hcju/isabelleps huggingface dataset:
continuous_fst: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c \<times> 'd" assumes "continuous F f" shows "continuous F (\<lambda>x. fst (f x))"
poly_tendsto_1: fixes p :: "real poly"   and x :: "real" shows "poly p \<midarrow>x\<rightarrow> poly p x"
MultiAppAux: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and k :: "nat"   and l1 :: "int"   and r :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and n :: "nat"   and l2 :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" assumes "(M, l, s, k, l1, r) \<in> MStep"   and "(M, l1, r, n, l2, t) \<in> Step" shows "(M, l, s, Suc k + n, l2, t) \<in> MStep"
bin_rsplit_aux_len_indep: fixes n :: "nat"   and bs :: "int list"   and cs :: "int list"   and nw :: "nat"   and v :: "int"   and w :: "int" assumes "n \<noteq> 0"   and "length bs = length cs" shows "length (bin_rsplit_aux n nw v bs) = length (bin_rsplit_aux n nw w cs)"
isMaxim_def: fixes A :: "'b set"   and b :: "'b" shows "s.isMaxim A b \<equiv> b \<in> A \<and> (\<forall>a\<in>A. (a, b) \<in> s)"
GcdI: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and A :: "'a set"   and b :: "'a" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize"   and "\<And>a. a \<in> A \<Longrightarrow> dvd.dvd times b a"   and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> dvd.dvd times c a) \<Longrightarrow> dvd.dvd times c b"   and "normalize b = b" shows "b = Gcd A"
P1: shows "Bet B A Y \<and> Cong A Y A C"
char_rv: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and A :: "'b set" assumes "measure_space M"   and "A \<in> measurable_sets M" shows "\<chi> A \<in> rv M"
set_plus_imp_minus: fixes a :: "'a"   and b :: "'a"   and C :: "'a set" assumes "a \<in> b +o C" shows "a - b \<in> C"
lunit_simp: fixes a :: "'c" assumes "C.ide a" shows "C'.lunit a = \<r>\<^sub>C[a]"
IH: shows "\<forall>m<n. \<forall>\<Gamma> \<delta>. ( \<Gamma> \<oplus> Compound F Fs \<Rightarrow>* \<delta>, m) \<in> derivable R* \<longrightarrow> (\<forall>r'\<in>R. leftPrincipal r' (Compound F Fs) \<longrightarrow> ( \<Gamma>' \<Rightarrow>* Em) \<in> set (fst r')) \<longrightarrow> (\<exists>m'\<le>m. ( \<Gamma> + \<Gamma>' \<Rightarrow>* \<delta>, m') \<in> derivable R*)"
pos_divide_le_eq: fixes c :: "'a"   and b :: "'a"   and a :: "'a" assumes "(0::'a) < c" shows "(b / c \<le> a) = (b \<le> a * c)"
bounded_countable_network_def: fixes \<Delta> :: "('a, 'b) network_scheme" shows "bounded_countable_network \<Delta> \<equiv> countable_network \<Delta> \<and> bounded_countable_network_axioms \<Delta>"
assignment_rhs_append_subset_2: fixes S' :: "('a, 'b) strand_step list"   and S :: "('a, 'b) strand_step list" shows "assignment_rhs\<^sub>s\<^sub>t S' \<subseteq> assignment_rhs\<^sub>s\<^sub>t (S @ S')"
assms: shows "\<forall>x. convex (at_least_as_good x carrier relation)"
step_def: shows "step = Step (\<lambda>x e S. {s(x := aval e s) |s. s \<in> S}) (\<lambda>b S. {s \<in> S. bval b s})"
\<mu>: fixes n :: "nat" shows "real_distribution (\<mu> n)"
UNION_fun_upd: fixes A :: "'d \<Rightarrow> 'c set"   and i :: "'d"   and B :: "'c set"   and J :: "'d set" shows "\<Union> (A(i := B) ` J) = \<Union> (A ` (J - {i})) \<union> (if i \<in> J then B else {})"
hyps: shows "y \<le> y \<cdot> x\<^bsup>Suc n\<^esup> + z \<cdot> x\<^sup>\<star>"
associativitySym: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and \<Psi> :: "'d"   and \<Psi>' :: "'d"   and \<Psi>'' :: "'d" assumes "assertion SCompose SImp SBottom SChanEq" shows "assertionAux.AssertionStatEq SImp (SCompose (SCompose \<Psi> \<Psi>') \<Psi>'') (SCompose (SCompose \<Psi> \<Psi>'') \<Psi>')"
