(*  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 ordered_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: prems 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 ordered_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: abs_minus_cancel power_abs) 

 lemma zero_less_power_abs_iff [simp, noatp]:

   "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 power_diff:

   fixes a :: "'a::field"

   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)

 text{*Perhaps these should be simprules.*}

 lemma power_inverse:

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

   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, power}) ^ n =  (1 / a) ^ n"

   by (simp add: divide_inverse) (rule power_inverse)

 lemma power_divide:

   "(a / b) ^ n = (a::'a::{field,division_by_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