hcju/isabelleps huggingface dataset:
finite_ordertype_eq_card: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and m :: "nat" assumes "small A"   and "wf r"   and "trans r"   and "total_on A r" shows "(ordertype A r = ord_of_nat m) = (finite A \<and> card A = m)"
l2_init_def: shows "l2_init \<equiv> {\<lparr>ik = {}, secret = {}, progress = Map.empty, signals = \<lambda>x. 0, chan = {}, bad = bad_init\<rparr>}"
IDFT_def: fixes n :: "nat"   and a :: "nat \<Rightarrow> complex" shows "IDFT n a \<equiv> \<lambda>i. \<Sum>k = 0..<n. a k / FFT.root n ^ (i * k)"
ideD_1: fixes a :: "('a, 'a set) Maps.arr" assumes "Maps.ide a" shows "Maps.arr a"
is_shadow_root_ptr_kind_none: fixes ptr :: "(_) object_ptr" assumes "\<not> is_shadow_root_ptr_kind ptr" shows "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^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 ptr = None"
hyps_3: assumes "finite B" shows "(\<Sum>x\<in>A'. sum (f x) B) = (\<Sum>y\<in>B. \<Sum>x\<in>A'. f x y)"
npI: fixes u :: "'b list"   and v :: "'b list" assumes "u \<noteq> \<epsilon>"   and "u \<le>p v" shows "u \<le>np v"
del_min_mset: fixes l :: "int"   and r :: "int"   and heap :: "int \<Rightarrow> int"   and mod_heap :: "int \<Rightarrow> int"   and new_heap :: "int \<Rightarrow> int" assumes "l < r"   and "is_heap (\<le>) heap l r"   and "mod_heap = heap(l := heap (r - 1))"   and "arr_mset mod_heap l (r - 1) = arr_mset new_heap l (r - 1)" shows "arr_mset new_heap l (r - 1) = arr_mset heap l r - {#Min_mset (arr_mset heap l r)#}"
divide_p64_def: fixes p :: "uint64"   and x :: "uint64"   and y :: "uint64" shows "divide_p64 p x y = mult_p64 p x (inverse_p64 p y)"
\<Phi>_def: fixes bs :: "bool list" shows "\<Phi> bs = real (length (filter id bs))"
disc_iff_2: fixes b :: "'b"   and xs :: "'a llist" shows "(\<not> is_TNil (tllist_of_llist b xs)) = (\<not> lnull xs)"
\<iota>_in_hom: shows "\<guillemotleft>\<iota> : C.tensor \<I> \<I> \<rightarrow>\<^sub>C \<I>\<guillemotright>"
bind_lift: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'m2"   and g :: "'a \<Rightarrow> 'm2" shows "bind2 (m2.lift f x) g = bind2 x (g \<circ> f)"
openin_subtopology: fixes U :: "'b topology"   and V :: "'b set"   and S :: "'b set" shows "openin (subtopology U V) S = (\<exists>T. openin U T \<and> S = T \<inter> V)"
is_universal: fixes a' :: "'c"   and \<chi>' :: "(bool, unit) J.arr \<Rightarrow> 'c" assumes "D.cone a' \<chi>'" shows "\<exists>!f. \<guillemotleft>f : a' \<rightarrow> C.dom p0\<guillemotright> \<and> D.cones_map f (D.mkCone p0 p1) = \<chi>'"
Invoke_handlers: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and pc :: "nat"   and xt :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list"   and pc' :: "nat"   and d' :: "nat"   and n :: "char list"   and M :: "nat" assumes "match_ex_table P C pc xt = \<lfloor>(pc', d')\<rfloor>" shows "\<exists>(f, t, D, h, d)\<in>set (relevant_entries P (Invoke n M) pc xt). P \<turnstile> C \<preceq>\<^sup>* D \<and> pc \<in> {f..<t} \<and> pc' = h \<and> d' = d"
nhds_generated_topology: fixes T :: "'a set set"   and x :: "'a" assumes "open = generate_topology T" shows "nhds x = Inf (principal ` {S \<in> T. x \<in> S})"
