hcju/isabelleps huggingface dataset:
GoTo: fixes A :: "'c set"   and H :: "(('d, 'c) fm list \<times> 'c) set"   and i :: "'c"   and p :: "('d, 'c) fm" assumes "Hintikka A H"   and "i \<in> nominals p"   and "\<exists>a. p at a in' H" shows "\<exists>ps. (ps, i) \<in> H"
prime_eq_prime_mult: fixes p :: "'a list" assumes "p \<in> carrier (K [X])" shows "prime (K [X]) p = prime (Ring_Divisibility.mult_of (K [X])) p"
horner_sum_append: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'a"   and xs :: "'b list"   and ys :: "'b list" shows "horner_sum f a (xs @ ys) = horner_sum f a xs + a ^ length xs * horner_sum f a ys"
the_NF_NF: fixes a :: "'a" shows "the_NF A a \<in> NF A"
antisym_conv2: fixes x :: "'a list"   and y :: "'a list" assumes "x \<le>lex y" shows "(\<not> x <lex y) = (x = y)"
set_empty_eq: fixes xo :: "'c option" shows "(set_option xo = {}) = (xo = None)"
Sup_downset_id: shows "Sup_prod \<circ> prod.downset = id"
P2: shows "Bet C B C' \<and> Cong B C' C B"
subtopology_eq_discrete_topology_gen_finite: fixes X :: "'c topology"   and S :: "'c set" assumes "t1_space X"   and "finite S" shows "subtopology X S = discrete_topology (topspace X \<inter> S)"
bernoulli_denom_even: fixes n :: "nat" assumes "0 < n"   and "even n" shows "bernoulli_denom n = \<Prod>{p. prime p \<and> p - 1 dvd n}"
1: fixes b :: "'a" assumes "b \<in> S" shows "local.inner_Basis x b = local.inner_Basis y b"
Ext_def: fixes R :: "('a \<times> 'b) set"   and y :: "'b" shows "Ext R y \<equiv> {x. (x, y) \<in> R}"
peq: fixes x :: "real"   and y :: "real" assumes "x \<in> T"   and "y \<in> T"   and "open_segment x y \<inter> T = {}" shows "p x = p y"
ax_finite_pointwise: fixes F :: "('b \<times> 'c) set"   and G :: "prog"   and A :: "'d triple set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd triple"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'd triple" assumes "finite F"   and "\<And>x. \<lbrakk>\<And>x. x \<in> F \<Longrightarrow> case x of (C, sig) \<Rightarrow> G,A\<turnstile>f C sig; x \<in> F\<rbrakk> \<Longrightarrow> case x of (C, sig) \<Rightarrow> G,A\<turnstile>g C sig"   and "G,A|\<turnstile>(\<lambda>(x, y). f x y) ` F" shows "G,A|\<turnstile>(\<lambda>(x, y). g x y) ` F"
smap_alt: fixes f :: "'d \<Rightarrow> 'c"   and s :: "'d stream"   and s' :: "'c stream" shows "(smap f s = s') = (\<forall>n. f (s !! n) = s' !! n)"
minus_less_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less (uminus a) b = less (uminus b) a"
weakly_bisimilar_equivp: shows "equivp (\<approx>\<cdot>\<^sub>S)"
lcf_monom_2: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "monom (UP R) a n (deg R (monom (UP R) a n)) = a"
p: shows "p = Polynomial.smult c (\<Prod>a\<in>A. [:- a, 1::'a:] ^ m a)"
NatMultiplesp_NatMultiples_eq: fixes F :: "nat set" shows "NatMultiplesp (\<lambda>x. x \<in> F) = (\<lambda>x. x \<in> NatMultiples F)"
f_Suc: fixes n :: "nat" shows "f (Suc n) = G ((F ^^ n) s) (f n)"
elims: fixes x :: "('c, 'd) sqmatrix"   and y :: "('c, 'd, 'd) matrix" assumes "matrix_of_sqmatrix x = y" obtains A :: "'d atMost \<Rightarrow> 'd atMost \<Rightarrow> 'c" where "x = SqMatrix A" and "y = Matrix A"
