
paulson@12224: (*  Title       : MacLaurin.thy
paulson@12224:     Author      : Jacques D. Fleuriot
paulson@12224:     Copyright   : 2001 University of Edinburgh
paulson@12224:     Description : MacLaurin series
paulson@15079:     Conversion to Isar and new proofs by Lawrence C Paulson, 2004
paulson@12224: *)
paulson@12224: 
nipkow@15131: theory MacLaurin
nipkow@15140: imports Log
nipkow@15131: begin
paulson@15079: 
paulson@15079: lemma sumr_offset: "sumr 0 n (%m. f (m+k)) = sumr 0 (n+k) f - sumr 0 k f"
paulson@15079: by (induct_tac "n", auto)
paulson@15079: 
paulson@15079: lemma sumr_offset2: "\<forall>f. sumr 0 n (%m. f (m+k)) = sumr 0 (n+k) f - sumr 0 k f"
paulson@15079: by (induct_tac "n", auto)
paulson@15079: 
paulson@15079: lemma sumr_offset3: "sumr 0 (n+k) f = sumr 0 n (%m. f (m+k)) + sumr 0 k f"
paulson@15079: by (simp  add: sumr_offset)
paulson@15079: 
paulson@15079: lemma sumr_offset4: "\<forall>n f. sumr 0 (n+k) f = sumr 0 n (%m. f (m+k)) + sumr 0 k f"
paulson@15079: by (simp add: sumr_offset)
paulson@15079: 
paulson@15079: lemma sumr_from_1_from_0: "0 < n ==>
paulson@15079:       sumr (Suc 0) (Suc n) (%n. (if even(n) then 0 else
paulson@15079:              ((- 1) ^ ((n - (Suc 0)) div 2))/(real (fact n))) * a ^ n) =
paulson@15079:       sumr 0 (Suc n) (%n. (if even(n) then 0 else
paulson@15079:              ((- 1) ^ ((n - (Suc 0)) div 2))/(real (fact n))) * a ^ n)"
paulson@15079: by (rule_tac n1 = 1 in sumr_split_add [THEN subst], auto)
paulson@15079: 
paulson@15079: 
paulson@15079: subsection{*Maclaurin's Theorem with Lagrange Form of Remainder*}
paulson@15079: 
paulson@15079: text{*This is a very long, messy proof even now that it's been broken down
paulson@15079: into lemmas.*}
paulson@15079: 
paulson@15079: lemma Maclaurin_lemma:
paulson@15079:     "0 < h ==>
paulson@15079:      \<exists>B. f h = sumr 0 n (%m. (j m / real (fact m)) * (h^m)) +
paulson@15079:                (B * ((h^n) / real(fact n)))"
paulson@15234: apply (rule_tac x = "(f h - sumr 0 n (%m. (j m / real (fact m)) * h^m)) *
paulson@15079:                  real(fact n) / (h^n)"
paulson@15234:        in exI)
paulson@15234: apply (simp add: times_divide_eq) 
paulson@15234: done
paulson@15079: 
paulson@15079: lemma eq_diff_eq': "(x = y - z) = (y = x + (z::real))"
paulson@15079: by arith
paulson@15079: 
paulson@15079: text{*A crude tactic to differentiate by proof.*}
paulson@15079: ML
paulson@15079: {*
paulson@15079: exception DERIV_name;
paulson@15079: fun get_fun_name (_ $ (Const ("Lim.deriv",_) $ Abs(_,_, Const (f,_) $ _) $ _ $ _)) = f
paulson@15079: |   get_fun_name (_ $ (_ $ (Const ("Lim.deriv",_) $ Abs(_,_, Const (f,_) $ _) $ _ $ _))) = f
paulson@15079: |   get_fun_name _ = raise DERIV_name;
paulson@15079: 
paulson@15079: val deriv_rulesI = [DERIV_Id,DERIV_const,DERIV_cos,DERIV_cmult,
paulson@15079:                     DERIV_sin, DERIV_exp, DERIV_inverse,DERIV_pow,
paulson@15079:                     DERIV_add, DERIV_diff, DERIV_mult, DERIV_minus,
paulson@15079:                     DERIV_inverse_fun,DERIV_quotient,DERIV_fun_pow,
paulson@15079:                     DERIV_fun_exp,DERIV_fun_sin,DERIV_fun_cos,
paulson@15079:                     DERIV_Id,DERIV_const,DERIV_cos];
paulson@15079: 
paulson@15079: val deriv_tac =
paulson@15079:   SUBGOAL (fn (prem,i) =>
paulson@15079:    (resolve_tac deriv_rulesI i) ORELSE
paulson@15079:     ((rtac (read_instantiate [("f",get_fun_name prem)]
paulson@15079:                      DERIV_chain2) i) handle DERIV_name => no_tac));;
paulson@15079: 
paulson@15079: val DERIV_tac = ALLGOALS(fn i => REPEAT(deriv_tac i));
paulson@15079: *}
paulson@15079: 
paulson@15079: lemma Maclaurin_lemma2:
paulson@15079:       "[| \<forall>m t. m < n \<and> 0\<le>t \<and> t\<le>h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t;
