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lemma finsum_group:
assumes "\<And>n. f n \<in> carrier R"
assumes "\<And>n. g n \<in> carrier R"
shows "finite S \<Longrightarrow> finsum R f S \<oplus> finsum R g S = finsum R (\<lambda>n. f n \<oplus> g n) S " |
lemma CallRedsFinal:
assumes wwf: "wwf_prog P"
and "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<rightarrow>* \<langle>ref(a,Cs),s\<^sub>1\<rangle>"
"P,E \<turnstile> \<langle>es,s\<^sub>1\<rangle> [\<rightarrow>]* \<langle>map Val vs,(h\<^sub>2,l\<^sub>2)\<rangle>"
and hp: "h\<^sub>2 a = Some(C,S)"
and "method": "P \<turnstile> last Cs has least M = (Ts',T',pns',body') via Ds"
and select: "P \<turnstile> (C,Cs@\<^sub>pDs) selects M = (Ts,T,pns,body) via Cs'"
and size: "size vs = size pns"
and casts: "P \<turnstile> Ts Casts vs to vs'"
and l\<^sub>2': "l\<^sub>2' = [this \<mapsto> Ref(a,Cs'), pns[\<mapsto>]vs']"
and body_case:"new_body = (case T' of Class D \<Rightarrow> \<lparr>D\<rparr>body | _ \<Rightarrow> body)"
and body: "P,E(this \<mapsto> Class (last Cs'), pns [\<mapsto>] Ts) \<turnstile> \<langle>new_body,(h\<^sub>2,l\<^sub>2')\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3,l\<^sub>3)\<rangle>"
and final:"final ef"
shows "P,E \<turnstile> \<langle>e\<bullet>M(es), s\<^sub>0\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3,l\<^sub>2)\<rangle>" |
lemma cf_proj_fst_is_functor:
assumes "i \<in>\<^sub>\<circ> I"
shows "\<pi>\<^sub>C\<^sub>.\<^sub>1 \<AA> \<BB> : \<AA> \<times>\<^sub>C \<BB> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<AA>" |
lemma roots0: assumes p: "p \<noteq> 0" and p0: "degree p = 0"
shows "{x. poly p x = 0} = {}" |
theorem
assumes "Re z > 1" and "a > (0::real)"
shows Gamma_times_hurwitz_zeta_integral: "Gamma z * hurwitz_zeta a z =
(\<integral>x\<in>{0<..}. (of_real x powr (z - 1) * of_real (exp (-a*x) / (1 - exp (-x)))) \<partial>lebesgue)"
and Gamma_times_hurwitz_zeta_integrable:
"set_integrable lebesgue {0<..}
(\<lambda>x. of_real x powr (z - 1) * of_real (exp (-a*x) / (1 - exp (-x))))" |
lemma rep_qbs_prob_space:
"\<exists>X \<alpha> \<mu>. p = qbs_prob_space (X, \<alpha>, \<mu>) \<and> qbs_prob X \<alpha> \<mu>" |
lemma singleton_sup_aux:
"singleton default (A \<squnion> B) = (if A = \<bottom> then singleton default B
else if B = \<bottom> then singleton default A
else singleton default
(single (singleton default A) \<squnion> single (singleton default B)))" for default |
theorem (in finite_field) finite_field_order:
"\<exists>n. order R = char R ^ n \<and> n > 0" |
lemma decompose_rec:
"ys \<noteq> [] \<Longrightarrow> decompose x (y#ys) =
(case bezout_coefficients y (prod_list ys) of
(a, b) \<Rightarrow> (b*x) # decompose (a*x) ys)" |
lemma padic_add_closed:
assumes "prime p"
shows "\<And>x y.
