hcju/isabelleps huggingface dataset:
nat_filtration_not_borel_info: fixes p :: "real"   and M :: "bool stream measure"   and f :: "bool stream \<Rightarrow> 'b"   and n :: "nat"   and N :: "'b measure" assumes "infinite_coin_toss_space p M"   and "f \<in> infinite_coin_toss_space.nat_filtration M n \<rightarrow>\<^sub>M N"   and "set_discriminating n f N" shows "f \<circ> pseudo_proj_True n = f"
first_time_def: fixes \<rho> :: "'b run"   and K :: "clock"   and n :: "nat"   and \<tau> :: "'b tag_const" shows "first_time \<rho> K n \<tau> \<equiv> time (Rep_run \<rho> n K) = \<tau> \<and> \<not> (\<exists>n'<n. time (Rep_run \<rho> n' K) = \<tau>)"
cases: fixes a1 :: "'a fset"   and a2 :: "'b \<rightharpoonup>\<^sub>f 'c \<rightharpoonup>\<^sub>f ('a \<times> nat \<times> enat \<times> bool \<times> bool)"   and a3 :: "'a"   and a4 :: "'c option"   and a5 :: "'c"   and a6 :: "'a"   and a7 :: "'a \<times> nat \<times> enat \<times> bool \<times> bool" assumes "unique_closest_association_end a1 a2 a3 a4 a5 a6 a7" obtains classes :: "'a fset" and associations :: "'b \<rightharpoonup>\<^sub>f 'c \<rightharpoonup>\<^sub>f ('a \<times> nat \<times> enat \<times> bool \<times> bool)" and \<C> :: "'a" and "from" :: "'c option" and role :: "'c" and \<D> :: "'a" and "end" :: "'a \<times> nat \<times> enat \<times> bool \<times> bool" where "a1 = classes" and "a2 = associations" and "a3 = \<C>" and "a4 = from" and "a5 = role" and "a6 = \<D>" and "a7 = end" and "closest_association_end classes associations \<C> from role \<D> end" and "\<not> closest_association_end_not_unique classes associations \<C> from role \<D> end"
assms_1: shows "u \<rightarrow>\<^sup>* v"
runit_hcomp_2: fixes f :: "'d"   and g :: "'d" assumes "D.ide f"   and "D.ide g"   and "src\<^sub>D f = trg\<^sub>D g" shows "\<r>\<^sub>D\<^sup>-\<^sup>1[f \<star>\<^sub>D g] = \<a>\<^sub>D\<^sup>-\<^sup>1[f, g, src\<^sub>D g] \<cdot>\<^sub>D (f \<star>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[g])"
pickT_unique: fixes cl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and n :: "nat"   and II :: "nat set" assumes "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n"   and "n < length cl"   and "II \<in> local.shift cl n ` P n" shows "n = pickT cl P II"
result_eq: shows "(rel, aux') = update_since I pos rel1 rel2 (\<tau> \<sigma> ne) aux"
set1_G_map: fixes l1 :: "'n \<Rightarrow> 'u"   and l2 :: "'o \<Rightarrow> 'v" shows "set1_G \<circ> mapl_G l1 l2 = (`) l1 \<circ> set1_G"
lm119: fixes l :: "'b list"   and X :: "'b set" shows "(!) l ` set (filterpositions2 (\<lambda>x. x \<in> X) l) \<subseteq> X \<inter> set l"
Some_eq_enforce_option_iff: fixes y :: "'c"   and P :: "'c \<Rightarrow> bool"   and x :: "'c option" shows "(Some y = enforce_option P x) = (x = Some y \<and> P y)"
elims_2: fixes x :: "'at set"   and xa :: "'at Literal" assumes "x \<Turnstile> xa" obtains I :: "'at set" and A :: "'at" where "x = I__" and "xa = Pos A__" and "A__ \<in> I__"   | I :: "'at set" and A :: "'at" where "x = I" and "xa = Neg A" and "A \<notin> I"
