author | huffman |
Wed, 04 Mar 2009 17:12:23 -0800 | |
changeset 30273 | ecd6f0ca62ea |
parent 30242 | aea5d7fa7ef5 |
child 31100 | 6a2e67fe4488 |
permissions | -rw-r--r-- |
28952
15a4b2cf8c34
made repository layout more coherent with logical distribution structure; stripped some $Id$s
haftmann
parents:
28906
diff
changeset
|
1 |
(* Title : HOL/RealDef.thy |
5588 | 2 |
Author : Jacques D. Fleuriot |
3 |
Copyright : 1998 University of Cambridge |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
4 |
Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4 |
16819 | 5 |
Additional contributions by Jeremy Avigad |
14269 | 6 |
*) |
7 |
||
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
8 |
header{*Defining the Reals from the Positive Reals*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
9 |
|
15131 | 10 |
theory RealDef |
15140 | 11 |
imports PReal |
28952
15a4b2cf8c34
made repository layout more coherent with logical distribution structure; stripped some $Id$s
haftmann
parents:
28906
diff
changeset
|
12 |
uses ("Tools/real_arith.ML") |
15131 | 13 |
begin |
5588 | 14 |
|
19765 | 15 |
definition |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20554
diff
changeset
|
16 |
realrel :: "((preal * preal) * (preal * preal)) set" where |
28562 | 17 |
[code del]: "realrel = {p. \<exists>x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}" |
14269 | 18 |
|
14484 | 19 |
typedef (Real) real = "UNIV//realrel" |
14269 | 20 |
by (auto simp add: quotient_def) |
5588 | 21 |
|
19765 | 22 |
definition |
14484 | 23 |
(** these don't use the overloaded "real" function: users don't see them **) |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20554
diff
changeset
|
24 |
real_of_preal :: "preal => real" where |
28562 | 25 |
[code del]: "real_of_preal m = Abs_Real (realrel `` {(m + 1, 1)})" |
14484 | 26 |
|
25762 | 27 |
instantiation real :: "{zero, one, plus, minus, uminus, times, inverse, ord, abs, sgn}" |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
28 |
begin |
5588 | 29 |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
30 |
definition |
28562 | 31 |
real_zero_def [code del]: "0 = Abs_Real(realrel``{(1, 1)})" |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
32 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
33 |
definition |
28562 | 34 |
real_one_def [code del]: "1 = Abs_Real(realrel``{(1 + 1, 1)})" |
5588 | 35 |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
36 |
definition |
28562 | 37 |
real_add_def [code del]: "z + w = |
14484 | 38 |
contents (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w). |
27964 | 39 |
{ Abs_Real(realrel``{(x+u, y+v)}) })" |
10606 | 40 |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
41 |
definition |
28562 | 42 |
real_minus_def [code del]: "- r = contents (\<Union>(x,y) \<in> Rep_Real(r). { Abs_Real(realrel``{(y,x)}) })" |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
43 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
44 |
definition |
28562 | 45 |
real_diff_def [code del]: "r - (s::real) = r + - s" |
14484 | 46 |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
47 |
definition |
28562 | 48 |
real_mult_def [code del]: |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
49 |
"z * w = |
14484 | 50 |
contents (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w). |
27964 | 51 |
{ Abs_Real(realrel``{(x*u + y*v, x*v + y*u)}) })" |
5588 | 52 |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
53 |
definition |
28562 | 54 |
real_inverse_def [code del]: "inverse (R::real) = (THE S. (R = 0 & S = 0) | S * R = 1)" |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
55 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
56 |
definition |
28562 | 57 |
real_divide_def [code del]: "R / (S::real) = R * inverse S" |
14269 | 58 |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
59 |
definition |
28562 | 60 |
real_le_def [code del]: "z \<le> (w::real) \<longleftrightarrow> |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
61 |
(\<exists>x y u v. x+v \<le> u+y & (x,y) \<in> Rep_Real z & (u,v) \<in> Rep_Real w)" |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
62 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
63 |
definition |
28562 | 64 |
real_less_def [code del]: "x < (y\<Colon>real) \<longleftrightarrow> x \<le> y \<and> x \<noteq> y" |
5588 | 65 |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
66 |
definition |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
67 |
real_abs_def: "abs (r::real) = (if r < 0 then - r else r)" |
14334 | 68 |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
69 |
definition |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
70 |
real_sgn_def: "sgn (x::real) = (if x=0 then 0 else if 0<x then 1 else - 1)" |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
71 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
72 |
instance .. |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
73 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
74 |
end |
14334 | 75 |
|
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
76 |
subsection {* Equivalence relation over positive reals *} |
14269 | 77 |
|
14270 | 78 |
lemma preal_trans_lemma: |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
79 |
assumes "x + y1 = x1 + y" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
80 |
and "x + y2 = x2 + y" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
81 |
shows "x1 + y2 = x2 + (y1::preal)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
82 |
proof - |
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
83 |
have "(x1 + y2) + x = (x + y2) + x1" by (simp add: add_ac) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
84 |
also have "... = (x2 + y) + x1" by (simp add: prems) |
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
85 |
also have "... = x2 + (x1 + y)" by (simp add: add_ac) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
86 |
also have "... = x2 + (x + y1)" by (simp add: prems) |
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
87 |
also have "... = (x2 + y1) + x" by (simp add: add_ac) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
88 |
finally have "(x1 + y2) + x = (x2 + y1) + x" . |
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
89 |
thus ?thesis by (rule add_right_imp_eq) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
90 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
91 |
|
14269 | 92 |
|
14484 | 93 |
lemma realrel_iff [simp]: "(((x1,y1),(x2,y2)) \<in> realrel) = (x1 + y2 = x2 + y1)" |
94 |
by (simp add: realrel_def) |
|
14269 | 95 |
|
96 |
lemma equiv_realrel: "equiv UNIV realrel" |
|
30198 | 97 |
apply (auto simp add: equiv_def refl_on_def sym_def trans_def realrel_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
98 |
apply (blast dest: preal_trans_lemma) |
14269 | 99 |
done |
100 |
||
14497 | 101 |
text{*Reduces equality of equivalence classes to the @{term realrel} relation: |
102 |
@{term "(realrel `` {x} = realrel `` {y}) = ((x,y) \<in> realrel)"} *} |
|
14269 | 103 |
lemmas equiv_realrel_iff = |
104 |
eq_equiv_class_iff [OF equiv_realrel UNIV_I UNIV_I] |
|
105 |
||
106 |
declare equiv_realrel_iff [simp] |
|
107 |
||
14497 | 108 |
|
14484 | 109 |
lemma realrel_in_real [simp]: "realrel``{(x,y)}: Real" |
110 |
by (simp add: Real_def realrel_def quotient_def, blast) |
|
14269 | 111 |
|
22958 | 112 |
declare Abs_Real_inject [simp] |
14484 | 113 |
declare Abs_Real_inverse [simp] |
14269 | 114 |
|
115 |
||
14484 | 116 |
text{*Case analysis on the representation of a real number as an equivalence |
117 |
class of pairs of positive reals.*} |
|
118 |
lemma eq_Abs_Real [case_names Abs_Real, cases type: real]: |
|
119 |
"(!!x y. z = Abs_Real(realrel``{(x,y)}) ==> P) ==> P" |
|
120 |
apply (rule Rep_Real [of z, unfolded Real_def, THEN quotientE]) |
|
121 |
apply (drule arg_cong [where f=Abs_Real]) |
|
122 |
apply (auto simp add: Rep_Real_inverse) |
|
14269 | 123 |
done |
124 |
||
125 |
||
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
126 |
subsection {* Addition and Subtraction *} |
14269 | 127 |
|
128 |
lemma real_add_congruent2_lemma: |
|
129 |
"[|a + ba = aa + b; ab + bc = ac + bb|] |
|
130 |
==> a + ab + (ba + bc) = aa + ac + (b + (bb::preal))" |
|
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
131 |
