hcju/isabelleps huggingface dataset:
normalize_quot_mult: fixes x :: "'b \<times> 'b"   and y :: "'b \<times> 'b" assumes "snd x \<noteq> (0::'b)"   and "snd y \<noteq> (0::'b)" shows "normalize_quot (fst x * fst y, snd x * snd y) = normalize_quot (fst (normalize_quot x) * fst (normalize_quot y), snd (normalize_quot x) * snd (normalize_quot y))"
C_eq_iD_aux2: fixes p :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "noDenyAll1 p"   and "separated p"   and "p \<noteq> []"   and "x \<in> dom (C (list2FWpolicy p))" shows "C (list2FWpolicy (insertDenies p)) x = C (list2FWpolicy p) x"
elims_3: fixes x :: "Ferrack.num" assumes "\<not> nozerocoeff x" obtains n :: "nat" and c :: "int" and t :: "Ferrack.num" where "x = CN n c t" and "\<not> (c \<noteq> 0 \<and> nozerocoeff t)"
p_prod_distr: fixes X :: "'a list set"   and Y :: "'a list set"   and Z :: "'a list set" shows "p_prod (X \<union> Y) Z = p_prod X Z \<union> p_prod Y Z"
simps: fixes as :: "nat"   and lm' :: "nat list"   and m :: "nat"   and n :: "nat"   and p :: "nat"   and lm :: "nat list" shows "addition_inv (as, lm') m n p lm = (let sn = lm ! n; sm = lm ! m in lm ! p = 0 \<and> (if as = 0 then \<exists>x\<le>lm ! m. lm' = lm[m := x, n := sn + sm - x, p := sm - x] else if as = 1 then \<exists>x<lm ! m. lm' = lm[m := x, n := sn + sm - x - 1, p := sm - x - 1] else if as = 2 then \<exists>x<lm ! m. lm' = lm[m := x, n := sn + sm - x, p := sm - x - 1] else if as = 3 then \<exists>x<lm ! m. lm' = lm[m := x, n := sn + sm - x, p := sm - x] else if as = 4 then \<exists>x\<le>lm ! m. lm' = lm[m := x, n := sn + sm, p := sm - x] else if as = 5 then \<exists>x<lm ! m. lm' = lm[m := x, n := sn + sm, p := sm - x - 1] else if as = 6 then \<exists>x<lm ! m. lm' = lm[m := Suc x, n := sn + sm, p := sm - x - 1] else if as = 7 then lm' = lm[m := sm, n := sn + sm] else False))"
measurable_envelope_eq1: fixes A :: "'b set"   and E :: "'b set"   and M :: "'b measure" assumes "A \<subseteq> E"   and "E \<in> sets M" shows "measurable_envelope M A E = (\<forall>F\<in>sets M. F \<subseteq> E - A \<longrightarrow> emeasure M F = 0)"
A82_DAcc_level1: shows "DAcc level1 sA82 = {sA93}"
solovay_strassen_witness_def: fixes n :: "nat"   and a :: "nat" shows "solovay_strassen_witness n a = (let x = Jacobi (int a) (int n) in x \<noteq> 0 \<and> [x = int a ^ ((n - 1) div 2)] (mod int n))"
pred_inject_1: fixes P :: "'a \<Rightarrow> bool" shows "pred_phantom P PH1"
c_def: shows "C = P1 - {Pos A} \<union> (P2 - {Neg A})"
conv_abscissa_def: fixes f :: "'a fds" shows "conv_abscissa f = (INF s\<in>{s. fds_converges f s}. ereal (s \<bullet> (1::'a)))"
butlast_snoc: fixes xs :: "'d list"   and x :: "'d" shows "butlast (xs @ [x]) = xs"
fconverse_vintersection: fixes r :: "V"   and s :: "V" shows "(r \<inter>\<^sub>\<circ> s)\<inverse>\<^sub>\<bullet> = r\<inverse>\<^sub>\<bullet> \<inter>\<^sub>\<circ> s\<inverse>\<^sub>\<bullet>"
differentiable_inner: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "f differentiable at x within s"   and "g differentiable at x within s" shows "(\<lambda>x. f x \<bullet> g x) differentiable at x within s"
elims_1: fixes x :: "'b aexp"   and y :: "bool" assumes "is_lit x = y" obtains uu_ :: "value" where "x = L uu_" and "y"   | v :: "'b" where "x = V v__" and "\<not> y"   | v :: "'b aexp" and va :: "'b aexp" where "x = Plus va__ vaa__" and "\<not> y"   | v :: "'b aexp" and va :: "'b aexp" where "x = Minus vb__ vab__" and "\<not> y"   | v :: "'b aexp" and va :: "'b aexp" where "x = Times v va" and "\<not> y"
