hcju/isabelleps huggingface dataset:
trans_prec_F: fixes C1 :: "'f"   and C2 :: "'f"   and C3 :: "'f" assumes "C1 \<prec>\<cdot> C2"   and "C2 \<prec>\<cdot> C3" shows "C1 \<prec>\<cdot> C3"
pth_7_1: fixes x :: "'b" shows "(0::'b) + x \<equiv> x"
get_ancestors_is_l_get_ancestors: shows "l_get_ancestors get_ancestors"
R29_bc_1: shows "pmf (sds R29) b = 0"
memoryless_on_def: fixes f :: "'a \<Rightarrow> 'a pmf"   and s :: "'a" shows "memoryless_on f s = cfg_corec s f (\<lambda>_ t. t) s"
cases: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c list" assumes "class.linorder less_eq less" obtains "x = []"   | xa :: "'c" and ys :: "'c list" where "x = xa # ys"
diff_list_def: fixes xs :: "'a list"   and ys :: "'a list" shows "diff_list xs ys \<equiv> fold removeAll ys xs"
const_is_polynomial: fixes a :: "'a"   and K :: "'a set" assumes "a \<in> K - {\<zero>}" shows "polynomial K [a]"
aplus_le_aminus: fixes a :: "ant"   and b :: "ant"   and c :: "ant" assumes "a \<in> Z\<^sub>-\<^sub>\<infinity>"   and "b \<in> Z\<^sub>-\<^sub>\<infinity>"   and "c \<in> Z\<^sub>-\<^sub>\<infinity>"   and "- b \<in> Z\<^sub>-\<^sub>\<infinity>" shows "(a + b \<le> c) = (a \<le> c - b)"
term_poly_list_rel_Cons_iff: fixes y :: "char list"   and p :: "char list list"   and p' :: "char list multiset" shows "((y # p, p') \<in> term_poly_list_rel) = ((p, remove1_mset y p') \<in> term_poly_list_rel \<and> y \<in># p' \<and> y \<notin> set p \<and> y \<notin># remove1_mset y p' \<and> (\<forall>x\<in>#mset p. (y, x) \<in> var_order_rel))"
PiE_Int: fixes I :: "'b set"   and A :: "'b \<Rightarrow> 'c set"   and B :: "'b \<Rightarrow> 'c set" shows "Pi\<^sub>E I A \<inter> Pi\<^sub>E I B = (\<Pi>\<^sub>E x\<in>I. A x \<inter> B x)"
ord_spmf_eqD_set_spmf: fixes p :: "'c spmf"   and q :: "'c spmf" assumes "ord_spmf (=) p q" shows "set_spmf p \<subseteq> set_spmf q"
take_root: fixes k :: "nat"   and r :: "'b list" assumes "k \<noteq> 0" shows "r = take \<^bold>|r\<^bold>| (r \<^sup>@ k)"
fresh_variable_types: fixes t :: "typ" shows "\<exists>n. new_tv n t"
right_neutral: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and z :: "'d"   and a :: "'d" assumes "monoid f z" shows "f a z = a"
rel_mapping_alt: fixes Q :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'c \<Rightarrow> 'a option"   and P' :: "'c \<Rightarrow> 'b option" shows "rel_mapping Q P P' = (dom P = dom P' \<and> (\<forall>p\<in>dom P. Q (the (P p)) (the (P' p))))"
cod_cod: fixes f :: "'a" assumes "hom.arr f" shows "hom.cod (hom.cod f) = hom.cod f"
local_lipschitzI: fixes T :: "'b set"   and X :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "\<And>t x. \<lbrakk>t \<in> T; x \<in> X\<rbrakk> \<Longrightarrow> \<exists>u>0. \<exists>L. \<forall>t\<in>cball t u \<inter> T. L-lipschitz_on (cball x u \<inter> X) (f t)" shows "local_lipschitz T X f"
commute: fixes a :: "'k"   and b :: "'k" shows "max a b = max b a"
\<Gamma>_wf': fixes t :: "('fun, 'var) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m t" shows "wf\<^sub>t\<^sub>r\<^sub>m (\<Gamma> t)"
prems_4: fixes i :: "'a" assumes "i \<in> insert i__ s" shows "0 \<le> a i"
smooth_compat_def: fixes k :: "enat"   and c1 :: "('e, 'f) chart"   and c2 :: "('e, 'f) chart" shows "k-smooth_compat c1 c2 = (k-smooth_on (apply_chart c1 ` (domain c1 \<inter> domain c2)) (apply_chart c2 \<circ> inv_chart c1) \<and> k-smooth_on (apply_chart c2 ` (domain c1 \<inter> domain c2)) (apply_chart c1 \<circ> inv_chart c2))"
