hcju/isabelleps huggingface dataset:
measurable_on_def: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" shows "f measurable_on S \<equiv> \<exists>N g. negligible N \<and> (\<forall>n. continuous_on UNIV (g n)) \<and> (\<forall>x. x \<notin> N \<longrightarrow> (\<lambda>n. g n x) \<longlonglongrightarrow> (if x \<in> S then f x else (0::'b)))"
clS: shows "closedin (top_of_set (S \<union> T)) S"
assms_7: fixes A :: "'a trm" assumes "P S A" shows "P S (Snd A)"
memb_imp_not_empty: fixes x :: "'d"   and S :: "'d set" assumes "x \<in> S" shows "S \<noteq> {}"
disjoint_zero_sym: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.sep_algebra plus zero sep_disj" shows "sep_disj zero x"
finally_until: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n F\<^sub>n (\<phi> U\<^sub>n \<psi>) = w \<Turnstile>\<^sub>n F\<^sub>n \<psi>"
zero_le_mult_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less_eq zero (times a b) = (less_eq zero a \<and> less_eq zero b \<or> less_eq a zero \<and> less_eq b zero)"
Fract_def: shows "Fract \<equiv> map_fun id (map_fun id abs_fract) (\<lambda>a b. if b = (0::'b) then (0::'b, 1::'b) else (a, b))"
lfp_in_Fix: fixes f :: "'b \<Rightarrow> 'b" assumes "mono f" shows "lfp f \<in> Fix f"
sets_into_space: fixes x :: "'a set" assumes "x \<in> {A. \<forall>t. {\<omega> \<in> A. T \<omega> \<le> t} \<in> sets (F t)}" shows "x \<subseteq> \<Omega>"
binary_meet_exists': fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.Finite_Meet_cpo below" shows "\<exists>l. po.is_glb below {x, y} l"
integral_restrict_UNIV: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'b" shows "integral UNIV (\<lambda>x. if x \<in> S then f x else (0::'b)) = integral S f"
division_ring_divide_zero: fixes a :: "'a" shows "a / (0::'a) = (0::'a)"
Cons_eq_filterD: fixes x :: "'g"   and xs :: "'g list"   and P :: "'g \<Rightarrow> bool"   and ys :: "'g list" assumes "x # xs = filter P ys" shows "\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
Abs_bit0_inject: fixes x :: "int"   and y :: "int" assumes "x \<in> {0..<2 * int CARD('a)}"   and "y \<in> {0..<2 * int CARD('a)}" shows "(Abs_bit0 x = Abs_bit0 y) = (x = y)"
abs_mult: fixes a :: "'a"   and b :: "'a" shows "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
0: shows "a \<in> carrier Zp \<and> s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"
sel_1: fixes x1 :: "real"   and x2 :: "real" shows "Re (Complex x1 x2) = x1"
is_processT9_S_swap: fixes s :: "'a event list"   and P :: "'a process" assumes "s \<notin> D P" shows "s @ [tick] \<notin> D P"
coboundedI: fixes A :: "'a set"   and a :: "'a" assumes "finite A"   and "a \<in> A" shows "a \<le> \<Squnion>\<^sub>f\<^sub>i\<^sub>n A"
affine_hull_3: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "affine hull {a, b, c} = {u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c |u v w. u + v + w = 1}"
False: shows "\<not> (m \<le> 1 \<or> n \<le> 1)"
dvw_2: shows "atom (decode_Var w) \<sharp> [decode_Var v]"
symrun_interp_decreases_setinc: fixes \<Gamma> :: "'a constr list"   and \<Gamma>' :: "'a constr list" assumes "set \<Gamma> \<subseteq> set \<Gamma>'" shows "\<lbrakk>\<lbrakk> \<Gamma>' \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<subseteq> \<lbrakk>\<lbrakk> \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m"
