UDACA/Split-Isa-AF-MMA · Datasets at Hugging Face

Statement: stringlengths 7 24.3k
lemma set_as_map_elem : assumes "y \<in> m2f (set_as_map xs) x" shows "(x,y) \<in> xs"
lemma pathfinish_rectpath [simp]: "pathfinish (rectpath a1 a3) = a1"
lemma lmeasurable_frontier: "bounded S \<Longrightarrow> frontier S \<in> lmeasurable"
lemma gcd_code [code]: "gcd a b = int_of_integer (gcd_integer (of_int a) (of_int b))"
lemma bounded_linear_Eps2: "bounded_linear Eps2"
lemma cbifi:"b^-1 O f^-1 \<subseteq> b^-1"
lemma poly_const_conv: fixes x :: "'a::comm_ring_1" shows "poly [:c:] x = y \<longleftrightarrow> c = y"
lemma isOK_forallM_index [simp]: "isOK (forallM_index P xs) \<longleftrightarrow> (\<forall>i < length xs. isOK (P (xs ! i) i))"
lemma Abs_ffilter: "(ffilter f s = s') = ({e \<in> (fset s). f e} = (fset s'))"
lemma upd_params_fm [simp]: \<open>f \<notin> params_fm p \<Longrightarrow> \<lbrakk>E, F(f := x), G\<rbrakk> p = \<lbrakk>E, F, G\<rbrakk> p\<close>
lemma is_factorization_of_roots: fixes a :: "'a :: idom" assumes "is_factorization_of (a, fctrs) p" "p \<noteq> 0" shows "set (map fst fctrs) = {x. poly p x = 0}"
lemma LIM_fun_gt_zero: "f \<midarrow>c\<rightarrow> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> 0 < f x)" for f :: "real \<Rightarrow> real"
lemma tarski_top_omega_2: "x * top = (x * top) \<star> bot"
lemma not_par_inter_uniqueness: assumes "A \<noteq> B" and "C \<noteq> D" and "\<not> A B Par C D" and "Col A B X" and "Col C D X" and "Col A B Y" and "Col C D Y" shows "X = Y"
lemma err_semilat_eslI_aux: assumes semilat: "semilat (A, r, f)" shows "err_semilat(esl(A,r,f))"
lemma maddux1c: " a \<lhd> x \<sqinter> -(y \<rhd> a) \<le> a \<lhd> x"
lemma vfinite_induct[consumes 1, case_names vempty vinsert]: assumes "vfinite F" and "P 0" and "\<And>x F. \<lbrakk>vfinite F; x \<notin>\<^sub>\<circ> F; P F\<rbrakk> \<Longrightarrow> P (vinsert x F)" shows "P F"
lemma(in ring_functions) function_ring_car_closed: assumes "a \<in> S" assumes "f \<in> carrier F" shows "f a \<in> carrier R"
lemma wp1_prec: fixes e::tbd shows "wp\<^sub>1 c1 Q l s \<Longrightarrow> l x = prec c1 e s \<Longrightarrow> wp\<^sub>1 c1 (\<lambda>l s. Q l s \<and> e s = l x) l s"
lemma projecting_Stable: "projecting (%G. reachable (extend h F\<squnion>G)) h F (Stable (extend_set h A)) (Stable A)"
lemma [import_const "/\\"]: "(\<and>) = (\<lambda>p q. (\<lambda>f. f p q :: bool) = (\<lambda>f. f True True))"
lemma ipurge_tr_rev_trace: "secure P I D \<Longrightarrow> xs \<in> traces P \<Longrightarrow> ipurge_tr_rev I D u xs \<in> traces P"
lemma establish_convergence_static_verifies_transition : assumes "\<And> q1 q2 . q1 \<in> states M1 \<Longrightarrow> q2 \<in> states M1 \<Longrightarrow> q1 \<noteq> q2 \<Longrightarrow> \<exists> io . \<forall> k1 k2 . io \<in> set (dist_fun k1 q1) \<inter> set (dist_fun k2 q2) \<and> distinguishes M1 q1 q2 io" and "\<And> q k . q \<in> states M1 \<Longrightarrow> finite_tree (dist_fun k q)" shows "verifies_transition (establish_convergence_static dist_fun) M1 M2 V (fst (handle_state_cover_static dist_fun M1 V cg_initial cg_insert cg_lookup)) cg_insert cg_lookup"
lemma OclIsEmpty_infinite: "\<tau> \<Turnstile> \<delta> X \<Longrightarrow> \<not> finite (Rep_Bag_base X \<tau>) \<Longrightarrow> \<not> \<tau> \<Turnstile> \<delta> (X->isEmpty\<^sub>B\<^sub>a\<^sub>g())"
lemma Amax_ge:"\<lbrakk>f \<in> {j. j \<le> n} \<rightarrow> Z\<^sub>-\<^sub>\<infinity>; j \<in> {j. j \<le> n}\<rbrakk> \<Longrightarrow> (f j) \<le> (Amax n f)"
