author  paulson 
Fri, 09 Jan 2004 10:46:18 +0100  
changeset 14348  744c868ee0b7 
parent 8844  db71c334e854 
child 14353  79f9fbef9106 
permissions  rwrr 
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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 (Semi)Rings*} 
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axclass ringpower \<subseteq> semiring, 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::ringpower) ^ (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::ringpower))" 
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by (induct_tac "n", auto) 
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lemma power_one [simp]: "1^n = (1::'a::ringpower)" 
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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::ringpower) ^ 1 = a" 
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by (simp add: power_Suc) 
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lemma power_add: "(a::'a::ringpower) ^ (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::ringpower) ^ (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::ringpower) * 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_semiring,ringpower}) ==> 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_semiring,ringpower}) ==> 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_semiring,ringpower}) ==> 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_semiring)" 
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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_semiring,ringpower})" 
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shows "1 < a * a^n" 
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proof  
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have "1*1 < a * a^n" 
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proof (rule order_less_le_trans) 
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show "1*1 < a*1" by (simp add: gt1) 
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show "a*1 \<le> a * a^n" 
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by (simp only: mult_mono gt1 gt1_imp_ge0 one_le_power order_less_imp_le 
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zero_le_one order_refl) 
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qed 
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thus ?thesis by simp 
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qed 
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lemma power_gt1: 
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"1 < (a::'a::{ordered_semiring,ringpower}) ==> 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::{ringpower,ordered_semiring}) < 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 0<a<1 too.*} 
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lemma power_inject_exp [simp]: 
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"1 < (a::'a::{ordered_semiring,ringpower}) ==> (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::{ringpower,ordered_semiring}) < 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::{ringpower,ordered_semiring}) \<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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131 
done 
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132 

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133 
lemma power_strict_mono [rule_format]: 
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"[a < b; (0::'a::{ringpower,ordered_semiring}) \<le> a] 
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==> 0 < n > a^n < b^n" 
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136 
apply (induct_tac "n") 
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137 
apply (auto simp add: mult_strict_mono zero_le_power power_Suc 
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138 
order_le_less_trans [of 0 a b]) 
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139 
done 
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140 

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141 
lemma power_eq_0_iff [simp]: 
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142 
"(a^n = 0) = (a = (0::'a::{ordered_ring,ringpower}) & 0<n)" 
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143 
apply (induct_tac "n") 
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144 
apply (auto simp add: power_Suc zero_neq_one [THEN not_sym]) 
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145 
done 
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146 

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147 
lemma field_power_eq_0_iff [simp]: 
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"(a^n = 0) = (a = (0::'a::{field,ringpower}) & 0<n)" 
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149 
apply (induct_tac "n") 
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150 
apply (auto simp add: power_Suc field_mult_eq_0_iff zero_neq_one[THEN not_sym]) 
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151 
done 
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152 

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153 
lemma field_power_not_zero: "a \<noteq> (0::'a::{field,ringpower}) ==> a^n \<noteq> 0" 
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154 
by force 
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155 

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text{*Perhaps these should be simprules.*} 
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157 
lemma power_inverse: 
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"inverse ((a::'a::{field,division_by_zero,ringpower}) ^ n) = (inverse a) ^ n" 
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159 
apply (induct_tac "n") 
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160 
apply (auto simp add: power_Suc inverse_mult_distrib) 
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161 
done 
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162 

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163 
lemma power_abs: "abs(a ^ n) = abs(a::'a::{ordered_field,ringpower}) ^ n" 
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164 
apply (induct_tac "n") 
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165 
apply (auto simp add: power_Suc abs_mult) 
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166 
done 
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167 

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168 
lemma power_minus: "(a) ^ n = ( 1)^n * (a::'a::{ring,ringpower}) ^ n" 
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169 
proof  
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have "a = ( 1) * a" by (simp add: minus_mult_left [symmetric]) 
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171 
thus ?thesis by (simp only: power_mult_distrib) 
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172 
qed 
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173 

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text{*Lemma for @{text power_strict_decreasing}*} 
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175 
lemma power_Suc_less: 
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176 
"[(0::'a::{ordered_semiring,ringpower}) < a; a < 1] 
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177 
==> a * a^n < a^n" 
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178 
apply (induct_tac n) 
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179 
apply (auto simp add: power_Suc mult_strict_left_mono) 
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180 
done 
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181 

