src/HOL/Integration.thy
author haftmann
Fri Jan 02 08:12:46 2009 +0100 (2009-01-02)
changeset 29332 edc1e2a56398
parent 28952 15a4b2cf8c34
child 29352 165e959721c2
permissions -rw-r--r--
named code theorem for Fract_norm
     1 (*  Author      : Jacques D. Fleuriot
     2     Copyright   : 2000  University of Edinburgh
     3     Conversion to Isar and new proofs by Lawrence C Paulson, 2004
     4 *)
     5 
     6 header{*Theory of Integration*}
     7 
     8 theory Integration
     9 imports MacLaurin
    10 begin
    11 
    12 text{*We follow John Harrison in formalizing the Gauge integral.*}
    13 
    14 definition
    15   --{*Partitions and tagged partitions etc.*}
    16 
    17   partition :: "[(real*real),nat => real] => bool" where
    18   [code del]: "partition = (%(a,b) D. D 0 = a &
    19                          (\<exists>N. (\<forall>n < N. D(n) < D(Suc n)) &
    20                               (\<forall>n \<ge> N. D(n) = b)))"
    21 
    22 definition
    23   psize :: "(nat => real) => nat" where
    24   [code del]:"psize D = (SOME N. (\<forall>n < N. D(n) < D(Suc n)) &
    25                       (\<forall>n \<ge> N. D(n) = D(N)))"
    26 
    27 definition
    28   tpart :: "[(real*real),((nat => real)*(nat =>real))] => bool" where
    29   [code del]:"tpart = (%(a,b) (D,p). partition(a,b) D &
    30                           (\<forall>n. D(n) \<le> p(n) & p(n) \<le> D(Suc n)))"
    31 
    32   --{*Gauges and gauge-fine divisions*}
    33 
    34 definition
    35   gauge :: "[real => bool, real => real] => bool" where
    36   [code del]:"gauge E g = (\<forall>x. E x --> 0 < g(x))"
    37 
    38 definition
    39   fine :: "[real => real, ((nat => real)*(nat => real))] => bool" where
    40   [code del]:"fine = (%g (D,p). \<forall>n. n < (psize D) --> D(Suc n) - D(n) < g(p n))"
    41 
    42   --{*Riemann sum*}
    43 
    44 definition
    45   rsum :: "[((nat=>real)*(nat=>real)),real=>real] => real" where
    46   "rsum = (%(D,p) f. \<Sum>n=0..<psize(D). f(p n) * (D(Suc n) - D(n)))"
    47 
    48   --{*Gauge integrability (definite)*}
    49 
    50 definition
    51   Integral :: "[(real*real),real=>real,real] => bool" where
    52   [code del]: "Integral = (%(a,b) f k. \<forall>e > 0.
    53                                (\<exists>g. gauge(%x. a \<le> x & x \<le> b) g &
    54                                (\<forall>D p. tpart(a,b) (D,p) & fine(g)(D,p) -->
    55                                          \<bar>rsum(D,p) f - k\<bar> < e)))"
    56 
    57 
    58 lemma partition_zero [simp]: "a = b ==> psize (%n. if n = 0 then a else b) = 0"
    59 by (auto simp add: psize_def)
    60 
    61 lemma partition_one [simp]: "a < b ==> psize (%n. if n = 0 then a else b) = 1"
    62 apply (simp add: psize_def)
    63 apply (rule some_equality, auto)
    64 apply (drule_tac x = 1 in spec, auto)
    65 done
    66 
    67 lemma partition_single [simp]:
    68      "a \<le> b ==> partition(a,b)(%n. if n = 0 then a else b)"
    69 by (auto simp add: partition_def order_le_less)
    70 
    71 lemma partition_lhs: "partition(a,b) D ==> (D(0) = a)"
    72 by (simp add: partition_def)
    73 
    74 lemma partition:
    75        "(partition(a,b) D) =
    76         ((D 0 = a) &
    77          (\<forall>n < psize D. D n < D(Suc n)) &
    78          (\<forall>n \<ge> psize D. D n = b))"
    79 apply (simp add: partition_def, auto)
    80 apply (subgoal_tac [!] "psize D = N", auto)
    81 apply (simp_all (no_asm) add: psize_def)
    82 apply (rule_tac [!] some_equality, blast)
    83   prefer 2 apply blast
    84 apply (rule_tac [!] ccontr)
    85 apply (simp_all add: linorder_neq_iff, safe)
    86 apply (drule_tac x = Na in spec)
    87 apply (rotate_tac 3)
    88 apply (drule_tac x = "Suc Na" in spec, simp)
    89 apply (rotate_tac 2)
    90 apply (drule_tac x = N in spec, simp)
    91 apply (drule_tac x = Na in spec)
    92 apply (drule_tac x = "Suc Na" and P = "%n. Na\<le>n \<longrightarrow> D n = D Na" in spec, auto)
    93 done
    94 
    95 lemma partition_rhs: "partition(a,b) D ==> (D(psize D) = b)"
    96 by (simp add: partition)
    97 
    98 lemma partition_rhs2: "[|partition(a,b) D; psize D \<le> n |] ==> (D n = b)"
    99 by (simp add: partition)
   100 
   101 lemma lemma_partition_lt_gen [rule_format]:
   102  "partition(a,b) D & m + Suc d \<le> n & n \<le> (psize D) --> D(m) < D(m + Suc d)"
   103 apply (induct "d", auto simp add: partition)
   104 apply (blast dest: Suc_le_lessD  intro: less_le_trans order_less_trans)
   105 done
   106 
   107 lemma less_eq_add_Suc: "m < n ==> \<exists>d. n = m + Suc d"
   108 by (auto simp add: less_iff_Suc_add)
   109 
   110 lemma partition_lt_gen:
   111      "[|partition(a,b) D; m < n; n \<le> (psize D)|] ==> D(m) < D(n)"
   112 by (auto dest: less_eq_add_Suc intro: lemma_partition_lt_gen)
   113 
   114 lemma partition_lt: "partition(a,b) D ==> n < (psize D) ==> D(0) < D(Suc n)"
   115 apply (induct "n")
   116 apply (auto simp add: partition)