\<Delta>_simps_5: shows "cod ((r \<star> tab\<^sub>0 r) \<cdot> r.tab) = r \<star> tab\<^sub>0 r"
simps_3: fixes x2 :: "'l" shows "Some x2 \<noteq> None"
vfsequence_vrange_vproduct: fixes xs :: "V"   and A :: "V \<Rightarrow> V" assumes "vfsequence xs"   and "\<And>i. i \<in>\<^sub>\<circ> vcard xs \<Longrightarrow> xs\<lparr>i\<rparr> \<in>\<^sub>\<circ> A i" shows "xs \<in>\<^sub>\<circ> vproduct (vcard xs) A"
to_fun_to_poly: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp" shows "to_polynomial Zp a \<bullet> b = a"
assms: shows "x * f $$ fls_subdegree f = (1::'a)"
one_right: fixes a :: "'a" shows "a * (1::'a) = a"
alloc_def: fixes s :: "Store"   and t :: "New" shows "s\<langle>t\<rangle> \<equiv> case t of new_instance C \<Rightarrow> Abs_Store (Rep_Store s\<lparr>newOID := \<lambda>D. if C = D then Suc (newOID (Rep_Store s) C) else newOID (Rep_Store s) D\<rparr>) | new_array T l \<Rightarrow> Abs_Store (Rep_Store s\<lparr>newAID := \<lambda>S. if T = S then Suc (newAID (Rep_Store s) T) else newAID (Rep_Store s) S, vals := (vals (Rep_Store s))(arrLenLoc T (newAID (Rep_Store s) T) := intgV (int l))\<rparr>)"
oreachable_init: fixes s :: "'d \<times> 'e"   and A :: "('d \<times> 'e, 'f) automaton"   and S :: "'d \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "s \<in> init A" shows "s \<in> oreachable A S U"
E: shows "sim_equiv_class a t \<in> sim_equiv_class a ` jkbpC"
SDF_def: fixes sdf :: "('c \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> ('d \<times> 'd) set"   and A :: "'d set"   and Is :: "'c set"   and Pcond :: "'d set \<Rightarrow> 'c set \<Rightarrow> ('c \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> bool" shows "SDF sdf A Is Pcond \<equiv> \<forall>P. Pcond A Is P \<longrightarrow> choiceFn A (sdf P)"
gen_gens: fixes x :: "'d"   and gens :: "'d set"   and G :: "('d, 'e) monoid_scheme" assumes "x \<in> gens" shows "x \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>"
filter_union_mset: fixes P :: "'k \<Rightarrow> bool"   and M :: "'k multiset"   and N :: "'k multiset" shows "filter_mset P (M + N) = filter_mset P M + filter_mset P N"
inject: fixes x1 :: "'a"   and x2 :: "'d"   and y1 :: "'a"   and y2 :: "'d" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
cmp_eq: fixes D :: "'a set"   and cmp :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res"   and x :: "'a"   and y :: "'a" assumes "eq_linorder_on D cmp"   and "x \<in> D"   and "y \<in> D" shows "(cmp x y = EQUAL) = (x = y)"
assms_2: shows "\<forall>y. y \<in># M \<longrightarrow> SN_on s {y}"
m_def: shows "m \<equiv> (f 0 - f (- 2)) div 2"
minus_minus: fixes x :: "'a" assumes "x \<in> carrier R" shows "\<ominus> (\<ominus> x) = x"
vector_meet_comp: fixes v :: "'a"   and w :: "'a" assumes "is_vector v"   and "is_vector w" shows "v ; w\<^sup>T = v \<cdot> w\<^sup>T"
distinct_7: fixes x2 :: "real mpoly"   and x3 :: "real mpoly" shows "Eq x2 \<noteq> Leq x3"
reduce_not_empty: fixes f :: "nat \<Rightarrow> real" shows "reduce f PR \<noteq> {}"
