author  nipkow 
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child 25140  273772abbea2 
permissions  rwrr 
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(* Title : Real/RealDef.thy 
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ID : $Id$ 
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Author : Jacques D. Fleuriot 
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Copyright : 1998 University of Cambridge 

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Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4 
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Additional contributions by Jeremy Avigad 
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*) 
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header{*Defining the Reals from the Positive Reals*} 
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theory RealDef 
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imports PReal 
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uses ("real_arith.ML") 
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begin 
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definition 
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realrel :: "((preal * preal) * (preal * preal)) set" where 
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"realrel = {p. \<exists>x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}" 
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14484  20 
typedef (Real) real = "UNIV//realrel" 
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by (auto simp add: quotient_def) 
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19765  23 
definition 
14484  24 
(** these don't use the overloaded "real" function: users don't see them **) 
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real_of_preal :: "preal => real" where 
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"real_of_preal m = Abs_Real(realrel``{(m + 1, 1)})" 
14484  27 

23879  28 
instance real :: zero 
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real_zero_def: "0 == Abs_Real(realrel``{(1, 1)})" .. 

24198  30 
lemmas [code func del] = real_zero_def 
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23879  32 
instance real :: one 
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real_one_def: "1 == Abs_Real(realrel``{(1 + 1, 1)})" .. 

24198  34 
lemmas [code func del] = real_one_def 
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23879  36 
instance real :: plus 
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real_add_def: "z + w == 

14484  38 
contents (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w). 
23879  39 
{ Abs_Real(realrel``{(x+u, y+v)}) })" .. 
24198  40 
lemmas [code func del] = real_add_def 
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23879  42 
instance real :: minus 
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real_minus_def: " r == contents (\<Union>(x,y) \<in> Rep_Real(r). { Abs_Real(realrel``{(y,x)}) })" 

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real_diff_def: "r  (s::real) == r +  s" .. 

24198  45 
lemmas [code func del] = real_minus_def real_diff_def 
14484  46 

23879  47 
instance real :: times 
14484  48 
real_mult_def: 
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"z * w == 

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contents (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w). 

23879  51 
{ Abs_Real(realrel``{(x*u + y*v, x*v + y*u)}) })" .. 
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lemmas [code func del] = real_mult_def 
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23879  54 
instance real :: inverse 
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real_inverse_def: "inverse (R::real) == (THE S. (R = 0 & S = 0)  S * R = 1)" 

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real_divide_def: "R / (S::real) == R * inverse S" .. 

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lemmas [code func del] = real_inverse_def real_divide_def 
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23879  59 
instance real :: ord 
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real_le_def: "z \<le> (w::real) == 

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\<exists>x y u v. x+v \<le> u+y & (x,y) \<in> Rep_Real z & (u,v) \<in> Rep_Real w" 
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real_less_def: "(x < (y::real)) == (x \<le> y & x \<noteq> y)" .. 
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lemmas [code func del] = real_le_def real_less_def 
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23879  65 
instance real :: abs 
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real_abs_def: "abs (r::real) == (if r < 0 then  r else r)" .. 

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24506  68 
instance real :: sgn 
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real_sgn_def: "sgn x == (if x=0 then 0 else if 0<x then 1 else  1)" .. 

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subsection {* Equivalence relation over positive reals *} 
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lemma preal_trans_lemma: 
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assumes "x + y1 = x1 + y" 
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and "x + y2 = x2 + y" 
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shows "x1 + y2 = x2 + (y1::preal)" 
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proof  
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have "(x1 + y2) + x = (x + y2) + x1" by (simp add: add_ac) 
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also have "... = (x2 + y) + x1" by (simp add: prems) 
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also have "... = x2 + (x1 + y)" by (simp add: add_ac) 
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also have "... = x2 + (x + y1)" by (simp add: prems) 
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also have "... = (x2 + y1) + x" by (simp add: add_ac) 
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finally have "(x1 + y2) + x = (x2 + y1) + x" . 
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thus ?thesis by (rule add_right_imp_eq) 
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qed 
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lemma realrel_iff [simp]: "(((x1,y1),(x2,y2)) \<in> realrel) = (x1 + y2 = x2 + y1)" 
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by (simp add: realrel_def) 

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lemma equiv_realrel: "equiv UNIV realrel" 

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apply (auto simp add: equiv_def refl_def sym_def trans_def realrel_def) 
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apply (blast dest: preal_trans_lemma) 
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done 
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14497  96 
text{*Reduces equality of equivalence classes to the @{term realrel} relation: 
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@{term "(realrel `` {x} = realrel `` {y}) = ((x,y) \<in> realrel)"} *} 

14269  98 
lemmas equiv_realrel_iff = 
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eq_equiv_class_iff [OF equiv_realrel UNIV_I UNIV_I] 

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101 
declare equiv_realrel_iff [simp] 

102 

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14484  104 
lemma realrel_in_real [simp]: "realrel``{(x,y)}: Real" 
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by (simp add: Real_def realrel_def quotient_def, blast) 

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22958  107 
declare Abs_Real_inject [simp] 
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declare Abs_Real_inverse [simp] 
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text{*Case analysis on the representation of a real number as an equivalence 
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class of pairs of positive reals.*} 

113 
lemma eq_Abs_Real [case_names Abs_Real, cases type: real]: 

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"(!!x y. z = Abs_Real(realrel``{(x,y)}) ==> P) ==> P" 

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apply (rule Rep_Real [of z, unfolded Real_def, THEN quotientE]) 

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apply (drule arg_cong [where f=Abs_Real]) 

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apply (auto simp add: Rep_Real_inverse) 

14269  118 
done 
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120 

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subsection {* Addition and Subtraction *} 
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123 
lemma real_add_congruent2_lemma: 

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"[a + ba = aa + b; ab + bc = ac + bb] 

125 
==> a + ab + (ba + bc) = aa + ac + (b + (bb::preal))" 

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apply (simp add: add_assoc) 
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apply (rule add_left_commute [of ab, THEN ssubst]) 
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apply (simp add: add_assoc [symmetric]) 
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apply (simp add: add_ac) 
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done 
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lemma real_add: 

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"Abs_Real (realrel``{(x,y)}) + Abs_Real (realrel``{(u,v)}) = 
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Abs_Real (realrel``{(x+u, y+v)})" 

