src/HOL/Hyperreal/Integration.thy
author webertj
Wed Jul 26 19:23:04 2006 +0200 (2006-07-26)
changeset 20217 25b068a99d2b
parent 19765 dfe940911617
child 20256 5024ba0831a6
permissions -rw-r--r--
linear arithmetic splits certain operators (e.g. min, max, abs)
     1 (*  ID          : $Id$
     2     Author      : Jacques D. Fleuriot
     3     Copyright   : 2000  University of Edinburgh
     4     Conversion to Isar and new proofs by Lawrence C Paulson, 2004
     5 *)
     6 
     7 header{*Theory of Integration*}
     8 
     9 theory Integration
    10 imports MacLaurin
    11 begin
    12 
    13 text{*We follow John Harrison in formalizing the Gauge integral.*}
    14 
    15 definition
    16 
    17   --{*Partitions and tagged partitions etc.*}
    18 
    19   partition :: "[(real*real),nat => real] => bool"
    20   "partition = (%(a,b) D. D 0 = a &
    21                          (\<exists>N. (\<forall>n < N. D(n) < D(Suc n)) &
    22                               (\<forall>n \<ge> N. D(n) = b)))"
    23 
    24   psize :: "(nat => real) => nat"
    25   "psize D = (SOME N. (\<forall>n < N. D(n) < D(Suc n)) &
    26                       (\<forall>n \<ge> N. D(n) = D(N)))"
    27 
    28   tpart :: "[(real*real),((nat => real)*(nat =>real))] => bool"
    29   "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   gauge :: "[real => bool, real => real] => bool"
    35   "gauge E g = (\<forall>x. E x --> 0 < g(x))"
    36 
    37   fine :: "[real => real, ((nat => real)*(nat => real))] => bool"
    38   "fine = (%g (D,p). \<forall>n. n < (psize D) --> D(Suc n) - D(n) < g(p n))"
    39 
    40   --{*Riemann sum*}
    41 
    42   rsum :: "[((nat=>real)*(nat=>real)),real=>real] => real"
    43   "rsum = (%(D,p) f. \<Sum>n=0..<psize(D). f(p n) * (D(Suc n) - D(n)))"
    44 
    45   --{*Gauge integrability (definite)*}
    46 
    47   Integral :: "[(real*real),real=>real,real] => bool"
    48   "Integral = (%(a,b) f k. \<forall>e > 0.
    49                                (\<exists>g. gauge(%x. a \<le> x & x \<le> b) g &
    50                                (\<forall>D p. tpart(a,b) (D,p) & fine(g)(D,p) -->
    51                                          \<bar>rsum(D,p) f - k\<bar> < e)))"
    52 
    53 
    54 lemma partition_zero [simp]: "a = b ==> psize (%n. if n = 0 then a else b) = 0"
    55 by (auto simp add: psize_def)
    56 
    57 lemma partition_one [simp]: "a < b ==> psize (%n. if n = 0 then a else b) = 1"
    58 apply (simp add: psize_def)
    59 apply (rule some_equality, auto)
    60 apply (drule_tac x = 1 in spec, auto)
    61 done
    62 
    63 lemma partition_single [simp]:
    64      "a \<le> b ==> partition(a,b)(%n. if n = 0 then a else b)"
    65 by (auto simp add: partition_def order_le_less)
    66 
    67 lemma partition_lhs: "partition(a,b) D ==> (D(0) = a)"
    68 by (simp add: partition_def)
    69 
    70 lemma partition:
    71        "(partition(a,b) D) =
    72         ((D 0 = a) &
    73          (\<forall>n < psize D. D n < D(Suc n)) &
    74          (\<forall>n \<ge> psize D. D n = b))"
    75 apply (simp add: partition_def, auto)
    76 apply (subgoal_tac [!] "psize D = N", auto)
    77 apply (simp_all (no_asm) add: psize_def)
    78 apply (rule_tac [!] some_equality, blast)
    79   prefer 2 apply blast
    80 apply (rule_tac [!] ccontr)
    81 apply (simp_all add: linorder_neq_iff, safe)
    82 apply (drule_tac x = Na in spec)
    83 apply (rotate_tac 3)
    84 apply (drule_tac x = "Suc Na" in spec, simp)
    85 apply (rotate_tac 2)
    86 apply (drule_tac x = N in spec, simp)
    87 apply (drule_tac x = Na in spec)
    88 apply (drule_tac x = "Suc Na" and P = "%n. Na\<le>n \<longrightarrow> D n = D Na" in spec, auto)
    89 done
    90 
    91 lemma partition_rhs: "partition(a,b) D ==> (D(psize D) = b)"
    92 by (simp add: partition)
    93 
    94 lemma partition_rhs2: "[|partition(a,b) D; psize D \<le> n |] ==> (D n = b)"
    95 by (simp add: partition)
    96 
    97 lemma lemma_partition_lt_gen [rule_format]:
    98  "partition(a,b) D & m + Suc d \<le> n & n \<le> (psize D) --> D(m) < D(m + Suc d)"
    99 apply (induct "d", auto simp add: partition)
   100 apply (blast dest: Suc_le_lessD  intro: less_le_trans order_less_trans)
   101 done
   102 
   103 lemma less_eq_add_Suc: "m < n ==> \<exists>d. n = m + Suc d"
   104 by (auto simp add: less_iff_Suc_add)
   105 
   106 lemma partition_lt_gen:
   107      "[|partition(a,b) D; m < n; n \<le> (psize D)|] ==> D(m) < D(n)"
   108 by (auto dest: less_eq_add_Suc intro: lemma_partition_lt_gen)
   109 
   110 lemma partition_lt: "partition(a,b) D ==> n < (psize D) ==> D(0) < D(Suc n)"
   111 apply (induct "n")
   112 apply (auto simp add: partition)
   113 done
   114 
   115 lemma partition_le: "partition(a,b) D ==> a \<le> b"
   116 apply (frule partition [THEN iffD1], safe)
   117 apply (drule_tac x = "psize D" and P="%n. psize D \<le> n --> ?P n" in spec, safe)
