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

Statement: stringlengths 7 24.3k
lemma GatewayReq_L2: assumes h1:"msg (Suc 0) req" and h2:"msg (Suc 0) stop" and h3:"msg (Suc 0) a" and h4:"ts lose" and h5:"GatewayReq req dt a stop lose d ack i vc" and h6:"req (t + 3 + k) = [send]" and h7:"inf_last_ti dt t \<noteq> []" and h8:"\<forall>j\<le>2 * d + (4 + k). lose (t + j) = [False]" and h9:"\<forall>m\<le>k. ack (t + 2 + m) = [connection_ok]" shows "i (t + 3 + k + d) \<noteq> []"
lemma hurwitz_formula_integral_semiannulus: fixes N :: nat and r :: real and s :: complex defines "R \<equiv> real (2 * N + 1) * pi" assumes "r > 0" and "r < 2" shows "exp (-\<i> * pi * s) * integral {r..R} (\<lambda>x. x powr (-s) * exp (-a * x) / (1 - exp (-x))) + integral {r..R} (\<lambda>x. x powr (-s) * exp (a * x) / (1 - exp x)) + contour_integral (part_circlepath 0 R 0 pi) (f s) + contour_integral (part_circlepath 0 r pi 0) (f s) = -2 * pi * \<i> * exp (- s * of_real pi * \<i> / 2) * (\<Sum>k\<in>{0<..N}. Res s k)" (is ?thesis1) and "f s contour_integrable_on hankel_semiannulus r N"
lemma (in group) subgroup_card_dvd_group_ord: assumes "subgroup H G" shows "card H dvd order G"
lemma use_implies_allDef: assumes "lookupDef g m (var g v) = v" "m \<in> set (\<alpha>n g)" "var g v \<in> uses g m'" "g \<turnstile> m-ms\<rightarrow>m'" "\<forall>x \<in> set (tl ms). var g v \<notin> defs g x" shows "v \<in> braun_ssa.allDefs g (defNode v)"
lemma irreducible\<^sub>d_dvd_eq: fixes a b :: "'a::{comm_semiring_1,semiring_no_zero_divisors} poly" assumes "irreducible\<^sub>d a" and "irreducible\<^sub>d b" and "a dvd b" and "monic a" and "monic b" shows "a = b"
lemma find_set_3: "find_set_invariant p x y \<and> y = p[[y]] \<Longrightarrow> find_set_postcondition p x y"
lemma Loeb_iff_box_Loeb: "Loebian x \<longleftrightarrow> (\<forall>y. \|x] (ad ( \|x] y ) + d y) \<le> \|x] y)"
lemma implications_cm_all[simp]: "c_imp (cm_all' c \<Delta>) = c_imp c"
lemma path_component_intermediate_subset: "path_component_set u a \<subseteq> t \<and> t \<subseteq> u \<Longrightarrow> path_component_set t a = path_component_set u a"
lemma godel_code_1_iff[elim]: "\<lbrakk>i < length nl; \<not> Suc 0 < godel_code nl\<rbrakk> \<Longrightarrow> nl ! i = 0"
lemma older_seniors_tower''2: assumes "x \<le> n" assumes "y \<le> n" assumes "\<not>sink (token_run x (n + m))" assumes "\<not>sink (token_run y (n + m))" assumes "older_seniors x n = older_seniors x (n + m)" assumes "older_seniors y n \<subseteq> older_seniors x n" shows "older_seniors y n = older_seniors y (n + m)"
theorem Compositionality_While: assumes dind: "b \<equiv>\<^bsub>d\<^esub> b'" assumes bodyrelated: "[c] \<approx>\<^bsub>d\<^esub> [c']" shows "[while b do c od] \<approx>\<^bsub>d\<^esub> [while b' do c' od]"
lemma lem10: "M e"
lemma tailR_correct: "t\<noteq>empty \<Longrightarrow> toList (tailR t) = butlast (toList t)"
lemma finite_backflows_state: "finite (backflows (flows_state \<T>))"
lemma finite_map_graph: "finite A \<Longrightarrow> finite (map_graph f \<inter> (A \<times> UNIV))"
lemma coeffs_eq_Nil [simp]: "coeffs p = [] \<longleftrightarrow> p = 0"
lemma fps_cos_nth_add_2: "fps_cos c $ (n + 2) = - (c * c * fps_cos c $ n / (of_nat (n + 1) * of_nat (n + 2)))"
lemma inorder_node32: "height r > 0 \<Longrightarrow> inorder (treeD (node32 l a m' b r)) = inorder l @ a # inorder (treeD m') @ b # inorder r"
lemma dom_mdeg_le1_ikkbz_sub: "ikkbz_sub t \<noteq> t \<Longrightarrow> dom_children (ikkbz_sub t) T"
lemma red_crit_lifting_family: assumes q_in: "q \<in> Q" shows "calculus Bot_F Inf_F (entails_\<G>_q q) (Red_I_\<G>_q q) (Red_F_\<G>_q q)"
lemma last_tl: "xs = [] \<or> tl xs \<noteq> [] \<Longrightarrow>last (tl xs) = last xs"