apply (simp add: add_assoc) |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
132 |
apply (rule add_left_commute [of ab, THEN ssubst]) |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
133 |
apply (simp add: add_assoc [symmetric]) |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
134 |
apply (simp add: add_ac) |
14269 | 135 |
done |
136 |
||
137 |
lemma real_add: |
|
14497 | 138 |
"Abs_Real (realrel``{(x,y)}) + Abs_Real (realrel``{(u,v)}) = |
139 |
Abs_Real (realrel``{(x+u, y+v)})" |
|
140 |
proof - |
|
15169 | 141 |
have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). {Abs_Real (realrel `` {(x+u, y+v)})}) w) z) |
142 |
respects2 realrel" |
|
14497 | 143 |
by (simp add: congruent2_def, blast intro: real_add_congruent2_lemma) |
144 |
thus ?thesis |
|
145 |
by (simp add: real_add_def UN_UN_split_split_eq |
|
14658 | 146 |
UN_equiv_class2 [OF equiv_realrel equiv_realrel]) |
14497 | 147 |
qed |
14269 | 148 |
|
14484 | 149 |
lemma real_minus: "- Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})" |
150 |
proof - |
|
15169 | 151 |
have "(\<lambda>(x,y). {Abs_Real (realrel``{(y,x)})}) respects realrel" |
23288 | 152 |
by (simp add: congruent_def add_commute) |
14484 | 153 |
thus ?thesis |
154 |
by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel]) |
|
155 |
qed |
|
14334 | 156 |
|
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
157 |
instance real :: ab_group_add |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
158 |
proof |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
159 |
fix x y z :: real |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
160 |
show "(x + y) + z = x + (y + z)" |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
161 |
by (cases x, cases y, cases z, simp add: real_add add_assoc) |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
162 |
show "x + y = y + x" |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
163 |
by (cases x, cases y, simp add: real_add add_commute) |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
164 |
show "0 + x = x" |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
165 |
by (cases x, simp add: real_add real_zero_def add_ac) |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
166 |
show "- x + x = 0" |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
167 |
by (cases x, simp add: real_minus real_add real_zero_def add_commute) |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
168 |
show "x - y = x + - y" |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
169 |
by (simp add: real_diff_def) |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
170 |
qed |
14269 | 171 |
|
172 |
||
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
173 |
subsection {* Multiplication *} |
14269 | 174 |
|
14329 | 175 |
lemma real_mult_congruent2_lemma: |
176 |
"!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==> |
|
14484 | 177 |
x * x1 + y * y1 + (x * y2 + y * x2) = |
178 |
x * x2 + y * y2 + (x * y1 + y * x1)" |
|
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
179 |
apply (simp add: add_left_commute add_assoc [symmetric]) |
23288 | 180 |
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. |
14484 | 186 |
(%(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) |
29667 | 205 |
apply (simp add: real_mult algebra_simps) |
14269 | 206 |
done |
207 |
||
208 |
lemma real_mult_1: "(1::real) * z = z" |
|
14484 | 209 |
apply (cases z) |
29667 | 210 |
apply (simp add: real_mult real_one_def algebra_simps) |
14269 | 211 |
done |
212 |
||
213 |
lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)" |
|
14484 | 214 |
apply (cases z1, cases z2, cases w) |
29667 | 215 |
apply (simp add: real_add real_mult algebra_simps) |
14269 | 216 |
done |
217 |
||
14329 | 218 |
text{*one and zero are distinct*} |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
219 |
lemma real_zero_not_eq_one: "0 \<noteq> (1::real)" |
14484 | 220 |
proof - |
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
221 |
have "(1::preal) < 1 + 1" |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
222 |
by (simp add: preal_self_less_add_left) |
14484 | 223 |
thus ?thesis |
23288 | 224 |
by (simp add: real_zero_def real_one_def) |
14484 | 225 |
qed |
14269 | 226 |
|
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
227 |
instance real :: comm_ring_1 |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
228 |
proof |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
229 |
fix x y z :: real |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
230 |
show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc) |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
231 |
show "x * y = y * x" by (rule real_mult_commute) |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
232 |
show "1 * x = x" by (rule real_mult_1) |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
233 |
show "(x + y) * z = x * z + y * z" by (rule real_add_mult_distrib) |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
234 |
show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one) |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
235 |
qed |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
236 |
|
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
237 |
subsection {* Inverse and Division *} |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
238 |
|
14484 | 239 |
lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0" |
23288 | 240 |
by (simp add: real_zero_def add_commute) |
14269 | 241 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
242 |
text{*Instead of using an existential quantifier and constructing the inverse |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
243 |
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
|
244 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
245 |
lemma real_mult_inverse_left_ex: "x \<noteq> 0 ==> \<exists>y. y*x = (1::real)" |
14484 | 246 |
apply (simp add: real_zero_def real_one_def, cases x) |
14269 | 247 |
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
|
248 |
apply (auto dest!: less_add_left_Ex simp add: real_zero_iff) |
14334 | 249 |
apply (rule_tac |
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
250 |
x = "Abs_Real (realrel``{(1, inverse (D) + 1)})" |
14334 | 251 |
in exI) |
252 |
apply (rule_tac [2] |
|
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
253 |
x = "Abs_Real (realrel``{(inverse (D) + 1, 1)})" |
14334 | 254 |
in exI) |
29667 | 255 |
apply (auto simp add: real_mult preal_mult_inverse_right algebra_simps) |
14269 | 256 |
done |
257 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
258 |
lemma real_mult_inverse_left: "x \<noteq> 0 ==> inverse(x)*x = (1::real)" |
14484 | 259 |
apply (simp add: real_inverse_def) |
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
260 |
apply (drule real_mult_inverse_left_ex, safe) |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
261 |
apply (rule theI, assumption, rename_tac z) |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
262 |
apply (subgoal_tac "(z * x) * y = z * (x * y)") |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
263 |
apply (simp add: mult_commute) |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset
|
264 |
apply (rule mult_assoc) |
14269 | 265 |
done |
14334 | 266 |
|
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
267 |
|
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
268 |
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
|
269 |
|
14334 | 270 |
instance real :: field |
271 |
proof |
|
272 |
fix x y z :: real |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
273 |
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
|
274 |
show "x / y = x * inverse y" by (simp add: real_divide_def) |
14334 | 275 |
qed |
276 |
||
277 |
||
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
278 |
text{*Inverse of zero! Useful to simplify certain equations*} |
14269 | 279 |
|
14334 | 280 |
lemma INVERSE_ZERO: "inverse 0 = (0::real)" |
14484 | 281 |
by (simp add: real_inverse_def) |
14334 | 282 |
|
283 |
instance real :: division_by_zero |
|
284 |
proof |
|
285 |
show "inverse 0 = (0::real)" by (rule INVERSE_ZERO) |
|
286 |
qed |
|
287 |
||
14269 | 288 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
289 |
subsection{*The @{text "\<le>"} Ordering*} |
14269 | 290 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
