hcju/isabelleps huggingface dataset:
Compl_greaterThan: fixes k :: "'b" shows "- {k<..} = {..k}"
fun_merge: fixes i :: "nat"   and as :: "'c set list"   and a :: "'c"   and fs :: "('c \<Rightarrow> 'd) list" assumes "i < length as"   and "a \<in> as ! i"   and "\<And>i j a. \<lbrakk>i < length as; j < length as; a \<in> as ! i; a \<in> as ! j\<rbrakk> \<Longrightarrow> (fs ! i) a = (fs ! j) a" shows "fun_merge fs as a = (fs ! i) a"
adv_P1_def: fixes m1 :: "bool \<times> bool"   and m2 :: "bool \<times> bool"   and D :: "((bool \<times> bool) \<times> bool \<times> 'v_14_OT1) \<times> bool \<times> bool \<Rightarrow> bool spmf" shows "and_secret_sharing.adv_P1 m1 m2 D \<equiv> \<bar>spmf (and_R1 m1 m2 \<bind> D) True - spmf (and_funct m1 m2 \<bind> (\<lambda>(o1, o2). and_secret_sharing.Ideal1 m1 m2 o1 \<bind> D)) True\<bar>"
finite_tinfinite: fixes V :: "Preliminaries.var set" assumes "finite V" shows "tinfinite (UNIV - V)"
less_Infinitesimal_less: fixes x :: "real star"   and e :: "real star" assumes "0 < x"   and "x \<notin> Infinitesimal"   and "e \<in> Infinitesimal" shows "e < x"
simps_1: fixes x1 :: "'c extended"   and x2 :: "'c"   and x3 :: "'c extended"   and x4 :: "'d"   and y1 :: "'c extended"   and y2 :: "'c"   and y3 :: "'c extended"   and y4 :: "'d" shows "(InsertMessage x1 x2 x3 x4 = InsertMessage y1 y2 y3 y4) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3 \<and> x4 = y4)"
simps_5: fixes F :: "'b formula"   and G :: "'b formula" shows "cnf (F \<^bold>\<and> G) = cnf F \<union> cnf G"
trim_fair: fixes s :: "'state"   and rs :: "'rule stream" assumes "s \<in> S"   and "fair rs" shows "fair (trim rs s)"
labels_det: fixes c :: "cmd"   and l :: "nat"   and c' :: "cmd"   and c'' :: "cmd" assumes "labels c l c'"   and "labels c l c''" shows "c' = c''"
gcd_div_unit2: 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 a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a one" shows "gcd b (divide c a) = gcd b c"
insert_is_Un: fixes a :: "'b::type"   and A :: "'b::type set" shows "insert (a::'b::type) (A::'b::type set) = {a} \<union> A"
int_simps_31: fixes P :: "'a \<Rightarrow> bool" shows "\<turnstile> (\<exists>x. P) = P"
"3_4": fixes i :: "nat" assumes "i < length (x # y # xs) - 1" shows "(x # y # xs) ! i \<le> (x # y # xs) ! (i + 1)"
da_elim_cases_44: fixes Env :: "env"   and B :: "lname set"   and c :: "expr"   and e1 :: "expr"   and e2 :: "expr"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In1l (c ? e1 : e2)\<guillemotright> A" obtains C :: "assigned" and E1 :: "assigned" and E2 :: "assigned" where "Env\<turnstile>\<langle>c ? e1 : e2\<rangle>\<Colon>Inl (PrimT Boolean)" and "Env\<turnstile> B \<guillemotright>\<langle>c\<rangle>\<guillemotright> Ca__" and "Env\<turnstile> B \<union> assigns_if True c \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1__" and "Env\<turnstile> B \<union> assigns_if False c \<guillemotright>\<langle>e2\<rangle>\<guillemotright> E2__" and "nrm A = B \<union> assigns_if True (c ? e1 : e2) \<inter> assigns_if False (c ? e1 : e2)" and "brk A = (\<lambda>l. UNIV)"   | C :: "assigned" and E1 :: "assigned" and E2 :: "assigned" where "\<not> Env\<turnstile>\<langle>c ? e1 : e2\<rangle>\<Colon>Inl (PrimT Boolean)" and "Env\<turnstile> B \<guillemotright>\<langle>c\<rangle>\<guillemotright> C" and "Env\<turnstile> B \<union> assigns_if True c \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1" and "Env\<turnstile> B \<union> assigns_if False c \<guillemotright>\<langle>e2\<rangle>\<guillemotright> E2" and "nrm A = nrm E1 \<inter> nrm E2" and "brk A = (\<lambda>l. UNIV)"
simps_1: fixes n :: "nat"   and x :: "'b" shows "update1 n x \<langle>\<rangle> = \<langle>\<langle>\<rangle>, x, \<langle>\<rangle>\<rangle>"