linearity_def: fixes E :: "(node fset \<times> command_gadget \<times> node fset) list" shows "linearity E \<equiv> distinct E \<and> (\<forall>e\<in>set E. \<forall>f\<in>set E. e \<noteq> f \<longrightarrow> fst3 e |\<inter>| fst3 f = {||} \<and> thd3 e |\<inter>| thd3 f = {||})"
clean_decreasing: fixes A :: "('d, 'c) nba"   and w :: "'d stream"   and V :: "(nat \<times> 'c) set" shows "clean A w V \<subseteq> V"
preserving: fixes u :: "complex_homo"   and \<phi> :: "real" assumes "u \<noteq> 0\<^sub>h"   and "u \<noteq> \<infinity>\<^sub>h"   and "P (moebius_pt (moebius_rotation \<phi>) u)" shows "P u"
farkas_minkowsky_weyl_theorem_1: fixes X :: "'b vec set"   and n :: "nat"   and Bnd :: "'b" assumes "X \<subseteq> carrier_vec n"   and "finite X" shows "\<exists>nr A. A \<in> carrier_mat nr n \<and> gram_schmidt.cone n X = gram_schmidt.polyhedral_cone n A \<and> (X \<subseteq> \<int>\<^sub>v \<inter> Bounded_vec Bnd \<longrightarrow> A \<in> \<int>\<^sub>m \<inter> Bounded_mat (det_bound (n - 1) (max (1::'b) Bnd)))"
subtensor_plus: fixes i :: "nat"   and A :: "'b tensor"   and B :: "'b tensor" assumes "i < hd (dims A)"   and "dims A = dims B"   and "dims A \<noteq> []" shows "subtensor (A + B) i = subtensor A i + subtensor B i"
boundeffect: fixes \<omega> :: "variable \<Rightarrow> real"   and I :: "interp"   and \<alpha> :: "game"   and X :: "(variable \<Rightarrow> real) set" shows "(\<omega> \<in> game_sem I \<alpha> X) = (\<omega> \<in> game_sem I \<alpha> (selectlike X \<omega> (- BVG \<alpha>)))"
conjugate_square_ge_0_vec: fixes v :: "'a vec" shows "(0::'a) \<le> v \<bullet>c v"
p_id_idem: fixes x :: "'a" shows "x \<cdot> 1\<^sub>\<pi> \<cdot> 1\<^sub>\<pi> = x \<cdot> 1\<^sub>\<pi>"
obtains_1_c2: fixes c1 :: "nat \<Rightarrow>f value option"   and t :: "(String.literal \<times> value list) list"   and c2 :: "nat \<Rightarrow>f value option" assumes "obtains 1 c1 drinks2 0 <> t"   and "obtains 1 c2 drinks 0 <> t" shows "c2 $ 2 = Some (Num 0)"
mahl: shows "local.mahler_approximation_main dd c g mm k kk = (let mmm = mahler_landau_graeffe_approximation kk dd g; new_mm = if k = 0 then mmm else min mm mmm in if bnd \<le> k then new_mm else local.mahler_approximation_main (dd * dd) c (graeffe_one_step c g) new_mm (Suc k) (2 * kk))"
equivpI: fixes R :: "'k \<Rightarrow> 'k \<Rightarrow> bool" assumes "reflp R"   and "symp R"   and "transp R" shows "equivp R"
cases: fixes r :: "'a gorder" obtains carrier :: "'a set" and eqa :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "r = \<lparr>carrier = carrier, eq = eqa, le = le\<rparr>"
rtrancl_trancl_trancl: fixes x :: "'a"   and y :: "'a"   and r :: "('a \<times> 'a) set"   and z :: "'a" assumes "(x, y) \<in> r\<^sup>*"   and "(y, z) \<in> r\<^sup>+" shows "(x, z) \<in> r\<^sup>+"
ih: fixes tt :: "('s, 'v) tm"   and ss :: "('s, 'v) tm" assumes "{#hsize tt, hsize ss#} < {#hsize t, hsize s#}"   and "tt >\<^sub>t ss" shows "subst \<rho> tt >\<^sub>t subst \<rho> ss"
lless_antisym: fixes a :: "'a"   and b :: "'a"   and P :: "bool" assumes "a \<in> carrier L"   and "b \<in> carrier L"   and "a \<sqsubset> b"   and "b \<sqsubset> a" shows "P"
