(*  Title:      HOL/Power.thy

    ID:         $Id$

    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

    Copyright   1997  University of Cambridge

*)

header{*Exponentiation*}

theory Power

imports Nat

begin

class power =

  fixes power :: "'a \<Rightarrow> nat \<Rightarrow> 'a"            (infixr "^" 80)

subsection{*Powers for Arbitrary Monoids*}

class recpower = monoid_mult + power +

  assumes power_0 [simp]: "a ^ 0       = 1"

  assumes power_Suc:      "a ^ Suc n = a * (a ^ n)"

lemma power_0_Suc [simp]: "(0::'a::{recpower,semiring_0}) ^ (Suc n) = 0"

  by (simp add: power_Suc)

text{*It looks plausible as a simprule, but its effect can be strange.*}

lemma power_0_left: "0^n = (if n=0 then 1 else (0::'a::{recpower,semiring_0}))"

  by (induct n) simp_all

lemma power_one [simp]: "1^n = (1::'a::recpower)"

  by (induct n) (simp_all add: power_Suc)

lemma power_one_right [simp]: "(a::'a::recpower) ^ 1 = a"

  unfolding One_nat_def by (simp add: power_Suc)

lemma power_commutes: "(a::'a::recpower) ^ n * a = a * a ^ n"

  by (induct n) (simp_all add: power_Suc mult_assoc)

lemma power_Suc2: "(a::'a::recpower) ^ Suc n = a ^ n * a"

  by (simp add: power_Suc power_commutes)

lemma power_add: "(a::'a::recpower) ^ (m+n) = (a^m) * (a^n)"

  by (induct m) (simp_all add: power_Suc mult_ac)

lemma power_mult: "(a::'a::recpower) ^ (m*n) = (a^m) ^ n"

  by (induct n) (simp_all add: power_Suc power_add)

lemma power_mult_distrib: "((a::'a::{recpower,comm_monoid_mult}) * b) ^ n = (a^n) * (b^n)"

  by (induct n) (simp_all add: power_Suc mult_ac)

lemma zero_less_power[simp]:

     "0 < (a::'a::{ordered_semidom,recpower}) ==> 0 < a^n"

apply (induct "n")

apply (simp_all add: power_Suc zero_less_one mult_pos_pos)

done

lemma zero_le_power[simp]:

     "0 \<le> (a::'a::{ordered_semidom,recpower}) ==> 0 \<le> a^n"

apply (simp add: order_le_less)

apply (erule disjE)

apply (simp_all add: zero_less_one power_0_left)

done

lemma one_le_power[simp]:

     "1 \<le> (a::'a::{ordered_semidom,recpower}) ==> 1 \<le> a^n"

apply (induct "n")

apply (simp_all add: power_Suc)

apply (rule order_trans [OF _ mult_mono [of 1 _ 1]])

apply (simp_all add: zero_le_one order_trans [OF zero_le_one])

done

lemma gt1_imp_ge0: "1 < a ==> 0 \<le> (a::'a::ordered_semidom)"

  by (simp add: order_trans [OF zero_le_one order_less_imp_le])

lemma power_gt1_lemma:

  assumes gt1: "1 < (a::'a::{ordered_semidom,recpower})"

  shows "1 < a * a^n"

proof -

  have "1*1 < a*1" using gt1 by simp

  also have "\<dots> \<le> a * a^n" using gt1

    by (simp only: mult_mono gt1_imp_ge0 one_le_power order_less_imp_le

        zero_le_one order_refl)

  finally show ?thesis by simp

qed

lemma one_less_power[simp]:

  "\<lbrakk>1 < (a::'a::{ordered_semidom,recpower}); 0 < n\<rbrakk> \<Longrightarrow> 1 < a ^ n"

by (cases n, simp_all add: power_gt1_lemma power_Suc)

lemma power_gt1:

     "1 < (a::'a::{ordered_semidom,recpower}) ==> 1 < a ^ (Suc n)"

by (simp add: power_gt1_lemma power_Suc)

lemma power_le_imp_le_exp:

  assumes gt1: "(1::'a::{recpower,ordered_semidom}) < a"

  shows "!!n. a^m \<le> a^n ==> m \<le> n"

proof (induct m)

  case 0

  show ?case by simp

next

  case (Suc m)

  show ?case

  proof (cases n)

    case 0

    from prems have "a * a^m \<le> 1" by (simp add: power_Suc)

    with gt1 show ?thesis

      by (force simp only: power_gt1_lemma

          linorder_not_less [symmetric])

  next

    case (Suc n)

    from prems show ?thesis

      by (force dest: mult_left_le_imp_le

          simp add: power_Suc order_less_trans [OF zero_less_one gt1])

  qed

qed

text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}

lemma power_inject_exp [simp]:

     "1 < (a::'a::{ordered_semidom,recpower}) ==> (a^m = a^n) = (m=n)"

  by (force simp add: order_antisym power_le_imp_le_exp)

text{*Can relax the first premise to @{term "0<a"} in the case of the

natural numbers.*}

lemma power_less_imp_less_exp:

     "[| (1::'a::{recpower,ordered_semidom}) < a; a^m < a^n |] ==> m < n"

by (simp add: order_less_le [of m n] order_less_le [of "a^m" "a^n"]

              power_le_imp_le_exp)

lemma power_mono:

     "[|a \<le> b; (0::'a::{recpower,ordered_semidom}) \<le> a|] ==> a^n \<le> b^n"

apply (induct "n")

apply (simp_all add: power_Suc)

apply (auto intro: mult_mono order_trans [of 0 a b])

done

lemma power_strict_mono [rule_format]:

     "[|a < b; (0::'a::{recpower,ordered_semidom}) \<le> a|]

      ==> 0 < n --> a^n < b^n"

apply (induct "n")

apply (auto simp add: mult_strict_mono power_Suc

                      order_le_less_trans [of 0 a b])

done

lemma power_eq_0_iff [simp]:

  "(a^n = 0) \<longleftrightarrow>

   (a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,recpower}) & n\<noteq>0)"

apply (induct "n")

apply (auto simp add: power_Suc zero_neq_one [THEN not_sym] no_zero_divisors)

done

lemma field_power_not_zero:

  "a \<noteq> (0::'a::{ring_1_no_zero_divisors,recpower}) ==> a^n \<noteq> 0"

by force

lemma nonzero_power_inverse:

  fixes a :: "'a::{division_ring,recpower}"

  shows "a \<noteq> 0 ==> inverse (a ^ n) = (inverse a) ^ n"

apply (induct "n")

apply (auto simp add: power_Suc nonzero_inverse_mult_distrib power_commutes)

done (* TODO: reorient or rename to nonzero_inverse_power *)

text{*Perhaps these should be simprules.*}

lemma power_inverse:

  fixes a :: "'a::{division_ring,division_by_zero,recpower}"

  shows "inverse (a ^ n) = (inverse a) ^ n"

apply (cases "a = 0")

apply (simp add: power_0_left)

apply (simp add: nonzero_power_inverse)

done (* TODO: reorient or rename to inverse_power *)

lemma power_one_over: "1 / (a::'a::{field,division_by_zero,recpower})^n = 

    (1 / a)^n"

apply (simp add: divide_inverse)

apply (rule power_inverse)

done

lemma nonzero_power_divide:

    "b \<noteq> 0 ==> (a/b) ^ n = ((a::'a::{field,recpower}) ^ n) / (b ^ n)"

by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)

lemma power_divide:

    "(a/b) ^ n = ((a::'a::{field,division_by_zero,recpower}) ^ n / b ^ n)"

apply (case_tac "b=0", simp add: power_0_left)

apply (rule nonzero_power_divide)

apply assumption

done

lemma power_abs: "abs(a ^ n) = abs(a::'a::{ordered_idom,recpower}) ^ n"

apply (induct "n")

apply (auto simp add: power_Suc abs_mult)

done

lemma zero_less_power_abs_iff [simp,noatp]:

     "(0 < (abs a)^n) = (a \<noteq> (0::'a::{ordered_idom,recpower}) | n=0)"

proof (induct "n")

  case 0

    show ?case by (simp add: zero_less_one)

next

  case (Suc n)

    show ?case by (auto simp add: prems power_Suc zero_less_mult_iff

      abs_zero)

qed

lemma zero_le_power_abs [simp]:

     "(0::'a::{ordered_idom,recpower}) \<le> (abs a)^n"

by (rule zero_le_power [OF abs_ge_zero])

lemma power_minus: "(-a) ^ n = (- 1)^n * (a::'a::{ring_1,recpower}) ^ n"

proof (induct n)

  case 0 show ?case by simp

next

  case (Suc n) then show ?case

    by (simp add: power_Suc2 mult_assoc)

qed

text{*Lemma for @{text power_strict_decreasing}*}

lemma power_Suc_less:

     "[|(0::'a::{ordered_semidom,recpower}) < a; a < 1|]

      ==> a * a^n < a^n"

apply (induct n)

apply (auto simp add: power_Suc mult_strict_left_mono)

done

lemma power_strict_decreasing:

     "[|n < N; 0 < a; a < (1::'a::{ordered_semidom,recpower})|]

