author  hoelzl 
Fri, 22 Mar 2013 10:41:43 +0100  
changeset 51483  dc39d69774bb 
parent 51478  270b21f3ae0a 
child 53015  a1119cf551e8 
permissions  rwrr 
12196  1 
(* Title : NthRoot.thy 
2 
Author : Jacques D. Fleuriot 

3 
Copyright : 1998 University of Cambridge 

14477  4 
Conversion to Isar and new proofs by Lawrence C Paulson, 2004 
12196  5 
*) 
6 

22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

7 
header {* Nth Roots of Real Numbers *} 
14324  8 

15131  9 
theory NthRoot 
28952
15a4b2cf8c34
made repository layout more coherent with logical distribution structure; stripped some $Id$s
haftmann
parents:
25875
diff
changeset

10 
imports Parity Deriv 
15131  11 
begin 
14324  12 

51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

13 
lemma abs_sgn_eq: "abs (sgn x :: real) = (if x = 0 then 0 else 1)" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

14 
by (simp add: sgn_real_def) 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

15 

dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

16 
lemma inverse_sgn: "sgn (inverse a) = inverse (sgn a :: real)" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

17 
by (simp add: sgn_real_def) 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

18 

dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

19 
lemma power_eq_iff_eq_base: 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

20 
fixes a b :: "_ :: linordered_semidom" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

21 
shows "0 < n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a ^ n = b ^ n \<longleftrightarrow> a = b" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

22 
using power_eq_imp_eq_base[of a n b] by auto 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

23 

22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

24 
subsection {* Existence of Nth Root *} 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

25 

23009
01c295dd4a36
Prove existence of nth roots using Intermediate Value Theorem
huffman
parents:
22968
diff
changeset

26 
text {* Existence follows from the Intermediate Value Theorem *} 
14324  27 

23009
01c295dd4a36
Prove existence of nth roots using Intermediate Value Theorem
huffman
parents:
22968
diff
changeset

28 
lemma realpow_pos_nth: 
01c295dd4a36
Prove existence of nth roots using Intermediate Value Theorem
huffman
parents:
22968
diff
changeset

29 
assumes n: "0 < n" 
01c295dd4a36
Prove existence of nth roots using Intermediate Value Theorem
huffman
parents:
22968
diff
changeset

30 
assumes a: "0 < a" 
01c295dd4a36
Prove existence of nth roots using Intermediate Value Theorem
huffman
parents:
22968
diff
changeset

31 
shows "\<exists>r>0. r ^ n = (a::real)" 
01c295dd4a36
Prove existence of nth roots using Intermediate Value Theorem
huffman
parents:
22968
diff
changeset

32 
proof  
01c295dd4a36
Prove existence of nth roots using Intermediate Value Theorem
huffman
parents:
22968
diff
changeset

33 
have "\<exists>r\<ge>0. r \<le> (max 1 a) \<and> r ^ n = a" 
01c295dd4a36
Prove existence of nth roots using Intermediate Value Theorem
huffman
parents:
22968
diff
changeset

34 
proof (rule IVT) 
01c295dd4a36
Prove existence of nth roots using Intermediate Value Theorem
huffman
parents:
22968
diff
changeset

35 
show "0 ^ n \<le> a" using n a by (simp add: power_0_left) 
01c295dd4a36
Prove existence of nth roots using Intermediate Value Theorem
huffman
parents:
22968
diff
changeset

36 
show "0 \<le> max 1 a" by simp 
01c295dd4a36
Prove existence of nth roots using Intermediate Value Theorem
huffman
parents:
22968
diff
changeset

37 
from n have n1: "1 \<le> n" by simp 
01c295dd4a36
Prove existence of nth roots using Intermediate Value Theorem
huffman
parents:
22968
diff
changeset

38 
have "a \<le> max 1 a ^ 1" by simp 
01c295dd4a36
Prove existence of nth roots using Intermediate Value Theorem
huffman
parents:
22968
diff
changeset

39 
also have "max 1 a ^ 1 \<le> max 1 a ^ n" 
01c295dd4a36
Prove existence of nth roots using Intermediate Value Theorem
huffman
parents:
22968
diff
changeset

40 
using n1 by (rule power_increasing, simp) 
01c295dd4a36
Prove existence of nth roots using Intermediate Value Theorem
huffman
parents:
22968
diff
changeset

41 
finally show "a \<le> max 1 a ^ n" . 
01c295dd4a36
Prove existence of nth roots using Intermediate Value Theorem
huffman
parents:
22968
diff
changeset

42 
show "\<forall>r. 0 \<le> r \<and> r \<le> max 1 a \<longrightarrow> isCont (\<lambda>x. x ^ n) r" 
44289  43 
by simp 
23009
01c295dd4a36
Prove existence of nth roots using Intermediate Value Theorem
huffman
parents:
22968
diff
changeset

44 
qed 
01c295dd4a36
Prove existence of nth roots using Intermediate Value Theorem
huffman
parents:
22968
diff
changeset

45 
then obtain r where r: "0 \<le> r \<and> r ^ n = a" by fast 
01c295dd4a36
Prove existence of nth roots using Intermediate Value Theorem
huffman
parents:
22968
diff
changeset

46 
with n a have "r \<noteq> 0" by (auto simp add: power_0_left) 
01c295dd4a36
Prove existence of nth roots using Intermediate Value Theorem
huffman
parents:
22968
diff
changeset