space: shows "\<Omega> \<in> {A \<in> sigma_sets \<Omega> G. P A}"
isolate_dist_Ex_iff: fixes x :: "'c"   and S :: "'c set" shows "x isolate S = (x \<in> S \<and> (\<exists>e>0. \<forall>y\<in>S. dist x y < e \<longrightarrow> y = x))"
INF_mono: fixes Inf :: "'h set \<Rightarrow> 'h"   and Sup :: "'h set \<Rightarrow> 'h"   and inf :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and less :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and sup :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and bot :: "'h"   and top :: "'h"   and B :: "'i set"   and A :: "'j set"   and f :: "'j \<Rightarrow> 'h"   and g :: "'i \<Rightarrow> 'h" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. less_eq (f n) (g m)" shows "less_eq (Inf (f ` A)) (Inf (g ` B))"
assms_2: shows "ring_homomorphism f A addA multA zeroA oneA B addB multB zeroB oneB"
Decomp: fixes A :: "('fun, 'var) extstrand_step list"   and t :: "('fun, 'var) Term.term" assumes "well_analyzed A"   and "t \<in> subterms\<^sub>s\<^sub>e\<^sub>t (ik\<^sub>e\<^sub>s\<^sub>t A \<union> assignment_rhs\<^sub>e\<^sub>s\<^sub>t A) - range Var" shows "well_analyzed (A @ [Decomp t])"
dijkstra_correct: shows "dijkstra \<le> SPEC (is_shortest_path_map v0)"
fps_mult_image: fixes f :: "('d, 'c) fps"   and g :: "('d, 'c) fps"   and n :: "'d list" shows "(f \<cdot> g) $ n = \<Sum> ((\<lambda>p. f $ fst p \<cdot> g $ snd p) ` splitset n)"
alternating_harmonic_series_sums': shows "(\<lambda>k. inverse (real (2 * k + 1)) - inverse (real (2 * k + 2))) sums ln 2"
op_set_npick_remove_def: fixes X :: "'c set" shows "op_set_npick_remove X = SPEC (\<lambda>(x, X'). x \<in> X \<and> (X' = X - {x} \<or> X' = X))"
rcoset0: fixes H :: "'g set" assumes "Subgroup H" shows "rcoset_rel H `` {0::'g} = H"
cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^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_def: fixes ptr :: "(_) document_ptr" shows "cast ptr = object_ptr.Ext (Inr (Inl ptr))"
assms: shows "(\<lambda>n. f (n + i)) sums l"
fgroup_byD: fixes b :: "'d"   and cs :: "'e fset"   and f :: "'f \<Rightarrow> 'd \<times> 'e"   and as :: "'f fset"   and a :: "'f"   and c :: "'e" assumes "(b, cs) |\<in>| fgroup_by f as"   and "f a = (b, c)"   and "a |\<in>| as" shows "c |\<in>| cs"
fsi': fixes i :: "nat" assumes "i < length fs" shows "dim_vec (fs ! i) = n"
weak_sup_of_singleton: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a" assumes "weak_partial_order L"   and "x \<in> carrier L" shows "\<Squnion>\<^bsub>L\<^esub>{x} .=\<^bsub>L\<^esub> x"
infsetsum_nat': fixes f :: "nat \<Rightarrow> 'd" assumes "f abs_summable_on UNIV" shows "infsetsum f UNIV = (\<Sum>n. f n)"
thin: fixes G' :: "'b triple set"   and ts :: "'b triple set"   and G :: "'b triple set" assumes "G'||-ts"   and "G' \<subseteq> G" shows "G||-ts"
delta_def: fixes A :: "'d \<times> ('c \<Rightarrow> 'd \<Rightarrow> 'd) \<times> ('d \<Rightarrow> bool)" shows "delta A = foldl2 (next A)"
restrict_idle_include: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and n :: "nat"   and A :: "nat set"   and v :: "'c option list"   and I :: "nat set" assumes "getIJ getIJ"   and "wf_tuple n A v"   and "A \<subseteq> I" shows "restrict I v = v"
