hcju/isabelleps huggingface dataset:
list_strip_comb: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and abs_pred :: "('c \<Rightarrow> bool) \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "'c" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "term.list_comb app (fst (term.strip_comb unapp t)) (snd (term.strip_comb unapp t)) = t"
recognises_prim: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and t :: "(String.literal \<times> value list) list" shows "recognises_execution e s r t = recognises_prim e s r t"
subreachableE: fixes A :: "(('a \<Rightarrow> 'b) \<times> 'c, 'd) automaton"   and U :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"   and J :: "'a set"   and s :: "('a \<Rightarrow> 'b) \<times> 'c"   and I :: "'d \<Rightarrow> bool" assumes "subreachable A U J"   and "s \<in> oreachable A (\<lambda>s s'. I) U" shows "\<exists>\<sigma>. (\<forall>j\<in>J. \<sigma> j = fst s j) \<and> (\<sigma>, snd s) \<in> reachable A I"
eval_length_2: fixes x :: "'a"   and xs :: "'a list" shows "length (x # xs) = Suc (length xs)"
generate_subgroup_incl: fixes K :: "(nat \<Rightarrow> nat) set"   and H :: "(nat \<Rightarrow> nat) set" assumes "K \<subseteq> H"   and "subgroup H (alt_group n)" shows "generate (alt_group n) K \<subseteq> H"
surjective: fixes r :: "('a, 'b, 'c) spec_scheme" shows "r = \<lparr>init = init r, trans = TS.trans r, obs = obs r, \<dots> = spec.more r\<rparr>"
prems_2: shows "hd xs \<noteq> (0::'a)"
in_homE: fixes f :: "('a, 'a set) Maps.arr"   and a :: "('a, 'a set) Maps.arr"   and b :: "('a, 'a set) Maps.arr" assumes "Maps.in_hom f a b" obtains "Maps.arr f" and "Maps.dom f = a" and "Maps.cod f = b"
cindex_polyE_mult_cancel: fixes p' :: "real poly"   and a :: "real"   and b :: "real"   and q :: "real poly"   and p :: "real poly" assumes "p' \<noteq> 0" shows "cindex_polyE a b (p' * q) (p' * p) = cindex_polyE a b q p"
Un_absorb2: fixes B :: "'c set"   and A :: "'c set" assumes "B \<subseteq> A" shows "A \<union> B = A"
ordIso_transitive: fixes r :: "('f \<times> 'f) set"   and r' :: "('g \<times> 'g) set"   and r'' :: "('h \<times> 'h) set" assumes "r =o r'"   and "r' =o r''" shows "r =o r''"
prems_3: shows "\<not> ps (S t i) q \<noteq> None"
mod_in_res_units: fixes m :: "int"   and a :: "int" assumes "residues m"   and "1 < m"   and "coprime a m" shows "a mod m \<in> Units (residue_ring m)"
mult_mat_vec_def: fixes A :: "'d mat"   and v :: "'d Matrix.vec" shows "A *\<^sub>v v \<equiv> Matrix.vec (dim_row A) (\<lambda>i. Matrix.row A i \<bullet> v)"
Q_axioms_def: shows "Q_axioms \<equiv> {uu_. \<exists>A X1 X2. uu_ = A \<and> X1 \<noteq> X2 \<and> (A = Robinson_Arithmetic.neg (zer EQ suc (Var X1)) \<or> A = suc (Var X1) EQ suc (Var X2) IMP Var X1 EQ Var X2 \<or> A = Var X2 EQ zer OR exi X1 (Var X2 EQ suc (Var X1)) \<or> A = pls (Var X1) zer EQ Var X1 \<or> A = pls (Var X1) (suc (Var X2)) EQ suc (pls (Var X1) (Var X2)) \<or> A = tms (Var X1) zer EQ zer \<or> A = tms (Var X1) (suc (Var X2)) EQ pls (tms (Var X1) (Var X2)) (Var X1))}"
ex_rcoset_replist: shows "\<exists>gs. Supgroup.is_rcoset_replist G gs"
prj_simps_1: shows "C.seq \<mu>\<nu>_\<pi>.prj\<^sub>1 \<p>\<^sub>1[\<pi>.leg0 \<cdot> \<mu>\<nu>_\<pi>.prj\<^sub>0, \<rho>.leg1]"
min_range_coeffs_ge: fixes l :: "real"   and x :: "real"   and u :: "real"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and d :: "real" assumes "l \<le> x"   and "x \<le> u"   and "\<And>y. y \<in> {l..u} \<Longrightarrow> (f has_real_derivative f' y) (at y)"   and "\<And>y. y \<in> {l..u} \<Longrightarrow> a \<le> f' y"   and "(f u - f l - a * (u - l)) / 2 + \<bar>(f l + f u - a * (l + u)) / 2 - b\<bar> \<le> d" shows "\<bar>f x - (a * x + b)\<bar> \<le> d"
