src/HOL/Real/RComplete.thy
 author paulson Fri Mar 19 10:50:06 2004 +0100 (2004-03-19) changeset 14476 758e7acdea2f parent 14387 e96d5c42c4b0 child 14641 79b7bd936264 permissions -rw-r--r--
removed redundant thms
```     1 (*  Title       : RComplete.thy
```
```     2     ID          : \$Id\$
```
```     3     Author      : Jacques D. Fleuriot
```
```     4     Copyright   : 1998  University of Cambridge
```
```     5     Description : Completeness theorems for positive
```
```     6                   reals and reals
```
```     7 *)
```
```     8
```
```     9 header{*Completeness Theorems for Positive Reals and Reals.*}
```
```    10
```
```    11 theory RComplete = Lubs + RealDef:
```
```    12
```
```    13 lemma real_sum_of_halves: "x/2 + x/2 = (x::real)"
```
```    14 by simp
```
```    15
```
```    16
```
```    17 subsection{*Completeness of Reals by Supremum Property of type @{typ preal}*}
```
```    18
```
```    19  (*a few lemmas*)
```
```    20 lemma real_sup_lemma1:
```
```    21      "\<forall>x \<in> P. 0 < x ==>
```
```    22       ((\<exists>x \<in> P. y < x) = (\<exists>X. real_of_preal X \<in> P & y < real_of_preal X))"
```
```    23 by (blast dest!: bspec real_gt_zero_preal_Ex [THEN iffD1])
```
```    24
```
```    25 lemma real_sup_lemma2:
```
```    26      "[| \<forall>x \<in> P. 0 < x;  a \<in> P;   \<forall>x \<in> P. x < y |]
```
```    27       ==> (\<exists>X. X\<in> {w. real_of_preal w \<in> P}) &
```
```    28           (\<exists>Y. \<forall>X\<in> {w. real_of_preal w \<in> P}. X < Y)"
```
```    29 apply (rule conjI)
```
```    30 apply (blast dest: bspec real_gt_zero_preal_Ex [THEN iffD1], auto)
```
```    31 apply (drule bspec, assumption)
```
```    32 apply (frule bspec, assumption)
```
```    33 apply (drule order_less_trans, assumption)
```
```    34 apply (drule real_gt_zero_preal_Ex [THEN iffD1], force)
```
```    35 done
```
```    36
```
```    37 (*-------------------------------------------------------------
```
```    38             Completeness of Positive Reals
```
```    39  -------------------------------------------------------------*)
```
```    40
```
```    41 (**
```
```    42  Supremum property for the set of positive reals
```
```    43  FIXME: long proof - should be improved
```
```    44 **)
```
```    45
```
```    46 (*Let P be a non-empty set of positive reals, with an upper bound y.
```
```    47   Then P has a least upper bound (written S).
```
```    48 FIXME: Can the premise be weakened to \<forall>x \<in> P. x\<le> y ??*)
```
```    49 lemma posreal_complete: "[| \<forall>x \<in> P. (0::real) < x;  \<exists>x. x \<in> P;  \<exists>y. \<forall>x \<in> P. x<y |]
```
```    50       ==> (\<exists>S. \<forall>y. (\<exists>x \<in> P. y < x) = (y < S))"
```
```    51 apply (rule_tac x = "real_of_preal (psup ({w. real_of_preal w \<in> P}))" in exI)
```
```    52 apply clarify
```
```    53 apply (case_tac "0 < ya", auto)
```
```    54 apply (frule real_sup_lemma2, assumption+)
```
```    55 apply (drule real_gt_zero_preal_Ex [THEN iffD1])
```
```    56 apply (drule_tac [3] real_less_all_real2, auto)
```
```    57 apply (rule preal_complete [THEN iffD1])
```
```    58 apply (auto intro: order_less_imp_le)
```
```    59 apply (frule real_gt_preal_preal_Ex, force)
```
```    60 (* second part *)
```
```    61 apply (rule real_sup_lemma1 [THEN iffD2], assumption)
```
```    62 apply (auto dest!: real_less_all_real2 real_gt_zero_preal_Ex [THEN iffD1])
```
```    63 apply (frule_tac [2] real_sup_lemma2)
```
```    64 apply (frule real_sup_lemma2, assumption+, clarify)
```
```    65 apply (rule preal_complete [THEN iffD2, THEN bexE])
```
```    66 prefer 3 apply blast
```
```    67 apply (blast intro!: order_less_imp_le)+
```
```    68 done
```
```    69
```
```    70 (*--------------------------------------------------------
```
```    71    Completeness properties using isUb, isLub etc.