pSolved_Ind: fixes t :: "nat \<Rightarrow> cnf"   and p :: "PROB" assumes "t \<in> arch"   and "\<forall>n. \<exists>n'\<ge>n. \<parallel>sKs (bbop (\<sigma>\<^bsub>the_bb\<^esub>t n))\<parallel>\<^bsub>t n'\<^esub>" shows "\<forall>n. (\<exists>P. sub P \<in> bbrp (\<sigma>\<^bsub>the_bb\<^esub>t n) \<and> p \<in> P) \<longrightarrow> (\<exists>m\<ge>n. (p, solve p) = bbcs (\<sigma>\<^bsub>the_bb\<^esub>t m))"
Conjunctive_distr_disj: fixes HC :: "'d upred \<Rightarrow> 'd upred"   and P :: "'d upred"   and Q :: "'d upred" assumes "Conjunctive HC" shows "HC (P \<or> Q) = (HC P \<or> HC Q)"
\<phi>_def: fixes n :: "nat" shows "\<phi> n = char (distr M borel (\<lambda>x. S n x / sqrt (real n * \<sigma>\<^sup>2)))"
subdivision_trans: fixes G :: "('a, 'b) pre_digraph"   and rev_G :: "'b \<Rightarrow> 'b"   and H :: "('a, 'b) pre_digraph"   and rev_H :: "'b \<Rightarrow> 'b"   and I :: "('a, 'b) pre_digraph"   and rev_I :: "'b \<Rightarrow> 'b" assumes "subdivision (G, rev_G) (H, rev_H)"   and "subdivision (H, rev_H) (I, rev_I)" shows "subdivision (G, rev_G) (I, rev_I)"
femptyE: fixes a :: "'a"   and P :: "bool" assumes "a |\<in>| {||}" shows "P"
reflect_factorization: fixes p :: "'a poly"   and fctrs :: "'a \<times> ('a \<times> nat) list" assumes "coeff p 0 \<noteq> (0::'a)"   and "is_factorization_of fctrs p" shows "is_alt_factorization_of fctrs (reflect_poly p)"
squarefree_asymptotics: shows "(\<lambda>x. real (card {n. real n \<le> x \<and> squarefree n}) - 6 / pi\<^sup>2 * x) \<in> O(sqrt)"
leq_sum: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<forall>k. f k \<le> g k" shows "sup_monoid.sum f {k. True} \<le> sup_monoid.sum g {k. True}"
that: fixes h :: "real \<Rightarrow> 'a" assumes "path h"   and "path_image h \<subseteq> C"   and "\<And>t. t \<in> {0..1} \<Longrightarrow> p (h t) = g t" shows "thesis"
is_Tv_def: fixes "typ" :: "typ" shows "is_Tv typ = (\<exists>x21 x22. typ = Tv x21 x22)"
HVZK_def: shows "HVZK \<equiv> \<forall>e\<in>challenge_space. (\<forall>(h, w)\<in>Rel. R h w e = S h e) \<and> (\<forall>h\<in>valid_pub. \<forall>(a, z)\<in>set_spmf (S_raw h e). check h a e z)"
decode_state_at_preconditions: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat"   and k :: "nat"   and op :: "'b strips_operator"   and v :: "'b" assumes "is_valid_problem_strips \<Pi>"   and "\<A> \<Turnstile> \<Phi> \<Pi> t"   and "k < t"   and "op \<in> set ((\<Phi>\<inverse> \<Pi> \<A> t) ! k)"   and "v \<in> set (precondition_of op)" shows "\<A> (State k (index (\<Pi>\<^sub>\<V>) v))"
set_minus_imp_plus: fixes a :: "'b"   and b :: "'b"   and C :: "'b set" assumes "a - b \<in> C" shows "a \<in> b +o C"
adds_termI: fixes v :: "'t"   and t :: "'a"   and u :: "'t" assumes "v = t \<oplus> u" shows "u adds\<^sub>t v"
rq_len: fixes r :: "'b list"   and s :: "'b list" assumes "r \<le>s s" shows "\<^bold>|r\<^bold>| + \<^bold>|s\<^sup><\<inverse>r \<^bold>| = \<^bold>|s\<^bold>|"
cases: fixes a :: "['b]" assumes "distinct a" obtains "a = []"   | x :: "'b" and xs :: "['b]" where "a = x : xs" and "distinct xs" and "x \<notin> set xs"
intros_1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and T :: "ty" shows "P \<turnstile> T \<le> T"
