Renamed Multivariate-Analysis/Integration to Multivariate-Analysis/Integration_MV to avoid name clash with Integration.
authorhoelzl
Wed Feb 17 18:56:34 2010 +0100 (2010-02-17)
changeset 351739b24bfca8044
parent 35172 579dd5570f96
child 35176 3b9762ad372d
Renamed Multivariate-Analysis/Integration to Multivariate-Analysis/Integration_MV to avoid name clash with Integration.
src/HOL/Multivariate_Analysis/Integration.cert
src/HOL/Multivariate_Analysis/Integration.thy
src/HOL/Multivariate_Analysis/Integration_MV.cert
src/HOL/Multivariate_Analysis/Integration_MV.thy
src/HOL/Multivariate_Analysis/Multivariate_Analysis.thy
     1.1 --- a/src/HOL/Multivariate_Analysis/Integration.cert	Wed Feb 17 18:33:45 2010 +0100
     1.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.3 @@ -1,3270 +0,0 @@
     1.4 -tB2Atlor9W4pSnrAz5nHpw 907 0
     1.5 -#2 := false
     1.6 -#299 := 0::real
     1.7 -decl uf_1 :: (-> T3 T2 real)
     1.8 -decl uf_10 :: (-> T4 T2)
     1.9 -decl uf_7 :: T4
    1.10 -#15 := uf_7
    1.11 -#22 := (uf_10 uf_7)
    1.12 -decl uf_2 :: (-> T1 T3)
    1.13 -decl uf_4 :: T1
    1.14 -#11 := uf_4
    1.15 -#91 := (uf_2 uf_4)
    1.16 -#902 := (uf_1 #91 #22)
    1.17 -#297 := -1::real
    1.18 -#1084 := (* -1::real #902)
    1.19 -decl uf_16 :: T1
    1.20 -#50 := uf_16
    1.21 -#78 := (uf_2 uf_16)
    1.22 -#799 := (uf_1 #78 #22)
    1.23 -#1267 := (+ #799 #1084)
    1.24 -#1272 := (>= #1267 0::real)
    1.25 -#1266 := (= #799 #902)
    1.26 -decl uf_9 :: T3
    1.27 -#21 := uf_9
    1.28 -#23 := (uf_1 uf_9 #22)
    1.29 -#905 := (= #23 #902)
    1.30 -decl uf_11 :: T3
    1.31 -#24 := uf_11
    1.32 -#850 := (uf_1 uf_11 #22)
    1.33 -#904 := (= #850 #902)
    1.34 -decl uf_6 :: (-> T2 T4)
    1.35 -#74 := (uf_6 #22)
    1.36 -#281 := (= uf_7 #74)
    1.37 -#922 := (ite #281 #905 #904)
    1.38 -decl uf_8 :: T3
    1.39 -#18 := uf_8
    1.40 -#848 := (uf_1 uf_8 #22)
    1.41 -#903 := (= #848 #902)
    1.42 -#60 := 0::int
    1.43 -decl uf_5 :: (-> T4 int)
    1.44 -#803 := (uf_5 #74)
    1.45 -#117 := -1::int
    1.46 -#813 := (* -1::int #803)
    1.47 -#16 := (uf_5 uf_7)
    1.48 -#916 := (+ #16 #813)
    1.49 -#917 := (<= #916 0::int)
    1.50 -#925 := (ite #917 #922 #903)
    1.51 -#6 := (:var 0 T2)
    1.52 -#19 := (uf_1 uf_8 #6)
    1.53 -#544 := (pattern #19)
    1.54 -#25 := (uf_1 uf_11 #6)
    1.55 -#543 := (pattern #25)
    1.56 -#92 := (uf_1 #91 #6)
    1.57 -#542 := (pattern #92)
    1.58 -#13 := (uf_6 #6)
    1.59 -#541 := (pattern #13)
    1.60 -#447 := (= #19 #92)
    1.61 -#445 := (= #25 #92)
    1.62 -#444 := (= #23 #92)
    1.63 -#20 := (= #13 uf_7)
    1.64 -#446 := (ite #20 #444 #445)
    1.65 -#120 := (* -1::int #16)
    1.66 -#14 := (uf_5 #13)
    1.67 -#121 := (+ #14 #120)
    1.68 -#119 := (>= #121 0::int)
    1.69 -#448 := (ite #119 #446 #447)
    1.70 -#545 := (forall (vars (?x3 T2)) (:pat #541 #542 #543 #544) #448)
    1.71 -#451 := (forall (vars (?x3 T2)) #448)
    1.72 -#548 := (iff #451 #545)
    1.73 -#546 := (iff #448 #448)
    1.74 -#547 := [refl]: #546
    1.75 -#549 := [quant-intro #547]: #548
    1.76 -#26 := (ite #20 #23 #25)
    1.77 -#127 := (ite #119 #26 #19)
    1.78 -#368 := (= #92 #127)
    1.79 -#369 := (forall (vars (?x3 T2)) #368)
    1.80 -#452 := (iff #369 #451)
    1.81 -#449 := (iff #368 #448)
    1.82 -#450 := [rewrite]: #449
    1.83 -#453 := [quant-intro #450]: #452
    1.84 -#392 := (~ #369 #369)
    1.85 -#390 := (~ #368 #368)
    1.86 -#391 := [refl]: #390
    1.87 -#366 := [nnf-pos #391]: #392
    1.88 -decl uf_3 :: (-> T1 T2 real)
    1.89 -#12 := (uf_3 uf_4 #6)
    1.90 -#132 := (= #12 #127)
    1.91 -#135 := (forall (vars (?x3 T2)) #132)
    1.92 -#370 := (iff #135 #369)
    1.93 -#4 := (:var 1 T1)
    1.94 -#8 := (uf_3 #4 #6)
    1.95 -#5 := (uf_2 #4)
    1.96 -#7 := (uf_1 #5 #6)
    1.97 -#9 := (= #7 #8)
    1.98 -#10 := (forall (vars (?x1 T1) (?x2 T2)) #9)
    1.99 -#113 := [asserted]: #10
   1.100 -#371 := [rewrite* #113]: #370
   1.101 -#17 := (< #14 #16)
   1.102 -#27 := (ite #17 #19 #26)
   1.103 -#28 := (= #12 #27)
   1.104 -#29 := (forall (vars (?x3 T2)) #28)
   1.105 -#136 := (iff #29 #135)
   1.106 -#133 := (iff #28 #132)
   1.107 -#130 := (= #27 #127)
   1.108 -#118 := (not #119)
   1.109 -#124 := (ite #118 #19 #26)
   1.110 -#128 := (= #124 #127)
   1.111 -#129 := [rewrite]: #128
   1.112 -#125 := (= #27 #124)
   1.113 -#122 := (iff #17 #118)
   1.114 -#123 := [rewrite]: #122
   1.115 -#126 := [monotonicity #123]: #125
   1.116 -#131 := [trans #126 #129]: #130
   1.117 -#134 := [monotonicity #131]: #133
   1.118 -#137 := [quant-intro #134]: #136
   1.119 -#114 := [asserted]: #29
   1.120 -#138 := [mp #114 #137]: #135
   1.121 -#372 := [mp #138 #371]: #369
   1.122 -#367 := [mp~ #372 #366]: #369
   1.123 -#454 := [mp #367 #453]: #451
   1.124 -#550 := [mp #454 #549]: #545
   1.125 -#738 := (not #545)
   1.126 -#928 := (or #738 #925)
   1.127 -#75 := (= #74 uf_7)
   1.128 -#906 := (ite #75 #905 #904)
   1.129 -#907 := (+ #803 #120)
   1.130 -#908 := (>= #907 0::int)
   1.131 -#909 := (ite #908 #906 #903)
   1.132 -#929 := (or #738 #909)
   1.133 -#931 := (iff #929 #928)
   1.134 -#933 := (iff #928 #928)
   1.135 -#934 := [rewrite]: #933
   1.136 -#926 := (iff #909 #925)
   1.137 -#923 := (iff #906 #922)
   1.138 -#283 := (iff #75 #281)
   1.139 -#284 := [rewrite]: #283
   1.140 -#924 := [monotonicity #284]: #923
   1.141 -#920 := (iff #908 #917)
   1.142 -#910 := (+ #120 #803)
   1.143 -#913 := (>= #910 0::int)
   1.144 -#918 := (iff #913 #917)
   1.145 -#919 := [rewrite]: #918
   1.146 -#914 := (iff #908 #913)
   1.147 -#911 := (= #907 #910)
   1.148 -#912 := [rewrite]: #911
   1.149 -#915 := [monotonicity #912]: #914
   1.150 -#921 := [trans #915 #919]: #920
   1.151 -#927 := [monotonicity #921 #924]: #926
   1.152 -#932 := [monotonicity #927]: #931
   1.153 -#935 := [trans #932 #934]: #931
   1.154 -#930 := [quant-inst]: #929
   1.155 -#936 := [mp #930 #935]: #928
   1.156 -#1300 := [unit-resolution #936 #550]: #925
   1.157 -#989 := (= #16 #803)
   1.158 -#1277 := (= #803 #16)
   1.159 -#280 := [asserted]: #75
   1.160 -#287 := [mp #280 #284]: #281
   1.161 -#1276 := [symm #287]: #75
   1.162 -#1278 := [monotonicity #1276]: #1277
   1.163 -#1301 := [symm #1278]: #989
   1.164 -#1302 := (not #989)
   1.165 -#1303 := (or #1302 #917)
   1.166 -#1304 := [th-lemma]: #1303
   1.167 -#1305 := [unit-resolution #1304 #1301]: #917
   1.168 -#950 := (not #917)
   1.169 -#949 := (not #925)
   1.170 -#951 := (or #949 #950 #922)
   1.171 -#952 := [def-axiom]: #951
   1.172 -#1306 := [unit-resolution #952 #1305 #1300]: #922
   1.173 -#937 := (not #922)
   1.174 -#1307 := (or #937 #905)
   1.175 -#938 := (not #281)
   1.176 -#939 := (or #937 #938 #905)
   1.177 -#940 := [def-axiom]: #939
   1.178 -#1308 := [unit-resolution #940 #287]: #1307
   1.179 -#1309 := [unit-resolution #1308 #1306]: #905
   1.180 -#1356 := (= #799 #23)
   1.181 -#800 := (= #23 #799)
   1.182 -decl uf_15 :: T4
   1.183 -#40 := uf_15
   1.184 -#41 := (uf_5 uf_15)
   1.185 -#814 := (+ #41 #813)
   1.186 -#815 := (<= #814 0::int)
   1.187 -#836 := (not #815)
   1.188 -#158 := (* -1::int #41)
   1.189 -#1270 := (+ #16 #158)
   1.190 -#1265 := (>= #1270 0::int)
   1.191 -#1339 := (not #1265)
   1.192 -#1269 := (= #16 #41)
   1.193 -#1298 := (not #1269)
   1.194 -#286 := (= uf_7 uf_15)
   1.195 -#44 := (uf_10 uf_15)
   1.196 -#72 := (uf_6 #44)
   1.197 -#73 := (= #72 uf_15)
   1.198 -#277 := (= uf_15 #72)
   1.199 -#278 := (iff #73 #277)
   1.200 -#279 := [rewrite]: #278
   1.201 -#276 := [asserted]: #73
   1.202 -#282 := [mp #276 #279]: #277
   1.203 -#1274 := [symm #282]: #73
   1.204 -#729 := (= uf_7 #72)
   1.205 -decl uf_17 :: (-> int T4)
   1.206 -#611 := (uf_5 #72)
   1.207 -#991 := (uf_17 #611)
   1.208 -#1289 := (= #991 #72)
   1.209 -#992 := (= #72 #991)
   1.210 -#55 := (:var 0 T4)
   1.211 -#56 := (uf_5 #55)
   1.212 -#574 := (pattern #56)
   1.213 -#57 := (uf_17 #56)
   1.214 -#177 := (= #55 #57)
   1.215 -#575 := (forall (vars (?x7 T4)) (:pat #574) #177)
   1.216 -#195 := (forall (vars (?x7 T4)) #177)
   1.217 -#578 := (iff #195 #575)
   1.218 -#576 := (iff #177 #177)
   1.219 -#577 := [refl]: #576
   1.220 -#579 := [quant-intro #577]: #578
   1.221 -#405 := (~ #195 #195)
   1.222 -#403 := (~ #177 #177)
   1.223 -#404 := [refl]: #403
   1.224 -#406 := [nnf-pos #404]: #405
   1.225 -#58 := (= #57 #55)
   1.226 -#59 := (forall (vars (?x7 T4)) #58)
   1.227 -#196 := (iff #59 #195)
   1.228 -#193 := (iff #58 #177)
   1.229 -#194 := [rewrite]: #193
   1.230 -#197 := [quant-intro #194]: #196
   1.231 -#155 := [asserted]: #59
   1.232 -#200 := [mp #155 #197]: #195
   1.233 -#407 := [mp~ #200 #406]: #195
   1.234 -#580 := [mp #407 #579]: #575
   1.235 -#995 := (not #575)
   1.236 -#996 := (or #995 #992)
   1.237 -#997 := [quant-inst]: #996
   1.238 -#1273 := [unit-resolution #997 #580]: #992
   1.239 -#1290 := [symm #1273]: #1289
   1.240 -#1293 := (= uf_7 #991)
   1.241 -#993 := (uf_17 #803)
   1.242 -#1287 := (= #993 #991)
   1.243 -#1284 := (= #803 #611)
   1.244 -#987 := (= #41 #611)
   1.245 -#1279 := (= #611 #41)
   1.246 -#1280 := [monotonicity #1274]: #1279
   1.247 -#1281 := [symm #1280]: #987
   1.248 -#1282 := (= #803 #41)
   1.249 -#1275 := [hypothesis]: #1269
   1.250 -#1283 := [trans #1278 #1275]: #1282
   1.251 -#1285 := [trans #1283 #1281]: #1284
   1.252 -#1288 := [monotonicity #1285]: #1287
   1.253 -#1291 := (= uf_7 #993)
   1.254 -#994 := (= #74 #993)
   1.255 -#1000 := (or #995 #994)
   1.256 -#1001 := [quant-inst]: #1000
   1.257 -#1286 := [unit-resolution #1001 #580]: #994
   1.258 -#1292 := [trans #287 #1286]: #1291
   1.259 -#1294 := [trans #1292 #1288]: #1293
   1.260 -#1295 := [trans #1294 #1290]: #729
   1.261 -#1296 := [trans #1295 #1274]: #286
   1.262 -#290 := (not #286)
   1.263 -#76 := (= uf_15 uf_7)
   1.264 -#77 := (not #76)
   1.265 -#291 := (iff #77 #290)
   1.266 -#288 := (iff #76 #286)
   1.267 -#289 := [rewrite]: #288
   1.268 -#292 := [monotonicity #289]: #291
   1.269 -#285 := [asserted]: #77
   1.270 -#295 := [mp #285 #292]: #290
   1.271 -#1297 := [unit-resolution #295 #1296]: false
   1.272 -#1299 := [lemma #1297]: #1298
   1.273 -#1342 := (or #1269 #1339)
   1.274 -#1271 := (<= #1270 0::int)
   1.275 -#621 := (* -1::int #611)
   1.276 -#723 := (+ #16 #621)
   1.277 -#724 := (<= #723 0::int)
   1.278 -decl uf_12 :: T1
   1.279 -#30 := uf_12
   1.280 -#88 := (uf_2 uf_12)
   1.281 -#771 := (uf_1 #88 #44)
   1.282 -#45 := (uf_1 uf_9 #44)
   1.283 -#772 := (= #45 #771)
   1.284 -#796 := (not #772)
   1.285 -decl uf_14 :: T1
   1.286 -#38 := uf_14
   1.287 -#83 := (uf_2 uf_14)
   1.288 -#656 := (uf_1 #83 #44)
   1.289 -#1239 := (= #656 #771)
   1.290 -#1252 := (not #1239)
   1.291 -#1324 := (iff #1252 #796)
   1.292 -#1322 := (iff #1239 #772)
   1.293 -#1320 := (= #656 #45)
   1.294 -#661 := (= #45 #656)
   1.295 -#659 := (uf_1 uf_11 #44)
   1.296 -#664 := (= #656 #659)
   1.297 -#667 := (ite #277 #661 #664)
   1.298 -#657 := (uf_1 uf_8 #44)
   1.299 -#670 := (= #656 #657)
   1.300 -#622 := (+ #41 #621)
   1.301 -#623 := (<= #622 0::int)
   1.302 -#673 := (ite #623 #667 #670)
   1.303 -#84 := (uf_1 #83 #6)
   1.304 -#560 := (pattern #84)
   1.305 -#467 := (= #19 #84)
   1.306 -#465 := (= #25 #84)
   1.307 -#464 := (= #45 #84)
   1.308 -#43 := (= #13 uf_15)
   1.309 -#466 := (ite #43 #464 #465)
   1.310 -#159 := (+ #14 #158)
   1.311 -#157 := (>= #159 0::int)
   1.312 -#468 := (ite #157 #466 #467)
   1.313 -#561 := (forall (vars (?x5 T2)) (:pat #541 #560 #543 #544) #468)
   1.314 -#471 := (forall (vars (?x5 T2)) #468)
   1.315 -#564 := (iff #471 #561)
   1.316 -#562 := (iff #468 #468)
   1.317 -#563 := [refl]: #562
   1.318 -#565 := [quant-intro #563]: #564
   1.319 -#46 := (ite #43 #45 #25)
   1.320 -#165 := (ite #157 #46 #19)
   1.321 -#378 := (= #84 #165)
   1.322 -#379 := (forall (vars (?x5 T2)) #378)
   1.323 -#472 := (iff #379 #471)
   1.324 -#469 := (iff #378 #468)
   1.325 -#470 := [rewrite]: #469
   1.326 -#473 := [quant-intro #470]: #472
   1.327 -#359 := (~ #379 #379)
   1.328 -#361 := (~ #378 #378)
   1.329 -#358 := [refl]: #361
   1.330 -#356 := [nnf-pos #358]: #359
   1.331 -#39 := (uf_3 uf_14 #6)
   1.332 -#170 := (= #39 #165)
   1.333 -#173 := (forall (vars (?x5 T2)) #170)
   1.334 -#380 := (iff #173 #379)
   1.335 -#381 := [rewrite* #113]: #380
   1.336 -#42 := (< #14 #41)
   1.337 -#47 := (ite #42 #19 #46)
   1.338 -#48 := (= #39 #47)
   1.339 -#49 := (forall (vars (?x5 T2)) #48)
   1.340 -#174 := (iff #49 #173)
   1.341 -#171 := (iff #48 #170)
   1.342 -#168 := (= #47 #165)
   1.343 -#156 := (not #157)
   1.344 -#162 := (ite #156 #19 #46)
   1.345 -#166 := (= #162 #165)
   1.346 -#167 := [rewrite]: #166
   1.347 -#163 := (= #47 #162)
   1.348 -#160 := (iff #42 #156)
   1.349 -#161 := [rewrite]: #160
   1.350 -#164 := [monotonicity #161]: #163
   1.351 -#169 := [trans #164 #167]: #168
   1.352 -#172 := [monotonicity #169]: #171
   1.353 -#175 := [quant-intro #172]: #174
   1.354 -#116 := [asserted]: #49
   1.355 -#176 := [mp #116 #175]: #173
   1.356 -#382 := [mp #176 #381]: #379
   1.357 -#357 := [mp~ #382 #356]: #379
   1.358 -#474 := [mp #357 #473]: #471
   1.359 -#566 := [mp #474 #565]: #561
   1.360 -#676 := (not #561)
   1.361 -#677 := (or #676 #673)
   1.362 -#658 := (= #657 #656)
   1.363 -#660 := (= #659 #656)
   1.364 -#662 := (ite #73 #661 #660)
   1.365 -#612 := (+ #611 #158)
   1.366 -#613 := (>= #612 0::int)
   1.367 -#663 := (ite #613 #662 #658)
   1.368 -#678 := (or #676 #663)
   1.369 -#680 := (iff #678 #677)
   1.370 -#682 := (iff #677 #677)
   1.371 -#683 := [rewrite]: #682
   1.372 -#674 := (iff #663 #673)
   1.373 -#671 := (iff #658 #670)
   1.374 -#672 := [rewrite]: #671
   1.375 -#668 := (iff #662 #667)
   1.376 -#665 := (iff #660 #664)
   1.377 -#666 := [rewrite]: #665
   1.378 -#669 := [monotonicity #279 #666]: #668
   1.379 -#626 := (iff #613 #623)
   1.380 -#615 := (+ #158 #611)
   1.381 -#618 := (>= #615 0::int)
   1.382 -#624 := (iff #618 #623)
   1.383 -#625 := [rewrite]: #624
   1.384 -#619 := (iff #613 #618)
   1.385 -#616 := (= #612 #615)
   1.386 -#617 := [rewrite]: #616
   1.387 -#620 := [monotonicity #617]: #619
   1.388 -#627 := [trans #620 #625]: #626
   1.389 -#675 := [monotonicity #627 #669 #672]: #674
   1.390 -#681 := [monotonicity #675]: #680
   1.391 -#684 := [trans #681 #683]: #680
   1.392 -#679 := [quant-inst]: #678
   1.393 -#685 := [mp #679 #684]: #677
   1.394 -#1311 := [unit-resolution #685 #566]: #673
   1.395 -#1312 := (not #987)
   1.396 -#1313 := (or #1312 #623)
   1.397 -#1314 := [th-lemma]: #1313
   1.398 -#1315 := [unit-resolution #1314 #1281]: #623
   1.399 -#645 := (not #623)
   1.400 -#698 := (not #673)
   1.401 -#699 := (or #698 #645 #667)
   1.402 -#700 := [def-axiom]: #699
   1.403 -#1316 := [unit-resolution #700 #1315 #1311]: #667
   1.404 -#686 := (not #667)
   1.405 -#1317 := (or #686 #661)
   1.406 -#687 := (not #277)
   1.407 -#688 := (or #686 #687 #661)
   1.408 -#689 := [def-axiom]: #688
   1.409 -#1318 := [unit-resolution #689 #282]: #1317
   1.410 -#1319 := [unit-resolution #1318 #1316]: #661
   1.411 -#1321 := [symm #1319]: #1320
   1.412 -#1323 := [monotonicity #1321]: #1322
   1.413 -#1325 := [monotonicity #1323]: #1324
   1.414 -#1145 := (* -1::real #771)
   1.415 -#1240 := (+ #656 #1145)
   1.416 -#1241 := (<= #1240 0::real)
   1.417 -#1249 := (not #1241)
   1.418 -#1243 := [hypothesis]: #1241
   1.419 -decl uf_18 :: T3
   1.420 -#80 := uf_18
   1.421 -#1040 := (uf_1 uf_18 #44)
   1.422 -#1043 := (* -1::real #1040)
   1.423 -#1156 := (+ #771 #1043)
   1.424 -#1157 := (>= #1156 0::real)
   1.425 -#1189 := (not #1157)
   1.426 -#708 := (uf_1 #91 #44)
   1.427 -#1168 := (+ #708 #1043)
   1.428 -#1169 := (<= #1168 0::real)
   1.429 -#1174 := (or #1157 #1169)
   1.430 -#1177 := (not #1174)
   1.431 -#89 := (uf_1 #88 #6)
   1.432 -#552 := (pattern #89)
   1.433 -#81 := (uf_1 uf_18 #6)
   1.434 -#594 := (pattern #81)
   1.435 -#324 := (* -1::real #92)
   1.436 -#325 := (+ #81 #324)
   1.437 -#323 := (>= #325 0::real)
   1.438 -#317 := (* -1::real #89)
   1.439 -#318 := (+ #81 #317)
   1.440 -#319 := (<= #318 0::real)
   1.441 -#436 := (or #319 #323)
   1.442 -#437 := (not #436)
   1.443 -#601 := (forall (vars (?x11 T2)) (:pat #594 #552 #542) #437)
   1.444 -#440 := (forall (vars (?x11 T2)) #437)
   1.445 -#604 := (iff #440 #601)
   1.446 -#602 := (iff #437 #437)
   1.447 -#603 := [refl]: #602
   1.448 -#605 := [quant-intro #603]: #604
   1.449 -#326 := (not #323)
   1.450 -#320 := (not #319)
   1.451 -#329 := (and #320 #326)
   1.452 -#332 := (forall (vars (?x11 T2)) #329)
   1.453 -#441 := (iff #332 #440)
   1.454 -#438 := (iff #329 #437)
   1.455 -#439 := [rewrite]: #438
   1.456 -#442 := [quant-intro #439]: #441
   1.457 -#425 := (~ #332 #332)
   1.458 -#423 := (~ #329 #329)
   1.459 -#424 := [refl]: #423
   1.460 -#426 := [nnf-pos #424]: #425
   1.461 -#306 := (* -1::real #84)
   1.462 -#307 := (+ #81 #306)
   1.463 -#305 := (>= #307 0::real)
   1.464 -#308 := (not #305)
   1.465 -#301 := (* -1::real #81)
   1.466 -#79 := (uf_1 #78 #6)
   1.467 -#302 := (+ #79 #301)
   1.468 -#300 := (>= #302 0::real)
   1.469 -#298 := (not #300)
   1.470 -#311 := (and #298 #308)
   1.471 -#314 := (forall (vars (?x10 T2)) #311)
   1.472 -#335 := (and #314 #332)
   1.473 -#93 := (< #81 #92)
   1.474 -#90 := (< #89 #81)
   1.475 -#94 := (and #90 #93)
   1.476 -#95 := (forall (vars (?x11 T2)) #94)
   1.477 -#85 := (< #81 #84)
   1.478 -#82 := (< #79 #81)
   1.479 -#86 := (and #82 #85)
   1.480 -#87 := (forall (vars (?x10 T2)) #86)
   1.481 -#96 := (and #87 #95)
   1.482 -#336 := (iff #96 #335)
   1.483 -#333 := (iff #95 #332)
   1.484 -#330 := (iff #94 #329)
   1.485 -#327 := (iff #93 #326)
   1.486 -#328 := [rewrite]: #327
   1.487 -#321 := (iff #90 #320)
   1.488 -#322 := [rewrite]: #321
   1.489 -#331 := [monotonicity #322 #328]: #330
   1.490 -#334 := [quant-intro #331]: #333
   1.491 -#315 := (iff #87 #314)
   1.492 -#312 := (iff #86 #311)
   1.493 -#309 := (iff #85 #308)
   1.494 -#310 := [rewrite]: #309
   1.495 -#303 := (iff #82 #298)
   1.496 -#304 := [rewrite]: #303
   1.497 -#313 := [monotonicity #304 #310]: #312
   1.498 -#316 := [quant-intro #313]: #315
   1.499 -#337 := [monotonicity #316 #334]: #336
   1.500 -#293 := [asserted]: #96
   1.501 -#338 := [mp #293 #337]: #335
   1.502 -#340 := [and-elim #338]: #332
   1.503 -#427 := [mp~ #340 #426]: #332
   1.504 -#443 := [mp #427 #442]: #440
   1.505 -#606 := [mp #443 #605]: #601
   1.506 -#1124 := (not #601)
   1.507 -#1180 := (or #1124 #1177)
   1.508 -#1142 := (* -1::real #708)
   1.509 -#1143 := (+ #1040 #1142)
   1.510 -#1144 := (>= #1143 0::real)
   1.511 -#1146 := (+ #1040 #1145)
   1.512 -#1147 := (<= #1146 0::real)
   1.513 -#1148 := (or #1147 #1144)
   1.514 -#1149 := (not #1148)
   1.515 -#1181 := (or #1124 #1149)
   1.516 -#1183 := (iff #1181 #1180)
   1.517 -#1185 := (iff #1180 #1180)
   1.518 -#1186 := [rewrite]: #1185
   1.519 -#1178 := (iff #1149 #1177)
   1.520 -#1175 := (iff #1148 #1174)
   1.521 -#1172 := (iff #1144 #1169)
   1.522 -#1162 := (+ #1142 #1040)
   1.523 -#1165 := (>= #1162 0::real)
   1.524 -#1170 := (iff #1165 #1169)
   1.525 -#1171 := [rewrite]: #1170
   1.526 -#1166 := (iff #1144 #1165)
   1.527 -#1163 := (= #1143 #1162)
   1.528 -#1164 := [rewrite]: #1163
   1.529 -#1167 := [monotonicity #1164]: #1166
   1.530 -#1173 := [trans #1167 #1171]: #1172
   1.531 -#1160 := (iff #1147 #1157)
   1.532 -#1150 := (+ #1145 #1040)
   1.533 -#1153 := (<= #1150 0::real)
   1.534 -#1158 := (iff #1153 #1157)
   1.535 -#1159 := [rewrite]: #1158
   1.536 -#1154 := (iff #1147 #1153)
   1.537 -#1151 := (= #1146 #1150)
   1.538 -#1152 := [rewrite]: #1151
   1.539 -#1155 := [monotonicity #1152]: #1154
   1.540 -#1161 := [trans #1155 #1159]: #1160
   1.541 -#1176 := [monotonicity #1161 #1173]: #1175
   1.542 -#1179 := [monotonicity #1176]: #1178
   1.543 -#1184 := [monotonicity #1179]: #1183
   1.544 -#1187 := [trans #1184 #1186]: #1183
   1.545 -#1182 := [quant-inst]: #1181
   1.546 -#1188 := [mp #1182 #1187]: #1180
   1.547 -#1244 := [unit-resolution #1188 #606]: #1177
   1.548 -#1190 := (or #1174 #1189)
   1.549 -#1191 := [def-axiom]: #1190
   1.550 -#1245 := [unit-resolution #1191 #1244]: #1189
   1.551 -#1054 := (+ #656 #1043)
   1.552 -#1055 := (<= #1054 0::real)
   1.553 -#1079 := (not #1055)
   1.554 -#607 := (uf_1 #78 #44)
   1.555 -#1044 := (+ #607 #1043)
   1.556 -#1045 := (>= #1044 0::real)
   1.557 -#1060 := (or #1045 #1055)
   1.558 -#1063 := (not #1060)
   1.559 -#567 := (pattern #79)
   1.560 -#428 := (or #300 #305)
   1.561 -#429 := (not #428)
   1.562 -#595 := (forall (vars (?x10 T2)) (:pat #567 #594 #560) #429)
   1.563 -#432 := (forall (vars (?x10 T2)) #429)
   1.564 -#598 := (iff #432 #595)
   1.565 -#596 := (iff #429 #429)
   1.566 -#597 := [refl]: #596
   1.567 -#599 := [quant-intro #597]: #598
   1.568 -#433 := (iff #314 #432)
   1.569 -#430 := (iff #311 #429)
   1.570 -#431 := [rewrite]: #430
   1.571 -#434 := [quant-intro #431]: #433
   1.572 -#420 := (~ #314 #314)
   1.573 -#418 := (~ #311 #311)
   1.574 -#419 := [refl]: #418
   1.575 -#421 := [nnf-pos #419]: #420
   1.576 -#339 := [and-elim #338]: #314
   1.577 -#422 := [mp~ #339 #421]: #314
   1.578 -#435 := [mp #422 #434]: #432
   1.579 -#600 := [mp #435 #599]: #595
   1.580 -#1066 := (not #595)
   1.581 -#1067 := (or #1066 #1063)
   1.582 -#1039 := (* -1::real #656)
   1.583 -#1041 := (+ #1040 #1039)
   1.584 -#1042 := (>= #1041 0::real)
   1.585 -#1046 := (or #1045 #1042)
   1.586 -#1047 := (not #1046)
   1.587 -#1068 := (or #1066 #1047)
   1.588 -#1070 := (iff #1068 #1067)
   1.589 -#1072 := (iff #1067 #1067)
   1.590 -#1073 := [rewrite]: #1072
   1.591 -#1064 := (iff #1047 #1063)
   1.592 -#1061 := (iff #1046 #1060)
   1.593 -#1058 := (iff #1042 #1055)
   1.594 -#1048 := (+ #1039 #1040)
   1.595 -#1051 := (>= #1048 0::real)
   1.596 -#1056 := (iff #1051 #1055)
   1.597 -#1057 := [rewrite]: #1056
   1.598 -#1052 := (iff #1042 #1051)
   1.599 -#1049 := (= #1041 #1048)
   1.600 -#1050 := [rewrite]: #1049
   1.601 -#1053 := [monotonicity #1050]: #1052
   1.602 -#1059 := [trans #1053 #1057]: #1058
   1.603 -#1062 := [monotonicity #1059]: #1061
   1.604 -#1065 := [monotonicity #1062]: #1064
   1.605 -#1071 := [monotonicity #1065]: #1070
   1.606 -#1074 := [trans #1071 #1073]: #1070
   1.607 -#1069 := [quant-inst]: #1068
   1.608 -#1075 := [mp #1069 #1074]: #1067
   1.609 -#1246 := [unit-resolution #1075 #600]: #1063
   1.610 -#1080 := (or #1060 #1079)
   1.611 -#1081 := [def-axiom]: #1080
   1.612 -#1247 := [unit-resolution #1081 #1246]: #1079
   1.613 -#1248 := [th-lemma #1247 #1245 #1243]: false
   1.614 -#1250 := [lemma #1248]: #1249
   1.615 -#1253 := (or #1252 #1241)
   1.616 -#1254 := [th-lemma]: #1253
   1.617 -#1310 := [unit-resolution #1254 #1250]: #1252
   1.618 -#1326 := [mp #1310 #1325]: #796
   1.619 -#1328 := (or #724 #772)
   1.620 -decl uf_13 :: T3
   1.621 -#33 := uf_13
   1.622 -#609 := (uf_1 uf_13 #44)
   1.623 -#773 := (= #609 #771)
   1.624 -#775 := (ite #724 #773 #772)
   1.625 -#32 := (uf_1 uf_9 #6)
   1.626 -#553 := (pattern #32)
   1.627 -#34 := (uf_1 uf_13 #6)
   1.628 -#551 := (pattern #34)
   1.629 -#456 := (= #32 #89)
   1.630 -#455 := (= #34 #89)
   1.631 -#457 := (ite #119 #455 #456)
   1.632 -#554 := (forall (vars (?x4 T2)) (:pat #541 #551 #552 #553) #457)
   1.633 -#460 := (forall (vars (?x4 T2)) #457)
   1.634 -#557 := (iff #460 #554)
   1.635 -#555 := (iff #457 #457)
   1.636 -#556 := [refl]: #555
   1.637 -#558 := [quant-intro #556]: #557
   1.638 -#143 := (ite #119 #34 #32)
   1.639 -#373 := (= #89 #143)
   1.640 -#374 := (forall (vars (?x4 T2)) #373)
   1.641 -#461 := (iff #374 #460)
   1.642 -#458 := (iff #373 #457)
   1.643 -#459 := [rewrite]: #458
   1.644 -#462 := [quant-intro #459]: #461
   1.645 -#362 := (~ #374 #374)
   1.646 -#364 := (~ #373 #373)
   1.647 -#365 := [refl]: #364
   1.648 -#363 := [nnf-pos #365]: #362
   1.649 -#31 := (uf_3 uf_12 #6)
   1.650 -#148 := (= #31 #143)
   1.651 -#151 := (forall (vars (?x4 T2)) #148)
   1.652 -#375 := (iff #151 #374)
   1.653 -#376 := [rewrite* #113]: #375
   1.654 -#35 := (ite #17 #32 #34)
   1.655 -#36 := (= #31 #35)
   1.656 -#37 := (forall (vars (?x4 T2)) #36)
   1.657 -#152 := (iff #37 #151)
   1.658 -#149 := (iff #36 #148)
   1.659 -#146 := (= #35 #143)
   1.660 -#140 := (ite #118 #32 #34)
   1.661 -#144 := (= #140 #143)
   1.662 -#145 := [rewrite]: #144
   1.663 -#141 := (= #35 #140)
   1.664 -#142 := [monotonicity #123]: #141
   1.665 -#147 := [trans #142 #145]: #146
   1.666 -#150 := [monotonicity #147]: #149
   1.667 -#153 := [quant-intro #150]: #152
   1.668 -#115 := [asserted]: #37
   1.669 -#154 := [mp #115 #153]: #151
   1.670 -#377 := [mp #154 #376]: #374
   1.671 -#360 := [mp~ #377 #363]: #374
   1.672 -#463 := [mp #360 #462]: #460
   1.673 -#559 := [mp #463 #558]: #554
   1.674 -#778 := (not #554)
   1.675 -#779 := (or #778 #775)
   1.676 -#714 := (+ #611 #120)
   1.677 -#715 := (>= #714 0::int)
   1.678 -#774 := (ite #715 #773 #772)
   1.679 -#780 := (or #778 #774)
   1.680 -#782 := (iff #780 #779)
   1.681 -#784 := (iff #779 #779)
   1.682 -#785 := [rewrite]: #784
   1.683 -#776 := (iff #774 #775)
   1.684 -#727 := (iff #715 #724)
   1.685 -#717 := (+ #120 #611)
   1.686 -#720 := (>= #717 0::int)
   1.687 -#725 := (iff #720 #724)
   1.688 -#726 := [rewrite]: #725
   1.689 -#721 := (iff #715 #720)
   1.690 -#718 := (= #714 #717)
   1.691 -#719 := [rewrite]: #718
   1.692 -#722 := [monotonicity #719]: #721
   1.693 -#728 := [trans #722 #726]: #727
   1.694 -#777 := [monotonicity #728]: #776
   1.695 -#783 := [monotonicity #777]: #782
   1.696 -#786 := [trans #783 #785]: #782
   1.697 -#781 := [quant-inst]: #780
   1.698 -#787 := [mp #781 #786]: #779
   1.699 -#1327 := [unit-resolution #787 #559]: #775
   1.700 -#788 := (not #775)
   1.701 -#791 := (or #788 #724 #772)
   1.702 -#792 := [def-axiom]: #791
   1.703 -#1329 := [unit-resolution #792 #1327]: #1328
   1.704 -#1330 := [unit-resolution #1329 #1326]: #724
   1.705 -#988 := (>= #622 0::int)
   1.706 -#1331 := (or #1312 #988)
   1.707 -#1332 := [th-lemma]: #1331
   1.708 -#1333 := [unit-resolution #1332 #1281]: #988
   1.709 -#761 := (not #724)
   1.710 -#1334 := (not #988)
   1.711 -#1335 := (or #1271 #1334 #761)
   1.712 -#1336 := [th-lemma]: #1335
   1.713 -#1337 := [unit-resolution #1336 #1333 #1330]: #1271
   1.714 -#1338 := (not #1271)
   1.715 -#1340 := (or #1269 #1338 #1339)
   1.716 -#1341 := [th-lemma]: #1340
   1.717 -#1343 := [unit-resolution #1341 #1337]: #1342
   1.718 -#1344 := [unit-resolution #1343 #1299]: #1339
   1.719 -#990 := (>= #916 0::int)
   1.720 -#1345 := (or #1302 #990)
   1.721 -#1346 := [th-lemma]: #1345
   1.722 -#1347 := [unit-resolution #1346 #1301]: #990
   1.723 -#1348 := (not #990)
   1.724 -#1349 := (or #836 #1348 #1265)
   1.725 -#1350 := [th-lemma]: #1349
   1.726 -#1351 := [unit-resolution #1350 #1347 #1344]: #836
   1.727 -#1353 := (or #815 #800)
   1.728 -#801 := (uf_1 uf_13 #22)
   1.729 -#820 := (= #799 #801)
   1.730 -#823 := (ite #815 #820 #800)
   1.731 -#476 := (= #32 #79)
   1.732 -#475 := (= #34 #79)
   1.733 -#477 := (ite #157 #475 #476)
   1.734 -#568 := (forall (vars (?x6 T2)) (:pat #541 #551 #567 #553) #477)
   1.735 -#480 := (forall (vars (?x6 T2)) #477)
   1.736 -#571 := (iff #480 #568)
   1.737 -#569 := (iff #477 #477)
   1.738 -#570 := [refl]: #569
   1.739 -#572 := [quant-intro #570]: #571
   1.740 -#181 := (ite #157 #34 #32)
   1.741 -#383 := (= #79 #181)
   1.742 -#384 := (forall (vars (?x6 T2)) #383)
   1.743 -#481 := (iff #384 #480)
   1.744 -#478 := (iff #383 #477)
   1.745 -#479 := [rewrite]: #478
   1.746 -#482 := [quant-intro #479]: #481
   1.747 -#352 := (~ #384 #384)
   1.748 -#354 := (~ #383 #383)
   1.749 -#355 := [refl]: #354
   1.750 -#353 := [nnf-pos #355]: #352
   1.751 -#51 := (uf_3 uf_16 #6)
   1.752 -#186 := (= #51 #181)
   1.753 -#189 := (forall (vars (?x6 T2)) #186)
   1.754 -#385 := (iff #189 #384)
   1.755 -#386 := [rewrite* #113]: #385
   1.756 -#52 := (ite #42 #32 #34)
   1.757 -#53 := (= #51 #52)
   1.758 -#54 := (forall (vars (?x6 T2)) #53)
   1.759 -#190 := (iff #54 #189)
   1.760 -#187 := (iff #53 #186)
   1.761 -#184 := (= #52 #181)
   1.762 -#178 := (ite #156 #32 #34)
   1.763 -#182 := (= #178 #181)
   1.764 -#183 := [rewrite]: #182
   1.765 -#179 := (= #52 #178)
   1.766 -#180 := [monotonicity #161]: #179
   1.767 -#185 := [trans #180 #183]: #184
   1.768 -#188 := [monotonicity #185]: #187
   1.769 -#191 := [quant-intro #188]: #190
   1.770 -#139 := [asserted]: #54
   1.771 -#192 := [mp #139 #191]: #189
   1.772 -#387 := [mp #192 #386]: #384
   1.773 -#402 := [mp~ #387 #353]: #384
   1.774 -#483 := [mp #402 #482]: #480
   1.775 -#573 := [mp #483 #572]: #568
   1.776 -#634 := (not #568)
   1.777 -#826 := (or #634 #823)
   1.778 -#802 := (= #801 #799)
   1.779 -#804 := (+ #803 #158)
   1.780 -#805 := (>= #804 0::int)
   1.781 -#806 := (ite #805 #802 #800)
   1.782 -#827 := (or #634 #806)
   1.783 -#829 := (iff #827 #826)
   1.784 -#831 := (iff #826 #826)
   1.785 -#832 := [rewrite]: #831
   1.786 -#824 := (iff #806 #823)
   1.787 -#821 := (iff #802 #820)
   1.788 -#822 := [rewrite]: #821
   1.789 -#818 := (iff #805 #815)
   1.790 -#807 := (+ #158 #803)
   1.791 -#810 := (>= #807 0::int)
   1.792 -#816 := (iff #810 #815)
   1.793 -#817 := [rewrite]: #816
   1.794 -#811 := (iff #805 #810)
   1.795 -#808 := (= #804 #807)
   1.796 -#809 := [rewrite]: #808
   1.797 -#812 := [monotonicity #809]: #811
   1.798 -#819 := [trans #812 #817]: #818
   1.799 -#825 := [monotonicity #819 #822]: #824
   1.800 -#830 := [monotonicity #825]: #829
   1.801 -#833 := [trans #830 #832]: #829
   1.802 -#828 := [quant-inst]: #827
   1.803 -#834 := [mp #828 #833]: #826
   1.804 -#1352 := [unit-resolution #834 #573]: #823
   1.805 -#835 := (not #823)
   1.806 -#839 := (or #835 #815 #800)
   1.807 -#840 := [def-axiom]: #839
   1.808 -#1354 := [unit-resolution #840 #1352]: #1353
   1.809 -#1355 := [unit-resolution #1354 #1351]: #800
   1.810 -#1357 := [symm #1355]: #1356
   1.811 -#1358 := [trans #1357 #1309]: #1266
   1.812 -#1359 := (not #1266)
   1.813 -#1360 := (or #1359 #1272)
   1.814 -#1361 := [th-lemma]: #1360
   1.815 -#1362 := [unit-resolution #1361 #1358]: #1272
   1.816 -#1085 := (uf_1 uf_18 #22)
   1.817 -#1099 := (* -1::real #1085)
   1.818 -#1112 := (+ #902 #1099)
   1.819 -#1113 := (<= #1112 0::real)
   1.820 -#1137 := (not #1113)
   1.821 -#960 := (uf_1 #88 #22)
   1.822 -#1100 := (+ #960 #1099)
   1.823 -#1101 := (>= #1100 0::real)
   1.824 -#1118 := (or #1101 #1113)
   1.825 -#1121 := (not #1118)
   1.826 -#1125 := (or #1124 #1121)
   1.827 -#1086 := (+ #1085 #1084)
   1.828 -#1087 := (>= #1086 0::real)
   1.829 -#1088 := (* -1::real #960)
   1.830 -#1089 := (+ #1085 #1088)
   1.831 -#1090 := (<= #1089 0::real)
   1.832 -#1091 := (or #1090 #1087)
   1.833 -#1092 := (not #1091)
   1.834 -#1126 := (or #1124 #1092)
   1.835 -#1128 := (iff #1126 #1125)
   1.836 -#1130 := (iff #1125 #1125)
   1.837 -#1131 := [rewrite]: #1130
   1.838 -#1122 := (iff #1092 #1121)
   1.839 -#1119 := (iff #1091 #1118)
   1.840 -#1116 := (iff #1087 #1113)
   1.841 -#1106 := (+ #1084 #1085)
   1.842 -#1109 := (>= #1106 0::real)
   1.843 -#1114 := (iff #1109 #1113)
   1.844 -#1115 := [rewrite]: #1114
   1.845 -#1110 := (iff #1087 #1109)
   1.846 -#1107 := (= #1086 #1106)
   1.847 -#1108 := [rewrite]: #1107
   1.848 -#1111 := [monotonicity #1108]: #1110
   1.849 -#1117 := [trans #1111 #1115]: #1116
   1.850 -#1104 := (iff #1090 #1101)
   1.851 -#1093 := (+ #1088 #1085)
   1.852 -#1096 := (<= #1093 0::real)
   1.853 -#1102 := (iff #1096 #1101)
   1.854 -#1103 := [rewrite]: #1102
   1.855 -#1097 := (iff #1090 #1096)
   1.856 -#1094 := (= #1089 #1093)
   1.857 -#1095 := [rewrite]: #1094
   1.858 -#1098 := [monotonicity #1095]: #1097
   1.859 -#1105 := [trans #1098 #1103]: #1104
   1.860 -#1120 := [monotonicity #1105 #1117]: #1119
   1.861 -#1123 := [monotonicity #1120]: #1122
   1.862 -#1129 := [monotonicity #1123]: #1128
   1.863 -#1132 := [trans #1129 #1131]: #1128
   1.864 -#1127 := [quant-inst]: #1126
   1.865 -#1133 := [mp #1127 #1132]: #1125
   1.866 -#1363 := [unit-resolution #1133 #606]: #1121
   1.867 -#1138 := (or #1118 #1137)
   1.868 -#1139 := [def-axiom]: #1138
   1.869 -#1364 := [unit-resolution #1139 #1363]: #1137
   1.870 -#1200 := (+ #799 #1099)
   1.871 -#1201 := (>= #1200 0::real)
   1.872 -#1231 := (not #1201)
   1.873 -#847 := (uf_1 #83 #22)
   1.874 -#1210 := (+ #847 #1099)
   1.875 -#1211 := (<= #1210 0::real)
   1.876 -#1216 := (or #1201 #1211)
   1.877 -#1219 := (not #1216)
   1.878 -#1222 := (or #1066 #1219)
   1.879 -#1197 := (* -1::real #847)
   1.880 -#1198 := (+ #1085 #1197)
   1.881 -#1199 := (>= #1198 0::real)
   1.882 -#1202 := (or #1201 #1199)
   1.883 -#1203 := (not #1202)
   1.884 -#1223 := (or #1066 #1203)
   1.885 -#1225 := (iff #1223 #1222)
   1.886 -#1227 := (iff #1222 #1222)
   1.887 -#1228 := [rewrite]: #1227
   1.888 -#1220 := (iff #1203 #1219)
   1.889 -#1217 := (iff #1202 #1216)
   1.890 -#1214 := (iff #1199 #1211)
   1.891 -#1204 := (+ #1197 #1085)
   1.892 -#1207 := (>= #1204 0::real)
   1.893 -#1212 := (iff #1207 #1211)
   1.894 -#1213 := [rewrite]: #1212
   1.895 -#1208 := (iff #1199 #1207)
   1.896 -#1205 := (= #1198 #1204)
   1.897 -#1206 := [rewrite]: #1205
   1.898 -#1209 := [monotonicity #1206]: #1208
   1.899 -#1215 := [trans #1209 #1213]: #1214
   1.900 -#1218 := [monotonicity #1215]: #1217
   1.901 -#1221 := [monotonicity #1218]: #1220
   1.902 -#1226 := [monotonicity #1221]: #1225
   1.903 -#1229 := [trans #1226 #1228]: #1225
   1.904 -#1224 := [quant-inst]: #1223
   1.905 -#1230 := [mp #1224 #1229]: #1222
   1.906 -#1365 := [unit-resolution #1230 #600]: #1219
   1.907 -#1232 := (or #1216 #1231)
   1.908 -#1233 := [def-axiom]: #1232
   1.909 -#1366 := [unit-resolution #1233 #1365]: #1231
   1.910 -[th-lemma #1366 #1364 #1362]: false
   1.911 -unsat
   1.912 -NQHwTeL311Tq3wf2s5BReA 419 0
   1.913 -#2 := false
   1.914 -#194 := 0::real
   1.915 -decl uf_4 :: (-> T2 T3 real)
   1.916 -decl uf_6 :: (-> T1 T3)
   1.917 -decl uf_3 :: T1
   1.918 -#21 := uf_3
   1.919 -#25 := (uf_6 uf_3)
   1.920 -decl uf_5 :: T2
   1.921 -#24 := uf_5
   1.922 -#26 := (uf_4 uf_5 #25)
   1.923 -decl uf_7 :: T2
   1.924 -#27 := uf_7
   1.925 -#28 := (uf_4 uf_7 #25)
   1.926 -decl uf_10 :: T1
   1.927 -#38 := uf_10
   1.928 -#42 := (uf_6 uf_10)
   1.929 -decl uf_9 :: T2
   1.930 -#33 := uf_9
   1.931 -#43 := (uf_4 uf_9 #42)
   1.932 -#41 := (= uf_3 uf_10)
   1.933 -#44 := (ite #41 #43 #28)
   1.934 -#9 := 0::int
   1.935 -decl uf_2 :: (-> T1 int)
   1.936 -#39 := (uf_2 uf_10)
   1.937 -#226 := -1::int
   1.938 -#229 := (* -1::int #39)
   1.939 -#22 := (uf_2 uf_3)
   1.940 -#230 := (+ #22 #229)
   1.941 -#228 := (>= #230 0::int)
   1.942 -#236 := (ite #228 #44 #26)
   1.943 -#192 := -1::real
   1.944 -#244 := (* -1::real #236)
   1.945 -#642 := (+ #26 #244)
   1.946 -#643 := (<= #642 0::real)
   1.947 -#567 := (= #26 #236)
   1.948 -#227 := (not #228)
   1.949 -decl uf_1 :: (-> int T1)
   1.950 -#593 := (uf_1 #39)
   1.951 -#660 := (= #593 uf_10)
   1.952 -#594 := (= uf_10 #593)
   1.953 -#4 := (:var 0 T1)
   1.954 -#5 := (uf_2 #4)
   1.955 -#546 := (pattern #5)
   1.956 -#6 := (uf_1 #5)
   1.957 -#93 := (= #4 #6)
   1.958 -#547 := (forall (vars (?x1 T1)) (:pat #546) #93)
   1.959 -#96 := (forall (vars (?x1 T1)) #93)
   1.960 -#550 := (iff #96 #547)
   1.961 -#548 := (iff #93 #93)
   1.962 -#549 := [refl]: #548
   1.963 -#551 := [quant-intro #549]: #550
   1.964 -#448 := (~ #96 #96)
   1.965 -#450 := (~ #93 #93)
   1.966 -#451 := [refl]: #450
   1.967 -#449 := [nnf-pos #451]: #448
   1.968 -#7 := (= #6 #4)
   1.969 -#8 := (forall (vars (?x1 T1)) #7)
   1.970 -#97 := (iff #8 #96)
   1.971 -#94 := (iff #7 #93)
   1.972 -#95 := [rewrite]: #94
   1.973 -#98 := [quant-intro #95]: #97
   1.974 -#92 := [asserted]: #8
   1.975 -#101 := [mp #92 #98]: #96
   1.976 -#446 := [mp~ #101 #449]: #96
   1.977 -#552 := [mp #446 #551]: #547
   1.978 -#595 := (not #547)
   1.979 -#600 := (or #595 #594)
   1.980 -#601 := [quant-inst]: #600
   1.981 -#654 := [unit-resolution #601 #552]: #594
   1.982 -#680 := [symm #654]: #660
   1.983 -#681 := (= uf_3 #593)
   1.984 -#591 := (uf_1 #22)
   1.985 -#658 := (= #591 #593)
   1.986 -#656 := (= #593 #591)
   1.987 -#652 := (= #39 #22)
   1.988 -#647 := (= #22 #39)
   1.989 -#290 := (<= #230 0::int)
   1.990 -#70 := (<= #22 #39)
   1.991 -#388 := (iff #70 #290)
   1.992 -#389 := [rewrite]: #388
   1.993 -#341 := [asserted]: #70
   1.994 -#390 := [mp #341 #389]: #290
   1.995 -#646 := [hypothesis]: #228
   1.996 -#648 := [th-lemma #646 #390]: #647
   1.997 -#653 := [symm #648]: #652
   1.998 -#657 := [monotonicity #653]: #656
   1.999 -#659 := [symm #657]: #658
  1.1000 -#592 := (= uf_3 #591)
  1.1001 -#596 := (or #595 #592)
  1.1002 -#597 := [quant-inst]: #596
  1.1003 -#655 := [unit-resolution #597 #552]: #592
  1.1004 -#682 := [trans #655 #659]: #681
  1.1005 -#683 := [trans #682 #680]: #41
  1.1006 -#570 := (not #41)
  1.1007 -decl uf_11 :: T2
  1.1008 -#47 := uf_11
  1.1009 -#59 := (uf_4 uf_11 #42)
  1.1010 -#278 := (ite #41 #26 #59)
  1.1011 -#459 := (* -1::real #278)
  1.1012 -#637 := (+ #26 #459)
  1.1013 -#639 := (>= #637 0::real)
  1.1014 -#585 := (= #26 #278)
  1.1015 -#661 := [hypothesis]: #41
  1.1016 -#587 := (or #570 #585)
  1.1017 -#588 := [def-axiom]: #587
  1.1018 -#662 := [unit-resolution #588 #661]: #585
  1.1019 -#663 := (not #585)
  1.1020 -#664 := (or #663 #639)
  1.1021 -#665 := [th-lemma]: #664
  1.1022 -#666 := [unit-resolution #665 #662]: #639
  1.1023 -decl uf_8 :: T2
  1.1024 -#30 := uf_8
  1.1025 -#56 := (uf_4 uf_8 #42)
  1.1026 -#357 := (* -1::real #56)
  1.1027 -#358 := (+ #43 #357)
  1.1028 -#356 := (>= #358 0::real)
  1.1029 -#355 := (not #356)
  1.1030 -#374 := (* -1::real #59)
  1.1031 -#375 := (+ #56 #374)
  1.1032 -#373 := (>= #375 0::real)
  1.1033 -#376 := (not #373)
  1.1034 -#381 := (and #355 #376)
  1.1035 -#64 := (< #39 #39)
  1.1036 -#67 := (ite #64 #43 #59)
  1.1037 -#68 := (< #56 #67)
  1.1038 -#53 := (uf_4 uf_5 #42)
  1.1039 -#65 := (ite #64 #53 #43)
  1.1040 -#66 := (< #65 #56)
  1.1041 -#69 := (and #66 #68)
  1.1042 -#382 := (iff #69 #381)
  1.1043 -#379 := (iff #68 #376)
  1.1044 -#370 := (< #56 #59)
  1.1045 -#377 := (iff #370 #376)
  1.1046 -#378 := [rewrite]: #377
  1.1047 -#371 := (iff #68 #370)
  1.1048 -#368 := (= #67 #59)
  1.1049 -#363 := (ite false #43 #59)
  1.1050 -#366 := (= #363 #59)
  1.1051 -#367 := [rewrite]: #366
  1.1052 -#364 := (= #67 #363)
  1.1053 -#343 := (iff #64 false)
  1.1054 -#344 := [rewrite]: #343
  1.1055 -#365 := [monotonicity #344]: #364
  1.1056 -#369 := [trans #365 #367]: #368
  1.1057 -#372 := [monotonicity #369]: #371
  1.1058 -#380 := [trans #372 #378]: #379
  1.1059 -#361 := (iff #66 #355)
  1.1060 -#352 := (< #43 #56)
  1.1061 -#359 := (iff #352 #355)
  1.1062 -#360 := [rewrite]: #359
  1.1063 -#353 := (iff #66 #352)
  1.1064 -#350 := (= #65 #43)
  1.1065 -#345 := (ite false #53 #43)
  1.1066 -#348 := (= #345 #43)
  1.1067 -#349 := [rewrite]: #348
  1.1068 -#346 := (= #65 #345)
  1.1069 -#347 := [monotonicity #344]: #346
  1.1070 -#351 := [trans #347 #349]: #350
  1.1071 -#354 := [monotonicity #351]: #353
  1.1072 -#362 := [trans #354 #360]: #361
  1.1073 -#383 := [monotonicity #362 #380]: #382
  1.1074 -#340 := [asserted]: #69
  1.1075 -#384 := [mp #340 #383]: #381
  1.1076 -#385 := [and-elim #384]: #355
  1.1077 -#394 := (* -1::real #53)
  1.1078 -#395 := (+ #43 #394)
  1.1079 -#393 := (>= #395 0::real)
  1.1080 -#54 := (uf_4 uf_7 #42)
  1.1081 -#402 := (* -1::real #54)
  1.1082 -#403 := (+ #53 #402)
  1.1083 -#401 := (>= #403 0::real)
