src/HOL/RComplete.thy
author huffman
Thu, 26 Feb 2009 06:21:31 -0800
changeset 30102 799b687e4aac
parent 30097 57df8626c23b
child 30122 1c912a9d8200
permissions -rw-r--r--
disable floor_minus and ceiling_minus [simp]
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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(*  Title       : HOL/RComplete.thy
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    Author      : Jacques D. Fleuriot, University of Edinburgh
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    Author      : Larry Paulson, University of Cambridge
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    Author      : Jeremy Avigad, Carnegie Mellon University
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    Author      : Florian Zuleger, Johannes Hoelzl, and Simon Funke, TU Muenchen
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*)
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header {* Completeness of the Reals; Floor and Ceiling Functions *}
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theory RComplete
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imports Lubs RealDef
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begin
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lemma real_sum_of_halves: "x/2 + x/2 = (x::real)"
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  by simp
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subsection {* Completeness of Positive Reals *}
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text {*
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  Supremum property for the set of positive reals
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  Let @{text "P"} be a non-empty set of positive reals, with an upper
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  bound @{text "y"}.  Then @{text "P"} has a least upper bound
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  (written @{text "S"}).
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  FIXME: Can the premise be weakened to @{text "\<forall>x \<in> P. x\<le> y"}?
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*}
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lemma posreal_complete:
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  assumes positive_P: "\<forall>x \<in> P. (0::real) < x"
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    and not_empty_P: "\<exists>x. x \<in> P"
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    and upper_bound_Ex: "\<exists>y. \<forall>x \<in> P. x<y"
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  shows "\<exists>S. \<forall>y. (\<exists>x \<in> P. y < x) = (y < S)"
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proof (rule exI, rule allI)
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  fix y
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  let ?pP = "{w. real_of_preal w \<in> P}"
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  show "(\<exists>x\<in>P. y < x) = (y < real_of_preal (psup ?pP))"
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  proof (cases "0 < y")
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    assume neg_y: "\<not> 0 < y"
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    show ?thesis
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    proof
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      assume "\<exists>x\<in>P. y < x"
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      have "\<forall>x. y < real_of_preal x"
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        using neg_y by (rule real_less_all_real2)
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      thus "y < real_of_preal (psup ?pP)" ..
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    next
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      assume "y < real_of_preal (psup ?pP)"
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      obtain "x" where x_in_P: "x \<in> P" using not_empty_P ..
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      hence "0 < x" using positive_P by simp
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      hence "y < x" using neg_y by simp
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      thus "\<exists>x \<in> P. y < x" using x_in_P ..
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    qed
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  next
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    assume pos_y: "0 < y"
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    then obtain py where y_is_py: "y = real_of_preal py"
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      by (auto simp add: real_gt_zero_preal_Ex)
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    obtain a where "a \<in> P" using not_empty_P ..
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    with positive_P have a_pos: "0 < a" ..
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    then obtain pa where "a = real_of_preal pa"
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      by (auto simp add: real_gt_zero_preal_Ex)
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    hence "pa \<in> ?pP" using `a \<in> P` by auto
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    hence pP_not_empty: "?pP \<noteq> {}" by auto
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    obtain sup where sup: "\<forall>x \<in> P. x < sup"
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      using upper_bound_Ex ..
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    from this and `a \<in> P` have "a < sup" ..
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    hence "0 < sup" using a_pos by arith
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    then obtain possup where "sup = real_of_preal possup"
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      by (auto simp add: real_gt_zero_preal_Ex)
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    hence "\<forall>X \<in> ?pP. X \<le> possup"
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      using sup by (auto simp add: real_of_preal_lessI)
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    with pP_not_empty have psup: "\<And>Z. (\<exists>X \<in> ?pP. Z < X) = (Z < psup ?pP)"
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      by (rule preal_complete)
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    show ?thesis
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    proof
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      assume "\<exists>x \<in> P. y < x"
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      then obtain x where x_in_P: "x \<in> P" and y_less_x: "y < x" ..
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      hence "0 < x" using pos_y by arith
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      then obtain px where x_is_px: "x = real_of_preal px"
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        by (auto simp add: real_gt_zero_preal_Ex)
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      have py_less_X: "\<exists>X \<in> ?pP. py < X"
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      proof
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        show "py < px" using y_is_py and x_is_px and y_less_x
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          by (simp add: real_of_preal_lessI)
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        show "px \<in> ?pP" using x_in_P and x_is_px by simp
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      qed
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      have "(\<exists>X \<in> ?pP. py < X) ==> (py < psup ?pP)"
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        using psup by simp
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      hence "py < psup ?pP" using py_less_X by simp
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      thus "y < real_of_preal (psup {w. real_of_preal w \<in> P})"
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        using y_is_py and pos_y by (simp add: real_of_preal_lessI)
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    next
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      assume y_less_psup: "y < real_of_preal (psup ?pP)"
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      hence "py < psup ?pP" using y_is_py
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        by (simp add: real_of_preal_lessI)
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      then obtain "X" where py_less_X: "py < X" and X_in_pP: "X \<in> ?pP"
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        using psup by auto
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      then obtain x where x_is_X: "x = real_of_preal X"
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        by (simp add: real_gt_zero_preal_Ex)
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      hence "y < x" using py_less_X and y_is_py
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        by (simp add: real_of_preal_lessI)
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      moreover have "x \<in> P" using x_is_X and X_in_pP by simp
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      ultimately show "\<exists> x \<in> P. y < x" ..
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    qed
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  qed
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qed
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text {*
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  \medskip Completeness properties using @{text "isUb"}, @{text "isLub"} etc.
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*}
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lemma real_isLub_unique: "[| isLub R S x; isLub R S y |] ==> x = (y::real)"
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  apply (frule isLub_isUb)
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  apply (frule_tac x = y in isLub_isUb)
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  apply (blast intro!: order_antisym dest!: isLub_le_isUb)
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  done
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text {*
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  \medskip Completeness theorem for the positive reals (again).
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*}
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lemma posreals_complete:
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  assumes positive_S: "\<forall>x \<in> S. 0 < x"
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    and not_empty_S: "\<exists>x. x \<in> S"
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    and upper_bound_Ex: "\<exists>u. isUb (UNIV::real set) S u"
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  shows "\<exists>t. isLub (UNIV::real set) S t"
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proof
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  let ?pS = "{w. real_of_preal w \<in> S}"
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  obtain u where "isUb UNIV S u" using upper_bound_Ex ..
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  hence sup: "\<forall>x \<in> S. x \<le> u" by (simp add: isUb_def setle_def)
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   143
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   144
  obtain x where x_in_S: "x \<in> S" using not_empty_S ..
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   145
  hence x_gt_zero: "0 < x" using positive_S by simp
0cc94e6f6ae5 some structured proofs on completeness;
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   146
  have  "x \<le> u" using sup and x_in_S ..