share_all_until_volatile_write_Write_commute: fixes i :: "nat"   and ls :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and p :: "'b"   and volatile :: "bool"   and a :: "nat"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and \<D> :: "'c"   and \<O> :: "'d \<times> 'e"   and t :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and \<D>' :: "'c"   and S :: "nat \<Rightarrow> bool option" assumes "i < length ls"   and "ls ! i = (p, Write volatile a (D, f) A L R W # is, \<theta>, sb, \<D>, \<O>)" shows "share_all_until_volatile_write (ls[i := (p, is, \<theta>, sb @ [Write\<^sub>s\<^sub>b volatile a t (f \<theta>) A L R W], \<D>', \<O>)]) S = share_all_until_volatile_write ls S"
\<alpha>_def: shows "\<alpha> = \<Union>\<^sub>\<circ> (set {\<gamma>. Limit \<gamma> \<and> \<gamma> \<subseteq>\<^sub>\<circ> \<alpha>}) + \<gamma>"
invertible_matrix'_imp_invertible_lf: fixes X :: "(('a, 'b) vec, 'b) vec"   and Y :: "(('a, 'b) vec, 'b) vec"   and f :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec" assumes "invertible (matrix' X Y f)"   and "is_basis (set_of_vector X)"   and "linear (*s) (*s) f"   and "is_basis (set_of_vector Y)" shows "invertible_lf (*s) (*s) f"
MST_valid_def: shows "MST_valid = (\<forall>C m par code l0 T MI Anno. mbody_is C m (par, code, l0) \<longrightarrow> MST\<down>(C, m) = Some (T, MI, Anno) \<longrightarrow>  \<Turnstile> \<lbrace> (\<lambda>s0 s. s = mkState s0) \<rbrace> C , m , l0 \<lbrace> mkPost T \<rbrace> mkInv MI)"
StaysEnabled: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and start :: "'a \<Rightarrow> 'b"   and exec :: "('a, 'c, 'b) configuration list"   and trace :: "('a, 'c) message list"   and cfg :: "('a, 'c, 'b) configuration"   and msg :: "('a, 'c) message"   and n :: "nat" assumes "execution trans sends start exec trace"   and "enabled cfg msg"   and "cfg = exec ! n"   and "n < length exec" shows "enabled (last exec) msg \<or> (\<exists>i\<ge>n. i < length exec - 1 \<and> msg = trace ! i)"
wall_crossings_min_gallery_betwI: fixes C :: "'a set"   and Cs :: "'a set list"   and D :: "'a set" assumes "gallery (C # Cs @ [D])"   and "distinct (wall_crossings (C # Cs @ [D]))"   and "\<forall>H\<in>set (wall_crossings (C # Cs @ [D])). separated_by H C D" shows "min_gallery (C # Cs @ [D])"
set_antichain: fixes x :: "'b antichain" shows "set_antichain x \<in> {A. finite A \<and> incomparable A}"
rep_eq: fixes x :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xa :: "('b, 'c) mapping"   and xb :: "('b, 'c) mapping" shows "rep (combine x xa xb) = (\<lambda>xc. combine_options x (rep xa xc) (rep xb xc))"
gtr_find_thread: fixes c :: "'f multiset"   and e :: "'g"   and c' :: "'h multiset"   and gtrs :: "(('f \<times> 'f multiset) \<times> 'g \<times> 'h \<times> 'h multiset) set" assumes "(c, e, c') \<in> gtr gtrs" obtains s :: "'f" and ce :: "'f multiset" and s' :: "'h" and ce' :: "'h multiset" where "c = add_mset s ce" and "c' = add_mset s' ce'" and "((s, ce), e, s', ce') \<in> gtrs"
glb_dual_lub: fixes S :: "'b set"   and cl :: "'b potype" shows "glb S cl = lub S (dual cl)"
poly_split_def: fixes m :: "'c monom"   and p :: "('c monom \<times> 'b) list" shows "poly_split m p \<equiv> case List.extract (\<lambda>(n, uu_). m = n) p of None \<Rightarrow> (0::'b, p) | Some (p1, (x, c), p2) \<Rightarrow> (c, p1 @ p2)"
