src/HOL/Hyperreal/Integration.thy
author nipkow
Mon Feb 21 15:04:10 2005 +0100 (2005-02-21)
changeset 15539 333a88244569
parent 15536 3ce1cb7a24f0
child 15944 9b00875e21f7
permissions -rw-r--r--
comprehensive cleanup, replacing sumr by setsum
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(*  Title       : Integration.thy
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    Author      : Jacques D. Fleuriot
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    Copyright   : 2000  University of Edinburgh
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
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*)
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header{*Theory of Integration*}
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theory Integration
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imports MacLaurin
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begin
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text{*We follow John Harrison in formalizing the Gauge integral.*}
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constdefs
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  --{*Partitions and tagged partitions etc.*}
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  partition :: "[(real*real),nat => real] => bool"
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  "partition == %(a,b) D. D 0 = a &
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                         (\<exists>N. (\<forall>n < N. D(n) < D(Suc n)) &
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                              (\<forall>n \<ge> N. D(n) = b))"
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  psize :: "(nat => real) => nat"
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  "psize D == SOME N. (\<forall>n < N. D(n) < D(Suc n)) &
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                      (\<forall>n \<ge> N. D(n) = D(N))"
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  tpart :: "[(real*real),((nat => real)*(nat =>real))] => bool"
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  "tpart == %(a,b) (D,p). partition(a,b) D &
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                          (\<forall>n. D(n) \<le> p(n) & p(n) \<le> D(Suc n))"
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  --{*Gauges and gauge-fine divisions*}
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  gauge :: "[real => bool, real => real] => bool"
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  "gauge E g == \<forall>x. E x --> 0 < g(x)"
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  fine :: "[real => real, ((nat => real)*(nat => real))] => bool"
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  "fine == % g (D,p). \<forall>n. n < (psize D) --> D(Suc n) - D(n) < g(p n)"
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  --{*Riemann sum*}
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  rsum :: "[((nat=>real)*(nat=>real)),real=>real] => real"
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  "rsum == %(D,p) f. \<Sum>n=0..<psize(D). f(p n) * (D(Suc n) - D(n))"
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  --{*Gauge integrability (definite)*}
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   Integral :: "[(real*real),real=>real,real] => bool"
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   "Integral == %(a,b) f k. \<forall>e > 0.
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                               (\<exists>g. gauge(%x. a \<le> x & x \<le> b) g &
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                               (\<forall>D p. tpart(a,b) (D,p) & fine(g)(D,p) -->
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                                         \<bar>rsum(D,p) f - k\<bar> < e))"
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lemma partition_zero [simp]: "a = b ==> psize (%n. if n = 0 then a else b) = 0"
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by (auto simp add: psize_def)
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lemma partition_one [simp]: "a < b ==> psize (%n. if n = 0 then a else b) = 1"
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apply (simp add: psize_def)
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apply (rule some_equality, auto)
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apply (drule_tac x = 1 in spec, auto)
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done
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lemma partition_single [simp]:
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     "a \<le> b ==> partition(a,b)(%n. if n = 0 then a else b)"
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by (auto simp add: partition_def order_le_less)
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lemma partition_lhs: "partition(a,b) D ==> (D(0) = a)"
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by (simp add: partition_def)
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lemma partition:
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       "(partition(a,b) D) =
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        ((D 0 = a) &
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         (\<forall>n < psize D. D n < D(Suc n)) &
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         (\<forall>n \<ge> psize D. D n = b))"
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apply (simp add: partition_def, auto)
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apply (subgoal_tac [!] "psize D = N", auto)
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apply (simp_all (no_asm) add: psize_def)
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apply (rule_tac [!] some_equality, blast)
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  prefer 2 apply blast
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apply (rule_tac [!] ccontr)
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apply (simp_all add: linorder_neq_iff, safe)
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apply (drule_tac x = Na in spec)
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apply (rotate_tac 3)
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apply (drule_tac x = "Suc Na" in spec, simp)
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apply (rotate_tac 2)
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apply (drule_tac x = N in spec, simp)
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apply (drule_tac x = Na in spec)
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apply (drule_tac x = "Suc Na" and P = "%n. Na\<le>n \<longrightarrow> D n = D Na" in spec, auto)
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done
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lemma partition_rhs: "partition(a,b) D ==> (D(psize D) = b)"
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by (simp add: partition)
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lemma partition_rhs2: "[|partition(a,b) D; psize D \<le> n |] ==> (D n = b)"
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by (simp add: partition)
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lemma lemma_partition_lt_gen [rule_format]:
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 "partition(a,b) D & m + Suc d \<le> n & n \<le> (psize D) --> D(m) < D(m + Suc d)"
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apply (induct "d", auto simp add: partition)
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apply (blast dest: Suc_le_lessD  intro: less_le_trans order_less_trans)
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done
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lemma less_eq_add_Suc: "m < n ==> \<exists>d. n = m + Suc d"
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by (auto simp add: less_iff_Suc_add)
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lemma partition_lt_gen:
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     "[|partition(a,b) D; m < n; n \<le> (psize D)|] ==> D(m) < D(n)"
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by (auto dest: less_eq_add_Suc intro: lemma_partition_lt_gen)
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lemma partition_lt: "partition(a,b) D ==> n < (psize D) ==> D(0) < D(Suc n)"
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apply (induct "n")
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apply (auto simp add: partition)
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done
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lemma partition_le: "partition(a,b) D ==> a \<le> b"
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apply (frule partition [THEN iffD1], safe)
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apply (drule_tac x = "psize D" and P="%n. psize D \<le> n --> ?P n" in spec, safe)
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apply (case_tac "psize D = 0")
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apply (drule_tac [2] n = "psize D - 1" in partition_lt, auto)
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done
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lemma partition_gt: "[|partition(a,b) D; n < (psize D)|] ==> D(n) < D(psize D)"
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by (auto intro: partition_lt_gen)
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lemma partition_eq: "partition(a,b) D ==> ((a = b) = (psize D = 0))"
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apply (frule partition [THEN iffD1], safe)
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apply (rotate_tac 2)
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apply (drule_tac x = "psize D" in spec)
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apply (rule ccontr)
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apply (drule_tac n = "psize D - 1" in partition_lt)
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apply auto
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done
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lemma partition_lb: "partition(a,b) D ==> a \<le> D(r)"
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apply (frule partition [THEN iffD1], safe)
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apply (induct "r")
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apply (cut_tac [2] y = "Suc r" and x = "psize D" in linorder_le_less_linear)
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apply (auto intro: partition_le)
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apply (drule_tac x = r in spec)
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apply arith; 
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done
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lemma partition_lb_lt: "[| partition(a,b) D; psize D ~= 0 |] ==> a < D(Suc n)"
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apply (rule_tac t = a in partition_lhs [THEN subst], assumption)
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apply (cut_tac x = "Suc n" and y = "psize D" in linorder_le_less_linear)
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apply (frule partition [THEN iffD1], safe)
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 apply (blast intro: partition_lt less_le_trans)
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apply (rotate_tac 3)
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apply (drule_tac x = "Suc n" in spec)
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apply (erule impE)
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apply (erule less_imp_le)
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apply (frule partition_rhs)
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apply (drule partition_gt, assumption)
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apply (simp (no_asm_simp))
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done
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lemma partition_ub: "partition(a,b) D ==> D(r) \<le> b"
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apply (frule partition [THEN iffD1])
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apply (cut_tac x = "psize D" and y = r in linorder_le_less_linear, safe, blast)
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apply (subgoal_tac "\<forall>x. D ((psize D) - x) \<le> b")
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apply (rotate_tac 4)
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apply (drule_tac x = "psize D - r" in spec)
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apply (subgoal_tac "psize D - (psize D - r) = r")
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apply simp
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apply arith
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apply safe
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apply (induct_tac "x")
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apply (simp (no_asm), blast)
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apply (case_tac "psize D - Suc n = 0")
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apply (erule_tac V = "\<forall>n. psize D \<le> n --> D n = b" in thin_rl)
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apply (simp (no_asm_simp) add: partition_le)
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apply (rule order_trans)
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 prefer 2 apply assumption
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apply (subgoal_tac "psize D - n = Suc (psize D - Suc n)")
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 prefer 2 apply arith
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apply (drule_tac x = "psize D - Suc n" in spec, simp) 
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done
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lemma partition_ub_lt: "[| partition(a,b) D; n < psize D |] ==> D(n) < b"
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by (blast intro: partition_rhs [THEN subst] partition_gt)
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lemma lemma_partition_append1:
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     "[| partition (a, b) D1; partition (b, c) D2 |]
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       ==> (\<forall>n < psize D1 + psize D2.
