hcju/isabelleps huggingface dataset:
measurable_pair_swap_iff: fixes f :: "'c \<times> 'd \<Rightarrow> 'e"   and M2 :: "'c measure"   and M1 :: "'d measure"   and M :: "'e measure" shows "(f \<in> M2 \<Otimes>\<^sub>M M1 \<rightarrow>\<^sub>M M) = ((\<lambda>(x, y). f (y, x)) \<in> M1 \<Otimes>\<^sub>M M2 \<rightarrow>\<^sub>M M)"
\<R>_to_\<R>\<^sub>3: fixes c\<^sub>1 :: "(Eg1.var, aexp, bexp) Stmt"   and mds :: "Mode \<Rightarrow> Eg1.var set"   and mem\<^sub>1 :: "Eg1.var \<Rightarrow> nat"   and c\<^sub>2 :: "(Eg1.var, aexp, bexp) Stmt"   and mem\<^sub>2 :: "Eg1.var \<Rightarrow> nat"   and \<Gamma> :: "Eg1.var \<Rightarrow> bexp set option"   and \<S> :: "Eg1.var set \<times> Eg1.var set"   and P :: "bexp set"   and c :: "(Eg1.var, aexp, bexp) Stmt"   and \<Gamma>' :: "Eg1.var \<Rightarrow> bexp set option"   and \<S>' :: "Eg1.var set \<times> Eg1.var set"   and P' :: "bexp set" assumes "A.\<R>_abv \<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle>\<^sub>A \<Gamma> \<S> P \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>\<^sub>A"   and "A.has_type \<Gamma> \<S> P c \<Gamma>' \<S>' P'" shows "A.\<R>\<^sub>3_aux \<langle>c\<^sub>1 ;; c, mds, mem\<^sub>1\<rangle>\<^sub>A \<Gamma>' \<S>' P' \<langle>c\<^sub>2 ;; c, mds, mem\<^sub>2\<rangle>\<^sub>A"
frontier_comparable_False: fixes x :: "'a"   and M :: "'a zmultiset"   and y :: "'a" assumes "x \<in>\<^sub>A frontier M"   and "y \<in>\<^sub>A frontier M"   and "x < y" shows "False"
simps: fixes P :: "'b mat"   and A :: "'b mat"   and Q :: "'b mat"   and P' :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and A' :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and Q' :: "(('b, 'd) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "HMA_M3 (P, A, Q) (P', A', Q') = (Mod_Type_Connect.HMA_M P P' \<and> Mod_Type_Connect.HMA_M A A' \<and> Mod_Type_Connect.HMA_M Q Q')"
chamber_system_simplices: shows "order.greater_eq X folding.\<C>"
check_instr'_ins_jump_ok: fixes ins :: "'a instr"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and h :: "'b"   and stk :: "'a val list"   and loc :: "'a val list"   and C :: "String.literal"   and M :: "String.literal"   and pc :: "nat"   and frs :: "'a frame list" assumes "check_instr' ins P h stk loc C M pc frs" shows "ins_jump_ok ins pc"
transC_MtransT: fixes cf :: "(test, atom) com \<times> (Concrete.var \<Rightarrow> nat)"   and cf' :: "(test, atom) com \<times> (Concrete.var \<Rightarrow> nat)"   and s'' :: "Concrete.var \<Rightarrow> nat" assumes "Example_PL.transC cf cf'"   and "Example_PL.MtransT cf' s''" shows "Example_PL.MtransT cf s''"
circ_back_loop_prefixpoint: fixes z :: "'a"   and y :: "'a" shows "z * y\<^sup>\<star> * y \<squnion> z \<le> z * y\<^sup>\<star>"
language_state_alt_def: fixes M :: "('d, 'e, 'f) FSM"   and q :: "'f" shows "LS M q = {uu_. \<exists>io tr. uu_ = io \<and> path M (io || tr) q \<and> length io = length tr}"
integral_setdiff: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'a set" assumes "f integrable_on S"   and "f integrable_on T"   and "negligible (T - S)" shows "integral (S - T) f = integral S f - integral T f"
K_gamma_def: fixes i :: "nat"   and j :: "nat" shows "\<gamma>\<^bsub>i j\<^esub> = (if i = j then 0 else 1)"
IH: fixes y :: "nat" assumes "y < m" shows "norm (natfun_inverse f y) \<le> inverse (\<delta> ^ y)"
Inf_option_def: fixes A :: "'b option set" shows "\<Sqinter> A = (if None \<in> A then None else Some (\<Sqinter> Option.these A))"