47 
with r have "0 < r \<and> r ^ n = a" by simp 
01c295dd4a36
Prove existence of nth roots using Intermediate Value Theorem
huffman
parents:
22968
diff
changeset

48 
thus ?thesis .. 
01c295dd4a36
Prove existence of nth roots using Intermediate Value Theorem
huffman
parents:
22968
diff
changeset

49 
qed 
14325  50 

23047  51 
(* Used by Integration/RealRandVar.thy in AFP *) 
52 
lemma realpow_pos_nth2: "(0::real) < a \<Longrightarrow> \<exists>r>0. r ^ Suc n = a" 

53 
by (blast intro: realpow_pos_nth) 

54 

23009
01c295dd4a36
Prove existence of nth roots using Intermediate Value Theorem
huffman
parents:
22968
diff
changeset

55 
text {* Uniqueness of nth positive root *} 
14324  56 

51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

57 
lemma realpow_pos_nth_unique: "\<lbrakk>0 < n; 0 < a\<rbrakk> \<Longrightarrow> \<exists>!r. 0 < r \<and> r ^ n = (a::real)" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

58 
by (auto intro!: realpow_pos_nth simp: power_eq_iff_eq_base) 
14324  59 

20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

60 
subsection {* Nth Root *} 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

61 

22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

62 
text {* We define roots of negative reals such that 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

63 
@{term "root n ( x) =  root n x"}. This allows 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

64 
us to omit side conditions from many theorems. *} 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

65 

51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

66 
lemma inj_sgn_power: assumes "0 < n" shows "inj (\<lambda>y. sgn y * \<bar>y\<bar>^n :: real)" (is "inj ?f") 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

67 
proof (rule injI) 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

68 
have x: "\<And>a b :: real. (0 < a \<and> b < 0) \<or> (a < 0 \<and> 0 < b) \<Longrightarrow> a \<noteq> b" by auto 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

69 
fix x y assume "?f x = ?f y" with power_eq_iff_eq_base[of n "\<bar>x\<bar>" "\<bar>y\<bar>"] `0<n` show "x = y" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

70 
by (cases rule: linorder_cases[of 0 x, case_product linorder_cases[of 0 y]]) 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

71 
(simp_all add: x) 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

72 
qed 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

73 

dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

74 
lemma sgn_power_injE: "sgn a * \<bar>a\<bar> ^ n = x \<Longrightarrow> x = sgn b * \<bar>b\<bar> ^ n \<Longrightarrow> 0 < n \<Longrightarrow> a = (b::real)" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

75 
using inj_sgn_power[THEN injD, of n a b] by simp 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

76 

dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

77 
definition root :: "nat \<Rightarrow> real \<Rightarrow> real" where 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

78 
"root n x = (if n = 0 then 0 else the_inv (\<lambda>y. sgn y * \<bar>y\<bar>^n) x)" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

79 

dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

80 
lemma root_0 [simp]: "root 0 x = 0" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

81 
by (simp add: root_def) 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

82 

dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

83 
lemma root_sgn_power: "0 < n \<Longrightarrow> root n (sgn y * \<bar>y\<bar>^n) = y" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

84 
using the_inv_f_f[OF inj_sgn_power] by (simp add: root_def) 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

85 

dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

86 
lemma sgn_power_root: 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

87 
assumes "0 < n" shows "sgn (root n x) * \<bar>(root n x)\<bar>^n = x" (is "?f (root n x) = x") 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

88 
proof cases 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

89 
assume "x \<noteq> 0" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

90 
with realpow_pos_nth[OF `0 < n`, of "\<bar>x\<bar>"] obtain r where "0 < r" "r ^ n = \<bar>x\<bar>" by auto 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

91 
with `x \<noteq> 0` have S: "x \<in> range ?f" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

92 
by (intro image_eqI[of _ _ "sgn x * r"]) 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

93 
(auto simp: abs_mult sgn_mult power_mult_distrib abs_sgn_eq mult_sgn_abs) 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

94 
from `0 < n` f_the_inv_into_f[OF inj_sgn_power[OF `0 < n`] this] show ?thesis 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

95 
by (simp add: root_def) 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

96 
qed (insert `0 < n` root_sgn_power[of n 0], simp) 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

97 

dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

98 
lemma split_root: "P (root n x) \<longleftrightarrow> (n = 0 \<longrightarrow> P 0) \<and> (0 < n \<longrightarrow> (\<forall>y. sgn y * \<bar>y\<bar>^n = x \<longrightarrow> P y))" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

99 
apply (cases "n = 0") 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

100 
apply simp_all 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

101 
apply (metis root_sgn_power sgn_power_root) 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

102 
done 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

103 

22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

104 
lemma real_root_zero [simp]: "root n 0 = 0" 
51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

105 
by (simp split: split_root add: sgn_zero_iff) 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

106 

dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

107 
lemma real_root_minus: "root n ( x) =  root n x" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

108 
by (clarsimp split: split_root elim!: sgn_power_injE simp: sgn_minus) 
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

109 

51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

110 
lemma real_root_less_mono: "\<lbrakk>0 < n; x < y\<rbrakk> \<Longrightarrow> root n x < root n y" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

111 
proof (clarsimp split: split_root) 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

112 
have x: "\<And>a b :: real. (0 < b \<and> a < 0) \<Longrightarrow> \<not> a > b" by auto 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