LoadElim1: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and os :: "Val list"   and S :: "(Var \<times> Val) list"   and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and n :: "nat"   and ll :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and x :: "Var" assumes "(M, l, (os, S, h), n, ll, t) \<in> Step"   and "get_ins M l = Some (load x)" shows "\<exists>v. S\<down>x = Some v \<and> n = Suc 0 \<and> t = (v # os, S, h) \<and> ll = l + 1"
less_fun_irrefl: fixes f :: "'c \<Rightarrow> 'd" shows "\<not> less_fun f f"
rep_eq: fixes x :: "complex_homo_coords"   and xa :: "complex_homo_coords" shows "dist_fs_hcoords x xa = dist_fs_cvec (Rep_complex_homo_coords x) (Rep_complex_homo_coords xa)"
DisN: fixes a :: "'c"   and ps :: "('b, 'c) fm list"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and p :: "('b, 'c) fm"   and q :: "('b, 'c) fm"   and A :: "'c set"   and n :: "nat" assumes "(\<^bold>\<not> (p \<^bold>\<or> q)) at a in' set ((ps, a) # branch)"   and "new (\<^bold>\<not> p) a ((ps, a) # branch) \<or> new (\<^bold>\<not> q) a ((ps, a) # branch)"   and "A, Suc n \<turnstile> ((\<^bold>\<not> q) # (\<^bold>\<not> p) # ps, a) # branch" shows "A, n \<turnstile> (ps, a) # branch"
productive_on_def: shows "productive_on \<equiv> \<lambda>g. {x. productive_onp g x}"
V_def: shows "V = (\<lambda>n. space M - U n)"
surjective: fixes r :: "('d, 'e, 'f) lens_scheme" shows "r = \<lparr>lens_get = get\<^bsub>r\<^esub>, lens_put = put\<^bsub>r\<^esub>, \<dots> = more r\<rparr>"
eqpoll_finite_iff: fixes A :: "'a set"   and B :: "'b set" assumes "A \<approx> B" shows "finite A = finite B"
less_log2_of_power: fixes n :: "nat"   and m :: "nat" assumes "2 ^ n < m" shows "real n < log 2 (real m)"
sec_prikey_secret: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "n \<notin> bad_sec_prik" shows "Sec_PriKey n \<notin> spied s"
a_1: fixes i :: "nat" assumes "i \<in> N" shows "W i \<subseteq> X (a i)"
closed: fixes D :: "('c, 'd) Order_scheme" assumes "Order D" shows "rel D \<subseteq> carrier D \<times> carrier D"
Int_atLeastAtMostR1: fixes b :: "'b word"   and c :: "'b word"   and d :: "'b word" shows "signed.atMost b \<inter> signed.atLeastAtMost c d = signed.atLeastAtMost c (signed.min b d)"
echelon_form_condition2_explicit: fixes A :: "(('d, 'e) vec, 'f) vec"   and i :: "'f"   and j :: "'f" assumes "echelon_form A"   and "i < j"   and "\<not> is_zero_row i A"   and "\<not> is_zero_row j A" shows "(LEAST n. A $ i $ n \<noteq> (0::'d)) < (LEAST n. A $ j $ n \<noteq> (0::'d))"
set_tag_name_get_disconnected_nodes: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) document_ptr" assumes "l_set_tag_name_get_disconnected_nodes type_wf set_tag_name set_tag_name_locs get_disconnected_nodes get_disconnected_nodes_locs" shows "\<forall>w\<in>set_tag_name_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_disconnected_nodes_locs ptr'. r h h')"
normalize_sort_eq_imp_sort_eqv: fixes cs :: "(String.literal \<times> String.literal) set"   and s1 :: "String.literal set"   and s2 :: "String.literal set" assumes "wf_subclass_loc cs"   and "sort_ex cs s1"   and "sort_ex cs s2"   and "finite s1"   and "finite s2"   and "normalize_sort cs s1 = normalize_sort cs s2" shows "sort_eqv cs s1 s2"
