hcju/isabelleps huggingface dataset:
borel_measurable_has_bochner_integral: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'e"   and x :: "'e" assumes "has_bochner_integral M f x" shows "f \<in> borel_measurable M"
line_integral_on_path_eq_line_integral_on_equiv_chain'_2: fixes \<gamma> :: "real \<Rightarrow> real \<times> real"   and one_chain :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and basis :: "(real \<times> real) set" assumes "chain_subdiv_path \<gamma> one_chain"   and "boundary_chain one_chain"   and "line_integral_exists F basis \<gamma>"   and "\<forall>(k, \<gamma>)\<in>one_chain. valid_path \<gamma>"   and "finite basis" shows "\<forall>(k, \<gamma>)\<in>one_chain. line_integral_exists F basis \<gamma>"
order_trans: fixes x :: "'b word"   and y :: "'b word"   and z :: "'b word" assumes "x \<le>s y"   and "y \<le>s z" shows "x \<le>s z"
l_create_element\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool" shows "l_create_element\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms known_ptr \<equiv> known_ptr = ShadowRootClass.known_ptr"
return_cong: fixes A :: "'b measure"   and B :: "'b measure"   and x :: "'b" assumes "sets A = sets B" shows "return A x = return B x"
ss423: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "is_map x" shows "(y ; x \<le> z) = (y \<le> z ; x\<^sup>T)"
quot_set_quot_set: fixes A :: "'a set"   and x :: "'a"   and y :: "'a" shows "A \<div> x \<div> y = A \<div> x * y"
of_bl_append_same: fixes X :: "bool list"   and w :: "'b word" shows "of_bl (X @ to_bl w) = w"
ide_leg0: shows "ide f\<^sub>0g\<^sub>1.p\<^sub>0"
lincomb_union2: fixes A :: "'a vec set"   and B :: "'a vec set"   and a :: "'a vec \<Rightarrow> 'a"   and b :: "'a vec \<Rightarrow> 'a" assumes "finite (A \<union> B)"   and "A \<union> B \<subseteq> carrier_vec (n + 1)"   and "A \<inter> B = {}" shows "next_dim.lincomb a A + next_dim.lincomb b B = next_dim.lincomb (\<lambda>v. if v \<in> A then a v else b v) (A \<union> B)"
assms_5: shows "R_I = genericJoin I Q_I_pos Q_I_neg"
power2_abs: fixes 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 times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "power.power one times (abs a) 2 = power.power one times a 2"
SN_relto_Un: fixes Q :: "('a \<times> 'a) set"   and R :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set" shows "SN (Q\<^sup>* O (R \<union> S) O Q\<^sup>*) = (SN ((S \<union> Q)\<^sup>* O R O (S \<union> Q)\<^sup>*) \<and> SN (Q\<^sup>* O S O Q\<^sup>*))"
comp_arr_ide: fixes a :: "('a, 'a set) Maps.arr"   and f :: "('a, 'a set) Maps.arr" assumes "Maps.ide a"   and "Maps.seq f a" shows "f \<odot> a = f"
first_edge_hd_tl: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and xs :: "'c list"   and w :: "'c" assumes "Digraph G"   and "v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w"   and "tl xs \<noteq> []" shows "first_edge_of_walk xs = (v, hd (tl xs))"
numeral_fps_const: fixes k :: "num" shows "numeral k = fps_const (numeral k)"
interior_cball: fixes x :: "'c"   and e :: "real" shows "interior (cball x e) = ball x e"
