hcju/isabelleps huggingface dataset:
eval_simps'_1: fixes f :: "'c local.term" assumes "Arr f" shows "arr \<lbrace>f\<rbrace>"
prems_1: shows "f \<in> insert x F \<rightarrow> UNIV"
assms_6: shows "r \<cdot> q \<le> q \<cdot> r"
fls_integral_one': assumes "inverse (1::'a) = (1::'a)" shows "fls_integral 1 = fls_X"
simps_1: fixes r :: "real" shows "det_discount r 0 = 1"
poly_divides_exp: fixes m :: "nat"   and n :: "nat"   and p :: "'a list" assumes "m \<le> n" shows "p %^ m divides p %^ n"
time_closest_pair_rec_conv_closest_pair_recurrence: fixes ps :: "(int \<times> int) list" assumes "distinct ps"   and "sorted_fst ps" shows "real (time (closest_pair_rec_tm ps)) \<le> closest_pair_recurrence (length ps)"
net_par_oreachable_is_subnet: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and st :: "'e net_state"   and np :: "nat \<Rightarrow> ((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and p1 :: "net_tree"   and p2 :: "net_tree"   and S :: "(nat \<Rightarrow> 'd) \<Rightarrow> (nat \<Rightarrow> 'd) \<Rightarrow> 'f node_action \<Rightarrow> bool"   and U :: "(nat \<Rightarrow> 'd) \<Rightarrow> (nat \<Rightarrow> 'd) \<Rightarrow> bool" assumes "(\<sigma>, st) \<in> oreachable (opnet np (p1 \<parallel> p2)) S U" shows "\<exists>s t. st = SubnetS s t"
hyps_5: fixes b :: "lam" assumes " t1 \<longrightarrow>\<^sub>1 b" shows " b \<longrightarrow>\<^sub>1 t1'"
fls_nth_of_fpxs: fixes f :: "'b fpxs"   and n :: "int" shows "fls_nth (fls_of_fpxs f) n = fpxs_nth f (rat_of_int n / rat_of_nat (fpxs_root_order f))"
induct: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "Wellfounded.accp r x"   and "\<And>x. \<lbrakk>\<And>y. r y x \<Longrightarrow> Wellfounded.accp r y; \<And>y. r y x \<Longrightarrow> P y\<rbrakk> \<Longrightarrow> P x" shows "P x"
hyps_1: shows "atom ` domA \<Delta> \<sharp>* (\<Gamma>, L)"
mul_ext_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and xs :: "'a list"   and ys :: "'a list" shows "mul_ext f xs ys \<equiv> let s = {(x, y). fst (f x y)}; ns = {(x, y). snd (f x y)} in ((mset xs, mset ys) \<in> s_mul_ext ns s, (mset xs, mset ys) \<in> ns_mul_ext ns s)"
simps: fixes s :: "nat"   and l :: "cell list"   and r :: "cell list"   and lm :: "nat list"   and n :: "nat"   and ires :: "cell list" shows "mopup_jump_over1 (s, l, r) lm n ires = (\<exists>ln m1 m2 rn. m1 + m2 = Suc (lm ! n) \<and> l = Oc \<up> m1 @ Bk \<up> ln @ Bk # Bk # ires \<and> (r = Oc \<up> m2 @ Bk # <drop (Suc n) lm> @ Bk \<up> rn \<or> r = Oc \<up> m2 \<and> drop (Suc n) lm = []))"
med23s_def: fixes t :: "l3_state" shows "med23s t \<equiv> \<lparr>ik = ik t \<inter> payload, secret = secret t, progress = progress t, signalsInit = signalsInit t, signalsResp = signalsResp t, chan = local.abs (ik t), bad = l3_state.bad t\<rparr>"
plus_adds: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and s :: "'c"   and t :: "'c"   and u :: "'c" assumes "class.comm_powerprod plus minus zero" shows "plus.adds plus (plus s t) u = (plus.adds plus t u \<and> plus.adds plus s (minus u t))"
det_bezout_iterate: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and A :: "(('b, 'c) vec, 'c) vec"   and i :: "'c"   and k :: "nat"   and n :: "nat" assumes "is_bezout_ext bezout"   and "A $ i $ mod_type_class.from_nat k \<noteq> (0::'b)"   and "n < ncols A" shows "det (bezout_iterate A n i (mod_type_class.from_nat k) bezout) = det A"
