hcju/isabelleps huggingface dataset:
space_T: fixes K :: "'a \<Rightarrow> 'a pmf"   and s :: "'a" shows "space (MC_syntax.T K s) = space (stream_space (count_space UNIV))"
lem1: fixes x :: "'a"   and y :: "'a"   and M :: "'a multiset"   and N :: "'a multiset" assumes "x \<noteq> y" shows "(x, M = y, N) = (x \<in># N \<and> M = y, N - (x, {#}))"
l0e: shows "e * c / (f2 * (e - 1)) \<le> real l0"
observable_move_preserves_stacks: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and m :: "'node"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and a :: "'edge"   and m' :: "'node"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and cs :: "'edge list"   and rs :: "'edge list" assumes "S,f \<turnstile> (m # ms,s) -a\<rightarrow> (m' # ms',s')"   and "valid_call_list cs m"   and "\<forall>i<length rs. rs ! i \<in> get_return_edges (cs ! i)"   and "valid_return_list rs m"   and "length rs = length cs"   and "ms = targetnodes rs" obtains cs' :: "'edge list" and rs' :: "'edge list" where "valid_node m'" and "valid_call_list cs' m'" and "\<forall>i<length rs'. rs' ! i \<in> get_return_edges (cs' ! i)" and "valid_return_list rs' m'" and "length rs' = length cs'" and "ms' = targetnodes rs'" and "upd_cs cs [a] = cs'"
qSwapAll_preserves_qFreshAll: fixes ys :: "'a"   and zs :: "'a"   and z1 :: "'b"   and z2 :: "'b"   and y :: "'b"   and X :: "('c, 'd, 'a, 'b, 'e) qTerm"   and A :: "('c, 'd, 'a, 'b, 'e) qAbs" shows "qFresh ys (y @ys[z1 \<and> z2]_zs) (X #[[z1 \<and> z2]]_zs) = qFresh ys y X \<and> qFreshAbs ys (y @ys[z1 \<and> z2]_zs) (A $[[z1 \<and> z2]]_zs) = qFreshAbs ys y A"
RES_RES_RETURN_RES: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'a set" shows "RES A \<bind> (\<lambda>T. RES (f T)) = RES (\<Union> (f ` A))"
hyps: assumes "x \<in> set xs" shows "\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys"
times_gt_right_mono: fixes y :: "'a"   and z :: "'a"   and x :: "'a" assumes "y \<succ> z" shows "x * y \<succ> x * z"
interior_of_openin: fixes X :: "'c topology"   and S :: "'c set" assumes "openin X S" shows "X interior_of S = S"
Cast_\<tau>red1t_xt: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and T :: "ty" assumes "\<tau>red1gt uf P t h (e, xs) (e', xs')" shows "\<tau>red1gt uf P t h (Cast T e, xs) (Cast T e', xs')"
hd_b: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i < j"   and "j \<le> k" shows "hd (list_of (b k (j, i))) = \<Sqinter> (b k (j, i))"
card_Pow: fixes A :: "'g set" assumes "finite A" shows "card (Pow A) = 2 ^ card A"
fmap_btree_simps_1: fixes f :: "'b \<rightarrow> 'a" shows "fmap\<cdot>f\<cdot>\<bottom> = \<bottom>"
length_enumerate: fixes n :: "nat"   and xs :: "'f list" shows "length (enumerate n xs) = length xs"
no_writes_unchanged0: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and k :: "nat"   and v :: "'b"   and \<sigma> :: "'b \<Rightarrow> 'c" assumes "IFC prob"   and "\<forall>l<k. v \<notin> IFC_def.writes prob (IFC_def.path prob \<sigma> l)" shows "IFC_def.kth_state prob \<sigma> k v = \<sigma> v"
select_return_top_exists: fixes cts :: "ct list"   and c1 :: "ct"   and c2 :: "ct"   and ctm :: "checker_type" assumes "select_return_top cts c1 c2 = ctm"   and "ctm \<noteq> Bot" shows "\<exists>xs. ctm = TopType xs"
