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lemma vec_plus_Cons:
shows "vec_plus (a # as) (b # bs) = (a+b) # vec_plus as bs" |
lemma permutes_induct [consumes 2, case_names id swap]:
\<open>P p\<close> if \<open>p permutes S\<close> \<open>finite S\<close>
and id: \<open>P id\<close>
and swap: \<open>\<And>a b p. a \<in> S \<Longrightarrow> b \<in> S \<Longrightarrow> p permutes S \<Longrightarrow> P p \<Longrightarrow> P (transpose a b \<circ> p)\<close> |
lemma param_opt_lowest_tops_lem: "param_opt i t e = None \<Longrightarrow> \<exists>y. lowest_tops [i,t,e] = Some y" |
lemma fps_neg_nth [simp]: "(- f) $ n = - (f $ n)" |
lemma pref_compE [elim]: assumes "u \<bowtie> v" obtains "u \<le>p v" | "v \<le>p u" |
lemma BSIA_implies_IA:
"(BSIA \<rho> \<V> Tr\<^bsub>ES\<^esub>) \<Longrightarrow> (IA \<rho> \<V> Tr\<^bsub>ES\<^esub>)" |
lemma Ide_imp_Ide_last [simp]:
assumes "Ide T"
shows "R.ide (last T)" |
lemma unique_favorites: "i \<in> agents \<Longrightarrow> favorites R i = {favorite R i}" |
lemma infiniteTr_strong_coind[consumes 1, case_names sub]:
assumes *: "phi tr" and
**: "\<And> tr. phi tr \<Longrightarrow> \<exists> tr' \<in> set (sub tr). phi tr' \<or> infiniteTr tr'"
shows "infiniteTr tr" |
theorem f_Exec_Stream_causal: "
xs \<down> n = ys \<down> n \<Longrightarrow>
(f_Exec_Comp_Stream trans_fun xs c) \<down> n = (f_Exec_Comp_Stream trans_fun ys c) \<down> n" |
lemma squarefreeD: "squarefree n \<Longrightarrow> x ^ 2 dvd n \<Longrightarrow> x dvd 1" |
lemma lookup_map_key_PP: "lookup (Poly_Mapping.map_key PP p) t = lookup p (PP t)" |
lemma dual_zero:
"bot\<^sup>d = top" |
lemma plus_eq_infty_iff_enat: "(m::enat) + n = \<infinity> \<longleftrightarrow> m=\<infinity> \<or> n=\<infinity>" |
lemma defNode_eq[intro]:
assumes "n \<in> set (\<alpha>n g)" "v \<in> allDefs g n"
shows "defNode g v = n" |
lemma "test (do {
TEST_ID \<leftarrow> return ''test4-should-not-exist'';
e \<leftarrow> Document_getElementById_document . createElement(''div'');
e . setAttribute(''id'', TEST_ID);
tmp0 \<leftarrow> Document_getElementById_document . getElementById(TEST_ID);
assert_equals(tmp0, None, ''should be null'');
tmp1 \<leftarrow> Document_getElementById_document . body;
tmp1 . appendChild(e);
tmp2 \<leftarrow> Document_getElementById_document . getElementById(TEST_ID);
assert_equals(tmp2, e, ''should be the appended element'')
}) Document_getElementById_heap" |
lemma empty_\<Z>[simp]: "k > length Doc \<Longrightarrow> \<Z> k u = {}" |
lemma (in Group) contain_commutator:"\<lbrakk>G \<guillemotright> H; (commutators G) \<subseteq> H\<rbrakk> \<Longrightarrow> G \<triangleright> H" |
lemma Prefixes_union [simp]: "Prefixes (A \<union> B) = Prefixes A \<union> Prefixes B" |
lemma "(sep_empty imp ((((p4 ** p1) \<longrightarrow>* ((p8 ** sep_empty ) \<longrightarrow>* p0)) imp
