hcju/isabelleps huggingface dataset:
compute_SCC_spec_def: shows "compute_SCC_spec \<equiv> SPEC (\<lambda>l. distinct l \<and> \<Union> (set l) = reachable \<and> (\<forall>U\<in>set l. is_scc E U) \<and> (\<forall>i j. i < j \<and> j < length l \<longrightarrow> l ! j \<times> l ! i \<inter> E\<^sup>* = {}))"
distinct_41: fixes x6 :: "'b ptrm"   and x7 :: "'b ptrm" shows "PFst x6 \<noteq> PSnd x7"
rejectD: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state"   and m :: "'c match_expr" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>r, s\<rangle> \<Rightarrow> t"   and "r = [Rule m Reject]"   and "matches \<gamma> m p"   and "s = Undecided" shows "t = Decision FinalDeny"
disc_12: fixes x2 :: "event_data" shows "trm.is_Const (trm.Const x2)"
finite_rhs_def: fixes ES :: "('b \<times> 'c set) set" shows "finite_rhs ES \<equiv> \<forall>(X, rhs)\<in>ES. finite rhs"
Y_1: shows "Complete_Partial_Order.chain (\<le>) Y"
well_ordered_strategy_def: shows "well_ordered_strategy \<equiv> override_on \<sigma>_arbitrary (\<lambda>v. choose v v) (S - W)"
projection_to_emptyset_is_empty_trace: fixes l :: "'a list" shows "l \<upharpoonleft> {} = []"
inject_1: fixes x2 :: "'a basic_type"   and y2 :: "'a basic_type" shows "(x2[1] = y2[1]) = (x2 = y2)"
less_eq_V_def: fixes x :: "V"   and y :: "V" shows "x \<subseteq>\<^sub>\<circ> y \<equiv> elts x \<subseteq> elts y"
hyps_7: shows "n' = Actual_out (targetnode a, x)"
1: shows "(\<lambda>z. if 0 \<le> Im z then f z else cnj (f (cnj z))) holomorphic_on S \<inter> {z. 0 < Im z}"
taints_def: fixes ts :: "SINVAR_TaintingTrusted.taints" shows "taints ts \<equiv> taints_raw.taints_raw (taints.raw_of_taints ts)"
simps_3: fixes F :: "'a formula"   and G :: "'a formula" shows "to_mini_formula (F \<^bold>\<rightarrow> G) = to_mini_formula F \<^bold>\<rightarrow> to_mini_formula G"
meet_prop: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "(z \<le> x \<and> z \<le> y) = (z \<le> x \<sqinter> y)"
subst_apply_left_idemp: fixes \<sigma> :: "'f \<Rightarrow> ('d, 'e) Term.term"   and x :: "'f"   and t :: "('d, 'f) Term.term"   and s :: "('d, 'f) Term.term" assumes "\<sigma> x = t \<cdot> \<sigma>" shows "s \<cdot> subst x t \<cdot> \<sigma> = s \<cdot> \<sigma>"
vrrestriction_vrange: shows "r \<restriction>\<^sup>r\<^sub>\<circ> \<R>\<^sub>\<circ> r = r"
except_eq_zeroE: fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and S :: "'a set" assumes "except p S = 0" shows "keys p \<subseteq> S"
widens_lengthD: fixes P :: "'a prog"   and xs :: "ty list"   and ys :: "ty list" assumes "P \<turnstile> xs [\<le>] ys" shows "length xs = length ys"
repr'_comb': fixes x :: "'basis euclidean_space" shows "repr' (comb' x) = x"
enum_set_to_list_simps: fixes S :: "'b set" shows "enum_set_to_list S = (case enum_set_get_one enum_class.enum S of None \<Rightarrow> [] | Some a \<Rightarrow> a # enum_set_to_list (S - {a}))"
in_model: fixes n :: "nat"   and n_vars :: "nat"   and M :: "nat \<Rightarrow> bool" assumes "0 < n"   and "n < n_vars" shows "int n \<in> set (model_to_dimacs_model M [1..<n_vars]) \<or> - int n \<in> set (model_to_dimacs_model M [1..<n_vars])"
ifte_thms_3: fixes e1 :: "'a"   and e2 :: "'a" shows "If TT then e1 else e2 = e1"
