hcju/isabelleps huggingface dataset:
c1_gen_check_conv: fixes ys :: "nat list"   and s :: "nat"   and a :: "nat list"   and b :: "nat list" assumes "(ys, s) \<in> set (c2_gen_check a b b)" shows "c1_gen_check b ys a = bounded_gen_check.gen_check (cond1 b ys) a"
analytic_hurwitz_zeta: fixes a :: "real"   and A :: "complex set" assumes "0 < a"   and "1 \<notin> A" shows "hurwitz_zeta a analytic_on A"
sel_4: fixes x1 :: "'d set"   and x2 :: "'c set"   and x3 :: "'d \<Rightarrow> 'c \<Rightarrow> 'c set"   and x4 :: "'c \<Rightarrow> bool" shows "accepting (nba x1 x2 x3 x4) = x4"
cint: fixes k :: "nat" shows "(\<lambda>u. f u / (u - z) ^ Suc k) contour_integrable_on circlepath z r"
override_on_bot_1: fixes m :: "'d \<Rightarrow> 'e"   and S :: "'d set" shows "\<bottom> ++\<^bsub>S\<^esub> m = m f|` S"
Cut_c_redu_elim: fixes a :: "coname"   and M :: "trm"   and x :: "name"   and N :: "trm"   and R :: "trm" assumes "Cut <a>.M x.N \<longrightarrow>\<^sub>c R" shows "R = M{a:=x.N} \<and> \<not> fic M a \<or> R = N{x:=<a>.M} \<and> \<not> fin N x"
simps_3: fixes x31 :: "'b local.term"   and x32 :: "'b local.term"   and y31 :: "'b local.term"   and y32 :: "'b local.term" shows "(x31 \<^bold>\<star> x32 = y31 \<^bold>\<star> y32) = (x31 = y31 \<and> x32 = y32)"
simps: fixes s1 :: "('a, 'b, 'c) SLin_state"   and s2 :: "('a, 'e, 'f, 'g) SLin_state_scheme" shows "P8a (s1, s2) = (\<forall>ivs\<in>initSets s2. \<exists>rs\<in>pendingSeqs s1. \<Sqinter> ivs = dstate s1 \<star> rs)"
optimize_matches_matches_fst: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and f :: "'b match_expr \<Rightarrow> 'b match_expr"   and m :: "'b match_expr"   and a :: "action"   and p :: "'c"   and rs :: "'b rule list" assumes "matches \<gamma> (f m) a p" shows "optimize_matches f (Rule m a # rs) = Rule (f m) a # optimize_matches f rs"
split_conc: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and xs :: "('b btree \<times> 'b) list"   and p :: "'b"   and ls :: "('b btree \<times> 'b) list"   and rs :: "('b btree \<times> 'b) list" assumes "split split"   and "split xs p = (ls, rs)" shows "xs = ls @ rs"
bY: shows "[[b a\<^sub>1 a\<^sub>n]]"
nxt: fixes q :: "hf"   and x :: "'a" assumes "q \<in> nfa.states \<lparr>nfa.states = HF.Inl ` dfa.states MS \<union> HF.Inr ` dfa.states MT, init = {HF.Inl (dfa.init MS)}, final = HF.Inr ` dfa.final MT, nxt = \<lambda>q x. sum_case (\<lambda>qs. {HF.Inl (dfa.nxt MS qs x)}) (\<lambda>qt. {HF.Inr (dfa.nxt MT qt x)}) q, eps = (\<lambda>q. (HF.Inl q, HF.Inr (dfa.init MT))) ` dfa.final MS\<rparr>" shows "nfa.nxt \<lparr>nfa.states = HF.Inl ` dfa.states MS \<union> HF.Inr ` dfa.states MT, init = {HF.Inl (dfa.init MS)}, final = HF.Inr ` dfa.final MT, nxt = \<lambda>q x. sum_case (\<lambda>qs. {HF.Inl (dfa.nxt MS qs x)}) (\<lambda>qt. {HF.Inr (dfa.nxt MT qt x)}) q, eps = (\<lambda>q. (HF.Inl q, HF.Inr (dfa.init MT))) ` dfa.final MS\<rparr> q x \<subseteq> nfa.states \<lparr>nfa.states = HF.Inl ` dfa.states MS \<union> HF.Inr ` dfa.states MT, init = {HF.Inl (dfa.init MS)}, final = HF.Inr ` dfa.final MT, nxt = \<lambda>q x. sum_case (\<lambda>qs. {HF.Inl (dfa.nxt MS qs x)}) (\<lambda>qt. {HF.Inr (dfa.nxt MT qt x)}) q, eps = (\<lambda>q. (HF.Inl q, HF.Inr (dfa.init MT))) ` dfa.final MS\<rparr>"
le_fin_length: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a list" assumes "traces ind"   and "traces.le_fin ind w\<^sub>1 w\<^sub>2" shows "length w\<^sub>1 \<le> length w\<^sub>2"
