src/HOL/Real/RealDef.thy
 author wenzelm Wed Sep 17 21:27:14 2008 +0200 (2008-09-17) changeset 28263 69eaa97e7e96 parent 28091 50f2d6ba024c child 28351 abfc66969d1f permissions -rw-r--r--
moved global ML bindings to global place;
```     1 (*  Title       : Real/RealDef.thy
```
```     2     ID          : \$Id\$
```
```     3     Author      : Jacques D. Fleuriot
```
```     4     Copyright   : 1998  University of Cambridge
```
```     5     Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
```
```     6     Additional contributions by Jeremy Avigad
```
```     7 *)
```
```     8
```
```     9 header{*Defining the Reals from the Positive Reals*}
```
```    10
```
```    11 theory RealDef
```
```    12 imports PReal
```
```    13 uses ("real_arith.ML")
```
```    14 begin
```
```    15
```
```    16 definition
```
```    17   realrel   ::  "((preal * preal) * (preal * preal)) set" where
```
```    18   [code func del]: "realrel = {p. \<exists>x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}"
```
```    19
```
```    20 typedef (Real)  real = "UNIV//realrel"
```
```    21   by (auto simp add: quotient_def)
```
```    22
```
```    23 definition
```
```    24   (** these don't use the overloaded "real" function: users don't see them **)
```
```    25   real_of_preal :: "preal => real" where
```
```    26   [code func del]: "real_of_preal m = Abs_Real (realrel `` {(m + 1, 1)})"
```
```    27
```
```    28 instantiation real :: "{zero, one, plus, minus, uminus, times, inverse, ord, abs, sgn}"
```
```    29 begin
```
```    30
```
```    31 definition
```
```    32   real_zero_def [code func del]: "0 = Abs_Real(realrel``{(1, 1)})"
```
```    33
```
```    34 definition
```
```    35   real_one_def [code func del]: "1 = Abs_Real(realrel``{(1 + 1, 1)})"
```
```    36
```
```    37 definition
```
```    38   real_add_def [code func del]: "z + w =
```
```    39        contents (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w).
```
```    40                  { Abs_Real(realrel``{(x+u, y+v)}) })"
```
```    41
```
```    42 definition
```
```    43   real_minus_def [code func del]: "- r =  contents (\<Union>(x,y) \<in> Rep_Real(r). { Abs_Real(realrel``{(y,x)}) })"
```
```    44
```
```    45 definition
```
```    46   real_diff_def [code func del]: "r - (s::real) = r + - s"
```
```    47
```
```    48 definition
```
```    49   real_mult_def [code func del]:
```
```    50     "z * w =
```
```    51        contents (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w).
```
```    52                  { Abs_Real(realrel``{(x*u + y*v, x*v + y*u)}) })"
```
```    53
```
```    54 definition
```
```    55   real_inverse_def [code func del]: "inverse (R::real) = (THE S. (R = 0 & S = 0) | S * R = 1)"
```
```    56
```
```    57 definition
```
```    58   real_divide_def [code func del]: "R / (S::real) = R * inverse S"
```
```    59
```
```    60 definition
```
```    61   real_le_def [code func del]: "z \<le> (w::real) \<longleftrightarrow>
```
```    62     (\<exists>x y u v. x+v \<le> u+y & (x,y) \<in> Rep_Real z & (u,v) \<in> Rep_Real w)"
```
```    63
```
```    64 definition
```
```    65   real_less_def [code func del]: "x < (y\<Colon>real) \<longleftrightarrow> x \<le> y \<and> x \<noteq> y"
```
```    66
```
```    67 definition
```
```    68   real_abs_def:  "abs (r::real) = (if r < 0 then - r else r)"
```
```    69
```
```    70 definition
```
```    71   real_sgn_def: "sgn (x::real) = (if x=0 then 0 else if 0<x then 1 else - 1)"
```
```    72
```
```    73 instance ..
```
```    74
```
```    75 end
```
```    76
```
```    77 subsection {* Equivalence relation over positive reals *}
```
```    78
```
```    79 lemma preal_trans_lemma:
```
```    80   assumes "x + y1 = x1 + y"
```
```    81       and "x + y2 = x2 + y"
```
```    82   shows "x1 + y2 = x2 + (y1::preal)"
```
```    83 proof -
```
```    84   have "(x1 + y2) + x = (x + y2) + x1" by (simp add: add_ac)
```
```    85   also have "... = (x2 + y) + x1"  by (simp add: prems)
```
```    86   also have "... = x2 + (x1 + y)"  by (simp add: add_ac)
```
```    87   also have "... = x2 + (x + y1)"  by (simp add: prems)
```
```    88   also have "... = (x2 + y1) + x"  by (simp add: add_ac)
```
```    89   finally have "(x1 + y2) + x = (x2 + y1) + x" .
```
```    90   thus ?thesis by (rule add_right_imp_eq)
```
```    91 qed
```
```    92
```
```    93
```
```    94 lemma realrel_iff [simp]: "(((x1,y1),(x2,y2)) \<in> realrel) = (x1 + y2 = x2 + y1)"
```
```    95 by (simp add: realrel_def)
```
```    96
```
```    97 lemma equiv_realrel: "equiv UNIV realrel"
```
```    98 apply (auto simp add: equiv_def refl_def sym_def trans_def realrel_def)
```
```    99 apply (blast dest: preal_trans_lemma)
```
```   100 done
```
```   101
```
```   102 text{*Reduces equality of equivalence classes to the @{term realrel} relation:
```
```   103   @{term "(realrel `` {x} = realrel `` {y}) = ((x,y) \<in> realrel)"} *}
```
```   104 lemmas equiv_realrel_iff =
```
```   105        eq_equiv_class_iff [OF equiv_realrel UNIV_I UNIV_I]
```
```   106
```
```   107 declare equiv_realrel_iff [simp]
```
```   108
```
```   109
```
```   110 lemma realrel_in_real [simp]: "realrel``{(x,y)}: Real"
```
```   111 by (simp add: Real_def realrel_def quotient_def, blast)
```
```   112
```
```   113 declare Abs_Real_inject [simp]
```
```   114 declare Abs_Real_inverse [simp]
```
```   115
```
```   116
```
```   117 text{*Case analysis on the representation of a real number as an equivalence
```
```   118       class of pairs of positive reals.*}
```
```   119 lemma eq_Abs_Real [case_names Abs_Real, cases type: real]:
```
```   120      "(!!x y. z = Abs_Real(realrel``{(x,y)}) ==> P) ==> P"
```
```   121 apply (rule Rep_Real [of z, unfolded Real_def, THEN quotientE])
```
```   122 apply (drule arg_cong [where f=Abs_Real])
```
```   123 apply (auto simp add: Rep_Real_inverse)
```
```   124 done
```
```   125
```
```   126
```
```   127 subsection {* Addition and Subtraction *}
```
```   128
```
```   129 lemma real_add_congruent2_lemma:
```
```   130      "[|a + ba = aa + b; ab + bc = ac + bb|]
```
```   131       ==> a + ab + (ba + bc) = aa + ac + (b + (bb::preal))"
```
```   132 apply (simp add: add_assoc)
```
```   133 apply (rule add_left_commute [of ab, THEN ssubst])
```
```   134 apply (simp add: add_assoc [symmetric])
```
```   135 apply (simp add: add_ac)
```
```   136 done
```
```   137
```
```   138 lemma real_add:
```
```   139      "Abs_Real (realrel``{(x,y)}) + Abs_Real (realrel``{(u,v)}) =
```
```   140       Abs_Real (realrel``{(x+u, y+v)})"
```
```   141 proof -
```
```   142   have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). {Abs_Real (realrel `` {(x+u, y+v)})}) w) z)
```
```   143         respects2 realrel"
```
```   144     by (simp add: congruent2_def, blast intro: real_add_congruent2_lemma)
```
```   145   thus ?thesis