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             (if n < psize D1 then D1 n else D2 (n - psize D1))
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             < (if Suc n < psize D1 then D1 (Suc n)
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                else D2 (Suc n - psize D1))) &
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         (\<forall>n \<ge> psize D1 + psize D2.
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             (if n < psize D1 then D1 n else D2 (n - psize D1)) =
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             (if psize D1 + psize D2 < psize D1 then D1 (psize D1 + psize D2)
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              else D2 (psize D1 + psize D2 - psize D1)))"
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apply (auto intro: partition_lt_gen)
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apply (subgoal_tac "psize D1 = Suc n")
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apply (auto intro!: partition_lt_gen simp add: partition_lhs partition_ub_lt)
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apply (auto intro!: partition_rhs2 simp add: partition_rhs
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            split: nat_diff_split)
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done
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lemma lemma_psize1:
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     "[| partition (a, b) D1; partition (b, c) D2; N < psize D1 |]
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      ==> D1(N) < D2 (psize D2)"
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apply (rule_tac y = "D1 (psize D1)" in order_less_le_trans)
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apply (erule partition_gt)
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apply (auto simp add: partition_rhs partition_le)
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done
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lemma lemma_partition_append2:
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     "[| partition (a, b) D1; partition (b, c) D2 |]
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      ==> psize (%n. if n < psize D1 then D1 n else D2 (n - psize D1)) =
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          psize D1 + psize D2" 
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apply (unfold psize_def 
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         [of "%n. if n < psize D1 then D1 n else D2 (n - psize D1)"])
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apply (rule some1_equality)
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 prefer 2 apply (blast intro!: lemma_partition_append1)
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apply (rule ex1I, rule lemma_partition_append1) 
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apply (simp_all split: split_if_asm)
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 txt{*The case @{term "N < psize D1"}*} 
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 apply (drule_tac x = "psize D1 + psize D2" and P="%n. ?P n & ?Q n" in spec) 
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 apply (force dest: lemma_psize1)
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apply (rule order_antisym);
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 txt{*The case @{term "psize D1 \<le> N"}: 
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       proving @{term "N \<le> psize D1 + psize D2"}*}
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 apply (drule_tac x = "psize D1 + psize D2" in spec)
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 apply (simp add: partition_rhs2)
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apply (case_tac "N - psize D1 < psize D2") 
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 prefer 2 apply arith
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 txt{*Proving @{term "psize D1 + psize D2 \<le> N"}*}
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apply (drule_tac x = "psize D1 + psize D2" and P="%n. N\<le>n --> ?P n" in spec, simp)
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apply (drule_tac a = b and b = c in partition_gt, assumption, simp)
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done
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lemma tpart_eq_lhs_rhs: "[|psize D = 0; tpart(a,b) (D,p)|] ==> a = b"
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by (auto simp add: tpart_def partition_eq)
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lemma tpart_partition: "tpart(a,b) (D,p) ==> partition(a,b) D"
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by (simp add: tpart_def)
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lemma partition_append:
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     "[| tpart(a,b) (D1,p1); fine(g) (D1,p1);
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         tpart(b,c) (D2,p2); fine(g) (D2,p2) |]
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       ==> \<exists>D p. tpart(a,c) (D,p) & fine(g) (D,p)"
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apply (rule_tac x = "%n. if n < psize D1 then D1 n else D2 (n - psize D1)"
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       in exI)
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apply (rule_tac x = "%n. if n < psize D1 then p1 n else p2 (n - psize D1)"
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       in exI)
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apply (case_tac "psize D1 = 0")
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apply (auto dest: tpart_eq_lhs_rhs)
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 prefer 2
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apply (simp add: fine_def
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                 lemma_partition_append2 [OF tpart_partition tpart_partition])
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  --{*But must not expand @{term fine} in other subgoals*}
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apply auto
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apply (subgoal_tac "psize D1 = Suc n")
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 prefer 2 apply arith
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apply (drule tpart_partition [THEN partition_rhs])
