hcju/isabelleps huggingface dataset:
pdiv_def: fixes p :: "'a list"   and q :: "'a list" shows "p pdiv q = (if q = [] then [] else fst (long_division p q))"
IDE_mkarr_Ide: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b local.term" assumes "free_monoidal_category C"   and "monoidal_language.Ide C a" shows "free_monoidal_category.IDE C (free_monoidal_category.mkarr C a)"
nxtAct_def: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "\<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub> \<equiv> THE n'. n \<le> n' \<and> \<parallel>c\<parallel>\<^bsub>t n'\<^esub> \<and> \<not> (\<exists>k\<ge>n. k < n' \<and> \<parallel>c\<parallel>\<^bsub>t k\<^esub>)"
intro: fixes agents :: "'a set"   and alts :: "'b set" assumes "finite agents"   and "finite alts"   and "agents \<noteq> {}"   and "alts \<noteq> {}" shows "election agents alts"
nn_integral_completion: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" shows "integral\<^sup>N (completion M) f = integral\<^sup>N M f"
measure_Un_le: fixes A :: "'b set"   and M :: "'b measure"   and B :: "'b set" assumes "A \<in> sets M"   and "B \<in> sets M" shows "Sigma_Algebra.measure M (A \<union> B) \<le> Sigma_Algebra.measure M A + Sigma_Algebra.measure M B"
freshEnv_psubst_ident: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option" assumes "good X"   and "goodEnv rho"   and "\<And>zs z. freshEnv zs z rho \<or> fresh zs z X" shows "X #[rho] = X"
discr_transT: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "('b, 'c) com"   and s :: "'a"   and s' :: "'a" assumes "PL_Indis indis"   and "PL_Indis.discr tval aval indis c"   and "PL.transT tval aval (c, s) s'" shows "(s, s') \<in> indis"
set_mset_filter: fixes P :: "'b \<Rightarrow> bool"   and M :: "'b multiset" shows "set_mset (filter_mset P M) = {a. a \<in># M \<and> P a}"
infs_coinduct: fixes R :: "'b stream \<Rightarrow> bool"   and w :: "'b stream"   and P :: "'b \<Rightarrow> bool" assumes "R w"   and "\<And>w. R w \<Longrightarrow> Bex (sset w) P \<and> ev R (stl w)" shows "infs P w"
ev_mono: fixes \<phi> :: "'c stream \<Rightarrow> bool"   and xs :: "'c stream"   and \<psi> :: "'c stream \<Rightarrow> bool" assumes "ev \<phi> xs"   and "\<And>xs. \<phi> xs \<Longrightarrow> \<psi> xs" shows "ev \<psi> xs"
vimage2p_def: fixes f :: "'g \<Rightarrow> 'j"   and g :: "'h \<Rightarrow> 'k"   and R :: "'j \<Rightarrow> 'k \<Rightarrow> 'i" shows "BNF_Def.vimage2p f g R = (\<lambda>x y. R (f x) (g y))"
coreflexive_injective: fixes x :: "'a" assumes "x \<le> top" shows "x \<sqinter> id x \<le> top"
no_loop: fixes x :: "'v vertex" shows "\<not> edge \<Psi> x x"
exists_twice: fixes x :: "'a \<Longrightarrow> 'b"   and P :: "'b upred" assumes "mwb_lens x" shows "(\<exists> x \<bullet> \<exists> x \<bullet> P) = (\<exists> x \<bullet> P)"
dec_ex: fixes u :: "'b list"   and G :: "'b list set" assumes "u \<in> \<langle>G\<rangle>" shows "\<exists>us. us \<in> lists G\<^sub>+ \<and> concat us = u"
set_ar_cos_def: fixes A :: "('c, 'd) aGroup_scheme"   and I :: "'c set" shows "set_ar_cos A I = {X. \<exists>a\<in>carrier A. X = a \<uplus>\<^bsub>A\<^esub> I}"
dom_minus: fixes f :: "'c \<Rightarrow> 'b option"   and x :: "'c"   and A :: "'c set" assumes "f x = None" shows "dom f - insert x A = dom f - A"