xor_inverse: fixes L :: "('c, 'd) boolean_algebra_scheme"   and x :: "'c" assumes "boolean_algebra L"   and "x \<in> carrier L" shows "x \<oplus>\<^bsub>L\<^esub> x = \<bottom>\<^bsub>L\<^esub>"
app_shift: fixes f :: "hf"   and k :: "hf"   and j :: "hf" shows "app (shift f k) (k @+ j) = app f j"
image_poly_is_field: fixes L :: "('c multiset \<Rightarrow> ('a list \<times> nat) multiset \<Rightarrow> 'a) ring"   and i :: "'c"   and q :: "('c multiset \<Rightarrow> ('a list \<times> nat) multiset \<Rightarrow> 'a) list" assumes "field L"   and "\<And>P. P \<in> carrier L \<Longrightarrow> M.carrier_coeff P"   and "\<And>P. P \<in> carrier L \<Longrightarrow> M.index_free P i"   and "\<zero>\<^bsub>L\<^esub> = M.indexed_const \<zero>\<^bsub>M\<^esub>"   and "q \<in> carrier (poly_ring L)"   and "pirreducible\<^bsub>L\<^esub> (carrier L) q" shows "field (image_ring (M.eval_pmod L i q) (poly_ring L))"
ffact_def: fixes n :: "nat"   and a :: "'a" shows "ffact n a = pochhammer (a + (1::'a) - of_nat n) n"
mod_add_self2: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and a :: "'d"   and b :: "'d" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus a b) b = modulo a b"
mayDiverge_not_transT: fixes c :: "('test, 'atom) com"   and s :: "'state"   and s' :: "'state" assumes "mayDiverge c s" shows "\<not> (c, s) \<rightarrow>t s'"
line_integral_dist: shows "one_chain_line_integral F {i, j} one_chain_typeI = one_chain_line_integral F {i} one_chain_typeI + one_chain_line_integral F {j} one_chain_typeI \<and> one_chain_line_integral F {i, j} one_chain_typeII = one_chain_line_integral F {i} one_chain_typeII + one_chain_line_integral F {j} one_chain_typeII"
bg_in: shows "BG \<in> \<G>_F_q q B"
inj_on_dec_above: fixes b :: "nat"   and i :: "nat" shows "inj_on (dec_above b i) {x. x < b \<or> b + i \<le> x}"
fmdrop_set_twice: fixes S :: "'c set"   and T :: "'c set"   and m :: "('c, 'd) fmap" shows "fmdrop_set S (fmdrop_set T m) = fmdrop_set (S \<union> T) m"
multiset_filter_mono: fixes A :: "'a multiset"   and B :: "'a multiset"   and f :: "'a \<Rightarrow> bool" assumes "A \<subseteq># B" shows "filter_mset f A \<subseteq># filter_mset f B"
eq: fixes x :: "'a"   and y :: "'a" assumes "le x y"   and "le y x" shows "u x + \<epsilon> * f x = u y + \<epsilon> * f y"
troeger_1: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y)\<^sup>\<star> = x\<^sup>\<star> * Rf y ((x \<squnion> y)\<^sup>\<star>)"
traverse_cast_Dyn_eq_norm_stack: fixes xs :: "('dyn, 'ubx1, 'ubx2) unboxed list" assumes "list_all (\<lambda>x. x = None) (map typeof xs)" shows "traverse cast_Dyn xs = Some (norm_stack xs)"
wf_trm_param: fixes f :: "'fun"   and T :: "('fun, 'b) Term.term list"   and t :: "('fun, 'b) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m (Fun f T)"   and "t \<in> set T" shows "wf\<^sub>t\<^sub>r\<^sub>m t"
set_zip_cart: fixes x :: "'a \<times> 'b"   and l :: "'a list"   and l' :: "'b list" assumes "x \<in> set (zip l l')" shows "x \<in> set l \<times> set l'"
deg_pp_single: fixes x :: "'b"   and k :: "'a" shows "deg_pp (single_pp x k) = k"
has_field_def: fixes P :: "'a prog"   and C :: "char list"   and F :: "char list"   and b :: "staticb"   and T :: "ty"   and D :: "char list" shows "P \<turnstile> C has F,b:T in D \<equiv> \<exists>FDTs. P \<turnstile> C has_fields FDTs \<and> map_of FDTs (F, D) = \<lfloor>(b, T)\<rfloor>"
r2_leq: shows "r2 \<le> r'\<^sub>2"
update_convs_16: fixes sip' :: "nat \<Rightarrow> nat"   and ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and rreqid :: "nat"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and more :: "'a" shows "sip_update sip' \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, rreqid = rreqid, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, \<dots> = more\<rparr> = \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, rreqid = rreqid, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip' sip, \<dots> = more\<rparr>"
