hcju/isabelleps huggingface dataset:
weakInputFreshDerivative: fixes \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and x :: "name" assumes "\<Psi> : Q \<rhd> P \<Longrightarrow>M\<lparr>N\<rparr> \<prec> P'"   and "x \<sharp> P"   and "x \<sharp> N" shows "x \<sharp> P'"
2: shows "trms\<^sub>s\<^sub>t (a # A) \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta> = (trms\<^sub>s\<^sub>t\<^sub>p a \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>) \<union> (trms\<^sub>s\<^sub>t A \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>)"
cinner_adj_left: fixes G :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'd"   and x :: "'d"   and y :: "'c" shows "(G* *\<^sub>V x) \<bullet>\<^sub>C y = x \<bullet>\<^sub>C (G *\<^sub>V y)"
nat_seqs_def: fixes R :: "('c, 'd) ring_scheme" shows "R\<^bsup>\<omega>\<^esup> \<equiv> function_ring UNIV R"
r1: fixes a :: "'a" assumes "a \<in> F" shows "\<langle>f x, f a\<rangle> = 0"
least_0: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "ord.Least less_eq (\<lambda>n. n \<in> UNIV) = zero"
add_cvec_def: fixes z :: "complex \<times> complex"   and w :: "complex \<times> complex" shows "z +\<^sub>v w = (let (z1, z2) = z; (w1, w2) = w in if z2 \<noteq> 0 \<or> w2 \<noteq> 0 then (z1 * w2 + w1 * z2, z2 * w2) else (1, 0))"
ex_countable_basis: fixes "open" :: "'b set \<Rightarrow> bool" assumes "class.second_countable_topology open" shows "\<exists>B. countable B \<and> topological_space.topological_basis open B"
rep_mkarr: fixes f :: "'c local.term" assumes "Arr f" shows "\<F>C.rep (\<F>C.mkarr f) = \<^bold>\<parallel>f\<^bold>\<parallel>"
lookup_pair_eq_value: fixes xs :: "('a \<times> 'c) list"   and v :: "'c"   and k :: "'a" assumes "cmp2.oalist_inv_raw xs"   and "v \<noteq> (0::'c)" shows "(cmp2.lookup_pair xs k = v) = ((k, v) \<in> set xs)"
IH: fixes xs :: "ct list" shows "ct_list_eq xs ys = (\<exists>zs. ct_list_eq xs (to_ct_list zs) \<and> ct_list_eq (to_ct_list zs) ys)"
jf: fixes x :: "'a" shows "(j (f (h x)) = x) = (f (h x) = h x)"
vec_to_iarray_nth: fixes i :: "nat"   and A :: "('e, 'd) vec" assumes "i < CARD('d)" shows "vec_to_iarray A !! i = A $ mod_type_class.from_nat i"
crypts_union: fixes H :: "msg set"   and H' :: "msg set" shows "crypts (H \<union> H') = crypts H \<union> crypts H'"
min_ground_head_def: fixes \<zeta> :: "('s, 'v) hd" shows "min_ground_head \<zeta> = (SOME f. f \<in> local.ground_heads \<zeta> \<and> (\<forall>g\<in>local.ground_heads \<zeta>. wt_sym f + \<delta>\<^sub>h * arity_sym\<^sub>h f \<le> wt_sym g + \<delta>\<^sub>h * arity_sym\<^sub>h g))"
l_cancel_one: fixes x :: "'a list"   and a :: "'a list" assumes "x \<in> carrier (K [X])"   and "a \<in> carrier (K [X])" shows "(x \<oplus>\<^bsub>K [X]\<^esub> a = x) = (a = \<zero>\<^bsub>K [X]\<^esub>)"
loxodromic_attracting_fixed_point_attracts_uniformly: fixes f :: "'b \<Rightarrow> 'b"   and K :: "real"   and M :: "real" assumes "loxodromic_isometry f" shows "\<exists>N. \<forall>n\<ge>N. \<forall>x. extended_Gromov_product_at basepoint x (repelling_fixed_point f) \<le> ereal K \<longrightarrow> ereal M \<le> extended_Gromov_product_at basepoint ((Gromov_extension f ^^ n) x) (attracting_fixed_point f)"
fin: shows "\<forall>A\<in>M. f A \<noteq> \<infinity>"
brc_invar_add_def: shows "brc_invar_add \<equiv> {(Q, W, rcm). hs.invar Q \<and> distinct W \<and> hm.invar rcm}"
P10: shows "Bet P' X R' \<and> Cong X R' X R"
