hcju/isabelleps huggingface dataset:
L_def: shows "L = (J \<div> f) \<inter> P[X]"
exec: shows "steps0 (s, l, r) B stp = (s', l', r')"
ntrp_stack_comp: fixes s :: "'n list"   and c :: "'n list multiset"   and ww :: "('p, 'ba) label list el_step list"   and s' :: "'n list"   and c' :: "'n list multiset"   and r :: "'n list" assumes "((s, c), ww, s', c') \<in> trcl (ntrp fg)"   and "mon_s fg r \<inter> mon_env fg ww = {}" shows "((s @ r, c), ww, s' @ r, c') \<in> trcl (ntrp fg)"
inv_f_f: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "inj f" shows "inv f (f x) = x"
ExecAssignOK: fixes v :: "char list"   and \<sigma> :: "char list \<Rightarrow> nat option"   and e :: "expr"   and u :: "nat" assumes "v \<in> dom \<sigma>"   and "eval \<sigma> e = Some u" shows "Assign v e \<rhd> \<sigma> \<leadsto> Some (\<sigma>(v \<mapsto> u))"
prefs_from_table_wfD_4: fixes agents :: "'c set"   and alts :: "'d set"   and xss :: "('c \<times> 'd set list) list" assumes "prefs_from_table_wf agents alts xss" shows "set (map fst xss) = agents"
assms: fixes y :: "'a" assumes "y \<in> set B" shows "monotone sum_bot.le_fun sum_bot_ord (C y)"
diffs_affine_hull_span: fixes a :: "'a"   and S :: "'a set" assumes "a \<in> S" shows "(\<lambda>x. x - a) ` (affine hull S) = span ((\<lambda>x. x - a) ` S)"
a1: shows "s' = snd (fst (div_comp instr rs1 rd operand2 s)) \<and> get_S (cpu_reg_val PSR s) = 0"
mem_fine: fixes \<delta> :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and D :: "(real \<times> real \<times> real) list"   and u :: "real"   and x :: "real"   and v :: "real" assumes "fine \<delta> (a, b) D"   and "(u, x, v) \<in> set D" shows "u < v \<and> u \<le> x \<and> x \<le> v"
l8_3: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p" assumes "Per A B C"   and "A \<noteq> B"   and "Col B A A'" shows "Per A' B C"
vec_to_iarray_row: fixes i :: "'f"   and A :: "(('d, 'e) vec, 'f) vec" shows "vec_to_iarray (row i A) = row_iarray (mod_type_class.to_nat i) (matrix_to_iarray A)"
wfrec_def: fixes R :: "('b \<times> 'b) set"   and F :: "('b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'c" shows "wfrec R F = (\<lambda>x. THE y. wfrec_rel R (\<lambda>f x. F (cut f R x) x) x y)"
intros_3: fixes A :: "fmla"   and H :: "fmla set" assumes "A \<in> boolean_axioms" shows "H \<turnstile> A"
simps_1: fixes "is" :: "'b set"   and t :: "'b" shows "redT_updI is (Interrupt t) = insert t is"
sum_over_product: fixes y :: "i"   and z :: "i"   and x :: "i" assumes "O y z" shows "x \<oplus> y \<otimes> z = (x \<oplus> y) \<otimes> (x \<oplus> z)"
gt_GreaterD: fixes u :: "'a \<Rightarrow> real"   and X :: "'a set"   and I :: "'a \<Rightarrow> Regions.intv"   and r :: "('a \<times> 'a) set"   and k :: "'a \<Rightarrow> nat"   and c :: "'a" assumes "u \<in> Regions.region X I r"   and "Regions.valid_region X k I r"   and "c \<in> X"   and "real (k c) < u c" shows "I c = Regions.intv.Greater (k c)"
MsgCJ: shows "msgs cJ = (\<lambda>m. if \<exists>p. m = <p, inM \<exists>i<j. procList ! i = p> then 1 else 0)"
measurable_component_update: fixes x :: "'c \<Rightarrow> 'd"   and I :: "'c set"   and M :: "'c \<Rightarrow> 'd measure"   and i :: "'c" assumes "x \<in> space (Pi\<^sub>M I M)"   and "i \<notin> I" shows "fun_upd x i \<in> M i \<rightarrow>\<^sub>M Pi\<^sub>M (insert i I) M"
arrow_of_spans_axioms: shows "arrow_of_spans (\<cdot>) \<lparr>Chn = C.inv \<mu>.chine, Dom = Cod \<mu>, Cod = Dom \<mu>\<rparr>"
ih_8: shows "InvariantWatchesDiffer (getF state') (getWatch1 state') (getWatch2 state')"