lemma disjoint_paths_new_path: assumes no_small_separations: "\<And>S. Separation G v0 v1 S \<Longrightarrow> card S \<ge> Suc (card paths)" shows "\<exists>P_new. v0\<leadsto>P_new\<leadsto>v1 \<and> set P_new \<inter> second_vertices = {}"
lemma strong_orbit_induce_same: "a \<sim>\<^sub>\<K> b \<longrightarrow> (a \<lhd> c \<longleftrightarrow> b \<lhd> c)"
lemma reachable_steps: "\<exists> xs. steps xs \<and> hd xs = s\<^sub>0 \<and> last xs = x" if "reachable x"
lemma mult_zero_sup_circ: "(x \<squnion> y * bot)\<^sup>\<circ> = x\<^sup>\<circ> * (y * bot)\<^sup>\<circ>"
lemma integrable_\<nu>_trace_fin: "integrable (\<T> p s) (\<lambda>t. \<nu>_trace_fin t N)"
lemma isom_gch_units_transpTr0:"\<lbrakk>Ugp E; Gchain n g; Gchain n h; i \<le> n; j \<le> n; i < j; isom_Gchains n (transpos i j) g h\<rbrakk> \<Longrightarrow> {i. i \<le> n \<and> g i \<cong> E} - {i, j} ={i. i \<le> n \<and> h i \<cong> E} - {i, j}"
lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
lemma decseq_convergent: fixes X :: "nat \<Rightarrow> real" assumes "decseq X" and "\<forall>i. B \<le> X i" obtains L where "X \<longlonglongrightarrow> L" "\<forall>i. L \<le> X i"
lemma sum_add_split_nat_ivl: assumes le: "m <= (k::nat)" "k <= n" and g: "!!i. [\| m <= i; i < k \|] ==> g i = f i" and h: "!!i. [\| k <= i; i < n \|] ==> h i = f i" shows "sum g {m..<k} + sum h {k..<n} = sum f {m..<n}"
lemma ap_state_rev_B: "B f \<diamondop> B x = B (State_Monad.return (\<lambda>x f. f x) \<diamondop> x \<diamondop> f)"
lemma pnopI[intro]: assumes "vsv f" and "n \<in>\<^sub>\<circ> \<omega>" and "\<D>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> A ^\<^sub>\<times> n" and "\<R>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> A" shows "pnop A n f"
lemma sorted_spmat_mult_spmat: "sorted_spmat (B::('a::lattice_ring) spmat) \<Longrightarrow> sorted_spmat (mult_spmat A B)"
lemma map_to_set_empty_iff: "map_to_set m = {} \<longleftrightarrow> m = Map.empty" "{} = map_to_set m \<longleftrightarrow> m = Map.empty"
lemma inverse_ereal_ge0I: "0 \<le> (x :: ereal) \<Longrightarrow> 0 \<le> inverse x"
lemma isCont_LCons[THEN isCont_o2[rotated]]: "isCont (LCons x) l"
lemma redT_updT_None: "redT_updT ts ta t = None \<Longrightarrow> ts t = None"
lemma fixes e :: "'addr expr1" and es :: "'addr expr1 list" shows compE2_not_Return: "Return \<notin> set (compE2 e)" and compEs2_not_Return: "Return \<notin> set (compEs2 es)"
lemma ainvf_bijec:"\<lbrakk>aGroup F; aGroup G; bijec\<^bsub>F,G\<^esub> f\<rbrakk> \<Longrightarrow> bijec\<^bsub>G,F\<^esub> (ainvf\<^bsub>F,G\<^esub> f)"
lemma bin_last_bl_to_bin: "odd (bl_to_bin bs) \<longleftrightarrow> bs \<noteq> [] \<and> last bs"
lemma limit_o [simp]: assumes a: "a \<in> limit w" shows "f a \<in> limit (f \<circ> w)"
lemma prob_space_eP: "x \<in> space (PiM {0..<n} M) \<Longrightarrow> prob_space (eP n x)"
lemma map_key_PP_monomial [simp]: "Poly_Mapping.map_key PP (monomial c t) = monomial c (mapping_of t)"
lemma subset_remdups'_append: "set (remdups' f (xs @ ys)) \<subseteq> set (remdups' f xs) \<union> set (remdups' f ys)"
lemma tan_cot: "tan(pi/2 - x) = inverse(tan x)"
lemma R_phase: assumes "set w \<subseteq> \<Sigma>" shows "\<exists> cs. (st.init_config (map INP w), cs) \<in> st.dstep^^(3 + 2 * length w) \<and> rel_S\<^sub>0 cs (init_config w)"
lemma inf_glb: assumes "A \<subseteq> carrier L" shows "greatest L (\<Sqinter>A) (Lower L A)"
lemma IF_rel: "Cl_2 \<C> \<Longrightarrow> \<I> \<^bold>\<equiv> \<I>\<^sub>F \<F>"
lemma set_pders_nzero: assumes "p\<noteq>0" "q\<in>set (pders p)" shows "q\<noteq>0"
lemma finite_cmax: assumes r: "Card_order r" and s: "Card_order s" shows "finite (Field (cmax r s)) \<longleftrightarrow> finite (Field r) \<and> finite (Field s)"