seq_lexord_trans: fixes xs :: "'b seq"   and ys :: "'b seq"   and R :: "('b \<times> 'b) set"   and zs :: "'b seq" assumes "(xs, ys) \<in> seq_lexord R"   and "(ys, zs) \<in> seq_lexord R"   and "trans R" shows "(xs, zs) \<in> seq_lexord R"
relcomp_UNION_distrib: fixes s :: "('b \<times> 'd) set"   and r :: "'e \<Rightarrow> ('d \<times> 'c) set"   and I :: "'e set" shows "s O \<Union> (r ` I) = (\<Union>i\<in>I. s O r i)"
iwlsFSwSTR_alphaAll_qInitAll: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and qX :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm"   and qA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs" assumes "iwlsFSwSTR MOD" shows "(\<forall>qX'. qX #= qX' \<longrightarrow> qInit MOD qX = qInit MOD qX') \<and> (\<forall>qA'. qA $= qA' \<longrightarrow> qInitAbs MOD qA = qInitAbs MOD qA')"
Digamma_LIMSEQ: fixes z :: "'a" assumes "z \<noteq> (0::'a)" shows "(\<lambda>m. of_real (ln (real m)) - (\<Sum>n<m. inverse (z + of_nat n))) \<longlonglongrightarrow> Digamma z"
hyps_1: shows "sh C = \<lfloor>(sfs, i)\<rfloor>"
fx_in_orb: shows "f x \<in> orbit f (f x)"
height_mkt_bal_l2: fixes l :: "'b tree"   and r :: "'b tree"   and n :: "'b" assumes "avl l"   and "avl r"   and "height l \<noteq> height r + 2" shows "height (mkt_bal_l n l r) = 1 + max (height l) (height r)"
vp_split_second: fixes n :: "'node"   and as :: "'edge list"   and a :: "'edge"   and as' :: "'edge list"   and n' :: "'node" assumes "n -as @ a # as'\<rightarrow>\<^sub>\<surd>* n'" shows "sourcenode a -a # as'\<rightarrow>\<^sub>\<surd>* n'"
attractor_has_strategy: fixes W :: "'a set"   and p :: "Player" assumes "W \<subseteq> V" shows "\<exists>\<sigma>. strategy p \<sigma> \<and> strategy_attracts p \<sigma> (attractor p W) W"
in_M_lt_Sn: shows "\<forall>m\<in>#M. m < Suc n"
seqauto_def: shows "seqauto = {(S, I, L, D) |S I L D. SeqAuto S I L D}"
inf_img_fin_dom: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "finite (f ` A)"   and "infinite A" shows "\<exists>y\<in>f ` A. infinite (f -` {y})"
a: shows "\<forall>x. x \<in> Q \<longrightarrow> (x, f x) \<in> A \<and> f x \<in> Q"
ldropn_eq_LConsD: fixes n :: "nat"   and xs :: "'b llist"   and y :: "'b"   and ys :: "'b llist" assumes "ldropn n xs = LCons y ys" shows "enat n < llength xs"
B: shows "P'' \<longmapsto> a[b] \<prec> P'''"
simps: fixes a1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and a2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" shows "\<tau>mredT a1 a2 = (\<exists>s s' tl. a1 = s \<and> a2 = s' \<and> redT s tl s' \<and> m\<tau>move s tl s')"
diff_apply_chart: fixes \<psi> :: "('b, 'b) chart" assumes "\<psi> \<in> a.dest.atlas" shows "diff k (a.dest.charts_submanifold (domain \<psi>)) charts_eucl (apply_chart \<psi>)"
Standard_harg: fixes x :: "complex star" assumes "x \<in> SComplex" shows "harg x \<in> Standard"
in_idem: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<in> A" shows "max x (Max A) = Max A"
assms_4: shows "dX (\<phi> Pd' Ch' ds) d = dX (\<phi> (Pd'(d := Pd'')) Ch' ds) d"