cnf_of_encode_problem_structure_3: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "cnf (encode_operators \<Pi> t) \<subseteq> cnf (\<Phi> \<Pi> t)"
assms_1: shows "Lower_Semicontinuous.convex_on UNIV f"
strict_trans1: fixes b :: "'b DBMEntry"   and a :: "'b DBMEntry"   and c :: "'b DBMEntry" assumes "b \<preceq> a"   and "c \<prec> b" shows "c \<prec> a"
n: shows "nrows_iarray (IArray (map (vec_to_iarray \<circ> ($) A \<circ> mod_type_class.from_nat) [0..<CARD('b)])) = CARD('b)"
FL_valid_Not: fixes P :: "'state"   and x :: "('a, 'pred, 'act, 'effect) formula" shows "P \<Turnstile> FL_Formula.Not x = (\<not> P \<Turnstile> x)"
scale_left_imp_eq: fixes a :: "'d"   and x :: "'d"   and y :: "'d" assumes "a \<noteq> (0::'d)"   and "a * x = a * y" shows "x = y"
iT_Plus_Mult_distrib: fixes I :: "nat set"   and n :: "nat"   and m :: "nat" shows "I \<oplus> n \<otimes> m = I \<otimes> m \<oplus> n * m"
ide_mkObj: fixes a :: "'a" assumes "C.ide a" shows "ide (mkObj a)"
While: fixes I :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and b :: "bexp"   and c :: "com" assumes "\<turnstile>\<^sub>2 {\<lambda>s. I s + \<up> (bval b s)} c {\<lambda>t. I t + 1}" shows "\<turnstile>\<^sub>2 {\<lambda>s. I s + 1} WHILE b DO c {\<lambda>s. I s + \<up> (\<not> bval b s)}"
mat_default_def: fixes zero :: "'b"   and d :: "'b"   and n :: "nat" shows "zero.mat_default zero d n = mat n n (\<lambda>(i, j). if i = j then d else zero)"
freess_empty: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.freess frees [] = {||}"
map_ident: fixes t :: "'b stream" shows "smap (\<lambda>x. x) t = t"
of_int_max: fixes 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 times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "int"   and y :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "ring_1.of_int one plus zero minus (max x y) = ord.max less_eq (ring_1.of_int one plus zero minus x) (ring_1.of_int one plus zero minus y)"
fst_snd_lens_indep: shows "fst\<^sub>L \<bowtie> snd\<^sub>L"
Suc_length_conv: fixes n :: "nat"   and xs :: "'d list" shows "(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
intro: fixes ops0 :: "('a, 'b, 'c) set_ops_scheme"   and ops1 :: "('d, 'e, 'f) set_ops_scheme"   and ops2 :: "('d, 'g, 'h) set_ops_scheme" assumes "StdSet ops0"   and "g_set_xx_loc ops1 ops2" shows "g_set_xyy_loc ops0 ops1 ops2"
not_Interp_to_interp_imp_less: fixes A :: "'a"   and C :: "'a literal multiset"   and D :: "'a literal multiset" assumes "A \<notin> Interp C"   and "A \<in> interp D" shows "C < D"
cases: fixes a1 :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a2 :: "'b language"   and a3 :: "'c language" assumes "rel_language a1 a2 a3" obtains L :: "'b language" and K :: "'c language" and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" where "a1 = R" and "a2 = L" and "a3 = K" and "\<oo> L = \<oo> K" and "\<And>a b. R a b \<Longrightarrow> rel_language R (\<dd> L a) (\<dd> K b)"
restrict_PiE: fixes f :: "'e \<Rightarrow> 'f"   and I :: "'e set"   and S :: "'e \<Rightarrow> 'f set" shows "(restrict f I \<in> Pi\<^sub>E I S) = (f \<in> Pi I S)"
runit_char_4: fixes f :: "'a" assumes "ide f" shows "\<exists>!\<mu>. \<guillemotleft>\<mu> : f \<star> src f \<Rightarrow> f\<guillemotright> \<and> \<mu> \<star> src f = (f \<star> \<i>[src f]) \<cdot> \<a>[f, src f, src f]"
read_only_reads_unowned_axioms: shows "read_only_reads_unowned ts\<^sub>s\<^sub>b"