113 
fix a b :: real assume "0 < n" "sgn a * \<bar>a\<bar> ^ n < sgn b * \<bar>b\<bar> ^ n" then show "a < b" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

114 
using power_less_imp_less_base[of a n b] power_less_imp_less_base[of "b" n "a"] 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

115 
by (simp add: sgn_real_def power_less_zero_eq x[of "a ^ n" " (( b) ^ n)"] split: split_if_asm) 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

116 
qed 
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

117 

617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

118 
lemma real_root_gt_zero: "\<lbrakk>0 < n; 0 < x\<rbrakk> \<Longrightarrow> 0 < root n x" 
51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

119 
using real_root_less_mono[of n 0 x] by simp 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

120 

dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

121 
lemma real_root_ge_zero: "0 \<le> x \<Longrightarrow> 0 \<le> root n x" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

122 
using real_root_gt_zero[of n x] by (cases "n = 0") (auto simp add: le_less) 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

123 

22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

124 
lemma real_root_pow_pos: (* TODO: rename *) 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

125 
"\<lbrakk>0 < n; 0 < x\<rbrakk> \<Longrightarrow> root n x ^ n = x" 
51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

126 
using sgn_power_root[of n x] real_root_gt_zero[of n x] by simp 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

127 

22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

128 
lemma real_root_pow_pos2 [simp]: (* TODO: rename *) 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

129 
"\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n x ^ n = x" 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

130 
by (auto simp add: order_le_less real_root_pow_pos) 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

131 

51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

132 
lemma sgn_root: "0 < n \<Longrightarrow> sgn (root n x) = sgn x" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

133 
by (auto split: split_root simp: sgn_real_def power_less_zero_eq) 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

134 

23046  135 
lemma odd_real_root_pow: "odd n \<Longrightarrow> root n x ^ n = x" 
51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

136 
using sgn_power_root[of n x] by (simp add: odd_pos sgn_real_def split: split_if_asm) 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

137 

22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

138 
lemma real_root_power_cancel: "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n (x ^ n) = x" 
51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

139 
using root_sgn_power[of n x] by (auto simp add: le_less power_0_left) 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

140 

23046  141 
lemma odd_real_root_power_cancel: "odd n \<Longrightarrow> root n (x ^ n) = x" 
51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

142 
using root_sgn_power[of n x] by (simp add: odd_pos sgn_real_def power_0_left split: split_if_asm) 
23046  143 

51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

144 
lemma real_root_pos_unique: "\<lbrakk>0 < n; 0 \<le> y; y ^ n = x\<rbrakk> \<Longrightarrow> root n x = y" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

145 
using root_sgn_power[of n y] by (auto simp add: le_less power_0_left) 
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

146 

23046  147 
lemma odd_real_root_unique: 
148 
"\<lbrakk>odd n; y ^ n = x\<rbrakk> \<Longrightarrow> root n x = y" 

149 
by (erule subst, rule odd_real_root_power_cancel) 

150 

22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

151 
lemma real_root_one [simp]: "0 < n \<Longrightarrow> root n 1 = 1" 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

152 
by (simp add: real_root_pos_unique) 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

153 

617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

154 
text {* Root function is strictly monotonic, hence injective *} 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

155 

617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

156 
lemma real_root_le_mono: "\<lbrakk>0 < n; x \<le> y\<rbrakk> \<Longrightarrow> root n x \<le> root n y" 
51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

157 
by (auto simp add: order_le_less real_root_less_mono) 
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

158 

22721
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

159 
lemma real_root_less_iff [simp]: 
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

160 
"0 < n \<Longrightarrow> (root n x < root n y) = (x < y)" 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

161 
apply (cases "x < y") 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

162 
apply (simp add: real_root_less_mono) 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

163 
apply (simp add: linorder_not_less real_root_le_mono) 
22721
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

164 
done 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

165 

d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

166 
lemma real_root_le_iff [simp]: 
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

167 
"0 < n \<Longrightarrow> (root n x \<le> root n y) = (x \<le> y)" 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

168 
apply (cases "x \<le> y") 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

169 
apply (simp add: real_root_le_mono) 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

170 
apply (simp add: linorder_not_le real_root_less_mono) 
22721
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

171 
done 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

172 

d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

173 
lemma real_root_eq_iff [simp]: 
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

174 
"0 < n \<Longrightarrow> (root n x = root n y) = (x = y)" 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

175 
by (simp add: order_eq_iff) 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

176 

617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

177 
lemmas real_root_gt_0_iff [simp] = real_root_less_iff [where x=0, simplified] 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

178 
lemmas real_root_lt_0_iff [simp] = real_root_less_iff [where y=0, simplified] 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

179 
lemmas real_root_ge_0_iff [simp] = real_root_le_iff [where x=0, simplified] 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

180 
lemmas real_root_le_0_iff [simp] = real_root_le_iff [where y=0, simplified] 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

181 
lemmas real_root_eq_0_iff [simp] = real_root_eq_iff [where y=0, simplified] 
22721
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

182 

23257  183 
lemma real_root_gt_1_iff [simp]: "0 < n \<Longrightarrow> (1 < root n y) = (1 < y)" 
184 
by (insert real_root_less_iff [where x=1], simp) 

185 

186 
lemma real_root_lt_1_iff [simp]: "0 < n \<Longrightarrow> (root n x < 1) = (x < 1)" 

187 
by (insert real_root_less_iff [where y=1], simp) 