291 |
lemma real_le_refl: "w \<le> (w::real)" |
14484 | 292 |
by (cases w, force simp add: real_le_def) |
14269 | 293 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
294 |
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
|
295 |
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
|
296 |
following two lemmas.*} |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
297 |
lemma preal_eq_le_imp_le: |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
298 |
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
|
299 |
shows "b \<le> (d::preal)" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
300 |
proof - |
23288 | 301 |
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
|
302 |
hence "a+b \<le> a+d" by (simp add: prems) |
23288 | 303 |
thus "b \<le> d" by simp |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
304 |
qed |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
305 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
306 |
lemma real_le_lemma: |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
307 |
assumes l: "u1 + v2 \<le> u2 + v1" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
308 |
and "x1 + v1 = u1 + y1" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
309 |
and "x2 + v2 = u2 + y2" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
310 |
shows "x1 + y2 \<le> x2 + (y1::preal)" |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
311 |
proof - |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
312 |
have "(x1+v1) + (u2+y2) = (u1+y1) + (x2+v2)" by (simp add: prems) |
23288 | 313 |
hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: add_ac) |
314 |
also have "... \<le> (x2+y1) + (u2+v1)" by (simp add: prems) |
|
315 |
finally show ?thesis by simp |
|
316 |
qed |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
317 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
318 |
lemma real_le: |
14484 | 319 |
"(Abs_Real(realrel``{(x1,y1)}) \<le> Abs_Real(realrel``{(x2,y2)})) = |
320 |
(x1 + y2 \<le> x2 + y1)" |
|
23288 | 321 |
apply (simp add: real_le_def) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
322 |
apply (auto intro: real_le_lemma) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
323 |
done |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
324 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
325 |
lemma real_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::real)" |
15542 | 326 |
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
|
327 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
328 |
lemma real_trans_lemma: |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
329 |
assumes "x + v \<le> u + y" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
330 |
and "u + v' \<le> u' + v" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
331 |
and "x2 + v2 = u2 + y2" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
332 |
shows "x + v' \<le> u' + (y::preal)" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
333 |
proof - |
23288 | 334 |
have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: add_ac) |
335 |
also have "... \<le> (u+y) + (u+v')" by (simp add: prems) |
|
336 |
also have "... \<le> (u+y) + (u'+v)" by (simp add: prems) |
|
337 |
also have "... = (u'+y) + (u+v)" by (simp add: add_ac) |
|
338 |
finally show ?thesis by simp |
|
15542 | 339 |
qed |
14269 | 340 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
341 |
lemma real_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::real)" |
14484 | 342 |
apply (cases i, cases j, cases k) |
343 |
apply (simp add: real_le) |
|
23288 | 344 |
apply (blast intro: real_trans_lemma) |
14334 | 345 |
done |
346 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
347 |
instance real :: order |
27682 | 348 |
proof |
349 |
fix u v :: real |
|
350 |
show "u < v \<longleftrightarrow> u \<le> v \<and> \<not> v \<le> u" |
|
351 |
by (auto simp add: real_less_def intro: real_le_anti_sym) |
|
352 |
qed (assumption | rule real_le_refl real_le_trans real_le_anti_sym)+ |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
353 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
354 |
(* Axiom 'linorder_linear' of class 'linorder': *) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
355 |
lemma real_le_linear: "(z::real) \<le> w | w \<le> z" |
23288 | 356 |
apply (cases z, cases w) |
357 |
apply (auto simp add: real_le real_zero_def add_ac) |
|
14334 | 358 |
done |
359 |
||
360 |
instance real :: linorder |
|
361 |
by (intro_classes, rule real_le_linear) |
|
362 |
||
363 |
||
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
364 |
lemma real_le_eq_diff: "(x \<le> y) = (x-y \<le> (0::real))" |
14484 | 365 |
apply (cases x, cases y) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
366 |
apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus |
23288 | 367 |
add_ac) |
368 |
apply (simp_all add: add_assoc [symmetric]) |
|
15542 | 369 |
done |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
370 |
|
14484 | 371 |
lemma real_add_left_mono: |
372 |
assumes le: "x \<le> y" shows "z + x \<le> z + (y::real)" |
|
373 |
proof - |
|
27668 | 374 |
have "z + x - (z + y) = (z + -z) + (x - y)" |
29667 | 375 |
by (simp add: algebra_simps) |
14484 | 376 |
with le show ?thesis |
14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
377 |
by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"] diff_minus) |
14484 | 378 |
qed |
14334 | 379 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
380 |
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
|
381 |
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
|
382 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
383 |
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
|
384 |
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus) |
14334 | 385 |
|
386 |
lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y" |
|
14484 | 387 |
apply (cases x, cases y) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
388 |
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
|
389 |
linorder_not_le [where 'a = preal] |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
390 |
real_zero_def real_le real_mult) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
391 |
--{*Reduce to the (simpler) @{text "\<le>"} relation *} |
16973 | 392 |
apply (auto dest!: less_add_left_Ex |
29667 | 393 |
simp add: algebra_simps preal_self_less_add_left) |
14334 | 394 |
done |
395 |
||
396 |
lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y" |
|
397 |
apply (rule real_sum_gt_zero_less) |
|
398 |
apply (drule real_less_sum_gt_zero [of x y]) |
|
399 |
apply (drule real_mult_order, assumption) |
|
400 |
apply (simp add: right_distrib) |
|
401 |
done |
|
402 |
||
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
403 |
instantiation real :: distrib_lattice |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
404 |
begin |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
405 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
406 |
definition |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
407 |
"(inf \<Colon> real \<Rightarrow> real \<Rightarrow> real) = min" |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
408 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
409 |
definition |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
410 |
"(sup \<Colon> real \<Rightarrow> real \<Rightarrow> real) = max" |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
411 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
412 |
instance |
22456 | 413 |
by default (auto simp add: inf_real_def sup_real_def min_max.sup_inf_distrib1) |
414 |
||
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
415 |
end |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
416 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
417 |
|
14334 | 418 |
subsection{*The Reals Form an Ordered Field*} |
419 |
||
420 |
instance real :: ordered_field |
|
421 |
proof |
|
422 |
fix x y z :: real |
|
423 |
show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono) |
|
22962 | 424 |
show "x < y ==> 0 < z ==> z * x < z * y" by (rule real_mult_less_mono2) |
425 |
show "\<bar>x\<bar> = (if x < 0 then -x else x)" by (simp only: real_abs_def) |
|
24506 | 426 |
show "sgn x = (if x=0 then 0 else if 0<x then 1 else - 1)" |
427 |
by (simp only: real_sgn_def) |
|