RPC_action_defs_2: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rst :: "state \<Rightarrow> PrIds \<Rightarrow> rpcState"   and p :: "PrIds" shows "RPCFwd send rcv rst p \<equiv> $Calling send p \<and> $(rst!p) = #rpcA \<and> IsLegalRcvArg<arg<$(send!p)>> \<and> Call rcv p RPCRelayArg<arg<send!p>> \<and> (rst!p)$ = #rpcB \<and> unchanged (rtrner send!p)"
int': fixes y :: "complex"   and a' :: "real"   and b :: "real" assumes "y \<in> U"   and "real_of_int a \<le> a'"   and "a' \<le> b" shows "(\<lambda>t. pbernpoly n t *\<^sub>R f y t) integrable_on {a'..b}"
prems_4: shows "InvariantWatchListsContainOnlyClausesFromF (getWatchList state') (getF state')"
assms_2: shows "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^sub>.\<^sub>w\<^sub>i\<^sub>d\<^sub>e\<^bsub>\<alpha>\<^esub> \<CC>"
elims: fixes x :: "val option"   and xa :: "proc \<times> val"   and y :: "val" assumes "\<gamma> x xa = y" obtains r :: "proc \<times> val" where "x = \<bottom>" and "xa = r__" and "y = snd r__"   | v :: "val" and r :: "proc \<times> val" where "x = Some v" and "xa = r" and "y = v"
nth_prime_double_sqrt_less: fixes \<epsilon> :: "real" assumes "0 < \<epsilon>" shows "\<forall>\<^sub>F n in sequentially. real (nth_prime (2 * n) - nth_prime n) / sqrt (real (nth_prime n)) < real n powr (1 / 2 + \<epsilon>)"
axioms_2: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "near_ordered_set A less_eq" shows "transitive A less_eq"
set_iterator_intro: fixes it :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S0 :: "'e set"   and R :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "set_iterator_genord it S0 R" shows "set_iterator it S0"
inject_2: fixes x3 :: "'a formula"   and y3 :: "'a formula" shows "(\<^bold>\<not> x3 = \<^bold>\<not> y3) = (x3 = y3)"
sum_less: fixes x :: "ordinal"   and y :: "ordinal" assumes "x < a"   and "y < a" shows "x + y < a"
poly_as_sum_of_monoms': fixes p :: "'c poly"   and n :: "nat" assumes "degree p \<le> n" shows "(\<Sum>i\<le>n. monom (coeff p i) i) = p"
map_poly_idI': fixes p :: "'b poly"   and f :: "'b \<Rightarrow> 'b" assumes "\<And>x. x \<in> set (coeffs p) \<Longrightarrow> f x = x" shows "p = map_poly f p"
assms_1: shows "i *\<^sub>R proj2_rep v + j *\<^sub>R proj2_rep w = 0"
obtains_visits: fixes s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and e :: "((nat \<times> nat) \<times> transition) fset"   and s' :: "nat"   and r' :: "nat \<Rightarrow>f value option"   and t :: "(String.literal \<times> value list) list" assumes "obtains s r e s' r' t" shows "visits s e s' r' t"
invc_baldL: fixes l :: "('c \<times> color) tree"   and r :: "('c \<times> color) tree"   and a :: "'c" assumes "invc2 l"   and "invc r"   and "color r = Black" shows "invc (baldL l a r)"
set_\<Phi>o: fixes a :: "'s" assumes "S.ide a" shows "S'.set (\<Phi>o a) = \<phi> ` S.set a"
return_state_def: fixes x :: "'a" shows "local.return_state x = StateT (\<lambda>s. return (x, s))"
sinits_not_snil: fixes xs :: "[:'a:]" shows "sinits\<cdot>xs \<noteq> [::]"
f': fixes n :: "nat"   and i :: "nat" shows "indicat_real {n} i *\<^sub>R f i = indicat_real {n} i *\<^sub>R f n"
no_step_final: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and c' :: "('b, 'a, 'c) com"   and s' :: "('b, 'c) xstate"   and P :: "bool" assumes "\<Gamma>\<turnstile> (c, s) \<rightarrow> (c', s')"   and "final (c, s)" shows "P"
eN\<tau>_2: shows "eN\<tau> \<in> fmla"
bind_eq_None_conv: fixes a :: "'f option"   and f :: "'f \<Rightarrow> 'e option" shows "(a \<bind> f = None) = (a = None \<or> f (the a) = None)"
