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

Statement: stringlengths 7 24.3k
lemma multiple_is_wellformed: "incidence_system \<V> (multiple_blocks n)"
lemma Subseqs_compower [simp]: "Subseqs (A ^^ n) = Subseqs A ^^ n"
lemma i_shrink_last_nth: "0 < k \<Longrightarrow> (f \<div>\<^bsub>il\<^esub> k) n = f (n * k + k - Suc 0)"
lemma gtrancl_rel_sig: assumes "R \<subseteq> \<T>\<^sub>G \<F> \<times> \<T>\<^sub>G \<F>" shows "gtrancl_rel \<F> R \<subseteq> \<T>\<^sub>G \<F> \<times> \<T>\<^sub>G \<F>"
lemma sd_power_prop1 [simp]: "sd_power (x,y) (Suc i) = (x ^ (Suc i), \<Squnion>{\<alpha> (x ^ k) y\|k. k \<le> i})"
lemma impl_safe: assumes "inv_imps_work_sum c" and "inv_implications_nonneg c" and "\<And>loc. c_work c loc = {#}\<^sub>z" shows "impl_safe c"
lemma le_list_appendI: "\<And>b c d. a <=[r] b \<Longrightarrow> c <=[r] d \<Longrightarrow> a@c <=[r] b@d"
lemma Distr_Bag_Follows_lemma: "[\| G \<in> preserves distr.Out; D \<squnion> G \<in> Always {s. \<forall>elt \<in> set (distr.iIn s). elt < Nclients} \|] ==> D \<squnion> G \<in> Always {s. (\<Sum>i \<in> lessThan Nclients. bag_of (nths (distr.In s) {k. k < length (iIn s) & iIn s ! k = i})) = bag_of (nths (distr.In s) (lessThan (length (iIn s))))}"
lemma numbers_of_marks_replicate_True [simp]: "numbers_of_marks n (replicate m True) = {n..<n+m}"
lemma (in PolynRg) pol_expr_unique2:"\<lbrakk>pol_coeff S c; pol_coeff S d; fst c = fst d\<rbrakk> \<Longrightarrow> (polyn_expr R X (fst c) c = polyn_expr R X (fst d) d ) = (\<forall>j \<le> (fst c). (snd c) j = (snd d) j)"
lemma Zero_eq_map_rexp_iff[simp]: "Zero = map_rexp f x \<longleftrightarrow> x = Zero" "map_rexp f x = Zero \<longleftrightarrow> x = Zero"
lemma compute_truncate_up[code]: "truncate_up p (Ratreal r) = (let (a, b) = quotient_of r in rapprox_rat p a b)"
lemma makes_compatible_intro [intro]: "\<lbrakk> length cms\<^sub>1 = length cms\<^sub>2 \<and> length cms\<^sub>1 = length mems; (\<And> i \<sigma>. \<lbrakk> i < length cms\<^sub>1; dom \<sigma> = differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i \<rbrakk> \<Longrightarrow> (cms\<^sub>1 ! i, (fst (mems ! i)) [\<mapsto> \<sigma>]) \<approx> (cms\<^sub>2 ! i, (snd (mems ! i)) [\<mapsto> \<sigma>])); (\<And> i x. \<lbrakk> i < length cms\<^sub>1; mem\<^sub>1 x = mem\<^sub>2 x \<or> dma mem\<^sub>1 x = High \<or> x \<in> \<C> \<rbrakk> \<Longrightarrow> x \<notin> differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i); (length cms\<^sub>1 = 0 \<and> mem\<^sub>1 =\<^sup>l mem\<^sub>2) \<or> (\<forall> x. \<exists> i. i < length cms\<^sub>1 \<and> x \<notin> differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i) \<rbrakk> \<Longrightarrow> makes_compatible (cms\<^sub>1, mem\<^sub>1) (cms\<^sub>2, mem\<^sub>2) mems"
lemma analz_insert_HPair [simp]: "analz (insert (Hash[X] Y) H) = insert (Hash[X] Y) (insert (Hash\<lbrace>X,Y\<rbrace>) (analz (insert Y H)))"
lemma (in Graph) isSimplePath_cons[split_path_simps]: "isSimplePath s (e#p) t \<longleftrightarrow> (\<exists>u. e=(s,u) \<and> s\<noteq>u \<and> (s,u)\<in>E \<and> isSimplePath u p t \<and> s\<notin>set (pathVertices_fwd u p))"
lemma Ord_mult [simp]: "\<lbrakk>Ord y; Ord x\<rbrakk> \<Longrightarrow> Ord (x*y)"
lemma has_field_derivative_higher_deriv: "\<lbrakk>f holomorphic_on S; open S; x \<in> S\<rbrakk> \<Longrightarrow> ((deriv ^^ n) f has_field_derivative (deriv ^^ (Suc n)) f x) (at x)"
lemma elements_matI [intro]: "A \<in> carrier_mat nr nc \<Longrightarrow> i < nr \<Longrightarrow> j < nc \<Longrightarrow> a = A $$ (i,j) \<Longrightarrow> a \<in> elements_mat A"
lemma diff_inf_eq_sup: "a - inf b c = a + sup (- b) (- c)"
lemma Pring_add_zero: assumes "a \<in> carrier (Pring R I)" shows "a \<oplus>\<^bsub>Pring R I\<^esub> \<zero>\<^bsub>Pring R I\<^esub> = a" "\<zero>\<^bsub>Pring R I\<^esub> \<oplus>\<^bsub>Pring R I\<^esub> a = a"
lemma cons_chain: assumes "(x, S 0) \<in> r" and "chain r S" shows "chain r (case_nat x S)"
lemma induced_id_kernel: "c \<in> FreeGroup S \<Longrightarrow> induced_id (\<lceil>FreeGroup S\|c\|Q\<rceil>) = 0 \<Longrightarrow> c\<in>Q"
