hcju/isabelleps huggingface dataset:
inR4': fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and G :: "'b formula"   and H :: "'b formula"   and I :: "'b formula"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "\<Gamma> \<Rightarrow> F, G, H, I, \<Delta> \<down> n" shows "\<Gamma> \<Rightarrow> H, I, F, G, \<Delta> \<down> n"
Inf: fixes f :: "'a \<Rightarrow> nat" assumes "infinite A"   and "bij_betw f A UNIV" shows "thesis"
orthogonal_gso: shows "orthogonal (map gs1.gso [0..<length (map of_int_hom.vec_hom fs)])"
add_assoc: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.semigroup_add plus" shows "plus (plus a b) c = plus a (plus b c)"
main: fixes i :: "nat" shows "F k = ffp Gk \<and> h k = ff hk \<and> (3 \<le> i \<longrightarrow> i \<le> k \<longrightarrow> F i \<in> range ffp \<and> \<beta> (Suc i) \<in> range ff)"
getFreshN: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and V :: "'a set"   and n :: "nat" assumes "Generic_Syntax var trm fmla Var FvarsT substT Fvars subst"   and "finite V" shows "set (Generic_Syntax.getFreshN var V n) \<subseteq> var \<and> set (Generic_Syntax.getFreshN var V n) \<inter> V = {} \<and> length (Generic_Syntax.getFreshN var V n) = n \<and> distinct (Generic_Syntax.getFreshN var V n)"
extreme_bound_iff: fixes A :: "'c set"   and X :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and s :: "'c" shows "extreme_bound A less_eq X s = (s \<in> A \<and> (\<forall>c\<in>A. (\<forall>x\<in>X. less_eq x c) \<longrightarrow> less_eq s c) \<and> (\<forall>x\<in>X. less_eq x s))"
card_of_ordIso: fixes A :: "'c set"   and B :: "'d set" shows "(\<exists>f. bij_betw f A B) = (|A| =o |B|)"
Rep_ty: fixes x :: "ty" shows "Rep_ty x \<in> Collect ty_Rep_set"
bernoulli_stream_element_prob_rec': fixes M :: "bool stream measure"   and p :: "real"   and n :: "nat"   and x :: "bool stream" assumes "M = bernoulli_stream p"   and "0 \<le> p"   and "p \<le> 1" shows "\<P>(w in M. stake (Suc n) w = stake (Suc n) x) = \<P>(w in M. stake n w = stake n x) * prob_component p x n"
PPs_closed_plus: fixes s :: "'c \<Rightarrow>\<^sub>0 nat"   and X :: "'c set"   and t :: "'c \<Rightarrow>\<^sub>0 nat" assumes "s \<in> .[X]"   and "t \<in> .[X]" shows "s + t \<in> .[X]"
fps_shift_diff: fixes n :: "nat"   and f :: "'a fps"   and g :: "'a fps" shows "fps_shift n (f - g) = fps_shift n f - fps_shift n g"
matrix_to_iarray_nth: fixes A :: "(('d, 'e) vec, 'f) vec"   and i :: "'f"   and j :: "'e" shows "matrix_to_iarray A !! mod_type_class.to_nat i !! mod_type_class.to_nat j = A $ i $ j"
final_Accessible_dfa: shows "final Accessible.Accessible_dfa = final Accessible_dfa \<inter> Accessible.accessible"
coeffs_reflect_poly: fixes p :: "'a poly" shows "coeffs (reflect_poly p) = rev (dropWhile ((=) (0::'a)) (coeffs p))"
ltln_of_dnf_prop_semantics: fixes \<A> :: "'b ltln fset set"   and \<I> :: "'b ltln set" assumes "finite \<A>" shows "\<I> \<Turnstile>\<^sub>P ltln_of_dnf \<A> = (\<exists>\<Phi>\<in>\<A>. \<forall>\<phi>. \<phi> |\<in>| \<Phi> \<longrightarrow> \<I> \<Turnstile>\<^sub>P \<phi>)"
