hcju/isabelleps huggingface dataset:
find_ce_code_def: fixes node_eq_impl :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and node_hash_impl :: "nat \<Rightarrow> 'c \<Rightarrow> nat"   and node_def_hash_size :: "nat"   and G_impl :: "('c \<Rightarrow> bool, 'c \<Rightarrow> 'c list, 'c list, 'c \<Rightarrow> integer, 'd) gen_igbg_impl_scheme" shows "find_ce_code node_eq_impl node_hash_impl node_def_hash_size G_impl \<equiv> stat_start_nres \<bind> (\<lambda>x. let y' = goinitial_code node_def_hash_size in LIST_FOREACH' (RETURN (id (gi_V0 G_impl))) go_is_no_brk_code (\<lambda>xa xb. if \<not> go_is_done_code node_eq_impl node_hash_impl xa xb then let y'a = (None, ginitial_code node_eq_impl node_hash_impl G_impl xa xb) in WHILE\<^sub>T (\<lambda>(xd, xe). is_None xd \<and> \<not> gpath_is_empty_code xe) (\<lambda>(xd, xe). gselect_edge_code node_eq_impl xe \<bind> (\<lambda>(xf, xg). case xf of None \<Rightarrow> gpop_code node_eq_impl node_hash_impl xg \<bind> (\<lambda>xh. RETURN (None, xh)) | Some xh \<Rightarrow> if gis_on_stack_code node_eq_impl node_hash_impl xh xg then gcollapse_code node_eq_impl node_hash_impl xh xg \<bind> (\<lambda>xi. last_is_acc_code G_impl xi \<bind> (\<lambda>xj. if xj then ce_code node_eq_impl node_hash_impl xi else RETURN (None, xi))) else if \<not> gis_done_code node_eq_impl node_hash_impl xh xg then RETURN (None, gpush_code node_eq_impl node_hash_impl G_impl xh xg) else RETURN (None, xg))) y'a \<bind> (\<lambda>(xc, xd). RETURN (gto_outer_code xc xd)) else RETURN xb) y' \<bind> (\<lambda>xa. stat_stop_nres \<bind> (\<lambda>xb. RETURN (goBrk_code xa))))"
composites_are_arrows_1: shows "hseq (cod \<mu>) (cod \<nu> \<star> cod \<pi>)"
foldr_conv_foldl: fixes f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and a :: "'d" shows "foldr f xs a = foldl (\<lambda>x y. f y x) a (rev xs)"
ccInf_empty: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf {} = top"
cyclic_on_perm_rem: fixes f :: "'c \<Rightarrow> 'c"   and S :: "'c set"   and x :: "'c" assumes "cyclic_on f S"   and "bij f"   and "S \<noteq> {x}" shows "cyclic_on (perm_rem x f) (S - {x})"
simps_3: fixes m :: "bool"   and x :: "'b" shows "strip3 (Atom3 m x) = Atom (m, x)"
matchrel_def: shows "matchrel = {(x, y). MATCH1 x y}"
padic_a_inv: fixes p :: "int"   and a :: "nat \<Rightarrow> int" assumes "prime p"   and "a \<in> carrier (padic_int p)" shows "\<ominus>\<^bsub>padic_int p\<^esub> a = (\<lambda>n. \<ominus>\<^bsub>residue_ring (p ^ n)\<^esub> a n)"
g_def: shows "g \<equiv> \<lambda>t. f t - ((\<Sum>m<n. diff m 0 / fact m * t ^ m) + B * (t ^ n / fact n))"
Ide_Nmlize_Can: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Can V src trg t" shows "bicategorical_language.Ide V src trg (bicategorical_language.Nmlize V t)"
smcf_comp_is_semifunctor: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<GG> :: "V"   and \<AA> :: "V"   and \<FF> :: "V" assumes "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^bsub>\<alpha>\<^esub> \<CC>"
to_mtx_inject: fixes x :: "((real, 'a) vec, 'a) vec"   and y :: "((real, 'a) vec, 'a) vec" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "(to_mtx x = to_mtx y) = (x = y)"
wt_wt': fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and T :: "ty" assumes "P,E,h \<turnstile> e : T" shows "P,E,h \<turnstile> e :' T"
fresh_prodD_1: fixes a :: "'b"   and x :: "'c"   and y :: "'d" assumes "a \<sharp> (x, y)" shows "a \<sharp> x"
