--- a/src/HOL/Deriv.thy Wed Feb 18 17:02:38 2009 -0800
+++ b/src/HOL/Deriv.thy Wed Feb 18 19:32:26 2009 -0800
@@ -1457,311 +1457,6 @@
qed
-subsection {* Derivatives of univariate polynomials *}
-
-definition
- pderiv :: "'a::real_normed_field poly \<Rightarrow> 'a poly" where
- "pderiv = poly_rec 0 (\<lambda>a p p'. p + pCons 0 p')"
-
-lemma pderiv_0 [simp]: "pderiv 0 = 0"
- unfolding pderiv_def by (simp add: poly_rec_0)
-
-lemma pderiv_pCons: "pderiv (pCons a p) = p + pCons 0 (pderiv p)"
- unfolding pderiv_def by (simp add: poly_rec_pCons)
-
-lemma coeff_pderiv: "coeff (pderiv p) n = of_nat (Suc n) * coeff p (Suc n)"
- apply (induct p arbitrary: n, simp)
- apply (simp add: pderiv_pCons coeff_pCons algebra_simps split: nat.split)
- done
-
-lemma pderiv_eq_0_iff: "pderiv p = 0 \<longleftrightarrow> degree p = 0"
- apply (rule iffI)
- apply (cases p, simp)
- apply (simp add: expand_poly_eq coeff_pderiv del: of_nat_Suc)
- apply (simp add: expand_poly_eq coeff_pderiv coeff_eq_0)
- done
-
-lemma degree_pderiv: "degree (pderiv p) = degree p - 1"
- apply (rule order_antisym [OF degree_le])
- apply (simp add: coeff_pderiv coeff_eq_0)
- apply (cases "degree p", simp)
- apply (rule le_degree)
- apply (simp add: coeff_pderiv del: of_nat_Suc)
- apply (rule subst, assumption)
- apply (rule leading_coeff_neq_0, clarsimp)
- done
-
-lemma pderiv_singleton [simp]: "pderiv [:a:] = 0"
-by (simp add: pderiv_pCons)
-
-lemma pderiv_add: "pderiv (p + q) = pderiv p + pderiv q"
-by (rule poly_ext, simp add: coeff_pderiv algebra_simps)
-
-lemma pderiv_minus: "pderiv (- p) = - pderiv p"
-by (rule poly_ext, simp add: coeff_pderiv)
-
-lemma pderiv_diff: "pderiv (p - q) = pderiv p - pderiv q"
-by (rule poly_ext, simp add: coeff_pderiv algebra_simps)
-
-lemma pderiv_smult: "pderiv (smult a p) = smult a (pderiv p)"
-by (rule poly_ext, simp add: coeff_pderiv algebra_simps)
-
-lemma pderiv_mult: "pderiv (p * q) = p * pderiv q + q * pderiv p"
-apply (induct p)
-apply simp
-apply (simp add: pderiv_add pderiv_smult pderiv_pCons algebra_simps)
-done
-
-lemma pderiv_power_Suc:
- "pderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p"
-apply (induct n)
-apply simp
-apply (subst power_Suc)
-apply (subst pderiv_mult)
-apply (erule ssubst)
-apply (simp add: smult_add_left algebra_simps)
-done
-
-lemma DERIV_cmult2: "DERIV f x :> D ==> DERIV (%x. (f x) * c :: real) x :> D * c"
-by (simp add: DERIV_cmult mult_commute [of _ c])
-
-lemma DERIV_pow2: "DERIV (%x. x ^ Suc n) x :> real (Suc n) * (x ^ n)"
-by (rule lemma_DERIV_subst, rule DERIV_pow, simp)
-declare DERIV_pow2 [simp] DERIV_pow [simp]
-
-lemma DERIV_add_const: "DERIV f x :> D ==> DERIV (%x. a + f x :: 'a::real_normed_field) x :> D"
-by (rule lemma_DERIV_subst, rule DERIV_add, auto)
-
-lemma poly_DERIV[simp]: "DERIV (%x. poly p x) x :> poly (pderiv p) x"
-apply (induct p)
-apply simp
-apply (simp add: pderiv_pCons)
-apply (rule lemma_DERIV_subst)
-apply (rule DERIV_add DERIV_mult DERIV_const DERIV_ident | assumption)+
-apply simp
-done
-
-text{* Consequences of the derivative theorem above*}
-
-lemma poly_differentiable[simp]: "(%x. poly p x) differentiable (x::real)"
-apply (simp add: differentiable_def)
-apply (blast intro: poly_DERIV)
-done
-
-lemma poly_isCont[simp]: "isCont (%x. poly p x) (x::real)"
-by (rule poly_DERIV [THEN DERIV_isCont])
-
-lemma poly_IVT_pos: "[| a < b; poly p (a::real) < 0; 0 < poly p b |]
- ==> \<exists>x. a < x & x < b & (poly p x = 0)"
-apply (cut_tac f = "%x. poly p x" and a = a and b = b and y = 0 in IVT_objl)
