hcju/isabelleps huggingface dataset:
list_all2_ConsI: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list"   and x :: "'a"   and y :: "'b" assumes "list_all2 P xs ys"   and "P x y" shows "list_all2 P (x # xs) (y # ys)"
J_M_J_transpose_K2_isometry: fixes k :: "real"   and repJ :: "((real, 3) vec, 3) vec" assumes "k \<noteq> 0"   and "repJ ** M ** Finite_Cartesian_Product.transpose repJ = k *\<^sub>R M" shows "is_K2_isometry (cltn2_abs repJ)"
hseqE: fixes \<nu> :: "'b"   and \<mu> :: "'b" assumes "hseq \<nu> \<mu>" obtains "arr \<mu>" and "arr \<nu>" and "src \<nu> = trg \<mu>"
mrDA_tl: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and p :: "((int \<times> int) set set, int) Combinators list" assumes "applied_rule_rev C x p = \<lfloor>DenyAll\<rfloor>"   and "wellformed_policy1_strong p" shows "applied_rule_rev C x (tl p) = \<bottom>"
hs_remove_max_def: fixes t :: "'c Tree" shows "hs_remove_max t \<equiv> let v' = fst (removeLeaf t); t' = snd (removeLeaf t) in if t' = E then (val t, E) else (val t, siftDown (set_val t' v'))"
\<eta>: shows "\<guillemotleft>\<eta> : src\<^sub>C f \<Rightarrow>\<^sub>C g \<star>\<^sub>C f\<guillemotright> \<and> F \<eta> = \<Phi> (g, f) \<cdot>\<^sub>D (F g \<star>\<^sub>D F f) \<cdot>\<^sub>D (\<mu> \<star>\<^sub>D F f) \<cdot>\<^sub>D \<phi> \<cdot>\<^sub>D D.inv (unit (src\<^sub>C f))"
select_convs_2: fixes ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) 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 pre :: "nat set"   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 "sn \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, pre = pre, rreqid = rreqid, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, \<dots> = more\<rparr> = sn"
\<b>_1: fixes X :: "'x set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and i :: "nat" assumes "finite X" shows "\<a> ps \<le> \<b> ps i"
new_last_to_v1: shows "\<exists>R. new_last \<leadsto>R\<leadsto>\<^bsub>remove_vertex v0\<^esub> v1"
finfun_upd_apply_same: fixes f :: "'b \<Rightarrow>f 'a"   and a :: "'b"   and b :: "'a" shows "f(a $:= b) $ a = b"
vec_simps_14: fixes af :: "'a"   and bf :: "'a"   and ce :: "'a" shows "af * bf * ce = af * (bf * ce)"
Rep_range: fixes Rep :: "'a \<Rightarrow> 'b"   and Abs :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "type_definition Rep Abs A" shows "range Rep = A"
counter_a_aux: fixes u :: "nat \<Rightarrow> nat"   and n :: "nat"   and x :: "nat \<Rightarrow> bool" assumes "u 0 = 0"   and "\<forall>i<n. (x i \<longrightarrow> u (Suc i) = Suc (u i)) \<and> (\<not> x i \<longrightarrow> u (Suc i) = u i)" shows "\<forall>i<n. count x i = u (Suc i)"
domain_block_non_negative: fixes xs :: "brick list" shows "0 \<le> domain_block xs"
assms: shows "upper_asymptotic_density (A \<Delta> B) = 0"
integrable_real_mult_indicator: fixes A :: "'b set"   and M :: "'b measure"   and f :: "'b \<Rightarrow> real" assumes "A \<in> sets M"   and "integrable M f" shows "integrable M (\<lambda>x. f x * indicat_real A x)"
strongSimWeakEqSim: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi" assumes "P \<leadsto>[Rel] Q" shows "P \<leadsto><Rel> Q"
inf_Sup_absorb: fixes 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 A :: "'a set"   and a :: "'a" assumes "class.lattice inf less_eq less sup"   and "finite A"   and "a \<in> A" shows "inf a (semilattice_sup.Sup_fin sup A) = a"
