# Theory Log

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theory Log
imports Transcendental
begin

    (*  Title       : Log.thy
        Author      : Jacques D. Fleuriot
                      Additional contributions by Jeremy Avigad
        Copyright   : 2000,2001 University of Edinburgh
    *)

    header{*Logarithms: Standard Version*}

    theory Log
    imports Transcendental
    begin

    constdefs

      powr  :: "[real,real] => real"     (infixr "powr" 80)
        --{*exponentation with real exponent*}
        "x powr a == exp(a * ln x)"

      log :: "[real,real] => real"
        --{*logarithm of @{term x} to base @{term a}*}
        "log a x == ln x / ln a"

    lemma powr_one_eq_one [simp]: "1 powr a = 1"
    by (simp add: powr_def)

    lemma powr_zero_eq_one [simp]: "x powr 0 = 1"
    by (simp add: powr_def)

    lemma powr_one_gt_zero_iff [simp]: "(x powr 1 = x) = (0 < x)"
    by (simp add: powr_def)
    declare powr_one_gt_zero_iff [THEN iffD2, simp]

    lemma powr_mult:
          "[| 0 < x; 0 < y |] ==> (x * y) powr a = (x powr a) * (y powr a)"
    by (simp add: powr_def exp_add [symmetric] ln_mult right_distrib)

    lemma powr_gt_zero [simp]: "0 < x powr a"
    by (simp add: powr_def)

    lemma powr_ge_pzero [simp]: "0 <= x powr y"
    by (rule order_less_imp_le, rule powr_gt_zero)

    lemma powr_not_zero [simp]: "x powr a ≠ 0"
    by (simp add: powr_def)

    lemma powr_divide:
         "[| 0 < x; 0 < y |] ==> (x / y) powr a = (x powr a)/(y powr a)"
    apply (simp add: divide_inverse positive_imp_inverse_positive powr_mult)
    apply (simp add: powr_def exp_minus [symmetric] exp_add [symmetric] ln_inverse)
    done

    lemma powr_divide2: "x powr a / x powr b = x powr (a - b)"
      apply (simp add: powr_def)
      apply (subst exp_diff [THEN sym])
      apply (simp add: left_diff_distrib)
    done

    lemma powr_add: "x powr (a + b) = (x powr a) * (x powr b)"
    by (simp add: powr_def exp_add [symmetric] left_distrib)

    lemma powr_powr: "(x powr a) powr b = x powr (a * b)"
    by (simp add: powr_def)

    lemma powr_powr_swap: "(x powr a) powr b = (x powr b) powr a"
    by (simp add: powr_powr real_mult_commute)

    lemma powr_minus: "x powr (-a) = inverse (x powr a)"
    by (simp add: powr_def exp_minus [symmetric])

    lemma powr_minus_divide: "x powr (-a) = 1/(x powr a)"
    by (simp add: divide_inverse powr_minus)

    lemma powr_less_mono: "[| a < b; 1 < x |] ==> x powr a < x powr b"
    by (simp add: powr_def)

    lemma powr_less_cancel: "[| x powr a < x powr b; 1 < x |] ==> a < b"
    by (simp add: powr_def)

    lemma powr_less_cancel_iff [simp]: "1 < x ==> (x powr a < x powr b) = (a < b)"
    by (blast intro: powr_less_cancel powr_less_mono)

    lemma powr_le_cancel_iff [simp]: "1 < x ==> (x powr a ≤ x powr b) = (a ≤ b)"
    by (simp add: linorder_not_less [symmetric])

    lemma log_ln: "ln x = log (exp(1)) x"
    by (simp add: log_def)

    lemma powr_log_cancel [simp]:
         "[| 0 < a; a ≠ 1; 0 < x |] ==> a powr (log a x) = x"
    by (simp add: powr_def log_def)

    lemma log_powr_cancel [simp]: "[| 0 < a; a ≠ 1 |] ==> log a (a powr y) = y"
    by (simp add: log_def powr_def)

    lemma log_mult:
         "[| 0 < a; a ≠ 1; 0 < x; 0 < y |]
          ==> log a (x * y) = log a x + log a y"
    by (simp add: log_def ln_mult divide_inverse left_distrib)

    lemma log_eq_div_ln_mult_log:
         "[| 0 < a; a ≠ 1; 0 < b; b ≠ 1; 0 < x |]
          ==> log a x = (ln b/ln a) * log b x"
    by (simp add: log_def divide_inverse)

