hcju/isabelleps huggingface dataset:
determ_additiveD: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "determ t" shows "Transformers.additive t"
hoare_relative_complete: fixes P :: "assn"   and c :: "stmt"   and Q :: "assn" assumes "\<Turnstile> {P} c {Q}" shows "{} \<turnstile> {P} c {Q}"
select_index_suc_even: fixes k :: "nat"   and n :: "nat"   and i :: "nat" assumes "k < 2 ^ n"   and "select_index n i k" shows "select_index (Suc n) i (2 * k)"
prognosis_Let: fixes \<Delta> :: "(Vars.var \<times> exp) list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and S :: "stack_elem list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and e :: "exp"   and a :: "Arity"   and as :: "Arity list" assumes "atom ` domA \<Delta> \<sharp>* \<Gamma>"   and "atom ` domA \<Delta> \<sharp>* S"   and "edom ae \<subseteq> domA \<Gamma> \<union> upds S" shows "prognosis (Aheap \<Delta> e\<cdot>a \<squnion> ae) as a (\<Delta> @ \<Gamma>, e, S) \<sqsubseteq> cHeap \<Delta> e\<cdot>a \<squnion> prognosis ae as a (\<Gamma>, Terms.Let \<Delta> e, S)"
bvars\<^sub>e\<^sub>s\<^sub>t_append: fixes A :: "('fun, 'var) extstrand_step list"   and B :: "('fun, 'var) extstrand_step list" shows "bvars\<^sub>e\<^sub>s\<^sub>t (A @ B) = bvars\<^sub>e\<^sub>s\<^sub>t A \<union> bvars\<^sub>e\<^sub>s\<^sub>t B"
minorM_eq: fixes M :: "(('a, 'b) vec, 'b) vec"   and i :: "'b"   and j :: "'b" shows "minorM M i j = to_vec (minor (from_vec M) i j)"
Cons_in_subseqsD: fixes y :: "'a"   and ys :: "'a list"   and xs :: "'a list" assumes "y # ys \<in> set (subseqs xs)" shows "ys \<in> set (subseqs xs)"
synthetic_div_altdef: fixes p :: "'a poly"   and c :: "'a" shows "synthetic_div p c = p div [:- c, 1::'a:]"
beyond_def: fixes k :: "natural"   and l :: "natural" shows "beyond k l = (if k < l then l else 0)"
trace_equivalent_symmetric: fixes e1 :: "((nat \<times> nat) \<times> transition) fset"   and e2 :: "((nat \<times> nat) \<times> transition) fset" shows "trace_equivalent e1 e2 = trace_equivalent e2 e1"
rightUniqueInjectiveOnFirstImplication: fixes P :: "('a \<times> 'b) set" assumes "runiq P" shows "inj_on fst P"
assms: shows "OSC L E \<and> matching V E M \<and> 1 < i"
fiter1: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice" assumes "finite_iteration (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)" shows "finite_iteration.fiter (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice) (c::'b::refinement_lattice) \<sqsubseteq> c"
Q''TransQ': shows "Q'' \<longmapsto> \<tau> \<prec> Q'"
weighted_RP_saturated: fixes Sts :: "(('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat) llist" assumes "full_chain (\<leadsto>\<^sub>w) Sts"   and "wrp.P_of_wstate (lhd Sts) = {}"   and "wrp.Q_of_wstate (lhd Sts) = {}" shows "redundancy_criterion.saturated_upto (ground_resolution_with_selection.ord_\<Gamma> (wrp.S_gQ Sts)) Rf (standard_redundancy_criterion.Ri (ground_resolution_with_selection.ord_\<Gamma> (wrp.S_gQ Sts))) (Liminf_llist (lmap (\<lambda>St. grounding_of_clss (wrp.N_of_wstate St \<union> wrp.P_of_wstate St \<union> wrp.Q_of_wstate St)) Sts))"
