hcju/isabelleps huggingface dataset:
F_def: fixes K :: "'a \<Rightarrow> 'a pmf"   and s :: "'a"   and t :: "'a" shows "MC_syntax.F K s t = \<P>(\<omega> in MC_syntax.T K s. ev (HLD {t}) (s ## \<omega>))"
qGoodInp_iff_goodInp_asInp: fixes qinp :: "'a \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm option" shows "goodInp (asInp qinp) = qGoodInp qinp"
cl_inI: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and \<E> :: "'b set"   and X :: "'b set"   and x :: "'b" assumes "matroid carrier indep"   and "\<E> \<subseteq> carrier"   and "X \<subseteq> \<E>"   and "x \<in> \<E>"   and "indep_system.lower_rank_of indep (insert x X) = indep_system.lower_rank_of indep X" shows "x \<in> matroid.cl_in indep \<E> X"
z: fixes z :: "ereal" assumes "0 < z"   and "z < 1" shows "z * x \<le> y"
simps: fixes c :: "char"   and s :: "char list" shows "string_cons c s = (if MAX_STR \<le> length s then s else c # s)"
assms_1: shows "cdens_ctxt_invar vs vs' \<Gamma> \<delta>"
ewp_alt1: fixes x :: "'a" shows "(\<forall>z. x\<^sup>\<omega> \<le> x\<^sup>\<star> \<cdot> z) = (\<forall>y z. y \<le> x \<cdot> y + z \<longrightarrow> y \<le> x\<^sup>\<star> \<cdot> z)"
prj_in_hom_1: shows "\<guillemotleft>\<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>1\<^sub>1 : HH\<mu>H\<nu>\<pi>\<rho>.chine \<rightarrow>\<^sub>C \<mu>.apex\<guillemotright>"
select_convs_2: fixes committed :: "nat set"   and justifying_exec :: "('a \<times> ('b, 'a) obs_event action) llist"   and justifying_ws :: "nat \<Rightarrow> nat"   and more :: "'c" shows "justifying_exec \<lparr>committed = committed, justifying_exec = justifying_exec, justifying_ws = justifying_ws, \<dots> = more\<rparr> = justifying_exec"
reachable_def: fixes step :: "'b \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b"   and s' :: "'b" shows "LTS.reachable step s s' \<equiv> \<exists>p. LTS.path step s p s'"
rec_right_def: shows "rec_right = Cn 1 rec_lo [recf.id 1 0, constn (Pi 2)]"
forall\<^sub>0_def: shows "forall\<^sub>0 \<equiv> map_fun (map_fun id eval\<o>) make\<o> (\<lambda>\<phi> s w. \<forall>x. \<phi> x s w)"
casI_nth: fixes p :: "'b list"   and u :: "'a"   and v :: "'a" assumes "p \<noteq> []"   and "u = tail G (hd p)"   and "v = head G (last p)"   and "\<And>i. Suc i < length p \<Longrightarrow> head G (p ! i) = tail G (p ! Suc i)" shows "cas u p v"
spanE: fixes p :: "'c"   and B :: "'c set" assumes "p \<in> ideal B" obtains A :: "'c set" and q :: "'c \<Rightarrow> 'c" where "finite A" and "A \<subseteq> B" and "p = (\<Sum>b\<in>A. q b * b)"
weakCongSimulation_def: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and \<Psi> :: "'e"   and P :: "('d, 'e, 'f) psi"   and Rel :: "('e \<times> ('d, 'e, 'f) psi \<times> ('d, 'e, 'f) psi) set"   and Q :: "('d, 'e, 'f) psi" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'" shows "env.weakCongSimulation substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> P Rel Q \<equiv> \<forall>Q'. env.semantics substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> Q (\<tau> \<prec> Q') \<longrightarrow> (\<exists>P'. (P, P') \<in> {(P, P'). env.semantics substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> P (\<tau> \<prec> P')}\<^sup>+ \<and> (\<Psi>, P', Q') \<in> Rel)"