sub_rel_of: fixes g :: "('c \<times> nat, 'd) rgraph_scheme"   and subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set" assumes "sub_rel_of g subs" shows "SubRel.vertices subs \<subseteq> Graph.vertices g"
inject_1: fixes x11 :: "nat"   and x12 :: "'c"   and x13 :: "nat set"   and y11 :: "nat"   and y12 :: "'c"   and y13 :: "nat set" shows "(NodeS x11 x12 x13 = NodeS y11 y12 y13) = (x11 = y11 \<and> x12 = y12 \<and> x13 = y13)"
uniqImpliesExclusiveTrueOrFalse: fixes e :: "'b"   and d :: "bool"   and M :: "('b \<times> bool) list" assumes "(e, d) \<in> set M"   and "uniq (elements M)" shows "(e, \<not> d) \<notin> set M"
assms: shows "(x, y) \<in> r\<^sup>*"
Eps_psimp: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "P x"   and "\<not> P y"   and "Eps P = y" shows "Eps P = x"
prveqlPT_def: fixes \<tau> :: "'fmla"   and t :: "'trm" shows "prveqlPT \<tau> t \<equiv> prv (subst \<tau> t out)"
simplicial_imp_singular_chain: fixes p :: "nat"   and S :: "('c \<Rightarrow> real) set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real) \<Rightarrow>\<^sub>0 int" assumes "simplicial_chain p S c" shows "singular_chain p (subtopology (powertop_real UNIV) S) c"
det_I: shows "Determinants.det (Finite_Cartesian_Product.mat (1::'a)) = (1::'a)"
simps_2: fixes x1 :: "'b"   and l1 :: "'b list" shows "subset_sorted (x1 # l1) [] = False"
lcr_trans_l: fixes R1 :: "('b \<times> 'b) set"   and R2 :: "('b \<times> 'b) set"   and A :: "'b multiset"   and B :: "'b multiset"   and C :: "'b multiset"   and D :: "'b multiset"   and E :: "'b multiset"   and F :: "'b multiset" assumes "locally_compatible_r R1 R2 (A + B) (C + D) (E + F)" shows "locally_compatible_r R1 R2 A C E"
t\<^sub>0_def: shows "t\<^sub>0 = (LEAST t. local.prefixes t j \<up>)"
extending_Always: fixes F :: "'a program"   and A :: "'a set" shows "extending (\<lambda>G. reachable (extend h F \<squnion> G)) h F (Always (extend_set h A)) (Always A)"
simps_6: fixes x :: "octo"   and y :: "octo" shows "Im5 (x + y) = Im5 x + Im5 y"
Inf_from_mono: fixes N :: "'b set"   and N' :: "'b set"   and Inf :: "'b inference set" assumes "N \<subseteq> N'" shows "inference_system.Inf_from Inf N \<subseteq> inference_system.Inf_from Inf N'"
order_zmset_exists_foundation': fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and M :: "'a zmultiset"   and t :: "'a" assumes "class.order less_eq less"   and "0 < zcount M t" shows "\<exists>s. less_eq s t \<and> 0 < zcount M s \<and> (\<forall>u. less u s \<longrightarrow> zcount M u \<le> 0)"
UNIV_rw: shows "UNIV = insert a (UNIV - {a})"
proper_map_def: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" shows "proper_map X Y f \<equiv> closed_map X Y f \<and> (\<forall>y\<in>topspace Y. compactin X {x \<in> topspace X. f x = y})"
snd_map_help_2: fixes x :: "nat"   and tsi :: "('b \<times> 'c) list" assumes "x < length tsi" shows "snd (tsi ! x) = separators tsi ! x"
pa_initial_def: fixes H1 :: "('a, 'c) hashedTa"   and H2 :: "('b, 'c) hashedTa" shows "pa_initial H1 H2 \<equiv> let Qip = hhh_cart.cart (hta_Qi H1) (hta_Qi H2) in (Qip, hs.to_list Qip, ls.empty ())"
eval_dioph_mod: fixes ks :: "int list"   and xs :: "nat list"   and n :: "nat" shows "eval_dioph ks xs mod int n = eval_dioph ks (map (\<lambda>x. x mod n) xs) mod int n"
assms_2: shows "SIDE_PRECOND_OPT (a' \<inter> b' = {})"