  1.1084 -#397 := (+ #43 #374)
  1.1085 -#398 := (<= #397 0::real)
  1.1086 -#412 := (and #393 #398 #401)
  1.1087 -#73 := (<= #43 #59)
  1.1088 -#72 := (<= #53 #43)
  1.1089 -#74 := (and #72 #73)
  1.1090 -#71 := (<= #54 #53)
  1.1091 -#75 := (and #71 #74)
  1.1092 -#415 := (iff #75 #412)
  1.1093 -#406 := (and #393 #398)
  1.1094 -#409 := (and #401 #406)
  1.1095 -#413 := (iff #409 #412)
  1.1096 -#414 := [rewrite]: #413
  1.1097 -#410 := (iff #75 #409)
  1.1098 -#407 := (iff #74 #406)
  1.1099 -#399 := (iff #73 #398)
  1.1100 -#400 := [rewrite]: #399
  1.1101 -#392 := (iff #72 #393)
  1.1102 -#396 := [rewrite]: #392
  1.1103 -#408 := [monotonicity #396 #400]: #407
  1.1104 -#404 := (iff #71 #401)
  1.1105 -#405 := [rewrite]: #404
  1.1106 -#411 := [monotonicity #405 #408]: #410
  1.1107 -#416 := [trans #411 #414]: #415
  1.1108 -#342 := [asserted]: #75
  1.1109 -#417 := [mp #342 #416]: #412
  1.1110 -#418 := [and-elim #417]: #393
  1.1111 -#650 := (+ #26 #394)
  1.1112 -#651 := (<= #650 0::real)
  1.1113 -#649 := (= #26 #53)
  1.1114 -#671 := (= #53 #26)
  1.1115 -#669 := (= #42 #25)
  1.1116 -#667 := (= #25 #42)
  1.1117 -#668 := [monotonicity #661]: #667
  1.1118 -#670 := [symm #668]: #669
  1.1119 -#672 := [monotonicity #670]: #671
  1.1120 -#673 := [symm #672]: #649
  1.1121 -#674 := (not #649)
  1.1122 -#675 := (or #674 #651)
  1.1123 -#676 := [th-lemma]: #675
  1.1124 -#677 := [unit-resolution #676 #673]: #651
  1.1125 -#462 := (+ #56 #459)
  1.1126 -#465 := (>= #462 0::real)
  1.1127 -#438 := (not #465)
  1.1128 -#316 := (ite #290 #278 #43)
  1.1129 -#326 := (* -1::real #316)
  1.1130 -#327 := (+ #56 #326)
  1.1131 -#325 := (>= #327 0::real)
  1.1132 -#324 := (not #325)
  1.1133 -#439 := (iff #324 #438)
  1.1134 -#466 := (iff #325 #465)
  1.1135 -#463 := (= #327 #462)
  1.1136 -#460 := (= #326 #459)
  1.1137 -#457 := (= #316 #278)
  1.1138 -#1 := true
  1.1139 -#452 := (ite true #278 #43)
  1.1140 -#455 := (= #452 #278)
  1.1141 -#456 := [rewrite]: #455
  1.1142 -#453 := (= #316 #452)
  1.1143 -#444 := (iff #290 true)
  1.1144 -#445 := [iff-true #390]: #444
  1.1145 -#454 := [monotonicity #445]: #453
  1.1146 -#458 := [trans #454 #456]: #457
  1.1147 -#461 := [monotonicity #458]: #460
  1.1148 -#464 := [monotonicity #461]: #463
  1.1149 -#467 := [monotonicity #464]: #466
  1.1150 -#468 := [monotonicity #467]: #439
  1.1151 -#297 := (ite #290 #54 #53)
  1.1152 -#305 := (* -1::real #297)
  1.1153 -#306 := (+ #56 #305)
  1.1154 -#307 := (<= #306 0::real)
  1.1155 -#308 := (not #307)
  1.1156 -#332 := (and #308 #324)
  1.1157 -#58 := (= uf_10 uf_3)
  1.1158 -#60 := (ite #58 #26 #59)
  1.1159 -#52 := (< #39 #22)
  1.1160 -#61 := (ite #52 #43 #60)
  1.1161 -#62 := (< #56 #61)
  1.1162 -#55 := (ite #52 #53 #54)
  1.1163 -#57 := (< #55 #56)
  1.1164 -#63 := (and #57 #62)
  1.1165 -#335 := (iff #63 #332)
  1.1166 -#281 := (ite #52 #43 #278)
  1.1167 -#284 := (< #56 #281)
  1.1168 -#287 := (and #57 #284)
  1.1169 -#333 := (iff #287 #332)
  1.1170 -#330 := (iff #284 #324)
  1.1171 -#321 := (< #56 #316)
  1.1172 -#328 := (iff #321 #324)
  1.1173 -#329 := [rewrite]: #328
  1.1174 -#322 := (iff #284 #321)
  1.1175 -#319 := (= #281 #316)
  1.1176 -#291 := (not #290)
  1.1177 -#313 := (ite #291 #43 #278)
  1.1178 -#317 := (= #313 #316)
  1.1179 -#318 := [rewrite]: #317
  1.1180 -#314 := (= #281 #313)
  1.1181 -#292 := (iff #52 #291)
  1.1182 -#293 := [rewrite]: #292
  1.1183 -#315 := [monotonicity #293]: #314
  1.1184 -#320 := [trans #315 #318]: #319
  1.1185 -#323 := [monotonicity #320]: #322
  1.1186 -#331 := [trans #323 #329]: #330
  1.1187 -#311 := (iff #57 #308)
  1.1188 -#302 := (< #297 #56)
  1.1189 -#309 := (iff #302 #308)
  1.1190 -#310 := [rewrite]: #309
  1.1191 -#303 := (iff #57 #302)
  1.1192 -#300 := (= #55 #297)
  1.1193 -#294 := (ite #291 #53 #54)
  1.1194 -#298 := (= #294 #297)
  1.1195 -#299 := [rewrite]: #298
  1.1196 -#295 := (= #55 #294)
  1.1197 -#296 := [monotonicity #293]: #295
  1.1198 -#301 := [trans #296 #299]: #300
  1.1199 -#304 := [monotonicity #301]: #303
  1.1200 -#312 := [trans #304 #310]: #311
  1.1201 -#334 := [monotonicity #312 #331]: #333
  1.1202 -#288 := (iff #63 #287)
  1.1203 -#285 := (iff #62 #284)
  1.1204 -#282 := (= #61 #281)
  1.1205 -#279 := (= #60 #278)
  1.1206 -#225 := (iff #58 #41)
  1.1207 -#277 := [rewrite]: #225
  1.1208 -#280 := [monotonicity #277]: #279
  1.1209 -#283 := [monotonicity #280]: #282
  1.1210 -#286 := [monotonicity #283]: #285
  1.1211 -#289 := [monotonicity #286]: #288
  1.1212 -#336 := [trans #289 #334]: #335
  1.1213 -#179 := [asserted]: #63
  1.1214 -#337 := [mp #179 #336]: #332
  1.1215 -#339 := [and-elim #337]: #324
  1.1216 -#469 := [mp #339 #468]: #438
  1.1217 -#678 := [th-lemma #469 #677 #418 #385 #666]: false
  1.1218 -#679 := [lemma #678]: #570
  1.1219 -#684 := [unit-resolution #679 #683]: false
  1.1220 -#685 := [lemma #684]: #227
  1.1221 -#577 := (or #228 #567)
  1.1222 -#578 := [def-axiom]: #577
  1.1223 -#645 := [unit-resolution #578 #685]: #567
  1.1224 -#686 := (not #567)
  1.1225 -#687 := (or #686 #643)
  1.1226 -#688 := [th-lemma]: #687
  1.1227 -#689 := [unit-resolution #688 #645]: #643
  1.1228 -#31 := (uf_4 uf_8 #25)
  1.1229 -#245 := (+ #31 #244)
  1.1230 -#246 := (<= #245 0::real)
  1.1231 -#247 := (not #246)
  1.1232 -#34 := (uf_4 uf_9 #25)
  1.1233 -#48 := (uf_4 uf_11 #25)
  1.1234 -#255 := (ite #228 #48 #34)
  1.1235 -#264 := (* -1::real #255)
  1.1236 -#265 := (+ #31 #264)
  1.1237 -#263 := (>= #265 0::real)
  1.1238 -#266 := (not #263)
  1.1239 -#271 := (and #247 #266)
  1.1240 -#40 := (< #22 #39)
  1.1241 -#49 := (ite #40 #34 #48)
  1.1242 -#50 := (< #31 #49)
  1.1243 -#45 := (ite #40 #26 #44)
  1.1244 -#46 := (< #45 #31)
  1.1245 -#51 := (and #46 #50)
  1.1246 -#272 := (iff #51 #271)
  1.1247 -#269 := (iff #50 #266)
  1.1248 -#260 := (< #31 #255)
  1.1249 -#267 := (iff #260 #266)
  1.1250 -#268 := [rewrite]: #267
  1.1251 -#261 := (iff #50 #260)
  1.1252 -#258 := (= #49 #255)
  1.1253 -#252 := (ite #227 #34 #48)
  1.1254 -#256 := (= #252 #255)
  1.1255 -#257 := [rewrite]: #256
  1.1256 -#253 := (= #49 #252)
  1.1257 -#231 := (iff #40 #227)
  1.1258 -#232 := [rewrite]: #231
  1.1259 -#254 := [monotonicity #232]: #253
  1.1260 -#259 := [trans #254 #257]: #258
  1.1261 -#262 := [monotonicity #259]: #261
  1.1262 -#270 := [trans #262 #268]: #269
  1.1263 -#250 := (iff #46 #247)
  1.1264 -#241 := (< #236 #31)
  1.1265 -#248 := (iff #241 #247)
  1.1266 -#249 := [rewrite]: #248
  1.1267 -#242 := (iff #46 #241)
  1.1268 -#239 := (= #45 #236)
  1.1269 -#233 := (ite #227 #26 #44)
  1.1270 -#237 := (= #233 #236)
  1.1271 -#238 := [rewrite]: #237
  1.1272 -#234 := (= #45 #233)
  1.1273 -#235 := [monotonicity #232]: #234
  1.1274 -#240 := [trans #235 #238]: #239
  1.1275 -#243 := [monotonicity #240]: #242
  1.1276 -#251 := [trans #243 #249]: #250
  1.1277 -#273 := [monotonicity #251 #270]: #272
  1.1278 -#178 := [asserted]: #51
  1.1279 -#274 := [mp #178 #273]: #271
  1.1280 -#275 := [and-elim #274]: #247
  1.1281 -#196 := (* -1::real #31)
  1.1282 -#212 := (+ #26 #196)
  1.1283 -#213 := (<= #212 0::real)
  1.1284 -#214 := (not #213)
  1.1285 -#197 := (+ #28 #196)
  1.1286 -#195 := (>= #197 0::real)
  1.1287 -#193 := (not #195)
  1.1288 -#219 := (and #193 #214)
  1.1289 -#23 := (< #22 #22)
  1.1290 -#35 := (ite #23 #34 #26)
  1.1291 -#36 := (< #31 #35)
  1.1292 -#29 := (ite #23 #26 #28)
  1.1293 -#32 := (< #29 #31)
  1.1294 -#37 := (and #32 #36)
  1.1295 -#220 := (iff #37 #219)
  1.1296 -#217 := (iff #36 #214)
  1.1297 -#209 := (< #31 #26)
  1.1298 -#215 := (iff #209 #214)
  1.1299 -#216 := [rewrite]: #215
  1.1300 -#210 := (iff #36 #209)
  1.1301 -#207 := (= #35 #26)
  1.1302 -#202 := (ite false #34 #26)
  1.1303 -#205 := (= #202 #26)
  1.1304 -#206 := [rewrite]: #205
  1.1305 -#203 := (= #35 #202)
  1.1306 -#180 := (iff #23 false)
  1.1307 -#181 := [rewrite]: #180
  1.1308 -#204 := [monotonicity #181]: #203
  1.1309 -#208 := [trans #204 #206]: #207
  1.1310 -#211 := [monotonicity #208]: #210
  1.1311 -#218 := [trans #211 #216]: #217
  1.1312 -#200 := (iff #32 #193)
  1.1313 -#189 := (< #28 #31)
  1.1314 -#198 := (iff #189 #193)
  1.1315 -#199 := [rewrite]: #198
  1.1316 -#190 := (iff #32 #189)
  1.1317 -#187 := (= #29 #28)
  1.1318 -#182 := (ite false #26 #28)
  1.1319 -#185 := (= #182 #28)
  1.1320 -#186 := [rewrite]: #185
  1.1321 -#183 := (= #29 #182)
  1.1322 -#184 := [monotonicity #181]: #183
  1.1323 -#188 := [trans #184 #186]: #187
  1.1324 -#191 := [monotonicity #188]: #190
  1.1325 -#201 := [trans #191 #199]: #200
  1.1326 -#221 := [monotonicity #201 #218]: #220
  1.1327 -#177 := [asserted]: #37
  1.1328 -#222 := [mp #177 #221]: #219
  1.1329 -#224 := [and-elim #222]: #214
  1.1330 -[th-lemma #224 #275 #689]: false
  1.1331 -unsat
  1.1332 -NX/HT1QOfbspC2LtZNKpBA 428 0
  1.1333 -#2 := false
  1.1334 -decl uf_10 :: T1
  1.1335 -#38 := uf_10
  1.1336 -decl uf_3 :: T1
  1.1337 -#21 := uf_3
  1.1338 -#45 := (= uf_3 uf_10)
  1.1339 -decl uf_1 :: (-> int T1)
  1.1340 -decl uf_2 :: (-> T1 int)
  1.1341 -#39 := (uf_2 uf_10)
  1.1342 -#588 := (uf_1 #39)
  1.1343 -#686 := (= #588 uf_10)
  1.1344 -#589 := (= uf_10 #588)
  1.1345 -#4 := (:var 0 T1)
  1.1346 -#5 := (uf_2 #4)
  1.1347 -#541 := (pattern #5)
  1.1348 -#6 := (uf_1 #5)
  1.1349 -#93 := (= #4 #6)
  1.1350 -#542 := (forall (vars (?x1 T1)) (:pat #541) #93)
  1.1351 -#96 := (forall (vars (?x1 T1)) #93)
  1.1352 -#545 := (iff #96 #542)
  1.1353 -#543 := (iff #93 #93)
  1.1354 -#544 := [refl]: #543
  1.1355 -#546 := [quant-intro #544]: #545
  1.1356 -#454 := (~ #96 #96)
  1.1357 -#456 := (~ #93 #93)
  1.1358 -#457 := [refl]: #456
  1.1359 -#455 := [nnf-pos #457]: #454
  1.1360 -#7 := (= #6 #4)
  1.1361 -#8 := (forall (vars (?x1 T1)) #7)
  1.1362 -#97 := (iff #8 #96)
  1.1363 -#94 := (iff #7 #93)
  1.1364 -#95 := [rewrite]: #94
  1.1365 -#98 := [quant-intro #95]: #97
  1.1366 -#92 := [asserted]: #8
  1.1367 -#101 := [mp #92 #98]: #96
  1.1368 -#452 := [mp~ #101 #455]: #96
  1.1369 -#547 := [mp #452 #546]: #542
  1.1370 -#590 := (not #542)
  1.1371 -#595 := (or #590 #589)
  1.1372 -#596 := [quant-inst]: #595
  1.1373 -#680 := [unit-resolution #596 #547]: #589
  1.1374 -#687 := [symm #680]: #686
  1.1375 -#688 := (= uf_3 #588)
  1.1376 -#22 := (uf_2 uf_3)
  1.1377 -#586 := (uf_1 #22)
  1.1378 -#684 := (= #586 #588)
  1.1379 -#682 := (= #588 #586)
  1.1380 -#678 := (= #39 #22)
  1.1381 -#676 := (= #22 #39)
  1.1382 -#9 := 0::int
  1.1383 -#227 := -1::int
  1.1384 -#230 := (* -1::int #39)
  1.1385 -#231 := (+ #22 #230)
  1.1386 -#296 := (<= #231 0::int)
  1.1387 -#70 := (<= #22 #39)
  1.1388 -#393 := (iff #70 #296)
  1.1389 -#394 := [rewrite]: #393
  1.1390 -#347 := [asserted]: #70
  1.1391 -#395 := [mp #347 #394]: #296
  1.1392 -#229 := (>= #231 0::int)
  1.1393 -decl uf_4 :: (-> T2 T3 real)
  1.1394 -decl uf_6 :: (-> T1 T3)
  1.1395 -#25 := (uf_6 uf_3)
  1.1396 -decl uf_7 :: T2
  1.1397 -#27 := uf_7
  1.1398 -#28 := (uf_4 uf_7 #25)
  1.1399 -decl uf_9 :: T2
  1.1400 -#33 := uf_9
  1.1401 -#34 := (uf_4 uf_9 #25)
  1.1402 -#46 := (uf_6 uf_10)
  1.1403 -decl uf_5 :: T2
  1.1404 -#24 := uf_5
  1.1405 -#47 := (uf_4 uf_5 #46)
  1.1406 -#48 := (ite #45 #47 #34)
  1.1407 -#256 := (ite #229 #48 #28)
  1.1408 -#568 := (= #28 #256)
  1.1409 -#648 := (not #568)
  1.1410 -#194 := 0::real
  1.1411 -#192 := -1::real
  1.1412 -#265 := (* -1::real #256)
  1.1413 -#640 := (+ #28 #265)
  1.1414 -#642 := (>= #640 0::real)
  1.1415 -#645 := (not #642)
  1.1416 -#643 := [hypothesis]: #642
  1.1417 -decl uf_8 :: T2
  1.1418 -#30 := uf_8
  1.1419 -#31 := (uf_4 uf_8 #25)
  1.1420 -#266 := (+ #31 #265)
  1.1421 -#264 := (>= #266 0::real)
  1.1422 -#267 := (not #264)
  1.1423 -#26 := (uf_4 uf_5 #25)
  1.1424 -decl uf_11 :: T2
  1.1425 -#41 := uf_11
  1.1426 -#42 := (uf_4 uf_11 #25)
  1.1427 -#237 := (ite #229 #42 #26)
  1.1428 -#245 := (* -1::real #237)
  1.1429 -#246 := (+ #31 #245)
  1.1430 -#247 := (<= #246 0::real)
  1.1431 -#248 := (not #247)
  1.1432 -#272 := (and #248 #267)
  1.1433 -#40 := (< #22 #39)
  1.1434 -#49 := (ite #40 #28 #48)
  1.1435 -#50 := (< #31 #49)
  1.1436 -#43 := (ite #40 #26 #42)
  1.1437 -#44 := (< #43 #31)
  1.1438 -#51 := (and #44 #50)
  1.1439 -#273 := (iff #51 #272)
  1.1440 -#270 := (iff #50 #267)
  1.1441 -#261 := (< #31 #256)
  1.1442 -#268 := (iff #261 #267)
  1.1443 -#269 := [rewrite]: #268
  1.1444 -#262 := (iff #50 #261)
  1.1445 -#259 := (= #49 #256)
  1.1446 -#228 := (not #229)
  1.1447 -#253 := (ite #228 #28 #48)
  1.1448 -#257 := (= #253 #256)
  1.1449 -#258 := [rewrite]: #257
  1.1450 -#254 := (= #49 #253)
  1.1451 -#232 := (iff #40 #228)
  1.1452 -#233 := [rewrite]: #232
  1.1453 -#255 := [monotonicity #233]: #254
  1.1454 -#260 := [trans #255 #258]: #259
  1.1455 -#263 := [monotonicity #260]: #262
  1.1456 -#271 := [trans #263 #269]: #270
  1.1457 -#251 := (iff #44 #248)
  1.1458 -#242 := (< #237 #31)
  1.1459 -#249 := (iff #242 #248)
  1.1460 -#250 := [rewrite]: #249
  1.1461 -#243 := (iff #44 #242)
  1.1462 -#240 := (= #43 #237)
  1.1463 -#234 := (ite #228 #26 #42)
  1.1464 -#238 := (= #234 #237)
  1.1465 -#239 := [rewrite]: #238
  1.1466 -#235 := (= #43 #234)
  1.1467 -#236 := [monotonicity #233]: #235
  1.1468 -#241 := [trans #236 #239]: #240
  1.1469 -#244 := [monotonicity #241]: #243
  1.1470 -#252 := [trans #244 #250]: #251
  1.1471 -#274 := [monotonicity #252 #271]: #273
  1.1472 -#178 := [asserted]: #51
  1.1473 -#275 := [mp #178 #274]: #272
  1.1474 -#277 := [and-elim #275]: #267
  1.1475 -#196 := (* -1::real #31)
  1.1476 -#197 := (+ #28 #196)
  1.1477 -#195 := (>= #197 0::real)
  1.1478 -#193 := (not #195)
  1.1479 -#213 := (* -1::real #34)
  1.1480 -#214 := (+ #31 #213)
  1.1481 -#212 := (>= #214 0::real)
  1.1482 -#215 := (not #212)
  1.1483 -#220 := (and #193 #215)
  1.1484 -#23 := (< #22 #22)
  1.1485 -#35 := (ite #23 #28 #34)
  1.1486 -#36 := (< #31 #35)
  1.1487 -#29 := (ite #23 #26 #28)
  1.1488 -#32 := (< #29 #31)
  1.1489 -#37 := (and #32 #36)
  1.1490 -#221 := (iff #37 #220)
  1.1491 -#218 := (iff #36 #215)
  1.1492 -#209 := (< #31 #34)
  1.1493 -#216 := (iff #209 #215)
  1.1494 -#217 := [rewrite]: #216
  1.1495 -#210 := (iff #36 #209)
  1.1496 -#207 := (= #35 #34)
  1.1497 -#202 := (ite false #28 #34)
  1.1498 -#205 := (= #202 #34)
  1.1499 -#206 := [rewrite]: #205
  1.1500 -#203 := (= #35 #202)
  1.1501 -#180 := (iff #23 false)
  1.1502 -#181 := [rewrite]: #180
  1.1503 -#204 := [monotonicity #181]: #203
  1.1504 -#208 := [trans #204 #206]: #207
  1.1505 -#211 := [monotonicity #208]: #210
  1.1506 -#219 := [trans #211 #217]: #218
  1.1507 -#200 := (iff #32 #193)
  1.1508 -#189 := (< #28 #31)
  1.1509 -#198 := (iff #189 #193)
  1.1510 -#199 := [rewrite]: #198
  1.1511 -#190 := (iff #32 #189)
  1.1512 -#187 := (= #29 #28)
  1.1513 -#182 := (ite false #26 #28)
  1.1514 -#185 := (= #182 #28)
  1.1515 -#186 := [rewrite]: #185
  1.1516 -#183 := (= #29 #182)
  1.1517 -#184 := [monotonicity #181]: #183
  1.1518 -#188 := [trans #184 #186]: #187
  1.1519 -#191 := [monotonicity #188]: #190
  1.1520 -#201 := [trans #191 #199]: #200
  1.1521 -#222 := [monotonicity #201 #219]: #221
  1.1522 -#177 := [asserted]: #37
  1.1523 -#223 := [mp #177 #222]: #220
  1.1524 -#224 := [and-elim #223]: #193
  1.1525 -#644 := [th-lemma #224 #277 #643]: false
  1.1526 -#646 := [lemma #644]: #645
  1.1527 -#647 := [hypothesis]: #568
  1.1528 -#649 := (or #648 #642)
  1.1529 -#650 := [th-lemma]: #649
  1.1530 -#651 := [unit-resolution #650 #647 #646]: false
  1.1531 -#652 := [lemma #651]: #648
  1.1532 -#578 := (or #229 #568)
  1.1533 -#579 := [def-axiom]: #578
  1.1534 -#675 := [unit-resolution #579 #652]: #229
  1.1535 -#677 := [th-lemma #675 #395]: #676
  1.1536 -#679 := [symm #677]: #678
  1.1537 -#683 := [monotonicity #679]: #682
  1.1538 -#685 := [symm #683]: #684
  1.1539 -#587 := (= uf_3 #586)
  1.1540 -#591 := (or #590 #587)
  1.1541 -#592 := [quant-inst]: #591
  1.1542 -#681 := [unit-resolution #592 #547]: #587
  1.1543 -#689 := [trans #681 #685]: #688
  1.1544 -#690 := [trans #689 #687]: #45
  1.1545 -#571 := (not #45)
  1.1546 -#54 := (uf_4 uf_11 #46)
  1.1547 -#279 := (ite #45 #28 #54)
  1.1548 -#465 := (* -1::real #279)
  1.1549 -#632 := (+ #28 #465)
  1.1550 -#633 := (<= #632 0::real)
  1.1551 -#580 := (= #28 #279)
  1.1552 -#656 := [hypothesis]: #45
  1.1553 -#582 := (or #571 #580)
  1.1554 -#583 := [def-axiom]: #582
  1.1555 -#657 := [unit-resolution #583 #656]: #580
  1.1556 -#658 := (not #580)
  1.1557 -#659 := (or #658 #633)
  1.1558 -#660 := [th-lemma]: #659
  1.1559 -#661 := [unit-resolution #660 #657]: #633
  1.1560 -#57 := (uf_4 uf_8 #46)
  1.1561 -#363 := (* -1::real #57)
  1.1562 -#379 := (+ #47 #363)
  1.1563 -#380 := (<= #379 0::real)
  1.1564 -#381 := (not #380)
  1.1565 -#364 := (+ #54 #363)
  1.1566 -#362 := (>= #364 0::real)
  1.1567 -#361 := (not #362)
  1.1568 -#386 := (and #361 #381)
  1.1569 -#59 := (uf_4 uf_7 #46)
  1.1570 -#64 := (< #39 #39)
  1.1571 -#67 := (ite #64 #59 #47)
  1.1572 -#68 := (< #57 #67)
  1.1573 -#65 := (ite #64 #47 #54)
  1.1574 -#66 := (< #65 #57)
  1.1575 -#69 := (and #66 #68)
  1.1576 -#387 := (iff #69 #386)
  1.1577 -#384 := (iff #68 #381)
  1.1578 -#376 := (< #57 #47)
  1.1579 -#382 := (iff #376 #381)
  1.1580 -#383 := [rewrite]: #382
  1.1581 -#377 := (iff #68 #376)
  1.1582 -#374 := (= #67 #47)
  1.1583 -#369 := (ite false #59 #47)
  1.1584 -#372 := (= #369 #47)
  1.1585 -#373 := [rewrite]: #372
  1.1586 -#370 := (= #67 #369)
  1.1587 -#349 := (iff #64 false)
  1.1588 -#350 := [rewrite]: #349
  1.1589 -#371 := [monotonicity #350]: #370
  1.1590 -#375 := [trans #371 #373]: #374
  1.1591 -#378 := [monotonicity #375]: #377
  1.1592 -#385 := [trans #378 #383]: #384
  1.1593 -#367 := (iff #66 #361)
  1.1594 -#358 := (< #54 #57)
  1.1595 -#365 := (iff #358 #361)
  1.1596 -#366 := [rewrite]: #365
  1.1597 -#359 := (iff #66 #358)
  1.1598 -#356 := (= #65 #54)
  1.1599 -#351 := (ite false #47 #54)
  1.1600 -#354 := (= #351 #54)
  1.1601 -#355 := [rewrite]: #354
  1.1602 -#352 := (= #65 #351)
  1.1603 -#353 := [monotonicity #350]: #352
  1.1604 -#357 := [trans #353 #355]: #356
  1.1605 -#360 := [monotonicity #357]: #359
  1.1606 -#368 := [trans #360 #366]: #367
  1.1607 -#388 := [monotonicity #368 #385]: #387
  1.1608 -#346 := [asserted]: #69
  1.1609 -#389 := [mp #346 #388]: #386
  1.1610 -#391 := [and-elim #389]: #381
  1.1611 -#397 := (* -1::real #59)
  1.1612 -#398 := (+ #47 #397)
  1.1613 -#399 := (<= #398 0::real)
  1.1614 -#409 := (* -1::real #54)
  1.1615 -#410 := (+ #47 #409)
  1.1616 -#408 := (>= #410 0::real)
  1.1617 -#60 := (uf_4 uf_9 #46)
  1.1618 -#402 := (* -1::real #60)
  1.1619 -#403 := (+ #59 #402)
  1.1620 -#404 := (<= #403 0::real)
  1.1621 -#418 := (and #399 #404 #408)
  1.1622 -#73 := (<= #59 #60)
  1.1623 -#72 := (<= #47 #59)
  1.1624 -#74 := (and #72 #73)
  1.1625 -#71 := (<= #54 #47)
  1.1626 -#75 := (and #71 #74)
  1.1627 -#421 := (iff #75 #418)
  1.1628 -#412 := (and #399 #404)
  1.1629 -#415 := (and #408 #412)
  1.1630 -#419 := (iff #415 #418)
  1.1631 -#420 := [rewrite]: #419
  1.1632 -#416 := (iff #75 #415)
  1.1633 -#413 := (iff #74 #412)
  1.1634 -#405 := (iff #73 #404)
  1.1635 -#406 := [rewrite]: #405
  1.1636 -#400 := (iff #72 #399)
  1.1637 -#401 := [rewrite]: #400
  1.1638 -#414 := [monotonicity #401 #406]: #413
  1.1639 -#407 := (iff #71 #408)
  1.1640 -#411 := [rewrite]: #407
  1.1641 -#417 := [monotonicity #411 #414]: #416
  1.1642 -#422 := [trans #417 #420]: #421
  1.1643 -#348 := [asserted]: #75
  1.1644 -#423 := [mp #348 #422]: #418
  1.1645 -#424 := [and-elim #423]: #399
  1.1646 -#637 := (+ #28 #397)
  1.1647 -#639 := (>= #637 0::real)
  1.1648 -#636 := (= #28 #59)
  1.1649 -#666 := (= #59 #28)
  1.1650 -#664 := (= #46 #25)
  1.1651 -#662 := (= #25 #46)
  1.1652 -#663 := [monotonicity #656]: #662
  1.1653 -#665 := [symm #663]: #664
  1.1654 -#667 := [monotonicity #665]: #666
  1.1655 -#668 := [symm #667]: #636
  1.1656 -#669 := (not #636)
  1.1657 -#670 := (or #669 #639)
  1.1658 -#671 := [th-lemma]: #670
  1.1659 -#672 := [unit-resolution #671 #668]: #639
  1.1660 -#468 := (+ #57 #465)
  1.1661 -#471 := (<= #468 0::real)
  1.1662 -#444 := (not #471)
  1.1663 -#322 := (ite #296 #279 #47)
  1.1664 -#330 := (* -1::real #322)
  1.1665 -#331 := (+ #57 #330)
  1.1666 -#332 := (<= #331 0::real)
  1.1667 -#333 := (not #332)
  1.1668 -#445 := (iff #333 #444)
  1.1669 -#472 := (iff #332 #471)
  1.1670 -#469 := (= #331 #468)
  1.1671 -#466 := (= #330 #465)
  1.1672 -#463 := (= #322 #279)
  1.1673 -#1 := true
  1.1674 -#458 := (ite true #279 #47)
  1.1675 -#461 := (= #458 #279)
  1.1676 -#462 := [rewrite]: #461
  1.1677 -#459 := (= #322 #458)
  1.1678 -#450 := (iff #296 true)
  1.1679 -#451 := [iff-true #395]: #450
  1.1680 -#460 := [monotonicity #451]: #459
  1.1681 -#464 := [trans #460 #462]: #463
  1.1682 -#467 := [monotonicity #464]: #466
  1.1683 -#470 := [monotonicity #467]: #469
  1.1684 -#473 := [monotonicity #470]: #472
  1.1685 -#474 := [monotonicity #473]: #445
  1.1686 -#303 := (ite #296 #60 #59)
  1.1687 -#313 := (* -1::real #303)
  1.1688 -#314 := (+ #57 #313)
  1.1689 -#312 := (>= #314 0::real)
  1.1690 -#311 := (not #312)
  1.1691 -#338 := (and #311 #333)
  1.1692 -#52 := (< #39 #22)
  1.1693 -#61 := (ite #52 #59 #60)
  1.1694 -#62 := (< #57 #61)
  1.1695 -#53 := (= uf_10 uf_3)
  1.1696 -#55 := (ite #53 #28 #54)
  1.1697 -#56 := (ite #52 #47 #55)
  1.1698 -#58 := (< #56 #57)
  1.1699 -#63 := (and #58 #62)
  1.1700 -#341 := (iff #63 #338)
  1.1701 -#282 := (ite #52 #47 #279)
  1.1702 -#285 := (< #282 #57)
  1.1703 -#291 := (and #62 #285)
  1.1704 -#339 := (iff #291 #338)
  1.1705 -#336 := (iff #285 #333)
  1.1706 -#327 := (< #322 #57)
  1.1707 -#334 := (iff #327 #333)
  1.1708 -#335 := [rewrite]: #334
  1.1709 -#328 := (iff #285 #327)
  1.1710 -#325 := (= #282 #322)
  1.1711 -#297 := (not #296)
  1.1712 -#319 := (ite #297 #47 #279)
  1.1713 -#323 := (= #319 #322)
  1.1714 -#324 := [rewrite]: #323
  1.1715 -#320 := (= #282 #319)
  1.1716 -#298 := (iff #52 #297)
  1.1717 -#299 := [rewrite]: #298
  1.1718 -#321 := [monotonicity #299]: #320
  1.1719 -#326 := [trans #321 #324]: #325
  1.1720 -#329 := [monotonicity #326]: #328
  1.1721 -#337 := [trans #329 #335]: #336
  1.1722 -#317 := (iff #62 #311)
  1.1723 -#308 := (< #57 #303)
  1.1724 -#315 := (iff #308 #311)
  1.1725 -#316 := [rewrite]: #315
  1.1726 -#309 := (iff #62 #308)
  1.1727 -#306 := (= #61 #303)
  1.1728 -#300 := (ite #297 #59 #60)
  1.1729 -#304 := (= #300 #303)
  1.1730 -#305 := [rewrite]: #304
  1.1731 -#301 := (= #61 #300)
  1.1732 -#302 := [monotonicity #299]: #301
  1.1733 -#307 := [trans #302 #305]: #306
  1.1734 -#310 := [monotonicity #307]: #309
  1.1735 -#318 := [trans #310 #316]: #317
  1.1736 -#340 := [monotonicity #318 #337]: #339
  1.1737 -#294 := (iff #63 #291)
  1.1738 -#288 := (and #285 #62)
  1.1739 -#292 := (iff #288 #291)
  1.1740 -#293 := [rewrite]: #292
  1.1741 -#289 := (iff #63 #288)
  1.1742 -#286 := (iff #58 #285)
  1.1743 -#283 := (= #56 #282)
  1.1744 -#280 := (= #55 #279)
  1.1745 -#226 := (iff #53 #45)
  1.1746 -#278 := [rewrite]: #226
  1.1747 -#281 := [monotonicity #278]: #280
  1.1748 -#284 := [monotonicity #281]: #283
  1.1749 -#287 := [monotonicity #284]: #286
  1.1750 -#290 := [monotonicity #287]: #289
  1.1751 -#295 := [trans #290 #293]: #294
  1.1752 -#342 := [trans #295 #340]: #341
  1.1753 -#179 := [asserted]: #63
  1.1754 -#343 := [mp #179 #342]: #338
  1.1755 -#345 := [and-elim #343]: #333
  1.1756 -#475 := [mp #345 #474]: #444
  1.1757 -#673 := [th-lemma #475 #672 #424 #391 #661]: false
  1.1758 -#674 := [lemma #673]: #571
  1.1759 -[unit-resolution #674 #690]: false
  1.1760 -unsat
  1.1761 -IL2powemHjRpCJYwmXFxyw 211 0
  1.1762 -#2 := false
  1.1763 -#33 := 0::real
  1.1764 -decl uf_11 :: (-> T5 T6 real)
  1.1765 -decl uf_15 :: T6
  1.1766 -#28 := uf_15
  1.1767 -decl uf_16 :: T5
  1.1768 -#30 := uf_16
  1.1769 -#31 := (uf_11 uf_16 uf_15)
  1.1770 -decl uf_12 :: (-> T7 T8 T5)
  1.1771 -decl uf_14 :: T8
  1.1772 -#26 := uf_14
  1.1773 -decl uf_13 :: (-> T1 T7)
  1.1774 -decl uf_8 :: T1
  1.1775 -#16 := uf_8
  1.1776 -#25 := (uf_13 uf_8)
  1.1777 -#27 := (uf_12 #25 uf_14)
  1.1778 -#29 := (uf_11 #27 uf_15)
  1.1779 -#73 := -1::real
  1.1780 -#84 := (* -1::real #29)
  1.1781 -#85 := (+ #84 #31)
  1.1782 -#74 := (* -1::real #31)
  1.1783 -#75 := (+ #29 #74)
  1.1784 -#112 := (>= #75 0::real)
  1.1785 -#119 := (ite #112 #75 #85)
  1.1786 -#127 := (* -1::real #119)
  1.1787 -decl uf_17 :: T5
  1.1788 -#37 := uf_17
  1.1789 -#38 := (uf_11 uf_17 uf_15)
  1.1790 -#102 := -1/3::real
  1.1791 -#103 := (* -1/3::real #38)
  1.1792 -#128 := (+ #103 #127)
  1.1793 -#100 := 1/3::real
  1.1794 -#101 := (* 1/3::real #31)
  1.1795 -#129 := (+ #101 #128)
  1.1796 -#130 := (<= #129 0::real)
  1.1797 -#131 := (not #130)
  1.1798 -#40 := 3::real
  1.1799 -#39 := (- #31 #38)
  1.1800 -#41 := (/ #39 3::real)
  1.1801 -#32 := (- #29 #31)
  1.1802 -#35 := (- #32)
  1.1803 -#34 := (< #32 0::real)
  1.1804 -#36 := (ite #34 #35 #32)
  1.1805 -#42 := (< #36 #41)
  1.1806 -#136 := (iff #42 #131)
  1.1807 -#104 := (+ #101 #103)
  1.1808 -#78 := (< #75 0::real)
  1.1809 -#90 := (ite #78 #85 #75)
  1.1810 -#109 := (< #90 #104)
  1.1811 -#134 := (iff #109 #131)
  1.1812 -#124 := (< #119 #104)
  1.1813 -#132 := (iff #124 #131)
  1.1814 -#133 := [rewrite]: #132
  1.1815 -#125 := (iff #109 #124)
  1.1816 -#122 := (= #90 #119)
  1.1817 -#113 := (not #112)
  1.1818 -#116 := (ite #113 #85 #75)
  1.1819 -#120 := (= #116 #119)
  1.1820 -#121 := [rewrite]: #120
  1.1821 -#117 := (= #90 #116)
  1.1822 -#114 := (iff #78 #113)
  1.1823 -#115 := [rewrite]: #114
  1.1824 -#118 := [monotonicity #115]: #117
  1.1825 -#123 := [trans #118 #121]: #122
  1.1826 -#126 := [monotonicity #123]: #125
  1.1827 -#135 := [trans #126 #133]: #134
  1.1828 -#110 := (iff #42 #109)
  1.1829 -#107 := (= #41 #104)
  1.1830 -#93 := (* -1::real #38)
  1.1831 -#94 := (+ #31 #93)
  1.1832 -#97 := (/ #94 3::real)
  1.1833 -#105 := (= #97 #104)
  1.1834 -#106 := [rewrite]: #105
  1.1835 -#98 := (= #41 #97)
  1.1836 -#95 := (= #39 #94)
  1.1837 -#96 := [rewrite]: #95
  1.1838 -#99 := [monotonicity #96]: #98
  1.1839 -#108 := [trans #99 #106]: #107
  1.1840 -#91 := (= #36 #90)
  1.1841 -#76 := (= #32 #75)
  1.1842 -#77 := [rewrite]: #76
  1.1843 -#88 := (= #35 #85)
  1.1844 -#81 := (- #75)
  1.1845 -#86 := (= #81 #85)
  1.1846 -#87 := [rewrite]: #86
  1.1847 -#82 := (= #35 #81)
  1.1848 -#83 := [monotonicity #77]: #82
  1.1849 -#89 := [trans #83 #87]: #88
  1.1850 -#79 := (iff #34 #78)
  1.1851 -#80 := [monotonicity #77]: #79
  1.1852 -#92 := [monotonicity #80 #89 #77]: #91
  1.1853 -#111 := [monotonicity #92 #108]: #110
  1.1854 -#137 := [trans #111 #135]: #136
  1.1855 -#72 := [asserted]: #42
  1.1856 -#138 := [mp #72 #137]: #131
  1.1857 -decl uf_1 :: T1
  1.1858 -#4 := uf_1
  1.1859 -#43 := (uf_13 uf_1)
  1.1860 -#44 := (uf_12 #43 uf_14)
  1.1861 -#45 := (uf_11 #44 uf_15)
  1.1862 -#149 := (* -1::real #45)
  1.1863 -#150 := (+ #38 #149)
  1.1864 -#140 := (+ #93 #45)
  1.1865 -#161 := (<= #150 0::real)
  1.1866 -#168 := (ite #161 #140 #150)
  1.1867 -#176 := (* -1::real #168)
  1.1868 -#177 := (+ #103 #176)
  1.1869 -#178 := (+ #101 #177)
  1.1870 -#179 := (<= #178 0::real)
  1.1871 -#180 := (not #179)
  1.1872 -#46 := (- #45 #38)
  1.1873 -#48 := (- #46)
  1.1874 -#47 := (< #46 0::real)
  1.1875 -#49 := (ite #47 #48 #46)
  1.1876 -#50 := (< #49 #41)
  1.1877 -#185 := (iff #50 #180)
  1.1878 -#143 := (< #140 0::real)
  1.1879 -#155 := (ite #143 #150 #140)
  1.1880 -#158 := (< #155 #104)
  1.1881 -#183 := (iff #158 #180)
  1.1882 -#173 := (< #168 #104)
  1.1883 -#181 := (iff #173 #180)
  1.1884 -#182 := [rewrite]: #181
  1.1885 -#174 := (iff #158 #173)
  1.1886 -#171 := (= #155 #168)
  1.1887 -#162 := (not #161)
  1.1888 -#165 := (ite #162 #150 #140)
  1.1889 -#169 := (= #165 #168)
  1.1890 -#170 := [rewrite]: #169
  1.1891 -#166 := (= #155 #165)
  1.1892 -#163 := (iff #143 #162)
  1.1893 -#164 := [rewrite]: #163
  1.1894 -#167 := [monotonicity #164]: #166
  1.1895 -#172 := [trans #167 #170]: #171
  1.1896 -#175 := [monotonicity #172]: #174
  1.1897 -#184 := [trans #175 #182]: #183
  1.1898 -#159 := (iff #50 #158)
  1.1899 -#156 := (= #49 #155)
  1.1900 -#141 := (= #46 #140)
  1.1901 -#142 := [rewrite]: #141
  1.1902 -#153 := (= #48 #150)
  1.1903 -#146 := (- #140)
  1.1904 -#151 := (= #146 #150)
  1.1905 -#152 := [rewrite]: #151
  1.1906 -#147 := (= #48 #146)
  1.1907 -#148 := [monotonicity #142]: #147
  1.1908 -#154 := [trans #148 #152]: #153
  1.1909 -#144 := (iff #47 #143)
  1.1910 -#145 := [monotonicity #142]: #144
  1.1911 -#157 := [monotonicity #145 #154 #142]: #156
  1.1912 -#160 := [monotonicity #157 #108]: #159
  1.1913 -#186 := [trans #160 #184]: #185
  1.1914 -#139 := [asserted]: #50
  1.1915 -#187 := [mp #139 #186]: #180
  1.1916 -#299 := (+ #140 #176)
  1.1917 -#300 := (<= #299 0::real)
  1.1918 -#290 := (= #140 #168)
  1.1919 -#329 := [hypothesis]: #162
  1.1920 -#191 := (+ #29 #149)
  1.1921 -#192 := (<= #191 0::real)
  1.1922 -#51 := (<= #29 #45)
  1.1923 -#193 := (iff #51 #192)
  1.1924 -#194 := [rewrite]: #193
  1.1925 -#188 := [asserted]: #51
  1.1926 -#195 := [mp #188 #194]: #192
  1.1927 -#298 := (+ #75 #127)
  1.1928 -#301 := (<= #298 0::real)
  1.1929 -#284 := (= #75 #119)
  1.1930 -#302 := [hypothesis]: #113
  1.1931 -#296 := (+ #85 #127)
  1.1932 -#297 := (<= #296 0::real)
  1.1933 -#285 := (= #85 #119)
  1.1934 -#288 := (or #112 #285)
  1.1935 -#289 := [def-axiom]: #288
  1.1936 -#303 := [unit-resolution #289 #302]: #285
  1.1937 -#304 := (not #285)
  1.1938 -#305 := (or #304 #297)
  1.1939 -#306 := [th-lemma]: #305
  1.1940 -#307 := [unit-resolution #306 #303]: #297
  1.1941 -#315 := (not #290)
  1.1942 -#310 := (not #300)
  1.1943 -#311 := (or #310 #112)
  1.1944 -#308 := [hypothesis]: #300
  1.1945 -#309 := [th-lemma #308 #307 #138 #302 #187 #195]: false
  1.1946 -#312 := [lemma #309]: #311
  1.1947 -#322 := [unit-resolution #312 #302]: #310
  1.1948 -#316 := (or #315 #300)
  1.1949 -#313 := [hypothesis]: #310
  1.1950 -#314 := [hypothesis]: #290
  1.1951 -#317 := [th-lemma]: #316
  1.1952 -#318 := [unit-resolution #317 #314 #313]: false
  1.1953 -#319 := [lemma #318]: #316
  1.1954 -#323 := [unit-resolution #319 #322]: #315
  1.1955 -#292 := (or #162 #290)
  1.1956 -#293 := [def-axiom]: #292
  1.1957 -#324 := [unit-resolution #293 #323]: #162
  1.1958 -#325 := [th-lemma #324 #307 #138 #302 #195]: false
  1.1959 -#326 := [lemma #325]: #112
  1.1960 -#286 := (or #113 #284)
  1.1961 -#287 := [def-axiom]: #286
  1.1962 -#330 := [unit-resolution #287 #326]: #284
  1.1963 -#331 := (not #284)
  1.1964 -#332 := (or #331 #301)
  1.1965 -#333 := [th-lemma]: #332
  1.1966 -#334 := [unit-resolution #333 #330]: #301
  1.1967 -#335 := [th-lemma #326 #334 #195 #329 #138]: false
  1.1968 -#336 := [lemma #335]: #161
  1.1969 -#327 := [unit-resolution #293 #336]: #290
  1.1970 -#328 := [unit-resolution #319 #327]: #300
  1.1971 -[th-lemma #326 #334 #195 #328 #187 #138]: false
  1.1972 -unsat
  1.1973 -GX51o3DUO/UBS3eNP2P9kA 285 0
  1.1974 -#2 := false
  1.1975 -#7 := 0::real
  1.1976 -decl uf_4 :: real
  1.1977 -#16 := uf_4
  1.1978 -#40 := -1::real
  1.1979 -#116 := (* -1::real uf_4)
  1.1980 -decl uf_3 :: real
  1.1981 -#11 := uf_3
  1.1982 -#117 := (+ uf_3 #116)
  1.1983 -#128 := (<= #117 0::real)
  1.1984 -#129 := (not #128)
  1.1985 -#220 := 2/3::real
  1.1986 -#221 := (* 2/3::real uf_3)
  1.1987 -#222 := (+ #221 #116)
  1.1988 -decl uf_2 :: real
  1.1989 -#5 := uf_2
  1.1990 -#67 := 1/3::real
  1.1991 -#68 := (* 1/3::real uf_2)
  1.1992 -#233 := (+ #68 #222)
  1.1993 -#243 := (<= #233 0::real)
  1.1994 -#268 := (not #243)
  1.1995 -#287 := [hypothesis]: #268
  1.1996 -#41 := (* -1::real uf_2)
  1.1997 -decl uf_1 :: real
  1.1998 -#4 := uf_1
  1.1999 -#42 := (+ uf_1 #41)
  1.2000 -#79 := (>= #42 0::real)
  1.2001 -#80 := (not #79)
  1.2002 -#297 := (or #80 #243)
  1.2003 -#158 := (+ uf_1 #116)
  1.2004 -#159 := (<= #158 0::real)
  1.2005 -#22 := (<= uf_1 uf_4)
  1.2006 -#160 := (iff #22 #159)
  1.2007 -#161 := [rewrite]: #160
  1.2008 -#155 := [asserted]: #22
  1.2009 -#162 := [mp #155 #161]: #159
  1.2010 -#200 := (* 1/3::real uf_3)
  1.2011 -#198 := -4/3::real
  1.2012 -#199 := (* -4/3::real uf_2)
  1.2013 -#201 := (+ #199 #200)
  1.2014 -#202 := (+ uf_1 #201)
  1.2015 -#203 := (>= #202 0::real)
  1.2016 -#258 := (not #203)
  1.2017 -#292 := [hypothesis]: #79
  1.2018 -#293 := (or #80 #258)
  1.2019 -#69 := -1/3::real
  1.2020 -#70 := (* -1/3::real uf_3)
  1.2021 -#186 := -2/3::real
  1.2022 -#187 := (* -2/3::real uf_2)
  1.2023 -#188 := (+ #187 #70)
  1.2024 -#189 := (+ uf_1 #188)
  1.2025 -#204 := (<= #189 0::real)
  1.2026 -#205 := (ite #79 #203 #204)
  1.2027 -#210 := (not #205)
  1.2028 -#51 := (* -1::real uf_1)
  1.2029 -#52 := (+ #51 uf_2)
  1.2030 -#86 := (ite #79 #42 #52)
  1.2031 -#94 := (* -1::real #86)
  1.2032 -#95 := (+ #70 #94)
  1.2033 -#96 := (+ #68 #95)
  1.2034 -#97 := (<= #96 0::real)
  1.2035 -#98 := (not #97)
  1.2036 -#211 := (iff #98 #210)
  1.2037 -#208 := (iff #97 #205)
  1.2038 -#182 := 4/3::real
  1.2039 -#183 := (* 4/3::real uf_2)
  1.2040 -#184 := (+ #183 #70)
  1.2041 -#185 := (+ #51 #184)
  1.2042 -#190 := (ite #79 #185 #189)
  1.2043 -#195 := (<= #190 0::real)
  1.2044 -#206 := (iff #195 #205)
  1.2045 -#207 := [rewrite]: #206
  1.2046 -#196 := (iff #97 #195)
  1.2047 -#193 := (= #96 #190)
  1.2048 -#172 := (+ #41 #70)
  1.2049 -#173 := (+ uf_1 #172)
  1.2050 -#170 := (+ uf_2 #70)
  1.2051 -#171 := (+ #51 #170)
  1.2052 -#174 := (ite #79 #171 #173)
  1.2053 -#179 := (+ #68 #174)
  1.2054 -#191 := (= #179 #190)
  1.2055 -#192 := [rewrite]: #191
  1.2056 -#180 := (= #96 #179)
  1.2057 -#177 := (= #95 #174)
  1.2058 -#164 := (ite #79 #52 #42)
  1.2059 -#167 := (+ #70 #164)
  1.2060 -#175 := (= #167 #174)
  1.2061 -#176 := [rewrite]: #175
  1.2062 -#168 := (= #95 #167)
  1.2063 -#156 := (= #94 #164)
  1.2064 -#165 := [rewrite]: #156
  1.2065 -#169 := [monotonicity #165]: #168
  1.2066 -#178 := [trans #169 #176]: #177
  1.2067 -#181 := [monotonicity #178]: #180
  1.2068 -#194 := [trans #181 #192]: #193
  1.2069 -#197 := [monotonicity #194]: #196
  1.2070 -#209 := [trans #197 #207]: #208
  1.2071 -#212 := [monotonicity #209]: #211
  1.2072 -#13 := 3::real
  1.2073 -#12 := (- uf_2 uf_3)
  1.2074 -#14 := (/ #12 3::real)
  1.2075 -#6 := (- uf_1 uf_2)
  1.2076 -#9 := (- #6)
  1.2077 -#8 := (< #6 0::real)
  1.2078 -#10 := (ite #8 #9 #6)
  1.2079 -#15 := (< #10 #14)
  1.2080 -#103 := (iff #15 #98)
  1.2081 -#71 := (+ #68 #70)
  1.2082 -#45 := (< #42 0::real)
  1.2083 -#57 := (ite #45 #52 #42)
  1.2084 -#76 := (< #57 #71)
  1.2085 -#101 := (iff #76 #98)
  1.2086 -#91 := (< #86 #71)
  1.2087 -#99 := (iff #91 #98)
  1.2088 -#100 := [rewrite]: #99
  1.2089 -#92 := (iff #76 #91)
  1.2090 -#89 := (= #57 #86)
  1.2091 -#83 := (ite #80 #52 #42)
  1.2092 -#87 := (= #83 #86)
  1.2093 -#88 := [rewrite]: #87
  1.2094 -#84 := (= #57 #83)
  1.2095 -#81 := (iff #45 #80)
  1.2096 -#82 := [rewrite]: #81
  1.2097 -#85 := [monotonicity #82]: #84
  1.2098 -#90 := [trans #85 #88]: #89
  1.2099 -#93 := [monotonicity #90]: #92
  1.2100 -#102 := [trans #93 #100]: #101
  1.2101 -#77 := (iff #15 #76)
  1.2102 -#74 := (= #14 #71)
  1.2103 -#60 := (* -1::real uf_3)
  1.2104 -#61 := (+ uf_2 #60)
  1.2105 -#64 := (/ #61 3::real)
  1.2106 -#72 := (= #64 #71)
  1.2107 -#73 := [rewrite]: #72
  1.2108 -#65 := (= #14 #64)
  1.2109 -#62 := (= #12 #61)
  1.2110 -#63 := [rewrite]: #62
  1.2111 -#66 := [monotonicity #63]: #65
  1.2112 -#75 := [trans #66 #73]: #74
  1.2113 -#58 := (= #10 #57)
  1.2114 -#43 := (= #6 #42)
  1.2115 -#44 := [rewrite]: #43
  1.2116 -#55 := (= #9 #52)
  1.2117 -#48 := (- #42)
  1.2118 -#53 := (= #48 #52)
  1.2119 -#54 := [rewrite]: #53
  1.2120 -#49 := (= #9 #48)
  1.2121 -#50 := [monotonicity #44]: #49
  1.2122 -#56 := [trans #50 #54]: #55
  1.2123 -#46 := (iff #8 #45)
  1.2124 -#47 := [monotonicity #44]: #46
  1.2125 -#59 := [monotonicity #47 #56 #44]: #58
  1.2126 -#78 := [monotonicity #59 #75]: #77
  1.2127 -#104 := [trans #78 #102]: #103
  1.2128 -#39 := [asserted]: #15
  1.2129 -#105 := [mp #39 #104]: #98
  1.2130 -#213 := [mp #105 #212]: #210
  1.2131 -#259 := (or #205 #80 #258)
  1.2132 -#260 := [def-axiom]: #259
  1.2133 -#294 := [unit-resolution #260 #213]: #293
  1.2134 -#295 := [unit-resolution #294 #292]: #258
  1.2135 -#296 := [th-lemma #287 #292 #295 #162]: false
  1.2136 -#298 := [lemma #296]: #297
  1.2137 -#299 := [unit-resolution #298 #287]: #80
  1.2138 -#261 := (not #204)
  1.2139 -#281 := (or #79 #261)
  1.2140 -#262 := (or #205 #79 #261)
  1.2141 -#263 := [def-axiom]: #262
  1.2142 -#282 := [unit-resolution #263 #213]: #281
  1.2143 -#300 := [unit-resolution #282 #299]: #261
  1.2144 -#290 := (or #79 #204 #243)
  1.2145 -#276 := [hypothesis]: #261
  1.2146 -#288 := [hypothesis]: #80
  1.2147 -#289 := [th-lemma #288 #276 #162 #287]: false
  1.2148 -#291 := [lemma #289]: #290
  1.2149 -#301 := [unit-resolution #291 #300 #299 #287]: false
  1.2150 -#302 := [lemma #301]: #243
  1.2151 -#303 := (or #129 #268)
  1.2152 -#223 := (* -4/3::real uf_3)
  1.2153 -#224 := (+ #223 uf_4)
  1.2154 -#234 := (+ #68 #224)
  1.2155 -#244 := (<= #234 0::real)
  1.2156 -#245 := (ite #128 #243 #244)
  1.2157 -#250 := (not #245)
  1.2158 -#107 := (+ #60 uf_4)
  1.2159 -#135 := (ite #128 #107 #117)
  1.2160 -#143 := (* -1::real #135)
  1.2161 -#144 := (+ #70 #143)
  1.2162 -#145 := (+ #68 #144)
  1.2163 -#146 := (<= #145 0::real)
  1.2164 -#147 := (not #146)
  1.2165 -#251 := (iff #147 #250)
  1.2166 -#248 := (iff #146 #245)
  1.2167 -#235 := (ite #128 #233 #234)
  1.2168 -#240 := (<= #235 0::real)
  1.2169 -#246 := (iff #240 #245)
  1.2170 -#247 := [rewrite]: #246
  1.2171 -#241 := (iff #146 #240)
  1.2172 -#238 := (= #145 #235)
  1.2173 -#225 := (ite #128 #222 #224)
  1.2174 -#230 := (+ #68 #225)
  1.2175 -#236 := (= #230 #235)
  1.2176 -#237 := [rewrite]: #236
  1.2177 -#231 := (= #145 #230)
  1.2178 -#228 := (= #144 #225)
  1.2179 -#214 := (ite #128 #117 #107)
  1.2180 -#217 := (+ #70 #214)
  1.2181 -#226 := (= #217 #225)
  1.2182 -#227 := [rewrite]: #226
  1.2183 -#218 := (= #144 #217)
  1.2184 -#215 := (= #143 #214)
  1.2185 -#216 := [rewrite]: #215
  1.2186 -#219 := [monotonicity #216]: #218
  1.2187 -#229 := [trans #219 #227]: #228
  1.2188 -#232 := [monotonicity #229]: #231
  1.2189 -#239 := [trans #232 #237]: #238
  1.2190 -#242 := [monotonicity #239]: #241
  1.2191 -#249 := [trans #242 #247]: #248
  1.2192 -#252 := [monotonicity #249]: #251
  1.2193 -#17 := (- uf_4 uf_3)
  1.2194 -#19 := (- #17)
  1.2195 -#18 := (< #17 0::real)
  1.2196 -#20 := (ite #18 #19 #17)
  1.2197 -#21 := (< #20 #14)
  1.2198 -#152 := (iff #21 #147)
  1.2199 -#110 := (< #107 0::real)
  1.2200 -#122 := (ite #110 #117 #107)
  1.2201 -#125 := (< #122 #71)
  1.2202 -#150 := (iff #125 #147)
  1.2203 -#140 := (< #135 #71)
  1.2204 -#148 := (iff #140 #147)
  1.2205 -#149 := [rewrite]: #148
  1.2206 -#141 := (iff #125 #140)
  1.2207 -#138 := (= #122 #135)
  1.2208 -#132 := (ite #129 #117 #107)
  1.2209 -#136 := (= #132 #135)
  1.2210 -#137 := [rewrite]: #136
  1.2211 -#133 := (= #122 #132)
  1.2212 -#130 := (iff #110 #129)
  1.2213 -#131 := [rewrite]: #130
  1.2214 -#134 := [monotonicity #131]: #133
  1.2215 -#139 := [trans #134 #137]: #138
  1.2216 -#142 := [monotonicity #139]: #141
  1.2217 -#151 := [trans #142 #149]: #150
  1.2218 -#126 := (iff #21 #125)
  1.2219 -#123 := (= #20 #122)
  1.2220 -#108 := (= #17 #107)
  1.2221 -#109 := [rewrite]: #108
  1.2222 -#120 := (= #19 #117)
  1.2223 -#113 := (- #107)
  1.2224 -#118 := (= #113 #117)
  1.2225 -#119 := [rewrite]: #118
  1.2226 -#114 := (= #19 #113)
  1.2227 -#115 := [monotonicity #109]: #114
  1.2228 -#121 := [trans #115 #119]: #120