\<Delta>\<^sub>d_def: fixes M :: "'a measure"   and I :: "'b set"   and A :: "'b \<Rightarrow> 'a set" assumes "prob_space_with_indicators M I A" shows "prob_space_with_indicators.\<Delta>\<^sub>d M I A = (\<Sum>i\<in>I. \<Sum>j | j \<in> I \<and> i \<noteq> j \<and> \<not> prob_space.indep M (A i) (A j). Sigma_Algebra.measure M (A i \<inter> A j))"
\<psi>\<^sub>3_is_state: shows "state (n + 1) (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then \<Sum>k<2 ^ n. (- 1) ^ (f k + i div 2 \<cdot>\<^bsub>n\<^esub>  k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else \<Sum>k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 \<cdot>\<^bsub>n\<^esub>  k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))"
intros_2: fixes S :: "'a set"   and n :: "nat"   and p :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "swapidseq_ext S n p"   and "a \<notin> S" shows "swapidseq_ext (insert a S) n p"
vimage_Diff: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'d set"   and B :: "'d set" shows "f -` (A - B) = f -` A - f -` B"
integrable_product_swap: fixes M1 :: "'b measure"   and M2 :: "'c measure"   and f :: "'b \<times> 'c \<Rightarrow> 'd" assumes "pair_sigma_finite M1 M2"   and "integrable (M1 \<Otimes>\<^sub>M M2) f" shows "integrable (M2 \<Otimes>\<^sub>M M1) (\<lambda>(x, y). f (y, x))"
cases: fixes a :: "'b llist"   and A :: "'b set" assumes "a \<in> A\<^sup>\<infinity>" obtains "a = LNil"   | l :: "'b llist" and aa :: "'b" where "a = aa ## l" and "l \<in> A\<^sup>\<infinity>" and "aa \<in> A"
tagged_division_ofD_6: fixes s :: "('c \<times> 'c set) set"   and i :: "'c set" assumes "s tagged_division_of i" shows "\<Union> {K. \<exists>x. (x, K) \<in> s} = i"
normalize_prod_mset_primes: fixes A :: "'a multiset" assumes "\<And>p. p \<in># A \<Longrightarrow> prime p" shows "normalize (\<Prod>\<^sub># A) = \<Prod>\<^sub># A"
a: shows "p \<succeq>[\<R>] q \<and> (q, p) \<notin> \<R>"
pnu_nu: fixes f :: "'a \<Rightarrow> 'a" assumes "has_greatest_postfixpoint f"   and "isotone f" shows "p\<nu> f = \<nu> f"
irreducible_mono: fixes b :: "'a"   and a :: "'a" assumes "irreducible b"   and "a dvd b"   and "\<not> is_unit a" shows "irreducible a"
set_map: fixes f :: "'h \<Rightarrow> 'g"   and xs :: "'h list" shows "set (map f xs) = f ` set xs"
fin: shows "finite {r. \<exists>I J. valid_region X k I J r}"
permute_swap_cancel: fixes permute :: "perm \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "atom"   and b :: "atom"   and x :: "'e" assumes "class.pt permute" shows "permute (a \<rightleftharpoons> b) (permute (a \<rightleftharpoons> b) x) = x"
f01: shows "f ` S \<subseteq> ball 0 1"
"1_2": shows "a \<notin> set w\<^sub>1"
tendsto_at_top_eq_left: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and x0 :: "'b"   and g :: "'b \<Rightarrow> 'c" assumes "(f \<longlongrightarrow> l) at_top"   and "\<And>x. x0 \<le> x \<Longrightarrow> f x = g x" shows "(g \<longlongrightarrow> l) at_top"
ar_coset_same3: fixes I :: "'a set"   and a :: "'a" assumes "ideal R I"   and "a \<in> carrier R"   and "a \<uplus>\<^bsub>R\<^esub> I = I" shows "a \<in> I"
subset_iff: fixes A :: "'l set"   and B :: "'l set" shows "(A \<subseteq> B) = (\<forall>t. t \<in> A \<longrightarrow> t \<in> B)"
prv_dsj_impL: fixes \<phi> :: "fm"   and \<chi> :: "fm" assumes "\<phi> \<in> UNIV"   and "\<chi> \<in> UNIV" shows "{} \<turnstile> \<phi> IMP \<phi> OR \<chi>"
a_rcosetsI: fixes H :: "'a set set"   and x :: "'a set" assumes "H \<subseteq> carrier (R Quot I)"   and "x \<in> carrier (R Quot I)" shows "H +>\<^bsub>R Quot I\<^esub> x \<in> a_rcosets\<^bsub>R Quot I\<^esub> H"