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182 
lemma power_strict_decreasing: 
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183 
"[n < N; 0 < a; a < (1::'a::{ordered_semiring,ringpower})] 
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184 
==> a^N < a^n" 
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185 
apply (erule rev_mp) 
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186 
apply (induct_tac "N") 
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187 
apply (auto simp add: power_Suc power_Suc_less less_Suc_eq) 
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188 
apply (rename_tac m) 
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189 
apply (subgoal_tac "a * a^m < 1 * a^n", simp) 
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190 
apply (rule mult_strict_mono) 
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191 
apply (auto simp add: zero_le_power zero_less_one order_less_imp_le) 
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192 
done 
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193 

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text{*Proof resembles that of @{text power_strict_decreasing}*} 
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195 
lemma power_decreasing: 
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196 
"[n \<le> N; 0 \<le> a; a \<le> (1::'a::{ordered_semiring,ringpower})] 
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197 
==> a^N \<le> a^n" 
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198 
apply (erule rev_mp) 
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199 
apply (induct_tac "N") 
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200 
apply (auto simp add: power_Suc le_Suc_eq) 
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201 
apply (rename_tac m) 
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202 
apply (subgoal_tac "a * a^m \<le> 1 * a^n", simp) 
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203 
apply (rule mult_mono) 
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204 
apply (auto simp add: zero_le_power zero_le_one) 
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205 
done 
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206 

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207 
lemma power_Suc_less_one: 
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208 
"[ 0 < a; a < (1::'a::{ordered_semiring,ringpower}) ] ==> a ^ Suc n < 1" 
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209 
apply (insert power_strict_decreasing [of 0 "Suc n" a], simp) 
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210 
done 
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211 

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212 
text{*Proof again resembles that of @{text power_strict_decreasing}*} 
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213 
lemma power_increasing: 
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214 
"[n \<le> N; (1::'a::{ordered_semiring,ringpower}) \<le> a] ==> a^n \<le> a^N" 
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215 
apply (erule rev_mp) 
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216 
apply (induct_tac "N") 
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217 
apply (auto simp add: power_Suc le_Suc_eq) 
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218 
apply (rename_tac m) 
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219 
apply (subgoal_tac "1 * a^n \<le> a * a^m", simp) 
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220 
apply (rule mult_mono) 
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221 
apply (auto simp add: order_trans [OF zero_le_one] zero_le_power) 
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222 
done 
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223 

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224 
text{*Lemma for @{text power_strict_increasing}*} 
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225 
lemma power_less_power_Suc: 
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226 
"(1::'a::{ordered_semiring,ringpower}) < a ==> a^n < a * a^n" 
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227 
apply (induct_tac n) 
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228 
apply (auto simp add: power_Suc mult_strict_left_mono order_less_trans [OF zero_less_one]) 
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229 
done 
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230 

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231 
lemma power_strict_increasing: 
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232 
"[n < N; (1::'a::{ordered_semiring,ringpower}) < a] ==> a^n < a^N" 
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233 
apply (erule rev_mp) 
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234 
apply (induct_tac "N") 
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235 
apply (auto simp add: power_less_power_Suc power_Suc less_Suc_eq) 
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236 
apply (rename_tac m) 
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237 
apply (subgoal_tac "1 * a^n < a * a^m", simp) 
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238 
apply (rule mult_strict_mono) 
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239 
apply (auto simp add: order_less_trans [OF zero_less_one] zero_le_power 
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240 
order_less_imp_le) 
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241 
done 
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242 

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243 
lemma power_le_imp_le_base: 
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244 
assumes le: "a ^ Suc n \<le> b ^ Suc n" 
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245 
and xnonneg: "(0::'a::{ordered_semiring,ringpower}) \<le> a" 
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246 
and ynonneg: "0 \<le> b" 
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247 
shows "a \<le> b" 
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248 
proof (rule ccontr) 
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249 
assume "~ a \<le> b" 
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250 
then have "b < a" by (simp only: linorder_not_le) 
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251 
then have "b ^ Suc n < a ^ Suc n" 
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252 
by (simp only: prems power_strict_mono) 
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253 
from le and this show "False" 
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254 
by (simp add: linorder_not_less [symmetric]) 
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255 
qed 
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256 

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257 
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_semiring,ringpower})" 
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by (blast intro: power_le_imp_le_base order_antisym order_eq_refl sym) 
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261 

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262 

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263 
subsection{*Exponentiation for the Natural Numbers*} 
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264 

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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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268 

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instance nat :: ringpower 
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proof 
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fix z :: nat 
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fix 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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275 
qed 
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276 

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lemma nat_one_le_power [simp]: "1 \<le> i ==> Suc 0 \<le> i^n" 
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278 
by (insert one_le_power [of i n], simp) 
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279 

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lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n" 
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281 
apply (unfold dvd_def) 
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282 
apply (erule not_less_iff_le [THEN iffD2, THEN add_diff_inverse, THEN subst]) 
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283 
apply (simp add: power_add) 
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284 
done 
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285 

