author bulwahn Wed Dec 15 08:34:01 2010 +0100 (2010-12-15) changeset 41120 74e41b2d48ea parent 41119 573f557ed716 child 41133 6c7c135a3270 child 41186 08a54d394e36
adding an Isar version of the MacLaurin theorem from some students' work in 2005
 src/HOL/MacLaurin.thy file | annotate | diff | revisions
```     1.1 --- a/src/HOL/MacLaurin.thy	Tue Dec 14 00:16:30 2010 +0100
1.2 +++ b/src/HOL/MacLaurin.thy	Wed Dec 15 08:34:01 2010 +0100
1.3 @@ -1,6 +1,7 @@
1.4  (*  Author      : Jacques D. Fleuriot
1.5      Copyright   : 2001 University of Edinburgh
1.6      Conversion to Isar and new proofs by Lawrence C Paulson, 2004
1.7 +    Conversion of Mac Laurin to Isar by Lukas Bulwahn and Bernhard Häupler, 2005
1.8  *)
1.9
1.11 @@ -18,11 +19,9 @@
1.12      "0 < h ==>
1.13       \<exists>B. f h = (\<Sum>m=0..<n. (j m / real (fact m)) * (h^m)) +
1.14                 (B * ((h^n) / real(fact n)))"
1.15 -apply (rule_tac x = "(f h - (\<Sum>m=0..<n. (j m / real (fact m)) * h^m)) *
1.16 +by (rule_tac x = "(f h - (\<Sum>m=0..<n. (j m / real (fact m)) * h^m)) *
1.17                   real(fact n) / (h^n)"
1.18 -       in exI)
1.19 -apply (simp)
1.20 -done
1.21 +       in exI, simp)
1.22
1.23  lemma eq_diff_eq': "(x = y - z) = (y = x + (z::real))"
1.24  by arith
1.25 @@ -32,46 +31,141 @@
1.26    by (subst fact_reduce_nat, auto)
1.27
1.28  lemma Maclaurin_lemma2:
1.29 -  assumes diff: "\<forall>m t. m < n \<and> 0\<le>t \<and> t\<le>h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
1.30 -  assumes n: "n = Suc k"
1.31 -  assumes difg: "difg =
1.32 -        (\<lambda>m t. diff m t -
1.33 -               ((\<Sum>p = 0..<n - m. diff (m + p) 0 / real (fact p) * t ^ p) +
1.34 -                B * (t ^ (n - m) / real (fact (n - m)))))"
1.35 -  shows
1.36 -      "\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (difg m) t :> difg (Suc m) t"
1.37 -unfolding difg
1.38 - apply clarify
1.39 - apply (rule DERIV_diff)
1.40 -  apply (simp add: diff)
1.41 - apply (simp only: n)
1.43 -  apply (rule_tac  DERIV_cmult)
1.44 -  apply (rule_tac  lemma_DERIV_subst)
1.45 -   apply (rule_tac  DERIV_quotient)
1.46 -     apply (rule_tac  DERIV_const)
1.47 -    apply (rule_tac  DERIV_pow)
1.48 -   prefer 3
1.49 +  assumes DERIV : "\<forall>m t. m < n \<and> 0\<le>t \<and> t\<le>h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
1.50 +  and INIT : "n = Suc k"
1.51 +  and DIFG_DEF: "difg = (\<lambda>m t. diff m t - ((\<Sum>p = 0..<n - m. diff (m + p) 0 / real (fact p) * t ^ p) +
1.52 +  B * (t ^ (n - m) / real (fact (n - m)))))"
1.53 +  shows "\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (difg m) t :> difg (Suc m) t"
1.54 +proof (rule allI)+
1.55 +  fix m
1.56 +  fix t
1.57 +  show "m < n \<and> 0 \<le> t \<and> t \<le> h \<longrightarrow> DERIV (difg m) t :> difg (Suc m) t"
1.58 +  proof
1.59 +    assume INIT2: "m < n & 0 \<le> t & t \<le> h"
1.60 +    hence INTERV: "0 \<le> t & t \<le> h" ..
