hcju/isabelleps huggingface dataset:
HDomain_Incl_Eats: fixes r :: "tm"   and d :: "tm"   and d' :: "tm" shows "{HDomain_Incl r d} \<turnstile> HDomain_Incl (Eats r (HPair d d')) (SUCC d)"
simple_def: fixes ct :: "'s \<Rightarrow> 's pmf" shows "simple ct = memoryless_on (\<lambda>s. if s \<in> S then ct s else arb_act s)"
convex_polychain_appendI: fixes xs :: "((real \<times> real) \<times> real \<times> real) list"   and ys :: "((real \<times> real) \<times> real \<times> real) list" assumes "convex_polychain xs"   and "convex_polychain ys"   and "polychain (xs @ ys)"   and "\<lbrakk>xs \<noteq> []; ys \<noteq> []\<rbrakk> \<Longrightarrow> 0 < det3 (fst (last xs)) (snd (last xs)) (snd (hd ys))" shows "convex_polychain (xs @ ys)"
assms: shows "\<exists>m. self_financing Mkt p \<and> trading_strategy p \<and> (\<forall>w\<in>space M. cls_val_process Mkt p 0 w = 0) \<and> (AE w in M. 0 \<le> cls_val_process Mkt p m w) \<and> 0 < prob {w \<in> space M. 0 < cls_val_process Mkt p m w}"
kzkki: fixes i :: "nat" assumes "i \<noteq> 0"   and "i < 2 ^ Suc (Suc k)" shows "ket_zero_k (Suc k) $ i = 0"
Bot_delR': fixes \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "\<Gamma> \<Rightarrow> \<Delta> \<down> n" shows "\<Gamma> \<Rightarrow> \<Delta> - (\<bottom>, {#}) \<down> n"
map_of_zip_is_None: fixes xs :: "'c list"   and ys :: "'d list"   and x :: "'c" assumes "length xs = length ys" shows "(map_of (zip xs ys) x = None) = (x \<notin> set xs)"
zero_less_norm_iff: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "(0 < norm x) = (x \<noteq> zero)"
lists_mono: fixes A :: "'b set"   and B :: "'b set" assumes "order.greater_eq B A" shows "order.greater_eq (lists B) (lists A)"
rel_fun_eq_eq_onp: fixes P :: "'c \<Rightarrow> bool" shows "rel_fun (=) (eq_onp P) = eq_onp (\<lambda>f. \<forall>x. P (f x))"
sorted_find_Min: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "sorted xs"   and "\<exists>x\<in>set xs. P x" shows "find P xs = Some (Min {x \<in> set xs. P x})"
ordered_keys_def: fixes m :: "('b, 'c) mapping" shows "ordered_keys m = (if finite (keys m) then sorted_list_of_set (keys m) else [])"
hyps: fixes l2 :: "'b list" assumes "lookup t1 f = Some e" shows "\<exists>i<length t1. e = t1 ! i \<and> (length t1 = length l2 \<longrightarrow> lookup2 t1 l2 f = Some (l2 ! i))"
inf_sup_aci_6: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.lattice inf less_eq less sup" shows "sup (sup x y) z = sup x (sup y z)"
ndiff_def: fixes n :: "nat"   and f :: "real \<Rightarrow> 'a"   and X :: "real set" shows "ndiff n f X = (\<forall>i<n. \<forall>x\<in>X. nderiv i f X differentiable at x within X)"
Stable_reachable: fixes F :: "'b program" shows "F \<in> Stable (reachable F)"
MonPon_PfP_implies_PfP: fixes \<alpha> :: "fm"   and \<beta> :: "fm" assumes "{} \<turnstile> \<alpha> IMP \<beta>"   and "ground_fm \<alpha>"   and "ground_fm \<beta>" shows "{PfP \<guillemotleft>\<alpha>\<guillemotright>} \<turnstile> PfP \<guillemotleft>\<beta>\<guillemotright>"
