hcju/isabelleps huggingface dataset:
elims: fixes x :: "('form, 'rule) graph_node"   and y :: "('form, 'var) antecedent fset" assumes "inPorts x = y" obtains r :: "'rule" where "x = Rule r__" and "y = f_antecedent r__"   | r :: "'form" where "x = Assumption ra__" and "y = {||}"   | r :: "'form" where "x = Conclusion r" and "y = {|plain_ant r|}"   | "x = Helper" and "y = {|plain_ant anyP|}"
stut_nextsuf_suc: fixes s :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "stutstep s n" shows "nextsuffix (s |\<^sub>s n) = nextsuffix (s |\<^sub>s Suc n)"
weak_nominal_ts_axioms: shows "weak_nominal_ts (\<turnstile>\<^sub>S) (\<rightarrow>\<^sub>S) (S_action.Act \<tau>)"
Levellist_def: fixes hds :: "ref list"   and "next" :: "ref \<Rightarrow> ref"   and ll :: "ref list list" shows "Levellist hds next ll = (map first ll = hds \<and> (\<forall>i<length hds. List (hds ! i) next (ll ! i)))"
char_power: fixes G :: "('c, 'd) monoid_scheme"   and \<chi> :: "'c \<Rightarrow> complex"   and a :: "'c"   and k :: "nat" assumes "character G \<chi>"   and "a \<in> carrier G" shows "\<chi> (a [^]\<^bsub>G\<^esub> k) = \<chi> a ^ k"
less_top: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and a :: "'b" assumes "class.order_top less_eq less top" shows "(a \<noteq> top) = less a top"
CAS_\<tau>red1t_xt2: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and v :: "'addr val"   and D :: "String.literal"   and F :: "String.literal"   and e3 :: "'addr expr1" assumes "\<tau>red1gt uf P t h (e, xs) (e', xs')" shows "\<tau>red1gt uf P t h (Val v\<bullet>compareAndSwap(D\<bullet>F, e, e3), xs) (Val v\<bullet>compareAndSwap(D\<bullet>F, e', e3), xs')"
lm_ball_Assoc_List_set: fixes m :: "('b, 'c) assoc_list"   and P :: "'b \<times> 'c \<Rightarrow> bool" shows "lm.ball m P = (\<forall>x\<in>Assoc_List.set m. P x)"
uint64_test_bit_def: fixes x :: "uint64"   and n :: "integer" shows "uint64_test_bit x n = (if n < 0 \<or> 63 < n then undefined bit x n else bit x (nat_of_integer n))"
doctor_separable_onI: fixes A :: "'x set"   and f :: "'x set \<Rightarrow> 'x set" assumes "\<And>B a b c. \<lbrakk>B \<subseteq> A; a \<in> A; b \<in> A; c \<in> A; Xd a \<noteq> Xd b; Xd b = Xd c; Xd a \<notin> Xd ` f (insert a (insert b B))\<rbrakk> \<Longrightarrow> Xd a \<notin> Xd ` f (insert a (insert b (insert c B)))" shows "doctor_separable_on A f"
fw_word': fixes p :: "nat"   and q :: "nat" assumes "\<not> p dvd q"   and "\<not> q dvd p" shows "\<^bold>|FW_word p q\<^bold>| = p + q - gcd p q - 1 \<and> periodN (FW_word p q) p \<and> periodN (FW_word p q) q \<and> \<not> periodN (FW_word p q) (gcd p q)"
assms_1: shows "xs @ [Broadcast (r, Delete ix e)] prefix of j"
Carmichael_root_exists: fixes n :: "nat" assumes "0 < n" obtains g :: "nat" where "g \<in> totatives n" and "ord n g = Carmichael n"
Ramsey2: fixes Z :: "'a set"   and f :: "'a set \<Rightarrow> nat"   and s :: "nat" assumes "infinite Z"   and "\<forall>x\<in>Z. \<forall>y\<in>Z. x \<noteq> y \<longrightarrow> f {x, y} < s" shows "\<exists>Y t. Y \<subseteq> Z \<and> infinite Y \<and> t < s \<and> (\<forall>x\<in>Y. \<forall>y\<in>Y. x \<noteq> y \<longrightarrow> f {x, y} = t)"
