hcju/isabelleps huggingface dataset:
borel_measurable_iff_greater: fixes f :: "'b \<Rightarrow> real"   and M :: "'b measure" shows "(f \<in> borel_measurable M) = (\<forall>a. {w \<in> space M. a < f w} \<in> sets M)"
kreturn: shows "(\<pi> \<guillemotleft> k) (n - k) = local.return"
lt_gr_keys: fixes u :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" assumes "u \<in> keys p"   and "v \<prec>\<^sub>t u" shows "v \<prec>\<^sub>t lt p"
rep_in_hom: shows "\<guillemotleft>trnr\<^sub>\<epsilon> (g \<star> f\<^sup>*) \<rho> : g \<star> f\<^sup>* \<Rightarrow> g \<star> f\<^sup>*\<guillemotright>"
is_normal_def: fixes a :: "('a, 'b) IEEE.float" shows "is_normal a = (0 < IEEE.exponent a \<and> IEEE.exponent a < emax TYPE(('a, 'b) IEEE.float))"
assms_1: shows "f \<longlonglongrightarrow> f0"
rep_eq: shows "blinfun_apply 0 = (\<lambda>x. 0::'b)"
hyps_8: assumes "monomial_decomp (fst (shift_list (h, U) x qs, B - {(h, U)}))" shows "monomial_decomp (fst (shift1 (shift_list (h, U) x qs, B - {(h, U)})))"
corestr_in_hom: fixes f :: "'s" assumes "arr f" shows "\<guillemotleft>corestr f : local.dom f \<rightarrow> img f\<guillemotright>"
hypreal_sqrt_pow2_iff: fixes x :: "real star" shows "(((*f* sqrt) x)\<^sup>2 = x) = (0 \<le> x)"
rep_eq: fixes x :: "'a ttree" shows "paths x = Rep_ttree x"
ptr_kinds_M_ptr_kinds: fixes h :: "'a"   and ptr_kinds :: "'a \<Rightarrow> 'b fset"   and xa :: "'b list" shows "h \<turnstile> l_ptr_kinds_M.a_ptr_kinds_M ptr_kinds \<rightarrow>\<^sub>r xa = (xa = sorted_list_of_set (fset (ptr_kinds h)))"
offs_enum_last: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and n :: "nat"   and xs :: "'c list"   and mi :: "'d"   and ma :: "'d"   and k :: "nat" assumes "index_less index key"   and "0 < n"   and "\<And>x. x \<in> set xs \<Longrightarrow> key x \<in> {mi..ma}" shows "offs (enum xs index key n mi ma) k ! (n - Suc 0) + offs_num n xs index key mi ma (n - Suc 0) = length xs + k"
is_hashmap'_def: fixes m :: "'a \<Rightarrow> 'b option"   and l :: "('a \<times> 'b) list list"   and ht :: "('a, 'b) hashtable" shows "is_hashmap' m l ht = is_hashtable l ht * \<up> (map_of (concat l) = m)"
prems_1: shows "\<O> \<inter> read_only S = {}"
infsetsum_finite: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite A" shows "infsetsum f A = sum f A"
Partition_diag: fixes n :: "nat" shows "Partition n n = 1"
simps_3: fixes v :: "'b heap" shows "pass\<^sub>1 [v] = [v]"
assms_1: shows "valid_mmsaux args cur (ot, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since) auxlist"
assms: fixes f :: "nat \<Rightarrow> 'a" assumes "f 0 \<in> A"   and "\<forall>i. (f i, f (Suc i)) \<in> r" shows "False"
the_all_deriv: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'a \<Rightarrow> 'b" assumes "\<forall>x. (f has_derivative F x) (at x)" shows "(THE G. \<forall>x. (f has_derivative G x) (at x)) = F"
hypnat_le0: fixes n :: "nat star" shows "0 \<le> n"
simps: fixes q :: "'b"   and l :: "'a"   and qs :: "'b list" shows "rhsl q \<rightarrow> l qs = l"
emeasure_lebesgue_ball_conv_unit_ball: fixes r :: "real"   and c :: "'c" assumes "0 \<le> r" shows "emeasure lebesgue (ball c r) = ennreal (r ^ DIM('c)) * emeasure lebesgue (ball (0::'c) 1)"