Rep_OrdSum_eq: fixes x :: "('e, 'f) OrdSum"   and y :: "('e, 'f) OrdSum" assumes "Rep_OrdSum x = Rep_OrdSum y" shows "x = y"
hyps_2: fixes xs :: "'a rexp list" shows "(Pluss s = Some xs) = (Full \<notin> set xs \<and> xs = sorted_list_of_set (toplevel_summands s - {Zero}))"
multiset_eqI: fixes A :: "'d multiset"   and B :: "'d multiset" assumes "\<And>x. count A x = count B x" shows "A = B"
cp: shows "char_poly (of_real_hom.mat_hom A) = map_poly of_real (char_poly A)"
gdd_index_non_zero_iff: fixes ps :: "'a set" assumes "ps \<subseteq> \<V>"   and "card ps = 2" shows "(int (\<B> index ps) = 1) = (size (filter_mset ((\<subseteq>) ps) (mset_set \<G>)) = 0)"
multi_member_split: fixes x :: "'c"   and M :: "'c multiset" assumes "x \<in># M" shows "\<exists>A. M = add_mset x A"
msubstlt: fixes c :: "poly"   and d :: "poly"   and a :: "poly"   and r :: "tm"   and vs :: "'b list"   and x :: "'b"   and bs :: "'b list"   and t :: "tm"   and s :: "tm" assumes "isnpoly c"   and "isnpoly d"   and "islin (Lt (CNP 0 a r))" shows "Ifm vs (x # bs) (msubstlt c t d s a r) = Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / (2::'b) # bs) (Lt (CNP 0 a r))"
bounded_inverse_sin_half: fixes d :: "real" assumes "0 < d" obtains B :: "real" where "0 < B" and "\<And>x. x \<in> {- pi..pi} - {- d<..<d} \<Longrightarrow> \<bar>inverse (sin (x / 2))\<bar> \<le> B"
assms_1: shows "\<sigma> =\<^sub>L \<sigma>'"
f: shows "\<guillemotleft>f : a' \<rightarrow> a\<guillemotright>"
Sum_def: fixes x2 :: "pi"   and x1 :: "pi" shows "x2 \<oplus> x1 \<equiv> Abs_pi (Sum_Rep (Rep_pi x2) (Rep_pi x1))"
ge_tpoly_refl: fixes p :: "('v pvar, hmultiset) tpoly" shows "p \<ge>\<^sub>p p"
elims: fixes x :: "complex Matrix.mat"   and xa :: "nat"   and y :: "complex Matrix.mat" assumes "x \<otimes>\<^bsup>xa\<^esup> = y" obtains A :: "complex Matrix.mat" where "x = A__" and "xa = Suc 0" and "y = A__"   | A :: "complex Matrix.mat" and v :: "nat" where "x = A" and "xa = Suc (Suc v)" and "y = A \<Otimes> A \<otimes>\<^bsup>Suc v\<^esup>"   | a :: "complex Matrix.mat" where "x = a" and "xa = 0" and "y = undefined"
assms_2: shows "(a, A) \<in> R \<cdot> S"
add_mset_subseteq_single_iff: fixes a :: "'c"   and M :: "'c multiset"   and b :: "'c" shows "(add_mset a M \<subseteq># {#b#}) = (M = {#} \<and> a = b)"
pred_transfer: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "((R ===> (=)) ===> llist_all2 R ===> (=)) llist_all llist_all"
int_of_integer_less_iff: fixes x :: "integer"   and y :: "integer" shows "(int_of_integer x < int_of_integer y) = (x < y)"
et_val: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and v :: "val" shows "\<Gamma> \<turnstile> Val v : val_type v"
subclsS_def: shows "subclsS \<equiv> \<lambda>P. {(xa, x). subclsSp P xa x}"
ty_cmd_no_internal: fixes \<Gamma> :: "(String.literal \<Rightarrow> unit option) \<times> (String.literal \<Rightarrow> unit option)"   and c :: "cmd"   and a :: "action + brk_ctd"   and c' :: "cmd" assumes "ty_cmd (\<Gamma>, False) c"   and "cfg c a c'" shows "isl a"
open_UNIV: fixes "open" :: "'c set \<Rightarrow> bool" assumes "class.topological_space open" shows "open UNIV"