lemma snapshot_stable_ver_3: shows "trace init t final \<Longrightarrow> ~ has_snapshotted (S t i) p \<Longrightarrow> i \<ge> j \<Longrightarrow> ~ has_snapshotted (S t j) p"
lemma op_cf_comma_ArrMap_v11[cat_comma_cs_intros]: assumes "\<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<CC>" and "\<HH> : \<BB> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<CC>" shows "v11 (op_cf_comma \<GG> \<HH>\<lparr>ArrMap\<rparr>)"
lemma dsjAssoc2: "H \<turnstile> (A OR B) OR C \<Longrightarrow> H \<turnstile> A OR (B OR C)"
lemma max_lin_independent_set_in_Span : assumes "set vs \<subseteq> V" "set us \<subseteq> Span vs" "lin_independent us" shows "length us \<le> length vs"
lemma (in monoid) prod_unit_r: assumes abunit[simp]: "a \<otimes> b \<in> Units G" and bunit[simp]: "b \<in> Units G" and carr[simp]: "a \<in> carrier G" "b \<in> carrier G" shows "a \<in> Units G"
lemma subst_cls_list_empty[simp]: "[] \<cdot>cl \<sigma> = []"
lemma assert_assert_comp: "{.p::'a::lattice.} o {.p'.} = {.p \<sqinter> p'.}"
lemma (in domain) degree_one_root: assumes "subfield K R" and "p \<in> carrier (K[X])" and "degree p = 1" shows "eval p (\<ominus> (inv (lead_coeff p) \<otimes> (const_term p))) = \<zero>" and "inv (lead_coeff p) \<otimes> (const_term p) \<in> K"
lemma ennreal_fact: "ennreal (fact n) = fact n"
lemma ri_rhs_neq [simp]: "e \<noteq> RI(C,e');Cs \<leftarrow> e"
lemma eta_induct[case_names bottom N F, consumes 1]: "\<lbrakk> \<eta>: x \<mapsto> x'; \<lbrakk> x = \<bottom>; x' = \<bottom> \<rbrakk> \<Longrightarrow> P x x'; \<And>n. \<lbrakk> x = ValTT; x' = unitK\<cdot>ValKTT \<rbrakk> \<Longrightarrow> P x x'; \<And>n. \<lbrakk> x = ValFF; x' = unitK\<cdot>ValKFF \<rbrakk> \<Longrightarrow> P x x'; \<And>n. \<lbrakk> x = ValN\<cdot>n; x' = unitK\<cdot>(ValKN\<cdot>n) \<rbrakk> \<Longrightarrow> P x x'; \<And>f f'. \<lbrakk> x = ValF\<cdot>f; x' = unitK\<cdot>(ValKF\<cdot>f'); \<theta>: f \<mapsto> f' \<rbrakk> \<Longrightarrow> P x x' \<rbrakk> \<Longrightarrow> P x x'"
lemma RepFun_iff [iff]: "v \<^bold>\<in> (RepFun A f) \<longleftrightarrow> (\<exists>u. u \<^bold>\<in> A \<and> v = f u)"
lemma tauChainSupportDerivative: fixes P :: pi and P' :: pi assumes "P \<Longrightarrow>\<^sub>\<tau> P'" shows "((supp P')::name set) \<subseteq> (supp P)"
lemma inv_after_clear_Suc_nonempty[simp]: "inv_after_clear (as, abc_lm_s lm n (Suc (abc_lm_v lm n))) (s, b, []) ires = False"
lemma eqButUID_recvOuterFriends_UIDs: assumes "eqButUID s s1" and "uid \<noteq> UID \<or> aid \<noteq> AID" shows "(aid, uid) \<in>\<in> recvOuterFriendIDs s uid' \<longleftrightarrow> (aid, uid) \<in>\<in> recvOuterFriendIDs s1 uid'"
lemma set_pmf_binomial[simp]: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (binomial_pmf n p) = {..n}"
lemma "L c X \<subseteq> Lb c X"
lemma [code]: "CEQ(unit) = Some (\<lambda>_ _. True)"
lemma maximal_exactly_one: assumes \<open>consistent S\<close> and \<open>maximal S\<close> shows \<open>p \<in> S \<longleftrightarrow> (\<^bold>\<not> p) \<notin> S\<close>
lemma test_aa_increasing: "test p \<Longrightarrow> p \<le> --p"
lemma antilinear_to_conjugate_space[simp]: \<open>antilinear to_conjugate_space\<close>
lemma (in Module) mimg_module:"\<lbrakk>R module N; f \<in> mHom R M N\<rbrakk> \<Longrightarrow> R module (mimg R M N f)"
lemma fbox_g_ode_subset: assumes "\<And>s. s \<in> S \<Longrightarrow> 0 \<in> U s \<and> is_interval (U s) \<and> U s \<subseteq> T" shows "\|x\<acute>= (\<lambda>t. f) & G on U S @ 0] Q = (\<lambda> s. s \<in> S \<longrightarrow> (\<forall>t\<in>(U s). (\<forall>\<tau>\<in>down (U s) t. G (\<phi> \<tau> s)) \<longrightarrow> Q (\<phi> t s)))"
