hcju/isabelleps huggingface dataset:
divide_left_mono: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq b a"   and "less_eq zero c"   and "less zero (times a b)" shows "less_eq (divide c a) (divide c b)"
create_gba__no_init: shows "create_gba \<phi> \<le> SPEC (\<lambda>\<A>. expand_init \<notin> name ` g_V \<A>)"
measurable_count_space_eq1: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and M :: "'c measure" shows "(f \<in> count_space A \<rightarrow>\<^sub>M M) = (f \<in> A \<rightarrow> space M)"
some_geodesic_endpoints_3: fixes x :: "'b"   and S :: "'b set"   and y :: "'b" shows "{x--S--y} \<noteq> {}"
le_enumerate: fixes S :: "nat set"   and n :: "nat" assumes "infinite S" shows "n \<le> wellorder_class.enumerate S n"
starfunNat_sumr: fixes f :: "nat \<Rightarrow> real"   and N :: "nat star" shows "(*f* (\<lambda>n. sum f {0..<n})) N = sumhr (0, N, f)"
kat_3_equiv': fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(d x \<cdot> y \<cdot> a z = \<bottom>) = (d x \<cdot> y = d x \<cdot> y \<cdot> d z)"
lunstream_append_raw: fixes g :: "'c \<Rightarrow> ('d, 'c) step"   and h :: "'e \<Rightarrow> ('d, 'e) step"   and sh :: "'e"   and sg :: "'c" assumes "productive g" shows "lunstream (append_raw g h sh) (Inl sg) = lappend (lunstream g sg) (lunstream h sh)"
sqrt_inverse_power2: fixes n :: "nat" shows "local.sqrt (n\<^sup>2) = n"
restrict_spmf_UNIV: fixes p :: "'c spmf" shows "p \<upharpoonleft> UNIV = p"
remark1_single: fixes e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau> :: "ty"   and \<pi>P :: "Syntax.term list"   and \<pi> :: "Syntax.term list"   and eP' :: "Syntax.term" assumes "{$$} \<turnstile> e, eP, eV : \<tau>"   and "\<lless>\<pi>P, eP\<ggreater> P\<rightarrow> \<lless>\<pi>P @ \<pi>, eP'\<ggreater>" obtains e' :: "Syntax.term" and eV' :: "Syntax.term" where "{$$} \<turnstile> e', eP', eV' : \<tau> \<and> \<lless>[], e\<ggreater> I\<rightarrow> \<lless>[], e'\<ggreater> \<and> \<lless>\<pi>, eV\<ggreater> V\<rightarrow> \<lless>[], eV'\<ggreater>"
Number_Tag: fixes X :: "nat"   and X' :: "tag" shows "Number X \<noteq> Tag X'"
closedin_compact: fixes S :: "'b set"   and T :: "'b set" assumes "compact S"   and "closedin (top_of_set S) T" shows "compact T"
H_noop: fixes \<rho>'' :: "'a \<Rightarrow> 'b"   and \<rho>' :: "'a \<Rightarrow> 'b" assumes "\<And>y. \<lbrakk>y \<in> S; y \<noteq> x\<rbrakk> \<Longrightarrow> (e1\<cdot>\<rho>'') y \<sqsubseteq> \<rho>' y" shows "H \<rho>'\<cdot>\<rho>'' \<sqsubseteq> \<rho>'"
f: fixes x :: "'d" assumes "x \<in> S" shows "f differentiable at x"
not_arr_null: shows "\<not> Right_a.arr Right_a.null"
affine_hull_convex_Int_open: fixes S :: "'a set"   and T :: "'a set" assumes "convex S"   and "open T"   and "S \<inter> T \<noteq> {}" shows "affine hull (S \<inter> T) = affine hull S"
prems_2: fixes s :: "'b set" assumes "s \<in> set (s__ # ss)" shows "local.subspace s"
prj0_in_hom: fixes f :: "'a"   and g :: "'a"   and a :: "'a"   and b :: "'a" assumes "C.cospan f g"   and "a = f \<down>\<down> g"   and "b = C.dom g" shows "\<guillemotleft>\<p>\<^sub>0[f, g] : a \<rightarrow> b\<guillemotright>"
