hcju/isabelleps huggingface dataset:
presCons_unique: fixes f :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and fA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'e, 'opSym, 'var, 'c, 'd, 'f) Recursion.model_scheme"   and g :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and gA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and us :: "'varSort"   and s' :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" assumes "presCons f fA MOD"   and "presCons g gA MOD" shows "(wls s X \<longrightarrow> f X = g X) \<and> (wlsAbs (us, s') A \<longrightarrow> fA A = gA A)"
satPB_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and \<Phi> :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) lit list set" shows "GE.satPB \<xi> \<Phi> \<equiv> \<forall>c\<in>\<Phi>. GE.satC \<xi> c"
wfE_min: fixes R :: "('c \<times> 'c) set"   and x :: "'c"   and Q :: "'c set" assumes "wf R"   and "x \<in> Q" obtains z :: "'c" where "z \<in> Q" and "\<And>y. (y, z) \<in> R \<Longrightarrow> y \<notin> Q"
assms_2: fixes s :: "'a set" shows "g (insert r s) = g s"
some_in_P_equiv: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "v_equiv K v (SOME w. w \<in> P\<^bsub> K v\<^esub>)"
mult_pos_neg2: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less zero a"   and "less b zero" shows "less (times b a) zero"
compatible_modes_def: fixes mdss :: "(Mode \<Rightarrow> 'Var set) list" shows "compatible_modes mdss = (\<forall>i x. i \<sqsubset> length mdss \<longrightarrow> (x \<in> (mdss ! i) AsmNoRead \<longrightarrow> (\<forall>j<length mdss. j \<noteq> i \<longrightarrow> x \<in> (mdss ! j) GuarNoRead)) \<and> (x \<in> (mdss ! i) AsmNoWrite \<longrightarrow> (\<forall>j<length mdss. j \<noteq> i \<longrightarrow> x \<in> (mdss ! j) GuarNoWrite)))"
Integer_arith_simps_2: fixes a :: "int"   and b :: "int" shows "MkI\<cdot>a * MkI\<cdot>b = MkI\<cdot>(a * b)"
cball_Pair_split_subset: fixes a :: "'b"   and b :: "'c"   and c :: "real" shows "cball (a, b) c \<subseteq> cball a c \<times> cball b c"
from_vec_eq_iff: fixes M :: "(('c, 'd) vec, 'd) vec"   and N :: "(('c, 'd) vec, 'd) vec" shows "(from_vec M = from_vec N) = (M = N)"
2: shows "compose (set_dom f) (set_func f) (\<lambda>x\<in>set_dom f. x) = set_func f"
gauss: shows "gauss_jordan B (0\<^sub>m (dim_row A) 0) = (C, D)"
src_in_sdnets: fixes x :: "((int \<times> int) set set, int) Combinators"   and p :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "\<not> member DenyAll x"   and "p \<in> dom (C x)" shows "subnetsOfAdr (src p) \<inter> fst_set (sdnets x) \<noteq> {}"
log_twice: fixes n :: "nat" assumes "n \<noteq> 0" shows "Discrete.log (2 * n) = Suc (Discrete.log n)"
SCallRedsParams: fixes es :: "char list exp list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and es' :: "char list exp list"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and M :: "char list" assumes "P \<turnstile> \<langle>es,s,b\<rangle> [\<rightarrow>]* \<langle>es',s',b'\<rangle>" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sM(es),s,b\<rangle> \<rightarrow>* \<langle>C\<bullet>\<^sub>sM(es'),s',b'\<rangle>"
simps_6: fixes a :: "MIR.num"   and b :: "MIR.num" shows "zsplit0 (Add a b) = (let (ia, a') = zsplit0 a; (ib, b') = zsplit0 b in (ia + ib, Add a' b'))"
iMOD_0_eq_conv: fixes r :: "nat"   and r' :: "nat"   and m' :: "nat" shows "([ r, mod 0 ] = [ r', mod m' ]) = (r = r' \<and> m' = 0)"