```
```   146     by (simp add: real_add_def UN_UN_split_split_eq
```
```   147                   UN_equiv_class2 [OF equiv_realrel equiv_realrel])
```
```   148 qed
```
```   149
```
```   150 lemma real_minus: "- Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})"
```
```   151 proof -
```
```   152   have "(\<lambda>(x,y). {Abs_Real (realrel``{(y,x)})}) respects realrel"
```
```   153     by (simp add: congruent_def add_commute)
```
```   154   thus ?thesis
```
```   155     by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel])
```
```   156 qed
```
```   157
```
```   158 instance real :: ab_group_add
```
```   159 proof
```
```   160   fix x y z :: real
```
```   161   show "(x + y) + z = x + (y + z)"
```
```   162     by (cases x, cases y, cases z, simp add: real_add add_assoc)
```
```   163   show "x + y = y + x"
```
```   164     by (cases x, cases y, simp add: real_add add_commute)
```
```   165   show "0 + x = x"
```
```   166     by (cases x, simp add: real_add real_zero_def add_ac)
```
```   167   show "- x + x = 0"
```
```   168     by (cases x, simp add: real_minus real_add real_zero_def add_commute)
```
```   169   show "x - y = x + - y"
```
```   170     by (simp add: real_diff_def)
```
```   171 qed
```
```   172
```
```   173
```
```   174 subsection {* Multiplication *}
```
```   175
```
```   176 lemma real_mult_congruent2_lemma:
```
```   177      "!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==>
```
```   178           x * x1 + y * y1 + (x * y2 + y * x2) =
```
```   179           x * x2 + y * y2 + (x * y1 + y * x1)"
```
```   180 apply (simp add: add_left_commute add_assoc [symmetric])
```
```   181 apply (simp add: add_assoc right_distrib [symmetric])
```
```   182 apply (simp add: add_commute)
```
```   183 done
```
```   184
```
```   185 lemma real_mult_congruent2:
```
```   186     "(%p1 p2.
```
```   187         (%(x1,y1). (%(x2,y2).
```
```   188           { Abs_Real (realrel``{(x1*x2 + y1*y2, x1*y2+y1*x2)}) }) p2) p1)
```
```   189      respects2 realrel"
```
```   190 apply (rule congruent2_commuteI [OF equiv_realrel], clarify)
```
```   191 apply (simp add: mult_commute add_commute)
```
```   192 apply (auto simp add: real_mult_congruent2_lemma)
```
```   193 done
```
```   194
```
```   195 lemma real_mult:
```
```   196       "Abs_Real((realrel``{(x1,y1)})) * Abs_Real((realrel``{(x2,y2)})) =
```
```   197        Abs_Real(realrel `` {(x1*x2+y1*y2,x1*y2+y1*x2)})"
```
```   198 by (simp add: real_mult_def UN_UN_split_split_eq
```
```   199          UN_equiv_class2 [OF equiv_realrel equiv_realrel real_mult_congruent2])
```
```   200
```
```   201 lemma real_mult_commute: "(z::real) * w = w * z"
```
```   202 by (cases z, cases w, simp add: real_mult add_ac mult_ac)
```
```   203
```
```   204 lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)"
```
```   205 apply (cases z1, cases z2, cases z3)
```
```   206 apply (simp add: real_mult right_distrib add_ac mult_ac)
```
```   207 done
```
```   208
```
```   209 lemma real_mult_1: "(1::real) * z = z"
```
```   210 apply (cases z)
```
```   211 apply (simp add: real_mult real_one_def right_distrib
```
```   212                   mult_1_right mult_ac add_ac)
```
```   213 done
```
```   214
```
```   215 lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)"
```
```   216 apply (cases z1, cases z2, cases w)
```
```   217 apply (simp add: real_add real_mult right_distrib add_ac mult_ac)
```
```   218 done
```
```   219
```
```   220 text{*one and zero are distinct*}
```
```   221 lemma real_zero_not_eq_one: "0 \<noteq> (1::real)"
```
```   222 proof -
```
```   223   have "(1::preal) < 1 + 1"
```
```   224     by (simp add: preal_self_less_add_left)
```
```   225   thus ?thesis
```
```   226     by (simp add: real_zero_def real_one_def)
```
```   227 qed
```
```   228
```
```   229 instance real :: comm_ring_1
```
```   230 proof
```
```   231   fix x y z :: real
```
```   232   show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc)
```
```   233   show "x * y = y * x" by (rule real_mult_commute)
```
```   234   show "1 * x = x" by (rule real_mult_1)
```
```   235   show "(x + y) * z = x * z + y * z" by (rule real_add_mult_distrib)
```
```   236   show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one)
```
```   237 qed
```
```   238
```
```   239 subsection {* Inverse and Division *}
```
```   240
```
```   241 lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0"
```
```   242 by (simp add: real_zero_def add_commute)
```
```   243
```
```   244 text{*Instead of using an existential quantifier and constructing the inverse
```
```   245 within the proof, we could define the inverse explicitly.*}
```
```   246
```
```   247 lemma real_mult_inverse_left_ex: "x \<noteq> 0 ==> \<exists>y. y*x = (1::real)"
```
```   248 apply (simp add: real_zero_def real_one_def, cases x)
```
```   249 apply (cut_tac x = xa and y = y in linorder_less_linear)
```
```   250 apply (auto dest!: less_add_left_Ex simp add: real_zero_iff)
```
```   251 apply (rule_tac
```
```   252         x = "Abs_Real (realrel``{(1, inverse (D) + 1)})"
```
```   253        in exI)
```
```   254 apply (rule_tac [2]
```
```   255         x = "Abs_Real (realrel``{(inverse (D) + 1, 1)})"
```
```   256        in exI)
```
```   257 apply (auto simp add: real_mult preal_mult_inverse_right ring_simps)
```
```   258 done
```
```   259
```
```   260 lemma real_mult_inverse_left: "x \<noteq> 0 ==> inverse(x)*x = (1::real)"
```
```   261 apply (simp add: real_inverse_def)
```
```   262 apply (drule real_mult_inverse_left_ex, safe)
```
```   263 apply (rule theI, assumption, rename_tac z)
```
```   264 apply (subgoal_tac "(z * x) * y = z * (x * y)")
```
```   265 apply (simp add: mult_commute)
```
```   266 apply (rule mult_assoc)
```
```   267 done
```
```   268
```
```   269
```
```   270 subsection{*The Real Numbers form a Field*}
```
```   271
```
```   272 instance real :: field
```
```   273 proof
```
```   274   fix x y z :: real
```
```   275   show "x \<noteq> 0 ==> inverse x * x = 1" by (rule real_mult_inverse_left)
```
```   276   show "x / y = x * inverse y" by (simp add: real_divide_def)
```
```   277 qed
```
```   278
```
```   279
```
```   280 text{*Inverse of zero!  Useful to simplify certain equations*}
```
```   281
```
```   282 lemma INVERSE_ZERO: "inverse 0 = (0::real)"
```
```   283 by (simp add: real_inverse_def)
```
```   284
```
```   285 instance real :: division_by_zero
```
```   286 proof
```
```   287   show "inverse 0 = (0::real)" by (rule INVERSE_ZERO)
```
```   288 qed
```
```   289
```
```   290
```
```   291 subsection{*The @{text "\<le>"} Ordering*}
```
```   292
```
```   293 lemma real_le_refl: "w \<le> (w::real)"
```
```   294 by (cases w, force simp add: real_le_def)
```
```   295
```
```   296 text{*The arithmetic decision procedure is not set up for type preal.