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   147
  hence "0 < u" using x_gt_zero by arith
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  then obtain pu where u_is_pu: "u = real_of_preal pu"
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   150
    by (auto simp add: real_gt_zero_preal_Ex)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   151
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   152
  have pS_less_pu: "\<forall>pa \<in> ?pS. pa \<le> pu"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   153
  proof
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   154
    fix pa
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   155
    assume "pa \<in> ?pS"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   156
    then obtain a where "a \<in> S" and "a = real_of_preal pa"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   157
      by simp
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   158
    moreover hence "a \<le> u" using sup by simp
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   159
    ultimately show "pa \<le> pu"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   160
      using sup and u_is_pu by (simp add: real_of_preal_le_iff)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   161
  qed
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 9429
diff changeset
   162
16893
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   163
  have "\<forall>y \<in> S. y \<le> real_of_preal (psup ?pS)"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   164
  proof
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   165
    fix y
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   166
    assume y_in_S: "y \<in> S"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   167
    hence "0 < y" using positive_S by simp
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   168
    then obtain py where y_is_py: "y = real_of_preal py"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   169
      by (auto simp add: real_gt_zero_preal_Ex)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   170
    hence py_in_pS: "py \<in> ?pS" using y_in_S by simp
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   171
    with pS_less_pu have "py \<le> psup ?pS"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   172
      by (rule preal_psup_le)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   173
    thus "y \<le> real_of_preal (psup ?pS)"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   174
      using y_is_py by (simp add: real_of_preal_le_iff)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   175
  qed
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   176
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   177
  moreover {
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   178
    fix x
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   179
    assume x_ub_S: "\<forall>y\<in>S. y \<le> x"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   180
    have "real_of_preal (psup ?pS) \<le> x"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   181
    proof -
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   182
      obtain "s" where s_in_S: "s \<in> S" using not_empty_S ..
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   183
      hence s_pos: "0 < s" using positive_S by simp
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   184
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   185
      hence "\<exists> ps. s = real_of_preal ps" by (simp add: real_gt_zero_preal_Ex)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   186
      then obtain "ps" where s_is_ps: "s = real_of_preal ps" ..
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   187
      hence ps_in_pS: "ps \<in> {w. real_of_preal w \<in> S}" using s_in_S by simp
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   188
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   189
      from x_ub_S have "s \<le> x" using s_in_S ..
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   190
      hence "0 < x" using s_pos by simp
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   191
      hence "\<exists> px. x = real_of_preal px" by (simp add: real_gt_zero_preal_Ex)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   192
      then obtain "px" where x_is_px: "x = real_of_preal px" ..
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   193
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   194
      have "\<forall>pe \<in> ?pS. pe \<le> px"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   195
      proof
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   196
	fix pe
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   197
	assume "pe \<in> ?pS"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   198
	hence "real_of_preal pe \<in> S" by simp
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   199
	hence "real_of_preal pe \<le> x" using x_ub_S by simp
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   200
	thus "pe \<le> px" using x_is_px by (simp add: real_of_preal_le_iff)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   201
      qed
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   202
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   203
      moreover have "?pS \<noteq> {}" using ps_in_pS by auto
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   204
      ultimately have "(psup ?pS) \<le> px" by (simp add: psup_le_ub)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   205
      thus "real_of_preal (psup ?pS) \<le> x" using x_is_px by (simp add: real_of_preal_le_iff)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   206
    qed
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   207
  }
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   208
  ultimately show "isLub UNIV S (real_of_preal (psup ?pS))"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   209
    by (simp add: isLub_def leastP_def isUb_def setle_def setge_def)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   210
qed
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   211
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   212
text {*
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   213
  \medskip reals Completeness (again!)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   214
*}
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 9429
diff changeset
   215
16893
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   216
lemma reals_complete:
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   217
  assumes notempty_S: "\<exists>X. X \<in> S"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   218
    and exists_Ub: "\<exists>Y. isUb (UNIV::real set) S Y"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   219
  shows "\<exists>t. isLub (UNIV :: real set) S t"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   220
proof -
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   221
  obtain X where X_in_S: "X \<in> S" using notempty_S ..
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   222
  obtain Y where Y_isUb: "isUb (UNIV::real set) S Y"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   223
    using exists_Ub ..
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   224
  let ?SHIFT = "{z. \<exists>x \<in>S. z = x + (-X) + 1} \<inter> {x. 0 < x}"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   225
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   226
  {
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   227
    fix x
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   228
    assume "isUb (UNIV::real set) S x"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   229
    hence S_le_x: "\<forall> y \<in> S. y <= x"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   230
      by (simp add: isUb_def setle_def)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   231
    {
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   232
      fix s
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   233
      assume "s \<in> {z. \<exists>x\<in>S. z = x + - X + 1}"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   234
      hence "\<exists> x \<in> S. s = x + -X + 1" ..
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   235
      then obtain x1 where "x1 \<in> S" and "s = x1 + (-X) + 1" ..
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   236
      moreover hence "x1 \<le> x" using S_le_x by simp
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   237
      ultimately have "s \<le> x + - X + 1" by arith
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   238
    }
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   239
    then have "isUb (UNIV::real set) ?SHIFT (x + (-X) + 1)"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   240
      by (auto simp add: isUb_def setle_def)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   241
  } note S_Ub_is_SHIFT_Ub = this
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   242
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   243
  hence "isUb UNIV ?SHIFT (Y + (-X) + 1)" using Y_isUb by simp
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   244
  hence "\<exists>Z. isUb UNIV ?SHIFT Z" ..
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   245
  moreover have "\<forall>y \<in> ?SHIFT. 0 < y" by auto
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   246
  moreover have shifted_not_empty: "\<exists>u. u \<in> ?SHIFT"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   247
    using X_in_S and Y_isUb by auto
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   248
  ultimately obtain t where t_is_Lub: "isLub UNIV ?SHIFT t"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   249
    using posreals_complete [of ?SHIFT] by blast
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   250
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   251
  show ?thesis
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   252
  proof
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   253
    show "isLub UNIV S (t + X + (-1))"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   254
    proof (rule isLubI2)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   255
      {
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   256
        fix x
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   257
        assume "isUb (UNIV::real set) S x"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   258
        hence "isUb (UNIV::real set) (?SHIFT) (x + (-X) + 1)"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   259
	  using S_Ub_is_SHIFT_Ub by simp
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   260
        hence "t \<le> (x + (-X) + 1)"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   261
	  using t_is_Lub by (simp add: isLub_le_isUb)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   262
        hence "t + X + -1 \<le> x" by arith
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   263
      }
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   264
      then show "(t + X + -1) <=* Collect (isUb UNIV S)"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   265
	by (simp add: setgeI)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   266
    next
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   267
      show "isUb UNIV S (t + X + -1)"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   268
      proof -
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   269
        {
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   270
          fix y
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   271
          assume y_in_S: "y \<in> S"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   272
          have "y \<le> t + X + -1"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   273
          proof -
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   274
            obtain "u" where u_in_shift: "u \<in> ?SHIFT" using shifted_not_empty ..
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   275
            hence "\<exists> x \<in> S. u = x + - X + 1" by simp
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   276
            then obtain "x" where x_and_u: "u = x + - X + 1" ..
