hcju/isabelleps huggingface dataset:
select_convs_2: fixes iiface :: "iface"   and oiface :: "iface"   and src :: "'a word \<times> nat"   and dst :: "'a word \<times> nat"   and proto :: "protocol"   and sports :: "16 word \<times> 16 word"   and dports :: "16 word \<times> 16 word"   and more :: "'b" shows "oiface \<lparr>iiface = iiface, oiface = oiface, src = src, dst = dst, proto = proto, sports = sports, dports = dports, \<dots> = more\<rparr> = oiface"
call_node_notin_slice_return_node_neither: fixes n :: "'node"   and n' :: "'node"   and S :: "'node SDG_node set" assumes "call_of_return_node n n'"   and "n' \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>" shows "n \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"
arr_char: fixes f :: "'a" shows "hom'.arr f = \<guillemotleft>f : trg e\<^sub>0 \<rightarrow> trg e\<^sub>1\<guillemotright>"
type_definition_JF1: shows "type_definition Rep_JF1 Abs_JF1 car_final1"
index_mult_mat_3: fixes A :: "'a Matrix.mat"   and B :: "'a Matrix.mat" shows "dim_col (A * B) = dim_col B"
app\<^sub>i_New: fixes C :: "char list"   and P :: "'a prog"   and pc :: "nat"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and ST :: "ty list"   and LT :: "ty err list" shows "app\<^sub>i (New C, P, pc, mxs, T\<^sub>r, ST, LT) = (is_class P C \<and> length ST < mxs)"
llist_all2_ltlI: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'d llist" assumes "llist_all2 P xs ys" shows "llist_all2 P (ltl xs) (ltl ys)"
conj_commute: fixes x :: "'c"   and y :: "'c" shows "x AND y = y AND x"
filterlim_inverse_at_iff: fixes g :: "'b \<Rightarrow> 'c"   and F :: "'b filter" shows "(LIM x F. inverse (g x) :> at (0::'c)) = filterlim g at_infinity F"
invariant3pD: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and I :: "'a set"   and s :: "'a"   and tl :: "'b"   and s' :: "'a" assumes "invariant3p r I"   and "r s tl s'"   and "s \<in> I" shows "s' \<in> I"
abs_fresh_1: fixes b :: "name"   and a :: "name"   and x :: "'d" shows "b \<sharp> [a].x = (b = a \<or> b \<sharp> x)"
0: shows "pderiv (q * (p div q)) = q * pderiv (p div q) + p div q * pderiv q"
no_start_end_points_path_iff: fixes x :: "'a" shows "no_start_end_points_path x = (no_start_points_path x \<and> no_end_points_path x)"
exhaust: fixes y :: "('e, 'f, 'd) ctree" obtains x1 :: "'d" where "y = Link x1"   | x21 :: "'e \<times> 'f" and x22 :: "('e, 'f, 'd) ctree fset" where "y = cNode x21 x22"
single_non_zero_def: shows "local.single_non_zero \<equiv> \<lambda>l k x A i j. (i \<notin> {k, l} \<longrightarrow> A $$ (i, k) = (0::'a)) \<and> A $$ (l, k) = x"
cases: fixes a1 :: "'a"   and a2 :: "('b \<times> 'a) list"   and ars :: "('a \<times> 'b \<times> 'a) set" assumes "(a1, a2) \<in> seq ars" obtains a :: "'a" where "a1 = a__" and "a2 = []"   | a :: "'a" and \<alpha> :: "'b" and b :: "'a" and ss :: "('b \<times> 'a) list" where "a1 = a" and "a2 = (\<alpha>, b) # ss" and "(a, \<alpha>, b) \<in> ars" and "(b, ss) \<in> seq ars"
assms_3: fixes k :: "'a"   and v :: "'b"   and xs :: "('a \<times> 'b) list" assumes "oalist_inv_raw ((k, v) # xs)"   and "oalist_inv_raw xs"   and "v \<noteq> (0::'b)"   and "\<And>k'. k' \<in> fst ` set xs \<Longrightarrow> lt k k'"   and "P xs" shows "P ((k, v) # xs)"
