src/HOL/RealPow.thy
author haftmann
Mon Apr 27 10:11:44 2009 +0200 (2009-04-27)
changeset 31001 7e6ffd8f51a9
parent 30960 fec1a04b7220
child 31014 79f0858d9d49
permissions -rw-r--r--
cleaned up theory power further
     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 instance real :: recpower ..
    16 
    17 lemma two_realpow_ge_one [simp]: "(1::real) \<le> 2 ^ n"
    18 by simp
    19 
    20 lemma two_realpow_gt [simp]: "real (n::nat) < 2 ^ n"
    21 apply (induct "n")
    22 apply (auto simp add: real_of_nat_Suc)
    23 apply (subst mult_2)
    24 apply (rule add_less_le_mono)
    25 apply (auto simp add: two_realpow_ge_one)
    26 done
    27 
    28 lemma realpow_Suc_le_self: "[| 0 \<le> r; r \<le> (1::real) |] ==> r ^ Suc n \<le> r"
    29 by (insert power_decreasing [of 1 "Suc n" r], simp)
    30 
    31 lemma realpow_minus_mult [rule_format]:
    32      "0 < n --> (x::real) ^ (n - 1) * x = x ^ n"
    33 apply (simp split add: nat_diff_split)
    34 done
    35 
    36 lemma realpow_two_mult_inverse [simp]:
    37      "r \<noteq> 0 ==> r * inverse r ^Suc (Suc 0) = inverse (r::real)"
    38 by (simp add:  real_mult_assoc [symmetric])
    39 
    40 lemma realpow_two_minus [simp]: "(-x)^Suc (Suc 0) = (x::real)^Suc (Suc 0)"
    41 by simp
    42 
    43 lemma realpow_two_diff:
    44      "(x::real)^Suc (Suc 0) - y^Suc (Suc 0) = (x - y) * (x + y)"
    45 apply (unfold real_diff_def)
    46 apply (simp add: algebra_simps)
    47 done
    48 
    49 lemma realpow_two_disj:
    50      "((x::real)^Suc (Suc 0) = y^Suc (Suc 0)) = (x = y | x = -y)"
    51 apply (cut_tac x = x and y = y in realpow_two_diff)
    52 apply auto
    53 done
    54 
    55 lemma realpow_real_of_nat: "real (m::nat) ^ n = real (m ^ n)"
    56 apply (induct "n")
    57 apply (auto simp add: real_of_nat_one real_of_nat_mult)
    58 done
    59 
    60 lemma realpow_real_of_nat_two_pos [simp] : "0 < real (Suc (Suc 0) ^ n)"
    61 apply (induct "n")
    62 apply (auto simp add: real_of_nat_mult zero_less_mult_iff)
    63 done
    64 
    65 (* used by AFP Integration theory *)
    66 lemma realpow_increasing:
    67      "[|(0::real) \<le> x; 0 \<le> y; x ^ Suc n \<le> y ^ Suc n|] ==> x \<le> y"
    68   by (rule power_le_imp_le_base)
    69 
    70 
    71 subsection{*Literal Arithmetic Involving Powers, Type @{typ real}*}
    72 
    73 lemma real_of_int_power: "real (x::int) ^ n = real (x ^ n)"
    74 apply (induct "n")
    75 apply (simp_all add: nat_mult_distrib)
    76 done
    77 declare real_of_int_power [symmetric, simp]
    78 
    79 lemma power_real_number_of:
    80      "(number_of v :: real) ^ n = real ((number_of v :: int) ^ n)"
    81 by (simp only: real_number_of [symmetric] real_of_int_power)
    82 
    83 declare power_real_number_of [of _ "number_of w", standard, simp]
    84 
    85 
    86 subsection {* Properties of Squares *}
    87 
    88 lemma sum_squares_ge_zero:
    89   fixes x y :: "'a::ordered_ring_strict"
    90   shows "0 \<le> x * x + y * y"
    91 by (intro add_nonneg_nonneg zero_le_square)
    92 
    93 lemma not_sum_squares_lt_zero:
    94   fixes x y :: "'a::ordered_ring_strict"
    95   shows "\<not> x * x + y * y < 0"
    96 by (simp add: linorder_not_less sum_squares_ge_zero)
    97 
    98 lemma sum_nonneg_eq_zero_iff:
    99   fixes x y :: "'a::pordered_ab_group_add"
   100   assumes x: "0 \<le> x" and y: "0 \<le> y"
   101   shows "(x + y = 0) = (x = 0 \<and> y = 0)"
   102 proof (auto)
   103   from y have "x + 0 \<le> x + y" by (rule add_left_mono)
   104   also assume "x + y = 0"
   105   finally have "x \<le> 0" by simp
   106   thus "x = 0" using x by (rule order_antisym)
   107 next
   108   from x have "0 + y \<le> x + y" by (rule add_right_mono)
   109   also assume "x + y = 0"
   110   finally have "y \<le> 0" by simp
   111   thus "y = 0" using y by (rule order_antisym)
   112 qed
   113 
   114 lemma sum_squares_eq_zero_iff:
   115   fixes x y :: "'a::ordered_ring_strict"
   116   shows "(x * x + y * y = 0) = (x = 0 \<and> y = 0)"
   117 by (simp add: sum_nonneg_eq_zero_iff)
   118 
   119 lemma sum_squares_le_zero_iff:
   120   fixes x y :: "'a::ordered_ring_strict"
   121   shows "(x * x + y * y \<le> 0) = (x = 0 \<and> y = 0)"
   122 by (simp add: order_le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
   123 
   124 lemma sum_squares_gt_zero_iff:
   125   fixes x y :: "'a::ordered_ring_strict"
   126   shows "(0 < x * x + y * y) = (x \<noteq> 0 \<or> y \<noteq> 0)"
   127 by (simp add: order_less_le sum_squares_ge_zero sum_squares_eq_zero_iff)
   128 
   129 lemma sum_power2_ge_zero:
   130   fixes x y :: "'a::{ordered_idom,recpower}"
   131   shows "0 \<le> x\<twosuperior> + y\<twosuperior>"
   132 unfolding power2_eq_square by (rule sum_squares_ge_zero)
   133 
   134 lemma not_sum_power2_lt_zero:
   135   fixes x y :: "'a::{ordered_idom,recpower}"
   136   shows "\<not> x\<twosuperior> + y\<twosuperior> < 0"
   137 unfolding power2_eq_square by (rule not_sum_squares_lt_zero)
   138 
   139 lemma sum_power2_eq_zero_iff:
   140   fixes x y :: "'a::{ordered_idom,recpower}"
   141   shows "(x\<twosuperior> + y\<twosuperior> = 0) = (x = 0 \<and> y = 0)"
   142 unfolding power2_eq_square by (rule sum_squares_eq_zero_iff)
   143 
   144 lemma sum_power2_le_zero_iff:
   145   fixes x y :: "'a::{ordered_idom,recpower}"
   146   shows "(x\<twosuperior> + y\<twosuperior> \<le> 0) = (x = 0 \<and> y = 0)"
   147 unfolding power2_eq_square by (rule sum_squares_le_zero_iff)
   148 
   149 lemma sum_power2_gt_zero_iff:
   150   fixes x y :: "'a::{ordered_idom,recpower}"
   151   shows "(0 < x\<twosuperior> + y\<twosuperior>) = (x \<noteq> 0 \<or> y \<noteq> 0)"
   152 unfolding power2_eq_square by (rule sum_squares_gt_zero_iff)
   153 
   154 
   155 subsection{* Squares of Reals *}
   156 
   157 lemma real_two_squares_add_zero_iff [simp]:
   158   "(x * x + y * y = 0) = ((x::real) = 0 \<and> y = 0)"
   159 by (rule sum_squares_eq_zero_iff)
   160 
   161 lemma real_sum_squares_cancel: "x * x + y * y = 0 ==> x = (0::real)"
   162 by simp
   163 
   164 lemma real_sum_squares_cancel2: "x * x + y * y = 0 ==> y = (0::real)"
   165 by simp
   166 
   167 lemma real_mult_self_sum_ge_zero: "(0::real) \<le> x*x + y*y"
   168 by (rule sum_squares_ge_zero)
   169 
   170 lemma real_sum_squares_cancel_a: "x * x = -(y * y) ==> x = (0::real) & y=0"
   171 by (simp add: real_add_eq_0_iff [symmetric])
   172 
   173 lemma real_squared_diff_one_factored: "x*x - (1::real) = (x + 1)*(x - 1)"
   174 by (simp add: left_distrib right_diff_distrib)
   175 
   176 lemma real_mult_is_one [simp]: "(x*x = (1::real)) = (x = 1 | x = - 1)"
