hcju/isabelleps huggingface dataset:
distinct_13: fixes x21 :: "nat"   and x22 :: "'a"   and x23 :: "'a"   and x24 :: "'b"   and x25 :: "'b"   and x26 :: "'c"   and x51 :: "nat"   and x52 :: "'a"   and x53 :: "'a" shows "Send x21 x22 x23 x24 x25 x26 \<noteq> RecvMarker x51 x52 x53"
simps_10: fixes Vs :: "char list list"   and V :: "char list"   and T :: "ty"   and e :: "char list exp" shows "compE\<^sub>1 Vs {V:T; e} = {length Vs:T; compE\<^sub>1 (Vs @ [V]) e}"
Nom_7: shows "Dup q i ((ps, a) # branch) xs"
released_simp: fixes p :: "'c"   and instrs :: "'d"   and \<theta> :: "'e"   and sb :: "'f"   and \<D> :: "'g"   and \<O> :: "'h"   and \<R> :: "'b" shows "released (p, instrs, \<theta>, sb, \<D>, \<O>, \<R>) = \<R>"
top_apply: fixes x :: "'b" shows "top_class.top x = top_class.top"
bounded_any_center: fixes S :: "'b set"   and a :: "'b" shows "bounded S = (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
dvdm_degree: fixes u :: "int poly"   and f :: "int poly" assumes "monic u"   and "pl.dvdm u f"   and "pl.Mp f \<noteq> 0" shows "degree u \<le> degree f"
inImage_ntsem: fixes a :: "univ"   and \<sigma> :: "'tp" assumes "a \<in> (eintF (Tag \<sigma>) \<circ> singl) ` {b. eintT \<sigma> b}" shows "ntsem \<sigma> a = a"
Nil_is_append_conv: fixes xs :: "'c list"   and ys :: "'c list" shows "([] = xs \<cdot> ys) = (xs = [] \<and> ys = [])"
id_arrow: fixes CC :: "('b, 'c, 'd) category_scheme"   and A :: "'b" assumes "category CC"   and "A \<in> Ob\<^bsub>CC\<^esub>" shows "Id\<^bsub>CC\<^esub> A \<in> Ar\<^bsub>CC\<^esub>"
uniformly_convergent_eval_fds_aux: fixes f :: "'a fds"   and s0 :: "'a"   and B :: "'a set" assumes "Bseq (\<lambda>n. \<Sum>k\<le>n. fds_nth f k / nat_power k s0)"   and "compact B"   and "\<And>z. z \<in> B \<Longrightarrow> s0 \<bullet> (1::'a) < z \<bullet> (1::'a)" shows "uniformly_convergent_on B (\<lambda>N z. \<Sum>n\<le>N. fds_nth f n / nat_power n z)"
TimesR_def: fixes R :: "'b regex set"   and s :: "'b regex" shows "TimesR R s = (\<lambda>r. Times r s) ` R"
geodesic_segment_expression: fixes x :: "'a"   and z :: "'a" shows "{x--z} = {y. dist x z = dist x y + dist y z}"
sup_bot_right: fixes x :: "'a" shows "x + (0::'a) = x"
simps_1: fixes l :: "'e strand_label"   and t :: "('f, 'g) Term.term" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p (l, send\<langle>t\<rangle>) = (l, receive\<langle>t\<rangle>)"
Units_r_inv_ex: fixes x :: "'c" assumes "x \<in> Units (add_monoid V)" shows "\<exists>y\<in>carrier V. x \<oplus>\<^bsub>V\<^esub> y = \<zero>\<^bsub>V\<^esub>"
i_def: fixes \<psi> :: "'a ltl" assumes "G \<psi> \<in> \<G>" shows "suffix i w \<Turnstile> G \<psi>"
sets_extend_measure: fixes G :: "'c \<Rightarrow> 'b set"   and I :: "'c set"   and \<Omega> :: "'b set"   and \<mu> :: "'c \<Rightarrow> ennreal" assumes "G ` I \<subseteq> Pow \<Omega>" shows "sets (extend_measure \<Omega> I G \<mu>) = sigma_sets \<Omega> (G ` I)"
e_app_intro: fixes f :: "(val \<times> val) fset"   and e1 :: "exp"   and \<rho> :: "(nat \<times> val) list"   and v2 :: "val"   and e2 :: "exp"   and v2' :: "val"   and v3' :: "val"   and v3 :: "val" assumes "VFun f \<in> E e1 \<rho>"   and "v2 \<in> E e2 \<rho>"   and "(v2', v3') \<in> fset f"   and "v2' \<sqsubseteq> v2"   and "v3 \<sqsubseteq> v3'" shows "v3 \<in> E (EApp e1 e2) \<rho>"