1.61 +    from INIT2 and INIT have mtok: "m < Suc k" by arith
1.62 +    have "DERIV (\<lambda>t. diff m t -
1.63 +    ((\<Sum>p = 0..<Suc k - m. diff (m + p) 0 / real (fact p) * t ^ p) +
1.64 +    B * (t ^ (Suc k - m) / real (fact (Suc k - m)))))
1.65 +    t :> diff (Suc m) t -
1.66 +    ((\<Sum>p = 0..<Suc k - Suc m. diff (Suc m + p) 0 / real (fact p) * t ^ p) +
1.67 +    B * (t ^ (Suc k - Suc m) / real (fact (Suc k - Suc m))))"
1.68 +    proof -
1.69 +      from DERIV and INIT2 have "DERIV (diff m) t :> diff (Suc m) t" by simp
1.70 +      moreover
1.71 +      have " DERIV (\<lambda>x. (\<Sum>p = 0..<Suc k - m. diff (m + p) 0 / real (fact p) * x ^ p) +
1.72 +	B * (x ^ (Suc k - m) / real (fact (Suc k - m))))
1.73 +	t :> (\<Sum>p = 0..<Suc k - Suc m. diff (Suc m + p) 0 / real (fact p) * t ^ p) +
1.74 +	B * (t ^ (Suc k - Suc m) / real (fact (Suc k - Suc m)))"
1.75 +      proof -
1.76 +	have "DERIV (\<lambda>x. \<Sum>p = 0..<Suc k - m. diff (m + p) 0 / real (fact p) * x ^ p) t
1.77 +	  :> (\<Sum>p = 0..<Suc k - Suc m. diff (Suc m + p) 0 / real (fact p) * t ^ p)"
1.78 +	proof -
1.79 +	  have "\<exists> d. k = m + d"
1.80 +	  proof -
1.81 +	    from INIT2 have "m < n" ..
1.82 +	    hence "\<exists> d. n = m + d + Suc 0" by arith
1.83 +	    with INIT show ?thesis by (simp del: setsum_op_ivl_Suc)
1.84 +	  qed
1.85 +	  from this obtain d where kmd: "k = m + d" ..
1.86 +	  have "DERIV (\<lambda>x. (\<Sum>ma = 0..<d. diff (Suc (m + ma)) 0 * x ^ Suc ma / real (fact (Suc ma))) +
1.87 +            diff m 0)
1.88 +	    t :> (\<Sum>p = 0..<d. diff (Suc (m + p)) 0 * t ^ p / real (fact p))"
1.89 +
1.90 +	  proof -
1.91 +	    have "DERIV (\<lambda>x. (\<Sum>ma = 0..<d. diff (Suc (m + ma)) 0 * x ^ Suc ma / real (fact (Suc ma))) + diff m 0) t :>  (\<Sum>r = 0..<d. diff (Suc (m + r)) 0 * t ^ r / real (fact r)) + 0"
1.92 +	    proof -
1.93 +	      from DERIV and INTERV have "DERIV (\<lambda>x. (\<Sum>ma = 0..<d. diff (Suc (m + ma)) 0 * x ^ Suc ma / real (fact (Suc ma)))) t :>  (\<Sum>r = 0..<d. diff (Suc (m + r)) 0 * t ^ r / real (fact r))"
1.94 +	      proof -
1.95 +		have "\<forall>r. 0 \<le> r \<and> r < 0 + d \<longrightarrow>
1.96 +		  DERIV (\<lambda>x. diff (Suc (m + r)) 0 * x ^ Suc r / real (fact (Suc r))) t
1.97 +		  :> diff (Suc (m + r)) 0 * t ^ r / real (fact r)"
1.98 +		proof (rule allI)
1.99 +		  fix r
1.100 +		  show " 0 \<le> r \<and> r < 0 + d \<longrightarrow>
1.101 +		    DERIV (\<lambda>x. diff (Suc (m + r)) 0 * x ^ Suc r / real (fact (Suc r))) t
1.102 +		    :> diff (Suc (m + r)) 0 * t ^ r / real (fact r)"
1.103 +		  proof
1.104 +		    assume "0 \<le> r & r < 0 + d"
1.105 +		    have "DERIV (\<lambda>x. diff (Suc (m + r)) 0 *
1.106 +                      (x ^ Suc r * inverse (real (fact (Suc r)))))
1.107 +		      t :> diff (Suc (m + r)) 0 * (t ^ r * inverse (real (fact r)))"
1.108 +		    proof -
1.109 +                      have "(1 + real r) * real (fact r) \<noteq> 0" by auto