```
```   297   This lemma is currently unused, but it could simplify the proofs of the
```
```   298   following two lemmas.*}
```
```   299 lemma preal_eq_le_imp_le:
```
```   300   assumes eq: "a+b = c+d" and le: "c \<le> a"
```
```   301   shows "b \<le> (d::preal)"
```
```   302 proof -
```
```   303   have "c+d \<le> a+d" by (simp add: prems)
```
```   304   hence "a+b \<le> a+d" by (simp add: prems)
```
```   305   thus "b \<le> d" by simp
```
```   306 qed
```
```   307
```
```   308 lemma real_le_lemma:
```
```   309   assumes l: "u1 + v2 \<le> u2 + v1"
```
```   310       and "x1 + v1 = u1 + y1"
```
```   311       and "x2 + v2 = u2 + y2"
```
```   312   shows "x1 + y2 \<le> x2 + (y1::preal)"
```
```   313 proof -
```
```   314   have "(x1+v1) + (u2+y2) = (u1+y1) + (x2+v2)" by (simp add: prems)
```
```   315   hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: add_ac)
```
```   316   also have "... \<le> (x2+y1) + (u2+v1)" by (simp add: prems)
```
```   317   finally show ?thesis by simp
```
```   318 qed
```
```   319
```
```   320 lemma real_le:
```
```   321      "(Abs_Real(realrel``{(x1,y1)}) \<le> Abs_Real(realrel``{(x2,y2)})) =
```
```   322       (x1 + y2 \<le> x2 + y1)"
```
```   323 apply (simp add: real_le_def)
```
```   324 apply (auto intro: real_le_lemma)
```
```   325 done
```
```   326
```
```   327 lemma real_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::real)"
```
```   328 by (cases z, cases w, simp add: real_le)
```
```   329
```
```   330 lemma real_trans_lemma:
```
```   331   assumes "x + v \<le> u + y"
```
```   332       and "u + v' \<le> u' + v"
```
```   333       and "x2 + v2 = u2 + y2"
```
```   334   shows "x + v' \<le> u' + (y::preal)"
```
```   335 proof -
```
```   336   have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: add_ac)
```
```   337   also have "... \<le> (u+y) + (u+v')" by (simp add: prems)
```
```   338   also have "... \<le> (u+y) + (u'+v)" by (simp add: prems)
```
```   339   also have "... = (u'+y) + (u+v)"  by (simp add: add_ac)
```
```   340   finally show ?thesis by simp
```
```   341 qed
```
```   342
```
```   343 lemma real_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::real)"
```
```   344 apply (cases i, cases j, cases k)
```
```   345 apply (simp add: real_le)
```
```   346 apply (blast intro: real_trans_lemma)
```
```   347 done
```
```   348
```
```   349 instance real :: order
```
```   350 proof
```
```   351   fix u v :: real
```
```   352   show "u < v \<longleftrightarrow> u \<le> v \<and> \<not> v \<le> u"
```
```   353     by (auto simp add: real_less_def intro: real_le_anti_sym)
```
```   354 qed (assumption | rule real_le_refl real_le_trans real_le_anti_sym)+
```
```   355
```
```   356 (* Axiom 'linorder_linear' of class 'linorder': *)
```
```   357 lemma real_le_linear: "(z::real) \<le> w | w \<le> z"
```
```   358 apply (cases z, cases w)
```
```   359 apply (auto simp add: real_le real_zero_def add_ac)
```
```   360 done
```
```   361
```
```   362 instance real :: linorder
```
```   363   by (intro_classes, rule real_le_linear)
```
```   364
```
```   365
```
```   366 lemma real_le_eq_diff: "(x \<le> y) = (x-y \<le> (0::real))"
```
```   367 apply (cases x, cases y)
```
```   368 apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus
```
```   369                       add_ac)
```
```   370 apply (simp_all add: add_assoc [symmetric])
```
```   371 done
```
```   372
```
```   373 lemma real_add_left_mono:
```
```   374   assumes le: "x \<le> y" shows "z + x \<le> z + (y::real)"
```
```   375 proof -
```
```   376   have "z + x - (z + y) = (z + -z) + (x - y)"
```
```   377     by (simp add: diff_minus add_ac)
```
```   378   with le show ?thesis
```
```   379     by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"] diff_minus)
```
```   380 qed
```
```   381
```
```   382 lemma real_sum_gt_zero_less: "(0 < S + (-W::real)) ==> (W < S)"
```
```   383 by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
```
```   384
```
```   385 lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (-W::real))"
```
```   386 by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
```
```   387
```
```   388 lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y"
```
```   389 apply (cases x, cases y)
```
```   390 apply (simp add: linorder_not_le [where 'a = real, symmetric]
```
```   391                  linorder_not_le [where 'a = preal]
```
```   392                   real_zero_def real_le real_mult)
```
```   393   --{*Reduce to the (simpler) @{text "\<le>"} relation *}
```
```   394 apply (auto dest!: less_add_left_Ex
```
```   395      simp add: add_ac mult_ac
```
```   396           right_distrib preal_self_less_add_left)
```
```   397 done
```
```   398
```
```   399 lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y"
```
```   400 apply (rule real_sum_gt_zero_less)
```
```   401 apply (drule real_less_sum_gt_zero [of x y])
```
```   402 apply (drule real_mult_order, assumption)
```
```   403 apply (simp add: right_distrib)
```
```   404 done
```
```   405
```
```   406 instantiation real :: distrib_lattice
```
```   407 begin
```
```   408
```
```   409 definition
```
```   410   "(inf \<Colon> real \<Rightarrow> real \<Rightarrow> real) = min"
```
```   411
```
```   412 definition
```
```   413   "(sup \<Colon> real \<Rightarrow> real \<Rightarrow> real) = max"
```
```   414
```
```   415 instance
```
```   416   by default (auto simp add: inf_real_def sup_real_def min_max.sup_inf_distrib1)
```
```   417
```
```   418 end
```
```   419
```
```   420
```
```   421 subsection{*The Reals Form an Ordered Field*}
```
```   422
```
```   423 instance real :: ordered_field
```
```   424 proof
```
```   425   fix x y z :: real
```
```   426   show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono)
```
```   427   show "x < y ==> 0 < z ==> z * x < z * y" by (rule real_mult_less_mono2)
```
```   428   show "\<bar>x\<bar> = (if x < 0 then -x else x)" by (simp only: real_abs_def)
```
```   429   show "sgn x = (if x=0 then 0 else if 0<x then 1 else - 1)"
```
```   430     by (simp only: real_sgn_def)
```
```   431 qed
```
```   432
```
```   433 instance real :: lordered_ab_group_add ..