lemma is_success_alt_def: \<open>is_success a \<longleftrightarrow> a = SUCCESS\<close>
lemma cone_ordLeq_cexp: "cone \<le>o r1 \<Longrightarrow> cone \<le>o r1 ^c r2"
lemma single_valued_D: "single_valued (D c)"
lemma blinop_eqI: "(\<And>i. x $ i = y $ i) \<Longrightarrow> x = y"
lemma find_pos_x_in_dverts: "(x,y) = find_pos_aux v p t1 \<Longrightarrow> x \<in> dverts t1 \<or> p=x"
lemma squarefree_imp_coprime_pderiv: fixes p :: "'a :: {factorial_ring_gcd,semiring_gcd_mult_normalize,semiring_char_0} poly" assumes "squarefree p" and "content p = 1" shows "Rings.coprime p (pderiv p)"
lemma lfp_while: assumes "mono f" and "!!X. X \<subseteq> C \<Longrightarrow> f X \<subseteq> C" and "finite C" shows "lfp f = while (\<lambda>A. f A \<noteq> A) f {}"
lemma euler_theorem: fixes a m :: nat assumes "coprime a m" shows "[a ^ totient m = 1] (mod m)"
lemma prefix_snocD: "prefix (xs@[x]) ys \<Longrightarrow> strict_prefix xs ys"
lemma Bleast_code [code]: "Bleast A P = (if finite A then case filter P (sorted_list_of_set A) of [] \<Rightarrow> abort_Bleast A P \| x # xs \<Rightarrow> x else abort_Bleast A P)"
lemma infsetsum_Un_Int: assumes "f abs_summable_on (A \<union> B)" shows "infsetsum f (A \<union> B) = infsetsum f A + infsetsum f B - infsetsum f (A \<inter> B)"
lemma conv_radius_cong_bigtheta: assumes "f \<in> \<Theta>(g)" shows "conv_radius f = conv_radius g"
lemma jmm_heap_conf: "heap_conf addr2thread_id thread_id2addr jmm_empty jmm_allocate jmm_typeof_addr jmm_heap_write (jmm_hconf P) P"
lemma mbisimI2: "\<lbrakk> finite (dom (thr s2)); locks s1 = locks s2; wset s1 = wset s2; interrupts s1 = interrupts s2; wset_thread_ok (wset s2) (thr s2); \<And>t. thr s2 t = None \<Longrightarrow> thr s1 t = None; \<And>t x2 ln. thr s2 t = \<lfloor>(x2, ln)\<rfloor> \<Longrightarrow> \<exists>x1. thr s1 t = \<lfloor>(x1, ln)\<rfloor> \<and> t \<turnstile> (x1, shr s1) \<approx> (x2, shr s2) \<and> (wset s2 t = None \<or> x1 \<approx>w x2) \<rbrakk> \<Longrightarrow> s1 \<approx>m s2"
lemma flush_data_cache_privilege: "(((get_S (cpu_reg_val PSR s)))::word1) = 0 \<Longrightarrow> s' = flush_data_cache s \<Longrightarrow> (((get_S (cpu_reg_val PSR s')))::word1) = 0"
lemma elem_index_inj: assumes "elem_index x = elem_index y" shows "x = y"
lemma d_states_index_properties : assumes "i < length (d_states M q)" shows "fst (d_states M q ! i) \<in> (states M - {q})" "fst (d_states M q ! i) \<noteq> q" "snd (d_states M q ! i) \<in> inputs M" "(\<forall> qx' \<in> set (take i (d_states M q)) . fst (d_states M q ! i) \<noteq> fst qx')" "(\<exists> t \<in> transitions M . t_source t = fst (d_states M q ! i) \<and> t_input t = snd (d_states M q ! i))" "(\<forall> t \<in> transitions M . (t_source t = fst (d_states M q ! i) \<and> t_input t = snd (d_states M q ! i)) \<longrightarrow> (t_target t = q \<or> (\<exists> qx' \<in> set (take i (d_states M q)) . fst qx' = (t_target t))))"
lemma lipschitz_on_compact: assumes "compact K" "K \<subseteq> T" assumes "compact Y" "Y \<subseteq> X" obtains L where "\<And>t. t \<in> K \<Longrightarrow> L-lipschitz_on Y (f t)"
lemma alphaBoundOutputChain': fixes yvec :: "name list" and xvec :: "name list" and B :: "('a, 'b, 'c) boundOutput" assumes "length xvec = length yvec" and "yvec \<sharp>* B" and "yvec \<sharp>* xvec" and "distinct yvec" shows "\<lparr>\<nu>xvec\<rparr>B = \<lparr>\<nu>yvec\<rparr>([xvec yvec] \<bullet>\<^sub>v B)"
lemma paths_belowI: "(\<And> x xs. x#xs \<in> paths t \<Longrightarrow> x#xs \<in> paths t') \<Longrightarrow> t \<sqsubseteq> t'"
lemma ca_lr_DBAbsNI: "\<lbrakk> P \<Down> DBAbsN M; closed P; \<And>x X. x \<triangleleft> X \<Longrightarrow> f\<cdot>x \<triangleleft> M<X/0> \<rbrakk> \<Longrightarrow> ValF\<cdot>f \<triangleleft> P"
