--- a/src/HOL/MacLaurin.thy Fri Jul 10 10:45:49 2009 -0400
+++ b/src/HOL/MacLaurin.thy Fri Jul 10 12:55:06 2009 -0400
@@ -27,6 +27,10 @@
lemma eq_diff_eq': "(x = y - z) = (y = x + (z::real))"
by arith
+lemma fact_diff_Suc [rule_format]:
+ "n < Suc m ==> fact (Suc m - n) = (Suc m - n) * fact (m - n)"
+ by (subst fact_reduce_nat, auto)
+
lemma Maclaurin_lemma2:
assumes diff: "\<forall>m t. m < n \<and> 0\<le>t \<and> t\<le>h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
assumes n: "n = Suc k"
@@ -47,22 +51,24 @@
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 3
+
+apply (simp add: fact_diff_Suc)
prefer 2 apply simp
apply (frule less_iff_Suc_add [THEN iffD1], clarify)
apply (simp del: setsum_op_ivl_Suc)
apply (insert sumr_offset4 [of "Suc 0"])
- apply (simp del: setsum_op_ivl_Suc fact_Suc power_Suc)
+ apply (simp del: setsum_op_ivl_Suc fact_Suc_nat power_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 power_Suc)
+ apply (simp (no_asm) add: divide_inverse mult_assoc del: fact_Suc_nat power_Suc)
apply (rule DERIV_cmult)
apply (rule lemma_DERIV_subst)
apply (best intro!: DERIV_intros)
- apply (subst fact_Suc)
+ apply (subst fact_Suc_nat)
apply (subst real_of_nat_mult)
apply (simp add: mult_ac)
done
@@ -114,7 +120,7 @@
apply (frule less_iff_Suc_add [THEN iffD1], clarify)
apply (simp del: setsum_op_ivl_Suc)
apply (insert sumr_offset4 [of "Suc 0"])
- apply (simp del: setsum_op_ivl_Suc fact_Suc)
+ apply (simp del: setsum_op_ivl_Suc fact_Suc_nat)
done
have isCont_difg: "\<And>m x. \<lbrakk>m < n; 0 \<le> x; x \<le> h\<rbrakk> \<Longrightarrow> isCont (difg m) x"
@@ -174,7 +180,7 @@
(\<Sum>m = 0..<n. diff m 0 / real (fact m) * h ^ m) +
diff n t / real (fact n) * h ^ n"
using `difg (Suc m) t = 0`
- by (simp add: m f_h difg_def del: fact_Suc)
+ by (simp add: m f_h difg_def del: fact_Suc_nat)
qed
qed