src/HOL/Real/RealPow.thy
 author paulson Tue Oct 05 15:30:50 2004 +0200 (2004-10-05) changeset 15229 1eb23f805c06 parent 15140 322485b816ac child 15251 bb6f072c8d10 permissions -rw-r--r--
new simprules for abs and for things like a/b<1
1 (*  Title       : HOL/Real/RealPow.thy
2     ID          : \$Id\$
3     Author      : Jacques D. Fleuriot
4     Copyright   : 1998  University of Cambridge
5     Description : Natural powers theory
7 *)
9 theory RealPow
10 imports RealDef
11 begin
13 declare abs_mult_self [simp]
15 instance real :: power ..
17 primrec (realpow)
18      realpow_0:   "r ^ 0       = 1"
19      realpow_Suc: "r ^ (Suc n) = (r::real) * (r ^ n)"
22 instance real :: recpower
23 proof
24   fix z :: real
25   fix n :: nat
26   show "z^0 = 1" by simp
27   show "z^(Suc n) = z * (z^n)" by simp
28 qed
31 lemma realpow_not_zero: "r \<noteq> (0::real) ==> r ^ n \<noteq> 0"
32   by (rule field_power_not_zero)
34 lemma realpow_zero_zero: "r ^ n = (0::real) ==> r = 0"
35 by simp
37 lemma realpow_two: "(r::real)^ (Suc (Suc 0)) = r * r"
38 by simp
40 text{*Legacy: weaker version of the theorem @{text power_strict_mono},
41 used 6 times in NthRoot and Transcendental*}
42 lemma realpow_less:
43      "[|(0::real) < x; x < y; 0 < n|] ==> x ^ n < y ^ n"
44 apply (rule power_strict_mono, auto)
45 done
47 lemma realpow_two_le [simp]: "(0::real) \<le> r^ Suc (Suc 0)"
50 lemma abs_realpow_two [simp]: "abs((x::real)^Suc (Suc 0)) = x^Suc (Suc 0)"
53 lemma realpow_two_abs [simp]: "abs(x::real)^Suc (Suc 0) = x^Suc (Suc 0)"
54 by (simp add: power_abs [symmetric] del: realpow_Suc)
56 lemma two_realpow_ge_one [simp]: "(1::real) \<le> 2 ^ n"
57 by (insert power_increasing [of 0 n "2::real"], simp)
59 lemma two_realpow_gt [simp]: "real (n::nat) < 2 ^ n"
60 apply (induct_tac "n")
61 apply (auto simp add: real_of_nat_Suc)
62 apply (subst mult_2)
64 apply (auto simp add: two_realpow_ge_one)
65 done
67 lemma realpow_Suc_le_self: "[| 0 \<le> r; r \<le> (1::real) |] ==> r ^ Suc n \<le> r"
68 by (insert power_decreasing [of 1 "Suc n" r], simp)
70 text{*Used ONCE in Transcendental*}
71 lemma realpow_Suc_less_one: "[| 0 < r; r < (1::real) |] ==> r ^ Suc n < 1"
72 by (insert power_strict_decreasing [of 0 "Suc n" r], simp)
74 text{*Used ONCE in Lim.ML*}
75 lemma realpow_minus_mult [rule_format]:
76      "0 < n --> (x::real) ^ (n - 1) * x = x ^ n"
77 apply (simp split add: nat_diff_split)
78 done
80 lemma realpow_two_mult_inverse [simp]:
81      "r \<noteq> 0 ==> r * inverse r ^Suc (Suc 0) = inverse (r::real)"
82 by (simp add: realpow_two real_mult_assoc [symmetric])
84 lemma realpow_two_minus [simp]: "(-x)^Suc (Suc 0) = (x::real)^Suc (Suc 0)"
85 by simp
87 lemma realpow_two_diff:
88      "(x::real)^Suc (Suc 0) - y^Suc (Suc 0) = (x - y) * (x + y)"
89 apply (unfold real_diff_def)
90 apply (simp add: right_distrib left_distrib mult_ac)
91 done
93 lemma realpow_two_disj:
94      "((x::real)^Suc (Suc 0) = y^Suc (Suc 0)) = (x = y | x = -y)"
95 apply (cut_tac x = x and y = y in realpow_two_diff)
96 apply (auto simp del: realpow_Suc)
97 done
99 lemma realpow_real_of_nat: "real (m::nat) ^ n = real (m ^ n)"
100 apply (induct_tac "n")
101 apply (auto simp add: real_of_nat_one real_of_nat_mult)
102 done
104 lemma realpow_real_of_nat_two_pos [simp] : "0 < real (Suc (Suc 0) ^ n)"
105 apply (induct_tac "n")
106 apply (auto simp add: real_of_nat_mult zero_less_mult_iff)
