hcju/isabelleps huggingface dataset:
llength_ldropn: fixes n :: "nat"   and xs :: "'c llist" shows "llength (ldropn n xs) = llength xs - enat n"
i: fixes a :: "real"   and b :: "real" assumes "a < 0"   and "0 \<le> b" shows "a / (2 * b) + 1 / 2 \<le> 1"
senior_states_older_seniors: fixes n :: "nat"   and x :: "nat" shows "(\<lambda>y. token_run y n) ` older_seniors x n = senior_states (token_run x n) n"
assms_1: fixes tset\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha> set['a]" assumes "\<And>x. x \<in> set_bset tset\<^sub>\<alpha> \<Longrightarrow> P x" shows "P (Conj\<^sub>\<alpha> tset\<^sub>\<alpha>)"
the_inv_into_f_eq: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and x :: "'b"   and y :: "'c" assumes "inj_on f A"   and "f x = y"   and "x \<in> A" shows "the_inv_into A f y = x"
i_expand_nth_if: fixes k :: "nat"   and f :: "nat \<Rightarrow> 'a message_af"   and n :: "nat" assumes "0 < k" shows "(f \<odot> k) n = (if n mod k = 0 then f (n div k) else NoMsg)"
f_def: fixes i :: "nat" shows "f i = insertion (nth_default 0 (L[var := sqrt SQ])) (isolate_variable_sparse p var i)"
assms: shows "z \<in> set (Term_Utils.map2 f xs ys)"
addr_of_sys_xcpt_def: fixes s :: "char list" shows "addr_of_sys_xcpt s \<equiv> if s = NullPointer then 0 else if s = ClassCast then 1 else if s = OutOfMemory then 2 else if s = NoClassDefFoundError then 3 else if s = IncompatibleClassChangeError then 4 else if s = NoSuchFieldError then 5 else if s = NoSuchMethodError then 6 else undefined"
rel_Grp: fixes A1 :: "'a set"   and f1 :: "'a \<Rightarrow> 'c"   and A2 :: "'b set"   and f2 :: "'b \<Rightarrow> 'd" shows "rel_prod (Grp A1 f1) (Grp A2 f2) = Grp {x. fsts x \<subseteq> A1 \<and> snds x \<subseteq> A2} (map_prod f1 f2)"
case_eq_if: fixes f :: "int \<Rightarrow> int \<Rightarrow> 'a"   and float10 :: "float10" shows "(case float10 of x \<e> xa \<Rightarrow> f x xa) = f (mantissa10 float10) (exponent10 float10)"
d_def: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "QR.d p q = card (QR.D p q)"
changes_itv_der_def: fixes a :: "real"   and b :: "real"   and p :: "real poly" shows "changes_itv_der a b p = (let ps = pders p in changes_poly_at ps a - changes_poly_at ps b)"
RedBlack_subpaths_from_def: fixes prb :: "('d, 'e, 'f, 'g) pre_RedBlack_scheme"   and rv :: "'d \<times> nat" shows "RedBlack_subpaths_from prb rv \<equiv> ui_es ` {res. \<exists>rv'. SubRel.subpath (red prb) rv res rv' (subs prb) \<and> \<not> marked prb rv'} \<union> {ui_es res1 @ bes2 |res1 bes2. \<exists>rv1. rv1 \<in> fringe prb \<and> SubRel.subpath (red prb) rv res1 rv1 (subs prb) \<and> (\<nexists>res21 bes22. bes2 = ui_es res21 @ bes22 \<and> res21 \<noteq> [] \<and> SubRel.subpath_from (red prb) rv1 res21 (subs prb)) \<and> Graph.subpath_from (black prb) (fst rv1) bes2}"
in_arcsD2: fixes u :: "'a"   and v :: "'a" assumes "(u, v) \<in> parcs sG" shows "v \<in> pverts sG"
sub_fields: fixes CT :: "nat \<Rightarrow> classDef option"   and C :: "nat"   and D :: "nat"   and Dg :: "varDef list" assumes "CT \<turnstile> C <: D"   and "fields(CT,D) = Dg" shows "\<exists>Cf. fields(CT,C) = (Dg @ Cf)"
inj_on_multc: fixes c :: "'a" assumes "c \<in> carrier G" shows "inj_on (\<lambda>x. x \<otimes> c) (carrier G)"