full_component_zeroE: fixes rep_nat_term :: "'a \<Rightarrow> (nat, nat) pp \<times> nat"   and splus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and is_scalar :: "'a itself \<Rightarrow> bool"   and lex_comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and deg_comp :: "('a \<Rightarrow> 'a \<Rightarrow> order) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> order"   and pot_comp :: "('a \<Rightarrow> 'a \<Rightarrow> order) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> order"   and t :: "(nat, nat) pp" assumes "class.nat_term_compare rep_nat_term splus less_eq less is_scalar lex_comp deg_comp pot_comp" obtains x :: "'a" where "rep_nat_term x = (t, 0)"
him: shows "h ` rel_frontier S = sphere z 1 \<inter> U"
Un_Diff_cancel2: fixes B :: "'d set"   and A :: "'d set" shows "B - A \<union> A = B \<union> A"
rev_nth_mem: fixes block :: "'b"   and branch :: "'b list" shows "(block \<in>. branch) = (\<exists>v. branch !. v = Some block)"
IH_2: fixes i :: "int"   and f :: "bool"   and n :: "int" assumes "0 \<le> i" shows "succs (bcomp b2 f i) n \<subseteq> {n..n + size (bcomp b2 f i)} \<union> {n + i + size (bcomp b2 f i)}"
Restr_iff: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set"   and A :: "'a set" shows "((a, b) \<in> Restr r A) = (a \<in> A \<and> b \<in> A \<and> (a, b) \<in> r)"
smult: fixes j :: "nat"   and i :: "nat"   and c :: "'a" assumes "j < n"   and "i < l" shows "(c \<cdot>\<^sub>v fs ! i) $ j = c * fs ! i $ j"
exhaust: fixes y :: "brick" obtains "y = vert"   | "y = cup"   | "y = cap"   | "y = over"   | "y = brick.under"
hyperpow_add: fixes r :: "'a star"   and n :: "nat star"   and m :: "nat star" shows "r pow (n + m) = r pow n * r pow m"
oriented_subsegment_scale: fixes x1 :: "'b"   and a :: "'b"   and b :: "'b"   and x2 :: "'b" assumes "x1 \<in> {a<--<b}"   and "x2 \<in> {x1<--<b}" obtains e :: "real" where "0 < e" and "b - a = e *\<^sub>R (x2 - x1)"
assms_2: shows "fst cf \<approx>01 fst df"
sc_a_0: fixes a :: "'b" assumes "a \<in> carrier R" shows "a \<cdot>\<^sub>s \<zero> = \<zero>"
open_p: fixes X :: "'a set" assumes "open X" shows "open (p ` X)"
mapAssoc: fixes S :: "('a, 'b) sequent"   and R :: "('a, 'b) sequent"   and c :: "('a, 'b) sequent list" shows "map (extend S) (map (extend R) c) = map (extend (extend S R)) c"
lemma_2_5_11: fixes a :: "'a"   and b :: "'a" shows "a * b \<le> a \<sqinter> b"
e: shows "(a, f\<^bsup>-1\<^esup> \<bullet> a) \<in> \<E> f'"
set_less_eq_trans: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A \<sqsubseteq> B"   and "B \<sqsubseteq> C" shows "A \<sqsubseteq> C"
function_sum_comp_is_seq_sum: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and f :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int"   and g :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "f \<in> carrier (function_ring (carrier Zp) Zp)"   and "g \<in> carrier (function_ring (carrier Zp) Zp)" shows "f \<oplus>\<^bsub>function_ring (carrier Zp) Zp\<^esub> g \<circ> s = (f \<circ> s) \<oplus>\<^bsub>Zp\<^bsup>\<omega>\<^esup>\<^esub> (g \<circ> s)"
