hcju/isabelleps huggingface dataset:
f: shows "f \<in> M \<rightarrow>\<^sub>M X"
inverse_functors_sym: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'d \<Rightarrow> 'c"   and G :: "'c \<Rightarrow> 'd" assumes "inverse_functors A B F G" shows "inverse_functors B A G F"
lossless_def: shows "local.ind_cpa.lossless = (\<lambda>(\<A>1, \<A>2). lossless_gpv \<I>_full \<A>1 \<and> (\<forall>cipher \<sigma>. lossless_gpv \<I>_full (\<A>2 cipher \<sigma>)))"
cJ1_def: shows "cJ1 = initCfgList ! Suc j"
RedWhile: fixes E :: "char list \<Rightarrow> ty option"   and b :: "expr"   and c :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   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" shows "P,E \<turnstile> \<langle>while (b) c,s\<rangle> \<rightarrow> \<langle>if (b) (c;; while (b) c) else unit,s\<rangle>"
size_hp_merge: fixes h1 :: "'b heap"   and h2 :: "'b heap" shows "size_hp (merge h1 h2) = size_hp h1 + size_hp h2"
lookup_zero: fixes k :: "'r" shows "lookup 0 k = (0::'q)"
pochhammer_Suc: fixes a :: "'b"   and n :: "nat" shows "pochhammer a (Suc n) = pochhammer a n * (a + of_nat n)"
IH: assumes "k \<le> n"   and "M k k < \<one>"   and "\<forall>c. 0 < v c"   and "\<forall>c\<in>set cs. v c \<le> n" shows "reset' M n cs v d k k < \<one>"
A1_def: shows "A1 = mat_of_cols (dim_row A) [col A 0]"
retraction_def: fixes g :: "'s" shows "retraction g \<equiv> \<exists>f. ide (g \<cdot> f)"
globally_sound_modes_invariant: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and cms :: "('a \<times> (Mode \<Rightarrow> 'b set)) list"   and mem :: "'b \<Rightarrow> 'c"   and cms' :: "('a \<times> (Mode \<Rightarrow> 'b set)) list"   and mem' :: "'b \<Rightarrow> 'c" assumes "sifum_security stop eval some_val some_val'"   and "sifum_security.globally_sound_mode_use eval (cms, mem)"   and "((cms, mem), cms', mem') \<in> sifum_security.meval eval" shows "sifum_security.globally_sound_mode_use eval (cms', mem')"
bipartite_V: shows "\<^bold>V \<subseteq> A \<Gamma> \<union> B \<Gamma>"
smooth_on_chart_inv: fixes k :: "enat"   and X :: "'c set"   and c :: "('d, 'd) chart"   and f :: "'c \<Rightarrow> 'd" assumes "k-smooth_on X (apply_chart c \<circ> f)"   and "continuous_on X f"   and "c \<in> manifold_eucl.atlas k"   and "open X"   and "f ` X \<subseteq> domain c" shows "k-smooth_on X f"
assms_6: shows "x - y2 \<in> orthogonal_complement W"
le: fixes h :: "'a" assumes "h \<noteq> (0::'a)"   and "norm h < k" shows "norm (f h) \<le> K * norm h"
null_space_orthogonal_complement_row_space_reals: fixes A :: "((real, 'a) vec, 'b) vec" shows "null_space A = real_matrix.v.orthogonal_complement (row_space A)"