I: shows "inst_ok \<Theta> (map (\<lambda>(v, S). ((v, S), \<rho> v S)) l)"
cd_def: shows "cd = [c, d]\<^sub>\<circ>"
rel_FGcontra_def: fixes L1 :: "'n \<Rightarrow> 'aa \<Rightarrow> bool"   and Co1 :: "'o \<Rightarrow> 'ab \<Rightarrow> bool"   and Co2 :: "'w \<Rightarrow> 'aj \<Rightarrow> bool"   and Co3 :: "'p \<Rightarrow> 'ac \<Rightarrow> bool"   and Co4 :: "'q \<Rightarrow> 'ad \<Rightarrow> bool"   and Co5 :: "'r \<Rightarrow> 'ae \<Rightarrow> bool"   and Contra1 :: "'s \<Rightarrow> 'af \<Rightarrow> bool"   and Contra2 :: "'t \<Rightarrow> 'ag \<Rightarrow> bool"   and Contra3 :: "'u \<Rightarrow> 'ah \<Rightarrow> bool"   and Contra4 :: "'v \<Rightarrow> 'ai \<Rightarrow> bool"   and Contra5 :: "'y \<Rightarrow> 'ak \<Rightarrow> bool" shows "rel_FGcontra L1 Co1 Co2 Co3 Co4 Co5 Contra1 Contra2 Contra3 Contra4 Contra5 = rel_F L1 Co1 Co3 Co1 Co4 Co5 (rel_G Contra1 Contra2 Contra3 Contra4 Co1 Co2) Contra1 Contra5"
g_diff_at: fixes x :: "real" assumes "x \<notin> k"   and "x \<in> {a<..<b}" shows "\<gamma> differentiable at x"
nat_mod_distrib: fixes x :: "int"   and y :: "int" assumes "0 \<le> x"   and "0 \<le> y" shows "nat (x mod y) = nat x mod nat y"
Join_unless: fixes F :: "'c program"   and G :: "'c program"   and A :: "'c set"   and B :: "'c set" shows "(F \<squnion> G \<in> A unless B) = (F \<in> A unless B \<and> G \<in> A unless B)"
other: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and \<zeta> :: "'b"   and R :: "nat set"   and \<sigma>' :: "nat \<Rightarrow> 'a" assumes "(\<sigma>, NodeS ii \<zeta> R) \<in> oreachable (\<langle>ii : T : R\<^sub>i\<rangle>\<^sub>o) S U"   and "U \<sigma> \<sigma>'"   and "P \<sigma> \<zeta> R" shows "P \<sigma>' \<zeta> R"
surjective: fixes r :: "('a, 'b) TS_scheme" shows "r = \<lparr>init = init r, trans = TS.trans r, \<dots> = TS.more r\<rparr>"
fps_const_0_eq_0: shows "fps_const (0::'a) = 0"
proj_comp_def: fixes cmp :: "(('a, 'b) pp \<times> nat) nat_term_order" shows "proj_comp cmp = (\<lambda>x y. nat_term_compare cmp (x, 0) (y, 0))"
absorb_other: fixes a :: "abopt"   and j :: "jump" assumes "a \<noteq> Some (Jump j)" shows "absorb j a = a"
assms_4: assumes "y \<sqsubseteq> x" shows "P"
affine_rel_open: fixes S :: "'b set" assumes "affine S" shows "rel_open S"
sup_continuous_def: fixes F :: "'d \<Rightarrow> 'e" shows "sup_continuous F = (\<forall>M. incseq M \<longrightarrow> F (Sup (range M)) = (SUP i. F (M i)))"
RETURN_refine_iff: fixes x :: "'g"   and R :: "('g \<times> 'h) set"   and y :: "'h" shows "(RETURN x \<le> \<Down> R (RETURN y)) = ((x, y) \<in> R)"
bit_cut_integer_def: fixes k :: "integer" shows "bit_cut_integer k = (k div 2, odd k)"
eq: shows "anonymous_profile R1 = image_mset (map ((`) \<sigma>)) (anonymous_profile R2)"
h: shows "h ` {l \<in> A. list.set l \<subseteq> M} \<subseteq> {..<2}"
class_add_eff_pc: fixes i :: "instr"   and P :: "jvm_prog"   and pc :: "nat"   and xt :: "ex_table"   and \<tau> :: "ty list \<times> ty err list"   and mpc :: "nat"   and wf_md :: "jvm_prog \<Rightarrow> char list \<Rightarrow> jvm_method mdecl \<Rightarrow> bool"   and C :: "char list"   and cdec :: "jvm_method class" assumes "\<forall>(pc', \<tau>')\<in>set (eff i P pc xt \<lfloor>\<tau>\<rfloor>). pc' < mpc"   and "wf_prog wf_md P"   and "\<not> is_class P C" shows "\<forall>(pc', \<tau>')\<in>set (eff i (class_add P (C, cdec)) pc xt \<lfloor>\<tau>\<rfloor>). pc' < mpc"
