hcju/isabelleps huggingface dataset:
ba: fixes i :: "nat"   and j :: "nat"   and k :: "nat"   and r :: "nat" assumes "i < j"   and "j \<le> k"   and "k < r" shows "b k (j, i) \<lless> a r"
Close2: fixes P :: "pi"   and a :: "name"   and y :: "name"   and P' :: "pi"   and Q :: "pi"   and x :: "name"   and Q' :: "pi" assumes "P \<longmapsto> a<\<nu>y> \<prec> P'"   and "Q \<longmapsto> a<x> \<prec> Q'"   and "y \<sharp> Q" shows "P \<parallel> Q \<longmapsto> \<tau> \<prec> <\<nu>y>(P' \<parallel> Q'[x::=y])"
subrel_runiq: fixes Q :: "('c \<times> 'd) set"   and P :: "('c \<times> 'd) set" assumes "runiq Q"   and "P \<subseteq> Q" shows "runiq P"
lunit_char_1: fixes f :: "'a" assumes "Left_a.ide f" shows "Left_a.in_hom (Left_a.lunit f) (Left_a.L f) f"
components_are_iso: fixes a :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" assumes "VVV.ide a" shows "local.iso \<a>[fst a, fst (snd a), snd (snd a)]"
has_equalizers_def: shows "J.has_equalizers = (\<forall>f0 f1. J.par f0 f1 \<longrightarrow> (\<exists>e. J.has_as_equalizer f0 f1 e))"
bounded_dist_comp: fixes f :: "'d \<Rightarrow> 'c"   and S :: "'d set"   and g :: "'d \<Rightarrow> 'c" assumes "bounded (f ` S)"   and "bounded (g ` S)" shows "bounded ((\<lambda>x. dist (f x) (g x)) ` S)"
SETmapsTo: fixes f :: "ZF"   and X :: "ZF"   and Y :: "ZF" assumes "f maps\<^bsub>SET\<^esub> X to Y" shows "isZFfun f \<and> |dom|f = X \<and> |cod|f = Y"
path_img: shows "path_image g \<subseteq> s - insert p pts"
consistent_sign_vectors_consistent_sign_vectors_r: fixes qs :: "rat poly list"   and S :: "real set" shows "consistent_sign_vectors_r (cast_rat_list qs) S = consistent_sign_vectors qs S"
paths_mono: fixes t :: "'a ttree"   and t' :: "'a ttree" assumes "t \<sqsubseteq> t'" shows "paths t \<sqsubseteq> paths t'"
eval_simps'_5: fixes f :: "'a local.term" assumes "Arr f" shows "cod \<lbrace>f\<rbrace> = \<lbrace>Cod f\<rbrace>"
nprv_impI: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and \<phi> :: "'c"   and F :: "'c set"   and \<psi> :: "'c" assumes "Deduct_with_False_Disj var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv"   and "Deduct_with_False_Disj.nprv cnj imp fls prv (insert \<phi> F) \<psi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "\<psi> \<in> fmla" shows "Deduct_with_False_Disj.nprv cnj imp fls prv F (imp \<phi> \<psi>)"
IH: assumes "\<forall>a\<in>set fs. is_literal_formula a" shows "cnf \<^bold>\<Or>fs = {{literal_formula_to_literal f |f. f \<in> set fs}}"
ICp_Suc: fixes p :: "nat"   and i :: "nat" assumes "correct p (te p (i + 1))" shows "IC p (i + 1) (te p (i + 1)) = cfn p (\<theta> p (i + 1))"
scalar_prodI_def: fixes ze :: "'b"   and pl :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and ti :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and v :: "'b list"   and w :: "'b list" shows "scalar_prodI ze pl ti v w = foldr (\<lambda>(x, y). pl (ti x y)) (zip v w) ze"
int_mult_closed: fixes x :: "nat \<Rightarrow> int"   and n :: "int" assumes "x \<in> carrier Zp" shows "[n] \<cdot> x \<in> carrier Zp"
a_nonneg: fixes n :: "nat" shows "0 \<le> a n"
cempty_not_cinsert: fixes a :: "'f"   and A :: "'f cset" shows "cempty \<noteq> cinsert a A"