   117 done
   118 
   119 lemma partition_le: "partition(a,b) D ==> a \<le> b"
   120 apply (frule partition [THEN iffD1], safe)
   121 apply (drule_tac x = "psize D" and P="%n. psize D \<le> n --> ?P n" in spec, safe)
   122 apply (case_tac "psize D = 0")
   123 apply (drule_tac [2] n = "psize D - 1" in partition_lt, auto)
   124 done
   125 
   126 lemma partition_gt: "[|partition(a,b) D; n < (psize D)|] ==> D(n) < D(psize D)"
   127 by (auto intro: partition_lt_gen)
   128 
   129 lemma partition_eq: "partition(a,b) D ==> ((a = b) = (psize D = 0))"
   130 apply (frule partition [THEN iffD1], safe)
   131 apply (rotate_tac 2)
   132 apply (drule_tac x = "psize D" in spec)
   133 apply (rule ccontr)
   134 apply (drule_tac n = "psize D - 1" in partition_lt)
   135 apply auto
   136 done
   137 
   138 lemma partition_lb: "partition(a,b) D ==> a \<le> D(r)"
   139 apply (frule partition [THEN iffD1], safe)
   140 apply (induct "r")
   141 apply (cut_tac [2] y = "Suc r" and x = "psize D" in linorder_le_less_linear)
   142 apply (auto intro: partition_le)
   143 apply (drule_tac x = r in spec)
   144 apply arith; 
   145 done
   146 
   147 lemma partition_lb_lt: "[| partition(a,b) D; psize D ~= 0 |] ==> a < D(Suc n)"
   148 apply (rule_tac t = a in partition_lhs [THEN subst], assumption)
   149 apply (cut_tac x = "Suc n" and y = "psize D" in linorder_le_less_linear)
   150 apply (frule partition [THEN iffD1], safe)
   151  apply (blast intro: partition_lt less_le_trans)
   152 apply (rotate_tac 3)
   153 apply (drule_tac x = "Suc n" in spec)
   154 apply (erule impE)
   155 apply (erule less_imp_le)
   156 apply (frule partition_rhs)
   157 apply (drule partition_gt[of _ _ _ 0], arith)
   158 apply (simp (no_asm_simp))
   159 done
   160 
   161 lemma partition_ub: "partition(a,b) D ==> D(r) \<le> b"
   162 apply (frule partition [THEN iffD1])
   163 apply (cut_tac x = "psize D" and y = r in linorder_le_less_linear, safe, blast)
   164 apply (subgoal_tac "\<forall>x. D ((psize D) - x) \<le> b")
   165 apply (rotate_tac 4)
   166 apply (drule_tac x = "psize D - r" in spec)
   167 apply (subgoal_tac "psize D - (psize D - r) = r")
   168 apply simp
   169 apply arith
   170 apply safe
   171 apply (induct_tac "x")
   172 apply (simp (no_asm), blast)
   173 apply (case_tac "psize D - Suc n = 0")
   174 apply (erule_tac V = "\<forall>n. psize D \<le> n --> D n = b" in thin_rl)
   175 apply (simp (no_asm_simp) add: partition_le)
   176 apply (rule order_trans)
   177  prefer 2 apply assumption
   178 apply (subgoal_tac "psize D - n = Suc (psize D - Suc n)")
   179  prefer 2 apply arith
   180 apply (drule_tac x = "psize D - Suc n" in spec, simp) 
   181 done
   182 
   183 lemma partition_ub_lt: "[| partition(a,b) D; n < psize D |] ==> D(n) < b"
   184 by (blast intro: partition_rhs [THEN subst] partition_gt)
   185 
   186 lemma lemma_partition_append1:
   187      "[| partition (a, b) D1; partition (b, c) D2 |]
   188        ==> (\<forall>n < psize D1 + psize D2.
   189              (if n < psize D1 then D1 n else D2 (n - psize D1))
   190              < (if Suc n < psize D1 then D1 (Suc n)
   191                 else D2 (Suc n - psize D1))) &
   192          (\<forall>n \<ge> psize D1 + psize D2.
   193              (if n < psize D1 then D1 n else D2 (n - psize D1)) =
   194              (if psize D1 + psize D2 < psize D1 then D1 (psize D1 + psize D2)
   195               else D2 (psize D1 + psize D2 - psize D1)))"
   196 apply (auto intro: partition_lt_gen)
   197 apply (subgoal_tac "psize D1 = Suc n")
   198 apply (auto intro!: partition_lt_gen simp add: partition_lhs partition_ub_lt)
   199 apply (auto intro!: partition_rhs2 simp add: partition_rhs
   200             split: nat_diff_split)
   201 done
   202 
   203 lemma lemma_psize1:
   204      "[| partition (a, b) D1; partition (b, c) D2; N < psize D1 |]
   205       ==> D1(N) < D2 (psize D2)"
   206 apply (rule_tac y = "D1 (psize D1)" in order_less_le_trans)
   207 apply (erule partition_gt)
   208 apply (auto simp add: partition_rhs partition_le)
   209 done
   210 
   211 lemma lemma_partition_append2:
   212      "[| partition (a, b) D1; partition (b, c) D2 |]
   213       ==> psize (%n. if n < psize D1 then D1 n else D2 (n - psize D1)) =
   214           psize D1 + psize D2" 
   215 apply (unfold psize_def 
   216          [of "%n. if n < psize D1 then D1 n else D2 (n - psize D1)"])
   217 apply (rule some1_equality)
   218  prefer 2 apply (blast intro!: lemma_partition_append1)
   219 apply (rule ex1I, rule lemma_partition_append1) 
   220 apply (simp_all split: split_if_asm)
   221  txt{*The case @{term "N < psize D1"}*} 
   222  apply (drule_tac x = "psize D1 + psize D2" and P="%n. ?P n & ?Q n" in spec) 
   223  apply (force dest: lemma_psize1)
   224 apply (rule order_antisym);
   225  txt{*The case @{term "psize D1 \<le> N"}: 
   226        proving @{term "N \<le> psize D1 + psize D2"}*}
   227  apply (drule_tac x = "psize D1 + psize D2" in spec)