simps_1: fixes P :: "Core.term \<Rightarrow> bool"   and t :: "Core.term"   and u :: "Core.term" shows "exists_subterm' P (t $ u) = (P (t $ u) \<or> exists_subterm' P t \<or> exists_subterm' P u)"
assms: shows "(pq, pq') \<in> sorted_poly_rel \<times>\<^sub>r sorted_poly_rel"
points_index_0_iff: fixes B :: "'a set multiset"   and ps :: "'a set" shows "(B index ps = 0) = (\<forall>b. b \<in># B \<longrightarrow> \<not> ps \<subseteq> b)"
eq: shows "Mf (lead_coeff f * lead_coeff g, fs + gs) = Mf (c, hs)"
spanning_def: fixes H :: "('c, 'd) pre_digraph"   and G :: "('c, 'd) pre_digraph" shows "spanning H G \<equiv> subgraph H G \<and> verts G = verts H"
sr_ext_Ri_def: fixes N :: "'a literal multiset set" shows "sr_ext_Ri N = sr.Ri N \<union> (ground_sound_\<Gamma> - gr.ord_\<Gamma>)"
distinct_1: shows "ht_NOOP \<noteq> ht_GetRoots"
coeff_iff_polynomial_cond: fixes K :: "'a set"   and p1 :: "'a list"   and p2 :: "'a list" assumes "polynomial K p1"   and "polynomial K p2" shows "(p1 = p2) = (R.coeff p1 = R.coeff p2)"
poly_induct5: fixes p :: "nat \<Rightarrow> 'a"   and Q :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "p \<in> carrier P"   and "\<And>p q. \<lbrakk>q \<in> carrier P; p \<in> carrier P; Q p; Q q\<rbrakk> \<Longrightarrow> Q (p \<oplus>\<^bsub>P\<^esub> q)"   and "\<And>n. Q (monom P \<one> n)"   and "\<And>p a. \<lbrakk>a \<in> carrier R; p \<in> carrier P; Q p\<rbrakk> \<Longrightarrow> Q (a \<odot>\<^bsub>P\<^esub> p)" shows "Q p"
\<mu>_cong: fixes j :: "nat"   and i :: "nat"   and f1 :: "rat vec list"   and f2 :: "rat vec list" assumes "\<And>k. \<lbrakk>j < i; k \<le> j\<rbrakk> \<Longrightarrow> f1 ! k = f2 ! k"   and "j < i \<Longrightarrow> f1 ! i = f2 ! i" shows "gram_schmidt_fs.\<mu> n f1 i j = gram_schmidt_fs.\<mu> n f2 i j"
Eps_cong: fixes P :: "'f \<Rightarrow> bool"   and Q :: "'f \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "Eps P = Eps Q"
Exz: shows "infdist (f w) {f x--f z} \<le> E"
diff_single_eq_union: fixes x :: "'d"   and M :: "'d multiset"   and N :: "'d multiset" assumes "x \<in># M" shows "(M - {#x#} = N) = (M = add_mset x N)"
cinner_same_CBasis: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and CBasis :: "'c set"   and u :: "'c" assumes "class.ceuclidean_space scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner CBasis"   and "u \<in> CBasis" shows "cinner u u = 1"
wf_trm_subterm_arity: fixes t :: "('fun, 'a) Term.term"   and f :: "'fun"   and T :: "('fun, 'a) Term.term list" assumes "wf\<^sub>t\<^sub>r\<^sub>m t"   and "Fun f T \<sqsubseteq> t" shows "length T = arity f"
region_def: shows "region \<equiv> \<lambda>X I J r. {x. regionp (\<lambda>x. x \<in> X) I J (\<lambda>x xa. (x, xa) \<in> r) x}"
unfold_accept_eq: fixes \<G> :: "'b ltl set"   and \<Sigma> :: "'b set set"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl" assumes "Only_G \<G>"   and "finite \<Sigma>"   and "range w \<subseteq> \<Sigma>" shows "mojmir_def.accept \<up>af\<^sub>G (Abs \<phi>) w {q. \<G> \<Turnstile>\<^sub>P Rep q} = mojmir_def.accept \<up>af\<^sub>G\<^sub>\<UU> (Abs (Unf\<^sub>G \<phi>)) w {q. \<G> \<Turnstile>\<^sub>P Rep q}"
