hcju/isabelleps huggingface dataset:
nonInterference_path_to_Low: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and backward_slice :: "'b set \<Rightarrow> 'b set"   and Exit :: "'b"   and H :: "'d set"   and L :: "'d set"   and High :: "'b"   and Low :: "'b"   and s :: "'c"   and s' :: "'c"   and S :: "'b set"   and as :: "'a list"   and as' :: "'a list" assumes "NonInterferenceIntraGraph sourcenode targetnode kind valid_edge Entry Def Use state_val backward_slice Exit H L High Low"   and "NonInterferenceIntraGraph.lowEquivalence state_val L s s'"   and "High \<notin> backward_slice S"   and "Low \<in> S"   and "CFG.path sourcenode targetnode valid_edge Entry as Low"   and "preds (CFG.kinds kind as) s"   and "CFG.path sourcenode targetnode valid_edge Entry as' Low"   and "preds (CFG.kinds kind as') s'" shows "NonInterferenceIntraGraph.lowEquivalence state_val L (transfers (CFG.kinds kind as) s) (transfers (CFG.kinds kind as') s')"
pmf_bind_pmf_of_set: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b pmf"   and x :: "'b" assumes "A \<noteq> {}"   and "finite A" shows "pmf (pmf_of_set A \<bind> f) x = (\<Sum>xa\<in>A. pmf (f xa) x) / real (card A)"
g_min_def: fixes m :: "'s"   and P :: "'k \<times> 'v \<Rightarrow> bool" shows "g_min m P \<equiv> iterateoi m (\<lambda>\<sigma>. \<sigma> = None) (\<lambda>x \<sigma>. if P x then Some x else None) None"
update_arg_wf_gives_cases: fixes r :: "nat \<times> k \<times> f \<times> nat \<times> nat" assumes "update_arg_wf r" shows "(\<pi>\<^sub>2 r = 0) = (\<pi>\<^sub>3 r = unk)"
assms_3: fixes p0 :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat" assumes "p0 \<in> set ps" shows "j \<le> component_of_term (sig_of_pair p0)"
entail_unions: fixes I :: "'a set"   and N :: "'f set"   and Ni :: "'a \<Rightarrow> 'f set" shows "(\<forall>i\<in>I. N \<Turnstile> Ni i) = (N \<Turnstile> \<Union> (Ni ` I))"
converged_same_icd: fixes \<pi> :: "nat \<Rightarrow> 'n"   and \<pi>' :: "nat \<Rightarrow> 'n"   and l :: "nat"   and n :: "nat"   and n' :: "nat"   and k :: "nat"   and k' :: "nat" assumes "is_path \<pi>"   and "is_path \<pi>'"   and "l < n"   and "cs\<^bsup>\<pi>\<^esup> n = cs\<^bsup>\<pi>'\<^esup> n'"   and "cs\<^bsup>\<pi>\<^esup> k = cs\<^bsup>\<pi>'\<^esup> k'"   and "l icd\<^bsup>\<pi>\<^esup>\<rightarrow> k"   and "\<pi> (Suc k) = \<pi>' (Suc k')" shows "\<exists>l'. cs\<^bsup>\<pi>\<^esup> l = cs\<^bsup>\<pi>'\<^esup> l'"
ClassI: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> E" shows "a \<in> Class b"
discI: fixes llist :: "'b llist"   and x21 :: "'b"   and x22 :: "'b llist" assumes "llist = LCons x21 x22" shows "llist \<noteq> LNil"
k_1: fixes p :: "nat" shows "k p 0 = 0"
concurrent_direct_steps_simulates_store_buffer_history_steps: fixes ts\<^sub>s\<^sub>b :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m\<^sub>s\<^sub>b :: "nat \<Rightarrow> nat"   and \<S>\<^sub>s\<^sub>b :: "nat \<Rightarrow> bool option"   and ts\<^sub>s\<^sub>b' :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m\<^sub>s\<^sub>b' :: "nat \<Rightarrow> nat"   and \<S>\<^sub>s\<^sub>b' :: "nat \<Rightarrow> bool option"   and ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option" assumes "ini.sbh_concurrent_steps program_step (ts\<^sub>s\<^sub>b, m\<^sub>s\<^sub>b, \<S>\<^sub>s\<^sub>b) (ts\<^sub>s\<^sub>b', m\<^sub>s\<^sub>b', \<S>\<^sub>s\<^sub>b')"   and "valid_ownership \<S>\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b"   and "valid_reads m\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b"   and "valid_history program_step ts\<^sub>s\<^sub>b"   and "valid_sharing \<S>\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b"   and "tmps_distinct ts\<^sub>s\<^sub>b"   and "valid_sops ts\<^sub>s\<^sub>b"   and "valid_data_dependency ts\<^sub>s\<^sub>b"   and "load_tmps_fresh ts\<^sub>s\<^sub>b"   and "enough_flushs ts\<^sub>s\<^sub>b"   and "valid_program_history ts\<^sub>s\<^sub>b"   and "valid ts\<^sub>s\<^sub>b"   and "(ts\<^sub>s\<^sub>b, m\<^sub>s\<^sub>b, \<S>\<^sub>s\<^sub>b) \<sim> (ts, m, \<S>)"   and "ini.safe_reach_direct program_step safe_delayed (ts, m, \<S>)" shows "valid_ownership \<S>\<^sub>s\<^sub>b' ts\<^sub>s\<^sub>b' \<and> valid_reads m\<^sub>s\<^sub>b' ts\<^sub>s\<^sub>b' \<and> valid_history program_step ts\<^sub>s\<^sub>b' \<and> valid_sharing \<S>\<^sub>s\<^sub>b' ts\<^sub>s\<^sub>b' \<and> tmps_distinct ts\<^sub>s\<^sub>b' \<and> valid_data_dependency ts\<^sub>s\<^sub>b' \<and> valid_sops ts\<^sub>s\<^sub>b' \<and> load_tmps_fresh ts\<^sub>s\<^sub>b' \<and> enough_flushs ts\<^sub>s\<^sub>b' \<and> valid_program_history ts\<^sub>s\<^sub>b' \<and> valid ts\<^sub>s\<^sub>b' \<and> (\<exists>ts' m' \<S>''. ini.direct_concurrent_steps program_step (ts, m, \<S>) (ts', m', \<S>'') \<and> (ts\<^sub>s\<^sub>b', m\<^sub>s\<^sub>b', \<S>\<^sub>s\<^sub>b') \<sim> (ts', m', \<S>''))"
x_2: shows "(\<theta>1 \<circ>\<^sub>s \<theta>2) x = t"
simps: fixes a1 :: "nat \<times> nat \<Rightarrow> 'capacity"   and a2 :: "nat \<Rightarrow> nat"   and a3 :: "pr_operation'"   and a4 :: "nat \<times> nat \<Rightarrow> 'capacity"   and a5 :: "nat \<Rightarrow> nat" shows "(((a1, a2), a3, a4, a5) \<in> pr_algo_lts') = ((\<exists>f l e. a1 = f \<and> a2 = l \<and> a3 = NONSAT_PUSH' \<and> a4 = push_effect f e \<and> a5 = l \<and> Height_Bounded_Labeling c s t f l \<and> nonsat_push_precond f l e) \<or> (\<exists>f l e. a1 = f \<and> a2 = l \<and> a3 = SAT_PUSH' e \<and> a4 = push_effect f e \<and> a5 = l \<and> Height_Bounded_Labeling c s t f l \<and> sat_push_precond f l e) \<or> (\<exists>f l u. a1 = f \<and> a2 = l \<and> a3 = RELABEL' \<and> a4 = f \<and> a5 = relabel_effect f l u \<and> Height_Bounded_Labeling c s t f l \<and> relabel_precond f l u))"
rhs: shows "{#a#} \<times># filter_mset Q (add_mset x B) = {#a#} \<times># filter_mset Q B + {#a#} \<times># filter_mset Q {#x#}"
cases: fixes x :: "'g list \<times> 'g list" obtains ys :: "'g list" where "x = ([], ys__)"   | xa :: "'g" and xs :: "'g list" and ys :: "'g list" where "x = (xa # xs, ys)"
H_def: fixes xy :: "'v \<times> 'v" shows "H xy = lim (\<lambda>n. h (k n) xy)"
has_derivative_at_within_iff_Ex: fixes x :: "'a"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b" assumes "x \<in> S"   and "open S" shows "(f has_derivative f') (at x within S) = (bounded_linear f' \<and> (\<exists>e. (\<forall>h. x + h \<in> S \<longrightarrow> f (x + h) = f x + f' h + e h) \<and> (\<lambda>h. norm (e h) / norm h) \<midarrow>0::'a\<rightarrow> 0))"