bind_def: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'b set" shows "A \<bind> f = {x. \<exists>B\<in>f ` A. x \<in> B}"
assms: shows "f $$ fls_subdegree f * g $$ fls_subdegree g \<noteq> (0::'a)"
MakeCatComp2: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e"   and g :: "'e" assumes "f \<approx>>\<^bsub>MakeCat C\<^esub> g" shows "f ;;\<^bsub>MakeCat C\<^esub> g = f ;;\<^bsub>C\<^esub> g"
weakBang: fixes \<Psi> :: "'b"   and R :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi" assumes "\<Psi> : R \<rhd> P \<parallel> !P \<Longrightarrow>\<alpha> \<prec> P'"   and "guarded P" shows "\<Psi> : R \<rhd> !P \<Longrightarrow>\<alpha> \<prec> P'"
elims: fixes x :: "atom"   and y :: "int" assumes "hd_coeff x = y" obtains i :: "int" and ks :: "int list" where "x = Le i__ ks__" and "y = (case ks__ of [] \<Rightarrow> 0 | k # x \<Rightarrow> k)"   | d :: "int" and i :: "int" and ks :: "int list" where "x = Dvd d__ ia__ ksa__" and "y = (case ksa__ of [] \<Rightarrow> 0 | k # x \<Rightarrow> k)"   | d :: "int" and i :: "int" and ks :: "int list" where "x = NDvd d i ks" and "y = (case ks of [] \<Rightarrow> 0 | k # x \<Rightarrow> k)"
rep_eq: fixes x :: "('a, 'b, 'c) formula" shows "Rep_formula (Formula.Not x) = Not\<^sub>\<alpha> (Rep_formula x)"
inf_hmultiset_def: fixes A :: "hmultiset"   and B :: "hmultiset" shows "inf A B = (if A < B then A else B)"
sum_in_smallo_1: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and h :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> o[F](h)"   and "g \<in> o[F](h)" shows "(\<lambda>x. f x + g x) \<in> o[F](h)"
simps_2: fixes q :: "'a"   and k :: "nat"   and l :: "nat"   and rA :: "'a \<times> 'a mat" shows "local.sub1 q (Suc k) l rA = local.mute q (l + Suc k) l (local.sub1 q k l rA)"
multi_member_split: fixes x :: "'b"   and M :: "'b multiset" assumes "x \<in># M" shows "\<exists>A. M = x, A"
closure_interior: fixes S :: "'a set" shows "closure S = - interior (- S)"
hyps_6: shows "l = (y\<^sub>2 - y\<^sub>1) / (x\<^sub>2 - x\<^sub>1)"
deriv_c1: shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P\<^sub>1 c\<^sub>1 I,A"
cpoly_of_def: fixes pR :: "real poly"   and pI :: "real poly" shows "cpoly_of pR pI = map_poly complex_of_real pR + smult \<i> (map_poly complex_of_real pI)"
diff_eq_eq: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.group_add minus plus zero uminus" shows "(minus a b = c) = (a = plus c b)"
matrix: shows "(V, E) = access_matrix \<lparr>pc_iiface = anyI, pc_oiface = anyO, pc_proto = p_proto p, pc_sport = p_sport p, pc_dport = p_dport p\<rparr> (to_simple_firewall_without_interfaces ipassmt rtblo rs)"
iso_iff_section_and_retraction: fixes f :: "'d" shows "D.iso f = (D.section f \<and> D.retraction f)"
b'_1: shows "b' \<in>\<^sub>A frontier (c_imp c loc2)"
converse_rtrancl_on_into_rtrancl_on: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set"   and c :: "'a"   and F :: "'a set" assumes "(a, b) \<in> r"   and "(b, c) \<in> rtrancl_on F r"   and "a \<in> F" shows "(a, c) \<in> rtrancl_on F r"
le_lt_comparator_of_1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "le_of_comp (linorder.comparator_of less) = less_eq"
assms_2: shows "Postdomination sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit"
Nmlize_Hcomp_Hcomp': fixes t :: "'a E.term"   and u :: "'a E.term"   and v :: "'a E.term" assumes "E.Arr t"   and "E.Arr u"   and "E.Arr v"   and "E.Src t = E.Trg u"   and "E.Src u = E.Trg v" shows "\<^bold>\<lfloor>t \<^bold>\<star> u \<^bold>\<star> v\<^bold>\<rfloor> = \<^bold>\<lfloor>\<^bold>\<lfloor>t\<^bold>\<rfloor> \<^bold>\<star> \<^bold>\<lfloor>u\<^bold>\<rfloor> \<^bold>\<star> \<^bold>\<lfloor>v\<^bold>\<rfloor>\<^bold>\<rfloor>"