l3_init_def: shows "l3_init \<equiv> {\<lparr>ik = ik_init, secret = {}, progress = Map.empty, signalsInit = \<lambda>x. 0, signalsResp = \<lambda>x. 0, signalsInit2 = \<lambda>x. 0, signalsResp2 = \<lambda>x. 0, l3_state.bad = bad_init\<rparr>}"
assms: shows "K\<^bsub>3,3\<^esub> G"
times_dimvec_def: fixes x :: "('a, 'b) dimvec"   and y :: "('a, 'b) dimvec" shows "x \<cdot> y = dim_lambda (\<lambda>i. dim_nth x i + dim_nth y i)"
ipresWlsAbs_def: fixes hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'b"   and MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'b, 'h) model_scheme" shows "ipresWlsAbs hA MOD \<equiv> \<forall>us s A. wlsAbs (us, s) A \<longrightarrow> igWlsAbs MOD (us, s) (hA A)"
inj_var_ran_unifiable_has_subst_match: fixes t :: "('a, 'b) Term.term"   and \<delta> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and s :: "('a, 'b) Term.term" assumes "Unifier \<delta> t s"   and "inj_on \<delta> (fv t)"   and "\<delta> ` fv t \<subseteq> range Var" shows "t = s \<cdot> \<delta> \<circ>\<^sub>s subst_var_inv \<delta> (fv t)"
dom_el_idem: fixes x :: "'a" shows "d x \<cdot> d x = d x"
IH_1: shows "insertion f (insertion (\<lambda>v. monom 0 (f v)) p1) = insertion f (reduce_nested_mpoly p1)"
rep_fmap_base: fixes P :: "('g \<Rightarrow> 'h option) \<Rightarrow> bool"   and F :: "'g \<Rightarrow> 'h option" shows "P F = pred_set_fmap P (set_fmap F)"
violations_0_implies_all_subterms_0: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list" assumes "computation channel trans send recv init final"   and "computation.count_violations channel trans send recv init t = 0" shows "\<forall>i\<in>{0..<length t}. (if computation.postrecording_event channel trans send recv init t i then card {j \<in> {i + 1..<length t}. computation.prerecording_event channel trans send recv init t j} else 0) = 0"
dynmethd_access_prop: fixes G :: "prog"   and statC :: "qtname"   and sig :: "sig"   and statM :: "qtname \<times> methd"   and accC :: "qtname"   and dynC :: "qtname"   and dynM :: "qtname \<times> methd" assumes "methd G statC sig = Some statM"   and "G \<turnstile>Methd sig statM of statC accessible_from accC"   and "dynmethd G statC dynC sig = Some dynM"   and "wf_prog G" shows "G \<turnstile>Methd sig dynM in dynC dyn_accessible_from accC"
simple_match_ip_Accept: fixes s :: "'a word" assumes "simple_match_ip (dst m) d" shows "(runFw s d c (SimpleRule m Accept # rs) = Decision FinalAllow) = (simple_match_ip (src m) s \<or> runFw s d c rs = Decision FinalAllow)"
abs_eq: shows "top = Abs_lifted_pair (\<lambda>f. (top, Filters.filter.Abs_filter {top}))"
finite_PiE: fixes S :: "'c set"   and T :: "'c \<Rightarrow> 'd set" assumes "finite S"   and "\<And>i. i \<in> S \<Longrightarrow> finite (T i)" shows "finite (Pi\<^sub>E S T)"
\<iota>_mapsto: shows "SC.UP \<in> UNIV \<rightarrow> SC.Univ"
cotrace_trace: fixes c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event llist"   and i :: "nat" assumes "cotrace c t" shows "\<exists>!c'. trace c (Co_Snapshot.ltake i t) c'"
assms_7: shows "unbounded c ([xs !! i]\<^sub>\<R>)"
integrable_combine: fixes a :: "real"   and c :: "real"   and f :: "real \<Rightarrow> 'a"   and b :: "real" assumes "integrable (lebesgue_on {a..c}) f"   and "integrable (lebesgue_on {c..b}) f"   and "a \<le> c"   and "c \<le> b" shows "integrable (lebesgue_on {a..b}) f"