135 
proof  

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have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). {Abs_Real (realrel `` {(x+u, y+v)})}) w) z) 
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respects2 realrel" 

14497  138 
by (simp add: congruent2_def, blast intro: real_add_congruent2_lemma) 
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thus ?thesis 

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by (simp add: real_add_def UN_UN_split_split_eq 

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UN_equiv_class2 [OF equiv_realrel equiv_realrel]) 
14497  142 
qed 
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14484  144 
lemma real_minus: " Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})" 
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proof  

15169  146 
have "(\<lambda>(x,y). {Abs_Real (realrel``{(y,x)})}) respects realrel" 
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by (simp add: congruent_def add_commute) 
14484  148 
thus ?thesis 
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by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel]) 

150 
qed 

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instance real :: ab_group_add 
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proof 
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fix x y z :: real 
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show "(x + y) + z = x + (y + z)" 
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by (cases x, cases y, cases z, simp add: real_add add_assoc) 
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show "x + y = y + x" 
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by (cases x, cases y, simp add: real_add add_commute) 
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show "0 + x = x" 
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by (cases x, simp add: real_add real_zero_def add_ac) 
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show " x + x = 0" 
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by (cases x, simp add: real_minus real_add real_zero_def add_commute) 
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show "x  y = x +  y" 
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by (simp add: real_diff_def) 
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qed 
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167 

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subsection {* Multiplication *} 
14269  169 

14329  170 
lemma real_mult_congruent2_lemma: 
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"!!(x1::preal). [ x1 + y2 = x2 + y1 ] ==> 

14484  172 
x * x1 + y * y1 + (x * y2 + y * x2) = 
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x * x2 + y * y2 + (x * y1 + y * x1)" 

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apply (simp add: add_left_commute add_assoc [symmetric]) 
23288  175 
apply (simp add: add_assoc right_distrib [symmetric]) 
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apply (simp add: add_commute) 

14269  177 
done 
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179 
lemma real_mult_congruent2: 

15169  180 
"(%p1 p2. 
14484  181 
(%(x1,y1). (%(x2,y2). 
15169  182 
{ Abs_Real (realrel``{(x1*x2 + y1*y2, x1*y2+y1*x2)}) }) p2) p1) 
183 
respects2 realrel" 

14658  184 
apply (rule congruent2_commuteI [OF equiv_realrel], clarify) 
23288  185 
apply (simp add: mult_commute add_commute) 
14269  186 
apply (auto simp add: real_mult_congruent2_lemma) 
187 
done 

188 

189 
lemma real_mult: 

14484  190 
"Abs_Real((realrel``{(x1,y1)})) * Abs_Real((realrel``{(x2,y2)})) = 
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Abs_Real(realrel `` {(x1*x2+y1*y2,x1*y2+y1*x2)})" 

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by (simp add: real_mult_def UN_UN_split_split_eq 

14658  193 
UN_equiv_class2 [OF equiv_realrel equiv_realrel real_mult_congruent2]) 
14269  194 

195 
lemma real_mult_commute: "(z::real) * w = w * z" 

23288  196 
by (cases z, cases w, simp add: real_mult add_ac mult_ac) 
14269  197 

198 
lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)" 

14484  199 
apply (cases z1, cases z2, cases z3) 
23288  200 
apply (simp add: real_mult right_distrib add_ac mult_ac) 
14269  201 
done 
202 

203 
lemma real_mult_1: "(1::real) * z = z" 

14484  204 
apply (cases z) 
23288  205 
apply (simp add: real_mult real_one_def right_distrib 
206 
mult_1_right mult_ac add_ac) 

14269  207 
done 
208 

209 
lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)" 

14484  210 
apply (cases z1, cases z2, cases w) 
23288  211 
apply (simp add: real_add real_mult right_distrib add_ac mult_ac) 
14269  212 
done 
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14329  214 
text{*one and zero are distinct*} 
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lemma real_zero_not_eq_one: "0 \<noteq> (1::real)" 
14484  216 
proof  
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have "(1::preal) < 1 + 1" 
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by (simp add: preal_self_less_add_left) 
14484  219 
thus ?thesis 
23288  220 
by (simp add: real_zero_def real_one_def) 
14484  221 
qed 
14269  222 

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instance real :: comm_ring_1 
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proof 
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fix x y z :: real 
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show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc) 
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show "x * y = y * x" by (rule real_mult_commute) 
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show "1 * x = x" by (rule real_mult_1) 
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show "(x + y) * z = x * z + y * z" by (rule real_add_mult_distrib) 
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show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one) 
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qed 
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subsection {* Inverse and Division *} 
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14484  235 
lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0" 
23288  236 
by (simp add: real_zero_def add_commute) 
14269  237 

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text{*Instead of using an existential quantifier and constructing the inverse 
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within the proof, we could define the inverse explicitly.*} 
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lemma real_mult_inverse_left_ex: "x \<noteq> 0 ==> \<exists>y. y*x = (1::real)" 
14484  242 
apply (simp add: real_zero_def real_one_def, cases x) 
14269  243 
apply (cut_tac x = xa and y = y in linorder_less_linear) 
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apply (auto dest!: less_add_left_Ex simp add: real_zero_iff) 
14334  245 
apply (rule_tac 
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x = "Abs_Real (realrel``{(1, inverse (D) + 1)})" 
14334  247 
in exI) 
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apply (rule_tac [2] 

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x = "Abs_Real (realrel``{(inverse (D) + 1, 1)})" 
14334  250 
in exI) 
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apply (auto simp add: real_mult preal_mult_inverse_right ring_simps) 
14269  252 
done 
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lemma real_mult_inverse_left: "x \<noteq> 0 ==> inverse(x)*x = (1::real)" 
14484  255 
apply (simp add: real_inverse_def) 
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apply (drule real_mult_inverse_left_ex, safe) 
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apply (rule theI, assumption, rename_tac z) 
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apply (subgoal_tac "(z * x) * y = z * (x * y)") 
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apply (simp add: mult_commute) 
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apply (rule mult_assoc) 
14269  261 
done 
14334  262 

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264 
subsection{*The Real Numbers form a Field*} 
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265 