orthogonal_comp_UNIV: shows "UNIV\<^sup>\<bottom> = {0::'a}"
zero_val_gen_whole: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier (Vr K v)" shows "(v x = 0) = (Vr K v \<diamondsuit>\<^sub>p x = carrier (Vr K v))"
Cauchy_Schwarz_ineq2: fixes x :: "'a"   and y :: "'a" shows "cmod (x \<bullet>\<^sub>C y) \<le> norm x * norm y"
pmf_integral_code_unfold: fixes p :: "'d pmf" shows "pmf_integral p = measure_pmf.expectation p"
elims: fixes x :: "int poly"   and xa :: "int poly"   and xb :: "(rat \<times> rat \<times> rat) \<times> rat \<times> rat \<times> rat"   and y :: "(rat \<times> rat \<times> rat) \<times> rat \<times> rat \<times> rat" assumes "tighten_poly_bounds_binary x xa xb = y" obtains cr1 :: "int poly" and cr2 :: "int poly" and l1 :: "rat" and r1 :: "rat" and sr1 :: "rat" and l2 :: "rat" and r2 :: "rat" and sr2 :: "rat" where "x = cr1" and "xa = cr2" and "xb = ((l1, r1, sr1), l2, r2, sr2)" and "y = (tighten_poly_bounds cr1 l1 r1 sr1, tighten_poly_bounds cr2 l2 r2 sr2)"
upper_mult_float_interval: fixes p :: "nat"   and x :: "float interval"   and y :: "float interval" shows "upper (mult_float_interval p x y) = snd (bnds_mult p (lower x) (upper x) (lower y) (upper y))"
cond_wf_Elem: fixes U :: "ZF"   and P :: "ZF \<Rightarrow> bool"   and a :: "ZF" assumes "\<forall>x. (\<forall>y. Elem y x \<longrightarrow> Elem y U \<longrightarrow> P y) \<longrightarrow> Elem x U \<longrightarrow> P x"   and "Elem a U" shows "P a"
csum_cong1: fixes p1 :: "('d \<times> 'd) set"   and r1 :: "('e \<times> 'e) set"   and q :: "('f \<times> 'f) set" assumes "p1 =o r1" shows "p1 +c q =o r1 +c q"
InlI: fixes a :: "'a"   and A :: "'a set"   and B :: "'b set" assumes "a \<in> A" shows "Inl a \<in> A <+> B"
assms_1: shows "e' mod order \<G> < e mod order \<G>"
nth_Cons_Suc: fixes x :: "'k"   and xs :: "'k list"   and n :: "nat" shows "(x # xs) ! Suc n = xs ! n"
absolutely_integrable_mult_Dirichlet_kernel_reflected_part2_2: fixes f :: "real \<Rightarrow> real"   and d :: "real"   and n :: "nat"   and t :: "real"   and c :: "real" assumes "f absolutely_integrable_on {- pi..pi}"   and "\<And>x. f (x + 2 * pi) = f x"   and "d \<le> pi" shows "(\<lambda>x. Dirichlet_kernel n x * (f (t + x) + f (t - x) - c)) absolutely_integrable_on {0..d}"
FF_def: shows "H.FF \<equiv> \<lambda>\<mu>\<nu>. if B.VV.arr \<mu>\<nu> then (H (fst \<mu>\<nu>), H (snd \<mu>\<nu>)) else C.VV.null"
pr_simps_2: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "local.dom (pr0 a b) = local.prod a b"
pwd_owner_ii_ex: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "pred_owner_ii n (pwd_spy_i n) (pwd_owner_ii n)"
codomains_char: fixes f :: "Arr" shows "codomains f = (if arr f then {cod f} else {})"
axioms_2: fixes \<eta> :: "'b \<Rightarrow> 'c"   and M :: "'b set"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit :: "'b"   and M' :: "'c set"   and composition' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit' :: "'c" assumes "monoid_homomorphism \<eta> M composition unit M' composition' unit'" shows "Group_Theory.monoid M composition unit"
jump_eq: shows "jumpF_pathstart (linepath a b) z = jumpF (\<lambda>t. f t / g t) (at_right 0)"
d_def: shows "d \<equiv> Max {degree (coeff p i) |i. i \<le> degree p}"
