src/HOL/Hyperreal/MacLaurin.thy
author nipkow
Fri, 18 Feb 2005 11:48:53 +0100
changeset 15536 3ce1cb7a24f0
parent 15481 fc075ae929e4
child 15539 333a88244569
permissions -rw-r--r--
starting to get rid of sumr
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
12224
02df7cbe7d25 even more theories from Jacques
paulson
parents:
diff changeset
     1
(*  Title       : MacLaurin.thy
02df7cbe7d25 even more theories from Jacques
paulson
parents:
diff changeset
     2
    Author      : Jacques D. Fleuriot
02df7cbe7d25 even more theories from Jacques
paulson
parents:
diff changeset
     3
    Copyright   : 2001 University of Edinburgh
02df7cbe7d25 even more theories from Jacques
paulson
parents:
diff changeset
     4
    Description : MacLaurin series
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
     5
    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
12224
02df7cbe7d25 even more theories from Jacques
paulson
parents:
diff changeset
     6
*)
02df7cbe7d25 even more theories from Jacques
paulson
parents:
diff changeset
     7
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15081
diff changeset
     8
theory MacLaurin
15140
322485b816ac import -> imports
nipkow
parents: 15131
diff changeset
     9
imports Log
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15081
diff changeset
    10
begin
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    11
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    12
lemma sumr_offset: "sumr 0 n (%m. f (m+k)) = sumr 0 (n+k) f - sumr 0 k f"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15234
diff changeset
    13
by (induct "n", auto)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    14
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    15
lemma sumr_offset2: "\<forall>f. sumr 0 n (%m. f (m+k)) = sumr 0 (n+k) f - sumr 0 k f"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15234
diff changeset
    16
by (induct "n", auto)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    17
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    18
lemma sumr_offset3: "sumr 0 (n+k) f = sumr 0 n (%m. f (m+k)) + sumr 0 k f"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    19
by (simp  add: sumr_offset)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    20
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    21
lemma sumr_offset4: "\<forall>n f. sumr 0 (n+k) f = sumr 0 n (%m. f (m+k)) + sumr 0 k f"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    22
by (simp add: sumr_offset)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    23
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    24
lemma sumr_from_1_from_0: "0 < n ==>
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    25
      sumr (Suc 0) (Suc n) (%n. (if even(n) then 0 else
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    26
             ((- 1) ^ ((n - (Suc 0)) div 2))/(real (fact n))) * a ^ n) =
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    27
      sumr 0 (Suc n) (%n. (if even(n) then 0 else
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    28
             ((- 1) ^ ((n - (Suc 0)) div 2))/(real (fact n))) * a ^ n)"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    29
by (rule_tac n1 = 1 in sumr_split_add [THEN subst], auto)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    30
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    31
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    32
subsection{*Maclaurin's Theorem with Lagrange Form of Remainder*}
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    33
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    34
text{*This is a very long, messy proof even now that it's been broken down
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    35
into lemmas.*}
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    36
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    37
lemma Maclaurin_lemma:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    38
    "0 < h ==>
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    39
     \<exists>B. f h = sumr 0 n (%m. (j m / real (fact m)) * (h^m)) +
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    40
               (B * ((h^n) / real(fact n)))"
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
    41
apply (rule_tac x = "(f h - sumr 0 n (%m. (j m / real (fact m)) * h^m)) *
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    42
                 real(fact n) / (h^n)"
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
    43
       in exI)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
    44
apply (simp add: times_divide_eq) 
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
    45
done
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    46
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    47
lemma eq_diff_eq': "(x = y - z) = (y = x + (z::real))"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    48
by arith
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    49
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    50
text{*A crude tactic to differentiate by proof.*}
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    51
ML
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    52
{*
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    53
exception DERIV_name;
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    54
fun get_fun_name (_ $ (Const ("Lim.deriv",_) $ Abs(_,_, Const (f,_) $ _) $ _ $ _)) = f
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    55
|   get_fun_name (_ $ (_ $ (Const ("Lim.deriv",_) $ Abs(_,_, Const (f,_) $ _) $ _ $ _))) = f
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    56
|   get_fun_name _ = raise DERIV_name;
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    57
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    58
val deriv_rulesI = [DERIV_Id,DERIV_const,DERIV_cos,DERIV_cmult,
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    59
                    DERIV_sin, DERIV_exp, DERIV_inverse,DERIV_pow,
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    60
                    DERIV_add, DERIV_diff, DERIV_mult, DERIV_minus,
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    61
                    DERIV_inverse_fun,DERIV_quotient,DERIV_fun_pow,
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    62
                    DERIV_fun_exp,DERIV_fun_sin,DERIV_fun_cos,
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    63
                    DERIV_Id,DERIV_const,DERIV_cos];
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    64
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    65
val deriv_tac =
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    66
  SUBGOAL (fn (prem,i) =>
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    67
   (resolve_tac deriv_rulesI i) ORELSE
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    68
    ((rtac (read_instantiate [("f",get_fun_name prem)]
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    69
                     DERIV_chain2) i) handle DERIV_name => no_tac));;
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    70
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    71
val DERIV_tac = ALLGOALS(fn i => REPEAT(deriv_tac i));
