src/HOL/Power.thy
author huffman
Wed Jun 20 05:18:39 2007 +0200 (2007-06-20)
changeset 23431 25ca91279a9b
parent 23305 8ae6f7b0903b
child 23544 4b4165cb3e0d
permissions -rw-r--r--
change simp rules for of_nat to work like int did previously (reorient of_nat_Suc, remove of_nat_mult [simp]); preserve original variable names in legacy int theorems
     1 (*  Title:      HOL/Power.thy
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1997  University of Cambridge
     5 
     6 *)
     7 
     8 header{*Exponentiation*}
     9 
    10 theory Power
    11 imports Nat
    12 begin
    13 
    14 subsection{*Powers for Arbitrary Monoids*}
    15 
    16 class recpower = monoid_mult + power +
    17   assumes power_0 [simp]: "a \<^loc>^ 0       = \<^loc>1"
    18   assumes power_Suc:      "a \<^loc>^ Suc n = a \<^loc>* (a \<^loc>^ n)"
    19 
    20 lemma power_0_Suc [simp]: "(0::'a::{recpower,semiring_0}) ^ (Suc n) = 0"
    21   by (simp add: power_Suc)
    22 
    23 text{*It looks plausible as a simprule, but its effect can be strange.*}
    24 lemma power_0_left: "0^n = (if n=0 then 1 else (0::'a::{recpower,semiring_0}))"
    25   by (induct n) simp_all
    26 
    27 lemma power_one [simp]: "1^n = (1::'a::recpower)"
    28   by (induct n) (simp_all add: power_Suc)
    29 
    30 lemma power_one_right [simp]: "(a::'a::recpower) ^ 1 = a"
    31   by (simp add: power_Suc)
    32 
    33 lemma power_commutes: "(a::'a::recpower) ^ n * a = a * a ^ n"
    34   by (induct n) (simp_all add: power_Suc mult_assoc)
    35 
    36 lemma power_add: "(a::'a::recpower) ^ (m+n) = (a^m) * (a^n)"
    37   by (induct m) (simp_all add: power_Suc mult_ac)
    38 
    39 lemma power_mult: "(a::'a::recpower) ^ (m*n) = (a^m) ^ n"
    40   by (induct n) (simp_all add: power_Suc power_add)
    41 
    42 lemma power_mult_distrib: "((a::'a::{recpower,comm_monoid_mult}) * b) ^ n = (a^n) * (b^n)"
    43   by (induct n) (simp_all add: power_Suc mult_ac)
    44 
    45 lemma zero_less_power:
    46      "0 < (a::'a::{ordered_semidom,recpower}) ==> 0 < a^n"
    47 apply (induct "n")
    48 apply (simp_all add: power_Suc zero_less_one mult_pos_pos)
    49 done
    50 
    51 lemma zero_le_power:
    52      "0 \<le> (a::'a::{ordered_semidom,recpower}) ==> 0 \<le> a^n"
    53 apply (simp add: order_le_less)
    54 apply (erule disjE)
    55 apply (simp_all add: zero_less_power zero_less_one power_0_left)
    56 done
    57 
    58 lemma one_le_power:
    59      "1 \<le> (a::'a::{ordered_semidom,recpower}) ==> 1 \<le> a^n"
    60 apply (induct "n")
    61 apply (simp_all add: power_Suc)
    62 apply (rule order_trans [OF _ mult_mono [of 1 _ 1]])
    63 apply (simp_all add: zero_le_one order_trans [OF zero_le_one])
    64 done
    65 
    66 lemma gt1_imp_ge0: "1 < a ==> 0 \<le> (a::'a::ordered_semidom)"
    67   by (simp add: order_trans [OF zero_le_one order_less_imp_le])
    68 
    69 lemma power_gt1_lemma:
    70   assumes gt1: "1 < (a::'a::{ordered_semidom,recpower})"
    71   shows "1 < a * a^n"
    72 proof -
    73   have "1*1 < a*1" using gt1 by simp
    74   also have "\<dots> \<le> a * a^n" using gt1
    75     by (simp only: mult_mono gt1_imp_ge0 one_le_power order_less_imp_le
    76         zero_le_one order_refl)
    77   finally show ?thesis by simp
    78 qed
    79 
    80 lemma power_gt1:
    81      "1 < (a::'a::{ordered_semidom,recpower}) ==> 1 < a ^ (Suc n)"
    82 by (simp add: power_gt1_lemma power_Suc)