poly_cnj_real: fixes p :: "complex poly"   and z :: "complex" assumes "\<And>n. coeff p n \<in> \<real>" shows "cnj (poly p z) = poly p (cnj z)"
post_meas0_bell00_fst: shows "post_meas0 2 |\<beta>\<^sub>0\<^sub>0\<rangle> 0 = |unit_vec 4 0\<rangle>"
trace_trans: fixes c :: "('p, 's, 'm) configuration"   and t :: "('p, 's, 'm) event list"   and c' :: "('p, 's, 'm) configuration"   and t' :: "('p, 's, 'm) event list"   and c'' :: "('p, 's, 'm) configuration" assumes "trace c t c'"   and "trace c' t' c''" shows "trace c (t @ t') c''"
i_def: shows "i = (\<lambda>n x. x powr (real n - 1 / 2) * sqrt (4 - x))"
hyps: fixes s1 :: "nat"   and r1 :: "nat \<Rightarrow>f value option"   and s2 :: "nat"   and r2 :: "nat \<Rightarrow>f value option"   and s3 :: "nat"   and r3 :: "nat \<Rightarrow>f value option" assumes "executionally_equivalent e1 s1 r1 e2 s2 r2 x"   and "executionally_equivalent e2 s2 r2 e3 s3 r3 x" shows "executionally_equivalent e1 s1 r1 e3 s3 r3 x"
of_int_divide_in_Ints: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "int"   and a :: "int" assumes "class.idom_divide times one plus zero minus uminus divide"   and "b dvd a" shows "divide (ring_1.of_int one plus zero minus a) (ring_1.of_int one plus zero minus b) \<in> ring_1.Ints one plus zero minus"
pow_comm_comm: fixes x :: "'b list"   and j :: "nat"   and y :: "'b list"   and k :: "nat" assumes "x \<^sup>@ j = y \<^sup>@ k"   and "j \<noteq> 0" shows "x \<cdot> y = y \<cdot> x"
open_Inter: fixes S :: "'a set set" assumes "finite S"   and "Ball S open" shows "open (\<Inter> S)"
rtrancl_listrel1_ConsI2: fixes x :: "'c"   and y :: "'c"   and r :: "('c \<times> 'c) set"   and xs :: "'c list"   and ys :: "'c list" assumes "(x, y) \<in> r\<^sup>*"   and "(xs, ys) \<in> (listrel1 r)\<^sup>*" shows "(x # xs, y # ys) \<in> (listrel1 r)\<^sup>*"
subtermeqI': fixes x :: "('a, 'b) Term.term"   and y :: "('a, 'b) Term.term" assumes "x = y" shows "x \<sqsubseteq> y"
DC_split: fixes P :: "'b \<Rightarrow> real"   and a :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and ab :: "bool"   and R :: "'b \<Rightarrow> real"   and Q :: "'b \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "P \<tturnstile> a ab R"   and "Q \<tturnstile> b ab R" shows "\<lambda>s. min (P s) (Q s) \<tturnstile> (a \<Sqinter> b) ab R"
assms_3: shows "\<not> min_gallery (map (\<lambda>w. w `\<rightarrow> C0) (sums ss))"
lexext_rev_hd_or_tl: fixes ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b" assumes "length ys = length xs"   and "lexext_rev gt (y # ys) (x # xs)" shows "gt y x \<or> lexext_rev gt ys xs"
mgu\<^sub>t\<^sub>s_def: fixes \<sigma> :: "char list \<Rightarrow> fterm"   and ts :: "fterm set" shows "mgu\<^sub>t\<^sub>s \<sigma> ts = (unifier\<^sub>t\<^sub>s \<sigma> ts \<and> (\<forall>u. unifier\<^sub>t\<^sub>s u ts \<longrightarrow> (\<exists>i. u = \<sigma> \<cdot> i)))"
model_to_dimacs_model_complete_disj: fixes f :: "nat formula"   and n_vars :: "nat"   and dimacs_M :: "int list" assumes "\<forall>v\<in>atoms f. 0 < v \<and> v < n_vars"   and "is_disj f"   and "distinct (map dimacs_lit_to_var dimacs_M)"   and "dimacs_model dimacs_M (cnf_to_dimacs f)" shows "dimacs_model_to_abs dimacs_M (\<lambda>_. False) \<Turnstile> f"