switch_hchop2: fixes v' :: "view"   and v1' :: "view"   and v2' :: "view"   and v :: "view"   and c :: "cars" shows "v'=v1'\<parallel>v2' \<and> v = c > v' \<longrightarrow> (\<exists>v1 v2. v1 = c > v1' \<and> v2 = c > v2' \<and> v=v1\<parallel>v2)"
c2: fixes x :: "'a set"   and y :: "'a set"   and u :: "'a" assumes "closed_csubspace x"   and "closed_csubspace y"   and "x \<subseteq> y"   and "u \<in> y" shows "u \<in> x +\<^sub>M orthogonal_complement x \<inter> y"
add: fixes b1 :: "'a"   and b2 :: "'a" shows "f'' (b1 + b2) = f'' b1 + f'' b2"
antisym: fixes a :: "'b list"   and b :: "'b list" assumes "prefix a b"   and "prefix b a" shows "a = b"
sort_def: fixes cmp :: "'b comparator"   and xs :: "'b list" shows "Sorting_Algorithms.sort cmp xs = foldr (Sorting_Algorithms.insort cmp) xs []"
vsv_appE: fixes a :: "V"   and b :: "V" assumes "r\<lparr>a\<rparr> = b"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" obtains "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"
cong_sym: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and b :: "'c"   and c :: "'c"   and a :: "'c" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "unique_euclidean_semiring.cong modulo b c a" shows "unique_euclidean_semiring.cong modulo c b a"
dghm_dag_Rel_ArrMap_vrange: fixes \<alpha> :: "V" shows "\<R>\<^sub>\<circ> (\<dagger>\<^sub>D\<^sub>G\<^sub>.\<^sub>R\<^sub>e\<^sub>l \<alpha>\<lparr>ArrMap\<rparr>) = dg_Rel \<alpha>\<lparr>Arr\<rparr>"
ev: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" shows "ev \<phi> \<pi>l = (\<exists>i. \<phi> (map (sdrop i) \<pi>l))"
vsv_vconverse: shows "vsv (r\<inverse>\<^sub>\<circ>)"
ipresIGWlsAll_def: fixes h :: "'w \<Rightarrow> 'x"   and hA :: "'y \<Rightarrow> 'z"   and MOD :: "('aa, 'ab, 'ac, 'ad, 'ae, 'af, 'w, 'y, 'ag) model_scheme"   and MOD' :: "('ah, 'ai, 'ac, 'ad, 'aj, 'ak, 'x, 'z, 'al) model_scheme" shows "ipresIGWlsAll h hA MOD MOD' \<equiv> ipresIGWls h MOD MOD' \<and> ipresIGWlsAbs hA MOD MOD'"
valid_last_prog: fixes i :: "nat"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'p memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "last_prog p sb = p"
Range_iff: fixes a :: "'d"   and r :: "('e \<times> 'd) set" shows "(a \<in> Range r) = (\<exists>y. (y, a) \<in> r)"
Image_closed_trancl: fixes succs :: "'c \<Rightarrow> 'c list"   and is_node :: "'c \<Rightarrow> bool"   and invariant :: "'d \<Rightarrow> bool"   and ins :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and memb :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and empt :: "'d"   and r :: "('e \<times> 'e) set"   and X :: "'e set" assumes "DFS succs is_node invariant ins memb empt"   and "r `` X \<subseteq> X" shows "r\<^sup>* `` X = X"
i_p: shows "(C, i) \<in># wP_of_wstate (lnth Sts t_C)"
19: fixes x :: "'a" shows "- x \<squnion> - - x = top"
cod_src_eq_dom_src: shows "assoc\<mu>\<nu>\<pi>.cod.src = assoc\<mu>\<nu>\<pi>.dsrc"
elims_2: fixes x :: "'b extT"   and xa :: "'b extT" assumes "x \<le> xa" obtains x :: "'b" and y :: "'b" where "xa__ = extT.Val x" and "xa = extT.Val y" and "x \<le> y"   | uv_ :: "'b extT" where "xa__ = uv_" and "xa = extT.Top"