SUP_least: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e \<times> 'f"   and u :: "'e \<times> 'f" assumes "\<And>i. i \<in> A \<Longrightarrow> less_eq_prod (f i) u" shows "less_eq_prod (Sup_prod (f ` A)) u"
one_less_power: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less one a"   and "0 < n" shows "less one (power.power one times a n)"
graph_ops_5: fixes G :: "'c graph"   and E :: "('c \<times> 'c) set" shows "delete_edges G E = \<lparr>nodes = nodes G, edges = edges G - E\<rparr>"
dg_dghm_id_is_dghm: shows "dghm_id (dg_Rel \<alpha>) : dg_Rel \<alpha> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> dg_Rel \<alpha>"
partList4_distinct: fixes ts :: "'b set list"   and s :: "'b set" assumes "{} \<notin> set ts"   and "disjoint_list ts" shows "distinct (partList4 s ts)"
disj_Domain_imp_disj_Image: fixes R :: "('a \<times> 'b) set"   and X :: "'a set"   and Y :: "'a set" assumes "Domain R \<inter> X \<inter> Y = {}"   and "runiq R"   and "runiq (R\<inverse>)" shows "R `` X \<inter> R `` Y = {}"
fbox_simp: fixes x :: "'a"   and y :: "'a" shows "antirange_semiring.bbox (\<cdot>) ar x y = antirange_semiring.bbox (\<cdot>) ar x (r y)"
intro: fixes M :: "'d measure"   and b :: "real" assumes "prob_space M"   and "information_space_axioms b" shows "information_space M b"
assms_3: shows "\<forall>ml. match_list \<gamma> (map (Match \<circ> C) (f ml)) a p = matches \<gamma> (alist_and (NegPos_map C ml)) a p"
simps_2: fixes n :: "nat"   and c :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val)"   and c' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val)" shows "c \<rightarrow>\<^bsub>Suc n\<^esub> c' = (\<exists>c''. c \<rightarrow>\<^bsub>n\<^esub> c'' \<and> c'' \<rightarrow> c')"
l_one: fixes x :: "'a vec" assumes "x \<in> carrier_vec n" shows "0\<^sub>v n + x = x"
bij: shows "bij_betw f (uverts G\<^sub>1) (uverts G\<^sub>2)"
sep_conj_impl: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and h :: "'a"   and P' :: "'a \<Rightarrow> bool"   and Q' :: "'a \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj"   and "sep_algebra.sep_conj plus sep_disj P Q h"   and "\<And>h. P h \<Longrightarrow> P' h"   and "\<And>h. Q h \<Longrightarrow> Q' h" shows "sep_algebra.sep_conj plus sep_disj P' Q' h"
thm_relation_negation_4: fixes v :: "i"   and p :: "\<o>" shows "[\<^bold>\<not>p\<^sup>- \<^bold>\<equiv> p in v]"
nonsimple_length_gt_1: fixes xs :: "'b list" assumes "xs \<noteq> []"   and "hd xs \<noteq> last xs" shows "1 < length xs"
DSourcesA81_L1: shows "DSources level1 sA81 = {sA71, sA91}"
sources_eq_domains: fixes \<mu> :: "'a" shows "sources \<mu> = H.domains \<mu>"
mask_AND_NOT_mask: fixes w :: "'b word"   and n :: "nat" shows "(w AND NOT (mask n)) AND mask n = 0"
connected_iff_interval: fixes U :: "'b set" shows "connected U = (\<forall>x\<in>U. \<forall>y\<in>U. \<forall>z\<ge>x. z \<le> y \<longrightarrow> z \<in> U)"
\<delta>_states_def: fixes \<delta> :: "('a, 'b) ta_rule set" shows "\<delta>_states \<delta> \<equiv> \<Union> (rule_states ` \<delta>)"
assms: shows "sgnx (poly p) (at_left a) \<noteq> sgnx (poly p) at_bot"
set_integrable_empty: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" shows "set_integrable M {} f"
g_sel_def: fixes m :: "'s"   and P :: "'k \<times> 'v \<Rightarrow> bool" shows "g_sel m P \<equiv> iteratei m (\<lambda>\<sigma>. \<sigma> = None) (\<lambda>x \<sigma>. if P x then Some x else None) None"
