src/HOL/Ln.thy
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```     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_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 (simp del: fact_Suc)
```
```    69         done
```
```    70       finally show ?thesis .
```
```    71     qed
```
```    72     moreover have "x ^ (Suc n + 2) <= x ^ (n + 2)"
```
```    73       by (simp add: mult_left_le_one_le mult_nonneg_nonneg a b)
```
```    74     ultimately have "inverse (fact (Suc n + 2)) *  x ^ (Suc n + 2) <=
```
```    75         (1 / 2 * inverse (fact (n + 2))) * x ^ (n + 2)"
```
```    76       apply (rule mult_mono)
```
```    77       apply (rule mult_nonneg_nonneg)
```
```    78       apply simp
```
```    79       apply (subst inverse_nonnegative_iff_nonnegative)
```
```    80       apply (rule real_of_nat_ge_zero)
```
```    81       apply (rule zero_le_power)
```
```    82       apply (rule a)
```
```    83       done
```
```    84     also have "... = 1 / 2 * (inverse (fact (n + 2)) * x ^ (n + 2))"
```
```    85       by simp
```
```    86     also have "... <= 1 / 2 * (x ^ 2 / 2 * (1 / 2) ^ n)"
```
```    87       apply (rule mult_left_mono)
```
```    88       apply (rule c)
```
```    89       apply simp
```
```    90       done
```
```    91     also have "... = x ^ 2 / 2 * (1 / 2 * (1 / 2) ^ n)"
```
```    92       by auto
```
```    93     also have "(1::real) / 2 * (1 / 2) ^ n = (1 / 2) ^ (Suc n)"
```
```    94       by (rule power_Suc [THEN sym])
```
```    95     finally show ?thesis .
```
```    96   qed
```
```    97 qed
```
```    98
```
```    99 lemma aux2: "(%n. (x::real) ^ 2 / 2 * (1 / 2) ^ n) sums x^2"
```
```   100 proof -
```
```   101   have "(%n. (1 / 2::real)^n) sums (1 / (1 - (1/2)))"
```
```   102     apply (rule geometric_sums)
```
```   103     by (simp add: abs_less_iff)
```
```   104   also have "(1::real) / (1 - 1/2) = 2"
```
```   105     by simp
```
```   106   finally have "(%n. (1 / 2::real)^n) sums 2" .
```
```   107   then have "(%n. x ^ 2 / 2 * (1 / 2) ^ n) sums (x^2 / 2 * 2)"
```
```   108     by (rule sums_mult)
```
```   109   also have "x^2 / 2 * 2 = x^2"
```
```   110     by simp
```
```   111   finally show ?thesis .
```
```   112 qed
```
```   113
```
```   114 lemma exp_bound: "0 <= (x::real) ==> x <= 1 ==> exp x <= 1 + x + x^2"
```
```   115 proof -
```
```   116   assume a: "0 <= x"
```
```   117   assume b: "x <= 1"
```
```   118   have c: "exp x = 1 + x + suminf (%n. inverse(fact (n+2)) *
```
```   119       (x ^ (n+2)))"
```
```   120     by (rule exp_first_two_terms)
```
```   121   moreover have "suminf (%n. inverse(fact (n+2)) * (x ^ (n+2))) <= x^2"
```
```   122   proof -
```
```   123     have "suminf (%n. inverse(fact (n+2)) * (x ^ (n+2))) <=
```
```   124         suminf (%n. (x^2/2) * ((1/2)^n))"
```
```   125       apply (rule summable_le)
```
```   126       apply (auto simp only: aux1 a b)
```
```   127       apply (rule exp_tail_after_first_two_terms_summable)
```
```   128       by (rule sums_summable, rule aux2)
```
```   129     also have "... = x^2"
```
```   130       by (rule sums_unique [THEN sym], rule aux2)
```
```   131     finally show ?thesis .