mem_CD_on_Cd: fixes x :: "'a"   and ds :: "'b set"   and X :: "'a set" shows "(x \<in> PdXXX.CD_on ds X) = (x \<in> PdXXX.Cd (Xd x) X \<and> Xd x \<in> ds)"
Linear_order_Restr: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" assumes "Linear_order r" shows "Linear_order (Restr r A)"
iT_Div_card: fixes d :: "nat"   and I :: "nat set"   and n :: "nat" assumes "0 < d"   and "finite I"   and "Max I div d \<le> n" shows "card (I \<oslash> d) = (\<Sum>k\<le>n. if I \<inter> [k * d\<dots>,d - Suc 0] = {} then 0 else Suc 0)"
in_sccs_imp_induced: fixes c :: "('a, 'b) pre_digraph" assumes "c \<in> sccs" shows "induced_subgraph c G"
cdf_def: fixes M :: "real measure" shows "cdf M \<equiv> \<lambda>x. Sigma_Algebra.measure M {..x}"
face_of_trans: fixes S :: "'a set"   and T :: "'a set"   and u :: "'a set" assumes "S face_of T"   and "T face_of u" shows "S face_of u"
finite_UN: fixes I :: "'b set"   and A :: "'b \<Rightarrow> 'c set"   and M :: "'c measure" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> null_sets M" shows "\<Union> (A ` I) \<in> null_sets M"
conv_abscissa_geI: fixes c :: "ereal"   and f :: "'a fds" assumes "\<And>c'. ereal c' < c \<Longrightarrow> \<exists>s. s \<bullet> (1::'a) = c' \<and> \<not> fds_converges f s" shows "c \<le> conv_abscissa f"
integral_smallo: fixes f :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real"   and a :: "real" assumes "f \<in> o(g')"   and "filterlim g at_top at_top"   and "\<And>a' x. \<lbrakk>a \<le> a'; a' \<le> x\<rbrakk> \<Longrightarrow> f integrable_on {a'..x}"   and "\<And>x. a \<le> x \<Longrightarrow> (g has_real_derivative g' x) (at x)"   and "continuous_on {a..} g'"   and "\<And>x. a \<le> x \<Longrightarrow> 0 \<le> g' x" shows "(\<lambda>x. integral {a..x} f) \<in> o(g)"
execute_heap: fixes f :: "heap \<Rightarrow> 'b \<times> heap" shows "execute (Heap_Monad.heap f) = Some \<circ> f"
weakCongSimulation_def: fixes P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs" shows "P \<leadsto><Rel> Q \<equiv> \<forall>a Q'. Q \<longmapsto>a \<prec> Q' \<longrightarrow> (\<exists>P'. P \<Longrightarrow>a \<prec> P' \<and> (P', Q') \<in> Rel)"
sorted_wrt_take_less_hd_drop: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and n :: "nat" assumes "sorted_wrt f xs"   and "n < length xs" shows "\<forall>x\<in>set (take n xs). f x (hd (drop n xs))"
all_not_in_conv: fixes A :: "'e set" shows "(\<forall>x. x \<notin> A) = (A = {})"
finite_fourier_poly_unique: fixes ws :: "complex list"   and p :: "complex poly" assumes "0 < length ws"   and "degree p \<le> length ws - 1"   and "\<forall>m\<le>length ws - 1. ws ! m = poly p (unity_root (length ws) (int m))" shows "p = finite_fourier_poly ws"
inv_def: fixes f :: "'c \<Rightarrow> 'b" shows "inv f = (\<lambda>y. SOME x. f x = y)"
f_Max\<^sub>0_equiv: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat" shows "f_Max\<^sub>0 f n = Max\<^sub>0 (f ` {1..n})"
assms: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> U"   and "y \<in> U"   and "x \<le> z"   and "z \<le> y" shows "z \<in> U"
run_fail_state: fixes fail :: "'c"   and s :: "'d" shows "run_state (fail_state fail) s = fail"
