src/HOL/Ln.thy
 author hoelzl Thu Jun 09 11:50:16 2011 +0200 (2011-06-09) changeset 43336 05aa7380f7fc parent 41959 b460124855b8 child 44289 d81d09cdab9c permissions -rw-r--r--
lemmas relating ln x and x - 1
```     1 (*  Title:      HOL/Ln.thy
```
```     2     Author:     Jeremy Avigad
```
```     3 *)
```
```     4
```
```     5 header {* Properties of ln *}
```
```     6
```
```     7 theory Ln
```
```     8 imports Transcendental
```
```     9 begin
```
```    10
```
```    11 lemma exp_first_two_terms: "exp x = 1 + x + suminf (%n.
```
```    12   inverse(fact (n+2)) * (x ^ (n+2)))"
```
```    13 proof -
```
```    14   have "exp x = suminf (%n. inverse(fact n) * (x ^ n))"
```
```    15     by (simp add: exp_def)
```
```    16   also from summable_exp have "... = (SUM n::nat : {0..<2}.
```
```    17       inverse(fact n) * (x ^ n)) + suminf (%n.
```
```    18       inverse(fact(n+2)) * (x ^ (n+2)))" (is "_ = ?a + _")
```
```    19     by (rule suminf_split_initial_segment)
```
```    20   also have "?a = 1 + x"
```
```    21     by (simp add: numerals)
```
```    22   finally show ?thesis .
```
```    23 qed
```
```    24
```
```    25 lemma exp_tail_after_first_two_terms_summable:
```
```    26   "summable (%n. inverse(fact (n+2)) * (x ^ (n+2)))"
```
```    27 proof -
```
```    28   note summable_exp
```
```    29   thus ?thesis
```
```    30     by (frule summable_ignore_initial_segment)
```
```    31 qed
```
```    32
```
```    33 lemma aux1: assumes a: "0 <= x" and b: "x <= 1"
```
```    34     shows "inverse (fact ((n::nat) + 2)) * x ^ (n + 2) <= (x^2/2) * ((1/2)^n)"
```
```    35 proof (induct n)
```
```    36   show "inverse (fact ((0::nat) + 2)) * x ^ (0 + 2) <=
```
```    37       x ^ 2 / 2 * (1 / 2) ^ 0"
```
```    38     by (simp add: real_of_nat_Suc power2_eq_square)
```
```    39 next
```
```    40   fix n :: nat
```
```    41   assume c: "inverse (fact (n + 2)) * x ^ (n + 2)
```
```    42        <= x ^ 2 / 2 * (1 / 2) ^ n"
```
```    43   show "inverse (fact (Suc n + 2)) * x ^ (Suc n + 2)
```
```    44            <= x ^ 2 / 2 * (1 / 2) ^ Suc n"
```
```    45   proof -
```
```    46     have "inverse(fact (Suc n + 2)) <= (1/2) * inverse (fact (n+2))"
```
```    47     proof -
```
```    48       have "Suc n + 2 = Suc (n + 2)" by simp
```
```    49       then have "fact (Suc n + 2) = Suc (n + 2) * fact (n + 2)"
```
```    50         by simp
```
```    51       then have "real(fact (Suc n + 2)) = real(Suc (n + 2) * fact (n + 2))"
```
```    52         apply (rule subst)
```
```    53         apply (rule refl)
```
```    54         done
```
```    55       also have "... = real(Suc (n + 2)) * real(fact (n + 2))"
```
```    56         by (rule real_of_nat_mult)
```
```    57       finally have "real (fact (Suc n + 2)) =
```
```    58          real (Suc (n + 2)) * real (fact (n + 2))" .
```
```    59       then have "inverse(fact (Suc n + 2)) =
```
```    60          inverse(Suc (n + 2)) * inverse(fact (n + 2))"
```
```    61         apply (rule ssubst)
```
```    62         apply (rule inverse_mult_distrib)
```
```    63         done
```
```    64       also have "... <= (1/2) * inverse(fact (n + 2))"
```
```    65         apply (rule mult_right_mono)
```
```    66         apply (subst inverse_eq_divide)
```
```    67         apply simp
```
```    68         apply (rule inv_real_of_nat_fact_ge_zero)
```
```    69         done
```
```    70       finally show ?thesis .