uadmit_prog_ntadjoint: fixes \<sigma> :: "'b \<Rightarrow> ('sf, 'sz) trm"   and \<alpha> :: "('sf + 'b, 'sc, 'sz) hp"   and U :: "('sz + 'sz) set"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and I :: "('sf, 'sc, 'sz) interp" assumes "PUadmitFO \<sigma> \<alpha> U"   and "Vagree \<nu> \<omega> (- U)"   and "\<And>i. dsafe (\<sigma> i)"   and "hpsafe \<alpha>"   and "is_interp I" shows "prog_sem (adjointFO I \<sigma> \<nu>) \<alpha> = prog_sem (adjointFO I \<sigma> \<omega>) \<alpha>"
assms_5: shows "\<I> \<turnstile>g \<A> \<eta> \<surd>"
to_nat_less_card: fixes j :: "'a" shows "mod_type_class.to_nat j < CARD('a)"
that: fixes p :: "(name \<times> name) list" assumes "set p \<subseteq> set xvec \<times> set (p \<bullet> xvec)"   and "distinctPerm p"   and "yvec = p \<bullet> xvec"   and "\<Psi>' = p \<bullet> \<Psi>" shows "thesis"
nsqn\<^sub>r_lte_dsn: fixes dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat"   and pre :: "nat set" shows "nsqn\<^sub>r (dsn, dsk, flag, hops, nhip, pre) \<le> dsn"
more_than_one_setD: fixes x :: "'b"   and p :: "'b list" assumes "\<not> one_call_in_path x p" shows "x \<in> set p"
tbisim: fixes t' :: "'t" assumes "t' \<noteq> t" shows "tbisim (ws t' = None) t' (ts1 t') m1 (ts2 t') m2"
prob_space: fixes i :: "'a" assumes "i \<in> I" shows "prob_space (M i)"
enum_less: fixes a :: "nat \<Rightarrow> nat"   and i :: "nat" assumes "a \<in> s"   and "i < n" shows "(enum i < a) = (enum (Suc i) \<le> a)"
bottom_lower: fixes x :: "'a" assumes "x \<in> carrier L" shows "\<bottom>\<^bsub>L\<^esub> \<sqsubseteq>\<^bsub>L\<^esub> x"
subst_apply_labeled_stateful_strand_def: fixes S :: "('a strand_label \<times> ('b, 'c) stateful_strand_step) list"   and \<theta> :: "'c \<Rightarrow> ('b, 'c) Term.term" shows "S \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta> \<equiv> map (\<lambda>x. x \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) S"
unique: shows "\<exists>id. \<forall>k id'. \<parallel>id'\<parallel>\<^bsub>k\<^esub> \<longrightarrow> id = id'"
merge_property: fixes t1 :: "'b list"   and A :: "'b set"   and t2 :: "'b list"   and B :: "'b set" assumes "set t1 \<subseteq> A"   and "set t2 \<subseteq> B"   and "t1 \<upharpoonleft> B = t2 \<upharpoonleft> A" shows "let t = merge A B t1 t2 in t \<upharpoonleft> A = t1 \<and> t \<upharpoonleft> B = t2 \<and> set t \<subseteq> set t1 \<union> set t2"
u_3: fixes y :: "'a" assumes "y \<in> cball x e" shows "s (flow0 y (u y)) = 0"
ccProd_mono1: fixes S' :: "Vars.var set"   and S'' :: "Vars.var set"   and S :: "Vars.var set" assumes "S' \<subseteq> S''" shows "S' G\<times> S \<sqsubseteq> S'' G\<times> S"
vcons_vdoubleton: fixes a :: "V"   and b :: "V" shows "[a, b]\<^sub>\<circ> = ZFC_in_HOL.set {\<langle>0\<^sub>\<nat>, a\<rangle>, \<langle>1\<^sub>\<nat>, b\<rangle>}"
one_reorient: fixes x :: "'a" shows "((1::'a) = x) = (x = (1::'a))"
CatIdCompId: fixes C :: "('a, 'b, 'c) Category_scheme"   and X :: "'a" assumes "Category C"   and "X \<in> obj\<^bsub>C\<^esub>" shows "id\<^bsub>C\<^esub> X ;;\<^bsub>C\<^esub> id\<^bsub>C\<^esub> X = id\<^bsub>C\<^esub> X"
closed_f: fixes A :: "'a set"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "Semilat A r f"   and "x \<in> A"   and "y \<in> A" shows "x \<squnion>\<^bsub>f\<^esub> y \<in> A"
gdpr_states_def: shows "gdpr_states \<equiv> {I. gdpr_scenario \<rightarrow>\<^sub>i* I}"
int_inc: fixes n :: "nat" shows "integral (cbox a b) (f n) \<le> integral (cbox a b) (f (Suc n))"
mset_subset_eq_mono_add_right_cancel: fixes A :: "'a multiset"   and C :: "'a multiset"   and B :: "'a multiset" shows "(A + C \<subseteq># B + C) = (A \<subseteq># B)"