   177 apply auto
   178 apply (drule right_minus_eq [THEN iffD2]) 
   179 apply (auto simp add: real_squared_diff_one_factored)
   180 done
   181 
   182 lemma real_sum_squares_not_zero: "x ~= 0 ==> x * x + y * y ~= (0::real)"
   183 by simp
   184 
   185 lemma real_sum_squares_not_zero2: "y ~= 0 ==> x * x + y * y ~= (0::real)"
   186 by simp
   187 
   188 lemma realpow_two_sum_zero_iff [simp]:
   189      "(x ^ 2 + y ^ 2 = (0::real)) = (x = 0 & y = 0)"
   190 by (rule sum_power2_eq_zero_iff)
   191 
   192 lemma realpow_two_le_add_order [simp]: "(0::real) \<le> u ^ 2 + v ^ 2"
   193 by (rule sum_power2_ge_zero)
   194 
   195 lemma realpow_two_le_add_order2 [simp]: "(0::real) \<le> u ^ 2 + v ^ 2 + w ^ 2"
   196 by (intro add_nonneg_nonneg zero_le_power2)
   197 
   198 lemma real_sum_square_gt_zero: "x ~= 0 ==> (0::real) < x * x + y * y"
   199 by (simp add: sum_squares_gt_zero_iff)
   200 
   201 lemma real_sum_square_gt_zero2: "y ~= 0 ==> (0::real) < x * x + y * y"
   202 by (simp add: sum_squares_gt_zero_iff)
   203 
   204 lemma real_minus_mult_self_le [simp]: "-(u * u) \<le> (x * (x::real))"
   205 by (rule_tac j = 0 in real_le_trans, auto)
   206 
   207 lemma realpow_square_minus_le [simp]: "-(u ^ 2) \<le> (x::real) ^ 2"
   208 by (auto simp add: power2_eq_square)
   209 
   210 (* The following theorem is by Benjamin Porter *)
   211 lemma real_sq_order:
   212   fixes x::real
   213   assumes xgt0: "0 \<le> x" and ygt0: "0 \<le> y" and sq: "x^2 \<le> y^2"
   214   shows "x \<le> y"
   215 proof -
   216   from sq have "x ^ Suc (Suc 0) \<le> y ^ Suc (Suc 0)"
   217     by (simp only: numeral_2_eq_2)
   218   thus "x \<le> y" using ygt0
   219     by (rule power_le_imp_le_base)
   220 qed
   221 
   222 
   223 subsection {*Various Other Theorems*}
   224 
   225 lemma real_le_add_half_cancel: "(x + y/2 \<le> (y::real)) = (x \<le> y /2)"
   226 by auto
   227 
   228 lemma real_minus_half_eq [simp]: "(x::real) - x/2 = x/2"
   229 by auto
   230 
   231 lemma real_mult_inverse_cancel:
   232      "[|(0::real) < x; 0 < x1; x1 * y < x * u |] 
   233       ==> inverse x * y < inverse x1 * u"
   234 apply (rule_tac c=x in mult_less_imp_less_left) 
   235 apply (auto simp add: real_mult_assoc [symmetric])
   236 apply (simp (no_asm) add: mult_ac)
   237 apply (rule_tac c=x1 in mult_less_imp_less_right) 
   238 apply (auto simp add: mult_ac)
   239 done
   240 
   241 lemma real_mult_inverse_cancel2:
   242      "[|(0::real) < x;0 < x1; x1 * y < x * u |] ==> y * inverse x < u * inverse x1"
   243 apply (auto dest: real_mult_inverse_cancel simp add: mult_ac)
   244 done
   245 
   246 lemma inverse_real_of_nat_gt_zero [simp]: "0 < inverse (real (Suc n))"
   247 by simp
   248 
   249 lemma inverse_real_of_nat_ge_zero [simp]: "0 \<le> inverse (real (Suc n))"
   250 by simp
   251 
   252 lemma realpow_num_eq_if: "(m::real) ^ n = (if n=0 then 1 else m * m ^ (n - 1))"
   253 by (case_tac "n", auto)
   254 
   255 subsection{* Float syntax *}
   256 
   257 syntax "_Float" :: "float_const \<Rightarrow> 'a"    ("_")
   258 
   259 use "Tools/float_syntax.ML"
   260 setup FloatSyntax.setup
   261 
   262 text{* Test: *}
   263 lemma "123.456 = -111.111 + 200 + 30 + 4 + 5/10 + 6/100 + (7/1000::real)"
   264 by simp
   265 
   266 end