author  haftmann 
Tue, 07 Oct 2008 16:07:25 +0200  
changeset 28520  376b9c083b04 
parent 28351  abfc66969d1f 
child 28562  4e74209f113e 
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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[code func del]: "realrel = {p. \<exists>x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}" 
14269  19 

14484  20 
typedef (Real) real = "UNIV//realrel" 
14269  21 
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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[code func del]: "real_of_preal m = Abs_Real (realrel `` {(m + 1, 1)})" 
14484  27 

25762  28 
instantiation real :: "{zero, one, plus, minus, uminus, times, inverse, ord, abs, sgn}" 
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begin 
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definition 
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real_zero_def [code func del]: "0 = Abs_Real(realrel``{(1, 1)})" 
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definition 
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real_one_def [code func del]: "1 = Abs_Real(realrel``{(1 + 1, 1)})" 
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definition 
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real_add_def [code func del]: "z + w = 
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contents (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w). 
27964  40 
{ Abs_Real(realrel``{(x+u, y+v)}) })" 
10606  41 

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definition 
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real_minus_def [code func del]: " r = contents (\<Union>(x,y) \<in> Rep_Real(r). { Abs_Real(realrel``{(y,x)}) })" 
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definition 
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real_diff_def [code func del]: "r  (s::real) = r +  s" 
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definition 
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real_mult_def [code func del]: 
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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). 
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{ Abs_Real(realrel``{(x*u + y*v, x*v + y*u)}) })" 
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definition 
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real_inverse_def [code func del]: "inverse (R::real) = (THE S. (R = 0 & S = 0)  S * R = 1)" 
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definition 
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real_divide_def [code func del]: "R / (S::real) = R * inverse S" 
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definition 
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real_le_def [code func del]: "z \<le> (w::real) \<longleftrightarrow> 
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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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definition 
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real_less_def [code func del]: "x < (y\<Colon>real) \<longleftrightarrow> x \<le> y \<and> x \<noteq> y" 
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definition 
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real_abs_def: "abs (r::real) = (if r < 0 then  r else r)" 
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definition 
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real_sgn_def: "sgn (x::real) = (if x=0 then 0 else if 0<x then 1 else  1)" 
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instance .. 
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end 
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subsection {* Equivalence relation over positive reals *} 
14269  78 

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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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14269  93 

14484  94 
lemma realrel_iff [simp]: "(((x1,y1),(x2,y2)) \<in> realrel) = (x1 + y2 = x2 + y1)" 
95 
by (simp add: realrel_def) 

14269  96 

97 
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) 
14269  100 
done 
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14497  102 
text{*Reduces equality of equivalence classes to the @{term realrel} relation: 
103 
@{term "(realrel `` {x} = realrel `` {y}) = ((x,y) \<in> realrel)"} *} 

14269  104 
lemmas equiv_realrel_iff = 
105 
eq_equiv_class_iff [OF equiv_realrel UNIV_I UNIV_I] 

106 

107 
declare equiv_realrel_iff [simp] 

108 

14497  109 

14484  110 
lemma realrel_in_real [simp]: "realrel``{(x,y)}: Real" 
111 
by (simp add: Real_def realrel_def quotient_def, blast) 

14269  112 

22958  113 
declare Abs_Real_inject [simp] 
14484  114 
declare Abs_Real_inverse [simp] 
14269  115 

116 

14484  117 
text{*Case analysis on the representation of a real number as an equivalence 
118 
class of pairs of positive reals.*} 

119 
lemma eq_Abs_Real [case_names Abs_Real, cases type: real]: 

120 
"(!!x y. z = Abs_Real(realrel``{(x,y)}) ==> P) ==> P" 

121 
apply (rule Rep_Real [of z, unfolded Real_def, THEN quotientE]) 

122 
apply (drule arg_cong [where f=Abs_Real]) 

123 
apply (auto simp add: Rep_Real_inverse) 

14269  124 
done 
125 

126 

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subsection {* Addition and Subtraction *} 
14269  128 

129 
lemma real_add_congruent2_lemma: 

130 
"[a + ba = aa + b; ab + bc = ac + bb] 

131 
==> a + ab + (ba + bc) = aa + ac + (b + (bb::preal))" 

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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) 
14269  136 
done 
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138 
lemma real_add: 

14497  139 
"Abs_Real (realrel``{(x,y)}) + Abs_Real (realrel``{(u,v)}) = 
140 
Abs_Real (realrel``{(x+u, y+v)})" 

141 
proof  

15169  142 
have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). {Abs_Real (realrel `` {(x+u, y+v)})}) w) z) 
143 
respects2 realrel" 

14497  144 
by (simp add: congruent2_def, blast intro: real_add_congruent2_lemma) 
145 
thus ?thesis 

146 
by (simp add: real_add_def UN_UN_split_split_eq 

14658  147 
UN_equiv_class2 [OF equiv_realrel equiv_realrel]) 
14497  148 
qed 
14269  149 

14484  150 
lemma real_minus: " Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})" 
151 
proof  

15169  152 
have "(\<lambda>(x,y). {Abs_Real (realrel``{(y,x)})}) respects realrel" 
23288  153 
by (simp add: congruent_def add_commute) 
14484  154 
thus ?thesis 
155 
by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel]) 

156 
qed 

14334  157 

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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 
14269  172 

173 

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subsection {* Multiplication *} 
14269  175 

14329  176 
lemma real_mult_congruent2_lemma: 
177 
"!!(x1::preal). [ x1 + y2 = x2 + y1 ] ==> 

14484  178 
x * x1 + y * y1 + (x * y2 + y * x2) = 
179 
x * x2 + y * y2 + (x * y1 + y * x1)" 

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apply (simp add: add_left_commute add_assoc [symmetric]) 
23288  181 
apply (simp add: add_assoc right_distrib [symmetric]) 
182 
apply (simp add: add_commute) 

14269  183 
done 
184 

185 
lemma real_mult_congruent2: 

15169  186 
"(%p1 p2. 
14484  187 
(%(x1,y1). (%(x2,y2). 
15169  188 
{ Abs_Real (realrel``{(x1*x2 + y1*y2, x1*y2+y1*x2)}) }) p2) p1) 
189 
respects2 realrel" 

