(* 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 power = one + times
begin
primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80) where
power_0: "a ^ 0 = 1"
| power_Suc: "a ^ Suc n = a * a ^ n"
notation (latex output)
power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
notation (HTML output)
power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
end
context monoid_mult
begin
subclass power .
lemma power_one [simp]:
"1 ^ n = 1"
by (induct n) simp_all
lemma power_one_right [simp]:
"a ^ 1 = a"
by simp
lemma power_commutes:
"a ^ n * a = a * a ^ n"
by (induct n) (simp_all add: mult_assoc)
lemma power_Suc2:
"a ^ Suc n = a ^ n * a"
by (simp add: power_commutes)
lemma power_add:
"a ^ (m + n) = a ^ m * a ^ n"
by (induct m) (simp_all add: algebra_simps)
lemma power_mult:
"a ^ (m * n) = (a ^ m) ^ n"
by (induct n) (simp_all add: power_add)
end
context comm_monoid_mult
begin
lemma power_mult_distrib:
"(a * b) ^ n = (a ^ n) * (b ^ n)"
by (induct n) (simp_all add: mult_ac)
end
context semiring_1
begin
lemma of_nat_power:
"of_nat (m ^ n) = of_nat m ^ n"
by (induct n) (simp_all add: of_nat_mult)
end
context comm_semiring_1
begin
text {* The divides relation *}
lemma le_imp_power_dvd:
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:
"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 dvd y \<Longrightarrow> x ^ n dvd y ^ n"
by (induct n) (auto simp add: mult_dvd_mono)
lemma dvd_power_le:
"x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m"
by (rule power_le_dvd [OF dvd_power_same])
lemma dvd_power [simp]:
assumes "n > (0::nat) \<or> x = 1"
shows "x dvd (x ^ n)"
using assms proof
assume "0 < n"
then have "x ^ n = x ^ Suc (n - 1)" by simp
then show "x dvd (x ^ n)" by simp
next
assume "x = 1"
then show "x dvd (x ^ n)" by simp
qed
end
context ring_1
begin
lemma power_minus:
"(- a) ^ n = (- 1) ^ n * a ^ 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
end
context linordered_semidom
begin
lemma zero_less_power [simp]:
"0 < a \<Longrightarrow> 0 < a ^ n"
by (induct n) (simp_all add: mult_pos_pos)
lemma zero_le_power [simp]:
"0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
by (induct n) (simp_all add: mult_nonneg_nonneg)
lemma one_le_power[simp]:
"1 \<le> a \<Longrightarrow> 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 power_gt1_lemma:
assumes gt1: "1 < a"
shows "1 < a * a ^ n"
proof -
from gt1 have "0 \<le> a"
by (fact order_trans [OF zero_le_one less_imp_le])
have "1 * 1 < a * 1" using gt1 by simp
also have "\<dots> \<le> a * a ^ n" using gt1
by (simp only: mult_mono `0 \<le> a` one_le_power order_less_imp_le
zero_le_one order_refl)
finally show ?thesis by simp
qed
lemma power_gt1:
"1 < a \<Longrightarrow> 1 < a ^ Suc n"
by (simp add: power_gt1_lemma)
lemma one_less_power [simp]:
"1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
by (cases n) (simp_all add: power_gt1_lemma)
lemma power_le_imp_le_exp:
assumes gt1: "1 < a"
shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
proof (induct m arbitrary: n)
case 0
show ?case by simp
next
case (Suc m)
show ?case
proof (cases n)
case 0
with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
with gt1 show ?thesis
by (force simp only: power_gt1_lemma
not_less [symmetric])
next
case (Suc n)
with Suc.prems Suc.hyps show ?thesis
by (force dest: mult_left_le_imp_le
simp add: 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 \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> 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 \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"]
power_le_imp_le_exp)
lemma power_mono:
"a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
by (induct n)
(auto intro: mult_mono order_trans [of 0 a b])
lemma power_strict_mono [rule_format]:
"a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
by (induct n)
(auto simp add: mult_strict_mono le_less_trans [of 0 a b])
text{*Lemma for @{text power_strict_decreasing}*}
lemma power_Suc_less:
"0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
by (induct n)
(auto simp add: mult_strict_left_mono)
lemma power_strict_decreasing [rule_format]:
"n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
proof (induct N)
case 0 then show ?case by simp
next
case (Suc N) then show ?case
apply (auto simp add: power_Suc_less less_Suc_eq)
apply (subgoal_tac "a * a^N < 1 * a^n")
apply simp
apply (rule mult_strict_mono) apply auto
done
qed
text{*Proof resembles that of @{text power_strict_decreasing}*}
lemma power_decreasing [rule_format]:
"n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n"
proof (induct N)
case 0 then show ?case by simp
next
case (Suc N) then show ?case
apply (auto simp add: le_Suc_eq)
apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp)
apply (rule mult_mono) apply auto
done
qed
lemma power_Suc_less_one:
"0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