    83 
    84 lemma power_le_imp_le_exp:
    85   assumes gt1: "(1::'a::{recpower,ordered_semidom}) < a"
    86   shows "!!n. a^m \<le> a^n ==> m \<le> n"
    87 proof (induct m)
    88   case 0
    89   show ?case by simp
    90 next
    91   case (Suc m)
    92   show ?case
    93   proof (cases n)
    94     case 0
    95     from prems have "a * a^m \<le> 1" by (simp add: power_Suc)
    96     with gt1 show ?thesis
    97       by (force simp only: power_gt1_lemma
    98           linorder_not_less [symmetric])
    99   next
   100     case (Suc n)
   101     from prems show ?thesis
   102       by (force dest: mult_left_le_imp_le
   103           simp add: power_Suc order_less_trans [OF zero_less_one gt1])
   104   qed
   105 qed
   106 
   107 text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}
   108 lemma power_inject_exp [simp]:
   109      "1 < (a::'a::{ordered_semidom,recpower}) ==> (a^m = a^n) = (m=n)"
   110   by (force simp add: order_antisym power_le_imp_le_exp)
   111 
   112 text{*Can relax the first premise to @{term "0<a"} in the case of the
   113 natural numbers.*}
   114 lemma power_less_imp_less_exp:
   115      "[| (1::'a::{recpower,ordered_semidom}) < a; a^m < a^n |] ==> m < n"
   116 by (simp add: order_less_le [of m n] order_less_le [of "a^m" "a^n"]
   117               power_le_imp_le_exp)
   118 
   119 
   120 lemma power_mono:
   121      "[|a \<le> b; (0::'a::{recpower,ordered_semidom}) \<le> a|] ==> a^n \<le> b^n"
   122 apply (induct "n")
   123 apply (simp_all add: power_Suc)
   124 apply (auto intro: mult_mono zero_le_power order_trans [of 0 a b])
   125 done
   126 
   127 lemma power_strict_mono [rule_format]:
   128      "[|a < b; (0::'a::{recpower,ordered_semidom}) \<le> a|]
   129       ==> 0 < n --> a^n < b^n"
   130 apply (induct "n")
   131 apply (auto simp add: mult_strict_mono zero_le_power power_Suc
   132                       order_le_less_trans [of 0 a b])
   133 done
   134 
   135 lemma power_eq_0_iff [simp]:
   136      "(a^n = 0) = (a = (0::'a::{dom,recpower}) & 0<n)"
   137 apply (induct "n")
   138 apply (auto simp add: power_Suc zero_neq_one [THEN not_sym])
   139 done
   140 
   141 lemma field_power_eq_0_iff:
   142      "(a^n = 0) = (a = (0::'a::{division_ring,recpower}) & 0<n)"
   143 by simp (* TODO: delete *)
   144 
   145 lemma field_power_not_zero: "a \<noteq> (0::'a::{dom,recpower}) ==> a^n \<noteq> 0"
   146 by force
   147 
   148 lemma nonzero_power_inverse:
   149   fixes a :: "'a::{division_ring,recpower}"
   150   shows "a \<noteq> 0 ==> inverse (a ^ n) = (inverse a) ^ n"
   151 apply (induct "n")
   152 apply (auto simp add: power_Suc nonzero_inverse_mult_distrib power_commutes)
   153 done (* TODO: reorient or rename to nonzero_inverse_power *)
   154 
   155 text{*Perhaps these should be simprules.*}
   156 lemma power_inverse:
   157   fixes a :: "'a::{division_ring,division_by_zero,recpower}"
   158   shows "inverse (a ^ n) = (inverse a) ^ n"
   159 apply (cases "a = 0")
   160 apply (simp add: power_0_left)
   161 apply (simp add: nonzero_power_inverse)
   162 done (* TODO: reorient or rename to inverse_power *)
   163 
   164 lemma power_one_over: "1 / (a::'a::{field,division_by_zero,recpower})^n = 
   165     (1 / a)^n"
   166 apply (simp add: divide_inverse)
   167 apply (rule power_inverse)
   168 done
   169 
   170 lemma nonzero_power_divide:
   171     "b \<noteq> 0 ==> (a/b) ^ n = ((a::'a::{field,recpower}) ^ n) / (b ^ n)"
   172 by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
   173 
   174 lemma power_divide:
   175     "(a/b) ^ n = ((a::'a::{field,division_by_zero,recpower}) ^ n / b ^ n)"
   176 apply (case_tac "b=0", simp add: power_0_left)
   177 apply (rule nonzero_power_divide)
   178 apply assumption
   179 done
   180 
   181 lemma power_abs: "abs(a ^ n) = abs(a::'a::{ordered_idom,recpower}) ^ n"
   182 apply (induct "n")
   183 apply (auto simp add: power_Suc abs_mult)
   184 done
   185 
   186 lemma zero_less_power_abs_iff [simp]:
   187      "(0 < (abs a)^n) = (a \<noteq> (0::'a::{ordered_idom,recpower}) | n=0)"
   188 proof (induct "n")
   189   case 0
   190     show ?case by (simp add: zero_less_one)
   191 next
   192   case (Suc n)
   193     show ?case by (force simp add: prems power_Suc zero_less_mult_iff)
   194 qed
   195 
   196 lemma zero_le_power_abs [simp]:
   197      "(0::'a::{ordered_idom,recpower}) \<le> (abs a)^n"
   198 by (rule zero_le_power [OF abs_ge_zero])
   199 
   200 lemma power_minus: "(-a) ^ n = (- 1)^n * (a::'a::{comm_ring_1,recpower}) ^ n"
   201 proof -
   202   have "-a = (- 1) * a"  by (simp add: minus_mult_left [symmetric])
   203   thus ?thesis by (simp only: power_mult_distrib)
   204 qed
   205 
   206 text{*Lemma for @{text power_strict_decreasing}*}
   207 lemma power_Suc_less:
   208      "[|(0::'a::{ordered_semidom,recpower}) < a; a < 1|]
   209       ==> a * a^n < a^n"
   210 apply (induct n)
   211 apply (auto simp add: power_Suc mult_strict_left_mono)
   212 done
   213 
   214 lemma power_strict_decreasing:
   215      "[|n < N; 0 < a; a < (1::'a::{ordered_semidom,recpower})|]
   216       ==> a^N < a^n"
   217 apply (erule rev_mp)
   218 apply (induct "N")
   219 apply (auto simp add: power_Suc power_Suc_less less_Suc_eq)
   220 apply (rename_tac m)
   221 apply (subgoal_tac "a * a^m < 1 * a^n", simp)
   222 apply (rule mult_strict_mono)
   223 apply (auto simp add: zero_le_power zero_less_one order_less_imp_le)
   224 done
   225 
   226 text{*Proof resembles that of @{text power_strict_decreasing}*}
   227 lemma power_decreasing:
   228      "[|n \<le> N; 0 \<le> a; a \<le> (1::'a::{ordered_semidom,recpower})|]
   229       ==> a^N \<le> a^n"
   230 apply (erule rev_mp)
   231 apply (induct "N")
   232 apply (auto simp add: power_Suc  le_Suc_eq)
   233 apply (rename_tac m)
   234 apply (subgoal_tac "a * a^m \<le> 1 * a^n", simp)
   235 apply (rule mult_mono)
   236 apply (auto simp add: zero_le_power zero_le_one)
   237 done
   238 
   239 lemma power_Suc_less_one:
   240      "[| 0 < a; a < (1::'a::{ordered_semidom,recpower}) |] ==> a ^ Suc n < 1"
   241 apply (insert power_strict_decreasing [of 0 "Suc n" a], simp)
   242 done
   243 
   244 text{*Proof again resembles that of @{text power_strict_decreasing}*}
   245 lemma power_increasing:
   246      "[|n \<le> N; (1::'a::{ordered_semidom,recpower}) \<le> a|] ==> a^n \<le> a^N"
   247 apply (erule rev_mp)
   248 apply (induct "N")
   249 apply (auto simp add: power_Suc le_Suc_eq)
   250 apply (rename_tac m)
   251 apply (subgoal_tac "1 * a^n \<le> a * a^m", simp)
   252 apply (rule mult_mono)
   253 apply (auto simp add: order_trans [OF zero_le_one] zero_le_power)
   254 done
   255 
   256 text{*Lemma for @{text power_strict_increasing}*}
   257 lemma power_less_power_Suc:
   258      "(1::'a::{ordered_semidom,recpower}) < a ==> a^n < a * a^n"