lemma ta_der_fmap_states_ta: assumes "q \|\<in>\| ta_der A t" shows "h q \|\<in>\| ta_der (fmap_states_ta h A) (map_vars_term h t)"
lemma iso_functorI: assumes "\<NN> : \<FF> \<mapsto>\<^sub>C\<^sub>F\<^sub>.\<^sub>i\<^sub>s\<^sub>o \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<FF> \<approx>\<^sub>C\<^sub>F\<^bsub>\<alpha>\<^esub> \<GG>"
lemma connected_verts_G_eq_T: assumes "graph G" and "connected G" and "Suc n = card (verts G)" shows "verts T = verts G"
lemma nfoldli_by_idx_gen: shows "nfoldli (drop k l) c f s = nfoldli [k..<length l] c (\<lambda>i s. do { ASSERT (i<length l); let x = l!i; f x s }) s"
lemma all_countings: "all_countings a b = (a + b) choose a"
lemma \<L>_complement_reg: assumes "ta_sig (ta \<A>) \|\<subseteq>\| \<F>" shows "\<L> (complement_reg \<A> \<F>) = \<T>\<^sub>G (fset \<F>) - \<L> \<A>"
lemma retraction_map_compose: "\<lbrakk>retraction_map X Y f; retraction_map Y Z g\<rbrakk> \<Longrightarrow> retraction_map X Z (g \<circ> f)"
lemma word_sless_msb_less: "x <s y \<longleftrightarrow> (msb y \<longrightarrow> msb x) \<and> ((msb x \<and> \<not> msb y) \<or> x < y)"
lemma wadjust_stop_Bk[simp]: "wadjust_backto_standard_pos m rs (c, Bk # list) \<Longrightarrow> wadjust_stop m rs (Bk # c, list)"
lemma empty_exposed_face_of [iff]: "{} exposed_face_of S"
lemma Lindelof_space_perfect_map_image: "\<lbrakk>Lindelof_space X; perfect_map X Y f\<rbrakk> \<Longrightarrow> Lindelof_space Y"
theorem serializable_encoding_decoded_plan_is_serializable: assumes "is_valid_problem_strips \<Pi>" and "\<A> \<Turnstile> \<Phi>\<^sub>\<forall> \<Pi> t" shows "is_serial_solution_for_problem \<Pi> (concat (\<Phi>\<inverse> \<Pi> \<A> t))"
lemma binrelchain_funcong_Cons_snoc: assumes "\<And>x y. P x y \<Longrightarrow> f y = f x" "binrelchain P (x#xs@[y])" shows "f y = f x"
lemma linorder_rank_singleton: "linorder_rank R {y} x = (if x \<noteq> y \<and> (y,x) \<in> R then 1 else 0)"
lemma sim12_sizes_map_Val [simp]: "sim12_sizes (map Val vs) = 0"
lemma ennreal_ineq_diff_add: "b \<le> a \<Longrightarrow> a = b + (a - b::ennreal)"
lemma sub_Un1:"B \<subseteq> B \<union> C"
lemma berlekamp_resulting_mat_i[transfer_rule]: "(poly_rel ===> mat_rel R) (berlekamp_resulting_mat_i p ff_ops) berlekamp_resulting_mat"
lemma nTgtOfTrFrom_nTgtOf_last: "tr \<noteq> [] \<Longrightarrow> nTgtOfTrFrom s tr = nTgtOf (last tr)"
lemma ni_diamond_star_induct_sup: "\<parallel>x\<guillemotright>ni(y) \<squnion> ni(z) \<le> ni(y) \<Longrightarrow> \<parallel>x\<^sup>\<star>\<guillemotright>ni(z) \<le> ni(y)"
lemma map_of_aset_update [rewrite]: "unique_keys_set S \<Longrightarrow> (k, v) \<in> S \<Longrightarrow> map_of_aset (S - {(k, v)} \<union> {(k, v')}) = (map_of_aset S) {k \<rightarrow> v'}"
lemma lgc_invar_lgc_step: assumes Si_lt: "enat (Suc i) < llength TNs" and invar: "lgc_invar TNs i" and step: "lnth TNs i \<leadsto>LGC lnth TNs (Suc i)" shows "lgc_invar TNs (Suc i)"
lemma measurable_PiM_finmap_of: assumes "finite J" shows "finmap_of J \<in> measurable (Pi\<^sub>M J M) (PiF {J} M)"
lemma xyxy_conj_yxxy: assumes "x \<cdot> y \<cdot> x \<cdot> y \<sim> y \<cdot> x \<cdot> x \<cdot> y" shows "x \<cdot> y = y \<cdot> x"
lemma list_ex_unfold: "list_ex P (x # y # xs) \<longleftrightarrow> P x \<or> list_ex P (y # xs)" "list_ex P [x] \<longleftrightarrow> P x"
lemma eoption_rule[refine_vcg]: "\<lbrakk> v=None \<Longrightarrow> S1 \<le> ESPEC \<Phi> \<Psi>; \<And>x. v=Some x \<Longrightarrow> f2 x \<le> ESPEC \<Phi> \<Psi>\<rbrakk> \<Longrightarrow> case_option S1 f2 v \<le> ESPEC \<Phi> \<Psi>"
lemma aux6 [simp]: "(x + y) \<cdot> -x = y \<cdot> -x"
lemma cont_nec_fact2_3[PLM]: "[\<^bold>\<not>(WeaklyContingent (E!)) in v]"