simps: fixes a1 :: "('a, 'b, 'c) com"   and a2 :: "('a, 'b, 'c) com" shows "((a1, a2) \<in> com_rel) = ((\<exists>c1 c2. a1 = c1 \<and> a2 = Seq c1 c2) \<or> (\<exists>c2 c1. a1 = c2 \<and> a2 = Seq c1 c2) \<or> (\<exists>c1 b c2. a1 = c1 \<and> a2 = Cond b c1 c2) \<or> (\<exists>c2 b c1. a1 = c2 \<and> a2 = Cond b c1 c2) \<or> (\<exists>c b. a1 = c \<and> a2 = While b c) \<or> (\<exists>c x. a1 = c x \<and> a2 = DynCom c) \<or> (\<exists>c f g. a1 = c \<and> a2 = Guard f g c) \<or> (\<exists>c1 c2. a1 = c1 \<and> a2 = Catch c1 c2) \<or> (\<exists>c2 c1. a1 = c2 \<and> a2 = Catch c1 c2))"
nrows_iarray_def: fixes A :: "'c iarray iarray" shows "nrows_iarray A = IArray.length A"
rem_closure: fixes K :: "'a set"   and A :: "'a set"   and B :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A"   and "B \<in> K .\<bottom>. \<phi>" shows "B = Cn B"
a: shows "rec_ci (Cn (Suc (Suc 0)) rec_right [Cn (Suc (Suc 0)) rec_conf [recf.id (Suc (Suc 0)) 0, recf.id (Suc (Suc 0)) (Suc 0), rec_halt]]) = (ap1, ar1, ft1)"
lexLessRestricted_def: fixes Vbl :: "nat set" shows "lexLessRestricted Vbl \<equiv> {(M1, M2). vars (elements M1) \<subseteq> Vbl \<and> consistent (elements M1) \<and> uniq (elements M1) \<and> vars (elements M2) \<subseteq> Vbl \<and> consistent (elements M2) \<and> uniq (elements M2) \<and> (M1, M2) \<in> lexLess}"
prefixToLevelAppend: fixes level :: "nat"   and a :: "('b \<times> bool) list"   and b :: "('b \<times> bool) list" shows "prefixToLevel level (a @ b) = (if level < currentLevel a then prefixToLevel level a else a @ prefixToLevel_aux b level (currentLevel a))"
UNIV_in_lattice: fixes L :: "'b set set" assumes "lattice L" shows "UNIV \<in> L"
fst_uminus_aform: fixes Y :: "'a \<times> 'b pdevs" shows "fst (uminus_aform Y) = - fst Y"
must_sync_preserved_d: fixes \<Phi> :: "String.literal \<Rightarrow> String.literal \<Rightarrow> (ty list \<times> ty err list) option list"   and t :: "'thread_id"   and xcp :: "'addr option"   and frs :: "'addr frame list"   and h :: "'heap"   and h' :: "'heap" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"   and "execd_mthr.must_sync P t (xcp, frs) h"   and "h \<unlhd> h'"   and "hconf h'"   and "\<Phi> \<turnstile> t:(xcp, h, frs) \<surd>" shows "execd_mthr.must_sync P t (xcp, frs) h'"
sum_my: fixes A :: "'c set"   and B :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "finite A"   and "finite B" shows "sum f A - sum g B = (\<Sum>x\<in>A \<inter> B. f x - g x) + sum f (A - B) - sum g (B - A)"
gauss_i_times_i: shows "gauss_i * gauss_i = - 1"
isomorphic_symmetric: fixes f :: "'a"   and g :: "'a" assumes "B.isomorphic f g" shows "B.isomorphic g f"
progress_e3: fixes \<S> :: "s_context"   and i :: "nat"   and vs :: "v list"   and cs_es :: "e list"   and ts' :: "t list"   and s :: "s" assumes "\<S>\<bullet>None \<tturnstile>_ i vs;cs_es : ts'"   and "cs_es \<noteq> [Trap]"   and "\<not> const_list cs_es"   and "store_typing s \<S>" shows "\<exists>a s' vs' es'. \<lparr>s;vs;cs_es\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"