N_\<mu>': fixes i :: "nat"   and j :: "nat" assumes "i < length fs"   and "j \<le> i" shows "(\<mu>' i j)\<^sup>2 \<le> N ^ (3 * Suc j)"
nt_mult_pm: fixes a :: "'a"   and m :: "int"   and n :: "int" assumes "a \<in> carrier A"   and "0 \<le> m"   and "n < 0" shows "m\<triangleright>n\<triangleright>a\<^bsub>A\<^esub>\<^bsub>A\<^esub> = (m * n)\<triangleright>a\<^bsub>A\<^esub>"
pth_3: fixes x :: "'a" shows "- x \<equiv> - 1 *\<^sub>R x"
rep_eq: fixes x :: "proper_base" shows "Rep_proper_base (square_base x) = Rep_proper_base x * Rep_proper_base x"
linear_inj_on_span_iff_independent_image: fixes f :: "'b \<Rightarrow> 'c"   and B :: "'b set" assumes "linear (*a) (*b) f"   and "vs2.independent (f ` B)" shows "inj_on f (vs1.span B) = inj_on f B"
poincare_distance_ge0: fixes u :: "complex_homo"   and v :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc" shows "0 \<le> poincare_distance u v"
distinct_count_atmost_1: fixes x :: "'c list" shows "distinct x = (\<forall>a. count (mset x) a = (if a \<in> set x then 1 else 0))"
satisfiableEntailedFormula: fixes formula0 :: "Literal list list"   and formula :: "Literal list list" assumes "formulaEntailsFormula formula0 formula" shows "satisfiable formula0 \<longrightarrow> satisfiable formula"
hyps: shows "\<bottom> \<star> append rs [r] = Some (snd r)"
arthan_aux1: fixes xs :: "'b list"   and x :: "'b"   and y :: "'b" assumes "xs \<noteq> []"   and "last xs \<noteq> (0::'b)"   and "x + y + sum_list xs = (0::'b)" shows "int (sign_changes (psums (x # y # xs))) - int (sign_changes (psums ((x + y) # xs))) \<le> int (sign_changes (x # y # xs)) - int (sign_changes ((x + y) # xs)) \<and> even (int (sign_changes (x # y # xs)) - int (sign_changes ((x + y) # xs)) - (int (sign_changes (psums (x # y # xs))) - int (sign_changes (psums ((x + y) # xs)))))"
wrmrm_decomp: fixes t :: "'b Tree"   and h :: "'b \<Rightarrow> int" shows "t \<noteq> Tip \<longrightarrow> wrmrm h t = (wrm h t, rm h t)"
path_independent_on_Chernoff_on: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set" assumes "path_independent_on A f"   and "f_range_on A f" shows "Chernoff_on A f"
f_Exec_Stream_hd: fixes xs :: "'d list"   and trans_fun :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e" assumes "0 < length xs" shows "hd (f_Exec_Comp_Stream trans_fun xs c) = trans_fun (hd xs) c"
pwd_spy_ii_def: fixes n :: "nat" shows "pwd_spy_ii n \<equiv> case pwd_owner_ii n of (s, S, A, x) \<Rightarrow> (insert (Spy, \<lbrace>Num 2, PubKey S\<rbrace>) s, S, A, S, 0, 0)"
inverse_of_nat_le: fixes n :: "nat"   and m :: "nat" assumes "n \<le> m"   and "n \<noteq> 0" shows "(1::'a) / of_nat m \<le> (1::'a) / of_nat n"
length_upt: fixes i :: "nat"   and j :: "nat" shows "length [i..<j] = j - i"
seqE: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "B.VxV.seq g f" obtains "B.seq (fst g) (fst f)" and "B.seq (snd g) (snd f)"
int_n_minus_1: shows "int (card X - Suc 0) = int (card X) - 1"
simps_31: fixes x9 :: "Term.var"   and x1 :: "qtname" shows "Acc x9 \<noteq> NewC x1"
bounded_antilinear_def: fixes f :: "'a \<Rightarrow> 'b" shows "bounded_antilinear f \<equiv> antilinear f \<and> bounded_antilinear_axioms f"
pdivides_iff_shell: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and p :: "'c list"   and q :: "'c list" assumes "domain R"   and "subfield K R"   and "p \<in> carrier (K [X]\<^bsub>R\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>R\<^esub>)" shows "p pdivides\<^bsub>R\<^esub> q = p divides\<^bsub>K [X]\<^bsub>R\<^esub>\<^esub> q"