      ==> a^N < a^n"

apply (erule rev_mp)

apply (induct "N")

apply (auto simp add: power_Suc power_Suc_less less_Suc_eq)

apply (rename_tac m)

apply (subgoal_tac "a * a^m < 1 * a^n", simp)

apply (rule mult_strict_mono)

apply (auto simp add: zero_less_one order_less_imp_le)

done

text{*Proof resembles that of @{text power_strict_decreasing}*}

lemma power_decreasing:

     "[|n \<le> N; 0 \<le> a; a \<le> (1::'a::{ordered_semidom,recpower})|]

      ==> a^N \<le> a^n"

apply (erule rev_mp)

apply (induct "N")

apply (auto simp add: power_Suc  le_Suc_eq)

apply (rename_tac m)

apply (subgoal_tac "a * a^m \<le> 1 * a^n", simp)

apply (rule mult_mono)

apply (auto simp add: zero_le_one)

done

lemma power_Suc_less_one:

     "[| 0 < a; a < (1::'a::{ordered_semidom,recpower}) |] ==> a ^ Suc n < 1"

apply (insert power_strict_decreasing [of 0 "Suc n" a], simp)

done

text{*Proof again resembles that of @{text power_strict_decreasing}*}

lemma power_increasing:

     "[|n \<le> N; (1::'a::{ordered_semidom,recpower}) \<le> a|] ==> a^n \<le> a^N"

apply (erule rev_mp)

apply (induct "N")

apply (auto simp add: power_Suc le_Suc_eq)

apply (rename_tac m)

apply (subgoal_tac "1 * a^n \<le> a * a^m", simp)

apply (rule mult_mono)

apply (auto simp add: order_trans [OF zero_le_one])

done

text{*Lemma for @{text power_strict_increasing}*}

lemma power_less_power_Suc:

     "(1::'a::{ordered_semidom,recpower}) < a ==> a^n < a * a^n"

apply (induct n)

apply (auto simp add: power_Suc mult_strict_left_mono order_less_trans [OF zero_less_one])

done

lemma power_strict_increasing:

     "[|n < N; (1::'a::{ordered_semidom,recpower}) < a|] ==> a^n < a^N"

apply (erule rev_mp)

apply (induct "N")

apply (auto simp add: power_less_power_Suc power_Suc less_Suc_eq)

apply (rename_tac m)

apply (subgoal_tac "1 * a^n < a * a^m", simp)

apply (rule mult_strict_mono)

apply (auto simp add: order_less_trans [OF zero_less_one] order_less_imp_le)

done

lemma power_increasing_iff [simp]:

  "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x \<le> b ^ y) = (x \<le> y)"

by (blast intro: power_le_imp_le_exp power_increasing order_less_imp_le) 

lemma power_strict_increasing_iff [simp]:

  "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x < b ^ y) = (x < y)"

by (blast intro: power_less_imp_less_exp power_strict_increasing) 

lemma power_le_imp_le_base:

assumes le: "a ^ Suc n \<le> b ^ Suc n"

    and ynonneg: "(0::'a::{ordered_semidom,recpower}) \<le> b"

shows "a \<le> b"

proof (rule ccontr)

  assume "~ a \<le> b"

  then have "b < a" by (simp only: linorder_not_le)

  then have "b ^ Suc n < a ^ Suc n"

    by (simp only: prems power_strict_mono)

  from le and this show "False"

    by (simp add: linorder_not_less [symmetric])

qed

lemma power_less_imp_less_base:

  fixes a b :: "'a::{ordered_semidom,recpower}"

  assumes less: "a ^ n < b ^ n"

  assumes nonneg: "0 \<le> b"

  shows "a < b"

proof (rule contrapos_pp [OF less])

  assume "~ a < b"

  hence "b \<le> a" by (simp only: linorder_not_less)

  hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)

  thus "~ a ^ n < b ^ n" by (simp only: linorder_not_less)

qed

lemma power_inject_base:

     "[| a ^ Suc n = b ^ Suc n; 0 \<le> a; 0 \<le> b |]

      ==> a = (b::'a::{ordered_semidom,recpower})"

by (blast intro: power_le_imp_le_base order_antisym order_eq_refl sym)

lemma power_eq_imp_eq_base:

  fixes a b :: "'a::{ordered_semidom,recpower}"

  shows "\<lbrakk>a ^ n = b ^ n; 0 \<le> a; 0 \<le> b; 0 < n\<rbrakk> \<Longrightarrow> a = b"

by (cases n, simp_all, rule power_inject_base)

text {* The divides relation *}

lemma le_imp_power_dvd:

  fixes a :: "'a::{comm_semiring_1,recpower}"

  assumes "m \<le> n" shows "a^m dvd a^n"

proof

  have "a^n = a^(m + (n - m))"

    using `m \<le> n` by simp

  also have "\<dots> = a^m * a^(n - m)"

    by (rule power_add)

  finally show "a^n = a^m * a^(n - m)" .