scf_automorphism: fixes agents :: "'a set"   and alts :: "'b set"   and scf :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b set"   and \<sigma> :: "'b \<Rightarrow> 'b"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "an_scf agents alts scf"   and "\<sigma> permutes alts"   and "pref_profile_wf agents alts R"   and "image_mset (map ((`) \<sigma>)) (anonymous_profile R) = anonymous_profile R" shows "\<sigma> ` scf R = scf R"
2: fixes y :: "(char list multiset \<times> int) list"   and s :: "(char list list \<times> int) list" assumes "(p, y) \<in> \<langle>unsorted_term_poly_list_rel \<times>\<^sub>r int_rel\<rangle>list_rel"   and "p' = mset y"   and "map (\<lambda>(a, y). (mset a, y)) (rev p) = map (\<lambda>(a, y). (mset a, y)) s"   and "\<forall>x\<in>set s. sorted_wrt var_order (fst x)" shows "mset y = {#case x of (a, x) \<Rightarrow> (mset a, x). x \<in># mset s#}"
rep_eq: fixes x :: "'b"   and xa :: "'b fset" shows "(x |\<in>| xa) = (x \<in> fset xa)"
run_put_env: fixes put :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and s :: "'c"   and m :: "('b, 'a) envT"   and r :: "'b" shows "run_env (put_env put s m) r = put s (run_env m r)"
ineq_dfa_accepting: fixes l :: "int"   and m :: "int"   and n :: "nat"   and ks :: "int list"   and bss :: "bool list list" assumes "(l, m) \<in> (succsr (dioph_ineq_succs n ks))\<^sup>*"   and "list_all (is_alph n) bss" shows "dfa_accepting (ineq_dfa n ks l) (dfa_steps (ineq_dfa n ks l) (the (fst (dioph_ineq_dfs n ks l) ! int_encode m)) bss) = (eval_dioph ks (nats_of_boolss n bss) \<le> m)"
const_ms_aux_def: fixes c :: "real" shows "const_ms_aux c = MSLCons (const_expansion c, 0) MSLNil"
fst_convol: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f" shows "fst \<circ> <f , g> = f"
star_of_inject: fixes x :: "'b"   and y :: "'b" shows "(star_of x = star_of y) = (x = y)"
pairs_phi_less_eq: fixes x :: "'a"   and y :: "'b Filters.filter" assumes "(x, y) \<in> pairs" shows "phi (- x) \<le> y"
ax_uniqueness: fixes P :: "Points"   and l :: "Lines"   and Q :: "Points"   and m :: "Lines" shows "incid P l \<longrightarrow> incid Q l \<longrightarrow> incid P m \<longrightarrow> incid Q m \<longrightarrow> P = Q \<or> l = m"
simps_7: fixes P :: "pi"   and Q :: "pi" shows "valid (P \<oplus> Q) = (valid P \<and> valid Q)"
is_ground_subst_list_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and \<sigma>s :: "'c list" shows "substitution_ops.is_ground_subst_list subst_atm \<sigma>s = (\<forall>\<sigma>\<in>set \<sigma>s. substitution_ops.is_ground_subst subst_atm \<sigma>)"
Suc_2: shows "y < output_size (shallow_model' Z M (Suc N))"
simps: fixes pre :: "'a word"   and len :: "nat" shows "ipcidr_to_interval_end (pre, len) = (let netmask = mask len << LENGTH('a) - len; network_prefix = pre && netmask in network_prefix || ~~ netmask)"
r2s_inter_hom_var: fixes R :: "('b \<times> 'b) set"   and S :: "('b \<times> 'b) set" assumes "R \<subseteq> Id"   and "S \<subseteq> Id" shows "r2s (R ; S) = r2s R \<inter> r2s S"
assms: fixes x :: "'a" shows "(x \<in> S) = (a + x \<in> T)"
basic_cong_1: fixes p :: "int" assumes "2 < p" shows "[- 1 \<noteq> 1] (mod p)"