lemma ball_trivial [simp]: "ball x 0 = {}"
lemma totient_code_naive [code]: "totient n = totient_naive n 0 n"
lemma b_e_check_single_top_not_bot_sound: assumes "type_update ts (to_ct_list t_in) (TopType []) = ts'" "ts \<noteq> Bot" "ts' \<noteq> Bot" shows "\<exists>tn. c_types_agree ts tn \<and> suffix t_in tn"
lemma connected_if_composable: assumes "\<nu> \<star> \<mu> \<noteq> null" shows "sources \<nu> = targets \<mu>"
lemma reduction_step: assumes "q \<in> nodes" "run w q" obtains (deferred) a where "ren a q" "[a] \<preceq>\<^sub>F\<^sub>I w" \| (omitted) "{a. ren a q} \<subseteq> invisible" "Ind {a. ren a q} (sset w)"
lemma sub_tm_semantics [simp]: \<open>\<lparr>E, F\<rparr> (sub_tm s t) = \<lparr>\<lambda>n. \<lparr>E, F\<rparr> (s n), F\<rparr> t\<close>
lemma totalize_transient_iff: "(totalize F \<in> transient A) = (\<exists>act\<in>Acts F. A \<subseteq> Domain act & act``A \<subseteq> -A)"
lemma fmdom'_fmsubset_restrict_set: fixes X1 X2 and s :: "('a, bool) fmap" assumes "X1 \<subseteq> X2" "fmdom' s = X2" shows "fmdom' (fmrestrict_set X1 s) = X1"
lemma fps_X_neq_one [simp]: "(fps_X :: 'a :: zero_neq_one fps) \<noteq> 1"
lemma numeral_neq_fps_zero [simp]: "(numeral f :: 'a :: field_char_0 fps) \<noteq> 0"
lemma reduction: "0 < reduction" "reduction \<le> 1" "\<alpha> > 4/3 \<Longrightarrow> reduction < 1" "\<alpha> = 4/3 \<Longrightarrow> reduction = 1"
theorem LeftDerivationLadder_cut_appendix: assumes LDLadder: "LeftDerivationLadder (\<alpha>@\<delta>) D L \<gamma>" assumes ladder_i_in_\<alpha>: "ladder_i L 0 < length \<alpha>" shows "\<exists> E F \<gamma>1 \<gamma>2 L'. D = E@F \<and> \<gamma> = \<gamma>1 @ \<gamma>2 \<and> LeftDerivationLadder \<alpha> E L' \<gamma>1 \<and> derivation_ge F (length \<gamma>1) \<and> LeftDerivation \<delta> (derivation_shift F (length \<gamma>1) 0) \<gamma>2 \<and> length L' = length L \<and> ladder_i L' 0 = ladder_i L 0 \<and> ladder_last_j L' = ladder_last_j L"
lemma "Fr_2 \<F> \<Longrightarrow> \<forall>A B. Cl(A) \<longrightarrow> lCoP1\<^sup>A\<^sup>B(\<^bold>\<rightarrow>) \<^bold>\<not>\<^sup>I"
lemma inner_divide_right: fixes a :: "'a :: {real_inner,real_div_algebra}" shows "inner a (b / of_real m) = (inner a b) / m"
lemma lift_CFG: assumes wf:"CFGExit_wf sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses" and pd:"Postdomination sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit" shows "CFG src trg knd (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry (lift_get_proc get_proc Main) (lift_get_return_edges get_return_edges valid_edge sourcenode targetnode kind) procs Main"
lemma gctxtex_onp_idem [simp]: assumes "P \<box>\<^sub>G" and "\<And> C D. P C \<Longrightarrow> Q D \<Longrightarrow> Q (C \<circ>\<^sub>G\<^sub>c D)" shows "gctxtex_onp P (gctxtex_onp Q \<R>) = gctxtex_onp Q \<R>" (is "?Ls = ?Rs")
lemma bool_to_ternary_simp4: "eval_ternary_Not (bool_to_ternary X) = TernaryFalse \<longleftrightarrow> X"
lemma eval_fps_mult: fixes f g :: "'a :: {banach, real_normed_div_algebra, comm_ring_1} fps" assumes "norm z < fps_conv_radius f" "norm z < fps_conv_radius g" shows "eval_fps (f * g) z = eval_fps f z * eval_fps g z"
lemma cong: assumes "[a = b] (mod n)" shows "\<chi> a = \<chi> b"
lemma embedded_eq: "(A = B) \<Longrightarrow> (embedded_style A) = (embedded_style B)"
lemma eval_fds_power: fixes s :: "'a :: {nat_power, real_normed_field, banach, second_countable_topology}" assumes "fds_abs_converges f s" shows "eval_fds (f ^ n) s = eval_fds f s ^ n"
