src/HOL/Power.thy
author paulson
Tue Jul 20 14:22:49 2004 +0200 (2004-07-20)
changeset 15066 d2f2b908e0a4
parent 15004 44ac09ba7213
child 15131 c69542757a4d
permissions -rw-r--r--
two new results
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(*  Title:      HOL/Power.thy
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1997  University of Cambridge
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*)
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header{*Exponentiation and Binomial Coefficients*}
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theory Power = Divides:
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subsection{*Powers for Arbitrary Semirings*}
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axclass recpower \<subseteq> comm_semiring_1_cancel, power
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  power_0 [simp]: "a ^ 0       = 1"
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  power_Suc:      "a ^ (Suc n) = a * (a ^ n)"
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lemma power_0_Suc [simp]: "(0::'a::recpower) ^ (Suc n) = 0"
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by (simp add: power_Suc)
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text{*It looks plausible as a simprule, but its effect can be strange.*}
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lemma power_0_left: "0^n = (if n=0 then 1 else (0::'a::recpower))"
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by (induct_tac "n", auto)
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lemma power_one [simp]: "1^n = (1::'a::recpower)"
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apply (induct_tac "n")
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apply (auto simp add: power_Suc)
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done
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lemma power_one_right [simp]: "(a::'a::recpower) ^ 1 = a"
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by (simp add: power_Suc)
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lemma power_add: "(a::'a::recpower) ^ (m+n) = (a^m) * (a^n)"
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apply (induct_tac "n")
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apply (simp_all add: power_Suc mult_ac)
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done
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lemma power_mult: "(a::'a::recpower) ^ (m*n) = (a^m) ^ n"
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apply (induct_tac "n")
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apply (simp_all add: power_Suc power_add)
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done
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lemma power_mult_distrib: "((a::'a::recpower) * b) ^ n = (a^n) * (b^n)"
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apply (induct_tac "n")
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apply (auto simp add: power_Suc mult_ac)
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done
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lemma zero_less_power:
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     "0 < (a::'a::{ordered_semidom,recpower}) ==> 0 < a^n"
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apply (induct_tac "n")
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apply (simp_all add: power_Suc zero_less_one mult_pos)
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done
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lemma zero_le_power:
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     "0 \<le> (a::'a::{ordered_semidom,recpower}) ==> 0 \<le> a^n"
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apply (simp add: order_le_less)
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apply (erule disjE)
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apply (simp_all add: zero_less_power zero_less_one power_0_left)
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done
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lemma one_le_power:
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     "1 \<le> (a::'a::{ordered_semidom,recpower}) ==> 1 \<le> a^n"
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apply (induct_tac "n")
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apply (simp_all add: power_Suc)
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apply (rule order_trans [OF _ mult_mono [of 1 _ 1]])
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apply (simp_all add: zero_le_one order_trans [OF zero_le_one])
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done
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lemma gt1_imp_ge0: "1 < a ==> 0 \<le> (a::'a::ordered_semidom)"
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  by (simp add: order_trans [OF zero_le_one order_less_imp_le])
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lemma power_gt1_lemma:
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  assumes gt1: "1 < (a::'a::{ordered_semidom,recpower})"
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  shows "1 < a * a^n"
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proof -
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  have "1*1 < a*1" using gt1 by simp
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  also have "\<dots> \<le> a * a^n" using gt1
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    by (simp only: mult_mono gt1_imp_ge0 one_le_power order_less_imp_le
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        zero_le_one order_refl)
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  finally show ?thesis by simp
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qed
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lemma power_gt1:
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     "1 < (a::'a::{ordered_semidom,recpower}) ==> 1 < a ^ (Suc n)"
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by (simp add: power_gt1_lemma power_Suc)
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lemma power_le_imp_le_exp:
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  assumes gt1: "(1::'a::{recpower,ordered_semidom}) < a"
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  shows "!!n. a^m \<le> a^n ==> m \<le> n"
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proof (induct m)
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  case 0
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  show ?case by simp
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next
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  case (Suc m)
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  show ?case
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  proof (cases n)
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    case 0
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    from prems have "a * a^m \<le> 1" by (simp add: power_Suc)
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    with gt1 show ?thesis
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      by (force simp only: power_gt1_lemma
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          linorder_not_less [symmetric])
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  next
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    case (Suc n)
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    from prems show ?thesis
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      by (force dest: mult_left_le_imp_le
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          simp add: power_Suc order_less_trans [OF zero_less_one gt1])
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  qed
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qed
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text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}
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lemma power_inject_exp [simp]:
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     "1 < (a::'a::{ordered_semidom,recpower}) ==> (a^m = a^n) = (m=n)"
