hcju/isabelleps huggingface dataset:
spies_Says_rev: fixes evs :: "event list"   and A :: "agent"   and B :: "agent"   and X :: "msg" shows "knows Spy (evs @ [Says A B X]) = insert X (knows Spy evs)"
nonempty_Euclidean_space: fixes n :: "nat" shows "topspace (Euclidean_space n) \<noteq> {}"
wpo_with_assms_axioms: shows "wpo_with_assms {} UNIV prc prl full_status False (\<lambda>_. False)"
ps'_def: shows "ps' = Time_Monad.val (filter_tm (\<lambda>x. dist (case x of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case (l, snd (case x of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) of (x, y) \<Rightarrow> (real_of_int x, y)) < dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) ps)"
right_def: fixes X :: "msg" shows "right X = Abs_Msg (\<Union>U\<in>Rep_Msg X. msgrel `` {freeright U})"
simps_3: fixes c :: "int"   and s :: "MIR.num"   and t :: "MIR.num" shows "nozerocoeff (CF c s t) = (c \<noteq> 0 \<and> nozerocoeff t)"
Body: fixes G :: "'b triple set"   and P :: "pname \<Rightarrow> 'b assn"   and Q :: "pname \<Rightarrow> 'b assn"   and Procs :: "pname set" assumes "G \<union> (\<lambda>p. {P p}. BODY p .{Q p}) ` Procs||-(\<lambda>p. {P p}. the (body p) .{Q p}) ` Procs" shows "G||-(\<lambda>p. {P p}. BODY p .{Q p}) ` Procs"
ket_zero_to_mat_of_cols_list: shows "|Deutsch.zero\<rangle> = Tensor.mat_of_cols_list 2 [[1, 0]]"
value_plus_flow: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and g :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "wf_residual_network"   and "flow \<Delta> f"   and "flow (residual_network f) g" shows "value_flow \<Delta> (f \<oplus> g) = value_flow \<Delta> f + value_flow \<Delta> g"
semiring_normalization_rules_7: fixes a :: "'a"   and b :: "'a" shows "a * b = b * a"
simps: fixes a :: "'a word"   and b :: "'a word" shows "word_upto' a b = (if a = b then [a] else a # word_upto' (a + 1) b)"
retraction_def: fixes g :: "'a" shows "A.retraction g \<equiv> \<exists>f. A.ide (g \<cdot>\<^sub>A f)"
1: shows "{z \<in> A. z < y} \<subset> {z \<in> A. z < x}"
l_one: fixes x :: "'b \<Rightarrow> 'e" assumes "x \<in> carrier (DirProds Hs J)" shows "\<one>\<^bsub>DirProds Hs J\<^esub> \<otimes>\<^bsub>DirProds Hs J\<^esub> x = x"
process_ret_split_fst_NeqReturn: fixes a :: "action"   and m :: "'c match_expr"   and rs :: "'c rule list" assumes "a \<noteq> Return" shows "process_ret (Rule m a # rs) = Rule m a # process_ret rs"
assms_3: shows "snd (rep_nat_term u) \<le> snd (rep_nat_term v)"
has_codomain_iff_arr: fixes f :: "'b" shows "(D.codomains f \<noteq> {}) = D.arr f"
word_log2_max: fixes w :: "'b word" shows "word_log2 w < size w"
diff_add_zmset_swap: fixes b :: "'b"   and M :: "'b zmultiset"   and A :: "'b zmultiset" shows "add_zmset b M - A = add_zmset b (M - A)"
s_1: shows "s \<sqsubseteq>\<^bsub>r\<^esub> \<tau>s ! 0"
mk_vec_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "mk_vec n f \<equiv> \<lambda>i. if i < n then f i else undef_vec (i - n)"
transfer_operations_rl_12: fixes fas :: "floatarith list" shows "(nres_of o3 approx_slp_dres ops optns, approx_slp_spec fas) \<in> nat_rel \<rightarrow> fas_rel \<rightarrow> appr_rell \<rightarrow> \<langle>\<langle>appr_rell\<rangle>option_rel\<rangle>nres_rel"