ipurge_ref_aux_less_last: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and U :: "'c set"   and xs :: "'b list"   and x :: "'b"   and X :: "'b set" shows "ipurge_ref_aux_less I D U (xs @ [x]) X = (if \<exists>v\<in>U \<union> sinks_aux_less I D U xs. (v, D x) \<in> I then ipurge_ref_aux_less I D U xs {x' \<in> X. (D x, D x') \<notin> I} else ipurge_ref_aux_less I D U xs X)"
objE: fixes a :: "'c" assumes "C.obj a" obtains "C.ide a" and "src\<^sub>C a = a" and "trg\<^sub>C a = a"
smaps_over_im: fixes f :: "'a"   and m :: "'b \<Rightarrow> 'a set"   and a :: "'b"   and A :: "'b set"   and B :: "'a set" assumes "f \<in> m a"   and "m \<in> smaps_over A B" shows "f \<in> B"
comp_map_def: fixes f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" shows "f \<parallel>\<^sub>m g = (\<forall>x\<in>dom f \<inter> dom g. the (f x) = the (g x))"
main_2: assumes "c = round (gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom fs) i j)"   and "\<mu>_small_row i fs (Suc j)" shows "\<mu>_small_row i fs' j"
j: shows "j = ne + length aux + length (fst st) - 1"
Elem_Ext_ZF_hull: fixes R :: "(ZF \<times> ZF) set"   and x :: "ZF"   and S :: "ZF" assumes "set_like R" shows "Elem x (Ext_ZF_hull R S) = (\<exists>n. Elem x (Ext_ZF_n R S n))"
scaleC_left_commute: fixes a :: "complex"   and b :: "complex"   and x :: "'d" shows "a *\<^sub>C b *\<^sub>C x = b *\<^sub>C a *\<^sub>C x"
splus_mono_left: fixes s :: "'a"   and t :: "'a"   and v :: "'a" assumes "s \<preceq> t" shows "local.punit.splus s v \<preceq> local.punit.splus t v"
absorb2: fixes b :: "'a multiset"   and a :: "'a multiset" assumes "b \<subseteq># a" shows "a \<inter># b = b"
reflect_poly_hom: fixes p :: "int poly" shows "reflect_poly (of_int_poly p) = of_int_poly (reflect_poly p)"
map_xs: shows "map (\<lambda>g. eval g (p # cs)) (map (\<lambda>i. Cn (Suc m) (r_code_constn n) [recf.Id (Suc m) i]) [0..<Suc m]) = map Some (map (code_constn n) (p # cs))"
abs: fixes X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and x :: "'var"   and p :: "'param"   and xs :: "'varSort" assumes "good X"   and "x \<notin> varsOf p xs"   and "phi X p" shows "phiAbs (Abs xs x X) p"
induct: fixes P :: "('b \<times> nat) tree \<Rightarrow> 'b \<Rightarrow> ('b \<times> nat) tree \<Rightarrow> 'b \<Rightarrow> ('b \<times> nat) tree \<Rightarrow> bool"   and a0 :: "('b \<times> nat) tree"   and a1 :: "'b"   and a2 :: "('b \<times> nat) tree"   and a3 :: "'b"   and a4 :: "('b \<times> nat) tree" assumes "\<And>A a B1 b uu_ B2 c C. P A a \<langle>B1, (b, uu_), B2\<rangle> c C"   and "\<And>a b d e. P a b \<langle>\<rangle> d e" shows "P a0 a1 a2 a3 a4"
InfinitesimalD: fixes x :: "'b star" assumes "x \<in> Infinitesimal" shows "\<forall>r\<in>\<real>. 0 < r \<longrightarrow> hnorm x < r"
subtypings_refl: fixes CT :: "nat \<Rightarrow> classDef option"   and Cs :: "nat list" shows "CT \<turnstile>+ Cs <: Cs"
strategyproofness_1: shows "pmf (sds R25) d + pmf (sds R25) b \<le> pmf (sds R26) d + pmf (sds R26) c"
congruent2_def: fixes r1 :: "('b \<times> 'b) set"   and r2 :: "('c \<times> 'c) set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" shows "congruent2 r1 r2 f = (\<forall>(y1, z1)\<in>r1. \<forall>(y2, z2)\<in>r2. f y1 y2 = f z1 z2)"
lappend_inf: fixes xs :: "'b llist"   and ys :: "'b llist" assumes "linfinite xs" shows "xs $ ys = xs"
total2I: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat option" assumes "\<And>x y. f x y \<down>" shows "total2 f"
map_def: fixes f :: "'j" shows "\<chi>.A.map f = (if J.arr f then a else C.null)"
