paulson@3390: (*  Title:      HOL/Power.thy
paulson@3390:     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
paulson@3390:     Copyright   1997  University of Cambridge
paulson@3390: *)
paulson@3390: 
haftmann@30960: header {* Exponentiation *}
paulson@14348: 
nipkow@15131: theory Power
haftmann@21413: imports Nat
nipkow@15131: begin
paulson@14348: 
haftmann@30960: subsection {* Powers for Arbitrary Monoids *}
haftmann@30960: 
haftmann@30996: class power = one + times
haftmann@30960: begin
haftmann@24996: 
haftmann@30960: primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80) where
haftmann@30960:     power_0: "a ^ 0 = 1"
haftmann@30960:   | power_Suc: "a ^ Suc n = a * a ^ n"
paulson@14348: 
haftmann@30996: notation (latex output)
haftmann@30996:   power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
haftmann@30996: 
haftmann@30996: notation (HTML output)
haftmann@30996:   power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
haftmann@30996: 
haftmann@30960: end
paulson@14348: 
haftmann@30996: context monoid_mult
haftmann@30996: begin
paulson@14348: 
haftmann@30996: subclass power ..
paulson@14348: 
haftmann@30996: lemma power_one [simp]:
haftmann@30996:   "1 ^ n = 1"
huffman@30273:   by (induct n) simp_all
paulson@14348: 
haftmann@30996: lemma power_one_right [simp]:
haftmann@30996:   "a ^ 1 = a * 1"
haftmann@30996:   by simp
paulson@14348: 
haftmann@30996: lemma power_commutes:
haftmann@30996:   "a ^ n * a = a * a ^ n"
huffman@30273:   by (induct n) (simp_all add: mult_assoc)
krauss@21199: 
haftmann@30996: lemma power_Suc2:
haftmann@30996:   "a ^ Suc n = a ^ n * a"
huffman@30273:   by (simp add: power_commutes)
huffman@28131: 
haftmann@30996: lemma power_add:
haftmann@30996:   "a ^ (m + n) = a ^ m * a ^ n"
haftmann@30996:   by (induct m) (simp_all add: algebra_simps)
paulson@14348: 
haftmann@30996: lemma power_mult:
haftmann@30996:   "a ^ (m * n) = (a ^ m) ^ n"
huffman@30273:   by (induct n) (simp_all add: power_add)
paulson@14348: 
haftmann@30996: end
haftmann@30996: 
haftmann@30996: context comm_monoid_mult
haftmann@30996: begin
haftmann@30996: 
haftmann@30996: lemma power_mult_distrib:
haftmann@30996:   "(a * b) ^ n = (a ^ n) * (b ^ n)"
huffman@30273:   by (induct n) (simp_all add: mult_ac)
paulson@14348: 
haftmann@30996: end
haftmann@30996: 
haftmann@30996: context semiring_1
haftmann@30996: begin
haftmann@30996: 
haftmann@30996: lemma of_nat_power:
haftmann@30996:   "of_nat (m ^ n) = of_nat m ^ n"
haftmann@30996:   by (induct n) (simp_all add: of_nat_mult)
haftmann@30996: 
haftmann@30996: end
haftmann@30996: 
haftmann@30996: context comm_semiring_1
haftmann@30996: begin
haftmann@30996: 
haftmann@30996: text {* The divides relation *}
haftmann@30996: 
haftmann@30996: lemma le_imp_power_dvd:
haftmann@30996:   assumes "m \<le> n" shows "a ^ m dvd a ^ n"
haftmann@30996: proof
haftmann@30996:   have "a ^ n = a ^ (m + (n - m))"
haftmann@30996:     using `m \<le> n` by simp
haftmann@30996:   also have "\<dots> = a ^ m * a ^ (n - m)"
haftmann@30996:     by (rule power_add)
haftmann@30996:   finally show "a ^ n = a ^ m * a ^ (n - m)" .