paulson@15079:           n = Suc k;
paulson@15079:         difg =
paulson@15079:         (\<lambda>m t. diff m t -
paulson@15079:                ((\<Sum>p = 0..<n - m. diff (m + p) 0 / real (fact p) * t ^ p) +
paulson@15079:                 B * (t ^ (n - m) / real (fact (n - m)))))|] ==>
paulson@15079:         \<forall>m t. m < n & 0 \<le> t & t \<le> h -->
paulson@15079:                     DERIV (difg m) t :> difg (Suc m) t"
paulson@15079: apply clarify
paulson@15079: apply (rule DERIV_diff)
paulson@15079: apply (simp (no_asm_simp))
paulson@15079: apply (tactic DERIV_tac)
paulson@15079: apply (tactic DERIV_tac)
paulson@15079: apply (rule_tac [2] lemma_DERIV_subst)
paulson@15079: apply (rule_tac [2] DERIV_quotient)
paulson@15079: apply (rule_tac [3] DERIV_const)
paulson@15079: apply (rule_tac [2] DERIV_pow)
paulson@15079:   prefer 3 apply (simp add: fact_diff_Suc)
paulson@15079:  prefer 2 apply simp
paulson@15079: apply (frule_tac m = m in less_add_one, clarify)
paulson@15079: apply (simp del: sumr_Suc)
paulson@15079: apply (insert sumr_offset4 [of 1])
paulson@15079: apply (simp del: sumr_Suc fact_Suc realpow_Suc)
paulson@15079: apply (rule lemma_DERIV_subst)
paulson@15079: apply (rule DERIV_add)
paulson@15079: apply (rule_tac [2] DERIV_const)
paulson@15079: apply (rule DERIV_sumr, clarify)
paulson@15079:  prefer 2 apply simp
paulson@15079: apply (simp (no_asm) add: divide_inverse mult_assoc del: fact_Suc realpow_Suc)
paulson@15079: apply (rule DERIV_cmult)
paulson@15079: apply (rule lemma_DERIV_subst)
paulson@15079: apply (best intro: DERIV_chain2 intro!: DERIV_intros)
paulson@15079: apply (subst fact_Suc)
paulson@15079: apply (subst real_of_nat_mult)
paulson@15079: apply (simp add: inverse_mult_distrib mult_ac)
paulson@15079: done
paulson@15079: 
paulson@15079: 
paulson@15079: lemma Maclaurin_lemma3:
paulson@15079:      "[|\<forall>k t. k < Suc m \<and> 0\<le>t & t\<le>h \<longrightarrow> DERIV (difg k) t :> difg (Suc k) t;
paulson@15079:         \<forall>k<Suc m. difg k 0 = 0; DERIV (difg n) t :> 0;  n < m; 0 < t;
paulson@15079:         t < h|]
paulson@15079:      ==> \<exists>ta. 0 < ta & ta < t & DERIV (difg (Suc n)) ta :> 0"
paulson@15079: apply (rule Rolle, assumption, simp)
paulson@15079: apply (drule_tac x = n and P="%k. k<Suc m --> difg k 0 = 0" in spec)
paulson@15079: apply (rule DERIV_unique)
paulson@15079: prefer 2 apply assumption
paulson@15079: apply force
paulson@15079: apply (subgoal_tac "\<forall>ta. 0 \<le> ta & ta \<le> t --> (difg (Suc n)) differentiable ta")
paulson@15079: apply (simp add: differentiable_def)
paulson@15079: apply (blast dest!: DERIV_isCont)
paulson@15079: apply (simp add: differentiable_def, clarify)
paulson@15079: apply (rule_tac x = "difg (Suc (Suc n)) ta" in exI)
paulson@15079: apply force
paulson@15079: apply (simp add: differentiable_def, clarify)
paulson@15079: apply (rule_tac x = "difg (Suc (Suc n)) x" in exI)
paulson@15079: apply force
paulson@15079: done
obua@14738: 
paulson@15079: lemma Maclaurin:
paulson@15079:    "[| 0 < h; 0 < n; diff 0 = f;
paulson@15079:        \<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t |]
paulson@15079:     ==> \<exists>t. 0 < t &
paulson@15079:               t < h &
paulson@15079:               f h =
paulson@15079:               sumr 0 n (%m. (diff m 0 / real (fact m)) * h ^ m) +
paulson@15079:               (diff n t / real (fact n)) * h ^ n"
paulson@15079: apply (case_tac "n = 0", force)
paulson@15079: apply (drule not0_implies_Suc)
paulson@15079: apply (erule exE)
paulson@15079: apply (frule_tac f=f and n=n and j="%m. diff m 0" in Maclaurin_lemma)
paulson@15079: apply (erule exE)
paulson@15079: apply (subgoal_tac "\<exists>g.