g12_carr: shows "g1 \<in> carrier G \<and> g2 \<in> carrier G"
assms_5: shows "(c, s) \<rightarrow>c (c', s')"
is_initial: fixes y' :: "'c"   and g :: "'d" assumes "arrow_to_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G x y' g" shows "\<exists>!f. is_ext y' g f"
foldseq_foldseq: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and n :: "nat"   and m :: "nat" assumes "associative f"   and "associative g"   and "\<forall>a b c d. g (f a b) (f c d) = f (g a c) (g b d)" shows "foldseq g (\<lambda>j. foldseq f (A j) n) m = foldseq f (\<lambda>j. foldseq g (transpose_infmatrix A j) m) n"
C_1: shows "rep_Bonk_Schramm_extension x \<in> extended_distance_set"
PDG_BS_s_def: fixes S :: "'node set" shows "PDG_BS S \<equiv> PDG.PDG_BS sourcenode targetnode valid_edge Def Use standard_control_dependence S"
cltn2_rep_def: fixes A :: "cltn2" shows "cltn2_rep A \<equiv> SOME B. B \<in> Rep_cltn2 A"
in_add: fixes e :: "'b environment"   and a :: "char list"   and X :: "'b" assumes "ok (e\<lparr>a:X\<rparr>)" shows "a \<in> env_dom (e\<lparr>a:X\<rparr>)"
thetaPar_Sretr: shows "thetaPar \<subseteq> Sretr (thetaPar \<union> Sbis)"
br_def: fixes \<alpha> :: "'b \<Rightarrow> 'c"   and I :: "'b \<Rightarrow> bool" shows "br \<alpha> I \<equiv> {(c, a). a = \<alpha> c \<and> I c}"
lt_mult_scalar_monomial_right: fixes c :: "'c"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and v :: "'t" assumes "c \<noteq> (0::'c)"   and "p \<noteq> 0" shows "lt (p \<odot> monomial c v) = punit.lt p \<oplus> v"
ideD_1: fixes a :: "'a" assumes "R.ide a" shows "R.arr a"
IH: assumes "pairwise (\<lambda>S T. S \<inter> T \<subseteq> K) \<F>"   and "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>a g. a \<notin> U \<and> continuous_on (S - {a}) g \<and> g ` (S - {a}) \<subseteq> T \<and> (\<forall>x\<in>S \<inter> K. g x = h x)"   and "\<And>S. S \<in> \<F> \<Longrightarrow> closed S" shows "\<exists>C g. finite C \<and> disjnt C U \<and> card C \<le> card \<F> \<and> continuous_on (\<Union> \<F> - C) g \<and> g ` (\<Union> \<F> - C) \<subseteq> T \<and> (\<forall>x\<in>(\<Union> \<F> - C) \<inter> K. g x = h x)"
order_less_le: fixes x :: "'a"   and y :: "'a" shows "(x < y) = (x \<le> y \<and> x \<noteq> y)"
simps_3: fixes x :: "'b"   and t :: "'b tree" shows "cost (Insert x) [t] = T_insert x t"
L_ide: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and D :: "'a \<times> 'b \<Rightarrow> 'd"   and l :: "'b \<Rightarrow> 'd"   and \<chi> :: "'b \<Rightarrow> 'a \<Rightarrow> 'd"   and a :: "'b" assumes "parametrized_diagram J A B D"   and "\<forall>a. partial_magma.ide A a \<longrightarrow> limit_cone J B (\<lambda>j. D (j, a)) (l a) (\<chi> a)"   and "partial_magma.ide A a" shows "(if partial_magma.arr A a then limit_cone.induced_arrow J B (\<lambda>j. D (j, partial_magma.cod A a)) (l (partial_magma.cod A a)) (\<chi> (partial_magma.cod A a)) (l (partial_magma.dom A a)) (vertical_composite.map J B (\<chi> (partial_magma.dom A a)) (\<lambda>j. D (j, a))) else partial_magma.null B) = l a"
simps: fixes a :: "int poly"   and b :: "int poly" shows "min_degree_poly a b = (if degree a \<le> degree b then a else b)"
