src/HOL/Library/Poly_Deriv.thy
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```     1 (*  Title:      HOL/Library/Poly_Deriv.thy
```
```     2     Author:     Amine Chaieb
```
```     3     Author:     Brian Huffman
```
```     4 *)
```
```     5
```
```     6 section\<open>Polynomials and Differentiation\<close>
```
```     7
```
```     8 theory Poly_Deriv
```
```     9 imports Deriv Polynomial
```
```    10 begin
```
```    11
```
```    12 subsection \<open>Derivatives of univariate polynomials\<close>
```
```    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   by (induct p arbitrary: n)
```
```    32      (auto simp add: pderiv_pCons coeff_pCons algebra_simps split: nat.split)
```
```    33
```
```    34 primrec pderiv_coeffs :: "'a::comm_monoid_add list \<Rightarrow> 'a list"
```
```    35 where
```
```    36   "pderiv_coeffs [] = []"
```
```    37 | "pderiv_coeffs (x # xs) = plus_coeffs xs (cCons 0 (pderiv_coeffs xs))"
```
```    38
```
```    39 lemma coeffs_pderiv [code abstract]:
```
```    40   "coeffs (pderiv p) = pderiv_coeffs (coeffs p)"
```
```    41   by (rule sym, induct p) (simp_all add: pderiv_pCons coeffs_plus_eq_plus_coeffs cCons_def)
```
```    42
```
```    43 lemma pderiv_eq_0_iff: "pderiv p = 0 \<longleftrightarrow> degree p = 0"
```
```    44   apply (rule iffI)
```
```    45   apply (cases p, simp)
```
```    46   apply (simp add: poly_eq_iff coeff_pderiv del: of_nat_Suc)
```
```    47   apply (simp add: poly_eq_iff coeff_pderiv coeff_eq_0)
```
```    48   done
```
```    49
```
```    50 lemma degree_pderiv: "degree (pderiv p) = degree p - 1"
```
```    51   apply (rule order_antisym [OF degree_le])
```
```    52   apply (simp add: coeff_pderiv coeff_eq_0)
```
```    53   apply (cases "degree p", simp)
```
```    54   apply (rule le_degree)
```
```    55   apply (simp add: coeff_pderiv del: of_nat_Suc)
```
```    56   apply (metis degree_0 leading_coeff_0_iff nat.distinct(1))
```
```    57   done
```
```    58
```
```    59 lemma pderiv_singleton [simp]: "pderiv [:a:] = 0"
```
```    60 by (simp add: pderiv_pCons)
```
```    61
```
```    62 lemma pderiv_add: "pderiv (p + q) = pderiv p + pderiv q"
```
```    63 by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
```
```    64
```
```    65 lemma pderiv_minus: "pderiv (- p) = - pderiv p"
```
```    66 by (rule poly_eqI, simp add: coeff_pderiv)
```
```    67
```
```    68 lemma pderiv_diff: "pderiv (p - q) = pderiv p - pderiv q"
```
```    69 by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
```
```    70
```
```    71 lemma pderiv_smult: "pderiv (smult a p) = smult a (pderiv p)"
```
```    72 by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
```
```    73
```
```    74 lemma pderiv_mult: "pderiv (p * q) = p * pderiv q + q * pderiv p"
```
```    75 by (induct p) (auto simp: pderiv_add pderiv_smult pderiv_pCons algebra_simps)
```
```    76
```
```    77 lemma pderiv_power_Suc:
```
```    78   "pderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p"
```
```    79 apply (induct n)
```
```    80 apply simp
```
```    81 apply (subst power_Suc)
```
```    82 apply (subst pderiv_mult)
```
```    83 apply (erule ssubst)
```
```    84 apply (simp only: of_nat_Suc smult_add_left smult_1_left)
```
```    85 apply (simp add: algebra_simps)
```
```    86 done
```
```    87
```
```    88 lemma DERIV_pow2: "DERIV (%x. x ^ Suc n) x :> real (Suc n) * (x ^ n)"
```
```    89 by (rule DERIV_cong, rule DERIV_pow, simp)
```
```    90 declare DERIV_pow2 [simp] DERIV_pow [simp]
```
