hcju/isabelleps huggingface dataset:
WTrtNil: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option" shows "P,E,h \<turnstile> [] [:] []"
negg: shows "negligible (g ` S - g ` {x \<in> S. det (matrix (g' x)) \<noteq> 0})"
assms_2: fixes \<xi> :: "'a" shows "S \<xi> \<xi>"
i: shows "(\<lambda>k. integral (cbox a b) (f k)) \<longlonglongrightarrow> i"
inversion_number_between_sorted_correct: fixes xs :: "'a list"   and ys :: "'a list" assumes "sorted_wrt (\<le>) xs"   and "sorted_wrt (\<le>) ys" shows "inversion_number_between_sorted xs ys = inversion_number_between xs ys"
vsubsetI: fixes a :: "V"   and b :: "V" assumes "\<And>x. x \<in> elts a \<Longrightarrow> x \<in> elts b" shows "a \<le> b"
su2_4: shows "dist (flow0 x su2) v < duv"
derivable_def: fixes C :: "'a eclause"   and P :: "'a eclause set"   and S :: "'a eclause set"   and \<sigma> :: "('a \<times> 'a trm) list"   and k :: "inferences"   and C' :: "'a literal set" shows "derivable C P S \<sigma> k C' = ((\<exists>P1 P2. P1 \<in> S \<and> P2 \<in> S \<and> P = {P1, P2} \<and> superposition P1 P2 C \<sigma> k C') \<or> (\<exists>P1. P1 \<in> S \<and> P = {P1} \<and> factorization P1 C \<sigma> k C') \<or> (\<exists>P1. P1 \<in> S \<and> P = {P1} \<and> reflexion P1 C \<sigma> k C'))"
gcd_diff_dvd_right2: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a dvd c" shows "gcd a (b - c) = gcd a b"
measurable_hld: fixes t :: "'a set"   and M :: "'a measure" assumes "t \<in> sets M" shows "Measurable.pred (stream_space M) (HLD t)"
restrict_current_IN_A: fixes a :: "'a"   and \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "a \<in> A (quotient_web \<Gamma> f)" shows "d_IN (g \<upharpoonleft> \<Gamma> / f) a = 0"
rep_eq: fixes x :: "\<Pi>\<^sub>2 \<Rightarrow> \<o>" shows "eval\<o> (forall\<^sub>2 x) = (\<lambda>s w. \<forall>xa. eval\<o> (x xa) s w)"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b val \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and f3 :: "'a"   and extCallRet :: "'b extCallRet" shows "P (case extCallRet of RetVal x \<Rightarrow> f1 x | RetExc x \<Rightarrow> f2 x | RetStaySame \<Rightarrow> f3) = (\<not> ((\<exists>x1. extCallRet = RetVal x1 \<and> \<not> P (f1 x1)) \<or> (\<exists>x2. extCallRet = RetExc x2 \<and> \<not> P (f2 x2)) \<or> extCallRet = RetStaySame \<and> \<not> P f3))"
deep_model_correct_params_axioms: shows "deep_model_correct_params rs"
in_HomD_4: fixes F :: "'a set"   and a :: "'a"   and b :: "'a"   and f :: "'a" assumes "F \<in> Hom a b"   and "f \<in> F" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>B b\<guillemotright>"
smc_SemiCAT_is_arrI: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<FF> : \<AA> \<mapsto>\<^bsub>smc_SemiCAT \<alpha>\<^esub> \<BB>"
max_stutter_sampler: fixes \<sigma> :: "nat \<Rightarrow> 'a" shows "stutter_sampler (max_stutter_sampler \<sigma>) \<sigma>"
cl_rank_of: fixes x :: "'a"   and X :: "'a set" assumes "x \<in> cl X" shows "rank_of (insert x X) = rank_of X"
from_bool_def: fixes b :: "bool" shows "from_bool b \<equiv> case b of True \<Rightarrow> word_of_nat 1 | False \<Rightarrow> word_of_nat 0"
sred: shows "gram_schmidt_fs.reduced n (map of_int_hom.vec_hom fs) \<alpha> (Suc i)"
rGamma_nonpos_Int: fixes z :: "'a" assumes "z \<in> \<int>\<^sub>\<le>\<^sub>0" shows "rGamma z = (0::'a)"
check_ipresIGSubstAll: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" shows "ipresIGSubstAll check check (errMOD MOD) MOD"
lnth_0: fixes x :: "'c"   and xs :: "'c llist" shows "LCons x xs $ 0 = x"
