src/HOL/Hyperreal/MacLaurin.thy

 (*  Title       : MacLaurin.thy
     Author      : Jacques D. Fleuriot
     Copyright   : 2001 University of Edinburgh
     Description : MacLaurin series
+    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
 *)
 
-theory MacLaurin = Log
-files ("MacLaurin_lemmas.ML"):
-
-use "MacLaurin_lemmas.ML"
-
-lemma Maclaurin_sin_bound: 
-  "abs(sin x - sumr 0 n (%m. (if even m then 0 else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) * 
-  x ^ m))  <= inverse(real (fact n)) * abs(x) ^ n"
+theory MacLaurin = Log:
+
+lemma sumr_offset: "sumr 0 n (%m. f (m+k)) = sumr 0 (n+k) f - sumr 0 k f"
+by (induct_tac "n", auto)
+
+lemma sumr_offset2: "\<forall>f. sumr 0 n (%m. f (m+k)) = sumr 0 (n+k) f - sumr 0 k f"
+by (induct_tac "n", auto)
+
+lemma sumr_offset3: "sumr 0 (n+k) f = sumr 0 n (%m. f (m+k)) + sumr 0 k f"
+by (simp  add: sumr_offset)
+
+lemma sumr_offset4: "\<forall>n f. sumr 0 (n+k) f = sumr 0 n (%m. f (m+k)) + sumr 0 k f"
+by (simp add: sumr_offset)
+
+lemma sumr_from_1_from_0: "0 < n ==>
+      sumr (Suc 0) (Suc n) (%n. (if even(n) then 0 else
+             ((- 1) ^ ((n - (Suc 0)) div 2))/(real (fact n))) * a ^ n) =
+      sumr 0 (Suc n) (%n. (if even(n) then 0 else
+             ((- 1) ^ ((n - (Suc 0)) div 2))/(real (fact n))) * a ^ n)"
+by (rule_tac n1 = 1 in sumr_split_add [THEN subst], auto)
+
+
+subsection{*Maclaurin's Theorem with Lagrange Form of Remainder*}
+
+text{*This is a very long, messy proof even now that it's been broken down
+into lemmas.*}
+
+lemma Maclaurin_lemma:
+    "0 < h ==>
+     \<exists>B. f h = sumr 0 n (%m. (j m / real (fact m)) * (h^m)) +
+               (B * ((h^n) / real(fact n)))"
+by (rule_tac x = "(f h - sumr 0 n (%m. (j m / real (fact m)) * h^m)) *
+                 real(fact n) / (h^n)"
+       in exI, auto)
+
+
+lemma eq_diff_eq': "(x = y - z) = (y = x + (z::real))"
+by arith
+
+text{*A crude tactic to differentiate by proof.*}
+ML
+{*
+exception DERIV_name;
+fun get_fun_name (_ $ (Const ("Lim.deriv",_) $ Abs(_,_, Const (f,_) $ _) $ _ $ _)) = f
+|   get_fun_name (_ $ (_ $ (Const ("Lim.deriv",_) $ Abs(_,_, Const (f,_) $ _) $ _ $ _))) = f
+|   get_fun_name _ = raise DERIV_name;
+
+val deriv_rulesI = [DERIV_Id,DERIV_const,DERIV_cos,DERIV_cmult,
+                    DERIV_sin, DERIV_exp, DERIV_inverse,DERIV_pow,
+                    DERIV_add, DERIV_diff, DERIV_mult, DERIV_minus,
+                    DERIV_inverse_fun,DERIV_quotient,DERIV_fun_pow,
+                    DERIV_fun_exp,DERIV_fun_sin,DERIV_fun_cos,
+                    DERIV_Id,DERIV_const,DERIV_cos];
+
+val deriv_tac =
+  SUBGOAL (fn (prem,i) =>
+   (resolve_tac deriv_rulesI i) ORELSE
+    ((rtac (read_instantiate [("f",get_fun_name prem)]
+                     DERIV_chain2) i) handle DERIV_name => no_tac));;
+
+val DERIV_tac = ALLGOALS(fn i => REPEAT(deriv_tac i));