consistency_def: fixes C :: "'b fm set set" shows "consistency C \<equiv> \<forall>S\<in>C. (\<forall>p. \<not> (Pro p \<in> S \<and> (\<^bold>\<not> Pro p) \<in> S)) \<and> \<^bold>\<bottom> \<notin> S \<and> (\<forall>Z. (\<^bold>\<not> \<^bold>\<not> Z) \<in> S \<longrightarrow> S \<union> {Z} \<in> C) \<and> (\<forall>A B. (A \<^bold>\<and> B) \<in> S \<longrightarrow> S \<union> {A, B} \<in> C) \<and> (\<forall>A B. (\<^bold>\<not> (A \<^bold>\<or> B)) \<in> S \<longrightarrow> S \<union> {\<^bold>\<not> A, \<^bold>\<not> B} \<in> C) \<and> (\<forall>A B. (A \<^bold>\<or> B) \<in> S \<longrightarrow> S \<union> {A} \<in> C \<or> S \<union> {B} \<in> C) \<and> (\<forall>A B. (\<^bold>\<not> (A \<^bold>\<and> B)) \<in> S \<longrightarrow> S \<union> {\<^bold>\<not> A} \<in> C \<or> S \<union> {\<^bold>\<not> B} \<in> C) \<and> (\<forall>A B. (A \<^bold>\<longrightarrow> B) \<in> S \<longrightarrow> S \<union> {\<^bold>\<not> A} \<in> C \<or> S \<union> {B} \<in> C) \<and> (\<forall>A B. (\<^bold>\<not> (A \<^bold>\<longrightarrow> B)) \<in> S \<longrightarrow> S \<union> {A, \<^bold>\<not> B} \<in> C) \<and> (\<forall>A. tautology A \<longrightarrow> S \<union> {A} \<in> C) \<and> (\<forall>A i. \<not> (K i A \<in> S \<and> (\<^bold>\<not> K i A) \<in> S))"
lexordp_linear: fixes xs :: "'a list"   and ys :: "'a list" shows "lexordp xs ys \<or> xs = ys \<or> lexordp ys xs"
CLSQPS: shows "card LSQPS \<le> 4"
lF_simps_4: fixes f :: "'a" assumes "C.ide f" shows "D.dom (lF f) = F (trg\<^sub>C f) \<star>\<^sub>D F f"
simps: fixes r :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and a1 :: "'d"   and a2 :: "'b" shows "r\<inverse>\<inverse> a1 a2 = (\<exists>a b. a1 = b \<and> a2 = a \<and> r a b)"
low_eq_def: fixes mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" shows "mem\<^sub>1 =\<^sup>l mem\<^sub>2 \<equiv> \<forall>x. dma x = Low \<longrightarrow> mem\<^sub>1 x = mem\<^sub>2 x"
ltl_prop_implies_def: fixes \<phi> :: "'a ltl"   and \<psi> :: "'a ltl" shows "\<phi> \<longrightarrow>\<^sub>P \<psi> \<equiv> \<forall>\<A>. \<A> \<Turnstile>\<^sub>P \<phi> \<longrightarrow> \<A> \<Turnstile>\<^sub>P \<psi>"
simps_1: fixes c :: "'b poly"   and p :: "'b poly" shows "head (CN c 0 p) = head p"
idem: fixes a :: "'c \<times> 'c \<Rightarrow> 'd" shows "a \<oplus> a = a"
simps_2: fixes cs :: "com \<times> (char list \<Rightarrow> int)"   and n :: "nat"   and cs'' :: "com \<times> (char list \<Rightarrow> int)" shows "cs \<rightarrow>(Suc n) cs'' = (\<exists>cs'. cs \<rightarrow> cs' \<and> cs' \<rightarrow>(n) cs'')"
rank_eq_dim_col_space': fixes A :: "(('a, 'c) vec, 'b) vec" shows "vec.dim (col_space A) = rank A"
sel_2: fixes x21 :: "'e"   and x22 :: "('e, 'f) Term.term list" shows "the_Fun (Fun x21 x22) = x21"
inner_mult_left': fixes a :: "'c"   and m :: "real"   and b :: "'c" shows "inner (a * of_real m) b = m * inner a b"
length_nGet1: fixes nReturn :: "'b \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and k :: "'b"   and recv :: "nat \<Rightarrow> 'a Frame list"   and nC :: "'b \<Rightarrow> Config"   and nStore :: "'b \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nSend :: "'b \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "'b \<Rightarrow> nat \<Rightarrow> nat list"   and t :: "nat" assumes "FlexRayController (nReturn k) recv (nC k) (nStore k) (nSend k) (nGet k)" shows "length (nGet k t) \<le> Suc 0"