ix_lt: fixes p :: "(nat \<times> int) list"   and dm :: "nat"   and lm :: "nat"   and d :: "nat" assumes "p \<in> sparsegrid dm lm"   and "d < dm" shows "ix p d < 2 ^ (lv p d + 1)"
scaleRat_QDelta_def: fixes r :: "rat"   and qd :: "QDelta" shows "r *R qd = QDelta (r * qdfst qd) (r * qdsnd qd)"
abs_integer_code: fixes k :: "integer" shows "\<bar>k\<bar> = (if k < 0 then - k else k)"
wqo_on_def: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" shows "wqo_on P A = (transp_on P A \<and> almost_full_on P A)"
card_of_def: fixes A :: "'b set" shows "|A| = (SOME r. card_order_on A r)"
VInter_def: fixes A :: "V" shows "\<Inter>\<^sub>\<circ> A = (if A = 0 \<Rightarrow> 0 | otherwise \<Rightarrow> set {a. \<forall>x\<in>elts A. a \<in>\<^sub>\<circ> x})"
bij_betw_inv: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "bij_betw f A B" shows "\<exists>g. bij_betw g B A"
all_left_spec_def: fixes lf :: "nat option list" shows "local.all_left_spec lf \<equiv> length lf = length xs \<and> (\<forall>i<length xs. lf ! i = local.left_spec i)"
split_splits_wrt: fixes U :: "'x set"   and S :: "('x \<Rightarrow>\<^sub>0 nat) set"   and t :: "'x \<Rightarrow>\<^sub>0 nat"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite X"   and "U \<subseteq> X"   and "finite S"   and "t \<in> .[X]"   and "ideal F \<div> monomial (1::'b) t = ideal (monomial (1::'b) ` S)" shows "splits_wrt (split t U S) (cone (monomial (1::'b) t, U)) F"
SC_discr: fixes c :: "(test, atom) com" assumes "Example_PL.SC_discr c" shows "discr c"
map_option_idI: fixes x :: "'c option"   and f :: "'c \<Rightarrow> 'c" assumes "\<And>y. y \<in> set_option x \<Longrightarrow> f y = y" shows "map_option f x = x"
F1set2_bd': fixes x :: "('e, 'd, 'f) F1"   and r2 :: "(bd_type_F2 \<times> bd_type_F2) set" shows "|F1set2 x| \<le>o dir_image (bd_F1 +c r2) Abs_bd_type_F"
unif_mono: fixes E :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) multiset"   and E' :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) multiset"   and M :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) multiset" assumes "(E, E') \<in> unif" shows "\<exists>\<sigma>. (E + M, E' + subst_mset \<sigma> M) \<in> unif"
comp_MkArr: fixes A :: "'c \<Rightarrow> 's"   and B :: "'c \<Rightarrow> 's"   and F :: "'c \<Rightarrow> 's"   and C :: "'c \<Rightarrow> 's"   and G :: "'c \<Rightarrow> 's" assumes "Cop_S.arr (Cop_S.MkArr A B F)"   and "Cop_S.arr (Cop_S.MkArr B C G)" shows "Cop_S.comp (Cop_S.MkArr B C G) (Cop_S.MkArr A B F) = Cop_S.MkArr A C (vertical_composite.map (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) F G)"
gprod_2_1_def: fixes R :: "('b \<times> 'b) set" shows "gprod_2_1 R \<equiv> {((a, b), c, d). a = d \<and> (b, c) \<in> R \<or> b = c \<and> (a, d) \<in> R}"
shrK_notin_image_publicKey: fixes x :: "agent"   and b :: "keymode"   and AA :: "agent set" shows "shrK x \<notin> publicKey b ` AA"
assms_2: fixes u :: "'a"   and v :: "'a" assumes "fst (rep_nat_term u) = fst (rep_nat_term v)"   and "snd (rep_nat_term u) < snd (rep_nat_term v)" shows "cmp u v = Lt"
readerror: shows "\<forall>p q i. correct p (te p (i + 1)) \<and> correct q (te p (i + 1)) \<longrightarrow> \<bar>\<theta> p (i + 1) q - IC q i (te p (i + 1))\<bar> \<le> \<Lambda>"
rGamma_plus1: fixes z :: "'a" shows "z * rGamma (z + (1::'a)) = rGamma z"
\<psi>_in_terms_of_\<epsilon>: fixes x :: "'d"   and g :: "'c"   and y :: "'c" assumes "D.ide x"   and "\<guillemotleft>g : y \<rightarrow>\<^sub>C G x\<guillemotright>" shows "\<psi> x g = Adj.\<epsilon> x \<cdot>\<^sub>D F g"