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    72
*}
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    73
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    74
lemma Maclaurin_lemma2:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    75
      "[| \<forall>m t. m < n \<and> 0\<le>t \<and> t\<le>h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t;
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    76
          n = Suc k;
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    77
        difg =
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    78
        (\<lambda>m t. diff m t -
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    79
               ((\<Sum>p = 0..<n - m. diff (m + p) 0 / real (fact p) * t ^ p) +
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    80
                B * (t ^ (n - m) / real (fact (n - m)))))|] ==>
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    81
        \<forall>m t. m < n & 0 \<le> t & t \<le> h -->
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    82
                    DERIV (difg m) t :> difg (Suc m) t"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    83
apply clarify
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    84
apply (rule DERIV_diff)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    85
apply (simp (no_asm_simp))
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    86
apply (tactic DERIV_tac)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    87
apply (tactic DERIV_tac)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    88
apply (rule_tac [2] lemma_DERIV_subst)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    89
apply (rule_tac [2] DERIV_quotient)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    90
apply (rule_tac [3] DERIV_const)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    91
apply (rule_tac [2] DERIV_pow)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    92
  prefer 3 apply (simp add: fact_diff_Suc)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    93
 prefer 2 apply simp
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    94
apply (frule_tac m = m in less_add_one, clarify)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    95
apply (simp del: sumr_Suc)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    96
apply (insert sumr_offset4 [of 1])
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    97
apply (simp del: sumr_Suc fact_Suc realpow_Suc)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    98
apply (rule lemma_DERIV_subst)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
    99
apply (rule DERIV_add)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   100
apply (rule_tac [2] DERIV_const)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   101
apply (rule DERIV_sumr, clarify)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   102
 prefer 2 apply simp
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   103
apply (simp (no_asm) add: divide_inverse mult_assoc del: fact_Suc realpow_Suc)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   104
apply (rule DERIV_cmult)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   105
apply (rule lemma_DERIV_subst)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   106
apply (best intro: DERIV_chain2 intro!: DERIV_intros)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   107
apply (subst fact_Suc)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   108
apply (subst real_of_nat_mult)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   109
apply (simp add: inverse_mult_distrib mult_ac)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   110
done
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   111
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   112
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   113
lemma Maclaurin_lemma3:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   114
     "[|\<forall>k t. k < Suc m \<and> 0\<le>t & t\<le>h \<longrightarrow> DERIV (difg k) t :> difg (Suc k) t;
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   115
        \<forall>k<Suc m. difg k 0 = 0; DERIV (difg n) t :> 0;  n < m; 0 < t;
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   116
        t < h|]
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   117
     ==> \<exists>ta. 0 < ta & ta < t & DERIV (difg (Suc n)) ta :> 0"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   118
apply (rule Rolle, assumption, simp)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   119
apply (drule_tac x = n and P="%k. k<Suc m --> difg k 0 = 0" in spec)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   120
apply (rule DERIV_unique)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   121
prefer 2 apply assumption
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   122
apply force
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   123
apply (subgoal_tac "\<forall>ta. 0 \<le> ta & ta \<le> t --> (difg (Suc n)) differentiable ta")
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   124
apply (simp add: differentiable_def)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   125
apply (blast dest!: DERIV_isCont)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   126
apply (simp add: differentiable_def, clarify)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   127
apply (rule_tac x = "difg (Suc (Suc n)) ta" in exI)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   128
apply force
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   129
apply (simp add: differentiable_def, clarify)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   130
apply (rule_tac x = "difg (Suc (Suc n)) x" in exI)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   131
apply force
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   132
done
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 12224
diff changeset
   133
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   134
lemma Maclaurin:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   135
   "[| 0 < h; 0 < n; diff 0 = f;
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   136
       \<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t |]
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   137
    ==> \<exists>t. 0 < t &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   138
              t < h &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   139
              f h =
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   140
              sumr 0 n (%m. (diff m 0 / real (fact m)) * h ^ m) +
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   141
              (diff n t / real (fact n)) * h ^ n"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   142
apply (case_tac "n = 0", force)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   143
apply (drule not0_implies_Suc)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   144
apply (erule exE)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   145
apply (frule_tac f=f and n=n and j="%m. diff m 0" in Maclaurin_lemma)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   146
apply (erule exE)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   147
apply (subgoal_tac "\<exists>g.
