haftmann@28952
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(* Author : Jacques D. Fleuriot
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paulson@12224
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Copyright : 2001 University of Edinburgh
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paulson@15079
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Conversion to Isar and new proofs by Lawrence C Paulson, 2004
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paulson@12224
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*)
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paulson@12224
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paulson@15944
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header{*MacLaurin Series*}
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paulson@15944
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nipkow@15131
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theory MacLaurin
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chaieb@26163
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imports Dense_Linear_Order Transcendental
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nipkow@15131
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begin
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obua@14738
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paulson@15079
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subsection{*Maclaurin's Theorem with Lagrange Form of Remainder*}
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paulson@15079
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paulson@15079
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text{*This is a very long, messy proof even now that it's been broken down
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paulson@15079
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into lemmas.*}
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paulson@15079
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paulson@15079
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lemma Maclaurin_lemma:
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paulson@15079
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"0 < h ==>
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nipkow@15539
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\<exists>B. f h = (\<Sum>m=0..<n. (j m / real (fact m)) * (h^m)) +
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paulson@15079
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(B * ((h^n) / real(fact n)))"
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nipkow@15539
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apply (rule_tac x = "(f h - (\<Sum>m=0..<n. (j m / real (fact m)) * h^m)) *
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paulson@15079
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real(fact n) / (h^n)"
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paulson@15234
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in exI)
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nipkow@15539
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apply (simp)
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paulson@15234
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done
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paulson@15079
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paulson@15079
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lemma eq_diff_eq': "(x = y - z) = (y = x + (z::real))"
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paulson@15079
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by arith
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paulson@15079
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paulson@15079
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text{*A crude tactic to differentiate by proof.*}
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wenzelm@24180
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wenzelm@24180
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lemmas deriv_rulesI =
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wenzelm@24180
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DERIV_ident DERIV_const DERIV_cos DERIV_cmult
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wenzelm@24180
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DERIV_sin DERIV_exp DERIV_inverse DERIV_pow
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wenzelm@24180
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DERIV_add DERIV_diff DERIV_mult DERIV_minus
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wenzelm@24180
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DERIV_inverse_fun DERIV_quotient DERIV_fun_pow
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wenzelm@24180
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DERIV_fun_exp DERIV_fun_sin DERIV_fun_cos
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wenzelm@24180
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DERIV_ident DERIV_const DERIV_cos
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wenzelm@24180
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paulson@15079
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ML
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paulson@15079
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{*
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wenzelm@19765
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local
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paulson@15079
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exception DERIV_name;
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paulson@15079
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fun get_fun_name (_ $ (Const ("Lim.deriv",_) $ Abs(_,_, Const (f,_) $ _) $ _ $ _)) = f
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paulson@15079
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| get_fun_name (_ $ (_ $ (Const ("Lim.deriv",_) $ Abs(_,_, Const (f,_) $ _) $ _ $ _))) = f
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paulson@15079
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| get_fun_name _ = raise DERIV_name;
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paulson@15079
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wenzelm@24180
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in
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wenzelm@24180
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wenzelm@27227
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fun deriv_tac ctxt = SUBGOAL (fn (prem, i) =>
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wenzelm@27227
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resolve_tac @{thms deriv_rulesI} i ORELSE
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wenzelm@27239
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((rtac (read_instantiate ctxt [(("f", 0), get_fun_name prem)]
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wenzelm@27227
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@{thm DERIV_chain2}) i) handle DERIV_name => no_tac));
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paulson@15079
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wenzelm@27227
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fun DERIV_tac ctxt = ALLGOALS (fn i => REPEAT (deriv_tac ctxt i));
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wenzelm@19765
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wenzelm@19765
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end
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paulson@15079
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*}
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paulson@15079
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paulson@15079
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lemma Maclaurin_lemma2:
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huffman@29187
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assumes diff: "\<forall>m t. m < n \<and> 0\<le>t \<and> t\<le>h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
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huffman@29187
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assumes n: "n = Suc k"
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huffman@29187
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assumes difg: "difg =
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paulson@15079
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(\<lambda>m t. diff m t -
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paulson@15079
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((\<Sum>p = 0..<n - m. diff (m + p) 0 / real (fact p) * t ^ p) +
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huffman@29187
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B * (t ^ (n - m) / real (fact (n - m)))))"
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huffman@29187
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shows
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huffman@29187
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"\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (difg m) t :> difg (Suc m) t"
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huffman@29187
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unfolding difg
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huffman@29187
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apply clarify
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huffman@29187
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apply (rule DERIV_diff)
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huffman@29187
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apply (simp add: diff)
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huffman@29187
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apply (simp only: n)
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huffman@29187
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apply (rule DERIV_add)
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huffman@29187
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apply (rule_tac [2] DERIV_cmult)
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huffman@29187
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apply (rule_tac [2] lemma_DERIV_subst)
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huffman@29187
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apply (rule_tac [2] DERIV_quotient)
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huffman@29187
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apply (rule_tac [3] DERIV_const)
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huffman@29187
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apply (rule_tac [2] DERIV_pow)
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huffman@29187