paulson@15079:      g = (%t. f t - (sumr 0 n (%m. (diff m 0 / real(fact m)) * t^m) + (B * (t^n / real(fact n)))))")
paulson@15079:  prefer 2 apply blast
paulson@15079: apply (erule exE)
paulson@15079: apply (subgoal_tac "g 0 = 0 & g h =0")
paulson@15079:  prefer 2
paulson@15079:  apply (simp del: sumr_Suc)
paulson@15079:  apply (cut_tac n = m and k = 1 in sumr_offset2)
paulson@15079:  apply (simp add: eq_diff_eq' del: sumr_Suc)
paulson@15079: 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))))))")
paulson@15079:  prefer 2 apply blast
paulson@15079: apply (erule exE)
paulson@15079: apply (subgoal_tac "difg 0 = g")
paulson@15079:  prefer 2 apply simp
paulson@15079: apply (frule Maclaurin_lemma2, assumption+)
paulson@15079: apply (subgoal_tac "\<forall>ma. ma < n --> (\<exists>t. 0 < t & t < h & difg (Suc ma) t = 0) ")
paulson@15234:  apply (drule_tac x = m and P="%m. m<n --> (\<exists>t. ?QQ m t)" in spec)
paulson@15234:  apply (erule impE)
paulson@15234:   apply (simp (no_asm_simp))
paulson@15234:  apply (erule exE)
paulson@15234:  apply (rule_tac x = t in exI)
paulson@15234:  apply (simp add: times_divide_eq del: realpow_Suc fact_Suc)
paulson@15079: apply (subgoal_tac "\<forall>m. m < n --> difg m 0 = 0")
paulson@15079:  prefer 2
paulson@15079:  apply clarify
paulson@15079:  apply simp
paulson@15079:  apply (frule_tac m = ma in less_add_one, clarify)
paulson@15079:  apply (simp del: sumr_Suc)
paulson@15079: apply (insert sumr_offset4 [of 1])
paulson@15079: apply (simp del: sumr_Suc fact_Suc realpow_Suc)
paulson@15079: apply (subgoal_tac "\<forall>m. m < n --> (\<exists>t. 0 < t & t < h & DERIV (difg m) t :> 0) ")
paulson@15079: apply (rule allI, rule impI)
paulson@15079: apply (drule_tac x = ma and P="%m. m<n --> (\<exists>t. ?QQ m t)" in spec)
paulson@15079: apply (erule impE, assumption)
paulson@15079: apply (erule exE)
paulson@15079: apply (rule_tac x = t in exI)
paulson@15079: (* do some tidying up *)
paulson@15079: 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)))))"
paulson@15079:        in thin_rl)
paulson@15079: 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))))"
paulson@15079:        in thin_rl)
paulson@15079: apply (erule_tac [!] V="f h = sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) + B * (h ^ n / real (fact n))"
paulson@15079:        in thin_rl)
paulson@15079: (* back to business *)
paulson@15079: apply (simp (no_asm_simp))
paulson@15079: apply (rule DERIV_unique)
paulson@15079: prefer 2 apply blast
paulson@15079: apply force
paulson@15079: apply (rule allI, induct_tac "ma")
paulson@15079: apply (rule impI, rule Rolle, assumption, simp, simp)
paulson@15079: apply (subgoal_tac "\<forall>t. 0 \<le> t & t \<le> h --> g differentiable t")
paulson@15079: apply (simp add: differentiable_def)
paulson@15079: apply (blast dest: DERIV_isCont)
paulson@15079: apply (simp add: differentiable_def, clarify)
paulson@15079: apply (rule_tac x = "difg (Suc 0) t" in exI)
paulson@15079: apply force
paulson@15079: apply (simp add: differentiable_def, clarify)
paulson@15079: apply (rule_tac x = "difg (Suc 0) x" in exI)
paulson@15079: apply force
paulson@15079: apply safe
paulson@15079: apply force
paulson@15079: apply (frule Maclaurin_lemma3, assumption+, safe)
paulson@15079: apply (rule_tac x = ta in exI, force)
paulson@15079: done
paulson@15079: 
paulson@15079: lemma Maclaurin_objl:
paulson@15079:      "0 < h & 0 < n & diff 0 = f &
paulson@15079:        (\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)
paulson@15079:     --> (\<exists>t. 0 < t &
paulson@15079:               t < h &
paulson@15079:               f h =
paulson@15079:               sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) +
paulson@15079:               diff n t / real (fact n) * h ^ n)"
paulson@15079: by (blast intro: Maclaurin)
paulson@15079: 
paulson@15079: 
paulson@15079: lemma Maclaurin2:
paulson@15079:    "[| 0 < h; diff 0 = f;
paulson@15079:        \<forall>m t.
