hcju/isabelleps huggingface dataset:
last_append: fixes p :: "'a"   and xs :: "('b \<times> 'a) list"   and ys :: "('b \<times> 'a) list" shows "Dioid_Models.last (p, xs @ ys) = Dioid_Models.last (Dioid_Models.last (p, xs), ys)"
exec1_def: fixes P :: "instr list"   and c :: "int \<times> (char list \<Rightarrow> int) \<times> int list"   and c' :: "int \<times> (char list \<Rightarrow> int) \<times> int list" shows "P \<turnstile> c \<rightarrow> c' = (\<exists>i s stk. c = (i, s, stk) \<and> c' = iexec (P !! i) (i, s, stk) \<and> 0 \<le> i \<and> i < size P)"
ODE_vars_lr: fixes x :: "'a"   and I :: "('b, 'c, 'a) interp"   and ODE :: "('b, 'a) ODE" shows "(Inl x \<in> semBV I ODE) = (Inr x \<in> semBV I ODE)"
sel_2: fixes x2 :: "'e" shows "the_Set (Set x2) = x2"
inorder_insert: fixes t :: "('b \<times> nat) tree"   and x :: "'b" assumes "avl t \<and> Sorted_Less.sorted (Tree2.inorder t)" shows "Tree2.inorder (AVL_Set_Code.insert x t) = ins_list x (Tree2.inorder t)"
positiveD1: fixes M :: "'b set set"   and f :: "'b set \<Rightarrow> ennreal" assumes "positive M f" shows "f {} = 0"
cases: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a1 :: "'c llist"   and a2 :: "'c llist" assumes "llexord r a1 a2" obtains xs :: "'c llist" and ys :: "'c llist" and x :: "'c" where "a1 = LCons x__ xs__" and "a2 = LCons x__ ys__" and "llexord r xs__ ys__"   | x :: "'c" and y :: "'c" and xs :: "'c llist" and ys :: "'c llist" where "a1 = LCons x xs" and "a2 = LCons y ysa__" and "r x y"   | ys :: "'c llist" where "a1 = LNil" and "a2 = ys"
a_subset_M: fixes k :: "nat" shows "a k \<subseteq> M k"
3: shows "{x. (x = p \<or> x \<in> P) \<and> x \<sqsubset> p} = P"
unit_imp_mod_eq_0: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and b :: "'a"   and a :: "'a" assumes "class.euclidean_semiring divide plus minus zero times one modulo euclidean_size"   and "dvd.dvd times b one" shows "modulo a b = zero"
set_intros_2: fixes y :: "('a, 'b) Term.term"   and x22 :: "('a, 'b) Term.term list"   and ya :: "'a"   and x21 :: "'a" assumes "y \<in> set x22"   and "ya \<in> funs_term y" shows "ya \<in> funs_term (Fun x21 x22)"
weak_sharing_consis: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "weak_sharing_consistent \<O> sb"
inject_3: fixes x31 :: "'c acom"   and x32 :: "'c acom"   and y31 :: "'c acom"   and y32 :: "'c acom" shows "(x31;; x32 = y31;; y32) = (x31 = y31 \<and> x32 = y32)"
hyps_3: shows "component_of_term ` keys q \<subseteq> component_of_term ` keys p \<union> component_of_term ` Keys F"
multisetLeListDiff: fixes r :: "('b \<times> 'b) set"   and a :: "'b list"   and b :: "'b list" assumes "trans r" shows "multiset_le (mset (list_diff a b)) (mset a) r"
stirling_integral_def: fixes n :: "nat"   and s :: "'a" shows "stirling_integral n s = lim (\<lambda>x. integral {0..real x} (\<lambda>x. of_real (pbernpoly n x) / (of_real x + s) ^ n))"