degree_minus: fixes p :: "'a poly" shows "Polynomial.degree (- p) = Polynomial.degree p"
bind_runit: fixes x :: "'b option" shows "x \<bind> Some = x"
efficient_code': fixes n :: "nat" shows "totient n = (if n = 0 then 0 else \<Prod>(p, d)\<leftarrow>prime_factorization_nat' n. p ^ Suc d - p ^ (Suc d - 1))"
basis_ex: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set" assumes "FinDimVectorSpace smult V" shows "\<exists>vs. set vs \<subseteq> V \<and> V = R_scalar_mult.RSpan UNIV smult vs \<and> R_scalar_mult.R_lin_independent UNIV smult vs"
u2: shows "subst_term (zip (zip fresh_idns (map snd (map fst insts))) (map snd insts)) (subst_term (zip (map fst insts) (map2 Fv fresh_idns (map snd (map fst insts)))) t) = fold (\<lambda>single. subst_term [single]) (zip (zip fresh_idns (map snd (map fst insts))) (map snd insts)) (subst_term (zip (map fst insts) (map2 Fv fresh_idns (map snd (map fst insts)))) t)"
init_tr_dist_nodes: fixes M :: "nat bdd list \<times> bool list"   and q :: "nat"   and p :: "nat"   and v :: "nat" assumes "dfa_is_node M q"   and "p < q" shows "tr_lookup (init_tr M) q p = dist_nodes M 0 v p q"
formulas_sup: fixes x :: "'a formula"   and S :: "'a set"   and y :: "'a formula" assumes "x \<in> formulas S"   and "y \<in> formulas S" shows "x \<squnion> y \<in> formulas S"
CondTrue: fixes s :: "'d"   and b :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com"   and cs :: "('d, 'e, 'f) com list"   and css :: "(('d, 'e, 'f) com list \<times> ('d, 'e, 'f) com list) list" assumes "s \<in> b" shows "\<Gamma>\<turnstile> (Cond b c\<^sub>1 c\<^sub>2 # cs, css, Normal s) \<rightarrow> (c\<^sub>1 # cs, css, Normal s)"
well_valued0_lchild_if_well_valued0: fixes t :: "'a tree" assumes "local.well_valued0 t" shows "local.well_valued0 (lchild t)"
finite_SigmaI: fixes A :: "'g set"   and B :: "'g \<Rightarrow> 'h set" assumes "finite A"   and "\<And>a. a \<in> A \<Longrightarrow> finite (B a)" shows "finite (Sigma A B)"
path0: fixes l1 :: "'vtx"   and l2 :: "'vtx" assumes "l1 = l2" shows "path l1 l2 []"
diamond_sem: fixes I :: "('e, 'f, 'd) interp"   and \<alpha> :: "('e, 'f, 'd) hp"   and \<phi> :: "('e, 'f, 'd) formula" shows "fml_sem I (\<langle> \<alpha> \<rangle> \<phi>) = {\<nu>. \<exists>\<omega>. (\<nu>, \<omega>) \<in> prog_sem I \<alpha> \<and> \<omega> \<in> fml_sem I \<phi>}"
holomorphic_Dirichlet_L_weak: fixes m :: "nat"   and A :: "complex set"   and \<chi> :: "nat \<Rightarrow> complex" assumes "0 < m"   and "1 \<notin> A" shows "Dirichlet_L m \<chi> holomorphic_on A"
mk_monoid_simps_1: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and x :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "(x \<in> carrier \<lparr>carrier = UNIV - {zero}, monoid.mult = times, one = one\<rparr>) = (x \<noteq> zero)"
zero_eq_add_iff_both_eq_0: fixes plus :: "'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.canonically_ordered_monoid_add plus zero less_eq less" shows "(zero = plus x y) = (x = zero \<and> y = zero)"
