author | huffman |
Wed, 20 Jun 2007 05:18:39 +0200 | |
changeset 23431 | 25ca91279a9b |
parent 23289 | 0cf371d943b1 |
child 23438 | dd824e86fa8a |
permissions | -rw-r--r-- |
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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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Polymorphic treatment of binary arithmetic using axclasses
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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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||
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header{*Defining the Reals from the Positive Reals*} |
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|
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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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|
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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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typedef (Real) real = "UNIV//realrel" |
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by (auto simp add: quotient_def) |
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instance real :: "{ord, zero, one, plus, times, minus, inverse}" .. |
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definition |
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|
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(** 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)})" |
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|
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consts |
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(*overloaded constant for injecting other types into "real"*) |
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real :: "'a => real" |
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||
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defs (overloaded) |
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real_zero_def: |
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"0 == Abs_Real(realrel``{(1, 1)})" |
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Numerals and simprocs for types real and hypreal. The abstract
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real_one_def: |
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"1 == Abs_Real(realrel``{(1 + 1, 1)})" |
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|
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real_minus_def: |
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"- r == contents (\<Union>(x,y) \<in> Rep_Real(r). { Abs_Real(realrel``{(y,x)}) })" |
47 |
||
48 |
real_add_def: |
|
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"z + w == |
|
50 |
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)}) })" |
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|
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real_diff_def: |
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"r - (s::real) == r + - s" |
55 |
||
56 |
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). |
|
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{ Abs_Real(realrel``{(x*u + y*v, x*v + y*u)}) })" |
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real_inverse_def: |
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"inverse (R::real) == (THE S. (R = 0 & S = 0) | S * R = 1)" |
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|
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real_divide_def: |
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"R / (S::real) == R * inverse S" |
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real_le_def: |
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"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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real_abs_def: "abs (r::real) == (if r < 0 then - r else r)" |
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75 |
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subsection {* Equivalence relation over positive reals *} |
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|
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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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82 |
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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91 |
|
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lemma realrel_iff [simp]: "(((x1,y1),(x2,y2)) \<in> realrel) = (x1 + y2 = x2 + y1)" |
94 |
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 |
100 |
||
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text{*Reduces equality of equivalence classes to the @{term realrel} relation: |
102 |
@{term "(realrel `` {x} = realrel `` {y}) = ((x,y) \<in> realrel)"} *} |
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lemmas equiv_realrel_iff = |
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eq_equiv_class_iff [OF equiv_realrel UNIV_I UNIV_I] |
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declare equiv_realrel_iff [simp] |
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lemma realrel_in_real [simp]: "realrel``{(x,y)}: Real" |
110 |
by (simp add: Real_def realrel_def quotient_def, blast) |
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declare Abs_Real_inject [simp] |
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declare Abs_Real_inverse [simp] |
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115 |
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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.*} |
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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) |
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done |
124 |
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125 |
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subsection {* Addition and Subtraction *} |
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lemma real_add_congruent2_lemma: |
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"[|a + ba = aa + b; ab + bc = ac + bb|] |
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==> 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)}) = |
139 |
Abs_Real (realrel``{(x+u, y+v)})" |
|
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proof - |
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have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). {Abs_Real (realrel `` {(x+u, y+v)})}) w) z) |