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   277
            have u_le_t: "u \<le> t" using u_in_shift and t_is_Lub by (simp add: isLubD2)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   278
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   279
            show ?thesis
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   280
            proof cases
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   281
              assume "y \<le> x"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   282
              moreover have "x = u + X + - 1" using x_and_u by arith
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   283
              moreover have "u + X + - 1  \<le> t + X + -1" using u_le_t by arith
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   284
              ultimately show "y  \<le> t + X + -1" by arith
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   285
            next
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   286
              assume "~(y \<le> x)"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   287
              hence x_less_y: "x < y" by arith
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   288
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   289
              have "x + (-X) + 1 \<in> ?SHIFT" using x_and_u and u_in_shift by simp
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   290
              hence "0 < x + (-X) + 1" by simp
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   291
              hence "0 < y + (-X) + 1" using x_less_y by arith
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   292
              hence "y + (-X) + 1 \<in> ?SHIFT" using y_in_S by simp
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   293
              hence "y + (-X) + 1 \<le> t" using t_is_Lub  by (simp add: isLubD2)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   294
              thus ?thesis by simp
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   295
            qed
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   296
          qed
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   297
        }
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   298
        then show ?thesis by (simp add: isUb_def setle_def)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   299
      qed
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   300
    qed
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   301
  qed
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   302
qed
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 9429
diff changeset
   303
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 9429
diff changeset
   304
16893
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   305
subsection {* The Archimedean Property of the Reals *}
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   306
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   307
theorem reals_Archimedean:
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   308
  assumes x_pos: "0 < x"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   309
  shows "\<exists>n. inverse (real (Suc n)) < x"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   310
proof (rule ccontr)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   311
  assume contr: "\<not> ?thesis"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   312
  have "\<forall>n. x * real (Suc n) <= 1"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   313
  proof
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   314
    fix n
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   315
    from contr have "x \<le> inverse (real (Suc n))"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   316
      by (simp add: linorder_not_less)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   317
    hence "x \<le> (1 / (real (Suc n)))"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   318
      by (simp add: inverse_eq_divide)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   319
    moreover have "0 \<le> real (Suc n)"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   320
      by (rule real_of_nat_ge_zero)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   321
    ultimately have "x * real (Suc n) \<le> (1 / real (Suc n)) * real (Suc n)"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   322
      by (rule mult_right_mono)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   323
    thus "x * real (Suc n) \<le> 1" by simp
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   324
  qed
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   325
  hence "{z. \<exists>n. z = x * (real (Suc n))} *<= 1"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   326
    by (simp add: setle_def, safe, rule spec)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   327
  hence "isUb (UNIV::real set) {z. \<exists>n. z = x * (real (Suc n))} 1"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   328
    by (simp add: isUbI)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   329
  hence "\<exists>Y. isUb (UNIV::real set) {z. \<exists>n. z = x* (real (Suc n))} Y" ..
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   330
  moreover have "\<exists>X. X \<in> {z. \<exists>n. z = x* (real (Suc n))}" by auto
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   331
  ultimately have "\<exists>t. isLub UNIV {z. \<exists>n. z = x * real (Suc n)} t"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   332
    by (simp add: reals_complete)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   333
  then obtain "t" where
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   334
    t_is_Lub: "isLub UNIV {z. \<exists>n. z = x * real (Suc n)} t" ..
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   335
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   336
  have "\<forall>n::nat. x * real n \<le> t + - x"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   337
  proof
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   338
    fix n
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   339
    from t_is_Lub have "x * real (Suc n) \<le> t"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   340
      by (simp add: isLubD2)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   341
    hence  "x * (real n) + x \<le> t"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   342
      by (simp add: right_distrib real_of_nat_Suc)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   343
    thus  "x * (real n) \<le> t + - x" by arith
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   344
  qed
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 9429
diff changeset
   345
16893
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   346
  hence "\<forall>m. x * real (Suc m) \<le> t + - x" by simp
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   347
  hence "{z. \<exists>n. z = x * (real (Suc n))}  *<= (t + - x)"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   348
    by (auto simp add: setle_def)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   349
  hence "isUb (UNIV::real set) {z. \<exists>n. z = x * (real (Suc n))} (t + (-x))"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   350
    by (simp add: isUbI)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   351
  hence "t \<le> t + - x"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   352
    using t_is_Lub by (simp add: isLub_le_isUb)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   353
  thus False using x_pos by arith
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   354
qed
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   355
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   356
text {*
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   357
  There must be other proofs, e.g. @{text "Suc"} of the largest
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   358
  integer in the cut representing @{text "x"}.
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   359
*}
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 9429
diff changeset
   360
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 9429
diff changeset
   361
lemma reals_Archimedean2: "\<exists>n. (x::real) < real (n::nat)"
16893
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   362
proof cases
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   363
  assume "x \<le> 0"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   364
  hence "x < real (1::nat)" by simp
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   365
  thus ?thesis ..
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   366
next
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   367
  assume "\<not> x \<le> 0"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   368
  hence x_greater_zero: "0 < x" by simp
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   369
  hence "0 < inverse x" by simp
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   370
  then obtain n where "inverse (real (Suc n)) < inverse x"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   371
    using reals_Archimedean by blast
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   372
  hence "inverse (real (Suc n)) * x < inverse x * x"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   373
    using x_greater_zero by (rule mult_strict_right_mono)
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   374
  hence "inverse (real (Suc n)) * x < 1"
23008
c4a259f3bbcc avoid using real_mult_inverse_left; cleaned up
huffman
parents: 22998
diff changeset
   375
    using x_greater_zero by simp
16893
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   376
  hence "real (Suc n) * (inverse (real (Suc n)) * x) < real (Suc n) * 1"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   377
    by (rule mult_strict_left_mono) simp
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   378
  hence "x < real (Suc n)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 28952
diff changeset
   379
    by (simp add: algebra_simps)
16893
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   380
  thus "\<exists>(n::nat). x < real n" ..
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   381
qed
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 9429
diff changeset
   382
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   383
instance real :: archimedean_field
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   384
proof
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   385
  fix r :: real
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   386
  obtain n :: nat where "r < real n"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   387
    using reals_Archimedean2 ..
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   388
  then have "r \<le> of_int (int n)"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   389
    unfolding real_eq_of_nat by simp
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   390
  then show "\<exists>z. r \<le> of_int z" ..
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   391
qed
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   392
16893
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   393
lemma reals_Archimedean3:
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   394
  assumes x_greater_zero: "0 < x"
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   395
  shows "\<forall>(y::real). \<exists>(n::nat). y < real n * x"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   396
  unfolding real_of_nat_def using `0 < x`
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   397
  by (auto intro: ex_less_of_nat_mult)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 9429
diff changeset
   398
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   399
lemma reals_Archimedean6:
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   400
     "0 \<le> r ==> \<exists>(n::nat). real (n - 1) \<le> r & r < real (n)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   401
unfolding real_of_nat_def
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   402
apply (rule exI [where x="nat (floor r + 1)"])
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   403
apply (insert floor_correct [of r])
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   404
apply (simp add: nat_add_distrib of_nat_nat)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   405
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   406
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   407
lemma reals_Archimedean6a: "0 \<le> r ==> \<exists>n. real (n) \<le> r & r < real (Suc n)"
16893
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   408
  by (drule reals_Archimedean6) auto
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   409
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   410
lemma reals_Archimedean_6b_int:
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   411
     "0 \<le> r ==> \<exists>n::int. real n \<le> r & r < real (n+1)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   412
  unfolding real_of_int_def by (rule floor_exists)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   413
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   414
lemma reals_Archimedean_6c_int:
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   415
     "r < 0 ==> \<exists>n::int. real n \<le> r & r < real (n+1)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   416
  unfolding real_of_int_def by (rule floor_exists)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   417
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   418
28091
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   419
subsection{*Density of the Rational Reals in the Reals*}
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   420
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   421
text{* This density proof is due to Stefan Richter and was ported by TN.  The
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   422
original source is \emph{Real Analysis} by H.L. Royden.