lemma GatewayReq_L2: assumes h1:"msg (Suc 0) req" and h2:"msg (Suc 0) stop" and h3:"msg (Suc 0) a" and h4:"ts lose" and h5:"GatewayReq req dt a stop lose d ack i vc" and h6:"req (t + 3 + k) = [send]" and h7:"inf_last_ti dt t \<noteq> []" and h8:"\<forall>j\<le>2 * d + (4 + k). lose (t + j) = [False]" and h9:"\<forall>m\<le>k. ack (t + 2 + m) = [connection_ok]" shows "i (t + 3 + k + d) \<noteq> []"

lemma hurwitz_formula_integral_semiannulus: fixes N :: nat and r :: real and s :: complex defines "R \<equiv> real (2 * N + 1) * pi" assumes "r > 0" and "r < 2" shows "exp (-\ * pi * s) * integral {r..R} (\<lambda>x. x powr (-s) * exp (-a * x) / (1 - exp (-x))) + integral {r..R} (\<lambda>x. x powr (-s) * exp (a * x) / (1 - exp x)) + contour_integral (part_circlepath 0 R 0 pi) (f s) + contour_integral (part_circlepath 0 r pi 0) (f s) = -2 * pi * \ * exp (- s * of_real pi * \ / 2) * (\<Sum>k\<in>{0<..N}. Res s k)" (is ?thesis1) and "f s contour_integrable_on hankel_semiannulus r N"

lemma (in group) subgroup_card_dvd_group_ord: assumes "subgroup H G" shows "card H dvd order G"

lemma use_implies_allDef: assumes "lookupDef g m (var g v) = v" "m \<in> set (\<alpha>n g)" "var g v \<in> uses g m'" "g \<turnstile> m-ms\<rightarrow>m'" "\<forall>x \<in> set (tl ms). var g v \<notin> defs g x" shows "v \<in> braun_ssa.allDefs g (defNode v)"

lemma irreducible\<^sub>d_dvd_eq: fixes a b :: "'a::{comm_semiring_1,semiring_no_zero_divisors} poly" assumes "irreducible\<^sub>d a" and "irreducible\<^sub>d b" and "a dvd b" and "monic a" and "monic b" shows "a = b"

lemma find_set_3: "find_set_invariant p x y \<and> y = p[[y]] \<Longrightarrow> find_set_postcondition p x y"

lemma Loeb_iff_box_Loeb: "Loebian x \<longleftrightarrow> (\<forall>y. |x] (ad ( |x] y ) + d y) \<le> |x] y)"

lemma implications_cm_all[simp]: "c_imp (cm_all' c \<Delta>) = c_imp c"

lemma path_component_intermediate_subset: "path_component_set u a \<subseteq> t \<and> t \<subseteq> u \<Longrightarrow> path_component_set t a = path_component_set u a"

lemma godel_code_1_iff[elim]: "\<lbrakk>i < length nl; \<not> Suc 0 < godel_code nl\<rbrakk> \<Longrightarrow> nl ! i = 0"

lemma older_seniors_tower''2: assumes "x \<le> n" assumes "y \<le> n" assumes "\<not>sink (token_run x (n + m))" assumes "\<not>sink (token_run y (n + m))" assumes "older_seniors x n = older_seniors x (n + m)" assumes "older_seniors y n \<subseteq> older_seniors x n" shows "older_seniors y n = older_seniors y (n + m)"

theorem Compositionality_While: assumes dind: "b \<equiv>\<^bsub>d\<^esub> b'" assumes bodyrelated: "[c] \<approx>\<^bsub>d\<^esub> [c']" shows "[while b do c od] \<approx>\<^bsub>d\<^esub> [while b' do c' od]"

lemma lem10: "M e"

lemma tailR_correct: "t\<noteq>empty \<Longrightarrow> toList (tailR t) = butlast (toList t)"

lemma finite_backflows_state: "finite (backflows (flows_state \<T>))"

lemma finite_map_graph: "finite A \<Longrightarrow> finite (map_graph f \<inter> (A \<times> UNIV))"

lemma coeffs_eq_Nil [simp]: "coeffs p = [] \<longleftrightarrow> p = 0"

lemma fps_cos_nth_add_2: "fps_cos c $ (n + 2) = - (c * c * fps_cos c $ n / (of_nat (n + 1) * of_nat (n + 2)))"

lemma inorder_node32: "height r > 0 \<Longrightarrow> inorder (treeD (node32 l a m' b r)) = inorder l @ a # inorder (treeD m') @ b # inorder r"

lemma dom_mdeg_le1_ikkbz_sub: "ikkbz_sub t \<noteq> t \<Longrightarrow> dom_children (ikkbz_sub t) T"

lemma red_crit_lifting_family: assumes q_in: "q \<in> Q" shows "calculus Bot_F Inf_F (entails_\<G>_q q) (Red_I_\<G>_q q) (Red_F_\<G>_q q)"

lemma last_tl: "xs = [] \<or> tl xs \<noteq> [] \<Longrightarrow>last (tl xs) = last xs"

