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+(* from Isabelle2013-2 src/HOL/Power.thy; BSD license *)
+
+(* Title: HOL/Power.thy
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1997 University of Cambridge
+*)
+
+header {* Exponentiation *}
+
+theory Power
+imports Num
+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_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
+
+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_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 power_zero_numeral [simp]: "(0::'a) ^ 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 power_neg_numeral_Bit0 [simp]:
+ "neg_numeral k ^ numeral (Num.Bit0 l) = numeral (Num.pow k (Num.Bit0 l))"
+ by (simp only: neg_numeral_def power_minus_Bit0 power_numeral)
+
+lemma power_neg_numeral_Bit1 [simp]:
+ "neg_numeral k ^ numeral (Num.Bit1 l) = neg_numeral (Num.pow k (Num.Bit1 l))"
+ by (simp only: neg_numeral_def power_minus_Bit1 power_numeral pow.simps)
+
+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
+
+context ring_1_no_zero_divisors
+begin
+
+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)
+
+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 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 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, 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])
+
+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 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: mult_ac 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: mult_ac 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)
+
+end
+
+
+subsection {* Miscellaneous rules *}
+
+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 power2_sum:
+ fixes x y :: "'a::comm_semiring_1"
+ shows "(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 power2_diff:
+ fixes x y :: "'a::comm_ring_1"
+ shows "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
+ by (simp add: ring_distribs power2_eq_square mult_2) (rule mult_commute)
+
+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
+
+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"
+ using assms by (cases m) (simp_all add: power2_eq_square)
+
+
+
+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
+
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