card_fcs_aa'_le: shows "card {G.face_cycle_set a, G.face_cycle_set a'} \<le> card G.face_cycle_sets"
assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<oplus>\<^sub>a b \<oplus>\<^sub>a c = a \<oplus>\<^sub>a (b \<oplus>\<^sub>a c)"
iMODb_0_eq_conv: fixes r :: "nat"   and m :: "nat"   and r' :: "nat"   and m' :: "nat"   and c' :: "nat" shows "([ r, mod m, 0 ] = [ r', mod m', c' ]) = (r = r' \<and> (m' = 0 \<or> c' = 0))"
ord0_less_induct: fixes P :: "ord0 \<Rightarrow> bool"   and a :: "ord0" assumes "\<And>x. \<forall>y. (y, x) \<in> {(x, y). x < y} \<longrightarrow> P y \<Longrightarrow> P x" shows "P a"
obs_singleton: fixes n :: "'node"   and S :: "'node set" shows "(\<exists>m. obs n (PDG_BS S) = {m}) \<or> obs n (PDG_BS S) = {}"
LeadsTo_empty: fixes F :: "'b program"   and A :: "'b set" assumes "F \<in> A \<longmapsto>w {}"   and "all_total F" shows "F \<in> Always (- A)"
assms_4: shows "automaton.trans A = seqp_sos \<Gamma>"
iprev_nth_surj_on: fixes I :: "nat set" assumes "finite I" shows "surj_on (iprev_nth I) UNIV I"
simps_3: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and V :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a> :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i> :: "'d \<Rightarrow> 'd"   and src :: "'d \<Rightarrow> 'd"   and trg :: "'d \<Rightarrow> 'd"   and E :: "'c \<Rightarrow> 'd"   and t :: "'c bicategorical_language.term"   and u :: "'c bicategorical_language.term" assumes "Coherence.evaluation_map C src\<^sub>C trg\<^sub>C V H \<a> \<i> src trg E" shows "Coherence.evaluation_map.eval V H \<a> \<i> src trg E (bicategorical_language.Hcomp t u) = H (Coherence.evaluation_map.eval V H \<a> \<i> src trg E t) (Coherence.evaluation_map.eval V H \<a> \<i> src trg E u)"
invKey_symmetric: shows "all_symmetric \<longrightarrow> invKey = id"
linear_irreducible_field: fixes p :: "'b poly" assumes "degree p = 1" shows "irreducible p"
simps: fixes a :: "'b llist" shows "lfinite a = (a = <> \<or> (\<exists>xs x. a = x % xs \<and> lfinite xs))"
matrix_left_right_inverse: fixes A :: "(('b, 'c) vec, 'c) vec"   and A' :: "(('b, 'c) vec, 'c) vec" shows "(A ** A' = mat (1::'b)) = (A' ** A = mat (1::'b))"
dimension_pair: shows "DIM('b \<times> 'c) = DIM('b) + DIM('c)"
subspace_sum1: fixes U :: "'c set"   and V :: "'c set" assumes "vectorspace U"   and "vectorspace V" shows "Subspace.subspace U (U + V)"
dbfm_abst_swap_subst: fixes name :: "name"   and name' :: "name"   and u :: "dbtm"   and j :: "nat"   and A :: "dbfm" assumes "name \<noteq> name'"   and "atom name' \<sharp> u" shows "subst_dbfm u name (abst_dbfm name' j A) = abst_dbfm name' j (subst_dbfm u name A)"
iIN_Max: fixes n :: "nat"   and d :: "nat" shows "Max [n\<dots>,d] = n + d"
bangPres: fixes P :: "pi"   and Q :: "pi" assumes "P \<approx> Q" shows "!P \<approx> !Q"
pure_def: fixes R :: "('e \<times> 'd) set" shows "pure R \<equiv> \<lambda>a c. \<up> ((c, a) \<in> R)"
assms_3: shows "(k, \<gamma>) \<in> one_chain"
lm136: fixes X :: "'c set"   and f :: "('c \<times> 'd) set" assumes "finite X"   and "runiq f" shows "MiscTools.sum' f X = sum (toFunction f) (X \<inter> Domain f)"
precondition_def: fixes s :: "'state_t"   and a :: "'action_t option" shows "precondition s a \<equiv> a \<rightharpoonup> kprecondition s"