assms_4: fixes i :: "nat"   and j :: "nat" assumes "i < m"   and "j < n" shows "polyfun N (\<lambda>x. A x $$ (i, j))"
powhr_as_starfun: fixes x :: "real star"   and a :: "real star" shows "x powhr a = (if x = 0 then 0 else (*f* exp) (a * (*f* real_ln) x))"
min_rvar_incdec_eq_None: fixes dir :: "('i, 'a) Direction"   and s :: "('i, 'a) state"   and eq :: "nat \<times> linear_poly"   and "is" :: "'i list" assumes "min_rvar_incdec_eq dir s eq = Inl is" shows "(\<forall>x\<in>rvars_eq eq. \<not> (0 < coeff (rhs eq) x \<and> \<lhd>\<^sub>u\<^sub>b (lt dir) (\<langle>\<V> s\<rangle> x) (UB dir s x) \<or> coeff (rhs eq) x < 0 \<and> \<rhd>\<^sub>l\<^sub>b (lt dir) (\<langle>\<V> s\<rangle> x) (LB dir s x))) \<and> set is = {LI dir s (lhs eq)} \<union> {LI dir s x |x. x \<in> rvars_eq eq \<and> coeff (rhs eq) x < 0} \<union> {UI dir s x |x. x \<in> rvars_eq eq \<and> 0 < coeff (rhs eq) x} \<and> (dir = Positive \<or> dir = Negative \<longrightarrow> LI dir s (lhs eq) \<in> indices_state s \<longrightarrow> set is \<subseteq> indices_state s)"
gen_contains_key_by_lookup_def: fixes is_map :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'c \<Rightarrow> assn"   and lookup :: "'a \<Rightarrow> 'c \<Rightarrow> 'b option Heap" shows "gen_contains_key_by_lookup is_map lookup \<equiv> imp_map_lookup is_map lookup"
lset_conv_lnth: fixes xs :: "'e llist" shows "lset xs = {lnth xs n |n. enat n < llength xs}"
Gromov_product_at_diff1: fixes a :: "'b"   and x :: "'b"   and y :: "'b"   and b :: "'b" shows "\<bar>Gromov_product_at a x y - Gromov_product_at b x y\<bar> \<le> dist a b"
is_pureE: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> assn" assumes "is_pure P" obtains P' :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where "\<And>x x'. P x x' = \<up> (P' x x')"
not_HFinite_HInfinite: fixes x :: "'a star" assumes "x \<notin> HFinite" shows "x \<in> HInfinite"
supp: fixes G :: "'a set"   and T :: "'a \<Rightarrow> 'b" assumes "GroupHom G T" shows "order.greater_eq G (supp T)"
GuardFault: fixes s :: "'d"   and g :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and f :: "'f"   and c :: "('d, 'e, 'f) com" assumes "s \<notin> g" shows "\<Gamma>\<turnstile> \<langle>Guard f g c,Normal s\<rangle> \<Rightarrow> Fault f"
preserves_cod: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "cod (\<F>C.D f) = \<F>C.D (\<F>C.cod f)"
cdp: shows "(\<alpha>\<^sup>2 + 3 * \<beta>\<^sup>2) ^ 3 = c\<^sup>2 + 3 * d\<^sup>2 \<and> coprime c (3 * d)"
strip_comb_simps_1: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "'b"   and u :: "'b" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "term.strip_comb unapp (app t u) = (let (f, args) = term.strip_comb unapp t in (f, args @ [u]))"
ll2f_Suc_Some: fixes t :: "'b llist"   and i :: "nat"   and x :: "'b" assumes "t !! Suc i = Some x" obtains x :: "'b" where "t !! i = Some x"
TunerMode_CTRL_States_def: shows "TunerMode_CTRL_States = {''1'', ''2'', ''3'', ''4''}"
finvimage_of_vempty: fixes r :: "V" shows "r -`\<^sub>\<bullet> []\<^sub>\<circ> = []\<^sub>\<circ>"
simps: fixes s :: "nat"   and l :: "cell list"   and r :: "cell list"   and p :: "(action \<times> nat) list"   and off :: "nat" shows "step (s, l, r) (p, off) = (let (a, s') = fetch p (s - off) (read r) in (s', update a (l, r)))"
mQmpTr0: fixes H :: "'a set"   and N :: "'a set"   and m :: "'a" assumes "submodule R M H"   and "submodule R M N"   and "H \<subseteq> N"   and "m \<in> carrier M" shows "(Mp\<^bsub>M  H,N\<^esub>) (m \<uplus>\<^bsub>M\<^esub> H) = m \<uplus>\<^bsub>M\<^esub> N"