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apply (drule tpart_partition [THEN partition_lhs])
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apply (auto split: nat_diff_split)
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apply (auto simp add: tpart_def)
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defer 1
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 apply (subgoal_tac "psize D1 = Suc n")
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  prefer 2 apply arith
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 apply (drule partition_rhs)
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 apply (drule partition_lhs, auto)
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apply (simp split: nat_diff_split)
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apply (subst partition)
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apply (simplesubst lemma_partition_append2, assumption+)
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   --{*Need to substitute the last occurrence*}
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apply (rule conjI)
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apply (drule_tac [2] lemma_partition_append1)
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apply (auto simp add: partition_lhs partition_rhs)
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done
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text{*We can always find a division that is fine wrt any gauge*}
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lemma partition_exists:
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     "[| a \<le> b; gauge(%x. a \<le> x & x \<le> b) g |]
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      ==> \<exists>D p. tpart(a,b) (D,p) & fine g (D,p)"
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apply (cut_tac P = "%(u,v). a \<le> u & v \<le> b --> 
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                   (\<exists>D p. tpart (u,v) (D,p) & fine (g) (D,p))" 
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       in lemma_BOLZANO2)
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apply safe
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apply (blast intro: order_trans)+
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apply (auto intro: partition_append)
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apply (case_tac "a \<le> x & x \<le> b")
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apply (rule_tac [2] x = 1 in exI, auto)
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apply (rule_tac x = "g x" in exI)
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apply (auto simp add: gauge_def)
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apply (rule_tac x = "%n. if n = 0 then aa else ba" in exI)
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apply (rule_tac x = "%n. if n = 0 then x else ba" in exI)
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apply (auto simp add: tpart_def fine_def)
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done
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text{*Lemmas about combining gauges*}
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lemma gauge_min:
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     "[| gauge(E) g1; gauge(E) g2 |]
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   298
      ==> gauge(E) (%x. if g1(x) < g2(x) then g1(x) else g2(x))"
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   299
by (simp add: gauge_def)
paulson@15093
   300
paulson@15093
   301
lemma fine_min:
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   302
      "fine (%x. if g1(x) < g2(x) then g1(x) else g2(x)) (D,p)
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   303
       ==> fine(g1) (D,p) & fine(g2) (D,p)"
paulson@15093
   304
by (auto simp add: fine_def split: split_if_asm)
paulson@15093
   305
paulson@15093
   306
paulson@15093
   307
text{*The integral is unique if it exists*}
paulson@15093
   308
paulson@15093
   309
lemma Integral_unique:
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   310
    "[| a \<le> b; Integral(a,b) f k1; Integral(a,b) f k2 |] ==> k1 = k2"
paulson@15093
   311
apply (simp add: Integral_def)
paulson@15093
   312
apply (drule_tac x = "\<bar>k1 - k2\<bar> /2" in spec)+
paulson@15093
   313
apply auto
paulson@15093
   314
apply (drule gauge_min, assumption)
paulson@15093
   315
apply (drule_tac g = "%x. if g x < ga x then g x else ga x" 
paulson@15093
   316
       in partition_exists, assumption, auto)
paulson@15093
   317
apply (drule fine_min)
paulson@15093
   318
apply (drule spec)+
paulson@15093
   319
apply auto
paulson@15094
   320
apply (subgoal_tac "\<bar>(rsum (D,p) f - k2) - (rsum (D,p) f - k1)\<bar> < \<bar>k1 - k2\<bar>")
paulson@15093
   321
apply arith
paulson@15093
   322
apply (drule add_strict_mono, assumption)
paulson@15093
   323
apply (auto simp only: left_distrib [symmetric] mult_2_right [symmetric] 
paulson@15093
   324
                mult_less_cancel_right, arith)
paulson@15093
   325
done
paulson@15093
   326
paulson@15093
   327
lemma Integral_zero [simp]: "Integral(a,a) f 0"
paulson@15093
   328
apply (auto simp add: Integral_def)
paulson@15093
   329
apply (rule_tac x = "%x. 1" in exI)
paulson@15093
   330
apply (auto dest: partition_eq simp add: gauge_def tpart_def rsum_def)
paulson@15093
   331
done
paulson@15093
   332
paulson@15093
   333
lemma sumr_partition_eq_diff_bounds [simp]:
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   334
     "(\<Sum>n=0..<m. D (Suc n) - D n::real) = D(m) - D 0"
paulson@15251
   335
by (induct "m", auto)
paulson@15093
   336
paulson@15093
   337
lemma Integral_eq_diff_bounds: "a \<le> b ==> Integral(a,b) (%x. 1) (b - a)"
paulson@15219
   338
apply (auto simp add: order_le_less rsum_def Integral_def)
paulson@15093
   339
apply (rule_tac x = "%x. b - a" in exI)
paulson@15093
   340
apply (auto simp add: gauge_def abs_interval_iff tpart_def partition)
paulson@15093
   341
done
paulson@15093
   342
paulson@15093
   343
lemma Integral_mult_const: "a \<le> b ==> Integral(a,b) (%x. c)  (c*(b - a))"
paulson@15219
   344
apply (auto simp add: order_le_less rsum_def Integral_def)
paulson@15093
   345
apply (rule_tac x = "%x. b - a" in exI)
nipkow@15536
   346
apply (auto simp add: setsum_mult [symmetric] gauge_def abs_interval_iff 
paulson@15093
   347
               right_diff_distrib [symmetric] partition tpart_def)
