hcju/isabelleps huggingface dataset:
successor_is_node: fixes n :: "'node"   and g :: "'g"   and n' :: "'node" assumes "n \<in> set (old.predecessors g n')"   and "n \<in> set (\<alpha>n g)"   and "invar g" shows "n' \<in> set (\<alpha>n g)"
cases: fixes a1 :: "char list \<Rightarrow> secLevel option"   and a2 :: "secLevel"   and a3 :: "com" assumes "a1,a2 \<turnstile> a3" obtains \<Gamma> :: "char list \<Rightarrow> secLevel option" and T :: "secLevel" where "a1 = \<Gamma>'__" and "a2 = Tb__" and "a3 = Skip"   | \<Gamma> :: "char list \<Rightarrow> secLevel option" and V :: "char list" and T :: "secLevel" and e :: "expr" where "a1 = \<Gamma>''__" and "a2 = Tc__" and "a3 = V__:=e__" and "\<Gamma>''__ V__ = Some High"   | \<Gamma> :: "char list \<Rightarrow> secLevel option" and e :: "expr" and V :: "char list" where "a1 = \<Gamma>'''__" and "a2 = Low" and "a3 = V:=e" and "\<Gamma>'''__ \<turnstile> e : Low" and "\<Gamma>'''__ V = Some Low"   | \<Gamma> :: "char list \<Rightarrow> secLevel option" and T :: "secLevel" and c1 :: "com" and c2 :: "com" where "a1 = \<Gamma>''''__" and "a2 = Td__" and "a3 = c1__;; c2__" and "\<Gamma>''''__,Td__ \<turnstile> c1__" and "\<Gamma>''''__,Td__ \<turnstile> c2__"   | \<Gamma> :: "char list \<Rightarrow> secLevel option" and b :: "expr" and T :: "secLevel" and c :: "com" where "a1 = \<Gamma>'''''__" and "a2 = Te__" and "a3 = while (ba__) ca__" and "\<Gamma>'''''__ \<turnstile> ba__ : Te__" and "\<Gamma>'''''__,Te__ \<turnstile> ca__"   | \<Gamma> :: "char list \<Rightarrow> secLevel option" and b :: "expr" and T :: "secLevel" and c1 :: "com" and c2 :: "com" where "a1 = \<Gamma>''''''__" and "a2 = T" and "a3 = if (b) c1 else c2" and "\<Gamma>''''''__ \<turnstile> b : T" and "\<Gamma>''''''__,T \<turnstile> c1" and "\<Gamma>''''''__,T \<turnstile> c2"   | \<Gamma> :: "char list \<Rightarrow> secLevel option" and c :: "com" where "a1 = \<Gamma>" and "a2 = Low" and "a3 = c" and "\<Gamma>,High \<turnstile> c"
OR4_analz_knows_Spy: fixes B :: "agent"   and N :: "msg"   and X :: "msg"   and X' :: "msg"   and evs :: "event list" assumes "Gets B \<lbrace>N, X, Crypt (shrK B) X'\<rbrace> \<in> set evs"   and "evs \<in> otway" shows "X \<in> analz (knows Spy evs)"
ipurge_tr_cons_same: fixes I :: "('e \<times> 'e) set"   and D :: "'f \<Rightarrow> 'e"   and x :: "'f"   and xs :: "'f list" assumes "refl I" shows "ipurge_tr I D (D x) (x # xs) = ipurge_tr I D (D x) xs"
assms: shows "x = of_int \<lfloor>x\<rfloor>"
elims_2: fixes x :: "atom"   and xa :: "real list" assumes "I\<^sub>R x xa" obtains r :: "real" and cs :: "real list" and xs :: "real list" where "x = Less r__ cs__" and "xa = xs__" and "r__ < \<langle>cs__,xs__\<rangle>"   | r :: "real" and cs :: "real list" and xs :: "real list" where "x = Eq r cs" and "xa = xs" and "r = \<langle>cs,xs\<rangle>"
cases: fixes a :: "'a list" assumes "Specialisation_Examples.sorted a" obtains "a = []"   | xs :: "'a list" and x :: "'a" where "a = x \<Colon> xs" and "\<forall>y\<in>set xs. x \<le> y" and "Specialisation_Examples.sorted xs"