+*}
+
+lemma Maclaurin_lemma2:
+      "[| \<forall>m t. m < n \<and> 0\<le>t \<and> t\<le>h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t;
+          n = Suc k;
+        difg =
+        (\<lambda>m t. diff m t -
+               ((\<Sum>p = 0..<n - m. diff (m + p) 0 / real (fact p) * t ^ p) +
+                B * (t ^ (n - m) / real (fact (n - m)))))|] ==>
+        \<forall>m t. m < n & 0 \<le> t & t \<le> h -->
+                    DERIV (difg m) t :> difg (Suc m) t"
+apply clarify
+apply (rule DERIV_diff)
+apply (simp (no_asm_simp))
+apply (tactic DERIV_tac)
+apply (tactic DERIV_tac)
+apply (rule_tac [2] lemma_DERIV_subst)
+apply (rule_tac [2] DERIV_quotient)
+apply (rule_tac [3] DERIV_const)
+apply (rule_tac [2] DERIV_pow)
+  prefer 3 apply (simp add: fact_diff_Suc)
+ prefer 2 apply simp
+apply (frule_tac m = m in less_add_one, clarify)
+apply (simp del: sumr_Suc)
+apply (insert sumr_offset4 [of 1])
+apply (simp del: sumr_Suc fact_Suc realpow_Suc)
+apply (rule lemma_DERIV_subst)
+apply (rule DERIV_add)
+apply (rule_tac [2] DERIV_const)
+apply (rule DERIV_sumr, clarify)
+ prefer 2 apply simp
+apply (simp (no_asm) add: divide_inverse mult_assoc del: fact_Suc realpow_Suc)
+apply (rule DERIV_cmult)
+apply (rule lemma_DERIV_subst)
+apply (best intro: DERIV_chain2 intro!: DERIV_intros)
+apply (subst fact_Suc)
+apply (subst real_of_nat_mult)
+apply (simp add: inverse_mult_distrib mult_ac)
+done
+
+
+lemma Maclaurin_lemma3:
+     "[|\<forall>k t. k < Suc m \<and> 0\<le>t & t\<le>h \<longrightarrow> DERIV (difg k) t :> difg (Suc k) t;
+        \<forall>k<Suc m. difg k 0 = 0; DERIV (difg n) t :> 0;  n < m; 0 < t;
+        t < h|]
+     ==> \<exists>ta. 0 < ta & ta < t & DERIV (difg (Suc n)) ta :> 0"
+apply (rule Rolle, assumption, simp)
+apply (drule_tac x = n and P="%k. k<Suc m --> difg k 0 = 0" in spec)
+apply (rule DERIV_unique)
+prefer 2 apply assumption
+apply force
+apply (subgoal_tac "\<forall>ta. 0 \<le> ta & ta \<le> t --> (difg (Suc n)) differentiable ta")
+apply (simp add: differentiable_def)
+apply (blast dest!: DERIV_isCont)
+apply (simp add: differentiable_def, clarify)
+apply (rule_tac x = "difg (Suc (Suc n)) ta" in exI)
+apply force
+apply (simp add: differentiable_def, clarify)
+apply (rule_tac x = "difg (Suc (Suc n)) x" in exI)
+apply force
+done
+
+lemma Maclaurin:
+   "[| 0 < h; 0 < n; diff 0 = f;
+       \<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t |]
+    ==> \<exists>t. 0 < t &
+              t < h &
+              f h =
+              sumr 0 n (%m. (diff m 0 / real (fact m)) * h ^ m) +
+              (diff n t / real (fact n)) * h ^ n"
+apply (case_tac "n = 0", force)
+apply (drule not0_implies_Suc)
+apply (erule exE)
+apply (frule_tac f=f and n=n and j="%m. diff m 0" in Maclaurin_lemma)
+apply (erule exE)
+apply (subgoal_tac "\<exists>g.