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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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289 
lemma nat_power_less_imp_less: "!!i::nat. [ 0 < i; i^m < i^n ] ==> m < n" 
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290 
apply (rule ccontr) 
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291 
apply (drule leI [THEN le_imp_power_dvd, THEN dvd_imp_le, THEN leD]) 
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292 
apply (erule zero_less_power, auto) 
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293 
done 
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294 

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295 
lemma nat_zero_less_power_iff [simp]: "(0 < x^n) = (x \<noteq> (0::nat)  n=0)" 
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296 
by (induct_tac "n", auto) 
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297 

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298 
lemma power_le_dvd [rule_format]: "k^j dvd n > i\<le>j > k^i dvd (n::nat)" 
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299 
apply (induct_tac "j") 
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300 
apply (simp_all add: le_Suc_eq) 
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301 
apply (blast dest!: dvd_mult_right) 
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302 
done 
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303 

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lemma power_dvd_imp_le: "[i^m dvd i^n; (1::nat) < i] ==> m \<le> n" 
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305 
apply (rule power_le_imp_le_exp, assumption) 
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306 
apply (erule dvd_imp_le, simp) 
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307 
done 
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308 

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309 

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310 
subsection{*Binomial Coefficients*} 
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311 

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text{*This development is based on the work of Andy Gordon and 
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313 
Florian Kammueller*} 
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314 

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consts 
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binomial :: "[nat,nat] => nat" (infixl "choose" 65) 
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317 

5183  318 
primrec 
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binomial_0: "(0 choose k) = (if k = 0 then 1 else 0)" 
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320 

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321 
binomial_Suc: "(Suc n choose k) = 
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322 
(if k = 0 then 1 else (n choose (k  1)) + (n choose k))" 
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323 

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324 
lemma binomial_n_0 [simp]: "(n choose 0) = 1" 
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325 
by (case_tac "n", simp_all) 
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326 

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327 
lemma binomial_0_Suc [simp]: "(0 choose Suc k) = 0" 
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328 
by simp 
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329 

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330 
lemma binomial_Suc_Suc [simp]: 
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331 
"(Suc n choose Suc k) = (n choose k) + (n choose Suc k)" 
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332 
by simp 
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333 

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334 
lemma binomial_eq_0 [rule_format]: "\<forall>k. n < k > (n choose k) = 0" 
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335 
apply (induct_tac "n", auto) 
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336 
apply (erule allE) 
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337 
apply (erule mp, arith) 
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338 
done 
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339 

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340 
declare binomial_0 [simp del] binomial_Suc [simp del] 
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341 

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342 
lemma binomial_n_n [simp]: "(n choose n) = 1" 
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343 
apply (induct_tac "n") 
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344 
apply (simp_all add: binomial_eq_0) 
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345 
done 
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346 

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347 
lemma binomial_Suc_n [simp]: "(Suc n choose n) = Suc n" 
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348 
by (induct_tac "n", simp_all) 
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349 

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350 
lemma binomial_1 [simp]: "(n choose Suc 0) = n" 
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351 
by (induct_tac "n", simp_all) 
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352 

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353 
lemma zero_less_binomial [rule_format]: "k \<le> n > 0 < (n choose k)" 
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354 
by (rule_tac m = n and n = k in diff_induct, simp_all) 
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355 

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356 
lemma binomial_eq_0_iff: "(n choose k = 0) = (n<k)" 
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357 
apply (safe intro!: binomial_eq_0) 
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358 
apply (erule contrapos_pp) 
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359 
apply (simp add: zero_less_binomial) 
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360 
done 
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361 

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362 
lemma zero_less_binomial_iff: "(0 < n choose k) = (k\<le>n)" 
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363 
by (simp add: linorder_not_less [symmetric] binomial_eq_0_iff [symmetric]) 
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364 

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365 
(*Might be more useful if reoriented*) 
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366 
lemma Suc_times_binomial_eq [rule_format]: 
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367 
"\<forall>k. k \<le> n > Suc n * (n choose k) = (Suc n choose Suc k) * Suc k" 
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368 
apply (induct_tac "n") 
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369 
apply (simp add: binomial_0, clarify) 
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370 
apply (case_tac "k") 
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371 
apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq 
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372 
binomial_eq_0) 
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373 
done 
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374 

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375 
text{*This is the wellknown version, but it's harder to use because of the 
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376 
need to reason about division.*} 
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377 
lemma binomial_Suc_Suc_eq_times: 
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378 
"k \<le> n ==> (Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k" 
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379 
by (simp add: Suc_times_binomial_eq div_mult_self_is_m zero_less_Suc 
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380 
del: mult_Suc mult_Suc_right) 
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381 