intros_12: fixes s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and D :: "char list"   and S :: "(char list list \<times> (char list \<Rightarrow> val option)) set"   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 C :: "char list"   and Cs' :: "char list list"   and E :: "char list \<Rightarrow> ty option"   and Cs :: "char list list" assumes "hp s a = \<lfloor>(D, S)\<rfloor>"   and "P \<turnstile> Path D to C via Cs' "   and "P \<turnstile> Path D to C unique" shows "P,E \<turnstile> \<langle>Cast C (ref (a, Cs)),s\<rangle> \<rightarrow> \<langle>ref (a, Cs'),s\<rangle>"
context_equiv_def: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and P :: "'BExp set"   and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option" shows "\<Gamma> =:\<^sub>P \<Gamma>' \<equiv> dom \<Gamma> = dom \<Gamma>' \<and> (\<forall>x\<in>dom \<Gamma>'. the (\<Gamma> x) =:\<^sub>P the (\<Gamma>' x))"
exhaust: fixes y :: "sort_dir" obtains "y = sort_dir.Left"   | "y = sort_dir.Right"
Abs_unit_inverse: fixes y :: "bool" assumes "y \<in> {True}" shows "Rep_unit (Abs_unit y) = y"
False: shows "\<not> (\<exists>v\<in>U. (v, D x) \<in> I)"
gamma_Step_subcomm: fixes f1 :: "char list \<Rightarrow> aexp \<Rightarrow> (char list \<Rightarrow> int) set \<Rightarrow> (char list \<Rightarrow> int) set"   and f2 :: "char list \<Rightarrow> aexp \<Rightarrow> 'av st option \<Rightarrow> 'av st option"   and g1 :: "bexp \<Rightarrow> (char list \<Rightarrow> int) set \<Rightarrow> (char list \<Rightarrow> int) set"   and g2 :: "bexp \<Rightarrow> 'av st option \<Rightarrow> 'av st option"   and S :: "'av st option"   and C :: "'av st option acom" assumes "\<And>x e S. f1 x e (\<gamma>\<^sub>o S) \<subseteq> \<gamma>\<^sub>o (f2 x e S)"   and "\<And>b S. g1 b (\<gamma>\<^sub>o S) \<subseteq> \<gamma>\<^sub>o (g2 b S)" shows "Step f1 g1 (\<gamma>\<^sub>o S) (\<gamma>\<^sub>c C) \<le> \<gamma>\<^sub>c (Step f2 g2 S C)"
noStuck_Call: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and p :: "'d"   and s :: "'e" assumes "\<Gamma>\<turnstile> \<langle>Call p,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}" shows "p \<in> dom \<Gamma>"
less_fun_asym: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "less_fun f g" shows "\<not> less_fun g f"
is_supI: fixes x :: "'b"   and sup :: "'b"   and y :: "'b" assumes "x \<sqsubseteq> sup"   and "y \<sqsubseteq> sup"   and "\<And>z. \<lbrakk>x \<sqsubseteq> z; y \<sqsubseteq> z\<rbrakk> \<Longrightarrow> sup \<sqsubseteq> z" shows "is_sup x y sup"
pt_perm_compose: fixes pi2 :: "('b \<times> 'b) list"   and pi1 :: "('b \<times> 'b) list"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "pi2 \<bullet> pi1 \<bullet> x = (pi2 \<bullet> pi1) \<bullet> pi2 \<bullet> x"
skip_hoare_r: fixes p :: "'a upred" shows "\<lbrace>p\<rbrace> II \<lbrace>p\<rbrace>\<^sub>u"
distinct_17: fixes x21 :: "nat set"   and x22 :: "'e"   and x4 :: "nat" shows "groupcast x21 x22 \<noteq> \<not>unicast x4"
intros_1: fixes s :: "char list \<Rightarrow> int" shows "(SKIP, s) \<Rightarrow> Suc 0 \<Down> s"
prems: shows "S \<Turnstile>\<^sub>P af \<phi> (xs @ [x])"