confluentpI': fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y1 y2. \<lbrakk>r\<^sup>*\<^sup>* x y1; r x y2\<rbrakk> \<Longrightarrow> \<exists>z. r\<^sup>*\<^sup>* y1 z \<and> r\<^sup>*\<^sup>* y2 z" shows "confluentp r"
carrier_sum_group: fixes I :: "'b set"   and G :: "'b \<Rightarrow> ('c, 'd) monoid_scheme" assumes "\<And>i. i \<in> I \<Longrightarrow> Group.group (G i)" shows "carrier (sum_group I G) = {x \<in> \<Pi>\<^sub>E i\<in>I. carrier (G i). finite {i \<in> I. x i \<noteq> \<one>\<^bsub>G i\<^esub>}}"
analz_insert_Crypt: fixes K :: "key"   and H :: "msg set"   and X :: "msg" assumes "Key (invKey K) \<notin> analz H" shows "analz (insert (Crypt K X) H) = insert (Crypt K X) (analz H)"
simps_2: fixes x :: "'d"   and e :: "'f"   and h :: "('d \<times> 'f) list"   and eval :: "'f \<Rightarrow> 'e" shows "evalHeap ((x, e) # h) eval = (evalHeap h eval)(x := eval e)"
projr_def: fixes sum :: "'b + 'a" shows "projr sum = (case sum of Inr x2 \<Rightarrow> x2)"
assms_1: shows "(inv_into M (ordermap M r) \<alpha>, inv_into M (ordermap M r) \<beta>) \<in> r"
ODrel_def: shows "ODrel = {Z \<in> ODnums \<times> ODnums. fst Z \<sqsubseteq> snd Z}"
less_le_not_le: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and y :: "'k" assumes "class.preorder less_eq less" shows "less x y = (less_eq x y \<and> \<not> less_eq y x)"
continuous_on_open_gen: fixes f :: "'c \<Rightarrow> 'b"   and S :: "'c set"   and T :: "'b set" assumes "f ` S \<subseteq> T" shows "continuous_on S f = (\<forall>U. openin (top_of_set T) U \<longrightarrow> openin (top_of_set S) (S \<inter> f -` U))"
fS_1: shows "t = \<langle>f S\<rangle>\<^sub>t"
mpj_def: fixes M :: "('e, 'f, 'g) Module_scheme"   and H :: "'e set" shows "mpj M H = (\<lambda>x\<in>carrier M. x \<uplus>\<^bsub>M\<^esub> H)"
vec: fixes x :: "real" assumes "x \<in> {a..b} - S" shows "(f has_vector_derivative f' x) (at x)"
C_eq_RD_aux: fixes p :: "((int \<times> int) set set, int) Combinators"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" shows "C p x = C (removeDuplicates p) x"
iSince_iWeakSince_conv: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t'. t' \<S> t I. Q t) = ((P t'. t' \<B> t I. Q t) \<and> (\<diamond> t I. Q t))"
iso_runit: fixes a :: "'a" assumes "ide a" shows "local.iso \<r>[a]"
permutation_face_cycle_succ: shows "permutation G.face_cycle_succ"
assms_2: shows "f \<in> M' \<rightarrow>\<^sub>M N'"
fun_dual1: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "(f = g \<circ> \<partial>) = (f \<circ> \<partial> = g)"
cutoff_internal: fixes T :: "tree"   and red :: "bool list \<Rightarrow> bool" assumes "anybranch T red" shows "anyinternal (cutoff red [] T) (\<lambda>p. \<not> red p)"
dgrad_p_set_le_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and G :: "('a \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "punit.dgrad_p_set_le d F G"   and "G \<subseteq> punit.dgrad_p_set d m" shows "F \<subseteq> punit.dgrad_p_set d m"
reachable_trans: fixes u :: "'a"   and v :: "'a"   and w :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v"   and "v \<rightarrow>\<^sup>*\<^bsub>G\<^esub> w" shows "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> w"