```
```    72  -------------------------------------------------------*)
```
```    73
```
```    74 lemma real_isLub_unique: "[| isLub R S x; isLub R S y |] ==> x = (y::real)"
```
```    75 apply (frule isLub_isUb)
```
```    76 apply (frule_tac x = y in isLub_isUb)
```
```    77 apply (blast intro!: order_antisym dest!: isLub_le_isUb)
```
```    78 done
```
```    79
```
```    80 lemma real_order_restrict: "[| (x::real) <=* S'; S <= S' |] ==> x <=* S"
```
```    81 by (unfold setle_def setge_def, blast)
```
```    82
```
```    83 (*----------------------------------------------------------------
```
```    84            Completeness theorem for the positive reals(again)
```
```    85  ----------------------------------------------------------------*)
```
```    86
```
```    87 lemma posreals_complete:
```
```    88      "[| \<forall>x \<in>S. 0 < x;
```
```    89          \<exists>x. x \<in>S;
```
```    90          \<exists>u. isUb (UNIV::real set) S u
```
```    91       |] ==> \<exists>t. isLub (UNIV::real set) S t"
```
```    92 apply (rule_tac x = "real_of_preal (psup ({w. real_of_preal w \<in> S}))" in exI)
```
```    93 apply (auto simp add: isLub_def leastP_def isUb_def)
```
```    94 apply (auto intro!: setleI setgeI dest!: real_gt_zero_preal_Ex [THEN iffD1])
```
```    95 apply (frule_tac x = y in bspec, assumption)
```
```    96 apply (drule real_gt_zero_preal_Ex [THEN iffD1])
```
```    97 apply (auto simp add: real_of_preal_le_iff)
```
```    98 apply (frule_tac y = "real_of_preal ya" in setleD, assumption)
```
```    99 apply (frule real_ge_preal_preal_Ex, safe)
```
```   100 apply (blast intro!: preal_psup_le dest!: setleD intro: real_of_preal_le_iff [THEN iffD1])
```
```   101 apply (frule_tac x = x in bspec, assumption)
```
```   102 apply (frule isUbD2)
```
```   103 apply (drule real_gt_zero_preal_Ex [THEN iffD1])
```
```   104 apply (auto dest!: isUbD real_ge_preal_preal_Ex simp add: real_of_preal_le_iff)
```
```   105 apply (blast dest!: setleD intro!: psup_le_ub intro: real_of_preal_le_iff [THEN iffD1])
```
```   106 done
```
```   107
```
```   108
```
```   109 (*-------------------------------
```
```   110     Lemmas
```
```   111  -------------------------------*)
```
```   112 lemma real_sup_lemma3: "\<forall>y \<in> {z. \<exists>x \<in> P. z = x + (-xa) + 1} Int {x. 0 < x}. 0 < y"
```
```   113 by auto
```
```   114
```
```   115 lemma lemma_le_swap2: "(xa <= S + X + (-Z)) = (xa + (-X) + Z <= (S::real))"
```
```   116 by auto
```
```   117
```
```   118 lemma lemma_real_complete2b: "[| (x::real) + (-X) + 1 <= S; xa <= x |] ==> xa <= S + X + (- 1)"
```
```   119 by arith
```
```   120
```
```   121 (*----------------------------------------------------------
```
```   122       reals Completeness (again!)
```
```   123  ----------------------------------------------------------*)
```
```   124 lemma reals_complete: "[| \<exists>X. X \<in>S;  \<exists>Y. isUb (UNIV::real set) S Y |]
```
```   125       ==> \<exists>t. isLub (UNIV :: real set) S t"
```
```   126 apply safe
```
```   127 apply (subgoal_tac "\<exists>u. u\<in> {z. \<exists>x \<in>S. z = x + (-X) + 1} Int {x. 0 < x}")
```
```   128 apply (subgoal_tac "isUb (UNIV::real set) ({z. \<exists>x \<in>S. z = x + (-X) + 1} Int {x. 0 < x}) (Y + (-X) + 1) ")
```
```   129 apply (cut_tac P = S and xa = X in real_sup_lemma3)
```
```   130 apply (frule posreals_complete [OF _ _ exI], blast, blast, safe)
```
```   131 apply (rule_tac x = "t + X + (- 1) " in exI)
```
```   132 apply (rule isLubI2)
```
```   133 apply (rule_tac [2] setgeI, safe)
```
```   134 apply (subgoal_tac [2] "isUb (UNIV:: real set) ({z. \<exists>x \<in>S. z = x + (-X) + 1} Int {x. 0 < x}) (y + (-X) + 1) ")
```
```   135 apply (drule_tac [2] y = " (y + (- X) + 1) " in isLub_le_isUb)
```
```   136  prefer 2 apply assumption
```
```   137  prefer 2
```
```   138 apply arith
```
```   139 apply (rule setleI [THEN isUbI], safe)
```
```   140 apply (rule_tac x = x and y = y in linorder_cases)
```
```   141 apply (subst lemma_le_swap2)
