author | paulson |
Thu, 29 Jul 2004 16:14:42 +0200 | |
changeset 15085 | 5693a977a767 |
parent 15077 | 89840837108e |
child 15086 | e6a2a98d5ef5 |
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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*) |
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||
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Polymorphic treatment of binary arithmetic using axclasses
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header{*Defining the Reals from the Positive Reals*} |
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Polymorphic treatment of binary arithmetic using axclasses
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|
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Polymorphic treatment of binary arithmetic using axclasses
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theory RealDef = PReal |
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files ("real_arith.ML"): |
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|
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constdefs |
|
14 |
realrel :: "((preal * preal) * (preal * preal)) set" |
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14269 | 15 |
"realrel == {p. \<exists>x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}" |
16 |
||
14484 | 17 |
typedef (Real) real = "UNIV//realrel" |
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by (auto simp add: quotient_def) |
5588 | 19 |
|
14691 | 20 |
instance real :: "{ord, zero, one, plus, times, minus, inverse}" .. |
14269 | 21 |
|
14484 | 22 |
constdefs |
23 |
||
24 |
(** these don't use the overloaded "real" function: users don't see them **) |
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25 |
||
26 |
real_of_preal :: "preal => real" |
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27 |
"real_of_preal m == |
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28 |
Abs_Real(realrel``{(m + preal_of_rat 1, preal_of_rat 1)})" |
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29 |
||
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consts |
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(*Overloaded constant denoting the Real subset of enclosing |
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types such as hypreal and complex*) |
33 |
Reals :: "'a set" |
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34 |
||
35 |
(*overloaded constant for injecting other types into "real"*) |
|
36 |
real :: "'a => real" |
|
5588 | 37 |
|
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syntax (xsymbols) |
39 |
Reals :: "'a set" ("\<real>") |
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40 |
||
5588 | 41 |
|
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defs (overloaded) |
5588 | 43 |
|
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real_zero_def: |
14484 | 45 |
"0 == Abs_Real(realrel``{(preal_of_rat 1, preal_of_rat 1)})" |
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|
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real_one_def: |
14484 | 48 |
"1 == Abs_Real(realrel`` |
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{(preal_of_rat 1 + preal_of_rat 1, |
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preal_of_rat 1)})" |
5588 | 51 |
|
14269 | 52 |
real_minus_def: |
14484 | 53 |
"- r == contents (\<Union>(x,y) \<in> Rep_Real(r). { Abs_Real(realrel``{(y,x)}) })" |
54 |
||
55 |
real_add_def: |
|
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"z + w == |
|
57 |
contents (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w). |
|
58 |
{ Abs_Real(realrel``{(x+u, y+v)}) })" |
|
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|
14269 | 60 |
real_diff_def: |
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"r - (s::real) == r + - s" |
62 |
||
63 |
real_mult_def: |
|
64 |
"z * w == |
|
65 |
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)}) })" |
|
5588 | 67 |
|
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real_inverse_def: |
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"inverse (R::real) == (SOME S. (R = 0 & S = 0) | S * R = 1)" |
5588 | 70 |
|
14269 | 71 |
real_divide_def: |
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"R / (S::real) == R * inverse S" |
14269 | 73 |
|
14484 | 74 |
real_le_def: |
75 |
"z \<le> (w::real) == |
|
76 |
\<exists>x y u v. x+v \<le> u+y & (x,y) \<in> Rep_Real z & (u,v) \<in> Rep_Real w" |
|
5588 | 77 |
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real_less_def: "(x < (y::real)) == (x \<le> y & x \<noteq> y)" |
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79 |
|
14334 | 80 |
real_abs_def: "abs (r::real) == (if 0 \<le> r then r else -r)" |
81 |
||
82 |
||
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83 |
|
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subsection{*Proving that realrel is an equivalence relation*} |
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|
14270 | 86 |
lemma preal_trans_lemma: |
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87 |
assumes "x + y1 = x1 + y" |
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and "x + y2 = x2 + y" |
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89 |
shows "x1 + y2 = x2 + (y1::preal)" |
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90 |
proof - |
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91 |
have "(x1 + y2) + x = (x + y2) + x1" by (simp add: preal_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: preal_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: preal_add_ac) |
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finally have "(x1 + y2) + x = (x2 + y1) + x" . |
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thus ?thesis by (simp add: preal_add_right_cancel_iff) |
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98 |
qed |
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99 |
|
14269 | 100 |
|
14484 | 101 |
lemma realrel_iff [simp]: "(((x1,y1),(x2,y2)) \<in> realrel) = (x1 + y2 = x2 + y1)" |
102 |
by (simp add: realrel_def) |
|
14269 | 103 |
|
104 |
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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106 |
apply (blast dest: preal_trans_lemma) |
14269 | 107 |
done |
108 |
||
14497 | 109 |
text{*Reduces equality of equivalence classes to the @{term realrel} relation: |
110 |
@{term "(realrel `` {x} = realrel `` {y}) = ((x,y) \<in> realrel)"} *} |
|
14269 | 111 |
lemmas equiv_realrel_iff = |
112 |
eq_equiv_class_iff [OF equiv_realrel UNIV_I UNIV_I] |
|
113 |
||
114 |
declare equiv_realrel_iff [simp] |
|
115 |
||
14497 | 116 |
|
14484 | 117 |
lemma realrel_in_real [simp]: "realrel``{(x,y)}: Real" |
118 |
by (simp add: Real_def realrel_def quotient_def, blast) |
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14269 | 119 |
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120 |
|
14484 | 121 |
lemma inj_on_Abs_Real: "inj_on Abs_Real Real" |
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apply (rule inj_on_inverseI) |
14484 | 123 |
apply (erule Abs_Real_inverse) |
