src/HOL/Real/RComplete.thy
 author wenzelm Fri Jun 02 23:22:29 2006 +0200 (2006-06-02) changeset 19765 dfe940911617 parent 16893 0cc94e6f6ae5 child 19850 29c125556d86 permissions -rw-r--r--
misc cleanup;
```     1 (*  Title       : HOL/Real/RComplete.thy
```
```     2     ID          : \$Id\$
```
```     3     Author      : Jacques D. Fleuriot, University of Edinburgh
```
```     4     Author      : Larry Paulson, University of Cambridge
```
```     5     Author      : Jeremy Avigad, Carnegie Mellon University
```
```     6     Author      : Florian Zuleger, Johannes Hoelzl, and Simon Funke, TU Muenchen
```
```     7 *)
```
```     8
```
```     9 header {* Completeness of the Reals; Floor and Ceiling Functions *}
```
```    10
```
```    11 theory RComplete
```
```    12 imports Lubs RealDef
```
```    13 begin
```
```    14
```
```    15 lemma real_sum_of_halves: "x/2 + x/2 = (x::real)"
```
```    16   by simp
```
```    17
```
```    18
```
```    19 subsection {* Completeness of Positive Reals *}
```
```    20
```
```    21 text {*
```
```    22   Supremum property for the set of positive reals
```
```    23
```
```    24   Let @{text "P"} be a non-empty set of positive reals, with an upper
```
```    25   bound @{text "y"}.  Then @{text "P"} has a least upper bound
```
```    26   (written @{text "S"}).
```
```    27
```
```    28   FIXME: Can the premise be weakened to @{text "\<forall>x \<in> P. x\<le> y"}?
```
```    29 *}
```
```    30
```
```    31 lemma posreal_complete:
```
```    32   assumes positive_P: "\<forall>x \<in> P. (0::real) < x"
```
```    33     and not_empty_P: "\<exists>x. x \<in> P"
```
```    34     and upper_bound_Ex: "\<exists>y. \<forall>x \<in> P. x<y"
```
```    35   shows "\<exists>S. \<forall>y. (\<exists>x \<in> P. y < x) = (y < S)"
```
```    36 proof (rule exI, rule allI)
```
```    37   fix y
```
```    38   let ?pP = "{w. real_of_preal w \<in> P}"
```
```    39
```
```    40   show "(\<exists>x\<in>P. y < x) = (y < real_of_preal (psup ?pP))"
```
```    41   proof (cases "0 < y")
```
```    42     assume neg_y: "\<not> 0 < y"
```
```    43     show ?thesis
```
```    44     proof
```
```    45       assume "\<exists>x\<in>P. y < x"
```
```    46       have "\<forall>x. y < real_of_preal x"
```
```    47         using neg_y by (rule real_less_all_real2)
```
```    48       thus "y < real_of_preal (psup ?pP)" ..
```
```    49     next
```
```    50       assume "y < real_of_preal (psup ?pP)"
```
```    51       obtain "x" where x_in_P: "x \<in> P" using not_empty_P ..
```
```    52       hence "0 < x" using positive_P by simp
```
```    53       hence "y < x" using neg_y by simp
```
```    54       thus "\<exists>x \<in> P. y < x" using x_in_P ..
```
```    55     qed
```
```    56   next
```
```    57     assume pos_y: "0 < y"
```
```    58
```
```    59     then obtain py where y_is_py: "y = real_of_preal py"
```
```    60       by (auto simp add: real_gt_zero_preal_Ex)
```
```    61
```
```    62     obtain a where a_in_P: "a \<in> P" using not_empty_P ..
```
```    63     have a_pos: "0 < a" using positive_P ..
```
```    64     then obtain pa where "a = real_of_preal pa"
```
```    65       by (auto simp add: real_gt_zero_preal_Ex)
```
```    66     hence "pa \<in> ?pP" using a_in_P by auto
```
```    67     hence pP_not_empty: "?pP \<noteq> {}" by auto
```
```    68
```
```    69     obtain sup where sup: "\<forall>x \<in> P. x < sup"
```
```    70       using upper_bound_Ex ..
```
```    71     hence  "a < sup" ..
```
```    72     hence "0 < sup" using a_pos by arith
```
```    73     then obtain possup where "sup = real_of_preal possup"
```
```    74       by (auto simp add: real_gt_zero_preal_Ex)
```
```    75     hence "\<forall>X \<in> ?pP. X \<le> possup"
```
```    76       using sup by (auto simp add: real_of_preal_lessI)
```
```    77     with pP_not_empty have psup: "\<And>Z. (\<exists>X \<in> ?pP. Z < X) = (Z < psup ?pP)"
```
```    78       by (rule preal_complete)
```
```    79
```
```    80     show ?thesis
```
```    81     proof
```
```    82       assume "\<exists>x \<in> P. y < x"
```
```    83       then obtain x where x_in_P: "x \<in> P" and y_less_x: "y < x" ..
```
```    84       hence "0 < x" using pos_y by arith
```
```    85       then obtain px where x_is_px: "x = real_of_preal px"
```
```    86         by (auto simp add: real_gt_zero_preal_Ex)
```
```    87
```
```    88       have py_less_X: "\<exists>X \<in> ?pP. py < X"
```
```    89       proof
```
```    90         show "py < px" using y_is_py and x_is_px and y_less_x
```
```    91           by (simp add: real_of_preal_lessI)
```
```    92         show "px \<in> ?pP" using x_in_P and x_is_px by simp
```
```    93       qed
```
```    94
```
```    95       have "(\<exists>X \<in> ?pP. py < X) ==> (py < psup ?pP)"
```
```    96         using psup by simp
```
```    97       hence "py < psup ?pP" using py_less_X by simp
```
```    98       thus "y < real_of_preal (psup {w. real_of_preal w \<in> P})"
```
```    99         using y_is_py and pos_y by (simp add: real_of_preal_lessI)
```
```   100     next
```
```   101       assume y_less_psup: "y < real_of_preal (psup ?pP)"
```
```   102
```
```   103       hence "py < psup ?pP" using y_is_py
```
```   104         by (simp add: real_of_preal_lessI)
```
```   105       then obtain "X" where py_less_X: "py < X" and X_in_pP: "X \<in> ?pP"
```
```   106         using psup by auto
```
```   107       then obtain x where x_is_X: "x = real_of_preal X"
```
```   108         by (simp add: real_gt_zero_preal_Ex)
```
```   109       hence "y < x" using py_less_X and y_is_py
```
```   110         by (simp add: real_of_preal_lessI)
```
```   111
```
```   112       moreover have "x \<in> P" using x_is_X and X_in_pP by simp
```
```   113
```
```   114       ultimately show "\<exists> x \<in> P. y < x" ..
```
```   115     qed
```
```   116   qed
```
```   117 qed
```
```   118
```
```   119 text {*
```
```   120   \medskip Completeness properties using @{text "isUb"}, @{text "isLub"} etc.
```
```   121 *}
```
```   122
```
```   123 lemma real_isLub_unique: "[| isLub R S x; isLub R S y |] ==> x = (y::real)"
```
```   124   apply (frule isLub_isUb)
```
```   125   apply (frule_tac x = y in isLub_isUb)
```
```   126   apply (blast intro!: order_antisym dest!: isLub_le_isUb)
```
```   127   done
```
```   128
```
```   129
```
```   130 text {*
```
```   131   \medskip Completeness theorem for the positive reals (again).
```
```   132 *}
```
```   133
```
```   134 lemma posreals_complete:
```
```   135   assumes positive_S: "\<forall>x \<in> S. 0 < x"
```
```   136     and not_empty_S: "\<exists>x. x \<in> S"
```
```   137     and upper_bound_Ex: "\<exists>u. isUb (UNIV::real set) S u"
```
```   138   shows "\<exists>t. isLub (UNIV::real set) S t"
```
```   139 proof
```
```   140   let ?pS = "{w. real_of_preal w \<in> S}"
```
```   141
```
```   142   obtain u where "isUb UNIV S u" using upper_bound_Ex ..
```
```   143   hence sup: "\<forall>x \<in> S. x \<le> u" by (simp add: isUb_def setle_def)
```
```   144
```
```   145   obtain x where x_in_S: "x \<in> S" using not_empty_S ..
```
```   146   hence x_gt_zero: "0 < x" using positive_S by simp
```
```   147   have  "x \<le> u" using sup and x_in_S ..