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   423
It employs the Archimedean property of the reals. *}
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   424
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   425
lemma Rats_dense_in_nn_real: fixes x::real
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   426
assumes "0\<le>x" and "x<y" shows "\<exists>r \<in> \<rat>.  x<r \<and> r<y"
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   427
proof -
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   428
  from `x<y` have "0 < y-x" by simp
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   429
  with reals_Archimedean obtain q::nat 
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   430
    where q: "inverse (real q) < y-x" and "0 < real q" by auto  
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   431
  def p \<equiv> "LEAST n.  y \<le> real (Suc n)/real q"  
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   432
  from reals_Archimedean2 obtain n::nat where "y * real q < real n" by auto
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   433
  with `0 < real q` have ex: "y \<le> real n/real q" (is "?P n")
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   434
    by (simp add: pos_less_divide_eq[THEN sym])
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   435
  also from assms have "\<not> y \<le> real (0::nat) / real q" by simp
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   436
  ultimately have main: "(LEAST n. y \<le> real n/real q) = Suc p"
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   437
    by (unfold p_def) (rule Least_Suc)
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   438
  also from ex have "?P (LEAST x. ?P x)" by (rule LeastI)
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   439
  ultimately have suc: "y \<le> real (Suc p) / real q" by simp
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   440
  def r \<equiv> "real p/real q"
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   441
  have "x = y-(y-x)" by simp
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   442
  also from suc q have "\<dots> < real (Suc p)/real q - inverse (real q)" by arith
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   443
  also have "\<dots> = real p / real q"
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   444
    by (simp only: inverse_eq_divide real_diff_def real_of_nat_Suc 
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   445
    minus_divide_left add_divide_distrib[THEN sym]) simp
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   446
  finally have "x<r" by (unfold r_def)
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   447
  have "p<Suc p" .. also note main[THEN sym]
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   448
  finally have "\<not> ?P p"  by (rule not_less_Least)
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   449
  hence "r<y" by (simp add: r_def)
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   450
  from r_def have "r \<in> \<rat>" by simp
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   451
  with `x<r` `r<y` show ?thesis by fast
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   452
qed
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   453
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   454
theorem Rats_dense_in_real: fixes x y :: real
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   455
assumes "x<y" shows "\<exists>r \<in> \<rat>.  x<r \<and> r<y"
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   456
proof -
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   457
  from reals_Archimedean2 obtain n::nat where "-x < real n" by auto
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   458
  hence "0 \<le> x + real n" by arith
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   459
  also from `x<y` have "x + real n < y + real n" by arith
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   460
  ultimately have "\<exists>r \<in> \<rat>. x + real n < r \<and> r < y + real n"
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   461
    by(rule Rats_dense_in_nn_real)
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   462
  then obtain r where "r \<in> \<rat>" and r2: "x + real n < r" 
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   463
    and r3: "r < y + real n"
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   464
    by blast
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   465
  have "r - real n = r + real (int n)/real (-1::int)" by simp
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   466
  also from `r\<in>\<rat>` have "r + real (int n)/real (-1::int) \<in> \<rat>" by simp
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   467
  also from r2 have "x < r - real n" by arith
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   468
  moreover from r3 have "r - real n < y" by arith
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   469
  ultimately show ?thesis by fast
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   470
qed
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   471
50f2d6ba024c Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents: 27435
diff changeset
   472
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   473
subsection{*Floor and Ceiling Functions from the Reals to the Integers*}
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   474
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   475
lemma number_of_less_real_of_int_iff [simp]:
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   476
     "((number_of n) < real (m::int)) = (number_of n < m)"
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   477
apply auto
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   478
apply (rule real_of_int_less_iff [THEN iffD1])
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   479
apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   480
done
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   481
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   482
lemma number_of_less_real_of_int_iff2 [simp]:
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   483
     "(real (m::int) < (number_of n)) = (m < number_of n)"
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   484
apply auto
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   485
apply (rule real_of_int_less_iff [THEN iffD1])
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   486
apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   487
done
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   488
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   489
lemma number_of_le_real_of_int_iff [simp]:
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   490
     "((number_of n) \<le> real (m::int)) = (number_of n \<le> m)"
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   491
by (simp add: linorder_not_less [symmetric])
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   492
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   493
lemma number_of_le_real_of_int_iff2 [simp]:
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   494
     "(real (m::int) \<le> (number_of n)) = (m \<le> number_of n)"
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   495
by (simp add: linorder_not_less [symmetric])
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   496
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   497
lemma floor_real_of_nat_zero: "floor (real (0::nat)) = 0"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   498
by auto (* delete? *)
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   499
24355
93d78fdeb55a remove int_of_nat
huffman
parents: 23477
diff changeset
   500
lemma floor_real_of_nat [simp]: "floor (real (n::nat)) = int n"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   501
unfolding real_of_nat_def by simp
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   502
24355
93d78fdeb55a remove int_of_nat
huffman
parents: 23477
diff changeset
   503
lemma floor_minus_real_of_nat [simp]: "floor (- real (n::nat)) = - int n"
30102
799b687e4aac disable floor_minus and ceiling_minus [simp]
huffman
parents: 30097
diff changeset
   504
unfolding real_of_nat_def by (simp add: floor_minus)
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   505
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   506
lemma floor_real_of_int [simp]: "floor (real (n::int)) = n"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   507
unfolding real_of_int_def by simp
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   508
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   509
lemma floor_minus_real_of_int [simp]: "floor (- real (n::int)) = - n"
30102
799b687e4aac disable floor_minus and ceiling_minus [simp]
huffman
parents: 30097
diff changeset
   510
unfolding real_of_int_def by (simp add: floor_minus)
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   511
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   512
lemma real_lb_ub_int: " \<exists>n::int. real n \<le> r & r < real (n+1)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   513
unfolding real_of_int_def by (rule floor_exists)
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   514
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   515
lemma lemma_floor:
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   516
  assumes a1: "real m \<le> r" and a2: "r < real n + 1"
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   517
  shows "m \<le> (n::int)"
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   518
proof -
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23309
diff changeset
   519
  have "real m < real n + 1" using a1 a2 by (rule order_le_less_trans)
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23309
diff changeset
   520
  also have "... = real (n + 1)" by simp
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23309
diff changeset
   521
  finally have "m < n + 1" by (simp only: real_of_int_less_iff)
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   522
  thus ?thesis by arith
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   523
qed
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   524
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   525
lemma real_of_int_floor_le [simp]: "real (floor r) \<le> r"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   526
unfolding real_of_int_def by (rule of_int_floor_le)
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   527
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   528
lemma lemma_floor2: "real n < real (x::int) + 1 ==> n \<le> x"
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   529
by (auto intro: lemma_floor)
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   530
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   531
lemma real_of_int_floor_cancel [simp]:
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   532
    "(real (floor x) = x) = (\<exists>n::int. x = real n)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   533
  using floor_real_of_int by metis
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   534
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   535
lemma floor_eq: "[| real n < x; x < real n + 1 |] ==> floor x = n"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   536
  unfolding real_of_int_def using floor_unique [of n x] by simp
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   537
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   538
lemma floor_eq2: "[| real n \<le> x; x < real n + 1 |] ==> floor x = n"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   539
  unfolding real_of_int_def by (rule floor_unique)
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   540
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   541
lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat(floor x) = n"
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   542
apply (rule inj_int [THEN injD])
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   543
apply (simp add: real_of_nat_Suc)
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15234
diff changeset
   544
apply (simp add: real_of_nat_Suc floor_eq floor_eq [where n = "int n"])
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   545
done
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   546
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   547
lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat(floor x) = n"
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   548
apply (drule order_le_imp_less_or_eq)
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   549
apply (auto intro: floor_eq3)
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   550
done
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   551
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   552
lemma floor_number_of_eq:
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   553
     "floor(number_of n :: real) = (number_of n :: int)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   554
  by (rule floor_number_of) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   555
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   556
lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real(floor r)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   557
  unfolding real_of_int_def using floor_correct [of r] by simp
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   558
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   559
lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real(floor r)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   560
  unfolding real_of_int_def using floor_correct [of r] by simp
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   561
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   562
lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real(floor r) + 1"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   563
  unfolding real_of_int_def using floor_correct [of r] by simp
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   564
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   565
lemma real_of_int_floor_add_one_gt [simp]: "r < real(floor r) + 1"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   566
  unfolding real_of_int_def using floor_correct [of r] by simp
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   567
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   568
lemma le_floor: "real a <= x ==> a <= floor x"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   569
  unfolding real_of_int_def by (simp add: le_floor_iff)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   570
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   571
lemma real_le_floor: "a <= floor x ==> real a <= x"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   572
  unfolding real_of_int_def by (simp add: le_floor_iff)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   573
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   574
lemma le_floor_eq: "(a <= floor x) = (real a <= x)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   575
  unfolding real_of_int_def by (rule le_floor_iff)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   576
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   577
lemma le_floor_eq_number_of:
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   578
    "(number_of n <= floor x) = (number_of n <= x)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   579
  by (rule number_of_le_floor) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   580
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   581
lemma le_floor_eq_zero: "(0 <= floor x) = (0 <= x)"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   582
  by (rule zero_le_floor) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   583
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   584
lemma le_floor_eq_one: "(1 <= floor x) = (1 <= x)"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   585
  by (rule one_le_floor) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   586
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   587
lemma floor_less_eq: "(floor x < a) = (x < real a)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   588
  unfolding real_of_int_def by (rule floor_less_iff)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   589
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   590
lemma floor_less_eq_number_of:
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   591
    "(floor x < number_of n) = (x < number_of n)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   592
  by (rule floor_less_number_of) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   593
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   594
lemma floor_less_eq_zero: "(floor x < 0) = (x < 0)"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   595
  by (rule floor_less_zero) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   596
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   597
lemma floor_less_eq_one: "(floor x < 1) = (x < 1)"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   598
  by (rule floor_less_one) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   599
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   600
lemma less_floor_eq: "(a < floor x) = (real a + 1 <= x)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   601
  unfolding real_of_int_def by (rule less_floor_iff)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   602
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   603
lemma less_floor_eq_number_of:
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   604
    "(number_of n < floor x) = (number_of n + 1 <= x)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   605
  by (rule number_of_less_floor) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   606
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   607
lemma less_floor_eq_zero: "(0 < floor x) = (1 <= x)"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   608
  by (rule zero_less_floor) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   609
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   610
lemma less_floor_eq_one: "(1 < floor x) = (2 <= x)"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   611
  by (rule one_less_floor) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   612
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   613
lemma floor_le_eq: "(floor x <= a) = (x < real a + 1)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   614
  unfolding real_of_int_def by (rule floor_le_iff)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   615
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   616
lemma floor_le_eq_number_of:
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   617
    "(floor x <= number_of n) = (x < number_of n + 1)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   618
  by (rule floor_le_number_of) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   619
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   620
lemma floor_le_eq_zero: "(floor x <= 0) = (x < 1)"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   621
  by (rule floor_le_zero) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   622
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   623
lemma floor_le_eq_one: "(floor x <= 1) = (x < 2)"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   624
  by (rule floor_le_one) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   625
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   626
lemma floor_add [simp]: "floor (x + real a) = floor x + a"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   627
  unfolding real_of_int_def by (rule floor_add_of_int)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   628
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   629
lemma floor_subtract [simp]: "floor (x - real a) = floor x - a"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   630
  unfolding real_of_int_def by (rule floor_diff_of_int)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   631
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   632
lemma floor_subtract_number_of: "floor (x - number_of n) =
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   633
    floor x - number_of n"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   634
  by (rule floor_diff_number_of) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   635
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   636
lemma floor_subtract_one: "floor (x - 1) = floor x - 1"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   637
  by (rule floor_diff_one) (* already declared [simp] *)
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   638
24355
93d78fdeb55a remove int_of_nat
huffman
parents: 23477
diff changeset
   639
lemma ceiling_real_of_nat [simp]: "ceiling (real (n::nat)) = int n"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   640
  unfolding real_of_nat_def by simp
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   641
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   642
lemma ceiling_real_of_nat_zero: "ceiling (real (0::nat)) = 0"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   643
by auto (* delete? *)
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   644
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   645
lemma ceiling_floor [simp]: "ceiling (real (floor r)) = floor r"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   646
  unfolding real_of_int_def by simp
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   647
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   648
lemma floor_ceiling [simp]: "floor (real (ceiling r)) = ceiling r"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   649
  unfolding real_of_int_def by simp
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   650
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   651
lemma real_of_int_ceiling_ge [simp]: "r \<le> real (ceiling r)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   652
  unfolding real_of_int_def by (rule le_of_int_ceiling)
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   653
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   654
lemma ceiling_real_of_int [simp]: "ceiling (real (n::int)) = n"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   655
  unfolding real_of_int_def by simp
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   656
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   657
lemma real_of_int_ceiling_cancel [simp]:
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   658
     "(real (ceiling x) = x) = (\<exists>n::int. x = real n)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   659
  using ceiling_real_of_int by metis
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   660
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   661
lemma ceiling_eq: "[| real n < x; x < real n + 1 |] ==> ceiling x = n + 1"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   662
  unfolding real_of_int_def using ceiling_unique [of "n + 1" x] by simp
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   663
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   664
lemma ceiling_eq2: "[| real n < x; x \<le> real n + 1 |] ==> ceiling x = n + 1"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   665
  unfolding real_of_int_def using ceiling_unique [of "n + 1" x] by simp
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   666
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   667
lemma ceiling_eq3: "[| real n - 1 < x; x \<le> real n  |] ==> ceiling x = n"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   668
  unfolding real_of_int_def using ceiling_unique [of n x] by simp
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   669
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   670
lemma ceiling_number_of_eq:
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   671
     "ceiling (number_of n :: real) = (number_of n)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   672
  by (rule ceiling_number_of) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   673
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   674
lemma real_of_int_ceiling_diff_one_le [simp]: "real (ceiling r) - 1 \<le> r"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   675
  unfolding real_of_int_def using ceiling_correct [of r] by simp
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   676
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   677
lemma real_of_int_ceiling_le_add_one [simp]: "real (ceiling r) \<le> r + 1"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   678
  unfolding real_of_int_def using ceiling_correct [of r] by simp
14641
79b7bd936264 moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
paulson