run_one_step_return: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and e :: "e"   and s' :: "s"   and vs' :: "v list"   and res :: "v list" assumes "run_one_step d i (s, vs, ves, e) = (s', vs', RSReturn res)" shows "e = $Return \<or> (\<exists>n les es. e = Label n les es)"
length_ins_sws: fixes k :: "'b \<Rightarrow> 'c"   and x :: "'b"   and xs :: "'b list" shows "length (ins_sws k x xs) \<le> length xs"
Rep_stone_phi_pair_cases: fixes y :: "'c regular \<times> 'c dense Filters.filter" assumes "y \<in> stone_phi.pairs" obtains x :: "'c stone_phi_pair" where "y = Rep_stone_phi_pair x"
fst_netgmap_netlift: fixes sr :: "'c \<Rightarrow> 'a \<times> 'b"   and s :: "'c net_state" shows "fst (netgmap sr s) = netlift sr s"
ind_sheaf_axioms_def: fixes is_open :: "'d set \<Rightarrow> bool"   and U :: "'d set" shows "ind_sheaf_axioms is_open U \<equiv> is_open U"
collapse_1: fixes resumption :: "('f, 'g, 'h) Resumption.resumption" assumes "Resumption.resumption.is_Done resumption" shows "Resumption.resumption.Done (Resumption.resumption.result resumption) = resumption"
cases: fixes a1 :: "'b \<Rightarrow> 'c"   and a2 :: "'b option"   and a3 :: "'b option"   and a4 :: "'c DBMEntry" assumes "dbm_entry_val a1 a2 a3 a4" obtains u :: "'b \<Rightarrow> 'c" and r :: "'b" and d :: "'c" where "a1 = u__" and "a2 = Some r__" and "a3 = None" and "a4 = Le d__" and "u__ r__ \<le> d__"   | u :: "'b \<Rightarrow> 'c" and c :: "'b" and d :: "'c" where "a1 = ua__" and "a2 = None" and "a3 = Some c__" and "a4 = Le da__" and "- ua__ c__ \<le> da__"   | u :: "'b \<Rightarrow> 'c" and r :: "'b" and d :: "'c" where "a1 = ub__" and "a2 = Some ra__" and "a3 = None" and "a4 = Lt db__" and "ub__ ra__ < db__"   | u :: "'b \<Rightarrow> 'c" and c :: "'b" and d :: "'c" where "a1 = uc__" and "a2 = None" and "a3 = Some ca__" and "a4 = Lt dc__" and "- uc__ ca__ < dc__"   | u :: "'b \<Rightarrow> 'c" and r :: "'b" and c :: "'b" and d :: "'c" where "a1 = ud__" and "a2 = Some rb__" and "a3 = Some cb__" and "a4 = Le dd__" and "ud__ rb__ - ud__ cb__ \<le> dd__"   | u :: "'b \<Rightarrow> 'c" and r :: "'b" and c :: "'b" and d :: "'c" where "a1 = u" and "a2 = Some r" and "a3 = Some c" and "a4 = Lt d" and "u r - u c < d"   | uu_ :: "'b \<Rightarrow> 'c" and uv_ :: "'b option" and uw_ :: "'b option" where "a1 = uu_" and "a2 = uv_" and "a3 = uw_" and "a4 = \<infinity>"
igSubstBinp_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'b, 'j) model_scheme"   and ys :: "'e"   and Y :: "'i"   and y :: "'h"   and binp :: "'a \<Rightarrow> 'b option" shows "igSubstBinp MOD ys Y y binp \<equiv> lift (igSubstAbs MOD ys Y y) binp"
borel_measurable_log: fixes f :: "'d \<Rightarrow> real"   and M :: "'d measure"   and g :: "'d \<Rightarrow> real" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. log (g x) (f x)) \<in> borel_measurable M"
poly_subst_uminus: fixes f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "poly_subst f (- p) = - poly_subst f p"
GEN_OP_D: fixes c :: "'a"   and a :: "'b"   and R :: "('a \<times> 'b) set" assumes "GEN_OP c a R" shows "(c, a) \<in> R"
sup_left_commute: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "semilattice_sup sup less_eq less" shows "sup x (sup y z) = sup y (sup x z)"