flush_all_until_volatile_write_append_non_volatile_write_commute: fixes sb :: "'b memref list"   and i :: "nat"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and p :: "'b"   and "is" :: "instr list"   and xs :: "nat \<Rightarrow> nat option"   and \<D> :: "'c"   and \<O> :: "'d"   and \<R> :: "'e"   and a :: "nat"   and p' :: "'b"   and is' :: "instr list"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and v :: "nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and \<D>' :: "'c"   and \<R>' :: "'e"   and m :: "nat \<Rightarrow> nat" assumes "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb = {}"   and "i < length ts"   and "ts ! i = (p, is, xs, sb, \<D>, \<O>, \<R>)"   and "\<forall>j<length ts. i \<noteq> j \<longrightarrow> (let (uu_, uu_, uu_, sb\<^sub>j, uu_, uu_, uu_) = ts ! j in a \<notin> outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>j))" shows "flush_all_until_volatile_write (ts[i := (p', is', xs, sb @ [Write\<^sub>s\<^sub>b False a sop v A L R W], \<D>', \<O>, \<R>')]) m = (flush_all_until_volatile_write ts m)(a := v)"
dim: fixes n :: "nat" shows "f n \<in> carrier_mat dim dim"
primeidealCD: assumes "cring R"   and "\<not> primeideal I R" shows "carrier R = I \<or> (\<exists>a b. a \<in> carrier R \<and> b \<in> carrier R \<and> a \<otimes> b \<in> I \<and> a \<notin> I \<and> b \<notin> I)"
sees_methods_is_class_Object: fixes P :: "'a prog"   and D :: "char list"   and Mm :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> 'a) \<times> char list) option" assumes "P \<turnstile> D sees_methods Mm" shows "is_class P Object"
u1: shows "x \<in> B \<inter> (*\<^sub>C) c ` B"
instrs_append_Read\<^sub>s\<^sub>b: fixes sb :: "'b memref list"   and volatile :: "bool"   and a :: "nat"   and t :: "nat"   and v :: "nat" shows "instrs (sb @ [Read\<^sub>s\<^sub>b volatile a t v]) = instrs sb @ [Read volatile a t]"
Follows_Un_lemma: fixes F :: "'b program"   and f :: "'b \<Rightarrow> 'c set"   and g :: "'b \<Rightarrow> 'c set"   and g' :: "'b \<Rightarrow> 'd"   and f' :: "'b \<Rightarrow> 'c set"   and k :: "'c set" assumes "F \<in> Increasing f"   and "F \<in> Increasing g"   and "F \<in> Increasing g'"   and "F \<in> Always {s. f' s \<subseteq> f s}"   and "\<forall>k. F \<in> {s. k \<subseteq> f s} \<longmapsto>w {s. k \<subseteq> f' s}" shows "F \<in> {s. k \<subseteq> f s \<union> g s} \<longmapsto>w {s. k \<subseteq> f' s \<union> g s}"
host_shadow_root_rel_shadow_root: fixes h :: "(_) heap"   and host :: "(_) element_ptr"   and shadow_root_option :: "(_) shadow_root_ptr option"   and shadow_root :: "(_) shadow_root_ptr" assumes "h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r shadow_root_option" shows "(shadow_root_option = Some shadow_root) = ((cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r host, cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root) \<in> local.a_host_shadow_root_rel h)"
mult_minus_left: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.ring plus zero minus uminus times" shows "times (uminus a) b = uminus (times a b)"
in_set_conv_decomp_first: fixes x :: "'a"   and xs :: "'a list" shows "(x \<in> list.set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> list.set ys)"
less_fract: fixes b :: "'b"   and d :: "'b"   and a :: "'b"   and c :: "'b" assumes "b \<noteq> (0::'b)"   and "d \<noteq> (0::'b)" shows "(Fract a b < Fract c d) = (a * d * (b * d) < c * b * (b * d))"