map_eq_conv: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and g :: "'b \<Rightarrow> 'a" shows "(map f xs = map g xs) = (\<forall>x\<in>set xs. f x = g x)"
cases: fixes E :: "('e \<times> 'e) set"   and a1 :: "'e"   and a2 :: "'e list"   and a3 :: "'e" assumes "path E a1 a2 a3" obtains u :: "'e" where "a1 = u__" and "a2 = []" and "a3 = u__"   | u :: "'e" and v :: "'e" and l :: "'e list" and w :: "'e" where "a1 = u" and "a2 = u # l" and "a3 = w" and "(u, v) \<in> E" and "path E v l w"
simps_2: fixes x :: "'sz"   and y :: "'sz"   and p :: "'sz"   and args :: "'sz \<Rightarrow> ('sf, 'sz) trm" shows "FUrename x y ($\<phi> p args) = $\<phi> p (\<lambda>i. TUrename x y (args i))"
from_nat_into_surj: fixes A :: "'b set"   and a :: "'b" assumes "countable A"   and "a \<in> A" shows "\<exists>n. from_nat_into A n = a"
mkarr_rep_ARR: fixes f :: "'c local.term set" assumes "ARR f" shows "mkarr (rep f) = f"
simple_path_circlepath: fixes z :: "complex"   and r :: "real" shows "simple_path (circlepath z r) = (r \<noteq> 0)"
current_leI: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "current \<Gamma> f"   and "\<And>e. g e \<le> f e"   and "\<And>x. x \<notin> A \<Gamma> \<Longrightarrow> d_OUT g x \<le> d_IN g x" shows "current \<Gamma> g"
nn_cond_exp_cong: fixes f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal" assumes "AE x in M. f x = g x"   and "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "AE x in M. nn_cond_exp M F f x = nn_cond_exp M F g x"
onorm_triangle: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "bounded_linear f"   and "bounded_linear g" shows "onorm (\<lambda>x. f x + g x) \<le> onorm f + onorm g"
except_keys_Int: fixes p :: "'c \<Rightarrow>\<^sub>0 'd"   and U :: "'c set" shows "except p (keys p \<inter> U) = except p U"
subcls_asym: fixes wf_md :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and D :: "char list" assumes "wf_prog wf_md P"   and "(C, D) \<in> (subcls1 P)\<^sup>+" shows "(D, C) \<notin> (subcls1 P)\<^sup>+"
qGood_iff_good_asTerm: fixes qX :: "('a, 'b, 'varSort, 'var, 'c) qTerm" shows "good (asTerm qX) = qGood qX"
is_scc_restrict3: fixes E :: "('b \<times> 'b) set"   and U :: "'b set" assumes "is_scc E U" shows "E\<^sup>* `` (E\<^sup>* `` U - U) \<inter> U = {}"
is_denormal_FloatD_1: fixes x :: "('a, 'b) IEEE.float itself"   and f :: "Float.float" assumes "is_denormal_Float x f" shows "mantissa f \<noteq> 0"
ceiling_divide_eq_div: fixes a :: "int"   and b :: "int" shows "\<lceil>of_int a / of_int b\<rceil> = - (- a div b)"
simps_5: fixes f1 :: "nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> 'a"   and x1 :: "nat" shows "(case I x1 of I x \<Rightarrow> f1 x | R x \<Rightarrow> f2 x) = f1 x1"
member_remove: fixes x :: "'b"   and y :: "'b"   and A :: "'b set" shows "((x::'b) \<in> Set.remove (y::'b) (A::'b set)) = (x \<in> A \<and> x \<noteq> y)"
wt_elim_cases_7: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and e :: "expr"   and T' :: "ref_ty"   and T :: "tys" assumes "E,dt\<Turnstile>In1l (e InstOf T')\<Colon>T" obtains T :: "ref_ty" where "Tb__ = Inl (PrimT Boolean)" and "E,dt\<Turnstile>e\<Colon>-RefT T" and "is_acc_type (prg E) (pkg E) (RefT T')" and "prg E\<turnstile>RefT T\<preceq>? RefT T'"
CatExp_def: fixes A :: "('d, 'f, 'h) Category_scheme"   and B :: "('e, 'g, 'i) Category_scheme" shows "CatExp A B \<equiv> MakeCat (CatExp' A B)"
sets_1: shows "A \<subseteq> Pi\<^sub>E J S"