inv_comp_right: fixes f :: "'a \<Rightarrow> 'a" assumes "bijection f" shows "f \<circ> inv f = id"
monad_envT: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad return bind" shows "monad (return_env return) (bind_env bind)"
inner_verts_def: fixes p :: "('a \<times> 'a) list" shows "S.inner_verts p \<equiv> tl (map fst p)"
vreal_impl_eqpoll: shows "range vreal_of_real_impl \<approx> elts vreal_impl"
card_lessThan: fixes u :: "nat" shows "card {..<u} = u"
simpme: fixes j :: "nat" assumes "j < length (col (Matrix_Legacy.transpose nr (v # m)) i)" shows "col (Matrix_Legacy.transpose nr (v # m)) i ! j = row (v # m) i ! j"
cInf_greatest: fixes X :: "'k multiset set"   and z :: "'k multiset" assumes "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> z \<subseteq># x" shows "z \<subseteq># Inf X"
nn_integral_pmf_eq_1: fixes p :: "'c pmf" shows "(\<Sum>\<^sup>+ x. ennreal (pmf p x)) = 1"
timpl_closure_finite: fixes t :: "(('fun, 'atom, 'sets) prot_fun, 'b) Term.term"   and TI :: "('sets set \<times> 'sets set) list" assumes "wf\<^sub>t\<^sub>r\<^sub>m t" shows "finite (timpl_closure t (set TI))"
T_on_rand'_nn: fixes A :: "('state \<Rightarrow> 'b pmf) \<times> ('state \<times> 'b \<Rightarrow> 'request \<Rightarrow> ('answer \<times> 'b) pmf)"   and s :: "('state \<times> 'b) pmf"   and qs :: "'request list" shows "0 \<le> T_on_rand' A s qs"
maybe_negate_true: fixes x :: "trilean" shows "(\<not>? x = true) = (x = false)"
r_def: shows "r = \<bar>ai\<bar> - 2 ^ nat k"
simps_2: fixes e :: "nat \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a list \<Rightarrow> bool" shows "eval e f g TT = True"
GSMP_wt_substI: fixes t :: "('fun, 'var) Term.term"   and M :: "('fun, 'var) Term.term set"   and I :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "t \<in> M"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t I"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range I)"   and "subst_domain I = \<V> \<and> ground (subst_range I)" shows "t \<cdot> I \<in> GSMP M"
convex_rel_interior: fixes S :: "'a set" assumes "convex S" shows "convex (rel_interior S)"
insert_ops_sorted_oids: fixes xs :: "('a \<times> 'a option) list"   and i1 :: "'a"   and r1 :: "'a option"   and ys :: "('a \<times> 'a option) list"   and i2 :: "'a"   and r2 :: "'a option" assumes "insert_ops (xs @ [(i1, r1)] @ ys @ [(i2, r2)])" shows "i1 < i2"
ZfunI: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "\<And>r. 0 < r \<Longrightarrow> \<forall>\<^sub>F x in F. norm (f x) < r" shows "Zfun f F"
pdivmod_monic_1: fixes g :: "'a poly"   and f :: "'a poly"   and q :: "'a poly"   and r :: "'a poly" assumes "monic g"   and "pdivmod_monic f g = (q, r)" shows "f = g * q + r"
mker_mzeromap: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and N :: "('i, 'f, 'j) Module_scheme" assumes "R module M"   and "R module N" shows "ker\<^bsub>M,N\<^esub> mzeromap M N = carrier M"
blinfun_apply_blinfun_compose: fixes a :: "'g \<Rightarrow>\<^sub>L 'e"   and b :: "'f \<Rightarrow>\<^sub>L 'g"   and c :: "'f" shows "blinfun_apply (a o\<^sub>L b) c = blinfun_apply a (blinfun_apply b c)"
