(*  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

 definition

   powr  :: "[real,real] => real"     (infixr "powr" 80) where

     --{*exponentation with real exponent*}

   "x powr a = exp(a * ln x)"

 definition

   log :: "[real,real] => real" where

     --{*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 \<noteq> 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 \<le> x powr b) = (a \<le> b)"

 by (simp add: linorder_not_less [symmetric])

 lemma log_ln: "ln x = log (exp(1)) x"

 by (simp add: log_def)

 lemma DERIV_log: "x > 0 ==> DERIV (%y. log b y) x :> 1 / (ln b * x)"

   apply (subst log_def)

   apply (subgoal_tac "(%y. ln y / ln b) = (%y. (1 / ln b) * ln y)")

   apply (erule ssubst)

   apply (subgoal_tac "1 / (ln b * x) = (1 / ln b) * (1 / x)")

   apply (erule ssubst)

   apply (rule DERIV_cmult)

   apply (erule DERIV_ln_divide)

   apply auto

 done

 lemma powr_log_cancel [simp]:

      "[| 0 < a; a \<noteq> 1; 0 < x |] ==> a powr (log a x) = x"

 by (simp add: powr_def log_def)

 lemma log_powr_cancel [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a (a powr y) = y"

 by (simp add: log_def powr_def)

 lemma log_mult: 

      "[| 0 < a; a \<noteq> 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 \<noteq> 1; 0 < b; b \<noteq> 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 \<noteq> 1 |] ==> log a a = 1"

 by (simp add: log_def)

 lemma log_inverse:

      "[| 0 < a; a \<noteq> 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 \<noteq> 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 \<le> log a y) = (x \<le> 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_powr: "0 < x ==> 0 < y ==> ln(x powr y) = y * ln x"

 by (unfold powr_def, simp)

 lemma log_powr: "0 < x ==> 0 \<le> y ==> log b (x powr y) = y * log b x"

   apply (case_tac "y = 0")

   apply force

   apply (auto simp add: log_def ln_powr field_simps)

 done

 lemma log_nat_power: "0 < x ==> log b (x^n) = real n * log b x"

   apply (subst powr_realpow [symmetric])

   apply (auto simp add: log_powr)

 done

 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_powr [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_iff)

   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

 end