fps_X_nth: fixes n :: "nat" shows "fps_X $ n = (if n = 1 then 1::'a else (0::'a))"
proj_def: fixes \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" shows "proj \<phi> x \<equiv> {y. \<phi> x y}"
eq: shows "\<forall>d\<in>ds. lv p d = lv b d"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   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_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_mode :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, shadow_root_mode) prog"   and get_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, char list option) prog"   and get_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and first_in_tree_order :: "(_) object_ptr \<Rightarrow> ((_) object_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog) \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and find_slot :: "bool \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and assigned_slot :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog" assumes "l_get_parent type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs"   and "l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs"   and "l_get_mode type_wf get_mode get_mode_locs"   and "l_get_attribute type_wf get_attribute get_attribute_locs"   and "l_get_tag_name type_wf get_tag_name get_tag_name_locs"   and "l_to_tree_order to_tree_order"   and "l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M to_tree_order first_in_tree_order"   and "l_find_slot\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_parent get_shadow_root get_mode get_attribute get_tag_name first_in_tree_order find_slot assigned_slot" shows "l_find_slot\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_parent get_parent_locs get_shadow_root get_shadow_root_locs get_mode get_mode_locs get_attribute get_attribute_locs get_tag_name get_tag_name_locs first_in_tree_order find_slot assigned_slot type_wf known_ptr known_ptrs get_child_nodes get_child_nodes_locs to_tree_order"
alts_ge_4: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf" assumes "sds_impossibility agents alts sds" shows "4 \<le> card alts"
assms_7: shows "ptr \<notin> set |h \<turnstile> get_dom_component (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^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 |h \<turnstile> get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 node_ptr)|\<^sub>r)|\<^sub>r"
R0_def: shows "R0 = R1 \<union> R2"
rbt_less_prop: fixes k :: "'a"   and t :: "('a, 'e) rbt" shows "(t |\<guillemotleft> k) = (\<forall>x\<in>set (keys t). x < k)"
sas_plus_problem_to_strips_problem_variable_set_element_iff: fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and v :: "'a"   and a :: "'b" assumes "is_valid_problem_sas_plus \<Psi>" shows "((v, a) \<in> set ((\<phi> \<Psi> )\<^sub>\<V>)) = (v \<in> set (\<Psi>\<^sub>\<V>\<^sub>+) \<and> a \<in> \<R>\<^sub>+ \<Psi> v)"
arc_reversepath: fixes g :: "real \<Rightarrow> 'b" assumes "arc g" shows "arc (reversepath g)"
IH: shows "local.hanoi n = repeat (2 ^ n - 1) (tick (return ()))"
power_aform_err_def: fixes p :: "nat"   and X :: "(real \<times> real pdevs) \<times> real"   and n :: "nat" shows "power_aform_err p X n = (if n = 0 then ((1, zero_pdevs), 0) else if n = 1 then X else let x0 = float_of (fst (fst X)); xs = snd (fst X); xe = float_of (snd X); C = the (approx p (Power (Num x0) n) []); (c, ce) = mid_err C; NX = the (approx p (Num (of_nat n) * Power (Num x0) (n - 1)) []); (nx, nxe) = mid_err NX; Y = scaleR_pdevs nx xs; (Y', Y_err) = trunc_bound_pdevs p Y; t = tdev' p xs; Ye = truncate_up p (nxe * t); err = the (approx p (Num (of_nat n) * Num xe * Power (Abs (Num x0)) (n - 1) + Sum\<^sub>e (\<lambda>k. Num (of_nat (n choose k)) * Power (Abs (Num x0)) (n - k) * Power (Num xe + Num (float_of t)) k) [2..<Suc n]) []); ERR = upper err in ((c, Y'), sum_list' p [ce, Y_err, Ye, real_of_float ERR]))"