14658  190 
apply (rule congruent2_commuteI [OF equiv_realrel], clarify) 
23288  191 
apply (simp add: mult_commute add_commute) 
14269  192 
apply (auto simp add: real_mult_congruent2_lemma) 
193 
done 

194 

195 
lemma real_mult: 

14484  196 
"Abs_Real((realrel``{(x1,y1)})) * Abs_Real((realrel``{(x2,y2)})) = 
197 
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  199 
UN_equiv_class2 [OF equiv_realrel equiv_realrel real_mult_congruent2]) 
14269  200 

201 
lemma real_mult_commute: "(z::real) * w = w * z" 

23288  202 
by (cases z, cases w, simp add: real_mult add_ac mult_ac) 
14269  203 

204 
lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)" 

14484  205 
apply (cases z1, cases z2, cases z3) 
23288  206 
apply (simp add: real_mult right_distrib add_ac mult_ac) 
14269  207 
done 
208 

209 
lemma real_mult_1: "(1::real) * z = z" 

14484  210 
apply (cases z) 
23288  211 
apply (simp add: real_mult real_one_def right_distrib 
212 
mult_1_right mult_ac add_ac) 

14269  213 
done 
214 

215 
lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)" 

14484  216 
apply (cases z1, cases z2, cases w) 
23288  217 
apply (simp add: real_add real_mult right_distrib add_ac mult_ac) 
14269  218 
done 
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14329  220 
text{*one and zero are distinct*} 
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lemma real_zero_not_eq_one: "0 \<noteq> (1::real)" 
14484  222 
proof  
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have "(1::preal) < 1 + 1" 
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by (simp add: preal_self_less_add_left) 
14484  225 
thus ?thesis 
23288  226 
by (simp add: real_zero_def real_one_def) 
14484  227 
qed 
14269  228 

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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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238 

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subsection {* Inverse and Division *} 
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14484  241 
lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0" 
23288  242 
by (simp add: real_zero_def add_commute) 
14269  243 

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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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246 

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lemma real_mult_inverse_left_ex: "x \<noteq> 0 ==> \<exists>y. y*x = (1::real)" 
14484  248 
apply (simp add: real_zero_def real_one_def, cases x) 
14269  249 
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  251 
apply (rule_tac 
23287
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x = "Abs_Real (realrel``{(1, inverse (D) + 1)})" 
14334  253 
in exI) 
254 
apply (rule_tac [2] 

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x = "Abs_Real (realrel``{(inverse (D) + 1, 1)})" 
14334  256 
in exI) 
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apply (auto simp add: real_mult preal_mult_inverse_right ring_simps) 
14269  258 
done 
259 

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lemma real_mult_inverse_left: "x \<noteq> 0 ==> inverse(x)*x = (1::real)" 
14484  261 
apply (simp add: real_inverse_def) 
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apply (drule real_mult_inverse_left_ex, safe) 
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263 
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  267 
done 
14334  268 

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subsection{*The Real Numbers form a Field*} 
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271 

14334  272 
instance real :: field 
273 
proof 

274 
fix x y z :: real 

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show "x \<noteq> 0 ==> inverse x * x = 1" by (rule real_mult_inverse_left) 
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show "x / y = x * inverse y" by (simp add: real_divide_def) 
14334  277 
qed 
278 

279 

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text{*Inverse of zero! Useful to simplify certain equations*} 
14269  281 

14334  282 
lemma INVERSE_ZERO: "inverse 0 = (0::real)" 
14484  283 
by (simp add: real_inverse_def) 
14334  284 

285 
instance real :: division_by_zero 

286 
proof 

287 
show "inverse 0 = (0::real)" by (rule INVERSE_ZERO) 

288 
qed 

289 

14269  290 

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subsection{*The @{text "\<le>"} Ordering*} 
14269  292 

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lemma real_le_refl: "w \<le> (w::real)" 
14484  294 
by (cases w, force simp add: real_le_def) 
14269  295 

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text{*The arithmetic decision procedure is not set up for type preal. 
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This lemma is currently unused, but it could simplify the proofs of the 
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298 
following two lemmas.*} 
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299 
lemma preal_eq_le_imp_le: 
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assumes eq: "a+b = c+d" and le: "c \<le> a" 
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shows "b \<le> (d::preal)" 
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302 
proof  
23288  303 
have "c+d \<le> a+d" by (simp add: prems) 
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hence "a+b \<le> a+d" by (simp add: prems) 
23288  305 
thus "b \<le> d" by simp 
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qed 
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307 

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lemma real_le_lemma: 
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assumes l: "u1 + v2 \<le> u2 + v1" 
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and "x1 + v1 = u1 + y1" 
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and "x2 + v2 = u2 + y2" 
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shows "x1 + y2 \<le> x2 + (y1::preal)" 
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proof  
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314 
have "(x1+v1) + (u2+y2) = (u1+y1) + (x2+v2)" by (simp add: prems) 
23288  315 
hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: add_ac) 
316 
also have "... \<le> (x2+y1) + (u2+v1)" by (simp add: prems) 

317 
finally show ?thesis by simp 

318 
qed 

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319 

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320 
lemma real_le: 
14484  321 
"(Abs_Real(realrel``{(x1,y1)}) \<le> Abs_Real(realrel``{(x2,y2)})) = 
322 
(x1 + y2 \<le> x2 + y1)" 

23288  323 
apply (simp add: real_le_def) 
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apply (auto intro: real_le_lemma) 
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325 
done 
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326 

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327 
lemma real_le_anti_sym: "[ z \<le> w; w \<le> z ] ==> z = (w::real)" 
15542  328 
by (cases z, cases w, simp add: real_le) 
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329 

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330 
lemma real_trans_lemma: 
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331 
assumes "x + v \<le> u + y" 
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332 
and "u + v' \<le> u' + v" 
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333 
and "x2 + v2 = u2 + y2" 
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334 
shows "x + v' \<le> u' + (y::preal)" 
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335 
proof  
23288  336 
have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: add_ac) 
337 
also have "... \<le> (u+y) + (u+v')" by (simp add: prems) 

338 
also have "... \<le> (u+y) + (u'+v)" by (simp add: prems) 