prime_imp_coprime: fixes one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and gcd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and lcm :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and p :: "'c"   and n :: "'c" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "\<not> dvd.dvd times p n" shows "algebraic_semidom.coprime times one p n"
simps: fixes N :: "('a literal list \<times> nat) list"   and P :: "('a literal list \<times> nat) list"   and Q :: "('a literal list \<times> nat) list"   and n :: "nat" shows "deterministic_RP_step (N, P, Q, n) = (if \<exists>Ci\<in>set (P @ Q). fst Ci = [] then ([], [], remdups_clss P @ Q, n + length (remdups_clss P)) else case N of [] \<Rightarrow> case P of [] \<Rightarrow> (N, P, Q, n) | P0 # P' \<Rightarrow> let (C, i) = select_min_weight_clause P0 P'; N = map (\<lambda>D. (D, n)) (remdups_gen mset (resolve_rename C C @ concat (map (resolve_rename_either_way C \<circ> fst) Q))); P = filter (\<lambda>(D, j). mset D \<noteq> mset C) P; Q = (C, i) # Q; n = Suc n in (N, P, Q, n) | (C, i) # N \<Rightarrow> let C = reduce (map fst (P @ Q)) [] C in if C = [] then ([], [], [([], i)], Suc n) else if is_tautology C \<or> subsume (map fst (P @ Q)) C then (N, P, Q, n) else let P = reduce_all C P; (back_to_P, Q) = reduce_all2 C Q; P = back_to_P @ P; Q = filter (Not \<circ> strictly_subsume [C] \<circ> fst) Q; P = filter (Not \<circ> strictly_subsume [C] \<circ> fst) P; P = (C, i) # P in (N, P, Q, n))"
eventually_nhds_x_imp_x: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "eventually P (nhds x)" shows "P x"
nsubst_sterm: fixes \<sigma> :: "'b \<Rightarrow> ('sf, 'sz) trm"   and \<theta> :: "('sf + 'b, 'sz) trm"   and I :: "('sf, 'sc, 'sz) interp"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "TadmitFFO \<sigma> \<theta>"   and "\<And>i. dsafe (\<sigma> i)" shows "sterm_sem I (TsubstFO \<theta> \<sigma>) (fst \<nu>) = sterm_sem (adjointFO I \<sigma> \<nu>) \<theta> (fst \<nu>)"
numeral_uexpr_simp: fixes x :: "num" shows "numeral x = \<guillemotleft>numeral x\<guillemotright>"
ptolemy_trigonometric: fixes \<omega>\<^sub>1 :: "real"   and \<omega>\<^sub>2 :: "real"   and \<omega>\<^sub>3 :: "real" shows "sin (\<omega>\<^sub>1 + \<omega>\<^sub>2) * sin (\<omega>\<^sub>2 + \<omega>\<^sub>3) = sin \<omega>\<^sub>1 * sin \<omega>\<^sub>3 + sin \<omega>\<^sub>2 * sin (\<omega>\<^sub>1 + \<omega>\<^sub>2 + \<omega>\<^sub>3)"
rep_eq: fixes x :: "'c \<Rightarrow> 'b"   and xa :: "'c fset" shows "fset (x |`| xa) = x ` fset xa"
Signature_axioms: shows "Signature TYPE('tp) TE_wtFsym wtPsym"
inverse_arrow_unique: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and g' :: "'a \<times> 'a" assumes "CC.inverse_arrows f g"   and "CC.inverse_arrows f g'" shows "g = g'"
is_processT6: fixes P :: "'c process" shows "\<forall>s X. (s @ [tick], {}) \<in> F P \<longrightarrow> (s, X - {tick}) \<in> F P"
swap_rows_transfer: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'b mat"   and B :: "'c mat"   and i :: "nat"   and j :: "nat" assumes "mat_rel R A B"   and "i < dim_row B"   and "j < dim_row B" shows "mat_rel R (swaprows i j A) (swaprows i j B)"