1.110 +		      from this have "real (fact r) + real r * real (fact r) \<noteq> 0"
1.111 +                        by (metis add_nonneg_eq_0_iff mult_nonneg_nonneg real_of_nat_fact_not_zero real_of_nat_ge_zero)
1.112 +                      from this have "DERIV (\<lambda>x. x ^ Suc r * inverse (real (fact (Suc r)))) t :> real (Suc r) * t ^ (Suc r - Suc 0) * inverse (real (fact (Suc r))) +
1.113 +			0 * t ^ Suc r"
1.114 +                        apply - by ( rule DERIV_intros | rule refl)+ auto
1.115 +		      moreover
1.116 +		      have "real (Suc r) * t ^ (Suc r - Suc 0) * inverse (real (fact (Suc r))) +
1.117 +			0 * t ^ Suc r =
1.118 +			t ^ r * inverse (real (fact r))"
1.119 +		      proof -
1.120 +
1.121 +			have " real (Suc r) * t ^ (Suc r - Suc 0) *
1.122 +			  inverse (real (Suc r) * real (fact r)) +
1.123 +			  0 * t ^ Suc r =
1.124 +			  t ^ r * inverse (real (fact r))" by (simp add: mult_ac)
1.125 +			hence "real (Suc r) * t ^ (Suc r - Suc 0) * inverse (real (Suc r * fact r)) +
1.126 +			  0 * t ^ Suc r =
1.127 +			  t ^ r * inverse (real (fact r))" by (subst real_of_nat_mult)
1.128 +			thus ?thesis by (subst fact_Suc)
1.129 +		      qed
1.130 +		      ultimately have " DERIV (\<lambda>x. x ^ Suc r * inverse (real (fact (Suc r)))) t
1.131 +			:> t ^ r * inverse (real (fact r))" by (rule lemma_DERIV_subst)
1.132 +		      thus ?thesis by (rule DERIV_cmult)
1.133 +		    qed
1.134 +		    thus "DERIV (\<lambda>x. diff (Suc (m + r)) 0 * x ^ Suc r /
1.135 +                      real (fact (Suc r)))
1.136 +		      t :> diff (Suc (m + r)) 0 * t ^ r / real (fact r)" by (simp (no_asm) add: divide_inverse mult_assoc del: fact_Suc power_Suc)
1.137 +		  qed
1.138 +		qed
1.139 +		thus ?thesis by (rule DERIV_sumr)
1.140 +	      qed
1.141 +	      moreover
1.142 +	      from DERIV_const have "DERIV (\<lambda>x. diff m 0) t :> 0" .
1.143 +	      ultimately show ?thesis by (rule DERIV_add)
1.144 +	    qed
1.145 +	    moreover
1.146 +	    have " (\<Sum>r = 0..<d. diff (Suc (m + r)) 0 * t ^ r / real (fact r)) + 0 =  (\<Sum>p = 0..<d. diff (Suc (m + p)) 0 * t ^ p / real (fact p))"  by simp
1.147 +	    ultimately show ?thesis by (rule lemma_DERIV_subst)
1.148 +	  qed
1.149 +	  with kmd and sumr_offset4 [of 1] show ?thesis by (simp del: setsum_op_ivl_Suc fact_Suc power_Suc)
1.150 +	qed
1.151 +	moreover
1.152 +	have " DERIV (\<lambda>x. B * (x ^ (Suc k - m) / real (fact (Suc k - m)))) t
1.153 +	  :> B * (t ^ (Suc k - Suc m) / real (fact (Suc k - Suc m)))"
1.154 +	proof -
1.155 +	  have " DERIV (\<lambda>x. x ^ (Suc k - m) / real (fact (Suc k - m))) t
1.156 +	    :> t ^ (Suc k - Suc m) / real (fact (Suc k - Suc m))"
1.157 +	  proof -
1.158 +	    have "DERIV (\<lambda>x. x ^ (Suc k - m)) t :> real (Suc k - m) * t ^ (Suc k - m - Suc 0)" by (rule DERIV_pow)
1.159 +	    moreover
1.160 +	    have "DERIV (\<lambda>x. real (fact (Suc k - m))) t :> 0" by (rule DERIV_const)
1.161 +	    moreover