list_of_oalist_shift_keys: fixes t :: "'b"   and f :: "'c \<Rightarrow> 'c"   and xs :: "('b, 'c) oalist_ntm" shows "list_of_oalist_ntm (shift_map_keys_punit t f xs) = map_raw (\<lambda>kv. (punit.splus t (fst kv), f (snd kv))) (list_of_oalist_ntm xs)"
G'_2: shows "VV p** \<subseteq> G'.VV p**"
distinct_13: fixes x2 :: "nat"   and x3 :: "fresh_t" shows "Number x2 \<noteq> Nonce x3"
assms_3: shows "cri.InvMsgInGlob (exchange_config c)"
restrict_relation_restrict_relation: fixes A :: "'b set"   and B :: "'b set"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "restrict_relation A (restrict_relation B R) = restrict_relation (A \<inter> B) R"
prems_3: shows "ref \<notin> set (interp_ins xs)"
dim_vec_def: shows "dim_vec \<equiv> map_fun Rep_vec id fst"
da_elim_cases_28: fixes Env :: "env"   and B :: "lname set"   and T :: "ty"   and e :: "expr"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In1l (Cast T e)\<guillemotright> A" obtains "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> A"
in_keys_plusI2: fixes t :: "'a"   and q :: "'a \<Rightarrow>\<^sub>0 'b"   and p :: "'a \<Rightarrow>\<^sub>0 'b" assumes "t \<in> keys q"   and "t \<notin> keys p" shows "t \<in> keys (p + q)"
dg_prod_is_arrD_2: fixes \<alpha> :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and a :: "V"   and b :: "V"   and f :: "V" assumes "pdigraph_base \<alpha> I \<AA>"   and "f : a \<mapsto>\<^bsub>dg_prod I \<AA>\<^esub> b" shows "\<D>\<^sub>\<circ> f = I"
non_pivot_base_gen_def: fixes uminus :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and A :: "'b mat"   and pivots :: "(nat \<times> nat) list" shows "non_pivot_base_gen uminus zero one A pivots \<equiv> let nr = dim_row A; nc = dim_col A; invers = map_of (map prod.swap pivots) in (\<lambda>qj. vec nc (\<lambda>i. if i = qj then one else case invers i of None \<Rightarrow> zero | Some j \<Rightarrow> uminus (A $$ (j, qj))))"
assms: shows "automaton.trans OA = oseqp_sos \<Gamma> i"
mS0: shows "Sigma_Algebra.measure lebesgue (frontier S) = 0"
ring_n_prod_primesTr1: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and I :: "'a set" assumes "0 < n"   and "distinct_pds K n P"   and "ideal (O\<^bsub>K P n\<^esub>) I"   and "I \<noteq> {\<zero>\<^bsub>O\<^bsub>K P n\<^esub>\<^esub>}"   and "I \<noteq> carrier (O\<^bsub>K P n\<^esub>)" shows "I = (O\<^bsub>K P n\<^esub>) \<diamondsuit>\<^sub>p mprod_exp K (mL K P I) (Kb\<^bsub>K n P\<^esub>) n"
disjoint_from_sum: fixes z :: "i"   and x :: "i"   and y :: "i" shows "(\<not> O z (x \<oplus> y)) = (\<not> O z x \<and> \<not> O z y)"
list_emb_append2: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "list_emb P xs ys" shows "list_emb P xs (zs @ ys)"
adj_sub_diff_assign: fixes \<sigma> :: "('b, 'c, 'a) subst"   and e :: "('b, 'a) trm"   and x :: "'a" shows "(\<Union>i\<in>SIGT e. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x) \<subseteq> \<Union> (SFV \<sigma> ` (SDom \<sigma> \<inter> SIGP (DiffAssign x e)))"
eval_simps'_4: fixes f :: "'c local.term" assumes "Arr f" shows "local.dom \<lbrace>f\<rbrace> = \<lbrace>Dom f\<rbrace>"
5: shows "v $ find_first (col A i) (cols A) \<noteq> (0::'a)"
special_cosets_has_bottom: shows "dual_order.has_bottom \<P>"
part_witness_interior_simps_2: shows "interior {part_witness.b} = {}"