K_\<Lambda>_GDD_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and sizes :: "int set"   and groups :: "'b set set"   and index :: "int" shows "K_\<Lambda>_GDD point_set block_collection sizes groups index \<equiv> K_block_design point_set block_collection sizes \<and> GDD point_set block_collection groups index"
permute_flip_at: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "(a \<leftrightarrow> b) \<bullet> c = (if c = a then b else if c = b then a else c)"
exp_conj_1_right: fixes A :: "'c \<Rightarrow> real" assumes "nneg A" shows "A && (\<lambda>_. 1) = A"
lin_indpt_subset_cols: fixes A :: "'a mat"   and B :: "'a vec set" assumes "A \<in> carrier_mat n n"   and "invertible_mat A"   and "B \<subseteq> set (cols A)" shows "lin_indpt B"
all_subst_rename_prv: fixes \<phi> :: "'fmla"   and x :: "'var"   and y :: "'var" assumes "\<phi> \<in> fmla"   and "x \<in> var"   and "y \<in> var"   and "y \<notin> Fvars \<phi>"   and "bprv (all x \<phi>)" shows "bprv (all y (subst \<phi> (Var y) x))"
subset_proj_absorb: fixes vs1 :: "'d set"   and vs2 :: "'d set"   and PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set" assumes "vs1 \<subseteq> vs2" shows "prob_proj (prob_proj PROB vs2) vs1 = prob_proj PROB vs1"
simps_113: fixes x141 :: "expr"   and x142 :: "expr"   and x4 :: "val" shows "while (x141) x142 \<noteq> Val x4"
and_not_mask_twice: fixes w :: "'b word"   and n :: "nat"   and m :: "nat" shows "(w AND NOT (mask n)) AND NOT (mask m) = w AND NOT (mask (max m n))"
\<alpha>0: fixes i :: "nat" assumes "1 < i" shows "(\<alpha> i = 0) = (k < i - 1)"
representant_exists: fixes A :: "('a \<times> 'a) set" assumes "A \<in> rel.Partition" shows "\<exists>a\<in>R \<times> S. a \<in> A \<and> A = rel.Class a"
1: fixes x :: "'a"   and y :: "'a" shows "- (- x \<squnion> y) \<squnion> x = x"
ivl_disj_un_one_4: fixes l :: "'e"   and u :: "'e" assumes "l \<le> u" shows "{..<l} \<union> {l..u} = {..u}"
label_wrt_eq_on_adjacent_vertex: fixes B :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and v :: "'a"   and v' :: "'a"   and C :: "'a set"   and z :: "'a set"   and z' :: "'a set" assumes "label_wrt B f"   and "f v = f v'"   and "chamber C"   and "chamber (insert v z)"   and "chamber (insert v' z')"   and "z \<lhd> C"   and "z' \<lhd> C"   and "insert v z \<noteq> C"   and "insert v' z' \<noteq> C" shows "insert v z = insert v' z'"
\<phi>_def: fixes t1 :: "'b tree"   and t2 :: "'b tree" shows "S34.\<phi> t1 t2 = 1 / 3 * log (root 3 4) (real (size1 t1 + size1 t2))"
irreflp_greater: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> comp" assumes "comparator cmp" shows "irreflp (\<lambda>a b. cmp a b = Greater)"
not_less_less_Suc_eq: fixes n :: "nat"   and m :: "nat" assumes "\<not> n < m" shows "(n < Suc m) = (n = m)"
