hcju/isabelleps huggingface dataset:
leI: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.linorder less_eq less"   and "\<not> less x y" shows "less_eq y x"
simps: fixes gpv :: "('c, 'out, 'd) gpv" shows "local.interaction_bound gpv = (\<Squnion>generat\<in>set_spmf (the_gpv gpv). case generat of Pure x \<Rightarrow> 0 | IO out c \<Rightarrow> if consider out then eSuc (\<Squnion>input. local.interaction_bound (c input)) else \<Squnion>input. local.interaction_bound (c input))"
intros_6: fixes s :: "'a"   and b :: "'a set"   and c1 :: "'a com"   and s' :: "'a option"   and c2 :: "'a com" assumes "s \<in> b"   and "Sem c1 (Some s) s'" shows "Sem (Cond b c1 c2) (Some s) s'"
validE_3: fixes b :: "binding"   and \<Gamma> :: "binding list" assumes "\<turnstile> (b # \<Gamma>) ok" obtains X :: "tyvrs" and T :: "ty" where "b = TVarB X Ta__" and "\<turnstile> \<Gamma> ok" and "X \<sharp> ty_dom \<Gamma>" and "Ta__ closed_in \<Gamma>"   | x :: "vrs" and T :: "ty" where "b = VarB x T" and "\<turnstile> \<Gamma> ok" and "x \<sharp> trm_dom \<Gamma>" and "T closed_in \<Gamma>"
inv_image_def: fixes r :: "('c \<times> 'c) set"   and f :: "'a \<Rightarrow> 'c" shows "inv_image r f = {(x, y). (f x, f y) \<in> r}"
fresh_star_Cons: fixes as :: "atom set"   and x :: "'b"   and xs :: "'b list" shows "as \<sharp>* (x # xs) = (as \<sharp>* x \<and> as \<sharp>* xs)"
simps_5: fixes x5 :: "('b, 'c) fm"   and y5 :: "('b, 'c) fm" shows "((\<^bold>\<diamond> x5) = (\<^bold>\<diamond> y5)) = (x5 = y5)"
Oadmit_Prod: fixes \<sigma> :: "('a, 'b, 'c) subst"   and ODE1 :: "('a, 'c) ODE"   and U :: "('c + 'c) set"   and ODE2 :: "('a, 'c) ODE" assumes "Oadmit \<sigma> ODE1 U"   and "Oadmit \<sigma> ODE2 U"   and "ODE_dom (Osubst ODE1 \<sigma>) \<inter> ODE_dom (Osubst ODE2 \<sigma>) = {}" shows "Oadmit \<sigma> (OProd ODE1 ODE2) U"
simps_2: fixes n :: "nat"   and m :: "nat" shows "bin_rep_aux (Suc n) m = m div 2 ^ n # bin_rep_aux n (m mod 2 ^ n)"
dvd_field_iff: 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 a :: "'a"   and b :: "'a" assumes "class.field divide inverse times one plus zero minus uminus" shows "dvd.dvd times a b = (a = zero \<longrightarrow> b = zero)"
finite_has_minimal2: fixes A :: "'a list set"   and a :: "'a list" assumes "finite A"   and "a \<in> A" shows "\<exists>m\<in>A. m \<le>lex a \<and> (\<forall>b\<in>A. b \<le>lex m \<longrightarrow> m = b)"
kernel_dim_def: fixes A :: "'b mat" shows "kernel_dim A = vectorspace.dim class_ring (module_vec TYPE('b) (dim_col A)\<lparr>carrier := mat_kernel A\<rparr>)"
commute: fixes a :: "'d upred"   and b :: "'d upred" shows "(a \<and> b) = (b \<and> a)"
coprime_iff_gcd_eq_1: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "algebraic_semidom.coprime times one a b = (gcd a b = one)"
twoCisTypeII_2: shows "\<forall>x\<in>{a..b}. g2 x \<le> g1 x"
simps_2: fixes guard :: "'a \<Rightarrow> bool"   and body :: "'a \<Rightarrow> 'a spmf"   and n :: "nat"   and s :: "'a" shows "loop_spmf.iter guard body (Suc n) s = (if guard s then body s \<bind> loop_spmf.iter guard body n else return_spmf s)"