ik_assignment_rhs_decomp_fv: fixes t :: "('fun, 'var) Term.term"   and A :: "('fun, 'var) extstrand_step list" assumes "t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t ik\<^sub>e\<^sub>s\<^sub>t A \<union> assignment_rhs\<^sub>e\<^sub>s\<^sub>t A" shows "fv\<^sub>e\<^sub>s\<^sub>t (A @ [Decomp t]) = fv\<^sub>e\<^sub>s\<^sub>t A"
small_asymmetric: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> \<omega>[F](g)"   and "g \<in> \<omega>[F](f)" shows "\<forall>\<^sub>F x in F. f x = (0::'c)"
inter_guards_Cond: fixes cnd :: "'d set"   and t1 :: "('d, 'e, 'f) com"   and e1 :: "('d, 'e, 'f) com"   and c2 :: "('d, 'e, 'f) com"   and c :: "('d, 'e, 'f) com" shows "((Cond cnd t1 e1 \<inter>\<^sub>g c2) = Some c) = (\<exists>t2 e2 t e. c2 = Cond cnd t2 e2 \<and> (t1 \<inter>\<^sub>g t2) = Some t \<and> (e1 \<inter>\<^sub>g e2) = Some e \<and> c = Cond cnd t e)"
lang\<^sub>M\<^sub>2\<^sub>L_FAnd: fixes n :: "nat"   and \<phi>\<^sub>1 :: "'a formula"   and \<phi>\<^sub>2 :: "'a formula" assumes "wf_formula n (FAnd \<phi>\<^sub>1 \<phi>\<^sub>2)" shows "lang\<^sub>M\<^sub>2\<^sub>L n (FAnd \<phi>\<^sub>1 \<phi>\<^sub>2) \<subseteq> lang\<^sub>M\<^sub>2\<^sub>L n \<phi>\<^sub>1 \<inter> lang\<^sub>M\<^sub>2\<^sub>L n \<phi>\<^sub>2 \<inter> {enc (w, I) |w I. length I = n \<and> wf_interp_for_formula (w, I) (FAnd \<phi>\<^sub>1 \<phi>\<^sub>2)}"
inverse_functor_unique: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'d \<Rightarrow> 'c"   and G :: "'c \<Rightarrow> 'd"   and G' :: "'c \<Rightarrow> 'd" assumes "inverse_functors C D F G"   and "inverse_functors C D F G'" shows "G = G'"
carrier_direct_lim_def: shows "carrier_direct_lim \<equiv> Partition"
wI_2: shows "wf_interp_for_formula (w, I) \<phi>"
knows1k: fixes A :: "specID"   and m :: "Keys" shows "know A (kKS m) = knows A [kE m]"
wfenv_good_ctx: fixes \<rho> :: "val list"   and \<eta> :: "val set list"   and \<Gamma> :: "(ty \<times> nat) list \<times> nat" assumes "\<turnstile> \<rho>,\<eta> : \<Gamma>" shows "good_ctx \<Gamma>"
KN_essential: shows "KN.essential_class (C \<times> Some ` C)"
simps_2: fixes css :: "nat \<Rightarrow> 'd list"   and i :: "nat" shows "seq css (Suc i) = css i @ seq css i"
irrefl: fixes a :: "('d, 'e) Term.term" shows "(a, a) \<notin> {\<rhd>}"
continuous_on_topological: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "continuous_on s f = (\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow> (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
list_of_oalist_shift_keys: fixes t :: "'a"   and f :: "'c \<Rightarrow> 'c"   and xs :: "('t, 'c) oalist_ntm" shows "list_of_oalist_ntm (shift_map_keys t f xs) = map_raw (\<lambda>kv. (t \<oplus> fst kv, f (snd kv))) (list_of_oalist_ntm xs)"
directional_derivative_def: fixes k :: "enat"   and a :: "'c"   and v :: "'c" shows "directional_derivative k a v = restrict0 (manifold_eucl.diff_fun_space k) (\<lambda>f. frechet_derivative f (at a) v)"
no_proots_line_def: fixes p :: "complex poly"   and st :: "complex"   and tt :: "complex" shows "no_proots_line p st tt = (proots_within p (closed_segment st tt) = {})"
dg_digraph_if_ge_Limit: fixes \<beta> :: "V" assumes "\<Z> \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "digraph \<beta> \<BB>"