recognises_step_equiv: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and t :: "(String.literal \<times> value list) list" shows "recognises_execution e s r ((l, i) # t) = (\<exists>(s', T)|\<in>|possible_steps e s r l i. recognises_execution e s' (evaluate_updates T i r) t)"
lookup_HSem_other: fixes y :: "Vars.var"   and h :: "(Vars.var \<times> exp) list"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" assumes "y \<notin> domA h" shows "(\<N>\<lbrace>h\<rbrace>\<rho>) y = \<rho> y"
PL: fixes \<phi> :: "'fmla" assumes "imp (PP \<langle>\<phi>L \<phi>\<rangle>) (PP \<langle>\<phi>\<rangle>) \<in> fmla"   and "Fvars (imp (PP \<langle>\<phi>L \<phi>\<rangle>) (PP \<langle>\<phi>\<rangle>)) = {}"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "prv (imp (PP \<langle>\<phi>L \<phi>\<rangle>) (PP \<langle>\<phi>\<rangle>))"
atms_of_subst_atms: fixes C :: "'a literal multiset"   and \<sigma> :: "'s" shows "atms_of C \<cdot>as \<sigma> = atms_of (C \<cdot> \<sigma>)"
functor_axioms: shows "functor B D (K \<circ> F)"
wl'_Inl: fixes ra :: "'a"   and s :: "'s"   and r' :: "'a"   and s' :: "'s" assumes "(Some (ra, s), Some (r', s')) \<in> whileLoop_results C (\<lambda>a. gets_the (B a))" shows "s = s' \<and> (Some ra, Some r') \<in> option_while' (\<lambda>a. C a s) (\<lambda>a. B a s)"
simps_5: fixes a :: "name"   and b :: "name"   and P :: "pi" shows "hnf ([a\<frown>b]P) = False"
valid_node_Exit_path: fixes prog :: "cmd"   and n :: "w_node" assumes "valid_node prog n" shows "\<exists>as. While_CFG.path prog n as (_Exit_)"
fold_keys_max_bot_eq: fixes t :: "('b, unit) RBT.rbt" assumes "\<not> RBT.is_empty t" shows "fold_keys max t bot = fold1_keys max t"
sel: fixes x :: "'d" shows "run_option (OptionT x) = x"
sup0: fixes a :: "'a" shows "a \<equiv> a \<squnion> \<bottom>"
ordertype_def: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" shows "ordertype A r \<equiv> set (ordermap A r ` A)"
permutep_single: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'b" assumes "bij f" shows "permutep f (Poly_Mapping.single a b) = Poly_Mapping.single (inv f a) b"
ideals_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "ord.ideals less_eq = {X. X \<noteq> {} \<and> ord.downclosed_set less_eq X \<and> ord.directed less_eq X}"
mk_next_def: fixes x :: "'a ltln" shows "mk_next x \<equiv> case x of true\<^sub>n \<Rightarrow> true\<^sub>n | false\<^sub>n \<Rightarrow> false\<^sub>n | _ \<Rightarrow> X\<^sub>n x"
nonzero_mult_div_cancel_left: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.semidom_divide divide plus minus zero times one"   and "a \<noteq> zero" shows "divide (times a b) a = b"
distinct_2: fixes int :: "int" shows "\<infinity> \<noteq> eint int"
prio_emb_pos_le: fixes p :: "dir list"   and t :: "('c, 'd) tm"   and s :: "('c, 'd) tm"   and prio :: "dir list \<Rightarrow> nat" assumes "p \<noteq> []"   and "position_of t p"   and "emb_step_at' p t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s" shows "prio (prio_emb_pos prio t s) \<le> prio p"
Polygamma_approx_2_complex: fixes x :: "complex"   and m :: "nat" assumes "0 < Re x"   and "0 < m" shows "Polygamma_approx (Suc (Suc 0)) m x = stirling_sum' 1 m x"
dom_map_of_conv_domA: fixes \<Gamma> :: "('d \<times> 'e) list" shows "dom (map_of \<Gamma>) = domA \<Gamma>"