beta: fixes x :: "name"   and t :: "lam"   and s :: "lam" assumes "x \<sharp> t" shows "(Lam [x].s) \<degree> t \<rightarrow>\<^sub>\<beta> s[x::=t]"
CodFunctor: fixes f :: "'c" assumes "f \<in> mor\<^bsub>CatDom F\<^esub>" shows "cod\<^bsub>CatCod F\<^esub> (F ## f) = F @@ cod\<^bsub>CatDom F\<^esub> f"
Block_self: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A"   and "A \<in> L.Partition" shows "L.Class a = A"
lt: shows "2 ^ n * (unat a div 2 ^ n + 1) < 2 ^ LENGTH('a)"
hext_typeof_addr_map_le: fixes h :: "'heap"   and h' :: "'heap" assumes "h \<unlhd> h'" shows "typeof_addr h \<subseteq>\<^sub>m typeof_addr h'"
lessThan_iff: fixes i :: "'b"   and k :: "'b" shows "(i \<in> {..<k}) = (i < k)"
dim: shows "dim_vec (M.sumlist (map (\<lambda>j. \<mu> i j \<cdot>\<^sub>v gso j) [0..<Suc i])) = n"
integrable_min: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> real"   and g :: "'b \<Rightarrow> real" assumes "integrable M f"   and "integrable M g" shows "integrable M (\<lambda>x. min (f x) (g x))"
bv_uminus_def: fixes w :: "bit list" shows "bv_uminus w = int_to_bv (- bv_to_int w)"
poly_add_inv: fixes p :: "('c monom \<times> 'd) list"   and q :: "('c monom \<times> 'd) list" assumes "poly_inv p"   and "poly_inv q" shows "poly_inv (poly_add p q)"
HSem_subset_below: fixes \<Gamma> :: "('b \<times> 'a) list"   and \<Delta> :: "('b \<times> 'a) list"   and \<rho> :: "'b \<Rightarrow> 'c" assumes "atom ` domA \<Gamma> \<sharp>* \<Delta>" shows "\<lbrace> \<Delta> \<rbrace>\<rho> f|` (- domA \<Gamma>) \<sqsubseteq> (\<lbrace> \<Delta> @ \<Gamma> \<rbrace>\<rho>) f|` (- domA \<Gamma>)"
Nmlize_Cod: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Arr V src trg t" shows "bicategorical_language.Nmlize V (bicategorical_language.Cod V src trg t) = bicategorical_language.Cod V src trg (bicategorical_language.Nmlize V t)"
height_Leaf: fixes t :: "'b btree" shows "(height t = 0) = (t = Leaf)"
sup_Sec_def: fixes d1 :: "Sec"   and d2 :: "Sec" shows "sup d1 d2 = (if d1 = High \<or> d2 = High then High else Low)"
assms: shows "\<turnstile> F \<longrightarrow> G"
weak_ranking_total_preorder_1: shows "is_weak_ranking (weak_ranking le)"
is_SDG_path_hd: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "n is-ns\<rightarrow>\<^sub>d* n'"   and "ns \<noteq> []" shows "hd ns = n"
lunit_char_2: fixes a :: "'d" assumes "D.ide a" shows "\<I>\<^sub>D \<otimes>\<^sub>D \<l>\<^sub>D[a] = (\<iota>\<^sub>D \<otimes>\<^sub>D a) \<cdot>\<^sub>D D.assoc' \<I>\<^sub>D \<I>\<^sub>D a"
induct: fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'b"   and a1 :: "'a list" assumes "\<And>f. P f []"   and "\<And>f x. P f [x]"   and "\<And>f v vb vc. \<lbrakk>\<And>x xa xb y. \<lbrakk>x = length (v # vb # vc) div 2; xa = split_at x (v # vb # vc); (xb, y) = xa\<rbrakk> \<Longrightarrow> P f xb; \<And>x xa xb y. \<lbrakk>x = length (v # vb # vc) div 2; xa = split_at x (v # vb # vc); (xb, y) = xa\<rbrakk> \<Longrightarrow> P f y\<rbrakk> \<Longrightarrow> P f (v # vb # vc)" shows "P a0 a1"
minus_diff_commute: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b" assumes "class.ab_group_add plus zero minus uminus" shows "minus (uminus b) a = minus (uminus a) b"