singular_relboundary_restrict: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set" shows "singular_relboundary p X (topspace X \<inter> S) = singular_relboundary p X S"
Atoms_Pderiv: fixes a :: "'b"   and R :: "'b rexp set" shows "Atoms (Pderiv a R) \<subseteq> Atoms R"
boxL: fixes A :: "(C, BD) form" shows "g3mod2p [ \<LM> A  \<RM> \<Rightarrow>* \<Empt>] ( \<LM> \<box> A  \<RM> \<Rightarrow>* \<Empt>)"
kill_step_smaller: fixes G :: "nat set \<times> nat set set"   and k :: "nat" assumes "short_cycles G k \<noteq> {}" shows "short_cycles (G -- choose_v G k) k \<subset> short_cycles G k"
igOpIPresIGWls_fromMOD: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h) Recursion.model" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.gOpPresGWls Delta MOD" shows "FixSyn.igOpIPresIGWls Delta (FixSyn.fromMOD Delta MOD)"
s_2: fixes n :: "nat"   and i :: "nat" shows "X (s n i) \<in> A (Suc n)"
lessThan_Suc_shift: fixes g :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "sum g {..<Suc n} = g 0 + (\<Sum>i<n. g (Suc i))"
u_1: shows "\<forall>i\<in>{1..k}. 0 \<le> u i \<and> v i \<in> S"
R2_S2_True: fixes \<sigma> :: "bool"   and b0 :: "bool"   and b1 :: "bool"   and D2 :: "bool \<times> (nat \<times> nat) \<times> bool \<times> bool \<Rightarrow> bool spmf" assumes "(if \<sigma> then b0 else b1) = True"   and "\<forall>a. lossless_spmf (D2 a)" shows "\<bar>spmf (etp.R2 (b0, b1) \<sigma> \<bind> D2) True - spmf (funct_OT_12 (b0, b1) \<sigma> \<bind> (\<lambda>(out1, out2). etp.S2 \<sigma> out2 \<bind> D2)) True\<bar> = \<bar>2 * (spmf (etp_rsa.HCP_game etp.\<A> \<sigma> (if \<sigma> then b1 else b0) D2) True - 1 / 2)\<bar>"
hyps: shows "(\<exists>\<^sub>\<infinity>i. P (i - n)) = (\<exists>\<^sub>\<infinity>i. P i)"
assms_3: shows "\<not> ps (S t i) q \<noteq> None"
matchT_M_def2: fixes d :: "('test, 'atom) com"   and c :: "('test, 'atom) com" shows "matchT_M d c = (\<forall>s t t'. s \<approx> t \<and> (d, t) \<rightarrow>t t' \<longrightarrow> (\<exists>c' s'. (c, s) \<rightarrow>*c (c', s') \<and> s' \<approx> t' \<and> discr c') \<or> (\<exists>s'. (c, s) \<rightarrow>*t s' \<and> s' \<approx> t'))"
set_tag_name_impl: shows "set_tag_name = CD.a_set_tag_name"
atlas_is_atlas: fixes a1 :: "('c, 'c) chart"   and k :: "enat"   and a2 :: "('c, 'c) chart" assumes "a1 \<in> manifold_eucl.atlas k"   and "a2 \<in> manifold_eucl.atlas k" shows "k-smooth_compat a1 a2"
S\<^sub>r_directed_towards_S2: fixes s :: "'s" assumes "s \<in> S\<^sub>r" shows "s \<in> directed_towards S2 {(s, t) |s t. s \<in> S\<^sub>r \<and> (s, t) \<in> E}"
IndP_fresh_iff: fixes a :: "atom"   and x :: "tm" shows "a \<sharp> IndP x = a \<sharp> x"
ideI: fixes a :: "'a" assumes "left (trg f) a"   and "ide a" shows "Left.ide a"
in_sccs_verts_conv: fixes S :: "'a set" shows "(S \<in> sccs_verts) = (G \<restriction> S \<in> sccs)"