set_eq_fold: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and xs :: "'b list" assumes "semilattice_set f" shows "semilattice_set.F f (set (x # xs)) = fold f xs x"
less_numeral_Suc: fixes k :: "num"   and n :: "nat" shows "(numeral k < Suc n) = (pred_numeral k < n)"
pdevs_apply_pdevs_of_list: fixes xs :: "'d list"   and i :: "nat" shows "pdevs_apply (pdevs_of_list xs) i = (if i < length xs then xs ! i else (0::'d))"
echelon_form_of_euclidean_def: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and sgn :: "'d \<Rightarrow> 'd"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and inner :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and Basis :: "'d set"   and A :: "(('e, 'f) vec, 'g) vec" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "euclidean_space.echelon_form_of_euclidean A = echelon_form_of A euclid_ext2"
add_mset: fixes x :: "'b"   and N :: "'b multiset" shows "\<Sum>\<^sub># (add_mset x N) = x + \<Sum>\<^sub># N"
inj_on_map_inv_f: fixes l :: "'a list"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "set l \<subseteq> A"   and "inj_on f A" shows "map (inv_on f A) (map f l) = l"
C_def: shows "C = k + 2 * nat \<bar>l\<bar> + 1"
v2: shows "V2 = \<lparr>ext = x2, lan = lan v, own = own v\<rparr>"
loop_orientable_def: fixes x :: "'a" shows "bounded_distrib_allegory.loop_orientable top (\<sqinter>) (\<le>) (\<squnion>) id x \<equiv> \<exists>y. y \<squnion> id y = x \<and> y \<sqinter> id y \<le> top"
inject: fixes x1 :: "'c"   and x2 :: "'d"   and x3 :: "('a, 'b, 'c, 'd, 'e) qTerm"   and y1 :: "'c"   and y2 :: "'d"   and y3 :: "('a, 'b, 'c, 'd, 'e) qTerm" shows "(qAbs x1 x2 x3 = qAbs y1 y2 y3) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3)"
intros_6: fixes a :: "'c"   and b :: "'c" assumes "a < b" shows "Lt a \<prec> Lt b"
ntrunc_def: fixes k :: "nat"   and N :: "('a, 'b) node set" shows "ntrunc k N \<equiv> {n \<in> N. ndepth n < k}"
seq_extract_ident: fixes xs :: "'b list" shows "dom\<^sub>l xs \<upharpoonleft>\<^sub>l xs = xs"
ereal_of_enat_Sup: fixes A :: "enat set" assumes "A \<noteq> {}" shows "ereal_of_enat (Sup A) = Sup (ereal_of_enat ` A)"
strict_iff_order: fixes a :: "'g"   and b :: "'g" shows "(a < b) = (a \<le> b \<and> a \<noteq> b)"
below_lub: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "nat \<Rightarrow> 'b"   and x :: "'b"   and i :: "nat" assumes "class.cpo below"   and "po.chain below S"   and "below x (S i)" shows "below x (po.lub below (range S))"
operative_approximableI: fixes e :: "real"   and f :: "'a \<Rightarrow> 'b" assumes "0 \<le> e" shows "operative (\<and>) True (\<lambda>i. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i)"
assms: shows "(am, adjacent_nodes) \<in> nat_rel \<rightarrow> \<langle>nat_rel\<rangle>list_set_rel"
cSup_upper: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and X :: "'c set" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "x \<in> X"   and "preorder.bdd_above less_eq X" shows "less_eq x (Sup X)"
rtrancl_into_trancl1: fixes a :: "'b"   and b :: "'b"   and r :: "('b \<times> 'b) set"   and c :: "'b" assumes "(a, b) \<in> r\<^sup>*"   and "(b, c) \<in> r" shows "(a, c) \<in> r\<^sup>+"
add_left: fixes a :: "'d"   and a' :: "'d"   and b :: "'b" shows "g (a + a') ** b = g a ** b + g a' ** b"
power_diff': fixes b :: "nat"   and a :: "nat"   and x :: "'b" assumes "b \<le> a" shows "x ^ (a - b) = (if x = (0::'b) \<and> a = b then 1::'b else x ^ a / x ^ b)"
