src/HOL/RealPow.thy
 author wenzelm Mon Mar 16 18:24:30 2009 +0100 (2009-03-16) changeset 30549 d2d7874648bd parent 30273 ecd6f0ca62ea child 30960 fec1a04b7220 permissions -rw-r--r--
simplified method setup;
1 (*  Title       : HOL/RealPow.thy
2     Author      : Jacques D. Fleuriot
3     Copyright   : 1998  University of Cambridge
4 *)
6 header {* Natural powers theory *}
8 theory RealPow
9 imports RealDef
10 uses ("Tools/float_syntax.ML")
11 begin
13 declare abs_mult_self [simp]
15 instantiation real :: recpower
16 begin
18 primrec power_real where
19   "r ^ 0     = (1\<Colon>real)"
20 | "r ^ Suc n = (r\<Colon>real) * r ^ n"
22 instance proof
23   fix z :: real
24   fix n :: nat
25   show "z^0 = 1" by simp
26   show "z^(Suc n) = z * (z^n)" by simp
27 qed
29 declare power_real.simps [simp del]
31 end
34 lemma two_realpow_ge_one [simp]: "(1::real) \<le> 2 ^ n"
35 by simp
37 lemma two_realpow_gt [simp]: "real (n::nat) < 2 ^ n"
38 apply (induct "n")
39 apply (auto simp add: real_of_nat_Suc)
40 apply (subst mult_2)
42 apply (auto simp add: two_realpow_ge_one)
43 done
45 lemma realpow_Suc_le_self: "[| 0 \<le> r; r \<le> (1::real) |] ==> r ^ Suc n \<le> r"
46 by (insert power_decreasing [of 1 "Suc n" r], simp)
48 lemma realpow_minus_mult [rule_format]:
49      "0 < n --> (x::real) ^ (n - 1) * x = x ^ n"
50 unfolding One_nat_def
51 apply (simp split add: nat_diff_split)
52 done
54 lemma realpow_two_mult_inverse [simp]:
55      "r \<noteq> 0 ==> r * inverse r ^Suc (Suc 0) = inverse (r::real)"
56 by (simp add:  real_mult_assoc [symmetric])
58 lemma realpow_two_minus [simp]: "(-x)^Suc (Suc 0) = (x::real)^Suc (Suc 0)"
59 by simp
61 lemma realpow_two_diff:
62      "(x::real)^Suc (Suc 0) - y^Suc (Suc 0) = (x - y) * (x + y)"
63 apply (unfold real_diff_def)
65 done
67 lemma realpow_two_disj:
68      "((x::real)^Suc (Suc 0) = y^Suc (Suc 0)) = (x = y | x = -y)"
69 apply (cut_tac x = x and y = y in realpow_two_diff)
70 apply auto
71 done
73 lemma realpow_real_of_nat: "real (m::nat) ^ n = real (m ^ n)"
74 apply (induct "n")
75 apply (auto simp add: real_of_nat_one real_of_nat_mult)
76 done
78 lemma realpow_real_of_nat_two_pos [simp] : "0 < real (Suc (Suc 0) ^ n)"
79 apply (induct "n")
80 apply (auto simp add: real_of_nat_mult zero_less_mult_iff)
81 done
83 (* used by AFP Integration theory *)
84 lemma realpow_increasing:
85      "[|(0::real) \<le> x; 0 \<le> y; x ^ Suc n \<le> y ^ Suc n|] ==> x \<le> y"
86   by (rule power_le_imp_le_base)
89 subsection{*Literal Arithmetic Involving Powers, Type @{typ real}*}
91 lemma real_of_int_power: "real (x::int) ^ n = real (x ^ n)"
92 apply (induct "n")
94 done
95 declare real_of_int_power [symmetric, simp]
97 lemma power_real_number_of:
98      "(number_of v :: real) ^ n = real ((number_of v :: int) ^ n)"
99 by (simp only: real_number_of [symmetric] real_of_int_power)
101 declare power_real_number_of [of _ "number_of w", standard, simp]
104 subsection {* Properties of Squares *}
106 lemma sum_squares_ge_zero:
107   fixes x y :: "'a::ordered_ring_strict"
108   shows "0 \<le> x * x + y * y"
111 lemma not_sum_squares_lt_zero:
112   fixes x y :: "'a::ordered_ring_strict"
113   shows "\<not> x * x + y * y < 0"
114 by (simp add: linorder_not_less sum_squares_ge_zero)
116 lemma sum_nonneg_eq_zero_iff:
117   fixes x y :: "'a::pordered_ab_group_add"
118   assumes x: "0 \<le> x" and y: "0 \<le> y"
119   shows "(x + y = 0) = (x = 0 \<and> y = 0)"
120 proof (auto)
121   from y have "x + 0 \<le> x + y" by (rule add_left_mono)
122   also assume "x + y = 0"
123   finally have "x \<le> 0" by simp
124   thus "x = 0" using x by (rule order_antisym)
125 next
126   from x have "0 + y \<le> x + y" by (rule add_right_mono)
127   also assume "x + y = 0"
128   finally have "y \<le> 0" by simp
129   thus "y = 0" using y by (rule order_antisym)
130 qed
132 lemma sum_squares_eq_zero_iff:
133   fixes x y :: "'a::ordered_ring_strict"
134   shows "(x * x + y * y = 0) = (x = 0 \<and> y = 0)"
137 lemma sum_squares_le_zero_iff:
138   fixes x y :: "'a::ordered_ring_strict"
139   shows "(x * x + y * y \<le> 0) = (x = 0 \<and> y = 0)"
140 by (simp add: order_le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
142 lemma sum_squares_gt_zero_iff:
143   fixes x y :: "'a::ordered_ring_strict"
144   shows "(0 < x * x + y * y) = (x \<noteq> 0 \<or> y \<noteq> 0)"
145 by (simp add: order_less_le sum_squares_ge_zero sum_squares_eq_zero_iff)
147 lemma sum_power2_ge_zero:
148   fixes x y :: "'a::{ordered_idom,recpower}"
149   shows "0 \<le> x\<twosuperior> + y\<twosuperior>"
150 unfolding power2_eq_square by (rule sum_squares_ge_zero)
152 lemma not_sum_power2_lt_zero:
153   fixes x y :: "'a::{ordered_idom,recpower}"
154   shows "\<not> x\<twosuperior> + y\<twosuperior> < 0"
155 unfolding power2_eq_square by (rule not_sum_squares_lt_zero)
157 lemma sum_power2_eq_zero_iff:
158   fixes x y :: "'a::{ordered_idom,recpower}"
159   shows "(x\<twosuperior> + y\<twosuperior> = 0) = (x = 0 \<and> y = 0)"
160 unfolding power2_eq_square by (rule sum_squares_eq_zero_iff)
162 lemma sum_power2_le_zero_iff:
163   fixes x y :: "'a::{ordered_idom,recpower}"
164   shows "(x\<twosuperior> + y\<twosuperior> \<le> 0) = (x = 0 \<and> y = 0)"
165 unfolding power2_eq_square by (rule sum_squares_le_zero_iff)
167 lemma sum_power2_gt_zero_iff:
168   fixes x y :: "'a::{ordered_idom,recpower}"
169   shows "(0 < x\<twosuperior> + y\<twosuperior>) = (x \<noteq> 0 \<or> y \<noteq> 0)"
170 unfolding power2_eq_square by (rule sum_squares_gt_zero_iff)
173 subsection{* Squares of Reals *}
176   "(x * x + y * y = 0) = ((x::real) = 0 \<and> y = 0)"
177 by (rule sum_squares_eq_zero_iff)
179 lemma real_sum_squares_cancel: "x * x + y * y = 0 ==> x = (0::real)"
180 by simp
182 lemma real_sum_squares_cancel2: "x * x + y * y = 0 ==> y = (0::real)"
183 by simp
185 lemma real_mult_self_sum_ge_zero: "(0::real) \<le> x*x + y*y"
186 by (rule sum_squares_ge_zero)
188 lemma real_sum_squares_cancel_a: "x * x = -(y * y) ==> x = (0::real) & y=0"