integral: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "a \<otimes> b = \<zero>"   and "a \<in> carrier Zp"   and "b \<in> carrier Zp" shows "a = \<zero> \<or> b = \<zero>"
decomp_closure_def: fixes M :: "('fun, 'var) Term.term set"   and M' :: "('fun, 'var) Term.term set" shows "decomp_closure M M' \<equiv> \<forall>t. (M \<turnstile> t \<and> (\<exists>t'\<in>M. t \<sqsubseteq> t')) = (t \<in> M')"
valid_unMulti: shows "valid_unMultigraph (del_unEdge v e v' G)"
Setcompr_subset: fixes f :: "'b \<Rightarrow> 'a"   and P :: "'b \<Rightarrow> bool"   and S :: "'a set" shows "({f x |x. P x} \<subseteq> S) = (\<forall>x. P x \<longrightarrow> f x \<in> S)"
f_dvd_ab: fixes a :: "'a mod_ring poly"   and b :: "'a mod_ring poly" assumes "f dvd a * b"   and "degree a < degree f"   and "degree b < degree f" shows "a = 0 \<or> b = 0"
preserves_arr: fixes f :: "'a arrow_of_spans_data" assumes "arr f" shows "arr (R f)"
qSkelBinp_def: fixes qbinp :: "'f \<Rightarrow> ('g, 'h, 'i, 'j, 'k) qAbs option" shows "qSkelBinp qbinp = lift qSkelAbs qbinp"
divide_less_eq_1_pos: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a" shows "less (divide b a) one = less b a"
dupe: shows "Hensel_Lifting_Type_Based.dupe_monic (#v) (#w) s t 1 = (a, b)"
lens_put_irr1: fixes X :: "'e \<Longrightarrow> 'f"   and Y :: "'g \<Longrightarrow> 'f"   and \<sigma> :: "'f"   and v :: "'g" assumes "X \<bowtie> Y" shows "get\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) = get\<^bsub>X\<^esub> \<sigma>"
min_aform_err_def: fixes p :: "nat"   and a1 :: "(real \<times> real pdevs) \<times> real"   and a2 :: "(real \<times> real pdevs) \<times> real" shows "min_aform_err p a1 a2 = (let ivl1 = ivl_of_aform_err p a1; ivl2 = ivl_of_aform_err p a2 in if upper ivl1 < lower ivl2 then a1 else if upper ivl2 < lower ivl1 then a2 else ivl_err (real_interval (min_interval ivl1 ivl2)))"
insert_not_empty: fixes a :: "'b"   and A :: "'b set" shows "a \<triangleright> A \<noteq> {}"
f_prime: fixes p :: "nat" assumes "prime p" shows "f p = f_prime p"
skolemize2': fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "\<forall>x. \<exists>y z. P x y z" shows "\<exists>f g. \<forall>x. P x (f x) (g x)"
not_one_less_zero: fixes one :: "'g"   and times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less one zero"
inv_image_betw_def: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" shows "inv_image_betw P f A B = (\<lambda>x y. x \<in> A \<and> y \<in> A \<and> f x \<in> B \<and> f y \<in> B \<and> P (f x) (f y))"
scaleR: fixes r :: "real"   and b :: "'a" shows "(if x \<in> T then g' x else (\<lambda>x. 0::'b)) (r *\<^sub>R b) = r *\<^sub>R (if x \<in> T then g' x else (\<lambda>x. 0::'b)) b"
strictly_maximal_literal_exists: fixes n :: "nat" shows "\<forall>C. finite C \<and> card C = n \<and> n \<noteq> 0 \<and> \<not> tautology C \<longrightarrow> (\<exists>A. strictly_maximal_literal C A)"
rel_cases: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and a :: "'g list"   and b :: "'h list" assumes "list_all2 R a b" obtains "a = []" and "b = []"   | x1 :: "'g" and x2 :: "'g list" and y1 :: "'h" and y2 :: "'h list" where "a = x1 # x2" and "b = y1 # y2" and "R x1 y1" and "list_all2 R x2 y2"
unifiable_imp_unif: fixes e :: "('c, 'd) Term.term \<times> ('c, 'd) Term.term" assumes "unifiable {e}" shows "\<exists>E'. ({#e#}, E') \<in> unif"
normalize_poly_i_def: fixes ops :: "'b arith_ops_record"   and xs :: "'b list" shows "normalize_poly_i ops xs = smult_i ops (arith_ops_record.inverse ops (arith_ops_record.unit_factor ops (lead_coeff_i ops xs))) xs"