incl_def: fixes f :: "'s" shows "incl f = (arr f \<and> Dom f \<subseteq> Cod f \<and> f = mkArr (Dom f) (Cod f) (\<lambda>x. x))"
MinkowskiSpacetime_axioms: shows "MinkowskiSpacetime \<E> \<P> betw"
surj_adjoint_iff_inj: fixes f :: "'a \<Rightarrow> 'b" assumes "linear f" shows "surj (adjoint f) = inj f"
independent_injective_image: fixes s :: "('b, 'n) vec set" assumes "vec.independent s"   and "inj_on f (vec.span s)" shows "vec.independent (f ` s)"
is_empty_def: fixes m :: "('b, 'c) mapping" shows "is_empty m = (keys m = {})"
flip_itself: fixes y :: "nat"   and X :: "bool list" assumes "y < length X" shows "flip y X ! y = (\<not> X ! y)"
inorder_1: shows "inorder t1 = [i..k - 1]"
sufI: fixes p :: "'b list"   and s :: "'b list"   and w :: "'b list" assumes "p \<cdot> s = w" shows "s \<le>s w"
pole_lemma_open: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and a :: "complex" assumes "f holomorphic_on S"   and "open S" shows "(\<lambda>z. if z = a then deriv f a else (f z - f a) / (z - a)) holomorphic_on S"
ext: fixes g :: "'a"   and f :: "'a" assumes "R.comp g f \<noteq> R.null" shows "R.seq g f"
\<psi>: shows "\<psi> \<in> Right_a.hom f g \<and> \<psi> \<star> a = \<phi>"
mult_imp_div_pos_le: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and y :: "'d"   and x :: "'d"   and z :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero y"   and "less_eq x (times z y)" shows "less_eq (divide x y) z"
that: assumes "porder_on A r"   and "total_onP A r" shows "thesis"
linear_injective_left_inverse: fixes f :: "('c, 'd) vec \<Rightarrow> ('c, 'e) vec" assumes "Vector_Spaces.linear (*s) (*s) f"   and "inj f" shows "\<exists>g. Vector_Spaces.linear (*s) (*s) g \<and> g \<circ> f = id"
rep_eq: fixes x :: "'b mpoly" shows "monomials x = keys (mapping_of x)"
simps_6: fixes y :: "nat" shows "pratt2_max \<infinity>\<^sub>2 [y]\<^sub>2 = \<infinity>\<^sub>2"
TS_yxyx': fixes x :: "nat"   and y :: "nat"   and qs :: "nat list"   and h :: "nat list"   and h0 :: "nat list"   and r :: "nat list" assumes "x \<noteq> y"   and "qs \<in> lang (seq [Times (Atom y) (Atom x), Star (Times (Atom y) (Atom x))])"   and "(\<exists>hs. h = [x, x] @ hs) \<or> index h y = length h" shows "Partial_Cost_Model.T_on' (rTS h0) ([x, y], h) (qs @ r) = length qs - 1 + Partial_Cost_Model.T_on' (rTS h0) ([x, y], rev qs @ h) r \<and> (\<exists>hs. rev qs @ h = [x, y] @ hs) \<and> Partial_Cost_Model.config' (rTS h0) ([x, y], h) qs = ([x, y], rev qs @ h)"
prv_cnj_imp_mono: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<chi>1 :: "'fmla"   and \<chi>2 :: "'fmla" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<chi>1 \<in> fmla"   and "\<chi>2 \<in> fmla" shows "prv (imp (imp \<phi>1 \<chi>1) (imp (imp \<phi>2 \<chi>2) (imp (cnj \<phi>1 \<phi>2) (cnj \<chi>1 \<chi>2))))"
