diff -r f707dbcf11e3 -r fc41a5650fb1 src/HOL/Library/Polynomial_FPS.thy --- a/src/HOL/Library/Polynomial_FPS.thy Thu Apr 06 08:33:37 2017 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,313 +0,0 @@ -(* Title: HOL/Library/Polynomial_FPS.thy - Author: Manuel Eberl, TU München - -Converting polynomials to formal power series. -*) - -section \Converting polynomials to formal power series\ - -theory Polynomial_FPS - imports Polynomial Formal_Power_Series -begin - -definition fps_of_poly where - "fps_of_poly p = Abs_fps (coeff p)" - -lemma fps_of_poly_eq_iff: "fps_of_poly p = fps_of_poly q \ p = q" - by (simp add: fps_of_poly_def poly_eq_iff fps_eq_iff) - -lemma fps_of_poly_nth [simp]: "fps_of_poly p $ n = coeff p n" - by (simp add: fps_of_poly_def) - -lemma fps_of_poly_const: "fps_of_poly [:c:] = fps_const c" -proof (subst fps_eq_iff, clarify) - fix n :: nat show "fps_of_poly [:c:] $ n = fps_const c $ n" - by (cases n) (auto simp: fps_of_poly_def) -qed - -lemma fps_of_poly_0 [simp]: "fps_of_poly 0 = 0" - by (subst fps_const_0_eq_0 [symmetric], subst fps_of_poly_const [symmetric]) simp - -lemma fps_of_poly_1 [simp]: "fps_of_poly 1 = 1" - by (subst fps_const_1_eq_1 [symmetric], subst fps_of_poly_const [symmetric]) - (simp add: one_poly_def) - -lemma fps_of_poly_1' [simp]: "fps_of_poly [:1:] = 1" - by (subst fps_const_1_eq_1 [symmetric], subst fps_of_poly_const [symmetric]) - (simp add: one_poly_def) - -lemma fps_of_poly_numeral [simp]: "fps_of_poly (numeral n) = numeral n" - by (simp add: numeral_fps_const fps_of_poly_const [symmetric] numeral_poly) - -lemma fps_of_poly_numeral' [simp]: "fps_of_poly [:numeral n:] = numeral n" - by (simp add: numeral_fps_const fps_of_poly_const [symmetric] numeral_poly) - -lemma fps_of_poly_X [simp]: "fps_of_poly [:0, 1:] = X" - by (auto simp add: fps_of_poly_def fps_eq_iff coeff_pCons split: nat.split) - -lemma fps_of_poly_add: "fps_of_poly (p + q) = fps_of_poly p + fps_of_poly q" - by (simp add: fps_of_poly_def plus_poly.rep_eq fps_plus_def) - -lemma fps_of_poly_diff: "fps_of_poly (p - q) = fps_of_poly p - fps_of_poly q" - by (simp add: fps_of_poly_def minus_poly.rep_eq fps_minus_def) - -lemma fps_of_poly_uminus: "fps_of_poly (-p) = -fps_of_poly p" - by (simp add: fps_of_poly_def uminus_poly.rep_eq fps_uminus_def) - -lemma fps_of_poly_mult: "fps_of_poly (p * q) = fps_of_poly p * fps_of_poly q" - by (simp add: fps_of_poly_def fps_times_def fps_eq_iff coeff_mult atLeast0AtMost) - -lemma fps_of_poly_smult: - "fps_of_poly (smult c p) = fps_const c * fps_of_poly p" - using fps_of_poly_mult[of "[:c:]" p] by (simp add: fps_of_poly_mult fps_of_poly_const) - -lemma fps_of_poly_sum: "fps_of_poly (sum f A) = sum (\x. fps_of_poly (f x)) A" - by (cases "finite A", induction rule: finite_induct) (simp_all add: fps_of_poly_add) - -lemma fps_of_poly_sum_list: "fps_of_poly (sum_list xs) = sum_list (map fps_of_poly xs)" - by (induction xs) (simp_all add: fps_of_poly_add) - -lemma fps_of_poly_prod: "fps_of_poly (prod f A) = prod (\x. fps_of_poly (f x)) A" - by (cases "finite A", induction rule: finite_induct) (simp_all add: fps_of_poly_mult) - -lemma fps_of_poly_prod_list: "fps_of_poly (prod_list xs) = prod_list (map fps_of_poly xs)" - by (induction xs) (simp_all add: fps_of_poly_mult) - -lemma fps_of_poly_pCons: - "fps_of_poly (pCons (c :: 'a :: semiring_1) p) = fps_const c + fps_of_poly p * X" - by (subst fps_mult_X_commute [symmetric], intro fps_ext) - (auto simp: fps_of_poly_def coeff_pCons split: nat.split) - -lemma fps_of_poly_pderiv: "fps_of_poly (pderiv p) = fps_deriv (fps_of_poly p)" - by (intro fps_ext) (simp add: fps_of_poly_nth coeff_pderiv) - -lemma fps_of_poly_power: "fps_of_poly (p ^ n) = fps_of_poly p ^ n" - by (induction n) (simp_all add: fps_of_poly_mult) - -lemma fps_of_poly_monom: "fps_of_poly (monom (c :: 'a :: comm_ring_1) n) = fps_const c * X ^ n" - by (intro fps_ext) simp_all - -lemma fps_of_poly_monom': "fps_of_poly (monom (1 :: 'a :: comm_ring_1) n) = X ^ n" - by (simp add: fps_of_poly_monom) - -lemma fps_of_poly_div: - assumes "(q :: 'a :: field poly) dvd p" - shows "fps_of_poly (p div q) = fps_of_poly p / fps_of_poly q" -proof (cases "q = 0") - case False - from False fps_of_poly_eq_iff[of q 0] have nz: "fps_of_poly q \ 0" by simp - from assms have "p = (p div q) * q" by simp - also have "fps_of_poly \ = fps_of_poly (p div q) * fps_of_poly q" - by (simp add: fps_of_poly_mult) - also from nz have "\ / fps_of_poly q = fps_of_poly (p div q)" - by (intro nonzero_mult_div_cancel_right) (auto simp: fps_of_poly_0) - finally show ?thesis .. -qed simp - -lemma fps_of_poly_divide_numeral: - "fps_of_poly (smult (inverse (numeral c :: 'a :: field)) p) = fps_of_poly p / numeral c" -proof - - have "smult (inverse (numeral c)) p = [:inverse (numeral c):] * p" by simp - also have "fps_of_poly \ = fps_of_poly p / numeral c" - by (subst fps_of_poly_mult) (simp add: numeral_fps_const fps_of_poly_pCons) - finally show ?thesis by simp -qed - - -lemma subdegree_fps_of_poly: - assumes "p \ 0" - defines "n \ Polynomial.order 0 p" - shows "subdegree (fps_of_poly p) = n" -proof (rule subdegreeI) - from assms have "monom 1 n dvd p" by (simp add: monom_1_dvd_iff) - thus zero: "fps_of_poly p $ i = 0" if "i < n" for i - using that by (simp add: monom_1_dvd_iff') - - from assms have "\monom 1 (Suc n) dvd p" - by (auto simp: monom_1_dvd_iff simp del: power_Suc) - then obtain k where k: "k \ n" "fps_of_poly p $ k \ 0" - by (auto simp: monom_1_dvd_iff' less_Suc_eq_le) - with zero[of k] have "k = n" by linarith - with k show "fps_of_poly p $ n \ 0" by simp -qed - -lemma fps_of_poly_dvd: - assumes "p dvd q" - shows "fps_of_poly (p :: 'a :: field poly) dvd fps_of_poly q" -proof (cases "p = 0 \ q = 0") - case False - with assms fps_of_poly_eq_iff[of p 0] fps_of_poly_eq_iff[of q 0] show ?thesis - by (auto simp: fps_dvd_iff subdegree_fps_of_poly dvd_imp_order_le) -qed (insert assms, auto) - - -lemmas fps_of_poly_simps = - fps_of_poly_0 fps_of_poly_1 fps_of_poly_numeral fps_of_poly_const fps_of_poly_X - fps_of_poly_add fps_of_poly_diff fps_of_poly_uminus fps_of_poly_mult fps_of_poly_smult - fps_of_poly_sum fps_of_poly_sum_list fps_of_poly_prod fps_of_poly_prod_list - fps_of_poly_pCons