less_minus_one_simps_1: fixes 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 times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (uminus one) zero"
dist_pos_lt: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.metric_space dist uniformity open"   and "x \<noteq> y" shows "0 < dist x y"
comp_eq_dest: fixes a :: "'s \<Rightarrow> 'r"   and b :: "'q \<Rightarrow> 's"   and c :: "'t \<Rightarrow> 'r"   and d :: "'q \<Rightarrow> 't"   and v :: "'q" assumes "a \<circ> b = c \<circ> d" shows "a (b v) = c (d v)"
cross_def: fixes p :: "real poly"   and a :: "real"   and b :: "real" shows "cross p a b = variation (poly p a) (poly p b)"
rd_H1_H2: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "R (\<not> H1 (\<lambda>(A, A'). P (A, A'\<lparr>ok := False\<rparr>)) \<turnstile> H1 (\<lambda>(A, A'). P (A, A'\<lparr>ok := True\<rparr>))) = R (\<not> (H1 \<circ> H2) (\<lambda>(A, A'). P (A, A'\<lparr>ok := False\<rparr>)) \<turnstile> (H1 \<circ> H2) (\<lambda>(A, A'). P (A, A'\<lparr>ok := True\<rparr>)))"
Gamma_times_hurwitz_zeta_integral: fixes z :: "complex"   and a :: "real" assumes "1 < Re z"   and "0 < a" shows "Gamma z * hurwitz_zeta a z = CLINT x:{0<..}|lebesgue. complex_of_real x powr (z - 1) * complex_of_real (exp (- a * x) / (1 - exp (- x)))"
f11: shows "word_cat (cpu_reg_val Y (snd (fst (get_curr_win () s2)))) (user_reg_val (fst (fst (get_curr_win () s2))) rs1 (snd (fst (get_curr_win () s2)))) = word_cat (cpu_reg_val Y (snd (fst (get_curr_win () s1)))) (user_reg_val (fst (fst (get_curr_win () s1))) rs1 (snd (fst (get_curr_win () s1))))"
simps_1: fixes y :: "'a"   and x :: "'a" shows "local.efficient_funpow y x 0 = y"
wallis: shows "(\<lambda>n. \<Prod>k = 1..n. 4 * (real k)\<^sup>2 / (4 * (real k)\<^sup>2 - 1)) \<longlonglongrightarrow> pi / 2"
for_rec2_for_rec2': fixes f :: "'b \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b nres"   and a :: "'b"   and n :: "nat"   and i :: "nat"   and j :: "nat" shows "for_rec2 f a n i j = for_rec2' f a n i j"
LPDs_code: fixes r :: "mregex" shows "LPDs r = saturate (\<lambda>S. insert r S \<union> (insert r S \<bind> LPD)) {}"
proj_image: fixes \<phi> :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool" shows "proj \<phi> ` Collect P = Collect (P /// \<phi>)"
FlexRayArch_FrameTransmission: fixes n :: "nat"   and nReturn :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nC :: "nat \<Rightarrow> Config"   and nStore :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "nat \<Rightarrow> nat \<Rightarrow> nat list" assumes "FlexRayArch n nReturn nC nStore nGet"   and "\<forall>i<n. msg (Suc 0) (nReturn i)"   and "DisjointSchedules n nC"   and "IdenticCycleLength n nC" shows "FrameTransmission n nStore nReturn nGet nC"
homeomorphic_imp_open_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "homeomorphic_map X Y f" shows "open_map X Y f"
ImageE: fixes b :: "'c"   and r :: "('d \<times> 'c) set"   and A :: "'d set" assumes "b \<in> r `` A" obtains x :: "'d" where "(x, b) \<in> r" and "x \<in> A"
identify_blocks_main_cong: fixes k :: "nat"   and A :: "'a mat"   and B :: "'a mat"   and xs :: "(nat \<times> nat) list" assumes "k < n"   and "local.same_upto k A B" shows "identify_blocks_main A k xs = identify_blocks_main B k xs"
A: shows "A \<in> carrier_mat (dim_row A) (dim_col A)"