next_execs_def: fixes current :: "'c \<Rightarrow> 'a"   and control :: "'c \<Rightarrow> 'a \<Rightarrow> 'b list list \<Rightarrow> 'b option \<times> 'b list list \<times> 'c"   and s :: "'c"   and execs :: "'a \<Rightarrow> 'b list list" shows "Kernel.next_execs current control s execs = execs(current s := fst (snd (control s (current s) (execs (current s)))))"
freshChainPerm: fixes xvec :: "name list"   and yvec :: "name list"   and C :: "'a"   and x :: "name" assumes "length xvec = length yvec"   and "yvec \<sharp>* C"   and "xvec \<sharp>* yvec"   and "x mem xvec"   and "distinct yvec" shows "x \<sharp> [xvec yvec] \<bullet>\<^sub>v C"
intros: fixes Enc :: "'a \<Rightarrow> 'b"   and S :: "'a" shows "(TargetTerm (Enc S), SourceTerm S) \<in> encoding.indRelL Enc"
hm_size_impl: shows "imp_map_size is_hashmap hm_size"
le_sum_encode_Inr: fixes x :: "nat"   and y :: "nat" assumes "x \<le> y" shows "x \<le> sum_encode (Inr y)"
p_3: fixes i :: "nat" assumes "i < nr"   and "Suc i < nr" shows "f i < f (Suc i) \<or> f (Suc i) = n"
wivs_overflow_def: shows "wivs_overflow \<equiv> 2 ^ (dflt_size - 1)"
ext_inject: fixes aActs :: "'b"   and aTrans :: "'c"   and more :: "'d"   and aActs' :: "'b"   and aTrans' :: "'c"   and more' :: "'d" shows "(\<lparr>aActs = aActs, aTrans = aTrans, \<dots> = more\<rparr> = \<lparr>aActs = aActs', aTrans = aTrans', \<dots> = more'\<rparr>) = (aActs = aActs' \<and> aTrans = aTrans' \<and> more = more')"
measure_translation: fixes a :: "'b"   and S :: "'b set" shows "Sigma_Algebra.measure lebesgue ((+) a ` S) = Sigma_Algebra.measure lebesgue S"
case_sum_map_sum: fixes l :: "'e \<Rightarrow> 'd"   and r :: "'f \<Rightarrow> 'd"   and f :: "'g \<Rightarrow> 'e"   and g :: "'h \<Rightarrow> 'f"   and x :: "'g + 'h" shows "(case map_sum f g x of Inl x \<Rightarrow> l x | Inr x \<Rightarrow> r x) = (case x of Inl x \<Rightarrow> (l \<circ> f) x | Inr x \<Rightarrow> (r \<circ> g) x)"
rep_eq: fixes x :: "('c, 'd) sq_matrix" shows "Square_Matrix.trace x = (\<Sum>i\<in>UNIV. to_fun x i i)"
tell_writerT_def: shows "tell = tell_writer Monad_Overloading.return Monad_Overloading.bind"
Image_iff: fixes b :: "'d"   and r :: "('e \<times> 'd) set"   and A :: "'e set" shows "(b \<in> r `` A) = (\<exists>x\<in>A. (x, b) \<in> r)"
apps_eq_Abs_conv: fixes s :: "lam"   and ss :: "lam list"   and x :: "name"   and r :: "lam" shows "(s \<degree>\<degree> ss = (Lam [x].r)) = (s = (Lam [x].r) \<and> ss = [])"
base_in_hom_1: shows "\<guillemotleft>F r : src\<^sub>D (F r) \<rightarrow>\<^sub>D trg\<^sub>D (F r)\<guillemotright>"
Plus: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and A :: "'c set"   and B :: "'d set"   and g :: "'c + 'd \<Rightarrow> 'b" assumes "comm_monoid_set f z"   and "finite A"   and "finite B" shows "comm_monoid_set.F f z g (A <+> B) = f (comm_monoid_set.F f z (g \<circ> Inl) A) (comm_monoid_set.F f z (g \<circ> Inr) B)"
Diff_insert: fixes A :: "'d set"   and a :: "'d"   and B :: "'d set" shows "A - insert a B = A - B - {a}"
cobounded1: fixes a :: "'k multiset"   and b :: "'k multiset" shows "a \<subseteq># a \<union># b"
index_minus_mat_2: fixes A :: "'a mat"   and B :: "'a mat" shows "dim_row (A - B) = dim_row B"