monad_state_stateT: fixes return :: "'a \<times> 'b \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "monad return bind" shows "monad_state (return_state return) (bind_state bind) get_state put_state"
stermsl_before_dtermsl: fixes p :: "('a, 'b, 'c, 'd) seqp" shows "\<Union> (dtermsl ` stermsl p) = dtermsl p"
obj_simps_2: fixes a :: "'a" assumes "obj a" shows "I.map a = a"
lookup_pair_ko_def: fixes ko :: "'a key_order" shows "lookup_pair_ko ko \<equiv> comparator.lookup_pair (key_compare ko)"
assms_3: shows "b \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"
has_integral_factor_content_real: fixes f :: "real \<Rightarrow> 'a"   and i :: "'a"   and a :: "real"   and b :: "real" shows "(f has_integral i) {a..b} = (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i) \<le> e * content {a..b}))"
assms: shows "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L"
ds: shows "(c', ictxt, fctxt) \<in> decompose_com c"
uminus_eq_mat: fixes A :: "'a mat"   and B :: "'a mat" shows "(- A = - B) = (A = B)"
axioms_4: fixes V\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and V\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>G :: "'c \<times> 'c \<Rightarrow> 'd"   and \<tau>\<^sub>0 :: "'c \<Rightarrow> 'd"   and \<tau>\<^sub>1 :: "'c \<Rightarrow> 'd" assumes "pseudonatural_transformation V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1" shows "pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G"
bot_filter_def: shows "bot = Filter.filter.Abs_filter (\<lambda>P. True)"
trail_connected: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c" assumes "wf_digraph G"   and "Digraph_Component.connected G"   and "pre_digraph.trail G u p v"   and "set p \<noteq> arcs G" shows "\<exists>e. e \<in> arcs G - set p \<and> (tail G e \<in> set (pre_digraph.awalk_verts G u p) \<or> head G e \<in> set (pre_digraph.awalk_verts G u p))"
inj_on_image: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set set" assumes "inj_on f (\<Union> A)" shows "inj_on ((`) f) A"
i_f_join_i_shrink_iT_Plus_iT_Div_mod_subst: fixes k :: "nat"   and I :: "nat set"   and A :: "nat set"   and B :: "nat set"   and f :: "nat \<Rightarrow> 'b message_af" assumes "0 < k"   and "finite I"   and "\<forall>x\<in>I. x mod k = 0"   and "A = I \<oplus> k - 1"   and "B = I \<oslash> k" shows "f \<longmapsto> k \<Join>\<^sub> A = f \<div> k \<Join>\<^sub> B"
that: fixes n :: "'a itrm" assumes "n \<simeq> x"   and "n \<in> CF" shows "thesis"
strip_pfp_wn: fixes f :: "'a acom \<Rightarrow> 'a acom"   and C :: "'a acom"   and C' :: "'a acom" assumes "\<forall>C. strip (f C) = strip C"   and "pfp_wn f C = Some C'" shows "strip C' = strip C"
that: fixes p :: "('a, 'out, ('a, 'out, 'in) rpv) generat spmf" assumes "gpv = GPV p"   and "lossless_spmf p"   and "\<And>out c input. \<lbrakk>IO out c \<in> set_spmf p; input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> local.gen_lossless_gpv (c input)" shows "thesis"
1: fixes \<tau>\<mu>\<nu> :: "('a E.term, 'a) arr \<times> ('a E.term, 'a) arr \<times> ('a E.term, 'a) arr" assumes "VVV.arr \<tau>\<mu>\<nu>" shows "cod (\<a> (fst \<tau>\<mu>\<nu>) (fst (snd \<tau>\<mu>\<nu>)) (snd (snd \<tau>\<mu>\<nu>))) = HoVH (VVV.cod \<tau>\<mu>\<nu>)"
in_SINK_restrict_current: fixes x :: "'a"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "x \<in> SINK g" shows "x \<in> SINK (g \<upharpoonleft> \<Gamma> / f)"
a_r_coset_def': fixes G :: "('a, 'b) ring_scheme"   and H :: "'a set"   and a :: "'a" shows "H +>\<^bsub>G\<^esub> a \<equiv> \<Union>h\<in>H. {h \<oplus>\<^bsub>G\<^esub> a}"