assms_3: shows "cfunc_usr core = cfunc_usr core' \<oplus>\<^sub>O (\<lambda>s i. map_spmf (Pair (h_out i)) (fn s i))"
Ord_linear: fixes k :: "V"   and l :: "V" assumes "Ord k"   and "Ord l" shows "k \<in>\<^sub>\<circ> l \<or> k = l \<or> l \<in>\<^sub>\<circ> k"
prv_neg_fls: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and \<phi> :: "'c" assumes "Deduct_with_False var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num prv"   and "\<phi> \<in> fmla"   and "prv \<phi>"   and "prv (Syntax_with_Connectives_False.neg imp fls \<phi>)" shows "prv fls"
card_sequences: fixes A :: "'b set"   and B :: "'c set" assumes "finite A" shows "card {xs. set xs \<subseteq> B \<and> length xs = card A} = card B ^ card A"
gen_length_code_1: fixes n :: "nat" shows "List.gen_length n \<epsilon> = n"
attractor_set_VVp: fixes v :: "'a"   and p :: "Player"   and w :: "'a"   and W :: "'a set" assumes "v \<in> G'.VV p"   and "v \<rightarrow>\<^bsub>G'\<^esub> w"   and "w \<in> G'.attractor p W" shows "v \<in> G'.attractor p W"
span_sum: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'd"   and S :: "'d set" assumes "module scale"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<in> module.span scale S" shows "sum f A \<in> module.span scale S"
deriv: fixes y :: "(nat \<times> form) list" shows "deriv y = insert (0, y) (inc ` \<Union> (deriv ` {w. \<not> is_axiom (s_of_ns y) \<and> w \<in> set (subs y)}))"
Igdpr_def: shows "Igdpr \<equiv> {gdpr_scenario}"
one_circ_mult_split: fixes x :: "'a" shows "(1::'a)\<^sup>\<circ> * x = L \<squnion> x"
mult_le_prts: fixes a1 :: "'b"   and a :: "'b"   and a2 :: "'b"   and b1 :: "'b"   and b :: "'b"   and b2 :: "'b" assumes "a1 \<le> a"   and "a \<le> a2"   and "b1 \<le> b"   and "b \<le> b2" shows "a * b \<le> pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
distributed_integrable: fixes M :: "'c measure"   and N :: "'d measure"   and X :: "'c \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> real"   and g :: "'d \<Rightarrow> real" assumes "distributed M N X (\<lambda>x. ennreal (f x))"   and "g \<in> borel_measurable N"   and "\<And>x. x \<in> space N \<Longrightarrow> 0 \<le> f x" shows "integrable N (\<lambda>x. f x * g x) = integrable M (\<lambda>x. g (X x))"
less_trans: fixes x :: "'b list"   and y :: "'b list"   and z :: "'b list" assumes "strict_suffix x y"   and "strict_suffix y z" shows "strict_suffix x z"
triangle: fixes x :: "'p"   and z :: "'p"   and y :: "'p" shows "dist x z \<le> dist y x + dist y z"
trans: fixes x :: "'alt"   and y :: "'alt"   and z :: "'alt" assumes "Ri' x y"   and "Ri' y z" shows "Ri' x z"
simps_2: fixes l :: "interval_tree"   and it :: "nat idx_interval"   and m :: "nat"   and r :: "interval_tree" shows "tree_max_inv (Node l it m r) = (tree_max_inv l \<and> tree_max_inv r \<and> m = max3 it (tmax l) (tmax r))"
matchPres: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and a :: "name"   and b :: "name" assumes "P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"   and "Rel \<subseteq> Rel'" shows "[a\<frown>b]P \<leadsto>\<guillemotleft>Rel'\<guillemotright> [a\<frown>b]Q"
closure_Union: fixes X :: "'b set set" assumes "finite X" shows "closure (\<Union> X) = \<Union> (closure ` X)"