less_add_same_cancel2: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less a (plus b a) = less zero b"
cube: fixes x :: "nat \<Rightarrow> nat"   and i :: "nat" assumes "\<forall>i<n. x i \<le> p"   and "i < n"   and "x i = p \<or> x i = 0" shows "(\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<in> cbox (0::'a) One"
nn_integral_0_iff_AE: fixes u :: "'k \<Rightarrow> ennreal"   and M :: "'k measure" assumes "u \<in> borel_measurable M" shows "(integral\<^sup>N M u = 0) = (AE x in M. u x = 0)"
hyps_1: fixes t :: "btrm"   and q :: "pat"   and u :: "btrm" assumes "(\<lambda>[p\<^sub>1]. t) = (\<lambda>[q]. u)"   and "pat_type p\<^sub>1 = pat_type q"   and "distinct (pat_vars p\<^sub>1)"   and "distinct (pat_vars q)" shows "\<exists>pi. p\<^sub>1 = pi \<bullet> q \<and> t = pi \<bullet> u \<and> set pi \<subseteq> (supp p\<^sub>1 \<union> supp q) \<times> (supp p\<^sub>1 \<union> supp q)"
ibind_mono: fixes m :: "'f dres"   and m' :: "'f dres"   and f :: "'f \<Rightarrow> 'f dres"   and f' :: "'f \<Rightarrow> 'f dres" assumes "m \<le> m'"   and "\<And>x. f x \<le> f' x" shows "m \<bind> f \<le> m' \<bind> f'"
Eq_FalseI: fixes P :: "bool" assumes "\<not> P" shows "P \<equiv> False"
jpTraces_init_inv: fixes s :: "'s"   and jp :: "'a \<Rightarrow> ('obs, 'aAct, 'c) Protocol" assumes "tInit s \<in> jpTraces jp" shows "s \<in> set envInit"
assms_2: shows "k = amalgamate (the (f 0)) (the (f 1))"
strategy_attractsI: fixes A :: "'a set"   and p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and W :: "'a set" assumes "\<And>P v. \<lbrakk>v \<in> A; vmc_path G P v p \<sigma>\<rbrakk> \<Longrightarrow> visits_via P A W" shows "strategy_attracts p \<sigma> A W"
pow_mult_distrib: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x \<otimes> y = y \<otimes> x"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "(x \<otimes> y) [^] n = x [^] n \<otimes> y [^] n"
simps_11: fixes b :: "expr"   and e1 :: "expr"   and e2 :: "expr" shows "constVal (b ? e1 : e2) = (case constVal b of None \<Rightarrow> None | Some bv \<Rightarrow> case the_Bool bv of True \<Rightarrow> case constVal e2 of None \<Rightarrow> None | Some v \<Rightarrow> constVal e1 | False \<Rightarrow> case constVal e1 of None \<Rightarrow> None | Some v \<Rightarrow> constVal e2)"
type_equiv_refl: fixes t :: "'BExp set"   and P :: "'BExp set" shows "t =:\<^sub>P t"
CK_card: shows "card {f. CK f} \<le> 14"
zip_simps_2: fixes x :: "'b"   and xs :: "['b]" shows "zip\<cdot>(x : xs)\<cdot>[] = []"
impl_OrdSum_def: fixes a :: "('a, 'b) OrdSum"   and b :: "('a, 'b) OrdSum" shows "a l\<rightarrow> b \<equiv> if second a = (1::'b) \<and> second b = (1::'b) then Abs_OrdSum (first a l\<rightarrow> first b, 1::'b) else if first a = (1::'a) \<and> first b = (1::'a) then Abs_OrdSum (1::'a, second a l\<rightarrow> second b) else if first a = (1::'a) \<and> second b = (1::'b) then b else 1"
S2: shows "b_least2 f x (Suc y) \<le> b_least2 f x y"
Base: fixes v :: "'a"   and W :: "'a set"   and p :: "Player" assumes "v \<in> W" shows "v \<in> attractor_inductive p W"
dom_def: fixes f :: "'a" shows "V.dom f = (if V.domains f \<noteq> {} then SOME a. a \<in> V.domains f else V.null)"