paulson@15079:           m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t |]
paulson@15079:     ==> \<exists>t. 0 < t &
paulson@15079:               t \<le> h &
paulson@15079:               f h =
paulson@15079:               sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) +
paulson@15079:               diff n t / real (fact n) * h ^ n"
paulson@15079: apply (case_tac "n", auto)
paulson@15079: apply (drule Maclaurin, auto)
paulson@15079: done
paulson@15079: 
paulson@15079: lemma Maclaurin2_objl:
paulson@15079:      "0 < h & diff 0 = f &
paulson@15079:        (\<forall>m t.
paulson@15079:           m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)
paulson@15079:     --> (\<exists>t. 0 < t &
paulson@15079:               t \<le> h &
paulson@15079:               f h =
paulson@15079:               sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) +
paulson@15079:               diff n t / real (fact n) * h ^ n)"
paulson@15079: by (blast intro: Maclaurin2)
paulson@15079: 
paulson@15079: lemma Maclaurin_minus:
paulson@15079:    "[| h < 0; 0 < n; diff 0 = f;
paulson@15079:        \<forall>m t. m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t |]
paulson@15079:     ==> \<exists>t. h < t &
paulson@15079:               t < 0 &
paulson@15079:               f h =
paulson@15079:               sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) +
paulson@15079:               diff n t / real (fact n) * h ^ n"
paulson@15079: apply (cut_tac f = "%x. f (-x)"
paulson@15079:         and diff = "%n x. ((- 1) ^ n) * diff n (-x)"
paulson@15079:         and h = "-h" and n = n in Maclaurin_objl)
paulson@15234: apply (simp add: times_divide_eq) 
paulson@15079: apply safe
paulson@15079: apply (subst minus_mult_right)
paulson@15079: apply (rule DERIV_cmult)
paulson@15079: apply (rule lemma_DERIV_subst)
paulson@15079: apply (rule DERIV_chain2 [where g=uminus])
paulson@15079: apply (rule_tac [2] DERIV_minus, rule_tac [2] DERIV_Id)
paulson@15079: prefer 2 apply force
paulson@15079: apply force
paulson@15079: apply (rule_tac x = "-t" in exI, auto)
paulson@15079: apply (subgoal_tac "(\<Sum>m = 0..<n. -1 ^ m * diff m 0 * (-h)^m / real(fact m)) =
paulson@15079:                     (\<Sum>m = 0..<n. diff m 0 * h ^ m / real(fact m))")
paulson@15079: apply (rule_tac [2] sumr_fun_eq)
paulson@15079: apply (auto simp add: divide_inverse power_mult_distrib [symmetric])
paulson@15079: done
paulson@15079: 
paulson@15079: lemma Maclaurin_minus_objl:
paulson@15079:      "(h < 0 & 0 < n & diff 0 = f &
paulson@15079:        (\<forall>m t.