cases: fixes a1 :: "value"   and a2 :: "value" assumes "a1 \<approx>\<^sub>e a2" obtains ts :: "value list" and us :: "value list" and name :: "name" where "a1 = Vconstr namea__ ts" and "a2 = Vconstr namea__ us" and "list_all2 erelated ts us"   | cs :: "(Term_Class.term \<times> sterm) list" and \<Gamma>\<^sub>1 :: "(name, value) fmap" and \<Gamma>\<^sub>2 :: "(name, value) fmap" where "a1 = Vabs cs \<Gamma>\<^sub>1__" and "a2 = Vabs cs \<Gamma>\<^sub>2__" and "fmrel_on_fset (ids (Sabs cs)) erelated \<Gamma>\<^sub>1__ \<Gamma>\<^sub>2__"   | \<Gamma>\<^sub>1 :: "(name, value) fmap" and \<Gamma>\<^sub>2 :: "(name, value) fmap" and css :: "(name, (Term_Class.term \<times> sterm) list) fmap" and name :: "name" where "a1 = Vrecabs css name \<Gamma>\<^sub>1" and "a2 = Vrecabs css name \<Gamma>\<^sub>2" and "pred_fmap (\<lambda>cs. fmrel_on_fset (ids (Sabs cs)) erelated \<Gamma>\<^sub>1 \<Gamma>\<^sub>2) css"
\<delta>complete_def: fixes TA :: "('a, 'b) tree_automaton_rec"   and A :: "'b \<Rightarrow> nat option" assumes "det_tree_automaton TA A" shows "det_tree_automaton.\<delta>complete TA A \<equiv> remap_rule Some ` ta_rules TA \<union> {None \<rightarrow> f qs |f qs. A f = Some (length qs) \<and> qs \<in> lists (det_tree_automaton.Qcomplete TA) \<and> (\<nexists>qo qso. qo \<rightarrow> f qso \<in> ta_rules TA \<and> qs = map Some qso)}"
reduction_words_inf: fixes q :: "'state"   and w :: "'action stream" assumes "q \<in> nodes"   and "R.run w q" shows "run w q"
check_poly_gt_def: fixes gt :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "('d monom \<times> 'c) list"   and q :: "('d monom \<times> 'c) list" shows "check_poly_gt gt p q \<equiv> let (a1, p1) = poly_split 1 p; (b1, q1) = poly_split 1 q in gt a1 b1 \<and> check_poly_ge p1 q1"
invariant_1_2_poly_cond2: fixes rai :: "int poly \<times> rat \<times> rat" assumes "invariant_1_2 rai" shows "poly_cond2 (poly_real_alg_1 rai)"
delete_first_simps_1: fixes P :: "'a \<Rightarrow> bool" shows "delete_first P [] = None"
existence_ivl_empty_iff: fixes x0 :: "'a" shows "(existence_ivl t0 x0 = {}) = (t0 \<notin> T \<or> x0 \<notin> X)"
lookup: fixes x :: "'c \<Rightarrow>\<^sub>0 'd" shows "lookup x \<in> {f. finite {x. f x \<noteq> (0::'d)}}"
simps_41: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "Stop \<noteq> x41 ;; x42"
dependent_with: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd" assumes "module scale" shows "module.dependent scale = dependent_with (+) (0::'d) scale"
subsequence_shift: fixes i :: "nat"   and w :: "nat \<Rightarrow> 'b"   and j :: "nat"   and k :: "nat" shows "suffix i w [j \<rightarrow> k] = w [i + j \<rightarrow> i + k]"
least: fixes A :: "'b set"   and x :: "'b"   and b :: "'b" assumes "lub A x"   and "\<And>a. a \<in> A \<Longrightarrow> a \<le> b" shows "x \<le> b"
F_LESS: fixes a :: "'b" assumes "a \<in> A" shows "|F a| <o r"
refl_ge_eq: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>x. R x x" shows "(=) \<le> R"
not_kD_sqn: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "dip \<notin> kD rt" shows "sqn rt dip = 0"
satPB_snvPB_transE: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "I.wtE \<xi>" shows "I.satPB \<xi> \<Phi>"
ars: fixes x :: "'a"   and y :: "'a" shows "r (r x \<cdot> y) = r (x \<cdot> y)"