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   148
     g = (%t. f t - (sumr 0 n (%m. (diff m 0 / real(fact m)) * t^m) + (B * (t^n / real(fact n)))))")
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   149
 prefer 2 apply blast
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   150
apply (erule exE)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   151
apply (subgoal_tac "g 0 = 0 & g h =0")
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   152
 prefer 2
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   153
 apply (simp del: sumr_Suc)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   154
 apply (cut_tac n = m and k = 1 in sumr_offset2)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   155
 apply (simp add: eq_diff_eq' del: sumr_Suc)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   156
apply (subgoal_tac "\<exists>difg. difg = (%m t. diff m t - (sumr 0 (n - m) (%p. (diff (m + p) 0 / real (fact p)) * (t ^ p)) + (B * ((t ^ (n - m)) / real (fact (n - m))))))")
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   157
 prefer 2 apply blast
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   158
apply (erule exE)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   159
apply (subgoal_tac "difg 0 = g")
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   160
 prefer 2 apply simp
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   161
apply (frule Maclaurin_lemma2, assumption+)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   162
apply (subgoal_tac "\<forall>ma. ma < n --> (\<exists>t. 0 < t & t < h & difg (Suc ma) t = 0) ")
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   163
 apply (drule_tac x = m and P="%m. m<n --> (\<exists>t. ?QQ m t)" in spec)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   164
 apply (erule impE)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   165
  apply (simp (no_asm_simp))
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   166
 apply (erule exE)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   167
 apply (rule_tac x = t in exI)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   168
 apply (simp add: times_divide_eq del: realpow_Suc fact_Suc)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   169
apply (subgoal_tac "\<forall>m. m < n --> difg m 0 = 0")
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   170
 prefer 2
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   171
 apply clarify
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   172
 apply simp
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   173
 apply (frule_tac m = ma in less_add_one, clarify)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   174
 apply (simp del: sumr_Suc)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   175
apply (insert sumr_offset4 [of 1])
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   176
apply (simp del: sumr_Suc fact_Suc realpow_Suc)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   177
apply (subgoal_tac "\<forall>m. m < n --> (\<exists>t. 0 < t & t < h & DERIV (difg m) t :> 0) ")
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   178
apply (rule allI, rule impI)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   179
apply (drule_tac x = ma and P="%m. m<n --> (\<exists>t. ?QQ m t)" in spec)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   180
apply (erule impE, assumption)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   181
apply (erule exE)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   182
apply (rule_tac x = t in exI)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   183
(* do some tidying up *)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   184
apply (erule_tac [!] V= "difg = (%m t. diff m t - (sumr 0 (n - m) (%p. diff (m + p) 0 / real (fact p) * t ^ p) + B * (t ^ (n - m) / real (fact (n - m)))))"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   185
       in thin_rl)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   186
apply (erule_tac [!] V="g = (%t. f t - (sumr 0 n (%m. diff m 0 / real (fact m) * t ^ m) + B * (t ^ n / real (fact n))))"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   187
       in thin_rl)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   188
apply (erule_tac [!] V="f h = sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) + B * (h ^ n / real (fact n))"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   189
       in thin_rl)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   190
(* back to business *)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   191
apply (simp (no_asm_simp))
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   192
apply (rule DERIV_unique)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   193
prefer 2 apply blast
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   194
apply force
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   195
apply (rule allI, induct_tac "ma")
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   196
apply (rule impI, rule Rolle, assumption, simp, simp)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   197
apply (subgoal_tac "\<forall>t. 0 \<le> t & t \<le> h --> g differentiable t")
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   198
apply (simp add: differentiable_def)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   199
apply (blast dest: DERIV_isCont)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   200
apply (simp add: differentiable_def, clarify)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   201
apply (rule_tac x = "difg (Suc 0) t" in exI)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   202
apply force
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   203
apply (simp add: differentiable_def, clarify)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   204
apply (rule_tac x = "difg (Suc 0) x" in exI)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   205
apply force
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   206
apply safe
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   207
apply force
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   208