F_mndet: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b process" assumes "A \<noteq> {}" shows "F (mndet A P) = (\<Union>x\<in>A. F (x \<rightarrow> P x))"
mod_nonneg_pos_int: fixes x :: "int"   and y :: "int" assumes "0 \<le> x"   and "0 < y" shows "mod_nonneg_pos_int x y = x mod y"
seq_imp_steps: fixes ss :: "'a \<times> ('b \<times> 'a) list"   and ars :: "('a \<times> 'b \<times> 'a) set"   and a :: "'a"   and b :: "'a" assumes "ss \<in> seq ars"   and "fst ss = a"   and "lst ss = b" shows "(a, b) \<in> (unlabel ars)\<^sup>*"
component_trans: fixes F :: "'c program"   and G :: "'c program"   and H :: "'c program" assumes "F \<le> G"   and "G \<le> H" shows "F \<le> H"
cdf_lim_at_bot: fixes k :: "nat" shows "(cdf (\<mu> k) \<longlongrightarrow> 0) at_bot"
upd_set'_aux2: fixes m :: "('c, 'a) mapping"   and k :: "'c"   and a :: "'b" assumes "Mapping.lookup m k = None" shows "upd_set' m d f {b. b = k \<or> (a, b) \<in> A} = Mapping.update k d (upd_set' m d f {b. (a, b) \<in> A})"
assms: shows "ccw'.sortedP 0 (u # v # w # xs)"
simps_2: fixes uu :: "dB" shows "opaque (Pure uu) = []"
Translations_E: fixes f :: "'a \<Rightarrow> 'a" assumes "f \<in> translation ` G" obtains x :: "'a" where "x \<in> G" and "f = (x)\<^sub>L"
euclidean_representation: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and x :: "'b" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "comm_monoid_add.sum plus zero (\<lambda>b. scaleR (inner x b) b) Basis = x"
card_eq_0_iff: fixes A :: "'g set" shows "(card A = 0) = (A = {} \<or> infinite A)"
comm_monoid_add_ow_Ball_def: fixes S :: "'b set"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b" shows "comm_monoid_add_ow S pls z = (ab_semigroup_add_ow S pls \<and> (\<forall>a\<in>S. pls z a = a) \<and> z \<in> S)"
mem_cartesian: fixes M :: "('g, 'a, 'h) Module_scheme"   and N :: "('i, 'a, 'h) Module_scheme"   and m :: "'g"   and n :: "'i" assumes "R module M"   and "R module N"   and "m \<in> carrier M"   and "n \<in> carrier N" shows "(m, n) \<in> M \<times>\<^sub>c N"
AssumeH_1: fixes A :: "fmla"   and H :: "fmla set" shows "insert A H \<turnstile> A"
s_prod_annil: fixes x :: "'a" shows "(0::'a) \<cdot> x = (0::'a)"
of_nat_less_0_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and m :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less (semiring_1.of_nat one plus zero m) zero"
that: fixes as' :: "'edge list" assumes "((ms\<^sub>1', s\<^sub>1'), ms\<^sub>1', transfer (slice_kind S (last as)) s\<^sub>2) \<in> WS S"   and "S,slice_kind S \<turnstile> (ms\<^sub>2,s\<^sub>2) =as' @ [last as]\<Rightarrow> (ms\<^sub>1',transfer (slice_kind S (last as)) s\<^sub>2)" shows "thesis"
get_disconnected_document_disconnected_document_in_heap: fixes h :: "(_) heap"   and child_node :: "(_) node_ptr"   and disconnected_document :: "(_) document_ptr" assumes "h \<turnstile> get_disconnected_document child_node \<rightarrow>\<^sub>r disconnected_document" shows "disconnected_document |\<in>| document_ptr_kinds h"
greatest_postfixpoint_fixpoint: fixes f :: "'a \<Rightarrow> 'a" assumes "has_greatest_postfixpoint f"   and "isotone f" shows "is_greatest_fixpoint f (p\<nu> f)"
