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lemma periodic_imp_uniform_discrete:
assumes "periodic_set S \<delta>"
shows "uniform_discrete S" |
theorem proportionality_equiv: "equiv local.non_zero_vectors local.proportionality" |
lemma outstanding_not_volatile_Read\<^sub>s\<^sub>b_refs_conv: "outstanding_refs (Not \<circ> is_volatile_Read\<^sub>s\<^sub>b) sb =
outstanding_refs is_Write\<^sub>s\<^sub>b sb \<union> outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b sb" |
lemma vec_index_vCons_Suc [simp]:
fixes v :: "'a vec"
shows "vCons a v $ Suc n = v $ n" |
lemma Spy_not_see_encrypted_key:
"[| Says Server B
\<lbrace>NA, Crypt (shrK A) \<lbrace>NA, Key K\<rbrace>,
Crypt (shrK B) \<lbrace>NB, Key K\<rbrace>\<rbrace> \<in> set evs;
Notes Spy \<lbrace>NA, NB, Key K\<rbrace> \<notin> set evs;
A \<notin> bad; B \<notin> bad; evs \<in> otway |]
==> Key K \<notin> analz (knows Spy evs)" |
theorem
"\<lbrakk>iface_packet_check ifl pii \<noteq> None;
mlf (case next_hop (routing_table_semantics rt (p_dst pii)) of None \<Rightarrow> p_dst pii | Some a \<Rightarrow> a) \<noteq> None\<rbrakk> \<Longrightarrow>
\<exists>x. map_option (\<lambda>p. p\<lparr>p_l2dst := x\<rparr>) (simple_linux_router_nol12 rt fw pii) = simple_linux_router rt fw mlf ifl pii" |
lemma subst_cv_id [simp]: "subst_cv A a (V_var a) = A" |
lemma l9_10:
assumes "\<not> Col A P Q"
shows "\<exists> C. P Q TS A C" |
lemma p0_sets[measurable_cong]: "x \<in> space Ms \<Longrightarrow> sets (p 0 (\<lambda>_. undefined,x)) = sets Ma" |
lemma
"{(1::nat, 2), (2, 3), (3, 4), (4, 5)}\<^sup>* `` {1} = {1, 2, 3, 4, 5}"
"{(1::nat, 2), (2, 3), (3, 4), (4, 5)}\<^sup>+ `` {1} = {2, 3, 4, 5}" |
lemma s_ns_mul_ext_trans:
assumes "trans s" "trans ns" "compatible_l ns s" "compatible_r ns s" "refl ns"
and "(A, B) \<in> s_mul_ext ns s"
and "(B, C) \<in> ns_mul_ext ns s"
shows "(A, C) \<in> s_mul_ext ns s" |
lemma equivD1: "x \<le> y" if "x \<approx> y" |
lemma distinct_eq_append: "distinct_eq eq (xs @ ys) = (distinct_eq eq xs \<and> distinct_eq eq ys \<and> (\<forall> x \<in> set xs. \<forall> y \<in> set ys. \<not> (eq y x)))" |
lemma \<psi>\<^sub>1_is_state:
assumes "n \<ge> 1"
shows "state (n+1) (\<psi>\<^sub>1 n)" |
lemma embed_ge_0[simp,intro]:
"0 \<le> \<guillemotleft>G\<guillemotright> s" |
lemma poly_scalar_mult_iter:
assumes "\<one> \<noteq>\<zero>"
assumes "P \<in> Pring_set R I"
assumes "k \<in> carrier R"
assumes "n \<in> carrier R"
shows "poly_scalar_mult R k (poly_scalar_mult R n P) = poly_scalar_mult R (k \<otimes> n) P" |
lemma borel_measurable_count_space[measurable (raw)]:
"f \<in> borel_measurable (count_space S)" |
lemma dverts_reachable1_if_dom_children_aux_root:
assumes "\<forall>v\<in>dverts (Node r xs). \<exists>x\<in>set r0 \<union> X \<union> path_lverts (Node r xs) (hd v). x \<rightarrow>\<^bsub>T\<^esub> hd v"
and "\<forall>y\<in>X. \<exists>x\<in>set r0. x \<rightarrow>\<^sup>+\<^bsub>T\<^esub> y"
and "forward r"
shows "\<forall>y\<in>set r. \<exists>x\<in>set r0. x \<rightarrow>\<^sup>+\<^bsub>T\<^esub> y" |
lemma ltakeWhile_K_False [simp]: "ltakeWhile (\<lambda>_. False) xs = LNil" |
lemma finite_intersection_of_idempot [simp]:
"finite intersection_of finite intersection_of P = finite intersection_of P" |
lemma one_side_symmetry:
assumes "P Q OS A B"
shows "P Q OS B A" |
lemma swapEnvIm_preserves_wls:
assumes "wlsEnv rho"
shows "wlsEnv (swapEnvIm xs x y rho)" |
lemma Arg2pi_Ln:
assumes "0 < Arg2pi z" shows "Arg2pi z = Im(Ln(-z)) + pi" |
lemma less_le_trans:
assumes "x <\<^sub>a y" and "y \<le>\<^sub>a z"
shows "x <\<^sub>a z" |
lemma af_imp_almost_full_on:
assumes "af A P"
shows "almost_full_on P A" |
lemma prog_sifum_secure_cont_def2:
"prog_sifum_secure_cont cmds \<equiv>
(\<forall> mem\<^sub>1 mem\<^sub>2. INIT mem\<^sub>1 \<and> INIT mem\<^sub>2 \<and> mem\<^sub>1 =\<^sup>l mem\<^sub>2 \<longrightarrow>
(\<forall> sched cms\<^sub>1' mem\<^sub>1'.