   118 apply (case_tac "psize D = 0")
   119 apply (drule_tac [2] n = "psize D - 1" in partition_lt, auto)
   120 done
   121 
   122 lemma partition_gt: "[|partition(a,b) D; n < (psize D)|] ==> D(n) < D(psize D)"
   123 by (auto intro: partition_lt_gen)
   124 
   125 lemma partition_eq: "partition(a,b) D ==> ((a = b) = (psize D = 0))"
   126 apply (frule partition [THEN iffD1], safe)
   127 apply (rotate_tac 2)
   128 apply (drule_tac x = "psize D" in spec)
   129 apply (rule ccontr)
   130 apply (drule_tac n = "psize D - 1" in partition_lt)
   131 apply auto
   132 done
   133 
   134 lemma partition_lb: "partition(a,b) D ==> a \<le> D(r)"
   135 apply (frule partition [THEN iffD1], safe)
   136 apply (induct "r")
   137 apply (cut_tac [2] y = "Suc r" and x = "psize D" in linorder_le_less_linear)
   138 apply (auto intro: partition_le)
   139 apply (drule_tac x = r in spec)
   140 apply arith; 
   141 done
   142 
   143 lemma partition_lb_lt: "[| partition(a,b) D; psize D ~= 0 |] ==> a < D(Suc n)"
   144 apply (rule_tac t = a in partition_lhs [THEN subst], assumption)
   145 apply (cut_tac x = "Suc n" and y = "psize D" in linorder_le_less_linear)
   146 apply (frule partition [THEN iffD1], safe)
   147  apply (blast intro: partition_lt less_le_trans)
   148 apply (rotate_tac 3)
   149 apply (drule_tac x = "Suc n" in spec)
   150 apply (erule impE)
   151 apply (erule less_imp_le)
   152 apply (frule partition_rhs)
   153 apply (drule partition_gt, assumption)
   154 apply (simp (no_asm_simp))
   155 done
   156 
   157 lemma partition_ub: "partition(a,b) D ==> D(r) \<le> b"
   158 apply (frule partition [THEN iffD1])
   159 apply (cut_tac x = "psize D" and y = r in linorder_le_less_linear, safe, blast)
   160 apply (subgoal_tac "\<forall>x. D ((psize D) - x) \<le> b")
   161 apply (rotate_tac 4)
   162 apply (drule_tac x = "psize D - r" in spec)
   163 apply (subgoal_tac "psize D - (psize D - r) = r")
   164 apply simp
   165 apply arith
   166 apply safe
   167 apply (induct_tac "x")
   168 apply (simp (no_asm), blast)
   169 apply (case_tac "psize D - Suc n = 0")
   170 apply (erule_tac V = "\<forall>n. psize D \<le> n --> D n = b" in thin_rl)
   171 apply (simp (no_asm_simp) add: partition_le)
   172 apply (rule order_trans)
   173  prefer 2 apply assumption
   174 apply (subgoal_tac "psize D - n = Suc (psize D - Suc n)")
   175  prefer 2 apply arith
   176 apply (drule_tac x = "psize D - Suc n" in spec, simp) 
   177 done
   178 
   179 lemma partition_ub_lt: "[| partition(a,b) D; n < psize D |] ==> D(n) < b"
   180 by (blast intro: partition_rhs [THEN subst] partition_gt)
   181 
   182 lemma lemma_partition_append1:
   183      "[| partition (a, b) D1; partition (b, c) D2 |]
   184        ==> (\<forall>n < psize D1 + psize D2.
   185              (if n < psize D1 then D1 n else D2 (n - psize D1))
   186              < (if Suc n < psize D1 then D1 (Suc n)
   187                 else D2 (Suc n - psize D1))) &
   188          (\<forall>n \<ge> psize D1 + psize D2.
   189              (if n < psize D1 then D1 n else D2 (n - psize D1)) =
   190              (if psize D1 + psize D2 < psize D1 then D1 (psize D1 + psize D2)
   191               else D2 (psize D1 + psize D2 - psize D1)))"
   192 apply (auto intro: partition_lt_gen)
   193 apply (subgoal_tac "psize D1 = Suc n")
   194 apply (auto intro!: partition_lt_gen simp add: partition_lhs partition_ub_lt)
   195 apply (auto intro!: partition_rhs2 simp add: partition_rhs
   196             split: nat_diff_split)
   197 done
   198 
   199 lemma lemma_psize1:
   200      "[| partition (a, b) D1; partition (b, c) D2; N < psize D1 |]
   201       ==> D1(N) < D2 (psize D2)"
   202 apply (rule_tac y = "D1 (psize D1)" in order_less_le_trans)
   203 apply (erule partition_gt)
   204 apply (auto simp add: partition_rhs partition_le)
   205 done
   206 
   207 lemma lemma_partition_append2:
   208      "[| partition (a, b) D1; partition (b, c) D2 |]
   209       ==> psize (%n. if n < psize D1 then D1 n else D2 (n - psize D1)) =
   210           psize D1 + psize D2" 
   211 apply (unfold psize_def 
   212          [of "%n. if n < psize D1 then D1 n else D2 (n - psize D1)"])
   213 apply (rule some1_equality)
   214  prefer 2 apply (blast intro!: lemma_partition_append1)
   215 apply (rule ex1I, rule lemma_partition_append1) 
   216 apply (simp_all split: split_if_asm)
   217  txt{*The case @{term "N < psize D1"}*} 
   218  apply (drule_tac x = "psize D1 + psize D2" and P="%n. ?P n & ?Q n" in spec) 
   219  apply (force dest: lemma_psize1)
   220 apply (rule order_antisym);
   221  txt{*The case @{term "psize D1 \<le> N"}: 
   222        proving @{term "N \<le> psize D1 + psize D2"}*}
   223  apply (drule_tac x = "psize D1 + psize D2" in spec)
   224  apply (simp add: partition_rhs2)
   225 apply (case_tac "N - psize D1 < psize D2") 
   226  prefer 2 apply arith