T2: fixes w :: "'a"   and w' :: "'a"   and \<theta> :: "'a"   and u :: "'a"   and \<theta>' :: "'a"   and \<beta> :: "'a" assumes "ide w"   and "ide w'"   and "\<guillemotleft>\<theta> : u\<^sub>0 \<star> w \<Rightarrow> u\<guillemotright>"   and "\<guillemotleft>\<theta>' : u\<^sub>0 \<star> w' \<Rightarrow> u\<guillemotright>"   and "\<guillemotleft>\<beta> : u\<^sub>1 \<star> w \<Rightarrow> u\<^sub>1 \<star> w'\<guillemotright>"   and "(u \<star> \<theta>) \<cdot> \<a>[u, u\<^sub>0, w] \<cdot> (\<mu> \<star> w) = ((u \<star> \<theta>') \<cdot> \<a>[u, u\<^sub>0, w'] \<cdot> (\<mu> \<star> w')) \<cdot> \<beta>" shows "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w \<Rightarrow> w'\<guillemotright> \<and> \<beta> = u\<^sub>1 \<star> \<gamma> \<and> \<theta> = \<theta>' \<cdot> (u\<^sub>0 \<star> \<gamma>)"
assms_1: shows "pz \<equiv> {w. f w = 0 \<or> w \<in> poles}"
span_finite: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and S :: "'d set" assumes "module scale"   and "finite S" shows "module.span scale S = range (\<lambda>u. \<Sum>v\<in>S. scale (u v) v)"
update_convs_17: fixes sip' :: "nat \<Rightarrow> nat"   and ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and pre :: "nat set"   and rreqid :: "nat"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and handled :: "bool"   and more :: "'a" shows "sip_update sip' \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, pre = pre, rreqid = rreqid, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, handled = handled, \<dots> = more\<rparr> = \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, pre = pre, rreqid = rreqid, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip' sip, handled = handled, \<dots> = more\<rparr>"
bernoulli_Suc: fixes n :: "nat" shows "bernoulli (Suc n) = - 1 / (real n + 2) * (\<Sum>k\<le>n. real (n + 2 choose k) * bernoulli k)"
refl: fixes x :: "'a" assumes "x \<in> alts'" shows "R x x"
measurable_snth: fixes n :: "nat"   and M :: "'b measure" shows "(\<lambda>\<omega>. \<omega> !! n) \<in> stream_space M \<rightarrow>\<^sub>M M"
assms_2: shows "c \<turnstile> ev \<mapsto> c'"
equiv_up_to_seq: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and d :: "com"   and d' :: "com" assumes "P \<Turnstile> c \<sim> c'"   and "Q \<Turnstile> d \<sim> d'"   and "\<And>s s'. \<lbrakk>(c, s) \<Rightarrow> s'; P s\<rbrakk> \<Longrightarrow> Q s'" shows "P \<Turnstile> c;; d \<sim> c';; d'"
Sup_aform'_def: fixes p :: "nat"   and X :: "'a \<times> 'a pdevs" shows "Sup_aform' p X = eucl_truncate_up p (fst X + Radius' p X)"
neg_prod_le: fixes c :: "'b"   and x :: "'b" assumes "c < (0::'b)" shows "c * x \<le> (0::'b) \<equiv> (0::'b) \<le> x"
right_neutral: fixes a :: "'b \<Rightarrow> bool" shows "(a \<and>* \<box>) = a"
circ_plus_one: shows "mtop = mtop \<oplus> mtop"
finter_commute: fixes A :: "'a fset"   and B :: "'a fset" shows "A |\<inter>| B = B |\<inter>| A"
rg_fo_inv: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and cf :: "nat \<times> nat \<Rightarrow> 'a" assumes "RPreGraph c s t cf" shows "residualGraph c (Network.flow_of_cf c cf) = cf"