weak_bisimilarity_implies_weak_equivalence_Pred: fixes x :: "('idx, 'pred, 'act) formula"   and P :: "'state"   and Q :: "'state"   and \<phi> :: "'pred" assumes "\<And>P Q. P \<approx>\<cdot> Q \<Longrightarrow> P \<Turnstile> x = Q \<Turnstile> x"   and "P \<approx>\<cdot> Q"   and "P \<Turnstile> \<langle>\<langle>\<tau>\<rangle>\<rangle>Conj (binsert (Pred \<phi>) (bsingleton x))" shows "Q \<Turnstile> \<langle>\<langle>\<tau>\<rangle>\<rangle>Conj (binsert (Pred \<phi>) (bsingleton x))"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'c \<Rightarrow> 'b"   and f2 :: "wake_up_status \<Rightarrow> 'b"   and wait_set_status :: "'c wait_set_status" shows "P (case wait_set_status of InWS x \<Rightarrow> f1 x | PostWS x \<Rightarrow> f2 x) = ((\<forall>x1. wait_set_status = InWS x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. wait_set_status = PostWS x2 \<longrightarrow> P (f2 x2)))"
comp_cod_arr: fixes f :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "H.VV.arr f"   and "H.VV.cod f = b" shows "H.VV.comp b f = f"
Exit_path: fixes n :: "'b" assumes "valid_node n" shows "\<exists>as. n -as\<rightarrow>* Exit"
PNT_implies_sum_moebius_mu_sublinear: assumes "\<psi> \<sim>[at_top] (\<lambda>x. x)" shows "sum_upto moebius_mu \<in> o(\<lambda>x. x)"
markedElementsAppend: fixes M1 :: "('a \<times> bool) list"   and M2 :: "('a \<times> bool) list" shows "markedElements (M1 @ M2) = markedElements M1 @ markedElements M2"
image_image: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "f ` g ` A = (\<lambda>x. f (g x)) ` A"
rep_eq: fixes x :: "integer" shows "bit x = bit (int_of_integer x)"
lifting_wf_init_fin_lift: shows "lifting_wf init_fin_final init_fin (init_fin_lift P)"
bernoulli'_conv_akiyama_tanigawa: fixes n :: "nat" shows "bernoulli' n = akiyama_tanigawa n 0"
measurable_count_space: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" shows "f \<in> count_space A \<rightarrow>\<^sub>M count_space UNIV"
ccBinds_reorder1: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and v :: "Vars.var"   and e :: "exp"   and ccExp :: "exp \<Rightarrow> Arity \<rightarrow> CoCalls" assumes "map_of \<Gamma> v = Some e" shows "CoCallAnalysis.ccBinds ccExp \<Gamma> = CoCallAnalysis.ccBind ccExp v e \<squnion> CoCallAnalysis.ccBinds ccExp (delete v \<Gamma>)"
typeof_simps_3: fixes sh :: "int" shows "\<tau> shortV sh = ShortT"
4: shows "concat (map (\<lambda>xs. concat (map (\<lambda>a. vec_smult a (vec B)) xs)) (map vec As)) = concat (map vec (map (\<lambda>A. tensor_from_vec (dims A @ dims B) (concat (map (\<lambda>a. vec_smult a (vec B)) (vec A)))) As))"
independent_explicit: fixes B :: "('a, 'b) vec set" shows "vec.independent B = (finite B \<and> (\<forall>c. (\<Sum>v\<in>B. c v *s v) = 0 \<longrightarrow> (\<forall>v\<in>B. c v = (0::'a))))"
polytope_def: fixes n :: "nat"   and P :: "'b vec set" shows "gram_schmidt.polytope n P = (\<exists>V\<subseteq>carrier_vec n. finite V \<and> P = gram_schmidt.convex_hull n V)"
uniformity_cblinfun_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
index_add_vec_1: fixes i :: "nat"   and v\<^sub>2 :: "'b vec"   and v\<^sub>1 :: "'b vec" assumes "i < dim_vec v\<^sub>2" shows "(v\<^sub>1 + v\<^sub>2) $ i = v\<^sub>1 $ i + v\<^sub>2 $ i"
ln_exp_cf2_lower_bound_pos: fixes x :: "real" assumes "0 \<le> x" shows "ln (exp_cf2 x) \<le> x"
thetaSPar_def: shows "thetaSPar \<equiv> {(Par cl, Par dl) |cl dl. properL cl \<and> properL dl \<and> SbisL cl dl}"
root_int_main_all: fixes n :: "int"   and p :: "nat"   and y :: "int"   and b :: "bool"   and x :: "int" assumes "0 \<le> n"   and "root_int_main p n = (y, b)" shows "0 \<le> y \<and> b = (y ^ p = n) \<and> (0 < p \<longrightarrow> y ^ p \<le> n \<and> n < (y + 1) ^ p) \<and> (0 < p \<longrightarrow> 0 \<le> x \<longrightarrow> x ^ p = n \<longrightarrow> y = x \<and> b)"