lemma simple_closed_path_norm_winding_number_inside: assumes "simple_path \<gamma>" "z \<in> inside(path_image \<gamma>)" shows "norm (winding_number \<gamma> z) = 1"
lemma update_set_swap: "Array.update a i v (set r v' h) = set r v' (Array.update a i v h)"
lemma unique_session_keys: "\<lbrakk>Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace>, X\<rbrace> \<in> set evs; Says Server A' \<lbrace>Crypt (shrK A') \<lbrace>Agent B', Key K, na', nb'\<rbrace>, X'\<rbrace> \<in> set evs; evs \<in> yahalom\<rbrakk> \<Longrightarrow> A=A' \<and> B=B' \<and> na=na' \<and> nb=nb'"
lemma Sigma_mset_Int_distrib2: "(SIGMAMSET i\<in>#I. A i \<inter># B i) = Sigma_mset I A \<inter># Sigma_mset I B"
lemma add_positive_le_reduce1: assumes dA: "A \<in> carrier_mat n n" and dB: "B \<in> carrier_mat n n" and dC: "C \<in> carrier_mat n n" and pB: "positive B" and le: "A + B \<le>\<^sub>L C" shows "A \<le>\<^sub>L C"
lemma client_progress_lemma: "[\| Client \<squnion> G \<in> Increasing giv; Client ok G \|] ==> Client \<squnion> G \<in> {s. h \<le> giv s & h pfixGe ask s} LeadsTo {s. h \<le> rel s}"
lemma bl_to_bin_ge2p_aux: "bl_to_bin_aux bs w \<ge> w * (2 ^ length bs)"
lemma scene_union_inter_minus: assumes "a \<in> scene_space" "b \<in> scene_space" shows "a \<squnion>\<^sub>S (b \<sqinter>\<^sub>S - a) = a \<squnion>\<^sub>S b"
lemma abs_summable_product: fixes x :: "'a \<Rightarrow> 'b::{real_normed_div_algebra,banach,second_countable_topology}" assumes x2_sum: "(\<lambda>i. (x i) * (x i)) abs_summable_on A" and y2_sum: "(\<lambda>i. (y i) * (y i)) abs_summable_on A" shows "(\<lambda>i. x i * y i) abs_summable_on A"
lemma ipurge_fail_aux_t_input_2: "ipurge_fail_aux_t_inv_2 I D U xs X \<lparr>Pol = I, Map = D, Doms = U, List = xs, ListOp = None, Set = X\<rparr>"
lemma "x = f3 p"
lemma enum_0_eq_Inf_finite: fixes N :: "nat set" assumes "finite N" "N \<noteq> {}" shows "enum N 0 = Inf N"
lemma [simp]: "gsi_\<alpha> (gsi_\<gamma> gs) = gs" "gsi_\<gamma> (gsi_\<alpha> gsi) = gsi"
lemma evimage_Compl: assumes "f \<in> carrier (R\<^bsup>n\<^esup>) \<rightarrow> carrier (R\<^bsup>m\<^esup>)" shows "(f \<inverse>\<^bsub>n\<^esub>(A\<^sup>c\<^bsub>m\<^esub>)) = ((f -` A)\<^sup>c\<^bsub>n\<^esub>) "
lemma parSym: fixes P :: pi and Q :: pi shows "P \<parallel> Q \<sim>\<^sub>e Q \<parallel> P"
lemma lookup_eq_None_iff: assumes invar: "invar_trie ((Trie vo kvs) :: ('key, 'val) trie)" shows "lookup_trie (Trie vo kvs) ks = None \<longleftrightarrow> (ks = [] \<and> vo = None) \<or> (\<exists>k ks'. ks = k # ks' \<and> (\<forall>t. (k, t) \<in> set kvs \<longrightarrow> lookup_trie t ks' = None))"
lemma Some_le [iff]: "(Some x <=_(le r) ox) = (\<exists>y. ox = Some y \<and> x <=_r y)"
lemma type_max_dma_type [simp]: "type_max (dma_type x) mem = dma mem x"
lemma compute_mult_scalar_pprod [code]: "mult_scalar_pprod (Pm_fmap (fmap_of_list xs)) q = (case xs of ((t, c) # ys) \<Rightarrow> (monom_mult_pprod c t q + mult_scalar_pprod (except (Pm_fmap (fmap_of_list ys)) {t}) q) \| _ \<Rightarrow> Pm_fmap fmempty)"
lemma bi_unique_union_r: assumes "bi_unique T" and "rel_set T a a'" and "rel_set T b b'" and "rel_set T (a \<union> b) xr" shows "a' \<union> b' = xr"
lemma reduced_run_length_1: assumes "reduced_run q v\<^sub>1 v\<^sub>2 l w w\<^sub>1 w\<^sub>2 u" shows "length v\<^sub>1 \<le> length w\<^sub>1"
lemma psubst_subst_fresh_switch: assumes "\<phi> \<in> fmla" "snd ` set txs \<subseteq> var" "fst ` set txs \<subseteq> trm" and "\<forall>x\<in>snd ` set txs. x \<notin> FvarsT s" "\<forall>t\<in>fst ` set txs. y \<notin> FvarsT t" and "distinct (map snd txs)" and "s \<in> trm" "y \<in> var" "y \<notin> snd ` set txs" shows "psubst (subst \<phi> s y) txs = subst (psubst \<phi> txs) s y"
lemma K: "\<lfloor>(\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<psi>)) \<^bold>\<rightarrow> (\<^bold>\<box>\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi>)\<rfloor>"
lemma update_kno_dsn_greater_zero: "\<And>rt dip ip dsn hops npre. 1 \<le> dsn \<Longrightarrow> 1 \<le> (sqn (update rt dip (dsn, kno, val, hops, ip, npre)) dip)"
lemma unify2EquivClasses_alt: assumes "R``{x} \<noteq> R``{y}" and inV: "y\<in>V" "x\<in>V" and "R\<subseteq>V\<times>V" and eq: "equiv V R" and [simp]: "finite V" shows "Suc (card (quotient V (per_union R x y))) = card (quotient V R)"