valid_index_list_all2_iff: fixes "is" :: "nat list"   and ds :: "nat list" shows "(is \<lhd> ds) = list_all2 (<) is ds"
less_eq_def: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x = y \<or> x < y)"
fps_of_ratfps_cutoff: fixes n :: "nat"   and x :: "'b ratfps" shows "fps_of_poly (ratfps_cutoff n x) = fps_cutoff n (fps_of_ratfps x)"
finsum_insert: fixes F :: "'c set"   and a :: "'c"   and f :: "'c \<Rightarrow> 'a" assumes "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> carrier R"   and "f a \<in> carrier R" shows "finsum R f (insert a F) = f a \<oplus> finsum R f F"
ccw_self_1: fixes a :: "real \<times> real"   and b :: "real \<times> real" shows "ccw a a b"
observable_tau_transition: fixes P :: "'state"   and P' :: "'state" assumes "P \<Rightarrow>{\<tau>} P'" shows "P \<Rightarrow> P'"
12: shows "{z. \<exists>qf. - Rep_regular pf \<le> Rep_dense qf \<and> z = get_dense qf f} \<subseteq> Filters.filter.Rep_filter (Rep_phi f (fst (Rep_lifted_pair (Rep_regular pf) f)))"
pred_sexpI2: fixes M :: "('c, unit) Old_Datatype.node set"   and N :: "('c, unit) Old_Datatype.node set" assumes "M \<in> sexp"   and "N \<in> sexp" shows "(N, Scons M N) \<in> pred_sexp"
simps: fixes xs :: "'b list" shows "IArray.list_of (IArray xs) = xs"
norm_le: fixes x :: "'b"   and y :: "'c" shows "(norm x \<le> norm y) = (x \<bullet> x \<le> y \<bullet> y)"
m_s2_rep: fixes X :: "'b set"   and S1 :: "'b \<Rightarrow> 'a"   and S2 :: "'b \<Rightarrow> 'a" assumes "finite X"   and "S1 = S2 on - X"   and "\<forall>x. S1 x \<le> S2 x"   and "S1 \<noteq> S2" shows "(\<Sum>x\<in>X. m (S2 x)) < (\<Sum>x\<in>X. m (S1 x))"
\<rho>'_ide_simp: fixes a :: "'a" assumes "ide a" shows "\<rho>' a = \<r>\<^sup>-\<^sup>1[a]"
euler_maclaurin_raw_strong_int: fixes a :: "int"   and b :: "int"   and N :: "nat"   and Y :: "real set"   and fs :: "nat \<Rightarrow> real \<Rightarrow> 'a"   and f :: "real \<Rightarrow> 'a" assumes "a \<le> b"   and "0 < N"   and "finite Y"   and "fs 0 = f"   and "\<And>k. k \<le> N \<Longrightarrow> continuous_on {real_of_int a..real_of_int b} (fs k)"   and "\<And>k x. \<lbrakk>k < N; x \<in> {real_of_int a..real_of_int b} - Y\<rbrakk> \<Longrightarrow> (fs k has_vector_derivative fs (Suc k) x) (at x)" shows "(\<Sum>i\<in>{a<..b}. f (real_of_int i)) - integral {real_of_int a..real_of_int b} f = (\<Sum>k<N. (bernoulli' (Suc k) / fact (Suc k)) *\<^sub>R (fs k (real_of_int b) - fs k (real_of_int a))) + EM_remainder' N (fs N) (real_of_int a) (real_of_int b)"
Domainp_iff: fixes T :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d" shows "Domainp T x = (\<exists>y. T x y)"
f2r_inj_iff: fixes R :: "('a \<times> 'b) set"   and S :: "('a \<times> 'b) set" shows "(\<F> R = \<F> S) = (R = S)"
P1_b: fixes x :: "'a" shows "x r\<rightarrow> x = (1::'a)"
freeC: shows "var \<notin> vars ((isolate_variable_sparse p var (Suc 0))\<^sup>2 - 4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)"
linear_def: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'f" shows "Vector_Spaces.linear s1 s2 f \<equiv> vector_space s1 \<and> vector_space s2 \<and> module_hom s1 s2 f"