max_x_le_take: fixes y :: "nat list"   and i :: "nat"   and l :: "nat" assumes "length y \<le> n" shows "max_x y i \<le> max_x (take l y) i"
list_decode_encode: fixes xs :: "nat list" shows "Partial_Recursive.list_decode (Partial_Recursive.list_encode xs) = xs"
simps_1: fixes uu :: "'b set" shows "intersection_list_opt uu [] = []"
unique: fixes x :: "'b"   and i :: "'a"   and j :: "'a" assumes "x \<in> space M"   and "P i x"   and "P j x" shows "i = j"
inject_6: fixes x7 :: "'a OCL_Types.type"   and y7 :: "'a OCL_Types.type" shows "(Bag x7 = Bag y7) = (x7 = y7)"
is_mgu_def: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) set" shows "is_mgu \<sigma> E = (\<sigma> \<in> unifiers E \<and> (\<forall>\<tau>\<in>unifiers E. \<sigma> \<preceq>\<^sub>\<circ> \<tau>))"
simps_1: fixes xs :: "'b SList.list" shows "SList.take xs 0 = []"
assms_1: shows "f \<in> space\<^sub>N (\<LL> p M)"
sorted_wrt_drop: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and n :: "nat" assumes "sorted_wrt f xs" shows "sorted_wrt f (drop n xs)"
u2: shows "u\<^sub>2 \<equiv>\<^sub>U r2"
locally_full: fixes f :: "'a"   and f' :: "'a"   and \<nu> :: "'b" assumes "B.ide f"   and "B.ide f'"   and "src\<^sub>B f = src\<^sub>B f'"   and "trg\<^sub>B f = trg\<^sub>B f'"   and "\<guillemotleft>\<nu> : F f \<Rightarrow>\<^sub>C F f'\<guillemotright>" shows "\<exists>\<mu>. \<guillemotleft>\<mu> : f \<rightarrow>\<^sub>B f'\<guillemotright> \<and> F \<mu> = \<nu>"
Nml_HcompD_7: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Nml (t \<^bold>\<star> u)" shows "Src t = Trg u"
A: fixes i :: "'a" shows "A i \<in> null_sets M \<and> (\<forall>x\<in>space M - A i. f i x = g i x)"
fdia_fbox_de_morgan_2: fixes x :: "'a"   and y :: "'a" shows "ar (\<langle>x| y) = [x| ar y"
inv_atI: fixes p :: "'a mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'a mat"   and m :: "nat" assumes "\<And>i. i < n \<Longrightarrow> p A i m" shows "local.inv_at p A m"
neq_fun_neq_init: fixes f :: "nat \<Rightarrow> nat option"   and g :: "nat \<Rightarrow> nat option" assumes "f \<in> \<R>"   and "g \<in> \<R>"   and "f \<noteq> g" shows "\<exists>n. f \<triangleright> n \<noteq> g \<triangleright> n"
PRoject_def: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and x :: "'c" shows "\<pi>\<^bsub>I,A,x\<^esub> = (\<lambda>f\<in>carr_prodag I A. f x)"
axiom11_F2: fixes x :: "('e, 'f, 'd) F2" shows "|F2set3 x| \<le>o bd_F2"
cancel_div_mod_rules_2: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "plus (plus (times b (divide a b)) (modulo a b)) c = plus a c"
uniform_discrete_subset: fixes S :: "'c set"   and T :: "'c set" assumes "uniform_discrete S"   and "T \<subseteq> S" shows "uniform_discrete T"
hyper_subset_def: fixes A :: "'b set option"   and B :: "'b set option" shows "A \<sqsubseteq> B \<equiv> case B of None \<Rightarrow> True | \<lfloor>B\<rfloor> \<Rightarrow> case A of None \<Rightarrow> False | \<lfloor>A\<rfloor> \<Rightarrow> A \<subseteq> B"
HoVH_def: fixes \<mu> :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" shows "HoVH \<mu> \<equiv> if VVV.arr \<mu> then fst \<mu> \<star> fst (snd \<mu>) \<star> snd (snd \<mu>) else null"