set_insert: fixes x :: "'b"   and A :: "'b set" assumes "(x::'b) \<in> (A::'b set)" obtains B :: "'b set" where "(A::'b set) = (x::'b) \<triangleright> (B::'b set)" and "(x::'b) \<notin> (B::'b set)"
simps_2: fixes X :: "freemsg"   and Y :: "freemsg" shows "freeright (MPAIR X Y) = Y"
cyc_free_diag_dest: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and i :: "nat"   and xs :: "nat list" assumes "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len M i i xs"   and "i \<le> n"   and "set xs \<subseteq> {0..n}" shows "\<one> \<le> len M i i xs"
Act\<^sub>\<alpha>: fixes t\<^sub>\<alpha> :: "('b, 'c, 'd, 'e) Tree\<^sub>\<alpha>"   and f :: "'e"   and \<alpha> :: "'d" assumes "hereditarily_fs t\<^sub>\<alpha>" shows "hereditarily_fs (Act\<^sub>\<alpha> f \<alpha> t\<^sub>\<alpha>)"
2: shows "arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F (Go (C.cod f)) (C.cod f') (f' \<cdot>\<^sub>C \<epsilon>o (C.cod f))"
ex_mset: fixes X :: "'c multiset" shows "\<exists>xs. mset xs = X"
Im_Arccos_bound: fixes w :: "complex" shows "\<bar>Im (Arccos w)\<bar> \<le> cmod w"
vandermonde: fixes m :: "nat"   and n :: "nat"   and r :: "nat" shows "(\<Sum>k\<le>r. (m choose k) * (n choose (r - k))) = m + n choose r"
uconnected_def: fixes E :: "'b uprod set" shows "uconnected E \<equiv> {(u, v). \<exists>p. epath E u p v}"
pair_loopfree_digraph_def: fixes G :: "'b pair_pre_digraph" shows "pair_loopfree_digraph G \<equiv> pair_wf_digraph G \<and> pair_loopfree_digraph_axioms G"
igFreshIGVar_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'j, 'k) model_scheme" shows "igFreshIGVar MOD \<equiv> \<forall>ys y xs x. ys \<noteq> xs \<or> y \<noteq> x \<longrightarrow> igFresh MOD ys y (igVar MOD xs x)"
dd_cc_ent_d: shows "DD \<union> set (side_prems_of \<iota>) \<Turnstile> {concl_of \<iota>}"
frac_of_int: fixes z :: "int" shows "frac (of_int z) = (0::'b)"
dense_repr_set_snd: fixes p :: "'a list list" shows "snd ` set (UP.dense_repr p) \<subseteq> {..<length p}"
subst_lit_def: fixes subst_atm :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and L :: "'a literal"   and \<sigma> :: "'b" shows "substitution_ops.subst_lit subst_atm L \<sigma> = map_literal (\<lambda>A. subst_atm A \<sigma>) L"
fix_eq: shows "P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr\<^sub>D n,Up\<^sub>D\<rbrakk> = (\<Lambda> X. (\<lambda>s. Mprefix (Tr\<^sub>D n s) (\<lambda>e. X (Up\<^sub>D s e))))\<cdot>(P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr\<^sub>D n,Up\<^sub>D\<rbrakk>)"
dm_def: fixes r :: "('b \<times> 'b) set"   and M :: "'b multiset" shows "r \<down>m M = r \<down>s set_mset M"
l_get_root_node_si_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_root_node_si_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs Shadow_DOM.get_parent Shadow_DOM.get_parent_locs Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs get_host get_host_locs get_ancestors_si get_ancestors_si_locs get_root_node_si get_root_node_si_locs get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name Shadow_DOM.heap_is_wellformed Shadow_DOM.parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_document get_disconnected_document_locs"