sigma_finite_measure_def: fixes M :: "'b measure" shows "sigma_finite_measure M \<equiv> \<exists>A. countable A \<and> A \<subseteq> sets M \<and> \<Union> A = space M \<and> (\<forall>a\<in>A. emeasure M a \<noteq> \<infinity>)"
is_monoidal_category: shows "monoidal_category S.comp (\<lambda>\<mu>\<nu>. S.hcomp (fst \<mu>\<nu>) (snd \<mu>\<nu>)) S.\<alpha> \<omega>"
wtE_pickE_pickE: fixes xl :: "Preliminaries.var list"   and al :: "univ list" assumes "length xl = length al"   and "distinct xl"   and "\<And>i. i < length xl \<Longrightarrow> eintT (TE.tpOfV (xl ! i)) (al ! i)" shows "wtE (pickE xl al) \<and> (\<forall>i<length xl. pickE xl al (xl ! i) = al ! i)"
gen_igbg_impl_rel_eext_def_internal: fixes Rm :: "('f \<times> 'h) set"   and Racc :: "('e \<times> ('g \<Rightarrow> nat set)) set" shows "gen_igbg_impl_rel_eext Rm Racc \<equiv> {(\<lparr>igbgi_num_acc = num_acci, igbgi_acc = acci, \<dots> = mi\<rparr>, \<lparr>igbg_num_acc = num_acc, igbg_acc = acc, \<dots> = m\<rparr>) |num_acci acci mi num_acc acc m. (num_acci, num_acc) \<in> nat_rel \<and> (acci, acc) \<in> Racc \<and> (mi, m) \<in> Rm}"
c12_3: shows "lst_conv (fst (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))), snd (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))) @ snd \<delta>3) = lst_conv \<delta>3"
solvable_imp_subgroup: fixes G :: "('a, 'b) monoid_scheme"   and H :: "'a set" assumes "Group.group G"   and "solvable_seq G H" shows "subgroup H G"
that: fixes Df :: "'a \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'b" assumes "\<And>a h. Df a 0 h = f a"   and "\<And>a h i. \<lbrakk>i < n; a \<in> X\<rbrakk> \<Longrightarrow> ((\<lambda>a. Df a i H) has_derivative Df a (Suc i)) (at a)"   and "\<And>a i. \<lbrakk>i \<le> n; a \<in> X\<rbrakk> \<Longrightarrow> Df a i H = nth_derivative i f a H" shows "thesis"
pp_fusion_assoc: fixes x :: "'a ppath"   and y :: "'a ppath"   and z :: "'a ppath" assumes "pp_last x = pp_first y"   and "pp_last y = pp_first z" shows "pp_fusion x (pp_fusion y z) = pp_fusion (pp_fusion x y) z"
incseq_imp_monoseq: fixes X :: "nat \<Rightarrow> 'b" assumes "incseq X" shows "monoseq X"
l14: fixes p :: "'node"   and Q :: "'node set" assumes "quorum_of p Q"   and "p \<in> W" shows "quorum (truncation p Q)"
porder_extend_to_linorder: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "partial_order_on A r" obtains s :: "('a \<times> 'a) set" where "linear_order_on A s" and "order_consistent r s"
lookup_def: shows "Mapping.lookup \<equiv> map_fun mapping.rep id (\<lambda>m. m)"
index_add_vec_2: fixes v\<^sub>1 :: "'b Matrix.vec"   and v\<^sub>2 :: "'b Matrix.vec" shows "dim_vec (v\<^sub>1 + v\<^sub>2) = dim_vec v\<^sub>2"
part2_su: fixes V :: "'a set" assumes "infinite V" shows "part2 V \<subseteq> V"
12376: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - (x \<sqinter> y) = - x"
nat_mult_max_right: fixes m :: "nat"   and n :: "nat"   and q :: "nat" shows "m * max n q = max (m * n) (m * q)"
