src/HOL/Library/Poly_Deriv.thy
 author hoelzl Thu Apr 03 17:56:08 2014 +0200 (2014-04-03) changeset 56381 0556204bc230 parent 56217 dc429a5b13c4 child 56383 8e7052e9fda4 permissions -rw-r--r--
merged DERIV_intros, has_derivative_intros into derivative_intros
```     1 (*  Title:      HOL/Library/Poly_Deriv.thy
```
```     2     Author:     Amine Chaieb
```
```     3     Author:     Brian Huffman
```
```     4 *)
```
```     5
```
```     6 header{* Polynomials and Differentiation *}
```
```     7
```
```     8 theory Poly_Deriv
```
```     9 imports Deriv Polynomial
```
```    10 begin
```
```    11
```
```    12 subsection {* Derivatives of univariate polynomials *}
```
```    13
```
```    14 function pderiv :: "'a::real_normed_field poly \<Rightarrow> 'a poly"
```
```    15 where
```
```    16   [simp del]: "pderiv (pCons a p) = (if p = 0 then 0 else p + pCons 0 (pderiv p))"
```
```    17   by (auto intro: pCons_cases)
```
```    18
```
```    19 termination pderiv
```
```    20   by (relation "measure degree") simp_all
```
```    21
```
```    22 lemma pderiv_0 [simp]:
```
```    23   "pderiv 0 = 0"
```
```    24   using pderiv.simps [of 0 0] by simp
```
```    25
```
```    26 lemma pderiv_pCons:
```
```    27   "pderiv (pCons a p) = p + pCons 0 (pderiv p)"
```
```    28   by (simp add: pderiv.simps)
```
```    29
```
```    30 lemma coeff_pderiv: "coeff (pderiv p) n = of_nat (Suc n) * coeff p (Suc n)"
```
```    31   apply (induct p arbitrary: n, simp)
```
```    32   apply (simp add: pderiv_pCons coeff_pCons algebra_simps split: nat.split)
```
```    33   done
```
```    34
```
```    35 primrec pderiv_coeffs :: "'a::comm_monoid_add list \<Rightarrow> 'a list"
```
```    36 where
```
```    37   "pderiv_coeffs [] = []"
```
```    38 | "pderiv_coeffs (x # xs) = plus_coeffs xs (cCons 0 (pderiv_coeffs xs))"
```
```    39
```
```    40 lemma coeffs_pderiv [code abstract]:
```
```    41   "coeffs (pderiv p) = pderiv_coeffs (coeffs p)"
```
```    42   by (rule sym, induct p) (simp_all add: pderiv_pCons coeffs_plus_eq_plus_coeffs cCons_def)
```
```    43
```
```    44 lemma pderiv_eq_0_iff: "pderiv p = 0 \<longleftrightarrow> degree p = 0"
```
```    45   apply (rule iffI)
```
```    46   apply (cases p, simp)
```
```    47   apply (simp add: poly_eq_iff coeff_pderiv del: of_nat_Suc)
```
```    48   apply (simp add: poly_eq_iff coeff_pderiv coeff_eq_0)
```
```    49   done
```
```    50
```
```    51 lemma degree_pderiv: "degree (pderiv p) = degree p - 1"
```
```    52   apply (rule order_antisym [OF degree_le])
```
```    53   apply (simp add: coeff_pderiv coeff_eq_0)
```
```    54   apply (cases "degree p", simp)
```
```    55   apply (rule le_degree)
```
```    56   apply (simp add: coeff_pderiv del: of_nat_Suc)
```
```    57   apply (rule subst, assumption)
```
```    58   apply (rule leading_coeff_neq_0, clarsimp)
```
```    59   done
```
```    60
```
```    61 lemma pderiv_singleton [simp]: "pderiv [:a:] = 0"
```
```    62 by (simp add: pderiv_pCons)
```
```    63
```
```    64 lemma pderiv_add: "pderiv (p + q) = pderiv p + pderiv q"
```
```    65 by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
```
```    66
```
```    67 lemma pderiv_minus: "pderiv (- p) = - pderiv p"
```
```    68 by (rule poly_eqI, simp add: coeff_pderiv)
```
```    69
```
```    70 lemma pderiv_diff: "pderiv (p - q) = pderiv p - pderiv q"
```
```    71 by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
```
```    72
```
```    73 lemma pderiv_smult: "pderiv (smult a p) = smult a (pderiv p)"
```
```    74 by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
```
```    75
```
```    76 lemma pderiv_mult: "pderiv (p * q) = p * pderiv q + q * pderiv p"
```
```    77 apply (induct p)
```
```    78 apply simp
```
```    79 apply (simp add: pderiv_add pderiv_smult pderiv_pCons algebra_simps)
```
```    80 done
```
```    81
```
```    82 lemma pderiv_power_Suc:
```
```    83   "pderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p"
```
```    84 apply (induct n)
```
```    85 apply simp
```
```    86 apply (subst power_Suc)
```
```    87 apply (subst pderiv_mult)
```
```    88 apply (erule ssubst)
```
```    89 apply (simp only: of_nat_Suc smult_add_left smult_1_left)
```
```    90 apply (simp add: algebra_simps) (* FIXME *)
```
```    91 done
```
```    92
```
```    93
```
```    94 lemma DERIV_cmult2: "DERIV f x :> D ==> DERIV (%x. (f x) * c :: real) x :> D * c"
```
```    95 by (simp add: DERIV_cmult mult_commute [of _ c])
```
```    96
```
```    97 lemma DERIV_pow2: "DERIV (%x. x ^ Suc n) x :> real (Suc n) * (x ^ n)"
```
```    98 by (rule DERIV_cong, rule DERIV_pow, simp)
```
```    99 declare DERIV_pow2 [simp] DERIV_pow [simp]
```
```   100
```
```   101 lemma DERIV_add_const: "DERIV f x :> D ==>  DERIV (%x. a + f x :: 'a::real_normed_field) x :> D"
```
```   102 by (rule DERIV_cong, rule DERIV_add, auto)
```
```   103
```
```   104 lemma poly_DERIV[simp]: "DERIV (%x. poly p x) x :> poly (pderiv p) x"
```
```   105   by (induct p, auto intro!: derivative_eq_intros simp add: pderiv_pCons)
```
```   106
```
```   107 text{* Consequences of the derivative theorem above*}
```
```   108
```
```   109 lemma poly_differentiable[simp]: "(%x. poly p x) differentiable (at x::real filter)"
```
```   110 apply (simp add: real_differentiable_def)
```
```   111 apply (blast intro: poly_DERIV)
```
```   112 done
```
```   113
```
```   114 lemma poly_isCont[simp]: "isCont (%x. poly p x) (x::real)"
```
```   115 by (rule poly_DERIV [THEN DERIV_isCont])
```
```   116
```
```   117 lemma poly_IVT_pos: "[| a < b; poly p (a::real) < 0; 0 < poly p b |]
```
```   118       ==> \<exists>x. a < x & x < b & (poly p x = 0)"
```
```   119 apply (cut_tac f = "%x. poly p x" and a = a and b = b and y = 0 in IVT_objl)
```
```   120 apply (auto simp add: order_le_less)
```
```   121 done
```
```   122
```
```   123 lemma poly_IVT_neg: "[| (a::real) < b; 0 < poly p a; poly p b < 0 |]
```
```   124       ==> \<exists>x. a < x & x < b & (poly p x = 0)"
```
```   125 by (insert poly_IVT_pos [where p = "- p" ]) simp
```
```   126
```
```   127 lemma poly_MVT: "(a::real) < b ==>
```
```   128      \<exists>x. a < x & x < b & (poly p b - poly p a = (b - a) * poly (pderiv p) x)"
```
```   129 apply (drule_tac f = "poly p" in MVT, auto)
```
```   130 apply (rule_tac x = z in exI)
```
```   131 apply (auto simp add: mult_left_cancel poly_DERIV [THEN DERIV_unique])
```
```   132 done
```
```   133
```
```   134 text{*Lemmas for Derivatives*}
```
```   135
```
```   136 lemma order_unique_lemma:
```
```   137   fixes p :: "'a::idom poly"
```
```   138   assumes "[:-a, 1:] ^ n dvd p \<and> \<not> [:-a, 1:] ^ Suc n dvd p"
```
```   139   shows "n = order a p"
```
```   140 unfolding Polynomial.order_def
```
```   141 apply (rule Least_equality [symmetric])
```
```   142 apply (rule assms [THEN conjunct2])
```
```   143 apply (erule contrapos_np)
```
```   144 apply (rule power_le_dvd)
```
```   145 apply (rule assms [THEN conjunct1])
```
```   146 apply simp
```
```   147 done
```
```   148
```
```   149 lemma lemma_order_pderiv1:
```
```   150   "pderiv ([:- a, 1:] ^ Suc n * q) = [:- a, 1:] ^ Suc n * pderiv q +
```
```   151     smult (of_nat (Suc n)) (q * [:- a, 1:] ^ n)"
```
```   152 apply (simp only: pderiv_mult pderiv_power_Suc)
```
```   153 apply (simp del: power_Suc of_nat_Suc add: pderiv_pCons)
```
```   154 done
```
```   155
```
```   156 lemma dvd_add_cancel1:
```
```   157   fixes a b c :: "'a::comm_ring_1"
```
```   158   shows "a dvd b + c \<Longrightarrow> a dvd b \<Longrightarrow> a dvd c"
```
```   159   by (drule (1) Rings.dvd_diff, simp)
```
```   160
```
```   161 lemma lemma_order_pderiv [rule_format]:
```
```   162      "\<forall>p q a. 0 < n &
```
```   163        pderiv p \<noteq> 0 &
```