nth_cons_def: fixes s :: "'s"   and n :: "nat" shows "local.nth_cons s n = unstream g s ! n"
zero_vec_zero: shows "0\<^sub>v 0 = vNil"
simps_2: fixes f :: "'h \<Rightarrow> 'g \<Rightarrow> 'g"   and x :: "'h"   and xs :: "'h list" shows "foldr f (x # xs) = f x \<circ> foldr f xs"
intros_1: fixes u :: "'b \<Rightarrow> real"   and x :: "'b"   and d :: "nat" assumes "u x = real d" shows "intv_elem x u (Const d)"
assms_1: shows "run (ref x) \<sigma> \<sigma>' r"
preSetpreList_eq: fixes xs :: "('a \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> 'b) list"   and C :: "acom"   and l :: "'a \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" shows "preList xs C l s = preSet (set xs) C l s"
Cast_\<tau>red1r_xt: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and T :: "ty" assumes "\<tau>red1gr uf P t h (e, xs) (e', xs')" shows "\<tau>red1gr uf P t h (Cast T e, xs) (Cast T e', xs')"
at_least_at_most_def: fixes A :: "('a, 'b) gorder_scheme"   and l :: "'a"   and u :: "'a" shows "\<lbrace>l..u\<rbrace>\<^bsub>A\<^esub> = {x \<in> carrier A. l \<sqsubseteq>\<^bsub>A\<^esub> x \<and> x \<sqsubseteq>\<^bsub>A\<^esub> u}"
T_subprob: fixes I :: "'s pmf" shows "measure_pmf.random_variable I (subprob_algebra (stream_space (count_space UNIV))) T"
K2_centre_on_equator_meridian_1: shows "proj2_incident K2_centre equator"
ge_0_from_bool: fixes P :: "bool" shows "(0 < from_bool P) = P"
bounded_byI2: fixes P :: "'c \<Rightarrow> real"   and b :: "real" assumes "P \<tturnstile> \<lambda>s. b" shows "bounded_by b P"
Number_Aenc: fixes X :: "nat"   and X' :: "msg"   and Y' :: "msg" shows "Number X \<noteq> Aenc X' Y'"
rtrancl_imp_singleton_seq2: fixes x :: "'a"   and y :: "'a" assumes "x \<rightarrow>\<^sub>i* y" shows "x = y \<or> (\<exists>s. s \<noteq> [] \<and> tl s \<noteq> [] \<and> s ! 0 = x \<and> s ! (length s - 1) = y \<and> (\<forall>i<length s - 1. s ! i \<rightarrow>\<^sub>i s ! Suc i))"
simps_1: fixes pi :: "('b \<times> 'b) list" shows "pi \<bullet> [] = []"
mul_instr_result: fixes i :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "fst i \<in> {arith_type UMUL, arith_type SMUL, arith_type SMULcc}" shows "snd (mul_instr i s) = False"
iso_char: fixes \<phi> :: "'a" assumes "Left_a.arr \<phi>" shows "Left_a.iso \<phi> = local.iso \<phi>"
Acc_to_mojmir_accept: fixes \<psi> :: "'a ltl"   and \<pi> :: "'a ltl \<Rightarrow> nat option"   and \<phi> :: "'a ltl" assumes "G \<psi> \<in> dom \<pi>"   and "dom \<pi> \<subseteq> \<^bold>G \<phi>"   and "accepting_pair\<^sub>R (\<delta>\<^sub>\<A> \<Sigma>) (\<iota>\<^sub>\<A> \<phi>) (Acc\<^sub>\<A> \<Sigma> \<pi> (G \<psi>)) w"   and "the (\<pi> (G \<psi>)) < semi_mojmir_def.max_rank \<Sigma> \<up>af\<^sub>G (Abs \<psi>)" shows "mojmir_def.accept \<up>af\<^sub>G (Abs \<psi>) w {q. dom \<pi> \<up>\<Turnstile>\<^sub>P q}"
cong: fixes b :: "'b"   and c :: "'b"   and a :: "'b" assumes "b = c" shows "a * b = a * c"
dirichlet_prod'_inversion1: fixes x :: "real" assumes "\<forall>x\<ge>1. g x = dirichlet_prod' a f x"   and "1 \<le> x"   and "dirichlet_prod a ainv = (\<lambda>n. if n = 1 then 1::'a else (0::'a))" shows "f x = dirichlet_prod' ainv g x"