continuous_on_cases: fixes s :: "'a set"   and t :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and P :: "'a \<Rightarrow> bool" assumes "closed s"   and "closed t"   and "continuous_on s f"   and "continuous_on t g"   and "\<forall>x. x \<in> s \<and> \<not> P x \<or> x \<in> t \<and> P x \<longrightarrow> f x = g x" shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
subcocycle_birkhoff: fixes f :: "'a \<Rightarrow> real" assumes "integrable M f" shows "subcocycle (birkhoff_sum f)"
HVZK_def: shows "\<Sigma>_AND.HVZK \<equiv> \<forall>e\<in>challenge_space. (\<forall>(h, w)\<in>Rel_AND. \<Sigma>_AND.R h w e = \<Sigma>_AND.S h e) \<and> (\<forall>h\<in>valid_pub_AND. \<forall>(a, z)\<in>set_spmf (S_AND h e). check_AND h a e z)"
SIGT: shows "x1a \<in> SIGT ($f x1a x2a)"
config_rand_induct: fixes P :: "'b list \<Rightarrow> bool"   and s0 :: "'b list"   and A :: "('b list \<Rightarrow> 'c pmf) \<times> ('b list \<times> 'c \<Rightarrow> 'b \<Rightarrow> ((nat \<times> nat list) \<times> 'c) pmf)"   and qs :: "'b list" assumes "P s0"   and "\<And>s q a. P s \<Longrightarrow> P (step s q a)" shows "\<forall>x\<in>set_pmf (config'_rand A (fst A s0 \<bind> (\<lambda>is. return_pmf (s0, is))) qs). P (fst x)"
1: assumes "distinct ms"   and "[l] \<langle>f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> ms \<preceq> [l] \<langle>g\<rangle> ms" shows "[l] \<langle>star \<circ> f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> ms \<preceq> [l] \<langle>g\<rangle> ms"
starfun_const_fun: fixes k :: "'a"   and x :: "'b star" shows "(*f* (\<lambda>x. k)) x = star_of k"
invar_empty: shows "braun \<langle>\<rangle> \<and> heap \<langle>\<rangle>"
bounded_measure: fixes A :: "'a set" shows "Sigma_Algebra.measure M A \<le> Sigma_Algebra.measure M (space M)"
mat_append_col_col: fixes A :: "'b mat"   and b :: "'b vec" assumes "dim_row A = dim_vec b" shows "col (mat_append_col A b) (dim_col A) = b"
trans: fixes b :: "'d"   and a :: "'d"   and c :: "'d" assumes "b \<le> a"   and "c \<le> b" shows "c \<le> a"
IH: shows "((t, s) \<in> setops\<^sub>s\<^sub>s\<^sub>t A) = (insert\<langle>t,s\<rangle> \<in> set A \<or> delete\<langle>t,s\<rangle> \<in> set A \<or> (\<exists>ac. \<langle>ac: t \<in> s\<rangle> \<in> set A) \<or> (\<exists>X F F'. \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: F'\<rangle> \<in> set A \<and> (t, s) \<in> set F'))"
\<K>_power_bound: fixes k :: "nat" assumes "0 < k" shows "\<K> ^ k \<le> real (k * max_len)"
step_a: fixes A :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and l :: "'d"   and u :: "'e \<Rightarrow> 'f"   and a :: "'g"   and l' :: "'d"   and u' :: "'e \<Rightarrow> 'f" assumes "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow>\<^bsub>a\<^esub> \<langle>l', u'\<rangle>" shows "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow> \<langle>l',u'\<rangle>"
abs_eq: fixes x :: "'c" shows "Pred\<^sub>\<alpha> x = abs_Tree\<^sub>\<alpha> (tPred x)"
multiplicity_geI: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b"   and p :: "'b"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "\<not> dvd.dvd times p one"   and "dvd.dvd times (power.power one times p n) x" shows "n \<le> factorial_semiring.multiplicity times one p x"