inR1': fixes \<Gamma> :: "'b formula multiset"   and G :: "'b formula"   and H :: "'b formula"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "\<Gamma> \<Rightarrow> G, H, \<Delta> \<down> n" shows "\<Gamma> \<Rightarrow> H, G, \<Delta> \<down> n"
Cd_range: fixes d :: "'b"   and X :: "'a set" shows "Cd d X \<subseteq> X \<inter> Field (Pd d)"
size_cartesian_product: fixes A :: "'a multiset"   and B :: "'b multiset" shows "size (A \<times># B) = size A * size B"
insert_body_contains_new_elem: fixes xs :: "('c \<times> 'd \<times> bool) list"   and e :: "'c \<times> 'd \<times> bool" shows "\<exists>p s. xs = p @ s \<and> insert_body xs e = p @ e # s"
inf_idem: fixes x :: "'b multiset" shows "x \<inter># x = x"
rel_refl_strong: fixes x :: "'b \<times> 'c"   and R1a :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and R2a :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>z1. z1 \<in> Basic_BNFs.fsts x \<Longrightarrow> R1a z1 z1"   and "\<And>z2. z2 \<in> Basic_BNFs.snds x \<Longrightarrow> R2a z2 z2" shows "rel_prod R1a R2a x x"
lookup_frag_cmul: fixes c :: "int"   and x :: "'b \<Rightarrow>\<^sub>0 int"   and i :: "'b" shows "lookup (frag_cmul c x) i = c * lookup x i"
acts_MapOps_lookup_def: shows "acts_MapOps_lookup \<equiv> \<lambda>m k. Option.bind (trie_odlist_lookup m (fst k)) (\<lambda>m. trie_odlist_lookup m (snd k))"
G3_def: shows "G3 = (- 1) ^ (degree G1 - degree G2 + 1) * pseudo_mod G1 G2"
sdrop_sappend: fixes i :: "Integer"   and xs :: "[:'a:]"   and ys :: "[:'a:]" shows "sdrop\<cdot>i\<cdot>(xs :@ ys) = sdrop\<cdot>i\<cdot>xs :@ sdrop\<cdot>(i - slength\<cdot>xs)\<cdot>ys"
gauss_int_code_post_3: shows "Gauss_Int 0 (- 1) = - \<i>\<^sub>\<int>"
take_add: fixes i :: "nat"   and j :: "nat"   and xs :: "'d list" shows "take (i + j) xs = take i xs @ take j (drop i xs)"
fun_upd_eqD: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'c"   and g :: "'b \<Rightarrow> 'c"   and z :: "'c" assumes "f(x := y) = g(x := z)" shows "y = z"
fmlookup_drop: fixes a :: "'b"   and m :: "('b, 'a) fmap"   and x :: "'b" shows "fmlookup (fmdrop a m) x = (if x \<noteq> a then fmlookup m x else None)"
Gromov_converging_at_infinity_quasi_isometry: fixes lambda :: "real"   and C :: "real"   and f :: "'c \<Rightarrow> 'd"   and u :: "nat \<Rightarrow> 'c" assumes "lambda C -quasi_isometry f" shows "Gromov_converging_at_boundary (\<lambda>n. f (u n)) = Gromov_converging_at_boundary u"
union_def: fixes xs :: "'b odlist"   and ys :: "'b odlist" shows "ODList.union xs ys = ODList (merge (toList xs) (toList ys))"
simps: fixes f :: "'a \<Rightarrow> 'a"   and b :: "bool"   and a1 :: "'a"   and a2 :: "'a" shows "mod2 f b (a1, a2) = (if b then (a1, f a2) else (f a1, a2))"
exhaust: fixes y :: "X4" obtains "y = X4.Xd1"   | "y = X4.Xd1'"   | "y = X4.Xd2"   | "y = Xd2'"