lemma disjoint_paths_new_path: assumes no_small_separations: "\<And>S. Separation G v0 v1 S \<Longrightarrow> card S \<ge> Suc (card paths)" shows "\<exists>P_new. v0\<leadsto>P_new\<leadsto>v1 \<and> set P_new \<inter> second_vertices = {}"

lemma strong_orbit_induce_same: "a \<sim>\<^sub>\<K> b \<longrightarrow> (a \<lhd> c \<longleftrightarrow> b \<lhd> c)"

lemma reachable_steps: "\<exists> xs. steps xs \<and> hd xs = s\<^sub>0 \<and> last xs = x" if "reachable x"

lemma mult_zero_sup_circ: "(x \<squnion> y * bot)\<^sup>\<circ> = x\<^sup>\<circ> * (y * bot)\<^sup>\<circ>"

lemma integrable_\<nu>_trace_fin: "integrable (\<T> p s) (\<lambda>t. \<nu>_trace_fin t N)"

lemma isom_gch_units_transpTr0:"\<lbrakk>Ugp E; Gchain n g; Gchain n h; i \<le> n; j \<le> n; i < j; isom_Gchains n (transpos i j) g h\<rbrakk> \<Longrightarrow> {i. i \<le> n \<and> g i \<cong> E} - {i, j} ={i. i \<le> n \<and> h i \<cong> E} - {i, j}"

lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"

lemma decseq_convergent: fixes X :: "nat \<Rightarrow> real" assumes "decseq X" and "\<forall>i. B \<le> X i" obtains L where "X \<longlonglongrightarrow> L" "\<forall>i. L \<le> X i"

lemma sum_add_split_nat_ivl: assumes le: "m <= (k::nat)" "k <= n" and g: "!!i. [| m <= i; i < k |] ==> g i = f i" and h: "!!i. [| k <= i; i < n |] ==> h i = f i" shows "sum g {m..<k} + sum h {k..<n} = sum f {m..<n}"

lemma ap_state_rev_B: "B f \<diamondop> B x = B (State_Monad.return (\<lambda>x f. f x) \<diamondop> x \<diamondop> f)"

lemma pnopI[intro]: assumes "vsv f" and "n \<in>\<^sub>\<circ> \<omega>" and "\<D>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> A ^\<^sub>\<times> n" and "\<R>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> A" shows "pnop A n f"

lemma sorted_spmat_mult_spmat: "sorted_spmat (B::('a::lattice_ring) spmat) \<Longrightarrow> sorted_spmat (mult_spmat A B)"

lemma map_to_set_empty_iff: "map_to_set m = {} \<longleftrightarrow> m = Map.empty" "{} = map_to_set m \<longleftrightarrow> m = Map.empty"

lemma inverse_ereal_ge0I: "0 \<le> (x :: ereal) \<Longrightarrow> 0 \<le> inverse x"

lemma isCont_LCons[THEN isCont_o2[rotated]]: "isCont (LCons x) l"

lemma redT_updT_None: "redT_updT ts ta t = None \<Longrightarrow> ts t = None"

lemma fixes e :: "'addr expr1" and es :: "'addr expr1 list" shows compE2_not_Return: "Return \<notin> set (compE2 e)" and compEs2_not_Return: "Return \<notin> set (compEs2 es)"

lemma ainvf_bijec:"\<lbrakk>aGroup F; aGroup G; bijec\<^bsub>F,G\<^esub> f\<rbrakk> \<Longrightarrow> bijec\<^bsub>G,F\<^esub> (ainvf\<^bsub>F,G\<^esub> f)"

lemma bin_last_bl_to_bin: "odd (bl_to_bin bs) \<longleftrightarrow> bs \<noteq> [] \<and> last bs"

lemma limit_o [simp]: assumes a: "a \<in> limit w" shows "f a \<in> limit (f \<circ> w)"

lemma prob_space_eP: "x \<in> space (PiM {0..<n} M) \<Longrightarrow> prob_space (eP n x)"

lemma map_key_PP_monomial [simp]: "Poly_Mapping.map_key PP (monomial c t) = monomial c (mapping_of t)"

lemma subset_remdups'_append: "set (remdups' f (xs @ ys)) \<subseteq> set (remdups' f xs) \<union> set (remdups' f ys)"

lemma tan_cot: "tan(pi/2 - x) = inverse(tan x)"

lemma R_phase: assumes "set w \<subseteq> \<Sigma>" shows "\<exists> cs. (st.init_config (map INP w), cs) \<in> st.dstep^^(3 + 2 * length w) \<and> rel_S\<^sub>0 cs (init_config w)"

lemma inf_glb: assumes "A \<subseteq> carrier L" shows "greatest L (\<Sqinter>A) (Lower L A)"

lemma IF_rel: "Cl_2 \<C> \<Longrightarrow> \ \<^bold>\<equiv> \\<^sub>F \<F>"

lemma set_pders_nzero: assumes "p\<noteq>0" "q\<in>set (pders p)" shows "q\<noteq>0"

lemma finite_cmax: assumes r: "Card_order r" and s: "Card_order s" shows "finite (Field (cmax r s)) \<longleftrightarrow> finite (Field r) \<and> finite (Field s)"

lemma is_success_alt_def: \<open>is_success a \<longleftrightarrow> a = SUCCESS\<close>

lemma cone_ordLeq_cexp: "cone \<le>o r1 \<Longrightarrow> cone \<le>o r1 ^c r2"

