author | huffman |
Mon, 11 Jun 2007 02:24:39 +0200 | |
changeset 23305 | 8ae6f7b0903b |
parent 23183 | af27d3ad9baf |
child 23431 | 25ca91279a9b |
permissions | -rw-r--r-- |
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New theory "Power" of exponentiation (and binomial coefficients)
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(* Title: HOL/Power.thy |
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New theory "Power" of exponentiation (and binomial coefficients)
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ID: $Id$ |
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New theory "Power" of exponentiation (and binomial coefficients)
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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New theory "Power" of exponentiation (and binomial coefficients)
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Copyright 1997 University of Cambridge |
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*) |
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linear arithmetic now takes "&" in assumptions apart.
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header{*Exponentiation*} |
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theory Power |
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imports Nat |
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begin |
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subsection{*Powers for Arbitrary Monoids*} |
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class recpower = monoid_mult + power + |
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assumes power_0 [simp]: "a \<^loc>^ 0 = \<^loc>1" |
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assumes power_Suc: "a \<^loc>^ Suc n = a \<^loc>* (a \<^loc>^ n)" |
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lemma power_0_Suc [simp]: "(0::'a::{recpower,semiring_0}) ^ (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,semiring_0}))" |
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by (induct n) simp_all |
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lemma power_one [simp]: "1^n = (1::'a::recpower)" |
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by (induct n) (simp_all add: power_Suc) |
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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_commutes: "(a::'a::recpower) ^ n * a = a * a ^ n" |
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by (induct n) (simp_all add: power_Suc mult_assoc) |
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lemma power_add: "(a::'a::recpower) ^ (m+n) = (a^m) * (a^n)" |
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by (induct m) (simp_all add: power_Suc mult_ac) |
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lemma power_mult: "(a::'a::recpower) ^ (m*n) = (a^m) ^ n" |
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by (induct n) (simp_all add: power_Suc power_add) |
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lemma power_mult_distrib: "((a::'a::{recpower,comm_monoid_mult}) * b) ^ n = (a^n) * (b^n)" |
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by (induct n) (simp_all add: power_Suc mult_ac) |
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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 "n") |
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apply (simp_all add: power_Suc zero_less_one mult_pos_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 "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 "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 "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::{dom,recpower}) & 0<n)" |
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apply (induct "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: |
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"(a^n = 0) = (a = (0::'a::{division_ring,recpower}) & 0<n)" |
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by simp (* TODO: delete *) |
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lemma field_power_not_zero: "a \<noteq> (0::'a::{dom,recpower}) ==> a^n \<noteq> 0" |
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by force |
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lemma nonzero_power_inverse: |
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fixes a :: "'a::{division_ring,recpower}" |
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shows "a \<noteq> 0 ==> inverse (a ^ n) = (inverse a) ^ n" |
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apply (induct "n") |
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apply (auto simp add: power_Suc nonzero_inverse_mult_distrib power_commutes) |
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done (* TODO: reorient or rename to nonzero_inverse_power *) |
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text{*Perhaps these should be simprules.*} |
