src/HOL/RealDef.thy
 author wenzelm Mon Mar 22 20:58:52 2010 +0100 (2010-03-22) changeset 35898 c890a3835d15 parent 35635 90fffd5ff996 child 36349 39be26d1bc28 permissions -rw-r--r--
1 (*  Title       : HOL/RealDef.thy
2     Author      : Jacques D. Fleuriot
3     Copyright   : 1998  University of Cambridge
4     Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
6 *)
8 header{*Defining the Reals from the Positive Reals*}
10 theory RealDef
11 imports PReal
12 begin
14 definition
15   realrel   ::  "((preal * preal) * (preal * preal)) set" where
16   [code del]: "realrel = {p. \<exists>x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}"
18 typedef (Real)  real = "UNIV//realrel"
19   by (auto simp add: quotient_def)
21 definition
22   (** these don't use the overloaded "real" function: users don't see them **)
23   real_of_preal :: "preal => real" where
24   [code del]: "real_of_preal m = Abs_Real (realrel `` {(m + 1, 1)})"
26 instantiation real :: "{zero, one, plus, minus, uminus, times, inverse, ord, abs, sgn}"
27 begin
29 definition
30   real_zero_def [code del]: "0 = Abs_Real(realrel``{(1, 1)})"
32 definition
33   real_one_def [code del]: "1 = Abs_Real(realrel``{(1 + 1, 1)})"
35 definition
36   real_add_def [code del]: "z + w =
37        contents (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w).
38                  { Abs_Real(realrel``{(x+u, y+v)}) })"
40 definition
41   real_minus_def [code del]: "- r =  contents (\<Union>(x,y) \<in> Rep_Real(r). { Abs_Real(realrel``{(y,x)}) })"
43 definition
44   real_diff_def [code del]: "r - (s::real) = r + - s"
46 definition
47   real_mult_def [code del]:
48     "z * w =
49        contents (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w).
50                  { Abs_Real(realrel``{(x*u + y*v, x*v + y*u)}) })"
52 definition
53   real_inverse_def [code del]: "inverse (R::real) = (THE S. (R = 0 & S = 0) | S * R = 1)"
55 definition
56   real_divide_def [code del]: "R / (S::real) = R * inverse S"
58 definition
59   real_le_def [code del]: "z \<le> (w::real) \<longleftrightarrow>
60     (\<exists>x y u v. x+v \<le> u+y & (x,y) \<in> Rep_Real z & (u,v) \<in> Rep_Real w)"
62 definition
63   real_less_def [code del]: "x < (y\<Colon>real) \<longleftrightarrow> x \<le> y \<and> x \<noteq> y"
65 definition
66   real_abs_def:  "abs (r::real) = (if r < 0 then - r else r)"
68 definition
69   real_sgn_def: "sgn (x::real) = (if x=0 then 0 else if 0<x then 1 else - 1)"
71 instance ..
73 end
75 subsection {* Equivalence relation over positive reals *}
77 lemma preal_trans_lemma:
78   assumes "x + y1 = x1 + y"
79       and "x + y2 = x2 + y"
80   shows "x1 + y2 = x2 + (y1::preal)"
81 proof -
82   have "(x1 + y2) + x = (x + y2) + x1" by (simp add: add_ac)
83   also have "... = (x2 + y) + x1"  by (simp add: prems)
84   also have "... = x2 + (x1 + y)"  by (simp add: add_ac)
85   also have "... = x2 + (x + y1)"  by (simp add: prems)
86   also have "... = (x2 + y1) + x"  by (simp add: add_ac)
87   finally have "(x1 + y2) + x = (x2 + y1) + x" .
88   thus ?thesis by (rule add_right_imp_eq)
89 qed
92 lemma realrel_iff [simp]: "(((x1,y1),(x2,y2)) \<in> realrel) = (x1 + y2 = x2 + y1)"
95 lemma equiv_realrel: "equiv UNIV realrel"
96 apply (auto simp add: equiv_def refl_on_def sym_def trans_def realrel_def)
97 apply (blast dest: preal_trans_lemma)
98 done
100 text{*Reduces equality of equivalence classes to the @{term realrel} relation:
101   @{term "(realrel `` {x} = realrel `` {y}) = ((x,y) \<in> realrel)"} *}
102 lemmas equiv_realrel_iff =
103        eq_equiv_class_iff [OF equiv_realrel UNIV_I UNIV_I]
105 declare equiv_realrel_iff [simp]
108 lemma realrel_in_real [simp]: "realrel``{(x,y)}: Real"
109 by (simp add: Real_def realrel_def quotient_def, blast)
111 declare Abs_Real_inject [simp]
112 declare Abs_Real_inverse [simp]
115 text{*Case analysis on the representation of a real number as an equivalence
116       class of pairs of positive reals.*}
117 lemma eq_Abs_Real [case_names Abs_Real, cases type: real]:
118      "(!!x y. z = Abs_Real(realrel``{(x,y)}) ==> P) ==> P"
119 apply (rule Rep_Real [of z, unfolded Real_def, THEN quotientE])
120 apply (drule arg_cong [where f=Abs_Real])
121 apply (auto simp add: Rep_Real_inverse)
122 done
125 subsection {* Addition and Subtraction *}
128      "[|a + ba = aa + b; ab + bc = ac + bb|]
129       ==> a + ab + (ba + bc) = aa + ac + (b + (bb::preal))"
131 apply (rule add_left_commute [of ab, THEN ssubst])
134 done
137      "Abs_Real (realrel``{(x,y)}) + Abs_Real (realrel``{(u,v)}) =
138       Abs_Real (realrel``{(x+u, y+v)})"
139 proof -
140   have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). {Abs_Real (realrel `` {(x+u, y+v)})}) w) z)
141         respects2 realrel"
143   thus ?thesis
145                   UN_equiv_class2 [OF equiv_realrel equiv_realrel])
146 qed
148 lemma real_minus: "- Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})"
149 proof -
150   have "(\<lambda>(x,y). {Abs_Real (realrel``{(y,x)})}) respects realrel"
152   thus ?thesis
153     by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel])
154 qed
157 proof
158   fix x y z :: real
159   show "(x + y) + z = x + (y + z)"
161   show "x + y = y + x"
163   show "0 + x = x"
165   show "- x + x = 0"
167   show "x - y = x + - y"
169 qed
172 subsection {* Multiplication *}
174 lemma real_mult_congruent2_lemma:
175      "!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==>
176           x * x1 + y * y1 + (x * y2 + y * x2) =
177           x * x2 + y * y2 + (x * y1 + y * x1)"
181 done
183 lemma real_mult_congruent2:
184     "(%p1 p2.