monic: fixes a :: "'a" assumes "a \<in> S" shows "monic (f a)"
p_altdef': fixes k :: "nat" shows "p k = x ^ k + y ^ k + z ^ k"
rn: fixes C :: "'a eclause"   and S :: "'a eclause set"   and D :: "'a eclause" assumes "derivable_ecl C S"   and "renaming_cl C D" shows "derivable_ecl D S"
Abort: fixes r :: "('a, 'b, 'c) Resumption.resumption"   and r' :: "('a, 'b, 'c) Resumption.resumption" assumes "X r r'"   and "\<not> Resumption.resumption.is_Done r'"   and "Resumption.resumption.is_Done r" shows "Resumption.resumption.result r = None"
viewLn_list: fixes t :: "('a, 'b) FingerTreeStruc"   and nd :: "('a, 'b) Node"   and s :: "('a, 'b) FingerTreeStruc" assumes "FingerTreeStruc.viewLn t = Some (nd, s)" shows "FingerTreeStruc.toList t = FingerTreeStruc.nodeToList nd @ FingerTreeStruc.toList s"
simps_2: fixes x :: "'a list"   and xs :: "'a list list" shows "reduct (x # xs) = xs"
nat_power_inject_base: fixes n :: "nat"   and a :: "nat"   and b :: "nat" assumes "n \<noteq> 0"   and "a ^ n = b ^ n" shows "a = b"
is_lb_def: fixes S :: "'a set"   and x :: "'a" shows "S >| x = (\<forall>y\<in>S. x \<sqsubseteq> y)"
ucast_ucast_mask: fixes x :: "'b word" shows "UCAST('c \<rightarrow> 'b) (UCAST('b \<rightarrow> 'c) x) = x AND mask LENGTH('c)"
f2: shows "(\<one> |\<^bsub>rel\<^esub> s) \<otimes>\<^bsub>rec_rng_of_frac\<^esub> (s |\<^bsub>rel\<^esub> \<one>) = (\<one> |\<^bsub>rel\<^esub> \<one>)"
same_fw_spec: fixes ip1 :: "'a word"   and ip2 :: "'a word"   and rs :: "'a simple_rule list"   and c :: "'b parts_connection_scheme" assumes "same_fw_behaviour ip1 ip2 rs" shows "same_fw_behaviour_one ip1 ip2 c rs"
times_subset_iff: fixes A :: "'c set"   and C :: "'d set"   and B :: "'c set"   and D :: "'d set" shows "(A \<times> C \<subseteq> B \<times> D) = (A = {} \<or> C = {} \<or> A \<subseteq> B \<and> C \<subseteq> D)"
domain_finmap_of: fixes inds :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "finite inds" shows "Fin_Map.domain (Fin_Map.finmap_of inds f) = inds"
residual_network_def: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" shows "RES.residual_network f = \<lparr>edge = \<lambda>x y. edge (residual_network f) x y \<or> edge (residual_network f) y x \<and> y \<noteq> source (residual_network f), capacity = \<lambda>(x, y). if edge (residual_network f) x y then capacity (residual_network f) (x, y) - f (x, y) else if y = source (residual_network f) then 0 else f (y, x), source = source (residual_network f), sink = sink (residual_network f), \<dots> = network.more (residual_network f)\<rparr>"
wf_lderiv: fixes n :: "nat"   and r :: "'b rexp"   and w :: "'a" assumes "local.wf n r" shows "local.wf n (lderiv w r)"
distinct_40: fixes x71 :: "nat"   and x72 :: "msg"   and x5 :: "msg" shows "Crypt x71 x72 \<noteq> Hash x5"
is_dghmI: fixes \<alpha> :: "V"   and \<FF> :: "V"   and \<AA> :: "V"   and \<BB> :: "V" assumes "\<Z> \<alpha>"   and "vfsequence \<FF>"   and "digraph \<alpha> \<AA>"   and "digraph \<alpha> \<BB>"   and "vcard \<FF> = 4\<^sub>\<nat>"   and "\<FF>\<lparr>HomDom\<rparr> = \<AA>"   and "\<FF>\<lparr>HomCod\<rparr> = \<BB>"   and "vsv (\<FF>\<lparr>ObjMap\<rparr>)"   and "vsv (\<FF>\<lparr>ArrMap\<rparr>)"   and "\<D>\<^sub>\<circ> (\<FF>\<lparr>ObjMap\<rparr>) = \<AA>\<lparr>Obj\<rparr>"   and "\<R>\<^sub>\<circ> (\<FF>\<lparr>ObjMap\<rparr>) \<subseteq>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"   and "\<D>\<^sub>\<circ> (\<FF>\<lparr>ArrMap\<rparr>) = \<AA>\<lparr>Arr\<rparr>"   and "\<And>a b f. f : a \<mapsto>\<^bsub>\<AA>\<^esub> b \<Longrightarrow> \<FF>\<lparr>ArrMap\<rparr>\<lparr>f\<rparr> : \<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> \<mapsto>\<^bsub>\<BB>\<^esub> \<FF>\<lparr>ObjMap\<rparr>\<lparr>b\<rparr>" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