m_s_anti_mono_rep: fixes S1 :: "'a \<Rightarrow> 'av"   and S2 :: "'a \<Rightarrow> 'av"   and X :: "'a set" assumes "\<forall>x. S1 x \<le> S2 x" shows "(\<Sum>x\<in>X. m (S2 x)) \<le> (\<Sum>x\<in>X. m (S1 x))"
update_correct_2: fixes m :: "'a"   and k :: "'b"   and v :: "'c" assumes "invar m" shows "invar (update k v m)"
npMulElmR: fixes R :: "('c, 'd) Ring_scheme"   and x :: "'c"   and n :: "nat" assumes "Ring R"   and "x \<in> carrier R" shows "x^\<^bsup>R n\<^esup> \<cdot>\<^sub>r\<^bsub>R\<^esub> x = x^\<^bsup>R Suc n\<^esup>"
A: fixes k :: "nat" shows "(\<Sum>d | d dvd k. \<chi> d) \<in> \<real>"
has_integral_unique_cbox: fixes f :: "'a \<Rightarrow> 'b"   and k1 :: "'b"   and a :: "'a"   and b :: "'a"   and k2 :: "'b" assumes "(f has_integral k1) (cbox a b)"   and "(f has_integral k2) (cbox a b)" shows "k1 = k2"
iTrigger_iSince_conv: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t'. t' \<T> t I. Q t) = ((\<box> t I. Q t) \<or> (Q t'. t' \<S> t I. Q t \<and> P t))"
one_not_zero: shows "\<chi>\<^sub>0 1 \<noteq> 0"
inverse_arrows_char: fixes f :: "'s"   and g :: "'s" shows "S.inverse_arrows f g = (S.antipar f g \<and> compose (S.Dom f) (S.Fun g) (S.Fun f) = (\<lambda>x\<in>S.Dom f. x) \<and> compose (S.Dom g) (S.Fun f) (S.Fun g) = (\<lambda>y\<in>S.Dom g. y))"
invar_pass1: fixes hs :: "'b hp list" assumes "\<forall>h\<in>set hs. php h" shows "\<forall>h\<in>set (pass\<^sub>1 hs). php h"
Kripke_def: fixes S :: "'d set"   and S0 :: "'d set"   and R :: "('d \<times> 'd) set"   and L :: "'d \<Rightarrow> 'e set option" shows "Kripke S S0 R L = (S0 \<subseteq> S \<and> R \<subseteq> S \<times> S \<and> Domain R = S \<and> dom L = S)"
a_comp_simp: fixes antidomain_op :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.antidomain_left_monoid antidomain_op one times" shows "times (antidomain_left_monoid.am_add_op antidomain_op times (antidomain_op x) (antidomain_op y)) (antidomain_left_monoid.am_d antidomain_op x) = times (antidomain_op y) (antidomain_left_monoid.am_d antidomain_op x)"
wo_aux_def: shows "wo_aux = (SOME r. Card_order r \<and> infinite (Field r) \<and> regularCard r \<and> |UNIV| <o r)"
no_output_none_if_empty: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and r :: "nat \<Rightarrow>f value option"   and t :: "(String.literal \<times> value list) stream" shows "alw (output_eq []) (make_full_observation e None r [] t)"
TESL_interp_set_lifting: fixes \<Phi> :: "'b TESL_atomic list"   and \<Phi>' :: "'b TESL_atomic list" assumes "set \<Phi> = set \<Phi>'" shows "\<lbrakk>\<lbrakk> \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = \<lbrakk>\<lbrakk> \<Phi>' \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
poly_to_row_def: fixes ts :: "'d list"   and p :: "'d \<Rightarrow>\<^sub>0 'c" shows "poly_to_row ts p = vec_of_list (map (lookup p) ts)"
determ_def: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "determ t \<equiv> Transformers.additive t \<and> maximal t"
rs'_props_1: shows "g \<turnstile> defNode g r-rs'\<rightarrow>pred\<^sub>\<phi>\<^sub>r"
weak_upper_semilattice_axioms: shows "weak_upper_semilattice L"