ideal_ele_sumTr2: fixes I :: "'a set"   and a :: "'a"   and b :: "'a" assumes "ideal R I"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "a \<plusminus> b \<in> I"   and "b \<in> I" shows "a \<in> I"
simps_4: shows "T_f32 \<noteq> T_i32"
reduce_column_aux_index: fixes H :: "'a mat"   and m :: "nat"   and n :: "nat"   and P_init :: "'a mat"   and K_init :: "'a mat"   and P :: "'a mat"   and K :: "'a mat"   and div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "nat list"   and i :: "nat" assumes "H \<in> carrier_mat m n"   and "P_init \<in> carrier_mat m m"   and "K_init \<in> carrier_mat m n"   and "P_init * H = K_init"   and "(P, K) = local.reduce_column_aux div_op xs H (P_init, K_init)"   and "0 < m"   and "invertible_mat P_init"   and "0 \<notin> set xs"   and "\<forall>x\<in>set xs. x < m"   and "distinct xs"   and "i \<in> set xs" shows "Matrix.row K i = Matrix.row (addrow (- div_op (H $$ (i, 0)) (H $$ (0, 0))) i 0 K_init) i"
times_Box_def: fixes x :: "'c Box"   and y :: "'c Box" shows "x * y \<equiv> bliftM2 (\<Lambda> a b. a * b)\<cdot>x\<cdot>y"
oalist_inv_raw_def: fixes xs :: "('a \<times> 'c) list" shows "oalist_inv_raw xs = ((0::'c) \<notin> snd ` set xs \<and> sorted_wrt lt (map fst xs))"
med_progress_dom: fixes r :: "fid_t \<Rightarrow> Runs.var set option" shows "dom (med_progress r) = dom r"
i_shrink_eq_Msg_iSince_conv: fixes k :: "nat"   and m :: "'b message_af"   and s :: "nat \<Rightarrow> 'b message_af"   and t :: "nat" assumes "0 < k"   and "m \<noteq> NoMsg" shows "((s \<div> k) t = m) = (s t2 = NoMsg. t2 \<S> t1 [t * k\<dots>,k - Suc 0]. s t1 = m)"
inv_greek_mono: fixes r :: "('a \<times> 'a) set"   and s :: "(accent \<times> 'a) list"   and t :: "(accent \<times> 'a) list" assumes "trans r"   and "(s, t) \<in> greek_less r" shows "(inv_greek s, inv_greek t) \<in> greek_less r"
sup_continuous_def: fixes F :: "'b \<Rightarrow> 'c" shows "sup_continuous F = (\<forall>M. incseq M \<longrightarrow> F (Sup (range M)) = (SUP i. F (M i)))"
from_conjugate_space_inverse: fixes x :: "'c conjugate_space" shows "to_conjugate_space (from_conjugate_space x) = x"
hseqI: fixes \<mu> :: "'a"   and a :: "'a"   and b :: "'a"   and \<nu> :: "'a"   and c :: "'a" assumes "\<guillemotleft>\<mu> : a \<rightarrow>\<^sub>B b\<guillemotright>"   and "\<guillemotleft>\<nu> : b \<rightarrow>\<^sub>B c\<guillemotright>" shows "B.hseq \<nu> \<mu>"
simps_2: fixes xs :: "'a list"   and n :: "nat" shows "feval xs (FVar n) = nth_el xs n"
fls_divide_uminus: fixes f :: "'a fls"   and g :: "'a fls" shows "- f / g = - (f / g)"
emb_step_at_is_App: fixes p :: "dir list"   and d :: "dir"   and u :: "('c, 'd) tm" assumes "emb_step_at p d u \<noteq> u" shows "is_App u"
accmodi_method_simp: fixes sig :: "sig"   and m :: "qtname \<times> methd" shows "accmodi (method sig m) = accmodi m"
pderiv_char_poly_mat_erase: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "monom (1::'b) 1 * pderiv (char_poly A) = (\<Sum>i<n. char_poly (mat_erase A i i))"
CodFunctor: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme"   and f :: "'c" assumes "Functor F"   and "f \<in> mor\<^bsub>CatDom F\<^esub>" shows "cod\<^bsub>CatCod F\<^esub> (F ## f) = F @@ cod\<^bsub>CatDom F\<^esub> f"
Bv_def: shows "Bv = map vec_of_basis_enum B"
three_composable_arrows_of_spans_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data"   and \<pi> :: "'b arrow_of_spans_data" shows "three_composable_arrows_of_spans C prj0 prj1 \<mu> \<nu> \<pi> \<equiv> (span_bicategory C prj0 prj1 \<and> arrow_of_spans C \<mu> \<and> arrow_of_spans C \<nu>) \<and> arrow_of_spans C \<pi> \<and> two_composable_arrows_of_spans C prj0 prj1 \<mu> \<nu> \<and> two_composable_arrows_of_spans C prj0 prj1 \<nu> \<pi>"