elims_3: fixes x :: "'a option"   and xa :: "'a option" assumes "\<not> x \<sqsubseteq> xa" obtains x :: "'a" and y :: "'a" where "xa__ = Some x" and "xa = Some y" and "\<not> x \<sqsubseteq> y"   | uu_ :: "'a" where "xa__ = Some uu_" and "xa = None"
n_distrib2_opp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "n x \<cdot> n y \<oplus> n z = (n x \<oplus> n z) \<cdot> (n y \<oplus> n z)"
full_gb_def: fixes bs :: "(('a \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list" shows "punit.full_gb bs = map (\<lambda>k. (monomial (1::'d) (fst (0::'a, k)), 0, default)) (remdups (map local.punit.component_of_term (punit.Keys_to_list (map fst bs))))"
7: shows "\<Sqinter> insert s ss = s0 \<star> rs''"
card_UNIV_word_def: shows "card_UNIV_class.card_UNIV = Phantom('a word) (2 ^ LENGTH('a))"
scale_minus_left: fixes a :: "real"   and x :: "'b" shows "- a *\<^sub>R x = - (a *\<^sub>R x)"
mult_exp_exp: fixes x :: "'b"   and y :: "'b" shows "exp x * exp y = exp (x + y)"
Sup_assertion: fixes X :: "'a set" assumes "X \<subseteq> assertion" shows "\<Squnion>X \<in> assertion"
C1: fixes R :: "ccs"   and T :: "ccs" shows "(R \<parallel> T, T \<parallel> R) \<in> Rel"
static_fld_simp: fixes f :: "'b \<times> 'a member_scheme" shows "static (fld f) = is_static f"
that: fixes a :: "'c" assumes "a \<in> as"   and "fst (f a) = b" shows "thesis"
keys_monom_mult: fixes c :: "'c"   and t :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "c \<noteq> (0::'c)" shows "keys (local.punit.monom_mult c t p) = local.punit.splus t ` keys p"
decr_grading_p_def: fixes d :: "'a \<Rightarrow> nat"   and n :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "decr_grading_p d n p = (\<Sum>v\<in>keys p. monomial (lookup p v) (decr_grading_term d n v))"
gcd_greatest: fixes c :: "'b"   and a :: "'b"   and b :: "'b" assumes "c dvd a"   and "c dvd b" shows "c dvd gcd a b"
r_3: shows "recfn 1 r \<and> r \<simeq> Cn 1 f [h]"
A7: shows "\<forall>TQ SR. TQ \<in>T Q \<and> SR \<in>S R \<longrightarrow> False"
cr_option_le_gpv_def: fixes x :: "'e option"   and gpv :: "('e, 'f, 'g) gpv" shows "cr_option_le_gpv x gpv = (gpv = (lift_spmf \<circ> return_pmf) x \<or> x = None)"
hyps_2: shows "\<Delta> @ \<Gamma> : body \<Down>\<^bsub>L\<^esub> \<Theta> : z"
rt_strictly_fresherE': fixes i :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt2 :: "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> (nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option) \<Rightarrow> nat \<Rightarrow> bool" assumes "rt1 \<sqsubset>\<^bsub>i\<^esub> rt2"   and "\<lbrakk>rt1 \<sqsubseteq>\<^bsub>i\<^esub> rt2; \<not> rt2 \<sqsubseteq>\<^bsub>i\<^esub> rt1\<rbrakk> \<Longrightarrow> P rt1 rt2 i" shows "P rt1 rt2 i"
double_dual_iso_apply: fixes G :: "('c, 'd) monoid_scheme"   and \<chi> :: "'c \<Rightarrow> complex"   and x :: "'c" assumes "finite_comm_group G"   and "character G \<chi>" shows "finite_comm_group.double_dual_iso G x \<chi> = \<chi> x"
finite_measure_Diff: fixes A :: "'a set"   and M :: "'a pmf"   and B :: "'a set" assumes "A \<in> measure_pmf.events M"   and "B \<in> measure_pmf.events M"   and "B \<subseteq> A" shows "measure_pmf.prob M (A - B) = measure_pmf.prob M A - measure_pmf.prob M B"
k12_1: shows "\<forall>x. k1 (1, x) = g x"