step_invariantE: fixes A :: "('a, 'b) automaton"   and I :: "'b \<Rightarrow> bool"   and P :: "'a \<times> 'b \<times> 'a \<Rightarrow> bool"   and s :: "'a"   and a :: "'b"   and s' :: "'a" assumes "A \<TTurnstile>\<^sub>A (I \<rightarrow>) P"   and "s \<in> reachable A I"   and "(s, a, s') \<in> automaton.trans A"   and "I a" obtains "P (s, a, s')"
integrate_delete_result: fixes M :: "('c, 'd) message set"   and m :: "'c delete_message"   and s :: "('c, 'd option) insert_message list" assumes "consistent (M \<union> {Delete m})"   and "is_associated_string M s" shows "is_certified_associated_string (M \<union> {Delete m}) (integrate_delete m s)"
rle: shows "r \<le> t * cmod (deriv f a) / 12"
simps_2: fixes e :: "('a, 'b, 'c) env"   and xs :: "'c list" shows "??.Nested_Environment.lookup_option (Some e) xs = lookup e xs"
indpt: fixes c :: "nat \<Rightarrow> 'a" assumes "lincomb_list c (map ((*\<^sub>v) A) ss) = 0\<^sub>v n" shows "\<forall>i<length ss. c i = (0::'a)"
Follows_union_lemma: fixes F :: "'b program"   and f :: "'b \<Rightarrow> 'c multiset"   and g :: "'b \<Rightarrow> 'c multiset"   and g' :: "'b \<Rightarrow> 'd"   and f' :: "'b \<Rightarrow> 'c multiset"   and k :: "'c multiset" assumes "F \<in> Increasing f"   and "F \<in> Increasing g"   and "F \<in> Increasing g'"   and "F \<in> Always {s. f' s \<le> f s}"   and "\<forall>k. F \<in> {s. k \<le> f s} \<longmapsto>w {s. k \<le> f' s}" shows "F \<in> {s. k \<le> f s + g s} \<longmapsto>w {s. k \<le> f' s + g s}"
cone_determines_arrow_1: fixes d :: "('b \<Rightarrow> 'a, 'b \<Rightarrow> 'a) concrete_category.arr"   and a :: "'a"   and \<chi> :: "'b \<Rightarrow> 'a" assumes "J_C.ide d"   and "Limit.cone J (\<cdot>) (J_C.Map d) a \<chi>" shows "arrow_from_functor (\<cdot>) (\<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>]) \<Delta>.map a d (J_C.MkArr (constant_functor.map J (\<cdot>) a) (J_C.Map d) \<chi>)"
row_vec_def: fixes v :: "complex vec" shows "row_vec v \<equiv> mat 1 (dim_vec v) (\<lambda>(i, j). v $ j)"
iterate_additive: fixes t\<^sub>1 :: "nat"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and t\<^sub>2 :: "nat"   and z :: "'b" assumes "iterate t\<^sub>1 f x = y"   and "iterate t\<^sub>2 f y = z" shows "iterate (t\<^sub>1 + t\<^sub>2) f x = z"
Rep_bit_iff_odd: fixes b :: "bit" shows "Rep_bit b = odd b"
step: shows "lnth Ns i \<leadsto>GC lnth Ns (Suc i)"
field_differentiable_at_tan: fixes z :: "'b" assumes "cos z \<noteq> (0::'b)" shows "tan field_differentiable at z"
deadlock1_imp_\<tau>s_deadlock2: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and bisim_wait :: "'a \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves"   and s1 :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and s2 :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('g \<times> 'd \<Rightarrow>f nat) option) \<times> 'h) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and convert_RA :: "'d \<Rightarrow>f nat \<Rightarrow> 'f list" assumes "FWdelay_bisimulation_diverge final1 r1 final2 r2 bisim bisim_wait \<tau>move1 \<tau>move2"   and "FWbisimulation_base.mbisim bisim bisim_wait s1 s2"   and "multithreaded_base.deadlock final1 r1 s1" shows "\<exists>s2'. (\<tau>trsys.silent_move (multithreaded_base.redT final2 r2 convert_RA) (\<tau>multithreaded.m\<tau>move \<tau>move2))\<^sup>*\<^sup>* s2 s2' \<and> multithreaded_base.deadlock final2 r2 s2' \<and> FWbisimulation_base.mbisim bisim bisim_wait s1 s2'"