paulson@15093
   348
done
paulson@15093
   349
paulson@15093
   350
lemma Integral_mult:
paulson@15093
   351
     "[| a \<le> b; Integral(a,b) f k |] ==> Integral(a,b) (%x. c * f x) (c * k)"
paulson@15221
   352
apply (auto simp add: order_le_less 
paulson@15221
   353
            dest: Integral_unique [OF order_refl Integral_zero])
nipkow@15536
   354
apply (auto simp add: rsum_def Integral_def setsum_mult[symmetric] mult_assoc)
paulson@15093
   355
apply (rule_tac a2 = c in abs_ge_zero [THEN real_le_imp_less_or_eq, THEN disjE])
paulson@15093
   356
 prefer 2 apply force
paulson@15093
   357
apply (drule_tac x = "e/abs c" in spec, auto)
paulson@15093
   358
apply (simp add: zero_less_mult_iff divide_inverse)
paulson@15093
   359
apply (rule exI, auto)
paulson@15093
   360
apply (drule spec)+
paulson@15093
   361
apply auto
paulson@15094
   362
apply (rule_tac z1 = "inverse (abs c)" in real_mult_less_iff1 [THEN iffD1])
paulson@15093
   363
apply (auto simp add: divide_inverse [symmetric] right_diff_distrib [symmetric])
paulson@15093
   364
done
paulson@15093
   365
paulson@15093
   366
text{*Fundamental theorem of calculus (Part I)*}
paulson@15093
   367
nipkow@15105
   368
text{*"Straddle Lemma" : Swartz and Thompson: AMM 95(7) 1988 *}
paulson@15093
   369
paulson@15093
   370
lemma choiceP: "\<forall>x. P(x) --> (\<exists>y. Q x y) ==> \<exists>f. (\<forall>x. P(x) --> Q x (f x))"
paulson@15093
   371
by meson
paulson@15093
   372
paulson@15093
   373
lemma choiceP2: "\<forall>x. P(x) --> (\<exists>y. R(y) & (\<exists>z. Q x y z)) ==>
paulson@15093
   374
      \<exists>f fa. (\<forall>x. P(x) --> R(f x) & Q x (f x) (fa x))"
paulson@15093
   375
by meson
paulson@15093
   376
paulson@15093
   377
(*UNUSED
paulson@15093
   378
lemma choice2: "\<forall>x. (\<exists>y. R(y) & (\<exists>z. Q x y z)) ==>
paulson@15093
   379
      \<exists>f fa. (\<forall>x. R(f x) & Q x (f x) (fa x))"
paulson@15093
   380
*)
paulson@15093
   381
paulson@15093
   382
paulson@15093
   383
(* new simplifications e.g. (y < x/n) = (y * n < x) are a real nuisance
paulson@15094
   384
   they break the original proofs and make new proofs longer!*)
paulson@15093
   385
lemma strad1:
paulson@15093
   386
       "\<lbrakk>\<forall>xa::real. xa \<noteq> x \<and> \<bar>xa + - x\<bar> < s \<longrightarrow>
paulson@15093
   387
             \<bar>(f xa - f x) / (xa - x) + - f' x\<bar> * 2 < e;
paulson@15093
   388
        0 < e; a \<le> x; x \<le> b; 0 < s\<rbrakk>
paulson@15093
   389
       \<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>"
paulson@15093
   390
apply auto
paulson@15094
   391
apply (case_tac "0 < \<bar>z - x\<bar>")
paulson@15093
   392
 prefer 2 apply (simp add: zero_less_abs_iff)
paulson@15093
   393
apply (drule_tac x = z in spec)
paulson@15093
   394
apply (rule_tac z1 = "\<bar>inverse (z - x)\<bar>" 
paulson@15093
   395
       in real_mult_le_cancel_iff2 [THEN iffD1])
paulson@15093
   396
 apply simp
paulson@15093
   397
apply (simp del: abs_inverse abs_mult add: abs_mult [symmetric]
paulson@15093
   398
          mult_assoc [symmetric])
paulson@15093
   399
apply (subgoal_tac "inverse (z - x) * (f z - f x - f' x * (z - x)) 
paulson@15093
   400
                    = (f z - f x) / (z - x) - f' x")
paulson@15093
   401
 apply (simp add: abs_mult [symmetric] mult_ac diff_minus)
paulson@15093
   402
apply (subst mult_commute)
paulson@15093
   403
apply (simp add: left_distrib diff_minus)
paulson@15093
   404
apply (simp add: mult_assoc divide_inverse)
paulson@15093
   405
apply (simp add: left_distrib)
paulson@15093
   406
done
paulson@15093
   407
paulson@15093
   408
lemma lemma_straddle:
paulson@15093
   409
     "[| \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x); 0 < e |]
paulson@15093
   410
      ==> \<exists>g. gauge(%x. a \<le> x & x \<le> b) g &
paulson@15093
   411
                (\<forall>x u v. a \<le> u & u \<le> x & x \<le> v & v \<le> b & (v - u) < g(x)
paulson@15094
   412
                  --> \<bar>(f(v) - f(u)) - (f'(x) * (v - u))\<bar> \<le> e * (v - u))"
paulson@15093
   413
apply (simp add: gauge_def)
paulson@15093
   414
apply (subgoal_tac "\<forall>x. a \<le> x & x \<le> b --> 
nipkow@15360
   415
        (\<exists>d > 0. \<forall>u v. u \<le> x & x \<le> v & (v - u) < d --> 
nipkow@15360
   416
                       \<bar>(f (v) - f (u)) - (f' (x) * (v - u))\<bar> \<le> e * (v - u))")
paulson@15093
   417
apply (drule choiceP, auto)
paulson@15093
   418
apply (drule spec, auto)
paulson@15093
   419
apply (auto simp add: DERIV_iff2 LIM_def)
paulson@15093
   420
apply (drule_tac x = "e/2" in spec, auto)
paulson@15093
   421
apply (frule strad1, assumption+)
paulson@15093
   422
apply (rule_tac x = s in exI, auto)
paulson@15093
   423
apply (rule_tac x = u and y = v in linorder_cases, auto)
paulson@15219
   424
apply (rule_tac y = "\<bar>(f (v) - f (x)) - (f' (x) * (v - x))\<bar> + 
paulson@15094
   425
                     \<bar>(f (x) - f (u)) - (f' (x) * (x - u))\<bar>"
paulson@15219
   426
       in order_trans)
paulson@15219
   427
apply (rule abs_triangle_ineq [THEN [2] order_trans])
paulson@15093
   428
apply (simp add: right_diff_distrib, arith)
paulson@15094
   429
apply (rule_tac t = "e* (v - u)" in real_sum_of_halves [THEN subst])
paulson@15093
   430
apply (rule add_mono)
paulson@15219
   431
apply (rule_tac y = "(e/2) * \<bar>v - x\<bar>" in order_trans)
paulson@15229
   432
 prefer 2 apply simp
paulson@15219
   433
apply (erule_tac [!] V= "\<forall>x'. x' ~= x & \<bar>x' + - x\<bar> < s --> ?P x'" in thin_rl)
paulson@15234
   434
apply (drule_tac x = v in spec, simp add: times_divide_eq)
paulson@15093
   435
apply (drule_tac x = u in spec, auto, arith)
paulson@15093
   436
apply (subgoal_tac "\<bar>f u - f x - f' x * (u - x)\<bar> = \<bar>f x - f u - f' x * (x - u)\<bar>")
paulson@15093
   437
apply (rule order_trans)
paulson@15093
   438
apply (auto simp add: abs_le_interval_iff)
paulson@15093
   439
apply (simp add: right_diff_distrib, arith)
paulson@15093
   440
done
paulson@15093
   441
paulson@15093
   442
lemma FTC1: "[|a \<le> b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
paulson@15219
   443
             ==> Integral(a,b) f' (f(b) - f(a))"
paulson@15219
   444
apply (drule order_le_imp_less_or_eq, auto) 
paulson@15093
   445
apply (auto simp add: Integral_def)
paulson@15093
   446
apply (rule ccontr)
nipkow@15360
   447
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)")
paulson@15093