card_lists_distinct_length_eq_union: fixes X :: "'a set"   and Y :: "'a set"   and n :: "nat" assumes "finite X"   and "finite Y"   and "X \<inter> Y = {}" shows "card {zs. length zs = n \<and> distinct zs \<and> set zs \<subseteq> X \<union> Y} = (\<Sum>k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))"
z_def: shows "z = (SOME z. z \<in> Y \<and> (\<forall>y\<in>Y. r y \<le> r z))"
grpHN: shows "Group.group (G\<lparr>carrier := H\<rparr> Mod N)"
3: fixes x :: "'a" shows "(x = (THE x. x \<in> set xs \<and> (\<forall>y. y \<in> set xs \<longrightarrow> y \<le> x))) = (x \<in> set xs \<and> (\<forall>y. y \<in> set xs \<longrightarrow> y \<le> x))"
irrefl_less_nmultiset: fixes X :: "'a nmultiset" assumes "X < X" shows "False"
nb\<^sub>5: fixes op\<^sub>1 :: "('a, 'b) sas_plus_operator"   and op\<^sub>2 :: "('a, 'b) sas_plus_operator" assumes "op\<^sub>1 \<in> set ops"   and "op\<^sub>2 \<in> set ops" shows "\<forall>(v, a)\<in>set (effect_of op\<^sub>1). \<forall>(v', a')\<in>set (effect_of op\<^sub>2). v \<noteq> v' \<or> a = a'"
case_1: fixes f1 :: "'f"   and f2 :: "'g \<Rightarrow> 'f" shows "(case None of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f1"
signs: fixes i :: "nat"   and x :: "real" assumes "i < length ps - 2"   and "poly (ps ! (i + 1)) x = 0" shows "poly (ps ! (i + 2)) x * poly (ps ! i) x < 0"
inverse_mod_coprime_exp: fixes m :: "int"   and p :: "int"   and n :: "nat"   and x :: "int" assumes "m = p ^ n"   and "prime p"   and "n \<noteq> 0"   and "comm_monoid_mult_class.coprime x p" shows "poly_mod.M m (inverse_mod x m * x) = 1"
basis_def: fixes A :: "'a mod_ring vec set" shows "V.basis A = (V.lin_indpt A \<and> V.gen_set A \<and> A \<subseteq> carrier local.V)"
that: fixes d :: "real" assumes "0 < d"   and "\<forall>t. c - d < t \<and> t \<le> c \<longrightarrow> norm (integral {a..c} f - integral {a..t} f) < e" shows "thesis"
G_holds_def: fixes \<phi> :: "('sf, 'sc, 'sz) formula"   and \<alpha> :: "('sf, 'sc, 'sz) hp" shows "G_holds \<phi> \<alpha> \<equiv> valid \<phi> \<longrightarrow> valid ([[\<alpha>]]\<phi>)"
f_1: shows "f' = of_int_poly_i ff_ops f"
foldl_length: fixes l :: "'a list" shows "foldl (\<lambda>i x. Suc i) 0 l = length l"
add_match_MatchAnd_foldr_empty_rs2: fixes ms :: "'c match_expr list" shows "add_match_MatchAnd_foldr ms [] = []"
pm_of_idx_pm_monomial: fixes xs :: "'c list"   and i :: "nat"   and c :: "'d" assumes "distinct xs" shows "pm_of_idx_pm xs (monomial c i) = (monomial c (xs ! i) when i < length xs)"
RedFAss: fixes h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) 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 Cs' :: "char list list"   and F :: "char list"   and T :: "ty"   and Cs :: "char list list"   and v :: "val"   and v' :: "val"   and Ds :: "char list list"   and fs :: "char list \<Rightarrow> val option"   and E :: "char list \<Rightarrow> ty option"   and l :: "char list \<Rightarrow> val option" assumes "h a = \<lfloor>(D, S)\<rfloor>"   and "P \<turnstile> last Cs' has least F:T via Cs"   and "P \<turnstile> T casts v to v' "   and "Ds = Cs' @\<^sub>p Cs"   and "(Ds, fs) \<in> S" shows "P,E \<turnstile> \<langle>ref (a, Cs')\<bullet>F{Cs} := Val v,(h, l)\<rangle> \<rightarrow> \<langle>Val v',(h(a \<mapsto> (D, insert (Ds, fs(F \<mapsto> v')) (S - {(Ds, fs)}))), l)\<rangle>"