mono_lift: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and app :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and n :: "nat"   and A :: "'a set"   and step :: "nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> 'a) list" assumes "order r"   and "app_mono r app n A"   and "bounded (err_step n app step) n"   and "\<forall>s p t. s \<in> A \<and> p < n \<and> s \<sqsubseteq>\<^bsub>r\<^esub> t \<longrightarrow> app p t \<longrightarrow> set (step p s) {\<sqsubseteq>\<^bsub>r\<^esub>} set (step p t)" shows "SemilatAlg.mono (Err.le r) (err_step n app step) n (err A)"
is_mapD: fixes M :: "('a \<times> 'b) set"   and a :: "'a"   and b\<^sub>1 :: "'b"   and b\<^sub>2 :: "'b" assumes "is_map M"   and "(a, b\<^sub>1) \<in> M"   and "(a, b\<^sub>2) \<in> M" shows "b\<^sub>1 = b\<^sub>2"
eqj: shows "matrix_sum d f (Suc j) = f j + matrix_sum d f' (Suc j)"
A\<^sub>C_sec: shows "\<forall>x. dma\<^sub>C mem\<^sub>1\<^sub>C [\<parallel>\<^sub>1 A\<^sub>C] x = Low \<and> (x \<notin> mds\<^sub>C AsmNoReadOrWrite \<or> x \<in> \<C>\<^sub>C) \<longrightarrow> mem\<^sub>1\<^sub>C [\<parallel>\<^sub>1 A\<^sub>C] x = mem\<^sub>2\<^sub>C [\<parallel>\<^sub>2 A\<^sub>C] x"
onorm_zero: shows "onorm (\<lambda>x. 0::'b) = 0"
PiE_cong: fixes I :: "'a set"   and A :: "'a \<Rightarrow> 'b set"   and B :: "'a \<Rightarrow> 'b set" assumes "\<And>i. i \<in> I \<Longrightarrow> A i = B i" shows "Pi\<^sub>E I A = Pi\<^sub>E I B"
Smith_normal_form_wrt_def: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and \<Q> :: "'b set" shows "Smith_normal_form_wrt A \<Q> = ((\<forall>a b. mod_type_class.to_nat a = mod_type_class.to_nat b \<and> mod_type_class.to_nat a + 1 < nrows A \<and> mod_type_class.to_nat b + 1 < ncols A \<longrightarrow> A $h a $h b dvd A $h (a + (1::'d)) $h (b + (1::'c))) \<and> isDiagonal A \<and> Complete_set_non_associates \<Q> \<and> (\<forall>a b. mod_type_class.to_nat a = mod_type_class.to_nat b \<and> mod_type_class.to_nat a < min (nrows A) (ncols A) \<and> mod_type_class.to_nat b < min (nrows A) (ncols A) \<longrightarrow> A $h a $h b \<in> \<Q>))"
finsum_scalar_prod_sum': fixes f :: "'b \<Rightarrow> 'a vec"   and U :: "'b set"   and w :: "'a vec" assumes "f \<in> U \<rightarrow> carrier_vec n"   and "w \<in> carrier_vec n" shows "w \<bullet> finsum V f U = (\<Sum>u\<in>U. w \<bullet> f u)"
all_trms_irreducible_def: fixes E :: "'b trm set"   and f :: "'b trm \<Rightarrow> 'b trm" shows "all_trms_irreducible E f = (\<forall>x y. x \<in> E \<longrightarrow> occurs_in y x \<longrightarrow> f y = y)"
resumption_ord_DONE1: fixes x :: "'f"   and r :: "('f, 'g, 'h) Resumption.resumption" shows "resumption_ord (DONE x) r = (r = DONE x)"
move_def: shows "move \<equiv> takeWhile (\<lambda>(t, X). left I \<le> nt - t) (linearize data_prev)"
monad_discard_def: fixes return :: "'d \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e" shows "monad_discard return bind \<equiv> monad return bind \<and> monad_discard_axioms bind"
Data_DataOverride: fixes P :: "'b pdata"   and Q :: "'b data" assumes "PDataSpace P = Data.DataSpace Q" shows "Data (map OptionOverride (zip (PDataValue P) (DataValue Q))) (Data.DataSpace Q)"