has_extend_witness: fixes n :: "nat"   and S :: "'c set"   and xs :: "'c list" assumes "has n S"   and "set xs \<subseteq> S"   and "length xs < n" shows "set xs \<subset> S"
pd_trans: fixes y :: "'n"   and x :: "'n"   and z :: "'n" assumes "y pd\<rightarrow> x"   and "z pd\<rightarrow> y" shows "z pd\<rightarrow> x"
orbit_altdef_permutation: fixes f :: "'b \<Rightarrow> 'b"   and s :: "'b" assumes "permutation f" shows "orbit f s = {(f ^^ n) s |n. True}"
mins_nonempty: fixes c_k :: "int \<Rightarrow> int \<Rightarrow> int \<Rightarrow> nat"   and c :: "int \<Rightarrow> int \<Rightarrow> nat"   and w :: "int \<Rightarrow> int \<Rightarrow> nat"   and i :: "int"   and j :: "int" assumes "QI c_k c w"   and "i < j" shows "Args_min_on (c_k i j) {i + 1..j} \<noteq> {}"
map_def: fixes f :: "'j" shows "Cod_f.map f = (if J.arr f then C.cod f__ else C.null)"
h6: shows "get_matrix (snd (combine_systems p (qs1, matrix1, subsets1, signs1) (qs2, matrix2, subsets2, signs2))) = M_mat (get_signs (snd (combine_systems p (qs1, matrix1, subsets1, signs1) (qs2, matrix2, subsets2, signs2)))) (get_subsets (snd (combine_systems p (qs1, matrix1, subsets1, signs1) (qs2, matrix2, subsets2, signs2))))"
from_nat_suc: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a"   and j :: "nat" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "mod_type.from_nat Abs (j + 1) = plus (mod_type.from_nat Abs j) one"
sum_list_closed_prod: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'r"   and xys :: "('c \<times> 'd) list" assumes "set (map (\<lambda>(x, y). f x y) xys) \<subseteq> R" shows "(\<Sum>(x, y)\<leftarrow>xys. f x y) \<in> R"
assms: shows "interpretation\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"
map_comp_lunit: fixes f :: "'c \<Rightarrow> 'd option" shows "Some \<circ>\<^sub>m f = f"
args_to_set_subset: fixes ap :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> nat \<times> 'h \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and ab :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> nat \<times> 'h \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and gs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and data :: "nat \<times> 'h"   and ps :: "((('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list" assumes "ap_spec ap"   and "ab_spec ab" shows "args_to_set (gs, ab gs bs hs data, ap gs bs ps hs data) \<subseteq> fst ` (set gs \<union> set bs \<union> fst ` set ps \<union> snd ` set ps \<union> set hs)"
fps_Lambert_W_def: shows "fps_Lambert_W = fps_inv (fps_X * fps_exp 1)"
LeastI: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and k :: "'c" assumes "class.wellorder less_eq less"   and "P k" shows "P (ord.Least less_eq P)"
simps: fixes a1 :: "'a list"   and a2 :: "'b sequent" shows "upRulesp a1 a2 = (\<exists>c R Fs ps. a1 = ps \<and> a2 = c \<and> MultiSequents.mset c = \<LM> Compound R Fs  \<RM> \<and> ps \<noteq> [])"
elims: fixes x :: "'b literal"   and y :: "sign" assumes "get_sign x = y" obtains xa :: "'b equation" where "x = Pos xa__" and "y = pos"   | xa :: "'b equation" where "x = Neg xa" and "y = neg"