simps_14: fixes p :: "nat"   and a :: "floatarith"   and vs :: "((real \<times> real pdevs) \<times> real) list" shows "approx_floatarith p (Ln a) vs = approx_floatarith p a vs \<bind> ln_aform_err p"
empty_Collect_eq: fixes P :: "'b \<Rightarrow> bool" shows "({} = Collect P) = (\<forall>x. \<not> P x)"
simps: fixes a :: "'a"   and r :: "('a \<times> 'b) set" shows "(a \<in> Domain r) = (\<exists>aa b. a = aa \<and> (aa, b) \<in> r)"
in_hhomE: fixes \<mu> :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>\<mu> : a \<rightarrow>\<^sub>B b\<guillemotright>" obtains "B.arr \<mu>" and "B.obj a" and "B.obj b" and "src \<mu> = a" and "trg \<mu> = b"
unifiers_def: fixes E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) set" shows "unifiers E = {\<sigma>. \<forall>p\<in>E. fst p \<cdot> \<sigma> = snd p \<cdot> \<sigma>}"
and_extract_pure_left_ctx_iff: fixes P :: "assn"   and b :: "bool"   and Q :: "assn" shows "P * \<up> b \<and>\<^sub>A Q = (P \<and>\<^sub>A Q) * \<up> b"
bind_spmf_pmf_assoc: fixes p :: "'c pmf"   and f :: "'c \<Rightarrow> 'b spmf"   and g :: "'b \<Rightarrow> 'a spmf" shows "p \<bind> f \<bind> g = p \<bind> (\<lambda>x. f x \<bind> g)"
ex_un_null: shows "\<exists>!n. \<forall>f. n \<cdot>\<^sub>2 f = n \<and> f \<cdot>\<^sub>2 n = n"
is_functor: shows "functor (\<cdot>\<^sub>B) (\<cdot>\<^sub>B) B.map"
assms: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<sqinter> y \<sqinter> 1\<^sub>\<sigma>) \<cdot> z = (x \<sqinter> 1\<^sub>\<sigma>) \<cdot> z \<sqinter> (y \<sqinter> 1\<^sub>\<sigma>) \<cdot> z"
assms_3: shows "f $ 0 \<noteq> (0::'a)"
vpeq_def: fixes u :: "partition_id_t"   and s :: "state_t"   and t :: "state_t" shows "vpeq u s t \<equiv> vpeq_obj u s t \<and> vpeq_subj_subj u s t \<and> vpeq_subj_obj u s t \<and> vpeq_local u s t"
ortho_sum_in_span: fixes W :: "'a vec set"   and X :: "'a vec set"   and l1 :: "'a vec \<Rightarrow> 'a"   and l2 :: "'a vec \<Rightarrow> 'a" assumes "W \<subseteq> carrier_vec n"   and "X \<subseteq> carrier_vec n"   and "\<And>w x. \<lbrakk>w \<in> W; x \<in> X\<rbrakk> \<Longrightarrow> x \<bullet> w = (0::'a)"   and "lincomb l1 X + lincomb l2 W \<in> local.span X" shows "lincomb l2 W = 0\<^sub>v n"
tendsto_Zfun_iff: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and F :: "'a filter" shows "(f \<longlongrightarrow> a) F = Zfun (\<lambda>x. f x - a) F"
plus_prod_def: fixes m1 :: "'c \<times> 'd"   and m2 :: "'c \<times> 'd" shows "m1 + m2 \<equiv> (fst m1 + fst m2, snd m1 + snd m2)"
fw_shortest_path_up_to: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "nat"   and k :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat" assumes "class.linordered_ab_monoid_add plus zero less_eq less"   and "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..k} \<longrightarrow> less_eq zero (linordered_ab_monoid_add.len plus m i i xs)"   and "i \<le> n"   and "j \<le> n"   and "k \<le> n" shows "linordered_ab_monoid_add.D plus less_eq m i j k = linordered_ab_monoid_add.fw plus less_eq m n k i j"
test_sup: fixes p :: "'a"   and q :: "'a" shows "\<iota> (sup p q) = \<iota> p + \<iota> q"
adds_fun: fixes s :: "'a \<Rightarrow> 'b"   and t :: "'a \<Rightarrow> 'b" shows "(s adds t) = (s \<le> t)"
