hcju/isabelleps huggingface dataset:
B_ne: assumes "i \<in> set (take j xs)" shows "{k. k < length xs \<and> k < j \<and> xs ! k = i} \<noteq> {}"
ns_mul_ext_IdI: fixes M :: "'a multiset"   and X :: "'a multiset"   and Z :: "'a multiset"   and N :: "'a multiset"   and Y :: "'a multiset"   and R :: "('a \<times> 'a) set" assumes "M = X + Z"   and "N = Y + Z"   and "\<forall>y\<in>#Y. \<exists>x\<in>#X. (x, y) \<in> R" shows "(M, N) \<in> ns_mul_ext Id R"
addS_def: shows "addS \<equiv> \<Lambda> x y. bindS\<cdot>x\<cdot>(\<Lambda> xv. bindS\<cdot>y\<cdot>(\<Lambda> yv. returnS\<cdot>(xv + yv)))"
tendsto_upperbound: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and "open" :: "'d set \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd"   and x :: "'d"   and F :: "'e filter"   and a :: "'d" assumes "class.linorder_topology less_eq less open"   and "filterlim f (topological_space.nhds open x) F"   and "\<forall>\<^sub>F i in F. less_eq (f i) a"   and "F \<noteq> bot" shows "less_eq x a"
hankel_semiannulus_def: fixes r :: "real"   and N :: "nat" shows "hankel_semiannulus r N = (let R = (2 * real N + 1) * pi in part_circlepath 0 R 0 pi +++ linepath (complex_of_real (- R)) (complex_of_real (- r)) +++ part_circlepath 0 r pi 0 +++ linepath (complex_of_real r) (complex_of_real R))"
contr_2: shows "(\<exists>p. \<gamma> (ofe_fields a) p) \<or> (\<exists>p. \<gamma> (ofe_fields b) p)"
tab_simps_5: shows "cod \<rho>' = r \<star> f'"
initialized3_def: fixes A :: "int \<times> day \<Rightarrow> int"   and i :: "int"   and k :: "int" shows "initialized3 A i k = (\<forall>j\<in>{0..<k}. A (i, val j) = 0)"
traverse_cast_Dyn_to_all_Dyn: fixes Finca_empty :: "'a"   and Finca_get :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'f) Inca.instr fundef option"   and Finca_add :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'f) Inca.instr fundef \<Rightarrow> 'a"   and Finca_to_list :: "'a \<Rightarrow> ('b \<times> ('c, 'd, 'b, 'e, 'f) Inca.instr fundef) list"   and Fubx_empty :: "'g"   and Fubx_get :: "'g \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'f, 'h, 'i, 'j) Ubx.instr fundef option"   and Fubx_add :: "'g \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'f, 'h, 'i, 'j) Ubx.instr fundef \<Rightarrow> 'g"   and Fubx_to_list :: "'g \<Rightarrow> ('b \<times> ('c, 'd, 'b, 'e, 'f, 'h, 'i, 'j) Ubx.instr fundef) list"   and heap_empty :: "'k"   and heap_get :: "'k \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c option"   and heap_add :: "'k \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c \<Rightarrow> 'k"   and heap_to_list :: "'k \<Rightarrow> (('d \<times> 'c) \<times> 'c) list"   and is_true :: "'c \<Rightarrow> bool"   and is_false :: "'c \<Rightarrow> bool"   and box_ubx1 :: "'i \<Rightarrow> 'c"   and unbox_ubx1 :: "'c \<Rightarrow> 'i option"   and box_ubx2 :: "'j \<Rightarrow> 'c"   and unbox_ubx2 :: "'c \<Rightarrow> 'j option"   and \<OO>\<pp> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<AA>\<rr>\<ii>\<tt>\<yy> :: "'e \<Rightarrow> nat"   and \<II>\<nn>\<ll>\<OO>\<pp> :: "'f \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<II>\<nn>\<ll> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'f option"   and \<II>\<ss>\<II>\<nn>\<ll> :: "'f \<Rightarrow> 'c list \<Rightarrow> bool"   and \<DD>\<ee>\<II>\<nn>\<ll> :: "'f \<Rightarrow> 'e"   and \<UU>\<bb>\<xx>\<OO>\<pp> :: "'h \<Rightarrow> ('c, 'i, 'j) unboxed list \<Rightarrow> ('c, 'i, 'j) unboxed option"   and \<UU>\<bb>\<xx> :: "'f \<Rightarrow> Unboxed.type option list \<Rightarrow> 'h option"   and \<BB>\<oo>\<xx> :: "'h \<Rightarrow> 'f"   and \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp> :: "'h \<Rightarrow> Unboxed.type option list \<times> Unboxed.type option"   and xs :: "('l, 'm, 'n) unboxed list"   and ys :: "'l list" assumes "inca_to_ubx_simulation Finca_empty Finca_get Finca_add Finca_to_list Fubx_empty Fubx_get Fubx_add Fubx_to_list heap_empty heap_get heap_add heap_to_list is_true is_false box_ubx1 unbox_ubx1 box_ubx2 unbox_ubx2 \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<II>\<nn>\<ll>\<OO>\<pp> \<II>\<nn>\<ll> \<II>\<ss>\<II>\<nn>\<ll> \<DD>\<ee>\<II>\<nn>\<ll> \<UU>\<bb>\<xx>\<OO>\<pp> \<UU>\<bb>\<xx> \<BB>\<oo>\<xx> \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp>"   and "traverse cast_Dyn xs = Some ys" shows "list_all (\<lambda>x. typeof x = None) xs"
simps_2: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree" shows "Tree.complete \<langle>l, x, r\<rangle> = (height l = height r \<and> Tree.complete l \<and> Tree.complete r)"
codomains_comp: fixes g :: "'a"   and f :: "'a" assumes "B.seq g f" shows "B.codomains (g \<cdot>\<^sub>B f) = B.codomains g"
not_sym: fixes t :: "'d"   and s :: "'d" assumes "t \<noteq> s" shows "s \<noteq> t"
test_eq: fixes p :: "'a"   and q :: "'a" shows "(p = q) = (\<iota> p = \<iota> q)"
rel_pmf_restrict_relp_iff: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool"   and x :: "'c pmf"   and y :: "'d pmf" shows "rel_pmf (R \<upharpoonleft> P \<otimes> Q) x y = (rel_pmf R x y \<and> pred_pmf P x \<and> pred_pmf Q y)"
of_bits_int_simps_1: shows "of_bits [] = 0"
not_call_simps_5: fixes l :: "'d"   and fmsg :: "'a \<Rightarrow> 'b"   and p :: "('a, 'b, 'c, 'd) seqp" shows "not_call ({l}broadcast(fmsg) . p)"
ivl_cong: fixes a :: "'b"   and c :: "'b"   and b :: "'b"   and d :: "'b"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "a = c"   and "b = d"   and "\<And>x. \<lbrakk>c \<le> x; x < d\<rbrakk> \<Longrightarrow> g x = h x" shows "prod g {a..<b} = prod h {c..<d}"
sum_fun_apply: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'b"   and x :: "'c" assumes "finite A" shows "sum f A x = (\<Sum>a\<in>A. f a x)"
\<omega>_simps_5: shows "S.cod \<i>[a] = a"
MOST_iff_cofinite: fixes P :: "'a \<Rightarrow> bool" shows "(\<forall>\<^sub>\<infinity>x. P x) = finite {x. \<not> P x}"
Q_coarser: shows "Q \<in> coarser_partitions_with elem P"
\<tau>2: shows "typ_of (Abs \<tau>' t) = Some (\<tau>' \<rightarrow> \<tau>)"
line_in'_1: fixes y :: "real" assumes "y \<in> cbox 0 1" shows "h * y + t \<in> T"
RPDs_aux1: fixes r :: "mregex" shows "lfp (\<lambda>S. insert r S \<union> (insert r S \<bind> RPD)) \<subseteq> RPDs r"
Vr_pOp_f_pOp: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and y :: "'a" assumes "valuation K v"   and "x \<in> carrier (Vr K v)"   and "y \<in> carrier (Vr K v)" shows "x \<plusminus>\<^bsub>Vr K v\<^esub> y = x \<plusminus> y"
simps_1: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'e"   and w :: "'e"   and xs :: "'e list" assumes "Digraph G" shows "H.first_edge_of_walk (v # w # xs) = (v, w)"