lemma incseq_of_rat_interlaced_seq: "\<lbrakk> (\<lambda>n. of_rat (r n)) \<longlonglongrightarrow> (x::real); (\<lambda>n. of_rat (s n)) \<longlonglongrightarrow> (x::real); \<forall>n. of_rat (r n) < x; \<forall>n. of_rat (s n) < x \<rbrakk> \<Longrightarrow> incseq (\<lambda>n. real_of_rat (interlaced_seq r s n))"
lemma scaleR_mono: "a \<le> b \<Longrightarrow> x \<le> y \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> x \<Longrightarrow> a \<^sub>R x \<le> b \<^sub>R y"
lemma "SML4 \<Longrightarrow> DM4 \<^bold>\<not>"
lemma cont_both[cont2cont,simp]: "cont f \<Longrightarrow> cont g \<Longrightarrow> cont (\<lambda> x. f x \<otimes>\<otimes> g x)"
lemma distinct_factor_complex_poly: "distinct (map fst (snd (factor_complex_poly p)))"
lemma ex_certain_iff_singleton_support: shows "(\<exists>x. pmf p x = 1) \<longleftrightarrow> card (set_pmf p) = 1"
lemma per_comp: assumes "r \<le>p (f w)\<^sup>\<omega>" shows "r \<bowtie> f w \<cdot> \<alpha>"
lemma freshChainAppend[simp]: fixes xvec :: "name list" and yvec :: "name list" and C :: "'a::fs_name" shows "(xvec@yvec) \<sharp>* C = ((xvec \<sharp>* C) \<and> (yvec \<sharp>* C))"
lemma le_arcsin_iff: assumes "x \<ge> -1" "x \<le> 1" "y \<ge> -pi/2" "y \<le> pi/2" shows "arcsin x \<ge> y \<longleftrightarrow> x \<ge> sin y"
lemma Completion: "[\| F \<in> A LeadsTo (A' \<union> C); F \<in> A' Co (A' \<union> C); F \<in> B LeadsTo (B' \<union> C); F \<in> B' Co (B' \<union> C) \|] ==> F \<in> (A \<inter> B) LeadsTo ((A' \<inter> B') \<union> C)"
lemma suf_less_Lyndon: assumes "w \<noteq> \<epsilon>" and "\<forall>s. (s \<le>ns w \<longrightarrow> s \<noteq> w \<longrightarrow> w <lex s)" shows "Lyndon w"
lemma polymul_norm: fixes p::"'a::field poly" shows "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polymul p q)"
lemma pre_below_pre_one: "x\<guillemotleft>-p \<le> x\<guillemotleft>1"
lemma word_minus_one_le_leq: "\<lbrakk> x - 1 < y \<rbrakk> \<Longrightarrow> x \<le> (y :: 'a :: len word)"
lemma return_pmf_bind_option: "return_pmf (Option.bind x f) = bind_spmf (return_pmf x) (return_pmf \<circ> f)"
lemma amor_delete: assumes "bst t" shows "T_delete a t + \<Phi>(Splay_Tree.delete a t) - \<Phi> t \<le> 6 * log 2 (size1 t) + 3"
lemma is_invar1: "A.is_invar invar1"
lemma wf_extend_oprod2: assumes "wf r" shows "wf {((x1,y), (x2,y)) . (x1, x2) \<in> r \<and> y \<in> A}"
lemma psi_convex2: assumes "psi a c b" assumes "psi a d b" assumes "0 \<le> u" "0 \<le> v" "u + v = 1" shows "psi a (u \<^sub>R c + v \<^sub>R d) b"
lemma compare_MkI_MkI [simp]: "compare\<cdot>(MkI\<cdot>x)\<cdot>(MkI\<cdot>y) = (if x < y then LT else if x > y then GT else EQ)"
lemma weakTransitiveCoinduct[case_names cStatEq cSim cExt cSym, case_conclusion bisim step, consumes 2]: fixes \<Psi> :: 'b and P :: "('a, 'b, 'c) psi" and Q :: "('a, 'b, 'c) psi" and X :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes p: "(\<Psi>, P, Q) \<in> X" and Eqvt: "eqvt X" and rStatEq: "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> insertAssertion (extractFrame P) \<Psi> \<simeq>\<^sub>F insertAssertion (extractFrame Q) \<Psi>" and rSim: "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> \<Psi> \<rhd> P \<leadsto>[({(\<Psi>, P, Q) \| \<Psi> P P' Q' Q. \<Psi> \<rhd> P \<sim> P' \<and> (\<Psi>, P', Q') \<in> X \<and> \<Psi> \<rhd> Q' \<sim> Q})] Q" and rExt: "\<And>\<Psi> P Q \<Psi>'. (\<Psi>, P, Q) \<in> X \<Longrightarrow> (\<Psi> \<otimes> \<Psi>', P, Q) \<in> X" and rSym: "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> (\<Psi>, Q, P) \<in> X" shows "\<Psi> \<rhd> P \<sim> Q"
lemma rel_interior_sum_gen: fixes S :: "'a \<Rightarrow> 'n::euclidean_space set" assumes "\<And>i. i\<in>I \<Longrightarrow> convex (S i)" shows "rel_interior (sum S I) = sum (\<lambda>i. rel_interior (S i)) I"
lemma resChainPerm[simp]: fixes perm :: "name prm" and lst :: "name list" and P :: pi shows "perm \<bullet> (resChain lst P) = resChain (perm \<bullet> lst) (perm \<bullet> P)"
lemma c_dom_sources: "c_sources I (c_dom D) u xps = c_sources I D u (map fst xps)"
lemma ccApprox_mono': "t \<sqsubseteq> t' \<Longrightarrow> ccApprox t \<sqsubseteq> ccApprox t'"
