hcju/isabelleps huggingface dataset:
trace_parallel_plan_strips_operator_preconditions: fixes k :: "nat"   and I :: "'b \<Rightarrow> bool option"   and \<pi> :: "'b strips_operator list list" assumes "k < length (trace_parallel_plan_strips I \<pi>) - 1" shows "are_all_operators_applicable (trace_parallel_plan_strips I \<pi> ! k) (\<pi> ! k) \<and> are_all_operator_effects_consistent (\<pi> ! k)"
IH: assumes "token_run x n \<in> F" shows "token_run x (n + m) \<in> F"
prime_odd_nat: fixes p :: "nat" assumes "prime p"   and "2 < p" shows "odd p"
Noetherian_def: fixes x :: "'a" shows "Noetherian x = (\<forall>y. \<Omega> x y = (zero_class.zero::'a) \<longrightarrow> d y = (zero_class.zero::'a))"
not_nrows: shows "Suc (mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A)) \<noteq> nrows A"
c0: shows "dim_row A \<Otimes> B\<^sup>\<dagger> = dim_row (A\<^sup>\<dagger> \<Otimes> B\<^sup>\<dagger>)"
cps23_1: fixes x :: "\<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>type\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and y :: "\<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>type\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "\<AA> state \<times> \<AA> state" shows "(x \<doteq> y) \<tau> = ((\<lambda>_. x \<tau>) \<doteq> (\<lambda>_. y \<tau>)) \<tau>"
uniformity_blinfun_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
meet_left: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<sqinter> y \<sqsubseteq> x"
prevs_def: fixes v :: "nat" shows "prevs v = (g_E G)\<inverse> `` {v}"
locally_compact: fixes s :: "'b set" shows "locally compact s = (\<forall>x\<in>s. \<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and> openin (top_of_set s) u \<and> compact v)"
assign_proj_refl: fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set"   and i :: "'b" assumes "Hintikka A H"   and "Nom i at i in' H"   and "i \<in> A" shows "assign A H i = proj (hequiv_rel A H) i"
some_img_def: shows "S.some_img = (SOME img. set_category_given_img (\<cdot>\<^sub>S) img (cardSuc (cmax |UNIV| natLeq)))"
assms_2: fixes a :: "'a" assumes "P a" shows "Q (f a)"
intro: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and t :: "'b \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and G :: "'b \<Rightarrow> 'c" assumes "\<And>a. partial_magma.ide A a \<Longrightarrow> partial_magma.in_hom B (t a) (F a) (G a)"   and "\<And>f. partial_magma.arr A f \<Longrightarrow> B (t (partial_magma.cod A f)) (F f) = B (G f) (t (partial_magma.dom A f))" shows "transformation_by_components_axioms A B F G t"
dl_monotone: fixes r :: "('a \<times> 'a) set"   and \<sigma> :: "'a list"   and \<tau> :: "'a list"   and \<tau>' :: "'a list" shows "r \<down>l (\<sigma> @ \<tau>) \<subseteq> r \<down>l (\<sigma> @ \<tau>' @ \<tau>)"
par_beta_ssubst: fixes t :: "sterm"   and t' :: "sterm"   and v :: "sterm"   and v' :: "sterm"   and x :: "char list" assumes "t => t'"   and "v => v'" shows "[x \<rightarrow> v] t => [x \<rightarrow> v'] t'"
sqrt_int_main_def: fixes x :: "int" shows "sqrt_int_main x = root_int_main 2 x"
unit_disc_fix_f_moebius_pt: fixes M :: "moebius" assumes "unit_disc_fix M" shows "unit_disc_fix_f (moebius_pt M)"
zero_less_real_of_ereal: fixes x :: "ereal" shows "(0 < real_of_ereal x) = (0 < x \<and> x \<noteq> \<infinity>)"