case_NegE: fixes P :: "formula \<Rightarrow> bool"   and \<phi> :: "formula" assumes "case \<phi> of Neg \<phi>' \<Rightarrow> P \<phi>' | _ \<Rightarrow> False" obtains \<phi>'' :: "formula" where "\<phi> = Neg \<phi>''" and "P \<phi>''"
prems: shows "prove_nonneg prnt (Suc i) p (slp_of_fas [fa]) XSS"
init_satisfies_bounds: fixes t :: "(nat \<times> linear_poly) list" shows "\<langle>\<V> (init t)\<rangle> \<Turnstile>\<^sub>b \<B> (init t)"
nullstellensatz_lemma: fixes p :: "complex poly"   and q :: "complex poly"   and n :: "nat" assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"   and "degree p = n"   and "n \<noteq> 0" shows "p dvd q ^ n"
1: fixes t' :: "'state" assumes "(d, t) \<rightarrow>*t t'"   and "s' \<approx> t'" shows "thesis"
left_adjoints_compose: fixes f :: "('a S.E.term, 'a) S.arr"   and f' :: "('a S.E.term, 'a) S.arr" assumes "S.is_left_adjoint f"   and "S.is_left_adjoint f'"   and "S.src f' = S.trg f" shows "S.is_left_adjoint (f' \<star>\<^sub>S f)"
vbrelation_vunion: fixes r :: "V"   and s :: "V" shows "vbrelation (r \<union>\<^sub>\<circ> s) = (vbrelation r \<and> vbrelation s)"
ba': fixes z :: "complex" assumes "z \<in> closed_segment b a'" shows "d \<bullet> z \<le> k"
c: fixes x :: "real" assumes "x \<in> {0..1} - (\<phi> -` (s' \<inter> {0..1}) \<union> s \<inter> {0..1})" shows "D x * (F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative \<gamma>2 (at (\<phi> x) within {0..1}) \<bullet> b)) = F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b)"
validFrom_Nil: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and s :: "'a" shows "IO_Automaton.validFrom step s []"
Set_filter_rbt_filter: fixes P :: "'b \<Rightarrow> bool"   and t :: "('b, unit) RBT.rbt" shows "Set.filter P (Set t) = rbt_filter P t"
Entry_path: fixes sourcenode :: "'g \<Rightarrow> 'h"   and targetnode :: "'g \<Rightarrow> 'h"   and kind :: "'g \<Rightarrow> ('i, 'j, 'k, 'l) edge_kind"   and valid_edge :: "'g \<Rightarrow> bool"   and Entry :: "'h"   and get_proc :: "'h \<Rightarrow> 'l"   and get_return_edges :: "'g \<Rightarrow> 'g set"   and procs :: "('l \<times> 'i list \<times> 'i list) list"   and Main :: "'l"   and Exit :: "'h"   and n :: "'h" assumes "Postdomination sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit"   and "CFG.valid_node sourcenode targetnode valid_edge n" shows "\<exists>as. CFG.valid_path' sourcenode targetnode kind valid_edge get_return_edges Entry as n"
Fn: shows "list_all (\<lambda>a. GE.int \<xi> (gT a) = Ik.int \<xi> a) Tl"
incr_bounds_eq: fixes k :: "nat"   and n :: "nat"   and t :: "Term_Class.term"   and i :: "int" assumes "k \<le> n"   and "wellformed' k t" shows "incr_bounds i n t = t"
Abs: fixes r :: "dB"   and r' :: "dB" assumes "r \<rightarrow>\<^sub>l r'" shows "Abs r \<rightarrow>\<^sub>l Abs r'"
step_Normal_elim_cases_1: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and f :: "'f"   and u :: "('e, 'd, 'f) com list \<times> (('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (c # cs, css, Fault f) \<rightarrow> u" obtains "u = (cs, css, Fault f)"
eq_upred_def: fixes x :: "('b, 'a) uexpr"   and y :: "('b, 'a) uexpr" shows "(x =\<^sub>u y) = bop (=) x y"
pmdl_closed_spoly: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<in> pmdl F"   and "q \<in> pmdl F" shows "spoly p q \<in> pmdl F"
delivery_has_a_cause: fixes m :: "'msg"   and i :: "nat" assumes "Deliver m \<in> set (history i)" shows "\<exists>j. Broadcast m \<in> set (history j)"
round_max_eq: fixes t :: "nat \<times> nat list \<times> 'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and p :: "nat"   and q :: "nat"   and r :: "nat" assumes "fst (snd t) \<noteq> []"   and "Max (set (fst (snd t))) = Suc 0" shows "Max (set (fst (snd (round index key p q r t)))) = Suc 0"