+     g = (%t. f t - (sumr 0 n (%m. (diff m 0 / real(fact m)) * t^m) + (B * (t^n / real(fact n)))))")
+ prefer 2 apply blast
+apply (erule exE)
+apply (subgoal_tac "g 0 = 0 & g h =0")
+ prefer 2
+ apply (simp del: sumr_Suc)
+ apply (cut_tac n = m and k = 1 in sumr_offset2)
+ apply (simp add: eq_diff_eq' del: sumr_Suc)
+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))))))")
+ prefer 2 apply blast
+apply (erule exE)
+apply (subgoal_tac "difg 0 = g")
+ prefer 2 apply simp
+apply (frule Maclaurin_lemma2, assumption+)
+apply (subgoal_tac "\<forall>ma. ma < n --> (\<exists>t. 0 < t & t < h & difg (Suc ma) t = 0) ")
+apply (drule_tac x = m and P="%m. m<n --> (\<exists>t. ?QQ m t)" in spec)
+apply (erule impE)
+apply (simp (no_asm_simp))
+apply (erule exE)
+apply (rule_tac x = t in exI)
+apply (simp del: realpow_Suc fact_Suc)
+apply (subgoal_tac "\<forall>m. m < n --> difg m 0 = 0")
+ prefer 2
+ apply clarify
+ apply simp
+ apply (frule_tac m = ma in less_add_one, clarify)
+ apply (simp del: sumr_Suc)
+apply (insert sumr_offset4 [of 1])
+apply (simp del: sumr_Suc fact_Suc realpow_Suc)
+apply (subgoal_tac "\<forall>m. m < n --> (\<exists>t. 0 < t & t < h & DERIV (difg m) t :> 0) ")
+apply (rule allI, rule impI)
+apply (drule_tac x = ma and P="%m. m<n --> (\<exists>t. ?QQ m t)" in spec)
+apply (erule impE, assumption)
+apply (erule exE)
+apply (rule_tac x = t in exI)
+(* do some tidying up *)
+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)))))"
+       in thin_rl)
+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))))"
+       in thin_rl)
+apply (erule_tac [!] V="f h = sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) + B * (h ^ n / real (fact n))"
+       in thin_rl)
+(* back to business *)
+apply (simp (no_asm_simp))
+apply (rule DERIV_unique)
+prefer 2 apply blast
+apply force
+apply (rule allI, induct_tac "ma")
+apply (rule impI, rule Rolle, assumption, simp, simp)
+apply (subgoal_tac "\<forall>t. 0 \<le> t & t \<le> h --> g differentiable t")
+apply (simp add: differentiable_def)
+apply (blast dest: DERIV_isCont)
+apply (simp add: differentiable_def, clarify)
+apply (rule_tac x = "difg (Suc 0) t" in exI)
+apply force
+apply (simp add: differentiable_def, clarify)
+apply (rule_tac x = "difg (Suc 0) x" in exI)
+apply force
+apply safe
+apply force
+apply (frule Maclaurin_lemma3, assumption+, safe)
+apply (rule_tac x = ta in exI, force)
+done
+
+lemma Maclaurin_objl:
+     "0 < h & 0 < n & diff 0 = f &
+       (\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)
+    --> (\<exists>t. 0 < t &
+              t < h &
+              f h =
+              sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) +
+              diff n t / real (fact n) * h ^ n)"
+by (blast intro: Maclaurin)
+
+
+lemma Maclaurin2:
+   "[| 0 < h; diff 0 = f;
+       \<forall>m t.
+          m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t |]
+    ==> \<exists>t. 0 < t &
+              t \<le> h &
+              f h =
+              sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) +
+              diff n t / real (fact n) * h ^ n"
+apply (case_tac "n", auto)
+apply (drule Maclaurin, auto)
+done
+
+lemma Maclaurin2_objl:
+     "0 < h & diff 0 = f &
+       (\<forall>m t.
+          m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)
+    --> (\<exists>t. 0 < t &
+              t \<le> h &
+              f h =
+              sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) +
+              diff n t / real (fact n) * h ^ n)"
+by (blast intro: Maclaurin2)
+
+lemma Maclaurin_minus:
+   "[| h < 0; 0 < n; diff 0 = f;
+       \<forall>m t. m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t |]
+    ==> \<exists>t. h < t &
+              t < 0 &
+              f h =
+              sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) +
+              diff n t / real (fact n) * h ^ n"
+apply (cut_tac f = "%x. f (-x)"
+        and diff = "%n x. ((- 1) ^ n) * diff n (-x)"
+        and h = "-h" and n = n in Maclaurin_objl)
+apply simp
+apply safe
+apply (subst minus_mult_right)
+apply (rule DERIV_cmult)
+apply (rule lemma_DERIV_subst)
+apply (rule DERIV_chain2 [where g=uminus])
+apply (rule_tac [2] DERIV_minus, rule_tac [2] DERIV_Id)
+prefer 2 apply force
+apply force
+apply (rule_tac x = "-t" in exI, auto)
+apply (subgoal_tac "(\<Sum>m = 0..<n. -1 ^ m * diff m 0 * (-h)^m / real(fact m)) =
+                    (\<Sum>m = 0..<n. diff m 0 * h ^ m / real(fact m))")
+apply (rule_tac [2] sumr_fun_eq)
+apply (auto simp add: divide_inverse power_mult_distrib [symmetric])
+done
+
+lemma Maclaurin_minus_objl:
+     "(h < 0 & 0 < n & diff 0 = f &
+       (\<forall>m t.