Exit_empty: shows "Def (_Exit_) = {} \<and> Use (_Exit_) = {}"
option_symp: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "symp R" shows "symp (rel_option R)"
inext_nth_cut_less_eq: fixes n :: "nat"   and I :: "nat set"   and t :: "nat" assumes "n < card (I \<down>< t)" shows "I \<down>< t \<rightarrow> n = I \<rightarrow> n"
reach\<^sub>t_def: fixes \<Sigma> :: "'d set"   and \<delta> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and q\<^sub>0 :: "'c" shows "reach\<^sub>t \<Sigma> \<delta> q\<^sub>0 = {run\<^sub>t \<delta> q\<^sub>0 w n |w n. range w \<subseteq> \<Sigma>}"
ll: fixes p :: "int"   and F :: "int poly"   and l :: "nat"   and u :: "int poly"   and f :: "int poly"   and n :: "nat"   and gs :: "int poly list"   and Degs :: "nat set"   and j' :: "nat" assumes "prime p"   and "0 < degree F"   and "2 ^ (degree F)\<^sup>2 * B2_LLL F ^ (2 * degree F) \<le> (p ^ l)\<^sup>2"   and "0 < degree u"   and "monic u"   and "poly_mod.irreducible_m p u"   and "poly_mod.dvdm p u f"   and "f dvd F"   and "n \<equiv> degree f"   and "poly_mod.unique_factorization_m (p ^ l) f (lead_coeff f, mset gs)"   and "comm_monoid_mult_class.coprime (lead_coeff f) p"   and "poly_mod.square_free_m p f"   and "square_free f"   and "u \<in> set gs"   and "map (poly_mod.Mp (p ^ l)) gs = gs"   and "\<And>factor. \<lbrakk>factor dvd f; poly_mod.dvdm p u factor\<rbrakk> \<Longrightarrow> degree factor \<in> Degs"   and "degree u \<le> j'"   and "j' < n"   and "\<And>factor. \<lbrakk>poly_mod.dvdm p u factor; factor dvd f\<rbrakk> \<Longrightarrow> j' \<le> degree factor" shows "Factorization_Algorithm_16_22.ll p l f j' \<le> l"
disc_iff: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and \<sigma>' :: "'a \<Rightarrow> 'a"   and v0 :: "'a" shows "\<not> lnull (greedy_conforming_path p \<sigma> \<sigma>' v0)"
tendsto_power_int: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and F :: "'a filter"   and n :: "int" assumes "(f \<longlongrightarrow> a) F"   and "a \<noteq> (0::'b)" shows "((\<lambda>x. f x powi n) \<longlongrightarrow> a powi n) F"
matchC_ZOC_def: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and theta :: "(('b, 'c) com \<times> ('b, 'c) com) set"   and c :: "('b, 'c) com"   and d :: "('b, 'c) com" assumes "PL_Indis indis" shows "PL_Indis.matchC_ZOC tval aval indis theta c d \<equiv> \<forall>s t c' s'. (s, t) \<in> indis \<and> PL.transC tval aval (c, s) (c', s') \<longrightarrow> (s', t) \<in> indis \<and> (c', d) \<in> theta \<or> (\<exists>d' t'. PL.transC tval aval (d, t) (d', t') \<and> (s', t') \<in> indis \<and> (c', d') \<in> theta)"
whileLoopE_def: fixes C :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and body :: "'c \<Rightarrow> 'd \<Rightarrow> (('e + 'c) \<times> 'd) set \<times> bool" shows "whileLoopE C   body \<equiv> \<lambda>r. whileLoop (\<lambda>r s. case r of Inl a \<Rightarrow> False | Inr v \<Rightarrow> C v s)   (lift body) (Inr r)"
IH: assumes "\<And>x. x \<in> A \<Longrightarrow> (0::'b) \<le> f x" shows "sum f A \<le> (\<Prod>x\<in>A. (1::'b) + f x)"
bit_minus_2_iff: fixes n :: "nat" shows "bit (- (2::'a)) n = ((2::'a) ^ n \<noteq> (0::'a) \<and> 0 < n)"