take_bit_horner_sum_bit_eq: 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 division_segment :: "'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and bs :: "bool list" assumes "class.unique_euclidean_semiring_with_bit_shifts divide plus minus zero times one modulo euclidean_size division_segment bit push_bit drop_bit take_bit" shows "take_bit n (comm_semiring_0.horner_sum plus zero times (zero_neq_one.of_bool one zero) (numeral.numeral one plus (num.Bit0 num.One)) bs) = comm_semiring_0.horner_sum plus zero times (zero_neq_one.of_bool one zero) (numeral.numeral one plus (num.Bit0 num.One)) (take n bs)"
regular_one_closed: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and conv :: "'c \<Rightarrow> 'c" assumes "class.stone_relation_algebra inf less_eq less sup bot top uminus one times conv" shows "one = uminus (uminus one)"
simps_1: fixes prec :: "nat"   and t :: "nat"   and lf :: "floatarith"   and rt :: "floatarith"   and s :: "nat"   and ivl :: "float interval"   and ivl' :: "float interval" shows "approx_tse_concl prec t (Less lf rt) s ivl ivl' = approx_tse_form' prec t (Add rt (Minus lf)) s (sup ivl ivl') (\<lambda>ivl. 0 < lower ivl)"
extend_JN: fixes h :: "'c \<times> 'd \<Rightarrow> 'e"   and I :: "'f set"   and F :: "'f \<Rightarrow> 'c program" assumes "Extend h" shows "extend h (JOIN I F) = (\<Squnion>i\<in>I. extend h (F i))"
ensuresD: fixes F :: "'c program"   and A :: "'c set"   and B :: "'c set" assumes "F \<in> A ensures B" shows "F \<in> A - B co A \<union> B \<and> F \<in> transient (A - B)"
ivl_rep_of_set_def: fixes X :: "'c set" shows "ivl_rep_of_set X = SPEC (\<lambda>(i, s). i \<le> s \<and> X \<subseteq> {i..s})"
cfs_X_plus: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "X_poly_plus R a n = (if n = 0 then a else if n = 1 then \<one> else \<zero>)"
elims: fixes rep_key_order :: "'c \<Rightarrow> 'a key_order"   and x :: "'a \<times> 'b"   and xa :: "('a \<times> 'b) list \<times> 'c"   and y :: "('a \<times> 'b) list \<times> 'c" assumes "oalist_raw.update_by_raw rep_key_order x xa = y" obtains kv :: "'a \<times> 'b" and xs :: "('a \<times> 'b) list" and ko :: "'c" where "x = kv" and "xa = (xs, ko)" and "y = (comparator.update_by_pair (key_compare (rep_key_order ko)) kv xs, ko)"
inject: fixes x2 :: "'f"   and y2 :: "'f" shows "(Some x2 = Some y2) = (x2 = y2)"
lookup4: fixes a :: "coname"   and \<Delta> :: "(coname \<times> ty) list"   and y :: "name"   and x :: "name" assumes "\<not> a \<sharp> idc \<Delta> y" shows "lookupa x a (idc \<Delta> y) = Cut <a>.Ax x a y.Ax y a"
distinct_keys: fixes t :: "('b, 'c) RBT.rbt" shows "distinct (RBT.keys t)"
exhaust_sel: fixes interval :: "'a interval" obtains "interval = Interval (interval.lower interval) (interval.upper interval)"
nhop_update_unk_val: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dip :: "nat"   and dsn :: "nat"   and hops :: "nat"   and ip :: "nat"   and npre :: "nat set" shows "the (nhop (update rt dip (dsn, unk, val, hops, ip, npre)) dip) = ip"
Fvars_fmla_simps_8: fixes a :: "name"   and A :: "fmla" shows "Fvars (all a A) = Fvars A - {a}"
mset_inter_filter: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and A :: "'a multiset" shows "{#a \<in># A. P a \<and> Q a#} = filter_mset P A \<inter># filter_mset Q A"