space_restrict_space2: fixes \<Omega> :: "'a set"   and M :: "'a measure" assumes "\<Omega> \<in> sets M" shows "space (restrict_space M \<Omega>) = \<Omega>"
\<epsilon>_def: shows "\<epsilon> \<equiv> integral\<^sup>N (count_space (B \<Gamma>)) u"
that: fixes a :: "'a" assumes "1 \<notin> ideal (poly_eval (\<lambda>_. monomial a 0) ` focus {x} ` F)" shows "thesis"
SumSym: fixes P :: "pi"   and Q :: "pi" shows "P \<oplus> Q \<equiv>\<^sub>e Q \<oplus> P"
right_neutral: fixes a :: "'a" shows "a \<^bold>\<sqinter> \<one> = a"
le_def: fixes P :: "'b prog"   and mxs :: "nat"   and mxl :: "nat" shows "JVM_SemiType.le P mxs mxl \<equiv> fst (snd (JVM_SemiType.sl P mxs mxl))"
incls_coherent: fixes f :: "'s"   and f' :: "'s" assumes "par f f'"   and "incl f"   and "incl f'" shows "f = f'"
SUP_bot: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup ((\<lambda>x. bot) ` A) = bot"
assms_3: shows "k < length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>)"
simps_5: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and v :: "'c match_expr"   and rs :: "'c rule list" shows "no_matching_Goto \<gamma> p (Rule v Log # rs) = no_matching_Goto \<gamma> p rs"
naturality: fixes f :: "'c" assumes "C.arr f" shows "\<epsilon> (C.cod f) \<cdot>\<^sub>C FG.map f = C.map f \<cdot>\<^sub>C \<epsilon> (C.dom f)"
Min_antimono: fixes M :: "'c Word.word set"   and N :: "'c Word.word set" assumes "M \<subseteq> N"   and "M \<noteq> {}"   and "finite N" shows "signed.Min N \<le>s signed.Min M"
subst_atms_single: fixes A :: "'a"   and \<sigma> :: "'s" shows "{A} \<cdot>as \<sigma> = {A \<cdot>a \<sigma>}"
segment_unsplit: shows "{t0--t_max} \<union> {t_max--tt} = {t0--tt}"
lookup_mult: fixes f :: "'c \<Rightarrow>\<^sub>0 'b"   and g :: "'c \<Rightarrow>\<^sub>0 'b"   and k :: "'c" shows "lookup (f * g) k = (\<Sum>l. lookup f l * (\<Sum>q. lookup g q when k = l + q))"
f3: shows "(r' |\<^bsub>rel\<^esub> s') \<otimes>\<^bsub>rec_rng_of_frac\<^esub> (r'' |\<^bsub>rel\<^esub> s'') = (r' \<otimes> r'' |\<^bsub>rel\<^esub> s' \<otimes> s'')"
vec_to_iarray_plus: fixes a :: "('a, 'b) vec"   and b :: "('a, 'b) vec" shows "vec_to_iarray (a + b) = vec_to_iarray a + vec_to_iarray b"
igSubstIGVar2_def: fixes MOD :: "('a, 'b, 'varSort, 'sort, 'c, 'd, 'e, 'f, 'g) model_scheme" shows "igSubstIGVar2 MOD \<equiv> \<forall>ys y Y. igWls MOD (asSort ys) Y \<longrightarrow> igSubst MOD ys Y y (igVar MOD ys y) = Y"
enum_v_def: fixes n :: "nat" shows "enum_v n = from_nat_into (\<^bold>V - {source \<Delta>, sink \<Delta>}) (fst (prod_decode n))"
openin_subtopology_Un: fixes X :: "'c topology"   and T :: "'c set"   and S :: "'c set"   and U :: "'c set" assumes "openin (subtopology X T) S"   and "openin (subtopology X U) S" shows "openin (subtopology X (T \<union> U)) S"
Key_analz_image_Key_lemma: fixes P :: "bool"   and K :: "nat"   and KK :: "nat set"   and H :: "msg set" assumes "P \<longrightarrow> Key K \<in> analz (Key ` KK \<union> H) \<longrightarrow> K \<in> KK \<or> Key K \<in> analz H" shows "P \<longrightarrow> (Key K \<in> analz (Key ` KK \<union> H)) = (K \<in> KK \<or> Key K \<in> analz H)"