conversion_to_lconv: fixes u :: "'a"   and v :: "'a"   and R :: "'c \<Rightarrow> ('a \<times> 'a) set"   and M :: "'c set"   and E :: "'c \<Rightarrow> ('a \<times> 'a) set"   and L :: "'c \<Rightarrow> ('a \<times> 'a) set" assumes "(u, v) \<in> conversion' L R E M" obtains xs :: "(accent \<times> 'c) list" where "snd ` set xs \<subseteq> M" and "(u, v) \<in> lconv L R E xs"
simps_7: fixes x3 :: "'a" shows "\<langle>x3\<rangle> \<noteq> vertex.SINK"
star_sqrt_HFinite: fixes x :: "real star" assumes "x \<in> HFinite"   and "0 \<le> x" shows "(*f* sqrt) x \<in> HFinite"
less_degree_pdevs_of_list_imp_less_length: fixes i :: "nat"   and xs :: "'a list" assumes "i < degree (pdevs_of_list xs)" shows "i < length xs"
rev_induct: fixes P :: "'b list \<Rightarrow> bool"   and xs :: "'b list" assumes "P \<epsilon>"   and "\<And>x xs. P xs \<Longrightarrow> P (xs \<cdot> [x])" shows "P xs"
InvariantConsistentAfterDecide: fixes M :: "(Literal \<times> bool) list"   and literal :: "Literal"   and M' :: "(Literal \<times> bool) list" assumes "InvariantConsistent M"   and "var literal \<notin> vars (elements M)"   and "M' = M @ [(literal, True)]" shows "InvariantConsistent M'"
indep_system_axioms: shows "indep_system carrier (indep_in carrier)"
1: shows "Can t1 \<and> Can t2 \<and> Can t3 \<and> Src t1 = Trg t2 \<and> Src t2 = Trg t3"
d_mu_Ints: fixes n :: "nat"   and fs :: "'a vec list"   and l :: "nat"   and k :: "nat" assumes "gram_schmidt_fs_int n fs"   and "l \<le> k"   and "k < length fs" shows "gram_schmidt.Gramian_determinant n fs (Suc l) * gram_schmidt_fs.\<mu> n fs k l \<in> \<int>"
subst_eq: shows "subst \<sigma> (fst (mems ! j)) x = fst (mems ! j) x \<and> subst \<sigma> (snd (mems ! j)) x = snd (mems ! j) x"
rrep_prrep: shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PRrep-:2..PRrep-:5} \<longrightarrow> dip \<xi> \<in> kD (rt \<xi>) \<and> sqn (rt \<xi>) (dip \<xi>) = dsn \<xi> \<and> the (dhops (rt \<xi>) (dip \<xi>)) = Suc (hops \<xi>) \<and> the (flag (rt \<xi>) (dip \<xi>)) = val \<and> the (nhop (rt \<xi>) (dip \<xi>)) \<in> kD (rt \<xi>))"
intros: fixes attributes :: "'a \<rightharpoonup>\<^sub>f 'b \<rightharpoonup>\<^sub>f 'c"   and \<D>' :: "'a"   and attr :: "'b"   and \<tau>' :: "'c"   and \<C> :: "'a"   and \<D> :: "'a"   and \<tau> :: "'d" assumes "owned_attribute' attributes \<D>' attr \<tau>'"   and "\<C> \<le> \<D>'"   and "\<D>' < \<D>" shows "attribute_not_closest attributes \<C> attr \<D> \<tau>"
annotate_normI: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and anno :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and c :: "('e, 'd, 'f) com" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P anno Q,A"   and "Language.normalize c = Language.normalize anno" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
word_and_le1: fixes y :: "'a word"   and a :: "'a word" shows "y && a \<le> a"
chamber_system_def: fixes X :: "'b set set" assumes "ChamberComplex X" shows "ChamberComplex.chamber_system X \<equiv> {C. SimplicialComplex.maxsimp X C}"
induct: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> ('c, 'd) SkewBinomialTree list \<Rightarrow> bool"   and a0 :: "'c"   and a1 :: "'d"   and a2 :: "('c, 'd) SkewBinomialTree list" assumes "\<And>e a. P e a []"   and "\<And>e a t. P e a [t]"   and "\<And>e a t t' bq. P e a (t # t' # bq)" shows "P a0 a1 a2"