map_iterator_rule_insert_P: fixes it :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<times> 'c \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and m :: "'b \<Rightarrow> 'c option"   and I :: "'b set \<Rightarrow> 'a \<Rightarrow> bool"   and \<sigma>0 :: "'a"   and c :: "'a \<Rightarrow> bool"   and f :: "'b \<times> 'c \<Rightarrow> 'a \<Rightarrow> 'a"   and P :: "'a \<Rightarrow> bool" assumes "map_iterator it m"   and "I {} \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>c \<sigma>; k \<in> dom m - it; m k = Some v; it \<subseteq> dom m; I it \<sigma>\<rbrakk> \<Longrightarrow> I (insert k it) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I (dom m) \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> dom m; it \<noteq> dom m; \<not> c \<sigma>; I it \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
external_invariant_translation: fixes Sa :: "('b, 'a) spec"   and Q :: "'a set"   and pi :: "'c \<Rightarrow> 'a"   and P :: "'c set"   and Sc :: "('d, 'c) spec" assumes "oreach Sa \<subseteq> Q"   and "pi -` Q \<subseteq> P"   and "implements pi Sa Sc" shows "oreach Sc \<subseteq> P"
diff_from_empty: fixes S :: "'a set" shows "{#} -s S = {#}"
invariant3p_ts_ok: fixes final :: "'a \<Rightarrow> bool"   and r :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and P :: "'b \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and convert_RA :: "'d \<Rightarrow>f nat \<Rightarrow> 'f list" assumes "lifting_wf final r P" shows "invariant3p (multithreaded_base.redT final r convert_RA) {s. ts_ok P (thr s) (shr s)}"
splitDigit_def: fixes p :: "'f \<Rightarrow> bool"   and i :: "'f"   and d :: "('d, 'f) Digit" shows "splitDigit p i d = splitNlist p i (digitToNlist d)"
quote_all_MonPon_PfP_ssubst: fixes \<alpha> :: "fm"   and \<beta> :: "fm"   and V :: "name set" assumes "{} \<turnstile> \<alpha> IMP \<beta>"   and "V \<subseteq> Vs"   and "supp \<alpha> \<subseteq> atom ` Vs"   and "supp \<beta> \<subseteq> atom ` Vs" shows "quote_all p V \<turnstile> PfP (ssubst \<lfloor>\<alpha>\<rfloor>V V F) IMP PfP (ssubst \<lfloor>\<beta>\<rfloor>V V F)"
sel_2: fixes x1 :: "'b literal multiset multiset"   and x2 :: "'b literal multiset"   and x3 :: "'b literal multiset" shows "main_prem_of (Infer x1 x2 x3) = x2"
objE: fixes a :: "'a" assumes "obj a" obtains "ide a" and "src a = a" and "trg a = a"
Iod_Torder: fixes X :: "'a set" assumes "X \<subseteq> carrier D" shows "Torder (Iod D X)"
bigo_minus: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> O(g)" shows "- f \<in> O(g)"
elims_3: fixes x :: "real_min_top"   and xa :: "real_min_top" assumes "\<not> x \<le> xa" obtains uv_ :: "real" where "xa__ = PInfty" and "xa = real_min_top.R uv_"   | x :: "real" and y :: "real" where "xa__ = real_min_top.R x" and "xa = real_min_top.R y" and "\<not> x \<le> y"
homogeneous_ideal': fixes F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and q :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "\<And>f. f \<in> F \<Longrightarrow> homogeneous f"   and "p \<in> ideal F"   and "q \<in> hom_components p" shows "q \<in> ideal F"
nat_term_comp_lex_comp_aux: shows "nat_term_comp lex_comp_aux"
stern_brocot_num_den: shows "pure Pair \<diamondop> num \<diamondop> den = stern_brocot_recurse"
atoms\<^sub>0_def: shows "atoms\<^sub>0 = ATOM.atoms\<^sub>0 depends\<^sub>R"
closure_greaterThan: fixes a :: "'a" shows "closure {a<..} = {a..}"
simps_28: fixes x31 :: "'b local.term"   and x32 :: "'b local.term" shows "\<^bold>\<I> \<noteq> x31 \<^bold>\<otimes> x32"
residual_index: fixes ps :: "'a set" assumes "ps \<subseteq> bl\<^sup>c"   and "card ps = 2" shows "int (\<B>\<^sup>R index ps) = \<Lambda>"
verts_H: fixes G :: "('a, 'b) pre_digraph"   and rev_G :: "'b \<Rightarrow> 'b"   and H :: "('a, 'b) pre_digraph"   and rev_H :: "'b \<Rightarrow> 'b"   and u :: "'a"   and v :: "'a"   and w :: "'a"   and uv :: "'b"   and uw :: "'b"   and vw :: "'b" assumes "subdiv_step G rev_G H rev_H u v w uv uw vw" shows "verts H = verts G \<union> {w}"