   228  apply (simp add: partition_rhs2)
   229 apply (case_tac "N - psize D1 < psize D2") 
   230  prefer 2 apply arith
   231  txt{*Proving @{term "psize D1 + psize D2 \<le> N"}*}
   232 apply (drule_tac x = "psize D1 + psize D2" and P="%n. N\<le>n --> ?P n" in spec, simp)
   233 apply (drule_tac a = b and b = c in partition_gt, assumption, simp)
   234 done
   235 
   236 lemma tpart_eq_lhs_rhs: "[|psize D = 0; tpart(a,b) (D,p)|] ==> a = b"
   237 by (auto simp add: tpart_def partition_eq)
   238 
   239 lemma tpart_partition: "tpart(a,b) (D,p) ==> partition(a,b) D"
   240 by (simp add: tpart_def)
   241 
   242 lemma partition_append:
   243      "[| tpart(a,b) (D1,p1); fine(g) (D1,p1);
   244          tpart(b,c) (D2,p2); fine(g) (D2,p2) |]
   245        ==> \<exists>D p. tpart(a,c) (D,p) & fine(g) (D,p)"
   246 apply (rule_tac x = "%n. if n < psize D1 then D1 n else D2 (n - psize D1)"
   247        in exI)
   248 apply (rule_tac x = "%n. if n < psize D1 then p1 n else p2 (n - psize D1)"
   249        in exI)
   250 apply (case_tac "psize D1 = 0")
   251 apply (auto dest: tpart_eq_lhs_rhs)
   252  prefer 2
   253 apply (simp add: fine_def
   254                  lemma_partition_append2 [OF tpart_partition tpart_partition])
   255   --{*But must not expand @{term fine} in other subgoals*}
   256 apply auto
   257 apply (subgoal_tac "psize D1 = Suc n")
   258  prefer 2 apply arith
   259 apply (drule tpart_partition [THEN partition_rhs])
   260 apply (drule tpart_partition [THEN partition_lhs])
   261 apply (auto split: nat_diff_split)
   262 apply (auto simp add: tpart_def)
   263 defer 1
   264  apply (subgoal_tac "psize D1 = Suc n")
   265   prefer 2 apply arith
   266  apply (drule partition_rhs)
   267  apply (drule partition_lhs, auto)
   268 apply (simp split: nat_diff_split)
   269 apply (subst partition) 
   270 apply (subst (1 2) lemma_partition_append2, assumption+)
   271 apply (rule conjI) 
   272 apply (simp add: partition_lhs)
   273 apply (drule lemma_partition_append1)
   274 apply assumption; 
   275 apply (simp add: partition_rhs)
   276 done
   277 
   278 
   279 text{*We can always find a division that is fine wrt any gauge*}
   280 
   281 lemma partition_exists:
   282      "[| a \<le> b; gauge(%x. a \<le> x & x \<le> b) g |]
   283       ==> \<exists>D p. tpart(a,b) (D,p) & fine g (D,p)"
   284 apply (cut_tac P = "%(u,v). a \<le> u & v \<le> b --> 
   285                    (\<exists>D p. tpart (u,v) (D,p) & fine (g) (D,p))" 
   286        in lemma_BOLZANO2)
   287 apply safe
   288 apply (blast intro: order_trans)+
   289 apply (auto intro: partition_append)
   290 apply (case_tac "a \<le> x & x \<le> b")
   291 apply (rule_tac [2] x = 1 in exI, auto)
   292 apply (rule_tac x = "g x" in exI)
   293 apply (auto simp add: gauge_def)
   294 apply (rule_tac x = "%n. if n = 0 then aa else ba" in exI)
   295 apply (rule_tac x = "%n. if n = 0 then x else ba" in exI)
   296 apply (auto simp add: tpart_def fine_def)
   297 done
   298 
   299 text{*Lemmas about combining gauges*}
   300 
   301 lemma gauge_min:
   302      "[| gauge(E) g1; gauge(E) g2 |]
   303       ==> gauge(E) (%x. if g1(x) < g2(x) then g1(x) else g2(x))"
   304 by (simp add: gauge_def)
   305 
   306 lemma fine_min:
   307       "fine (%x. if g1(x) < g2(x) then g1(x) else g2(x)) (D,p)
   308        ==> fine(g1) (D,p) & fine(g2) (D,p)"
   309 by (auto simp add: fine_def split: split_if_asm)
   310 
   311 
   312 text{*The integral is unique if it exists*}
   313 
   314 lemma Integral_unique:
   315     "[| a \<le> b; Integral(a,b) f k1; Integral(a,b) f k2 |] ==> k1 = k2"
   316 apply (simp add: Integral_def)
   317 apply (drule_tac x = "\<bar>k1 - k2\<bar> /2" in spec)+
   318 apply auto
   319 apply (drule gauge_min, assumption)
   320 apply (drule_tac g = "%x. if g x < ga x then g x else ga x" 
   321        in partition_exists, assumption, auto)
   322 apply (drule fine_min)
   323 apply (drule spec)+
   324 apply auto
   325 apply (subgoal_tac "\<bar>(rsum (D,p) f - k2) - (rsum (D,p) f - k1)\<bar> < \<bar>k1 - k2\<bar>")
   326 apply arith
   327 apply (drule add_strict_mono, assumption)
   328 apply (auto simp only: left_distrib [symmetric] mult_2_right [symmetric] 
   329                 mult_less_cancel_right)
   330 done
   331 
   332 lemma Integral_zero [simp]: "Integral(a,a) f 0"
   333 apply (auto simp add: Integral_def)
   334 apply (rule_tac x = "%x. 1" in exI)
   335 apply (auto dest: partition_eq simp add: gauge_def tpart_def rsum_def)
   336 done
   337 
   338 lemma sumr_partition_eq_diff_bounds [simp]:
   339      "(\<Sum>n=0..<m. D (Suc n) - D n::real) = D(m) - D 0"
   340 by (induct "m", auto)
   341 
   342 lemma Integral_eq_diff_bounds: "a \<le> b ==> Integral(a,b) (%x. 1) (b - a)"
   343 apply (auto simp add: order_le_less rsum_def Integral_def)
   344 apply (rule_tac x = "%x. b - a" in exI)
   345 apply (auto simp add: gauge_def abs_less_iff tpart_def partition)
   346 done
   347 