+          m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t))
+    --> (\<exists>t. h < t &
+              t < 0 &
+              f h =
+              sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) +
+              diff n t / real (fact n) * h ^ n)"
+by (blast intro: Maclaurin_minus)
+
+
+subsection{*More Convenient "Bidirectional" Version.*}
+
+(* not good for PVS sin_approx, cos_approx *)
+
+lemma Maclaurin_bi_le_lemma [rule_format]:
+     "0 < n \<longrightarrow>
+       diff 0 0 =
+       (\<Sum>m = 0..<n. diff m 0 * 0 ^ m / real (fact m)) +
+       diff n 0 * 0 ^ n / real (fact n)"
+by (induct_tac "n", auto)
+
+lemma Maclaurin_bi_le:
+   "[| diff 0 = f;
+       \<forall>m t. m < n & abs t \<le> abs x --> DERIV (diff m) t :> diff (Suc m) t |]
+    ==> \<exists>t. abs t \<le> abs x &
+              f x =
+              sumr 0 n (%m. diff m 0 / real (fact m) * x ^ m) +
+              diff n t / real (fact n) * x ^ n"
+apply (case_tac "n = 0", force)
+apply (case_tac "x = 0")
+apply (rule_tac x = 0 in exI)
+apply (force simp add: Maclaurin_bi_le_lemma)
+apply (cut_tac x = x and y = 0 in linorder_less_linear, auto)
+txt{*Case 1, where @{term "x < 0"}*}
+apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_minus_objl, safe)
+apply (simp add: abs_if)
+apply (rule_tac x = t in exI)
+apply (simp add: abs_if)
+txt{*Case 2, where @{term "0 < x"}*}
+apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_objl, safe)
+apply (simp add: abs_if)
+apply (rule_tac x = t in exI)
+apply (simp add: abs_if)
+done
+
+lemma Maclaurin_all_lt:
+     "[| diff 0 = f;
+         \<forall>m x. DERIV (diff m) x :> diff(Suc m) x;
+        x ~= 0; 0 < n
+      |] ==> \<exists>t. 0 < abs t & abs t < abs x &
+               f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) +
+                     (diff n t / real (fact n)) * x ^ n"
+apply (rule_tac x = x and y = 0 in linorder_cases)
+prefer 2 apply blast
+apply (drule_tac [2] diff=diff in Maclaurin)
+apply (drule_tac diff=diff in Maclaurin_minus, simp_all, safe)
+apply (rule_tac [!] x = t in exI, auto, arith+)
+done
+
+lemma Maclaurin_all_lt_objl:
+     "diff 0 = f &
+      (\<forall>m x. DERIV (diff m) x :> diff(Suc m) x) &
+      x ~= 0 & 0 < n
+      --> (\<exists>t. 0 < abs t & abs t < abs x &
+               f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) +
+                     (diff n t / real (fact n)) * x ^ n)"
+by (blast intro: Maclaurin_all_lt)
+
+lemma Maclaurin_zero [rule_format]:
+     "x = (0::real)
+      ==> 0 < n -->
+          sumr 0 n (%m. (diff m (0::real) / real (fact m)) * x ^ m) =
+          diff 0 0"
+by (induct n, auto)
+
+lemma Maclaurin_all_le: "[| diff 0 = f;
+        \<forall>m x. DERIV (diff m) x :> diff (Suc m) x
+      |] ==> \<exists>t. abs t \<le> abs x &
+              f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) +
+                    (diff n t / real (fact n)) * x ^ n"
+apply (insert linorder_le_less_linear [of n 0])
+apply (erule disjE, force)
+apply (case_tac "x = 0")
+apply (frule_tac diff = diff and n = n in Maclaurin_zero, assumption)
+apply (drule gr_implies_not0 [THEN not0_implies_Suc])
+apply (rule_tac x = 0 in exI, force)
+apply (frule_tac diff = diff and n = n in Maclaurin_all_lt, auto)
+apply (rule_tac x = t in exI, auto)
+done
+
+lemma Maclaurin_all_le_objl: "diff 0 = f &
+      (\<forall>m x. DERIV (diff m) x :> diff (Suc m) x)