is_rbt_rbt_sorted: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "('b, 'c) RBT_Impl.rbt" assumes "ord.is_rbt less t" shows "ord.rbt_sorted less t"
degree_map_poly_le: fixes f :: "'c \<Rightarrow> 'b"   and p :: "'c poly" shows "degree (map_poly f p) \<le> degree p"
width_lower_bound_1: fixes G :: "('c, 'd) Graph_scheme"   and T :: "('e, 'f) Graph_scheme"   and bag :: "'e \<Rightarrow> 'c set"   and v :: "'c"   and w :: "'c" assumes "TreeDecomposition G T bag"   and "v \<rightarrow>\<^bsub>G\<^esub> w" shows "1 \<le> TreeDecomposition.width T bag"
ee_is_fmset: fixes as :: "'a list"   and bs :: "'a list" assumes "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "essentially_equal G as bs = (fmset G as = fmset G bs)"
assms_1: shows "(\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> xi"
cobounded1: fixes a :: "'e"   and b :: "'e" shows "a \<le> max a b"
vars_add: fixes p1 :: "'b mpoly"   and p2 :: "'b mpoly" shows "vars (p1 + p2) \<subseteq> vars p1 \<union> vars p2"
list_all2_inj_map_eq: fixes as :: "'a list"   and bs :: "'a list" assumes "list_all2 (\<lambda>a b. a = f b) (map f as) bs"   and "list_all PB (map f as)"   and "list_all PA bs" shows "as = bs"
map_ident: fixes t :: "'b \<Rightarrow> 'c" shows "(\<lambda>x. x) \<circ> t = t"
mat_of_cols_def: fixes n :: "nat"   and cs :: "'a vec list" shows "mat_of_cols n cs = mat n (length cs) (\<lambda>(i, j). cs ! j $ i)"
dirichlet_inverse_noninvertible: fixes f :: "nat \<Rightarrow> 'b"   and i :: "'b"   and n :: "nat" assumes "f (Suc 0) = (0::'b)"   and "i = (0::'b)" shows "dirichlet_inverse f i n = (0::'b)"
simps: fixes a :: "('state, 'act, 'out) trans list" shows "valid2 a = ((\<exists>trn. a = [trn] \<and> validTrans trn) \<or> (\<exists>tr trn. a = tr ## trn \<and> valid2 tr \<and> tgtOf (last tr) = srcOf trn \<and> validTrans trn))"
ordinal_times_strict_monoR: fixes y :: "ordinal"   and y' :: "ordinal"   and x :: "ordinal" assumes "y < y'"   and "0 < x" shows "x * y < x * y'"
intros_3: fixes c :: "nat"   and u :: "'a \<Rightarrow> real"   and x :: "'a" assumes "real c < u x" shows "Regions.intv_elem x u (Regions.intv.Greater c)"
assms_1: shows "P \<parallel> Q \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> PQ'"
sestoftUnGC: fixes r :: "Vars.var list"   and c :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and d :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" assumes "to_gc_conf r c \<Rightarrow>\<^sub>G\<^sup>* d"   and "heap_upds_ok_conf c"   and "closed c"   and "r_ok r c" shows "\<exists>r' c'. c \<Rightarrow>\<^sup>* c' \<and> d = to_gc_conf r' c' \<and> r_ok r' c'"
domain: shows "Domainp pcr_rat = (\<lambda>x. \<exists>y. rel_prod (=) (=) x y \<and> y \<approx> y)"
nat_cut_ge_greater_conv: fixes I :: "nat set"   and t :: "nat" shows "I \<down>\<ge> Suc t = I \<down>> t"
finite_Ej: fixes a :: "nat list"   and b :: "nat list"   and j :: "nat"   and x :: "nat list" shows "finite (hlde_ops.Ej a b j x)"
conv_mirror: fixes \<sigma> :: "'a \<times> (bool \<times> 'b \<times> 'a) list"   and ars :: "('a \<times> 'b \<times> 'a) set" assumes "\<sigma> \<in> conv ars" shows "conv_mirror \<sigma> \<in> conv ars \<and> set (labels_conv (conv_mirror \<sigma>)) = set (labels_conv \<sigma>) \<and> fst \<sigma> = lst_conv (conv_mirror \<sigma>) \<and> lst_conv \<sigma> = fst (conv_mirror \<sigma>)"