int_and_Bits: fixes x :: "int"   and b :: "bool"   and y :: "int"   and c :: "bool" shows "x BIT b AND y BIT c = (x AND y) BIT (b \<and> c)"
R_consts_3: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" shows "w \<Turnstile>\<^sub>n true\<^sub>n R\<^sub>n \<phi> = w \<Turnstile>\<^sub>n \<phi>"
h2: shows "|~ P \<and> _actrans N f \<longrightarrow> \<circle>P"
lastDecisionThenUIP: fixes M :: "(Literal \<times> bool) list"   and l :: "Literal"   and c :: "Literal list" assumes "uniq (elements M)"   and "opposite l el decisions M"   and "clauseFalse c (elements M)"   and "isLastAssertedLiteral (opposite l) (oppositeLiteralList c) (elements M)" shows "isUIP l c M"
B_def: shows "B = {x \<in> space M. f' x < f x}"
A: shows "set_pmf p \<inter> A \<noteq> {}"
rel_envirD: fixes p :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'f"   and g :: "'h \<Rightarrow> 'g"   and x :: "'h" assumes "rel_envir p f g" shows "p (f x) (g x)"
lim_null_mult_left_bounded: fixes g :: "'a \<Rightarrow> 'b"   and B :: "real"   and F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in F. norm (g x) \<le> B"   and "(f \<longlongrightarrow> (0::'b)) F" shows "((\<lambda>z. g z * f z) \<longlongrightarrow> (0::'b)) F"
solves_ode_on_subset_domain: fixes x :: "real \<Rightarrow> 'a"   and S :: "real set"   and Y :: "'a set" assumes "(x solves_ode f) S Y"   and "x t_max = x t_max"   and "t_max \<in> S"   and "is_interval S"   and "S \<subseteq> cball t_max et" shows "(x solves_ode f) S (cball (x t_max) ex)"
lotteries_on_def: fixes A :: "'b set" shows "lotteries_on A = {p. set_pmf p \<subseteq> A}"
ifp_curr: shows "ifp\<^sup>*\<^sup>* (current s) u"
mds\<^sub>s_def: fixes x :: "Mode" shows "mds\<^sub>s x = {}"
hcomp_assoc: fixes \<mu> :: "('a E.term, 'a) arr"   and \<nu> :: "('a E.term, 'a) arr"   and \<tau> :: "('a E.term, 'a) arr" shows "(\<mu> \<star> \<nu>) \<star> \<tau> = \<mu> \<star> \<nu> \<star> \<tau>"
less_eq_prod_def: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "(x \<le> y) = (fst x \<le> fst y \<and> snd x \<le> snd y)"
heap_is_wellformed_one_disc_parent: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list"   and document_ptr' :: "(_) document_ptr"   and disc_nodes' :: "(_) node_ptr list" assumes "Shadow_DOM.heap_is_wellformed h"   and "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes"   and "h \<turnstile> get_disconnected_nodes document_ptr' \<rightarrow>\<^sub>r disc_nodes'"   and "set disc_nodes \<inter> set disc_nodes' \<noteq> {}" shows "document_ptr = document_ptr'"
wcode_jump1: fixes m :: "nat"   and ires :: "cell list"   and rs :: "nat"   and n :: "nat" shows "\<exists>stp ln rn. steps0 (Suc t_twice_len + length t_wcode_main_first_part div 2, Bk \<up> m @ Bk # Bk # ires, Oc \<up> Suc (2 * rs) @ Bk \<up> n) t_wcode_main stp = (Suc 0, Bk \<up> ln @ Bk # ires, Bk # Oc \<up> Suc (2 * rs) @ Bk \<up> rn)"
bilinear_form_scale_v2: fixes v1 :: "complex \<times> complex"   and k :: "complex"   and v2 :: "complex \<times> complex"   and H :: "complex \<times> complex \<times> complex \<times> complex" shows "bilinear_form v1 (k *\<^sub>s\<^sub>v v2) H = k * bilinear_form v1 v2 H"
1: shows "\<forall>m\<in>pair ` setops\<^sub>s\<^sub>s\<^sub>t S. is_Fun m"
choose_arc_def: fixes x :: "'a" shows "choose_arc x \<equiv> if x = bot then bot else SOME y. arc y \<and> y \<le> - - x"