142 |
respects2 realrel" |
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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]) |
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qed |
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lemma real_minus: "- Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})" |
150 |
proof - |
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have "(\<lambda>(x,y). {Abs_Real (realrel``{(y,x)})}) respects realrel" |
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by (simp add: congruent_def add_commute) |
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thus ?thesis |
154 |
by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel]) |
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155 |
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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172 |
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subsection {* Multiplication *} |
14269 | 174 |
|
14329 | 175 |
lemma real_mult_congruent2_lemma: |
176 |
"!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==> |
|
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x * x1 + y * y1 + (x * y2 + y * x2) = |
178 |
x * x2 + y * y2 + (x * y1 + y * x1)" |
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apply (simp add: add_left_commute add_assoc [symmetric]) |
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apply (simp add: add_assoc right_distrib [symmetric]) |
181 |
apply (simp add: add_commute) |
|
14269 | 182 |
done |
183 |
||
184 |
lemma real_mult_congruent2: |
|
15169 | 185 |
"(%p1 p2. |
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(%(x1,y1). (%(x2,y2). |
15169 | 187 |
{ Abs_Real (realrel``{(x1*x2 + y1*y2, x1*y2+y1*x2)}) }) p2) p1) |
188 |
respects2 realrel" |
|
14658 | 189 |
apply (rule congruent2_commuteI [OF equiv_realrel], clarify) |
23288 | 190 |
apply (simp add: mult_commute add_commute) |
14269 | 191 |
apply (auto simp add: real_mult_congruent2_lemma) |
192 |
done |
|
193 |
||
194 |
lemma real_mult: |
|
14484 | 195 |
"Abs_Real((realrel``{(x1,y1)})) * Abs_Real((realrel``{(x2,y2)})) = |
196 |
Abs_Real(realrel `` {(x1*x2+y1*y2,x1*y2+y1*x2)})" |
|
197 |
by (simp add: real_mult_def UN_UN_split_split_eq |
|
14658 | 198 |
UN_equiv_class2 [OF equiv_realrel equiv_realrel real_mult_congruent2]) |
14269 | 199 |
|
200 |
lemma real_mult_commute: "(z::real) * w = w * z" |
|
23288 | 201 |
by (cases z, cases w, simp add: real_mult add_ac mult_ac) |
14269 | 202 |
|
203 |
lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)" |
|
14484 | 204 |
apply (cases z1, cases z2, cases z3) |
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apply (simp add: real_mult right_distrib add_ac mult_ac) |
14269 | 206 |
done |
207 |
||
208 |
lemma real_mult_1: "(1::real) * z = z" |
|
14484 | 209 |
apply (cases z) |
23288 | 210 |
apply (simp add: real_mult real_one_def right_distrib |
211 |
mult_1_right mult_ac add_ac) |
|
14269 | 212 |
done |
213 |
||
214 |
lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)" |
|
14484 | 215 |
apply (cases z1, cases z2, cases w) |
23288 | 216 |
apply (simp add: real_add real_mult right_distrib add_ac mult_ac) |
14269 | 217 |
done |
218 |
||
14329 | 219 |
text{*one and zero are distinct*} |
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220 |
lemma real_zero_not_eq_one: "0 \<noteq> (1::real)" |
14484 | 221 |
proof - |
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222 |
have "(1::preal) < 1 + 1" |
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223 |
by (simp add: preal_self_less_add_left) |
14484 | 224 |
thus ?thesis |
23288 | 225 |
by (simp add: real_zero_def real_one_def) |
14484 | 226 |
qed |
14269 | 227 |
|
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228 |
instance real :: comm_ring_1 |
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229 |
proof |
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230 |
fix x y z :: real |
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231 |
show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc) |
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232 |
show "x * y = y * x" by (rule real_mult_commute) |
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233 |
show "1 * x = x" by (rule real_mult_1) |
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234 |
show "(x + y) * z = x * z + y * z" by (rule real_add_mult_distrib) |
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235 |
show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one) |
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236 |
qed |
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237 |
|
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238 |
subsection {* Inverse and Division *} |
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239 |
|
14484 | 240 |
lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0" |
23288 | 241 |
by (simp add: real_zero_def add_commute) |
14269 | 242 |
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243 |
text{*Instead of using an existential quantifier and constructing the inverse |
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|
244 |
within the proof, we could define the inverse explicitly.*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
245 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
246 |
lemma real_mult_inverse_left_ex: "x \<noteq> 0 ==> \<exists>y. y*x = (1::real)" |
14484 | 247 |
apply (simp add: real_zero_def real_one_def, cases x) |
14269 | 248 |
apply (cut_tac x = xa and y = y in linorder_less_linear) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
249 |
apply (auto dest!: less_add_left_Ex simp add: real_zero_iff) |
14334 | 250 |
apply (rule_tac |
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
251 |
x = "Abs_Real (realrel``{(1, inverse (D) + 1)})" |
14334 | 252 |
in exI) |
253 |
apply (rule_tac [2] |
|
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
254 |
x = "Abs_Real (realrel``{(inverse (D) + 1, 1)})" |
14334 | 255 |
in exI) |
23288 | 256 |
apply (auto simp add: real_mult ring_distrib |
257 |
preal_mult_inverse_right add_ac mult_ac) |
|
14269 | 258 |
done |
259 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
260 |
lemma real_mult_inverse_left: "x \<noteq> 0 ==> inverse(x)*x = (1::real)" |
14484 | 261 |
apply (simp add: real_inverse_def) |
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
262 |
apply (drule real_mult_inverse_left_ex, safe) |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
263 |
