(*  Title:      HOL/Power.thy

     ID:         $Id$

     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

     Copyright   1997  University of Cambridge

 *)

 header{*Exponentiation and Binomial Coefficients*}

 theory Power = Divides:

 subsection{*Powers for Arbitrary (Semi)Rings*}

 axclass recpower \<subseteq> comm_semiring_1_cancel, power

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

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

 lemma power_0_Suc [simp]: "(0::'a::recpower) ^ (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))"

 by (induct_tac "n", auto)

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

 apply (induct_tac "n")

 apply (auto simp add: power_Suc)

 done

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

 by (simp add: power_Suc)

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

 apply (induct_tac "n")

 apply (simp_all add: power_Suc mult_ac)

 done

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

 apply (induct_tac "n")

 apply (simp_all add: power_Suc power_add)

 done

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

 apply (induct_tac "n")

 apply (auto simp add: power_Suc mult_ac)

 done

 lemma zero_less_power:

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

 apply (induct_tac "n")

 apply (simp_all add: power_Suc zero_less_one mult_pos)

 done

 lemma zero_le_power:

      "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_power zero_less_one power_0_left)

 done

 lemma one_le_power:

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

 apply (induct_tac "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 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_tac "n")

 apply (simp_all add: power_Suc)

 apply (auto intro: mult_mono zero_le_power 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_tac "n")

 apply (auto simp add: mult_strict_mono zero_le_power power_Suc

                       order_le_less_trans [of 0 a b])

 done

 lemma power_eq_0_iff [simp]:

      "(a^n = 0) = (a = (0::'a::{ordered_idom,recpower}) & 0<n)"

 apply (induct_tac "n")

 apply (auto simp add: power_Suc zero_neq_one [THEN not_sym])

 done

 lemma field_power_eq_0_iff [simp]:

      "(a^n = 0) = (a = (0::'a::{field,recpower}) & 0<n)"

 apply (induct_tac "n")

 apply (auto simp add: power_Suc field_mult_eq_0_iff zero_neq_one[THEN not_sym])

 done

 lemma field_power_not_zero: "a \<noteq> (0::'a::{field,recpower}) ==> a^n \<noteq> 0"

 by force

 lemma nonzero_power_inverse:

   "a \<noteq> 0 ==> inverse ((a::'a::{field,recpower}) ^ n) = (inverse a) ^ n"

 apply (induct_tac "n")

 apply (auto simp add: power_Suc nonzero_inverse_mult_distrib mult_commute)

 done

 text{*Perhaps these should be simprules.*}

 lemma power_inverse:

   "inverse ((a::'a::{field,division_by_zero,recpower}) ^ n) = (inverse a) ^ n"

 apply (induct_tac "n")

 apply (auto simp add: power_Suc inverse_mult_distrib)

 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_tac "n")

 apply (auto simp add: power_Suc abs_mult)

 done

 lemma zero_less_power_abs_iff [simp]:

      "(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 (force simp add: prems power_Suc zero_less_mult_iff)

 qed

 lemma zero_le_power_abs [simp]:

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

 apply (induct_tac "n")

 apply (auto simp add: zero_le_one zero_le_power)

 done

 lemma power_minus: "(-a) ^ n = (- 1)^n * (a::'a::{comm_ring_1,recpower}) ^ n"

 proof -

   have "-a = (- 1) * a"  by (simp add: minus_mult_left [symmetric])

   thus ?thesis by (simp only: power_mult_distrib)

 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_tac 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_tac "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_le_power 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_tac "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_power 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_tac "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] zero_le_power)

 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_tac 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_tac "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] zero_le_power

                  order_less_imp_le)

 done

 lemma power_le_imp_le_base:

   assumes le: "a ^ Suc n \<le> b ^ Suc n"

       and xnonneg: "(0::'a::{ordered_semidom,recpower}) \<le> a"

       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_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)

 subsection{*Exponentiation for the Natural Numbers*}

 primrec (power)

   "p ^ 0 = 1"

   "p ^ (Suc n) = (p::nat) * (p ^ n)"

 instance nat :: recpower

 proof

   fix z n :: nat

   show "z^0 = 1" by simp

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

 qed

 lemma nat_one_le_power [simp]: "1 \<le> i ==> Suc 0 \<le> i^n"

 by (insert one_le_power [of i n], simp)

 lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n"

 apply (unfold dvd_def)

 apply (erule not_less_iff_le [THEN iffD2, THEN add_diff_inverse, THEN subst])

 apply (simp add: power_add)

 done

 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: "!!i::nat. [| 0 < i; i^m < i^n |] ==> m < n"

 apply (rule ccontr)

 apply (drule leI [THEN le_imp_power_dvd, THEN dvd_imp_le, THEN leD])

 apply (erule zero_less_power, auto)

 done

 lemma nat_zero_less_power_iff [simp]: "(0 < x^n) = (x \<noteq> (0::nat) | n=0)"