191 lemma real_squared_diff_one_factored: "x*x - (1::real) = (x + 1)*(x - 1)"
192 by (simp add: left_distrib right_diff_distrib)
194 lemma real_mult_is_one [simp]: "(x*x = (1::real)) = (x = 1 | x = - 1)"
195 apply auto
196 apply (drule right_minus_eq [THEN iffD2])
197 apply (auto simp add: real_squared_diff_one_factored)
198 done
200 lemma real_sum_squares_not_zero: "x ~= 0 ==> x * x + y * y ~= (0::real)"
201 by simp
203 lemma real_sum_squares_not_zero2: "y ~= 0 ==> x * x + y * y ~= (0::real)"
204 by simp
206 lemma realpow_two_sum_zero_iff [simp]:
207      "(x ^ 2 + y ^ 2 = (0::real)) = (x = 0 & y = 0)"
208 by (rule sum_power2_eq_zero_iff)
210 lemma realpow_two_le_add_order [simp]: "(0::real) \<le> u ^ 2 + v ^ 2"
211 by (rule sum_power2_ge_zero)
213 lemma realpow_two_le_add_order2 [simp]: "(0::real) \<le> u ^ 2 + v ^ 2 + w ^ 2"
216 lemma real_sum_square_gt_zero: "x ~= 0 ==> (0::real) < x * x + y * y"
219 lemma real_sum_square_gt_zero2: "y ~= 0 ==> (0::real) < x * x + y * y"
222 lemma real_minus_mult_self_le [simp]: "-(u * u) \<le> (x * (x::real))"
223 by (rule_tac j = 0 in real_le_trans, auto)
225 lemma realpow_square_minus_le [simp]: "-(u ^ 2) \<le> (x::real) ^ 2"
226 by (auto simp add: power2_eq_square)
228 (* The following theorem is by Benjamin Porter *)
229 lemma real_sq_order:
230   fixes x::real
231   assumes xgt0: "0 \<le> x" and ygt0: "0 \<le> y" and sq: "x^2 \<le> y^2"
232   shows "x \<le> y"
233 proof -
234   from sq have "x ^ Suc (Suc 0) \<le> y ^ Suc (Suc 0)"
235     by (simp only: numeral_2_eq_2)
236   thus "x \<le> y" using ygt0
237     by (rule power_le_imp_le_base)
238 qed
241 subsection {*Various Other Theorems*}
243 lemma real_le_add_half_cancel: "(x + y/2 \<le> (y::real)) = (x \<le> y /2)"
244 by auto
246 lemma real_minus_half_eq [simp]: "(x::real) - x/2 = x/2"
247 by auto
249 lemma real_mult_inverse_cancel:
250      "[|(0::real) < x; 0 < x1; x1 * y < x * u |]
251       ==> inverse x * y < inverse x1 * u"
252 apply (rule_tac c=x in mult_less_imp_less_left)
253 apply (auto simp add: real_mult_assoc [symmetric])
254 apply (simp (no_asm) add: mult_ac)
255 apply (rule_tac c=x1 in mult_less_imp_less_right)
256 apply (auto simp add: mult_ac)
257 done
259 lemma real_mult_inverse_cancel2:
260      "[|(0::real) < x;0 < x1; x1 * y < x * u |] ==> y * inverse x < u * inverse x1"
261 apply (auto dest: real_mult_inverse_cancel simp add: mult_ac)
262 done
264 lemma inverse_real_of_nat_gt_zero [simp]: "0 < inverse (real (Suc n))"
265 by simp
267 lemma inverse_real_of_nat_ge_zero [simp]: "0 \<le> inverse (real (Suc n))"
268 by simp
270 lemma realpow_num_eq_if: "(m::real) ^ n = (if n=0 then 1 else m * m ^ (n - 1))"
271 by (case_tac "n", auto)
273 subsection{* Float syntax *}
275 syntax "_Float" :: "float_const \<Rightarrow> 'a"    ("_")
277 use "Tools/float_syntax.ML"
278 setup FloatSyntax.setup
280 text{* Test: *}
281 lemma "123.456 = -111.111 + 200 + 30 + 4 + 5/10 + 6/100 + (7/1000::real)"
282 by simp
284 end