paulson@15079:           m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t))
paulson@15079:     --> (\<exists>t. h < t &
paulson@15079:               t < 0 &
paulson@15079:               f h =
paulson@15079:               sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) +
paulson@15079:               diff n t / real (fact n) * h ^ n)"
paulson@15079: by (blast intro: Maclaurin_minus)
paulson@15079: 
paulson@15079: 
paulson@15079: subsection{*More Convenient "Bidirectional" Version.*}
paulson@15079: 
paulson@15079: (* not good for PVS sin_approx, cos_approx *)
paulson@15079: 
paulson@15079: lemma Maclaurin_bi_le_lemma [rule_format]:
paulson@15079:      "0 < n \<longrightarrow>
paulson@15079:        diff 0 0 =
paulson@15079:        (\<Sum>m = 0..<n. diff m 0 * 0 ^ m / real (fact m)) +
paulson@15079:        diff n 0 * 0 ^ n / real (fact n)"
paulson@15079: by (induct_tac "n", auto)
obua@14738: 
paulson@15079: lemma Maclaurin_bi_le:
paulson@15079:    "[| diff 0 = f;
paulson@15079:        \<forall>m t. m < n & abs t \<le> abs x --> DERIV (diff m) t :> diff (Suc m) t |]
paulson@15079:     ==> \<exists>t. abs t \<le> abs x &
paulson@15079:               f x =
paulson@15079:               sumr 0 n (%m. diff m 0 / real (fact m) * x ^ m) +
paulson@15079:               diff n t / real (fact n) * x ^ n"
paulson@15079: apply (case_tac "n = 0", force)
paulson@15079: apply (case_tac "x = 0")
paulson@15079: apply (rule_tac x = 0 in exI)
paulson@15234: apply (force simp add: Maclaurin_bi_le_lemma times_divide_eq)
paulson@15079: apply (cut_tac x = x and y = 0 in linorder_less_linear, auto)
paulson@15079: txt{*Case 1, where @{term "x < 0"}*}
paulson@15079: apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_minus_objl, safe)
paulson@15079: apply (simp add: abs_if)
paulson@15079: apply (rule_tac x = t in exI)
paulson@15079: apply (simp add: abs_if)
paulson@15079: txt{*Case 2, where @{term "0 < x"}*}
paulson@15079: apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_objl, safe)
paulson@15079: apply (simp add: abs_if)
paulson@15079: apply (rule_tac x = t in exI)
paulson@15079: apply (simp add: abs_if)
paulson@15079: done
paulson@15079: 
paulson@15079: lemma Maclaurin_all_lt:
paulson@15079:      "[| diff 0 = f;
paulson@15079:          \<forall>m x. DERIV (diff m) x :> diff(Suc m) x;
paulson@15079:         x ~= 0; 0 < n
paulson@15079:       |] ==> \<exists>t. 0 < abs t & abs t < abs x &
paulson@15079:                f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) +
paulson@15079:                      (diff n t / real (fact n)) * x ^ n"
paulson@15079: apply (rule_tac x = x and y = 0 in linorder_cases)
paulson@15079: prefer 2 apply blast
paulson@15079: apply (drule_tac [2] diff=diff in Maclaurin)
paulson@15079: apply (drule_tac diff=diff in Maclaurin_minus, simp_all, safe)
paulson@15229: apply (rule_tac [!] x = t in exI, auto)
paulson@15079: done
paulson@15079: 
paulson@15079: lemma Maclaurin_all_lt_objl:
paulson@15079:      "diff 0 = f &
paulson@15079:       (\<forall>m x. DERIV (diff m) x :> diff(Suc m) x) &
paulson@15079:       x ~= 0 & 0 < n
paulson@15079:       --> (\<exists>t. 0 < abs t & abs t < abs x &
paulson@15079:                f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) +
paulson@15079:                      (diff n t / real (fact n)) * x ^ n)"
paulson@15079: by (blast intro: Maclaurin_all_lt)
paulson@15079: 
paulson@15079: lemma Maclaurin_zero [rule_format]:
paulson@15079:      "x = (0::real)
paulson@15079:       ==> 0 < n -->
paulson@15079:           sumr 0 n (%m. (diff m (0::real) / real (fact m)) * x ^ m) =
paulson@15079:           diff 0 0"
paulson@15079: by (induct n, auto)
paulson@15079: 
paulson@15079: lemma Maclaurin_all_le: "[| diff 0 = f;
paulson@15079:         \<forall>m x. DERIV (diff m) x :> diff (Suc m) x
paulson@15079:       |] ==> \<exists>t. abs t \<le> abs x &
paulson@15079:               f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) +
paulson@15079:                     (diff n t / real (fact n)) * x ^ n"