ts_ord_antisym: fixes x :: "'a"   and y :: "'a" assumes "n x \<sqsubseteq> n y"   and "n y \<sqsubseteq> n x" shows "n x = n y"
Confidentiality_A: fixes A :: "agent"   and Tk :: "nat"   and B :: "agent"   and K :: "nat"   and X :: "msg"   and evs :: "event list" assumes "Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, X\<rbrace> \<in> parts (knows Spy evs)"   and "Notes Spy \<lbrace>Number Tk, Key K\<rbrace> \<notin> set evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> bankerberos" shows "Key K \<notin> analz (knows Spy evs)"
simps: fixes x :: "'a"   and y :: "'b Filters.filter"   and z :: "'a"   and w :: "'b Filters.filter" shows "pairs_inf (x, y) (z, w) = (x \<sqinter> z, y \<squnion> w)"
transitionE_3: fixes P :: "pi"   and u :: "name"   and P'' :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>a<x> \<prec> P'" shows "\<exists>P'''. P \<Longrightarrow>\<^sub>\<tau> P''' \<and> P''' \<longmapsto> a<x> \<prec> P'' \<and> P''[x::=u] \<Longrightarrow>\<^sub>\<tau> P'"
gamma2_def: shows "gamma2 \<equiv> $pc2 = #g \<and> pc2$ = #a \<and> sem$ = Suc<$sem> \<and> Unchanged (x, y, pc1)"
uminus_closed: fixes a :: "'m" assumes "a \<in> U\<^sub>M" shows "-\<^sub>M a \<in> U\<^sub>M"
dimension_zero: fixes K :: "'a set"   and E :: "'a set" assumes "subfield K R"   and "dimension 0 K E" shows "E = {\<zero>}"
fun_dual3: fixes f :: "'b dual set \<Rightarrow> 'c"   and g :: "'b set \<Rightarrow> 'c" shows "(f \<circ> (`) \<partial> = g) = (f = g \<circ> (`) \<partial>\<^sup>-)"
bit_not_int_iff': fixes k :: "int"   and n :: "nat" shows "bit (- k - 1) n = (\<not> bit k n)"
sat_precond_drest_sat_precond: fixes vs :: "'d set"   and s :: "('d, bool) fmap"   and as :: "(('d, bool) fmap \<times> ('d, bool) fmap) list" assumes "sat_precond_as (fmrestrict_set vs s) as" shows "sat_precond_as s as"
inv_5: fixes i :: "nat" assumes "i < m" shows "gs1.gso i \<in> Rn"
bind_eq_Ok_conv: fixes x :: "('k, 'm) result"   and f :: "'m \<Rightarrow> ('k, 'l) result"   and z :: "'l" shows "(x \<bind> f = Ok z) = (\<exists>y. x = Ok y \<and> f y = Ok z)"
genby_empty': fixes w :: "'b" assumes "w \<in> \<langle>{}\<rangle>" shows "w = (0::'b)"
face_cycle_sets_def: shows "G'.face_cycle_sets = G'.face_cycle_set ` arcs G"
splay_max_Leaf_iff: fixes t :: "'b tree" shows "(splay_max t = \<langle>\<rangle>) = (t = \<langle>\<rangle>)"
IDirProds_is_subgroup: fixes G :: "('c, 'd) monoid_scheme"   and S :: "'e \<Rightarrow> 'c set"   and I :: "'e set" assumes "Group.group G"   and "\<Union> (S ` I) \<subseteq> carrier G" shows "subgroup (IDirProds G S I) G"
FunctorCodDom_2: 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"
icd_cs: fixes l :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat" assumes "l icd\<^bsup>\<pi>\<^esup>\<rightarrow> k" shows "cs\<^bsup>\<pi>\<^esup> l = cs\<^bsup>\<pi>\<^esup> k @ [\<pi> l]"
short_count_def: fixes g :: "nat set \<times> nat set set" shows "short_count g = card (short_cycles g k)"
summable_f: fixes z :: "complex" shows "summable (\<lambda>n. f n * z ^ n)"
Norm_def: fixes N :: "'c quasinorm"   and x :: "'c" shows "Norm N x = enn2real (eNorm N x)"