(p1 \<longrightarrow>* (p1 ** ((p4 ** p1) \<longrightarrow>* ((p8 ** sep_empty ) \<longrightarrow>* p0))))) \<longrightarrow>*
(((p4 ** p1) \<longrightarrow>* ((p8 ** sep_empty ) \<longrightarrow>* p0)) imp (p1 \<longrightarrow>* (((p1 ** p4) \<longrightarrow>*
((p8 ** sep_empty ) \<longrightarrow>* p0)) ** p1)))))
(h::'a::heap_sep_algebra)" |
lemma NR\<^sub>N_0[simp]: "NR\<^sub>N 0 u l = 0" |
lemma restr_empty [simp]:
"restrict {} al = []"
"restrict A [] = []" |
lemma outputSupportDerivative:
fixes P :: pi
and a :: name
and b :: name
and P' :: pi
assumes "P \<longmapsto>a[b] \<prec> P'"
shows "(supp P') \<subseteq> ((supp P)::name set)" |
lemma T_Assign_helper:
"\<lbrakk> eval_bool p \<tau> ; l' = (l \<tau>) (Inl v := eval_word e \<tau>) ; \<tau>' = \<tau> (| l := l' |) \<rbrakk>
\<Longrightarrow> step_t a (\<tau>, (Assign (Var v) e, p)) \<tau>'" |
lemma bounded_bilinear_sq_mtx_vec_mult: "bounded_bilinear (\<lambda>A s. A *\<^sub>V s)" |
lemma list_empty: "list idle = [] \<longleftrightarrow> is_empty idle" |
theorem ratfps_nth_bigo:
fixes q :: "complex poly"
assumes "R > 0"
assumes roots1: "\<And>z. z \<in> ball 0 (1 / R) \<Longrightarrow> poly q z \<noteq> 0"
assumes roots2: "\<And>z. z \<in> sphere 0 (1 / R) \<Longrightarrow> poly q z = 0 \<Longrightarrow> order z q \<le> Suc k"
shows "fps_nth (fps_of_poly p / fps_of_poly q) \<in> O(\<lambda>n. of_nat n ^ k * of_real R ^ n)" |
lemma inside_iter:
"inside z n = (0 < Iter pdec2 n z)" |
lemma prime_Esigma_mult: assumes "prime m" and "prime n" and "m \<noteq> n"
shows "Esigma (m*n) = (Esigma n)*(Esigma m)" |
lemma (in bibd) bibd_to_pbdI[intro]:
assumes "\<Lambda> = 1"
shows "k_PBD \<V> \<B> \<k>" |
lemma spec_imp_exchange_spec: "FullSpec s \<Longrightarrow> cri.spec (smap exchange_config s)" |
lemma r_result_converg':
assumes "eval r_univ [i, x] \<down>= v"
shows "\<exists>t. (\<forall>t'\<ge>t. eval r_result [t', i, x] \<down>= Suc v) \<and> (\<forall>t'<t. eval r_result [t', i, x] \<down>= 0)" |
lemma degree_poly_mult_rat[simp]: assumes "r \<noteq> 0" shows "degree (poly_mult_rat r p) = degree p" |
lemma le_convert:"\<lbrakk>a = b; a \<le> c\<rbrakk> \<Longrightarrow> b \<le> c" |
lemma b_least2_aux2: "b_least2 f x y < y \<Longrightarrow> b_least2 f x (Suc y) = b_least2 f x y" |
lemma e_type_local:
assumes "\<S>\<bullet>\<C> \<turnstile> [Local n i vs es] : (ts _> ts')"
shows "\<exists>tls. i < length (s_inst \<S>)
\<and> length tls = n
\<and> (\<S>\<bullet>((s_inst \<S>)!i)\<lparr>local := (local ((s_inst \<S>)!i)) @ (map typeof vs), return := Some tls\<rparr> \<turnstile> es : ([] _> tls))
\<and> ts' = ts @ tls" |
lemma ref_WHILEIT_invarI:
assumes "I s \<Longrightarrow> M \<le> \<Down>R (WHILEIT I b f s)"
shows "M \<le> \<Down>R (WHILEIT I b f s)" |
