---input---
(* from Isabelle2013-2 src/HOL/Power.thy; BSD license *)

(*  Title:      HOL/Power.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1997  University of Cambridge
*)

header {* Exponentiation *}

theory Power
imports Num
begin

subsection {* Powers for Arbitrary Monoids *}

class power = one + times
begin

primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80) where
    power_0: "a ^ 0 = 1"
  | power_Suc: "a ^ Suc n = a * a ^ n"

notation (latex output)
  power ("(_\<^bsup>_\<^esup>)" [1000] 1000)

notation (HTML output)
  power ("(_\<^bsup>_\<^esup>)" [1000] 1000)

text {* Special syntax for squares. *}

abbreviation (xsymbols)
  power2 :: "'a \<Rightarrow> 'a"  ("(_\<^sup>2)" [1000] 999) where
  "x\<^sup>2 \<equiv> x ^ 2"

notation (latex output)
  power2  ("(_\<^sup>2)" [1000] 999)

notation (HTML output)
  power2  ("(_\<^sup>2)" [1000] 999)

end

context monoid_mult
begin

subclass power .

lemma power_one [simp]:
  "1 ^ n = 1"
  by (induct n) simp_all

lemma power_one_right [simp]:
  "a ^ 1 = a"
  by simp

lemma power_commutes:
  "a ^ n * a = a * a ^ n"
  by (induct n) (simp_all add: mult_assoc)

lemma power_Suc2:
  "a ^ Suc n = a ^ n * a"
  by (simp add: power_commutes)

lemma power_add:
  "a ^ (m + n) = a ^ m * a ^ n"
  by (induct m) (simp_all add: algebra_simps)

lemma power_mult:
  "a ^ (m * n) = (a ^ m) ^ n"
  by (induct n) (simp_all add: power_add)

lemma power2_eq_square: "a\<^sup>2 = a * a"
  by (simp add: numeral_2_eq_2)

lemma power3_eq_cube: "a ^ 3 = a * a * a"
  by (simp add: numeral_3_eq_3 mult_assoc)

lemma power_even_eq:
  "a ^ (2 * n) = (a ^ n)\<^sup>2"
  by (subst mult_commute) (simp add: power_mult)

lemma power_odd_eq:
  "a ^ Suc (2*n) = a * (a ^ n)\<^sup>2"
  by (simp add: power_even_eq)

lemma power_numeral_even:
  "z ^ numeral (Num.Bit0 w) = (let w = z ^ (numeral w) in w * w)"
  unfolding numeral_Bit0 power_add Let_def ..

lemma power_numeral_odd:
  "z ^ numeral (Num.Bit1 w) = (let w = z ^ (numeral w) in z * w * w)"
  unfolding numeral_Bit1 One_nat_def add_Suc_right add_0_right
  unfolding power_Suc power_add Let_def mult_assoc ..

lemma funpow_times_power:
  "(times x ^^ f x) = times (x ^ f x)"
proof (induct "f x" arbitrary: f)
  case 0 then show ?case by (simp add: fun_eq_iff)
next
  case (Suc n)
  def g \<equiv> "\<lambda>x. f x - 1"
  with Suc have "n = g x" by simp
  with Suc have "times x ^^ g x = times (x ^ g x)" by simp
  moreover from Suc g_def have "f x = g x + 1" by simp
  ultimately show ?case by (simp add: power_add funpow_add fun_eq_iff mult_assoc)
qed

end

context comm_monoid_mult
begin

lemma power_mult_distrib:
  "(a * b) ^ n = (a ^ n) * (b ^ n)"
  by (induct n) (simp_all add: mult_ac)

end

context semiring_numeral
begin

lemma numeral_sqr: "numeral (Num.sqr k) = numeral k * numeral k"
  by (simp only: sqr_conv_mult numeral_mult)

lemma numeral_pow: "numeral (Num.pow k l) = numeral k ^ numeral l"
  by (induct l, simp_all only: numeral_class.numeral.simps pow.simps
    numeral_sqr numeral_mult power_add power_one_right)

lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)"
  by (rule numeral_pow [symmetric])

end

context semiring_1
begin

lemma of_nat_power:
  "of_nat (m ^ n) = of_nat m ^ n"
  by (induct n) (simp_all add: of_nat_mult)

lemma power_zero_numeral [simp]: "(0::'a) ^ numeral k = 0"
  by (simp add: numeral_eq_Suc)

lemma zero_power2: "0\<^sup>2 = 0" (* delete? *)
  by (rule power_zero_numeral)

lemma one_power2: "1\<^sup>2 = 1" (* delete? *)
  by (rule power_one)

end

context comm_semiring_1
begin

text {* The divides relation *}

lemma le_imp_power_dvd:
  assumes "m \<le> n" shows "a ^ m dvd a ^ n"
proof
  have "a ^ n = a ^ (m + (n - m))"
    using `m \<le> n` by simp
  also have "\<dots> = a ^ m * a ^ (n - m)"
    by (rule power_add)
  finally show "a ^ n = a ^ m * a ^ (n - m)" .
qed

lemma power_le_dvd:
  "a ^ n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a ^ m dvd b"
  by (rule dvd_trans [OF le_imp_power_dvd])

lemma dvd_power_same:
  "x dvd y \<Longrightarrow> x ^ n dvd y ^ n"
  by (induct n) (auto simp add: mult_dvd_mono)

lemma dvd_power_le:
  "x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m"
  by (rule power_le_dvd [OF dvd_power_same])

lemma dvd_power [simp]:
  assumes "n > (0::nat) \<or> x = 1"
  shows "x dvd (x ^ n)"
using assms proof
  assume "0 < n"
  then have "x ^ n = x ^ Suc (n - 1)" by simp
  then show "x dvd (x ^ n)" by simp
next
  assume "x = 1"
  then show "x dvd (x ^ n)" by simp
qed

end

context ring_1
begin

lemma power_minus:
  "(- a) ^ n = (- 1) ^ n * a ^ n"
proof (induct n)
  case 0 show ?case by simp
next
  case (Suc n) then show ?case
    by (simp del: power_Suc add: power_Suc2 mult_assoc)
qed

lemma power_minus_Bit0:
  "(- x) ^ numeral (Num.Bit0 k) = x ^ numeral (Num.Bit0 k)"
  by (induct k, simp_all only: numeral_class.numeral.simps power_add
    power_one_right mult_minus_left mult_minus_right minus_minus)

lemma power_minus_Bit1:
  "(- x) ^ numeral (Num.Bit1 k) = - (x ^ numeral (Num.Bit1 k))"
  by (simp only: eval_nat_numeral(3) power_Suc power_minus_Bit0 mult_minus_left)

lemma power_neg_numeral_Bit0 [simp]:
  "neg_numeral k ^ numeral (Num.Bit0 l) = numeral (Num.pow k (Num.Bit0 l))"
  by (simp only: neg_numeral_def power_minus_Bit0 power_numeral)

lemma power_neg_numeral_Bit1 [simp]:
  "neg_numeral k ^ numeral (Num.Bit1 l) = neg_numeral (Num.pow k (Num.Bit1 l))"
  by (simp only: neg_numeral_def power_minus_Bit1 power_numeral pow.simps)

lemma power2_minus [simp]:
  "(- a)\<^sup>2 = a\<^sup>2"
  by (rule power_minus_Bit0)

lemma power_minus1_even [simp]:
  "-1 ^ (2*n) = 1"
proof (induct n)
  case 0 show ?case by simp
next
  case (Suc n) then show ?case by (simp add: power_add power2_eq_square)
qed

lemma power_minus1_odd:
  "-1 ^ Suc (2*n) = -1"
  by simp

lemma power_minus_even [simp]:
  "(-a) ^ (2*n) = a ^ (2*n)"
  by (simp add: power_minus [of a])

end

context ring_1_no_zero_divisors
begin

lemma field_power_not_zero:
  "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
  by (induct n) auto

lemma zero_eq_power2 [simp]:
  "a\<^sup>2 = 0 \<longleftrightarrow> a = 0"
  unfolding power2_eq_square by simp

lemma power2_eq_1_iff:
  "a\<^sup>2 = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
  unfolding power2_eq_square by (rule square_eq_1_iff)

end

context idom
begin

lemma power2_eq_iff: "x\<^sup>2 = y\<^sup>2 \<longleftrightarrow> x = y \<or> x = - y"
  unfolding power2_eq_square by (rule square_eq_iff)

end

context division_ring
begin

text {* FIXME reorient or rename to @{text nonzero_inverse_power} *}
lemma nonzero_power_inverse:
  "a \<noteq> 0 \<Longrightarrow> inverse (a ^ n) = (inverse a) ^ n"
  by (induct n)
    (simp_all add: nonzero_inverse_mult_distrib power_commutes field_power_not_zero)