```    91
```
```    92 lemma DERIV_add_const: "DERIV f x :> D ==>  DERIV (%x. a + f x :: 'a::real_normed_field) x :> D"
```
```    93 by (rule DERIV_cong, rule DERIV_add, auto)
```
```    94
```
```    95 lemma poly_DERIV[simp]: "DERIV (%x. poly p x) x :> poly (pderiv p) x"
```
```    96   by (induct p, auto intro!: derivative_eq_intros simp add: pderiv_pCons)
```
```    97
```
```    98 text\<open>Consequences of the derivative theorem above\<close>
```
```    99
```
```   100 lemma poly_differentiable[simp]: "(%x. poly p x) differentiable (at x::real filter)"
```
```   101 apply (simp add: real_differentiable_def)
```
```   102 apply (blast intro: poly_DERIV)
```
```   103 done
```
```   104
```
```   105 lemma poly_isCont[simp]: "isCont (%x. poly p x) (x::real)"
```
```   106 by (rule poly_DERIV [THEN DERIV_isCont])
```
```   107
```
```   108 lemma poly_IVT_pos: "[| a < b; poly p (a::real) < 0; 0 < poly p b |]
```
```   109       ==> \<exists>x. a < x & x < b & (poly p x = 0)"
```
```   110 using IVT_objl [of "poly p" a 0 b]
```
```   111 by (auto simp add: order_le_less)
```
```   112
```
```   113 lemma poly_IVT_neg: "[| (a::real) < b; 0 < poly p a; poly p b < 0 |]
```
```   114       ==> \<exists>x. a < x & x < b & (poly p x = 0)"
```
```   115 by (insert poly_IVT_pos [where p = "- p" ]) simp
```
```   116
```
```   117 lemma poly_MVT: "(a::real) < b ==>
```
```   118      \<exists>x. a < x & x < b & (poly p b - poly p a = (b - a) * poly (pderiv p) x)"
```
```   119 using MVT [of a b "poly p"]
```
```   120 apply auto
```
```   121 apply (rule_tac x = z in exI)
```
```   122 apply (auto simp add: mult_left_cancel poly_DERIV [THEN DERIV_unique])
```
```   123 done
```
```   124
```
```   125 text\<open>Lemmas for Derivatives\<close>
```
```   126
```
```   127 lemma order_unique_lemma:
```
```   128   fixes p :: "'a::idom poly"
```
```   129   assumes "[:-a, 1:] ^ n dvd p" "\<not> [:-a, 1:] ^ Suc n dvd p"
```
```   130   shows "n = order a p"
```
```   131 unfolding Polynomial.order_def
```
```   132 apply (rule Least_equality [symmetric])
```
```   133 apply (fact assms)
```
```   134 apply (rule classical)
```
```   135 apply (erule notE)
```
```   136 unfolding not_less_eq_eq
```
```   137 using assms(1) apply (rule power_le_dvd)
```
```   138 apply assumption
```
```   139 done
```
```   140
```
```   141 lemma lemma_order_pderiv1:
```
```   142   "pderiv ([:- a, 1:] ^ Suc n * q) = [:- a, 1:] ^ Suc n * pderiv q +
```
```   143     smult (of_nat (Suc n)) (q * [:- a, 1:] ^ n)"
```
```   144 apply (simp only: pderiv_mult pderiv_power_Suc)
```
```   145 apply (simp del: power_Suc of_nat_Suc add: pderiv_pCons)
```
```   146 done
```
```   147
```
```   148 lemma dvd_add_cancel1:
```
```   149   fixes a b c :: "'a::comm_ring_1"
```
```   150   shows "a dvd b + c \<Longrightarrow> a dvd b \<Longrightarrow> a dvd c"
```
```   151   by (drule (1) Rings.dvd_diff, simp)
```
```   152
```
```   153 lemma lemma_order_pderiv:
```
```   154   assumes n: "0 < n"
```
```   155       and pd: "pderiv p \<noteq> 0"
```
```   156       and pe: "p = [:- a, 1:] ^ n * q"
```
```   157       and nd: "~ [:- a, 1:] dvd q"
```
```   158     shows "n = Suc (order a (pderiv p))"
```
```   159 using n
```
```   160 proof -
```
```   161   have "pderiv ([:- a, 1:] ^ n * q) \<noteq> 0"
```
```   162     using assms by auto
```
```   163   obtain n' where "n = Suc n'" "0 < Suc n'" "pderiv ([:- a, 1:] ^ Suc n' * q) \<noteq> 0"
```
```   164     using assms by (cases n) auto
```