107 done
109 lemma realpow_increasing:
110      "[|(0::real) \<le> x; 0 \<le> y; x ^ Suc n \<le> y ^ Suc n|] ==> x \<le> y"
111   by (rule power_le_imp_le_base)
114 lemma zero_less_realpow_abs_iff [simp]:
115      "(0 < (abs x)^n) = (x \<noteq> (0::real) | n=0)"
116 apply (induct_tac "n")
117 apply (auto simp add: zero_less_mult_iff)
118 done
120 lemma zero_le_realpow_abs [simp]: "(0::real) \<le> (abs x)^n"
121 apply (induct_tac "n")
122 apply (auto simp add: zero_le_mult_iff)
123 done
126 subsection{*Literal Arithmetic Involving Powers, Type @{typ real}*}
128 lemma real_of_int_power: "real (x::int) ^ n = real (x ^ n)"
129 apply (induct_tac "n")
131 done
132 declare real_of_int_power [symmetric, simp]
134 lemma power_real_number_of:
135      "(number_of v :: real) ^ n = real ((number_of v :: int) ^ n)"
136 by (simp only: real_number_of [symmetric] real_of_int_power)
138 declare power_real_number_of [of _ "number_of w", standard, simp]
141 subsection{*Various Other Theorems*}
143 text{*Used several times in Hyperreal/Transcendental.ML*}
144 lemma real_sum_squares_cancel_a: "x * x = -(y * y) ==> x = (0::real) & y=0"
147   done
149 lemma real_squared_diff_one_factored: "x*x - (1::real) = (x + 1)*(x - 1)"
150 by (auto simp add: left_distrib right_distrib real_diff_def)
152 lemma real_mult_is_one [simp]: "(x*x = (1::real)) = (x = 1 | x = - 1)"
153 apply auto
154 apply (drule right_minus_eq [THEN iffD2])
155 apply (auto simp add: real_squared_diff_one_factored)
156 done
158 lemma real_le_add_half_cancel: "(x + y/2 \<le> (y::real)) = (x \<le> y /2)"
159 by auto
161 lemma real_minus_half_eq [simp]: "(x::real) - x/2 = x/2"
162 by auto
164 lemma real_mult_inverse_cancel:
165      "[|(0::real) < x; 0 < x1; x1 * y < x * u |]
166       ==> inverse x * y < inverse x1 * u"
167 apply (rule_tac c=x in mult_less_imp_less_left)
168 apply (auto simp add: real_mult_assoc [symmetric])
169 apply (simp (no_asm) add: mult_ac)
170 apply (rule_tac c=x1 in mult_less_imp_less_right)
171 apply (auto simp add: mult_ac)
172 done
174 text{*Used once: in Hyperreal/Transcendental.ML*}
175 lemma real_mult_inverse_cancel2:
176      "[|(0::real) < x;0 < x1; x1 * y < x * u |] ==> y * inverse x < u * inverse x1"
177 apply (auto dest: real_mult_inverse_cancel simp add: mult_ac)
178 done
180 lemma inverse_real_of_nat_gt_zero [simp]: "0 < inverse (real (Suc n))"
181 by auto
183 lemma inverse_real_of_nat_ge_zero [simp]: "0 \<le> inverse (real (Suc n))"
184 by auto
186 lemma real_sum_squares_not_zero: "x ~= 0 ==> x * x + y * y ~= (0::real)"
187 by (blast dest!: real_sum_squares_cancel)
189 lemma real_sum_squares_not_zero2: "y ~= 0 ==> x * x + y * y ~= (0::real)"
190 by (blast dest!: real_sum_squares_cancel2)
193 subsection {*Various Other Theorems*}
195 lemma realpow_divide:
196     "(x/y) ^ n = ((x::real) ^ n/ y ^ n)"
197 apply (unfold real_divide_def)
198 apply (auto simp add: power_mult_distrib power_inverse)
199 done
201 lemma realpow_two_sum_zero_iff [simp]:
202      "(x ^ 2 + y ^ 2 = (0::real)) = (x = 0 & y = 0)"
203 apply (auto intro: real_sum_squares_cancel real_sum_squares_cancel2
205 done
207 lemma realpow_two_le_add_order [simp]: "(0::real) \<le> u ^ 2 + v ^ 2"
209 apply (auto simp add: power2_eq_square)
210 done
212 lemma realpow_two_le_add_order2 [simp]: "(0::real) \<le> u ^ 2 + v ^ 2 + w ^ 2"
214 apply (auto simp add: power2_eq_square)