apply (frule Maclaurin_lemma3, assumption+, safe)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   209
apply (rule_tac x = ta in exI, force)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   210
done
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   211
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   212
lemma Maclaurin_objl:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   213
     "0 < h & 0 < n & diff 0 = f &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   214
       (\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   215
    --> (\<exists>t. 0 < t &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   216
              t < h &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   217
              f h =
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   218
              sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) +
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   219
              diff n t / real (fact n) * h ^ n)"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   220
by (blast intro: Maclaurin)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   221
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   222
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   223
lemma Maclaurin2:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   224
   "[| 0 < h; diff 0 = f;
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   225
       \<forall>m t.
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   226
          m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t |]
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   227
    ==> \<exists>t. 0 < t &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   228
              t \<le> h &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   229
              f h =
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   230
              sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) +
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   231
              diff n t / real (fact n) * h ^ n"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   232
apply (case_tac "n", auto)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   233
apply (drule Maclaurin, auto)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   234
done
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   235
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   236
lemma Maclaurin2_objl:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   237
     "0 < h & diff 0 = f &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   238
       (\<forall>m t.
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   239
          m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   240
    --> (\<exists>t. 0 < t &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   241
              t \<le> h &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   242
              f h =
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   243
              sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) +
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   244
              diff n t / real (fact n) * h ^ n)"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   245
by (blast intro: Maclaurin2)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   246
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   247
lemma Maclaurin_minus:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   248
   "[| h < 0; 0 < n; diff 0 = f;
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   249
       \<forall>m t. m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t |]
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   250
    ==> \<exists>t. h < t &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   251
              t < 0 &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   252
              f h =
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   253
              sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) +
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   254
              diff n t / real (fact n) * h ^ n"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   255
apply (cut_tac f = "%x. f (-x)"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   256
        and diff = "%n x. ((- 1) ^ n) * diff n (-x)"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   257
        and h = "-h" and n = n in Maclaurin_objl)
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   258
apply (simp add: times_divide_eq) 
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   259
apply safe
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   260
apply (subst minus_mult_right)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   261
apply (rule DERIV_cmult)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   262
apply (rule lemma_DERIV_subst)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   263
apply (rule DERIV_chain2 [where g=uminus])
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   264
apply (rule_tac [2] DERIV_minus, rule_tac [2] DERIV_Id)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   265
prefer 2 apply force
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   266
apply force
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   267
apply (rule_tac x = "-t" in exI, auto)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   268
apply (subgoal_tac "(\<Sum>m = 0..<n. -1 ^ m * diff m 0 * (-h)^m / real(fact m)) =
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   269
                    (\<Sum>m = 0..<n. diff m 0 * h ^ m / real(fact m))")
15536
3ce1cb7a24f0 starting to get rid of sumr
nipkow
parents: 15481
diff changeset
   270
apply (rule_tac [2] setsum_cong[OF refl])
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   271
apply (auto simp add: divide_inverse power_mult_distrib [symmetric])
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   272
done
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   273
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   274
lemma Maclaurin_minus_objl:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   275
     "(h < 0 & 0 < n & diff 0 = f &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   276
       (\<forall>m t.