```
```    71     qed
```
```    72     moreover have "x ^ (Suc n + 2) <= x ^ (n + 2)"
```
```    73       apply (simp add: mult_compare_simps)
```
```    74       apply (simp add: assms)
```
```    75       apply (subgoal_tac "0 <= x * (x * x^n)")
```
```    76       apply force
```
```    77       apply (rule mult_nonneg_nonneg, rule a)+
```
```    78       apply (rule zero_le_power, rule a)
```
```    79       done
```
```    80     ultimately have "inverse (fact (Suc n + 2)) *  x ^ (Suc n + 2) <=
```
```    81         (1 / 2 * inverse (fact (n + 2))) * x ^ (n + 2)"
```
```    82       apply (rule mult_mono)
```
```    83       apply (rule mult_nonneg_nonneg)
```
```    84       apply simp
```
```    85       apply (subst inverse_nonnegative_iff_nonnegative)
```
```    86       apply (rule real_of_nat_ge_zero)
```
```    87       apply (rule zero_le_power)
```
```    88       apply (rule a)
```
```    89       done
```
```    90     also have "... = 1 / 2 * (inverse (fact (n + 2)) * x ^ (n + 2))"
```
```    91       by simp
```
```    92     also have "... <= 1 / 2 * (x ^ 2 / 2 * (1 / 2) ^ n)"
```
```    93       apply (rule mult_left_mono)
```
```    94       apply (rule c)
```
```    95       apply simp
```
```    96       done
```
```    97     also have "... = x ^ 2 / 2 * (1 / 2 * (1 / 2) ^ n)"
```
```    98       by auto
```
```    99     also have "(1::real) / 2 * (1 / 2) ^ n = (1 / 2) ^ (Suc n)"
```
```   100       by (rule power_Suc [THEN sym])
```
```   101     finally show ?thesis .
```
```   102   qed
```
```   103 qed
```
```   104
```
```   105 lemma aux2: "(%n. (x::real) ^ 2 / 2 * (1 / 2) ^ n) sums x^2"
```
```   106 proof -
```
```   107   have "(%n. (1 / 2::real)^n) sums (1 / (1 - (1/2)))"
```
```   108     apply (rule geometric_sums)
```
```   109     by (simp add: abs_less_iff)
```
```   110   also have "(1::real) / (1 - 1/2) = 2"
```
```   111     by simp
```
```   112   finally have "(%n. (1 / 2::real)^n) sums 2" .
```
```   113   then have "(%n. x ^ 2 / 2 * (1 / 2) ^ n) sums (x^2 / 2 * 2)"
```
```   114     by (rule sums_mult)
```
```   115   also have "x^2 / 2 * 2 = x^2"
```
```   116     by simp
```
```   117   finally show ?thesis .
```
```   118 qed
```
```   119
```
```   120 lemma exp_bound: "0 <= (x::real) ==> x <= 1 ==> exp x <= 1 + x + x^2"
```
```   121 proof -
```
```   122   assume a: "0 <= x"
```
```   123   assume b: "x <= 1"
```
```   124   have c: "exp x = 1 + x + suminf (%n. inverse(fact (n+2)) *
```
```   125       (x ^ (n+2)))"
```
```   126     by (rule exp_first_two_terms)
```
```   127   moreover have "suminf (%n. inverse(fact (n+2)) * (x ^ (n+2))) <= x^2"
```
```   128   proof -
```
```   129     have "suminf (%n. inverse(fact (n+2)) * (x ^ (n+2))) <=
```
```   130         suminf (%n. (x^2/2) * ((1/2)^n))"
```
```   131       apply (rule summable_le)
```
```   132       apply (auto simp only: aux1 a b)
```
```   133       apply (rule exp_tail_after_first_two_terms_summable)
```
```   134       by (rule sums_summable, rule aux2)
```
```   135     also have "... = x^2"
```
```   136       by (rule sums_unique [THEN sym], rule aux2)
```
```   137     finally show ?thesis .