Confidentiality_Tgs: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and evs :: "event list" assumes "Says Tgs A \<lbrace>Crypt authK \<lbrace>Key servK, Agent B, Number Ts\<rbrace>, servTicket\<rbrace> \<in> set evs"   and "Key authK \<notin> analz (knows Spy evs)"   and "\<not> expiredSK Ts evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> kerbV" shows "Key servK \<notin> analz (knows Spy evs)"
simps_2: fixes P :: "pi" shows "summands (\<tau>.P) = {\<tau>.P}"
Rel_app: fixes A :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'f"   and g :: "'g \<Rightarrow> 'h"   and x :: "'e"   and y :: "'g" assumes "Transfer.Rel (A ===> B) f g"   and "Transfer.Rel A x y" shows "Transfer.Rel B (f x) (g y)"
listset_ConsE: fixes ys :: "'b list"   and x :: "'b set"   and xs :: "'b set list" assumes "ys \<in> listset (x # xs)" obtains y :: "'b" and ys' :: "'b list" where "y \<in> x" and "ys' \<in> listset xs" and "ys = y # ys'"
hypnat_add_one_gt_one: fixes N :: "nat star" assumes "0 < N" shows "1 < N + 1"
is_sorted_list_of_set_def: fixes A :: "'b set"   and xs :: "'b list" shows "is_sorted_list_of_set A xs = ((\<forall>i<length xs - 1. xs ! i < xs ! (i + 1)) \<and> ran\<^sub>l xs = A)"
gOr_def: fixes v :: "'a gexp"   and va :: "'a gexp" shows "gOr v va \<equiv> Nor (Nor v va) (Nor v va)"
comp_swap: fixes l\<^sub>1 :: "'a literal"   and l\<^sub>2 :: "'a literal" shows "(l\<^sub>1\<^sup>c = l\<^sub>2) = (l\<^sub>1 = l\<^sub>2\<^sup>c)"
vimage_image: fixes A :: "V"   and Aa :: "V"   and f :: "V \<Rightarrow> V" assumes "A \<subseteq>\<^sub>\<circ> \<D>\<^sub>\<circ> (VLambda Aa f)" shows "elts (VLambda Aa f `\<^sub>\<circ> A) = app (VLambda Aa f) ` elts A"
lm053: fixes a :: "('c \<times> 'd set) set"   and G :: "'d set"   and N :: "'c set"   and i :: "'c"   and X :: "'c set" assumes "a \<in> allAllocations N G"   and "i \<in> N - X"   and "Domain a \<inter> X \<noteq> {}" shows "Outside' (X \<union> {i}) a \<union> {i} \<times> {\<Union> (a `` (X \<union> {i}))} \<in> allAllocations (N - X) (\<Union> (Range a))"
direct_decompI: fixes ss :: "'b set list"   and A :: "'b set" assumes "inj_on sum_list (listset ss)"   and "sum_list ` listset ss = A" shows "direct_decomp A ss"
prems_1: fixes k :: "nat" assumes "k < Suc n" shows "f k \<in> carrier_mat d d"
preserves_reflects_arr: fixes f :: "'d" shows "D.arr (trg\<^sub>D f) = D.arr f"
lookup_plus: fixes A :: "'b tensor"   and B :: "'b tensor"   and "is" :: "nat list" assumes "dims A = dims B"   and "is \<lhd> dims A" shows "lookup (A + B) is = lookup A is + lookup B is"
nlen_zero: fixes x :: "nat" assumes "0 = nlen x" shows "x = 0"
subgraph_is_subgraph_isomorphic: fixes G' :: "nat set \<times> nat set set"   and G :: "nat set \<times> nat set set" assumes "uwellformed G'"   and "uwellformed G"   and "subgraph G' G" shows "G' \<sqsubseteq> G"
ide_cod: fixes f :: "'a" assumes "hom.arr f" shows "hom.ide (hom.cod f)"
fps_ln_nth: fixes c :: "'a"   and n :: "nat" shows "fps_nth (fps_ln c) n = (if n = 0 then 0::'a else (1::'a) / c * ((- (1::'a)) ^ (n - 1) / of_nat n))"