simps_17: fixes v :: "nat"   and M :: "(floatarith, nat) mapping"   and slp :: "floatarith list"   and slpl :: "nat" shows "slp_of_fa_rev (Var v) M slp slpl = slp_of_fa_rev_cnst (Var v) (Var (v + slpl)) M slp slpl"
from_conjugate_space_inverse: fixes x :: "'d conjugate_space" shows "to_conjugate_space (from_conjugate_space x) = x"
A_notempty: fixes i :: "'a" assumes "i \<in> domain x" shows "A i \<noteq> {}"
axioms: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "span_vertical_category C" shows "category C"
infsetsum_0: fixes A :: "'d set" shows "(\<Sum>\<^sub>a_\<in>A. (0::'c)) = (0::'c)"
disj: fixes i :: "nat"   and j :: "nat" assumes "i \<noteq> j" shows "A i \<inter> A j = {}"
feasible_unitaryD: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and P :: "'c \<Rightarrow> real" assumes "feasible t"   and "unitary P" shows "unitary (t P)"
map_def: fixes f :: "'c local.term set" shows "local.map f \<equiv> if \<F>C.arr f then \<lbrace>\<F>C.rep f\<rbrace> else D.null"
axioms_1: fixes f :: "'d \<Rightarrow> 'e"   and I :: "'d set"   and R :: "('d, 'f) ring_scheme" assumes "weak_ring_morphism f I R" shows "ideal I R"
lcs_unique: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and lcs :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and s :: "'a"   and l :: "'a"   and t :: "'a" assumes "class.ulcs_powerprod plus minus zero lcs"   and "plus.adds plus s l"   and "plus.adds plus t l"   and "\<And>u. \<lbrakk>plus.adds plus s u; plus.adds plus t u\<rbrakk> \<Longrightarrow> plus.adds plus l u" shows "l = lcs s t"
phi_translate_adverse: fixes z :: "nat" shows "\<exists>f\<in>\<R>\<^sup>2. \<forall>i j. \<phi> (the (f i j)) = local.adverse z i j"
map_spmf_conv_bind_spmf: fixes f :: "'c \<Rightarrow> 'b"   and p :: "'c spmf" shows "map_spmf f p = p \<bind> (\<lambda>x. return_spmf (f x))"
simps: fixes B :: "ty"   and C :: "ty"   and X :: "ntrm set" shows "ANDLEFT2 (B AND C) X = {y:AndL2 x.M y |x y M. fin (AndL2 x.M y) y \<and> x:M \<in> X}"
length_pos_if_in_set: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set xs" shows "0 < \<^bold>|xs\<^bold>|"
integral_restrict_space: fixes \<Omega> :: "'d set"   and M :: "'d measure"   and f :: "'d \<Rightarrow> 'e" assumes "\<Omega> \<inter> space M \<in> sets M" shows "integral\<^sup>L (restrict_space M \<Omega>) f = LINT x|M. indicat_real \<Omega> x *\<^sub>R f x"
ladder_cut_prefix_ix: fixes index :: "nat"   and L :: "(nat \<times> nat \<times> nat) list"   and d :: "nat" assumes "index < length L"   and "0 < length L" shows "ladder_ix (ladder_cut_prefix d L) index = ladder_ix L index"
cut_le_bound: fixes n :: "'a"   and I :: "'a set"   and t :: "'a" assumes "n \<in> I \<down>\<le> t" shows "n \<le> t"
source_size: fixes c :: "'a word \<Rightarrow> 'b" shows "source_size c = LENGTH('a)"
NegPos_map_map_Neg: fixes C :: "'c \<Rightarrow> 'b"   and as :: "'c list" shows "NegPos_map C (map Neg as) = map Neg (map C as)"
g_def: shows "g = (\<lambda>i. monom P (p i) i)"
Tp_darstellung: fixes qs :: "'c list"   and Strat :: "(nat \<times> nat list) list"   and init :: "'c list" assumes "length qs = length Strat" shows "T\<^sub>p init qs Strat = (\<Sum>i<length qs. t\<^sub>p (steps' init qs Strat i) (qs ! i) (Strat ! i))"