paulson@15079: apply (insert linorder_le_less_linear [of n 0])
paulson@15079: apply (erule disjE, force)
paulson@15079: apply (case_tac "x = 0")
paulson@15079: apply (frule_tac diff = diff and n = n in Maclaurin_zero, assumption)
paulson@15079: apply (drule gr_implies_not0 [THEN not0_implies_Suc])
paulson@15079: apply (rule_tac x = 0 in exI, force)
paulson@15079: apply (frule_tac diff = diff and n = n in Maclaurin_all_lt, auto)
paulson@15079: apply (rule_tac x = t in exI, auto)
paulson@15079: done
paulson@15079: 
paulson@15079: lemma Maclaurin_all_le_objl: "diff 0 = f &
paulson@15079:       (\<forall>m x. DERIV (diff m) x :> diff (Suc m) x)
paulson@15079:       --> (\<exists>t. abs t \<le> abs x &
paulson@15079:               f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) +
paulson@15079:                     (diff n t / real (fact n)) * x ^ n)"
paulson@15079: by (blast intro: Maclaurin_all_le)
paulson@15079: 
paulson@15079: 
paulson@15079: subsection{*Version for Exponential Function*}
paulson@15079: 
paulson@15079: lemma Maclaurin_exp_lt: "[| x ~= 0; 0 < n |]
paulson@15079:       ==> (\<exists>t. 0 < abs t &
paulson@15079:                 abs t < abs x &
paulson@15079:                 exp x = sumr 0 n (%m. (x ^ m) / real (fact m)) +
paulson@15079:                         (exp t / real (fact n)) * x ^ n)"
paulson@15079: by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_lt_objl, auto)
paulson@15079: 
paulson@15079: 
paulson@15079: lemma Maclaurin_exp_le:
paulson@15079:      "\<exists>t. abs t \<le> abs x &
paulson@15079:             exp x = sumr 0 n (%m. (x ^ m) / real (fact m)) +
paulson@15079:                        (exp t / real (fact n)) * x ^ n"
paulson@15079: by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_le_objl, auto)
paulson@15079: 
paulson@15079: 
paulson@15079: subsection{*Version for Sine Function*}
paulson@15079: 
paulson@15079: lemma MVT2:
paulson@15079:      "[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
paulson@15079:       ==> \<exists>z. a < z & z < b & (f b - f a = (b - a) * f'(z))"
paulson@15079: apply (drule MVT)
paulson@15079: apply (blast intro: DERIV_isCont)
paulson@15079: apply (force dest: order_less_imp_le simp add: differentiable_def)
paulson@15079: apply (blast dest: DERIV_unique order_less_imp_le)
paulson@15079: done
paulson@15079: 
paulson@15079: lemma mod_exhaust_less_4:
paulson@15079:      "m mod 4 = 0 | m mod 4 = 1 | m mod 4 = 2 | m mod 4 = (3::nat)"
paulson@15079: by (case_tac "m mod 4", auto, arith)
paulson@15079: 
paulson@15079: lemma Suc_Suc_mult_two_diff_two [rule_format, simp]:
paulson@15079:      "0 < n --> Suc (Suc (2 * n - 2)) = 2*n"
paulson@15079: by (induct_tac "n", auto)
paulson@15079: 
paulson@15079: lemma lemma_Suc_Suc_4n_diff_2 [rule_format, simp]:
paulson@15079:      "0 < n --> Suc (Suc (4*n - 2)) = 4*n"
paulson@15079: by (induct_tac "n", auto)
paulson@15079: 
paulson@15079: lemma Suc_mult_two_diff_one [rule_format, simp]:
paulson@15079:       "0 < n --> Suc (2 * n - 1) = 2*n"
paulson@15079: by (induct_tac "n", auto)
paulson@15079: 
paulson@15234: 
paulson@15234: text{*It is unclear why so many variant results are needed.*}
paulson@15079: 
paulson@15079: lemma Maclaurin_sin_expansion2:
paulson@15079:      "\<exists>t. abs t \<le> abs x &
paulson@15079:        sin x =
paulson@15079:        (sumr 0 n (%m. (if even m then 0
paulson@15079:                        else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
paulson@15079:                        x ^ m))
paulson@15079:       + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
paulson@15079: apply (cut_tac f = sin and n = n and x = x
paulson@15079:         and diff = "%n x. sin (x + 1/2*real n * pi)" in Maclaurin_all_lt_objl)
paulson@15079: apply safe
paulson@15079: apply (simp (no_asm))
paulson@15234: apply (simp (no_asm) add: times_divide_eq)
paulson@15079: apply (case_tac "n", clarify, simp, simp)
paulson@15079: apply (rule ccontr, simp)
paulson@15079: apply (drule_tac x = x in spec, simp)
paulson@15079: apply (erule ssubst)
paulson@15079: apply (rule_tac x = t in exI, simp)