185         (%(x1,y1). (%(x2,y2).
186           { Abs_Real (realrel``{(x1*x2 + y1*y2, x1*y2+y1*x2)}) }) p2) p1)
187      respects2 realrel"
188 apply (rule congruent2_commuteI [OF equiv_realrel], clarify)
190 apply (auto simp add: real_mult_congruent2_lemma)
191 done
193 lemma real_mult:
194       "Abs_Real((realrel``{(x1,y1)})) * Abs_Real((realrel``{(x2,y2)})) =
195        Abs_Real(realrel `` {(x1*x2+y1*y2,x1*y2+y1*x2)})"
196 by (simp add: real_mult_def UN_UN_split_split_eq
197          UN_equiv_class2 [OF equiv_realrel equiv_realrel real_mult_congruent2])
199 lemma real_mult_commute: "(z::real) * w = w * z"
202 lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)"
203 apply (cases z1, cases z2, cases z3)
204 apply (simp add: real_mult algebra_simps)
205 done
207 lemma real_mult_1: "(1::real) * z = z"
208 apply (cases z)
209 apply (simp add: real_mult real_one_def algebra_simps)
210 done
212 lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)"
213 apply (cases z1, cases z2, cases w)
215 done
217 text{*one and zero are distinct*}
218 lemma real_zero_not_eq_one: "0 \<noteq> (1::real)"
219 proof -
220   have "(1::preal) < 1 + 1"
222   thus ?thesis
223     by (simp add: real_zero_def real_one_def)
224 qed
226 instance real :: comm_ring_1
227 proof
228   fix x y z :: real
229   show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc)
230   show "x * y = y * x" by (rule real_mult_commute)
231   show "1 * x = x" by (rule real_mult_1)
232   show "(x + y) * z = x * z + y * z" by (rule real_add_mult_distrib)
233   show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one)
234 qed
236 subsection {* Inverse and Division *}
238 lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0"
241 text{*Instead of using an existential quantifier and constructing the inverse
242 within the proof, we could define the inverse explicitly.*}
244 lemma real_mult_inverse_left_ex: "x \<noteq> 0 ==> \<exists>y. y*x = (1::real)"
245 apply (simp add: real_zero_def real_one_def, cases x)
246 apply (cut_tac x = xa and y = y in linorder_less_linear)
248 apply (rule_tac
249         x = "Abs_Real (realrel``{(1, inverse (D) + 1)})"
250        in exI)
251 apply (rule_tac [2]
252         x = "Abs_Real (realrel``{(inverse (D) + 1, 1)})"
253        in exI)
254 apply (auto simp add: real_mult preal_mult_inverse_right algebra_simps)
255 done
257 lemma real_mult_inverse_left: "x \<noteq> 0 ==> inverse(x)*x = (1::real)"
259 apply (drule real_mult_inverse_left_ex, safe)
260 apply (rule theI, assumption, rename_tac z)
261 apply (subgoal_tac "(z * x) * y = z * (x * y)")
263 apply (rule mult_assoc)
264 done
267 subsection{*The Real Numbers form a Field*}
269 instance real :: field
270 proof
271   fix x y z :: real
272   show "x \<noteq> 0 ==> inverse x * x = 1" by (rule real_mult_inverse_left)
273   show "x / y = x * inverse y" by (simp add: real_divide_def)
274 qed
277 text{*Inverse of zero!  Useful to simplify certain equations*}
279 lemma INVERSE_ZERO: "inverse 0 = (0::real)"
282 instance real :: division_by_zero
283 proof
284   show "inverse 0 = (0::real)" by (rule INVERSE_ZERO)
285 qed
288 subsection{*The @{text "\<le>"} Ordering*}
290 lemma real_le_refl: "w \<le> (w::real)"
291 by (cases w, force simp add: real_le_def)
293 text{*The arithmetic decision procedure is not set up for type preal.
