src/HOL/Ln.thy
 author huffman Sun Apr 01 16:09:58 2012 +0200 (2012-04-01) changeset 47255 30a1692557b0 parent 47244 a7f85074c169 child 50326 b5afeccab2db permissions -rw-r--r--
removed Nat_Numeral.thy, moving all theorems elsewhere
```     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: numeral_2_eq_2)
```
```    22   finally show ?thesis .
```
```    23 qed
```
```    24
```
```    25 lemma exp_bound: "0 <= (x::real) ==> x <= 1 ==> exp x <= 1 + x + x^2"
```
```    26 proof -
```
```    27   assume a: "0 <= x"
```
```    28   assume b: "x <= 1"
```
```    29   { fix n :: nat
```
```    30     have "2 * 2 ^ n \<le> fact (n + 2)"
```
```    31       by (induct n, simp, simp)
```
```    32     hence "real ((2::nat) * 2 ^ n) \<le> real (fact (n + 2))"
```
```    33       by (simp only: real_of_nat_le_iff)
```
```    34     hence "2 * 2 ^ n \<le> real (fact (n + 2))"
```
```    35       by simp
```
```    36     hence "inverse (fact (n + 2)) \<le> inverse (2 * 2 ^ n)"
```
```    37       by (rule le_imp_inverse_le) simp
```
```    38     hence "inverse (fact (n + 2)) \<le> 1/2 * (1/2)^n"
```
```    39       by (simp add: inverse_mult_distrib power_inverse)
```
```    40     hence "inverse (fact (n + 2)) * (x^n * x\<twosuperior>) \<le> 1/2 * (1/2)^n * (1 * x\<twosuperior>)"
```
```    41       by (rule mult_mono)
```
```    42         (rule mult_mono, simp_all add: power_le_one a b mult_nonneg_nonneg)
```
```    43     hence "inverse (fact (n + 2)) * x ^ (n + 2) \<le> (x\<twosuperior>/2) * ((1/2)^n)"
```
```    44       unfolding power_add by (simp add: mult_ac del: fact_Suc) }
```
```    45   note aux1 = this
```
```    46   have "(\<lambda>n. x\<twosuperior> / 2 * (1 / 2) ^ n) sums (x\<twosuperior> / 2 * (1 / (1 - 1 / 2)))"
```
```    47     by (intro sums_mult geometric_sums, simp)
```
```    48   hence aux2: "(\<lambda>n. (x::real) ^ 2 / 2 * (1 / 2) ^ n) sums x^2"
```
```    49     by simp
```
```    50   have "suminf (%n. inverse(fact (n+2)) * (x ^ (n+2))) <= x^2"
```
```    51   proof -
```
```    52     have "suminf (%n. inverse(fact (n+2)) * (x ^ (n+2))) <=
```
```    53         suminf (%n. (x^2/2) * ((1/2)^n))"
```
```    54       apply (rule summable_le)
```
```    55       apply (rule allI, rule aux1)
```
```    56       apply (rule summable_exp [THEN summable_ignore_initial_segment])
```
```    57       by (rule sums_summable, rule aux2)
```
```    58     also have "... = x^2"
```
```    59       by (rule sums_unique [THEN sym], rule aux2)
```
```    60     finally show ?thesis .
```
```    61   qed
```
```    62   thus ?thesis unfolding exp_first_two_terms by auto
```
```    63 qed
```
```    64
```
```    65 lemma ln_one_plus_pos_lower_bound: "0 <= x ==> x <= 1 ==>
```
```    66     x - x^2 <= ln (1 + x)"
```
```    67 proof -
```
```    68   assume a: "0 <= x" and b: "x <= 1"
```
```    69   have "exp (x - x^2) = exp x / exp (x^2)"
```
```    70     by (rule exp_diff)
```
```    71   also have "... <= (1 + x + x^2) / exp (x ^2)"
```
```    72     apply (rule divide_right_mono)
```
```    73     apply (rule exp_bound)
```
```    74     apply (rule a, rule b)
```
```    75     apply simp
```
```    76     done
```
```    77   also have "... <= (1 + x + x^2) / (1 + x^2)"
```
```    78     apply (rule divide_left_mono)
```
```    79     apply (simp add: exp_ge_add_one_self_aux)
```
```    80     apply (simp add: a)
```
```    81     apply (simp add: mult_pos_pos add_pos_nonneg)
```
```    82     done
```
```    83   also from a have "... <= 1 + x"
```
```    84     by (simp add: field_simps add_strict_increasing zero_le_mult_iff)
```
```    85   finally have "exp (x - x^2) <= 1 + x" .