open_euler2: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "euler_trail u p v"   and "u \<noteq> v" shows "Digraph_Component.connected G \<and> (\<forall>w\<in>verts G. u \<noteq> w \<longrightarrow> v \<noteq> w \<longrightarrow> in_degree G w = out_degree G w) \<and> in_degree G u + 1 = out_degree G u \<and> out_degree G v + 1 = in_degree G v"
cast_shadow_root_ptr_not_node_ptr_1: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and node_ptr :: "(_) node_ptr" shows "cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr \<noteq> cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr"
Pair_mono: fixes x :: "'c"   and x' :: "'c"   and y :: "'d"   and y' :: "'d" assumes "x \<le> x'"   and "y \<le> y'" shows "(x, y) \<le> (x', y')"
binomial_strict_mono: fixes k :: "nat"   and k' :: "nat"   and n :: "nat" assumes "k < k'"   and "2 * k' \<le> n" shows "n choose k < n choose k'"
distributivity: fixes id :: "'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zer :: "'a"   and g :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inver :: "'a \<Rightarrow> 'a"   and A1 :: "'a list list"   and A2 :: "'a list list"   and B1 :: "'a list list"   and B2 :: "'a list list" assumes "plus_mult id f zer g inver"   and "plus_mult.matrix_match A1 A2 B1 B2" shows "mult.Tensor f (mat_multI zer g f (mult.row_length A1) A1 A2) (mat_multI zer g f (mult.row_length B1) B1 B2) = mat_multI zer g f (mult.row_length (mult.Tensor f A1 B1)) (mult.Tensor f A1 B1) (mult.Tensor f A2 B2)"
issue_expr_load_tmps_range: fixes t :: "nat"   and e :: "expr" shows "\<forall>i\<in>load_tmps (issue_expr t e). t \<le> i \<and> i < t + used_tmps e"
fin_c\<sigma>: shows "finite (C \<cdot>\<^sub>l\<^sub>s sub_of_denot E)"
poss_subset_filterD: fixes AA :: "'a multiset"   and \<rho> :: "'s"   and C :: "'a literal list" assumes "poss AA \<subseteq># {#L \<cdot>l \<rho>. L \<in># mset C#}" shows "\<exists>AA0. poss AA0 \<subseteq># mset C \<and> AA = AA0 \<cdot>am \<rho>"
is_root_imp_pdivides: fixes p :: "'a list list"   and x :: "'a list" assumes "p \<in> carrier (poly_ring (poly_ring R))"   and "UP.is_root p x" shows "[\<one>\<^bsub>poly_ring R\<^esub>, \<ominus>\<^bsub>poly_ring R\<^esub> x] pdivides\<^bsub>poly_ring R\<^esub> p"
cong_left_commutativity: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Cong A B C D" shows "Cong B A C D"
part_6: fixes i :: "nat" assumes "i < lo" shows "xs' ! i = xs ! i"
a_def: shows "a = insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var 2)"
composition_series_snd_simple: assumes "2 \<le> length \<GG>" shows "simple_group (G\<lparr>carrier := \<GG> ! 1\<rparr>)"
i_less_suc: shows "mod_type_class.to_nat i \<le> Suc n"
Sup_upper: fixes x :: "'al"   and A :: "'al set" assumes "x \<in> A"   and "A \<subseteq> U" shows "x \<le>\<^sub>o\<^sub>w \<Squnion>\<^sub>o\<^sub>w A"
M_2: shows "Args_min_on f {i..j} \<noteq> {}"
scaleR_right: fixes a :: "real"   and r :: "real"   and b :: "'a" shows "a *\<^sub>R r *\<^sub>R b = r *\<^sub>R a *\<^sub>R b"
spmf_of_set_empty: shows "spmf_of_set {} = return_pmf None"
ireturn_rule: fixes \<Phi> :: "'e \<Rightarrow> bool"   and x :: "'e" assumes "\<Phi> x" shows "RETURN x \<le> SPEC \<Phi>"