floor_sum_times_2_powr_sgn_eq: fixes a :: "real"   and p :: "int"   and ai :: "int"   and b :: "real"   and q :: "int" assumes "a * 2 powr real_of_int p = real_of_int ai"   and "\<bar>b * 2 powr real_of_int (p + 1)\<bar> \<le> 1"   and "q \<le> p" shows "\<lfloor>(a + b) * 2 powr real_of_int q\<rfloor> = \<lfloor>(real_of_int (2 * ai) + sgn b) * 2 powr real_of_int (q - p - 1)\<rfloor>"
tell_writer_def: fixes return :: "'h \<times> 'g list \<Rightarrow> 'i"   and bind :: "'i \<Rightarrow> ('h \<times> 'g list \<Rightarrow> 'i) \<Rightarrow> 'i"   and w :: "'g"   and m :: "('g, 'h, 'i) writerT" shows "tell_writer return bind w m = WriterT (bind (run_writer m) (\<lambda>(a, ws). return (a, w # ws)))"
open_segment_PairD: fixes x :: "'b"   and x' :: "'c"   and a :: "'b"   and a' :: "'c"   and b :: "'b"   and b' :: "'c" assumes "(x, x') \<in> open_segment (a, a') (b, b')" shows "(x \<in> open_segment a b \<or> a = b) \<and> (x' \<in> open_segment a' b' \<or> a' = b')"
w_simps_6: shows "cod \<chi>.chine = \<chi>.chine"
map_of_remap_SomeD2: fixes t :: "(('d \<times> 'a) \<times> 'b \<times> 'c) list"   and k :: "'d"   and k' :: "'a"   and x :: "'b"   and x' :: "'c" assumes "map_of (map (\<lambda>((k, k'), x, x'). (k, k', x, x')) t) k = \<lfloor>(k', x, x')\<rfloor>" shows "map_of t (k, k') = \<lfloor>(x, x')\<rfloor>"
mc: fixes \<sigma> :: "'tp" shows "\<sigma> \<turnstile>2 c"
rev_subsetD: fixes c :: "'g"   and A :: "'g set"   and B :: "'g set" assumes "c \<in> A"   and "A \<subseteq> B" shows "c \<in> B"
Rel2Mon_if: fixes b :: "'a \<Rightarrow> bool"   and c :: "'a \<Rightarrow> (unit \<times> 'a) option"   and \<sigma> :: "'a" shows "Rel2Mon {(s, t). if b s then (s, t) \<in> Mon2Rel c O lfp (\<Gamma> b (Mon2Rel c)) else s = t} \<sigma> = (if b \<sigma> then Rel2Mon (Mon2Rel c O lfp (\<Gamma> b (Mon2Rel c))) \<sigma> else Some ((), \<sigma>))"
reachable_succs: fixes x :: "bool list" shows "reachable (set (subset_succs A x)) \<subseteq> reachable {x}"
is_sublist_is_nextElem: fixes vs :: "'a list"   and x :: "'a"   and y :: "'a"   and as :: "'a list" assumes "distinct vs"   and "is_nextElem vs x y"   and "is_sublist as vs"   and "x \<in> set as"   and "x \<noteq> last as" shows "is_sublist [x, y] as"
intros: fixes xs :: "'b list list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x \<noteq> []" shows "map_head_side xs"
has_field_def: fixes 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 F :: "char list"   and T :: "ty"   and D :: "char list" shows "P \<turnstile> C has F:T in D \<equiv> \<exists>FDTs. P \<turnstile> C has_fields FDTs \<and> map_of FDTs (F, D) = \<lfloor>T\<rfloor>"
th2_5: fixes a :: "hyp2"   and b :: "hyp2"   and c :: "hyp2"   and d :: "hyp2" assumes "a b \<congruent>\<^sub>K c d" shows "a b \<congruent>\<^sub>K d c"
CD_on_def: fixes ds :: "D2 set"   and X :: "X5 set" shows "BSI.CD_on ds X = (\<Union>d\<in>ds. BSI.Cd d X)"
span_span: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'b set" assumes "Modules.module scale" shows "module.span scale (module.span scale A) = module.span scale A"
IH: shows "\<forall>m<n. \<forall>\<Gamma> \<Delta>. ( \<Gamma> \<oplus> Compound F Fs \<Rightarrow>* \<Delta>, m) \<in> derivable R* \<longrightarrow> (\<forall>r'\<in>R. leftPrincipal r' (Compound F Fs) \<longrightarrow> ( \<Gamma>' \<Rightarrow>* \<Delta>') \<in> set (fst r')) \<longrightarrow> (\<exists>m'\<le>m. ( \<Gamma> + \<Gamma>' \<Rightarrow>* \<Delta> + \<Delta>', m') \<in> derivable R*)"