mono2mono_ennreal: fixes orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "'c \<Rightarrow> real" assumes "monotone orda (\<le>) t" shows "monotone orda (\<le>) (\<lambda>x. ennreal (t x))"
a: assumes "x = y" shows "filter P (merge_wrt rel xs ys) = filter P ys"
poly_minimum_modulus_disc: fixes r :: "real"   and p :: "complex poly" shows "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
Follows: fixes n :: "nat" shows "(case three.f (Suc n) of (x', y', z') \<Rightarrow> \<lambda>(x, y, z). Follows x' y' z' x y z) (three.f n)"
dghm_ObjMap_vrange: shows "\<R>\<^sub>\<circ> (\<FF>\<lparr>ObjMap\<rparr>) \<subseteq>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
padd_residue_eq: fixes a :: "int"   and p\<^sub>1 :: "int \<times> int \<times> int"   and p\<^sub>2 :: "int \<times> int \<times> int" shows "padd a p\<^sub>1 p\<^sub>2 = mpadd (int p) a p\<^sub>1 p\<^sub>2"
fail_butlast_simp_2: shows "butlast (io_init @ [io_last] || tr1_init @ [snd FAIL]) = io_init || tr1_init"
lcm_proj1_if_dvd: 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 b :: "'b"   and a :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times b a" shows "lcm a b = normalize a"
iIN_not_empty: fixes n :: "nat"   and d :: "nat" shows "[n\<dots>,d] \<noteq> {}"
assms: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> r" shows "\<exists>a b. x = [a, b]\<^sub>\<circ>"
card_of_Plus_empty1: fixes A :: "'c set" shows "|A| =o |A <+> {}|"
hyps: fixes used :: "'b set" assumes "finite used"   and "i \<notin> list_nominals ps" shows "i \<notin> list_nominals (witness_list ps ({i} \<union> used))"
assoc_simps_4: fixes f :: "'a arrow_of_spans_data"   and g :: "'a arrow_of_spans_data"   and h :: "'a arrow_of_spans_data" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "src \<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc (\<cdot>) prj0 prj1 f g h, Dom = Dom ((f \<star> g) \<star> h), Cod = Cod (f \<star> g \<star> h)\<rparr> = src h"
convex_connected: fixes S :: "'b set" assumes "convex S" shows "connected S"
\<Gamma>_def: shows "\<Gamma> \<equiv> web_of_network \<Delta>"
gen_iapath_co_path: fixes e :: "'a \<times> 'a"   and w :: "'a"   and V :: "'a set"   and u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "e \<in> parcs G"   and "w \<notin> pverts G"   and "V \<subseteq> pverts G"   and "pre_digraph.gen_iapath (with_proj (subdivide G e w)) V u p v" shows "gen_iapath V u (co_path e w p) v"
step_pos: fixes i :: "nat"   and x :: "real" assumes "i < k"   and "x\<^sub>1 \<le> x" shows "0 < bs ! i * x + (hs ! i) x"
total_onD: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "total_on A r"   and "x \<in> A"   and "y \<in> A" shows "(x, y) \<in> r \<or> x = y \<or> (y, x) \<in> r"
rel_simps_49: fixes n :: "num" shows "((1::'a) < numeral n) = (num.One < n)"
powr_nat_of_nat: fixes x :: "real"   and n :: "nat" shows "powr_nat x (real n) = x ^ n"
fim: shows "f ` (X - {a}) \<subseteq> T"
simps_3: fixes p :: "perm"   and f :: "'e"   and \<phi> :: "'c" shows "p \<bullet> tPred f \<phi> = tPred (p \<bullet> f) (p \<bullet> \<phi>)"