lemma single_valued_D: "single_valued (D c)"

lemma blinop_eqI: "(\<And>i. x $ i = y $ i) \<Longrightarrow> x = y"

lemma find_pos_x_in_dverts: "(x,y) = find_pos_aux v p t1 \<Longrightarrow> x \<in> dverts t1 \<or> p=x"

lemma squarefree_imp_coprime_pderiv: fixes p :: "'a :: {factorial_ring_gcd,semiring_gcd_mult_normalize,semiring_char_0} poly" assumes "squarefree p" and "content p = 1" shows "Rings.coprime p (pderiv p)"

lemma lfp_while: assumes "mono f" and "!!X. X \<subseteq> C \<Longrightarrow> f X \<subseteq> C" and "finite C" shows "lfp f = while (\<lambda>A. f A \<noteq> A) f {}"

lemma euler_theorem: fixes a m :: nat assumes "coprime a m" shows "[a ^ totient m = 1] (mod m)"

lemma prefix_snocD: "prefix (xs@[x]) ys \<Longrightarrow> strict_prefix xs ys"

lemma Bleast_code [code]: "Bleast A P = (if finite A then case filter P (sorted_list_of_set A) of [] \<Rightarrow> abort_Bleast A P | x # xs \<Rightarrow> x else abort_Bleast A P)"

lemma infsetsum_Un_Int: assumes "f abs_summable_on (A \<union> B)" shows "infsetsum f (A \<union> B) = infsetsum f A + infsetsum f B - infsetsum f (A \<inter> B)"

lemma conv_radius_cong_bigtheta: assumes "f \<in> \<Theta>(g)" shows "conv_radius f = conv_radius g"

lemma jmm_heap_conf: "heap_conf addr2thread_id thread_id2addr jmm_empty jmm_allocate jmm_typeof_addr jmm_heap_write (jmm_hconf P) P"

lemma mbisimI2: "\<lbrakk> finite (dom (thr s2)); locks s1 = locks s2; wset s1 = wset s2; interrupts s1 = interrupts s2; wset_thread_ok (wset s2) (thr s2); \<And>t. thr s2 t = None \<Longrightarrow> thr s1 t = None; \<And>t x2 ln. thr s2 t = \<lfloor>(x2, ln)\<rfloor> \<Longrightarrow> \<exists>x1. thr s1 t = \<lfloor>(x1, ln)\<rfloor> \<and> t \<turnstile> (x1, shr s1) \<approx> (x2, shr s2) \<and> (wset s2 t = None \<or> x1 \<approx>w x2) \<rbrakk> \<Longrightarrow> s1 \<approx>m s2"

lemma flush_data_cache_privilege: "(((get_S (cpu_reg_val PSR s)))::word1) = 0 \<Longrightarrow> s' = flush_data_cache s \<Longrightarrow> (((get_S (cpu_reg_val PSR s')))::word1) = 0"

lemma elem_index_inj: assumes "elem_index x = elem_index y" shows "x = y"

lemma d_states_index_properties : assumes "i < length (d_states M q)" shows "fst (d_states M q ! i) \<in> (states M - {q})" "fst (d_states M q ! i) \<noteq> q" "snd (d_states M q ! i) \<in> inputs M" "(\<forall> qx' \<in> set (take i (d_states M q)) . fst (d_states M q ! i) \<noteq> fst qx')" "(\<exists> t \<in> transitions M . t_source t = fst (d_states M q ! i) \<and> t_input t = snd (d_states M q ! i))" "(\<forall> t \<in> transitions M . (t_source t = fst (d_states M q ! i) \<and> t_input t = snd (d_states M q ! i)) \<longrightarrow> (t_target t = q \<or> (\<exists> qx' \<in> set (take i (d_states M q)) . fst qx' = (t_target t))))"

lemma lipschitz_on_compact: assumes "compact K" "K \<subseteq> T" assumes "compact Y" "Y \<subseteq> X" obtains L where "\<And>t. t \<in> K \<Longrightarrow> L-lipschitz_on Y (f t)"

lemma alphaBoundOutputChain': fixes yvec :: "name list" and xvec :: "name list" and B :: "('a, 'b, 'c) boundOutput" assumes "length xvec = length yvec" and "yvec \<sharp>* B" and "yvec \<sharp>* xvec" and "distinct yvec" shows "\<lparr>\<nu>*xvec\<rparr>B = \<lparr>\<nu>*yvec\<rparr>([xvec yvec] \<bullet>\<^sub>v B)"

lemma paths_belowI: "(\<And> x xs. x#xs \<in> paths t \<Longrightarrow> x#xs \<in> paths t') \<Longrightarrow> t \<sqsubseteq> t'"

lemma ca_lr_DBAbsNI: "\<lbrakk> P \<Down> DBAbsN M; closed P; \<And>x X. x \<triangleleft> X \<Longrightarrow> f\<cdot>x \<triangleleft> M<X/0> \<rbrakk> \<Longrightarrow> ValF\<cdot>f \<triangleleft> P"

lemma ta_der_fmap_states_ta: assumes "q |\<in>| ta_der A t" shows "h q |\<in>| ta_der (fmap_states_ta h A) (map_vars_term h t)"