   259 apply (induct n)
   260 apply (auto simp add: power_Suc mult_strict_left_mono order_less_trans [OF zero_less_one])
   261 done
   262 
   263 lemma power_strict_increasing:
   264      "[|n < N; (1::'a::{ordered_semidom,recpower}) < a|] ==> a^n < a^N"
   265 apply (erule rev_mp)
   266 apply (induct "N")
   267 apply (auto simp add: power_less_power_Suc power_Suc less_Suc_eq)
   268 apply (rename_tac m)
   269 apply (subgoal_tac "1 * a^n < a * a^m", simp)
   270 apply (rule mult_strict_mono)
   271 apply (auto simp add: order_less_trans [OF zero_less_one] zero_le_power
   272                  order_less_imp_le)
   273 done
   274 
   275 lemma power_increasing_iff [simp]: 
   276      "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x \<le> b ^ y) = (x \<le> y)"
   277   by (blast intro: power_le_imp_le_exp power_increasing order_less_imp_le) 
   278 
   279 lemma power_strict_increasing_iff [simp]:
   280      "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x < b ^ y) = (x < y)"
   281   by (blast intro: power_less_imp_less_exp power_strict_increasing) 
   282 
   283 lemma power_le_imp_le_base:
   284   assumes le: "a ^ Suc n \<le> b ^ Suc n"
   285       and ynonneg: "(0::'a::{ordered_semidom,recpower}) \<le> b"
   286   shows "a \<le> b"
   287  proof (rule ccontr)
   288    assume "~ a \<le> b"
   289    then have "b < a" by (simp only: linorder_not_le)
   290    then have "b ^ Suc n < a ^ Suc n"
   291      by (simp only: prems power_strict_mono)
   292    from le and this show "False"
   293       by (simp add: linorder_not_less [symmetric])
   294  qed
   295 
   296 lemma power_less_imp_less_base:
   297   fixes a b :: "'a::{ordered_semidom,recpower}"
   298   assumes less: "a ^ n < b ^ n"
   299   assumes nonneg: "0 \<le> b"
   300   shows "a < b"
   301 proof (rule contrapos_pp [OF less])
   302   assume "~ a < b"
   303   hence "b \<le> a" by (simp only: linorder_not_less)
   304   hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
   305   thus "~ a ^ n < b ^ n" by (simp only: linorder_not_less)
   306 qed
   307 
   308 lemma power_inject_base:
   309      "[| a ^ Suc n = b ^ Suc n; 0 \<le> a; 0 \<le> b |]
   310       ==> a = (b::'a::{ordered_semidom,recpower})"
   311 by (blast intro: power_le_imp_le_base order_antisym order_eq_refl sym)
   312 
   313 lemma power_eq_imp_eq_base:
   314   fixes a b :: "'a::{ordered_semidom,recpower}"
   315   shows "\<lbrakk>a ^ n = b ^ n; 0 \<le> a; 0 \<le> b; 0 < n\<rbrakk> \<Longrightarrow> a = b"
   316 by (cases n, simp_all, rule power_inject_base)
   317 
   318 
   319 subsection{*Exponentiation for the Natural Numbers*}
   320 
   321 instance nat :: power ..
   322 
   323 primrec (power)
   324   "p ^ 0 = 1"
   325   "p ^ (Suc n) = (p::nat) * (p ^ n)"
   326 
   327 instance nat :: recpower
   328 proof
   329   fix z n :: nat
   330   show "z^0 = 1" by simp
   331   show "z^(Suc n) = z * (z^n)" by simp
   332 qed
   333 
   334 lemma of_nat_power:
   335   "of_nat (m ^ n) = (of_nat m::'a::{semiring_1,recpower}) ^ n"
   336 by (induct n, simp_all add: power_Suc of_nat_mult)
   337 
   338 lemma nat_one_le_power [simp]: "1 \<le> i ==> Suc 0 \<le> i^n"
   339 by (insert one_le_power [of i n], simp)
   340 
   341 lemma nat_zero_less_power_iff [simp]: "(0 < x^n) = (x \<noteq> (0::nat) | n=0)"
   342 by (induct "n", auto)
   343 
   344 text{*Valid for the naturals, but what if @{text"0<i<1"}?