A: shows "\<tau> \<Turnstile> H .allInstances()->includes\<^sub>S\<^sub>e\<^sub>t(x)"
concat_conv_foldr: fixes xss :: "'b list list" shows "concat xss = foldr (@) xss []"
ring_axioms: shows "ring stx.carrier_direct_lim stx.add_rel stx.mult_rel (stx.class_of X (zero_str\<^sub>X X)) (stx.class_of X (one_str\<^sub>X X))"
distinct_63: fixes x6 :: "'a OCL_Types.type"   and x8 :: "'a OCL_Types.type" shows "OrderedSet x6 \<noteq> Sequence x8"
except_eq_zeroE: fixes p :: "'b \<Rightarrow>\<^sub>0 'c"   and S :: "'b set" assumes "except p S = 0" shows "keys p \<subseteq> S"
stable_ordIso2: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "stable r"   and "r =o r'" shows "stable r'"
orthogonal_complement_def: fixes W :: "'b set" shows "orthogonal_complement W = {x. \<forall>y\<in>W. orthogonal x y}"
ordinal_le_plusL: fixes y :: "ordinal"   and x :: "ordinal" shows "y \<le> x + y"
bounded_WHILE_shift: fixes m :: "'c nres"   and cond :: "'c \<Rightarrow> bool"   and n :: "nat"   and step :: "'c \<Rightarrow> 'c nres" shows "m \<bind> (\<lambda>x. if cond x then bounded_WHILE n cond step (step x) else RETURN x) = bounded_WHILE n cond step m \<bind> (\<lambda>x. if cond x then step x else RETURN x)"
a_rcosI: fixes h :: "'c"   and H :: "'c set"   and x :: "'c" assumes "h \<in> H"   and "H \<subseteq> carrier S"   and "x \<in> carrier S" shows "h \<oplus>\<^bsub>S\<^esub> x \<in> H +>\<^bsub>S\<^esub> x"
induced_automorphism_selfcomp_halftrivial_f: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "fixespointwise (OpposedThinChamberComplexFoldings.induced_automorphism X f g \<circ> OpposedThinChamberComplexFoldings.induced_automorphism X f g) (\<Union> (f \<turnstile> X))"
restrict_ext: fixes A :: "'h set"   and f :: "'h \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'i" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "restrict f A = restrict g A"
assms: shows "as1 @ as2 @ as3 = as_proj p vs"
visits_via_LCons: fixes P :: "'c llist"   and A :: "'c set"   and W :: "'c set"   and v0 :: "'c" assumes "visits_via P A W" shows "visits_via (LCons v0 P) (insert v0 A) W"
oalist_inv_map_val_raw: fixes xs :: "('a \<times> 'e) list \<times> 'o"   and f :: "'a \<Rightarrow> 'e \<Rightarrow> 'f" assumes "oalist_inv xs" shows "oalist_inv (map_val_raw f xs)"
INT_I: fixes A :: "'k set"   and b :: "'l"   and B :: "'k \<Rightarrow> 'l set" assumes "\<And>x. x \<in> A \<Longrightarrow> b \<in> B x" shows "b \<in> \<Inter> (B ` A)"
loc1_a_s_2: shows "s \<in>\<^sub>A path_summary loc1 loc2"
cases: fixes a1 :: "'a sequent list \<times> 'a sequent"   and a2 :: "'a form" assumes "leftPrincipal a1 a2" obtains C :: "'a sequent" and F :: "'a" and Fs :: "'a form list" and Ps :: "'a sequent list" where "a1 = (Ps, C)" and "a2 = Compound F Fs" and "C = ( \<LM> Compound F Fs  \<RM> \<Rightarrow>* \<Empt>)"
Hoelder_Fourier_convergence_periodic: fixes f :: "real \<Rightarrow> real"   and d :: "real"   and a :: "real"   and t :: "real"   and M :: "real" assumes "f absolutely_integrable_on {- pi..pi}"   and "0 < d"   and "0 < a"   and "\<And>x. \<bar>x - t\<bar> < d \<Longrightarrow> \<bar>f x - f t\<bar> \<le> M * \<bar>x - t\<bar> powr a"   and "\<And>x. f (x + 2 * pi) = f x" shows "(\<lambda>n. \<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k t) \<longlonglongrightarrow> f t"
\<zeta>q: shows "\<zeta> \<notin> path_image q"