end

context field
begin

lemma nonzero_power_divide:
  "b \<noteq> 0 \<Longrightarrow> (a / b) ^ n = a ^ n / b ^ n"
  by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)

end


subsection {* Exponentiation on ordered types *}

context linordered_ring (* TODO: move *)
begin

lemma sum_squares_ge_zero:
  "0 \<le> x * x + y * y"
  by (intro add_nonneg_nonneg zero_le_square)

lemma not_sum_squares_lt_zero:
  "\<not> x * x + y * y < 0"
  by (simp add: not_less sum_squares_ge_zero)

end

context linordered_semidom
begin

lemma zero_less_power [simp]:
  "0 < a \<Longrightarrow> 0 < a ^ n"
  by (induct n) (simp_all add: mult_pos_pos)

lemma zero_le_power [simp]:
  "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
  by (induct n) (simp_all add: mult_nonneg_nonneg)

lemma power_mono:
  "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
  by (induct n) (auto intro: mult_mono order_trans [of 0 a b])

lemma one_le_power [simp]: "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
  using power_mono [of 1 a n] by simp

lemma power_le_one: "\<lbrakk>0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> a ^ n \<le> 1"
  using power_mono [of a 1 n] by simp

lemma power_gt1_lemma:
  assumes gt1: "1 < a"
  shows "1 < a * a ^ n"
proof -
  from gt1 have "0 \<le> a"
    by (fact order_trans [OF zero_le_one less_imp_le])
  have "1 * 1 < a * 1" using gt1 by simp
  also have "\<dots> \<le> a * a ^ n" using gt1
    by (simp only: mult_mono `0 \<le> a` one_le_power order_less_imp_le
        zero_le_one order_refl)
  finally show ?thesis by simp
qed

lemma power_gt1:
  "1 < a \<Longrightarrow> 1 < a ^ Suc n"
  by (simp add: power_gt1_lemma)

lemma one_less_power [simp]:
  "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
  by (cases n) (simp_all add: power_gt1_lemma)

lemma power_le_imp_le_exp:
  assumes gt1: "1 < a"
  shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
proof (induct m arbitrary: n)
  case 0
  show ?case by simp
next
  case (Suc m)
  show ?case
  proof (cases n)
    case 0
    with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
    with gt1 show ?thesis
      by (force simp only: power_gt1_lemma
          not_less [symmetric])
  next
    case (Suc n)
    with Suc.prems Suc.hyps show ?thesis
      by (force dest: mult_left_le_imp_le
          simp add: less_trans [OF zero_less_one gt1])
  qed
qed

text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}
lemma power_inject_exp [simp]:
  "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
  by (force simp add: order_antisym power_le_imp_le_exp)

text{*Can relax the first premise to @{term "0<a"} in the case of the
natural numbers.*}
lemma power_less_imp_less_exp:
  "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
  by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"]
    power_le_imp_le_exp)

lemma power_strict_mono [rule_format]:
  "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
  by (induct n)
   (auto simp add: mult_strict_mono le_less_trans [of 0 a b])

text{*Lemma for @{text power_strict_decreasing}*}
lemma power_Suc_less:
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
  by (induct n)
    (auto simp add: mult_strict_left_mono)

lemma power_strict_decreasing [rule_format]:
  "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
proof (induct N)
  case 0 then show ?case by simp
next
  case (Suc N) then show ?case 
  apply (auto simp add: power_Suc_less less_Suc_eq)
  apply (subgoal_tac "a * a^N < 1 * a^n")
  apply simp
  apply (rule mult_strict_mono) apply auto
  done
qed

text{*Proof resembles that of @{text power_strict_decreasing}*}
lemma power_decreasing [rule_format]:
  "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n"
proof (induct N)
  case 0 then show ?case by simp
next
  case (Suc N) then show ?case 
  apply (auto simp add: le_Suc_eq)
  apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp)
  apply (rule mult_mono) apply auto
  done
qed

lemma power_Suc_less_one:
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
  using power_strict_decreasing [of 0 "Suc n" a] by simp

text{*Proof again resembles that of @{text power_strict_decreasing}*}
lemma power_increasing [rule_format]:
  "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
proof (induct N)
  case 0 then show ?case by simp
next
  case (Suc N) then show ?case 
  apply (auto simp add: le_Suc_eq)
  apply (subgoal_tac "1 * a^n \<le> a * a^N", simp)
  apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one])
  done
qed

text{*Lemma for @{text power_strict_increasing}*}
lemma power_less_power_Suc:
  "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
  by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one])

lemma power_strict_increasing [rule_format]:
  "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
proof (induct N)
  case 0 then show ?case by simp
next
  case (Suc N) then show ?case 
  apply (auto simp add: power_less_power_Suc less_Suc_eq)
  apply (subgoal_tac "1 * a^n < a * a^N", simp)
  apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
  done
qed

lemma power_increasing_iff [simp]:
  "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
  by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)

lemma power_strict_increasing_iff [simp]:
  "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
by (blast intro: power_less_imp_less_exp power_strict_increasing) 

lemma power_le_imp_le_base:
  assumes le: "a ^ Suc n \<le> b ^ Suc n"
    and ynonneg: "0 \<le> b"
  shows "a \<le> b"
proof (rule ccontr)
  assume "~ a \<le> b"
  then have "b < a" by (simp only: linorder_not_le)
  then have "b ^ Suc n < a ^ Suc n"
    by (simp only: assms power_strict_mono)
  from le and this show False
    by (simp add: linorder_not_less [symmetric])
qed

lemma power_less_imp_less_base:
  assumes less: "a ^ n < b ^ n"
  assumes nonneg: "0 \<le> b"
  shows "a < b"
proof (rule contrapos_pp [OF less])
  assume "~ a < b"
  hence "b \<le> a" by (simp only: linorder_not_less)
  hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
  thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
qed

lemma power_inject_base:
  "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
by (blast intro: power_le_imp_le_base antisym eq_refl sym)

lemma power_eq_imp_eq_base:
  "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
  by (cases n) (simp_all del: power_Suc, rule power_inject_base)

lemma power2_le_imp_le:
  "x\<^sup>2 \<le> y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
  unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)

lemma power2_less_imp_less:
  "x\<^sup>2 < y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
  by (rule power_less_imp_less_base)

lemma power2_eq_imp_eq:
  "x\<^sup>2 = y\<^sup>2 \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
  unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp

end

context linordered_ring_strict
begin

lemma sum_squares_eq_zero_iff:
  "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
  by (simp add: add_nonneg_eq_0_iff)

lemma sum_squares_le_zero_iff:
  "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
  by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)

lemma sum_squares_gt_zero_iff:
  "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
  by (simp add: not_le [symmetric] sum_squares_le_zero_iff)

end

context linordered_idom
begin

lemma power_abs:
  "abs (a ^ n) = abs a ^ n"
  by (induct n) (auto simp add: abs_mult)

lemma abs_power_minus [simp]:
  "abs ((-a) ^ n) = abs (a ^ n)"
  by (simp add: power_abs)

lemma zero_less_power_abs_iff [simp, no_atp]:
  "0 < abs a ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
proof (induct n)
  case 0 show ?case by simp
next
  case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff)
qed

lemma zero_le_power_abs [simp]:
  "0 \<le> abs a ^ n"
  by (rule zero_le_power [OF abs_ge_zero])

lemma zero_le_power2 [simp]:
  "0 \<le> a\<^sup>2"
  by (simp add: power2_eq_square)

lemma zero_less_power2 [simp]:
  "0 < a\<^sup>2 \<longleftrightarrow> a \<noteq> 0"
  by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)

lemma power2_less_0 [simp]:
  "\<not> a\<^sup>2 < 0"
  by (force simp add: power2_eq_square mult_less_0_iff)

lemma abs_power2 [simp]:
  "abs (a\<^sup>2) = a\<^sup>2"
  by (simp add: power2_eq_square abs_mult abs_mult_self)

lemma power2_abs [simp]:
  "(abs a)\<^sup>2 = a\<^sup>2"
  by (simp add: power2_eq_square abs_mult_self)

lemma odd_power_less_zero:
  "a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0"
proof (induct n)
  case 0
  then show ?case by simp
next
  case (Suc n)
  have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
    by (simp add: mult_ac power_add power2_eq_square)
  thus ?case
    by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
qed

lemma odd_0_le_power_imp_0_le:
  "0 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a"
  using odd_power_less_zero [of a n]
    by (force simp add: linorder_not_less [symmetric]) 