1.162 +	    have "(\<lambda>x. real (fact (Suc k - m))) t \<noteq> 0" by simp
1.163 +	    ultimately have " DERIV (\<lambda>y. y ^ (Suc k - m) / real (fact (Suc k - m))) t
1.164 +	      :>  ( real (Suc k - m) * t ^ (Suc k - m - Suc 0) * real (fact (Suc k - m)) + - (0 * t ^ (Suc k - m))) /
1.165 +	      real (fact (Suc k - m)) ^ Suc (Suc 0)"
1.166 +              apply -
1.167 +              apply (rule DERIV_cong) by (rule DERIV_intros | rule refl)+ auto
1.168 +	    moreover
1.169 +	    from mtok and INIT have "( real (Suc k - m) * t ^ (Suc k - m - Suc 0) * real (fact (Suc k - m)) + - (0 * t ^ (Suc k - m))) /
1.170 +	      real (fact (Suc k - m)) ^ Suc (Suc 0) =  t ^ (Suc k - Suc m) / real (fact (Suc k - Suc m))" by (simp add: fact_diff_Suc)
1.171 +	    ultimately show ?thesis by (rule lemma_DERIV_subst)
1.172 +	  qed
1.173 +	  moreover
1.174 +	  thus ?thesis by (rule DERIV_cmult)
1.175 +	qed
1.176 +	ultimately show ?thesis by (rule DERIV_add)
1.177 +      qed
1.178 +      ultimately show ?thesis by (rule DERIV_diff)
1.179 +    qed
1.180 +    from INIT and this and DIFG_DEF show "DERIV (difg m) t :> difg (Suc m) t" by clarify
1.181 +  qed
1.182 +qed
1.183
1.185 -  prefer 2 apply simp
1.186 - apply (frule less_iff_Suc_add [THEN iffD1], clarify)
1.187 - apply (simp del: setsum_op_ivl_Suc)
1.188 - apply (insert sumr_offset4 [of "Suc 0"])
1.189 - apply (simp del: setsum_op_ivl_Suc fact_Suc power_Suc)
1.190 - apply (rule lemma_DERIV_subst)
1.192 -   apply (rule_tac  DERIV_const)
1.193 -  apply (rule DERIV_sumr, clarify)
1.194 -  prefer 2 apply simp
1.195 - apply (simp (no_asm) add: divide_inverse mult_assoc del: fact_Suc power_Suc)
1.196 - apply (rule DERIV_cmult)
1.197 - apply (rule lemma_DERIV_subst)
1.198 -  apply (best intro!: DERIV_intros)
1.199 - apply (subst fact_Suc)
1.200 - apply (subst real_of_nat_mult)
1.201 - apply (simp add: mult_ac)
1.202 -done
1.203
1.204  lemma Maclaurin:
1.205    assumes h: "0 < h"
1.206 @@ -83,7 +177,7 @@
1.207      "\<exists>t. 0 < t & t < h &
1.208                f h =
1.209                setsum (%m. (diff m 0 / real (fact m)) * h ^ m) {0..<n} +
1.210 -              (diff n t / real (fact n)) * h ^ n"
1.211 +              (diff n t / real (fact n)) * h ^ n"
1.212  proof -
1.213    from n obtain m where m: "n = Suc m"
1.214      by (cases n, simp add: n)
1.215 @@ -195,17 +289,23 @@
1.216
1.217
1.218  lemma Maclaurin2:
1.219 -   "[| 0 < h; diff 0 = f;
1.220 -       \<forall>m t.
1.221 -          m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t |]
1.222 -    ==> \<exists>t. 0 < t &
1.223 -              t \<le> h &
1.224 -              f h =
1.225 -              (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
1.226 -              diff n t / real (fact n) * h ^ n"
1.227 -apply (case_tac "n", auto)
1.228 -apply (drule Maclaurin, auto)
1.229 -done
1.230 +  assumes INIT1: "0 < h " and INIT2: "diff 0 = f"
1.231 +  and DERIV: "\<forall>m t.