cyc_free_not_empty: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and v :: "'e \<Rightarrow> nat" assumes "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len M i i xs"   and "\<forall>c. v c \<le> n \<longrightarrow> 0 < v c" shows "[M]\<^bsub>v,n\<^esub> \<noteq> {}"
se_invkrecv: fixes e :: "exp"   and m :: "nat"   and es :: "exp list" shows "e \<in> isubexprs(MethodInvk e m es)"
C_eq_if_mr_eq: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and b :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "applied_rule_rev Cp x b = applied_rule_rev Cp x a"   and "a \<noteq> []"   and "b \<noteq> []" shows "Cp (list2FWpolicy a) x = Cp (list2FWpolicy b) x"
simps_4: fixes bs :: "'b list"   and a :: "'b poly"   and b :: "'b poly" shows "Ipoly bs (Add a b) = Ipoly bs a + Ipoly bs b"
Q_def: shows "Q = mult_row (mat (1::'a)) (mod_type_class.from_nat i) ((1::'a) / A_interchange $ mod_type_class.from_nat i $ mod_type_class.from_nat k)"
update_cases: fixes r :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set"   and ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and P :: "(nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option) \<Rightarrow> bool" assumes "(\<pi>\<^sub>2 r = 0) = (\<pi>\<^sub>3 r = unk)"   and "ip \<notin> kD rt \<Longrightarrow> P (rt(ip \<mapsto> r))"   and "\<lbrakk>ip \<in> kD rt; sqn rt ip < \<pi>\<^sub>2 r\<rbrakk> \<Longrightarrow> P (rt(ip \<mapsto> addpre r (\<pi>\<^sub>7 (the (rt ip)))))"   and "\<lbrakk>ip \<in> kD rt; sqn rt ip = \<pi>\<^sub>2 r; \<pi>\<^sub>5 r < the (dhops rt ip)\<rbrakk> \<Longrightarrow> P (rt(ip \<mapsto> addpre r (\<pi>\<^sub>7 (the (rt ip)))))"   and "\<lbrakk>ip \<in> kD rt; sqn rt ip = \<pi>\<^sub>2 r; the (flag rt ip) = Aodv_Basic.inv\<rbrakk> \<Longrightarrow> P (rt(ip \<mapsto> addpre r (\<pi>\<^sub>7 (the (rt ip)))))"   and "\<lbrakk>ip \<in> kD rt; \<pi>\<^sub>3 r = unk\<rbrakk> \<Longrightarrow> P (rt(ip \<mapsto> (\<pi>\<^sub>2 (the (rt ip)), \<pi>\<^sub>3 r, \<pi>\<^sub>4 r, \<pi>\<^sub>5 r, \<pi>\<^sub>6 r, \<pi>\<^sub>7 (addpre r (\<pi>\<^sub>7 (the (rt ip)))))))"   and "\<lbrakk>ip \<in> kD rt; \<pi>\<^sub>2 r \<le> sqn rt ip; \<pi>\<^sub>3 r = kno; sqn rt ip = \<pi>\<^sub>2 r \<Longrightarrow> the (dhops rt ip) \<le> \<pi>\<^sub>5 r \<and> the (flag rt ip) = val\<rbrakk> \<Longrightarrow> P (rt(ip \<mapsto> addpre (the (rt ip)) (\<pi>\<^sub>7 r)))" shows "P (update rt ip r)"
empty_eq_map_of_iff: fixes xys :: "('a \<times> 'b) list" shows "(Map.empty = map_of xys) = (xys = [])"
NDHC: fixes \<Gamma> :: "'a formula set"   and F :: "'a formula" assumes "\<Gamma> \<turnstile> F" shows "AX10 \<union> \<Gamma> \<turnstile>\<^sub>H F"
sheaf_of_rings_axioms_def: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd" shows "sheaf_of_rings_axioms S is_open \<FF> \<rho> zero_str \<equiv> (\<forall>U I V s. open_cover_of_open_subset S is_open U I V \<longrightarrow> (\<forall>i. i \<in> I \<longrightarrow> V i \<subseteq> U) \<longrightarrow> s \<in> \<FF> U \<longrightarrow> (\<forall>i. i \<in> I \<longrightarrow> \<rho> U (V i) s = zero_str (V i)) \<longrightarrow> s = zero_str U) \<and> (\<forall>U I V s. open_cover_of_open_subset S is_open U I V \<longrightarrow> (\<forall>i. i \<in> I \<longrightarrow> V i \<subseteq> U \<and> s i \<in> \<FF> (V i)) \<longrightarrow> (\<forall>i j. i \<in> I \<longrightarrow> j \<in> I \<longrightarrow> \<rho> (V i) (V i \<inter> V j) (s i) = \<rho> (V j) (V i \<inter> V j) (s j)) \<longrightarrow> (\<exists>t. t \<in> \<FF> U \<and> (\<forall>i. i \<in> I \<longrightarrow> \<rho> U (V i) t = s i)))"