14334 | 428 |
qed |
429 |
||
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25162
diff
changeset
|
430 |
instance real :: lordered_ab_group_add .. |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25162
diff
changeset
|
431 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
432 |
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
|
433 |
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
|
434 |
positive reals.*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
435 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
436 |
lemma real_of_preal_add: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
437 |
"real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y" |
29667 | 438 |
by (simp add: real_of_preal_def real_add algebra_simps) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
439 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
440 |
lemma real_of_preal_mult: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
441 |
"real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y" |
29667 | 442 |
by (simp add: real_of_preal_def real_mult algebra_simps) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
443 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
444 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
445 |
text{*Gleason prop 9-4.4 p 127*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
446 |
lemma real_of_preal_trichotomy: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
447 |
"\<exists>m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)" |
14484 | 448 |
apply (simp add: real_of_preal_def real_zero_def, cases x) |
23288 | 449 |
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
|
450 |
apply (cut_tac x = x and y = y in linorder_less_linear) |
23288 | 451 |
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
|
452 |
done |
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_leD: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
455 |
"real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2" |
23288 | 456 |
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
|
457 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
458 |
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
|
459 |
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
|
460 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
461 |
lemma real_of_preal_lessD: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
462 |
"real_of_preal m1 < real_of_preal m2 ==> m1 < m2" |
23288 | 463 |
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
|
464 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
465 |
lemma real_of_preal_less_iff [simp]: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
466 |
"(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
|
467 |
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
|
468 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
469 |
lemma real_of_preal_le_iff: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
470 |
"(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)" |
23288 | 471 |
by (simp add: linorder_not_less [symmetric]) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
472 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
473 |
lemma real_of_preal_zero_less: "0 < real_of_preal m" |
23288 | 474 |
apply (insert preal_self_less_add_left [of 1 m]) |
475 |
apply (auto simp add: real_zero_def real_of_preal_def |
|
476 |
real_less_def real_le_def add_ac) |
|
477 |
apply (rule_tac x="m + 1" in exI, rule_tac x="1" in exI) |
|
478 |
apply (simp add: add_ac) |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
479 |
done |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
480 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
481 |
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
|
482 |
by (simp add: real_of_preal_zero_less) |
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_not_minus_gt_zero: "~ 0 < - real_of_preal m" |
14484 | 485 |
proof - |
486 |
from real_of_preal_minus_less_zero |
|
487 |
show ?thesis by (blast dest: order_less_trans) |
|
488 |
qed |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
489 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
490 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
491 |
subsection{*Theorems About the Ordering*} |
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_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
|
494 |
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
|
495 |
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
|
496 |
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
|
497 |
done |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
498 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
499 |
lemma real_gt_preal_preal_Ex: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
500 |
"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
|
501 |
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
|
502 |
intro: real_gt_zero_preal_Ex [THEN iffD1]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
503 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
504 |
lemma real_ge_preal_preal_Ex: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
505 |
"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
|
506 |
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
|
507 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
508 |
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
|
509 |
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
|
510 |
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
|
511 |
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_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
|
514 |
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
|
515 |
|
14334 | 516 |
|
517 |
subsection{*More Lemmas*} |
|
518 |
||
519 |
lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)" |
|
520 |
by auto |
|
521 |
||
522 |
lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)" |
|
523 |
by auto |
|
524 |
||
525 |
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)" |
|
526 |
by (force elim: order_less_asym |
|
527 |
simp add: Ring_and_Field.mult_less_cancel_right) |
|
528 |
||
529 |
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
|
530 |
apply (simp add: mult_le_cancel_right) |
23289 | 531 |
apply (blast intro: elim: order_less_asym) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
532 |
done |
14334 | 533 |
|
534 |
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)" |
|
15923 | 535 |
by(simp add:mult_commute) |
14334 | 536 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
537 |
lemma real_inverse_gt_one: "[| (0::real) < x; x < 1 |] ==> 1 < inverse x" |
23289 | 538 |
by (simp add: one_less_inverse_iff) (* TODO: generalize/move *) |
14334 | 539 |
|
540 |
||
24198 | 541 |
subsection {* Embedding numbers into the Reals *} |
542 |
||
543 |
abbreviation |
|
544 |
real_of_nat :: "nat \<Rightarrow> real" |
|
545 |
where |
|
546 |
"real_of_nat \<equiv> of_nat" |
|
547 |
||
548 |
abbreviation |
|
549 |
real_of_int :: "int \<Rightarrow> real" |
|
550 |
where |
|
551 |
"real_of_int \<equiv> of_int" |
|
552 |
||
553 |
abbreviation |
|
554 |
real_of_rat :: "rat \<Rightarrow> real" |
|
555 |
where |
|
556 |
"real_of_rat \<equiv> of_rat" |
|
557 |
||
558 |
consts |
|
559 |
(*overloaded constant for injecting other types into "real"*) |
|
560 |
real :: "'a => real" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
561 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
562 |
defs (overloaded) |
28520 | 563 |
real_of_nat_def [code unfold]: "real == real_of_nat" |
564 |
real_of_int_def [code unfold]: "real == real_of_int" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
565 |
|
16819 | 566 |
lemma real_eq_of_nat: "real = of_nat" |
24198 | 567 |
unfolding real_of_nat_def .. |
16819 | 568 |
|
569 |
lemma real_eq_of_int: "real = of_int" |
|
24198 | 570 |
unfolding real_of_int_def .. |
16819 | 571 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
572 |
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
|
573 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
574 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
575 |
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
|
576 |
by (simp add: real_of_int_def) |
14334 | 577 |
|
16819 | 578 |
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
|