   227  txt{*Proving @{term "psize D1 + psize D2 \<le> N"}*}
   228 apply (drule_tac x = "psize D1 + psize D2" and P="%n. N\<le>n --> ?P n" in spec, simp)
   229 apply (drule_tac a = b and b = c in partition_gt, assumption, simp)
   230 done
   231 
   232 lemma tpart_eq_lhs_rhs: "[|psize D = 0; tpart(a,b) (D,p)|] ==> a = b"
   233 by (auto simp add: tpart_def partition_eq)
   234 
   235 lemma tpart_partition: "tpart(a,b) (D,p) ==> partition(a,b) D"
   236 by (simp add: tpart_def)
   237 
   238 lemma partition_append:
   239      "[| tpart(a,b) (D1,p1); fine(g) (D1,p1);
   240          tpart(b,c) (D2,p2); fine(g) (D2,p2) |]
   241        ==> \<exists>D p. tpart(a,c) (D,p) & fine(g) (D,p)"
   242 apply (rule_tac x = "%n. if n < psize D1 then D1 n else D2 (n - psize D1)"
   243        in exI)
   244 apply (rule_tac x = "%n. if n < psize D1 then p1 n else p2 (n - psize D1)"
   245        in exI)
   246 apply (case_tac "psize D1 = 0")
   247 apply (auto dest: tpart_eq_lhs_rhs)
   248  prefer 2
   249 apply (simp add: fine_def
   250                  lemma_partition_append2 [OF tpart_partition tpart_partition])
   251   --{*But must not expand @{term fine} in other subgoals*}
   252 apply auto
   253 apply (subgoal_tac "psize D1 = Suc n")
   254  prefer 2 apply arith
   255 apply (drule tpart_partition [THEN partition_rhs])
   256 apply (drule tpart_partition [THEN partition_lhs])
   257 apply (auto split: nat_diff_split)
   258 apply (auto simp add: tpart_def)
   259 defer 1
   260  apply (subgoal_tac "psize D1 = Suc n")
   261   prefer 2 apply arith
   262  apply (drule partition_rhs)
   263  apply (drule partition_lhs, auto)
   264 apply (simp split: nat_diff_split)
   265 apply (subst partition) 
   266 apply (subst (1 2) lemma_partition_append2, assumption+)
   267 apply (rule conjI) 
   268 apply (simp add: partition_lhs)
   269 apply (drule lemma_partition_append1)
   270 apply assumption; 
   271 apply (simp add: partition_rhs)
   272 done
   273 
   274 
   275 text{*We can always find a division that is fine wrt any gauge*}
   276 
   277 lemma partition_exists:
   278      "[| a \<le> b; gauge(%x. a \<le> x & x \<le> b) g |]
   279       ==> \<exists>D p. tpart(a,b) (D,p) & fine g (D,p)"
   280 apply (cut_tac P = "%(u,v). a \<le> u & v \<le> b --> 
   281                    (\<exists>D p. tpart (u,v) (D,p) & fine (g) (D,p))" 
   282        in lemma_BOLZANO2)
   283 apply safe
   284 apply (blast intro: order_trans)+
   285 apply (auto intro: partition_append)
   286 apply (case_tac "a \<le> x & x \<le> b")
   287 apply (rule_tac [2] x = 1 in exI, auto)
   288 apply (rule_tac x = "g x" in exI)
   289 apply (auto simp add: gauge_def)
   290 apply (rule_tac x = "%n. if n = 0 then aa else ba" in exI)
   291 apply (rule_tac x = "%n. if n = 0 then x else ba" in exI)
   292 apply (auto simp add: tpart_def fine_def)
   293 done
   294 
   295 text{*Lemmas about combining gauges*}
   296 
   297 lemma gauge_min:
   298      "[| gauge(E) g1; gauge(E) g2 |]
   299       ==> gauge(E) (%x. if g1(x) < g2(x) then g1(x) else g2(x))"
   300 by (simp add: gauge_def)
   301 
   302 lemma fine_min:
   303       "fine (%x. if g1(x) < g2(x) then g1(x) else g2(x)) (D,p)
   304        ==> fine(g1) (D,p) & fine(g2) (D,p)"
   305 by (auto simp add: fine_def split: split_if_asm)
   306 
   307 
   308 text{*The integral is unique if it exists*}
   309 
   310 lemma Integral_unique:
   311     "[| a \<le> b; Integral(a,b) f k1; Integral(a,b) f k2 |] ==> k1 = k2"
   312 apply (simp add: Integral_def)
   313 apply (drule_tac x = "\<bar>k1 - k2\<bar> /2" in spec)+
   314 apply auto
   315 apply (drule gauge_min, assumption)
   316 apply (drule_tac g = "%x. if g x < ga x then g x else ga x" 
   317        in partition_exists, assumption, auto)
   318 apply (drule fine_min)
   319 apply (drule spec)+
   320 apply auto
   321 apply (subgoal_tac "\<bar>(rsum (D,p) f - k2) - (rsum (D,p) f - k1)\<bar> < \<bar>k1 - k2\<bar>")
   322 apply arith
   323 apply (drule add_strict_mono, assumption)
   324 apply (auto simp only: left_distrib [symmetric] mult_2_right [symmetric] 
   325                 mult_less_cancel_right)
   326 done
   327 
   328 lemma Integral_zero [simp]: "Integral(a,a) f 0"
   329 apply (auto simp add: Integral_def)
   330 apply (rule_tac x = "%x. 1" in exI)
   331 apply (auto dest: partition_eq simp add: gauge_def tpart_def rsum_def)
   332 done
   333 
   334 lemma sumr_partition_eq_diff_bounds [simp]:
   335      "(\<Sum>n=0..<m. D (Suc n) - D n::real) = D(m) - D 0"
   336 by (induct "m", auto)
   337 
   338 lemma Integral_eq_diff_bounds: "a \<le> b ==> Integral(a,b) (%x. 1) (b - a)"
   339 apply (auto simp add: order_le_less rsum_def Integral_def)
   340 apply (rule_tac x = "%x. b - a" in exI)
   341 apply (auto simp add: gauge_def abs_interval_iff tpart_def partition)
   342 done
   343 
   344 lemma Integral_mult_const: "a \<le> b ==> Integral(a,b) (%x. c)  (c*(b - a))"