apply (rule theI, assumption, rename_tac z) |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
264 |
apply (subgoal_tac "(z * x) * y = z * (x * y)") |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
265 |
apply (simp add: mult_commute) |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
266 |
apply (rule mult_assoc) |
14269 | 267 |
done |
14334 | 268 |
|
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
269 |
|
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
270 |
subsection{*The Real Numbers form a Field*} |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
271 |
|
14334 | 272 |
instance real :: field |
273 |
proof |
|
274 |
fix x y z :: real |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
275 |
show "x \<noteq> 0 ==> inverse x * x = 1" by (rule real_mult_inverse_left) |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14426
diff
changeset
|
276 |
show "x / y = x * inverse y" by (simp add: real_divide_def) |
14334 | 277 |
qed |
278 |
||
279 |
||
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
280 |
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 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
291 |
subsection{*The @{text "\<le>"} Ordering*} |
14269 | 292 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
293 |
lemma real_le_refl: "w \<le> (w::real)" |
14484 | 294 |
by (cases w, force simp add: real_le_def) |
14269 | 295 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
296 |
text{*The arithmetic decision procedure is not set up for type preal. |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
297 |
This lemma is currently unused, but it could simplify the proofs of the |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
298 |
following two lemmas.*} |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
299 |
lemma preal_eq_le_imp_le: |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
300 |
assumes eq: "a+b = c+d" and le: "c \<le> a" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
301 |
shows "b \<le> (d::preal)" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
302 |
proof - |
23288 | 303 |
have "c+d \<le> a+d" by (simp add: prems) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
304 |
hence "a+b \<le> a+d" by (simp add: prems) |
23288 | 305 |
thus "b \<le> d" by simp |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
306 |
qed |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
307 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
308 |
lemma real_le_lemma: |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
309 |
assumes l: "u1 + v2 \<le> u2 + v1" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
310 |
and "x1 + v1 = u1 + y1" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
311 |
and "x2 + v2 = u2 + y2" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
312 |
shows "x1 + y2 \<le> x2 + (y1::preal)" |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
313 |
proof - |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
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 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
319 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
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) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
324 |
apply (auto intro: real_le_lemma) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
325 |
done |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
326 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
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) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
329 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
330 |
lemma real_trans_lemma: |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
331 |
assumes "x + v \<le> u + y" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
332 |
and "u + v' \<le> u' + v" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
333 |
and "x2 + v2 = u2 + y2" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
334 |
shows "x + v' \<le> u' + (y::preal)" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
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 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
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 |
||
349 |
(* Axiom 'order_less_le' of class 'order': *) |
|
350 |
lemma real_less_le: "((w::real) < z) = (w \<le> z & w \<noteq> z)" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
351 |
by (simp add: real_less_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
352 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
353 |
instance real :: order |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
354 |
proof qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
355 |
(assumption | |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
356 |
rule real_le_refl real_le_trans real_le_anti_sym real_less_le)+ |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
357 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
358 |
(* Axiom 'linorder_linear' of class 'linorder': *) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
359 |
lemma real_le_linear: "(z::real) \<le> w | w \<le> z" |
23288 | 360 |
apply (cases z, cases w) |
361 |
apply (auto simp add: real_le real_zero_def add_ac) |
|
14334 | 362 |
done |
363 |
||
364 |
||
365 |
instance real :: linorder |
|
366 |
by (intro_classes, rule real_le_linear) |
|
367 |
||
368 |
||
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
369 |
lemma real_le_eq_diff: "(x \<le> y) = (x-y \<le> (0::real))" |
14484 | 370 |
apply (cases x, cases y) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
371 |
apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus |
23288 | 372 |
add_ac) |
373 |
apply (simp_all add: add_assoc [symmetric]) |
|
15542 | 374 |
done |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
375 |
|
14484 | 376 |
lemma real_add_left_mono: |
377 |
assumes le: "x \<le> y" shows "z + x \<le> z + (y::real)" |
|
378 |
proof - |
|
379 |
have "z + x - (z + y) = (z + -z) + (x - y)" |
|
380 |
by (simp add: diff_minus add_ac) |
|
381 |
with le show ?thesis |
|
14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
382 |
by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"] diff_minus) |
14484 | 383 |
qed |
14334 | 384 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
385 |
lemma real_sum_gt_zero_less: "(0 < S + (-W::real)) ==> (W < S)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
386 |
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