infinite_growing: fixes X :: "complex set" assumes "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> \<exists>y\<in>X. x <\<^sub>\<complex> y" shows "infinite X"
Total_Id_Field: fixes r :: "('d \<times> 'd) set" assumes "Total r"   and "\<not> r \<subseteq> Id" shows "Field r = Field (r - Id)"
all_tries_def: fixes P :: "'b \<Rightarrow> bool" shows "all_tries P = all_trie (list_all P)"
P_s\<^sub>0: shows "P s\<^sub>0"
nb\<^sub>3: shows "cnf (encode_state_variable t (index (\<Pi>\<^sub>\<V>) v) ((\<Pi>\<^sub>G) v)) \<subseteq> \<Union> {cnf (encode_state_variable t (index (\<Pi>\<^sub>\<V>) v) ((\<Pi>\<^sub>G) v)) |v. v \<in> set (\<Pi>\<^sub>\<V>) \<and> (\<Pi>\<^sub>G) v \<noteq> None}"
disc_2: fixes x21 :: "('c, 'd) tm"   and x22 :: "('c, 'd) tm" shows "is_App (App x21 x22)"
LN_5: shows "\<S>\<bullet>\<C> \<turnstile> vs @ [Label n es' lfilledk] @ es'' : ts _> ts'"
fst_initmissing_netgmap_default_toy_init_netlift: fixes sr :: "'b \<Rightarrow> state \<times> 'a"   and s :: "'b net_state" shows "fst (initmissing (netgmap sr s)) = Lib.default toy_init (netlift sr s)"
Suc: fixes F :: "(atom list \<times> atom fm list \<times> nat) list"   and \<Gamma> :: "real list" shows "(\<exists>xs. length xs = amount \<and> eval (list_disj (map (\<lambda>a. case a of (L, F, n) \<Rightarrow> ExN n (list_conj (map fm.Atom L @ F))) F)) (xs @ \<Gamma>)) = eval (the_real_step_augment step amount F) \<Gamma>"
ax_InitS: fixes G :: "prog"   and C :: "qtname"   and c :: "class"   and A :: "'b triple set"   and Q :: "'b assn"   and R :: "'b assn"   and P :: "'b assn" assumes "the (class G C) = c"   and "C \<noteq> Object"   and "\<forall>l. G,A\<turnstile>{Q \<and>. (\<lambda>s. l = locals (snd s)) ;. set_lvars Map.empty} .init c. {set_lvars l .; R}"   and "G,A\<turnstile>{Normal (P \<and>. Not \<circ> initd C ;. supd (init_class_obj G C))} .Init (super c). {Q}" shows "G,A\<turnstile>{Normal (P \<and>. Not \<circ> initd C)} .Init C. {R}"
finite_listE: fixes x :: "'b"   and xs :: "['b]" assumes "finite_list (x : xs)" obtains "finite_list xs"
param_out_slice2: fixes n'' :: "'node SDG_node"   and V :: "'var"   and p :: "'pname"   and n' :: "'node SDG_node"   and n :: "'node SDG_node" assumes "n'' s-p:V\<rightarrow>\<^bsub>out\<^esub> n'"   and "n' \<in> sum_SDG_slice2 n" shows "n'' \<in> sum_SDG_slice2 n"
simps: fixes l :: "('b \<times> nat) tree"   and a :: "'b"   and r :: "('b \<times> nat) tree" shows "N l a r = \<langle>l, (a, size_wbt l + size_wbt r + 1), r\<rangle>"
heap_upds_ok_to_stack: fixes x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> 'b) list"   and S :: "stack_elem list" assumes "x \<in> domA \<Gamma>"   and "heap_upds_ok (\<Gamma>, S)" shows "heap_upds_ok (delete x \<Gamma>, Upd x # S)"
sel_1: fixes r :: "real"   and x :: "quat" shows "quat.Re (r *\<^sub>R x) = r * quat.Re x"
supp_fun_eqvt: fixes f :: "'c" assumes "eqvt f" shows "supp f = {}"
le_minus_one_simps_3: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "\<not> less_eq zero (uminus one)"