   345 apply (auto simp add: order_le_less rsum_def Integral_def)
   346 apply (rule_tac x = "%x. b - a" in exI)
   347 apply (auto simp add: setsum_right_distrib [symmetric] gauge_def abs_interval_iff 
   348                right_diff_distrib [symmetric] partition tpart_def)
   349 done
   350 
   351 lemma Integral_mult:
   352      "[| a \<le> b; Integral(a,b) f k |] ==> Integral(a,b) (%x. c * f x) (c * k)"
   353 apply (auto simp add: order_le_less 
   354             dest: Integral_unique [OF order_refl Integral_zero])
   355 apply (auto simp add: rsum_def Integral_def setsum_right_distrib[symmetric] mult_assoc)
   356 apply (rule_tac a2 = c in abs_ge_zero [THEN real_le_imp_less_or_eq, THEN disjE])
   357  prefer 2 apply force
   358 apply (drule_tac x = "e/abs c" in spec, auto)
   359 apply (simp add: zero_less_mult_iff divide_inverse)
   360 apply (rule exI, auto)
   361 apply (drule spec)+
   362 apply auto
   363 apply (rule_tac z1 = "inverse (abs c)" in real_mult_less_iff1 [THEN iffD1])
   364 apply (auto simp add: abs_mult divide_inverse [symmetric] right_diff_distrib [symmetric])
   365 done
   366 
   367 text{*Fundamental theorem of calculus (Part I)*}
   368 
   369 text{*"Straddle Lemma" : Swartz and Thompson: AMM 95(7) 1988 *}
   370 
   371 lemma choiceP: "\<forall>x. P(x) --> (\<exists>y. Q x y) ==> \<exists>f. (\<forall>x. P(x) --> Q x (f x))" 
   372 by (insert bchoice [of "Collect P" Q], simp) 
   373 
   374 (*UNUSED
   375 lemma choice2: "\<forall>x. (\<exists>y. R(y) & (\<exists>z. Q x y z)) ==>
   376       \<exists>f fa. (\<forall>x. R(f x) & Q x (f x) (fa x))"
   377 *)
   378 
   379 
   380 (* new simplifications e.g. (y < x/n) = (y * n < x) are a real nuisance
   381    they break the original proofs and make new proofs longer!*)
   382 lemma strad1:
   383        "\<lbrakk>\<forall>xa::real. xa \<noteq> x \<and> \<bar>xa + - x\<bar> < s \<longrightarrow>
   384              \<bar>(f xa - f x) / (xa - x) + - f' x\<bar> * 2 < e;
   385         0 < e; a \<le> x; x \<le> b; 0 < s\<rbrakk>
   386        \<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>"
   387 apply auto
   388 apply (case_tac "0 < \<bar>z - x\<bar>")
   389  prefer 2 apply (simp add: zero_less_abs_iff)
   390 apply (drule_tac x = z in spec)
   391 apply (rule_tac z1 = "\<bar>inverse (z - x)\<bar>" 
   392        in real_mult_le_cancel_iff2 [THEN iffD1])
   393  apply simp
   394 apply (simp del: abs_inverse abs_mult add: abs_mult [symmetric]
   395           mult_assoc [symmetric])
   396 apply (subgoal_tac "inverse (z - x) * (f z - f x - f' x * (z - x)) 
   397                     = (f z - f x) / (z - x) - f' x")
   398  apply (simp add: abs_mult [symmetric] mult_ac diff_minus)
   399 apply (subst mult_commute)
   400 apply (simp add: left_distrib diff_minus)
   401 apply (simp add: mult_assoc divide_inverse)
   402 apply (simp add: left_distrib)
   403 done
   404 
   405 lemma lemma_straddle:
   406      "[| \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x); 0 < e |]
   407       ==> \<exists>g. gauge(%x. a \<le> x & x \<le> b) g &
   408                 (\<forall>x u v. a \<le> u & u \<le> x & x \<le> v & v \<le> b & (v - u) < g(x)
   409                   --> \<bar>(f(v) - f(u)) - (f'(x) * (v - u))\<bar> \<le> e * (v - u))"
   410 apply (simp add: gauge_def)
   411 apply (subgoal_tac "\<forall>x. a \<le> x & x \<le> b --> 
   412         (\<exists>d > 0. \<forall>u v. u \<le> x & x \<le> v & (v - u) < d --> 
   413                        \<bar>(f (v) - f (u)) - (f' (x) * (v - u))\<bar> \<le> e * (v - u))")
   414 apply (drule choiceP, auto)
   415 apply (drule spec, auto)
   416 apply (auto simp add: DERIV_iff2 LIM_def)
   417 apply (drule_tac x = "e/2" in spec, auto)
   418 apply (frule strad1, assumption+)
   419 apply (rule_tac x = s in exI, auto)
   420 apply (rule_tac x = u and y = v in linorder_cases, auto)
   421 apply (rule_tac y = "\<bar>(f (v) - f (x)) - (f' (x) * (v - x))\<bar> + 
   422                      \<bar>(f (x) - f (u)) - (f' (x) * (x - u))\<bar>"
   423        in order_trans)
   424 apply (rule abs_triangle_ineq [THEN [2] order_trans])
   425 apply (simp add: right_diff_distrib)
   426 apply (rule_tac t = "e* (v - u)" in real_sum_of_halves [THEN subst])
   427 apply (rule add_mono)
   428 apply (rule_tac y = "(e/2) * \<bar>v - x\<bar>" in order_trans)
   429  prefer 2 apply simp
   430 apply (erule_tac [!] V= "\<forall>x'. x' ~= x & \<bar>x' + - x\<bar> < s --> ?P x'" in thin_rl)
   431 apply (drule_tac x = v in spec, simp add: times_divide_eq)
   432 apply (drule_tac x = u in spec, auto)
   433 apply (subgoal_tac "\<bar>f u - f x - f' x * (u - x)\<bar> = \<bar>f x - f u - f' x * (x - u)\<bar>")
   434 apply (rule order_trans)
   435 apply (auto simp add: abs_le_interval_iff)
   436 apply (simp add: right_diff_distrib)
   437 done
   438 
   439 lemma FTC1: "[|a \<le> b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