inner_prod_with_itself_Im: fixes u :: "complex Matrix.vec" shows "Im \<langle>u|u\<rangle> = 0"
disc_1: fixes x1 :: "'a" shows "is_Terminates (Terminates x1)"
partition_def: shows "partition \<equiv> \<lambda>uu uua. rec_list (\<lambda>P. ([], [])) (\<lambda>x xs xsa P. let (yes, no) = xsa P in if P x then (x # yes, no) else (yes, x # no)) uua uu"
d_1: shows "d = rat_of_int (lead_coeff (gcd F G))"
lub_upper: fixes cl :: "'b potype"   and S :: "'b set"   and x :: "'b" assumes "CL cl"   and "S \<subseteq> pset cl"   and "x \<in> S" shows "(x, lub S cl) \<in> order cl"
hom_lead_coeff: fixes hom :: "'b \<Rightarrow> 'c"   and p :: "'b poly" assumes "zero_hom_0 hom" shows "lead_coeff (map_poly hom p) = hom (lead_coeff p)"
surj_if_then_else: fixes f :: "'b \<Rightarrow> 'a"   and X :: "'b set"   and g' :: "'b \<Rightarrow> 'a" assumes "- f ` X = g' ` (- X)" shows "surj (\<lambda>z. if z \<in> X then f z else g' z)"
except_keys: fixes p :: "'c \<Rightarrow>\<^sub>0 'd" shows "except p (keys p) = 0"
member_Id_on: fixes t :: "'b set_rbt" assumes "ID ccompare \<noteq> None" shows "RBT_Set2.member (RBT_Set2.Id_on t) = (\<lambda>(k, k'). k = k' \<and> RBT_Set2.member t k)"
type_definition_integer: shows "type_definition int_of_integer integer_of_int UNIV"
lunit_tensor: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "\<l>\<^sub>1[a \<otimes>\<^sub>1 b] = (\<l>\<^sub>1[a] \<otimes>\<^sub>1 b) \<cdot> \<a>\<^sub>1\<^sup>-\<^sup>1[\<I>\<^sub>1, a, b]"
CAND_NotR_elim: fixes a :: "coname"   and x :: "name"   and M :: "trm"   and B :: "ty" assumes "<a>:NotR x.M a \<in> \<parallel><B>\<parallel>"   and "<a>:NotR x.M a \<notin> BINDINGc B (\<parallel>(B)\<parallel>)" shows "\<exists>B'. B = NOT B' \<and> x:M \<in> \<parallel>(B')\<parallel>"
sigma_finite_measure_count_space_countable: fixes A :: "'b set" assumes "countable A" shows "sigma_finite_measure (count_space A)"
open_segment_eq_empty': fixes a :: "'b"   and b :: "'b" shows "({} = open_segment a b) = (a = b)"
after_failures: fixes xs :: "'e list"   and P :: "'e process" assumes "xs \<in> traces P" shows "failures (P \<setminus> xs) = futures P xs"
filter_sup_least_upper_bound: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and H :: "'b set"   and F :: "'b set"   and G :: "'b set" assumes "class.semilattice_inf inf less_eq less"   and "ord.filter less_eq H"   and "F \<subseteq> H"   and "G \<subseteq> H" shows "semilattice_inf.filter_sup inf less_eq F G \<subseteq> H"
strict_iff_order: fixes b :: "'a list"   and a :: "'a list" shows "strict_prefix b a = (prefix b a \<and> a \<noteq> b)"
E_wfD_2: fixes v :: "'v"   and v' :: "'v" assumes "(v, v') \<in> E" shows "v' \<in> V"
assms_3: shows "k \<le> 46 * n - 1"
to_nat_on_surj: fixes A :: "'b set"   and n :: "nat" assumes "countable A"   and "infinite A" shows "\<exists>a\<in>A. to_nat_on A a = n"
Gmult_def: fixes smult :: "('b, 'c) aezfun \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "'c"   and v :: "'a" shows "aezfun_scalar_mult.Gmult smult g v \<equiv> smult ((1::'b) \<delta>\<delta> g) v"