self_mult_gauss_cnj: fixes z :: "gauss_int" shows "z * gauss_cnj z = of_nat (gauss_int_norm z)"
cSUP_least: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c multiset"   and M :: "'c multiset" assumes "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq># M" shows "Sup (f ` A) \<subseteq># M"
shiftr_mask_eq: fixes x :: "'b word"   and n :: "nat" shows "(x >> n) AND mask (size x - n) = x >> n"
bi_uniqueI: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "left_unique R"   and "right_unique R" shows "bi_unique R"
LeadsTo_weaken_R: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B' :: "'c set" assumes "F \<in> A \<longmapsto>w A'"   and "A' \<subseteq> B'" shows "F \<in> A \<longmapsto>w B'"
winning_strategy_updates: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and v0 :: "'a"   and v :: "'a"   and w :: "'a" assumes "G'.strategy p \<sigma>"   and "G'.winning_strategy p \<sigma> v0"   and "v \<notin> G'.winning_region p"   and "v \<rightarrow>\<^bsub>G'\<^esub> w" shows "G'.winning_strategy p (\<sigma>(v := w)) v0"
zero_right: fixes prod :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and a :: "'d" assumes "bounded_cbilinear prod" shows "prod a (0::'e) = (0::'f)"
extend_set_def: fixes h :: "'e \<times> 'f \<Rightarrow> 'd"   and A :: "'e set" shows "extend_set h A = h ` (A \<times> UNIV)"
arr_cod: fixes f :: "'a" assumes "B.arr f" shows "B.arr (B.cod f)"
div_const_poly: fixes p :: "'b poly"   and c :: "'b" shows "p div [:c:] = smult (inverse c) p"
proj_fst_k_is_projection: fixes k :: "nat" shows "proj_fst_k k * proj_fst_k k = proj_fst_k k"
call_mono: fixes ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "'a" shows "monotone (fun_ord ord) ord (\<lambda>f. f t)"
le_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and e\<^sub>1 :: "'b err"   and e\<^sub>2 :: "'b err" shows "Err.le r e\<^sub>1 e\<^sub>2 = (case e\<^sub>2 of Err \<Rightarrow> True | OK y \<Rightarrow> case e\<^sub>1 of Err \<Rightarrow> False | OK x \<Rightarrow> x \<sqsubseteq>\<^bsub>r\<^esub> y)"
lfilter_idem: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "lfilter P (lfilter P xs) = lfilter P xs"
assms_3: fixes I :: "'b"   and a :: "'a"   and b :: "'a"   and J :: "'b"   and c :: "'a"   and d :: "'a" assumes "Q I a b"   and "Q J c d"   and "[[a c d b]]" shows "P I J"
card_idx_nat': fixes \<tau> :: "'b"   and satisfies :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and transition :: "'c \<Rightarrow> ('b, 'c) residual \<Rightarrow> bool" assumes "indexed_weak_nominal_ts TYPE('a) \<tau> satisfies transition" shows "|UNIV| <o natLeq +c |UNIV|"
hyps_3: assumes "p \<in> P[X]" shows "P p"
Union_image_insert: fixes f :: "'c \<Rightarrow> 'b set"   and a :: "'c"   and B :: "'c set" shows "\<Union> (f ` insert a B) = f a \<union> \<Union> (f ` B)"
inv_ide: fixes a :: "'b" assumes "D.ide a" shows "D.inv a = a"
EX_MR_aux: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "applied_rule_rev Cp x (DenyAll # p) \<noteq> \<lfloor>DenyAll\<rfloor>" shows "\<exists>y. applied_rule_rev Cp x p = \<lfloor>y\<rfloor>"
iso_\<gamma>\<^sub>\<tau>: shows "local.iso \<gamma>\<^sub>\<tau>"
Zero_fract_def: shows "0 = Fraction_Field.Fract (0::'a) (1::'a)"
relax_outgoing_def: fixes u :: "'v"   and D :: "'v \<Rightarrow> enat" shows "relax_outgoing u D \<equiv> \<lambda>v. min (D v) (D u + w (u, v))"