+      --> (\<exists>t. abs t \<le> abs x &
+              f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) +
+                    (diff n t / real (fact n)) * x ^ n)"
+by (blast intro: Maclaurin_all_le)
+
+
+subsection{*Version for Exponential Function*}
+
+lemma Maclaurin_exp_lt: "[| x ~= 0; 0 < n |]
+      ==> (\<exists>t. 0 < abs t &
+                abs t < abs x &
+                exp x = sumr 0 n (%m. (x ^ m) / real (fact m)) +
+                        (exp t / real (fact n)) * x ^ n)"
+by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_lt_objl, auto)
+
+
+lemma Maclaurin_exp_le:
+     "\<exists>t. abs t \<le> abs x &
+            exp x = sumr 0 n (%m. (x ^ m) / real (fact m)) +
+                       (exp t / real (fact n)) * x ^ n"
+by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_le_objl, auto)
+
+
+subsection{*Version for Sine Function*}
+
+lemma MVT2:
+     "[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
+      ==> \<exists>z. a < z & z < b & (f b - f a = (b - a) * f'(z))"
+apply (drule MVT)
+apply (blast intro: DERIV_isCont)
+apply (force dest: order_less_imp_le simp add: differentiable_def)
+apply (blast dest: DERIV_unique order_less_imp_le)
+done
+
+lemma mod_exhaust_less_4:
+     "m mod 4 = 0 | m mod 4 = 1 | m mod 4 = 2 | m mod 4 = (3::nat)"
+by (case_tac "m mod 4", auto, arith)
+
+lemma Suc_Suc_mult_two_diff_two [rule_format, simp]:
+     "0 < n --> Suc (Suc (2 * n - 2)) = 2*n"
+by (induct_tac "n", auto)
+
+lemma lemma_Suc_Suc_4n_diff_2 [rule_format, simp]:
+     "0 < n --> Suc (Suc (4*n - 2)) = 4*n"
+by (induct_tac "n", auto)
+
+lemma Suc_mult_two_diff_one [rule_format, simp]:
+      "0 < n --> Suc (2 * n - 1) = 2*n"
+by (induct_tac "n", auto)
+
+lemma Maclaurin_sin_expansion:
+     "\<exists>t. sin x =
+       (sumr 0 n (%m. (if even m then 0
+                       else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
+                       x ^ m))
+      + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
+apply (cut_tac f = sin and n = n and x = x and diff = "%n x. sin (x + 1/2*real (n) *pi)" in Maclaurin_all_lt_objl)
+apply safe
+apply (simp (no_asm))
+apply (simp (no_asm))
+apply (case_tac "n", clarify, simp)
+apply (drule_tac x = 0 in spec, simp, simp)
+apply (rule ccontr, simp)
+apply (drule_tac x = x in spec, simp)
+apply (erule ssubst)
+apply (rule_tac x = t in exI, simp)
+apply (rule sumr_fun_eq)
+apply (auto simp add: odd_Suc_mult_two_ex)
+apply (auto simp add: even_mult_two_ex simp del: fact_Suc realpow_Suc)
+(*Could sin_zero_iff help?*)
+done
+
+lemma Maclaurin_sin_expansion2:
+     "\<exists>t. abs t \<le> abs x &
+       sin x =
+       (sumr 0 n (%m. (if even m then 0
+                       else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
+                       x ^ m))
+      + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
+apply (cut_tac f = sin and n = n and x = x
+        and diff = "%n x. sin (x + 1/2*real n * pi)" in Maclaurin_all_lt_objl)
+apply safe
+apply (simp (no_asm))
+apply (simp (no_asm))
+apply (case_tac "n", clarify, simp, simp)
+apply (rule ccontr, simp)
+apply (drule_tac x = x in spec, simp)
+apply (erule ssubst)
+apply (rule_tac x = t in exI, simp)
+apply (rule sumr_fun_eq)
+apply (auto simp add: odd_Suc_mult_two_ex)
+apply (auto simp add: even_mult_two_ex simp del: fact_Suc realpow_Suc)
+done
+
