src/HOL/RComplete.thy
 changeset 36795 e05e1283c550 parent 35578 384ad08a1d1b child 36826 4d4462d644ae
```     1.1 --- a/src/HOL/RComplete.thy	Mon May 10 11:47:56 2010 -0700
1.2 +++ b/src/HOL/RComplete.thy	Mon May 10 12:12:58 2010 -0700
1.3 @@ -30,92 +30,27 @@
1.4    FIXME: Can the premise be weakened to @{text "\<forall>x \<in> P. x\<le> y"}?
1.5  *}
1.6
1.7 +text {* Only used in HOL/Import/HOL4Compat.thy; delete? *}
1.8 +
1.9  lemma posreal_complete:
1.10    assumes positive_P: "\<forall>x \<in> P. (0::real) < x"
1.11      and not_empty_P: "\<exists>x. x \<in> P"
1.12      and upper_bound_Ex: "\<exists>y. \<forall>x \<in> P. x<y"
1.13    shows "\<exists>S. \<forall>y. (\<exists>x \<in> P. y < x) = (y < S)"
1.14 -proof (rule exI, rule allI)
1.15 -  fix y
1.16 -  let ?pP = "{w. real_of_preal w \<in> P}"
1.17 -
1.18 -  show "(\<exists>x\<in>P. y < x) = (y < real_of_preal (psup ?pP))"
1.19 -  proof (cases "0 < y")
1.20 -    assume neg_y: "\<not> 0 < y"
1.21 -    show ?thesis
1.22 -    proof
1.23 -      assume "\<exists>x\<in>P. y < x"
1.24 -      have "\<forall>x. y < real_of_preal x"
1.25 -        using neg_y by (rule real_less_all_real2)
1.26 -      thus "y < real_of_preal (psup ?pP)" ..
1.27 -    next
1.28 -      assume "y < real_of_preal (psup ?pP)"
1.29 -      obtain "x" where x_in_P: "x \<in> P" using not_empty_P ..
1.30 -      hence "0 < x" using positive_P by simp
1.31 -      hence "y < x" using neg_y by simp
1.32 -      thus "\<exists>x \<in> P. y < x" using x_in_P ..
1.33 -    qed
1.34 -  next
1.35 -    assume pos_y: "0 < y"
1.36 -
1.37 -    then obtain py where y_is_py: "y = real_of_preal py"
1.38 -      by (auto simp add: real_gt_zero_preal_Ex)
1.39 -
1.40 -    obtain a where "a \<in> P" using not_empty_P ..
1.41 -    with positive_P have a_pos: "0 < a" ..
1.42 -    then obtain pa where "a = real_of_preal pa"
1.43 -      by (auto simp add: real_gt_zero_preal_Ex)
1.44 -    hence "pa \<in> ?pP" using `a \<in> P` by auto
1.45 -    hence pP_not_empty: "?pP \<noteq> {}" by auto
1.46 -
1.47 -    obtain sup where sup: "\<forall>x \<in> P. x < sup"
1.48 -      using upper_bound_Ex ..
1.49 -    from this and `a \<in> P` have "a < sup" ..
1.50 -    hence "0 < sup" using a_pos by arith
1.51 -    then obtain possup where "sup = real_of_preal possup"
1.52 -      by (auto simp add: real_gt_zero_preal_Ex)
1.53 -    hence "\<forall>X \<in> ?pP. X \<le> possup"
1.54 -      using sup by (auto simp add: real_of_preal_lessI)
1.55 -    with pP_not_empty have psup: "\<And>Z. (\<exists>X \<in> ?pP. Z < X) = (Z < psup ?pP)"
1.56 -      by (rule preal_complete)
1.57 -
1.58 -    show ?thesis
1.59 -    proof
1.60 -      assume "\<exists>x \<in> P. y < x"
1.61 -      then obtain x where x_in_P: "x \<in> P" and y_less_x: "y < x" ..