188 

189 
lemma real_root_ge_1_iff [simp]: "0 < n \<Longrightarrow> (1 \<le> root n y) = (1 \<le> y)" 

190 
by (insert real_root_le_iff [where x=1], simp) 

191 

192 
lemma real_root_le_1_iff [simp]: "0 < n \<Longrightarrow> (root n x \<le> 1) = (x \<le> 1)" 

193 
by (insert real_root_le_iff [where y=1], simp) 

194 

195 
lemma real_root_eq_1_iff [simp]: "0 < n \<Longrightarrow> (root n x = 1) = (x = 1)" 

196 
by (insert real_root_eq_iff [where y=1], simp) 

197 

51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

198 
text {* Roots of multiplication and division *} 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

199 

dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

200 
lemma real_root_mult: "root n (x * y) = root n x * root n y" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

201 
by (auto split: split_root elim!: sgn_power_injE simp: sgn_mult abs_mult power_mult_distrib) 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

202 

dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

203 
lemma real_root_inverse: "root n (inverse x) = inverse (root n x)" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

204 
by (auto split: split_root elim!: sgn_power_injE simp: inverse_sgn power_inverse) 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

205 

dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

206 
lemma real_root_divide: "root n (x / y) = root n x / root n y" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

207 
by (simp add: divide_inverse real_root_mult real_root_inverse) 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

208 

dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

209 
lemma real_root_abs: "0 < n \<Longrightarrow> root n \<bar>x\<bar> = \<bar>root n x\<bar>" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

210 
by (simp add: abs_if real_root_minus) 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

211 

dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

212 
lemma real_root_power: "0 < n \<Longrightarrow> root n (x ^ k) = root n x ^ k" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

213 
by (induct k) (simp_all add: real_root_mult) 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

214 

23257  215 
text {* Roots of roots *} 
216 

217 
lemma real_root_Suc_0 [simp]: "root (Suc 0) x = x" 

218 
by (simp add: odd_real_root_unique) 

219 

51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

220 
lemma real_root_mult_exp: "root (m * n) x = root m (root n x)" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

221 
by (auto split: split_root elim!: sgn_power_injE 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

222 
simp: sgn_zero_iff sgn_mult power_mult[symmetric] abs_mult power_mult_distrib abs_sgn_eq) 
23257  223 

51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

224 
lemma real_root_commute: "root m (root n x) = root n (root m x)" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

225 
by (simp add: real_root_mult_exp [symmetric] mult_commute) 
23257  226 

227 
text {* Monotonicity in first argument *} 

228 

229 
lemma real_root_strict_decreasing: 

230 
"\<lbrakk>0 < n; n < N; 1 < x\<rbrakk> \<Longrightarrow> root N x < root n x" 

231 
apply (subgoal_tac "root n (root N x) ^ n < root N (root n x) ^ N", simp) 

232 
apply (simp add: real_root_commute power_strict_increasing 

233 
del: real_root_pow_pos2) 

234 
done 

235 

236 
lemma real_root_strict_increasing: 

237 
"\<lbrakk>0 < n; n < N; 0 < x; x < 1\<rbrakk> \<Longrightarrow> root n x < root N x" 

238 
apply (subgoal_tac "root N (root n x) ^ N < root n (root N x) ^ n", simp) 

239 
apply (simp add: real_root_commute power_strict_decreasing 

240 
del: real_root_pow_pos2) 

241 
done 

242 

243 
lemma real_root_decreasing: 

244 
"\<lbrakk>0 < n; n < N; 1 \<le> x\<rbrakk> \<Longrightarrow> root N x \<le> root n x" 

245 
by (auto simp add: order_le_less real_root_strict_decreasing) 

246 

247 
lemma real_root_increasing: 

248 
"\<lbrakk>0 < n; n < N; 0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> root n x \<le> root N x" 

249 
by (auto simp add: order_le_less real_root_strict_increasing) 

250 

23042
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

251 
text {* Continuity and derivatives *} 
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

252 

51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

253 
lemma isCont_real_root: "isCont (root n) x" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

254 
proof cases 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

255 
assume n: "0 < n" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

256 
let ?f = "\<lambda>y::real. sgn y * \<bar>y\<bar>^n" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

257 
have "continuous_on ({0..} \<union> {.. 0}) (\<lambda>x. if 0 < x then x ^ n else  ((x) ^ n) :: real)" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

258 
using n by (intro continuous_on_If continuous_on_intros) auto 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

259 
then have "continuous_on UNIV ?f" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

260 
by (rule continuous_on_cong[THEN iffD1, rotated 2]) (auto simp: not_less real_sgn_neg le_less n) 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

261 
then have [simp]: "\<And>x. isCont ?f x" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

262 
by (simp add: continuous_on_eq_continuous_at) 
23042
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

263 

51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

264 
have "isCont (root n) (?f (root n x))" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

265 
by (rule isCont_inverse_function [where f="?f" and d=1]) (auto simp: root_sgn_power n) 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

266 
then show ?thesis 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

267 
by (simp add: sgn_power_root n) 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

268 
qed (simp add: root_def[abs_def]) 
23042
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

269 

51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
49962
diff
changeset

270 
lemma tendsto_real_root[tendsto_intros]: 
51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

271 
"(f > x) F \<Longrightarrow> ((\<lambda>x. root n (f x)) > root n x) F" 
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

272 
using isCont_tendsto_compose[OF isCont_real_root, of f x F] . 
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
49962
diff
changeset