le_approx3: fixes P :: "'b process"   and Q :: "'b process" assumes "P \<sqsubseteq> Q" shows "min_elems (D P) \<subseteq> T Q"
lcos_module_rev: fixes H :: "'d set"   and G :: "('d, 'e) monoid_scheme"   and x :: "'d"   and x' :: "'d" assumes "subgroup H G"   and "Group.group G"   and "x \<in> carrier G"   and "x' \<in> carrier G"   and "inv\<^bsub>G\<^esub> x \<otimes>\<^bsub>G\<^esub> x' \<in> H" shows "x' \<in> x <#\<^bsub>G\<^esub> H"
Taylor: fixes n :: "nat"   and diff :: "nat \<Rightarrow> real \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and c :: "real"   and x :: "real" assumes "0 < n"   and "diff 0 = f"   and "\<forall>m t. m < n \<and> a \<le> t \<and> t \<le> b \<longrightarrow> (diff m has_real_derivative diff (Suc m) t) (at t)"   and "a \<le> c"   and "c \<le> b"   and "a \<le> x"   and "x \<le> b"   and "x \<noteq> c" shows "\<exists>t. (if x < c then x < t \<and> t < c else c < t \<and> t < x) \<and> f x = (\<Sum>m<n. diff m c / fact m * (x - c) ^ m) + diff n t / fact n * (x - c) ^ n"
infsetsum_cmult_left: fixes c :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "c \<noteq> (0::'c) \<Longrightarrow> f abs_summable_on A" shows "(\<Sum>\<^sub>ax\<in>A. f x * c) = infsetsum f A * c"
mtx_vec_mult_minus_rdistrib: fixes A :: "'a sq_mtx"   and B :: "'a sq_mtx"   and x :: "(real, 'a) vec" shows "(A - B) *\<^sub>V x = A *\<^sub>V x - B *\<^sub>V x"
antisym: fixes b :: "'b list"   and a :: "'b list" assumes "Sublist.prefix b a"   and "Sublist.prefix a b" shows "a = b"
step: fixes x :: "'a node \<times> 'a node set" shows "f x \<le> SPEC (\<lambda>a. expand_assm_exist \<xi> x \<longrightarrow> expand_rslt_exist \<xi> x a)"
Disj_eq_iff: fixes xset1 :: "('a, 'b, 'c) formula set['a]"   and xset2 :: "('a, 'b, 'c) formula set['a]" assumes "finite (supp xset1)"   and "finite (supp xset2)" shows "(Disj xset1 = Disj xset2) = (xset1 = xset2)"
dep_imp_supset_circuit: fixes X :: "'a set" assumes "X \<subseteq> carrier"   and "\<not> indep X" shows "\<exists>C. circuit C \<and> C \<subseteq> X"
rcD_5: shows "poly_cond (poly_real_alg_1 (p, l, r))"
LowNon0: shows "low_proc_non0\<^sub>2 s\<^sub>L"
hd_append: fixes xs :: "'d list"   and ys :: "'d list" shows "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
S2: shows "nat_to_sch (c_pair 3 0) = (let u = mod7 (c_fst (c_pair 3 0)); v = c_snd (c_pair 3 0); v1 = c_fst v; v2 = c_snd v; sch1 = nat_to_sch v1 in Let (nat_to_sch v2) (loc_f u sch1))"
less_imp_decode_less: fixes M :: "hmultiset"   and e :: "nat"   and N :: "hmultiset" assumes "local.well_base\<^sub>h M"   and "local.aligned\<^sub>d e M"   and "local.aligned\<^sub>d e N"   and "M < N" shows "local.decode e M < local.decode e N"
P_Pf: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "let PPf = \<lambda>t1 t2. psubst Pf [(t1, yy), (t2, xx)] in bprv (eqv (subst P \<langle>\<phi>\<rangle> xx) (exi yy (PPf (Var yy) \<langle>\<phi>\<rangle>)))"
row_carrier: fixes A :: "'b mat"   and i :: "nat" shows "row A i \<in> carrier_vec (dim_col A)"
mult_one_right: fixes P :: "real mpoly" shows "P * Const 1 = P"
asInp_equal_iff_alphaInp: fixes qinp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm option"   and qinp' :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm option" assumes "qGoodInp qinp" shows "(asInp qinp = asInp qinp') = ((qinp, qinp') \<in> alphaInp)"