294   This lemma is currently unused, but it could simplify the proofs of the
295   following two lemmas.*}
296 lemma preal_eq_le_imp_le:
297   assumes eq: "a+b = c+d" and le: "c \<le> a"
298   shows "b \<le> (d::preal)"
299 proof -
300   have "c+d \<le> a+d" by (simp add: prems)
301   hence "a+b \<le> a+d" by (simp add: prems)
302   thus "b \<le> d" by simp
303 qed
305 lemma real_le_lemma:
306   assumes l: "u1 + v2 \<le> u2 + v1"
307       and "x1 + v1 = u1 + y1"
308       and "x2 + v2 = u2 + y2"
309   shows "x1 + y2 \<le> x2 + (y1::preal)"
310 proof -
311   have "(x1+v1) + (u2+y2) = (u1+y1) + (x2+v2)" by (simp add: prems)
312   hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: add_ac)
313   also have "... \<le> (x2+y1) + (u2+v1)" by (simp add: prems)
314   finally show ?thesis by simp
315 qed
317 lemma real_le:
318      "(Abs_Real(realrel``{(x1,y1)}) \<le> Abs_Real(realrel``{(x2,y2)})) =
319       (x1 + y2 \<le> x2 + y1)"
321 apply (auto intro: real_le_lemma)
322 done
324 lemma real_le_antisym: "[| z \<le> w; w \<le> z |] ==> z = (w::real)"
325 by (cases z, cases w, simp add: real_le)
327 lemma real_trans_lemma:
328   assumes "x + v \<le> u + y"
329       and "u + v' \<le> u' + v"
330       and "x2 + v2 = u2 + y2"
331   shows "x + v' \<le> u' + (y::preal)"
332 proof -
333   have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: add_ac)
334   also have "... \<le> (u+y) + (u+v')" by (simp add: prems)
335   also have "... \<le> (u+y) + (u'+v)" by (simp add: prems)
336   also have "... = (u'+y) + (u+v)"  by (simp add: add_ac)
337   finally show ?thesis by simp
338 qed
340 lemma real_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::real)"
341 apply (cases i, cases j, cases k)
343 apply (blast intro: real_trans_lemma)
344 done
346 instance real :: order
347 proof
348   fix u v :: real
349   show "u < v \<longleftrightarrow> u \<le> v \<and> \<not> v \<le> u"
350     by (auto simp add: real_less_def intro: real_le_antisym)
351 qed (assumption | rule real_le_refl real_le_trans real_le_antisym)+
353 (* Axiom 'linorder_linear' of class 'linorder': *)
354 lemma real_le_linear: "(z::real) \<le> w | w \<le> z"
355 apply (cases z, cases w)
357 done
359 instance real :: linorder
360   by (intro_classes, rule real_le_linear)
363 lemma real_le_eq_diff: "(x \<le> y) = (x-y \<le> (0::real))"
364 apply (cases x, cases y)
368 done
371   assumes le: "x \<le> y" shows "z + x \<le> z + (y::real)"
372 proof -
373   have "z + x - (z + y) = (z + -z) + (x - y)"
375   with le show ?thesis
376     by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"] diff_minus)
377 qed
379 lemma real_sum_gt_zero_less: "(0 < S + (-W::real)) ==> (W < S)"
380 by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
382 lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (-W::real))"
383 by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
385 lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y"
386 apply (cases x, cases y)
387 apply (simp add: linorder_not_le [where 'a = real, symmetric]
388                  linorder_not_le [where 'a = preal]
389                   real_zero_def real_le real_mult)
390   --{*Reduce to the (simpler) @{text "\<le>"} relation *}
393 done
395 lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y"
396 apply (rule real_sum_gt_zero_less)
397 apply (drule real_less_sum_gt_zero [of x y])
398 apply (drule real_mult_order, assumption)
400 done
402 instantiation real :: distrib_lattice
403 begin
405 definition
406   "(inf \<Colon> real \<Rightarrow> real \<Rightarrow> real) = min"
408 definition
409   "(sup \<Colon> real \<Rightarrow> real \<Rightarrow> real) = max"
411 instance
412   by default (auto simp add: inf_real_def sup_real_def min_max.sup_inf_distrib1)
414 end
417 subsection{*The Reals Form an Ordered Field*}
419 instance real :: linordered_field
420 proof
421   fix x y z :: real
422   show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono)
423   show "x < y ==> 0 < z ==> z * x < z * y" by (rule real_mult_less_mono2)
424   show "\<bar>x\<bar> = (if x < 0 then -x else x)" by (simp only: real_abs_def)
425   show "sgn x = (if x=0 then 0 else if 0<x then 1 else - 1)"
426     by (simp only: real_sgn_def)
427 qed
429 text{*The function @{term real_of_preal} requires many proofs, but it seems