haftmann@30996: qed
haftmann@30996: 
haftmann@30996: lemma power_le_dvd:
haftmann@30996:   "a ^ n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a ^ m dvd b"
haftmann@30996:   by (rule dvd_trans [OF le_imp_power_dvd])
haftmann@30996: 
haftmann@30996: lemma dvd_power_same:
haftmann@30996:   "x dvd y \<Longrightarrow> x ^ n dvd y ^ n"
haftmann@30996:   by (induct n) (auto simp add: mult_dvd_mono)
haftmann@30996: 
haftmann@30996: lemma dvd_power_le:
haftmann@30996:   "x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m"
haftmann@30996:   by (rule power_le_dvd [OF dvd_power_same])
paulson@14348: 
haftmann@30996: lemma dvd_power [simp]:
haftmann@30996:   assumes "n > (0::nat) \<or> x = 1"
haftmann@30996:   shows "x dvd (x ^ n)"
haftmann@30996: using assms proof
haftmann@30996:   assume "0 < n"
haftmann@30996:   then have "x ^ n = x ^ Suc (n - 1)" by simp
haftmann@30996:   then show "x dvd (x ^ n)" by simp
haftmann@30996: next
haftmann@30996:   assume "x = 1"
haftmann@30996:   then show "x dvd (x ^ n)" by simp
haftmann@30996: qed
haftmann@30996: 
haftmann@30996: end
haftmann@30996: 
haftmann@30996: context ring_1
haftmann@30996: begin
haftmann@30996: 
haftmann@30996: lemma power_minus:
haftmann@30996:   "(- a) ^ n = (- 1) ^ n * a ^ n"
haftmann@30996: proof (induct n)
haftmann@30996:   case 0 show ?case by simp
haftmann@30996: next
haftmann@30996:   case (Suc n) then show ?case
haftmann@30996:     by (simp del: power_Suc add: power_Suc2 mult_assoc)
haftmann@30996: qed
haftmann@30996: 
haftmann@30996: end
haftmann@30996: 
haftmann@30996: context ordered_semidom
haftmann@30996: begin
haftmann@30996: 
haftmann@30996: lemma zero_less_power [simp]:
haftmann@30996:   "0 < a \<Longrightarrow> 0 < a ^ n"
haftmann@30996:   by (induct n) (simp_all add: mult_pos_pos)
haftmann@30996: 
haftmann@30996: lemma zero_le_power [simp]:
haftmann@30996:   "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
haftmann@30996:   by (induct n) (simp_all add: mult_nonneg_nonneg)
paulson@14348: 
nipkow@25874: lemma one_le_power[simp]:
haftmann@30996:   "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
haftmann@30996:   apply (induct n)
haftmann@30996:   apply simp_all
haftmann@30996:   apply (rule order_trans [OF _ mult_mono [of 1 _ 1]])
haftmann@30996:   apply (simp_all add: order_trans [OF zero_le_one])
haftmann@30996:   done
paulson@14348: 
paulson@14348: lemma power_gt1_lemma:
haftmann@30996:   assumes gt1: "1 < a"
haftmann@30996:   shows "1 < a * a ^ n"
paulson@14348: proof -
haftmann@30996:   from gt1 have "0 \<le> a"
haftmann@30996:     by (fact order_trans [OF zero_le_one less_imp_le])
haftmann@30996:   have "1 * 1 < a * 1" using gt1 by simp
haftmann@30996:   also have "\<dots> \<le> a * a ^ n" using gt1
haftmann@30996:     by (simp only: mult_mono `0 \<le> a` one_le_power order_less_imp_le
wenzelm@14577:         zero_le_one order_refl)
wenzelm@14577:   finally show ?thesis by simp
paulson@14348: qed
paulson@14348: 
haftmann@30996: lemma power_gt1:
haftmann@30996:   "1 < a \<Longrightarrow> 1 < a ^ Suc n"
haftmann@30996:   by (simp add: power_gt1_lemma)
huffman@24376: 
haftmann@30996: lemma one_less_power [simp]:
haftmann@30996:   "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
haftmann@30996:   by (cases n) (simp_all add: power_gt1_lemma)
paulson@14348: 
paulson@14348: lemma power_le_imp_le_exp:
haftmann@30996:   assumes gt1: "1 < a"
haftmann@30996:   shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
haftmann@30996: proof (induct m arbitrary: n)
paulson@14348:   case 0
wenzelm@14577:   show ?case by simp
paulson@14348: next
paulson@14348:   case (Suc m)
wenzelm@14577:   show ?case
wenzelm@14577:   proof (cases n)
wenzelm@14577:     case 0
haftmann@30996:     with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
wenzelm@14577:     with gt1 show ?thesis
wenzelm@14577:       by (force simp only: power_gt1_lemma
haftmann@30996:           not_less [symmetric])
wenzelm@14577:   next
wenzelm@14577:     case (Suc n)
haftmann@30996:     with Suc.prems Suc.hyps show ?thesis