```
```   138   qed
```
```   139   ultimately show ?thesis
```
```   140     by auto
```
```   141 qed
```
```   142
```
```   143 lemma aux4: "0 <= (x::real) ==> x <= 1 ==> exp (x - x^2) <= 1 + x"
```
```   144 proof -
```
```   145   assume a: "0 <= x" and b: "x <= 1"
```
```   146   have "exp (x - x^2) = exp x / exp (x^2)"
```
```   147     by (rule exp_diff)
```
```   148   also have "... <= (1 + x + x^2) / exp (x ^2)"
```
```   149     apply (rule divide_right_mono)
```
```   150     apply (rule exp_bound)
```
```   151     apply (rule a, rule b)
```
```   152     apply simp
```
```   153     done
```
```   154   also have "... <= (1 + x + x^2) / (1 + x^2)"
```
```   155     apply (rule divide_left_mono)
```
```   156     apply (auto simp add: exp_ge_add_one_self_aux)
```
```   157     apply (rule add_nonneg_nonneg)
```
```   158     using a apply auto
```
```   159     apply (rule mult_pos_pos)
```
```   160     apply auto
```
```   161     apply (rule add_pos_nonneg)
```
```   162     apply auto
```
```   163     done
```
```   164   also from a have "... <= 1 + x"
```
```   165     by (simp add: field_simps zero_compare_simps)
```
```   166   finally show ?thesis .
```
```   167 qed
```
```   168
```
```   169 lemma ln_one_plus_pos_lower_bound: "0 <= x ==> x <= 1 ==>
```
```   170     x - x^2 <= ln (1 + x)"
```
```   171 proof -
```
```   172   assume a: "0 <= x" and b: "x <= 1"
```
```   173   then have "exp (x - x^2) <= 1 + x"
```
```   174     by (rule aux4)
```
```   175   also have "... = exp (ln (1 + x))"
```
```   176   proof -
```
```   177     from a have "0 < 1 + x" by auto
```
```   178     thus ?thesis
```
```   179       by (auto simp only: exp_ln_iff [THEN sym])
```
```   180   qed
```
```   181   finally have "exp (x - x ^ 2) <= exp (ln (1 + x))" .
```
```   182   thus ?thesis by (auto simp only: exp_le_cancel_iff)
```
```   183 qed
```
```   184
```
```   185 lemma ln_one_minus_pos_upper_bound: "0 <= x ==> x < 1 ==> ln (1 - x) <= - x"
```
```   186 proof -
```
```   187   assume a: "0 <= (x::real)" and b: "x < 1"
```
```   188   have "(1 - x) * (1 + x + x^2) = (1 - x^3)"
```
```   189     by (simp add: algebra_simps power2_eq_square power3_eq_cube)
```
```   190   also have "... <= 1"
```
```   191     by (auto simp add: a)
```
```   192   finally have "(1 - x) * (1 + x + x ^ 2) <= 1" .
```
```   193   moreover have "0 < 1 + x + x^2"
```
```   194     apply (rule add_pos_nonneg)
```
```   195     using a apply auto
```
```   196     done
```
```   197   ultimately have "1 - x <= 1 / (1 + x + x^2)"
```
```   198     by (elim mult_imp_le_div_pos)
```
```   199   also have "... <= 1 / exp x"
```
```   200     apply (rule divide_left_mono)
```
```   201     apply (rule exp_bound, rule a)
```
```   202     using a b apply auto
```
```   203     apply (rule mult_pos_pos)
```
```   204     apply (rule add_pos_nonneg)
```
```   205     apply auto
```
```   206     done
```
```   207   also have "... = exp (-x)"
```
```   208     by (auto simp add: exp_minus divide_inverse)
```
```   209   finally have "1 - x <= exp (- x)" .