430 to be essential for proving completeness of the reals from that of the
431 positive reals.*}
434      "real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y"
437 lemma real_of_preal_mult:
438      "real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y"
439 by (simp add: real_of_preal_def real_mult algebra_simps)
442 text{*Gleason prop 9-4.4 p 127*}
443 lemma real_of_preal_trichotomy:
444       "\<exists>m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)"
445 apply (simp add: real_of_preal_def real_zero_def, cases x)
447 apply (cut_tac x = x and y = y in linorder_less_linear)
449 done
451 lemma real_of_preal_leD:
452       "real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2"
453 by (simp add: real_of_preal_def real_le)
455 lemma real_of_preal_lessI: "m1 < m2 ==> real_of_preal m1 < real_of_preal m2"
456 by (auto simp add: real_of_preal_leD linorder_not_le [symmetric])
458 lemma real_of_preal_lessD:
459       "real_of_preal m1 < real_of_preal m2 ==> m1 < m2"
460 by (simp add: real_of_preal_def real_le linorder_not_le [symmetric])
462 lemma real_of_preal_less_iff [simp]:
463      "(real_of_preal m1 < real_of_preal m2) = (m1 < m2)"
464 by (blast intro: real_of_preal_lessI real_of_preal_lessD)
466 lemma real_of_preal_le_iff:
467      "(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)"
468 by (simp add: linorder_not_less [symmetric])
470 lemma real_of_preal_zero_less: "0 < real_of_preal m"
471 apply (insert preal_self_less_add_left [of 1 m])
472 apply (auto simp add: real_zero_def real_of_preal_def
474 apply (rule_tac x="m + 1" in exI, rule_tac x="1" in exI)
476 done
478 lemma real_of_preal_minus_less_zero: "- real_of_preal m < 0"
481 lemma real_of_preal_not_minus_gt_zero: "~ 0 < - real_of_preal m"
482 proof -
483   from real_of_preal_minus_less_zero
484   show ?thesis by (blast dest: order_less_trans)
485 qed
490 lemma real_gt_zero_preal_Ex: "(0 < x) = (\<exists>y. x = real_of_preal y)"
491 apply (auto simp add: real_of_preal_zero_less)
492 apply (cut_tac x = x in real_of_preal_trichotomy)
493 apply (blast elim!: real_of_preal_not_minus_gt_zero [THEN notE])
494 done
496 lemma real_gt_preal_preal_Ex:
497      "real_of_preal z < x ==> \<exists>y. x = real_of_preal y"
498 by (blast dest!: real_of_preal_zero_less [THEN order_less_trans]
499              intro: real_gt_zero_preal_Ex [THEN iffD1])
501 lemma real_ge_preal_preal_Ex:
502      "real_of_preal z \<le> x ==> \<exists>y. x = real_of_preal y"
503 by (blast dest: order_le_imp_less_or_eq real_gt_preal_preal_Ex)
505 lemma real_less_all_preal: "y \<le> 0 ==> \<forall>x. y < real_of_preal x"
506 by (auto elim: order_le_imp_less_or_eq [THEN disjE]
507             intro: real_of_preal_zero_less [THEN [2] order_less_trans]
510 lemma real_less_all_real2: "~ 0 < y ==> \<forall>x. y < real_of_preal x"
511 by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1])
514 subsection{*More Lemmas*}
516 lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
517 by auto
519 lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)"
520 by auto
522 lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)"
523   by (force elim: order_less_asym
526 lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)"
528 apply (blast intro: elim: order_less_asym)
529 done
531 lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)"
534 lemma real_inverse_gt_one: "[| (0::real) < x; x < 1 |] ==> 1 < inverse x"
535 by (simp add: one_less_inverse_iff) (* TODO: generalize/move *)
538 subsection {* Embedding numbers into the Reals *}
540 abbreviation
541   real_of_nat :: "nat \<Rightarrow> real"
542 where
543   "real_of_nat \<equiv> of_nat"
545 abbreviation
546   real_of_int :: "int \<Rightarrow> real"
547 where
548   "real_of_int \<equiv> of_int"
550 abbreviation
551   real_of_rat :: "rat \<Rightarrow> real"
552 where
553   "real_of_rat \<equiv> of_rat"
555 consts
556   (*overloaded constant for injecting other types into "real"*)
557   real :: "'a => real"
560   real_of_nat_def [code_unfold]: "real == real_of_nat"
561   real_of_int_def [code_unfold]: "real == real_of_int"
563 lemma real_eq_of_nat: "real = of_nat"
564   unfolding real_of_nat_def ..
566 lemma real_eq_of_int: "real = of_int"
567   unfolding real_of_int_def ..