   345 Premises cannot be weakened: consider the case where @{term "i=0"},
   346 @{term "m=1"} and @{term "n=0"}.*}
   347 lemma nat_power_less_imp_less:
   348   assumes nonneg: "0 < (i\<Colon>nat)"
   349   assumes less: "i^m < i^n"
   350   shows "m < n"
   351 proof (cases "i = 1")
   352   case True with less power_one [where 'a = nat] show ?thesis by simp
   353 next
   354   case False with nonneg have "1 < i" by auto
   355   from power_strict_increasing_iff [OF this] less show ?thesis ..
   356 qed
   357 
   358 lemma power_diff:
   359   assumes nz: "a ~= 0"
   360   shows "n <= m ==> (a::'a::{recpower, field}) ^ (m-n) = (a^m) / (a^n)"
   361   by (induct m n rule: diff_induct)
   362     (simp_all add: power_Suc nonzero_mult_divide_cancel_left nz)
   363 
   364 
   365 text{*ML bindings for the general exponentiation theorems*}
   366 ML
   367 {*
   368 val power_0 = thm"power_0";
   369 val power_Suc = thm"power_Suc";
   370 val power_0_Suc = thm"power_0_Suc";
   371 val power_0_left = thm"power_0_left";
   372 val power_one = thm"power_one";
   373 val power_one_right = thm"power_one_right";
   374 val power_add = thm"power_add";
   375 val power_mult = thm"power_mult";
   376 val power_mult_distrib = thm"power_mult_distrib";
   377 val zero_less_power = thm"zero_less_power";
   378 val zero_le_power = thm"zero_le_power";
   379 val one_le_power = thm"one_le_power";
   380 val gt1_imp_ge0 = thm"gt1_imp_ge0";
   381 val power_gt1_lemma = thm"power_gt1_lemma";
   382 val power_gt1 = thm"power_gt1";
   383 val power_le_imp_le_exp = thm"power_le_imp_le_exp";
   384 val power_inject_exp = thm"power_inject_exp";
   385 val power_less_imp_less_exp = thm"power_less_imp_less_exp";
   386 val power_mono = thm"power_mono";
   387 val power_strict_mono = thm"power_strict_mono";
   388 val power_eq_0_iff = thm"power_eq_0_iff";
   389 val field_power_eq_0_iff = thm"field_power_eq_0_iff";
   390 val field_power_not_zero = thm"field_power_not_zero";
   391 val power_inverse = thm"power_inverse";
   392 val nonzero_power_divide = thm"nonzero_power_divide";
   393 val power_divide = thm"power_divide";
   394 val power_abs = thm"power_abs";
   395 val zero_less_power_abs_iff = thm"zero_less_power_abs_iff";
   396 val zero_le_power_abs = thm "zero_le_power_abs";
   397 val power_minus = thm"power_minus";
   398 val power_Suc_less = thm"power_Suc_less";
   399 val power_strict_decreasing = thm"power_strict_decreasing";
   400 val power_decreasing = thm"power_decreasing";
   401 val power_Suc_less_one = thm"power_Suc_less_one";
   402 val power_increasing = thm"power_increasing";
   403 val power_strict_increasing = thm"power_strict_increasing";
   404 val power_le_imp_le_base = thm"power_le_imp_le_base";
   405 val power_inject_base = thm"power_inject_base";
   406 *}
   407 
   408 text{*ML bindings for the remaining theorems*}
   409 ML
   410 {*
   411 val nat_one_le_power = thm"nat_one_le_power";
   412 val nat_power_less_imp_less = thm"nat_power_less_imp_less";
   413 val nat_zero_less_power_iff = thm"nat_zero_less_power_iff";
   414 *}
   415 
   416 end
   417