```
```    86   also have "... = exp (ln (1 + x))"
```
```    87   proof -
```
```    88     from a have "0 < 1 + x" by auto
```
```    89     thus ?thesis
```
```    90       by (auto simp only: exp_ln_iff [THEN sym])
```
```    91   qed
```
```    92   finally have "exp (x - x ^ 2) <= exp (ln (1 + x))" .
```
```    93   thus ?thesis by (auto simp only: exp_le_cancel_iff)
```
```    94 qed
```
```    95
```
```    96 lemma ln_one_minus_pos_upper_bound: "0 <= x ==> x < 1 ==> ln (1 - x) <= - x"
```
```    97 proof -
```
```    98   assume a: "0 <= (x::real)" and b: "x < 1"
```
```    99   have "(1 - x) * (1 + x + x^2) = (1 - x^3)"
```
```   100     by (simp add: algebra_simps power2_eq_square power3_eq_cube)
```
```   101   also have "... <= 1"
```
```   102     by (auto simp add: a)
```
```   103   finally have "(1 - x) * (1 + x + x ^ 2) <= 1" .
```
```   104   moreover have c: "0 < 1 + x + x\<twosuperior>"
```
```   105     by (simp add: add_pos_nonneg a)
```
```   106   ultimately have "1 - x <= 1 / (1 + x + x^2)"
```
```   107     by (elim mult_imp_le_div_pos)
```
```   108   also have "... <= 1 / exp x"
```
```   109     apply (rule divide_left_mono)
```
```   110     apply (rule exp_bound, rule a)
```
```   111     apply (rule b [THEN less_imp_le])
```
```   112     apply simp
```
```   113     apply (rule mult_pos_pos)
```
```   114     apply (rule c)
```
```   115     apply simp
```
```   116     done
```
```   117   also have "... = exp (-x)"
```
```   118     by (auto simp add: exp_minus divide_inverse)
```
```   119   finally have "1 - x <= exp (- x)" .
```
```   120   also have "1 - x = exp (ln (1 - x))"
```
```   121   proof -
```
```   122     have "0 < 1 - x"
```
```   123       by (insert b, auto)
```
```   124     thus ?thesis
```
```   125       by (auto simp only: exp_ln_iff [THEN sym])
```
```   126   qed
```
```   127   finally have "exp (ln (1 - x)) <= exp (- x)" .
```
```   128   thus ?thesis by (auto simp only: exp_le_cancel_iff)
```
```   129 qed
```
```   130
```
```   131 lemma aux5: "x < 1 ==> ln(1 - x) = - ln(1 + x / (1 - x))"
```
```   132 proof -
```
```   133   assume a: "x < 1"
```
```   134   have "ln(1 - x) = - ln(1 / (1 - x))"
```
```   135   proof -
```
```   136     have "ln(1 - x) = - (- ln (1 - x))"
```
```   137       by auto
```
```   138     also have "- ln(1 - x) = ln 1 - ln(1 - x)"
```
```   139       by simp
```
```   140     also have "... = ln(1 / (1 - x))"
```
```   141       apply (rule ln_div [THEN sym])
```
```   142       by (insert a, auto)
```
```   143     finally show ?thesis .
```
```   144   qed
```
```   145   also have " 1 / (1 - x) = 1 + x / (1 - x)" using a by(simp add:field_simps)
```
```   146   finally show ?thesis .
```
```   147 qed
```
```   148
```
```   149 lemma ln_one_minus_pos_lower_bound: "0 <= x ==> x <= (1 / 2) ==>
```
```   150     - x - 2 * x^2 <= ln (1 - x)"
```
```   151 proof -
```
```   152   assume a: "0 <= x" and b: "x <= (1 / 2)"
```
```   153   from b have c: "x < 1"
```
```   154     by auto
```
```   155   then have "ln (1 - x) = - ln (1 + x / (1 - x))"
```
```   156     by (rule aux5)
```
```   157   also have "- (x / (1 - x)) <= ..."
```
```   158   proof -
```
```   159     have "ln (1 + x / (1 - x)) <= x / (1 - x)"
```
```   160       apply (rule ln_add_one_self_le_self)
```
```   161       apply (rule divide_nonneg_pos)
```
```   162       by (insert a c, auto)
```
```   163     thus ?thesis
```
```   164       by auto
```
```   165   qed
```
```   166   also have "- (x / (1 - x)) = -x / (1 - x)"
```
```   167     by auto
```
```   168   finally have d: "- x / (1 - x) <= ln (1 - x)" .