387 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
388 |
lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (-W::real))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
389 |
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus) |
14334 | 390 |
|
391 |
lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y" |
|
14484 | 392 |
apply (cases x, cases y) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
393 |
apply (simp add: linorder_not_le [where 'a = real, symmetric] |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
394 |
linorder_not_le [where 'a = preal] |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
395 |
real_zero_def real_le real_mult) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
396 |
--{*Reduce to the (simpler) @{text "\<le>"} relation *} |
16973 | 397 |
apply (auto dest!: less_add_left_Ex |
23288 | 398 |
simp add: add_ac mult_ac |
399 |
right_distrib preal_self_less_add_left) |
|
14334 | 400 |
done |
401 |
||
402 |
lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y" |
|
403 |
apply (rule real_sum_gt_zero_less) |
|
404 |
apply (drule real_less_sum_gt_zero [of x y]) |
|
405 |
apply (drule real_mult_order, assumption) |
|
406 |
apply (simp add: right_distrib) |
|
407 |
done |
|
408 |
||
22456 | 409 |
instance real :: distrib_lattice |
410 |
"inf x y \<equiv> min x y" |
|
411 |
"sup x y \<equiv> max x y" |
|
412 |
by default (auto simp add: inf_real_def sup_real_def min_max.sup_inf_distrib1) |
|
413 |
||
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
414 |
|
14334 | 415 |
subsection{*The Reals Form an Ordered Field*} |
416 |
||
417 |
instance real :: ordered_field |
|
418 |
proof |
|
419 |
fix x y z :: real |
|
420 |
show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono) |
|
22962 | 421 |
show "x < y ==> 0 < z ==> z * x < z * y" by (rule real_mult_less_mono2) |
422 |
show "\<bar>x\<bar> = (if x < 0 then -x else x)" by (simp only: real_abs_def) |
|
14334 | 423 |
qed |
424 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
425 |
text{*The function @{term real_of_preal} requires many proofs, but it seems |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
426 |
to be essential for proving completeness of the reals from that of the |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
427 |
positive reals.*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
428 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
429 |
lemma real_of_preal_add: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
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_add left_distrib add_ac) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
432 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
433 |
lemma real_of_preal_mult: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
434 |
"real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y" |
23288 | 435 |
by (simp add: real_of_preal_def real_mult right_distrib add_ac mult_ac) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
436 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
437 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
438 |
text{*Gleason prop 9-4.4 p 127*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
439 |
lemma real_of_preal_trichotomy: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
440 |
"\<exists>m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)" |
14484 | 441 |
apply (simp add: real_of_preal_def real_zero_def, cases x) |
23288 | 442 |
apply (auto simp add: real_minus add_ac) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
443 |
apply (cut_tac x = x and y = y in linorder_less_linear) |
23288 | 444 |
apply (auto dest!: less_add_left_Ex simp add: add_assoc [symmetric]) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
445 |
done |
3d4df8c166ae
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:
14348
diff
changeset
|
447 |
lemma real_of_preal_leD: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
448 |
"real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2" |
23288 | 449 |
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
|
450 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
451 |
lemma real_of_preal_lessI: "m1 < m2 ==> real_of_preal m1 < real_of_preal m2" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
452 |
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:
14348
diff
changeset
|
453 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
454 |
lemma real_of_preal_lessD: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
455 |
"real_of_preal m1 < real_of_preal m2 ==> m1 < m2" |
23288 | 456 |
by (simp add: real_of_preal_def real_le linorder_not_le [symmetric]) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
457 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
458 |
lemma real_of_preal_less_iff [simp]: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
459 |
"(real_of_preal m1 < real_of_preal m2) = (m1 < m2)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
460 |
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
|
461 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
462 |
lemma real_of_preal_le_iff: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
463 |
"(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)" |
23288 | 464 |
by (simp add: linorder_not_less [symmetric]) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
465 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
466 |
lemma real_of_preal_zero_less: "0 < real_of_preal m" |
23288 | 467 |
apply (insert preal_self_less_add_left [of 1 m]) |
468 |
apply (auto simp add: real_zero_def real_of_preal_def |
|
469 |
real_less_def real_le_def add_ac) |
|
470 |
apply (rule_tac x="m + 1" in exI, rule_tac x="1" in exI) |
|
471 |
apply (simp add: add_ac) |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
472 |
done |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
473 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
474 |
lemma real_of_preal_minus_less_zero: "- real_of_preal m < 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
475 |
by (simp add: real_of_preal_zero_less) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
476 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
477 |
lemma real_of_preal_not_minus_gt_zero: "~ 0 < - real_of_preal m" |
14484 | 478 |
proof - |
479 |
from real_of_preal_minus_less_zero |
|
480 |
show ?thesis by (blast dest: order_less_trans) |
|
481 |
qed |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
482 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