bind_eq_return_pmf: fixes p :: "'b pmf"   and f :: "'b \<Rightarrow> 'a pmf"   and x :: "'a" shows "(p \<bind> f = return_pmf x) = (\<forall>y\<in>set_pmf p. f y = return_pmf x)"
heads_True: fixes xs :: "bool list" shows "heads (True # xs) = 1 + heads xs"
Cd_greatest: fixes d :: "'b"   and X :: "'a set" shows "TruePref_tax.Cd d X = greatest (Pd'_tax d) (X \<inter> Field (Pd'_tax d))"
L2_set_def: fixes f :: "'a \<Rightarrow> real"   and A :: "'a set" shows "L2_set f A = sqrt (\<Sum>i\<in>A. (f i)\<^sup>2)"
card_bij_eq: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'c" assumes "inj_on f A"   and "f ` A \<subseteq> B"   and "inj_on g B"   and "g ` B \<subseteq> A"   and "finite A"   and "finite B" shows "card A = card B"
fps_cutoff_zero_iff: fixes n :: "nat"   and f :: "'a fps" shows "(fps_cutoff n f = 0) = (f = 0 \<or> n \<le> subdegree f)"
eventually_conj: fixes P :: "'c \<Rightarrow> bool"   and F :: "'c filter"   and Q :: "'c \<Rightarrow> bool" assumes "eventually P F"   and "eventually Q F" shows "\<forall>\<^sub>F x in F. P x \<and> Q x"
distance_det: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and n :: "'b"   and n' :: "'b"   and x :: "nat"   and x' :: "nat" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.distance sourcenode targetnode kind valid_edge n n' x"   and "CFG.distance sourcenode targetnode kind valid_edge n n' x'" shows "x = x'"
simps_2: fixes x :: "num"   and y :: "num" shows "Num.pow x (num.Bit0 y) = Num.sqr (Num.pow x y)"
coperm_def: fixes l :: "'d llist"   and r :: "'d llist" shows "coperm l r = (\<exists>xs ys z. xs <~~> ys \<and> l = prepend xs z \<and> r = prepend ys z)"
not_arr_Null: shows "\<not> arrow_of_spans (\<odot>) Span.Null"
le_bot: fixes a :: "'c" assumes "top \<le> a" shows "a = top"
lb_cos: fixes x :: "float"   and prec :: "nat" assumes "0 \<le> real_of_float x"   and "real_of_float x \<le> pi" shows "cos (real_of_float x) \<in> {real_of_float (lb_cos prec x)..real_of_float (ub_cos prec x)}"
integrable_iexp: fixes f :: "'a \<Rightarrow> complex" assumes "random_variable borel f"   and "\<And>x. Im (f x) = 0" shows "complex_integrable M (\<lambda>x. exp (\<i> * f x))"
simps_1: fixes A :: "'b \<times> ('c option \<Rightarrow> 'b \<Rightarrow> 'b set) \<times> ('b \<Rightarrow> bool)" shows "NAe.steps A [] = (eps A)\<^sup>*"
cinner_CBasis: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and CBasis :: "'c set"   and u :: "'c"   and v :: "'c" assumes "class.ceuclidean_space scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner CBasis"   and "u \<in> CBasis"   and "v \<in> CBasis" shows "cinner u v = (if u = v then 1 else 0)"
prems: shows "flatten (Language.normalize (Seq c1 c2)) = x # xs"
T_sync: fixes P :: "'a process"   and A :: "'a set"   and Q :: "'a process" shows "T (P \<lbrakk>A\<rbrakk> Q) = {s. \<exists>t u. t \<in> T P \<and> u \<in> T Q \<and> s setinterleaves ((t, u), ev ` A \<union> {tick})} \<union> {s. \<exists>t u r v. front_tickFree v \<and> (tickFree r \<or> v = []) \<and> s = r @ v \<and> r setinterleaves ((t, u), ev ` A \<union> {tick}) \<and> (t \<in> D P \<and> u \<in> T Q \<or> t \<in> D Q \<and> u \<in> T P)}"
