(*  Title:      HOL/Power.thy

     ID:         $Id$

     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

     Copyright   1997  University of Cambridge

 *)

 header{*Exponentiation*}

 theory Power

 imports Nat

 begin

 class power =

   fixes power :: "'a \<Rightarrow> nat \<Rightarrow> 'a"            (infixr "^" 80)

 subsection{*Powers for Arbitrary Monoids*}

 class recpower = monoid_mult + power +

   assumes power_0 [simp]: "a ^ 0       = 1"

   assumes power_Suc:      "a ^ Suc n = a * (a ^ n)"

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

   by (simp add: power_Suc)

 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 add: power_Suc)

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

   unfolding One_nat_def by (simp add: power_Suc)

 lemma power_commutes: "(a::'a::recpower) ^ n * a = a * a ^ n"

   by (induct n) (simp_all add: power_Suc mult_assoc)

 lemma power_Suc2: "(a::'a::recpower) ^ Suc n = a ^ n * a"

   by (simp add: power_Suc power_commutes)

 lemma power_add: "(a::'a::recpower) ^ (m+n) = (a^m) * (a^n)"

   by (induct m) (simp_all add: power_Suc mult_ac)

 lemma power_mult: "(a::'a::recpower) ^ (m*n) = (a^m) ^ n"

   by (induct n) (simp_all add: power_Suc power_add)

 lemma power_mult_distrib: "((a::'a::{recpower,comm_monoid_mult}) * b) ^ n = (a^n) * (b^n)"

   by (induct n) (simp_all add: power_Suc mult_ac)

 lemma zero_less_power[simp]:

      "0 < (a::'a::{ordered_semidom,recpower}) ==> 0 < a^n"

 apply (induct "n")

 apply (simp_all add: power_Suc zero_less_one mult_pos_pos)

 done

 lemma zero_le_power[simp]:

      "0 \<le> (a::'a::{ordered_semidom,recpower}) ==> 0 \<le> a^n"

 apply (simp add: order_le_less)

 apply (erule disjE)

 apply (simp_all add: zero_less_one power_0_left)

 done

 lemma one_le_power[simp]:

      "1 \<le> (a::'a::{ordered_semidom,recpower}) ==> 1 \<le> a^n"

 apply (induct "n")

 apply (simp_all add: power_Suc)

 apply (rule order_trans [OF _ mult_mono [of 1 _ 1]])

 apply (simp_all add: zero_le_one 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 power_Suc)

 lemma power_gt1:

      "1 < (a::'a::{ordered_semidom,recpower}) ==> 1 < a ^ (Suc n)"

 by (simp add: power_gt1_lemma power_Suc)

 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 add: power_Suc)

     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: power_Suc 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 add: power_Suc)

 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 power_Suc

                       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: power_Suc zero_neq_one [THEN not_sym] 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: power_Suc 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: power_Suc abs_mult)

 done

 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 add: zero_less_one)

 next

   case (Suc n)

     show ?case by (auto simp add: prems power_Suc zero_less_mult_iff

       abs_zero)

 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 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: power_Suc 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 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: zero_less_one 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: power_Suc  le_Suc_eq)

 apply (rename_tac m)

 apply (subgoal_tac "a * a^m \<le> 1 * a^n", simp)

 apply (rule mult_mono)

 apply (auto simp add: zero_le_one)

 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: power_Suc 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: power_Suc 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 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, 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])

 subsection{*Exponentiation for the Natural Numbers*}

 instantiation nat :: recpower

 begin

 primrec power_nat where

   "p ^ 0 = (1\<Colon>nat)"

   | "p ^ (Suc n) = (p\<Colon>nat) * (p ^ n)"

 instance proof

   fix z n :: nat

   show "z^0 = 1" by simp

   show "z^(Suc n) = z * (z^n)" by simp

 qed

 end

 lemma of_nat_power:

   "of_nat (m ^ n) = (of_nat m::'a::{semiring_1,recpower}) ^ n"

 by (induct n, simp_all add: power_Suc 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_tac 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: power_Suc nonzero_mult_divide_cancel_left nz)

 text{*ML bindings for the general exponentiation theorems*}

 ML

 {*

 val power_0 = thm"power_0";

 val power_Suc = thm"power_Suc";

 val power_0_Suc = thm"power_0_Suc";

 val power_0_left = thm"power_0_left";

 val power_one = thm"power_one";

 val power_one_right = thm"power_one_right";

 val power_add = thm"power_add";

 val power_mult = thm"power_mult";

 val power_mult_distrib = thm"power_mult_distrib";

 val zero_less_power = thm"zero_less_power";

 val zero_le_power = thm"zero_le_power";

 val one_le_power = thm"one_le_power";

 val gt1_imp_ge0 = thm"gt1_imp_ge0";

 val power_gt1_lemma = thm"power_gt1_lemma";

 val power_gt1 = thm"power_gt1";

 val power_le_imp_le_exp = thm"power_le_imp_le_exp";

 val power_inject_exp = thm"power_inject_exp";

 val power_less_imp_less_exp = thm"power_less_imp_less_exp";

 val power_mono = thm"power_mono";

 val power_strict_mono = thm"power_strict_mono";

 val power_eq_0_iff = thm"power_eq_0_iff";

 val field_power_eq_0_iff = thm"power_eq_0_iff";

 val field_power_not_zero = thm"field_power_not_zero";

 val power_inverse = thm"power_inverse";

 val nonzero_power_divide = thm"nonzero_power_divide";

 val power_divide = thm"power_divide";

 val power_abs = thm"power_abs";

 val zero_less_power_abs_iff = thm"zero_less_power_abs_iff";

 val zero_le_power_abs = thm "zero_le_power_abs";

 val power_minus = thm"power_minus";

 val power_Suc_less = thm"power_Suc_less";

 val power_strict_decreasing = thm"power_strict_decreasing";

 val power_decreasing = thm"power_decreasing";

 val power_Suc_less_one = thm"power_Suc_less_one";

 val power_increasing = thm"power_increasing";

 val power_strict_increasing = thm"power_strict_increasing";

 val power_le_imp_le_base = thm"power_le_imp_le_base";

 val power_inject_base = thm"power_inject_base";

 *}

 text{*ML bindings for the remaining theorems*}

 ML

 {*

 val nat_one_le_power = thm"nat_one_le_power";

 val nat_power_less_imp_less = thm"nat_power_less_imp_less";

 val nat_zero_less_power_iff = thm"nat_zero_less_power_iff";

 *}

 end