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   277
          m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t))
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   278
    --> (\<exists>t. h < t &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   279
              t < 0 &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   280
              f h =
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   281
              sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) +
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   282
              diff n t / real (fact n) * h ^ n)"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   283
by (blast intro: Maclaurin_minus)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   284
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   285
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   286
subsection{*More Convenient "Bidirectional" Version.*}
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   287
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   288
(* not good for PVS sin_approx, cos_approx *)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   289
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   290
lemma Maclaurin_bi_le_lemma [rule_format]:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   291
     "0 < n \<longrightarrow>
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   292
       diff 0 0 =
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   293
       (\<Sum>m = 0..<n. diff m 0 * 0 ^ m / real (fact m)) +
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   294
       diff n 0 * 0 ^ n / real (fact n)"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15234
diff changeset
   295
by (induct "n", auto)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 12224
diff changeset
   296
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   297
lemma Maclaurin_bi_le:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   298
   "[| diff 0 = f;
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   299
       \<forall>m t. m < n & abs t \<le> abs x --> DERIV (diff m) t :> diff (Suc m) t |]
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   300
    ==> \<exists>t. abs t \<le> abs x &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   301
              f x =
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   302
              sumr 0 n (%m. diff m 0 / real (fact m) * x ^ m) +
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   303
              diff n t / real (fact n) * x ^ n"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   304
apply (case_tac "n = 0", force)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   305
apply (case_tac "x = 0")
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   306
apply (rule_tac x = 0 in exI)
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   307
apply (force simp add: Maclaurin_bi_le_lemma times_divide_eq)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   308
apply (cut_tac x = x and y = 0 in linorder_less_linear, auto)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   309
txt{*Case 1, where @{term "x < 0"}*}
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   310
apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_minus_objl, safe)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   311
apply (simp add: abs_if)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   312
apply (rule_tac x = t in exI)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   313
apply (simp add: abs_if)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   314
txt{*Case 2, where @{term "0 < x"}*}
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   315
apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_objl, safe)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   316
apply (simp add: abs_if)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   317
apply (rule_tac x = t in exI)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   318
apply (simp add: abs_if)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   319
done
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   320
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   321
lemma Maclaurin_all_lt:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   322
     "[| diff 0 = f;
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   323
         \<forall>m x. DERIV (diff m) x :> diff(Suc m) x;
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   324
        x ~= 0; 0 < n
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   325
      |] ==> \<exists>t. 0 < abs t & abs t < abs x &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   326
               f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) +
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   327
                     (diff n t / real (fact n)) * x ^ n"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   328
apply (rule_tac x = x and y = 0 in linorder_cases)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   329
prefer 2 apply blast
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   330
apply (drule_tac [2] diff=diff in Maclaurin)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   331
apply (drule_tac diff=diff in Maclaurin_minus, simp_all, safe)
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15140
diff changeset
   332
apply (rule_tac [!] x = t in exI, auto)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   333
done
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   334
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   335
lemma Maclaurin_all_lt_objl:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   336
     "diff 0 = f &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   337
      (\<forall>m x. DERIV (diff m) x :> diff(Suc m) x) &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   338
      x ~= 0 & 0 < n
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   339
      --> (\<exists>t. 0 < abs t & abs t < abs x &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   340
               f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) +
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   341
                     (diff n t / real (fact n)) * x ^ n)"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   342
by (blast intro: Maclaurin_all_lt)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   343
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   344
lemma Maclaurin_zero [rule_format]:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   345
     "x = (0::real)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   346
      ==> 0 < n -->
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   347
          sumr 0 n (%m. (diff m (0::real) / real (fact m)) * x ^ m) =
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   348
          diff 0 0"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   349
by (induct n, auto)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   350
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   351
lemma Maclaurin_all_le: "[| diff 0 = f;
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   352
        \<forall>m x. DERIV (diff m) x :> diff (Suc m) x
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   353
      |] ==> \<exists>t. abs t \<le> abs x &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   354
              f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) +
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   355
                    (diff n t / real (fact n)) * x ^ n"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   356
apply (insert linorder_le_less_linear [of n 0])
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   357
apply (erule disjE, force)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   358