set_eq_iff: fixes A :: "'a set"   and B :: "'a set" shows "(A = B) = (\<forall>x. (x \<in> A) = (x \<in> B))"
wellformed_induce_subgraph: fixes vs :: "'a set" shows "wf_digraph ((G \<restriction> {w. u \<rightarrow>\<^sup>* w}) \<restriction> vs)"
sig_red_top_tailI: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "sig_red sing_reg top_tail F p q" shows "sig_red sing_reg (\<preceq>) F p q"
permutes_inv_eq: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and y :: "'a"   and x :: "'a" assumes "p permutes S" shows "(inv p y = x) = (p x = y)"
delta_def: fixes a :: "'a"   and s :: "'s" shows "delta a s = sim_delta s ! index (\<sigma> n) a"
assms_6: shows "f \<in> {real_of_float fl..real_of_float fu}"
cring_ltrm_mult: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P"   and "p (deg R p) \<otimes> q (deg R q) \<noteq> \<zero>" shows "monom P ((p \<otimes>\<^bsub>P\<^esub> q) (deg R (p \<otimes>\<^bsub>P\<^esub> q))) (deg R (p \<otimes>\<^bsub>P\<^esub> q)) = monom P (p (deg R p)) (deg R p) \<otimes>\<^bsub>P\<^esub> monom P (q (deg R q)) (deg R q)"
subspace_hyperplane: fixes a :: "'c" shows "subspace {x. a \<bullet> x = 0}"
internal_iface_name_wildcard_longest_commute: fixes i1 :: "char list"   and i2 :: "char list" assumes "iface_name_is_wildcard i1"   and "iface_name_is_wildcard i2" shows "internal_iface_name_wildcard_longest i1 i2 = internal_iface_name_wildcard_longest i2 i1"
received_messages_eff: fixes i :: "'a"   and j :: "nat" assumes "is_valid_state_id (i, j)" shows "set (received_messages (i, j)) = make_set j (\<lambda>k x. \<exists>s. event_at (i, k) (Receive s x))"
name_upd_incoming_f: fixes x :: "('b, 'c) node_scheme"   and n :: "('b, 'd) node_scheme" shows "name (upd_incoming_f n x) = name x"
induct: fixes P :: "((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 \<Rightarrow> (variable \<times> typ) list \<Rightarrow> variable set \<Rightarrow> Core.term list \<Rightarrow> proofterm \<Rightarrow> bool"   and a0 :: "((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 a1 :: "(variable \<times> typ) list"   and a2 :: "variable set"   and a3 :: "Core.term list"   and a4 :: "proofterm" assumes "\<And>thy uu_ uv_ Hs t Tis. P thy uu_ uv_ Hs (PAxm t Tis)"   and "\<And>thy uw_ ux_ Hs n. P thy uw_ ux_ Hs (PBound n)"   and "\<And>thy vs ns Hs T p. (\<And>x xa y. \<lbrakk>typ_ok thy T; x = variant_variable (Free STR ''default'') ns; (xa, y) = x\<rbrakk> \<Longrightarrow> P thy ((xa, T) # vs) y Hs p) \<Longrightarrow> P thy vs ns Hs (Abst T p)"   and "\<And>thy vs ns Hs p t. P thy vs ns Hs p \<Longrightarrow> P thy vs ns Hs (Appt p t)"   and "\<And>thy vs ns Hs t p. (\<And>x. x = subst_bvs (map (\<lambda>(x, y). Fv x y) vs) t \<Longrightarrow> P thy vs ns (x # Hs) p) \<Longrightarrow> P thy vs ns Hs (AbsP t p)"   and "\<And>thy vs ns Hs p1 p2. \<lbrakk>P thy vs ns Hs p1; \<And>x. x = replay'' thy vs ns Hs p1 \<Longrightarrow> P thy vs ns Hs p2\<rbrakk> \<Longrightarrow> P thy vs ns Hs (AppP p1 p2)"   and "\<And>thy vs ns Hs ty c. P thy vs ns Hs (OfClass ty c)"   and "\<And>thy vs ns Hs t. P thy vs ns Hs (Hyp t)" shows "P a0 a1 a2 a3 a4"