proj_mult_scalar: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and k :: "'c"   and p :: "'b \<Rightarrow>\<^sub>0 'd"   and q :: "'a \<Rightarrow>\<^sub>0 'd" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.proj_poly term_of_pair k (term_powerprod.mult_scalar pair_of_term term_of_pair p q) = p * term_powerprod.proj_poly term_of_pair k q"
ScopeExt: fixes xvec :: "name list"   and \<Psi>' :: "'b"   and T :: "('a, 'b, 'c) psi"   and S :: "('a, 'b, 'c) psi" assumes "xvec \<sharp>* \<Psi>'"   and "xvec \<sharp>* T" shows "(\<Psi>', \<lparr>\<nu>*xvec\<rparr>S \<parallel> T, (\<lparr>\<nu>*xvec\<rparr>S) \<parallel> T) \<in> Rel"
map2_val_compat'D2: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and f :: "('b \<times> 'c) list \<times> 'a \<Rightarrow> ('b \<times> 'd) list \<times> 'a"   and zs :: "('b \<times> 'c) list \<times> 'a" assumes "oalist_raw.map2_val_compat' rep_key_order f"   and "oalist_raw.oalist_inv rep_key_order zs" shows "snd (f zs) = snd zs"
recurrent_iffI_communicating: fixes x :: "'a"   and y :: "'a"   and K :: "'a \<Rightarrow> 'a pmf" assumes "(x, y) \<in> MC_syntax.communicating K" shows "MC_syntax.recurrent K x = MC_syntax.recurrent K y"
simps: 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 a :: "('d, 'c, 'a) cmd" assumes "PL_Indis cval indis" shows "PL_Indis.siso aval tval cval indis a = (\<exists>c. a = c \<and> (\<forall>x xa. xa < PL.brn c \<longrightarrow> PL_Indis.siso aval tval cval indis (PL.cont aval tval cval c x xa)) \<and> (\<forall>x xa xb. xb < PL.brn c \<and> (x, xa) \<in> indis \<longrightarrow> (PL.eff aval tval cval c x xb, PL.eff aval tval cval c xa xb) \<in> indis \<and> PL.wt aval tval cval c x xb = PL.wt aval tval cval c xa xb \<and> PL.cont aval tval cval c x xb = PL.cont aval tval cval c xa xb))"
incremental_atom_ops: shows "Incremental_Atom_Ops init_state assert_s check_s \<V> checkpoint_s backtrack_s \<triangle> invariant_s checked_s"
Rep_perm_simps_2: fixes p1 :: "perm"   and p2 :: "perm" shows "Rep_perm (p1 + p2) = Rep_perm p1 \<circ> Rep_perm p2"
new_edge_subdivFace': fixes g :: "graph"   and f :: "face"   and u :: "nat"   and v :: "nat"   and n :: "nat"   and ovs :: "nat option list" assumes "pre_subdivFace' g f u v n ovs"   and "minGraphProps g"   and "f \<in> \<F> g" shows "subdivFace' g f v n ovs = makeFaceFinal f g \<or> (\<forall>f'\<in>\<F> (subdivFace' g f v n ovs) - (\<F> g - {f}). \<exists>e\<in>\<E> f'. e \<notin> \<E> g)"
mask_split_sum_twice: fixes b :: "nat"   and a :: "nat"   and p :: "'b word" assumes "b \<le> a" shows "(p AND NOT (mask a)) + ((p AND mask a) AND NOT (mask b)) + (p AND mask b) = p"
map_F_comp: fixes l1 :: "'ag \<Rightarrow> 'x"   and l2 :: "'ah \<Rightarrow> 'y"   and l3 :: "'ai \<Rightarrow> 'z"   and co1 :: "'aj \<Rightarrow> 'aa"   and co2 :: "'ak \<Rightarrow> 'ab"   and co3 :: "'al \<Rightarrow> 'ac"   and contra1 :: "'ad \<Rightarrow> 'am"   and contra2 :: "'ae \<Rightarrow> 'an"   and contra3 :: "'af \<Rightarrow> 'ao"   and l1' :: "'n \<Rightarrow> 'ag"   and l2' :: "'o \<Rightarrow> 'ah"   and l3' :: "'p \<Rightarrow> 'ai"   and co1' :: "'q \<Rightarrow> 'aj"   and co2' :: "'r \<Rightarrow> 'ak"   and co3' :: "'s \<Rightarrow> 'al"   and contra1' :: "'am \<Rightarrow> 't"   and contra2' :: "'an \<Rightarrow> 'u"   and contra3' :: "'ao \<Rightarrow> 'v" shows "map_F l1 l2 l3 co1 co2 co3 contra1 contra2 contra3 \<circ> map_F l1' l2' l3' co1' co2' co3' contra1' contra2' contra3' = map_F (l1 \<circ> l1') (l2 \<circ> l2') (l3 \<circ> l3') (co1 \<circ> co1') (co2 \<circ> co2') (co3 \<circ> co3') (contra1' \<circ> contra1) (contra2' \<circ> contra2) (contra3' \<circ> contra3)"
