src/HOL/RealPow.thy
 author nipkow Fri Mar 06 17:38:47 2009 +0100 (2009-03-06) changeset 30313 b2441b0c8d38 parent 30273 ecd6f0ca62ea child 30960 fec1a04b7220 permissions -rw-r--r--
```     1 (*  Title       : HOL/RealPow.thy
```
```     2     Author      : Jacques D. Fleuriot
```
```     3     Copyright   : 1998  University of Cambridge
```
```     4 *)
```
```     5
```
```     6 header {* Natural powers theory *}
```
```     7
```
```     8 theory RealPow
```
```     9 imports RealDef
```
```    10 uses ("Tools/float_syntax.ML")
```
```    11 begin
```
```    12
```
```    13 declare abs_mult_self [simp]
```
```    14
```
```    15 instantiation real :: recpower
```
```    16 begin
```
```    17
```
```    18 primrec power_real where
```
```    19   "r ^ 0     = (1\<Colon>real)"
```
```    20 | "r ^ Suc n = (r\<Colon>real) * r ^ n"
```
```    21
```
```    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
```
```    28
```
```    29 declare power_real.simps [simp del]
```
```    30
```
```    31 end
```
```    32
```
```    33
```
```    34 lemma two_realpow_ge_one [simp]: "(1::real) \<le> 2 ^ n"
```
```    35 by simp
```
```    36
```
```    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)
```
```    41 apply (rule add_less_le_mono)
```
```    42 apply (auto simp add: two_realpow_ge_one)
```
```    43 done
```
```    44
```
```    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)
```
```    47
```
```    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
```
```    53
```
```    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])
```
```    57
```
```    58 lemma realpow_two_minus [simp]: "(-x)^Suc (Suc 0) = (x::real)^Suc (Suc 0)"
```
```    59 by simp
```
```    60
```
```    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)
```
```    64 apply (simp add: algebra_simps)
```
```    65 done
```
```    66
```
```    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
```
```    72
```
```    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
```
```    77
```
```    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
```
```    82
```
```    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)
```
```    87
```
```    88
```
```    89 subsection{*Literal Arithmetic Involving Powers, Type @{typ real}*}
```
```    90
```
```    91 lemma real_of_int_power: "real (x::int) ^ n = real (x ^ n)"
```
```    92 apply (induct "n")
```
```    93 apply (simp_all add: nat_mult_distrib)
```
```    94 done
```
```    95 declare real_of_int_power [symmetric, simp]
```
```    96
```
```    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)
```
```   100
```
```   101 declare power_real_number_of [of _ "number_of w", standard, simp]
```
```   102
```
```   103
```
```   104 subsection {* Properties of Squares *}
```
```   105
```
```   106 lemma sum_squares_ge_zero:
```
```   107   fixes x y :: "'a::ordered_ring_strict"
```
```   108   shows "0 \<le> x * x + y * y"
```
```   109 by (intro add_nonneg_nonneg zero_le_square)
```
```   110
```
```   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)
```
```   115
```
```   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
```
```   131
```
```   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)"
```
```   135 by (simp add: sum_nonneg_eq_zero_iff)
```
```   136
```
```   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)
```
```   141
```
```   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)
```
```   146
```
```   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)
```
```   151
```
```   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)
```
```   156
```
```   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)
```
```   161
```
```   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)
```
```   166
```
```   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)
```
```   171
```
```   172
```
```   173 subsection{* Squares of Reals *}
```
```   174
```
```   175 lemma real_two_squares_add_zero_iff [simp]:
```
```   176   "(x * x + y * y = 0) = ((x::real) = 0 \<and> y = 0)"
```
```   177 by (rule sum_squares_eq_zero_iff)
```
```   178
```
```   179 lemma real_sum_squares_cancel: "x * x + y * y = 0 ==> x = (0::real)"
```
```   180 by simp
```
```   181
```
```   182 lemma real_sum_squares_cancel2: "x * x + y * y = 0 ==> y = (0::real)"
```
```   183 by simp
```
```   184
```
```   185 lemma real_mult_self_sum_ge_zero: "(0::real) \<le> x*x + y*y"
```
```   186 by (rule sum_squares_ge_zero)
```
```   187
```
```   188 lemma real_sum_squares_cancel_a: "x * x = -(y * y) ==> x = (0::real) & y=0"
```
```   189 by (simp add: real_add_eq_0_iff [symmetric])
```
```   190
```
```   191 lemma real_squared_diff_one_factored: "x*x - (1::real) = (x + 1)*(x - 1)"
```
```   192 by (simp add: left_distrib right_diff_distrib)
```
```   193
```
```   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
```
```   199
```
```   200 lemma real_sum_squares_not_zero: "x ~= 0 ==> x * x + y * y ~= (0::real)"
```
```   201 by simp
```
```   202
```
```   203 lemma real_sum_squares_not_zero2: "y ~= 0 ==> x * x + y * y ~= (0::real)"
```
```   204 by simp
```
```   205
```
```   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)
```
```   209
```
```   210 lemma realpow_two_le_add_order [simp]: "(0::real) \<le> u ^ 2 + v ^ 2"
```
```   211 by (rule sum_power2_ge_zero)
```
```   212
```
```   213 lemma realpow_two_le_add_order2 [simp]: "(0::real) \<le> u ^ 2 + v ^ 2 + w ^ 2"
```
```   214 by (intro add_nonneg_nonneg zero_le_power2)
```
```   215
```
```   216 lemma real_sum_square_gt_zero: "x ~= 0 ==> (0::real) < x * x + y * y"
```
```   217 by (simp add: sum_squares_gt_zero_iff)
```
```   218
```
```   219 lemma real_sum_square_gt_zero2: "y ~= 0 ==> (0::real) < x * x + y * y"
```
```   220 by (simp add: sum_squares_gt_zero_iff)
```
```   221
```
```   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)
```
```   224
```
```   225 lemma realpow_square_minus_le [simp]: "-(u ^ 2) \<le> (x::real) ^ 2"
```
```   226 by (auto simp add: power2_eq_square)
```
```   227
```
```   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
```
```   239
```
```   240
```
```   241 subsection {*Various Other Theorems*}
```
```   242
```
```   243 lemma real_le_add_half_cancel: "(x + y/2 \<le> (y::real)) = (x \<le> y /2)"
```
```   244 by auto
```
```   245
```
```   246 lemma real_minus_half_eq [simp]: "(x::real) - x/2 = x/2"
```
```   247 by auto
```
```   248
```
```   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
```
```   258
```
```   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
```
```   263
```
```   264 lemma inverse_real_of_nat_gt_zero [simp]: "0 < inverse (real (Suc n))"
```
```   265 by simp
```
```   266
```
```   267 lemma inverse_real_of_nat_ge_zero [simp]: "0 \<le> inverse (real (Suc n))"
```
```   268 by simp
```
```   269
```
```   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)
```
```   272
```
```   273 subsection{* Float syntax *}
```
```   274
```
```   275 syntax "_Float" :: "float_const \<Rightarrow> 'a"    ("_")
```
```   276
```
```   277 use "Tools/float_syntax.ML"
```
```   278 setup FloatSyntax.setup
```
```   279
```
```   280 text{* Test: *}
```
```   281 lemma "123.456 = -111.111 + 200 + 30 + 4 + 5/10 + 6/100 + (7/1000::real)"
```
```   282 by simp
```
```   283
```
```   284 end
```