separating_TER_plus_current: fixes x :: "'v"   and y :: "'v"   and p :: "'v list" assumes "x \<in> RF (TER f)"   and "y \<in> B \<Gamma>"   and "path \<Gamma> x p y" shows "(\<exists>z\<in>set p. z \<in> TER (plus_current f g)) \<or> x \<in> TER (plus_current f g)"
Abs_pdata_inverse: fixes y :: "'b option list \<times> 'b dataspace" assumes "y \<in> pdata" shows "Rep_pdata (Abs_pdata y) = y"
subcls1I: fixes C :: "cname"   and m :: "class"   and D :: "cname" assumes "class C = Some m"   and "super m = D"   and "C \<noteq> Object" shows "C \<prec>C1 D"
Invariants_cond_exp_induced_map: fixes A :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "A \<in> sets M"   and "integrable M f" shows "AE x in restrict_space M A. real_cond_exp (restrict_space M A) (qmpt.Invariants (restrict_space M A) (induced_map A)) (induced_function A f) x = real_cond_exp M Invariants f x * real_cond_exp (restrict_space M A) (qmpt.Invariants (restrict_space M A) (induced_map A)) (\<lambda>x. real (return_time_function A x)) x"
assms: shows "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (r \<restriction>\<^sup>r\<^sub>\<circ> A)"
fcard_finsert_if: fixes x :: "'a"   and A :: "'a fset" shows "fcard (finsert x A) = (if x |\<in>| A then fcard A else Suc (fcard A))"
cases: fixes G :: "prog"   and a1 :: "ty"   and a2 :: "ty" assumes "G\<turnstile>a1\<preceq>a2" obtains T :: "ty" where "a1 = T__" and "a2 = T__"   | I :: "qtname" and J :: "qtname" where "a1 = Iface I__" and "a2 = Iface J" and "G\<turnstile>I__\<preceq>I J"   | I :: "qtname" where "a1 = Iface Ia__" and "a2 = Class Object"   | C :: "qtname" and D :: "qtname" where "a1 = Class C__" and "a2 = Class D" and "G\<turnstile>C__\<preceq>\<^sub>C D"   | C :: "qtname" and I :: "qtname" where "a1 = Class C" and "a2 = Iface I" and "G\<turnstile>C\<leadsto>I"   | R :: "ref_ty" where "a1 = NT" and "a2 = RefT R"   | T :: "ty" where "a1 = Ta__.[]" and "a2 = Class Object"   | S :: "ref_ty" and T :: "ref_ty" where "a1 = RefT S.[]" and "a2 = RefT T.[]" and "G\<turnstile>RefT S\<preceq>RefT T"
get_child_nodes_pure: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap" shows "pure (Shadow_DOM.get_child_nodes ptr) h"
imirror_cut_greater': fixes I :: "nat set"   and t :: "nat" assumes "finite I"   and "t \<in> I" shows "imirror I \<down>> mirror_elem t I = imirror_bounds (I \<down>< t) (iMin I) (Max I)"
dominatesI: fixes a :: "'b"   and u :: "'a"   and v :: "'a" assumes "arc_to_ends H a = (u, v)"   and "a \<in> arcs H" shows "u \<rightarrow>\<^bsub>H\<^esub> v"
ring_of_empty: shows "local.im_sheaf {} = {\<O>b}"
safe: shows "SIDE_PRECOND (\<forall>x\<in>X. x \<subseteq> d)"
dehomogenize_sum: fixes x :: "'c"   and f :: "'e \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and A :: "'e set" shows "dehomogenize x (sum f A) = (\<Sum>a\<in>A. dehomogenize x (f a))"
sq_norm_poly_ge_0: fixes p :: "'a poly" shows "(0::'a) \<le> \<parallel>p\<parallel>\<^sup>2"
semantics_pltl_sugar_3: fixes w :: "nat \<Rightarrow> 'b"   and \<phi> :: "'b pltl"   and \<psi> :: "'b pltl" shows "w \<Turnstile>\<^sub>p \<phi> or\<^sub>p \<psi> = (w \<Turnstile>\<^sub>p \<phi> \<or> w \<Turnstile>\<^sub>p \<psi>)"
sifum_refinement_axioms: shows "sifum_refinement dma\<^sub>A dma\<^sub>C \<C>_vars\<^sub>A \<C>_vars\<^sub>C eval\<^sub>A eval\<^sub>C var\<^sub>C_of"