\<phi>: shows "bij_betw \<phi> {k. k < n} I"
iT_Div_Max: fixes I :: "nat set"   and k :: "nat" assumes "finite I"   and "I \<noteq> {}" shows "Max (I \<oslash> k) = Max I div k"
LANDAU_PROD_def: fixes L :: "(real \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> 'b) set"   and c1 :: "'b"   and c2 :: "real"   and fs :: "(primfun \<times> real) list" shows "LANDAU_PROD L c1 c2 fs = ((\<lambda>_. c1) \<in> L (\<lambda>x. c2 * eval_primfuns fs x))"
highBodies: fixes \<Gamma> :: "char list \<Rightarrow> secLevel option"   and c :: "com"   and s1 :: "char list \<Rightarrow> val option"   and s2 :: "char list \<Rightarrow> val option" assumes "\<Gamma>,High \<turnstile> c"   and "\<langle>c,s1\<rangle> \<rightarrow>* \<langle>Skip,s2\<rangle>" shows "\<Gamma> \<turnstile> s1 \<approx>\<^sub>L s2"
a2': shows "s |\<^sub>s n \<approx> t |\<^sub>s m \<and> (s |\<^sub>s Suc n \<approx> t |\<^sub>s Suc m \<or> s |\<^sub>s Suc n \<approx> t |\<^sub>s m)"
fundantivertex_unstable: fixes s :: "'a permutation" assumes "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})" shows "s \<rightarrow> fundantivertex s \<noteq> fundantivertex s"
of_int_def: shows "Rat.of_int = rat_of_int"
1: assumes "Arr t" shows "t \<in> HHom (Src t) (Trg t) \<and> t \<in> VHom (Dom t) (Cod t)"
assms_2: shows "\<forall>v. dim_vec v = dim_col U \<longrightarrow> \<parallel>col_fst (U * |v\<rangle>)\<parallel> = \<parallel>v\<parallel>"
boolean_II_is_boolean: shows "class.boolean_algebra (-) uminus (\<sqinter>) (\<sqsubseteq>) (\<sqsubset>) (\<squnion>) \<bottom> \<top>"
interval_neqs_1: fixes r :: "real"   and s :: "real"   and t :: "real" shows "{r<..<s} \<noteq> {t<..}"
col_permutation_5: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet A B C" shows "Tarski_neutral_dimensionless.Col Bet A C B"
pwd_spy_sesk_msg: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "case pwd_spy_sesk n of (s, S, A, B, C, D) \<Rightarrow> pwd_spy_sesk_state n (S, A, B, C, D) \<subseteq> s \<and> {Key (Auth_ShaKey n)} \<in> key_sets (PriKey S) (crypts (Log -` spied s))"
simps_5: fixes f1 :: "'d"   and f2 :: "'e \<Rightarrow> 'd"   and x2 :: "'e" shows "(case Some x2 of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f2 x2"
bind_eq_OkD: fixes x :: "('l, 'n) result"   and f :: "'n \<Rightarrow> ('l, 'm) result"   and z :: "'m" assumes "x \<bind> f = Ok z" shows "\<exists>y. x = Ok y \<and> f y = Ok z"
change_respecting_subst: fixes cms :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and mem :: "'Var \<Rightarrow> 'Val"   and cms' :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and mem' :: "'Var \<Rightarrow> 'Val"   and X :: "'Var set" assumes "change_respecting (cms, mem) (cms', mem') X" shows "\<forall>\<sigma>. dom \<sigma> = X \<longrightarrow> (cms, subst \<sigma> mem) \<leadsto> (cms', subst \<sigma> mem')"
assms: shows "fds_nth f (Suc 0) \<noteq> (0::'a)"
eexp_mono: fixes x :: "ereal"   and y :: "ereal" shows "(eexp x \<le> eexp y) = (x \<le> y)"
fds_moebius_inversion: fixes f :: "'b fds"   and g :: "'b fds" shows "(f = fds moebius_mu * g) = (g = f * fds_zeta)"
lbutlast_LCons: fixes r :: "'b llist"   and A :: "'b set"   and a :: "'b" assumes "r \<in> A\<^sup>\<star>" shows "lbutlast (a ## r) = (if r = LNil then LNil else a ## lbutlast r)"