Decomp: fixes \<D> :: "('fun, 'var) extstrand_step list"   and \<M> :: "('fun, 'var) Term.term set"   and \<N> :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and f :: "'fun"   and T :: "('fun, 'var) Term.term list"   and K :: "('fun, 'var) Term.term list"   and M :: "('fun, 'var) Term.term list" assumes "\<D> \<in> decomps\<^sub>e\<^sub>s\<^sub>t \<M> \<N> \<I>"   and "Fun f T \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t \<M> \<union> \<N>"   and "Ana (Fun f T) = (K, M)"   and "M \<noteq> []"   and "\<M> \<union> ik\<^sub>e\<^sub>s\<^sub>t \<D> \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile>\<^sub>c Fun f T \<cdot> \<I>"   and "\<And>k. k \<in> set K \<Longrightarrow> \<M> \<union> ik\<^sub>e\<^sub>s\<^sub>t \<D> \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile>\<^sub>c k \<cdot> \<I>" shows "\<D> @ [Decomp (Fun f T)] \<in> decomps\<^sub>e\<^sub>s\<^sub>t \<M> \<N> \<I>"
common_factor_via_short: fixes f :: "int poly"   and g :: "int poly"   and u :: "int poly"   and m :: "int" assumes "0 < degree f"   and "0 < degree g"   and "monic u"   and "0 < degree u"   and "poly_mod.dvdm m u f"   and "poly_mod.dvdm m u g"   and "\<parallel>f\<parallel>\<^sup>2 ^ degree g * \<parallel>g\<parallel>\<^sup>2 ^ degree f < m\<^sup>2"   and "0 \<le> m" shows "0 < degree (gcd f g)"
pairs_bot_closed: fixes phi :: "'c \<Rightarrow> 'd Filters.filter" assumes "triple phi" shows "(bot, Filters.filter.Abs_filter UNIV) \<in> triple.pairs phi"
pick_eqv': fixes a :: "'b" assumes "a \<in> domain" shows "pick \<lfloor>a\<rfloor> \<sim> a"
assm_2: fixes i :: "'a set"   and x :: "'a"   and K :: "'a set" assumes "i \<in> I"   and "(x, K) \<in> pfn i" shows "x \<in> K"
\<UU>\<bb>\<xx>\<OO>\<pp>_correct: fixes opubx :: "'opubx"   and \<Sigma> :: "('dyn, 'ubx1, 'ubx2) unboxed list"   and z :: "('dyn, 'ubx1, 'ubx2) unboxed" assumes "\<UU>\<bb>\<xx>\<OO>\<pp> opubx \<Sigma> = Some z" shows "\<II>\<nn>\<ll>\<OO>\<pp> (\<BB>\<oo>\<xx> opubx) (map Subx.norm_unboxed \<Sigma>) = Subx.norm_unboxed z"
l_inv_ex: fixes x :: "Vertex \<Rightarrow> Vertex" assumes "x \<in> carrier tetrahedral_group" shows "\<exists>y\<in>carrier tetrahedral_group. y \<otimes>\<^bsub>tetrahedral_group\<^esub> x = \<one>\<^bsub>tetrahedral_group\<^esub>"
triangle_inequality: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and D :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B D"   and "Cong B C B D" shows "Tarski_neutral_dimensionless.Le Bet Cong A C A D"
moebius_pt_comp: fixes M1 :: "moebius"   and M2 :: "moebius"   and z :: "complex_homo" shows "moebius_pt (moebius_comp M1 M2) z = moebius_pt M1 (moebius_pt M2 z)"
delete_add_edge: fixes a :: "'a"   and H :: "('a, 'b) graph"   and c :: "'a"   and w :: "'b" assumes "a \<in> nodes H"   and "c \<in> nodes H"   and "(a, w, c) \<notin> edges H" shows "delete_edge a w c (add_edge a w c H) = H"
last_appendR: fixes ys :: "'a list"   and xs :: "'a list" assumes "ys \<noteq> []" shows "List.last (xs @ ys) = List.last ys"
simps: fixes a1 :: "value"   and a2 :: "value" shows "a1 \<approx>\<^sub>e a2 = ((\<exists>ts us name. a1 = Vconstr name ts \<and> a2 = Vconstr name us \<and> list_all2 erelated ts us) \<or> (\<exists>cs \<Gamma>\<^sub>1 \<Gamma>\<^sub>2. a1 = Vabs cs \<Gamma>\<^sub>1 \<and> a2 = Vabs cs \<Gamma>\<^sub>2 \<and> fmrel_on_fset (ids (Sabs cs)) erelated \<Gamma>\<^sub>1 \<Gamma>\<^sub>2) \<or> (\<exists>\<Gamma>\<^sub>1 \<Gamma>\<^sub>2 css name. a1 = Vrecabs css name \<Gamma>\<^sub>1 \<and> a2 = Vrecabs css name \<Gamma>\<^sub>2 \<and> pred_fmap (\<lambda>cs. fmrel_on_fset (ids (Sabs cs)) erelated \<Gamma>\<^sub>1 \<Gamma>\<^sub>2) css))"