ANALYZE_def: fixes x :: "'a" shows "ANALYZE x \<equiv> x"
image_Collect: fixes f :: "'e \<Rightarrow> 'd"   and P :: "'e \<Rightarrow> bool" shows "f ` {x. P x} = {f x |x. P x}"
PO_obs_consistent: shows "obs_consistent R0n0i med0n0i a0n a0i"
induct: fixes x :: "'g list"   and A :: "'g set"   and P :: "'g list \<Rightarrow> bool" assumes "x \<in> lists A"   and "P []"   and "\<And>a l. \<lbrakk>a \<in> A; l \<in> lists A; P l\<rbrakk> \<Longrightarrow> P (a # l)" shows "P x"
ThreadFinishAction: fixes x :: "'x1"   and t :: "'t"   and m :: "'m1" assumes "final1 x" shows "r1.init_fin t ((Running, x), m) \<lbrace>ThreadFinishAction\<rbrace> ((Finished, x), m)"
z_1: shows "z \<in> \<omega>_limit_set y \<union> \<alpha>_limit_set y"
L2_3: fixes L :: "'b list set"   and R :: "('b list \<times> 'b list) set" assumes "MyhillNerode L R" obtains "finite (UNIV // eq_app_right L)" and "card (UNIV // eq_app_right L) \<le> card (UNIV // R)"
g_def: shows "g \<equiv> \<lambda>z. ((f \<circ> fst) z, snd z)"
ex_max_flow: shows "\<exists>f. flow \<Psi> f \<and> value_flow \<Psi> f = \<Squnion> (value_flow \<Psi> ` {g. flow \<Psi> g}) \<and> (\<forall>x. d_IN f x \<le> value_flow \<Psi> f)"
dg_singleton_is_arrD: fixes j :: "V"   and a :: "V"   and b :: "V"   and f :: "V" assumes "ZFC_in_HOL.set {\<langle>j, f\<rangle>} : ZFC_in_HOL.set {\<langle>j, a\<rangle>} \<mapsto>\<^bsub>\<Prod>\<^sub>D\<^sub>Gj\<in>\<^sub>\<circ>ZFC_in_HOL.set {j}. \<CC>\<^esub> ZFC_in_HOL.set {\<langle>j, b\<rangle>}"   and "j \<in>\<^sub>\<circ> Vset \<alpha>" shows "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b"
fact_expand_psi: fixes n :: "nat" shows "ln (fact n) - 2 * ln (fact (n div 2)) = (\<Sum>d = 1..n. (- 1) ^ (d + 1) * psi (n div d))"
atomize_all: fixes P :: "'d \<Rightarrow> bool" shows "(\<And>x. P x) \<equiv> \<forall>x. P x"
Gauss_Jordan_in_ij_iarrays_def: fixes A :: "'a iarray iarray"   and i :: "nat"   and j :: "nat" shows "Gauss_Jordan_in_ij_iarrays A i j = (let n = least_non_zero_position_of_vector_from_index (column_iarray j A) i; interchange_A = interchange_rows_iarray A i n; A' = mult_row_iarray interchange_A i ((1::'a) / interchange_A !! i !! j) in IArray.of_fun (\<lambda>s. if s = i then A' !! s else row_add_iarray A' s i (- interchange_A !! s !! j) !! s) (nrows_iarray A))"
prime_nat_naive: fixes p :: "nat" shows "Primes.prime_nat p = (1 < p \<and> (\<forall>n\<in>{1<..<p}. \<not> n dvd p))"
B: shows "(Asset m, Token m (Auth_PriK m) B C' SK') \<in> s"
simps_1: shows "FingerTreeStruc.gmnl [] = (0::'a)"
prems_4: shows "\<forall>a\<in>set (a # l). case a of (k, a) \<Rightarrow> line_integral_exists F basis a"
bounded_linear_image: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "bounded S"   and "bounded_linear f" shows "bounded (f ` S)"
card_Diff_insert: fixes A :: "'c set"   and a :: "'c"   and B :: "'c set" assumes "finite A"   and "a \<in> A"   and "a \<notin> B" shows "card (A - insert a B) = card (A - B) - 1"