14334  266 
instance real :: field 
267 
proof 

268 
fix x y z :: real 

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269 
show "x \<noteq> 0 ==> inverse x * x = 1" by (rule real_mult_inverse_left) 
14430
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270 
show "x / y = x * inverse y" by (simp add: real_divide_def) 
14334  271 
qed 
272 

273 

14341
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274 
text{*Inverse of zero! Useful to simplify certain equations*} 
14269  275 

14334  276 
lemma INVERSE_ZERO: "inverse 0 = (0::real)" 
14484  277 
by (simp add: real_inverse_def) 
14334  278 

279 
instance real :: division_by_zero 

280 
proof 

281 
show "inverse 0 = (0::real)" by (rule INVERSE_ZERO) 

282 
qed 

283 

14269  284 

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285 
subsection{*The @{text "\<le>"} Ordering*} 
14269  286 

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287 
lemma real_le_refl: "w \<le> (w::real)" 
14484  288 
by (cases w, force simp add: real_le_def) 
14269  289 

14378
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290 
text{*The arithmetic decision procedure is not set up for type preal. 
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291 
This lemma is currently unused, but it could simplify the proofs of the 
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292 
following two lemmas.*} 
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293 
lemma preal_eq_le_imp_le: 
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294 
assumes eq: "a+b = c+d" and le: "c \<le> a" 
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295 
shows "b \<le> (d::preal)" 
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296 
proof  
23288  297 
have "c+d \<le> a+d" by (simp add: prems) 
14378
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298 
hence "a+b \<le> a+d" by (simp add: prems) 
23288  299 
thus "b \<le> d" by simp 
14378
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300 
qed 
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301 

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302 
lemma real_le_lemma: 
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303 
assumes l: "u1 + v2 \<le> u2 + v1" 
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304 
and "x1 + v1 = u1 + y1" 
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305 
and "x2 + v2 = u2 + y2" 
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306 
shows "x1 + y2 \<le> x2 + (y1::preal)" 
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307 
proof  
14378
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308 
have "(x1+v1) + (u2+y2) = (u1+y1) + (x2+v2)" by (simp add: prems) 
23288  309 
hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: add_ac) 
310 
also have "... \<le> (x2+y1) + (u2+v1)" by (simp add: prems) 

311 
finally show ?thesis by simp 

312 
qed 

14378
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313 

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314 
lemma real_le: 
14484  315 
"(Abs_Real(realrel``{(x1,y1)}) \<le> Abs_Real(realrel``{(x2,y2)})) = 
316 
(x1 + y2 \<le> x2 + y1)" 

23288  317 
apply (simp add: real_le_def) 
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318 
apply (auto intro: real_le_lemma) 
14378
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319 
done 
69c4d5997669
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320 

69c4d5997669
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321 
lemma real_le_anti_sym: "[ z \<le> w; w \<le> z ] ==> z = (w::real)" 
15542  322 
by (cases z, cases w, simp add: real_le) 
14378
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323 

69c4d5997669
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324 
lemma real_trans_lemma: 
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325 
assumes "x + v \<le> u + y" 
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326 
and "u + v' \<le> u' + v" 
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327 
and "x2 + v2 = u2 + y2" 
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328 
shows "x + v' \<le> u' + (y::preal)" 
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329 
proof  
23288  330 
have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: add_ac) 
331 
also have "... \<le> (u+y) + (u+v')" by (simp add: prems) 

332 
also have "... \<le> (u+y) + (u'+v)" by (simp add: prems) 

333 
also have "... = (u'+y) + (u+v)" by (simp add: add_ac) 

334 
finally show ?thesis by simp 

15542  335 
qed 
14269  336 

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337 
lemma real_le_trans: "[ i \<le> j; j \<le> k ] ==> i \<le> (k::real)" 
14484  338 
apply (cases i, cases j, cases k) 
339 
apply (simp add: real_le) 

23288  340 
apply (blast intro: real_trans_lemma) 
14334  341 
done 
342 

343 
(* Axiom 'order_less_le' of class 'order': *) 

344 
lemma real_less_le: "((w::real) < z) = (w \<le> z & w \<noteq> z)" 

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345 
by (simp add: real_less_def) 
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346 

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347 
instance real :: order 
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348 
proof qed 
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349 
(assumption  
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350 
rule real_le_refl real_le_trans real_le_anti_sym real_less_le)+ 
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351 

14378
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352 
(* Axiom 'linorder_linear' of class 'linorder': *) 
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353 
lemma real_le_linear: "(z::real) \<le> w  w \<le> z" 
23288  354 
apply (cases z, cases w) 
355 
apply (auto simp add: real_le real_zero_def add_ac) 

14334  356 
done 
357 

358 

359 
instance real :: linorder 

360 
by (intro_classes, rule real_le_linear) 

361 

362 

14378
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363 
lemma real_le_eq_diff: "(x \<le> y) = (xy \<le> (0::real))" 
14484  364 
apply (cases x, cases y) 
14378
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365 
apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus 
23288  366 
add_ac) 
367 
apply (simp_all add: add_assoc [symmetric]) 

15542  368 
done 
14378
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369 

14484  370 
lemma real_add_left_mono: 
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)" 

374 
by (simp add: diff_minus add_ac) 

375 
with le show ?thesis 

14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

376 
by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"] diff_minus) 
14484  377 
qed 
14334  378 

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379 
lemma real_sum_gt_zero_less: "(0 < S + (W::real)) ==> (W < S)" 
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380 
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus) 
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changeset

381 

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382 
lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (W::real))" 
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383 
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus) 
14334  384 

385 
lemma real_mult_order: "[ 0 < x; 0 < y ] ==> (0::real) < x * y" 

14484  386 
apply (cases x, cases y) 
14378
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changeset

387 
apply (simp add: linorder_not_le [where 'a = real, symmetric] 
69c4d5997669
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388 
linorder_not_le [where 'a = preal] 
69c4d5997669
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389 
real_zero_def real_le real_mult) 
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390 
{*Reduce to the (simpler) @{text "\<le>"} relation *} 
16973  391 
apply (auto dest!: less_add_left_Ex 
23288  392 
simp add: add_ac mult_ac 
393 
right_distrib preal_self_less_add_left) 

14334  394 
done 
395 

396 
lemma real_mult_less_mono2: "[ (0::real) < z; x < y ] ==> z * x < z * y" 