```   165   then have *: "!!k l. k dvd k * pderiv q + smult (of_nat (Suc n')) l \<Longrightarrow> k dvd l"
```
```   166     by (metis dvd_add_cancel1 dvd_smult_iff dvd_triv_left of_nat_eq_0_iff old.nat.distinct(2))
```
```   167   have "n' = order a (pderiv ([:- a, 1:] ^ Suc n' * q))"
```
```   168   proof (rule order_unique_lemma)
```
```   169     show "[:- a, 1:] ^ n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)"
```
```   170       apply (subst lemma_order_pderiv1)
```
```   171       apply (rule dvd_add)
```
```   172       apply (metis dvdI dvd_mult2 power_Suc2)
```
```   173       apply (metis dvd_smult dvd_triv_right)
```
```   174       done
```
```   175   next
```
```   176     show "\<not> [:- a, 1:] ^ Suc n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)"
```
```   177      apply (subst lemma_order_pderiv1)
```
```   178      by (metis * nd dvd_mult_cancel_right power_not_zero pCons_eq_0_iff power_Suc zero_neq_one)
```
```   179   qed
```
```   180   then show ?thesis
```
```   181     by (metis \<open>n = Suc n'\<close> pe)
```
```   182 qed
```
```   183
```
```   184 lemma order_decomp:
```
```   185   assumes "p \<noteq> 0"
```
```   186   shows "\<exists>q. p = [:- a, 1:] ^ order a p * q \<and> \<not> [:- a, 1:] dvd q"
```
```   187 proof -
```
```   188   from assms have A: "[:- a, 1:] ^ order a p dvd p"
```
```   189     and B: "\<not> [:- a, 1:] ^ Suc (order a p) dvd p" by (auto dest: order)
```
```   190   from A obtain q where C: "p = [:- a, 1:] ^ order a p * q" ..
```
```   191   with B have "\<not> [:- a, 1:] ^ Suc (order a p) dvd [:- a, 1:] ^ order a p * q"
```
```   192     by simp
```
```   193   then have "\<not> [:- a, 1:] ^ order a p * [:- a, 1:] dvd [:- a, 1:] ^ order a p * q"
```
```   194     by simp
```
```   195   then have D: "\<not> [:- a, 1:] dvd q"
```
```   196     using idom_class.dvd_mult_cancel_left [of "[:- a, 1:] ^ order a p" "[:- a, 1:]" q]
```
```   197     by auto
```
```   198   from C D show ?thesis by blast
```
```   199 qed
```
```   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 (erule dvdE)+
```
```   223     apply (rule order_unique_lemma [symmetric], fold t_def)
```
```   224     apply (simp_all add: power_add t_dvd_iff)
```
```   225     done
```
```   226 qed
```
```   227
```
```   228 text\<open>Now justify the standard squarefree decomposition, i.e. f / gcd(f,f').\<close>
```
```   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 (metis not_less_eq_eq power_le_dvd)
```
```   234 apply (erule power_le_dvd [OF order_1])
```
```   235 done
```
```   236
```
```   237 lemma poly_squarefree_decomp_order:
```
```   238   assumes "pderiv p \<noteq> 0"
```
```   239   and p: "p = q * d"
```
```   240   and p': "pderiv p = e * d"
```
```   241   and d: "d = r * p + s * pderiv p"
```
```   242   shows "order a q = (if order a p = 0 then 0 else 1)"
```
```   243 proof (rule classical)
```
```   244   assume 1: "order a q \<noteq> (if order a p = 0 then 0 else 1)"
```
```   245   from \<open>pderiv p \<noteq> 0\<close> have "p \<noteq> 0" by auto
```
```   246   with p have "order a p = order a q + order a d"
```
```   247     by (simp add: order_mult)
```
```   248   with 1 have "order a p \<noteq> 0" by (auto split: if_splits)
```
```   249   have "order a (pderiv p) = order a e + order a d"
```
```   250     using \<open>pderiv p \<noteq> 0\<close> \<open>pderiv p = e * d\<close> by (simp add: order_mult)
```
```   251   have "order a p = Suc (order a (pderiv p))"
```