queue_invar_simps_3: fixes q :: "('a, 'b) BinomialHeapStruc.BinomialTree list"   and q' :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.queue_invar (q @ q') = (BinomialHeapStruc.queue_invar q \<and> BinomialHeapStruc.queue_invar q')"
commute: fixes a :: "'a"   and b :: "'a" shows "a \<squnion> b = b \<squnion> a"
assms_3: shows "list_all (\<lambda>((idn, S), T). has_sort (osig (sig \<Theta>)) T S) insts"
floor_divide_upper: fixes q :: "'c"   and p :: "'c" assumes "(0::'c) < q" shows "p < (of_int \<lfloor>p / q\<rfloor> + (1::'c)) * q"
distinct_1: fixes x2 :: "'a" shows "None \<noteq> pure x2"
rel_cases: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b message_af"   and b :: "'c message_af" assumes "rel_message_af R a b" obtains "a = NoMsg" and "b = NoMsg"   | x :: "'b" and y :: "'c" where "a = Msg x" and "b = Msg y" and "R x y"
rspR_on_def: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" shows "rspR_on A f = {(x, y). {x, y} \<subseteq> A \<and> (y, x) \<notin> rsp_on A f}"
simps_5: fixes f1 :: "'x"   and f2 :: "'y \<Rightarrow> 'x"   and x2 :: "'y" shows "(case Some x2 of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f2 x2"
hd0_imp_R_lin_dependent: fixes ms :: "'v list" shows "\<not> R_lin_independent ((0::'v) # ms)"
before_xor: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" assumes "pre_between vs ram1 ram2" shows "(\<not> before vs ram1 ram2) = before vs ram2 ram1"
comparator_sum_simps_2: fixes comp\<^sub>'\<^sub>a :: "'c \<Rightarrow> 'c \<Rightarrow> order"   and comp\<^sub>'\<^sub>b :: "'d \<Rightarrow> 'd \<Rightarrow> order"   and x :: "'c"   and ya :: "'d" shows "comparator_sum comp\<^sub>'\<^sub>a comp\<^sub>'\<^sub>b (Inl x) (Inr ya) = Lt"
assms_2: fixes n :: "nat" shows "emeasure M (A n) < \<infinity>"
emeasure_pair_measure_Times: fixes A :: "'c set"   and N :: "'c measure"   and B :: "'a set" assumes "A \<in> sets N"   and "B \<in> sets M" shows "emeasure (N \<Otimes>\<^sub>M M) (A \<times> B) = emeasure N A * emeasure M B"
assms_5: fixes A :: "'a trm"   and B :: "'a trm" assumes "P S A"   and "P S B" shows "P S (SimplyTyped.Pair A B)"
strict_coboundedI2: fixes c :: "'t"   and b :: "'t"   and a :: "'t" assumes "c \<prec>\<^sub>t b" shows "c \<prec>\<^sub>t ord_term_lin.max a b"
match_ex_table_stack_xliftD: fixes P :: "'a prog"   and C :: "String.literal"   and pc :: "nat"   and n :: "nat"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and pc' :: "nat"   and d :: "nat" assumes "match_ex_table P C pc (stack_xlift n xt) = \<lfloor>(pc', d)\<rfloor>" shows "n \<le> d \<and> match_ex_table P C pc xt = \<lfloor>(pc', d - n)\<rfloor>"
simps_6: fixes opinl :: "'opinl" shows "norm_instr (IOpInl opinl) = Std.instr.IOp (\<DD>\<ee>\<II>\<nn>\<ll> opinl)"
length_enumerate: fixes n :: "nat"   and xs :: "'a list" shows "length (List.enumerate n xs) = length xs"
predK_def: shows "predK \<equiv> \<Lambda> nK. bindK\<cdot>nK\<cdot>(\<Lambda> (ValKN\<cdot>n). case n of 0 \<Rightarrow> \<bottom> | Suc n \<Rightarrow> unitK\<cdot>(ValKN\<cdot>n))"
len_add: fixes m :: "nat"   and P :: "'v" shows "len (add m P) = max (Suc m) (len P)"
rep_eq: shows "impl_of RBT_Mapping2.empty = rbt.Empty"
cinsert_def: shows "cinsert \<equiv> map_fun id (map_fun rcset acset) insert"