mprod_expR_memTr: fixes e :: "nat \<Rightarrow> nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "e \<in> {j. j \<le> n} \<rightarrow> {j. 0 \<le> j} \<and> f \<in> {j. j \<le> n} \<rightarrow> carrier R \<longrightarrow> mprod_expR R e f n \<in> carrier R"
is_path_def: fixes r :: "nat \<Rightarrow> nat \<Rightarrow> b"   and p :: "nat \<times> nat list \<times> nat"   and i :: "nat"   and j :: "nat"   and k :: "nat" shows "is_path r p i j k = (fst p = j \<and> snd (snd p) = k \<and> list_all (\<lambda>x. x < i) (fst (snd p)) \<and> is_path' r (fst p) (fst (snd p)) (snd (snd p)))"
neq: assumes "p - 1 \<noteq> int n" shows "trivial_group (reduced_homology_group (p - 1) (nsphere n))"
combine_eq_val_combine_tm: fixes p\<^sub>0\<^sub>L :: "int \<times> int"   and p\<^sub>1\<^sub>L :: "int \<times> int"   and p\<^sub>0\<^sub>R :: "int \<times> int"   and p\<^sub>1\<^sub>R :: "int \<times> int"   and l :: "int"   and ps :: "(int \<times> int) list" shows "Time_Monad.val (combine_tm (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) l ps) = combine (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) l ps"
plus_except: fixes p :: "'b \<Rightarrow>\<^sub>0 'c"   and t :: "'b" shows "p = monomial (lookup p t) t + except p {t}"
spanset_reduce_Cons: fixes ms :: "'m list"   and m :: "'m" assumes "set ms \<subseteq> M"   and "m \<in> RSpan ms" shows "RSpan (m # ms) = RSpan ms"
Cons_1: assumes "xs = [] \<Longrightarrow> thesis"   and "\<And>ys x. \<lbrakk>\<not> List.member ys x; xs \<approx> x # ys\<rbrakk> \<Longrightarrow> thesis" shows "thesis"
iso_imp_homomorphism: fixes h :: "'b \<Rightarrow> 'c"   and G :: "('b, 'd) monoid_scheme"   and H :: "('c, 'e) monoid_scheme" assumes "h \<in> Group.iso G H" shows "h \<in> hom G H"
"case": fixes f :: "nat \<Rightarrow> 'c \<Rightarrow> 'c tree list \<Rightarrow> 'b"   and x1 :: "nat"   and x2 :: "'c"   and x3 :: "'c tree list" shows "(case Node x1 x2 x3 of Node x xa xb \<Rightarrow> f x xa xb) = f x1 x2 x3"
qItem_versus_item_simps_4: fixes fAbs :: "('c, 'd, 'varSort, 'var, 'e) qAbs \<Rightarrow> 'f"   and Aa :: "('c, 'd, 'varSort, 'var, 'e) qAbs" assumes "fAbs respectsP ($=)"   and "qGoodAbs Aa" shows "Equiv_Relation2.univ fAbs (asAbs Aa) = fAbs Aa"
grey_protects_whiteE_1: fixes w :: "'g"   and g :: "'g"   and s :: "'h process_name \<Rightarrow> ('e, 'h, 'f, 'g) Model.local_state" assumes "(g points_to w) s"   and "grey g s"   and "white w s" shows "(g grey_protects_white w) s"
prerecording_event: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat" shows "prerecording_event t i \<equiv> i < length t \<and> regular_event (t ! i) \<and> \<not> ps (S t i) (occurs_on (t ! i)) \<noteq> None"
NotL: fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and \<Delta> :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow> F, \<Delta>" shows "\<^bold>\<not> F, \<Gamma> \<Rightarrow> \<Delta>"
a2: fixes x :: "'a vec" assumes "x \<in> set R" shows "dim_vec x = d"
Call_\<tau>red1t_param: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and es :: "'addr expr1 list"   and xs :: "'addr val list"   and es' :: "'addr expr1 list"   and xs' :: "'addr val list"   and v :: "'addr val"   and M :: "String.literal" assumes "\<tau>reds1gt uf P t h (es, xs) (es', xs')" shows "\<tau>red1gt uf P t h (Val v\<bullet>M(es), xs) (Val v\<bullet>M(es'), xs')"