339 
also have "... = (u'+y) + (u+v)" by (simp add: add_ac) 

340 
finally show ?thesis by simp 

15542  341 
qed 
14269  342 

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343 
lemma real_le_trans: "[ i \<le> j; j \<le> k ] ==> i \<le> (k::real)" 
14484  344 
apply (cases i, cases j, cases k) 
345 
apply (simp add: real_le) 

23288  346 
apply (blast intro: real_trans_lemma) 
14334  347 
done 
348 

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349 
instance real :: order 
27682  350 
proof 
351 
fix u v :: real 

352 
show "u < v \<longleftrightarrow> u \<le> v \<and> \<not> v \<le> u" 

353 
by (auto simp add: real_less_def intro: real_le_anti_sym) 

354 
qed (assumption  rule real_le_refl real_le_trans real_le_anti_sym)+ 

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355 

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356 
(* Axiom 'linorder_linear' of class 'linorder': *) 
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357 
lemma real_le_linear: "(z::real) \<le> w  w \<le> z" 
23288  358 
apply (cases z, cases w) 
359 
apply (auto simp add: real_le real_zero_def add_ac) 

14334  360 
done 
361 

362 
instance real :: linorder 

363 
by (intro_classes, rule real_le_linear) 

364 

365 

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366 
lemma real_le_eq_diff: "(x \<le> y) = (xy \<le> (0::real))" 
14484  367 
apply (cases x, cases y) 
14378
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368 
apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus 
23288  369 
add_ac) 
370 
apply (simp_all add: add_assoc [symmetric]) 

15542  371 
done 
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372 

14484  373 
lemma real_add_left_mono: 
374 
assumes le: "x \<le> y" shows "z + x \<le> z + (y::real)" 

375 
proof  

27668  376 
have "z + x  (z + y) = (z + z) + (x  y)" 
14484  377 
by (simp add: diff_minus add_ac) 
378 
with le show ?thesis 

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379 
by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"] diff_minus) 
14484  380 
qed 
14334  381 

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

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385 
lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (W::real))" 
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386 
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus) 
14334  387 

388 
lemma real_mult_order: "[ 0 < x; 0 < y ] ==> (0::real) < x * y" 

14484  389 
apply (cases x, cases y) 
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390 
apply (simp add: linorder_not_le [where 'a = real, symmetric] 
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391 
linorder_not_le [where 'a = preal] 
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392 
real_zero_def real_le real_mult) 
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393 
{*Reduce to the (simpler) @{text "\<le>"} relation *} 
16973  394 
apply (auto dest!: less_add_left_Ex 
23288  395 
simp add: add_ac mult_ac 
396 
right_distrib preal_self_less_add_left) 

14334  397 
done 
398 

399 
lemma real_mult_less_mono2: "[ (0::real) < z; x < y ] ==> z * x < z * y" 

400 
apply (rule real_sum_gt_zero_less) 

401 
apply (drule real_less_sum_gt_zero [of x y]) 

402 
apply (drule real_mult_order, assumption) 

403 
apply (simp add: right_distrib) 

404 
done 

405 

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406 
instantiation real :: distrib_lattice 
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407 
begin 
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408 

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409 
definition 
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410 
"(inf \<Colon> real \<Rightarrow> real \<Rightarrow> real) = min" 
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411 

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412 
definition 
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413 
"(sup \<Colon> real \<Rightarrow> real \<Rightarrow> real) = max" 
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414 

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415 
instance 
22456  416 
by default (auto simp add: inf_real_def sup_real_def min_max.sup_inf_distrib1) 
417 

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418 
end 
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419 

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420 

14334  421 
subsection{*The Reals Form an Ordered Field*} 
422 

423 
instance real :: ordered_field 

424 
proof 

425 
fix x y z :: real 

426 
show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono) 

22962  427 
show "x < y ==> 0 < z ==> z * x < z * y" by (rule real_mult_less_mono2) 
428 
show "\<bar>x\<bar> = (if x < 0 then x else x)" by (simp only: real_abs_def) 

24506  429 
show "sgn x = (if x=0 then 0 else if 0<x then 1 else  1)" 
430 
by (simp only: real_sgn_def) 

14334  431 
qed 
432 

25303
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433 
instance real :: lordered_ab_group_add .. 
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434 

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435 
text{*The function @{term real_of_preal} requires many proofs, but it seems 
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436 
to be essential for proving completeness of the reals from that of the 
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437 
positive reals.*} 
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438 

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439 
lemma real_of_preal_add: 
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440 
"real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y" 
23288  441 
by (simp add: real_of_preal_def real_add left_distrib add_ac) 
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442 

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443 
lemma real_of_preal_mult: 
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444 
"real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y" 
23288  445 
by (simp add: real_of_preal_def real_mult right_distrib add_ac mult_ac) 
14365
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

446 

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

447 

3d4df8c166ae
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parents:
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diff
changeset

448 
text{*Gleason prop 94.4 p 127*} 
3d4df8c166ae
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paulson
parents:
14348
diff
changeset

449 
lemma real_of_preal_trichotomy: 
3d4df8c166ae
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paulson
parents:
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diff
changeset

450 
"\<exists>m. (x::real) = real_of_preal m  x = 0  x = (real_of_preal m)" 
14484  451 
apply (simp add: real_of_preal_def real_zero_def, cases x) 
23288  452 
apply (auto simp add: real_minus add_ac) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

453 
apply (cut_tac x = x and y = y in linorder_less_linear) 
23288  454 
apply (auto dest!: less_add_left_Ex simp add: add_assoc [symmetric]) 
14365
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paulson
parents:
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diff
changeset

455 
done 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

456 

3d4df8c166ae
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paulson
parents:
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diff
changeset

457 
lemma real_of_preal_leD: 
3d4df8c166ae
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paulson
parents:
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diff
changeset

458 
"real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2" 
23288  459 
by (simp add: real_of_preal_def real_le) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

460 

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

461 
lemma real_of_preal_lessI: "m1 < m2 ==> real_of_preal m1 < real_of_preal m2" 
3d4df8c166ae
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parents:
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diff
changeset

462 
by (auto simp add: real_of_preal_leD linorder_not_le [symmetric]) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