coeff_add: fixes p :: "'b mpoly"   and m :: "nat \<Rightarrow>\<^sub>0 nat"   and q :: "'b mpoly" shows "MPoly_Type.coeff p m + MPoly_Type.coeff q m = MPoly_Type.coeff (p + q) m"
cod_simps_2: shows "f.cod.leg0 = f.leg0"
abs_eq: fixes xa :: "('c, 'd) phi \<Rightarrow> 'd"   and x :: "('c, 'd) phi \<Rightarrow> 'd" assumes "eq_onp (\<lambda>xf. True) xa xa"   and "eq_onp (\<lambda>xf. True) x x" shows "Abs_lifted_distrib_lattice_top xa \<sqinter> Abs_lifted_distrib_lattice_top x = Abs_lifted_distrib_lattice_top (\<lambda>f. xa f \<sqinter> x f)"
interior_halfspace_le: fixes a :: "'b"   and b :: "real" assumes "a \<noteq> (0::'b)" shows "interior {x. a \<bullet> x \<le> b} = {x. a \<bullet> x < b}"
e0_2: fixes s :: "real" assumes "s \<in> {a..b}"   and "dist u s < e0" shows "dist (f u) (f s) < delta - deltaG TYPE('a)"
compare_Maybe_def: shows "compare = maybe\<cdot>(maybe\<cdot>EQ\<cdot>(\<Lambda> y. LT))\<cdot>(\<Lambda> x. maybe\<cdot>GT\<cdot>(\<Lambda> y. compare\<cdot>x\<cdot>y))"
lowner_le_swap: fixes A :: "complex mat"   and n :: "nat"   and B :: "complex mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "A \<le>\<^sub>L B" shows "- B \<le>\<^sub>L - A"
Busemann_function_xy_yx: fixes xi :: "'b Gromov_completion"   and x :: "'b"   and y :: "'b" shows "Busemann_function_at xi x y + Busemann_function_at xi y x \<le> 2 * deltaG TYPE('b)"
assms: fixes x :: "'a" shows "fmlookup m x = fmlookup n x"
gen_dfs_simps_2: fixes succs :: "'d \<Rightarrow> 'd list"   and ins :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and memb :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c"   and x :: "'d"   and xs :: "'d list" shows "gen_dfs succs ins memb S (x # xs) = (if memb x S then gen_dfs succs ins memb S xs else gen_dfs succs ins memb (ins x S) (succs x @ xs))"
31: shows "bprv (all yy' (imp (SS \<langle>\<chi>\<rangle> \<langle>\<psi>\<rangle>) (imp (SS \<langle>\<chi>\<rangle> (Var yy')) (eql \<langle>\<psi>\<rangle> (Var yy')))))"
length_take: fixes n :: "nat"   and xs :: "'b list" shows "\<^bold>|take n xs\<^bold>| = min \<^bold>|xs\<^bold>| n"
igSubstAllIPresIGWlsAllSTR_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igSubstAllIPresIGWlsAllSTR MOD \<equiv> igSubstIPresIGWlsSTR MOD \<and> igSubstAbsIPresIGWlsAbsSTR MOD"
simps_9: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "is_\<nu>LTL (\<phi> R\<^sub>n \<psi>) = (is_\<nu>LTL \<phi> \<and> is_\<nu>LTL \<psi>)"
QR_decomposition_def: fixes A :: "(('c, 'd) vec, 'e) vec" shows "QR_decomposition A = (let Q = divide_by_norm (Gram_Schmidt_matrix A) in (Q, Finite_Cartesian_Product.transpose Q ** A))"
paint_inv1l: fixes t :: "('a, 'b) rbt"   and c :: "color" assumes "inv1l t" shows "inv1l (paint c t)"
isSp_Nil: fixes n :: "'node SDG_node" assumes "valid_SDG_node n" shows "n is-[]\<rightarrow>\<^sub>d* n"
integral_nonneg_AE: fixes f :: "'a \<Rightarrow> real"   and M :: "'a measure" assumes "AE x in M. 0 \<le> f x" shows "0 \<le> integral\<^sup>L M f"