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prefer 3 apply (simp add: fact_diff_Suc)
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huffman@29187
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prefer 2 apply simp
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huffman@29187
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apply (frule less_iff_Suc_add [THEN iffD1], clarify)
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huffman@29187
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apply (simp del: setsum_op_ivl_Suc)
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huffman@29187
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apply (insert sumr_offset4 [of 1])
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huffman@29187
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apply (simp del: setsum_op_ivl_Suc fact_Suc realpow_Suc)
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huffman@29187
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apply (rule lemma_DERIV_subst)
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huffman@29187
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apply (rule DERIV_add)
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huffman@29187
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apply (rule_tac [2] DERIV_const)
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huffman@29187
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apply (rule DERIV_sumr, clarify)
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huffman@29187
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prefer 2 apply simp
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huffman@29187
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apply (simp (no_asm) add: divide_inverse mult_assoc del: fact_Suc realpow_Suc)
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huffman@29187
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apply (rule DERIV_cmult)
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huffman@29187
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apply (rule lemma_DERIV_subst)
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huffman@29187
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apply (best intro: DERIV_chain2 intro!: DERIV_intros)
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huffman@29187
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apply (subst fact_Suc)
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huffman@29187
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apply (subst real_of_nat_mult)
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huffman@29187
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apply (simp add: mult_ac)
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paulson@15079
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done
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paulson@15079
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paulson@15079
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paulson@15079
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lemma Maclaurin:
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huffman@29187
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assumes h: "0 < h"
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huffman@29187
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assumes n: "0 < n"
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huffman@29187
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assumes diff_0: "diff 0 = f"
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huffman@29187
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assumes diff_Suc:
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huffman@29187
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"\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t"
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huffman@29187
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shows
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huffman@29187
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"\<exists>t. 0 < t & t < h &
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paulson@15079
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f h =
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nipkow@15539
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setsum (%m. (diff m 0 / real (fact m)) * h ^ m) {0..<n} +
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paulson@15079
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(diff n t / real (fact n)) * h ^ n"
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huffman@29187
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proof -
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huffman@29187
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from n obtain m where m: "n = Suc m"
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huffman@29187
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by (cases n, simp add: n)
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huffman@29187
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huffman@29187
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obtain B where f_h: "f h =
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huffman@29187
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(\<Sum>m = 0..<n. diff m (0\<Colon>real) / real (fact m) * h ^ m) +
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huffman@29187
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B * (h ^ n / real (fact n))"
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huffman@29187
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using Maclaurin_lemma [OF h] ..
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huffman@29187
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huffman@29187
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obtain g where g_def: "g = (%t. f t -
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huffman@29187
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(setsum (%m. (diff m 0 / real(fact m)) * t^m) {0..<n}
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huffman@29187
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+ (B * (t^n / real(fact n)))))" by blast
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huffman@29187
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huffman@29187
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have g2: "g 0 = 0 & g h = 0"
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huffman@29187
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apply (simp add: m f_h g_def del: setsum_op_ivl_Suc)
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huffman@29187
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apply (cut_tac n = m and k = 1 in sumr_offset2)
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huffman@29187
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apply (simp add: eq_diff_eq' diff_0 del: setsum_op_ivl_Suc)
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huffman@29187
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done
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huffman@29187
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huffman@29187
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obtain difg where difg_def: "difg = (%m t. diff m t -
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huffman@29187
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(setsum (%p. (diff (m + p) 0 / real (fact p)) * (t ^ p)) {0..<n-m}
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huffman@29187
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+ (B * ((t ^ (n - m)) / real (fact (n - m))))))" by blast
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huffman@29187
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huffman@29187
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have difg_0: "difg 0 = g"
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huffman@29187
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unfolding difg_def g_def by (simp add: diff_0)
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huffman@29187
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huffman@29187
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have difg_Suc: "\<forall>(m\<Colon>nat) t\<Colon>real.
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huffman@29187
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m < n \<and> (0\<Colon>real) \<le> t \<and> t \<le> h \<longrightarrow> DERIV (difg m) t :> difg (Suc m) t"
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huffman@29187
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using diff_Suc m difg_def by (rule Maclaurin_lemma2)
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huffman@29187
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141 |
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huffman@29187
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have difg_eq_0: "\<forall>m. m < n --> difg m 0 = 0"
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huffman@29187
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apply clarify
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huffman@29187
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apply (simp add: m difg_def)
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huffman@29187
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apply (frule less_iff_Suc_add [THEN iffD1], clarify)
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huffman@29187
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apply (simp del: setsum_op_ivl_Suc)
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huffman@29187
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apply (insert sumr_offset4 [of 1])
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huffman@29187
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apply (simp del: setsum_op_ivl_Suc fact_Suc realpow_Suc)
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huffman@29187
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done
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huffman@29187
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150 |
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huffman@29187
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have isCont_difg: "\<And>m x. \<lbrakk>m < n; 0 \<le> x; x \<le> h\<rbrakk> \<Longrightarrow> isCont (difg m) x"
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huffman@29187
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by (rule DERIV_isCont [OF difg_Suc [rule_format]]) simp
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huffman@29187
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153 |
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huffman@29187
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have differentiable_difg:
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huffman@29187