that: fixes \<chi> :: "'a \<Rightarrow> complex" assumes "character G \<chi>"   and "\<chi> x = z" shows "thesis"
surjective_mult_closed: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" assumes "matrix_bounded_idempotent_semiring.surjective x"   and "matrix_bounded_idempotent_semiring.surjective y" shows "matrix_bounded_idempotent_semiring.surjective (x \<odot> y)"
fun_cong: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "f = g" shows "f x = g x"
simps_1: fixes n :: "int"   and s :: "char list \<Rightarrow> int option" shows "paval' (N n) s = Some n"
x: fixes x :: "'a pmf"   and y :: "'b pmf" assumes "(x, y) \<in> set_pmf pq" shows "map_pmf fst (PQ x y) = x"
inject_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(In1 x1 = In1 y1) = (x1 = y1)"
adm_wo_def: fixes H :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" shows "adm_wo H \<equiv> \<forall>f g x. (\<forall>y\<in>local.underS x. f y = g y) \<longrightarrow> H f x = H g x"
scale_one: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d" assumes "module scale" shows "scale (1::'c) x = x"
nat_filtration_AE_eq: fixes f :: "bool stream \<Rightarrow> real"   and g :: "bool stream \<Rightarrow> real"   and n :: "nat"   and w :: "bool stream" assumes "AEeq M f g"   and "0 < p"   and "p < 1"   and "f \<in> borel_measurable (nat_filtration n)"   and "g \<in> borel_measurable (nat_filtration n)" shows "f w = g w"
nth_partition: fixes i :: "nat"   and xs :: "'a list"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "i < length xs"   and "\<And>i'. i' < length (filter P xs) \<Longrightarrow> Q (filter P xs ! i')"   and "\<And>i'. i' < length (filter (Not \<circ> P) xs) \<Longrightarrow> Q (filter (Not \<circ> P) xs ! i')" shows "Q (xs ! i)"
simps_1: fixes uu :: "'b infty" shows "uu + Infty = Infty"
get_root_node_root_in_heap: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and h :: "(_) heap"   and ptr :: "(_) object_ptr"   and root :: "(_) object_ptr" assumes "l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf known_ptrs heap_is_wellformed parent_child_rel get_child_nodes get_child_nodes_locs get_disconnected_nodes get_parent get_parent_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs"   and "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_root_node ptr \<rightarrow>\<^sub>r root" shows "root |\<in>| object_ptr_kinds h"
ARR_iff_has_codomain: fixes f :: "'c local.term set" shows "ARR f = (codomains f \<noteq> {})"
HomFtorMor_2: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and f :: "'e" assumes "LSCategory C"   and "X \<in> obj\<^bsub>C\<^esub>"   and "f \<in> mor\<^bsub>C\<^esub>" shows "dom\<^bsub>SET\<^esub> (Hom\<^bsub>C\<^esub>[X,f]) = Hom\<^bsub>C\<^esub> X dom\<^bsub>C\<^esub> f"
gcd_greatest: 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 c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times c a"   and "dvd.dvd times c b" shows "dvd.dvd times c (gcd a b)"
S6_successors: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> $S6 rmhist p \<and> ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<longrightarrow> S1 rmhist p$ \<or> S3 rmhist p$ \<or> S6 rmhist p$"
inverse_prod_list_field: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "(\<Prod>x\<leftarrow>xs. inverse (f x)) = inverse (prod_list (map f xs))"
obtainable_def: fixes s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and e :: "((nat \<times> nat) \<times> transition) fset" shows "obtainable s r e = (\<exists>t. obtains s r e 0 <> t)"