lookup_minus_higher: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t"   and u :: "'t" shows "lookup (p - higher p v) u = (lookup p u when u \<preceq>\<^sub>t v)"
monofun_pair: fixes x1 :: "'a"   and x2 :: "'a"   and y1 :: "'b"   and y2 :: "'b" assumes "x1 \<sqsubseteq> x2"   and "y1 \<sqsubseteq> y2" shows "(x1, y1) \<sqsubseteq> (x2, y2)"
simps: fixes r :: "nat rexp"   and s :: "nat rexp"   and vs :: "('b \<times> 'b) set list" shows "rel_eq (r, s) vs = (rel_of_regexp vs r = rel_of_regexp vs s)"
iterate_to_list_foldli: fixes xs :: "'a list" shows "iterate_to_list (foldli xs) = rev xs"
finite_top_fun: shows "top = \<Squnion>\<^sub>f\<^sub>i\<^sub>n UNIV"
dot_norm: fixes x :: "'a"   and y :: "'a" shows "x \<bullet> y = ((norm (x + y))\<^sup>2 - (norm x)\<^sup>2 - (norm y)\<^sup>2) / 2"
naturality: fixes f :: "'a" assumes "B.arr f" shows "trg\<^sub>B (B.cod f) \<cdot>\<^sub>B trg\<^sub>B f = trg\<^sub>B f \<cdot>\<^sub>B trg\<^sub>B (B.dom f)"
preprocess_def: fixes rs :: "'i common_primitive rule list" shows "preprocess rs \<equiv> upper_closure (optimize_matches abstract_for_simple_firewall (upper_closure (packet_assume_new rs)))"
PAC_checker_step_PAC_checker_specification2: fixes \<V> :: "nat set"   and A :: "(nat, int mpoly) fmap"   and \<V>\<^sub>B :: "nat set"   and B :: "int mpoly multiset"   and a :: "status"   and spec :: "int mpoly"   and A\<^sub>0 :: "int mpoly multiset"   and \<V>\<^sub>0 :: "nat set"   and st :: "(int mpoly, nat, nat) pac_step" assumes "((\<V>, A), \<V>\<^sub>B, B) \<in> polys_rel_full"   and "\<not> is_failed a"   and "a = FOUND \<Longrightarrow> spec \<in> pac_ideal (set_mset A\<^sub>0)"   and "PAC_Format\<^sup>*\<^sup>* (\<V>\<^sub>0, A\<^sub>0) (\<V>, B)"   and "vars spec \<subseteq> \<V>\<^sub>0"   and "\<Union> (vars ` set_mset A\<^sub>0) \<subseteq> \<V>\<^sub>0" shows "PAC_checker_step spec (a, \<V>, A) st \<le> \<Down> (status_rel \<times>\<^sub>r polys_rel_full) (PAC_checker_specification_step2 (\<V>\<^sub>0, A\<^sub>0) spec (\<V>, B))"
pochhammer_pos: fixes x :: "'a"   and n :: "nat" assumes "(0::'a) < x" shows "(0::'a) < pochhammer x n"
wfp_onI_min: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x Q. \<lbrakk>x \<in> Q; Q \<subseteq> A\<rbrakk> \<Longrightarrow> \<exists>z\<in>Q. \<forall>y\<in>A. r y z \<longrightarrow> y \<notin> Q" shows "wfp_on r A"
Inf_countable_INF: fixes A :: "ereal set" assumes "A \<noteq> {}" shows "\<exists>f. decseq f \<and> range f \<subseteq> A \<and> Inf A = Inf (range f)"
bounded_clinear_cinner_right: fixes a :: "'a" shows "bounded_clinear ((\<bullet>\<^sub>C) a)"
star_inductr_eq: fixes z :: "'a"   and y :: "'a"   and x :: "'a" assumes "z + y ; x = y" shows "z ; x\<^sup>\<star> \<le> y"
simps_2: fixes x :: "char list"   and a :: "aexp"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "pre (x ::= a) Q = (\<lambda>l s. Q l (s[a/x]))"
shadow_root_ptr_kinds_eq_h2: shows "shadow_root_ptr_kinds h3 = shadow_root_ptr_kinds h2"
ts_process_next_events: fixes T :: "'a list set"   and x :: "'a"   and xs :: "'a list" assumes "trace_set T" shows "(x \<in> next_events (ts_process T) xs) = (xs @ [x] \<in> T)"
B_def: shows "B \<equiv> \<lambda>r. {x \<in> topspace (Euclidean_space n). enorm x < r}"
emeasure_stat_not_C: fixes y :: "'s"   and C :: "'s set" assumes "y \<notin> C" shows "emeasure (stat C) {y} = 0"
dghm_op_dghm_op_dghm: shows "op_dghm (op_dghm \<FF>) = \<FF>"