```
```   132   qed
```
```   133   ultimately show ?thesis
```
```   134     by auto
```
```   135 qed
```
```   136
```
```   137 lemma aux4: "0 <= (x::real) ==> x <= 1 ==> exp (x - x^2) <= 1 + x"
```
```   138 proof -
```
```   139   assume a: "0 <= x" and b: "x <= 1"
```
```   140   have "exp (x - x^2) = exp x / exp (x^2)"
```
```   141     by (rule exp_diff)
```
```   142   also have "... <= (1 + x + x^2) / exp (x ^2)"
```
```   143     apply (rule divide_right_mono)
```
```   144     apply (rule exp_bound)
```
```   145     apply (rule a, rule b)
```
```   146     apply simp
```
```   147     done
```
```   148   also have "... <= (1 + x + x^2) / (1 + x^2)"
```
```   149     apply (rule divide_left_mono)
```
```   150     apply (auto simp add: exp_ge_add_one_self_aux)
```
```   151     apply (rule add_nonneg_nonneg)
```
```   152     using a apply auto
```
```   153     apply (rule mult_pos_pos)
```
```   154     apply auto
```
```   155     apply (rule add_pos_nonneg)
```
```   156     apply auto
```
```   157     done
```
```   158   also from a have "... <= 1 + x"
```
```   159     by (simp add: field_simps add_strict_increasing zero_le_mult_iff)
```
```   160   finally show ?thesis .
```
```   161 qed
```
```   162
```
```   163 lemma ln_one_plus_pos_lower_bound: "0 <= x ==> x <= 1 ==>
```
```   164     x - x^2 <= ln (1 + x)"
```
```   165 proof -
```
```   166   assume a: "0 <= x" and b: "x <= 1"
```
```   167   then have "exp (x - x^2) <= 1 + x"
```
```   168     by (rule aux4)
```
```   169   also have "... = exp (ln (1 + x))"
```
```   170   proof -
```
```   171     from a have "0 < 1 + x" by auto
```
```   172     thus ?thesis
```
```   173       by (auto simp only: exp_ln_iff [THEN sym])
```
```   174   qed
```
```   175   finally have "exp (x - x ^ 2) <= exp (ln (1 + x))" .
```
```   176   thus ?thesis by (auto simp only: exp_le_cancel_iff)
```
```   177 qed
```
```   178
```
```   179 lemma ln_one_minus_pos_upper_bound: "0 <= x ==> x < 1 ==> ln (1 - x) <= - x"
```
```   180 proof -
```
```   181   assume a: "0 <= (x::real)" and b: "x < 1"
```
```   182   have "(1 - x) * (1 + x + x^2) = (1 - x^3)"
```
```   183     by (simp add: algebra_simps power2_eq_square power3_eq_cube)
```
```   184   also have "... <= 1"
```
```   185     by (auto simp add: a)
```
```   186   finally have "(1 - x) * (1 + x + x ^ 2) <= 1" .
```
```   187   moreover have "0 < 1 + x + x^2"
```
```   188     apply (rule add_pos_nonneg)
```
```   189     using a apply auto
```
```   190     done
```
```   191   ultimately have "1 - x <= 1 / (1 + x + x^2)"
```
```   192     by (elim mult_imp_le_div_pos)
```
```   193   also have "... <= 1 / exp x"
```
```   194     apply (rule divide_left_mono)
```
```   195     apply (rule exp_bound, rule a)
```
```   196     using a b apply auto
```
```   197     apply (rule mult_pos_pos)
```
```   198     apply (rule add_pos_nonneg)
```
```   199     apply auto
```
```   200     done
```
```   201   also have "... = exp (-x)"
```
```   202     by (auto simp add: exp_minus divide_inverse)
```
```   203   finally have "1 - x <= exp (- x)" .