subst_implies_prv_SS: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {xx}" shows "bprv (SS \<langle>\<phi>\<rangle> \<langle>subst \<phi> \<langle>\<phi>\<rangle> xx\<rangle>)"
simps_4: fixes f :: "'a"   and v :: "'a local.term"   and va :: "'a local.term" shows "\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<Down> v \<^bold>\<otimes> va = \<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<otimes> v \<^bold>\<otimes> va"
step_invatiant_unfold: fixes w :: "'a list \<times> 's \<times> 's"   and ws :: "('a list \<times> 's \<times> 's) list"   and ps :: "('s \<times> 's) list"   and N :: "('s \<times> 's) set"   and ws' :: "('a list \<times> 's \<times> 's) list"   and ps' :: "('s \<times> 's) list"   and N' :: "('s \<times> 's) set" assumes "step_invariant (w # ws, ps, N) = (ws', ps', N')" shows "\<exists>xs r s. w = (xs, r, s) \<and> ps' = (r, s) # ps \<and> ws' = ws @ remdups' (map_prod post post \<circ> snd) (filter (\<lambda>(uu_, p). map_prod post post p \<notin> N) (map (\<lambda>a. (a # xs, delta a r, delta a s)) (\<sigma> n))) \<and> N' = set (map (\<lambda>a. (post (delta a r), post (delta a s))) (\<sigma> n)) \<union> N"
cvector_fraction_eq_iff: fixes u :: "complex"   and v :: "complex"   and a :: "'d"   and x :: "'d" shows "((u / v) *\<^sub>C a = x) = (if v = 0 then x = (0::'d) else u *\<^sub>C a = v *\<^sub>C x)"
map_of_Cons_code_1: fixes k :: "'b" shows "map_of [] k = None"
assms_5: shows "local.eij i j \<le> y ! j"
exenormalize_sort_def: fixes cs :: "(String.literal \<times> String.literal) list"   and s :: "String.literal set" shows "exenormalize_sort cs s = {c \<in> s. \<not> (\<exists>c'\<in>s. execlass_les cs c' c)}"
only_horizontal_division_def: fixes one_chain :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and two_chain :: "(real \<times> real \<Rightarrow> real \<times> real) set" shows "only_horizontal_division one_chain two_chain \<equiv> \<exists>\<H> \<V>. finite \<H> \<and> finite \<V> \<and> (\<forall>(k, \<gamma>)\<in>\<H>. \<exists>(k', \<gamma>')\<in>two_chain_horizontal_boundary two_chain. \<exists>a\<in>{0..1}. \<exists>b\<in>{0..1}. a \<le> b \<and> subpath a b \<gamma>' = \<gamma>) \<and> (common_sudiv_exists (two_chain_vertical_boundary two_chain) \<V> \<or> common_reparam_exists \<V> (two_chain_vertical_boundary two_chain)) \<and> boundary_chain \<V> \<and> one_chain = \<H> \<union> \<V> \<and> (\<forall>(k, \<gamma>)\<in>\<V>. valid_path \<gamma>)"
can_exec_def: fixes root :: "(att \<times> char list, att, nat) env"   and xs :: "operation list" shows "can_exec root xs = (\<exists>root'. root \<Midarrow>xs\<Rightarrow> root')"
vcons_eqD'_2: fixes xs\<^sub>1 :: "V"   and xs\<^sub>2 :: "V"   and x\<^sub>1 :: "V"   and x\<^sub>2 :: "V" assumes "vfsequence xs\<^sub>1"   and "vfsequence xs\<^sub>2"   and "xs\<^sub>1 #\<^sub>\<circ> x\<^sub>1 = xs\<^sub>2 #\<^sub>\<circ> x\<^sub>2" shows "x\<^sub>1 = x\<^sub>2"
succ_in_verts: fixes s :: "nat"   and n :: "nat" assumes "s \<in> succs n" shows "{s, n} \<subseteq> set (g_V G)"
equalityCE: fixes A :: "'e set"   and B :: "'e set"   and c :: "'e" assumes "A = B" obtains "c \<in> A" and "c \<in> B"   | "c \<notin> A" and "c \<notin> B"