principal_g3mod2: fixes r :: "(C, BD) sequent list \<times> (C, BD) sequent"   and R :: "((C, BD) sequent list \<times> (C, BD) sequent) set"   and A :: "(C, BD) form" assumes "r \<in> g3mod2"   and "R = Ax \<union> g3up \<union> g3mod1 \<union> g3mod2"   and "rightPrincipal r (\<box> A) R" shows "( \<Empt> \<Rightarrow>* \<LM> A  \<RM>) \<in> set (fst r)"
A_3: shows "set (rhsq r) \<subseteq> Q - W"
hcomp_reassoc_1: fixes \<tau> :: "'a"   and \<mu> :: "'a"   and \<nu> :: "'a" assumes "arr \<tau>"   and "arr \<mu>"   and "arr \<nu>"   and "src \<tau> = trg \<mu>"   and "src \<mu> = trg \<nu>" shows "(\<tau> \<star> \<mu>) \<star> \<nu> = \<a>\<^sup>-\<^sup>1[cod \<tau>, cod \<mu>, cod \<nu>] \<cdot> (\<tau> \<star> \<mu> \<star> \<nu>) \<cdot> \<a>[local.dom \<tau>, local.dom \<mu>, local.dom \<nu>]"
iT_Plus_Min: fixes I :: "nat set"   and k :: "nat" assumes "I \<noteq> {}" shows "iMin (I \<oplus> k) = iMin I + k"
intros_2: fixes c :: "i64"   and iop :: "unop_i" shows "\<lparr>[$C ConstInt64 c, $Unop_i T_i64 iop]\<rparr> \<leadsto> \<lparr>[$C ConstInt64 (app_unop_i iop c)]\<rparr>"
initState_into_used: fixes c :: "msg"   and B :: "agent"   and evs :: "event list" assumes "c \<in> parts (initState B)" shows "c \<in> used evs"
sum_image_le: fixes I :: "'c set"   and g :: "'e \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'e" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> (0::'d) \<le> g (f i)" shows "sum g (f ` I) \<le> sum (g \<circ> f) I"
phis'_def: fixes u :: "'g \<Rightarrow> 'node \<Rightarrow> 'val set"   and p :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option"   and g :: "'g"   and x :: "'node \<times> 'val" shows "uninst.phis' u p g x \<equiv> case x of (n, v) \<Rightarrow> if v = uninst.chooseNext u p g then None else map_option (map (uninst.substNext u p g)) (p g (n, v))"
length_map: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "\<^bold>|map f xs\<^bold>| = \<^bold>|xs\<^bold>|"
oseq_step_invariant_sterms: fixes A :: "((nat \<Rightarrow> 'e) \<times> ('e, 'f, 'g, 'h) seqp, 'f seq_action) automaton"   and S :: "(nat \<Rightarrow> 'e) \<Rightarrow> (nat \<Rightarrow> 'e) \<Rightarrow> 'f seq_action \<Rightarrow> bool"   and U :: "(nat \<Rightarrow> 'e) \<Rightarrow> (nat \<Rightarrow> 'e) \<Rightarrow> bool"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and P :: "((nat \<Rightarrow> 'e) \<times> 'h) \<times> 'f seq_action \<times> (nat \<Rightarrow> 'e) \<times> 'h \<Rightarrow> bool"   and i :: "nat"   and l' :: "'h"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<sigma> :: "nat \<Rightarrow> 'e"   and p :: "('e, 'f, 'g, 'h) seqp"   and p' :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'e" assumes "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) onll \<Gamma> P"   and "wellformed \<Gamma>"   and "automaton.trans A = oseqp_sos \<Gamma> i"   and "l' \<in> labels \<Gamma> q"   and "(\<sigma>, p) \<in> oreachable A S U"   and "((\<sigma>, p'), a, \<sigma>', q) \<in> automaton.trans A"   and "S \<sigma> \<sigma>' a"   and "p' \<in> sterms \<Gamma> p" shows "\<forall>l\<in>labels \<Gamma> p'. P ((\<sigma>, l), a, \<sigma>', l')"
linxM: shows "u \<in> {y \<in> S. x \<le> y}"
map_comp: fixes g :: "'s \<Rightarrow> 'r"   and f :: "'t \<Rightarrow> 's"   and v :: "'q \<Rightarrow> 't" shows "g \<circ> (f \<circ> v) = g \<circ> f \<circ> v"