```
```   148   hence "0 < u" using x_gt_zero by arith
```
```   149
```
```   150   then obtain pu where u_is_pu: "u = real_of_preal pu"
```
```   151     by (auto simp add: real_gt_zero_preal_Ex)
```
```   152
```
```   153   have pS_less_pu: "\<forall>pa \<in> ?pS. pa \<le> pu"
```
```   154   proof
```
```   155     fix pa
```
```   156     assume "pa \<in> ?pS"
```
```   157     then obtain a where "a \<in> S" and "a = real_of_preal pa"
```
```   158       by simp
```
```   159     moreover hence "a \<le> u" using sup by simp
```
```   160     ultimately show "pa \<le> pu"
```
```   161       using sup and u_is_pu by (simp add: real_of_preal_le_iff)
```
```   162   qed
```
```   163
```
```   164   have "\<forall>y \<in> S. y \<le> real_of_preal (psup ?pS)"
```
```   165   proof
```
```   166     fix y
```
```   167     assume y_in_S: "y \<in> S"
```
```   168     hence "0 < y" using positive_S by simp
```
```   169     then obtain py where y_is_py: "y = real_of_preal py"
```
```   170       by (auto simp add: real_gt_zero_preal_Ex)
```
```   171     hence py_in_pS: "py \<in> ?pS" using y_in_S by simp
```
```   172     with pS_less_pu have "py \<le> psup ?pS"
```
```   173       by (rule preal_psup_le)
```
```   174     thus "y \<le> real_of_preal (psup ?pS)"
```
```   175       using y_is_py by (simp add: real_of_preal_le_iff)
```
```   176   qed
```
```   177
```
```   178   moreover {
```
```   179     fix x
```
```   180     assume x_ub_S: "\<forall>y\<in>S. y \<le> x"
```
```   181     have "real_of_preal (psup ?pS) \<le> x"
```
```   182     proof -
```
```   183       obtain "s" where s_in_S: "s \<in> S" using not_empty_S ..
```
```   184       hence s_pos: "0 < s" using positive_S by simp
```
```   185
```
```   186       hence "\<exists> ps. s = real_of_preal ps" by (simp add: real_gt_zero_preal_Ex)
```
```   187       then obtain "ps" where s_is_ps: "s = real_of_preal ps" ..
```
```   188       hence ps_in_pS: "ps \<in> {w. real_of_preal w \<in> S}" using s_in_S by simp
```
```   189
```
```   190       from x_ub_S have "s \<le> x" using s_in_S ..
```
```   191       hence "0 < x" using s_pos by simp
```
```   192       hence "\<exists> px. x = real_of_preal px" by (simp add: real_gt_zero_preal_Ex)
```
```   193       then obtain "px" where x_is_px: "x = real_of_preal px" ..
```
```   194
```
```   195       have "\<forall>pe \<in> ?pS. pe \<le> px"
```
```   196       proof
```
```   197 	fix pe
```
```   198 	assume "pe \<in> ?pS"
```
```   199 	hence "real_of_preal pe \<in> S" by simp
```
```   200 	hence "real_of_preal pe \<le> x" using x_ub_S by simp
```
```   201 	thus "pe \<le> px" using x_is_px by (simp add: real_of_preal_le_iff)
```
```   202       qed
```
```   203
```
```   204       moreover have "?pS \<noteq> {}" using ps_in_pS by auto
```
```   205       ultimately have "(psup ?pS) \<le> px" by (simp add: psup_le_ub)
```
```   206       thus "real_of_preal (psup ?pS) \<le> x" using x_is_px by (simp add: real_of_preal_le_iff)
```
```   207     qed
```
```   208   }
```
```   209   ultimately show "isLub UNIV S (real_of_preal (psup ?pS))"
```
```   210     by (simp add: isLub_def leastP_def isUb_def setle_def setge_def)
```
```   211 qed
```
```   212
```
```   213 text {*
```
```   214   \medskip reals Completeness (again!)
```
```   215 *}
```
```   216
```
```   217 lemma reals_complete:
```
```   218   assumes notempty_S: "\<exists>X. X \<in> S"
```
```   219     and exists_Ub: "\<exists>Y. isUb (UNIV::real set) S Y"
```
```   220   shows "\<exists>t. isLub (UNIV :: real set) S t"
```
```   221 proof -
```
```   222   obtain X where X_in_S: "X \<in> S" using notempty_S ..
```
```   223   obtain Y where Y_isUb: "isUb (UNIV::real set) S Y"
```
```   224     using exists_Ub ..
```
```   225   let ?SHIFT = "{z. \<exists>x \<in>S. z = x + (-X) + 1} \<inter> {x. 0 < x}"
```
```   226
```
```   227   {
```
```   228     fix x
```
```   229     assume "isUb (UNIV::real set) S x"
```
```   230     hence S_le_x: "\<forall> y \<in> S. y <= x"
```
```   231       by (simp add: isUb_def setle_def)
```
```   232     {
```
```   233       fix s
```
```   234       assume "s \<in> {z. \<exists>x\<in>S. z = x + - X + 1}"
```
```   235       hence "\<exists> x \<in> S. s = x + -X + 1" ..
```
```   236       then obtain x1 where "x1 \<in> S" and "s = x1 + (-X) + 1" ..
```
```   237       moreover hence "x1 \<le> x" using S_le_x by simp
```
```   238       ultimately have "s \<le> x + - X + 1" by arith
```
```   239     }
```
```   240     then have "isUb (UNIV::real set) ?SHIFT (x + (-X) + 1)"
```
```   241       by (auto simp add: isUb_def setle_def)
```
```   242   } note S_Ub_is_SHIFT_Ub = this
```
```   243
```
```   244   hence "isUb UNIV ?SHIFT (Y + (-X) + 1)" using Y_isUb by simp
```
```   245   hence "\<exists>Z. isUb UNIV ?SHIFT Z" ..
```
```   246   moreover have "\<forall>y \<in> ?SHIFT. 0 < y" by auto
```
```   247   moreover have shifted_not_empty: "\<exists>u. u \<in> ?SHIFT"
```
```   248     using X_in_S and Y_isUb by auto
```
```   249   ultimately obtain t where t_is_Lub: "isLub UNIV ?SHIFT t"
```
```   250     using posreals_complete [of ?SHIFT] by blast
```
```   251
```
```   252   show ?thesis
```
```   253   proof
```
```   254     show "isLub UNIV S (t + X + (-1))"
```
```   255     proof (rule isLubI2)
```
```   256       {
```
```   257         fix x
```
```   258         assume "isUb (UNIV::real set) S x"
```
```   259         hence "isUb (UNIV::real set) (?SHIFT) (x + (-X) + 1)"
```
```   260 	  using S_Ub_is_SHIFT_Ub by simp
```
```   261         hence "t \<le> (x + (-X) + 1)"
```
```   262 	  using t_is_Lub by (simp add: isLub_le_isUb)
```
```   263         hence "t + X + -1 \<le> x" by arith
```
```   264       }
```
```   265       then show "(t + X + -1) <=* Collect (isUb UNIV S)"
```
```   266 	by (simp add: setgeI)
```
```   267     next
```
```   268       show "isUb UNIV S (t + X + -1)"
```
```   269       proof -
```
```   270         {
```
```   271           fix y
```
```   272           assume y_in_S: "y \<in> S"
```
```   273           have "y \<le> t + X + -1"
```
```   274           proof -
```
```   275             obtain "u" where u_in_shift: "u \<in> ?SHIFT" using shifted_not_empty ..
```
```   276             hence "\<exists> x \<in> S. u = x + - X + 1" by simp
```
```   277             then obtain "x" where x_and_u: "u = x + - X + 1" ..