sat_slice_strong: fixes v :: "event_data list"   and S :: "event_data list set"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and V' :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and \<phi> :: "formula"   and E :: "(String.literal \<times> event_data list) set"   and \<sigma> :: "(String.literal \<times> event_data list) Trace.trace"   and i :: "nat" assumes "v \<in> S"   and "dom V = dom V'"   and "\<And>p v i. \<lbrakk>p \<in> dom V; (p, v) \<in> relevant_events \<phi> S\<rbrakk> \<Longrightarrow> (v \<in> the (V p) i) = (v \<in> the (V' p) i)"   and "relevant_events \<phi> S - {e. fst e \<in> dom V} \<subseteq> E" shows "sat \<sigma> V v i \<phi> = sat (map_\<Gamma> (\<lambda>D. D \<inter> E) \<sigma>) V' v i \<phi>"
prveqlPT_nprv_instInpP_instInp: fixes \<phi> :: "'fmla"   and \<tau> :: "'fmla"   and t :: "'trm" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> \<subseteq> {inp}"   and "\<tau> \<in> ptrm 0"   and "t \<in> trm"   and "FvarsT t = {}"   and "prveqlPT \<tau> t" shows "nprv {instInp \<phi> t} (instInpP \<phi> 0 \<tau>)"
skip: fixes P :: "'a \<Rightarrow> bool" shows "\<lbrace>P\<rbrace> skip\<^sub>S\<^sub>E \<lbrace>\<lambda>_. P\<rbrace>"
round_add_inv: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and p :: "nat"   and q :: "nat"   and t :: "nat \<times> nat list \<times> 'c list"   and n :: "nat"   and r :: "nat" assumes "index_less index key"   and "bn_inv p q t"   and "add_inv n t" shows "add_inv n (round index key p q r t)"
assm: shows "consec i j \<and> n \<notin> Rep_nat_int i \<union> Rep_nat_int j"
disc_eq_case_2: fixes llist :: "'c llist" shows "(\<not> lnull llist) = (case llist of LNil \<Rightarrow> False | LCons uu_ uua_ \<Rightarrow> True)"
rev_eqvt: fixes p :: "perm"   and xs :: "'a list" shows "p \<bullet> rev xs = rev (p \<bullet> xs)"
rotate13: fixes A13 :: "fm"   and A1 :: "fm"   and A2 :: "fm"   and A3 :: "fm"   and A4 :: "fm"   and A5 :: "fm"   and A6 :: "fm"   and A7 :: "fm"   and A8 :: "fm"   and A9 :: "fm"   and A10 :: "fm"   and A11 :: "fm"   and A12 :: "fm"   and H :: "fm set"   and B :: "fm" assumes "insert A13 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 (insert A12 H)))))))))))) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 (insert A12 (insert A13 H)))))))))))) \<turnstile> B"
simps_1: fixes C :: "String.literal"   and A :: "'f set option" shows "\<D> (new C) A = True"
supp_fmap_update: fixes x :: "'a"   and y :: "'b"   and f :: "('a, 'b) fmap" shows "supp f(x $$:= y) \<subseteq> supp (f, x, y)"
filter_id_conv: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
induct: fixes P :: "ctx \<Rightarrow> bool"   and ctx :: "ctx" assumes "P \<box>"   and "\<And>x1 x2. P x1 \<Longrightarrow> P (CAPPL x1 x2)"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (CAPPR x1 x2)"   and "\<And>x1 x2. P x1 \<Longrightarrow> P (CDIFFL x1 x2)"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (CDIFFR x1 x2)"   and "\<And>x1 x2. P x1 \<Longrightarrow> P (CPLUSL x1 x2)"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (CPLUSR x1 x2)"   and "\<And>x1 x2 x3. P x1 \<Longrightarrow> P (CIF x1 x2 x3)"   and "\<And>x. P x \<Longrightarrow> P (CZET x)"   and "\<And>x1 x2. P x1 \<Longrightarrow> P (CEQIL x1 x2)"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (CEQIR x1 x2)" shows "P ctx"