   448
apply (rotate_tac 3)
paulson@15093
   449
apply (drule_tac x = "e/2" in spec, auto)
paulson@15093
   450
apply (drule spec, auto)
paulson@15093
   451
apply ((drule spec)+, auto)
paulson@15094
   452
apply (drule_tac e = "ea/ (b - a)" in lemma_straddle)
paulson@15093
   453
apply (auto simp add: zero_less_divide_iff)
paulson@15093
   454
apply (rule exI)
paulson@15093
   455
apply (auto simp add: tpart_def rsum_def)
nipkow@15539
   456
apply (subgoal_tac "(\<Sum>n=0..<psize D. f(D(Suc n)) - f(D n)) = f b - f a")
paulson@15093
   457
 prefer 2
paulson@15094
   458
 apply (cut_tac D = "%n. f (D n)" and m = "psize D"
paulson@15093
   459
        in sumr_partition_eq_diff_bounds)
paulson@15093
   460
 apply (simp add: partition_lhs partition_rhs)
paulson@15093
   461
apply (drule sym, simp)
nipkow@15536
   462
apply (simp (no_asm) add: setsum_subtractf[symmetric])
nipkow@15536
   463
apply (rule setsum_abs [THEN order_trans])
nipkow@15539
   464
apply (subgoal_tac "ea = (\<Sum>n=0..<psize D. (ea / (b - a)) * (D (Suc n) - (D n)))")
paulson@15093
   465
apply (simp add: abs_minus_commute)
paulson@15093
   466
apply (rule_tac t = ea in ssubst, assumption)
nipkow@15539
   467
apply (rule setsum_mono)
nipkow@15536
   468
apply (rule_tac [2] setsum_mult [THEN subst])
paulson@15093
   469
apply (auto simp add: partition_rhs partition_lhs partition_lb partition_ub
paulson@15093
   470
          fine_def)
paulson@15093
   471
done
paulson@13958
   472
paulson@13958
   473
paulson@15093
   474
lemma Integral_subst: "[| Integral(a,b) f k1; k2=k1 |] ==> Integral(a,b) f k2"
paulson@15093
   475
by simp
paulson@15093
   476
paulson@15093
   477
lemma Integral_add:
paulson@15093
   478
     "[| a \<le> b; b \<le> c; Integral(a,b) f' k1; Integral(b,c) f' k2;
paulson@15093
   479
         \<forall>x. a \<le> x & x \<le> c --> DERIV f x :> f' x |]
paulson@15093
   480
     ==> Integral(a,c) f' (k1 + k2)"
paulson@15093
   481
apply (rule FTC1 [THEN Integral_subst], auto)
paulson@15093
   482
apply (frule FTC1, auto)
paulson@15093
   483
apply (frule_tac a = b in FTC1, auto)
paulson@15093
   484
apply (drule_tac x = x in spec, auto)
paulson@15093
   485
apply (drule_tac ?k2.0 = "f b - f a" in Integral_unique)
paulson@15093
   486
apply (drule_tac [3] ?k2.0 = "f c - f b" in Integral_unique, auto)
paulson@15093
   487
done
paulson@15093
   488
paulson@15093
   489
lemma partition_psize_Least:
paulson@15093
   490
     "partition(a,b) D ==> psize D = (LEAST n. D(n) = b)"
paulson@15093
   491
apply (auto intro!: Least_equality [symmetric] partition_rhs)
paulson@15219
   492
apply (auto dest: partition_ub_lt simp add: linorder_not_less [symmetric])
paulson@15093
   493
done
paulson@15093
   494
paulson@15093
   495
lemma lemma_partition_bounded: "partition (a, c) D ==> ~ (\<exists>n. c < D(n))"
paulson@15093
   496
apply safe
paulson@15093
   497
apply (drule_tac r = n in partition_ub, auto)
paulson@15093
   498
done
paulson@15093
   499
paulson@15093
   500
lemma lemma_partition_eq:
paulson@15093
   501
     "partition (a, c) D ==> D = (%n. if D n < c then D n else c)"
paulson@15093
   502
apply (rule ext, auto)
paulson@15093
   503
apply (auto dest!: lemma_partition_bounded)
paulson@15093
   504
apply (drule_tac x = n in spec, auto)
paulson@15093
   505
done
paulson@15093
   506
paulson@15093
   507
lemma lemma_partition_eq2:
paulson@15093
   508
     "partition (a, c) D ==> D = (%n. if D n \<le> c then D n else c)"
paulson@15093
   509
apply (rule ext, auto)
paulson@15093
   510
apply (auto dest!: lemma_partition_bounded)
paulson@15093
   511
apply (drule_tac x = n in spec, auto)
paulson@15093
   512
done
paulson@15093
   513
paulson@15093
   514
lemma partition_lt_Suc:
paulson@15093
   515
     "[| partition(a,b) D; n < psize D |] ==> D n < D (Suc n)"
paulson@15093
   516
by (auto simp add: partition)
paulson@15093
   517
paulson@15093
   518
lemma tpart_tag_eq: "tpart(a,c) (D,p) ==> p = (%n. if D n < c then p n else c)"
paulson@15093
   519
apply (rule ext)
paulson@15093
   520
apply (auto simp add: tpart_def)
paulson@15093
   521
apply (drule linorder_not_less [THEN iffD1])
paulson@15093
   522
apply (drule_tac r = "Suc n" in partition_ub)
paulson@15093
   523
apply (drule_tac x = n in spec, auto)
paulson@15093
   524
done
paulson@15093
   525
paulson@15093
   526
subsection{*Lemmas for Additivity Theorem of Gauge Integral*}
paulson@15093
   527
paulson@15093
   528
lemma lemma_additivity1:
paulson@15093
   529
     "[| a \<le> D n; D n < b; partition(a,b) D |] ==> n < psize D"
paulson@15093
   530
by (auto simp add: partition linorder_not_less [symmetric])
paulson@15093
   531
paulson@15093
   532
lemma lemma_additivity2: "[| a \<le> D n; partition(a,D n) D |] ==> psize D \<le> n"
paulson@15093
   533
apply (rule ccontr, drule not_leE)
paulson@15093
   534
apply (frule partition [THEN iffD1], safe)
paulson@15093
   535
apply (frule_tac r = "Suc n" in partition_ub)
paulson@15093
   536
apply (auto dest!: spec)
paulson@15093
   537
done
paulson@15093
   538
paulson@15093
   539
lemma partition_eq_bound:
paulson@15093
   540
     "[| partition(a,b) D; psize D < m |] ==> D(m) = D(psize D)"
paulson@15093
   541
by (auto simp add: partition)
paulson@15093
   542
paulson@15093
   543
lemma partition_ub2: "[| partition(a,b) D; psize D < m |] ==> D(r) \<le> D(m)"
paulson@15093
   544
by (simp add: partition partition_ub)
paulson@15093
   545
paulson@15093
   546
lemma tag_point_eq_partition_point:
paulson@15093
   547
    "[| tpart(a,b) (D,p); psize D \<le> m |] ==> p(m) = D(m)"
paulson@15093
   548
apply (simp add: tpart_def, auto)
paulson@15093
   549
apply (drule_tac x = m in spec)
paulson@15093
   550
apply (auto simp add: partition_rhs2)
paulson@15093
   551
done
paulson@15093
   552
paulson@15093
   553
lemma partition_lt_cancel: "[| partition(a,b) D; D m < D n |] ==> m < n"
paulson@15093
   554
apply (cut_tac m = n and n = "psize D" in less_linear, auto)
paulson@15219
   555
apply (cut_tac m = m and n = n in less_linear)
paulson@15093
   556
apply (cut_tac m = m and n = "psize D" in less_linear)
paulson@15093
   557
apply (auto dest: partition_gt)
paulson@15093
   558
apply (drule_tac n = m in partition_lt_gen, auto)
paulson@15093
   559
apply (frule partition_eq_bound)
paulson@15093
   560
apply (drule_tac [2] partition_gt, auto)
paulson@15093
   561
apply (rule ccontr, drule leI, drule le_imp_less_or_eq)
paulson@15093
   562
apply (auto dest: partition_eq_bound)