```
```   278             have u_le_t: "u \<le> t" using u_in_shift and t_is_Lub by (simp add: isLubD2)
```
```   279
```
```   280             show ?thesis
```
```   281             proof cases
```
```   282               assume "y \<le> x"
```
```   283               moreover have "x = u + X + - 1" using x_and_u by arith
```
```   284               moreover have "u + X + - 1  \<le> t + X + -1" using u_le_t by arith
```
```   285               ultimately show "y  \<le> t + X + -1" by arith
```
```   286             next
```
```   287               assume "~(y \<le> x)"
```
```   288               hence x_less_y: "x < y" by arith
```
```   289
```
```   290               have "x + (-X) + 1 \<in> ?SHIFT" using x_and_u and u_in_shift by simp
```
```   291               hence "0 < x + (-X) + 1" by simp
```
```   292               hence "0 < y + (-X) + 1" using x_less_y by arith
```
```   293               hence "y + (-X) + 1 \<in> ?SHIFT" using y_in_S by simp
```
```   294               hence "y + (-X) + 1 \<le> t" using t_is_Lub  by (simp add: isLubD2)
```
```   295               thus ?thesis by simp
```
```   296             qed
```
```   297           qed
```
```   298         }
```
```   299         then show ?thesis by (simp add: isUb_def setle_def)
```
```   300       qed
```
```   301     qed
```
```   302   qed
```
```   303 qed
```
```   304
```
```   305
```
```   306 subsection {* The Archimedean Property of the Reals *}
```
```   307
```
```   308 theorem reals_Archimedean:
```
```   309   assumes x_pos: "0 < x"
```
```   310   shows "\<exists>n. inverse (real (Suc n)) < x"
```
```   311 proof (rule ccontr)
```
```   312   assume contr: "\<not> ?thesis"
```
```   313   have "\<forall>n. x * real (Suc n) <= 1"
```
```   314   proof
```
```   315     fix n
```
```   316     from contr have "x \<le> inverse (real (Suc n))"
```
```   317       by (simp add: linorder_not_less)
```
```   318     hence "x \<le> (1 / (real (Suc n)))"
```
```   319       by (simp add: inverse_eq_divide)
```
```   320     moreover have "0 \<le> real (Suc n)"
```
```   321       by (rule real_of_nat_ge_zero)
```
```   322     ultimately have "x * real (Suc n) \<le> (1 / real (Suc n)) * real (Suc n)"
```
```   323       by (rule mult_right_mono)
```
```   324     thus "x * real (Suc n) \<le> 1" by simp
```
```   325   qed
```
```   326   hence "{z. \<exists>n. z = x * (real (Suc n))} *<= 1"
```
```   327     by (simp add: setle_def, safe, rule spec)
```
```   328   hence "isUb (UNIV::real set) {z. \<exists>n. z = x * (real (Suc n))} 1"
```
```   329     by (simp add: isUbI)
```
```   330   hence "\<exists>Y. isUb (UNIV::real set) {z. \<exists>n. z = x* (real (Suc n))} Y" ..
```
```   331   moreover have "\<exists>X. X \<in> {z. \<exists>n. z = x* (real (Suc n))}" by auto
```
```   332   ultimately have "\<exists>t. isLub UNIV {z. \<exists>n. z = x * real (Suc n)} t"
```
```   333     by (simp add: reals_complete)
```
```   334   then obtain "t" where
```
```   335     t_is_Lub: "isLub UNIV {z. \<exists>n. z = x * real (Suc n)} t" ..
```
```   336
```
```   337   have "\<forall>n::nat. x * real n \<le> t + - x"
```
```   338   proof
```
```   339     fix n
```
```   340     from t_is_Lub have "x * real (Suc n) \<le> t"
```
```   341       by (simp add: isLubD2)
```
```   342     hence  "x * (real n) + x \<le> t"
```
```   343       by (simp add: right_distrib real_of_nat_Suc)
```
```   344     thus  "x * (real n) \<le> t + - x" by arith
```
```   345   qed
```
```   346
```
```   347   hence "\<forall>m. x * real (Suc m) \<le> t + - x" by simp
```
```   348   hence "{z. \<exists>n. z = x * (real (Suc n))}  *<= (t + - x)"
```
```   349     by (auto simp add: setle_def)
```
```   350   hence "isUb (UNIV::real set) {z. \<exists>n. z = x * (real (Suc n))} (t + (-x))"
```
```   351     by (simp add: isUbI)
```
```   352   hence "t \<le> t + - x"
```
```   353     using t_is_Lub by (simp add: isLub_le_isUb)
```
```   354   thus False using x_pos by arith
```
```   355 qed
```
```   356
```
```   357 text {*
```
```   358   There must be other proofs, e.g. @{text "Suc"} of the largest
```
```   359   integer in the cut representing @{text "x"}.
```
```   360 *}
```
```   361
```
```   362 lemma reals_Archimedean2: "\<exists>n. (x::real) < real (n::nat)"
```
```   363 proof cases
```
```   364   assume "x \<le> 0"
```
```   365   hence "x < real (1::nat)" by simp
```
```   366   thus ?thesis ..
```
```   367 next
```
```   368   assume "\<not> x \<le> 0"
```
```   369   hence x_greater_zero: "0 < x" by simp
```
```   370   hence "0 < inverse x" by simp
```
```   371   then obtain n where "inverse (real (Suc n)) < inverse x"
```
```   372     using reals_Archimedean by blast
```
```   373   hence "inverse (real (Suc n)) * x < inverse x * x"
```
```   374     using x_greater_zero by (rule mult_strict_right_mono)
```
```   375   hence "inverse (real (Suc n)) * x < 1"
```
```   376     using x_greater_zero by (simp add: real_mult_inverse_left mult_commute)
```
```   377   hence "real (Suc n) * (inverse (real (Suc n)) * x) < real (Suc n) * 1"
```
```   378     by (rule mult_strict_left_mono) simp
```
```   379   hence "x < real (Suc n)"
```
```   380     by (simp add: mult_commute ring_eq_simps real_mult_inverse_left)
```
```   381   thus "\<exists>(n::nat). x < real n" ..
```
```   382 qed
```
```   383
```
```   384 lemma reals_Archimedean3:
```
```   385   assumes x_greater_zero: "0 < x"
```
```   386   shows "\<forall>(y::real). \<exists>(n::nat). y < real n * x"
```
```   387 proof
```
```   388   fix y
```
```   389   have x_not_zero: "x \<noteq> 0" using x_greater_zero by simp
```
```   390   obtain n where "y * inverse x < real (n::nat)"
```
```   391     using reals_Archimedean2 ..
```
```   392   hence "y * inverse x * x < real n * x"
```
```   393     using x_greater_zero by (simp add: mult_strict_right_mono)
```
```   394   hence "x * inverse x * y < x * real n"
```
```   395     by (simp add: mult_commute ring_eq_simps)
```
```   396   hence "y < real (n::nat) * x"
```
```   397     using x_not_zero by (simp add: real_mult_inverse_left ring_eq_simps)
```
```   398   thus "\<exists>(n::nat). y < real n * x" ..