OstrowskiTr32: fixes v :: "'a \<Rightarrow> ant"   and s :: "'a" assumes "valuation K v"   and "s \<in> carrier K"   and "0 < v (1\<^sub>r \<plusminus> -\<^sub>a s)" shows "0 \<le> v s"
divideR_right: fixes r :: "real"   and y :: "'c"   and x :: "'c" assumes "r \<noteq> 0" shows "(y = x /\<^sub>R r) = (r *\<^sub>R y = x)"
lfinite_induct: fixes xs :: "'c llist"   and P :: "'c llist \<Rightarrow> bool" assumes "lfinite xs"   and "\<And>xs. lnull xs \<Longrightarrow> P xs"   and "\<And>xs. \<lbrakk>lfinite xs; \<not> lnull xs; P (ltl xs)\<rbrakk> \<Longrightarrow> P xs" shows "P xs"
op_norm_def: fixes A :: "(('a, 'b) vec, 'c) vec" shows "\<parallel>A\<parallel>\<^sub>o\<^sub>p = Sup {\<parallel>A *v x\<parallel> |x. \<parallel>x\<parallel> = 1}"
x_split_var: fixes x :: "('a \<times> 'a set) set" shows "x \<inter> NC \<union> x \<cdot> {} = x"
lnth_ldropn: fixes n :: "nat"   and m :: "nat"   and xs :: "'c llist" assumes "enat (n + m) < llength xs" shows "ldropn n xs $ m = xs $ m + n"
elims_3: fixes x :: "Runs.var set option"   and xa :: "Runs.var" assumes "\<not> in_progress x xa" obtains S :: "Runs.var set" and x :: "Runs.var" where "xa__ = Some S" and "xa = xb__" and "xb__ \<notin> S"   | x :: "Runs.var" where "xa__ = None" and "xa = x"
simps: fixes a :: "msg"   and H :: "msg set" shows "(a \<in> parts H) = ((\<exists>X. a = X \<and> X \<in> H) \<or> (\<exists>X Y. a = X \<and> \<lbrace>X, Y\<rbrace> \<in> parts H) \<or> (\<exists>X Y. a = Y \<and> \<lbrace>X, Y\<rbrace> \<in> parts H) \<or> (\<exists>K X. a = X \<and> Crypt K X \<in> parts H))"
evaldjf_bound0: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> fm" assumes "\<forall>x\<in>set xs. bound0 (f x)" shows "bound0 (evaldjf f xs)"
kmp_SPEC_altdefs_3: fixes s :: "'a list"   and t :: "'a list" shows "kmp_SPEC s t = RETURN (if sublist s t then Some (LEAST i. sublist_at s t i) else None)"
fresh_subst: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y :: "'var" assumes "good X"   and "good Y" shows "fresh zs z (X #[Y / y]_ys) = ((zs = ys \<and> z = y \<or> fresh zs z X) \<and> (fresh ys y X \<or> fresh zs z Y))"
yun_factorization_def: fixes Gcd :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"   and p :: "'a poly" shows "yun_factorization Gcd p = (if p = 0 then (0::'a, []) else let c = lead_coeff p; q = smult (inverse c) p in (c, yun_gcd.yun_monic_factorization Gcd q))"
Sup_distr: fixes X :: "('c \<times> 'd) set"   and y :: "'c \<times> 'd" shows "times_prod' (Sup_prod X) y = Sup_prod {times_prod' x y |x. x \<in> X}"
ParComm: fixes P :: "pi"   and Q :: "pi" shows "P \<parallel> Q \<equiv>\<^sub>s Q \<parallel> P"
app_lb_def: fixes K :: "('c, 'd) Ring_scheme"   and n :: "nat"   and vv :: "nat \<Rightarrow> 'c \<Rightarrow> ant"   and x :: "nat \<Rightarrow> 'c"   and m :: "nat" shows "\<Psi>\<^bsub>K n vv x m\<^esub> = (\<lambda>j\<in>{h. h \<le> n}. SOME L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>j\<in>{h. h \<le> n}. x j \<cdot>\<^sub>r\<^bsub>K\<^esub> (1\<^sub>r\<^bsub>K\<^esub> \<plusminus>\<^bsub>K\<^esub> -\<^sub>a\<^bsub>K\<^esub> (1\<^sub>r\<^bsub>K\<^esub> \<plusminus>\<^bsub>K\<^esub> -\<^sub>a\<^bsub>K\<^esub> (\<Omega>\<^bsub>K vv n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) n \<plusminus>\<^bsub>K\<^esub> -\<^sub>a\<^bsub>K\<^esub> x j))"