preserves_ide: fixes a :: "'b" assumes "D.ide a" shows "D.ide (src\<^sub>D a)"
rel_compp_Grp: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_sctn R = (BNF_Def.Grp {x. set_sctn x \<subseteq> {(x, y). R x y}} (map_sctn fst))\<inverse>\<inverse> OO BNF_Def.Grp {x. set_sctn x \<subseteq> {(x, y). R x y}} (map_sctn snd)"
string2_rel_def: shows "string2_rel \<equiv> \<langle>Id\<rangle>list_rel"
simps_2: fixes M :: "(('fun, 'b, 'sets) prot_fun, 'c) Term.term list"   and TI :: "('sets set \<times> 'sets set) list"   and f :: "('fun, 'b, 'sets) prot_fun"   and T :: "(('fun, 'b, 'sets) prot_fun, 'c) Term.term list" shows "intruder_synth_mod_timpls' M TI (Fun f T) = (list_ex (\<lambda>t. timpls_transformable_to' TI t (Fun f T)) M \<or> public f \<and> length T = arity f \<and> list_all (intruder_synth_mod_timpls' M TI) T)"
simps_2: fixes \<nu> :: "'a"   and \<mu> :: "'a" shows "\<^bold>\<langle>\<nu>\<^bold>\<rangle> \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> \<^bold>\<langle>\<mu>\<^bold>\<rangle> = \<^bold>\<langle>\<nu> \<cdot> \<mu>\<^bold>\<rangle>"
finals_no_step: fixes es :: "char list exp list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and es' :: "char list exp list"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "finals es" shows "((es, s, b), es', s', b') \<notin> reds P"
rep_eq: fixes x :: "'b fset" shows "fBall x = Ball (fset x)"
normalize_content: fixes p :: "'a poly" shows "normalize (content p) = content p"
Fls_E: fixes H :: "fm set"   and A :: "fm" shows "insert Fls H \<turnstile> A"
finite_supp_abs_residual_pair: fixes \<alpha> :: "'a"   and P :: "'b" shows "finite (supp \<langle>\<alpha>,P\<rangle>)"
c_types_agree_imp_ct_list_eq: fixes cts :: "ct list"   and ts :: "t list" assumes "c_types_agree (TopType cts) ts" shows "\<exists>ts' ts''. ts = ts' @ ts'' \<and> ct_list_eq cts (to_ct_list ts'')"
pathfinish_linepath: fixes a :: "'a"   and b :: "'a" shows "pathfinish (linepath a b) = b"
similar_mat_witD2_4: fixes A :: "'a mat"   and n :: "nat"   and m :: "nat"   and B :: "'a mat"   and P :: "'a mat"   and Q :: "'a mat" assumes "A \<in> carrier_mat n m"   and "similar_mat_wit A B P Q" shows "A \<in> carrier_mat n n"
cda_add_ax: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "d ((x + y) \<cdot> z) = d (x \<cdot> z) + d (y \<cdot> z)"
finfun_Diag_update_update: fixes f :: "('a, 'b) finfun"   and a :: "'a"   and b :: "'b"   and g :: "('a, 'c) finfun"   and a' :: "'a"   and c :: "'c" shows "finfun_Diag (finfun_update f a b) (finfun_update g a' c) = (if a = a' then finfun_update (finfun_Diag f g) a (b, c) else finfun_update (finfun_update (finfun_Diag f g) a (b, finfun_apply g a)) a' (finfun_apply f a', c))"
contour_integrable_div: fixes f :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex"   and c :: "complex" assumes "f contour_integrable_on g" shows "(\<lambda>x. f x / c) contour_integrable_on g"