```
```   399 qed
```
```   400
```
```   401 lemma reals_Archimedean6:
```
```   402      "0 \<le> r ==> \<exists>(n::nat). real (n - 1) \<le> r & r < real (n)"
```
```   403 apply (insert reals_Archimedean2 [of r], safe)
```
```   404 apply (frule_tac P = "%k. r < real k" and k = n and m = "%x. x"
```
```   405        in ex_has_least_nat, auto)
```
```   406 apply (rule_tac x = x in exI)
```
```   407 apply (case_tac x, simp)
```
```   408 apply (rename_tac x')
```
```   409 apply (drule_tac x = x' in spec, simp)
```
```   410 done
```
```   411
```
```   412 lemma reals_Archimedean6a: "0 \<le> r ==> \<exists>n. real (n) \<le> r & r < real (Suc n)"
```
```   413   by (drule reals_Archimedean6) auto
```
```   414
```
```   415 lemma reals_Archimedean_6b_int:
```
```   416      "0 \<le> r ==> \<exists>n::int. real n \<le> r & r < real (n+1)"
```
```   417 apply (drule reals_Archimedean6a, auto)
```
```   418 apply (rule_tac x = "int n" in exI)
```
```   419 apply (simp add: real_of_int_real_of_nat real_of_nat_Suc)
```
```   420 done
```
```   421
```
```   422 lemma reals_Archimedean_6c_int:
```
```   423      "r < 0 ==> \<exists>n::int. real n \<le> r & r < real (n+1)"
```
```   424 apply (rule reals_Archimedean_6b_int [of "-r", THEN exE], simp, auto)
```
```   425 apply (rename_tac n)
```
```   426 apply (drule real_le_imp_less_or_eq, auto)
```
```   427 apply (rule_tac x = "- n - 1" in exI)
```
```   428 apply (rule_tac [2] x = "- n" in exI, auto)
```
```   429 done
```
```   430
```
```   431
```
```   432 subsection{*Floor and Ceiling Functions from the Reals to the Integers*}
```
```   433
```
```   434 definition
```
```   435   floor :: "real => int"
```
```   436   "floor r = (LEAST n::int. r < real (n+1))"
```
```   437
```
```   438   ceiling :: "real => int"
```
```   439   "ceiling r = - floor (- r)"
```
```   440
```
```   441 const_syntax (xsymbols)
```
```   442   floor  ("\<lfloor>_\<rfloor>")
```
```   443   ceiling  ("\<lceil>_\<rceil>")
```
```   444
```
```   445 const_syntax (HTML output)
```
```   446   floor  ("\<lfloor>_\<rfloor>")
```
```   447   ceiling  ("\<lceil>_\<rceil>")
```
```   448
```
```   449
```
```   450 lemma number_of_less_real_of_int_iff [simp]:
```
```   451      "((number_of n) < real (m::int)) = (number_of n < m)"
```
```   452 apply auto
```
```   453 apply (rule real_of_int_less_iff [THEN iffD1])
```
```   454 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
```
```   455 done
```
```   456
```
```   457 lemma number_of_less_real_of_int_iff2 [simp]:
```
```   458      "(real (m::int) < (number_of n)) = (m < number_of n)"
```
```   459 apply auto
```
```   460 apply (rule real_of_int_less_iff [THEN iffD1])
```
```   461 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
```
```   462 done
```
```   463
```
```   464 lemma number_of_le_real_of_int_iff [simp]:
```
```   465      "((number_of n) \<le> real (m::int)) = (number_of n \<le> m)"
```
```   466 by (simp add: linorder_not_less [symmetric])
```
```   467
```
```   468 lemma number_of_le_real_of_int_iff2 [simp]:
```
```   469      "(real (m::int) \<le> (number_of n)) = (m \<le> number_of n)"
```
```   470 by (simp add: linorder_not_less [symmetric])
```
```   471
```
```   472 lemma floor_zero [simp]: "floor 0 = 0"
```
```   473 apply (simp add: floor_def del: real_of_int_add)
```
```   474 apply (rule Least_equality)
```
```   475 apply simp_all
```
```   476 done
```
```   477
```
```   478 lemma floor_real_of_nat_zero [simp]: "floor (real (0::nat)) = 0"
```
```   479 by auto
```
```   480
```
```   481 lemma floor_real_of_nat [simp]: "floor (real (n::nat)) = int n"
```
```   482 apply (simp only: floor_def)
```
```   483 apply (rule Least_equality)
```
```   484 apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
```
```   485 apply (drule_tac [2] real_of_int_less_iff [THEN iffD1])
```
```   486 apply (simp_all add: real_of_int_real_of_nat)
```
```   487 done
```
```   488
```
```   489 lemma floor_minus_real_of_nat [simp]: "floor (- real (n::nat)) = - int n"
```
```   490 apply (simp only: floor_def)
```
```   491 apply (rule Least_equality)
```
```   492 apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
```
```   493 apply (drule_tac [2] real_of_int_minus [THEN sym, THEN subst])
```
```   494 apply (drule_tac [2] real_of_int_less_iff [THEN iffD1])
```
```   495 apply (simp_all add: real_of_int_real_of_nat)
```
```   496 done
```
```   497
```
```   498 lemma floor_real_of_int [simp]: "floor (real (n::int)) = n"
```
```   499 apply (simp only: floor_def)
```
```   500 apply (rule Least_equality)
```
```   501 apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
```
```   502 apply (drule_tac [2] real_of_int_less_iff [THEN iffD1], auto)
```
```   503 done
```
```   504
```
```   505 lemma floor_minus_real_of_int [simp]: "floor (- real (n::int)) = - n"
```
```   506 apply (simp only: floor_def)
```
```   507 apply (rule Least_equality)
```
```   508 apply (drule_tac [2] real_of_int_minus [THEN sym, THEN subst])
```
```   509 apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
```
```   510 apply (drule_tac [2] real_of_int_less_iff [THEN iffD1], auto)
```
```   511 done
```
```   512
```
```   513 lemma real_lb_ub_int: " \<exists>n::int. real n \<le> r & r < real (n+1)"
```
```   514 apply (case_tac "r < 0")
```
```   515 apply (blast intro: reals_Archimedean_6c_int)
```
```   516 apply (simp only: linorder_not_less)
```
```   517 apply (blast intro: reals_Archimedean_6b_int reals_Archimedean_6c_int)
```
```   518 done
```
```   519
```
```   520 lemma lemma_floor:
```
```   521   assumes a1: "real m \<le> r" and a2: "r < real n + 1"
```
```   522   shows "m \<le> (n::int)"
```
```   523 proof -
```
```   524   have "real m < real n + 1" by (rule order_le_less_trans)
```
```   525   also have "... = real(n+1)" by simp
```
```   526   finally have "m < n+1" by (simp only: real_of_int_less_iff)
```
```   527   thus ?thesis by arith
```
```   528 qed
```
```   529
```
```   530 lemma real_of_int_floor_le [simp]: "real (floor r) \<le> r"
```
```   531 apply (simp add: floor_def Least_def)
```
```   532 apply (insert real_lb_ub_int [of r], safe)
```
```   533 apply (rule theI2)
```
```   534 apply auto
```
```   535 done
```
```   536
```
```   537 lemma floor_mono: "x < y ==> floor x \<le> floor y"
```
```   538 apply (simp add: floor_def Least_def)
```
```   539 apply (insert real_lb_ub_int [of x])
```
```   540 apply (insert real_lb_ub_int [of y], safe)
```
```   541 apply (rule theI2)
```
```   542 apply (rule_tac [3] theI2)
```
```   543 apply simp
```
```   544 apply (erule conjI)
```
```   545 apply (auto simp add: order_eq_iff int_le_real_less)
```
```   546 done
```
```   547
```
```   548 lemma floor_mono2: "x \<le> y ==> floor x \<le> floor y"
```
```   549 by (auto dest: real_le_imp_less_or_eq simp add: floor_mono)
```
```   550
```
```   551 lemma lemma_floor2: "real n < real (x::int) + 1 ==> n \<le> x"
```
```   552 by (auto intro: lemma_floor)
```
```   553
```
```   554 lemma real_of_int_floor_cancel [simp]:
```
```   555     "(real (floor x) = x) = (\<exists>n::int. x = real n)"
```
```   556 apply (simp add: floor_def Least_def)
```
```   557 apply (insert real_lb_ub_int [of x], erule exE)
```
```   558 apply (rule theI2)
```
```   559 apply (auto intro: lemma_floor)
```
```   560 done
```
```   561
```
```   562 lemma floor_eq: "[| real n < x; x < real n + 1 |] ==> floor x = n"
```
```   563 apply (simp add: floor_def)
```
```   564 apply (rule Least_equality)
```
```   565 apply (auto intro: lemma_floor)
```
```   566 done
```