Cod_comp: fixes f :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and g :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.arr f"   and "J_C.arr g"   and "J_C.Dom g = J_C.Cod f" shows "J_C.Cod (g \<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] f) = J_C.Cod g"
axioms_1: fixes R :: "('e, 'f) ring_scheme"   and M :: "('e, 'g, 'h) module_scheme" assumes "module R M" shows "cring R"
llist_all2_LNil2: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c llist" shows "llist_all2 P xs LNil = (xs = LNil)"
ide_char: fixes a :: "'a" shows "L.ide a = (L.arr a \<and> ide a)"
simps_1: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and s1 :: "'e stream"   and s2 :: "'f stream" shows "shd (smap2 f s1 s2) = f (shd s1) (shd s2)"
minus_equality: fixes y :: "'c \<Rightarrow> 'a"   and x :: "'c \<Rightarrow> 'a" assumes "y \<oplus>\<^bsub>F\<^esub> x = \<zero>\<^bsub>F\<^esub>"   and "x \<in> carrier F"   and "y \<in> carrier F" shows "\<ominus>\<^bsub>F\<^esub> x = y"
span_li_not_depend_2: fixes S :: "rat vec set"   and N :: "rat vec set" assumes "S \<subseteq> N"   and "LinearCombinations.submodule class_ring N gs.V" shows "module.lin_dep class_ring (gs.vs N) S = gs.lin_dep S"
states_def: fixes p :: "prog" shows "states p \<equiv> {\<sigma>. match \<sigma> (ctxt p)}"
strong_fermat_witness_lower_bound: fixes n :: "nat" assumes "odd n"   and "2 < n"   and "\<not> prime n" shows "real (card {a. 2 \<le> a \<and> a < n \<and> strong_fermat_liar a n}) < real (n - 2) / 2"
fbrelation_vsubset: fixes s :: "V"   and r :: "V" assumes "fbrelation s"   and "r \<subseteq>\<^sub>\<circ> s" shows "fbrelation r"
T_integral_preserving_1: fixes f :: "'a \<Rightarrow> 'b" assumes "integrable M f" shows "integrable M (\<lambda>x. f (T x))"
Gin: fixes x :: "'a" assumes "x \<in> S" shows "x \<in> G x"
small_imp_big: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> o[F](g)" shows "f \<in> O[F](g)"
h1: shows "((\<Sum>i = 0..<n + 1. f i * B ^ (n - i)) * B ^ (n mod 2) < 0) = ((\<Sum>i = 0..<n + 1. f i / B ^ i) * B ^ n * B ^ (n mod 2) < 0)"
eq_neutr_iff: fixes a :: "'c"   and b :: "'c" shows "(a \<squnion> b = \<bottom>) = (a = \<bottom> \<and> b = \<bottom>)"
assms_1: fixes x :: "'a" assumes "x \<in> s" shows "f x = g x"
chart_last_inv_continuous: shows "continuous_on UNIV chart_last_inv"
when_mult: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and P :: "bool"   and b :: "'b" assumes "class.mult_zero times zero" shows "times (zero.when zero a P) b = zero.when zero (times a b) P"
Rep_inverse: fixes a :: "'b" shows "Abs (Rep a) = a"
empty_eq_affine_hull: fixes S :: "'a set" shows "({} = affine hull S) = (S = {})"
keys_pp_iff: fixes x :: "'a"   and t :: "('a, 'b) pp" shows "(x \<in> keys_pp t) = (lookup_pp t x \<noteq> (0::'b))"
square_free_multD_2: fixes f :: "'b poly"   and g :: "'b poly" assumes "square_free (f * g)" shows "square_free f"
interaction_bounded_by_FailI: fixes "consider" :: "'d \<Rightarrow> bool" shows "interaction_bounded_by consider Fail 0"
snd_o_Pair: fixes x :: "'b" shows "snd \<circ> Pair x = (\<lambda>x. x)"
list_set_rel_def: fixes R :: "('e \<times> 'f) set" shows "\<langle>R\<rangle>list_set_rel \<equiv> \<langle>R\<rangle>list_rel O br set distinct"