finite_baseClasses: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Bs :: "base list"   and fs :: "(char list \<times> ty) list"   and ms :: "(char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list" assumes "class P C = \<lfloor>(Bs, fs, ms)\<rfloor>" shows "finite (baseClasses Bs)"
1: shows "pfp (step' \<top>) (bot c) = Some C"
k_mkAlgAuto_ec: fixes t :: "'s Trace"   and a :: "'a" assumes "t \<in> jkbpC" shows "simAbs (runJP k_mkAlgAuto t a) = sim_equiv_class a t"
fresh_def: fixes fresh_in :: "'b set \<Rightarrow> 'b" shows "class.fresh fresh_in \<equiv> \<forall>S. finite S \<longrightarrow> fresh_in S \<notin> S"
simps_982: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "tf \<Rightarrow> b_e list \<Rightarrow> 'a"   and f6 :: "tf \<Rightarrow> b_e list \<Rightarrow> 'a"   and f7 :: "tf \<Rightarrow> b_e list \<Rightarrow> b_e list \<Rightarrow> 'a"   and f8 :: "nat \<Rightarrow> 'a"   and f9 :: "nat \<Rightarrow> 'a"   and f10 :: "nat list \<Rightarrow> nat \<Rightarrow> 'a"   and f11 :: "'a"   and f12 :: "nat \<Rightarrow> 'a"   and f13 :: "nat \<Rightarrow> 'a"   and f14 :: "nat \<Rightarrow> 'a"   and f15 :: "nat \<Rightarrow> 'a"   and f16 :: "nat \<Rightarrow> 'a"   and f17 :: "nat \<Rightarrow> 'a"   and f18 :: "nat \<Rightarrow> 'a"   and f19 :: "t \<Rightarrow> (tp \<times> sx) option \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f20 :: "t \<Rightarrow> tp option \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f21 :: "'a"   and f22 :: "'a"   and f23 :: "v \<Rightarrow> 'a"   and f24 :: "t \<Rightarrow> unop_i \<Rightarrow> 'a"   and f25 :: "t \<Rightarrow> unop_f \<Rightarrow> 'a"   and f26 :: "t \<Rightarrow> binop_i \<Rightarrow> 'a"   and f27 :: "t \<Rightarrow> binop_f \<Rightarrow> 'a"   and f28 :: "t \<Rightarrow> testop \<Rightarrow> 'a"   and f29 :: "t \<Rightarrow> relop_i \<Rightarrow> 'a"   and f30 :: "t \<Rightarrow> relop_f \<Rightarrow> 'a"   and f31 :: "t \<Rightarrow> cvtop \<Rightarrow> t \<Rightarrow> sx option \<Rightarrow> 'a"   and x281 :: "t"   and x282 :: "testop" shows "(case Testop x281 x282 of Unreachable \<Rightarrow> f1 | Nop \<Rightarrow> f2 | Drop \<Rightarrow> f3 | Select \<Rightarrow> f4 | Block x xa \<Rightarrow> f5 x xa | Loop x xa \<Rightarrow> f6 x xa | b_e.If x xa xb \<Rightarrow> f7 x xa xb | Br x \<Rightarrow> f8 x | Br_if x \<Rightarrow> f9 x | Br_table x xa \<Rightarrow> f10 x xa | Return \<Rightarrow> f11 | Call x \<Rightarrow> f12 x | Call_indirect x \<Rightarrow> f13 x | Get_local x \<Rightarrow> f14 x | Set_local x \<Rightarrow> f15 x | Tee_local x \<Rightarrow> f16 x | Get_global x \<Rightarrow> f17 x | Set_global x \<Rightarrow> f18 x | Load x xa xb xc \<Rightarrow> f19 x xa xb xc | Store x xa xb xc \<Rightarrow> f20 x xa xb xc | Current_memory \<Rightarrow> f21 | Grow_memory \<Rightarrow> f22 | C x \<Rightarrow> f23 x | Unop_i x xa \<Rightarrow> f24 x xa | Unop_f x xa \<Rightarrow> f25 x xa | Binop_i x xa \<Rightarrow> f26 x xa | Binop_f x xa \<Rightarrow> f27 x xa | Testop x xa \<Rightarrow> f28 x xa | Relop_i x xa \<Rightarrow> f29 x xa | Relop_f x xa \<Rightarrow> f30 x xa | Cvtop x xa xb xc \<Rightarrow> f31 x xa xb xc) = f28 x281 x282"