1.62 -      hence "0 < x" using pos_y by arith
1.63 -      then obtain px where x_is_px: "x = real_of_preal px"
1.64 -        by (auto simp add: real_gt_zero_preal_Ex)
1.65 -
1.66 -      have py_less_X: "\<exists>X \<in> ?pP. py < X"
1.67 -      proof
1.68 -        show "py < px" using y_is_py and x_is_px and y_less_x
1.69 -          by (simp add: real_of_preal_lessI)
1.70 -        show "px \<in> ?pP" using x_in_P and x_is_px by simp
1.71 -      qed
1.72 -
1.73 -      have "(\<exists>X \<in> ?pP. py < X) ==> (py < psup ?pP)"
1.74 -        using psup by simp
1.75 -      hence "py < psup ?pP" using py_less_X by simp
1.76 -      thus "y < real_of_preal (psup {w. real_of_preal w \<in> P})"
1.77 -        using y_is_py and pos_y by (simp add: real_of_preal_lessI)
1.78 -    next
1.79 -      assume y_less_psup: "y < real_of_preal (psup ?pP)"
1.80 -
1.81 -      hence "py < psup ?pP" using y_is_py
1.82 -        by (simp add: real_of_preal_lessI)
1.83 -      then obtain "X" where py_less_X: "py < X" and X_in_pP: "X \<in> ?pP"
1.84 -        using psup by auto
1.85 -      then obtain x where x_is_X: "x = real_of_preal X"
1.86 -        by (simp add: real_gt_zero_preal_Ex)
1.87 -      hence "y < x" using py_less_X and y_is_py
1.88 -        by (simp add: real_of_preal_lessI)
1.89 -
1.90 -      moreover have "x \<in> P" using x_is_X and X_in_pP by simp
1.91 -
1.92 -      ultimately show "\<exists> x \<in> P. y < x" ..
1.93 -    qed
1.94 +proof -
1.95 +  from upper_bound_Ex have "\<exists>z. \<forall>x\<in>P. x \<le> z"
1.96 +    by (auto intro: less_imp_le)
1.97 +  from complete_real [OF not_empty_P this] obtain S
1.98 +  where S1: "\<And>x. x \<in> P \<Longrightarrow> x \<le> S" and S2: "\<And>z. \<forall>x\<in>P. x \<le> z \<Longrightarrow> S \<le> z" by fast
1.99 +  have "\<forall>y. (\<exists>x \<in> P. y < x) = (y < S)"
1.100 +  proof
1.101 +    fix y show "(\<exists>x\<in>P. y < x) = (y < S)"
1.102 +      apply (cases "\<exists>x\<in>P. y < x", simp_all)
1.103 +      apply (clarify, drule S1, simp)
1.104 +      apply (simp add: not_less S2)
1.105 +      done
1.106    qed
1.107 +  thus ?thesis ..
1.108  qed
1.109
1.110  text {*
1.111 @@ -130,89 +65,6 @@
1.112
1.113
1.114  text {*
1.115 -  \medskip Completeness theorem for the positive reals (again).
1.116 -*}
1.117 -
1.118 -lemma posreals_complete:
1.119 -  assumes positive_S: "\<forall>x \<in> S. 0 < x"
1.120 -    and not_empty_S: "\<exists>x. x \<in> S"
1.121 -    and upper_bound_Ex: "\<exists>u. isUb (UNIV::real set) S u"
1.122 -  shows "\<exists>t. isLub (UNIV::real set) S t"
1.123 -proof
1.124 -  let ?pS = "{w. real_of_preal w \<in> S}"
1.125 -
1.126 -  obtain u where "isUb UNIV S u" using upper_bound_Ex ..
1.127 -  hence sup: "\<forall>x \<in> S. x \<le> u" by (simp add: isUb_def setle_def)
1.128 -
1.129 -  obtain x where x_in_S: "x \<in> S" using not_empty_S ..
1.130 -  hence x_gt_zero: "0 < x" using positive_S by simp
1.131 -  have  "x \<le> u" using sup and x_in_S ..