```
```   169   have "0 < 1 - x" using a b by simp
```
```   170   hence e: "-x - 2 * x^2 <= - x / (1 - x)"
```
```   171     using mult_right_le_one_le[of "x*x" "2*x"] a b
```
```   172     by (simp add:field_simps power2_eq_square)
```
```   173   from e d show "- x - 2 * x^2 <= ln (1 - x)"
```
```   174     by (rule order_trans)
```
```   175 qed
```
```   176
```
```   177 lemma exp_ge_add_one_self [simp]: "1 + (x::real) <= exp x"
```
```   178   apply (case_tac "0 <= x")
```
```   179   apply (erule exp_ge_add_one_self_aux)
```
```   180   apply (case_tac "x <= -1")
```
```   181   apply (subgoal_tac "1 + x <= 0")
```
```   182   apply (erule order_trans)
```
```   183   apply simp
```
```   184   apply simp
```
```   185   apply (subgoal_tac "1 + x = exp(ln (1 + x))")
```
```   186   apply (erule ssubst)
```
```   187   apply (subst exp_le_cancel_iff)
```
```   188   apply (subgoal_tac "ln (1 - (- x)) <= - (- x)")
```
```   189   apply simp
```
```   190   apply (rule ln_one_minus_pos_upper_bound)
```
```   191   apply auto
```
```   192 done
```
```   193
```
```   194 lemma ln_add_one_self_le_self2: "-1 < x ==> ln(1 + x) <= x"
```
```   195   apply (subgoal_tac "ln (1 + x) \<le> ln (exp x)", simp)
```
```   196   apply (subst ln_le_cancel_iff)
```
```   197   apply auto
```
```   198 done
```
```   199
```
```   200 lemma abs_ln_one_plus_x_minus_x_bound_nonneg:
```
```   201     "0 <= x ==> x <= 1 ==> abs(ln (1 + x) - x) <= x^2"
```
```   202 proof -
```
```   203   assume x: "0 <= x"
```
```   204   assume x1: "x <= 1"
```
```   205   from x have "ln (1 + x) <= x"
```
```   206     by (rule ln_add_one_self_le_self)
```
```   207   then have "ln (1 + x) - x <= 0"
```
```   208     by simp
```
```   209   then have "abs(ln(1 + x) - x) = - (ln(1 + x) - x)"
```
```   210     by (rule abs_of_nonpos)
```
```   211   also have "... = x - ln (1 + x)"
```
```   212     by simp
```
```   213   also have "... <= x^2"
```
```   214   proof -
```
```   215     from x x1 have "x - x^2 <= ln (1 + x)"
```
```   216       by (intro ln_one_plus_pos_lower_bound)
```
```   217     thus ?thesis
```
```   218       by simp
```
```   219   qed
```
```   220   finally show ?thesis .
```
```   221 qed
```
```   222
```
```   223 lemma abs_ln_one_plus_x_minus_x_bound_nonpos:
```
```   224     "-(1 / 2) <= x ==> x <= 0 ==> abs(ln (1 + x) - x) <= 2 * x^2"
```
```   225 proof -
```
```   226   assume a: "-(1 / 2) <= x"
```
```   227   assume b: "x <= 0"
```
```   228   have "abs(ln (1 + x) - x) = x - ln(1 - (-x))"
```
```   229     apply (subst abs_of_nonpos)
```
```   230     apply simp
```
```   231     apply (rule ln_add_one_self_le_self2)
```
```   232     using a apply auto
```
```   233     done
```
```   234   also have "... <= 2 * x^2"
```
```   235     apply (subgoal_tac "- (-x) - 2 * (-x)^2 <= ln (1 - (-x))")
```
```   236     apply (simp add: algebra_simps)
```
```   237     apply (rule ln_one_minus_pos_lower_bound)
```
```   238     using a b apply auto
```
```   239     done
```
```   240   finally show ?thesis .