neutr_eq_iff: fixes a :: "'c multiset"   and b :: "'c multiset" shows "({#} = a \<union># b) = (a = {#} \<and> b = {#})"
mem_val_alt_8_unchanged1: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36"   and addr :: "word36"   and val :: "word8" assumes "mem_equal s1 s2 pa" shows "mem_val_alt 8 ((pa AND 68719476732) + 1) (s1\<lparr>mem := (mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))\<rparr>) = mem_val_alt 8 ((pa AND 68719476732) + 1) (s2\<lparr>mem := (mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))\<rparr>)"
fsingleton_iff: fixes b :: "'a"   and a :: "'a" shows "(b |\<in>| {|a|}) = (b = a)"
compact_eq_seq_compact_metric: fixes S :: "'a set" shows "compact S = seq_compact S"
utp_theory_lattice_axioms_def: fixes hcond :: "'a hrel \<Rightarrow> 'a hrel" shows "utp_theory_lattice_axioms hcond \<equiv> complete_lattice (utp_order hcond)"
simps_3: fixes t :: "'a local.term"   and u :: "'a local.term" shows "Cod (t \<^bold>\<otimes> u) = Cod t \<^bold>\<otimes> Cod u"
ivl_disj_int_2: fixes l :: "'b"   and u :: "'b" shows "{..<l} \<inter> {l..<u} = {}"
sorted_wrt_filter: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and P :: "'c \<Rightarrow> bool" assumes "sorted_wrt f xs" shows "sorted_wrt f (filter P xs)"
eq_iff: fixes x :: "'t"   and y :: "'t" shows "(x = y) = (x \<preceq>\<^sub>t y \<and> y \<preceq>\<^sub>t x)"
add_diff_cancel_right': fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and a :: "'g"   and b :: "'g" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus a b) b = a"
fold_A_chamber_system_image: shows "fold_A \<turnstile> ChamberComplex.chamber_system A = {F \<in> ChamberComplex.chamber_system A. face_distance z F = chamber_distance C F}"
ennsqrt_square2: fixes x :: "ennreal" shows "ennsqrt (x * x) = x"
divmod_abs'_def: fixes x :: "integer"   and y :: "integer" assumes "0 \<le> x"   and "0 \<le> y" shows "divmod_abs' x y = Code_Numeral.divmod_abs x y"
preal_downwards_closed': fixes A :: "rat set"   and y :: "rat"   and z :: "rat" assumes "Dedekind_Real.cut A"   and "y \<in> A"   and "0 < z"   and "z \<le> y" shows "z \<in> A"
inj_hom_imp_solvable: assumes "inj_on id (carrier (alt_group n))"   and "solvable (sym_group n)" shows "solvable (alt_group n)"
path_append: fixes A :: "('c, 'd) nba"   and r :: "('c \<times> 'd) list"   and p :: "'d"   and s :: "('c \<times> 'd) list" assumes "NBA.path A r p"   and "NBA.path A s (target r p)" shows "NBA.path A (r @ s) p"
add: fixes x :: "'b"   and y :: "'b"   and ns :: "('b \<times> 'b) set"   and X :: "'b multiset"   and Y :: "'b multiset" assumes "(x, y) \<in> ns"   and "(X, Y) \<in> multpw ns" shows "(add_mset x X, add_mset y Y) \<in> multpw ns"
rcosI: fixes h :: "'a"   and H :: "'a set"   and x :: "'a" assumes "h \<in> H"   and "H \<subseteq> carrier H'"   and "x \<in> carrier H'" shows "h \<otimes>\<^bsub>H'\<^esub> x \<in> H #>\<^bsub>H'\<^esub> x"
intro: assumes "\<And>x y. PP x y = (P x y \<and> x \<noteq> y)"   and "\<And>x y. (x = y) = (\<forall>z. P z x = P z y)" shows "M3_axioms"
fls_times_conv_fps_times: fixes f :: "'a fls"   and g :: "'a fls" assumes "0 \<le> fls_subdegree f"   and "0 \<le> fls_subdegree g" shows "f * g = fps_to_fls (fls_regpart f * fls_regpart g)"
irreducible_mult_unit_left: fixes a :: "'b"   and p :: "'b" assumes "a dvd (1::'b)" shows "irreducible (a * p) = irreducible p"