lemma nDASC[rule_format]: "wellformed_policy1 p \<longrightarrow> singleCombinators p \<longrightarrow> noDenyAll1 p"
lemma sup_state_opt_err : "(Err.le (sup_state_opt P)) s s"
lemma osp_bet__osp: assumes "A B C OSP P R" and "Bet P Q R" shows "A B C OSP P Q"
lemma mec_bot_imp_bot: assumes "mec \<phi> bot" shows "\<phi> = bot"
lemma (in weak_upper_semilattice) join_cong_r: assumes carr: "x \<in> carrier L" "y \<in> carrier L" "y' \<in> carrier L" and yy': "y .= y'" shows "x \<squnion> y .= x \<squnion> y'"
theorem (in finite_comm_group) Characters_iso: shows "G \<cong> Characters G"
lemma TokensAt_items_le[symmetric]: "TokensAt k I = TokensAt k (items_le k I)"
lemma Spy_see_shrK_D [dest!]: "[\|Key (shrK A) \<in> parts (knows Spy evs); evs \<in> woolam\|] ==> A \<in> bad"
lemma invertible_mat_mult_int: assumes "A = P * B" and "P \<in> carrier_mat n n" and "B \<in> carrier_mat n n" and "invertible_mat P" and "invertible_mat (map_mat rat_of_int B)" shows "invertible_mat (map_mat rat_of_int A)"
lemma list_succ_rotate[simp]: assumes "distinct xs" shows "list_succ (rotate n xs) = list_succ xs"
lemma OrL_inv': assumes "Or F G, \<Gamma> \<Rightarrow> \<Delta> \<down> n" shows "F,\<Gamma> \<Rightarrow> \<Delta> \<down> n \<and> G,\<Gamma> \<Rightarrow> \<Delta> \<down> n"
lemma ipresIGVar_termMOD[simp]: "ipresIGVar h termMOD MOD = ipresVar h MOD"
lemma takeWhile_take_has_property_nth: "\<lbrakk> n < length (takeWhile P xs) \<rbrakk> \<Longrightarrow> P (xs ! n)"
lemma randomElLemma: assumes "set list \<noteq> {}" shows "randomEl list random \<in> set list"
lemma rivest_perfect_hiding: "rivest_commit.perfect_hiding_ind_cpa \<A>"
lemma SeqReds: "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle> \<Longrightarrow> P \<turnstile> \<langle>e;;e\<^sub>2, s\<rangle> \<rightarrow>* \<langle>e';;e\<^sub>2, s'\<rangle>"
lemma Derives1_bound': "Derives1 a i r b \<Longrightarrow> i \<le> length b"
lemma ht_size_replicate[simp, intro!]: "ht_size (replicate n []) 0"
lemma scene_top_greatest: "X \<le> \<top>\<^sub>S"
lemma SDG_Def_parent_Def: "V \<in> Def\<^bsub>SDG\<^esub> n \<Longrightarrow> V \<in> Def (parent_node n)"
lemma nn_integral_less: assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M" assumes f: "(\<integral>\<^sup>+x. f x \<partial>M) \<noteq> \<infinity>" assumes ord: "AE x in M. f x \<le> g x" "\<not> (AE x in M. g x \<le> f x)" shows "(\<integral>\<^sup>+x. f x \<partial>M) < (\<integral>\<^sup>+x. g x \<partial>M)"
lemma fmrel_trancl_fmdom_eq: "(fmrel f)\<^sup>+\<^sup>+ xm ym \<Longrightarrow> fmdom xm = fmdom ym"
lemma E_null[simp]: "E (return_pmf 0) = 0"
lemma ccSUP_inf_distrib2: "countable A \<Longrightarrow> countable B \<Longrightarrow> inf (SUP a\<in>A. f a) (SUP b\<in>B. g b) = (SUP a\<in>A. SUP b\<in>B. inf (f a) (g b))"
lemma l2_step1_refines_step1: "{R12s} l1_step1 Ra A B, l2_step1 Ra A B {>R12s}"
lemma implements_oreach: "implements pi Sa Sc \<Longrightarrow> pi`(oreach Sc) \<subseteq> oreach Sa"
lemma typeof_ka: "typeof v \<noteq> None \<Longrightarrow> ka_Val v = {}"
lemma (in lbvc) cert_approx [intro?]: "\<lbrakk> pc < length \<phi>; c!pc \<noteq> \<bottom> \<rbrakk> \<Longrightarrow> c!pc = \<phi>!pc"
lemma vdistinct_vempty[intro, simp]: "vdistinct []\<^sub>\<circ>"
lemma to_fun_monom: assumes "c \<in> carrier R" assumes "x \<in> carrier R" shows "to_fun (monom P c n) x = c \<otimes> x [^] n"
lemma w_addrs_lift_start_heap_obs: "w_addrs (w_values P vs (map snd (lift_start_obs start_tid start_heap_obs))) \<subseteq> w_addrs vs"