p_antitone_inf: fixes x :: "'a"   and y :: "'a" shows "- x \<le> - (x \<sqinter> y)"
lhd_ldropn: fixes n :: "nat"   and xs :: "'b llist" assumes "enat n < llength xs" shows "lhd (ldropn n xs) = xs $ n"
case_1: fixes f1 :: "'h"   and f2 :: "'i \<Rightarrow> 'h" shows "(case None of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f1"
iso_inv_iso: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "category.iso C f" shows "category.iso C (category.inv C f)"
conf_def: fixes typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and P :: "'c prog"   and h :: "'a"   and v :: "'b val"   and T :: "ty" shows "heap_base.conf typeof_addr P h v T \<equiv> \<exists>T'. heap_base.typeof_h typeof_addr h v = \<lfloor>T'\<rfloor> \<and> P \<turnstile> T' \<le> T"
id_in_auto: shows "(\<lambda>x\<in>carrier G. x) \<in> auto G"
lcos_mult_one: fixes G :: "('c, 'd) monoid_scheme"   and M :: "'c set" assumes "Group.group G"   and "M \<subseteq> carrier G" shows "\<one>\<^bsub>G\<^esub> <#\<^bsub>G\<^esub> M = M"
nodes_of_uses_eq_SomeD: fixes v :: "'val"   and nodes :: "'node set"   and n :: "'node"   and args :: "'val list" assumes "Mapping.lookup nodes_of_uses v = Some nodes"   and "Mapping.lookup (phis g) (n, v) = Some args" obtains "nodes = {n \<in> set (\<alpha>n g). \<exists>vs. Mapping.lookup (uses g) n = Some vs \<and> v \<in> vs}"
suminf_ereal_minus: fixes g :: "nat \<Rightarrow> ereal"   and f :: "nat \<Rightarrow> ereal" assumes "\<And>i. g i \<le> f i"   and "\<And>i. 0 \<le> g i"   and "suminf f \<noteq> \<infinity>"   and "suminf g \<noteq> \<infinity>" shows "(\<Sum>i. f i - g i) = suminf f - suminf g"
residue_mult_group_def: fixes n :: "nat" shows "residue_mult_group n = \<lparr>carrier = totatives n, monoid.mult = \<lambda>x y. x * y mod n, one = 1\<rparr>"
fvar_def2: fixes C :: "qtname"   and stat :: "bool"   and fn :: "vname"   and a' :: "val"   and s :: "state" shows "fvar C stat fn a' s = ((the (values (the (globs (snd s) (if stat then Inr C else Inl (the_Addr a')))) (Inl (fn, C))), \<lambda>v. supd (upd_gobj (if stat then Inr C else Inl (the_Addr a')) (Inl (fn, C)) v)), abupd (if stat then id else np a') s)"
in_set_of_interval_of: fixes x :: "'a"   and y :: "'a" shows "(x \<in>\<^sub>i interval_of y) = (x = y)"
simps_5: fixes I :: "interp"   and \<theta> :: "trm"   and \<eta> :: "trm" shows "term_sem I (Plus \<theta> \<eta>) = (\<lambda>\<omega>. term_sem I \<theta> \<omega> + term_sem I \<eta> \<omega>)"
winding_number_zero_at_infinity: fixes \<gamma> :: "real \<Rightarrow> complex" assumes "path \<gamma>"   and "pathfinish \<gamma> = pathstart \<gamma>" shows "\<exists>B. \<forall>z. B \<le> cmod z \<longrightarrow> winding_number \<gamma> z = 0"
diff_nat_numeral: fixes v :: "num"   and v' :: "num" shows "numeral v - numeral v' = nat (numeral v - numeral v')"
ir_Seq: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and d :: "com"   and d' :: "com"   and R :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "ir_hoare P c c' Q"   and "ir_hoare Q d d' R" shows "ir_hoare P (c;; d) (c';; d') R"
CombineStrip: 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 c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/{} \<^esub>P strip_guards (- F) c UNIV,UNIV" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/{}\<^esub> P c Q,A"