```
```   142 apply (frule isLubD2)
```
```   143  prefer 2 apply assumption
```
```   144 apply safe
```
```   145 apply blast
```
```   146 apply arith
```
```   147 apply (subst lemma_le_swap2)
```
```   148 apply (frule isLubD2)
```
```   149  prefer 2 apply assumption
```
```   150 apply blast
```
```   151 apply (rule lemma_real_complete2b)
```
```   152 apply (erule_tac [2] order_less_imp_le)
```
```   153 apply (blast intro!: isLubD2, blast)
```
```   154 apply (simp (no_asm_use) add: real_add_assoc)
```
```   155 apply (blast dest: isUbD intro!: setleI [THEN isUbI] intro: add_right_mono)
```
```   156 apply (blast dest: isUbD intro!: setleI [THEN isUbI] intro: add_right_mono, auto)
```
```   157 done
```
```   158
```
```   159
```
```   160 subsection{*Corollary: the Archimedean Property of the Reals*}
```
```   161
```
```   162 lemma reals_Archimedean: "0 < x ==> \<exists>n. inverse (real(Suc n)) < x"
```
```   163 apply (rule ccontr)
```
```   164 apply (subgoal_tac "\<forall>n. x * real (Suc n) <= 1")
```
```   165  prefer 2
```
```   166 apply (simp add: linorder_not_less inverse_eq_divide, clarify)
```
```   167 apply (drule_tac x = n in spec)
```
```   168 apply (drule_tac c = "real (Suc n)"  in mult_right_mono)
```
```   169 apply (rule real_of_nat_ge_zero)
```
```   170 apply (simp add: real_of_nat_Suc_gt_zero [THEN real_not_refl2, THEN not_sym] real_mult_commute)
```
```   171 apply (subgoal_tac "isUb (UNIV::real set) {z. \<exists>n. z = x* (real (Suc n))} 1")
```
```   172 apply (subgoal_tac "\<exists>X. X \<in> {z. \<exists>n. z = x* (real (Suc n))}")
```
```   173 apply (drule reals_complete)
```
```   174 apply (auto intro: isUbI setleI)
```
```   175 apply (subgoal_tac "\<forall>m. x* (real (Suc m)) <= t")
```
```   176 apply (simp add: real_of_nat_Suc right_distrib)
```
```   177 prefer 2 apply (blast intro: isLubD2)
```
```   178 apply (simp add: le_diff_eq [symmetric] real_diff_def)
```
```   179 apply (subgoal_tac "isUb (UNIV::real set) {z. \<exists>n. z = x* (real (Suc n))} (t + (-x))")
```
```   180 prefer 2 apply (blast intro!: isUbI setleI)
```
```   181 apply (drule_tac y = "t+ (-x) " in isLub_le_isUb)
```
```   182 apply (auto simp add: real_of_nat_Suc right_distrib)
```
```   183 done
```
```   184
```
```   185 (*There must be other proofs, e.g. Suc of the largest integer in the
```
```   186   cut representing x*)
```
```   187 lemma reals_Archimedean2: "\<exists>n. (x::real) < real (n::nat)"
```
```   188 apply (rule_tac x = x and y = 0 in linorder_cases)
```
```   189 apply (rule_tac x = 0 in exI)
```
```   190 apply (rule_tac [2] x = 1 in exI)
```
```   191 apply (auto elim: order_less_trans simp add: real_of_nat_one)
```
```   192 apply (frule positive_imp_inverse_positive [THEN reals_Archimedean], safe)
```
```   193 apply (rule_tac x = "Suc n" in exI)
```
```   194 apply (frule_tac b = "inverse x" in mult_strict_right_mono, auto)
```
```   195 done
```
```   196
```
```   197 lemma reals_Archimedean3: "0 < x ==> \<forall>y. \<exists>(n::nat). y < real n * x"
```
```   198 apply safe
```
```   199 apply (cut_tac x = "y*inverse (x) " in reals_Archimedean2)
```
```   200 apply safe
```
```   201 apply (frule_tac a = "y * inverse x" in mult_strict_right_mono)
```
```   202 apply (auto simp add: mult_assoc real_of_nat_def)
```
```   203 done
```
```   204
```
```   205 ML
```
```   206 {*
```
```   207 val real_sum_of_halves = thm "real_sum_of_halves";
```
```   208 val posreal_complete = thm "posreal_complete";
```
```   209 val real_isLub_unique = thm "real_isLub_unique";
```
```   210 val real_order_restrict = thm "real_order_restrict";
```
```   211 val posreals_complete = thm "posreals_complete";
```
```   212 val reals_complete = thm "reals_complete";
```
```   213 val reals_Archimedean = thm "reals_Archimedean";
```
```   214 val reals_Archimedean2 = thm "reals_Archimedean2";
```
```   215 val reals_Archimedean3 = thm "reals_Archimedean3";
```
```   216 *}
```
```   217
```
```   218 end
```
```   219
```
```   220
```
```   221
```