theFTOne_def: fixes cl :: "('d, 'e, 'f) cmd list"   and dl :: "('g, 'h, 'i) cmd list" shows "theFTOne cl dl \<equiv> theFT cl \<union> theFT dl"
assms_1: shows "(u, v) \<in> (edges g)\<^sup>*"
ide_in_hom: fixes a :: "('obj, 'obj) concrete_category.arr" shows "ide a = \<guillemotleft>a : a \<rightarrow> a\<guillemotright>"
pref_hd_eq: fixes u :: "'b list"   and v :: "'b list" assumes "u \<le>p v"   and "u \<noteq> \<epsilon>" shows "hd u = hd v"
inj_basic_econst: shows "inj (\<lambda>v. $C v)"
Abs_x: shows "[\<lparr>A!,x\<^sup>P\<rparr> in v]"
pderiv_deg_0: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P"   and "deg R f = 0" shows "pderiv f = \<zero>\<^bsub>P\<^esub>"
not_less_iff_gr_or_eq: fixes x :: "'b"   and y :: "'b" shows "(\<not> x < y) = (y < x \<or> x = y)"
chainTransitionAppend_1: fixes P :: "pi"   and P' :: "pi"   and Rs :: "residual" assumes "P \<Longrightarrow>\<^sub>\<tau> P'"   and "P' \<Longrightarrow>\<^sub>l Rs" shows "P \<Longrightarrow>\<^sub>l Rs"
finite_group_axioms: shows "finite_group (DirProds (\<lambda>p. Z (p ^ multiplicity p n)) (prime_factors n))"
that: fixes x :: "'a" assumes "x \<in> carrier"   and "f x \<noteq> (0::'b)" shows "g x \<noteq> (0::'c)"
supp_1: fixes x2 :: "nat"   and x1 :: "NominalSequents.var list" shows "supp (At x2 x1) = supp x2 \<union> supp x1"
IH: fixes y :: "'a" assumes "y \<in> A" shows "y \<sqsubseteq>\<^bsub>r\<^esub> l \<Squnion>\<^bsub>f\<^esub> y"
strict_boundedE: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "b \<squnion> c < a" obtains "b < a" and "c < a"
constructive_security_obsf'D: fixes \<eta> :: "nat" shows "constructive_security_obsf (real2 \<eta>) (ideal2 \<eta>) (sim2 \<eta>) (\<I>_real2 \<eta>) (\<I>_inner2 \<eta>) (\<I>_common2 \<eta>) (absorb (\<A> \<eta>) (obsf_converter (parallel_wiring \<odot> (converter_of_resource (real1 \<eta>) |\<^sub>\<propto> 1\<^sub>C)))) (advantage (absorb (\<A> \<eta>) (obsf_converter (parallel_wiring \<odot> (converter_of_resource (real1 \<eta>) |\<^sub>\<propto> 1\<^sub>C)))) (obsf_resource (sim2 \<eta> |\<^sub>= 1\<^sub>C \<rhd> ideal2 \<eta>)) (obsf_resource (real2 \<eta>)))"
trivial_D2: fixes \<sigma> :: "'c \<times> ('d \<times> 'c) list"   and ars :: "('c \<times> 'd \<times> 'c) set"   and r :: "('d \<times> 'd) set" assumes "\<sigma> \<in> seq ars" shows "D2 r (\<sigma>, (fst \<sigma>, []), (lst \<sigma>, []), \<sigma>)"
assms_3: shows "List.member (alice_meas \<phi>) (p, q)"
stalk_is_ring: fixes V :: "'a set" assumes "is_open V"   and "x \<in> V" shows "ring stk2.carrier_stalk stk2.add_stalk stk2.mult_stalk (stk2.zero_stalk V) (stk2.one_stalk V)"
igWlsDisj_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'j, 'k) Iteration.model_scheme" shows "igWlsDisj MOD \<equiv> \<forall>s s' X. igWls MOD s X \<and> igWls MOD s' X \<longrightarrow> s = s'"