 by (induct_tac "n", auto)

 lemma power_le_dvd [rule_format]: "k^j dvd n --> i\<le>j --> k^i dvd (n::nat)"

 apply (induct_tac "j")

 apply (simp_all add: le_Suc_eq)

 apply (blast dest!: dvd_mult_right)

 done

 lemma power_dvd_imp_le: "[|i^m dvd i^n;  (1::nat) < i|] ==> m \<le> n"

 apply (rule power_le_imp_le_exp, assumption)

 apply (erule dvd_imp_le, simp)

 done

 subsection{*Binomial Coefficients*}

 text{*This development is based on the work of Andy Gordon and

 Florian Kammueller*}

 consts

   binomial :: "[nat,nat] => nat"      (infixl "choose" 65)

 primrec

   binomial_0:   "(0     choose k) = (if k = 0 then 1 else 0)"

   binomial_Suc: "(Suc n choose k) =

                  (if k = 0 then 1 else (n choose (k - 1)) + (n choose k))"

 lemma binomial_n_0 [simp]: "(n choose 0) = 1"

 by (case_tac "n", simp_all)

 lemma binomial_0_Suc [simp]: "(0 choose Suc k) = 0"

 by simp

 lemma binomial_Suc_Suc [simp]:

      "(Suc n choose Suc k) = (n choose k) + (n choose Suc k)"

 by simp

 lemma binomial_eq_0 [rule_format]: "\<forall>k. n < k --> (n choose k) = 0"

 apply (induct_tac "n", auto)

 apply (erule allE)

 apply (erule mp, arith)

 done

 declare binomial_0 [simp del] binomial_Suc [simp del]

 lemma binomial_n_n [simp]: "(n choose n) = 1"

 apply (induct_tac "n")

 apply (simp_all add: binomial_eq_0)

 done

 lemma binomial_Suc_n [simp]: "(Suc n choose n) = Suc n"

 by (induct_tac "n", simp_all)

 lemma binomial_1 [simp]: "(n choose Suc 0) = n"

 by (induct_tac "n", simp_all)

 lemma zero_less_binomial [rule_format]: "k \<le> n --> 0 < (n choose k)"

 by (rule_tac m = n and n = k in diff_induct, simp_all)

 lemma binomial_eq_0_iff: "(n choose k = 0) = (n<k)"

 apply (safe intro!: binomial_eq_0)

 apply (erule contrapos_pp)

 apply (simp add: zero_less_binomial)

 done

 lemma zero_less_binomial_iff: "(0 < n choose k) = (k\<le>n)"

 by (simp add: linorder_not_less [symmetric] binomial_eq_0_iff [symmetric])

 (*Might be more useful if re-oriented*)

 lemma Suc_times_binomial_eq [rule_format]:

      "\<forall>k. k \<le> n --> Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"

 apply (induct_tac "n")

 apply (simp add: binomial_0, clarify)

 apply (case_tac "k")

 apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq

                       binomial_eq_0)

 done

 text{*This is the well-known version, but it's harder to use because of the

   need to reason about division.*}

 lemma binomial_Suc_Suc_eq_times:

      "k \<le> n ==> (Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k"

 by (simp add: Suc_times_binomial_eq div_mult_self_is_m zero_less_Suc

         del: mult_Suc mult_Suc_right)

 text{*Another version, with -1 instead of Suc.*}

 lemma times_binomial_minus1_eq:

      "[|k \<le> n;  0<k|] ==> (n choose k) * k = n * ((n - 1) choose (k - 1))"

 apply (cut_tac n = "n - 1" and k = "k - 1" in Suc_times_binomial_eq)

 apply (simp split add: nat_diff_split, auto)

 done

 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"field_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 le_imp_power_dvd = thm"le_imp_power_dvd";

 val nat_power_less_imp_less = thm"nat_power_less_imp_less";

 val nat_zero_less_power_iff = thm"nat_zero_less_power_iff";

 val power_le_dvd = thm"power_le_dvd";

 val power_dvd_imp_le = thm"power_dvd_imp_le";

 val binomial_n_0 = thm"binomial_n_0";

 val binomial_0_Suc = thm"binomial_0_Suc";

 val binomial_Suc_Suc = thm"binomial_Suc_Suc";

 val binomial_eq_0 = thm"binomial_eq_0";

 val binomial_n_n = thm"binomial_n_n";

 val binomial_Suc_n = thm"binomial_Suc_n";

 val binomial_1 = thm"binomial_1";

 val zero_less_binomial = thm"zero_less_binomial";

 val binomial_eq_0_iff = thm"binomial_eq_0_iff";

 val zero_less_binomial_iff = thm"zero_less_binomial_iff";

 val Suc_times_binomial_eq = thm"Suc_times_binomial_eq";

 val binomial_Suc_Suc_eq_times = thm"binomial_Suc_Suc_eq_times";

 val times_binomial_minus1_eq = thm"times_binomial_minus1_eq";

 *}

 end