apply (case_tac "x = 0")
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   359
apply (frule_tac diff = diff and n = n in Maclaurin_zero, assumption)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   360
apply (drule gr_implies_not0 [THEN not0_implies_Suc])
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   361
apply (rule_tac x = 0 in exI, force)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   362
apply (frule_tac diff = diff and n = n in Maclaurin_all_lt, auto)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   363
apply (rule_tac x = t in exI, auto)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   364
done
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   365
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   366
lemma Maclaurin_all_le_objl: "diff 0 = f &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   367
      (\<forall>m x. DERIV (diff m) x :> diff (Suc m) x)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   368
      --> (\<exists>t. abs t \<le> abs x &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   369
              f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) +
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   370
                    (diff n t / real (fact n)) * x ^ n)"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   371
by (blast intro: Maclaurin_all_le)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   372
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   373
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   374
subsection{*Version for Exponential Function*}
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   375
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   376
lemma Maclaurin_exp_lt: "[| x ~= 0; 0 < n |]
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   377
      ==> (\<exists>t. 0 < abs t &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   378
                abs t < abs x &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   379
                exp x = sumr 0 n (%m. (x ^ m) / real (fact m)) +
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   380
                        (exp t / real (fact n)) * x ^ n)"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   381
by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_lt_objl, auto)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   382
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   383
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   384
lemma Maclaurin_exp_le:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   385
     "\<exists>t. abs t \<le> abs x &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   386
            exp x = sumr 0 n (%m. (x ^ m) / real (fact m)) +
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   387
                       (exp t / real (fact n)) * x ^ n"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   388
by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_le_objl, auto)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   389
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   390
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   391
subsection{*Version for Sine Function*}
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   392
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   393
lemma MVT2:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   394
     "[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   395
      ==> \<exists>z. a < z & z < b & (f b - f a = (b - a) * f'(z))"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   396
apply (drule MVT)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   397
apply (blast intro: DERIV_isCont)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   398
apply (force dest: order_less_imp_le simp add: differentiable_def)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   399
apply (blast dest: DERIV_unique order_less_imp_le)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   400
done
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   401
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   402
lemma mod_exhaust_less_4:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   403
     "m mod 4 = 0 | m mod 4 = 1 | m mod 4 = 2 | m mod 4 = (3::nat)"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   404
by (case_tac "m mod 4", auto, arith)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   405
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   406
lemma Suc_Suc_mult_two_diff_two [rule_format, simp]:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   407
     "0 < n --> Suc (Suc (2 * n - 2)) = 2*n"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15234
diff changeset
   408
by (induct "n", auto)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   409
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   410
lemma lemma_Suc_Suc_4n_diff_2 [rule_format, simp]:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   411
     "0 < n --> Suc (Suc (4*n - 2)) = 4*n"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15234
diff changeset
   412
by (induct "n", auto)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   413
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   414
lemma Suc_mult_two_diff_one [rule_format, simp]:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   415
      "0 < n --> Suc (2 * n - 1) = 2*n"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15234
diff changeset
   416
by (induct "n", auto)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   417
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   418
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   419
text{*It is unclear why so many variant results are needed.*}
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   420
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   421
lemma Maclaurin_sin_expansion2:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   422
     "\<exists>t. abs t \<le> abs x &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   423
       sin x =
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   424
       (sumr 0 n (%m. (if even m then 0
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   425
                       else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   426
                       x ^ m))
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   427
      + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   428
apply (cut_tac f = sin and n = n and x = x
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   429
        and diff = "%n x. sin (x + 1/2*real n * pi)" in Maclaurin_all_lt_objl)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   430
apply safe
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   431
apply (simp (no_asm))
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   432
apply (simp (no_asm) add: times_divide_eq)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   433
apply (case_tac "n", clarify, simp, simp)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   434
apply (rule ccontr, simp)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   435
apply (drule_tac x = x in spec, simp)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   436
apply (erule ssubst)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   437
apply (rule_tac x = t in exI, simp)
15536
3ce1cb7a24f0 starting to get rid of sumr
nipkow
parents: 15481
diff changeset
   438
apply (rule setsum_cong[OF refl])
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   439
apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex times_divide_eq)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   440
done
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   441
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   442