\<tau>diverge_neq_wfP_silent_move_from: fixes t :: "'t"   and s :: "'x \<times> 'm" shows "\<tau>diverge t s \<noteq> wfP (flip (silent_move_from t s))"
know2knows_neg_k: fixes A :: "specID"   and m :: "Keys" assumes "\<not> know A (kKS m)" shows "\<not> knows A [kE m]"
lappend_Lazy_llist: fixes xs :: "unit \<Rightarrow> ('a \<times> 'a llist) option"   and ys :: "'a llist" shows "lappend (Lazy_llist xs) ys = Lazy_llist (\<lambda>_. case xs () of None \<Rightarrow> force ys | Some (x, xs') \<Rightarrow> Some (x, lappend xs' ys))"
distinct_13: fixes x21 :: "'c"   and x22 :: "'c \<Rightarrow> ('a, 'c) trm"   and x3 :: "('a, 'b, 'c) formula" shows "$\<phi> x21 x22 \<noteq> ! x3"
h2: shows "\<forall>x. (\<forall>i. P i x) \<longrightarrow> Q x"
span_span: fixes A :: "'b set" shows "span (span A) = span A"
homotopic_with_def: fixes P :: "('d \<Rightarrow> 'e) \<Rightarrow> bool"   and X :: "'d topology"   and Y :: "'e topology"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" shows "homotopic_with P X Y f g \<equiv> \<exists>h. continuous_map (prod_topology (top_of_set {0..1}) X) Y h \<and> (\<forall>x. h (0, x) = f x) \<and> (\<forall>x. h (1, x) = g x) \<and> (\<forall>t\<in>{0..1}. P (\<lambda>x. h (t, x)))"
assms_1: fixes f :: "three \<Rightarrow> three \<Rightarrow> three"   and h :: "nat \<Rightarrow> three" assumes "associative f" shows "map (foldl1 f \<circ> map h) jss = scanl1 f (map h [0..<n + 1])"
af_append_congruent: fixes \<phi> :: "'a ltln"   and w :: "'a set list"   and \<psi> :: "'a ltln"   and w' :: "'a set list" assumes "af \<phi> w \<sim> af \<psi> w" shows "af \<phi> (w @ w') \<sim> af \<psi> (w @ w')"
pythagorean_theorem: fixes x :: "'b"   and y :: "'b" assumes "is_orthogonal x y" shows "(norm (x + y))\<^sup>2 = (norm x)\<^sup>2 + (norm y)\<^sup>2"
Uhyp_2: fixes s :: "'a"   and t :: "real" assumes "s \<in> S"   and "t \<in> U s" shows "t\<^sub>0 \<le> t"
complex_half: fixes a :: "real"   and b :: "real" shows "Complex a b / 2 = Complex (a / 2) (b / 2)"
e: shows "eff (shd (trim (sdrop n rs) s')) s' sl''"
inject: fixes x21 :: "'e"   and x22 :: "'e list"   and y21 :: "'e"   and y22 :: "'e list" shows "(x21 # x22 = y21 # y22) = (x21 = y21 \<and> x22 = y22)"
weaken_local: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and I :: "'e set"   and \<sigma> :: "'e \<Rightarrow> 'd"   and \<sigma>' :: "'e \<Rightarrow> 'd"   and Q :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "other P I \<sigma> \<sigma>'"   and "\<And>\<xi> \<xi>'. P \<xi> \<xi>' \<Longrightarrow> Q \<xi> \<xi>'" shows "other Q I \<sigma> \<sigma>'"
mprod_expR_def: fixes R :: "('d, 'e) Ring_scheme"   and e :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'd"   and n :: "nat" shows "mprod_expR R e f n = e\<Pi>\<^bsub>R,n\<^esub> (\<lambda>j. f j^\<^bsup>R e j\<^esup>)"
get_owner_document_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "h \<turnstile> ok Shadow_DOM.get_owner_document ptr" shows "ptr |\<in>| object_ptr_kinds h"
wf_exec: shows "P \<turnstile> (E, ws) \<surd>"
inject: fixes x1 :: "real"   and y1 :: "real" shows "(real_min_top.R x1 = real_min_top.R y1) = (x1 = y1)"
cod_in_codomains: fixes f :: "(bool, 'arr list) concrete_category.arr" assumes "codomains f \<noteq> {}" shows "cod f \<in> codomains f"