all_resolvents_upon_def: fixes S :: "'at Literal set set"   and A :: "'at" shows "all_resolvents_upon S A = {C. \<exists>P1 P2. P1 \<in> S \<and> P2 \<in> S \<and> C = resolvent_upon P1 P2 A}"
map_add_SomeD: fixes m :: "'e \<Rightarrow> 'd option"   and n :: "'e \<Rightarrow> 'd option"   and k :: "'e"   and x :: "'d" assumes "(m ++ n) k = Some x" shows "n k = Some x \<or> n k = None \<and> m k = Some x"
dg_prod_2_Dom_vsv: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V" assumes "digraph \<alpha> \<AA>"   and "digraph \<alpha> \<BB>" shows "vsv ((\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Dom\<rparr>)"
induct: fixes P :: "tm \<Rightarrow> bool"   and a0 :: "tm" assumes "\<And>j. P (CP j)"   and "\<And>n c a. (\<And>x. P a) \<Longrightarrow> P (CNP n c a)"   and "\<And>v. P (tm.Bound v)"   and "\<And>v va. P (tm.Add v va)"   and "\<And>v va. P (tm.Mul v va)"   and "\<And>v. P (tm.Neg v)"   and "\<And>v va. P (tm.Sub v va)" shows "P a0"
dests_iD_invalidate: fixes dests :: "nat \<Rightarrow> nat option"   and ip :: "nat"   and rsn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "dests ip = Some rsn"   and "ip \<in> kD rt" shows "ip \<in> iD (invalidate rt dests)"
assms_1: shows "\<lbrace>\<alpha> u \<cdot> \<tau> v\<rbrace>x\<lbrace>z\<rbrace>"
inj_Inl: fixes A :: "'b set" shows "inj_on Inl A"
add: fixes b1 :: "'m"   and b2 :: "'m" shows "f (b1 + b2) = f b1 + f b2"
convol_def: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f" shows "BNF_Def.convol f g \<equiv> \<lambda>a. (f a, g a)"
charslength_0: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and ts :: "(symbol \<times> character list) list" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "(\<forall>t\<in>set ts. chars_of_token t = []) = (charslength ts = 0)"
hoare_plus: fixes p :: "'a"   and x :: "'b"   and q :: "'a"   and y :: "'b" assumes "\<lbrace>\<iota> p\<rbrace> x \<lbrace>\<iota> q\<rbrace>"   and "\<lbrace>\<iota> p\<rbrace> y \<lbrace>\<iota> q\<rbrace>" shows "\<lbrace>\<iota> p\<rbrace> x + y \<lbrace>\<iota> q\<rbrace>"
Star_mono: fixes r :: "'a language"   and s :: "'a language" assumes "r \<le> s" shows "Star r \<le> Star s"
uint32_mod_def: fixes x :: "uint32"   and y :: "uint32" shows "uint32_mod x y = snd (uint32_divmod x y)"
mono: fixes i :: "'b" assumes "i \<in> I" shows "G i \<subseteq> F i"
nth_Cons: fixes x :: "'b"   and xs :: "'b list"   and n :: "nat" shows "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
empty: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and g :: "'c \<Rightarrow> 'b" assumes "abel_semigroup_set f z" shows "abel_semigroup_set.F f z g {} = z"
llist_le_refl: fixes s :: "'b llist" shows "s \<le> s"
user_reg_val_def: fixes window :: "'b word"   and ur :: "word5"   and state :: "'b sparc_state" shows "user_reg_val window ur state \<equiv> if ur = 0 then 0 else user_reg state window ur"
aubs_ex_n_maxTr: fixes a :: "ant"   and A :: "ant set"   and n :: "nat" assumes "a \<in> A"   and "\<nexists>m. m \<in> A \<and> (\<forall>x\<in>A. x \<le> m)" shows "a + an n \<le> aasc_seq A a n"
zero_smallest: fixes j :: "nat + 'a Bonk_Schramm_extension_unfolded set" assumes "j \<in> Field wo_aux" shows "(zero wo_aux, j) \<in> wo_aux"
simps_1: fixes h :: "'b \<Rightarrow> 'b ref"   and x :: "'b ref"   and y :: "'b ref" shows "Path h x [] y = (x = y)"
insert_rga_set_ins: fixes i :: "'a"   and xs :: "'a list"   and oid :: "'a" assumes "i \<in> set xs" shows "set (insert_rga xs (oid, Some i)) = insert oid (set xs)"