enumerate_simps_1: fixes n :: "nat" shows "enumerate n [] = []"
is_aligned_imp_not_bit: fixes w :: "'b word"   and n :: "nat"   and m :: "nat" assumes "is_aligned w n"   and "m < n" shows "\<not> bit w m"
typing_alt_simps_22: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and k :: "call_kind"   and elem :: "String.literal"   and \<tau> :: "'a OCL_Types.type" shows "(\<Gamma> \<turnstile>\<^sub>E TupleElementCall src k elem : \<tau>) = (k = DotCall \<and> (\<exists>\<pi>. \<Gamma> \<turnstile>\<^sub>E src : Tuple \<pi> \<and> fmlookup \<pi> elem = Some \<tau>))"
group_left_neutral: fixes a :: "'a perm" shows "1 * a = a"
inter_fset: fixes x :: "'a fset"   and xa :: "'a fset" shows "fset (x |\<inter>| xa) = fset x \<inter> fset xa"
relation_rules_15: fixes P :: "'c \<times> 'd \<Rightarrow> bool"   and Q :: "'d \<times> 'e \<Rightarrow> bool"   and ac :: "'c \<times> 'e" assumes "(P ;; Q) ac" obtains a :: "'c" and b :: "'d" and c :: "'e" where "ac = (a, c)" and "P (a, b)" and "Q (b, c)"
pathfinish_rectpath: fixes a1 :: "complex"   and a3 :: "complex" shows "pathfinish (rectpath a1 a3) = a1"
assms_2: shows "subs1 \<subseteq> subs2"
Trg_Cod: fixes t :: "'c local.term" assumes "Arr t" shows "Trg (Cod t) = Trg t"
jlength1: fixes f :: "nat \<Rightarrow> real" shows "j < dim_vec (evaluate_net (insert_weights s m1 (\<lambda>i. f (i + a))) (take (length (input_sizes (insert_weights s m1 (\<lambda>i. f (i + a))))) inputs))"
perm_is_ub_simp: fixes \<pi> :: "perm"   and S :: "'c set"   and x :: "'c" shows "\<pi> \<bullet> S <| \<pi> \<bullet> x = S <| x"
sel_2: fixes x21 :: "nat"   and x22 :: "'a"   and x23 :: "'a"   and x24 :: "'b"   and x25 :: "'b"   and x26 :: "'c" shows "occurs_on (Send x21 x22 x23 x24 x25 x26) = x22"
simps: shows "typeof_opubx OpAddNumUbx = ([Some Ubx1, Some Ubx1], Some Ubx1)"
subst_dom_update1: fixes v :: "'a"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "v \<notin> subst_domain \<sigma>" shows "subst_domain (rm_var v \<sigma>) = subst_domain \<sigma>"
swap_involutive2: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and zs :: "'varSort"   and y :: "'var"   and x :: "'var" assumes "good X" shows "X #[x \<and> y]_zs #[y \<and> x]_zs = X"
mlex_leq: fixes f :: "'b \<Rightarrow> nat"   and x :: "'b"   and y :: "'b"   and R :: "('b \<times> 'b) set" assumes "f x \<le> f y"   and "(x, y) \<in> R" shows "(x, y) \<in> f <*mlex*> R"
that: fixes h :: "nat \<Rightarrow> 'a" assumes "bij_betw h {..<card M} M"   and "strict_mono_on h {..<card M}" shows "thesis"
seq_compactI: fixes S :: "'b set" assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l" shows "seq_compact S"
n_z: shows "0 < (1 + 1 / (k - 1)) powr k"
map_Gmult_closed: fixes g :: "'g"   and vs :: "'v list" assumes "g \<in> G"   and "set vs \<subseteq> V" shows "set (map ((*\<cdot>) g) vs) \<subseteq> V"
dim_row_echelon_1: fixes A :: "'c mat" shows "dim_row (row_echelon A) = dim_row A"
card_sccs_dv: shows "card sccs = Suc (card H.sccs)"