simps_2: fixes l :: "('b \<times> nat) tree"   and uu :: "'b \<times> nat"   and r :: "('b \<times> nat) tree" shows "del_min \<langle>l, uu, r\<rangle> = merge l r"
induct: fixes P :: "('b \<Rightarrow> 'c) \<Rightarrow> 'b negation_type list \<Rightarrow> bool"   and a0 :: "'b \<Rightarrow> 'c"   and a1 :: "'b negation_type list" assumes "\<And>uu_. P uu_ []"   and "\<And>f a as. P f as \<Longrightarrow> P f (Pos a # as)"   and "\<And>f a as. P f as \<Longrightarrow> P f (Neg a # as)" shows "P a0 a1"
span_scale: fixes x :: "'b poly"   and S :: "'b poly set"   and c :: "'b" assumes "x \<in> poly_vs.span S" shows "smult c x \<in> poly_vs.span S"
shr_init_fin_lift_state: fixes s :: "status"   and \<sigma> :: "('b, 'c) locks \<times> (('c \<Rightarrow> ('d \<times> 'b \<Rightarrow>f nat) option) \<times> 'a) \<times> ('c \<Rightarrow> 'e wait_set_status option) \<times> 'c set" shows "shr (init_fin_lift_state s \<sigma>) = shr \<sigma>"
weak_map_of_SomeI: fixes k :: "'d"   and x :: "'e"   and l :: "('d \<times> 'e) list" assumes "(k, x) \<in> set l" shows "\<exists>x. map_of l k = Some x"
least_consequence_graph_def: fixes t :: "'b itself"   and Rs :: "(('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) set"   and S :: "('d, 'f) labeled_graph"   and G :: "('d, 'f) labeled_graph" shows "least_consequence_graph t Rs S G \<equiv> consequence_graph Rs G \<and> least t Rs S G"
additive: fixes g :: "'m"   and g' :: "'m" assumes "g \<in> M"   and "g' \<in> M" shows "T (g + g') = T g + T g'"
hash_of_assoc_map_def: fixes xs :: "('c \<times> 'd) list" shows "hash_of_assoc_map xs = fold (\<lambda>(k, v) m. if m k \<noteq> None then m else m(k \<mapsto> v)) xs Map.empty"
gdd_index_options: fixes ps :: "'a set" assumes "ps \<subseteq> \<V>"   and "card ps = 2" shows "\<B> index ps = 0 \<or> int (\<B> index ps) = \<Lambda>"
perron_frobenius: fixes A :: "((complex, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" assumes "real_non_neg_mat A" shows "\<exists>v. eigen_vector A v (complex_of_real (HMA_Connect.spectral_radius A)) \<and> real_non_neg_vec v"
JF2rel_def: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "JF2rel R = (BNF_Def.Grp (JF2in {(x, y). R x y}) (JF2map fst))\<inverse>\<inverse> OO BNF_Def.Grp (JF2in {(x, y). R x y}) (JF2map snd)"
mfinal_def: fixes final :: "'a \<Rightarrow> bool"   and s :: "('b, 'c) locks \<times> (('c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('c \<Rightarrow> 'e wait_set_status option) \<times> 'c set" shows "final_thread.mfinal final s = (\<forall>t x ln. thr s t = \<lfloor>(x, ln)\<rfloor> \<longrightarrow> final x \<and> ln = no_wait_locks \<and> wset s t = None)"
mfinalI: fixes s :: "('b, 'a) locks \<times> (('a \<Rightarrow> ('x2 \<times> 'b \<Rightarrow>f nat) option) \<times> 'c) \<times> ('a \<Rightarrow> 'd wait_set_status option) \<times> 'a set" assumes "\<And>t x ln. thr s t = \<lfloor>(x, ln)\<rfloor> \<Longrightarrow> final2 x \<and> ln = no_wait_locks \<and> wset s t = None" shows "r2.mfinal s"
floor_diff_of_int: fixes x :: "'b"   and z :: "int" shows "\<lfloor>x - of_int z\<rfloor> = \<lfloor>x\<rfloor> - z"
alpha_sym_1: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and p :: "perm"   and x :: "'e"   and y :: "'e"   and bs :: "atom set"   and f :: "'e \<Rightarrow> atom set"   and cs :: "atom set" assumes "R (p \<bullet> x) y \<Longrightarrow> R (- p \<bullet> y) x"   and "(bs, x) \<approx>set R f p (cs, y)" shows "(cs, y) \<approx>set R f (- p) (bs, x)"