DBM_canonical_subset_le: fixes X :: "'b set"   and v :: "'b \<Rightarrow> nat"   and n :: "nat"   and not_in_X :: "'b"   and M :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and M' :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and i :: "nat"   and j :: "nat" assumes "Beta_Regions' X v n not_in_X"   and "canonical M n"   and "[M]\<^bsub>v,n\<^esub> \<subseteq> [M']\<^bsub>v,n\<^esub>"   and "[M]\<^bsub>v,n\<^esub> \<noteq> {}"   and "i \<le> n"   and "j \<le> n"   and "i \<noteq> j" shows "M i j \<le> M' i j"
smult_vec_def: fixes a :: "'c"   and v :: "'c vec" shows "a \<cdot>\<^sub>v v \<equiv> vec (dim_vec v) (\<lambda>i. a * v $ i)"
cobounded2: fixes b :: "'b"   and a :: "'b" shows "b \<le> a \<squnion> b"
nsum_eq_i: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "\<forall>j\<le>n. f j \<in> carrier A"   and "\<forall>j\<le>n. g j \<in> carrier A"   and "i \<le> n"   and "\<forall>l\<le>i. f l = g l" shows "\<Sigma>\<^sub>e A f i = \<Sigma>\<^sub>e A g i"
sort_descending_set_inv: fixes k :: "'e \<Rightarrow> 'f"   and t :: "'e list" shows "set (sort_descending_key k t) = set t"
hsub: fixes s :: "'s" shows "healthy (wp (p (f s)))"
integral_substitution_2: fixes g :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real" assumes "set_integrable lborel {g a..g b} f"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> (g has_real_derivative g' x) (at x)"   and "continuous_on {a..b} g'"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> 0 \<le> g' x"   and "a \<le> b" shows "LBINT x. f x * indicat_real {g a..g b} x = LBINT x. f (g x) * g' x * indicat_real {a..b} x"
sorted_map_remove1: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and x :: "'b" assumes "sorted (map f xs)" shows "sorted (map f (remove1 x xs))"
matchers_map_empty: shows "matchers_map Map.empty = UNIV"
measure_space_closed: fixes \<Omega> :: "'a set"   and M :: "'a set set"   and \<mu> :: "'a set \<Rightarrow> ennreal" assumes "measure_space \<Omega> M \<mu>" shows "M \<subseteq> Pow \<Omega>"
Base: fixes f :: "nat \<Rightarrow> 'c" shows "shift 0 f = f"
intros_2: fixes a :: "'c"   and b :: "'c"   and r :: "('c \<times> 'c) set"   and c :: "'c" assumes "(a, b) \<in> r\<^sup>*"   and "(b, c) \<in> r" shows "(a, c) \<in> r\<^sup>*"
simple_function_count_space: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "simple_function (count_space A) f = finite (f ` A)"
is_pole_inverse_holomorphic: fixes s :: "complex set"   and f :: "complex \<Rightarrow> complex"   and z :: "complex" assumes "open s"   and "f holomorphic_on s - {z}"   and "is_pole f z"   and "\<forall>x\<in>s - {z}. f x \<noteq> 0" shows "(\<lambda>x. if x = z then 0 else inverse (f x)) holomorphic_on s"
fun_relI: fixes A :: "('c \<times> 'd) set"   and f :: "'c \<Rightarrow> 'e"   and f' :: "'d \<Rightarrow> 'f"   and B :: "('e \<times> 'f) set" assumes "\<And>a a'. (a, a') \<in> A \<Longrightarrow> (f a, f' a') \<in> B" shows "(f, f') \<in> A \<rightarrow> B"
dgrad_sig_set_closed_monom_mult_zero: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and j :: "nat"   and d :: "'a \<Rightarrow> nat"   and c :: "'b" assumes "p \<in> dgrad_sig_set' j d" shows "monom_mult c (0::'a) p \<in> dgrad_sig_set' j d"
assms_3: fixes t :: "'a" shows "snd (rep_nat_term (term_of_pair (t, the_min))) = 0"