483 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
484 |
subsection{*Theorems About the Ordering*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
485 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
486 |
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
paulson
parents:
14348
diff
changeset
|
487 |
apply (auto simp add: real_of_preal_zero_less) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
488 |
apply (cut_tac x = x in real_of_preal_trichotomy) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
489 |
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:
14348
diff
changeset
|
490 |
done |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
491 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
492 |
lemma real_gt_preal_preal_Ex: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
493 |
"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
|
494 |
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:
14348
diff
changeset
|
495 |
intro: real_gt_zero_preal_Ex [THEN iffD1]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
496 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
497 |
lemma real_ge_preal_preal_Ex: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
498 |
"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:
14348
diff
changeset
|
499 |
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
|
500 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
501 |
lemma real_less_all_preal: "y \<le> 0 ==> \<forall>x. y < real_of_preal x" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
502 |
by (auto elim: order_le_imp_less_or_eq [THEN disjE] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
503 |
intro: real_of_preal_zero_less [THEN [2] order_less_trans] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
504 |
simp add: real_of_preal_zero_less) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
505 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
506 |
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
paulson
parents:
14348
diff
changeset
|
507 |
by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
508 |
|
14334 | 509 |
|
510 |
subsection{*More Lemmas*} |
|
511 |
||
512 |
lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)" |
|
513 |
by auto |
|
514 |
||
515 |
lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)" |
|
516 |
by auto |
|
517 |
||
518 |
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)" |
|
519 |
by (force elim: order_less_asym |
|
520 |
simp add: Ring_and_Field.mult_less_cancel_right) |
|
521 |
||
522 |
lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
523 |
apply (simp add: mult_le_cancel_right) |
23289 | 524 |
apply (blast intro: elim: order_less_asym) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
525 |
done |
14334 | 526 |
|
527 |
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)" |
|
15923 | 528 |
by(simp add:mult_commute) |
14334 | 529 |
|
23289 | 530 |
(* FIXME: redundant, but used by Integration/Integral.thy in AFP *) |
14334 | 531 |
lemma real_le_add_order: "[| 0 \<le> x; 0 \<le> y |] ==> (0::real) \<le> x + y" |
22958 | 532 |
by (rule add_nonneg_nonneg) |
14334 | 533 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
534 |
lemma real_inverse_gt_one: "[| (0::real) < x; x < 1 |] ==> 1 < inverse x" |
23289 | 535 |
by (simp add: one_less_inverse_iff) (* TODO: generalize/move *) |
14334 | 536 |
|
537 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
538 |
subsection{*Embedding the Integers into the Reals*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
539 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
540 |
defs (overloaded) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
541 |
real_of_nat_def: "real z == of_nat z" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
542 |
real_of_int_def: "real z == of_int z" |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
543 |
|
16819 | 544 |
lemma real_eq_of_nat: "real = of_nat" |
545 |
apply (rule ext) |
|
546 |
apply (unfold real_of_nat_def) |
|
547 |
apply (rule refl) |
|
548 |
done |
|
549 |
||
550 |
lemma real_eq_of_int: "real = of_int" |
|
551 |
apply (rule ext) |
|
552 |
apply (unfold real_of_int_def) |
|
553 |
apply (rule refl) |
|
554 |
done |
|
555 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
556 |
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
|
557 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
558 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
559 |
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
|
560 |
by (simp add: real_of_int_def) |
14334 | 561 |
|
16819 | 562 |
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
|
563 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
564 |
|
16819 | 565 |
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
|
566 |
by (simp add: real_of_int_def) |
16819 | 567 |
|
568 |
lemma real_of_int_diff [simp]: "real(x - y) = real (x::int) - real y" |
|
569 |
by (simp add: real_of_int_def) |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
570 |
|
16819 | 571 |
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
|
572 |
by (simp add: real_of_int_def) |
14334 | 573 |
|
16819 | 574 |
lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))" |
575 |
apply (subst real_eq_of_int)+ |
|
576 |
apply (rule of_int_setsum) |
|
577 |
done |
|
578 |
||
579 |
lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) = |
|
580 |
(PROD x:A. real(f x))" |
|
581 |
apply (subst real_eq_of_int)+ |
|
582 |
apply (rule of_int_setprod) |
|
583 |
done |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
584 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
585 |
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
|
586 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
587 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
588 |
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
|
589 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
590 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
591 |
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
|
592 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
593 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
594 |
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
|
595 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
596 |
|
16819 | 597 |
lemma real_of_int_gt_zero_cancel_iff [simp]: "(0 < real (n::int)) = (0 < n)" |
598 |
by (simp add: real_of_int_def) |
|
599 |
||
600 |
lemma real_of_int_ge_zero_cancel_iff [simp]: "(0 <= real (n::int)) = (0 <= n)" |
|
601 |