569 lemma real_of_int_zero [simp]: "real (0::int) = 0"
572 lemma real_of_one [simp]: "real (1::int) = (1::real)"
575 lemma real_of_int_add [simp]: "real(x + y) = real (x::int) + real y"
578 lemma real_of_int_minus [simp]: "real(-x) = -real (x::int)"
581 lemma real_of_int_diff [simp]: "real(x - y) = real (x::int) - real y"
584 lemma real_of_int_mult [simp]: "real(x * y) = real (x::int) * real y"
587 lemma real_of_int_power [simp]: "real (x ^ n) = real (x::int) ^ n"
588 by (simp add: real_of_int_def of_int_power)
590 lemmas power_real_of_int = real_of_int_power [symmetric]
592 lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))"
593   apply (subst real_eq_of_int)+
594   apply (rule of_int_setsum)
595 done
597 lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) =
598     (PROD x:A. real(f x))"
599   apply (subst real_eq_of_int)+
600   apply (rule of_int_setprod)
601 done
603 lemma real_of_int_zero_cancel [simp, algebra, presburger]: "(real x = 0) = (x = (0::int))"
606 lemma real_of_int_inject [iff, algebra, presburger]: "(real (x::int) = real y) = (x = y)"
609 lemma real_of_int_less_iff [iff, presburger]: "(real (x::int) < real y) = (x < y)"
612 lemma real_of_int_le_iff [simp, presburger]: "(real (x::int) \<le> real y) = (x \<le> y)"
615 lemma real_of_int_gt_zero_cancel_iff [simp, presburger]: "(0 < real (n::int)) = (0 < n)"
618 lemma real_of_int_ge_zero_cancel_iff [simp, presburger]: "(0 <= real (n::int)) = (0 <= n)"
621 lemma real_of_int_lt_zero_cancel_iff [simp, presburger]: "(real (n::int) < 0) = (n < 0)"
624 lemma real_of_int_le_zero_cancel_iff [simp, presburger]: "(real (n::int) <= 0) = (n <= 0)"
627 lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))"
628 by (auto simp add: abs_if)
630 lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)"
631   apply (subgoal_tac "real n + 1 = real (n + 1)")
633   apply auto
634 done
636 lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)"
637   apply (subgoal_tac "real m + 1 = real (m + 1)")
639   apply simp
640 done
642 lemma real_of_int_div_aux: "d ~= 0 ==> (real (x::int)) / (real d) =
643     real (x div d) + (real (x mod d)) / (real d)"
644 proof -
645   assume "d ~= 0"
646   have "x = (x div d) * d + x mod d"
647     by auto
648   then have "real x = real (x div d) * real d + real(x mod d)"
649     by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym])
650   then have "real x / real d = ... / real d"
651     by simp
652   then show ?thesis
654 qed
656 lemma real_of_int_div: "(d::int) ~= 0 ==> d dvd n ==>
657     real(n div d) = real n / real d"
658   apply (frule real_of_int_div_aux [of d n])
659   apply simp
661 done
663 lemma real_of_int_div2:
664   "0 <= real (n::int) / real (x) - real (n div x)"
665   apply (case_tac "x = 0")
666   apply simp
667   apply (case_tac "0 < x")
669   apply (subst real_of_int_div_aux)
670   apply simp
671   apply simp
672   apply (subst zero_le_divide_iff)
673   apply auto
675   apply (subst real_of_int_div_aux)
676   apply simp
677   apply simp
678   apply (subst zero_le_divide_iff)
679   apply auto
680 done
682 lemma real_of_int_div3:
683   "real (n::int) / real (x) - real (n div x) <= 1"
684   apply(case_tac "x = 0")
685   apply simp
687   apply (subst real_of_int_div_aux)
688   apply assumption
689   apply simp
690   apply (subst divide_le_eq)
691   apply clarsimp
692   apply (rule conjI)
693   apply (rule impI)
694   apply (rule order_less_imp_le)
695   apply simp
696   apply (rule impI)
697   apply (rule order_less_imp_le)
698   apply simp
699 done
701 lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x"
702 by (insert real_of_int_div2 [of n x], simp)
704 lemma Ints_real_of_int [simp]: "real (x::int) \<in> Ints"
705 unfolding real_of_int_def by (rule Ints_of_int)
708 subsection{*Embedding the Naturals into the Reals*}
710 lemma real_of_nat_zero [simp]: "real (0::nat) = 0"
713 lemma real_of_nat_1 [simp]: "real (1::nat) = 1"
716 lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)"
719 lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n"
722 (*Not for addsimps: often the LHS is used to represent a positive natural*)
723 lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)"
726 lemma real_of_nat_less_iff [iff]:
727      "(real (n::nat) < real m) = (n < m)"
730 lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)"
733 lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)"
734 by (simp add: real_of_nat_def zero_le_imp_of_nat)
736 lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"
737 by (simp add: real_of_nat_def del: of_nat_Suc)
739 lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"
740 by (simp add: real_of_nat_def of_nat_mult)
742 lemma real_of_nat_power [simp]: "real (m ^ n) = real (m::nat) ^ n"
743 by (simp add: real_of_nat_def of_nat_power)
745 lemmas power_real_of_nat = real_of_nat_power [symmetric]
747 lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) =
748     (SUM x:A. real(f x))"
749   apply (subst real_eq_of_nat)+
750   apply (rule of_nat_setsum)
751 done
753 lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) =
754     (PROD x:A. real(f x))"
755   apply (subst real_eq_of_nat)+
756   apply (rule of_nat_setprod)
757 done
759 lemma real_of_card: "real (card A) = setsum (%x.1) A"
760   apply (subst card_eq_setsum)
761   apply (subst real_of_nat_setsum)
762   apply simp
763 done
765 lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"
768 lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)"
771 lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n"