split_scaleC_neg_le: fixes a :: "complex"   and x :: "'b" assumes "0 \<le> a \<and> x \<le> (0::'b) \<or> a \<le> 0 \<and> (0::'b) \<le> x" shows "a *\<^sub>C x \<le> (0::'b)"
index_take: fixes i :: "nat"   and xs :: "'a list"   and x :: "'a" assumes "i \<le> index xs x" shows "x \<notin> set (take i xs)"
cp_defined: fixes X :: "'d state \<times> 'd state \<Rightarrow> 'e"   and \<tau> :: "'d state \<times> 'd state" shows "(\<delta> X) \<tau> = (\<delta> (\<lambda>_. X \<tau>)) \<tau>"
exhaust: fixes y :: "pol" obtains x1 :: "int" where "y = Pc x1"   | x21 :: "nat" and x22 :: "pol" where "y = Pinj x21 x22"   | x31 :: "pol" and x32 :: "nat" and x33 :: "pol" where "y = PX x31 x32 x33"
shift1_inv_some_snd_2: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and B :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set" assumes "shift1_inv (qs, B)"   and "1 < card B"   and "(h, U) = (SOME b. b \<in> B \<and> card (snd b) = Suc m)" shows "(h, U) \<in> set qs"
vrestr_Inter: fixes A :: "'a set set"   and f :: "'a \<Rightarrow> 'a" assumes "A \<noteq> {}" shows "f --` \<Inter> A = \<Inter> ((--`) f ` A)"
fls_subdegree_lt_fls_regpart_subdegree: fixes f :: "'a fls" shows "fls_subdegree f \<le> int (subdegree (fls_regpart f))"
wf_permute_alts: fixes \<sigma> :: "'alt \<Rightarrow> 'alt" assumes "\<sigma> permutes alts" shows "is_pref_profile (permute_profile \<sigma> R2)"
takeWhile_queue_rep: fixes f :: "'a \<Rightarrow> bool"   and q :: "'a queue" shows "linearize (takeWhile_queue f q) = takeWhile f (linearize q)"
assms_3: assumes "\<not> rword a b"   and "rword b a" shows "thesis"
pdevs_val_of_list_of_pdevs2: fixes e :: "nat \<Rightarrow> real"   and I :: "real set"   and X :: "'a pdevs" assumes "e \<in> funcset UNIV I" obtains e' :: "nat \<Rightarrow> real" where "pdevs_val e X = pdevs_val e' (pdevs_of_list (map snd (list_of_pdevs X)))" and "e' \<in> funcset UNIV I"
plus_eq: fixes x :: "'a"   and y :: "'a" shows "x + y = abs_nat (rep_nat x + rep_nat y)"
variants_sym: fixes D :: "'a literal multiset"   and D' :: "'a literal multiset" shows "variants D D' = variants D' D"
bleq0: fixes x :: "real" assumes "\<forall>y<x. b * y + c < 0" shows "0 \<le> b"
replicate_empty: fixes n :: "nat"   and x :: "'b" shows "(replicate n x = \<epsilon>) = (n = 0)"
continuous_onE: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and e :: "real" assumes "continuous_on s f"   and "x \<in> s"   and "0 < e" obtains d :: "real" where "0 < d" and "\<And>x'. \<lbrakk>x' \<in> s; dist x' x \<le> d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
sup_set_def: fixes A :: "'c set"   and B :: "'c set" shows "A \<union> B = Collect ((\<lambda>x. x \<in> A) \<squnion> (\<lambda>x. x \<in> B))"
strongBisimWeakPsiCong: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<sim> Q" shows "\<Psi> \<rhd> P \<doteq> Q"
are_operator_effects_consistent_set: fixes op\<^sub>1 :: "('c, 'd) strips_operator_scheme"   and ops :: "('c, 'd) strips_operator_scheme list"   and op\<^sub>2 :: "('c, 'd) strips_operator_scheme" assumes "op\<^sub>1 \<in> set ops"   and "op\<^sub>2 \<in> set ops" shows "STRIPS_Semantics.are_operator_effects_consistent op\<^sub>1 op\<^sub>2 = (set (add_effects_of op\<^sub>1) \<inter> set (delete_effects_of op\<^sub>2) = {} \<and> set (delete_effects_of op\<^sub>1) \<inter> set (add_effects_of op\<^sub>2) = {})"