wenzelm@14577:       by (force dest: mult_left_le_imp_le
haftmann@30996:           simp add: less_trans [OF zero_less_one gt1])
wenzelm@14577:   qed
paulson@14348: qed
paulson@14348: 
wenzelm@14577: text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}
paulson@14348: lemma power_inject_exp [simp]:
haftmann@30996:   "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
wenzelm@14577:   by (force simp add: order_antisym power_le_imp_le_exp)
paulson@14348: 
paulson@14348: text{*Can relax the first premise to @{term "0<a"} in the case of the
paulson@14348: natural numbers.*}
paulson@14348: lemma power_less_imp_less_exp:
haftmann@30996:   "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
haftmann@30996:   by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"]
haftmann@30996:     power_le_imp_le_exp)
paulson@14348: 
paulson@14348: lemma power_mono:
haftmann@30996:   "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
haftmann@30996:   by (induct n)
haftmann@30996:     (auto intro: mult_mono order_trans [of 0 a b])
paulson@14348: 
paulson@14348: lemma power_strict_mono [rule_format]:
haftmann@30996:   "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
haftmann@30996:   by (induct n)
haftmann@30996:    (auto simp add: mult_strict_mono le_less_trans [of 0 a b])
paulson@14348: 
paulson@14348: text{*Lemma for @{text power_strict_decreasing}*}
paulson@14348: lemma power_Suc_less:
haftmann@30996:   "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
haftmann@30996:   by (induct n)
haftmann@30996:     (auto simp add: mult_strict_left_mono)
paulson@14348: 
haftmann@30996: lemma power_strict_decreasing [rule_format]:
haftmann@30996:   "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
haftmann@30996: proof (induct N)
haftmann@30996:   case 0 then show ?case by simp
haftmann@30996: next
haftmann@30996:   case (Suc N) then show ?case 
haftmann@30996:   apply (auto simp add: power_Suc_less less_Suc_eq)
haftmann@30996:   apply (subgoal_tac "a * a^N < 1 * a^n")
haftmann@30996:   apply simp
haftmann@30996:   apply (rule mult_strict_mono) apply auto
haftmann@30996:   done
haftmann@30996: qed
paulson@14348: 
paulson@14348: text{*Proof resembles that of @{text power_strict_decreasing}*}
haftmann@30996: lemma power_decreasing [rule_format]:
haftmann@30996:   "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n"
haftmann@30996: proof (induct N)
haftmann@30996:   case 0 then show ?case by simp
haftmann@30996: next
haftmann@30996:   case (Suc N) then show ?case 
haftmann@30996:   apply (auto simp add: le_Suc_eq)
haftmann@30996:   apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp)
haftmann@30996:   apply (rule mult_mono) apply auto
haftmann@30996:   done
haftmann@30996: qed
paulson@14348: 
paulson@14348: lemma power_Suc_less_one:
haftmann@30996:   "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
haftmann@30996:   using power_strict_decreasing [of 0 "Suc n" a] by simp
paulson@14348: 
paulson@14348: text{*Proof again resembles that of @{text power_strict_decreasing}*}
haftmann@30996: lemma power_increasing [rule_format]:
haftmann@30996:   "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
haftmann@30996: proof (induct N)
haftmann@30996:   case 0 then show ?case by simp
haftmann@30996: next
haftmann@30996:   case (Suc N) then show ?case 
haftmann@30996:   apply (auto simp add: le_Suc_eq)
haftmann@30996:   apply (subgoal_tac "1 * a^n \<le> a * a^N", simp)
haftmann@30996:   apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one])
haftmann@30996:   done
haftmann@30996: qed
paulson@14348: 
paulson@14348: text{*Lemma for @{text power_strict_increasing}*}
paulson@14348: lemma power_less_power_Suc:
haftmann@30996:   "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
haftmann@30996:   by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one])
paulson@14348: 
haftmann@30996: lemma power_strict_increasing [rule_format]:
haftmann@30996:   "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
haftmann@30996: proof (induct N)