   440              ==> Integral(a,b) f' (f(b) - f(a))"
   441 apply (drule order_le_imp_less_or_eq, auto) 
   442 apply (auto simp add: Integral_def)
   443 apply (rule ccontr)
   444 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)")
   445 apply (rotate_tac 3)
   446 apply (drule_tac x = "e/2" in spec, auto)
   447 apply (drule spec, auto)
   448 apply ((drule spec)+, auto)
   449 apply (drule_tac e = "ea/ (b - a)" in lemma_straddle)
   450 apply (auto simp add: zero_less_divide_iff)
   451 apply (rule exI)
   452 apply (auto simp add: tpart_def rsum_def)
   453 apply (subgoal_tac "(\<Sum>n=0..<psize D. f(D(Suc n)) - f(D n)) = f b - f a")
   454  prefer 2
   455  apply (cut_tac D = "%n. f (D n)" and m = "psize D"
   456         in sumr_partition_eq_diff_bounds)
   457  apply (simp add: partition_lhs partition_rhs)
   458 apply (drule sym, simp)
   459 apply (simp (no_asm) add: setsum_subtractf[symmetric])
   460 apply (rule setsum_abs [THEN order_trans])
   461 apply (subgoal_tac "ea = (\<Sum>n=0..<psize D. (ea / (b - a)) * (D (Suc n) - (D n)))")
   462 apply (simp add: abs_minus_commute)
   463 apply (rule_tac t = ea in ssubst, assumption)
   464 apply (rule setsum_mono)
   465 apply (rule_tac [2] setsum_right_distrib [THEN subst])
   466 apply (auto simp add: partition_rhs partition_lhs partition_lb partition_ub
   467           fine_def)
   468 done
   469 
   470 
   471 lemma Integral_subst: "[| Integral(a,b) f k1; k2=k1 |] ==> Integral(a,b) f k2"
   472 by simp
   473 
   474 lemma Integral_add:
   475      "[| a \<le> b; b \<le> c; Integral(a,b) f' k1; Integral(b,c) f' k2;
   476          \<forall>x. a \<le> x & x \<le> c --> DERIV f x :> f' x |]
   477      ==> Integral(a,c) f' (k1 + k2)"
   478 apply (rule FTC1 [THEN Integral_subst], auto)
   479 apply (frule FTC1, auto)
   480 apply (frule_tac a = b in FTC1, auto)
   481 apply (drule_tac x = x in spec, auto)
   482 apply (drule_tac ?k2.0 = "f b - f a" in Integral_unique)
   483 apply (drule_tac [3] ?k2.0 = "f c - f b" in Integral_unique, auto)
   484 done
   485 
   486 lemma partition_psize_Least:
   487      "partition(a,b) D ==> psize D = (LEAST n. D(n) = b)"
   488 apply (auto intro!: Least_equality [symmetric] partition_rhs)
   489 apply (auto dest: partition_ub_lt simp add: linorder_not_less [symmetric])
   490 done
   491 
   492 lemma lemma_partition_bounded: "partition (a, c) D ==> ~ (\<exists>n. c < D(n))"
   493 apply safe
   494 apply (drule_tac r = n in partition_ub, auto)
   495 done
   496 
   497 lemma lemma_partition_eq:
   498      "partition (a, c) D ==> D = (%n. if D n < c then D n else c)"
   499 apply (rule ext, auto)
   500 apply (auto dest!: lemma_partition_bounded)
   501 apply (drule_tac x = n in spec, auto)
   502 done
   503 
   504 lemma lemma_partition_eq2:
   505      "partition (a, c) D ==> D = (%n. if D n \<le> c then D n else c)"
   506 apply (rule ext, auto)
   507 apply (auto dest!: lemma_partition_bounded)
   508 apply (drule_tac x = n in spec, auto)
   509 done
   510 
   511 lemma partition_lt_Suc:
   512      "[| partition(a,b) D; n < psize D |] ==> D n < D (Suc n)"
   513 by (auto simp add: partition)
   514 
   515 lemma tpart_tag_eq: "tpart(a,c) (D,p) ==> p = (%n. if D n < c then p n else c)"
   516 apply (rule ext)
   517 apply (auto simp add: tpart_def)
   518 apply (drule linorder_not_less [THEN iffD1])
   519 apply (drule_tac r = "Suc n" in partition_ub)
   520 apply (drule_tac x = n in spec, auto)
   521 done
   522 
   523 subsection{*Lemmas for Additivity Theorem of Gauge Integral*}
   524 
   525 lemma lemma_additivity1:
   526      "[| a \<le> D n; D n < b; partition(a,b) D |] ==> n < psize D"
   527 by (auto simp add: partition linorder_not_less [symmetric])
   528 
   529 lemma lemma_additivity2: "[| a \<le> D n; partition(a,D n) D |] ==> psize D \<le> n"
   530 apply (rule ccontr, drule not_leE)
   531 apply (frule partition [THEN iffD1], safe)
   532 apply (frule_tac r = "Suc n" in partition_ub)
   533 apply (auto dest!: spec)
   534 done
   535 
   536 lemma partition_eq_bound:
   537      "[| partition(a,b) D; psize D < m |] ==> D(m) = D(psize D)"
   538 by (auto simp add: partition)
   539 
   540 lemma partition_ub2: "[| partition(a,b) D; psize D < m |] ==> D(r) \<le> D(m)"
   541 by (simp add: partition partition_ub)
   542 
   543 lemma tag_point_eq_partition_point:
   544     "[| tpart(a,b) (D,p); psize D \<le> m |] ==> p(m) = D(m)"
   545 apply (simp add: tpart_def, auto)
   546 apply (drule_tac x = m in spec)
   547 apply (auto simp add: partition_rhs2)
   548 done
   549 
   550 lemma partition_lt_cancel: "[| partition(a,b) D; D m < D n |] ==> m < n"
   551 apply (cut_tac m = n and n = "psize D" in less_linear, auto)
   552 apply (cut_tac m = m and n = n in less_linear)
   553 apply (cut_tac m = m and n = "psize D" in less_linear)
   554 apply (auto dest: partition_gt)