emeasure_bind: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd measure"   and N :: "'d measure"   and X :: "'d set" assumes "space M \<noteq> {}"   and "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N"   and "X \<in> sets N" shows "emeasure (M \<bind> f) X = \<integral>\<^sup>+ x. emeasure (f x) X \<partial>M"
borel_measurable_induct_real: fixes u :: "'c \<Rightarrow> real"   and M :: "'c measure"   and P :: "('c \<Rightarrow> real) \<Rightarrow> bool" assumes "u \<in> borel_measurable M"   and "\<And>x. 0 \<le> u x"   and "\<And>A. A \<in> sets M \<Longrightarrow> P (indicat_real A)"   and "\<And>u c. \<lbrakk>0 \<le> c; u \<in> borel_measurable M; \<And>x. 0 \<le> u x; P u\<rbrakk> \<Longrightarrow> P (\<lambda>x. c * u x)"   and "\<And>u v. \<lbrakk>u \<in> borel_measurable M; \<And>x. 0 \<le> u x; P u; v \<in> borel_measurable M; \<And>x. 0 \<le> v x; \<And>x. x \<in> space M \<Longrightarrow> u x = 0 \<or> v x = 0; P v\<rbrakk> \<Longrightarrow> P (\<lambda>x. v x + u x)"   and "\<And>U. \<lbrakk>\<And>i. U i \<in> borel_measurable M; \<And>i x. 0 \<le> U i x; \<And>i. P (U i); incseq U; \<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. U i x) \<longlonglongrightarrow> u x\<rbrakk> \<Longrightarrow> P u" shows "P u"
le_fin_le_fininf_transp: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list"   and w\<^sub>3 :: "'item stream" assumes "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2"   and "w\<^sub>2 \<preceq>\<^sub>F\<^sub>I w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>3"
extensional0_def: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "extensional0 A f = (\<forall>x. x \<notin> A \<longrightarrow> f x = (0::'d))"
AT: shows "A\<^sup>T \<in> carrier_mat m n"
assms_2: assumes "otherbot \<noteq> default_value" shows "\<exists>G nP i F. wf_graph G \<and> \<not> sinvar G nP \<and> F \<in> SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP \<and> sinvar (delete_edges G F) nP \<and> i \<in> fst ` F \<and> sinvar G (nP(i := otherbot))"
arc_top_arc: fixes x :: "'a" assumes "arc x" shows "x * top * x = x"
microstep_choiceD: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and p1 :: "('b, 'c, 'a, 'd) seqp"   and p2 :: "('b, 'c, 'a, 'd) seqp"   and p :: "('b, 'c, 'a, 'd) seqp" assumes "(p1 \<oplus> p2) \<leadsto>\<^bsub>\<Gamma>\<^esub> p" shows "p = p1 \<or> p = p2"
degen_def: fixes cs :: "('a \<Rightarrow> bool) list" shows "degen cs \<equiv> \<lambda>(a, k). length cs \<le> k \<or> (cs ! k) a"
integer_less_iff: fixes k :: "integer"   and l :: "integer" shows "(k < l) = (int_of_integer k < int_of_integer l)"
zero_in_succ: fixes i :: "V" assumes "Ord i" shows "0 \<in> elts (ZFC_in_HOL.succ i)"
sse_term_rel_def: fixes \<Sigma>i :: "'b set"   and R :: "('b \<times> 'b) set" shows "sse_term_rel \<Sigma>i R \<equiv> {(\<sigma>', \<sigma>). \<sigma> \<in> sse_invar \<Sigma>i R \<and> (\<sigma>, \<sigma>') \<in> sse_step R}"
ZassenhausTr2_4: fixes H :: "'a set"   and H1 :: "'a set"   and h :: "'a"   and h1 :: "'a" assumes "G \<guillemotright> H "   and "G \<guillemotright> H1 "   and "\<natural>H \<triangleright> H1"   and "h \<in> H"   and "h1 \<in> H1" shows "h \<cdot> h1 \<cdot> \<rho> h \<in> H1"