(cmds, mem\<^sub>1) \<rightarrow>\<^bsub>sched\<^esub> (cms\<^sub>1', mem\<^sub>1') \<longrightarrow>
(\<exists> cms\<^sub>2' mem\<^sub>2'. (cmds, mem\<^sub>2) \<rightarrow>\<^bsub>sched\<^esub> (cms\<^sub>2', mem\<^sub>2')) \<and>
(\<forall> cms\<^sub>2' mem\<^sub>2'. (cmds, mem\<^sub>2) \<rightarrow>\<^bsub>sched\<^esub> (cms\<^sub>2', mem\<^sub>2') \<longrightarrow>
map snd cms\<^sub>1' = map snd cms\<^sub>2' \<and>
length cms\<^sub>2' = length cms\<^sub>1' \<and>
(\<forall> x. dma mem\<^sub>1' x = Low \<and> (x \<in> \<C> \<or> (\<forall> i < length cms\<^sub>1'.
x \<notin> snd (cms\<^sub>1' ! i) AsmNoReadOrWrite)) \<longrightarrow> mem\<^sub>1' x = mem\<^sub>2' x))))" |
lemma eqExcPID2_imp2:
assumes "reach s" and "eqExcPID2 s s1" and "pid \<noteq> PID \<or> PID \<noteq> pid"
shows "getReviewersReviews s cid pid = getReviewersReviews s1 cid pid" |
lemma analz_image_freshK [rule_format (no_asm)]:
"evs \<in> bankerberos \<Longrightarrow>
\<forall>K KK. KK \<subseteq> - (range shrK) \<longrightarrow>
(Key K \<in> analz (Key`KK \<union> (spies evs))) =
(K \<in> KK | Key K \<in> analz (spies evs))" |
lemma cblinfun_of_mat_plus:
defines "nA \<equiv> length (canonical_basis :: 'a::{basis_enum,complex_normed_vector} list)"
and "nB \<equiv> length (canonical_basis :: 'b::{basis_enum,complex_normed_vector} list)"
assumes [simp,intro]: "M \<in> carrier_mat nB nA" and [simp,intro]: "N \<in> carrier_mat nB nA"
shows "(cblinfun_of_mat (M + N) :: 'a \<Rightarrow>\<^sub>C\<^sub>L 'b) = ((cblinfun_of_mat M + cblinfun_of_mat N))" |
lemma ennreal_right_diff_distrib:
fixes a b c :: ennreal
assumes "a \<noteq> top"
shows "a * (b - c) = a * b - a * c" |
lemma zero_vector_sup_distributive:
"zero_vector x \<Longrightarrow> sup_distributive x" |
lemma refresh_1_apply[simp]: "fst (the (refresh_1 f p)) = fst (the (f p))" |
lemma n_eq_0: "s \<in> S \<Longrightarrow> cfg \<in> cfg_on s \<Longrightarrow> v cfg = 0 \<Longrightarrow> n s = 0" |
theorem complement_of_product_is_sum_of_complements:
"O x y \<Longrightarrow> x \<oplus> y \<noteq> u \<Longrightarrow> \<midarrow>(x \<otimes> y) = (\<midarrow>x) \<oplus> (\<midarrow>y)" |
lemma "filterlim (\<lambda>x::real. x powr (1 / sqrt x)) (at_right 1) at_top" |
lemma wf_filterQuery:
assumes "I \<subseteq> V"
assumes "card I \<ge> 1"
assumes "rwf_query n V Qp Qn"
assumes "QQp = filterQuery I Qp"
assumes "QQn = filterQueryNeg I Qn"
shows "wf_query n I QQp QQn" "non_empty_query QQp" "covering I QQp" |
lemma is_unit_i [simp]:
\<open>\<i> dvd 1\<close> |
lemma b_e_check_single_weaken_type:
assumes "check_single \<C> e (Type tn) = (Type tm)"
shows "check_single \<C> e (Type (ts@tn)) = Type (ts@tm)" |
lemma contour_integrable_subpath_refl [iff]: "f contour_integrable_on (subpath u u g)" |
lemma le_combine_matrix:
assumes
"f 0 0 = 0"
"\<forall>a b c d. a <= b & c <= d \<longrightarrow> f a c <= f b d"
"A <= B"
"C <= D"
shows