CallThrowParams: fixes es :: "expr list"   and vs :: "val list"   and r :: "nat \<times> char list list"   and es' :: "expr list"   and E :: "char list \<Rightarrow> ty option"   and v :: "val"   and Copt :: "char list option"   and M :: "char list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" assumes "es = map Val vs @ Throw r # es'" shows "P,E \<turnstile> \<langle>Call (Val v) Copt M es,s\<rangle> \<rightarrow> \<langle>Throw r,s\<rangle>"
intro: fixes return :: "int \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> (int \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "(('c \<Rightarrow> int option) \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "('c \<Rightarrow> int option) \<Rightarrow> 'b \<Rightarrow> 'b"   and altc :: "int cset \<Rightarrow> (int \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_state_altc return bind get put altc"   and "memoization return bind get put" shows "nondet_exp return bind get put altc"
bounded_set_imp_lmeasurable: fixes S :: "'a set" assumes "bounded S"   and "S \<in> sets lebesgue" shows "S \<in> lmeasurable"
eucl_less_le_not_le: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and abs :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less" shows "less x y = (less_eq x y \<and> \<not> less_eq y x)"
inner_verts_conv: fixes p :: "('a \<times> 'a) list"   and u :: "'a" shows "G.inner_verts p = butlast (tl (pawalk_verts u p))"
H: shows "\<Delta> @ TVarB Q \<Colon> \<Gamma> \<turnstile> t : T"
axioms_2: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "isomorphic_categories A B" shows "category B"
residue_add_comm: fixes p :: "int"   and k :: "nat"   and x :: "int"   and y :: "int" assumes "padic_integers p" shows "x \<oplus>\<^bsub>residue_ring (p ^ k)\<^esub> y = y \<oplus>\<^bsub>residue_ring (p ^ k)\<^esub> x"
lossless_weight_spmfD: fixes p :: "'l spmf" assumes "lossless_spmf p" shows "weight_spmf p = 1"
Bounded_matD: fixes A :: "'b mat"   and b :: "'b"   and nr :: "nat"   and nc :: "nat"   and i :: "nat"   and j :: "nat" assumes "A \<in> Bounded_mat b"   and "A \<in> carrier_mat nr nc"   and "i < nr"   and "j < nc" shows "\<bar>A $$ (i, j)\<bar> \<le> b"
remdups_rev_removeAll: fixes r :: "'b"   and rs :: "'b list" shows "remdups_rev (removeAll r rs) = removeAll r (remdups_rev rs)"
add_vert_simps_3: fixes u :: "'a" shows "tail (add_vert u) = tail G"
eqOnUC_def: fixes i :: "nat"   and i' :: "nat"   and u :: "'U"   and c :: "'C" shows "eqOnUC i i' u c \<equiv> eq (atom (Last u c) i) (atom (Last u c) i')"
antimonoD: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "class.order less_eq less"   and "order.antimono less_eq f"   and "less_eq x y" shows "f y \<le> f x"
def_hashmap_size_array_def: shows "def_hashmap_size = (\<lambda>_. 10)"
complex_box_ne_empty_2: fixes a :: "complex"   and b :: "complex" shows "(box a b \<noteq> {}) = (Re a < Re b \<and> Im a < Im b)"
assms_1: shows "InvariantVarsQ Q F0 Vbl"
secure_termination_def: fixes I :: "('a \<times> 'a) set"   and D :: "'b option \<Rightarrow> 'a" shows "secure_termination I D \<equiv> \<forall>x. (D x, D None) \<in> I \<and> x \<noteq> None \<longrightarrow> (\<forall>u\<in>range D. (D x, u) \<in> I)"