hom_deg_set_closed_sum: fixes A :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set"   and I :: "'d set"   and f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and z :: "nat" assumes "0 \<in> A"   and "\<And>a1 a2. \<lbrakk>a1 \<in> A; a2 \<in> A\<rbrakk> \<Longrightarrow> a1 + a2 \<in> A"   and "\<And>i. i \<in> I \<Longrightarrow> f i \<in> hom_deg_set z A" shows "sum f I \<in> hom_deg_set z A"
assms: fixes y :: "'a" assumes "y \<in> S"   and "y \<noteq> x" shows "(e1\<cdot>\<rho>'') y \<sqsubseteq> \<rho>' y"
simps: fixes prec :: "nat"   and x :: "float" shows "lb_exp prec x = (if 0 < x then float_divl prec 1 (ub_exp prec (- x)) else let horner = \<lambda>x. let y = lb_exp_horner prec (get_even (prec + 2)) 1 1 x in if y \<le> 0 then Float 1 (- 2) else y in if x < - 1 then power_down_fl prec (horner (float_divl prec x (- floor_fl x))) (nat (- int_floor_fl x)) else horner x)"
leq_add_left: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.add_linorder_min plus minus zero less_eq less" shows "less_eq x (plus y x)"
ran_nodes: fixes fP :: "('c \<Rightarrow> nat option) \<times> 'c set"   and A :: "('d, 'c) nba" assumes "fP \<in> nodes (complement A)" shows "ran (fst fP) \<subseteq> {0..2 * card (nodes A)}"
BS3: fixes f :: "'a"   and f' :: "'a"   and \<mu> :: "'a"   and \<mu>' :: "'a" assumes "is_left_adjoint f"   and "is_left_adjoint f'"   and "\<guillemotleft>\<mu> : f \<rightarrow> f'\<guillemotright>"   and "\<guillemotleft>\<mu>' : f \<rightarrow> f'\<guillemotright>" shows "local.iso \<mu> \<and> local.iso \<mu>' \<and> \<mu> = \<mu>'"
wfp_onI_chain: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<nexists>f. \<forall>i. f i \<in> A \<and> r (f (Suc i)) (f i)" shows "wfp_on r A"
nsumTailTr: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "(\<forall>j\<le>Suc n. f j \<in> carrier A) \<longrightarrow> \<Sigma>\<^sub>e A f Suc n = \<Sigma>\<^sub>e A (\<lambda>i. f (Suc i)) n \<plusminus> f 0"
zero: fixes f :: "nat \<Rightarrow> 'a" assumes "multiplicative_function f" shows "f 0 = (0::'a)"
assms: shows "\<forall>i\<in>A. f i \<noteq> 0"
simps_1: fixes v :: "'b \<Rightarrow> ('a \<times> 'a) set" shows "word_rel v [] = Id"
relto_Id: fixes S :: "('a \<times> 'a) set"   and R :: "('a \<times> 'a) set" shows "(S\<^sup>=)\<^sup>* O R O (S\<^sup>=)\<^sup>* = S\<^sup>* O R O S\<^sup>*"
asym: fixes a :: "'x \<Rightarrow>\<^sub>0 nat"   and b :: "'x \<Rightarrow>\<^sub>0 nat" assumes "a \<prec> b"   and "b \<prec> a" shows "False"
measure_distr: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and S :: "'c set" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "S \<in> sets N" shows "Sigma_Algebra.measure (distr M N f) S = Sigma_Algebra.measure M (f -` S \<inter> space M)"
child_other: fixes d :: "nat"   and d' :: "nat"   and p :: "(nat \<times> int) list"   and dir :: "dir" assumes "d \<noteq> d'" shows "child p dir d ! d' = p ! d'"
events_in_local_order: fixes history :: "nat \<Rightarrow> 'c list"   and pre :: "'c list"   and e2 :: "'c"   and i :: "nat"   and e1 :: "'c" assumes "node_histories history"   and "node_histories.prefix_of_node_history history (pre @ [e2]) i"   and "e1 \<in> set pre" shows "node_histories.history_order history e1 i e2"