+lemma Maclaurin_sin_expansion3:
+     "[| 0 < n; 0 < x |] ==>
+       \<exists>t. 0 < t & t < x &
+       sin x =
+       (sumr 0 n (%m. (if even m then 0
+                       else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
+                       x ^ m))
+      + ((sin(t + 1/2 * real(n) *pi) / real (fact n)) * x ^ n)"
+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)
+apply safe
+apply simp
+apply (simp (no_asm))
+apply (erule ssubst)
+apply (rule_tac x = t in exI, simp)
+apply (rule sumr_fun_eq)
+apply (auto simp add: odd_Suc_mult_two_ex)
+apply (auto simp add: even_mult_two_ex simp del: fact_Suc realpow_Suc)
+done
+
+lemma Maclaurin_sin_expansion4:
+     "0 < x ==>
+       \<exists>t. 0 < t & t \<le> x &
+       sin x =
+       (sumr 0 n (%m. (if even m then 0
+                       else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
+                       x ^ m))
+      + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
+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)
+apply safe
+apply simp
+apply (simp (no_asm))
+apply (erule ssubst)
+apply (rule_tac x = t in exI, simp)
+apply (rule sumr_fun_eq)
+apply (auto simp add: odd_Suc_mult_two_ex)
+apply (auto simp add: even_mult_two_ex simp del: fact_Suc realpow_Suc)
+done
+
+
+subsection{*Maclaurin Expansion for Cosine Function*}
+
+lemma sumr_cos_zero_one [simp]:
+     "sumr 0 (Suc n)
+         (%m. (if even m
+               then (- 1) ^ (m div 2)/(real  (fact m))
+               else 0) *
+              0 ^ m) = 1"
+by (induct_tac "n", auto)
+
+lemma Maclaurin_cos_expansion:
+     "\<exists>t. abs t \<le> abs x &
+       cos x =
+       (sumr 0 n (%m. (if even m
+                       then (- 1) ^ (m div 2)/(real (fact m))
+                       else 0) *
+                       x ^ m))
+      + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
+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)
+apply safe
+apply (simp (no_asm))
+apply (simp (no_asm))
+apply (case_tac "n", simp)
+apply (simp del: sumr_Suc)
+apply (rule ccontr, simp)
+apply (drule_tac x = x in spec, simp)
+apply (erule ssubst)
+apply (rule_tac x = t in exI, simp)
+apply (rule sumr_fun_eq)
+apply (auto simp add: odd_Suc_mult_two_ex)
+apply (auto simp add: even_mult_two_ex left_distrib cos_add simp del: fact_Suc realpow_Suc)
+apply (simp add: mult_commute [of _ pi])
+done
+
+lemma Maclaurin_cos_expansion2:
+     "[| 0 < x; 0 < n |] ==>
+       \<exists>t. 0 < t & t < x &
+       cos x =
+       (sumr 0 n (%m. (if even m
+                       then (- 1) ^ (m div 2)/(real (fact m))
+                       else 0) *
+                       x ^ m))
+      + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
+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)
+apply safe
+apply simp
+apply (simp (no_asm))
+apply (erule ssubst)
+apply (rule_tac x = t in exI, simp)
+apply (rule sumr_fun_eq)
+apply (auto simp add: odd_Suc_mult_two_ex)
+apply (auto simp add: even_mult_two_ex left_distrib cos_add simp del: fact_Suc realpow_Suc)
+apply (simp add: mult_commute [of _ pi])
+done
+
+lemma Maclaurin_minus_cos_expansion: "[| x < 0; 0 < n |] ==>
+       \<exists>t. x < t & t < 0 &
+       cos x =
+       (sumr 0 n (%m. (if even m
+                       then (- 1) ^ (m div 2)/(real (fact m))
+                       else 0) *
+                       x ^ m))
+      + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