w\<^sub>g'_def: shows "w\<^sub>g' = g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine"
mix_pmf_in_lotteries: fixes p :: "'a pmf"   and A :: "'a set"   and q :: "'a pmf"   and a :: "real" assumes "p \<in> \<P> A"   and "q \<in> \<P> A"   and "a \<in> {0<..<1}" shows "mix_pmf a p q \<in> \<P> A"
assms_1: shows "\<delta>\<^sub>L' = dist_code p\<^sub>0\<^sub>L p\<^sub>1\<^sub>L"
bind_SUCCEED: fixes f :: "'c \<Rightarrow> 'b nres" shows "SUCCEED \<bind> f = SUCCEED"
mprefix_Par_Int: fixes B :: "'a set"   and C :: "'a set"   and A :: "'a set"   and P :: "'a \<Rightarrow> 'a process"   and Q :: "'a \<Rightarrow> 'a process" assumes "B \<inter> C = {}"   and "A \<inter> C = {}" shows "(Mprefix A P \<lbrakk>C\<rbrakk> Mprefix B Q) = (Mprefix A (\<lambda>x. P x \<lbrakk>C\<rbrakk> Mprefix B Q) [+] Mprefix B (\<lambda>y. Mprefix A P \<lbrakk>C\<rbrakk> Q y))"
fv\<^sub>e\<^sub>q_def: fixes ac :: "poscheckvariant"   and x :: "('d, 'c) strand_step" shows "fv\<^sub>e\<^sub>q ac x \<equiv> case x of \<langle>ac': s \<doteq> t\<rangle>\<^sub>s\<^sub>t \<Rightarrow> if ac = ac' then fv s \<union> fv t else {} | _ \<Rightarrow> {}"
simps_3: fixes x1 :: "'b"   and l1 :: "'b list"   and x2 :: "'b"   and l2 :: "'b list" shows "diff_sorted (x1 # l1) (x2 # l2) = (if x1 < x2 then x1 # diff_sorted l1 (x2 # l2) else if x1 = x2 then diff_sorted l1 l2 else diff_sorted (x1 # l1) l2)"
real_non_neg_mat_def: fixes A :: "((complex, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "real_non_neg_mat A \<equiv> \<forall>a\<in>elements_mat_h A. a \<in> \<real> \<and> 0 \<le> Re a"
compose_conv: fixes xs :: "('e \<times> 'f) list"   and ys :: "('f \<times> 'd) list"   and k :: "'e" shows "map_of (compose xs ys) k = (map_of ys \<circ>\<^sub>m map_of xs) k"
ground_head_var_def: fixes x :: "nat" shows "ground_head_var x = UNIV"
set_diff_eq: fixes A :: "'d set"   and B :: "'d set" shows "A - B = {x \<in> A. x \<notin> B}"
open_vimage: fixes s :: "'c set"   and f :: "'d \<Rightarrow> 'c" assumes "open s"   and "continuous_on UNIV f" shows "open (f -` s)"
synth_subset_iff: fixes G :: "msg set"   and H :: "msg set" shows "(synth G \<subseteq> synth H) = (G \<subseteq> synth H)"
disc_1: fixes x1 :: "('e, 'f) Term.term" shows "stateful_strand_step.is_Send (send\<langle>x1\<rangle>)"
comp_assoc: fixes h :: "'s"   and g :: "'s"   and f :: "'s" shows "(h \<cdot> g) \<cdot> f = h \<cdot> g \<cdot> f"
HC_intros_11: fixes F :: "'b formula" shows "AX10 \<turnstile>\<^sub>H ((F::'b formula) \<^bold>\<rightarrow> \<bottom>) \<^bold>\<rightarrow> \<^bold>\<not> F"
le_funI: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>x. f x \<le> g x" shows "f \<le> g"
rel_fun_def: fixes A :: "'q \<Rightarrow> 's \<Rightarrow> bool"   and B :: "'r \<Rightarrow> 't \<Rightarrow> bool" shows "rel_fun A B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))"
that: fixes r :: "'alt pmf" assumes "r \<in> lotteries_on alts"   and "Pareto (SD \<circ> R) q r"   and "set_pmf r \<subset> set_pmf p" shows "thesis"
ordinal_of_nat_eq: fixes x :: "nat"   and y :: "nat" shows "(ordinal_of_nat x = ordinal_of_nat y) = (x = y)"
subalgebra_def: fixes M :: "'c measure"   and F :: "'c measure" shows "subalgebra M F = (space F = space M \<and> sets F \<subseteq> sets M)"
zero_pdevs_def: shows "zero_pdevs \<equiv> Abs_pdev (\<lambda>_. 0::'a)"