that: fixes t :: "tm" assumes "x = \<lbrakk>\<guillemotleft>t\<guillemotright>\<rbrakk>e" shows "thesis"
Rep_perm: fixes x :: "'b perm" shows "Rep_perm x \<in> perms"
onorm_id_le: shows "onorm (\<lambda>x. x) \<le> 1"
measurable_count_space_insert: fixes s :: "'d"   and S :: "'d set"   and A :: "'d set" assumes "s \<in> S"   and "A \<in> sets (count_space S)" shows "insert s A \<in> sets (count_space S)"
the_deriv: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "(f has_derivative F) (at x)" shows "(THE G. (f has_derivative G) (at x)) = F"
ldec_zero: fixes n :: "nat" shows "ldec 0 n = 0"
R2_idem2: fixes x :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "R2 (R2 x) = R2 x"
share_read_only_mono: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<S>' :: "nat \<Rightarrow> bool option"   and sb :: "'b memref list" assumes "read_only \<S> \<subseteq> read_only \<S>'" shows "read_only (share sb \<S>) \<subseteq> read_only (share sb \<S>')"
discI_1: fixes literal :: "'a literal"   and x1 :: "'a" assumes "literal = Pos x1" shows "is_pos literal"
exhaust: fixes y :: "'a llist" obtains "y = LNil"   | x21 :: "'a" and x22 :: "'a llist" where "y = LCons x21 x22"
AndE1: fixes G :: "('a, 'b) form list"   and a :: "('a, 'b) form"   and b :: "('a, 'b) form" assumes "G \<turnstile> And a b" shows "G \<turnstile> a"
True_False_eps_concI: fixes L :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and p :: "bool list"   and R :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)" assumes "fin L p" shows "(True # p, False # start R) \<in> eps (RegExp2NAe.conc L R)"
Not_Inter: fixes r :: "'a language"   and s :: "'a language" shows "Coinductive_Language.Not (Inter r s) = Plus (Coinductive_Language.Not r) (Coinductive_Language.Not s)"
thetaSD_Sbis: shows "thetaSD \<subseteq> Sbis"
ref_in_rel: shows "(ref, n) \<in> succ_rel (ref # list)"
of_bool_eq_iff: fixes one :: "'d"   and zero :: "'d"   and p :: "bool"   and q :: "bool" assumes "class.zero_neq_one one zero" shows "(zero_neq_one.of_bool one zero p = zero_neq_one.of_bool one zero q) = (p = q)"
rpr_def: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" shows "rpr A r \<equiv> complete A r \<and> refl_on A r \<and> trans r"
conf: shows "Partial_Cost_Model.config' (rTS h0) ([x, y], h) v = ([x, y], rev v @ h)"
simps: fixes f :: "'op"   and ts :: "'op rose_tree list" shows "acost_sum (T f ts) = acost f (map state ts) + sum_list (map acost_sum ts)"
big_step_t3_gt0: fixes C :: "com"   and ps :: "char list \<Rightarrow> int option"   and x :: "nat"   and ps' :: "char list \<Rightarrow> int option" assumes "(C, ps) \<Rightarrow>\<^sub>A x \<Down> ps'" shows "0 < x"
LeadsTo_def: fixes A :: "'c set"   and B :: "'c set" shows "A \<longmapsto>w B \<equiv> {F. F \<in> reachable F \<inter> A \<longmapsto> B}"
subst_dom_update3: fixes t :: "('a, 'b) Term.term"   and v :: "'b"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "t = Var v" shows "subst_domain (\<sigma>(v := t)) = subst_domain \<sigma> - {v}"
natural_isomorphisms_compose: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<sigma> :: "'d \<Rightarrow> 'e"   and H :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" assumes "natural_isomorphism A B F G \<sigma>"   and "natural_isomorphism A B G H \<tau>" shows "natural_isomorphism A B F H (vertical_composite.map A B \<sigma> \<tau>)"