   348 lemma Integral_mult_const: "a \<le> b ==> Integral(a,b) (%x. c)  (c*(b - a))"
   349 apply (auto simp add: order_le_less rsum_def Integral_def)
   350 apply (rule_tac x = "%x. b - a" in exI)
   351 apply (auto simp add: setsum_right_distrib [symmetric] gauge_def abs_less_iff 
   352                right_diff_distrib [symmetric] partition tpart_def)
   353 done
   354 
   355 lemma Integral_mult:
   356      "[| a \<le> b; Integral(a,b) f k |] ==> Integral(a,b) (%x. c * f x) (c * k)"
   357 apply (auto simp add: order_le_less 
   358             dest: Integral_unique [OF order_refl Integral_zero])
   359 apply (auto simp add: rsum_def Integral_def setsum_right_distrib[symmetric] mult_assoc)
   360 apply (rule_tac a2 = c in abs_ge_zero [THEN order_le_imp_less_or_eq, THEN disjE])
   361  prefer 2 apply force
   362 apply (drule_tac x = "e/abs c" in spec, auto)
   363 apply (simp add: zero_less_mult_iff divide_inverse)
   364 apply (rule exI, auto)
   365 apply (drule spec)+
   366 apply auto
   367 apply (rule_tac z1 = "inverse (abs c)" in real_mult_less_iff1 [THEN iffD1])
   368 apply (auto simp add: abs_mult divide_inverse [symmetric] right_diff_distrib [symmetric])
   369 done
   370 
   371 text{*Fundamental theorem of calculus (Part I)*}
   372 
   373 text{*"Straddle Lemma" : Swartz and Thompson: AMM 95(7) 1988 *}
   374 
   375 lemma choiceP: "\<forall>x. P(x) --> (\<exists>y. Q x y) ==> \<exists>f. (\<forall>x. P(x) --> Q x (f x))" 
   376 by (insert bchoice [of "Collect P" Q], simp) 
   377 
   378 (*UNUSED
   379 lemma choice2: "\<forall>x. (\<exists>y. R(y) & (\<exists>z. Q x y z)) ==>
   380       \<exists>f fa. (\<forall>x. R(f x) & Q x (f x) (fa x))"
   381 *)
   382 
   383 
   384 (* new simplifications e.g. (y < x/n) = (y * n < x) are a real nuisance
   385    they break the original proofs and make new proofs longer!*)
   386 lemma strad1:
   387        "\<lbrakk>\<forall>xa::real. xa \<noteq> x \<and> \<bar>xa - x\<bar> < s \<longrightarrow>
   388              \<bar>(f xa - f x) / (xa - x) - f' x\<bar> * 2 < e;
   389         0 < e; a \<le> x; x \<le> b; 0 < s\<rbrakk>
   390        \<Longrightarrow> \<forall>z. \<bar>z - x\<bar> < s -->\<bar>f z - f x - f' x * (z - x)\<bar> * 2 \<le> e * \<bar>z - x\<bar>"
   391 apply auto
   392 apply (case_tac "0 < \<bar>z - x\<bar>")
   393  prefer 2 apply (simp add: zero_less_abs_iff)
   394 apply (drule_tac x = z in spec)
   395 apply (rule_tac z1 = "\<bar>inverse (z - x)\<bar>" 
   396        in real_mult_le_cancel_iff2 [THEN iffD1])
   397  apply simp
   398 apply (simp del: abs_inverse abs_mult add: abs_mult [symmetric]
   399           mult_assoc [symmetric])
   400 apply (subgoal_tac "inverse (z - x) * (f z - f x - f' x * (z - x)) 
   401                     = (f z - f x) / (z - x) - f' x")
   402  apply (simp add: abs_mult [symmetric] mult_ac diff_minus)
   403 apply (subst mult_commute)
   404 apply (simp add: left_distrib diff_minus)
   405 apply (simp add: mult_assoc divide_inverse)
   406 apply (simp add: left_distrib)
   407 done
   408 
   409 lemma lemma_straddle:
   410      "[| \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x); 0 < e |]
   411       ==> \<exists>g. gauge(%x. a \<le> x & x \<le> b) g &
   412                 (\<forall>x u v. a \<le> u & u \<le> x & x \<le> v & v \<le> b & (v - u) < g(x)
   413                   --> \<bar>(f(v) - f(u)) - (f'(x) * (v - u))\<bar> \<le> e * (v - u))"
   414 apply (simp add: gauge_def)
   415 apply (subgoal_tac "\<forall>x. a \<le> x & x \<le> b --> 
   416         (\<exists>d > 0. \<forall>u v. u \<le> x & x \<le> v & (v - u) < d --> 
   417                        \<bar>(f (v) - f (u)) - (f' (x) * (v - u))\<bar> \<le> e * (v - u))")
   418 apply (drule choiceP, auto)
   419 apply (drule spec, auto)
   420 apply (auto simp add: DERIV_iff2 LIM_def)
   421 apply (drule_tac x = "e/2" in spec, auto)
   422 apply (frule strad1, assumption+)
   423 apply (rule_tac x = s in exI, auto)
   424 apply (rule_tac x = u and y = v in linorder_cases, auto)
   425 apply (rule_tac y = "\<bar>(f (v) - f (x)) - (f' (x) * (v - x))\<bar> + 
   426                      \<bar>(f (x) - f (u)) - (f' (x) * (x - u))\<bar>"
   427        in order_trans)
   428 apply (rule abs_triangle_ineq [THEN [2] order_trans])
   429 apply (simp add: right_diff_distrib)
   430 apply (rule_tac t = "e* (v - u)" in real_sum_of_halves [THEN subst])
   431 apply (rule add_mono)
   432 apply (rule_tac y = "(e/2) * \<bar>v - x\<bar>" in order_trans)
   433  prefer 2 apply simp
   434 apply (erule_tac [!] V= "\<forall>x'. x' ~= x & \<bar>x' - x\<bar> < s --> ?P x'" in thin_rl)
   435 apply (drule_tac x = v in spec, simp add: times_divide_eq)
   436 apply (drule_tac x = u in spec, auto)
   437 apply (subgoal_tac "\<bar>f u - f x - f' x * (u - x)\<bar> = \<bar>f x - f u - f' x * (x - u)\<bar>")
   438 apply (rule order_trans)
   439 apply (auto simp add: abs_le_iff)
   440 apply (simp add: right_diff_distrib)
   441 done
   442 