simps_2: fixes A :: "'b \<times> ('c \<Rightarrow> 'b \<Rightarrow> 'b set) \<times> ('b \<Rightarrow> bool)"   and a :: "'c"   and w :: "'c list" shows "steps A (a # w) = step A a O steps A w"
ediv_cancel2: fixes k :: "nat"   and x :: "enat" assumes "0 < k" shows "ediv (enat k * x) k = x"
disc_2: fixes x21 :: "'c"   and x22 :: "'d" shows "\<not> is_Get (Put x21 x22)"
simps_7: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and x2 :: "'b" shows "rec_option f1 f2 (Some x2) = f2 x2"
norm_f_n: shows "\<exists>M. \<forall>n. norm (f n) \<le> M"
det3_nonneg_scaleR1: fixes e :: "real"   and xr :: "real \<times> real"   and P :: "real \<times> real" assumes "0 < e"   and "0 \<le> det3 0 xr P" shows "0 \<le> det3 0 (e *\<^sub>R xr) P"
1: shows "snd (sub2_wit [] fs) = map gso [0..<length fs]"
assms_2: shows "U *\<^sub>S \<top> = \<top>"
Int_Un_eq_3: fixes S :: "'b set"   and T :: "'b set" shows "S \<union> S \<inter> T = S"
card_Diff_singleton_if: fixes A :: "'c set"   and x :: "'c" assumes "finite A" shows "card (A - {x}) = (if x \<in> A then card A - 1 else card A)"
Polygamma_Real: fixes z :: "'a"   and n :: "nat" assumes "z \<in> \<real>"   and "z \<noteq> (0::'a)" shows "Polygamma n z \<in> \<real>"
check_no_overlap_alt: fixes \<gamma> :: "'d set \<Rightarrow> 'e \<Rightarrow> bool"   and ft :: "('d, 'f) flow_entry_match list" shows "check_no_overlap \<gamma> ft = check_no_overlap2 \<gamma> ft"
count_mset_set_if: fixes A :: "'b set"   and a :: "'b" shows "count (mset_set A) a = (if a \<in> A \<and> finite A then 1 else 0)"
numeral_eq_iff: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and m :: "num"   and n :: "num" assumes "class.semiring_char_0 one times plus zero" shows "(numeral.numeral one plus m = numeral.numeral one plus n) = (m = n)"
MapO_def: fixes F :: "('b, 'a, 'c, 'd, 'e, 'f, 'g) Functor_scheme"   and X :: "'b" shows "F @@ X \<equiv> THE Y. Y \<in> obj\<^bsub>CatCod F\<^esub> \<and> F ## id\<^bsub>CatDom F\<^esub> X = id\<^bsub>CatCod F\<^esub> Y"
cr_ltl_prop_equiv_quotient_def: shows "cr_ltl_prop_equiv_quotient \<equiv> \<lambda>x. (=) (Abs x)"
simps_2: fixes acc :: "nat"   and l :: "'a list"   and ls :: "'a list list" shows "acc_lengths acc (l # ls) = (acc + length l) # acc_lengths (acc + length l) ls"
hs_memb_impl: shows "imp_set_memb is_hashset hs_memb"
newQuery_equiv_def: fixes V :: "nat set"   and Q :: "(nat set \<times> 'b option list set) set"   and st :: "nat set"   and t :: "'b option list" shows "New_max.newQuery V Q (st, t) = New_max.projectQuery V ((\<lambda>tab. New_max.semiJoin tab (st, t)) ` Q)"
le_cSup_finite: fixes X :: "'c multiset set"   and x :: "'c multiset" assumes "finite X"   and "x \<in> X" shows "x \<subseteq># Sup X"
assms: shows "r \<subseteq>\<^sub>\<circ> A \<times>\<^sub>\<bullet> B"
ordinal_rec0_oLimit: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and lub :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a"   and s :: "ordinal \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and f :: "nat \<Rightarrow> ordinal" assumes "omega_complete le lub"   and "\<forall>p x. le x (s p x)" shows "ordinal_rec0 z s lub (oLimit f) = lub (\<lambda>n. ordinal_rec0 z s lub (f n))"
ex_un_null: shows "\<exists>!n. \<forall>f. Right_a.comp n f = n \<and> Right_a.comp f n = n"
fun_ord_def: fixes ord :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'f"   and g :: "'h \<Rightarrow> 'g" shows "fun_ord ord f g = (\<forall>x. ord (f x) (g x))"
hom: shows "homeomorphic_maps (top_of_set U) (Euclidean_space (dim U)) \<phi> \<phi>'"