simps: fixes ap :: "abc_inst list" shows "layout_of ap = map length_of ap"
wf_update_matchF_invar: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and I :: "\<I>"   and r :: "Formula.formula Regex.regex"   and aux :: "(nat \<times> event_data option list set list \<times> event_data option list set) list"   and ne :: "nat"   and st :: "(nat \<times> event_data option list set list \<times> event_data option list set) list \<times> (mregex, event_data option list set) mapping"   and mr :: "mregex"   and \<phi>s :: "Formula.formula list"   and mrs :: "mregex list"   and j :: "nat" assumes "wf_matchF_aux \<sigma> V n R I r aux ne (length (fst st) - 1)"   and "wf_matchF_invar \<sigma> V n R I r st (ne + length aux)"   and "safe_regex Futu Strict r"   and "to_mregex r = (mr, \<phi>s)"   and "mrs = sorted_list_of_set (LPDs mr)"   and "j = ne + length aux + length (fst st) - 1" shows "wf_matchF_invar \<sigma> V n R I r (foldr (update_matchF_step I mr mrs (\<tau> \<sigma> j)) aux st) ne"
le_parity_def: fixes x :: "parity"   and y :: "parity" shows "(x \<sqsubseteq> y) = (y = Either \<or> x = y)"
mult_est_spmat_def: fixes r1 :: "(nat \<times> (nat \<times> 'a) list) list"   and r2 :: "(nat \<times> (nat \<times> 'a) list) list"   and s1 :: "(nat \<times> (nat \<times> 'a) list) list"   and s2 :: "(nat \<times> (nat \<times> 'a) list) list" shows "mult_est_spmat r1 r2 s1 s2 = add_spmat (mult_spmat (pprt_spmat s2) (pprt_spmat r2)) (add_spmat (mult_spmat (pprt_spmat s1) (nprt_spmat r2)) (add_spmat (mult_spmat (nprt_spmat s2) (pprt_spmat r1)) (mult_spmat (nprt_spmat s1) (nprt_spmat r1))))"
bot_edge_is_compos_of_scal_and_g2: shows "(\<lambda>x. twoC (x, 0)) = (\<lambda>x. (x, g2 x)) \<circ> (\<lambda>x. a + (b - a) * x)"
inv_comm: fixes x :: "'a"   and y :: "'a" assumes "x \<oplus> y = \<zero>"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "y \<oplus> x = \<zero>"
a_gt_plus: shows "\<forall>\<^sub>F n in sequentially. c (n + 1) < c n \<longrightarrow> real_of_int (a n) + real_of_int (a n - c (n + 1) - 1) / real_of_int (c (n + 1)) < real_of_int (a (n + 1))"
src_def: fixes F :: "('a E.term, 'a) arr" shows "src F \<equiv> if arr F then MkIde (E.Src (Dom F)) else null"
valid_subgraph: fixes H :: "('v, 'w) graph" assumes "subgraph H G" shows "valid_graph H"
smult_mat_def: fixes a :: "'d"   and A :: "'d mat" shows "a \<cdot>\<^sub>m A \<equiv> map_mat ((*) a) A"
assms_6: shows "P \<turnstile> \<langle>INIT D ([D],False) \<leftarrow> unit,(h\<^sub>1, l\<^sub>1, sh\<^sub>1),False\<rangle> \<rightarrow>* \<langle>Val v',(h\<^sub>2, l\<^sub>2, sh\<^sub>2),b\<^sub>2\<rangle>"
mk_disjoint_insert: fixes a :: "'c"   and A :: "'c set" assumes "a \<in> A" shows "\<exists>B. A = insert a B \<and> a \<notin> B"
finfun_upd_triv: fixes f :: "'c \<Rightarrow>f 'd"   and x :: "'c" shows "f(x $:= f $ x) = f"
lconjby_closed: fixes g :: "'g"   and x :: "'g" assumes "g \<in> G"   and "x \<in> G" shows "lconjby g x \<in> G"
\<epsilon>_in_terms_of_\<psi>: fixes x :: "'b" assumes "B.ide x" shows "F'G'.\<epsilon> x = \<psi>' x (G' x)"