```
```   210   also have "1 - x = exp (ln (1 - x))"
```
```   211   proof -
```
```   212     have "0 < 1 - x"
```
```   213       by (insert b, auto)
```
```   214     thus ?thesis
```
```   215       by (auto simp only: exp_ln_iff [THEN sym])
```
```   216   qed
```
```   217   finally have "exp (ln (1 - x)) <= exp (- x)" .
```
```   218   thus ?thesis by (auto simp only: exp_le_cancel_iff)
```
```   219 qed
```
```   220
```
```   221 lemma aux5: "x < 1 ==> ln(1 - x) = - ln(1 + x / (1 - x))"
```
```   222 proof -
```
```   223   assume a: "x < 1"
```
```   224   have "ln(1 - x) = - ln(1 / (1 - x))"
```
```   225   proof -
```
```   226     have "ln(1 - x) = - (- ln (1 - x))"
```
```   227       by auto
```
```   228     also have "- ln(1 - x) = ln 1 - ln(1 - x)"
```
```   229       by simp
```
```   230     also have "... = ln(1 / (1 - x))"
```
```   231       apply (rule ln_div [THEN sym])
```
```   232       by (insert a, auto)
```
```   233     finally show ?thesis .
```
```   234   qed
```
```   235   also have " 1 / (1 - x) = 1 + x / (1 - x)" using a by(simp add:field_simps)
```
```   236   finally show ?thesis .
```
```   237 qed
```
```   238
```
```   239 lemma ln_one_minus_pos_lower_bound: "0 <= x ==> x <= (1 / 2) ==>
```
```   240     - x - 2 * x^2 <= ln (1 - x)"
```
```   241 proof -
```
```   242   assume a: "0 <= x" and b: "x <= (1 / 2)"
```
```   243   from b have c: "x < 1"
```
```   244     by auto
```
```   245   then have "ln (1 - x) = - ln (1 + x / (1 - x))"
```
```   246     by (rule aux5)
```
```   247   also have "- (x / (1 - x)) <= ..."
```
```   248   proof -
```
```   249     have "ln (1 + x / (1 - x)) <= x / (1 - x)"
```
```   250       apply (rule ln_add_one_self_le_self)
```
```   251       apply (rule divide_nonneg_pos)
```
```   252       by (insert a c, auto)
```
```   253     thus ?thesis
```
```   254       by auto
```
```   255   qed
```
```   256   also have "- (x / (1 - x)) = -x / (1 - x)"
```
```   257     by auto
```
```   258   finally have d: "- x / (1 - x) <= ln (1 - x)" .
```
```   259   have "0 < 1 - x" using a b by simp
```
```   260   hence e: "-x - 2 * x^2 <= - x / (1 - x)"
```
```   261     using mult_right_le_one_le[of "x*x" "2*x"] a b
```
```   262     by (simp add:field_simps power2_eq_square)
```
```   263   from e d show "- x - 2 * x^2 <= ln (1 - x)"
```
```   264     by (rule order_trans)
```
```   265 qed
```
```   266
```
```   267 lemma exp_ge_add_one_self [simp]: "1 + (x::real) <= exp x"
```
```   268   apply (case_tac "0 <= x")
```
```   269   apply (erule exp_ge_add_one_self_aux)
```
```   270   apply (case_tac "x <= -1")
```
```   271   apply (subgoal_tac "1 + x <= 0")
```
```   272   apply (erule order_trans)
```
```   273   apply simp
```
```   274   apply simp
```
```   275   apply (subgoal_tac "1 + x = exp(ln (1 + x))")
```
```   276   apply (erule ssubst)
```
```   277   apply (subst exp_le_cancel_iff)
```
```   278   apply (subgoal_tac "ln (1 - (- x)) <= - (- x)")
```
```   279   apply simp
```
```   280   apply (rule ln_one_minus_pos_upper_bound)
```
```   281   apply auto
```
```   282 done
```
```   283
```
```   284 lemma ln_add_one_self_le_self2: "-1 < x ==> ln(1 + x) <= x"
```
```   285   apply (subgoal_tac "x = ln (exp x)")