emeasure_pair_measure_alt: fixes X :: "('e \<times> 'c) set"   and N :: "'e measure" assumes "X \<in> sets (N \<Otimes>\<^sub>M N)" shows "emeasure (N \<Otimes>\<^sub>M N) X = \<integral>\<^sup>+ x. emeasure N (Pair x -` X) \<partial>N"
lborel_has_bochner_integral_real_affine_iff: fixes c :: "real"   and f :: "real \<Rightarrow> 'b"   and x :: "'b"   and t :: "real" assumes "c \<noteq> 0" shows "has_bochner_integral lborel f x = has_bochner_integral lborel (\<lambda>x. f (t + c * x)) (x /\<^sub>R \<bar>c\<bar>)"
le_nhds_metric_le: fixes F :: "'a filter"   and a :: "'a" shows "(F \<le> nhds a) = (\<forall>e>0. \<forall>\<^sub>F x in F. dist x a \<le> e)"
that: fixes x :: "'v" assumes "x \<in> \<^bold>V"   and "x \<noteq> source \<Delta>" shows "d_OUT (capacity \<Delta>) x < \<top>"
gWlsDisj_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'j, 'k) Recursion.model_scheme" shows "gWlsDisj MOD \<equiv> \<forall>s s' X. gWls MOD s X \<and> gWls MOD s' X \<longrightarrow> s = s'"
even_push_bit_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and n :: "nat"   and a :: "'c" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (push_bit n a) = (n \<noteq> 0 \<or> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a)"
that: shows "prove_nonneg prnt i p (slp_of_fas [fa]) (YS # YSS)"
ex_un_null: shows "\<exists>!n. \<forall>f. VoV.comp n f = n \<and> VoV.comp f n = n"
s: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'e"   and k :: "enat"   and S :: "'f set"   and f :: "'f \<Rightarrow> 'a"   and g :: "'f \<Rightarrow> 'b" assumes "bounded_bilinear prod"   and "k-smooth_on S f"   and "k-smooth_on S g"   and "open S" shows "k-smooth_on S (\<lambda>x. prod (f x) (g x))"
BIND_def: fixes name :: "char list"   and inp :: "nat"   and P :: "'a" shows "Case_Labeling.BIND name inp P \<equiv> P"
covector_inf_comp_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "vector x" shows "(y \<sqinter> x\<^sup>T) * z = (y \<sqinter> x\<^sup>T) * (x \<sqinter> z)"
measurable_iff_measure_of: fixes N :: "'c set set"   and \<Omega> :: "'c set"   and f :: "'d \<Rightarrow> 'c"   and M :: "'d measure"   and \<mu> :: "'c set \<Rightarrow> ennreal" assumes "N \<subseteq> Pow \<Omega>"   and "f \<in> space M \<rightarrow> \<Omega>" shows "(f \<in> M \<rightarrow>\<^sub>M measure_of \<Omega> N \<mu>) = (\<forall>A\<in>N. f -` A \<inter> space M \<in> sets M)"
zero_p_id_prop: fixes x :: "'a" shows "x \<cdot> (0::'a) \<cdot> 1\<^sub>\<pi> = x \<cdot> (0::'a)"
not_const_intv: fixes u :: "'c \<Rightarrow> real" assumes "u \<in> V"   and "\<forall>c\<in>\<X>. \<not> (\<exists>d\<le>k c. u c = real d)" shows "\<forall>c\<in>\<X>. \<forall>u\<in>[u]\<^sub>\<R>. \<not> (\<exists>d\<le>k c. u c = real d)"
SubspaceD1: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and U :: "'b set" assumes "VectorSpace smult V"   and "RModule UNIV smult U \<and> U \<subseteq> V" shows "VectorSpace smult U"
subst_bv2_subst_lt: fixes j :: "nat"   and i :: "nat"   and u :: "Core.term" assumes "j < i" shows "subst_bv2 (Bv j) i u = Bv j"
IH: fixes i :: "nat"   and j :: "nat" assumes "\<And>k z. \<lbrakk>k \<in> {i..<j + 1}; z \<in> atms r\<rbrakk> \<Longrightarrow> t k z = t' k z" shows "match t r i j = match t' r i j"
fin_dim_def: fixes K :: "('b, 'c) ring_scheme"   and V :: "('b, 'd, 'e) module_scheme" assumes "vectorspace K V" shows "vectorspace.fin_dim K V = (\<exists>A. finite A \<and> A \<subseteq> carrier V \<and> LinearCombinations.module.span K V A = carrier V)"