   443 lemma FTC1: "[|a \<le> b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
   444              ==> Integral(a,b) f' (f(b) - f(a))"
   445 apply (drule order_le_imp_less_or_eq, auto) 
   446 apply (auto simp add: Integral_def)
   447 apply (rule ccontr)
   448 apply (subgoal_tac "\<forall>e > 0. \<exists>g. gauge (%x. a \<le> x & x \<le> b) g & (\<forall>D p. tpart (a, b) (D, p) & fine g (D, p) --> \<bar>rsum (D, p) f' - (f b - f a)\<bar> \<le> e)")
   449 apply (rotate_tac 3)
   450 apply (drule_tac x = "e/2" in spec, auto)
   451 apply (drule spec, auto)
   452 apply ((drule spec)+, auto)
   453 apply (drule_tac e = "ea/ (b - a)" in lemma_straddle)
   454 apply (auto simp add: zero_less_divide_iff)
   455 apply (rule exI)
   456 apply (auto simp add: tpart_def rsum_def)
   457 apply (subgoal_tac "(\<Sum>n=0..<psize D. f(D(Suc n)) - f(D n)) = f b - f a")
   458  prefer 2
   459  apply (cut_tac D = "%n. f (D n)" and m = "psize D"
   460         in sumr_partition_eq_diff_bounds)
   461  apply (simp add: partition_lhs partition_rhs)
   462 apply (drule sym, simp)
   463 apply (simp (no_asm) add: setsum_subtractf[symmetric])
   464 apply (rule setsum_abs [THEN order_trans])
   465 apply (subgoal_tac "ea = (\<Sum>n=0..<psize D. (ea / (b - a)) * (D (Suc n) - (D n)))")
   466 apply (simp add: abs_minus_commute)
   467 apply (rule_tac t = ea in ssubst, assumption)
   468 apply (rule setsum_mono)
   469 apply (rule_tac [2] setsum_right_distrib [THEN subst])
   470 apply (auto simp add: partition_rhs partition_lhs partition_lb partition_ub
   471           fine_def)
   472 done
   473 
   474 
   475 lemma Integral_subst: "[| Integral(a,b) f k1; k2=k1 |] ==> Integral(a,b) f k2"
   476 by simp
   477 
   478 lemma Integral_add:
   479      "[| a \<le> b; b \<le> c; Integral(a,b) f' k1; Integral(b,c) f' k2;
   480          \<forall>x. a \<le> x & x \<le> c --> DERIV f x :> f' x |]
   481      ==> Integral(a,c) f' (k1 + k2)"
   482 apply (rule FTC1 [THEN Integral_subst], auto)
   483 apply (frule FTC1, auto)
   484 apply (frule_tac a = b in FTC1, auto)
   485 apply (drule_tac x = x in spec, auto)
   486 apply (drule_tac ?k2.0 = "f b - f a" in Integral_unique)
   487 apply (drule_tac [3] ?k2.0 = "f c - f b" in Integral_unique, auto)
   488 done
   489 
   490 lemma partition_psize_Least:
   491      "partition(a,b) D ==> psize D = (LEAST n. D(n) = b)"
   492 apply (auto intro!: Least_equality [symmetric] partition_rhs)
   493 apply (auto dest: partition_ub_lt simp add: linorder_not_less [symmetric])
   494 done
   495 
   496 lemma lemma_partition_bounded: "partition (a, c) D ==> ~ (\<exists>n. c < D(n))"
   497 apply safe
   498 apply (drule_tac r = n in partition_ub, auto)
   499 done
   500 
   501 lemma lemma_partition_eq:
   502      "partition (a, c) D ==> D = (%n. if D n < c then D n else c)"
   503 apply (rule ext, auto)
   504 apply (auto dest!: lemma_partition_bounded)
   505 apply (drule_tac x = n in spec, auto)
   506 done
   507 
   508 lemma lemma_partition_eq2:
   509      "partition (a, c) D ==> D = (%n. if D n \<le> c then D n else c)"
   510 apply (rule ext, auto)
   511 apply (auto dest!: lemma_partition_bounded)
   512 apply (drule_tac x = n in spec, auto)
   513 done
   514 
   515 lemma partition_lt_Suc:
   516      "[| partition(a,b) D; n < psize D |] ==> D n < D (Suc n)"
   517 by (auto simp add: partition)
   518 
   519 lemma tpart_tag_eq: "tpart(a,c) (D,p) ==> p = (%n. if D n < c then p n else c)"
   520 apply (rule ext)
   521 apply (auto simp add: tpart_def)
   522 apply (drule linorder_not_less [THEN iffD1])
   523 apply (drule_tac r = "Suc n" in partition_ub)
   524 apply (drule_tac x = n in spec, auto)
   525 done
   526 
   527 subsection{*Lemmas for Additivity Theorem of Gauge Integral*}
   528 
   529 lemma lemma_additivity1:
   530      "[| a \<le> D n; D n < b; partition(a,b) D |] ==> n < psize D"
   531 by (auto simp add: partition linorder_not_less [symmetric])
   532 
   533 lemma lemma_additivity2: "[| a \<le> D n; partition(a,D n) D |] ==> psize D \<le> n"
   534 apply (rule ccontr, drule not_leE)
   535 apply (frule partition [THEN iffD1], safe)
   536 apply (frule_tac r = "Suc n" in partition_ub)
   537 apply (auto dest!: spec)
   538 done
   539 
   540 lemma partition_eq_bound:
   541      "[| partition(a,b) D; psize D < m |] ==> D(m) = D(psize D)"
   542 by (auto simp add: partition)
   543 
   544 lemma partition_ub2: "[| partition(a,b) D; psize D < m |] ==> D(r) \<le> D(m)"
   545 by (simp add: partition partition_ub)
   546 
   547 lemma tag_point_eq_partition_point:
   548     "[| tpart(a,b) (D,p); psize D \<le> m |] ==> p(m) = D(m)"
   549 apply (simp add: tpart_def, auto)
   550 apply (drule_tac x = m in spec)
   551 apply (auto simp add: partition_rhs2)
   552 done
   553 
   554 lemma partition_lt_cancel: "[| partition(a,b) D; D m < D n |] ==> m < n"
   555 apply (cut_tac less_linear [of n "psize D"], auto)
   556 apply (cut_tac less_linear [of m n])