subset_singleton_iff: fixes X :: "'b set"   and a :: "'b" shows "(X \<subseteq> {a}) = (X = {} \<or> X = {a})"
smult_closed: fixes x :: "rat vec"   and a :: "rat" assumes "True"   and "x \<in> Rn" shows "a \<cdot>\<^sub>v x \<in> Rn"
se_sat_imp_sat: fixes c :: "('c, 'd) conf"   and l :: "('c, 'd) label"   and c' :: "('c, 'd) conf" assumes "se c l c'"   and "Conf.sat c'" shows "Conf.sat c"
fimage_vsingleton_in: fixes a :: "V"   and A :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> A" shows "set {[a, b]\<^sub>\<circ>} `\<^sub>\<bullet> A = set {b}"
2: shows "dense (Abs_lifted_pair (\<lambda>f. (top, up_filter (Rep_lifted_distrib_lattice_top xf f))))"
abs_eq: fixes x :: "'a rexp" shows "ACI_lang (ACI_class x) = lang x"
m_ivl_anti_mono: fixes y :: "ivl"   and x :: "ivl" assumes "y \<sqsubseteq> x" shows "m_ivl x \<le> m_ivl y"
lub_fun: fixes S :: "nat \<Rightarrow> 'c \<Rightarrow> 'd" assumes "chain S" shows "(\<Squnion>i. S i) = (\<lambda>x. \<Squnion>i. S i x)"
derived_minimal: fixes H :: "'a set" assumes "H \<lhd> G"   and "comm_group (G Mod H)" shows "derived G (carrier G) \<subseteq> H"
lcosSubset: shows "H #> inv g \<subseteq> carrier G"
inv_def: fixes g :: "graph" shows "Invariants.inv g \<equiv> minGraphProps g \<and> one_final g \<and> 2 \<le> |faces g|"
2: shows "S' = P2 * D * Q2 \<and> invertible_mat P2 \<and> invertible_mat Q2 \<and> Smith_normal_form_mat S' \<and> P2 \<in> carrier_mat 2 2 \<and> S' \<in> carrier_mat 2 2 \<and> Q2 \<in> carrier_mat 2 2"
CDPlayer_CTRL_Delta_def: shows "CDPlayer_CTRL_Delta = {(''ReadTracks'', (En ''LastTrack'', defaultguard, CDPlayer_CTRL_Action1), ''CDFull''), (''CDFull'', (And (En ''CDEject'') (In ''On''), defaultguard, CDPlayer_CTRL_Action3), ''CDEmpty''), (''ReadTracks'', (En ''NewTrack'', defaultguard, CDPlayer_CTRL_Action2), ''ReadTracks''), (''CDEmpty'', (En ''CDIn'', defaultguard, defaultaction), ''ReadTracks'')}"
ne: shows "{x. \<omega> \<le> f x} \<noteq> {}"
transpose_cnj_is_dagger: fixes M :: "complex mat" shows "M\<^sup>t\<^sup>\<star> = M\<^sup>\<dagger>"
replicate_not_True: fixes xs :: "bool list"   and n :: "nat" assumes "xs = replicate n False" shows "True \<notin> set xs"
a_mpm: shows "- \<infinity> + - \<infinity> = - \<infinity>"
prefix_length_le: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs \<le> ys" shows "length xs \<le> length ys"
left_minus_one_mult_self: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "times (power.power one times (uminus one) n) (times (power.power one times (uminus one) n) a) = a"
rep_eq: fixes x :: "'b fpxs" shows "fpxs_supp x = Formal_Puiseux_Series.supp (fpxs_nth x)"
one_im_sheaf_def: shows "one_im_sheaf \<equiv> \<lambda>V. one_str\<^sub>X (f \<^sup>\<inverse> X V)"
eq_iff_2: fixes ty1 :: "ty"   and ty2 :: "ty"   and ty1a :: "ty"   and ty2a :: "ty" shows "(Fun ty1 ty2 = Fun ty1a ty2a) = (ty1 = ty1a \<and> ty2 = ty2a)"
assms: shows "ops \<in> set (\<phi>\<^sub>P\<inverse> \<Pi> List_Supplement.embed \<pi>)"
d_repr_def: fixes ds :: "int iarray"   and fs :: "int vec list" shows "d_repr ds fs = (ds = IArray.of_fun (d fs) (Suc m))"
is_ortho_set_def: fixes S :: "'c set" shows "is_ortho_set S = ((\<forall>x\<in>S. \<forall>y\<in>S. x \<noteq> y \<longrightarrow> is_orthogonal x y) \<and> (0::'c) \<notin> S)"