Sup_sigma: fixes M :: "'c set set set"   and \<Omega> :: "'c set" assumes "M \<noteq> {}"   and "\<And>m. m \<in> M \<Longrightarrow> m \<subseteq> Pow \<Omega>" shows "Sup (sigma \<Omega> ` M) = sigma \<Omega> (\<Union> M)"
rename_formula_def: fixes A :: "'at set"   and S :: "'at Literal set set" shows "rename_formula A S = {C. \<exists>CC. CC \<in> S \<and> C = rename_clause A CC}"
simps_13: fixes a :: "floatarith" shows "fold_const_fa (Exp a) = Exp (fold_const_fa a)"
subset_trans: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A \<subseteq> B"   and "B \<subseteq> C" shows "A \<subseteq> C"
scount_eq_emeasure: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" shows "ennreal_of_enat (scount P \<omega>) = emeasure (count_space UNIV) {i. P (sdrop i \<omega>)}"
lookup_map_entry_neq: fixes x :: "'b"   and y :: "'b"   and f :: "'c \<Rightarrow> 'c"   and m :: "('b, 'c) mapping" assumes "x \<noteq> y" shows "lookup (map_entry x f m) y = lookup m y"
list_comb_cases: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and abs_pred :: "('c \<Rightarrow> bool) \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "'c"   and f :: "'c"   and xs :: "'c list" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred"   and "t = term.list_comb app f xs" obtains "t = f" and "xs = []"   | u :: "'c" and v :: "'c" where "t = app u v"
self_cscalar_prod_geq_0: fixes v :: "'a vec" shows "(0::'a) \<le> inner_prod v v"
resPres: fixes P :: "ccs"   and Q :: "ccs"   and x :: "name" assumes "P \<approx> Q" shows "\<lparr>\<nu>x\<rparr>P \<approx> \<lparr>\<nu>x\<rparr>Q"
pre_id_unch: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "STUTINV F" shows "|~ F \<and> Unchanged id \<longrightarrow> \<circle>F"
hx': shows "x' \<in> upper_circle_01 \<union> lower_circle_01"
less_asym': fixes a :: "'t \<Rightarrow>\<^sub>0 'c"   and b :: "'t \<Rightarrow>\<^sub>0 'c"   and P :: "bool" assumes "a \<prec>\<^sub>p b"   and "b \<prec>\<^sub>p a" shows "P"
axioms_2: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd" assumes "FGModuleHom G smult V smult' T" shows "RModuleHom (aezfun_setspan G) smult V smult' T"
fmadd_assoc: fixes m :: "('a, 'b) fmap"   and n :: "('a, 'b) fmap"   and p :: "('a, 'b) fmap" shows "m ++\<^sub>f (n ++\<^sub>f p) = m ++\<^sub>f n ++\<^sub>f p"
spr_ii_sim_tFirst_tLast: fixes t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and s :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState set \<times> ('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState" assumes "spr_ii_sim t = s"   and "t \<in> SPR.jkbpC" shows "snd s \<in> fst s"
prj_simps_13: shows "Maps.dom SPN_fgh.Prj\<^sub>1 = Chn (SPN f \<circ> SPN g \<circ> SPN h)"
W2b: fixes b :: "'a"   and a :: "'a" shows "(b l\<rightarrow> a) r\<rightarrow> a = (b r\<rightarrow> a) l\<rightarrow> a"