```   567
```
```   568 lemma floor_eq2: "[| real n \<le> x; x < real n + 1 |] ==> floor x = n"
```
```   569 apply (simp add: floor_def)
```
```   570 apply (rule Least_equality)
```
```   571 apply (auto intro: lemma_floor)
```
```   572 done
```
```   573
```
```   574 lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat(floor x) = n"
```
```   575 apply (rule inj_int [THEN injD])
```
```   576 apply (simp add: real_of_nat_Suc)
```
```   577 apply (simp add: real_of_nat_Suc floor_eq floor_eq [where n = "int n"])
```
```   578 done
```
```   579
```
```   580 lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat(floor x) = n"
```
```   581 apply (drule order_le_imp_less_or_eq)
```
```   582 apply (auto intro: floor_eq3)
```
```   583 done
```
```   584
```
```   585 lemma floor_number_of_eq [simp]:
```
```   586      "floor(number_of n :: real) = (number_of n :: int)"
```
```   587 apply (subst real_number_of [symmetric])
```
```   588 apply (rule floor_real_of_int)
```
```   589 done
```
```   590
```
```   591 lemma floor_one [simp]: "floor 1 = 1"
```
```   592   apply (rule trans)
```
```   593   prefer 2
```
```   594   apply (rule floor_real_of_int)
```
```   595   apply simp
```
```   596 done
```
```   597
```
```   598 lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real(floor r)"
```
```   599 apply (simp add: floor_def Least_def)
```
```   600 apply (insert real_lb_ub_int [of r], safe)
```
```   601 apply (rule theI2)
```
```   602 apply (auto intro: lemma_floor)
```
```   603 done
```
```   604
```
```   605 lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real(floor r)"
```
```   606 apply (simp add: floor_def Least_def)
```
```   607 apply (insert real_lb_ub_int [of r], safe)
```
```   608 apply (rule theI2)
```
```   609 apply (auto intro: lemma_floor)
```
```   610 done
```
```   611
```
```   612 lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real(floor r) + 1"
```
```   613 apply (insert real_of_int_floor_ge_diff_one [of r])
```
```   614 apply (auto simp del: real_of_int_floor_ge_diff_one)
```
```   615 done
```
```   616
```
```   617 lemma real_of_int_floor_add_one_gt [simp]: "r < real(floor r) + 1"
```
```   618 apply (insert real_of_int_floor_gt_diff_one [of r])
```
```   619 apply (auto simp del: real_of_int_floor_gt_diff_one)
```
```   620 done
```
```   621
```
```   622 lemma le_floor: "real a <= x ==> a <= floor x"
```
```   623   apply (subgoal_tac "a < floor x + 1")
```
```   624   apply arith
```
```   625   apply (subst real_of_int_less_iff [THEN sym])
```
```   626   apply simp
```
```   627   apply (insert real_of_int_floor_add_one_gt [of x])
```
```   628   apply arith
```
```   629 done
```
```   630
```
```   631 lemma real_le_floor: "a <= floor x ==> real a <= x"
```
```   632   apply (rule order_trans)
```
```   633   prefer 2
```
```   634   apply (rule real_of_int_floor_le)
```
```   635   apply (subst real_of_int_le_iff)
```
```   636   apply assumption
```
```   637 done
```
```   638
```
```   639 lemma le_floor_eq: "(a <= floor x) = (real a <= x)"
```
```   640   apply (rule iffI)
```
```   641   apply (erule real_le_floor)
```
```   642   apply (erule le_floor)
```
```   643 done
```
```   644
```
```   645 lemma le_floor_eq_number_of [simp]:
```
```   646     "(number_of n <= floor x) = (number_of n <= x)"
```
```   647 by (simp add: le_floor_eq)
```
```   648
```
```   649 lemma le_floor_eq_zero [simp]: "(0 <= floor x) = (0 <= x)"
```
```   650 by (simp add: le_floor_eq)
```
```   651
```
```   652 lemma le_floor_eq_one [simp]: "(1 <= floor x) = (1 <= x)"
```
```   653 by (simp add: le_floor_eq)
```
```   654
```
```   655 lemma floor_less_eq: "(floor x < a) = (x < real a)"
```
```   656   apply (subst linorder_not_le [THEN sym])+
```
```   657   apply simp
```
```   658   apply (rule le_floor_eq)
```
```   659 done
```
```   660
```
```   661 lemma floor_less_eq_number_of [simp]:
```
```   662     "(floor x < number_of n) = (x < number_of n)"
```
```   663 by (simp add: floor_less_eq)
```
```   664
```
```   665 lemma floor_less_eq_zero [simp]: "(floor x < 0) = (x < 0)"
```
```   666 by (simp add: floor_less_eq)
```
```   667
```
```   668 lemma floor_less_eq_one [simp]: "(floor x < 1) = (x < 1)"
```
```   669 by (simp add: floor_less_eq)
```
```   670
```
```   671 lemma less_floor_eq: "(a < floor x) = (real a + 1 <= x)"
```
```   672   apply (insert le_floor_eq [of "a + 1" x])
```
```   673   apply auto
```
```   674 done
```
```   675
```
```   676 lemma less_floor_eq_number_of [simp]:
```
```   677     "(number_of n < floor x) = (number_of n + 1 <= x)"
```
```   678 by (simp add: less_floor_eq)
```
```   679
```
```   680 lemma less_floor_eq_zero [simp]: "(0 < floor x) = (1 <= x)"
```
```   681 by (simp add: less_floor_eq)
```
```   682
```
```   683 lemma less_floor_eq_one [simp]: "(1 < floor x) = (2 <= x)"
```
```   684 by (simp add: less_floor_eq)
```
```   685
```
```   686 lemma floor_le_eq: "(floor x <= a) = (x < real a + 1)"
```
```   687   apply (insert floor_less_eq [of x "a + 1"])
```
```   688   apply auto
```
```   689 done
```
```   690
```
```   691 lemma floor_le_eq_number_of [simp]:
```
```   692     "(floor x <= number_of n) = (x < number_of n + 1)"
```
```   693 by (simp add: floor_le_eq)
```
```   694
```
```   695 lemma floor_le_eq_zero [simp]: "(floor x <= 0) = (x < 1)"
```
```   696 by (simp add: floor_le_eq)
```
```   697
```
```   698 lemma floor_le_eq_one [simp]: "(floor x <= 1) = (x < 2)"
```
```   699 by (simp add: floor_le_eq)
```
```   700
```
```   701 lemma floor_add [simp]: "floor (x + real a) = floor x + a"
```
```   702   apply (subst order_eq_iff)
```
```   703   apply (rule conjI)
```
```   704   prefer 2
```
```   705   apply (subgoal_tac "floor x + a < floor (x + real a) + 1")
```
```   706   apply arith
```
```   707   apply (subst real_of_int_less_iff [THEN sym])
```
```   708   apply simp
```
```   709   apply (subgoal_tac "x + real a < real(floor(x + real a)) + 1")
```
```   710   apply (subgoal_tac "real (floor x) <= x")
```
```   711   apply arith
```
```   712   apply (rule real_of_int_floor_le)
```
```   713   apply (rule real_of_int_floor_add_one_gt)
```
```   714   apply (subgoal_tac "floor (x + real a) < floor x + a + 1")
```
```   715   apply arith
```
```   716   apply (subst real_of_int_less_iff [THEN sym])
```
```   717   apply simp
```
```   718   apply (subgoal_tac "real(floor(x + real a)) <= x + real a")
```
```   719   apply (subgoal_tac "x < real(floor x) + 1")
```
```   720   apply arith
```
```   721   apply (rule real_of_int_floor_add_one_gt)
```
```   722   apply (rule real_of_int_floor_le)
```
```   723 done
```
```   724
```
```   725 lemma floor_add_number_of [simp]:
```
```   726     "floor (x + number_of n) = floor x + number_of n"
```
```   727   apply (subst floor_add [THEN sym])
```
```   728   apply simp
```
```   729 done
```
```   730
```
```   731 lemma floor_add_one [simp]: "floor (x + 1) = floor x + 1"
```
```   732   apply (subst floor_add [THEN sym])
```
```   733   apply simp
```
```   734 done
```
```   735
```
```   736 lemma floor_subtract [simp]: "floor (x - real a) = floor x - a"
```
```   737   apply (subst diff_minus)+
```
```   738   apply (subst real_of_int_minus [THEN sym])
```
```   739   apply (rule floor_add)
```
```   740 done
```
```   741
```
```   742 lemma floor_subtract_number_of [simp]: "floor (x - number_of n) =
```
```   743     floor x - number_of n"
```
```   744   apply (subst floor_subtract [THEN sym])
```
```   745   apply simp
```
```   746 done
```
```   747
```