Interleaves_ipurge_tr_aux: fixes I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and U :: "'b set"   and xs :: "'a list" shows "xs \<cong> {ipurge_tr_rev_aux I D U xs, rev (ipurge_tr_aux (I\<inverse>) D U (rev xs)), \<lambda>y ys. \<exists>v\<in>sinks_aux (I\<inverse>) D U (rev ys). (D y, v) \<in> I}"
funpow_swap1: fixes f :: "'c \<Rightarrow> 'c"   and n :: "nat"   and x :: "'c" shows "f ((f ^^ n) x) = (f ^^ n) (f x)"
ss: fixes s :: "'e"   and s' :: "'e"   and x0 :: "'g"   and x1 :: "'a"   and a0 :: "'j"   and a1 :: "'d"   and z0 :: "'k"   and z1 :: "'f"   and z0' :: "'k"   and z1' :: "'f" assumes "s \<noteq> s'"   and "(x0, x1) \<in> valid_pub_AND"   and "s \<in> challenge_space"   and "s' \<in> challenge_space"   and "check_AND (x0, x1) (a0, a1) s (z0, z1)"   and "check_AND (x0, x1) (a0, a1) s' (z0', z1')" shows "lossless_spmf (\<A>ss_AND (x0, x1) ((a0, a1), s, z0, z1) ((a0, a1), s', z0', z1')) \<and> (\<forall>w'\<in>set_spmf (\<A>ss_AND (x0, x1) ((a0, a1), s, z0, z1) ((a0, a1), s', z0', z1')). ((x0, x1), w') \<in> Rel_AND)"
neq0_conv: fixes n :: "nat" shows "(n \<noteq> 0) = (0 \<sqsubset> n)"
verify_def: fixes x :: "'pub_input \<times> 'witness"   and e :: "'challenge"   and a :: "'msg"   and z :: "'response" shows "verify x e a z = check (fst x) a e z"
sub_m_comm: fixes H :: "'e set"   and R :: "('e, 'f) ring_scheme"   and h1 :: "'e"   and h2 :: "'e" assumes "subcring H R"   and "h1 \<in> H"   and "h2 \<in> H" shows "h1 \<otimes>\<^bsub>R\<^esub> h2 = h2 \<otimes>\<^bsub>R\<^esub> h1"
A1: shows "set (divisors_nat 12285) = divisor_set 12285"
mismatchPres: fixes P :: "pi"   and Q :: "pi"   and a :: "name"   and b :: "name" assumes "P \<approx> Q" shows "[a\<noteq>b]P \<approx> [a\<noteq>b]Q"
n': shows "enat n' < llength (lnth (LCons xs' xss'') m)"
nat_diff_left_cancel_le1: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "k - m \<le> k - n"   and "m < k" shows "n \<le> m"
simps_1: fixes all :: "String.literal"   and S :: "typ"   and T :: "typ"   and t :: "Core.term" shows "strip_all_single_body (Ct all S $ Abs T t) = (if all = STR ''Pure.all'' \<and> S = (T \<rightarrow> propT) \<rightarrow> propT then t else Ct all S $ Abs T t)"
SV_inv4_def: shows "SV_inv4 = {s. \<forall>v a r. votes s r a = Some v \<longrightarrow> safe s (Suc r) v}"
path_obs_subset: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node"   and S :: "'node set" assumes "n -as\<rightarrow>* n'"   and "\<forall>n'\<in>set (sourcenodes as). n' \<notin> S" shows "obs n' S \<subseteq> obs n S"
simps_4: fixes ts' :: "ct list"   and ts :: "ct list" shows "produce (TopType ts') (TopType ts) = TopType ts"
elims_2: fixes x :: "'edge list"   and xa :: "'edge list" assumes "valid_path_aux x xa" obtains cs :: "'edge list" where "x = csa__" and "xa = []"   | cs :: "'edge list" and a :: "'edge" and as :: "'edge list" where "x = cs" and "xa = a # as" and "case kind a of Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<Rightarrow> valid_path_aux (a # cs) as | Q\<hookleftarrow>\<^bsub>p\<^esub>f \<Rightarrow> case cs of [] \<Rightarrow> valid_path_aux [] as | c' # cs' \<Rightarrow> a \<in> get_return_edges c' \<and> valid_path_aux cs' as | _ \<Rightarrow> valid_path_aux cs as"