by (simp add: real_of_int_def) |
|
602 |
||
603 |
lemma real_of_int_lt_zero_cancel_iff [simp]: "(real (n::int) < 0) = (n < 0)" |
|
604 |
by (simp add: real_of_int_def) |
|
605 |
||
606 |
lemma real_of_int_le_zero_cancel_iff [simp]: "(real (n::int) <= 0) = (n <= 0)" |
|
607 |
by (simp add: real_of_int_def) |
|
608 |
||
16888 | 609 |
lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))" |
610 |
by (auto simp add: abs_if) |
|
611 |
||
16819 | 612 |
lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)" |
613 |
apply (subgoal_tac "real n + 1 = real (n + 1)") |
|
614 |
apply (simp del: real_of_int_add) |
|
615 |
apply auto |
|
616 |
done |
|
617 |
||
618 |
lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)" |
|
619 |
apply (subgoal_tac "real m + 1 = real (m + 1)") |
|
620 |
apply (simp del: real_of_int_add) |
|
621 |
apply simp |
|
622 |
done |
|
623 |
||
624 |
lemma real_of_int_div_aux: "d ~= 0 ==> (real (x::int)) / (real d) = |
|
625 |
real (x div d) + (real (x mod d)) / (real d)" |
|
626 |
proof - |
|
627 |
assume "d ~= 0" |
|
628 |
have "x = (x div d) * d + x mod d" |
|
629 |
by auto |
|
630 |
then have "real x = real (x div d) * real d + real(x mod d)" |
|
631 |
by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym]) |
|
632 |
then have "real x / real d = ... / real d" |
|
633 |
by simp |
|
634 |
then show ?thesis |
|
635 |
by (auto simp add: add_divide_distrib ring_eq_simps prems) |
|
636 |
qed |
|
637 |
||
638 |
lemma real_of_int_div: "(d::int) ~= 0 ==> d dvd n ==> |
|
639 |
real(n div d) = real n / real d" |
|
640 |
apply (frule real_of_int_div_aux [of d n]) |
|
641 |
apply simp |
|
642 |
apply (simp add: zdvd_iff_zmod_eq_0) |
|
643 |
done |
|
644 |
||
645 |
lemma real_of_int_div2: |
|
646 |
"0 <= real (n::int) / real (x) - real (n div x)" |
|
647 |
apply (case_tac "x = 0") |
|
648 |
apply simp |
|
649 |
apply (case_tac "0 < x") |
|
650 |
apply (simp add: compare_rls) |
|
651 |
apply (subst real_of_int_div_aux) |
|
652 |
apply simp |
|
653 |
apply simp |
|
654 |
apply (subst zero_le_divide_iff) |
|
655 |
apply auto |
|
656 |
apply (simp add: compare_rls) |
|
657 |
apply (subst real_of_int_div_aux) |
|
658 |
apply simp |
|
659 |
apply simp |
|
660 |
apply (subst zero_le_divide_iff) |
|
661 |
apply auto |
|
662 |
done |
|
663 |
||
664 |
lemma real_of_int_div3: |
|
665 |
"real (n::int) / real (x) - real (n div x) <= 1" |
|
666 |
apply(case_tac "x = 0") |
|
667 |
apply simp |
|
668 |
apply (simp add: compare_rls) |
|
669 |
apply (subst real_of_int_div_aux) |
|
670 |
apply assumption |
|
671 |
apply simp |
|
672 |
apply (subst divide_le_eq) |
|
673 |
apply clarsimp |
|
674 |
apply (rule conjI) |
|
675 |
apply (rule impI) |
|
676 |
apply (rule order_less_imp_le) |
|
677 |
apply simp |
|
678 |
apply (rule impI) |
|
679 |
apply (rule order_less_imp_le) |
|
680 |
apply simp |
|
681 |
done |
|
682 |
||
683 |
lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x" |
|
684 |
by (insert real_of_int_div2 [of n x], simp) |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
685 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
686 |
subsection{*Embedding the Naturals into the Reals*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
687 |
|
14334 | 688 |
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
|
689 |
by (simp add: real_of_nat_def) |
14334 | 690 |
|
691 |
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
|
692 |
by (simp add: real_of_nat_def) |
14334 | 693 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
694 |
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
|
695 |
by (simp add: real_of_nat_def) |
14334 | 696 |
|
697 |
(*Not for addsimps: often the LHS is used to represent a positive natural*) |
|
698 |
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
|
699 |
by (simp add: real_of_nat_def) |
14334 | 700 |
|
701 |
lemma real_of_nat_less_iff [iff]: |
|
702 |
"(real (n::nat) < real m) = (n < m)" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
703 |
by (simp add: real_of_nat_def) |
14334 | 704 |
|
705 |
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
|
706 |
by (simp add: real_of_nat_def) |
14334 | 707 |
|
708 |
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
|
709 |
by (simp add: real_of_nat_def zero_le_imp_of_nat) |
14334 | 710 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
711 |
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
|
712 |
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
|
713 |
|
14334 | 714 |
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
|
715 |
by (simp add: real_of_nat_def of_nat_mult) |
14334 | 716 |
|
16819 | 717 |
lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) = |
718 |
(SUM x:A. real(f x))" |
|
719 |
apply (subst real_eq_of_nat)+ |
|
720 |
apply (rule of_nat_setsum) |
|
721 |
done |
|
722 |
||
723 |
lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) = |
|
724 |
(PROD x:A. real(f x))" |
|
725 |
apply (subst real_eq_of_nat)+ |
|
726 |
apply (rule of_nat_setprod) |
|
727 |
done |
|
728 |
||
729 |
lemma real_of_card: "real (card A) = setsum (%x.1) A" |
|
730 |
apply (subst card_eq_setsum) |
|
731 |
apply (subst real_of_nat_setsum) |
|
732 |
apply simp |
|
733 |
done |
|
734 |
||
14334 | 735 |
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
|
736 |
by (simp add: real_of_nat_def) |
14334 | 737 |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
738 |
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
|
739 |
by (simp add: real_of_nat_def) |
14334 | 740 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
741 |
lemma real_of_nat_diff: "n \<le> m ==> 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
|
742 |
by (simp add: add: real_of_nat_def) |
14334 | 743 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
744 |
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
|
745 |
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
|
746 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
747 |
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
|
748 |
by (simp add: add: real_of_nat_def) |
14334 | 749 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
750 |
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
|
751 |
by (simp add: add: real_of_nat_def) |
14334 | 752 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
753 |
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
|
754 |
by (simp add: add: real_of_nat_def) |
14334 | 755 |
|
16819 | 756 |
lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)" |
757 |
apply (subgoal_tac "real n + 1 = real (Suc n)") |