lemma wf_until_aux_Cons: "wf_until_aux \<sigma> n R pos \<phi> I \<psi> (a # aux) ne \<Longrightarrow>
wf_until_aux \<sigma> n R pos \<phi> I \<psi> aux (Suc ne)" |
lemma PO_l2_inv7 [iff]: "reach l2 \<subseteq> l2_inv7" |
lemma (in Group) r_unit_sg:"\<lbrakk>G \<guillemotright> H; h \<in> H\<rbrakk> \<Longrightarrow> h \<cdot> \<one> = h" |
lemma append_fun_equiv:
"\<lbrakk> t1' \<sim> t1; t2' \<sim> t2 \<rbrakk> \<Longrightarrow> t1' @ t2' \<sim> t1 @ t2" |
lemma has_fps_expansion_exp_neg1 [fps_expansion_intros]:
"(\<lambda>x::'a :: {banach, real_normed_field}. exp (-x)) has_fps_expansion fps_exp (-1)" |
lemma hd_o_singl[simp]: "hd o singl = id" |
lemma subst_is_fresh[simp]:
assumes "atom y \<sharp> z"
shows
"atom y \<sharp> e[y ::= z]"
and
"atom ` domA \<Gamma> \<sharp>* y \<Longrightarrow> atom y \<sharp> \<Gamma>[y::h=z]" |
lemma Iagree_sub:"\<And>I J A B . A \<subseteq> B \<Longrightarrow> Iagree I J B \<Longrightarrow> Iagree I J A" |
lemma owhile_NF [wp]:
"\<lbrakk>\<And>a. ovalidNF (\<lambda>s. I a s \<and> C a s) (B a) I;
\<And>a m. ovalid (\<lambda>s. I a s \<and> C a s \<and> M a s = m) (B a) (\<lambda>r s. M r s < m);
\<And>a s. \<lbrakk>I a s; \<not> C a s\<rbrakk> \<Longrightarrow> Q a s\<rbrakk>
\<Longrightarrow> ovalidNF (I a) (owhile_inv C B a I M) Q" |
lemma set_of_times: "set_of (X * Y) = {x * y | x y. x \<in> set_of X \<and> y \<in> set_of Y}"
for X Y::"'a :: {linordered_ring, real_normed_algebra, linear_continuum_topology} interval" |
lemma language_state_elim[elim]:
assumes "w \<in> LS M q"
obtains r
where "path M (w || r) q" "length w = length r" |
lemma "\<lfloor>\<^bold>\<box>\<^bold>\<exists>\<^sup>E G \<^bold>\<leftrightarrow> \<^bold>\<box>\<^bold>\<exists>\<^sup>E \<^bold>\<down>G\<rfloor>" |
lemma silent_moves_preds_transfers_path:
"\<lbrakk>S,f \<turnstile> (n,s) =as\<Rightarrow>\<^sub>\<tau> (n',s'); valid_node n\<rbrakk>
\<Longrightarrow> preds (map f as) s \<and> transfers (map f as) s = s' \<and> n -as\<rightarrow>* n'" |
lemma g1_leadsto_a1:
"\<turnstile> \<box>[(N1 \<or> N2) \<and> \<not>beta1]_(x,y,sem,pc1,pc2) \<and> SF(N1)_(x,y,sem,pc1,pc2) \<and> \<box>#True
\<longrightarrow> (pc1 = #g \<leadsto> pc1 = #a)" |
lemma HF_Ord [simp]: "Ord i \<Longrightarrow> HF i = W i" |
lemma comp_one_vector:
"one_vector x \<Longrightarrow> one_vector y \<Longrightarrow> one_vector (x * y)" |
lemma inv_register_tensor[simp]:
assumes [simp]: \<open>iso_register F\<close> \<open>iso_register G\<close>
shows \<open>inv (F \<otimes>\<^sub>r G) = inv F \<otimes>\<^sub>r inv G\<close> |
lemma reduce_system_matrix_equation_preserved_R:
fixes p:: "real poly"
fixes qs :: "real poly list"
fixes subsets :: "(nat list*nat list) list"
fixes signs :: "rat list list"
assumes nonzero: "p \<noteq> 0"
assumes welldefined_signs: "well_def_signs (length qs) signs"