simps_12: fixes x :: "nat"   and a :: "floatarith"   and vs :: "real list" shows "isDERIV x (Exp a) vs = isDERIV x a vs"
sylvester_index_mat: fixes i :: "nat"   and p :: "'b poly"   and q :: "'b poly"   and j :: "nat" assumes "i < degree p + degree q"   and "j < degree p + degree q" shows "sylvester_mat p q $$ (i, j) = (if i < degree q then if i \<le> j \<and> j - i \<le> degree p then coeff p (degree p + i - j) else (0::'b) else if i - degree q \<le> j \<and> j \<le> i then coeff q (i - j) else (0::'b))"
inject_5: fixes x61 :: "(((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> char list set) \<times> ((char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int) \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> char list set \<times> (char list \<Rightarrow> char list set)"   and x62 :: "bexp"   and x63 :: "acom"   and y61 :: "(((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> char list set) \<times> ((char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int) \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> char list set \<times> (char list \<Rightarrow> char list set)"   and y62 :: "bexp"   and y63 :: "acom" shows "({x61} WHILE x62 DO x63 = {y61} WHILE y62 DO y63) = (x61 = y61 \<and> x62 = y62 \<and> x63 = y63)"
rcos_module_imp: fixes x :: "'a"   and x' :: "'a" assumes "Group.group G"   and "x \<in> carrier G"   and "x' \<in> H #> x" shows "x' \<otimes> inv x \<in> H"
scaleR_fun_def: shows "(*\<^sub>R) = (\<lambda>c f x. c *\<^sub>R f x)"
arr_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "partial_magma C" shows "partial_magma.arr C f \<equiv> partial_magma.domains C f \<noteq> {} \<or> partial_magma.codomains C f \<noteq> {}"
2: shows "path_image ((\<lambda>z. (x1, h z)) +++ (\<lambda>z. (g z, y2))) \<subseteq> (\<Union>x\<in>s. \<Union>x1\<in>t. {(x, x1)})"
ex_less_of_int: fixes x :: "'a" shows "\<exists>z. x < of_int z"
linear_iff: fixes f :: "'c \<Rightarrow> 'd" shows "linear f = ((\<forall>x y. f (x + y) = f x + f y) \<and> (\<forall>c x. f (c *\<^sub>R x) = c *\<^sub>R f x))"
count_prod: fixes B :: "'c set"   and N :: "'c measure"   and Y :: "'a \<Rightarrow> 'c"   and X :: "'a \<Rightarrow> real" assumes "B \<subseteq> space N"   and "point_measurable M (space N) Y"   and "integrable M X"   and "\<forall>w\<in>space M. 0 \<le> X w" shows "\<forall>i. LINT y|M. X y * indicat_real (countable_preimages B Y i \<inter> space M) y = LINT y|M. expl_cond_expect M Y X y * indicat_real (countable_preimages B Y i \<inter> space M) y"
assms_3: shows "pfinite_converter \<I>_real \<I>_ideal sim'"
bounded_linear: fixes f :: "'c \<Rightarrow> 'd" assumes "bounded_linear f" shows "bounded_linear f"
implies_not_e2: fixes \<Gamma> :: "'b constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and \<Psi> :: "'b TESL_atomic list"   and \<Phi> :: "'b TESL_atomic list" shows "\<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^sub>e K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi>"
remove_vertex_path_from_to: fixes v :: "'a"   and xs :: "'a list"   and w :: "'a"   and x :: "'a" assumes "v \<leadsto>xs\<leadsto>\<^bsub>remove_vertex v\<^esub> w"   and "x \<in> V\<^bsub>remove_vertex v\<^esub>"   and "x \<notin> set xs" shows "v \<leadsto>xs\<leadsto>\<^bsub>G.remove_vertex x\<^esub> w"