14269 | 124 |
done |
125 |
||
14484 | 126 |
declare inj_on_Abs_Real [THEN inj_on_iff, simp] |
127 |
declare Abs_Real_inverse [simp] |
|
14269 | 128 |
|
129 |
||
14484 | 130 |
text{*Case analysis on the representation of a real number as an equivalence |
131 |
class of pairs of positive reals.*} |
|
132 |
lemma eq_Abs_Real [case_names Abs_Real, cases type: real]: |
|
133 |
"(!!x y. z = Abs_Real(realrel``{(x,y)}) ==> P) ==> P" |
|
134 |
apply (rule Rep_Real [of z, unfolded Real_def, THEN quotientE]) |
|
135 |
apply (drule arg_cong [where f=Abs_Real]) |
|
136 |
apply (auto simp add: Rep_Real_inverse) |
|
14269 | 137 |
done |
138 |
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139 |
||
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subsection{*Congruence property for addition*} |
14269 | 141 |
|
142 |
lemma real_add_congruent2_lemma: |
|
143 |
"[|a + ba = aa + b; ab + bc = ac + bb|] |
|
144 |
==> a + ab + (ba + bc) = aa + ac + (b + (bb::preal))" |
|
145 |
apply (simp add: preal_add_assoc) |
|
146 |
apply (rule preal_add_left_commute [of ab, THEN ssubst]) |
|
147 |
apply (simp add: preal_add_assoc [symmetric]) |
|
148 |
apply (simp add: preal_add_ac) |
|
149 |
done |
|
150 |
||
151 |
lemma real_add: |
|
14497 | 152 |
"Abs_Real (realrel``{(x,y)}) + Abs_Real (realrel``{(u,v)}) = |
153 |
Abs_Real (realrel``{(x+u, y+v)})" |
|
154 |
proof - |
|
14658 | 155 |
have "congruent2 realrel realrel |
14497 | 156 |
(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). {Abs_Real (realrel `` {(x+u, y+v)})}) w) z)" |
157 |
by (simp add: congruent2_def, blast intro: real_add_congruent2_lemma) |
|
158 |
thus ?thesis |
|
159 |
by (simp add: real_add_def UN_UN_split_split_eq |
|
14658 | 160 |
UN_equiv_class2 [OF equiv_realrel equiv_realrel]) |
14497 | 161 |
qed |
14269 | 162 |
|
163 |
lemma real_add_commute: "(z::real) + w = w + z" |
|
14497 | 164 |
by (cases z, cases w, simp add: real_add preal_add_ac) |
14269 | 165 |
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166 |
lemma real_add_assoc: "((z1::real) + z2) + z3 = z1 + (z2 + z3)" |
|
14497 | 167 |
by (cases z1, cases z2, cases z3, simp add: real_add preal_add_assoc) |
14269 | 168 |
|
169 |
lemma real_add_zero_left: "(0::real) + z = z" |
|
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by (cases z, simp add: real_add real_zero_def preal_add_ac) |
14269 | 171 |
|
14738 | 172 |
instance real :: comm_monoid_add |
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by (intro_classes, |
174 |
(assumption | |
|
175 |
rule real_add_commute real_add_assoc real_add_zero_left)+) |
|
176 |
||
177 |
||
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subsection{*Additive Inverse on real*} |
179 |
||
14484 | 180 |
lemma real_minus: "- Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})" |
181 |
proof - |
|
182 |
have "congruent realrel (\<lambda>(x,y). {Abs_Real (realrel``{(y,x)})})" |
|
183 |
by (simp add: congruent_def preal_add_commute) |
|
184 |
thus ?thesis |
|
185 |
by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel]) |
|
186 |
qed |
|
14334 | 187 |
|
188 |
lemma real_add_minus_left: "(-z) + z = (0::real)" |
|
14497 | 189 |
by (cases z, simp add: real_minus real_add real_zero_def preal_add_commute) |
14269 | 190 |
|
191 |
||
14329 | 192 |
subsection{*Congruence property for multiplication*} |
14269 | 193 |
|
14329 | 194 |
lemma real_mult_congruent2_lemma: |
195 |
"!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==> |
|
14484 | 196 |
x * x1 + y * y1 + (x * y2 + y * x2) = |
197 |
x * x2 + y * y2 + (x * y1 + y * x1)" |
|
198 |
apply (simp add: preal_add_left_commute preal_add_assoc [symmetric]) |
|
14269 | 199 |
apply (simp add: preal_add_assoc preal_add_mult_distrib2 [symmetric]) |
200 |
apply (simp add: preal_add_commute) |
|
201 |
done |
|
202 |
||
203 |
lemma real_mult_congruent2: |
|
14658 | 204 |
"congruent2 realrel realrel (%p1 p2. |
14484 | 205 |
(%(x1,y1). (%(x2,y2). |
206 |
{ Abs_Real (realrel``{(x1*x2 + y1*y2, x1*y2+y1*x2)}) }) p2) p1)" |
|
14658 | 207 |
apply (rule congruent2_commuteI [OF equiv_realrel], clarify) |
14269 | 208 |
apply (simp add: preal_mult_commute preal_add_commute) |
209 |
apply (auto simp add: real_mult_congruent2_lemma) |
|
210 |
done |
|
211 |
||
212 |
lemma real_mult: |
|
14484 | 213 |
"Abs_Real((realrel``{(x1,y1)})) * Abs_Real((realrel``{(x2,y2)})) = |
214 |
Abs_Real(realrel `` {(x1*x2+y1*y2,x1*y2+y1*x2)})" |
|
215 |
by (simp add: real_mult_def UN_UN_split_split_eq |
|
14658 | 216 |
UN_equiv_class2 [OF equiv_realrel equiv_realrel real_mult_congruent2]) |
14269 | 217 |
|
218 |
lemma real_mult_commute: "(z::real) * w = w * z" |
|
14497 | 219 |
by (cases z, cases w, simp add: real_mult preal_add_ac preal_mult_ac) |
14269 | 220 |
|
221 |
lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)" |
|
14484 | 222 |
apply (cases z1, cases z2, cases z3) |
223 |
apply (simp add: real_mult preal_add_mult_distrib2 preal_add_ac preal_mult_ac) |
|
14269 | 224 |
done |
225 |
||
226 |
lemma real_mult_1: "(1::real) * z = z" |
|
14484 | 227 |
apply (cases z) |
228 |
apply (simp add: real_mult real_one_def preal_add_mult_distrib2 |
|
229 |
preal_mult_1_right preal_mult_ac preal_add_ac) |
|
14269 | 230 |
done |
231 |
||
232 |
lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)" |
|
14484 | 233 |
apply (cases z1, cases z2, cases w) |
234 |
apply (simp add: real_add real_mult preal_add_mult_distrib2 |
|
235 |
preal_add_ac preal_mult_ac) |
|
14269 | 236 |
done |
237 |
||
14329 | 238 |
text{*one and zero are distinct*} |
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239 |
lemma real_zero_not_eq_one: "0 \<noteq> (1::real)" |
14484 | 240 |
proof - |
241 |
have "preal_of_rat 1 < preal_of_rat 1 + preal_of_rat 1" |
|
242 |
by (simp add: preal_self_less_add_left) |
|
243 |
thus ?thesis |
|
244 |
by (simp add: real_zero_def real_one_def preal_add_right_cancel_iff) |
|
245 |
qed |
|
14269 | 246 |
|
14329 | 247 |
subsection{*existence of inverse*} |
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248 |
|
14484 | 249 |
lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0" |
14497 | 250 |
by (simp add: real_zero_def preal_add_commute) |
14269 | 251 |
|
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252 |
text{*Instead of using an existential quantifier and constructing the inverse |
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253 |
within the proof, we could define the inverse explicitly.*} |
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254 |
|
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|
255 |
lemma real_mult_inverse_left_ex: "x \<noteq> 0 ==> \<exists>y. y*x = (1::real)" |
14484 | 256 |
apply (simp add: real_zero_def real_one_def, cases x) |
14269 | 257 |
apply (cut_tac x = xa and y = y in linorder_less_linear) |
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258 |
apply (auto dest!: less_add_left_Ex simp add: real_zero_iff) |
14334 | 259 |
apply (rule_tac |
14484 | 260 |
x = "Abs_Real (realrel `` { (preal_of_rat 1, |
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|