|
758 |
apply simp |
|
759 |
apply (auto simp add: real_of_nat_Suc) |
|
760 |
done |
|
761 |
||
762 |
lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)" |
|
763 |
apply (subgoal_tac "real m + 1 = real (Suc m)") |
|
764 |
apply (simp add: less_Suc_eq_le) |
|
765 |
apply (simp add: real_of_nat_Suc) |
|
766 |
done |
|
767 |
||
768 |
lemma real_of_nat_div_aux: "0 < d ==> (real (x::nat)) / (real d) = |
|
769 |
real (x div d) + (real (x mod d)) / (real d)" |
|
770 |
proof - |
|
771 |
assume "0 < d" |
|
772 |
have "x = (x div d) * d + x mod d" |
|
773 |
by auto |
|
774 |
then have "real x = real (x div d) * real d + real(x mod d)" |
|
775 |
by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym]) |
|
776 |
then have "real x / real d = \<dots> / real d" |
|
777 |
by simp |
|
778 |
then show ?thesis |
|
779 |
by (auto simp add: add_divide_distrib ring_eq_simps prems) |
|
780 |
qed |
|
781 |
||
782 |
lemma real_of_nat_div: "0 < (d::nat) ==> d dvd n ==> |
|
783 |
real(n div d) = real n / real d" |
|
784 |
apply (frule real_of_nat_div_aux [of d n]) |
|
785 |
apply simp |
|
786 |
apply (subst dvd_eq_mod_eq_0 [THEN sym]) |
|
787 |
apply assumption |
|
788 |
done |
|
789 |
||
790 |
lemma real_of_nat_div2: |
|
791 |
"0 <= real (n::nat) / real (x) - real (n div x)" |
|
792 |
apply(case_tac "x = 0") |
|
793 |
apply simp |
|
794 |
apply (simp add: compare_rls) |
|
795 |
apply (subst real_of_nat_div_aux) |
|
796 |
apply assumption |
|
797 |
apply simp |
|
798 |
apply (subst zero_le_divide_iff) |
|
799 |
apply simp |
|
800 |
done |
|
801 |
||
802 |
lemma real_of_nat_div3: |
|
803 |
"real (n::nat) / real (x) - real (n div x) <= 1" |
|
804 |
apply(case_tac "x = 0") |
|
805 |
apply simp |
|
806 |
apply (simp add: compare_rls) |
|
807 |
apply (subst real_of_nat_div_aux) |
|
808 |
apply assumption |
|
809 |
apply simp |
|
810 |
done |
|
811 |
||
812 |
lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x" |
|
813 |
by (insert real_of_nat_div2 [of n x], simp) |
|
814 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
815 |
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
|
816 |
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
|
817 |
|
14426 | 818 |
lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n" |
819 |
by (simp add: real_of_int_def real_of_nat_def) |
|
14334 | 820 |
|
16819 | 821 |
lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x" |
822 |
apply (subgoal_tac "real(int(nat x)) = real(nat x)") |
|
823 |
apply force |
|
824 |
apply (simp only: real_of_int_real_of_nat) |
|
825 |
done |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
826 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
827 |
subsection{*Numerals and Arithmetic*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
828 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
829 |
instance real :: number .. |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
830 |
|
15013 | 831 |
defs (overloaded) |
20485 | 832 |
real_number_of_def: "(number_of w :: real) == of_int w" |
15013 | 833 |
--{*the type constraint is essential!*} |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
834 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
835 |
instance real :: number_ring |
15013 | 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 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
838 |
text{*Collapse applications of @{term real} to @{term number_of}*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
839 |
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
|
840 |
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
|
841 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
842 |
lemma real_of_nat_number_of [simp]: |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
843 |
"real (number_of v :: nat) = |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
844 |
(if neg (number_of v :: int) then 0 |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
845 |
else (number_of v :: real))" |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
846 |
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
|
847 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
848 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
849 |
use "real_arith.ML" |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
850 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
851 |
setup real_arith_setup |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
852 |
|
19023
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
16973
diff
changeset
|
853 |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
854 |
subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
855 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
856 |
text{*Needed in this non-standard form by Hyperreal/Transcendental*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
857 |
lemma real_0_le_divide_iff: |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
858 |
"((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
|
859 |
by (simp add: real_divide_def zero_le_mult_iff, auto) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
860 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
861 |
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
|
862 |
by arith |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
863 |
|
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15077
diff
changeset
|
864 |
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
|
865 |
by auto |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
866 |
|
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15077
diff
changeset
|
867 |
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
|
868 |
by auto |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
869 |
|
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15077
diff
changeset
|
870 |
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
|
871 |
by auto |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
872 |
|
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15077
diff
changeset
|
873 |
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
|
874 |
by auto |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
875 |
|
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15077
diff
changeset
|
876 |
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
|
877 |
by auto |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
878 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
879 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
880 |
(* |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