463 

3d4df8c166ae
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parents:
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diff
changeset

464 
lemma real_of_preal_lessD: 
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parents:
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diff
changeset

465 
"real_of_preal m1 < real_of_preal m2 ==> m1 < m2" 
23288  466 
by (simp add: real_of_preal_def real_le linorder_not_le [symmetric]) 
14365
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paulson
parents:
14348
diff
changeset

467 

3d4df8c166ae
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parents:
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diff
changeset

468 
lemma real_of_preal_less_iff [simp]: 
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parents:
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diff
changeset

469 
"(real_of_preal m1 < real_of_preal m2) = (m1 < m2)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

470 
by (blast intro: real_of_preal_lessI real_of_preal_lessD) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

471 

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

472 
lemma real_of_preal_le_iff: 
3d4df8c166ae
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parents:
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diff
changeset

473 
"(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)" 
23288  474 
by (simp add: linorder_not_less [symmetric]) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
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parents:
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diff
changeset

475 

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

476 
lemma real_of_preal_zero_less: "0 < real_of_preal m" 
23288  477 
apply (insert preal_self_less_add_left [of 1 m]) 
478 
apply (auto simp add: real_zero_def real_of_preal_def 

479 
real_less_def real_le_def add_ac) 

480 
apply (rule_tac x="m + 1" in exI, rule_tac x="1" in exI) 

481 
apply (simp add: add_ac) 

14365
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parents:
14348
diff
changeset

482 
done 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

483 

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

484 
lemma real_of_preal_minus_less_zero: " real_of_preal m < 0" 
3d4df8c166ae
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paulson
parents:
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diff
changeset

485 
by (simp add: real_of_preal_zero_less) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

486 

3d4df8c166ae
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parents:
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diff
changeset

487 
lemma real_of_preal_not_minus_gt_zero: "~ 0 <  real_of_preal m" 
14484  488 
proof  
489 
from real_of_preal_minus_less_zero 

490 
show ?thesis by (blast dest: order_less_trans) 

491 
qed 

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

492 

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

493 

3d4df8c166ae
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paulson
parents:
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diff
changeset

494 
subsection{*Theorems About the Ordering*} 
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parents:
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diff
changeset

495 

3d4df8c166ae
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paulson
parents:
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diff
changeset

496 
lemma real_gt_zero_preal_Ex: "(0 < x) = (\<exists>y. x = real_of_preal y)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
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parents:
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diff
changeset

497 
apply (auto simp add: real_of_preal_zero_less) 
3d4df8c166ae
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parents:
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diff
changeset

498 
apply (cut_tac x = x in real_of_preal_trichotomy) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

499 
apply (blast elim!: real_of_preal_not_minus_gt_zero [THEN notE]) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

500 
done 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

501 

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

502 
lemma real_gt_preal_preal_Ex: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

503 
"real_of_preal z < x ==> \<exists>y. x = real_of_preal y" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

504 
by (blast dest!: real_of_preal_zero_less [THEN order_less_trans] 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

505 
intro: real_gt_zero_preal_Ex [THEN iffD1]) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

506 

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

507 
lemma real_ge_preal_preal_Ex: 
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parents:
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diff
changeset

508 
"real_of_preal z \<le> x ==> \<exists>y. x = real_of_preal y" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

509 
by (blast dest: order_le_imp_less_or_eq real_gt_preal_preal_Ex) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

510 

3d4df8c166ae
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paulson
parents:
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diff
changeset

511 
lemma real_less_all_preal: "y \<le> 0 ==> \<forall>x. y < real_of_preal x" 
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replacing HOL/Real/PRat, PNat by the rational number development
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parents:
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diff
changeset

512 
by (auto elim: order_le_imp_less_or_eq [THEN disjE] 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

513 
intro: real_of_preal_zero_less [THEN [2] order_less_trans] 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

514 
simp add: real_of_preal_zero_less) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

515 

3d4df8c166ae
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paulson
parents:
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diff
changeset

516 
lemma real_less_all_real2: "~ 0 < y ==> \<forall>x. y < real_of_preal x" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
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parents:
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diff
changeset

517 
by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1]) 
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paulson
parents:
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diff
changeset

518 

14334  519 

520 
subsection{*More Lemmas*} 

521 

522 
lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)" 

523 
by auto 

524 

525 
lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)" 

526 
by auto 

527 

528 
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)" 

529 
by (force elim: order_less_asym 

530 
simp add: Ring_and_Field.mult_less_cancel_right) 

531 

532 
lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)" 

14365
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paulson
parents:
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diff
changeset

533 
apply (simp add: mult_le_cancel_right) 
23289  534 
apply (blast intro: elim: order_less_asym) 
14365
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replacing HOL/Real/PRat, PNat by the rational number development
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parents:
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diff
changeset

535 
done 
14334  536 

537 
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)" 

15923  538 
by(simp add:mult_commute) 
14334  539 

14365
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paulson
parents:
14348
diff
changeset

540 
lemma real_inverse_gt_one: "[ (0::real) < x; x < 1 ] ==> 1 < inverse x" 
23289  541 
by (simp add: one_less_inverse_iff) (* TODO: generalize/move *) 
14334  542 

543 

24198  544 
subsection {* Embedding numbers into the Reals *} 
545 

546 
abbreviation 

547 
real_of_nat :: "nat \<Rightarrow> real" 

548 
where 

549 
"real_of_nat \<equiv> of_nat" 

550 

551 
abbreviation 

552 
real_of_int :: "int \<Rightarrow> real" 

553 
where 

554 
"real_of_int \<equiv> of_int" 

555 

556 
abbreviation 

557 
real_of_rat :: "rat \<Rightarrow> real" 

558 
where 

559 
"real_of_rat \<equiv> of_rat" 

560 

561 
consts 

562 
(*overloaded constant for injecting other types into "real"*) 

563 
real :: "'a => real" 

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

564 

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

565 
defs (overloaded) 
28520  566 
real_of_nat_def [code unfold]: "real == real_of_nat" 
567 
real_of_int_def [code unfold]: "real == real_of_int" 

14365
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paulson
parents:
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diff
changeset

568 

16819  569 
lemma real_eq_of_nat: "real = of_nat" 
24198  570 
unfolding real_of_nat_def .. 
16819  571 