exec_bind_SE_failure2: fixes A :: "'b \<Rightarrow> ('a \<times> 'b) option"   and \<sigma> :: "'b"   and M :: "'b \<Rightarrow> (bool \<times> 'b) option" assumes "A \<sigma> = None" shows "\<not> (\<sigma> \<Turnstile> A ;- M)"
fun_upd_restrict_conv: fixes x :: "'a"   and D :: "'a set"   and m :: "'a \<Rightarrow> 'b option"   and y :: "'b option" assumes "x \<in> D" shows "(m |` D)(x := y) = (m |` (D - {x}))(x := y)"
iop_i_i: fixes a :: "'a" assumes "a \<in> carrier G" shows "\<rho> (\<rho> a) = a"
rec_exec_pr_Suc_simps: fixes n :: "nat"   and f :: "recf"   and g :: "recf"   and xs :: "nat list"   and y :: "nat" shows "rec_exec (Pr n f g) (xs @ [Suc y]) = rec_exec g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])])"
distinct_mset_def: fixes S :: "'a multiset" shows "distinct_mset S = (\<forall>a. a \<in># S \<longrightarrow> count S a = 1)"
insert_union_subset_iff: fixes a :: "'k"   and A :: "'k multiset"   and B :: "'k multiset" shows "(add_mset a A \<subset># B) = (a \<in># B \<and> A \<subset># B - {#a#})"
pwd_spy_i_msg: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "case pwd_spy_i n of (s, S, A, B, C, D) \<Rightarrow> pwd_spy_i_state n (S, A, B, C, D) \<subseteq> s"
simps_4: fixes \<C> :: "t_context"   and t :: "t"   and uw :: "binop_i"   and ts :: "checker_type" shows "check_single \<C> (Binop_i t uw) ts = (if is_int_t t then type_update ts [TSome t, TSome t] (Type [t]) else Bot)"
add_scale_eq_noteq: fixes r :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "r \<noteq> (0::'a)"   and "a = b \<and> c \<noteq> d" shows "a + r * c \<noteq> b + r * d"
cyclic_group_axioms: fixes \<eta> :: "nat" shows "cyclic_group (\<G> \<eta>)"
bounded_linear_scaleR_left: fixes x :: "'b" shows "bounded_linear (\<lambda>r. r *\<^sub>R x)"
mult_1_right: fixes a :: "'a" shows "a \<cdot> (1::'a) = a"
intros_3: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set" assumes "F \<triangleright> \<Gamma> \<turnstile> \<bottom>" shows "\<Gamma> \<turnstile> \<^bold>\<not> F"
Gen_subset_\<J>_0_0_helper: fixes x :: "item"   and \<gamma> :: "symbol list" assumes "wellformed_item x"   and "item_origin x = 0"   and "item_end x = 0"   and "derives (item_\<alpha> x) []"   and "is_derivation (item_nonterminal x # \<gamma>)" shows "x \<in> \<pi> 0 {} Init"
\<eta>\<epsilon>: shows "internal_equivalence f g \<eta> \<epsilon>"
bezout_matrix_not_zero: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and a :: "'c"   and b :: "'c"   and A :: "(('b, 'd) vec, 'c) vec"   and j :: "'d" assumes "is_bezout_ext bezout"   and "a \<noteq> b"   and "A $ a $ j \<noteq> (0::'b)" shows "(bezout_matrix A a b j bezout ** A) $ a $ j \<noteq> (0::'b)"
ord_isom_trans: fixes E :: "('g, 'h) Order_scheme"   and F :: "('i, 'j) Order_scheme"   and f :: "'a \<Rightarrow> 'g"   and g :: "'g \<Rightarrow> 'i" assumes "Order E"   and "Order F"   and "ord_isom D E f"   and "ord_isom E F g" shows "ord_isom D F (compose (carrier D) g f)"
trees_args: fixes t :: "'a"   and s :: "'a" assumes "t \<in> trees"   and "s \<in> set (args t)" shows "s \<in> trees"
4: assumes "a = c"   and "b \<noteq> c" shows "\<exists>f. (\<Sum>i\<in>{c, b, c}. f i * i) = x * c + y * b + z * c"
return_bind: fixes x :: "'f"   and f :: "'f \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'd) \<times> bool" shows "return x >>= f = f x"
diff_add_eq: fixes plus :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and zero :: "'r"   and minus :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and uminus :: "'r \<Rightarrow> 'r"   and a :: "'r"   and b :: "'r"   and c :: "'r" assumes "class.ab_group_add plus zero minus uminus" shows "plus (minus a b) c = minus (plus a c) b"