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lemma power_inverse: |
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fixes a :: "'a::{division_ring,division_by_zero,recpower}" |
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shows "inverse (a ^ n) = (inverse a) ^ n" |
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apply (cases "a = 0") |
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apply (simp add: power_0_left) |
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apply (simp add: nonzero_power_inverse) |
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done (* TODO: reorient or rename to inverse_power *) |
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lemma power_one_over: "1 / (a::'a::{field,division_by_zero,recpower})^n = |
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(1 / a)^n" |
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apply (simp add: divide_inverse) |
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apply (rule power_inverse) |
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168 |
done |
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169 |
|
14577 | 170 |
lemma nonzero_power_divide: |
15004 | 171 |
"b \<noteq> 0 ==> (a/b) ^ n = ((a::'a::{field,recpower}) ^ n) / (b ^ n)" |
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172 |
by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse) |
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173 |
|
14577 | 174 |
lemma power_divide: |
15004 | 175 |
"(a/b) ^ n = ((a::'a::{field,division_by_zero,recpower}) ^ n / b ^ n)" |
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176 |
apply (case_tac "b=0", simp add: power_0_left) |
14577 | 177 |
apply (rule nonzero_power_divide) |
178 |
apply assumption |
|
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179 |
done |
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180 |
|
15004 | 181 |
lemma power_abs: "abs(a ^ n) = abs(a::'a::{ordered_idom,recpower}) ^ n" |
15251 | 182 |
apply (induct "n") |
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183 |
apply (auto simp add: power_Suc abs_mult) |
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184 |
done |
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185 |
|
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186 |
lemma zero_less_power_abs_iff [simp]: |
15004 | 187 |
"(0 < (abs a)^n) = (a \<noteq> (0::'a::{ordered_idom,recpower}) | n=0)" |
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188 |
proof (induct "n") |
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189 |
case 0 |
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190 |
show ?case by (simp add: zero_less_one) |
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191 |
next |
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192 |
case (Suc n) |
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193 |
show ?case by (force simp add: prems power_Suc zero_less_mult_iff) |
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194 |
qed |
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195 |
|
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196 |
lemma zero_le_power_abs [simp]: |
15004 | 197 |
"(0::'a::{ordered_idom,recpower}) \<le> (abs a)^n" |
22957 | 198 |
by (rule zero_le_power [OF abs_ge_zero]) |
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199 |
|
15004 | 200 |
lemma power_minus: "(-a) ^ n = (- 1)^n * (a::'a::{comm_ring_1,recpower}) ^ n" |
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201 |
proof - |
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202 |
have "-a = (- 1) * a" by (simp add: minus_mult_left [symmetric]) |
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203 |
thus ?thesis by (simp only: power_mult_distrib) |
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204 |
qed |
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205 |
|
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206 |
text{*Lemma for @{text power_strict_decreasing}*} |
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207 |
lemma power_Suc_less: |
15004 | 208 |
"[|(0::'a::{ordered_semidom,recpower}) < a; a < 1|] |
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209 |
==> a * a^n < a^n" |
15251 | 210 |
apply (induct n) |
14577 | 211 |
apply (auto simp add: power_Suc mult_strict_left_mono) |
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212 |
done |
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213 |
|
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214 |
lemma power_strict_decreasing: |
15004 | 215 |
"[|n < N; 0 < a; a < (1::'a::{ordered_semidom,recpower})|] |
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216 |
==> a^N < a^n" |
14577 | 217 |
apply (erule rev_mp) |
15251 | 218 |
apply (induct "N") |
14577 | 219 |
apply (auto simp add: power_Suc power_Suc_less less_Suc_eq) |