under_incl_iff: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a" assumes "trans r"   and "Refl r"   and "a \<in> Field r" shows "(under r a \<subseteq> under r b) = ((a, b) \<in> r)"
uabc_uhalt': fixes tp :: "(action \<times> nat) list"   and l :: "nat"   and lm :: "'a"   and ap :: "abc_inst list"   and pos :: "nat"   and md :: "nat" assumes "tm_wf0 tp"   and "\<forall>stp. \<not> TSTD (steps0 (Suc 0, Bk \<up> l, <lm>) tp stp)"   and "rec_ci rec_F = (ap, pos, md)" shows "{\<lambda>nl. nl = [code tp, bl2wc (<lm>)]} ap \<up>"
n_mult_lb2: fixes x :: "'a"   and y :: "'a" shows "!x \<cdot> !y \<sqsubseteq> !y"
Lower_antimono: fixes A :: "'c set"   and B :: "'c set"   and L :: "('c, 'd) gorder_scheme" assumes "A \<subseteq> B" shows "Lower L B \<subseteq> Lower L A"
subgroupE_4: fixes H :: "'a set"   and a :: "'a"   and b :: "'a" assumes "subgroup H \<G>"   and "a \<in> H"   and "b \<in> H" shows "a \<otimes> b \<in> H"
longest_common_prefix_prefix1: fixes xs :: "'b list"   and ys :: "'b list" shows "prefix (longest_common_prefix xs ys) xs"
iwlsFSwSb_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'd, 'e, 'f) model_scheme" shows "iwlsFSwSb MOD \<equiv> iwlsFSw MOD \<and> igSubstAllIPresIGWlsAll MOD \<and> igSubstCls MOD"
align': shows "\<forall>m\<in>#hmsetmset M. e < local.decode 0 m"
fds_nth_truncate: fixes m :: "nat"   and f :: "'b fds"   and n :: "nat" shows "fds_nth (fds_truncate m f) n = (if n \<le> m then fds_nth f n else (0::'b))"
da_elim_cases_50: fixes Env :: "env"   and B :: "lname set"   and D :: "qtname"   and c :: "stmt"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In1l (Body D c)\<guillemotright> A" obtains C :: "assigned" where "Env\<turnstile> B \<guillemotright>\<langle>c\<rangle>\<guillemotright> C" and "jumpNestingOkS {Ret} c" and "Result \<in> nrm C" and "nrm A = B" and "brk A = (\<lambda>l. UNIV)"
extf_compat_append_right: fixes f :: "'s"   and gt :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool"   and ys :: "('s, 'v) tm list"   and xs :: "('s, 'v) tm list"   and zs :: "('s, 'v) tm list" assumes "extf f gt ys xs" shows "extf f gt (ys @ zs) (xs @ zs)"
intros_1: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a" shows "path_conforms_with_strategy p LNil \<sigma>"
norm_EM_remainder_le_strong_nat': fixes Y :: "real set"   and n :: "nat"   and D :: "real"   and a :: "real"   and f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real" assumes "finite Y"   and "\<forall>x. \<bar>pbernpoly n x\<bar> \<le> D"   and "continuous_on {a..} f"   and "continuous_on {a..} g"   and "continuous_on {a..} g'"   and "(g \<longlongrightarrow> 0) at_top"   and "\<And>x. a \<le> x \<Longrightarrow> norm (f x) \<le> g' x"   and "\<And>x. x \<in> {a..} - Y \<Longrightarrow> (g has_real_derivative - g' x) (at x)" shows "\<forall>x. a \<le> real x \<longrightarrow> norm (EM_remainder n f (int x)) \<le> D / fact n * g (real x)"
distinct_7: fixes x51 :: "bexp"   and x52 :: "com" shows "SKIP \<noteq> WHILE x51 DO x52"
wf_list_graph_axioms_def: fixes G :: "'a list_graph" shows "wf_list_graph_axioms G = (fst ` set (edgesL G) \<subseteq> set (nodesL G) \<and> snd ` set (edgesL G) \<subseteq> set (nodesL G))"