1.232 +  m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t"
1.233 +  shows "\<exists>t. 0 < t \<and> t \<le> h \<and> f h =
1.234 +  (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
1.235 +  diff n t / real (fact n) * h ^ n"
1.236 +proof (cases "n")
1.237 +  case 0 with INIT1 INIT2 show ?thesis by fastsimp
1.238 +next
1.239 +  case Suc
1.240 +  hence "n > 0" by simp
1.241 +  from INIT1 this INIT2 DERIV have "\<exists>t>0. t < h \<and>
1.242 +    f h =
1.243 +    (\<Sum>m = 0..<n. diff m 0 / real (fact m) * h ^ m) + diff n t / real (fact n) * h ^ n"
1.244 +    by (rule Maclaurin)
1.245 +  thus ?thesis by fastsimp
1.246 +qed
1.247
1.248  lemma Maclaurin2_objl:
1.249       "0 < h & diff 0 = f &
1.250 @@ -219,31 +319,62 @@
1.251  by (blast intro: Maclaurin2)
1.252
1.253  lemma Maclaurin_minus:
1.254 -   "[| h < 0; n > 0; diff 0 = f;
1.255 -       \<forall>m t. m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t |]
1.256 -    ==> \<exists>t. h < t &
1.257 -              t < 0 &
1.258 -              f h =
1.259 -              (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
1.260 -              diff n t / real (fact n) * h ^ n"
1.261 -apply (cut_tac f = "%x. f (-x)"
1.262 -        and diff = "%n x. (-1 ^ n) * diff n (-x)"
1.263 -        and h = "-h" and n = n in Maclaurin_objl)
1.264 -apply (simp)
1.265 -apply safe
1.266 -apply (subst minus_mult_right)
1.267 -apply (rule DERIV_cmult)
1.268 -apply (rule lemma_DERIV_subst)
1.269 -apply (rule DERIV_chain2 [where g=uminus])
1.270 -apply (rule_tac  DERIV_minus, rule_tac  DERIV_ident)
1.271 -prefer 2 apply force
1.272 -apply force
1.273 -apply (rule_tac x = "-t" in exI, auto)
1.274 -apply (subgoal_tac "(\<Sum>m = 0..<n. -1 ^ m * diff m 0 * (-h)^m / real(fact m)) =
1.275 -                    (\<Sum>m = 0..<n. diff m 0 * h ^ m / real(fact m))")
1.276 -apply (rule_tac  setsum_cong[OF refl])
1.277 -apply (auto simp add: divide_inverse power_mult_distrib [symmetric])
1.278 -done
1.279 +  assumes INTERV : "h < 0" and
1.280 +  INIT : "0 < n" "diff 0 = f" and
1.281 +             ABL : "\<forall>m t. m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t"
1.282 +  shows "\<exists>t. h < t & t < 0 &
1.283 +         f h = (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
1.284 +         diff n t / real (fact n) * h ^ n"
1.285 +proof -
1.286 +  from INTERV have "0 < -h" by simp
1.287 +  moreover
1.288 +  from INIT have "0 < n" by simp
1.289 +  moreover
1.290 +  from INIT have "(% x. ( - 1) ^ 0 * diff 0 (- x)) = (% x. f (- x))" by simp
1.291 +  moreover
1.292 +  have "\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> - h \<longrightarrow>
1.293 +    DERIV (\<lambda>x. (- 1) ^ m * diff m (- x)) t :> (- 1) ^ Suc m * diff (Suc m) (- t)"
1.294 +  proof (rule allI impI)+
1.295 +    fix m t
1.296 +    assume tINTERV:" m < n \<and> 0 \<le> t \<and> t \<le> - h"
1.297 +    with ABL show "DERIV (\<lambda>x. (- 1) ^ m * diff m (- x)) t :> (- 1) ^ Suc m * diff (Suc m) (- t)"
1.298 +    proof -
1.299 +
1.300 +      from ABL and tINTERV have "DERIV (\<lambda>x. diff m (- x)) t :> - diff (Suc m) (- t)" (is ?tABL)
1.301 +      proof -
1.302 +	from ABL and tINTERV have "DERIV (diff m) (- t) :> diff (Suc m) (- t)" by force
1.303 +	moreover
1.304 +	from DERIV_ident[of t] have "DERIV uminus t :> (- 1)" by (rule DERIV_minus)
1.305 +	ultimately have "DERIV (\<lambda>x. diff m (- x)) t :> diff (Suc m) (- t) * - 1" by (rule DERIV_chain2)
1.306 +	thus ?thesis by simp
1.307 +      qed
1.308 +      thus ?thesis
1.309 +      proof -
1.310 +	assume ?tABL hence "DERIV (\<lambda>x. -1 ^ m * diff m (- x)) t :> -1 ^ m * - diff (Suc m) (- t)" by (rule DERIV_cmult)
1.311 +	hence "DERIV (\<lambda>x. -1 ^ m * diff m (- x)) t :> - (-1 ^ m * diff (Suc m) (- t))" by (subst minus_mult_right)
1.312 +	thus ?thesis by simp
1.313 +      qed
1.314 +    qed
1.315 +  qed
1.316 +  ultimately have t_exists: "\<exists>t>0. t < - h \<and>
1.317 +    f (- (- h)) =
1.318 +    (\<Sum>m = 0..<n.