397 
apply (rule real_sum_gt_zero_less) 

398 
apply (drule real_less_sum_gt_zero [of x y]) 

399 
apply (drule real_mult_order, assumption) 

400 
apply (simp add: right_distrib) 

401 
done 

402 

22456  403 
instance real :: distrib_lattice 
404 
"inf x y \<equiv> min x y" 

405 
"sup x y \<equiv> max x y" 

406 
by default (auto simp add: inf_real_def sup_real_def min_max.sup_inf_distrib1) 

407 

14378
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paulson
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408 

14334  409 
subsection{*The Reals Form an Ordered Field*} 
410 

411 
instance real :: ordered_field 

412 
proof 

413 
fix x y z :: real 

414 
show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono) 

22962  415 
show "x < y ==> 0 < z ==> z * x < z * y" by (rule real_mult_less_mono2) 
416 
show "\<bar>x\<bar> = (if x < 0 then x else x)" by (simp only: real_abs_def) 

24506  417 
show "sgn x = (if x=0 then 0 else if 0<x then 1 else  1)" 
418 
by (simp only: real_sgn_def) 

14334  419 
qed 
420 

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421 
text{*The function @{term real_of_preal} requires many proofs, but it seems 
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422 
to be essential for proving completeness of the reals from that of the 
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423 
positive reals.*} 
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424 

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425 
lemma real_of_preal_add: 
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426 
"real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y" 
23288  427 
by (simp add: real_of_preal_def real_add left_distrib add_ac) 
14365
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changeset

428 

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429 
lemma real_of_preal_mult: 
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430 
"real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y" 
23288  431 
by (simp add: real_of_preal_def real_mult right_distrib add_ac mult_ac) 
14365
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changeset

432 

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changeset

433 

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434 
text{*Gleason prop 94.4 p 127*} 
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changeset

435 
lemma real_of_preal_trichotomy: 
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diff
changeset

436 
"\<exists>m. (x::real) = real_of_preal m  x = 0  x = (real_of_preal m)" 
14484  437 
apply (simp add: real_of_preal_def real_zero_def, cases x) 
23288  438 
apply (auto simp add: real_minus add_ac) 
14365
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14348
diff
changeset

439 
apply (cut_tac x = x and y = y in linorder_less_linear) 
23288  440 
apply (auto dest!: less_add_left_Ex simp add: add_assoc [symmetric]) 
14365
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changeset

441 
done 
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diff
changeset

442 

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443 
lemma real_of_preal_leD: 
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changeset

444 
"real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2" 
23288  445 
by (simp add: real_of_preal_def real_le) 
14365
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changeset

446 

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changeset

447 
lemma real_of_preal_lessI: "m1 < m2 ==> real_of_preal m1 < real_of_preal m2" 
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changeset

448 
by (auto simp add: real_of_preal_leD linorder_not_le [symmetric]) 
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changeset

449 

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changeset

450 
lemma real_of_preal_lessD: 
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changeset

451 
"real_of_preal m1 < real_of_preal m2 ==> m1 < m2" 
23288  452 
by (simp add: real_of_preal_def real_le linorder_not_le [symmetric]) 
14365
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paulson
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diff
changeset

453 

3d4df8c166ae
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changeset

454 
lemma real_of_preal_less_iff [simp]: 
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changeset

455 
"(real_of_preal m1 < real_of_preal m2) = (m1 < m2)" 
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
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changeset

456 
by (blast intro: real_of_preal_lessI real_of_preal_lessD) 
3d4df8c166ae
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paulson
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diff
changeset

457 

3d4df8c166ae
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changeset

458 
lemma real_of_preal_le_iff: 
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paulson
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changeset

459 
"(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)" 
23288  460 
by (simp add: linorder_not_less [symmetric]) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

461 

3d4df8c166ae
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changeset

462 
lemma real_of_preal_zero_less: "0 < real_of_preal m" 
23288  463 
apply (insert preal_self_less_add_left [of 1 m]) 
464 
apply (auto simp add: real_zero_def real_of_preal_def 

465 
real_less_def real_le_def add_ac) 

466 
apply (rule_tac x="m + 1" in exI, rule_tac x="1" in exI) 

467 
apply (simp add: add_ac) 

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diff
changeset

468 
done 
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parents:
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diff
changeset

469 

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diff
changeset

470 
lemma real_of_preal_minus_less_zero: " real_of_preal m < 0" 
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parents:
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diff
changeset

471 
by (simp add: real_of_preal_zero_less) 
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diff
changeset

472 

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diff
changeset

473 
lemma real_of_preal_not_minus_gt_zero: "~ 0 <  real_of_preal m" 
14484  474 
proof  
475 
from real_of_preal_minus_less_zero 

476 
show ?thesis by (blast dest: order_less_trans) 

477 
qed 

14365
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diff
changeset

478 

3d4df8c166ae
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parents:
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diff
changeset

479 

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changeset

480 
subsection{*Theorems About the Ordering*} 
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changeset

481 

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changeset

482 
lemma real_gt_zero_preal_Ex: "(0 < x) = (\<exists>y. x = real_of_preal y)" 
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parents:
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changeset

483 
apply (auto simp add: real_of_preal_zero_less) 
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diff
changeset

484 
apply (cut_tac x = x in real_of_preal_trichotomy) 
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parents:
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changeset

485 
apply (blast elim!: real_of_preal_not_minus_gt_zero [THEN notE]) 
3d4df8c166ae
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changeset

486 
done 
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parents:
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diff
changeset

487 

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diff
changeset

488 
lemma real_gt_preal_preal_Ex: 
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changeset

489 
"real_of_preal z < x ==> \<exists>y. x = real_of_preal y" 
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parents:
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changeset

490 
by (blast dest!: real_of_preal_zero_less [THEN order_less_trans] 
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parents:
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changeset

491 
intro: real_gt_zero_preal_Ex [THEN iffD1]) 
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changeset

492 

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changeset

493 
lemma real_ge_preal_preal_Ex: 
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494 
"real_of_preal z \<le> x ==> \<exists>y. x = real_of_preal y" 
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parents:
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diff
changeset

495 
by (blast dest: order_le_imp_less_or_eq real_gt_preal_preal_Ex) 
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changeset