```   748 lemma floor_subtract_one [simp]: "floor (x - 1) = floor x - 1"
```
```   749   apply (subst floor_subtract [THEN sym])
```
```   750   apply simp
```
```   751 done
```
```   752
```
```   753 lemma ceiling_zero [simp]: "ceiling 0 = 0"
```
```   754 by (simp add: ceiling_def)
```
```   755
```
```   756 lemma ceiling_real_of_nat [simp]: "ceiling (real (n::nat)) = int n"
```
```   757 by (simp add: ceiling_def)
```
```   758
```
```   759 lemma ceiling_real_of_nat_zero [simp]: "ceiling (real (0::nat)) = 0"
```
```   760 by auto
```
```   761
```
```   762 lemma ceiling_floor [simp]: "ceiling (real (floor r)) = floor r"
```
```   763 by (simp add: ceiling_def)
```
```   764
```
```   765 lemma floor_ceiling [simp]: "floor (real (ceiling r)) = ceiling r"
```
```   766 by (simp add: ceiling_def)
```
```   767
```
```   768 lemma real_of_int_ceiling_ge [simp]: "r \<le> real (ceiling r)"
```
```   769 apply (simp add: ceiling_def)
```
```   770 apply (subst le_minus_iff, simp)
```
```   771 done
```
```   772
```
```   773 lemma ceiling_mono: "x < y ==> ceiling x \<le> ceiling y"
```
```   774 by (simp add: floor_mono ceiling_def)
```
```   775
```
```   776 lemma ceiling_mono2: "x \<le> y ==> ceiling x \<le> ceiling y"
```
```   777 by (simp add: floor_mono2 ceiling_def)
```
```   778
```
```   779 lemma real_of_int_ceiling_cancel [simp]:
```
```   780      "(real (ceiling x) = x) = (\<exists>n::int. x = real n)"
```
```   781 apply (auto simp add: ceiling_def)
```
```   782 apply (drule arg_cong [where f = uminus], auto)
```
```   783 apply (rule_tac x = "-n" in exI, auto)
```
```   784 done
```
```   785
```
```   786 lemma ceiling_eq: "[| real n < x; x < real n + 1 |] ==> ceiling x = n + 1"
```
```   787 apply (simp add: ceiling_def)
```
```   788 apply (rule minus_equation_iff [THEN iffD1])
```
```   789 apply (simp add: floor_eq [where n = "-(n+1)"])
```
```   790 done
```
```   791
```
```   792 lemma ceiling_eq2: "[| real n < x; x \<le> real n + 1 |] ==> ceiling x = n + 1"
```
```   793 by (simp add: ceiling_def floor_eq2 [where n = "-(n+1)"])
```
```   794
```
```   795 lemma ceiling_eq3: "[| real n - 1 < x; x \<le> real n  |] ==> ceiling x = n"
```
```   796 by (simp add: ceiling_def floor_eq2 [where n = "-n"])
```
```   797
```
```   798 lemma ceiling_real_of_int [simp]: "ceiling (real (n::int)) = n"
```
```   799 by (simp add: ceiling_def)
```
```   800
```
```   801 lemma ceiling_number_of_eq [simp]:
```
```   802      "ceiling (number_of n :: real) = (number_of n)"
```
```   803 apply (subst real_number_of [symmetric])
```
```   804 apply (rule ceiling_real_of_int)
```
```   805 done
```
```   806
```
```   807 lemma ceiling_one [simp]: "ceiling 1 = 1"
```
```   808   by (unfold ceiling_def, simp)
```
```   809
```
```   810 lemma real_of_int_ceiling_diff_one_le [simp]: "real (ceiling r) - 1 \<le> r"
```
```   811 apply (rule neg_le_iff_le [THEN iffD1])
```
```   812 apply (simp add: ceiling_def diff_minus)
```
```   813 done
```
```   814
```
```   815 lemma real_of_int_ceiling_le_add_one [simp]: "real (ceiling r) \<le> r + 1"
```
```   816 apply (insert real_of_int_ceiling_diff_one_le [of r])
```
```   817 apply (simp del: real_of_int_ceiling_diff_one_le)
```
```   818 done
```
```   819
```
```   820 lemma ceiling_le: "x <= real a ==> ceiling x <= a"
```
```   821   apply (unfold ceiling_def)
```
```   822   apply (subgoal_tac "-a <= floor(- x)")
```
```   823   apply simp
```
```   824   apply (rule le_floor)
```
```   825   apply simp
```
```   826 done
```
```   827
```
```   828 lemma ceiling_le_real: "ceiling x <= a ==> x <= real a"
```
```   829   apply (unfold ceiling_def)
```
```   830   apply (subgoal_tac "real(- a) <= - x")
```
```   831   apply simp
```
```   832   apply (rule real_le_floor)
```
```   833   apply simp
```
```   834 done
```
```   835
```
```   836 lemma ceiling_le_eq: "(ceiling x <= a) = (x <= real a)"
```
```   837   apply (rule iffI)
```
```   838   apply (erule ceiling_le_real)
```
```   839   apply (erule ceiling_le)
```
```   840 done
```
```   841
```
```   842 lemma ceiling_le_eq_number_of [simp]:
```
```   843     "(ceiling x <= number_of n) = (x <= number_of n)"
```
```   844 by (simp add: ceiling_le_eq)
```
```   845
```
```   846 lemma ceiling_le_zero_eq [simp]: "(ceiling x <= 0) = (x <= 0)"
```
```   847 by (simp add: ceiling_le_eq)
```
```   848
```
```   849 lemma ceiling_le_eq_one [simp]: "(ceiling x <= 1) = (x <= 1)"
```
```   850 by (simp add: ceiling_le_eq)
```
```   851
```
```   852 lemma less_ceiling_eq: "(a < ceiling x) = (real a < x)"
```
```   853   apply (subst linorder_not_le [THEN sym])+
```
```   854   apply simp
```
```   855   apply (rule ceiling_le_eq)
```
```   856 done
```
```   857
```
```   858 lemma less_ceiling_eq_number_of [simp]:
```
```   859     "(number_of n < ceiling x) = (number_of n < x)"
```
```   860 by (simp add: less_ceiling_eq)
```
```   861
```
```   862 lemma less_ceiling_eq_zero [simp]: "(0 < ceiling x) = (0 < x)"
```
```   863 by (simp add: less_ceiling_eq)
```
```   864
```
```   865 lemma less_ceiling_eq_one [simp]: "(1 < ceiling x) = (1 < x)"
```
```   866 by (simp add: less_ceiling_eq)
```
```   867
```
```   868 lemma ceiling_less_eq: "(ceiling x < a) = (x <= real a - 1)"
```
```   869   apply (insert ceiling_le_eq [of x "a - 1"])
```
```   870   apply auto
```
```   871 done
```
```   872
```
```   873 lemma ceiling_less_eq_number_of [simp]:
```
```   874     "(ceiling x < number_of n) = (x <= number_of n - 1)"
```
```   875 by (simp add: ceiling_less_eq)
```
```   876
```
```   877 lemma ceiling_less_eq_zero [simp]: "(ceiling x < 0) = (x <= -1)"
```
```   878 by (simp add: ceiling_less_eq)
```
```   879
```
```   880 lemma ceiling_less_eq_one [simp]: "(ceiling x < 1) = (x <= 0)"
```
```   881 by (simp add: ceiling_less_eq)
```
```   882
```
```   883 lemma le_ceiling_eq: "(a <= ceiling x) = (real a - 1 < x)"
```
```   884   apply (insert less_ceiling_eq [of "a - 1" x])
```
```   885   apply auto
```
```   886 done
```
```   887
```
```   888 lemma le_ceiling_eq_number_of [simp]:
```
```   889     "(number_of n <= ceiling x) = (number_of n - 1 < x)"
```
```   890 by (simp add: le_ceiling_eq)
```
```   891
```
```   892 lemma le_ceiling_eq_zero [simp]: "(0 <= ceiling x) = (-1 < x)"
```
```   893 by (simp add: le_ceiling_eq)
```
```   894
```
```   895 lemma le_ceiling_eq_one [simp]: "(1 <= ceiling x) = (0 < x)"
```
```   896 by (simp add: le_ceiling_eq)
```
```   897
```
```   898 lemma ceiling_add [simp]: "ceiling (x + real a) = ceiling x + a"
```
```   899   apply (unfold ceiling_def, simp)
```
```   900   apply (subst real_of_int_minus [THEN sym])
```
```   901   apply (subst floor_add)
```
```   902   apply simp
```
```   903 done
```
```   904
```
```   905 lemma ceiling_add_number_of [simp]: "ceiling (x + number_of n) =
```
```   906     ceiling x + number_of n"
```
```   907   apply (subst ceiling_add [THEN sym])
```
```   908   apply simp
```
```   909 done
```
```   910
```
```   911 lemma ceiling_add_one [simp]: "ceiling (x + 1) = ceiling x + 1"
```
```   912   apply (subst ceiling_add [THEN sym])
```
```   913   apply simp
```
```   914 done
```
```   915
```
```   916 lemma ceiling_subtract [simp]: "ceiling (x - real a) = ceiling x - a"
```
```   917   apply (subst diff_minus)+
```
```   918   apply (subst real_of_int_minus [THEN sym])
```
```   919   apply (rule ceiling_add)
```
```   920 done
```
```   921
```
```   922 lemma ceiling_subtract_number_of [simp]: "ceiling (x - number_of n) =
```
```   923     ceiling x - number_of n"
```
```   924   apply (subst ceiling_subtract [THEN sym])