ide_img: fixes f :: "'s" assumes "arr f" shows "ide (img f)"
simps_1: fixes a :: "'a"   and as :: "'a list"   and b :: "'a"   and bs :: "'a list"   and accs :: "'a list" shows "merge_tailrec_inefficient (a # as) (b # bs) accs = (if a < b then merge_tailrec_inefficient as (b # bs) (accs @ [a]) else if a = b then merge_tailrec_inefficient as bs (accs @ [a]) else merge_tailrec_inefficient (a # as) bs (accs @ [b]))"
zero_powr: fixes x :: "real" shows "0 powr x = 0"
exhaust: fixes y :: "msg" obtains x11 :: "nat" and x12 :: "nat" and x13 :: "nat" and x14 :: "nat" and x15 :: "k" and x16 :: "nat" and x17 :: "nat" and x18 :: "nat" and x19 :: "bool" where "y = Rreq x11 x12 x13 x14 x15 x16 x17 x18 x19"   | x21 :: "nat" and x22 :: "nat" and x23 :: "nat" and x24 :: "nat" and x25 :: "nat" where "y = Rrep x21 x22 x23 x24 x25"   | x31 :: "nat \<Rightarrow> nat option" and x32 :: "nat" where "y = Rerr x31 x32"   | x41 :: "nat" and x42 :: "nat" where "y = Newpkt x41 x42"   | x51 :: "nat" and x52 :: "nat" and x53 :: "nat" where "y = Pkt x51 x52 x53"
assms_2: fixes a :: "'a \<times> 'b"   and b :: "'a \<times> 'b" shows "comp (fst (f a)) (fst (f b)) = comp (fst a) (fst b)"
indep_var_rv2: fixes S :: "'c measure"   and X :: "'b \<Rightarrow> 'c"   and T :: "'c measure"   and Y :: "'b \<Rightarrow> 'c" assumes "X.indep_var S X T Y" shows "X.random_variable T Y"
at_begin_norm_Bk: fixes as :: "nat"   and am :: "nat list"   and q :: "nat"   and aaa :: "cell list"   and ires :: "cell list" assumes "at_begin_norm (as, am) (q, aaa, []) ires" shows "at_begin_norm (as, am) (q, aaa, [Bk]) ires"
inj_Cons: fixes xs :: "'b list" shows "inj (\<lambda>x. x # xs)"
e2lenls_def: fixes e :: "nat" shows "e2lenls e \<equiv> e_length (e2ls e)"
T_eval1: fixes x :: "dB" shows "\<T> \<degree> x \<rightarrow>\<^sub>\<beta> Abs (Var 0 \<degree> lift x 0)"
lowner_lub_form: shows "local.lowner_is_lub (mat dim dim (\<lambda>(i, j). lim (\<lambda>n. f n $$ (i, j))))"
r_def: shows "r = (if l = \<infinity> then 1 else real_of_ereal l / 2)"
fds_abs_converges_altdef': fixes f :: "'a fds"   and s :: "'a" shows "fds_abs_converges f s = ((\<lambda>n. fds_nth f n / nat_power n s) abs_summable_on UNIV)"
transfer: shows "pcr_scene (=) (\<lambda>x y. x) \<bottom>\<^sub>S"
suff_open: fixes r :: "'a llist" assumes "r \<in> A\<^sup>\<star>" shows "suff A r open"
polyhedron_def: fixes A :: "'b mat"   and b :: "'b vec" shows "polyhedron A b = {x \<in> carrier_vec n. A *\<^sub>v x \<le> b}"
Nil_elem_T: fixes P :: "'c process" shows "[] \<in> T P"
set_intros_1: fixes x21 :: "'x"   and x22 :: "'x list" shows "x21 \<in> set (x21 # x22)"