assumes welldefined_subsets: "all_list_constr_R (subsets) (length qs)"
assumes distinct_signs: "distinct signs"
assumes all_info: "set (characterize_consistent_signs_at_roots p qs) \<subseteq> set(signs)"
assumes match: "satisfy_equation_R p qs subsets signs"
assumes invertible_mat: "invertible_mat (matrix_A_R signs subsets)"
shows "satisfy_equation_R p qs (get_subsets_R (reduce_system_R p (qs, ((matrix_A_R signs subsets), (subsets, signs)))))
(get_signs_R (reduce_system_R p (qs, ((matrix_A_R signs subsets), (subsets, signs)))))" |
lemma sylow_greater_zero:
shows "card (subgroups_of_size (p ^ a)) > 0" |
lemma Converter_parametric [transfer_rule]:
"((A ===> rel_gpv'' (rel_prod B (rel_converter A B C R)) C R) ===> rel_converter A B C R) Converter Converter" |
lemma zero_less_arctan_iff [simp]: "0 < arctan x \<longleftrightarrow> 0 < x" |
lemma qtable_wf_tupleD: "qtable n A P Q X \<Longrightarrow> \<forall>x\<in>X. wf_tuple n A x" |
lemma similar_nice_def: "x \<triangleleft>\<triangleright> y \<longleftrightarrow> (x = \<bottom> \<and> y = \<bottom> \<or> (\<exists> b. x = B\<cdot>(Discr b) \<and> y = CB\<cdot>(Discr b)) \<or> (\<exists> f g. x = Fn\<cdot>f \<and> y = CFn\<cdot>g \<and> (\<forall> a b. a \<triangleleft>\<triangleright> b\<cdot>C\<^sup>\<infinity> \<longrightarrow> f\<cdot>a \<triangleleft>\<triangleright> g\<cdot>b\<cdot>C\<^sup>\<infinity>)))"
(is "?L \<longleftrightarrow> ?R") |
lemma fold_A_comp_fixespointwise:
"fixespointwise (fold_A \<circ> opp_fold_A) (\<Union> (fold_A \<turnstile> A))" |
lemma abs_ik_append: "(ik\<^sub>s\<^sub>s\<^sub>t (A@B) \<cdot>\<^sub>s\<^sub>e\<^sub>t I) \<cdot>\<^sub>\<alpha>\<^sub>s\<^sub>e\<^sub>t a = (ik\<^sub>s\<^sub>s\<^sub>t A \<cdot>\<^sub>s\<^sub>e\<^sub>t I) \<cdot>\<^sub>\<alpha>\<^sub>s\<^sub>e\<^sub>t a \<union> (ik\<^sub>s\<^sub>s\<^sub>t B \<cdot>\<^sub>s\<^sub>e\<^sub>t I) \<cdot>\<^sub>\<alpha>\<^sub>s\<^sub>e\<^sub>t a" |
lemma CAS_\<tau>red1r_xt3:
"\<tau>red1gr uf P t h (e, xs) (e', xs') \<Longrightarrow> \<tau>red1gr uf P t h (Val v\<bullet>compareAndSwap(D\<bullet>F, Val v', e), xs) (Val v\<bullet>compareAndSwap(D\<bullet>F, Val v', e'), xs')" |
lemma order_listI2[intro!] : "order r A \<Longrightarrow> order(Listn.le r) (\<Union>{nlists n A |n. n \<le> mxs})" |
lemma inessential_eq_continuous_logarithm:
fixes f :: "'a::real_normed_vector \<Rightarrow> complex"
shows "(\<exists>a. homotopic_with_canon (\<lambda>h. True) S (-{0}) f (\<lambda>t. a)) \<longleftrightarrow>
(\<exists>g. continuous_on S g \<and> (\<forall>x \<in> S. f x = exp(g x)))"
(is "?lhs \<longleftrightarrow> ?rhs") |
lemma finite_tvsT[simp]: "finite (tvsT T)" |
lemma polyneg0:
fixes p::"'a::ring_1 poly"
shows "isnpolyh p n \<Longrightarrow> (~\<^sub>p p) = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p" |