arcsin: fixes y :: "real" assumes "- 1 \<le> y"   and "y \<le> 1" shows "- (pi / 2) \<le> arcsin y \<and> arcsin y \<le> pi / 2 \<and> sin (arcsin y) = y"
func_lemma: fixes I :: "('d, 'e, 'f) interp"   and args :: "'f \<Rightarrow> ('d, 'f) trm"   and \<nu> :: "(real, 'f) vec"   and f :: "'d" assumes "is_interp I"   and "\<And>\<theta>. \<theta> \<in> range args \<Longrightarrow> (sterm_sem I \<theta> has_derivative frechet I \<theta> \<nu>) (at \<nu>)" shows "(sterm_sem I ($f f args) has_derivative frechet I ($f f args) \<nu>) (at \<nu>)"
prefixes_converg_le: fixes t :: "nat"   and j :: "nat"   and t' :: "nat" assumes "local.prefixes t j \<down>"   and "t' \<le> t" shows "local.prefixes t' j \<down>"
mult_of_int_commute: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "int"   and y :: "'a" assumes "class.ring_1 one times plus zero minus uminus" shows "times (ring_1.of_int one plus zero minus x) y = times y (ring_1.of_int one plus zero minus x)"
swapVal_preserves_sWls1: fixes SEM :: "('b, 'c, 'd, 'sort, 'e, 'f, 'g) semDom_scheme"   and val :: "'varSort \<Rightarrow> 'var \<Rightarrow> 'f"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "sWlsVal SEM val" shows "sWlsVal SEM (val ^[z1 \<and> z2]_zs)"
prime_elem_nat_iff: fixes n :: "nat" shows "prime_elem n = prime n"
orderE: fixes b :: "'b::semilattice_sup"   and a :: "'b::semilattice_sup" assumes "(b::'b::semilattice_sup) \<le> (a::'b::semilattice_sup)" obtains "(a::'b::semilattice_sup) = a \<squnion> (b::'b::semilattice_sup)"
fold_ohearn2: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "ir_valid P SKIP c' Q = (\<forall>t. ohearn (P t) c' (Q t))"
shifted_def: fixes I :: "\<I>" shows "shifted I = (\<lambda>n. subtract n I) ` {0..case right I of enat n \<Rightarrow> n | \<infinity> \<Rightarrow> left I}"
less_append: fixes xs :: "nat list"   and vs :: "nat list"   and ys :: "nat list"   and ws :: "nat list" assumes "xs <\<^sub>v vs"   and "ys \<le>\<^sub>v ws" shows "xs @ ys <\<^sub>v vs @ ws"
compBGood_def: fixes f :: "Val \<Rightarrow> Val \<Rightarrow> bool" shows "compBGood f = (\<forall>\<beta> v v' w w'. (\<beta>, v, v') \<in> twiddleVal \<longrightarrow> (\<beta>, w, w') \<in> twiddleVal \<longrightarrow> f v w = f v' w')"
map_def: fixes f :: "'s" shows "S.map f = (if S.arr f then f else S.null)"
length_summarize_aforms: fixes a :: "nat"   and b :: "(real \<times> real pdevs) list \<Rightarrow> nat \<Rightarrow> real list \<Rightarrow> bool"   and e :: "nat"   and d :: "(real \<times> real pdevs) list" shows "length (summarize_aforms a b e d) = length d"
inject_2: fixes x2 :: "lam"   and x1 :: "lam"   and y2 :: "lam"   and y1 :: "lam" shows "(APP x2 x1 = APP y2 y1) = (x2 = y2 \<and> x1 = y1)"
integral_linear: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and h :: "'b \<Rightarrow> 'c" assumes "f integrable_on S"   and "bounded_linear h" shows "integral S (h \<circ> f) = h (integral S f)"
left_right_inverse_power: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c"   and n :: "nat" assumes "class.monoid_mult one times"   and "times x y = one" shows "times (power.power one times x n) (power.power one times y n) = one"