1.132 -  hence "0 < u" using x_gt_zero by arith
1.133 -
1.134 -  then obtain pu where u_is_pu: "u = real_of_preal pu"
1.135 -    by (auto simp add: real_gt_zero_preal_Ex)
1.136 -
1.137 -  have pS_less_pu: "\<forall>pa \<in> ?pS. pa \<le> pu"
1.138 -  proof
1.139 -    fix pa
1.140 -    assume "pa \<in> ?pS"
1.141 -    then obtain a where "a \<in> S" and "a = real_of_preal pa"
1.142 -      by simp
1.143 -    moreover hence "a \<le> u" using sup by simp
1.144 -    ultimately show "pa \<le> pu"
1.145 -      using sup and u_is_pu by (simp add: real_of_preal_le_iff)
1.146 -  qed
1.147 -
1.148 -  have "\<forall>y \<in> S. y \<le> real_of_preal (psup ?pS)"
1.149 -  proof
1.150 -    fix y
1.151 -    assume y_in_S: "y \<in> S"
1.152 -    hence "0 < y" using positive_S by simp
1.153 -    then obtain py where y_is_py: "y = real_of_preal py"
1.154 -      by (auto simp add: real_gt_zero_preal_Ex)
1.155 -    hence py_in_pS: "py \<in> ?pS" using y_in_S by simp
1.156 -    with pS_less_pu have "py \<le> psup ?pS"
1.157 -      by (rule preal_psup_le)
1.158 -    thus "y \<le> real_of_preal (psup ?pS)"
1.159 -      using y_is_py by (simp add: real_of_preal_le_iff)
1.160 -  qed
1.161 -
1.162 -  moreover {
1.163 -    fix x
1.164 -    assume x_ub_S: "\<forall>y\<in>S. y \<le> x"
1.165 -    have "real_of_preal (psup ?pS) \<le> x"
1.166 -    proof -
1.167 -      obtain "s" where s_in_S: "s \<in> S" using not_empty_S ..
1.168 -      hence s_pos: "0 < s" using positive_S by simp
1.169 -
1.170 -      hence "\<exists> ps. s = real_of_preal ps" by (simp add: real_gt_zero_preal_Ex)
1.171 -      then obtain "ps" where s_is_ps: "s = real_of_preal ps" ..
1.172 -      hence ps_in_pS: "ps \<in> {w. real_of_preal w \<in> S}" using s_in_S by simp
1.173 -
1.174 -      from x_ub_S have "s \<le> x" using s_in_S ..
1.175 -      hence "0 < x" using s_pos by simp
1.176 -      hence "\<exists> px. x = real_of_preal px" by (simp add: real_gt_zero_preal_Ex)
1.177 -      then obtain "px" where x_is_px: "x = real_of_preal px" ..
1.178 -
1.179 -      have "\<forall>pe \<in> ?pS. pe \<le> px"
1.180 -      proof
1.181 -        fix pe
1.182 -        assume "pe \<in> ?pS"
1.183 -        hence "real_of_preal pe \<in> S" by simp
1.184 -        hence "real_of_preal pe \<le> x" using x_ub_S by simp
1.185 -        thus "pe \<le> px" using x_is_px by (simp add: real_of_preal_le_iff)
1.186 -      qed
1.187 -
1.188 -      moreover have "?pS \<noteq> {}" using ps_in_pS by auto
1.189 -      ultimately have "(psup ?pS) \<le> px" by (simp add: psup_le_ub)
1.190 -      thus "real_of_preal (psup ?pS) \<le> x" using x_is_px by (simp add: real_of_preal_le_iff)
1.191 -    qed
1.192 -  }
1.193 -  ultimately show "isLub UNIV S (real_of_preal (psup ?pS))"
1.194 -    by (simp add: isLub_def leastP_def isUb_def setle_def setge_def)
1.195 -qed
1.196 -
1.197 -text {*
1.198    \medskip reals Completeness (again!)
1.199  *}
1.200
1.201 @@ -221,87 +73,11 @@
1.202      and exists_Ub: "\<exists>Y. isUb (UNIV::real set) S Y"
1.203    shows "\<exists>t. isLub (UNIV :: real set) S t"
1.204  proof -
1.205 -  obtain X where X_in_S: "X \<in> S" using notempty_S ..