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"\<And>m x. \<lbrakk>m < n; 0 \<le> x; x \<le> h\<rbrakk> \<Longrightarrow> difg m differentiable x"
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huffman@29187
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by (rule differentiableI [OF difg_Suc [rule_format]]) simp
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huffman@29187
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157 |
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huffman@29187
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have difg_Suc_eq_0: "\<And>m t. \<lbrakk>m < n; 0 \<le> t; t \<le> h; DERIV (difg m) t :> 0\<rbrakk>
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huffman@29187
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\<Longrightarrow> difg (Suc m) t = 0"
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huffman@29187
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by (rule DERIV_unique [OF difg_Suc [rule_format]]) simp
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huffman@29187
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161 |
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huffman@29187
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have "m < n" using m by simp
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huffman@29187
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163 |
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huffman@29187
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have "\<exists>t. 0 < t \<and> t < h \<and> DERIV (difg m) t :> 0"
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huffman@29187
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using `m < n`
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huffman@29187
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proof (induct m)
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huffman@29187
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case 0
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huffman@29187
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show ?case
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huffman@29187
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proof (rule Rolle)
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huffman@29187
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show "0 < h" by fact
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huffman@29187
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show "difg 0 0 = difg 0 h" by (simp add: difg_0 g2)
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huffman@29187
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show "\<forall>x. 0 \<le> x \<and> x \<le> h \<longrightarrow> isCont (difg (0\<Colon>nat)) x"
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huffman@29187
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by (simp add: isCont_difg n)
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huffman@29187
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show "\<forall>x. 0 < x \<and> x < h \<longrightarrow> difg (0\<Colon>nat) differentiable x"
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huffman@29187
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by (simp add: differentiable_difg n)
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huffman@29187
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qed
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huffman@29187
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next
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huffman@29187
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case (Suc m')
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huffman@29187
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hence "\<exists>t. 0 < t \<and> t < h \<and> DERIV (difg m') t :> 0" by simp
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huffman@29187
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then obtain t where t: "0 < t" "t < h" "DERIV (difg m') t :> 0" by fast
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huffman@29187
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have "\<exists>t'. 0 < t' \<and> t' < t \<and> DERIV (difg (Suc m')) t' :> 0"
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huffman@29187
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proof (rule Rolle)
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huffman@29187
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show "0 < t" by fact
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huffman@29187
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show "difg (Suc m') 0 = difg (Suc m') t"
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huffman@29187
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using t `Suc m' < n` by (simp add: difg_Suc_eq_0 difg_eq_0)
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huffman@29187
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186 |
show "\<forall>x. 0 \<le> x \<and> x \<le> t \<longrightarrow> isCont (difg (Suc m')) x"
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huffman@29187
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187 |
using `t < h` `Suc m' < n` by (simp add: isCont_difg)
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huffman@29187
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188 |
show "\<forall>x. 0 < x \<and> x < t \<longrightarrow> difg (Suc m') differentiable x"
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huffman@29187
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189 |
using `t < h` `Suc m' < n` by (simp add: differentiable_difg)
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huffman@29187
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190 |
qed
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huffman@29187
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191 |
thus ?case
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huffman@29187
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192 |
using `t < h` by auto
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huffman@29187
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193 |
qed
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huffman@29187
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194 |
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huffman@29187
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195 |
then obtain t where "0 < t" "t < h" "DERIV (difg m) t :> 0" by fast
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huffman@29187
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196 |
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huffman@29187
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197 |
hence "difg (Suc m) t = 0"
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huffman@29187
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198 |
using `m < n` by (simp add: difg_Suc_eq_0)
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huffman@29187
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199 |
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huffman@29187
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200 |
show ?thesis
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huffman@29187
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201 |
proof (intro exI conjI)
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huffman@29187
|
202 |
show "0 < t" by fact
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huffman@29187
|
203 |
show "t < h" by fact
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huffman@29187
|
204 |
show "f h =
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huffman@29187
|
205 |
(\<Sum>m = 0..<n. diff m 0 / real (fact m) * h ^ m) +
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huffman@29187
|
206 |
diff n t / real (fact n) * h ^ n"
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huffman@29187
|
207 |
using `difg (Suc m) t = 0`
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huffman@29187
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208 |
by (simp add: m f_h difg_def del: realpow_Suc fact_Suc)
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huffman@29187
|
209 |
qed
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huffman@29187
|
210 |
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huffman@29187
|
211 |
qed
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paulson@15079
|
212 |
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paulson@15079
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213 |
lemma Maclaurin_objl:
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nipkow@25162
|
214 |
"0 < h & n>0 & diff 0 = f &
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nipkow@25134
|
215 |
(\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)
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nipkow@25134
|
216 |
--> (\<exists>t. 0 < t & t < h &
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nipkow@25134
|
217 |
f h = (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
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nipkow@25134
|
218 |
diff n t / real (fact n) * h ^ n)"
|
paulson@15079
|
219 |
by (blast intro: Maclaurin)
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paulson@15079
|
220 |
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paulson@15079
|
221 |
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paulson@15079
|
222 |
lemma Maclaurin2:
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paulson@15079
|
223 |
"[| 0 < h; diff 0 = f;
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paulson@15079
|
224 |
\<forall>m t.
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paulson@15079
|
225 |
m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t |]
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paulson@15079
|
226 |
==> \<exists>t. 0 < t &
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paulson@15079
|
227 |
t \<le> h &
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paulson@15079
|
228 |
f h =
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nipkow@15539
|
229 |
(\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
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paulson@15079
|
230 |
diff n t / real (fact n) * h ^ n"
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paulson@15079
|
231 |
apply (case_tac "n", auto)
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paulson@15079
|
232 |
apply (drule Maclaurin, auto)
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paulson@15079
|
233 |
done
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paulson@15079
|
234 |
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paulson@15079
|
235 |
lemma Maclaurin2_objl:
|
paulson@15079
|
236 |
"0 < h & diff 0 = f &
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paulson@15079
|
237 |
(\<forall>m t.