paulson@15093
   563
apply (rule ccontr, drule leI, drule le_imp_less_or_eq)
paulson@15093
   564
apply (frule partition_eq_bound, assumption)
paulson@15093
   565
apply (drule_tac m = m in partition_eq_bound, auto)
paulson@15093
   566
done
paulson@15093
   567
paulson@15093
   568
lemma lemma_additivity4_psize_eq:
paulson@15093
   569
     "[| a \<le> D n; D n < b; partition (a, b) D |]
paulson@15093
   570
      ==> psize (%x. if D x < D n then D(x) else D n) = n"
paulson@15093
   571
apply (unfold psize_def)
paulson@15093
   572
apply (frule lemma_additivity1)
paulson@15093
   573
apply (assumption, assumption)
paulson@15093
   574
apply (rule some_equality)
paulson@15093
   575
apply (auto intro: partition_lt_Suc)
paulson@15219
   576
apply (drule_tac n = n in partition_lt_gen, assumption)
paulson@15219
   577
apply (arith, arith)
paulson@15093
   578
apply (cut_tac m = na and n = "psize D" in less_linear)
paulson@15093
   579
apply (auto dest: partition_lt_cancel)
paulson@15093
   580
apply (rule_tac x=N and y=n in linorder_cases)
paulson@15219
   581
apply (drule_tac x = n and P="%m. N \<le> m --> ?f m = ?g m" in spec, simp)
paulson@15093
   582
apply (drule_tac n = n in partition_lt_gen, auto, arith)
paulson@15093
   583
apply (drule_tac x = n in spec)
paulson@15093
   584
apply (simp split: split_if_asm)
paulson@15093
   585
done
paulson@15093
   586
paulson@15093
   587
lemma lemma_psize_left_less_psize:
paulson@15093
   588
     "partition (a, b) D
paulson@15093
   589
      ==> psize (%x. if D x < D n then D(x) else D n) \<le> psize D"
paulson@15093
   590
apply (frule_tac r = n in partition_ub)
paulson@15219
   591
apply (drule_tac x = "D n" in order_le_imp_less_or_eq)
paulson@15093
   592
apply (auto simp add: lemma_partition_eq [symmetric])
paulson@15093
   593
apply (frule_tac r = n in partition_lb)
paulson@15219
   594
apply (drule (2) lemma_additivity4_psize_eq)  
paulson@15219
   595
apply (rule ccontr, auto)
paulson@15093
   596
apply (frule_tac not_leE [THEN [2] partition_eq_bound])
paulson@15093
   597
apply (auto simp add: partition_rhs)
paulson@15093
   598
done
paulson@15093
   599
paulson@15093
   600
lemma lemma_psize_left_less_psize2:
paulson@15093
   601
     "[| partition(a,b) D; na < psize (%x. if D x < D n then D(x) else D n) |]
paulson@15093
   602
      ==> na < psize D"
paulson@15219
   603
by (erule lemma_psize_left_less_psize [THEN [2] less_le_trans])
paulson@15093
   604
paulson@15093
   605
paulson@15093
   606
lemma lemma_additivity3:
paulson@15093
   607
     "[| partition(a,b) D; D na < D n; D n < D (Suc na);
paulson@15093
   608
         n < psize D |]
paulson@15093
   609
      ==> False"
paulson@15093
   610
apply (cut_tac m = n and n = "Suc na" in less_linear, auto)
paulson@15093
   611
apply (drule_tac [2] n = n in partition_lt_gen, auto)
paulson@15093
   612
apply (cut_tac m = "psize D" and n = na in less_linear)
paulson@15093
   613
apply (auto simp add: partition_rhs2 less_Suc_eq)
paulson@15093
   614
apply (drule_tac n = na in partition_lt_gen, auto)
paulson@15093
   615
done
paulson@15093
   616
paulson@15093
   617
lemma psize_const [simp]: "psize (%x. k) = 0"
paulson@15219
   618
by (auto simp add: psize_def)
paulson@15093
   619
paulson@15093
   620
lemma lemma_additivity3a:
paulson@15093
   621
     "[| partition(a,b) D; D na < D n; D n < D (Suc na);
paulson@15093
   622
         na < psize D |]
paulson@15093
   623
      ==> False"
paulson@15093
   624
apply (frule_tac m = n in partition_lt_cancel)
paulson@15093
   625
apply (auto intro: lemma_additivity3)
paulson@15093
   626
done
paulson@15093
   627
paulson@15093
   628
lemma better_lemma_psize_right_eq1:
paulson@15093
   629
     "[| partition(a,b) D; D n < b |] ==> psize (%x. D (x + n)) \<le> psize D - n"
paulson@15093
   630
apply (simp add: psize_def [of "(%x. D (x + n))"]);
paulson@15093
   631
apply (rule_tac a = "psize D - n" in someI2, auto)
paulson@15093
   632
  apply (simp add: partition less_diff_conv)
paulson@15219
   633
 apply (simp add: le_diff_conv partition_rhs2 split: nat_diff_split)
paulson@15093
   634
apply (drule_tac x = "psize D - n" in spec, auto)
paulson@15093
   635
apply (frule partition_rhs, safe)
paulson@15093
   636
apply (frule partition_lt_cancel, assumption)
paulson@15093
   637
apply (drule partition [THEN iffD1], safe)
paulson@15093
   638
apply (subgoal_tac "~ D (psize D - n + n) < D (Suc (psize D - n + n))")
paulson@15093
   639
 apply blast
paulson@15093
   640
apply (drule_tac x = "Suc (psize D)" and P="%n. ?P n \<longrightarrow> D n = D (psize D)"
paulson@15093
   641
       in spec)
paulson@15219
   642
apply simp
paulson@15093
   643
done
paulson@15093
   644
paulson@15219
   645
lemma psize_le_n: "partition (a, D n) D ==> psize D \<le> n" 
paulson@15093
   646
apply (rule ccontr, drule not_leE)
paulson@15093
   647
apply (frule partition_lt_Suc, assumption)
paulson@15093
   648
apply (frule_tac r = "Suc n" in partition_ub, auto)
paulson@15093
   649
done
paulson@15093
   650
paulson@15093
   651
lemma better_lemma_psize_right_eq1a:
paulson@15093
   652
     "partition(a,D n) D ==> psize (%x. D (x + n)) \<le> psize D - n"
paulson@15093
   653
apply (simp add: psize_def [of "(%x. D (x + n))"]);
paulson@15093
   654
apply (rule_tac a = "psize D - n" in someI2, auto)
paulson@15093
   655
  apply (simp add: partition less_diff_conv)
paulson@15093
   656
 apply (simp add: le_diff_conv)
paulson@15093
   657
apply (case_tac "psize D \<le> n")
paulson@15093
   658
  apply (force intro: partition_rhs2)
paulson@15093
   659
 apply (simp add: partition linorder_not_le)
paulson@15093
   660
apply (rule ccontr, drule not_leE)
paulson@15093
   661
apply (frule psize_le_n)
paulson@15093
   662
apply (drule_tac x = "psize D - n" in spec, simp)
paulson@15093
   663
apply (drule partition [THEN iffD1], safe)
paulson@15219
   664
apply (drule_tac x = "Suc n" and P="%na. ?s \<le> na \<longrightarrow> D na = D n" in spec, auto)
paulson@15093
   665
done
paulson@15093
   666
paulson@15093
   667
lemma better_lemma_psize_right_eq:
paulson@15093
   668
     "partition(a,b) D ==> psize (%x. D (x + n)) \<le> psize D - n"
paulson@15219
   669
apply (frule_tac r1 = n in partition_ub [THEN order_le_imp_less_or_eq])
paulson@15093
   670
apply (blast intro: better_lemma_psize_right_eq1a better_lemma_psize_right_eq1)
paulson@15093
   671
done
paulson@15093
   672
paulson@15093
   673
lemma lemma_psize_right_eq1:
paulson@15093