1.206 -  obtain Y where Y_isUb: "isUb (UNIV::real set) S Y"
1.207 -    using exists_Ub ..
1.208 -  let ?SHIFT = "{z. \<exists>x \<in>S. z = x + (-X) + 1} \<inter> {x. 0 < x}"
1.209 -
1.210 -  {
1.211 -    fix x
1.212 -    assume "isUb (UNIV::real set) S x"
1.213 -    hence S_le_x: "\<forall> y \<in> S. y <= x"
1.214 -      by (simp add: isUb_def setle_def)
1.215 -    {
1.216 -      fix s
1.217 -      assume "s \<in> {z. \<exists>x\<in>S. z = x + - X + 1}"
1.218 -      hence "\<exists> x \<in> S. s = x + -X + 1" ..
1.219 -      then obtain x1 where "x1 \<in> S" and "s = x1 + (-X) + 1" ..
1.220 -      moreover hence "x1 \<le> x" using S_le_x by simp
1.221 -      ultimately have "s \<le> x + - X + 1" by arith
1.222 -    }
1.223 -    then have "isUb (UNIV::real set) ?SHIFT (x + (-X) + 1)"
1.224 -      by (auto simp add: isUb_def setle_def)
1.225 -  } note S_Ub_is_SHIFT_Ub = this
1.226 -
1.227 -  hence "isUb UNIV ?SHIFT (Y + (-X) + 1)" using Y_isUb by simp
1.228 -  hence "\<exists>Z. isUb UNIV ?SHIFT Z" ..
1.229 -  moreover have "\<forall>y \<in> ?SHIFT. 0 < y" by auto
1.230 -  moreover have shifted_not_empty: "\<exists>u. u \<in> ?SHIFT"
1.231 -    using X_in_S and Y_isUb by auto
1.232 -  ultimately obtain t where t_is_Lub: "isLub UNIV ?SHIFT t"
1.233 -    using posreals_complete [of ?SHIFT] by blast
1.234 -
1.235 -  show ?thesis
1.236 -  proof
1.237 -    show "isLub UNIV S (t + X + (-1))"
1.238 -    proof (rule isLubI2)
1.239 -      {
1.240 -        fix x
1.241 -        assume "isUb (UNIV::real set) S x"
1.242 -        hence "isUb (UNIV::real set) (?SHIFT) (x + (-X) + 1)"
1.243 -          using S_Ub_is_SHIFT_Ub by simp
1.244 -        hence "t \<le> (x + (-X) + 1)"
1.245 -          using t_is_Lub by (simp add: isLub_le_isUb)
1.246 -        hence "t + X + -1 \<le> x" by arith
1.247 -      }
1.248 -      then show "(t + X + -1) <=* Collect (isUb UNIV S)"
1.249 -        by (simp add: setgeI)
1.250 -    next
1.251 -      show "isUb UNIV S (t + X + -1)"
1.252 -      proof -
1.253 -        {
1.254 -          fix y
1.255 -          assume y_in_S: "y \<in> S"
1.256 -          have "y \<le> t + X + -1"
1.257 -          proof -
1.258 -            obtain "u" where u_in_shift: "u \<in> ?SHIFT" using shifted_not_empty ..
1.259 -            hence "\<exists> x \<in> S. u = x + - X + 1" by simp
1.260 -            then obtain "x" where x_and_u: "u = x + - X + 1" ..