subst_Var_simp2: fixes Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort"   and x :: "'var" assumes "good Y" shows "Var xs x #[Y / x]_xs = Y"
nat_one_le_power: fixes i :: "nat"   and n :: "nat" assumes "Suc 0 \<le> i" shows "Suc 0 \<le> i ^ n"
x_in_trapC: shows "(2, 0) \<in> trapC"
Sigma_Diff_distrib1: fixes I :: "'a set"   and J :: "'a set"   and C :: "'a \<Rightarrow> 'b set" shows "Sigma (I - J) C = Sigma I C - Sigma J C"
less_irrefl: fixes x :: "'a word" shows "\<not> x <s x"
Int_Diff: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and B :: "'b set" assumes "finite A" shows "sum g A = sum g (A \<inter> B) + sum g (A - B)"
val_simps_7: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b tm"   and prod :: "'c \<times> 'd" shows "Time_Monad.val (case prod of (x, xa) \<Rightarrow> f x xa) = (case prod of (x1, x2) \<Rightarrow> Time_Monad.val (f x1 x2))"
graph_Entry_axioms_def: fixes \<alpha>n :: "'d \<Rightarrow> 'e list"   and invar :: "'d \<Rightarrow> bool"   and inEdges' :: "'d \<Rightarrow> 'e \<Rightarrow> ('e \<times> 'f) list"   and Entry :: "'d \<Rightarrow> 'e" shows "graph_Entry_axioms \<alpha>n invar inEdges' Entry \<equiv> (\<forall>g. Entry g \<in> set (\<alpha>n g)) \<and> (\<forall>g. invar g \<longrightarrow> graph_path_base.inEdges inEdges' g (Entry g) = []) \<and> (\<forall>n g. n \<in> set (\<alpha>n g) \<longrightarrow> invar g \<longrightarrow> (\<exists>ns. graph_path_base.path2 \<alpha>n invar inEdges' g (Entry g) ns n))"
tendsto_minus_cancel_right: fixes g :: "'b \<Rightarrow> 'c"   and l :: "'c"   and F :: "'b filter" shows "((\<lambda>x. - g x) \<longlongrightarrow> l) F = (g \<longlongrightarrow> - l) F"
negative_x_axis_def: shows "negative_x_axis = {z \<in> circline_set x_axis. z \<noteq> \<infinity>\<^sub>h \<and> Re (to_complex z) < 0}"
add_rat_unique_root: fixes m :: "rat"   and p :: "int poly"   and l :: "rat"   and r :: "rat" shows "(\<exists>!x. root_cond (cf_pos_poly (poly_add_rat m p), l, r) x) = (\<exists>!x. root_cond (p, l - m, r - m) x)"
lookup_alt_def: fixes x :: "'b" shows "RFI_lookup x = ask_env (\<lambda>s. case apply s x of None \<Rightarrow> fail_env (fail_option return_id) | Some y \<Rightarrow> return_env (return_option return_id) y)"
absorb_iff2: fixes b :: "'a"   and a :: "'a" shows "(b \<le> a) = (min a b = b)"
as: shows "wordinterval_CIDR_split1 r = (Some s, u)"
mbuf2t_take_add'_2: fixes f :: "event_data option list set \<Rightarrow> event_data option list set \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and xs :: "event_data option list set list"   and ys :: "event_data option list set list"   and buf :: "event_data option list set list \<times> event_data option list set list"   and nts :: "nat list"   and z' :: "'a"   and buf' :: "event_data option list set list \<times> event_data option list set list"   and nts' :: "nat list"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and j' :: "nat"   and P' :: "String.literal \<Rightarrow> nat option"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and \<phi> :: "Formula.formula"   and \<psi> :: "Formula.formula" assumes "mbuf2t_take f z (mbuf2_add xs ys buf) nts = (z', buf', nts')"   and "pred_mapping (\<lambda>x. x \<le> j) P"   and "pred_mapping (\<lambda>x. x \<le> j') P'"   and "rel_mapping (\<le>) P P'"   and "wf_mbuf2' \<sigma> P V j n R \<phi> \<psi> buf"   and "list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [min (Monitor.progress \<sigma> P \<phi> j) (Monitor.progress \<sigma> P \<psi> j)..