```
```   204   also have "1 - x = exp (ln (1 - x))"
```
```   205   proof -
```
```   206     have "0 < 1 - x"
```
```   207       by (insert b, auto)
```
```   208     thus ?thesis
```
```   209       by (auto simp only: exp_ln_iff [THEN sym])
```
```   210   qed
```
```   211   finally have "exp (ln (1 - x)) <= exp (- x)" .
```
```   212   thus ?thesis by (auto simp only: exp_le_cancel_iff)
```
```   213 qed
```
```   214
```
```   215 lemma aux5: "x < 1 ==> ln(1 - x) = - ln(1 + x / (1 - x))"
```
```   216 proof -
```
```   217   assume a: "x < 1"
```
```   218   have "ln(1 - x) = - ln(1 / (1 - x))"
```
```   219   proof -
```
```   220     have "ln(1 - x) = - (- ln (1 - x))"
```
```   221       by auto
```
```   222     also have "- ln(1 - x) = ln 1 - ln(1 - x)"
```
```   223       by simp
```
```   224     also have "... = ln(1 / (1 - x))"
```
```   225       apply (rule ln_div [THEN sym])
```
```   226       by (insert a, auto)
```
```   227     finally show ?thesis .
```
```   228   qed
```
```   229   also have " 1 / (1 - x) = 1 + x / (1 - x)" using a by(simp add:field_simps)
```
```   230   finally show ?thesis .
```
```   231 qed
```
```   232
```
```   233 lemma ln_one_minus_pos_lower_bound: "0 <= x ==> x <= (1 / 2) ==>
```
```   234     - x - 2 * x^2 <= ln (1 - x)"
```
```   235 proof -
```
```   236   assume a: "0 <= x" and b: "x <= (1 / 2)"
```
```   237   from b have c: "x < 1"
```
```   238     by auto
```
```   239   then have "ln (1 - x) = - ln (1 + x / (1 - x))"
```
```   240     by (rule aux5)
```
```   241   also have "- (x / (1 - x)) <= ..."
```
```   242   proof -
```
```   243     have "ln (1 + x / (1 - x)) <= x / (1 - x)"
```
```   244       apply (rule ln_add_one_self_le_self)
```
```   245       apply (rule divide_nonneg_pos)
```
```   246       by (insert a c, auto)
```
```   247     thus ?thesis
```
```   248       by auto
```
```   249   qed
```
```   250   also have "- (x / (1 - x)) = -x / (1 - x)"
```
```   251     by auto
```
```   252   finally have d: "- x / (1 - x) <= ln (1 - x)" .
```
```   253   have "0 < 1 - x" using a b by simp
```
```   254   hence e: "-x - 2 * x^2 <= - x / (1 - x)"
```
```   255     using mult_right_le_one_le[of "x*x" "2*x"] a b
```
```   256     by (simp add:field_simps power2_eq_square)
```
```   257   from e d show "- x - 2 * x^2 <= ln (1 - x)"
```
```   258     by (rule order_trans)
```
```   259 qed
```
```   260
```
```   261 lemma exp_ge_add_one_self [simp]: "1 + (x::real) <= exp x"
```
```   262   apply (case_tac "0 <= x")
```
```   263   apply (erule exp_ge_add_one_self_aux)
```
```   264   apply (case_tac "x <= -1")
```
```   265   apply (subgoal_tac "1 + x <= 0")
```
```   266   apply (erule order_trans)
```
```   267   apply simp
```
```   268   apply simp
```
```   269   apply (subgoal_tac "1 + x = exp(ln (1 + x))")
```
```   270   apply (erule ssubst)
```
```   271   apply (subst exp_le_cancel_iff)
```
```   272   apply (subgoal_tac "ln (1 - (- x)) <= - (- x)")
```
```   273   apply simp
```
```   274   apply (rule ln_one_minus_pos_upper_bound)
```
```   275   apply auto
```
```   276 done
```
```   277
```
```   278 lemma ln_add_one_self_le_self2: "-1 < x ==> ln(1 + x) <= x"