reach_update: fixes s :: "Store"   and l :: "Location"   and x :: "Value"   and y :: "Value"   and k :: "Location" assumes "\<not> s\<turnstile> l reachable_from x" shows "s\<langle>l := y\<rangle>\<turnstile> k reachable_from x = s\<turnstile> k reachable_from x"
lipschitz_on_transform: fixes C :: "real"   and U :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "C-lipschitz_on U f"   and "\<And>x. x \<in> U \<Longrightarrow> g x = f x" shows "C-lipschitz_on U g"
singleton_set_def: fixes X :: "'a set" shows "singleton_set X \<equiv> {Y. \<exists>x\<in>X. Y = {x}}"
PO_hoare_defs_2: fixes r :: "('f \<times> 'e) set"   and s :: "'f set" shows "r `` s = {y. \<exists>x\<in>s. (x, y) \<in> r}"
x: shows "x \<in> {fs ! i $ j |i j. i < m \<and> j < n} \<union> {d\<mu> fs i j |i j. j < i \<and> i < m} \<union> {d fs i |i. i \<le> m}"
tt_lower: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" assumes "tt p \<prec>\<^sub>t v" shows "tt (lower p v) = tt p"
smap_alt: fixes f :: "'e \<Rightarrow> 'd"   and s :: "'e stream"   and s' :: "'d stream" shows "(smap f s = s') = (\<forall>n. f (s !! n) = s' !! n)"
simps_2: fixes x :: "'d"   and f :: "'d \<Rightarrow> 'c option" shows "Option.bind (Some x) f = f x"
le_multiset_right_total: fixes M :: "'b multiset"   and x :: "'b" shows "M < add_mset x M"
covector_complement_closed: fixes x :: "'a" assumes "covector x" shows "covector (- x)"
strict_implies_order: fixes b :: "'c list"   and a :: "'c list" assumes "strict_suffix b a" shows "suffix b a"
sgn1_nz_eq_arg_pos: fixes x :: "nat" shows "(sgn1 x \<noteq> 0) = (0 < x)"
has_integral_separate_sides: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'b"   and a :: "'a"   and b :: "'a"   and e :: "real"   and k :: "'a"   and c :: "real" assumes "(f has_integral i) (cbox a b)"   and "0 < e"   and "k \<in> Basis" obtains d :: "'a \<Rightarrow> 'a set" where "gauge d" and "\<forall>p1 p2. p1 tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<le> c} \<and> d fine p1 \<and> p2 tagged_division_of cbox a b \<inter> {x. c \<le> x \<bullet> k} \<and> d fine p2 \<longrightarrow> norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) < e"
Ds_mono: fixes A :: "'b set option"   and A' :: "'b set option"   and es :: "('b, 'c, 'd) exp list" assumes "A \<sqsubseteq> A'"   and "\<D>s es A" shows "\<D>s es A'"
simps_3: fixes f :: "'a"   and v :: "'a" shows "\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<Down> \<^bold>\<langle>v\<^bold>\<rangle> = \<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<otimes> \<^bold>\<langle>v\<^bold>\<rangle>"
strong_induct: fixes P :: "'a \<Rightarrow> fmla \<Rightarrow> bool"   and c :: "'a"   and fmla :: "fmla" assumes "\<And>x1 x2 c. P c (x1 EQ x2)"   and "\<And>x1 x2 c. \<lbrakk>\<And>c. P c x1; \<And>c. P c x2\<rbrakk> \<Longrightarrow> P c (x1 OR x2)"   and "\<And>x c. (\<And>c. P c x) \<Longrightarrow> P c (Robinson_Arithmetic.neg x)"   and "\<And>x1 x2 c. \<lbrakk>{atom x1} \<sharp>* c; \<And>c. P c x2\<rbrakk> \<Longrightarrow> P c (exi x1 x2)" shows "P c fmla"