```   164        p = [:- a, 1:] ^ n * q & ~ [:- a, 1:] dvd q
```
```   165        --> n = Suc (order a (pderiv p))"
```
```   166  apply (cases "n", safe, rename_tac n p q a)
```
```   167  apply (rule order_unique_lemma)
```
```   168  apply (rule conjI)
```
```   169   apply (subst lemma_order_pderiv1)
```
```   170   apply (rule dvd_add)
```
```   171    apply (rule dvd_mult2)
```
```   172    apply (rule le_imp_power_dvd, simp)
```
```   173   apply (rule dvd_smult)
```
```   174   apply (rule dvd_mult)
```
```   175   apply (rule dvd_refl)
```
```   176  apply (subst lemma_order_pderiv1)
```
```   177  apply (erule contrapos_nn) back
```
```   178  apply (subgoal_tac "[:- a, 1:] ^ Suc n dvd q * [:- a, 1:] ^ n")
```
```   179   apply (simp del: mult_pCons_left)
```
```   180  apply (drule dvd_add_cancel1)
```
```   181   apply (simp del: mult_pCons_left)
```
```   182  apply (drule dvd_smult_cancel, simp del: of_nat_Suc)
```
```   183  apply assumption
```
```   184 done
```
```   185
```
```   186 lemma order_decomp:
```
```   187      "p \<noteq> 0
```
```   188       ==> \<exists>q. p = [:-a, 1:] ^ (order a p) * q &
```
```   189                 ~([:-a, 1:] dvd q)"
```
```   190 apply (drule order [where a=a])
```
```   191 apply (erule conjE)
```
```   192 apply (erule dvdE)
```
```   193 apply (rule exI)
```
```   194 apply (rule conjI, assumption)
```
```   195 apply (erule contrapos_nn)
```
```   196 apply (erule ssubst) back
```
```   197 apply (subst power_Suc2)
```
```   198 apply (erule mult_dvd_mono [OF dvd_refl])
```
```   199 done
```
```   200
```
```   201 lemma order_pderiv: "[| pderiv p \<noteq> 0; order a p \<noteq> 0 |]
```
```   202       ==> (order a p = Suc (order a (pderiv p)))"
```
```   203 apply (case_tac "p = 0", simp)
```
```   204 apply (drule_tac a = a and p = p in order_decomp)
```
```   205 using neq0_conv
```
```   206 apply (blast intro: lemma_order_pderiv)
```
```   207 done
```
```   208
```
```   209 lemma order_mult: "p * q \<noteq> 0 \<Longrightarrow> order a (p * q) = order a p + order a q"
```
```   210 proof -
```
```   211   def i \<equiv> "order a p"
```
```   212   def j \<equiv> "order a q"
```
```   213   def t \<equiv> "[:-a, 1:]"
```
```   214   have t_dvd_iff: "\<And>u. t dvd u \<longleftrightarrow> poly u a = 0"
```
```   215     unfolding t_def by (simp add: dvd_iff_poly_eq_0)
```
```   216   assume "p * q \<noteq> 0"
```
```   217   then show "order a (p * q) = i + j"
```
```   218     apply clarsimp
```
```   219     apply (drule order [where a=a and p=p, folded i_def t_def])
```
```   220     apply (drule order [where a=a and p=q, folded j_def t_def])
```
```   221     apply clarify
```
```   222     apply (rule order_unique_lemma [symmetric], fold t_def)
```
```   223     apply (erule dvdE)+
```
```   224     apply (simp add: power_add t_dvd_iff)
```
```   225     done
```
```   226 qed
```
```   227
```
```   228 text{*Now justify the standard squarefree decomposition, i.e. f / gcd(f,f'). *}
```
```   229
```
```   230 lemma order_divides: "[:-a, 1:] ^ n dvd p \<longleftrightarrow> p = 0 \<or> n \<le> order a p"
```
```   231 apply (cases "p = 0", auto)
```
```   232 apply (drule order_2 [where a=a and p=p])
```
```   233 apply (erule contrapos_np)
```
```   234 apply (erule power_le_dvd)
```
```   235 apply simp
```
```   236 apply (erule power_le_dvd [OF order_1])
```
```   237 done
```
```   238
```
```   239 lemma poly_squarefree_decomp_order:
```
```   240   assumes "pderiv p \<noteq> 0"
```
```   241   and p: "p = q * d"
```
```   242   and p': "pderiv p = e * d"
```
```   243   and d: "d = r * p + s * pderiv p"
```
```   244   shows "order a q = (if order a p = 0 then 0 else 1)"
```
```   245 proof (rule classical)
```
```   246   assume 1: "order a q \<noteq> (if order a p = 0 then 0 else 1)"
```
```   247   from `pderiv p \<noteq> 0` have "p \<noteq> 0" by auto
```