axioms_2: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" assumes "conservative M T" shows "conservative_axioms M T"
Abs_ExpList_def: fixes Xs :: "freeExp list" shows "Abs_ExpList Xs = map (\<lambda>U. Abs_Exp (exprel `` {U})) Xs"
invar2_init_init: fixes r :: "'a" shows "Prim2.prim_invar2_init r (Prim2.initQ r) Prim2.init\<pi>"
LV_LV: shows "LVE \<union> LVQ \<subseteq> \<Union> (funStar SS ` (LVE \<union> LVQ))"
induced_imp_subgraph: fixes H :: "('e, 'f) pre_digraph"   and G :: "('e, 'f) pre_digraph" assumes "induced_subgraph H G" shows "subgraph H G"
f_subset: shows "f \<subseteq> \<Union> (range g)"
foldr_aux_1_3: fixes us :: "'var list"   and g :: "'g"   and n :: "'node"   and u :: "('node, ('var \<times> 'node \<times> Def) set) mapping"   and p :: "('node \<times> 'var, ('var \<times> 'node \<times> Def) list) mapping"   and u' :: "('node, ('var \<times> 'node \<times> Def) set) mapping"   and p' :: "('node \<times> 'var, ('var \<times> 'node \<times> Def) list) mapping"   and m :: "'node"   and v :: "'var" assumes "set us \<subseteq> uses g n"   and "Mapping.lookup u n = None"   and "foldr (aux_1 g n) us (u, p) = (u', p')"   and "finite (Mapping.keys p)"   and "\<And>n v. (n, v) \<in> Mapping.keys p \<Longrightarrow> length (predecessors g n) \<noteq> 1"   and "\<And>v. Mapping.lookup p (Entry g, v) \<in> {None, Some []}" shows "(if m \<in> phiDefNodes_aux g v (filter (\<lambda>n. (n, v) \<notin> Mapping.keys p) (\<alpha>n g)) n \<and> v \<in> set us then Some (map (\<lambda>m. lookupDef g m v) (predecessors g m)) else Mapping.lookup p (m, v)) = Mapping.lookup p' (m, v)"
less_irrefl: fixes x :: "'b zmultiset" shows "\<not> x \<subset>#\<^sub>z x"
sigma_finite_countable: fixes i :: "'i" shows "\<exists>A. countable A \<and> A \<subseteq> sets (M i) \<and> \<Union> A = space (M i) \<and> (\<forall>a\<in>A. emeasure (M i) a \<noteq> \<infinity>)"
assms_2: shows "\<forall>y. f (g y) = y"
cases: fixes a :: "('test, 'atom) com" assumes "siso0 a" obtains c :: "('test, 'atom) com" where "a = c" and "\<And>s c' s'. \<lbrakk>mustT c s; (c, s) \<rightarrow>c (c', s')\<rbrakk> \<Longrightarrow> siso0 c'" and "\<And>s t c' s'. \<lbrakk>mustT c s; mustT c t; s \<approx> t; (c, s) \<rightarrow>c (c', s')\<rbrakk> \<Longrightarrow> \<exists>t'. s' \<approx> t' \<and> (c, t) \<rightarrow>c (c', t')" and "\<And>s t s'. \<lbrakk>mustT c s; mustT c t; s \<approx> t; (c, s) \<rightarrow>t s'\<rbrakk> \<Longrightarrow> \<exists>t'. s' \<approx> t' \<and> (c, t) \<rightarrow>t t'"
cyclic_group: fixes G :: "('e, 'f) monoid_scheme" assumes "Group.group G" shows "cyclic_group G = (\<exists>x\<in>carrier G. carrier G = range (([^]\<^bsub>G\<^esub>) x))"
vertices_def: fixes g :: "('d, 'e) rgraph_scheme" shows "Graph.vertices g = {root g} \<union> src ` edges g \<union> tgt ` edges g"
forall_in_division: fixes d :: "'a set set"   and i :: "'a set"   and P :: "'a set \<Rightarrow> bool" assumes "d division_of i" shows "(\<forall>x\<in>d. P x) = (\<forall>a b. cbox a b \<in> d \<longrightarrow> P (cbox a b))"
simps_2: fixes t :: "(val \<times> val) fset" shows "vsize (VFun t) = fsize t"