containsE: fixes A :: "'a set"   and x :: "'a" assumes "contains A x" obtains A' :: "'a set" and x' :: "'a" where "A = A'" and "x = x'" and "x \<in> A"
intro: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set" assumes "RModule UNIV smult M" shows "Module smult M"
DBM_reset_not_cyc_free_preservation: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and k :: "nat"   and d :: "'c"   and M' :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry" assumes "\<not> (\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len M i i xs)"   and "DBM_reset M n k d M'"   and "k \<le> n" shows "\<not> (\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len M' i i xs)"
apath_decomp_disjoint: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and q :: "'b list"   and r :: "'b list"   and x :: "'a" assumes "G.apath u p v"   and "p = q @ r"   and "x \<in> set (G.awalk_verts u q)"   and "x \<in> set (tl (G.awalk_verts (G.awlast u q) r))" shows "False"
ih: shows "X = X @@ A ^^ Suc n \<union> (\<Union>m\<le>n. B @@ A ^^ m)"
f4: shows "s \<otimes> s' = s' \<otimes> s"
prems: fixes k :: "nat" assumes "k < Suc n" shows "f k \<in> carrier_mat d d"
assms_5: shows "\<exists>p. f n = Neg (Forall p)"
map_2: fixes f :: "'d \<Rightarrow> 'c"   and x2 :: "'d" shows "map_option f (Some x2) = Some (f x2)"
ts_ord_def: fixes x :: "'a"   and y :: "'a" shows "(x \<sqsubseteq> y) = (!x \<cdot> !y = !y)"
comp_cod_arr: fixes f :: "'c"   and b :: "'c" assumes "C.arr f"   and "C.cod f = b" shows "b \<cdot>\<^sub>C f = f"
assms_5: shows "r\<^sup>T * g\<^sup>\<star> \<le> r\<^sup>T * w\<^sup>\<star>"
indRelSTEQ_impl_SRel_and_TRel_weakly_reflect_barbs_2: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_weakly_reflects_barbs (indRelSTEQ SRel TRel) (STCalWB SWB TWB)" shows "rel_weakly_reflects_barbs TRel TWB"
ms_of_greek_shorter: fixes x :: "accent \<times> 'a"   and t :: "(accent \<times> 'a) list"   and s :: "(accent \<times> 'a) list" assumes "(x, t) \<in># ms_of_greek s" shows "length t < length s"
padr_image: fixes U :: "'b vec set"   and n :: "nat"   and m :: "nat" assumes "U \<subseteq> carrier_vec n" shows "(\<lambda>v. v @\<^sub>v 0\<^sub>v m) ` U \<subseteq> carrier_vec (n + m)"
infiter_def: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice" assumes "infinite_iteration (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)" shows "infinite_iteration.infiter (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (c::'b::refinement_lattice) \<equiv> lfp (seq c)"
Pool_1: fixes w1 :: "nat \<Rightarrow> real"   and w2 :: "nat \<Rightarrow> real" assumes "\<And>i. i < count_weights s m1 \<Longrightarrow> w1 i = w2 i" shows "insert_weights s m1 w1 = insert_weights s m1 w2"
simps_2: fixes x :: "'a"   and xs :: "'a list" shows "head (x # xs) = x"
has_codomain_iff_arr: fixes f :: "'b" shows "(B.codomains f \<noteq> {}) = B.arr f"
fpxs_as_fls: fixes f :: "'b fpxs" shows "fpxs_compose_power (fpxs_of_fls (fls_of_fpxs f)) (1 / rat_of_nat (fpxs_root_order f)) = f"
real_distr_M: fixes n :: "nat" shows "real_distribution (M n)"
right_idem: fixes a :: "'b multiset"   and b :: "'b multiset" shows "a \<union># b \<union># b = a \<union># b"
gu1: fixes l :: "'a Standard_Constant"   and x :: "'b"   and y :: "'b" assumes "(l, x, y) \<in> edges G"   and "x \<in> vertices G"   and "y \<in> vertices G" shows "(l, f x, f y) \<in> edges G"
mf_measurable: fixes M :: "'d measure" assumes "space M = UNIV" shows "mf \<in> Pi\<^sub>F {f ` J} (\<lambda>_. M) \<rightarrow>\<^sub>M Pi\<^sub>M J (\<lambda>_. M)"