```
```   434
```
```   435 text{*The function @{term real_of_preal} requires many proofs, but it seems
```
```   436 to be essential for proving completeness of the reals from that of the
```
```   437 positive reals.*}
```
```   438
```
```   439 lemma real_of_preal_add:
```
```   440      "real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y"
```
```   441 by (simp add: real_of_preal_def real_add left_distrib add_ac)
```
```   442
```
```   443 lemma real_of_preal_mult:
```
```   444      "real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y"
```
```   445 by (simp add: real_of_preal_def real_mult right_distrib add_ac mult_ac)
```
```   446
```
```   447
```
```   448 text{*Gleason prop 9-4.4 p 127*}
```
```   449 lemma real_of_preal_trichotomy:
```
```   450       "\<exists>m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)"
```
```   451 apply (simp add: real_of_preal_def real_zero_def, cases x)
```
```   452 apply (auto simp add: real_minus add_ac)
```
```   453 apply (cut_tac x = x and y = y in linorder_less_linear)
```
```   454 apply (auto dest!: less_add_left_Ex simp add: add_assoc [symmetric])
```
```   455 done
```
```   456
```
```   457 lemma real_of_preal_leD:
```
```   458       "real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2"
```
```   459 by (simp add: real_of_preal_def real_le)
```
```   460
```
```   461 lemma real_of_preal_lessI: "m1 < m2 ==> real_of_preal m1 < real_of_preal m2"
```
```   462 by (auto simp add: real_of_preal_leD linorder_not_le [symmetric])
```
```   463
```
```   464 lemma real_of_preal_lessD:
```
```   465       "real_of_preal m1 < real_of_preal m2 ==> m1 < m2"
```
```   466 by (simp add: real_of_preal_def real_le linorder_not_le [symmetric])
```
```   467
```
```   468 lemma real_of_preal_less_iff [simp]:
```
```   469      "(real_of_preal m1 < real_of_preal m2) = (m1 < m2)"
```
```   470 by (blast intro: real_of_preal_lessI real_of_preal_lessD)
```
```   471
```
```   472 lemma real_of_preal_le_iff:
```
```   473      "(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)"
```
```   474 by (simp add: linorder_not_less [symmetric])
```
```   475
```
```   476 lemma real_of_preal_zero_less: "0 < real_of_preal m"
```
```   477 apply (insert preal_self_less_add_left [of 1 m])
```
```   478 apply (auto simp add: real_zero_def real_of_preal_def
```
```   479                       real_less_def real_le_def add_ac)
```
```   480 apply (rule_tac x="m + 1" in exI, rule_tac x="1" in exI)
```
```   481 apply (simp add: add_ac)
```
```   482 done
```
```   483
```
```   484 lemma real_of_preal_minus_less_zero: "- real_of_preal m < 0"
```
```   485 by (simp add: real_of_preal_zero_less)
```
```   486
```
```   487 lemma real_of_preal_not_minus_gt_zero: "~ 0 < - real_of_preal m"
```
```   488 proof -
```
```   489   from real_of_preal_minus_less_zero
```
```   490   show ?thesis by (blast dest: order_less_trans)
```
```   491 qed
```
```   492
```
```   493
```
```   494 subsection{*Theorems About the Ordering*}
```
```   495
```
```   496 lemma real_gt_zero_preal_Ex: "(0 < x) = (\<exists>y. x = real_of_preal y)"
```
```   497 apply (auto simp add: real_of_preal_zero_less)
```
```   498 apply (cut_tac x = x in real_of_preal_trichotomy)
```
```   499 apply (blast elim!: real_of_preal_not_minus_gt_zero [THEN notE])
```
```   500 done
```
```   501
```
```   502 lemma real_gt_preal_preal_Ex:
```
```   503      "real_of_preal z < x ==> \<exists>y. x = real_of_preal y"
```
```   504 by (blast dest!: real_of_preal_zero_less [THEN order_less_trans]
```
```   505              intro: real_gt_zero_preal_Ex [THEN iffD1])
```
```   506
```
```   507 lemma real_ge_preal_preal_Ex:
```
```   508      "real_of_preal z \<le> x ==> \<exists>y. x = real_of_preal y"
```
```   509 by (blast dest: order_le_imp_less_or_eq real_gt_preal_preal_Ex)
```
```   510
```
```   511 lemma real_less_all_preal: "y \<le> 0 ==> \<forall>x. y < real_of_preal x"
```
```   512 by (auto elim: order_le_imp_less_or_eq [THEN disjE]
```
```   513             intro: real_of_preal_zero_less [THEN [2] order_less_trans]
```
```   514             simp add: real_of_preal_zero_less)
```
```   515
```
```   516 lemma real_less_all_real2: "~ 0 < y ==> \<forall>x. y < real_of_preal x"
```
```   517 by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1])
```
```   518
```
```   519
```
```   520 subsection{*More Lemmas*}
```
```   521
```
```   522 lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
```
```   523 by auto
```
```   524
```
```   525 lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)"
```
```   526 by auto
```
```   527
```
```   528 lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)"
```
```   529   by (force elim: order_less_asym
```
```   530             simp add: Ring_and_Field.mult_less_cancel_right)
```
```   531
```
```   532 lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)"
```
```   533 apply (simp add: mult_le_cancel_right)
```
```   534 apply (blast intro: elim: order_less_asym)
```
```   535 done
```
```   536
```
```   537 lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)"
```
```   538 by(simp add:mult_commute)
```
```   539
```
```   540 lemma real_inverse_gt_one: "[| (0::real) < x; x < 1 |] ==> 1 < inverse x"
```
```   541 by (simp add: one_less_inverse_iff) (* TODO: generalize/move *)
```
```   542
```
```   543
```
```   544 subsection {* Embedding numbers into the Reals *}
```
```   545
```
```   546 abbreviation
```
```   547   real_of_nat :: "nat \<Rightarrow> real"
```
```   548 where
```
```   549   "real_of_nat \<equiv> of_nat"
```
```   550
```
```   551 abbreviation
```
```   552   real_of_int :: "int \<Rightarrow> real"
```
```   553 where
```
```   554   "real_of_int \<equiv> of_int"
```
```   555
```
```   556 abbreviation
```
```   557   real_of_rat :: "rat \<Rightarrow> real"
```
```   558 where
```
```   559   "real_of_rat \<equiv> of_rat"
```
```   560
```
```   561 consts
```
```   562   (*overloaded constant for injecting other types into "real"*)
```
```   563   real :: "'a => real"
```
```   564
```
```   565 defs (overloaded)
```
```   566   real_of_nat_def [code inline]: "real == real_of_nat"
```
```   567   real_of_int_def [code inline]: "real == real_of_int"
```
```   568
```
```   569 lemma real_eq_of_nat: "real = of_nat"
```
```   570   unfolding real_of_nat_def ..