   674
     "[| partition(a,b) D; D n < b |] ==> psize (%x. D (x + n)) \<le> psize D"
paulson@15219
   675
apply (simp add: psize_def [of "(%x. D (x + n))"])
paulson@15093
   676
apply (rule_tac a = "psize D - n" in someI2, auto)
paulson@15093
   677
  apply (simp add: partition less_diff_conv)
paulson@15093
   678
 apply (subgoal_tac "n \<le> psize D")
paulson@15093
   679
  apply (simp add: partition le_diff_conv)
paulson@15093
   680
 apply (rule ccontr, drule not_leE)
paulson@15219
   681
 apply (drule_tac less_imp_le [THEN [2] partition_rhs2], assumption, simp)
paulson@15093
   682
apply (drule_tac x = "psize D" in spec)
paulson@15093
   683
apply (simp add: partition)
paulson@15093
   684
done
paulson@15093
   685
paulson@15093
   686
(* should be combined with previous theorem; also proof has redundancy *)
paulson@15093
   687
lemma lemma_psize_right_eq1a:
paulson@15093
   688
     "partition(a,D n) D ==> psize (%x. D (x + n)) \<le> psize D"
paulson@15093
   689
apply (simp add: psize_def [of "(%x. D (x + n))"]);
paulson@15093
   690
apply (rule_tac a = "psize D - n" in someI2, auto)
paulson@15093
   691
  apply (simp add: partition less_diff_conv)
paulson@15093
   692
 apply (case_tac "psize D \<le> n")
paulson@15093
   693
  apply (force intro: partition_rhs2 simp add: le_diff_conv)
paulson@15093
   694
 apply (simp add: partition le_diff_conv)
paulson@15093
   695
apply (rule ccontr, drule not_leE)
paulson@15093
   696
apply (drule_tac x = "psize D" in spec)
paulson@15093
   697
apply (simp add: partition)
paulson@15093
   698
done
paulson@15093
   699
paulson@15093
   700
lemma lemma_psize_right_eq:
paulson@15093
   701
     "[| partition(a,b) D |] ==> psize (%x. D (x + n)) \<le> psize D"
paulson@15219
   702
apply (frule_tac r1 = n in partition_ub [THEN order_le_imp_less_or_eq])
paulson@15093
   703
apply (blast intro: lemma_psize_right_eq1a lemma_psize_right_eq1)
paulson@15093
   704
done
paulson@15093
   705
paulson@15093
   706
lemma tpart_left1:
paulson@15093
   707
     "[| a \<le> D n; tpart (a, b) (D, p) |]
paulson@15093
   708
      ==> tpart(a, D n) (%x. if D x < D n then D(x) else D n,
paulson@15093
   709
          %x. if D x < D n then p(x) else D n)"
paulson@15093
   710
apply (frule_tac r = n in tpart_partition [THEN partition_ub])
paulson@15219
   711
apply (drule_tac x = "D n" in order_le_imp_less_or_eq)
paulson@15093
   712
apply (auto simp add: tpart_partition [THEN lemma_partition_eq, symmetric] tpart_tag_eq [symmetric])
paulson@15093
   713
apply (frule_tac tpart_partition [THEN [3] lemma_additivity1])
paulson@15093
   714
apply (auto simp add: tpart_def)
paulson@15219
   715
apply (drule_tac [2] linorder_not_less [THEN iffD1, THEN order_le_imp_less_or_eq], auto)
paulson@15219
   716
  prefer 3 apply (drule_tac x=na in spec, arith)
paulson@15093
   717
 prefer 2 apply (blast dest: lemma_additivity3)
paulson@15219
   718
apply (frule (2) lemma_additivity4_psize_eq)
paulson@15093
   719
apply (rule partition [THEN iffD2])
paulson@15093
   720
apply (frule partition [THEN iffD1])
paulson@15219
   721
apply safe 
paulson@15219
   722
apply (auto simp add: partition_lt_gen)  
nipkow@15197
   723
apply (drule (1) partition_lt_cancel, arith)
paulson@15093
   724
done
paulson@15093
   725
paulson@15093
   726
lemma fine_left1:
paulson@15093
   727
     "[| a \<le> D n; tpart (a, b) (D, p); gauge (%x. a \<le> x & x \<le> D n) g;
paulson@15093
   728
         fine (%x. if x < D n then min (g x) ((D n - x)/ 2)
paulson@15093
   729
                 else if x = D n then min (g (D n)) (ga (D n))
paulson@15093
   730
                      else min (ga x) ((x - D n)/ 2)) (D, p) |]
paulson@15093
   731
      ==> fine g
paulson@15093
   732
           (%x. if D x < D n then D(x) else D n,
paulson@15093
   733
            %x. if D x < D n then p(x) else D n)"
paulson@15093
   734
apply (auto simp add: fine_def tpart_def gauge_def)
paulson@15093
   735
apply (frule_tac [!] na=na in lemma_psize_left_less_psize2)
paulson@15093
   736
apply (drule_tac [!] x = na in spec, auto)
paulson@15093
   737
apply (drule_tac [!] x = na in spec, auto)
paulson@15093
   738
apply (auto dest: lemma_additivity3a simp add: split_if_asm)
paulson@15093
   739
done
paulson@15093
   740
paulson@15093
   741
lemma tpart_right1:
paulson@15093
   742
     "[| a \<le> D n; tpart (a, b) (D, p) |]
paulson@15093
   743
      ==> tpart(D n, b) (%x. D(x + n),%x. p(x + n))"
paulson@15093
   744
apply (simp add: tpart_def partition_def, safe)
paulson@15093
   745
apply (rule_tac x = "N - n" in exI, auto)
paulson@15221
   746
apply (drule_tac x = "na + n" in spec, arith)+
paulson@15093
   747
done
paulson@15093
   748
paulson@15093
   749
lemma fine_right1:
paulson@15093
   750
     "[| a \<le> D n; tpart (a, b) (D, p); gauge (%x. D n \<le> x & x \<le> b) ga;
paulson@15093
   751
         fine (%x. if x < D n then min (g x) ((D n - x)/ 2)
paulson@15093
   752
                 else if x = D n then min (g (D n)) (ga (D n))
paulson@15093
   753
                      else min (ga x) ((x - D n)/ 2)) (D, p) |]
paulson@15093
   754
      ==> fine ga (%x. D(x + n),%x. p(x + n))"
paulson@15093
   755
apply (auto simp add: fine_def gauge_def)
paulson@15093
   756
apply (drule_tac x = "na + n" in spec)
paulson@15093
   757
apply (frule_tac n = n in tpart_partition [THEN better_lemma_psize_right_eq], auto, arith)
paulson@15093
   758
apply (simp add: tpart_def, safe)
paulson@15094
   759
apply (subgoal_tac "D n \<le> p (na + n)")
paulson@15219
   760
apply (drule_tac y = "p (na + n)" in order_le_imp_less_or_eq)
paulson@15093
   761
apply safe
paulson@15093
   762
apply (simp split: split_if_asm, simp)
paulson@15093
   763
apply (drule less_le_trans, assumption)
paulson@15093
   764
apply (rotate_tac 5)
paulson@15093
   765
apply (drule_tac x = "na + n" in spec, safe)
paulson@15093
   766
apply (rule_tac y="D (na + n)" in order_trans)
paulson@15093
   767
apply (case_tac "na = 0", auto)
paulson@15093
   768
apply (erule partition_lt_gen [THEN order_less_imp_le], arith+)
paulson@15093
   769
done
paulson@15093
   770
paulson@15093
   771
lemma rsum_add: "rsum (D, p) (%x. f x + g x) =  rsum (D, p) f + rsum(D, p) g"
nipkow@15536
   772
by (simp add: rsum_def setsum_addf left_distrib)
paulson@15093
   773
paulson@15094
   774
text{* Bartle/Sherbert: Theorem 10.1.5 p. 278 *}
paulson@15093
   775
lemma Integral_add_fun:
paulson@15093
   776
    "[| a \<le> b; Integral(a,b) f k1; Integral(a,b) g k2 |]