open_Diff: fixes "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and T :: "'b set" assumes "class.topological_space open"   and "open S"   and "topological_space.closed open T" shows "open (S - T)"
filter_univ: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.semilattice_inf inf less_eq less" shows "ord.filter less_eq UNIV"
WTrtFAssNT: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp"   and T\<^sub>2 :: "ty"   and F :: "char list"   and D :: "char list" assumes "P,E,h \<turnstile> e\<^sub>1 : NT"   and "P,E,h \<turnstile> e\<^sub>2 : T\<^sub>2" shows "P,E,h \<turnstile> e\<^sub>1\<bullet>F{D} := e\<^sub>2 : Void"
sum_le_card_Max: fixes A :: "'a set"   and f :: "'a \<Rightarrow> nat" assumes "finite A"   and "A \<noteq> {}" shows "sum f A \<le> card A * Max (f ` A)"
prim_rootI: fixes u :: "'b list"   and r :: "'b list" assumes "u \<noteq> \<epsilon>"   and "u \<in> r*"   and "primitive r" shows "u \<in>\<^sub>p r *"
bprv_prv': fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "bprv \<phi>" shows "prv \<phi>"
less_not_sym: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.preorder less_eq less"   and "less x y" shows "\<not> less y x"
ex1_eqI: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "\<exists>!x. P x"   and "P a"   and "P b" shows "a = b"
Reds_preserves_bconf: fixes P :: "(char list list \<times> char list exp) prog"   and 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" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>es,s,b\<rangle> [\<rightarrow>]* \<langle>es',s',b'\<rangle>"   and "iconfs (shp s) es"   and "P,shp s \<turnstile>\<^sub>b (es,b) \<surd>" shows "P,shp s' \<turnstile>\<^sub>b (es',b') \<surd>"
interval_def: fixes a :: "'a"   and b :: "'a" shows "interval a b \<equiv> insert b (insert a (segment a b))"
mult_cancel: fixes s :: "('k \<times> 'k) set"   and X :: "'k multiset"   and Z :: "'k multiset"   and Y :: "'k multiset" assumes "trans s"   and "irrefl s" shows "((X + Z, Y + Z) \<in> mult s) = ((X, Y) \<in> mult s)"
le_numeral_extra_3: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq zero zero"
omit_branch_def: fixes xs :: "(nat \<times> nat) set"   and branch :: "(('c, 'd) fm list \<times> 'd) list" shows "omit_branch xs branch \<equiv> mapi (\<lambda>v. omit_block (proj xs v)) branch"
tuple_in_hom: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "commutative_square f g h k" shows "\<guillemotleft>ECP.tuple h f g k : local.dom h \<rightarrow> ECP.pbdom f g\<guillemotright>"
star_contl: fixes X :: "'a set"   and Y :: "'a set" shows "X \<cdot> Y\<^sup>\<star> = (\<Union>n. X \<cdot> Y ^ n)"
gs: shows "set_mset (image_mset card (mset_set \<G>)) = group_sizes"
enabled_disjD: fixes F :: "state \<times> state \<Rightarrow> bool"   and G :: "state \<times> state \<Rightarrow> bool" shows "\<turnstile> Enabled (F \<or> G) \<longrightarrow> Enabled F \<or> Enabled G"
arcs_ends_blocks: fixes G :: "('a, 'b) pre_digraph"   and x :: "'a"   and y :: "'a" assumes "graph G"   and "{x, y} \<in># graph.arcs_blocks G" shows "x \<rightarrow>\<^bsub>G\<^esub> y \<and> y \<rightarrow>\<^bsub>G\<^esub> x"
Ra_def: shows "Ra = (\<lambda>x. - sum_upto (\<lambda>xa. real xa powr - \<alpha> * frac (x / real xa)) x)"
3: fixes u :: "'b" shows "(0::'b) \<notin> snd ` set (update_by_pair (k, u) xs)"