parents: 14476
diff changeset
   679
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   680
lemma ceiling_le: "x <= real a ==> ceiling x <= a"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   681
  unfolding real_of_int_def by (simp add: ceiling_le_iff)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   682
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   683
lemma ceiling_le_real: "ceiling x <= a ==> x <= real a"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   684
  unfolding real_of_int_def by (simp add: ceiling_le_iff)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   685
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   686
lemma ceiling_le_eq: "(ceiling x <= a) = (x <= real a)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   687
  unfolding real_of_int_def by (rule ceiling_le_iff)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   688
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   689
lemma ceiling_le_eq_number_of:
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   690
    "(ceiling x <= number_of n) = (x <= number_of n)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   691
  by (rule ceiling_le_number_of) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   692
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   693
lemma ceiling_le_zero_eq: "(ceiling x <= 0) = (x <= 0)"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   694
  by (rule ceiling_le_zero) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   695
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   696
lemma ceiling_le_eq_one: "(ceiling x <= 1) = (x <= 1)"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   697
  by (rule ceiling_le_one) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   698
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   699
lemma less_ceiling_eq: "(a < ceiling x) = (real a < x)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   700
  unfolding real_of_int_def by (rule less_ceiling_iff)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   701
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   702
lemma less_ceiling_eq_number_of:
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   703
    "(number_of n < ceiling x) = (number_of n < x)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   704
  by (rule number_of_less_ceiling) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   705
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   706
lemma less_ceiling_eq_zero: "(0 < ceiling x) = (0 < x)"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   707
  by (rule zero_less_ceiling) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   708
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   709
lemma less_ceiling_eq_one: "(1 < ceiling x) = (1 < x)"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   710
  by (rule one_less_ceiling) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   711
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   712
lemma ceiling_less_eq: "(ceiling x < a) = (x <= real a - 1)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   713
  unfolding real_of_int_def by (rule ceiling_less_iff)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   714
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   715
lemma ceiling_less_eq_number_of:
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   716
    "(ceiling x < number_of n) = (x <= number_of n - 1)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   717
  by (rule ceiling_less_number_of) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   718
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   719
lemma ceiling_less_eq_zero: "(ceiling x < 0) = (x <= -1)"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   720
  by (rule ceiling_less_zero) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   721
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   722
lemma ceiling_less_eq_one: "(ceiling x < 1) = (x <= 0)"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   723
  by (rule ceiling_less_one) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   724
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   725
lemma le_ceiling_eq: "(a <= ceiling x) = (real a - 1 < x)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   726
  unfolding real_of_int_def by (rule le_ceiling_iff)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   727
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   728
lemma le_ceiling_eq_number_of:
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   729
    "(number_of n <= ceiling x) = (number_of n - 1 < x)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   730
  by (rule number_of_le_ceiling) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   731
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   732
lemma le_ceiling_eq_zero: "(0 <= ceiling x) = (-1 < x)"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   733
  by (rule zero_le_ceiling) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   734
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   735
lemma le_ceiling_eq_one: "(1 <= ceiling x) = (0 < x)"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   736
  by (rule one_le_ceiling) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   737
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   738
lemma ceiling_add [simp]: "ceiling (x + real a) = ceiling x + a"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   739
  unfolding real_of_int_def by (rule ceiling_add_of_int)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   740
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   741
lemma ceiling_subtract [simp]: "ceiling (x - real a) = ceiling x - a"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   742
  unfolding real_of_int_def by (rule ceiling_diff_of_int)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   743
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   744
lemma ceiling_subtract_number_of: "ceiling (x - number_of n) =
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   745
    ceiling x - number_of n"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   746
  by (rule ceiling_diff_number_of) (* already declared [simp] *)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   747
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   748
lemma ceiling_subtract_one: "ceiling (x - 1) = ceiling x - 1"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   749
  by (rule ceiling_diff_one) (* already declared [simp] *)
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   750
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   751
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   752
subsection {* Versions for the natural numbers *}
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   753
19765
dfe940911617 misc cleanup;
wenzelm
parents: 16893
diff changeset
   754
definition
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21210
diff changeset
   755
  natfloor :: "real => nat" where
19765
dfe940911617 misc cleanup;
wenzelm
parents: 16893
diff changeset
   756
  "natfloor x = nat(floor x)"
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21210
diff changeset
   757
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21210
diff changeset
   758
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21210
diff changeset
   759
  natceiling :: "real => nat" where
19765
dfe940911617 misc cleanup;
wenzelm
parents: 16893
diff changeset
   760
  "natceiling x = nat(ceiling x)"
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   761
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   762
lemma natfloor_zero [simp]: "natfloor 0 = 0"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   763
  by (unfold natfloor_def, simp)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   764
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   765
lemma natfloor_one [simp]: "natfloor 1 = 1"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   766
  by (unfold natfloor_def, simp)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   767
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   768
lemma zero_le_natfloor [simp]: "0 <= natfloor x"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   769
  by (unfold natfloor_def, simp)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   770
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   771
lemma natfloor_number_of_eq [simp]: "natfloor (number_of n) = number_of n"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   772
  by (unfold natfloor_def, simp)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   773
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   774
lemma natfloor_real_of_nat [simp]: "natfloor(real n) = n"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   775
  by (unfold natfloor_def, simp)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   776
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   777
lemma real_natfloor_le: "0 <= x ==> real(natfloor x) <= x"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   778
  by (unfold natfloor_def, simp)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   779
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   780
lemma natfloor_neg: "x <= 0 ==> natfloor x = 0"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   781
  apply (unfold natfloor_def)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   782
  apply (subgoal_tac "floor x <= floor 0")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   783
  apply simp
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   784
  apply (erule floor_mono)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   785
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   786
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   787
lemma natfloor_mono: "x <= y ==> natfloor x <= natfloor y"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   788
  apply (case_tac "0 <= x")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   789
  apply (subst natfloor_def)+
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   790
  apply (subst nat_le_eq_zle)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   791
  apply force
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   792
  apply (erule floor_mono)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   793
  apply (subst natfloor_neg)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   794
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   795
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   796
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   797
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   798
lemma le_natfloor: "real x <= a ==> x <= natfloor a"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   799
  apply (unfold natfloor_def)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   800
  apply (subst nat_int [THEN sym])
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   801
  apply (subst nat_le_eq_zle)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   802
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   803
  apply (rule le_floor)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   804
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   805
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   806
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   807
lemma le_natfloor_eq: "0 <= x ==> (a <= natfloor x) = (real a <= x)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   808
  apply (rule iffI)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   809
  apply (rule order_trans)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   810
  prefer 2
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   811
  apply (erule real_natfloor_le)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   812
  apply (subst real_of_nat_le_iff)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   813
  apply assumption
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   814
  apply (erule le_natfloor)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   815
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   816
16893
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   817
lemma le_natfloor_eq_number_of [simp]:
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   818
    "~ neg((number_of n)::int) ==> 0 <= x ==>
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   819