map_poly_coeff_lift_hom: fixes p :: "'a poly" shows "map_poly ((\<lambda>a. [:a:]) \<circ> hom) p = map_poly (map_poly hom) (map_poly (\<lambda>a. [:a:]) p)"
borel_measurable_LIMSEQ_real: fixes M :: "'c measure"   and u :: "nat \<Rightarrow> 'c \<Rightarrow> real"   and u' :: "'c \<Rightarrow> real" assumes "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) \<longlonglongrightarrow> u' x"   and "\<And>i. u i \<in> borel_measurable M" shows "u' \<in> borel_measurable M"
servK_notin_authKeysD: fixes authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "msg"   and A :: "agent"   and evs :: "event list" assumes "Crypt authK \<lbrace>Key servK, Agent B, Ts, Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Ts\<rbrace>\<rbrace> \<in> parts (knows Spy evs)"   and "Key servK \<notin> analz (knows Spy evs)"   and "B \<noteq> Tgs"   and "evs \<in> kerbIV_gets" shows "servK \<notin> authKeys evs"
cong: fixes A :: "'c fset"   and B :: "'c fset"   and g :: "'c \<Rightarrow> 'a"   and h :: "'c \<Rightarrow> 'a" assumes "A = B"   and "\<And>x. x |\<in>| B \<Longrightarrow> g x = h x" shows "F g A = F h B"
ran_restrict_map_def: fixes f :: "'a \<Rightarrow> 'b option"   and B :: "'b set" shows "f\<upharpoonleft>\<^bsub>B\<^esub> = (\<lambda>x. f x \<bind> (\<lambda>v. if v \<in> B then Some v else None))"
chine_hcomp_props_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data" assumes "span_bicategory C prj0 prj1"   and "partial_magma.arr (span_vertical_category.vcomp C) \<mu>"   and "partial_magma.arr (span_vertical_category.vcomp C) \<nu>"   and "span_vertical_category.src C \<nu> = span_vertical_category.trg C \<mu>" shows "category.commutative_square C (Leg0 (Cod \<nu>)) (Leg1 (Cod \<mu>)) (C (Chn \<nu>) (prj1 (Leg0 (Dom \<nu>)) (Leg1 (Dom \<mu>)))) (C (Chn \<mu>) (prj0 (Leg0 (Dom \<nu>)) (Leg1 (Dom \<mu>))))"
valid_Disj: fixes xset :: "('a, 'pred, 'act) formula set['a]"   and P :: "'state" assumes "finite (supp xset)" shows "P \<Turnstile> Disj xset = (\<exists>x\<in>set_bset xset. P \<Turnstile> x)"
conv_involutive: fixes x :: "'a" shows "x\<^sup>T\<^sup>T = x"
fps_mult_of_nat_nth_2: fixes f :: "'a fps"   and k :: "nat"   and n :: "nat" shows "fps_nth (f * of_nat k) n = fps_nth f n * of_nat k"
max_less_iff_conj: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.linorder less_eq less" shows "less (ord.max less_eq x y) z = (less x z \<and> less y z)"
unwrapForall': fixes j :: "nat"   and A :: "atom fm"   and L :: "real list" shows "eval (AllN (j + 1) A) L = eval (AllN j (AllQ A)) L"
init_reds_sync_unit': fixes e\<^sub>0 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and v' :: "val"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and e' :: "char list exp" assumes "P \<turnstile> \<langle>e\<^sub>0,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v',s\<^sub>1,b\<^sub>1\<rangle>"   and "init_exp_of e\<^sub>0 = \<lfloor>unit\<rfloor>"   and "INIT_class e\<^sub>0 = \<lfloor>C\<rfloor>"   and "\<not> sub_RI e'" shows "P \<turnstile> \<langle>init_switch e\<^sub>0 e',s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>e',s\<^sub>1,icheck P (the (INIT_class e\<^sub>0)) e'\<rangle>"