220 |
apply (rename_tac m) |
|
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221 |
apply (subgoal_tac "a * a^m < 1 * a^n", simp) |
14577 | 222 |
apply (rule mult_strict_mono) |
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223 |
apply (auto simp add: zero_le_power zero_less_one order_less_imp_le) |
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224 |
done |
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225 |
|
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226 |
text{*Proof resembles that of @{text power_strict_decreasing}*} |
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227 |
lemma power_decreasing: |
15004 | 228 |
"[|n \<le> N; 0 \<le> a; a \<le> (1::'a::{ordered_semidom,recpower})|] |
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229 |
==> a^N \<le> a^n" |
14577 | 230 |
apply (erule rev_mp) |
15251 | 231 |
apply (induct "N") |
14577 | 232 |
apply (auto simp add: power_Suc le_Suc_eq) |
233 |
apply (rename_tac m) |
|
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234 |
apply (subgoal_tac "a * a^m \<le> 1 * a^n", simp) |
14577 | 235 |
apply (rule mult_mono) |
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236 |
apply (auto simp add: zero_le_power zero_le_one) |
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237 |
done |
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238 |
|
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239 |
lemma power_Suc_less_one: |
15004 | 240 |
"[| 0 < a; a < (1::'a::{ordered_semidom,recpower}) |] ==> a ^ Suc n < 1" |
14577 | 241 |
apply (insert power_strict_decreasing [of 0 "Suc n" a], simp) |
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242 |
done |
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243 |
|
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244 |
text{*Proof again resembles that of @{text power_strict_decreasing}*} |
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245 |
lemma power_increasing: |
15004 | 246 |
"[|n \<le> N; (1::'a::{ordered_semidom,recpower}) \<le> a|] ==> a^n \<le> a^N" |
14577 | 247 |
apply (erule rev_mp) |
15251 | 248 |
apply (induct "N") |
14577 | 249 |
apply (auto simp add: power_Suc le_Suc_eq) |
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250 |
apply (rename_tac m) |
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251 |
apply (subgoal_tac "1 * a^n \<le> a * a^m", simp) |
14577 | 252 |
apply (rule mult_mono) |
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253 |
apply (auto simp add: order_trans [OF zero_le_one] zero_le_power) |
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254 |
done |
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255 |
|
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256 |
text{*Lemma for @{text power_strict_increasing}*} |
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257 |
lemma power_less_power_Suc: |
15004 | 258 |
"(1::'a::{ordered_semidom,recpower}) < a ==> a^n < a * a^n" |
15251 | 259 |
apply (induct n) |
14577 | 260 |
apply (auto simp add: power_Suc mult_strict_left_mono order_less_trans [OF zero_less_one]) |
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261 |
done |
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262 |
|
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263 |
lemma power_strict_increasing: |
15004 | 264 |
"[|n < N; (1::'a::{ordered_semidom,recpower}) < a|] ==> a^n < a^N" |
14577 | 265 |
apply (erule rev_mp) |
15251 | 266 |
apply (induct "N") |
14577 | 267 |
apply (auto simp add: power_less_power_Suc power_Suc less_Suc_eq) |
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268 |
apply (rename_tac m) |
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269 |
apply (subgoal_tac "1 * a^n < a * a^m", simp) |
14577 | 270 |
apply (rule mult_strict_mono) |
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271 |
apply (auto simp add: order_less_trans [OF zero_less_one] zero_le_power |
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272 |
order_less_imp_le) |
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273 |
done |
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274 |
|
15066 | 275 |
lemma power_increasing_iff [simp]: |
276 |
"1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x \<le> b ^ y) = (x \<le> y)" |
|
277 |
by (blast intro: power_le_imp_le_exp power_increasing order_less_imp_le) |
|
278 |
||
279 |
lemma power_strict_increasing_iff [simp]: |
|
280 |
"1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x < b ^ y) = (x < y)" |
|
281 |
by (blast intro: power_less_imp_less_exp power_strict_increasing) |
|
282 |
||
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283 |
lemma power_le_imp_le_base: |
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284 |
assumes le: "a ^ Suc n \<le> b ^ Suc n" |
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285 |
and ynonneg: "(0::'a::{ordered_semidom,recpower}) \<le> b" |