opt_bst_wpl2_eq_pair: fixes w :: "int \<Rightarrow> int \<Rightarrow> nat"   and i :: "int"   and j :: "int" assumes "Optimal_BST2 w" shows "Optimal_BST2.opt_bst_wpl2 w i j = (Optimal_BST2.opt_bst2 w i j, wpl.wpl w i j (Optimal_BST2.opt_bst2 w i j))"
assms_3: shows "f \<approx>>\<^bsub>C\<^esub> g"
id1: shows "eval_poly (\<lambda>_. x) (mc # p) = x ^ monom_degree m * c + eval_poly (\<lambda>_. x) p"
mod_star_trueE: fixes P :: "assn"   and h :: "heap \<times> nat set" assumes "h \<Turnstile> P * true" obtains h' :: "heap \<times> nat set" where "h' \<Turnstile> P"
ts_id_def_2: shows "ts' = ts_id @ [arity_2_result e]"
a_repr_independenceD: fixes H :: "'e set"   and G :: "('e, 'f) ring_scheme"   and y :: "'e"   and x :: "'e" assumes "abelian_subgroup H G"   and "y \<in> carrier G"   and "H +>\<^bsub>G\<^esub> x = H +>\<^bsub>G\<^esub> y" shows "y \<in> H +>\<^bsub>G\<^esub> x"
exhaust: fixes y :: "CValue" obtains "y = \<bottom>"   | cfun :: "(C \<rightarrow> CValue) \<rightarrow> C \<rightarrow> CValue" where "y = CFn\<cdot>cfun"   | discr :: "bool discr" where "y = CB\<cdot>discr"
update_convs_1: fixes pc_iiface' :: "char list \<Rightarrow> char list"   and pc_iiface :: "char list"   and pc_oiface :: "char list"   and pc_proto :: "8 word"   and pc_sport :: "16 word"   and pc_dport :: "16 word"   and more :: "'a" shows "pc_iiface_update pc_iiface' \<lparr>pc_iiface = pc_iiface, pc_oiface = pc_oiface, pc_proto = pc_proto, pc_sport = pc_sport, pc_dport = pc_dport, \<dots> = more\<rparr> = \<lparr>pc_iiface = pc_iiface' pc_iiface, pc_oiface = pc_oiface, pc_proto = pc_proto, pc_sport = pc_sport, pc_dport = pc_dport, \<dots> = more\<rparr>"
convex_contains_segment: fixes S :: "'b set" shows "convex S = (\<forall>a\<in>S. \<forall>b\<in>S. closed_segment a b \<subseteq> S)"
some_geodesic_is_geodesic_segment_1: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.geodesic_space dist uniformity open" shows "geodesic_segment_between {x--y} x y"
span_empty: fixes K :: "('b, 'c) ring_scheme"   and V :: "('b, 'd, 'e) module_scheme" assumes "vectorspace K V" shows "LinearCombinations.module.span K V {} = {\<zero>\<^bsub>V\<^esub>}"
ide_g': shows "D.ide (\<sigma>.map\<^sub>0 a' \<star>\<^sub>D g \<star>\<^sub>D \<tau>\<^sub>0 a)"
intro: fixes \<epsilon> :: "'a"   and n :: "'a" assumes "(0::'a) < \<epsilon>"   and "(0::'a) < n" shows "sqrt_approximation \<epsilon> n"
ext_inject: fixes pset :: "'b set"   and order :: "('b \<times> 'b) set"   and more :: "'c"   and pset' :: "'b set"   and order' :: "('b \<times> 'b) set"   and more' :: "'c" shows "(\<lparr>pset = pset, order = order, \<dots> = more\<rparr> = \<lparr>pset = pset', order = order', \<dots> = more'\<rparr>) = (pset = pset' \<and> order = order' \<and> more = more')"
hb_map_def: fixes f :: "'b \<Rightarrow> 'a"   and xq :: "'b option lazy_sequence" shows "hb_map f xq = Lazy_Sequence.map (map_option f) xq"
continuous_on_polymonial_function: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "polynomial_function f" shows "continuous_on S f"
succ_inject: fixes zero :: "'a"   and succ :: "'a \<Rightarrow> 'a"   and m :: "'a"   and n :: "'a" assumes "peano zero succ" shows "(succ m = succ n) = (m = n)"
prime_elemD2: fixes p :: "'a"   and a :: "'a" assumes "prime_elem p"   and "a dvd p"   and "\<not> is_unit a" shows "p dvd a"