last_prog_append_Read\<^sub>s\<^sub>b: fixes p :: "'b"   and sb :: "'b memref list"   and volatile :: "bool"   and a :: "nat"   and t :: "nat"   and v :: "nat" shows "last_prog p (sb @ [Read\<^sub>s\<^sub>b volatile a t v]) = last_prog p sb"
subspace_trans: fixes V :: "'c set"   and U :: "'c set"   and W :: "'c set" assumes "vectorspace V"   and "Subspace.subspace U V"   and "Subspace.subspace V W" shows "Subspace.subspace U W"
sran_singleton: fixes k :: "'d"   and vs :: "'c set" shows "sran { k := vs}. = vs"
measure_ge_1_iff: fixes M :: "'d measure"   and A :: "'d set" assumes "prob_space M" shows "(1 \<le> Sigma_Algebra.measure M A) = (Sigma_Algebra.measure M A = 1)"
greaterThan_def: fixes less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and l :: "'b::type" shows "ord.greaterThan (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (l::'b::type) \<equiv> {x::'b::type. less l x}"
trivial_isomorphism: shows "ChamberComplexIsomorphism X X id"
BIT_special_simps_3: shows "1 BIT False = 2"
that: fixes C :: "'a set" assumes "C \<in> path_components_of X"   and "S \<subseteq> C" shows "thesis"
cobounded2: fixes a :: "'a"   and b :: "'a" assumes "a \<in> U"   and "b \<in> U" shows "local.min a b \<le>\<^sub>o\<^sub>w b"
s'_D_3: shows "(\<forall>x\<in>{0..1} - s'. (\<gamma>2 has_vector_derivative D__ x) (at x)) \<and> continuous_on ({0..1} - s') D__"
ccw_system12_axioms: shows "ccw_system12 ccw S"
inv4: shows "\<forall>v1 v2. v1 \<in> V - X \<and> v2 \<in> S \<longrightarrow> {v1, v2} \<notin> E"
neq_Nil_conv2: fixes xs :: "'a list" shows "(xs \<noteq> []) = (\<exists>x xs'. xs = xs' @ [x])"
neg_numeral_eq_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and m :: "num"   and n :: "num" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(uminus (numeral.numeral one plus m) = uminus (numeral.numeral one plus n)) = (m = n)"
simps: fixes iif1 :: "iface"   and oif1 :: "iface"   and sip1 :: "'a word \<times> nat"   and dip1 :: "'a word \<times> nat"   and p1 :: "protocol"   and sps1 :: "16 word \<times> 16 word"   and dps1 :: "16 word \<times> 16 word"   and iif2 :: "iface"   and oif2 :: "iface"   and sip2 :: "'a word \<times> nat"   and dip2 :: "'a word \<times> nat"   and p2 :: "protocol"   and sps2 :: "16 word \<times> 16 word"   and dps2 :: "16 word \<times> 16 word" shows "simple_match_and \<lparr>iiface = iif1, oiface = oif1, src = sip1, dst = dip1, proto = p1, sports = sps1, dports = dps1\<rparr> \<lparr>iiface = iif2, oiface = oif2, src = sip2, dst = dip2, proto = p2, sports = sps2, dports = dps2\<rparr> = (case ipcidr_conjunct sip1 sip2 of None \<Rightarrow> None | Some sip \<Rightarrow> case ipcidr_conjunct dip1 dip2 of None \<Rightarrow> None | Some dip \<Rightarrow> case iface_conjunct iif1 iif2 of None \<Rightarrow> None | Some iif \<Rightarrow> case iface_conjunct oif1 oif2 of None \<Rightarrow> None | Some oif \<Rightarrow> case simple_proto_conjunct p1 p2 of None \<Rightarrow> None | Some p \<Rightarrow> Some \<lparr>iiface = iif, oiface = oif, src = sip, dst = dip, proto = p, sports = simpl_ports_conjunct sps1 sps2, dports = simpl_ports_conjunct dps1 dps2\<rparr>)"
l: shows "2 \<cdot>\<^sub>m proj_psi * (2 \<cdot>\<^sub>m proj_psi - 1\<^sub>m N) = 4 \<cdot>\<^sub>m proj_psi - 2 \<cdot>\<^sub>m proj_psi"