    text{*Base 10 logarithms*}
    lemma log_base_10_eq1: "0 < x ==> log 10 x = (ln (exp 1) / ln 10) * ln x"
    by (simp add: log_def)

    lemma log_base_10_eq2: "0 < x ==> log 10 x = (log 10 (exp 1)) * ln x"
    by (simp add: log_def)

    lemma log_one [simp]: "log a 1 = 0"
    by (simp add: log_def)

    lemma log_eq_one [simp]: "[| 0 < a; a ≠ 1 |] ==> log a a = 1"
    by (simp add: log_def)

    lemma log_inverse:
         "[| 0 < a; a ≠ 1; 0 < x |] ==> log a (inverse x) = - log a x"
    apply (rule_tac a1 = "log a x" in add_left_cancel [THEN iffD1])
    apply (simp add: log_mult [symmetric])
    done

    lemma log_divide:
         "[|0 < a; a ≠ 1; 0 < x; 0 < y|] ==> log a (x/y) = log a x - log a y"
    by (simp add: log_mult divide_inverse log_inverse)

    lemma log_less_cancel_iff [simp]:
         "[| 1 < a; 0 < x; 0 < y |] ==> (log a x < log a y) = (x < y)"
    apply safe
    apply (rule_tac [2] powr_less_cancel)
    apply (drule_tac a = "log a x" in powr_less_mono, auto)
    done

    lemma log_le_cancel_iff [simp]:
         "[| 1 < a; 0 < x; 0 < y |] ==> (log a x ≤ log a y) = (x ≤ y)"
    by (simp add: linorder_not_less [symmetric])

    lemma powr_realpow: "0 < x ==> x powr (real n) = x^n"
      apply (induct n, simp)
      apply (subgoal_tac "real(Suc n) = real n + 1")
      apply (erule ssubst)
      apply (subst powr_add, simp, simp)
    done

    lemma powr_realpow2: "0 <= x ==> 0 < n ==> x^n = (if (x = 0) then 0
      else x powr (real n))"
      apply (case_tac "x = 0", simp, simp)
      apply (rule powr_realpow [THEN sym], simp)
    done

    lemma ln_pwr: "0 < x ==> 0 < y ==> ln(x powr y) = y * ln x"
    by (unfold powr_def, simp)

    lemma ln_bound: "1 <= x ==> ln x <= x"
      apply (subgoal_tac "ln(1 + (x - 1)) <= x - 1")
      apply simp
      apply (rule ln_add_one_self_le_self, simp)
    done

    lemma powr_mono: "a <= b ==> 1 <= x ==> x powr a <= x powr b"
      apply (case_tac "x = 1", simp)
      apply (case_tac "a = b", simp)
      apply (rule order_less_imp_le)
      apply (rule powr_less_mono, auto)
    done

    lemma ge_one_powr_ge_zero: "1 <= x ==> 0 <= a ==> 1 <= x powr a"
      apply (subst powr_zero_eq_one [THEN sym])
      apply (rule powr_mono, assumption+)
    done

    lemma powr_less_mono2: "0 < a ==> 0 < x ==> x < y ==> x powr a <
        y powr a"
      apply (unfold powr_def)
      apply (rule exp_less_mono)
      apply (rule mult_strict_left_mono)
      apply (subst ln_less_cancel_iff, assumption)
      apply (rule order_less_trans)
      prefer 2
      apply assumption+
    done

    lemma powr_less_mono2_neg: "a < 0 ==> 0 < x ==> x < y ==> y powr a <
        x powr a"
      apply (unfold powr_def)
      apply (rule exp_less_mono)
      apply (rule mult_strict_left_mono_neg)
      apply (subst ln_less_cancel_iff)
      apply assumption
      apply (rule order_less_trans)
      prefer 2
      apply assumption+
    done

    lemma powr_mono2: "0 <= a ==> 0 < x ==> x <= y ==> x powr a <= y powr a"
      apply (case_tac "a = 0", simp)
      apply (case_tac "x = y", simp)
      apply (rule order_less_imp_le)
      apply (rule powr_less_mono2, auto)
    done

    lemma ln_powr_bound: "1 <= x ==> 0 < a ==> ln x <= (x powr a) / a"
      apply (rule mult_imp_le_div_pos)
      apply (assumption)
      apply (subst mult_commute)
      apply (subst ln_pwr [THEN sym])
      apply auto
      apply (rule ln_bound)
      apply (erule ge_one_powr_ge_zero)
      apply (erule order_less_imp_le)
    done