proper_def: fixes S :: "'a set" shows "proper S \<equiv> \<forall>x r. compact (cball x r \<inter> S)"
cos_odd_kpi: fixes k :: "int" assumes "odd k" shows "cos (real_of_int k * pi) = - 1"
norm_ket_k_ge_K: fixes k :: "nat" assumes "K \<le> k" shows "inner_prod (ket_k k) (ket_k k) = 0"
AssumeH_3: fixes A1 :: "fmla"   and A2 :: "fmla"   and A3 :: "fmla"   and H :: "fmla set" shows "insert A1 (insert A2 (insert A3 H)) \<turnstile> A3"
normalize_mult_unit_factor: fixes a :: "'b poly" shows "smult (inverse (lead_coeff a)) a * [:lead_coeff a:] = a"
upd_cols_empty: fixes M :: "('c, 'd) sq_matrix"   and f :: "'d \<Rightarrow> ('c, 'd) vec" shows "upd_cols M {} f = M"
eq_trans: fixes eq :: "'d \<rightarrow> 'd \<rightarrow> tr"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "class.Eq_equiv eq"   and "eq\<cdot>x\<cdot>y = TT"   and "eq\<cdot>y\<cdot>z = TT" shows "eq\<cdot>x\<cdot>z = TT"
homf_preserves_id: fixes B :: "'o"   and A :: "'o" assumes "B \<in> Ob" shows "Hom(A,_) \<^bsub>\<a>\<^esub> (Id B) = Id\<^bsub>Set\<^esub> Hom(A,_) \<^bsub>\<o>\<^esub> B"
mkM_def: fixes T :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace set" shows "SPR.mkM T \<equiv> \<lparr>worlds = T, relations = \<lambda>a. {(t, t'). {t, t'} \<subseteq> T \<and> spr_jview a t = spr_jview a t'}, valuation = envVal \<circ> tLast\<rparr>"
le_step_down_word: fixes i :: "'b word"   and n :: "'b word"   and P :: "bool" assumes "i \<le> n"   and "i = n \<longrightarrow> P"   and "i \<le> n - 1 \<longrightarrow> P" shows "P"
sep_disj_addD1: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.sep_algebra plus zero sep_disj"   and "sep_disj x (plus y z)"   and "sep_disj y z" shows "sep_disj x y"
upclosed_mono: fixes f :: "'d \<Rightarrow> 'c" assumes "\<forall>Y. upclosed_set Y \<longrightarrow> upclosed_set (f -` Y)" shows "mono f"
circlepath_minus: fixes z :: "complex"   and r :: "real"   and x :: "real" shows "circlepath z (- r) x = circlepath z r (x + 1 / 2)"
sconjunctive_simp: fixes x :: "'d"   and Q :: "'d set"   and S :: "'d \<Rightarrow> 'e" assumes "x \<in> Q"   and "sconjunctive S" shows "S (Inf Q) = Inf (S ` Q)"
ex_points_eng: fixes a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool" assumes "QCongA a" shows "\<exists>A B C. a A B C"
sm: fixes x :: "'a" assumes "norm x < s" shows "(\<lambda>n. a n * x ^ n) sums f x"
simps: fixes \<Sigma> :: "'b set set"   and \<phi> :: "'b ltl" shows "ltl_to_generalized_rabin_af \<Sigma> \<phi> = ltl_to_rabin_base_def.ltl_to_generalized_rabin \<up>af \<up>af\<^sub>G Abs Abs M_fin \<Sigma> \<phi>"
intTI_def: shows "intTI \<equiv> MonotProblem.intTI GE_wtFsym GE_wtPsym GE_arOf GE_resOf GE_parOf GE_\<Phi>"
open_imp_subset: fixes S :: "'b set"   and is_open :: "'b set \<Rightarrow> bool"   and U :: "'b set" assumes "topological_space S is_open"   and "is_open U" shows "U \<subseteq> S"
lim_increasing_cl: fixes f :: "nat \<Rightarrow> 'b" assumes "\<And>n m. m \<le> n \<Longrightarrow> f m \<le> f n" obtains l :: "'b" where "f \<longlonglongrightarrow> l"