572 
lemma real_eq_of_int: "real = of_int" 

24198  573 
unfolding real_of_int_def .. 
16819  574 

14365
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paulson
parents:
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diff
changeset

575 
lemma real_of_int_zero [simp]: "real (0::int) = 0" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

576 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

577 

3d4df8c166ae
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paulson
parents:
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diff
changeset

578 
lemma real_of_one [simp]: "real (1::int) = (1::real)" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

579 
by (simp add: real_of_int_def) 
14334  580 

16819  581 
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

582 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

583 

16819  584 
lemma real_of_int_minus [simp]: "real(x) = real (x::int)" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

585 
by (simp add: real_of_int_def) 
16819  586 

587 
lemma real_of_int_diff [simp]: "real(x  y) = real (x::int)  real y" 

588 
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

589 

16819  590 
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:
14369
diff
changeset

591 
by (simp add: real_of_int_def) 
14334  592 

16819  593 
lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))" 
594 
apply (subst real_eq_of_int)+ 

595 
apply (rule of_int_setsum) 

596 
done 

597 

598 
lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) = 

599 
(PROD x:A. real(f x))" 

600 
apply (subst real_eq_of_int)+ 

601 
apply (rule of_int_setprod) 

602 
done 

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

603 

27668  604 
lemma real_of_int_zero_cancel [simp, algebra, presburger]: "(real x = 0) = (x = (0::int))" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

605 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

606 

27668  607 
lemma real_of_int_inject [iff, algebra, presburger]: "(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

608 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

609 

27668  610 
lemma real_of_int_less_iff [iff, presburger]: "(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

611 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

612 

27668  613 
lemma real_of_int_le_iff [simp, presburger]: "(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

614 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

615 

27668  616 
lemma real_of_int_gt_zero_cancel_iff [simp, presburger]: "(0 < real (n::int)) = (0 < n)" 
16819  617 
by (simp add: real_of_int_def) 
618 

27668  619 
lemma real_of_int_ge_zero_cancel_iff [simp, presburger]: "(0 <= real (n::int)) = (0 <= n)" 
16819  620 
by (simp add: real_of_int_def) 
621 

27668  622 
lemma real_of_int_lt_zero_cancel_iff [simp, presburger]: "(real (n::int) < 0) = (n < 0)" 
16819  623 
by (simp add: real_of_int_def) 
624 

27668  625 
lemma real_of_int_le_zero_cancel_iff [simp, presburger]: "(real (n::int) <= 0) = (n <= 0)" 
16819  626 
by (simp add: real_of_int_def) 
627 

16888  628 
lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))" 
629 
by (auto simp add: abs_if) 

630 

16819  631 
lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)" 
632 
apply (subgoal_tac "real n + 1 = real (n + 1)") 

633 
apply (simp del: real_of_int_add) 

634 
apply auto 

635 
done 

636 

637 
lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)" 

638 
apply (subgoal_tac "real m + 1 = real (m + 1)") 

639 
apply (simp del: real_of_int_add) 

640 
apply simp 

641 
done 

642 

643 
lemma real_of_int_div_aux: "d ~= 0 ==> (real (x::int)) / (real d) = 

644 
real (x div d) + (real (x mod d)) / (real d)" 

645 
proof  

646 
assume "d ~= 0" 

647 
have "x = (x div d) * d + x mod d" 

648 
by auto 

649 
then have "real x = real (x div d) * real d + real(x mod d)" 

650 
by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym]) 

651 
then have "real x / real d = ... / real d" 

652 
by simp 

653 
then show ?thesis 

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

654 
by (auto simp add: add_divide_distrib ring_simps prems) 
16819  655 
qed 
656 

657 
lemma real_of_int_div: "(d::int) ~= 0 ==> d dvd n ==> 

658 
real(n div d) = real n / real d" 

659 
apply (frule real_of_int_div_aux [of d n]) 

660 
apply simp 

661 
apply (simp add: zdvd_iff_zmod_eq_0) 

662 
done 

663 

664 
lemma real_of_int_div2: 

665 
"0 <= real (n::int) / real (x)  real (n div x)" 

666 
apply (case_tac "x = 0") 

667 
apply simp 

668 
apply (case_tac "0 < x") 

669 
apply (simp add: compare_rls) 

670 
apply (subst real_of_int_div_aux) 

671 
apply simp 

672 
apply simp 

673 
apply (subst zero_le_divide_iff) 

674 
apply auto 

675 
apply (simp add: compare_rls) 

676 
apply (subst real_of_int_div_aux) 

677 
apply simp 

678 
apply simp 

679 
apply (subst zero_le_divide_iff) 

680 
apply auto 

681 
done 

682 

683 
lemma real_of_int_div3: 

684 
"real (n::int) / real (x)  real (n div x) <= 1" 

685 
apply(case_tac "x = 0") 

686 
apply simp 

687 
apply (simp add: compare_rls) 

688 
apply (subst real_of_int_div_aux) 

689 
apply assumption 

690 
apply simp 

691 
apply (subst divide_le_eq) 

692 
apply clarsimp 

693 
apply (rule conjI) 

694 
apply (rule impI) 

695 
apply (rule order_less_imp_le) 

696 
apply simp 

697 
apply (rule impI) 

698 
apply (rule order_less_imp_le) 

699 
apply simp 

700 
done 

701 

702 
lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x" 

27964  703 
by (insert real_of_int_div2 [of n x], simp) 
704 

705 

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

706 
subsection{*Embedding the Naturals into the Reals*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

707 

14334  708 
lemma real_of_nat_zero [simp]: "real (0::nat) = 0" 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

709 
by (simp add: real_of_nat_def) 
14334  710 

711 
lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)" 

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

712 
by (simp add: real_of_nat_def) 
14334  713 

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

714 
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

715 
by (simp add: real_of_nat_def) 
14334  716 

717 
(*Not for addsimps: often the LHS is used to represent a positive natural*) 

718 
lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)" 

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

719 
by (simp add: real_of_nat_def) 
14334  720 

721 
lemma real_of_nat_less_iff [iff]: 

722 
"(real (n::nat) < real m) = (n < m)" 