lemma set_as_map_elem : assumes "y \<in> m2f (set_as_map xs) x" shows "(x,y) \<in> xs"

lemma pathfinish_rectpath [simp]: "pathfinish (rectpath a1 a3) = a1"

lemma lmeasurable_frontier: "bounded S \<Longrightarrow> frontier S \<in> lmeasurable"

lemma gcd_code [code]: "gcd a b = int_of_integer (gcd_integer (of_int a) (of_int b))"

lemma bounded_linear_Eps2: "bounded_linear Eps2"

lemma cbifi:"b^-1 O f^-1 \<subseteq> b^-1"

lemma poly_const_conv: fixes x :: "'a::comm_ring_1" shows "poly [:c:] x = y \<longleftrightarrow> c = y"

lemma isOK_forallM_index [simp]: "isOK (forallM_index P xs) \<longleftrightarrow> (\<forall>i < length xs. isOK (P (xs ! i) i))"

lemma Abs_ffilter: "(ffilter f s = s') = ({e \<in> (fset s). f e} = (fset s'))"

lemma upd_params_fm [simp]: \<open>f \<notin> params_fm p \<Longrightarrow> \<lbrakk>E, F(f := x), G\<rbrakk> p = \<lbrakk>E, F, G\<rbrakk> p\<close>

lemma is_factorization_of_roots: fixes a :: "'a :: idom" assumes "is_factorization_of (a, fctrs) p" "p \<noteq> 0" shows "set (map fst fctrs) = {x. poly p x = 0}"

lemma LIM_fun_gt_zero: "f \<midarrow>c\<rightarrow> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> 0 < f x)" for f :: "real \<Rightarrow> real"

lemma tarski_top_omega_2: "x * top = (x * top) \<star> bot"

lemma not_par_inter_uniqueness: assumes "A \<noteq> B" and "C \<noteq> D" and "\<not> A B Par C D" and "Col A B X" and "Col C D X" and "Col A B Y" and "Col C D Y" shows "X = Y"

lemma err_semilat_eslI_aux: assumes semilat: "semilat (A, r, f)" shows "err_semilat(esl(A,r,f))"

lemma maddux1c: " a \<lhd> x \<sqinter> -(y \<rhd> a) \<le> a \<lhd> x"

lemma vfinite_induct[consumes 1, case_names vempty vinsert]: assumes "vfinite F" and "P 0" and "\<And>x F. \<lbrakk>vfinite F; x \<notin>\<^sub>\<circ> F; P F\<rbrakk> \<Longrightarrow> P (vinsert x F)" shows "P F"

lemma(in ring_functions) function_ring_car_closed: assumes "a \<in> S" assumes "f \<in> carrier F" shows "f a \<in> carrier R"

lemma wp1_prec: fixes e::tbd shows "wp\<^sub>1 c1 Q l s \<Longrightarrow> l x = prec c1 e s \<Longrightarrow> wp\<^sub>1 c1 (\<lambda>l s. Q l s \<and> e s = l x) l s"

lemma projecting_Stable: "projecting (%G. reachable (extend h F\<squnion>G)) h F (Stable (extend_set h A)) (Stable A)"

lemma [import_const "/\\"]: "(\<and>) = (\<lambda>p q. (\<lambda>f. f p q :: bool) = (\<lambda>f. f True True))"

lemma ipurge_tr_rev_trace: "secure P I D \<Longrightarrow> xs \<in> traces P \<Longrightarrow> ipurge_tr_rev I D u xs \<in> traces P"

lemma establish_convergence_static_verifies_transition : assumes "\<And> q1 q2 . q1 \<in> states M1 \<Longrightarrow> q2 \<in> states M1 \<Longrightarrow> q1 \<noteq> q2 \<Longrightarrow> \<exists> io . \<forall> k1 k2 . io \<in> set (dist_fun k1 q1) \<inter> set (dist_fun k2 q2) \<and> distinguishes M1 q1 q2 io" and "\<And> q k . q \<in> states M1 \<Longrightarrow> finite_tree (dist_fun k q)" shows "verifies_transition (establish_convergence_static dist_fun) M1 M2 V (fst (handle_state_cover_static dist_fun M1 V cg_initial cg_insert cg_lookup)) cg_insert cg_lookup"

lemma OclIsEmpty_infinite: "\<tau> \<Turnstile> \<delta> X \<Longrightarrow> \<not> finite (Rep_Bag_base X \<tau>) \<Longrightarrow> \<not> \<tau> \<Turnstile> \<delta> (X->isEmpty\<^sub>B\<^sub>a\<^sub>g())"

lemma Amax_ge:"\<lbrakk>f \<in> {j. j \<le> n} \<rightarrow> Z\<^sub>-\<^sub>\<infinity>; j \<in> {j. j \<le> n}\<rbrakk> \<Longrightarrow> (f j) \<le> (Amax n f)"