273 

270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
49962
diff
changeset

274 
lemma continuous_real_root[continuous_intros]: 
51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

275 
"continuous F f \<Longrightarrow> continuous F (\<lambda>x. root n (f x))" 
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
49962
diff
changeset

276 
unfolding continuous_def by (rule tendsto_real_root) 
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
49962
diff
changeset

277 

270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
49962
diff
changeset

278 
lemma continuous_on_real_root[continuous_on_intros]: 
51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

279 
"continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. root n (f x))" 
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
49962
diff
changeset

280 
unfolding continuous_on_def by (auto intro: tendsto_real_root) 
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
49962
diff
changeset

281 

23042
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

282 
lemma DERIV_real_root: 
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

283 
assumes n: "0 < n" 
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

284 
assumes x: "0 < x" 
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

285 
shows "DERIV (root n) x :> inverse (real n * root n x ^ (n  Suc 0))" 
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

286 
proof (rule DERIV_inverse_function) 
23044  287 
show "0 < x" using x . 
288 
show "x < x + 1" by simp 

289 
show "\<forall>y. 0 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y" 

23042
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

290 
using n by simp 
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

291 
show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n  Suc 0)" 
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

292 
by (rule DERIV_pow) 
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

293 
show "real n * root n x ^ (n  Suc 0) \<noteq> 0" 
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

294 
using n x by simp 
51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

295 
qed (rule isCont_real_root) 
23042
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

296 

23046  297 
lemma DERIV_odd_real_root: 
298 
assumes n: "odd n" 

299 
assumes x: "x \<noteq> 0" 

300 
shows "DERIV (root n) x :> inverse (real n * root n x ^ (n  Suc 0))" 

301 
proof (rule DERIV_inverse_function) 

302 
show "x  1 < x" by simp 

303 
show "x < x + 1" by simp 

304 
show "\<forall>y. x  1 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y" 

305 
using n by (simp add: odd_real_root_pow) 

306 
show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n  Suc 0)" 

307 
by (rule DERIV_pow) 

308 
show "real n * root n x ^ (n  Suc 0) \<noteq> 0" 

309 
using odd_pos [OF n] x by simp 

51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

310 
qed (rule isCont_real_root) 
23046  311 

31880  312 
lemma DERIV_even_real_root: 
313 
assumes n: "0 < n" and "even n" 

314 
assumes x: "x < 0" 

315 
shows "DERIV (root n) x :> inverse ( real n * root n x ^ (n  Suc 0))" 

316 
proof (rule DERIV_inverse_function) 

317 
show "x  1 < x" by simp 

318 
show "x < 0" using x . 

319 
next 

320 
show "\<forall>y. x  1 < y \<and> y < 0 \<longrightarrow>  (root n y ^ n) = y" 

321 
proof (rule allI, rule impI, erule conjE) 

322 
fix y assume "x  1 < y" and "y < 0" 

323 
hence "root n (y) ^ n = y" using `0 < n` by simp 

51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

324 
with real_root_minus and `even n` 
31880  325 
show " (root n y ^ n) = y" by simp 
326 
qed 

327 
next 

328 
show "DERIV (\<lambda>x.  (x ^ n)) (root n x) :>  real n * root n x ^ (n  Suc 0)" 

329 
by (auto intro!: DERIV_intros) 

330 
show " real n * root n x ^ (n  Suc 0) \<noteq> 0" 

331 
using n x by simp 

51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

332 
qed (rule isCont_real_root) 
31880  333 

334 
lemma DERIV_real_root_generic: 

335 
assumes "0 < n" and "x \<noteq> 0" 

49753  336 
and "\<lbrakk> even n ; 0 < x \<rbrakk> \<Longrightarrow> D = inverse (real n * root n x ^ (n  Suc 0))" 
337 
and "\<lbrakk> even n ; x < 0 \<rbrakk> \<Longrightarrow> D =  inverse (real n * root n x ^ (n  Suc 0))" 

338 
and "odd n \<Longrightarrow> D = inverse (real n * root n x ^ (n  Suc 0))" 

31880  339 
shows "DERIV (root n) x :> D" 
340 
using assms by (cases "even n", cases "0 < x", 

341 
auto intro: DERIV_real_root[THEN DERIV_cong] 

342 
DERIV_odd_real_root[THEN DERIV_cong] 

343 
DERIV_even_real_root[THEN DERIV_cong]) 

344 

22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

345 
subsection {* Square Root *} 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

346 

51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

347 
definition sqrt :: "real \<Rightarrow> real" where 
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

348 
"sqrt = root 2" 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

349 

22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

350 
lemma pos2: "0 < (2::nat)" by simp 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

351 

617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

352 
lemma real_sqrt_unique: "\<lbrakk>y\<twosuperior> = x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt x = y" 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

353 
unfolding sqrt_def by (rule real_root_pos_unique [OF pos2]) 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

354 

22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

355 
lemma real_sqrt_abs [simp]: "sqrt (x\<twosuperior>) = \<bar>x\<bar>" 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

356 
apply (rule real_sqrt_unique) 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

357 
apply (rule power2_abs) 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

358 
apply (rule abs_ge_zero) 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

359 
done 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

360 

22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

361 
lemma real_sqrt_pow2 [simp]: "0 \<le> x \<Longrightarrow> (sqrt x)\<twosuperior> = x" 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