qed

lemma power_le_dvd:

  fixes a b :: "'a::{comm_semiring_1,recpower}"

  shows "a^n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a^m dvd b"

  by (rule dvd_trans [OF le_imp_power_dvd])

subsection{*Exponentiation for the Natural Numbers*}

instantiation nat :: recpower

begin

primrec power_nat where

  "p ^ 0 = (1\<Colon>nat)"

  | "p ^ (Suc n) = (p\<Colon>nat) * (p ^ n)"

instance proof

  fix z n :: nat

  show "z^0 = 1" by simp

  show "z^(Suc n) = z * (z^n)" by simp

qed

end

lemma of_nat_power:

  "of_nat (m ^ n) = (of_nat m::'a::{semiring_1,recpower}) ^ n"

by (induct n, simp_all add: power_Suc of_nat_mult)

lemma nat_one_le_power [simp]: "Suc 0 \<le> i ==> Suc 0 \<le> i^n"

by (rule one_le_power [of i n, unfolded One_nat_def])

lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat) | n=0)"

by (induct "n", auto)

lemma nat_power_eq_Suc_0_iff [simp]: 

  "((x::nat)^m = Suc 0) = (m = 0 | x = Suc 0)"

by (induct_tac m, auto)

lemma power_Suc_0[simp]: "(Suc 0)^n = Suc 0"

by simp

text{*Valid for the naturals, but what if @{text"0<i<1"}?

Premises cannot be weakened: consider the case where @{term "i=0"},

@{term "m=1"} and @{term "n=0"}.*}

lemma nat_power_less_imp_less:

  assumes nonneg: "0 < (i\<Colon>nat)"

  assumes less: "i^m < i^n"

  shows "m < n"

proof (cases "i = 1")

  case True with less power_one [where 'a = nat] show ?thesis by simp

next

  case False with nonneg have "1 < i" by auto

  from power_strict_increasing_iff [OF this] less show ?thesis ..

qed

lemma power_diff:

  assumes nz: "a ~= 0"

  shows "n <= m ==> (a::'a::{recpower, field}) ^ (m-n) = (a^m) / (a^n)"

  by (induct m n rule: diff_induct)

    (simp_all add: power_Suc nonzero_mult_divide_cancel_left nz)

text{*ML bindings for the general exponentiation theorems*}

ML

{*

val power_0 = thm"power_0";

val power_Suc = thm"power_Suc";

val power_0_Suc = thm"power_0_Suc";

val power_0_left = thm"power_0_left";

val power_one = thm"power_one";

val power_one_right = thm"power_one_right";

val power_add = thm"power_add";

val power_mult = thm"power_mult";

val power_mult_distrib = thm"power_mult_distrib";

val zero_less_power = thm"zero_less_power";

val zero_le_power = thm"zero_le_power";

val one_le_power = thm"one_le_power";

val gt1_imp_ge0 = thm"gt1_imp_ge0";

val power_gt1_lemma = thm"power_gt1_lemma";

val power_gt1 = thm"power_gt1";

val power_le_imp_le_exp = thm"power_le_imp_le_exp";

val power_inject_exp = thm"power_inject_exp";

val power_less_imp_less_exp = thm"power_less_imp_less_exp";

val power_mono = thm"power_mono";

val power_strict_mono = thm"power_strict_mono";

val power_eq_0_iff = thm"power_eq_0_iff";

val field_power_eq_0_iff = thm"power_eq_0_iff";

val field_power_not_zero = thm"field_power_not_zero";

val power_inverse = thm"power_inverse";

val nonzero_power_divide = thm"nonzero_power_divide";

val power_divide = thm"power_divide";

val power_abs = thm"power_abs";

val zero_less_power_abs_iff = thm"zero_less_power_abs_iff";

val zero_le_power_abs = thm "zero_le_power_abs";

val power_minus = thm"power_minus";

val power_Suc_less = thm"power_Suc_less";

val power_strict_decreasing = thm"power_strict_decreasing";

val power_decreasing = thm"power_decreasing";

val power_Suc_less_one = thm"power_Suc_less_one";

val power_increasing = thm"power_increasing";

val power_strict_increasing = thm"power_strict_increasing";

val power_le_imp_le_base = thm"power_le_imp_le_base";

val power_inject_base = thm"power_inject_base";

*}

text{*ML bindings for the remaining theorems*}

ML

{*

val nat_one_le_power = thm"nat_one_le_power";

val nat_power_less_imp_less = thm"nat_power_less_imp_less";

val nat_zero_less_power_iff = thm"nat_zero_less_power_iff";

*}

end