579 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
580 |
|
16819 | 581 |
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
|
582 |
by (simp add: real_of_int_def) |
16819 | 583 |
|
584 |
lemma real_of_int_diff [simp]: "real(x - y) = real (x::int) - real y" |
|
585 |
by (simp add: real_of_int_def) |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
586 |
|
16819 | 587 |
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
|
588 |
by (simp add: real_of_int_def) |
14334 | 589 |
|
16819 | 590 |
lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))" |
591 |
apply (subst real_eq_of_int)+ |
|
592 |
apply (rule of_int_setsum) |
|
593 |
done |
|
594 |
||
595 |
lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) = |
|
596 |
(PROD x:A. real(f x))" |
|
597 |
apply (subst real_eq_of_int)+ |
|
598 |
apply (rule of_int_setprod) |
|
599 |
done |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
600 |
|
27668 | 601 |
lemma real_of_int_zero_cancel [simp, algebra, presburger]: "(real x = 0) = (x = (0::int))" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
602 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
603 |
|
27668 | 604 |
lemma real_of_int_inject [iff, algebra, presburger]: "(real (x::int) = real y) = (x = y)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
605 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
606 |
|
27668 | 607 |
lemma real_of_int_less_iff [iff, presburger]: "(real (x::int) < real y) = (x < y)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
608 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
609 |
|
27668 | 610 |
lemma real_of_int_le_iff [simp, presburger]: "(real (x::int) \<le> real y) = (x \<le> y)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
611 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
612 |
|
27668 | 613 |
lemma real_of_int_gt_zero_cancel_iff [simp, presburger]: "(0 < real (n::int)) = (0 < n)" |
16819 | 614 |
by (simp add: real_of_int_def) |
615 |
||
27668 | 616 |
lemma real_of_int_ge_zero_cancel_iff [simp, presburger]: "(0 <= real (n::int)) = (0 <= n)" |
16819 | 617 |
by (simp add: real_of_int_def) |
618 |
||
27668 | 619 |
lemma real_of_int_lt_zero_cancel_iff [simp, presburger]: "(real (n::int) < 0) = (n < 0)" |
16819 | 620 |
by (simp add: real_of_int_def) |
621 |
||
27668 | 622 |
lemma real_of_int_le_zero_cancel_iff [simp, presburger]: "(real (n::int) <= 0) = (n <= 0)" |
16819 | 623 |
by (simp add: real_of_int_def) |
624 |
||
16888 | 625 |
lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))" |
626 |
by (auto simp add: abs_if) |
|
627 |
||
16819 | 628 |
lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)" |
629 |
apply (subgoal_tac "real n + 1 = real (n + 1)") |
|
630 |
apply (simp del: real_of_int_add) |
|
631 |
apply auto |
|
632 |
done |
|
633 |
||
634 |
lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)" |
|
635 |
apply (subgoal_tac "real m + 1 = real (m + 1)") |
|
636 |
apply (simp del: real_of_int_add) |
|
637 |
apply simp |
|
638 |
done |
|
639 |
||
640 |
lemma real_of_int_div_aux: "d ~= 0 ==> (real (x::int)) / (real d) = |
|
641 |
real (x div d) + (real (x mod d)) / (real d)" |
|
642 |
proof - |
|
643 |
assume "d ~= 0" |
|
644 |
have "x = (x div d) * d + x mod d" |
|
645 |
by auto |
|
646 |
then have "real x = real (x div d) * real d + real(x mod d)" |
|
647 |
by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym]) |
|
648 |
then have "real x / real d = ... / real d" |
|
649 |
by simp |
|
650 |
then show ?thesis |
|
29667 | 651 |
by (auto simp add: add_divide_distrib algebra_simps prems) |
16819 | 652 |
qed |
653 |
||
654 |
lemma real_of_int_div: "(d::int) ~= 0 ==> d dvd n ==> |
|
655 |
real(n div d) = real n / real d" |
|
656 |
apply (frule real_of_int_div_aux [of d n]) |
|
657 |
apply simp |
|
30042 | 658 |
apply (simp add: dvd_eq_mod_eq_0) |
16819 | 659 |
done |
660 |
||
661 |
lemma real_of_int_div2: |
|
662 |
"0 <= real (n::int) / real (x) - real (n div x)" |
|
663 |
apply (case_tac "x = 0") |
|
664 |
apply simp |
|
665 |
apply (case_tac "0 < x") |
|
29667 | 666 |
apply (simp add: algebra_simps) |
16819 | 667 |
apply (subst real_of_int_div_aux) |
668 |
apply simp |
|
669 |
apply simp |
|
670 |
apply (subst zero_le_divide_iff) |
|
671 |
apply auto |
|
29667 | 672 |
apply (simp add: algebra_simps) |
16819 | 673 |
apply (subst real_of_int_div_aux) |
674 |
apply simp |
|
675 |
apply simp |
|
676 |
apply (subst zero_le_divide_iff) |
|
677 |
apply auto |
|
678 |
done |
|
679 |
||
680 |
lemma real_of_int_div3: |
|
681 |
"real (n::int) / real (x) - real (n div x) <= 1" |
|
682 |
apply(case_tac "x = 0") |
|
683 |
apply simp |
|
29667 | 684 |
apply (simp add: algebra_simps) |
16819 | 685 |
apply (subst real_of_int_div_aux) |
686 |
apply assumption |
|
687 |
apply simp |
|
688 |
apply (subst divide_le_eq) |
|
689 |
apply clarsimp |
|
690 |
apply (rule conjI) |
|
691 |
apply (rule impI) |
|
692 |
apply (rule order_less_imp_le) |
|
693 |
apply simp |
|
694 |
apply (rule impI) |
|
695 |
apply (rule order_less_imp_le) |
|
696 |
apply simp |
|
697 |
done |
|
698 |
||
699 |
lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x" |
|
27964 | 700 |
by (insert real_of_int_div2 [of n x], simp) |
701 |
||
702 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
703 |
subsection{*Embedding the Naturals into the Reals*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
704 |
|
14334 | 705 |
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
|
706 |
by (simp add: real_of_nat_def) |
14334 | 707 |
|
30082
43c5b7bfc791
make more proofs work whether or not One_nat_def is a simp rule
huffman
parents:
30042
diff
changeset
|
708 |
lemma real_of_nat_1 [simp]: "real (1::nat) = 1" |
43c5b7bfc791
make more proofs work whether or not One_nat_def is a simp rule
huffman
parents:
30042
diff
changeset
|
709 |
by (simp add: real_of_nat_def) |
43c5b7bfc791
make more proofs work whether or not One_nat_def is a simp rule
huffman
parents:
30042
diff
changeset
|
710 |
|
14334 | 711 |
lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)" |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
712 |
by (simp add: real_of_nat_def) |
14334 | 713 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
714 |
lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
715 |
by (simp add: real_of_nat_def) |
14334 | 716 |
|
717 |
(*Not for addsimps: often the LHS is used to represent a positive natural*) |
|
718 |
lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)" |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
719 |
by (simp add: real_of_nat_def) |
14334 | 720 |
|
721 |
lemma real_of_nat_less_iff [iff]: |
|
722 |
"(real (n::nat) < real m) = (n < m)" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
723 |
by (simp add: real_of_nat_def) |
14334 | 724 |
|
725 |
lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)" |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
726 |
by (simp add: real_of_nat_def) |
14334 | 727 |
|
728 |
lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)" |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
729 |
by (simp add: real_of_nat_def zero_le_imp_of_nat) |
14334 | 730 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
731 |
lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
732 |
by (simp add: real_of_nat_def del: of_nat_Suc) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
733 |
|
14334 | 734 |
lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n" |
23431
25ca91279a9b
change simp rules for of_nat to work like int did previously (reorient of_nat_Suc, remove of_nat_mult [simp]); preserve original variable names in legacy int theorems
huffman
parents:
23289
diff
changeset
|
735 |
by (simp add: real_of_nat_def of_nat_mult) |
14334 | 736 |
|
16819 | 737 |
lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) = |
738 |
(SUM x:A. real(f x))" |
|
739 |
apply (subst real_eq_of_nat)+ |
|
740 |
apply (rule of_nat_setsum) |
|
741 |
done |
|
742 |
||
743 |
lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) = |
|
744 |
(PROD x:A. real(f x))" |
|
745 |
apply (subst real_eq_of_nat)+ |
|
746 |
apply (rule of_nat_setprod) |
|
747 |
done |
|
748 |
||
749 |
lemma real_of_card: "real (card A) = setsum (%x.1) A" |
|
750 |
apply (subst card_eq_setsum) |
|
751 |
apply (subst real_of_nat_setsum) |
|
752 |
apply simp |
|
753 |
done |
|