trg_trg: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "trg\<^sub>B (trg\<^sub>B \<mu>) = trg\<^sub>B \<mu>"
that: assumes "CSP1 P = P"   and "CSP2 P = P"   and "R P = P" shows "thesis"
isomorphic_reflexive: fixes f :: "('a, 'a set) Maps.arr" assumes "Maps.ide f" shows "Maps.isomorphic f f"
neg_add2: shows "Negligible.negligible (\<lambda>n. local.ddh.advantage n (P1_DDH_mal_adv_\<sigma>_false n M z \<A> D) + local.ddh.advantage n (local.ddh.DDH_\<A>' n (P1_DDH_mal_adv_\<sigma>_false n M z \<A> D)))"
to_fract_eq_0_iff: fixes x :: "'a" shows "(to_fract x = 0) = (x = (0::'a))"
is_red_indI1: fixes f :: "'t \<Rightarrow>\<^sub>0 'c"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "f \<in> F"   and "f \<noteq> 0"   and "p \<noteq> 0"   and "lt f adds\<^sub>t lt p" shows "is_red F p"
four_two_ind: fixes \<delta>S :: "real"   and q :: "nat"   and i :: "nat"   and p :: "nat"   and t :: "real" assumes "\<beta> \<le> rmin"   and "\<mu> \<le> \<delta>S"   and "\<gamma>1 \<delta>S \<le> \<delta>S"   and "\<gamma>2 \<delta>S \<le> \<delta>"   and "\<gamma>3 \<delta>S \<le> \<delta>"   and "te q (i + 1) \<le> te p (i + 1)"   and "okClocks p q i"   and "0 \<le> t"   and "t < max (te p (i + 1)) (te q (i + 1))"   and "max (te p i) (te q i) \<le> t"   and "max (te p i) (te q i) < max (te p (i + 1)) (te q (i + 1))"   and "correct p t"   and "correct q t" shows "\<bar>VC p t - VC q t\<bar> \<le> \<delta>"
simps_14: fixes e1 :: "'a aexp"   and e2 :: "'a aexp"   and v :: "'a"   and va :: "value list" shows "local.less_gexp_aux (Gt e1 e2) (In v va) = True"
edom_fun_upd_subset: fixes h :: "'d \<Rightarrow> 'e"   and x :: "'d"   and v :: "'e" shows "edom (h(x := v)) \<subseteq> insert x (edom h)"
ntChanEq: fixes \<Psi> :: "'b"   and x :: "name" shows "\<Psi> \<turnstile> nameTerm x \<leftrightarrow> nameTerm x"
rev_finite_subset: fixes B :: "'d set"   and A :: "'d set" assumes "finite B"   and "A \<subseteq> B" shows "finite A"
cgenideal_eq_genideal: fixes i :: "'a" assumes "i \<in> carrier R" shows "PIdl i = Idl {i}"
is_aligned_drop: fixes w :: "'b word"   and n :: "nat" assumes "is_aligned w n"   and "n \<le> LENGTH('b)" shows "drop (LENGTH('b) - n) (to_bl w) = replicate n False"
sum_left: fixes g :: "'c \<Rightarrow> 'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and S :: "'c set"   and x :: "'b" shows "sum g S *\<^sub>V x = (\<Sum>i\<in>S. g i *\<^sub>V x)"
intro: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and design_rep_number :: "int" assumes "proper_design point_set block_collection"   and "constant_rep_design_axioms point_set block_collection design_rep_number" shows "constant_rep_design point_set block_collection design_rep_number"
typeof_IncompatibleClassChangeError: fixes h :: "heap" assumes "preallocated h" shows "typeof\<^bsub>h\<^esub> (Addr (addr_of_sys_xcpt IncompatibleClassChangeError)) = \<lfloor>Class IncompatibleClassChangeError\<rfloor>"
md2: shows "local.cont c s i \<approx>s local.cont d s j"
H: fixes x :: "nat"   and y :: "nat" shows "(\<lambda>n. h' (k n) (x, y)) \<longlonglongrightarrow> H x y"
inf_idem: fixes x :: "'d multiset" shows "x \<inter># x = x"
modifies_spec: shows "\<forall>\<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/UNIV \<^esub>{\<sigma>} Call p (Modif \<sigma>),(ModifAbr \<sigma>)"
that: fixes c :: "'a fract"   and p' :: "'a poly" assumes "p = smult c (map_poly to_fract p')"   and "content p' = (1::'a)" shows "thesis"
part: shows "partn_lst_VWF \<beta> \<alpha> n"