261 |
inverse (D) + preal_of_rat 1)}) " |
14334 | 262 |
in exI) |
263 |
apply (rule_tac [2] |
|
14484 | 264 |
x = "Abs_Real (realrel `` { (inverse (D) + preal_of_rat 1, |
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|
265 |
preal_of_rat 1)})" |
14334 | 266 |
in exI) |
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|
267 |
apply (auto simp add: real_mult preal_mult_1_right |
14329 | 268 |
preal_add_mult_distrib2 preal_add_mult_distrib preal_mult_1 |
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|
269 |
preal_mult_inverse_right preal_add_ac preal_mult_ac) |
14269 | 270 |
done |
271 |
||
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|
272 |
lemma real_mult_inverse_left: "x \<noteq> 0 ==> inverse(x)*x = (1::real)" |
14484 | 273 |
apply (simp add: real_inverse_def) |
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|
274 |
apply (frule real_mult_inverse_left_ex, safe) |
14269 | 275 |
apply (rule someI2, auto) |
276 |
done |
|
14334 | 277 |
|
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
278 |
|
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
279 |
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
|
280 |
|
14334 | 281 |
instance real :: field |
282 |
proof |
|
283 |
fix x y z :: real |
|
284 |
show "- x + x = 0" by (rule real_add_minus_left) |
|
285 |
show "x - y = x + (-y)" by (simp add: real_diff_def) |
|
286 |
show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc) |
|
287 |
show "x * y = y * x" by (rule real_mult_commute) |
|
288 |
show "1 * x = x" by (rule real_mult_1) |
|
289 |
show "(x + y) * z = x * z + y * z" by (simp add: real_add_mult_distrib) |
|
290 |
show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one) |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
291 |
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
|
292 |
show "x / y = x * inverse y" by (simp add: real_divide_def) |
14334 | 293 |
qed |
294 |
||
295 |
||
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
296 |
text{*Inverse of zero! Useful to simplify certain equations*} |
14269 | 297 |
|
14334 | 298 |
lemma INVERSE_ZERO: "inverse 0 = (0::real)" |
14484 | 299 |
by (simp add: real_inverse_def) |
14334 | 300 |
|
301 |
instance real :: division_by_zero |
|
302 |
proof |
|
303 |
show "inverse 0 = (0::real)" by (rule INVERSE_ZERO) |
|
304 |
qed |
|
305 |
||
306 |
||
307 |
(*Pull negations out*) |
|
308 |
declare minus_mult_right [symmetric, simp] |
|
309 |
minus_mult_left [symmetric, simp] |
|
310 |
||
311 |
lemma real_mult_1_right: "z * (1::real) = z" |
|
14738 | 312 |
by (rule OrderedGroup.mult_1_right) |
14269 | 313 |
|
314 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
315 |
subsection{*The @{text "\<le>"} Ordering*} |
14269 | 316 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
317 |
lemma real_le_refl: "w \<le> (w::real)" |
14484 | 318 |
by (cases w, force simp add: real_le_def) |
14269 | 319 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
320 |
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
|
321 |
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
|
322 |
following two lemmas.*} |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
323 |
lemma preal_eq_le_imp_le: |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
324 |
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
|
325 |
shows "b \<le> (d::preal)" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
326 |
proof - |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
327 |
have "c+d \<le> a+d" by (simp add: prems preal_cancels) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
328 |
hence "a+b \<le> a+d" by (simp add: prems) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
329 |
thus "b \<le> d" by (simp add: preal_cancels) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
330 |
qed |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
331 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
332 |
lemma real_le_lemma: |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
333 |
assumes l: "u1 + v2 \<le> u2 + v1" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
334 |
and "x1 + v1 = u1 + y1" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
335 |
and "x2 + v2 = u2 + y2" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
336 |
shows "x1 + y2 \<le> x2 + (y1::preal)" |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
337 |
proof - |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
338 |
have "(x1+v1) + (u2+y2) = (u1+y1) + (x2+v2)" by (simp add: prems) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
339 |
hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: preal_add_ac) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
340 |
also have "... \<le> (x2+y1) + (u2+v1)" |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
341 |
by (simp add: prems preal_add_le_cancel_left) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
342 |
finally show ?thesis by (simp add: preal_add_le_cancel_right) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
343 |
qed |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
344 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
345 |
lemma real_le: |
14484 | 346 |
"(Abs_Real(realrel``{(x1,y1)}) \<le> Abs_Real(realrel``{(x2,y2)})) = |
347 |
(x1 + y2 \<le> x2 + y1)" |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
348 |
apply (simp add: real_le_def) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
349 |
apply (auto intro: real_le_lemma) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
350 |
done |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
351 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
352 |
lemma real_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::real)" |
14497 | 353 |
by (cases z, cases w, simp add: real_le order_antisym) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
354 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
355 |
lemma real_trans_lemma: |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
356 |
assumes "x + v \<le> u + y" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
357 |
and "u + v' \<le> u' + v" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
358 |
and "x2 + v2 = u2 + y2" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
359 |
shows "x + v' \<le> u' + (y::preal)" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
360 |
proof - |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
361 |
have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: preal_add_ac) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
362 |
also have "... \<le> (u+y) + (u+v')" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
363 |
by (simp add: preal_add_le_cancel_right prems) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
364 |
also have "... \<le> (u+y) + (u'+v)" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
365 |
by (simp add: preal_add_le_cancel_left prems) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
366 |
also have "... = (u'+y) + (u+v)" by (simp add: preal_add_ac) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