754 |
||
14334 | 755 |
lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
756 |
by (simp add: real_of_nat_def) |
14334 | 757 |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
758 |
lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
759 |
by (simp add: real_of_nat_def) |
14334 | 760 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
761 |
lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n" |
23438
dd824e86fa8a
remove simp attribute from of_nat_diff, for backward compatibility with zdiff_int
huffman
parents:
23431
diff
changeset
|
762 |
by (simp add: add: real_of_nat_def of_nat_diff) |
14334 | 763 |
|
25162 | 764 |
lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)" |
25140 | 765 |
by (auto simp: real_of_nat_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
766 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
767 |
lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
768 |
by (simp add: add: real_of_nat_def) |
14334 | 769 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
770 |
lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
771 |
by (simp add: add: real_of_nat_def) |
14334 | 772 |
|
25140 | 773 |
lemma real_of_nat_ge_zero_cancel_iff [simp]: "(0 \<le> real (n::nat))" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
774 |
by (simp add: add: real_of_nat_def) |
14334 | 775 |
|
16819 | 776 |
lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)" |
777 |
apply (subgoal_tac "real n + 1 = real (Suc n)") |
|
778 |
apply simp |
|
779 |
apply (auto simp add: real_of_nat_Suc) |
|
780 |
done |
|
781 |
||
782 |
lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)" |
|
783 |
apply (subgoal_tac "real m + 1 = real (Suc m)") |
|
784 |
apply (simp add: less_Suc_eq_le) |
|
785 |
apply (simp add: real_of_nat_Suc) |
|
786 |
done |
|
787 |
||
788 |
lemma real_of_nat_div_aux: "0 < d ==> (real (x::nat)) / (real d) = |
|
789 |
real (x div d) + (real (x mod d)) / (real d)" |
|
790 |
proof - |
|
791 |
assume "0 < d" |
|
792 |
have "x = (x div d) * d + x mod d" |
|
793 |
by auto |
|
794 |
then have "real x = real (x div d) * real d + real(x mod d)" |
|
795 |
by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym]) |
|
796 |
then have "real x / real d = \<dots> / real d" |
|
797 |
by simp |
|
798 |
then show ?thesis |
|
29667 | 799 |
by (auto simp add: add_divide_distrib algebra_simps prems) |
16819 | 800 |
qed |
801 |
||
802 |
lemma real_of_nat_div: "0 < (d::nat) ==> d dvd n ==> |
|
803 |
real(n div d) = real n / real d" |
|
804 |
apply (frule real_of_nat_div_aux [of d n]) |
|
805 |
apply simp |
|
806 |
apply (subst dvd_eq_mod_eq_0 [THEN sym]) |
|
807 |
apply assumption |
|
808 |
done |
|
809 |
||
810 |
lemma real_of_nat_div2: |
|
811 |
"0 <= real (n::nat) / real (x) - real (n div x)" |
|
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
812 |
apply(case_tac "x = 0") |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
813 |
apply (simp) |
29667 | 814 |
apply (simp add: algebra_simps) |
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
815 |
apply (subst real_of_nat_div_aux) |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
816 |
apply simp |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
817 |
apply simp |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
818 |
apply (subst zero_le_divide_iff) |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
819 |
apply simp |
16819 | 820 |
done |
821 |
||
822 |
lemma real_of_nat_div3: |
|
823 |
"real (n::nat) / real (x) - real (n div x) <= 1" |
|
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
824 |
apply(case_tac "x = 0") |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
825 |
apply (simp) |
29667 | 826 |
apply (simp add: algebra_simps) |
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
827 |
apply (subst real_of_nat_div_aux) |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
828 |
apply simp |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
829 |
apply simp |
16819 | 830 |
done |
831 |
||
832 |
lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x" |
|
29667 | 833 |
by (insert real_of_nat_div2 [of n x], simp) |
16819 | 834 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
835 |
lemma real_of_int_real_of_nat: "real (int n) = real n" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
836 |
by (simp add: real_of_nat_def real_of_int_def int_eq_of_nat) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
837 |
|
14426 | 838 |
lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n" |
839 |
by (simp add: real_of_int_def real_of_nat_def) |
|
14334 | 840 |
|
16819 | 841 |
lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x" |
842 |
apply (subgoal_tac "real(int(nat x)) = real(nat x)") |
|
843 |
apply force |
|
844 |
apply (simp only: real_of_int_real_of_nat) |
|
845 |
done |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
846 |
|
28001 | 847 |
|
848 |
subsection{* Rationals *} |
|
849 |
||
28091
50f2d6ba024c
Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents:
28001
diff
changeset
|
850 |
lemma Rats_real_nat[simp]: "real(n::nat) \<in> \<rat>" |
50f2d6ba024c
Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents:
28001
diff
changeset
|
851 |
by (simp add: real_eq_of_nat) |
50f2d6ba024c
Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents:
28001
diff
changeset
|
852 |
|
50f2d6ba024c
Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents:
28001
diff
changeset
|
853 |
|
28001 | 854 |
lemma Rats_eq_int_div_int: |
28091
50f2d6ba024c
Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents:
28001
diff
changeset
|
855 |
"\<rat> = { real(i::int)/real(j::int) |i j. j \<noteq> 0}" (is "_ = ?S") |
28001 | 856 |
proof |
28091
50f2d6ba024c
Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents:
28001
diff
changeset
|
857 |
show "\<rat> \<subseteq> ?S" |
28001 | 858 |
proof |
28091
50f2d6ba024c
Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents:
28001
diff
changeset
|
859 |
fix x::real assume "x : \<rat>" |
28001 | 860 |
then obtain r where "x = of_rat r" unfolding Rats_def .. |
861 |
have "of_rat r : ?S" |
|
862 |
by (cases r)(auto simp add:of_rat_rat real_eq_of_int) |
|
863 |
thus "x : ?S" using `x = of_rat r` by simp |
|
864 |
qed |
|
865 |
next |
|
28091
50f2d6ba024c
Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents:
28001
diff
changeset
|
866 |
show "?S \<subseteq> \<rat>" |
28001 | 867 |
proof(auto simp:Rats_def) |
868 |
fix i j :: int assume "j \<noteq> 0" |
|
869 |
hence "real i / real j = of_rat(Fract i j)" |
|
870 |
by (simp add:of_rat_rat real_eq_of_int) |
|
871 |
thus "real i / real j \<in> range of_rat" by blast |
|
872 |
qed |
|
873 |
qed |
|
874 |
||
875 |
lemma Rats_eq_int_div_nat: |
|
28091
50f2d6ba024c
Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and
nipkow
parents:
28001
diff
changeset
|
876 |
"\<rat> = { real(i::int)/real(n::nat) |i n. n \<noteq> 0}" |
28001 | 877 |
proof(auto simp:Rats_eq_int_div_int) |
878 |
fix i j::int assume "j \<noteq> 0" |
|
879 |
show "EX (i'::int) (n::nat). real i/real j = real i'/real n \<and> 0<n" |
|
880 |
proof cases |
|
881 |
assume "j>0" |
|
882 |
hence "real i/real j = real i/real(nat j) \<and> 0<nat j" |
|
883 |
by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat) |
|
884 |
thus ?thesis by blast |
|
885 |
next |
|
886 |
assume "~ j>0" |
|
887 |
hence "real i/real j = real(-i)/real(nat(-j)) \<and> 0<nat(-j)" using `j\<noteq>0` |
|
888 |
by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat) |
|
889 |
thus ?thesis by blast |
|
890 |
qed |
|
891 |
next |
|
892 |
fix i::int and n::nat assume "0 < n" |
|
893 |
hence "real i/real n = real i/real(int n) \<and> int n \<noteq> 0" by simp |
|
894 |
thus "\<exists>(i'::int) j::int. real i/real n = real i'/real j \<and> j \<noteq> 0" by blast |
|
895 |
qed |
|
896 |
||
897 |
lemma Rats_abs_nat_div_natE: |
|
898 |
assumes "x \<in> \<rat>" |
|
899 |
obtains m n where "n \<noteq> 0" and "\<bar>x\<bar> = real m / real n" and "gcd m n = 1" |
|
900 |
proof - |
|
901 |
from `x \<in> \<rat>` obtain i::int and n::nat where "n \<noteq> 0" and "x = real i / real n" |
|
902 |
by(auto simp add: Rats_eq_int_div_nat) |
|
903 |
hence "\<bar>x\<bar> = real(nat(abs i)) / real n" by simp |
|
904 |
then obtain m :: nat where x_rat: "\<bar>x\<bar> = real m / real n" by blast |
|
905 |
let ?gcd = "gcd m n" |
|