"combine_matrix f A C <= combine_matrix f B D" |
lemma propositions[PLM]:
"[\<^bold>\<exists> p . \<^bold>\<box>(p \<^bold>\<equiv> p') in v]" |
lemma (in Ring) r_apow_Suc:"ideal R I \<Longrightarrow> I\<^bsup>R (an (Suc 0))\<^esup> = I" |
lemma retract_cancel1_aux:
assumes "cancel1 ys (map f xs)"
shows "\<exists>zs. cancel1 zs xs \<and> ys = map f zs \<and> set zs \<subseteq> set xs" |
lemma sing_pow_fac': assumes "a \<noteq> b" and "w \<in> [a]*" shows "\<not> ([b] \<le>f w)" |
lemma arctan_upper_12:
assumes "x \<le> 0" shows "arctan(x) \<le> arctan_upper_12 x" |
lemma rbt_comp_union_with_key: "rbt_comp_union_with_key = ord.rbt_union_with_key (lt_of_comp c)" |
lemma component_edges_subset: "edges (component_of g r) \<subseteq> edges g" |
lemma last_prog_append_Prog\<^sub>s\<^sub>b:
"\<And>x. last_prog x (sb@[Prog\<^sub>s\<^sub>b p p' mis]) = p'" |
lemma(in ring_functions) function_smult_assoc1:
assumes "a \<in> carrier R"
assumes "b \<in> carrier R"
assumes "f \<in> carrier F"
shows "b \<odot>\<^bsub>F\<^esub> (a \<odot>\<^bsub>F\<^esub> f) = (b \<otimes> a)\<odot>\<^bsub>F\<^esub>f" |
lemma unfold_plussub_lift2:
"e1 +_(lift2 f) e2 == lift2 f e1 e2" |
lemma convex_condition_imp_convex:
assumes "is_convex_condition I"
shows "is_convex (I \<alpha> \<beta>)" |
lemma Jacobi_mult_left [simp]: "Jacobi (a * b) n = Jacobi a n * Jacobi b n" |
lemma (in group) r_inv [simp]:
"x \<in> carrier G ==> x \<otimes> inv x = \<one>" |
lemma "(12::nat) * 11 = 132" |
lemma (in graph) graph_block_size: assumes "bl \<in># arcs_blocks" shows "card bl = 2" |
lemma G_af_simp[simp]:
"\<^bold>G (af \<phi> w) = \<^bold>G \<phi>" |
lemma WHILET_refine[refine]:
assumes R0: "(x,x')\<in>R"
assumes COND_REF: "\<And>x x'. \<lbrakk> (x,x')\<in>R \<rbrakk> \<Longrightarrow> b x = b' x'"
assumes STEP_REF:
"\<And>x x'. \<lbrakk> (x,x')\<in>R; b x; b' x' \<rbrakk> \<Longrightarrow> f x \<le> \<Down>R (f' x')"
shows "WHILET b f x \<le>\<Down>R (WHILET b' f' x')" |
lemma invpst_baliL: "invpst l \<Longrightarrow> invpst r \<Longrightarrow> invpst (baliL l a r)" |
lemma disjE2: "\<lbrakk> P \<or> Q; \<lbrakk>P; \<not>Q\<rbrakk> \<Longrightarrow> R; Q \<Longrightarrow> R \<rbrakk> \<Longrightarrow> R" |
lemma (in lbv) bottom_le [simp, intro!]: "\<bottom> \<sqsubseteq>\<^sub>r x" |
lemma push_bit_Suc [simp]:
"push_bit (Suc n) a = push_bit n (a * 2)" |
lemma normalize_smult:
fixes c :: "'a :: {normalization_semidom_multiplicative, idom_divide}"
shows "normalize (smult c p) = smult (normalize c) (normalize p)" |
lemma has_derivative_sequence_Lipschitz:
fixes f :: "nat \<Rightarrow> 'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
assumes "convex S"
and "\<And>n x. x \<in> S \<Longrightarrow> ((f n) has_derivative (f' n x)) (at x within S)"
and nle: "\<And>e. e > 0 \<Longrightarrow> \<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"
and "e > 0"
shows "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>y\<in>S.