inj_iff_trivial_ker: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and h :: "'b \<Rightarrow> 'd" assumes "group_hom G H h" shows "inj_on h (carrier G) = (kernel G H h = {\<one>\<^bsub>G\<^esub>})"
2: shows "covector (r\<^sup>T * g\<^sup>\<star>)"
C\<pi>x: fixes x :: "'b" shows "C\<pi> *\<^sub>V ket x = (case \<pi> x of None \<Rightarrow> 0 | Some i \<Rightarrow> ket i)"
map_upd_Some_unfold: fixes m :: "'d \<Rightarrow> 'c option"   and a :: "'d"   and b :: "'c"   and x :: "'d"   and y :: "'c" shows "((m(a \<mapsto> b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)"
unstream_foldrS: fixes a :: "('b, 'c) Stream"   and f :: "'b \<rightarrow> 'd \<rightarrow> 'd"   and z :: "'d" assumes "a \<noteq> \<bottom>" shows "foldrS\<cdot>f\<cdot>z\<cdot>a = foldrL\<cdot>f\<cdot>z\<cdot>(unstream\<cdot>a)"
get_child_nodes_split: fixes P :: "((_) heap, exception, (_) node_ptr list) prog \<Rightarrow> bool"   and xs :: "(((_) object_ptr \<Rightarrow> bool) \<times> ((_) object_ptr \<Rightarrow> unit \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog)) list"   and ptr :: "(_) object_ptr" shows "P (invoke (a_get_child_nodes_tups @ xs) ptr ()) = ((known_ptr ptr \<longrightarrow> P (get_child_nodes ptr)) \<and> (\<not> known_ptr ptr \<longrightarrow> P (invoke xs ptr ())))"
eq_f: shows "functions_of A B N = domain_and_range_permutation A B `` {f}"
iT_Div_image_conv: fixes I :: "nat set"   and k :: "nat" shows "I \<oslash> k = (\<lambda>n. n div k) ` I"
min_ground_head_min: fixes f :: "'s"   and \<zeta> :: "('s, 'v) hd" assumes "f \<in> local.ground_heads \<zeta>" shows "wt_sym (min_ground_head \<zeta>) + \<delta>\<^sub>h * arity_sym\<^sub>h (min_ground_head \<zeta>) \<le> wt_sym f + \<delta>\<^sub>h * arity_sym\<^sub>h f"
le_iff_plus_unchanged2: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "(x \<sqsubseteq>\<^bsub>r\<^esub> y) = (y \<squnion>\<^bsub>f\<^esub> x = y)"
isCont_P: fixes x :: "real" assumes "0 < x" shows "isCont P x"
intros_2: fixes E :: "'b \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> 'a"   and e :: "'b"   and m :: "'c \<Rightarrow> 'a"   and v :: "'a"   and x :: "'c"   and \<iota> :: "nat"   and BMap :: "'a \<Rightarrow> bool" assumes "E e m = v" shows "((x:=\<^bsub>\<iota>\<^esub> e, m), [], None, m(x := v)) \<in> MWLs_semantics.MWLsSteps_det E BMap"
eq_finfun_def: fixes f :: "'a \<Rightarrow>f 'b"   and g :: "'a \<Rightarrow>f 'b" shows "equal_class.equal f g = finfun_All ((\<lambda>(x, y). x = y) \<circ>$ ($f, g$))"
bisim1TryFail: fixes P :: "'b prog"   and h :: "'heap"   and e :: "'addr expr1"   and a :: "'addr"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and C' :: "String.literal"   and C :: "String.literal"   and V :: "nat"   and e2 :: "'addr expr1" assumes "P,e,h \<turnstile> (Throw a, xs) \<leftrightarrow> (stk, loc, pc, \<lfloor>a\<rfloor>)"   and "typeof_addr h a = \<lfloor>Class_type C'\<rfloor>"   and "\<not> P \<turnstile> C' \<preceq>\<^sup>* C" shows "P,try e catch(C V) e2,h \<turnstile> (Throw a, xs) \<leftrightarrow> (stk, loc, pc, \<lfloor>a\<rfloor>)"