      (number_of n <= natfloor x) = (number_of n <= x)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   820
  apply (subst le_natfloor_eq, assumption)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   821
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   822
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   823
16820
5c9d597e4cb9 fixed typos in theorem names
avigad
parents: 16819
diff changeset
   824
lemma le_natfloor_eq_one [simp]: "(1 <= natfloor x) = (1 <= x)"
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   825
  apply (case_tac "0 <= x")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   826
  apply (subst le_natfloor_eq, assumption, simp)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   827
  apply (rule iffI)
16893
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   828
  apply (subgoal_tac "natfloor x <= natfloor 0")
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   829
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   830
  apply (rule natfloor_mono)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   831
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   832
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   833
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   834
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   835
lemma natfloor_eq: "real n <= x ==> x < real n + 1 ==> natfloor x = n"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   836
  apply (unfold natfloor_def)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   837
  apply (subst nat_int [THEN sym]);back;
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   838
  apply (subst eq_nat_nat_iff)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   839
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   840
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   841
  apply (rule floor_eq2)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   842
  apply auto
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   843
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   844
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   845
lemma real_natfloor_add_one_gt: "x < real(natfloor x) + 1"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   846
  apply (case_tac "0 <= x")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   847
  apply (unfold natfloor_def)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   848
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   849
  apply simp_all
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   850
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   851
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   852
lemma real_natfloor_gt_diff_one: "x - 1 < real(natfloor x)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 28952
diff changeset
   853
using real_natfloor_add_one_gt by (simp add: algebra_simps)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   854
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   855
lemma ge_natfloor_plus_one_imp_gt: "natfloor z + 1 <= n ==> z < real n"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   856
  apply (subgoal_tac "z < real(natfloor z) + 1")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   857
  apply arith
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   858
  apply (rule real_natfloor_add_one_gt)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   859
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   860
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   861
lemma natfloor_add [simp]: "0 <= x ==> natfloor (x + real a) = natfloor x + a"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   862
  apply (unfold natfloor_def)
24355
93d78fdeb55a remove int_of_nat
huffman
parents: 23477
diff changeset
   863
  apply (subgoal_tac "real a = real (int a)")
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   864
  apply (erule ssubst)
23309
678ee30499d2 remove references to constant int::nat=>int
huffman
parents: 23012
diff changeset
   865
  apply (simp add: nat_add_distrib del: real_of_int_of_nat_eq)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   866
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   867
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   868
16893
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   869
lemma natfloor_add_number_of [simp]:
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   870
    "~neg ((number_of n)::int) ==> 0 <= x ==>
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   871
      natfloor (x + number_of n) = natfloor x + number_of n"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   872
  apply (subst natfloor_add [THEN sym])
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   873
  apply simp_all
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   874
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   875
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   876
lemma natfloor_add_one: "0 <= x ==> natfloor(x + 1) = natfloor x + 1"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   877
  apply (subst natfloor_add [THEN sym])
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   878
  apply assumption
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   879
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   880
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   881
16893
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   882
lemma natfloor_subtract [simp]: "real a <= x ==>
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   883
    natfloor(x - real a) = natfloor x - a"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   884
  apply (unfold natfloor_def)
24355
93d78fdeb55a remove int_of_nat
huffman
parents: 23477
diff changeset
   885
  apply (subgoal_tac "real a = real (int a)")
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   886
  apply (erule ssubst)
23309
678ee30499d2 remove references to constant int::nat=>int
huffman
parents: 23012
diff changeset
   887
  apply (simp del: real_of_int_of_nat_eq)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   888
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   889
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   890
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   891
lemma natceiling_zero [simp]: "natceiling 0 = 0"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   892
  by (unfold natceiling_def, simp)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   893
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   894
lemma natceiling_one [simp]: "natceiling 1 = 1"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   895
  by (unfold natceiling_def, simp)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   896
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   897
lemma zero_le_natceiling [simp]: "0 <= natceiling x"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   898
  by (unfold natceiling_def, simp)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   899
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   900
lemma natceiling_number_of_eq [simp]: "natceiling (number_of n) = number_of n"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   901
  by (unfold natceiling_def, simp)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   902
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   903
lemma natceiling_real_of_nat [simp]: "natceiling(real n) = n"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   904
  by (unfold natceiling_def, simp)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   905
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   906
lemma real_natceiling_ge: "x <= real(natceiling x)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   907
  apply (unfold natceiling_def)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   908
  apply (case_tac "x < 0")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   909
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   910
  apply (subst real_nat_eq_real)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   911
  apply (subgoal_tac "ceiling 0 <= ceiling x")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   912
  apply simp
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   913
  apply (rule ceiling_mono)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   914
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   915
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   916
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   917
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   918
lemma natceiling_neg: "x <= 0 ==> natceiling x = 0"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   919
  apply (unfold natceiling_def)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   920
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   921
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   922
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   923
lemma natceiling_mono: "x <= y ==> natceiling x <= natceiling y"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   924
  apply (case_tac "0 <= x")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   925
  apply (subst natceiling_def)+
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   926
  apply (subst nat_le_eq_zle)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   927
  apply (rule disjI2)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   928
  apply (subgoal_tac "real (0::int) <= real(ceiling y)")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   929
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   930
  apply (rule order_trans)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   931
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   932
  apply (erule order_trans)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   933
  apply simp
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   934
  apply (erule ceiling_mono)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   935
  apply (subst natceiling_neg)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   936
  apply simp_all
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   937
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   938
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   939
lemma natceiling_le: "x <= real a ==> natceiling x <= a"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   940
  apply (unfold natceiling_def)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   941
  apply (case_tac "x < 0")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   942
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   943
  apply (subst nat_int [THEN sym]);back;
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   944
  apply (subst nat_le_eq_zle)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   945
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   946
  apply (rule ceiling_le)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   947
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   948
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   949
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   950
lemma natceiling_le_eq: "0 <= x ==> (natceiling x <= a) = (x <= real a)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   951
  apply (rule iffI)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   952
  apply (rule order_trans)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   953
  apply (rule real_natceiling_ge)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   954
  apply (subst real_of_nat_le_iff)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   955
  apply assumption
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   956
  apply (erule natceiling_le)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   957
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   958
16893
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   959
lemma natceiling_le_eq_number_of [simp]:
16820
5c9d597e4cb9 fixed typos in theorem names
avigad
parents: 16819
diff changeset
   960
    "~ neg((number_of n)::int) ==> 0 <= x ==>
5c9d597e4cb9 fixed typos in theorem names
avigad
parents: 16819
diff changeset
   961
      (natceiling x <= number_of n) = (x <= number_of n)"
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   962
  apply (subst natceiling_le_eq, assumption)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   963
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   964