le_iff_plus_unchanged: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "Semilat A r f"   and "x \<in> A"   and "y \<in> A" shows "(x \<sqsubseteq>\<^bsub>r\<^esub> y) = (x \<squnion>\<^bsub>f\<^esub> y = y)"
distributed_cong_density: fixes f :: "'c \<Rightarrow> ennreal"   and g :: "'c \<Rightarrow> ennreal"   and N :: "'c measure"   and M :: "'d measure"   and X :: "'d \<Rightarrow> 'c" assumes "AE x in N. f x = g x"   and "g \<in> borel_measurable N"   and "f \<in> borel_measurable N" shows "distributed M N X f = distributed M N X g"
prj_simps_1: shows "C.seq \<mu>_\<nu>\<pi>.prj\<^sub>1 \<p>\<^sub>1[\<nu>\<pi>.leg0 \<cdot> \<mu>_\<nu>\<pi>.prj\<^sub>0, \<rho>.leg1]"
ring_hom_axioms: shows "Ring_Hom.ring_hom poly_y_x"
round_robin_scheduler_spec: fixes I :: "(('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set) set"   and n0 :: "nat" assumes "\<alpha>.deterministic I" shows "scheduler_spec final r (local.round_robin n0) local.round_robin_invar thr_\<alpha> thr_invar ws_\<alpha> ws_invar is_\<alpha> is_invar I"
ratio_of_consecutive_primes: shows "(\<lambda>n. real (nth_prime (n + 1)) / real (nth_prime n)) \<longlonglongrightarrow> 1"
phiDefs_def: fixes g :: "'g"   and n :: "'node" shows "phiDefs g n \<equiv> {v. (n, v) \<in> dom (phis g)}"
chine_eq_apex: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "identity_arrow_of_spans C \<mu>" shows "Chn \<mu> = span_in_category.apex C (Dom \<mu>)"
le_SucE: fixes m :: "nat"   and n :: "nat" assumes "(m::nat) \<le> Suc (n::nat)" obtains "(m::nat) \<le> (n::nat)"   | "(m::nat) = Suc (n::nat)"
build_ip_partition_no_empty_elems: fixes wi :: "'b wordinterval"   and c :: "parts_connection"   and rs :: "'b simple_rule list" assumes "wi \<in> set (build_ip_partition c rs)" shows "\<not> wordinterval_empty wi"
int_in_range_abs: fixes n :: "nat" shows "int n \<in> range abs"
Case: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rs :: "('a, 'b, 'c) residual"   and \<phi> :: "'c"   and Cs :: "('c \<times> ('a, 'b, 'c) psi) list" assumes "\<Psi> \<rhd> P \<longmapsto> Rs"   and "(\<phi>, P) mem Cs"   and "\<Psi> \<turnstile> \<phi>"   and "guarded P" shows "\<Psi> \<rhd> Cases Cs \<longmapsto> Rs"
lift_character: fixes \<chi> :: "'a \<Rightarrow> complex"   and z :: "complex" assumes "character (G\<lparr>carrier := H'\<rparr>) \<chi>"   and "z ^ subgroup_indicator G H' a = \<chi> (a [^] subgroup_indicator G H' a)" shows "character (G\<lparr>carrier := adjoin G H' a\<rparr>) (H'.lift_character (\<chi>, z))"
sphere_trivial: fixes x :: "'b" shows "sphere x 0 = {x}"
assms_8: shows "g2 ` T \<subseteq> U"
rsiA: shows "recurrent_subset_infty A \<in> sets (restrict_space M A)"
star_denest: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "(x \<union> y)\<^sup>* = (x\<^sup>* O y\<^sup>*)\<^sup>*"
decomp_2: shows "tr = take (Suc n - n) (drop n t)"
invertibleE: fixes u :: "'a"   and thesis :: "bool" assumes "additive.invertible u"   and "\<And>v. \<lbrakk>u + v = \<zero> \<and> v + u = \<zero>; v \<in> R\<rbrakk> \<Longrightarrow> thesis"   and "u \<in> R" shows "thesis"
eq_vecI: fixes w :: "'c vec"   and v :: "'c vec" assumes "\<And>i. i < dim_vec w \<Longrightarrow> v $ i = w $ i"   and "dim_vec v = dim_vec w" shows "v = w"