paulson@15093
   777
     ==> Integral(a,b) (%x. f x + g x) (k1 + k2)"
paulson@15093
   778
apply (simp add: Integral_def, auto)
paulson@15093
   779
apply ((drule_tac x = "e/2" in spec)+)
paulson@15093
   780
apply auto
paulson@15093
   781
apply (drule gauge_min, assumption)
paulson@15094
   782
apply (rule_tac x = " (%x. if ga x < gaa x then ga x else gaa x)" in exI)
paulson@15093
   783
apply auto
paulson@15093
   784
apply (drule fine_min)
paulson@15093
   785
apply ((drule spec)+, auto)
paulson@15093
   786
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)
paulson@15093
   787
apply (auto simp only: rsum_add left_distrib [symmetric]
paulson@15093
   788
                mult_2_right [symmetric] real_mult_less_iff1, arith)
paulson@15093
   789
done
paulson@15093
   790
paulson@15093
   791
lemma partition_lt_gen2:
paulson@15093
   792
     "[| partition(a,b) D; r < psize D |] ==> 0 < D (Suc r) - D r"
paulson@15093
   793
by (auto simp add: partition)
paulson@15093
   794
paulson@15093
   795
lemma lemma_Integral_le:
paulson@15093
   796
     "[| \<forall>x. a \<le> x & x \<le> b --> f x \<le> g x;
paulson@15093
   797
         tpart(a,b) (D,p)
nipkow@15360
   798
      |] ==> \<forall>n \<le> psize D. f (p n) \<le> g (p n)"
paulson@15093
   799
apply (simp add: tpart_def)
paulson@15093
   800
apply (auto, frule partition [THEN iffD1], auto)
paulson@15093
   801
apply (drule_tac x = "p n" in spec, auto)
paulson@15093
   802
apply (case_tac "n = 0", simp)
paulson@15093
   803
apply (rule partition_lt_gen [THEN order_less_le_trans, THEN order_less_imp_le], auto)
paulson@15093
   804
apply (drule le_imp_less_or_eq, auto)
paulson@15093
   805
apply (drule_tac [2] x = "psize D" in spec, auto)
paulson@15093
   806
apply (drule_tac r = "Suc n" in partition_ub)
paulson@15093
   807
apply (drule_tac x = n in spec, auto)
paulson@15093
   808
done
paulson@15093
   809
paulson@15093
   810
lemma lemma_Integral_rsum_le:
paulson@15093
   811
     "[| \<forall>x. a \<le> x & x \<le> b --> f x \<le> g x;
paulson@15093
   812
         tpart(a,b) (D,p)
paulson@15093
   813
      |] ==> rsum(D,p) f \<le> rsum(D,p) g"
paulson@15093
   814
apply (simp add: rsum_def)
nipkow@15539
   815
apply (auto intro!: setsum_mono dest: tpart_partition [THEN partition_lt_gen2]
paulson@15093
   816
               dest!: lemma_Integral_le)
paulson@15093
   817
done
paulson@15093
   818
paulson@15093
   819
lemma Integral_le:
paulson@15093
   820
    "[| a \<le> b;
paulson@15093
   821
        \<forall>x. a \<le> x & x \<le> b --> f(x) \<le> g(x);
paulson@15093
   822
        Integral(a,b) f k1; Integral(a,b) g k2
paulson@15093
   823
     |] ==> k1 \<le> k2"
paulson@15093
   824
apply (simp add: Integral_def)
paulson@15093
   825
apply (rotate_tac 2)
paulson@15093
   826
apply (drule_tac x = "\<bar>k1 - k2\<bar> /2" in spec)
paulson@15221
   827
apply (drule_tac x = "\<bar>k1 - k2\<bar> /2" in spec, auto)
paulson@15093
   828
apply (drule gauge_min, assumption)
paulson@15093
   829
apply (drule_tac g = "%x. if ga x < gaa x then ga x else gaa x" 
paulson@15093
   830
       in partition_exists, assumption, auto)
paulson@15093
   831
apply (drule fine_min)
paulson@15093
   832
apply (drule_tac x = D in spec, drule_tac x = D in spec)
paulson@15093
   833
apply (drule_tac x = p in spec, drule_tac x = p in spec, auto)
paulson@15093
   834
apply (frule lemma_Integral_rsum_le, assumption)
paulson@15094
   835
apply (subgoal_tac "\<bar>(rsum (D,p) f - k1) - (rsum (D,p) g - k2)\<bar> < \<bar>k1 - k2\<bar>")
paulson@15093
   836
apply arith
paulson@15093
   837
apply (drule add_strict_mono, assumption)
paulson@15093
   838
apply (auto simp only: left_distrib [symmetric] mult_2_right [symmetric]
paulson@15093
   839
                       real_mult_less_iff1, arith)
paulson@15093
   840
done
paulson@15093
   841
paulson@15093
   842
lemma Integral_imp_Cauchy:
paulson@15093
   843
     "(\<exists>k. Integral(a,b) f k) ==>
nipkow@15360
   844
      (\<forall>e > 0. \<exists>g. gauge (%x. a \<le> x & x \<le> b) g &
paulson@15093
   845
                       (\<forall>D1 D2 p1 p2.
paulson@15093
   846
                            tpart(a,b) (D1, p1) & fine g (D1,p1) &
paulson@15093
   847
                            tpart(a,b) (D2, p2) & fine g (D2,p2) -->
nipkow@15360
   848
                            \<bar>rsum(D1,p1) f - rsum(D2,p2) f\<bar> < e))"
paulson@15093
   849
apply (simp add: Integral_def, auto)
paulson@15093
   850
apply (drule_tac x = "e/2" in spec, auto)
paulson@15093
   851
apply (rule exI, auto)
paulson@15093
   852
apply (frule_tac x = D1 in spec)
paulson@15093
   853
apply (frule_tac x = D2 in spec)
paulson@15093
   854
apply ((drule spec)+, auto)
paulson@15093
   855
apply (erule_tac V = "0 < e" in thin_rl)
paulson@15093
   856
apply (drule add_strict_mono, assumption)
paulson@15093
   857
apply (auto simp only: left_distrib [symmetric] mult_2_right [symmetric]
paulson@15093
   858
                       real_mult_less_iff1, arith)
paulson@15093
   859
done
paulson@15093
   860
paulson@15093
   861
lemma Cauchy_iff2:
paulson@15093
   862
     "Cauchy X =
nipkow@15360
   863
      (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. \<bar>X m + - X n\<bar> < inverse(real (Suc j))))"
paulson@15093
   864
apply (simp add: Cauchy_def, auto)
paulson@15093
   865
apply (drule reals_Archimedean, safe)
paulson@15093
   866
apply (drule_tac x = n in spec, auto)
paulson@15093
   867
apply (rule_tac x = M in exI, auto)
nipkow@15360
   868
apply (drule_tac x = m in spec, simp)
paulson@15093
   869
apply (drule_tac x = na in spec, auto)
paulson@15093
   870
done
paulson@15093
   871
paulson@15093
   872
lemma partition_exists2:
paulson@15093
   873
     "[| a \<le> b; \<forall>n. gauge (%x. a \<le> x & x \<le> b) (fa n) |]
paulson@15093
   874
      ==> \<forall>n. \<exists>D p. tpart (a, b) (D, p) & fine (fa n) (D, p)"
paulson@15219
   875
by (blast dest: partition_exists) 
paulson@15093
   876
paulson@15093
   877
lemma monotonic_anti_derivative:
paulson@15093
   878
     "[| a \<le> b; \<forall>c. a \<le> c & c \<le> b --> f' c \<le> g' c;
paulson@15093
   879
         \<forall>x. DERIV f x :> f' x; \<forall>x. DERIV g x :> g' x |]
paulson@15093
   880
      ==> f b - f a \<le> g b - g a"
paulson@15093
   881
apply (rule Integral_le, assumption)
paulson@15219
   882
apply (auto intro: FTC1) 
paulson@15093
   883
done
paulson@15093
   884
paulson@15093
   885
end