mignotte_helper_coeff_int: fixes lst :: "complex list"   and i :: "int" shows "cmod (coeff_int (\<Prod>a\<leftarrow>lst. [:- a, 1:]) i) \<le> real (choose_int (length lst - 1) i) * (\<Prod>a\<leftarrow>lst. max 1 (cmod a)) + real (choose_int (length lst - 1) (i - 1))"
4: shows "supp \<alpha>1 - bn \<alpha>1 = supp \<alpha>2 - bn \<alpha>2"
real_mon_conv_add: fixes x :: "nat \<Rightarrow> real"   and y :: "real"   and a :: "nat \<Rightarrow> real"   and b :: "real" assumes "x\<up>y"   and "a\<up>b" shows "(\<lambda>m. x m + a m)\<up>y + b"
good_ruleset_def: fixes rs :: "'b rule list" shows "good_ruleset rs \<equiv> \<forall>r\<in>set rs. (\<nexists>chain. get_action r = Call chain) \<and> get_action r \<noteq> Return \<and> (\<nexists>chain. get_action r = Goto chain) \<and> get_action r \<noteq> Unknown"
BPd_def: shows "BPd \<equiv> map_of_default {} [(Xd d1h1, linord_of_list [d1h3, d1h2, d1h1]), (Xd d2h1, linord_of_list [d2h3, d2h2, d2h1]), (Xd d3h1, linord_of_list [d3h1, d3h2, d3h3])]"
map_resource_resource_of_oracle: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'b"   and "oracle" :: "'e \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'e) spmf"   and s :: "'e" shows "map_resource f g (RES oracle s) = RES (map_fun id (map_fun f (map_spmf (map_prod g id))) oracle) s"
fst_scaleC: fixes r :: "complex"   and A :: "'d \<times> 'e" shows "fst (r *\<^sub>C A) = r *\<^sub>C fst A"
pdevs_val_sum_pdevs: fixes e :: "nat \<Rightarrow> real"   and f :: "'b \<Rightarrow> 'a pdevs"   and X :: "'b set" shows "pdevs_val e (sum_pdevs f X) = (\<Sum>x\<in>X. pdevs_val e (f x))"
ignore_def: fixes \<gamma> :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and \<Delta> :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and s :: "'a"   and s1 :: "'e"   and vl1 :: "'f"   and a :: "'b"   and ou :: "'c"   and s' :: "'a"   and vl' :: "'d" shows "BD_Security.ignore \<gamma> \<Delta> s s1 vl1 a ou s' vl' \<equiv> \<not> \<gamma> (Trans s a ou s') \<and> \<Delta> s' vl' s1 vl1"
dirichlet_prod_inverse': fixes f :: "nat \<Rightarrow> 'a"   and i :: "'a" assumes "f 1 * i = (1::'a)" shows "dirichlet_prod (dirichlet_inverse f i) f = (\<lambda>n. if n = 1 then 1::'a else (0::'a))"
induct: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and x1 :: "'a list"   and x2 :: "'b list"   and x3 :: "'c list"   and Pa :: "'a list \<Rightarrow> 'b list \<Rightarrow> 'c list \<Rightarrow> bool" assumes "list_all3 P x1 x2 x3"   and "Pa [] [] []"   and "\<And>a1 a2 a3 q1 q2 q3. \<lbrakk>P a1 a2 a3; list_all3 P q1 q2 q3; Pa q1 q2 q3\<rbrakk> \<Longrightarrow> Pa (a1 # q1) (a2 # q2) (a3 # q3)" shows "Pa x1 x2 x3"
word_sint_msb_eq: fixes x :: "'b word" shows "sint x = uint x - (if msb x then 2 ^ size x else 0)"
thin_triangles: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and deltaG :: "'b itself \<Rightarrow> real"   and Gxy :: "'b set"   and x :: "'b"   and y :: "'b"   and Gxz :: "'b set"   and z :: "'b"   and Gyz :: "'b set"   and w :: "'b" assumes "class.Gromov_hyperbolic_space dist uniformity open deltaG"   and "geodesic_segment_between Gxy x y"   and "geodesic_segment_between Gxz x z"   and "geodesic_segment_between Gyz y z"   and "w \<in> Gyz" shows "infdist w (Gxy \<union> Gxz) \<le> 4 * deltaG TYPE('b)"
step: shows "(name, rhs) |\<in>| Rewriting_Pterm.compile rs"