   555 apply (drule_tac n = m in partition_lt_gen, auto)
   556 apply (frule partition_eq_bound)
   557 apply (drule_tac [2] partition_gt, auto)
   558 apply (rule ccontr, drule leI, drule le_imp_less_or_eq)
   559 apply (auto dest: partition_eq_bound)
   560 apply (rule ccontr, drule leI, drule le_imp_less_or_eq)
   561 apply (frule partition_eq_bound, assumption)
   562 apply (drule_tac m = m in partition_eq_bound, auto)
   563 done
   564 
   565 lemma lemma_additivity4_psize_eq:
   566      "[| a \<le> D n; D n < b; partition (a, b) D |]
   567       ==> psize (%x. if D x < D n then D(x) else D n) = n"
   568 apply (unfold psize_def)
   569 apply (frule lemma_additivity1)
   570 apply (assumption, assumption)
   571 apply (rule some_equality)
   572 apply (auto intro: partition_lt_Suc)
   573 apply (drule_tac n = n in partition_lt_gen, assumption)
   574 apply (arith, arith)
   575 apply (cut_tac m = na and n = "psize D" in less_linear)
   576 apply (auto dest: partition_lt_cancel)
   577 apply (rule_tac x=N and y=n in linorder_cases)
   578 apply (drule_tac x = n and P="%m. N \<le> m --> ?f m = ?g m" in spec, simp)
   579 apply (drule_tac n = n in partition_lt_gen, auto)
   580 apply (drule_tac x = n in spec)
   581 apply (simp split: split_if_asm)
   582 done
   583 
   584 lemma lemma_psize_left_less_psize:
   585      "partition (a, b) D
   586       ==> psize (%x. if D x < D n then D(x) else D n) \<le> psize D"
   587 apply (frule_tac r = n in partition_ub)
   588 apply (drule_tac x = "D n" in order_le_imp_less_or_eq)
   589 apply (auto simp add: lemma_partition_eq [symmetric])
   590 apply (frule_tac r = n in partition_lb)
   591 apply (drule (2) lemma_additivity4_psize_eq)  
   592 apply (rule ccontr, auto)
   593 apply (frule_tac not_leE [THEN [2] partition_eq_bound])
   594 apply (auto simp add: partition_rhs)
   595 done
   596 
   597 lemma lemma_psize_left_less_psize2:
   598      "[| partition(a,b) D; na < psize (%x. if D x < D n then D(x) else D n) |]
   599       ==> na < psize D"
   600 by (erule lemma_psize_left_less_psize [THEN [2] less_le_trans])
   601 
   602 
   603 lemma lemma_additivity3:
   604      "[| partition(a,b) D; D na < D n; D n < D (Suc na);
   605          n < psize D |]
   606       ==> False"
   607 apply (cut_tac m = n and n = "Suc na" in less_linear, auto)
   608 apply (drule_tac [2] n = n in partition_lt_gen, auto)
   609 apply (cut_tac m = "psize D" and n = na in less_linear)
   610 apply (auto simp add: partition_rhs2 less_Suc_eq)
   611 apply (drule_tac n = na in partition_lt_gen, auto)
   612 done
   613 
   614 lemma psize_const [simp]: "psize (%x. k) = 0"
   615 by (auto simp add: psize_def)
   616 
   617 lemma lemma_additivity3a:
   618      "[| partition(a,b) D; D na < D n; D n < D (Suc na);
   619          na < psize D |]
   620       ==> False"
   621 apply (frule_tac m = n in partition_lt_cancel)
   622 apply (auto intro: lemma_additivity3)
   623 done
   624 
   625 lemma better_lemma_psize_right_eq1:
   626      "[| partition(a,b) D; D n < b |] ==> psize (%x. D (x + n)) \<le> psize D - n"
   627 apply (simp add: psize_def [of "(%x. D (x + n))"]);
   628 apply (rule_tac a = "psize D - n" in someI2, auto)
   629   apply (simp add: partition less_diff_conv)
   630  apply (simp add: le_diff_conv partition_rhs2 split: nat_diff_split)
   631 apply (drule_tac x = "psize D - n" in spec, auto)
   632 apply (frule partition_rhs, safe)
   633 apply (frule partition_lt_cancel, assumption)
   634 apply (drule partition [THEN iffD1], safe)
   635 apply (subgoal_tac "~ D (psize D - n + n) < D (Suc (psize D - n + n))")
   636  apply blast
   637 apply (drule_tac x = "Suc (psize D)" and P="%n. ?P n \<longrightarrow> D n = D (psize D)"
   638        in spec)
   639 apply simp
   640 done
   641 
   642 lemma psize_le_n: "partition (a, D n) D ==> psize D \<le> n" 
   643 apply (rule ccontr, drule not_leE)
   644 apply (frule partition_lt_Suc, assumption)
   645 apply (frule_tac r = "Suc n" in partition_ub, auto)
   646 done
   647 
   648 lemma better_lemma_psize_right_eq1a:
   649      "partition(a,D n) D ==> psize (%x. D (x + n)) \<le> psize D - n"
   650 apply (simp add: psize_def [of "(%x. D (x + n))"]);
   651 apply (rule_tac a = "psize D - n" in someI2, auto)
   652   apply (simp add: partition less_diff_conv)
   653  apply (simp add: le_diff_conv)
   654 apply (case_tac "psize D \<le> n")
   655   apply (force intro: partition_rhs2)
   656  apply (simp add: partition linorder_not_le)
   657 apply (rule ccontr, drule not_leE)
   658 apply (frule psize_le_n)
   659 apply (drule_tac x = "psize D - n" in spec, simp)
   660 apply (drule partition [THEN iffD1], safe)
   661 apply (drule_tac x = "Suc n" and P="%na. ?s \<le> na \<longrightarrow> D na = D n" in spec, auto)
   662 done
   663 
   664 lemma better_lemma_psize_right_eq:
   665      "partition(a,b) D ==> psize (%x. D (x + n)) \<le> psize D - n"