red_rtrancl_mult: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and c :: "'c"   and t :: "'a" assumes "(red F)\<^sup>*\<^sup>* p q" shows "(red F)\<^sup>*\<^sup>* (monom_mult c t p) (monom_mult c t q)"
vsubstAbs_psubstAbs_compose: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "goodAbs A"   and "goodEnv rho" shows "A $[rho] $[y1 // y]_ys = A $[rho &[y1 // y]_ys]"
ideal_contained_maxid: fixes I :: "'a set" assumes "\<not> zeroring R"   and "ideal R I"   and "1\<^sub>r \<notin> I" shows "\<exists>mx. maximal_ideal R mx \<and> I \<subseteq> mx"
L_def: shows "L \<equiv> \<^bold>\<lambda>x. \<lparr>E!,x\<^sup>P\<rparr> \<^bold>\<rightarrow> \<lparr>E!,x\<^sup>P\<rparr>"
bigI: fixes c :: "real"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "0 < c"   and "\<forall>\<^sub>F x in F. norm (f x) \<le> c * norm (g x)" shows "f \<in> O[F](g)"
inpt_always: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" shows "inpt (\<lambda>(x, y). (\<box> lft_rel_st r) x y) = (\<lambda>(x, y). (\<box> lft_pred_st (inpt_st r)) x y)"
simps_2: fixes v :: "'c" shows "is_heap (T v E E) = True"
induct: fixes P :: "'b up\<^sub>d \<Rightarrow> 'b \<Rightarrow> 'b tree234 \<Rightarrow> bool"   and a0 :: "'b up\<^sub>d"   and a1 :: "'b"   and a2 :: "'b tree234" assumes "\<And>l a r. P (T\<^sub>d l) a r"   and "\<And>l a lr b rr. P (Up\<^sub>d l) a \<langle>lr, b, rr\<rangle>"   and "\<And>l a lr b mr c rr. P (Up\<^sub>d l) a \<langle>lr, b, mr, c, rr\<rangle>"   and "\<And>t1 a t2 b t3 c t4 d t5. P (Up\<^sub>d t1) a \<langle>t2, b, t3, c, t4, d, t5\<rangle>"   and "\<And>v b. P (Up\<^sub>d v) b \<langle>\<rangle>" shows "P a0 a1 a2"
contractible_convex_tweak_boundary_points: fixes S :: "'a set"   and T :: "'a set" assumes "convex S"   and "rel_interior S \<subseteq> T"   and "T \<subseteq> closure S" shows "contractible T"
hyps_2: shows "Q'' \<longmapsto>\<tau> \<prec> Q'"
inverse_unique: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b" assumes "category C"   and "category.inverse_arrows C f g" shows "category.inv C f = g"
take: fixes m :: "nat" shows "C_take m\<cdot>r = C\<^bsup>f m\<^esup>"
open_halfspace_gt: fixes b :: "real"   and a :: "'a" shows "open {x. b < a \<bullet> x}"
\<tau>move1TryRed: fixes P :: "'a prog"   and h :: "'heap"   and v :: "'addr val"   and C :: "String.literal"   and V :: "'b"   and e :: "('b, 'c, 'addr) exp" shows "\<tau>move1 P h (try Val v catch(C V) e)"
absorb1: fixes b :: "'b multiset"   and a :: "'b multiset" assumes "b \<subseteq># a" shows "a \<union># b = a"
memFreshChain_1: fixes p :: "'d"   and l :: "'d list"   and xvec :: "name list" assumes "p mem l"   and "xvec \<sharp>* l" shows "xvec \<sharp>* p"
is_ring_from_is_homomorphism: fixes U :: "'a set set" assumes "is_zariski_open U" shows "ring (\<O> U) (add_sheaf_spec U) (mult_sheaf_spec U) (zero_sheaf_spec U) (one_sheaf_spec U)"
dvd_result: shows "p ^ n dvd k * (p ^ n choose k)"
skl1_inv5_trans: shows "{skl1_inv5} TS.trans skl1 {> skl1_inv5}"
add_Suc_shift: fixes m :: "nat"   and n :: "nat" shows "Suc m + n = m + Suc n"