direct_decomp_Int_zero: fixes A :: "'b set"   and ss :: "'b set list"   and i :: "nat"   and j :: "nat" assumes "direct_decomp A ss"   and "i < j"   and "j < length ss"   and "\<And>s. s \<in> set ss \<Longrightarrow> (0::'b) \<in> s" shows "ss ! i \<inter> ss ! j = {0::'b}"
negatively_invariant_complD: fixes M :: "'a set" assumes "negatively_invariant (X - M)" shows "positively_invariant M"
msii_def: fixes I :: "'b \<Rightarrow> bool"   and m :: "'b nres"   and c :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b nres" shows "msii I m c f \<equiv> lfp (WHILEI_lfp_body I m c f)"
ineq: fixes x :: "real" assumes "x \<in> {0<..<1}" shows "1 \<le> x - ln x"
distinct_3: fixes x11 :: "agent"   and x12 :: "agent"   and x13 :: "msg"   and x31 :: "agent"   and x32 :: "msg" shows "Says x11 x12 x13 \<noteq> Notes x31 x32"
poslsts_UNIV: fixes s :: "'b llist" shows "(s \<in> UNIV\<^sup>\<spadesuit>) = (s \<noteq> LNil)"
finite_arcs: shows "finite (arcs (app_iso iso_f G))"
mod_eq_0_iff_dvd: 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "(modulo a b = zero) = dvd.dvd times b a"
wf_arrow: fixes \<Gamma> :: "binding list"   and T :: "POPLmark.type"   and U :: "POPLmark.type" assumes "\<Gamma> \<turnstile>\<^sub>w\<^sub>f T"   and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f U" shows "\<Gamma> \<turnstile>\<^sub>w\<^sub>f T \<rightarrow> U"
seqE: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and g :: "'d"   and f :: "'d" assumes "category C"   and "partial_magma.arr C (C g f)" obtains "partial_magma.arr C f" and "partial_magma.arr C g" and "partial_magma.dom C g = partial_magma.cod C f"
elims_2: fixes x :: "'a extC"   and xa :: "'a extC" assumes "x \<le> xa" obtains uu_ :: "'a extC" where "xa__ = extC.Bot" and "xa = uu_"   | x :: "'a" and y :: "'a" where "xa__ = extC.Val x" and "xa = extC.Val y" and "x \<le> y"   | uw_ :: "'a" where "xa__ = extC.Val uw_" and "xa = extC.Top"   | "xa__ = extC.Top" and "xa = extC.Top"
prob_eq: fixes A :: "nat set set set" shows "prob A = (if A \<subseteq> Pow S_edges then \<Sum>edges\<in>A. p ^ card edges * (1 - p) ^ card (S_edges - edges) else 0)"
character_data_ptr_casts_commute: fixes node_ptr :: "(_) node_ptr"   and character_data_ptr :: "(_) character_data_ptr" shows "(cast node_ptr = Some character_data_ptr) = (cast character_data_ptr = node_ptr)"
qSwapAbs_preserves_qGoodAbs: fixes zs :: "'varSort"   and x :: "'var"   and y :: "'var"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs" shows "qGoodAbs (A $[[x \<and> y]]_zs) = qGoodAbs A"
fresh_star_eqvt: fixes p :: "perm"   and as :: "atom set"   and x :: "'a" shows "p \<bullet> as \<sharp>* x = (p \<bullet> as) \<sharp>* (p \<bullet> x)"
perm_fun_def: fixes pi :: "('e \<times> 'e) list"   and f :: "'c \<Rightarrow> 'd" shows "pi \<bullet> f = (\<lambda>x. pi \<bullet> f (rev pi \<bullet> x))"
invariant_1_def: fixes tup :: "int poly \<times> rat \<times> rat" shows "invariant_1 tup \<equiv> case tup of (p, l, r) \<Rightarrow> (\<exists>!x. root_cond (p, l, r) x) \<and> sgn l = sgn r \<and> poly_cond p"
tminus_right_antimono: fixes c :: "real"   and b :: "real"   and a :: "real" assumes "c \<le> b" shows "a \<ominus> b \<le> a \<ominus> c"
exist_nj: shows "\<forall>j\<in>{0..<m}. \<exists>nj. enat (Suc nj) < llength Ns \<and> prems_of \<iota> ! j \<notin> active_subset (snd (lnth Ns nj)) \<and> (\<forall>k>nj. enat k < llength Ns \<longrightarrow> prems_of \<iota> ! j \<in> active_subset (snd (lnth Ns k)))"
Q_\<pi>_consistent: fixes u :: "'v"   and d :: "nat"   and v :: "'v" assumes "Q' Q \<pi> u u = enat d"   and "\<pi>' Q \<pi> u u = Some v" shows "d = w {u, v}"