BinOp: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and e1 :: "expr"   and Q :: "'b assn"   and binop :: "binop"   and e2 :: "expr"   and R :: "'b assn" assumes "G,A\<turnstile>{Normal P} e1-\<succ> {Q}"   and "\<forall>v1. G,A\<turnstile>{Q\<leftarrow>In1 v1} (if need_second_arg binop v1 then In1l e2 else In1r Skip)\<succ> {\<lambda>Val:v2:. R\<leftarrow>In1 (eval_binop binop v1 v2)}" shows "G,A\<turnstile>{Normal P} BinOp binop e1 e2-\<succ> {R}"
semiring_1_def: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" shows "class.semiring_1 one times plus zero \<equiv> class.monoid_mult one times \<and> class.semiring_0 plus zero times \<and> class.zero_neq_one one zero"
test_def: fixes A :: "natural set"   and P :: "natural \<Rightarrow> bool" shows "test A P = card (A \<inter> {x. P x})"
\<gamma>_def: shows "\<gamma> \<equiv> ordermap A VWF x"
H2_J_2_a: fixes P :: "'b \<times> 'c alpha_d_scheme \<Rightarrow> bool"   and a :: "'b"   and b :: "'c alpha_d_scheme" shows "P (a, b) \<longrightarrow> (P ;; J) (a, b)"
diag_undiag_fatom: fixes ts :: "fterm list"   and p :: "char list" assumes "ground\<^sub>t\<^sub>s ts" shows "fatom_of_nat (nat_of_fatom (p, ts)) = (p, ts)"
isSubobjs_subclsR_rev: 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 Cs :: "char list list"   and D :: "char list"   and D' :: "char list"   and Cs' :: "char list list" assumes "is_subobj P (C, Cs @ [D, D'] @ rev Cs')" shows "P \<turnstile> D \<prec>\<^sub>R D'"
lossless: fixes x :: "'call"   and s :: "'s" assumes "x \<in> outs_\<I> \<I>" shows "lossless_spmf (the_gpv (callee s x))"
Archive_def: shows "Archive \<equiv> set (Tri @ Quad @ Pent @ Hex)"
terminates_list_generator: shows "terminates (list_has_next, list_next)"
factor_greater_one1: fixes n :: "nat"   and m :: "nat"   and k :: "nat" assumes "n = m * k"   and "m < n"   and "k < n" shows "Suc 0 < m"
language_def: shows "R.language \<equiv> {smap int (p ## trace w p) |p w. init p \<and> R.run w p}"
l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes 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 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_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   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"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool" 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 \<equiv> (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"
filterlim_iff: fixes f :: "'a \<Rightarrow> 'b"   and F2 :: "'b filter"   and F1 :: "'a filter" shows "filterlim f F2 F1 = (\<forall>P. eventually P F2 \<longrightarrow> (\<forall>\<^sub>F x in F1. P (f x)))"
inf_left_idem: fixes x :: "'b multiset"   and y :: "'b multiset" shows "x \<inter># (x \<inter># y) = x \<inter># y"
r2: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "x = y"   and "\<not> suffix xs ys"   and "P xs ys" shows "P (xs @ [x]) (ys @ [y])"
edc_add: fixes vs :: "nat list"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr"   and e :: "expr"   and f :: "cexpr"   and t :: "pdf_type" assumes "(vs, vs', \<Gamma>, \<delta>) \<turnstile>\<^sub>c e \<Rightarrow> f"   and "\<Gamma> \<turnstile> e : PRODUCT t t" shows "(vs, vs', \<Gamma>, \<delta>) \<turnstile>\<^sub>c Add $$ e \<Rightarrow> \<integral>\<^sub>c (map_vars (case_nat 0 (\<lambda>x. x + 2)) f \<circ>\<^sub>c <CVar 0, CVar 1 -\<^sub>c CVar 0>\<^sub>c) \<partial>t"