496 

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changeset

497 
lemma real_less_all_preal: "y \<le> 0 ==> \<forall>x. y < real_of_preal x" 
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changeset

498 
by (auto elim: order_le_imp_less_or_eq [THEN disjE] 
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parents:
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diff
changeset

499 
intro: real_of_preal_zero_less [THEN [2] order_less_trans] 
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changeset

500 
simp add: real_of_preal_zero_less) 
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diff
changeset

501 

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changeset

502 
lemma real_less_all_real2: "~ 0 < y ==> \<forall>x. y < real_of_preal x" 
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changeset

503 
by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1]) 
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parents:
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changeset

504 

14334  505 

506 
subsection{*More Lemmas*} 

507 

508 
lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)" 

509 
by auto 

510 

511 
lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)" 

512 
by auto 

513 

514 
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)" 

515 
by (force elim: order_less_asym 

516 
simp add: Ring_and_Field.mult_less_cancel_right) 

517 

518 
lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)" 

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519 
apply (simp add: mult_le_cancel_right) 
23289  520 
apply (blast intro: elim: order_less_asym) 
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changeset

521 
done 
14334  522 

523 
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)" 

15923  524 
by(simp add:mult_commute) 
14334  525 

14365
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changeset

526 
lemma real_inverse_gt_one: "[ (0::real) < x; x < 1 ] ==> 1 < inverse x" 
23289  527 
by (simp add: one_less_inverse_iff) (* TODO: generalize/move *) 
14334  528 

529 

24198  530 
subsection {* Embedding numbers into the Reals *} 
531 

532 
abbreviation 

533 
real_of_nat :: "nat \<Rightarrow> real" 

534 
where 

535 
"real_of_nat \<equiv> of_nat" 

536 

537 
abbreviation 

538 
real_of_int :: "int \<Rightarrow> real" 

539 
where 

540 
"real_of_int \<equiv> of_int" 

541 

542 
abbreviation 

543 
real_of_rat :: "rat \<Rightarrow> real" 

544 
where 

545 
"real_of_rat \<equiv> of_rat" 

546 

547 
consts 

548 
(*overloaded constant for injecting other types into "real"*) 

549 
real :: "'a => real" 

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changeset

550 

14378
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generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
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diff
changeset

551 
defs (overloaded) 
24534
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New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

552 
real_of_nat_def [code inline]: "real == real_of_nat" 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
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diff
changeset

553 
real_of_int_def [code inline]: "real == real_of_int" 
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changeset

554 

16819  555 
lemma real_eq_of_nat: "real = of_nat" 
24198  556 
unfolding real_of_nat_def .. 
16819  557 

558 
lemma real_eq_of_int: "real = of_int" 

24198  559 
unfolding real_of_int_def .. 
16819  560 

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changeset

561 
lemma real_of_int_zero [simp]: "real (0::int) = 0" 
14378
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generic of_nat and of_int functions, and generalization of iszero
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diff
changeset

562 
by (simp add: real_of_int_def) 
14365
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paulson
parents:
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diff
changeset

563 

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changeset

564 
lemma real_of_one [simp]: "real (1::int) = (1::real)" 
14378
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paulson
parents:
14369
diff
changeset

565 
by (simp add: real_of_int_def) 
14334  566 

16819  567 
lemma real_of_int_add [simp]: "real(x + y) = real (x::int) + real y" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

568 
by (simp add: real_of_int_def) 
14365
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paulson
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diff
changeset

569 

16819  570 
lemma real_of_int_minus [simp]: "real(x) = real (x::int)" 
14378
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generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
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diff
changeset

571 
by (simp add: real_of_int_def) 
16819  572 

573 
lemma real_of_int_diff [simp]: "real(x  y) = real (x::int)  real y" 

574 
by (simp add: real_of_int_def) 

14365
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changeset

575 

16819  576 
lemma real_of_int_mult [simp]: "real(x * y) = real (x::int) * real y" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
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diff
changeset

577 
by (simp add: real_of_int_def) 
14334  578 

16819  579 
lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))" 
580 
apply (subst real_eq_of_int)+ 

581 
apply (rule of_int_setsum) 

582 
done 

583 

584 
lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) = 

585 
(PROD x:A. real(f x))" 

586 
apply (subst real_eq_of_int)+ 

587 
apply (rule of_int_setprod) 

588 
done 

14365
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parents:
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diff
changeset

589 

3d4df8c166ae
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diff
changeset

590 
lemma real_of_int_zero_cancel [simp]: "(real x = 0) = (x = (0::int))" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

591 
by (simp add: real_of_int_def) 
14365
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

592 

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changeset

593 
lemma real_of_int_inject [iff]: "(real (x::int) = real y) = (x = y)" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

594 
by (simp add: real_of_int_def) 
14365
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

595 

3d4df8c166ae
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diff
changeset

596 
lemma real_of_int_less_iff [iff]: "(real (x::int) < real y) = (x < y)" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

597 
by (simp add: real_of_int_def) 
14365
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

598 

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paulson
parents:
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diff
changeset

599 
lemma real_of_int_le_iff [simp]: "(real (x::int) \<le> real y) = (x \<le> y)" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

600 
by (simp add: real_of_int_def) 
14365
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paulson
parents:
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diff
changeset

601 

16819  602 
lemma real_of_int_gt_zero_cancel_iff [simp]: "(0 < real (n::int)) = (0 < n)" 
603 
by (simp add: real_of_int_def) 

604 

605 
lemma real_of_int_ge_zero_cancel_iff [simp]: "(0 <= real (n::int)) = (0 <= n)" 

606 
by (simp add: real_of_int_def) 

607 

608 
lemma real_of_int_lt_zero_cancel_iff [simp]: "(real (n::int) < 0) = (n < 0)" 

609 
by (simp add: real_of_int_def) 

610 

611 
lemma real_of_int_le_zero_cancel_iff [simp]: "(real (n::int) <= 0) = (n <= 0)" 

612 
by (simp add: real_of_int_def) 

613 

16888  614 
lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))" 
615 
by (auto simp add: abs_if) 

616 

16819  617 
lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)" 
618 
apply (subgoal_tac "real n + 1 = real (n + 1)") 

619 
apply (simp del: real_of_int_add) 

620 
apply auto 

621 
done 

622 

623 
lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)" 

624 
apply (subgoal_tac "real m + 1 = real (m + 1)") 