```
```   571
```
```   572 lemma real_eq_of_int: "real = of_int"
```
```   573   unfolding real_of_int_def ..
```
```   574
```
```   575 lemma real_of_int_zero [simp]: "real (0::int) = 0"
```
```   576 by (simp add: real_of_int_def)
```
```   577
```
```   578 lemma real_of_one [simp]: "real (1::int) = (1::real)"
```
```   579 by (simp add: real_of_int_def)
```
```   580
```
```   581 lemma real_of_int_add [simp]: "real(x + y) = real (x::int) + real y"
```
```   582 by (simp add: real_of_int_def)
```
```   583
```
```   584 lemma real_of_int_minus [simp]: "real(-x) = -real (x::int)"
```
```   585 by (simp add: real_of_int_def)
```
```   586
```
```   587 lemma real_of_int_diff [simp]: "real(x - y) = real (x::int) - real y"
```
```   588 by (simp add: real_of_int_def)
```
```   589
```
```   590 lemma real_of_int_mult [simp]: "real(x * y) = real (x::int) * real y"
```
```   591 by (simp add: real_of_int_def)
```
```   592
```
```   593 lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))"
```
```   594   apply (subst real_eq_of_int)+
```
```   595   apply (rule of_int_setsum)
```
```   596 done
```
```   597
```
```   598 lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) =
```
```   599     (PROD x:A. real(f x))"
```
```   600   apply (subst real_eq_of_int)+
```
```   601   apply (rule of_int_setprod)
```
```   602 done
```
```   603
```
```   604 lemma real_of_int_zero_cancel [simp, algebra, presburger]: "(real x = 0) = (x = (0::int))"
```
```   605 by (simp add: real_of_int_def)
```
```   606
```
```   607 lemma real_of_int_inject [iff, algebra, presburger]: "(real (x::int) = real y) = (x = y)"
```
```   608 by (simp add: real_of_int_def)
```
```   609
```
```   610 lemma real_of_int_less_iff [iff, presburger]: "(real (x::int) < real y) = (x < y)"
```
```   611 by (simp add: real_of_int_def)
```
```   612
```
```   613 lemma real_of_int_le_iff [simp, presburger]: "(real (x::int) \<le> real y) = (x \<le> y)"
```
```   614 by (simp add: real_of_int_def)
```
```   615
```
```   616 lemma real_of_int_gt_zero_cancel_iff [simp, presburger]: "(0 < real (n::int)) = (0 < n)"
```
```   617 by (simp add: real_of_int_def)
```
```   618
```
```   619 lemma real_of_int_ge_zero_cancel_iff [simp, presburger]: "(0 <= real (n::int)) = (0 <= n)"
```
```   620 by (simp add: real_of_int_def)
```
```   621
```
```   622 lemma real_of_int_lt_zero_cancel_iff [simp, presburger]: "(real (n::int) < 0) = (n < 0)"
```
```   623 by (simp add: real_of_int_def)
```
```   624
```
```   625 lemma real_of_int_le_zero_cancel_iff [simp, presburger]: "(real (n::int) <= 0) = (n <= 0)"
```
```   626 by (simp add: real_of_int_def)
```
```   627
```
```   628 lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))"
```
```   629 by (auto simp add: abs_if)
```
```   630
```
```   631 lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)"
```
```   632   apply (subgoal_tac "real n + 1 = real (n + 1)")
```
```   633   apply (simp del: real_of_int_add)
```
```   634   apply auto
```
```   635 done
```
```   636
```
```   637 lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)"
```
```   638   apply (subgoal_tac "real m + 1 = real (m + 1)")
```
```   639   apply (simp del: real_of_int_add)
```
```   640   apply simp
```
```   641 done
```
```   642
```
```   643 lemma real_of_int_div_aux: "d ~= 0 ==> (real (x::int)) / (real d) =
```
```   644     real (x div d) + (real (x mod d)) / (real d)"
```
```   645 proof -
```
```   646   assume "d ~= 0"
```
```   647   have "x = (x div d) * d + x mod d"
```
```   648     by auto
```
```   649   then have "real x = real (x div d) * real d + real(x mod d)"
```
```   650     by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym])
```
```   651   then have "real x / real d = ... / real d"
```
```   652     by simp
```
```   653   then show ?thesis
```
```   654     by (auto simp add: add_divide_distrib ring_simps prems)
```
```   655 qed
```
```   656
```
```   657 lemma real_of_int_div: "(d::int) ~= 0 ==> d dvd n ==>
```
```   658     real(n div d) = real n / real d"
```
```   659   apply (frule real_of_int_div_aux [of d n])
```
```   660   apply simp
```
```   661   apply (simp add: zdvd_iff_zmod_eq_0)
```
```   662 done
```
```   663
```
```   664 lemma real_of_int_div2:
```
```   665   "0 <= real (n::int) / real (x) - real (n div x)"
```
```   666   apply (case_tac "x = 0")
```
```   667   apply simp
```
```   668   apply (case_tac "0 < x")
```
```   669   apply (simp add: compare_rls)
```
```   670   apply (subst real_of_int_div_aux)
```
```   671   apply simp
```
```   672   apply simp
```
```   673   apply (subst zero_le_divide_iff)
```
```   674   apply auto
```
```   675   apply (simp add: compare_rls)
```
```   676   apply (subst real_of_int_div_aux)
```
```   677   apply simp
```
```   678   apply simp
```
```   679   apply (subst zero_le_divide_iff)
```
```   680   apply auto
```
```   681 done
```
```   682
```
```   683 lemma real_of_int_div3:
```
```   684   "real (n::int) / real (x) - real (n div x) <= 1"
```
```   685   apply(case_tac "x = 0")
```
```   686   apply simp
```
```   687   apply (simp add: compare_rls)
```
```   688   apply (subst real_of_int_div_aux)
```
```   689   apply assumption
```
```   690   apply simp
```
```   691   apply (subst divide_le_eq)
```
```   692   apply clarsimp
```
```   693   apply (rule conjI)
```
```   694   apply (rule impI)
```
```   695   apply (rule order_less_imp_le)
```
```   696   apply simp
```
```   697   apply (rule impI)
```
```   698   apply (rule order_less_imp_le)
```
```   699   apply simp
```
```   700 done
```
```   701
```
```   702 lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x"
```
```   703 by (insert real_of_int_div2 [of n x], simp)
```
```   704
```
```   705
```
```   706 subsection{*Embedding the Naturals into the Reals*}
```
```   707
```
```   708 lemma real_of_nat_zero [simp]: "real (0::nat) = 0"
```
```   709 by (simp add: real_of_nat_def)
```
```   710
```
```   711 lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)"
```
```   712 by (simp add: real_of_nat_def)
```
```   713
```
```   714 lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n"
```