RDirProd_iso4: fixes f :: "'a \<Rightarrow> 'b"   and R :: "('a, 'c) ring_scheme"   and S :: "('b, 'd) ring_scheme"   and T :: "('e, 'f) ring_scheme" assumes "f \<in> ring_iso R S" shows "(\<lambda>(r, t). (f r, t)) \<in> ring_iso (RDirProd R T) (RDirProd S T)"
special_cf_def: fixes S :: "('d, 'e) Ring_scheme" shows "C\<^sub>0 S = (0, \<lambda>j. 1\<^sub>r\<^bsub>S\<^esub>)"
wary_nth_ss: fixes i :: "nat" assumes "i < length ss" shows "wary (ss ! i)"
uniform_limit_on_subset: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and J :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and F :: "'a filter"   and I :: "'b set" assumes "uniform_limit J f g F"   and "I \<subseteq> J" shows "uniform_limit I f g F"
ev_blocks_partD: fixes m :: "nat"   and A :: "'b mat"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "ev_blocks_part m A"   and "i < j"   and "j < k"   and "k < m"   and "A $$ (k, k) = A $$ (i, i)" shows "A $$ (j, j) = A $$ (i, i)"
insrt: fixes dxs :: "'a odlist"   and x :: "'a"   and xs :: "'a list" assumes "dxs = fromList (x # xs)"   and "distinct (x # xs)"   and "sorted (x # xs)"   and "P (fromList xs)" shows "P dxs"
assms_2: shows "uniform_limit S g (\<lambda>_. 0) F"
a2: shows "x:N \<in> \<parallel>(B)\<parallel>"
19: shows "(w \<sqinter> - (top * selected_edge h j g * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * selected_edge h j g * w\<^sup>T\<^sup>\<star>)\<^sup>T) \<sqinter> - forest_components f * selected_edge h j g * top \<sqinter> top * selected_edge h j g\<^sup>T * forest_components f \<le> top * selected_edge h j g\<^sup>T * (w \<sqinter> - (top * selected_edge h j g * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * selected_edge h j g * w\<^sup>T\<^sup>\<star>)\<^sup>T)\<^sup>T\<^sup>\<star>"
left_exact_injec: fixes R :: "('e, 'f) Ring_scheme"   and M :: "('g, 'e) Module"   and N :: "('h, 'e) Module"   and z :: "'i \<Rightarrow> 'g"   and e :: "'i"   and f :: "'g \<Rightarrow> 'h" assumes "Ring R"   and "R module M"   and "R module N"   and "z \<in> mHom R (Zm R e) M"   and "f \<in> mHom R M N"   and "exact3 R (Zm R e) z M f N" shows "injec\<^bsub>M,N\<^esub> f"
simps_30: shows "ShiftRightZeros \<noteq> Eq"
NS_Spy_see_staticSecret: fixes m :: "msg"   and A :: "agent"   and evs :: "event list" assumes "m \<in> staticSecret A"   and "evs \<in> ns_public" shows "(m \<in> parts (knows Spy evs)) = (A \<in> bad)"
neqneg2: shows "w \<noteq> 2 ^ 31 - 1"
substAll_def: fixes g :: "'g" shows "uninst.substAll uses phis g \<equiv> while (cond g) (uninst.step uses phis g) (uses g, phis g)"
fun_ordD: fixes ord :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'f"   and g :: "'h \<Rightarrow> 'g"   and x :: "'h" assumes "fun_ord ord f g" shows "ord (f x) (g x)"
l4_6: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "Bet A B C"   and "A B C Cong3 A' B' C'" shows "Bet A' B' C'"
mult_vec_minus_mult_left: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "- p ** q = - (p ** q)"
absolutely_integrable_mult_Fejer_kernel_reflected1: fixes f :: "real \<Rightarrow> real"   and n :: "nat"   and t :: "real" assumes "f absolutely_integrable_on {- pi..pi}"   and "\<And>x. f (x + 2 * pi) = f x" shows "(\<lambda>x. Fejer_kernel n x * f (t + x)) absolutely_integrable_on {- pi..pi}"