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paulson@15079
|
238 |
m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)
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paulson@15079
|
239 |
--> (\<exists>t. 0 < t &
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paulson@15079
|
240 |
t \<le> h &
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paulson@15079
|
241 |
f h =
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nipkow@15539
|
242 |
(\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
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paulson@15079
|
243 |
diff n t / real (fact n) * h ^ n)"
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paulson@15079
|
244 |
by (blast intro: Maclaurin2)
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paulson@15079
|
245 |
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paulson@15079
|
246 |
lemma Maclaurin_minus:
|
nipkow@25162
|
247 |
"[| h < 0; n > 0; diff 0 = f;
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paulson@15079
|
248 |
\<forall>m t. m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t |]
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paulson@15079
|
249 |
==> \<exists>t. h < t &
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paulson@15079
|
250 |
t < 0 &
|
paulson@15079
|
251 |
f h =
|
nipkow@15539
|
252 |
(\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
|
paulson@15079
|
253 |
diff n t / real (fact n) * h ^ n"
|
paulson@15079
|
254 |
apply (cut_tac f = "%x. f (-x)"
|
huffman@23177
|
255 |
and diff = "%n x. (-1 ^ n) * diff n (-x)"
|
paulson@15079
|
256 |
and h = "-h" and n = n in Maclaurin_objl)
|
nipkow@15539
|
257 |
apply (simp)
|
paulson@15079
|
258 |
apply safe
|
paulson@15079
|
259 |
apply (subst minus_mult_right)
|
paulson@15079
|
260 |
apply (rule DERIV_cmult)
|
paulson@15079
|
261 |
apply (rule lemma_DERIV_subst)
|
paulson@15079
|
262 |
apply (rule DERIV_chain2 [where g=uminus])
|
huffman@23069
|
263 |
apply (rule_tac [2] DERIV_minus, rule_tac [2] DERIV_ident)
|
paulson@15079
|
264 |
prefer 2 apply force
|
paulson@15079
|
265 |
apply force
|
paulson@15079
|
266 |
apply (rule_tac x = "-t" in exI, auto)
|
paulson@15079
|
267 |
apply (subgoal_tac "(\<Sum>m = 0..<n. -1 ^ m * diff m 0 * (-h)^m / real(fact m)) =
|
paulson@15079
|
268 |
(\<Sum>m = 0..<n. diff m 0 * h ^ m / real(fact m))")
|
nipkow@15536
|
269 |
apply (rule_tac [2] setsum_cong[OF refl])
|
paulson@15079
|
270 |
apply (auto simp add: divide_inverse power_mult_distrib [symmetric])
|
paulson@15079
|
271 |
done
|
paulson@15079
|
272 |
|
paulson@15079
|
273 |
lemma Maclaurin_minus_objl:
|
nipkow@25162
|
274 |
"(h < 0 & n > 0 & diff 0 = f &
|
paulson@15079
|
275 |
(\<forall>m t.
|
paulson@15079
|
276 |
m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t))
|
paulson@15079
|
277 |
--> (\<exists>t. h < t &
|
paulson@15079
|
278 |
t < 0 &
|
paulson@15079
|
279 |
f h =
|
nipkow@15539
|
280 |
(\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
|
paulson@15079
|
281 |
diff n t / real (fact n) * h ^ n)"
|
paulson@15079
|
282 |
by (blast intro: Maclaurin_minus)
|
paulson@15079
|
283 |
|
paulson@15079
|
284 |
|
paulson@15079
|
285 |
subsection{*More Convenient "Bidirectional" Version.*}
|
paulson@15079
|
286 |
|
paulson@15079
|
287 |
(* not good for PVS sin_approx, cos_approx *)
|
paulson@15079
|
288 |
|
paulson@15079
|
289 |
lemma Maclaurin_bi_le_lemma [rule_format]:
|
nipkow@25162
|
290 |
"n>0 \<longrightarrow>
|
nipkow@25134
|
291 |
diff 0 0 =
|
nipkow@25134
|
292 |
(\<Sum>m = 0..<n. diff m 0 * 0 ^ m / real (fact m)) +
|
nipkow@25134
|
293 |
diff n 0 * 0 ^ n / real (fact n)"
|
paulson@15251
|
294 |
by (induct "n", auto)
|
paulson@15079
|
295 |
|
paulson@15079
|
296 |
lemma Maclaurin_bi_le:
|
paulson@15079
|
297 |
"[| diff 0 = f;
|
paulson@15079
|
298 |
\<forall>m t. m < n & abs t \<le> abs x --> DERIV (diff m) t :> diff (Suc m) t |]
|
paulson@15079
|
299 |
==> \<exists>t. abs t \<le> abs x &
|
paulson@15079