367 |
finally show ?thesis by (simp add: preal_add_le_cancel_right) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
368 |
qed |
14269 | 369 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
370 |
lemma real_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::real)" |
14484 | 371 |
apply (cases i, cases j, cases k) |
372 |
apply (simp add: real_le) |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
373 |
apply (blast intro: real_trans_lemma) |
14334 | 374 |
done |
375 |
||
376 |
(* Axiom 'order_less_le' of class 'order': *) |
|
377 |
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
|
378 |
by (simp add: real_less_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
379 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
380 |
instance real :: order |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
381 |
proof qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
382 |
(assumption | |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
383 |
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
|
384 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
385 |
(* Axiom 'linorder_linear' of class 'linorder': *) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
386 |
lemma real_le_linear: "(z::real) \<le> w | w \<le> z" |
14484 | 387 |
apply (cases z, cases w) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
388 |
apply (auto simp add: real_le real_zero_def preal_add_ac preal_cancels) |
14334 | 389 |
done |
390 |
||
391 |
||
392 |
instance real :: linorder |
|
393 |
by (intro_classes, rule real_le_linear) |
|
394 |
||
395 |
||
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
396 |
lemma real_le_eq_diff: "(x \<le> y) = (x-y \<le> (0::real))" |
14484 | 397 |
apply (cases x, cases y) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
398 |
apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
399 |
preal_add_ac) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
400 |
apply (simp_all add: preal_add_assoc [symmetric] preal_cancels) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
401 |
done |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
402 |
|
14484 | 403 |
lemma real_add_left_mono: |
404 |
assumes le: "x \<le> y" shows "z + x \<le> z + (y::real)" |
|
405 |
proof - |
|
406 |
have "z + x - (z + y) = (z + -z) + (x - y)" |
|
407 |
by (simp add: diff_minus add_ac) |
|
408 |
with le show ?thesis |
|
14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
409 |
by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"] diff_minus) |
14484 | 410 |
qed |
14334 | 411 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
412 |
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
|
413 |
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
|
414 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
415 |
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
|
416 |
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus) |
14334 | 417 |
|
418 |
lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y" |
|
14484 | 419 |
apply (cases x, cases y) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
420 |
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
|
421 |
linorder_not_le [where 'a = preal] |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
422 |
real_zero_def real_le real_mult) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
423 |
--{*Reduce to the (simpler) @{text "\<le>"} relation *} |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
424 |
apply (auto dest!: less_add_left_Ex |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
425 |
simp add: preal_add_ac preal_mult_ac |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
426 |
preal_add_mult_distrib2 preal_cancels preal_self_less_add_right) |
14334 | 427 |
done |
428 |
||
429 |
lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y" |
|
430 |
apply (rule real_sum_gt_zero_less) |
|
431 |
apply (drule real_less_sum_gt_zero [of x y]) |
|
432 |
apply (drule real_mult_order, assumption) |
|
433 |
apply (simp add: right_distrib) |
|
434 |
done |
|
435 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
436 |
text{*lemma for proving @{term "0<(1::real)"}*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
437 |
lemma real_zero_le_one: "0 \<le> (1::real)" |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
438 |
by (simp add: real_zero_def real_one_def real_le |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
439 |
preal_self_less_add_left order_less_imp_le) |
14334 | 440 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
441 |
|
14334 | 442 |
subsection{*The Reals Form an Ordered Field*} |
443 |
||
444 |
instance real :: ordered_field |
|
445 |
proof |
|
446 |
fix x y z :: real |
|
447 |
show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono) |
|
448 |
show "x < y ==> 0 < z ==> z * x < z * y" by (simp add: real_mult_less_mono2) |
|
449 |
show "\<bar>x\<bar> = (if x < 0 then -x else x)" |
|
450 |
by (auto dest: order_le_less_trans simp add: real_abs_def linorder_not_le) |
|
451 |
qed |
|
452 |
||
14365
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 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
455 |
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
|
456 |
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
|
457 |
positive reals.*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
458 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
459 |
lemma real_of_preal_add: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
460 |
"real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
461 |
by (simp add: real_of_preal_def real_add preal_add_mult_distrib preal_mult_1 |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
462 |
preal_add_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
463 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
464 |
lemma real_of_preal_mult: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
465 |
"real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
466 |
by (simp add: real_of_preal_def real_mult preal_add_mult_distrib2 |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
467 |
preal_mult_1 preal_mult_1_right preal_add_ac preal_mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
468 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
469 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
470 |
text{*Gleason prop 9-4.4 p 127*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
471 |
lemma real_of_preal_trichotomy: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
472 |
"\<exists>m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)" |
14484 | 473 |
apply (simp add: real_of_preal_def real_zero_def, cases x) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
474 |
apply (auto simp add: real_minus preal_add_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
475 |
apply (cut_tac x = x and y = y in linorder_less_linear) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
476 |
apply (auto dest!: less_add_left_Ex simp add: preal_add_assoc [symmetric]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