mset_le_decr_left1: fixes a :: "'b multiset"   and c :: "'b multiset"   and b :: "'b multiset" assumes "a + c \<subseteq># b" shows "a \<subseteq># b"
option_rel_Some1: fixes A :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and x :: "'l"   and y :: "'m option" shows "rel_option A (Some x) y = (\<exists>y'. y = Some y' \<and> A x y')"
rel_refl_strong: fixes x :: "'a behaviour"   and Ra :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>z. z \<in> set_behaviour x \<Longrightarrow> Ra z z" shows "rel_behaviour Ra x x"
to_fun_diff_factor: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and f :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp"   and "f \<in> carrier (UP Zp)" shows "\<exists>c. c \<in> carrier Zp \<and> f \<bullet> a \<ominus> f \<bullet> b = (a \<ominus> b) \<otimes> c"
borel_measurable_indicator_iff: fixes A :: "'b set"   and M :: "'b measure" shows "(indicator A \<in> borel_measurable M) = (A \<inter> space M \<in> sets M)"
otherassms_7: shows "p \<in> sterms \<Gamma> pp"
aA: shows "a \<squnion> \<Squnion>A \<in> carrier L"
invertible_row_add: fixes a :: "'a"   and b :: "'a"   and q :: "'b" assumes "a \<noteq> b" shows "invertible (row_add (mat (1::'b)) a b q)"
fps_right_inverse: fixes f :: "'a fps"   and y :: "'a" assumes "fps_nth f 0 * y = (1::'a)" shows "f * fps_right_inverse f y = 1"
preprocess_index: fixes cs :: "('i \<times> 'a ns_constraint) list"   and t :: "(nat \<times> linear_poly) list"   and as :: "('i \<times> 'a atom) list"   and trans_v :: "(nat, 'a) mapping \<Rightarrow> (nat, 'a) mapping"   and U :: "'i list" assumes "preprocess cs = (t, as, trans_v, U)" shows "fst ` set as \<union> set U \<subseteq> fst ` set cs"
L_bins_lower_bound_card: fixes P :: "'a set set"   and P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and V :: "'a set" assumes "bp P"   and "inv\<^sub>1 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V"   and "\<forall>B\<in>P\<^sub>1 \<union> wrap B\<^sub>1. B \<inter> L \<noteq> {}" shows "card (P\<^sub>1 \<union> wrap B\<^sub>1 \<union> {{v} |v. v \<in> V \<inter> L}) \<le> card P"
interp_ins_last_None: fixes oid :: "'a"   and ops :: "('a \<times> 'a option) list" shows "oid \<in> set (interp_ins (ops @ [(oid, None)]))"
Ord_mk_doubleton_image_vsubset_Vset_succ: fixes \<alpha> :: "V"   and X :: "V"   and Y :: "V" assumes "Ord \<alpha>"   and "X \<subseteq>\<^sub>\<circ> Y"   and "Y \<in>\<^sub>\<circ> Vset \<alpha>" shows "mk_doubleton_image X Y \<subseteq>\<^sub>\<circ> Vset (ZFC_in_HOL.succ \<alpha>)"
HPhase0Read_HInv4b: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase0Read s s' p d"   and "HInv4b s q" shows "HInv4b s' q"
var_asm_not_written_def: fixes mds :: "Mode \<Rightarrow> 'c set"   and x :: "'c" shows "var_asm_not_written mds x \<equiv> x \<in> mds AsmNoWrite \<or> x \<in> mds AsmNoReadOrWrite"