verticesFrom_v: fixes f :: "face"   and a :: "nat list"   and v :: "nat"   and b :: "nat list" assumes "distinct (vertices f)"   and "vertices f = a @ v # b" shows "verticesFrom f v = v # b @ a"
closedin_union_complement_components: fixes U :: "'b set"   and S :: "'b set"   and c :: "'b set set" assumes "locally connected U"   and "closedin (top_of_set U) S"   and "c \<subseteq> components (U - S)" shows "closedin (top_of_set U) (S \<union> \<Union> c)"
distinct_65: fixes x41 :: "('a, 'b, 'c) com"   and x42 :: "('a, 'b, 'c) com" shows "Seq x41 x42 \<noteq> Throw"
list2policy_def: fixes l :: "('a \<Rightarrow> 'b decision option) list" shows "list2policy l = foldr (\<lambda>x y. x \<Oplus> y) l \<emptyset>"
smallo_power: fixes n :: "nat"   and f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "0 < n"   and "f \<in> o[F](g)" shows "(\<lambda>x. f x ^ n) \<in> o[F](\<lambda>x. g x ^ n)"
simps_2: fixes x :: "Vars.var"   and e :: "exp" shows "isVal (Lam [x]. e) = True"
l_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM_SC_DOM_Components.get_scdom_component Shadow_DOM_SC_DOM_Components.is_strongly_scdom_component_safe Shadow_DOM_SC_DOM_Components.is_weakly_scdom_component_safe Shadow_DOM.get_owner_document get_disconnected_nodes Shadow_DOM.to_tree_order"
emptyProc_pState_inv: fixes prog :: "program" assumes "program_inv prog" shows "pState_inv prog emptyProc"
assms: fixes i :: "'e" assumes "i \<in> Basis" shows "continuous_on s (\<lambda>x. blinfun_apply (f x) i)"
simps: fixes S :: "'a literal multiset \<Rightarrow> 'a literal multiset"   and a1 :: "'s"   and a2 :: "'a list"   and a3 :: "'a literal multiset" shows "eligible S a1 a2 a3 = (\<exists>DA As \<sigma>. a1 = \<sigma> \<and> a2 = As \<and> a3 = DA \<and> (S DA = negs (mset As) \<or> S DA = {#} \<and> length As = 1 \<and> maximal_wrt (As ! 0 \<cdot>a \<sigma>) (DA \<cdot> \<sigma>)))"
is_ide: shows "ide (k\<^sup>* \<star> h)"
induct: fixes P :: "dag \<Rightarrow> bool"   and dag :: "dag" assumes "P Tip"   and "\<And>x1 x2 x3. \<lbrakk>P x1; P x3\<rbrakk> \<Longrightarrow> P (Node x1 x2 x3)" shows "P dag"
axioms_3: fixes A1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'a \<times> 'c \<Rightarrow> 'd" assumes "binary_functor A1 A2 B F" shows "category B"
strategyproofness: shows "pmf (sds R9) c + pmf (sds R9) d \<le> pmf (sds R12) d + pmf (sds R12) c"
addSync: fixes sa :: "'c event list"   and t :: "'c event list"   and A :: "'c event set"   and u :: "'c event list"   and y1 :: "'c event" shows "sa setinterleaves ((t, u), A) \<and> y1 \<in> A \<longrightarrow> sa @ [y1] setinterleaves ((t @ [y1], u @ [y1]), A)"
mult_mod_left: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "times (modulo a b) c = modulo (times a c) (times b c)"
impl_mp1: fixes P :: "'a upred"   and Q :: "'a upred" shows "(P \<and> (P \<Rightarrow> Q)) = (P \<and> Q)"
in_face_cycle_setsD: fixes x :: "'b set" assumes "x \<in> G.face_cycle_sets" shows "x \<subseteq> arcs G"
subformulae_self: fixes F :: "'a formula" shows "F \<in> set (subformulae F)"
val_Integer_simps_1: fixes i :: "int" shows "\<lbrakk>MkI\<cdot>i\<rbrakk> = i"