    lemma ln_powr_bound2: "1 < x ==> 0 < a ==> (ln x) powr a <= (a powr a) * x"
    proof -
      assume "1 < x" and "0 < a"
      then have "ln x <= (x powr (1 / a)) / (1 / a)"
        apply (intro ln_powr_bound)
        apply (erule order_less_imp_le)
        apply (rule divide_pos_pos)
        apply simp_all
        done
      also have "... = a * (x powr (1 / a))"
        by simp
      finally have "(ln x) powr a <= (a * (x powr (1 / a))) powr a"
        apply (intro powr_mono2)
        apply (rule order_less_imp_le, rule prems)
        apply (rule ln_gt_zero)
        apply (rule prems)
        apply assumption
        done
      also have "... = (a powr a) * ((x powr (1 / a)) powr a)"
        apply (rule powr_mult)
        apply (rule prems)
        apply (rule powr_gt_zero)
        done
      also have "(x powr (1 / a)) powr a = x powr ((1 / a) * a)"
        by (rule powr_powr)
      also have "... = x"
        apply simp
        apply (subgoal_tac "a ~= 0")
        apply (insert prems, auto)
        done
      finally show ?thesis .
    qed

    lemma LIMSEQ_neg_powr: "0 < s ==> (%x. (real x) powr - s) ----> 0"
      apply (unfold LIMSEQ_def)
      apply clarsimp
      apply (rule_tac x = "natfloor(r powr (1 / - s)) + 1" in exI)
      apply clarify
      proof -
        fix r fix n
        assume "0 < s" and "0 < r" and "natfloor (r powr (1 / - s)) + 1 <= n"
        have "r powr (1 / - s) < real(natfloor(r powr (1 / - s))) + 1"
          by (rule real_natfloor_add_one_gt)
        also have "... = real(natfloor(r powr (1 / -s)) + 1)"
          by simp
        also have "... <= real n"
          apply (subst real_of_nat_le_iff)
          apply (rule prems)
          done
        finally have "r powr (1 / - s) < real n".
        then have "real n powr (- s) < (r powr (1 / - s)) powr - s"
          apply (intro powr_less_mono2_neg)
          apply (auto simp add: prems)
          done
        also have "... = r"
          by (simp add: powr_powr prems less_imp_neq [THEN not_sym])
        finally show "real n powr - s < r" .
      qed

    ML
    {*
    val powr_one_eq_one = thm "powr_one_eq_one";
    val powr_zero_eq_one = thm "powr_zero_eq_one";
    val powr_one_gt_zero_iff = thm "powr_one_gt_zero_iff";
    val powr_mult = thm "powr_mult";
    val powr_gt_zero = thm "powr_gt_zero";
    val powr_not_zero = thm "powr_not_zero";
    val powr_divide = thm "powr_divide";
    val powr_add = thm "powr_add";
    val powr_powr = thm "powr_powr";
    val powr_powr_swap = thm "powr_powr_swap";
    val powr_minus = thm "powr_minus";
    val powr_minus_divide = thm "powr_minus_divide";
    val powr_less_mono = thm "powr_less_mono";
    val powr_less_cancel = thm "powr_less_cancel";
    val powr_less_cancel_iff = thm "powr_less_cancel_iff";
    val powr_le_cancel_iff = thm "powr_le_cancel_iff";
    val log_ln = thm "log_ln";
    val powr_log_cancel = thm "powr_log_cancel";
    val log_powr_cancel = thm "log_powr_cancel";
    val log_mult = thm "log_mult";
    val log_eq_div_ln_mult_log = thm "log_eq_div_ln_mult_log";
    val log_base_10_eq1 = thm "log_base_10_eq1";
    val log_base_10_eq2 = thm "log_base_10_eq2";
    val log_one = thm "log_one";
    val log_eq_one = thm "log_eq_one";
    val log_inverse = thm "log_inverse";
    val log_divide = thm "log_divide";
    val log_less_cancel_iff = thm "log_less_cancel_iff";
    val log_le_cancel_iff = thm "log_le_cancel_iff";
    *}

    end

lemma powr_one_eq_one:

      1 powr a = 1

lemma powr_zero_eq_one:

      x powr 0 = 1

lemma powr_one_gt_zero_iff:

      (x powr 1 = x) = (0 < x)

lemma powr_mult:

      [| 0 < x; 0 < y |] ==> (x * y) powr a = x powr a * y powr a

lemma powr_gt_zero:

      0 < x powr a

lemma powr_ge_pzero:

      0 ≤ x powr y

lemma powr_not_zero:

      x powr a ≠ 0

lemma powr_divide:

      [| 0 < x; 0 < y |] ==> (x / y) powr a = x powr a / y powr a

lemma powr_divide2:

      x powr a / x powr b = x powr (a - b)

lemma powr_add:

      x powr (a + b) = x powr a * x powr b

lemma powr_powr:

      (x powr a) powr b = x powr (a * b)

lemma powr_powr_swap:

      (x powr a) powr b = (x powr b) powr a

lemma powr_minus:

      x powr - a = inverse (x powr a)

lemma powr_minus_divide:

      x powr - a = 1 / x powr a

lemma powr_less_mono:

      [| a < b; 1 < x |] ==> x powr a < x powr b

lemma powr_less_cancel:

      [| x powr a < x powr b; 1 < x |] ==> a < b

lemma powr_less_cancel_iff:

      1 < x ==> (x powr a < x powr b) = (a < b)

lemma powr_le_cancel_iff:

      1 < x ==> (x powr a ≤ x powr b) = (a ≤ b)

lemma log_ln:

      ln x = log (exp 1) x

lemma powr_log_cancel:

      [| 0 < a; a ≠ 1; 0 < x |] ==> a powr log a x = x

lemma log_powr_cancel:

      [| 0 < a; a ≠ 1 |] ==> log a (a powr y) = y

lemma log_mult:

      [| 0 < a; a ≠ 1; 0 < x; 0 < y |] ==> log a (x * y) = log a x + log a y

lemma log_eq_div_ln_mult_log:

      [| 0 < a; a ≠ 1; 0 < b; b ≠ 1; 0 < x |] ==> log a x = ln b / ln a * log b x

lemma log_base_10_eq1:

      0 < x ==> log 10 x = ln (exp 1) / ln 10 * ln x

lemma log_base_10_eq2:

      0 < x ==> log 10 x = log 10 (exp 1) * ln x

lemma log_one:

      log a 1 = 0

lemma log_eq_one:

      [| 0 < a; a ≠ 1 |] ==> log a a = 1

lemma log_inverse:

      [| 0 < a; a ≠ 1; 0 < x |] ==> log a (inverse x) = - log a x

lemma log_divide:

      [| 0 < a; a ≠ 1; 0 < x; 0 < y |] ==> log a (x / y) = log a x - log a y

lemma log_less_cancel_iff:

      [| 1 < a; 0 < x; 0 < y |] ==> (log a x < log a y) = (x < y)

lemma log_le_cancel_iff:

      [| 1 < a; 0 < x; 0 < y |] ==> (log a x ≤ log a y) = (x ≤ y)

lemma powr_realpow:

      0 < x ==> x powr real n = x ^ n

lemma powr_realpow2:

      [| 0 ≤ x; 0 < n |] ==> x ^ n = (if x = 0 then 0 else x powr real n)

lemma ln_pwr:

      [| 0 < x; 0 < y |] ==> ln (x powr y) = y * ln x

lemma ln_bound:

      1 ≤ x ==> ln x ≤ x

lemma powr_mono:

      [| a ≤ b; 1 ≤ x |] ==> x powr a ≤ x powr b

lemma ge_one_powr_ge_zero:

      [| 1 ≤ x; 0 ≤ a |] ==> 1 ≤ x powr a

lemma powr_less_mono2:

      [| 0 < a; 0 < x; x < y |] ==> x powr a < y powr a

lemma powr_less_mono2_neg:

      [| a < 0; 0 < x; x < y |] ==> y powr a < x powr a

lemma powr_mono2:

      [| 0 ≤ a; 0 < x; x ≤ y |] ==> x powr a ≤ y powr a

lemma ln_powr_bound:

      [| 1 ≤ x; 0 < a |] ==> ln x ≤ x powr a / a

lemma ln_powr_bound2:

      [| 1 < x; 0 < a |] ==> ln x powr a ≤ a powr a * x

lemma LIMSEQ_neg_powr:

      0 < s ==> (%x. real x powr - s) ----> 0