--- 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 \<open>Converting polynomials to formal power series\<close>
-
-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 \<longleftrightarrow> 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 (\<lambda>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 (\<lambda>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 \<noteq> 0" by simp
- from assms have "p = (p div q) * q" by simp
- also have "fps_of_poly \<dots> = fps_of_poly (p div q) * fps_of_poly q"
- by (simp add: fps_of_poly_mult)
- also from nz have "\<dots> / 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 \<dots> = 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 \<noteq> 0"
- defines "n \<equiv> 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 "\<not>monom 1 (Suc n) dvd p"
- by (auto simp: monom_1_dvd_iff simp del: power_Suc)
- then obtain k where k: "k \<le> n" "fps_of_poly p $ k \<noteq> 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 \<noteq> 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 \<or> 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 \<open>
- 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 \<open>p\<close>.
-
- This may sound trivial, but it covers a number of annoying side conditions like
- @{term "1 + X \<noteq> 0"} that would otherwise not be solved automatically.
-\<close>
-
-ML \<open>
-
-(* 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
-\<close>
-
-simproc_setup poly_fps_eq ("(f :: 'a fps) = g") = \<open>K (K Poly_Fps.eq_simproc)\<close>
-
-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 \<Rightarrow> nat" where
- "poly_subdegree p = subdegree (fps_of_poly p)"
-
-lemma coeff_less_poly_subdegree:
- "k < poly_subdegree p \<Longrightarrow> 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 \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> nat" where
- "prefix_length P xs = length (takeWhile P xs)"
-
-primrec prefix_length_aux :: "('a \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> 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 \<le> length xs"
- by (induction xs) (simp_all add: prefix_length_def)
-
-lemma prefix_length_less_length: "(\<exists>x\<in>set xs. \<not>P x) \<Longrightarrow> prefix_length P xs < length xs"
- by (induction xs) (simp_all add: prefix_length_def)
-
-lemma nth_prefix_length:
- "(\<exists>x\<in>set xs. \<not>P x) \<Longrightarrow> \<not>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 \<Longrightarrow> 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 "\<exists>k. coeff p k \<noteq> 0" by (auto simp: poly_eq_iff)
- hence ex: "\<exists>x\<in>set (coeffs p). x \<noteq> 0" by (auto simp: coeffs_def)
- hence n_less: "n < length (coeffs p)" and nonzero: "coeffs p ! n \<noteq> 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) \<noteq> 0"
- unfolding n_def by simp
- next
- fix k assume A: "k < prefix_length (op = 0) (coeffs p)"
- also have "\<dots> \<le> 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