tauTransitionChain: fixes P :: "pi"   and P' :: "pi" shows "P \<Longrightarrow>\<^sup>^\<tau> \<prec> P' = ((P, P') \<in> {(P, P') |P P'. P \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P'}\<^sup>*)"
otherwith_syncD: fixes Q :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and I :: "'e set"   and P :: "('e \<Rightarrow> 'd) \<Rightarrow> 'f \<Rightarrow> bool"   and \<sigma> :: "'e \<Rightarrow> 'd"   and \<sigma>' :: "'e \<Rightarrow> 'd"   and a :: "'f" assumes "otherwith Q I P \<sigma> \<sigma>' a" shows "\<forall>j. j \<notin> I \<longrightarrow> Q (\<sigma> j) (\<sigma>' j)"
simps_1: fixes X :: "nat" shows "RED (TVar X) = {t. SN t}"
gb_red_aux_def: fixes bs :: "(('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) \<times> ('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list" shows "gb_red_aux bs ps = (let bs' = map fst bs in filter (\<lambda>h. h \<noteq> 0) (map (trdsp bs') ps))"
N: fixes n :: "nat" assumes "N \<le> n" shows "emeasure M (K n) < epsilon"
simps_2: fixes nothing :: "unit"   and nothinga :: "unit"   and doctype :: "char list"   and document_element :: "(_) element_ptr option"   and disconnected_nodes :: "(_) node_ptr list"   and more :: "'p" shows "doctype \<lparr>RObject.nothing = nothing, RDocument.nothing = nothinga, doctype = doctype, document_element = document_element, disconnected_nodes = disconnected_nodes, \<dots> = more\<rparr> = doctype"
simps: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and C :: "(nat \<Rightarrow> 'b) set"   and i :: "nat" assumes "minimal_element P A" shows "minimal_element.lexmin P A C i = minimal_element.min_elt P A (ith (eq_upto C (minimal_element.lexmin P A C) i) i)"
assms_2: shows "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"
e2_cball: shows "cball z e2 \<subseteq> s"
tendsto_imp_eventually_ne: fixes "open" :: "'a set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and c :: "'a"   and F :: "'b filter"   and c' :: "'a" assumes "class.t1_space open"   and "filterlim f (topological_space.nhds open c) F"   and "c \<noteq> c'" shows "\<forall>\<^sub>F z in F. f z \<noteq> c'"
nn_integral_C: fixes m :: "nat"   and m' :: "nat"   and f :: "(nat \<Rightarrow> 'a) \<Rightarrow> ennreal"   and n :: "nat"   and x :: "nat \<Rightarrow> 'a" assumes "m \<le> m'"   and "f \<in> borel_measurable (Pi\<^sub>M {0..<n + m} M)"   and "\<And>x. x \<in> space (Pi\<^sub>M {0..<n + m} M) \<Longrightarrow> 0 \<le> f x"   and "x \<in> space (Pi\<^sub>M {0..<n} M)" shows "integral\<^sup>N (C n m x) f = \<integral>\<^sup>+ x. f (restrict x {0..<n + m}) \<partial>C n m' x"
exists_best_and_degenerate: shows "local.degenerate_lotteries \<inter> local.best \<noteq> {}"
belong_ar_coset1: fixes I :: "'a set"   and a :: "'a"   and x :: "'a" assumes "ideal R I"   and "a \<in> carrier R"   and "x \<in> carrier R"   and "x \<plusminus> -\<^sub>a a \<in> I" shows "x \<in> a \<uplus>\<^bsub>R\<^esub> I"
angle_commute: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "angle a b c = angle c b a"
\<Delta>''_sel_3: shows "source \<Delta>'' = Vertex (source \<Delta>)"