finite_subset_induct: fixes F :: "'b set"   and A :: "'b set"   and P :: "'b set \<Rightarrow> bool" assumes "finite F"   and "F \<subseteq> A"   and "P {}"   and "\<And>a F. \<lbrakk>finite F; a \<in> A; a \<notin> F; P F\<rbrakk> \<Longrightarrow> P (insert a F)" shows "P F"
l_new_shadow_root_get_shadow_root_axioms_def: fixes get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_shadow_root_get_shadow_root_axioms get_shadow_root_locs \<equiv> \<forall>h new_shadow_root_ptr h' r ptr'. h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr \<longrightarrow> h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_shadow_root_locs ptr' \<longrightarrow> r h h'"
assms_6: shows "cls_val_process Mkt pf n w < 0"
key_map_axioms: shows "key_map R (+) (\<cdot>) \<zero> \<one> \<pp>"
closest_point_exists_1: fixes S :: "'a set"   and a :: "'a" assumes "closed S"   and "S \<noteq> {}" shows "closest_point S a \<in> S"
budan_fourier_aux_right: fixes c :: "real"   and d2 :: "real"   and p :: "real poly" assumes "c < d2"   and "p \<noteq> 0"   and "\<forall>x. c < x \<and> x \<le> d2 \<longrightarrow> (\<forall>q\<in>set (pders p). poly q x \<noteq> 0)" shows "changes_itv_der c d2 p = 0"
\<Phi>\<^sub>d_case: fixes tndl :: "'a tree \<times> nat \<times> nat" shows "\<Phi>\<^sub>d tndl = (case case tndl of (x, y) \<Rightarrow> (x, case y of (x, y) \<Rightarrow> (x, real y)) of (t, n, dl) \<Rightarrow> \<Phi> t + 4 * dl / cd)"
1: shows "injective (w \<sqinter> - prim_EP w v e)"
tendsto_Im1: fixes g :: "'a \<Rightarrow> octo"   and a :: "octo"   and F :: "'a filter" assumes "(g \<longlongrightarrow> a) F" shows "((\<lambda>x. Im1 (g x)) \<longlongrightarrow> Im1 a) F"
g1_scale_i_contin: shows "continuous_on (cbox a b) (\<lambda>x. (g1 x, 0::'a))"
case_rews_1: fixes f :: "'g \<rightarrow> 'f \<rightarrow> 'e" shows "writer_case\<cdot>f\<cdot>\<bottom> = \<bottom>"
less_vec_def: fixes v :: "'b vec"   and w :: "'b vec" shows "(v < w) = (v \<le> w \<and> \<not> w \<le> v)"
acquired_reads_append: fixes pending_write :: "bool"   and xs :: "'b memref list"   and ys :: "'b memref list"   and A :: "nat set" shows "acquired_reads pending_write (xs @ ys) A = acquired_reads pending_write xs A \<union> acquired_reads (pending_write \<or> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b xs \<noteq> {}) ys (acquired pending_write xs A)"
monad_commute_envT: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_commute return bind" shows "monad_commute (return_env return) (bind_env bind)"
assms_2: shows "nprv (insert (subst \<phi> t x) F) \<psi>"
is_document_ptr_kind_def: fixes ptr :: "(_) object_ptr" shows "is_document_ptr_kind ptr = (case cast ptr of None \<Rightarrow> False | Some x \<Rightarrow> True)"
simps_11: fixes w :: "'a"   and x :: "'b"   and b :: "('a, 'b) rbt"   and s :: "'a"   and t :: "'b"   and c :: "('a, 'b) rbt"   and y :: "'a"   and z :: "'b"   and va :: "('a, 'b) rbt"   and vb :: "'a"   and vc :: "'b"   and vd :: "('a, 'b) rbt" shows "balance Empty w x (Branch R (Branch R b s t c) y z (Branch B va vb vc vd)) = Branch R (Branch B Empty w x b) s t (Branch B c y z (Branch B va vb vc vd))"
bn_fresh_Act: fixes \<alpha> :: "'g"   and x :: "('e, 'f, 'g) formula" shows "bn \<alpha> \<sharp>* Act \<alpha> x"
size_neq: fixes x :: "'c dual" shows "size x \<noteq> 0"