haftmann@30996:   case 0 then show ?case by simp
haftmann@30996: next
haftmann@30996:   case (Suc N) then show ?case 
haftmann@30996:   apply (auto simp add: power_less_power_Suc less_Suc_eq)
haftmann@30996:   apply (subgoal_tac "1 * a^n < a * a^N", simp)
haftmann@30996:   apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
haftmann@30996:   done
haftmann@30996: qed
paulson@14348: 
nipkow@25134: lemma power_increasing_iff [simp]:
haftmann@30996:   "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
haftmann@30996:   by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
paulson@15066: 
paulson@15066: lemma power_strict_increasing_iff [simp]:
haftmann@30996:   "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
nipkow@25134: by (blast intro: power_less_imp_less_exp power_strict_increasing) 
paulson@15066: 
paulson@14348: lemma power_le_imp_le_base:
haftmann@30996:   assumes le: "a ^ Suc n \<le> b ^ Suc n"
haftmann@30996:     and ynonneg: "0 \<le> b"
haftmann@30996:   shows "a \<le> b"
nipkow@25134: proof (rule ccontr)
nipkow@25134:   assume "~ a \<le> b"
nipkow@25134:   then have "b < a" by (simp only: linorder_not_le)
nipkow@25134:   then have "b ^ Suc n < a ^ Suc n"
nipkow@25134:     by (simp only: prems power_strict_mono)
haftmann@30996:   from le and this show False
nipkow@25134:     by (simp add: linorder_not_less [symmetric])
nipkow@25134: qed
wenzelm@14577: 
huffman@22853: lemma power_less_imp_less_base:
huffman@22853:   assumes less: "a ^ n < b ^ n"
huffman@22853:   assumes nonneg: "0 \<le> b"
huffman@22853:   shows "a < b"
huffman@22853: proof (rule contrapos_pp [OF less])
huffman@22853:   assume "~ a < b"
huffman@22853:   hence "b \<le> a" by (simp only: linorder_not_less)
huffman@22853:   hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
haftmann@30996:   thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
huffman@22853: qed
huffman@22853: 
paulson@14348: lemma power_inject_base:
haftmann@30996:   "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
haftmann@30996: by (blast intro: power_le_imp_le_base antisym eq_refl sym)
paulson@14348: 
huffman@22955: lemma power_eq_imp_eq_base:
haftmann@30996:   "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
haftmann@30996:   by (cases n) (simp_all del: power_Suc, rule power_inject_base)
huffman@22955: 
haftmann@30996: end
haftmann@30996: 
haftmann@30996: context ordered_idom
haftmann@30996: begin
huffman@29978: 
haftmann@30996: lemma power_abs:
haftmann@30996:   "abs (a ^ n) = abs a ^ n"
haftmann@30996:   by (induct n) (auto simp add: abs_mult)
haftmann@30996: 
haftmann@30996: lemma abs_power_minus [simp]:
haftmann@30996:   "abs ((-a) ^ n) = abs (a ^ n)"
haftmann@30996:   by (simp add: abs_minus_cancel power_abs) 
haftmann@30996: 
haftmann@30996: lemma zero_less_power_abs_iff [simp, noatp]:
haftmann@30996:   "0 < abs a ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
haftmann@30996: proof (induct n)
haftmann@30996:   case 0 show ?case by simp
haftmann@30996: next
haftmann@30996:   case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff)
huffman@29978: qed
huffman@29978: 
haftmann@30996: lemma zero_le_power_abs [simp]:
haftmann@30996:   "0 \<le> abs a ^ n"
haftmann@30996:   by (rule zero_le_power [OF abs_ge_zero])
haftmann@30996: 
haftmann@30996: end
haftmann@30996: 
haftmann@30996: context ring_1_no_zero_divisors
haftmann@30996: begin
haftmann@30996: 
haftmann@30996: lemma field_power_not_zero:
haftmann@30996:   "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
haftmann@30996:   by (induct n) auto
haftmann@30996: 
haftmann@30996: end
haftmann@30996: 
haftmann@30996: context division_ring
haftmann@30996: begin
huffman@29978: 
haftmann@30997: text {* FIXME reorient or rename to @{text nonzero_inverse_power} *}
haftmann@30996: lemma nonzero_power_inverse:
haftmann@30996:   "a \<noteq> 0 \<Longrightarrow> inverse (a ^ n) = (inverse a) ^ n"
haftmann@30996:   by (induct n)
haftmann@30996:     (simp_all add: nonzero_inverse_mult_distrib power_commutes field_power_not_zero)