b: shows "\<forall>u. b u \<in> \<rat> \<and> x \<bullet> u < b u \<and> b u - x \<bullet> u < e'"
landau_flip_4: fixes g :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" shows "(g \<in> \<omega>[F](f)) = (f \<in> o[F](g))"
cdi_prefix: fixes n :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and m :: "nat"   and n' :: "nat" assumes "n cd\<^bsup>\<pi>\<^esup>\<rightarrow> m"   and "m < n'"   and "n' \<le> n" shows "n' cd\<^bsup>\<pi>\<^esup>\<rightarrow> m"
iT_Plus_card: fixes I :: "nat set"   and k :: "nat" shows "card (I \<oplus> k) = card I"
has_density_embed_measure'': fixes f :: "'a \<Rightarrow> 'b"   and N :: "'a measure"   and f' :: "'b \<Rightarrow> 'a"   and M :: "'b measure"   and \<delta> :: "'b \<Rightarrow> ennreal" assumes "inj f"   and "\<And>x. x \<in> space N \<Longrightarrow> f' (f x) = x"   and "has_density M (embed_measure N f) \<delta>" shows "has_density (distr M N f') N (\<delta> \<circ> f)"
prems_2: shows "length m2 = Suc nn \<and> (\<forall>a\<in>set m2. vec nr a)"
synth_idem: fixes H :: "msg set" shows "synth (synth H) = synth H"
index_set_closed_in: fixes I :: "nat set"   and i :: "nat"   and j :: "nat" assumes "index_set I"   and "i \<in> I"   and "\<forall>x. eval r_phi [i, x] = eval r_phi [j, x]" shows "j \<in> I"
strict_mono_f_make_mono: fixes f :: "nat \<Rightarrow> ordinal" assumes "\<forall>n. f n < oLimit f" shows "OrdinalInduct.strict_mono (\<lambda>n. f (make_mono f n))"
count_roots_between: fixes p :: "real poly"   and a :: "real"   and b :: "real" shows "count_roots_between p a b = (let q = pderiv p in if b < a \<or> p = 0 then 0 else if (poly p a \<noteq> 0 \<or> poly q a \<noteq> 0) \<and> (poly p b \<noteq> 0 \<or> poly q b \<noteq> 0) then let ps = sturm p in sign_changes ps a - sign_changes ps b else let ps = sturm_squarefree p in sign_changes ps a - sign_changes ps b)"
justifying_execution_ext_def: fixes action_translation :: "nat \<Rightarrow> nat"   and more :: "'a" shows "\<lparr>action_translation = action_translation, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons justifying_execution_ext_Tuple_Iso action_translation more"
that: fixes l :: "nat" assumes "i < l"   and "l < j"   and "is_HOLD (tts (s l) t')" shows "thesis"
comp_min_basis_distinct: fixes xs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "distinct (comp_min_basis xs)"
pgen_lossless_gpv_lift_spmf: fixes fail :: "ennreal"   and \<I> :: "('a, 'b) \<I>"   and p :: "'c spmf" shows "pgen_lossless_gpv fail \<I> (lift_spmf p) = (lossless_spmf p \<or> fail = 1)"
map_of_distinct_upd2: fixes x :: "'a"   and xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'b) list"   and y :: "'b" assumes "x \<notin> set (map fst xs)"   and "x \<notin> set (map fst ys)" shows "map_of (xs @ (x, y) # ys) = map_of (xs @ ys)(x \<mapsto> y)"
periodN_drop: fixes w :: "'b list"   and p :: "nat" assumes "periodN w p"   and "p < \<^bold>|w\<^bold>|" shows "periodN (drop p w) p"
and_neg_mask_eq_iff_not_mask_le: fixes w :: "'b word"   and n :: "nat" shows "(w AND NOT (mask n) = NOT (mask n)) = (NOT (mask n) \<le> w)"
T2: fixes w :: "'a"   and w' :: "'a"   and \<theta> :: "'a"   and u :: "'a"   and \<theta>' :: "'a"   and \<beta> :: "'a" assumes "ide w"   and "ide w'"   and "\<guillemotleft>\<theta> : tab\<^sub>0 h \<star> w \<rightarrow> u\<guillemotright>"   and "\<guillemotleft>\<theta>' : tab\<^sub>0 h \<star> w' \<rightarrow> u\<guillemotright>"   and "\<guillemotleft>\<beta> : tab\<^sub>1 h \<star> w \<rightarrow> tab\<^sub>1 h \<star> w'\<guillemotright>"   and "(h \<star> \<theta>) \<cdot> \<a>[h, tab\<^sub>0 h, w] \<cdot> (h.tab \<star> w) = ((h \<star> \<theta>') \<cdot> \<a>[h, tab\<^sub>0 h, w'] \<cdot> (h.tab \<star> w')) \<cdot> \<beta>" shows "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w \<rightarrow> w'\<guillemotright> \<and> \<beta> = tab\<^sub>1 h \<star> \<gamma> \<and> \<theta> = \<theta>' \<cdot> (tab\<^sub>0 h \<star> \<gamma>)"