jointfun_mem: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'd"   and A :: "'d set"   and m :: "nat"   and g :: "nat \<Rightarrow> 'd"   and B :: "'d set"   and l :: "nat" assumes "\<forall>j\<le>n. f j \<in> A"   and "\<forall>j\<le>m. g j \<in> B"   and "l \<le> Suc (n + m)" shows "jointfun n f m g l \<in> A \<union> B"
converg: fixes t :: "nat" assumes "t < t\<^sub>0" shows "local.prefixes t j \<down>"
order_refl: fixes x :: "'f multiset" shows "x \<subseteq># x"
pure_pure: fixes P :: "('f \<times> 'e) set" shows "is_pure (pure P)"
seg_path: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<in> segment a b" obtains ab :: "'a set" where "path ab a b" and "segment a b \<subseteq> ab"
assms_1: shows "typeof (sglob_val s i j) = tg_t (global \<C> ! j)"
assertLiteralQEffect: fixes state :: "State"   and literal :: "Literal"   and decision :: "bool" assumes "InvariantConsistent (getM state @ [(literal, decision)])"   and "InvariantUniq (getM state @ [(literal, decision)])"   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)" shows "let state' = assertLiteral literal decision state in set (getQ state') = set (getQ state) \<union> {ul. \<exists>uc. uc el getF state \<and> opposite literal el uc \<and> isUnitClause uc ul (elements (getM state) @ [literal])}"
atms_simps_1: fixes n :: "nat" shows "atms (regex.Skip n) = {}"
inv_solve_right: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "(a = b \<otimes> inv c) = (b = a \<otimes> c)"
NSLIM_D: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and L :: "'b"   and x :: "'a star" assumes "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L"   and "x \<noteq> star_of a"   and "x \<approx> star_of a" shows "(*f* f) x \<approx> star_of L"
map_append_inv: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and ys :: "'a list"   and zs :: "'a list" assumes "map f xs = ys @ zs" shows "\<exists>vs ws. xs = vs @ ws \<and> map f vs = ys \<and> map f ws = zs"
norm_smult: fixes a :: "real"   and x :: "(real, 'a) Finite_Cartesian_Product.vec" shows "norm (a *s x) = \<bar>a\<bar> * norm x"
cosine_law_triangle'': fixes b :: "'b"   and a :: "'b"   and c :: "'b" shows "cos (angle b a c) = ((dist a b)\<^sup>2 + (dist a c)\<^sup>2 - (dist b c)\<^sup>2) / (2 * dist a b * dist a c)"
simps: fixes a :: "'b formula multiset" shows "a \<Rightarrow>\<^sub>n = ((\<exists>k \<Gamma>. a = \<^bold>\<not> (Atom k), Atom k, \<Gamma>) \<or> (\<exists>\<Gamma>. a = \<bottom>, \<Gamma>) \<or> (\<exists>F G \<Gamma>. a = F \<^bold>\<and> G, \<Gamma> \<and> F, G, \<Gamma> \<Rightarrow>\<^sub>n) \<or> (\<exists>F \<Gamma> G. a = F \<^bold>\<or> G, \<Gamma> \<and> F, \<Gamma> \<Rightarrow>\<^sub>n \<and> G, \<Gamma> \<Rightarrow>\<^sub>n) \<or> (\<exists>F G \<Gamma>. a = \<^bold>\<not> (F \<^bold>\<or> G), \<Gamma> \<and> \<^bold>\<not> F, \<^bold>\<not> G, \<Gamma> \<Rightarrow>\<^sub>n) \<or> (\<exists>F \<Gamma> G. a = \<^bold>\<not> (F \<^bold>\<and> G), \<Gamma> \<and> \<^bold>\<not> F, \<Gamma> \<Rightarrow>\<^sub>n \<and> \<^bold>\<not> G, \<Gamma> \<Rightarrow>\<^sub>n) \<or> (\<exists>F \<Gamma> G. a = F \<^bold>\<rightarrow> G, \<Gamma> \<and> \<^bold>\<not> F, \<Gamma> \<Rightarrow>\<^sub>n \<and> G, \<Gamma> \<Rightarrow>\<^sub>n) \<or> (\<exists>F G \<Gamma>. a = \<^bold>\<not> (F \<^bold>\<rightarrow> G), \<Gamma> \<and> F, \<^bold>\<not> G, \<Gamma> \<Rightarrow>\<^sub>n) \<or> (\<exists>F \<Gamma>. a = \<^bold>\<not> (\<^bold>\<not> F), \<Gamma> \<and> F, \<Gamma> \<Rightarrow>\<^sub>n))"