encR: fixes S :: "'procS"   and TRel :: "('procT \<times> 'procT) set" shows "SourceTerm S \<R>\<lbrakk>\<cdot>\<rbrakk>T<TRel> TargetTerm (\<lbrakk>S\<rbrakk>)"
subfield_m_inv_1: fixes K :: "'a list set"   and k :: "'a list" assumes "subfield K (poly_ring R)"   and "k \<in> K - {\<zero>\<^bsub>poly_ring R\<^esub>}" shows "inv\<^bsub>poly_ring R\<^esub> k \<in> K - {\<zero>\<^bsub>poly_ring R\<^esub>}"
assms: shows "(x, y) \<in> set (non_special_solutions a b)"
bne: shows "ball z 1 \<inter> U \<noteq> {}"
VSEquality_3_times_def: shows "VSEquality_3_times = VSEquality \<circ> VSEquality \<circ> VSEquality"
IH: fixes x0 :: "real \<times> real" assumes "list_all (\<lambda>x. x \<noteq> 0) xs"   and "list_all (\<lambda>x. Counterclockwise_2D_Arbitrary.lex x 0) xs" shows "distinct (map fst (polychain_of x0 xs))"
poly_eval_times: fixes a :: "'b \<Rightarrow> 'a"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "poly_eval a (p * q) = poly_eval a p * poly_eval a q"
division_inter: fixes p1 :: "'a set set"   and s1 :: "'a set"   and p2 :: "'a set set"   and s2 :: "'a set" assumes "p1 division_of s1"   and "p2 division_of s2" shows "{k1 \<inter> k2 |k1 k2. k1 \<in> p1 \<and> k2 \<in> p2 \<and> k1 \<inter> k2 \<noteq> {}} division_of s1 \<inter> s2"
him: shows "h ` (U \<times> {0..1}) \<subseteq> S"
winding_number_offset: fixes p :: "real \<Rightarrow> complex"   and z :: "complex" shows "winding_number p z = winding_number (\<lambda>w. p w - z) 0"
Iod_not_le_less: fixes T :: "'a set"   and a :: "'a"   and b :: "'a" assumes "T \<subseteq> carrier D"   and "a \<in> T"   and "b \<in> T"   and "Torder (Iod D T)" shows "(\<not> a \<preceq>\<^bsub>Iod D T\<^esub> b) = b \<prec>\<^bsub>Iod D T\<^esub> a"
post_before_pre_different_processes: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and i :: "nat"   and j :: "nat"   and t :: "('d, 'e, 'f) event list" assumes "computation channel trans send recv init final"   and "i < j"   and "j < length t"   and "\<forall>k. i < k \<and> k < j \<longrightarrow> \<not> regular_event (t ! k)"   and "computation.postrecording_event channel trans send recv init t i"   and "computation.prerecording_event channel trans send recv init t j"   and "distributed_system.trace channel trans send recv init t final" shows "occurs_on (t ! i) \<noteq> occurs_on (t ! j)"
bad_prik_def: shows "bad_prik \<equiv> SOME U. U \<subseteq> range Auth_PriK"
vend_nothing_def: shows "vend_nothing \<equiv> \<lparr>Label = STR ''vend'', Arity = 0, Guards = [], Outputs = [], Updates = [(1, V (R 1)), (2, V (R 2))]\<rparr>"
failures_def: fixes P :: "'e process" shows "failures P \<equiv> fst (Rep_process P)"
bigtheta_mult_eq: fixes F :: "'c filter"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "\<Theta>[F](\<lambda>x. f x * g x) = \<Theta>[F](f) * \<Theta>[F](g)"
F_def: fixes P :: "'c process" shows "F P = FAILURES (Rep_process P)"
pre_Base: fixes R :: "'a language \<Rightarrow> 'a language \<Rightarrow> bool"   and x :: "'a language"   and y :: "'a language" assumes "R x y" shows "Plus_Times_pre_cong R x y"
lunit_in_hom: fixes a :: "'a" assumes "ide a" shows "\<guillemotleft>\<l>[a] : \<one> \<otimes> a \<rightarrow> a\<guillemotright>"
import_param_3_1: fixes P :: "bool"   and Q :: "bool" shows "\<up> (P \<and> Q) = \<up> P * \<up> Q"