spair_alt_spair_sigs: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" shows "spair p q = monom_mult ((1::'b) / punit.lc (rep_list p)) (pp_of_term (fst (spair_sigs p q)) - lp p) p - monom_mult ((1::'b) / punit.lc (rep_list q)) (pp_of_term (snd (spair_sigs p q)) - lp q) q"
simps_3: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and D :: "char list"   and E :: "char list"   and Xs :: "char list list" shows "is_subobj P (C, D # E # Xs) = (let Ys = butlast (D # E # Xs); Y = last (D # E # Xs); X = last Ys in is_subobj P (C, Ys) \<and> P \<turnstile> X \<prec>\<^sub>R Y)"
LNil_le: fixes s :: "'a llist" shows "LNil \<le> s"
step: fixes a :: "'a" assumes "\<And>b. (a, b) \<in> r \<Longrightarrow> P b" shows "P a"
FAssReds1: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and F :: "char list"   and Cs :: "char list list"   and e\<^sub>2 :: "expr" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>e\<bullet>F{Cs} := e\<^sub>2,s\<rangle> \<rightarrow>* \<langle>e'\<bullet>F{Cs} := e\<^sub>2,s'\<rangle>"
restrict_merge_3: fixes J :: "'d set"   and I :: "'d set"   and x :: "'d \<Rightarrow> 'e"   and y :: "'d \<Rightarrow> 'e" assumes "J \<inter> I = {}" shows "restrict (merge I J (x, y)) I = restrict x I"
cong_4312: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Cong A B C D" shows "Cong D C A B"
has_integral: fixes f :: "'a \<Rightarrow> 'b"   and y :: "'b"   and a :: "'a"   and b :: "'a" shows "(f has_integral y) (cbox a b) = (\<forall>e>0. \<exists>\<gamma>. gauge \<gamma> \<and> (\<forall>\<D>. \<D> tagged_division_of cbox a b \<and> \<gamma> fine \<D> \<longrightarrow> norm ((\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R f x) - y) < e))"
contr_is_subgraph_subdivision: shows "\<exists>H. subgraph (with_proj H) (with_proj K) \<and> subdivision_pair (gen_contr_graph (with_proj K) (verts3 (with_proj K))) H"
inj_idom_hom_axioms: shows "inj_idom_hom fps_to_fls"
asset_ii_spied_start: fixes s :: "(agent \<times> msg) set"   and s' :: "(agent \<times> msg) set"   and B :: "nat"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "s \<turnstile> s'"   and "PriKey B \<in> spied s'"   and "PriKey B \<notin> spied s"   and "(Asset n, \<lbrace>Num 2, PubKey B\<rbrace>) \<in> s" shows "Auth_PriKey n \<in> spied s \<and> (\<exists>C SK. (Asset n, Token n (Auth_PriK n) B C SK) \<in> s)"
order_char_2: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x \<squnion> - x \<le> y \<squnion> - x)"
that_5: assumes "\<not> ed_ou X \<and> \<not> part X \<and> extremally_disconnected X" shows "thesis"
CallRedsNull: fixes e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and es :: "char list exp list"   and vs :: "val list"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and M :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<rightarrow>* \<langle>null,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>es,s\<^sub>1\<rangle> [\<rightarrow>]* \<langle>map Val vs,s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>M(es),s\<^sub>0\<rangle> \<rightarrow>* \<langle>THROW NullPointer,s\<^sub>2\<rangle>"
unary_wt_sym_0_imp_\<delta>_eq_\<epsilon>: fixes f :: "'s" assumes "arity_sym f = 1"   and "wt_sym f = 0" shows "\<delta> = \<epsilon>"
Rep_formula_eqvt: fixes x :: "perm"   and xa :: "('a, 'b, 'c, 'd) FL_Formula.formula" shows "x \<bullet> FL_Formula.formula.Rep_formula xa = FL_Formula.formula.Rep_formula (x \<bullet> xa)"
f_def: shows "f = (\<lambda>n p. complex_of_real (ln (real p) / real p) / of_nat n powr s)"