lemma analyzed_if_all_analyzed_in: assumes M: "\<forall>t \<in> M. analyzed_in t M" shows "analyzed M"
lemma si ([simp]):"s \<inter> s^-1 = {}"
lemma inc_vec_elems_max_two: "card (set\<^sub>v (inc_vec_of Vs bl)) \<le> 2"
lemma f_Exec_Stream_Acc_Output_nth: " \<lbrakk> 0 < k; n < length xs \<rbrakk> \<Longrightarrow> f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun xs c ! n = last_message (map output_fun ( f_Exec_Comp_Stream trans_fun (xs ! n # \<NoMsg>\<^bsup>k - Suc 0\<^esup>) ( f_Exec_Comp trans_fun (xs \<down> n \<odot>\<^sub>f k) c)))"
lemma Extend_empty_satisfies0: "\<lbrakk>Length \<AA> = 0; len P = 0\<rbrakk> \<Longrightarrow> Extend k i \<AA> P \<Turnstile>\<^sub>0 a \<longleftrightarrow> \<AA> \<Turnstile>\<^sub>0 a"
lemma runErrorT_bindET [simp]: "runErrorT\<cdot>(bindET\<cdot>m\<cdot>k) = bind\<cdot>(runErrorT\<cdot>m)\<cdot> (\<Lambda> n. case n of Err\<cdot>e \<Rightarrow> return\<cdot>(Err\<cdot>e) \| Ok\<cdot>x \<Rightarrow> runErrorT\<cdot>(k\<cdot>x))"
lemma tdghm_0_NTMap_vdomain[dg_cs_simps]: "\<D>\<^sub>\<circ> (tdghm_0 \<CC>\<lparr>NTMap\<rparr>) = 0"
lemma fps_integral0_sum: "fps_integral0 (sum f S) = sum (\<lambda>i. fps_integral0 (f i)) S"
lemma fds_commutes: assumes "\<And>m n. m > 0 \<Longrightarrow> n > 0 \<Longrightarrow> fds_nth f m * fds_nth g n = fds_nth g n * fds_nth f m" shows "f * g = g * f"
lemma primitive_part_smult_int: fixes f :: "int poly" shows "primitive_part (smult d f) = smult (sgn d) (primitive_part f)"
lemma utp_order_le [simp]: "le (utp_order T) = (\<sqsubseteq>)"
lemma "inj_on encode_fk_state (dom encode_fk_state)"
lemma D_seq : "D(P `;` Q) = {t1 @ t2 \|t1 t2. t1 \<in> D P \<and> tickFree t1 \<and> front_tickFree t2} \<union> {t1 @ t2 \|t1 t2. t1 @ [tick] \<in> T P \<and> t2 \<in> D Q}"
lemma field_differentiable_const [simp,derivative_intros]: "(\<lambda>z. c) field_differentiable F"
lemma "wf_sub_rel {}"
lemma generate_valid_stateful_policy_IFSACS_2_filter_compliant_stateful_ACS: assumes validReqs: "valid_reqs M" and wfG: "wf_graph G" and high_level_policy_valid: "all_security_requirements_fulfilled M G" and edgesList: "(set edgesList) \<subseteq> edges G" and Tau: "\<T> = generate_valid_stateful_policy_IFSACS_2 G M edgesList" shows "\<forall>F\<in>get_offending_flows (get_ACS M) (stateful_policy_to_network_graph \<T>). F \<subseteq> backflows (filternew_flows_state \<T>)"
lemma BIT_c2: assumes A: "x \<noteq> y" "init \<in> {[x,y],[y,x]}" "v \<in> lang (seq [Atom x, Times (Atom y) (Atom x), Star (Times (Atom y) (Atom x)), Atom x])" shows "T\<^sub>p_on_rand' BIT (type0 init x y) v = 0.75 * (length v - 1) - 0.5" (is ?T) and "config'_rand BIT (type0 init x y) v = (type0 init x y)" (is ?C)
lemma LIMSEQ_powreal_minus_nat: "a > 1 \<Longrightarrow> (\<lambda>n. a pow\<^sub>\<real> (-real n)) \<longlonglongrightarrow> 0"
lemma rel_spmf_mono: "\<lbrakk>rel_spmf A f g; \<And>x y. A x y \<Longrightarrow> B x y \<rbrakk> \<Longrightarrow> rel_spmf B f g"
lemma MakeCatComp: "f \<approx>>\<^bsub>C\<^esub> g \<Longrightarrow> f ;;\<^bsub>MakeCat C\<^esub> g = f ;;\<^bsub>C\<^esub> g"
lemma mk_next_pow_simp [simp, code_unfold]: "mk_next_pow 0 x = x" "mk_next_pow 1 x = mk_next x"
lemma db\<^sub>s\<^sub>s\<^sub>t_in_cases: assumes "(t,s) \<in> set (db'\<^sub>s\<^sub>s\<^sub>t A I D)" shows "(t,s) \<in> set D \<or> (\<exists>t' s'. insert\<langle>t',s'\<rangle> \<in> set A \<and> t = t' \<cdot> I \<and> s = s' \<cdot> I)"
lemma "let x = (1::nat) + y in let P = (if x > 0 then True else False) in False \<or> P = (x - 1 = y) \<or> (\<not>P \<longrightarrow> False)"
lemma integral_indicator: assumes "(S \<inter> T) \<in> lmeasurable" shows "integral T (indicator S) = measure lebesgue (S \<inter> T)"
lemma cong2_conga_obtuse__cong_conga2: assumes "Obtuse A B C" and "A B C CongA A' B' C'" and "Cong A C A' C'" and "Cong B C B' C'" shows "Cong B A B' A' \<and> B A C CongA B' A' C' \<and> B C A CongA B' C' A'"
lemma int_of_integer_sub [simp]: "int_of_integer (Num.sub k l) = Num.sub k l"
lemma prefix_up: "prefix (up a xs) (up a (xs @ ys))"
lemma gtt_only_prod_syms: "gtt_syms (gtt_only_prod \<G>) \|\<subseteq>\| gtt_syms \<G>"