774 lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)"
775 by (auto simp: real_of_nat_def)
777 lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)"
780 lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0"
783 (* FIXME: duplicates real_of_nat_ge_zero *)
784 lemma real_of_nat_ge_zero_cancel_iff: "(0 \<le> real (n::nat))"
787 lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)"
788   apply (subgoal_tac "real n + 1 = real (Suc n)")
789   apply simp
790   apply (auto simp add: real_of_nat_Suc)
791 done
793 lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)"
794   apply (subgoal_tac "real m + 1 = real (Suc m)")
797 done
799 lemma real_of_nat_div_aux: "0 < d ==> (real (x::nat)) / (real d) =
800     real (x div d) + (real (x mod d)) / (real d)"
801 proof -
802   assume "0 < d"
803   have "x = (x div d) * d + x mod d"
804     by auto
805   then have "real x = real (x div d) * real d + real(x mod d)"
806     by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym])
807   then have "real x / real d = \<dots> / real d"
808     by simp
809   then show ?thesis
811 qed
813 lemma real_of_nat_div: "0 < (d::nat) ==> d dvd n ==>
814     real(n div d) = real n / real d"
815   apply (frule real_of_nat_div_aux [of d n])
816   apply simp
817   apply (subst dvd_eq_mod_eq_0 [THEN sym])
818   apply assumption
819 done
821 lemma real_of_nat_div2:
822   "0 <= real (n::nat) / real (x) - real (n div x)"
823 apply(case_tac "x = 0")
824  apply (simp)
826 apply (subst real_of_nat_div_aux)
827  apply simp
828 apply simp
829 apply (subst zero_le_divide_iff)
830 apply simp
831 done
833 lemma real_of_nat_div3:
834   "real (n::nat) / real (x) - real (n div x) <= 1"
835 apply(case_tac "x = 0")
836 apply (simp)
838 apply (subst real_of_nat_div_aux)
839  apply simp
840 apply simp
841 done
843 lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x"
844 by (insert real_of_nat_div2 [of n x], simp)
846 lemma real_of_int_real_of_nat: "real (int n) = real n"
847 by (simp add: real_of_nat_def real_of_int_def int_eq_of_nat)
849 lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n"
850 by (simp add: real_of_int_def real_of_nat_def)
852 lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x"
853   apply (subgoal_tac "real(int(nat x)) = real(nat x)")
854   apply force
855   apply (simp only: real_of_int_real_of_nat)
856 done
858 lemma Nats_real_of_nat [simp]: "real (n::nat) \<in> Nats"
859 unfolding real_of_nat_def by (rule of_nat_in_Nats)
861 lemma Ints_real_of_nat [simp]: "real (n::nat) \<in> Ints"
862 unfolding real_of_nat_def by (rule Ints_of_nat)
865 subsection{* Rationals *}
867 lemma Rats_real_nat[simp]: "real(n::nat) \<in> \<rat>"
871 lemma Rats_eq_int_div_int:
872   "\<rat> = { real(i::int)/real(j::int) |i j. j \<noteq> 0}" (is "_ = ?S")
873 proof
874   show "\<rat> \<subseteq> ?S"
875   proof
876     fix x::real assume "x : \<rat>"
877     then obtain r where "x = of_rat r" unfolding Rats_def ..
878     have "of_rat r : ?S"
879       by (cases r)(auto simp add:of_rat_rat real_eq_of_int)
880     thus "x : ?S" using `x = of_rat r` by simp
881   qed
882 next
883   show "?S \<subseteq> \<rat>"
884   proof(auto simp:Rats_def)
885     fix i j :: int assume "j \<noteq> 0"
886     hence "real i / real j = of_rat(Fract i j)"
888     thus "real i / real j \<in> range of_rat" by blast
889   qed
890 qed
892 lemma Rats_eq_int_div_nat:
893   "\<rat> = { real(i::int)/real(n::nat) |i n. n \<noteq> 0}"
894 proof(auto simp:Rats_eq_int_div_int)
895   fix i j::int assume "j \<noteq> 0"
896   show "EX (i'::int) (n::nat). real i/real j = real i'/real n \<and> 0<n"
897   proof cases
898     assume "j>0"
899     hence "real i/real j = real i/real(nat j) \<and> 0<nat j"
900       by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat)
901     thus ?thesis by blast
902   next
903     assume "~ j>0"
904     hence "real i/real j = real(-i)/real(nat(-j)) \<and> 0<nat(-j)" using `j\<noteq>0`
905       by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat)
906     thus ?thesis by blast
907   qed
908 next
909   fix i::int and n::nat assume "0 < n"
910   hence "real i/real n = real i/real(int n) \<and> int n \<noteq> 0" by simp
911   thus "\<exists>(i'::int) j::int. real i/real n = real i'/real j \<and> j \<noteq> 0" by blast
912 qed
914 lemma Rats_abs_nat_div_natE:
915   assumes "x \<in> \<rat>"
916   obtains m n :: nat
917   where "n \<noteq> 0" and "\<bar>x\<bar> = real m / real n" and "gcd m n = 1"
918 proof -
919   from `x \<in> \<rat>` obtain i::int and n::nat where "n \<noteq> 0" and "x = real i / real n"
921   hence "\<bar>x\<bar> = real(nat(abs i)) / real n" by simp
922   then obtain m :: nat where x_rat: "\<bar>x\<bar> = real m / real n" by blast
923   let ?gcd = "gcd m n"
924   from `n\<noteq>0` have gcd: "?gcd \<noteq> 0" by simp
925   let ?k = "m div ?gcd"
926   let ?l = "n div ?gcd"
927   let ?gcd' = "gcd ?k ?l"
928   have "?gcd dvd m" .. then have gcd_k: "?gcd * ?k = m"
929     by (rule dvd_mult_div_cancel)
930   have "?gcd dvd n" .. then have gcd_l: "?gcd * ?l = n"
931     by (rule dvd_mult_div_cancel)
932   from `n\<noteq>0` and gcd_l have "?l \<noteq> 0" by (auto iff del: neq0_conv)
933   moreover
934   have "\<bar>x\<bar> = real ?k / real ?l"
935   proof -
936     from gcd have "real ?k / real ?l =
937         real (?gcd * ?k) / real (?gcd * ?l)" by simp
938     also from gcd_k and gcd_l have "\<dots> = real m / real n" by simp
939     also from x_rat have "\<dots> = \<bar>x\<bar>" ..