intro: fixes \<eta> :: "'b \<Rightarrow> 'c"   and M :: "'b set"   and M' :: "'c set"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit :: "'b"   and composition' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit' :: "'c" assumes "map \<eta> M M'"   and "Group_Theory.monoid M composition unit"   and "Group_Theory.monoid M' composition' unit'"   and "monoid_homomorphism_axioms \<eta> M composition unit composition' unit'" shows "monoid_homomorphism \<eta> M composition unit M' composition' unit'"
filter_eq_ConsD: fixes P :: "'a \<Rightarrow> bool"   and ys :: "'a list"   and x :: "'a"   and xs :: "'a list" assumes "filter P ys = x # xs" shows "\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
thin1: fixes H :: "fmla set"   and B :: "fmla"   and A :: "fmla" assumes "H \<turnstile> B" shows "insert A H \<turnstile> B"
is_unit_normalize: fixes a :: "'b poly" assumes "is_unit a" shows "smult (inverse (lead_coeff a)) a = 1"
find_first_prefix: fixes x :: "'a"   and xs :: "'a list"   and i :: "nat"   and n :: "nat" assumes "find_first x xs = Some i"   and "i < n" shows "find_first x (take n xs) = Some i"
rank_col_rank: fixes A :: "(('a, 'b) vec, 'c) vec" shows "rank A = col_rank A"
unique_idx_Nil: fixes data :: "nat \<times> 'h" shows "unique_idx [] data"
simps_2: fixes x :: "bool"   and xs :: "bool list"   and y :: "bool"   and ys :: "bool list" shows "bv_or (x # xs) (y # ys) = (x \<or> y) # bv_or xs ys"
preserves_iso: fixes f :: "'b \<times> 'b \<times> 'b" assumes "VVV.iso f" shows "local.iso (\<alpha>' f)"
simps: fixes l :: "nat"   and tss :: "(symbol \<times> character list) list set" shows "by_length l tss = {ts \<in> tss. length (chars ts) = l}"
ex_conj_sylow_group: fixes G :: "('c, 'd) monoid_scheme"   and p :: "nat"   and a :: "nat"   and m :: "nat"   and H :: "'c set"   and b :: "nat"   and P :: "'c set" assumes "snd_sylow G p a m"   and "H \<in> group.subgroups_of_size G (p ^ b)"   and "P \<in> group.subgroups_of_size G (p ^ a)" obtains g :: "'c" where "g \<in> carrier G" and "H \<subseteq> g <#\<^bsub>G\<^esub> (P #>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> g)"
the_\<nu>_def: shows "the_\<nu> \<equiv> THE \<nu>. \<guillemotleft>\<nu> : r\<^sub>1 \<Rightarrow> s\<^sub>1 \<star> w\<guillemotright>"
take_encode: shows "take k (digit_encode (replicate (Suc k) 2) i) = digit_encode (replicate k 2) i"
Joints_aforms_of_ivls: fixes lX0 :: "real list"   and uX0 :: "real list" assumes "list_all2 (\<le>) lX0 uX0" shows "Joints (aforms_of_ivls lX0 uX0) = list_interval lX0 uX0"
cross_mult_right: fixes x :: "(real, 3) vec"   and c :: "real"   and y :: "(real, 3) vec" shows "x \<times> (c *\<^sub>R y) = c *\<^sub>R x \<times> y"
CFGExit_wf_axioms: shows "CFGExit_wf src trg knd lve NewEntry lDef lUse state_val NewExit"
s2'_def: shows "s2' \<equiv> activate_cond_actions2 (ls, (ts1, m1), ws, is) (ls, (ts2, m2), ws, is) cts"
FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m_is_FORK: fixes i :: "nat" shows "FORK i = FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m i 0"
infdist_Un_min: fixes A :: "'c set"   and B :: "'c set"   and x :: "'c" assumes "A \<noteq> {}"   and "B \<noteq> {}" shows "infdist x (A \<union> B) = min (infdist x A) (infdist x B)"
map_prod_imageI: fixes a :: "'c"   and b :: "'d"   and R :: "('c \<times> 'd) set"   and f :: "'c \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f" assumes "(a, b) \<in> R" shows "(f a, g b) \<in> map_prod f g ` R"
pderiv_monom: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "pderiv (monom P a n) = monom P ([n] \<cdot> a) (n - 1)"