elem_le_ffold: fixes k :: "'e"   and x :: "'e \<rightharpoonup>\<^sub>f 'f"   and f :: "'f \<Rightarrow> nat" assumes "k |\<in>| fmdom x" shows "f (the (fmlookup x k)) < Suc (ffold tcf 0 (fset_of_fmap (fmmap f x)))"
that: fixes d :: "nat" assumes "z = (Pc + sum_list (take d Ps), Pc + sum_list (take (Suc d) Ps))" shows "thesis"
simps_5: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c llist \<Rightarrow> 'b"   and x21 :: "'c"   and x22 :: "'c llist" shows "(case x21 ## x22 of LNil \<Rightarrow> f1 | x ## xa \<Rightarrow> f2 x xa) = f2 x21 x22"
simple_Fourier_convergence_periodic: fixes f :: "real \<Rightarrow> real"   and t :: "real" assumes "f absolutely_integrable_on {- pi..pi}"   and "(\<lambda>x. (f (x + t) - f t) / sin (x / 2)) absolutely_integrable_on {- pi..pi}"   and "\<And>x. f (x + 2 * pi) = f x" shows "(\<lambda>n. \<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k t) \<longlonglongrightarrow> f t"
Card_ord_of_nat: fixes n :: "nat" shows "Card (ord_of_nat n)"
exhaust: fixes y :: "'a msllist" obtains "y = MSLNil"   | x21 :: "'a" and x22 :: "'a msllist" where "y = MSLCons x21 x22"
inf_sup_aci_8: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.lattice inf less_eq less sup" shows "sup x (sup x y) = sup x y"
h_diff_le_g: fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and i :: "nat"   and e :: "'a \<times> 'a" assumes "flow_network \<Delta> g" shows "flow_network.h_diff \<Delta> g i e \<le> g e"
that: fixes D :: "'a \<times> 'a \<Rightarrow> bool" assumes "eventually D uniformity"   and "\<And>x y z. \<lbrakk>D (x, y); D (y, z)\<rbrakk> \<Longrightarrow> E (x, z)" shows "thesis"
mk_disjoint_finsert: fixes a :: "'c"   and A :: "'c fset" assumes "a |\<in>| A" shows "\<exists>B. A = finsert a B \<and> a |\<notin>| B"
represents_nth_root_neg_real: fixes n :: "nat"   and p :: "int poly"   and x :: "real" assumes "n \<noteq> 0"   and "p represents x"   and "x < 0" shows "poly_uminus (poly_nth_root n (poly_uminus p)) represents root n x"
assms: fixes e :: "real"   and x :: "'a" assumes "0 < e"   and "x \<in> s" shows "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) \<le> e"
assms: shows "(X, Y) \<in> mult2_alt b ns s"
div_field_poly_i_def: fixes ops :: "'b arith_ops_record"   and cf :: "'b list"   and cg :: "'b list" shows "div_field_poly_i ops cf cg = (if cg = [] then [] else let ilc = arith_ops_record.inverse ops (last cg); ch = map (arith_ops_record.times ops ilc) cg; q = fst (divmod_poly_one_main_i ops [] (rev cf) (rev ch) (1 + length cf - length cg)) in poly_of_list_i ops (map (arith_ops_record.times ops ilc) q))"
preserves_iso_in_hom: fixes f :: "'c \<times> 'c"   and a :: "'c \<times> 'c"   and b :: "'c \<times> 'c" assumes "C.VV.iso_in_hom f a b" shows "C.iso_in_hom (fst f \<star>\<^sub>C snd f) (fst a \<star>\<^sub>C snd a) (fst b \<star>\<^sub>C snd b)"
Nil_in_star: fixes A :: "'a list set" shows "[] \<in> A\<star>"
sum_list_lists_in_genby: fixes ss :: "'b list"   and S :: "'b set" assumes "ss \<in> lists (S \<union> uminus ` S)" shows "sum_list ss \<in> \<langle>S\<rangle>"
fcomp_frange: fixes r :: "V"   and s :: "V" shows "\<R>\<^sub>\<bullet> (r \<circ>\<^sub>\<bullet> s) = r `\<^sub>\<bullet> \<R>\<^sub>\<bullet> s"