906 |
from `n\<noteq>0` have gcd: "?gcd \<noteq> 0" by (simp add: gcd_zero) |
|
907 |
let ?k = "m div ?gcd" |
|
908 |
let ?l = "n div ?gcd" |
|
909 |
let ?gcd' = "gcd ?k ?l" |
|
910 |
have "?gcd dvd m" .. then have gcd_k: "?gcd * ?k = m" |
|
911 |
by (rule dvd_mult_div_cancel) |
|
912 |
have "?gcd dvd n" .. then have gcd_l: "?gcd * ?l = n" |
|
913 |
by (rule dvd_mult_div_cancel) |
|
914 |
from `n\<noteq>0` and gcd_l have "?l \<noteq> 0" by (auto iff del: neq0_conv) |
|
915 |
moreover |
|
916 |
have "\<bar>x\<bar> = real ?k / real ?l" |
|
917 |
proof - |
|
918 |
from gcd have "real ?k / real ?l = |
|
919 |
real (?gcd * ?k) / real (?gcd * ?l)" by simp |
|
920 |
also from gcd_k and gcd_l have "\<dots> = real m / real n" by simp |
|
921 |
also from x_rat have "\<dots> = \<bar>x\<bar>" .. |
|
922 |
finally show ?thesis .. |
|
923 |
qed |
|
924 |
moreover |
|
925 |
have "?gcd' = 1" |
|
926 |
proof - |
|
927 |
have "?gcd * ?gcd' = gcd (?gcd * ?k) (?gcd * ?l)" |
|
928 |
by (rule gcd_mult_distrib2) |
|
929 |
with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp |
|
930 |
with gcd show ?thesis by simp |
|
931 |
qed |
|
932 |
ultimately show ?thesis .. |
|
933 |
qed |
|
934 |
||
935 |
||
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
936 |
subsection{*Numerals and Arithmetic*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
937 |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
938 |
instantiation real :: number_ring |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
939 |
begin |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
940 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
941 |
definition |
28562 | 942 |
real_number_of_def [code del]: "number_of w = real_of_int w" |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
943 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
944 |
instance |
24198 | 945 |
by intro_classes (simp add: real_number_of_def) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
946 |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
947 |
end |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset
|
948 |
|
25965 | 949 |
lemma [code unfold, symmetric, code post]: |
24198 | 950 |
"number_of k = real_of_int (number_of k)" |
951 |
unfolding number_of_is_id real_number_of_def .. |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
952 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
953 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
954 |
text{*Collapse applications of @{term real} to @{term number_of}*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
955 |
lemma real_number_of [simp]: "real (number_of v :: int) = number_of v" |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
956 |
by (simp add: real_of_int_def of_int_number_of_eq) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
957 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
958 |
lemma real_of_nat_number_of [simp]: |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
959 |
"real (number_of v :: nat) = |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
960 |
(if neg (number_of v :: int) then 0 |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
961 |
else (number_of v :: real))" |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
962 |
by (simp add: real_of_int_real_of_nat [symmetric] int_nat_number_of) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
963 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
964 |
|
28952
15a4b2cf8c34
made repository layout more coherent with logical distribution structure; stripped some $Id$s
haftmann
parents:
28906
diff
changeset
|
965 |
use "Tools/real_arith.ML" |
24075 | 966 |
declaration {* K real_arith_setup *} |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
967 |
|
19023
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
16973
diff
changeset
|
968 |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
969 |
subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
970 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
971 |
text{*Needed in this non-standard form by Hyperreal/Transcendental*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
972 |
lemma real_0_le_divide_iff: |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
973 |
"((0::real) \<le> x/y) = ((x \<le> 0 | 0 \<le> y) & (0 \<le> x | y \<le> 0))" |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
974 |
by (simp add: real_divide_def zero_le_mult_iff, auto) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
975 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
976 |
lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)" |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
977 |
by arith |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
978 |
|
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15077
diff
changeset
|
979 |
lemma real_add_eq_0_iff: "(x+y = (0::real)) = (y = -x)" |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
980 |
by auto |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
981 |
|
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15077
diff
changeset
|
982 |
lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)" |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
983 |
by auto |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
984 |
|
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15077
diff
changeset
|
985 |
lemma real_0_less_add_iff: "((0::real) < x+y) = (-x < y)" |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
986 |
by auto |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
987 |
|
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15077
diff
changeset
|
988 |
lemma real_add_le_0_iff: "(x+y \<le> (0::real)) = (y \<le> -x)" |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
989 |
by auto |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
990 |
|
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15077
diff
changeset
|
991 |
lemma real_0_le_add_iff: "((0::real) \<le> x+y) = (-x \<le> y)" |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
992 |
by auto |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
993 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
994 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
995 |
(* |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
996 |
FIXME: we should have this, as for type int, but many proofs would break. |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
997 |
It replaces x+-y by x-y. |
15086 | 998 |
declare real_diff_def [symmetric, simp] |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
999 |
*) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1000 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1001 |
subsubsection{*Density of the Reals*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1002 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1003 |
lemma real_lbound_gt_zero: |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1004 |
"[| (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
|
1005 |
apply (rule_tac x = " (min d1 d2) /2" in exI) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1006 |
apply (simp add: min_def) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1007 |
done |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1008 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1009 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1010 |
text{*Similar results are proved in @{text Ring_and_Field}*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1011 |
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
|
1012 |
by auto |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1013 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1014 |
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
|
1015 |
by auto |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1016 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1017 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1018 |
subsection{*Absolute Value Function for the Reals*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1019 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1020 |
lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))" |
15003 | 1021 |
by (simp add: abs_if) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1022 |
|
23289 | 1023 |