select_convs_1: fixes hosts :: "'a set"   and flows_fix :: "('a \<times> 'a) set"   and flows_state :: "('a \<times> 'a) set"   and more :: "'b" shows "hosts \<lparr>hosts = hosts, flows_fix = flows_fix, flows_state = flows_state, \<dots> = more\<rparr> = hosts"
set_minus_singleton_eq: fixes x :: "'a"   and X :: "'a set" assumes "x \<notin> X" shows "X - {x} = X"
length_Cons: fixes x :: "'e"   and xs :: "'e list" shows "length (x # xs) = Suc (length xs)"
fin_digraph_axioms: shows "fin_digraph (app_iso hom G)"
mk_rtrancl_list_def: fixes subsumes :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'b \<Rightarrow> 'b list"   and init :: "'b list" shows "mk_rtrancl_list subsumes r init \<equiv> mk_rtrancl_list_main subsumes r init []"
orderI: fixes a :: "'g multiset"   and b :: "'g multiset" assumes "a = a \<union># b" shows "b \<subseteq># a"
FR: fixes x :: "'a"   and s :: "'b" shows "nres_of (f x s) \<le> f' x s"
assms_4: shows "\<R>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> A"
call_only_one_intra_edge: fixes a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs" shows "\<exists>!a'. valid_edge a' \<and> sourcenode a' = sourcenode a \<and> intra_kind (kind a')"
that_2: assumes "h2 \<in> o[F](h1)" shows "thesis"
assms_2: shows "geodesic_segment_between G (f a) (f b)"
uses'_codem_correct: fixes g :: "'g"   and nodes_of_uses :: "('val, 'node set) mapping"   and "next" :: "'node \<times> 'val" assumes "g \<turnstile> nodes_of_uses \<approx>\<^sub>\<phi> uninst_code.ssa.useNodes_of uses g"   and "next \<in> Mapping.keys (phis g)" shows "usesOf (uses'_codem g next (uninst_code.substitution_code phis g next) nodes_of_uses) = uninst_code.uses'_code uses phis g next"
sets_PiM_cong: fixes I :: "'b set"   and J :: "'b set"   and M :: "'b \<Rightarrow> 'c measure"   and N :: "'b \<Rightarrow> 'c measure" assumes "I = J"   and "\<And>i. i \<in> J \<Longrightarrow> sets (M i) = sets (N i)" shows "sets (Pi\<^sub>M I M) = sets (Pi\<^sub>M J N)"
assms_3: shows "g \<in> carrier (function_ring (carrier R) R)"
intros_1: fixes l :: "nat" shows "l \<turnstile> SKIP"
A6: shows "(\<sigma>, pp) \<in> oreachable A S U"
conv: fixes x :: "'a"   and y :: "'a" shows "x \<rhd> y = (x \<rhd> (1::'a)) \<cdot> y"
sets_PiM_empty: fixes M :: "'d \<Rightarrow> 'e measure" shows "sets (Pi\<^sub>M {} M) = {{}, {\<lambda>k. undefined}}"
ddh_0_def: fixes \<A> :: "'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> bool spmf" shows "ddh_0 \<A> = sample_uniform (order \<G>) \<bind> (\<lambda>x. sample_uniform (order \<G>) \<bind> (\<lambda>y. \<A> (\<^bold>g [^] x) (\<^bold>g [^] y) (\<^bold>g [^] (x * y))))"
seq_char: fixes g :: "(bool, bool list) concrete_category.arr"   and f :: "(bool, bool list) concrete_category.arr" shows "J.seq g f = (J.arr g \<and> J.arr f \<and> (f = J.Zero \<and> g \<noteq> J.One \<or> f \<noteq> J.Zero \<and> g = J.One))"
thenOrdering_simps_3: fixes y :: "Ordering" shows "thenOrdering\<cdot>GT\<cdot>y = GT"