```   715 by (simp add: real_of_nat_def)
```
```   716
```
```   717 (*Not for addsimps: often the LHS is used to represent a positive natural*)
```
```   718 lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)"
```
```   719 by (simp add: real_of_nat_def)
```
```   720
```
```   721 lemma real_of_nat_less_iff [iff]:
```
```   722      "(real (n::nat) < real m) = (n < m)"
```
```   723 by (simp add: real_of_nat_def)
```
```   724
```
```   725 lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)"
```
```   726 by (simp add: real_of_nat_def)
```
```   727
```
```   728 lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)"
```
```   729 by (simp add: real_of_nat_def zero_le_imp_of_nat)
```
```   730
```
```   731 lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"
```
```   732 by (simp add: real_of_nat_def del: of_nat_Suc)
```
```   733
```
```   734 lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"
```
```   735 by (simp add: real_of_nat_def of_nat_mult)
```
```   736
```
```   737 lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) =
```
```   738     (SUM x:A. real(f x))"
```
```   739   apply (subst real_eq_of_nat)+
```
```   740   apply (rule of_nat_setsum)
```
```   741 done
```
```   742
```
```   743 lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) =
```
```   744     (PROD x:A. real(f x))"
```
```   745   apply (subst real_eq_of_nat)+
```
```   746   apply (rule of_nat_setprod)
```
```   747 done
```
```   748
```
```   749 lemma real_of_card: "real (card A) = setsum (%x.1) A"
```
```   750   apply (subst card_eq_setsum)
```
```   751   apply (subst real_of_nat_setsum)
```
```   752   apply simp
```
```   753 done
```
```   754
```
```   755 lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"
```
```   756 by (simp add: real_of_nat_def)
```
```   757
```
```   758 lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)"
```
```   759 by (simp add: real_of_nat_def)
```
```   760
```
```   761 lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n"
```
```   762 by (simp add: add: real_of_nat_def of_nat_diff)
```
```   763
```
```   764 lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)"
```
```   765 by (auto simp: real_of_nat_def)
```
```   766
```
```   767 lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)"
```
```   768 by (simp add: add: real_of_nat_def)
```
```   769
```
```   770 lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0"
```
```   771 by (simp add: add: real_of_nat_def)
```
```   772
```
```   773 lemma real_of_nat_ge_zero_cancel_iff [simp]: "(0 \<le> real (n::nat))"
```
```   774 by (simp add: add: real_of_nat_def)
```
```   775
```
```   776 lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)"
```
```   777   apply (subgoal_tac "real n + 1 = real (Suc n)")
```
```   778   apply simp
```
```   779   apply (auto simp add: real_of_nat_Suc)
```
```   780 done
```
```   781
```
```   782 lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)"
```
```   783   apply (subgoal_tac "real m + 1 = real (Suc m)")
```
```   784   apply (simp add: less_Suc_eq_le)
```
```   785   apply (simp add: real_of_nat_Suc)
```
```   786 done
```
```   787
```
```   788 lemma real_of_nat_div_aux: "0 < d ==> (real (x::nat)) / (real d) =
```
```   789     real (x div d) + (real (x mod d)) / (real d)"
```
```   790 proof -
```
```   791   assume "0 < d"
```
```   792   have "x = (x div d) * d + x mod d"
```
```   793     by auto
```
```   794   then have "real x = real (x div d) * real d + real(x mod d)"
```
```   795     by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym])
```
```   796   then have "real x / real d = \<dots> / real d"
```
```   797     by simp
```
```   798   then show ?thesis
```
```   799     by (auto simp add: add_divide_distrib ring_simps prems)
```
```   800 qed
```
```   801
```
```   802 lemma real_of_nat_div: "0 < (d::nat) ==> d dvd n ==>
```
```   803     real(n div d) = real n / real d"
```
```   804   apply (frule real_of_nat_div_aux [of d n])
```
```   805   apply simp
```
```   806   apply (subst dvd_eq_mod_eq_0 [THEN sym])
```
```   807   apply assumption
```
```   808 done
```
```   809
```
```   810 lemma real_of_nat_div2:
```
```   811   "0 <= real (n::nat) / real (x) - real (n div x)"
```
```   812 apply(case_tac "x = 0")
```
```   813  apply (simp)
```
```   814 apply (simp add: compare_rls)
```
```   815 apply (subst real_of_nat_div_aux)
```
```   816  apply simp
```
```   817 apply simp
```
```   818 apply (subst zero_le_divide_iff)
```
```   819 apply simp
```
```   820 done
```
```   821
```
```   822 lemma real_of_nat_div3:
```
```   823   "real (n::nat) / real (x) - real (n div x) <= 1"
```
```   824 apply(case_tac "x = 0")
```
```   825 apply (simp)
```
```   826 apply (simp add: compare_rls)
```
```   827 apply (subst real_of_nat_div_aux)
```
```   828  apply simp
```
```   829 apply simp
```
```   830 done
```
```   831
```
```   832 lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x"
```
```   833   by (insert real_of_nat_div2 [of n x], simp)
```
```   834
```
```   835 lemma real_of_int_real_of_nat: "real (int n) = real n"
```
```   836 by (simp add: real_of_nat_def real_of_int_def int_eq_of_nat)
```
```   837
```
```   838 lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n"
```
```   839 by (simp add: real_of_int_def real_of_nat_def)
```
```   840
```
```   841 lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x"
```
```   842   apply (subgoal_tac "real(int(nat x)) = real(nat x)")
```
```   843   apply force
```
```   844   apply (simp only: real_of_int_real_of_nat)
```
```   845 done
```
```   846
```
```   847
```
```   848 subsection{* Rationals *}
```
```   849
```
```   850 lemma Rats_real_nat[simp]: "real(n::nat) \<in> \<rat>"
```
```   851 by (simp add: real_eq_of_nat)
```
```   852
```
```   853
```
```   854 lemma Rats_eq_int_div_int:
```
```   855   "\<rat> = { real(i::int)/real(j::int) |i j. j \<noteq> 0}" (is "_ = ?S")
```
```   856 proof
```
```   857   show "\<rat> \<subseteq> ?S"
```
```   858   proof
```
```   859     fix x::real assume "x : \<rat>"
```
```   860     then obtain r where "x = of_rat r" unfolding Rats_def ..