i_aggregate_1: fixes ag :: "'b list \<Rightarrow> 'b"   and f :: "nat \<Rightarrow> 'b" assumes "\<And>x. ag [x] = x" shows "i_aggregate f (Suc 0) ag = f"
c1: shows "dim_col A \<Otimes> B\<^sup>\<dagger> = dim_col (A\<^sup>\<dagger> \<Otimes> B\<^sup>\<dagger>)"
l4_17: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p"   and Q :: "'p" assumes "A \<noteq> B"   and "Col A B C"   and "Cong A P A Q"   and "Cong B P B Q" shows "Cong C P C Q"
seqr_inf_distl: fixes P :: "('a, 'c) urel"   and Q :: "('a, 'c) urel"   and R :: "('c, 'b) urel" shows "P \<sqinter> Q ;; R = (P ;; R) \<sqinter> (Q ;; R)"
eq_nextl_class_in_left_lang_im: fixes u :: "'a list" shows "eq_nextl `` {u} \<in> left_lang ` states M"
assms: shows "{0..} \<subseteq> existence_ivl0 x"
simps_1: fixes s1 :: "'s1"   and s2 :: "'s2"   and a :: "'a" shows "local.parallel_oracle (s1, s2) (Inl a) = map_spmf (map_prod Inl (\<lambda>s1'. (s1', s2))) (left s1 a)"
rerr_def: shows "rerr \<equiv> \<lambda>(dests, sip). Rerr dests sip"
insert_remove: fixes A :: "'a word set"   and x :: "'a word" assumes "finite A" shows "signed.Min (insert x A) = (if A - {x} = {} then x else signed.min x (signed.Min (A - {x})))"
from_hma\<^sub>v_def: fixes v :: "('d, 'e) Finite_Cartesian_Product.vec" shows "HMA_Connect.from_hma\<^sub>v v = Matrix.vec CARD('e) (\<lambda>i. v $h Bij_Nat.from_nat i)"
assms_4: shows "((\<xi>, p), a, \<xi>', q) \<in> automaton.trans A"
assms: shows "\<turnstile> $P \<and> P$ \<longrightarrow> A"
attractor_step_mono: fixes W :: "'a set"   and p :: "Player" shows "mono (\<lambda>S. W \<union> S \<union> directly_attracted p S)"
distinct_227: fixes x91 :: "floatarith"   and x92 :: "floatarith"   and x151 :: "floatarith"   and x152 :: "nat" shows "floatarith.Min x91 x92 \<noteq> Power x151 x152"
step: fixes s :: "'b" assumes "list.has_next g s"   and "\<not> P (fst (list.next g s))"   and "Q (snd (list.next g s))" shows "Q s"
compat_NS_S_point: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "(x, y) \<in> NS"   and "(y, z) \<in> S" shows "(x, z) \<in> S"
toFull_def: fixes \<sigma> :: "'tp" shows "toFull \<sigma> \<equiv> SOME F. bij_betw F {a. intT \<sigma> a} UNIV"
walk_rev: fixes G :: "('c, 'd) Graph_scheme"   and xs :: "'c list" assumes "Graph G"   and "Graph.walk G xs" shows "Graph.walk G (rev xs)"
vc_sound: fixes C :: "acom"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and upds :: "(char list \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat)) list" assumes "vc C Q"   and "finite (support Q)"   and "finite (varacom C)"   and "lesvars upds \<inter> varacom C = {}"   and "distinct (map fst upds)" shows "\<turnstile>\<^sub>1 {\<lambda>l s. pre C Q l s \<and> preList upds C l s} strip C { time C \<Down> \<lambda>l s. Q l s \<and> postList upds l s} \<and> (\<forall>l s. pre C Q l s \<longrightarrow> Q l (postQ C s))"