fps_of_poly_pderiv fps_of_poly_power fps_of_poly_monom - fps_of_poly_divide_numeral - -lemma fps_of_poly_pcompose: - assumes "coeff q 0 = (0 :: 'a :: idom)" - shows "fps_of_poly (pcompose p q) = fps_compose (fps_of_poly p) (fps_of_poly q)" - using assms by (induction p rule: pCons_induct) - (auto simp: pcompose_pCons fps_of_poly_simps fps_of_poly_pCons - fps_compose_add_distrib fps_compose_mult_distrib) - -lemmas reify_fps_atom = - fps_of_poly_0 fps_of_poly_1' fps_of_poly_numeral' fps_of_poly_const fps_of_poly_X - - -text \ - The following simproc can reduce the equality of two polynomial FPSs two equality of the - respective polynomials. A polynomial FPS is one that only has finitely many non-zero - coefficients and can therefore be written as @{term "fps_of_poly p"} for some - polynomial \p\. - - This may sound trivial, but it covers a number of annoying side conditions like - @{term "1 + X \ 0"} that would otherwise not be solved automatically. -\ - -ML \ - -(* TODO: Support for division *) -signature POLY_FPS = sig - -val reify_conv : conv -val eq_conv : conv -val eq_simproc : cterm -> thm option - -end - - -structure Poly_Fps = struct - -fun const_binop_conv s conv ct = - case Thm.term_of ct of - (Const (s', _) $ _ $ _) => - if s = s' then - Conv.binop_conv conv ct - else - raise CTERM ("const_binop_conv", [ct]) - | _ => raise CTERM ("const_binop_conv", [ct]) - -fun reify_conv ct = - let - val rewr = Conv.rewrs_conv o map (fn thm => thm RS @{thm eq_reflection}) - val un = Conv.arg_conv reify_conv - val bin = Conv.binop_conv reify_conv - in - case Thm.term_of ct of - (Const (@{const_name "fps_of_poly"}, _) $ _) => ct |> Conv.all_conv - | (Const (@{const_name "Groups.plus"}, _) $ _ $ _) => ct |> ( - bin then_conv rewr @{thms fps_of_poly_add [symmetric]}) - | (Const (@{const_name "Groups.uminus"}, _) $ _) => ct |> ( - un then_conv rewr @{thms fps_of_poly_uminus [symmetric]}) - | (Const (@{const_name "Groups.minus"}, _) $ _ $ _) => ct |> ( - bin then_conv rewr @{thms fps_of_poly_diff [symmetric]}) - | (Const (@{const_name "Groups.times"}, _) $ _ $ _) => ct |> ( - bin then_conv rewr @{thms fps_of_poly_mult [symmetric]}) - | (Const (@{const_name "Rings.divide"}, _) $ _ $ (Const (@{const_name "Num.numeral"}, _) $ _)) - => ct |> (Conv.fun_conv (Conv.arg_conv reify_conv) - then_conv rewr @{thms fps_of_poly_divide_numeral [symmetric]}) - | (Const (@{const_name "Power.power"}, _) $ Const (@{const_name "X"},_) $ _) => ct |> ( - rewr @{thms fps_of_poly_monom' [symmetric]}) - | (Const (@{const_name "Power.power"}, _) $ _ $ _) => ct |> ( - Conv.fun_conv (Conv.arg_conv reify_conv) - then_conv rewr @{thms fps_of_poly_power [symmetric]}) - | _ => ct |> ( - rewr @{thms reify_fps_atom [symmetric]}) - end - - -fun eq_conv ct = - case Thm.term_of ct of - (Const (@{const_name "HOL.eq"}, _) $ _ $ _) => ct |> ( - Conv.binop_conv reify_conv - then_conv Conv.rewr_conv @{thm fps_of_poly_eq_iff[THEN eq_reflection]}) - | _ => raise CTERM ("poly_fps_eq_conv", [ct]) - -val eq_simproc = try eq_conv - -end -\ - -simproc_setup poly_fps_eq ("(f :: 'a fps) = g") = \K (K Poly_Fps.eq_simproc)\ - -lemma fps_of_poly_linear: "fps_of_poly [:a,1 :: 'a :: field:] = X + fps_const a" - by simp - -lemma fps_of_poly_linear': "fps_of_poly [:1,a :: 'a :: field:] = 1 + fps_const a * X" - by simp - -lemma fps_of_poly_cutoff [simp]: - "fps_of_poly (poly_cutoff n p) = fps_cutoff n (fps_of_poly p)" - by (simp add: fps_eq_iff coeff_poly_cutoff) - -lemma fps_of_poly_shift [simp]: "fps_of_poly (poly_shift n p) = fps_shift n (fps_of_poly p)" - by (simp add: fps_eq_iff coeff_poly_shift) - - -definition poly_subdegree :: "'a::zero poly \ nat" where - "poly_subdegree p = subdegree (fps_of_poly p)" - -lemma coeff_less_poly_subdegree: - "k < poly_subdegree p \ coeff p k = 0" - unfolding poly_subdegree_def using nth_less_subdegree_zero[of k "fps_of_poly p"] by simp - -(* TODO: Move ? *) -definition prefix_length :: "('a \ bool) \ 'a list \ nat" where - "prefix_length P xs = length (takeWhile P xs)" - -primrec prefix_length_aux :: "('a \ bool) \ nat \ 'a list \ nat" where - "prefix_length_aux P acc [] = acc" -| "prefix_length_aux P acc (x#xs) = (if P x then prefix_length_aux P (Suc acc) xs else acc)" - -lemma prefix_length_aux_correct: "prefix_length_aux P acc xs = prefix_length P xs + acc" - by (induction xs arbitrary: acc) (simp_all add: prefix_length_def) - -lemma prefix_length_code [code]: "prefix_length P xs = prefix_length_aux P 0 xs" - by (simp add: prefix_length_aux_correct) - -lemma prefix_length_le_length: "prefix_length P xs \ length xs" - by (induction xs) (simp_all add: prefix_length_def) - -lemma prefix_length_less_length: "(\x\set xs. \P x) \ prefix_length P xs < length xs" - by (induction xs) (simp_all add: prefix_length_def) - -lemma nth_prefix_length: - "(\x\set xs. \P x) \ \P (xs ! prefix_length P xs)" - by (induction xs) (simp_all add: prefix_length_def) - -lemma nth_less_prefix_length: - "n < prefix_length P xs \ P (xs ! n)" - by (induction xs arbitrary: n) - (auto simp: prefix_length_def nth_Cons split: if_splits nat.splits) -(* END TODO *) - -lemma poly_subdegree_code [code]: "poly_subdegree p = prefix_length (op = 0) (coeffs p)" -proof (cases "p = 0") - case False - note [simp] = this - define n where "n = prefix_length (op = 0) (coeffs p)" - from False have "\k. coeff p k \ 0" by (auto simp: poly_eq_iff) - hence ex: "\x\set (coeffs p). x \ 0" by (auto simp: coeffs_def) - hence n_less: "n < length (coeffs p)" and nonzero: "coeffs p ! n \ 0" - unfolding n_def by (auto intro!: prefix_length_less_length nth_prefix_length) - show ?thesis unfolding poly_subdegree_def - proof (intro subdegreeI) - from n_less have "fps_of_poly p $ n = coeffs p ! n" - by (subst coeffs_nth) (simp_all add: degree_eq_length_coeffs) - with nonzero show "fps_of_poly p $ prefix_length (op = 0) (coeffs p) \ 0" - unfolding n_def by simp - next - fix k assume A: "k < prefix_length (op = 0) (coeffs p)" - also have "\ \ length (coeffs p)" by (rule prefix_length_le_length) - finally show "fps_of_poly p $ k = 0" - using nth_less_prefix_length[OF A] - by (simp add: coeffs_nth degree_eq_length_coeffs) - qed -qed (simp_all add: poly_subdegree_def prefix_length_def) - -end