x \<in> carrier (padic_int p) \<Longrightarrow>
y \<in> carrier (padic_int p) \<Longrightarrow>
x \<oplus>\<^bsub>(padic_int p)\<^esub> y \<in> carrier (padic_int p)" |
lemma pr_trans[trans]:
fixes A::"'a prog"
assumes prAB: "A \<sqsubseteq> B"
and prBC: "B \<sqsubseteq> C"
shows "A \<sqsubseteq> C" |
lemma homeomorphic_ball01_UNIV:
"ball (0::'a::real_normed_vector) 1 homeomorphic (UNIV:: 'a set)"
(is "?B homeomorphic ?U") |
lemma Source_union : "Source s \<union> Source t = Source (s \<union> t)" |
lemma eeqButPID_trans:
assumes "eeqButPID s s1" and "eeqButPID s1 s2" shows "eeqButPID s s2" |
lemma product_simps[simp]:
"alphabet\<^sub>2 (product AA) = \<Inter> (alphabet\<^sub>1 ` set AA)"
"initial\<^sub>2 (product AA) = listset (map initial\<^sub>1 AA)"
"transition\<^sub>2 (product AA) a ps = listset (map2 (\<lambda> A p. transition\<^sub>1 A a p) AA ps)"
"condition\<^sub>2 (product AA) = condition (map condition\<^sub>1 AA)" |
lemma gen_model_tc_rels[dest]:
assumes M: "kripke M"
and R: "(w', w'') \<in> (\<Union>a \<in> as. relations (gen_model M w) a)\<^sup>+"
shows "(w', w'') \<in> (\<Union>a \<in> as. relations M a)\<^sup>+" |
lemma fimageI1:
assumes "x \<in>\<^sub>\<circ> \<R>\<^sub>\<bullet> (r \<restriction>\<^sup>l\<^sub>\<bullet> A)"
shows "x \<in>\<^sub>\<circ> r `\<^sub>\<bullet> A" |
lemma sameDom_refl: "sameDom inp inp" |
lemma tendsto_Lambert_W_1:
assumes "(f \<longlongrightarrow> L) F" "eventually (\<lambda>x. f x \<ge> -exp (-1)) F"
shows "((\<lambda>x. Lambert_W (f x)) \<longlongrightarrow> Lambert_W L) F" |
lemma bij_betw_characters_dcharacters:
"bij_betw c2dc (characters G) (dcharacters n)" |
lemma closed_le_abs [simp]:
"closed {x. (C::real) \<le> abs x}" |
lemma dom_prog_interpretation: "partial_function_definitions dom_prog_ord dom_prog_lub" |
lemma vcg_arraycpy_unfolds[named_ss vcg_bb]:
"wlp \<pi> (x[] ::= a) Q s = Q (UPD_STATE s x (s a))"
"wp \<pi> (x[] ::= a) Q s = Q (UPD_STATE s x (s a))" |
lemma mult_float_interval_mono:
"mult_float_interval prec A B \<le> mult_float_interval prec X Y"
if "A \<le> X" "B \<le> Y" |
lemma ConjI[PLM_intro]:
"\<lbrakk>[\<phi> in v]; [\<psi> in v]\<rbrakk> \<Longrightarrow> [\<phi> \<^bold>& \<psi> in v]" |
lemma mod_ring_uminus[transfer_rule]: "(mod_ring_rel ===> mod_ring_rel) (uminus_p p) uminus" |
lemma tm_weak_copy_correct6:
"\<lbrace>\<lambda>tap. \<exists>z4. tap = (Bk \<up> z4, <[n::nat]> @[Bk])\<rbrace> tm_weak_copy \<lbrace>\<lambda>tap. \<exists>k l. tap = (Bk \<up> k, <[n::nat, n]> @ Bk \<up> l) \<rbrace>" |
lemma Diag_implies_coherent:
assumes "Diag t"
shows "coherent t" |
lemma octo_inner_1 [simp]: "inner 1 x = Ree x"
and octo_inner_1_right [simp]: "inner x 1 = Ree x" |
lemma these_tiny_tdghms_iff: (*not simp*)
"\<NN> \<in>\<^sub>\<circ> these_tiny_tdghms \<alpha> \<AA> \<BB> \<FF> \<GG> \<longleftrightarrow>
\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>" |
lemma resolvable_design_num_res_classes: "size \<P> = \<r>" |
lemma "zip\<cdot>[]\<cdot>[] = []" |
lemma monom_eval_comp_fun:
fixes g:: "'c \<Rightarrow> 'a"
assumes "closed_fun R g"
shows "comp_fun_commute (\<lambda> x . \<lambda>y. if y \<in> carrier R then (g x) \<otimes> y else \<zero>)" |