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286 |
shows "a \<le> b" |
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287 |
proof (rule ccontr) |
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288 |
assume "~ a \<le> b" |
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289 |
then have "b < a" by (simp only: linorder_not_le) |
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290 |
then have "b ^ Suc n < a ^ Suc n" |
14577 | 291 |
by (simp only: prems power_strict_mono) |
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292 |
from le and this show "False" |
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293 |
by (simp add: linorder_not_less [symmetric]) |
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294 |
qed |
14577 | 295 |
|
22853 | 296 |
lemma power_less_imp_less_base: |
297 |
fixes a b :: "'a::{ordered_semidom,recpower}" |
|
298 |
assumes less: "a ^ n < b ^ n" |
|
299 |
assumes nonneg: "0 \<le> b" |
|
300 |
shows "a < b" |
|
301 |
proof (rule contrapos_pp [OF less]) |
|
302 |
assume "~ a < b" |
|
303 |
hence "b \<le> a" by (simp only: linorder_not_less) |
|
304 |
hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono) |
|
305 |
thus "~ a ^ n < b ^ n" by (simp only: linorder_not_less) |
|
306 |
qed |
|
307 |
||
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308 |
lemma power_inject_base: |
14577 | 309 |
"[| a ^ Suc n = b ^ Suc n; 0 \<le> a; 0 \<le> b |] |
15004 | 310 |
==> a = (b::'a::{ordered_semidom,recpower})" |
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311 |
by (blast intro: power_le_imp_le_base order_antisym order_eq_refl sym) |
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312 |
|
22955 | 313 |
lemma power_eq_imp_eq_base: |
314 |
fixes a b :: "'a::{ordered_semidom,recpower}" |
|
315 |
shows "\<lbrakk>a ^ n = b ^ n; 0 \<le> a; 0 \<le> b; 0 < n\<rbrakk> \<Longrightarrow> a = b" |
|
316 |
by (cases n, simp_all, rule power_inject_base) |
|
317 |
||
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318 |
|
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319 |
subsection{*Exponentiation for the Natural Numbers*} |
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320 |
|
21456 | 321 |
instance nat :: power .. |
322 |
||
8844 | 323 |
primrec (power) |
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324 |
"p ^ 0 = 1" |
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325 |
"p ^ (Suc n) = (p::nat) * (p ^ n)" |
14577 | 326 |
|
15004 | 327 |
instance nat :: recpower |
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328 |
proof |
14438 | 329 |
fix z n :: nat |
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330 |
show "z^0 = 1" by simp |
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331 |
show "z^(Suc n) = z * (z^n)" by simp |
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332 |
qed |
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333 |
|
23305 | 334 |
lemma of_nat_power: |
335 |
"of_nat (m ^ n) = (of_nat m::'a::{semiring_1,recpower}) ^ n" |
|
336 |
by (induct n, simp_all add: power_Suc) |
|
337 |
||
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338 |
lemma nat_one_le_power [simp]: "1 \<le> i ==> Suc 0 \<le> i^n" |
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339 |
by (insert one_le_power [of i n], simp) |
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340 |
|
21413 | 341 |
lemma nat_zero_less_power_iff [simp]: "(0 < x^n) = (x \<noteq> (0::nat) | n=0)" |
342 |
by (induct "n", auto) |
|
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343 |
|
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|
344 |
text{*Valid for the naturals, but what if @{text"0<i<1"}? |
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345 |
Premises cannot be weakened: consider the case where @{term "i=0"}, |
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346 |
@{term "m=1"} and @{term "n=0"}.*} |
21413 | 347 |
lemma nat_power_less_imp_less: |
348 |
assumes nonneg: "0 < (i\<Colon>nat)" |
|
349 |
assumes less: "i^m < i^n" |
|
350 |
shows "m < n" |
|
351 |
proof (cases "i = 1") |
|
352 |
case True with less power_one [where 'a = nat] show ?thesis by simp |
|
353 |
next |
|
354 |
case False with nonneg have "1 < i" by auto |
|
355 |
from power_strict_increasing_iff [OF this] less show ?thesis .. |
|
356 |
qed |
|
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|
357 |
|
17149
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
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|
358 |
lemma power_diff: |
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|
359 |
assumes nz: "a ~= 0" |
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|
360 |
shows "n <= m ==> (a::'a::{recpower, field}) ^ (m-n) = (a^m) / (a^n)" |
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Lemmas on dvd, power and finite summation added or strengthened.