lemma multiple_is_wellformed: "incidence_system \<V> (multiple_blocks n)"

lemma Subseqs_compower [simp]: "Subseqs (A ^^ n) = Subseqs A ^^ n"

lemma i_shrink_last_nth: "0 < k \<Longrightarrow> (f \<div>\<^bsub>il\<^esub> k) n = f (n * k + k - Suc 0)"

lemma gtrancl_rel_sig: assumes "R \<subseteq> \<T>\<^sub>G \<F> \<times> \<T>\<^sub>G \<F>" shows "gtrancl_rel \<F> R \<subseteq> \<T>\<^sub>G \<F> \<times> \<T>\<^sub>G \<F>"

lemma sd_power_prop1 [simp]: "sd_power (x,y) (Suc i) = (x ^ (Suc i), \<Squnion>{\<alpha> (x ^ k) y|k. k \<le> i})"

lemma impl_safe: assumes "inv_imps_work_sum c" and "inv_implications_nonneg c" and "\<And>loc. c_work c loc = {#}\<^sub>z" shows "impl_safe c"

lemma le_list_appendI: "\<And>b c d. a <=[r] b \<Longrightarrow> c <=[r] d \<Longrightarrow> a@c <=[r] b@d"

lemma Distr_Bag_Follows_lemma: "[| G \<in> preserves distr.Out; D \<squnion> G \<in> Always {s. \<forall>elt \<in> set (distr.iIn s). elt < Nclients} |] ==> D \<squnion> G \<in> Always {s. (\<Sum>i \<in> lessThan Nclients. bag_of (nths (distr.In s) {k. k < length (iIn s) & iIn s ! k = i})) = bag_of (nths (distr.In s) (lessThan (length (iIn s))))}"

lemma numbers_of_marks_replicate_True [simp]: "numbers_of_marks n (replicate m True) = {n..<n+m}"

lemma (in PolynRg) pol_expr_unique2:"\<lbrakk>pol_coeff S c; pol_coeff S d; fst c = fst d\<rbrakk> \<Longrightarrow> (polyn_expr R X (fst c) c = polyn_expr R X (fst d) d ) = (\<forall>j \<le> (fst c). (snd c) j = (snd d) j)"

lemma Zero_eq_map_rexp_iff[simp]: "Zero = map_rexp f x \<longleftrightarrow> x = Zero" "map_rexp f x = Zero \<longleftrightarrow> x = Zero"

lemma compute_truncate_up[code]: "truncate_up p (Ratreal r) = (let (a, b) = quotient_of r in rapprox_rat p a b)"

lemma makes_compatible_intro [intro]: "\<lbrakk> length cms\<^sub>1 = length cms\<^sub>2 \<and> length cms\<^sub>1 = length mems; (\<And> i \<sigma>. \<lbrakk> i < length cms\<^sub>1; dom \<sigma> = differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i \<rbrakk> \<Longrightarrow> (cms\<^sub>1 ! i, (fst (mems ! i)) [\<mapsto> \<sigma>]) \<approx> (cms\<^sub>2 ! i, (snd (mems ! i)) [\<mapsto> \<sigma>])); (\<And> i x. \<lbrakk> i < length cms\<^sub>1; mem\<^sub>1 x = mem\<^sub>2 x \<or> dma mem\<^sub>1 x = High \<or> x \<in> \<C> \<rbrakk> \<Longrightarrow> x \<notin> differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i); (length cms\<^sub>1 = 0 \<and> mem\<^sub>1 =\<^sup>l mem\<^sub>2) \<or> (\<forall> x. \<exists> i. i < length cms\<^sub>1 \<and> x \<notin> differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i) \<rbrakk> \<Longrightarrow> makes_compatible (cms\<^sub>1, mem\<^sub>1) (cms\<^sub>2, mem\<^sub>2) mems"

lemma analz_insert_HPair [simp]: "analz (insert (Hash[X] Y) H) = insert (Hash[X] Y) (insert (Hash\<lbrace>X,Y\<rbrace>) (analz (insert Y H)))"

lemma (in Graph) isSimplePath_cons[split_path_simps]: "isSimplePath s (e#p) t \<longleftrightarrow> (\<exists>u. e=(s,u) \<and> s\<noteq>u \<and> (s,u)\<in>E \<and> isSimplePath u p t \<and> s\<notin>set (pathVertices_fwd u p))"

lemma Ord_mult [simp]: "\<lbrakk>Ord y; Ord x\<rbrakk> \<Longrightarrow> Ord (x*y)"

lemma has_field_derivative_higher_deriv: "\<lbrakk>f holomorphic_on S; open S; x \<in> S\<rbrakk> \<Longrightarrow> ((deriv ^^ n) f has_field_derivative (deriv ^^ (Suc n)) f x) (at x)"

lemma elements_matI [intro]: "A \<in> carrier_mat nr nc \<Longrightarrow> i < nr \<Longrightarrow> j < nc \<Longrightarrow> a = A $$ (i,j) \<Longrightarrow> a \<in> elements_mat A"

lemma diff_inf_eq_sup: "a - inf b c = a + sup (- b) (- c)"

lemma Pring_add_zero: assumes "a \<in> carrier (Pring R I)" shows "a \<oplus>\<^bsub>Pring R I\<^esub> \<zero>\<^bsub>Pring R I\<^esub> = a" "\<zero>\<^bsub>Pring R I\<^esub> \<oplus>\<^bsub>Pring R I\<^esub> a = a"

lemma cons_chain: assumes "(x, S 0) \<in> r" and "chain r S" shows "chain r (case_nat x S)"

lemma induced_id_kernel: "c \<in> FreeGroup S \<Longrightarrow> induced_id (\<lceil>FreeGroup S|c|Q\<rceil>) = 0 \<Longrightarrow> c\<in>Q"