   666 apply (frule_tac r1 = n in partition_ub [THEN order_le_imp_less_or_eq])
   667 apply (blast intro: better_lemma_psize_right_eq1a better_lemma_psize_right_eq1)
   668 done
   669 
   670 lemma lemma_psize_right_eq1:
   671      "[| partition(a,b) D; D n < b |] ==> psize (%x. D (x + n)) \<le> psize D"
   672 apply (simp add: psize_def [of "(%x. D (x + n))"])
   673 apply (rule_tac a = "psize D - n" in someI2, auto)
   674   apply (simp add: partition less_diff_conv)
   675  apply (subgoal_tac "n \<le> psize D")
   676   apply (simp add: partition le_diff_conv)
   677  apply (rule ccontr, drule not_leE)
   678  apply (drule_tac less_imp_le [THEN [2] partition_rhs2], assumption, simp)
   679 apply (drule_tac x = "psize D" in spec)
   680 apply (simp add: partition)
   681 done
   682 
   683 (* should be combined with previous theorem; also proof has redundancy *)
   684 lemma lemma_psize_right_eq1a:
   685      "partition(a,D n) D ==> psize (%x. D (x + n)) \<le> psize D"
   686 apply (simp add: psize_def [of "(%x. D (x + n))"]);
   687 apply (rule_tac a = "psize D - n" in someI2, auto)
   688   apply (simp add: partition less_diff_conv)
   689  apply (case_tac "psize D \<le> n")
   690   apply (force intro: partition_rhs2 simp add: le_diff_conv)
   691  apply (simp add: partition le_diff_conv)
   692 apply (rule ccontr, drule not_leE)
   693 apply (drule_tac x = "psize D" in spec)
   694 apply (simp add: partition)
   695 done
   696 
   697 lemma lemma_psize_right_eq:
   698      "[| partition(a,b) D |] ==> psize (%x. D (x + n)) \<le> psize D"
   699 apply (frule_tac r1 = n in partition_ub [THEN order_le_imp_less_or_eq])
   700 apply (blast intro: lemma_psize_right_eq1a lemma_psize_right_eq1)
   701 done
   702 
   703 lemma tpart_left1:
   704      "[| a \<le> D n; tpart (a, b) (D, p) |]
   705       ==> tpart(a, D n) (%x. if D x < D n then D(x) else D n,
   706           %x. if D x < D n then p(x) else D n)"
   707 apply (frule_tac r = n in tpart_partition [THEN partition_ub])
   708 apply (drule_tac x = "D n" in order_le_imp_less_or_eq)
   709 apply (auto simp add: tpart_partition [THEN lemma_partition_eq, symmetric] tpart_tag_eq [symmetric])
   710 apply (frule_tac tpart_partition [THEN [3] lemma_additivity1])
   711 apply (auto simp add: tpart_def)
   712 apply (drule_tac [2] linorder_not_less [THEN iffD1, THEN order_le_imp_less_or_eq], auto)
   713   prefer 3 apply (drule_tac x=na in spec, arith)
   714  prefer 2 apply (blast dest: lemma_additivity3)
   715 apply (frule (2) lemma_additivity4_psize_eq)
   716 apply (rule partition [THEN iffD2])
   717 apply (frule partition [THEN iffD1])
   718 apply safe 
   719 apply (auto simp add: partition_lt_gen)  
   720 apply (drule (1) partition_lt_cancel, arith)
   721 done
   722 
   723 lemma fine_left1:
   724      "[| a \<le> D n; tpart (a, b) (D, p); gauge (%x. a \<le> x & x \<le> D n) g;
   725          fine (%x. if x < D n then min (g x) ((D n - x)/ 2)
   726                  else if x = D n then min (g (D n)) (ga (D n))
   727                       else min (ga x) ((x - D n)/ 2)) (D, p) |]
   728       ==> fine g
   729            (%x. if D x < D n then D(x) else D n,
   730             %x. if D x < D n then p(x) else D n)"
   731 apply (auto simp add: fine_def tpart_def gauge_def)
   732 apply (frule_tac [!] na=na in lemma_psize_left_less_psize2)
   733 apply (drule_tac [!] x = na in spec, auto)
   734 apply (drule_tac [!] x = na in spec, auto)
   735 apply (auto dest: lemma_additivity3a simp add: split_if_asm)
   736 done
   737 
   738 lemma tpart_right1:
   739      "[| a \<le> D n; tpart (a, b) (D, p) |]
   740       ==> tpart(D n, b) (%x. D(x + n),%x. p(x + n))"
   741 apply (simp add: tpart_def partition_def, safe)
   742 apply (rule_tac x = "N - n" in exI, auto)
   743 done
   744 
   745 lemma fine_right1:
   746      "[| a \<le> D n; tpart (a, b) (D, p); gauge (%x. D n \<le> x & x \<le> b) ga;
   747          fine (%x. if x < D n then min (g x) ((D n - x)/ 2)
   748                  else if x = D n then min (g (D n)) (ga (D n))
   749                       else min (ga x) ((x - D n)/ 2)) (D, p) |]
   750       ==> fine ga (%x. D(x + n),%x. p(x + n))"
   751 apply (auto simp add: fine_def gauge_def)
   752 apply (drule_tac x = "na + n" in spec)
   753 apply (frule_tac n = n in tpart_partition [THEN better_lemma_psize_right_eq], auto)
   754 apply (simp add: tpart_def, safe)
   755 apply (subgoal_tac "D n \<le> p (na + n)")
   756 apply (drule_tac y = "p (na + n)" in order_le_imp_less_or_eq)
   757 apply safe
   758 apply (simp split: split_if_asm, simp)
   759 apply (drule less_le_trans, assumption)
   760 apply (rotate_tac 5)
   761 apply (drule_tac x = "na + n" in spec, safe)
   762 apply (rule_tac y="D (na + n)" in order_trans)
   763 apply (case_tac "na = 0", auto)
   764 apply (erule partition_lt_gen [THEN order_less_imp_le], arith+)
   765 done
   766 