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_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and remove_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_child_locs :: "(_) object_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and remove :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, unit) 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_owner_document get_owner_document"   and "l_set_child_nodes_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs"   and "l_set_child_nodes_get_disconnected_nodes type_wf set_child_nodes set_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs"   and "l_set_disconnected_nodes_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs"   and "l_set_disconnected_nodes_get_child_nodes set_disconnected_nodes_locs get_child_nodes_locs"   and "l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes set_child_nodes set_child_nodes_locs get_parent get_owner_document get_disconnected_nodes set_disconnected_nodes set_disconnected_nodes_locs remove_child remove_child_locs remove" shows "l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs get_parent get_parent_locs get_owner_document get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs remove_child remove_child_locs remove type_wf known_ptr known_ptrs"
integrable_bounded_linear: fixes T :: "'c \<Rightarrow> 'd"   and M :: "'e measure"   and f :: "'e \<Rightarrow> 'c" assumes "bounded_linear T"   and "integrable M f" shows "integrable M (\<lambda>x. T (f x))"
annotate_normI: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and anno :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and c :: "('e, 'd, 'f) com" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P anno Q,A"   and "Language.normalize c = Language.normalize anno" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
nth_list_update_neq: fixes i :: "nat"   and j :: "nat"   and xs :: "'c list"   and x :: "'c" assumes "i \<noteq> j" shows "xs[i := x] ! j = xs ! j"
directed: fixes i :: "'c"   and j :: "'c" assumes "i \<in> I"   and "j \<in> I" shows "\<exists>k\<in>I. f i + g j \<le> f k + g k"
SUPP: fixes y :: "'b" assumes "y \<in> Field s" shows "rs.SUPP (\<lambda>b. if b \<in> Field s then if b = y then x else r.zero else undefined) = {y}"
Inf_greatest: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'a set"   and z :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq z x" shows "less_eq z (Inf A)"
clean_successors: fixes v :: "nat \<times> 'a"   and V :: "(nat \<times> 'a) set"   and u :: "nat \<times> 'a"   and w :: "'b stream"   and A :: "('b, 'a) nba" assumes "v \<in> V"   and "u \<in> gusuccessors A w v"   and "u \<in> clean A w V" shows "v \<in> clean A w V"
hyps_1: fixes p :: "real poly"   and q :: "real poly" assumes "xs = smods p q"   and "coprime p q" shows "cindex_polyE a b q p = real_of_int (changes_alt_itv_smods a b p q) / 2"
join_upper2: fixes y :: "'c"   and x :: "'c" shows "y \<sqsubseteq> x \<squnion> y"
in_IF2rel: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d IF2"   and y :: "'e IF2" shows "IF2rel R x y = (\<exists>z. z \<in> IF2in {(x, y). R x y} \<and> IF2map fst z = x \<and> IF2map snd z = y)"
funpow_simps_right_1: fixes f :: "'g \<Rightarrow> 'g" shows "f ^^ 0 = id"
FVG_def: fixes \<alpha> :: "game" shows "FVG \<alpha> = {x. \<exists>I \<nu> \<omega> X. Vagree \<nu> \<omega> (- {x}) \<and> \<nu> \<in> game_sem I \<alpha> (restrictto X (- {x})) \<and> \<omega> \<notin> game_sem I \<alpha> (restrictto X (- {x}))}"
unit_factor_self: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "dvd.dvd times (unit_factor a) a"
omega_fbox: fixes x :: "'a"   and y :: "'a" shows "\<Omega> x y = d y \<cdot>  |x] ad y"
basis_in_finite: fixes \<E> :: "'a set"   and A :: "'a set" assumes "\<E> \<subseteq> \<E>"   and "\<E>.basis_in \<E> A" shows "finite A"