362 
unfolding sqrt_def by (rule real_root_pow_pos2 [OF pos2]) 
22856  363 

22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

364 
lemma real_sqrt_pow2_iff [simp]: "((sqrt x)\<twosuperior> = x) = (0 \<le> x)" 
22856  365 
apply (rule iffI) 
366 
apply (erule subst) 

367 
apply (rule zero_le_power2) 

368 
apply (erule real_sqrt_pow2) 

20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

369 
done 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

370 

22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

371 
lemma real_sqrt_zero [simp]: "sqrt 0 = 0" 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

372 
unfolding sqrt_def by (rule real_root_zero) 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

373 

617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

374 
lemma real_sqrt_one [simp]: "sqrt 1 = 1" 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

375 
unfolding sqrt_def by (rule real_root_one [OF pos2]) 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

376 

617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

377 
lemma real_sqrt_minus: "sqrt ( x) =  sqrt x" 
51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

378 
unfolding sqrt_def by (rule real_root_minus) 
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

379 

617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

380 
lemma real_sqrt_mult: "sqrt (x * y) = sqrt x * sqrt y" 
51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

381 
unfolding sqrt_def by (rule real_root_mult) 
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

382 

617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

383 
lemma real_sqrt_inverse: "sqrt (inverse x) = inverse (sqrt x)" 
51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

384 
unfolding sqrt_def by (rule real_root_inverse) 
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

385 

617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

386 
lemma real_sqrt_divide: "sqrt (x / y) = sqrt x / sqrt y" 
51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

387 
unfolding sqrt_def by (rule real_root_divide) 
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

388 

617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

389 
lemma real_sqrt_power: "sqrt (x ^ k) = sqrt x ^ k" 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

390 
unfolding sqrt_def by (rule real_root_power [OF pos2]) 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

391 

617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

392 
lemma real_sqrt_gt_zero: "0 < x \<Longrightarrow> 0 < sqrt x" 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

393 
unfolding sqrt_def by (rule real_root_gt_zero [OF pos2]) 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

394 

617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

395 
lemma real_sqrt_ge_zero: "0 \<le> x \<Longrightarrow> 0 \<le> sqrt x" 
51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

396 
unfolding sqrt_def by (rule real_root_ge_zero) 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

397 

22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

398 
lemma real_sqrt_less_mono: "x < y \<Longrightarrow> sqrt x < sqrt y" 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

399 
unfolding sqrt_def by (rule real_root_less_mono [OF pos2]) 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

400 

617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

401 
lemma real_sqrt_le_mono: "x \<le> y \<Longrightarrow> sqrt x \<le> sqrt y" 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

402 
unfolding sqrt_def by (rule real_root_le_mono [OF pos2]) 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

403 

617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

404 
lemma real_sqrt_less_iff [simp]: "(sqrt x < sqrt y) = (x < y)" 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

405 
unfolding sqrt_def by (rule real_root_less_iff [OF pos2]) 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

406 

617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

407 
lemma real_sqrt_le_iff [simp]: "(sqrt x \<le> sqrt y) = (x \<le> y)" 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

408 
unfolding sqrt_def by (rule real_root_le_iff [OF pos2]) 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

409 

617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

410 
lemma real_sqrt_eq_iff [simp]: "(sqrt x = sqrt y) = (x = y)" 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

411 
unfolding sqrt_def by (rule real_root_eq_iff [OF pos2]) 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

412 

617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

413 
lemmas real_sqrt_gt_0_iff [simp] = real_sqrt_less_iff [where x=0, simplified] 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

414 
lemmas real_sqrt_lt_0_iff [simp] = real_sqrt_less_iff [where y=0, simplified] 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

415 
lemmas real_sqrt_ge_0_iff [simp] = real_sqrt_le_iff [where x=0, simplified] 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

416 
lemmas real_sqrt_le_0_iff [simp] = real_sqrt_le_iff [where y=0, simplified] 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

417 
lemmas real_sqrt_eq_0_iff [simp] = real_sqrt_eq_iff [where y=0, simplified] 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

418 

617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

419 
lemmas real_sqrt_gt_1_iff [simp] = real_sqrt_less_iff [where x=1, simplified] 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

420 
lemmas real_sqrt_lt_1_iff [simp] = real_sqrt_less_iff [where y=1, simplified] 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

421 
lemmas real_sqrt_ge_1_iff [simp] = real_sqrt_le_iff [where x=1, simplified] 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

422 
lemmas real_sqrt_le_1_iff [simp] = real_sqrt_le_iff [where y=1, simplified] 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

423 
lemmas real_sqrt_eq_1_iff [simp] = real_sqrt_eq_iff [where y=1, simplified] 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

424 

23042
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

425 
lemma isCont_real_sqrt: "isCont sqrt x" 
51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

426 
unfolding sqrt_def by (rule isCont_real_root) 
23042
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

427 

51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
49962
diff
changeset

428 
lemma tendsto_real_sqrt[tendsto_intros]: 
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
49962
diff
changeset

429 
"(f > x) F \<Longrightarrow> ((\<lambda>x. sqrt (f x)) > sqrt x) F" 
51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

430 
unfolding sqrt_def by (rule tendsto_real_root) 
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
49962
diff
changeset

431 

270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
49962
diff
changeset

432 
lemma continuous_real_sqrt[continuous_intros]: 
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
49962
diff
changeset

433 
"continuous F f \<Longrightarrow> continuous F (\<lambda>x. sqrt (f x))" 
51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