lemma zero_le_even_power'[simp]:
  "0 \<le> a ^ (2*n)"
proof (induct n)
  case 0
    show ?case by simp
next
  case (Suc n)
    have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)" 
      by (simp add: mult_ac power_add power2_eq_square)
    thus ?case
      by (simp add: Suc zero_le_mult_iff)
qed

lemma sum_power2_ge_zero:
  "0 \<le> x\<^sup>2 + y\<^sup>2"
  by (intro add_nonneg_nonneg zero_le_power2)

lemma not_sum_power2_lt_zero:
  "\<not> x\<^sup>2 + y\<^sup>2 < 0"
  unfolding not_less by (rule sum_power2_ge_zero)

lemma sum_power2_eq_zero_iff:
  "x\<^sup>2 + y\<^sup>2 = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
  unfolding power2_eq_square by (simp add: add_nonneg_eq_0_iff)

lemma sum_power2_le_zero_iff:
  "x\<^sup>2 + y\<^sup>2 \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
  by (simp add: le_less sum_power2_eq_zero_iff not_sum_power2_lt_zero)

lemma sum_power2_gt_zero_iff:
  "0 < x\<^sup>2 + y\<^sup>2 \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
  unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)

end


subsection {* Miscellaneous rules *}

lemma power_eq_if: "p ^ m = (if m=0 then 1 else p * (p ^ (m - 1)))"
  unfolding One_nat_def by (cases m) simp_all

lemma power2_sum:
  fixes x y :: "'a::comm_semiring_1"
  shows "(x + y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 + 2 * x * y"
  by (simp add: algebra_simps power2_eq_square mult_2_right)

lemma power2_diff:
  fixes x y :: "'a::comm_ring_1"
  shows "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
  by (simp add: ring_distribs power2_eq_square mult_2) (rule mult_commute)

lemma power_0_Suc [simp]:
  "(0::'a::{power, semiring_0}) ^ Suc n = 0"
  by simp

text{*It looks plausible as a simprule, but its effect can be strange.*}
lemma power_0_left:
  "0 ^ n = (if n = 0 then 1 else (0::'a::{power, semiring_0}))"
  by (induct n) simp_all

lemma power_eq_0_iff [simp]:
  "a ^ n = 0 \<longleftrightarrow>
     a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,power}) \<and> n \<noteq> 0"
  by (induct n)
    (auto simp add: no_zero_divisors elim: contrapos_pp)

lemma (in field) power_diff:
  assumes nz: "a \<noteq> 0"
  shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
  by (induct m n rule: diff_induct) (simp_all add: nz field_power_not_zero)

text{*Perhaps these should be simprules.*}
lemma power_inverse:
  fixes a :: "'a::division_ring_inverse_zero"
  shows "inverse (a ^ n) = inverse a ^ n"
apply (cases "a = 0")
apply (simp add: power_0_left)
apply (simp add: nonzero_power_inverse)
done (* TODO: reorient or rename to inverse_power *)

lemma power_one_over:
  "1 / (a::'a::{field_inverse_zero, power}) ^ n =  (1 / a) ^ n"
  by (simp add: divide_inverse) (rule power_inverse)

lemma power_divide:
  "(a / b) ^ n = (a::'a::field_inverse_zero) ^ n / b ^ n"
apply (cases "b = 0")
apply (simp add: power_0_left)
apply (rule nonzero_power_divide)
apply assumption
done

text {* Simprules for comparisons where common factors can be cancelled. *}

lemmas zero_compare_simps =
    add_strict_increasing add_strict_increasing2 add_increasing
    zero_le_mult_iff zero_le_divide_iff 
    zero_less_mult_iff zero_less_divide_iff 
    mult_le_0_iff divide_le_0_iff 
    mult_less_0_iff divide_less_0_iff 
    zero_le_power2 power2_less_0


subsection {* Exponentiation for the Natural Numbers *}

lemma nat_one_le_power [simp]:
  "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
  by (rule one_le_power [of i n, unfolded One_nat_def])

lemma nat_zero_less_power_iff [simp]:
  "x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0"
  by (induct n) auto

lemma nat_power_eq_Suc_0_iff [simp]: 
  "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
  by (induct m) auto

lemma power_Suc_0 [simp]:
  "Suc 0 ^ n = Suc 0"
  by simp

text{*Valid for the naturals, but what if @{text"0<i<1"}?
Premises cannot be weakened: consider the case where @{term "i=0"},
@{term "m=1"} and @{term "n=0"}.*}
lemma nat_power_less_imp_less:
  assumes nonneg: "0 < (i\<Colon>nat)"
  assumes less: "i ^ m < i ^ n"
  shows "m < n"
proof (cases "i = 1")
  case True with less power_one [where 'a = nat] show ?thesis by simp
next
  case False with nonneg have "1 < i" by auto
  from power_strict_increasing_iff [OF this] less show ?thesis ..
qed

lemma power_dvd_imp_le:
  "i ^ m dvd i ^ n \<Longrightarrow> (1::nat) < i \<Longrightarrow> m \<le> n"
  apply (rule power_le_imp_le_exp, assumption)
  apply (erule dvd_imp_le, simp)
  done

lemma power2_nat_le_eq_le:
  fixes m n :: nat
  shows "m\<^sup>2 \<le> n\<^sup>2 \<longleftrightarrow> m \<le> n"
  by (auto intro: power2_le_imp_le power_mono)

lemma power2_nat_le_imp_le:
  fixes m n :: nat
  assumes "m\<^sup>2 \<le> n"
  shows "m \<le> n"
  using assms by (cases m) (simp_all add: power2_eq_square)



subsection {* Code generator tweak *}

lemma power_power_power [code]:
  "power = power.power (1::'a::{power}) (op *)"
  unfolding power_def power.power_def ..

declare power.power.simps [code]

code_identifier
  code_module Power \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith

end


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'text'        Keyword
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'context'     Keyword
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'begin'       Keyword
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'subsection'  Generic.Subheading
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'context'     Keyword
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'begin'       Keyword
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'lemma'       Keyword.Namespace
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'end'         Keyword
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'context'     Keyword
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'begin'       Keyword
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'text'        Keyword
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'text'        Keyword
'{*'          Comment
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'text'        Keyword
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'\\<longrightarrow>' Literal.String.Symbol
' a ^ n < a ^ N' Literal.String
'"'           Literal.String
'\n'          Text

'proof'       Keyword
' '           Text
'('           Operator
'induct'      Name
' '           Text
'N'           Name
')'           Operator
'\n  '        Text
'case'        Keyword
' '           Text
'0'           Name
' '           Text
'then'        Keyword
' '           Text
'show'        Keyword
' '           Text
'?'           Operator
'case'        Keyword
' '           Text
'by'          Keyword
' '           Text
'simp'        Name
'\n'          Text

'next'        Keyword
'\n  '        Text
'case'        Keyword
' '           Text
'('           Operator
'Suc'         Name
' '           Text
'N'           Name
')'           Operator
' '           Text
'then'        Keyword
' '           Text
'show'        Keyword
' '           Text
'?'           Operator
'case'        Keyword
' \n  '       Text
'apply'       Keyword.Pseudo
' '           Text
'('           Operator
'auto'        Name
' '           Text
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'power_less_power_Suc' Name
' '           Text
'less_Suc_eq' Name
')'           Operator
'\n  '        Text
'apply'       Keyword.Pseudo
' '           Text
'('           Operator
'subgoal_tac' Name
' '           Text
'"'           Literal.String
'1 * a^n < a * a^N' Literal.String
'"'           Literal.String
','           Operator
' '           Text
'simp'        Name
')'           Operator
'\n  '        Text
'apply'       Keyword.Pseudo
' '           Text
'('           Operator
'rule'        Name
' '           Text
'mult_strict_mono' Name
')'           Operator
' '           Text
'apply'       Keyword.Pseudo
' '           Text
'('           Operator
'auto'        Name
' '           Text
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'less_trans'  Name
' '           Text
'['           Operator
'OF'          Name
' '           Text
'zero_less_one' Name
']'           Operator
' '           Text
'less_imp_le' Name
')'           Operator
'\n  '        Text
'done'        Keyword
'\n'          Text

'qed'         Keyword
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'power_increasing_iff' Name
' '           Text
'['           Operator
'simp'        Name
']'           Operator
':'           Operator
'\n  '        Text
'"'           Literal.String
'1 < b '      Literal.String
'\\<Longrightarrow>' Literal.String.Symbol
' b ^ x '     Literal.String
'\\<le>'      Literal.String.Symbol
' b ^ y '     Literal.String
'\\<longleftrightarrow>' Literal.String.Symbol
' x '         Literal.String
'\\<le>'      Literal.String.Symbol
' y'          Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'('           Operator
'blast'       Name
' '           Text
'intro'       Name
':'           Operator
' '           Text
'power_le_imp_le_exp' Name
' '           Text
'power_increasing' Name
' '           Text
'less_imp_le' Name
')'           Operator
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'power_strict_increasing_iff' Name
' '           Text
'['           Operator
'simp'        Name
']'           Operator
':'           Operator
'\n  '        Text
'"'           Literal.String
'1 < b '      Literal.String
'\\<Longrightarrow>' Literal.String.Symbol
' b ^ x < b ^ y ' Literal.String
'\\<longleftrightarrow>' Literal.String.Symbol
' x < y'      Literal.String
'"'           Literal.String
'\n'          Text