two: shows "x\<^sup>\<star> \<cdot> x\<^sup>\<star> = x\<^sup>\<star>"
os_prepend_def: fixes a :: "'b"   and n :: "'b node ref option" shows "os_prepend a n = ref (Node a n) \<bind> (\<lambda>p. return (Some p))"
upd_1: fixes j :: "nat"   and n :: "nat"   and M1 :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and f :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and m :: "nat"   and N :: "nat" assumes "j \<le> n" shows "DBM_upd M1 f (Suc m) n N (Suc m) j = DBM_upd M1 f (Suc m) j N (Suc m) j"
supports_def: fixes S :: "'d set"   and x :: "'e" shows "S supports x = (\<forall>a b. a \<notin> S \<and> b \<notin> S \<longrightarrow> [(a, b)] \<bullet> x = x)"
simps_1: fixes s :: "'a stream" shows "stake 0 s = []"
wf: fixes t :: "('state \<times> 'rule) tree" assumes "snd (tree.root t) \<in> R"   and "effStep (tree.root t) ((fst \<circ> tree.root) |`| cont t)"   and "\<And>t'. t' |\<in>| cont t \<Longrightarrow> local.wf t'" shows "local.wf t"
elims_3: fixes x :: "('c, 'd) trie" assumes "\<not> is_empty_trie x" obtains v :: "'d option" and m :: "('c \<times> ('c, 'd) trie) list" where "x = Trie v m" and "\<not> (v = None \<and> m = [])"
intros_7: fixes xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and P :: "('a \<times> real) msllist \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool" assumes "is_expansion_aux xs f basis" shows "ms_closure P xs f basis"
IH: fixes cs :: "'edge list" shows "upd_rev_cs cs (a # as') = a # upd_rev_cs cs as'"
D: shows "Crypt (sesK (d * e)) (Passwd m') \<in> parts (A \<union> spies evsFC5)"
lprefix_lfiniteD: fixes xs :: "'a llist"   and ys :: "'a llist" assumes "lprefix xs ys"   and "lfinite ys" shows "lfinite xs"
unit_div_commute: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "is_unit b" shows "a div b * c = a * c div b"
bal_tree_def2: fixes n :: "nat"   and t :: "'b tree" shows "Root_Balanced_Tree.bal_tree n t = Let (Root_Balanced_Tree.inorder2 t []) (Root_Balanced_Tree.bal_list n)"
num_params_tm_pi: fixes prec :: "nat" shows "num_params (tm_poly (tm_pi prec)) = 0"
one_set_greater: fixes A :: "'c set"   and A' :: "'d set" shows "(\<exists>f. f ` A \<subseteq> A' \<and> inj_on f A) \<or> (\<exists>g. g ` A' \<subseteq> A \<and> inj_on g A')"
generate_fieldp_generate_field_eq: fixes R :: "('a, 'b) ring_scheme"   and H :: "'a set" shows "generate_fieldp R (\<lambda>x. x \<in> H) = (\<lambda>x. x \<in> generate_field R H)"
greatest_eq_card: shows "Suc (mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A)) = nrows A"
intros: fixes a :: "'h"   and b :: "'i"   and r :: "('h \<times> 'i) set" assumes "(a, b) \<in> r" shows "(b, a) \<in> r\<inverse>"
authTicket_authentic: fixes A :: "agent"   and authK :: "nat"   and Ta :: "nat"   and evs :: "event list" assumes "Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace> \<in> parts (knows Spy evs)"   and "evs \<in> kerbIV" shows "Says Kas A (Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace>\<rbrace>) \<in> set evs"
sel_18: fixes x31 :: "nat"   and x32 :: "'b"   and x33 :: "'b"   and x34 :: "'a"   and x35 :: "'a"   and x36 :: "'c" shows "un_Recv4 (Recv x31 x32 x33 x34 x35 x36) = x34"