434 
unfolding sqrt_def by (rule continuous_real_root) 
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
49962
diff
changeset

435 

270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
49962
diff
changeset

436 
lemma continuous_on_real_sqrt[continuous_on_intros]: 
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
49962
diff
changeset

437 
"continuous_on s f \<Longrightarrow> 0 < n \<Longrightarrow> continuous_on s (\<lambda>x. sqrt (f x))" 
51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

438 
unfolding sqrt_def by (rule continuous_on_real_root) 
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
49962
diff
changeset

439 

31880  440 
lemma DERIV_real_sqrt_generic: 
441 
assumes "x \<noteq> 0" 

442 
assumes "x > 0 \<Longrightarrow> D = inverse (sqrt x) / 2" 

443 
assumes "x < 0 \<Longrightarrow> D =  inverse (sqrt x) / 2" 

444 
shows "DERIV sqrt x :> D" 

445 
using assms unfolding sqrt_def 

446 
by (auto intro!: DERIV_real_root_generic) 

447 

23042
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

448 
lemma DERIV_real_sqrt: 
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

449 
"0 < x \<Longrightarrow> DERIV sqrt x :> inverse (sqrt x) / 2" 
31880  450 
using DERIV_real_sqrt_generic by simp 
451 

452 
declare 

453 
DERIV_real_sqrt_generic[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros] 

454 
DERIV_real_root_generic[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros] 

23042
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

455 

20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

456 
lemma not_real_square_gt_zero [simp]: "(~ (0::real) < x*x) = (x = 0)" 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

457 
apply auto 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

458 
apply (cut_tac x = x and y = 0 in linorder_less_linear) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

459 
apply (simp add: zero_less_mult_iff) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

460 
done 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

461 

fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

462 
lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>" 
22856  463 
apply (subst power2_eq_square [symmetric]) 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

464 
apply (rule real_sqrt_abs) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

465 
done 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

466 

fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

467 
lemma real_inv_sqrt_pow2: "0 < x ==> inverse (sqrt(x)) ^ 2 = inverse x" 
22856  468 
by (simp add: power_inverse [symmetric]) 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

469 

fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

470 
lemma real_sqrt_eq_zero_cancel: "[ 0 \<le> x; sqrt(x) = 0] ==> x = 0" 
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

471 
by simp 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

472 

fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

473 
lemma real_sqrt_ge_one: "1 \<le> x ==> 1 \<le> sqrt x" 
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

474 
by simp 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

475 

22443  476 
lemma sqrt_divide_self_eq: 
477 
assumes nneg: "0 \<le> x" 

478 
shows "sqrt x / x = inverse (sqrt x)" 

479 
proof cases 

480 
assume "x=0" thus ?thesis by simp 

481 
next 

482 
assume nz: "x\<noteq>0" 

483 
hence pos: "0<x" using nneg by arith 

484 
show ?thesis 

485 
proof (rule right_inverse_eq [THEN iffD1, THEN sym]) 

486 
show "sqrt x / x \<noteq> 0" by (simp add: divide_inverse nneg nz) 

487 
show "inverse (sqrt x) / (sqrt x / x) = 1" 

488 
by (simp add: divide_inverse mult_assoc [symmetric] 

489 
power2_eq_square [symmetric] real_inv_sqrt_pow2 pos nz) 

490 
qed 

491 
qed 

492 

22721
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

493 
lemma real_divide_square_eq [simp]: "(((r::real) * a) / (r * r)) = a / r" 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

494 
apply (simp add: divide_inverse) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

495 
apply (case_tac "r=0") 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

496 
apply (auto simp add: mult_ac) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

497 
done 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

498 

23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

499 
lemma lemma_real_divide_sqrt_less: "0 < u ==> u / sqrt 2 < u" 
35216  500 
by (simp add: divide_less_eq) 
23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

501 

11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

502 
lemma four_x_squared: 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

503 
fixes x::real 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

504 
shows "4 * x\<twosuperior> = (2 * x)\<twosuperior>" 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

505 
by (simp add: power2_eq_square) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

506 

22856  507 
subsection {* Square Root of Sum of Squares *} 
508 

44320  509 
lemma real_sqrt_sum_squares_ge_zero: "0 \<le> sqrt (x\<twosuperior> + y\<twosuperior>)" 
510 
by simp (* TODO: delete *) 

22856  511 

23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

512 
declare real_sqrt_sum_squares_ge_zero [THEN abs_of_nonneg, simp] 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

513 

22856  514 
lemma real_sqrt_sum_squares_mult_ge_zero [simp]: 
515 
"0 \<le> sqrt ((x\<twosuperior> + y\<twosuperior>)*(xa\<twosuperior> + ya\<twosuperior>))" 

44320  516 
by (simp add: zero_le_mult_iff) 
22856  517 

518 
lemma real_sqrt_sum_squares_mult_squared_eq [simp]: 

519 
"sqrt ((x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)) ^ 2 = (x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)" 

44320  520 
by (simp add: zero_le_mult_iff) 
22856  521 

23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

522 
lemma real_sqrt_sum_squares_eq_cancel: "sqrt (x\<twosuperior> + y\<twosuperior>) = x \<Longrightarrow> y = 0" 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

523 
by (drule_tac f = "%x. x\<twosuperior>" in arg_cong, simp) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