  1.2229 -#111 := (iff #18 #110)
  1.2230 -#112 := [monotonicity #109]: #111
  1.2231 -#124 := [monotonicity #112 #121 #109]: #123
  1.2232 -#127 := [monotonicity #124 #75]: #126
  1.2233 -#153 := [trans #127 #151]: #152
  1.2234 -#106 := [asserted]: #21
  1.2235 -#154 := [mp #106 #153]: #147
  1.2236 -#253 := [mp #154 #252]: #250
  1.2237 -#269 := (or #245 #129 #268)
  1.2238 -#270 := [def-axiom]: #269
  1.2239 -#304 := [unit-resolution #270 #253]: #303
  1.2240 -#305 := [unit-resolution #304 #302]: #129
  1.2241 -#271 := (not #244)
  1.2242 -#306 := (or #128 #271)
  1.2243 -#272 := (or #245 #128 #271)
  1.2244 -#273 := [def-axiom]: #272
  1.2245 -#307 := [unit-resolution #273 #253]: #306
  1.2246 -#308 := [unit-resolution #307 #305]: #271
  1.2247 -#285 := (or #128 #244)
  1.2248 -#274 := [hypothesis]: #271
  1.2249 -#275 := [hypothesis]: #129
  1.2250 -#278 := (or #204 #128 #244)
  1.2251 -#277 := [th-lemma #276 #275 #274 #162]: false
  1.2252 -#279 := [lemma #277]: #278
  1.2253 -#280 := [unit-resolution #279 #275 #274]: #204
  1.2254 -#283 := [unit-resolution #282 #280]: #79
  1.2255 -#284 := [th-lemma #275 #274 #283 #162]: false
  1.2256 -#286 := [lemma #284]: #285
  1.2257 -[unit-resolution #286 #308 #305]: false
  1.2258 -unsat
  1.2259 -cebG074uorSr8ODzgTmcKg 97 0
  1.2260 -#2 := false
  1.2261 -#18 := 0::real
  1.2262 -decl uf_1 :: (-> T2 T1 real)
  1.2263 -decl uf_5 :: T1
  1.2264 -#11 := uf_5
  1.2265 -decl uf_2 :: T2
  1.2266 -#4 := uf_2
  1.2267 -#20 := (uf_1 uf_2 uf_5)
  1.2268 -#42 := -1::real
  1.2269 -#53 := (* -1::real #20)
  1.2270 -decl uf_3 :: T2
  1.2271 -#7 := uf_3
  1.2272 -#19 := (uf_1 uf_3 uf_5)
  1.2273 -#54 := (+ #19 #53)
  1.2274 -#63 := (<= #54 0::real)
  1.2275 -#21 := (- #19 #20)
  1.2276 -#22 := (< 0::real #21)
  1.2277 -#23 := (not #22)
  1.2278 -#74 := (iff #23 #63)
  1.2279 -#57 := (< 0::real #54)
  1.2280 -#60 := (not #57)
  1.2281 -#72 := (iff #60 #63)
  1.2282 -#64 := (not #63)
  1.2283 -#67 := (not #64)
  1.2284 -#70 := (iff #67 #63)
  1.2285 -#71 := [rewrite]: #70
  1.2286 -#68 := (iff #60 #67)
  1.2287 -#65 := (iff #57 #64)
  1.2288 -#66 := [rewrite]: #65
  1.2289 -#69 := [monotonicity #66]: #68
  1.2290 -#73 := [trans #69 #71]: #72
  1.2291 -#61 := (iff #23 #60)
  1.2292 -#58 := (iff #22 #57)
  1.2293 -#55 := (= #21 #54)
  1.2294 -#56 := [rewrite]: #55
  1.2295 -#59 := [monotonicity #56]: #58
  1.2296 -#62 := [monotonicity #59]: #61
  1.2297 -#75 := [trans #62 #73]: #74
  1.2298 -#41 := [asserted]: #23
  1.2299 -#76 := [mp #41 #75]: #63
  1.2300 -#5 := (:var 0 T1)
  1.2301 -#8 := (uf_1 uf_3 #5)
  1.2302 -#141 := (pattern #8)
  1.2303 -#6 := (uf_1 uf_2 #5)
  1.2304 -#140 := (pattern #6)
  1.2305 -#45 := (* -1::real #8)
  1.2306 -#46 := (+ #6 #45)
  1.2307 -#44 := (>= #46 0::real)
  1.2308 -#43 := (not #44)
  1.2309 -#142 := (forall (vars (?x1 T1)) (:pat #140 #141) #43)
  1.2310 -#49 := (forall (vars (?x1 T1)) #43)
  1.2311 -#145 := (iff #49 #142)
  1.2312 -#143 := (iff #43 #43)
  1.2313 -#144 := [refl]: #143
  1.2314 -#146 := [quant-intro #144]: #145
  1.2315 -#80 := (~ #49 #49)
  1.2316 -#82 := (~ #43 #43)
  1.2317 -#83 := [refl]: #82
  1.2318 -#81 := [nnf-pos #83]: #80
  1.2319 -#9 := (< #6 #8)
  1.2320 -#10 := (forall (vars (?x1 T1)) #9)
  1.2321 -#50 := (iff #10 #49)
  1.2322 -#47 := (iff #9 #43)
  1.2323 -#48 := [rewrite]: #47
  1.2324 -#51 := [quant-intro #48]: #50
  1.2325 -#39 := [asserted]: #10
  1.2326 -#52 := [mp #39 #51]: #49
  1.2327 -#79 := [mp~ #52 #81]: #49
  1.2328 -#147 := [mp #79 #146]: #142
  1.2329 -#164 := (not #142)
  1.2330 -#165 := (or #164 #64)
  1.2331 -#148 := (* -1::real #19)
  1.2332 -#149 := (+ #20 #148)
  1.2333 -#150 := (>= #149 0::real)
  1.2334 -#151 := (not #150)
  1.2335 -#166 := (or #164 #151)
  1.2336 -#168 := (iff #166 #165)
  1.2337 -#170 := (iff #165 #165)
  1.2338 -#171 := [rewrite]: #170
  1.2339 -#162 := (iff #151 #64)
  1.2340 -#160 := (iff #150 #63)
  1.2341 -#152 := (+ #148 #20)
  1.2342 -#155 := (>= #152 0::real)
  1.2343 -#158 := (iff #155 #63)
  1.2344 -#159 := [rewrite]: #158
  1.2345 -#156 := (iff #150 #155)
  1.2346 -#153 := (= #149 #152)
  1.2347 -#154 := [rewrite]: #153
  1.2348 -#157 := [monotonicity #154]: #156
  1.2349 -#161 := [trans #157 #159]: #160
  1.2350 -#163 := [monotonicity #161]: #162
  1.2351 -#169 := [monotonicity #163]: #168
  1.2352 -#172 := [trans #169 #171]: #168
  1.2353 -#167 := [quant-inst]: #166
  1.2354 -#173 := [mp #167 #172]: #165
  1.2355 -[unit-resolution #173 #147 #76]: false
  1.2356 -unsat
  1.2357 -DKRtrJ2XceCkITuNwNViRw 57 0
  1.2358 -#2 := false
  1.2359 -#4 := 0::real
  1.2360 -decl uf_1 :: (-> T2 real)
  1.2361 -decl uf_2 :: (-> T1 T1 T2)
  1.2362 -decl uf_12 :: (-> T4 T1)
  1.2363 -decl uf_4 :: T4
  1.2364 -#11 := uf_4
  1.2365 -#39 := (uf_12 uf_4)
  1.2366 -decl uf_10 :: T4
  1.2367 -#27 := uf_10
  1.2368 -#38 := (uf_12 uf_10)
  1.2369 -#40 := (uf_2 #38 #39)
  1.2370 -#41 := (uf_1 #40)
  1.2371 -#264 := (>= #41 0::real)
  1.2372 -#266 := (not #264)
  1.2373 -#43 := (= #41 0::real)
  1.2374 -#44 := (not #43)
  1.2375 -#131 := [asserted]: #44
  1.2376 -#272 := (or #43 #266)
  1.2377 -#42 := (<= #41 0::real)
  1.2378 -#130 := [asserted]: #42
  1.2379 -#265 := (not #42)
  1.2380 -#270 := (or #43 #265 #266)
  1.2381 -#271 := [th-lemma]: #270
  1.2382 -#273 := [unit-resolution #271 #130]: #272
  1.2383 -#274 := [unit-resolution #273 #131]: #266
  1.2384 -#6 := (:var 0 T1)
  1.2385 -#5 := (:var 1 T1)
  1.2386 -#7 := (uf_2 #5 #6)
  1.2387 -#241 := (pattern #7)
  1.2388 -#8 := (uf_1 #7)
  1.2389 -#65 := (>= #8 0::real)
  1.2390 -#242 := (forall (vars (?x1 T1) (?x2 T1)) (:pat #241) #65)
  1.2391 -#66 := (forall (vars (?x1 T1) (?x2 T1)) #65)
  1.2392 -#245 := (iff #66 #242)
  1.2393 -#243 := (iff #65 #65)
  1.2394 -#244 := [refl]: #243
  1.2395 -#246 := [quant-intro #244]: #245
  1.2396 -#149 := (~ #66 #66)
  1.2397 -#151 := (~ #65 #65)
  1.2398 -#152 := [refl]: #151
  1.2399 -#150 := [nnf-pos #152]: #149
  1.2400 -#9 := (<= 0::real #8)
  1.2401 -#10 := (forall (vars (?x1 T1) (?x2 T1)) #9)
  1.2402 -#67 := (iff #10 #66)
  1.2403 -#63 := (iff #9 #65)
  1.2404 -#64 := [rewrite]: #63
  1.2405 -#68 := [quant-intro #64]: #67
  1.2406 -#60 := [asserted]: #10
  1.2407 -#69 := [mp #60 #68]: #66
  1.2408 -#147 := [mp~ #69 #150]: #66
  1.2409 -#247 := [mp #147 #246]: #242
  1.2410 -#267 := (not #242)
  1.2411 -#268 := (or #267 #264)
  1.2412 -#269 := [quant-inst]: #268
  1.2413 -[unit-resolution #269 #247 #274]: false
  1.2414 -unsat
  1.2415 -97KJAJfUio+nGchEHWvgAw 91 0
  1.2416 -#2 := false
  1.2417 -#38 := 0::real
  1.2418 -decl uf_1 :: (-> T1 T2 real)
  1.2419 -decl uf_3 :: T2
  1.2420 -#5 := uf_3
  1.2421 -decl uf_4 :: T1
  1.2422 -#7 := uf_4
  1.2423 -#8 := (uf_1 uf_4 uf_3)
  1.2424 -#35 := -1::real
  1.2425 -#36 := (* -1::real #8)
  1.2426 -decl uf_2 :: T1
  1.2427 -#4 := uf_2
  1.2428 -#6 := (uf_1 uf_2 uf_3)
  1.2429 -#37 := (+ #6 #36)
  1.2430 -#130 := (>= #37 0::real)
  1.2431 -#155 := (not #130)
  1.2432 -#43 := (= #6 #8)
  1.2433 -#55 := (not #43)
  1.2434 -#15 := (= #8 #6)
  1.2435 -#16 := (not #15)
  1.2436 -#56 := (iff #16 #55)
  1.2437 -#53 := (iff #15 #43)
  1.2438 -#54 := [rewrite]: #53
  1.2439 -#57 := [monotonicity #54]: #56
  1.2440 -#34 := [asserted]: #16
  1.2441 -#60 := [mp #34 #57]: #55
  1.2442 -#158 := (or #43 #155)
  1.2443 -#39 := (<= #37 0::real)
  1.2444 -#9 := (<= #6 #8)
  1.2445 -#40 := (iff #9 #39)
  1.2446 -#41 := [rewrite]: #40
  1.2447 -#32 := [asserted]: #9
  1.2448 -#42 := [mp #32 #41]: #39
  1.2449 -#154 := (not #39)
  1.2450 -#156 := (or #43 #154 #155)
  1.2451 -#157 := [th-lemma]: #156
  1.2452 -#159 := [unit-resolution #157 #42]: #158
  1.2453 -#160 := [unit-resolution #159 #60]: #155
  1.2454 -#10 := (:var 0 T2)
  1.2455 -#12 := (uf_1 uf_2 #10)
  1.2456 -#123 := (pattern #12)
  1.2457 -#11 := (uf_1 uf_4 #10)
  1.2458 -#122 := (pattern #11)
  1.2459 -#44 := (* -1::real #12)
  1.2460 -#45 := (+ #11 #44)
  1.2461 -#46 := (<= #45 0::real)
  1.2462 -#124 := (forall (vars (?x1 T2)) (:pat #122 #123) #46)
  1.2463 -#49 := (forall (vars (?x1 T2)) #46)
  1.2464 -#127 := (iff #49 #124)
  1.2465 -#125 := (iff #46 #46)
  1.2466 -#126 := [refl]: #125
  1.2467 -#128 := [quant-intro #126]: #127
  1.2468 -#62 := (~ #49 #49)
  1.2469 -#64 := (~ #46 #46)
  1.2470 -#65 := [refl]: #64
  1.2471 -#63 := [nnf-pos #65]: #62
  1.2472 -#13 := (<= #11 #12)
  1.2473 -#14 := (forall (vars (?x1 T2)) #13)
  1.2474 -#50 := (iff #14 #49)
  1.2475 -#47 := (iff #13 #46)
  1.2476 -#48 := [rewrite]: #47
  1.2477 -#51 := [quant-intro #48]: #50
  1.2478 -#33 := [asserted]: #14
  1.2479 -#52 := [mp #33 #51]: #49
  1.2480 -#61 := [mp~ #52 #63]: #49
  1.2481 -#129 := [mp #61 #128]: #124
  1.2482 -#144 := (not #124)
  1.2483 -#145 := (or #144 #130)
  1.2484 -#131 := (* -1::real #6)
  1.2485 -#132 := (+ #8 #131)
  1.2486 -#133 := (<= #132 0::real)
  1.2487 -#146 := (or #144 #133)
  1.2488 -#148 := (iff #146 #145)
  1.2489 -#150 := (iff #145 #145)
  1.2490 -#151 := [rewrite]: #150
  1.2491 -#142 := (iff #133 #130)
  1.2492 -#134 := (+ #131 #8)
  1.2493 -#137 := (<= #134 0::real)
  1.2494 -#140 := (iff #137 #130)
  1.2495 -#141 := [rewrite]: #140
  1.2496 -#138 := (iff #133 #137)
  1.2497 -#135 := (= #132 #134)
  1.2498 -#136 := [rewrite]: #135
  1.2499 -#139 := [monotonicity #136]: #138
  1.2500 -#143 := [trans #139 #141]: #142
  1.2501 -#149 := [monotonicity #143]: #148
  1.2502 -#152 := [trans #149 #151]: #148
  1.2503 -#147 := [quant-inst]: #146
  1.2504 -#153 := [mp #147 #152]: #145
  1.2505 -[unit-resolution #153 #129 #160]: false
  1.2506 -unsat
  1.2507 -flJYbeWfe+t2l/zsRqdujA 149 0
  1.2508 -#2 := false
  1.2509 -#19 := 0::real
  1.2510 -decl uf_1 :: (-> T1 T2 real)
  1.2511 -decl uf_3 :: T2
  1.2512 -#5 := uf_3
  1.2513 -decl uf_4 :: T1
  1.2514 -#7 := uf_4
  1.2515 -#8 := (uf_1 uf_4 uf_3)
  1.2516 -#44 := -1::real
  1.2517 -#156 := (* -1::real #8)
  1.2518 -decl uf_2 :: T1
  1.2519 -#4 := uf_2
  1.2520 -#6 := (uf_1 uf_2 uf_3)
  1.2521 -#203 := (+ #6 #156)
  1.2522 -#205 := (>= #203 0::real)
  1.2523 -#9 := (= #6 #8)
  1.2524 -#40 := [asserted]: #9
  1.2525 -#208 := (not #9)
  1.2526 -#209 := (or #208 #205)
  1.2527 -#210 := [th-lemma]: #209
  1.2528 -#211 := [unit-resolution #210 #40]: #205
  1.2529 -decl uf_5 :: T1
  1.2530 -#12 := uf_5
  1.2531 -#22 := (uf_1 uf_5 uf_3)
  1.2532 -#160 := (* -1::real #22)
  1.2533 -#161 := (+ #6 #160)
  1.2534 -#207 := (>= #161 0::real)
  1.2535 -#222 := (not #207)
  1.2536 -#206 := (= #6 #22)
  1.2537 -#216 := (not #206)
  1.2538 -#62 := (= #8 #22)
  1.2539 -#70 := (not #62)
  1.2540 -#217 := (iff #70 #216)
  1.2541 -#214 := (iff #62 #206)
  1.2542 -#212 := (iff #206 #62)
  1.2543 -#213 := [monotonicity #40]: #212
  1.2544 -#215 := [symm #213]: #214
  1.2545 -#218 := [monotonicity #215]: #217
  1.2546 -#23 := (= #22 #8)
  1.2547 -#24 := (not #23)
  1.2548 -#71 := (iff #24 #70)
  1.2549 -#68 := (iff #23 #62)
  1.2550 -#69 := [rewrite]: #68
  1.2551 -#72 := [monotonicity #69]: #71
  1.2552 -#43 := [asserted]: #24
  1.2553 -#75 := [mp #43 #72]: #70
  1.2554 -#219 := [mp #75 #218]: #216
  1.2555 -#225 := (or #206 #222)
  1.2556 -#162 := (<= #161 0::real)
  1.2557 -#172 := (+ #8 #160)
  1.2558 -#173 := (>= #172 0::real)
  1.2559 -#178 := (not #173)
  1.2560 -#163 := (not #162)
  1.2561 -#181 := (or #163 #178)
  1.2562 -#184 := (not #181)
  1.2563 -#10 := (:var 0 T2)
  1.2564 -#15 := (uf_1 uf_4 #10)
  1.2565 -#149 := (pattern #15)
  1.2566 -#13 := (uf_1 uf_5 #10)
  1.2567 -#148 := (pattern #13)
  1.2568 -#11 := (uf_1 uf_2 #10)
  1.2569 -#147 := (pattern #11)
  1.2570 -#50 := (* -1::real #15)
  1.2571 -#51 := (+ #13 #50)
  1.2572 -#52 := (<= #51 0::real)
  1.2573 -#76 := (not #52)
  1.2574 -#45 := (* -1::real #13)
  1.2575 -#46 := (+ #11 #45)
  1.2576 -#47 := (<= #46 0::real)
  1.2577 -#78 := (not #47)
  1.2578 -#73 := (or #78 #76)
  1.2579 -#83 := (not #73)
  1.2580 -#150 := (forall (vars (?x1 T2)) (:pat #147 #148 #149) #83)
  1.2581 -#86 := (forall (vars (?x1 T2)) #83)
  1.2582 -#153 := (iff #86 #150)
  1.2583 -#151 := (iff #83 #83)
  1.2584 -#152 := [refl]: #151
  1.2585 -#154 := [quant-intro #152]: #153
  1.2586 -#55 := (and #47 #52)
  1.2587 -#58 := (forall (vars (?x1 T2)) #55)
  1.2588 -#87 := (iff #58 #86)
  1.2589 -#84 := (iff #55 #83)
  1.2590 -#85 := [rewrite]: #84
  1.2591 -#88 := [quant-intro #85]: #87
  1.2592 -#79 := (~ #58 #58)
  1.2593 -#81 := (~ #55 #55)
  1.2594 -#82 := [refl]: #81
  1.2595 -#80 := [nnf-pos #82]: #79
  1.2596 -#16 := (<= #13 #15)
  1.2597 -#14 := (<= #11 #13)
  1.2598 -#17 := (and #14 #16)
  1.2599 -#18 := (forall (vars (?x1 T2)) #17)
  1.2600 -#59 := (iff #18 #58)
  1.2601 -#56 := (iff #17 #55)
  1.2602 -#53 := (iff #16 #52)
  1.2603 -#54 := [rewrite]: #53
  1.2604 -#48 := (iff #14 #47)
  1.2605 -#49 := [rewrite]: #48
  1.2606 -#57 := [monotonicity #49 #54]: #56
  1.2607 -#60 := [quant-intro #57]: #59
  1.2608 -#41 := [asserted]: #18
  1.2609 -#61 := [mp #41 #60]: #58
  1.2610 -#77 := [mp~ #61 #80]: #58
  1.2611 -#89 := [mp #77 #88]: #86
  1.2612 -#155 := [mp #89 #154]: #150
  1.2613 -#187 := (not #150)
  1.2614 -#188 := (or #187 #184)
  1.2615 -#157 := (+ #22 #156)
  1.2616 -#158 := (<= #157 0::real)
  1.2617 -#159 := (not #158)
  1.2618 -#164 := (or #163 #159)
  1.2619 -#165 := (not #164)
  1.2620 -#189 := (or #187 #165)
  1.2621 -#191 := (iff #189 #188)
  1.2622 -#193 := (iff #188 #188)
  1.2623 -#194 := [rewrite]: #193
  1.2624 -#185 := (iff #165 #184)
  1.2625 -#182 := (iff #164 #181)
  1.2626 -#179 := (iff #159 #178)
  1.2627 -#176 := (iff #158 #173)
  1.2628 -#166 := (+ #156 #22)
  1.2629 -#169 := (<= #166 0::real)
  1.2630 -#174 := (iff #169 #173)
  1.2631 -#175 := [rewrite]: #174
  1.2632 -#170 := (iff #158 #169)
  1.2633 -#167 := (= #157 #166)
  1.2634 -#168 := [rewrite]: #167
  1.2635 -#171 := [monotonicity #168]: #170
  1.2636 -#177 := [trans #171 #175]: #176
  1.2637 -#180 := [monotonicity #177]: #179
  1.2638 -#183 := [monotonicity #180]: #182
  1.2639 -#186 := [monotonicity #183]: #185
  1.2640 -#192 := [monotonicity #186]: #191
  1.2641 -#195 := [trans #192 #194]: #191
  1.2642 -#190 := [quant-inst]: #189
  1.2643 -#196 := [mp #190 #195]: #188
  1.2644 -#220 := [unit-resolution #196 #155]: #184
  1.2645 -#197 := (or #181 #162)
  1.2646 -#198 := [def-axiom]: #197
  1.2647 -#221 := [unit-resolution #198 #220]: #162
  1.2648 -#223 := (or #206 #163 #222)
  1.2649 -#224 := [th-lemma]: #223
  1.2650 -#226 := [unit-resolution #224 #221]: #225
  1.2651 -#227 := [unit-resolution #226 #219]: #222
  1.2652 -#199 := (or #181 #173)
  1.2653 -#200 := [def-axiom]: #199
  1.2654 -#228 := [unit-resolution #200 #220]: #173
  1.2655 -[th-lemma #228 #227 #211]: false
  1.2656 -unsat
  1.2657 -rbrrQuQfaijtLkQizgEXnQ 222 0
  1.2658 -#2 := false
  1.2659 -#4 := 0::real
  1.2660 -decl uf_2 :: (-> T2 T1 real)
  1.2661 -decl uf_5 :: T1
  1.2662 -#15 := uf_5
  1.2663 -decl uf_3 :: T2
  1.2664 -#7 := uf_3
  1.2665 -#20 := (uf_2 uf_3 uf_5)
  1.2666 -decl uf_6 :: T2
  1.2667 -#17 := uf_6
  1.2668 -#18 := (uf_2 uf_6 uf_5)
  1.2669 -#59 := -1::real
  1.2670 -#73 := (* -1::real #18)
  1.2671 -#106 := (+ #73 #20)
  1.2672 -decl uf_1 :: real
  1.2673 -#5 := uf_1
  1.2674 -#78 := (* -1::real #20)
  1.2675 -#79 := (+ #18 #78)
  1.2676 -#144 := (+ uf_1 #79)
  1.2677 -#145 := (<= #144 0::real)
  1.2678 -#148 := (ite #145 uf_1 #106)
  1.2679 -#279 := (* -1::real #148)
  1.2680 -#280 := (+ uf_1 #279)
  1.2681 -#281 := (<= #280 0::real)
  1.2682 -#289 := (not #281)
  1.2683 -#72 := 1/2::real
  1.2684 -#151 := (* 1/2::real #148)
  1.2685 -#248 := (<= #151 0::real)
  1.2686 -#162 := (= #151 0::real)
  1.2687 -#24 := 2::real
  1.2688 -#27 := (- #20 #18)
  1.2689 -#28 := (<= uf_1 #27)
  1.2690 -#29 := (ite #28 uf_1 #27)
  1.2691 -#30 := (/ #29 2::real)
  1.2692 -#31 := (+ #18 #30)
  1.2693 -#32 := (= #31 #18)
  1.2694 -#33 := (not #32)
  1.2695 -#34 := (not #33)
  1.2696 -#165 := (iff #34 #162)
  1.2697 -#109 := (<= uf_1 #106)
  1.2698 -#112 := (ite #109 uf_1 #106)
  1.2699 -#118 := (* 1/2::real #112)
  1.2700 -#123 := (+ #18 #118)
  1.2701 -#129 := (= #18 #123)
  1.2702 -#163 := (iff #129 #162)
  1.2703 -#154 := (+ #18 #151)
  1.2704 -#157 := (= #18 #154)
  1.2705 -#160 := (iff #157 #162)
  1.2706 -#161 := [rewrite]: #160
  1.2707 -#158 := (iff #129 #157)
  1.2708 -#155 := (= #123 #154)
  1.2709 -#152 := (= #118 #151)
  1.2710 -#149 := (= #112 #148)
  1.2711 -#146 := (iff #109 #145)
  1.2712 -#147 := [rewrite]: #146
  1.2713 -#150 := [monotonicity #147]: #149
  1.2714 -#153 := [monotonicity #150]: #152
  1.2715 -#156 := [monotonicity #153]: #155
  1.2716 -#159 := [monotonicity #156]: #158
  1.2717 -#164 := [trans #159 #161]: #163
  1.2718 -#142 := (iff #34 #129)
  1.2719 -#134 := (not #129)
  1.2720 -#137 := (not #134)
  1.2721 -#140 := (iff #137 #129)
  1.2722 -#141 := [rewrite]: #140
  1.2723 -#138 := (iff #34 #137)
  1.2724 -#135 := (iff #33 #134)
  1.2725 -#132 := (iff #32 #129)
  1.2726 -#126 := (= #123 #18)
  1.2727 -#130 := (iff #126 #129)
  1.2728 -#131 := [rewrite]: #130
  1.2729 -#127 := (iff #32 #126)
  1.2730 -#124 := (= #31 #123)
  1.2731 -#121 := (= #30 #118)
  1.2732 -#115 := (/ #112 2::real)
  1.2733 -#119 := (= #115 #118)
  1.2734 -#120 := [rewrite]: #119
  1.2735 -#116 := (= #30 #115)
  1.2736 -#113 := (= #29 #112)
  1.2737 -#107 := (= #27 #106)
  1.2738 -#108 := [rewrite]: #107
  1.2739 -#110 := (iff #28 #109)
  1.2740 -#111 := [monotonicity #108]: #110
  1.2741 -#114 := [monotonicity #111 #108]: #113
  1.2742 -#117 := [monotonicity #114]: #116
  1.2743 -#122 := [trans #117 #120]: #121
  1.2744 -#125 := [monotonicity #122]: #124
  1.2745 -#128 := [monotonicity #125]: #127
  1.2746 -#133 := [trans #128 #131]: #132
  1.2747 -#136 := [monotonicity #133]: #135
  1.2748 -#139 := [monotonicity #136]: #138
  1.2749 -#143 := [trans #139 #141]: #142
  1.2750 -#166 := [trans #143 #164]: #165
  1.2751 -#105 := [asserted]: #34
  1.2752 -#167 := [mp #105 #166]: #162
  1.2753 -#283 := (not #162)
  1.2754 -#284 := (or #283 #248)
  1.2755 -#285 := [th-lemma]: #284
  1.2756 -#286 := [unit-resolution #285 #167]: #248
  1.2757 -#287 := [hypothesis]: #281
  1.2758 -#53 := (<= uf_1 0::real)
  1.2759 -#54 := (not #53)
  1.2760 -#6 := (< 0::real uf_1)
  1.2761 -#55 := (iff #6 #54)
  1.2762 -#56 := [rewrite]: #55
  1.2763 -#50 := [asserted]: #6
  1.2764 -#57 := [mp #50 #56]: #54
  1.2765 -#288 := [th-lemma #57 #287 #286]: false
  1.2766 -#290 := [lemma #288]: #289
  1.2767 -#241 := (= uf_1 #148)
  1.2768 -#242 := (= #106 #148)
  1.2769 -#299 := (not #242)
  1.2770 -#282 := (+ #106 #279)
  1.2771 -#291 := (<= #282 0::real)
  1.2772 -#296 := (not #291)
  1.2773 -decl uf_4 :: T2
  1.2774 -#10 := uf_4
  1.2775 -#16 := (uf_2 uf_4 uf_5)
  1.2776 -#260 := (+ #16 #78)
  1.2777 -#261 := (>= #260 0::real)
  1.2778 -#266 := (not #261)
  1.2779 -#8 := (:var 0 T1)
  1.2780 -#11 := (uf_2 uf_4 #8)
  1.2781 -#234 := (pattern #11)
  1.2782 -#9 := (uf_2 uf_3 #8)
  1.2783 -#233 := (pattern #9)
  1.2784 -#60 := (* -1::real #11)
  1.2785 -#61 := (+ #9 #60)
  1.2786 -#62 := (<= #61 0::real)
  1.2787 -#179 := (not #62)
  1.2788 -#235 := (forall (vars (?x1 T1)) (:pat #233 #234) #179)
  1.2789 -#178 := (forall (vars (?x1 T1)) #179)
  1.2790 -#238 := (iff #178 #235)
  1.2791 -#236 := (iff #179 #179)
  1.2792 -#237 := [refl]: #236
  1.2793 -#239 := [quant-intro #237]: #238
  1.2794 -#65 := (exists (vars (?x1 T1)) #62)
  1.2795 -#68 := (not #65)
  1.2796 -#175 := (~ #68 #178)
  1.2797 -#180 := (~ #179 #179)
  1.2798 -#177 := [refl]: #180
  1.2799 -#176 := [nnf-neg #177]: #175
  1.2800 -#12 := (<= #9 #11)
  1.2801 -#13 := (exists (vars (?x1 T1)) #12)
  1.2802 -#14 := (not #13)
  1.2803 -#69 := (iff #14 #68)
  1.2804 -#66 := (iff #13 #65)
  1.2805 -#63 := (iff #12 #62)
  1.2806 -#64 := [rewrite]: #63
  1.2807 -#67 := [quant-intro #64]: #66
  1.2808 -#70 := [monotonicity #67]: #69
  1.2809 -#51 := [asserted]: #14
  1.2810 -#71 := [mp #51 #70]: #68
  1.2811 -#173 := [mp~ #71 #176]: #178
  1.2812 -#240 := [mp #173 #239]: #235
  1.2813 -#269 := (not #235)
  1.2814 -#270 := (or #269 #266)
  1.2815 -#250 := (* -1::real #16)
  1.2816 -#251 := (+ #20 #250)
  1.2817 -#252 := (<= #251 0::real)
  1.2818 -#253 := (not #252)
  1.2819 -#271 := (or #269 #253)
  1.2820 -#273 := (iff #271 #270)
  1.2821 -#275 := (iff #270 #270)
  1.2822 -#276 := [rewrite]: #275
  1.2823 -#267 := (iff #253 #266)
  1.2824 -#264 := (iff #252 #261)
  1.2825 -#254 := (+ #250 #20)
  1.2826 -#257 := (<= #254 0::real)
  1.2827 -#262 := (iff #257 #261)
  1.2828 -#263 := [rewrite]: #262
  1.2829 -#258 := (iff #252 #257)
  1.2830 -#255 := (= #251 #254)
  1.2831 -#256 := [rewrite]: #255
  1.2832 -#259 := [monotonicity #256]: #258
  1.2833 -#265 := [trans #259 #263]: #264
  1.2834 -#268 := [monotonicity #265]: #267
  1.2835 -#274 := [monotonicity #268]: #273
  1.2836 -#277 := [trans #274 #276]: #273
  1.2837 -#272 := [quant-inst]: #271
  1.2838 -#278 := [mp #272 #277]: #270
  1.2839 -#293 := [unit-resolution #278 #240]: #266
  1.2840 -#90 := (* 1/2::real #20)
  1.2841 -#102 := (+ #73 #90)
  1.2842 -#89 := (* 1/2::real #16)
  1.2843 -#103 := (+ #89 #102)
  1.2844 -#100 := (>= #103 0::real)
  1.2845 -#23 := (+ #16 #20)
  1.2846 -#25 := (/ #23 2::real)
  1.2847 -#26 := (<= #18 #25)
  1.2848 -#98 := (iff #26 #100)
  1.2849 -#91 := (+ #89 #90)
  1.2850 -#94 := (<= #18 #91)
  1.2851 -#97 := (iff #94 #100)
  1.2852 -#99 := [rewrite]: #97
  1.2853 -#95 := (iff #26 #94)
  1.2854 -#92 := (= #25 #91)
  1.2855 -#93 := [rewrite]: #92
  1.2856 -#96 := [monotonicity #93]: #95
  1.2857 -#101 := [trans #96 #99]: #98
  1.2858 -#58 := [asserted]: #26
  1.2859 -#104 := [mp #58 #101]: #100
  1.2860 -#294 := [hypothesis]: #291
  1.2861 -#295 := [th-lemma #294 #104 #293 #286]: false
  1.2862 -#297 := [lemma #295]: #296
  1.2863 -#298 := [hypothesis]: #242
  1.2864 -#300 := (or #299 #291)
  1.2865 -#301 := [th-lemma]: #300
  1.2866 -#302 := [unit-resolution #301 #298 #297]: false
  1.2867 -#303 := [lemma #302]: #299
  1.2868 -#246 := (or #145 #242)
  1.2869 -#247 := [def-axiom]: #246
  1.2870 -#304 := [unit-resolution #247 #303]: #145
  1.2871 -#243 := (not #145)
  1.2872 -#244 := (or #243 #241)
  1.2873 -#245 := [def-axiom]: #244
  1.2874 -#305 := [unit-resolution #245 #304]: #241
  1.2875 -#306 := (not #241)
  1.2876 -#307 := (or #306 #281)
  1.2877 -#308 := [th-lemma]: #307
  1.2878 -[unit-resolution #308 #305 #290]: false
  1.2879 -unsat
  1.2880 -hwh3oeLAWt56hnKIa8Wuow 248 0
  1.2881 -#2 := false
  1.2882 -#4 := 0::real
  1.2883 -decl uf_2 :: (-> T2 T1 real)
  1.2884 -decl uf_5 :: T1
  1.2885 -#15 := uf_5
  1.2886 -decl uf_6 :: T2
  1.2887 -#17 := uf_6
  1.2888 -#18 := (uf_2 uf_6 uf_5)
  1.2889 -decl uf_4 :: T2
  1.2890 -#10 := uf_4
  1.2891 -#16 := (uf_2 uf_4 uf_5)
  1.2892 -#66 := -1::real
  1.2893 -#137 := (* -1::real #16)
  1.2894 -#138 := (+ #137 #18)
  1.2895 -decl uf_1 :: real
  1.2896 -#5 := uf_1
  1.2897 -#80 := (* -1::real #18)
  1.2898 -#81 := (+ #16 #80)
  1.2899 -#201 := (+ uf_1 #81)
  1.2900 -#202 := (<= #201 0::real)
  1.2901 -#205 := (ite #202 uf_1 #138)
  1.2902 -#352 := (* -1::real #205)
  1.2903 -#353 := (+ uf_1 #352)
  1.2904 -#354 := (<= #353 0::real)
  1.2905 -#362 := (not #354)
  1.2906 -#79 := 1/2::real
  1.2907 -#244 := (* 1/2::real #205)
  1.2908 -#322 := (<= #244 0::real)
  1.2909 -#245 := (= #244 0::real)
  1.2910 -#158 := -1/2::real
  1.2911 -#208 := (* -1/2::real #205)
  1.2912 -#211 := (+ #18 #208)
  1.2913 -decl uf_3 :: T2
  1.2914 -#7 := uf_3
  1.2915 -#20 := (uf_2 uf_3 uf_5)
  1.2916 -#117 := (+ #80 #20)
  1.2917 -#85 := (* -1::real #20)
  1.2918 -#86 := (+ #18 #85)
  1.2919 -#188 := (+ uf_1 #86)
  1.2920 -#189 := (<= #188 0::real)
  1.2921 -#192 := (ite #189 uf_1 #117)
  1.2922 -#195 := (* 1/2::real #192)
  1.2923 -#198 := (+ #18 #195)
  1.2924 -#97 := (* 1/2::real #20)
  1.2925 -#109 := (+ #80 #97)
  1.2926 -#96 := (* 1/2::real #16)
  1.2927 -#110 := (+ #96 #109)
  1.2928 -#107 := (>= #110 0::real)
  1.2929 -#214 := (ite #107 #198 #211)
  1.2930 -#217 := (= #18 #214)
  1.2931 -#248 := (iff #217 #245)
  1.2932 -#241 := (= #18 #211)
  1.2933 -#246 := (iff #241 #245)
  1.2934 -#247 := [rewrite]: #246
  1.2935 -#242 := (iff #217 #241)
  1.2936 -#239 := (= #214 #211)
  1.2937 -#234 := (ite false #198 #211)
  1.2938 -#237 := (= #234 #211)
  1.2939 -#238 := [rewrite]: #237
  1.2940 -#235 := (= #214 #234)
  1.2941 -#232 := (iff #107 false)
  1.2942 -#104 := (not #107)
  1.2943 -#24 := 2::real
  1.2944 -#23 := (+ #16 #20)
  1.2945 -#25 := (/ #23 2::real)
  1.2946 -#26 := (< #25 #18)
  1.2947 -#108 := (iff #26 #104)
  1.2948 -#98 := (+ #96 #97)
  1.2949 -#101 := (< #98 #18)
  1.2950 -#106 := (iff #101 #104)
  1.2951 -#105 := [rewrite]: #106
  1.2952 -#102 := (iff #26 #101)
  1.2953 -#99 := (= #25 #98)
  1.2954 -#100 := [rewrite]: #99
  1.2955 -#103 := [monotonicity #100]: #102
  1.2956 -#111 := [trans #103 #105]: #108
  1.2957 -#65 := [asserted]: #26
  1.2958 -#112 := [mp #65 #111]: #104
  1.2959 -#233 := [iff-false #112]: #232
  1.2960 -#236 := [monotonicity #233]: #235
  1.2961 -#240 := [trans #236 #238]: #239
  1.2962 -#243 := [monotonicity #240]: #242
  1.2963 -#249 := [trans #243 #247]: #248
  1.2964 -#33 := (- #18 #16)
  1.2965 -#34 := (<= uf_1 #33)
  1.2966 -#35 := (ite #34 uf_1 #33)
  1.2967 -#36 := (/ #35 2::real)
  1.2968 -#37 := (- #18 #36)
  1.2969 -#28 := (- #20 #18)
  1.2970 -#29 := (<= uf_1 #28)
  1.2971 -#30 := (ite #29 uf_1 #28)
  1.2972 -#31 := (/ #30 2::real)
  1.2973 -#32 := (+ #18 #31)
  1.2974 -#27 := (<= #18 #25)
  1.2975 -#38 := (ite #27 #32 #37)
  1.2976 -#39 := (= #38 #18)
  1.2977 -#40 := (not #39)
  1.2978 -#41 := (not #40)
  1.2979 -#220 := (iff #41 #217)
  1.2980 -#141 := (<= uf_1 #138)
  1.2981 -#144 := (ite #141 uf_1 #138)
  1.2982 -#159 := (* -1/2::real #144)
  1.2983 -#160 := (+ #18 #159)
  1.2984 -#120 := (<= uf_1 #117)
  1.2985 -#123 := (ite #120 uf_1 #117)
  1.2986 -#129 := (* 1/2::real #123)
  1.2987 -#134 := (+ #18 #129)
  1.2988 -#114 := (<= #18 #98)
  1.2989 -#165 := (ite #114 #134 #160)
  1.2990 -#171 := (= #18 #165)
  1.2991 -#218 := (iff #171 #217)
  1.2992 -#215 := (= #165 #214)
  1.2993 -#212 := (= #160 #211)
  1.2994 -#209 := (= #159 #208)
  1.2995 -#206 := (= #144 #205)
  1.2996 -#203 := (iff #141 #202)
  1.2997 -#204 := [rewrite]: #203
  1.2998 -#207 := [monotonicity #204]: #206
  1.2999 -#210 := [monotonicity #207]: #209
  1.3000 -#213 := [monotonicity #210]: #212
  1.3001 -#199 := (= #134 #198)
  1.3002 -#196 := (= #129 #195)
  1.3003 -#193 := (= #123 #192)
  1.3004 -#190 := (iff #120 #189)
  1.3005 -#191 := [rewrite]: #190
  1.3006 -#194 := [monotonicity #191]: #193
  1.3007 -#197 := [monotonicity #194]: #196
  1.3008 -#200 := [monotonicity #197]: #199
  1.3009 -#187 := (iff #114 #107)
  1.3010 -#186 := [rewrite]: #187
  1.3011 -#216 := [monotonicity #186 #200 #213]: #215
  1.3012 -#219 := [monotonicity #216]: #218
  1.3013 -#184 := (iff #41 #171)
  1.3014 -#176 := (not #171)
  1.3015 -#179 := (not #176)
  1.3016 -#182 := (iff #179 #171)
  1.3017 -#183 := [rewrite]: #182
  1.3018 -#180 := (iff #41 #179)
  1.3019 -#177 := (iff #40 #176)
  1.3020 -#174 := (iff #39 #171)
  1.3021 -#168 := (= #165 #18)
  1.3022 -#172 := (iff #168 #171)
  1.3023 -#173 := [rewrite]: #172
  1.3024 -#169 := (iff #39 #168)
  1.3025 -#166 := (= #38 #165)
  1.3026 -#163 := (= #37 #160)
  1.3027 -#150 := (* 1/2::real #144)
  1.3028 -#155 := (- #18 #150)
  1.3029 -#161 := (= #155 #160)
  1.3030 -#162 := [rewrite]: #161
  1.3031 -#156 := (= #37 #155)
  1.3032 -#153 := (= #36 #150)
  1.3033 -#147 := (/ #144 2::real)
  1.3034 -#151 := (= #147 #150)
  1.3035 -#152 := [rewrite]: #151
  1.3036 -#148 := (= #36 #147)
  1.3037 -#145 := (= #35 #144)
  1.3038 -#139 := (= #33 #138)
  1.3039 -#140 := [rewrite]: #139
  1.3040 -#142 := (iff #34 #141)
  1.3041 -#143 := [monotonicity #140]: #142
  1.3042 -#146 := [monotonicity #143 #140]: #145
  1.3043 -#149 := [monotonicity #146]: #148
  1.3044 -#154 := [trans #149 #152]: #153
  1.3045 -#157 := [monotonicity #154]: #156
  1.3046 -#164 := [trans #157 #162]: #163
  1.3047 -#135 := (= #32 #134)
  1.3048 -#132 := (= #31 #129)
  1.3049 -#126 := (/ #123 2::real)
  1.3050 -#130 := (= #126 #129)
  1.3051 -#131 := [rewrite]: #130
  1.3052 -#127 := (= #31 #126)
  1.3053 -#124 := (= #30 #123)
  1.3054 -#118 := (= #28 #117)
  1.3055 -#119 := [rewrite]: #118
  1.3056 -#121 := (iff #29 #120)
  1.3057 -#122 := [monotonicity #119]: #121
  1.3058 -#125 := [monotonicity #122 #119]: #124
  1.3059 -#128 := [monotonicity #125]: #127
  1.3060 -#133 := [trans #128 #131]: #132
  1.3061 -#136 := [monotonicity #133]: #135
  1.3062 -#115 := (iff #27 #114)
  1.3063 -#116 := [monotonicity #100]: #115
  1.3064 -#167 := [monotonicity #116 #136 #164]: #166
  1.3065 -#170 := [monotonicity #167]: #169
  1.3066 -#175 := [trans #170 #173]: #174
  1.3067 -#178 := [monotonicity #175]: #177
  1.3068 -#181 := [monotonicity #178]: #180
  1.3069 -#185 := [trans #181 #183]: #184
  1.3070 -#221 := [trans #185 #219]: #220
  1.3071 -#113 := [asserted]: #41
  1.3072 -#222 := [mp #113 #221]: #217
  1.3073 -#250 := [mp #222 #249]: #245
  1.3074 -#356 := (not #245)
  1.3075 -#357 := (or #356 #322)
  1.3076 -#358 := [th-lemma]: #357
  1.3077 -#359 := [unit-resolution #358 #250]: #322
  1.3078 -#360 := [hypothesis]: #354
  1.3079 -#60 := (<= uf_1 0::real)
  1.3080 -#61 := (not #60)
  1.3081 -#6 := (< 0::real uf_1)
  1.3082 -#62 := (iff #6 #61)
  1.3083 -#63 := [rewrite]: #62
  1.3084 -#57 := [asserted]: #6
  1.3085 -#64 := [mp #57 #63]: #61
  1.3086 -#361 := [th-lemma #64 #360 #359]: false
  1.3087 -#363 := [lemma #361]: #362
  1.3088 -#315 := (= uf_1 #205)
  1.3089 -#316 := (= #138 #205)
  1.3090 -#371 := (not #316)
  1.3091 -#355 := (+ #138 #352)
  1.3092 -#364 := (<= #355 0::real)
  1.3093 -#368 := (not #364)
  1.3094 -#87 := (<= #86 0::real)
  1.3095 -#82 := (<= #81 0::real)
  1.3096 -#90 := (and #82 #87)
  1.3097 -#21 := (<= #18 #20)
  1.3098 -#19 := (<= #16 #18)
  1.3099 -#22 := (and #19 #21)
  1.3100 -#91 := (iff #22 #90)
  1.3101 -#88 := (iff #21 #87)
  1.3102 -#89 := [rewrite]: #88
  1.3103 -#83 := (iff #19 #82)
  1.3104 -#84 := [rewrite]: #83
  1.3105 -#92 := [monotonicity #84 #89]: #91
  1.3106 -#59 := [asserted]: #22
  1.3107 -#93 := [mp #59 #92]: #90
  1.3108 -#95 := [and-elim #93]: #87
  1.3109 -#366 := [hypothesis]: #364
  1.3110 -#367 := [th-lemma #366 #95 #112 #359]: false
  1.3111 -#369 := [lemma #367]: #368
  1.3112 -#370 := [hypothesis]: #316
  1.3113 -#372 := (or #371 #364)
  1.3114 -#373 := [th-lemma]: #372
  1.3115 -#374 := [unit-resolution #373 #370 #369]: false
  1.3116 -#375 := [lemma #374]: #371
  1.3117 -#320 := (or #202 #316)
  1.3118 -#321 := [def-axiom]: #320
  1.3119 -#376 := [unit-resolution #321 #375]: #202
  1.3120 -#317 := (not #202)
  1.3121 -#318 := (or #317 #315)
  1.3122 -#319 := [def-axiom]: #318
  1.3123 -#377 := [unit-resolution #319 #376]: #315
  1.3124 -#378 := (not #315)
  1.3125 -#379 := (or #378 #354)
  1.3126 -#380 := [th-lemma]: #379
  1.3127 -[unit-resolution #380 #377 #363]: false
  1.3128 -unsat
  1.3129 -WdMJH3tkMv/rps8y9Ukq5Q 86 0
  1.3130 -#2 := false
  1.3131 -#37 := 0::real
  1.3132 -decl uf_2 :: (-> T2 T1 real)
  1.3133 -decl uf_4 :: T1
  1.3134 -#12 := uf_4
  1.3135 -decl uf_3 :: T2
  1.3136 -#5 := uf_3
  1.3137 -#13 := (uf_2 uf_3 uf_4)
  1.3138 -#34 := -1::real
  1.3139 -#140 := (* -1::real #13)
  1.3140 -decl uf_1 :: real
  1.3141 -#4 := uf_1
  1.3142 -#141 := (+ uf_1 #140)
  1.3143 -#143 := (>= #141 0::real)
  1.3144 -#6 := (:var 0 T1)
  1.3145 -#7 := (uf_2 uf_3 #6)
  1.3146 -#127 := (pattern #7)
  1.3147 -#35 := (* -1::real #7)
  1.3148 -#36 := (+ uf_1 #35)
  1.3149 -#47 := (>= #36 0::real)
  1.3150 -#134 := (forall (vars (?x2 T1)) (:pat #127) #47)
  1.3151 -#49 := (forall (vars (?x2 T1)) #47)
  1.3152 -#137 := (iff #49 #134)
  1.3153 -#135 := (iff #47 #47)
  1.3154 -#136 := [refl]: #135
  1.3155 -#138 := [quant-intro #136]: #137
  1.3156 -#67 := (~ #49 #49)
  1.3157 -#58 := (~ #47 #47)
  1.3158 -#66 := [refl]: #58
  1.3159 -#68 := [nnf-pos #66]: #67
  1.3160 -#10 := (<= #7 uf_1)
  1.3161 -#11 := (forall (vars (?x2 T1)) #10)
  1.3162 -#50 := (iff #11 #49)
  1.3163 -#46 := (iff #10 #47)
  1.3164 -#48 := [rewrite]: #46
  1.3165 -#51 := [quant-intro #48]: #50
  1.3166 -#32 := [asserted]: #11
  1.3167 -#52 := [mp #32 #51]: #49
  1.3168 -#69 := [mp~ #52 #68]: #49
  1.3169 -#139 := [mp #69 #138]: #134
  1.3170 -#149 := (not #134)
  1.3171 -#150 := (or #149 #143)
  1.3172 -#151 := [quant-inst]: #150
  1.3173 -#144 := [unit-resolution #151 #139]: #143
  1.3174 -#142 := (<= #141 0::real)
  1.3175 -#38 := (<= #36 0::real)
  1.3176 -#128 := (forall (vars (?x1 T1)) (:pat #127) #38)
  1.3177 -#41 := (forall (vars (?x1 T1)) #38)
  1.3178 -#131 := (iff #41 #128)
  1.3179 -#129 := (iff #38 #38)
  1.3180 -#130 := [refl]: #129
  1.3181 -#132 := [quant-intro #130]: #131
  1.3182 -#62 := (~ #41 #41)
  1.3183 -#64 := (~ #38 #38)
  1.3184 -#65 := [refl]: #64
  1.3185 -#63 := [nnf-pos #65]: #62
  1.3186 -#8 := (<= uf_1 #7)
  1.3187 -#9 := (forall (vars (?x1 T1)) #8)
  1.3188 -#42 := (iff #9 #41)
  1.3189 -#39 := (iff #8 #38)
  1.3190 -#40 := [rewrite]: #39
  1.3191 -#43 := [quant-intro #40]: #42
  1.3192 -#31 := [asserted]: #9
  1.3193 -#44 := [mp #31 #43]: #41
  1.3194 -#61 := [mp~ #44 #63]: #41
  1.3195 -#133 := [mp #61 #132]: #128
  1.3196 -#145 := (not #128)
  1.3197 -#146 := (or #145 #142)
  1.3198 -#147 := [quant-inst]: #146
  1.3199 -#148 := [unit-resolution #147 #133]: #142
  1.3200 -#45 := (= uf_1 #13)
  1.3201 -#55 := (not #45)
  1.3202 -#14 := (= #13 uf_1)
  1.3203 -#15 := (not #14)
  1.3204 -#56 := (iff #15 #55)
  1.3205 -#53 := (iff #14 #45)
  1.3206 -#54 := [rewrite]: #53
  1.3207 -#57 := [monotonicity #54]: #56
  1.3208 -#33 := [asserted]: #15
  1.3209 -#60 := [mp #33 #57]: #55
  1.3210 -#153 := (not #143)
  1.3211 -#152 := (not #142)
  1.3212 -#154 := (or #45 #152 #153)
  1.3213 -#155 := [th-lemma]: #154
  1.3214 -[unit-resolution #155 #60 #148 #144]: false
  1.3215 -unsat
  1.3216 -V+IAyBZU/6QjYs6JkXx8LQ 57 0
  1.3217 -#2 := false
  1.3218 -#4 := 0::real
  1.3219 -decl uf_1 :: (-> T2 real)
  1.3220 -decl uf_2 :: (-> T1 T1 T2)
  1.3221 -decl uf_12 :: (-> T4 T1)
  1.3222 -decl uf_4 :: T4
  1.3223 -#11 := uf_4
  1.3224 -#39 := (uf_12 uf_4)
  1.3225 -decl uf_10 :: T4
  1.3226 -#27 := uf_10
  1.3227 -#38 := (uf_12 uf_10)
  1.3228 -#40 := (uf_2 #38 #39)
  1.3229 -#41 := (uf_1 #40)
  1.3230 -#264 := (>= #41 0::real)
  1.3231 -#266 := (not #264)
  1.3232 -#43 := (= #41 0::real)
  1.3233 -#44 := (not #43)
  1.3234 -#131 := [asserted]: #44
  1.3235 -#272 := (or #43 #266)
  1.3236 -#42 := (<= #41 0::real)
  1.3237 -#130 := [asserted]: #42
  1.3238 -#265 := (not #42)
  1.3239 -#270 := (or #43 #265 #266)
  1.3240 -#271 := [th-lemma]: #270
  1.3241 -#273 := [unit-resolution #271 #130]: #272
  1.3242 -#274 := [unit-resolution #273 #131]: #266
  1.3243 -#6 := (:var 0 T1)
  1.3244 -#5 := (:var 1 T1)
  1.3245 -#7 := (uf_2 #5 #6)
  1.3246 -#241 := (pattern #7)
  1.3247 -#8 := (uf_1 #7)
  1.3248 -#65 := (>= #8 0::real)
  1.3249 -#242 := (forall (vars (?x1 T1) (?x2 T1)) (:pat #241) #65)
  1.3250 -#66 := (forall (vars (?x1 T1) (?x2 T1)) #65)
  1.3251 -#245 := (iff #66 #242)
  1.3252 -#243 := (iff #65 #65)
  1.3253 -#244 := [refl]: #243
  1.3254 -#246 := [quant-intro #244]: #245
  1.3255 -#149 := (~ #66 #66)
  1.3256 -#151 := (~ #65 #65)
  1.3257 -#152 := [refl]: #151
  1.3258 -#150 := [nnf-pos #152]: #149
  1.3259 -#9 := (<= 0::real #8)
  1.3260 -#10 := (forall (vars (?x1 T1) (?x2 T1)) #9)
  1.3261 -#67 := (iff #10 #66)
  1.3262 -#63 := (iff #9 #65)
  1.3263 -#64 := [rewrite]: #63
  1.3264 -#68 := [quant-intro #64]: #67
  1.3265 -#60 := [asserted]: #10
  1.3266 -#69 := [mp #60 #68]: #66
  1.3267 -#147 := [mp~ #69 #150]: #66
  1.3268 -#247 := [mp #147 #246]: #242
  1.3269 -#267 := (not #242)
  1.3270 -#268 := (or #267 #264)
  1.3271 -#269 := [quant-inst]: #268
  1.3272 -[unit-resolution #269 #247 #274]: false
  1.3273 -unsat
     2.1 --- a/src/HOL/Multivariate_Analysis/Integration.thy	Wed Feb 17 18:33:45 2010 +0100
     2.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
     2.3 @@ -1,3465 +0,0 @@
     2.4 -
     2.5 -header {* Kurzweil-Henstock gauge integration in many dimensions. *}
     2.6 -(*  Author:                     John Harrison
     2.7 -    Translation from HOL light: Robert Himmelmann, TU Muenchen *)
     2.8 -
     2.9 -theory Integration_Aleph
    2.10 -  imports Derivative SMT
    2.11 -begin
    2.12 -
    2.13 -declare [[smt_certificates="~~/src/HOL/Multivariate_Analysis/Integration.cert"]]
    2.14 -declare [[smt_record=true]]
    2.15 -declare [[z3_proofs=true]]
    2.16 -
    2.17 -lemma conjunctD2: assumes "a \<and> b" shows a b using assms by auto
    2.18 -lemma conjunctD3: assumes "a \<and> b \<and> c" shows a b c using assms by auto
    2.19 -lemma conjunctD4: assumes "a \<and> b \<and> c \<and> d" shows a b c d using assms by auto
    2.20 -lemma conjunctD5: assumes "a \<and> b \<and> c \<and> d \<and> e" shows a b c d e using assms by auto
    2.21 -
    2.22 -declare smult_conv_scaleR[simp]
    2.23 -
    2.24 -subsection {* Some useful lemmas about intervals. *}
    2.25 -
    2.26 -lemma empty_as_interval: "{} = {1..0::real^'n}"
    2.27 -  apply(rule set_ext,rule) defer unfolding vector_le_def mem_interval
    2.28 -  using UNIV_witness[where 'a='n] apply(erule_tac exE,rule_tac x=x in allE) by auto
    2.29 -
    2.30 -lemma interior_subset_union_intervals: 
    2.31 -  assumes "i = {a..b::real^'n}" "j = {c..d}" "interior j \<noteq> {}" "i \<subseteq> j \<union> s" "interior(i) \<inter> interior(j) = {}"
    2.32 -  shows "interior i \<subseteq> interior s" proof-
    2.33 -  have "{a<..<b} \<inter> {c..d} = {}" using inter_interval_mixed_eq_empty[of c d a b] and assms(3,5)
    2.34 -    unfolding assms(1,2) interior_closed_interval by auto
    2.35 -  moreover have "{a<..<b} \<subseteq> {c..d} \<union> s" apply(rule order_trans,rule interval_open_subset_closed)
    2.36 -    using assms(4) unfolding assms(1,2) by auto
    2.37 -  ultimately show ?thesis apply-apply(rule interior_maximal) defer apply(rule open_interior)
    2.38 -    unfolding assms(1,2) interior_closed_interval by auto qed
    2.39 -
    2.40 -lemma inter_interior_unions_intervals: fixes f::"(real^'n) set set"
    2.41 -  assumes "finite f" "open s" "\<forall>t\<in>f. \<exists>a b. t = {a..b}" "\<forall>t\<in>f. s \<inter> (interior t) = {}"
    2.42 -  shows "s \<inter> interior(\<Union>f) = {}" proof(rule ccontr,unfold ex_in_conv[THEN sym]) case goal1
    2.43 -  have lem1:"\<And>x e s U. ball x e \<subseteq> s \<inter> interior U \<longleftrightarrow> ball x e \<subseteq> s \<inter> U" apply rule  defer apply(rule_tac Int_greatest)
    2.44 -    unfolding open_subset_interior[OF open_ball]  using interior_subset by auto
    2.45 -  have lem2:"\<And>x s P. \<exists>x\<in>s. P x \<Longrightarrow> \<exists>x\<in>insert x s. P x" by auto
    2.46 -  have "\<And>f. finite f \<Longrightarrow> (\<forall>t\<in>f. \<exists>a b. t = {a..b}) \<Longrightarrow> (\<exists>x. x \<in> s \<inter> interior (\<Union>f)) \<Longrightarrow> (\<exists>t\<in>f. \<exists>x. \<exists>e>0. ball x e \<subseteq> s \<inter> t)" proof- case goal1
    2.47 -  thus ?case proof(induct rule:finite_induct) 
    2.48 -    case empty from this(2) guess x .. hence False unfolding Union_empty interior_empty by auto thus ?case by auto next
    2.49 -    case (insert i f) guess x using insert(5) .. note x = this
    2.50 -    then guess e unfolding open_contains_ball_eq[OF open_Int[OF assms(2) open_interior],rule_format] .. note e=this
    2.51 -    guess a using insert(4)[rule_format,OF insertI1] .. then guess b .. note ab = this
    2.52 -    show ?case proof(cases "x\<in>i") case False hence "x \<in> UNIV - {a..b}" unfolding ab by auto
    2.53 -      then guess d unfolding open_contains_ball_eq[OF open_Diff[OF open_UNIV closed_interval],rule_format] ..