625 
apply (simp del: real_of_int_add) 

626 
apply simp 

627 
done 

628 

629 
lemma real_of_int_div_aux: "d ~= 0 ==> (real (x::int)) / (real d) = 

630 
real (x div d) + (real (x mod d)) / (real d)" 

631 
proof  

632 
assume "d ~= 0" 

633 
have "x = (x div d) * d + x mod d" 

634 
by auto 

635 
then have "real x = real (x div d) * real d + real(x mod d)" 

636 
by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym]) 

637 
then have "real x / real d = ... / real d" 

638 
by simp 

639 
then show ?thesis 

23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23438
diff
changeset

640 
by (auto simp add: add_divide_distrib ring_simps prems) 
16819  641 
qed 
642 

643 
lemma real_of_int_div: "(d::int) ~= 0 ==> d dvd n ==> 

644 
real(n div d) = real n / real d" 

645 
apply (frule real_of_int_div_aux [of d n]) 

646 
apply simp 

647 
apply (simp add: zdvd_iff_zmod_eq_0) 

648 
done 

649 

650 
lemma real_of_int_div2: 

651 
"0 <= real (n::int) / real (x)  real (n div x)" 

652 
apply (case_tac "x = 0") 

653 
apply simp 

654 
apply (case_tac "0 < x") 

655 
apply (simp add: compare_rls) 

656 
apply (subst real_of_int_div_aux) 

657 
apply simp 

658 
apply simp 

659 
apply (subst zero_le_divide_iff) 

660 
apply auto 

661 
apply (simp add: compare_rls) 

662 
apply (subst real_of_int_div_aux) 

663 
apply simp 

664 
apply simp 

665 
apply (subst zero_le_divide_iff) 

666 
apply auto 

667 
done 

668 

669 
lemma real_of_int_div3: 

670 
"real (n::int) / real (x)  real (n div x) <= 1" 

671 
apply(case_tac "x = 0") 

672 
apply simp 

673 
apply (simp add: compare_rls) 

674 
apply (subst real_of_int_div_aux) 

675 
apply assumption 

676 
apply simp 

677 
apply (subst divide_le_eq) 

678 
apply clarsimp 

679 
apply (rule conjI) 

680 
apply (rule impI) 

681 
apply (rule order_less_imp_le) 

682 
apply simp 

683 
apply (rule impI) 

684 
apply (rule order_less_imp_le) 

685 
apply simp 

686 
done 

687 

688 
lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x" 

689 
by (insert real_of_int_div2 [of n x], simp) 

14365
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paulson
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diff
changeset

690 

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changeset

691 
subsection{*Embedding the Naturals into the Reals*} 
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changeset

692 

14334  693 
lemma real_of_nat_zero [simp]: "real (0::nat) = 0" 
14365
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

694 
by (simp add: real_of_nat_def) 
14334  695 

696 
lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)" 

14365
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

697 
by (simp add: real_of_nat_def) 
14334  698 

14365
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paulson
parents:
14348
diff
changeset

699 
lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

700 
by (simp add: real_of_nat_def) 
14334  701 

702 
(*Not for addsimps: often the LHS is used to represent a positive natural*) 

703 
lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)" 

14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

704 
by (simp add: real_of_nat_def) 
14334  705 

706 
lemma real_of_nat_less_iff [iff]: 

707 
"(real (n::nat) < real m) = (n < m)" 

14365
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

708 
by (simp add: real_of_nat_def) 
14334  709 

710 
lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)" 

14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

711 
by (simp add: real_of_nat_def) 
14334  712 

713 
lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)" 

14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

714 
by (simp add: real_of_nat_def zero_le_imp_of_nat) 
14334  715 

14365
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

716 
lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

717 
by (simp add: real_of_nat_def del: of_nat_Suc) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

718 

14334  719 
lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n" 
23431
25ca91279a9b
change simp rules for of_nat to work like int did previously (reorient of_nat_Suc, remove of_nat_mult [simp]); preserve original variable names in legacy int theorems
huffman
parents:
23289
diff
changeset

720 
by (simp add: real_of_nat_def of_nat_mult) 
14334  721 

16819  722 
lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) = 
723 
(SUM x:A. real(f x))" 

724 
apply (subst real_eq_of_nat)+ 

725 
apply (rule of_nat_setsum) 

726 
done 

727 

728 
lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) = 

729 
(PROD x:A. real(f x))" 

730 
apply (subst real_eq_of_nat)+ 

731 
apply (rule of_nat_setprod) 

732 
done 

733 

734 
lemma real_of_card: "real (card A) = setsum (%x.1) A" 

735 
apply (subst card_eq_setsum) 

736 
apply (subst real_of_nat_setsum) 

737 
apply simp 

738 
done 

739 

14334  740 
lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

741 
by (simp add: real_of_nat_def) 
14334  742 

14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

743 
lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

744 
by (simp add: real_of_nat_def) 
14334  745 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

746 
lemma real_of_nat_diff: "n \<le> m ==> real (m  n) = real (m::nat)  real n" 
23438
dd824e86fa8a
remove simp attribute from of_nat_diff, for backward compatibility with zdiff_int
huffman
parents:
23431
diff
changeset

747 
by (simp add: add: real_of_nat_def of_nat_diff) 
14334  748 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

749 
lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

750 
by (simp add: add: real_of_nat_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

751 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

752 
lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

753 
by (simp add: add: real_of_nat_def) 
14334  754 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

755 
lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

756 
by (simp add: add: real_of_nat_def) 
14334  757 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

758 
lemma real_of_nat_ge_zero_cancel_iff [simp]: "(0 \<le> real (n::nat)) = (0 \<le> n)" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

759 
by (simp add: add: real_of_nat_def) 
14334  760 

16819  761 
lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)" 
762 
apply (subgoal_tac "real n + 1 = real (Suc n)") 

763 
apply simp 

764 
apply (auto simp add: real_of_nat_Suc) 

765 
done 

766 

767 
lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)" 

768 
apply (subgoal_tac "real m + 1 = real (Suc m)") 

769 
apply (simp add: less_Suc_eq_le) 

770 
apply (simp add: real_of_nat_Suc) 