s: shows "sgn (lead_coeff f) \<in> {- 1, 1}"
cis_one: fixes \<phi> :: "real" assumes "sin \<phi> = 0"   and "cos \<phi> = 1" shows "cis \<phi> = 1"
Cons_2: shows "sum_list (map2 (+) xs ys) = sum_list xs + sum_list ys"
project_set_def: fixes h :: "'c \<times> 'd \<Rightarrow> 'e"   and C :: "'e set" shows "project_set h C = {x. \<exists>y. h (x, y) \<in> C}"
P: shows "\<forall>x. x \<in> s \<longrightarrow> x \<le> k"
assms_2: shows "\<S>' = update\<^sub>s\<^sub>t \<S> (\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S)"
A: fixes u :: "nat \<Rightarrow> real" assumes "(\<lambda>x. ereal (u x)) \<longlonglongrightarrow> \<infinity>" shows "Gromov_converging_at_boundary u"
rep_eq: fixes x :: "nat"   and xa :: "nat"   and xb :: "(nat, 'b) fmap" shows "fmlookup (lowerPowers\<^sub>f x xa xb) = (\<lambda>xc. if xc \<in> {x..<x + xa} then None else fmlookup xb (dec_above x xa xc))"
S: fixes i :: "nat" assumes "i < length ss"   and "i < length ts"   and "fst (r (ss ! i) (ts ! i))" shows "fst (r (map f ss ! i) (map f ts ! i))"
interaction_bounded_by'_exec_gpv_count: fixes "consider" :: "bool list \<Rightarrow> bool"   and gpv :: "('b, bool list, bool list \<times> bool list) gpv"   and n :: "nat"   and x :: "'b"   and s' :: "bool list \<Rightarrow> bool list option"   and s :: "bool list \<Rightarrow> bool list option"   and count :: "(bool list \<Rightarrow> bool list option) \<Rightarrow> nat" assumes "interaction_bounded_by' consider gpv n"   and "(x, s') \<in> set_spmf (exec_gpv rf_encrypt gpv s)"   and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (rf_encrypt s x); finite (dom s); consider x; x \<in> outs_\<I> \<I>_full\<rbrakk> \<Longrightarrow> count s' \<le> Suc (count s)"   and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (rf_encrypt s x); finite (dom s); \<not> consider x; x \<in> outs_\<I> \<I>_full\<rbrakk> \<Longrightarrow> count s' \<le> count s"   and "\<I>_full \<turnstile>g gpv \<surd>"   and "finite (dom s)" shows "count s' \<le> n + count s"
assms_1: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>)"
abs_eq: shows "1 = Abs_poly_mapping (\<lambda>k. (1::'c) when k = (0::'b))"
discI: fixes option :: "'c option"   and x2 :: "'c" assumes "option = Some x2" shows "option \<noteq> None"
exec_step_ind_Init_Processing: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and Cs :: "char list list"   and P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list" assumes "sh C = \<lfloor>(sfs, Processing)\<rfloor>" shows "exec_step_ind (StepC C Cs) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, Called Cs) # frs, sh)"
invertible_lf: shows "\<exists>g. linear (*s) (*s) g \<and> g \<circ> f = id \<and> f \<circ> g = id"
Units_closed: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b" assumes "Group.monoid G"   and "x \<in> Units G" shows "x \<in> carrier G"
slice_kind_Return: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and a :: "'a"   and S :: "'b SDG_node set"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and p :: "'f"   and f :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "sourcenode a \<notin> SDG.SDG_to_CFG_set (SDG.HRB_slice sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S)"   and "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f" shows "SDG.slice_kind sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S a = \<lambda>cf. True\<hookleftarrow>\<^bsub>p\<^esub>\<lambda>cf cf'. cf'"
