(* Title: HOL/Power.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1997 University of Cambridge
*)
section {* Exponentiation *}
theory Power
imports Num Equiv_Relations
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)
text {* Special syntax for squares. *}
abbreviation (xsymbols)
power2 :: "'a \<Rightarrow> 'a" ("(_\<^sup>2)" [1000] 999) where
"x\<^sup>2 \<equiv> x ^ 2"
notation (latex output)
power2 ("(_\<^sup>2)" [1000] 999)
notation (HTML output)
power2 ("(_\<^sup>2)" [1000] 999)
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_Suc0_right [simp]:
"a ^ Suc 0 = 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)
lemma power2_eq_square: "a\<^sup>2 = a * a"
by (simp add: numeral_2_eq_2)
lemma power3_eq_cube: "a ^ 3 = a * a * a"
by (simp add: numeral_3_eq_3 mult.assoc)
lemma power_even_eq:
"a ^ (2 * n) = (a ^ n)\<^sup>2"
by (subst mult.commute) (simp add: power_mult)
lemma power_odd_eq:
"a ^ Suc (2*n) = a * (a ^ n)\<^sup>2"
by (simp add: power_even_eq)
lemma power_numeral_even:
"z ^ numeral (Num.Bit0 w) = (let w = z ^ (numeral w) in w * w)"
unfolding numeral_Bit0 power_add Let_def ..
lemma power_numeral_odd:
"z ^ numeral (Num.Bit1 w) = (let w = z ^ (numeral w) in z * w * w)"
unfolding numeral_Bit1 One_nat_def add_Suc_right add_0_right
unfolding power_Suc power_add Let_def mult.assoc ..
lemma funpow_times_power:
"(times x ^^ f x) = times (x ^ f x)"
proof (induct "f x" arbitrary: f)
case 0 then show ?case by (simp add: fun_eq_iff)
next
case (Suc n)
def g \<equiv> "\<lambda>x. f x - 1"
with Suc have "n = g x" by simp
with Suc have "times x ^^ g x = times (x ^ g x)" by simp
moreover from Suc g_def have "f x = g x + 1" by simp
ultimately show ?case by (simp add: power_add funpow_add fun_eq_iff mult.assoc)
qed
lemma power_commuting_commutes:
assumes "x * y = y * x"
shows "x ^ n * y = y * x ^n"
proof (induct n)
case (Suc n)
have "x ^ Suc n * y = x ^ n * y * x"
by (subst power_Suc2) (simp add: assms ac_simps)
also have "\<dots> = y * x ^ Suc n"
unfolding Suc power_Suc2
by (simp add: ac_simps)
finally show ?case .
qed simp
end
context comm_monoid_mult
begin
lemma power_mult_distrib [field_simps]:
"(a * b) ^ n = (a ^ n) * (b ^ n)"
by (induct n) (simp_all add: ac_simps)
end
declare power_mult_distrib [where a = "numeral w" for w, simp]
declare power_mult_distrib [where b = "numeral w" for w, simp]
context semiring_numeral
begin
lemma numeral_sqr: "numeral (Num.sqr k) = numeral k * numeral k"
by (simp only: sqr_conv_mult numeral_mult)
lemma numeral_pow: "numeral (Num.pow k l) = numeral k ^ numeral l"
by (induct l, simp_all only: numeral_class.numeral.simps pow.simps
numeral_sqr numeral_mult power_add power_one_right)
lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)"
by (rule numeral_pow [symmetric])
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)
lemma zero_power:
"0 < n \<Longrightarrow> 0 ^ n = 0"
by (cases n) simp_all
lemma power_zero_numeral [simp]:
"0 ^ numeral k = 0"
by (simp add: numeral_eq_Suc)
lemma zero_power2: "0\<^sup>2 = 0" (* delete? *)
by (rule power_zero_numeral)
lemma one_power2: "1\<^sup>2 = 1" (* delete? *)
by (rule power_one)
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
lemma power_minus_Bit0:
"(- x) ^ numeral (Num.Bit0 k) = x ^ numeral (Num.Bit0 k)"
by (induct k, simp_all only: numeral_class.numeral.simps power_add
power_one_right mult_minus_left mult_minus_right minus_minus)
lemma power_minus_Bit1:
"(- x) ^ numeral (Num.Bit1 k) = - (x ^ numeral (Num.Bit1 k))"
by (simp only: eval_nat_numeral(3) power_Suc power_minus_Bit0 mult_minus_left)
lemma power2_minus [simp]:
"(- a)\<^sup>2 = a\<^sup>2"
by (rule power_minus_Bit0)
lemma power_minus1_even [simp]:
"(- 1) ^ (2*n) = 1"
proof (induct n)
case 0 show ?case by simp
next
case (Suc n) then show ?case by (simp add: power_add power2_eq_square)
qed
lemma power_minus1_odd:
"(- 1) ^ Suc (2*n) = -1"
by simp
lemma power_minus_even [simp]:
"(-a) ^ (2*n) = a ^ (2*n)"
by (simp add: power_minus [of a])
end
lemma power_eq_0_nat_iff [simp]:
fixes m n :: nat
shows "m ^ n = 0 \<longleftrightarrow> m = 0 \<and> n > 0"
by (induct n) auto
context ring_1_no_zero_divisors
begin
lemma power_eq_0_iff [simp]:
"a ^ n = 0 \<longleftrightarrow> a = 0 \<and> n > 0"
by (induct n) auto
lemma field_power_not_zero:
"a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
by (induct n) auto
lemma zero_eq_power2 [simp]:
"a\<^sup>2 = 0 \<longleftrightarrow> a = 0"
unfolding power2_eq_square by simp
lemma power2_eq_1_iff:
"a\<^sup>2 = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
unfolding power2_eq_square by (rule square_eq_1_iff)
end
context idom
begin
lemma power2_eq_iff: "x\<^sup>2 = y\<^sup>2 \<longleftrightarrow> x = y \<or> x = - y"
unfolding power2_eq_square by (rule square_eq_iff)
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)
declare nonzero_power_divide [where b = "numeral w" for w, simp]
end
subsection {* Exponentiation on ordered types *}
context linordered_ring (* TODO: move *)
begin
lemma sum_squares_ge_zero:
"0 \<le> x * x + y * y"
by (intro add_nonneg_nonneg zero_le_square)
lemma not_sum_squares_lt_zero:
"\<not> x * x + y * y < 0"
by (simp add: not_less sum_squares_ge_zero)
end
context linordered_semidom
begin
lemma zero_less_power [simp]:
"0 < a \<Longrightarrow> 0 < a ^ n"
by (induct n) simp_all
lemma zero_le_power [simp]:
"0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
by (induct n) simp_all
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 one_le_power [simp]: "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
using power_mono [of 1 a n] by simp
lemma power_le_one: "\<lbrakk>0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> a ^ n \<le> 1"
using power_mono [of a 1 n] by simp
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_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)
lemma power2_le_imp_le:
"x\<^sup>2 \<le> y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
lemma power2_less_imp_less:
"x\<^sup>2 < y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
by (rule power_less_imp_less_base)
lemma power2_eq_imp_eq:
"x\<^sup>2 = y\<^sup>2 \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp
end
context linordered_ring_strict
begin
lemma sum_squares_eq_zero_iff:
"x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
by (simp add: add_nonneg_eq_0_iff)
lemma sum_squares_le_zero_iff:
"x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
lemma sum_squares_gt_zero_iff:
"0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
by (simp add: not_le [symmetric] sum_squares_le_zero_iff)
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]:
"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])
lemma zero_le_power2 [simp]:
"0 \<le> a\<^sup>2"
by (simp add: power2_eq_square)
lemma zero_less_power2 [simp]:
"0 < a\<^sup>2 \<longleftrightarrow> a \<noteq> 0"
by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
lemma power2_less_0 [simp]:
"\<not> a\<^sup>2 < 0"
by (force simp add: power2_eq_square mult_less_0_iff)
lemma power2_less_eq_zero_iff [simp]:
"a\<^sup>2 \<le> 0 \<longleftrightarrow> a = 0"
by (simp add: le_less)
lemma abs_power2 [simp]:
"abs (a\<^sup>2) = a\<^sup>2"
by (simp add: power2_eq_square abs_mult abs_mult_self)
lemma power2_abs [simp]:
"(abs a)\<^sup>2 = a\<^sup>2"
by (simp add: power2_eq_square abs_mult_self)
lemma odd_power_less_zero:
"a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0"
proof (induct n)
case 0
then show ?case by simp
next
case (Suc n)
have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
by (simp add: ac_simps power_add power2_eq_square)
thus ?case
by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
qed
lemma odd_0_le_power_imp_0_le:
"0 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a"
using odd_power_less_zero [of a n]
by (force simp add: linorder_not_less [symmetric])
lemma zero_le_even_power'[simp]:
"0 \<le> a ^ (2*n)"
proof (induct n)
case 0
show ?case by simp
next
case (Suc n)
have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)"
by (simp add: ac_simps power_add power2_eq_square)
thus ?case
by (simp add: Suc zero_le_mult_iff)
qed
lemma sum_power2_ge_zero:
"0 \<le> x\<^sup>2 + y\<^sup>2"
by (intro add_nonneg_nonneg zero_le_power2)
lemma not_sum_power2_lt_zero:
"\<not> x\<^sup>2 + y\<^sup>2 < 0"
unfolding not_less by (rule sum_power2_ge_zero)
lemma sum_power2_eq_zero_iff:
"x\<^sup>2 + y\<^sup>2 = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
unfolding power2_eq_square by (simp add: add_nonneg_eq_0_iff)
lemma sum_power2_le_zero_iff:
"x\<^sup>2 + y\<^sup>2 \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
by (simp add: le_less sum_power2_eq_zero_iff not_sum_power2_lt_zero)
lemma sum_power2_gt_zero_iff:
"0 < x\<^sup>2 + y\<^sup>2 \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)
lemma abs_le_square_iff:
"\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> x\<^sup>2 \<le> y\<^sup>2"
proof
assume "\<bar>x\<bar> \<le> \<bar>y\<bar>"
then have "\<bar>x\<bar>\<^sup>2 \<le> \<bar>y\<bar>\<^sup>2" by (rule power_mono, simp)
then show "x\<^sup>2 \<le> y\<^sup>2" by simp
next
assume "x\<^sup>2 \<le> y\<^sup>2"
then show "\<bar>x\<bar> \<le> \<bar>y\<bar>"
by (auto intro!: power2_le_imp_le [OF _ abs_ge_zero])
qed
lemma abs_square_le_1:"x\<^sup>2 \<le> 1 \<longleftrightarrow> abs(x) \<le> 1"
using abs_le_square_iff [of x 1]
by simp
lemma abs_square_eq_1: "x\<^sup>2 = 1 \<longleftrightarrow> abs(x) = 1"
by (auto simp add: abs_if power2_eq_1_iff)
lemma abs_square_less_1: "x\<^sup>2 < 1 \<longleftrightarrow> abs(x) < 1"
using abs_square_eq_1 [of x] abs_square_le_1 [of x]
by (auto simp add: le_less)
end
subsection {* Miscellaneous rules *}
lemma self_le_power:
fixes x::"'a::linordered_semidom"
shows "1 \<le> x \<Longrightarrow> 0 < n \<Longrightarrow> x \<le> x ^ n"
using power_increasing[of 1 n x] power_one_right[of x] by auto
lemma power_eq_if: "p ^ m = (if m=0 then 1 else p * (p ^ (m - 1)))"
unfolding One_nat_def by (cases m) simp_all
lemma (in comm_semiring_1) power2_sum:
"(x + y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 + 2 * x * y"
by (simp add: algebra_simps power2_eq_square mult_2_right)
lemma (in comm_ring_1) power2_diff:
"(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
by (simp add: algebra_simps power2_eq_square mult_2_right)
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 (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"
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, power}) ^ n = (1 / a) ^ n"
by (simp add: divide_inverse) (rule power_inverse)
lemma power_divide [field_simps, divide_simps]:
"(a / b) ^ n = (a::'a::field) ^ n / b ^ n"
apply (cases "b = 0")
apply (simp add: power_0_left)
apply (rule nonzero_power_divide)
apply assumption
done
text {* Simprules for comparisons where common factors can be cancelled. *}
lemmas zero_compare_simps =
add_strict_increasing add_strict_increasing2 add_increasing
zero_le_mult_iff zero_le_divide_iff
zero_less_mult_iff zero_less_divide_iff
mult_le_0_iff divide_le_0_iff
mult_less_0_iff divide_less_0_iff
zero_le_power2 power2_less_0