940     finally show ?thesis ..
941   qed
942   moreover
943   have "?gcd' = 1"
944   proof -
945     have "?gcd * ?gcd' = gcd (?gcd * ?k) (?gcd * ?l)"
946       by (rule gcd_mult_distrib_nat)
947     with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp
948     with gcd show ?thesis by auto
949   qed
950   ultimately show ?thesis ..
951 qed
954 subsection{*Numerals and Arithmetic*}
956 instantiation real :: number_ring
957 begin
959 definition
960   real_number_of_def [code del]: "number_of w = real_of_int w"
962 instance
963   by intro_classes (simp add: real_number_of_def)
965 end
967 lemma [code_unfold_post]:
968   "number_of k = real_of_int (number_of k)"
969   unfolding number_of_is_id real_number_of_def ..
972 text{*Collapse applications of @{term real} to @{term number_of}*}
973 lemma real_number_of [simp]: "real (number_of v :: int) = number_of v"
976 lemma real_of_nat_number_of [simp]:
977      "real (number_of v :: nat) =
978         (if neg (number_of v :: int) then 0
979          else (number_of v :: real))"
980 by (simp add: real_of_int_real_of_nat [symmetric])
982 declaration {*
983   K (Lin_Arith.add_inj_thms [@{thm real_of_nat_le_iff} RS iffD2, @{thm real_of_nat_inject} RS iffD2]
984     (* not needed because x < (y::nat) can be rewritten as Suc x <= y: real_of_nat_less_iff RS iffD2 *)
985   #> Lin_Arith.add_inj_thms [@{thm real_of_int_le_iff} RS iffD2, @{thm real_of_int_inject} RS iffD2]
986     (* not needed because x < (y::int) can be rewritten as x + 1 <= y: real_of_int_less_iff RS iffD2 *)
988       @{thm real_of_nat_mult}, @{thm real_of_int_zero}, @{thm real_of_one},
989       @{thm real_of_int_add}, @{thm real_of_int_minus}, @{thm real_of_int_diff},
990       @{thm real_of_int_mult}, @{thm real_of_int_of_nat_eq},
991       @{thm real_of_nat_number_of}, @{thm real_number_of}]
992   #> Lin_Arith.add_inj_const (@{const_name real}, HOLogic.natT --> HOLogic.realT)
993   #> Lin_Arith.add_inj_const (@{const_name real}, HOLogic.intT --> HOLogic.realT))
994 *}
997 subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*}
999 text{*Needed in this non-standard form by Hyperreal/Transcendental*}
1000 lemma real_0_le_divide_iff:
1001      "((0::real) \<le> x/y) = ((x \<le> 0 | 0 \<le> y) & (0 \<le> x | y \<le> 0))"
1002 by (simp add: real_divide_def zero_le_mult_iff, auto)
1004 lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)"
1005 by arith
1007 lemma real_add_eq_0_iff: "(x+y = (0::real)) = (y = -x)"
1008 by auto
1010 lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)"
1011 by auto
1013 lemma real_0_less_add_iff: "((0::real) < x+y) = (-x < y)"
1014 by auto
1016 lemma real_add_le_0_iff: "(x+y \<le> (0::real)) = (y \<le> -x)"
1017 by auto
1019 lemma real_0_le_add_iff: "((0::real) \<le> x+y) = (-x \<le> y)"
1020 by auto
1023 (*
1024 FIXME: we should have this, as for type int, but many proofs would break.
1025 It replaces x+-y by x-y.
1026 declare real_diff_def [symmetric, simp]
1027 *)
1029 subsubsection{*Density of the Reals*}
1031 lemma real_lbound_gt_zero:
1032      "[| (0::real) < d1; 0 < d2 |] ==> \<exists>e. 0 < e & e < d1 & e < d2"
1033 apply (rule_tac x = " (min d1 d2) /2" in exI)
1035 done
1038 text{*Similar results are proved in @{text Fields}*}
1039 lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)"
1040   by auto
1042 lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y"
1043   by auto
1046 subsection{*Absolute Value Function for the Reals*}
1048 lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))"
1051 (* FIXME: redundant, but used by Integration/RealRandVar.thy in AFP *)
1052 lemma abs_le_interval_iff: "(abs x \<le> r) = (-r\<le>x & x\<le>(r::real))"
1053 by (force simp add: abs_le_iff)
1055 lemma abs_add_one_gt_zero [simp]: "(0::real) < 1 + abs(x)"
1058 lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)"
1059 by (rule abs_of_nonneg [OF real_of_nat_ge_zero])
1061 lemma abs_add_one_not_less_self [simp]: "~ abs(x) + (1::real) < x"
1062 by simp
1064 lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \<le> abs(x + -l) + abs(y + -m)"
1065 by simp
1068 subsection {* Implementation of rational real numbers *}
1070 definition Ratreal :: "rat \<Rightarrow> real" where
1071   [simp]: "Ratreal = of_rat"
1073 code_datatype Ratreal
1075 lemma Ratreal_number_collapse [code_post]:
1076   "Ratreal 0 = 0"
1077   "Ratreal 1 = 1"
1078   "Ratreal (number_of k) = number_of k"
1079 by simp_all
1081 lemma zero_real_code [code, code_unfold]:
1082   "0 = Ratreal 0"
1083 by simp
1085 lemma one_real_code [code, code_unfold]:
1086   "1 = Ratreal 1"
1087 by simp
1089 lemma number_of_real_code [code_unfold]:
1090   "number_of k = Ratreal (number_of k)"
1091 by simp
1093 lemma Ratreal_number_of_quotient [code_post]:
1094   "Ratreal (number_of r) / Ratreal (number_of s) = number_of r / number_of s"
1095 by simp
1097 lemma Ratreal_number_of_quotient2 [code_post]:
1098   "Ratreal (number_of r / number_of s) = number_of r / number_of s"
1099 unfolding Ratreal_number_of_quotient [symmetric] Ratreal_def of_rat_divide ..