simps_11: fixes v :: "fm"   and va :: "fm" shows "not (Or v va) = NOT (Or v va)"
\<theta>': shows "\<guillemotleft>\<l>[f] \<cdot> (\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, f\<^sub>a, f] : f \<star> f\<^sub>a \<star> f \<Rightarrow> f\<guillemotright>"
not_trivial_limit_within: fixes x :: "'b"   and S :: "'b set" shows "(at x within S = bot) \<noteq> (x \<in> closure (S - {x}))"
is_valid_problem_sas_plus_then_2: fixes \<Psi> :: "('c, 'd) sas_plus_problem" assumes "is_valid_problem_sas_plus \<Psi>" shows "\<forall>op\<in>set (\<Psi>\<^sub>\<O>\<^sub>+). is_valid_operator_sas_plus \<Psi> op"
inSetOfComponents_def: fixes i :: "AbstrLevelsID"   and chSet :: "chanID set" shows "inSetOfComponents i chSet \<equiv> {X. IN X \<inter> chSet \<noteq> {} \<and> X \<in> AbstrLevel i}"
clset_instances_def: fixes S :: "('a eclause \<times> ('a \<times> 'a trm) list) set" shows "clset_instances S = {C. \<exists>x. x \<in> S \<and> C = subst_cl (cl_ecl (fst x)) (snd x)}"
that: fixes \<psi> :: "'a pltl" assumes "next_free \<psi>"   and "atoms_pltl \<psi> \<subseteq> atoms_pltl \<phi>"   and "\<forall>\<sigma>. \<sigma> \<Turnstile>\<^sub>p \<psi> = \<sigma> \<Turnstile>\<^sub>p \<phi>" shows "thesis"
cmp_simps_5: fixes f :: "'c"   and g :: "'c" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.cod (\<Phi>\<^sub>G (f, g)) = G (f \<star>\<^sub>C g)"
Kleene_iter_gpfp: fixes f :: "'a \<Rightarrow> 'a"   and p :: "'a"   and k :: "nat" assumes "mono f"   and "p \<le> f p" shows "p \<le> (f ^^ k) top"
\<pi>\<rho>': shows "\<pi>' =\<^bsub>n'\<^esub> \<rho>'"
assms_4: shows "H \<turnstile> HaddP k1 k2 k"
dests_inv: shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). (l \<in> {PAodv-:15, PPkt-:7, PRreq-:9, PRreq-:21, PRrep-:9, PRerr-:1} \<longrightarrow> (\<forall>ip\<in>dom (dests \<xi>). ip \<in> vD (rt \<xi>))) \<and> (l \<in> {PAodv-:16..PAodv-:19} \<union> {PPkt-:8..PPkt-:11} \<union> {PRreq-:10..PRreq-:13} \<union> {PRreq-:22..PRreq-:25} \<union> {PRrep-:10..PRrep-:13} \<union> {PRerr-:2..PRerr-:5} \<longrightarrow> (\<forall>ip\<in>dom (dests \<xi>). ip \<in> iD (rt \<xi>) \<and> the (dests \<xi> ip) = sqn (rt \<xi>) ip)) \<and> (l = PPkt-:14 \<longrightarrow> dip \<xi> \<in> iD (rt \<xi>)))"
simulation_axioms: shows "simulation as\<^sub>0 alstep bs\<^sub>0 blstep sim_rel1"
Ra_def: fixes P :: "'c process"   and s :: "'c event list" shows "Ra P s = {X. (s, X) \<in> F P}"
simps_2: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and M :: "'f"   and I :: "('f, 'd, 'e) input" assumes "assertionAux SCompose SImp SBottom SChanEq" shows "assertionAux.extractFrame SCompose SBottom (M\<lparr>I) = \<langle>\<epsilon>, SBottom\<rangle>"
cosh_real_abs: fixes x :: "real" shows "cosh \<bar>x\<bar> = cosh x"
intro: fixes M :: "('a, 'b) message set"   and a :: "'a extended \<Rightarrow> 'a sort_key extended"   and s :: "('a, 'b option) insert_message list"   and m :: "'a delete_message" assumes "integrate_insert_commute M a s"   and "integrate_insert_commute_delete_axioms M m" shows "integrate_insert_commute_delete M a s m"