im_of_ideal_is_ideal: fixes I :: "'c set"   and A :: "'c set"   and addA :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multA :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zeroA :: "'c"   and oneA :: "'c"   and f :: "'c \<Rightarrow> 'd"   and B :: "'d set"   and addB :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multB :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zeroB :: "'d"   and oneB :: "'d" assumes "ideal I A addA multA zeroA oneA"   and "ring_epimorphism f A addA multA zeroA oneA B addB multB zeroB oneB" shows "ideal (f ` I) B addB multB zeroB oneB"
one_neq_zero: fixes one :: "'k"   and zero :: "'k" assumes "class.zero_neq_one one zero" shows "one \<noteq> zero"
Ball_False: fixes P :: "'b \<Rightarrow> bool"   and t :: "('b, 'c) RBT_Impl.rbt" shows "RBT_Impl.fold (\<lambda>k v s. s \<and> P k) t False = False"
preSetpreList: fixes xs :: "('a \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat)) list"   and C :: "acom"   and l :: "'a \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" assumes "preList xs C l s" shows "preSet (set xs) C l s"
simps_171: fixes x101 :: "Term.var"   and x102 :: "expr"   and x71 :: "binop"   and x72 :: "expr"   and x73 :: "expr" shows "x101:=x102 \<noteq> BinOp x71 x72 x73"
B_def: shows "B = (\<Union>n. (T ^^ n) --` A)"
r_zero: fixes x :: "'c" assumes "x \<in> carrier V" shows "x \<oplus>\<^bsub>V\<^esub> \<zero>\<^bsub>V\<^esub> = x"
assms_1: shows "s1 \<in> syzygy_module B"
simps_1: fixes n :: "nat"   and n1 :: "('a, 'b) Node" shows "FingerTreeStruc.is_leveln_digit n (One n1) = FingerTreeStruc.is_leveln_node n n1"
seq_S: fixes x :: "'a"   and i :: "nat" shows "seq x i \<in> S"
bdd_below_mono: fixes B :: "'b multiset set"   and A :: "'b multiset set" assumes "subset_mset.bdd_below B"   and "A \<subseteq> B" shows "subset_mset.bdd_below A"
simps: fixes hp :: "'b tree" shows "is_root hp = (case hp of \<langle>\<rangle> \<Rightarrow> True | \<langle>l, x, r\<rangle> \<Rightarrow> r = \<langle>\<rangle>)"
set_deterministic: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and l :: "'l"   and v :: "('r, 'l, 'v) val"   and s' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "s r = Some (\<sigma>, \<tau>, \<E> [Assign (VE (Loc l)) (VE v)])" shows "revision_step r s s' = (l \<in> dom (\<sigma>;;\<tau>) \<and> s' = s(r \<mapsto> (\<sigma>, \<tau>(l \<mapsto> v), \<E> [VE (CV Unit)])))"
is_lasso_def: fixes m :: "'b"   and L :: "'Q lasso" shows "gb_graph.is_lasso (to_gbg_ext m) L \<equiv> graph.is_lasso_pre (to_gbg_ext m) L \<and> (\<forall>A\<in>gbg_F (to_gbg_ext m). set (lasso_cycle L) \<inter> A \<noteq> {})"
induct: fixes x :: "'a llist"   and P :: "'a llist \<Rightarrow> bool" assumes "lfinite x"   and "P LNil"   and "\<And>xs x. \<lbrakk>lfinite xs; P xs\<rbrakk> \<Longrightarrow> P (LCons x xs)" shows "P x"
scaleR_right_mono: 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 scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "real"   and b :: "real"   and x :: "'a" assumes "class.ordered_real_vector plus zero minus uminus less_eq less scaleR"   and "a \<le> b"   and "less_eq zero x" shows "less_eq (scaleR a x) (scaleR b x)"
before_filter: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list"   and u :: "'a"   and v :: "'a" assumes "filter P xs = ys"   and "distinct xs"   and "before ys u v" shows "before xs u v"