enumerate_in_set: fixes S :: "'a set"   and n :: "nat" assumes "infinite S" shows "wellorder_class.enumerate S n \<in> S"
simps_2: shows "real_of_ereal \<infinity> = 0"
pWlsInps_1: fixes delta :: "'opSym"   and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option" assumes "wlsInp delta inp" shows "gWlsInp MOD delta (lift (rec MOD) inp)"
set_of_seq_code_1: shows "set_of_seq seq.Empty = Set_Monad []"
local_unique_solutions: fixes t0 :: "real"   and x0 :: "'a" assumes "t0 \<in> UNIV"   and "x0 \<in> X" obtains t :: "real" and u :: "real" and L :: "real" where "0 < t" and "0 < u" and "cball t0 t \<subseteq> (+) t0 ` existence_ivl0 x0" and "cball x0 (2 * u) \<subseteq> X" and "\<And>t'. t' \<in> cball t0 t \<Longrightarrow> L-lipschitz_on (cball x0 (2 * u)) f" and "\<And>x. x \<in> cball x0 u \<Longrightarrow> ((\<lambda>t. flow0 x (t - t0)) usolves_ode (\<lambda>_. f) from t0) (cball t0 t) (cball x u)" and "\<And>x. x \<in> cball x0 u \<Longrightarrow> cball x u \<subseteq> X"
regular_Derivs_finite: fixes r :: "'b rexp" shows "finite (range (\<lambda>w. Derivs w (lang r)))"
sum_I: shows "(\<lambda>i. \<Sum>\<^sub>ax\<in>S' i. norm (f x)) abs_summable_on I"
dest_port_datatype_def: fixes x :: "int \<times> (DatatypeAddress \<times> int) \<times> (DatatypeAddress \<times> int) \<times> 'a" shows "dest_port x \<equiv> (snd \<circ> fst \<circ> snd \<circ> snd) x"
assms_1: shows "support_set pf1 \<subseteq> stocks Mkt \<union> {x}"
rep_eq: fixes x :: "'b word" shows "to_bl x = bin_to_bl LENGTH('b) (Word.rep x)"
prod_prime_factors: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.factorial_semiring_multiplicative divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero" shows "comm_monoid_mult.prod times one (\<lambda>p. power.power one times p (factorial_semiring.multiplicity times one p x)) (set_mset (factorial_semiring.prime_factorization zero times one normalize x)) = normalize x"
prob_space_distr: fixes f :: "(int \<Rightarrow> 'a) \<Rightarrow> 'b"   and M' :: "'b measure" assumes "I.random_variable M' f" shows "prob_space (distr natural_extension_measure M' f)"
Strongly_Secure_def: fixes V :: "('exp, 'id) MWLfCom list" shows "Strongly_Secure V \<equiv> \<forall>d. V \<approx>\<^bsub>d\<^esub> V"
rref_Suc: shows "reduced_row_echelon_form_upt_k A (Suc k)"
compatible_one: shows "h \<one> = \<one>\<^bsub>S\<^esub>"
sqn_invalidate_in_dests: fixes dests :: "nat \<Rightarrow> nat option"   and ipa :: "nat"   and rsn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "dests ipa = Some rsn"   and "ipa \<in> kD rt" shows "sqn (invalidate rt dests) ipa = rsn"
RETURN_refine: fixes x :: "'b"   and x' :: "'c"   and R :: "('b \<times> 'c) set" assumes "(x, x') \<in> R" shows "RETURN x \<le> \<Down> R (RETURN x')"
of_int_eq: fixes z :: "int" shows "of_int z = Abs_bit0 (z mod int CARD('b bit0))"
Reception: fixes evsr :: "event list"   and A :: "agent"   and B :: "agent"   and X :: "msg" assumes "evsr \<in> set_mr"   and "Says A B X \<in> set evsr" shows "Gets B X # evsr \<in> set_mr"
inf_prefix_length_take_correct: fixes n :: "nat"   and xs :: "'b list"   and ys :: "'b list" assumes "n \<le> inf_prefix_length xs ys" shows "xs \<down> n = ys \<down> n"
ofilter_AboveS_Field: fixes A :: "'a set" assumes "local.ofilter A" shows "A \<union> local.AboveS A = Field r"
fps_to_fls_power: fixes f :: "'b fps"   and n :: "nat" shows "fps_to_fls (f ^ n) = fps_to_fls f ^ n"