added_files_def: fixes es :: "('d \<times> ('d, 'b) operation) event list"   and p :: "'b" shows "added_files es p \<equiv> List.map_filter (\<lambda>x. case x of Broadcast x \<Rightarrow> Map.empty x | Deliver (i, Create id x) \<Rightarrow> Map.empty x | Deliver (i, Delete set x) \<Rightarrow> Map.empty x | Deliver (i, Append j e) \<Rightarrow> if e = p then Some j else None | Deliver (i, Expunge aa id1 x) \<Rightarrow> Map.empty x | Deliver (i, Store e mo j) \<Rightarrow> if e = p then Some j else None) es"
in_Ints_imp_algebraic: fixes x :: "'b" assumes "x \<in> \<int>" shows "algebraic x"
prj_in_hom_2: shows "\<guillemotleft>\<mu>_\<nu>\<pi>.prj\<^sub>0 : \<mu>.leg0 \<down>\<down> \<nu>\<pi>.leg1 \<rightarrow>\<^sub>C \<nu>\<pi>.apex\<guillemotright>"
replicate_merge_anywhere: fixes a :: "nat"   and x :: "'a"   and b :: "nat"   and ys :: "'a list" shows "x \<up> a @ x \<up> b @ ys = x \<up> (a + b) @ ys"
uminus_in_nonpos_Ints_iff: fixes x :: "'a" shows "(- x \<in> \<int>\<^sub>\<le>\<^sub>0) = (x \<in> \<nat>)"
continuous_on_bernpoly: fixes A :: "'a set"   and n :: "nat" shows "continuous_on A (bernpoly n)"
bot_nat_int_def: shows "bot \<equiv> Abs_nat_int {}"
ordIso_natLeq_infinite1: fixes A :: "'a set" assumes "|A| =o natLeq" shows "infinite A"
monoE: fixes g :: "'a"   and f :: "'a"   and f' :: "'a" assumes "B.mono g"   and "B.seq g f"   and "B.seq g f'"   and "g \<cdot>\<^sub>B f = g \<cdot>\<^sub>B f'" shows "f' = f"
tendsto_zero_ennreal: fixes f :: "'b \<Rightarrow> ennreal"   and F :: "'b filter" assumes "\<And>r. 0 < r \<Longrightarrow> \<forall>\<^sub>F x in F. f x < ennreal r" shows "(f \<longlongrightarrow> 0) F"
map_of_eq_NoneI: fixes x :: "'b"   and xs :: "('b \<times> 'c) list" assumes "x \<notin> fst ` set xs" shows "map_of xs x = None"
lcons_list: fixes a :: "'c \<times> 'd"   and t :: "('c, 'd) FingerTreeStruc" shows "toList (a \<lhd> t) = a # toList t"
ae_filter_bot: fixes M :: "'b pmf" shows "ae_filter (measure_pmf M) \<noteq> bot"
permut_aux_length: fixes f :: "nat \<Rightarrow> nat"   and as :: "'c list"   and bs :: "'c list" assumes "bij_betw f {..<length as} {..<length bs}" shows "length (permut_aux as f bs) = length as"
fromFull_toFull: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and \<sigma> :: "'c"   and a :: "univ" assumes "InfModel wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP"   and "intT \<sigma> a" shows "InfModel.fromFull intT \<sigma> (InfModel.toFull intT \<sigma> a) = a"
cases: fixes a :: "trm" assumes "a \<in> num" obtains "a = zer"   | t :: "trm" where "a = suc t" and "t \<in> num"
WT_calleeI: fixes \<I> :: "('c, 'd) \<I>"   and callee :: "'c \<Rightarrow> ('d \<times> 'e) spmf" assumes "\<And>call ret s. \<lbrakk>call \<in> outs_\<I> \<I>; (ret, s) \<in> set_spmf (callee call)\<rbrakk> \<Longrightarrow> ret \<in> responses_\<I> \<I> call" shows "\<I> \<turnstile>c callee \<surd>"
aligned_sub_aligned': fixes a :: "'b word"   and n :: "nat"   and b :: "'b word" assumes "is_aligned a n"   and "is_aligned b n"   and "n < LENGTH('b)" shows "is_aligned (a - b) n"