lookup_operator'_correct: fixes name :: "char list" shows "lookup_operator' problem name = lookup_operator name"
inf_idem: fixes x :: "'a upred" shows "(x \<and> x) = x"
cond_mono: fixes P\<^sub>2 :: "('d, 'e) uexpr"   and P\<^sub>1 :: "('d, 'e) uexpr"   and Q\<^sub>2 :: "('d, 'e) uexpr"   and Q\<^sub>1 :: "('d, 'e) uexpr"   and b :: "'e upred" assumes "P\<^sub>1 \<sqsubseteq> P\<^sub>2"   and "Q\<^sub>1 \<sqsubseteq> Q\<^sub>2" shows "P\<^sub>1 \<triangleleft> b \<triangleright> Q\<^sub>1 \<sqsubseteq> P\<^sub>2 \<triangleleft> b \<triangleright> Q\<^sub>2"
ev: shows "(cms ! k, mem) \<leadsto> (cms\<^sub>k', mem') \<and> k \<sqsubset> length cms \<and> cms' = cms[k := cms\<^sub>k']"
has_derivative_within_alt: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd"   and x :: "'c"   and s :: "'c set" shows "(f has_derivative f') (at x within s) = (bounded_linear f' \<and> (\<forall>e>0. \<exists>d>0. \<forall>y\<in>s. norm (y - x) < d \<longrightarrow> norm (f y - f x - f' (y - x)) \<le> e * norm (y - x)))"
assms_2: fixes i :: "'a" assumes "i \<in> I" shows "closed (U i)"
aform_val_zero_pdevs: fixes e :: "nat \<Rightarrow> real"   and x :: "'a" shows "aform_val e (x, zero_pdevs) = x"
dvd_imp_degree_le: fixes p :: "'d poly"   and q :: "'d poly" assumes "p dvd q"   and "q \<noteq> 0" shows "degree p \<le> degree q"
access_matrix_def: fixes c :: "parts_connection"   and rs :: "'a simple_rule list" shows "access_matrix c rs \<equiv> let W = build_ip_partition c rs; R = map getOneIp W in (zip R W, filter (\<lambda>(s, d). runFw s d c rs = Decision FinalAllow) (all_pairs R))"
add_order_preserving_left: fixes a :: "'a"   and b :: "'a"   and u :: "'a" assumes "a \<le> b" shows "u + a \<le> u + b"
grey_protects_whiteI: fixes g :: "'e"   and s :: "'f process_name \<Rightarrow> ('g, 'f, 'h, 'e) Model.local_state" assumes "grey g s" shows "(g grey_protects_white g) s"
IH: assumes "periodN w p"   and "p \<le> \<^bold>|w\<^bold>|" shows "periodN (take p w \<^sup>@ k \<cdot> w) p"
Anno_cases_6: fixes is_lub' :: "ty \<Rightarrow> ty \<Rightarrow> ty \<Rightarrow> bool"   and P :: "'b J_prog"   and E :: "String.literal \<Rightarrow> ty option"   and V :: "String.literal"   and e' :: "'b expr" assumes "is_lub',P,E \<turnstile> Var V \<leadsto> e'" obtains T :: "ty" where "e' = Var V" and "E V = \<lfloor>T__\<rfloor>" and "V \<noteq> Annotate.super"   | C :: "String.literal" and T :: "ty" and fm :: "fmod" and D :: "String.literal" where "e' = Var this\<bullet>V{D}" and "E V = None" and "V \<noteq> Annotate.super" and "E this = \<lfloor>Class C\<rfloor>" and "P \<turnstile> C sees V:T (fm) in D"
assms: shows "xs \<simeq> {ys, zs, P}"
rev_update: fixes k :: "nat"   and xs :: "'a list"   and y :: "'a" assumes "k < length xs" shows "rev (xs[k := y]) = (rev xs)[length xs - k - 1 := y]"
arity: fixes i :: "nat" shows "length (a i) = snd (r i)"