contracting_inverse: fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run" assumes "dilating f sub r" shows "contracting (dil_inverse f) r sub f"
mult_ceiling_le: fixes a :: "'a"   and b :: "'a" assumes "(0::'a) \<le> a"   and "(0::'a) \<le> b" shows "\<lceil>a * b\<rceil> \<le> \<lceil>a\<rceil> * \<lceil>b\<rceil>"
simps_1: fixes n :: "int" shows "\<gamma>_const (Const n) = {n}"
infer_valid: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and M :: "'a trm"   and \<tau> :: "PreSimplyTyped.type" shows "\<Gamma> \<turnstile> M : \<tau> = (infer \<Gamma> M = Some \<tau>)"
StepC: fixes P :: "jvm_prog"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and C' :: "char list"   and Cs :: "char list list" assumes "exec_step_input P C M pc ics = StepC C' Cs" shows "ics = Calling C' Cs"
finvimage_flrestriction: fixes r :: "V"   and A :: "V"   and B :: "V" shows "(r \<restriction>\<^sup>l\<^sub>\<bullet> A) -`\<^sub>\<bullet> B = A \<inter>\<^sub>\<circ> r -`\<^sub>\<bullet> B"
bij_betw_congI: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and A' :: "'b set"   and g :: "'a \<Rightarrow> 'b" assumes "bij_betw f A A'"   and "\<forall>a\<in>A. f a = g a" shows "bij_betw g A A'"
is_offending_flows_def: fixes f :: "('v \<times> 'v) set"   and G :: "'v graph"   and nP :: "'v \<Rightarrow> 'a" shows "is_offending_flows f G nP \<equiv> \<not> sinvar G nP \<and> sinvar (delete_edges G f) nP"
openin_Int_open: fixes U :: "'a set"   and S :: "'a set"   and T :: "'a set" assumes "openin (top_of_set U) S"   and "open T" shows "openin (top_of_set U) (S \<inter> T)"
dvd_add_times_triv_left_iff: fixes a :: "'a"   and c :: "'a"   and b :: "'a" shows "(a dvd c * a + b) = (a dvd b)"
ternary_lift6: fixes t1 :: "ternaryvalue"   and t2 :: "ternaryvalue" shows "(eval_ternary_And t1 t2 = TernaryFalse) = (t1 = TernaryFalse \<or> t2 = TernaryFalse)"
inf_sup_distrib2: fixes y :: "'b multiset"   and z :: "'b multiset"   and x :: "'b multiset" shows "y \<union># z \<inter># x = y \<inter># x \<union># (z \<inter># x)"
C: shows "\<forall>X\<in>A. (xs, X) \<in> seq_comp_failures P Q"
advantage_def: fixes \<A> :: "(bool, 'domain, 'range) gpv" shows "advantage \<A> = \<bar>spmf (game_0 \<A>) True - spmf (game_1 \<A>) True\<bar>"
fps_compose_def: fixes a :: "'b fps"   and b :: "'b fps" shows "a oo b = Abs_fps (\<lambda>n. \<Sum>i = 0..n. a $ i * b ^ i $ n)"
nat_mult_max_left: fixes m :: "nat"   and n :: "nat"   and q :: "nat" shows "max m n * q = max (m * q) (n * q)"
iwlsFSbSwTR_defs_1: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'd, 'e, 'f) model_scheme" shows "iwlsFSbSwTR MOD \<equiv> igWlsAllDisj MOD \<and> igWlsAbsIsInBar MOD \<and> igConsIPresIGWlsSTR MOD \<and> igSubstAllIPresIGWlsAllSTR MOD \<and> igFreshClsSTR MOD \<and> igSubstClsSTR MOD \<and> igAbsRenSTR MOD"
simps_63: fixes x121 :: "expr"   and x122 :: "expr"   and x21 :: "char list"   and x22 :: "expr" shows "x121;; x122 \<noteq> Cast x21 x22"
proj_preserves: fixes x :: "'c"   and A :: "'c set"   and r :: "('c \<times> 'c) set" assumes "x \<in> A" shows "Equiv_Relations.proj r x \<in> A // r"
commit_correct: shows "chaum_ped_\<Sigma>_commit.abstract_com.correct"
path2_forget_hd: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m" shows "g \<turnstile> hd ns-ns\<rightarrow>m"