(* FIXME: redundant, but used by Integration/RealRandVar.thy in AFP *) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1024 |
lemma abs_le_interval_iff: "(abs x \<le> r) = (-r\<le>x & x\<le>(r::real))" |
14738 | 1025 |
by (force simp add: OrderedGroup.abs_le_iff) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1026 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1027 |
lemma abs_add_one_gt_zero [simp]: "(0::real) < 1 + abs(x)" |
15003 | 1028 |
by (simp add: abs_if) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1029 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1030 |
lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)" |
22958 | 1031 |
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
|
1032 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1033 |
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
|
1034 |
by simp |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1035 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1036 |
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
|
1037 |
by simp |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
1038 |
|
26732 | 1039 |
instance real :: lordered_ring |
1040 |
proof |
|
1041 |
fix a::real |
|
1042 |
show "abs a = sup a (-a)" |
|
1043 |
by (auto simp add: real_abs_def sup_real_def) |
|
1044 |
qed |
|
1045 |
||
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1046 |
|
27544 | 1047 |
subsection {* Implementation of rational real numbers *} |
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1048 |
|
27544 | 1049 |
definition Ratreal :: "rat \<Rightarrow> real" where |
1050 |
[simp]: "Ratreal = of_rat" |
|
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1051 |
|
24623 | 1052 |
code_datatype Ratreal |
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1053 |
|
27544 | 1054 |
lemma Ratreal_number_collapse [code post]: |
1055 |
"Ratreal 0 = 0" |
|
1056 |
"Ratreal 1 = 1" |
|
1057 |
"Ratreal (number_of k) = number_of k" |
|
1058 |
by simp_all |
|
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1059 |
|
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1060 |
lemma zero_real_code [code, code unfold]: |
27544 | 1061 |
"0 = Ratreal 0" |
1062 |
by simp |
|
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1063 |
|
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1064 |
lemma one_real_code [code, code unfold]: |
27544 | 1065 |
"1 = Ratreal 1" |
1066 |
by simp |
|
1067 |
||
1068 |
lemma number_of_real_code [code unfold]: |
|
1069 |
"number_of k = Ratreal (number_of k)" |
|
1070 |
by simp |
|
1071 |
||
1072 |
lemma Ratreal_number_of_quotient [code post]: |
|
1073 |
"Ratreal (number_of r) / Ratreal (number_of s) = number_of r / number_of s" |
|
1074 |
by simp |
|
1075 |
||
1076 |
lemma Ratreal_number_of_quotient2 [code post]: |
|
1077 |
"Ratreal (number_of r / number_of s) = number_of r / number_of s" |
|
1078 |
unfolding Ratreal_number_of_quotient [symmetric] Ratreal_def of_rat_divide .. |
|
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1079 |
|
26513 | 1080 |
instantiation real :: eq |
1081 |
begin |
|
1082 |
||
27544 | 1083 |
definition "eq_class.eq (x\<Colon>real) y \<longleftrightarrow> x - y = 0" |
26513 | 1084 |
|
1085 |
instance by default (simp add: eq_real_def) |
|
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1086 |
|
27544 | 1087 |
lemma real_eq_code [code]: "eq_class.eq (Ratreal x) (Ratreal y) \<longleftrightarrow> eq_class.eq x y" |
1088 |
by (simp add: eq_real_def eq) |
|
26513 | 1089 |
|
28351 | 1090 |
lemma real_eq_refl [code nbe]: |
1091 |
"eq_class.eq (x::real) x \<longleftrightarrow> True" |
|
1092 |
by (rule HOL.eq_refl) |
|
1093 |
||
26513 | 1094 |
end |
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1095 |
|
27544 | 1096 |
lemma real_less_eq_code [code]: "Ratreal x \<le> Ratreal y \<longleftrightarrow> x \<le> y" |
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27544
diff
changeset
|
1097 |
by (simp add: of_rat_less_eq) |
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1098 |
|
27544 | 1099 |
lemma real_less_code [code]: "Ratreal x < Ratreal y \<longleftrightarrow> x < y" |
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27544
diff
changeset
|
1100 |
by (simp add: of_rat_less) |
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1101 |
|
27544 | 1102 |
lemma real_plus_code [code]: "Ratreal x + Ratreal y = Ratreal (x + y)" |
1103 |
by (simp add: of_rat_add) |
|
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1104 |
|
27544 | 1105 |
lemma real_times_code [code]: "Ratreal x * Ratreal y = Ratreal (x * y)" |
1106 |
by (simp add: of_rat_mult) |
|
1107 |
||
1108 |
lemma real_uminus_code [code]: "- Ratreal x = Ratreal (- x)" |
|
1109 |
by (simp add: of_rat_minus) |
|
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1110 |
|
27544 | 1111 |
lemma real_minus_code [code]: "Ratreal x - Ratreal y = Ratreal (x - y)" |
1112 |
by (simp add: of_rat_diff) |
|
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1113 |
|
27544 | 1114 |
lemma real_inverse_code [code]: "inverse (Ratreal x) = Ratreal (inverse x)" |
1115 |
by (simp add: of_rat_inverse) |
|
1116 |
||
1117 |
lemma real_divide_code [code]: "Ratreal x / Ratreal y = Ratreal (x / y)" |
|
1118 |
by (simp add: of_rat_divide) |
|
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1119 |
|
24623 | 1120 |
text {* Setup for SML code generator *} |
23031
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
1121 |
|
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
1122 |
types_code |
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1123 |
real ("(int */ int)") |
23031
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
1124 |
attach (term_of) {* |
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1125 |
fun term_of_real (p, q) = |
24623 | 1126 |
let |
1127 |
val rT = HOLogic.realT |
|
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1128 |
in |
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1129 |
if q = 1 orelse p = 0 then HOLogic.mk_number rT p |
24623 | 1130 |
else @{term "op / \<Colon> real \<Rightarrow> real \<Rightarrow> real"} $ |
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1131 |
HOLogic.mk_number rT p $ HOLogic.mk_number rT q |
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1132 |
end; |
23031
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
1133 |
*} |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
1134 |
attach (test) {* |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
1135 |
fun gen_real i = |
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1136 |
let |
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1137 |
val p = random_range 0 i; |
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1138 |
val q = random_range 1 (i + 1); |
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1139 |
val g = Integer.gcd p q; |
24630
351a308ab58d
simplified type int (eliminated IntInf.int, integer);
wenzelm
parents:
24623
diff
changeset
|
1140 |
val p' = p div g; |
351a308ab58d
simplified type int (eliminated IntInf.int, integer);
wenzelm
parents:
24623
diff
changeset
|
1141 |
val q' = q div g; |
25885 | 1142 |
val r = (if one_of [true, false] then p' else ~ p', |
1143 |
if p' = 0 then 0 else q') |
|
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1144 |
in |
25885 | 1145 |
(r, fn () => term_of_real r) |
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1146 |
end; |
23031
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
1147 |
*} |
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
1148 |
|
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset
|
1149 |
consts_code |
24623 | 1150 |
Ratreal ("(_)") |
24534
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1151 |
|
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1152 |
consts_code |
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1153 |
"of_int :: int \<Rightarrow> real" ("\<module>real'_of'_int") |
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1154 |
attach {* |
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1155 |
fun real_of_int 0 = (0, 0) |
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1156 |
| real_of_int i = (i, 1); |
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1157 |
*} |
09b9a47904b7
New code generator setup (taken from Library/Executable_Real.thy,
berghofe
parents:
24506
diff
changeset
|
1158 |
|
5588 | 1159 |
end |