class_leq_def: fixes sub :: "('a \<times> 'b) set"   and c1 :: "'a"   and c2 :: "'b" shows "class_leq sub c1 c2 = ((c1, c2) \<in> sub)"
visible_fields_exist: fixes P :: "'a prog"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list"   and F :: "char list"   and D :: "char list"   and b :: "staticb"   and T :: "ty" assumes "P \<turnstile> C has_fields FDTs"   and "map_of FDTs (F, D) = \<lfloor>(b, T)\<rfloor>" shows "\<exists>D' fs ms. class P D = \<lfloor>(D', fs, ms)\<rfloor> \<and> map_of fs F = \<lfloor>(b, T)\<rfloor>"
similar_mat_witD_4: fixes n :: "nat"   and A :: "'a mat"   and B :: "'a mat"   and P :: "'a mat"   and Q :: "'a mat" assumes "n = dim_row A"   and "similar_mat_wit A B P Q" shows "A \<in> carrier_mat n n"
pdivides_def: fixes R :: "('d, 'e) ring_scheme"   and p :: "'d list"   and q :: "'d list" shows "p pdivides\<^bsub>R\<^esub> q = p divides\<^bsub>poly_ring R\<^esub> q"
atms_of_poss: fixes AA :: "'b multiset" shows "atms_of (poss AA) = set_mset AA"
b_2: shows "\<forall>y\<in>local.degenerate_lotteries. b \<succeq>[\<R>] y"
is_processT5_S6: fixes c :: "'c event"   and P :: "'c process" assumes "([], {c}) \<notin> F P" shows "([c], {}) \<in> F P"
u_t: shows "T\<^sub>p_on_rand' BIT (type0 init x y) (u @ v) = T\<^sub>p_on_rand' BIT (type0 init x y) v"
\<tau>inv_lift: shows "\<tau>inv r1.redT r2.redT mbisim mta_bisim m\<tau>move1 m\<tau>move2"
algebraic_imp_non_trivial_ker: fixes K :: "'a set"   and x :: "'a" assumes "subring K R"   and "x \<in> carrier R"   and "(algebraic over K) x" shows "a_kernel (K [X]) R (\<lambda>p. local.eval p x) \<noteq> {[]}"
not_iMOD_iTILL_subset: fixes m' :: "nat"   and r' :: "nat"   and n :: "nat" assumes "0 < m'" shows "\<not> [ r', mod m' ] \<subseteq> [\<dots>n]"
simps_4: fixes f1 :: "'a"   and f2 :: "tree \<Rightarrow> tree \<Rightarrow> 'a" shows "(case Leaf of Leaf \<Rightarrow> f1 | Branching x xa \<Rightarrow> f2 x xa) = f1"
inj_mod_ideal: fixes f :: "'c \<Rightarrow> 'd"   and I :: "'c set"   and R :: "('c, 'e) ring_scheme"   and a :: "'c"   and b :: "'c" assumes "weak_ring_morphism f I R"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "(f a = f b) = (a \<ominus>\<^bsub>R\<^esub> b \<in> I)"
f'_def: shows "f' \<equiv> augment (augmentingFlow p)"
cost: shows "cost (basis_reduction_main_cost upw i state c) \<le> c + body_cost * LLL_measure i (fs_state state)"
lmap_llist_of: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "lmap f (llist_of xs) = llist_of (map f xs)"
isomorphic_implies_hpar_3: fixes f :: "'a"   and f' :: "'a" assumes "f \<cong>\<^sub>C f'" shows "src\<^sub>C f = src\<^sub>C f'"
Borsukian_Un_lemma: fixes S :: "'a set"   and T :: "'a set" assumes "Borsukian S"   and "Borsukian T"   and "connected (S \<inter> T)"   and "\<And>f g. \<lbrakk>continuous_on S f; continuous_on T g; \<And>x. x \<in> S \<and> x \<in> T \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> continuous_on (S \<union> T) (\<lambda>x. if x \<in> S then f x else g x)" shows "Borsukian (S \<union> T)"
pair_of_map_component: fixes f :: "'k \<Rightarrow> 'k"   and v :: "'t" shows "pair_of_term (map_component f v) = (pp_of_term v, f (component_of_term v))"