simps_1: fixes ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) 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 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 "ip \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, handled = handled, \<dots> = more\<rparr> = ip"
sub_induction: fixes u :: "trm"   and \<tau> :: "ty"   and t :: "trm"   and \<sigma> :: "ty" assumes "SN u"   and "u \<in> RED \<tau>"   and "NEUT t"   and "CR2 \<tau>"   and "CR3 \<sigma>"   and "CR3_RED t (\<tau> \<rightarrow> \<sigma>)" shows "App t u \<in> RED \<sigma>"
T_Sub_5: fixes S' :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> U <: S"   and "TVarB U \<Colon> \<Gamma> \<turnstile> s : S'"   and "TVarB U \<Colon> \<Gamma> \<turnstile> S' <: U'" shows "P"
sup_sharp_def: fixes R :: "('h, 'i) Ring_scheme"   and M :: "('g, 'h, 'j) Module_scheme"   and N :: "('e, 'h, 'k) Module_scheme"   and L :: "('f, 'h, 'l) Module_scheme"   and u :: "'g \<Rightarrow> 'e" shows "sup_sharp R M N L u = (\<lambda>f\<in>mHom R N L. compos M f u)"
assms_3: shows "inj_on (g \<circ> f) (carrier G)"
hcomp_simps\<^sub>W\<^sub>C_3: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<nu> :: "'b"   and \<mu> :: "'b" assumes "weak_composition V H"   and "H \<nu> \<mu> \<noteq> partial_magma.null V" shows "partial_magma.cod V (H \<nu> \<mu>) = H (partial_magma.cod V \<nu>) (partial_magma.cod V \<mu>)"
pr_pri_agrk_unused: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and x :: "rat" assumes "(evs, S, A, U) \<in> protocol"   and "Pri_AgrK x \<notin> U" shows "analz (insert (Pri_AgrK x) (A \<union> spies evs)) = insert (Pri_AgrK x) (analz (A \<union> spies evs))"
induct: fixes P :: "term list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "term list"   and a1 :: "'a list" assumes "P [] []"   and "\<And>t ts u us. (\<And>a. match t u = Some a \<Longrightarrow> P ts us) \<Longrightarrow> P (t # ts) (u # us)"   and "\<And>v va. P (v # va) []"   and "\<And>v va. P [] (v # va)" shows "P a0 a1"
exhaust: fixes y :: "'c\<cdot>maybe" obtains "y = \<bottom>"   | "y = Nothing"   | a :: "'c" where "y = Just\<cdot>a"
dom_uniq_1: fixes \<sigma> :: "'Var \<Rightarrow> 'Val option" assumes "dom \<sigma> = differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems k" shows "dom_g1 = dom (g1 \<sigma>)"
match_list_append: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and m1 :: "'c match_expr list"   and m2 :: "'c match_expr list"   and a :: "action"   and p :: "'d" shows "match_list \<gamma> (m1 @ m2) a p = (\<not> match_list \<gamma> m1 a p \<longrightarrow> match_list \<gamma> m2 a p)"
prod_dfs_statetrans: fixes bs :: "bool list"   and i :: "nat"   and j :: "nat"   and s1 :: "nat"   and s2 :: "nat"   and k :: "nat" assumes "length bs = n"   and "dfa_is_node A i"   and "dfa_is_node B j"   and "dfa_is_node A s1"   and "dfa_is_node B s2"   and "fst (prod_dfs A B (s1, s2)) ! i ! j = Some k" obtains k' :: "nat" where "fst (prod_dfs A B (s1, s2)) ! dfa_trans A i bs ! dfa_trans B j bs = Some k'" and "dfa_is_node A (dfa_trans A i bs)" and "dfa_is_node B (dfa_trans B j bs)" and "k' < length (snd (prod_dfs A B (s1, s2)))"