'by'          Keyword
' '           Text
'('           Operator
'blast'       Name
' '           Text
'intro'       Name
':'           Operator
' '           Text
'power_less_imp_less_exp' Name
' '           Text
'power_strict_increasing' Name
')'           Operator
' \n\n'       Text

'lemma'       Keyword.Namespace
' '           Text
'power_le_imp_le_base' Name
':'           Operator
'\n  '        Text
'assumes'     Keyword.Pseudo
' '           Text
'le'          Name
':'           Operator
' '           Text
'"'           Literal.String
'a ^ Suc n '  Literal.String
'\\<le>'      Literal.String.Symbol
' b ^ Suc n'  Literal.String
'"'           Literal.String
'\n    '      Text
'and'         Keyword.Pseudo
' '           Text
'ynonneg'     Name
':'           Operator
' '           Text
'"'           Literal.String
'0 '          Literal.String
'\\<le>'      Literal.String.Symbol
' b'          Literal.String
'"'           Literal.String
'\n  '        Text
'shows'       Keyword.Pseudo
' '           Text
'"'           Literal.String
'a '          Literal.String
'\\<le>'      Literal.String.Symbol
' b'          Literal.String
'"'           Literal.String
'\n'          Text

'proof'       Keyword
' '           Text
'('           Operator
'rule'        Name
' '           Text
'ccontr'      Name
')'           Operator
'\n  '        Text
'assume'      Keyword
' '           Text
'"'           Literal.String
'~ a '        Literal.String
'\\<le>'      Literal.String.Symbol
' b'          Literal.String
'"'           Literal.String
'\n  '        Text
'then'        Keyword
' '           Text
'have'        Keyword
' '           Text
'"'           Literal.String
'b < a'       Literal.String
'"'           Literal.String
' '           Text
'by'          Keyword
' '           Text
'('           Operator
'simp'        Name
' '           Text
'only'        Name
':'           Operator
' '           Text
'linorder_not_le' Name
')'           Operator
'\n  '        Text
'then'        Keyword
' '           Text
'have'        Keyword
' '           Text
'"'           Literal.String
'b ^ Suc n < a ^ Suc n' Literal.String
'"'           Literal.String
'\n    '      Text
'by'          Keyword
' '           Text
'('           Operator
'simp'        Name
' '           Text
'only'        Name
':'           Operator
' '           Text
'assms'       Name
' '           Text
'power_strict_mono' Name
')'           Operator
'\n  '        Text
'from'        Keyword
' '           Text
'le'          Name
' '           Text
'and'         Keyword.Pseudo
' '           Text
'this'        Name
' '           Text
'show'        Keyword
' '           Text
'False'       Name
'\n    '      Text
'by'          Keyword
' '           Text
'('           Operator
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'linorder_not_less' Name
' '           Text
'['           Operator
'symmetric'   Name
']'           Operator
')'           Operator
'\n'          Text

'qed'         Keyword
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'power_less_imp_less_base' Name
':'           Operator
'\n  '        Text
'assumes'     Keyword.Pseudo
' '           Text
'less'        Name
':'           Operator
' '           Text
'"'           Literal.String
'a ^ n < b ^ n' Literal.String
'"'           Literal.String
'\n  '        Text
'assumes'     Keyword.Pseudo
' '           Text
'nonneg'      Name
':'           Operator
' '           Text
'"'           Literal.String
'0 '          Literal.String
'\\<le>'      Literal.String.Symbol
' b'          Literal.String
'"'           Literal.String
'\n  '        Text
'shows'       Keyword.Pseudo
' '           Text
'"'           Literal.String
'a < b'       Literal.String
'"'           Literal.String
'\n'          Text

'proof'       Keyword
' '           Text
'('           Operator
'rule'        Name
' '           Text
'contrapos_pp' Name
' '           Text
'['           Operator
'OF'          Name
' '           Text
'less'        Name
']'           Operator
')'           Operator
'\n  '        Text
'assume'      Keyword
' '           Text
'"'           Literal.String
'~ a < b'     Literal.String
'"'           Literal.String
'\n  '        Text
'hence'       Keyword
' '           Text
'"'           Literal.String
'b '          Literal.String
'\\<le>'      Literal.String.Symbol
' a'          Literal.String
'"'           Literal.String
' '           Text
'by'          Keyword
' '           Text
'('           Operator
'simp'        Name
' '           Text
'only'        Name
':'           Operator
' '           Text
'linorder_not_less' Name
')'           Operator
'\n  '        Text
'hence'       Keyword
' '           Text
'"'           Literal.String
'b ^ n '      Literal.String
'\\<le>'      Literal.String.Symbol
' a ^ n'      Literal.String
'"'           Literal.String
' '           Text
'using'       Keyword
' '           Text
'nonneg'      Name
' '           Text
'by'          Keyword
' '           Text
'('           Operator
'rule'        Name
' '           Text
'power_mono'  Name
')'           Operator
'\n  '        Text
'thus'        Keyword
' '           Text
'"'           Literal.String
'\\<not>'     Literal.String.Symbol
' a ^ n < b ^ n' Literal.String
'"'           Literal.String
' '           Text
'by'          Keyword
' '           Text
'('           Operator
'simp'        Name
' '           Text
'only'        Name
':'           Operator
' '           Text
'linorder_not_less' Name
')'           Operator
'\n'          Text

'qed'         Keyword
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'power_inject_base' Name
':'           Operator
'\n  '        Text
'"'           Literal.String
'a ^ Suc n = b ^ Suc n ' Literal.String
'\\<Longrightarrow>' Literal.String.Symbol
' 0 '         Literal.String
'\\<le>'      Literal.String.Symbol
' a '         Literal.String
'\\<Longrightarrow>' Literal.String.Symbol
' 0 '         Literal.String
'\\<le>'      Literal.String.Symbol
' b '         Literal.String
'\\<Longrightarrow>' Literal.String.Symbol
' a = b'      Literal.String
'"'           Literal.String
'\n'          Text

'by'          Keyword
' '           Text
'('           Operator
'blast'       Name
' '           Text
'intro'       Name
':'           Operator
' '           Text
'power_le_imp_le_base' Name
' '           Text
'antisym'     Name
' '           Text
'eq_refl'     Name
' '           Text
'sym'         Name
')'           Operator
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'power_eq_imp_eq_base' Name
':'           Operator
'\n  '        Text
'"'           Literal.String
'a ^ n = b ^ n ' Literal.String
'\\<Longrightarrow>' Literal.String.Symbol
' 0 '         Literal.String
'\\<le>'      Literal.String.Symbol
' a '         Literal.String
'\\<Longrightarrow>' Literal.String.Symbol
' 0 '         Literal.String
'\\<le>'      Literal.String.Symbol
' b '         Literal.String
'\\<Longrightarrow>' Literal.String.Symbol
' 0 < n '     Literal.String
'\\<Longrightarrow>' Literal.String.Symbol
' a = b'      Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'('           Operator
'cases'       Name
' '           Text
'n'           Name
')'           Operator
' '           Text
'('           Operator
'simp_all'    Name
' '           Text
'del'         Name
':'           Operator
' '           Text
'power_Suc'   Name
','           Operator
' '           Text
'rule'        Name
' '           Text
'power_inject_base' Name
')'           Operator
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'power2_le_imp_le' Name
':'           Operator
'\n  '        Text
'"'           Literal.String
'x'           Literal.String
'\\'          Literal.String
'<^sup>2 '    Literal.String
'\\<le>'      Literal.String.Symbol
' y'          Literal.String
'\\'          Literal.String
'<^sup>2 '    Literal.String
'\\<Longrightarrow>' Literal.String.Symbol
' 0 '         Literal.String
'\\<le>'      Literal.String.Symbol
' y '         Literal.String
'\\<Longrightarrow>' Literal.String.Symbol
' x '         Literal.String
'\\<le>'      Literal.String.Symbol
' y'          Literal.String
'"'           Literal.String
'\n  '        Text
'unfolding'   Keyword
' '           Text
'numeral_2_eq_2' Name
' '           Text
'by'          Keyword
' '           Text
'('           Operator
'rule'        Name
' '           Text
'power_le_imp_le_base' Name
')'           Operator
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'power2_less_imp_less' Name
':'           Operator
'\n  '        Text
'"'           Literal.String
'x'           Literal.String
'\\'          Literal.String
'<^sup>2 < y' Literal.String
'\\'          Literal.String
'<^sup>2 '    Literal.String
'\\<Longrightarrow>' Literal.String.Symbol
' 0 '         Literal.String
'\\<le>'      Literal.String.Symbol
' y '         Literal.String
'\\<Longrightarrow>' Literal.String.Symbol
' x < y'      Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'('           Operator
'rule'        Name
' '           Text
'power_less_imp_less_base' Name
')'           Operator
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'power2_eq_imp_eq' Name
':'           Operator
'\n  '        Text
'"'           Literal.String
'x'           Literal.String
'\\'          Literal.String
'<^sup>2 = y' Literal.String
'\\'          Literal.String
'<^sup>2 '    Literal.String
'\\<Longrightarrow>' Literal.String.Symbol
' 0 '         Literal.String
'\\<le>'      Literal.String.Symbol
' x '         Literal.String
'\\<Longrightarrow>' Literal.String.Symbol
' 0 '         Literal.String
'\\<le>'      Literal.String.Symbol
' y '         Literal.String
'\\<Longrightarrow>' Literal.String.Symbol
' x = y'      Literal.String
'"'           Literal.String
'\n  '        Text
'unfolding'   Keyword
' '           Text
'numeral_2_eq_2' Name
' '           Text
'by'          Keyword
' '           Text
'('           Operator
'erule'       Name
' '           Text
'('           Operator
'2'           Name
')'           Operator
' '           Text
'power_eq_imp_eq_base' Name
')'           Operator
' '           Text
'simp'        Name
'\n\n'        Text