-apply (auto simp add: order_le_less)
-done
-
-lemma poly_IVT_neg: "[| (a::real) < b; 0 < poly p a; poly p b < 0 |]
- ==> \<exists>x. a < x & x < b & (poly p x = 0)"
-by (insert poly_IVT_pos [where p = "- p" ]) simp
-
-lemma poly_MVT: "(a::real) < b ==>
- \<exists>x. a < x & x < b & (poly p b - poly p a = (b - a) * poly (pderiv p) x)"
-apply (drule_tac f = "poly p" in MVT, auto)
-apply (rule_tac x = z in exI)
-apply (auto simp add: real_mult_left_cancel poly_DERIV [THEN DERIV_unique])
-done
-
-text{*Lemmas for Derivatives*}
-
-lemma order_unique_lemma:
- fixes p :: "'a::idom poly"
- assumes "[:-a, 1:] ^ n dvd p \<and> \<not> [:-a, 1:] ^ Suc n dvd p"
- shows "n = order a p"
-unfolding Polynomial.order_def
-apply (rule Least_equality [symmetric])
-apply (rule assms [THEN conjunct2])
-apply (erule contrapos_np)
-apply (rule power_le_dvd)
-apply (rule assms [THEN conjunct1])
-apply simp
-done
-
-lemma lemma_order_pderiv1:
- "pderiv ([:- a, 1:] ^ Suc n * q) = [:- a, 1:] ^ Suc n * pderiv q +
- smult (of_nat (Suc n)) (q * [:- a, 1:] ^ n)"
-apply (simp only: pderiv_mult pderiv_power_Suc)
-apply (simp del: power_poly_Suc of_nat_Suc add: pderiv_pCons)
-done
-
-lemma dvd_add_cancel1:
- fixes a b c :: "'a::comm_ring_1"
- shows "a dvd b + c \<Longrightarrow> a dvd b \<Longrightarrow> a dvd c"
- by (drule (1) Ring_and_Field.dvd_diff, simp)
-
-lemma lemma_order_pderiv [rule_format]:
- "\<forall>p q a. 0 < n &
- pderiv p \<noteq> 0 &
- p = [:- a, 1:] ^ n * q & ~ [:- a, 1:] dvd q
- --> n = Suc (order a (pderiv p))"
- apply (cases "n", safe, rename_tac n p q a)
- apply (rule order_unique_lemma)
- apply (rule conjI)
- apply (subst lemma_order_pderiv1)
- apply (rule dvd_add)
- apply (rule dvd_mult2)
- apply (rule le_imp_power_dvd, simp)
- apply (rule dvd_smult)
- apply (rule dvd_mult)
- apply (rule dvd_refl)
- apply (subst lemma_order_pderiv1)
- apply (erule contrapos_nn) back
- apply (subgoal_tac "[:- a, 1:] ^ Suc n dvd q * [:- a, 1:] ^ n")
- apply (simp del: mult_pCons_left)
- apply (drule dvd_add_cancel1)
- apply (simp del: mult_pCons_left)
- apply (drule dvd_smult_cancel, simp del: of_nat_Suc)
- apply assumption
-done
-
-lemma order_decomp:
- "p \<noteq> 0
- ==> \<exists>q. p = [:-a, 1:] ^ (order a p) * q &
- ~([:-a, 1:] dvd q)"
-apply (drule order [where a=a])
-apply (erule conjE)
-apply (erule dvdE)
-apply (rule exI)
-apply (rule conjI, assumption)
-apply (erule contrapos_nn)
-apply (erule ssubst) back
-apply (subst power_Suc2)
-apply (erule mult_dvd_mono [OF dvd_refl])
-done
-
-lemma order_pderiv: "[| pderiv p \<noteq> 0; order a p \<noteq> 0 |]
- ==> (order a p = Suc (order a (pderiv p)))"
-apply (case_tac "p = 0", simp)
-apply (drule_tac a = a and p = p in order_decomp)
-using neq0_conv
-apply (blast intro: lemma_order_pderiv)
-done
-
-lemma order_mult: "p * q \<noteq> 0 \<Longrightarrow> order a (p * q) = order a p + order a q"
-proof -
- def i \<equiv> "order a p"
- def j \<equiv> "order a q"
- def t \<equiv> "[:-a, 1:]"
- have t_dvd_iff: "\<And>u. t dvd u \<longleftrightarrow> poly u a = 0"
- unfolding t_def by (simp add: dvd_iff_poly_eq_0)
- assume "p * q \<noteq> 0"
- then show "order a (p * q) = i + j"
- apply clarsimp
- apply (drule order [where a=a and p=p, folded i_def t_def])
- apply (drule order [where a=a and p=q, folded j_def t_def])
- apply clarify
- apply (rule order_unique_lemma [symmetric], fold t_def)
- apply (erule dvdE)+
- apply (simp add: power_add t_dvd_iff)
- done
-qed
-
-text{*Now justify the standard squarefree decomposition, i.e. f / gcd(f,f'). *}