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   965
16820
5c9d597e4cb9 fixed typos in theorem names
avigad
parents: 16819
diff changeset
   966
lemma natceiling_le_eq_one: "(natceiling x <= 1) = (x <= 1)"
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   967
  apply (case_tac "0 <= x")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   968
  apply (subst natceiling_le_eq)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   969
  apply assumption
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   970
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   971
  apply (subst natceiling_neg)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   972
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   973
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   974
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   975
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   976
lemma natceiling_eq: "real n < x ==> x <= real n + 1 ==> natceiling x = n + 1"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   977
  apply (unfold natceiling_def)
19850
29c125556d86 fixed subst step;
wenzelm
parents: 19765
diff changeset
   978
  apply (simplesubst nat_int [THEN sym]) back back
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   979
  apply (subgoal_tac "nat(int n) + 1 = nat(int n + 1)")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   980
  apply (erule ssubst)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   981
  apply (subst eq_nat_nat_iff)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   982
  apply (subgoal_tac "ceiling 0 <= ceiling x")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   983
  apply simp
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
   984
  apply (rule ceiling_mono)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   985
  apply force
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   986
  apply force
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   987
  apply (rule ceiling_eq2)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   988
  apply (simp, simp)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   989
  apply (subst nat_add_distrib)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   990
  apply auto
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   991
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   992
16893
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
   993
lemma natceiling_add [simp]: "0 <= x ==>
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   994
    natceiling (x + real a) = natceiling x + a"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   995
  apply (unfold natceiling_def)
24355
93d78fdeb55a remove int_of_nat
huffman
parents: 23477
diff changeset
   996
  apply (subgoal_tac "real a = real (int a)")
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   997
  apply (erule ssubst)
23309
678ee30499d2 remove references to constant int::nat=>int
huffman
parents: 23012
diff changeset
   998
  apply (simp del: real_of_int_of_nat_eq)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
   999
  apply (subst nat_add_distrib)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1000
  apply (subgoal_tac "0 = ceiling 0")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1001
  apply (erule ssubst)
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 29667
diff changeset
  1002
  apply (erule ceiling_mono)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1003
  apply simp_all
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1004
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1005
16893
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
  1006
lemma natceiling_add_number_of [simp]:
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
  1007
    "~ neg ((number_of n)::int) ==> 0 <= x ==>
16820
5c9d597e4cb9 fixed typos in theorem names
avigad
parents: 16819
diff changeset
  1008
      natceiling (x + number_of n) = natceiling x + number_of n"
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1009
  apply (subst natceiling_add [THEN sym])
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1010
  apply simp_all
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1011
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1012
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1013
lemma natceiling_add_one: "0 <= x ==> natceiling(x + 1) = natceiling x + 1"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1014
  apply (subst natceiling_add [THEN sym])
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1015
  apply assumption
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1016
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1017
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1018
16893
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
  1019
lemma natceiling_subtract [simp]: "real a <= x ==>
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1020
    natceiling(x - real a) = natceiling x - a"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1021
  apply (unfold natceiling_def)
24355
93d78fdeb55a remove int_of_nat
huffman
parents: 23477
diff changeset
  1022
  apply (subgoal_tac "real a = real (int a)")
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1023
  apply (erule ssubst)
23309
678ee30499d2 remove references to constant int::nat=>int
huffman
parents: 23012
diff changeset
  1024
  apply (simp del: real_of_int_of_nat_eq)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1025
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1026
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1027
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25140
diff changeset
  1028
lemma natfloor_div_nat: "1 <= x ==> y > 0 ==>
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1029
  natfloor (x / real y) = natfloor x div y"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1030
proof -
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25140
diff changeset
  1031
  assume "1 <= (x::real)" and "(y::nat) > 0"
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1032
  have "natfloor x = (natfloor x) div y * y + (natfloor x) mod y"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1033
    by simp
16893
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
  1034
  then have a: "real(natfloor x) = real ((natfloor x) div y) * real y +
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1035
    real((natfloor x) mod y)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1036
    by (simp only: real_of_nat_add [THEN sym] real_of_nat_mult [THEN sym])
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1037
  have "x = real(natfloor x) + (x - real(natfloor x))"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1038
    by simp
16893
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
  1039
  then have "x = real ((natfloor x) div y) * real y +
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1040
      real((natfloor x) mod y) + (x - real(natfloor x))"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1041
    by (simp add: a)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1042
  then have "x / real y = ... / real y"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1043
    by simp
16893
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
  1044
  also have "... = real((natfloor x) div y) + real((natfloor x) mod y) /
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1045
    real y + (x - real(natfloor x)) / real y"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 28952
diff changeset
  1046
    by (auto simp add: algebra_simps add_divide_distrib
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1047
      diff_divide_distrib prems)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1048
  finally have "natfloor (x / real y) = natfloor(...)" by simp
16893
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
  1049
  also have "... = natfloor(real((natfloor x) mod y) /
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1050
    real y + (x - real(natfloor x)) / real y + real((natfloor x) div y))"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1051
    by (simp add: add_ac)
16893
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
  1052
  also have "... = natfloor(real((natfloor x) mod y) /
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1053
    real y + (x - real(natfloor x)) / real y) + (natfloor x) div y"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1054
    apply (rule natfloor_add)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1055
    apply (rule add_nonneg_nonneg)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1056
    apply (rule divide_nonneg_pos)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1057
    apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1058
    apply (simp add: prems)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1059
    apply (rule divide_nonneg_pos)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 28952
diff changeset
  1060
    apply (simp add: algebra_simps)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1061
    apply (rule real_natfloor_le)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1062
    apply (insert prems, auto)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1063
    done
16893
0cc94e6f6ae5 some structured proofs on completeness;
wenzelm
parents: 16827
diff changeset
  1064
  also have "natfloor(real((natfloor x) mod y) /
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1065
    real y + (x - real(natfloor x)) / real y) = 0"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1066
    apply (rule natfloor_eq)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1067
    apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1068
    apply (rule add_nonneg_nonneg)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1069
    apply (rule divide_nonneg_pos)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1070
    apply force
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1071
    apply (force simp add: prems)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1072
    apply (rule divide_nonneg_pos)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 28952
diff changeset
  1073
    apply (simp add: algebra_simps)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1074
    apply (rule real_natfloor_le)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1075
    apply (auto simp add: prems)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1076
    apply (insert prems, arith)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1077
    apply (simp add: add_divide_distrib [THEN sym])
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1078
    apply (subgoal_tac "real y = real y - 1 + 1")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1079
    apply (erule ssubst)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1080
    apply (rule add_le_less_mono)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 28952
diff changeset
  1081
    apply (simp add: algebra_simps)
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 28952
diff changeset
  1082
    apply (subgoal_tac "1 + real(natfloor x mod y) =
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1083
      real(natfloor x mod y + 1)")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1084
    apply (erule ssubst)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1085
    apply (subst real_of_nat_le_iff)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1086
    apply (subgoal_tac "natfloor x mod y < y")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1087
    apply arith
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1088
    apply (rule mod_less_divisor)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1089
    apply auto
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 28952
diff changeset
  1090
    using real_natfloor_add_one_gt
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 28952
diff changeset
  1091
    apply (simp add: algebra_simps)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1092
    done
25140
273772abbea2 More changes from >0 to ~=0::nat
nipkow
parents: 24355
diff changeset
  1093
  finally show ?thesis by simp
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1094
qed
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15539
diff changeset
  1095
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 9429
diff changeset
  1096
end