(*  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 ringpower \<subseteq> semiring, power

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

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

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

by (induct_tac "n", auto)

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

apply (induct_tac "n")

apply (auto simp add: power_Suc)  

done

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

by (simp add: power_Suc)

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

apply (induct_tac "n")

apply (simp_all add: power_Suc mult_ac)

done

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

apply (induct_tac "n")

apply (simp_all add: power_Suc power_add)

done

lemma power_mult_distrib: "((a::'a::ringpower) * 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_semiring,ringpower}) ==> 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_semiring,ringpower}) ==> 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_semiring,ringpower}) ==> 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_semiring)"

  by (simp add: order_trans [OF zero_le_one order_less_imp_le])

lemma power_gt1_lemma:

     assumes gt1: "1 < (a::'a::{ordered_semiring,ringpower})"

        shows     "1 < a * a^n"

proof -

  have "1*1 < a * a^n"

    proof (rule order_less_le_trans) 

      show "1*1 < a*1" by (simp add: gt1)

      show  "a*1 \<le> a * a^n"

   by (simp only: mult_mono gt1 gt1_imp_ge0 one_le_power order_less_imp_le 

                  zero_le_one order_refl)

    qed

  thus ?thesis by simp

qed

lemma power_gt1:

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

by (simp add: power_gt1_lemma power_Suc)

lemma power_le_imp_le_exp:

     assumes gt1: "(1::'a::{ringpower,ordered_semiring}) < 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 0<a<1 too.*}

lemma power_inject_exp [simp]:

     "1 < (a::'a::{ordered_semiring,ringpower}) ==> (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::{ringpower,ordered_semiring}) < 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::{ringpower,ordered_semiring}) \<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::{ringpower,ordered_semiring}) \<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_ring,ringpower}) & 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,ringpower}) & 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,ringpower}) ==> a^n \<noteq> 0"

by force

lemma nonzero_power_inverse:

  "a \<noteq> 0 ==> inverse ((a::'a::{field,ringpower}) ^ 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,ringpower}) ^ 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,ringpower}) ^ 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,ringpower}) ^ 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_ring,ringpower}) ^ 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_ring,ringpower}) | 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_ring,ringpower}) \<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::{ring,ringpower}) ^ 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_semiring,ringpower}) < 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_semiring,ringpower})|]

      ==> 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_semiring,ringpower})|]

      ==> 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_semiring,ringpower}) |] ==> 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_semiring,ringpower}) \<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_semiring,ringpower}) < 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_semiring,ringpower}) < 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_semiring,ringpower}) \<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_semiring,ringpower})"

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 :: ringpower

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