```
```   861     have "of_rat r : ?S"
```
```   862       by (cases r)(auto simp add:of_rat_rat real_eq_of_int)
```
```   863     thus "x : ?S" using `x = of_rat r` by simp
```
```   864   qed
```
```   865 next
```
```   866   show "?S \<subseteq> \<rat>"
```
```   867   proof(auto simp:Rats_def)
```
```   868     fix i j :: int assume "j \<noteq> 0"
```
```   869     hence "real i / real j = of_rat(Fract i j)"
```
```   870       by (simp add:of_rat_rat real_eq_of_int)
```
```   871     thus "real i / real j \<in> range of_rat" by blast
```
```   872   qed
```
```   873 qed
```
```   874
```
```   875 lemma Rats_eq_int_div_nat:
```
```   876   "\<rat> = { real(i::int)/real(n::nat) |i n. n \<noteq> 0}"
```
```   877 proof(auto simp:Rats_eq_int_div_int)
```
```   878   fix i j::int assume "j \<noteq> 0"
```
```   879   show "EX (i'::int) (n::nat). real i/real j = real i'/real n \<and> 0<n"
```
```   880   proof cases
```
```   881     assume "j>0"
```
```   882     hence "real i/real j = real i/real(nat j) \<and> 0<nat j"
```
```   883       by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat)
```
```   884     thus ?thesis by blast
```
```   885   next
```
```   886     assume "~ j>0"
```
```   887     hence "real i/real j = real(-i)/real(nat(-j)) \<and> 0<nat(-j)" using `j\<noteq>0`
```
```   888       by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat)
```
```   889     thus ?thesis by blast
```
```   890   qed
```
```   891 next
```
```   892   fix i::int and n::nat assume "0 < n"
```
```   893   hence "real i/real n = real i/real(int n) \<and> int n \<noteq> 0" by simp
```
```   894   thus "\<exists>(i'::int) j::int. real i/real n = real i'/real j \<and> j \<noteq> 0" by blast
```
```   895 qed
```
```   896
```
```   897 lemma Rats_abs_nat_div_natE:
```
```   898   assumes "x \<in> \<rat>"
```
```   899   obtains m n where "n \<noteq> 0" and "\<bar>x\<bar> = real m / real n" and "gcd m n = 1"
```
```   900 proof -
```
```   901   from `x \<in> \<rat>` obtain i::int and n::nat where "n \<noteq> 0" and "x = real i / real n"
```
```   902     by(auto simp add: Rats_eq_int_div_nat)
```
```   903   hence "\<bar>x\<bar> = real(nat(abs i)) / real n" by simp
```
```   904   then obtain m :: nat where x_rat: "\<bar>x\<bar> = real m / real n" by blast
```
```   905   let ?gcd = "gcd m n"
```
```   906   from `n\<noteq>0` have gcd: "?gcd \<noteq> 0" by (simp add: gcd_zero)
```
```   907   let ?k = "m div ?gcd"
```
```   908   let ?l = "n div ?gcd"
```
```   909   let ?gcd' = "gcd ?k ?l"
```
```   910   have "?gcd dvd m" .. then have gcd_k: "?gcd * ?k = m"
```
```   911     by (rule dvd_mult_div_cancel)
```
```   912   have "?gcd dvd n" .. then have gcd_l: "?gcd * ?l = n"
```
```   913     by (rule dvd_mult_div_cancel)
```
```   914   from `n\<noteq>0` and gcd_l have "?l \<noteq> 0" by (auto iff del: neq0_conv)
```
```   915   moreover
```
```   916   have "\<bar>x\<bar> = real ?k / real ?l"
```
```   917   proof -
```
```   918     from gcd have "real ?k / real ?l =
```
```   919         real (?gcd * ?k) / real (?gcd * ?l)" by simp
```
```   920     also from gcd_k and gcd_l have "\<dots> = real m / real n" by simp
```
```   921     also from x_rat have "\<dots> = \<bar>x\<bar>" ..
```
```   922     finally show ?thesis ..
```
```   923   qed
```
```   924   moreover
```
```   925   have "?gcd' = 1"
```
```   926   proof -
```
```   927     have "?gcd * ?gcd' = gcd (?gcd * ?k) (?gcd * ?l)"
```
```   928       by (rule gcd_mult_distrib2)
```
```   929     with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp
```
```   930     with gcd show ?thesis by simp
```
```   931   qed
```
```   932   ultimately show ?thesis ..
```
```   933 qed
```
```   934
```
```   935
```
```   936 subsection{*Numerals and Arithmetic*}
```
```   937
```
```   938 instantiation real :: number_ring
```
```   939 begin
```
```   940
```
```   941 definition
```
```   942   real_number_of_def [code func del]: "number_of w = real_of_int w"
```
```   943
```
```   944 instance
```
```   945   by intro_classes (simp add: real_number_of_def)
```
```   946
```
```   947 end
```
```   948
```
```   949 lemma [code unfold, symmetric, code post]:
```
```   950   "number_of k = real_of_int (number_of k)"
```
```   951   unfolding number_of_is_id real_number_of_def ..
```
```   952
```
```   953
```
```   954 text{*Collapse applications of @{term real} to @{term number_of}*}
```
```   955 lemma real_number_of [simp]: "real (number_of v :: int) = number_of v"
```
```   956 by (simp add:  real_of_int_def of_int_number_of_eq)
```
```   957
```
```   958 lemma real_of_nat_number_of [simp]:
```
```   959      "real (number_of v :: nat) =
```
```   960         (if neg (number_of v :: int) then 0
```
```   961          else (number_of v :: real))"
```
```   962 by (simp add: real_of_int_real_of_nat [symmetric] int_nat_number_of)
```
```   963
```
```   964
```
```   965 use "real_arith.ML"
```
```   966 declaration {* K real_arith_setup *}
```
```   967
```
```   968
```
```   969 subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*}
```
```   970
```
```   971 text{*Needed in this non-standard form by Hyperreal/Transcendental*}
```
```   972 lemma real_0_le_divide_iff:
```
```   973      "((0::real) \<le> x/y) = ((x \<le> 0 | 0 \<le> y) & (0 \<le> x | y \<le> 0))"
```
```   974 by (simp add: real_divide_def zero_le_mult_iff, auto)
```
```   975
```
```   976 lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)"
```
```   977 by arith
```
```   978
```
```   979 lemma real_add_eq_0_iff: "(x+y = (0::real)) = (y = -x)"
```
```   980 by auto
```
```   981
```
```   982 lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)"
```
```   983 by auto
```
```   984
```
```   985 lemma real_0_less_add_iff: "((0::real) < x+y) = (-x < y)"
```
```   986 by auto
```
```   987
```
```   988 lemma real_add_le_0_iff: "(x+y \<le> (0::real)) = (y \<le> -x)"
```
```   989 by auto
```
```   990
```
```   991 lemma real_0_le_add_iff: "((0::real) \<le> x+y) = (-x \<le> y)"
```
```   992 by auto
```
```   993
```
```   994
```
```   995 (*
```
```   996 FIXME: we should have this, as for type int, but many proofs would break.
```
```   997 It replaces x+-y by x-y.
