite_path_weightp_distinct: fixes l1 :: "'vtx"   and l2 :: "'vtx" shows "finite {xs. path_weightp_distinct l1 l2 xs}"
can_take_empty: fixes i :: "value list"   and a :: "nat"   and c :: "nat \<Rightarrow>f value option" assumes "length i = a" shows "can_take a [] i c"
numer_cf5_deriv_def: shows "numer_cf5_deriv \<equiv> \<lambda>x. 5 * x ^ 4 + 120 * x ^ 3 + 1260 * x\<^sup>2 + 6720 * x + 15120"
directed_def: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "directed A r \<equiv> \<forall>x\<in>A. \<forall>y\<in>A. \<exists>z\<in>A. r x z \<and> r y z"
simps_1: fixes uu :: "real_max_bot" shows "(MInfty \<le> uu) = True"
rel_spmf_bindI2: fixes q :: "'a spmf"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'b spmf"   and f :: "'a \<Rightarrow> 'c spmf" assumes "\<And>x. x \<in> set_spmf q \<Longrightarrow> rel_spmf R p (f x)"   and "lossless_spmf q" shows "rel_spmf R p (q \<bind> f)"
int_pow_mult: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and i :: "int"   and j :: "int" assumes "Group.group G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> (i + j) = x [^]\<^bsub>G\<^esub> i \<otimes>\<^bsub>G\<^esub> x [^]\<^bsub>G\<^esub> j"
Zfun_right: fixes f :: "'e \<Rightarrow> 'b"   and F :: "'e filter"   and a :: "'a" assumes "Zfun f F" shows "Zfun (\<lambda>x. a ** f x) F"
mojmir_accept_iff_token_set_accept: fixes \<Sigma> :: "'b set"   and \<delta> :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'b"   and F :: "'c set" assumes "mojmir \<Sigma> \<delta> q\<^sub>0 w F" shows "mojmir_def.accept \<delta> q\<^sub>0 w F = (\<exists>i<semi_mojmir_def.max_rank \<Sigma> \<delta> q\<^sub>0. finite (mojmir_def.fail \<Sigma> \<delta> q\<^sub>0 w F) \<and> finite (mojmir_def.merge \<Sigma> \<delta> q\<^sub>0 w F i) \<and> infinite (mojmir_def.succeed \<Sigma> \<delta> q\<^sub>0 w F i))"
degree_m_eq: fixes f :: "int poly" assumes "lead_coeff f mod p \<noteq> 0"   and "1 < p" shows "degree_m f = degree f"
connex_axioms: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "well_ordered_set A less_eq" shows "connex A less_eq"
mem_carrier: fixes H :: "'d set"   and G :: "('d, 'e) monoid_scheme"   and x :: "'d" assumes "subgroup H G"   and "x \<in> H" shows "x \<in> carrier G"
nat_descend_induct: fixes n :: "nat"   and P :: "nat \<Rightarrow> bool"   and m :: "nat" assumes "\<And>k. n < k \<Longrightarrow> P k"   and "\<And>k. \<lbrakk>k \<le> n; \<And>i. k < i \<Longrightarrow> P i\<rbrakk> \<Longrightarrow> P k" shows "P m"
makes_compatible_eval_k: fixes cms\<^sub>1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and cms\<^sub>2 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and mems :: "(('Var \<Rightarrow> 'Val) \<times> ('Var \<Rightarrow> 'Val)) list"   and k :: "nat"   and cms\<^sub>1' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1' :: "'Var \<Rightarrow> 'Val" assumes "makes_compatible (cms\<^sub>1, mem\<^sub>1) (cms\<^sub>2, mem\<^sub>2) mems"   and "map snd cms\<^sub>1 = map snd cms\<^sub>2"   and "sound_mode_use (cms\<^sub>1, mem\<^sub>1)"   and "sound_mode_use (cms\<^sub>2, mem\<^sub>2)"   and "(cms\<^sub>1, mem\<^sub>1) \<rightarrow>\<^bsub>k\<^esub> (cms\<^sub>1', mem\<^sub>1')" shows "\<exists>cms\<^sub>2' mem\<^sub>2' mems'. sound_mode_use (cms\<^sub>1', mem\<^sub>1') \<and> sound_mode_use (cms\<^sub>2', mem\<^sub>2') \<and> map snd cms\<^sub>1' = map snd cms\<^sub>2' \<and> (cms\<^sub>2, mem\<^sub>2) \<rightarrow>\<^bsub>k\<^esub> (cms\<^sub>2', mem\<^sub>2') \<and> makes_compatible (cms\<^sub>1', mem\<^sub>1') (cms\<^sub>2', mem\<^sub>2') mems'"
projs_2: fixes dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat" shows "\<pi>\<^sub>3 (dsn, dsk, flag, hops, nhip) = dsk"
acc_iff: fixes x :: "'s"   and y :: "'s" shows "((x, y) \<in> acc) = (\<exists>n. 0 < p x y n)"
take_Suc_Cons: fixes n :: "nat"   and x :: "'a"   and xs :: "'a list" shows "(x # xs) \<down> Suc n = x # xs \<down> n"
cases: fixes a1 :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> Arity \<times> Arity list"   and a2 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" assumes "a_consistent a1 a2" obtains ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" and \<Gamma> :: "(Vars.var \<times> exp) list" and S :: "stack_elem list" and a :: "Arity" and e :: "exp" and as :: "Arity list" where "a1 = (ae, a, as)" and "a2 = (\<Gamma>, e, S)" and "edom ae \<subseteq> domA \<Gamma> \<union> upds S" and "Astack S \<sqsubseteq> a" and "(ABinds \<Gamma>\<cdot>ae \<squnion> \<A>\<^bsub>a\<^esub> e \<squnion> AEstack as S) f|` (domA \<Gamma> \<union> upds S) \<sqsubseteq> ae" and "stack_consistent as S"
two_vdiff_zero: shows "set {0, 1\<^sub>\<nat>} -\<^sub>\<circ> set {0} = set {1\<^sub>\<nat>}"
assms_2: shows "fds_nth f 1 \<noteq> (0::'a)"
Immediate: fixes G :: "prog"   and membr :: "qtname \<times> memberdecl"   and "class" :: "qtname"   and accclass :: "qtname" assumes "G \<turnstile> membr member_of class"   and "G \<turnstile> Class class accessible_in pid accclass"   and "G \<turnstile> membr in class permits_acc_from accclass" shows "G \<turnstile> membr of class accessible_from accclass"
subterm_of_ex_set_iff_subtermeqset: fixes t :: "('c, 'd) Term.term"   and M :: "('c, 'd) Term.term set" shows "(t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t M) = (\<exists>t'\<in>M. subterm_of t t')"