'end'         Keyword
'\n\n'        Text

'context'     Keyword
' '           Text
'linordered_ring_strict' Name
'\n'          Text

'begin'       Keyword
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'sum_squares_eq_zero_iff' Name
':'           Operator
'\n  '        Text
'"'           Literal.String
'x * x + y * y = 0 ' Literal.String
'\\<longleftrightarrow>' Literal.String.Symbol
' x = 0 '     Literal.String
'\\<and>'     Literal.String.Symbol
' y = 0'      Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'('           Operator
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'add_nonneg_eq_0_iff' Name
')'           Operator
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'sum_squares_le_zero_iff' Name
':'           Operator
'\n  '        Text
'"'           Literal.String
'x * x + y * y ' Literal.String
'\\<le>'      Literal.String.Symbol
' 0 '         Literal.String
'\\<longleftrightarrow>' Literal.String.Symbol
' x = 0 '     Literal.String
'\\<and>'     Literal.String.Symbol
' y = 0'      Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'('           Operator
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'le_less'     Name
' '           Text
'not_sum_squares_lt_zero' Name
' '           Text
'sum_squares_eq_zero_iff' Name
')'           Operator
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'sum_squares_gt_zero_iff' Name
':'           Operator
'\n  '        Text
'"'           Literal.String
'0 < x * x + y * y ' Literal.String
'\\<longleftrightarrow>' Literal.String.Symbol
' x '         Literal.String
'\\<noteq>'   Literal.String.Symbol
' 0 '         Literal.String
'\\<or>'      Literal.String.Symbol
' y '         Literal.String
'\\<noteq>'   Literal.String.Symbol
' 0'          Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'('           Operator
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'not_le'      Name
' '           Text
'['           Operator
'symmetric'   Name
']'           Operator
' '           Text
'sum_squares_le_zero_iff' Name
')'           Operator
'\n\n'        Text

'end'         Keyword
'\n\n'        Text

'context'     Keyword
' '           Text
'linordered_idom' Name
'\n'          Text

'begin'       Keyword
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'power_abs'   Name
':'           Operator
'\n  '        Text
'"'           Literal.String
'abs (a ^ n) = abs a ^ n' Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'('           Operator
'induct'      Name
' '           Text
'n'           Name
')'           Operator
' '           Text
'('           Operator
'auto'        Name
' '           Text
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'abs_mult'    Name
')'           Operator
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'abs_power_minus' Name
' '           Text
'['           Operator
'simp'        Name
']'           Operator
':'           Operator
'\n  '        Text
'"'           Literal.String
'abs ((-a) ^ n) = abs (a ^ n)' Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'('           Operator
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'power_abs'   Name
')'           Operator
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'zero_less_power_abs_iff' Name
' '           Text
'['           Operator
'simp'        Name
','           Operator
' '           Text
'no_atp'      Name
']'           Operator
':'           Operator
'\n  '        Text
'"'           Literal.String
'0 < abs a ^ n ' Literal.String
'\\<longleftrightarrow>' Literal.String.Symbol
' a '         Literal.String
'\\<noteq>'   Literal.String.Symbol
' 0 '         Literal.String
'\\<or>'      Literal.String.Symbol
' n = 0'      Literal.String
'"'           Literal.String
'\n'          Text

'proof'       Keyword
' '           Text
'('           Operator
'induct'      Name
' '           Text
'n'           Name
')'           Operator
'\n  '        Text
'case'        Keyword
' '           Text
'0'           Name
' '           Text
'show'        Keyword
' '           Text
'?'           Operator
'case'        Keyword
' '           Text
'by'          Keyword
' '           Text
'simp'        Name
'\n'          Text

'next'        Keyword
'\n  '        Text
'case'        Keyword
' '           Text
'('           Operator
'Suc'         Name
' '           Text
'n'           Name
')'           Operator
' '           Text
'show'        Keyword
' '           Text
'?'           Operator
'case'        Keyword
' '           Text
'by'          Keyword
' '           Text
'('           Operator
'auto'        Name
' '           Text
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'Suc'         Name
' '           Text
'zero_less_mult_iff' Name
')'           Operator
'\n'          Text

'qed'         Keyword
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'zero_le_power_abs' Name
' '           Text
'['           Operator
'simp'        Name
']'           Operator
':'           Operator
'\n  '        Text
'"'           Literal.String
'0 '          Literal.String
'\\<le>'      Literal.String.Symbol
' abs a ^ n'  Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'('           Operator
'rule'        Name
' '           Text
'zero_le_power' Name
' '           Text
'['           Operator
'OF'          Name
' '           Text
'abs_ge_zero' Name
']'           Operator
')'           Operator
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'zero_le_power2' Name
' '           Text
'['           Operator
'simp'        Name
']'           Operator
':'           Operator
'\n  '        Text
'"'           Literal.String
'0 '          Literal.String
'\\<le>'      Literal.String.Symbol
' a'          Literal.String
'\\'          Literal.String
'<^sup>2'     Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'('           Operator
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'power2_eq_square' Name
')'           Operator
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'zero_less_power2' Name
' '           Text
'['           Operator
'simp'        Name
']'           Operator
':'           Operator
'\n  '        Text
'"'           Literal.String
'0 < a'       Literal.String
'\\'          Literal.String
'<^sup>2 '    Literal.String
'\\<longleftrightarrow>' Literal.String.Symbol
' a '         Literal.String
'\\<noteq>'   Literal.String.Symbol
' 0'          Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'('           Operator
'force'       Name
' '           Text
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'power2_eq_square' Name
' '           Text
'zero_less_mult_iff' Name
' '           Text
'linorder_neq_iff' Name
')'           Operator
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'power2_less_0' Name
' '           Text
'['           Operator
'simp'        Name
']'           Operator
':'           Operator
'\n  '        Text
'"'           Literal.String
'\\<not>'     Literal.String.Symbol
' a'          Literal.String
'\\'          Literal.String
'<^sup>2 < 0' Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'('           Operator
'force'       Name
' '           Text
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'power2_eq_square' Name
' '           Text
'mult_less_0_iff' Name
')'           Operator
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'abs_power2'  Name
' '           Text
'['           Operator
'simp'        Name
']'           Operator
':'           Operator
'\n  '        Text
'"'           Literal.String
'abs (a'      Literal.String
'\\'          Literal.String
'<^sup>2) = a' Literal.String
'\\'          Literal.String
'<^sup>2'     Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'('           Operator
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'power2_eq_square' Name
' '           Text
'abs_mult'    Name
' '           Text
'abs_mult_self' Name
')'           Operator
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'power2_abs'  Name
' '           Text
'['           Operator
'simp'        Name
']'           Operator
':'           Operator
'\n  '        Text
'"'           Literal.String
'(abs a)'     Literal.String
'\\'          Literal.String
'<^sup>2 = a' Literal.String
'\\'          Literal.String
'<^sup>2'     Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'('           Operator
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'power2_eq_square' Name
' '           Text
'abs_mult_self' Name
')'           Operator
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'odd_power_less_zero' Name
':'           Operator
'\n  '        Text
'"'           Literal.String
'a < 0 '      Literal.String
'\\<Longrightarrow>' Literal.String.Symbol
' a ^ Suc (2*n) < 0' Literal.String
'"'           Literal.String
'\n'          Text