continuous_pre_zeta': fixes a :: "real"   and s :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> complex" assumes "0 < a"   and "continuous (at s within A) f" shows "continuous (at s within A) (\<lambda>s. pre_zeta a (f s))"
IA: shows "Iagree I J {Inl x |x. x \<in> SIGT (Times \<theta>\<^sub>1 \<theta>\<^sub>2)}"
span_swap: fixes E :: "'a vec set"   and u :: "'a vec"   and v :: "'a vec" assumes "finite E"   and "E \<subseteq> carrier_vec n"   and "u \<in> carrier_vec n"   and "u \<notin> local.span E"   and "v \<in> carrier_vec n"   and "u \<in> local.span (insert v E)" shows "local.span (insert u E) \<subseteq> local.span (insert v E)"
set_map: fixes f :: "'c \<Rightarrow> 'a"   and v :: "'c set['b]" shows "set_bset (map_bset f v) = f ` set_bset v"
is_standard_operator_def: shows "is_standard_operator \<equiv> \<lambda>(uu_, uu_, effects, uu_). list_all is_standard_effect effects"
saturated_binary_rule_def: fixes atom_ordering :: "('a \<times> 'a) set"   and Rule :: "'b Literal set \<Rightarrow> 'b Literal set \<Rightarrow> 'b Literal set \<Rightarrow> bool"   and S :: "'b Literal set set" assumes "propositional_atoms atom_ordering" shows "saturated_binary_rule Rule S \<equiv> \<forall>P1 P2 C. P1 \<in> S \<and> P2 \<in> S \<and> Rule P1 P2 C \<longrightarrow> redundant C S"
equiintegrable_on_subset: fixes F :: "('a \<Rightarrow> 'b) set"   and I :: "'a set"   and G :: "('a \<Rightarrow> 'b) set" assumes "F equiintegrable_on I"   and "G \<subseteq> F" shows "G equiintegrable_on I"
ntr_reverse_split_atU: fixes c :: "'n list multiset"   and U :: "'n set"   and c' :: "'n list multiset"   and w :: "('p, 'ba) label list list" assumes "valid fg c"   and "atU U c'"   and "(c, w, c') \<in> trcl (ntr fg)" shows "\<exists>s w' c1'. s \<in># c \<and> w' \<preceq> w \<and> c1' \<subseteq># c' \<and> atU U c1' \<and> ({#s#}, w', c1') \<in> trcl (ntr fg)"
KBasic_3: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<box>(\<phi> \<^bold>& \<psi>) \<^bold>\<equiv> \<^bold>\<box>\<phi> \<^bold>& \<^bold>\<box>\<psi> in v]"
r_def: shows "r = p - monom_mult (lookup p (t \<oplus> lt q) / lc q) t q"
simps: fixes x :: "'c"   and v :: "'c \<Rightarrow> 'd"   and lb :: "'c \<Rightarrow> 'd option"   and ub :: "'c \<Rightarrow> 'd option" shows "in_bounds x v (lb, ub) = (v x \<ge>\<^sub>l\<^sub>b lb x \<and> v x \<le>\<^sub>u\<^sub>b ub x)"
bij_betw_UNION_disjoint: fixes A' :: "'b \<Rightarrow> 'c set"   and I :: "'b set"   and f :: "'d \<Rightarrow> 'c"   and A :: "'b \<Rightarrow> 'd set" assumes "disjoint_family_on A' I"   and "\<And>i. i \<in> I \<Longrightarrow> bij_betw f (A i) (A' i)" shows "bij_betw f (\<Union> (A ` I)) (\<Union> (A' ` I))"
simps_3: fixes V :: "'c set"   and t :: "('d, 'c) Term.term"   and S :: "('d, 'c) strand_step list" shows "wf\<^sub>s\<^sub>t V (send\<langle>t\<rangle>\<^sub>s\<^sub>t # S) = wf\<^sub>s\<^sub>t (V \<union> fv t) S"
less_imp_not_eq2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.order less_eq less"   and "less x y" shows "(y = x) = False"
remdups_adj_Cons_alt: fixes x :: "'c"   and xs :: "'c list" shows "x # tl (remdups_adj (x # xs)) = remdups_adj (x # xs)"