lr_fundamental: fixes R :: "('a \<Rightarrow> ValD) set"   and \<rho> :: "'a \<Rightarrow> nat \<rightarrow> ValD"   and M :: "expr" assumes "PCF_lr R"   and "\<forall>v. (\<lambda>i. \<rho> i\<cdot>v) \<in> R" shows "(\<lambda>i. \<lbrakk>M\<rbrakk>(\<rho> i)) \<in> R"
trg_hcomp: fixes \<mu> :: "'a arrow_of_spans_data"   and \<nu> :: "'a arrow_of_spans_data" assumes "arr \<mu>"   and "arr \<nu>"   and "src \<nu> = trg \<mu>" shows "trg (\<nu> \<star> \<mu>) = trg \<nu>"
Lim_within: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and a :: "'a"   and S :: "'a set" shows "(f \<longlongrightarrow> l) (at a within S) = (\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e)"
T_on_rand_as_sum: fixes A :: "('b list \<Rightarrow> 'c pmf) \<times> ('b list \<times> 'c \<Rightarrow> 'b \<Rightarrow> ((nat \<times> nat list) \<times> 'c) pmf)"   and s0 :: "'b list"   and rs :: "'b list" shows "T_on_rand' A (fst A s0 \<bind> (\<lambda>is. return_pmf (s0, is))) rs = sum (T_on_rand'_n A (fst A s0 \<bind> (\<lambda>is. return_pmf (s0, is))) rs) {..<length rs}"
\<phi>f: shows "\<guillemotleft>\<phi> \<star> f : r \<star> f \<Rightarrow> r' \<star> f\<guillemotright>"
mrg_prior_vwb_lens: shows "vwb_lens mrg_prior"
cis_pm_pi_2: fixes x :: "real" shows "cis (x + pi) = - cis x"
\<omega>_simps_5: shows "S.cod \<omega> = w"
shadows_consts_sterm_simps_1: fixes C_info :: "(name, dt_def) fmap"   and heads :: "name fset"   and t\<^sub>1 :: "sterm"   and t\<^sub>2 :: "sterm" shows "pre_constants.shadows_consts C_info heads (t\<^sub>1 $\<^sub>s t\<^sub>2) = (pre_constants.shadows_consts C_info heads t\<^sub>1 \<or> pre_constants.shadows_consts C_info heads t\<^sub>2)"
antipar_2: shows "src r\<^sub>0\<^sup>* = trg r\<^sub>0"
PBInv_def: shows "PBInv \<equiv> \<lambda>s ind. obc s \<subset> Blacks (M s) \<or> (\<forall>i<ind. \<not> BtoW (E s ! i, M s) \<or> \<not> z s \<and> i = R \<and> snd (E s ! R) = T \<and> (\<exists>r\<ge>ind. r < length (E s) \<and> BtoW (E s ! r, M s)))"
in_lnth_in_Supremum_ldrop: fixes i :: "nat"   and xs :: "'b multiset llist"   and x :: "'b" assumes "enat i < llength xs"   and "x \<in># lnth xs i" shows "x \<in> Sup_llist (lmap set_mset (ldrop (enat i) xs))"
OP_def: fixes x :: "'c" shows "Autoref_Tagging.OP x \<equiv> x"
fags_mem: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and n :: "int" assumes "fags_cond f z i a" shows "n\<Odot>a\<^bsub>f,i,z\<^esub> \<in> fags_carr f z i a"
homeomorphism_apply2: fixes S :: "'c set"   and T :: "'d set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c"   and x :: "'d" assumes "homeomorphism S T f g"   and "x \<in> T" shows "f (g x) = x"
b: shows " M \<longrightarrow>\<^sub>1 M2"
pochhammer_of_nat_eq_0_iff: fixes n :: "nat"   and k :: "nat" shows "(pochhammer (- of_nat n) k = (0::'a)) = (n < k)"
len_view_hchop_left: fixes v :: "view"   and ts :: "traffic"   and c :: "cars"   and v1 :: "view"   and v2 :: "view" shows "len v ts c = ext v \<and> v=v1\<parallel>v2 \<longrightarrow> len v1 ts c = ext v1"
fac_acc_body1_def: shows "fac_acc_body1 \<equiv> \<Lambda> r n. if n = 0 then n2a\<cdot>1 else wrapA\<cdot>r\<cdot>(n -\<^sub># 1) >>= (\<Lambda> res. n2a\<cdot>(n *\<^sub># res))"