lemma simple_closed_path_norm_winding_number_inside: assumes "simple_path \<gamma>" "z \<in> inside(path_image \<gamma>)" shows "norm (winding_number \<gamma> z) = 1"

lemma update_set_swap: "Array.update a i v (set r v' h) = set r v' (Array.update a i v h)"

lemma unique_session_keys: "\<lbrakk>Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace>, X\<rbrace> \<in> set evs; Says Server A' \<lbrace>Crypt (shrK A') \<lbrace>Agent B', Key K, na', nb'\<rbrace>, X'\<rbrace> \<in> set evs; evs \<in> yahalom\<rbrakk> \<Longrightarrow> A=A' \<and> B=B' \<and> na=na' \<and> nb=nb'"

lemma Sigma_mset_Int_distrib2: "(SIGMAMSET i\<in>#I. A i \<inter># B i) = Sigma_mset I A \<inter># Sigma_mset I B"

lemma add_positive_le_reduce1: assumes dA: "A \<in> carrier_mat n n" and dB: "B \<in> carrier_mat n n" and dC: "C \<in> carrier_mat n n" and pB: "positive B" and le: "A + B \<le>\<^sub>L C" shows "A \<le>\<^sub>L C"

lemma client_progress_lemma: "[| Client \<squnion> G \<in> Increasing giv; Client ok G |] ==> Client \<squnion> G \<in> {s. h \<le> giv s & h pfixGe ask s} LeadsTo {s. h \<le> rel s}"

lemma bl_to_bin_ge2p_aux: "bl_to_bin_aux bs w \<ge> w * (2 ^ length bs)"

lemma scene_union_inter_minus: assumes "a \<in> scene_space" "b \<in> scene_space" shows "a \<squnion>\<^sub>S (b \<sqinter>\<^sub>S - a) = a \<squnion>\<^sub>S b"

lemma abs_summable_product: fixes x :: "'a \<Rightarrow> 'b::{real_normed_div_algebra,banach,second_countable_topology}" assumes x2_sum: "(\<lambda>i. (x i) * (x i)) abs_summable_on A" and y2_sum: "(\<lambda>i. (y i) * (y i)) abs_summable_on A" shows "(\<lambda>i. x i * y i) abs_summable_on A"

lemma ipurge_fail_aux_t_input_2: "ipurge_fail_aux_t_inv_2 I D U xs X \<lparr>Pol = I, Map = D, Doms = U, List = xs, ListOp = None, Set = X\<rparr>"

lemma "x = f3 p"

lemma enum_0_eq_Inf_finite: fixes N :: "nat set" assumes "finite N" "N \<noteq> {}" shows "enum N 0 = Inf N"

lemma [simp]: "gsi_\<alpha> (gsi_\<gamma> gs) = gs" "gsi_\<gamma> (gsi_\<alpha> gsi) = gsi"

lemma evimage_Compl: assumes "f \<in> carrier (R\<^bsup>n\<^esup>) \<rightarrow> carrier (R\<^bsup>m\<^esup>)" shows "(f \<inverse>\<^bsub>n\<^esub>(A\<^sup>c\<^bsub>m\<^esub>)) = ((f -` A)\<^sup>c\<^bsub>n\<^esub>) "

lemma parSym: fixes P :: pi and Q :: pi shows "P \<parallel> Q \<sim>\<^sub>e Q \<parallel> P"

lemma lookup_eq_None_iff: assumes invar: "invar_trie ((Trie vo kvs) :: ('key, 'val) trie)" shows "lookup_trie (Trie vo kvs) ks = None \<longleftrightarrow> (ks = [] \<and> vo = None) \<or> (\<exists>k ks'. ks = k # ks' \<and> (\<forall>t. (k, t) \<in> set kvs \<longrightarrow> lookup_trie t ks' = None))"

lemma Some_le [iff]: "(Some x <=_(le r) ox) = (\<exists>y. ox = Some y \<and> x <=_r y)"

lemma type_max_dma_type [simp]: "type_max (dma_type x) mem = dma mem x"

lemma compute_mult_scalar_pprod [code]: "mult_scalar_pprod (Pm_fmap (fmap_of_list xs)) q = (case xs of ((t, c) # ys) \<Rightarrow> (monom_mult_pprod c t q + mult_scalar_pprod (except (Pm_fmap (fmap_of_list ys)) {t}) q) | _ \<Rightarrow> Pm_fmap fmempty)"

lemma bi_unique_union_r: assumes "bi_unique T" and "rel_set T a a'" and "rel_set T b b'" and "rel_set T (a \<union> b) xr" shows "a' \<union> b' = xr"

lemma reduced_run_length_1: assumes "reduced_run q v\<^sub>1 v\<^sub>2 l w w\<^sub>1 w\<^sub>2 u" shows "length v\<^sub>1 \<le> length w\<^sub>1"

lemma psubst_subst_fresh_switch: assumes "\<phi> \<in> fmla" "snd ` set txs \<subseteq> var" "fst ` set txs \<subseteq> trm" and "\<forall>x\<in>snd ` set txs. x \<notin> FvarsT s" "\<forall>t\<in>fst ` set txs. y \<notin> FvarsT t" and "distinct (map snd txs)" and "s \<in> trm" "y \<in> var" "y \<notin> snd ` set txs" shows "psubst (subst \<phi> s y) txs = subst (psubst \<phi> txs) s y"

lemma K: "\<lfloor>(\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<psi>)) \<^bold>\<rightarrow> (\<^bold>\<box>\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi>)\<rfloor>"

lemma update_kno_dsn_greater_zero: "\<And>rt dip ip dsn hops npre. 1 \<le> dsn \<Longrightarrow> 1 \<le> (sqn (update rt dip (dsn, kno, val, hops, ip, npre)) dip)"

lemma unify2EquivClasses_alt: assumes "R``{x} \<noteq> R``{y}" and inV: "y\<in>V" "x\<in>V" and "R\<subseteq>V\<times>V" and eq: "equiv V R" and [simp]: "finite V" shows "Suc (card (quotient V (per_union R x y))) = card (quotient V R)"