'proof'       Keyword
' '           Text
'('           Operator
'induct'      Name
' '           Text
'n'           Name
')'           Operator
'\n  '        Text
'case'        Keyword
' '           Text
'0'           Name
'\n  '        Text
'then'        Keyword
' '           Text
'show'        Keyword
' '           Text
'?'           Operator
'case'        Keyword
' '           Text
'by'          Keyword
' '           Text
'simp'        Name
'\n'          Text

'next'        Keyword
'\n  '        Text
'case'        Keyword
' '           Text
'('           Operator
'Suc'         Name
' '           Text
'n'           Name
')'           Operator
'\n  '        Text
'have'        Keyword
' '           Text
'"'           Literal.String
'a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)' Literal.String
'"'           Literal.String
'\n    '      Text
'by'          Keyword
' '           Text
'('           Operator
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'mult_ac'     Name
' '           Text
'power_add'   Name
' '           Text
'power2_eq_square' Name
')'           Operator
'\n  '        Text
'thus'        Keyword
' '           Text
'?'           Operator
'case'        Keyword
'\n    '      Text
'by'          Keyword
' '           Text
'('           Operator
'simp'        Name
' '           Text
'del'         Name
':'           Operator
' '           Text
'power_Suc'   Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'Suc'         Name
' '           Text
'mult_less_0_iff' Name
' '           Text
'mult_neg_neg' Name
')'           Operator
'\n'          Text

'qed'         Keyword
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'odd_0_le_power_imp_0_le' Name
':'           Operator
'\n  '        Text
'"'           Literal.String
'0 '          Literal.String
'\\<le>'      Literal.String.Symbol
' a ^ Suc (2*n) ' Literal.String
'\\<Longrightarrow>' Literal.String.Symbol
' 0 '         Literal.String
'\\<le>'      Literal.String.Symbol
' a'          Literal.String
'"'           Literal.String
'\n  '        Text
'using'       Keyword
' '           Text
'odd_power_less_zero' Name
' '           Text
'['           Operator
'of'          Name
' '           Text
'a'           Name
' '           Text
'n'           Name
']'           Operator
'\n    '      Text
'by'          Keyword
' '           Text
'('           Operator
'force'       Name
' '           Text
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'linorder_not_less' Name
' '           Text
'['           Operator
'symmetric'   Name
']'           Operator
')'           Operator
' \n\n'       Text

'lemma'       Keyword.Namespace
' '           Text
"zero_le_even_power'" Name
'['           Operator
'simp'        Name
']'           Operator
':'           Operator
'\n  '        Text
'"'           Literal.String
'0 '          Literal.String
'\\<le>'      Literal.String.Symbol
' a ^ (2*n)'  Literal.String
'"'           Literal.String
'\n'          Text

'proof'       Keyword
' '           Text
'('           Operator
'induct'      Name
' '           Text
'n'           Name
')'           Operator
'\n  '        Text
'case'        Keyword
' '           Text
'0'           Name
'\n    '      Text
'show'        Keyword
' '           Text
'?'           Operator
'case'        Keyword
' '           Text
'by'          Keyword
' '           Text
'simp'        Name
'\n'          Text

'next'        Keyword
'\n  '        Text
'case'        Keyword
' '           Text
'('           Operator
'Suc'         Name
' '           Text
'n'           Name
')'           Operator
'\n    '      Text
'have'        Keyword
' '           Text
'"'           Literal.String
'a ^ (2 * Suc n) = (a*a) * a ^ (2*n)' Literal.String
'"'           Literal.String
' \n      '   Text
'by'          Keyword
' '           Text
'('           Operator
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'mult_ac'     Name
' '           Text
'power_add'   Name
' '           Text
'power2_eq_square' Name
')'           Operator
'\n    '      Text
'thus'        Keyword
' '           Text
'?'           Operator
'case'        Keyword
'\n      '    Text
'by'          Keyword
' '           Text
'('           Operator
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'Suc'         Name
' '           Text
'zero_le_mult_iff' Name
')'           Operator
'\n'          Text

'qed'         Keyword
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'sum_power2_ge_zero' Name
':'           Operator
'\n  '        Text
'"'           Literal.String
'0 '          Literal.String
'\\<le>'      Literal.String.Symbol
' x'          Literal.String
'\\'          Literal.String
'<^sup>2 + y' Literal.String
'\\'          Literal.String
'<^sup>2'     Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'('           Operator
'intro'       Name
' '           Text
'add_nonneg_nonneg' Name
' '           Text
'zero_le_power2' Name
')'           Operator
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'not_sum_power2_lt_zero' Name
':'           Operator
'\n  '        Text
'"'           Literal.String
'\\<not>'     Literal.String.Symbol
' x'          Literal.String
'\\'          Literal.String
'<^sup>2 + y' Literal.String
'\\'          Literal.String
'<^sup>2 < 0' Literal.String
'"'           Literal.String
'\n  '        Text
'unfolding'   Keyword
' '           Text
'not_less'    Name
' '           Text
'by'          Keyword
' '           Text
'('           Operator
'rule'        Name
' '           Text
'sum_power2_ge_zero' Name
')'           Operator
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'sum_power2_eq_zero_iff' Name
':'           Operator
'\n  '        Text
'"'           Literal.String
'x'           Literal.String
'\\'          Literal.String
'<^sup>2 + y' Literal.String
'\\'          Literal.String
'<^sup>2 = 0 ' Literal.String
'\\<longleftrightarrow>' Literal.String.Symbol
' x = 0 '     Literal.String
'\\<and>'     Literal.String.Symbol
' y = 0'      Literal.String
'"'           Literal.String
'\n  '        Text
'unfolding'   Keyword
' '           Text
'power2_eq_square' Name
' '           Text
'by'          Keyword
' '           Text
'('           Operator
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'add_nonneg_eq_0_iff' Name
')'           Operator
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'sum_power2_le_zero_iff' Name
':'           Operator
'\n  '        Text
'"'           Literal.String
'x'           Literal.String
'\\'          Literal.String
'<^sup>2 + y' Literal.String
'\\'          Literal.String
'<^sup>2 '    Literal.String
'\\<le>'      Literal.String.Symbol
' 0 '         Literal.String
'\\<longleftrightarrow>' Literal.String.Symbol
' x = 0 '     Literal.String
'\\<and>'     Literal.String.Symbol
' y = 0'      Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'('           Operator
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'le_less'     Name
' '           Text
'sum_power2_eq_zero_iff' Name
' '           Text
'not_sum_power2_lt_zero' Name
')'           Operator
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'sum_power2_gt_zero_iff' Name
':'           Operator
'\n  '        Text
'"'           Literal.String
'0 < x'       Literal.String
'\\'          Literal.String
'<^sup>2 + y' Literal.String
'\\'          Literal.String
'<^sup>2 '    Literal.String
'\\<longleftrightarrow>' Literal.String.Symbol
' x '         Literal.String
'\\<noteq>'   Literal.String.Symbol
' 0 '         Literal.String
'\\<or>'      Literal.String.Symbol
' y '         Literal.String
'\\<noteq>'   Literal.String.Symbol
' 0'          Literal.String
'"'           Literal.String
'\n  '        Text
'unfolding'   Keyword
' '           Text
'not_le'      Name
' '           Text
'['           Operator
'symmetric'   Name
']'           Operator
' '           Text
'by'          Keyword
' '           Text
'('           Operator
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'sum_power2_le_zero_iff' Name
')'           Operator
'\n\n'        Text