881 |
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
|
882 |
It replaces x+-y by x-y. |
15086 | 883 |
declare real_diff_def [symmetric, simp] |
14387
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 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
886 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
887 |
subsubsection{*Density of the Reals*} |
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 |
lemma real_lbound_gt_zero: |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
890 |
"[| (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
|
891 |
apply (rule_tac x = " (min d1 d2) /2" in exI) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
892 |
apply (simp add: min_def) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
893 |
done |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
894 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
895 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
896 |
text{*Similar results are proved in @{text Ring_and_Field}*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
897 |
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
|
898 |
by auto |
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 |
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
|
901 |
by auto |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
902 |
|
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 |
subsection{*Absolute Value Function for the Reals*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
905 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
906 |
lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))" |
15003 | 907 |
by (simp add: abs_if) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
908 |
|
23289 | 909 |
(* FIXME: redundant, but used by Integration/RealRandVar.thy in AFP *) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
910 |
lemma abs_le_interval_iff: "(abs x \<le> r) = (-r\<le>x & x\<le>(r::real))" |
14738 | 911 |
by (force simp add: OrderedGroup.abs_le_iff) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
912 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
913 |
lemma abs_add_one_gt_zero [simp]: "(0::real) < 1 + abs(x)" |
15003 | 914 |
by (simp add: abs_if) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
915 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
916 |
lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)" |
22958 | 917 |
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
|
918 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
919 |
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
|
920 |
by simp |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
921 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
922 |
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
|
923 |
by simp |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
924 |
|
23031
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
925 |
subsection{*Code generation using Isabelle's rats*} |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
926 |
|
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
927 |
types_code |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
928 |
real ("Rat.rat") |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
929 |
attach (term_of) {* |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
930 |
fun term_of_real x = |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
931 |
let |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
932 |
val rT = HOLogic.realT |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
933 |
val (p, q) = Rat.quotient_of_rat x |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
934 |
in if q = 1 then HOLogic.mk_number rT p |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
935 |
else Const("HOL.divide",[rT,rT] ---> rT) $ |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
936 |
(HOLogic.mk_number rT p) $ (HOLogic.mk_number rT q) |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
937 |
end; |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
938 |
*} |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
939 |
attach (test) {* |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
940 |
fun gen_real i = |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
941 |
let val p = random_range 0 i; val q = random_range 0 i; |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
942 |
val r = if q=0 then Rat.rat_of_int i else Rat.rat_of_quotient(p,q) |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
943 |
in if one_of [true,false] then r else Rat.neg r end; |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
944 |
*} |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
945 |
|
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
946 |
consts_code |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
947 |
"0 :: real" ("Rat.zero") |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
948 |
"1 :: real" ("Rat.one") |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
949 |
"uminus :: real \<Rightarrow> real" ("Rat.neg") |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
950 |
"op + :: real \<Rightarrow> real \<Rightarrow> real" ("Rat.add") |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
951 |
"op * :: real \<Rightarrow> real \<Rightarrow> real" ("Rat.mult") |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
952 |
"inverse :: real \<Rightarrow> real" ("Rat.inv") |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
953 |
"op \<le> :: real \<Rightarrow> real \<Rightarrow> bool" ("Rat.le") |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
954 |
"op < :: real \<Rightarrow> real \<Rightarrow> bool" ("(Rat.ord (_, _) = LESS)") |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
955 |
"op = :: real \<Rightarrow> real \<Rightarrow> bool" ("curry Rat.eq") |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
956 |
"real :: int \<Rightarrow> real" ("Rat.rat'_of'_int") |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
957 |
"real :: nat \<Rightarrow> real" ("(Rat.rat'_of'_int o {*int*})") |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
958 |
|
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
959 |
|
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
960 |
lemma [code, code unfold]: |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
961 |
"number_of k = real (number_of k :: int)" |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
962 |
by simp |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
963 |
|
5588 | 964 |
end |