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  by (force simp add: order_antisym power_le_imp_le_exp)
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text{*Can relax the first premise to @{term "0<a"} in the case of the
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natural numbers.*}
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lemma power_less_imp_less_exp:
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     "[| (1::'a::{recpower,ordered_semidom}) < a; a^m < a^n |] ==> m < n"
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by (simp add: order_less_le [of m n] order_less_le [of "a^m" "a^n"]
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              power_le_imp_le_exp)
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lemma power_mono:
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     "[|a \<le> b; (0::'a::{recpower,ordered_semidom}) \<le> a|] ==> a^n \<le> b^n"
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apply (induct_tac "n")
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apply (simp_all add: power_Suc)
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apply (auto intro: mult_mono zero_le_power order_trans [of 0 a b])
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done
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lemma power_strict_mono [rule_format]:
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     "[|a < b; (0::'a::{recpower,ordered_semidom}) \<le> a|]
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      ==> 0 < n --> a^n < b^n"
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apply (induct_tac "n")
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apply (auto simp add: mult_strict_mono zero_le_power power_Suc
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                      order_le_less_trans [of 0 a b])
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done
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lemma power_eq_0_iff [simp]:
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     "(a^n = 0) = (a = (0::'a::{ordered_idom,recpower}) & 0<n)"
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apply (induct_tac "n")
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apply (auto simp add: power_Suc zero_neq_one [THEN not_sym])
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done
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lemma field_power_eq_0_iff [simp]:
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     "(a^n = 0) = (a = (0::'a::{field,recpower}) & 0<n)"
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apply (induct_tac "n")
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apply (auto simp add: power_Suc field_mult_eq_0_iff zero_neq_one[THEN not_sym])
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done
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lemma field_power_not_zero: "a \<noteq> (0::'a::{field,recpower}) ==> a^n \<noteq> 0"
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by force
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lemma nonzero_power_inverse:
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  "a \<noteq> 0 ==> inverse ((a::'a::{field,recpower}) ^ n) = (inverse a) ^ n"
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apply (induct_tac "n")
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apply (auto simp add: power_Suc nonzero_inverse_mult_distrib mult_commute)
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done
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text{*Perhaps these should be simprules.*}
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lemma power_inverse:
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  "inverse ((a::'a::{field,division_by_zero,recpower}) ^ n) = (inverse a) ^ n"
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apply (induct_tac "n")
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apply (auto simp add: power_Suc inverse_mult_distrib)
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done
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lemma nonzero_power_divide:
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    "b \<noteq> 0 ==> (a/b) ^ n = ((a::'a::{field,recpower}) ^ n) / (b ^ n)"
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by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
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lemma power_divide:
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    "(a/b) ^ n = ((a::'a::{field,division_by_zero,recpower}) ^ n / b ^ n)"
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apply (case_tac "b=0", simp add: power_0_left)
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apply (rule nonzero_power_divide)
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apply assumption
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done
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lemma power_abs: "abs(a ^ n) = abs(a::'a::{ordered_idom,recpower}) ^ n"
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apply (induct_tac "n")
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apply (auto simp add: power_Suc abs_mult)
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done
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lemma zero_less_power_abs_iff [simp]:
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     "(0 < (abs a)^n) = (a \<noteq> (0::'a::{ordered_idom,recpower}) | n=0)"
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proof (induct "n")
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  case 0
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    show ?case by (simp add: zero_less_one)
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next
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  case (Suc n)
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    show ?case by (force simp add: prems power_Suc zero_less_mult_iff)
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qed
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lemma zero_le_power_abs [simp]:
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     "(0::'a::{ordered_idom,recpower}) \<le> (abs a)^n"
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apply (induct_tac "n")
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apply (auto simp add: zero_le_one zero_le_power)
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done
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lemma power_minus: "(-a) ^ n = (- 1)^n * (a::'a::{comm_ring_1,recpower}) ^ n"
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proof -
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  have "-a = (- 1) * a"  by (simp add: minus_mult_left [symmetric])
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  thus ?thesis by (simp only: power_mult_distrib)
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qed
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text{*Lemma for @{text power_strict_decreasing}*}
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lemma power_Suc_less:
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     "[|(0::'a::{ordered_semidom,recpower}) < a; a < 1|]
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      ==> a * a^n < a^n"
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apply (induct_tac n)
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apply (auto simp add: power_Suc mult_strict_left_mono)
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done
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lemma power_strict_decreasing:
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     "[|n < N; 0 < a; a < (1::'a::{ordered_semidom,recpower})|]
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      ==> a^N < a^n"
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apply (erule rev_mp)
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apply (induct_tac "N")
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apply (auto simp add: power_Suc power_Suc_less less_Suc_eq)
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apply (rename_tac m)
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apply (subgoal_tac "a * a^m < 1 * a^n", simp)