that: fixes n :: "int" assumes "x + real_of_int n * (2 * pi) \<in> aform_err e Y" shows "thesis"
it_to_it_map_linord_correct: fixes it :: "(('a \<times> 'b) list \<Rightarrow> bool) \<Rightarrow> ('a \<times> 'b \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('a \<times> 'b) list) \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('a \<times> 'b) list"   and S :: "'a \<Rightarrow> 'b option" assumes "map_iterator_linord it S" shows "map_iterator_linord (it_to_it it) S"
f2_def: shows "f2 \<equiv> \<lambda>A. THE x. \<exists>a b c d. A = {a, b, c, d} \<and> a < b \<and> b < c \<and> c < d \<and> x = f {(a, d), (b, c)}"
matchId: fixes a :: "name"   and P :: "pi" shows "[a\<frown>a]P \<approx> P"
det_not_0_imp_lin_indpt_rows: fixes A :: "'a mat" assumes "A \<in> carrier_mat n n"   and "det A \<noteq> (0::'a)" shows "lin_indpt (set (rows A))"
trivial_def: fixes g :: "'g"   and v :: "'val" shows "step.trivial g v \<equiv> \<exists>v'\<in>step.allVars g. step.isTrivialPhi g v v'"
continuous_rabs: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> real" assumes "continuous F f" shows "continuous F (\<lambda>x. \<bar>f x\<bar>)"
Ssegment_not_inc_iff: fixes D :: "'c Order"   and a :: "'c"   and b :: "'c" assumes "Torder D"   and "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<preceq>\<^bsub>D\<^esub> b = (b \<notin> Ssegment D a)"
convex_cball: fixes x :: "'a"   and e :: "real" shows "convex (cball x e)"
cong_eq_0_I: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and n :: "'b" assumes "\<forall>i\<in>A. [f i mod n = 0::'b] (mod n)" shows "[sum f A = 0::'b] (mod n)"
axioms_1: fixes G :: "('e, 'f) monoid_scheme" assumes "finite_comm_group G" shows "finite_group G"
get_dom_component_ptr: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_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_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_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and is_strongly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and is_weakly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_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_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_element_by_id :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_elements_by_tag_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog"   and h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list" assumes "l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_tag_name"   and "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_component ptr \<rightarrow>\<^sub>r c" shows "ptr \<in> set c"
f_Exec_Stream_Init_nth_Suc_calc: fixes n :: "nat"   and xs :: "'d list"   and trans_fun :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e" assumes "n < length xs" shows "f_Exec_Comp_Stream_Init trans_fun xs c ! Suc n = trans_fun (xs ! n) (f_Exec_Comp_Stream_Init trans_fun xs c ! n)"
map\<^sub>0_simps_1: fixes a :: "'a" assumes "C.obj a" shows "D.obj (G.map\<^sub>0 a)"
length_init: shows "all_list_constr_R subsets (length qs)"
simps: fixes v1 :: "'v"   and p1 :: "'outPort"   and v2 :: "'v"   and p2 :: "'inPort" shows "edge_end ((v1, p1), v2, p2) = v2"
filterlim_nat_sequentially: shows "filterlim nat sequentially at_top"
ex3: fixes C :: "'b formula set set" assumes "pcp (C::'b formula set set)"   and "subset_closed (C::'b formula set set)" shows "\<exists>C'::'b formula set set. (C::'b formula set set) \<subseteq> C' \<and> pcp C' \<and> finite_character C'"