mono_ndet_FD: fixes P :: "'a process"   and P' :: "'a process"   and S :: "'a process"   and S' :: "'a process" assumes "P \<sqsubseteq>\<^sub>F\<^sub>D P'"   and "S \<sqsubseteq>\<^sub>F\<^sub>D S'" shows "(P |-| S) \<sqsubseteq>\<^sub>F\<^sub>D (P' |-| S')"
valid_Tree_eqvt': fixes P :: "'state"   and t :: "('b, 'pred, 'act) Tree"   and p :: "perm" shows "valid_Tree P t = valid_Tree (p \<bullet> P) (p \<bullet> t)"
rat_number_collapse_1: fixes k :: "int" shows "Fract 0 k = 0"
get_g: shows "Fubx_get (Fubx_add F f (Subx.generalize_fundef fd)) g = Some gd"
prems: fixes x :: "'a" assumes "x \<in> insert a A" shows "\<exists>y. y \<in> insert a A \<and> R x y"
HPhase1or2Write_blocksOf: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and r :: "Proc" assumes "HPhase1or2Write s s' p d" shows "blocksOf s' r \<subseteq> blocksOf s r"
ass_adv_OT12: fixes funct_OT12 :: "bool \<times> bool \<Rightarrow> bool \<Rightarrow> (unit \<times> bool) spmf"   and m0 :: "bool"   and m1 :: "bool"   and c :: "bool"   and D :: "'v_OT121 \<Rightarrow> bool spmf" shows "sim_det_def.adv_P1 R1_OT12 S1_OT12 funct_OT12 (m0, m1) c D \<le> adv_OT12"
assms_1: shows "h \<turnstile> f \<bind> g \<rightarrow>\<^sub>h h''"
added_ids_Deliver_Expunge_same_collapse: fixes i :: "'c"   and e :: "'b"   and mo :: "'c"   and j :: "'c" shows "added_ids [Deliver (i, Expunge e mo j)] e = [j]"
igSubstIGAbsSTR_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'j, 'k) model_scheme" shows "igSubstIGAbsSTR MOD \<equiv> \<forall>ys y Y xs x X. (xs \<noteq> ys \<or> x \<noteq> y) \<and> igFresh MOD xs x Y \<longrightarrow> igSubstAbs MOD ys Y y (igAbs MOD xs x X) = igAbs MOD xs x (igSubst MOD ys Y y X)"
t: assumes "i < length w" shows "is_terminal ((w @ a) ! i)"
ABinds_delete_bot: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list" assumes "ae x = \<bottom>" shows "ABinds (delete x \<Gamma>)\<cdot>ae = ABinds \<Gamma>\<cdot>ae"
rb_Nil_sp: fixes prb :: "('d, 'e, 'f) pre_RedBlack"   and rv1 :: "'d \<times> nat"   and rv2 :: "'d \<times> nat" assumes "RedBlack prb" shows "SubRel.subpath (red prb) rv1 [] rv2 (subs prb) = (rv1 \<in> red_vertices prb \<and> (rv1 = rv2 \<or> (rv1, rv2) \<in> subs prb))"
cases: fixes x :: "rat vec \<times> rat vec" obtains c :: "rat vec" and b :: "rat vec" where "x = (c, b)"
assms_2: shows "\<not> \<parallel>c\<parallel>\<^bsub>lnth t n\<^esub>"
x: shows "x \<in> set (take (Suc i) BL) - set (take i FL)"
igSubstIGVar1STR_imp_igSubstIGVar1: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('g, 'h, 'd, 'e, 'i, 'j, 'k, 'l, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.igSubstIGVar1STR Delta MOD" shows "FixSyn.igSubstIGVar1 Delta MOD"
inf': shows "\<forall>i. \<Gamma>\<turnstile>the (\<Gamma> (p i)) \<down> Normal (s i) \<and> (\<exists>css. \<Gamma>\<turnstile> ([the (\<Gamma> (p i))], [], Normal (s i)) \<rightarrow>\<^sup>+ ([the (\<Gamma> (p (Suc i)))], css, Normal (s (Suc i))))"
semiring_simprules_15: fixes x :: "'a" assumes "x \<in> carrier R" shows "x \<otimes> \<zero> = \<zero>"
pos: shows "\<forall>\<^sub>F x in at_top. 0 < hd basis x"
lat_dist4: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x + y) \<sqinter> z = x \<sqinter> z + y \<sqinter> z"
mu_G_spec: fixes I :: "'b set"   and P :: "'b set \<Rightarrow> ('b \<Rightarrow> 'c) measure"   and M :: "'b \<Rightarrow> 'c measure"   and J :: "'b set"   and X :: "('b \<Rightarrow> 'c) set" assumes "projective_family I P M"   and "finite J"   and "J \<subseteq> I"   and "X \<in> sets (Pi\<^sub>M J M)" shows "projective_family.mu_G I P M (prod_emb I M J X) = emeasure (P J) X"