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
lemma power2_nat_le_eq_le:
fixes m n :: nat
shows "m\<^sup>2 \<le> n\<^sup>2 \<longleftrightarrow> m \<le> n"
by (auto intro: power2_le_imp_le power_mono)
lemma power2_nat_le_imp_le:
fixes m n :: nat
assumes "m\<^sup>2 \<le> n"
shows "m \<le> n"
proof (cases m)
case 0 then show ?thesis by simp
next
case (Suc k)
show ?thesis
proof (rule ccontr)
assume "\<not> m \<le> n"
then have "n < m" by simp
with assms Suc show False
by (auto simp add: algebra_simps) (simp add: power2_eq_square)
qed
qed
subsubsection {* Cardinality of the Powerset *}
lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
unfolding UNIV_bool by simp
lemma card_Pow: "finite A \<Longrightarrow> card (Pow A) = 2 ^ card A"
proof (induct rule: finite_induct)
case empty
show ?case by auto
next
case (insert x A)
then have "inj_on (insert x) (Pow A)"
unfolding inj_on_def by (blast elim!: equalityE)
then have "card (Pow A) + card (insert x ` Pow A) = 2 * 2 ^ card A"
by (simp add: mult_2 card_image Pow_insert insert.hyps)
then show ?case using insert
apply (simp add: Pow_insert)
apply (subst card_Un_disjoint, auto)
done
qed
subsubsection {* Generalized sum over a set *}
lemma setsum_zero_power [simp]:
fixes c :: "nat \<Rightarrow> 'a::division_ring"
shows "(\<Sum>i\<in>A. c i * 0^i) = (if finite A \<and> 0 \<in> A then c 0 else 0)"
apply (cases "finite A")
by (induction A rule: finite_induct) auto
lemma setsum_zero_power' [simp]:
fixes c :: "nat \<Rightarrow> 'a::field"
shows "(\<Sum>i\<in>A. c i * 0^i / d i) = (if finite A \<and> 0 \<in> A then c 0 / d 0 else 0)"
using setsum_zero_power [of "\<lambda>i. c i / d i" A]
by auto
subsubsection {* Generalized product over a set *}
lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"
apply (erule finite_induct)
apply auto
done
lemma setprod_power_distrib:
fixes f :: "'a \<Rightarrow> 'b::comm_semiring_1"
shows "setprod f A ^ n = setprod (\<lambda>x. (f x) ^ n) A"
proof (cases "finite A")
case True then show ?thesis
by (induct A rule: finite_induct) (auto simp add: power_mult_distrib)
next
case False then show ?thesis
by simp
qed
lemma power_setsum:
"c ^ (\<Sum>a\<in>A. f a) = (\<Prod>a\<in>A. c ^ f a)"
by (induct A rule: infinite_finite_induct) (simp_all add: power_add)
lemma setprod_gen_delta:
assumes fS: "finite S"
shows "setprod (\<lambda>k. if k=a then b k else c) S = (if a \<in> S then (b a ::'a::comm_monoid_mult) * c^ (card S - 1) else c^ card S)"
proof-
let ?f = "(\<lambda>k. if k=a then b k else c)"
{assume a: "a \<notin> S"
hence "\<forall> k\<in> S. ?f k = c" by simp
hence ?thesis using a setprod_constant[OF fS, of c] by simp }
moreover
{assume a: "a \<in> S"
let ?A = "S - {a}"
let ?B = "{a}"
have eq: "S = ?A \<union> ?B" using a by blast
have dj: "?A \<inter> ?B = {}" by simp
from fS have fAB: "finite ?A" "finite ?B" by auto
have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
apply (rule setprod.cong) by auto
have cA: "card ?A = card S - 1" using fS a by auto
have fA1: "setprod ?f ?A = c ^ card ?A" unfolding fA0 apply (rule setprod_constant) using fS by auto
have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
using setprod.union_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
by simp
then have ?thesis using a cA
by (simp add: fA1 field_simps cong add: setprod.cong cong del: if_weak_cong)}
ultimately show ?thesis by blast
qed
subsection {* Code generator tweak *}
lemma power_power_power [code]:
"power = power.power (1::'a::{power}) (op *)"
unfolding power_def power.power_def ..
declare power.power.simps [code]
code_identifier
code_module Power \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
end