lemma Maclaurin_sin_expansion:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   443
     "\<exists>t. sin x =
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   444
       (sumr 0 n (%m. (if even m then 0
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   445
                       else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   446
                       x ^ m))
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   447
      + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   448
apply (insert Maclaurin_sin_expansion2 [of x n]) 
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   449
apply (blast intro: elim:); 
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   450
done
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   451
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   452
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   453
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   454
lemma Maclaurin_sin_expansion3:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   455
     "[| 0 < n; 0 < x |] ==>
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   456
       \<exists>t. 0 < t & t < x &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   457
       sin x =
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   458
       (sumr 0 n (%m. (if even m then 0
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   459
                       else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   460
                       x ^ m))
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   461
      + ((sin(t + 1/2 * real(n) *pi) / real (fact n)) * x ^ n)"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   462
apply (cut_tac f = sin and n = n and h = x and diff = "%n x. sin (x + 1/2*real (n) *pi)" in Maclaurin_objl)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   463
apply safe
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   464
apply simp
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   465
apply (simp (no_asm) add: times_divide_eq)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   466
apply (erule ssubst)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   467
apply (rule_tac x = t in exI, simp)
15536
3ce1cb7a24f0 starting to get rid of sumr
nipkow
parents: 15481
diff changeset
   468
apply (rule setsum_cong[OF refl])
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   469
apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex times_divide_eq)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   470
done
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   471
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   472
lemma Maclaurin_sin_expansion4:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   473
     "0 < x ==>
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   474
       \<exists>t. 0 < t & t \<le> x &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   475
       sin x =
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   476
       (sumr 0 n (%m. (if even m then 0
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   477
                       else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   478
                       x ^ m))
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   479
      + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   480
apply (cut_tac f = sin and n = n and h = x and diff = "%n x. sin (x + 1/2*real (n) *pi)" in Maclaurin2_objl)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   481
apply safe
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   482
apply simp
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   483
apply (simp (no_asm) add: times_divide_eq)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   484
apply (erule ssubst)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   485
apply (rule_tac x = t in exI, simp)
15536
3ce1cb7a24f0 starting to get rid of sumr
nipkow
parents: 15481
diff changeset
   486
apply (rule setsum_cong[OF refl])
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   487
apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex times_divide_eq)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   488
done
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   489
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   490
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   491
subsection{*Maclaurin Expansion for Cosine Function*}
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   492
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   493
lemma sumr_cos_zero_one [simp]:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   494
     "sumr 0 (Suc n)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   495
         (%m. (if even m
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   496
               then (- 1) ^ (m div 2)/(real  (fact m))
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   497
               else 0) *
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   498
              0 ^ m) = 1"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15234
diff changeset
   499
by (induct "n", auto)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   500
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   501
lemma Maclaurin_cos_expansion:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   502
     "\<exists>t. abs t \<le> abs x &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   503
       cos x =
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   504
       (sumr 0 n (%m. (if even m
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   505
                       then (- 1) ^ (m div 2)/(real (fact m))
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   506
                       else 0) *
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   507
                       x ^ m))
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   508
      + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   509
apply (cut_tac f = cos and n = n and x = x and diff = "%n x. cos (x + 1/2*real (n) *pi)" in Maclaurin_all_lt_objl)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   510
apply safe
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   511
apply (simp (no_asm))
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   512
apply (simp (no_asm) add: times_divide_eq)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   513
apply (case_tac "n", simp)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   514
apply (simp del: sumr_Suc)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   515
apply (rule ccontr, simp)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   516
apply (drule_tac x = x in spec, simp)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   517
apply (erule ssubst)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   518
apply (rule_tac x = t in exI, simp)
15536
3ce1cb7a24f0 starting to get rid of sumr
nipkow
parents: 15481
diff changeset
   519
apply (rule setsum_cong[OF refl])
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   520
apply (auto simp add: cos_zero_iff even_mult_two_ex)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   521
done
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   522
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   523
lemma Maclaurin_cos_expansion2:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   524
     "[| 0 < x; 0 < n |] ==>
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   525
       \<exists>t. 0 < t & t < x &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   526
       cos x =
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   527
       (sumr 0 n (%m. (if even m
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   528
                       then (- 1) ^ (m div 2)/(real (fact m))
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   529
