hcju/isabelleps huggingface dataset:
i_Exec_Stream_output_causal: fixes n :: "nat"   and input1 :: "nat \<Rightarrow> 'd"   and input2 :: "nat \<Rightarrow> 'd"   and output1 :: "nat \<Rightarrow> 'e"   and output_fun :: "'f \<Rightarrow> 'e"   and trans_fun :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and c :: "'f"   and output2 :: "nat \<Rightarrow> 'e" assumes "input1 \<Down> n = input2 \<Down> n"   and "output1 = output_fun \<circ> i_Exec_Comp_Stream trans_fun input1 c"   and "output2 = output_fun \<circ> i_Exec_Comp_Stream trans_fun input2 c" shows "output1 \<Down> n = output2 \<Down> n"
norm_28: fixes P :: "(nat \<Rightarrow> 'g) \<Rightarrow> bool" shows "((\<lambda>s. False) \<R> P) = \<box>P"
that: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"   and "b = \<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>" shows "thesis"
pderivs_lang_UNIV: fixes r :: "'b rexp" shows "pderivs_lang UNIV r = pderivs [] r \<union> pderivs_lang UNIV1 r"
at_within_union: fixes "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and S :: "'b set"   and T :: "'b set" assumes "class.topological_space open" shows "topological_space.at_within open x (S \<union> T) = sup (topological_space.at_within open x S) (topological_space.at_within open x T)"
keys_sum_subset: fixes b :: "'d \<Rightarrow> 'b \<Rightarrow>\<^sub>0 'c"   and I :: "'d set" shows "keys (sum b I) \<subseteq> (\<Union>i\<in>I. keys (b i))"
simps_13: fixes v :: "fm"   and va :: "fm" shows "not (Or v va) = NOT (Or v va)"
set_eq_iff: fixes A :: "'g set"   and B :: "'g set" shows "(A = B) = (\<forall>x. (x \<in> A) = (x \<in> B))"
consis: shows "history_consistent \<theta> (hd_prog p (x # xs)) (x # xs)"
strict_border_suffix: fixes xs :: "'a list"   and ys :: "'a list" assumes "strict_border xs ys" shows "strict_suffix xs ys"
closure_of_complement: fixes X :: "'b topology"   and S :: "'b set" shows "X closure_of (topspace X - S) = topspace X - X interior_of S"
assms_3: fixes c :: "'a \<Rightarrow> 'c"   and d :: "'a \<Rightarrow> 'c" assumes "c \<sim>[F] d" shows "f c \<sim>[F] f d"
DTBranch_def: fixes f :: "('g \<Rightarrow> ('h, 'g) Old_Datatype.node set) \<Rightarrow> 'f"   and M :: "('h, 'g) Old_Datatype.node set" shows "DTBranch f M = (THE u. \<exists>x. M = Old_Datatype.Lim x \<and> u = f x)"
isNSCont_def: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a" shows "isNSCont f a = (\<forall>y. y \<approx> star_of a \<longrightarrow> (*f* f) y \<approx> star_of (f a))"
representsE: fixes p :: "int poly"   and x :: "'b" assumes "p represents x" obtains "p \<noteq> 0" and "ipoly p x = (0::'b)"
valid_states_def: fixes prob :: "(('a, 'c) fmap \<times> ('a, 'd) fmap) set" shows "valid_states prob \<equiv> {s. fmdom' s = prob_dom prob}"
assms_2: shows "s \<in> {s. fmdom' s = X2}"
dj_perm_forget: fixes pi :: "('b \<times> 'b) list"   and x :: "'a" assumes "disjoint TYPE('a) TYPE('b)" shows "pi \<bullet> x = x"
converse_unfold: fixes r :: "('c \<times> 'b) set" shows "r\<inverse> = {(y, x). (x, y) \<in> r}"
strict_implies_not_eq: fixes b :: "'t"   and a :: "'t" assumes "b \<prec>\<^sub>t a" shows "a \<noteq> b"