lemma A_authenticates_B:
"\<lbrakk> Crypt K (Number Ta) \<in> parts (spies evs);
Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, X\<rbrace> \<in> parts (spies evs);
Key K \<notin> analz (spies evs);
A \<notin> bad; B \<notin> bad; evs \<in> bankerb_gets \<rbrakk>
\<Longrightarrow> Says B A (Crypt K (Number Ta)) \<in> set evs" |
lemma mat_O_mult_beta:
"mat_O *\<^sub>v \<beta> = - \<beta>" |
lemma most_recent_write_recent:
"\<lbrakk> P,E \<turnstile> r \<leadsto>mrw w; adal \<in> action_loc P E r; w' \<in> write_actions E; adal \<in> action_loc P E w' \<rbrakk>
\<Longrightarrow> E \<turnstile> w' \<le>a w \<or> E \<turnstile> r \<le>a w'" |
lemma pairs_stone:
"(x,y) \<in> pairs \<Longrightarrow> pairs_sup (pairs_uminus (x,y)) (pairs_uminus (pairs_uminus (x,y))) = pairs_top" |
lemma act_sup_distr: "\<alpha> (x \<squnion> y) p = (\<alpha> x p) \<squnion> (\<alpha> y p)" |
lemma wadjust_loop_start_Oc_via_Bk_move[simp]:
"wadjust_loop_right_move2 m rs (c, Bk # list) \<Longrightarrow> wadjust_loop_start m rs (c, Oc # list)" |
lemma g_diff_alt:
"g_diff s1 s2 = iterate_diff_set s1 s1.delete (s2.iteratei s2)" |
lemma spmf_cond_bind_spmf_fst [simp]:
"spmf (cond_bind_spmf_fst p f x) i = spmf p i * spmf (f i) x / spmf (bind_spmf p f) x" |
lemma path_edge_rev_cases[case_names no_use split]:
assumes "path (insert (u,v) E) w p x"
obtains
"path E w p x"
| p1 p2 where "path (insert (u,v) E) w p1 u" "path E v p2 x" |
lemma bless_eq_trans [trans]:
assumes "s \<le>\<^sub>b t" and "t \<le>\<^sub>b u"
shows "s \<le>\<^sub>b u" |
lemma compl_set [code]:
"- set xs = List.coset xs" |
lemma alive_arr_loc [simp]:
"isArrV x \<Longrightarrow> alive (ref (x.[i])) s = alive x s" |
lemma inst_output_consistency:
assumes rvpeq: "rvpeq (rcurrent s) s t"
and current_eq: "rcurrent s = rcurrent t"
shows "routput_f s a = routput_f t a" |
lemma [trans] : "P' \<subseteq> P \<Longrightarrow> \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A \<Longrightarrow> \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P' c Q,A" |
lemma Arcsin_unique:
"\<lbrakk>sin z = w; \<bar>Re z\<bar> < pi/2 \<or> (\<bar>Re z\<bar> = pi/2 \<and> Im z = 0)\<rbrakk> \<Longrightarrow> Arcsin w = z" |
lemma val_ineq_cancel_le':
assumes "a \<in> nonzero Q\<^sub>p"
assumes "b \<in> carrier Q\<^sub>p"
assumes "c \<in> carrier Q\<^sub>p"
assumes "val b < val c"
shows "val (a \<otimes> b) < val (a \<otimes> c)" |
lemma basis_hull_sub: "\<BB> \<langle>G\<rangle> \<subseteq> G" |
lemma transpos_id_1:"\<lbrakk>i \<le> n; j \<le> n; i \<noteq> j; x \<le> n; x \<noteq> i; x \<noteq> j\<rbrakk> \<Longrightarrow>
transpos i j x = x" |
lemma matchPres:
fixes P :: pi
and Q :: pi
and a :: name
and b :: name
and Rel :: "(pi \<times> pi) set"
and Rel' :: "(pi \<times> pi) set"
assumes PSimQ: "P \<leadsto>\<^sup>^<Rel> Q"