|
300 |
f x =
|
nipkow@15539
|
301 |
(\<Sum>m=0..<n. diff m 0 / real (fact m) * x ^ m) +
|
paulson@15079
|
302 |
diff n t / real (fact n) * x ^ n"
|
paulson@15079
|
303 |
apply (case_tac "n = 0", force)
|
paulson@15079
|
304 |
apply (case_tac "x = 0")
|
nipkow@25134
|
305 |
apply (rule_tac x = 0 in exI)
|
nipkow@25134
|
306 |
apply (force simp add: Maclaurin_bi_le_lemma)
|
nipkow@25134
|
307 |
apply (cut_tac x = x and y = 0 in linorder_less_linear, auto)
|
nipkow@25134
|
308 |
txt{*Case 1, where @{term "x < 0"}*}
|
nipkow@25134
|
309 |
apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_minus_objl, safe)
|
nipkow@25134
|
310 |
apply (simp add: abs_if)
|
nipkow@25134
|
311 |
apply (rule_tac x = t in exI)
|
nipkow@25134
|
312 |
apply (simp add: abs_if)
|
paulson@15079
|
313 |
txt{*Case 2, where @{term "0 < x"}*}
|
paulson@15079
|
314 |
apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_objl, safe)
|
nipkow@25134
|
315 |
apply (simp add: abs_if)
|
paulson@15079
|
316 |
apply (rule_tac x = t in exI)
|
paulson@15079
|
317 |
apply (simp add: abs_if)
|
paulson@15079
|
318 |
done
|
paulson@15079
|
319 |
|
paulson@15079
|
320 |
lemma Maclaurin_all_lt:
|
paulson@15079
|
321 |
"[| diff 0 = f;
|
paulson@15079
|
322 |
\<forall>m x. DERIV (diff m) x :> diff(Suc m) x;
|
nipkow@25162
|
323 |
x ~= 0; n > 0
|
paulson@15079
|
324 |
|] ==> \<exists>t. 0 < abs t & abs t < abs x &
|
nipkow@15539
|
325 |
f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
|
paulson@15079
|
326 |
(diff n t / real (fact n)) * x ^ n"
|
paulson@15079
|
327 |
apply (rule_tac x = x and y = 0 in linorder_cases)
|
paulson@15079
|
328 |
prefer 2 apply blast
|
paulson@15079
|
329 |
apply (drule_tac [2] diff=diff in Maclaurin)
|
paulson@15079
|
330 |
apply (drule_tac diff=diff in Maclaurin_minus, simp_all, safe)
|
paulson@15229
|
331 |
apply (rule_tac [!] x = t in exI, auto)
|
paulson@15079
|
332 |
done
|
paulson@15079
|
333 |
|
paulson@15079
|
334 |
lemma Maclaurin_all_lt_objl:
|
paulson@15079
|
335 |
"diff 0 = f &
|
paulson@15079
|
336 |
(\<forall>m x. DERIV (diff m) x :> diff(Suc m) x) &
|
nipkow@25162
|
337 |
x ~= 0 & n > 0
|
paulson@15079
|
338 |
--> (\<exists>t. 0 < abs t & abs t < abs x &
|
nipkow@15539
|
339 |
f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
|
paulson@15079
|
340 |
(diff n t / real (fact n)) * x ^ n)"
|
paulson@15079
|
341 |
by (blast intro: Maclaurin_all_lt)
|
paulson@15079
|
342 |
|
paulson@15079
|
343 |
lemma Maclaurin_zero [rule_format]:
|
paulson@15079
|
344 |
"x = (0::real)
|
nipkow@25134
|
345 |
==> n \<noteq> 0 -->
|
nipkow@15539
|
346 |
(\<Sum>m=0..<n. (diff m (0::real) / real (fact m)) * x ^ m) =
|
paulson@15079
|
347 |
diff 0 0"
|
paulson@15079
|
348 |
by (induct n, auto)
|
paulson@15079
|
349 |
|
paulson@15079
|
350 |
lemma Maclaurin_all_le: "[| diff 0 = f;
|
paulson@15079
|
351 |
\<forall>m x. DERIV (diff m) x :> diff (Suc m) x
|
paulson@15079
|
352 |
|] ==> \<exists>t. abs t \<le> abs x &
|
nipkow@15539
|
353 |
f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
|
paulson@15079
|
354 |
(diff n t / real (fact n)) * x ^ n"
|
nipkow@25134
|
355 |
apply(cases "n=0")
|
nipkow@25134
|
356 |
apply (force)
|
paulson@15079
|
357 |
apply (case_tac "x = 0")
|
paulson@15079
|
358 |
apply (frule_tac diff = diff and n = n in Maclaurin_zero, assumption)
|
nipkow@25134
|
359 |
apply (drule not0_implies_Suc)
|
paulson@15079
|
360 |
apply (rule_tac x = 0 in exI, force)
|
paulson@15079
|
361 |
apply (frule_tac diff = diff and n = n in Maclaurin_all_lt, auto)
|
paulson@15079
|
362 |
apply (rule_tac x = t in exI, auto)
|
paulson@15079
|
363 |
done
|
paulson@15079
|
364 |
|
paulson@15079
|
365 |
lemma Maclaurin_all_le_objl: "diff 0 = f &
|
paulson@15079
|
366 |
(\<forall>m x. DERIV (diff m) x :> diff (Suc m) x)
|
paulson@15079
|
367 |
--> (\<exists>t. abs t \<le> abs x &
|
nipkow@15539
|
368 |
f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
|
paulson@15079
|
369 |
(diff n t / real (fact n)) * x ^ n)"
|