accfield_def: fixes G :: "prog"   and S :: "qtname"   and C :: "qtname" shows "accfield G S C = Let (table_of (map (\<lambda>((n, d), f). (n, d, f)) (DeclConcepts.fields G C))) (filter_tab (\<lambda>n (declC, f). G \<turnstile> (declC, fdecl (n, f)) of C accessible_from S))"
insert_set_delete_index: fixes n :: "nat"   and xs :: "'b list" assumes "n < length xs" shows "insert (xs ! n) (set (delete_index n xs)) = set xs"
\<T>o_def: fixes e :: "'s"   and b :: "'c" shows "\<T>o e b = S.mkArr (Hom.set (b, a)) (Ya'.SET b) (\<lambda>x. Ya'.FUN (\<psi> (b, a) x) e)"
axioms_1: fixes F :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "idem_overrider F" shows "overrider F"
compxE2_stack_xlift_convs: fixes d :: "nat"   and e :: "'a expr1"   and pc :: "nat" assumes "0 < d" shows "compxE2 e pc d = stack_xlift d (compxE2 e pc 0)"
cross_zero_right: fixes x :: "(real, 3) vec" shows "x \<times> 0 = 0"
dual_anti: fixes x :: "'b"   and y :: "'b" assumes "x \<le> y" shows "\<partial> y \<le> \<partial> x"
cases: fixes a1 :: "init_state"   and a2 :: "init_state" assumes "a1 \<le>\<^sub>i a2" obtains i :: "init_state" where "a1 = Prepared" and "a2 = i__"   | "a1 = Processing" and "a2 = Done"   | "a1 = Processing" and "a2 = Error"   | i :: "init_state" where "a1 = i" and "a2 = i"
e_hs_aux: shows "0 < e \<and> length hs = k \<and> (\<forall>h\<in>set hs. h \<in> O(\<lambda>x. real x / ln (real x) powr (1 + e))) \<and> (\<forall>t\<in>set ts. \<forall>x\<ge>x\<^sub>1. x\<^sub>0 \<le> t x \<and> t x < x) \<and> (\<forall>i<k. \<forall>x\<ge>x\<^sub>1. bs ! i * real x + (hs ! i) x = real ((ts ! i) x))"
add_diff_cancel: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.group_add minus plus zero uminus" shows "minus (plus a b) b = a"
C_def: shows "C \<equiv> \<lambda>r. {x \<in> topspace (Euclidean_space n). enorm x \<le> r}"
Union_least: fixes A :: "'c set set"   and C :: "'c set" assumes "\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C" shows "\<Union> A \<subseteq> C"
less2: shows "\<^bold>|v \<and>\<^sub>p u\<^bold>| < \<^bold>|v\<^bold>|"
A: shows "limsup (\<lambda>n. ereal (root n (norm (c ^ n * f n)))) = ereal (norm c) * limsup (\<lambda>n. ereal (root n (norm (f n))))"
eventually_at_infinity_pos: fixes p :: "'a \<Rightarrow> bool" shows "eventually p at_infinity = (\<exists>b>0. \<forall>x. b \<le> norm x \<longrightarrow> p x)"
eval_nembed_false: fixes P :: "'c \<Rightarrow> bool"   and s :: "'c" assumes "\<not> P s" shows "\<guillemotleft> \<N> P \<guillemotright> s = 1"
local_steps_def: fixes T :: "((('g \<Rightarrow> 'h) \<times> 'i) \<times> 'j \<times> ('g \<Rightarrow> 'k) \<times> 'l) set"   and J :: "'g set" shows "local_steps T J \<equiv> \<forall>\<sigma> \<zeta> s a \<sigma>' s'. ((\<sigma>, s), a, \<sigma>', s') \<in> T \<and> (\<forall>j\<in>J. \<zeta> j = \<sigma> j) \<longrightarrow> (\<exists>\<zeta>'. (\<forall>j\<in>J. \<zeta>' j = \<sigma>' j) \<and> ((\<zeta>, s), a, \<zeta>', s') \<in> T)"