apply_adaptation_def: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and first :: "bool"   and mem :: "'b \<Rightarrow> 'c"   and A :: "'b \<Rightarrow> ('c \<times> 'c) option" assumes "sifum_security stop eval some_val some_val'" shows "sifum_security.apply_adaptation first mem A = (\<lambda>x. case A x of None \<Rightarrow> mem x | Some (v\<^sub>1, v\<^sub>2) \<Rightarrow> if first then v\<^sub>1 else v\<^sub>2)"
nres_rel_mono: fixes A :: "('a \<times> 'b) set"   and B :: "('a \<times> 'b) set" assumes "A \<subseteq> B" shows "\<langle>A\<rangle>nres_rel \<subseteq> \<langle>B\<rangle>nres_rel"
gen_nth0: fixes B :: "nat"   and n :: "nat" shows "gen B n ! 0 = zeroes n"
max_repU2: fixes w1 :: "32 Word.word"   and y :: "real"   and w2 :: "32 Word.word"   and x :: "real" assumes "w1 \<equiv>\<^sub>U y"   and "w2 \<equiv>\<^sub>U x" shows "wmax w1 w2 \<equiv>\<^sub>U x"
vwalk_join_def2: fixes p :: "'a list"   and q :: "'a list" assumes "joinable p q" shows "p \<oplus> q = butlast p @ q"
forall: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "\<forall>x. P x" shows "P x"
ex_unique_moebius_three_points: fixes z1 :: "complex_homo"   and z2 :: "complex_homo"   and z3 :: "complex_homo"   and w1 :: "complex_homo"   and w2 :: "complex_homo"   and w3 :: "complex_homo" assumes "z1 \<noteq> z2"   and "z1 \<noteq> z3"   and "z2 \<noteq> z3"   and "w1 \<noteq> w2"   and "w1 \<noteq> w3"   and "w2 \<noteq> w3" shows "\<exists>!M. moebius_pt M z1 = w1 \<and> moebius_pt M z2 = w2 \<and> moebius_pt M z3 = w3"
lfinite_LCons: fixes x :: "'c"   and xs :: "'c llist" shows "lfinite (LCons x xs) = lfinite xs"
ideal_condition: fixes R :: "('e, 'f) Ring_scheme"   and I :: "'e set" assumes "Ring R"   and "I \<subseteq> carrier R"   and "I \<noteq> {}"   and "\<forall>x\<in>I. \<forall>y\<in>I. x \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> y \<in> I"   and "\<forall>r\<in>carrier R. \<forall>x\<in>I. r \<cdot>\<^sub>r\<^bsub>R\<^esub> x \<in> I" shows "ideal R I"
path0: fixes l1 :: "'loc"   and l2 :: "'loc" assumes "l1 = l2" shows "path l1 l2 []"
HSem_restr: fixes h :: "('var \<times> 'exp) list"   and \<rho> :: "'var \<Rightarrow> 'value" shows "\<lbrace> h \<rbrace>\<rho> f|` (- domA h) = \<lbrace> h \<rbrace>\<rho>"
struct_spec_red_supset: fixes sel :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ap :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ab :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and sps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data' :: "nat \<times> 'g" assumes "struct_spec sel ap ab compl"   and "ps \<noteq> []"   and "sps = sel gs bs ps data"   and "hs = fst (add_indices (compl gs bs (ps -- sps) sps data) data)"   and "hs \<noteq> []" shows "fst ` set (ab gs bs hs data') \<sqsupset>p fst ` set bs"
mkArr_Fun: fixes f :: "'s" assumes "S.arr f" shows "S.mkArr (S.Dom f) (S.Cod f) (S.Fun f) = f"
asymptotics4: fixes x :: "real" assumes "x\<^sub>0 \<le> x" shows "1 \<le> (1 - ln x powr (- e / 2)) * 2"
impl_def: shows "(\<Rightarrow>) \<equiv> map_fun Rep_uexpr (map_fun Rep_uexpr mk\<^sub>e) (\<lambda>P Q A. P A \<longrightarrow> Q A)"
implies_antitone: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "y \<leadsto> z \<le> x \<leadsto> z"
pnet_maintains_dom: fixes s :: "'c net_state"   and a :: "'d node_action"   and s' :: "'c net_state"   and np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and p :: "net_tree" assumes "(s, a, s') \<in> automaton.trans (pnet np p)" shows "net_ips s = net_ips s'"