R_def: fixes s :: "'s"   and t :: "'t" shows "R s t = (s \<in> S \<and> f s = t)"
wp_While_If: fixes b :: "bexp"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int" shows "wp (WHILE b DO c) Q s = wp (IF b THEN c;; WHILE b DO c ELSE SKIP) Q s"
dropWhile_append1: fixes x :: "'a"   and xs :: "'a list"   and P :: "'a \<Rightarrow> bool"   and ys :: "'a list" assumes "x \<in> set xs"   and "\<not> P x" shows "dropWhile P (xs @ ys) = dropWhile P xs @ ys"
simps_2: fixes P :: "interface"   and c :: "command"   and Q :: "interface" shows "com_cell (Basic P c Q) = c"
range_composition: fixes f :: "'e \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'e" shows "range (\<lambda>x. f (g x)) = f ` range g"
q0: shows "of_int (lead_coeff q) \<noteq> (0::'a)"
VSigmaE: fixes c :: "V"   and A :: "V"   and B :: "V \<Rightarrow> V" assumes "c \<in>\<^sub>\<circ> VSigma A B" obtains x :: "V" and y :: "V" where "x \<in>\<^sub>\<circ> A" and "y \<in>\<^sub>\<circ> B x" and "c = \<langle>x, y\<rangle>"
pre_hensel_2: shows "\<exists>N. \<forall>n>N. val_Zp (a \<ominus> ns n) = val_Zp (local.divide (f \<bullet> a) (f' \<bullet> a))"
residue_fieldTr: fixes mx :: "'a set"   and x :: "'a set" assumes "maximal_ideal R mx"   and "x \<in> carrier (R /\<^sub>r mx)"   and "x \<noteq> \<zero>\<^bsub>R /\<^sub>r mx\<^esub>" shows "\<exists>y\<in>carrier (R /\<^sub>r mx). y \<cdot>\<^sub>r\<^bsub>R /\<^sub>r mx\<^esub> x = 1\<^sub>r\<^bsub>R /\<^sub>r mx\<^esub>"
simps_6: shows "1\<^sub>r \<noteq> 0\<^sub>r"
m_st_widen: fixes S2 :: "ivl Abs_State.st"   and S1 :: "ivl Abs_State.st" assumes "\<not> S2 \<sqsubseteq> S1" shows "m_st m_ivl (S1 \<nabla> S2) < m_st m_ivl S1"
STEP: fixes i :: "nat"   and s :: "'a" assumes "l \<le> i"   and "i < u" shows "<I i s> f i s <I (i + 1)>"
sum_ennreal: fixes I :: "'b set"   and f :: "'b \<Rightarrow> real" assumes "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i" shows "(\<Sum>i\<in>I. ennreal (f i)) = ennreal (sum f I)"
open_imp_locally_compact: fixes S :: "'b set" assumes "open S" shows "locally compact S"
singular_relcycle_def: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set" shows "singular_relcycle p X S \<equiv> \<lambda>c. singular_chain p X c \<and> (chain_boundary p c, 0) \<in> mod_subset (p - 1) (subtopology X S)"
runit_char_3: fixes f :: "'a" assumes "ide f" shows "\<exists>!\<mu>. \<guillemotleft>\<mu> : R f \<Rightarrow> f\<guillemotright> \<and> R \<mu> = (f \<star> \<i>[src f]) \<cdot> \<a>[f, src f, src f]"
simp_el_hull: fixes b :: "'b list"   and G :: "'b list set" assumes " b \<in>B G " shows " b \<in>B \<langle>G\<rangle> "
term_ok_subst_bv_no_change: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and t :: "Core.term"   and u :: "Core.term" assumes "term_ok \<Theta> t" shows "subst_bv u t = t"
hyps: fixes out :: "'b list"   and state :: "'a \<times> 'a iarray \<times> (nat \<times> 'b) list" assumes "local.create_subseqs base elements n = (out, state)" shows "invariant base elements n state \<and> set out = S base elements n"
compl_sets: fixes a :: "'b set"   and M :: "'b measure" assumes "a \<in> sets M" shows "space M - a \<in> sets M"
assms_1: fixes x :: "'a"   and y :: "'a" assumes "x \<in> F"   and "y \<in> F" shows "x + y \<in> F"
integrableI_nn_integral_finite: fixes f :: "'a \<Rightarrow> real"   and M :: "'a measure"   and x :: "real" assumes "f \<in> borel_measurable M"   and "AE x in M. 0 \<le> f x"   and "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M = ennreal x" shows "integrable M f"