using power_strict_decreasing [of 0 "Suc n" a] by simp
text{*Proof again resembles that of @{text power_strict_decreasing}*}
lemma power_increasing [rule_format]:
"n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
proof (induct N)
case 0 then show ?case by simp
next
case (Suc N) then show ?case
apply (auto simp add: le_Suc_eq)
apply (subgoal_tac "1 * a^n \<le> a * a^N", simp)
apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one])
done
qed
text{*Lemma for @{text power_strict_increasing}*}
lemma power_less_power_Suc:
"1 < a \<Longrightarrow> a ^ n < a * a ^ n"
by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one])
lemma power_strict_increasing [rule_format]:
"n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
proof (induct N)
case 0 then show ?case by simp
next
case (Suc N) then show ?case
apply (auto simp add: power_less_power_Suc less_Suc_eq)
apply (subgoal_tac "1 * a^n < a * a^N", simp)
apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
done
qed
lemma power_increasing_iff [simp]:
"1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
lemma power_strict_increasing_iff [simp]:
"1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> 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 \<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: assms power_strict_mono)
from le and this show False
by (simp add: linorder_not_less [symmetric])
qed
lemma power_less_imp_less_base:
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 "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
qed
lemma power_inject_base:
"a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
by (blast intro: power_le_imp_le_base antisym eq_refl sym)
lemma power_eq_imp_eq_base:
"a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
by (cases n) (simp_all del: power_Suc, rule power_inject_base)
end
context linordered_idom
begin
lemma power_abs:
"abs (a ^ n) = abs a ^ n"
by (induct n) (auto simp add: abs_mult)
lemma abs_power_minus [simp]:
"abs ((-a) ^ n) = abs (a ^ n)"
by (simp add: power_abs)
lemma zero_less_power_abs_iff [simp, no_atp]:
"0 < abs a ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
proof (induct n)
case 0 show ?case by simp
next
case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff)
qed
lemma zero_le_power_abs [simp]:
"0 \<le> abs a ^ n"
by (rule zero_le_power [OF abs_ge_zero])
end
context ring_1_no_zero_divisors
begin
lemma field_power_not_zero:
"a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
by (induct n) auto
end
context division_ring
begin
text {* FIXME reorient or rename to @{text nonzero_inverse_power} *}
lemma nonzero_power_inverse:
"a \<noteq> 0 \<Longrightarrow> inverse (a ^ n) = (inverse a) ^ n"
by (induct n)
(simp_all add: nonzero_inverse_mult_distrib power_commutes field_power_not_zero)
end
context field
begin
lemma nonzero_power_divide:
"b \<noteq> 0 \<Longrightarrow> (a / b) ^ n = a ^ n / b ^ n"
by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
end
lemma power_0_Suc [simp]:
"(0::'a::{power, 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::{power, semiring_0}))"
by (induct n) simp_all
lemma power_eq_0_iff [simp]:
"a ^ n = 0 \<longleftrightarrow>
a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,power}) \<and> n \<noteq> 0"
by (induct n)
(auto simp add: no_zero_divisors elim: contrapos_pp)
lemma (in field) power_diff:
assumes nz: "a \<noteq> 0"
shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
by (induct m n rule: diff_induct) (simp_all add: nz field_power_not_zero)
text{*Perhaps these should be simprules.*}
lemma power_inverse:
fixes a :: "'a::division_ring_inverse_zero"
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_inverse_zero, power}) ^ n = (1 / a) ^ n"
by (simp add: divide_inverse) (rule power_inverse)
lemma power_divide:
"(a / b) ^ n = (a::'a::field_inverse_zero) ^ n / b ^ n"
apply (cases "b = 0")
apply (simp add: power_0_left)
apply (rule nonzero_power_divide)
apply assumption
done
subsection {* Exponentiation for the Natural Numbers *}
lemma nat_one_le_power [simp]:
"Suc 0 \<le> i \<Longrightarrow> 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 \<longleftrightarrow> x > (0::nat) \<or> n = 0"
by (induct n) auto
lemma nat_power_eq_Suc_0_iff [simp]:
"x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> 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_dvd_imp_le:
"i ^ m dvd i ^ n \<Longrightarrow> (1::nat) < i \<Longrightarrow> m \<le> n"
apply (rule power_le_imp_le_exp, assumption)
apply (erule dvd_imp_le, simp)
done
subsection {* Code generator tweak *}
lemma power_power_power [code, code_unfold, code_inline del]:
"power = power.power (1::'a::{power}) (op *)"
unfolding power_def power.power_def ..
declare power.power.simps [code]
code_modulename SML
Power Arith
code_modulename OCaml
Power Arith
code_modulename Haskell
Power Arith
end