```
```   286   apply (erule ssubst)back
```
```   287   apply (subst ln_le_cancel_iff)
```
```   288   apply auto
```
```   289 done
```
```   290
```
```   291 lemma abs_ln_one_plus_x_minus_x_bound_nonneg:
```
```   292     "0 <= x ==> x <= 1 ==> abs(ln (1 + x) - x) <= x^2"
```
```   293 proof -
```
```   294   assume x: "0 <= x"
```
```   295   assume x1: "x <= 1"
```
```   296   from x have "ln (1 + x) <= x"
```
```   297     by (rule ln_add_one_self_le_self)
```
```   298   then have "ln (1 + x) - x <= 0"
```
```   299     by simp
```
```   300   then have "abs(ln(1 + x) - x) = - (ln(1 + x) - x)"
```
```   301     by (rule abs_of_nonpos)
```
```   302   also have "... = x - ln (1 + x)"
```
```   303     by simp
```
```   304   also have "... <= x^2"
```
```   305   proof -
```
```   306     from x x1 have "x - x^2 <= ln (1 + x)"
```
```   307       by (intro ln_one_plus_pos_lower_bound)
```
```   308     thus ?thesis
```
```   309       by simp
```
```   310   qed
```
```   311   finally show ?thesis .
```
```   312 qed
```
```   313
```
```   314 lemma abs_ln_one_plus_x_minus_x_bound_nonpos:
```
```   315     "-(1 / 2) <= x ==> x <= 0 ==> abs(ln (1 + x) - x) <= 2 * x^2"
```
```   316 proof -
```
```   317   assume a: "-(1 / 2) <= x"
```
```   318   assume b: "x <= 0"
```
```   319   have "abs(ln (1 + x) - x) = x - ln(1 - (-x))"
```
```   320     apply (subst abs_of_nonpos)
```
```   321     apply simp
```
```   322     apply (rule ln_add_one_self_le_self2)
```
```   323     using a apply auto
```
```   324     done
```
```   325   also have "... <= 2 * x^2"
```
```   326     apply (subgoal_tac "- (-x) - 2 * (-x)^2 <= ln (1 - (-x))")
```
```   327     apply (simp add: algebra_simps)
```
```   328     apply (rule ln_one_minus_pos_lower_bound)
```
```   329     using a b apply auto
```
```   330     done
```
```   331   finally show ?thesis .
```
```   332 qed
```
```   333
```
```   334 lemma abs_ln_one_plus_x_minus_x_bound:
```
```   335     "abs x <= 1 / 2 ==> abs(ln (1 + x) - x) <= 2 * x^2"
```
```   336   apply (case_tac "0 <= x")
```
```   337   apply (rule order_trans)
```
```   338   apply (rule abs_ln_one_plus_x_minus_x_bound_nonneg)
```
```   339   apply auto
```
```   340   apply (rule abs_ln_one_plus_x_minus_x_bound_nonpos)
```
```   341   apply auto
```
```   342 done
```
```   343
```
```   344 lemma ln_x_over_x_mono: "exp 1 <= x ==> x <= y ==> (ln y / y) <= (ln x / x)"
```
```   345 proof -
```
```   346   assume x: "exp 1 <= x" "x <= y"
```
```   347   have a: "0 < x" and b: "0 < y"
```
```   348     apply (insert x)
```
```   349     apply (subgoal_tac "0 < exp (1::real)")
```
```   350     apply arith
```
```   351     apply auto
```
```   352     apply (subgoal_tac "0 < exp (1::real)")
```
```   353     apply arith
```
```   354     apply auto
```
```   355     done
```
```   356   have "x * ln y - x * ln x = x * (ln y - ln x)"
```
```   357     by (simp add: algebra_simps)
```
```   358   also have "... = x * ln(y / x)"
```
```   359     apply (subst ln_div)
```
```   360     apply (rule b, rule a, rule refl)
```
```   361     done
```
```   362   also have "y / x = (x + (y - x)) / x"
```
```   363     by simp
```
```   364   also have "... = 1 + (y - x) / x" using x a by (simp add: field_simps)