naturally_isomorphic_def: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e" shows "naturally_isomorphic A B F G = (\<exists>\<tau>. natural_isomorphism A B F G \<tau>)"
arrI: fixes f :: "'a" assumes "right \<omega> f" shows "S.arr f"
inverse: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in F. f x \<noteq> (0::'b)"   and "\<forall>\<^sub>F x in F. g x \<noteq> (0::'b)"   and "f \<in> L F g" shows "(\<lambda>x. inverse (g x)) \<in> L F (\<lambda>x. inverse (f x))"
SR3nMT: fixes C :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and p :: "'a list" assumes "\<not> not_MT C p" shows "rm_MT_rules C p = []"
periodic_arithmetic_homothecy: fixes f :: "nat \<Rightarrow> complex"   and k :: "nat"   and a :: "nat" assumes "periodic_arithmetic f k" shows "periodic_arithmetic (\<lambda>l. f (l * a)) k"
fold_congs_1: fixes r :: "('b, 'c) partial_object_scheme"   and r' :: "('b, 'c) partial_object_scheme"   and v' :: "'b set"   and f :: "'b set \<Rightarrow> 'b set"   and f' :: "'b set \<Rightarrow> 'b set" assumes "r = r'"   and "carrier r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "carrier_update f r = carrier_update f' r'"
soundI_mem: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list" assumes "\<And>I. \<lbrakk>is_interp I; \<And>\<phi>. List.member SG \<phi> \<Longrightarrow> seq_sem I \<phi> = UNIV\<rbrakk> \<Longrightarrow> seq_sem I C = UNIV" shows "sound (SG, C)"
embed_in_Field: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set"   and f :: "'d \<Rightarrow> 'e"   and a :: "'d" assumes "embed r r' f"   and "a \<in> Field r" shows "f a \<in> Field r'"
A_2: shows "(\<lambda>_. UNIT) \<turnstile> e : t"
UNCLAMP'_5: fixes a :: "real"   and b :: "real"   and c :: "real" shows "\<Theta>(CLAMP' (MASTER_BOUND a b c)) \<equiv> \<Theta>(MASTER_BOUND a b c)"
indspaceD_supp': fixes f :: "('f, 'g) aezfun \<Rightarrow> 'v"   and G :: "'g set"   and FH :: "('f, 'g) aezfun set"   and V :: "'v set"   and x :: "('f, 'g) aezfun" assumes "f \<in> BaseFGMod.indspace G FH V"   and "x \<notin> FH" shows "f x = (0::'v)"
orthogonal_self: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and x :: "'c" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "real_inner.orthogonal inner x x = (x = zero)"
odlist_Abs_2: shows "xs \<in> rel_ext (\<lambda>x. sorted x \<and> distinct x)"
assms_2: shows "bijective (e * top) \<and> bijective (e[[top]])"
52: shows "bprv (all yy (imp (cnj (subst \<phi> (Var yy) xx) (SS \<langle>\<chi>\<rangle> (Var yy))) (subst \<phi> \<langle>\<psi>\<rangle> xx)))"
a: shows "a \<in> A (bipartite_web_of \<Gamma>)"
valid_pratt_tree_imp_prime: fixes t :: "pratt_tree" assumes "valid_pratt_tree t" shows "prime (pratt_tree_number t)"
numeral_less_real_of_nat_iff: fixes w :: "num"   and n :: "nat" shows "(numeral w < real n) = (numeral w < n)"
justified: shows "P \<turnstile> (E, ws) weakly_justified_by J"
ty: shows "[] \<turnstile> v : T\<^sub>1 \<rightarrow> T\<^sub>2"
map_spmf_idI: fixes p :: "'g spmf"   and f :: "'g \<Rightarrow> 'g" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> f x = x" shows "map_spmf f p = p"