paulson@15079: apply (rule sumr_fun_eq)
paulson@15234: apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex times_divide_eq)
paulson@15079: done
paulson@15079: 
paulson@15234: lemma Maclaurin_sin_expansion:
paulson@15234:      "\<exists>t. sin x =
paulson@15234:        (sumr 0 n (%m. (if even m then 0
paulson@15234:                        else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
paulson@15234:                        x ^ m))
paulson@15234:       + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
paulson@15234: apply (insert Maclaurin_sin_expansion2 [of x n]) 
paulson@15234: apply (blast intro: elim:); 
paulson@15234: done
paulson@15234: 
paulson@15234: 
paulson@15234: 
paulson@15079: lemma Maclaurin_sin_expansion3:
paulson@15079:      "[| 0 < n; 0 < x |] ==>
paulson@15079:        \<exists>t. 0 < t & t < x &
paulson@15079:        sin x =
paulson@15079:        (sumr 0 n (%m. (if even m then 0
paulson@15079:                        else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
paulson@15079:                        x ^ m))
paulson@15079:       + ((sin(t + 1/2 * real(n) *pi) / real (fact n)) * x ^ n)"
paulson@15079: 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)
paulson@15079: apply safe
paulson@15079: apply simp
paulson@15234: apply (simp (no_asm) add: times_divide_eq)
paulson@15079: apply (erule ssubst)
paulson@15079: apply (rule_tac x = t in exI, simp)
paulson@15079: apply (rule sumr_fun_eq)
paulson@15234: apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex times_divide_eq)
paulson@15079: done
paulson@15079: 
paulson@15079: lemma Maclaurin_sin_expansion4:
paulson@15079:      "0 < x ==>
paulson@15079:        \<exists>t. 0 < t & t \<le> x &
paulson@15079:        sin x =
paulson@15079:        (sumr 0 n (%m. (if even m then 0
paulson@15079:                        else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
paulson@15079:                        x ^ m))
paulson@15079:       + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
paulson@15079: 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)
paulson@15079: apply safe
paulson@15079: apply simp
paulson@15234: apply (simp (no_asm) add: times_divide_eq)
paulson@15079: apply (erule ssubst)
paulson@15079: apply (rule_tac x = t in exI, simp)
paulson@15079: apply (rule sumr_fun_eq)
paulson@15234: apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex times_divide_eq)
paulson@15079: done
paulson@15079: 
paulson@15079: 
paulson@15079: subsection{*Maclaurin Expansion for Cosine Function*}
paulson@15079: 
paulson@15079: lemma sumr_cos_zero_one [simp]:
paulson@15079:      "sumr 0 (Suc n)
paulson@15079:          (%m. (if even m
paulson@15079:                then (- 1) ^ (m div 2)/(real  (fact m))
paulson@15079:                else 0) *
paulson@15079:               0 ^ m) = 1"
paulson@15079: by (induct_tac "n", auto)
paulson@15079: 
paulson@15079: lemma Maclaurin_cos_expansion:
paulson@15079:      "\<exists>t. abs t \<le> abs x &
paulson@15079:        cos x =
paulson@15079:        (sumr 0 n (%m. (if even m
paulson@15079:                        then (- 1) ^ (m div 2)/(real (fact m))
paulson@15079:                        else 0) *
paulson@15079:                        x ^ m))
paulson@15079:       + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
paulson@15079: 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)
paulson@15079: apply safe
paulson@15079: apply (simp (no_asm))
paulson@15234: apply (simp (no_asm) add: times_divide_eq)
paulson@15079: apply (case_tac "n", simp)
paulson@15079: apply (simp del: sumr_Suc)
paulson@15079: apply (rule ccontr, simp)
paulson@15079: apply (drule_tac x = x in spec, simp)
paulson@15079: apply (erule ssubst)
paulson@15079: apply (rule_tac x = t in exI, simp)
paulson@15079: apply (rule sumr_fun_eq)
paulson@15234: apply (auto simp add: cos_zero_iff even_mult_two_ex)
paulson@15079: done
paulson@15079: 
paulson@15079: lemma Maclaurin_cos_expansion2:
paulson@15079:      "[| 0 < x; 0 < n |] ==>
paulson@15079:        \<exists>t. 0 < t & t < x &