Inf_less_iff: fixes S :: "'b set"   and a :: "'b" shows "(\<Sqinter> S < a) = (\<exists>x\<in>S. x < a)"
effect_apply_eqvt': fixes satisfies :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and transition :: "'b \<Rightarrow> ('d, 'b) residual \<Rightarrow> bool"   and effect_apply :: "'e \<Rightarrow> 'b \<Rightarrow> 'b"   and L :: "'d \<times> 'e fs_set \<times> 'e \<Rightarrow> 'e fs_set"   and p :: "perm"   and f :: "'e"   and P :: "'b" assumes "effect_nominal_ts satisfies transition effect_apply L" shows "p \<bullet> effect_apply f P = effect_apply (p \<bullet> f) (p \<bullet> P)"
5: shows "(e, fe) \<in> {(e, f e) |e. f e \<noteq> bot} \<and> (\<forall>y. y \<in> {(e, f e) |e. f e \<noteq> bot} \<longrightarrow> (case (e, fe) of (e, fe) \<Rightarrow> \<lambda>(d, fd). fe + bot \<le> fd + bot) y)"
D_not_in: shows "D \<notin> Red_F_G (\<G>_Fset N)"
varSort_lt_var_def: fixes uu :: "'a"   and uv :: "'b" shows "varSort_lt_var uu uv \<equiv> |UNIV| <o |UNIV|"
MC_action_defs_3: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and cst :: "state \<Rightarrow> PrIds \<Rightarrow> mClkState"   and p :: "PrIds" shows "MClkRetry send rcv cst p = ($(cst!p) = #clkB \<and> res<$(rcv!p)> = #RPCFailure \<and> Call rcv p MClkRelayArg<arg<send!p>> \<and> unchanged (cst!p, rtrner send!p))"
poly_mapping_except_induct: fixes P :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and p :: "'a \<Rightarrow>\<^sub>0 'b" assumes "P 0"   and "\<And>p t. \<lbrakk>p \<noteq> 0; t \<in> keys p; P (except p {t})\<rbrakk> \<Longrightarrow> P p" shows "P p"
that: fixes r :: "(nat \<times> 'a) stream"   and v :: "nat \<times> 'a" assumes "grun A w V r v"   and "sset (gtrace r v) \<subseteq> clean A w V"   and "sset (gtrace r v) \<subseteq> - prune A w (clean A w V)" shows "thesis"
r_neg: fixes G :: "('e, 'f) ring_scheme"   and x :: "'e" assumes "abelian_group G"   and "x \<in> carrier G" shows "x \<oplus>\<^bsub>G\<^esub> \<ominus>\<^bsub>G\<^esub> x = \<zero>\<^bsub>G\<^esub>"
l7_9: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and P :: "'a"   and X :: "'a"   and Q :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong A P X"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong A Q X" shows "P = Q"
extended_distance_middle_2: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and a :: "'c Bonk_Schramm_extension_unfolded"   and b :: "'c Bonk_Schramm_extension_unfolded" assumes "class.metric_space dist uniformity open"   and "a \<in> metric_space.extended_distance_set dist"   and "b \<in> metric_space.extended_distance_set dist" shows "metric_space.extended_distance dist b (middle a b) = metric_space.extended_distance dist a b / 2"
map_ide: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and b :: "'a"   and a :: "'a" assumes "hom_functor C S \<phi>"   and "partial_magma.ide C b"   and "partial_magma.ide C a" shows "hom_functor.map C S \<phi> (b, a) = set_category_given_img.mkIde S (set_category.some_img S (cardSuc (cmax |UNIV| natLeq))) (cardSuc (cmax |UNIV| natLeq)) (hom_functor.set C \<phi> (b, a))"
sset_streams: fixes s :: "'d stream"   and A :: "'d set" assumes "sset s \<subseteq> A" shows "s \<in> streams A"