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apply (rule mult_strict_mono)
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apply (auto simp add: zero_le_power zero_less_one order_less_imp_le)
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done
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text{*Proof resembles that of @{text power_strict_decreasing}*}
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lemma power_decreasing:
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     "[|n \<le> N; 0 \<le> a; a \<le> (1::'a::{ordered_semidom,recpower})|]
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      ==> a^N \<le> a^n"
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apply (erule rev_mp)
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apply (induct_tac "N")
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apply (auto simp add: power_Suc  le_Suc_eq)
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apply (rename_tac m)
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apply (subgoal_tac "a * a^m \<le> 1 * a^n", simp)
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apply (rule mult_mono)
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apply (auto simp add: zero_le_power zero_le_one)
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done
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lemma power_Suc_less_one:
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     "[| 0 < a; a < (1::'a::{ordered_semidom,recpower}) |] ==> a ^ Suc n < 1"
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apply (insert power_strict_decreasing [of 0 "Suc n" a], simp)
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done
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text{*Proof again resembles that of @{text power_strict_decreasing}*}
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lemma power_increasing:
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     "[|n \<le> N; (1::'a::{ordered_semidom,recpower}) \<le> a|] ==> a^n \<le> a^N"
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apply (erule rev_mp)
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apply (induct_tac "N")
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apply (auto simp add: power_Suc le_Suc_eq)
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apply (rename_tac m)
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apply (subgoal_tac "1 * a^n \<le> a * a^m", simp)
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apply (rule mult_mono)
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apply (auto simp add: order_trans [OF zero_le_one] zero_le_power)
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done
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text{*Lemma for @{text power_strict_increasing}*}
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lemma power_less_power_Suc:
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     "(1::'a::{ordered_semidom,recpower}) < a ==> a^n < a * a^n"
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apply (induct_tac n)
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apply (auto simp add: power_Suc mult_strict_left_mono order_less_trans [OF zero_less_one])
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done
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lemma power_strict_increasing:
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     "[|n < N; (1::'a::{ordered_semidom,recpower}) < a|] ==> a^n < a^N"
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apply (erule rev_mp)
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apply (induct_tac "N")
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apply (auto simp add: power_less_power_Suc power_Suc less_Suc_eq)
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apply (rename_tac m)
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apply (subgoal_tac "1 * a^n < a * a^m", simp)
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apply (rule mult_strict_mono)
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apply (auto simp add: order_less_trans [OF zero_less_one] zero_le_power
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                 order_less_imp_le)
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done
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lemma power_increasing_iff [simp]: 
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     "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x \<le> b ^ y) = (x \<le> y)"
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  by (blast intro: power_le_imp_le_exp power_increasing order_less_imp_le) 
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lemma power_strict_increasing_iff [simp]:
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     "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x < b ^ y) = (x < y)"
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  by (blast intro: power_less_imp_less_exp power_strict_increasing) 
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lemma power_le_imp_le_base:
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  assumes le: "a ^ Suc n \<le> b ^ Suc n"
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      and xnonneg: "(0::'a::{ordered_semidom,recpower}) \<le> a"
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      and ynonneg: "0 \<le> b"
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  shows "a \<le> b"
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 proof (rule ccontr)
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   assume "~ a \<le> b"
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   then have "b < a" by (simp only: linorder_not_le)
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   then have "b ^ Suc n < a ^ Suc n"
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     by (simp only: prems power_strict_mono)
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   from le and this show "False"
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      by (simp add: linorder_not_less [symmetric])
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 qed
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lemma power_inject_base:
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     "[| a ^ Suc n = b ^ Suc n; 0 \<le> a; 0 \<le> b |]
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      ==> a = (b::'a::{ordered_semidom,recpower})"
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by (blast intro: power_le_imp_le_base order_antisym order_eq_refl sym)
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subsection{*Exponentiation for the Natural Numbers*}
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primrec (power)
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  "p ^ 0 = 1"
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  "p ^ (Suc n) = (p::nat) * (p ^ n)"
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instance nat :: recpower
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proof
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  fix z n :: nat
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  show "z^0 = 1" by simp
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  show "z^(Suc n) = z * (z^n)" by simp
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qed
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lemma nat_one_le_power [simp]: "1 \<le> i ==> Suc 0 \<le> i^n"
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by (insert one_le_power [of i n], simp)
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lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n"
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apply (unfold dvd_def)
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apply (erule not_less_iff_le [THEN iffD2, THEN add_diff_inverse, THEN subst])
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apply (simp add: power_add)
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done
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text{*Valid for the naturals, but what if @{text"0<i<1"}?