map_seq: fixes f :: "'j" assumes "J.arr f" shows "C.seq (Fun_g.map (J.cod f)) (Fun_f.map f)"
update_convs_2: fixes edges' :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"   and nodes :: "'a set"   and edges :: "('a \<times> 'a) set"   and more :: "'b" shows "edges_update edges' \<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = \<lparr>nodes = nodes, edges = edges' edges, \<dots> = more\<rparr>"
connected_diff_open_from_closed: fixes s :: "'a set"   and t :: "'a set"   and u :: "'a set" assumes "s \<subseteq> t"   and "t \<subseteq> u"   and "open s"   and "closed t"   and "connected u"   and "connected (t - s)" shows "connected (u - s)"
pos_decomp_def: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) \<times> 'x set) list" shows "ps\<^sub>+ = filter (\<lambda>p. snd p \<noteq> {}) ps"
62: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> (x \<sqinter> - y \<squnion> x \<sqinter> - - y) = x"
absorb2: fixes b :: "'c"   and a :: "'c" assumes "b \<le> a" shows "inf a b = b"
zero_divisor_def: fixes x :: "'b"   and y :: "'b" shows "zw.target.zero_divisor x y \<equiv> x \<noteq> \<zero>\<^bsub>w\<^esub> \<and> y \<noteq> \<zero>\<^bsub>w\<^esub> \<and> \<cdot>\<^bsub>w\<^esub> x y = \<zero>\<^bsub>w\<^esub>"
prj_simps_1: shows "C.arr src_src.prj\<^sub>1"
cos_paired: fixes x :: "real" shows "(\<lambda>n. (- 1) ^ n / fact (2 * n) * x ^ (2 * n)) sums cos x"
Obj: shows "\<AA>\<lparr>Obj\<rparr> = []\<^sub>\<circ>"
OV_inv2_def: shows "OV_inv2 = {s. \<forall>r\<ge>next_round s. obsv_state.obs s r = Map.empty}"
substT2_fresh_switch: fixes r :: "'trm"   and t :: "'trm"   and s :: "'trm"   and x :: "'var"   and y :: "'var" assumes "r \<in> atrm"   and "t \<in> trm"   and "s \<in> trm"   and "x \<in> var"   and "y \<in> var"   and "x \<noteq> y"   and "x \<notin> FvarsT s"   and "y \<notin> FvarsT t" shows "substT (substT r s y) t x = substT (substT r t x) s y"
S3LegalRcvArg: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> S3 rmhist p \<longrightarrow> IsLegalRcvArg<arg<crCh!p>>"
rsp: shows "((\<approx>) ===> (=)) List.member List.member"
sumlist_rev: fixes xs :: "'a list" assumes "set xs \<subseteq> carrier G" shows "sumlist (rev xs) = sumlist xs"
bind_assoc: fixes x :: "'m"   and f :: "'s \<Rightarrow> 'm"   and g :: "'s \<Rightarrow> 'm" shows "bind (bind x f) g = bind x (\<lambda>y. bind (f y) g)"
is_extensional: fixes f :: "'a" assumes "\<not> arr f" shows "\<Delta>o\<delta>_\<pi>o\<Delta>.map f = CC.null"
dnorm_cont: fixes Tr :: "'c \<Rightarrow> 'b set"   and Ur :: "'c \<Rightarrow> 'b \<Rightarrow> 'a" shows "cont (\<lambda>X s. Mprefix (Tr s) (\<lambda>e. X (Ur s e)))"
position_of_left: fixes t :: "('c, 'd) tm"   and s :: "('c, 'd) tm"   and ds :: "dir list" shows "position_of (App t s) (dir.Left # ds) = position_of t ds"
inverts_mat_def: fixes A :: "'b mat"   and B :: "'b mat" shows "inverts_mat A B \<equiv> A * B = 1\<^sub>m (dim_row A)"
independent_append: fixes Us :: "'a list"   and Vs :: "'a list" assumes "local.independent K Us"   and "local.independent K Vs"   and "Span K Us \<inter> Span K Vs = {\<zero>}" shows "local.independent K (Us @ Vs)"
sha1: fixes x :: "bit list" shows "sha1 x = (let y = sha1padd x in sha1block (getblock y) (delblock y) IV1 IV2 IV3 IV4 IV5)"