lemma incseq_of_rat_interlaced_seq: "\<lbrakk> (\<lambda>n. of_rat (r n)) \<longlonglongrightarrow> (x::real); (\<lambda>n. of_rat (s n)) \<longlonglongrightarrow> (x::real); \<forall>n. of_rat (r n) < x; \<forall>n. of_rat (s n) < x \<rbrakk> \<Longrightarrow> incseq (\<lambda>n. real_of_rat (interlaced_seq r s n))"

lemma scaleR_mono: "a \<le> b \<Longrightarrow> x \<le> y \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> x \<Longrightarrow> a *\<^sub>R x \<le> b *\<^sub>R y"

lemma "SML4 \<Longrightarrow> DM4 \<^bold>\<not>"

lemma cont_both[cont2cont,simp]: "cont f \<Longrightarrow> cont g \<Longrightarrow> cont (\<lambda> x. f x \<otimes>\<otimes> g x)"

lemma distinct_factor_complex_poly: "distinct (map fst (snd (factor_complex_poly p)))"

lemma ex_certain_iff_singleton_support: shows "(\<exists>x. pmf p x = 1) \<longleftrightarrow> card (set_pmf p) = 1"

lemma per_comp: assumes "r \<le>p (f w)\<^sup>\<omega>" shows "r \<bowtie> f w \<cdot> \<alpha>"

lemma freshChainAppend[simp]: fixes xvec :: "name list" and yvec :: "name list" and C :: "'a::fs_name" shows "(xvec@yvec) \<sharp>* C = ((xvec \<sharp>* C) \<and> (yvec \<sharp>* C))"

lemma le_arcsin_iff: assumes "x \<ge> -1" "x \<le> 1" "y \<ge> -pi/2" "y \<le> pi/2" shows "arcsin x \<ge> y \<longleftrightarrow> x \<ge> sin y"

lemma Completion: "[| F \<in> A LeadsTo (A' \<union> C); F \<in> A' Co (A' \<union> C); F \<in> B LeadsTo (B' \<union> C); F \<in> B' Co (B' \<union> C) |] ==> F \<in> (A \<inter> B) LeadsTo ((A' \<inter> B') \<union> C)"

lemma suf_less_Lyndon: assumes "w \<noteq> \<epsilon>" and "\<forall>s. (s \<le>ns w \<longrightarrow> s \<noteq> w \<longrightarrow> w <lex s)" shows "Lyndon w"

lemma polymul_norm: fixes p::"'a::field poly" shows "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polymul p q)"

lemma pre_below_pre_one: "x\<guillemotleft>-p \<le> x\<guillemotleft>1"

lemma word_minus_one_le_leq: "\<lbrakk> x - 1 < y \<rbrakk> \<Longrightarrow> x \<le> (y :: 'a :: len word)"

lemma return_pmf_bind_option: "return_pmf (Option.bind x f) = bind_spmf (return_pmf x) (return_pmf \<circ> f)"

lemma amor_delete: assumes "bst t" shows "T_delete a t + \<Phi>(Splay_Tree.delete a t) - \<Phi> t \<le> 6 * log 2 (size1 t) + 3"

lemma is_invar1: "A.is_invar invar1"

lemma wf_extend_oprod2: assumes "wf r" shows "wf {((x1,y), (x2,y)) . (x1, x2) \<in> r \<and> y \<in> A}"

lemma psi_convex2: assumes "psi a c b" assumes "psi a d b" assumes "0 \<le> u" "0 \<le> v" "u + v = 1" shows "psi a (u *\<^sub>R c + v *\<^sub>R d) b"

lemma compare_MkI_MkI [simp]: "compare\<cdot>(MkI\<cdot>x)\<cdot>(MkI\<cdot>y) = (if x < y then LT else if x > y then GT else EQ)"

lemma weakTransitiveCoinduct[case_names cStatEq cSim cExt cSym, case_conclusion bisim step, consumes 2]: fixes \<Psi> :: 'b and P :: "('a, 'b, 'c) psi" and Q :: "('a, 'b, 'c) psi" and X :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes p: "(\<Psi>, P, Q) \<in> X" and Eqvt: "eqvt X" and rStatEq: "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> insertAssertion (extractFrame P) \<Psi> \<simeq>\<^sub>F insertAssertion (extractFrame Q) \<Psi>" and rSim: "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> \<Psi> \<rhd> P \<leadsto>[({(\<Psi>, P, Q) | \<Psi> P P' Q' Q. \<Psi> \<rhd> P \<sim> P' \<and> (\<Psi>, P', Q') \<in> X \<and> \<Psi> \<rhd> Q' \<sim> Q})] Q" and rExt: "\<And>\<Psi> P Q \<Psi>'. (\<Psi>, P, Q) \<in> X \<Longrightarrow> (\<Psi> \<otimes> \<Psi>', P, Q) \<in> X" and rSym: "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> (\<Psi>, Q, P) \<in> X" shows "\<Psi> \<rhd> P \<sim> Q"

lemma rel_interior_sum_gen: fixes S :: "'a \<Rightarrow> 'n::euclidean_space set" assumes "\<And>i. i\<in>I \<Longrightarrow> convex (S i)" shows "rel_interior (sum S I) = sum (\<lambda>i. rel_interior (S i)) I"

lemma resChainPerm[simp]: fixes perm :: "name prm" and lst :: "name list" and P :: pi shows "perm \<bullet> (resChain lst P) = resChain (perm \<bullet> lst) (perm \<bullet> P)"

lemma c_dom_sources: "c_sources I (c_dom D) u xps = c_sources I D u (map fst xps)"

lemma ccApprox_mono': "t \<sqsubseteq> t' \<Longrightarrow> ccApprox t \<sqsubseteq> ccApprox t'"

lemma analyzed_if_all_analyzed_in: assumes M: "\<forall>t \<in> M. analyzed_in t M" shows "analyzed M"