atLeast1_atMost_eq: fixes g :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "prod g {Suc 0..n} = (\<Prod>k<n. g (Suc k))"
compass_reflect_compass_4: shows "apply_cltn2 south meridian_reflect = south"
DefX: shows "[\<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall>F. \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> (\<^bold>\<exists>p. p \<^bold>& F \<^bold>= (\<^bold>\<lambda>x. p))) in v]"
le_ASSERTI: fixes \<Phi> :: "bool"   and M :: "'maa"   and M' :: "'maa" assumes "\<Phi> \<Longrightarrow> M \<le> M'" shows "M \<le> abind (aASSERT \<Phi>) (\<lambda>_. M')"
mark_guards_simps_7: fixes f :: "'f"   and p :: "'e" shows "mark_guards f (Call p) = Call p"
2: shows "v\<^sup>T \<le> r\<^sup>T * components ((v \<squnion> e\<^sup>T * top) * (v \<squnion> e\<^sup>T * top)\<^sup>T \<sqinter> g) \<and> v \<le> (r\<^sup>T * components ((v \<squnion> e\<^sup>T * top) * (v \<squnion> e\<^sup>T * top)\<^sup>T \<sqinter> g))\<^sup>T"
Re_half: fixes x :: "complex" shows "Re (x / 2) = Re x / 2"
vector_diff_chain_within: fixes f :: "real \<Rightarrow> real"   and f' :: "real"   and x :: "real"   and s :: "real set"   and g :: "real \<Rightarrow> 'b"   and g' :: "'b" assumes "(f has_vector_derivative f') (at x within s)"   and "(g has_vector_derivative g') (at (f x) within \<P> f s)" shows "(g \<circ> f has_vector_derivative f' *\<^sub>R g') (at x within s)"
pseudo_rationalizable_on_def2: fixes A :: "'a set"   and Rs :: "('a \<times> 'a) set set"   and f :: "'a set \<Rightarrow> 'a set" shows "pseudo_rationalizable_on A Rs f = ((\<forall>r\<in>Rs. Linear_order r) \<and> (\<forall>B\<subseteq>A. f B = (\<Union>r\<in>Rs. set_option (MaxR.MaxR_opt r B))))"
simps_1: fixes n :: "nat" shows "FingerTreeStruc.is_leveln_ftree n Empty = True"
pop_impl_def_opt: fixes s :: "'b list \<times> nat list \<times> ('b \<Rightarrow> node_state option) \<times> (nat \<times> 'b set) list" shows "pop_impl s \<equiv> case s of (S, B, I, P) \<Rightarrow> let lsi = length B - 1 in ASSERT (lsi < length B) \<bind> (\<lambda>_. WHILE\<^sub>T (\<lambda>(l, I). l < (if lsi + 1 = length B then length S else B ! (lsi + 1))) (\<lambda>(l, I). ASSERT (l < length S) \<bind> (\<lambda>_. RETURN (Suc l, I(S ! l \<mapsto> DONE)))) (B ! lsi, I) \<bind> (\<lambda>(uu_, I). RETURN I) \<bind> (\<lambda>I. ASSERT (B \<noteq> []) \<bind> (\<lambda>_. let S = take (last B) S in ASSERT (B \<noteq> []) \<bind> (\<lambda>_. let B = butlast B in RETURN (S, B, I, P)))))"
assms_2: shows "f \<cdot> g = f' \<cdot> g"
quicksort_def: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "ord.quicksort less = ord.quicksort_acc less []"
assms_2: shows "v \<noteq> \<zero>\<^bsub>V\<^esub>"