1.261 -            have u_le_t: "u \<le> t" using u_in_shift and t_is_Lub by (simp add: isLubD2)
1.262 -
1.263 -            show ?thesis
1.264 -            proof cases
1.265 -              assume "y \<le> x"
1.266 -              moreover have "x = u + X + - 1" using x_and_u by arith
1.267 -              moreover have "u + X + - 1  \<le> t + X + -1" using u_le_t by arith
1.268 -              ultimately show "y  \<le> t + X + -1" by arith
1.269 -            next
1.270 -              assume "~(y \<le> x)"
1.271 -              hence x_less_y: "x < y" by arith
1.272 -
1.273 -              have "x + (-X) + 1 \<in> ?SHIFT" using x_and_u and u_in_shift by simp
1.274 -              hence "0 < x + (-X) + 1" by simp
1.275 -              hence "0 < y + (-X) + 1" using x_less_y by arith
1.276 -              hence "y + (-X) + 1 \<in> ?SHIFT" using y_in_S by simp
1.277 -              hence "y + (-X) + 1 \<le> t" using t_is_Lub  by (simp add: isLubD2)
1.278 -              thus ?thesis by simp
1.279 -            qed
1.280 -          qed
1.281 -        }
1.282 -        then show ?thesis by (simp add: isUb_def setle_def)
1.283 -      qed
1.284 -    qed
1.285 -  qed
1.286 +  from assms have "\<exists>X. X \<in> S" and "\<exists>Y. \<forall>x\<in>S. x \<le> Y"
1.287 +    unfolding isUb_def setle_def by simp_all
1.288 +  from complete_real [OF this] show ?thesis
1.289 +    unfolding isLub_def leastP_def setle_def setge_def Ball_def
1.290 +      Collect_def mem_def isUb_def UNIV_def by simp
1.291  qed
1.292
1.293  text{*A version of the same theorem without all those predicates!*}
1.294 @@ -310,13 +86,7 @@
1.295    assumes "\<exists>y. y\<in>S" and "\<exists>(x::real). \<forall>y\<in>S. y \<le> x"
1.296    shows "\<exists>x. (\<forall>y\<in>S. y \<le> x) &
1.297                 (\<forall>z. ((\<forall>y\<in>S. y \<le> z) --> x \<le> z))"
1.298 -proof -
1.299 -  have "\<exists>x. isLub UNIV S x"
1.300 -    by (rule reals_complete)
1.301 -       (auto simp add: isLub_def isUb_def leastP_def setle_def setge_def prems)
1.302 -  thus ?thesis
1.303 -    by (metis UNIV_I isLub_isUb isLub_le_isUb isUbD isUb_def setleI)
1.304 -qed
1.305 +using assms by (rule complete_real)
1.306
1.307
1.308  subsection {* The Archimedean Property of the Reals *}
1.309 @@ -324,88 +94,11 @@
1.310  theorem reals_Archimedean:
1.311    assumes x_pos: "0 < x"
1.312    shows "\<exists>n. inverse (real (Suc n)) < x"
1.313 -proof (rule ccontr)
1.314 -  assume contr: "\<not> ?thesis"
1.315 -  have "\<forall>n. x * real (Suc n) <= 1"
1.316 -  proof
1.317 -    fix n
1.318 -    from contr have "x \<le> inverse (real (Suc n))"
1.319 -      by (simp add: linorder_not_less)
1.320 -    hence "x \<le> (1 / (real (Suc n)))"
1.321 -      by (simp add: inverse_eq_divide)
1.322 -    moreover have "0 \<le> real (Suc n)"
1.323 -      by (rule real_of_nat_ge_zero)
1.324 -    ultimately have "x * real (Suc n) \<le> (1 / real (Suc n)) * real (Suc n)"
1.325 -      by (rule mult_right_mono)
1.326 -    thus "x * real (Suc n) \<le> 1" by simp
1.327 -  qed
1.328 -  hence "{z. \<exists>n. z = x * (real (Suc n))} *<= 1"
1.329 -    by (simp add: setle_def, safe, rule spec)
1.330 -  hence "isUb (UNIV::real set) {z. \<exists>n. z = x * (real (Suc n))} 1"
1.331 -    by (simp add: isUbI)
1.332 -  hence "\<exists>Y. isUb (UNIV::real set) {z. \<exists>n. z = x* (real (Suc n))} Y" ..