lemma snapshot_stable_ver_3: shows "trace init t final \<Longrightarrow> ~ has_snapshotted (S t i) p \<Longrightarrow> i \<ge> j \<Longrightarrow> ~ has_snapshotted (S t j) p"

lemma op_cf_comma_ArrMap_v11[cat_comma_cs_intros]: assumes "\<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<CC>" and "\<HH> : \<BB> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<CC>" shows "v11 (op_cf_comma \<GG> \<HH>\<lparr>ArrMap\<rparr>)"

lemma dsjAssoc2: "H \<turnstile> (A OR B) OR C \<Longrightarrow> H \<turnstile> A OR (B OR C)"

lemma max_lin_independent_set_in_Span : assumes "set vs \<subseteq> V" "set us \<subseteq> Span vs" "lin_independent us" shows "length us \<le> length vs"

lemma (in monoid) prod_unit_r: assumes abunit[simp]: "a \<otimes> b \<in> Units G" and bunit[simp]: "b \<in> Units G" and carr[simp]: "a \<in> carrier G" "b \<in> carrier G" shows "a \<in> Units G"

lemma subst_cls_list_empty[simp]: "[] \<cdot>cl \<sigma> = []"

lemma assert_assert_comp: "{.p::'a::lattice.} o {.p'.} = {.p \<sqinter> p'.}"

lemma (in domain) degree_one_root: assumes "subfield K R" and "p \<in> carrier (K[X])" and "degree p = 1" shows "eval p (\<ominus> (inv (lead_coeff p) \<otimes> (const_term p))) = \<zero>" and "inv (lead_coeff p) \<otimes> (const_term p) \<in> K"

lemma ennreal_fact: "ennreal (fact n) = fact n"

lemma ri_rhs_neq [simp]: "e \<noteq> RI(C,e');Cs \<leftarrow> e"

lemma eta_induct[case_names bottom N F, consumes 1]: "\<lbrakk> \<eta>: x \<mapsto> x'; \<lbrakk> x = \<bottom>; x' = \<bottom> \<rbrakk> \<Longrightarrow> P x x'; \<And>n. \<lbrakk> x = ValTT; x' = unitK\<cdot>ValKTT \<rbrakk> \<Longrightarrow> P x x'; \<And>n. \<lbrakk> x = ValFF; x' = unitK\<cdot>ValKFF \<rbrakk> \<Longrightarrow> P x x'; \<And>n. \<lbrakk> x = ValN\<cdot>n; x' = unitK\<cdot>(ValKN\<cdot>n) \<rbrakk> \<Longrightarrow> P x x'; \<And>f f'. \<lbrakk> x = ValF\<cdot>f; x' = unitK\<cdot>(ValKF\<cdot>f'); \<theta>: f \<mapsto> f' \<rbrakk> \<Longrightarrow> P x x' \<rbrakk> \<Longrightarrow> P x x'"

lemma RepFun_iff [iff]: "v \<^bold>\<in> (RepFun A f) \<longleftrightarrow> (\<exists>u. u \<^bold>\<in> A \<and> v = f u)"

lemma tauChainSupportDerivative: fixes P :: pi and P' :: pi assumes "P \<Longrightarrow>\<^sub>\<tau> P'" shows "((supp P')::name set) \<subseteq> (supp P)"

lemma inv_after_clear_Suc_nonempty[simp]: "inv_after_clear (as, abc_lm_s lm n (Suc (abc_lm_v lm n))) (s, b, []) ires = False"

lemma eqButUID_recvOuterFriends_UIDs: assumes "eqButUID s s1" and "uid \<noteq> UID \<or> aid \<noteq> AID" shows "(aid, uid) \<in>\<in> recvOuterFriendIDs s uid' \<longleftrightarrow> (aid, uid) \<in>\<in> recvOuterFriendIDs s1 uid'"

lemma set_pmf_binomial[simp]: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (binomial_pmf n p) = {..n}"

lemma "L c X \<subseteq> Lb c X"

lemma [code]: "CEQ(unit) = Some (\<lambda>_ _. True)"

lemma maximal_exactly_one: assumes \<open>consistent S\<close> and \<open>maximal S\<close> shows \<open>p \<in> S \<longleftrightarrow> (\<^bold>\<not> p) \<notin> S\<close>

lemma test_aa_increasing: "test p \<Longrightarrow> p \<le> --p"

lemma antilinear_to_conjugate_space[simp]: \<open>antilinear to_conjugate_space\<close>

lemma (in Module) mimg_module:"\<lbrakk>R module N; f \<in> mHom R M N\<rbrakk> \<Longrightarrow> R module (mimg R M N f)"

lemma fbox_g_ode_subset: assumes "\<And>s. s \<in> S \<Longrightarrow> 0 \<in> U s \<and> is_interval (U s) \<and> U s \<subseteq> T" shows "|x\<acute>= (\<lambda>t. f) & G on U S @ 0] Q = (\<lambda> s. s \<in> S \<longrightarrow> (\<forall>t\<in>(U s). (\<forall>\<tau>\<in>down (U s) t. G (\<phi> \<tau> s)) \<longrightarrow> Q (\<phi> t s)))"