transfer_arrive': fixes np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and onp :: "nat \<Rightarrow> ((nat \<Rightarrow> 'e) \<times> 'f, 'd seq_action) automaton"   and sr :: "'c \<Rightarrow> 'e \<times> 'f"   and s :: "'c net_state"   and H :: "nat set"   and K :: "nat set"   and m :: "'d"   and s' :: "'c net_state"   and n :: "net_tree"   and \<sigma> :: "nat \<Rightarrow> 'e"   and \<zeta> :: "'f net_state"   and \<sigma>' :: "nat \<Rightarrow> 'e"   and \<zeta>' :: "'f net_state" assumes "openproc np onp sr"   and "(s, H\<not>K:arrive(m), s') \<in> automaton.trans (pnet np n)"   and "s \<in> reachable (pnet np n) TT"   and "netgmap sr s = netmask (net_tree_ips n) (\<sigma>, \<zeta>)"   and "netgmap sr s' = netmask (net_tree_ips n) (\<sigma>', \<zeta>')"   and "wf_net_tree n" shows "((\<sigma>, \<zeta>), H\<not>K:arrive(m), \<sigma>', \<zeta>') \<in> automaton.trans (opnet onp n)"
true_dsij_zero: fixes P :: "'b \<Rightarrow> bool" shows "(P \<or> true) = true"
norm_unsigned_idem: fixes w :: "bit list" shows "norm_unsigned (norm_unsigned w) = norm_unsigned w"
fv_list_is_fv: fixes t :: "('d, 'c) Term.term" shows "fv t = set (fv_list t)"
int_of_uint32_plus: fixes x :: "uint32"   and y :: "uint32" shows "int_of_uint32 (x + y) = (int_of_uint32 x + int_of_uint32 y) mod 4294967296"
different_arc_in_sup_arc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "bijective (x * top) \<and> bijective (x[[top]])"   and "bijective (y * top) \<and> bijective (y[[top]])"   and "x \<le> z \<squnion> y"   and "x \<noteq> y" shows "x \<le> z"
hyps_6: fixes left :: "(('b, 'a) fm list \<times> 'a) list"   and right :: "(('b, 'a) fm list \<times> 'a) list" assumes "(p # ps, a) # branch = left @ right" shows "A, Suc m \<turnstile> filter nonempty left @ right"
map_iterator_genord_product_correct: fixes it_a :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and m :: "'e \<Rightarrow> 'f option"   and R_a :: "'e \<times> 'f \<Rightarrow> 'e \<times> 'f \<Rightarrow> bool"   and it_b :: "'f \<Rightarrow> ('d \<Rightarrow> bool) \<Rightarrow> ('g \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S_b :: "'f \<Rightarrow> 'g set"   and R_b :: "'f \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> bool"   and R_ab :: "'e \<times> 'g \<Rightarrow> 'e \<times> 'g \<Rightarrow> bool" assumes "map_iterator_genord it_a m R_a"   and "\<And>k v. m k = Some v \<Longrightarrow> set_iterator_genord (it_b v) (S_b v) (R_b v)"   and "\<And>k v u k' v' u'. \<lbrakk>m k = Some v; u \<in> S_b v; m k' = Some v'; u' \<in> S_b v'; if k = k' then R_b v u u' else R_a (k, v) (k', v')\<rbrakk> \<Longrightarrow> R_ab (k, u) (k', u')" shows "set_iterator_genord (map_iterator_product it_a it_b) {(k, e). \<exists>v. m k = Some v \<and> e \<in> S_b v} R_ab"
chine_simps_2: shows "C.dom assoc\<nu>\<pi>\<rho>.chine = assoc\<nu>\<pi>\<rho>.dom.apex"
d1_weak: fixes x :: "'a"   and y :: "'a" shows "(apd_d x + apd_d y) \<cdot> x = x"
star_Arbitrary_ExceptD: fixes v :: "('a \<times> bool list) list"   and n :: "nat"   and r :: "nat"   and b :: "bool"   and i :: "nat" assumes "v \<in> star (lang n (Atom (Arbitrary_Except r b)))"   and "r < n"   and "i < length v" shows "snd (v ! i) ! r = b"
admits_triangular_reduction_def: fixes A :: "'b mat" shows "admits_triangular_reduction A = (\<exists>U. U \<in> carrier_mat (dim_col A) (dim_col A) \<and> invertible_mat U \<and> lower_triangular (A * U))"
star_of_of_real: fixes r :: "real" shows "star_of (of_real r) = of_real r"