1101 instantiation real :: eq
1102 begin
1104 definition "eq_class.eq (x\<Colon>real) y \<longleftrightarrow> x - y = 0"
1106 instance by default (simp add: eq_real_def)
1108 lemma real_eq_code [code]: "eq_class.eq (Ratreal x) (Ratreal y) \<longleftrightarrow> eq_class.eq x y"
1109   by (simp add: eq_real_def eq)
1111 lemma real_eq_refl [code nbe]:
1112   "eq_class.eq (x::real) x \<longleftrightarrow> True"
1113   by (rule HOL.eq_refl)
1115 end
1117 lemma real_less_eq_code [code]: "Ratreal x \<le> Ratreal y \<longleftrightarrow> x \<le> y"
1120 lemma real_less_code [code]: "Ratreal x < Ratreal y \<longleftrightarrow> x < y"
1123 lemma real_plus_code [code]: "Ratreal x + Ratreal y = Ratreal (x + y)"
1126 lemma real_times_code [code]: "Ratreal x * Ratreal y = Ratreal (x * y)"
1129 lemma real_uminus_code [code]: "- Ratreal x = Ratreal (- x)"
1132 lemma real_minus_code [code]: "Ratreal x - Ratreal y = Ratreal (x - y)"
1135 lemma real_inverse_code [code]: "inverse (Ratreal x) = Ratreal (inverse x)"
1138 lemma real_divide_code [code]: "Ratreal x / Ratreal y = Ratreal (x / y)"
1141 definition (in term_syntax)
1142   valterm_ratreal :: "rat \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> real \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
1143   [code_unfold]: "valterm_ratreal k = Code_Evaluation.valtermify Ratreal {\<cdot>} k"
1145 notation fcomp (infixl "o>" 60)
1146 notation scomp (infixl "o\<rightarrow>" 60)
1148 instantiation real :: random
1149 begin
1151 definition
1152   "Quickcheck.random i = Quickcheck.random i o\<rightarrow> (\<lambda>r. Pair (valterm_ratreal r))"
1154 instance ..
1156 end
1158 no_notation fcomp (infixl "o>" 60)
1159 no_notation scomp (infixl "o\<rightarrow>" 60)
1161 text {* Setup for SML code generator *}
1163 types_code
1164   real ("(int */ int)")
1165 attach (term_of) {*
1166 fun term_of_real (p, q) =
1167   let
1168     val rT = HOLogic.realT
1169   in
1170     if q = 1 orelse p = 0 then HOLogic.mk_number rT p
1171     else @{term "op / \<Colon> real \<Rightarrow> real \<Rightarrow> real"} \$
1172       HOLogic.mk_number rT p \$ HOLogic.mk_number rT q
1173   end;
1174 *}
1175 attach (test) {*
1176 fun gen_real i =
1177   let
1178     val p = random_range 0 i;
1179     val q = random_range 1 (i + 1);
1180     val g = Integer.gcd p q;
1181     val p' = p div g;
1182     val q' = q div g;
1183     val r = (if one_of [true, false] then p' else ~ p',
1184       if p' = 0 then 1 else q')
1185   in
1186     (r, fn () => term_of_real r)
1187   end;
1188 *}
1190 consts_code
1191   Ratreal ("(_)")
1193 consts_code
1194   "of_int :: int \<Rightarrow> real" ("\<module>real'_of'_int")
1195 attach {*
1196 fun real_of_int i = (i, 1);
1197 *}
1199 setup {*
1200   Nitpick.register_frac_type @{type_name real}
1201    [(@{const_name zero_real_inst.zero_real}, @{const_name Nitpick.zero_frac}),
1202     (@{const_name one_real_inst.one_real}, @{const_name Nitpick.one_frac}),
1203     (@{const_name plus_real_inst.plus_real}, @{const_name Nitpick.plus_frac}),
1204     (@{const_name times_real_inst.times_real}, @{const_name Nitpick.times_frac}),
1205     (@{const_name uminus_real_inst.uminus_real}, @{const_name Nitpick.uminus_frac}),
1206     (@{const_name number_real_inst.number_of_real}, @{const_name Nitpick.number_of_frac}),
1207     (@{const_name inverse_real_inst.inverse_real}, @{const_name Nitpick.inverse_frac}),
1208     (@{const_name ord_real_inst.less_eq_real}, @{const_name Nitpick.less_eq_frac})]
1209 *}
1211 lemmas [nitpick_def] = inverse_real_inst.inverse_real
1212     number_real_inst.number_of_real one_real_inst.one_real
1213     ord_real_inst.less_eq_real plus_real_inst.plus_real
1214     times_real_inst.times_real uminus_real_inst.uminus_real
1215     zero_real_inst.zero_real
1217 end