set_mult_card_le: fixes H :: "'a set"   and J :: "'a set" assumes "finite H"   and "H \<subseteq> carrier G"   and "J \<subseteq> carrier G" shows "card (H <#> J) \<le> card H * card J"
QR_lemma_10: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "QR.a p q = QR.c p q"
max_add_distrib_left: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "plus (ord.max less_eq x y) z = ord.max less_eq (plus x z) (plus y z)"
lebesgue_affine_euclidean: fixes c :: "'b \<Rightarrow> real"   and t :: "'b" assumes "\<And>j. j \<in> Basis \<Longrightarrow> c j \<noteq> 0" shows "lebesgue = density (distr lebesgue lebesgue (\<lambda>x. t + (\<Sum>j\<in>Basis. (c j * (x \<bullet> j)) *\<^sub>R j))) (\<lambda>x. ennreal (\<Prod>j\<in>Basis. \<bar>c j\<bar>))"
unit_simps_3: fixes a :: "'c" assumes "C.obj a" shows "trg\<^sub>D (G.unit a) = G.map\<^sub>0 a"
ex_iso_to_coxeter_complex': shows "\<exists>\<psi>. ChamberComplexIsomorphism X (CoxeterComplex.TheComplex (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})) \<psi>"
intros_3: assumes "\<S>\<bullet>\<C> \<turnstile> [$e1, $e2] : t1s _> t2s" shows "\<S>\<bullet>\<C> \<turnstile> [$e1, $e2] : ts @ t1s _> ts @ t2s"
map2_tl: fixes t :: "'b list"   and s :: "'c list"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "length t = length s" shows "Map2.map2 f (tl t) (tl s) = tl (Map2.map2 f t s)"
linorder_rank_le_iff: fixes B :: "'b set"   and R :: "('b \<times> 'b) set"   and A :: "'b set"   and x :: "'b"   and y :: "'b" assumes "linorder_on B R"   and "finite A"   and "A \<subseteq> B"   and "x \<in> A"   and "y \<in> A" shows "(linorder_rank R A x \<le> linorder_rank R A y) = ((x, y) \<in> R)"
Times_subset_cancel2: fixes x :: "'c"   and C :: "'c set"   and A :: "'d set"   and B :: "'d set" assumes "x \<in> C" shows "(A \<times> C \<subseteq> B \<times> C) = (A \<subseteq> B)"
quo_lemma1: fixes x :: "nat"   and y :: "nat" shows "rec_exec rec_quo [x, y] = quo [x, y]"
leg1_simps_2: shows "hseq (tab\<^sub>1 f \<star> Tfg.p\<^sub>1) TTfgh.p\<^sub>1"
returns_to_congI1: fixes Y :: "'a set"   and g :: "'a \<Rightarrow> 'a"   and A :: "'a set"   and x :: "'a" assumes "Y = Y"   and "\<And>x t. x \<in> Y \<Longrightarrow> g x = g x"   and "Y.returns_to A x" shows "auto_ll_on_open.returns_to g Y A x"
abs_eq: fixes xa :: "'b"   and x :: "'b" shows "Interval' xa x = map_option Interval (if xa \<le> x then Some (xa, x) else None)"
enum_0_bot: fixes a :: "nat \<Rightarrow> nat" assumes "a \<in> s" shows "(a = s.enum 0) = (\<forall>a'\<in>s. a \<le> a')"
resPres: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and x :: "name" assumes "P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"   and "\<And>R S x. (R, S) \<in> Rel \<Longrightarrow> (<\<nu>x>R, <\<nu>x>S) \<in> Rel'"   and "Rel \<subseteq> Rel'"   and "eqvt Rel"   and "eqvt Rel'" shows "<\<nu>x>P \<leadsto>\<guillemotleft>Rel'\<guillemotright> <\<nu>x>Q"
euclerian_path_ex: fixes v :: "kon_node"   and ps :: "(kon_node \<times> kon_bridge \<times> kon_node) list"   and v' :: "kon_node" assumes "kon_graph.is_Eulerian_trail v ps v'"   and "finite kon_graph.V"   and "finite kon_graph.E" shows "(\<forall>v\<in>kon_graph.V. even (degree v kon_graph)) \<or> num_of_odd_nodes kon_graph = 2"