cycle_free_loop_dest: fixes i :: "nat"   and n :: "nat"   and xs :: "nat list"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry" assumes "i \<le> n"   and "set xs \<subseteq> {0..n}"   and "linordered_monoid.cycle_free m n" shows "Le (0::'d) \<preceq> linordered_monoid.len m i i xs"
simps_2: fixes M :: "'d"   and xvec :: "name list"   and N :: "'d" shows "subject (M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle>) = Some M"
one_closed: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "Group.submonoid H G" shows "\<one>\<^bsub>G\<^esub> \<in> H"
right_gpv_bind_gpv: fixes gpv :: "('f, 'c, 'e) gpv"   and f :: "'f \<Rightarrow> ('a, 'c, 'e) gpv" shows "right_gpv (gpv \<bind> f) = right_gpv gpv \<bind> right_gpv \<circ> f"
assms: shows "\<turnstile> A \<and> $P \<longrightarrow> Q$"
trss_modify_context: fixes s :: "'n list"   and c :: "'n list multiset"   and w :: "('p, 'ba) label list"   and s' :: "'n list"   and c' :: "'n list multiset"   and cn :: "'n list multiset" assumes "((s, c), w, s', c') \<in> trcl (trss fg)"   and "mon_w fg w \<inter> mon_c fg cn = {}" shows "\<exists>csp. c' = csp + c \<and> mon_c fg csp = {} \<and> ((s, cn), w, s', csp + cn) \<in> trcl (trss fg)"
that: fixes r :: "recf" assumes "recfn 2 r"   and "Partial_Recursive.total r"   and "f = (\<lambda>x\<^sub>1 x\<^sub>2. eval r [x\<^sub>1, x\<^sub>2])" shows "thesis"
convex_translation: fixes S :: "'c set"   and a :: "'c" assumes "convex S" shows "convex ((+) a ` S)"
cxt_inv_await: fixes E :: "('Var, 'AExp, 'BExp) Stmt list"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and e :: "'BExp"   and p :: "('Var, 'AExp, 'BExp) Stmt" assumes "cxt_to_stmt E c = Await e p" shows "c = Await e p \<and> E = []"
t2_def: shows "t2 = (- (v\<^sub>o - v\<^sub>e) - sqrt D2) / (a\<^sub>o - a\<^sub>e)"
ideD_2: fixes a :: "('o, 'a) arr" assumes "ide a" shows "local.dom a = a"
g_def: shows "g = dirichlet_prod (ind squarefree) moebius_mu"
refine: fixes fi :: "'e \<Rightarrow> 'f nres"   and f :: "'g \<Rightarrow> 'h nres"   and A :: "('e \<times> 'g) set"   and B :: "('f \<times> 'h) set"   and Xi :: "'e list"   and X :: "'g set"   and P :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "(fi, f) \<in> A \<rightarrow> \<langle>B\<rangle>nres_rel"   and "(Xi, X) \<in> \<langle>A\<rangle>list_wset_rel"   and "PREFER single_valued A"   and "PREFER single_valued B" shows "(local.PAR_IMAGE_nres fi Xi, PAR_IMAGE P $ f $ X) \<in> \<langle>\<langle>A \<times>\<^sub>r B\<rangle>list_wset_rel\<rangle>nres_rel"
HPhase0Read_Inv2a: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk" assumes "Inv2a s"   and "HPhase0Read s s' p d" shows "Inv2a s'"
D: shows "distributed M lborel X (\<lambda>x. ennreal (indicat_real {a..b} x / content {a..b}))"
PUren_FUren: fixes I :: "('sf, 'sc, 'sz) interp"   and \<alpha> :: "('sf, 'sc, 'sz) hp"   and x :: "'sz"   and y :: "'sz"   and \<phi> :: "('sf, 'sc, 'sz) formula" assumes "is_interp I" shows "(PRadmit \<alpha> \<longrightarrow> hpsafe \<alpha> \<longrightarrow> (\<forall>\<nu> \<omega>. ((\<nu>, \<omega>) \<in> prog_sem I (PUrename x y \<alpha>)) = ((Radj x y \<nu>, Radj x y \<omega>) \<in> prog_sem I \<alpha>))) \<and> (FRadmit \<phi> \<longrightarrow> fsafe \<phi> \<longrightarrow> (\<forall>\<nu>. (\<nu> \<in> fml_sem I (FUrename x y \<phi>)) = (Radj x y \<nu> \<in> fml_sem I \<phi>)))"
inf_add_S_right: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le>\<^sub>\<S> y" shows "x \<le>\<^sub>\<S> y \<oplus> z"