unsat_state_core_def: fixes s :: "('a, 'b) state" shows "unsat_state_core s = (set (the (\<U>\<^sub>c s)) \<subseteq> indices_state s \<and> (\<nexists>v. (set (the (\<U>\<^sub>c s)), v) \<Turnstile>\<^sub>i\<^sub>s s))"
src_inv: fixes \<mu> :: "'a" assumes "S.iso \<mu>" shows "S.src (S.inv \<mu>) = S.src \<mu>"
connectedD: fixes g :: "'b ugraph"   and u :: "'b"   and v :: "'b" assumes "Undirected_Graph.connected g"   and "u \<in> nodes g"   and "v \<in> nodes g" shows "(u, v) \<in> (edges g)\<^sup>*"
ex_P: shows "\<forall>X. \<exists>\<delta>. subst_domain \<delta> = set X \<and> ground (subst_range \<delta>)"
sg_subset_elem: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set"   and h :: "'c" assumes "Group G"   and "G \<guillemotright> H "   and "h \<in> H" shows "h \<in> carrier G"
l7_17: fixes A :: "'p"   and P :: "'p"   and P' :: "'p"   and B :: "'p" assumes "A Midpoint P P'"   and "B Midpoint P P'" shows "A = B"
Diag_TensorE_2: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Diag (t \<^bold>\<otimes> u)" shows "C.arr (un_Prim t)"
transitionI_2: fixes P :: "pi"   and P''' :: "pi"   and a :: "name"   and x :: "name"   and P'' :: "pi"   and P' :: "pi" assumes "P \<Longrightarrow>\<^sub>\<tau> P'''"   and "P''' \<longmapsto> a<\<nu>x> \<prec> P''"   and "P'' \<Longrightarrow>\<^sub>\<tau> P'" shows "P \<Longrightarrow>a<\<nu>x> \<prec> P'"
IH_1: fixes i :: "int"   and j :: "int" assumes "inorder l = [i..j]"   and "i \<le> j + 1" shows "(\<Sum>x\<in>leaves i l. f (fst x)) = sum f {i..j + 1}"
select_convs_4: fixes states :: "hf"   and init :: "hf"   and final :: "hf"   and "next" :: "hf \<Rightarrow> 'b \<Rightarrow> hf \<Rightarrow> bool"   and more :: "'c" shows "next \<lparr>states = states, init = init, final = final, next = next, \<dots> = more\<rparr> = next"
is_rewrite_ordD5: fixes rword :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and b :: "'t \<Rightarrow>\<^sub>0 'b"   and a :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_rewrite_ord rword"   and "dickson_grading d"   and "is_sig_GB_upt d G (lt b)"   and "a \<in> G"   and "b \<in> G"   and "a \<noteq> 0"   and "b \<noteq> 0"   and "lt a adds\<^sub>t lt b"   and "\<not> is_sig_red (\<prec>\<^sub>t) (=) G b" shows "rword (spp_of a) (spp_of b)"
T_diff_eq: shows "T x - 2 * T (x / 2) = sum_upto (\<lambda>n. a n * real_of_int (\<lfloor>x / real n\<rfloor> - 2 * \<lfloor>x / real (2 * n)\<rfloor>)) (x / 2) + (\<Sum>n | 0 < n \<and> real n \<in> {x / 2<..x}. a n * real_of_int \<lfloor>x / real n\<rfloor>)"
simps_1: fixes h :: "'a \<Rightarrow> 'b"   and x :: "'b blindable\<^sub>h" shows "blind_blindable h (Blinded x) = Blinded x"
rep_dom: fixes f :: "'c local.term set" assumes "arr f" shows "rep (local.dom f) = \<^bold>\<parallel>DOM f\<^bold>\<parallel>"
q01_2: shows "q 1 = a - complex_of_real d"
swap_simps_2: fixes a :: "atom"   and b :: "atom" shows "[(a, b)] \<bullet> b = a"
exp_2dist_def: fixes a :: "proj2"   and b :: "proj2" shows "exp_2dist a b \<equiv> if a = b then 1 else cross_ratio (endpoint_in_S a b) (endpoint_in_S b a) a b"