```   248   with p have "order a p = order a q + order a d"
```
```   249     by (simp add: order_mult)
```
```   250   with 1 have "order a p \<noteq> 0" by (auto split: if_splits)
```
```   251   have "order a (pderiv p) = order a e + order a d"
```
```   252     using `pderiv p \<noteq> 0` `pderiv p = e * d` by (simp add: order_mult)
```
```   253   have "order a p = Suc (order a (pderiv p))"
```
```   254     using `pderiv p \<noteq> 0` `order a p \<noteq> 0` by (rule order_pderiv)
```
```   255   have "d \<noteq> 0" using `p \<noteq> 0` `p = q * d` by simp
```
```   256   have "([:-a, 1:] ^ (order a (pderiv p))) dvd d"
```
```   257     apply (simp add: d)
```
```   258     apply (rule dvd_add)
```
```   259     apply (rule dvd_mult)
```
```   260     apply (simp add: order_divides `p \<noteq> 0`
```
```   261            `order a p = Suc (order a (pderiv p))`)
```
```   262     apply (rule dvd_mult)
```
```   263     apply (simp add: order_divides)
```
```   264     done
```
```   265   then have "order a (pderiv p) \<le> order a d"
```
```   266     using `d \<noteq> 0` by (simp add: order_divides)
```
```   267   show ?thesis
```
```   268     using `order a p = order a q + order a d`
```
```   269     using `order a (pderiv p) = order a e + order a d`
```
```   270     using `order a p = Suc (order a (pderiv p))`
```
```   271     using `order a (pderiv p) \<le> order a d`
```
```   272     by auto
```
```   273 qed
```
```   274
```
```   275 lemma poly_squarefree_decomp_order2: "[| pderiv p \<noteq> 0;
```
```   276          p = q * d;
```
```   277          pderiv p = e * d;
```
```   278          d = r * p + s * pderiv p
```
```   279       |] ==> \<forall>a. order a q = (if order a p = 0 then 0 else 1)"
```
```   280 apply (blast intro: poly_squarefree_decomp_order)
```
```   281 done
```
```   282
```
```   283 lemma order_pderiv2: "[| pderiv p \<noteq> 0; order a p \<noteq> 0 |]
```
```   284       ==> (order a (pderiv p) = n) = (order a p = Suc n)"
```
```   285 apply (auto dest: order_pderiv)
```
```   286 done
```
```   287
```
```   288 definition
```
```   289   rsquarefree :: "'a::idom poly => bool" where
```
```   290   "rsquarefree p = (p \<noteq> 0 & (\<forall>a. (order a p = 0) | (order a p = 1)))"
```
```   291
```
```   292 lemma pderiv_iszero: "pderiv p = 0 \<Longrightarrow> \<exists>h. p = [:h:]"
```
```   293 apply (simp add: pderiv_eq_0_iff)
```
```   294 apply (case_tac p, auto split: if_splits)
```
```   295 done
```
```   296
```
```   297 lemma rsquarefree_roots:
```
```   298   "rsquarefree p = (\<forall>a. ~(poly p a = 0 & poly (pderiv p) a = 0))"
```
```   299 apply (simp add: rsquarefree_def)
```
```   300 apply (case_tac "p = 0", simp, simp)
```
```   301 apply (case_tac "pderiv p = 0")
```
```   302 apply simp
```
```   303 apply (drule pderiv_iszero, clarify)
```
```   304 apply simp
```
```   305 apply (rule allI)
```
```   306 apply (cut_tac p = "[:h:]" and a = a in order_root)
```
```   307 apply simp
```
```   308 apply (auto simp add: order_root order_pderiv2)
```
```   309 apply (erule_tac x="a" in allE, simp)
```
```   310 done
```
```   311
```
```   312 lemma poly_squarefree_decomp:
```
```   313   assumes "pderiv p \<noteq> 0"
```
```   314     and "p = q * d"
```
```   315     and "pderiv p = e * d"
```
```   316     and "d = r * p + s * pderiv p"
```
```   317   shows "rsquarefree q & (\<forall>a. (poly q a = 0) = (poly p a = 0))"
```
```   318 proof -
```
```   319   from `pderiv p \<noteq> 0` have "p \<noteq> 0" by auto
```
```   320   with `p = q * d` have "q \<noteq> 0" by simp
```
```   321   have "\<forall>a. order a q = (if order a p = 0 then 0 else 1)"
```
```   322     using assms by (rule poly_squarefree_decomp_order2)
```
```   323   with `p \<noteq> 0` `q \<noteq> 0` show ?thesis
```
```   324     by (simp add: rsquarefree_def order_root)
```
```   325 qed
```
```   326
```
```   327 end
```