infsetsum_0D: fixes f :: "'c \<Rightarrow> real"   and A :: "'c set"   and x :: "'c" assumes "infsetsum f A = 0"   and "f abs_summable_on A"   and "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x"   and "x \<in> A" shows "f x = 0"
t_ag_def_3: shows "c_types_agree (TopType x1) (t_ag @ t_in)"
simps_1: fixes f :: "'c \<rightarrow> 'b" shows "lmap\<cdot>f\<cdot>lnil = lnil"
sign_def: fixes p :: "'b \<Rightarrow> 'b" shows "sign p = (if evenperm p then 1 else - 1)"
union_with_outside: fixes S :: "'a set" shows "S \<union> outside S = - inside S"
step_elim_cases_6: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and b :: "'e set"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and u :: "('e, 'd, 'f) com \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (Cond b c1 c2, s) \<rightarrow> u" obtains sa :: "'e" where "s = Normal sa__" and "u = (c1, Normal sa__)" and "sa__ \<in> b"   | sa :: "'e" where "s = Normal sa" and "u = (c2, Normal sa)" and "sa \<notin> b"   | f :: "'f" where "s = Fault f" and "u = (Skip, Fault f)"   | "s = Stuck" and "u = (Skip, Stuck)"
List_rec_def: fixes M :: "('c, unit) Old_Datatype.node set"   and c :: "'b"   and d :: "('c, unit) Old_Datatype.node set \<Rightarrow> ('c, unit) Old_Datatype.node set \<Rightarrow> 'b \<Rightarrow> 'b" shows "List_rec M c d = wfrec (pred_sexp\<^sup>+) (\<lambda>g. List_case c (\<lambda>x y. d x y (g y))) M"
fps_conv_radius_inverse_pos_aux: fixes f :: "'b fps" assumes "fps_nth f 0 = (1::'b)"   and "0 < fps_conv_radius f" shows "0 < fps_conv_radius (inverse f)"
all_compatible: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.Finite_Join_cpo below" shows "cpo.compatible below x y"
isomorphic_to_irr_right': fixes U :: "'v set" assumes "\<And>U. RModule FG (\<star>) U \<and> U \<subseteq> W \<Longrightarrow> U = 0 \<or> U = W"   and "RModule FG (\<cdot>) U \<and> U \<subseteq> V" shows "U = 0 \<or> U = V"
splitAt_dist_ram12: fixes vs :: "'a list"   and a :: "'a list"   and ram1 :: "'a"   and b :: "'a list"   and ram2 :: "'a"   and c :: "'a list" assumes "distinct vs"   and "vs = a @ ram1 # b @ ram2 # c" shows "(b, c) = splitAt ram2 (snd (splitAt ram1 vs))"
pi2: shows "\<pi> ` b2 \<in># remove1_mset (\<pi> ` b1) \<B>'"
assms_2: shows "w\<^sub>2 =\<^sub>F w\<^sub>3"
set_takeWhileD: fixes x :: "'c"   and P :: "'c \<Rightarrow> bool"   and xs :: "'c list" assumes "x \<in> set (takeWhile P xs)" shows "x \<in> set xs \<and> P x"
c3: fixes x :: "'a"   and y :: "'a" shows "x \<parallel> y \<cdot> 1\<^sub>\<pi> = x \<cdot> 1\<^sub>\<pi> \<parallel> (y \<cdot> 1\<^sub>\<pi>)"
"3_2": shows "gurun A w (fromN 1 ||| r) (0, p)"
simps_28: fixes x7 :: "'b ptrm"   and x2 :: "'b" shows "PSnd x7 \<noteq> PVar x2"
distinct_9: fixes x1 :: "'a"   and x6 :: "nat" shows "broadcast x1 \<noteq> deliver x6"
is_natural_2: fixes f :: "'d \<times> 'd \<times> 'd" assumes "CCC.arr f" shows "\<alpha> (CCC.cod f) \<cdot> T.ToTC f = \<alpha> f"
Un_def: fixes A :: "'f set"   and B :: "'f set" shows "A \<union> B = {x. x \<in> A \<or> x \<in> B}"
less_eq_set_def: fixes A :: "'e set"   and B :: "'e set" shows "(A \<subseteq> B) = ((\<lambda>x. x \<in> A) \<sqsubseteq> (\<lambda>x. x \<in> B))"