same: shows "b\<^sup>2 - 4 * a * c = (2 * a * r)\<^sup>2 - 4 * a * (a * r\<^sup>2)"
orientable_symmetric: fixes x :: "'a" assumes "orientable x" shows "symmetric x"
comp_assoc: fixes a :: "('b \<times> 'b trm) list"   and b :: "('b \<times> 'b trm) list"   and c :: "('b \<times> 'b trm) list" shows "a \<lozenge> b \<lozenge> c \<doteq> a \<lozenge> (b \<lozenge> c)"
mset_lt_single_right_iff: fixes M :: "'a multiset"   and y :: "'a" shows "(M < {#y#}) = (\<forall>x\<in>#M. x < y)"
r_rev: fixes e :: "nat" shows "eval r_rev [e] \<down>= Partial_Recursive.list_encode (rev (Partial_Recursive.list_decode e))"
GodExImpliesNecEx_v1: shows "\<lfloor>(\<lambda>w. (\<exists>x. \<forall>xa. \<P> xa w \<longrightarrow> xa x) \<longrightarrow> (\<^bold>\<box>(\<lambda>v. \<exists>x. existsAt x v \<and> (\<forall>xa. \<P> xa v \<longrightarrow> xa x))) w)\<rfloor>"
assms: shows "eval r_univ [i, x] \<down>= v"
that: fixes x :: "'b" assumes "(x, y) \<in> S" shows "thesis"
is_ground_atms_def: fixes AA :: "'a set" shows "is_ground_atms AA = Ball AA is_ground_atm"
hcomp_ide_isomorphic: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "B.ide f"   and "g \<cong>\<^sub>B h"   and "src f = trg g" shows "f \<star> g \<cong>\<^sub>B f \<star> h"
Pinv_2: fixes x :: "'a acom" assumes "P x" shows "P (f x)"
lepoll_trans2: fixes A :: "'b set"   and B :: "'c set"   and C :: "'d set" assumes "A \<lesssim> B"   and "B \<approx> C" shows "A \<lesssim> C"
map: shows "\<beta> \<in> contained.Partition \<rightarrow>\<^sub>E G'"
nprod_excTr: fixes P :: "'a set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "prime_ideal R P" shows "(\<forall>i\<le>n. f i \<in> carrier R) \<and> (\<forall>l\<le>n. f l \<notin> P) \<longrightarrow> e\<Pi>\<^bsub>R,n\<^esub> f \<notin> P"
length_concat: fixes xss :: "'b list list" shows "length (concat xss) = sum_list (map length xss)"
assms_3: shows "Sig aS\<^bsub>T\<^esub> \<triangleright> Vx : B \<turnstile> d : C"
box_equals: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a = b"   and "a = c"   and "b = d" shows "c = d"
assms_2: assumes "concurrent_set a xs"   and "x \<parallel> a" shows "G"
lhd_stakeWhile: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c stream" assumes "P (shd xs)" shows "lhd (stakeWhile P xs) = shd xs"
lfinite_llength_enat: fixes xs :: "'g llist" assumes "lfinite xs" shows "\<exists>n. llength xs = enat n"
embranch_lemma: fixes rgs :: "recf list"   and n :: "nat"   and rcs :: "recf list"   and xs :: "nat list" assumes "length rgs = n"   and "length rcs = n"   and "0 < n"   and "\<exists>i<n. rec_exec (rcs ! i) xs = 1 \<and> (\<forall>j<n. j \<noteq> i \<longrightarrow> rec_exec (rcs ! j) xs = 0)"   and "list_all (\<lambda>rf. primerec rf (length xs)) (rgs @ rcs)" shows "rec_exec (rec_embranch (zip rgs rcs)) xs = Embranch (zip (map rec_exec rgs) (map (\<lambda>r args. 0 < rec_exec r args) rcs)) xs"