prems: shows "containsCall procs (Call q es' rets') ps p"
elims_part: fixes \<Gamma>\<^sub>1 :: "'c constr list"   and n\<^sub>1 :: "nat"   and \<Psi>\<^sub>1 :: "'c TESL_atomic list"   and \<Phi>\<^sub>1 :: "'c TESL_atomic list"   and \<Gamma>\<^sub>2 :: "'c constr list"   and n\<^sub>2 :: "nat"   and \<Psi>\<^sub>2 :: "'c TESL_atomic list"   and \<Phi>\<^sub>2 :: "'c TESL_atomic list" assumes "\<Gamma>\<^sub>1, n\<^sub>1 \<turnstile> \<Psi>\<^sub>1 \<triangleright> \<Phi>\<^sub>1 \<hookrightarrow>\<^sub>e \<Gamma>\<^sub>2, n\<^sub>2 \<turnstile> \<Psi>\<^sub>2 \<triangleright> \<Phi>\<^sub>2" shows "\<Gamma>\<^sub>1, n\<^sub>1 \<turnstile> \<Psi>\<^sub>1 \<triangleright> \<Phi>\<^sub>1 \<hookrightarrow> \<Gamma>\<^sub>2, n\<^sub>2 \<turnstile> \<Psi>\<^sub>2 \<triangleright> \<Phi>\<^sub>2"
small: shows "small ((\<lambda>x. x\<lparr>1\<^sub>\<nat>\<rparr>) ` elts r)"
simps_1: fixes P :: "'a" shows "post \<lbrace> P \<rbrace> = P"
cases: fixes a1 :: "(any_unop + boolean_unop + numeric_unop + string_unop + collection_unop) + (super_binop + boolean_binop + numeric_binop + string_binop + collection_binop) + (string_ternop + collection_ternop) + String.literal"   and a2 :: "call_kind"   and a3 :: "'a OCL_Types.type"   and a4 :: "'a OCL_Types.type list"   and a5 :: "'a OCL_Types.type" assumes "op_type a1 a2 a3 a4 a5" obtains op :: "any_unop + boolean_unop + numeric_unop + string_unop + collection_unop" and k :: "call_kind" and \<tau> :: "'a OCL_Types.type" and \<upsilon> :: "'a OCL_Types.type" where "a1 = Inl op__" and "a2 = k__" and "a3 = \<tau>__" and "a4 = []" and "a5 = \<upsilon>__" and "unop_type op__ k__ \<tau>__ \<upsilon>__"   | op :: "super_binop + boolean_binop + numeric_binop + string_binop + collection_binop" and k :: "call_kind" and \<tau> :: "'a OCL_Types.type" and \<sigma> :: "'a OCL_Types.type" and \<upsilon> :: "'a OCL_Types.type" where "a1 = Inr (Inl opa__)" and "a2 = ka__" and "a3 = \<tau>'__" and "a4 = [\<sigma>__]" and "a5 = \<upsilon>'__" and "binop_type opa__ ka__ \<tau>'__ \<sigma>__ \<upsilon>'__"   | op :: "string_ternop + collection_ternop" and k :: "call_kind" and \<tau> :: "'a OCL_Types.type" and \<sigma> :: "'a OCL_Types.type" and \<rho> :: "'a OCL_Types.type" and \<upsilon> :: "'a OCL_Types.type" where "a1 = Inr (Inr (Inl opb__))" and "a2 = k" and "a3 = \<tau>''__" and "a4 = [\<sigma>, \<rho>]" and "a5 = \<upsilon>" and "ternop_type opb__ k \<tau>''__ \<sigma> \<rho> \<upsilon>"   | \<tau> :: "'a OCL_Types.type" and op :: "String.literal" and \<pi> :: "'a OCL_Types.type list" and oper :: "String.literal \<times> 'a OCL_Types.type \<times> (String.literal \<times> 'a OCL_Types.type \<times> param_dir) list \<times> 'a OCL_Types.type \<times> bool \<times> 'a expr option" where "a1 = Inr (Inr (Inr op))" and "a2 = DotCall" and "a3 = \<tau>" and "a4 = \<pi>" and "a5 = oper_type oper" and "operation \<tau> op \<pi> oper"
poly_eval_prod: fixes a :: "'c \<Rightarrow> 'b"   and f :: "'d \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and I :: "'d set" shows "poly_eval a (prod f I) = (\<Prod>i\<in>I. poly_eval a (f i))"