477 |
apply (auto simp add: preal_add_commute) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
478 |
done |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
479 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
480 |
lemma real_of_preal_leD: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
481 |
"real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2" |
14484 | 482 |
by (simp add: real_of_preal_def real_le preal_cancels) |
14365
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 |
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
|
485 |
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
|
486 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
487 |
lemma real_of_preal_lessD: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
488 |
"real_of_preal m1 < real_of_preal m2 ==> m1 < m2" |
14484 | 489 |
by (simp add: real_of_preal_def real_le linorder_not_le [symmetric] |
490 |
preal_cancels) |
|
491 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
492 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
493 |
lemma real_of_preal_less_iff [simp]: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
494 |
"(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
|
495 |
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
|
496 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
497 |
lemma real_of_preal_le_iff: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
498 |
"(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
499 |
by (simp add: linorder_not_less [symmetric]) |
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_of_preal_zero_less: "0 < real_of_preal m" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
502 |
apply (auto simp add: real_zero_def real_of_preal_def real_less_def real_le_def |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
503 |
preal_add_ac preal_cancels) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
504 |
apply (simp_all add: preal_add_assoc [symmetric] preal_cancels) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
505 |
apply (blast intro: preal_self_less_add_left order_less_imp_le) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
506 |
apply (insert preal_not_eq_self [of "preal_of_rat 1" m]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
507 |
apply (simp add: preal_add_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
508 |
done |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
509 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
510 |
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
|
511 |
by (simp add: real_of_preal_zero_less) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
512 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
513 |
lemma real_of_preal_not_minus_gt_zero: "~ 0 < - real_of_preal m" |
14484 | 514 |
proof - |
515 |
from real_of_preal_minus_less_zero |
|
516 |
show ?thesis by (blast dest: order_less_trans) |
|
517 |
qed |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
518 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
519 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
520 |
subsection{*Theorems About the Ordering*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
521 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
522 |
text{*obsolete but used a lot*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
523 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
524 |
lemma real_not_refl2: "x < y ==> x \<noteq> (y::real)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
525 |
by blast |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
526 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
527 |
lemma real_le_imp_less_or_eq: "!!(x::real). x \<le> y ==> x < y | x = y" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
528 |
by (simp add: order_le_less) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
529 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
530 |
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
|
531 |
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
|
532 |
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
|
533 |
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
|
534 |
done |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
535 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
536 |
lemma real_gt_preal_preal_Ex: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
537 |
"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
|
538 |
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
|
539 |
intro: real_gt_zero_preal_Ex [THEN iffD1]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
540 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
541 |
lemma real_ge_preal_preal_Ex: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
542 |
"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
|
543 |
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
|
544 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
545 |
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
|
546 |
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
|
547 |
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
|
548 |
simp add: real_of_preal_zero_less) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
549 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
550 |
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
|
551 |
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
|
552 |
|
14334 | 553 |
lemma real_add_less_le_mono: "[| w'<w; z'\<le>z |] ==> w' + z' < w + (z::real)" |
14738 | 554 |
by (rule OrderedGroup.add_less_le_mono) |
14334 | 555 |
|
556 |
lemma real_add_le_less_mono: |
|
557 |
"!!z z'::real. [| w'\<le>w; z'<z |] ==> w' + z' < w + z" |
|
14738 | 558 |
by (rule OrderedGroup.add_le_less_mono) |
14334 | 559 |
|
560 |
lemma real_le_square [simp]: "(0::real) \<le> x*x" |
|
561 |
by (rule Ring_and_Field.zero_le_square) |
|
562 |
||
563 |
||
564 |
subsection{*More Lemmas*} |
|
565 |
||
566 |
lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)" |
|
567 |
by auto |
|
568 |
||
569 |
lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)" |
|
570 |
by auto |
|
571 |
||
572 |
text{*The precondition could be weakened to @{term "0\<le>x"}*} |
|
573 |
lemma real_mult_less_mono: |
|
574 |
"[| u<v; x<y; (0::real) < v; 0 < x |] ==> u*x < v* y" |
|
575 |
by (simp add: Ring_and_Field.mult_strict_mono order_less_imp_le) |
|
576 |
||
577 |
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)" |
|
578 |
by (force elim: order_less_asym |
|
579 |
simp add: Ring_and_Field.mult_less_cancel_right) |
|
580 |
||
581 |
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
|
582 |
apply (simp add: mult_le_cancel_right) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
583 |
apply (blast intro: elim: order_less_asym) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
584 |
done |
14334 | 585 |
|
586 |
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)" |
|
587 |
by (force elim: order_less_asym |
|
588 |
simp add: Ring_and_Field.mult_le_cancel_left) |
|