arity_simps_2: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" shows "arity (App s t) = arity s - 1"
f11: fixes x :: "'a"   and gpv :: "('a, 'f, 'h) gpv" assumes "Pure x \<in> set_spmf (the_gpv gpv)"   and "Domainp (rel_generat A C (rel_fun R (rel_gpv'' A C R))) (Pure x)" shows "Domainp A x"
IH: fixes z :: "complex" assumes "z \<in> S" shows "(deriv ^^ n) (\<lambda>w. f (u * w)) z = u ^ n * (deriv ^^ n) f (u * z)"
l2_same: fixes p :: "(nat \<times> int) list"   and d :: "nat" shows "l2_\<phi> (p ! d) (p ! d) = 1 / 3 / 2 ^ lv p d"
l_set_disconnected_nodes_get_ancestors_si\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes 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 get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   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_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors_si :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_si_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node_si :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_si_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool" shows "l_set_disconnected_nodes_get_ancestors_si\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_disconnected_nodes set_disconnected_nodes_locs get_parent get_parent_locs type_wf known_ptr known_ptrs get_child_nodes get_child_nodes_locs get_host get_host_locs get_ancestors_si get_ancestors_si_locs get_root_node_si get_root_node_si_locs type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M \<equiv> (l_set_disconnected_nodes_get_parent set_disconnected_nodes_locs get_parent_locs \<and> l_get_root_node_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_host get_host_locs get_ancestors_si get_ancestors_si_locs get_root_node_si get_root_node_si_locs) \<and> l_set_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_disconnected_nodes set_disconnected_nodes_locs \<and> l_set_disconnected_nodes_get_host set_disconnected_nodes_locs get_host_locs"
toVariable_channels_inv': fixes g :: "'a gState_scheme"   and p :: "pState"   and v :: "varDecl"   and a :: "String.literal"   and b :: "variable"   and c :: "channel list" assumes "toVariable g p v = (a, b, c)" shows "\<forall>x\<in>set c. channel_inv x"
gram_schmidt0_result_1: fixes ws :: "'b vec list"   and n :: "nat" assumes "set ws \<subseteq> carrier_vec n" shows "set (gram_schmidt0 n ws) \<subseteq> carrier_vec n"
isomorphic_transitive: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "f \<cong>\<^sub>B g"   and "g \<cong>\<^sub>B h" shows "f \<cong>\<^sub>B h"
check_poly_weak_mono_discrete_def: fixes p :: "('c monom \<times> 'd) list"   and v :: "'c" shows "check_poly_weak_mono_discrete p v \<equiv> check_poly_ge (poly_subst (\<lambda>w. poly_of (if w = v then PSum [PNum (1::'d), PVar v] else PVar w)) p) p"
af_ltl_continuation: fixes w :: "'b set list"   and w' :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "(w \<frown> w') \<Turnstile>\<^sub>n \<phi> = w' \<Turnstile>\<^sub>n af \<phi> w"
nnonzero_annonzero: fixes N :: "nat" assumes "0 < N" shows "an N \<noteq> 0"
distinct_verts1: shows "distinct [0..<length (g_V G)]"