<j'] nts"   and "list_all2 (\<lambda>i. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<phi>)) [Monitor.progress \<sigma> P \<phi> j..<Monitor.progress \<sigma> P' \<phi> j'] xs"   and "list_all2 (\<lambda>i. qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<psi>)) [Monitor.progress \<sigma> P \<psi> j..<Monitor.progress \<sigma> P' \<psi> j'] ys"   and "j \<le> j'" shows "wf_ts \<sigma> P' j' \<phi> \<psi> nts'"
\<Z>_\<omega>_\<alpha>\<omega>: shows "\<omega> \<in>\<^sub>\<circ> \<alpha> + \<omega> + \<omega>"
le_\<I>_def: fixes \<I> :: "('y, 'z) \<I>"   and \<I>' :: "('y, 'z) \<I>" shows "(\<I> \<le> \<I>') = (outs_\<I> \<I> \<subseteq> outs_\<I> \<I>' \<and> (\<forall>x\<in>outs_\<I> \<I>. responses_\<I> \<I>' x \<subseteq> responses_\<I> \<I> x))"
zeros: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool" assumes "\<forall>n'\<ge>n. \<not> f n'" shows "wf_set_bits_int f"
abs_if: fixes abs :: "'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.abs_if abs uminus zero less" shows "abs a = (if less a zero then uminus a else a)"
aux_split_2: shows "case hd_aux of (t, a1, a2) \<Rightarrow> (t, a1, a2) = (ts, filter_a1_map pos (tp - len) a1_map, filter_a2_map I ts (tp - len) a2_map)"
Sup_prod_def: fixes A :: "('c \<times> 'd) set" shows "Sup A = (Sup (fst ` A), Sup (snd ` A))"
pcompose_altdef: fixes p :: "'b poly"   and q :: "'b poly" shows "p \<circ>\<^sub>p q = poly (map_poly (\<lambda>x. [:x:]) p) q"
of_real_eq_numeral_power_int_cancel_iff: fixes y :: "real"   and x :: "num"   and n :: "int" shows "(of_real y = numeral x powi n) = (y = numeral x powi n)"
cone_mono: fixes Vs :: "'a vec set"   and Ws :: "'a vec set" assumes "Vs \<subseteq> Ws" shows "next_dim.cone Vs \<subseteq> next_dim.cone Ws"
set_q_a: shows "set (map ((\<otimes>) a) q @ replicate (length p) \<zero>) \<subseteq> carrier R"
exhaust: fixes y :: "('d, 'e, 'f) residual_Rep" obtains x11 :: "'d" and x12 :: "'d" and x13 :: "('d, 'e, 'f) psi" where "y = RIn_Rep x11 x12 x13"   | x21 :: "'d" and x22 :: "('d, 'e, 'f) boundOutput" where "y = ROut_Rep x21 x22"   | x3 :: "('d, 'e, 'f) psi" where "y = RTau_Rep x3"
le_supI2: fixes x :: "'c multiset"   and b :: "'c multiset"   and a :: "'c multiset" assumes "x \<subseteq># b" shows "x \<subseteq># a \<union># b"
has_domain_iff_arr: fixes f :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" shows "(Cop_S.domains f \<noteq> {}) = Cop_S.arr f"
assms: fixes i :: "'a" shows "Rep_ell2 x i = 0 \<or> Rep_ell2 y i = 0"
forall_ex_fun: fixes S :: "'a set"   and P :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "finite S" shows "(\<forall>x\<in>S. \<exists>y. P y x) \<longrightarrow> (\<exists>f. \<forall>x\<in>S. P (f x) x)"
insert_in_sets: fixes x :: "'a"   and M :: "'a measure"   and A :: "'a set" assumes "{x} \<in> sets M"   and "A \<in> sets M" shows "insert x A \<in> sets M"
index_row_1: fixes i :: "nat"   and A :: "'a mat"   and j :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "row A i $ j = A $$ (i, j)"