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382 
text{*Another version, with 1 instead of Suc.*} 
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383 
lemma times_binomial_minus1_eq: 
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384 
"[k \<le> n; 0<k] ==> (n choose k) * k = n * ((n  1) choose (k  1))" 
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385 
apply (cut_tac n = "n  1" and k = "k  1" in Suc_times_binomial_eq) 
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386 
apply (simp split add: nat_diff_split, auto) 
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387 
done 
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388 

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389 
text{*ML bindings for the general exponentiation theorems*} 
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390 
ML 
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391 
{* 
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392 
val power_0 = thm"power_0"; 
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393 
val power_Suc = thm"power_Suc"; 
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394 
val power_0_Suc = thm"power_0_Suc"; 
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395 
val power_0_left = thm"power_0_left"; 
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396 
val power_one = thm"power_one"; 
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397 
val power_one_right = thm"power_one_right"; 
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398 
val power_add = thm"power_add"; 
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399 
val power_mult = thm"power_mult"; 
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400 
val power_mult_distrib = thm"power_mult_distrib"; 
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401 
val zero_less_power = thm"zero_less_power"; 
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402 
val zero_le_power = thm"zero_le_power"; 
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403 
val one_le_power = thm"one_le_power"; 
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404 
val gt1_imp_ge0 = thm"gt1_imp_ge0"; 
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405 
val power_gt1_lemma = thm"power_gt1_lemma"; 
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406 
val power_gt1 = thm"power_gt1"; 
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407 
val power_le_imp_le_exp = thm"power_le_imp_le_exp"; 
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408 
val power_inject_exp = thm"power_inject_exp"; 
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409 
val power_less_imp_less_exp = thm"power_less_imp_less_exp"; 
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410 
val power_mono = thm"power_mono"; 
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411 
val power_strict_mono = thm"power_strict_mono"; 
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412 
val power_eq_0_iff = thm"power_eq_0_iff"; 
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413 
val field_power_eq_0_iff = thm"field_power_eq_0_iff"; 
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414 
val field_power_not_zero = thm"field_power_not_zero"; 
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415 
val power_inverse = thm"power_inverse"; 
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416 
val power_abs = thm"power_abs"; 
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417 
val power_minus = thm"power_minus"; 
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418 
val power_Suc_less = thm"power_Suc_less"; 
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419 
val power_strict_decreasing = thm"power_strict_decreasing"; 
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420 
val power_decreasing = thm"power_decreasing"; 
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421 
val power_Suc_less_one = thm"power_Suc_less_one"; 
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422 
val power_increasing = thm"power_increasing"; 
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423 
val power_strict_increasing = thm"power_strict_increasing"; 
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424 
val power_le_imp_le_base = thm"power_le_imp_le_base"; 
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425 
val power_inject_base = thm"power_inject_base"; 
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426 
*} 
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427 

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428 
text{*ML bindings for the remaining theorems*} 
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429 
ML 
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430 
{* 
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431 
val nat_one_le_power = thm"nat_one_le_power"; 
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432 
val le_imp_power_dvd = thm"le_imp_power_dvd"; 
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433 
val nat_power_less_imp_less = thm"nat_power_less_imp_less"; 
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434 
val nat_zero_less_power_iff = thm"nat_zero_less_power_iff"; 
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435 
val power_le_dvd = thm"power_le_dvd"; 
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436 
val power_dvd_imp_le = thm"power_dvd_imp_le"; 
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437 
val binomial_n_0 = thm"binomial_n_0"; 
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438 
val binomial_0_Suc = thm"binomial_0_Suc"; 
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439 
val binomial_Suc_Suc = thm"binomial_Suc_Suc"; 
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440 
val binomial_eq_0 = thm"binomial_eq_0"; 
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441 
val binomial_n_n = thm"binomial_n_n"; 
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442 
val binomial_Suc_n = thm"binomial_Suc_n"; 
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443 
val binomial_1 = thm"binomial_1"; 
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444 
val zero_less_binomial = thm"zero_less_binomial"; 
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445 
val binomial_eq_0_iff = thm"binomial_eq_0_iff"; 
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446 
val zero_less_binomial_iff = thm"zero_less_binomial_iff"; 
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447 
val Suc_times_binomial_eq = thm"Suc_times_binomial_eq"; 
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448 
val binomial_Suc_Suc_eq_times = thm"binomial_Suc_Suc_eq_times"; 
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449 
val times_binomial_minus1_eq = thm"times_binomial_minus1_eq"; 
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450 
*} 
3390
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New theory "Power" of exponentiation (and binomial coefficients)
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451 

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452 
end 
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453 