lemma ball_trivial [simp]: "ball x 0 = {}"

lemma totient_code_naive [code]: "totient n = totient_naive n 0 n"

lemma b_e_check_single_top_not_bot_sound: assumes "type_update ts (to_ct_list t_in) (TopType []) = ts'" "ts \<noteq> Bot" "ts' \<noteq> Bot" shows "\<exists>tn. c_types_agree ts tn \<and> suffix t_in tn"

lemma connected_if_composable: assumes "\<nu> \<star> \<mu> \<noteq> null" shows "sources \<nu> = targets \<mu>"

lemma reduction_step: assumes "q \<in> nodes" "run w q" obtains (deferred) a where "ren a q" "[a] \<preceq>\<^sub>F\<^sub>I w" | (omitted) "{a. ren a q} \<subseteq> invisible" "Ind {a. ren a q} (sset w)"

lemma sub_tm_semantics [simp]: \<open>\<lparr>E, F\<rparr> (sub_tm s t) = \<lparr>\<lambda>n. \<lparr>E, F\<rparr> (s n), F\<rparr> t\<close>

lemma totalize_transient_iff: "(totalize F \<in> transient A) = (\<exists>act\<in>Acts F. A \<subseteq> Domain act & act``A \<subseteq> -A)"

lemma fmdom'_fmsubset_restrict_set: fixes X1 X2 and s :: "('a, bool) fmap" assumes "X1 \<subseteq> X2" "fmdom' s = X2" shows "fmdom' (fmrestrict_set X1 s) = X1"

lemma fps_X_neq_one [simp]: "(fps_X :: 'a :: zero_neq_one fps) \<noteq> 1"

lemma numeral_neq_fps_zero [simp]: "(numeral f :: 'a :: field_char_0 fps) \<noteq> 0"

lemma reduction: "0 < reduction" "reduction \<le> 1" "\<alpha> > 4/3 \<Longrightarrow> reduction < 1" "\<alpha> = 4/3 \<Longrightarrow> reduction = 1"

theorem LeftDerivationLadder_cut_appendix: assumes LDLadder: "LeftDerivationLadder (\<alpha>@\<delta>) D L \<gamma>" assumes ladder_i_in_\<alpha>: "ladder_i L 0 < length \<alpha>" shows "\<exists> E F \<gamma>1 \<gamma>2 L'. D = E@F \<and> \<gamma> = \<gamma>1 @ \<gamma>2 \<and> LeftDerivationLadder \<alpha> E L' \<gamma>1 \<and> derivation_ge F (length \<gamma>1) \<and> LeftDerivation \<delta> (derivation_shift F (length \<gamma>1) 0) \<gamma>2 \<and> length L' = length L \<and> ladder_i L' 0 = ladder_i L 0 \<and> ladder_last_j L' = ladder_last_j L"

lemma "Fr_2 \<F> \<Longrightarrow> \<forall>A B. Cl(A) \<longrightarrow> lCoP1\<^sup>A\<^sup>B(\<^bold>\<rightarrow>) \<^bold>\<not>\<^sup>I"

lemma inner_divide_right: fixes a :: "'a :: {real_inner,real_div_algebra}" shows "inner a (b / of_real m) = (inner a b) / m"

lemma lift_CFG: assumes wf:"CFGExit_wf sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses" and pd:"Postdomination sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit" shows "CFG src trg knd (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry (lift_get_proc get_proc Main) (lift_get_return_edges get_return_edges valid_edge sourcenode targetnode kind) procs Main"

lemma gctxtex_onp_idem [simp]: assumes "P \<box>\<^sub>G" and "\<And> C D. P C \<Longrightarrow> Q D \<Longrightarrow> Q (C \<circ>\<^sub>G\<^sub>c D)" shows "gctxtex_onp P (gctxtex_onp Q \<R>) = gctxtex_onp Q \<R>" (is "?Ls = ?Rs")

lemma bool_to_ternary_simp4: "eval_ternary_Not (bool_to_ternary X) = TernaryFalse \<longleftrightarrow> X"

lemma eval_fps_mult: fixes f g :: "'a :: {banach, real_normed_div_algebra, comm_ring_1} fps" assumes "norm z < fps_conv_radius f" "norm z < fps_conv_radius g" shows "eval_fps (f * g) z = eval_fps f z * eval_fps g z"

lemma cong: assumes "[a = b] (mod n)" shows "\<chi> a = \<chi> b"

lemma embedded_eq: "(A = B) \<Longrightarrow> (embedded_style A) = (embedded_style B)"

lemma eval_fds_power: fixes s :: "'a :: {nat_power, real_normed_field, banach, second_countable_topology}" assumes "fds_abs_converges f s" shows "eval_fds (f ^ n) s = eval_fds f s ^ n"