FP_def: fixes F :: "'c program" shows "FP F \<equiv> {s. F \<in> stable {s}}"
get_roots_UN_get_work_locs_subseteq_gc_W_empty_locs: shows "get_roots_UN_get_work_locs \<subseteq> gc_W_empty_locs"
big_sum_in_bigo: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> O[F](g)" shows "(\<lambda>x. \<Sum>y\<in>A. f y x) \<in> O[F](g)"
ident: fixes D :: "'e set"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and e :: "'e"   and x :: "'e" assumes "ACeD D f e"   and "x \<in> D" shows "f x e = x"
nnf_semantics: fixes \<A> :: "'b \<Rightarrow> bool"   and F :: "'b formula" shows "\<A> \<Turnstile> nnf F = \<A> \<Turnstile> F"
simps: fixes a1 :: "'b prog"   and a2 :: "ty"   and a3 :: "bop"   and a4 :: "ty"   and a5 :: "ty" shows "(a1 \<turnstile> a2\<guillemotleft>a3\<guillemotright>a4 : a5) = ((\<exists>P T1 T2. a1 = P \<and> a2 = T1 \<and> a3 = Eq \<and> a4 = T2 \<and> a5 = Boolean) \<or> (\<exists>P T1 T2. a1 = P \<and> a2 = T1 \<and> a3 = NotEq \<and> a4 = T2 \<and> a5 = Boolean) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = LessThan \<and> a4 = Integer \<and> a5 = Boolean) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = LessOrEqual \<and> a4 = Integer \<and> a5 = Boolean) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = GreaterThan \<and> a4 = Integer \<and> a5 = Boolean) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = GreaterOrEqual \<and> a4 = Integer \<and> a5 = Boolean) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = Add \<and> a4 = Integer \<and> a5 = Integer) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = Subtract \<and> a4 = Integer \<and> a5 = Integer) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = Mult \<and> a4 = Integer \<and> a5 = Integer) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = Div \<and> a4 = Integer \<and> a5 = Integer) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = Mod \<and> a4 = Integer \<and> a5 = Integer) \<or> (\<exists>P. a1 = P \<and> a2 = Boolean \<and> a3 = BinAnd \<and> a4 = Boolean \<and> a5 = Boolean) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = BinAnd \<and> a4 = Integer \<and> a5 = Integer) \<or> (\<exists>P. a1 = P \<and> a2 = Boolean \<and> a3 = BinOr \<and> a4 = Boolean \<and> a5 = Boolean) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = BinOr \<and> a4 = Integer \<and> a5 = Integer) \<or> (\<exists>P. a1 = P \<and> a2 = Boolean \<and> a3 = BinXor \<and> a4 = Boolean \<and> a5 = Boolean) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = BinXor \<and> a4 = Integer \<and> a5 = Integer) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = ShiftLeft \<and> a4 = Integer \<and> a5 = Integer) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = ShiftRightZeros \<and> a4 = Integer \<and> a5 = Integer) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = ShiftRightSigned \<and> a4 = Integer \<and> a5 = Integer))"
left_unit: fixes a :: "'a" assumes "a \<in> A" shows "multA oneA a = a"
emeasure_eq_measure: fixes A :: "('a \<Rightarrow> 'b) set" shows "emeasure (Pi\<^sub>M I M) A = ennreal (local.prob A)"
zeta_partial_sum_bigo_neg: fixes s :: "real" assumes "0 < s" shows "(\<lambda>n. (\<Sum>i = 1..n. real i powr s) - real n powr (1 + s) / (1 + s)) \<in> O(\<lambda>x. real x powr s)"
reduction_subsets_is_get_subsets: fixes p :: "real poly"   and qs :: "real poly list"   and signs :: "rat list list"   and subsets :: "nat list list"   and m :: "rat mat" shows "reduction_subsets p qs signs subsets m = get_subsets (reduce_system p (qs, m, subsets, signs))"