LeftDerivationIntro_finish: fixes \<alpha> :: "symbol list"   and \<delta> :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and ix :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and j :: "nat"   and \<gamma> :: "symbol list" assumes "LeftDerivationIntro (\<alpha> @ \<delta>) i r ix D j \<gamma>"   and "i < length \<alpha>" shows "\<exists>k \<omega> \<delta>''. k \<le> length D \<and> LeftDerivation \<alpha> ((i, r) # take k D) \<omega> \<and> LeftDerivation (\<alpha> @ \<delta>) ((i, r) # take k D) (\<omega> @ \<delta>) \<and> derivation_ge (drop k D) (length \<omega>) \<and> LeftDerivation \<delta> (derivation_shift (drop k D) (length \<omega>) 0) \<delta>'' \<and> \<gamma> = \<omega> @ \<delta>'' \<and> j < length \<omega>"
fun_star_inductl: fixes h :: "'b nd_fun"   and f :: "'b nd_fun"   and g :: "'b nd_fun" assumes "h \<squnion> f \<cdot> g \<le> g" shows "qstar f \<cdot> h \<le> g"
assms_9: fixes A :: "'a"   and B :: "'a" assumes "reflexive_rewrite A B" shows "equalish A B \<or> rewrite A B"
hyps: assumes "list_all (coll 0 x) ys"   and "x \<noteq> 0"   and "list_all (\<lambda>x. Counterclockwise_2D_Arbitrary.lex x 0) ys" shows "\<bar>sum_list ys\<bar> = sum_list (map abs ys)"
cbelow_on_intro: fixes A :: "'b set"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and z :: "'b"   and rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "cbelow_on A ord z rel a b"   and "c \<in> A"   and "rel b c"   and "ord c z" shows "cbelow_on A ord z rel a c"
a_known_ptr_def: fixes ptr :: "(_) object_ptr" shows "CharacterDataClass.known_ptr ptr = (ElementClass.known_ptr ptr \<or> is_character_data_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)"
coeff_def: fixes p :: "'a mpoly" shows "coeff p = poly_mapping.lookup (mapping_of p)"
pow_2_64: shows "2 ^ 64 = 18446744073709551616"
add_rel_def: fixes X :: "('a set \<times> 'b) set"   and Y :: "('a set \<times> 'b) set" shows "add_rel X Y \<equiv> let x = SOME x. x \<in> X; y = SOME y. y \<in> Y; w = get_lower_bound (fst x) (fst y) in \<lfloor>w, +\<^bsub>w\<^esub> (\<rho> (fst x) w (snd x)) (\<rho> (fst y) w (snd y))\<rfloor>"
exhaust: fixes y :: "'b tree" obtains "y = \<langle>\<rangle>"   | x21 :: "'b tree" and x22 :: "'b" and x23 :: "'b tree" where "y = \<langle>x21, x22, x23\<rangle>"
remove1Prio_xlate: fixes a :: "'d"   and q :: "('c, 'd) BsSkewBinomialTree list" shows "map bsmapt (remove1Prio a q) = SkewBinomialHeapStruc.remove1Prio a (map bsmapt q)"
assms: shows "t \<rightarrow>\<^sub>\<beta> t'"
typeof_OutOfMemory: fixes h :: "heap" assumes "preallocated h" shows "typeof\<^bsub>h\<^esub> (Addr (addr_of_sys_xcpt OutOfMemory)) = \<lfloor>Class OutOfMemory\<rfloor>"
deterministic_THE: fixes I :: "(('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set) set"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and x :: "'x"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m" assumes "deterministic I"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "t \<turnstile> \<langle>x, shr s\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>"   and "actions_ok s t ta"   and "s \<in> I" shows "(THE (ta, x', m'). t \<turnstile> \<langle>x, shr s\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle> \<and> actions_ok s t ta) = (ta, x', m')"