771 
done 

772 

773 
lemma real_of_nat_div_aux: "0 < d ==> (real (x::nat)) / (real d) = 

774 
real (x div d) + (real (x mod d)) / (real d)" 

775 
proof  

776 
assume "0 < d" 

777 
have "x = (x div d) * d + x mod d" 

778 
by auto 

779 
then have "real x = real (x div d) * real d + real(x mod d)" 

780 
by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym]) 

781 
then have "real x / real d = \<dots> / real d" 

782 
by simp 

783 
then show ?thesis 

23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23438
diff
changeset

784 
by (auto simp add: add_divide_distrib ring_simps prems) 
16819  785 
qed 
786 

787 
lemma real_of_nat_div: "0 < (d::nat) ==> d dvd n ==> 

788 
real(n div d) = real n / real d" 

789 
apply (frule real_of_nat_div_aux [of d n]) 

790 
apply simp 

791 
apply (subst dvd_eq_mod_eq_0 [THEN sym]) 

792 
apply assumption 

793 
done 

794 

795 
lemma real_of_nat_div2: 

796 
"0 <= real (n::nat) / real (x)  real (n div x)" 

25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

797 
apply(case_tac "x = 0") 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

798 
apply (simp) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

799 
apply (simp add: compare_rls) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

800 
apply (subst real_of_nat_div_aux) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

801 
apply simp 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

802 
apply simp 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

803 
apply (subst zero_le_divide_iff) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

804 
apply simp 
16819  805 
done 
806 

807 
lemma real_of_nat_div3: 

808 
"real (n::nat) / real (x)  real (n div x) <= 1" 

25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

809 
apply(case_tac "x = 0") 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

810 
apply (simp) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

811 
apply (simp add: compare_rls) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

812 
apply (subst real_of_nat_div_aux) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

813 
apply simp 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

814 
apply simp 
16819  815 
done 
816 

817 
lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x" 

818 
by (insert real_of_nat_div2 [of n x], simp) 

819 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

820 
lemma real_of_int_real_of_nat: "real (int n) = real n" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

821 
by (simp add: real_of_nat_def real_of_int_def int_eq_of_nat) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

822 

14426  823 
lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n" 
824 
by (simp add: real_of_int_def real_of_nat_def) 

14334  825 

16819  826 
lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x" 
827 
apply (subgoal_tac "real(int(nat x)) = real(nat x)") 

828 
apply force 

829 
apply (simp only: real_of_int_real_of_nat) 

830 
done 

14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

831 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

832 
subsection{*Numerals and Arithmetic*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

833 

24198  834 
instance real :: number_ring 
835 
real_number_of_def: "number_of w \<equiv> real_of_int w" 

836 
by intro_classes (simp add: real_number_of_def) 

14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

837 

24198  838 
lemma [code, code unfold]: 
839 
"number_of k = real_of_int (number_of k)" 

840 
unfolding number_of_is_id real_number_of_def .. 

14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

841 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

842 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

843 
text{*Collapse applications of @{term real} to @{term number_of}*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

844 
lemma real_number_of [simp]: "real (number_of v :: int) = number_of v" 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

845 
by (simp add: real_of_int_def of_int_number_of_eq) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

846 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

847 
lemma real_of_nat_number_of [simp]: 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

848 
"real (number_of v :: nat) = 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

849 
(if neg (number_of v :: int) then 0 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

850 
else (number_of v :: real))" 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

851 
by (simp add: real_of_int_real_of_nat [symmetric] int_nat_number_of) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

852 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

853 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

854 
use "real_arith.ML" 
24075  855 
declaration {* K real_arith_setup *} 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

856 

19023
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
16973
diff
changeset

857 

14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

858 
subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

859 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

860 
text{*Needed in this nonstandard form by Hyperreal/Transcendental*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

861 
lemma real_0_le_divide_iff: 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

862 
"((0::real) \<le> x/y) = ((x \<le> 0  0 \<le> y) & (0 \<le> x  y \<le> 0))" 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

863 
by (simp add: real_divide_def zero_le_mult_iff, auto) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

864 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

865 
lemma real_add_minus_iff [simp]: "(x +  a = (0::real)) = (x=a)" 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

866 
by arith 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

867 

15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15077
diff
changeset

868 
lemma real_add_eq_0_iff: "(x+y = (0::real)) = (y = x)" 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

869 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

870 

15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15077
diff
changeset

871 
lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < x)" 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

872 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

873 

15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15077
diff
changeset

874 
lemma real_0_less_add_iff: "((0::real) < x+y) = (x < y)" 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

875 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

876 

15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15077
diff
changeset

877 
lemma real_add_le_0_iff: "(x+y \<le> (0::real)) = (y \<le> x)" 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

878 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

879 

15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15077
diff
changeset

880 
lemma real_0_le_add_iff: "((0::real) \<le> x+y) = (x \<le> y)" 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

881 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

882 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

883 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

884 
(* 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

885 
FIXME: we should have this, as for type int, but many proofs would break. 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

886 
It replaces x+y by xy. 
15086  887 
declare real_diff_def [symmetric, simp] 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

888 
*) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

889 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

890 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

891 
subsubsection{*Density of the Reals*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

892 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

893 
lemma real_lbound_gt_zero: 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

894 
"[ (0::real) < d1; 0 < d2 ] ==> \<exists>e. 0 < e & e < d1 & e < d2" 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

895 
apply (rule_tac x = " (min d1 d2) /2" in exI) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

896 
apply (simp add: min_def) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

897 
done 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

898 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

899 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

900 
text{*Similar results are proved in @{text Ring_and_Field}*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

901 
lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)" 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

902 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

903 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

904 
lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y" 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

905 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

906 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

907 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

908 
subsection{*Absolute Value Function for the Reals*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

909 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

910 
lemma abs_minus_add_cancel: "abs(x + (y)) = abs (y + ((x::real)))" 
15003  911 
by (simp add: abs_if) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

912 

23289  913 
(* FIXME: redundant, but used by Integration/RealRandVar.thy in AFP *) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

914 
lemma abs_le_interval_iff: "(abs x \<le> r) = (r\<le>x & x\<le>(r::real))" 
14738  915 
by (force simp add: OrderedGroup.abs_le_iff) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

916 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

917 
lemma abs_add_one_gt_zero [simp]: "(0::real) < 1 + abs(x)" 
15003  918 
by (simp add: abs_if) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