lemma nDASC[rule_format]: "wellformed_policy1 p \<longrightarrow> singleCombinators p \<longrightarrow> noDenyAll1 p"

lemma sup_state_opt_err : "(Err.le (sup_state_opt P)) s s"

lemma osp_bet__osp: assumes "A B C OSP P R" and "Bet P Q R" shows "A B C OSP P Q"

lemma mec_bot_imp_bot: assumes "mec \<phi> bot" shows "\<phi> = bot"

lemma (in weak_upper_semilattice) join_cong_r: assumes carr: "x \<in> carrier L" "y \<in> carrier L" "y' \<in> carrier L" and yy': "y .= y'" shows "x \<squnion> y .= x \<squnion> y'"

theorem (in finite_comm_group) Characters_iso: shows "G \<cong> Characters G"

lemma TokensAt_items_le[symmetric]: "TokensAt k I = TokensAt k (items_le k I)"

lemma Spy_see_shrK_D [dest!]: "[|Key (shrK A) \<in> parts (knows Spy evs); evs \<in> woolam|] ==> A \<in> bad"

lemma invertible_mat_mult_int: assumes "A = P * B" and "P \<in> carrier_mat n n" and "B \<in> carrier_mat n n" and "invertible_mat P" and "invertible_mat (map_mat rat_of_int B)" shows "invertible_mat (map_mat rat_of_int A)"

lemma list_succ_rotate[simp]: assumes "distinct xs" shows "list_succ (rotate n xs) = list_succ xs"

lemma OrL_inv': assumes "Or F G, \<Gamma> \<Rightarrow> \<Delta> \<down> n" shows "F,\<Gamma> \<Rightarrow> \<Delta> \<down> n \<and> G,\<Gamma> \<Rightarrow> \<Delta> \<down> n"

lemma ipresIGVar_termMOD[simp]: "ipresIGVar h termMOD MOD = ipresVar h MOD"

lemma takeWhile_take_has_property_nth: "\<lbrakk> n < length (takeWhile P xs) \<rbrakk> \<Longrightarrow> P (xs ! n)"

lemma randomElLemma: assumes "set list \<noteq> {}" shows "randomEl list random \<in> set list"

lemma rivest_perfect_hiding: "rivest_commit.perfect_hiding_ind_cpa \<A>"

lemma SeqReds: "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle> \<Longrightarrow> P \<turnstile> \<langle>e;;e\<^sub>2, s\<rangle> \<rightarrow>* \<langle>e';;e\<^sub>2, s'\<rangle>"

lemma Derives1_bound': "Derives1 a i r b \<Longrightarrow> i \<le> length b"

lemma ht_size_replicate[simp, intro!]: "ht_size (replicate n []) 0"

lemma scene_top_greatest: "X \<le> \<top>\<^sub>S"

lemma SDG_Def_parent_Def: "V \<in> Def\<^bsub>SDG\<^esub> n \<Longrightarrow> V \<in> Def (parent_node n)"

lemma nn_integral_less: assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M" assumes f: "(\<integral>\<^sup>+x. f x \<partial>M) \<noteq> \<infinity>" assumes ord: "AE x in M. f x \<le> g x" "\<not> (AE x in M. g x \<le> f x)" shows "(\<integral>\<^sup>+x. f x \<partial>M) < (\<integral>\<^sup>+x. g x \<partial>M)"

lemma fmrel_trancl_fmdom_eq: "(fmrel f)\<^sup>+\<^sup>+ xm ym \<Longrightarrow> fmdom xm = fmdom ym"

lemma E_null[simp]: "E (return_pmf 0) = 0"

lemma ccSUP_inf_distrib2: "countable A \<Longrightarrow> countable B \<Longrightarrow> inf (SUP a\<in>A. f a) (SUP b\<in>B. g b) = (SUP a\<in>A. SUP b\<in>B. inf (f a) (g b))"

lemma l2_step1_refines_step1: "{R12s} l1_step1 Ra A B, l2_step1 Ra A B {>R12s}"

lemma implements_oreach: "implements pi Sa Sc \<Longrightarrow> pi`(oreach Sc) \<subseteq> oreach Sa"

lemma typeof_ka: "typeof v \<noteq> None \<Longrightarrow> ka_Val v = {}"

lemma (in lbvc) cert_approx [intro?]: "\<lbrakk> pc < length \<phi>; c!pc \<noteq> \<bottom> \<rbrakk> \<Longrightarrow> c!pc = \<phi>!pc"

lemma vdistinct_vempty[intro, simp]: "vdistinct []\<^sub>\<circ>"

lemma to_fun_monom: assumes "c \<in> carrier R" assumes "x \<in> carrier R" shows "to_fun (monom P c n) x = c \<otimes> x [^] n"

lemma w_addrs_lift_start_heap_obs: "w_addrs (w_values P vs (map snd (lift_start_obs start_tid start_heap_obs))) \<subseteq> w_addrs vs"