augment_edge_def: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity" shows "augment_edge f \<equiv> \<lambda>(u, v) \<Delta>. if (u, v) \<in> E then f((u, v) := f (u, v) + \<Delta>) else if (v, u) \<in> E then f((v, u) := f (v, u) - \<Delta>) else f"
bit_push_bit_iff_int: fixes m :: "nat"   and k :: "int"   and n :: "nat" shows "bit (push_bit m k) n = (m \<le> n \<and> bit k (n - m))"
equivp_reflp_symp_transp: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "equivp R = (reflp R \<and> symp R \<and> transp R)"
high_proc_no_low_output_change_s\<^sub>H\<^sub>0: shows "high_proc_no_low_output_change s\<^sub>H\<^sub>0"
map_le_override_bothD: fixes h\<^sub>0' :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option"   and h\<^sub>0 :: "'c \<Rightarrow> 'd option" assumes "h\<^sub>0' ++ h\<^sub>1 \<subseteq>\<^sub>m h\<^sub>0 ++ h\<^sub>1"   and "h\<^sub>0' \<bottom> h\<^sub>1"   and "h\<^sub>0 \<bottom> h\<^sub>1" shows "h\<^sub>0' \<subseteq>\<^sub>m h\<^sub>0"
vertices_residual_network: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" shows "vertex (residual_network f) = vertex \<Delta>"
Diff: fixes A :: "'a set" assumes "A \<in> M" shows "\<Omega> - A \<in> M"
projs_4: fixes dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat"   and pre :: "nat set" shows "\<pi>\<^sub>5 (dsn, dsk, flag, hops, nhip, pre) = hops"
Fail'_def: fixes uu :: "unit" shows "Fail' uu = Fail"
zero_le_divide_abs_iff: fixes a :: "'a"   and b :: "'a" shows "((0::'a) \<le> a / \<bar>b\<bar>) = ((0::'a) \<le> a \<or> b = (0::'a))"
eSubst_simp3: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and X :: "'c"   and ys :: "'varSort"   and eY :: "'c withERR"   and y :: "'var" assumes "\<forall>s. \<not> igWls MOD s X" shows "eSubst MOD ys eY y (OK X) = ERR"
avoid: fixes t :: "'b" assumes "t \<in> T" shows "l t \<noteq> b"
simps: fixes V :: "'e list set"   and M1 :: "('e, 'f, 'g) FSM"   and M2 :: "('e, 'f, 'g) FSM"   and v' :: "('e \<times> 'f) list"   and io :: "('e \<times> 'f) list" shows "minimal_sequence_to_failure_extending V M1 M2 v' io = (v' \<in> L\<^sub>i\<^sub>n M1 V \<and> sequence_to_failure M1 M2 (v' @ io) \<and> (\<nexists>io'. \<exists>w'\<in>L\<^sub>i\<^sub>n M1 V. sequence_to_failure M1 M2 (w' @ io') \<and> length io' < length io))"
concurrent_comm: fixes s1 :: "'a \<times> ('a, 'b) operation"   and s2 :: "'a \<times> ('a, 'b) operation" shows "hb.concurrent s1 s2 = hb.concurrent s2 s1"
integrable_continuous_interval: fixes a :: "'b"   and b :: "'b"   and f :: "'b \<Rightarrow> 'c" assumes "continuous_on {a..b} f" shows "f integrable_on {a..b}"
tfinite: fixes t :: "'a tree" assumes "\<And>t'. t' |\<in>| cont t \<Longrightarrow> tfinite t'" shows "tfinite t"
weight_scale_spmf: fixes r :: "real"   and p :: "'c spmf" shows "weight_spmf (scale_spmf r p) = min 1 (max 0 r * weight_spmf p)"
that: fixes t :: "nat \<Rightarrow> real" assumes "filterlim t at_top sequentially"   and "(f \<circ> t) \<longlonglongrightarrow> l"   and "\<forall>n. b \<le> t n" shows "thesis"