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

723 
by (simp add: real_of_nat_def) 
14334  724 

725 
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

726 
by (simp add: real_of_nat_def) 
14334  727 

728 
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

729 
by (simp add: real_of_nat_def zero_le_imp_of_nat) 
14334  730 

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

731 
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

732 
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

733 

14334  734 
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

735 
by (simp add: real_of_nat_def of_nat_mult) 
14334  736 

16819  737 
lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) = 
738 
(SUM x:A. real(f x))" 

739 
apply (subst real_eq_of_nat)+ 

740 
apply (rule of_nat_setsum) 

741 
done 

742 

743 
lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) = 

744 
(PROD x:A. real(f x))" 

745 
apply (subst real_eq_of_nat)+ 

746 
apply (rule of_nat_setprod) 

747 
done 

748 

749 
lemma real_of_card: "real (card A) = setsum (%x.1) A" 

750 
apply (subst card_eq_setsum) 

751 
apply (subst real_of_nat_setsum) 

752 
apply simp 

753 
done 

754 

14334  755 
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

756 
by (simp add: real_of_nat_def) 
14334  757 

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

758 
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

759 
by (simp add: real_of_nat_def) 
14334  760 

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

761 
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

762 
by (simp add: add: real_of_nat_def of_nat_diff) 
14334  763 

25162  764 
lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)" 
25140  765 
by (auto simp: real_of_nat_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

766 

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

767 
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

768 
by (simp add: add: real_of_nat_def) 
14334  769 

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

770 
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

771 
by (simp add: add: real_of_nat_def) 
14334  772 

25140  773 
lemma real_of_nat_ge_zero_cancel_iff [simp]: "(0 \<le> real (n::nat))" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

774 
by (simp add: add: real_of_nat_def) 
14334  775 

16819  776 
lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)" 
777 
apply (subgoal_tac "real n + 1 = real (Suc n)") 

778 
apply simp 

779 
apply (auto simp add: real_of_nat_Suc) 

780 
done 

781 

782 
lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)" 

783 
apply (subgoal_tac "real m + 1 = real (Suc m)") 

784 
apply (simp add: less_Suc_eq_le) 

785 
apply (simp add: real_of_nat_Suc) 

786 
done 

787 

788 
lemma real_of_nat_div_aux: "0 < d ==> (real (x::nat)) / (real d) = 

789 
real (x div d) + (real (x mod d)) / (real d)" 

790 
proof  

791 
assume "0 < d" 

792 
have "x = (x div d) * d + x mod d" 

793 
by auto 

794 
then have "real x = real (x div d) * real d + real(x mod d)" 

795 
by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym]) 

796 
then have "real x / real d = \<dots> / real d" 

797 
by simp 

798 
then show ?thesis 

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

799 
by (auto simp add: add_divide_distrib ring_simps prems) 
16819  800 
qed 
801 

802 
lemma real_of_nat_div: "0 < (d::nat) ==> d dvd n ==> 

803 
real(n div d) = real n / real d" 

804 
apply (frule real_of_nat_div_aux [of d n]) 

805 
apply simp 

806 
apply (subst dvd_eq_mod_eq_0 [THEN sym]) 

807 
apply assumption 

808 
done 

809 

810 
lemma real_of_nat_div2: 

811 
"0 <= real (n::nat) / real (x)  real (n div x)" 

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

812 
apply(case_tac "x = 0") 
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 add: compare_rls) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

815 
apply (subst real_of_nat_div_aux) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

816 
apply simp 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

817 
apply simp 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

818 
apply (subst zero_le_divide_iff) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

819 
apply simp 
16819  820 
done 
821 

822 
lemma real_of_nat_div3: 

823 
"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

824 
apply(case_tac "x = 0") 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

825 
apply (simp) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

826 
apply (simp add: compare_rls) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

827 
apply (subst real_of_nat_div_aux) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

828 
apply simp 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

829 
apply simp 
16819  830 
done 
831 

832 
lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x" 

833 
by (insert real_of_nat_div2 [of n x], simp) 

834 

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

835 
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

836 
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

837 

14426  838 
lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n" 
839 
by (simp add: real_of_int_def real_of_nat_def) 

14334  840 

16819  841 
lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x" 
842 
apply (subgoal_tac "real(int(nat x)) = real(nat x)") 

843 
apply force 

844 
apply (simp only: real_of_int_real_of_nat) 

845 
done 

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

846 

28001  847 

848 
subsection{* Rationals *} 

849 

28091
50f2d6ba024c
Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents:
28001
diff
changeset

850 
lemma Rats_real_nat[simp]: "real(n::nat) \<in> \<rat>" 
50f2d6ba024c
Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents:
28001
diff
changeset

851 
by (simp add: real_eq_of_nat) 
50f2d6ba024c
Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents:
28001
diff
changeset

852 

50f2d6ba024c
Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents:
28001
diff
changeset

853 

28001  854 
lemma Rats_eq_int_div_int: 
28091
50f2d6ba024c
Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents:
28001
diff
changeset

855 
"\<rat> = { real(i::int)/real(j::int) i j. j \<noteq> 0}" (is "_ = ?S") 
28001  856 
proof 
28091
50f2d6ba024c
Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents:
28001
diff
changeset

857 
show "\<rat> \<subseteq> ?S" 
28001  858 
proof 
28091
50f2d6ba024c
Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents:
28001
diff
changeset

859 
fix x::real assume "x : \<rat>" 
28001  860 
then obtain r where "x = of_rat r" unfolding Rats_def .. 
861 
have "of_rat r : ?S" 

862 
by (cases r)(auto simp add:of_rat_rat real_eq_of_int) 

863 
thus "x : ?S" using `x = of_rat r` by simp 

864 
qed 

865 
next 

28091
50f2d6ba024c
Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents:
28001
diff
changeset

866 
show "?S \<subseteq> \<rat>" 
28001  867 
proof(auto simp:Rats_def) 
868 
fix i j :: int assume "j \<noteq> 0" 

869 
hence "real i / real j = of_rat(Fract i j)" 

870 
by (simp add:of_rat_rat real_eq_of_int) 

871 
thus "real i / real j \<in> range of_rat" by blast 

872 
qed 

873 
qed 

874 

875 
lemma Rats_eq_int_div_nat: 