    2.54 -      hence "0 < d" "ball x (min d e) \<subseteq> UNIV - i" using e unfolding ab by auto
    2.55 -      hence "ball x (min d e) \<subseteq> s \<inter> interior (\<Union>f)" using e unfolding lem1 by auto hence "x \<in> s \<inter> interior (\<Union>f)" using `d>0` e by auto
    2.56 -      hence "\<exists>t\<in>f. \<exists>x e. 0 < e \<and> ball x e \<subseteq> s \<inter> t" apply-apply(rule insert(3)) using insert(4) by auto thus ?thesis by auto next
    2.57 -    case True show ?thesis proof(cases "x\<in>{a<..<b}")
    2.58 -      case True then guess d unfolding open_contains_ball_eq[OF open_interval,rule_format] ..
    2.59 -      thus ?thesis apply(rule_tac x=i in bexI,rule_tac x=x in exI,rule_tac x="min d e" in exI)
    2.60 -	unfolding ab using interval_open_subset_closed[of a b] and e by fastsimp+ next
    2.61 -    case False then obtain k where "x$k \<le> a$k \<or> x$k \<ge> b$k" unfolding mem_interval by(auto simp add:not_less) 
    2.62 -    hence "x$k = a$k \<or> x$k = b$k" using True unfolding ab and mem_interval apply(erule_tac x=k in allE) by auto
    2.63 -    hence "\<exists>x. ball x (e/2) \<subseteq> s \<inter> (\<Union>f)" proof(erule_tac disjE)
    2.64 -      let ?z = "x - (e/2) *\<^sub>R basis k" assume as:"x$k = a$k" have "ball ?z (e / 2) \<inter> i = {}" apply(rule ccontr) unfolding ex_in_conv[THEN sym] proof(erule exE)
    2.65 -	fix y assume "y \<in> ball ?z (e / 2) \<inter> i" hence "dist ?z y < e/2" and yi:"y\<in>i" by auto
    2.66 -	hence "\<bar>(?z - y) $ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding vector_dist_norm by auto
    2.67 -	hence "y$k < a$k" unfolding vector_component_simps vector_scaleR_component as using e[THEN conjunct1] by(auto simp add:field_simps)
    2.68 -	hence "y \<notin> i" unfolding ab mem_interval not_all by(rule_tac x=k in exI,auto) thus False using yi by auto qed
    2.69 -      moreover have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)" apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]]) proof
    2.70 -	fix y assume as:"y\<in> ball ?z (e/2)" have "norm (x - y) \<le> \<bar>e\<bar> / 2 + norm (x - y - (e / 2) *\<^sub>R basis k)"
    2.71 -	   apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "(e/2) *\<^sub>R basis k"])
    2.72 -	  unfolding norm_scaleR norm_basis by auto
    2.73 -	also have "\<dots> < \<bar>e\<bar> / 2 + \<bar>e\<bar> / 2" apply(rule add_strict_left_mono) using as unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps)
    2.74 -	finally show "y\<in>ball x e" unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps) qed
    2.75 -      ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto
    2.76 -    next let ?z = "x + (e/2) *\<^sub>R basis k" assume as:"x$k = b$k" have "ball ?z (e / 2) \<inter> i = {}" apply(rule ccontr) unfolding ex_in_conv[THEN sym] proof(erule exE)
    2.77 -	fix y assume "y \<in> ball ?z (e / 2) \<inter> i" hence "dist ?z y < e/2" and yi:"y\<in>i" by auto
    2.78 -	hence "\<bar>(?z - y) $ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding vector_dist_norm by auto
    2.79 -	hence "y$k > b$k" unfolding vector_component_simps vector_scaleR_component as using e[THEN conjunct1] by(auto simp add:field_simps)
    2.80 -	hence "y \<notin> i" unfolding ab mem_interval not_all by(rule_tac x=k in exI,auto) thus False using yi by auto qed
    2.81 -      moreover have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)" apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]]) proof
    2.82 -	fix y assume as:"y\<in> ball ?z (e/2)" have "norm (x - y) \<le> \<bar>e\<bar> / 2 + norm (x - y + (e / 2) *\<^sub>R basis k)"
    2.83 -	   apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "- (e/2) *\<^sub>R basis k"])
    2.84 -	  unfolding norm_scaleR norm_basis by auto
    2.85 -	also have "\<dots> < \<bar>e\<bar> / 2 + \<bar>e\<bar> / 2" apply(rule add_strict_left_mono) using as unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps)
    2.86 -	finally show "y\<in>ball x e" unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps) qed
    2.87 -      ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto qed 
    2.88 -    then guess x .. hence "x \<in> s \<inter> interior (\<Union>f)" unfolding lem1[where U="\<Union>f",THEN sym] using centre_in_ball e[THEN conjunct1] by auto
    2.89 -    thus ?thesis apply-apply(rule lem2,rule insert(3)) using insert(4) by auto qed qed qed qed note * = this
    2.90 -  guess t using *[OF assms(1,3) goal1]  .. from this(2) guess x .. then guess e ..
    2.91 -  hence "x \<in> s" "x\<in>interior t" defer using open_subset_interior[OF open_ball, of x e t] by auto
    2.92 -  thus False using `t\<in>f` assms(4) by auto qed
    2.93 -subsection {* Bounds on intervals where they exist. *}
    2.94 -
    2.95 -definition "interval_upperbound (s::(real^'n) set) = (\<chi> i. Sup {a. \<exists>x\<in>s. x$i = a})"
    2.96 -
    2.97 -definition "interval_lowerbound (s::(real^'n) set) = (\<chi> i. Inf {a. \<exists>x\<in>s. x$i = a})"
    2.98 -
    2.99 -lemma interval_upperbound[simp]: assumes "\<forall>i. a$i \<le> b$i" shows "interval_upperbound {a..b} = b"
   2.100 -  using assms unfolding interval_upperbound_def Cart_eq Cart_lambda_beta apply-apply(rule,erule_tac x=i in allE)
   2.101 -  apply(rule Sup_unique) unfolding setle_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer
   2.102 -  apply(rule,rule) apply(rule_tac x="b$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=b in bexI)
   2.103 -  unfolding mem_interval using assms by auto
   2.104 -
   2.105 -lemma interval_lowerbound[simp]: assumes "\<forall>i. a$i \<le> b$i" shows "interval_lowerbound {a..b} = a"
   2.106 -  using assms unfolding interval_lowerbound_def Cart_eq Cart_lambda_beta apply-apply(rule,erule_tac x=i in allE)
   2.107 -  apply(rule Inf_unique) unfolding setge_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer
   2.108 -  apply(rule,rule) apply(rule_tac x="a$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=a in bexI)
   2.109 -  unfolding mem_interval using assms by auto
   2.110 -
   2.111 -lemmas interval_bounds = interval_upperbound interval_lowerbound
   2.112 -
   2.113 -lemma interval_bounds'[simp]: assumes "{a..b}\<noteq>{}" shows "interval_upperbound {a..b} = b" "interval_lowerbound {a..b} = a"
   2.114 -  using assms unfolding interval_ne_empty by auto
   2.115 -
   2.116 -lemma interval_upperbound_1[simp]: "dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> interval_upperbound {a..b} = (b::real^1)"
   2.117 -  apply(rule interval_upperbound) by auto
   2.118 -
   2.119 -lemma interval_lowerbound_1[simp]: "dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> interval_lowerbound {a..b} = (a::real^1)"
   2.120 -  apply(rule interval_lowerbound) by auto
   2.121 -
   2.122 -lemmas interval_bound_1 = interval_upperbound_1 interval_lowerbound_1
   2.123 -
   2.124 -subsection {* Content (length, area, volume...) of an interval. *}
   2.125 -
   2.126 -definition "content (s::(real^'n) set) =
   2.127 -       (if s = {} then 0 else (\<Prod>i\<in>UNIV. (interval_upperbound s)$i - (interval_lowerbound s)$i))"
   2.128 -
   2.129 -lemma interval_not_empty:"\<forall>i. a$i \<le> b$i \<Longrightarrow> {a..b::real^'n} \<noteq> {}"
   2.130 -  unfolding interval_eq_empty unfolding not_ex not_less by assumption
   2.131 -
   2.132 -lemma content_closed_interval: assumes "\<forall>i. a$i \<le> b$i"
   2.133 -  shows "content {a..b} = (\<Prod>i\<in>UNIV. b$i - a$i)"
   2.134 -  using interval_not_empty[OF assms] unfolding content_def interval_upperbound[OF assms] interval_lowerbound[OF assms] by auto
   2.135 -
   2.136 -lemma content_closed_interval': assumes "{a..b}\<noteq>{}" shows "content {a..b} = (\<Prod>i\<in>UNIV. b$i - a$i)"
   2.137 -  apply(rule content_closed_interval) using assms unfolding interval_ne_empty .
   2.138 -
   2.139 -lemma content_1:"dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> content {a..b} = dest_vec1 b - dest_vec1 a"
   2.140 -  using content_closed_interval[of a b] by auto
   2.141 -
   2.142 -lemma content_1':"a \<le> b \<Longrightarrow> content {vec1 a..vec1 b} = b - a" using content_1[of "vec a" "vec b"] by auto
   2.143 -
   2.144 -lemma content_unit[intro]: "content{0..1::real^'n} = 1" proof-
   2.145 -  have *:"\<forall>i. 0$i \<le> (1::real^'n::finite)$i" by auto
   2.146 -  have "0 \<in> {0..1::real^'n::finite}" unfolding mem_interval by auto
   2.147 -  thus ?thesis unfolding content_def interval_bounds[OF *] using setprod_1 by auto qed
   2.148 -
   2.149 -lemma content_pos_le[intro]: "0 \<le> content {a..b}" proof(cases "{a..b}={}")
   2.150 -  case False hence *:"\<forall>i. a $ i \<le> b $ i" unfolding interval_ne_empty by assumption
   2.151 -  have "(\<Prod>i\<in>UNIV. interval_upperbound {a..b} $ i - interval_lowerbound {a..b} $ i) \<ge> 0"
   2.152 -    apply(rule setprod_nonneg) unfolding interval_bounds[OF *] using * apply(erule_tac x=x in allE) by auto
   2.153 -  thus ?thesis unfolding content_def by(auto simp del:interval_bounds') qed(unfold content_def, auto)
   2.154 -
   2.155 -lemma content_pos_lt: assumes "\<forall>i. a$i < b$i" shows "0 < content {a..b}"
   2.156 -proof- have help_lemma1: "\<forall>i. a$i < b$i \<Longrightarrow> \<forall>i. a$i \<le> ((b$i)::real)" apply(rule,erule_tac x=i in allE) by auto
   2.157 -  show ?thesis unfolding content_closed_interval[OF help_lemma1[OF assms]] apply(rule setprod_pos)
   2.158 -    using assms apply(erule_tac x=x in allE) by auto qed
   2.159 -
   2.160 -lemma content_pos_lt_1: "dest_vec1 a < dest_vec1 b \<Longrightarrow> 0 < content({a..b})"
   2.161 -  apply(rule content_pos_lt) by auto
   2.162 -
   2.163 -lemma content_eq_0: "content({a..b::real^'n}) = 0 \<longleftrightarrow> (\<exists>i. b$i \<le> a$i)" proof(cases "{a..b} = {}")
   2.164 -  case True thus ?thesis unfolding content_def if_P[OF True] unfolding interval_eq_empty apply-
   2.165 -    apply(rule,erule exE) apply(rule_tac x=i in exI) by auto next
   2.166 -  guess a using UNIV_witness[where 'a='n] .. case False note as=this[unfolded interval_eq_empty not_ex not_less]
   2.167 -  show ?thesis unfolding content_def if_not_P[OF False] setprod_zero_iff[OF finite_UNIV]
   2.168 -    apply(rule) apply(erule_tac[!] exE bexE) unfolding interval_bounds[OF as] apply(rule_tac x=x in exI) defer
   2.169 -    apply(rule_tac x=i in bexI) using as apply(erule_tac x=i in allE) by auto qed
   2.170 -
   2.171 -lemma cond_cases:"(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)" by auto
   2.172 -
   2.173 -lemma content_closed_interval_cases:
   2.174 -  "content {a..b} = (if \<forall>i. a$i \<le> b$i then setprod (\<lambda>i. b$i - a$i) UNIV else 0)" apply(rule cond_cases) 
   2.175 -  apply(rule content_closed_interval) unfolding content_eq_0 not_all not_le defer apply(erule exE,rule_tac x=x in exI) by auto
   2.176 -
   2.177 -lemma content_eq_0_interior: "content {a..b} = 0 \<longleftrightarrow> interior({a..b}) = {}"
   2.178 -  unfolding content_eq_0 interior_closed_interval interval_eq_empty by auto
   2.179 -
   2.180 -lemma content_eq_0_1: "content {a..b::real^1} = 0 \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a"
   2.181 -  unfolding content_eq_0 by auto
   2.182 -
   2.183 -lemma content_pos_lt_eq: "0 < content {a..b} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
   2.184 -  apply(rule) defer apply(rule content_pos_lt,assumption) proof- assume "0 < content {a..b}"
   2.185 -  hence "content {a..b} \<noteq> 0" by auto thus "\<forall>i. a$i < b$i" unfolding content_eq_0 not_ex not_le by auto qed
   2.186 -
   2.187 -lemma content_empty[simp]: "content {} = 0" unfolding content_def by auto
   2.188 -
   2.189 -lemma content_subset: assumes "{a..b} \<subseteq> {c..d}" shows "content {a..b::real^'n} \<le> content {c..d}" proof(cases "{a..b}={}")
   2.190 -  case True thus ?thesis using content_pos_le[of c d] by auto next
   2.191 -  case False hence ab_ne:"\<forall>i. a $ i \<le> b $ i" unfolding interval_ne_empty by auto
   2.192 -  hence ab_ab:"a\<in>{a..b}" "b\<in>{a..b}" unfolding mem_interval by auto
   2.193 -  have "{c..d} \<noteq> {}" using assms False by auto
   2.194 -  hence cd_ne:"\<forall>i. c $ i \<le> d $ i" using assms unfolding interval_ne_empty by auto
   2.195 -  show ?thesis unfolding content_def unfolding interval_bounds[OF ab_ne] interval_bounds[OF cd_ne]
   2.196 -    unfolding if_not_P[OF False] if_not_P[OF `{c..d} \<noteq> {}`] apply(rule setprod_mono,rule) proof fix i::'n
   2.197 -    show "0 \<le> b $ i - a $ i" using ab_ne[THEN spec[where x=i]] by auto
   2.198 -    show "b $ i - a $ i \<le> d $ i - c $ i"
   2.199 -      using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(2),of i]
   2.200 -      using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(1),of i] by auto qed qed
   2.201 -
   2.202 -lemma content_lt_nz: "0 < content {a..b} \<longleftrightarrow> content {a..b} \<noteq> 0"
   2.203 -  unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by auto
   2.204 -
   2.205 -subsection {* The notion of a gauge --- simply an open set containing the point. *}
   2.206 -
   2.207 -definition gauge where "gauge d \<longleftrightarrow> (\<forall>x. x\<in>(d x) \<and> open(d x))"
   2.208 -
   2.209 -lemma gaugeI:assumes "\<And>x. x\<in>g x" "\<And>x. open (g x)" shows "gauge g"
   2.210 -  using assms unfolding gauge_def by auto
   2.211 -
   2.212 -lemma gaugeD[dest]: assumes "gauge d" shows "x\<in>d x" "open (d x)" using assms unfolding gauge_def by auto
   2.213 -
   2.214 -lemma gauge_ball_dependent: "\<forall>x. 0 < e x \<Longrightarrow> gauge (\<lambda>x. ball x (e x))"
   2.215 -  unfolding gauge_def by auto 
   2.216 -
   2.217 -lemma gauge_ball[intro?]: "0 < e \<Longrightarrow> gauge (\<lambda>x. ball x e)" unfolding gauge_def by auto 
   2.218 -
   2.219 -lemma gauge_trivial[intro]: "gauge (\<lambda>x. ball x 1)" apply(rule gauge_ball) by auto
   2.220 -
   2.221 -lemma gauge_inter: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. (d1 x) \<inter> (d2 x))"
   2.222 -  unfolding gauge_def by auto 
   2.223 -
   2.224 -lemma gauge_inters: assumes "finite s" "\<forall>d\<in>s. gauge (f d)" shows "gauge(\<lambda>x. \<Inter> {f d x | d. d \<in> s})" proof-
   2.225 -  have *:"\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s" by auto show ?thesis
   2.226 -  unfolding gauge_def unfolding * 
   2.227 -  using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto qed
   2.228 -
   2.229 -lemma gauge_existence_lemma: "(\<forall>x. \<exists>d::real. p x \<longrightarrow> 0 < d \<and> q d x) \<longleftrightarrow> (\<forall>x. \<exists>d>0. p x \<longrightarrow> q d x)" by(meson zero_less_one)
   2.230 -
   2.231 -subsection {* Divisions. *}
   2.232 -
   2.233 -definition division_of (infixl "division'_of" 40) where
   2.234 -  "s division_of i \<equiv>
   2.235 -        finite s \<and>
   2.236 -        (\<forall>k\<in>s. k \<subseteq> i \<and> k \<noteq> {} \<and> (\<exists>a b. k = {a..b})) \<and>
   2.237 -        (\<forall>k1\<in>s. \<forall>k2\<in>s. k1 \<noteq> k2 \<longrightarrow> interior(k1) \<inter> interior(k2) = {}) \<and>
   2.238 -        (\<Union>s = i)"
   2.239 -
   2.240 -lemma division_ofD[dest]: assumes  "s division_of i"
   2.241 -  shows"finite s" "\<And>k. k\<in>s \<Longrightarrow> k \<subseteq> i" "\<And>k. k\<in>s \<Longrightarrow>  k \<noteq> {}" "\<And>k. k\<in>s \<Longrightarrow> (\<exists>a b. k = {a..b})"
   2.242 -  "\<And>k1 k2. \<lbrakk>k1\<in>s; k2\<in>s; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}" "\<Union>s = i" using assms unfolding division_of_def by auto
   2.243 -
   2.244 -lemma division_ofI:
   2.245 -  assumes "finite s" "\<And>k. k\<in>s \<Longrightarrow> k \<subseteq> i" "\<And>k. k\<in>s \<Longrightarrow>  k \<noteq> {}" "\<And>k. k\<in>s \<Longrightarrow> (\<exists>a b. k = {a..b})"
   2.246 -  "\<And>k1 k2. \<lbrakk>k1\<in>s; k2\<in>s; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}" "\<Union>s = i"
   2.247 -  shows "s division_of i" using assms unfolding division_of_def by auto
   2.248 -
   2.249 -lemma division_of_finite: "s division_of i \<Longrightarrow> finite s"
   2.250 -  unfolding division_of_def by auto
   2.251 -
   2.252 -lemma division_of_self[intro]: "{a..b} \<noteq> {} \<Longrightarrow> {{a..b}} division_of {a..b}"
   2.253 -  unfolding division_of_def by auto
   2.254 -
   2.255 -lemma division_of_trivial[simp]: "s division_of {} \<longleftrightarrow> s = {}" unfolding division_of_def by auto 
   2.256 -
   2.257 -lemma division_of_sing[simp]: "s division_of {a..a::real^'n} \<longleftrightarrow> s = {{a..a}}" (is "?l = ?r") proof
   2.258 -  assume ?r moreover { assume "s = {{a}}" moreover fix k assume "k\<in>s" 
   2.259 -    ultimately have"\<exists>x y. k = {x..y}" apply(rule_tac x=a in exI)+ unfolding interval_sing[THEN conjunct1] by auto }
   2.260 -  ultimately show ?l unfolding division_of_def interval_sing[THEN conjunct1] by auto next
   2.261 -  assume ?l note as=conjunctD4[OF this[unfolded division_of_def interval_sing[THEN conjunct1]]]
   2.262 -  { fix x assume x:"x\<in>s" have "x={a}" using as(2)[rule_format,OF x] by auto }
   2.263 -  moreover have "s \<noteq> {}" using as(4) by auto ultimately show ?r unfolding interval_sing[THEN conjunct1] by auto qed
   2.264 -
   2.265 -lemma elementary_empty: obtains p where "p division_of {}"
   2.266 -  unfolding division_of_trivial by auto
   2.267 -
   2.268 -lemma elementary_interval: obtains p where  "p division_of {a..b}"
   2.269 -  by(metis division_of_trivial division_of_self)
   2.270 -
   2.271 -lemma division_contains: "s division_of i \<Longrightarrow> \<forall>x\<in>i. \<exists>k\<in>s. x \<in> k"
   2.272 -  unfolding division_of_def by auto
   2.273 -
   2.274 -lemma forall_in_division:
   2.275 - "d division_of i \<Longrightarrow> ((\<forall>x\<in>d. P x) \<longleftrightarrow> (\<forall>a b. {a..b} \<in> d \<longrightarrow> P {a..b}))"
   2.276 -  unfolding division_of_def by fastsimp
   2.277 -
   2.278 -lemma division_of_subset: assumes "p division_of (\<Union>p)" "q \<subseteq> p" shows "q division_of (\<Union>q)"
   2.279 -  apply(rule division_ofI) proof- note as=division_ofD[OF assms(1)]
   2.280 -  show "finite q" apply(rule finite_subset) using as(1) assms(2) by auto
   2.281 -  { fix k assume "k \<in> q" hence kp:"k\<in>p" using assms(2) by auto show "k\<subseteq>\<Union>q" using `k \<in> q` by auto
   2.282 -  show "\<exists>a b. k = {a..b}" using as(4)[OF kp] by auto show "k \<noteq> {}" using as(3)[OF kp] by auto }
   2.283 -  fix k1 k2 assume "k1 \<in> q" "k2 \<in> q" "k1 \<noteq> k2" hence *:"k1\<in>p" "k2\<in>p" "k1\<noteq>k2" using assms(2) by auto
   2.284 -  show "interior k1 \<inter> interior k2 = {}" using as(5)[OF *] by auto qed auto
   2.285 -
   2.286 -lemma division_of_union_self[intro]: "p division_of s \<Longrightarrow> p division_of (\<Union>p)" unfolding division_of_def by auto
   2.287 -
   2.288 -lemma division_of_content_0: assumes "content {a..b} = 0" "d division_of {a..b}" shows "\<forall>k\<in>d. content k = 0"
   2.289 -  unfolding forall_in_division[OF assms(2)] apply(rule,rule,rule) apply(drule division_ofD(2)[OF assms(2)])
   2.290 -  apply(drule content_subset) unfolding assms(1) proof- case goal1 thus ?case using content_pos_le[of a b] by auto qed
   2.291 -
   2.292 -lemma division_inter: assumes "p1 division_of s1" "p2 division_of (s2::(real^'a) set)"
   2.293 -  shows "{k1 \<inter> k2 | k1 k2 .k1 \<in> p1 \<and> k2 \<in> p2 \<and> k1 \<inter> k2 \<noteq> {}} division_of (s1 \<inter> s2)" (is "?A' division_of _") proof-
   2.294 -let ?A = "{s. s \<in>  (\<lambda>(k1,k2). k1 \<inter> k2) ` (p1 \<times> p2) \<and> s \<noteq> {}}" have *:"?A' = ?A" by auto
   2.295 -show ?thesis unfolding * proof(rule division_ofI) have "?A \<subseteq> (\<lambda>(x, y). x \<inter> y) ` (p1 \<times> p2)" by auto
   2.296 -  moreover have "finite (p1 \<times> p2)" using assms unfolding division_of_def by auto ultimately show "finite ?A" by auto
   2.297 -  have *:"\<And>s. \<Union>{x\<in>s. x \<noteq> {}} = \<Union>s" by auto show "\<Union>?A = s1 \<inter> s2" apply(rule set_ext) unfolding * and Union_image_eq UN_iff
   2.298 -    using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)] by auto
   2.299 -  { fix k assume "k\<in>?A" then obtain k1 k2 where k:"k = k1 \<inter> k2" "k1\<in>p1" "k2\<in>p2" "k\<noteq>{}" by auto thus "k \<noteq> {}" by auto
   2.300 -  show "k \<subseteq> s1 \<inter> s2" using division_ofD(2)[OF assms(1) k(2)] and division_ofD(2)[OF assms(2) k(3)] unfolding k by auto
   2.301 -  guess a1 using division_ofD(4)[OF assms(1) k(2)] .. then guess b1 .. note ab1=this
   2.302 -  guess a2 using division_ofD(4)[OF assms(2) k(3)] .. then guess b2 .. note ab2=this
   2.303 -  show "\<exists>a b. k = {a..b}" unfolding k ab1 ab2 unfolding inter_interval by auto } fix k1 k2
   2.304 -  assume "k1\<in>?A" then obtain x1 y1 where k1:"k1 = x1 \<inter> y1" "x1\<in>p1" "y1\<in>p2" "k1\<noteq>{}" by auto
   2.305 -  assume "k2\<in>?A" then obtain x2 y2 where k2:"k2 = x2 \<inter> y2" "x2\<in>p1" "y2\<in>p2" "k2\<noteq>{}" by auto
   2.306 -  assume "k1 \<noteq> k2" hence th:"x1\<noteq>x2 \<or> y1\<noteq>y2" unfolding k1 k2 by auto
   2.307 -  have *:"(interior x1 \<inter> interior x2 = {} \<or> interior y1 \<inter> interior y2 = {}) \<Longrightarrow>
   2.308 -      interior(x1 \<inter> y1) \<subseteq> interior(x1) \<Longrightarrow> interior(x1 \<inter> y1) \<subseteq> interior(y1) \<Longrightarrow>
   2.309 -      interior(x2 \<inter> y2) \<subseteq> interior(x2) \<Longrightarrow> interior(x2 \<inter> y2) \<subseteq> interior(y2)
   2.310 -      \<Longrightarrow> interior(x1 \<inter> y1) \<inter> interior(x2 \<inter> y2) = {}" by auto
   2.311 -  show "interior k1 \<inter> interior k2 = {}" unfolding k1 k2 apply(rule *) defer apply(rule_tac[1-4] subset_interior)
   2.312 -    using division_ofD(5)[OF assms(1) k1(2) k2(2)]
   2.313 -    using division_ofD(5)[OF assms(2) k1(3) k2(3)] using th by auto qed qed
   2.314 -
   2.315 -lemma division_inter_1: assumes "d division_of i" "{a..b::real^'n} \<subseteq> i"
   2.316 -  shows "{ {a..b} \<inter> k |k. k \<in> d \<and> {a..b} \<inter> k \<noteq> {} } division_of {a..b}" proof(cases "{a..b} = {}")
   2.317 -  case True show ?thesis unfolding True and division_of_trivial by auto next
   2.318 -  have *:"{a..b} \<inter> i = {a..b}" using assms(2) by auto 
   2.319 -  case False show ?thesis using division_inter[OF division_of_self[OF False] assms(1)] unfolding * by auto qed
   2.320 -
   2.321 -lemma elementary_inter: assumes "p1 division_of s" "p2 division_of (t::(real^'n) set)"
   2.322 -  shows "\<exists>p. p division_of (s \<inter> t)"
   2.323 -  by(rule,rule division_inter[OF assms])
   2.324 -
   2.325 -lemma elementary_inters: assumes "finite f" "f\<noteq>{}" "\<forall>s\<in>f. \<exists>p. p division_of (s::(real^'n) set)"
   2.326 -  shows "\<exists>p. p division_of (\<Inter> f)" using assms apply-proof(induct f rule:finite_induct)
   2.327 -case (insert x f) show ?case proof(cases "f={}")
   2.328 -  case True thus ?thesis unfolding True using insert by auto next
   2.329 -  case False guess p using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] ..
   2.330 -  moreover guess px using insert(5)[rule_format,OF insertI1] .. ultimately
   2.331 -  show ?thesis unfolding Inter_insert apply(rule_tac elementary_inter) by assumption+ qed qed auto
   2.332 -
   2.333 -lemma division_disjoint_union:
   2.334 -  assumes "p1 division_of s1" "p2 division_of s2" "interior s1 \<inter> interior s2 = {}"
   2.335 -  shows "(p1 \<union> p2) division_of (s1 \<union> s2)" proof(rule division_ofI) 
   2.336 -  note d1 = division_ofD[OF assms(1)] and d2 = division_ofD[OF assms(2)]
   2.337 -  show "finite (p1 \<union> p2)" using d1(1) d2(1) by auto
   2.338 -  show "\<Union>(p1 \<union> p2) = s1 \<union> s2" using d1(6) d2(6) by auto
   2.339 -  { fix k1 k2 assume as:"k1 \<in> p1 \<union> p2" "k2 \<in> p1 \<union> p2" "k1 \<noteq> k2" moreover let ?g="interior k1 \<inter> interior k2 = {}"
   2.340 -  { assume as:"k1\<in>p1" "k2\<in>p2" have ?g using subset_interior[OF d1(2)[OF as(1)]] subset_interior[OF d2(2)[OF as(2)]]
   2.341 -      using assms(3) by blast } moreover
   2.342 -  { assume as:"k1\<in>p2" "k2\<in>p1" have ?g using subset_interior[OF d1(2)[OF as(2)]] subset_interior[OF d2(2)[OF as(1)]]
   2.343 -      using assms(3) by blast} ultimately
   2.344 -  show ?g using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto }
   2.345 -  fix k assume k:"k \<in> p1 \<union> p2"  show "k \<subseteq> s1 \<union> s2" using k d1(2) d2(2) by auto
   2.346 -  show "k \<noteq> {}" using k d1(3) d2(3) by auto show "\<exists>a b. k = {a..b}" using k d1(4) d2(4) by auto qed
   2.347 -
   2.348 -lemma partial_division_extend_1:
   2.349 -  assumes "{c..d} \<subseteq> {a..b::real^'n}" "{c..d} \<noteq> {}"
   2.350 -  obtains p where "p division_of {a..b}" "{c..d} \<in> p"
   2.351 -proof- def n \<equiv> "CARD('n)" have n:"1 \<le> n" "0 < n" "n \<noteq> 0" unfolding n_def by auto
   2.352 -  guess \<pi> using ex_bij_betw_nat_finite_1[OF finite_UNIV[where 'a='n]] .. note \<pi>=this
   2.353 -  def \<pi>' \<equiv> "inv_into {1..n} \<pi>"
   2.354 -  have \<pi>':"bij_betw \<pi>' UNIV {1..n}" using bij_betw_inv_into[OF \<pi>] unfolding \<pi>'_def n_def by auto
   2.355 -  hence \<pi>'i:"\<And>i. \<pi>' i \<in> {1..n}" unfolding bij_betw_def by auto 
   2.356 -  have \<pi>\<pi>'[simp]:"\<And>i. \<pi> (\<pi>' i) = i" unfolding \<pi>'_def apply(rule f_inv_into_f) unfolding n_def using \<pi> unfolding bij_betw_def by auto
   2.357 -  have \<pi>'\<pi>[simp]:"\<And>i. i\<in>{1..n} \<Longrightarrow> \<pi>' (\<pi> i) = i" unfolding \<pi>'_def apply(rule inv_into_f_eq) using \<pi> unfolding n_def bij_betw_def by auto
   2.358 -  have "{c..d} \<noteq> {}" using assms by auto
   2.359 -  let ?p1 = "\<lambda>l. {(\<chi> i. if \<pi>' i < l then c$i else a$i) .. (\<chi> i. if \<pi>' i < l then d$i else if \<pi>' i = l then c$\<pi> l else b$i)}"
   2.360 -  let ?p2 = "\<lambda>l. {(\<chi> i. if \<pi>' i < l then c$i else if \<pi>' i = l then d$\<pi> l else a$i) .. (\<chi> i. if \<pi>' i < l then d$i else b$i)}"
   2.361 -  let ?p =  "{?p1 l |l. l \<in> {1..n+1}} \<union> {?p2 l |l. l \<in> {1..n+1}}"
   2.362 -  have abcd:"\<And>i. a $ i \<le> c $ i \<and> c$i \<le> d$i \<and> d $ i \<le> b $ i" using assms unfolding subset_interval interval_eq_empty by(auto simp add:not_le not_less)
   2.363 -  show ?thesis apply(rule that[of ?p]) apply(rule division_ofI)
   2.364 -  proof- have "\<And>i. \<pi>' i < Suc n"
   2.365 -    proof(rule ccontr,unfold not_less) fix i assume "Suc n \<le> \<pi>' i"
   2.366 -      hence "\<pi>' i \<notin> {1..n}" by auto thus False using \<pi>' unfolding bij_betw_def by auto
   2.367 -    qed hence "c = (\<chi> i. if \<pi>' i < Suc n then c $ i else a $ i)"
   2.368 -        "d = (\<chi> i. if \<pi>' i < Suc n then d $ i else if \<pi>' i = n + 1 then c $ \<pi> (n + 1) else b $ i)"
   2.369 -      unfolding Cart_eq Cart_lambda_beta using \<pi>' unfolding bij_betw_def by auto
   2.370 -    thus cdp:"{c..d} \<in> ?p" apply-apply(rule UnI1) unfolding mem_Collect_eq apply(rule_tac x="n + 1" in exI) by auto
   2.371 -    have "\<And>l. l\<in>{1..n+1} \<Longrightarrow> ?p1 l \<subseteq> {a..b}"  "\<And>l. l\<in>{1..n+1} \<Longrightarrow> ?p2 l \<subseteq> {a..b}"
   2.372 -      unfolding subset_eq apply(rule_tac[!] ballI,rule_tac[!] ccontr)
   2.373 -    proof- fix l assume l:"l\<in>{1..n+1}" fix x assume "x\<notin>{a..b}"
   2.374 -      then guess i unfolding mem_interval not_all .. note i=this
   2.375 -      show "x \<in> ?p1 l \<Longrightarrow> False" "x \<in> ?p2 l \<Longrightarrow> False" unfolding mem_interval apply(erule_tac[!] x=i in allE)
   2.376 -        apply(case_tac[!] "\<pi>' i < l", case_tac[!] "\<pi>' i = l") using abcd[of i] i by auto 
   2.377 -    qed moreover have "\<And>x. x \<in> {a..b} \<Longrightarrow> x \<in> \<Union>?p"
   2.378 -    proof- fix x assume x:"x\<in>{a..b}"
   2.379 -      { presume "x\<notin>{c..d} \<Longrightarrow> x \<in> \<Union>?p" thus "x \<in> \<Union>?p" using cdp by blast }
   2.380 -      let ?M = "{i. i\<in>{1..n+1} \<and> \<not> (c $ \<pi> i \<le> x $ \<pi> i \<and> x $ \<pi> i \<le> d $ \<pi> i)}"
   2.381 -      assume "x\<notin>{c..d}" then guess i0 unfolding mem_interval not_all ..
   2.382 -      hence "\<pi>' i0 \<in> ?M" using \<pi>' unfolding bij_betw_def by(auto intro!:le_SucI)
   2.383 -      hence M:"finite ?M" "?M \<noteq> {}" by auto
   2.384 -      def l \<equiv> "Min ?M" note l = Min_less_iff[OF M,unfolded l_def[symmetric]] Min_in[OF M,unfolded mem_Collect_eq l_def[symmetric]]
   2.385 -        Min_gr_iff[OF M,unfolded l_def[symmetric]]
   2.386 -      have "x\<in>?p1 l \<or> x\<in>?p2 l" using l(2)[THEN conjunct2] unfolding de_Morgan_conj not_le
   2.387 -        apply- apply(erule disjE) apply(rule disjI1) defer apply(rule disjI2)
   2.388 -      proof- assume as:"x $ \<pi> l < c $ \<pi> l"
   2.389 -        show "x \<in> ?p1 l" unfolding mem_interval Cart_lambda_beta
   2.390 -        proof case goal1 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le by auto
   2.391 -          thus ?case using as x[unfolded mem_interval,rule_format,of i]
   2.392 -            apply auto using l(3)[of "\<pi>' i"] by(auto elim!:ballE[where x="\<pi>' i"])
   2.393 -        qed
   2.394 -      next assume as:"x $ \<pi> l > d $ \<pi> l"
   2.395 -        show "x \<in> ?p2 l" unfolding mem_interval Cart_lambda_beta
   2.396 -        proof case goal1 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le by auto
   2.397 -          thus ?case using as x[unfolded mem_interval,rule_format,of i]
   2.398 -            apply auto using l(3)[of "\<pi>' i"] by(auto elim!:ballE[where x="\<pi>' i"])
   2.399 -        qed qed
   2.400 -      thus "x \<in> \<Union>?p" using l(2) by blast 
   2.401 -    qed ultimately show "\<Union>?p = {a..b}" apply-apply(rule) defer apply(rule) by(assumption,blast)
   2.402 -    
   2.403 -    show "finite ?p" by auto
   2.404 -    fix k assume k:"k\<in>?p" then obtain l where l:"k = ?p1 l \<or> k = ?p2 l" "l \<in> {1..n + 1}" by auto
   2.405 -    show "k\<subseteq>{a..b}" apply(rule,unfold mem_interval,rule,rule) 
   2.406 -    proof- fix i::'n and x assume "x \<in> k" moreover have "\<pi>' i < l \<or> \<pi>' i = l \<or> \<pi>' i > l" by auto
   2.407 -      ultimately show "a$i \<le> x$i" "x$i \<le> b$i" using abcd[of i] using l by(auto elim:disjE elim!:allE[where x=i] simp add:vector_le_def)
   2.408 -    qed have "\<And>l. ?p1 l \<noteq> {}" "\<And>l. ?p2 l \<noteq> {}" unfolding interval_eq_empty not_ex apply(rule_tac[!] allI)
   2.409 -    proof- case goal1 thus ?case using abcd[of x] by auto
   2.410 -    next   case goal2 thus ?case using abcd[of x] by auto
   2.411 -    qed thus "k \<noteq> {}" using k by auto
   2.412 -    show "\<exists>a b. k = {a..b}" using k by auto
   2.413 -    fix k' assume k':"k' \<in> ?p" "k \<noteq> k'" then obtain l' where l':"k' = ?p1 l' \<or> k' = ?p2 l'" "l' \<in> {1..n + 1}" by auto
   2.414 -    { fix k k' l l'
   2.415 -      assume k:"k\<in>?p" and l:"k = ?p1 l \<or> k = ?p2 l" "l \<in> {1..n + 1}" 
   2.416 -      assume k':"k' \<in> ?p" "k \<noteq> k'" and  l':"k' = ?p1 l' \<or> k' = ?p2 l'" "l' \<in> {1..n + 1}" 
   2.417 -      assume "l \<le> l'" fix x
   2.418 -      have "x \<notin> interior k \<inter> interior k'" 
   2.419 -      proof(rule,cases "l' = n+1") assume x:"x \<in> interior k \<inter> interior k'"
   2.420 -        case True hence "\<And>i. \<pi>' i < l'" using \<pi>'i by(auto simp add:less_Suc_eq_le)
   2.421 -        hence k':"k' = {c..d}" using l'(1) \<pi>'i by(auto simp add:Cart_nth_inverse)
   2.422 -        have ln:"l < n + 1" 
   2.423 -        proof(rule ccontr) case goal1 hence l2:"l = n+1" using l by auto
   2.424 -          hence "\<And>i. \<pi>' i < l" using \<pi>'i by(auto simp add:less_Suc_eq_le)
   2.425 -          hence "k = {c..d}" using l(1) \<pi>'i by(auto simp add:Cart_nth_inverse)
   2.426 -          thus False using `k\<noteq>k'` k' by auto
   2.427 -        qed have **:"\<pi>' (\<pi> l) = l" using \<pi>'\<pi>[of l] using l ln by auto
   2.428 -        have "x $ \<pi> l < c $ \<pi> l \<or> d $ \<pi> l < x $ \<pi> l" using l(1) apply-
   2.429 -        proof(erule disjE)
   2.430 -          assume as:"k = ?p1 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
   2.431 -          show ?thesis using *[of "\<pi> l"] using ln unfolding Cart_lambda_beta ** by auto
   2.432 -        next assume as:"k = ?p2 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
   2.433 -          show ?thesis using *[of "\<pi> l"] using ln unfolding Cart_lambda_beta ** by auto
   2.434 -        qed thus False using x unfolding k' unfolding Int_iff interior_closed_interval mem_interval
   2.435 -          by(auto elim!:allE[where x="\<pi> l"])
   2.436 -      next case False hence "l < n + 1" using l'(2) using `l\<le>l'` by auto
   2.437 -        hence ln:"l \<in> {1..n}" "l' \<in> {1..n}" using l l' False by auto
   2.438 -        note \<pi>l = \<pi>'\<pi>[OF ln(1)] \<pi>'\<pi>[OF ln(2)]
   2.439 -        assume x:"x \<in> interior k \<inter> interior k'"
   2.440 -        show False using l(1) l'(1) apply-
   2.441 -        proof(erule_tac[!] disjE)+
   2.442 -          assume as:"k = ?p1 l" "k' = ?p1 l'"
   2.443 -          note * = x[unfolded as Int_iff interior_closed_interval mem_interval]
   2.444 -          have "l \<noteq> l'" using k'(2)[unfolded as] by auto
   2.445 -          thus False using * by(smt Cart_lambda_beta \<pi>l)
   2.446 -        next assume as:"k = ?p2 l" "k' = ?p2 l'"
   2.447 -          note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
   2.448 -          have "l \<noteq> l'" apply(rule) using k'(2)[unfolded as] by auto
   2.449 -          thus False using *[of "\<pi> l"] *[of "\<pi> l'"]
   2.450 -            unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` by auto
   2.451 -        next assume as:"k = ?p1 l" "k' = ?p2 l'"
   2.452 -          note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
   2.453 -          show False using *[of "\<pi> l"] *[of "\<pi> l'"]
   2.454 -            unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` using abcd[of "\<pi> l'"] by smt 
   2.455 -        next assume as:"k = ?p2 l" "k' = ?p1 l'"
   2.456 -          note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
   2.457 -          show False using *[of "\<pi> l"] *[of "\<pi> l'"]
   2.458 -            unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` using abcd[of "\<pi> l'"] by smt
   2.459 -        qed qed } 
   2.460 -    from this[OF k l k' l'] this[OF k'(1) l' k _ l] have "\<And>x. x \<notin> interior k \<inter> interior k'"
   2.461 -      apply - apply(cases "l' \<le> l") using k'(2) by auto            
   2.462 -    thus "interior k \<inter> interior k' = {}" by auto        
   2.463 -qed qed
   2.464 -
   2.465 -lemma partial_division_extend_interval: assumes "p division_of (\<Union>p)" "(\<Union>p) \<subseteq> {a..b}"
   2.466 -  obtains q where "p \<subseteq> q" "q division_of {a..b::real^'n}" proof(cases "p = {}")
   2.467 -  case True guess q apply(rule elementary_interval[of a b]) .
   2.468 -  thus ?thesis apply- apply(rule that[of q]) unfolding True by auto next
   2.469 -  case False note p = division_ofD[OF assms(1)]
   2.470 -  have *:"\<forall>k\<in>p. \<exists>q. q division_of {a..b} \<and> k\<in>q" proof case goal1
   2.471 -    guess c using p(4)[OF goal1] .. then guess d .. note cd_ = this
   2.472 -    have *:"{c..d} \<subseteq> {a..b}" "{c..d} \<noteq> {}" using p(2,3)[OF goal1, unfolded cd_] using assms(2) by auto
   2.473 -    guess q apply(rule partial_division_extend_1[OF *]) . thus ?case unfolding cd_ by auto qed
   2.474 -  guess q using bchoice[OF *] .. note q = conjunctD2[OF this[rule_format]]
   2.475 -  have "\<And>x. x\<in>p \<Longrightarrow> \<exists>d. d division_of \<Union>(q x - {x})" apply(rule,rule_tac p="q x" in division_of_subset) proof-
   2.476 -    fix x assume x:"x\<in>p" show "q x division_of \<Union>q x" apply-apply(rule division_ofI)
   2.477 -      using division_ofD[OF q(1)[OF x]] by auto show "q x - {x} \<subseteq> q x" by auto qed
   2.478 -  hence "\<exists>d. d division_of \<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)" apply- apply(rule elementary_inters)
   2.479 -    apply(rule finite_imageI[OF p(1)]) unfolding image_is_empty apply(rule False) by auto
   2.480 -  then guess d .. note d = this
   2.481 -  show ?thesis apply(rule that[of "d \<union> p"]) proof-
   2.482 -    have *:"\<And>s f t. s \<noteq> {} \<Longrightarrow> (\<forall>i\<in>s. f i \<union> i = t) \<Longrightarrow> t = \<Inter> (f ` s) \<union> (\<Union>s)" by auto
   2.483 -    have *:"{a..b} = \<Inter> (\<lambda>i. \<Union>(q i - {i})) ` p \<union> \<Union>p" apply(rule *[OF False]) proof fix i assume i:"i\<in>p"
   2.484 -      show "\<Union>(q i - {i}) \<union> i = {a..b}" using division_ofD(6)[OF q(1)[OF i]] using q(2)[OF i] by auto qed
   2.485 -    show "d \<union> p division_of {a..b}" unfolding * apply(rule division_disjoint_union[OF d assms(1)])
   2.486 -      apply(rule inter_interior_unions_intervals) apply(rule p open_interior ballI)+ proof(assumption,rule)
   2.487 -      fix k assume k:"k\<in>p" have *:"\<And>u t s. u \<subseteq> s \<Longrightarrow> s \<inter> t = {} \<Longrightarrow> u \<inter> t = {}" by auto
   2.488 -      show "interior (\<Inter>(\<lambda>i. \<Union>(q i - {i})) ` p) \<inter> interior k = {}" apply(rule *[of _ "interior (\<Union>(q k - {k}))"])
   2.489 -	defer apply(subst Int_commute) apply(rule inter_interior_unions_intervals) proof- note qk=division_ofD[OF q(1)[OF k]]
   2.490 -	show "finite (q k - {k})" "open (interior k)"  "\<forall>t\<in>q k - {k}. \<exists>a b. t = {a..b}" using qk by auto
   2.491 -	show "\<forall>t\<in>q k - {k}. interior k \<inter> interior t = {}" using qk(5) using q(2)[OF k] by auto
   2.492 -	have *:"\<And>x s. x \<in> s \<Longrightarrow> \<Inter>s \<subseteq> x" by auto show "interior (\<Inter>(\<lambda>i. \<Union>(q i - {i})) ` p) \<subseteq> interior (\<Union>(q k - {k}))"
   2.493 -	  apply(rule subset_interior *)+ using k by auto qed qed qed auto qed
   2.494 -
   2.495 -lemma elementary_bounded[dest]: "p division_of s \<Longrightarrow> bounded (s::(real^'n) set)"
   2.496 -  unfolding division_of_def by(metis bounded_Union bounded_interval) 
   2.497 -
   2.498 -lemma elementary_subset_interval: "p division_of s \<Longrightarrow> \<exists>a b. s \<subseteq> {a..b::real^'n}"
   2.499 -  by(meson elementary_bounded bounded_subset_closed_interval)
   2.500 -
   2.501 -lemma division_union_intervals_exists: assumes "{a..b::real^'n} \<noteq> {}"
   2.502 -  obtains p where "(insert {a..b} p) division_of ({a..b} \<union> {c..d})" proof(cases "{c..d} = {}")
   2.503 -  case True show ?thesis apply(rule that[of "{}"]) unfolding True using assms by auto next
   2.504 -  case False note false=this show ?thesis proof(cases "{a..b} \<inter> {c..d} = {}")
   2.505 -  have *:"\<And>a b. {a,b} = {a} \<union> {b}" by auto
   2.506 -  case True show ?thesis apply(rule that[of "{{c..d}}"]) unfolding * apply(rule division_disjoint_union)
   2.507 -    using false True assms using interior_subset by auto next
   2.508 -  case False obtain u v where uv:"{a..b} \<inter> {c..d} = {u..v}" unfolding inter_interval by auto
   2.509 -  have *:"{u..v} \<subseteq> {c..d}" using uv by auto
   2.510 -  guess p apply(rule partial_division_extend_1[OF * False[unfolded uv]]) . note p=this division_ofD[OF this(1)]
   2.511 -  have *:"{a..b} \<union> {c..d} = {a..b} \<union> \<Union>(p - {{u..v}})" "\<And>x s. insert x s = {x} \<union> s" using p(8) unfolding uv[THEN sym] by auto
   2.512 -  show thesis apply(rule that[of "p - {{u..v}}"]) unfolding *(1) apply(subst *(2)) apply(rule division_disjoint_union)
   2.513 -    apply(rule,rule assms) apply(rule division_of_subset[of p]) apply(rule division_of_union_self[OF p(1)]) defer
   2.514 -    unfolding interior_inter[THEN sym] proof-
   2.515 -    have *:"\<And>cd p uv ab. p \<subseteq> cd \<Longrightarrow> ab \<inter> cd = uv \<Longrightarrow> ab \<inter> p = uv \<inter> p" by auto
   2.516 -    have "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = interior({u..v} \<inter> \<Union>(p - {{u..v}}))" 
   2.517 -      apply(rule arg_cong[of _ _ interior]) apply(rule *[OF _ uv]) using p(8) by auto
   2.518 -    also have "\<dots> = {}" unfolding interior_inter apply(rule inter_interior_unions_intervals) using p(6) p(7)[OF p(2)] p(3) by auto
   2.519 -    finally show "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = {}" by assumption qed auto qed qed
   2.520 -
   2.521 -lemma division_of_unions: assumes "finite f"  "\<And>p. p\<in>f \<Longrightarrow> p division_of (\<Union>p)"
   2.522 -  "\<And>k1 k2. \<lbrakk>k1 \<in> \<Union>f; k2 \<in> \<Union>f; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
   2.523 -  shows "\<Union>f division_of \<Union>\<Union>f" apply(rule division_ofI) prefer 5 apply(rule assms(3)|assumption)+
   2.524 -  apply(rule finite_Union assms(1))+ prefer 3 apply(erule UnionE) apply(rule_tac s=X in division_ofD(3)[OF assms(2)])
   2.525 -  using division_ofD[OF assms(2)] by auto
   2.526 -  
   2.527 -lemma elementary_union_interval: assumes "p division_of \<Union>p"
   2.528 -  obtains q where "q division_of ({a..b::real^'n} \<union> \<Union>p)" proof-
   2.529 -  note assm=division_ofD[OF assms]
   2.530 -  have lem1:"\<And>f s. \<Union>\<Union> (f ` s) = \<Union>(\<lambda>x.\<Union>(f x)) ` s" by auto
   2.531 -  have lem2:"\<And>f s. f \<noteq> {} \<Longrightarrow> \<Union>{s \<union> t |t. t \<in> f} = s \<union> \<Union>f" by auto
   2.532 -{ presume "p={} \<Longrightarrow> thesis" "{a..b} = {} \<Longrightarrow> thesis" "{a..b} \<noteq> {} \<Longrightarrow> interior {a..b} = {} \<Longrightarrow> thesis"
   2.533 -    "p\<noteq>{} \<Longrightarrow> interior {a..b}\<noteq>{} \<Longrightarrow> {a..b} \<noteq> {} \<Longrightarrow> thesis"
   2.534 -  thus thesis by auto
   2.535 -next assume as:"p={}" guess p apply(rule elementary_interval[of a b]) .
