equal
deleted
inserted
replaced
26 |
26 |
27 subsection {* Gauges *} |
27 subsection {* Gauges *} |
28 |
28 |
29 definition |
29 definition |
30 gauge :: "[real set, real => real] => bool" where |
30 gauge :: "[real set, real => real] => bool" where |
31 [code del]: "gauge E g = (\<forall>x\<in>E. 0 < g(x))" |
31 "gauge E g = (\<forall>x\<in>E. 0 < g(x))" |
32 |
32 |
33 |
33 |
34 subsection {* Gauge-fine divisions *} |
34 subsection {* Gauge-fine divisions *} |
35 |
35 |
36 inductive |
36 inductive |
257 |
257 |
258 subsection {* Gauge integrability (definite) *} |
258 subsection {* Gauge integrability (definite) *} |
259 |
259 |
260 definition |
260 definition |
261 Integral :: "[(real*real),real=>real,real] => bool" where |
261 Integral :: "[(real*real),real=>real,real] => bool" where |
262 [code del]: "Integral = (%(a,b) f k. \<forall>e > 0. |
262 "Integral = (%(a,b) f k. \<forall>e > 0. |
263 (\<exists>\<delta>. gauge {a .. b} \<delta> & |
263 (\<exists>\<delta>. gauge {a .. b} \<delta> & |
264 (\<forall>D. fine \<delta> (a,b) D --> |
264 (\<forall>D. fine \<delta> (a,b) D --> |
265 \<bar>rsum D f - k\<bar> < e)))" |
265 \<bar>rsum D f - k\<bar> < e)))" |
266 |
266 |
267 lemma Integral_eq: |
267 lemma Integral_eq: |