28091
50f2d6ba024c
Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents:
28001
diff
changeset

876 
"\<rat> = { real(i::int)/real(n::nat) i n. n \<noteq> 0}" 
28001  877 
proof(auto simp:Rats_eq_int_div_int) 
878 
fix i j::int assume "j \<noteq> 0" 

879 
show "EX (i'::int) (n::nat). real i/real j = real i'/real n \<and> 0<n" 

880 
proof cases 

881 
assume "j>0" 

882 
hence "real i/real j = real i/real(nat j) \<and> 0<nat j" 

883 
by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat) 

884 
thus ?thesis by blast 

885 
next 

886 
assume "~ j>0" 

887 
hence "real i/real j = real(i)/real(nat(j)) \<and> 0<nat(j)" using `j\<noteq>0` 

888 
by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat) 

889 
thus ?thesis by blast 

890 
qed 

891 
next 

892 
fix i::int and n::nat assume "0 < n" 

893 
hence "real i/real n = real i/real(int n) \<and> int n \<noteq> 0" by simp 

894 
thus "\<exists>(i'::int) j::int. real i/real n = real i'/real j \<and> j \<noteq> 0" by blast 

895 
qed 

896 

897 
lemma Rats_abs_nat_div_natE: 

898 
assumes "x \<in> \<rat>" 

899 
obtains m n where "n \<noteq> 0" and "\<bar>x\<bar> = real m / real n" and "gcd m n = 1" 

900 
proof  

901 
from `x \<in> \<rat>` obtain i::int and n::nat where "n \<noteq> 0" and "x = real i / real n" 

902 
by(auto simp add: Rats_eq_int_div_nat) 

903 
hence "\<bar>x\<bar> = real(nat(abs i)) / real n" by simp 

904 
then obtain m :: nat where x_rat: "\<bar>x\<bar> = real m / real n" by blast 

905 
let ?gcd = "gcd m n" 

906 
from `n\<noteq>0` have gcd: "?gcd \<noteq> 0" by (simp add: gcd_zero) 

907 
let ?k = "m div ?gcd" 

908 
let ?l = "n div ?gcd" 

909 
let ?gcd' = "gcd ?k ?l" 

910 
have "?gcd dvd m" .. then have gcd_k: "?gcd * ?k = m" 

911 
by (rule dvd_mult_div_cancel) 

912 
have "?gcd dvd n" .. then have gcd_l: "?gcd * ?l = n" 

913 
by (rule dvd_mult_div_cancel) 

914 
from `n\<noteq>0` and gcd_l have "?l \<noteq> 0" by (auto iff del: neq0_conv) 

915 
moreover 

916 
have "\<bar>x\<bar> = real ?k / real ?l" 

917 
proof  

918 
from gcd have "real ?k / real ?l = 

919 
real (?gcd * ?k) / real (?gcd * ?l)" by simp 

920 
also from gcd_k and gcd_l have "\<dots> = real m / real n" by simp 

921 
also from x_rat have "\<dots> = \<bar>x\<bar>" .. 

922 
finally show ?thesis .. 

923 
qed 

924 
moreover 

925 
have "?gcd' = 1" 

926 
proof  

927 
have "?gcd * ?gcd' = gcd (?gcd * ?k) (?gcd * ?l)" 

928 
by (rule gcd_mult_distrib2) 

929 
with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp 

930 
with gcd show ?thesis by simp 

931 
qed 

932 
ultimately show ?thesis .. 

933 
qed 

934 

935 

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

936 
subsection{*Numerals and Arithmetic*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

937 

25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

938 
instantiation real :: number_ring 
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

939 
begin 
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

940 

c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

941 
definition 
25965  942 
real_number_of_def [code func del]: "number_of w = real_of_int w" 
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

943 

c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

944 
instance 
24198  945 
by intro_classes (simp add: real_number_of_def) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

946 

25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

947 
end 
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

948 

25965  949 
lemma [code unfold, symmetric, code post]: 
24198  950 
"number_of k = real_of_int (number_of k)" 
951 
unfolding number_of_is_id real_number_of_def .. 

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

952 

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

953 

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

954 
text{*Collapse applications of @{term real} to @{term number_of}*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

955 
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

956 
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

957 

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

958 
lemma real_of_nat_number_of [simp]: 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

959 
"real (number_of v :: nat) = 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

960 
(if neg (number_of v :: int) then 0 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

961 
else (number_of v :: real))" 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

962 
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

963 

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

964 

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

965 
use "real_arith.ML" 
24075  966 
declaration {* K real_arith_setup *} 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

967 

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

968 

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

969 
subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

970 

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

971 
text{*Needed in this nonstandard form by Hyperreal/Transcendental*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

972 
lemma real_0_le_divide_iff: 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

973 
"((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

974 
by (simp add: real_divide_def zero_le_mult_iff, auto) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

975 

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

976 
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

977 
by arith 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

978 

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

979 
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

980 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

981 

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

982 
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

983 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

984 

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

985 
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

986 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

987 

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

988 
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

989 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

990 

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

991 
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

992 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

993 

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

994 

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

995 
(* 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

996 
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

997 
It replaces x+y by xy. 
15086  998 
declare real_diff_def [symmetric, simp] 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

999 
*) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1000 

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

1001 

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

1002 
subsubsection{*Density of the Reals*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1003 

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

1004 
lemma real_lbound_gt_zero: 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1005 
"[ (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

1006 
apply (rule_tac x = " (min d1 d2) /2" in exI) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1007 
apply (simp add: min_def) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1008 
done 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1009 

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

1010 

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

1011 
text{*Similar results are proved in @{text Ring_and_Field}*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1012 
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

1013 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1014 

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

1015 
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

1016 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1017 

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

1018 

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

1019 
subsection{*Absolute Value Function for the Reals*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1020 

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

1021 
lemma abs_minus_add_cancel: "abs(x + (y)) = abs (y + ((x::real)))" 
15003  1022 
by (simp add: abs_if) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1023 

23289  1024 
(* FIXME: redundant, but used by Integration/RealRandVar.thy in AFP *) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1025 
lemma abs_le_interval_iff: "(abs x \<le> r) = (r\<le>x & x\<le>(r::real))" 
14738  1026 
by (force simp add: OrderedGroup.abs_le_iff) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1027 