589 |
||
590 |
text{*Only two uses?*} |
|
591 |
lemma real_mult_less_mono': |
|
592 |
"[| x < y; r1 < r2; (0::real) \<le> r1; 0 \<le> x|] ==> r1 * x < r2 * y" |
|
593 |
by (rule Ring_and_Field.mult_strict_mono') |
|
594 |
||
595 |
text{*FIXME: delete or at least combine the next two lemmas*} |
|
596 |
lemma real_sum_squares_cancel: "x * x + y * y = 0 ==> x = (0::real)" |
|
14738 | 597 |
apply (drule OrderedGroup.equals_zero_I [THEN sym]) |
14334 | 598 |
apply (cut_tac x = y in real_le_square) |
14476 | 599 |
apply (auto, drule order_antisym, auto) |
14334 | 600 |
done |
601 |
||
602 |
lemma real_sum_squares_cancel2: "x * x + y * y = 0 ==> y = (0::real)" |
|
603 |
apply (rule_tac y = x in real_sum_squares_cancel) |
|
14476 | 604 |
apply (simp add: add_commute) |
14334 | 605 |
done |
606 |
||
607 |
lemma real_add_order: "[| 0 < x; 0 < y |] ==> (0::real) < x + y" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
608 |
by (drule add_strict_mono [of concl: 0 0], assumption, simp) |
14334 | 609 |
|
610 |
lemma real_le_add_order: "[| 0 \<le> x; 0 \<le> y |] ==> (0::real) \<le> x + y" |
|
611 |
apply (drule order_le_imp_less_or_eq)+ |
|
612 |
apply (auto intro: real_add_order order_less_imp_le) |
|
613 |
done |
|
614 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
615 |
lemma real_inverse_unique: "x*y = (1::real) ==> y = inverse x" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
616 |
apply (case_tac "x \<noteq> 0") |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
617 |
apply (rule_tac c1 = x in real_mult_left_cancel [THEN iffD1], auto) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
618 |
done |
14334 | 619 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
620 |
lemma real_inverse_gt_one: "[| (0::real) < x; x < 1 |] ==> 1 < inverse x" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
621 |
by (auto dest: less_imp_inverse_less) |
14334 | 622 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
623 |
lemma real_mult_self_sum_ge_zero: "(0::real) \<le> x*x + y*y" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
624 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
625 |
have "0 + 0 \<le> x*x + y*y" by (blast intro: add_mono zero_le_square) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
626 |
thus ?thesis by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
627 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
628 |
|
14334 | 629 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
630 |
subsection{*Embedding the Integers into the Reals*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
631 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
632 |
defs (overloaded) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
633 |
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
|
634 |
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
|
635 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
636 |
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
|
637 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
638 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
639 |
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
|
640 |
by (simp add: real_of_int_def) |
14334 | 641 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
642 |
lemma real_of_int_add: "real (x::int) + real y = real (x + y)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
643 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
644 |
declare real_of_int_add [symmetric, simp] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
645 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
646 |
lemma real_of_int_minus: "-real (x::int) = real (-x)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
647 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
648 |
declare real_of_int_minus [symmetric, simp] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
649 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
650 |
lemma real_of_int_diff: "real (x::int) - real y = real (x - y)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
651 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
652 |
declare real_of_int_diff [symmetric, simp] |
14334 | 653 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
654 |
lemma real_of_int_mult: "real (x::int) * real y = real (x * y)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
655 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
656 |
declare real_of_int_mult [symmetric, simp] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
657 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
658 |
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
|
659 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
660 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
661 |
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
|
662 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
663 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
664 |
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
|
665 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
666 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
667 |
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
|
668 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
669 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
670 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
671 |
subsection{*Embedding the Naturals into the Reals*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
672 |
|
14334 | 673 |
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
|
674 |
by (simp add: real_of_nat_def) |
14334 | 675 |
|
676 |
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
|
677 |
by (simp add: real_of_nat_def) |
14334 | 678 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
679 |
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
|
680 |
by (simp add: real_of_nat_def) |
14334 | 681 |
|
682 |
(*Not for addsimps: often the LHS is used to represent a positive natural*) |
|
683 |
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
|
684 |
by (simp add: real_of_nat_def) |
14334 | 685 |
|
686 |
lemma real_of_nat_less_iff [iff]: |
|
687 |
"(real (n::nat) < real m) = (n < m)" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
688 |
by (simp add: real_of_nat_def) |
14334 | 689 |
|
690 |
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
|
691 |
by (simp add: real_of_nat_def) |
14334 | 692 |
|
693 |
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
|
694 |
by (simp add: real_of_nat_def zero_le_imp_of_nat) |
14334 | 695 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
696 |
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
|
697 |
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
|
698 |
|
14334 | 699 |
lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
700 |
by (simp add: real_of_nat_def) |
14334 | 701 |
|
702 |
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
|
703 |
by (simp add: real_of_nat_def) |
14334 | 704 |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
705 |
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
|
706 |