   2.536 -  thus thesis apply(rule_tac that[of p]) unfolding as by auto 
   2.537 -next assume as:"{a..b}={}" show thesis apply(rule that) unfolding as using assms by auto
   2.538 -next assume as:"interior {a..b} = {}" "{a..b} \<noteq> {}"
   2.539 -  show thesis apply(rule that[of "insert {a..b} p"],rule division_ofI)
   2.540 -    unfolding finite_insert apply(rule assm(1)) unfolding Union_insert  
   2.541 -    using assm(2-4) as apply- by(fastsimp dest: assm(5))+
   2.542 -next assume as:"p \<noteq> {}" "interior {a..b} \<noteq> {}" "{a..b}\<noteq>{}"
   2.543 -  have "\<forall>k\<in>p. \<exists>q. (insert {a..b} q) division_of ({a..b} \<union> k)" proof case goal1
   2.544 -    from assm(4)[OF this] guess c .. then guess d ..
   2.545 -    thus ?case apply-apply(rule division_union_intervals_exists[OF as(3),of c d]) by auto
   2.546 -  qed from bchoice[OF this] guess q .. note q=division_ofD[OF this[rule_format]]
   2.547 -  let ?D = "\<Union>{insert {a..b} (q k) | k. k \<in> p}"
   2.548 -  show thesis apply(rule that[of "?D"]) proof(rule division_ofI)
   2.549 -    have *:"{insert {a..b} (q k) |k. k \<in> p} = (\<lambda>k. insert {a..b} (q k)) ` p" by auto
   2.550 -    show "finite ?D" apply(rule finite_Union) unfolding * apply(rule finite_imageI) using assm(1) q(1) by auto
   2.551 -    show "\<Union>?D = {a..b} \<union> \<Union>p" unfolding * lem1 unfolding lem2[OF as(1), of "{a..b}",THEN sym]
   2.552 -      using q(6) by auto
   2.553 -    fix k assume k:"k\<in>?D" thus " k \<subseteq> {a..b} \<union> \<Union>p" using q(2) by auto
   2.554 -    show "k \<noteq> {}" using q(3) k by auto show "\<exists>a b. k = {a..b}" using q(4) k by auto
   2.555 -    fix k' assume k':"k'\<in>?D" "k\<noteq>k'"
   2.556 -    obtain x  where x: "k \<in>insert {a..b} (q x)"  "x\<in>p"  using k  by auto
   2.557 -    obtain x' where x':"k'\<in>insert {a..b} (q x')" "x'\<in>p" using k' by auto
   2.558 -    show "interior k \<inter> interior k' = {}" proof(cases "x=x'")
   2.559 -      case True show ?thesis apply(rule q(5)) using x x' k' unfolding True by auto
   2.560 -    next case False 
   2.561 -      { presume "k = {a..b} \<Longrightarrow> ?thesis" "k' = {a..b} \<Longrightarrow> ?thesis" 
   2.562 -        "k \<noteq> {a..b} \<Longrightarrow> k' \<noteq> {a..b} \<Longrightarrow> ?thesis"
   2.563 -        thus ?thesis by auto }
   2.564 -      { assume as':"k  = {a..b}" show ?thesis apply(rule q(5)) using x' k'(2) unfolding as' by auto }
   2.565 -      { assume as':"k' = {a..b}" show ?thesis apply(rule q(5)) using x  k'(2) unfolding as' by auto }
   2.566 -      assume as':"k \<noteq> {a..b}" "k' \<noteq> {a..b}"
   2.567 -      guess c using q(4)[OF x(2,1)] .. then guess d .. note c_d=this
   2.568 -      have "interior k  \<inter> interior {a..b} = {}" apply(rule q(5)) using x  k'(2) using as' by auto
   2.569 -      hence "interior k \<subseteq> interior x" apply-
   2.570 -        apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x(2,1)]]) by auto moreover
   2.571 -      guess c using q(4)[OF x'(2,1)] .. then guess d .. note c_d=this
   2.572 -      have "interior k' \<inter> interior {a..b} = {}" apply(rule q(5)) using x' k'(2) using as' by auto
   2.573 -      hence "interior k' \<subseteq> interior x'" apply-
   2.574 -        apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x'(2,1)]]) by auto
   2.575 -      ultimately show ?thesis using assm(5)[OF x(2) x'(2) False] by auto
   2.576 -    qed qed } qed
   2.577 -
   2.578 -lemma elementary_unions_intervals:
   2.579 -  assumes "finite f" "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = {a..b::real^'n}"
   2.580 -  obtains p where "p division_of (\<Union>f)" proof-
   2.581 -  have "\<exists>p. p division_of (\<Union>f)" proof(induct_tac f rule:finite_subset_induct) 
   2.582 -    show "\<exists>p. p division_of \<Union>{}" using elementary_empty by auto
   2.583 -    fix x F assume as:"finite F" "x \<notin> F" "\<exists>p. p division_of \<Union>F" "x\<in>f"
   2.584 -    from this(3) guess p .. note p=this
   2.585 -    from assms(2)[OF as(4)] guess a .. then guess b .. note ab=this
   2.586 -    have *:"\<Union>F = \<Union>p" using division_ofD[OF p] by auto
   2.587 -    show "\<exists>p. p division_of \<Union>insert x F" using elementary_union_interval[OF p[unfolded *], of a b]
   2.588 -      unfolding Union_insert ab * by auto
   2.589 -  qed(insert assms,auto) thus ?thesis apply-apply(erule exE,rule that) by auto qed
   2.590 -
   2.591 -lemma elementary_union: assumes "ps division_of s" "pt division_of (t::(real^'n) set)"
   2.592 -  obtains p where "p division_of (s \<union> t)"
   2.593 -proof- have "s \<union> t = \<Union>ps \<union> \<Union>pt" using assms unfolding division_of_def by auto
   2.594 -  hence *:"\<Union>(ps \<union> pt) = s \<union> t" by auto
   2.595 -  show ?thesis apply-apply(rule elementary_unions_intervals[of "ps\<union>pt"])
   2.596 -    unfolding * prefer 3 apply(rule_tac p=p in that)
   2.597 -    using assms[unfolded division_of_def] by auto qed
   2.598 -
   2.599 -lemma partial_division_extend: fixes t::"(real^'n) set"
   2.600 -  assumes "p division_of s" "q division_of t" "s \<subseteq> t"
   2.601 -  obtains r where "p \<subseteq> r" "r division_of t" proof-
   2.602 -  note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)]
   2.603 -  obtain a b where ab:"t\<subseteq>{a..b}" using elementary_subset_interval[OF assms(2)] by auto
   2.604 -  guess r1 apply(rule partial_division_extend_interval) apply(rule assms(1)[unfolded divp(6)[THEN sym]])
   2.605 -    apply(rule subset_trans) by(rule ab assms[unfolded divp(6)[THEN sym]])+  note r1 = this division_ofD[OF this(2)]
   2.606 -  guess p' apply(rule elementary_unions_intervals[of "r1 - p"]) using r1(3,6) by auto 
   2.607 -  then obtain r2 where r2:"r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)" 
   2.608 -    apply- apply(drule elementary_inter[OF _ assms(2)[unfolded divq(6)[THEN sym]]]) by auto
   2.609 -  { fix x assume x:"x\<in>t" "x\<notin>s"
   2.610 -    hence "x\<in>\<Union>r1" unfolding r1 using ab by auto
   2.611 -    then guess r unfolding Union_iff .. note r=this moreover
   2.612 -    have "r \<notin> p" proof assume "r\<in>p" hence "x\<in>s" using divp(2) r by auto
   2.613 -      thus False using x by auto qed
   2.614 -    ultimately have "x\<in>\<Union>(r1 - p)" by auto }
   2.615 -  hence *:"t = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)" unfolding divp divq using assms(3) by auto
   2.616 -  show ?thesis apply(rule that[of "p \<union> r2"]) unfolding * defer apply(rule division_disjoint_union)
   2.617 -    unfolding divp(6) apply(rule assms r2)+
   2.618 -  proof- have "interior s \<inter> interior (\<Union>(r1-p)) = {}"
   2.619 -    proof(rule inter_interior_unions_intervals)
   2.620 -      show "finite (r1 - p)" "open (interior s)" "\<forall>t\<in>r1-p. \<exists>a b. t = {a..b}" using r1 by auto
   2.621 -      have *:"\<And>s. (\<And>x. x \<in> s \<Longrightarrow> False) \<Longrightarrow> s = {}" by auto
   2.622 -      show "\<forall>t\<in>r1-p. interior s \<inter> interior t = {}" proof(rule)
   2.623 -        fix m x assume as:"m\<in>r1-p"
   2.624 -        have "interior m \<inter> interior (\<Union>p) = {}" proof(rule inter_interior_unions_intervals)
   2.625 -          show "finite p" "open (interior m)" "\<forall>t\<in>p. \<exists>a b. t = {a..b}" using divp by auto
   2.626 -          show "\<forall>t\<in>p. interior m \<inter> interior t = {}" apply(rule, rule r1(7)) using as using r1 by auto
   2.627 -        qed thus "interior s \<inter> interior m = {}" unfolding divp by auto
   2.628 -      qed qed        
   2.629 -    thus "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}" using interior_subset by auto
   2.630 -  qed auto qed
   2.631 -
   2.632 -subsection {* Tagged (partial) divisions. *}
   2.633 -
   2.634 -definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40) where
   2.635 -  "(s tagged_partial_division_of i) \<equiv>
   2.636 -        finite s \<and>
   2.637 -        (\<forall>x k. (x,k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
   2.638 -        (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ((x1,k1) \<noteq> (x2,k2))
   2.639 -                       \<longrightarrow> (interior(k1) \<inter> interior(k2) = {}))"
   2.640 -
   2.641 -lemma tagged_partial_division_ofD[dest]: assumes "s tagged_partial_division_of i"
   2.642 -  shows "finite s" "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k" "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
   2.643 -  "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
   2.644 -  "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2,k2) \<in> s \<Longrightarrow> (x1,k1) \<noteq> (x2,k2) \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}"
   2.645 -  using assms unfolding tagged_partial_division_of_def  apply- by blast+ 
   2.646 -
   2.647 -definition tagged_division_of (infixr "tagged'_division'_of" 40) where
   2.648 -  "(s tagged_division_of i) \<equiv>
   2.649 -        (s tagged_partial_division_of i) \<and> (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
   2.650 -
   2.651 -lemma tagged_division_of_finite[dest]: "s tagged_division_of i \<Longrightarrow> finite s"
   2.652 -  unfolding tagged_division_of_def tagged_partial_division_of_def by auto
   2.653 -
   2.654 -lemma tagged_division_of:
   2.655 - "(s tagged_division_of i) \<longleftrightarrow>
   2.656 -        finite s \<and>
   2.657 -        (\<forall>x k. (x,k) \<in> s
   2.658 -               \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
   2.659 -        (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ~((x1,k1) = (x2,k2))
   2.660 -                       \<longrightarrow> (interior(k1) \<inter> interior(k2) = {})) \<and>
   2.661 -        (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
   2.662 -  unfolding tagged_division_of_def tagged_partial_division_of_def by auto
   2.663 -
   2.664 -lemma tagged_division_ofI: assumes
   2.665 -  "finite s" "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k" "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"  "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
   2.666 -  "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2,k2) \<in> s \<Longrightarrow> ~((x1,k1) = (x2,k2)) \<Longrightarrow> (interior(k1) \<inter> interior(k2) = {})"
   2.667 -  "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
   2.668 -  shows "s tagged_division_of i"
   2.669 -  unfolding tagged_division_of apply(rule) defer apply rule
   2.670 -  apply(rule allI impI conjI assms)+ apply assumption
   2.671 -  apply(rule, rule assms, assumption) apply(rule assms, assumption)
   2.672 -  using assms(1,5-) apply- by blast+
   2.673 -
   2.674 -lemma tagged_division_ofD[dest]: assumes "s tagged_division_of i"
   2.675 -  shows "finite s" "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k" "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"  "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
   2.676 -  "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2,k2) \<in> s \<Longrightarrow> ~((x1,k1) = (x2,k2)) \<Longrightarrow> (interior(k1) \<inter> interior(k2) = {})"
   2.677 -  "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)" using assms unfolding tagged_division_of apply- by blast+
   2.678 -
   2.679 -lemma division_of_tagged_division: assumes"s tagged_division_of i"  shows "(snd ` s) division_of i"
   2.680 -proof(rule division_ofI) note assm=tagged_division_ofD[OF assms]
   2.681 -  show "\<Union>snd ` s = i" "finite (snd ` s)" using assm by auto
   2.682 -  fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto
   2.683 -  thus  "k \<subseteq> i" "k \<noteq> {}" "\<exists>a b. k = {a..b}" using assm apply- by fastsimp+
   2.684 -  fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto
   2.685 -  thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto
   2.686 -qed
   2.687 -
   2.688 -lemma partial_division_of_tagged_division: assumes "s tagged_partial_division_of i"
   2.689 -  shows "(snd ` s) division_of \<Union>(snd ` s)"
   2.690 -proof(rule division_ofI) note assm=tagged_partial_division_ofD[OF assms]
   2.691 -  show "finite (snd ` s)" "\<Union>snd ` s = \<Union>snd ` s" using assm by auto
   2.692 -  fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto
   2.693 -  thus "k\<noteq>{}" "\<exists>a b. k = {a..b}" "k \<subseteq> \<Union>snd ` s" using assm by auto
   2.694 -  fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto
   2.695 -  thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto
   2.696 -qed
   2.697 -
   2.698 -lemma tagged_partial_division_subset: assumes "s tagged_partial_division_of i" "t \<subseteq> s"
   2.699 -  shows "t tagged_partial_division_of i"
   2.700 -  using assms unfolding tagged_partial_division_of_def using finite_subset[OF assms(2)] by blast
   2.701 -
   2.702 -lemma setsum_over_tagged_division_lemma: fixes d::"(real^'m) set \<Rightarrow> 'a::real_normed_vector"
   2.703 -  assumes "p tagged_division_of i" "\<And>u v. {u..v} \<noteq> {} \<Longrightarrow> content {u..v} = 0 \<Longrightarrow> d {u..v} = 0"
   2.704 -  shows "setsum (\<lambda>(x,k). d k) p = setsum d (snd ` p)"
   2.705 -proof- note assm=tagged_division_ofD[OF assms(1)]
   2.706 -  have *:"(\<lambda>(x,k). d k) = d \<circ> snd" unfolding o_def apply(rule ext) by auto
   2.707 -  show ?thesis unfolding * apply(subst eq_commute) proof(rule setsum_reindex_nonzero)
   2.708 -    show "finite p" using assm by auto
   2.709 -    fix x y assume as:"x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y" 
   2.710 -    obtain a b where ab:"snd x = {a..b}" using assm(4)[of "fst x" "snd x"] as(1) by auto
   2.711 -    have "(fst x, snd y) \<in> p" "(fst x, snd y) \<noteq> y" unfolding as(4)[THEN sym] using as(1-3) by auto
   2.712 -    hence "interior (snd x) \<inter> interior (snd y) = {}" apply-apply(rule assm(5)[of "fst x" _ "fst y"]) using as by auto 
   2.713 -    hence "content {a..b} = 0" unfolding as(4)[THEN sym] ab content_eq_0_interior by auto
   2.714 -    hence "d {a..b} = 0" apply-apply(rule assms(2)) using assm(2)[of "fst x" "snd x"] as(1) unfolding ab[THEN sym] by auto
   2.715 -    thus "d (snd x) = 0" unfolding ab by auto qed qed
   2.716 -
   2.717 -lemma tag_in_interval: "p tagged_division_of i \<Longrightarrow> (x,k) \<in> p \<Longrightarrow> x \<in> i" by auto
   2.718 -
   2.719 -lemma tagged_division_of_empty: "{} tagged_division_of {}"
   2.720 -  unfolding tagged_division_of by auto
   2.721 -
   2.722 -lemma tagged_partial_division_of_trivial[simp]:
   2.723 - "p tagged_partial_division_of {} \<longleftrightarrow> p = {}"
   2.724 -  unfolding tagged_partial_division_of_def by auto
   2.725 -
   2.726 -lemma tagged_division_of_trivial[simp]:
   2.727 - "p tagged_division_of {} \<longleftrightarrow> p = {}"
   2.728 -  unfolding tagged_division_of by auto
   2.729 -
   2.730 -lemma tagged_division_of_self:
   2.731 - "x \<in> {a..b} \<Longrightarrow> {(x,{a..b})} tagged_division_of {a..b}"
   2.732 -  apply(rule tagged_division_ofI) by auto
   2.733 -
   2.734 -lemma tagged_division_union:
   2.735 -  assumes "p1 tagged_division_of s1"  "p2 tagged_division_of s2" "interior s1 \<inter> interior s2 = {}"
   2.736 -  shows "(p1 \<union> p2) tagged_division_of (s1 \<union> s2)"
   2.737 -proof(rule tagged_division_ofI) note p1=tagged_division_ofD[OF assms(1)] and p2=tagged_division_ofD[OF assms(2)]
   2.738 -  show "finite (p1 \<union> p2)" using p1(1) p2(1) by auto
   2.739 -  show "\<Union>{k. \<exists>x. (x, k) \<in> p1 \<union> p2} = s1 \<union> s2" using p1(6) p2(6) by blast
   2.740 -  fix x k assume xk:"(x,k)\<in>p1\<union>p2" show "x\<in>k" "\<exists>a b. k = {a..b}" using xk p1(2,4) p2(2,4) by auto
   2.741 -  show "k\<subseteq>s1\<union>s2" using xk p1(3) p2(3) by blast
   2.742 -  fix x' k' assume xk':"(x',k')\<in>p1\<union>p2" "(x,k) \<noteq> (x',k')"
   2.743 -  have *:"\<And>a b. a\<subseteq> s1 \<Longrightarrow> b\<subseteq> s2 \<Longrightarrow> interior a \<inter> interior b = {}" using assms(3) subset_interior by blast
   2.744 -  show "interior k \<inter> interior k' = {}" apply(cases "(x,k)\<in>p1", case_tac[!] "(x',k')\<in>p1")
   2.745 -    apply(rule p1(5)) prefer 4 apply(rule *) prefer 6 apply(subst Int_commute,rule *) prefer 8 apply(rule p2(5))
   2.746 -    using p1(3) p2(3) using xk xk' by auto qed 
   2.747 -
   2.748 -lemma tagged_division_unions:
   2.749 -  assumes "finite iset" "\<forall>i\<in>iset. (pfn(i) tagged_division_of i)"
   2.750 -  "\<forall>i1 \<in> iset. \<forall>i2 \<in> iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})"
   2.751 -  shows "\<Union>(pfn ` iset) tagged_division_of (\<Union>iset)"
   2.752 -proof(rule tagged_division_ofI)
   2.753 -  note assm = tagged_division_ofD[OF assms(2)[rule_format]]
   2.754 -  show "finite (\<Union>pfn ` iset)" apply(rule finite_Union) using assms by auto
   2.755 -  have "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>pfn ` iset} = \<Union>(\<lambda>i. \<Union>{k. \<exists>x. (x, k) \<in> pfn i}) ` iset" by blast 
   2.756 -  also have "\<dots> = \<Union>iset" using assm(6) by auto
   2.757 -  finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>pfn ` iset} = \<Union>iset" . 
   2.758 -  fix x k assume xk:"(x,k)\<in>\<Union>pfn ` iset" then obtain i where i:"i \<in> iset" "(x, k) \<in> pfn i" by auto
   2.759 -  show "x\<in>k" "\<exists>a b. k = {a..b}" "k \<subseteq> \<Union>iset" using assm(2-4)[OF i] using i(1) by auto
   2.760 -  fix x' k' assume xk':"(x',k')\<in>\<Union>pfn ` iset" "(x, k) \<noteq> (x', k')" then obtain i' where i':"i' \<in> iset" "(x', k') \<in> pfn i'" by auto
   2.761 -  have *:"\<And>a b. i\<noteq>i' \<Longrightarrow> a\<subseteq> i \<Longrightarrow> b\<subseteq> i' \<Longrightarrow> interior a \<inter> interior b = {}" using i(1) i'(1)
   2.762 -    using assms(3)[rule_format] subset_interior by blast
   2.763 -  show "interior k \<inter> interior k' = {}" apply(cases "i=i'")
   2.764 -    using assm(5)[OF i _ xk'(2)]  i'(2) using assm(3)[OF i] assm(3)[OF i'] defer apply-apply(rule *) by auto
   2.765 -qed
   2.766 -
   2.767 -lemma tagged_partial_division_of_union_self:
   2.768 -  assumes "p tagged_partial_division_of s" shows "p tagged_division_of (\<Union>(snd ` p))"
   2.769 -  apply(rule tagged_division_ofI) using tagged_partial_division_ofD[OF assms] by auto
   2.770 -
   2.771 -lemma tagged_division_of_union_self: assumes "p tagged_division_of s"
   2.772 -  shows "p tagged_division_of (\<Union>(snd ` p))"
   2.773 -  apply(rule tagged_division_ofI) using tagged_division_ofD[OF assms] by auto
   2.774 -
   2.775 -subsection {* Fine-ness of a partition w.r.t. a gauge. *}
   2.776 -
   2.777 -definition fine (infixr "fine" 46) where
   2.778 -  "d fine s \<longleftrightarrow> (\<forall>(x,k) \<in> s. k \<subseteq> d(x))"
   2.779 -
   2.780 -lemma fineI: assumes "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x"
   2.781 -  shows "d fine s" using assms unfolding fine_def by auto
   2.782 -
   2.783 -lemma fineD[dest]: assumes "d fine s"
   2.784 -  shows "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x" using assms unfolding fine_def by auto
   2.785 -
   2.786 -lemma fine_inter: "(\<lambda>x. d1 x \<inter> d2 x) fine p \<longleftrightarrow> d1 fine p \<and> d2 fine p"
   2.787 -  unfolding fine_def by auto
   2.788 -
   2.789 -lemma fine_inters:
   2.790 - "(\<lambda>x. \<Inter> {f d x | d.  d \<in> s}) fine p \<longleftrightarrow> (\<forall>d\<in>s. (f d) fine p)"
   2.791 -  unfolding fine_def by blast
   2.792 -
   2.793 -lemma fine_union:
   2.794 -  "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)"
   2.795 -  unfolding fine_def by blast
   2.796 -
   2.797 -lemma fine_unions:"(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)"
   2.798 -  unfolding fine_def by auto
   2.799 -
   2.800 -lemma fine_subset:  "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p"
   2.801 -  unfolding fine_def by blast
   2.802 -
   2.803 -subsection {* Gauge integral. Define on compact intervals first, then use a limit. *}
   2.804 -
   2.805 -definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46) where
   2.806 -  "(f has_integral_compact_interval y) i \<equiv>
   2.807 -        (\<forall>e>0. \<exists>d. gauge d \<and>
   2.808 -          (\<forall>p. p tagged_division_of i \<and> d fine p
   2.809 -                        \<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - y) < e))"
   2.810 -
   2.811 -definition has_integral (infixr "has'_integral" 46) where 
   2.812 -"((f::(real^'n \<Rightarrow> 'b::real_normed_vector)) has_integral y) i \<equiv>
   2.813 -        if (\<exists>a b. i = {a..b}) then (f has_integral_compact_interval y) i
   2.814 -        else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b}
   2.815 -              \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral_compact_interval z) {a..b} \<and>
   2.816 -                                       norm(z - y) < e))"
   2.817 -
   2.818 -lemma has_integral:
   2.819 - "(f has_integral y) ({a..b}) \<longleftrightarrow>
   2.820 -        (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p
   2.821 -                        \<longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
   2.822 -  unfolding has_integral_def has_integral_compact_interval_def by auto
   2.823 -
   2.824 -lemma has_integralD[dest]: assumes
   2.825 - "(f has_integral y) ({a..b})" "e>0"
   2.826 -  obtains d where "gauge d" "\<And>p. p tagged_division_of {a..b} \<Longrightarrow> d fine p
   2.827 -                        \<Longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f(x)) p - y) < e"
   2.828 -  using assms unfolding has_integral by auto
   2.829 -
   2.830 -lemma has_integral_alt:
   2.831 - "(f has_integral y) i \<longleftrightarrow>
   2.832 -      (if (\<exists>a b. i = {a..b}) then (f has_integral y) i
   2.833 -       else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b}
   2.834 -                               \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0)
   2.835 -                                        has_integral z) ({a..b}) \<and>
   2.836 -                                       norm(z - y) < e)))"
   2.837 -  unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto
   2.838 -
   2.839 -lemma has_integral_altD:
   2.840 -  assumes "(f has_integral y) i" "\<not> (\<exists>a b. i = {a..b})" "e>0"
   2.841 -  obtains B where "B>0" "\<forall>a b. ball 0 B \<subseteq> {a..b}\<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0) has_integral z) ({a..b}) \<and> norm(z - y) < e)"
   2.842 -  using assms unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto
   2.843 -
   2.844 -definition integrable_on (infixr "integrable'_on" 46) where
   2.845 -  "(f integrable_on i) \<equiv> \<exists>y. (f has_integral y) i"
   2.846 -
   2.847 -definition "integral i f \<equiv> SOME y. (f has_integral y) i"
   2.848 -
   2.849 -lemma integrable_integral[dest]:
   2.850 - "f integrable_on i \<Longrightarrow> (f has_integral (integral i f)) i"
   2.851 -  unfolding integrable_on_def integral_def by(rule someI_ex)
   2.852 -
   2.853 -lemma has_integral_integrable[intro]: "(f has_integral i) s \<Longrightarrow> f integrable_on s"
   2.854 -  unfolding integrable_on_def by auto
   2.855 -
   2.856 -lemma has_integral_integral:"f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s"
   2.857 -  by auto
   2.858 -
   2.859 -lemma setsum_content_null:
   2.860 -  assumes "content({a..b}) = 0" "p tagged_division_of {a..b}"
   2.861 -  shows "setsum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)"
   2.862 -proof(rule setsum_0',rule) fix y assume y:"y\<in>p"
   2.863 -  obtain x k where xk:"y = (x,k)" using surj_pair[of y] by blast
   2.864 -  note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]]
   2.865 -  from this(2) guess c .. then guess d .. note c_d=this
   2.866 -  have "(\<lambda>(x, k). content k *\<^sub>R f x) y = content k *\<^sub>R f x" unfolding xk by auto
   2.867 -  also have "\<dots> = 0" using content_subset[OF assm(1)[unfolded c_d]] content_pos_le[of c d]
   2.868 -    unfolding assms(1) c_d by auto
   2.869 -  finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" .
   2.870 -qed
   2.871 -
   2.872 -subsection {* Some basic combining lemmas. *}
   2.873 -
   2.874 -lemma tagged_division_unions_exists:
   2.875 -  assumes "finite iset" "\<forall>i \<in> iset. \<exists>p. p tagged_division_of i \<and> d fine p"
   2.876 -  "\<forall>i1\<in>iset. \<forall>i2\<in>iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})" "(\<Union>iset = i)"
   2.877 -   obtains p where "p tagged_division_of i" "d fine p"
   2.878 -proof- guess pfn using bchoice[OF assms(2)] .. note pfn = conjunctD2[OF this[rule_format]]
   2.879 -  show thesis apply(rule_tac p="\<Union>(pfn ` iset)" in that) unfolding assms(4)[THEN sym]
   2.880 -    apply(rule tagged_division_unions[OF assms(1) _ assms(3)]) defer 
   2.881 -    apply(rule fine_unions) using pfn by auto
   2.882 -qed
   2.883 -
   2.884 -subsection {* The set we're concerned with must be closed. *}
   2.885 -
   2.886 -lemma division_of_closed: "s division_of i \<Longrightarrow> closed (i::(real^'n) set)"
   2.887 -  unfolding division_of_def by(fastsimp intro!: closed_Union closed_interval)
   2.888 -
   2.889 -subsection {* General bisection principle for intervals; might be useful elsewhere. *}
   2.890 -
   2.891 -lemma interval_bisection_step:
   2.892 -  assumes "P {}" "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))" "~(P {a..b::real^'n})"
   2.893 -  obtains c d where "~(P{c..d})"
   2.894 -  "\<forall>i. a$i \<le> c$i \<and> c$i \<le> d$i \<and> d$i \<le> b$i \<and> 2 * (d$i - c$i) \<le> b$i - a$i"
   2.895 -proof- have "{a..b} \<noteq> {}" using assms(1,3) by auto
   2.896 -  note ab=this[unfolded interval_eq_empty not_ex not_less]
   2.897 -  { fix f have "finite f \<Longrightarrow>
   2.898 -        (\<forall>s\<in>f. P s) \<Longrightarrow>
   2.899 -        (\<forall>s\<in>f. \<exists>a b. s = {a..b}) \<Longrightarrow>
   2.900 -        (\<forall>s\<in>f.\<forall>t\<in>f. ~(s = t) \<longrightarrow> interior(s) \<inter> interior(t) = {}) \<Longrightarrow> P(\<Union>f)"
   2.901 -    proof(induct f rule:finite_induct)
   2.902 -      case empty show ?case using assms(1) by auto
   2.903 -    next case (insert x f) show ?case unfolding Union_insert apply(rule assms(2)[rule_format])
   2.904 -        apply rule defer apply rule defer apply(rule inter_interior_unions_intervals)
   2.905 -        using insert by auto
   2.906 -    qed } note * = this
   2.907 -  let ?A = "{{c..d} | c d. \<forall>i. (c$i = a$i) \<and> (d$i = (a$i + b$i) / 2) \<or> (c$i = (a$i + b$i) / 2) \<and> (d$i = b$i)}"
   2.908 -  let ?PP = "\<lambda>c d. \<forall>i. a$i \<le> c$i \<and> c$i \<le> d$i \<and> d$i \<le> b$i \<and> 2 * (d$i - c$i) \<le> b$i - a$i"
   2.909 -  { presume "\<forall>c d. ?PP c d \<longrightarrow> P {c..d} \<Longrightarrow> False"
   2.910 -    thus thesis unfolding atomize_not not_all apply-apply(erule exE)+ apply(rule_tac c=x and d=xa in that) by auto }
   2.911 -  assume as:"\<forall>c d. ?PP c d \<longrightarrow> P {c..d}"
   2.912 -  have "P (\<Union> ?A)" proof(rule *, rule_tac[2-] ballI, rule_tac[4] ballI, rule_tac[4] impI) 
   2.913 -    let ?B = "(\<lambda>s.{(\<chi> i. if i \<in> s then a$i else (a$i + b$i) / 2) ..
   2.914 -      (\<chi> i. if i \<in> s then (a$i + b$i) / 2 else b$i)}) ` {s. s \<subseteq> UNIV}"
   2.915 -    have "?A \<subseteq> ?B" proof case goal1
   2.916 -      then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+) note c_d=this[rule_format]
   2.917 -      have *:"\<And>a b c d. a = c \<Longrightarrow> b = d \<Longrightarrow> {a..b} = {c..d}" by auto
   2.918 -      show "x\<in>?B" unfolding image_iff apply(rule_tac x="{i. c$i = a$i}" in bexI)
   2.919 -        unfolding c_d apply(rule * ) unfolding Cart_eq cond_component Cart_lambda_beta
   2.920 -      proof(rule_tac[1-2] allI) fix i show "c $ i = (if i \<in> {i. c $ i = a $ i} then a $ i else (a $ i + b $ i) / 2)"
   2.921 -          "d $ i = (if i \<in> {i. c $ i = a $ i} then (a $ i + b $ i) / 2 else b $ i)"
   2.922 -          using c_d(2)[of i] ab[THEN spec[where x=i]] by(auto simp add:field_simps)
   2.923 -      qed auto qed
   2.924 -    thus "finite ?A" apply(rule finite_subset[of _ ?B]) by auto
   2.925 -    fix s assume "s\<in>?A" then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+)
   2.926 -    note c_d=this[rule_format]
   2.927 -    show "P s" unfolding c_d apply(rule as[rule_format]) proof- case goal1 show ?case 
   2.928 -        using c_d(2)[of i] using ab[THEN spec[where x=i]] by auto qed
   2.929 -    show "\<exists>a b. s = {a..b}" unfolding c_d by auto
   2.930 -    fix t assume "t\<in>?A" then guess e unfolding mem_Collect_eq .. then guess f apply- by(erule exE,(erule conjE)+)
   2.931 -    note e_f=this[rule_format]
   2.932 -    assume "s \<noteq> t" hence "\<not> (c = e \<and> d = f)" unfolding c_d e_f by auto
   2.933 -    then obtain i where "c$i \<noteq> e$i \<or> d$i \<noteq> f$i" unfolding de_Morgan_conj Cart_eq by auto
   2.934 -    hence i:"c$i \<noteq> e$i" "d$i \<noteq> f$i" apply- apply(erule_tac[!] disjE)
   2.935 -    proof- assume "c$i \<noteq> e$i" thus "d$i \<noteq> f$i" using c_d(2)[of i] e_f(2)[of i] by fastsimp
   2.936 -    next   assume "d$i \<noteq> f$i" thus "c$i \<noteq> e$i" using c_d(2)[of i] e_f(2)[of i] by fastsimp
   2.937 -    qed have *:"\<And>s t. (\<And>a. a\<in>s \<Longrightarrow> a\<in>t \<Longrightarrow> False) \<Longrightarrow> s \<inter> t = {}" by auto
   2.938 -    show "interior s \<inter> interior t = {}" unfolding e_f c_d interior_closed_interval proof(rule *)
   2.939 -      fix x assume "x\<in>{c<..<d}" "x\<in>{e<..<f}"
   2.940 -      hence x:"c$i < d$i" "e$i < f$i" "c$i < f$i" "e$i < d$i" unfolding mem_interval apply-apply(erule_tac[!] x=i in allE)+ by auto
   2.941 -      show False using c_d(2)[of i] apply- apply(erule_tac disjE)
   2.942 -      proof(erule_tac[!] conjE) assume as:"c $ i = a $ i" "d $ i = (a $ i + b $ i) / 2"
   2.943 -        show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps)
   2.944 -      next assume as:"c $ i = (a $ i + b $ i) / 2" "d $ i = b $ i"
   2.945 -        show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps)
   2.946 -      qed qed qed
   2.947 -  also have "\<Union> ?A = {a..b}" proof(rule set_ext,rule)
   2.948 -    fix x assume "x\<in>\<Union>?A" then guess Y unfolding Union_iff ..
   2.949 -    from this(1) guess c unfolding mem_Collect_eq .. then guess d ..
   2.950 -    note c_d = this[THEN conjunct2,rule_format] `x\<in>Y`[unfolded this[THEN conjunct1]]
   2.951 -    show "x\<in>{a..b}" unfolding mem_interval proof 
   2.952 -      fix i show "a $ i \<le> x $ i \<and> x $ i \<le> b $ i"
   2.953 -        using c_d(1)[of i] c_d(2)[unfolded mem_interval,THEN spec[where x=i]] by auto qed
   2.954 -  next fix x assume x:"x\<in>{a..b}"
   2.955 -    have "\<forall>i. \<exists>c d. (c = a$i \<and> d = (a$i + b$i) / 2 \<or> c = (a$i + b$i) / 2 \<and> d = b$i) \<and> c\<le>x$i \<and> x$i \<le> d"
   2.956 -      (is "\<forall>i. \<exists>c d. ?P i c d") unfolding mem_interval proof fix i
   2.957 -      have "?P i (a$i) ((a $ i + b $ i) / 2) \<or> ?P i ((a $ i + b $ i) / 2) (b$i)"
   2.958 -        using x[unfolded mem_interval,THEN spec[where x=i]] by auto thus "\<exists>c d. ?P i c d" by blast
   2.959 -    qed thus "x\<in>\<Union>?A" unfolding Union_iff lambda_skolem unfolding Bex_def mem_Collect_eq
   2.960 -      apply-apply(erule exE)+ apply(rule_tac x="{xa..xaa}" in exI) unfolding mem_interval by auto
   2.961 -  qed finally show False using assms by auto qed
   2.962 -
   2.963 -lemma interval_bisection:
   2.964 -  assumes "P {}" "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))" "\<not> P {a..b::real^'n}"
   2.965 -  obtains x where "x \<in> {a..b}" "\<forall>e>0. \<exists>c d. x \<in> {c..d} \<and> {c..d} \<subseteq> ball x e \<and> {c..d} \<subseteq> {a..b} \<and> ~P({c..d})"
   2.966 -proof-
   2.967 -  have "\<forall>x. \<exists>y. \<not> P {fst x..snd x} \<longrightarrow> (\<not> P {fst y..snd y} \<and> (\<forall>i. fst x$i \<le> fst y$i \<and> fst y$i \<le> snd y$i \<and> snd y$i \<le> snd x$i \<and>
   2.968 -                           2 * (snd y$i - fst y$i) \<le> snd x$i - fst x$i))" proof case goal1 thus ?case proof-
   2.969 -      presume "\<not> P {fst x..snd x} \<Longrightarrow> ?thesis"
   2.970 -      thus ?thesis apply(cases "P {fst x..snd x}") by auto
   2.971 -    next assume as:"\<not> P {fst x..snd x}" from interval_bisection_step[of P, OF assms(1-2) as] guess c d . 
   2.972 -      thus ?thesis apply- apply(rule_tac x="(c,d)" in exI) by auto
   2.973 -    qed qed then guess f apply-apply(drule choice) by(erule exE) note f=this
   2.974 -  def AB \<equiv> "\<lambda>n. (f ^^ n) (a,b)" def A \<equiv> "\<lambda>n. fst(AB n)" and B \<equiv> "\<lambda>n. snd(AB n)" note ab_def = this AB_def
   2.975 -  have "A 0 = a" "B 0 = b" "\<And>n. \<not> P {A(Suc n)..B(Suc n)} \<and>
   2.976 -    (\<forall>i. A(n)$i \<le> A(Suc n)$i \<and> A(Suc n)$i \<le> B(Suc n)$i \<and> B(Suc n)$i \<le> B(n)$i \<and> 
   2.977 -    2 * (B(Suc n)$i - A(Suc n)$i) \<le> B(n)$i - A(n)$i)" (is "\<And>n. ?P n")
   2.978 -  proof- show "A 0 = a" "B 0 = b" unfolding ab_def by auto
   2.979 -    case goal3 note S = ab_def funpow.simps o_def id_apply show ?case
   2.980 -    proof(induct n) case 0 thus ?case unfolding S apply(rule f[rule_format]) using assms(3) by auto
   2.981 -    next case (Suc n) show ?case unfolding S apply(rule f[rule_format]) using Suc unfolding S by auto
   2.982 -    qed qed note AB = this(1-2) conjunctD2[OF this(3),rule_format]
   2.983 -
   2.984 -  have interv:"\<And>e. 0 < e \<Longrightarrow> \<exists>n. \<forall>x\<in>{A n..B n}. \<forall>y\<in>{A n..B n}. dist x y < e"
   2.985 -  proof- case goal1 guess n using real_arch_pow2[of "(setsum (\<lambda>i. b$i - a$i) UNIV) / e"] .. note n=this
   2.986 -    show ?case apply(rule_tac x=n in exI) proof(rule,rule)
   2.987 -      fix x y assume xy:"x\<in>{A n..B n}" "y\<in>{A n..B n}"
   2.988 -      have "dist x y \<le> setsum (\<lambda>i. abs((x - y)$i)) UNIV" unfolding vector_dist_norm by(rule norm_le_l1)
   2.989 -      also have "\<dots> \<le> setsum (\<lambda>i. B n$i - A n$i) UNIV"
   2.990 -      proof(rule setsum_mono) fix i show "\<bar>(x - y) $ i\<bar> \<le> B n $ i - A n $ i"
   2.991 -          using xy[unfolded mem_interval,THEN spec[where x=i]]
   2.992 -          unfolding vector_minus_component by auto qed
   2.993 -      also have "\<dots> \<le> setsum (\<lambda>i. b$i - a$i) UNIV / 2^n" unfolding setsum_divide_distrib
   2.994 -      proof(rule setsum_mono) case goal1 thus ?case
   2.995 -        proof(induct n) case 0 thus ?case unfolding AB by auto
   2.996 -        next case (Suc n) have "B (Suc n) $ i - A (Suc n) $ i \<le> (B n $ i - A n $ i) / 2" using AB(4)[of n i] by auto
   2.997 -          also have "\<dots> \<le> (b $ i - a $ i) / 2 ^ Suc n" using Suc by(auto simp add:field_simps) finally show ?case .
   2.998 -        qed qed
   2.999 -      also have "\<dots> < e" using n using goal1 by(auto simp add:field_simps) finally show "dist x y < e" .
  2.1000 -    qed qed
  2.1001 -  { fix n m ::nat assume "m \<le> n" then guess d unfolding le_Suc_ex_iff .. note d=this
  2.1002 -    have "{A n..B n} \<subseteq> {A m..B m}" unfolding d 
  2.1003 -    proof(induct d) case 0 thus ?case by auto
  2.1004 -    next case (Suc d) show ?case apply(rule subset_trans[OF _ Suc])
  2.1005 -        apply(rule) unfolding mem_interval apply(rule,erule_tac x=i in allE)
  2.1006 -      proof- case goal1 thus ?case using AB(4)[of "m + d" i] by(auto simp add:field_simps)
  2.1007 -      qed qed } note ABsubset = this 
  2.1008 -  have "\<exists>a. \<forall>n. a\<in>{A n..B n}" apply(rule decreasing_closed_nest[rule_format,OF closed_interval _ ABsubset interv])
  2.1009 -  proof- fix n show "{A n..B n} \<noteq> {}" apply(cases "0<n") using AB(3)[of "n - 1"] assms(1,3) AB(1-2) by auto qed auto
  2.1010 -  then guess x0 .. note x0=this[rule_format]
  2.1011 -  show thesis proof(rule that[rule_format,of x0])
  2.1012 -    show "x0\<in>{a..b}" using x0[of 0] unfolding AB .
  2.1013 -    fix e assume "0 < (e::real)" from interv[OF this] guess n .. note n=this
  2.1014 -    show "\<exists>c d. x0 \<in> {c..d} \<and> {c..d} \<subseteq> ball x0 e \<and> {c..d} \<subseteq> {a..b} \<and> \<not> P {c..d}"
  2.1015 -      apply(rule_tac x="A n" in exI,rule_tac x="B n" in exI) apply(rule,rule x0) apply rule defer 
  2.1016 -    proof show "\<not> P {A n..B n}" apply(cases "0<n") using AB(3)[of "n - 1"] assms(3) AB(1-2) by auto
  2.1017 -      show "{A n..B n} \<subseteq> ball x0 e" using n using x0[of n] by auto
  2.1018 -      show "{A n..B n} \<subseteq> {a..b}" unfolding AB(1-2)[symmetric] apply(rule ABsubset) by auto
  2.1019 -    qed qed qed 
  2.1020 -
  2.1021 -subsection {* Cousin's lemma. *}
  2.1022 -
  2.1023 -lemma fine_division_exists: assumes "gauge g" 
  2.1024 -  obtains p where "p tagged_division_of {a..b::real^'n}" "g fine p"
  2.1025 -proof- presume "\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p) \<Longrightarrow> False"
  2.1026 -  then guess p unfolding atomize_not not_not .. thus thesis apply-apply(rule that[of p]) by auto
  2.1027 -next assume as:"\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p)"
  2.1028 -  guess x apply(rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p",rule_format,OF _ _ as])
  2.1029 -    apply(rule_tac x="{}" in exI) defer apply(erule conjE exE)+
  2.1030 -  proof- show "{} tagged_division_of {} \<and> g fine {}" unfolding fine_def by auto
  2.1031 -    fix s t p p' assume "p tagged_division_of s" "g fine p" "p' tagged_division_of t" "g fine p'" "interior s \<inter> interior t = {}"
  2.1032 -    thus "\<exists>p. p tagged_division_of s \<union> t \<and> g fine p" apply-apply(rule_tac x="p \<union> p'" in exI) apply rule
  2.1033 -      apply(rule tagged_division_union) prefer 4 apply(rule fine_union) by auto
  2.1034 -  qed note x=this
  2.1035 -  obtain e where e:"e>0" "ball x e \<subseteq> g x" using gaugeD[OF assms, of x] unfolding open_contains_ball by auto
  2.1036 -  from x(2)[OF e(1)] guess c d apply-apply(erule exE conjE)+ . note c_d = this
  2.1037 -  have "g fine {(x, {c..d})}" unfolding fine_def using e using c_d(2) by auto
  2.1038 -  thus False using tagged_division_of_self[OF c_d(1)] using c_d by auto qed
  2.1039 -
  2.1040 -subsection {* Basic theorems about integrals. *}
  2.1041 -
  2.1042 -lemma has_integral_unique: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
  2.1043 -  assumes "(f has_integral k1) i" "(f has_integral k2) i" shows "k1 = k2"
  2.1044 -proof(rule ccontr) let ?e = "norm(k1 - k2) / 2" assume as:"k1 \<noteq> k2" hence e:"?e > 0" by auto
  2.1045 -  have lem:"\<And>f::real^'n \<Rightarrow> 'a.  \<And> a b k1 k2.
  2.1046 -    (f has_integral k1) ({a..b}) \<Longrightarrow> (f has_integral k2) ({a..b}) \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> False"
  2.1047 -  proof- case goal1 let ?e = "norm(k1 - k2) / 2" from goal1(3) have e:"?e > 0" by auto
  2.1048 -    guess d1 by(rule has_integralD[OF goal1(1) e]) note d1=this
  2.1049 -    guess d2 by(rule has_integralD[OF goal1(2) e]) note d2=this
  2.1050 -    guess p by(rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)],of a b]) note p=this
  2.1051 -    let ?c = "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" have "norm (k1 - k2) \<le> norm (?c - k2) + norm (?c - k1)"
  2.1052 -      using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"] by(auto simp add:group_simps norm_minus_commute)
  2.1053 -    also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
  2.1054 -      apply(rule add_strict_mono) apply(rule_tac[!] d2(2) d1(2)) using p unfolding fine_def by auto
  2.1055 -    finally show False by auto
  2.1056 -  qed { presume "\<not> (\<exists>a b. i = {a..b}) \<Longrightarrow> False"
  2.1057 -    thus False apply-apply(cases "\<exists>a b. i = {a..b}")
  2.1058 -      using assms by(auto simp add:has_integral intro:lem[OF _ _ as]) }
  2.1059 -  assume as:"\<not> (\<exists>a b. i = {a..b})"
  2.1060 -  guess B1 by(rule has_integral_altD[OF assms(1) as,OF e]) note B1=this[rule_format]
  2.1061 -  guess B2 by(rule has_integral_altD[OF assms(2) as,OF e]) note B2=this[rule_format]
  2.1062 -  have "\<exists>a b::real^'n. ball 0 B1 \<union> ball 0 B2 \<subseteq> {a..b}" apply(rule bounded_subset_closed_interval)
  2.1063 -    using bounded_Un bounded_ball by auto then guess a b apply-by(erule exE)+
  2.1064 -  note ab=conjunctD2[OF this[unfolded Un_subset_iff]]
  2.1065 -  guess w using B1(2)[OF ab(1)] .. note w=conjunctD2[OF this]
  2.1066 -  guess z using B2(2)[OF ab(2)] .. note z=conjunctD2[OF this]
  2.1067 -  have "z = w" using lem[OF w(1) z(1)] by auto
  2.1068 -  hence "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)"
  2.1069 -    using norm_triangle_ineq4[of "k1 - w" "k2 - z"] by(auto simp add: norm_minus_commute) 
  2.1070 -  also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2" apply(rule add_strict_mono) by(rule_tac[!] z(2) w(2))
  2.1071 -  finally show False by auto qed
  2.1072 -
  2.1073 -lemma integral_unique[intro]:
  2.1074 -  "(f has_integral y) k \<Longrightarrow> integral k f = y"
  2.1075 -  unfolding integral_def apply(rule some_equality) by(auto intro: has_integral_unique) 
  2.1076 -
  2.1077 -lemma has_integral_is_0: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector" 
  2.1078 -  assumes "\<forall>x\<in>s. f x = 0" shows "(f has_integral 0) s"
  2.1079 -proof- have lem:"\<And>a b. \<And>f::real^'n \<Rightarrow> 'a.
  2.1080 -    (\<forall>x\<in>{a..b}. f(x) = 0) \<Longrightarrow> (f has_integral 0) ({a..b})" unfolding has_integral
  2.1081 -  proof(rule,rule) fix a b e and f::"real^'n \<Rightarrow> 'a"
  2.1082 -    assume as:"\<forall>x\<in>{a..b}. f x = 0" "0 < (e::real)"
  2.1083 -    show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e)"
  2.1084 -      apply(rule_tac x="\<lambda>x. ball x 1" in exI)  apply(rule,rule gaugeI) unfolding centre_in_ball defer apply(rule open_ball)
  2.1085 -    proof(rule,rule,erule conjE) case goal1
  2.1086 -      have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) = 0" proof(rule setsum_0',rule)
  2.1087 -        fix x assume x:"x\<in>p" have "f (fst x) = 0" using tagged_division_ofD(2-3)[OF goal1(1), of "fst x" "snd x"] using as x by auto
  2.1088 -        thus "(\<lambda>(x, k). content k *\<^sub>R f x) x = 0" apply(subst surjective_pairing[of x]) unfolding split_conv by auto
  2.1089 -      qed thus ?case using as by auto
  2.1090 -    qed auto qed  { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
  2.1091 -    thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}")
  2.1092 -      using assms by(auto simp add:has_integral intro:lem) }
  2.1093 -  have *:"(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. 0)" apply(rule ext) using assms by auto
  2.1094 -  assume "\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P *
  2.1095 -  apply(rule,rule,rule_tac x=1 in exI,rule) defer apply(rule,rule,rule)
  2.1096 -  proof- fix e::real and a b assume "e>0"
  2.1097 -    thus "\<exists>z. ((\<lambda>x::real^'n. 0::'a) has_integral z) {a..b} \<and> norm (z - 0) < e"
  2.1098 -      apply(rule_tac x=0 in exI) apply(rule,rule lem) by auto
  2.1099 -  qed auto qed
  2.1100 -
  2.1101 -lemma has_integral_0[simp]: "((\<lambda>x::real^'n. 0) has_integral 0) s"
  2.1102 -  apply(rule has_integral_is_0) by auto 
  2.1103 -
  2.1104 -lemma has_integral_0_eq[simp]: "((\<lambda>x. 0) has_integral i) s \<longleftrightarrow> i = 0"
  2.1105 -  using has_integral_unique[OF has_integral_0] by auto
  2.1106 -
  2.1107 -lemma has_integral_linear: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
  2.1108 -  assumes "(f has_integral y) s" "bounded_linear h" shows "((h o f) has_integral ((h y))) s"
  2.1109 -proof- interpret bounded_linear h using assms(2) . from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format]
  2.1110 -  have lem:"\<And>f::real^'n \<Rightarrow> 'a. \<And> y a b.
  2.1111 -    (f has_integral y) ({a..b}) \<Longrightarrow> ((h o f) has_integral h(y)) ({a..b})"
  2.1112 -  proof(subst has_integral,rule,rule) case goal1
  2.1113 -    from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format]
  2.1114 -    have *:"e / B > 0" apply(rule divide_pos_pos) using goal1(2) B by auto
  2.1115 -    guess g using has_integralD[OF goal1(1) *] . note g=this
  2.1116 -    show ?case apply(rule_tac x=g in exI) apply(rule,rule g(1))
  2.1117 -    proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "g fine p" 
  2.1118 -      have *:"\<And>x k. h ((\<lambda>(x, k). content k *\<^sub>R f x) x) = (\<lambda>(x, k). h (content k *\<^sub>R f x)) x" by auto
  2.1119 -      have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) = setsum (h \<circ> (\<lambda>(x, k). content k *\<^sub>R f x)) p"
  2.1120 -        unfolding o_def unfolding scaleR[THEN sym] * by simp
  2.1121 -      also have "\<dots> = h (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" using setsum[of "\<lambda>(x,k). content k *\<^sub>R f x" p] using as by auto
  2.1122 -      finally have *:"(\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) = h (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" .
  2.1123 -      show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) - h y) < e" unfolding * diff[THEN sym]
  2.1124 -        apply(rule le_less_trans[OF B(2)]) using g(2)[OF as] B(1) by(auto simp add:field_simps)
  2.1125 -    qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
  2.1126 -    thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) }
  2.1127 -  assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P
  2.1128 -  proof(rule,rule) fix e::real  assume e:"0<e"
  2.1129 -    have *:"0 < e/B" by(rule divide_pos_pos,rule e,rule B(1))
  2.1130 -    guess M using has_integral_altD[OF assms(1) as *,rule_format] . note M=this
  2.1131 -    show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> s then (h \<circ> f) x else 0) has_integral z) {a..b} \<and> norm (z - h y) < e)"
  2.1132 -      apply(rule_tac x=M in exI) apply(rule,rule M(1))
  2.1133 -    proof(rule,rule,rule) case goal1 guess z using M(2)[OF goal1(1)] .. note z=conjunctD2[OF this]
  2.1134 -      have *:"(\<lambda>x. if x \<in> s then (h \<circ> f) x else 0) = h \<circ> (\<lambda>x. if x \<in> s then f x else 0)"
  2.1135 -        unfolding o_def apply(rule ext) using zero by auto
  2.1136 -      show ?case apply(rule_tac x="h z" in exI,rule) unfolding * apply(rule lem[OF z(1)]) unfolding diff[THEN sym]
  2.1137 -        apply(rule le_less_trans[OF B(2)]) using B(1) z(2) by(auto simp add:field_simps)
  2.1138 -    qed qed qed
  2.1139 -
  2.1140 -lemma has_integral_cmul:
  2.1141 -  shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_integral (c *\<^sub>R k)) s"
  2.1142 -  unfolding o_def[THEN sym] apply(rule has_integral_linear,assumption)
  2.1143 -  by(rule scaleR.bounded_linear_right)
  2.1144 -
  2.1145 -lemma has_integral_neg:
  2.1146 -  shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. -(f x)) has_integral (-k)) s"
  2.1147 -  apply(drule_tac c="-1" in has_integral_cmul) by auto
  2.1148 -
  2.1149 -lemma has_integral_add: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector" 
  2.1150 -  assumes "(f has_integral k) s" "(g has_integral l) s"
  2.1151 -  shows "((\<lambda>x. f x + g x) has_integral (k + l)) s"
  2.1152 -proof- have lem:"\<And>f g::real^'n \<Rightarrow> 'a. \<And>a b k l.