paulson@15079
|
370 |
by (blast intro: Maclaurin_all_le)
|
paulson@15079
|
371 |
|
paulson@15079
|
372 |
|
paulson@15079
|
373 |
subsection{*Version for Exponential Function*}
|
paulson@15079
|
374 |
|
nipkow@25162
|
375 |
lemma Maclaurin_exp_lt: "[| x ~= 0; n > 0 |]
|
paulson@15079
|
376 |
==> (\<exists>t. 0 < abs t &
|
paulson@15079
|
377 |
abs t < abs x &
|
nipkow@15539
|
378 |
exp x = (\<Sum>m=0..<n. (x ^ m) / real (fact m)) +
|
paulson@15079
|
379 |
(exp t / real (fact n)) * x ^ n)"
|
paulson@15079
|
380 |
by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_lt_objl, auto)
|
paulson@15079
|
381 |
|
paulson@15079
|
382 |
|
paulson@15079
|
383 |
lemma Maclaurin_exp_le:
|
paulson@15079
|
384 |
"\<exists>t. abs t \<le> abs x &
|
nipkow@15539
|
385 |
exp x = (\<Sum>m=0..<n. (x ^ m) / real (fact m)) +
|
paulson@15079
|
386 |
(exp t / real (fact n)) * x ^ n"
|
paulson@15079
|
387 |
by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_le_objl, auto)
|
paulson@15079
|
388 |
|
paulson@15079
|
389 |
|
paulson@15079
|
390 |
subsection{*Version for Sine Function*}
|
paulson@15079
|
391 |
|
paulson@15079
|
392 |
lemma mod_exhaust_less_4:
|
nipkow@25134
|
393 |
"m mod 4 = 0 | m mod 4 = 1 | m mod 4 = 2 | m mod 4 = (3::nat)"
|
webertj@20217
|
394 |
by auto
|
paulson@15079
|
395 |
|
paulson@15079
|
396 |
lemma Suc_Suc_mult_two_diff_two [rule_format, simp]:
|
nipkow@25134
|
397 |
"n\<noteq>0 --> Suc (Suc (2 * n - 2)) = 2*n"
|
paulson@15251
|
398 |
by (induct "n", auto)
|
paulson@15079
|
399 |
|
paulson@15079
|
400 |
lemma lemma_Suc_Suc_4n_diff_2 [rule_format, simp]:
|
nipkow@25134
|
401 |
"n\<noteq>0 --> Suc (Suc (4*n - 2)) = 4*n"
|
paulson@15251
|
402 |
by (induct "n", auto)
|
paulson@15079
|
403 |
|
paulson@15079
|
404 |
lemma Suc_mult_two_diff_one [rule_format, simp]:
|
nipkow@25134
|
405 |
"n\<noteq>0 --> Suc (2 * n - 1) = 2*n"
|
paulson@15251
|
406 |
by (induct "n", auto)
|
paulson@15079
|
407 |
|
paulson@15234
|
408 |
|
paulson@15234
|
409 |
text{*It is unclear why so many variant results are needed.*}
|
paulson@15079
|
410 |
|
paulson@15079
|
411 |
lemma Maclaurin_sin_expansion2:
|
paulson@15079
|
412 |
"\<exists>t. abs t \<le> abs x &
|
paulson@15079
|
413 |
sin x =
|
nipkow@15539
|
414 |
(\<Sum>m=0..<n. (if even m then 0
|
huffman@23177
|
415 |
else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
|
nipkow@15539
|
416 |
x ^ m)
|
paulson@15079
|
417 |
+ ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
|
paulson@15079
|
418 |
apply (cut_tac f = sin and n = n and x = x
|
paulson@15079
|
419 |
and diff = "%n x. sin (x + 1/2*real n * pi)" in Maclaurin_all_lt_objl)
|
paulson@15079
|
420 |
apply safe
|
paulson@15079
|
421 |
apply (simp (no_asm))
|
nipkow@15539
|
422 |
apply (simp (no_asm))
|
huffman@23242
|
423 |
apply (case_tac "n", clarify, simp, simp add: lemma_STAR_sin)
|
paulson@15079
|
424 |
apply (rule ccontr, simp)
|
paulson@15079
|
425 |
apply (drule_tac x = x in spec, simp)
|
paulson@15079
|
426 |
apply (erule ssubst)
|
paulson@15079
|
427 |
apply (rule_tac x = t in exI, simp)
|
nipkow@15536
|
428 |
apply (rule setsum_cong[OF refl])
|
nipkow@15539
|
429 |
apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
|
paulson@15079
|
430 |
done
|
paulson@15079
|
431 |
|
paulson@15234
|
432 |
lemma Maclaurin_sin_expansion:
|
paulson@15234
|
433 |
"\<exists>t. sin x =
|
nipkow@15539
|
434 |
(\<Sum>m=0..<n. (if even m then 0
|
huffman@23177
|
435 |
else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
|
nipkow@15539
|
436 |
x ^ m)
|
paulson@15234
|
437 |
+ ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
|
paulson@15234
|
438 |
apply (insert Maclaurin_sin_expansion2 [of x n])
|
paulson@15234
|
439 |
apply (blast intro: elim:);
|
paulson@15234
|
440 |
done
|
paulson@15234
|
441 |
|
paulson@15234
|
442 |
|
paulson@15079
|
443 |
lemma Maclaurin_sin_expansion3:
|
nipkow@25162
|
444 |
"[| n > 0; 0 < x |] ==>
|
paulson@15079
|
445 |
\<exists>t. 0 < t & t < x &
|
paulson@15079
|
446 |
sin x =