enum_idx_enum: fixes i :: "nat" assumes "i < CARD('b)" shows "enum_idx (enum_class.enum ! i) = i"
both_contains_arg1: fixes t :: "'b ttree"   and t' :: "'b ttree" shows "paths t \<subseteq> paths (t \<otimes>\<otimes> t')"
reach\<^sub>t_foldl_def: fixes \<Sigma> :: "'b set"   and \<delta> :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and q\<^sub>0 :: "'a" shows "reach\<^sub>t \<Sigma> \<delta> q\<^sub>0 = {(foldl \<delta> q\<^sub>0 w, \<nu>, foldl \<delta> q\<^sub>0 (w @ [\<nu>])) |w \<nu>. set w \<subseteq> \<Sigma> \<and> \<nu> \<in> \<Sigma>}"
simps: fixes state :: "State" shows "applyExplainUIP state = (if getCn state = 1 then state else applyExplainUIP (applyExplain (getCl state) state))"
limit_vimage: fixes a :: "'b"   and x :: "nat \<Rightarrow> 'b" shows "(a \<in> limit x) = infinite (x -` {a})"
subdiv_ends_H_rev_2: shows "tail H (rev_H vw) = w"
huntington_is_boolean_II: shows "class.boolean_algebra_II uminus (\<sqinter>) (\<squnion>) \<bottom> \<top>"
SUP_identity_eq: fixes Sup :: "'e set \<Rightarrow> 'e"   and A :: "'e set" shows "Sup ((\<lambda>x. x) ` A) = Sup A"
Suci_div_s_2: fixes r :: "nat"   and s :: "nat"   and i :: "nat" assumes "0 < r"   and "0 < s"   and "i \<le> r * s - Suc 0"   and "i mod s < s - Suc 0" shows "Suc i div s = i div s"
path_set_induct: fixes E :: "('b \<times> 'b) set"   and u :: "'b"   and p :: "'b list"   and v :: "'b"   and I :: "'b set" assumes "path E u p v"   and "u \<in> I"   and "E `` I \<subseteq> I" shows "set p \<subseteq> I"
leadsTo_eq_subset_wlt: fixes F :: "'c program"   and A :: "'c set"   and B :: "'c set" shows "(F \<in> A \<longmapsto> B) = (A \<subseteq> wlt F B)"
wprepare_loop_start_on_rightmost_Oc2: fixes m :: "nat"   and a :: "nat"   and b :: "cell list"   and list :: "cell list" assumes "wprepare_goto_start_pos m [a] (b, Oc # list)" shows "wprepare_loop_start_on_rightmost m [a] (Oc # b, list)"
subdegree_mult: fixes f :: "'a fps"   and g :: "'a fps" assumes "f \<noteq> 0"   and "g \<noteq> 0" shows "subdegree (f * g) = subdegree f + subdegree g"
disc_37: fixes x31 :: "trm"   and x32 :: "trm" shows "is_Eq (formula.Eq x31 x32)"
real_zeta_ge_one_over_minus_one: fixes z :: "real" assumes "1 < z" shows "1 / (z - 1) \<le> Re (zeta (complex_of_real z))"
cyclic_on_alldef: fixes f :: "'c \<Rightarrow> 'c"   and S :: "'c set" shows "cyclic_on f S = (S \<noteq> {} \<and> (\<forall>s\<in>S. S = orbit f s))"
oprod_zero_1: fixes r :: "('e \<times> 'e) set" shows "{} *o r = {}"
set_of_ivl_def: fixes x :: "'d \<times> 'd" shows "set_of_ivl x = {fst x..snd x}"
Act\<^sub>\<alpha>_eqvt: fixes p :: "perm"   and \<alpha> :: "'d"   and t\<^sub>\<alpha> :: "('b, 'c, 'd) Tree\<^sub>\<alpha>" shows "p \<bullet> Act\<^sub>\<alpha> \<alpha> t\<^sub>\<alpha> = Act\<^sub>\<alpha> (p \<bullet> \<alpha>) (p \<bullet> t\<^sub>\<alpha>)"
K: shows "K \<equiv> {(x, y). rel_set (\<lambda>y x. T x y) x y} `` K'"
comp_def: fixes g :: "'c"   and f :: "'c" shows "g \<cdot>\<^sup>o\<^sup>p f \<equiv> f \<cdot> g"
nneg_bounded_by_nneg: fixes b :: "real"   and P :: "'c \<Rightarrow> real" assumes "bounded_by b P"   and "nneg P" shows "0 \<le> b"
tllist_all2_tfinite1_terminalD: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "('a, 'c) tllist"   and ys :: "('b, 'd) tllist" assumes "tllist_all2 P Q xs ys"   and "tfinite xs" shows "Q (terminal xs) (terminal ys)"