valid_path_imp_path: fixes g :: "real \<Rightarrow> 'b" assumes "valid_path g" shows "path g"
l_inv_ex: fixes x :: "'c" assumes "x \<in> carrier H" shows "\<exists>y\<in>carrier H. y \<oplus>\<^bsub>H\<^esub> x = \<zero>\<^bsub>H\<^esub>"
stable_rank_def: fixes x :: "nat"   and i :: "nat" shows "stable_rank x i = (\<forall>\<^sub>\<infinity>n. rank x n = Some i)"
frac_le_self: fixes x :: "'b" assumes "(0::'b) \<le> x" shows "frac x \<le> x"
add_left: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and a :: "'a"   and a' :: "'a"   and b :: "'b" assumes "bounded_bilinear prod" shows "prod (a + a') b = prod a b + prod a' b"
measurable_const: fixes c :: "'c"   and M' :: "'c measure"   and M :: "'d measure" assumes "c \<in> space M'" shows "(\<lambda>x. c) \<in> M \<rightarrow>\<^sub>M M'"
comp_in_homE: fixes g :: "'c"   and f :: "'c"   and a :: "'c"   and c :: "'c" assumes "\<guillemotleft>g \<cdot> f : a \<rightarrow> c\<guillemotright>" obtains b :: "'c" where "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" and "\<guillemotleft>g : b \<rightarrow> c\<guillemotright>"
vector_space_with_def: fixes plus\<^sub>V\<^sub>S :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero\<^sub>V\<^sub>S :: "'c"   and minus\<^sub>V\<^sub>S :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus\<^sub>V\<^sub>S :: "'c \<Rightarrow> 'c"   and scale :: "'d \<Rightarrow> 'c \<Rightarrow> 'c" shows "vector_space_with plus\<^sub>V\<^sub>S zero\<^sub>V\<^sub>S minus\<^sub>V\<^sub>S uminus\<^sub>V\<^sub>S scale \<equiv> class.ab_group_add plus\<^sub>V\<^sub>S zero\<^sub>V\<^sub>S minus\<^sub>V\<^sub>S uminus\<^sub>V\<^sub>S \<and> vector_space_with_axioms plus\<^sub>V\<^sub>S scale"
nn_integral_cmult: fixes f :: "'d \<Rightarrow> ennreal"   and M :: "'d measure"   and c :: "ennreal" assumes "f \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. c * f x \<partial>M = c * integral\<^sup>N M f"
WT_implies_WTrt: fixes P :: "(char list list \<times> char list exp) prog"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and T :: "ty"   and h :: "heap"   and sh :: "sheap" assumes "P,E \<turnstile> e :: T" shows "P,E,h,sh \<turnstile> e : T"
match_mono: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and \<phi> :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and f :: "('a, 'b, 'c) trans \<Rightarrow> 'd"   and \<gamma> :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and g :: "('a, 'b, 'c) trans \<Rightarrow> 'e"   and \<Delta> :: "'a \<Rightarrow> 'f \<Rightarrow> 'a \<Rightarrow> 'd list \<Rightarrow> bool"   and s :: "'a"   and s1 :: "'a"   and vl1 :: "'d list"   and a :: "'b"   and ou :: "'c"   and s' :: "'a"   and vl' :: "'f"   and \<Delta>' :: "'a \<Rightarrow> 'f \<Rightarrow> 'a \<Rightarrow> 'd list \<Rightarrow> bool" assumes "BD_Security.match step \<phi> f \<gamma> g \<Delta> s s1 vl1 a ou s' vl'"   and "\<And>s vl s1 vl1. \<Delta> s vl s1 vl1 \<Longrightarrow> \<Delta>' s vl s1 vl1" shows "BD_Security.match step \<phi> f \<gamma> g \<Delta>' s s1 vl1 a ou s' vl'"
ctr_transfer_1: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_option R None None"
numbers_of_marks_def: fixes n :: "nat"   and bs :: "bool list" shows "numbers_of_marks n bs = fst ` {x \<in> set (enumerate n bs). snd x}"
P1: shows "Col T1 A B \<and> Bet C T1 D"
enum_DF_le_iff: fixes j :: "nat"   and k :: "nat"   and i :: "nat"   and i' :: "nat" assumes "j \<le> k" shows "(enum (DF k i) j \<le> enum (DF k i') j) = (i \<le> i')"
set_bd_2: fixes x :: "('l, 'm, 'k) F2" shows "|F2set2 x| \<le>o bd_F2"