comp_min_basis_pmdl: fixes xs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "is_Groebner_basis (set xs)" shows "pmdl (set (comp_min_basis xs)) = pmdl (set xs)"
tab\<^sub>0_simps_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and r :: "'b" assumes "bicategory_of_spans V H \<a> \<i> src trg"   and "partial_magma.ide V r" shows "partial_magma.ide V (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg r)"
tedious_reasoning: fixes M0 :: "'a multiset"   and a :: "'a" assumes "accessible_on (mulex1 P) (multisets A) M0"   and "a \<in> A"   and "M0 \<in> multisets A"   and "\<And>b. \<lbrakk>b \<in> A; P b a\<rbrakk> \<Longrightarrow> \<forall>M. accessible_on (mulex1 P) (multisets A) M \<longrightarrow> accessible_on (mulex1 P) (multisets A) (M + {#b#})"   and "\<forall>M. M \<in> multisets A \<and> mulex1 P M M0 \<longrightarrow> accessible_on (mulex1 P) (multisets A) (M + {#a#})" shows "accessible_on (mulex1 P) (multisets A) (add_mset a M0)"
assms_3: shows "real x\<^sub>0 \<le> b * real x\<^sub>1 + c"
ord_spmf_bindI: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'a spmf"   and q :: "'a spmf"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b spmf"   and g :: "'a \<Rightarrow> 'b spmf" assumes "ord_spmf R p q"   and "\<And>x y. R x y \<Longrightarrow> ord_spmf P (f x) (g y)" shows "ord_spmf P (p \<bind> f) (q \<bind> g)"
oalist_inv_raw_map2_val_pair: fixes x :: "'o"   and xs :: "('a \<times> 'e) list"   and ys :: "('a \<times> 'f) list"   and g :: "('a \<times> 'e) list \<Rightarrow> ('a \<times> 'g) list"   and h :: "('a \<times> 'f) list \<Rightarrow> ('a \<times> 'g) list"   and f :: "'a \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'g" assumes "oalist_inv_raw x xs"   and "oalist_inv_raw x ys"   and "map2_val_compat x g"   and "map2_val_compat x h" shows "oalist_inv_raw x (map2_val_pair x f g h xs ys)"
assms: shows "(k, \<gamma>) \<in> boundary rot_circle_cube"
deg_sect_def: fixes X :: "'c set"   and d :: "nat" shows "deg_sect X d = .[X] \<inter> {t. deg_pm t = d}"
sets_lim_sequence: fixes M :: "'b measure"   and K :: "'b \<Rightarrow> 'b measure"   and x :: "'b" assumes "discrete_Markov_process M K"   and "x \<in> space M" shows "sets (discrete_Markov_process.lim_sequence M K x) = sets (Pi\<^sub>M UNIV (\<lambda>i. M))"
psubsetI: fixes A :: "'d set"   and B :: "'d set" assumes "A \<subseteq> B"   and "A \<noteq> B" shows "A \<subset> B"
id_converter_eq_self: fixes \<I> :: "('y, 'z) \<I>"   and \<I>' :: "('y, 'z) \<I>" assumes "\<I> \<le> \<I>'" shows "\<I>,\<I>' \<turnstile>\<^sub>C 1\<^sub>C \<sim> 1\<^sub>C"
outs_bind_gpv: fixes x :: "('g, 'd, 'f) gpv"   and f :: "'g \<Rightarrow> ('e, 'd, 'f) gpv" shows "outs'_gpv (x \<bind> f) = outs'_gpv x \<union> (\<Union>x\<in>results'_gpv x. outs'_gpv (f x))"
AE_component: fixes i :: "nat"   and P :: "real \<Rightarrow> bool" assumes "i \<in> UNIV"   and "almost_everywhere (exponential (R i)) P" shows "AE x in Pi\<^sub>M UNIV (\<lambda>n. exponential (R n)). P (x i)"