and RelStay: "\<And>P Q a. (P, Q) \<in> Rel \<Longrightarrow> ([a\<frown>a]P, Q) \<in> Rel"
and RelRel': "Rel \<subseteq> Rel'"
shows "[a\<frown>b]P \<leadsto>\<^sup>^<Rel'> [a\<frown>b]Q" |
lemma "preorder R \<Longrightarrow> \<lfloor>T\<rfloor> \<and> \<lfloor>IV\<rfloor>" |
lemma wf_subtrOf:
assumes "wf tr" and "Inr n \<in> prodOf tr"
shows "wf (subtrOf tr n)" |
lemma unsat_state_core_unsat: "unsat_state_core s \<Longrightarrow> \<not> (\<exists> v. v \<Turnstile>\<^sub>s s)" |
lemma maximal_distinct_prefix :
assumes "\<not> distinct xs"
obtains n where "distinct (take (Suc n) xs)"
and "\<not> (distinct (take (Suc (Suc n)) xs))" |
lemma ring_cfs_to_univ_poly_top_coeff:
assumes "as \<in> carrier (R\<^bsup>Suc n\<^esup>)"
shows "(ring_cfs_to_univ_poly n as) n = as ! n" |
lemma trnl\<^sub>\<epsilon>_comp:
assumes "ide u" and "seq \<mu> \<nu>" and "src f = trg \<mu>"
shows "trnl\<^sub>\<epsilon> u (\<mu> \<cdot> \<nu>) = trnl\<^sub>\<epsilon> u \<mu> \<cdot> (f \<star> \<nu>)" |
lemma Suc_mk_markovian[simp]: "\<pi>_Suc (mk_markovian p) x = mk_markovian (\<lambda>n. p (Suc n))" |
lemma accepting_pair\<^sub>G\<^sub>R_abstract:
assumes "finite (reach\<^sub>t \<Sigma> \<delta> q\<^sub>0)"
and "finite (reach\<^sub>t \<Sigma> \<delta>' q\<^sub>0')"
assumes "range w \<subseteq> \<Sigma>"
assumes "run\<^sub>t \<delta> q\<^sub>0 w = f o (run\<^sub>t \<delta>' q\<^sub>0' w)"
assumes "\<And>t. t \<in> reach\<^sub>t \<Sigma> \<delta>' q\<^sub>0' \<Longrightarrow> f t \<in> F \<longleftrightarrow> t \<in> F'"
assumes "\<And>t i. i \<in> \<I> \<Longrightarrow> t \<in> reach\<^sub>t \<Sigma> \<delta>' q\<^sub>0' \<Longrightarrow> f t \<in> I i \<longleftrightarrow> t \<in> I' i"
shows "accepting_pair\<^sub>G\<^sub>R \<delta> q\<^sub>0 (F, {I i | i. i \<in> \<I>}) w \<longleftrightarrow> accepting_pair\<^sub>G\<^sub>R \<delta>' q\<^sub>0' (F', {I' i | i. i \<in> \<I>}) w" |
lemma knights_path_min1: "knights_path (board n m) ps \<Longrightarrow> min1 ps = 1" |
lemma bisimReflexive:
fixes \<Psi> :: 'b
and P :: "('a, 'b, 'c) psi"
shows "\<Psi> \<rhd> P \<sim> P" |
lemma anti_mono: "l \<turnstile> c \<Longrightarrow> l' \<le> l \<Longrightarrow> l' \<turnstile> c" |
lemma proper_it_mono_dres_dom_flat:
assumes PR: "proper_it' it it'"
assumes A: "\<And>kv x. flat_ge (f kv x) (f' kv x)"
shows "flat_ge
((map_iterator_dom o it') s (case_dres False False c) (\<lambda>kv s. s \<bind> f kv) \<sigma>)
((map_iterator_dom o it') s (case_dres False False c) (\<lambda>kv s. s \<bind> f' kv) \<sigma>)" |
lemma d_strict:
"d(x) = bot \<longleftrightarrow> x \<le> Z" |
lemma ASSERT_refine_left:
assumes "\<Phi>"
assumes "\<Phi> \<Longrightarrow> S \<le> \<Down>R S'"
shows "do{ASSERT \<Phi>; S} \<le> \<Down>R S'" |
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