by (simp add: real_of_nat_def) |
14334 | 707 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
708 |
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
|
709 |
by (simp add: add: real_of_nat_def) |
14334 | 710 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
711 |
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
|
712 |
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
|
713 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
714 |
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
|
715 |
by (simp add: add: real_of_nat_def) |
14334 | 716 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
717 |
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
|
718 |
by (simp add: add: real_of_nat_def) |
14334 | 719 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
720 |
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
|
721 |
by (simp add: add: real_of_nat_def) |
14334 | 722 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
723 |
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
|
724 |
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
|
725 |
|
14426 | 726 |
lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n" |
727 |
by (simp add: real_of_int_def real_of_nat_def) |
|
14334 | 728 |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
729 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
730 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
731 |
subsection{*Numerals and Arithmetic*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
732 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
733 |
instance real :: number .. |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
734 |
|
15013 | 735 |
defs (overloaded) |
736 |
real_number_of_def: "(number_of w :: real) == of_int (Rep_Bin w)" |
|
737 |
--{*the type constraint is essential!*} |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
738 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
739 |
instance real :: number_ring |
15013 | 740 |
by (intro_classes, simp add: real_number_of_def) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
741 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
742 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
743 |
text{*Collapse applications of @{term real} to @{term number_of}*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
744 |
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
|
745 |
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
|
746 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
747 |
lemma real_of_nat_number_of [simp]: |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
748 |
"real (number_of v :: nat) = |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
749 |
(if neg (number_of v :: int) then 0 |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
750 |
else (number_of v :: real))" |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
751 |
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
|
752 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
753 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
754 |
use "real_arith.ML" |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
755 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
756 |
setup real_arith_setup |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
757 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
758 |
subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
759 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
760 |
text{*Needed in this non-standard form by Hyperreal/Transcendental*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
761 |
lemma real_0_le_divide_iff: |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
762 |
"((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
|
763 |
by (simp add: real_divide_def zero_le_mult_iff, auto) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
764 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
765 |
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
|
766 |
by arith |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
767 |
|
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15077
diff
changeset
|
768 |
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
|
769 |
by auto |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
770 |
|
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15077
diff
changeset
|
771 |
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
|
772 |
by auto |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
773 |
|
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15077
diff
changeset
|
774 |
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
|
775 |
by auto |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
776 |
|
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15077
diff
changeset
|
777 |
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
|
778 |
by auto |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
779 |
|
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15077
diff
changeset
|
780 |
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
|
781 |
by auto |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
782 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
783 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
784 |
(** Simprules combining x-y and 0 (needed??) **) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
785 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
786 |
lemma real_0_less_diff_iff [iff]: "((0::real) < x-y) = (y < x)" |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
787 |
by auto |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
788 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
789 |
lemma real_0_le_diff_iff [iff]: "((0::real) \<le> x-y) = (y \<le> x)" |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
790 |
by auto |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
791 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
792 |
(* |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
793 |
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
|
794 |
It replaces x+-y by x-y. |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
795 |
Addsimps [symmetric real_diff_def] |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
796 |
*) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
797 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
798 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
799 |
subsubsection{*Density of the Reals*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
800 |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
801 |
(*????FIXME: rename d1, d2 to x, y*) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
802 |
lemma real_lbound_gt_zero: |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
803 |
"[| (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
|
804 |
apply (rule_tac x = " (min d1 d2) /2" in exI) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
805 |
apply (simp add: min_def) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
806 |