```
```   279   apply (subgoal_tac "x = ln (exp x)")
```
```   280   apply (erule ssubst)back
```
```   281   apply (subst ln_le_cancel_iff)
```
```   282   apply auto
```
```   283 done
```
```   284
```
```   285 lemma abs_ln_one_plus_x_minus_x_bound_nonneg:
```
```   286     "0 <= x ==> x <= 1 ==> abs(ln (1 + x) - x) <= x^2"
```
```   287 proof -
```
```   288   assume x: "0 <= x"
```
```   289   assume x1: "x <= 1"
```
```   290   from x have "ln (1 + x) <= x"
```
```   291     by (rule ln_add_one_self_le_self)
```
```   292   then have "ln (1 + x) - x <= 0"
```
```   293     by simp
```
```   294   then have "abs(ln(1 + x) - x) = - (ln(1 + x) - x)"
```
```   295     by (rule abs_of_nonpos)
```
```   296   also have "... = x - ln (1 + x)"
```
```   297     by simp
```
```   298   also have "... <= x^2"
```
```   299   proof -
```
```   300     from x x1 have "x - x^2 <= ln (1 + x)"
```
```   301       by (intro ln_one_plus_pos_lower_bound)
```
```   302     thus ?thesis
```
```   303       by simp
```
```   304   qed
```
```   305   finally show ?thesis .
```
```   306 qed
```
```   307
```
```   308 lemma abs_ln_one_plus_x_minus_x_bound_nonpos:
```
```   309     "-(1 / 2) <= x ==> x <= 0 ==> abs(ln (1 + x) - x) <= 2 * x^2"
```
```   310 proof -
```
```   311   assume a: "-(1 / 2) <= x"
```
```   312   assume b: "x <= 0"
```
```   313   have "abs(ln (1 + x) - x) = x - ln(1 - (-x))"
```
```   314     apply (subst abs_of_nonpos)
```
```   315     apply simp
```
```   316     apply (rule ln_add_one_self_le_self2)
```
```   317     using a apply auto
```
```   318     done
```
```   319   also have "... <= 2 * x^2"
```
```   320     apply (subgoal_tac "- (-x) - 2 * (-x)^2 <= ln (1 - (-x))")
```
```   321     apply (simp add: algebra_simps)
```
```   322     apply (rule ln_one_minus_pos_lower_bound)
```
```   323     using a b apply auto
```
```   324     done
```
```   325   finally show ?thesis .
```
```   326 qed
```
```   327
```
```   328 lemma abs_ln_one_plus_x_minus_x_bound:
```
```   329     "abs x <= 1 / 2 ==> abs(ln (1 + x) - x) <= 2 * x^2"
```
```   330   apply (case_tac "0 <= x")
```
```   331   apply (rule order_trans)
```
```   332   apply (rule abs_ln_one_plus_x_minus_x_bound_nonneg)
```
```   333   apply auto
```
```   334   apply (rule abs_ln_one_plus_x_minus_x_bound_nonpos)
```
```   335   apply auto
```
```   336 done
```
```   337
```
```   338 lemma ln_x_over_x_mono: "exp 1 <= x ==> x <= y ==> (ln y / y) <= (ln x / x)"
```
```   339 proof -
```
```   340   assume x: "exp 1 <= x" "x <= y"
```
```   341   moreover have "0 < exp (1::real)" by simp
```
```   342   ultimately have a: "0 < x" and b: "0 < y"
```
```   343     by (fast intro: less_le_trans order_trans)+
```
```   344   have "x * ln y - x * ln x = x * (ln y - ln x)"
```
```   345     by (simp add: algebra_simps)
```
```   346   also have "... = x * ln(y / x)"
```
```   347     by (simp only: ln_div a b)
```
```   348   also have "y / x = (x + (y - x)) / x"
```
```   349     by simp
```
```   350   also have "... = 1 + (y - x) / x"
```
```   351     using x a by (simp add: field_simps)
```
```   352   also have "x * ln(1 + (y - x) / x) <= x * ((y - x) / x)"