```   252     using \<open>pderiv p \<noteq> 0\<close> \<open>order a p \<noteq> 0\<close> by (rule order_pderiv)
```
```   253   have "d \<noteq> 0" using \<open>p \<noteq> 0\<close> \<open>p = q * d\<close> by simp
```
```   254   have "([:-a, 1:] ^ (order a (pderiv p))) dvd d"
```
```   255     apply (simp add: d)
```
```   256     apply (rule dvd_add)
```
```   257     apply (rule dvd_mult)
```
```   258     apply (simp add: order_divides \<open>p \<noteq> 0\<close>
```
```   259            \<open>order a p = Suc (order a (pderiv p))\<close>)
```
```   260     apply (rule dvd_mult)
```
```   261     apply (simp add: order_divides)
```
```   262     done
```
```   263   then have "order a (pderiv p) \<le> order a d"
```
```   264     using \<open>d \<noteq> 0\<close> by (simp add: order_divides)
```
```   265   show ?thesis
```
```   266     using \<open>order a p = order a q + order a d\<close>
```
```   267     using \<open>order a (pderiv p) = order a e + order a d\<close>
```
```   268     using \<open>order a p = Suc (order a (pderiv p))\<close>
```
```   269     using \<open>order a (pderiv p) \<le> order a d\<close>
```
```   270     by auto
```
```   271 qed
```
```   272
```
```   273 lemma poly_squarefree_decomp_order2: "[| pderiv p \<noteq> 0;
```
```   274          p = q * d;
```
```   275          pderiv p = e * d;
```
```   276          d = r * p + s * pderiv p
```
```   277       |] ==> \<forall>a. order a q = (if order a p = 0 then 0 else 1)"
```
```   278 by (blast intro: poly_squarefree_decomp_order)
```
```   279
```
```   280 lemma order_pderiv2: "[| pderiv p \<noteq> 0; order a p \<noteq> 0 |]
```
```   281       ==> (order a (pderiv p) = n) = (order a p = Suc n)"
```
```   282 by (auto dest: order_pderiv)
```
```   283
```
```   284 definition
```
```   285   rsquarefree :: "'a::idom poly => bool" where
```
```   286   "rsquarefree p = (p \<noteq> 0 & (\<forall>a. (order a p = 0) | (order a p = 1)))"
```
```   287
```
```   288 lemma pderiv_iszero: "pderiv p = 0 \<Longrightarrow> \<exists>h. p = [:h:]"
```
```   289 apply (simp add: pderiv_eq_0_iff)
```
```   290 apply (case_tac p, auto split: if_splits)
```
```   291 done
```
```   292
```
```   293 lemma rsquarefree_roots:
```
```   294   "rsquarefree p = (\<forall>a. ~(poly p a = 0 & poly (pderiv p) a = 0))"
```
```   295 apply (simp add: rsquarefree_def)
```
```   296 apply (case_tac "p = 0", simp, simp)
```
```   297 apply (case_tac "pderiv p = 0")
```
```   298 apply simp
```
```   299 apply (drule pderiv_iszero, clarsimp)
```
```   300 apply (metis coeff_0 coeff_pCons_0 degree_pCons_0 le0 le_antisym order_degree)
```
```   301 apply (force simp add: order_root order_pderiv2)
```
```   302 done
```
```   303
```
```   304 lemma poly_squarefree_decomp:
```
```   305   assumes "pderiv p \<noteq> 0"
```
```   306     and "p = q * d"
```
```   307     and "pderiv p = e * d"
```
```   308     and "d = r * p + s * pderiv p"
```
```   309   shows "rsquarefree q & (\<forall>a. (poly q a = 0) = (poly p a = 0))"
```
```   310 proof -
```
```   311   from \<open>pderiv p \<noteq> 0\<close> have "p \<noteq> 0" by auto
```
```   312   with \<open>p = q * d\<close> have "q \<noteq> 0" by simp
```
```   313   have "\<forall>a. order a q = (if order a p = 0 then 0 else 1)"
```
```   314     using assms by (rule poly_squarefree_decomp_order2)
```
```   315   with \<open>p \<noteq> 0\<close> \<open>q \<noteq> 0\<close> show ?thesis
```
```   316     by (simp add: rsquarefree_def order_root)
```
```   317 qed
```
```   318
```
```   319 end
```