```
```   998 declare real_diff_def [symmetric, simp]
```
```   999 *)
```
```  1000
```
```  1001
```
```  1002 subsubsection{*Density of the Reals*}
```
```  1003
```
```  1004 lemma real_lbound_gt_zero:
```
```  1005      "[| (0::real) < d1; 0 < d2 |] ==> \<exists>e. 0 < e & e < d1 & e < d2"
```
```  1006 apply (rule_tac x = " (min d1 d2) /2" in exI)
```
```  1007 apply (simp add: min_def)
```
```  1008 done
```
```  1009
```
```  1010
```
```  1011 text{*Similar results are proved in @{text Ring_and_Field}*}
```
```  1012 lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)"
```
```  1013   by auto
```
```  1014
```
```  1015 lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y"
```
```  1016   by auto
```
```  1017
```
```  1018
```
```  1019 subsection{*Absolute Value Function for the Reals*}
```
```  1020
```
```  1021 lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))"
```
```  1022 by (simp add: abs_if)
```
```  1023
```
```  1024 (* FIXME: redundant, but used by Integration/RealRandVar.thy in AFP *)
```
```  1025 lemma abs_le_interval_iff: "(abs x \<le> r) = (-r\<le>x & x\<le>(r::real))"
```
```  1026 by (force simp add: OrderedGroup.abs_le_iff)
```
```  1027
```
```  1028 lemma abs_add_one_gt_zero [simp]: "(0::real) < 1 + abs(x)"
```
```  1029 by (simp add: abs_if)
```
```  1030
```
```  1031 lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)"
```
```  1032 by (rule abs_of_nonneg [OF real_of_nat_ge_zero])
```
```  1033
```
```  1034 lemma abs_add_one_not_less_self [simp]: "~ abs(x) + (1::real) < x"
```
```  1035 by simp
```
```  1036
```
```  1037 lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \<le> abs(x + -l) + abs(y + -m)"
```
```  1038 by simp
```
```  1039
```
```  1040 instance real :: lordered_ring
```
```  1041 proof
```
```  1042   fix a::real
```
```  1043   show "abs a = sup a (-a)"
```
```  1044     by (auto simp add: real_abs_def sup_real_def)
```
```  1045 qed
```
```  1046
```
```  1047
```
```  1048 subsection {* Implementation of rational real numbers *}
```
```  1049
```
```  1050 definition Ratreal :: "rat \<Rightarrow> real" where
```
```  1051   [simp]: "Ratreal = of_rat"
```
```  1052
```
```  1053 code_datatype Ratreal
```
```  1054
```
```  1055 lemma Ratreal_number_collapse [code post]:
```
```  1056   "Ratreal 0 = 0"
```
```  1057   "Ratreal 1 = 1"
```
```  1058   "Ratreal (number_of k) = number_of k"
```
```  1059 by simp_all
```
```  1060
```
```  1061 lemma zero_real_code [code, code unfold]:
```
```  1062   "0 = Ratreal 0"
```
```  1063 by simp
```
```  1064
```
```  1065 lemma one_real_code [code, code unfold]:
```
```  1066   "1 = Ratreal 1"
```
```  1067 by simp
```
```  1068
```
```  1069 lemma number_of_real_code [code unfold]:
```
```  1070   "number_of k = Ratreal (number_of k)"
```
```  1071 by simp
```
```  1072
```
```  1073 lemma Ratreal_number_of_quotient [code post]:
```
```  1074   "Ratreal (number_of r) / Ratreal (number_of s) = number_of r / number_of s"
```
```  1075 by simp
```
```  1076
```
```  1077 lemma Ratreal_number_of_quotient2 [code post]:
```
```  1078   "Ratreal (number_of r / number_of s) = number_of r / number_of s"
```
```  1079 unfolding Ratreal_number_of_quotient [symmetric] Ratreal_def of_rat_divide ..
```
```  1080
```
```  1081 instantiation real :: eq
```
```  1082 begin
```
```  1083
```
```  1084 definition "eq_class.eq (x\<Colon>real) y \<longleftrightarrow> x - y = 0"
```
```  1085
```
```  1086 instance by default (simp add: eq_real_def)
```
```  1087
```
```  1088 lemma real_eq_code [code]: "eq_class.eq (Ratreal x) (Ratreal y) \<longleftrightarrow> eq_class.eq x y"
```
```  1089   by (simp add: eq_real_def eq)
```
```  1090
```
```  1091 end
```
```  1092
```
```  1093 lemma real_less_eq_code [code]: "Ratreal x \<le> Ratreal y \<longleftrightarrow> x \<le> y"
```
```  1094   by (simp add: of_rat_less_eq)
```
```  1095
```
```  1096 lemma real_less_code [code]: "Ratreal x < Ratreal y \<longleftrightarrow> x < y"
```
```  1097   by (simp add: of_rat_less)
```
```  1098
```
```  1099 lemma real_plus_code [code]: "Ratreal x + Ratreal y = Ratreal (x + y)"
```
```  1100   by (simp add: of_rat_add)
```
```  1101
```
```  1102 lemma real_times_code [code]: "Ratreal x * Ratreal y = Ratreal (x * y)"
```
```  1103   by (simp add: of_rat_mult)
```
```  1104
```
```  1105 lemma real_uminus_code [code]: "- Ratreal x = Ratreal (- x)"
```
```  1106   by (simp add: of_rat_minus)
```
```  1107
```
```  1108 lemma real_minus_code [code]: "Ratreal x - Ratreal y = Ratreal (x - y)"
```
```  1109   by (simp add: of_rat_diff)
```
```  1110
```
```  1111 lemma real_inverse_code [code]: "inverse (Ratreal x) = Ratreal (inverse x)"
```
```  1112   by (simp add: of_rat_inverse)
```
```  1113
```
```  1114 lemma real_divide_code [code]: "Ratreal x / Ratreal y = Ratreal (x / y)"
```
```  1115   by (simp add: of_rat_divide)
```
```  1116
```
```  1117 text {* Setup for SML code generator *}
```
```  1118
```
```  1119 types_code
```
```  1120   real ("(int */ int)")
```
```  1121 attach (term_of) {*
```
```  1122 fun term_of_real (p, q) =
```
```  1123   let
```
```  1124     val rT = HOLogic.realT
```
```  1125   in
```
```  1126     if q = 1 orelse p = 0 then HOLogic.mk_number rT p
```
```  1127     else @{term "op / \<Colon> real \<Rightarrow> real \<Rightarrow> real"} \$
```
```  1128       HOLogic.mk_number rT p \$ HOLogic.mk_number rT q
```
```  1129   end;
```
```  1130 *}
```
```  1131 attach (test) {*
```
```  1132 fun gen_real i =
```
```  1133   let
```
```  1134     val p = random_range 0 i;
```
```  1135     val q = random_range 1 (i + 1);
```
```  1136     val g = Integer.gcd p q;
```
```  1137     val p' = p div g;
```
```  1138     val q' = q div g;
```
```  1139     val r = (if one_of [true, false] then p' else ~ p',
```
```  1140       if p' = 0 then 0 else q')
```
```  1141   in
```
```  1142     (r, fn () => term_of_real r)
```
```  1143   end;
```
```  1144 *}
```
```  1145
```
```  1146 consts_code
```
```  1147   Ratreal ("(_)")
```
```  1148
```
```  1149 consts_code
```
```  1150   "of_int :: int \<Rightarrow> real" ("\<module>real'_of'_int")
```
```  1151 attach {*
```
```  1152 fun real_of_int 0 = (0, 0)
```
```  1153   | real_of_int i = (i, 1);
```
```  1154 *}
```
```  1155
```
```  1156 declare real_of_int_of_nat_eq [symmetric, code]
```
```  1157
```
```  1158 end
```