   557 apply (cut_tac less_linear [of m "psize D"])
   558 apply (auto dest: partition_gt)
   559 apply (drule_tac n = m in partition_lt_gen, auto)
   560 apply (frule partition_eq_bound)
   561 apply (drule_tac [2] partition_gt, auto)
   562 apply (metis dense_linear_order_class.dlo_simps(8) not_less partition_rhs partition_rhs2)
   563 apply (metis le_less_trans dense_linear_order_class.dlo_simps(8) nat_le_linear partition_eq_bound partition_rhs2)
   564 done
   565 
   566 lemma lemma_additivity4_psize_eq:
   567      "[| a \<le> D n; D n < b; partition (a, b) D |]
   568       ==> psize (%x. if D x < D n then D(x) else D n) = n"
   569 apply (unfold psize_def)
   570 apply (frule lemma_additivity1)
   571 apply (assumption, assumption)
   572 apply (rule some_equality)
   573 apply (auto intro: partition_lt_Suc)
   574 apply (drule_tac n = n in partition_lt_gen, assumption)
   575 apply (arith, arith)
   576 apply (cut_tac x = na and y = "psize D" in less_linear)
   577 apply (auto dest: partition_lt_cancel)
   578 apply (rule_tac x=N and y=n in linorder_cases)
   579 apply (drule_tac x = n and P="%m. N \<le> m --> ?f m = ?g m" in spec, simp)
   580 apply (drule_tac n = n in partition_lt_gen, auto)
   581 apply (drule_tac x = n in spec)
   582 apply (simp split: split_if_asm)
   583 done
   584 
   585 lemma lemma_psize_left_less_psize:
   586      "partition (a, b) D
   587       ==> psize (%x. if D x < D n then D(x) else D n) \<le> psize D"
   588 apply (frule_tac r = n in partition_ub)
   589 apply (drule_tac x = "D n" in order_le_imp_less_or_eq)
   590 apply (auto simp add: lemma_partition_eq [symmetric])
   591 apply (frule_tac r = n in partition_lb)
   592 apply (drule (2) lemma_additivity4_psize_eq)  
   593 apply (rule ccontr, auto)
   594 apply (frule_tac not_leE [THEN [2] partition_eq_bound])
   595 apply (auto simp add: partition_rhs)
   596 done
   597 
   598 lemma lemma_psize_left_less_psize2:
   599      "[| partition(a,b) D; na < psize (%x. if D x < D n then D(x) else D n) |]
   600       ==> na < psize D"
   601 by (erule lemma_psize_left_less_psize [THEN [2] less_le_trans])
   602 
   603 
   604 lemma lemma_additivity3:
   605      "[| partition(a,b) D; D na < D n; D n < D (Suc na);
   606          n < psize D |]
   607       ==> False"
   608 by (metis not_less_eq partition_lt_cancel real_of_nat_less_iff)
   609 
   610 
   611 lemma psize_const [simp]: "psize (%x. k) = 0"
   612 by (auto simp add: psize_def)
   613 
   614 lemma lemma_additivity3a:
   615      "[| partition(a,b) D; D na < D n; D n < D (Suc na);
   616          na < psize D |]
   617       ==> False"
   618 apply (frule_tac m = n in partition_lt_cancel)
   619 apply (auto intro: lemma_additivity3)
   620 done
   621 
   622 lemma better_lemma_psize_right_eq1:
   623      "[| partition(a,b) D; D n < b |] ==> psize (%x. D (x + n)) \<le> psize D - n"
   624 apply (simp add: psize_def [of "(%x. D (x + n))"]);
   625 apply (rule_tac a = "psize D - n" in someI2, auto)
   626   apply (simp add: partition less_diff_conv)
   627  apply (simp add: le_diff_conv partition_rhs2 split: nat_diff_split)
   628 apply (drule_tac x = "psize D - n" in spec, auto)
   629 apply (frule partition_rhs, safe)
   630 apply (frule partition_lt_cancel, assumption)
   631 apply (drule partition [THEN iffD1], safe)
   632 apply (subgoal_tac "~ D (psize D - n + n) < D (Suc (psize D - n + n))")
   633  apply blast
   634 apply (drule_tac x = "Suc (psize D)" and P="%n. ?P n \<longrightarrow> D n = D (psize D)"
   635        in spec)
   636 apply simp
   637 done
   638 
   639 lemma psize_le_n: "partition (a, D n) D ==> psize D \<le> n" 
   640 apply (rule ccontr, drule not_leE)
   641 apply (frule partition_lt_Suc, assumption)
   642 apply (frule_tac r = "Suc n" in partition_ub, auto)
   643 done
   644 
   645 lemma better_lemma_psize_right_eq1a:
   646      "partition(a,D n) D ==> psize (%x. D (x + n)) \<le> psize D - n"
   647 apply (simp add: psize_def [of "(%x. D (x + n))"]);
   648 apply (rule_tac a = "psize D - n" in someI2, auto)
   649   apply (simp add: partition less_diff_conv)
   650  apply (simp add: le_diff_conv)
   651 apply (case_tac "psize D \<le> n")
   652   apply (force intro: partition_rhs2)
   653  apply (simp add: partition linorder_not_le)
   654 apply (rule ccontr, drule not_leE)
   655 apply (frule psize_le_n)
   656 apply (drule_tac x = "psize D - n" in spec, simp)
   657 apply (drule partition [THEN iffD1], safe)
   658 apply (drule_tac x = "Suc n" and P="%na. ?s \<le> na \<longrightarrow> D na = D n" in spec, auto)
   659 done
   660 
   661 lemma better_lemma_psize_right_eq:
   662      "partition(a,b) D ==> psize (%x. D (x + n)) \<le> psize D - n"
   663 apply (frule_tac r1 = n in partition_ub [THEN order_le_imp_less_or_eq])
   664 apply (blast intro: better_lemma_psize_right_eq1a better_lemma_psize_right_eq1)
   665 done
   666 
   667 lemma lemma_psize_right_eq1:
   668      "[| partition(a,b) D; D n < b |] ==> psize (%x. D (x + n)) \<le> psize D"