  2.1153 -    (f has_integral k) ({a..b}) \<Longrightarrow> (g has_integral l) ({a..b}) \<Longrightarrow>
  2.1154 -     ((\<lambda>x. f(x) + g(x)) has_integral (k + l)) ({a..b})" proof- case goal1
  2.1155 -    show ?case unfolding has_integral proof(rule,rule) fix e::real assume e:"e>0" hence *:"e/2>0" by auto
  2.1156 -      guess d1 using has_integralD[OF goal1(1) *] . note d1=this
  2.1157 -      guess d2 using has_integralD[OF goal1(2) *] . note d2=this
  2.1158 -      show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e)"
  2.1159 -        apply(rule_tac x="\<lambda>x. (d1 x) \<inter> (d2 x)" in exI) apply(rule,rule gauge_inter[OF d1(1) d2(1)])
  2.1160 -      proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d1 x \<inter> d2 x) fine p"
  2.1161 -        have *:"(\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) = (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p. content k *\<^sub>R g x)"
  2.1162 -          unfolding scaleR_right_distrib setsum_addf[of "\<lambda>(x,k). content k *\<^sub>R f x" "\<lambda>(x,k). content k *\<^sub>R g x" p,THEN sym]
  2.1163 -          by(rule setsum_cong2,auto)
  2.1164 -        have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) = norm (((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k) + ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - l))"
  2.1165 -          unfolding * by(auto simp add:group_simps) also let ?res = "\<dots>"
  2.1166 -        from as have *:"d1 fine p" "d2 fine p" unfolding fine_inter by auto
  2.1167 -        have "?res < e/2 + e/2" apply(rule le_less_trans[OF norm_triangle_ineq])
  2.1168 -          apply(rule add_strict_mono) using d1(2)[OF as(1) *(1)] and d2(2)[OF as(1) *(2)] by auto
  2.1169 -        finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e" by auto
  2.1170 -      qed qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
  2.1171 -    thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) }
  2.1172 -  assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P
  2.1173 -  proof(rule,rule) case goal1 hence *:"e/2 > 0" by auto
  2.1174 -    from has_integral_altD[OF assms(1) as *] guess B1 . note B1=this[rule_format]
  2.1175 -    from has_integral_altD[OF assms(2) as *] guess B2 . note B2=this[rule_format]
  2.1176 -    show ?case apply(rule_tac x="max B1 B2" in exI) apply(rule,rule min_max.less_supI1,rule B1)
  2.1177 -    proof(rule,rule,rule) fix a b assume "ball 0 (max B1 B2) \<subseteq> {a..b::real^'n}"
  2.1178 -      hence *:"ball 0 B1 \<subseteq> {a..b::real^'n}" "ball 0 B2 \<subseteq> {a..b::real^'n}" by auto
  2.1179 -      guess w using B1(2)[OF *(1)] .. note w=conjunctD2[OF this]
  2.1180 -      guess z using B2(2)[OF *(2)] .. note z=conjunctD2[OF this]
  2.1181 -      have *:"\<And>x. (if x \<in> s then f x + g x else 0) = (if x \<in> s then f x else 0) + (if x \<in> s then g x else 0)" by auto
  2.1182 -      show "\<exists>z. ((\<lambda>x. if x \<in> s then f x + g x else 0) has_integral z) {a..b} \<and> norm (z - (k + l)) < e"
  2.1183 -        apply(rule_tac x="w + z" in exI) apply(rule,rule lem[OF w(1) z(1), unfolded *[THEN sym]])
  2.1184 -        using norm_triangle_ineq[of "w - k" "z - l"] w(2) z(2) by(auto simp add:field_simps)
  2.1185 -    qed qed qed
  2.1186 -
  2.1187 -lemma has_integral_sub:
  2.1188 -  shows "(f has_integral k) s \<Longrightarrow> (g has_integral l) s \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) has_integral (k - l)) s"
  2.1189 -  using has_integral_add[OF _ has_integral_neg,of f k s g l] unfolding group_simps by auto
  2.1190 -
  2.1191 -lemma integral_0: "integral s (\<lambda>x::real^'n. 0::real^'m) = 0"
  2.1192 -  by(rule integral_unique has_integral_0)+
  2.1193 -
  2.1194 -lemma integral_add:
  2.1195 -  shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow>
  2.1196 -   integral s (\<lambda>x. f x + g x) = integral s f + integral s g"
  2.1197 -  apply(rule integral_unique) apply(drule integrable_integral)+
  2.1198 -  apply(rule has_integral_add) by assumption+
  2.1199 -
  2.1200 -lemma integral_cmul:
  2.1201 -  shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. c *\<^sub>R f x) = c *\<^sub>R integral s f"
  2.1202 -  apply(rule integral_unique) apply(drule integrable_integral)+
  2.1203 -  apply(rule has_integral_cmul) by assumption+
  2.1204 -
  2.1205 -lemma integral_neg:
  2.1206 -  shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. - f x) = - integral s f"
  2.1207 -  apply(rule integral_unique) apply(drule integrable_integral)+
  2.1208 -  apply(rule has_integral_neg) by assumption+
  2.1209 -
  2.1210 -lemma integral_sub:
  2.1211 -  shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> integral s (\<lambda>x. f x - g x) = integral s f - integral s g"
  2.1212 -  apply(rule integral_unique) apply(drule integrable_integral)+
  2.1213 -  apply(rule has_integral_sub) by assumption+
  2.1214 -
  2.1215 -lemma integrable_0: "(\<lambda>x. 0) integrable_on s"
  2.1216 -  unfolding integrable_on_def using has_integral_0 by auto
  2.1217 -
  2.1218 -lemma integrable_add:
  2.1219 -  shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x + g x) integrable_on s"
  2.1220 -  unfolding integrable_on_def by(auto intro: has_integral_add)
  2.1221 -
  2.1222 -lemma integrable_cmul:
  2.1223 -  shows "f integrable_on s \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) integrable_on s"
  2.1224 -  unfolding integrable_on_def by(auto intro: has_integral_cmul)
  2.1225 -
  2.1226 -lemma integrable_neg:
  2.1227 -  shows "f integrable_on s \<Longrightarrow> (\<lambda>x. -f(x)) integrable_on s"
  2.1228 -  unfolding integrable_on_def by(auto intro: has_integral_neg)
  2.1229 -
  2.1230 -lemma integrable_sub:
  2.1231 -  shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x - g x) integrable_on s"
  2.1232 -  unfolding integrable_on_def by(auto intro: has_integral_sub)
  2.1233 -
  2.1234 -lemma integrable_linear:
  2.1235 -  shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> (h o f) integrable_on s"
  2.1236 -  unfolding integrable_on_def by(auto intro: has_integral_linear)
  2.1237 -
  2.1238 -lemma integral_linear:
  2.1239 -  shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> integral s (h o f) = h(integral s f)"
  2.1240 -  apply(rule has_integral_unique) defer unfolding has_integral_integral 
  2.1241 -  apply(drule has_integral_linear,assumption,assumption) unfolding has_integral_integral[THEN sym]
  2.1242 -  apply(rule integrable_linear) by assumption+
  2.1243 -
  2.1244 -lemma has_integral_setsum:
  2.1245 -  assumes "finite t" "\<forall>a\<in>t. ((f a) has_integral (i a)) s"
  2.1246 -  shows "((\<lambda>x. setsum (\<lambda>a. f a x) t) has_integral (setsum i t)) s"
  2.1247 -proof(insert assms(1) subset_refl[of t],induct rule:finite_subset_induct)
  2.1248 -  case (insert x F) show ?case unfolding setsum_insert[OF insert(1,3)]
  2.1249 -    apply(rule has_integral_add) using insert assms by auto
  2.1250 -qed auto
  2.1251 -
  2.1252 -lemma integral_setsum:
  2.1253 -  shows "finite t \<Longrightarrow> \<forall>a\<in>t. (f a) integrable_on s \<Longrightarrow>
  2.1254 -  integral s (\<lambda>x. setsum (\<lambda>a. f a x) t) = setsum (\<lambda>a. integral s (f a)) t"
  2.1255 -  apply(rule integral_unique) apply(rule has_integral_setsum)
  2.1256 -  using integrable_integral by auto
  2.1257 -
  2.1258 -lemma integrable_setsum:
  2.1259 -  shows "finite t \<Longrightarrow> \<forall>a \<in> t.(f a) integrable_on s \<Longrightarrow> (\<lambda>x. setsum (\<lambda>a. f a x) t) integrable_on s"
  2.1260 -  unfolding integrable_on_def apply(drule bchoice) using has_integral_setsum[of t] by auto
  2.1261 -
  2.1262 -lemma has_integral_eq:
  2.1263 -  assumes "\<forall>x\<in>s. f x = g x" "(f has_integral k) s" shows "(g has_integral k) s"
  2.1264 -  using has_integral_sub[OF assms(2), of "\<lambda>x. f x - g x" 0]
  2.1265 -  using has_integral_is_0[of s "\<lambda>x. f x - g x"] using assms(1) by auto
  2.1266 -
  2.1267 -lemma integrable_eq:
  2.1268 -  shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> f integrable_on s \<Longrightarrow> g integrable_on s"
  2.1269 -  unfolding integrable_on_def using has_integral_eq[of s f g] by auto
  2.1270 -
  2.1271 -lemma has_integral_eq_eq:
  2.1272 -  shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> ((f has_integral k) s \<longleftrightarrow> (g has_integral k) s)"
  2.1273 -  using has_integral_eq[of s f g] has_integral_eq[of s g f] by auto 
  2.1274 -
  2.1275 -lemma has_integral_null[dest]:
  2.1276 -  assumes "content({a..b}) = 0" shows  "(f has_integral 0) ({a..b})"
  2.1277 -  unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI,rule) defer
  2.1278 -proof(rule,rule,erule conjE) fix e::real assume e:"e>0" thus "gauge (\<lambda>x. ball x 1)" by auto
  2.1279 -  fix p assume p:"p tagged_division_of {a..b}" (*"(\<lambda>x. ball x 1) fine p"*)
  2.1280 -  have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) = 0" unfolding norm_eq_zero diff_0_right
  2.1281 -    using setsum_content_null[OF assms(1) p, of f] . 
  2.1282 -  thus "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e" using e by auto qed
  2.1283 -
  2.1284 -lemma has_integral_null_eq[simp]:
  2.1285 -  shows "content({a..b}) = 0 \<Longrightarrow> ((f has_integral i) ({a..b}) \<longleftrightarrow> i = 0)"
  2.1286 -  apply rule apply(rule has_integral_unique,assumption) 
  2.1287 -  apply(drule has_integral_null,assumption)
  2.1288 -  apply(drule has_integral_null) by auto
  2.1289 -
  2.1290 -lemma integral_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> integral({a..b}) f = 0"
  2.1291 -  by(rule integral_unique,drule has_integral_null)
  2.1292 -
  2.1293 -lemma integrable_on_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> f integrable_on {a..b}"
  2.1294 -  unfolding integrable_on_def apply(drule has_integral_null) by auto
  2.1295 -
  2.1296 -lemma has_integral_empty[intro]: shows "(f has_integral 0) {}"
  2.1297 -  unfolding empty_as_interval apply(rule has_integral_null) 
  2.1298 -  using content_empty unfolding empty_as_interval .
  2.1299 -
  2.1300 -lemma has_integral_empty_eq[simp]: shows "(f has_integral i) {} \<longleftrightarrow> i = 0"
  2.1301 -  apply(rule,rule has_integral_unique,assumption) by auto
  2.1302 -
  2.1303 -lemma integrable_on_empty[intro]: shows "f integrable_on {}" unfolding integrable_on_def by auto
  2.1304 -
  2.1305 -lemma integral_empty[simp]: shows "integral {} f = 0"
  2.1306 -  apply(rule integral_unique) using has_integral_empty .
  2.1307 -
  2.1308 -lemma has_integral_refl[intro]: shows "(f has_integral 0) {a..a}"
  2.1309 -  apply(rule has_integral_null) unfolding content_eq_0_interior
  2.1310 -  unfolding interior_closed_interval using interval_sing by auto
  2.1311 -
  2.1312 -lemma integrable_on_refl[intro]: shows "f integrable_on {a..a}" unfolding integrable_on_def by auto
  2.1313 -
  2.1314 -lemma integral_refl: shows "integral {a..a} f = 0" apply(rule integral_unique) by auto
  2.1315 -
  2.1316 -subsection {* Cauchy-type criterion for integrability. *}
  2.1317 -
  2.1318 -lemma integrable_cauchy: fixes f::"real^'n \<Rightarrow> 'a::{real_normed_vector,complete_space}" 
  2.1319 -  shows "f integrable_on {a..b} \<longleftrightarrow>
  2.1320 -  (\<forall>e>0.\<exists>d. gauge d \<and> (\<forall>p1 p2. p1 tagged_division_of {a..b} \<and> d fine p1 \<and>
  2.1321 -                            p2 tagged_division_of {a..b} \<and> d fine p2
  2.1322 -                            \<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 -
  2.1323 -                                     setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) < e))" (is "?l = (\<forall>e>0. \<exists>d. ?P e d)")
  2.1324 -proof assume ?l
  2.1325 -  then guess y unfolding integrable_on_def has_integral .. note y=this
  2.1326 -  show "\<forall>e>0. \<exists>d. ?P e d" proof(rule,rule) case goal1 hence "e/2 > 0" by auto
  2.1327 -    then guess d apply- apply(drule y[rule_format]) by(erule exE,erule conjE) note d=this[rule_format]
  2.1328 -    show ?case apply(rule_tac x=d in exI,rule,rule d) apply(rule,rule,rule,(erule conjE)+)
  2.1329 -    proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b}" "d fine p1" "p2 tagged_division_of {a..b}" "d fine p2"
  2.1330 -      show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e"
  2.1331 -        apply(rule dist_triangle_half_l[where y=y,unfolded vector_dist_norm])
  2.1332 -        using d(2)[OF conjI[OF as(1-2)]] d(2)[OF conjI[OF as(3-4)]] .
  2.1333 -    qed qed
  2.1334 -next assume "\<forall>e>0. \<exists>d. ?P e d" hence "\<forall>n::nat. \<exists>d. ?P (inverse(real (n + 1))) d" by auto
  2.1335 -  from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
  2.1336 -  have "\<And>n. gauge (\<lambda>x. \<Inter>{d i x |i. i \<in> {0..n}})" apply(rule gauge_inters) using d(1) by auto
  2.1337 -  hence "\<forall>n. \<exists>p. p tagged_division_of {a..b} \<and> (\<lambda>x. \<Inter>{d i x |i. i \<in> {0..n}}) fine p" apply-
  2.1338 -  proof case goal1 from this[of n] show ?case apply(drule_tac fine_division_exists) by auto qed
  2.1339 -  from choice[OF this] guess p .. note p = conjunctD2[OF this[rule_format]]
  2.1340 -  have dp:"\<And>i n. i\<le>n \<Longrightarrow> d i fine p n" using p(2) unfolding fine_inters by auto
  2.1341 -  have "Cauchy (\<lambda>n. setsum (\<lambda>(x,k). content k *\<^sub>R (f x)) (p n))"
  2.1342 -  proof(rule CauchyI) case goal1 then guess N unfolding real_arch_inv[of e] .. note N=this
  2.1343 -    show ?case apply(rule_tac x=N in exI)
  2.1344 -    proof(rule,rule,rule,rule) fix m n assume mn:"N \<le> m" "N \<le> n" have *:"N = (N - 1) + 1" using N by auto
  2.1345 -      show "norm ((\<Sum>(x, k)\<in>p m. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p n. content k *\<^sub>R f x)) < e"
  2.1346 -        apply(rule less_trans[OF _ N[THEN conjunct2,THEN conjunct2]]) apply(subst *) apply(rule d(2))
  2.1347 -        using dp p(1) using mn by auto 
  2.1348 -    qed qed
  2.1349 -  then guess y unfolding convergent_eq_cauchy[THEN sym] .. note y=this[unfolded Lim_sequentially,rule_format]
  2.1350 -  show ?l unfolding integrable_on_def has_integral apply(rule_tac x=y in exI)
  2.1351 -  proof(rule,rule) fix e::real assume "e>0" hence *:"e/2 > 0" by auto
  2.1352 -    then guess N1 unfolding real_arch_inv[of "e/2"] .. note N1=this hence N1':"N1 = N1 - 1 + 1" by auto
  2.1353 -    guess N2 using y[OF *] .. note N2=this
  2.1354 -    show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - y) < e)"
  2.1355 -      apply(rule_tac x="d (N1 + N2)" in exI) apply rule defer 
  2.1356 -    proof(rule,rule,erule conjE) show "gauge (d (N1 + N2))" using d by auto
  2.1357 -      fix q assume as:"q tagged_division_of {a..b}" "d (N1 + N2) fine q"
  2.1358 -      have *:"inverse (real (N1 + N2 + 1)) < e / 2" apply(rule less_trans) using N1 by auto
  2.1359 -      show "norm ((\<Sum>(x, k)\<in>q. content k *\<^sub>R f x) - y) < e" apply(rule norm_triangle_half_r)
  2.1360 -        apply(rule less_trans[OF _ *]) apply(subst N1', rule d(2)[of "p (N1+N2)"]) defer
  2.1361 -        using N2[rule_format,unfolded vector_dist_norm,of "N1+N2"]
  2.1362 -        using as dp[of "N1 - 1 + 1 + N2" "N1 + N2"] using p(1)[of "N1 + N2"] using N1 by auto qed qed qed
  2.1363 -
  2.1364 -subsection {* Additivity of integral on abutting intervals. *}
  2.1365 -
  2.1366 -lemma interval_split:
  2.1367 -  "{a..b::real^'n} \<inter> {x. x$k \<le> c} = {a .. (\<chi> i. if i = k then min (b$k) c else b$i)}"
  2.1368 -  "{a..b} \<inter> {x. x$k \<ge> c} = {(\<chi> i. if i = k then max (a$k) c else a$i) .. b}"
  2.1369 -  apply(rule_tac[!] set_ext) unfolding Int_iff mem_interval mem_Collect_eq
  2.1370 -  unfolding Cart_lambda_beta by auto
  2.1371 -
  2.1372 -lemma content_split:
  2.1373 -  "content {a..b::real^'n} = content({a..b} \<inter> {x. x$k \<le> c}) + content({a..b} \<inter> {x. x$k >= c})"
  2.1374 -proof- note simps = interval_split content_closed_interval_cases Cart_lambda_beta vector_le_def
  2.1375 -  { presume "a\<le>b \<Longrightarrow> ?thesis" thus ?thesis apply(cases "a\<le>b") unfolding simps by auto }
  2.1376 -  have *:"UNIV = insert k (UNIV - {k})" "\<And>x. finite (UNIV-{x::'n})" "\<And>x. x\<notin>UNIV-{x}" by auto
  2.1377 -  have *:"\<And>X Y Z. (\<Prod>i\<in>UNIV. Z i (if i = k then X else Y i)) = Z k X * (\<Prod>i\<in>UNIV-{k}. Z i (Y i))"
  2.1378 -    "(\<Prod>i\<in>UNIV. b$i - a$i) = (\<Prod>i\<in>UNIV-{k}. b$i - a$i) * (b$k - a$k)" 
  2.1379 -    apply(subst *(1)) defer apply(subst *(1)) unfolding setprod_insert[OF *(2-)] by auto
  2.1380 -  assume as:"a\<le>b" moreover have "\<And>x. min (b $ k) c = max (a $ k) c
  2.1381 -    \<Longrightarrow> x* (b$k - a$k) = x*(max (a $ k) c - a $ k) + x*(b $ k - max (a $ k) c)"
  2.1382 -    by  (auto simp add:field_simps)
  2.1383 -  moreover have "\<not> a $ k \<le> c \<Longrightarrow> \<not> c \<le> b $ k \<Longrightarrow> False"
  2.1384 -    unfolding not_le using as[unfolded vector_le_def,rule_format,of k] by auto
  2.1385 -  ultimately show ?thesis 
  2.1386 -    unfolding simps unfolding *(1)[of "\<lambda>i x. b$i - x"] *(1)[of "\<lambda>i x. x - a$i"] *(2) by(auto)
  2.1387 -qed
  2.1388 -
  2.1389 -lemma division_split_left_inj:
  2.1390 -  assumes "d division_of i" "k1 \<in> d" "k2 \<in> d"  "k1 \<noteq> k2"
  2.1391 -  "k1 \<inter> {x::real^'n. x$k \<le> c} = k2 \<inter> {x. x$k \<le> c}"
  2.1392 -  shows "content(k1 \<inter> {x. x$k \<le> c}) = 0"
  2.1393 -proof- note d=division_ofD[OF assms(1)]
  2.1394 -  have *:"\<And>a b::real^'n. \<And> c k. (content({a..b} \<inter> {x. x$k \<le> c}) = 0 \<longleftrightarrow> interior({a..b} \<inter> {x. x$k \<le> c}) = {})"
  2.1395 -    unfolding interval_split content_eq_0_interior by auto
  2.1396 -  guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this
  2.1397 -  guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this
  2.1398 -  have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto
  2.1399 -  show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]])
  2.1400 -    defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed
  2.1401 -
  2.1402 -lemma division_split_right_inj:
  2.1403 -  assumes "d division_of i" "k1 \<in> d" "k2 \<in> d"  "k1 \<noteq> k2"
  2.1404 -  "k1 \<inter> {x::real^'n. x$k \<ge> c} = k2 \<inter> {x. x$k \<ge> c}"
  2.1405 -  shows "content(k1 \<inter> {x. x$k \<ge> c}) = 0"
  2.1406 -proof- note d=division_ofD[OF assms(1)]
  2.1407 -  have *:"\<And>a b::real^'n. \<And> c k. (content({a..b} \<inter> {x. x$k >= c}) = 0 \<longleftrightarrow> interior({a..b} \<inter> {x. x$k >= c}) = {})"
  2.1408 -    unfolding interval_split content_eq_0_interior by auto
  2.1409 -  guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this
  2.1410 -  guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this
  2.1411 -  have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto
  2.1412 -  show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]])
  2.1413 -    defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed
  2.1414 -
  2.1415 -lemma tagged_division_split_left_inj:
  2.1416 -  assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2"  "k1 \<inter> {x. x$k \<le> c} = k2 \<inter> {x. x$k \<le> c}" 
  2.1417 -  shows "content(k1 \<inter> {x. x$k \<le> c}) = 0"
  2.1418 -proof- have *:"\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c" unfolding image_iff apply(rule_tac x="(a,b)" in bexI) by auto
  2.1419 -  show ?thesis apply(rule division_split_left_inj[OF division_of_tagged_division[OF assms(1)]])
  2.1420 -    apply(rule_tac[1-2] *) using assms(2-) by auto qed
  2.1421 -
  2.1422 -lemma tagged_division_split_right_inj:
  2.1423 -  assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2"  "k1 \<inter> {x. x$k \<ge> c} = k2 \<inter> {x. x$k \<ge> c}" 
  2.1424 -  shows "content(k1 \<inter> {x. x$k \<ge> c}) = 0"
  2.1425 -proof- have *:"\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c" unfolding image_iff apply(rule_tac x="(a,b)" in bexI) by auto
  2.1426 -  show ?thesis apply(rule division_split_right_inj[OF division_of_tagged_division[OF assms(1)]])
  2.1427 -    apply(rule_tac[1-2] *) using assms(2-) by auto qed
  2.1428 -
  2.1429 -lemma division_split:
  2.1430 -  assumes "p division_of {a..b::real^'n}"
  2.1431 -  shows "{l \<inter> {x. x$k \<le> c} | l. l \<in> p \<and> ~(l \<inter> {x. x$k \<le> c} = {})} division_of ({a..b} \<inter> {x. x$k \<le> c})" (is "?p1 division_of ?I1") and 
  2.1432 -        "{l \<inter> {x. x$k \<ge> c} | l. l \<in> p \<and> ~(l \<inter> {x. x$k \<ge> c} = {})} division_of ({a..b} \<inter> {x. x$k \<ge> c})" (is "?p2 division_of ?I2")
  2.1433 -proof(rule_tac[!] division_ofI) note p=division_ofD[OF assms]
  2.1434 -  show "finite ?p1" "finite ?p2" using p(1) by auto show "\<Union>?p1 = ?I1" "\<Union>?p2 = ?I2" unfolding p(6)[THEN sym] by auto
  2.1435 -  { fix k assume "k\<in>?p1" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this
  2.1436 -    guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this
  2.1437 -    show "k\<subseteq>?I1" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l
  2.1438 -      using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split) by auto
  2.1439 -    fix k' assume "k'\<in>?p1" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this
  2.1440 -    assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto }
  2.1441 -  { fix k assume "k\<in>?p2" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this
  2.1442 -    guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this
  2.1443 -    show "k\<subseteq>?I2" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l
  2.1444 -      using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split) by auto
  2.1445 -    fix k' assume "k'\<in>?p2" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this
  2.1446 -    assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto }
  2.1447 -qed
  2.1448 -
  2.1449 -lemma has_integral_split: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
  2.1450 -  assumes "(f has_integral i) ({a..b} \<inter> {x. x$k \<le> c})"  "(f has_integral j) ({a..b} \<inter> {x. x$k \<ge> c})"
  2.1451 -  shows "(f has_integral (i + j)) ({a..b})"
  2.1452 -proof(unfold has_integral,rule,rule) case goal1 hence e:"e/2>0" by auto
  2.1453 -  guess d1 using has_integralD[OF assms(1)[unfolded interval_split] e] . note d1=this[unfolded interval_split[THEN sym]]
  2.1454 -  guess d2 using has_integralD[OF assms(2)[unfolded interval_split] e] . note d2=this[unfolded interval_split[THEN sym]]
  2.1455 -  let ?d = "\<lambda>x. if x$k = c then (d1 x \<inter> d2 x) else ball x (abs(x$k - c)) \<inter> d1 x \<inter> d2 x"
  2.1456 -  show ?case apply(rule_tac x="?d" in exI,rule) defer apply(rule,rule,(erule conjE)+)
  2.1457 -  proof- show "gauge ?d" using d1(1) d2(1) unfolding gauge_def by auto
  2.1458 -    fix p assume "p tagged_division_of {a..b}" "?d fine p" note p = this tagged_division_ofD[OF this(1)]
  2.1459 -    have lem0:"\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x$k \<le> c} = {}) \<Longrightarrow> x$k \<le> c"
  2.1460 -         "\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x$k \<ge> c} = {}) \<Longrightarrow> x$k \<ge> c"
  2.1461 -    proof- fix x kk assume as:"(x,kk)\<in>p"
  2.1462 -      show "~(kk \<inter> {x. x$k \<le> c} = {}) \<Longrightarrow> x$k \<le> c"
  2.1463 -      proof(rule ccontr) case goal1
  2.1464 -        from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x $ k - c\<bar>"
  2.1465 -          using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
  2.1466 -        hence "\<exists>y. y \<in> ball x \<bar>x $ k - c\<bar> \<inter> {x. x $ k \<le> c}" using goal1(1) by blast 
  2.1467 -        then guess y .. hence "\<bar>x $ k - y $ k\<bar> < \<bar>x $ k - c\<bar>" "y$k \<le> c" apply-apply(rule le_less_trans)
  2.1468 -          using component_le_norm[of "x - y" k,unfolded vector_minus_component] by(auto simp add:vector_dist_norm)
  2.1469 -        thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
  2.1470 -      qed
  2.1471 -      show "~(kk \<inter> {x. x$k \<ge> c} = {}) \<Longrightarrow> x$k \<ge> c"
  2.1472 -      proof(rule ccontr) case goal1
  2.1473 -        from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x $ k - c\<bar>"
  2.1474 -          using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
  2.1475 -        hence "\<exists>y. y \<in> ball x \<bar>x $ k - c\<bar> \<inter> {x. x $ k \<ge> c}" using goal1(1) by blast 
  2.1476 -        then guess y .. hence "\<bar>x $ k - y $ k\<bar> < \<bar>x $ k - c\<bar>" "y$k \<ge> c" apply-apply(rule le_less_trans)
  2.1477 -          using component_le_norm[of "x - y" k,unfolded vector_minus_component] by(auto simp add:vector_dist_norm)
  2.1478 -        thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
  2.1479 -      qed
  2.1480 -    qed
  2.1481 -
  2.1482 -    have lem1: "\<And>f P Q. (\<forall>x k. (x,k) \<in> {(x,f k) | x k. P x k} \<longrightarrow> Q x k) \<longleftrightarrow> (\<forall>x k. P x k \<longrightarrow> Q x (f k))" by auto
  2.1483 -    have lem2: "\<And>f s P f. finite s \<Longrightarrow> finite {(x,f k) | x k. (x,k) \<in> s \<and> P x k}"
  2.1484 -    proof- case goal1 thus ?case apply-apply(rule finite_subset[of _ "(\<lambda>(x,k). (x,f k)) ` s"]) by auto qed
  2.1485 -    have lem3: "\<And>g::(real ^ 'n \<Rightarrow> bool) \<Rightarrow> real ^ 'n \<Rightarrow> bool. finite p \<Longrightarrow>
  2.1486 -      setsum (\<lambda>(x,k). content k *\<^sub>R f x) {(x,g k) |x k. (x,k) \<in> p \<and> ~(g k = {})}
  2.1487 -               = setsum (\<lambda>(x,k). content k *\<^sub>R f x) ((\<lambda>(x,k). (x,g k)) ` p)"
  2.1488 -      apply(rule setsum_mono_zero_left) prefer 3
  2.1489 -    proof fix g::"(real ^ 'n \<Rightarrow> bool) \<Rightarrow> real ^ 'n \<Rightarrow> bool" and i::"(real^'n) \<times> ((real^'n) set)"
  2.1490 -      assume "i \<in> (\<lambda>(x, k). (x, g k)) ` p - {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}"
  2.1491 -      then obtain x k where xk:"i=(x,g k)" "(x,k)\<in>p" "(x,g k) \<notin> {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}" by auto
  2.1492 -      have "content (g k) = 0" using xk using content_empty by auto
  2.1493 -      thus "(\<lambda>(x, k). content k *\<^sub>R f x) i = 0" unfolding xk split_conv by auto
  2.1494 -    qed auto
  2.1495 -    have lem4:"\<And>g. (\<lambda>(x,l). content (g l) *\<^sub>R f x) = (\<lambda>(x,l). content l *\<^sub>R f x) o (\<lambda>(x,l). (x,g l))" apply(rule ext) by auto
  2.1496 -
  2.1497 -    let ?M1 = "{(x,kk \<inter> {x. x$k \<le> c}) |x kk. (x,kk) \<in> p \<and> kk \<inter> {x. x$k \<le> c} \<noteq> {}}"
  2.1498 -    have "norm ((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) < e/2" apply(rule d1(2),rule tagged_division_ofI)
  2.1499 -      apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)
  2.1500 -    proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M1} = {a..b} \<inter> {x. x$k \<le> c}" unfolding p(8)[THEN sym] by auto
  2.1501 -      fix x l assume xl:"(x,l)\<in>?M1"
  2.1502 -      then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note xl'=this
  2.1503 -      have "l' \<subseteq> d1 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
  2.1504 -      thus "l \<subseteq> d1 x" unfolding xl' by auto
  2.1505 -      show "x\<in>l" "l \<subseteq> {a..b} \<inter> {x. x $ k \<le> c}" unfolding xl' using p(4-6)[OF xl'(3)] using xl'(4)
  2.1506 -        using lem0(1)[OF xl'(3-4)] by auto
  2.1507 -      show  "\<exists>a b. l = {a..b}" unfolding xl' using p(6)[OF xl'(3)] by(fastsimp simp add: interval_split[where c=c and k=k])
  2.1508 -      fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M1"
  2.1509 -      then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note yr'=this
  2.1510 -      assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}"
  2.1511 -      proof(cases "l' = r' \<longrightarrow> x' = y'")
  2.1512 -        case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
  2.1513 -      next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto
  2.1514 -        thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
  2.1515 -      qed qed moreover
  2.1516 -
  2.1517 -    let ?M2 = "{(x,kk \<inter> {x. x$k \<ge> c}) |x kk. (x,kk) \<in> p \<and> kk \<inter> {x. x$k \<ge> c} \<noteq> {}}" 
  2.1518 -    have "norm ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j) < e/2" apply(rule d2(2),rule tagged_division_ofI)
  2.1519 -      apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)
  2.1520 -    proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M2} = {a..b} \<inter> {x. x$k \<ge> c}" unfolding p(8)[THEN sym] by auto
  2.1521 -      fix x l assume xl:"(x,l)\<in>?M2"
  2.1522 -      then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note xl'=this
  2.1523 -      have "l' \<subseteq> d2 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
  2.1524 -      thus "l \<subseteq> d2 x" unfolding xl' by auto
  2.1525 -      show "x\<in>l" "l \<subseteq> {a..b} \<inter> {x. x $ k \<ge> c}" unfolding xl' using p(4-6)[OF xl'(3)] using xl'(4)
  2.1526 -        using lem0(2)[OF xl'(3-4)] by auto
  2.1527 -      show  "\<exists>a b. l = {a..b}" unfolding xl' using p(6)[OF xl'(3)] by(fastsimp simp add: interval_split[where c=c and k=k])
  2.1528 -      fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M2"
  2.1529 -      then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note yr'=this
  2.1530 -      assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}"
  2.1531 -      proof(cases "l' = r' \<longrightarrow> x' = y'")
  2.1532 -        case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
  2.1533 -      next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto
  2.1534 -        thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
  2.1535 -      qed qed ultimately
  2.1536 -
  2.1537 -    have "norm (((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) + ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j)) < e/2 + e/2"
  2.1538 -      apply- apply(rule norm_triangle_lt) by auto
  2.1539 -    also { have *:"\<And>x y. x = (0::real) \<Longrightarrow> x *\<^sub>R (y::'a) = 0" using scaleR_zero_left by auto
  2.1540 -      have "((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) + ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j)
  2.1541 -       = (\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - (i + j)" by auto
  2.1542 -      also have "\<dots> = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. x $ k \<le> c}) *\<^sub>R f x) + (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. c \<le> x $ k}) *\<^sub>R f x) - (i + j)"
  2.1543 -        unfolding lem3[OF p(3)] apply(subst setsum_reindex_nonzero[OF p(3)]) defer apply(subst setsum_reindex_nonzero[OF p(3)])
  2.1544 -        defer unfolding lem4[THEN sym] apply(rule refl) unfolding split_paired_all split_conv apply(rule_tac[!] *)
  2.1545 -      proof- case goal1 thus ?case apply- apply(rule tagged_division_split_left_inj [OF p(1), of a b aa ba]) by auto
  2.1546 -      next case   goal2 thus ?case apply- apply(rule tagged_division_split_right_inj[OF p(1), of a b aa ba]) by auto
  2.1547 -      qed also note setsum_addf[THEN sym]
  2.1548 -      also have *:"\<And>x. x\<in>p \<Longrightarrow> (\<lambda>(x, ka). content (ka \<inter> {x. x $ k \<le> c}) *\<^sub>R f x) x + (\<lambda>(x, ka). content (ka \<inter> {x. c \<le> x $ k}) *\<^sub>R f x) x
  2.1549 -        = (\<lambda>(x,ka). content ka *\<^sub>R f x) x" unfolding split_paired_all split_conv
  2.1550 -      proof- fix a b assume "(a,b) \<in> p" from p(6)[OF this] guess u v apply-by(erule exE)+ note uv=this
  2.1551 -        thus "content (b \<inter> {x. x $ k \<le> c}) *\<^sub>R f a + content (b \<inter> {x. c \<le> x $ k}) *\<^sub>R f a = content b *\<^sub>R f a"
  2.1552 -          unfolding scaleR_left_distrib[THEN sym] unfolding uv content_split[of u v k c] by auto
  2.1553 -      qed note setsum_cong2[OF this]
  2.1554 -      finally have "(\<Sum>(x, k)\<in>{(x, kk \<inter> {x. x $ k \<le> c}) |x kk. (x, kk) \<in> p \<and> kk \<inter> {x. x $ k \<le> c} \<noteq> {}}. content k *\<^sub>R f x) - i +
  2.1555 -        ((\<Sum>(x, k)\<in>{(x, kk \<inter> {x. c \<le> x $ k}) |x kk. (x, kk) \<in> p \<and> kk \<inter> {x. c \<le> x $ k} \<noteq> {}}. content k *\<^sub>R f x) - j) =
  2.1556 -        (\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f x) - (i + j)" by auto }
  2.1557 -    finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (i + j)) < e" by auto qed qed
  2.1558 -
  2.1559 -subsection {* A sort of converse, integrability on subintervals. *}
  2.1560 -
  2.1561 -lemma tagged_division_union_interval:
  2.1562 -  assumes "p1 tagged_division_of ({a..b} \<inter> {x::real^'n. x$k \<le> (c::real)})"  "p2 tagged_division_of ({a..b} \<inter> {x. x$k \<ge> c})"
  2.1563 -  shows "(p1 \<union> p2) tagged_division_of ({a..b})"
  2.1564 -proof- have *:"{a..b} = ({a..b} \<inter> {x. x$k \<le> c}) \<union> ({a..b} \<inter> {x. x$k \<ge> c})" by auto
  2.1565 -  show ?thesis apply(subst *) apply(rule tagged_division_union[OF assms])
  2.1566 -    unfolding interval_split interior_closed_interval
  2.1567 -    by(auto simp add: vector_less_def Cart_lambda_beta elim!:allE[where x=k]) qed
  2.1568 -
  2.1569 -lemma has_integral_separate_sides: fixes f::"real^'m \<Rightarrow> 'a::real_normed_vector"
  2.1570 -  assumes "(f has_integral i) ({a..b})" "e>0"
  2.1571 -  obtains d where "gauge d" "(\<forall>p1 p2. p1 tagged_division_of ({a..b} \<inter> {x. x$k \<le> c}) \<and> d fine p1 \<and>
  2.1572 -                                p2 tagged_division_of ({a..b} \<inter> {x. x$k \<ge> c}) \<and> d fine p2
  2.1573 -                                \<longrightarrow> norm((setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 +
  2.1574 -                                          setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) - i) < e)"
  2.1575 -proof- guess d using has_integralD[OF assms] . note d=this
  2.1576 -  show ?thesis apply(rule that[of d]) apply(rule d) apply(rule,rule,rule,(erule conjE)+)
  2.1577 -  proof- fix p1 p2 assume "p1 tagged_division_of {a..b} \<inter> {x. x $ k \<le> c}" "d fine p1" note p1=tagged_division_ofD[OF this(1)] this
  2.1578 -                   assume "p2 tagged_division_of {a..b} \<inter> {x. c \<le> x $ k}" "d fine p2" note p2=tagged_division_ofD[OF this(1)] this
  2.1579 -    note tagged_division_union_interval[OF p1(7) p2(7)] note p12 = tagged_division_ofD[OF this] this
  2.1580 -    have "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) = norm ((\<Sum>(x, k)\<in>p1 \<union> p2. content k *\<^sub>R f x) - i)"
  2.1581 -      apply(subst setsum_Un_zero) apply(rule p1 p2)+ apply(rule) unfolding split_paired_all split_conv
  2.1582 -    proof- fix a b assume ab:"(a,b) \<in> p1 \<inter> p2"
  2.1583 -      have "(a,b) \<in> p1" using ab by auto from p1(4)[OF this] guess u v apply-by(erule exE)+ note uv =this
  2.1584 -      have "b \<subseteq> {x. x$k = c}" using ab p1(3)[of a b] p2(3)[of a b] by fastsimp
  2.1585 -      moreover have "interior {x. x $ k = c} = {}" 
  2.1586 -      proof(rule ccontr) case goal1 then obtain x where x:"x\<in>interior {x. x$k = c}" by auto
  2.1587 -        then guess e unfolding mem_interior .. note e=this
  2.1588 -        have x:"x$k = c" using x interior_subset by fastsimp
  2.1589 -        have *:"\<And>i. \<bar>(x - (x + (\<chi> i. if i = k then e / 2 else 0))) $ i\<bar> = (if i = k then e/2 else 0)" using e by auto
  2.1590 -        have "x + (\<chi> i. if i = k then e/2 else 0) \<in> ball x e" unfolding mem_ball vector_dist_norm 
  2.1591 -          apply(rule le_less_trans[OF norm_le_l1]) unfolding * 
  2.1592 -          unfolding setsum_delta[OF finite_UNIV] using e by auto 
  2.1593 -        hence "x + (\<chi> i. if i = k then e/2 else 0) \<in> {x. x$k = c}" using e by auto
  2.1594 -        thus False unfolding mem_Collect_eq using e x by auto
  2.1595 -      qed ultimately have "content b = 0" unfolding uv content_eq_0_interior apply-apply(drule subset_interior) by auto
  2.1596 -      thus "content b *\<^sub>R f a = 0" by auto
  2.1597 -    qed auto
  2.1598 -    also have "\<dots> < e" by(rule d(2) p12 fine_union p1 p2)+
  2.1599 -    finally show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) < e" . qed qed
  2.1600 -
  2.1601 -lemma integrable_split[intro]: fixes f::"real^'n \<Rightarrow> 'a::{real_normed_vector,complete_space}" assumes "f integrable_on {a..b}"
  2.1602 -  shows "f integrable_on ({a..b} \<inter> {x. x$k \<le> c})" (is ?t1) and "f integrable_on ({a..b} \<inter> {x. x$k \<ge> c})" (is ?t2) 
  2.1603 -proof- guess y using assms unfolding integrable_on_def .. note y=this
  2.1604 -  def b' \<equiv> "(\<chi> i. if i = k then min (b$k) c else b$i)::real^'n"
  2.1605 -  and a' \<equiv> "(\<chi> i. if i = k then max (a$k) c else a$i)::real^'n"
  2.1606 -  show ?t1 ?t2 unfolding interval_split integrable_cauchy unfolding interval_split[THEN sym]
  2.1607 -  proof(rule_tac[!] allI impI)+ fix e::real assume "e>0" hence "e/2>0" by auto
  2.1608 -    from has_integral_separate_sides[OF y this,of k c] guess d . note d=this[rule_format]
  2.1609 -    let ?P = "\<lambda>A. \<exists>d. gauge d \<and> (\<forall>p1 p2. p1 tagged_division_of {a..b} \<inter> A \<and> d fine p1 \<and> p2 tagged_division_of {a..b} \<inter> A \<and> d fine p2 \<longrightarrow>
  2.1610 -                              norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e)"
  2.1611 -    show "?P {x. x $ k \<le> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
  2.1612 -    proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b} \<inter> {x. x $ k \<le> c} \<and> d fine p1 \<and> p2 tagged_division_of {a..b} \<inter> {x. x $ k \<le> c} \<and> d fine p2"
  2.1613 -      show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e"
  2.1614 -      proof- guess p using fine_division_exists[OF d(1), of a' b] . note p=this
  2.1615 -        show ?thesis using norm_triangle_half_l[OF d(2)[of p1 p] d(2)[of p2 p]]
  2.1616 -          using as unfolding interval_split b'_def[symmetric] a'_def[symmetric]
  2.1617 -          using p using assms by(auto simp add:group_simps)
  2.1618 -      qed qed  
  2.1619 -    show "?P {x. x $ k \<ge> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
  2.1620 -    proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b} \<inter> {x. x $ k \<ge> c} \<and> d fine p1 \<and> p2 tagged_division_of {a..b} \<inter> {x. x $ k \<ge> c} \<and> d fine p2"
  2.1621 -      show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e"
  2.1622 -      proof- guess p using fine_division_exists[OF d(1), of a b'] . note p=this
  2.1623 -        show ?thesis using norm_triangle_half_l[OF d(2)[of p p1] d(2)[of p p2]]
  2.1624 -          using as unfolding interval_split b'_def[symmetric] a'_def[symmetric]
  2.1625 -          using p using assms by(auto simp add:group_simps) qed qed qed qed
  2.1626 -
  2.1627 -subsection {* Generalized notion of additivity. *}
  2.1628 -
  2.1629 -definition "neutral opp = (SOME x. \<forall>y. opp x y = y \<and> opp y x = y)"
  2.1630 -
  2.1631 -definition operative :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ((real^'n) set \<Rightarrow> 'a) \<Rightarrow> bool" where
  2.1632 -  "operative opp f \<equiv> 
  2.1633 -    (\<forall>a b. content {a..b} = 0 \<longrightarrow> f {a..b} = neutral(opp)) \<and>
  2.1634 -    (\<forall>a b c k. f({a..b}) =
  2.1635 -                   opp (f({a..b} \<inter> {x. x$k \<le> c}))
  2.1636 -                       (f({a..b} \<inter> {x. x$k \<ge> c})))"
  2.1637 -
  2.1638 -lemma operativeD[dest]: assumes "operative opp f"
  2.1639 -  shows "\<And>a b. content {a..b} = 0 \<Longrightarrow> f {a..b} = neutral(opp)"
  2.1640 -  "\<And>a b c k. f({a..b}) = opp (f({a..b} \<inter> {x. x$k \<le> c})) (f({a..b} \<inter> {x. x$k \<ge> c}))"
  2.1641 -  using assms unfolding operative_def by auto
  2.1642 -
  2.1643 -lemma operative_trivial:
  2.1644 - "operative opp f \<Longrightarrow> content({a..b}) = 0 \<Longrightarrow> f({a..b}) = neutral opp"
  2.1645 -  unfolding operative_def by auto
  2.1646 -
  2.1647 -lemma property_empty_interval:
  2.1648 - "(\<forall>a b. content({a..b}) = 0 \<longrightarrow> P({a..b})) \<Longrightarrow> P {}" 
  2.1649 -  using content_empty unfolding empty_as_interval by auto
  2.1650 -
  2.1651 -lemma operative_empty: "operative opp f \<Longrightarrow> f {} = neutral opp"
  2.1652 -  unfolding operative_def apply(rule property_empty_interval) by auto
  2.1653 -
  2.1654 -subsection {* Using additivity of lifted function to encode definedness. *}
  2.1655 -
  2.1656 -lemma forall_option: "(\<forall>x. P x) \<longleftrightarrow> P None \<and> (\<forall>x. P(Some x))"
  2.1657 -  by (metis map_of.simps option.nchotomy)
  2.1658 -
  2.1659 -lemma exists_option:
  2.1660 - "(\<exists>x. P x) \<longleftrightarrow> P None \<or> (\<exists>x. P(Some x))" 
  2.1661 -  by (metis map_of.simps option.nchotomy)
  2.1662 -
  2.1663 -fun lifted where 
  2.1664 -  "lifted (opp::'a\<Rightarrow>'a\<Rightarrow>'b) (Some x) (Some y) = Some(opp x y)" |
  2.1665 -  "lifted opp None _ = (None::'b option)" |
  2.1666 -  "lifted opp _ None = None"
  2.1667 -
  2.1668 -lemma lifted_simp_1[simp]: "lifted opp v None = None"
  2.1669 -  apply(induct v) by auto
  2.1670 -
  2.1671 -definition "monoidal opp \<equiv>  (\<forall>x y. opp x y = opp y x) \<and>
  2.1672 -                   (\<forall>x y z. opp x (opp y z) = opp (opp x y) z) \<and>
  2.1673 -                   (\<forall>x. opp (neutral opp) x = x)"
  2.1674 -
  2.1675 -lemma monoidalI: assumes "\<And>x y. opp x y = opp y x"
  2.1676 -  "\<And>x y z. opp x (opp y z) = opp (opp x y) z"
  2.1677 -  "\<And>x. opp (neutral opp) x = x" shows "monoidal opp"
  2.1678 -  unfolding monoidal_def using assms by fastsimp
  2.1679 -
  2.1680 -lemma monoidal_ac: assumes "monoidal opp"
  2.1681 -  shows "opp (neutral opp) a = a" "opp a (neutral opp) = a" "opp a b = opp b a"
  2.1682 -  "opp (opp a b) c = opp a (opp b c)"  "opp a (opp b c) = opp b (opp a c)"
  2.1683 -  using assms unfolding monoidal_def apply- by metis+
  2.1684 -
  2.1685 -lemma monoidal_simps[simp]: assumes "monoidal opp"
  2.1686 -  shows "opp (neutral opp) a = a" "opp a (neutral opp) = a"
  2.1687 -  using monoidal_ac[OF assms] by auto
  2.1688 -
  2.1689 -lemma neutral_lifted[cong]: assumes "monoidal opp"
  2.1690 -  shows "neutral (lifted opp) = Some(neutral opp)"
  2.1691 -  apply(subst neutral_def) apply(rule some_equality) apply(rule,induct_tac y) prefer 3
  2.1692 -proof- fix x assume "\<forall>y. lifted opp x y = y \<and> lifted opp y x = y"
  2.1693 -  thus "x = Some (neutral opp)" apply(induct x) defer
  2.1694 -    apply rule apply(subst neutral_def) apply(subst eq_commute,rule some_equality)
  2.1695 -    apply(rule,erule_tac x="Some y" in allE) defer apply(erule_tac x="Some x" in allE) by auto
  2.1696 -qed(auto simp add:monoidal_ac[OF assms])
  2.1697 -
  2.1698 -lemma monoidal_lifted[intro]: assumes "monoidal opp" shows "monoidal(lifted opp)"
  2.1699 -  unfolding monoidal_def forall_option neutral_lifted[OF assms] using monoidal_ac[OF assms] by auto
  2.1700 -
  2.1701 -definition "support opp f s = {x. x\<in>s \<and> f x \<noteq> neutral opp}"
  2.1702 -definition "fold' opp e s \<equiv> (if finite s then fold opp e s else e)"
  2.1703 -definition "iterate opp s f \<equiv> fold' (\<lambda>x a. opp (f x) a) (neutral opp) (support opp f s)"
  2.1704 -
  2.1705 -lemma support_subset[intro]:"support opp f s \<subseteq> s" unfolding support_def by auto
  2.1706 -lemma support_empty[simp]:"support opp f {} = {}" using support_subset[of opp f "{}"] by auto
  2.1707 -
  2.1708 -lemma fun_left_comm_monoidal[intro]: assumes "monoidal opp" shows "fun_left_comm opp"
  2.1709 -  unfolding fun_left_comm_def using monoidal_ac[OF assms] by auto
  2.1710 -
  2.1711 -lemma support_clauses:
  2.1712 -  "\<And>f g s. support opp f {} = {}"
  2.1713 -  "\<And>f g s. support opp f (insert x s) = (if f(x) = neutral opp then support opp f s else insert x (support opp f s))"
  2.1714 -  "\<And>f g s. support opp f (s - {x}) = (support opp f s) - {x}"
  2.1715 -  "\<And>f g s. support opp f (s \<union> t) = (support opp f s) \<union> (support opp f t)"
  2.1716 -  "\<And>f g s. support opp f (s \<inter> t) = (support opp f s) \<inter> (support opp f t)"
  2.1717 -  "\<And>f g s. support opp f (s - t) = (support opp f s) - (support opp f t)"
  2.1718 -  "\<And>f g s. support opp g (f ` s) = f ` (support opp (g o f) s)"
  2.1719 -unfolding support_def by auto
  2.1720 -
  2.1721 -lemma finite_support[intro]:"finite s \<Longrightarrow> finite (support opp f s)"
  2.1722 -  unfolding support_def by auto
  2.1723 -
  2.1724 -lemma iterate_empty[simp]:"iterate opp {} f = neutral opp"
  2.1725 -  unfolding iterate_def fold'_def by auto 
  2.1726 -
  2.1727 -lemma iterate_insert[simp]: assumes "monoidal opp" "finite s"
  2.1728 -  shows "iterate opp (insert x s) f = (if x \<in> s then iterate opp s f else opp (f x) (iterate opp s f))" 
  2.1729 -proof(cases "x\<in>s") case True hence *:"insert x s = s" by auto
  2.1730 -  show ?thesis unfolding iterate_def if_P[OF True] * by auto
  2.1731 -next case False note x=this
  2.1732 -  note * = fun_left_comm.fun_left_comm_apply[OF fun_left_comm_monoidal[OF assms(1)]]
  2.1733 -  show ?thesis proof(cases "f x = neutral opp")
  2.1734 -    case True show ?thesis unfolding iterate_def if_not_P[OF x] support_clauses if_P[OF True]
  2.1735 -      unfolding True monoidal_simps[OF assms(1)] by auto
  2.1736 -  next case False show ?thesis unfolding iterate_def fold'_def  if_not_P[OF x] support_clauses if_not_P[OF False]
  2.1737 -      apply(subst fun_left_comm.fold_insert[OF * finite_support])
  2.1738 -      using `finite s` unfolding support_def using False x by auto qed qed 
  2.1739 -
  2.1740 -lemma iterate_some:
  2.1741 -  assumes "monoidal opp"  "finite s"
  2.1742 -  shows "iterate (lifted opp) s (\<lambda>x. Some(f x)) = Some (iterate opp s f)" using assms(2)
  2.1743 -proof(induct s) case empty thus ?case using assms by auto
  2.1744 -next case (insert x F) show ?case apply(subst iterate_insert) prefer 3 apply(subst if_not_P)
  2.1745 -    defer unfolding insert(3) lifted.simps apply rule using assms insert by auto qed
  2.1746 -
  2.1747 -subsection {* Two key instances of additivity. *}
  2.1748 -
  2.1749 -lemma neutral_add[simp]:
  2.1750 -  "neutral op + = (0::_::comm_monoid_add)" unfolding neutral_def 
  2.1751 -  apply(rule some_equality) defer apply(erule_tac x=0 in allE) by auto
  2.1752 -
  2.1753 -lemma operative_content[intro]: "operative (op +) content"
  2.1754 -  unfolding operative_def content_split[THEN sym] neutral_add by auto
  2.1755 -
  2.1756 -lemma neutral_monoid[simp]: "neutral ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a) = 0"
  2.1757 -  unfolding neutral_def apply(rule some_equality) defer
  2.1758 -  apply(erule_tac x=0 in allE) by auto
  2.1759 -
  2.1760 -lemma monoidal_monoid[intro]:
  2.1761 -  shows "monoidal ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a)"
  2.1762 -  unfolding monoidal_def neutral_monoid by(auto simp add: group_simps) 
  2.1763 -
  2.1764 -lemma operative_integral: fixes f::"real^'n \<Rightarrow> 'a::banach"
  2.1765 -  shows "operative (lifted(op +)) (\<lambda>i. if f integrable_on i then Some(integral i f) else None)"
  2.1766 -  unfolding operative_def unfolding neutral_lifted[OF monoidal_monoid] neutral_add
  2.1767 -  apply(rule,rule,rule,rule) defer apply(rule allI)+
  2.1768 -proof- fix a b c k show "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) =
  2.1769 -              lifted op + (if f integrable_on {a..b} \<inter> {x. x $ k \<le> c} then Some (integral ({a..b} \<inter> {x. x $ k \<le> c}) f) else None)
  2.1770 -               (if f integrable_on {a..b} \<inter> {x. c \<le> x $ k} then Some (integral ({a..b} \<inter> {x. c \<le> x $ k}) f) else None)"
  2.1771 -  proof(cases "f integrable_on {a..b}") 
  2.1772 -    case True show ?thesis unfolding if_P[OF True]
  2.1773 -      unfolding if_P[OF integrable_split(1)[OF True]] if_P[OF integrable_split(2)[OF True]]
  2.1774 -      unfolding lifted.simps option.inject apply(rule integral_unique) apply(rule has_integral_split) 
  2.1775 -      apply(rule_tac[!] integrable_integral integrable_split)+ using True by assumption+
  2.1776 -  next case False have "(\<not> (f integrable_on {a..b} \<inter> {x. x $ k \<le> c})) \<or> (\<not> ( f integrable_on {a..b} \<inter> {x. c \<le> x $ k}))"
  2.1777 -    proof(rule ccontr) case goal1 hence "f integrable_on {a..b}" apply- unfolding integrable_on_def
  2.1778 -        apply(rule_tac x="integral ({a..b} \<inter> {x. x $ k \<le> c}) f + integral ({a..b} \<inter> {x. x $ k \<ge> c}) f" in exI)
  2.1779 -        apply(rule has_integral_split) apply(rule_tac[!] integrable_integral) by auto
  2.1780 -      thus False using False by auto
  2.1781 -    qed thus ?thesis using False by auto 
  2.1782 -  qed next 
  2.1783 -  fix a b assume as:"content {a..b::real^'n} = 0"
  2.1784 -  thus "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) = Some 0"
  2.1785 -    unfolding if_P[OF integrable_on_null[OF as]] using has_integral_null_eq[OF as] by auto qed
  2.1786 -
  2.1787 -subsection {* Points of division of a partition. *}
  2.1788 -
  2.1789 -definition "division_points (k::(real^'n) set) d = 
  2.1790 -    {(j,x). (interval_lowerbound k)$j < x \<and> x < (interval_upperbound k)$j \<and>
  2.1791 -           (\<exists>i\<in>d. (interval_lowerbound i)$j = x \<or> (interval_upperbound i)$j = x)}"
  2.1792 -
  2.1793 -lemma division_points_finite: assumes "d division_of i"
  2.1794 -  shows "finite (division_points i d)"
  2.1795 -proof- note assm = division_ofD[OF assms]
  2.1796 -  let ?M = "\<lambda>j. {(j,x)|x. (interval_lowerbound i)$j < x \<and> x < (interval_upperbound i)$j \<and>
  2.1797 -           (\<exists>i\<in>d. (interval_lowerbound i)$j = x \<or> (interval_upperbound i)$j = x)}"
  2.1798 -  have *:"division_points i d = \<Union>(?M ` UNIV)"
  2.1799 -    unfolding division_points_def by auto
  2.1800 -  show ?thesis unfolding * using assm by auto qed
  2.1801 -
  2.1802 -lemma division_points_subset:
  2.1803 -  assumes "d division_of {a..b}" "\<forall>i. a$i < b$i"  "a$k < c" "c < b$k"
  2.1804 -  shows "division_points ({a..b} \<inter> {x. x$k \<le> c}) {l \<inter> {x. x$k \<le> c} | l . l \<in> d \<and> ~(l \<inter> {x. x$k \<le> c} = {})}
  2.1805 -                  \<subseteq> division_points ({a..b}) d" (is ?t1) and
  2.1806 -        "division_points ({a..b} \<inter> {x. x$k \<ge> c}) {l \<inter> {x. x$k \<ge> c} | l . l \<in> d \<and> ~(l \<inter> {x. x$k \<ge> c} = {})}
  2.1807 -                  \<subseteq> division_points ({a..b}) d" (is ?t2)
  2.1808 -proof- note assm = division_ofD[OF assms(1)]
  2.1809 -  have *:"\<forall>i. a$i \<le> b$i"   "\<forall>i. a$i \<le> (\<chi> i. if i = k then min (b $ k) c else b $ i) $ i"
  2.1810 -    "\<forall>i. (\<chi> i. if i = k then max (a $ k) c else a $ i) $ i \<le> b$i"  "min (b $ k) c = c" "max (a $ k) c = c"
  2.1811 -    using assms using less_imp_le by auto
  2.1812 -  show ?t1 unfolding division_points_def interval_split[of a b]
  2.1813 -    unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] Cart_lambda_beta unfolding *
  2.1814 -    unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+ unfolding mem_Collect_eq apply(erule exE conjE)+
  2.1815 -  proof- fix i l x assume as:"a $ fst x < snd x" "snd x < (if fst x = k then c else b $ fst x)"
  2.1816 -      "interval_lowerbound i $ fst x = snd x \<or> interval_upperbound i $ fst x = snd x"  "i = l \<inter> {x. x $ k \<le> c}" "l \<in> d" "l \<inter> {x. x $ k \<le> c} \<noteq> {}"
  2.1817 -    from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
  2.1818 -    have *:"\<forall>i. u $ i \<le> (\<chi> i. if i = k then min (v $ k) c else v $ i) $ i" using as(6) unfolding l interval_split interval_ne_empty as .
  2.1819 -    have **:"\<forall>i. u$i \<le> v$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
  2.1820 -    show "a $ fst x < snd x \<and> snd x < b $ fst x \<and> (\<exists>i\<in>d. interval_lowerbound i $ fst x = snd x \<or> interval_upperbound i $ fst x = snd x)"
  2.1821 -      using as(1-3,5) unfolding l interval_split interval_ne_empty as interval_bounds[OF *] Cart_lambda_beta apply-
  2.1822 -      apply(rule,assumption,rule) defer apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **]
  2.1823 -      apply(case_tac[!] "fst x = k") using assms by auto
  2.1824 -  qed
  2.1825 -  show ?t2 unfolding division_points_def interval_split[of a b]
  2.1826 -    unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] Cart_lambda_beta unfolding *
  2.1827 -    unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+ unfolding mem_Collect_eq apply(erule exE conjE)+
  2.1828 -  proof- fix i l x assume as:"(if fst x = k then c else a $ fst x) < snd x" "snd x < b $ fst x" "interval_lowerbound i $ fst x = snd x \<or> interval_upperbound i $ fst x = snd x"
  2.1829 -      "i = l \<inter> {x. c \<le> x $ k}" "l \<in> d" "l \<inter> {x. c \<le> x $ k} \<noteq> {}"
  2.1830 -    from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
  2.1831 -    have *:"\<forall>i. (\<chi> i. if i = k then max (u $ k) c else u $ i) $ i \<le> v $ i" using as(6) unfolding l interval_split interval_ne_empty as .
