src/HOL/Power.thy

power operation defined generic

(*  Title:      HOL/Power.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1997  University of Cambridge
*)

header {* Exponentiation *}

theory Power
imports Nat
begin

subsection {* Powers for Arbitrary Monoids *}

class recpower = monoid_mult
begin

primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80) where
    power_0: "a ^ 0 = 1"
  | power_Suc: "a ^ Suc n = a * a ^ n"

end

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

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

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

lemma power_commutes: "(a::'a::recpower) ^ n * a = a * a ^ n"
  by (induct n) (simp_all add: mult_assoc)

lemma power_Suc2: "(a::'a::recpower) ^ Suc n = a ^ n * a"
  by (simp add: power_commutes)

lemma power_add: "(a::'a::recpower) ^ (m+n) = (a^m) * (a^n)"
  by (induct m) (simp_all add: mult_ac)

lemma power_mult: "(a::'a::recpower) ^ (m*n) = (a^m) ^ n"
  by (induct n) (simp_all add: power_add)

lemma power_mult_distrib: "((a::'a::{recpower,comm_monoid_mult}) * b) ^ n = (a^n) * (b^n)"
  by (induct n) (simp_all add: mult_ac)

lemma zero_less_power[simp]:
     "0 < (a::'a::{ordered_semidom,recpower}) ==> 0 < a^n"
by (induct n) (simp_all add: mult_pos_pos)

lemma zero_le_power[simp]:
     "0 \<le> (a::'a::{ordered_semidom,recpower}) ==> 0 \<le> a^n"
by (induct n) (simp_all add: mult_nonneg_nonneg)

lemma one_le_power[simp]:
     "1 \<le> (a::'a::{ordered_semidom,recpower}) ==> 1 \<le> a^n"
apply (induct "n")
apply simp_all
apply (rule order_trans [OF _ mult_mono [of 1 _ 1]])
apply (simp_all add: 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)

lemma power_gt1:
     "1 < (a::'a::{ordered_semidom,recpower}) ==> 1 < a ^ (Suc n)"
by (simp add: power_gt1_lemma)

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
    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: 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
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 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: 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: 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: abs_mult)
done

lemma abs_power_minus [simp]:
  fixes a:: "'a::{ordered_idom,recpower}" shows "abs((-a) ^ n) = abs(a ^ n)"
  by (simp add: abs_minus_cancel power_abs) 

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
next
  case (Suc n)
    show ?case by (auto simp add: prems zero_less_mult_iff)
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 del: power_Suc 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: 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_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: 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: le_Suc_eq)
apply (rename_tac m)
apply (subgoal_tac "a * a^m \<le> 1 * a^n", simp)
apply (rule mult_mono)
apply auto
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: 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: 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 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 del: power_Suc, 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])


lemma dvd_power_same:
  "(x::'a::{comm_semiring_1,recpower}) dvd y \<Longrightarrow> x^n dvd y^n"
by (induct n) (auto simp add: mult_dvd_mono)

lemma dvd_power_le:
  "(x::'a::{comm_semiring_1,recpower}) dvd y \<Longrightarrow> m >= n \<Longrightarrow> x^n dvd y^m"
by(rule power_le_dvd[OF dvd_power_same])

lemma dvd_power [simp]:
  "n > 0 | (x::'a::{comm_semiring_1,recpower}) = 1 \<Longrightarrow> x dvd x^n"
apply (erule disjE)
 apply (subgoal_tac "x ^ n = x^(Suc (n - 1))")
  apply (erule ssubst)
  apply (subst power_Suc)
  apply auto
done


subsection {* Exponentiation for the Natural Numbers *}

instance nat :: recpower ..

lemma of_nat_power:
  "of_nat (m ^ n) = (of_nat m::'a::{semiring_1,recpower}) ^ n"
by (induct n, simp_all add: 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 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: nonzero_mult_divide_cancel_left nz)

end