   669 apply (simp add: psize_def [of "(%x. D (x + n))"])
   670 apply (rule_tac a = "psize D - n" in someI2, auto)
   671   apply (simp add: partition less_diff_conv)
   672  apply (subgoal_tac "n \<le> psize D")
   673   apply (simp add: partition le_diff_conv)
   674  apply (rule ccontr, drule not_leE)
   675  apply (drule_tac less_imp_le [THEN [2] partition_rhs2], assumption, simp)
   676 apply (drule_tac x = "psize D" in spec)
   677 apply (simp add: partition)
   678 done
   679 
   680 (* should be combined with previous theorem; also proof has redundancy *)
   681 lemma lemma_psize_right_eq1a:
   682      "partition(a,D n) D ==> psize (%x. D (x + n)) \<le> psize D"
   683 apply (simp add: psize_def [of "(%x. D (x + n))"]);
   684 apply (rule_tac a = "psize D - n" in someI2, auto)
   685   apply (simp add: partition less_diff_conv)
   686  apply (case_tac "psize D \<le> n")
   687   apply (force intro: partition_rhs2 simp add: le_diff_conv)
   688  apply (simp add: partition le_diff_conv)
   689 apply (rule ccontr, drule not_leE)
   690 apply (drule_tac x = "psize D" in spec)
   691 apply (simp add: partition)
   692 done
   693 
   694 lemma lemma_psize_right_eq:
   695      "[| partition(a,b) D |] ==> psize (%x. D (x + n)) \<le> psize D"
   696 apply (frule_tac r1 = n in partition_ub [THEN order_le_imp_less_or_eq])
   697 apply (blast intro: lemma_psize_right_eq1a lemma_psize_right_eq1)
   698 done
   699 
   700 lemma tpart_left1:
   701      "[| a \<le> D n; tpart (a, b) (D, p) |]
   702       ==> tpart(a, D n) (%x. if D x < D n then D(x) else D n,
   703           %x. if D x < D n then p(x) else D n)"
   704 apply (frule_tac r = n in tpart_partition [THEN partition_ub])
   705 apply (drule_tac x = "D n" in order_le_imp_less_or_eq)
   706 apply (auto simp add: tpart_partition [THEN lemma_partition_eq, symmetric] tpart_tag_eq [symmetric])
   707 apply (frule_tac tpart_partition [THEN [3] lemma_additivity1])
   708 apply (auto simp add: tpart_def)
   709 apply (drule_tac [2] linorder_not_less [THEN iffD1, THEN order_le_imp_less_or_eq], auto)
   710   prefer 3 apply (drule_tac x=na in spec, arith)
   711  prefer 2 apply (blast dest: lemma_additivity3)
   712 apply (frule (2) lemma_additivity4_psize_eq)
   713 apply (rule partition [THEN iffD2])
   714 apply (frule partition [THEN iffD1])
   715 apply safe 
   716 apply (auto simp add: partition_lt_gen)  
   717 apply (drule (1) partition_lt_cancel, arith)
   718 done
   719 
   720 lemma fine_left1:
   721      "[| a \<le> D n; tpart (a, b) (D, p); gauge (%x. a \<le> x & x \<le> D n) g;
   722          fine (%x. if x < D n then min (g x) ((D n - x)/ 2)
   723                  else if x = D n then min (g (D n)) (ga (D n))
   724                       else min (ga x) ((x - D n)/ 2)) (D, p) |]
   725       ==> fine g
   726            (%x. if D x < D n then D(x) else D n,
   727             %x. if D x < D n then p(x) else D n)"
   728 apply (auto simp add: fine_def tpart_def gauge_def)
   729 apply (frule_tac [!] na=na in lemma_psize_left_less_psize2)
   730 apply (drule_tac [!] x = na in spec, auto)
   731 apply (drule_tac [!] x = na in spec, auto)
   732 apply (auto dest: lemma_additivity3a simp add: split_if_asm)
   733 done
   734 
   735 lemma tpart_right1:
   736      "[| a \<le> D n; tpart (a, b) (D, p) |]
   737       ==> tpart(D n, b) (%x. D(x + n),%x. p(x + n))"
   738 apply (simp add: tpart_def partition_def, safe)
   739 apply (rule_tac x = "N - n" in exI, auto)
   740 done
   741 
   742 lemma fine_right1:
   743      "[| a \<le> D n; tpart (a, b) (D, p); gauge (%x. D n \<le> x & x \<le> b) ga;
   744          fine (%x. if x < D n then min (g x) ((D n - x)/ 2)
   745                  else if x = D n then min (g (D n)) (ga (D n))
   746                       else min (ga x) ((x - D n)/ 2)) (D, p) |]
   747       ==> fine ga (%x. D(x + n),%x. p(x + n))"
   748 apply (auto simp add: fine_def gauge_def)
   749 apply (drule_tac x = "na + n" in spec)
   750 apply (frule_tac n = n in tpart_partition [THEN better_lemma_psize_right_eq], auto)
   751 apply (simp add: tpart_def, safe)
   752 apply (subgoal_tac "D n \<le> p (na + n)")
   753 apply (drule_tac y = "p (na + n)" in order_le_imp_less_or_eq)
   754 apply safe
   755 apply (simp split: split_if_asm, simp)
   756 apply (drule less_le_trans, assumption)
   757 apply (rotate_tac 5)
   758 apply (drule_tac x = "na + n" in spec, safe)
   759 apply (rule_tac y="D (na + n)" in order_trans)
   760 apply (case_tac "na = 0", auto)
   761 apply (erule partition_lt_gen [THEN order_less_imp_le])
   762 apply arith
   763 apply arith
   764 done
   765 
   766 lemma rsum_add: "rsum (D, p) (%x. f x + g x) =  rsum (D, p) f + rsum(D, p) g"
   767 by (simp add: rsum_def setsum_addf left_distrib)
   768 
   769 text{* Bartle/Sherbert: Theorem 10.1.5 p. 278 *}
   770 lemma Integral_add_fun:
   771     "[| a \<le> b; Integral(a,b) f k1; Integral(a,b) g k2 |]
   772      ==> Integral(a,b) (%x. f x + g x) (k1 + k2)"