inj_on_filter_key_eq: fixes f :: "'k \<Rightarrow> 'l"   and y :: "'k"   and xs :: "'k list" assumes "inj_on f (insert y (set xs))" shows "filter (\<lambda>x. f y = f x) xs = filter ((=) y) xs"
not_gr_zero_hmset: fixes M :: "hmultiset" shows "(\<not> 0 < M) = (M = 0)"
cases: fixes a :: "'b formula" assumes "a \<in> AX10" obtains F :: "'b formula" where "a = Fb__" and "Fb__ \<in> AX0"   | F :: "'b formula" and G :: "'b formula" where "a = Fc__ \<^bold>\<rightarrow> Fc__ \<^bold>\<or> Ga__"   | G :: "'b formula" and F :: "'b formula" where "a = Gb__ \<^bold>\<rightarrow> Fd__ \<^bold>\<or> Gb__"   | F :: "'b formula" and H :: "'b formula" and G :: "'b formula" where "a = (Fe__ \<^bold>\<rightarrow> H) \<^bold>\<rightarrow> (Gc__ \<^bold>\<rightarrow> H) \<^bold>\<rightarrow> (Fe__ \<^bold>\<or> Gc__) \<^bold>\<rightarrow> H"   | F :: "'b formula" and G :: "'b formula" where "a = (Ff__ \<^bold>\<and> Gd__) \<^bold>\<rightarrow> Ff__"   | F :: "'b formula" and G :: "'b formula" where "a = (Fg__ \<^bold>\<and> Ge__) \<^bold>\<rightarrow> Ge__"   | F :: "'b formula" and G :: "'b formula" where "a = Fh__ \<^bold>\<rightarrow> G \<^bold>\<rightarrow> Fh__ \<^bold>\<and> G"   | F :: "'b formula" where "a = (Fi__ \<^bold>\<rightarrow> \<bottom>) \<^bold>\<rightarrow> \<^bold>\<not> Fi__"   | F :: "'b formula" where "a = \<^bold>\<not> Fj__ \<^bold>\<rightarrow> Fj__ \<^bold>\<rightarrow> \<bottom>"   | F :: "'b formula" where "a = (\<^bold>\<not> F \<^bold>\<rightarrow> \<bottom>) \<^bold>\<rightarrow> F"
rcoset_rel_def: fixes G :: "'a set"   and H :: "'a set" assumes "Group G" shows "Group.rcoset_rel G H \<equiv> {(g, g'). g \<in> G \<and> g' \<in> G \<and> g - g' \<in> H}"
A1'_def: shows "A1' \<equiv> \<lambda>view (m0, m1). coin_spmf \<bind> (\<lambda>S2. coin_spmf \<bind> (\<lambda>S3. coin_spmf \<bind> (\<lambda>S4. coin_spmf \<bind> (\<lambda>S5. S1_OT12 (S2, S3) () \<bind> (\<lambda>a. R1_OT12 (S4, S5) c1 \<bind> (\<lambda>c. Let (M, (S2, S3, m0, m1, S4, S5), a, view, c) D))))))"
closest_point_def: fixes S :: "'a set"   and a :: "'a" shows "closest_point S a = (SOME x. x \<in> S \<and> (\<forall>y\<in>S. dist a x \<le> dist a y))"
holomorphic_factor_zero_nonconstant: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and \<xi> :: "complex" assumes "f holomorphic_on S"   and "open S"   and "connected S"   and "\<xi> \<in> S"   and "f \<xi> = 0"   and "\<not> f constant_on S" obtains g :: "complex \<Rightarrow> complex" and r :: "real" and n :: "nat" where "0 < n" and "0 < r" and "ball \<xi> r \<subseteq> S" and "g holomorphic_on ball \<xi> r" and "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w = (w - \<xi>) ^ n * g w" and "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> g w \<noteq> 0"
owner_seskey_prikey: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and m :: "nat"   and SK :: "nat option \<times> nat set \<times> nat set"   and B :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "n \<notin> bad_prikey"   and "(Owner m, SesKey SK) \<in> s"   and "(Asset n, \<lbrace>Num 2, PubKey B\<rbrace>) \<in> s"   and "B \<in> fst (snd SK)" shows "SesKey SK \<notin> spied s"
mult_mat_vec_hom: fixes A :: "real mat"   and nr :: "nat"   and n :: "nat"   and v :: "real Matrix.vec" assumes "A \<in> carrier_mat nr n"   and "v \<in> carrier_vec n" shows "of_real_hom.vec_hom (A *\<^sub>v v) = of_real_hom.mat_hom A *\<^sub>v of_real_hom.vec_hom v"