|
nipkow@15539
|
447 |
(\<Sum>m=0..<n. (if even m then 0
|
huffman@23177
|
448 |
else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
|
nipkow@15539
|
449 |
x ^ m)
|
paulson@15079
|
450 |
+ ((sin(t + 1/2 * real(n) *pi) / real (fact n)) * x ^ n)"
|
paulson@15079
|
451 |
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
|
452 |
apply safe
|
paulson@15079
|
453 |
apply simp
|
nipkow@15539
|
454 |
apply (simp (no_asm))
|
paulson@15079
|
455 |
apply (erule ssubst)
|
paulson@15079
|
456 |
apply (rule_tac x = t in exI, simp)
|
nipkow@15536
|
457 |
apply (rule setsum_cong[OF refl])
|
nipkow@15539
|
458 |
apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
|
paulson@15079
|
459 |
done
|
paulson@15079
|
460 |
|
paulson@15079
|
461 |
lemma Maclaurin_sin_expansion4:
|
paulson@15079
|
462 |
"0 < x ==>
|
paulson@15079
|
463 |
\<exists>t. 0 < t & t \<le> x &
|
paulson@15079
|
464 |
sin x =
|
nipkow@15539
|
465 |
(\<Sum>m=0..<n. (if even m then 0
|
huffman@23177
|
466 |
else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
|
nipkow@15539
|
467 |
x ^ m)
|
paulson@15079
|
468 |
+ ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
|
paulson@15079
|
469 |
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
|
470 |
apply safe
|
paulson@15079
|
471 |
apply simp
|
nipkow@15539
|
472 |
apply (simp (no_asm))
|
paulson@15079
|
473 |
apply (erule ssubst)
|
paulson@15079
|
474 |
apply (rule_tac x = t in exI, simp)
|
nipkow@15536
|
475 |
apply (rule setsum_cong[OF refl])
|
nipkow@15539
|
476 |
apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
|
paulson@15079
|
477 |
done
|
paulson@15079
|
478 |
|
paulson@15079
|
479 |
|
paulson@15079
|
480 |
subsection{*Maclaurin Expansion for Cosine Function*}
|
paulson@15079
|
481 |
|
paulson@15079
|
482 |
lemma sumr_cos_zero_one [simp]:
|
nipkow@15539
|
483 |
"(\<Sum>m=0..<(Suc n).
|
huffman@23177
|
484 |
(if even m then -1 ^ (m div 2)/(real (fact m)) else 0) * 0 ^ m) = 1"
|
paulson@15251
|
485 |
by (induct "n", auto)
|
paulson@15079
|
486 |
|
paulson@15079
|
487 |
lemma Maclaurin_cos_expansion:
|
paulson@15079
|
488 |
"\<exists>t. abs t \<le> abs x &
|
paulson@15079
|
489 |
cos x =
|
nipkow@15539
|
490 |
(\<Sum>m=0..<n. (if even m
|
huffman@23177
|
491 |
then -1 ^ (m div 2)/(real (fact m))
|
paulson@15079
|
492 |
else 0) *
|
nipkow@15539
|
493 |
x ^ m)
|
paulson@15079
|
494 |
+ ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
|
paulson@15079
|
495 |
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
|
496 |
apply safe
|
paulson@15079
|
497 |
apply (simp (no_asm))
|
nipkow@15539
|
498 |
apply (simp (no_asm))
|
paulson@15079
|
499 |
apply (case_tac "n", simp)
|
nipkow@15561
|
500 |
apply (simp del: setsum_op_ivl_Suc)
|
paulson@15079
|
501 |
apply (rule ccontr, simp)
|
paulson@15079
|
502 |
apply (drule_tac x = x in spec, simp)
|
paulson@15079
|
503 |
apply (erule ssubst)
|
paulson@15079
|
504 |
apply (rule_tac x = t in exI, simp)
|
nipkow@15536
|
505 |
apply (rule setsum_cong[OF refl])
|
paulson@15234
|
506 |
apply (auto simp add: cos_zero_iff even_mult_two_ex)
|
paulson@15079
|
507 |
done
|
paulson@15079
|
508 |
|
paulson@15079
|
509 |
lemma Maclaurin_cos_expansion2:
|
nipkow@25162
|
510 |
"[| 0 < x; n > 0 |] ==>
|
paulson@15079
|
511 |
\<exists>t. 0 < t & t < x &
|
paulson@15079
|
512 |
cos x =
|
nipkow@15539
|
513 |
(\<Sum>m=0..<n. (if even m
|
huffman@23177
|
514 |
then -1 ^ (m div 2)/(real (fact m))
|
paulson@15079
|
515 |
else 0) *
|
nipkow@15539
|
516 |
x ^ m)
|
paulson@15079
|
517 |
+ ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
|
paulson@15079
|
518 |
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
|
519 |
apply safe
|
paulson@15079
|
520 |
apply simp
|
nipkow@15539
|
521 |
apply (simp (no_asm))
|
paulson@15079
|
522 |
apply (erule ssubst)
|
paulson@15079
|
523 |
apply (rule_tac x = t in exI, simp)
|
nipkow@15536
|
524 |
apply (rule setsum_cong[OF refl])
|
paulson@15234
|
525 |
apply (auto simp add: cos_zero_iff even_mult_two_ex)