```
```   353     apply (rule mult_left_mono)
```
```   354     apply (rule ln_add_one_self_le_self)
```
```   355     apply (rule divide_nonneg_pos)
```
```   356     using x a apply simp_all
```
```   357     done
```
```   358   also have "... = y - x" using a by simp
```
```   359   also have "... = (y - x) * ln (exp 1)" by simp
```
```   360   also have "... <= (y - x) * ln x"
```
```   361     apply (rule mult_left_mono)
```
```   362     apply (subst ln_le_cancel_iff)
```
```   363     apply fact
```
```   364     apply (rule a)
```
```   365     apply (rule x)
```
```   366     using x apply simp
```
```   367     done
```
```   368   also have "... = y * ln x - x * ln x"
```
```   369     by (rule left_diff_distrib)
```
```   370   finally have "x * ln y <= y * ln x"
```
```   371     by arith
```
```   372   then have "ln y <= (y * ln x) / x" using a by (simp add: field_simps)
```
```   373   also have "... = y * (ln x / x)" by simp
```
```   374   finally show ?thesis using b by (simp add: field_simps)
```
```   375 qed
```
```   376
```
```   377 lemma ln_le_minus_one:
```
```   378   "0 < x \<Longrightarrow> ln x \<le> x - 1"
```
```   379   using exp_ge_add_one_self[of "ln x"] by simp
```
```   380
```
```   381 lemma ln_eq_minus_one:
```
```   382   assumes "0 < x" "ln x = x - 1" shows "x = 1"
```
```   383 proof -
```
```   384   let "?l y" = "ln y - y + 1"
```
```   385   have D: "\<And>x. 0 < x \<Longrightarrow> DERIV ?l x :> (1 / x - 1)"
```
```   386     by (auto intro!: DERIV_intros)
```
```   387
```
```   388   show ?thesis
```
```   389   proof (cases rule: linorder_cases)
```
```   390     assume "x < 1"
```
```   391     from dense[OF `x < 1`] obtain a where "x < a" "a < 1" by blast
```
```   392     from `x < a` have "?l x < ?l a"
```
```   393     proof (rule DERIV_pos_imp_increasing, safe)
```
```   394       fix y assume "x \<le> y" "y \<le> a"
```
```   395       with `0 < x` `a < 1` have "0 < 1 / y - 1" "0 < y"
```
```   396         by (auto simp: field_simps)
```
```   397       with D show "\<exists>z. DERIV ?l y :> z \<and> 0 < z"
```
```   398         by auto
```
```   399     qed
```
```   400     also have "\<dots> \<le> 0"
```
```   401       using ln_le_minus_one `0 < x` `x < a` by (auto simp: field_simps)
```
```   402     finally show "x = 1" using assms by auto
```
```   403   next
```
```   404     assume "1 < x"
```
```   405     from dense[OF `1 < x`] obtain a where "1 < a" "a < x" by blast
```
```   406     from `a < x` have "?l x < ?l a"
```
```   407     proof (rule DERIV_neg_imp_decreasing, safe)
```
```   408       fix y assume "a \<le> y" "y \<le> x"
```
```   409       with `1 < a` have "1 / y - 1 < 0" "0 < y"
```
```   410         by (auto simp: field_simps)
```
```   411       with D show "\<exists>z. DERIV ?l y :> z \<and> z < 0"
```
```   412         by blast
```
```   413     qed
```
```   414     also have "\<dots> \<le> 0"
```
```   415       using ln_le_minus_one `1 < a` by (auto simp: field_simps)
```
```   416     finally show "x = 1" using assms by auto
```
```   417   qed simp
```
```   418 qed
```
```   419
```
```   420 end
```