215 done
217 lemma real_sum_square_gt_zero: "x ~= 0 ==> (0::real) < x * x + y * y"
218 apply (cut_tac x = x and y = y in real_mult_self_sum_ge_zero)
219 apply (drule real_le_imp_less_or_eq)
220 apply (drule_tac y = y in real_sum_squares_not_zero, auto)
221 done
223 lemma real_sum_square_gt_zero2: "y ~= 0 ==> (0::real) < x * x + y * y"
224 apply (rule real_add_commute [THEN subst])
225 apply (erule real_sum_square_gt_zero)
226 done
228 lemma real_minus_mult_self_le [simp]: "-(u * u) \<le> (x * (x::real))"
229 by (rule_tac j = 0 in real_le_trans, auto)
231 lemma realpow_square_minus_le [simp]: "-(u ^ 2) \<le> (x::real) ^ 2"
232 by (auto simp add: power2_eq_square)
234 lemma realpow_num_eq_if: "(m::real) ^ n = (if n=0 then 1 else m * m ^ (n - 1))"
235 by (case_tac "n", auto)
237 lemma real_num_zero_less_two_pow [simp]: "0 < (2::real) ^ (4*d)"
238 apply (induct_tac "d")
239 apply (auto simp add: realpow_num_eq_if)
240 done
242 lemma lemma_realpow_num_two_mono:
243      "x * (4::real)   < y ==> x * (2 ^ 8) < y * (2 ^ 6)"
244 apply (subgoal_tac " (2::real) ^ 8 = 4 * (2 ^ 6) ")
245 apply (simp (no_asm_simp) add: real_mult_assoc [symmetric])
246 apply (auto simp add: realpow_num_eq_if)
247 done
250 ML
251 {*
252 val realpow_0 = thm "realpow_0";
253 val realpow_Suc = thm "realpow_Suc";
255 val realpow_not_zero = thm "realpow_not_zero";
256 val realpow_zero_zero = thm "realpow_zero_zero";
257 val realpow_two = thm "realpow_two";
258 val realpow_less = thm "realpow_less";
259 val realpow_two_le = thm "realpow_two_le";
260 val abs_realpow_two = thm "abs_realpow_two";
261 val realpow_two_abs = thm "realpow_two_abs";
262 val two_realpow_ge_one = thm "two_realpow_ge_one";
263 val two_realpow_gt = thm "two_realpow_gt";
264 val realpow_Suc_le_self = thm "realpow_Suc_le_self";
265 val realpow_Suc_less_one = thm "realpow_Suc_less_one";
266 val realpow_minus_mult = thm "realpow_minus_mult";
267 val realpow_two_mult_inverse = thm "realpow_two_mult_inverse";
268 val realpow_two_minus = thm "realpow_two_minus";
269 val realpow_two_disj = thm "realpow_two_disj";
270 val realpow_real_of_nat = thm "realpow_real_of_nat";
271 val realpow_real_of_nat_two_pos = thm "realpow_real_of_nat_two_pos";
272 val realpow_increasing = thm "realpow_increasing";
273 val zero_less_realpow_abs_iff = thm "zero_less_realpow_abs_iff";
274 val zero_le_realpow_abs = thm "zero_le_realpow_abs";
275 val real_of_int_power = thm "real_of_int_power";
276 val power_real_number_of = thm "power_real_number_of";
277 val real_sum_squares_cancel_a = thm "real_sum_squares_cancel_a";
278 val real_mult_inverse_cancel2 = thm "real_mult_inverse_cancel2";
279 val real_squared_diff_one_factored = thm "real_squared_diff_one_factored";
280 val real_mult_is_one = thm "real_mult_is_one";
282 val real_minus_half_eq = thm "real_minus_half_eq";
283 val real_mult_inverse_cancel = thm "real_mult_inverse_cancel";
284 val real_mult_inverse_cancel2 = thm "real_mult_inverse_cancel2";
285 val inverse_real_of_nat_gt_zero = thm "inverse_real_of_nat_gt_zero";
286 val inverse_real_of_nat_ge_zero = thm "inverse_real_of_nat_ge_zero";
287 val real_sum_squares_not_zero = thm "real_sum_squares_not_zero";
288 val real_sum_squares_not_zero2 = thm "real_sum_squares_not_zero2";
290 val realpow_divide = thm "realpow_divide";
291 val realpow_two_sum_zero_iff = thm "realpow_two_sum_zero_iff";