```
```   925   apply simp
```
```   926 done
```
```   927
```
```   928 lemma ceiling_subtract_one [simp]: "ceiling (x - 1) = ceiling x - 1"
```
```   929   apply (subst ceiling_subtract [THEN sym])
```
```   930   apply simp
```
```   931 done
```
```   932
```
```   933 subsection {* Versions for the natural numbers *}
```
```   934
```
```   935 definition
```
```   936   natfloor :: "real => nat"
```
```   937   "natfloor x = nat(floor x)"
```
```   938   natceiling :: "real => nat"
```
```   939   "natceiling x = nat(ceiling x)"
```
```   940
```
```   941 lemma natfloor_zero [simp]: "natfloor 0 = 0"
```
```   942   by (unfold natfloor_def, simp)
```
```   943
```
```   944 lemma natfloor_one [simp]: "natfloor 1 = 1"
```
```   945   by (unfold natfloor_def, simp)
```
```   946
```
```   947 lemma zero_le_natfloor [simp]: "0 <= natfloor x"
```
```   948   by (unfold natfloor_def, simp)
```
```   949
```
```   950 lemma natfloor_number_of_eq [simp]: "natfloor (number_of n) = number_of n"
```
```   951   by (unfold natfloor_def, simp)
```
```   952
```
```   953 lemma natfloor_real_of_nat [simp]: "natfloor(real n) = n"
```
```   954   by (unfold natfloor_def, simp)
```
```   955
```
```   956 lemma real_natfloor_le: "0 <= x ==> real(natfloor x) <= x"
```
```   957   by (unfold natfloor_def, simp)
```
```   958
```
```   959 lemma natfloor_neg: "x <= 0 ==> natfloor x = 0"
```
```   960   apply (unfold natfloor_def)
```
```   961   apply (subgoal_tac "floor x <= floor 0")
```
```   962   apply simp
```
```   963   apply (erule floor_mono2)
```
```   964 done
```
```   965
```
```   966 lemma natfloor_mono: "x <= y ==> natfloor x <= natfloor y"
```
```   967   apply (case_tac "0 <= x")
```
```   968   apply (subst natfloor_def)+
```
```   969   apply (subst nat_le_eq_zle)
```
```   970   apply force
```
```   971   apply (erule floor_mono2)
```
```   972   apply (subst natfloor_neg)
```
```   973   apply simp
```
```   974   apply simp
```
```   975 done
```
```   976
```
```   977 lemma le_natfloor: "real x <= a ==> x <= natfloor a"
```
```   978   apply (unfold natfloor_def)
```
```   979   apply (subst nat_int [THEN sym])
```
```   980   apply (subst nat_le_eq_zle)
```
```   981   apply simp
```
```   982   apply (rule le_floor)
```
```   983   apply simp
```
```   984 done
```
```   985
```
```   986 lemma le_natfloor_eq: "0 <= x ==> (a <= natfloor x) = (real a <= x)"
```
```   987   apply (rule iffI)
```
```   988   apply (rule order_trans)
```
```   989   prefer 2
```
```   990   apply (erule real_natfloor_le)
```
```   991   apply (subst real_of_nat_le_iff)
```
```   992   apply assumption
```
```   993   apply (erule le_natfloor)
```
```   994 done
```
```   995
```
```   996 lemma le_natfloor_eq_number_of [simp]:
```
```   997     "~ neg((number_of n)::int) ==> 0 <= x ==>
```
```   998       (number_of n <= natfloor x) = (number_of n <= x)"
```
```   999   apply (subst le_natfloor_eq, assumption)
```
```  1000   apply simp
```
```  1001 done
```
```  1002
```
```  1003 lemma le_natfloor_eq_one [simp]: "(1 <= natfloor x) = (1 <= x)"
```
```  1004   apply (case_tac "0 <= x")
```
```  1005   apply (subst le_natfloor_eq, assumption, simp)
```
```  1006   apply (rule iffI)
```
```  1007   apply (subgoal_tac "natfloor x <= natfloor 0")
```
```  1008   apply simp
```
```  1009   apply (rule natfloor_mono)
```
```  1010   apply simp
```
```  1011   apply simp
```
```  1012 done
```
```  1013
```
```  1014 lemma natfloor_eq: "real n <= x ==> x < real n + 1 ==> natfloor x = n"
```
```  1015   apply (unfold natfloor_def)
```
```  1016   apply (subst nat_int [THEN sym]);back;
```
```  1017   apply (subst eq_nat_nat_iff)
```
```  1018   apply simp
```
```  1019   apply simp
```
```  1020   apply (rule floor_eq2)
```
```  1021   apply auto
```
```  1022 done
```
```  1023
```
```  1024 lemma real_natfloor_add_one_gt: "x < real(natfloor x) + 1"
```
```  1025   apply (case_tac "0 <= x")
```
```  1026   apply (unfold natfloor_def)
```
```  1027   apply simp
```
```  1028   apply simp_all
```
```  1029 done
```
```  1030
```
```  1031 lemma real_natfloor_gt_diff_one: "x - 1 < real(natfloor x)"
```
```  1032   apply (simp add: compare_rls)
```
```  1033   apply (rule real_natfloor_add_one_gt)
```
```  1034 done
```
```  1035
```
```  1036 lemma ge_natfloor_plus_one_imp_gt: "natfloor z + 1 <= n ==> z < real n"
```
```  1037   apply (subgoal_tac "z < real(natfloor z) + 1")
```
```  1038   apply arith
```
```  1039   apply (rule real_natfloor_add_one_gt)
```
```  1040 done
```
```  1041
```
```  1042 lemma natfloor_add [simp]: "0 <= x ==> natfloor (x + real a) = natfloor x + a"
```
```  1043   apply (unfold natfloor_def)
```
```  1044   apply (subgoal_tac "real a = real (int a)")
```
```  1045   apply (erule ssubst)
```
```  1046   apply (simp add: nat_add_distrib)
```
```  1047   apply simp
```
```  1048 done
```
```  1049
```
```  1050 lemma natfloor_add_number_of [simp]:
```
```  1051     "~neg ((number_of n)::int) ==> 0 <= x ==>
```
```  1052       natfloor (x + number_of n) = natfloor x + number_of n"
```
```  1053   apply (subst natfloor_add [THEN sym])
```
```  1054   apply simp_all
```
```  1055 done
```
```  1056
```
```  1057 lemma natfloor_add_one: "0 <= x ==> natfloor(x + 1) = natfloor x + 1"
```
```  1058   apply (subst natfloor_add [THEN sym])
```
```  1059   apply assumption
```
```  1060   apply simp
```
```  1061 done
```
```  1062
```
```  1063 lemma natfloor_subtract [simp]: "real a <= x ==>
```
```  1064     natfloor(x - real a) = natfloor x - a"
```
```  1065   apply (unfold natfloor_def)
```
```  1066   apply (subgoal_tac "real a = real (int a)")
```
```  1067   apply (erule ssubst)
```
```  1068   apply simp
```
```  1069   apply (subst nat_diff_distrib)
```
```  1070   apply simp
```
```  1071   apply (rule le_floor)
```
```  1072   apply simp_all
```
```  1073 done
```
```  1074
```
```  1075 lemma natceiling_zero [simp]: "natceiling 0 = 0"
```
```  1076   by (unfold natceiling_def, simp)
```
```  1077
```
```  1078 lemma natceiling_one [simp]: "natceiling 1 = 1"
```
```  1079   by (unfold natceiling_def, simp)
```
```  1080
```
```  1081 lemma zero_le_natceiling [simp]: "0 <= natceiling x"
```
```  1082   by (unfold natceiling_def, simp)
```
```  1083
```
```  1084 lemma natceiling_number_of_eq [simp]: "natceiling (number_of n) = number_of n"
```
```  1085   by (unfold natceiling_def, simp)
```
```  1086
```
```  1087 lemma natceiling_real_of_nat [simp]: "natceiling(real n) = n"
```
```  1088   by (unfold natceiling_def, simp)
```
```  1089
```
```  1090 lemma real_natceiling_ge: "x <= real(natceiling x)"
```
```  1091   apply (unfold natceiling_def)
```
```  1092   apply (case_tac "x < 0")
```
```  1093   apply simp
```
```  1094   apply (subst real_nat_eq_real)
```
```  1095   apply (subgoal_tac "ceiling 0 <= ceiling x")
```
```  1096   apply simp
```
```  1097   apply (rule ceiling_mono2)
```
```  1098   apply simp
```
```  1099   apply simp
```
```  1100 done
```
```  1101
```
```  1102 lemma natceiling_neg: "x <= 0 ==> natceiling x = 0"
```
```  1103   apply (unfold natceiling_def)
```
```  1104   apply simp
```
```  1105 done
```
```  1106
```
```  1107 lemma natceiling_mono: "x <= y ==> natceiling x <= natceiling y"
```
```  1108   apply (case_tac "0 <= x")
```
```  1109   apply (subst natceiling_def)+
```
```  1110   apply (subst nat_le_eq_zle)
```
```  1111   apply (rule disjI2)
```
```  1112   apply (subgoal_tac "real (0::int) <= real(ceiling y)")
```
```  1113   apply simp
```
```  1114   apply (rule order_trans)
```
```  1115   apply simp