'end'         Keyword
'\n\n\n'      Text

'subsection'  Generic.Subheading
' '           Text
'{*'          Comment
' Miscellaneous rules ' Comment
'*}'          Comment
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'power_eq_if' Name
':'           Operator
' '           Text
'"'           Literal.String
'p ^ m = (if m=0 then 1 else p * (p ^ (m - 1)))' Literal.String
'"'           Literal.String
'\n  '        Text
'unfolding'   Keyword
' '           Text
'One_nat_def' Name
' '           Text
'by'          Keyword
' '           Text
'('           Operator
'cases'       Name
' '           Text
'm'           Name
')'           Operator
' '           Text
'simp_all'    Name
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'power2_sum'  Name
':'           Operator
'\n  '        Text
'fixes'       Keyword.Pseudo
' '           Text
'x'           Name
' '           Text
'y'           Name
' '           Text
'::'          Operator
' '           Text
'"'           Literal.String
"'a::comm_semiring_1" Literal.String
'"'           Literal.String
'\n  '        Text
'shows'       Keyword.Pseudo
' '           Text
'"'           Literal.String
'(x + y)'     Literal.String
'\\'          Literal.String
'<^sup>2 = x' Literal.String
'\\'          Literal.String
'<^sup>2 + y' Literal.String
'\\'          Literal.String
'<^sup>2 + 2 * x * y' Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'('           Operator
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'algebra_simps' Name
' '           Text
'power2_eq_square' Name
' '           Text
'mult_2_right' Name
')'           Operator
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'power2_diff' Name
':'           Operator
'\n  '        Text
'fixes'       Keyword.Pseudo
' '           Text
'x'           Name
' '           Text
'y'           Name
' '           Text
'::'          Operator
' '           Text
'"'           Literal.String
"'a::comm_ring_1" Literal.String
'"'           Literal.String
'\n  '        Text
'shows'       Keyword.Pseudo
' '           Text
'"'           Literal.String
'(x - y)'     Literal.String
'\\'          Literal.String
'<^sup>2 = x' Literal.String
'\\'          Literal.String
'<^sup>2 + y' Literal.String
'\\'          Literal.String
'<^sup>2 - 2 * x * y' Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'('           Operator
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'ring_distribs' Name
' '           Text
'power2_eq_square' Name
' '           Text
'mult_2'      Name
')'           Operator
' '           Text
'('           Operator
'rule'        Name
' '           Text
'mult_commute' Name
')'           Operator
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'power_0_Suc' Name
' '           Text
'['           Operator
'simp'        Name
']'           Operator
':'           Operator
'\n  '        Text
'"'           Literal.String
"(0::'a::{power, semiring_0}) ^ Suc n = 0" Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'simp'        Name
'\n\n'        Text

'text'        Keyword
'{*'          Comment
'It looks plausible as a simprule, but its effect can be strange.' Comment
'*}'          Comment
'\n'          Text

'lemma'       Keyword.Namespace
' '           Text
'power_0_left' Name
':'           Operator
'\n  '        Text
'"'           Literal.String
"0 ^ n = (if n = 0 then 1 else (0::'a::{power, semiring_0}))" Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'('           Operator
'induct'      Name
' '           Text
'n'           Name
')'           Operator
' '           Text
'simp_all'    Name
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'power_eq_0_iff' Name
' '           Text
'['           Operator
'simp'        Name
']'           Operator
':'           Operator
'\n  '        Text
'"'           Literal.String
'a ^ n = 0 '  Literal.String
'\\<longleftrightarrow>' Literal.String.Symbol
"\n     a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,power}) " Literal.String
'\\<and>'     Literal.String.Symbol
' n '         Literal.String
'\\<noteq>'   Literal.String.Symbol
' 0'          Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'('           Operator
'induct'      Name
' '           Text
'n'           Name
')'           Operator
'\n    '      Text
'('           Operator
'auto'        Name
' '           Text
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'no_zero_divisors' Name
' '           Text
'elim'        Name
':'           Operator
' '           Text
'contrapos_pp' Name
')'           Operator
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'('           Operator
'in'          Keyword.Pseudo
' '           Text
'field'       Name
')'           Operator
' '           Text
'power_diff'  Name
':'           Operator
'\n  '        Text
'assumes'     Keyword.Pseudo
' '           Text
'nz'          Name
':'           Operator
' '           Text
'"'           Literal.String
'a '          Literal.String
'\\<noteq>'   Literal.String.Symbol
' 0'          Literal.String
'"'           Literal.String
'\n  '        Text
'shows'       Keyword.Pseudo
' '           Text
'"'           Literal.String
'n '          Literal.String
'\\<le>'      Literal.String.Symbol
' m '         Literal.String
'\\<Longrightarrow>' Literal.String.Symbol
' a ^ (m - n) = a ^ m / a ^ n' Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'('           Operator
'induct'      Name
' '           Text
'm'           Name
' '           Text
'n'           Name
' '           Text
'rule'        Name
':'           Operator
' '           Text
'diff_induct' Name
')'           Operator
' '           Text
'('           Operator
'simp_all'    Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'nz'          Name
' '           Text
'field_power_not_zero' Name
')'           Operator
'\n\n'        Text

'text'        Keyword
'{*'          Comment
'Perhaps these should be simprules.' Comment
'*}'          Comment
'\n'          Text

'lemma'       Keyword.Namespace
' '           Text
'power_inverse' Name
':'           Operator
'\n  '        Text
'fixes'       Keyword.Pseudo
' '           Text
'a'           Name
' '           Text
'::'          Operator
' '           Text
'"'           Literal.String
"'a::division_ring_inverse_zero" Literal.String
'"'           Literal.String
'\n  '        Text
'shows'       Keyword.Pseudo
' '           Text
'"'           Literal.String
'inverse (a ^ n) = inverse a ^ n' Literal.String
'"'           Literal.String
'\n'          Text

'apply'       Keyword.Pseudo
' '           Text
'('           Operator
'cases'       Name
' '           Text
'"'           Literal.String
'a = 0'       Literal.String
'"'           Literal.String
')'           Operator
'\n'          Text

'apply'       Keyword.Pseudo
' '           Text
'('           Operator
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'power_0_left' Name
')'           Operator
'\n'          Text

'apply'       Keyword.Pseudo
' '           Text
'('           Operator
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'nonzero_power_inverse' Name
')'           Operator
'\n'          Text

'done'        Keyword
' '           Text
'(*'          Comment
' TODO: reorient or rename to inverse_power ' Comment
'*)'          Comment
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'power_one_over' Name
':'           Operator
'\n  '        Text
'"'           Literal.String
"1 / (a::'a::{field_inverse_zero, power}) ^ n =  (1 / a) ^ n" Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'('           Operator
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'divide_inverse' Name
')'           Operator
' '           Text
'('           Operator
'rule'        Name
' '           Text
'power_inverse' Name
')'           Operator
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'power_divide' Name
':'           Operator
'\n  '        Text
'"'           Literal.String
"(a / b) ^ n = (a::'a::field_inverse_zero) ^ n / b ^ n" Literal.String
'"'           Literal.String
'\n'          Text

'apply'       Keyword.Pseudo
' '           Text
'('           Operator
'cases'       Name
' '           Text
'"'           Literal.String
'b = 0'       Literal.String
'"'           Literal.String
')'           Operator
'\n'          Text

'apply'       Keyword.Pseudo
' '           Text
'('           Operator
'simp'        Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'power_0_left' Name
')'           Operator
'\n'          Text

'apply'       Keyword.Pseudo
' '           Text
'('           Operator
'rule'        Name
' '           Text
'nonzero_power_divide' Name
')'           Operator
'\n'          Text

'apply'       Keyword.Pseudo
' '           Text
'assumption'  Name
'\n'          Text

'done'        Keyword
'\n\n'        Text

'text'        Keyword
' '           Text
'{*'          Comment
' Simprules for comparisons where common factors can be cancelled. ' Comment
'*}'          Comment
'\n\n'        Text

'lemmas'      Keyword
' '           Text
'zero_compare_simps' Name
' '           Text
'='           Operator
'\n    '      Text
'add_strict_increasing' Name
' '           Text
'add_strict_increasing2' Name
' '           Text
'add_increasing' Name
'\n    '      Text
'zero_le_mult_iff' Name
' '           Text
'zero_le_divide_iff' Name
' \n    '     Text
'zero_less_mult_iff' Name
' '           Text
'zero_less_divide_iff' Name
' \n    '     Text
'mult_le_0_iff' Name
' '           Text
'divide_le_0_iff' Name
' \n    '     Text
'mult_less_0_iff' Name
' '           Text
'divide_less_0_iff' Name
' \n    '     Text
'zero_le_power2' Name
' '           Text
'power2_less_0' Name
'\n\n\n'      Text