RID\<^sub>LE: fixes r :: "'a"   and s :: "('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr" assumes "r \<in> RID\<^sub>L s" obtains "\<And>\<sigma> \<tau> e. \<lbrakk>s = (\<sigma>, \<tau>, e); r \<in> RID\<^sub>S \<sigma> \<Longrightarrow> thesis; r \<in> RID\<^sub>S \<tau> \<Longrightarrow> thesis; r \<in> RID\<^sub>E e \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis"
minus_mod_self1: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and uminus :: "'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "modulo (minus b a) b = modulo (uminus a) b"
map_fst_zip: fixes xs :: "'c list"   and ys :: "'d list" assumes "length xs = length ys" shows "map fst (xs || ys) = xs"
in_Inf_multiset_iff: fixes A :: "'k multiset set"   and x :: "'k" assumes "A \<noteq> {}" shows "(x \<in># Inf A) = (\<forall>X\<in>A. x \<in># X)"
abs_eq: fixes xa :: "'c \<Rightarrow> 'd"   and x :: "'c \<Rightarrow> 'd" assumes "eq_onp (\<lambda>f. finite {x. f x \<noteq> (0::'d)}) xa xa"   and "eq_onp (\<lambda>f. finite {x. f x \<noteq> (0::'d)}) x x" shows "Abs_poly_mapping xa + Abs_poly_mapping x = Abs_poly_mapping (\<lambda>k. xa k + x k)"
table_of_filter_unique_SomeD: fixes P :: "'b \<times> 'a \<Rightarrow> bool"   and xs :: "('b \<times> 'a) list"   and k :: "'b"   and z :: "'a" assumes "table_of (filter P xs) k = Some z"   and "unique xs" shows "table_of xs k = Some z"
completion_ex_borel_measurable: fixes g :: "'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "g \<in> borel_measurable (completion M)" shows "\<exists>g'\<in>borel_measurable M. AE x in M. g x = g' x"
simps_1: fixes \<delta> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'd" shows "run \<delta> q\<^sub>0 w 0 = q\<^sub>0"
strictly_generalizes_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and C :: "'b literal multiset"   and D :: "'b literal multiset" shows "substitution_ops.strictly_generalizes subst_atm C D = (substitution_ops.generalizes subst_atm C D \<and> \<not> substitution_ops.generalizes subst_atm D C)"
path_components_of_subset_singleton: fixes X :: "'a topology"   and S :: "'a set" shows "(path_components_of X \<subseteq> {S}) = (path_connected_space X \<and> (topspace X = {} \<or> topspace X = S))"
finite_subint1: fixes G :: "prog" shows "finite (subint1 G)"
cone_iff_finite_cone: fixes A :: "'a vec set" assumes "A \<subseteq> carrier_vec (n + 1)"   and "finite A" shows "next_dim.cone A = next_dim.finite_cone A"
rel_intros_3: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x3 :: "'b"   and y3 :: "'c" assumes "R x3 y3" shows "rel_rexp R (Atom x3) (Atom y3)"
case_prod_curry: fixes f :: "'d \<times> 'e \<Rightarrow> 'f" shows "(\<lambda>(x, y). curry f x y) = f"
cases: fixes x :: "'c regular \<times> 'c dense Filters.filter" obtains xa :: "'c regular" and y :: "'c dense Filters.filter" where "x = (xa, y)"
lessThan_subset_iff: fixes x :: "'e"   and y :: "'e" shows "({..<x} \<subseteq> {..<y}) = (x \<le> y)"
close_eq_def: fixes s :: "predicate \<times> 'a list \<Rightarrow> bool" shows "close_eq s = (\<lambda>x. case x of predAtm p xs \<Rightarrow> s (p, xs) | Eq a b \<Rightarrow> a = b)"
bin_lcp_fst_mismatch_pref: shows "i.\<alpha> \<cdot> [i.c\<^sub>0] \<le>p x \<cdot> i.\<alpha>"
valid_prefixes_alt_def: fixes r :: "('b, 'c) routing_rule_scheme list" shows "valid_prefixes r = (\<forall>e\<in>set r. valid_prefix (routing_match e))"
countable_Un_Int_2: fixes I :: "'c set"   and N :: "'c \<Rightarrow> 'd set"   and M :: "'d measure" assumes "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> N i \<in> sets M" shows "\<Inter> (N ` I) \<in> sets M"