+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)
+apply safe
+apply simp
+apply (simp (no_asm))
+apply (erule ssubst)
+apply (rule_tac x = t in exI, simp)
+apply (rule sumr_fun_eq)
+apply (auto simp add: odd_Suc_mult_two_ex)
+apply (auto simp add: even_mult_two_ex left_distrib cos_add simp del: fact_Suc realpow_Suc)
+apply (simp add: mult_commute [of _ pi])
+done
+
+(* ------------------------------------------------------------------------- *)
+(* Version for ln(1 +/- x). Where is it??                                    *)
+(* ------------------------------------------------------------------------- *)
+
+lemma sin_bound_lemma:
+    "[|x = y; abs u \<le> (v::real) |] ==> abs ((x + u) - y) \<le> v"
+by auto
+
+lemma Maclaurin_sin_bound:
+  "abs(sin x - sumr 0 n (%m. (if even m then 0 else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
+  x ^ m))  \<le> inverse(real (fact n)) * abs(x) ^ n"
 proof -
-  have "!! x (y::real). x <= 1 \<Longrightarrow> 0 <= y \<Longrightarrow> x * y \<le> 1 * y" 
+  have "!! x (y::real). x \<le> 1 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x * y \<le> 1 * y"
     by (rule_tac mult_right_mono,simp_all)
   note est = this[simplified]
   show ?thesis
-    apply (cut_tac f=sin and n=n and x=x and 
+    apply (cut_tac f=sin and n=n and x=x and
       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)"
       in Maclaurin_all_le_objl)
-    apply (tactic{* (Step_tac 1) *})
-    apply (simp)
+    apply safe
+    apply simp
     apply (subst mod_Suc_eq_Suc_mod)
-    apply (tactic{* cut_inst_tac [("m1","m")] (CLAIM "0 < (4::nat)" RS mod_less_divisor RS lemma_exhaust_less_4) 1*})
-    apply (tactic{* Step_tac 1 *})
-    apply (simp)+
+    apply (cut_tac m=m in mod_exhaust_less_4, safe, simp+)
     apply (rule DERIV_minus, simp+)
     apply (rule lemma_DERIV_subst, rule DERIV_minus, rule DERIV_cos, simp)
-    apply (tactic{* dtac ssubst 1 THEN assume_tac 2 *})
-    apply (tactic {* rtac (ARITH_PROVE "[|x = y; abs u <= (v::real) |] ==> abs ((x + u) - y) <= v") 1 *})
-    apply (rule sumr_fun_eq)
-    apply (tactic{* Step_tac 1 *})
-    apply (tactic{*rtac (CLAIM "x = y ==> x * z = y * (z::real)") 1*})
+    apply (erule ssubst)
+    apply (rule sin_bound_lemma)
+    apply (rule sumr_fun_eq, safe)
+    apply (rule_tac f = "%u. u * (x^r)" in arg_cong)
     apply (subst even_even_mod_4_iff)
-    apply (tactic{* cut_inst_tac [("m1","r")] (CLAIM "0 < (4::nat)" RS mod_less_divisor RS lemma_exhaust_less_4) 1 *})
-    apply (tactic{* Step_tac 1 *})
-    apply (simp)
+    apply (cut_tac m=r in mod_exhaust_less_4, simp, safe)
     apply (simp_all add:even_num_iff)
     apply (drule lemma_even_mod_4_div_2[simplified])
-    apply(simp add: numeral_2_eq_2 real_divide_def)
-    apply (drule lemma_odd_mod_4_div_2 );
-    apply (simp add: numeral_2_eq_2 real_divide_def)
-    apply (auto intro: real_mult_le_lemma mult_right_mono simp add: est mult_pos_le mult_ac real_divide_def abs_mult abs_inverse power_abs[symmetric])
+    apply(simp add: numeral_2_eq_2 divide_inverse)
+    apply (drule lemma_odd_mod_4_div_2)
+    apply (simp add: numeral_2_eq_2 divide_inverse)
+    apply (auto intro: mult_right_mono [where b=1, simplified] mult_right_mono
+                   simp add: est mult_pos_le mult_ac divide_inverse
+                          power_abs [symmetric])
     done
 qed
 
 end