poly_ge_def: fixes p :: "('b monom \<times> 'c) list"   and q :: "('b monom \<times> 'c) list" shows "p \<ge>p q = (\<forall>\<alpha>. pos_assign \<alpha> \<longrightarrow> eval_poly \<alpha> q \<le> eval_poly \<alpha> p)"
simps: fixes a :: "'b multiset" shows "local.well_base a = (\<exists>M. a = M \<and> (\<forall>n. count M n < base))"
image_is_empty: fixes f :: "'f \<Rightarrow> 'e"   and A :: "'f set" shows "(f ` A = {}) = (A = {})"
inject_2: fixes x2 :: "'d"   and y2 :: "'d" shows "(deny x2 = deny y2) = (x2 = y2)"
Inf_greatest: fixes A :: "'al set"   and z :: "'al" assumes "A \<subseteq> U"   and "z \<in> U"   and "\<And>x. x \<in> A \<Longrightarrow> z \<le>\<^sub>o\<^sub>w x" shows "z \<le>\<^sub>o\<^sub>w \<Sqinter>\<^sub>o\<^sub>w A"
l2_inv6'_def: shows "l2_inv6' \<equiv> {s. \<forall>Rb A B gnx. guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<longrightarrow> in_progress (progress s Rb) xsk \<longrightarrow> sklvl1.guessed_frame Rb xgnx = Some gnx \<longrightarrow> sklvl1.guessed_frame Rb xsk = Some (Exp gnx (NonceF (Rb $ ny)))}"
refinesD: fixes prog :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and prog' :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and P :: "'c \<Rightarrow> real" assumes "prog \<sqsubseteq> prog'"   and "sound P" shows "wp prog P \<tturnstile> wp prog' P"
dvd_0_right: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and a :: "'c" assumes "class.comm_semiring_1 times one plus zero" shows "dvd.dvd times a zero"
extended_distance_set_basepoint: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c" assumes "class.metric_space dist uniformity open" shows "basepoint x \<in> metric_space.extended_distance_set dist"
D_mndet: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a process" assumes "A \<noteq> {}" shows "D (mndet A P) = (\<Union>x\<in>A. D (x \<rightarrow> P x))"
secret_parts_Spy: fixes m :: "msg"   and evs :: "event list" assumes "m \<in> parts (knows Spy evs)" shows "m \<in> initState Spy \<or> (\<exists>C B X. Says C B X \<in> set evs \<and> m \<in> parts {X}) \<or> (\<exists>C Y. Notes C Y \<in> set evs \<and> C \<in> bad \<and> m \<in> parts {Y})"
trancl_converse: fixes r :: "('b \<times> 'b) set" shows "(r\<inverse>)\<^sup>+ = (r\<^sup>+)\<inverse>"
cblinfun_apply_cblinfun_compose: fixes a :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'a"   and b :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'c"   and c :: "'b" shows "(a o\<^sub>C\<^sub>L b) *\<^sub>V c = a *\<^sub>V b *\<^sub>V c"
invar_def: fixes ts :: "'b tree list" shows "invar ts = ((\<forall>t\<in>set ts. invar_tree t) \<and> sorted_wrt (<) (map rank ts))"
fresh_3: fixes a :: "atom"   and nat :: "nat" shows "a \<sharp> DBInd nat = a \<sharp> nat"
lgR_ok: fixes y :: "nat"   and x :: "nat"   and xa :: "nat" assumes "Suc 0 < y"   and "lgR [x, y, xa]" shows "xa \<le> x"
Pbefore_in_def: fixes x :: "'b"   and y :: "'b"   and A :: "('b list \<Rightarrow> 'c pmf) \<times> ('b list \<times> 'c \<Rightarrow> 'b \<Rightarrow> ((nat \<times> nat list) \<times> 'c) pmf)"   and qs :: "'b list"   and init :: "'b list" shows "Pbefore_in x y A qs init = map_pmf (\<lambda>p. x < y in fst p) (Partial_Cost_Model.config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) qs)"