subject_reduction: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and E :: "char list \<Rightarrow> ty option"   and T :: "ty" assumes "wf_J_prog P"   and "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow> \<langle>e',s',b'\<rangle>"   and "P,E \<turnstile> s \<surd>"   and "iconf (shp s) e"   and "P,E,hp s,shp s \<turnstile> e : T" shows "\<exists>T'. P,E,hp s',shp s' \<turnstile> e' : T' \<and> P \<turnstile> T' \<le> T"
real_inner_1_right: fixes x :: "real" shows "x \<bullet> 1 = x"
monoE: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a"   and f' :: "'a \<times> 'a" assumes "VV.mono g"   and "VV.seq g f"   and "VV.seq g f'"   and "VV.comp g f = VV.comp g f'" shows "f' = f"
that: fixes x :: "'a" assumes "x \<in>#\<^sub>z A" shows "thesis"
kD_nsqn_quality_increases_trans: fixes i :: "nat"   and \<xi> :: "state"   and s :: "nat"   and \<xi>' :: "state" assumes "i \<in> kD (rt \<xi>)"   and "s \<le> nsqn (rt \<xi>) i"   and "quality_increases \<xi> \<xi>'" shows "i \<in> kD (rt \<xi>') \<and> s \<le> nsqn (rt \<xi>') i"
order_iff_strict: fixes a :: "'t"   and b :: "'t" shows "(a \<preceq>\<^sub>t b) = (a \<prec>\<^sub>t b \<or> a = b)"
outgoing_alt: fixes u :: "nat" shows "outgoing u = Pair u ` E `` {u}"
AA_p: shows "AA \<in> AAA \<and> (pairs \<circ> compare_linorder.sorted_list_of_set) AA = AB_pairs"
boundedI: fixes a :: "'b \<times> 'c"   and b :: "'b \<times> 'c"   and c :: "'b \<times> 'c" assumes "less_eq_prod a b"   and "less_eq_prod a c" shows "less_eq_prod a (inf_prod b c)"
mustT_MtransC: fixes c :: "('test, 'atom) com"   and s :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "mustT c s"   and "(c, s) \<rightarrow>*c (c', s')" shows "mustT c' s'"
ineqs_1: shows "(max 0 b + max 0 d + 2) / (1 - a - c) < real n\<^sub>0"
lincomb_list_Cons: fixes c :: "nat \<Rightarrow> 'a"   and v :: "'c"   and vs :: "'c list" shows "lincomb_list c (v # vs) = c 0 \<odot>\<^bsub>M\<^esub> v \<oplus>\<^bsub>M\<^esub> lincomb_list (c \<circ> Suc) vs"
rHom_one: fixes A :: "('c, 'd) Ring_scheme"   and R :: "('e, 'f) Ring_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "Ring A"   and "Ring R"   and "f \<in> rHom A R" shows "f 1\<^sub>r\<^bsub>A\<^esub> = 1\<^sub>r\<^bsub>R\<^esub>"
exesort_ex_def: fixes rel :: "('a \<times> 'a) list"   and S :: "'a set" shows "exesort_ex rel S = (\<forall>x\<in>S. List.member (exefield rel) x)"
x': shows "x' \<in> (\<Union>loc'. set_zmset (a loc'))"
isTrue_neg_excl: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "isTrue \<phi>"   and "isTrue (neg \<phi>)" shows "False"
of_int_and_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and k :: "int"   and l :: "int" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "ring_1.of_int one plus zero minus (k AND l) = and (ring_1.of_int one plus zero minus k) (ring_1.of_int one plus zero minus l)"
cmp_fun_sub_image: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set"   and g :: "'f \<Rightarrow> 'g"   and C :: "'g set"   and A1 :: "'e set" assumes "f \<in> A \<rightarrow> B"   and "g \<in> B \<rightarrow> C"   and "A1 \<subseteq> A" shows "cmp g f ` A1 = g ` f ` A1"