919 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

920 
lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)" 
22958  921 
by (rule abs_of_nonneg [OF real_of_nat_ge_zero]) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

922 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

923 
lemma abs_add_one_not_less_self [simp]: "~ abs(x) + (1::real) < x" 
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19765
diff
changeset

924 
by simp 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

925 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

926 
lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (l + m)) \<le> abs(x + l) + abs(y + m)" 
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19765
diff
changeset

927 
by simp 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

928 

24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

929 

09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

930 
subsection {* Implementation of rational real numbers as pairs of integers *} 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

931 

09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

932 
definition 
24623  933 
Ratreal :: "int \<times> int \<Rightarrow> real" 
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

934 
where 
24623  935 
"Ratreal = INum" 
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

936 

24623  937 
code_datatype Ratreal 
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

938 

24623  939 
lemma Ratreal_simp: 
940 
"Ratreal (k, l) = real_of_int k / real_of_int l" 

941 
unfolding Ratreal_def INum_def by simp 

24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

942 

24623  943 
lemma Ratreal_zero [simp]: "Ratreal 0\<^sub>N = 0" 
944 
by (simp add: Ratreal_simp) 

24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

945 

24623  946 
lemma Ratreal_lit [simp]: "Ratreal i\<^sub>N = real_of_int i" 
947 
by (simp add: Ratreal_simp) 

24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

948 

09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

949 
lemma zero_real_code [code, code unfold]: 
24623  950 
"0 = Ratreal 0\<^sub>N" by simp 
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

951 

09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

952 
lemma one_real_code [code, code unfold]: 
24623  953 
"1 = Ratreal 1\<^sub>N" by simp 
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

954 

09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

955 
instance real :: eq .. 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

956 

24623  957 
lemma real_eq_code [code]: "Ratreal x = Ratreal y \<longleftrightarrow> normNum x = normNum y" 
958 
unfolding Ratreal_def INum_normNum_iff .. 

24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

959 

24623  960 
lemma real_less_eq_code [code]: "Ratreal x \<le> Ratreal y \<longleftrightarrow> normNum x \<le>\<^sub>N normNum y" 
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

961 
proof  
24623  962 
have "normNum x \<le>\<^sub>N normNum y \<longleftrightarrow> Ratreal (normNum x) \<le> Ratreal (normNum y)" 
963 
by (simp add: Ratreal_def del: normNum) 

964 
also have "\<dots> = (Ratreal x \<le> Ratreal y)" by (simp add: Ratreal_def) 

24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

965 
finally show ?thesis by simp 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

966 
qed 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

967 

24623  968 
lemma real_less_code [code]: "Ratreal x < Ratreal y \<longleftrightarrow> normNum x <\<^sub>N normNum y" 
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

969 
proof  
24623  970 
have "normNum x <\<^sub>N normNum y \<longleftrightarrow> Ratreal (normNum x) < Ratreal (normNum y)" 
971 
by (simp add: Ratreal_def del: normNum) 

972 
also have "\<dots> = (Ratreal x < Ratreal y)" by (simp add: Ratreal_def) 

24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

973 
finally show ?thesis by simp 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

974 
qed 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

975 

24623  976 
lemma real_add_code [code]: "Ratreal x + Ratreal y = Ratreal (x +\<^sub>N y)" 
977 
unfolding Ratreal_def by simp 

24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

978 

24623  979 
lemma real_mul_code [code]: "Ratreal x * Ratreal y = Ratreal (x *\<^sub>N y)" 
980 
unfolding Ratreal_def by simp 

24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

981 

24623  982 
lemma real_neg_code [code]: " Ratreal x = Ratreal (~\<^sub>N x)" 
983 
unfolding Ratreal_def by simp 

24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

984 

24623  985 
lemma real_sub_code [code]: "Ratreal x  Ratreal y = Ratreal (x \<^sub>N y)" 
986 
unfolding Ratreal_def by simp 

24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

987 

24623  988 
lemma real_inv_code [code]: "inverse (Ratreal x) = Ratreal (Ninv x)" 
989 
unfolding Ratreal_def Ninv real_divide_def by simp 

24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

990 

24623  991 
lemma real_div_code [code]: "Ratreal x / Ratreal y = Ratreal (x \<div>\<^sub>N y)" 
992 
unfolding Ratreal_def by simp 

24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

993 

24623  994 
text {* Setup for SML code generator *} 
23031
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

995 

9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

996 
types_code 
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

997 
real ("(int */ int)") 
23031
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

998 
attach (term_of) {* 
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

999 
fun term_of_real (p, q) = 
24623  1000 
let 
1001 
val rT = HOLogic.realT 

24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1002 
in 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1003 
if q = 1 orelse p = 0 then HOLogic.mk_number rT p 
24623  1004 
else @{term "op / \<Colon> real \<Rightarrow> real \<Rightarrow> real"} $ 
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1005 
HOLogic.mk_number rT p $ HOLogic.mk_number rT q 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1006 
end; 
23031
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

1007 
*} 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

1008 
attach (test) {* 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

1009 
fun gen_real i = 
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1010 
let 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1011 
val p = random_range 0 i; 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1012 
val q = random_range 1 (i + 1); 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1013 
val g = Integer.gcd p q; 
24630
351a308ab58d
simplified type int (eliminated IntInf.int, integer);
wenzelm
parents:
24623
diff
changeset

1014 
val p' = p div g; 
351a308ab58d
simplified type int (eliminated IntInf.int, integer);
wenzelm
parents:
24623
diff
changeset

1015 
val q' = q div g; 
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1016 
in 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1017 
(if one_of [true, false] then p' else ~ p', 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1018 
if p' = 0 then 0 else q') 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1019 
end; 
23031
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

1020 
*} 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

1021 

9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

1022 
consts_code 
24623  1023 
Ratreal ("(_)") 
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1024 

09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1025 
consts_code 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1026 
"of_int :: int \<Rightarrow> real" ("\<module>real'_of'_int") 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1027 
attach {* 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1028 
fun real_of_int 0 = (0, 0) 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1029 
 real_of_int i = (i, 1); 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1030 
*} 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1031 

09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1032 
declare real_of_int_of_nat_eq [symmetric, code] 
23031
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

1033 

5588  1034 
end 