run_one_step_basic_binop_i_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and x261 :: "t"   and x262 :: "binop_i"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Binop_i x261 x262) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
zero_less_measure_iff: fixes M :: "'b measure"   and A :: "'b set" shows "(0 < Sigma_Algebra.measure M A) = (Sigma_Algebra.measure M A \<noteq> 0)"
Span_mem_iff_length_version: fixes K :: "'a set"   and Us :: "'a list"   and a :: "'a" assumes "subfield K R"   and "set Us \<subseteq> carrier R" shows "(a \<in> Span K Us) = (\<exists>Ks. set Ks \<subseteq> K \<and> length Ks = length Us \<and> a = combine Ks Us)"
valid_append_valid_suff: fixes as1 :: "'b list"   and as2 :: "'b list"   and PROB :: "'b set" assumes "as1 @ as2 \<in> valid_plans PROB" shows "as2 \<in> valid_plans PROB"
ltrm_closed: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P" shows "monom P (f (deg R f)) (deg R f) \<in> carrier P"
injects: fixes int :: "int"   and int' :: "int" shows "(MkI\<cdot>int = MkI\<cdot>int') = (int = int')"
l_set_child_nodes_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool"   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 set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog 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" shows "l_set_child_nodes_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes set_child_nodes_locs set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_shadow_root get_shadow_root_locs \<equiv> l_set_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes set_child_nodes_locs set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M \<and> l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs"
B: fixes i :: "nat" shows "\<bar>X i\<bar> \<le> B"
term_equivI: fixes s :: "dB"   and u :: "dB"   and t :: "dB" assumes "s \<rightarrow>\<^sub>\<beta>\<^sub>\<eta>\<^sup>* u"   and "t \<rightarrow>\<^sub>\<beta>\<^sub>\<eta>\<^sup>* u" shows "s \<leftrightarrow> t"
zero_uexpr_def: shows "0 = \<guillemotleft>0::'a\<guillemotright>"
substBinp_def: fixes xs :: "'varSort"   and X :: "('f, 'g, 'varSort, 'var, 'h) qTerm set"   and x :: "'var" shows "substBinp xs X x \<equiv> psubstBinp (idEnv [x \<leftarrow> X]_xs)"
l1_step3_def: fixes Ra :: "fid_t"   and A :: "agent"   and B :: "agent"   and K :: "msg" shows "l1_step3 Ra A B K \<equiv> {(s, s'). guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> progress s Ra = Some {xpkE, xskE} \<and> guessed_frame Ra xsk = Some K \<and> (can_signal s A B \<longrightarrow> (\<exists>Rb. guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> progress s Rb = Some {xpkE, xsk} \<and> guessed_frame Rb xpkE = Some (epubKF (Ra $ kE)) \<and> guessed_frame Rb xsk = Some K)) \<and> (Ra = test \<longrightarrow> K \<notin> synth (analz (ik s))) \<and> s' = s\<lparr>progress := progress s(Ra \<mapsto> {xpkE, xskE, xsk}), secret := {x. x = K \<and> Ra = test} \<union> secret s, l1_state.signals := if can_signal s A B then addSignal (l1_state.signals s) (Commit A B \<langle>epubKF (Ra $ kE), K\<rangle>) else l1_state.signals s\<rparr>}"