left_closed: fixes S :: "'c set"   and E :: "('c \<times> 'c) set"   and a :: "'c"   and b :: "'c" assumes "equivalence S E"   and "(a, b) \<in> E" shows "a \<in> S"
seq_char: fixes g :: "'a \<times> 'a \<times> 'a"   and f :: "'a \<times> 'a \<times> 'a" shows "VVV.seq g f = (VVV.arr f \<and> VVV.arr g \<and> VxVxV.seq g f)"
termi: shows "\<Gamma>\<turnstile>w \<down> fk"
Pi_I': fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and B :: "'a \<Rightarrow> 'b set" assumes "\<And>x. x \<in> A \<longrightarrow> f x \<in> B x" shows "f \<in> Pi A B"
position_if_emb_step_at: fixes p :: "dir list"   and d :: "dir"   and t :: "('c, 'd) tm"   and u :: "('c, 'd) tm" assumes "emb_step_at p d t = u"   and "t \<noteq> u" shows "position_of t (p @ [d])"
normalize_idem_imp_is_unit_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalize a = a" shows "dvd.dvd times a one = (a = one)"
ffun_plus_pos: fixes x :: "('a, 'b) ffun"   and y :: "('a, 'b) ffun" assumes "x + y = {}\<^sub>f" shows "x = {}\<^sub>f"
retract_ofE: fixes T :: "'c set"   and S :: "'c set" assumes "T retract_of S" obtains r :: "'c \<Rightarrow> 'c" where "T = r ` S" and "r ` S \<subseteq> S" and "continuous_on S r" and "\<And>x. x \<in> S \<Longrightarrow> r (r x) = r x"
inres_def: fixes S :: "'a nres"   and x :: "'a" shows "inres S x \<equiv> RETURN x \<le> S"
transfer_rule_numeral_uint: shows "rel_fun (=) cr_uint32 numeral numeral"
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_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" 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_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_parent get_parent_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs" shows "l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^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_ancestors get_ancestors_locs get_root_node get_root_node_locs"
emb_step_at_is_App: fixes p :: "dir list"   and d :: "dir"   and u :: "('a, 'b) tm" assumes "emb_step_at p d u \<noteq> u" shows "is_App u"
cil_set: fixes w :: "'d list"   and w1 :: "'d list"   and \<alpha> :: "'d \<Rightarrow> 'e set \<times> 'e set"   and w2 :: "'d list" assumes "w \<in> w1 \<otimes>\<^bsub>\<alpha>\<^esub> w2" shows "set w = set w1 \<union> set w2"
double_dual_iso_mult: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c" assumes "finite_comm_group G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "finite_comm_group.double_dual_iso G (x \<otimes>\<^bsub>G\<^esub> y) = finite_comm_group.double_dual_iso G x \<otimes>\<^bsub>Characters (Characters G)\<^esub> finite_comm_group.double_dual_iso G y"
bisim1BlockThrowNone: fixes P :: "'b prog"   and h :: "'heap"   and e :: "'addr expr1"   and a :: "'addr"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and V :: "nat"   and T :: "ty" assumes "P,e,h \<turnstile> (Throw a, xs) \<leftrightarrow> (stk, loc, pc, \<lfloor>a\<rfloor>)" shows "P,{V:T=None; e},h \<turnstile> (Throw a, xs) \<leftrightarrow> (stk, loc, pc, \<lfloor>a\<rfloor>)"
orderE: fixes a :: "'a Word.word"   and b :: "'a Word.word" assumes "a \<le>s b" obtains "a = signed.min a b"
assms_3: shows "borel_adapt_stoch_proc F (prices Mkt asset)"
bet__sams: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p" assumes "A \<noteq> B"   and "B \<noteq> C"   and "P \<noteq> B"   and "Bet A B C" shows "SAMS A B P P B C"