1.319 +    (- 1) ^ m * diff m (- 0) / real (fact m) * (- h) ^ m) +
1.320 +    (- 1) ^ n * diff n (- t) / real (fact n) * (- h) ^ n" (is "\<exists> t. ?P t") by (rule Maclaurin)
1.321 +  from this obtain t where t_def: "?P t" ..
1.322 +  moreover
1.323 +  have "-1 ^ n * diff n (- t) * (- h) ^ n / real (fact n) = diff n (- t) * h ^ n / real (fact n)"
1.324 +    by (auto simp add: power_mult_distrib[symmetric])
1.325 +  moreover
1.326 +  have "(SUM m = 0..<n. -1 ^ m * diff m 0 * (- h) ^ m / real (fact m)) = (SUM m = 0..<n. diff m 0 * h ^ m / real (fact m))"
1.327 +    by (auto intro: setsum_cong simp add: power_mult_distrib[symmetric])
1.328 +  ultimately have " h < - t \<and>
1.329 +    - t < 0 \<and>
1.330 +    f h =
1.331 +    (\<Sum>m = 0..<n. diff m 0 / real (fact m) * h ^ m) + diff n (- t) / real (fact n) * h ^ n"
1.332 +    by auto
1.333 +  thus ?thesis ..
1.334 +qed
1.335
1.336  lemma Maclaurin_minus_objl:
1.337       "(h < 0 & n > 0 & diff 0 = f &
1.338 @@ -269,42 +400,109 @@
1.339  by (induct "n", auto)
1.340
1.341  lemma Maclaurin_bi_le:
1.342 -   "[| diff 0 = f;
1.343 -       \<forall>m t. m < n & abs t \<le> abs x --> DERIV (diff m) t :> diff (Suc m) t |]
1.344 -    ==> \<exists>t. abs t \<le> abs x &
1.345 +   assumes INIT : "diff 0 = f"
1.346 +   and DERIV : "\<forall>m t. m < n & abs t \<le> abs x --> DERIV (diff m) t :> diff (Suc m) t"
1.347 +   shows "\<exists>t. abs t \<le> abs x &
1.348                f x =
1.349                (\<Sum>m=0..<n. diff m 0 / real (fact m) * x ^ m) +
1.350                diff n t / real (fact n) * x ^ n"
1.351 -apply (case_tac "n = 0", force)
1.352 -apply (case_tac "x = 0")
1.353 - apply (rule_tac x = 0 in exI)
1.354 - apply (force simp add: Maclaurin_bi_le_lemma)
1.355 -apply (cut_tac x = x and y = 0 in linorder_less_linear, auto)
1.356 - txt{*Case 1, where @{term "x < 0"}*}
1.357 - apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_minus_objl, safe)
1.358 -  apply (simp add: abs_if)
1.359 - apply (rule_tac x = t in exI)
1.360 - apply (simp add: abs_if)
1.361 -txt{*Case 2, where @{term "0 < x"}*}
1.362 -apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_objl, safe)
1.363 - apply (simp add: abs_if)
1.364 -apply (rule_tac x = t in exI)
1.366 -done
1.367 +proof (cases "n = 0")
1.368 +  case True from INIT True show ?thesis by force
1.369 +next
1.370 +  case False
1.371 +  from this have n_not_zero:"n \<noteq> 0" .
1.372 +  from False INIT DERIV show ?thesis
1.373 +  proof (cases "x = 0")
1.374 +    case True show ?thesis
1.375 +    proof -
1.376 +      from n_not_zero True INIT DERIV have "\<bar>0\<bar> \<le> \<bar>x\<bar> \<and>
1.377 +	f x = (\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) + diff n 0 / real (fact n) * x ^ n" by(force simp add: Maclaurin_bi_le_lemma)
1.378 +      thus ?thesis ..
1.379 +    qed
1.380 +  next
1.381 +    case False
1.382 +    note linorder_less_linear [of "x" "0"]
1.383 +    thus ?thesis
1.384 +    proof (elim disjE)
1.385 +      assume "x = 0" with False show ?thesis ..
