(* Title : NthRoot.thy
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
Conversion to Isar and new proofs by Lawrence C Paulson, 2004
*)
header{*Existence of Nth Root*}
theory NthRoot
imports SEQ
begin
text {*
Various lemmas needed for this result. We follow the proof given by
John Lindsay Orr (\texttt{jorr@math.unl.edu}) in his Analysis
Webnotes available at \url{http://www.math.unl.edu/~webnotes}.
Lemmas about sequences of reals are used to reach the result.
*}
lemma lemma_nth_realpow_non_empty:
"[| (0::real) < a; 0 < n |] ==> \<exists>s. s : {x. x ^ n <= a & 0 < x}"
apply (case_tac "1 <= a")
apply (rule_tac x = 1 in exI)
apply (drule_tac [2] linorder_not_le [THEN iffD1])
apply (drule_tac [2] less_not_refl2 [THEN not0_implies_Suc], simp)
apply (force intro!: realpow_Suc_le_self simp del: realpow_Suc)
done
text{*Used only just below*}
lemma realpow_ge_self2: "[| (1::real) \<le> r; 0 < n |] ==> r \<le> r ^ n"
by (insert power_increasing [of 1 n r], simp)
lemma lemma_nth_realpow_isUb_ex:
"[| (0::real) < a; 0 < n |]
==> \<exists>u. isUb (UNIV::real set) {x. x ^ n <= a & 0 < x} u"
apply (case_tac "1 <= a")
apply (rule_tac x = a in exI)
apply (drule_tac [2] linorder_not_le [THEN iffD1])
apply (rule_tac [2] x = 1 in exI)
apply (rule_tac [!] setleI [THEN isUbI], safe)
apply (simp_all (no_asm))
apply (rule_tac [!] ccontr)
apply (drule_tac [!] linorder_not_le [THEN iffD1])
apply (drule realpow_ge_self2, assumption)
apply (drule_tac n = n in realpow_less)
apply (assumption+)
apply (drule real_le_trans, assumption)
apply (drule_tac y = "y ^ n" in order_less_le_trans, assumption, simp)
apply (drule_tac n = n in zero_less_one [THEN realpow_less], auto)
done
lemma nth_realpow_isLub_ex:
"[| (0::real) < a; 0 < n |]
==> \<exists>u. isLub (UNIV::real set) {x. x ^ n <= a & 0 < x} u"
by (blast intro: lemma_nth_realpow_isUb_ex lemma_nth_realpow_non_empty reals_complete)
subsection{*First Half -- Lemmas First*}
lemma lemma_nth_realpow_seq:
"isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u
==> u + inverse(real (Suc k)) ~: {x. x ^ n <= a & 0 < x}"
apply (safe, drule isLubD2, blast)
apply (simp add: linorder_not_less [symmetric])
done
lemma lemma_nth_realpow_isLub_gt_zero:
"[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;
0 < a; 0 < n |] ==> 0 < u"
apply (drule lemma_nth_realpow_non_empty, auto)
apply (drule_tac y = s in isLub_isUb [THEN isUbD])
apply (auto intro: order_less_le_trans)
done
lemma lemma_nth_realpow_isLub_ge:
"[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;
0 < a; 0 < n |] ==> ALL k. a <= (u + inverse(real (Suc k))) ^ n"
apply safe
apply (frule lemma_nth_realpow_seq, safe)
apply (auto elim: order_less_asym simp add: linorder_not_less [symmetric]
iff: real_0_less_add_iff) --{*legacy iff rule!*}
apply (simp add: linorder_not_less)
apply (rule order_less_trans [of _ 0])
apply (auto intro: lemma_nth_realpow_isLub_gt_zero)
done
text{*First result we want*}