D0_def: shows "D0 = denote_while_n M0 M1 DS"
zero_order_3: fixes n :: "'a multiset" shows "\<not> n \<subset># {#}"
w2: shows "(WHILE b DO c, ps) \<Rightarrow>\<^sub>A 1 \<Down> ps"
degree_0_id: fixes p :: "'a poly" assumes "degree p = 0" shows "[:poly.coeff p 0:] = p"
mahler_approximation: fixes d :: "nat"   and f :: "int poly"   and bnd :: "nat" shows "\<lfloor>real d * mahler_measure f\<rfloor> \<le> mahler_approximation bnd d f"
eval_work_def: shows "eval_work \<equiv> fix\<cdot>(unwrap oo evalS_body oo wrap)"
rInput: fixes K :: "'a"   and Tvec :: "'a list" assumes "\<Psi> \<turnstile> M \<leftrightarrow> K"   and "set xvec \<subseteq> supp N"   and "length xvec = length Tvec"   and "distinct xvec"   and "\<alpha> = K\<lparr>N[xvec::=Tvec]\<rparr>"   and "P' = P[xvec::=Tvec]" shows "Prop (K\<lparr>N[xvec::=Tvec]\<rparr>) (P[xvec::=Tvec])"
pp_of_Keys_init_syzygy_list_superset: fixes bs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "pp_of_term ` Keys (set bs) \<subseteq> pp_of_term ` Keys (set (init_syzygy_list bs))"
findMin_def: fixes bq :: "('c, 'd) BinomialTree list" shows "findMin bq = (let min = getMinTree bq in (val min, prio min))"
ivl_disj_int_two_8: fixes l :: "'a"   and m :: "'a"   and u :: "'a" shows "{l..m} \<inter> {m<..u} = {}"
sums_integral_count_space_nat: fixes f :: "nat \<Rightarrow> 'd" assumes "integrable (count_space UNIV) f" shows "f sums integral\<^sup>L (count_space UNIV) f"
assms_4: shows "((\<lambda>x. ln (f x) * g x) expands_to E) basis"
edge_weight_alt: fixes G :: "('b, 'a) graph" shows "edge_weight G = (\<Sum>(u, w, v)\<in>edges G. w)"
assms_4: shows "\<not> (b \<and> (x \<or> \<not> x)) \<or> c"
Func_empty: fixes B :: "'b set" shows "Func {} B = {\<lambda>x. undefined}"
axioms: fixes R :: "('c, 'd) ring_scheme" assumes "UP_cring R" shows "cring R"
tminus_le: fixes b :: "real"   and a :: "real" assumes "b \<le> a" shows "a \<ominus> b = a - b"
distinct_17: fixes x11 :: "'g"   and x12 :: "'d \<Rightarrow> 'd set"   and x13 :: "('d, 'e, 'f, 'g) seqp"   and x10 :: "'f" shows "{x11}\<langle>x12\<rangle> x13 \<noteq> call(x10)"
frontier_real_greaterThan: fixes a :: "real" shows "frontier {a<..} = {a}"
reachable_s_finite_thm: fixes PROB :: "(('a, bool) fmap \<times> ('a, bool) fmap) set"   and s :: "('a, bool) fmap" assumes "finite PROB"   and "s \<in> valid_states PROB" shows "finite (reachable_s PROB s)"
strand_sem_Send_split_2: fixes M :: "('fun, 'var) Term.term set"   and T :: "('fun, 'var) Term.term list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and t :: "('fun, 'var) Term.term" assumes "\<lbrakk>M; map Send T\<rbrakk>\<^sub>d \<theta>"   and "t \<in> set T" shows "\<lbrakk>M; [send\<langle>t\<rangle>\<^sub>s\<^sub>t]\<rbrakk>\<^sub>d \<theta>"
Brzozowski_def: shows "Brzozowski \<equiv> dfa.Accessible_dfa (nfa.Power_dfa (Reverse_nfa (dfa.Accessible_dfa (nfa.Power_dfa (Reverse_nfa M)))))"
simps_2: fixes e1 :: "exp"   and e2 :: "exp"   and S :: "stack_elem list" shows "upds_list (Alts e1 e2 # S) = upds_list S"
distinct_82: fixes x8 :: "shift_type"   and x5 :: "load_store_type" shows "shift_type x8 \<noteq> load_store_type x5"