fixp_iterates_above: fixes lub :: "'b set \<Rightarrow> 'b"   and leq :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "bourbaki_witt_fixpoint lub leq f" shows "bourbaki_witt_fixpoint.fixp_above lub leq f a \<in> bourbaki_witt_fixpoint.iterates_above lub leq f a"
simps: fixes x :: "primfun"   and a :: "'a"   and e :: "'a" shows "powr_primfun (x, a) e = (x, e * a)"
R'_def: shows "R' = R \<inter> {x. 0 < f x} \<times> {y. 0 < g y}"
acyclicP_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "acyclicP r = (\<forall>x. \<not> r\<^sup>+\<^sup>+ x x)"
F_conv_def: fixes b_of_a :: "'d \<Rightarrow> 'c"   and F :: "char list \<Rightarrow> 'd list \<Rightarrow> 'd" shows "F_conv b_of_a F \<equiv> \<lambda>f bs. b_of_a (F f (map (inv b_of_a) bs))"
discrL_discr: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and cl :: "('d, 'c, 'a) cmd list"   and c :: "('d, 'c, 'a) cmd" assumes "PL_Indis cval indis"   and "PL_Indis.discrL aval tval cval indis cl"   and "c \<in> set cl" shows "PL_Indis.discr aval tval cval indis c"
diff_mod_pred: fixes a :: "nat"   and b :: "nat"   and m :: "nat" assumes "a < b" shows "(b - Suc a) mod m = (if a mod m = b mod m then m - Suc 0 else (b - a) mod m - Suc 0)"
divides_prod_l: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "Group.comm_monoid G"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G"   and "a divides\<^bsub>G\<^esub> b" shows "a divides\<^bsub>G\<^esub> c \<otimes>\<^bsub>G\<^esub> b"
not_Err_eq: fixes x :: "'a err" shows "(x \<noteq> Err) = (\<exists>a. x = OK a)"
mask_numeral: fixes 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 divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and n :: "num" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "mask (numeral n) = plus one (times (numeral.numeral one plus (num.Bit0 num.One)) (mask (pred_numeral n)))"
induct: fixes P :: "'b list \<Rightarrow> 'b trie \<Rightarrow> bool"   and a0 :: "'b list"   and a1 :: "'b trie" assumes "\<And>b m. P [] (trie.Nd b m)"   and "\<And>x xs b m. (\<And>x2. m x = Some x2 \<Longrightarrow> P xs x2) \<Longrightarrow> P (x # xs) (trie.Nd b m)" shows "P a0 a1"
distinct_9: fixes x2 :: "'b \<Rightarrow> bool"   and x31 :: "'b pltl"   and x32 :: "'b pltl" shows "atom\<^sub>p(x2) \<noteq> x31 implies\<^sub>p x32"
ring_isomorphism_def: fixes \<eta> :: "'c \<Rightarrow> 'd"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and R' :: "'d set"   and addition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero' :: "'d"   and unit' :: "'d" shows "ring_isomorphism \<eta> R addition multiplication zero unit R' addition' multiplication' zero' unit' \<equiv> ring_homomorphism \<eta> R addition multiplication zero unit R' addition' multiplication' zero' unit' \<and> bijective_map \<eta> R R'"
bin_join_table: fixes n :: "nat"   and A :: "nat set"   and t :: "'b option list set"   and A' :: "nat set"   and t' :: "'b option list set"   and pos :: "bool" assumes "table n A t"   and "table n A' t'" shows "bin_join n A t pos A' t' = join t pos t'"
val_LI_pos: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and I :: "'a set"   and j :: "nat" assumes "distinct_pds K n P"   and "ideal (O\<^bsub>K P n\<^esub>) I"   and "I \<noteq> {\<zero>\<^bsub>O\<^bsub>K P n\<^esub>\<^esub>}"   and "j \<le> n" shows "0 \<le> LI K (\<nu>\<^bsub>K P j\<^esub>) I"
fps_shift_mult_both: fixes m :: "nat"   and f :: "'a fps"   and n :: "nat"   and g :: "'a fps" assumes "m \<le> subdegree f"   and "n \<le> subdegree g" shows "fps_shift m f * fps_shift n g = fps_shift (m + n) (f * g)"