1.386 +      next
1.387 +      assume x_less_zero: "x < 0" moreover
1.388 +      from n_not_zero have "0 < n" by simp moreover
1.389 +      have "diff 0 = diff 0" by simp moreover
1.390 +      have "\<forall>m t. m < n \<and> x \<le> t \<and> t \<le> 0 \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
1.391 +      proof (rule allI, rule allI, rule impI)
1.392 +	fix m t
1.393 +	assume "m < n & x \<le> t & t \<le> 0"
1.394 +	with DERIV show " DERIV (diff m) t :> diff (Suc m) t" by (fastsimp simp add: abs_if)
1.395 +      qed
1.396 +      ultimately have t_exists:"\<exists>t>x. t < 0 \<and>
1.397 +        diff 0 x =
1.398 +        (\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) + diff n t / real (fact n) * x ^ n" (is "\<exists> t. ?P t") by (rule Maclaurin_minus)
1.399 +      from this obtain t where t_def: "?P t" ..
1.400 +      from t_def x_less_zero INIT  have "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and>
1.401 +	f x = (\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) + diff n t / real (fact n) * x ^ n"
1.402 +	by (simp add: abs_if order_less_le)
1.403 +      thus ?thesis by (rule exI)
1.404 +    next
1.405 +    assume x_greater_zero: "x > 0" moreover
1.406 +    from n_not_zero have "0 < n" by simp moreover
1.407 +    have "diff 0 = diff 0" by simp moreover
1.408 +    have "\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> x \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
1.409 +      proof (rule allI, rule allI, rule impI)
1.410 +	fix m t
1.411 +	assume "m < n & 0 \<le> t & t \<le> x"
1.412 +	with DERIV show " DERIV (diff m) t :> diff (Suc m) t" by fastsimp
1.413 +      qed
1.414 +      ultimately have t_exists:"\<exists>t>0. t < x \<and>
1.415 +        diff 0 x =
1.416 +        (\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) + diff n t / real (fact n) * x ^ n" (is "\<exists> t. ?P t") by (rule Maclaurin)
1.417 +      from this obtain t where t_def: "?P t" ..
1.418 +      from t_def x_greater_zero INIT  have "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and>
1.419 +	f x = (\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) + diff n t / real (fact n) * x ^ n"
1.420 +	by fastsimp
1.421 +      thus ?thesis ..
1.422 +    qed
1.423 +  qed
1.424 +qed
1.425 +
1.426
1.427  lemma Maclaurin_all_lt:
1.428 -     "[| diff 0 = f;
1.429 -         \<forall>m x. DERIV (diff m) x :> diff(Suc m) x;
1.430 -        x ~= 0; n > 0
1.431 -      |] ==> \<exists>t. 0 < abs t & abs t < abs x &
1.432 +  assumes INIT1: "diff 0 = f" and INIT2: "0 < n" and INIT3: "x \<noteq> 0"
1.433 +  and DERIV: "\<forall>m x. DERIV (diff m) x :> diff(Suc m) x"
1.434 +  shows "\<exists>t. 0 < abs t & abs t < abs x &
1.435                 f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
1.436 -                     (diff n t / real (fact n)) * x ^ n"
1.437 -apply (rule_tac x = x and y = 0 in linorder_cases)
1.438 -prefer 2 apply blast
1.439 -apply (drule_tac  diff=diff in Maclaurin)
1.440 -apply (drule_tac diff=diff in Maclaurin_minus, simp_all, safe)
1.441 -apply (rule_tac [!] x = t in exI, auto)
1.442 -done
1.443 +                     (diff n t / real (fact n)) * x ^ n"
1.444 +proof -
1.445 +  have "(x = 0) \<Longrightarrow> ?thesis"
1.446 +  proof -
1.447 +    assume "x = 0"
1.448 +    with INIT3 show "(x = 0) \<Longrightarrow> ?thesis"..
1.449 +  qed
1.450 +  moreover have "(x < 0) \<Longrightarrow> ?thesis"
1.451 +  proof -
1.452 +    assume x_less_zero: "x < 0"
1.453 +    from DERIV have "\<forall>m t. m < n \<and> x \<le> t \<and> t \<le> 0 \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t" by simp
1.454 +    with x_less_zero INIT2 INIT1 have "\<exists>t>x. t < 0 \<and> f x = (\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) + diff n t / real (fact n) * x ^ n" (is "\<exists> t. ?P t") by (rule Maclaurin_minus)
1.455 +    from this obtain t where "?P t" ..
1.456 +    with x_less_zero have "0 < \<bar>t\<bar> \<and>
1.457 +      \<bar>t\<bar> < \<bar>x\<bar> \<and>
1.458 +      f x = (\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) + diff n t / real (fact n) * x ^ n" by simp
1.459 +    thus ?thesis ..