   773 apply (simp add: Integral_def, auto)
   774 apply ((drule_tac x = "e/2" in spec)+)
   775 apply auto
   776 apply (drule gauge_min, assumption)
   777 apply (rule_tac x = " (%x. if ga x < gaa x then ga x else gaa x)" in exI)
   778 apply auto
   779 apply (drule fine_min)
   780 apply ((drule spec)+, auto)
   781 apply (drule_tac a = "\<bar>rsum (D, p) f - k1\<bar> * 2" and c = "\<bar>rsum (D, p) g - k2\<bar> * 2" in add_strict_mono, assumption)
   782 apply (auto simp only: rsum_add left_distrib [symmetric]
   783                 mult_2_right [symmetric] real_mult_less_iff1)
   784 done
   785 
   786 lemma partition_lt_gen2:
   787      "[| partition(a,b) D; r < psize D |] ==> 0 < D (Suc r) - D r"
   788 by (auto simp add: partition)
   789 
   790 lemma lemma_Integral_le:
   791      "[| \<forall>x. a \<le> x & x \<le> b --> f x \<le> g x;
   792          tpart(a,b) (D,p)
   793       |] ==> \<forall>n \<le> psize D. f (p n) \<le> g (p n)"
   794 apply (simp add: tpart_def)
   795 apply (auto, frule partition [THEN iffD1], auto)
   796 apply (drule_tac x = "p n" in spec, auto)
   797 apply (case_tac "n = 0", simp)
   798 apply (rule partition_lt_gen [THEN order_less_le_trans, THEN order_less_imp_le], auto)
   799 apply (drule le_imp_less_or_eq, auto)
   800 apply (drule_tac [2] x = "psize D" in spec, auto)
   801 apply (drule_tac r = "Suc n" in partition_ub)
   802 apply (drule_tac x = n in spec, auto)
   803 done
   804 
   805 lemma lemma_Integral_rsum_le:
   806      "[| \<forall>x. a \<le> x & x \<le> b --> f x \<le> g x;
   807          tpart(a,b) (D,p)
   808       |] ==> rsum(D,p) f \<le> rsum(D,p) g"
   809 apply (simp add: rsum_def)
   810 apply (auto intro!: setsum_mono dest: tpart_partition [THEN partition_lt_gen2]
   811                dest!: lemma_Integral_le)
   812 done
   813 
   814 lemma Integral_le:
   815     "[| a \<le> b;
   816         \<forall>x. a \<le> x & x \<le> b --> f(x) \<le> g(x);
   817         Integral(a,b) f k1; Integral(a,b) g k2
   818      |] ==> k1 \<le> k2"
   819 apply (simp add: Integral_def)
   820 apply (rotate_tac 2)
   821 apply (drule_tac x = "\<bar>k1 - k2\<bar> /2" in spec)
   822 apply (drule_tac x = "\<bar>k1 - k2\<bar> /2" in spec, auto)
   823 apply (drule gauge_min, assumption)
   824 apply (drule_tac g = "%x. if ga x < gaa x then ga x else gaa x" 
   825        in partition_exists, assumption, auto)
   826 apply (drule fine_min)
   827 apply (drule_tac x = D in spec, drule_tac x = D in spec)
   828 apply (drule_tac x = p in spec, drule_tac x = p in spec, auto)
   829 apply (frule lemma_Integral_rsum_le, assumption)
   830 apply (subgoal_tac "\<bar>(rsum (D,p) f - k1) - (rsum (D,p) g - k2)\<bar> < \<bar>k1 - k2\<bar>")
   831 apply arith
   832 apply (drule add_strict_mono, assumption)
   833 apply (auto simp only: left_distrib [symmetric] mult_2_right [symmetric]
   834                        real_mult_less_iff1)
   835 done
   836 
   837 lemma Integral_imp_Cauchy:
   838      "(\<exists>k. Integral(a,b) f k) ==>
   839       (\<forall>e > 0. \<exists>g. gauge (%x. a \<le> x & x \<le> b) g &
   840                        (\<forall>D1 D2 p1 p2.
   841                             tpart(a,b) (D1, p1) & fine g (D1,p1) &
   842                             tpart(a,b) (D2, p2) & fine g (D2,p2) -->
   843                             \<bar>rsum(D1,p1) f - rsum(D2,p2) f\<bar> < e))"
   844 apply (simp add: Integral_def, auto)
   845 apply (drule_tac x = "e/2" in spec, auto)
   846 apply (rule exI, auto)
   847 apply (frule_tac x = D1 in spec)
   848 apply (frule_tac x = D2 in spec)
   849 apply ((drule spec)+, auto)
   850 apply (erule_tac V = "0 < e" in thin_rl)
   851 apply (drule add_strict_mono, assumption)
   852 apply (auto simp only: left_distrib [symmetric] mult_2_right [symmetric]
   853                        real_mult_less_iff1)
   854 done
   855 
   856 lemma Cauchy_iff2:
   857      "Cauchy X =
   858       (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. \<bar>X m - X n\<bar> < inverse(real (Suc j))))"
   859 apply (simp add: Cauchy_def, auto)
   860 apply (drule reals_Archimedean, safe)
   861 apply (drule_tac x = n in spec, auto)
   862 apply (rule_tac x = M in exI, auto)
   863 apply (drule_tac x = m in spec, simp)
   864 apply (drule_tac x = na in spec, auto)
   865 done
   866 
   867 lemma partition_exists2:
   868      "[| a \<le> b; \<forall>n. gauge (%x. a \<le> x & x \<le> b) (fa n) |]
   869       ==> \<forall>n. \<exists>D p. tpart (a, b) (D, p) & fine (fa n) (D, p)"
   870 by (blast dest: partition_exists) 
   871 
   872 lemma monotonic_anti_derivative:
   873   fixes f g :: "real => real" shows
   874      "[| a \<le> b; \<forall>c. a \<le> c & c \<le> b --> f' c \<le> g' c;
   875          \<forall>x. DERIV f x :> f' x; \<forall>x. DERIV g x :> g' x |]
   876       ==> f b - f a \<le> g b - g a"
   877 apply (rule Integral_le, assumption)
   878 apply (auto intro: FTC1) 
   879 done
   880 
   881 end