lemma length_snoc: "length (xs @ [x]) = Suc (length xs)" |
lemma assign_twice: "\<lbrakk> mwb_lens x; x \<sharp> f \<rbrakk> \<Longrightarrow> (x := e ;; x := f) = (x := f)" |
lemma (in topological_space) filterlim_within_subset:
"filterlim f l (at x within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> filterlim f l (at x within T)" |
lemma vsv_vdomain_vrange_vsingleton:
assumes "\<D>\<^sub>\<circ> r = set {a}" and "\<R>\<^sub>\<circ> r = set{b}"
shows "r = set {\<langle>a, b\<rangle>}" |
lemma eq_poly_rel_eq[sepref_import_param]:
\<open>((=), (=)) \<in> poly_rel \<rightarrow> poly_rel \<rightarrow> bool_rel\<close> |
lemma "\<lbrakk>test p; test q; p\<cdot>x = p\<cdot>x\<cdot>q\<rbrakk> \<Longrightarrow> p\<cdot>x\<cdot>!q = 0" |
lemma mtp_Raise:
assumes "l \<le> d"
shows "mtp (x + d + k) (Raise l k t) = mtp (x + d) t" |
lemma class_rec_code [code]:
"class_rec G C t f =
(if wf_class G then
(case class G C of
None \<Rightarrow> class_rec G C t f
| Some (D, fs, ms) \<Rightarrow>
if C = Object then f Object fs ms t else f C fs ms (class_rec G D t f))
else class_rec G C t f)" |
lemma det_four_block_mat_upper_right_zero_col: assumes A1: "A1 \<in> carrier_mat n n"
and A20: "A2 = (0\<^sub>m n 1)" and A3: "A3 \<in> carrier_mat 1 n"
and A4: "A4 \<in> carrier_mat 1 1"
shows "det (four_block_mat A1 A2 A3 A4) = det A1 * det A4" (is "det ?A = _") |
lemma staticSecret_in_initState [simp]:
"staticSecret A \<subseteq> initState A" |
lemma per_distinct_1:
assumes "Per A B C" and
"B \<noteq> C"
shows "A \<noteq> C" |
lemma smult_dvd_cancel:
assumes "smult a p dvd q"
shows "p dvd q" |
lemma floor_conv_div_nat:
"of_int (floor (real m / real n)) = real (m div n)" |
lemma remove_writes:
"writes (remove_child_locs (the |h \<turnstile> get_parent child|\<^sub>r) |h \<turnstile> get_owner_document (cast child)|\<^sub>r)
(remove child) h h'" |
lemma min_zpassign_simps[simp]:
"min_zpassign (PWt x) = zhmset_of (wt_sym (min_ground_head (Var x)))"
"min_zpassign (PCoef x i) = 1" |
lemma map_of_delete [simp]: "k' \<noteq> k \<Longrightarrow> map_of (delete k al) k' = map_of al k'" |
lemma distr_uminus_real:
assumes "has_density M lborel (f :: real \<Rightarrow> ennreal)"
shows "has_density (distr M borel uminus) lborel (\<lambda>x. f (- x))" |
lemma SEval_def2: "SEval m e s = (\<exists>f. f \<in> set s & FEval m e f)" |
lemma Infinitesimal_less_SReal: "x \<in> \<real> \<Longrightarrow> y \<in> Infinitesimal \<Longrightarrow> 0 < x \<Longrightarrow> y < x"
for x y :: hypreal |
lemma autoref_ABS:
"\<lbrakk> \<And>x x'. (x,x')\<in>Ra \<Longrightarrow> (c x, a x')\<in>Rr \<rbrakk> \<Longrightarrow> (c, \<lambda>'x. a x)\<in>Ra\<rightarrow>Rr" |
lemma assumes v: "valid_tmap t"
shows mapset_some: "(mapOf t i = Some a) = ((i,a) : setOf t)" |
lemma defs_uses_disjoint'[simp]: "n \<in> set (\<alpha>n g) \<Longrightarrow> v \<in> defs g n \<Longrightarrow> v \<in> uses g n \<Longrightarrow> False" |
lemma Keys_init_syzygy_list:
"Keys (set (init_syzygy_list bs)) =
map_component (\<lambda>k. k + length bs) ` Keys (set bs) \<union> (\<lambda>i. term_of_pair (0, i)) ` {0..<length bs}" |
lemma prod_monom_1_1:
defines "P == (\<lambda> x n. (x[^]\<^bsub>(R)\<^esub> (CARD('a) ^ n) = x))"
assumes m: "monom 1 1 \<in> carrier R"
and eq: "P (monom 1 1) n"
shows "P ((\<Prod>i = 0..<b::nat. monom 1 1) mod f) n" |
lemma lambda\<Pi>\<^sub>3_aux[meta_aux]:
"make\<Pi>\<^sub>3 (\<lambda>u v r s w. \<exists>x. \<nu>\<upsilon> x = u \<and> (\<exists>y. \<nu>\<upsilon> y = v \<and>
(\<exists>z. \<nu>\<upsilon> z = r \<and> eval\<Pi>\<^sub>3 F (\<nu>\<upsilon> x) (\<nu>\<upsilon> y) (\<nu>\<upsilon> z) s w))) = F" |
lemma to_cnf2_eq_Cons: "to_cnf2 x = (a,b) # list \<Longrightarrow> a = oLog \<omega> x" |
lemma elimination_sing:
"(\<forall>m \<in> M. F \<in> {m} co (B m)) ==> F \<in> M co (\<Union>m \<in> M. B m)" |
lemma scalar_matrix_mult_row_code [code abstract]:
"vec_nth (scalar_matrix_mult_row c A i) =(% j. c * (A $ i $ j))" |
lemma from_digits_digit:
assumes "x < base ^ n"
shows "from_digits n (digit x) = x" |
lemma match_correct: "match s w \<longleftrightarrow> w \<in> lang s" |
lemma max_next_pcs_not_empty:
"pc<length bp \<Longrightarrow> x : set (exec (bp!pc) (pc,s)) \<Longrightarrow> max_next_pcs bp \<noteq> {}" |
lemma extend_preserves_model: (* only for ground *)
assumes f_infpath: "wf_infpath (f :: nat \<Rightarrow> partial_pred_denot)"
assumes C_ground: "ground\<^sub>l\<^sub>s C"
assumes C_sat: "\<not>falsifies\<^sub>c (f (Suc n)) C"
assumes n_max: "\<forall>l\<in>C. nat_of_fatom (get_atom l) \<le> n"
shows "eval\<^sub>c HFun (extend f) C" |
lemma "map\<cdot>(uncurry\<cdot>f)\<cdot>(zip\<cdot>x\<cdot>y) = zipWith\<cdot>f\<cdot>x\<cdot>y" |
lemma fpxs_nth_shift [simp]: "fpxs_nth (fpxs_shift r f) n = fpxs_nth f (n + r)" |
lemma DERIV_real_root_generic:
assumes "0 < n"
and "x \<noteq> 0"
and "even n \<Longrightarrow> 0 < x \<Longrightarrow> D = inverse (real n * root n x ^ (n - Suc 0))"
and "even n \<Longrightarrow> x < 0 \<Longrightarrow> D = - inverse (real n * root n x ^ (n - Suc 0))"
and "odd n \<Longrightarrow> D = inverse (real n * root n x ^ (n - Suc 0))"
shows "DERIV (root n) x :> D" |
lemma WT_red_external_aggr_imp_red_external:
"\<lbrakk> wf_prog wf_md P; (ta, va, h') \<in> red_external_aggr P t a M vs h; P,h \<turnstile> a\<bullet>M(vs) : U; P,h \<turnstile> t \<surd>t \<rbrakk>
\<Longrightarrow> P,t \<turnstile> \<langle>a\<bullet>M(vs), h\<rangle> -ta\<rightarrow>ext \<langle>va, h'\<rangle>" |
lemma complex_eq_0: "z=0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0" |
theorem ivls_of_aforms: "xs \<in> Joints XS \<Longrightarrow> bounded_by xs (ivls_of_aforms prec XS)" |
lemma c_tl_c_drop: "c_tl (c_drop y x) = c_drop y (c_tl x)" |
lemma
flow_has_flowderiv:
assumes "t \<in> existence_ivl0 x0"
shows "((\<lambda>(x0, t). flow0 x0 t) has_derivative flowderiv x0 t) (at (x0, t) within S)" |
lemma filtermap_append: "filtermap pred func (tr @ tr1) = filtermap pred func tr @ filtermap pred func tr1" |
lemma subtopology_Times:
shows "subtopology (prod_topology X Y) (S \<times> T) = prod_topology (subtopology X S) (subtopology Y T)" |
lemma p_preserves_inv_imps_work_sum:
assumes "next_propagate' c0 c1 loc t"
and "inv_imps_work_sum c0"
shows "inv_imps_work_sum c1" |
lemma swap_self [simp]:
"\<langle>a \<leftrightarrow> b\<rangle> * \<langle>a \<leftrightarrow> b\<rangle> = 1" |
lemma Cod_Cod [simp]:
shows "Arr t \<Longrightarrow> Cod (Cod t) = Cod t" |
lemma interval_eq_iff: "a = b \<longleftrightarrow> lower a = lower b \<and> upper a = upper b" |
lemma Serv_fresh_not_AKcryptSK:
"Key servK \<notin> used evs \<Longrightarrow> \<not> AKcryptSK authK servK evs" |
lemma path_conforms_with_strategy_ltl [intro]:
"path_conforms_with_strategy p P \<sigma> \<Longrightarrow> path_conforms_with_strategy p (ltl P) \<sigma>" |
lemma run_shift_elim[elim!]:
assumes "run (r @- s) p"
obtains "path r p" "run s (target r p)" |
lemma observable_io_targets_language :
assumes "io1 @ io2 \<in> LS M q1"
and "observable M"
and "q2 \<in> io_targets M io1 q1"
shows "io2 \<in> LS M q2" |
lemma assumes "invar_vebt t n " shows
"perInsTrans (vebt_buildup n) t" |
lemma sorted_list_of_set_keys:
"sorted_list_of_set (keys f) = filter (\<lambda>k. k \<in> keys f) [0..<degree f]" (is "_ = ?r") |
lemma valid_path_drop: "valid_path P \<Longrightarrow> valid_path (ldropn n P)" |
lemma asBinp_equal_imp_alphaBinp:
assumes "qGoodBinp qbinp" and "asBinp qbinp = asBinp qbinp'"
shows "qbinp %%= qbinp'" |
lemma ran_m_fmdrop:
\<open>C \<in># dom_m N \<Longrightarrow> ran_m (fmdrop C N) = remove1_mset (the (fmlookup N C)) (ran_m N)\<close> |
lemma add_top_left_ennreal [simp]: "top + x = (top :: ennreal)" |
lemma vec_plus[simp,intro]: "\<lbrakk>vec nr u; vec nr v\<rbrakk> \<Longrightarrow> vec nr (vec_plusI pl u v)" |
lemma LI_in_measure_trans: "(S\<^sub>1,\<theta>\<^sub>1) \<leadsto>\<^sup>+ (S\<^sub>2,\<theta>\<^sub>2) \<Longrightarrow> ((S\<^sub>2,\<theta>\<^sub>2),(S\<^sub>1,\<theta>\<^sub>1)) \<in> measure\<^sub>s\<^sub>t" |
lemma subtype_wfT:
fixes t1::\<tau> and t2::\<tau>
assumes "\<Theta>; \<B>; \<Gamma> \<turnstile> t1 \<lesssim> t2"
shows "\<Theta>; \<B>; \<Gamma> \<turnstile>\<^sub>w\<^sub>f t1 \<and> \<Theta>; \<B>; \<Gamma> \<turnstile>\<^sub>w\<^sub>f t2" |
lemma sopen_preserves_body[simp]:
fixes t s p
assumes "body t" and "lc s" and "lc p"
shows "body ({n \<rightarrow> [s,p]} t)" |
lemma memval_size_s16:
"|memval_type (Sint16_v v)|\<^sub>\<tau> = 2" |
lemma valid_line_split:
"valid_line l \<longleftrightarrow>
l = [] \<or>
(l!0 = B \<and> valid_line (tl l)) \<or>
length l \<ge> 3 \<and> (\<forall> i < length l. l ! i = R) \<or>
(\<exists> j < length l. j \<ge> 3 \<and> (\<forall> i < j. l ! i = R) \<and> l ! j = B \<and> valid_line (drop (j + 1) l))" |
lemma ffb_loopI: "P \<le> {s. I s} \<Longrightarrow> {s. I s} \<le> Q \<Longrightarrow> {s. I s} \<le> fb\<^sub>\<F> F {s. I s} \<Longrightarrow> P \<le> fb\<^sub>\<F> (LOOP F INV I) Q" |
lemma \<Delta>\<^sub>\<epsilon>_statesD':
"q |\<in>| eps_states (\<Delta>\<^sub>\<epsilon> \<A> \<B>) \<Longrightarrow> q |\<in>| \<Q> \<A> |\<union>| \<Q> \<B>" |
lemma subseq_strict_length:
assumes a: "subseq x y" "x \<noteq> y"
shows "length x < length y" |
lemma aristo_conversion2 :
assumes "A I B" shows "B I A" |
lemma set_mult_inclusion:
assumes H:"subgroup H G"
assumes Q:"P \<subseteq> carrier G"
assumes PQ:"H <#> P \<subseteq> H"
shows "P \<subseteq> H" |
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