lemma iso_functorI: assumes "\<NN> : \<FF> \<mapsto>\<^sub>C\<^sub>F\<^sub>.\<^sub>i\<^sub>s\<^sub>o \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<FF> \<approx>\<^sub>C\<^sub>F\<^bsub>\<alpha>\<^esub> \<GG>"

lemma connected_verts_G_eq_T: assumes "graph G" and "connected G" and "Suc n = card (verts G)" shows "verts T = verts G"

lemma nfoldli_by_idx_gen: shows "nfoldli (drop k l) c f s = nfoldli [k..<length l] c (\<lambda>i s. do { ASSERT (i<length l); let x = l!i; f x s }) s"

lemma all_countings: "all_countings a b = (a + b) choose a"

lemma \<L>_complement_reg: assumes "ta_sig (ta \<A>) |\<subseteq>| \<F>" shows "\<L> (complement_reg \<A> \<F>) = \<T>\<^sub>G (fset \<F>) - \<L> \<A>"

lemma retraction_map_compose: "\<lbrakk>retraction_map X Y f; retraction_map Y Z g\<rbrakk> \<Longrightarrow> retraction_map X Z (g \<circ> f)"

lemma word_sless_msb_less: "x <s y \<longleftrightarrow> (msb y \<longrightarrow> msb x) \<and> ((msb x \<and> \<not> msb y) \<or> x < y)"

lemma wadjust_stop_Bk[simp]: "wadjust_backto_standard_pos m rs (c, Bk # list) \<Longrightarrow> wadjust_stop m rs (Bk # c, list)"

lemma empty_exposed_face_of [iff]: "{} exposed_face_of S"

lemma Lindelof_space_perfect_map_image: "\<lbrakk>Lindelof_space X; perfect_map X Y f\<rbrakk> \<Longrightarrow> Lindelof_space Y"

theorem serializable_encoding_decoded_plan_is_serializable: assumes "is_valid_problem_strips \<Pi>" and "\<A> \<Turnstile> \<Phi>\<^sub>\<forall> \<Pi> t" shows "is_serial_solution_for_problem \<Pi> (concat (\<Phi>\<inverse> \<Pi> \<A> t))"

lemma binrelchain_funcong_Cons_snoc: assumes "\<And>x y. P x y \<Longrightarrow> f y = f x" "binrelchain P (x#xs@[y])" shows "f y = f x"

lemma linorder_rank_singleton: "linorder_rank R {y} x = (if x \<noteq> y \<and> (y,x) \<in> R then 1 else 0)"

lemma sim12_sizes_map_Val [simp]: "sim12_sizes (map Val vs) = 0"

lemma ennreal_ineq_diff_add: "b \<le> a \<Longrightarrow> a = b + (a - b::ennreal)"

lemma sub_Un1:"B \<subseteq> B \<union> C"

lemma berlekamp_resulting_mat_i[transfer_rule]: "(poly_rel ===> mat_rel R) (berlekamp_resulting_mat_i p ff_ops) berlekamp_resulting_mat"

lemma nTgtOfTrFrom_nTgtOf_last: "tr \<noteq> [] \<Longrightarrow> nTgtOfTrFrom s tr = nTgtOf (last tr)"

lemma ni_diamond_star_induct_sup: "\<parallel>x\<guillemotright>ni(y) \<squnion> ni(z) \<le> ni(y) \<Longrightarrow> \<parallel>x\<^sup>\<star>\<guillemotright>ni(z) \<le> ni(y)"

lemma map_of_aset_update [rewrite]: "unique_keys_set S \<Longrightarrow> (k, v) \<in> S \<Longrightarrow> map_of_aset (S - {(k, v)} \<union> {(k, v')}) = (map_of_aset S) {k \<rightarrow> v'}"

lemma lgc_invar_lgc_step: assumes Si_lt: "enat (Suc i) < llength TNs" and invar: "lgc_invar TNs i" and step: "lnth TNs i \<leadsto>LGC lnth TNs (Suc i)" shows "lgc_invar TNs (Suc i)"

lemma measurable_PiM_finmap_of: assumes "finite J" shows "finmap_of J \<in> measurable (Pi\<^sub>M J M) (PiF {J} M)"

lemma xyxy_conj_yxxy: assumes "x \<cdot> y \<cdot> x \<cdot> y \<sim> y \<cdot> x \<cdot> x \<cdot> y" shows "x \<cdot> y = y \<cdot> x"

lemma list_ex_unfold: "list_ex P (x # y # xs) \<longleftrightarrow> P x \<or> list_ex P (y # xs)" "list_ex P [x] \<longleftrightarrow> P x"

lemma eoption_rule[refine_vcg]: "\<lbrakk> v=None \<Longrightarrow> S1 \<le> ESPEC \<Phi> \<Psi>; \<And>x. v=Some x \<Longrightarrow> f2 x \<le> ESPEC \<Phi> \<Psi>\<rbrakk> \<Longrightarrow> case_option S1 f2 v \<le> ESPEC \<Phi> \<Psi>"

lemma aux6 [simp]: "(x + y) \<cdot> -x = y \<cdot> -x"

lemma cont_nec_fact2_3[PLM]: "[\<^bold>\<not>(WeaklyContingent (E!)) in v]"