'subsection'  Generic.Subheading
' '           Text
'{*'          Comment
' Exponentiation for the Natural Numbers ' Comment
'*}'          Comment
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'nat_one_le_power' Name
' '           Text
'['           Operator
'simp'        Name
']'           Operator
':'           Operator
'\n  '        Text
'"'           Literal.String
'Suc 0 '      Literal.String
'\\<le>'      Literal.String.Symbol
' i '         Literal.String
'\\<Longrightarrow>' Literal.String.Symbol
' Suc 0 '     Literal.String
'\\<le>'      Literal.String.Symbol
' i ^ n'      Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'('           Operator
'rule'        Name
' '           Text
'one_le_power' Name
' '           Text
'['           Operator
'of'          Name
' '           Text
'i'           Name
' '           Text
'n'           Name
','           Operator
' '           Text
'unfolded'    Name
' '           Text
'One_nat_def' Name
']'           Operator
')'           Operator
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'nat_zero_less_power_iff' Name
' '           Text
'['           Operator
'simp'        Name
']'           Operator
':'           Operator
'\n  '        Text
'"'           Literal.String
'x ^ n > 0 '  Literal.String
'\\<longleftrightarrow>' Literal.String.Symbol
' x > (0::nat) ' Literal.String
'\\<or>'      Literal.String.Symbol
' n = 0'      Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'('           Operator
'induct'      Name
' '           Text
'n'           Name
')'           Operator
' '           Text
'auto'        Name
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'nat_power_eq_Suc_0_iff' Name
' '           Text
'['           Operator
'simp'        Name
']'           Operator
':'           Operator
' \n  '       Text
'"'           Literal.String
'x ^ m = Suc 0 ' Literal.String
'\\<longleftrightarrow>' Literal.String.Symbol
' m = 0 '     Literal.String
'\\<or>'      Literal.String.Symbol
' x = Suc 0'  Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'('           Operator
'induct'      Name
' '           Text
'm'           Name
')'           Operator
' '           Text
'auto'        Name
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'power_Suc_0' Name
' '           Text
'['           Operator
'simp'        Name
']'           Operator
':'           Operator
'\n  '        Text
'"'           Literal.String
'Suc 0 ^ n = Suc 0' Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'simp'        Name
'\n\n'        Text

'text'        Keyword
'{*'          Comment
'Valid for the naturals, but what if @{text"0<i<1"' Comment
'}'           Comment
'?\nPremises cannot be weakened: consider the case where @{term "i=0"' Comment
'}'           Comment
',\n@{term "m=1"' Comment
'}'           Comment
' and @{term "n=0"' Comment
'}'           Comment
'.'           Comment
'*}'          Comment
'\n'          Text

'lemma'       Keyword.Namespace
' '           Text
'nat_power_less_imp_less' Name
':'           Operator
'\n  '        Text
'assumes'     Keyword.Pseudo
' '           Text
'nonneg'      Name
':'           Operator
' '           Text
'"'           Literal.String
'0 < (i'      Literal.String
'\\<Colon>'   Literal.String.Symbol
'nat)'        Literal.String
'"'           Literal.String
'\n  '        Text
'assumes'     Keyword.Pseudo
' '           Text
'less'        Name
':'           Operator
' '           Text
'"'           Literal.String
'i ^ m < i ^ n' Literal.String
'"'           Literal.String
'\n  '        Text
'shows'       Keyword.Pseudo
' '           Text
'"'           Literal.String
'm < n'       Literal.String
'"'           Literal.String
'\n'          Text

'proof'       Keyword
' '           Text
'('           Operator
'cases'       Name
' '           Text
'"'           Literal.String
'i = 1'       Literal.String
'"'           Literal.String
')'           Operator
'\n  '        Text
'case'        Keyword
' '           Text
'True'        Name
' '           Text
'with'        Keyword
' '           Text
'less'        Name
' '           Text
'power_one'   Name
' '           Text
'['           Operator
'where'       Keyword.Pseudo
' '           Text
"'a"          Name.Type
' '           Text
'='           Operator
' '           Text
'nat'         Name
']'           Operator
' '           Text
'show'        Keyword
' '           Text
'?'           Operator
'thesis'      Name
' '           Text
'by'          Keyword
' '           Text
'simp'        Name
'\n'          Text

'next'        Keyword
'\n  '        Text
'case'        Keyword
' '           Text
'False'       Name
' '           Text
'with'        Keyword
' '           Text
'nonneg'      Name
' '           Text
'have'        Keyword
' '           Text
'"'           Literal.String
'1 < i'       Literal.String
'"'           Literal.String
' '           Text
'by'          Keyword
' '           Text
'auto'        Name
'\n  '        Text
'from'        Keyword
' '           Text
'power_strict_increasing_iff' Name
' '           Text
'['           Operator
'OF'          Name
' '           Text
'this'        Name
']'           Operator
' '           Text
'less'        Name
' '           Text
'show'        Keyword
' '           Text
'?'           Operator
'thesis'      Name
' '           Text
'..'          Operator.Word
'\n'          Text

'qed'         Keyword
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'power_dvd_imp_le' Name
':'           Operator
'\n  '        Text
'"'           Literal.String
'i ^ m dvd i ^ n ' Literal.String
'\\<Longrightarrow>' Literal.String.Symbol
' (1::nat) < i ' Literal.String
'\\<Longrightarrow>' Literal.String.Symbol
' m '         Literal.String
'\\<le>'      Literal.String.Symbol
' n'          Literal.String
'"'           Literal.String
'\n  '        Text
'apply'       Keyword.Pseudo
' '           Text
'('           Operator
'rule'        Name
' '           Text
'power_le_imp_le_exp' Name
','           Operator
' '           Text
'assumption'  Name
')'           Operator
'\n  '        Text
'apply'       Keyword.Pseudo
' '           Text
'('           Operator
'erule'       Name
' '           Text
'dvd_imp_le'  Name
','           Operator
' '           Text
'simp'        Name
')'           Operator
'\n  '        Text
'done'        Keyword
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'power2_nat_le_eq_le' Name
':'           Operator
'\n  '        Text
'fixes'       Keyword.Pseudo
' '           Text
'm'           Name
' '           Text
'n'           Name
' '           Text
'::'          Operator
' '           Text
'nat'         Name
'\n  '        Text
'shows'       Keyword.Pseudo
' '           Text
'"'           Literal.String
'm'           Literal.String
'\\'          Literal.String
'<^sup>2 '    Literal.String
'\\<le>'      Literal.String.Symbol
' n'          Literal.String
'\\'          Literal.String
'<^sup>2 '    Literal.String
'\\<longleftrightarrow>' Literal.String.Symbol
' m '         Literal.String
'\\<le>'      Literal.String.Symbol
' n'          Literal.String
'"'           Literal.String
'\n  '        Text
'by'          Keyword
' '           Text
'('           Operator
'auto'        Name
' '           Text
'intro'       Name
':'           Operator
' '           Text
'power2_le_imp_le' Name
' '           Text
'power_mono'  Name
')'           Operator
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'power2_nat_le_imp_le' Name
':'           Operator
'\n  '        Text
'fixes'       Keyword.Pseudo
' '           Text
'm'           Name
' '           Text
'n'           Name
' '           Text
'::'          Operator
' '           Text
'nat'         Name
'\n  '        Text
'assumes'     Keyword.Pseudo
' '           Text
'"'           Literal.String
'm'           Literal.String
'\\'          Literal.String
'<^sup>2 '    Literal.String
'\\<le>'      Literal.String.Symbol
' n'          Literal.String
'"'           Literal.String
'\n  '        Text
'shows'       Keyword.Pseudo
' '           Text
'"'           Literal.String
'm '          Literal.String
'\\<le>'      Literal.String.Symbol
' n'          Literal.String
'"'           Literal.String
'\n  '        Text
'using'       Keyword
' '           Text
'assms'       Name
' '           Text
'by'          Keyword
' '           Text
'('           Operator
'cases'       Name
' '           Text
'm'           Name
')'           Operator
' '           Text
'('           Operator
'simp_all'    Name
' '           Text
'add'         Name
':'           Operator
' '           Text
'power2_eq_square' Name
')'           Operator
'\n\n\n\n'    Text

'subsection'  Generic.Subheading
' '           Text
'{*'          Comment
' Code generator tweak ' Comment
'*}'          Comment
'\n\n'        Text

'lemma'       Keyword.Namespace
' '           Text
'power_power_power' Name
' '           Text
'['           Operator
'code'        Name
']'           Operator
':'           Operator
'\n  '        Text
'"'           Literal.String
"power = power.power (1::'a::{power}) (op *)" Literal.String
'"'           Literal.String
'\n  '        Text
'unfolding'   Keyword
' '           Text
'power_def'   Name
' '           Text
'power.power_def' Name
' '           Text
'..'          Operator.Word
'\n\n'        Text

'declare'     Keyword
' '           Text
'power.power.simps' Name
' '           Text
'['           Operator
'code'        Name
']'           Operator
'\n\n'        Text

'code_identifier' Keyword
'\n  '        Text
'code_module' Keyword.Pseudo
' '           Text
'Power'       Name
' '           Text
'\\<rightharpoonup>' Text.Symbol
' '           Text
'('           Operator
'SML'         Name
')'           Operator
' '           Text
'Arith'       Name
' '           Text
'and'         Keyword.Pseudo
' '           Text
'('           Operator
'OCaml'       Name
')'           Operator
' '           Text
'Arith'       Name
' '           Text
'and'         Keyword.Pseudo
' '           Text
'('           Operator
'Haskell'     Name
')'           Operator
' '           Text
'Arith'       Name
'\n\n'        Text

'end'         Keyword
'\n'          Text