paulson@14348: 
haftmann@30996: end
haftmann@30996: 
haftmann@30996: context field
haftmann@30996: begin
haftmann@30996: 
haftmann@30996: lemma nonzero_power_divide:
haftmann@30996:   "b \<noteq> 0 \<Longrightarrow> (a / b) ^ n = a ^ n / b ^ n"
haftmann@30996:   by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
haftmann@30996: 
haftmann@30996: end
haftmann@30996: 
haftmann@30996: lemma power_0_Suc [simp]:
haftmann@30996:   "(0::'a::{power, semiring_0}) ^ Suc n = 0"
haftmann@30996:   by simp
nipkow@30313: 
haftmann@30996: text{*It looks plausible as a simprule, but its effect can be strange.*}
haftmann@30996: lemma power_0_left:
haftmann@30996:   "0 ^ n = (if n = 0 then 1 else (0::'a::{power, semiring_0}))"
haftmann@30996:   by (induct n) simp_all
haftmann@30996: 
haftmann@30996: lemma power_eq_0_iff [simp]:
haftmann@30996:   "a ^ n = 0 \<longleftrightarrow>
haftmann@30996:      a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,power}) \<and> n \<noteq> 0"
haftmann@30996:   by (induct n)
haftmann@30996:     (auto simp add: no_zero_divisors elim: contrapos_pp)
haftmann@30996: 
haftmann@30996: lemma power_diff:
haftmann@30996:   fixes a :: "'a::field"
haftmann@30996:   assumes nz: "a \<noteq> 0"
haftmann@30996:   shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
haftmann@30996:   by (induct m n rule: diff_induct) (simp_all add: nz)
nipkow@30313: 
haftmann@30996: text{*Perhaps these should be simprules.*}
haftmann@30996: lemma power_inverse:
haftmann@30996:   fixes a :: "'a::{division_ring,division_by_zero,power}"
haftmann@30996:   shows "inverse (a ^ n) = (inverse a) ^ n"
haftmann@30996: apply (cases "a = 0")
haftmann@30996: apply (simp add: power_0_left)
haftmann@30996: apply (simp add: nonzero_power_inverse)
haftmann@30996: done (* TODO: reorient or rename to inverse_power *)
haftmann@30996: 
haftmann@30996: lemma power_one_over:
haftmann@30996:   "1 / (a::'a::{field,division_by_zero, power}) ^ n =  (1 / a) ^ n"
haftmann@30996:   by (simp add: divide_inverse) (rule power_inverse)
haftmann@30996: 
haftmann@30996: lemma power_divide:
haftmann@30996:   "(a / b) ^ n = (a::'a::{field,division_by_zero}) ^ n / b ^ n"
haftmann@30996: apply (cases "b = 0")
haftmann@30996: apply (simp add: power_0_left)
haftmann@30996: apply (rule nonzero_power_divide)
haftmann@30996: apply assumption
nipkow@30313: done
nipkow@30313: 
haftmann@30996: class recpower = monoid_mult
haftmann@30996: 
nipkow@30313: 
haftmann@30960: subsection {* Exponentiation for the Natural Numbers *}
wenzelm@14577: 
haftmann@30960: instance nat :: recpower ..
haftmann@25836: 
haftmann@30996: lemma nat_one_le_power [simp]:
haftmann@30996:   "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
haftmann@30996:   by (rule one_le_power [of i n, unfolded One_nat_def])
huffman@23305: 
haftmann@30996: lemma nat_zero_less_power_iff [simp]:
haftmann@30996:   "x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0"
haftmann@30996:   by (induct n) auto
paulson@14348: 
nipkow@30056: lemma nat_power_eq_Suc_0_iff [simp]: 
haftmann@30996:   "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
haftmann@30996:   by (induct m) auto
nipkow@30056: 
haftmann@30996: lemma power_Suc_0 [simp]:
haftmann@30996:   "Suc 0 ^ n = Suc 0"
haftmann@30996:   by simp
nipkow@30056: 
paulson@14348: text{*Valid for the naturals, but what if @{text"0<i<1"}?
paulson@14348: Premises cannot be weakened: consider the case where @{term "i=0"},
paulson@14348: @{term "m=1"} and @{term "n=0"}.*}
haftmann@21413: lemma nat_power_less_imp_less:
haftmann@21413:   assumes nonneg: "0 < (i\<Colon>nat)"
haftmann@30996:   assumes less: "i ^ m < i ^ n"
haftmann@21413:   shows "m < n"
haftmann@21413: proof (cases "i = 1")
haftmann@21413:   case True with less power_one [where 'a = nat] show ?thesis by simp
haftmann@21413: next
haftmann@21413:   case False with nonneg have "1 < i" by auto
haftmann@21413:   from power_strict_increasing_iff [OF this] less show ?thesis ..
haftmann@21413: qed
paulson@14348: 
paulson@3390: end