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

1028 
lemma abs_add_one_gt_zero [simp]: "(0::real) < 1 + abs(x)" 
15003  1029 
by (simp add: abs_if) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1030 

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

1031 
lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)" 
22958  1032 
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

1033 

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

1034 
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

1035 
by simp 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1036 

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

1037 
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

1038 
by simp 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1039 

26732  1040 
instance real :: lordered_ring 
1041 
proof 

1042 
fix a::real 

1043 
show "abs a = sup a (a)" 

1044 
by (auto simp add: real_abs_def sup_real_def) 

1045 
qed 

1046 

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

1047 

27544  1048 
subsection {* Implementation of rational real numbers *} 
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1049 

27544  1050 
definition Ratreal :: "rat \<Rightarrow> real" where 
1051 
[simp]: "Ratreal = of_rat" 

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

1052 

24623  1053 
code_datatype Ratreal 
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1054 

27544  1055 
lemma Ratreal_number_collapse [code post]: 
1056 
"Ratreal 0 = 0" 

1057 
"Ratreal 1 = 1" 

1058 
"Ratreal (number_of k) = number_of k" 

1059 
by simp_all 

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

1060 

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

1061 
lemma zero_real_code [code, code unfold]: 
27544  1062 
"0 = Ratreal 0" 
1063 
by simp 

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

1064 

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

1065 
lemma one_real_code [code, code unfold]: 
27544  1066 
"1 = Ratreal 1" 
1067 
by simp 

1068 

1069 
lemma number_of_real_code [code unfold]: 

1070 
"number_of k = Ratreal (number_of k)" 

1071 
by simp 

1072 

1073 
lemma Ratreal_number_of_quotient [code post]: 

1074 
"Ratreal (number_of r) / Ratreal (number_of s) = number_of r / number_of s" 

1075 
by simp 

1076 

1077 
lemma Ratreal_number_of_quotient2 [code post]: 

1078 
"Ratreal (number_of r / number_of s) = number_of r / number_of s" 

1079 
unfolding Ratreal_number_of_quotient [symmetric] Ratreal_def of_rat_divide .. 

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

1080 

26513  1081 
instantiation real :: eq 
1082 
begin 

1083 

27544  1084 
definition "eq_class.eq (x\<Colon>real) y \<longleftrightarrow> x  y = 0" 
26513  1085 

1086 
instance by default (simp add: eq_real_def) 

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

1087 

27544  1088 
lemma real_eq_code [code]: "eq_class.eq (Ratreal x) (Ratreal y) \<longleftrightarrow> eq_class.eq x y" 
1089 
by (simp add: eq_real_def eq) 

26513  1090 

28351  1091 
lemma real_eq_refl [code nbe]: 
1092 
"eq_class.eq (x::real) x \<longleftrightarrow> True" 

1093 
by (rule HOL.eq_refl) 

1094 

26513  1095 
end 
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1096 

27544  1097 
lemma real_less_eq_code [code]: "Ratreal x \<le> Ratreal y \<longleftrightarrow> x \<le> y" 
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27544
diff
changeset

1098 
by (simp add: of_rat_less_eq) 
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1099 

27544  1100 
lemma real_less_code [code]: "Ratreal x < Ratreal y \<longleftrightarrow> x < y" 
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27544
diff
changeset

1101 
by (simp add: of_rat_less) 
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1102 

27544  1103 
lemma real_plus_code [code]: "Ratreal x + Ratreal y = Ratreal (x + y)" 
1104 
by (simp add: of_rat_add) 

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

1105 

27544  1106 
lemma real_times_code [code]: "Ratreal x * Ratreal y = Ratreal (x * y)" 
1107 
by (simp add: of_rat_mult) 

1108 

1109 
lemma real_uminus_code [code]: " Ratreal x = Ratreal ( x)" 

1110 
by (simp add: of_rat_minus) 

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

1111 

27544  1112 
lemma real_minus_code [code]: "Ratreal x  Ratreal y = Ratreal (x  y)" 
1113 
by (simp add: of_rat_diff) 

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

1114 

27544  1115 
lemma real_inverse_code [code]: "inverse (Ratreal x) = Ratreal (inverse x)" 
1116 
by (simp add: of_rat_inverse) 

1117 

1118 
lemma real_divide_code [code]: "Ratreal x / Ratreal y = Ratreal (x / y)" 

1119 
by (simp add: of_rat_divide) 

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

1120 

24623  1121 
text {* Setup for SML code generator *} 
23031
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

1122 

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

1123 
types_code 
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1124 
real ("(int */ int)") 
23031
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

1125 
attach (term_of) {* 
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1126 
fun term_of_real (p, q) = 
24623  1127 
let 
1128 
val rT = HOLogic.realT 

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

1129 
in 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1130 
if q = 1 orelse p = 0 then HOLogic.mk_number rT p 
24623  1131 
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

1132 
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

1133 
end; 
23031
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

1134 
*} 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

1135 
attach (test) {* 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

1136 
fun gen_real i = 
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1137 
let 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1138 
val p = random_range 0 i; 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1139 
val q = random_range 1 (i + 1); 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1140 
val g = Integer.gcd p q; 
24630
351a308ab58d
simplified type int (eliminated IntInf.int, integer);
wenzelm
parents:
24623
diff
changeset

1141 
val p' = p div g; 
351a308ab58d
simplified type int (eliminated IntInf.int, integer);
wenzelm
parents:
24623
diff
changeset

1142 
val q' = q div g; 
25885  1143 
val r = (if one_of [true, false] then p' else ~ p', 
1144 
if p' = 0 then 0 else q') 

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

1145 
in 
25885  1146 
(r, fn () => term_of_real r) 
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1147 
end; 
23031
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

1148 
*} 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

1149 

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

1150 
consts_code 
24623  1151 
Ratreal ("(_)") 
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1152 

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

1153 
consts_code 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1154 
"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

1155 
attach {* 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1156 
fun real_of_int 0 = (0, 0) 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1157 
 real_of_int i = (i, 1); 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1158 
*} 
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset

1159 

5588  1160 
end 