1.333 -  moreover have "\<exists>X. X \<in> {z. \<exists>n. z = x* (real (Suc n))}" by auto
1.334 -  ultimately have "\<exists>t. isLub UNIV {z. \<exists>n. z = x * real (Suc n)} t"
1.335 -    by (simp add: reals_complete)
1.336 -  then obtain "t" where
1.337 -    t_is_Lub: "isLub UNIV {z. \<exists>n. z = x * real (Suc n)} t" ..
1.338 -
1.339 -  have "\<forall>n::nat. x * real n \<le> t + - x"
1.340 -  proof
1.341 -    fix n
1.342 -    from t_is_Lub have "x * real (Suc n) \<le> t"
1.343 -      by (simp add: isLubD2)
1.344 -    hence  "x * (real n) + x \<le> t"
1.345 -      by (simp add: right_distrib real_of_nat_Suc)
1.346 -    thus  "x * (real n) \<le> t + - x" by arith
1.347 -  qed
1.348 -
1.349 -  hence "\<forall>m. x * real (Suc m) \<le> t + - x" by simp
1.350 -  hence "{z. \<exists>n. z = x * (real (Suc n))}  *<= (t + - x)"
1.351 -    by (auto simp add: setle_def)
1.352 -  hence "isUb (UNIV::real set) {z. \<exists>n. z = x * (real (Suc n))} (t + (-x))"
1.353 -    by (simp add: isUbI)
1.354 -  hence "t \<le> t + - x"
1.355 -    using t_is_Lub by (simp add: isLub_le_isUb)
1.356 -  thus False using x_pos by arith
1.357 -qed
1.358 -
1.359 -text {*
1.360 -  There must be other proofs, e.g. @{text "Suc"} of the largest
1.361 -  integer in the cut representing @{text "x"}.
1.362 -*}
1.363 +  unfolding real_of_nat_def using x_pos
1.364 +  by (rule ex_inverse_of_nat_Suc_less)
1.365
1.366  lemma reals_Archimedean2: "\<exists>n. (x::real) < real (n::nat)"
1.367 -proof cases
1.368 -  assume "x \<le> 0"
1.369 -  hence "x < real (1::nat)" by simp
1.370 -  thus ?thesis ..
1.371 -next
1.372 -  assume "\<not> x \<le> 0"
1.373 -  hence x_greater_zero: "0 < x" by simp
1.374 -  hence "0 < inverse x" by simp
1.375 -  then obtain n where "inverse (real (Suc n)) < inverse x"
1.376 -    using reals_Archimedean by blast
1.377 -  hence "inverse (real (Suc n)) * x < inverse x * x"
1.378 -    using x_greater_zero by (rule mult_strict_right_mono)
1.379 -  hence "inverse (real (Suc n)) * x < 1"
1.380 -    using x_greater_zero by simp
1.381 -  hence "real (Suc n) * (inverse (real (Suc n)) * x) < real (Suc n) * 1"
1.382 -    by (rule mult_strict_left_mono) simp
1.383 -  hence "x < real (Suc n)"
1.384 -    by (simp add: algebra_simps)
1.385 -  thus "\<exists>(n::nat). x < real n" ..
1.386 -qed
1.387 -
1.388 -instance real :: archimedean_field
1.389 -proof
1.390 -  fix r :: real
1.391 -  obtain n :: nat where "r < real n"
1.392 -    using reals_Archimedean2 ..
1.393 -  then have "r \<le> of_int (int n)"
1.394 -    unfolding real_eq_of_nat by simp
1.395 -  then show "\<exists>z. r \<le> of_int z" ..
1.396 -qed
1.397 +  unfolding real_of_nat_def by (rule ex_less_of_nat)
1.398
1.399  lemma reals_Archimedean3:
1.400    assumes x_greater_zero: "0 < x"
1.401 @@ -458,7 +151,7 @@
1.402    have "x = y-(y-x)" by simp
1.403    also from suc q have "\<dots> < real (Suc p)/real q - inverse (real q)" by arith
1.404    also have "\<dots> = real p / real q"
1.405 -    by (simp only: inverse_eq_divide real_diff_def real_of_nat_Suc
1.406 +    by (simp only: inverse_eq_divide diff_def real_of_nat_Suc