G_ide: fixes x :: "'c" assumes "C.ide x" shows "G x = Go x"
normalize_gcd: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "normalize (gcd a b) = gcd a b"
hom_into_subgroup: fixes h :: "'e \<Rightarrow> 'f"   and G :: "('e, 'g) monoid_scheme"   and G' :: "('f, 'h) monoid_scheme"   and H :: "'f set" assumes "h \<in> hom G G'"   and "h ` carrier G \<subseteq> H" shows "h \<in> hom G (subgroup_generated G' H)"
fv_Pair: fixes x :: "'b"   and y :: "'c" shows "fv (x, y) = fv x \<union> fv y"
path_from_to_ends: fixes v :: "'a"   and xs :: "'a list"   and w :: "'a"   and ys :: "'a list" assumes "v \<leadsto>(xs @ w # ys)\<leadsto>\<^bsub>G\<^esub> w" shows "ys = []"
mhead_ext_type_resTy_def: fixes m :: "'a mhead_scheme" shows "resTy m \<equiv> resT m"
sdp_SetDC: fixes S :: "'a \<Rightarrow> 'b set"   and p :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>s a. a \<in> S s \<Longrightarrow> sub_distrib_pconj (p a)"   and "\<And>s a. a \<in> S s \<Longrightarrow> healthy (wp (p a))"   and "\<And>s a. a \<in> S s \<Longrightarrow> nearly_healthy (wlp (p a))"   and "\<And>s. S s \<noteq> {}" shows "sub_distrib_pconj (SetDC p S)"
dim_gen_eigenspace_def: fixes A :: "'a mat"   and ev :: "'a"   and k :: "nat" shows "dim_gen_eigenspace A ev k = kernel_dim (char_matrix A ev ^\<^sub>m k)"
set_less_eq_finite_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and B :: "'b set" assumes "class.linorder less_eq less"   and "finite A"   and "finite B" shows "ord.set_less_eq less_eq A B = ord.set_less_eq_aux less_eq A B"
while_least: fixes L :: "'a set"   and f :: "'a \<Rightarrow> 'a"   and b :: "'a"   and q :: "'a"   and P :: "'a \<Rightarrow> bool"   and p :: "'a" assumes "\<forall>x\<in>L. \<forall>y\<in>L. x \<le> y \<longrightarrow> f x \<le> f y"   and "\<forall>x. x \<in> L \<longrightarrow> f x \<in> L"   and "\<forall>x\<in>L. b \<le> x"   and "b \<in> L"   and "f q \<le> q"   and "q \<in> L"   and "while_option P f b = Some p" shows "p \<le> q"
c_4: fixes stateful_strand_step :: "('e, 'f) stateful_strand_step" assumes "is_Insert stateful_strand_step" shows "insert\<langle>the_elem_term stateful_strand_step,the_set_term stateful_strand_step\<rangle> = stateful_strand_step"
intros_10: fixes v :: "Syntax.term"   and va :: "hash" assumes "value v" shows "value (Hashed va v)"
simps_2: fixes n :: "nat"   and b :: "'b set"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b"   and s' :: "'b" shows "iter (Suc n) b S s s' = (s \<in> b \<and> (\<exists>s''. S s s'' \<and> iter n b S s'' s'))"
diagonal_functor_axioms: shows "diagonal_functor (\<cdot>\<^sub>J) (\<cdot>)"
assms_1: shows "\<FF>\<lparr>ArrMap\<rparr>\<lparr>f\<rparr> = \<FF>\<lparr>ArrMap\<rparr>\<lparr>g\<rparr>"
cases: fixes r :: "point3d" obtains xc :: "int" and yc :: "int" and zc :: "int" where "r = \<lparr>xc = xc, yc = yc, zc = zc\<rparr>"
simps: fixes xp :: "nat option"   and h :: "heap"   and frs :: "frame list"   and sh :: "sheap" shows "frames_of (xp, h, frs, sh) = frs"