paulson@15079:        cos x =
paulson@15079:        (sumr 0 n (%m. (if even m
paulson@15079:                        then (- 1) ^ (m div 2)/(real (fact m))
paulson@15079:                        else 0) *
paulson@15079:                        x ^ m))
paulson@15079:       + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
paulson@15079: 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)
paulson@15079: apply safe
paulson@15079: apply simp
paulson@15234: apply (simp (no_asm) add: times_divide_eq)
paulson@15079: apply (erule ssubst)
paulson@15079: apply (rule_tac x = t in exI, simp)
paulson@15079: apply (rule sumr_fun_eq)
paulson@15234: apply (auto simp add: cos_zero_iff even_mult_two_ex)
paulson@15079: done
paulson@15079: 
paulson@15234: lemma Maclaurin_minus_cos_expansion:
paulson@15234:      "[| x < 0; 0 < n |] ==>
paulson@15079:        \<exists>t. x < t & t < 0 &
paulson@15079:        cos x =
paulson@15079:        (sumr 0 n (%m. (if even m
paulson@15079:                        then (- 1) ^ (m div 2)/(real (fact m))
paulson@15079:                        else 0) *
paulson@15079:                        x ^ m))
paulson@15079:       + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
paulson@15079: 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)
paulson@15079: apply safe
paulson@15079: apply simp
paulson@15234: apply (simp (no_asm) add: times_divide_eq)
paulson@15079: apply (erule ssubst)
paulson@15079: apply (rule_tac x = t in exI, simp)
paulson@15079: apply (rule sumr_fun_eq)
paulson@15234: apply (auto simp add: cos_zero_iff even_mult_two_ex)
paulson@15079: done
paulson@15079: 
paulson@15079: (* ------------------------------------------------------------------------- *)
paulson@15079: (* Version for ln(1 +/- x). Where is it??                                    *)
paulson@15079: (* ------------------------------------------------------------------------- *)
paulson@15079: 
paulson@15079: lemma sin_bound_lemma:
paulson@15081:     "[|x = y; abs u \<le> (v::real) |] ==> \<bar>(x + u) - y\<bar> \<le> v"
paulson@15079: by auto
paulson@15079: 
paulson@15079: lemma Maclaurin_sin_bound:
paulson@15079:   "abs(sin x - sumr 0 n (%m. (if even m then 0 else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
paulson@15081:   x ^ m))  \<le> inverse(real (fact n)) * \<bar>x\<bar> ^ n"
obua@14738: proof -
paulson@15079:   have "!! x (y::real). x \<le> 1 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x * y \<le> 1 * y"
obua@14738:     by (rule_tac mult_right_mono,simp_all)
obua@14738:   note est = this[simplified]
obua@14738:   show ?thesis
paulson@15079:     apply (cut_tac f=sin and n=n and x=x and
obua@14738:       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)"
obua@14738:       in Maclaurin_all_le_objl)
paulson@15079:     apply safe
paulson@15079:     apply simp
obua@14738:     apply (subst mod_Suc_eq_Suc_mod)
paulson@15079:     apply (cut_tac m=m in mod_exhaust_less_4, safe, simp+)
obua@14738:     apply (rule DERIV_minus, simp+)
obua@14738:     apply (rule lemma_DERIV_subst, rule DERIV_minus, rule DERIV_cos, simp)
paulson@15079:     apply (erule ssubst)
paulson@15079:     apply (rule sin_bound_lemma)
paulson@15079:     apply (rule sumr_fun_eq, safe)
paulson@15079:     apply (rule_tac f = "%u. u * (x^r)" in arg_cong)
obua@14738:     apply (subst even_even_mod_4_iff)
paulson@15079:     apply (cut_tac m=r in mod_exhaust_less_4, simp, safe)
obua@14738:     apply (simp_all add:even_num_iff)
obua@14738:     apply (drule lemma_even_mod_4_div_2[simplified])
paulson@15079:     apply(simp add: numeral_2_eq_2 divide_inverse)
paulson@15079:     apply (drule lemma_odd_mod_4_div_2)
paulson@15079:     apply (simp add: numeral_2_eq_2 divide_inverse)
paulson@15079:     apply (auto intro: mult_right_mono [where b=1, simplified] mult_right_mono
paulson@15079:                    simp add: est mult_pos_le mult_ac divide_inverse
paulson@15079:                           power_abs [symmetric])
obua@14738:     done
obua@14738: qed
obua@14738: 
paulson@15079: end