ga_gstep_def: shows "ga_gstep \<equiv> rel_of_enex (ga_gen, ga_gex)"
consequence_relation_family_axioms: shows "consequence_relation_family Bot_G Q entails_q"
closed_cball: fixes x :: "'c"   and e :: "real" shows "closed (cball x e)"
WT_gpv_OutD: fixes \<Gamma> :: "('b, 'd) \<I>"   and gpv :: "('e, 'b, 'd) gpv"   and out :: "'b"   and c :: "('e, 'b, 'd) rpv" assumes "\<Gamma> \<turnstile>g gpv \<surd>"   and "IO out c \<in> set_spmf (the_gpv gpv)" shows "out \<in> outs_\<I> \<Gamma>"
assms: fixes x :: "'b" assumes "x \<in> A" shows "abs_conv_abscissa (f x) \<le> d"
assms_1: shows "\<forall>x y. (x, y) \<in> rel \<longrightarrow> (x, y) \<in> rel'"
vec_to_iarray_minus: fixes a :: "('c, 'd) vec"   and b :: "('c, 'd) vec" shows "vec_to_iarray (a - b) = vec_to_iarray a - vec_to_iarray b"
power_less_128_helper: fixes n :: "nat" assumes "n \<le> 128 - 16" shows "2 ^ n * unat (of_bl (to_bl b)) < 2 ^ LENGTH(128)"
inv_f_eq: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'c" assumes "inj f"   and "f x = y" shows "inv f y = x"
rel_with_target_impl_transC_TRel_is_weak_reduction_simulation: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and TRel :: "('procT \<times> 'procT) set" assumes "weak_reduction_simulation Rel (STCal Source Target)"   and "\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel"   and "\<forall>T1 T2. (TargetTerm T1, TargetTerm T2) \<in> Rel \<longrightarrow> (T1, T2) \<in> TRel\<^sup>+" shows "weak_reduction_simulation (TRel\<^sup>+) Target"
term_brackets_Nil: fixes t :: "dB" shows "term_brackets [] t = Some t"
simps_2: fixes i :: "char list \<Rightarrow> tv" shows "eval i Truth = Det True"
node_props_spec_impl: fixes P :: "('a, SINVAR_BLPtrusted.node_config) TopoS_Params" shows "BLPtrusted.node_props_formaldef P = SINVAR_BLPtrusted_impl.NetModel_node_props P"
pop_def: fixes q :: "('c, 'd) pq" shows "pop q = (if is_empty q then None else Some (PQ.min q, remove_min q))"
mod_Suc': fixes n :: "nat"   and m :: "nat" assumes "0 < n" shows "Suc m mod n = (if m mod n < n - Suc 0 then Suc (m mod n) else 0)"
simps: fixes a1 :: "('test, 'atom) com \<times> 'state"   and a2 :: "'state" shows "a1 \<rightarrow>*t a2 = (\<exists>cf cf' s''. a1 = cf \<and> a2 = s'' \<and> cf \<rightarrow>*c cf' \<and> cf' \<rightarrow>t s'')"
containsConsiders: fixes gamma :: "(signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and iA :: "nat \<times> formula" assumes "branch subs gamma f"   and "\<forall>n. \<not> proofTree (tree subs (f n))"   and "EV (contains f iA)" shows "EV (considers f iA)"
subst_cls_mono: fixes C :: "'a literal multiset"   and D :: "'a literal multiset"   and \<sigma> :: "'s" assumes "set_mset C \<subseteq> set_mset D" shows "set_mset (C \<cdot> \<sigma>) \<subseteq> set_mset (D \<cdot> \<sigma>)"
freshEnv_idEnv: fixes xs :: "'d"   and x :: "'e" shows "freshEnv xs x idEnv"
coprime_add_one_right: fixes a :: "'a" shows "coprime a (a + (1::'a))"