if_false: fixes mem :: "'Var \<Rightarrow> 'Val"   and b :: "'BExp"   and t :: "('Var, 'AExp, 'BExp) Stmt"   and e :: "('Var, 'AExp, 'BExp) Stmt" assumes "\<not> ev\<^sub>B mem b" shows "eval\<^sub>w_simple_abv (Stmt.If b t e, mem) (e, mem)"
ac_Zp_factors_x_2: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "x \<noteq> \<zero>" shows "x = [p] \<cdot> \<one> [^] ord_Zp x \<otimes> ac_Zp x"
nf_ACI_D_3: fixes \<phi> :: "('a, 'b) aformula" assumes "nf_ACI \<phi>" shows "distinct (disjuncts_list \<phi>)"
cont2contlubE: fixes f :: "'d \<Rightarrow> 'e"   and Y :: "nat \<Rightarrow> 'd" assumes "cont f"   and "chain Y" shows "f (\<Squnion>i. Y i) = (\<Squnion>i. f (Y i))"
seq_if_composable: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "\<nu> \<star> \<mu> \<noteq> null" shows "src \<nu> = trg \<mu>"
pure_def: fixes P :: "'a \<Rightarrow> bool" shows "pure P \<equiv> \<forall>h h'. P h = P h'"
Fourier_coefficient_minus: fixes f :: "real \<Rightarrow> real"   and i :: "nat" assumes "f absolutely_integrable_on {- pi..pi}" shows "Fourier_coefficient (\<lambda>x. - f x) i = - Fourier_coefficient f i"
neg: shows "negligible {x + y |x y. x \<in> S - {0::'a} \<and> y \<in> T'}"
dom_def: fixes x :: "'a" shows "local.dom x = (\<exists>k. Q ((h ^^ k) x))"
a_Rec_inv_I: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and x :: "Syntax.var"   and e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau>\<^sub>1 :: "ty"   and \<tau>\<^sub>2 :: "ty" assumes "\<Gamma> \<turnstile> Rec x e, eP, eV : Fun \<tau>\<^sub>1 \<tau>\<^sub>2"   and "atom x \<sharp> \<Gamma>" obtains y :: "Syntax.var" and e' :: "Syntax.term" and eP' :: "Syntax.term" and eV' :: "Syntax.term" where "e = Lam y e'" and "eP = Rec x (Lam y eP')" and "eV = Rec x (Lam y eV')" and "atom y \<sharp> (\<Gamma>, x)" and "\<Gamma>(x $$:= Fun \<tau>\<^sub>1 \<tau>\<^sub>2) \<turnstile> Lam y e', Lam y eP', Lam y eV' : Fun \<tau>\<^sub>1 \<tau>\<^sub>2"
gen_igbg_impl_rel_eext_def: fixes Rm :: "('i \<times> 'k) set"   and Racc :: "('h \<times> ('j \<Rightarrow> nat set)) set" shows "\<langle>Rm, Racc\<rangle>gen_igbg_impl_rel_eext = {(\<lparr>igbgi_num_acc = num_acci, igbgi_acc = acci, \<dots> = mi\<rparr>, \<lparr>igbg_num_acc = num_acc, igbg_acc = acc, \<dots> = m\<rparr>) |num_acci acci mi num_acc acc m. (num_acci, num_acc) \<in> nat_rel \<and> (acci, acc) \<in> Racc \<and> (mi, m) \<in> Rm}"
right: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme"   and x :: "'d"   and y :: "'e" assumes "galois_connection G"   and "x \<in> carrier \<X>\<^bsub>G\<^esub>"   and "y \<in> carrier \<Y>\<^bsub>G\<^esub>"   and "\<pi>\<^sup>*\<^bsub>G\<^esub> x \<sqsubseteq>\<^bsub>\<Y>\<^bsub>G\<^esub>\<^esub> y" shows "x \<sqsubseteq>\<^bsub>\<X>\<^bsub>G\<^esub>\<^esub> \<pi>\<^sub>*\<^bsub>G\<^esub> y"
iso_point_subset_block_iff: fixes \<V> :: "'c set"   and \<B> :: "'c set multiset"   and \<V>' :: "'d set"   and \<B>' :: "'d set multiset"   and bij_map :: "'c \<Rightarrow> 'd"   and p :: "'c set"   and bl :: "'c set" assumes "incidence_system_isomorphism \<V> \<B> \<V>' \<B>' bij_map"   and "p \<subseteq> \<V>"   and "bl \<in># \<B>" shows "(p \<subseteq> bl) = (bij_map ` p \<subseteq> bij_map ` bl)"