1.460 +  qed
1.461 +  moreover have "(x > 0) \<Longrightarrow> ?thesis"
1.462 +  proof -
1.463 +    assume x_greater_zero: "x > 0"
1.464 +    from DERIV have "\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> x \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t" by simp
1.465 +    with x_greater_zero INIT2 INIT1 have "\<exists>t>0. t < x \<and> f x = (\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) + diff n t / real (fact n) * x ^ n" (is "\<exists> t. ?P t") by (rule Maclaurin)
1.466 +    from this obtain t where "?P t" ..
1.467 +    with x_greater_zero have "0 < \<bar>t\<bar> \<and>
1.468 +      \<bar>t\<bar> < \<bar>x\<bar> \<and>
1.469 +      f x = (\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) + diff n t / real (fact n) * x ^ n" by fastsimp
1.470 +    thus ?thesis ..
1.471 +  qed
1.472 +  ultimately show ?thesis by (fastsimp)
1.473 +qed
1.474 +
1.475
1.476  lemma Maclaurin_all_lt_objl:
1.477       "diff 0 = f &
1.478 @@ -322,20 +520,42 @@
1.479            diff 0 0"
1.480  by (induct n, auto)
1.481
1.482 -lemma Maclaurin_all_le: "[| diff 0 = f;
1.483 -        \<forall>m x. DERIV (diff m) x :> diff (Suc m) x
1.484 -      |] ==> \<exists>t. abs t \<le> abs x &
1.485 +
1.486 +lemma Maclaurin_all_le:
1.487 +  assumes INIT: "diff 0 = f"
1.488 +  and DERIV: "\<forall>m x. DERIV (diff m) x :> diff (Suc m) x"
1.489 +  shows "\<exists>t. abs t \<le> abs x &
1.490                f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
1.491                      (diff n t / real (fact n)) * x ^ n"
1.492 -apply(cases "n=0")
1.493 -apply (force)
1.494 -apply (case_tac "x = 0")
1.495 -apply (frule_tac diff = diff and n = n in Maclaurin_zero, assumption)
1.496 -apply (drule not0_implies_Suc)
1.497 -apply (rule_tac x = 0 in exI, force)
1.498 -apply (frule_tac diff = diff and n = n in Maclaurin_all_lt, auto)
1.499 -apply (rule_tac x = t in exI, auto)
1.500 -done
1.501 +proof -
1.502 +  note linorder_le_less_linear [of n 0]
1.503 +  thus ?thesis
1.504 +  proof
1.505 +    assume "n\<le> 0" with INIT show ?thesis by force
1.506 +  next
1.507 +    assume n_greater_zero: "n > 0" show ?thesis
1.508 +    proof (cases "x = 0")
1.509 +      case True
1.510 +      from n_greater_zero have "n \<noteq> 0" by auto
1.511 +      from True this  have f_0:"(\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) = diff 0 0" by (rule Maclaurin_zero)
1.512 +      from n_greater_zero have "n \<noteq> 0" by (rule gr_implies_not0)
1.513 +      hence "\<exists> m. n = Suc m" by (rule not0_implies_Suc)
1.514 +      with f_0 True INIT have " \<bar>0\<bar> \<le> \<bar>x\<bar> \<and>
1.515 +       f x = (\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) + diff n 0 / real (fact n) * x ^ n"
1.516 +	by force
1.517 +      thus ?thesis ..
1.518 +    next
1.519 +      case False
1.520 +      from INIT n_greater_zero this DERIV have "\<exists>t. 0 < \<bar>t\<bar> \<and>
1.521 +	\<bar>t\<bar> < \<bar>x\<bar> \<and> f x = (\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) + diff n t / real (fact n) * x ^ n" (is "\<exists> t. ?P t") by (rule Maclaurin_all_lt)
1.522 +      from this obtain t where "?P t" ..
1.523 +      hence "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and>
1.524 +       f x = (\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) + diff n t / real (fact n) * x ^ n" by (simp add: order_less_le)
1.525 +      thus ?thesis ..
1.526 +    qed
1.527 +  qed
1.528 +qed
1.529 +
1.530
1.531  lemma Maclaurin_all_le_objl: "diff 0 = f &
1.532        (\<forall>m x. DERIV (diff m) x :> diff (Suc m) x)
```