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Premises cannot be weakened: consider the case where @{term "i=0"},
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@{term "m=1"} and @{term "n=0"}.*}
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lemma nat_power_less_imp_less: "!!i::nat. [| 0 < i; i^m < i^n |] ==> m < n"
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apply (rule ccontr)
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apply (drule leI [THEN le_imp_power_dvd, THEN dvd_imp_le, THEN leD])
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apply (erule zero_less_power, auto)
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done
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lemma nat_zero_less_power_iff [simp]: "(0 < x^n) = (x \<noteq> (0::nat) | n=0)"
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by (induct_tac "n", auto)
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lemma power_le_dvd [rule_format]: "k^j dvd n --> i\<le>j --> k^i dvd (n::nat)"
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apply (induct_tac "j")
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apply (simp_all add: le_Suc_eq)
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apply (blast dest!: dvd_mult_right)
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done
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lemma power_dvd_imp_le: "[|i^m dvd i^n;  (1::nat) < i|] ==> m \<le> n"
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apply (rule power_le_imp_le_exp, assumption)
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apply (erule dvd_imp_le, simp)
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done
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subsection{*Binomial Coefficients*}
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text{*This development is based on the work of Andy Gordon and
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Florian Kammueller*}
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consts
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  binomial :: "[nat,nat] => nat"      (infixl "choose" 65)
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primrec
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  binomial_0:   "(0     choose k) = (if k = 0 then 1 else 0)"
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  binomial_Suc: "(Suc n choose k) =
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                 (if k = 0 then 1 else (n choose (k - 1)) + (n choose k))"
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lemma binomial_n_0 [simp]: "(n choose 0) = 1"
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by (case_tac "n", simp_all)
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lemma binomial_0_Suc [simp]: "(0 choose Suc k) = 0"
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by simp
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lemma binomial_Suc_Suc [simp]:
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     "(Suc n choose Suc k) = (n choose k) + (n choose Suc k)"
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by simp
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lemma binomial_eq_0 [rule_format]: "\<forall>k. n < k --> (n choose k) = 0"
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apply (induct_tac "n", auto)
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apply (erule allE)
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apply (erule mp, arith)
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done
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declare binomial_0 [simp del] binomial_Suc [simp del]
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lemma binomial_n_n [simp]: "(n choose n) = 1"
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apply (induct_tac "n")
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apply (simp_all add: binomial_eq_0)
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done
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lemma binomial_Suc_n [simp]: "(Suc n choose n) = Suc n"
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by (induct_tac "n", simp_all)
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lemma binomial_1 [simp]: "(n choose Suc 0) = n"
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by (induct_tac "n", simp_all)
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lemma zero_less_binomial [rule_format]: "k \<le> n --> 0 < (n choose k)"
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by (rule_tac m = n and n = k in diff_induct, simp_all)
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lemma binomial_eq_0_iff: "(n choose k = 0) = (n<k)"
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apply (safe intro!: binomial_eq_0)
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apply (erule contrapos_pp)
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apply (simp add: zero_less_binomial)
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done
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lemma zero_less_binomial_iff: "(0 < n choose k) = (k\<le>n)"
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by (simp add: linorder_not_less [symmetric] binomial_eq_0_iff [symmetric])
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(*Might be more useful if re-oriented*)
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lemma Suc_times_binomial_eq [rule_format]:
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     "\<forall>k. k \<le> n --> Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"
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apply (induct_tac "n")
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apply (simp add: binomial_0, clarify)
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apply (case_tac "k")
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apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq
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                      binomial_eq_0)
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done
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text{*This is the well-known version, but it's harder to use because of the
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  need to reason about division.*}
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lemma binomial_Suc_Suc_eq_times:
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     "k \<le> n ==> (Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k"
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by (simp add: Suc_times_binomial_eq div_mult_self_is_m zero_less_Suc
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        del: mult_Suc mult_Suc_right)
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text{*Another version, with -1 instead of Suc.*}
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lemma times_binomial_minus1_eq:
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     "[|k \<le> n;  0<k|] ==> (n choose k) * k = n * ((n - 1) choose (k - 1))"