l2_inv6_def: shows "l2_inv6 \<equiv> {s. \<forall>Ra A B gny. guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<longrightarrow> in_progress (progress s Ra) xsk \<longrightarrow> sklvl1.guessed_frame Ra xgny = Some gny \<longrightarrow> sklvl1.guessed_frame Ra xsk = Some (Exp gny (NonceF (Ra $ nx)))}"
sdp_DC: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "sub_distrib_pconj a"   and "sub_distrib_pconj b"   and "healthy (wp a)"   and "healthy (wp b)"   and "nearly_healthy (wlp b)" shows "sub_distrib_pconj (a \<Sqinter> b)"
c5_1: fixes x :: "real" assumes "x\<^sub>0 \<le> x"   and "x \<le> x\<^sub>1" shows "2 * (c5 * f_approx x) \<le> f x"
capacity_outside: fixes e :: "'a \<times> 'a" assumes "e \<notin> \<^bold>E" shows "capacity \<Delta> e = 0"
WHILE_Its: fixes b :: "bexp"   and c :: "com"   and s :: "char list \<Rightarrow> int"   and t :: "char list \<Rightarrow> int" assumes "(WHILE b DO c, s) \<Rightarrow> t" shows "\<exists>n. Its b c s n"
invar_pres: fixes s :: "nat \<times> nat"   and f :: "op\<^sub>t\<^sub>b" assumes "invar s" shows "invar (nxt f s)"
map2_val_def: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'd"   and xs :: "'x"   and ys :: "'y" shows "map2_val f xs ys = oalist_of_listz (map2_val_raw f (map_val_raw (\<lambda>k v. f k v (0::'c))) (map_val_raw (\<lambda>k. f k (0::'b))) (list_of_oalistx xs) (list_of_oalisty ys))"
sel_3: fixes x1 :: "'b"   and x2 :: "'b"   and x3 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x4 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x5 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x6 :: "'b \<Rightarrow> 'b"   and x7 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x8 :: "'b \<Rightarrow> 'b"   and x9 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x10 :: "'b \<Rightarrow> 'b"   and x11 :: "'b \<Rightarrow> 'b"   and x12 :: "int \<Rightarrow> 'b"   and x13 :: "'b \<Rightarrow> int"   and x14 :: "'b \<Rightarrow> bool" shows "arith_ops_record.plus (Arith_Ops_Record x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14) = x3"
known_ptrs_known_ptr: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "l_known_ptrs\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t.a_known_ptrs known_ptr h"   and "ptr |\<in>| object_ptr_kinds h" shows "known_ptr ptr"
absorb2: fixes a :: "'b"   and b :: "'b" assumes "a \<le> b" shows "max a b = b"
brauns1_def: fixes xs :: "'b list" shows "brauns1 xs = (if xs = [] then \<langle>\<rangle> else brauns 0 xs ! 0)"
directed_ub: fixes X :: "'a set" assumes "ord.directed (\<lambda>x y. y \<le> x) X" shows "\<forall>x\<in>X. \<forall>y\<in>X. \<exists>z\<in>X. z \<le> x \<and> z \<le> y"
omega_induct: fixes n :: "V"   and P :: "V \<Rightarrow> bool" assumes "n \<in>\<^sub>\<circ> \<omega>"   and "P 0"   and "\<And>n. \<lbrakk>n \<in>\<^sub>\<circ> \<omega>; P n\<rbrakk> \<Longrightarrow> P (ZFC_in_HOL.succ n)" shows "P n"
distrib: fixes g :: "'e \<Rightarrow> 'd"   and h :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "(\<Sum>x\<in>A. g x + h x) = sum g A + sum h A"
subst_Pair: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b"   and a :: "'a \<times> 'b" assumes "P x y"   and "a = (x, y)" shows "P (fst a) (snd a)"
extstrand_subst_hom_1: fixes S :: "('a, 'b) extstrand_step list"   and S' :: "('a, 'b) extstrand_step list"   and \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "S @ S' \<cdot>\<^sub>e\<^sub>s\<^sub>t \<theta> = (S \<cdot>\<^sub>e\<^sub>s\<^sub>t \<theta>) @ (S' \<cdot>\<^sub>e\<^sub>s\<^sub>t \<theta>)"