\<Sigma>_protocols_base_def: fixes Rel :: "('c \<times> 'd) set"   and valid_pub :: "'c set" shows "\<Sigma>_protocols_base Rel valid_pub \<equiv> Domain Rel \<subseteq> valid_pub"
A: shows "density M (\<lambda>x. ennreal (D x)) \<noteq> M"
blocking_on_imp_not_stable: fixes ds :: "'b set"   and X :: "'a set"   and h :: "'c"   and X'' :: "'a set" assumes "Pds'.blocking_on ds X h X''" shows "\<not> Pds'.stable_on ds X"
is_valid_problem_sas_plus_then_3: fixes \<Psi> :: "('c, 'd) sas_plus_problem" assumes "is_valid_problem_sas_plus \<Psi>" shows "dom (\<Psi>\<^sub>I\<^sub>+) = set (\<Psi>\<^sub>\<V>\<^sub>+)"
List_conv_islist_list: fixes h :: "'c \<Rightarrow> 'c ref"   and p :: "'c ref"   and as :: "'c list" shows "List h p as = (islist h p \<and> as = list h p)"
start_heap_def: 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" shows "start_heap P \<equiv> [addr_of_sys_xcpt NullPointer \<mapsto> blank P NullPointer, addr_of_sys_xcpt ClassCast \<mapsto> blank P ClassCast, addr_of_sys_xcpt OutOfMemory \<mapsto> blank P OutOfMemory]"
cases: fixes r :: "('d, 'g, 'e, 'h, 'f, 'i) Functor" obtains CatDom :: "('d, 'e, 'f) Category_scheme" and CatCod :: "('g, 'h, 'i) Category_scheme" and MapM :: "'e \<Rightarrow> 'h" where "r = \<lparr>CatDom = CatDom, CatCod = CatCod, MapM = MapM\<rparr>"
unat_bintrunc_neg: fixes bin :: "num" shows "unat (- numeral bin) = nat (take_bit LENGTH('b) (- numeral bin))"
SumRepl: fixes A :: "ZF"   and f :: "ZF \<Rightarrow> ZF" shows "HOLZF.Sum (Repl A (Singleton \<circ> f)) = Repl A f"
isOK_check_subseteq: fixes xs :: "'a list"   and ys :: "'a list" shows "isOK (check_subseteq xs ys) = (set xs \<subseteq> set ys)"
less_asym: fixes x :: "'a list"   and y :: "'a list"   and P :: "bool" assumes "strict_border x y"   and "\<not> P \<Longrightarrow> strict_border y x" shows "P"
ct_list_eq_def: fixes ct1s :: "ct list"   and ct2s :: "ct list" shows "ct_list_eq ct1s ct2s = list_all2 ct_eq ct1s ct2s"
length2_equiv: fixes l :: "tm list" shows "length2 l = ([hd l, hd (tl l)] = l)"
add_less_le_mono: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset"   and d :: "'c multiset" assumes "a \<subset># b"   and "c \<subseteq># d" shows "a + c \<subset># b + d"
start_of_Suc2: fixes ly :: "nat list"   and ap :: "abc_inst list"   and as :: "nat"   and n :: "nat"   and e :: "nat" assumes "ly = layout_of ap"   and "abc_fetch as ap = Some (Dec n e)" shows "start_of ly (Suc as) = start_of ly as + 2 * n + 16"
indexle_trans: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and k'' :: "nat"   and u'' :: "nat"   and k' :: "nat"   and u' :: "nat"   and k :: "nat"   and u :: "nat" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "indexle k'' u'' k' u'"   and "indexle k' u' k u" shows "indexle k'' u'' k u"
safe_max_empty: fixes X :: "'a set" shows "(safe_max X = None) = (X = {})"
weakStepSimTauChain: fixes Q :: "('a, 'b, 'c) psi"   and Q' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<Psi> \<rhd> Q \<Longrightarrow>\<^sub>\<tau> Q'"   and "\<Psi> \<rhd> P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> Rel \<Longrightarrow> \<Psi> \<rhd> P \<leadsto><Rel> Q" obtains P' :: "('a, 'b, 'c) psi" where "\<Psi> \<rhd> P \<Longrightarrow>\<^sub>\<tau> P'" and "(\<Psi>, P', Q') \<in> Rel"