   767 lemma rsum_add: "rsum (D, p) (%x. f x + g x) =  rsum (D, p) f + rsum(D, p) g"
   768 by (simp add: rsum_def setsum_addf left_distrib)
   769 
   770 text{* Bartle/Sherbert: Theorem 10.1.5 p. 278 *}
   771 lemma Integral_add_fun:
   772     "[| a \<le> b; Integral(a,b) f k1; Integral(a,b) g k2 |]
   773      ==> Integral(a,b) (%x. f x + g x) (k1 + k2)"
   774 apply (simp add: Integral_def, auto)
   775 apply ((drule_tac x = "e/2" in spec)+)
   776 apply auto
   777 apply (drule gauge_min, assumption)
   778 apply (rule_tac x = " (%x. if ga x < gaa x then ga x else gaa x)" in exI)
   779 apply auto
   780 apply (drule fine_min)
   781 apply ((drule spec)+, auto)
   782 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)
   783 apply (auto simp only: rsum_add left_distrib [symmetric]
   784                 mult_2_right [symmetric] real_mult_less_iff1)
   785 done
   786 
   787 lemma partition_lt_gen2:
   788      "[| partition(a,b) D; r < psize D |] ==> 0 < D (Suc r) - D r"
   789 by (auto simp add: partition)
   790 
   791 lemma lemma_Integral_le:
   792      "[| \<forall>x. a \<le> x & x \<le> b --> f x \<le> g x;
   793          tpart(a,b) (D,p)
   794       |] ==> \<forall>n \<le> psize D. f (p n) \<le> g (p n)"
   795 apply (simp add: tpart_def)
   796 apply (auto, frule partition [THEN iffD1], auto)
   797 apply (drule_tac x = "p n" in spec, auto)
   798 apply (case_tac "n = 0", simp)
   799 apply (rule partition_lt_gen [THEN order_less_le_trans, THEN order_less_imp_le], auto)
   800 apply (drule le_imp_less_or_eq, auto)
   801 apply (drule_tac [2] x = "psize D" in spec, auto)
   802 apply (drule_tac r = "Suc n" in partition_ub)
   803 apply (drule_tac x = n in spec, auto)
   804 done
   805 
   806 lemma lemma_Integral_rsum_le:
   807      "[| \<forall>x. a \<le> x & x \<le> b --> f x \<le> g x;
   808          tpart(a,b) (D,p)
   809       |] ==> rsum(D,p) f \<le> rsum(D,p) g"
   810 apply (simp add: rsum_def)
   811 apply (auto intro!: setsum_mono dest: tpart_partition [THEN partition_lt_gen2]
   812                dest!: lemma_Integral_le)
   813 done
   814 
   815 lemma Integral_le:
   816     "[| a \<le> b;
   817         \<forall>x. a \<le> x & x \<le> b --> f(x) \<le> g(x);
   818         Integral(a,b) f k1; Integral(a,b) g k2
   819      |] ==> k1 \<le> k2"
   820 apply (simp add: Integral_def)
   821 apply (rotate_tac 2)
   822 apply (drule_tac x = "\<bar>k1 - k2\<bar> /2" in spec)
   823 apply (drule_tac x = "\<bar>k1 - k2\<bar> /2" in spec, auto)
   824 apply (drule gauge_min, assumption)
   825 apply (drule_tac g = "%x. if ga x < gaa x then ga x else gaa x" 
   826        in partition_exists, assumption, auto)
   827 apply (drule fine_min)
   828 apply (drule_tac x = D in spec, drule_tac x = D in spec)
   829 apply (drule_tac x = p in spec, drule_tac x = p in spec, auto)
   830 apply (frule lemma_Integral_rsum_le, assumption)
   831 apply (subgoal_tac "\<bar>(rsum (D,p) f - k1) - (rsum (D,p) g - k2)\<bar> < \<bar>k1 - k2\<bar>")
   832 apply arith
   833 apply (drule add_strict_mono, assumption)
   834 apply (auto simp only: left_distrib [symmetric] mult_2_right [symmetric]
   835                        real_mult_less_iff1)
   836 done
   837 
   838 lemma Integral_imp_Cauchy:
   839      "(\<exists>k. Integral(a,b) f k) ==>
   840       (\<forall>e > 0. \<exists>g. gauge (%x. a \<le> x & x \<le> b) g &
   841                        (\<forall>D1 D2 p1 p2.
   842                             tpart(a,b) (D1, p1) & fine g (D1,p1) &
   843                             tpart(a,b) (D2, p2) & fine g (D2,p2) -->
   844                             \<bar>rsum(D1,p1) f - rsum(D2,p2) f\<bar> < e))"
   845 apply (simp add: Integral_def, auto)
   846 apply (drule_tac x = "e/2" in spec, auto)
   847 apply (rule exI, auto)
   848 apply (frule_tac x = D1 in spec)
   849 apply (frule_tac x = D2 in spec)
   850 apply ((drule spec)+, auto)
   851 apply (erule_tac V = "0 < e" in thin_rl)
   852 apply (drule add_strict_mono, assumption)
   853 apply (auto simp only: left_distrib [symmetric] mult_2_right [symmetric]
   854                        real_mult_less_iff1)
   855 done
   856 
   857 lemma Cauchy_iff2:
   858      "Cauchy X =
   859       (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. \<bar>X m + - X n\<bar> < inverse(real (Suc j))))"
   860 apply (simp add: Cauchy_def, auto)
   861 apply (drule reals_Archimedean, safe)
   862 apply (drule_tac x = n in spec, auto)
   863 apply (rule_tac x = M in exI, auto)
   864 apply (drule_tac x = m in spec, simp)
   865 apply (drule_tac x = na in spec, auto)
   866 done
   867 
   868 lemma partition_exists2:
   869      "[| a \<le> b; \<forall>n. gauge (%x. a \<le> x & x \<le> b) (fa n) |]
   870       ==> \<forall>n. \<exists>D p. tpart (a, b) (D, p) & fine (fa n) (D, p)"
   871 by (blast dest: partition_exists) 
   872 
   873 lemma monotonic_anti_derivative:
   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