ide_cod: fixes f :: "'c" assumes "C.arr f" shows "C.ide (C.cod f)"
mult_sv_one: fixes x :: "real \<times> real \<times> real \<times> real" shows "1 *\<^sub>s\<^sub>v\<^sub>4 x = x"
poly_roots_factorization: fixes c :: "'b"   and A :: "'b multiset" assumes "c \<noteq> (0::'b)" shows "poly_roots (Polynomial.smult c (\<Prod>x\<in>#A. [:- x, 1::'b:])) = A"
Units_l_inv_ex: fixes x :: "'a vec" assumes "x \<in> Units \<lparr>carrier = carrier_vec n, mult = (+), one = 0\<^sub>v n, \<dots> = undefined\<rparr>" shows "\<exists>y\<in>carrier_vec n. y + x = 0\<^sub>v n"
bisim1_pc_length_compE2D: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and xcp :: "'addr option" assumes "P,e,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, length (compE2 e), xcp)" shows "xcp = None \<and> call1 e' = None \<and> (\<exists>v. stk = [v] \<and> (is_val e' \<longrightarrow> e' = Val v \<and> xs = loc))"
cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes ptr :: "(_) shadow_root_ptr" shows "cast ptr = object_ptr.Ext (Inr (Inr (Inl ptr)))"
bi_total_alt_def: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "bi_total A = (left_total A \<and> right_total A)"
not_less: fixes x :: "'a"   and y :: "'a" shows "(\<not> x < y) = (y \<le> x)"
is_rcoset_replist_def: fixes H :: "'g set"   and gs :: "'g list" shows "is_rcoset_replist H gs \<equiv> set gs \<subseteq> G \<and> distinct (map (\<lambda>g. rcoset_rel H `` {g}) gs) \<and> G = (\<Union>g\<in>set gs. rcoset_rel H `` {g})"
mcont_const: fixes luba :: "'d set \<Rightarrow> 'd"   and orda :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and c :: "'a" shows "mcont luba orda Sup (\<le>) (\<lambda>x. c)"
linear_split_req: fixes xs :: "('b \<times> 'c) list"   and p :: "'c"   and ls :: "('b \<times> 'c) list"   and sub :: "'b"   and sep :: "'c"   and rs :: "('b \<times> 'c) list" assumes "linear_split xs p = (ls, (sub, sep) # rs)"   and "sorted_less (separators xs)" shows "p \<le> sep"
last_in_tl: fixes xs :: "'a list" assumes "1 < length xs" shows "last xs \<in> set (tl xs)"
h_is_map: shows "is_left_adjoint s\<^sub>1"
period_root_def: fixes u :: "'b list"   and r :: "'b list" shows "u \<le>p r\<^sup>\<omega> = (u \<le>p r \<cdot> u \<and> r \<noteq> \<epsilon>)"
is_simple_eq_def: fixes t1 :: "Formula.trm"   and t2 :: "Formula.trm" shows "is_simple_eq t1 t2 = (trm.is_Const t1 \<and> (trm.is_Const t2 \<or> trm.is_Var t2) \<or> trm.is_Var t1 \<and> trm.is_Const t2)"
HMA_diagonal_step_PQ: fixes i :: "nat"   and k :: "nat" assumes "i < min CARD('b) CARD('c)"   and "k < min CARD('b) CARD('c)" shows "(Mod_Type_Connect.HMA_M ===> (=) ===> rel_prod Mod_Type_Connect.HMA_M Mod_Type_Connect.HMA_M) (\<lambda>A. diagonal_step_PQ_JNF A i k) (\<lambda>A. diagonal_step_PQ A i k)"
antisym: fixes x :: "'b list"   and y :: "'b list" assumes "subseq x y"   and "subseq y x" shows "x = y"
paths_Either: fixes ts :: "'a ttree set" shows "paths (Either ts) = insert [] (\<Union> (paths ` ts))"
rel_set_conversep: fixes A :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_set A\<inverse>\<inverse> = (rel_set A)\<inverse>\<inverse>"
zip_map_fst_snd: fixes zs :: "('a \<times> 'b) list" shows "zip (map fst zs) (map snd zs) = zs"
events_div: fixes P :: "'a process" assumes "D P \<noteq> {}" shows "events_of P = UNIV"
vals_pos_LBset: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "{x \<in> v ` carrier K. 0 < x} \<subseteq> LBset 1"