fpsmodrel_def: fixes p :: "'a" shows "fpsmodrel p \<equiv> {(f, g). \<forall>n. f $ n mod p = g $ n mod p}"
is_inf_fun: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "is_inf f g (\<lambda>x. f x \<sqinter> g x)"
obj_simps_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "horizontal_homs V src trg"   and "horizontal_homs.obj V src a" shows "src a = a"
assms_2: shows "i < length (sturm_squarefree' p) - 1"
bv_or_nth: fixes l :: "bool list"   and r :: "bool list"   and i :: "nat" assumes "length l = length r"   and "i < length l" shows "bv_or l r ! i = (l ! i \<or> r ! i)"
erHomTr0: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and h :: "'a \<Rightarrow> 'g"   and x :: "'a" assumes "PolynRg A B Y"   and "h \<in> rHom S B"   and "x \<in> carrier R" shows "erH R S X A B Y h (-\<^sub>a x) = -\<^sub>a\<^bsub>A\<^esub> erH R S X A B Y h x"
Read: fixes a :: "nat"   and \<O> :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option"   and volatile :: "bool"   and \<D> :: "bool"   and \<O>s :: "nat set list"   and i :: "nat"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and m :: "nat \<Rightarrow> nat" assumes "a \<in> \<O> \<or> a \<in> read_only \<S> \<or> volatile \<and> a \<in> dom \<S>"   and "volatile \<longrightarrow> \<not> \<D>" shows "\<O>s,i\<turnstile> (Read volatile a t # is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
sup_red_inf_in_red_liminf: fixes Ns :: "'a set llist" assumes "chain (\<rhd>) Ns" shows "Sup_llist (lmap Red_I Ns) \<subseteq> Red_I (Liminf_llist Ns)"
add_col_sub_row_same_upto: fixes i :: "nat"   and j :: "nat"   and A :: "'a mat"   and v :: "'a" assumes "i < j"   and "j < n"   and "A \<in> carrier_mat n n"   and "local.inv_upto local.uppert A j" shows "local.same_upto j A (add_col_sub_row v i j A)"
term_cases: fixes t :: "'c" obtains name :: "name" where "t = free name__"   | name :: "name" where "t = const name"   | u\<^sub>1 :: "'c" and u\<^sub>2 :: "'c" where "t = app u\<^sub>1 u\<^sub>2"   | "unfree t = None" and "unapp t = None" and "unconst t = None"
subst_bvs_decr: fixes us :: "Core.term list"   and t :: "Core.term" assumes "\<forall>x\<in>set us. is_closed x"   and "\<not> is_dependent t" shows "subst_bvs us (decr 0 t) = decr 0 (subst_bvs1' t 1 us)"
fddg_smult_decomp: fixes g :: "'g"   and v :: "'v"   and f :: "'f" assumes "g \<in> G"   and "v \<in> V" shows "(f \<delta>\<delta> g) \<cdot> v = f \<sharp>\<cdot> g *\<cdot> v"
scalar_prod_left_mono: fixes u :: "'a vec"   and n :: "nat"   and v :: "'a vec"   and w :: "'a vec" assumes "u \<in> carrier_vec n"   and "v \<in> carrier_vec n"   and "w \<in> carrier_vec n"   and "vec_comp_all (\<succ>) u v" shows "u \<bullet> w \<succ> v \<bullet> w"
eval_ring_hom: fixes a :: "'a" assumes "a \<in> carrier R" shows "eval R R (\<lambda>x. x) a \<in> ring_hom P R"
ih_9: shows "InvariantWatchListsCharacterization (getWatchList state') (getWatch1 state') (getWatch2 state')"
obj_at_split: fixes Q :: "bool \<Rightarrow> bool"   and P :: "('a, 'b, 'c) object \<Rightarrow> bool"   and r :: "'c"   and s :: "'d process_name \<Rightarrow> ('a, 'd, 'b, 'c) Model.local_state" shows "Q (obj_at P r s) = ((sys_heap s r = None \<longrightarrow> Q False) \<and> (\<forall>obj. sys_heap s r = Some obj \<longrightarrow> Q (P obj)))"