  2.1832 -    have **:"\<forall>i. u$i \<le> v$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
  2.1833 -    show "a $ fst x < snd x \<and> snd x < b $ fst x \<and> (\<exists>i\<in>d. interval_lowerbound i $ fst x = snd x \<or> interval_upperbound i $ fst x = snd x)"
  2.1834 -      using as(1-3,5) unfolding l interval_split interval_ne_empty as interval_bounds[OF *] Cart_lambda_beta apply-
  2.1835 -      apply rule defer apply(rule,assumption) apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **]
  2.1836 -      apply(case_tac[!] "fst x = k") using assms by auto qed qed
  2.1837 -
  2.1838 -lemma division_points_psubset:
  2.1839 -  assumes "d division_of {a..b}"  "\<forall>i. a$i < b$i"  "a$k < c" "c < b$k"
  2.1840 -  "l \<in> d" "interval_lowerbound l$k = c \<or> interval_upperbound l$k = c"
  2.1841 -  shows "division_points ({a..b} \<inter> {x. x$k \<le> c}) {l \<inter> {x. x$k \<le> c} | l. l\<in>d \<and> l \<inter> {x. x$k \<le> c} \<noteq> {}} \<subset> division_points ({a..b}) d" (is "?D1 \<subset> ?D") 
  2.1842 -        "division_points ({a..b} \<inter> {x. x$k \<ge> c}) {l \<inter> {x. x$k \<ge> c} | l. l\<in>d \<and> l \<inter> {x. x$k \<ge> c} \<noteq> {}} \<subset> division_points ({a..b}) d" (is "?D2 \<subset> ?D") 
  2.1843 -proof- have ab:"\<forall>i. a$i \<le> b$i" using assms(2) by(auto intro!:less_imp_le)
  2.1844 -  guess u v using division_ofD(4)[OF assms(1,5)] apply-by(erule exE)+ note l=this
  2.1845 -  have uv:"\<forall>i. u$i \<le> v$i" "\<forall>i. a$i \<le> u$i \<and> v$i \<le> b$i" using division_ofD(2,2,3)[OF assms(1,5)] unfolding l interval_ne_empty
  2.1846 -    unfolding subset_eq apply- defer apply(erule_tac x=u in ballE, erule_tac x=v in ballE) unfolding mem_interval by auto
  2.1847 -  have *:"interval_upperbound ({a..b} \<inter> {x. x $ k \<le> interval_upperbound l $ k}) $ k = interval_upperbound l $ k"
  2.1848 -         "interval_upperbound ({a..b} \<inter> {x. x $ k \<le> interval_lowerbound l $ k}) $ k = interval_lowerbound l $ k"
  2.1849 -    unfolding interval_split apply(subst interval_bounds) prefer 3 apply(subst interval_bounds)
  2.1850 -    unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab by auto
  2.1851 -  have "\<exists>x. x \<in> ?D - ?D1" using assms(2-) apply-apply(erule disjE)
  2.1852 -    apply(rule_tac x="(k,(interval_lowerbound l)$k)" in exI) defer
  2.1853 -    apply(rule_tac x="(k,(interval_upperbound l)$k)" in exI)
  2.1854 -    unfolding division_points_def unfolding interval_bounds[OF ab]
  2.1855 -    apply (auto simp add:interval_bounds) unfolding * by auto
  2.1856 -  thus "?D1 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) by auto
  2.1857 -
  2.1858 -  have *:"interval_lowerbound ({a..b} \<inter> {x. x $ k \<ge> interval_lowerbound l $ k}) $ k = interval_lowerbound l $ k"
  2.1859 -         "interval_lowerbound ({a..b} \<inter> {x. x $ k \<ge> interval_upperbound l $ k}) $ k = interval_upperbound l $ k"
  2.1860 -    unfolding interval_split apply(subst interval_bounds) prefer 3 apply(subst interval_bounds)
  2.1861 -    unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab by auto
  2.1862 -  have "\<exists>x. x \<in> ?D - ?D2" using assms(2-) apply-apply(erule disjE)
  2.1863 -    apply(rule_tac x="(k,(interval_lowerbound l)$k)" in exI) defer
  2.1864 -    apply(rule_tac x="(k,(interval_upperbound l)$k)" in exI)
  2.1865 -    unfolding division_points_def unfolding interval_bounds[OF ab]
  2.1866 -    apply (auto simp add:interval_bounds) unfolding * by auto
  2.1867 -  thus "?D2 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) by auto qed
  2.1868 -
  2.1869 -subsection {* Preservation by divisions and tagged divisions. *}
  2.1870 -
  2.1871 -lemma support_support[simp]:"support opp f (support opp f s) = support opp f s"
  2.1872 -  unfolding support_def by auto
  2.1873 -
  2.1874 -lemma iterate_support[simp]: "iterate opp (support opp f s) f = iterate opp s f"
  2.1875 -  unfolding iterate_def support_support by auto
  2.1876 -
  2.1877 -lemma iterate_expand_cases:
  2.1878 -  "iterate opp s f = (if finite(support opp f s) then iterate opp (support opp f s) f else neutral opp)"
  2.1879 -  apply(cases) apply(subst if_P,assumption) unfolding iterate_def support_support fold'_def by auto 
  2.1880 -
  2.1881 -lemma iterate_image: assumes "monoidal opp"  "inj_on f s"
  2.1882 -  shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)"
  2.1883 -proof- have *:"\<And>s. finite s \<Longrightarrow>  \<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<longrightarrow> x = y \<Longrightarrow>
  2.1884 -     iterate opp (f ` s) g = iterate opp s (g \<circ> f)"
  2.1885 -  proof- case goal1 show ?case using goal1
  2.1886 -    proof(induct s) case empty thus ?case using assms(1) by auto
  2.1887 -    next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)]
  2.1888 -        unfolding if_not_P[OF insert(2)] apply(subst insert(3)[THEN sym])
  2.1889 -        unfolding image_insert defer apply(subst iterate_insert[OF assms(1)])
  2.1890 -        apply(rule finite_imageI insert)+ apply(subst if_not_P)
  2.1891 -        unfolding image_iff o_def using insert(2,4) by auto
  2.1892 -    qed qed
  2.1893 -  show ?thesis 
  2.1894 -    apply(cases "finite (support opp g (f ` s))")
  2.1895 -    apply(subst (1) iterate_support[THEN sym],subst (2) iterate_support[THEN sym])
  2.1896 -    unfolding support_clauses apply(rule *)apply(rule finite_imageD,assumption) unfolding inj_on_def[symmetric]
  2.1897 -    apply(rule subset_inj_on[OF assms(2) support_subset])+
  2.1898 -    apply(subst iterate_expand_cases) unfolding support_clauses apply(simp only: if_False)
  2.1899 -    apply(subst iterate_expand_cases) apply(subst if_not_P) by auto qed
  2.1900 -
  2.1901 -
  2.1902 -(* This lemma about iterations comes up in a few places.                     *)
  2.1903 -lemma iterate_nonzero_image_lemma:
  2.1904 -  assumes "monoidal opp" "finite s" "g(a) = neutral opp"
  2.1905 -  "\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = neutral opp"
  2.1906 -  shows "iterate opp {f x | x. x \<in> s \<and> f x \<noteq> a} g = iterate opp s (g \<circ> f)"
  2.1907 -proof- have *:"{f x |x. x \<in> s \<and> ~(f x = a)} = f ` {x. x \<in> s \<and> ~(f x = a)}" by auto
  2.1908 -  have **:"support opp (g \<circ> f) {x \<in> s. f x \<noteq> a} = support opp (g \<circ> f) s"
  2.1909 -    unfolding support_def using assms(3) by auto
  2.1910 -  show ?thesis unfolding *
  2.1911 -    apply(subst iterate_support[THEN sym]) unfolding support_clauses
  2.1912 -    apply(subst iterate_image[OF assms(1)]) defer
  2.1913 -    apply(subst(2) iterate_support[THEN sym]) apply(subst **)
  2.1914 -    unfolding inj_on_def using assms(3,4) unfolding support_def by auto qed
  2.1915 -
  2.1916 -lemma iterate_eq_neutral:
  2.1917 -  assumes "monoidal opp"  "\<forall>x \<in> s. (f(x) = neutral opp)"
  2.1918 -  shows "(iterate opp s f = neutral opp)"
  2.1919 -proof- have *:"support opp f s = {}" unfolding support_def using assms(2) by auto
  2.1920 -  show ?thesis apply(subst iterate_support[THEN sym]) 
  2.1921 -    unfolding * using assms(1) by auto qed
  2.1922 -
  2.1923 -lemma iterate_op: assumes "monoidal opp" "finite s"
  2.1924 -  shows "iterate opp s (\<lambda>x. opp (f x) (g x)) = opp (iterate opp s f) (iterate opp s g)" using assms(2)
  2.1925 -proof(induct s) case empty thus ?case unfolding iterate_insert[OF assms(1)] using assms(1) by auto
  2.1926 -next case (insert x F) show ?case unfolding iterate_insert[OF assms(1) insert(1)] if_not_P[OF insert(2)] insert(3)
  2.1927 -    unfolding monoidal_ac[OF assms(1)] by(rule refl) qed
  2.1928 -
  2.1929 -lemma iterate_eq: assumes "monoidal opp" "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
  2.1930 -  shows "iterate opp s f = iterate opp s g"
  2.1931 -proof- have *:"support opp g s = support opp f s"
  2.1932 -    unfolding support_def using assms(2) by auto
  2.1933 -  show ?thesis
  2.1934 -  proof(cases "finite (support opp f s)")
  2.1935 -    case False thus ?thesis apply(subst iterate_expand_cases,subst(2) iterate_expand_cases)
  2.1936 -      unfolding * by auto
  2.1937 -  next def su \<equiv> "support opp f s"
  2.1938 -    case True note support_subset[of opp f s] 
  2.1939 -    thus ?thesis apply- apply(subst iterate_support[THEN sym],subst(2) iterate_support[THEN sym]) unfolding * using True
  2.1940 -      unfolding su_def[symmetric]
  2.1941 -    proof(induct su) case empty show ?case by auto
  2.1942 -    next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)] 
  2.1943 -        unfolding if_not_P[OF insert(2)] apply(subst insert(3))
  2.1944 -        defer apply(subst assms(2)[of x]) using insert by auto qed qed qed
  2.1945 -
  2.1946 -lemma nonempty_witness: assumes "s \<noteq> {}" obtains x where "x \<in> s" using assms by auto
  2.1947 -
  2.1948 -lemma operative_division: fixes f::"(real^'n) set \<Rightarrow> 'a"
  2.1949 -  assumes "monoidal opp" "operative opp f" "d division_of {a..b}"
  2.1950 -  shows "iterate opp d f = f {a..b}"
  2.1951 -proof- def C \<equiv> "card (division_points {a..b} d)" thus ?thesis using assms
  2.1952 -  proof(induct C arbitrary:a b d rule:full_nat_induct)
  2.1953 -    case goal1
  2.1954 -    { presume *:"content {a..b} \<noteq> 0 \<Longrightarrow> ?case"
  2.1955 -      thus ?case apply-apply(cases) defer apply assumption
  2.1956 -      proof- assume as:"content {a..b} = 0"
  2.1957 -        show ?case unfolding operativeD(1)[OF assms(2) as] apply(rule iterate_eq_neutral[OF goal1(2)])
  2.1958 -        proof fix x assume x:"x\<in>d"
  2.1959 -          then guess u v apply(drule_tac division_ofD(4)[OF goal1(4)]) by(erule exE)+
  2.1960 -          thus "f x = neutral opp" using division_of_content_0[OF as goal1(4)] 
  2.1961 -            using operativeD(1)[OF assms(2)] x by auto
  2.1962 -        qed qed }
  2.1963 -    assume "content {a..b} \<noteq> 0" note ab = this[unfolded content_lt_nz[THEN sym] content_pos_lt_eq]
  2.1964 -    hence ab':"\<forall>i. a$i \<le> b$i" by (auto intro!: less_imp_le) show ?case 
  2.1965 -    proof(cases "division_points {a..b} d = {}")
  2.1966 -      case True have d':"\<forall>i\<in>d. \<exists>u v. i = {u..v} \<and>
  2.1967 -        (\<forall>j. u$j = a$j \<and> v$j = a$j \<or> u$j = b$j \<and> v$j = b$j \<or> u$j = a$j \<and> v$j = b$j)"
  2.1968 -        unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule)
  2.1969 -        apply(rule_tac x=a in exI,rule_tac x=b in exI) apply(rule,rule refl) apply(rule)
  2.1970 -      proof- fix u v j assume as:"{u..v} \<in> d" note division_ofD(3)[OF goal1(4) this]
  2.1971 -        hence uv:"\<forall>i. u$i \<le> v$i" "u$j \<le> v$j" unfolding interval_ne_empty by auto
  2.1972 -        have *:"\<And>p r Q. p \<or> r \<or> (\<forall>x\<in>d. Q x) \<Longrightarrow> p \<or> r \<or> (Q {u..v})" using as by auto
  2.1973 -        have "(j, u$j) \<notin> division_points {a..b} d"
  2.1974 -          "(j, v$j) \<notin> division_points {a..b} d" using True by auto
  2.1975 -        note this[unfolded de_Morgan_conj division_points_def mem_Collect_eq split_conv interval_bounds[OF ab'] bex_simps]
  2.1976 -        note *[OF this(1)] *[OF this(2)] note this[unfolded interval_bounds[OF uv(1)]]
  2.1977 -        moreover have "a$j \<le> u$j" "v$j \<le> b$j" using division_ofD(2,2,3)[OF goal1(4) as] 
  2.1978 -          unfolding subset_eq apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE)
  2.1979 -          unfolding interval_ne_empty mem_interval by auto
  2.1980 -        ultimately show "u$j = a$j \<and> v$j = a$j \<or> u$j = b$j \<and> v$j = b$j \<or> u$j = a$j \<and> v$j = b$j"
  2.1981 -          unfolding not_less de_Morgan_disj using ab[rule_format,of j] uv(2) by auto
  2.1982 -      qed have "(1/2) *\<^sub>R (a+b) \<in> {a..b}" unfolding mem_interval using ab by(auto intro!:less_imp_le)
  2.1983 -      note this[unfolded division_ofD(6)[OF goal1(4),THEN sym] Union_iff]
  2.1984 -      then guess i .. note i=this guess u v using d'[rule_format,OF i(1)] apply-by(erule exE conjE)+ note uv=this
  2.1985 -      have "{a..b} \<in> d"
  2.1986 -      proof- { presume "i = {a..b}" thus ?thesis using i by auto }
  2.1987 -        { presume "u = a" "v = b" thus "i = {a..b}" using uv by auto }
  2.1988 -        show "u = a" "v = b" unfolding Cart_eq
  2.1989 -        proof(rule_tac[!] allI) fix j note i(2)[unfolded uv mem_interval,rule_format,of j]
  2.1990 -          thus "u $ j = a $ j" "v $ j = b $ j" using uv(2)[rule_format,of j] by auto
  2.1991 -        qed qed
  2.1992 -      hence *:"d = insert {a..b} (d - {{a..b}})" by auto
  2.1993 -      have "iterate opp (d - {{a..b}}) f = neutral opp" apply(rule iterate_eq_neutral[OF goal1(2)])
  2.1994 -      proof fix x assume x:"x \<in> d - {{a..b}}" hence "x\<in>d" by auto note d'[rule_format,OF this]
  2.1995 -        then guess u v apply-by(erule exE conjE)+ note uv=this
  2.1996 -        have "u\<noteq>a \<or> v\<noteq>b" using x[unfolded uv] by auto  
  2.1997 -        then obtain j where "u$j \<noteq> a$j \<or> v$j \<noteq> b$j" unfolding Cart_eq by auto
  2.1998 -        hence "u$j = v$j" using uv(2)[rule_format,of j] by auto
  2.1999 -        hence "content {u..v} = 0"  unfolding content_eq_0 apply(rule_tac x=j in exI) by auto
  2.2000 -        thus "f x = neutral opp" unfolding uv(1) by(rule operativeD(1)[OF goal1(3)])
  2.2001 -      qed thus "iterate opp d f = f {a..b}" apply-apply(subst *) 
  2.2002 -        apply(subst iterate_insert[OF goal1(2)]) using goal1(2,4) by auto
  2.2003 -    next case False hence "\<exists>x. x\<in>division_points {a..b} d" by auto
  2.2004 -      then guess k c unfolding split_paired_Ex apply- unfolding division_points_def mem_Collect_eq split_conv
  2.2005 -        by(erule exE conjE)+ note kc=this[unfolded interval_bounds[OF ab']]
  2.2006 -      from this(3) guess j .. note j=this
  2.2007 -      def d1 \<equiv> "{l \<inter> {x. x$k \<le> c} | l. l \<in> d \<and> l \<inter> {x. x$k \<le> c} \<noteq> {}}"
  2.2008 -      def d2 \<equiv> "{l \<inter> {x. x$k \<ge> c} | l. l \<in> d \<and> l \<inter> {x. x$k \<ge> c} \<noteq> {}}"
  2.2009 -      def cb \<equiv> "(\<chi> i. if i = k then c else b$i)" and ca \<equiv> "(\<chi> i. if i = k then c else a$i)"
  2.2010 -      note division_points_psubset[OF goal1(4) ab kc(1-2) j]
  2.2011 -      note psubset_card_mono[OF _ this(1)] psubset_card_mono[OF _ this(2)]
  2.2012 -      hence *:"(iterate opp d1 f) = f ({a..b} \<inter> {x. x$k \<le> c})" "(iterate opp d2 f) = f ({a..b} \<inter> {x. x$k \<ge> c})"
  2.2013 -        apply- unfolding interval_split apply(rule_tac[!] goal1(1)[rule_format])
  2.2014 -        using division_split[OF goal1(4), where k=k and c=c]
  2.2015 -        unfolding interval_split d1_def[symmetric] d2_def[symmetric] unfolding goal1(2) Suc_le_mono
  2.2016 -        using goal1(2-3) using division_points_finite[OF goal1(4)] by auto
  2.2017 -      have "f {a..b} = opp (iterate opp d1 f) (iterate opp d2 f)" (is "_ = ?prev")
  2.2018 -        unfolding * apply(rule operativeD(2)) using goal1(3) .
  2.2019 -      also have "iterate opp d1 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x$k \<le> c}))"
  2.2020 -        unfolding d1_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])
  2.2021 -        unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+
  2.2022 -        unfolding empty_as_interval[THEN sym] apply(rule content_empty)
  2.2023 -      proof(rule,rule,rule,erule conjE) fix l y assume as:"l \<in> d" "y \<in> d" "l \<inter> {x. x $ k \<le> c} = y \<inter> {x. x $ k \<le> c}" "l \<noteq> y" 
  2.2024 -        from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this
  2.2025 -        show "f (l \<inter> {x. x $ k \<le> c}) = neutral opp" unfolding l interval_split
  2.2026 -          apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym] apply(rule division_split_left_inj)
  2.2027 -          apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as)+
  2.2028 -      qed also have "iterate opp d2 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x$k \<ge> c}))"
  2.2029 -        unfolding d2_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])
  2.2030 -        unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+
  2.2031 -        unfolding empty_as_interval[THEN sym] apply(rule content_empty)
  2.2032 -      proof(rule,rule,rule,erule conjE) fix l y assume as:"l \<in> d" "y \<in> d" "l \<inter> {x. c \<le> x $ k} = y \<inter> {x. c \<le> x $ k}" "l \<noteq> y" 
  2.2033 -        from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this
  2.2034 -        show "f (l \<inter> {x. x $ k \<ge> c}) = neutral opp" unfolding l interval_split
  2.2035 -          apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym] apply(rule division_split_right_inj)
  2.2036 -          apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as)+
  2.2037 -      qed also have *:"\<forall>x\<in>d. f x = opp (f (x \<inter> {x. x $ k \<le> c})) (f (x \<inter> {x. c \<le> x $ k}))"
  2.2038 -        unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule,rule operativeD(2)) using goal1(3) .
  2.2039 -      have "opp (iterate opp d (\<lambda>l. f (l \<inter> {x. x $ k \<le> c}))) (iterate opp d (\<lambda>l. f (l \<inter> {x. c \<le> x $ k})))
  2.2040 -        = iterate opp d f" apply(subst(3) iterate_eq[OF _ *[rule_format]]) prefer 3
  2.2041 -        apply(rule iterate_op[THEN sym]) using goal1 by auto
  2.2042 -      finally show ?thesis by auto
  2.2043 -    qed qed qed 
  2.2044 -
  2.2045 -lemma iterate_image_nonzero: assumes "monoidal opp"
  2.2046 -  "finite s" "\<forall>x\<in>s. \<forall>y\<in>s. ~(x = y) \<and> f x = f y \<longrightarrow> g(f x) = neutral opp"
  2.2047 -  shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)" using assms
  2.2048 -proof(induct rule:finite_subset_induct[OF assms(2) subset_refl])
  2.2049 -  case goal1 show ?case using assms(1) by auto
  2.2050 -next case goal2 have *:"\<And>x y. y = neutral opp \<Longrightarrow> x = opp y x" using assms(1) by auto
  2.2051 -  show ?case unfolding image_insert apply(subst iterate_insert[OF assms(1)])
  2.2052 -    apply(rule finite_imageI goal2)+
  2.2053 -    apply(cases "f a \<in> f ` F") unfolding if_P if_not_P apply(subst goal2(4)[OF assms(1) goal2(1)]) defer
  2.2054 -    apply(subst iterate_insert[OF assms(1) goal2(1)]) defer
  2.2055 -    apply(subst iterate_insert[OF assms(1) goal2(1)])
  2.2056 -    unfolding if_not_P[OF goal2(3)] defer unfolding image_iff defer apply(erule bexE)
  2.2057 -    apply(rule *) unfolding o_def apply(rule_tac y=x in goal2(7)[rule_format])
  2.2058 -    using goal2 unfolding o_def by auto qed 
  2.2059 -
  2.2060 -lemma operative_tagged_division: assumes "monoidal opp" "operative opp f" "d tagged_division_of {a..b}"
  2.2061 -  shows "iterate(opp) d (\<lambda>(x,l). f l) = f {a..b}"
  2.2062 -proof- have *:"(\<lambda>(x,l). f l) = (f o snd)" unfolding o_def by(rule,auto) note assm = tagged_division_ofD[OF assms(3)]
  2.2063 -  have "iterate(opp) d (\<lambda>(x,l). f l) = iterate opp (snd ` d) f" unfolding *
  2.2064 -    apply(rule iterate_image_nonzero[THEN sym,OF assms(1)]) apply(rule tagged_division_of_finite assms)+ 
  2.2065 -    unfolding Ball_def split_paired_All snd_conv apply(rule,rule,rule,rule,rule,rule,rule,erule conjE)
  2.2066 -  proof- fix a b aa ba assume as:"(a, b) \<in> d" "(aa, ba) \<in> d" "(a, b) \<noteq> (aa, ba)" "b = ba"
  2.2067 -    guess u v using assm(4)[OF as(1)] apply-by(erule exE)+ note uv=this
  2.2068 -    show "f b = neutral opp" unfolding uv apply(rule operativeD(1)[OF assms(2)])
  2.2069 -      unfolding content_eq_0_interior using tagged_division_ofD(5)[OF assms(3) as(1-3)]
  2.2070 -      unfolding as(4)[THEN sym] uv by auto
  2.2071 -  qed also have "\<dots> = f {a..b}" 
  2.2072 -    using operative_division[OF assms(1-2) division_of_tagged_division[OF assms(3)]] .
  2.2073 -  finally show ?thesis . qed
  2.2074 -
  2.2075 -subsection {* Additivity of content. *}
  2.2076 -
  2.2077 -lemma setsum_iterate:assumes "finite s" shows "setsum f s = iterate op + s f"
  2.2078 -proof- have *:"setsum f s = setsum f (support op + f s)"
  2.2079 -    apply(rule setsum_mono_zero_right)
  2.2080 -    unfolding support_def neutral_monoid using assms by auto
  2.2081 -  thus ?thesis unfolding * setsum_def iterate_def fold_image_def fold'_def
  2.2082 -    unfolding neutral_monoid . qed
  2.2083 -
  2.2084 -lemma additive_content_division: assumes "d division_of {a..b}"
  2.2085 -  shows "setsum content d = content({a..b})"
  2.2086 -  unfolding operative_division[OF monoidal_monoid operative_content assms,THEN sym]
  2.2087 -  apply(subst setsum_iterate) using assms by auto
  2.2088 -
  2.2089 -lemma additive_content_tagged_division:
  2.2090 -  assumes "d tagged_division_of {a..b}"
  2.2091 -  shows "setsum (\<lambda>(x,l). content l) d = content({a..b})"
  2.2092 -  unfolding operative_tagged_division[OF monoidal_monoid operative_content assms,THEN sym]
  2.2093 -  apply(subst setsum_iterate) using assms by auto
  2.2094 -
  2.2095 -subsection {* Finally, the integral of a constant\<forall> *}
  2.2096 -
  2.2097 -lemma has_integral_const[intro]:
  2.2098 -  "((\<lambda>x. c) has_integral (content({a..b::real^'n}) *\<^sub>R c)) ({a..b})"
  2.2099 -  unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI)
  2.2100 -  apply(rule,rule gauge_trivial)apply(rule,rule,erule conjE)
  2.2101 -  unfolding split_def apply(subst scaleR_left.setsum[THEN sym, unfolded o_def])
  2.2102 -  defer apply(subst additive_content_tagged_division[unfolded split_def]) apply assumption by auto
  2.2103 -
  2.2104 -subsection {* Bounds on the norm of Riemann sums and the integral itself. *}
  2.2105 -
  2.2106 -lemma dsum_bound: assumes "p division_of {a..b}" "norm(c) \<le> e"
  2.2107 -  shows "norm(setsum (\<lambda>l. content l *\<^sub>R c) p) \<le> e * content({a..b})" (is "?l \<le> ?r")
  2.2108 -  apply(rule order_trans,rule setsum_norm) defer unfolding norm_scaleR setsum_left_distrib[THEN sym]
  2.2109 -  apply(rule order_trans[OF mult_left_mono],rule assms,rule setsum_abs_ge_zero)
  2.2110 -  apply(subst real_mult_commute) apply(rule mult_left_mono)
  2.2111 -  apply(rule order_trans[of _ "setsum content p"]) apply(rule eq_refl,rule setsum_cong2)
  2.2112 -  apply(subst abs_of_nonneg) unfolding additive_content_division[OF assms(1)]
  2.2113 -proof- from order_trans[OF norm_ge_zero[of c] assms(2)] show "0 \<le> e" .
  2.2114 -  fix x assume "x\<in>p" from division_ofD(4)[OF assms(1) this] guess u v apply-by(erule exE)+
  2.2115 -  thus "0 \<le> content x" using content_pos_le by auto
  2.2116 -qed(insert assms,auto)
  2.2117 -
  2.2118 -lemma rsum_bound: assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. norm(f x) \<le> e"
  2.2119 -  shows "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p) \<le> e * content({a..b})"
  2.2120 -proof(cases "{a..b} = {}") case True
  2.2121 -  show ?thesis using assms(1) unfolding True tagged_division_of_trivial by auto
  2.2122 -next case False show ?thesis
  2.2123 -    apply(rule order_trans,rule setsum_norm) defer unfolding split_def norm_scaleR
  2.2124 -    apply(rule order_trans[OF setsum_mono]) apply(rule mult_left_mono[OF _ abs_ge_zero, of _ e]) defer
  2.2125 -    unfolding setsum_left_distrib[THEN sym] apply(subst real_mult_commute) apply(rule mult_left_mono)
  2.2126 -    apply(rule order_trans[of _ "setsum (content \<circ> snd) p"]) apply(rule eq_refl,rule setsum_cong2)
  2.2127 -    apply(subst o_def, rule abs_of_nonneg)
  2.2128 -  proof- show "setsum (content \<circ> snd) p \<le> content {a..b}" apply(rule eq_refl)
  2.2129 -      unfolding additive_content_tagged_division[OF assms(1),THEN sym] split_def by auto
  2.2130 -    guess w using nonempty_witness[OF False] .
  2.2131 -    thus "e\<ge>0" apply-apply(rule order_trans) defer apply(rule assms(2)[rule_format],assumption) by auto
  2.2132 -    fix xk assume *:"xk\<in>p" guess x k  using surj_pair[of xk] apply-by(erule exE)+ note xk = this *[unfolded this]
  2.2133 -    from tagged_division_ofD(4)[OF assms(1) xk(2)] guess u v apply-by(erule exE)+ note uv=this
  2.2134 -    show "0\<le> content (snd xk)" unfolding xk snd_conv uv by(rule content_pos_le)
  2.2135 -    show "norm (f (fst xk)) \<le> e" unfolding xk fst_conv using tagged_division_ofD(2,3)[OF assms(1) xk(2)] assms(2) by auto
  2.2136 -  qed(insert assms,auto) qed
  2.2137 -
  2.2138 -lemma rsum_diff_bound:
  2.2139 -  assumes "p tagged_division_of {a..b}"  "\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e"
  2.2140 -  shows "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - setsum (\<lambda>(x,k). content k *\<^sub>R g x) p) \<le> e * content({a..b})"
  2.2141 -  apply(rule order_trans[OF _ rsum_bound[OF assms]]) apply(rule eq_refl) apply(rule arg_cong[where f=norm])
  2.2142 -  unfolding setsum_subtractf[THEN sym] apply(rule setsum_cong2) unfolding scaleR.diff_right by auto
  2.2143 -
  2.2144 -lemma has_integral_bound: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
  2.2145 -  assumes "0 \<le> B" "(f has_integral i) ({a..b})" "\<forall>x\<in>{a..b}. norm(f x) \<le> B"
  2.2146 -  shows "norm i \<le> B * content {a..b}"
  2.2147 -proof- let ?P = "content {a..b} > 0" { presume "?P \<Longrightarrow> ?thesis"
  2.2148 -    thus ?thesis proof(cases ?P) case False
  2.2149 -      hence *:"content {a..b} = 0" using content_lt_nz by auto
  2.2150 -      hence **:"i = 0" using assms(2) apply(subst has_integral_null_eq[THEN sym]) by auto
  2.2151 -      show ?thesis unfolding * ** using assms(1) by auto
  2.2152 -    qed auto } assume ab:?P
  2.2153 -  { presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }
  2.2154 -  assume "\<not> ?thesis" hence *:"norm i - B * content {a..b} > 0" by auto
  2.2155 -  from assms(2)[unfolded has_integral,rule_format,OF *] guess d apply-by(erule exE conjE)+ note d=this[rule_format]
  2.2156 -  from fine_division_exists[OF this(1), of a b] guess p . note p=this
  2.2157 -  have *:"\<And>s B. norm s \<le> B \<Longrightarrow> \<not> (norm (s - i) < norm i - B)"
  2.2158 -  proof- case goal1 thus ?case unfolding not_less
  2.2159 -    using norm_triangle_sub[of i s] unfolding norm_minus_commute by auto
  2.2160 -  qed show False using d(2)[OF conjI[OF p]] *[OF rsum_bound[OF p(1) assms(3)]] by auto qed
  2.2161 -
  2.2162 -subsection {* Similar theorems about relationship among components. *}
  2.2163 -
  2.2164 -lemma rsum_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
  2.2165 -  assumes "p tagged_division_of {a..b}"  "\<forall>x\<in>{a..b}. (f x)$i \<le> (g x)$i"
  2.2166 -  shows "(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p)$i \<le> (setsum (\<lambda>(x,k). content k *\<^sub>R g x) p)$i"
  2.2167 -  unfolding setsum_component apply(rule setsum_mono)
  2.2168 -  apply(rule mp) defer apply assumption apply(induct_tac x,rule) unfolding split_conv
  2.2169 -proof- fix a b assume ab:"(a,b) \<in> p" note assm = tagged_division_ofD(2-4)[OF assms(1) ab]
  2.2170 -  from this(3) guess u v apply-by(erule exE)+ note b=this
  2.2171 -  show "(content b *\<^sub>R f a) $ i \<le> (content b *\<^sub>R g a) $ i" unfolding b
  2.2172 -    unfolding Cart_nth.scaleR real_scaleR_def apply(rule mult_left_mono)
  2.2173 -    defer apply(rule content_pos_le,rule assms(2)[rule_format]) using assm by auto qed
  2.2174 -
  2.2175 -lemma has_integral_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
  2.2176 -  assumes "(f has_integral i) s" "(g has_integral j) s"  "\<forall>x\<in>s. (f x)$k \<le> (g x)$k"
  2.2177 -  shows "i$k \<le> j$k"
  2.2178 -proof- have lem:"\<And>a b g i j. \<And>f::real^'n \<Rightarrow> real^'m. (f has_integral i) ({a..b}) \<Longrightarrow> 
  2.2179 -    (g has_integral j) ({a..b}) \<Longrightarrow> \<forall>x\<in>{a..b}. (f x)$k \<le> (g x)$k \<Longrightarrow> i$k \<le> j$k"
  2.2180 -  proof(rule ccontr) case goal1 hence *:"0 < (i$k - j$k) / 3" by auto
  2.2181 -    guess d1 using goal1(1)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d1=this[rule_format]
  2.2182 -    guess d2 using goal1(2)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d2=this[rule_format]
  2.2183 -    guess p using fine_division_exists[OF gauge_inter[OF d1(1) d2(1)], of a b] unfolding fine_inter .
  2.2184 -    note p = this(1) conjunctD2[OF this(2)]  note le_less_trans[OF component_le_norm, of _ _ k]
  2.2185 -    note this[OF d1(2)[OF conjI[OF p(1,2)]]] this[OF d2(2)[OF conjI[OF p(1,3)]]]
  2.2186 -    thus False unfolding Cart_nth.diff using rsum_component_le[OF p(1) goal1(3)] by smt
  2.2187 -  qed let ?P = "\<exists>a b. s = {a..b}"
  2.2188 -  { presume "\<not> ?P \<Longrightarrow> ?thesis" thus ?thesis proof(cases ?P)
  2.2189 -      case True then guess a b apply-by(erule exE)+ note s=this
  2.2190 -      show ?thesis apply(rule lem) using assms[unfolded s] by auto
  2.2191 -    qed auto } assume as:"\<not> ?P"
  2.2192 -  { presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }
  2.2193 -  assume "\<not> i$k \<le> j$k" hence ij:"(i$k - j$k) / 3 > 0" by auto
  2.2194 -  note has_integral_altD[OF _ as this] from this[OF assms(1)] this[OF assms(2)] guess B1 B2 . note B=this[rule_format]
  2.2195 -  have "bounded (ball 0 B1 \<union> ball (0::real^'n) B2)" unfolding bounded_Un by(rule conjI bounded_ball)+
  2.2196 -  from bounded_subset_closed_interval[OF this] guess a b apply- by(erule exE)+
  2.2197 -  note ab = conjunctD2[OF this[unfolded Un_subset_iff]]
  2.2198 -  guess w1 using B(2)[OF ab(1)] .. note w1=conjunctD2[OF this]
  2.2199 -  guess w2 using B(4)[OF ab(2)] .. note w2=conjunctD2[OF this]
  2.2200 -  have *:"\<And>w1 w2 j i::real .\<bar>w1 - i\<bar> < (i - j) / 3 \<Longrightarrow> \<bar>w2 - j\<bar> < (i - j) / 3 \<Longrightarrow> w1 \<le> w2 \<Longrightarrow> False" by smt(*SMTSMT*)
  2.2201 -  note le_less_trans[OF component_le_norm[of _ k]] note this[OF w1(2)] this[OF w2(2)] moreover
  2.2202 -  have "w1$k \<le> w2$k" apply(rule lem[OF w1(1) w2(1)]) using assms by auto ultimately
  2.2203 -  show False unfolding Cart_nth.diff by(rule *) qed
  2.2204 -
  2.2205 -lemma integral_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
  2.2206 -  assumes "f integrable_on s" "g integrable_on s"  "\<forall>x\<in>s. (f x)$k \<le> (g x)$k"
  2.2207 -  shows "(integral s f)$k \<le> (integral s g)$k"
  2.2208 -  apply(rule has_integral_component_le) using integrable_integral assms by auto
  2.2209 -
  2.2210 -lemma has_integral_dest_vec1_le: fixes f::"real^'n \<Rightarrow> real^1"
  2.2211 -  assumes "(f has_integral i) s"  "(g has_integral j) s" "\<forall>x\<in>s. f x \<le> g x"
  2.2212 -  shows "dest_vec1 i \<le> dest_vec1 j" apply(rule has_integral_component_le[OF assms(1-2)])
  2.2213 -  using assms(3) unfolding vector_le_def by auto
  2.2214 -
  2.2215 -lemma integral_dest_vec1_le: fixes f::"real^'n \<Rightarrow> real^1"
  2.2216 -  assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. f x \<le> g x"
  2.2217 -  shows "dest_vec1(integral s f) \<le> dest_vec1(integral s g)"
  2.2218 -  apply(rule has_integral_dest_vec1_le) apply(rule_tac[1-2] integrable_integral) using assms by auto
  2.2219 -
  2.2220 -lemma has_integral_component_pos: fixes f::"real^'n \<Rightarrow> real^'m"
  2.2221 -  assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> (f x)$k" shows "0 \<le> i$k"
  2.2222 -  using has_integral_component_le[OF has_integral_0 assms(1)] using assms(2) by auto
  2.2223 -
  2.2224 -lemma integral_component_pos: fixes f::"real^'n \<Rightarrow> real^'m"
  2.2225 -  assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> (f x)$k" shows "0 \<le> (integral s f)$k"
  2.2226 -  apply(rule has_integral_component_pos) using assms by auto
  2.2227 -
  2.2228 -lemma has_integral_dest_vec1_pos: fixes f::"real^'n \<Rightarrow> real^1"
  2.2229 -  assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> i"
  2.2230 -  using has_integral_component_pos[OF assms(1), of 1]
  2.2231 -  using assms(2) unfolding vector_le_def by auto
  2.2232 -
  2.2233 -lemma integral_dest_vec1_pos: fixes f::"real^'n \<Rightarrow> real^1"
  2.2234 -  assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> integral s f"
  2.2235 -  apply(rule has_integral_dest_vec1_pos) using assms by auto
  2.2236 -
  2.2237 -lemma has_integral_component_neg: fixes f::"real^'n \<Rightarrow> real^'m"
  2.2238 -  assumes "(f has_integral i) s" "\<forall>x\<in>s. (f x)$k \<le> 0" shows "i$k \<le> 0"
  2.2239 -  using has_integral_component_le[OF assms(1) has_integral_0] assms(2) by auto
  2.2240 -
  2.2241 -lemma has_integral_dest_vec1_neg: fixes f::"real^'n \<Rightarrow> real^1"
  2.2242 -  assumes "(f has_integral i) s" "\<forall>x\<in>s. f x \<le> 0" shows "i \<le> 0"
  2.2243 -  using has_integral_component_neg[OF assms(1),of 1] using assms(2) by auto
  2.2244 -
  2.2245 -lemma has_integral_component_lbound:
  2.2246 -  assumes "(f has_integral i) {a..b}"  "\<forall>x\<in>{a..b}. B \<le> f(x)$k" shows "B * content {a..b} \<le> i$k"
  2.2247 -  using has_integral_component_le[OF has_integral_const assms(1),of "(\<chi> i. B)" k] assms(2)
  2.2248 -  unfolding Cart_lambda_beta vector_scaleR_component by(auto simp add:field_simps)
  2.2249 -
  2.2250 -lemma has_integral_component_ubound: 
  2.2251 -  assumes "(f has_integral i) {a..b}" "\<forall>x\<in>{a..b}. f x$k \<le> B"
  2.2252 -  shows "i$k \<le> B * content({a..b})"
  2.2253 -  using has_integral_component_le[OF assms(1) has_integral_const, of k "vec B"]
  2.2254 -  unfolding vec_component Cart_nth.scaleR using assms(2) by(auto simp add:field_simps)
  2.2255 -
  2.2256 -lemma integral_component_lbound:
  2.2257 -  assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. B \<le> f(x)$k"
  2.2258 -  shows "B * content({a..b}) \<le> (integral({a..b}) f)$k"
  2.2259 -  apply(rule has_integral_component_lbound) using assms unfolding has_integral_integral by auto
  2.2260 -
  2.2261 -lemma integral_component_ubound:
  2.2262 -  assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. f(x)$k \<le> B"
  2.2263 -  shows "(integral({a..b}) f)$k \<le> B * content({a..b})"
  2.2264 -  apply(rule has_integral_component_ubound) using assms unfolding has_integral_integral by auto
  2.2265 -
  2.2266 -subsection {* Uniform limit of integrable functions is integrable. *}
  2.2267 -
  2.2268 -lemma real_arch_invD:
  2.2269 -  "0 < (e::real) \<Longrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
  2.2270 -  by(subst(asm) real_arch_inv)
  2.2271 -
  2.2272 -lemma integrable_uniform_limit: fixes f::"real^'n \<Rightarrow> 'a::banach"
  2.2273 -  assumes "\<forall>e>0. \<exists>g. (\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e) \<and> g integrable_on {a..b}"
  2.2274 -  shows "f integrable_on {a..b}"
  2.2275 -proof- { presume *:"content {a..b} > 0 \<Longrightarrow> ?thesis"
  2.2276 -    show ?thesis apply cases apply(rule *,assumption)
  2.2277 -      unfolding content_lt_nz integrable_on_def using has_integral_null by auto }
  2.2278 -  assume as:"content {a..b} > 0"
  2.2279 -  have *:"\<And>P. \<forall>e>(0::real). P e \<Longrightarrow> \<forall>n::nat. P (inverse (real n+1))" by auto
  2.2280 -  from choice[OF *[OF assms]] guess g .. note g=conjunctD2[OF this[rule_format],rule_format]
  2.2281 -  from choice[OF allI[OF g(2)[unfolded integrable_on_def], of "\<lambda>x. x"]] guess i .. note i=this[rule_format]
  2.2282 -  
  2.2283 -  have "Cauchy i" unfolding Cauchy_def
  2.2284 -  proof(rule,rule) fix e::real assume "e>0"
  2.2285 -    hence "e / 4 / content {a..b} > 0" using as by(auto simp add:field_simps)
  2.2286 -    then guess M apply-apply(subst(asm) real_arch_inv) by(erule exE conjE)+ note M=this
  2.2287 -    show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (i m) (i n) < e" apply(rule_tac x=M in exI,rule,rule,rule,rule)
  2.2288 -    proof- case goal1 have "e/4>0" using `e>0` by auto note * = i[unfolded has_integral,rule_format,OF this]
  2.2289 -      from *[of m] guess gm apply-by(erule conjE exE)+ note gm=this[rule_format]
  2.2290 -      from *[of n] guess gn apply-by(erule conjE exE)+ note gn=this[rule_format]
  2.2291 -      from fine_division_exists[OF gauge_inter[OF gm(1) gn(1)], of a b] guess p . note p=this
  2.2292 -      have lem2:"\<And>s1 s2 i1 i2. norm(s2 - s1) \<le> e/2 \<Longrightarrow> norm(s1 - i1) < e / 4 \<Longrightarrow> norm(s2 - i2) < e / 4 \<Longrightarrow>norm(i1 - i2) < e"
  2.2293 -      proof- case goal1 have "norm (i1 - i2) \<le> norm (i1 - s1) + norm (s1 - s2) + norm (s2 - i2)"
  2.2294 -          using norm_triangle_ineq[of "i1 - s1" "s1 - i2"]
  2.2295 -          using norm_triangle_ineq[of "s1 - s2" "s2 - i2"] by(auto simp add:group_simps)
  2.2296 -        also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:group_simps)
  2.2297 -        finally show ?case .
  2.2298 -      qed
  2.2299 -      show ?case unfolding vector_dist_norm apply(rule lem2) defer
  2.2300 -        apply(rule gm(2)[OF conjI[OF p(1)]],rule_tac[2] gn(2)[OF conjI[OF p(1)]])
  2.2301 -        using conjunctD2[OF p(2)[unfolded fine_inter]] apply- apply assumption+ apply(rule order_trans)
  2.2302 -        apply(rule rsum_diff_bound[OF p(1), where e="2 / real M"])
  2.2303 -      proof show "2 / real M * content {a..b} \<le> e / 2" unfolding divide_inverse 
  2.2304 -          using M as by(auto simp add:field_simps)
  2.2305 -        fix x assume x:"x \<in> {a..b}"
  2.2306 -        have "norm (f x - g n x) + norm (f x - g m x) \<le> inverse (real n + 1) + inverse (real m + 1)" 
  2.2307 -            using g(1)[OF x, of n] g(1)[OF x, of m] by auto
  2.2308 -        also have "\<dots> \<le> inverse (real M) + inverse (real M)" apply(rule add_mono)
  2.2309 -          apply(rule_tac[!] le_imp_inverse_le) using goal1 M by auto
  2.2310 -        also have "\<dots> = 2 / real M" unfolding real_divide_def by auto
  2.2311 -        finally show "norm (g n x - g m x) \<le> 2 / real M"
  2.2312 -          using norm_triangle_le[of "g n x - f x" "f x - g m x" "2 / real M"]
  2.2313 -          by(auto simp add:group_simps simp add:norm_minus_commute)
  2.2314 -      qed qed qed
  2.2315 -  from this[unfolded convergent_eq_cauchy[THEN sym]] guess s .. note s=this
  2.2316 -
  2.2317 -  show ?thesis unfolding integrable_on_def apply(rule_tac x=s in exI) unfolding has_integral
  2.2318 -  proof(rule,rule)  
  2.2319 -    case goal1 hence *:"e/3 > 0" by auto
  2.2320 -    from s[unfolded Lim_sequentially,rule_format,OF this] guess N1 .. note N1=this
  2.2321 -    from goal1 as have "e / 3 / content {a..b} > 0" by(auto simp add:field_simps)
  2.2322 -    from real_arch_invD[OF this] guess N2 apply-by(erule exE conjE)+ note N2=this
  2.2323 -    from i[of "N1 + N2",unfolded has_integral,rule_format,OF *] guess g' .. note g'=conjunctD2[OF this,rule_format]
  2.2324 -    have lem:"\<And>sf sg i. norm(sf - sg) \<le> e / 3 \<Longrightarrow> norm(i - s) < e / 3 \<Longrightarrow> norm(sg - i) < e / 3 \<Longrightarrow> norm(sf - s) < e"
  2.2325 -    proof- case goal1 have "norm (sf - s) \<le> norm (sf - sg) + norm (sg - i) + norm (i - s)"
  2.2326 -        using norm_triangle_ineq[of "sf - sg" "sg - s"]
  2.2327 -        using norm_triangle_ineq[of "sg -  i" " i - s"] by(auto simp add:group_simps)
  2.2328 -      also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:group_simps)
  2.2329 -      finally show ?case .
  2.2330 -    qed
  2.2331 -    show ?case apply(rule_tac x=g' in exI) apply(rule,rule g')
  2.2332 -    proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> g' fine p" note * = g'(2)[OF this]
  2.2333 -      show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - s) < e" apply-apply(rule lem[OF _ _ *])
  2.2334 -        apply(rule order_trans,rule rsum_diff_bound[OF p[THEN conjunct1]]) apply(rule,rule g,assumption)
  2.2335 -      proof- have "content {a..b} < e / 3 * (real N2)"
  2.2336 -          using N2 unfolding inverse_eq_divide using as by(auto simp add:field_simps)
  2.2337 -        hence "content {a..b} < e / 3 * (real (N1 + N2) + 1)"
  2.2338 -          apply-apply(rule less_le_trans,assumption) using `e>0` by auto 
  2.2339 -        thus "inverse (real (N1 + N2) + 1) * content {a..b} \<le> e / 3"
  2.2340 -          unfolding inverse_eq_divide by(auto simp add:field_simps)
  2.2341 -        show "norm (i (N1 + N2) - s) < e / 3" by(rule N1[rule_format,unfolded vector_dist_norm],auto)
  2.2342 -      qed qed qed qed
  2.2343 -
  2.2344 -subsection {* Negligible sets. *}
  2.2345 -
  2.2346 -definition "indicator s \<equiv> (\<lambda>x. if x \<in> s then 1 else (0::real))"
  2.2347 -
  2.2348 -lemma dest_vec1_indicator:
  2.2349 - "indicator s x = (if x \<in> s then 1 else 0)" unfolding indicator_def by auto
  2.2350 -
  2.2351 -lemma indicator_pos_le[intro]: "0 \<le> (indicator s x)" unfolding indicator_def by auto
  2.2352 -
  2.2353 -lemma indicator_le_1[intro]: "(indicator s x) \<le> 1" unfolding indicator_def by auto
  2.2354 -
  2.2355 -lemma dest_vec1_indicator_abs_le_1: "abs(indicator s x) \<le> 1"
  2.2356 -  unfolding indicator_def by auto
  2.2357 -
  2.2358 -definition "negligible (s::(real^'n) set) \<equiv> (\<forall>a b. ((indicator s) has_integral 0) {a..b})"
  2.2359 -
  2.2360 -lemma indicator_simps[simp]:"x\<in>s \<Longrightarrow> indicator s x = 1" "x\<notin>s \<Longrightarrow> indicator s x = 0"
  2.2361 -  unfolding indicator_def by auto
  2.2362 -
  2.2363 -subsection {* Negligibility of hyperplane. *}
  2.2364 -
  2.2365 -lemma vsum_nonzero_image_lemma: 
  2.2366 -  assumes "finite s" "g(a) = 0"
  2.2367 -  "\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = 0"
  2.2368 -  shows "setsum g {f x |x. x \<in> s \<and> f x \<noteq> a} = setsum (g o f) s"
  2.2369 -  unfolding setsum_iterate[OF assms(1)] apply(subst setsum_iterate) defer
  2.2370 -  apply(rule iterate_nonzero_image_lemma) apply(rule assms monoidal_monoid)+
  2.2371 -  unfolding assms using neutral_add unfolding neutral_add using assms by auto 
  2.2372 -
  2.2373 -lemma interval_doublesplit: shows "{a..b} \<inter> {x . abs(x$k - c) \<le> (e::real)} =
  2.2374 -  {(\<chi> i. if i = k then max (a$k) (c - e) else a$i) .. (\<chi> i. if i = k then min (b$k) (c + e) else b$i)}"
  2.2375 -proof- have *:"\<And>x c e::real. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
  2.2376 -  have **:"\<And>s P Q. s \<inter> {x. P x \<and> Q x} = (s \<inter> {x. Q x}) \<inter> {x. P x}" by blast
  2.2377 -  show ?thesis unfolding * ** interval_split by(rule refl) qed
  2.2378 -
  2.2379 -lemma division_doublesplit: assumes "p division_of {a..b::real^'n}" 
  2.2380 -  shows "{l \<inter> {x. abs(x$k - c) \<le> e} |l. l \<in> p \<and> l \<inter> {x. abs(x$k - c) \<le> e} \<noteq> {}} division_of ({a..b} \<inter> {x. abs(x$k - c) \<le> e})"
  2.2381 -proof- have *:"\<And>x c. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
  2.2382 -  have **:"\<And>p q p' q'. p division_of q \<Longrightarrow> p = p' \<Longrightarrow> q = q' \<Longrightarrow> p' division_of q'" by auto
  2.2383 -  note division_split(1)[OF assms, where c="c+e" and k=k,unfolded interval_split]
  2.2384 -  note division_split(2)[OF this, where c="c-e" and k=k] 
  2.2385 -  thus ?thesis apply(rule **) unfolding interval_doublesplit unfolding * unfolding interval_split interval_doublesplit
  2.2386 -    apply(rule set_ext) unfolding mem_Collect_eq apply rule apply(erule conjE exE)+ apply(rule_tac x=la in exI) defer
  2.2387 -    apply(erule conjE exE)+ apply(rule_tac x="l \<inter> {x. c + e \<ge> x $ k}" in exI) apply rule defer apply rule
  2.2388 -    apply(rule_tac x=l in exI) by blast+ qed
  2.2389 -
  2.2390 -lemma content_doublesplit: assumes "0 < e"
  2.2391 -  obtains d where "0 < d" "content({a..b} \<inter> {x. abs(x$k - c) \<le> d}) < e"
  2.2392 -proof(cases "content {a..b} = 0")
  2.2393 -  case True show ?thesis apply(rule that[of 1]) defer unfolding interval_doublesplit
  2.2394 -    apply(rule le_less_trans[OF content_subset]) defer apply(subst True)
  2.2395 -    unfolding interval_doublesplit[THEN sym] using assms by auto 
  2.2396 -next case False def d \<equiv> "e / 3 / setprod (\<lambda>i. b$i - a$i) (UNIV - {k})"
  2.2397 -  note False[unfolded content_eq_0 not_ex not_le, rule_format]
  2.2398 -  hence prod0:"0 < setprod (\<lambda>i. b$i - a$i) (UNIV - {k})" apply-apply(rule setprod_pos) by smt
  2.2399 -  hence "d > 0" unfolding d_def using assms by(auto simp add:field_simps) thus ?thesis
  2.2400 -  proof(rule that[of d]) have *:"UNIV = insert k (UNIV - {k})" by auto
  2.2401 -    have **:"{a..b} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d} \<noteq> {} \<Longrightarrow> 
  2.2402 -      (\<Prod>i\<in>UNIV - {k}. interval_upperbound ({a..b} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) $ i - interval_lowerbound ({a..b} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) $ i)
  2.2403 -      = (\<Prod>i\<in>UNIV - {k}. b$i - a$i)" apply(rule setprod_cong,rule refl)
  2.2404 -      unfolding interval_doublesplit interval_eq_empty not_ex not_less unfolding interval_bounds by auto
  2.2405 -    show "content ({a..b} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) < e" apply(cases) unfolding content_def apply(subst if_P,assumption,rule assms)
  2.2406 -      unfolding if_not_P apply(subst *) apply(subst setprod_insert) unfolding **
  2.2407 -      unfolding interval_doublesplit interval_eq_empty not_ex not_less unfolding interval_bounds unfolding Cart_lambda_beta if_P[OF refl]
  2.2408 -    proof- have "(min (b $ k) (c + d) - max (a $ k) (c - d)) \<le> 2 * d" by auto
  2.2409 -      also have "... < e / (\<Prod>i\<in>UNIV - {k}. b $ i - a $ i)" unfolding d_def using assms prod0 by(auto simp add:field_simps)
  2.2410 -      finally show "(min (b $ k) (c + d) - max (a $ k) (c - d)) * (\<Prod>i\<in>UNIV - {k}. b $ i - a $ i) < e"
  2.2411 -        unfolding pos_less_divide_eq[OF prod0] . qed auto qed qed
  2.2412 -
  2.2413 -lemma negligible_standard_hyperplane[intro]: "negligible {x. x$k = (c::real)}" 
  2.2414 -  unfolding negligible_def has_integral apply(rule,rule,rule,rule)
  2.2415 -proof- case goal1 from content_doublesplit[OF this,of a b k c] guess d . note d=this let ?i = "indicator {x. x$k = c}"
  2.2416 -  show ?case apply(rule_tac x="\<lambda>x. ball x d" in exI) apply(rule,rule gauge_ball,rule d)
  2.2417 -  proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> (\<lambda>x. ball x d) fine p"
  2.2418 -    have *:"(\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. abs(x$k - c) \<le> d}) *\<^sub>R ?i x)"
  2.2419 -      apply(rule setsum_cong2) unfolding split_paired_all real_scaleR_def mult_cancel_right split_conv
  2.2420 -      apply(cases,rule disjI1,assumption,rule disjI2)
  2.2421 -    proof- fix x l assume as:"(x,l)\<in>p" "?i x \<noteq> 0" hence xk:"x$k = c" unfolding indicator_def apply-by(rule ccontr,auto)
  2.2422 -      show "content l = content (l \<inter> {x. \<bar>x $ k - c\<bar> \<le> d})" apply(rule arg_cong[where f=content])
  2.2423 -        apply(rule set_ext,rule,rule) unfolding mem_Collect_eq
  2.2424 -      proof- fix y assume y:"y\<in>l" note p[THEN conjunct2,unfolded fine_def,rule_format,OF as(1),unfolded split_conv]
  2.2425 -        note this[unfolded subset_eq mem_ball vector_dist_norm,rule_format,OF y] note le_less_trans[OF component_le_norm[of _ k] this]
  2.2426 -        thus "\<bar>y $ k - c\<bar> \<le> d" unfolding Cart_nth.diff xk by auto
  2.2427 -      qed auto qed
  2.2428 -    note p'= tagged_division_ofD[OF p[THEN conjunct1]] and p''=division_of_tagged_division[OF p[THEN conjunct1]]
  2.2429 -    show "norm ((\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) - 0) < e" unfolding diff_0_right * unfolding real_scaleR_def real_norm_def
  2.2430 -      apply(subst abs_of_nonneg) apply(rule setsum_nonneg,rule) unfolding split_paired_all split_conv
  2.2431 -      apply(rule mult_nonneg_nonneg) apply(drule p'(4)) apply(erule exE)+ apply(rule_tac b=b in back_subst)
  2.2432 -      prefer 2 apply(subst(asm) eq_commute) apply assumption
  2.2433 -      apply(subst interval_doublesplit) apply(rule content_pos_le) apply(rule indicator_pos_le)
  2.2434 -    proof- have "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) * ?i x) \<le> (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}))"
  2.2435 -        apply(rule setsum_mono) unfolding split_paired_all split_conv 
  2.2436 -        apply(rule mult_right_le_one_le) apply(drule p'(4)) by(auto simp add:interval_doublesplit intro!:content_pos_le)
  2.2437 -      also have "... < e" apply(subst setsum_over_tagged_division_lemma[OF p[THEN conjunct1]])
  2.2438 -      proof- case goal1 have "content ({u..v} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) \<le> content {u..v}"
  2.2439 -          unfolding interval_doublesplit apply(rule content_subset) unfolding interval_doublesplit[THEN sym] by auto
  2.2440 -        thus ?case unfolding goal1 unfolding interval_doublesplit using content_pos_le by smt
  2.2441 -      next have *:"setsum content {l \<inter> {x. \<bar>x $ k - c\<bar> \<le> d} |l. l \<in> snd ` p \<and> l \<inter> {x. \<bar>x $ k - c\<bar> \<le> d} \<noteq> {}} \<ge> 0"
  2.2442 -          apply(rule setsum_nonneg,rule) unfolding mem_Collect_eq image_iff apply(erule exE bexE conjE)+ unfolding split_paired_all 
  2.2443 -        proof- fix x l a b assume as:"x = l \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}" "(a, b) \<in> p" "l = snd (a, b)"
  2.2444 -          guess u v using p'(4)[OF as(2)] apply-by(erule exE)+ note * = this
  2.2445 -          show "content x \<ge> 0" unfolding as snd_conv * interval_doublesplit by(rule content_pos_le)
  2.2446 -        qed have **:"norm (1::real) \<le> 1" by auto note division_doublesplit[OF p'',unfolded interval_doublesplit]
  2.2447 -        note dsum_bound[OF this **,unfolded interval_doublesplit[THEN sym]]
  2.2448 -        note this[unfolded real_scaleR_def real_norm_def class_semiring.semiring_rules, of k c d] note le_less_trans[OF this d(2)]
  2.2449 -        from this[unfolded abs_of_nonneg[OF *]] show "(\<Sum>ka\<in>snd ` p. content (ka \<inter> {x. \<bar>x $ k - c\<bar> \<le> d})) < e"
  2.2450 -          apply(subst vsum_nonzero_image_lemma[of "snd ` p" content "{}", unfolded o_def,THEN sym])
  2.2451 -          apply(rule finite_imageI p' content_empty)+ unfolding forall_in_division[OF p'']
  2.2452 -        proof(rule,rule,rule,rule,rule,rule,rule,erule conjE) fix m n u v
  2.2453 -          assume as:"{m..n} \<in> snd ` p" "{u..v} \<in> snd ` p" "{m..n} \<noteq> {u..v}"  "{m..n} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d} = {u..v} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}"
  2.2454 -          have "({m..n} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) \<inter> ({u..v} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) \<subseteq> {m..n} \<inter> {u..v}" by blast
  2.2455 -          note subset_interior[OF this, unfolded division_ofD(5)[OF p'' as(1-3)] interior_inter[of "{m..n}"]]
  2.2456 -          hence "interior ({m..n} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) = {}" unfolding as Int_absorb by auto
  2.2457 -          thus "content ({m..n} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) = 0" unfolding interval_doublesplit content_eq_0_interior[THEN sym] .