int_f: fixes c :: "'M"   and d :: "'M" assumes "cbox c d \<subseteq> box a b" shows "f integrable_on cbox c d"
poincare_map_returns: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and P :: "'b set"   and x :: "'b" assumes "auto_ll_on_open f X"   and "auto_ll_on_open.returns_to f X P x"   and "closed P" shows "auto_ll_on_open.poincare_map f X P x \<in> P"
LINEAR_mono: fixes r :: "('f + 'g + 'h, 'i) Old_Datatype.node set"   and r' :: "('f + 'g + 'h, 'i) Old_Datatype.node set"   and b :: "'g" assumes "r \<subseteq> r'" shows "LINEAR b r \<subseteq> LINEAR b r'"
thmD7_helper: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and N :: "symbol"   and v :: "symbol list" assumes "LeftDerivation [\<SS>] D (N # v)"   and "is_nonterminal N"   and "\<SS> \<noteq> N" shows "\<exists>n M a a1 a2 w. n < length D \<and> (M, a) \<in> \<RR> \<and> LeftDerivation [\<SS>] (take n D) (M # w) \<and> a = a1 @ [N] @ a2 \<and> derives a1 []"
Memop: fixes i :: "nat"   and ts :: "('p \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "unit"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and sb' :: "unit"   and m' :: "nat \<Rightarrow> nat"   and \<D>' :: "bool"   and \<O>' :: "nat set"   and \<R>' :: "nat \<Rightarrow> bool option"   and \<S>' :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "(is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow> (is', \<theta>', sb', m', \<D>', \<O>', \<R>', \<S>')" shows "direct_computation.concurrent_step (ts, m, \<S>) (ts[i := (p, is', \<theta>', sb', \<D>', \<O>', \<R>')], m', \<S>')"
rebase_rebase_poly: fixes f :: "'pq mod_ring poly" shows "##f = #f"
EmptyMap_ran: fixes S :: "'d set" assumes "S \<noteq> {}" shows "ran (EmptyMap S) = {{}}"
mult_comp_cls_val_process_Suc: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and Mkt :: "('d, 'c) discrete_market"   and qty :: "nat \<Rightarrow> 'd \<Rightarrow> real" assumes "portfolio pf1" shows "\<forall>n w. cls_val_process Mkt (qty_mult_comp pf1 qty) (Suc n) w = cls_val_process Mkt pf1 (Suc n) w * qty (Suc n) w"
fBex_triv_one_point1: fixes A :: "'b fset"   and a :: "'b" shows "fBex A (\<lambda>x. x = a) = (a |\<in>| A)"
length_berlekamp_basis_numbers_factors: fixes u :: "'b mod_ring poly"   and P :: "'b mod_ring poly set" assumes "0 < degree u"   and "finite P"   and "u = \<Prod>P"   and "P \<subseteq> {q. irreducible q \<and> monic q}" shows "length (berlekamp_basis u) = card P"
simps_1: fixes s :: "val option" shows "s \<star> [] = s"
strict_trans: fixes b :: "'c Word.word"   and a :: "'c Word.word"   and c :: "'c Word.word" assumes "b <s a"   and "c <s b" shows "c <s a"
assms: shows "P \<subseteq> A\<^sup>\<spadesuit>"
monic_factorization_uniqueness: fixes P :: "'b poly set"   and Q :: "'b poly set" assumes "finite P"   and "\<Prod>P = \<Prod>Q"   and "P \<subseteq> {q. irreducible\<^sub>d q \<and> monic q}"   and "finite Q"   and "Q \<subseteq> {q. irreducible\<^sub>d q \<and> monic q}" shows "P = Q"
vfsequence_vinsert: fixes a :: "V" shows "vfsequence (vinsert \<langle>vcard xs, a\<rangle> xs)"
set_2: fixes x21 :: "'c"   and x22 :: "'c list" shows "set (x21 # x22) = insert x21 (set x22)"