```
```  1116   apply (erule order_trans)
```
```  1117   apply simp
```
```  1118   apply (erule ceiling_mono2)
```
```  1119   apply (subst natceiling_neg)
```
```  1120   apply simp_all
```
```  1121 done
```
```  1122
```
```  1123 lemma natceiling_le: "x <= real a ==> natceiling x <= a"
```
```  1124   apply (unfold natceiling_def)
```
```  1125   apply (case_tac "x < 0")
```
```  1126   apply simp
```
```  1127   apply (subst nat_int [THEN sym]);back;
```
```  1128   apply (subst nat_le_eq_zle)
```
```  1129   apply simp
```
```  1130   apply (rule ceiling_le)
```
```  1131   apply simp
```
```  1132 done
```
```  1133
```
```  1134 lemma natceiling_le_eq: "0 <= x ==> (natceiling x <= a) = (x <= real a)"
```
```  1135   apply (rule iffI)
```
```  1136   apply (rule order_trans)
```
```  1137   apply (rule real_natceiling_ge)
```
```  1138   apply (subst real_of_nat_le_iff)
```
```  1139   apply assumption
```
```  1140   apply (erule natceiling_le)
```
```  1141 done
```
```  1142
```
```  1143 lemma natceiling_le_eq_number_of [simp]:
```
```  1144     "~ neg((number_of n)::int) ==> 0 <= x ==>
```
```  1145       (natceiling x <= number_of n) = (x <= number_of n)"
```
```  1146   apply (subst natceiling_le_eq, assumption)
```
```  1147   apply simp
```
```  1148 done
```
```  1149
```
```  1150 lemma natceiling_le_eq_one: "(natceiling x <= 1) = (x <= 1)"
```
```  1151   apply (case_tac "0 <= x")
```
```  1152   apply (subst natceiling_le_eq)
```
```  1153   apply assumption
```
```  1154   apply simp
```
```  1155   apply (subst natceiling_neg)
```
```  1156   apply simp
```
```  1157   apply simp
```
```  1158 done
```
```  1159
```
```  1160 lemma natceiling_eq: "real n < x ==> x <= real n + 1 ==> natceiling x = n + 1"
```
```  1161   apply (unfold natceiling_def)
```
```  1162   apply (subst nat_int [THEN sym]);back;
```
```  1163   apply (subgoal_tac "nat(int n) + 1 = nat(int n + 1)")
```
```  1164   apply (erule ssubst)
```
```  1165   apply (subst eq_nat_nat_iff)
```
```  1166   apply (subgoal_tac "ceiling 0 <= ceiling x")
```
```  1167   apply simp
```
```  1168   apply (rule ceiling_mono2)
```
```  1169   apply force
```
```  1170   apply force
```
```  1171   apply (rule ceiling_eq2)
```
```  1172   apply (simp, simp)
```
```  1173   apply (subst nat_add_distrib)
```
```  1174   apply auto
```
```  1175 done
```
```  1176
```
```  1177 lemma natceiling_add [simp]: "0 <= x ==>
```
```  1178     natceiling (x + real a) = natceiling x + a"
```
```  1179   apply (unfold natceiling_def)
```
```  1180   apply (subgoal_tac "real a = real (int a)")
```
```  1181   apply (erule ssubst)
```
```  1182   apply simp
```
```  1183   apply (subst nat_add_distrib)
```
```  1184   apply (subgoal_tac "0 = ceiling 0")
```
```  1185   apply (erule ssubst)
```
```  1186   apply (erule ceiling_mono2)
```
```  1187   apply simp_all
```
```  1188 done
```
```  1189
```
```  1190 lemma natceiling_add_number_of [simp]:
```
```  1191     "~ neg ((number_of n)::int) ==> 0 <= x ==>
```
```  1192       natceiling (x + number_of n) = natceiling x + number_of n"
```
```  1193   apply (subst natceiling_add [THEN sym])
```
```  1194   apply simp_all
```
```  1195 done
```
```  1196
```
```  1197 lemma natceiling_add_one: "0 <= x ==> natceiling(x + 1) = natceiling x + 1"
```
```  1198   apply (subst natceiling_add [THEN sym])
```
```  1199   apply assumption
```
```  1200   apply simp
```
```  1201 done
```
```  1202
```
```  1203 lemma natceiling_subtract [simp]: "real a <= x ==>
```
```  1204     natceiling(x - real a) = natceiling x - a"
```
```  1205   apply (unfold natceiling_def)
```
```  1206   apply (subgoal_tac "real a = real (int a)")
```
```  1207   apply (erule ssubst)
```
```  1208   apply simp
```
```  1209   apply (subst nat_diff_distrib)
```
```  1210   apply simp
```
```  1211   apply (rule order_trans)
```
```  1212   prefer 2
```
```  1213   apply (rule ceiling_mono2)
```
```  1214   apply assumption
```
```  1215   apply simp_all
```
```  1216 done
```
```  1217
```
```  1218 lemma natfloor_div_nat: "1 <= x ==> 0 < y ==>
```
```  1219   natfloor (x / real y) = natfloor x div y"
```
```  1220 proof -
```
```  1221   assume "1 <= (x::real)" and "0 < (y::nat)"
```
```  1222   have "natfloor x = (natfloor x) div y * y + (natfloor x) mod y"
```
```  1223     by simp
```
```  1224   then have a: "real(natfloor x) = real ((natfloor x) div y) * real y +
```
```  1225     real((natfloor x) mod y)"
```
```  1226     by (simp only: real_of_nat_add [THEN sym] real_of_nat_mult [THEN sym])
```
```  1227   have "x = real(natfloor x) + (x - real(natfloor x))"
```
```  1228     by simp
```
```  1229   then have "x = real ((natfloor x) div y) * real y +
```
```  1230       real((natfloor x) mod y) + (x - real(natfloor x))"
```
```  1231     by (simp add: a)
```
```  1232   then have "x / real y = ... / real y"
```
```  1233     by simp
```
```  1234   also have "... = real((natfloor x) div y) + real((natfloor x) mod y) /
```
```  1235     real y + (x - real(natfloor x)) / real y"
```
```  1236     by (auto simp add: ring_distrib ring_eq_simps add_divide_distrib
```
```  1237       diff_divide_distrib prems)
```
```  1238   finally have "natfloor (x / real y) = natfloor(...)" by simp
```
```  1239   also have "... = natfloor(real((natfloor x) mod y) /
```
```  1240     real y + (x - real(natfloor x)) / real y + real((natfloor x) div y))"
```
```  1241     by (simp add: add_ac)
```
```  1242   also have "... = natfloor(real((natfloor x) mod y) /
```
```  1243     real y + (x - real(natfloor x)) / real y) + (natfloor x) div y"
```
```  1244     apply (rule natfloor_add)
```
```  1245     apply (rule add_nonneg_nonneg)
```
```  1246     apply (rule divide_nonneg_pos)
```
```  1247     apply simp
```
```  1248     apply (simp add: prems)
```
```  1249     apply (rule divide_nonneg_pos)
```
```  1250     apply (simp add: compare_rls)
```
```  1251     apply (rule real_natfloor_le)
```
```  1252     apply (insert prems, auto)
```
```  1253     done
```
```  1254   also have "natfloor(real((natfloor x) mod y) /
```
```  1255     real y + (x - real(natfloor x)) / real y) = 0"
```
```  1256     apply (rule natfloor_eq)
```
```  1257     apply simp
```
```  1258     apply (rule add_nonneg_nonneg)
```
```  1259     apply (rule divide_nonneg_pos)
```
```  1260     apply force
```
```  1261     apply (force simp add: prems)
```
```  1262     apply (rule divide_nonneg_pos)
```
```  1263     apply (simp add: compare_rls)
```
```  1264     apply (rule real_natfloor_le)
```
```  1265     apply (auto simp add: prems)
```
```  1266     apply (insert prems, arith)
```
```  1267     apply (simp add: add_divide_distrib [THEN sym])
```
```  1268     apply (subgoal_tac "real y = real y - 1 + 1")
```
```  1269     apply (erule ssubst)
```
```  1270     apply (rule add_le_less_mono)
```
```  1271     apply (simp add: compare_rls)
```
```  1272     apply (subgoal_tac "real(natfloor x mod y) + 1 =
```
```  1273       real(natfloor x mod y + 1)")
```
```  1274     apply (erule ssubst)
```
```  1275     apply (subst real_of_nat_le_iff)
```
```  1276     apply (subgoal_tac "natfloor x mod y < y")
```
```  1277     apply arith
```
```  1278     apply (rule mod_less_divisor)
```
```  1279     apply assumption
```
```  1280     apply auto
```
```  1281     apply (simp add: compare_rls)
```
```  1282     apply (subst add_commute)
```
```  1283     apply (rule real_natfloor_add_one_gt)
```
```  1284     done
```
```  1285   finally show ?thesis
```
```  1286     by simp
```
```  1287 qed
```
```  1288
```
```  1289 end
```