```
```   241 qed
```
```   242
```
```   243 lemma abs_ln_one_plus_x_minus_x_bound:
```
```   244     "abs x <= 1 / 2 ==> abs(ln (1 + x) - x) <= 2 * x^2"
```
```   245   apply (case_tac "0 <= x")
```
```   246   apply (rule order_trans)
```
```   247   apply (rule abs_ln_one_plus_x_minus_x_bound_nonneg)
```
```   248   apply auto
```
```   249   apply (rule abs_ln_one_plus_x_minus_x_bound_nonpos)
```
```   250   apply auto
```
```   251 done
```
```   252
```
```   253 lemma ln_x_over_x_mono: "exp 1 <= x ==> x <= y ==> (ln y / y) <= (ln x / x)"
```
```   254 proof -
```
```   255   assume x: "exp 1 <= x" "x <= y"
```
```   256   moreover have "0 < exp (1::real)" by simp
```
```   257   ultimately have a: "0 < x" and b: "0 < y"
```
```   258     by (fast intro: less_le_trans order_trans)+
```
```   259   have "x * ln y - x * ln x = x * (ln y - ln x)"
```
```   260     by (simp add: algebra_simps)
```
```   261   also have "... = x * ln(y / x)"
```
```   262     by (simp only: ln_div a b)
```
```   263   also have "y / x = (x + (y - x)) / x"
```
```   264     by simp
```
```   265   also have "... = 1 + (y - x) / x"
```
```   266     using x a by (simp add: field_simps)
```
```   267   also have "x * ln(1 + (y - x) / x) <= x * ((y - x) / x)"
```
```   268     apply (rule mult_left_mono)
```
```   269     apply (rule ln_add_one_self_le_self)
```
```   270     apply (rule divide_nonneg_pos)
```
```   271     using x a apply simp_all
```
```   272     done
```
```   273   also have "... = y - x" using a by simp
```
```   274   also have "... = (y - x) * ln (exp 1)" by simp
```
```   275   also have "... <= (y - x) * ln x"
```
```   276     apply (rule mult_left_mono)
```
```   277     apply (subst ln_le_cancel_iff)
```
```   278     apply fact
```
```   279     apply (rule a)
```
```   280     apply (rule x)
```
```   281     using x apply simp
```
```   282     done
```
```   283   also have "... = y * ln x - x * ln x"
```
```   284     by (rule left_diff_distrib)
```
```   285   finally have "x * ln y <= y * ln x"
```
```   286     by arith
```
```   287   then have "ln y <= (y * ln x) / x" using a by (simp add: field_simps)
```
```   288   also have "... = y * (ln x / x)" by simp
```
```   289   finally show ?thesis using b by (simp add: field_simps)
```
```   290 qed
```
```   291
```
```   292 lemma ln_le_minus_one:
```
```   293   "0 < x \<Longrightarrow> ln x \<le> x - 1"
```
```   294   using exp_ge_add_one_self[of "ln x"] by simp
```
```   295
```
```   296 lemma ln_eq_minus_one:
```
```   297   assumes "0 < x" "ln x = x - 1" shows "x = 1"
```
```   298 proof -
```
```   299   let "?l y" = "ln y - y + 1"
```
```   300   have D: "\<And>x. 0 < x \<Longrightarrow> DERIV ?l x :> (1 / x - 1)"
```
```   301     by (auto intro!: DERIV_intros)
```
```   302
```
```   303   show ?thesis
```
```   304   proof (cases rule: linorder_cases)
```
```   305     assume "x < 1"
```
```   306     from dense[OF `x < 1`] obtain a where "x < a" "a < 1" by blast
```
```   307     from `x < a` have "?l x < ?l a"
```
```   308     proof (rule DERIV_pos_imp_increasing, safe)
```
```   309       fix y assume "x \<le> y" "y \<le> a"
```
```   310       with `0 < x` `a < 1` have "0 < 1 / y - 1" "0 < y"
```
```   311         by (auto simp: field_simps)
```
```   312       with D show "\<exists>z. DERIV ?l y :> z \<and> 0 < z"
```
```   313         by auto
```
```   314     qed
```
```   315     also have "\<dots> \<le> 0"
```
```   316       using ln_le_minus_one `0 < x` `x < a` by (auto simp: field_simps)
```
```   317     finally show "x = 1" using assms by auto
```
```   318   next
```
```   319     assume "1 < x"
```
```   320     from dense[OF `1 < x`] obtain a where "1 < a" "a < x" by blast
```
```   321     from `a < x` have "?l x < ?l a"
```
```   322     proof (rule DERIV_neg_imp_decreasing, safe)
```
```   323       fix y assume "a \<le> y" "y \<le> x"
```
```   324       with `1 < a` have "1 / y - 1 < 0" "0 < y"
```
```   325         by (auto simp: field_simps)
```
```   326       with D show "\<exists>z. DERIV ?l y :> z \<and> z < 0"
```
```   327         by blast
```
```   328     qed
```
```   329     also have "\<dots> \<le> 0"
```
```   330       using ln_le_minus_one `1 < a` by (auto simp: field_simps)
```
```   331     finally show "x = 1" using assms by auto
```
```   332   qed simp
```
```   333 qed
```
```   334
```
```   335 end
```