declassified_proj_ik_subset: fixes A :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and I :: "'var \<Rightarrow> ('fun, 'a) Term.term"   and n :: "'lbl" shows "declassified\<^sub>l\<^sub>s\<^sub>t A I \<subseteq> ik\<^sub>s\<^sub>t (proj_unl n A) \<cdot>\<^sub>s\<^sub>e\<^sub>t I"
lincomb_same: fixes S :: "'a vec set"   and a :: "'a vec \<Rightarrow> 'a" assumes "S \<subseteq> mat_kernel A" shows "A.lincomb a S = AB.NC.lincomb a S"
DirProds_is_finite_group: fixes I :: "'e set"   and G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme" assumes "\<And>i. i \<in> I \<Longrightarrow> finite_group (G i)"   and "finite I" shows "finite_group (DirProds G I)"
ccsubspace_add_right_incr: fixes a :: "'d ccsubspace"   and c :: "'d ccsubspace" shows "a \<le> a + c"
fresh_1: fixes a :: "name"   and x1 :: "name" shows "a \<sharp> Var x1 = a \<sharp> x1"
eq: fixes z :: "complex" assumes "z \<in> S" shows "\<exists>d>0. \<forall>w\<in>ball z d - {a}. g w = (w - a) * f w"
minus_eq: fixes G :: "('b, 'c) ring_scheme"   and x :: "'b"   and y :: "'b" assumes "abelian_group G" shows "x \<ominus>\<^bsub>G\<^esub> y = x \<oplus>\<^bsub>G\<^esub> \<ominus>\<^bsub>G\<^esub> y"
psi_converg_imp_prefix: fixes j :: "nat"   and x :: "nat" assumes "\<psi> j x \<down>" shows "\<exists>t b. local.prefixes t j \<down>= b \<and> x < e_length b"
permute_boolE: fixes p :: "perm"   and P :: "bool" assumes "p \<bullet> P" shows "P"
nonneg_eq_int: fixes k :: "int" assumes "0 \<le> k" obtains n :: "nat" where "k = int n"
ZHMSet_diff: fixes xa :: "hmultiset zmultiset"   and x :: "hmultiset zmultiset" shows "ZHMSet (xa - x) = ZHMSet xa - ZHMSet x"
maps_over_upd: fixes m :: "'a \<Rightarrow> 'b option"   and A :: "'a set"   and B :: "'b set"   and v :: "'a"   and k :: "'b" assumes "m \<in> maps_over A B"   and "v \<in> A"   and "k \<in> B" shows "m(v \<mapsto> k) \<in> maps_over A B"
sum_distrib_right: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and r :: "'b" shows "sum f A * r = (\<Sum>n\<in>A. f n * r)"
invD_1: fixes C :: "nat set"   and R :: "'a set" assumes "local.inv C R" shows "sc C (U - R)"
trans: fixes carrier :: "'a set"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "Order_Predicates.preorder_on carrier le"   and "le x y"   and "le y z" shows "le x z"
remdups_removeAll_commute: fixes x :: "'b"   and l :: "'b list" shows "remdups (removeAll x l) = removeAll x (remdups l)"
subset_iff: fixes A :: "'q set"   and B :: "'q set" shows "(A \<subseteq> B) = (\<forall>t. t \<in> A \<longrightarrow> t \<in> B)"
leq_antisym: fixes x :: "'b"   and y :: "'b" assumes "x \<le> y"   and "y \<le> x" shows "x = y"
equalityD2: fixes A :: "'b set"   and B :: "'b set" assumes "A = B" shows "order.greater_eq A B"
alphaAbs_qAbs_imp_alphaAbs_all_equal_or_qFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood X"   and "qAbs xs x X $= qAbs xs' x' X'" shows "alphaAbs_all_equal_or_qFresh xs x X xs' x' X'"
trans: fixes b :: "'a list"   and a :: "'a list"   and c :: "'a list" assumes "subseq b a"   and "subseq c b" shows "subseq c a"
AbstrLevels_S11opt_A82: fixes i :: "AbstrLevelsID" assumes "sS11opt \<in> AbstrLevel i" shows "sA82 \<notin> AbstrLevel i"
imp_def: fixes p :: "'b set"   and q :: "'b set" shows "p \<rightarrow> q = - p \<union> q"