set_of_vector_def: fixes A :: "('c, 'd) vec" shows "set_of_vector A = {A $ i |i. i \<in> UNIV}"
lt_monom_mult_le: fixes c :: "'c"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "lt (monom_mult c t p) \<preceq>\<^sub>t t \<oplus> lt p"
nth_mem_mset: fixes i :: "nat"   and ls :: "'b list" assumes "i < length ls" shows "ls ! i \<in># mset ls"
injectivity: fixes x :: "'a poly"   and y :: "'a poly" assumes "map_poly to_fract x = map_poly to_fract y" shows "x = y"
bbw_ao_absorb: fixes x :: "int"   and y :: "int" shows "x AND (y OR x) = x \<and> x OR y AND x = x"
downset_set_prop_var: fixes X :: "'a set" shows "\<Down> X = \<Union> (\<down> ` X)"
NS_O_S_2: fixes S :: "('a \<times> 'a) set"   and NS :: "('a \<times> 'a) set"   and T :: "('a \<times> 'b) set" assumes "order_pair S NS" shows "NS O S O T = S O T"
condition: fixes a :: "'d set"   and i :: "'c set"   and t :: "'d \<Rightarrow> 'c \<Rightarrow> 'c set"   and c :: "'c \<Rightarrow> bool" shows "accepting (nba a i t c) = c"
a0: shows "i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))"
finite_reach: fixes \<phi> :: "'a ltl" shows "finite (reach \<Sigma> (\<delta>\<^sub>\<A> \<Sigma>) (\<iota>\<^sub>\<A> \<phi>))"
simps: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a1 :: "'c list"   and a2 :: "'c list" shows "ord.lexordp_eq less a1 a2 = ((\<exists>ys. a1 = [] \<and> a2 = ys) \<or> (\<exists>x y xs ys. a1 = x # xs \<and> a2 = y # ys \<and> less x y) \<or> (\<exists>x y xs ys. a1 = x # xs \<and> a2 = y # ys \<and> \<not> less x y \<and> \<not> less y x \<and> ord.lexordp_eq less xs ys))"
simps_1: fixes c :: "'a" shows "num_params (c)\<^sub>p = 0"
a1: shows "s' = snd (fst (nop_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
elims_3: fixes x :: "atom set \<times> 'a"   and xa :: "atom set \<times> 'a" assumes "\<not> x \<approx>abs_res xa" obtains bs :: "atom set" and xb :: "'a" and cs :: "atom set" and y :: "'a" where "x = (bs, xb)" and "xa = (cs, y)" and "\<nexists>p. (bs, xb) \<approx>res (=) supp p (cs, y)"
extract_weights_Pool: fixes shared :: "bool"   and m1 :: "real mat convnet"   and m2 :: "real mat convnet" shows "extract_weights shared (Pool m1 m2) = (\<lambda>x. if x < count_weights shared (remove_weights m1) then extract_weights shared m1 x else extract_weights shared m2 (x - count_weights shared (remove_weights m1)))"
prems_2: shows "fst ` set (tx # txs) \<subseteq> trm"
conversepI: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b"   and b :: "'c" assumes "r a b" shows "r\<inverse>\<inverse> b a"
Derivation_count_terminals_leq: fixes a :: "symbol list"   and E :: "(nat \<times> symbol \<times> symbol list) list"   and b :: "symbol list" assumes "Derivation a E b" shows "count_terminals a \<le> count_terminals b"
nop_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (nop_instr instr s1)) \<and> t2 = snd (fst (nop_instr instr s2))" shows "low_equal t1 t2"
HMA_non_neg_mat: shows "rel_fun HMA_M (=) nonneg_mat non_neg_mat"
polyadd: fixes bs :: "'a list"   and p :: "poly"   and q :: "poly" shows "\<lparr>p +\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> + \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
assms_4: shows "g ` {0..1} \<subseteq> cbox (- 1) 1"
var_not_in_subst_dom: fixes v :: "'c"   and s :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "v \<notin> subst_domain s" shows "s v = Var v"