norm ((f m x - f n x) - (f m y - f n y)) \<le> e * norm (x - y)" |
lemma vint_struct_\<A>[vint_struct_simps]: "\<SS>\<^sub>\<int>\<lparr>\<A>\<rparr> = \<int>\<^sub>\<circ>" |
lemma power_diff':
assumes "m \<ge> n" "x \<noteq> 0"
shows "x ^ (m - n) = (x ^ m div x ^ n :: 'a :: unique_euclidean_semiring)" |
lemma vec_count_MV_Rel_direct:
assumes "MV_Rel v1 v2"
shows "count_vec v2 i = count_vec (vec_mod v1) (to_int_mod_ring i)" |
lemma omega_sum_aux_less: "omega_sum_aux k ms < \<omega> \<up> k" |
lemma not_\<S>_reps:
"(l, R) \<notin> \<S> \<Longrightarrow> reps (l, R) \<notin> S" |
lemma is_unit_hom_iff[simp]: "hom x dvd 1 \<longleftrightarrow> x dvd 1" |
lemma pathVertices_edge_old: "isPath u p v \<Longrightarrow> e \<in> set p \<Longrightarrow>
\<exists>vs1 vs2. pathVertices u p = vs1 @ fst e # snd e # vs2" |
lemma disc_I [simp]:
assumes "in_ocircline H z"
shows "z \<in> disc H" |
lemma (in cring)
assumes "x \<in> carrier R" "y \<in> carrier R"
shows "(x \<oplus> y) [^] (3::nat) =
x [^] (3::nat) \<oplus> y [^] (3::nat) \<oplus> \<guillemotleft>3\<guillemotright> \<otimes> x [^] (2::nat) \<otimes> y \<oplus> \<guillemotleft>3\<guillemotright> \<otimes> y [^] (2::nat) \<otimes> x" |
lemma noDAsep[rule_format]: "noDenyAll p \<Longrightarrow> noDenyAll (separate p)" |
lemma row_to_poly_zeroD:
assumes "distinct ts" and "dim_vec r = length ts" and "row_to_poly ts r = 0"
shows "r = 0\<^sub>v (length ts)" |
lemma fixes e :: "('a, 'b, 'addr) exp" and es :: "('a, 'b, 'addr) exp list"
shows D_None [iff]: "\<D> e None"
and Ds_None [iff]: "\<D>s es None" |
lemma substitutes_on_antimono:
assumes "substitutes_on B f"
assumes "A \<subseteq> B"
shows "substitutes_on A f" |
lemma (in Interpretation) FunctionalExpr:
"\<And>i j A B.\<lbrakk>L\<lbrakk>Vx : A \<turnstile> e : B\<rbrakk> \<rightarrow> i; L\<lbrakk>Vx : A \<turnstile> e : B\<rbrakk> \<rightarrow> j\<rbrakk> \<Longrightarrow> i = j" |
lemma point_in_vector_sup:
assumes "point p"
and "vector v"
and "regular v"
and "p \<le> v \<squnion> w"
shows "p \<le> v \<or> p \<le> w" |
lemma tkt_thread_eq:
assumes R: "A.reachable (c,n,ts)"
assumes A: "t<N" "has_ticket (ts t) k"
shows "tkt_thread ts k = t" |
lemma comp_lassocr\<^sub>C: "((conv1 |\<^sub>= conv2) |\<^sub>= conv3) \<odot> lassocr\<^sub>C = lassocr\<^sub>C \<odot> (conv1 |\<^sub>= conv2 |\<^sub>= conv3)" |
lemma bidual_embedding_apply[simp]: \<open>(bidual_embedding *\<^sub>V x) *\<^sub>V f = f *\<^sub>V x\<close> |
lemma uint_word_of_int_eq:
\<open>uint (word_of_int k :: 'a::len word) = take_bit LENGTH('a) k\<close> |
lemma ord_elems:
assumes "finite (carrier G)" "a \<in> carrier G"
shows "{a[^]x | x. x \<in> (UNIV :: nat set)} = {a[^]x | x. x \<in> {0 .. ord a - 1}}" (is "?L = ?R") |
lemma (in group) generate_one_switched_eqI:
assumes "A \<subseteq> carrier G" "a \<in> A" "B = (A - {a}) \<union> {b}"
and "b \<in> generate G A" "a \<in> generate G B"
shows "generate G A = generate G B" |
lemma good_context_2 :
"good_context (s::(('a::len) sparc_state)) \<and>
fetch_instruction (delayed_pool_write s) = Inr v1 \<and>
\<not>(\<exists>v2. (decode_instruction v1::(Exception list + instruction)) = Inr v2)
\<Longrightarrow> False" |
lemma clop_Inf_closed_var:
fixes f :: "'a::complete_lattice \<Rightarrow> 'a"
shows "clop f \<Longrightarrow> f \<circ> Inf \<circ> (`) f = Inf \<circ> (`) f" |