524 

11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

525 
lemma real_sqrt_sum_squares_eq_cancel2: "sqrt (x\<twosuperior> + y\<twosuperior>) = y \<Longrightarrow> x = 0" 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

526 
by (drule_tac f = "%x. x\<twosuperior>" in arg_cong, simp) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

527 

11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

528 
lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt (x\<twosuperior> + y\<twosuperior>)" 
22856  529 
by (rule power2_le_imp_le, simp_all) 
530 

23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

531 
lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt (x\<twosuperior> + y\<twosuperior>)" 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

532 
by (rule power2_le_imp_le, simp_all) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

533 

11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

534 
lemma real_sqrt_ge_abs1 [simp]: "\<bar>x\<bar> \<le> sqrt (x\<twosuperior> + y\<twosuperior>)" 
22856  535 
by (rule power2_le_imp_le, simp_all) 
536 

23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

537 
lemma real_sqrt_ge_abs2 [simp]: "\<bar>y\<bar> \<le> sqrt (x\<twosuperior> + y\<twosuperior>)" 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

538 
by (rule power2_le_imp_le, simp_all) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

539 

11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

540 
lemma le_real_sqrt_sumsq [simp]: "x \<le> sqrt (x * x + y * y)" 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

541 
by (simp add: power2_eq_square [symmetric]) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

542 

22858  543 
lemma real_sqrt_sum_squares_triangle_ineq: 
544 
"sqrt ((a + c)\<twosuperior> + (b + d)\<twosuperior>) \<le> sqrt (a\<twosuperior> + b\<twosuperior>) + sqrt (c\<twosuperior> + d\<twosuperior>)" 

545 
apply (rule power2_le_imp_le, simp) 

546 
apply (simp add: power2_sum) 

49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
49753
diff
changeset

547 
apply (simp only: mult_assoc distrib_left [symmetric]) 
22858  548 
apply (rule mult_left_mono) 
549 
apply (rule power2_le_imp_le) 

550 
apply (simp add: power2_sum power_mult_distrib) 

23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23475
diff
changeset

551 
apply (simp add: ring_distribs) 
22858  552 
apply (subgoal_tac "0 \<le> b\<twosuperior> * c\<twosuperior> + a\<twosuperior> * d\<twosuperior>  2 * (a * c) * (b * d)", simp) 
553 
apply (rule_tac b="(a * d  b * c)\<twosuperior>" in ord_le_eq_trans) 

554 
apply (rule zero_le_power2) 

555 
apply (simp add: power2_diff power_mult_distrib) 

556 
apply (simp add: mult_nonneg_nonneg) 

557 
apply simp 

558 
apply (simp add: add_increasing) 

559 
done 

560 

23122  561 
lemma real_sqrt_sum_squares_less: 
562 
"\<lbrakk>\<bar>x\<bar> < u / sqrt 2; \<bar>y\<bar> < u / sqrt 2\<rbrakk> \<Longrightarrow> sqrt (x\<twosuperior> + y\<twosuperior>) < u" 

563 
apply (rule power2_less_imp_less, simp) 

564 
apply (drule power_strict_mono [OF _ abs_ge_zero pos2]) 

565 
apply (drule power_strict_mono [OF _ abs_ge_zero pos2]) 

566 
apply (simp add: power_divide) 

567 
apply (drule order_le_less_trans [OF abs_ge_zero]) 

568 
apply (simp add: zero_less_divide_iff) 

569 
done 

570 

23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

571 
text{*Needed for the infinitely close relation over the nonstandard 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

572 
complex numbers*} 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

573 
lemma lemma_sqrt_hcomplex_capprox: 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

574 
"[ 0 < u; x < u/2; y < u/2; 0 \<le> x; 0 \<le> y ] ==> sqrt (x\<twosuperior> + y\<twosuperior>) < u" 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

575 
apply (rule_tac y = "u/sqrt 2" in order_le_less_trans) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

576 
apply (erule_tac [2] lemma_real_divide_sqrt_less) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

577 
apply (rule power2_le_imp_le) 
44349
f057535311c5
remove redundant lemma real_0_le_divide_iff in favor or zero_le_divide_iff
huffman
parents:
44320
diff
changeset

578 
apply (auto simp add: zero_le_divide_iff power_divide) 
23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

579 
apply (rule_tac t = "u\<twosuperior>" in real_sum_of_halves [THEN subst]) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

580 
apply (rule add_mono) 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
28952
diff
changeset

581 
apply (auto simp add: four_x_squared intro: power_mono) 
23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

582 
done 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

583 

22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

584 
text "Legacy theorem names:" 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

585 
lemmas real_root_pos2 = real_root_power_cancel 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

586 
lemmas real_root_pos_pos = real_root_gt_zero [THEN order_less_imp_le] 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

587 
lemmas real_root_pos_pos_le = real_root_ge_zero 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

588 
lemmas real_sqrt_mult_distrib = real_sqrt_mult 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

589 
lemmas real_sqrt_mult_distrib2 = real_sqrt_mult 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

590 
lemmas real_sqrt_eq_zero_cancel_iff = real_sqrt_eq_0_iff 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

591 

617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

592 
(* needed for CauchysMeanTheorem.het_base from AFP *) 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

593 
lemma real_root_pos: "0 < x \<Longrightarrow> root (Suc n) (x ^ (Suc n)) = x" 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

594 
by (rule real_root_power_cancel [OF zero_less_Suc order_less_imp_le]) 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

595 

14324  596 
end 