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apply (cut_tac n = "n - 1" and k = "k - 1" in Suc_times_binomial_eq)
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apply (simp split add: nat_diff_split, auto)
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done
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   426
text{*ML bindings for the general exponentiation theorems*}
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ML
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{*
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val power_0 = thm"power_0";
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   430
val power_Suc = thm"power_Suc";
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   431
val power_0_Suc = thm"power_0_Suc";
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val power_0_left = thm"power_0_left";
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   433
val power_one = thm"power_one";
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   434
val power_one_right = thm"power_one_right";
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   435
val power_add = thm"power_add";
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   436
val power_mult = thm"power_mult";
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   437
val power_mult_distrib = thm"power_mult_distrib";
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   438
val zero_less_power = thm"zero_less_power";
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   439
val zero_le_power = thm"zero_le_power";
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   440
val one_le_power = thm"one_le_power";
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   441
val gt1_imp_ge0 = thm"gt1_imp_ge0";
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   442
val power_gt1_lemma = thm"power_gt1_lemma";
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   443
val power_gt1 = thm"power_gt1";
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   444
val power_le_imp_le_exp = thm"power_le_imp_le_exp";
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   445
val power_inject_exp = thm"power_inject_exp";
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   446
val power_less_imp_less_exp = thm"power_less_imp_less_exp";
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   447
val power_mono = thm"power_mono";
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   448
val power_strict_mono = thm"power_strict_mono";
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   449
val power_eq_0_iff = thm"power_eq_0_iff";
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   450
val field_power_eq_0_iff = thm"field_power_eq_0_iff";
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   451
val field_power_not_zero = thm"field_power_not_zero";
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   452
val power_inverse = thm"power_inverse";
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   453
val nonzero_power_divide = thm"nonzero_power_divide";
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   454
val power_divide = thm"power_divide";
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   455
val power_abs = thm"power_abs";
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   456
val zero_less_power_abs_iff = thm"zero_less_power_abs_iff";
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   457
val zero_le_power_abs = thm "zero_le_power_abs";
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   458
val power_minus = thm"power_minus";
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   459
val power_Suc_less = thm"power_Suc_less";
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   460
val power_strict_decreasing = thm"power_strict_decreasing";
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   461
val power_decreasing = thm"power_decreasing";
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   462
val power_Suc_less_one = thm"power_Suc_less_one";
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   463
val power_increasing = thm"power_increasing";
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   464
val power_strict_increasing = thm"power_strict_increasing";
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   465
val power_le_imp_le_base = thm"power_le_imp_le_base";
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val power_inject_base = thm"power_inject_base";
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   467
*}
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   469
text{*ML bindings for the remaining theorems*}
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ML
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   471
{*
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   472
val nat_one_le_power = thm"nat_one_le_power";
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   473
val le_imp_power_dvd = thm"le_imp_power_dvd";
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   474
val nat_power_less_imp_less = thm"nat_power_less_imp_less";
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   475
val nat_zero_less_power_iff = thm"nat_zero_less_power_iff";
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   476
val power_le_dvd = thm"power_le_dvd";
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   477
val power_dvd_imp_le = thm"power_dvd_imp_le";
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   478
val binomial_n_0 = thm"binomial_n_0";
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   479
val binomial_0_Suc = thm"binomial_0_Suc";
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   480
val binomial_Suc_Suc = thm"binomial_Suc_Suc";
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   481
val binomial_eq_0 = thm"binomial_eq_0";
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   482
val binomial_n_n = thm"binomial_n_n";
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   483
val binomial_Suc_n = thm"binomial_Suc_n";
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   484
val binomial_1 = thm"binomial_1";
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   485
val zero_less_binomial = thm"zero_less_binomial";
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   486
val binomial_eq_0_iff = thm"binomial_eq_0_iff";
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   487
val zero_less_binomial_iff = thm"zero_less_binomial_iff";
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   488
val Suc_times_binomial_eq = thm"Suc_times_binomial_eq";
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   489
val binomial_Suc_Suc_eq_times = thm"binomial_Suc_Suc_eq_times";
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   490
val times_binomial_minus1_eq = thm"times_binomial_minus1_eq";
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   491
*}
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   492
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   493
end
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   494