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changeset
|
361 |
by (induct m n rule: diff_induct) |
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parents:
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diff
changeset
|
362 |
(simp_all add: power_Suc nonzero_mult_divide_cancel_left nz) |
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Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
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|
363 |
|
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Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
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changeset
|
364 |
|
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Defining the type class "ringpower" and deleting superseded theorems for
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|
365 |
text{*ML bindings for the general exponentiation theorems*} |
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|
366 |
ML |
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|
367 |
{* |
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Defining the type class "ringpower" and deleting superseded theorems for
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|
368 |
val power_0 = thm"power_0"; |
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|
369 |
val power_Suc = thm"power_Suc"; |
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|
370 |
val power_0_Suc = thm"power_0_Suc"; |
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Defining the type class "ringpower" and deleting superseded theorems for
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371 |
val power_0_left = thm"power_0_left"; |
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|
372 |
val power_one = thm"power_one"; |
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|
373 |
val power_one_right = thm"power_one_right"; |
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|
374 |
val power_add = thm"power_add"; |
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|
375 |
val power_mult = thm"power_mult"; |
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|
376 |
val power_mult_distrib = thm"power_mult_distrib"; |
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|
377 |
val zero_less_power = thm"zero_less_power"; |
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|
378 |
val zero_le_power = thm"zero_le_power"; |
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|
379 |
val one_le_power = thm"one_le_power"; |
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|
380 |
val gt1_imp_ge0 = thm"gt1_imp_ge0"; |
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|
381 |
val power_gt1_lemma = thm"power_gt1_lemma"; |
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|
382 |
val power_gt1 = thm"power_gt1"; |
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Defining the type class "ringpower" and deleting superseded theorems for
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parents:
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|
383 |
val power_le_imp_le_exp = thm"power_le_imp_le_exp"; |
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|
384 |
val power_inject_exp = thm"power_inject_exp"; |
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Defining the type class "ringpower" and deleting superseded theorems for
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changeset
|
385 |
val power_less_imp_less_exp = thm"power_less_imp_less_exp"; |
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|
386 |
val power_mono = thm"power_mono"; |
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|
387 |
val power_strict_mono = thm"power_strict_mono"; |
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|
388 |
val power_eq_0_iff = thm"power_eq_0_iff"; |
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|
389 |
val field_power_eq_0_iff = thm"field_power_eq_0_iff"; |
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|
390 |
val field_power_not_zero = thm"field_power_not_zero"; |
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|
391 |
val power_inverse = thm"power_inverse"; |
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|
392 |
val nonzero_power_divide = thm"nonzero_power_divide"; |
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Added lemmas to Ring_and_Field with slightly modified simplification rules
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changeset
|
393 |
val power_divide = thm"power_divide"; |
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|
394 |
val power_abs = thm"power_abs"; |
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changeset
|
395 |
val zero_less_power_abs_iff = thm"zero_less_power_abs_iff"; |
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Added lemmas to Ring_and_Field with slightly modified simplification rules
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parents:
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changeset
|
396 |
val zero_le_power_abs = thm "zero_le_power_abs"; |
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|
397 |
val power_minus = thm"power_minus"; |
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changeset
|
398 |
val power_Suc_less = thm"power_Suc_less"; |
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Defining the type class "ringpower" and deleting superseded theorems for
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changeset
|
399 |
val power_strict_decreasing = thm"power_strict_decreasing"; |
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parents:
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changeset
|
400 |
val power_decreasing = thm"power_decreasing"; |
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Defining the type class "ringpower" and deleting superseded theorems for
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parents:
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diff
changeset
|
401 |
val power_Suc_less_one = thm"power_Suc_less_one"; |
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parents:
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changeset
|
402 |
val power_increasing = thm"power_increasing"; |
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Defining the type class "ringpower" and deleting superseded theorems for
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parents:
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changeset
|
403 |
val power_strict_increasing = thm"power_strict_increasing"; |
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Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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diff
changeset
|
404 |
val power_le_imp_le_base = thm"power_le_imp_le_base"; |
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Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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diff
changeset
|
405 |
val power_inject_base = thm"power_inject_base"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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changeset
|
406 |
*} |
14577 | 407 |
|
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|
408 |
text{*ML bindings for the remaining theorems*} |
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Defining the type class "ringpower" and deleting superseded theorems for
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parents:
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changeset
|
409 |
ML |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset
|
410 |
{* |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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changeset
|
411 |
val nat_one_le_power = thm"nat_one_le_power"; |
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Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
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changeset
|
412 |
val nat_power_less_imp_less = thm"nat_power_less_imp_less"; |
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Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
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changeset
|
413 |
val nat_zero_less_power_iff = thm"nat_zero_less_power_iff"; |
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Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
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changeset
|
414 |
*} |
3390
0c7625196d95
New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff
changeset
|
415 |
|
0c7625196d95
New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff
changeset
|
416 |
end |
0c7625196d95
New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff
changeset
|
417 |