-
-lemma order_divides: "[:-a, 1:] ^ n dvd p \<longleftrightarrow> p = 0 \<or> n \<le> order a p"
-apply (cases "p = 0", auto)
-apply (drule order_2 [where a=a and p=p])
-apply (erule contrapos_np)
-apply (erule power_le_dvd)
-apply simp
-apply (erule power_le_dvd [OF order_1])
-done
-
-lemma poly_squarefree_decomp_order:
- assumes "pderiv p \<noteq> 0"
- and p: "p = q * d"
- and p': "pderiv p = e * d"
- and d: "d = r * p + s * pderiv p"
- shows "order a q = (if order a p = 0 then 0 else 1)"
-proof (rule classical)
- assume 1: "order a q \<noteq> (if order a p = 0 then 0 else 1)"
- from `pderiv p \<noteq> 0` have "p \<noteq> 0" by auto
- with p have "order a p = order a q + order a d"
- by (simp add: order_mult)
- with 1 have "order a p \<noteq> 0" by (auto split: if_splits)
- have "order a (pderiv p) = order a e + order a d"
- using `pderiv p \<noteq> 0` `pderiv p = e * d` by (simp add: order_mult)
- have "order a p = Suc (order a (pderiv p))"
- using `pderiv p \<noteq> 0` `order a p \<noteq> 0` by (rule order_pderiv)
- have "d \<noteq> 0" using `p \<noteq> 0` `p = q * d` by simp
- have "([:-a, 1:] ^ (order a (pderiv p))) dvd d"
- apply (simp add: d)
- apply (rule dvd_add)
- apply (rule dvd_mult)
- apply (simp add: order_divides `p \<noteq> 0`
- `order a p = Suc (order a (pderiv p))`)
- apply (rule dvd_mult)
- apply (simp add: order_divides)
- done
- then have "order a (pderiv p) \<le> order a d"
- using `d \<noteq> 0` by (simp add: order_divides)
- show ?thesis
- using `order a p = order a q + order a d`
- using `order a (pderiv p) = order a e + order a d`
- using `order a p = Suc (order a (pderiv p))`
- using `order a (pderiv p) \<le> order a d`
- by auto
-qed
-
-lemma poly_squarefree_decomp_order2: "[| pderiv p \<noteq> 0;
- p = q * d;
- pderiv p = e * d;
- d = r * p + s * pderiv p
- |] ==> \<forall>a. order a q = (if order a p = 0 then 0 else 1)"
-apply (blast intro: poly_squarefree_decomp_order)
-done
-
-lemma order_pderiv2: "[| pderiv p \<noteq> 0; order a p \<noteq> 0 |]
- ==> (order a (pderiv p) = n) = (order a p = Suc n)"
-apply (auto dest: order_pderiv)
-done
-
-definition
- rsquarefree :: "'a::idom poly => bool" where
- "rsquarefree p = (p \<noteq> 0 & (\<forall>a. (order a p = 0) | (order a p = 1)))"
-
-lemma pderiv_iszero: "pderiv p = 0 \<Longrightarrow> \<exists>h. p = [:h:]"
-apply (simp add: pderiv_eq_0_iff)
-apply (case_tac p, auto split: if_splits)
-done
-
-lemma rsquarefree_roots:
- "rsquarefree p = (\<forall>a. ~(poly p a = 0 & poly (pderiv p) a = 0))"
-apply (simp add: rsquarefree_def)
-apply (case_tac "p = 0", simp, simp)
-apply (case_tac "pderiv p = 0")
-apply simp
-apply (drule pderiv_iszero, clarify)
-apply simp
-apply (rule allI)
-apply (cut_tac p = "[:h:]" and a = a in order_root)
-apply simp
-apply (auto simp add: order_root order_pderiv2)
-apply (erule_tac x="a" in allE, simp)
-done
-
-lemma poly_squarefree_decomp:
- assumes "pderiv p \<noteq> 0"
- and "p = q * d"
- and "pderiv p = e * d"
- and "d = r * p + s * pderiv p"
- shows "rsquarefree q & (\<forall>a. (poly q a = 0) = (poly p a = 0))"
-proof -
- from `pderiv p \<noteq> 0` have "p \<noteq> 0" by auto
- with `p = q * d` have "q \<noteq> 0" by simp
- have "\<forall>a. order a q = (if order a p = 0 then 0 else 1)"
- using assms by (rule poly_squarefree_decomp_order2)
- with `p \<noteq> 0` `q \<noteq> 0` show ?thesis
- by (simp add: rsquarefree_def order_root)
-qed
-
-
subsection {* Theorems about Limits *}
(* need to rename second isCont_inverse *)