Rep_vector_inverse: fixes x :: "'b vector" shows "Abs_vector (Rep_vector x) = x"
algebraic_inverse_iff: fixes y :: "'b" shows "algebraic (inverse y) = algebraic y"
fls_lr_inverse_starting0_1: fixes f :: "'a fls" shows "fls_left_inverse f (0::'a) = 0"
VSLuckyBlocks: fixes \<phi> :: "atom fm" shows "\<forall>xs. eval (VSLuckyBlocks \<phi>) xs = eval \<phi> xs"
tendsto_Re: fixes g :: "'a \<Rightarrow> octo"   and a :: "octo"   and F :: "'a filter" assumes "(g \<longlongrightarrow> a) F" shows "((\<lambda>x. Ree (g x)) \<longlongrightarrow> Ree a) F"
Plus_idem_assoc: fixes r :: "'a language"   and s :: "'a language" shows "Plus r (Plus r s) = Plus r s"
is_Sup_Join: fixes A :: "'b set" shows "is_Sup A (\<Squnion>A)"
simps_1: fixes xs :: "'c llist" shows "lnth xs 0 = (case xs of LNil \<Rightarrow> undefined 0 | LCons x xs' \<Rightarrow> x)"
vector_star_1: fixes x :: "'a" assumes "vector x" shows "x\<^sup>T * (x * x\<^sup>T)\<^sup>\<star> \<le> x\<^sup>T"
enabled_imp_alw: fixes X :: "'s set"   and x :: "'s"   and \<omega> :: "'s stream" assumes "(\<Union>s\<in>X. set_pmf (K s)) \<subseteq> X"   and "x \<in> X"   and "enabled x \<omega>" shows "alw (HLD X) \<omega>"
domD: fixes a :: "'a"   and m :: "'a \<Rightarrow> 'b option" assumes "a \<in> dom m" shows "\<exists>b. m a = \<lfloor>b\<rfloor>"
basic_just_sp_pred_axioms_def: fixes G :: "('a, 'b) pre_digraph"   and dist :: "'a \<Rightarrow> ereal"   and c :: "'b \<Rightarrow> real"   and s :: "'a"   and num :: "'a \<Rightarrow> enat"   and pred :: "'a \<Rightarrow> 'b option" shows "basic_just_sp_pred_axioms G dist c s num pred \<equiv> \<forall>v. v \<in> verts G \<longrightarrow> v \<noteq> s \<longrightarrow> num v \<noteq> \<infinity> \<longrightarrow> (\<exists>e\<in>arcs G. e = the (pred v) \<and> v = head G e \<and> dist v = dist (tail G e) + ereal (c e) \<and> num v = num (tail G e) + enat 1)"
sqUn_lem: fixes A :: "'f set option"   and A' :: "'f set option"   and B :: "'f set option" assumes "A \<sqsubseteq> A'" shows "A \<squnion> B \<sqsubseteq> A' \<squnion> B"
zero_stalk_def: fixes V :: "'c set" shows "stfx.zero_stalk V \<equiv> stfx.class_of V (zero_str\<^sub>Y V)"
digit_decode_lt: fixes "is" :: "nat list"   and ds :: "nat list" assumes "is \<lhd> ds" shows "digit_decode ds is < prod_list ds"
p_LIMSEQ: fixes x :: "real" assumes "0 < x" shows "(\<lambda>n. p n x) \<longlonglongrightarrow> P x"
token_is_not_in_sink: fixes x :: "nat"   and n :: "nat"   and A :: "'b set" assumes "token_run x n \<notin> A"   and "token_run x (Suc n) \<in> A" shows "\<not> sink (token_run x n)"
symKeys_def: shows "symKeys \<equiv> {K. invKey K = K}"
assms_3: shows "seqll i P ((\<sigma>\<^sub>1', s), a, \<sigma>\<^sub>2', s')"
chamber_map: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and C :: "'b set" assumes "ChamberComplexRetraction X f"   and "SimplicialComplex.maxsimp X C" shows "SimplicialComplex.maxsimp X (f ` C)"
gso_cong: fixes x :: "nat"   and f1 :: "rat vec list"   and f2 :: "rat vec list" assumes "\<And>i. i \<le> x \<Longrightarrow> f1 ! i = f2 ! i" shows "gram_schmidt_fs.gso n f1 x = gram_schmidt_fs.gso n f2 x"
group_axioms_def: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and inverse :: "'c \<Rightarrow> 'c" shows "Groups.group_axioms f z inverse \<equiv> (\<forall>a. f z a = a) \<and> (\<forall>a. f (inverse a) a = z)"