done |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
807 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
808 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
809 |
text{*Similar results are proved in @{text Ring_and_Field}*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
810 |
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
|
811 |
by auto |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
812 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
813 |
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
|
814 |
by auto |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
815 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
816 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
817 |
subsection{*Absolute Value Function for the Reals*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
818 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
819 |
text{*FIXME: these should go!*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
820 |
lemma abs_eqI1: "(0::real)\<le>x ==> abs x = x" |
15003 | 821 |
by (simp add: abs_if) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
822 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
823 |
lemma abs_eqI2: "(0::real) < x ==> abs x = x" |
15003 | 824 |
by (simp add: abs_if) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
825 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
826 |
lemma abs_minus_eqI2: "x < (0::real) ==> abs x = -x" |
15003 | 827 |
by (simp add: abs_if linorder_not_less [symmetric]) |
14387
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 |
lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))" |
15003 | 830 |
by (simp add: abs_if) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
831 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
832 |
lemma abs_interval_iff: "(abs x < r) = (-r < x & x < (r::real))" |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
833 |
by (force simp add: Ring_and_Field.abs_less_iff) |
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 |
lemma abs_le_interval_iff: "(abs x \<le> r) = (-r\<le>x & x\<le>(r::real))" |
14738 | 836 |
by (force simp add: OrderedGroup.abs_le_iff) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
837 |
|
14484 | 838 |
(*FIXME: used only once, in SEQ.ML*) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
839 |
lemma abs_add_one_gt_zero [simp]: "(0::real) < 1 + abs(x)" |
15003 | 840 |
by (simp add: abs_if) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
841 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
842 |
lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)" |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
843 |
by (auto intro: abs_eqI1 simp add: real_of_nat_ge_zero) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
844 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
845 |
lemma abs_add_one_not_less_self [simp]: "~ abs(x) + (1::real) < x" |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
846 |
apply (simp add: linorder_not_less) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
847 |
apply (auto intro: abs_ge_self [THEN order_trans]) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
848 |
done |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
849 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
850 |
text{*Used only in Hyperreal/Lim.ML*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
851 |
lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \<le> abs(x + -l) + abs(y + -m)" |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
852 |
apply (simp add: real_add_assoc) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
853 |
apply (rule_tac a1 = y in add_left_commute [THEN ssubst]) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
854 |
apply (rule real_add_assoc [THEN subst]) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
855 |
apply (rule abs_triangle_ineq) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
856 |
done |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
857 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
858 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
859 |
|
14334 | 860 |
ML |
861 |
{* |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
862 |
val real_0_le_divide_iff = thm"real_0_le_divide_iff"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
863 |
val real_0_less_diff_iff = thm"real_0_less_diff_iff"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
864 |
val real_0_le_diff_iff = thm"real_0_le_diff_iff"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
865 |
val real_lbound_gt_zero = thm"real_lbound_gt_zero"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
866 |
val real_less_half_sum = thm"real_less_half_sum"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
867 |
val real_gt_half_sum = thm"real_gt_half_sum"; |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
868 |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
869 |
val abs_eqI1 = thm"abs_eqI1"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
870 |
val abs_eqI2 = thm"abs_eqI2"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
871 |
val abs_minus_eqI2 = thm"abs_minus_eqI2"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
872 |
val abs_ge_zero = thm"abs_ge_zero"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
873 |
val abs_idempotent = thm"abs_idempotent"; |
14738 | 874 |
val abs_eq_0 = thm"abs_eq_0"; |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
875 |
val abs_ge_self = thm"abs_ge_self"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
876 |
val abs_ge_minus_self = thm"abs_ge_minus_self"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
877 |
val abs_mult = thm"abs_mult"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
878 |
val abs_inverse = thm"abs_inverse"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
879 |
val abs_triangle_ineq = thm"abs_triangle_ineq"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
880 |
val abs_minus_cancel = thm"abs_minus_cancel"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
881 |
val abs_minus_add_cancel = thm"abs_minus_add_cancel"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
882 |
val abs_interval_iff = thm"abs_interval_iff"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
883 |
val abs_le_interval_iff = thm"abs_le_interval_iff"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
884 |
val abs_add_one_gt_zero = thm"abs_add_one_gt_zero"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
885 |
val abs_le_zero_iff = thm"abs_le_zero_iff"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
886 |
val abs_add_one_not_less_self = thm"abs_add_one_not_less_self"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
887 |
val abs_sum_triangle_ineq = thm"abs_sum_triangle_ineq"; |
14334 | 888 |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
889 |
val abs_mult_less = thm"abs_mult_less"; |
14334 | 890 |
*} |
10752
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10648
diff
changeset
|
891 |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
892 |
|
5588 | 893 |
end |