CapSS: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and Ops :: "('b \<Rightarrow> 'b) set"   and tenc :: "('b \<Rightarrow> 'b) \<Rightarrow> 'b"   and F :: "('b \<Rightarrow> 'b) set"   and encF :: "('b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b"   and N :: "'b \<Rightarrow> 'b"   and ssap :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and \<psi> :: "'c"   and f :: "'b \<Rightarrow> 'b" assumes "Jeroslow_Diagonalization var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num prv enc Ops tenc F encF N ssap"   and "\<psi> \<in> fmla"   and "Fvars \<psi> = {Generic_Syntax.Variable var 10}"   and "f \<in> F" shows "ssap \<psi> (tenc (encF f)) = enc (Generic_Syntax.inst var subst \<psi> (encF f (tenc (encF f))))"
rearrange_real_euclid_B: fixes y :: "(real, 'a) vec"   and w :: "(real, 'a) vec"   and h :: "real"   and z :: "(real, 'a) vec" shows "(y - w = h *\<^sub>R (z - w)) = (y = h *\<^sub>R z + (1 - h) *\<^sub>R w)"
cs_order_le: fixes \<pi> :: "nat \<Rightarrow> 'n"   and \<pi>' :: "nat \<Rightarrow> 'n"   and i :: "nat"   and i' :: "nat"   and j :: "nat"   and j' :: "nat" assumes "is_path \<pi>"   and "is_path \<pi>'"   and "cs\<^bsup>\<pi>\<^esup> i = cs\<^bsup>\<pi>'\<^esup> i'"   and "cs\<^bsup>\<pi>\<^esup> j = cs\<^bsup>\<pi>'\<^esup> j'"   and "\<pi> i \<noteq> local.return"   and "i \<le> j" shows "i' \<le> j'"
foldseq_zerotail2: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and s :: "nat \<Rightarrow> 'b"   and m :: "nat" assumes "\<forall>x. f x (0::'b) = x"   and "\<forall>i>n. s i = (0::'b)"   and "n \<le> m" shows "foldseq f s n = foldseq f s m"
lconjseq_reflections: fixes ss :: "'w list" assumes "ss \<in> lists S" shows "order.greater_eq (\<Union>w\<in>W. lconjby w ` S) (set (lconjseq ss))"
conj_chaos_left: fixes a :: "'a::refinement_lattice" shows "chaos \<iinter> (a::'a::refinement_lattice) = a"
construct_rhs_vector_def: fixes p :: "real poly"   and qs :: "real poly list"   and Is :: "nat list list" shows "construct_rhs_vector p qs Is = vec_of_list (map (\<lambda>I. construct_NofI p (retrieve_polys qs I)) Is)"
prob_\<mu>_\<Delta>\<^sub>d: fixes M :: "'a measure"   and I :: "'b set"   and A :: "'b \<Rightarrow> 'a set" assumes "prob_space_with_indicators M I A" shows "\<P>(a in M. prob_space_with_indicators.Y I A a = 0) \<le> 1 / prob_space_with_indicators.\<mu> M I A + prob_space_with_indicators.\<Delta>\<^sub>d M I A / (prob_space_with_indicators.\<mu> M I A)\<^sup>2"
smult_monom_mult: fixes a :: "'a"   and b :: "'a"   and n :: "nat"   and f :: "'a poly" shows "Polynomial.smult a (Polynomial.monom b n * f) = Polynomial.monom (a * b) n * f"
step: fixes x :: "'a node \<times> 'a node set" assumes "all_subfrmls (fst x) \<subseteq> B \<and> (\<forall>nd\<in>snd x. all_subfrmls nd \<subseteq> B)" shows "f x \<le> SPEC (\<lambda>r. \<forall>nd\<in>snd r. all_subfrmls nd \<subseteq> B)"
differentiable_on_eq_differentiable_at: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "open s" shows "(f differentiable_on s) = (\<forall>x\<in>s. f differentiable at x)"
simps_2: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v"   and y' :: "'v"   and y :: "'v" shows "split f (Edge x y', Vertex y) = (if y' = y then f (x, y) else 0)"
wp_def: fixes c :: "com"   and Q :: "state \<Rightarrow> bool" shows "wp c Q = (\<lambda>s. \<forall>t. s -c\<rightarrow> t \<longrightarrow> Q t)"
some_eq_trivial: fixes x :: "'a" shows "(SOME y. y = x) = x"
nn_integral_suminf: fixes f :: "nat \<Rightarrow> 'c \<Rightarrow> ennreal"   and M :: "'c measure" assumes "\<And>i. f i \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. (\<Sum>i. f i x) \<partial>M = (\<Sum>i. integral\<^sup>N M (f i))"