invariantQCharacterizationAfterApplyBackjump_2: fixes state :: "State"   and F0 :: "Literal list list" assumes "InvariantConsistent (getM state)"   and "InvariantUniq (getM state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"   and "InvariantConflictFlagCharacterization (getConflictFlag state) (getF state) (getM state)"   and "InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)"   and "InvariantUniqC (getC state)"   and "getC state \<noteq> [opposite (getCl state)]"   and "InvariantNoDecisionsWhenUnit (butlast (getF state)) (getM state) (currentLevel (getM state))"   and "InvariantNoDecisionsWhenConflict (butlast (getF state)) (getM state) (currentLevel (getM state))"   and "getF state \<noteq> []"   and "last (getF state) = getC state"   and "getConflictFlag state"   and "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"   and "InvariantCEntailed (getConflictFlag state) F0 (getC state)"   and "InvariantClCharacterization (getCl state) (getC state) (getM state)"   and "InvariantCllCharacterization (getCl state) (getCll state) (getC state) (getM state)"   and "InvariantClCurrentLevel (getCl state) (getM state)"   and "0 < currentLevel (getM state)"   and "isUIP (opposite (getCl state)) (getC state) (getM state)" shows "let state'' = applyBackjump state in InvariantQCharacterization (getConflictFlag state'') (getQ state'') (getF state'') (getM state'')"
cos_arg_i_mult_zero: fixes y :: "complex" assumes "y \<noteq> 0"   and "is_imag y" shows "cos (arg y) = 0"
trivial_RModule: assumes "\<And>r. r \<in> FG \<Longrightarrow> r \<cdot> (0::'v) = (0::'v)" shows "RModule FG (\<cdot>) 0"
padic_integers_def: fixes p :: "int" shows "padic_integers p \<equiv> prime p"
nonzero_divide_mult_cancel_right: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c" assumes "class.field divide inverse times one plus zero minus uminus"   and "b \<noteq> zero" shows "divide b (times a b) = divide one a"
binrelchain_append_reduce1: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "binrelchain P (xs @ ys)" shows "binrelchain P xs"
unique_factorization_m_alt_def: fixes f :: "int poly"   and cfs :: "int \<times> int poly multiset" shows "pl.unique_factorization_m f cfs = (pl.factorization_m f cfs \<and> (\<forall>dgs. pl.factorization_m f dgs \<longrightarrow> pl.Mf dgs = pl.Mf cfs))"
l_get_root_node_wf_def: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog" shows "l_get_root_node_wf heap_is_wellformed get_root_node type_wf known_ptr known_ptrs get_ancestors get_parent \<equiv> l_known_ptrs known_ptr known_ptrs \<and> l_get_root_node_wf_axioms heap_is_wellformed get_root_node type_wf known_ptrs get_ancestors get_parent"
VSGeneralBlocksLimited_def: shows "VSGeneralBlocksLimited = opt_group \<circ> QE_dnf opt_group (step_augment gen_qe IdentityHeuristic) \<circ> VSLuckiestBlocks \<circ> opt_group"
simps_4: shows "c\<^sub>c \<noteq> a\<^sub>c"
fin_set_lexord_def: fixes R :: "('b \<times> 'b) set" shows "fin_set_lexord R = {(A, B). finite A \<and> finite B \<and> (\<exists>xs ys. is_sorted_list_of_set_by R A xs \<and> is_sorted_list_of_set_by R B ys \<and> (xs, ys) \<in> lexord R)}"