intros_2: fixes \<sigma> :: "'a \<Rightarrow> 'b"   and k :: "nat"   and \<sigma>' :: "'a \<Rightarrow> 'b"   and k' :: "nat"   and prob :: "('c, 'a, 'b, 'd) ifc_problem"   and n :: "nat" assumes "((\<sigma>, k), \<sigma>', k') \<in> IFC_def.cp prob"   and "IFC_def.is_cdi prob n (IFC_def.path prob \<sigma>) k"   and "IFC_def.path prob \<sigma> (Suc k) \<noteq> IFC_def.path prob \<sigma>' (Suc k')"   and "IFC_def.path prob \<sigma> n \<in> dom (IFC_def.att prob)" shows "((\<sigma>, n), \<sigma>', k') \<in> IFC_def.cop prob"
abs_rat_def: fixes a :: "rat" shows "\<bar>a\<bar> = (if a < 0 then - a else a)"
argmin_Min: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "xs \<noteq> []" shows "f (argmin f xs) = Min (f ` set xs)"
connected_component_subset: fixes S :: "'b set"   and x :: "'b" shows "connected_component_set S x \<subseteq> S"
fractrel_normalize_quot_right: fixes x :: "'b \<times> 'b"   and y :: "'b \<times> 'b" assumes "snd x \<noteq> (0::'b)" shows "fractrel y (normalize_quot x) = fractrel y x"
Sup_Un_less: fixes Sup :: "'f set \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and A :: "'f set"   and B :: "'f set" assumes "class.ccpo Sup less_eq less"   and "Complete_Partial_Order.chain less_eq (A \<union> B)"   and "\<forall>x\<in>A. \<exists>y\<in>B. less_eq x y" shows "Sup (A \<union> B) = Sup B"
assms_4: shows "src\<^sub>C f = trg\<^sub>C g"
assms: shows "x \<cdot> (x + y)\<^sup>\<omega> + z = (x + y)\<^sup>\<omega>"
filter_conv_filter_it: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "filter P xs = filter_it P xs"
all_security_requirements_fulfilled_complies: fixes M :: "('b SecurityInvariant \<times> 'b SecurityInvariant_configured) list"   and G :: "'b list_graph" assumes "\<forall>(m_impl, m_spec)\<in>set M. SecurityInvariant_complies_formal_def m_impl m_spec"   and "wf_list_graph G" shows "TopoS_Composition_Theory_impl.all_security_requirements_fulfilled (get_impl M) G = TopoS_Composition_Theory.all_security_requirements_fulfilled (get_spec M) (list_graph_to_graph G)"
indRelLTPO_impl_TRel_is_weak_reduction_correspondence_simulation: fixes TRel :: "('procT \<times> 'procT) set" assumes "weak_reduction_correspondence_simulation (indRelLTPO TRel) (STCal Source Target)" shows "weak_reduction_correspondence_simulation (TRel\<^sup>+) Target"
map_comp: fixes f :: "'f \<Rightarrow> 'e"   and g :: "'g \<Rightarrow> 'f"   and p :: "'g spmf" shows "map_spmf f (map_spmf g p) = map_spmf (f \<circ> g) p"
const_OclValid1: fixes x :: "'a state \<times> 'a state \<Rightarrow> 'b"   and \<tau> :: "'a state \<times> 'a state"   and \<tau>' :: "'a state \<times> 'a state" assumes "const x" shows "(\<tau> \<Turnstile> \<delta> x) = (\<tau>' \<Turnstile> \<delta> x)"
mem\<^sub>A_def: shows "mem\<^sub>A = mem\<^sub>A_of mem\<^sub>C"
\<alpha>_eval: fixes i :: "nat" assumes "i < N" shows "\<alpha> $ i = complex_of_real (if f i then 0 else 1 / sqrt (real (N - M)))"
measurable_integrable: fixes S :: "'a set" shows "(S \<in> lmeasurable) = (indicat_real S integrable_on UNIV)"
fls_lr_inverse_one_one_2: shows "fls_right_inverse 1 (1::'b) = 1"
is_word_Derives1_index: fixes w :: "symbol list"   and a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "is_word w"   and "Derives1 (w @ a) i r b" shows "length w \<le> i"