rroots2_def: fixes p :: "real poly" shows "rroots2 p = (let a = coeff p 2; b = coeff p 1; c = coeff p 0; b2a = b / (2 * a); bac = b2a\<^sup>2 - c / a in if bac = 0 then [- b2a] else if bac < 0 then [] else let e = sqrt bac in [- b2a + e, - b2a - e])"
simps_1: fixes x :: "'s"   and \<omega> :: "'s stream" shows "shd (force_enabled x \<omega>) = (if shd \<omega> \<in> set_pmf (K x) then shd \<omega> else SOME y. y \<in> set_pmf (K x))"
Sigma_mset_plus_distrib2: fixes A :: "'c multiset"   and B :: "'c \<Rightarrow> 'd multiset"   and C :: "'c \<Rightarrow> 'd multiset" shows "(SIGMAMSET i\<in>#A. B i + C i) = Sigma_mset A B + Sigma_mset A C"
suf_less_Lyndon: fixes w :: "'a list" assumes "w \<noteq> \<epsilon>"   and "\<forall>s. s \<le>ns w \<longrightarrow> s \<noteq> w \<longrightarrow> w <lex s" shows "Lyndon w"
subset_eq_diff_conv: fixes A :: "'d multiset"   and C :: "'d multiset"   and B :: "'d multiset" shows "(A - C \<subseteq># B) = (A \<subseteq># B + C)"
st_vec_of_pmf_pmf_of_st_vec: fixes x :: "'a st_vec" shows "st_vec_of_pmf (pmf_of_st_vec x) = x"
has_finite_products': fixes I :: "'b set" assumes "I \<noteq> UNIV"   and "finite I" shows "has_products I"
selection_def: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset" shows "selection S \<equiv> (\<forall>C. S C \<subseteq># C) \<and> (\<forall>L C. L \<in># S C \<longrightarrow> is_neg L)"
tag_name_eq_h2: fixes ptr' :: "(_) element_ptr"   and disc_nodes :: "char list" assumes "ptr' \<noteq> new_element_ptr" shows "h2 \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r disc_nodes = h3 \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r disc_nodes"
simple_distributed_finite: fixes M :: "'c measure"   and X :: "'c \<Rightarrow> 'd"   and P :: "'d \<Rightarrow> real" assumes "simple_distributed M X P" shows "finite (X ` space M)"
W_automorphism: fixes w :: "'a permutation" assumes "w \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>" shows "ChamberComplexAutomorphism X ((\<rightarrow>) w)"
primorial_def: fixes x :: "real" shows "primorial x = \<Prod>{p. prime p \<and> real p \<le> x}"
set_mono_prefix: fixes xs :: "'a list"   and ys :: "'a list" assumes "prefix xs ys" shows "set xs \<subseteq> set ys"
snoc_2: fixes y :: "'a"   and x :: "'a" shows "lexext_rev gt (y # ys) (x # xs) = (lexext_rev gt ys xs \<or> ys = xs \<and> gt y x)"
Abs_var_inject: fixes x :: "atom"   and y :: "atom" assumes "x \<in> {a. sort_of a \<in> range sort_of_ty}"   and "y \<in> {a. sort_of a \<in> range sort_of_ty}" shows "(Abs_var x = Abs_var y) = (x = y)"
geodesic_subsegment_2: fixes G :: "'b set"   and x :: "'b"   and y :: "'b"   and s :: "real"   and t :: "real" assumes "geodesic_segment_between G x y"   and "0 \<le> s"   and "s \<le> t"   and "t \<le> dist x y" shows "geodesic_segment_between (geodesic_subsegment G x s t) (geodesic_segment_param G x s) (geodesic_segment_param G x t)"
compute_float_numeral: fixes k :: "num" shows "Float (numeral k) 0 = numeral k"
advantage_def: fixes \<A> :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool spmf" shows "advantage \<A> = \<bar>spmf (ddh_0 \<A>) True - spmf (ddh_1 \<A>) True\<bar>"
right_closed: fixes S :: "'d set"   and E :: "('d \<times> 'd) set"   and a :: "'d"   and b :: "'d" assumes "equivalence S E"   and "(a, b) \<in> E" shows "b \<in> S"