lemma_3_5_iii_1: fixes a :: "'a" shows "d1 a (1::'a) = a"
singleton_bex: fixes a :: "'a"   and P :: "'a \<Rightarrow> bool" shows "(\<exists>x\<in>{a}. P x) = P a"
bas': shows "A.basis ((*\<^sub>v) B ` bas)"
Nil: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and ys :: "'a list" shows "ord.lexordp less [] (y # ys)"
subset_spec_plane_def: fixes X :: "'c set"   and sctn :: "'c sctn" shows "subset_spec_plane X sctn = CHECK (\<lambda>_. print_msg ''subset_spec_plane: not in Basis'') (\<bar>normal sctn\<bar> \<in> set Basis_list) \<bind> (\<lambda>_. ivl_rep X \<bind> (\<lambda>(i, s). RETURN (i \<bullet> normal sctn = pstn sctn \<and> s \<bullet> normal sctn = pstn sctn)))"
sn_forget: fixes x :: "name"   and v :: "trm"   and t :: "trm" assumes "SN (t[x::=v])" shows "SN t"
measurable_bounded_by_integrable_imp_lebesgue_integrable: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and g :: "'c \<Rightarrow> real" assumes "f \<in> borel_measurable (lebesgue_on S)"   and "integrable (lebesgue_on S) g"   and "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> g x"   and "S \<in> sets lebesgue" shows "integrable (lebesgue_on S) f"
A_def: shows "A = (\<lambda>w. mat_of_rows n (ws @ [w]))"
ty_local_def: fixes c :: "cmd"   and ls :: "local_state"   and gs :: "global_state" shows "ty_local c ls gs \<equiv> \<exists>\<Gamma>. ty_cmd (\<Gamma>, False) c \<and> ty_fs \<Gamma> (ls, gs)"
rep_eq: fixes x :: "'q mpoly"   and xa :: "'q mpoly" shows "mapping_of (x + xa) = mapping_of x + mapping_of xa"
\<b>_1: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and i :: "nat" shows "\<a> ps \<le> \<b> ps i"
givenBy_eq_Collect: fixes v :: "'d \<Rightarrow> 'e" shows "givenBy v = {A. \<exists>P. A = {s. P (v s)}}"
tuple_encode_decode: fixes xs :: "'a list"   and i :: "nat" shows "tuple_encode (length xs - 1) (tuple_decode (length xs - 1) i) = i"
node_addressD: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and i :: "nat"   and \<zeta> :: "'e"   and R :: "nat set"   and ii :: "nat"   and T :: "((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and R\<^sub>i :: "nat set"   and S :: "(nat \<Rightarrow> 'd) \<Rightarrow> (nat \<Rightarrow> 'd) \<Rightarrow> 'f node_action \<Rightarrow> bool"   and U :: "(nat \<Rightarrow> 'd) \<Rightarrow> (nat \<Rightarrow> 'd) \<Rightarrow> bool" assumes "(\<sigma>, NodeS i \<zeta> R) \<in> oreachable (\<langle>ii : T : R\<^sub>i\<rangle>\<^sub>o) S U" shows "i = ii"
the_apartment_iso_chamber_map: fixes A :: "'a set set"   and B :: "'a set set"   and C :: "'a set"   and D :: "'a set" assumes "A \<in> \<A>"   and "B \<in> \<A>"   and "chamber C"   and "C \<in> A \<inter> B"   and "chamber D"   and "D \<in> A" shows "chamber (the_apartment_iso A B ` D)"
find_recfun_someD: fixes n :: "char list"   and funs :: "(char list \<times> 'a \<times> 'b) list"   and x :: "'a"   and e :: "'b" assumes "find_recfun n funs = Some (x, e)" shows "(n, x, e) \<in> set funs"
before_dist_not_r1: fixes vs :: "'a list"   and r1 :: "'a"   and r2 :: "'a" assumes "distinct vs"   and "before vs r1 r2" shows "r1 \<notin> set (snd (splitAt r2 vs))"
eq_upto_BAD_mem: fixes f :: "nat \<Rightarrow> 'b"   and A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and g :: "nat \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat" assumes "f \<in> eq_upto (mbs.BAD A P) g i" shows "f j \<in> A"