lemma si (*[simp]*):"s \<inter> s^-1 = {}"

lemma inc_vec_elems_max_two: "card (set\<^sub>v (inc_vec_of Vs bl)) \<le> 2"

lemma f_Exec_Stream_Acc_Output_nth: " \<lbrakk> 0 < k; n < length xs \<rbrakk> \<Longrightarrow> f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun xs c ! n = last_message (map output_fun ( f_Exec_Comp_Stream trans_fun (xs ! n # \<NoMsg>\<^bsup>k - Suc 0\<^esup>) ( f_Exec_Comp trans_fun (xs \<down> n \<odot>\<^sub>f k) c)))"

lemma Extend_empty_satisfies0: "\<lbrakk>Length \<AA> = 0; len P = 0\<rbrakk> \<Longrightarrow> Extend k i \<AA> P \<Turnstile>\<^sub>0 a \<longleftrightarrow> \<AA> \<Turnstile>\<^sub>0 a"

lemma runErrorT_bindET [simp]: "runErrorT\<cdot>(bindET\<cdot>m\<cdot>k) = bind\<cdot>(runErrorT\<cdot>m)\<cdot> (\<Lambda> n. case n of Err\<cdot>e \<Rightarrow> return\<cdot>(Err\<cdot>e) | Ok\<cdot>x \<Rightarrow> runErrorT\<cdot>(k\<cdot>x))"

lemma tdghm_0_NTMap_vdomain[dg_cs_simps]: "\<D>\<^sub>\<circ> (tdghm_0 \<CC>\<lparr>NTMap\<rparr>) = 0"

lemma fps_integral0_sum: "fps_integral0 (sum f S) = sum (\<lambda>i. fps_integral0 (f i)) S"

lemma fds_commutes: assumes "\<And>m n. m > 0 \<Longrightarrow> n > 0 \<Longrightarrow> fds_nth f m * fds_nth g n = fds_nth g n * fds_nth f m" shows "f * g = g * f"

lemma primitive_part_smult_int: fixes f :: "int poly" shows "primitive_part (smult d f) = smult (sgn d) (primitive_part f)"

lemma utp_order_le [simp]: "le (utp_order T) = (\<sqsubseteq>)"

lemma "inj_on encode_fk_state (dom encode_fk_state)"

lemma D_seq : "D(P `;` Q) = {t1 @ t2 |t1 t2. t1 \<in> D P \<and> tickFree t1 \<and> front_tickFree t2} \<union> {t1 @ t2 |t1 t2. t1 @ [tick] \<in> T P \<and> t2 \<in> D Q}"

lemma field_differentiable_const [simp,derivative_intros]: "(\<lambda>z. c) field_differentiable F"

lemma "wf_sub_rel {}"

lemma generate_valid_stateful_policy_IFSACS_2_filter_compliant_stateful_ACS: assumes validReqs: "valid_reqs M" and wfG: "wf_graph G" and high_level_policy_valid: "all_security_requirements_fulfilled M G" and edgesList: "(set edgesList) \<subseteq> edges G" and Tau: "\<T> = generate_valid_stateful_policy_IFSACS_2 G M edgesList" shows "\<forall>F\<in>get_offending_flows (get_ACS M) (stateful_policy_to_network_graph \<T>). F \<subseteq> backflows (filternew_flows_state \<T>)"

lemma BIT_c2: assumes A: "x \<noteq> y" "init \<in> {[x,y],[y,x]}" "v \<in> lang (seq [Atom x, Times (Atom y) (Atom x), Star (Times (Atom y) (Atom x)), Atom x])" shows "T\<^sub>p_on_rand' BIT (type0 init x y) v = 0.75 * (length v - 1) - 0.5" (is ?T) and "config'_rand BIT (type0 init x y) v = (type0 init x y)" (is ?C)

lemma LIMSEQ_powreal_minus_nat: "a > 1 \<Longrightarrow> (\<lambda>n. a pow\<^sub>\<real> (-real n)) \<longlonglongrightarrow> 0"

lemma rel_spmf_mono: "\<lbrakk>rel_spmf A f g; \<And>x y. A x y \<Longrightarrow> B x y \<rbrakk> \<Longrightarrow> rel_spmf B f g"

lemma MakeCatComp: "f \<approx>>\<^bsub>C\<^esub> g \<Longrightarrow> f ;;\<^bsub>MakeCat C\<^esub> g = f ;;\<^bsub>C\<^esub> g"

lemma mk_next_pow_simp [simp, code_unfold]: "mk_next_pow 0 x = x" "mk_next_pow 1 x = mk_next x"

lemma db\<^sub>s\<^sub>s\<^sub>t_in_cases: assumes "(t,s) \<in> set (db'\<^sub>s\<^sub>s\<^sub>t A I D)" shows "(t,s) \<in> set D \<or> (\<exists>t' s'. insert\<langle>t',s'\<rangle> \<in> set A \<and> t = t' \<cdot> I \<and> s = s' \<cdot> I)"

lemma "let x = (1::nat) + y in let P = (if x > 0 then True else False) in False \<or> P = (x - 1 = y) \<or> (\<not>P \<longrightarrow> False)"

lemma integral_indicator: assumes "(S \<inter> T) \<in> lmeasurable" shows "integral T (indicator S) = measure lebesgue (S \<inter> T)"

lemma cong2_conga_obtuse__cong_conga2: assumes "Obtuse A B C" and "A B C CongA A' B' C'" and "Cong A C A' C'" and "Cong B C B' C'" shows "Cong B A B' A' \<and> B A C CongA B' A' C' \<and> B C A CongA B' C' A'"

lemma int_of_integer_sub [simp]: "int_of_integer (Num.sub k l) = Num.sub k l"

lemma prefix_up: "prefix (up a xs) (up a (xs @ ys))"

lemma gtt_only_prod_syms: "gtt_syms (gtt_only_prod \<G>) |\<subseteq>| gtt_syms \<G>"