lemma(in padic_fields) prod_equal_val_imp_equal_val:
assumes "a \<in> nonzero Q\<^sub>p"
assumes "b \<in> carrier Q\<^sub>p"
assumes "c \<in> carrier Q\<^sub>p"
assumes "val (a \<otimes> b) = val (a \<otimes> c)"
shows "val b = val c" |
lemma incomplete_block_proper_subset: "incomplete_block bl \<Longrightarrow> bl \<subset> \<V>" |
lemma G_in_rell: "rell_G L1 L2 = (\<lambda>x y. \<exists>z. (set1_G z \<subseteq> {(x, y). L1 x y} \<and>
set2_G z \<subseteq> {(x, y). L2 x y}) \<and> mapl_G fst fst z = x \<and> mapl_G snd snd z = y)" |
lemma no_zero_height_Exec_derivs1: "(M,l,(os,S,H),0,h,v):Exec \<Longrightarrow> False" |
lemma extended_hyperb_ineq_4_points' [mono_intros]:
"Min {extended_Gromov_product_at (e::'a::Gromov_hyperbolic_space) x y, extended_Gromov_product_at e y z, extended_Gromov_product_at e z t} \<le> extended_Gromov_product_at e x t + 2 * deltaG(TYPE('a))" |
lemma fresh_star_set_eq: "set xs \<sharp>* c = xs \<sharp>* c" |
lemma B_id_secure:
assumes "\<And>tr vl vl1. B (V tr) vl1 \<Longrightarrow> validSystemTrace tr \<Longrightarrow> B_id (V tr) vl1"
shows "secure" |
lemma spmf_bind: "spmf (p \<bind> f) y = \<integral> x. spmf (f x) y \<partial>measure_spmf p" |
lemma \<Z>_srngE[elim]:
assumes "\<Z>_srng \<alpha> \<SS>"
obtains "\<Z>_vfsequence \<alpha> \<SS>"
and "vcard \<SS> = 3\<^sub>\<nat>"
and "\<Z>_csgrp \<alpha> [\<SS>\<lparr>\<A>\<rparr>, \<SS>\<lparr>vplus\<rparr>]\<^sub>\<circ>"
and "\<Z>_sgrp \<alpha> [\<SS>\<lparr>\<A>\<rparr>, \<SS>\<lparr>vmult\<rparr>]\<^sub>\<circ>"
and "\<And>a b c. \<lbrakk> a \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>; b \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>; c \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr> \<rbrakk> \<Longrightarrow>
(a +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b) *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c = (a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c) +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> (b *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c)"
and "\<And>a b c. \<lbrakk> a \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>; b \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>; c \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr> \<rbrakk> \<Longrightarrow>
a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> (b +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c) = (a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b) +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> (a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c)" |
lemma change_eval_lt:
fixes x y:: "real"
assumes "((aEvalUni (LessUni (a,b,c)) x \<noteq> aEvalUni (LessUni (a,b,c)) y) \<and> x < y)"
shows "(\<exists>w. x \<le> w \<and> w \<le> y \<and> a*w^2 + b*w + c = 0)" |
lemma GF_advice_a2:
"\<lbrakk>X \<subseteq> \<G>\<F> \<phi> w; w \<Turnstile>\<^sub>n \<phi>[X]\<^sub>\<nu>\<rbrakk> \<Longrightarrow> w \<Turnstile>\<^sub>n \<phi>" |
lemma NSBseq_isUb: "NSBseq X \<Longrightarrow> \<exists>U::real. isUb UNIV {x. \<exists>n. X n = x} U" |
lemma t_heap_of_A_log_bound:
"t_heap_of_A xs \<le> length xs * (nlog2 (length xs + 1) + 1)" |
lemma L\<^sub>a_bounded:
"bounded (range (\<lambda>a. L\<^sub>a a (apply_bfun v) s))" |
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