wlsAll_inversion: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs_s :: "'varSort \<times> 'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option" shows "(wls s X \<longrightarrow> (\<forall>xs x. X = Var xs x \<longrightarrow> s = asSort xs) \<and> (\<forall>delta inp binp. goodInp inp \<and> goodBinp binp \<and> X = Op delta inp binp \<longrightarrow> stOf delta = s \<and> wlsInp delta inp \<and> wlsBinp delta binp)) \<and> (wlsAbs xs_s A \<longrightarrow> isInBar xs_s \<and> (\<forall>x X. good X \<and> A = Abs (fst xs_s) x X \<longrightarrow> wls (snd xs_s) X)) \<and> (wlsInp delta inp \<longrightarrow> True) \<and> (wlsBinp delta binp \<longrightarrow> True)"
lprefix_trans: fixes xs :: "'f llist"   and ys :: "'f llist"   and zs :: "'f llist" assumes "lprefix xs ys"   and "lprefix ys zs" shows "lprefix xs zs"
irrefl: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "ext_irrefl ext"   and "\<forall>x\<in>set xs. \<not> gt x x" shows "\<not> ext gt xs xs"
assms_2: shows "(v0, w, v1) \<in> E"
finite_digraphI: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>"   and "vfinite (\<CC>\<lparr>Obj\<rparr>)"   and "vfinite (\<CC>\<lparr>Arr\<rparr>)" shows "finite_digraph \<alpha> \<CC>"
iprev_imirror_inext_conv: fixes I :: "nat set"   and n :: "nat" assumes "finite I"   and "n \<le> iMin I + Max I" shows "iprev (mirror_elem n I) (imirror I) = mirror_elem (inext n I) I"
comp_types: fixes B :: "'b"   and C :: "'b"   and A :: "'b" shows "(\<bullet>) \<in> Hom B C \<rightarrow> Hom A B \<rightarrow> Hom A C"
arr_cod_iff_arr: fixes f :: "'j" shows "J.arr (J.cod f) = J.arr f"
keys_monom_mult_subset: fixes c :: "'c"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "keys (monom_mult c t p) \<subseteq> (\<oplus>) t ` keys p"
fterms_of_hterms_hterms_of_fterms: fixes ts :: "fterm list" assumes "ground\<^sub>t\<^sub>s ts" shows "fterms_of_hterms (hterms_of_fterms ts) = ts"
code: fixes abort :: "bool"   and x :: "real"   and n :: "real"   and acc :: "real" shows "gbinomial_series_aux abort x n acc = (if abort \<and> acc = 0 then MSLNil else MSLCons acc (gbinomial_series_aux abort x (n + 1) ((x - n) / (n + 1) * acc)))"
CRR_hyps_axioms_def: fixes p :: "real"   and geom_proc :: "nat \<Rightarrow> bool stream \<Rightarrow> real"   and u :: "real"   and d :: "real"   and init :: "real"   and Mkt :: "(bool stream, 'b) discrete_market"   and risk_free_asset :: "'b"   and stk :: "'b" shows "CRR_hyps_axioms p geom_proc u d init Mkt risk_free_asset stk \<equiv> (stocks Mkt = {stk, risk_free_asset} \<and> prices Mkt stk = geom_proc \<and> 0 < init) \<and> (0 < d \<and> d < u) \<and> 0 < p \<and> p < 1"
map_of_rat_of_int_poly: fixes p :: "int poly" shows "map_poly of_rat (of_int_poly p) = of_int_poly p"
map_graph_homo_simp: fixes G :: "('e, 'f) labeled_graph"   and f :: "('f \<times> 'g) set" shows "graph_homomorphism G (map_graph f G) f = (univalent f \<and> vertices G = Domain f \<and> graph G)"
rell_G_def: fixes L1 :: "'l \<Rightarrow> 's \<Rightarrow> bool"   and L2 :: "'m \<Rightarrow> 't \<Rightarrow> bool" shows "rell_G L1 L2 = rel_G L1 L2 (=) (=) (=) (=)"
size_Diff_singleton_if: fixes A :: "'a multiset"   and x :: "'a" shows "size (A - {#x#}) = (if x \<in># A then size A - 1 else size A)"
WF_alt: fixes A :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "\<turnstile> WF(A)_v = (\<box>\<diamond>\<not> Enabled \<langle>A\<rangle>_v \<or> \<box>\<diamond>\<langle>A\<rangle>_v)"
maxR_def: fixes d :: "D2"   and x :: "X3"   and y :: "X3" shows "Theorem_9.maxR d x y = (if (x, y) \<in> PX3d d then y else x)"
app_vfieldI3: fixes a :: "V"   and b :: "V"   and r :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r" shows "b \<in>\<^sub>\<circ> \<F>\<^sub>\<circ> r"