diff: fixes x :: "real"   and y :: "real"   and xa :: "'a" shows "(x - y) *\<^sub>R xa = x *\<^sub>R xa - y *\<^sub>R xa"
lower_set_imp_not_SN_on: fixes s :: "'a"   and X :: "'a set"   and R :: "('a \<times> 'a) set" assumes "s \<in> X"   and "\<forall>t\<in>X. \<exists>u\<in>X. (t, u) \<in> R" shows "\<not> SN_on R {s}"
two_D_integral_to_one_D: shows "integral UNIV f = integral UNIV (\<lambda>x. integral UNIV (\<lambda>y. f (x, y)))"
length_list_update: fixes xs :: "'e list"   and i :: "nat"   and x :: "'e" shows "length (xs[i := x]) = length xs"
sum_mult_scalar_distrib_left: fixes r :: "'a \<Rightarrow>\<^sub>0 'c"   and f :: "'d \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'c"   and A :: "'d set" shows "local.punit.mult_scalar r (sum f A) = (\<Sum>a\<in>A. local.punit.mult_scalar r (f a))"
Fail: fixes m :: "nat"   and x :: "nat" assumes "n \<le> m"   and "x \<le> m" shows "sink (token_run x m) \<or> \<not> sink (token_run x (Suc m)) \<or> \<not> token_run x (Suc m) \<notin> F"
neighborhoods_def: fixes x :: "'a" shows "neighborhoods x \<equiv> {U. is_open\<^sub>X U \<and> x \<in> U}"
complex_mod_cnj: fixes z :: "complex" shows "cmod (cnj z) = cmod z"
vector_conv_covector: fixes v :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_bounded_idempotent_semiring.vector v = matrix_bounded_idempotent_semiring.covector (v\<^sup>t)"
list_mono: fixes A :: "('b, unit) Old_Datatype.node set set"   and B :: "('b, unit) Old_Datatype.node set set" assumes "A \<subseteq> B" shows "list A \<subseteq> list B"
lhd_subset_Sup_llist: fixes Xs :: "'a set llist" assumes "\<not> lnull Xs" shows "lhd Xs \<subseteq> Sup_llist Xs"
step_def: fixes A :: "'a \<times> ('b \<Rightarrow> 'a \<Rightarrow> 'a set) \<times> ('a \<Rightarrow> bool)"   and a :: "'b" shows "step A a = {(p, q). q \<in> next A a p}"
core_nhds: fixes U :: "'b set"   and T :: "'b set set"   and x :: "'b" assumes "U \<subseteq> carrier\<^bsub>T\<^esub>"   and "x \<in> core\<^bsub>T\<^esub> U" shows "U \<in> nhds\<^bsub>T\<^esub> x"
hermitian_is_normal: fixes A :: "'c mat" assumes "hermitian A" shows "A * adjoint A = adjoint A * A"
star_inductl_var: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and star :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.left_near_kleene_algebra plus times one less_eq less star"   and "less_eq (times x y) y" shows "less_eq (times (star x) y) y"
trms_of_deg_leq_zero_is_ctrm: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P"   and "0 < deg R f" shows "trms_of_deg_leq 0 f = monom P (f 0) 0"
inject_2: fixes x2 :: "('c, 'd) Term.term"   and y2 :: "('c, 'd) Term.term" shows "(receive\<langle>x2\<rangle>\<^sub>s\<^sub>t = receive\<langle>y2\<rangle>\<^sub>s\<^sub>t) = (x2 = y2)"
1: shows "[(p + 1) ^ n = p + 1] (mod p\<^sup>2)"
berlekamp_resulting_mat_def: fixes u :: "'a mod_ring poly" shows "berlekamp_resulting_mat u = (let Q = berlekamp_mat u; n = dim_row Q; QI = mat n n (\<lambda>(i, j). if i = j then Q $$ (i, j) - 1 else Q $$ (i, j)) in gauss_jordan_single QI\<^sup>T)"