                       else 0) *
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   530
                       x ^ m))
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   531
      + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   532
apply (cut_tac f = cos and n = n and h = x and diff = "%n x. cos (x + 1/2*real (n) *pi)" in Maclaurin_objl)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   533
apply safe
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   534
apply simp
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   535
apply (simp (no_asm) add: times_divide_eq)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   536
apply (erule ssubst)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   537
apply (rule_tac x = t in exI, simp)
15536
3ce1cb7a24f0 starting to get rid of sumr
nipkow
parents: 15481
diff changeset
   538
apply (rule setsum_cong[OF refl])
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   539
apply (auto simp add: cos_zero_iff even_mult_two_ex)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   540
done
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   541
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   542
lemma Maclaurin_minus_cos_expansion:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   543
     "[| x < 0; 0 < n |] ==>
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   544
       \<exists>t. x < t & t < 0 &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   545
       cos x =
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   546
       (sumr 0 n (%m. (if even m
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   547
                       then (- 1) ^ (m div 2)/(real (fact m))
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   548
                       else 0) *
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   549
                       x ^ m))
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   550
      + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   551
apply (cut_tac f = cos and n = n and h = x and diff = "%n x. cos (x + 1/2*real (n) *pi)" in Maclaurin_minus_objl)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   552
apply safe
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   553
apply simp
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   554
apply (simp (no_asm) add: times_divide_eq)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   555
apply (erule ssubst)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   556
apply (rule_tac x = t in exI, simp)
15536
3ce1cb7a24f0 starting to get rid of sumr
nipkow
parents: 15481
diff changeset
   557
apply (rule setsum_cong[OF refl])
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   558
apply (auto simp add: cos_zero_iff even_mult_two_ex)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   559
done
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   560
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   561
(* ------------------------------------------------------------------------- *)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   562
(* Version for ln(1 +/- x). Where is it??                                    *)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   563
(* ------------------------------------------------------------------------- *)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   564
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   565
lemma sin_bound_lemma:
15081
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
   566
    "[|x = y; abs u \<le> (v::real) |] ==> \<bar>(x + u) - y\<bar> \<le> v"
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   567
by auto
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   568
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   569
lemma Maclaurin_sin_bound:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   570
  "abs(sin x - sumr 0 n (%m. (if even m then 0 else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
15081
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
   571
  x ^ m))  \<le> inverse(real (fact n)) * \<bar>x\<bar> ^ n"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 12224
diff changeset
   572
proof -
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   573
  have "!! x (y::real). x \<le> 1 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x * y \<le> 1 * y"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 12224
diff changeset
   574
    by (rule_tac mult_right_mono,simp_all)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 12224
diff changeset
   575
  note est = this[simplified]
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 12224
diff changeset
   576
  show ?thesis
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   577
    apply (cut_tac f=sin and n=n and x=x and
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 12224
diff changeset
   578
      diff = "%n x. if n mod 4 = 0 then sin(x) else if n mod 4 = 1 then cos(x) else if n mod 4 = 2 then -sin(x) else -cos(x)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 12224
diff changeset
   579
      in Maclaurin_all_le_objl)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   580
    apply safe
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   581
    apply simp
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15251
diff changeset
   582
    apply (simplesubst mod_Suc_eq_Suc_mod)  --{*simultaneous substitution*}
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   583
    apply (cut_tac m=m in mod_exhaust_less_4, safe, simp+)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 12224
diff changeset
   584
    apply (rule DERIV_minus, simp+)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 12224
diff changeset
   585
    apply (rule lemma_DERIV_subst, rule DERIV_minus, rule DERIV_cos, simp)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   586
    apply (erule ssubst)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   587
    apply (rule sin_bound_lemma)
15536
3ce1cb7a24f0 starting to get rid of sumr
nipkow
parents: 15481
diff changeset
   588
    apply (rule setsum_cong[OF refl])
3ce1cb7a24f0 starting to get rid of sumr
nipkow
parents: 15481
diff changeset
   589
    apply (rule_tac f = "%u. u * (x^xa)" in arg_cong)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 12224
diff changeset
   590
    apply (subst even_even_mod_4_iff)
15536
3ce1cb7a24f0 starting to get rid of sumr
nipkow
parents: 15481
diff changeset
   591
    apply (cut_tac m=xa in mod_exhaust_less_4, simp, safe)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 12224
diff changeset
   592
    apply (simp_all add:even_num_iff)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 12224
diff changeset
   593
    apply (drule lemma_even_mod_4_div_2[simplified])
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   594
    apply(simp add: numeral_2_eq_2 divide_inverse)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   595
    apply (drule lemma_odd_mod_4_div_2)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   596
    apply (simp add: numeral_2_eq_2 divide_inverse)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   597
    apply (auto intro: mult_right_mono [where b=1, simplified] mult_right_mono
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   598
                   simp add: est mult_pos_le mult_ac divide_inverse
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   599
                          power_abs [symmetric])
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 12224
diff changeset
   600
    done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 12224
diff changeset
   601
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 12224
diff changeset
   602
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   603
end