elims_3: fixes x :: "nat \<Rightarrow> nat"   and xa :: "'b memref list" assumes "\<not> volatile_reads_consistent x xa" obtains m :: "nat \<Rightarrow> nat" and r :: "'b memref" and rs :: "'b memref list" where "x = m" and "xa = r # rs" and "\<not> (case r of Write\<^sub>s\<^sub>b volatile a sop v A L R W \<Rightarrow> volatile_reads_consistent (m(a := v)) rs | Read\<^sub>s\<^sub>b volatile a t v \<Rightarrow> (volatile \<longrightarrow> v = m a) \<and> volatile_reads_consistent m rs | _ \<Rightarrow> volatile_reads_consistent m rs)"
indep_events_def_alt: fixes A :: "'b \<Rightarrow> 'a set"   and I :: "'b set" shows "indep_events A I = indep_sets (\<lambda>i. {A i}) I"
exchange_lemma: fixes T :: "'b set"   and S :: "'b set" assumes "finite T"   and "independent S"   and "S \<subseteq> span T" shows "\<exists>t'. card t' = card T \<and> finite t' \<and> S \<subseteq> t' \<and> t' \<subseteq> S \<union> T \<and> S \<subseteq> span t'"
lbound_of_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and b :: "'a" assumes "ordering less_eq less" shows "ordering.lbound_of less_eq x y b \<equiv> less_eq b x \<and> less_eq b y"
Gch_bridge_def: fixes n :: "nat"   and g :: "nat \<Rightarrow> ('e set, 'f) Group_scheme"   and h :: "nat \<Rightarrow> ('e set, 'f) Group_scheme"   and f :: "nat \<Rightarrow> nat" shows "Gch_bridge n g h f = ((\<forall>l\<le>n. f l \<le> n) \<and> inj_on f {i. i \<le> n} \<and> isom_Gchains n f g h)"
natfun_inverse_bound: fixes f :: "'b fps"   and \<delta> :: "real"   and n :: "nat" assumes "fps_nth f 0 = (1::'b)"   and "0 < \<delta>"   and "summable (\<lambda>n. norm (fps_nth f (Suc n)) * \<delta> ^ Suc n)"   and "(\<Sum>n. norm (fps_nth f (Suc n)) * \<delta> ^ Suc n) \<le> 1" shows "norm (natfun_inverse f n) \<le> inverse (\<delta> ^ n)"
him: shows "h ` \<Union> \<G> \<subseteq> rel_frontier T"
ss_p18: fixes x :: "'a" shows "(x ; 1 = 0) = (x = 0)"
M_def: shows "M \<equiv> stock_measure (PRODUCT (\<Gamma> x) (\<Gamma> y))"
mset_subst_atm_list_subst_atm_mset: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and As :: "'b list"   and \<sigma> :: "'c" assumes "substitution subst_atm id_subst comp_subst renamings_apart atm_of_atms" shows "mset (substitution_ops.subst_atm_list subst_atm As \<sigma>) = substitution_ops.subst_atm_mset subst_atm (mset As) \<sigma>"
IH: assumes "Ex (is_icdi k \<pi>)"   and "is_path \<pi>"   and "\<pi> =\<^bsub>The (is_icdi k \<pi>)\<^esub> \<pi>'" shows "cs\<^bsup>\<pi>\<^esup> The (is_icdi k \<pi>) = cs\<^bsup>\<pi>'\<^esup> The (is_icdi k \<pi>)"
normalize_poly_def: fixes p :: "'a poly" shows "normalize p = p div [:unit_factor (lead_coeff p):]"
take_bit_minus_small_eq: fixes k :: "int"   and n :: "nat" assumes "0 < k"   and "k \<le> 2 ^ n" shows "take_bit n (- k) = 2 ^ n - k"
enumerate_eq_zip: fixes n :: "nat"   and xs :: "'a list" shows "enumerate n xs = zip [n..<n + length xs] xs"
measurable_ident_sets: fixes M :: "'b measure"   and M' :: "'b measure" assumes "sets M = sets M'" shows "(\<lambda>x. x) \<in> M \<rightarrow>\<^sub>M M'"
assms_3: shows "revision_step r' (s\<^sub>1(r \<mapsto> x)) (s\<^sub>1(r \<mapsto> x, r' \<mapsto> y))"
permute_eq_iff: fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a"   and p :: "perm"   and x :: "'a"   and y :: "'a" assumes "class.pt permute" shows "(permute p x = permute p y) = (x = y)"