lemma realpow_nth_ge:
"[| (0::real) < a; 0 < n;
isLub (UNIV::real set)
{x. x ^ n <= a & 0 < x} u |] ==> a <= u ^ n"
apply (frule lemma_nth_realpow_isLub_ge, safe)
apply (rule LIMSEQ_inverse_real_of_nat_add [THEN LIMSEQ_pow, THEN LIMSEQ_le_const])
apply (auto simp add: real_of_nat_def)
done
subsection{*Second Half*}
lemma less_isLub_not_isUb:
"[| isLub (UNIV::real set) S u; x < u |]
==> ~ isUb (UNIV::real set) S x"
apply safe
apply (drule isLub_le_isUb, assumption)
apply (drule order_less_le_trans, auto)
done
lemma not_isUb_less_ex:
"~ isUb (UNIV::real set) S u ==> \<exists>x \<in> S. u < x"
apply (rule ccontr, erule contrapos_np)
apply (rule setleI [THEN isUbI])
apply (auto simp add: linorder_not_less [symmetric])
done
lemma real_mult_less_self: "0 < r ==> r * (1 + -inverse(real (Suc n))) < r"
apply (simp (no_asm) add: right_distrib)
apply (rule add_less_cancel_left [of "-r", THEN iffD1])
apply (auto intro: mult_pos_pos
simp add: add_assoc [symmetric] neg_less_0_iff_less)
done
lemma real_mult_add_one_minus_ge_zero:
"0 < r ==> 0 <= r*(1 + -inverse(real (Suc n)))"
by (simp add: zero_le_mult_iff real_of_nat_inverse_le_iff real_0_le_add_iff)
lemma lemma_nth_realpow_isLub_le:
"[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;
0 < a; 0 < n |] ==> ALL k. (u*(1 + -inverse(real (Suc k)))) ^ n <= a"
apply safe
apply (frule less_isLub_not_isUb [THEN not_isUb_less_ex])
apply (rule_tac n = k in real_mult_less_self)
apply (blast intro: lemma_nth_realpow_isLub_gt_zero, safe)
apply (drule_tac n = k in
lemma_nth_realpow_isLub_gt_zero [THEN real_mult_add_one_minus_ge_zero], assumption+)
apply (blast intro: order_trans order_less_imp_le power_mono)
done
text{*Second result we want*}
lemma realpow_nth_le:
"[| (0::real) < a; 0 < n;
isLub (UNIV::real set)
{x. x ^ n <= a & 0 < x} u |] ==> u ^ n <= a"
apply (frule lemma_nth_realpow_isLub_le, safe)
apply (rule LIMSEQ_inverse_real_of_nat_add_minus_mult
[THEN LIMSEQ_pow, THEN LIMSEQ_le_const2])
apply (auto simp add: real_of_nat_def)
done
text{*The theorem at last!*}
lemma realpow_nth: "[| (0::real) < a; 0 < n |] ==> \<exists>r. r ^ n = a"
apply (frule nth_realpow_isLub_ex, auto)
apply (auto intro: realpow_nth_le realpow_nth_ge order_antisym)
done
(* positive only *)
lemma realpow_pos_nth: "[| (0::real) < a; 0 < n |] ==> \<exists>r. 0 < r & r ^ n = a"
apply (frule nth_realpow_isLub_ex, auto)
apply (auto intro: realpow_nth_le realpow_nth_ge order_antisym lemma_nth_realpow_isLub_gt_zero)
done
lemma realpow_pos_nth2: "(0::real) < a ==> \<exists>r. 0 < r & r ^ Suc n = a"
by (blast intro: realpow_pos_nth)
(* uniqueness of nth positive root *)
lemma realpow_pos_nth_unique:
"[| (0::real) < a; 0 < n |] ==> EX! r. 0 < r & r ^ n = a"
apply (auto intro!: realpow_pos_nth)
apply (cut_tac x = r and y = y in linorder_less_linear, auto)
apply (drule_tac x = r in realpow_less)
apply (drule_tac [4] x = y in realpow_less, auto)
done
end