word_suicide_l: fixes X :: "'a list set"   and Y :: "'a list set" assumes "\<not> l_ewp X"   and "Y \<noteq> {}" shows "\<forall>w\<in>Y. \<exists>n. w \<notin> X\<^bsup>Suc n\<^esup> \<cdot> Y"
line_in: fixes s :: "real" assumes "0 \<le> s"   and "s \<le> 1" shows "t + s * h \<in> {t..u}"
connected_component_of_sym: fixes X :: "'b topology"   and x :: "'b"   and y :: "'b" shows "connected_component_of X x y = connected_component_of X y x"
create_element_preserves_wellformedness_3: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and tag :: "char list"   and h' :: "(_) heap" assumes "heap_is_wellformed h"   and "h \<turnstile> create_element document_ptr tag \<rightarrow>\<^sub>h h'"   and "type_wf h"   and "known_ptrs h" shows "known_ptrs h'"
insert: fixes G :: "'a triple set"   and t :: "'a triple"   and ts :: "'a triple set" assumes "G|-t"   and "G||-ts" shows "G||-insert t ts"
simps_10: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" shows "Can \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[t, u, v\<^bold>] = (Can t \<and> Can u \<and> Can v)"
entails_transitive: fixes S1 :: "'at Literal set set"   and S2 :: "'at Literal set set"   and S3 :: "'at Literal set set" assumes "entails_formula S1 S2"   and "entails_formula S2 S3" shows "entails_formula S1 S3"
interleavings_set: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "interleavings xs ys zs" shows "set xs \<union> set ys = set zs"
pop_hd: fixes a :: "'a"   and u :: "'a list" shows "f (a # u) = f [a] \<cdot> f u"
simps_21: fixes t :: "'b local.term"   and u :: "'b local.term"   and va :: "'b local.term"   and vb :: "'b local.term" shows "(t \<^bold>\<star> u) \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> va \<^bold>\<star> vb = t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> va \<^bold>\<star> vb"
isUCont_Cauchy: fixes f :: "'a \<Rightarrow> 'b"   and X :: "nat \<Rightarrow> 'a" assumes "isUCont f"   and "Cauchy X" shows "Cauchy (\<lambda>n. f (X n))"
rn_rev_expl_cond_expect: fixes der :: "bool stream \<Rightarrow> real"   and matur :: "nat"   and N :: "bool stream measure"   and q :: "real"   and n :: "nat"   and w :: "bool stream" assumes "der \<in> borel_measurable (G matur)"   and "N = bernoulli_stream q"   and "0 < q"   and "q < 1"   and "n \<le> matur" shows "rn_rev_price N der matur n w = expl_cond_expect N (proj_stoch_proc geom_proc (matur - n)) (discounted_value r (\<lambda>m. der) n) w"
assms_1: shows "InvariantEquivalentZL (getF state) (getM state) F0"
Immediate: fixes G :: "prog"   and membr :: "qtname \<times> memberdecl"   and "class" :: "qtname"   and accclass :: "qtname" assumes "G \<turnstile> membr member_in class"   and "G \<turnstile> membr in class permits_acc_from accclass" shows "G \<turnstile> membr in class dyn_accessible_from accclass"
partial_order_onD_1: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" assumes "partial_order_on A r" shows "refl_on A r"
entails_trans: fixes N1 :: "'g set"   and N2 :: "'g set"   and N3 :: "'g set" assumes "N1 \<Turnstile>G N2"   and "N2 \<Turnstile>G N3" shows "N1 \<Turnstile>G N3"