tail_invar_app_single: fixes bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and t' :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" assumes "SkewBinomialHeapStruc.tail_invar bq"   and "\<forall>t\<in>set bq. SkewBinomialHeapStruc.rank t < SkewBinomialHeapStruc.rank t'"   and "SkewBinomialHeapStruc.tree_invar t'" shows "SkewBinomialHeapStruc.tail_invar (bq @ [t'])"
cobounded2: fixes b :: "'c"   and a :: "'c" shows "b \<le> a \<squnion> b"
arb_vp: shows "\<forall>n w. val_process Mkt2 arb_pf n w = prices Mkt2 pos_stock n w * diff_inv - pr n w"
prognosis_App: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and as :: "Arity list"   and a :: "Arity"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and x :: "Vars.var"   and S :: "stack_elem list" shows "prognosis ae as (inc\<cdot>a) (\<Gamma>, e, Arg x # S) \<sqsubseteq> prognosis ae as a (\<Gamma>, App e x, S)"
rerr_invalid_3: fixes crt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dests :: "nat \<Rightarrow> nat option"   and sip :: "nat" shows "rerr_invalid crt (Rerr dests sip) = (\<forall>rip\<in>dom dests. rip \<in> iD crt \<and> the (dests rip) = sqn crt rip)"
not_distinct_conv_prefix: fixes as :: "'b list" shows "(\<not> distinct as) = (\<exists>xs y ys. y \<in> set xs \<and> distinct xs \<and> as = xs @ y # ys)"
poly_y_x_fix_y_deg: fixes p :: "'c poly poly"   and d :: "nat" assumes "\<forall>i\<le>degree p. degree (coeff p i) \<le> d" shows "poly_y_x p = (\<Sum>i\<le>degree p. \<Sum>j\<le>d. monom (monom (coeff (coeff p i) j) i) j)"
mapping_of_monom: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'a" shows "mapping_of (MPoly_Type.monom m a) = monomial a m"
rep_plus: fixes x :: "'a"   and y :: "'a" shows "rep_nat (x + y) = rep_nat x + rep_nat y"
non_zero_vectors_def: shows "vec.non_zero_vectors \<equiv> {x. x \<noteq> 0}"
diffs_def: shows "diffs = insert 1 {\<bar>x - y\<bar> |x y. x \<in> rts \<and> y \<in> rts \<and> x \<noteq> y}"
assms_1: shows "u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> S\<^esub> v"
distinct_6: fixes x31 :: "'d"   and x32 :: "'d ifex"   and x33 :: "'d ifex" shows "IF x31 x32 x33 \<noteq> Falseif"
F2: shows "val_Zp (f' \<bullet> a) < val_Zp (a \<ominus> \<alpha>) \<and> val_Zp (a \<ominus> \<alpha>) = val_Zp (local.divide (f \<bullet> a) (f' \<bullet> a))"
e: shows "NTDom \<eta> ## f maps\<^bsub>CatCod (NTDom \<eta>)\<^esub> NTDom \<eta> @@ X to NTDom \<eta> @@ Y"
Un_empty: fixes A :: "'d set"   and B :: "'d set" shows "(A \<union> B = {}) = (A = {} \<and> B = {})"
f21_def: shows "f21 = snd (split_face oldF ram1 ram2 [])"
wp_while_unfold: fixes \<pi> :: "char list \<Rightarrow> com option"   and b :: "bexp"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "wp \<pi> (WHILE b DO c) Q s = (if bval b s then wp \<pi> c (wp \<pi> (WHILE b DO c) Q) s else Q s)"
bij_from_nat: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "bij_betw (mod_type.from_nat Abs) {0..<CARD('a)} UNIV"
UN_extend_simps_10: fixes B :: "'d \<Rightarrow> 'b set"   and f :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "(\<Union>a\<in>A. B (f a)) = \<Union> (B ` f ` A)"
IrrFinGroupRepresentation_axioms_def: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" shows "IrrFinGroupRepresentation_axioms G smult V \<equiv> \<forall>U. RModule (aezfun_setspan G) smult U \<and> U \<subseteq> V \<longrightarrow> U = 0 \<or> U = V"
mark_out_code: fixes n :: "nat"   and bs :: "bool list" shows "mark_out n bs = mark_out_aux n n bs"
Fun: fixes x2a :: "('b, 'a) Term.term" assumes "x2a \<in> set ss" shows "vars_term (scf_term scf x2a) \<subseteq> vars_term x2a"