|
paulson@15079
|
526 |
done
|
paulson@15079
|
527 |
|
paulson@15234
|
528 |
lemma Maclaurin_minus_cos_expansion:
|
nipkow@25162
|
529 |
"[| x < 0; n > 0 |] ==>
|
paulson@15079
|
530 |
\<exists>t. x < t & t < 0 &
|
paulson@15079
|
531 |
cos x =
|
nipkow@15539
|
532 |
(\<Sum>m=0..<n. (if even m
|
huffman@23177
|
533 |
then -1 ^ (m div 2)/(real (fact m))
|
paulson@15079
|
534 |
else 0) *
|
nipkow@15539
|
535 |
x ^ m)
|
paulson@15079
|
536 |
+ ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
|
paulson@15079
|
537 |
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
|
538 |
apply safe
|
paulson@15079
|
539 |
apply simp
|
nipkow@15539
|
540 |
apply (simp (no_asm))
|
paulson@15079
|
541 |
apply (erule ssubst)
|
paulson@15079
|
542 |
apply (rule_tac x = t in exI, simp)
|
nipkow@15536
|
543 |
apply (rule setsum_cong[OF refl])
|
paulson@15234
|
544 |
apply (auto simp add: cos_zero_iff even_mult_two_ex)
|
paulson@15079
|
545 |
done
|
paulson@15079
|
546 |
|
paulson@15079
|
547 |
(* ------------------------------------------------------------------------- *)
|
paulson@15079
|
548 |
(* Version for ln(1 +/- x). Where is it?? *)
|
paulson@15079
|
549 |
(* ------------------------------------------------------------------------- *)
|
paulson@15079
|
550 |
|
paulson@15079
|
551 |
lemma sin_bound_lemma:
|
paulson@15081
|
552 |
"[|x = y; abs u \<le> (v::real) |] ==> \<bar>(x + u) - y\<bar> \<le> v"
|
paulson@15079
|
553 |
by auto
|
paulson@15079
|
554 |
|
paulson@15079
|
555 |
lemma Maclaurin_sin_bound:
|
huffman@23177
|
556 |
"abs(sin x - (\<Sum>m=0..<n. (if even m then 0 else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
|
paulson@15081
|
557 |
x ^ m)) \<le> inverse(real (fact n)) * \<bar>x\<bar> ^ n"
|
obua@14738
|
558 |
proof -
|
paulson@15079
|
559 |
have "!! x (y::real). x \<le> 1 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x * y \<le> 1 * y"
|
obua@14738
|
560 |
by (rule_tac mult_right_mono,simp_all)
|
obua@14738
|
561 |
note est = this[simplified]
|
huffman@22985
|
562 |
let ?diff = "\<lambda>(n::nat) 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)"
|
huffman@22985
|
563 |
have diff_0: "?diff 0 = sin" by simp
|
huffman@22985
|
564 |
have DERIV_diff: "\<forall>m x. DERIV (?diff m) x :> ?diff (Suc m) x"
|
huffman@22985
|
565 |
apply (clarify)
|
huffman@22985
|
566 |
apply (subst (1 2 3) mod_Suc_eq_Suc_mod)
|
huffman@22985
|
567 |
apply (cut_tac m=m in mod_exhaust_less_4)
|
huffman@22985
|
568 |
apply (safe, simp_all)
|
huffman@22985
|
569 |
apply (rule DERIV_minus, simp)
|
huffman@22985
|
570 |
apply (rule lemma_DERIV_subst, rule DERIV_minus, rule DERIV_cos, simp)
|
huffman@22985
|
571 |
done
|
huffman@22985
|
572 |
from Maclaurin_all_le [OF diff_0 DERIV_diff]
|
huffman@22985
|
573 |
obtain t where t1: "\<bar>t\<bar> \<le> \<bar>x\<bar>" and
|
huffman@22985
|
574 |
t2: "sin x = (\<Sum>m = 0..<n. ?diff m 0 / real (fact m) * x ^ m) +
|
huffman@22985
|
575 |
?diff n t / real (fact n) * x ^ n" by fast
|
huffman@22985
|
576 |
have diff_m_0:
|
huffman@22985
|
577 |
"\<And>m. ?diff m 0 = (if even m then 0
|
huffman@23177
|
578 |
else -1 ^ ((m - Suc 0) div 2))"
|
huffman@22985
|
579 |
apply (subst even_even_mod_4_iff)
|
huffman@22985
|
580 |
apply (cut_tac m=m in mod_exhaust_less_4)
|
huffman@22985
|
581 |
apply (elim disjE, simp_all)
|
huffman@22985
|
582 |
apply (safe dest!: mod_eqD, simp_all)
|
huffman@22985
|
583 |
done
|
obua@14738
|
584 |
show ?thesis
|
huffman@22985
|
585 |
apply (subst t2)
|
paulson@15079
|
586 |
apply (rule sin_bound_lemma)
|
nipkow@15536
|
587 |
apply (rule setsum_cong[OF refl])
|
huffman@22985
|
588 |
apply (subst diff_m_0, simp)
|
paulson@15079
|
589 |
apply (auto intro: mult_right_mono [where b=1, simplified] mult_right_mono
|
avigad@16775
|
590 |
simp add: est mult_nonneg_nonneg mult_ac divide_inverse
|
paulson@16924
|
591 |
power_abs [symmetric] abs_mult)
|
obua@14738
|
592 |
done
|
obua@14738
|
593 |
qed
|
obua@14738
|
594 |
|
paulson@15079
|
595 |
end
|