rel_F_conversep: fixes L1 :: "'x \<Rightarrow> 'n \<Rightarrow> bool"   and L2 :: "'y \<Rightarrow> 'o \<Rightarrow> bool"   and L3 :: "'z \<Rightarrow> 'p \<Rightarrow> bool"   and Co1 :: "'aa \<Rightarrow> 'q \<Rightarrow> bool"   and Co2 :: "'ab \<Rightarrow> 'r \<Rightarrow> bool"   and Co3 :: "'ac \<Rightarrow> 's \<Rightarrow> bool"   and Contra1 :: "'ad \<Rightarrow> 't \<Rightarrow> bool"   and Contra2 :: "'ae \<Rightarrow> 'u \<Rightarrow> bool"   and Contra3 :: "'af \<Rightarrow> 'v \<Rightarrow> bool" shows "rel_F L1\<inverse>\<inverse> L2\<inverse>\<inverse> L3\<inverse>\<inverse> Co1\<inverse>\<inverse> Co2\<inverse>\<inverse> Co3\<inverse>\<inverse> Contra1\<inverse>\<inverse> Contra2\<inverse>\<inverse> Contra3\<inverse>\<inverse> = (rel_F L1 L2 L3 Co1 Co2 Co3 Contra1 Contra2 Contra3)\<inverse>\<inverse>"
f_Exec_Stream_Cons_rev: fixes xs :: "'d list"   and trans_fun :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e" assumes "xs \<noteq> []" shows "trans_fun (hd xs) c # f_Exec_Comp_Stream trans_fun (tl xs) (trans_fun (hd xs) c) = f_Exec_Comp_Stream trans_fun xs c"
src_src: fixes \<mu> :: "'a" assumes "S.arr \<mu>" shows "S.src (S.src \<mu>) = S.src \<mu>"
sim_val_def: fixes M :: "('e, 'f, 'g) KripkeStructure"   and M' :: "('e, 'f, 'h) KripkeStructure"   and f :: "'g \<Rightarrow> 'h" shows "sim_val M M' f \<equiv> \<forall>u\<in>worlds M. valuation M u = valuation M' (f u)"
cons_post_rulet: fixes P :: "assn"   and c :: "'a Heap"   and Q :: "'a \<Rightarrow> assn"   and Q' :: "'a \<Rightarrow> assn" assumes "<P> c <Q>\<^sub>t"   and "\<And>x. Q x \<Longrightarrow>\<^sub>t Q' x" shows "<P> c <Q'>\<^sub>t"
poly_deg_homogenize: fixes x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "x \<notin> indets p" shows "poly_deg (homogenize x p) = poly_deg p"
components_are_iso: fixes a :: "'a" assumes "\<phi>.A.ide a" shows "\<phi>.B.iso (\<phi> a)"
mono_neutral_left: fixes T :: "'k set"   and S :: "'k set"   and g :: "'k \<Rightarrow> 'l" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = (0::'l)" shows "sum g S = sum g T"
length_start_of_tm: fixes ap :: "abc_inst list" shows "start_of (layout_of ap) (length ap) = Suc (length (tm_of ap) div 2)"
sup_complement: fixes x :: "'b" shows "- x \<squnion> - - x = top"
mult_1_right: fixes a :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "a \<circ>\<^sub>s Var = a"
Red1InstanceOf: fixes s :: "'heap \<times> 'addr val list"   and v :: "'addr val"   and U :: "ty"   and b :: "bool"   and P :: "'addr J1_prog"   and T :: "ty"   and uf :: "bool"   and t :: "'thread_id" assumes "typeof\<^bsub>hp s\<^esub> v = \<lfloor>U\<rfloor>"   and "b = (v \<noteq> Null \<and> P \<turnstile> U \<le> T)" shows "uf,P,t \<turnstile>1 \<langle>Val v instanceof T,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Val (Bool b),s\<rangle>"
mpow_def: fixes m :: "int"   and x :: "int"   and n :: "nat" shows "x ^^^\<^bsub>m\<^esub> n = x ^ n mod m"
limp_lan_def: fixes Z :: "'a list set"   and Y :: "'a list set" shows "limp_lan Z Y = {x. \<forall>y\<in>Y. x @ y \<in> Z}"
t_fusion_def: fixes x :: "'a \<times> ('b \<times> 'a) list"   and y :: "'a \<times> ('b \<times> 'a) list" shows "t_fusion x y \<equiv> if Dioid_Models_New.last x = first y then (fst x, snd x @ snd y) else undefined"
irreducible\<^sub>d_primitive_part: fixes p :: "int poly" shows "irreducible\<^sub>d (primitive_part p) = irreducible\<^sub>d p"