```
```   365   also have "x * ln(1 + (y - x) / x) <= x * ((y - x) / x)"
```
```   366     apply (rule mult_left_mono)
```
```   367     apply (rule ln_add_one_self_le_self)
```
```   368     apply (rule divide_nonneg_pos)
```
```   369     using x a apply simp_all
```
```   370     done
```
```   371   also have "... = y - x" using a by simp
```
```   372   also have "... = (y - x) * ln (exp 1)" by simp
```
```   373   also have "... <= (y - x) * ln x"
```
```   374     apply (rule mult_left_mono)
```
```   375     apply (subst ln_le_cancel_iff)
```
```   376     apply force
```
```   377     apply (rule a)
```
```   378     apply (rule x)
```
```   379     using x apply simp
```
```   380     done
```
```   381   also have "... = y * ln x - x * ln x"
```
```   382     by (rule left_diff_distrib)
```
```   383   finally have "x * ln y <= y * ln x"
```
```   384     by arith
```
```   385   then have "ln y <= (y * ln x) / x" using a by (simp add: field_simps)
```
```   386   also have "... = y * (ln x / x)" by simp
```
```   387   finally show ?thesis using b by (simp add: field_simps)
```
```   388 qed
```
```   389
```
```   390 lemma ln_le_minus_one:
```
```   391   "0 < x \<Longrightarrow> ln x \<le> x - 1"
```
```   392   using exp_ge_add_one_self[of "ln x"] by simp
```
```   393
```
```   394 lemma ln_eq_minus_one:
```
```   395   assumes "0 < x" "ln x = x - 1" shows "x = 1"
```
```   396 proof -
```
```   397   let "?l y" = "ln y - y + 1"
```
```   398   have D: "\<And>x. 0 < x \<Longrightarrow> DERIV ?l x :> (1 / x - 1)"
```
```   399     by (auto intro!: DERIV_intros)
```
```   400
```
```   401   show ?thesis
```
```   402   proof (cases rule: linorder_cases)
```
```   403     assume "x < 1"
```
```   404     from dense[OF `x < 1`] obtain a where "x < a" "a < 1" by blast
```
```   405     from `x < a` have "?l x < ?l a"
```
```   406     proof (rule DERIV_pos_imp_increasing, safe)
```
```   407       fix y assume "x \<le> y" "y \<le> a"
```
```   408       with `0 < x` `a < 1` have "0 < 1 / y - 1" "0 < y"
```
```   409         by (auto simp: field_simps)
```
```   410       with D show "\<exists>z. DERIV ?l y :> z \<and> 0 < z"
```
```   411         by auto
```
```   412     qed
```
```   413     also have "\<dots> \<le> 0"
```
```   414       using ln_le_minus_one `0 < x` `x < a` by (auto simp: field_simps)
```
```   415     finally show "x = 1" using assms by auto
```
```   416   next
```
```   417     assume "1 < x"
```
```   418     from dense[OF `1 < x`] obtain a where "1 < a" "a < x" by blast
```
```   419     from `a < x` have "?l x < ?l a"
```
```   420     proof (rule DERIV_neg_imp_decreasing, safe)
```
```   421       fix y assume "a \<le> y" "y \<le> x"
```
```   422       with `1 < a` have "1 / y - 1 < 0" "0 < y"
```
```   423         by (auto simp: field_simps)
```
```   424       with D show "\<exists>z. DERIV ?l y :> z \<and> z < 0"
```
```   425         by blast
```
```   426     qed
```
```   427     also have "\<dots> \<le> 0"
```
```   428       using ln_le_minus_one `1 < a` by (auto simp: field_simps)
```
```   429     finally show "x = 1" using assms by auto
```
```   430   qed simp
```
```   431 qed
```
```   432
```
```   433 end
```