simps: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and a1 :: "('b, 'c, 'a, 'd) seqp"   and a2 :: "('b, 'c, 'a, 'd) seqp" shows "(a1 \<leadsto>\<^bsub>\<Gamma>\<^esub> a2) = ((\<exists>p1 p2. a1 = p1 \<oplus> p2 \<and> a2 = p1) \<or> (\<exists>p1 p2. a1 = p1 \<oplus> p2 \<and> a2 = p2) \<or> (\<exists>pn. a1 = call(pn) \<and> a2 = \<Gamma> pn))"
closed: fixes a :: "'al"   and b :: "'bl" assumes "a \<in> \<UU>A"   and "b \<in> \<UU>B" shows "f a b \<in> \<UU>C"
trancl_onp_trancl_on_eq: fixes U :: "'c set"   and r :: "('c \<times> 'c) set" shows "on (\<lambda>x. x \<in> U) (\<lambda>x xa. (x, xa) \<in> r)\<^sup>+\<^sup>+ = (\<lambda>x xa. (x, xa) \<in> on U r\<^sup>+)"
defined_offending: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and G :: "'c graph"   and nP :: "'c \<Rightarrow> 'd" assumes "SecurityInvariant_preliminaries sinvar"   and "wf_graph G"   and "\<not> sinvar G nP" shows "SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP \<noteq> {}"
init_def: fixes h :: "'a \<times> 'a"   and w :: "nat" shows "init h w = (let (h, h') = h in sample_uniform (order \<G>) \<bind> (\<lambda>r. return_spmf (r, \<^bold>g [^] r, g' [^] r)))"
strict_assoc: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "ide \<a>[f, g, h]"
retract_of_singleton: fixes x :: "'a"   and S :: "'a set" shows "({x} retract_of S) = (x \<in> S)"
f_Exec_Stream_expand_map_aggregate_append: fixes f :: "'e \<Rightarrow> 'd"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and xs :: "'f message_af list"   and ys :: "'f message_af list"   and k :: "nat"   and c :: "'e"   and ag :: "'d list \<Rightarrow> 'd" shows "f_aggregate (map f (f_Exec_Comp_Stream trans_fun ((xs @ ys) \<odot> k) c)) k ag = f_aggregate (map f (f_Exec_Comp_Stream trans_fun (xs \<odot> k) c)) k ag @ f_aggregate (map f (f_Exec_Comp_Stream trans_fun (ys \<odot> k) (f_Exec_Comp trans_fun (xs \<odot> k) c))) k ag"
linepath_refl: fixes a :: "'a" shows "linepath a a = (\<lambda>x. a)"
l2_oreach_dy_fake_msg: fixes s :: "l2_state"   and x :: "msg" assumes "s \<in> oreach l2"   and "x \<in> dy_fake_msg (bad s) (ik s) (chan s)" shows "s\<lparr>ik := insert x (ik s)\<rparr> \<in> oreach l2"
constant_functor_axioms: shows "constant_functor (\<cdot>\<^sub>J) (\<cdot>) (C.cod f)"
connectedD: fixes "open" :: "'c set \<Rightarrow> bool"   and A :: "'c set"   and U :: "'c set"   and V :: "'c set" assumes "class.topological_space open"   and "topological_space.connected open A"   and "open U"   and "open V"   and "U \<inter> V \<inter> A = {}"   and "A \<subseteq> U \<union> V" shows "U \<inter> A = {} \<or> V \<inter> A = {}"
unprot_def: fixes \<sigma> :: "'tp" shows "unprot \<sigma> \<equiv> \<not> prot \<sigma> \<and> \<not> protFw \<sigma>"
param: shows "((A ===> rel_spmf R) ===> A ===> rel_spmf R) F G"
partition_on_msetD1: fixes A :: "'b multiset"   and P :: "'b multiset multiset" assumes "partition_on_mset A P" shows "\<Sum>\<^sub># P = A"
ord_of_nat_of_ord: fixes x :: "hf" assumes "Ord x" shows "ord_of (nat_of_ord x) = x"
intro: fixes K :: "('c, 'd) Ring_scheme" assumes "Ring K"   and "\<forall>x\<in>carrier K - {\<zero>\<^bsub>K\<^esub>}. \<exists>x'\<in>carrier K. x' \<cdot>\<^sub>r\<^bsub>K\<^esub> x = 1\<^sub>r\<^bsub>K\<^esub>" shows "Corps K"
existence_ivl_initial_time: fixes t0 :: "real"   and x0 :: "'a" assumes "t0 \<in> T"   and "x0 \<in> S" shows "t0 \<in> existence_ivl t0 x0"