  2.2458 -        qed qed
  2.2459 -      finally show "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) * ?i x) < e" .
  2.2460 -    qed qed qed
  2.2461 -
  2.2462 -subsection {* A technical lemma about "refinement" of division. *}
  2.2463 -
  2.2464 -lemma tagged_division_finer: fixes p::"((real^'n) \<times> ((real^'n) set)) set"
  2.2465 -  assumes "p tagged_division_of {a..b}" "gauge d"
  2.2466 -  obtains q where "q tagged_division_of {a..b}" "d fine q" "\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q"
  2.2467 -proof-
  2.2468 -  let ?P = "\<lambda>p. p tagged_partial_division_of {a..b} \<longrightarrow> gauge d \<longrightarrow>
  2.2469 -    (\<exists>q. q tagged_division_of (\<Union>{k. \<exists>x. (x,k) \<in> p}) \<and> d fine q \<and>
  2.2470 -                   (\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q))"
  2.2471 -  { have *:"finite p" "p tagged_partial_division_of {a..b}" using assms(1) unfolding tagged_division_of_def by auto
  2.2472 -    presume "\<And>p. finite p \<Longrightarrow> ?P p" from this[rule_format,OF * assms(2)] guess q .. note q=this
  2.2473 -    thus ?thesis apply-apply(rule that[of q]) unfolding tagged_division_ofD[OF assms(1)] by auto
  2.2474 -  } fix p::"((real^'n) \<times> ((real^'n) set)) set" assume as:"finite p"
  2.2475 -  show "?P p" apply(rule,rule) using as proof(induct p) 
  2.2476 -    case empty show ?case apply(rule_tac x="{}" in exI) unfolding fine_def by auto
  2.2477 -  next case (insert xk p) guess x k using surj_pair[of xk] apply- by(erule exE)+ note xk=this
  2.2478 -    note tagged_partial_division_subset[OF insert(4) subset_insertI]
  2.2479 -    from insert(3)[OF this insert(5)] guess q1 .. note q1 = conjunctD3[OF this]
  2.2480 -    have *:"\<Union>{l. \<exists>y. (y,l) \<in> insert xk p} = k \<union> \<Union>{l. \<exists>y. (y,l) \<in> p}" unfolding xk by auto
  2.2481 -    note p = tagged_partial_division_ofD[OF insert(4)]
  2.2482 -    from p(4)[unfolded xk, OF insertI1] guess u v apply-by(erule exE)+ note uv=this
  2.2483 -
  2.2484 -    have "finite {k. \<exists>x. (x, k) \<in> p}" 
  2.2485 -      apply(rule finite_subset[of _ "snd ` p"],rule) unfolding subset_eq image_iff mem_Collect_eq
  2.2486 -      apply(erule exE,rule_tac x="(xa,x)" in bexI) using p by auto
  2.2487 -    hence int:"interior {u..v} \<inter> interior (\<Union>{k. \<exists>x. (x, k) \<in> p}) = {}"
  2.2488 -      apply(rule inter_interior_unions_intervals) apply(rule open_interior) apply(rule_tac[!] ballI)
  2.2489 -      unfolding mem_Collect_eq apply(erule_tac[!] exE) apply(drule p(4)[OF insertI2],assumption)      
  2.2490 -      apply(rule p(5))  unfolding uv xk apply(rule insertI1,rule insertI2) apply assumption
  2.2491 -      using insert(2) unfolding uv xk by auto
  2.2492 -
  2.2493 -    show ?case proof(cases "{u..v} \<subseteq> d x")
  2.2494 -      case True thus ?thesis apply(rule_tac x="{(x,{u..v})} \<union> q1" in exI) apply rule
  2.2495 -        unfolding * uv apply(rule tagged_division_union,rule tagged_division_of_self)
  2.2496 -        apply(rule p[unfolded xk uv] insertI1)+  apply(rule q1,rule int) 
  2.2497 -        apply(rule,rule fine_union,subst fine_def) defer apply(rule q1)
  2.2498 -        unfolding Ball_def split_paired_All split_conv apply(rule,rule,rule,rule)
  2.2499 -        apply(erule insertE) defer apply(rule UnI2) apply(drule q1(3)[rule_format]) unfolding xk uv by auto
  2.2500 -    next case False from fine_division_exists[OF assms(2), of u v] guess q2 . note q2=this
  2.2501 -      show ?thesis apply(rule_tac x="q2 \<union> q1" in exI)
  2.2502 -        apply rule unfolding * uv apply(rule tagged_division_union q2 q1 int fine_union)+
  2.2503 -        unfolding Ball_def split_paired_All split_conv apply rule apply(rule fine_union)
  2.2504 -        apply(rule q1 q2)+ apply(rule,rule,rule,rule) apply(erule insertE)
  2.2505 -        apply(rule UnI2) defer apply(drule q1(3)[rule_format])using False unfolding xk uv by auto
  2.2506 -    qed qed qed
  2.2507 -
  2.2508 -subsection {* Hence the main theorem about negligible sets. *}
  2.2509 -
  2.2510 -lemma finite_product_dependent: assumes "finite s" "\<And>x. x\<in>s\<Longrightarrow> finite (t x)"
  2.2511 -  shows "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" using assms
  2.2512 -proof(induct) case (insert x s) 
  2.2513 -  have *:"{(i, j) |i j. i \<in> insert x s \<and> j \<in> t i} = (\<lambda>y. (x,y)) ` (t x) \<union> {(i, j) |i j. i \<in> s \<and> j \<in> t i}" by auto
  2.2514 -  show ?case unfolding * apply(rule finite_UnI) using insert by auto qed auto
  2.2515 -
  2.2516 -lemma sum_sum_product: assumes "finite s" "\<forall>i\<in>s. finite (t i)"
  2.2517 -  shows "setsum (\<lambda>i. setsum (x i) (t i)::real) s = setsum (\<lambda>(i,j). x i j) {(i,j) | i j. i \<in> s \<and> j \<in> t i}" using assms
  2.2518 -proof(induct) case (insert a s)
  2.2519 -  have *:"{(i, j) |i j. i \<in> insert a s \<and> j \<in> t i} = (\<lambda>y. (a,y)) ` (t a) \<union> {(i, j) |i j. i \<in> s \<and> j \<in> t i}" by auto
  2.2520 -  show ?case unfolding * apply(subst setsum_Un_disjoint) unfolding setsum_insert[OF insert(1-2)]
  2.2521 -    prefer 4 apply(subst insert(3)) unfolding add_right_cancel
  2.2522 -  proof- show "setsum (x a) (t a) = (\<Sum>(xa, y)\<in>Pair a ` t a. x xa y)" apply(subst setsum_reindex) unfolding inj_on_def by auto
  2.2523 -    show "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" apply(rule finite_product_dependent) using insert by auto
  2.2524 -  qed(insert insert, auto) qed auto
  2.2525 -
  2.2526 -lemma has_integral_negligible: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
  2.2527 -  assumes "negligible s" "\<forall>x\<in>(t - s). f x = 0"
  2.2528 -  shows "(f has_integral 0) t"
  2.2529 -proof- presume P:"\<And>f::real^'n \<Rightarrow> 'a. \<And>a b. (\<forall>x. ~(x \<in> s) \<longrightarrow> f x = 0) \<Longrightarrow> (f has_integral 0) ({a..b})"
  2.2530 -  let ?f = "(\<lambda>x. if x \<in> t then f x else 0)"
  2.2531 -  show ?thesis apply(rule_tac f="?f" in has_integral_eq) apply(rule) unfolding if_P apply(rule refl)
  2.2532 -    apply(subst has_integral_alt) apply(cases,subst if_P,assumption) unfolding if_not_P
  2.2533 -  proof- assume "\<exists>a b. t = {a..b}" then guess a b apply-by(erule exE)+ note t = this
  2.2534 -    show "(?f has_integral 0) t" unfolding t apply(rule P) using assms(2) unfolding t by auto
  2.2535 -  next show "\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> t then ?f x else 0) has_integral z) {a..b} \<and> norm (z - 0) < e)"
  2.2536 -      apply(safe,rule_tac x=1 in exI,rule) apply(rule zero_less_one,safe) apply(rule_tac x=0 in exI)
  2.2537 -      apply(rule,rule P) using assms(2) by auto
  2.2538 -  qed
  2.2539 -next fix f::"real^'n \<Rightarrow> 'a" and a b::"real^'n" assume assm:"\<forall>x. x \<notin> s \<longrightarrow> f x = 0" 
  2.2540 -  show "(f has_integral 0) {a..b}" unfolding has_integral
  2.2541 -  proof(safe) case goal1
  2.2542 -    hence "\<And>n. e / 2 / ((real n+1) * (2 ^ n)) > 0" 
  2.2543 -      apply-apply(rule divide_pos_pos) defer apply(rule mult_pos_pos) by(auto simp add:field_simps)
  2.2544 -    note assms(1)[unfolded negligible_def has_integral,rule_format,OF this,of a b] note allI[OF this,of "\<lambda>x. x"] 
  2.2545 -    from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format]]
  2.2546 -    show ?case apply(rule_tac x="\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x" in exI) 
  2.2547 -    proof safe show "gauge (\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x)" using d(1) unfolding gauge_def by auto
  2.2548 -      fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x) fine p" 
  2.2549 -      let ?goal = "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e"
  2.2550 -      { presume "p\<noteq>{} \<Longrightarrow> ?goal" thus ?goal apply(cases "p={}") using goal1 by auto  }
  2.2551 -      assume as':"p \<noteq> {}" from real_arch_simple[of "Sup((\<lambda>(x,k). norm(f x)) ` p)"] guess N ..
  2.2552 -      hence N:"\<forall>x\<in>(\<lambda>(x, k). norm (f x)) ` p. x \<le> real N" apply(subst(asm) Sup_finite_le_iff) using as as' by auto
  2.2553 -      have "\<forall>i. \<exists>q. q tagged_division_of {a..b} \<and> (d i) fine q \<and> (\<forall>(x, k)\<in>p. k \<subseteq> (d i) x \<longrightarrow> (x, k) \<in> q)"
  2.2554 -        apply(rule,rule tagged_division_finer[OF as(1) d(1)]) by auto
  2.2555 -      from choice[OF this] guess q .. note q=conjunctD3[OF this[rule_format]]
  2.2556 -      have *:"\<And>i. (\<Sum>(x, k)\<in>q i. content k *\<^sub>R indicator s x) \<ge> 0" apply(rule setsum_nonneg,safe) 
  2.2557 -        unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) apply(drule tagged_division_ofD(4)[OF q(1)]) by auto
  2.2558 -      have **:"\<And>f g s t. finite s \<Longrightarrow> finite t \<Longrightarrow> (\<forall>(x,y) \<in> t. (0::real) \<le> g(x,y)) \<Longrightarrow> (\<forall>y\<in>s. \<exists>x. (x,y) \<in> t \<and> f(y) \<le> g(x,y)) \<Longrightarrow> setsum f s \<le> setsum g t"
  2.2559 -      proof- case goal1 thus ?case apply-apply(rule setsum_le_included[of s t g snd f]) prefer 4
  2.2560 -          apply safe apply(erule_tac x=x in ballE) apply(erule exE) apply(rule_tac x="(xa,x)" in bexI) by auto qed
  2.2561 -      have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) \<le> setsum (\<lambda>i. (real i + 1) *
  2.2562 -                     norm(setsum (\<lambda>(x,k). content k *\<^sub>R indicator s x) (q i))) {0..N+1}"
  2.2563 -        unfolding real_norm_def setsum_right_distrib abs_of_nonneg[OF *] diff_0_right
  2.2564 -        apply(rule order_trans,rule setsum_norm) defer apply(subst sum_sum_product) prefer 3 
  2.2565 -      proof(rule **,safe) show "finite {(i, j) |i j. i \<in> {0..N + 1} \<and> j \<in> q i}" apply(rule finite_product_dependent) using q by auto
  2.2566 -        fix i a b assume as'':"(a,b) \<in> q i" show "0 \<le> (real i + 1) * (content b *\<^sub>R indicator s a)"
  2.2567 -          unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) defer apply(rule mult_nonneg_nonneg)
  2.2568 -          using tagged_division_ofD(4)[OF q(1) as''] by auto
  2.2569 -      next fix i::nat show "finite (q i)" using q by auto
  2.2570 -      next fix x k assume xk:"(x,k) \<in> p" def n \<equiv> "nat \<lfloor>norm (f x)\<rfloor>"
  2.2571 -        have *:"norm (f x) \<in> (\<lambda>(x, k). norm (f x)) ` p" using xk by auto
  2.2572 -        have nfx:"real n \<le> norm(f x)" "norm(f x) \<le> real n + 1" unfolding n_def by auto
  2.2573 -        hence "n \<in> {0..N + 1}" using N[rule_format,OF *] by auto
  2.2574 -        moreover  note as(2)[unfolded fine_def,rule_format,OF xk,unfolded split_conv]
  2.2575 -        note q(3)[rule_format,OF xk,unfolded split_conv,rule_format,OF this] note this[unfolded n_def[symmetric]]
  2.2576 -        moreover have "norm (content k *\<^sub>R f x) \<le> (real n + 1) * (content k * indicator s x)"
  2.2577 -        proof(cases "x\<in>s") case False thus ?thesis using assm by auto
  2.2578 -        next case True have *:"content k \<ge> 0" using tagged_division_ofD(4)[OF as(1) xk] by auto
  2.2579 -          moreover have "content k * norm (f x) \<le> content k * (real n + 1)" apply(rule mult_mono) using nfx * by auto
  2.2580 -          ultimately show ?thesis unfolding abs_mult using nfx True by(auto simp add:field_simps)
  2.2581 -        qed ultimately show "\<exists>y. (y, x, k) \<in> {(i, j) |i j. i \<in> {0..N + 1} \<and> j \<in> q i} \<and> norm (content k *\<^sub>R f x) \<le> (real y + 1) * (content k *\<^sub>R indicator s x)"
  2.2582 -          apply(rule_tac x=n in exI,safe) apply(rule_tac x=n in exI,rule_tac x="(x,k)" in exI,safe) by auto
  2.2583 -      qed(insert as, auto)
  2.2584 -      also have "... \<le> setsum (\<lambda>i. e / 2 / 2 ^ i) {0..N+1}" apply(rule setsum_mono) 
  2.2585 -      proof- case goal1 thus ?case apply(subst mult_commute, subst pos_le_divide_eq[THEN sym])
  2.2586 -          using d(2)[rule_format,of "q i" i] using q[rule_format] by(auto simp add:field_simps)
  2.2587 -      qed also have "... < e * inverse 2 * 2" unfolding real_divide_def setsum_right_distrib[THEN sym]
  2.2588 -        apply(rule mult_strict_left_mono) unfolding power_inverse atLeastLessThanSuc_atLeastAtMost[THEN sym]
  2.2589 -        apply(subst sumr_geometric) using goal1 by auto
  2.2590 -      finally show "?goal" by auto qed qed qed
  2.2591 -
  2.2592 -lemma has_integral_spike: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
  2.2593 -  assumes "negligible s" "(\<forall>x\<in>(t - s). g x = f x)" "(f has_integral y) t"
  2.2594 -  shows "(g has_integral y) t"
  2.2595 -proof- { fix a b::"real^'n" and f g ::"real^'n \<Rightarrow> 'a" and y::'a
  2.2596 -    assume as:"\<forall>x \<in> {a..b} - s. g x = f x" "(f has_integral y) {a..b}"
  2.2597 -    have "((\<lambda>x. f x + (g x - f x)) has_integral (y + 0)) {a..b}" apply(rule has_integral_add[OF as(2)])
  2.2598 -      apply(rule has_integral_negligible[OF assms(1)]) using as by auto
  2.2599 -    hence "(g has_integral y) {a..b}" by auto } note * = this
  2.2600 -  show ?thesis apply(subst has_integral_alt) using assms(2-) apply-apply(rule cond_cases,safe)
  2.2601 -    apply(rule *, assumption+) apply(subst(asm) has_integral_alt) unfolding if_not_P
  2.2602 -    apply(erule_tac x=e in allE,safe,rule_tac x=B in exI,safe) apply(erule_tac x=a in allE,erule_tac x=b in allE,safe)
  2.2603 -    apply(rule_tac x=z in exI,safe) apply(rule *[where fa2="\<lambda>x. if x\<in>t then f x else 0"]) by auto qed
  2.2604 -
  2.2605 -lemma has_integral_spike_eq:
  2.2606 -  assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x"
  2.2607 -  shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
  2.2608 -  apply rule apply(rule_tac[!] has_integral_spike[OF assms(1)]) using assms(2) by auto
  2.2609 -
  2.2610 -lemma integrable_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x" "f integrable_on t"
  2.2611 -  shows "g integrable_on  t"
  2.2612 -  using assms unfolding integrable_on_def apply-apply(erule exE)
  2.2613 -  apply(rule,rule has_integral_spike) by fastsimp+
  2.2614 -
  2.2615 -lemma integral_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x"
  2.2616 -  shows "integral t f = integral t g"
  2.2617 -  unfolding integral_def using has_integral_spike_eq[OF assms] by auto
  2.2618 -
  2.2619 -subsection {* Some other trivialities about negligible sets. *}
  2.2620 -
  2.2621 -lemma negligible_subset[intro]: assumes "negligible s" "t \<subseteq> s" shows "negligible t" unfolding negligible_def 
  2.2622 -proof(safe) case goal1 show ?case using assms(1)[unfolded negligible_def,rule_format,of a b]
  2.2623 -    apply-apply(rule has_integral_spike[OF assms(1)]) defer apply assumption
  2.2624 -    using assms(2) unfolding indicator_def by auto qed
  2.2625 -
  2.2626 -lemma negligible_diff[intro?]: assumes "negligible s" shows "negligible(s - t)" using assms by auto
  2.2627 -
  2.2628 -lemma negligible_inter: assumes "negligible s \<or> negligible t" shows "negligible(s \<inter> t)" using assms by auto
  2.2629 -
  2.2630 -lemma negligible_union: assumes "negligible s" "negligible t" shows "negligible (s \<union> t)" unfolding negligible_def 
  2.2631 -proof safe case goal1 note assm = assms[unfolded negligible_def,rule_format,of a b]
  2.2632 -  thus ?case apply(subst has_integral_spike_eq[OF assms(2)])
  2.2633 -    defer apply assumption unfolding indicator_def by auto qed
  2.2634 -
  2.2635 -lemma negligible_union_eq[simp]: "negligible (s \<union> t) \<longleftrightarrow> (negligible s \<and> negligible t)"
  2.2636 -  using negligible_union by auto
  2.2637 -
  2.2638 -lemma negligible_sing[intro]: "negligible {a::real^'n}" 
  2.2639 -proof- guess x using UNIV_witness[where 'a='n] ..
  2.2640 -  show ?thesis using negligible_standard_hyperplane[of x "a$x"] by auto qed
  2.2641 -
  2.2642 -lemma negligible_insert[simp]: "negligible(insert a s) \<longleftrightarrow> negligible s"
  2.2643 -  apply(subst insert_is_Un) unfolding negligible_union_eq by auto
  2.2644 -
  2.2645 -lemma negligible_empty[intro]: "negligible {}" by auto
  2.2646 -
  2.2647 -lemma negligible_finite[intro]: assumes "finite s" shows "negligible s"
  2.2648 -  using assms apply(induct s) by auto
  2.2649 -
  2.2650 -lemma negligible_unions[intro]: assumes "finite s" "\<forall>t\<in>s. negligible t" shows "negligible(\<Union>s)"
  2.2651 -  using assms by(induct,auto) 
  2.2652 -
  2.2653 -lemma negligible:  "negligible s \<longleftrightarrow> (\<forall>t::(real^'n) set. (indicator s has_integral 0) t)"
  2.2654 -  apply safe defer apply(subst negligible_def)
  2.2655 -proof- fix t::"(real^'n) set" assume as:"negligible s"
  2.2656 -  have *:"(\<lambda>x. if x \<in> s \<inter> t then 1 else 0) = (\<lambda>x. if x\<in>t then if x\<in>s then 1 else 0 else 0)" by(rule ext,auto)  
  2.2657 -  show "(indicator s has_integral 0) t" apply(subst has_integral_alt)
  2.2658 -    apply(cases,subst if_P,assumption) unfolding if_not_P apply(safe,rule as[unfolded negligible_def,rule_format])
  2.2659 -    apply(rule_tac x=1 in exI) apply(safe,rule zero_less_one) apply(rule_tac x=0 in exI)
  2.2660 -    using negligible_subset[OF as,of "s \<inter> t"] unfolding negligible_def indicator_def unfolding * by auto qed auto
  2.2661 -
  2.2662 -subsection {* Finite case of the spike theorem is quite commonly needed. *}
  2.2663 -
  2.2664 -lemma has_integral_spike_finite: assumes "finite s" "\<forall>x\<in>t-s. g x = f x" 
  2.2665 -  "(f has_integral y) t" shows "(g has_integral y) t"
  2.2666 -  apply(rule has_integral_spike) using assms by auto
  2.2667 -
  2.2668 -lemma has_integral_spike_finite_eq: assumes "finite s" "\<forall>x\<in>t-s. g x = f x"
  2.2669 -  shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
  2.2670 -  apply rule apply(rule_tac[!] has_integral_spike_finite) using assms by auto
  2.2671 -
  2.2672 -lemma integrable_spike_finite:
  2.2673 -  assumes "finite s" "\<forall>x\<in>t-s. g x = f x" "f integrable_on t" shows "g integrable_on  t"
  2.2674 -  using assms unfolding integrable_on_def apply safe apply(rule_tac x=y in exI)
  2.2675 -  apply(rule has_integral_spike_finite) by auto
  2.2676 -
  2.2677 -subsection {* In particular, the boundary of an interval is negligible. *}
  2.2678 -
  2.2679 -lemma negligible_frontier_interval: "negligible({a..b} - {a<..<b})"
  2.2680 -proof- let ?A = "\<Union>((\<lambda>k. {x. x$k = a$k} \<union> {x. x$k = b$k}) ` UNIV)"
  2.2681 -  have "{a..b} - {a<..<b} \<subseteq> ?A" apply rule unfolding Diff_iff mem_interval not_all
  2.2682 -    apply(erule conjE exE)+ apply(rule_tac X="{x. x $ xa = a $ xa} \<union> {x. x $ xa = b $ xa}" in UnionI)
  2.2683 -    apply(erule_tac[!] x=xa in allE) by auto
  2.2684 -  thus ?thesis apply-apply(rule negligible_subset[of ?A]) apply(rule negligible_unions[OF finite_imageI]) by auto qed
  2.2685 -
  2.2686 -lemma has_integral_spike_interior:
  2.2687 -  assumes "\<forall>x\<in>{a<..<b}. g x = f x" "(f has_integral y) ({a..b})" shows "(g has_integral y) ({a..b})"
  2.2688 -  apply(rule has_integral_spike[OF negligible_frontier_interval _ assms(2)]) using assms(1) by auto
  2.2689 -
  2.2690 -lemma has_integral_spike_interior_eq:
  2.2691 -  assumes "\<forall>x\<in>{a<..<b}. g x = f x" shows "((f has_integral y) ({a..b}) \<longleftrightarrow> (g has_integral y) ({a..b}))"
  2.2692 -  apply rule apply(rule_tac[!] has_integral_spike_interior) using assms by auto
  2.2693 -
  2.2694 -lemma integrable_spike_interior: assumes "\<forall>x\<in>{a<..<b}. g x = f x" "f integrable_on {a..b}" shows "g integrable_on {a..b}"
  2.2695 -  using  assms unfolding integrable_on_def using has_integral_spike_interior[OF assms(1)] by auto
  2.2696 -
  2.2697 -subsection {* Integrability of continuous functions. *}
  2.2698 -
  2.2699 -lemma neutral_and[simp]: "neutral op \<and> = True"
  2.2700 -  unfolding neutral_def apply(rule some_equality) by auto
  2.2701 -
  2.2702 -lemma monoidal_and[intro]: "monoidal op \<and>" unfolding monoidal_def by auto
  2.2703 -
  2.2704 -lemma iterate_and[simp]: assumes "finite s" shows "(iterate op \<and>) s p \<longleftrightarrow> (\<forall>x\<in>s. p x)" using assms
  2.2705 -apply induct unfolding iterate_insert[OF monoidal_and] by auto
  2.2706 -
  2.2707 -lemma operative_division_and: assumes "operative op \<and> P" "d division_of {a..b}"
  2.2708 -  shows "(\<forall>i\<in>d. P i) \<longleftrightarrow> P {a..b}"
  2.2709 -  using operative_division[OF monoidal_and assms] division_of_finite[OF assms(2)] by auto
  2.2710 -
  2.2711 -lemma operative_approximable: assumes "0 \<le> e" fixes f::"real^'n \<Rightarrow> 'a::banach"
  2.2712 -  shows "operative op \<and> (\<lambda>i. \<exists>g. (\<forall>x\<in>i. norm (f x - g (x::real^'n)) \<le> e) \<and> g integrable_on i)" unfolding operative_def neutral_and
  2.2713 -proof safe fix a b::"real^'n" { assume "content {a..b} = 0"
  2.2714 -    thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" 
  2.2715 -      apply(rule_tac x=f in exI) using assms by(auto intro!:integrable_on_null) }
  2.2716 -  { fix c k g assume as:"\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}"
  2.2717 -    show "\<exists>g. (\<forall>x\<in>{a..b} \<inter> {x. x $ k \<le> c}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b} \<inter> {x. x $ k \<le> c}"
  2.2718 -      "\<exists>g. (\<forall>x\<in>{a..b} \<inter> {x. c \<le> x $ k}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b} \<inter> {x. c \<le> x $ k}"
  2.2719 -      apply(rule_tac[!] x=g in exI) using as(1) integrable_split[OF as(2)] by auto }
  2.2720 -  fix c k g1 g2 assume as:"\<forall>x\<in>{a..b} \<inter> {x. x $ k \<le> c}. norm (f x - g1 x) \<le> e" "g1 integrable_on {a..b} \<inter> {x. x $ k \<le> c}"
  2.2721 -                          "\<forall>x\<in>{a..b} \<inter> {x. c \<le> x $ k}. norm (f x - g2 x) \<le> e" "g2 integrable_on {a..b} \<inter> {x. c \<le> x $ k}"
  2.2722 -  let ?g = "\<lambda>x. if x$k = c then f x else if x$k \<le> c then g1 x else g2 x"
  2.2723 -  show "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" apply(rule_tac x="?g" in exI)
  2.2724 -  proof safe case goal1 thus ?case apply- apply(cases "x$k=c", case_tac "x$k < c") using as assms by auto
  2.2725 -  next case goal2 presume "?g integrable_on {a..b} \<inter> {x. x $ k \<le> c}" "?g integrable_on {a..b} \<inter> {x. x $ k \<ge> c}"
  2.2726 -    then guess h1 h2 unfolding integrable_on_def by auto from has_integral_split[OF this]
  2.2727 -    show ?case unfolding integrable_on_def by auto
  2.2728 -  next show "?g integrable_on {a..b} \<inter> {x. x $ k \<le> c}" "?g integrable_on {a..b} \<inter> {x. x $ k \<ge> c}"
  2.2729 -      apply(rule_tac[!] integrable_spike[OF negligible_standard_hyperplane[of k c]]) using as(2,4) by auto qed qed
  2.2730 -
  2.2731 -lemma approximable_on_division: fixes f::"real^'n \<Rightarrow> 'a::banach"
  2.2732 -  assumes "0 \<le> e" "d division_of {a..b}" "\<forall>i\<in>d. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
  2.2733 -  obtains g where "\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}"
  2.2734 -proof- note * = operative_division[OF monoidal_and operative_approximable[OF assms(1)] assms(2)]
  2.2735 -  note this[unfolded iterate_and[OF division_of_finite[OF assms(2)]]] from assms(3)[unfolded this[of f]]
  2.2736 -  guess g .. thus thesis apply-apply(rule that[of g]) by auto qed
  2.2737 -
  2.2738 -lemma integrable_continuous: fixes f::"real^'n \<Rightarrow> 'a::banach"
  2.2739 -  assumes "continuous_on {a..b} f" shows "f integrable_on {a..b}"
  2.2740 -proof(rule integrable_uniform_limit,safe) fix e::real assume e:"0 < e"
  2.2741 -  from compact_uniformly_continuous[OF assms compact_interval,unfolded uniformly_continuous_on_def,rule_format,OF e] guess d ..
  2.2742 -  note d=conjunctD2[OF this,rule_format]
  2.2743 -  from fine_division_exists[OF gauge_ball[OF d(1)], of a b] guess p . note p=this
  2.2744 -  note p' = tagged_division_ofD[OF p(1)]
  2.2745 -  have *:"\<forall>i\<in>snd ` p. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
  2.2746 -  proof(safe,unfold snd_conv) fix x l assume as:"(x,l) \<in> p" 
  2.2747 -    from p'(4)[OF this] guess a b apply-by(erule exE)+ note l=this
  2.2748 -    show "\<exists>g. (\<forall>x\<in>l. norm (f x - g x) \<le> e) \<and> g integrable_on l" apply(rule_tac x="\<lambda>y. f x" in exI)
  2.2749 -    proof safe show "(\<lambda>y. f x) integrable_on l" unfolding integrable_on_def l by(rule,rule has_integral_const)
  2.2750 -      fix y assume y:"y\<in>l" note fineD[OF p(2) as,unfolded subset_eq,rule_format,OF this]
  2.2751 -      note d(2)[OF _ _ this[unfolded mem_ball]]
  2.2752 -      thus "norm (f y - f x) \<le> e" using y p'(2-3)[OF as] unfolding vector_dist_norm l norm_minus_commute by fastsimp qed qed
  2.2753 -  from e have "0 \<le> e" by auto from approximable_on_division[OF this division_of_tagged_division[OF p(1)] *] guess g .
  2.2754 -  thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" by auto qed 
  2.2755 -
  2.2756 -subsection {* Specialization of additivity to one dimension. *}
  2.2757 -
  2.2758 -lemma operative_1_lt: assumes "monoidal opp"
  2.2759 -  shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real^1} = neutral opp) \<and>
  2.2760 -                (\<forall>a b c. a < c \<and> c < b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))"
  2.2761 -  unfolding operative_def content_eq_0_1 forall_1 vector_le_def vector_less_def
  2.2762 -proof safe fix a b c::"real^1" assume as:"\<forall>a b c. f {a..b} = opp (f ({a..b} \<inter> {x. x $ 1 \<le> c})) (f ({a..b} \<inter> {x. c \<le> x $ 1}))" "a $ 1 < c $ 1" "c $ 1 < b $ 1"
  2.2763 -    from this(2-) have "{a..b} \<inter> {x. x $ 1 \<le> c $ 1} = {a..c}" "{a..b} \<inter> {x. x $ 1 \<ge> c $ 1} = {c..b}" by auto
  2.2764 -    thus "opp (f {a..c}) (f {c..b}) = f {a..b}" unfolding as(1)[rule_format,of a b "c$1"] by auto
  2.2765 -next fix a b::"real^1" and c::real
  2.2766 -  assume as:"\<forall>a b. b $ 1 \<le> a $ 1 \<longrightarrow> f {a..b} = neutral opp" "\<forall>a b c. a $ 1 < c $ 1 \<and> c $ 1 < b $ 1 \<longrightarrow> opp (f {a..c}) (f {c..b}) = f {a..b}"
  2.2767 -  show "f {a..b} = opp (f ({a..b} \<inter> {x. x $ 1 \<le> c})) (f ({a..b} \<inter> {x. c \<le> x $ 1}))"
  2.2768 -  proof(cases "c \<in> {a$1 .. b$1}")
  2.2769 -    case False hence "c<a$1 \<or> c>b$1" by auto
  2.2770 -    thus ?thesis apply-apply(erule disjE)
  2.2771 -    proof- assume "c<a$1" hence *:"{a..b} \<inter> {x. x $ 1 \<le> c} = {1..0}"  "{a..b} \<inter> {x. c \<le> x $ 1} = {a..b}" by auto
  2.2772 -      show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto
  2.2773 -    next   assume "b$1<c" hence *:"{a..b} \<inter> {x. x $ 1 \<le> c} = {a..b}"  "{a..b} \<inter> {x. c \<le> x $ 1} = {1..0}" by auto
  2.2774 -      show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto
  2.2775 -    qed
  2.2776 -  next case True hence *:"min (b $ 1) c = c" "max (a $ 1) c = c" by auto
  2.2777 -    show ?thesis unfolding interval_split num1_eq_iff if_True * vec_def[THEN sym]
  2.2778 -    proof(cases "c = a$1 \<or> c = b$1")
  2.2779 -      case False thus "f {a..b} = opp (f {a..vec1 c}) (f {vec1 c..b})"
  2.2780 -        apply-apply(subst as(2)[rule_format]) using True by auto
  2.2781 -    next case True thus "f {a..b} = opp (f {a..vec1 c}) (f {vec1 c..b})" apply-
  2.2782 -      proof(erule disjE) assume "c=a$1" hence *:"a = vec1 c" unfolding Cart_eq by auto 
  2.2783 -        hence "f {a..vec1 c} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
  2.2784 -        thus ?thesis using assms unfolding * by auto
  2.2785 -      next assume "c=b$1" hence *:"b = vec1 c" unfolding Cart_eq by auto 
  2.2786 -        hence "f {vec1 c..b} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
  2.2787 -        thus ?thesis using assms unfolding * by auto qed qed qed qed
  2.2788 -
  2.2789 -lemma operative_1_le: assumes "monoidal opp"
  2.2790 -  shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real^1} = neutral opp) \<and>
  2.2791 -                (\<forall>a b c. a \<le> c \<and> c \<le> b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))"
  2.2792 -unfolding operative_1_lt[OF assms]
  2.2793 -proof safe fix a b c::"real^1" assume as:"\<forall>a b c. a \<le> c \<and> c \<le> b \<longrightarrow> opp (f {a..c}) (f {c..b}) = f {a..b}" "a < c" "c < b"
  2.2794 -  show "opp (f {a..c}) (f {c..b}) = f {a..b}" apply(rule as(1)[rule_format]) using as(2-) unfolding vector_le_def vector_less_def by auto
  2.2795 -next fix a b c ::"real^1"
  2.2796 -  assume "\<forall>a b. b \<le> a \<longrightarrow> f {a..b} = neutral opp" "\<forall>a b c. a < c \<and> c < b \<longrightarrow> opp (f {a..c}) (f {c..b}) = f {a..b}" "a \<le> c" "c \<le> b"
  2.2797 -  note as = this[rule_format]
  2.2798 -  show "opp (f {a..c}) (f {c..b}) = f {a..b}"
  2.2799 -  proof(cases "c = a \<or> c = b")
  2.2800 -    case False thus ?thesis apply-apply(subst as(2)) using as(3-) unfolding vector_le_def vector_less_def Cart_eq by(auto simp del:dest_vec1_eq)
  2.2801 -    next case True thus ?thesis apply-
  2.2802 -      proof(erule disjE) assume *:"c=a" hence "f {a..c} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
  2.2803 -        thus ?thesis using assms unfolding * by auto
  2.2804 -      next               assume *:"c=b" hence "f {c..b} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
  2.2805 -        thus ?thesis using assms unfolding * by auto qed qed qed 
  2.2806 -
  2.2807 -subsection {* Special case of additivity we need for the FCT. *}
  2.2808 -
  2.2809 -lemma additive_tagged_division_1: fixes f::"real^1 \<Rightarrow> 'a::real_normed_vector"
  2.2810 -  assumes "dest_vec1 a \<le> dest_vec1 b" "p tagged_division_of {a..b}"
  2.2811 -  shows "setsum (\<lambda>(x,k). f(interval_upperbound k) - f(interval_lowerbound k)) p = f b - f a"
  2.2812 -proof- let ?f = "(\<lambda>k::(real^1) set. if k = {} then 0 else f(interval_upperbound k) - f(interval_lowerbound k))"
  2.2813 -  have *:"operative op + ?f" unfolding operative_1_lt[OF monoidal_monoid] interval_eq_empty_1
  2.2814 -    by(auto simp add:not_less interval_bound_1 vector_less_def)
  2.2815 -  have **:"{a..b} \<noteq> {}" using assms(1) by auto note operative_tagged_division[OF monoidal_monoid * assms(2)]
  2.2816 -  note * = this[unfolded if_not_P[OF **] interval_bound_1[OF assms(1)],THEN sym ]
  2.2817 -  show ?thesis unfolding * apply(subst setsum_iterate[THEN sym]) defer
  2.2818 -    apply(rule setsum_cong2) unfolding split_paired_all split_conv using assms(2) by auto qed
  2.2819 -
  2.2820 -subsection {* A useful lemma allowing us to factor out the content size. *}
  2.2821 -
  2.2822 -lemma has_integral_factor_content:
  2.2823 -  "(f has_integral i) {a..b} \<longleftrightarrow> (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p
  2.2824 -    \<longrightarrow> norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content {a..b}))"
  2.2825 -proof(cases "content {a..b} = 0")
  2.2826 -  case True show ?thesis unfolding has_integral_null_eq[OF True] apply safe
  2.2827 -    apply(rule,rule,rule gauge_trivial,safe) unfolding setsum_content_null[OF True] True defer 
  2.2828 -    apply(erule_tac x=1 in allE,safe) defer apply(rule fine_division_exists[of _ a b],assumption)
  2.2829 -    apply(erule_tac x=p in allE) unfolding setsum_content_null[OF True] by auto
  2.2830 -next case False note F = this[unfolded content_lt_nz[THEN sym]]
  2.2831 -  let ?P = "\<lambda>e opp. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> opp (norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i)) e)"
  2.2832 -  show ?thesis apply(subst has_integral)
  2.2833 -  proof safe fix e::real assume e:"e>0"
  2.2834 -    { assume "\<forall>e>0. ?P e op <" thus "?P (e * content {a..b}) op \<le>" apply(erule_tac x="e * content {a..b}" in allE)
  2.2835 -        apply(erule impE) defer apply(erule exE,rule_tac x=d in exI)
  2.2836 -        using F e by(auto simp add:field_simps intro:mult_pos_pos) }
  2.2837 -    {  assume "\<forall>e>0. ?P (e * content {a..b}) op \<le>" thus "?P e op <" apply(erule_tac x="e / 2 / content {a..b}" in allE)
  2.2838 -        apply(erule impE) defer apply(erule exE,rule_tac x=d in exI)
  2.2839 -        using F e by(auto simp add:field_simps intro:mult_pos_pos) } qed qed
  2.2840 -
  2.2841 -subsection {* Fundamental theorem of calculus. *}
  2.2842 -
  2.2843 -lemma fundamental_theorem_of_calculus: fixes f::"real^1 \<Rightarrow> 'a::banach"
  2.2844 -  assumes "a \<le> b"  "\<forall>x\<in>{a..b}. ((f o vec1) has_vector_derivative f'(vec1 x)) (at x within {a..b})"
  2.2845 -  shows "(f' has_integral (f(vec1 b) - f(vec1 a))) ({vec1 a..vec1 b})"
  2.2846 -unfolding has_integral_factor_content
  2.2847 -proof safe fix e::real assume e:"e>0" have ab:"dest_vec1 (vec1 a) \<le> dest_vec1 (vec1 b)" using assms(1) by auto
  2.2848 -  note assm = assms(2)[unfolded has_vector_derivative_def has_derivative_within_alt]
  2.2849 -  have *:"\<And>P Q. \<forall>x\<in>{a..b}. P x \<and> (\<forall>e>0. \<exists>d>0. Q x e d) \<Longrightarrow> \<forall>x. \<exists>(d::real)>0. x\<in>{a..b} \<longrightarrow> Q x e d" using e by blast
  2.2850 -  note this[OF assm,unfolded gauge_existence_lemma] from choice[OF this,unfolded Ball_def[symmetric]]
  2.2851 -  guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
  2.2852 -  show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {vec1 a..vec1 b} \<and> d fine p \<longrightarrow>
  2.2853 -                 norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f (vec1 b) - f (vec1 a))) \<le> e * content {vec1 a..vec1 b})"
  2.2854 -    apply(rule_tac x="\<lambda>x. ball x (d (dest_vec1 x))" in exI,safe)
  2.2855 -    apply(rule gauge_ball_dependent,rule,rule d(1))
  2.2856 -  proof- fix p assume as:"p tagged_division_of {vec1 a..vec1 b}" "(\<lambda>x. ball x (d (dest_vec1 x))) fine p"
  2.2857 -    show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f (vec1 b) - f (vec1 a))) \<le> e * content {vec1 a..vec1 b}" 
  2.2858 -      unfolding content_1[OF ab] additive_tagged_division_1[OF ab as(1),of f,THEN sym]
  2.2859 -      unfolding vector_minus_component[THEN sym] additive_tagged_division_1[OF ab as(1),of "\<lambda>x. x",THEN sym]
  2.2860 -      apply(subst dest_vec1_setsum) unfolding setsum_right_distrib defer unfolding setsum_subtractf[THEN sym] 
  2.2861 -    proof(rule setsum_norm_le,safe) fix x k assume "(x,k)\<in>p"
  2.2862 -      note xk = tagged_division_ofD(2-4)[OF as(1) this] from this(3) guess u v apply-by(erule exE)+ note k=this
  2.2863 -      have *:"dest_vec1 u \<le> dest_vec1 v" using xk unfolding k by auto
  2.2864 -      have ball:"\<forall>xa\<in>k. xa \<in> ball x (d (dest_vec1 x))" using as(2)[unfolded fine_def,rule_format,OF `(x,k)\<in>p`,unfolded split_conv subset_eq] .
  2.2865 -      have "norm ((v$1 - u$1) *\<^sub>R f' x - (f v - f u)) \<le> norm (f u - f x - (u$1 - x$1) *\<^sub>R f' x) + norm (f v - f x - (v$1 - x$1) *\<^sub>R f' x)"
  2.2866 -        apply(rule order_trans[OF _ norm_triangle_ineq4]) apply(rule eq_refl) apply(rule arg_cong[where f=norm])
  2.2867 -        unfolding scaleR.diff_left by(auto simp add:group_simps)
  2.2868 -      also have "... \<le> e * norm (dest_vec1 u - dest_vec1 x) + e * norm (dest_vec1 v - dest_vec1 x)"
  2.2869 -        apply(rule add_mono) apply(rule d(2)[of "x$1" "u$1",unfolded o_def vec1_dest_vec1]) prefer 4
  2.2870 -        apply(rule d(2)[of "x$1" "v$1",unfolded o_def vec1_dest_vec1])
  2.2871 -        using ball[rule_format,of u] ball[rule_format,of v] 
  2.2872 -        using xk(1-2) unfolding k subset_eq by(auto simp add:vector_dist_norm norm_real)
  2.2873 -      also have "... \<le> e * dest_vec1 (interval_upperbound k - interval_lowerbound k)"
  2.2874 -        unfolding k interval_bound_1[OF *] using xk(1) unfolding k by(auto simp add:vector_dist_norm norm_real field_simps)
  2.2875 -      finally show "norm (content k *\<^sub>R f' x - (f (interval_upperbound k) - f (interval_lowerbound k))) \<le>
  2.2876 -        e * dest_vec1 (interval_upperbound k - interval_lowerbound k)" unfolding k interval_bound_1[OF *] content_1[OF *] .
  2.2877 -    qed(insert as, auto) qed qed
  2.2878 -
  2.2879 -subsection {* Attempt a systematic general set of "offset" results for components. *}
  2.2880 -
  2.2881 -lemma gauge_modify:
  2.2882 -  assumes "(\<forall>s. open s \<longrightarrow> open {x. f(x) \<in> s})" "gauge d"
  2.2883 -  shows "gauge (\<lambda>x y. d (f x) (f y))"
  2.2884 -  using assms unfolding gauge_def apply safe defer apply(erule_tac x="f x" in allE)
  2.2885 -  apply(erule_tac x="d (f x)" in allE) unfolding mem_def Collect_def by auto
  2.2886 -
  2.2887 -subsection {* Only need trivial subintervals if the interval itself is trivial. *}
  2.2888 -
  2.2889 -lemma division_of_nontrivial: fixes s::"(real^'n) set set"
  2.2890 -  assumes "s division_of {a..b}" "content({a..b}) \<noteq> 0"
  2.2891 -  shows "{k. k \<in> s \<and> content k \<noteq> 0} division_of {a..b}" using assms(1) apply-
  2.2892 -proof(induct "card s" arbitrary:s rule:nat_less_induct)
  2.2893 -  fix s::"(real^'n) set set" assume assm:"s division_of {a..b}"
  2.2894 -    "\<forall>m<card s. \<forall>x. m = card x \<longrightarrow> x division_of {a..b} \<longrightarrow> {k \<in> x. content k \<noteq> 0} division_of {a..b}" 
  2.2895 -  note s = division_ofD[OF assm(1)] let ?thesis = "{k \<in> s. content k \<noteq> 0} division_of {a..b}"
  2.2896 -  { presume *:"{k \<in> s. content k \<noteq> 0} \<noteq> s \<Longrightarrow> ?thesis"
  2.2897 -    show ?thesis apply cases defer apply(rule *,assumption) using assm(1) by auto }
  2.2898 -  assume noteq:"{k \<in> s. content k \<noteq> 0} \<noteq> s"
  2.2899 -  then obtain k where k:"k\<in>s" "content k = 0" by auto
  2.2900 -  from s(4)[OF k(1)] guess c d apply-by(erule exE)+ note k=k this
  2.2901 -  from k have "card s > 0" unfolding card_gt_0_iff using assm(1) by auto
  2.2902 -  hence card:"card (s - {k}) < card s" using assm(1) k(1) apply(subst card_Diff_singleton_if) by auto
  2.2903 -  have *:"closed (\<Union>(s - {k}))" apply(rule closed_Union) defer apply rule apply(drule DiffD1,drule s(4))
  2.2904 -    apply safe apply(rule closed_interval) using assm(1) by auto
  2.2905 -  have "k \<subseteq> \<Union>(s - {k})" apply safe apply(rule *[unfolded closed_limpt,rule_format]) unfolding islimpt_approachable
  2.2906 -  proof safe fix x and e::real assume as:"x\<in>k" "e>0"
  2.2907 -    from k(2)[unfolded k content_eq_0] guess i .. 
  2.2908 -    hence i:"c$i = d$i" using s(3)[OF k(1),unfolded k] unfolding interval_ne_empty by smt
  2.2909 -    hence xi:"x$i = d$i" using as unfolding k mem_interval by smt
  2.2910 -    def y \<equiv> "(\<chi> j. if j = i then if c$i \<le> (a$i + b$i) / 2 then c$i + min e (b$i - c$i) / 2 else c$i - min e (c$i - a$i) / 2 else x$j)"
  2.2911 -    show "\<exists>x'\<in>\<Union>(s - {k}). x' \<noteq> x \<and> dist x' x < e" apply(rule_tac x=y in bexI) 
  2.2912 -    proof have "d \<in> {c..d}" using s(3)[OF k(1)] unfolding k interval_eq_empty mem_interval by(fastsimp simp add: not_less)
  2.2913 -      hence "d \<in> {a..b}" using s(2)[OF k(1)] unfolding k by auto note di = this[unfolded mem_interval,THEN spec[where x=i]]
  2.2914 -      hence xyi:"y$i \<noteq> x$i" unfolding y_def unfolding i xi Cart_lambda_beta if_P[OF refl]
  2.2915 -        apply(cases) apply(subst if_P,assumption) unfolding if_not_P not_le using as(2) using assms(2)[unfolded content_eq_0] by smt+ 
  2.2916 -      thus "y \<noteq> x" unfolding Cart_eq by auto
  2.2917 -      have *:"UNIV = insert i (UNIV - {i})" by auto
  2.2918 -      have "norm (y - x) < e + setsum (\<lambda>i. 0) (UNIV::'n set)" apply(rule le_less_trans[OF norm_le_l1])
  2.2919 -        apply(subst *,subst setsum_insert) prefer 3 apply(rule add_less_le_mono)
  2.2920 -      proof- show "\<bar>(y - x) $ i\<bar> < e" unfolding y_def Cart_lambda_beta vector_minus_component if_P[OF refl]
  2.2921 -          apply(cases) apply(subst if_P,assumption) unfolding if_not_P unfolding i xi using di as(2) by auto
  2.2922 -        show "(\<Sum>i\<in>UNIV - {i}. \<bar>(y - x) $ i\<bar>) \<le> (\<Sum>i\<in>UNIV. 0)" unfolding y_def by auto 
  2.2923 -      qed auto thus "dist y x < e" unfolding vector_dist_norm by auto
  2.2924 -      have "y\<notin>k" unfolding k mem_interval apply rule apply(erule_tac x=i in allE) using xyi unfolding k i xi by auto
  2.2925 -      moreover have "y \<in> \<Union>s" unfolding s mem_interval
  2.2926 -      proof note simps = y_def Cart_lambda_beta if_not_P
  2.2927 -        fix j::'n show "a $ j \<le> y $ j \<and> y $ j \<le> b $ j" 
  2.2928 -        proof(cases "j = i") case False have "x \<in> {a..b}" using s(2)[OF k(1)] as(1) by auto
  2.2929 -          thus ?thesis unfolding simps if_not_P[OF False] unfolding mem_interval by auto
  2.2930 -        next case True note T = this show ?thesis
  2.2931 -          proof(cases "c $ i \<le> (a $ i + b $ i) / 2")
  2.2932 -            case True show ?thesis unfolding simps if_P[OF T] if_P[OF True] unfolding i
  2.2933 -              using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps) 
  2.2934 -          next case False thus ?thesis unfolding simps if_P[OF T] if_not_P[OF False] unfolding i
  2.2935 -              using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps)
  2.2936 -          qed qed qed
  2.2937 -      ultimately show "y \<in> \<Union>(s - {k})" by auto
  2.2938 -    qed qed hence "\<Union>(s - {k}) = {a..b}" unfolding s(6)[THEN sym] by auto
  2.2939 -  hence  "{ka \<in> s - {k}. content ka \<noteq> 0} division_of {a..b}" apply-apply(rule assm(2)[rule_format,OF card refl])
  2.2940 -    apply(rule division_ofI) defer apply(rule_tac[1-4] s) using assm(1) by auto
  2.2941 -  moreover have "{ka \<in> s - {k}. content ka \<noteq> 0} = {k \<in> s. content k \<noteq> 0}" using k by auto ultimately show ?thesis by auto qed
  2.2942 -
  2.2943 -subsection {* Integrabibility on subintervals. *}
  2.2944 -
  2.2945 -lemma operative_integrable: fixes f::"real^'n \<Rightarrow> 'a::banach" shows
  2.2946 -  "operative op \<and> (\<lambda>i. f integrable_on i)"
  2.2947 -  unfolding operative_def neutral_and apply safe apply(subst integrable_on_def)
  2.2948 -  unfolding has_integral_null_eq apply(rule,rule refl) apply(rule,assumption)+
  2.2949 -  unfolding integrable_on_def by(auto intro: has_integral_split)
  2.2950 -
  2.2951 -lemma integrable_subinterval: fixes f::"real^'n \<Rightarrow> 'a::banach" 
  2.2952 -  assumes "f integrable_on {a..b}" "{c..d} \<subseteq> {a..b}" shows "f integrable_on {c..d}" 
  2.2953 -  apply(cases "{c..d} = {}") defer apply(rule partial_division_extend_1[OF assms(2)],assumption)
  2.2954 -  using operative_division_and[OF operative_integrable,THEN sym,of _ _ _ f] assms(1) by auto
  2.2955 -
  2.2956 -subsection {* Combining adjacent intervals in 1 dimension. *}
  2.2957 -
  2.2958 -lemma has_integral_combine: assumes "(a::real^1) \<le> c" "c \<le> b"
  2.2959 -  "(f has_integral i) {a..c}" "(f has_integral (j::'a::banach)) {c..b}"
  2.2960 -  shows "(f has_integral (i + j)) {a..b}"
  2.2961 -proof- note operative_integral[of f, unfolded operative_1_le[OF monoidal_lifted[OF monoidal_monoid]]]
  2.2962 -  note conjunctD2[OF this,rule_format] note * = this(2)[OF conjI[OF assms(1-2)],unfolded if_P[OF assms(3)]]
  2.2963 -  hence "f integrable_on {a..b}" apply- apply(rule ccontr) apply(subst(asm) if_P) defer
  2.2964 -    apply(subst(asm) if_P) using assms(3-) by auto
  2.2965 -  with * show ?thesis apply-apply(subst(asm) if_P) defer apply(subst(asm) if_P) defer apply(subst(asm) if_P)
  2.2966 -    unfolding lifted.simps using assms(3-) by(auto simp add: integrable_on_def integral_unique) qed
  2.2967 -
  2.2968 -lemma integral_combine: fixes f::"real^1 \<Rightarrow> 'a::banach"
  2.2969 -  assumes "a \<le> c" "c \<le> b" "f integrable_on ({a..b})"
  2.2970 -  shows "integral {a..c} f + integral {c..b} f = integral({a..b}) f"
  2.2971 -  apply(rule integral_unique[THEN sym]) apply(rule has_integral_combine[OF assms(1-2)])
  2.2972 -  apply(rule_tac[!] integrable_integral integrable_subinterval[OF assms(3)])+ using assms(1-2) by auto
  2.2973 -
  2.2974 -lemma integrable_combine: fixes f::"real^1 \<Rightarrow> 'a::banach"
  2.2975 -  assumes "a \<le> c" "c \<le> b" "f integrable_on {a..c}" "f integrable_on {c..b}"
  2.2976 -  shows "f integrable_on {a..b}" using assms unfolding integrable_on_def by(fastsimp intro!:has_integral_combine)
  2.2977 -
  2.2978 -subsection {* Reduce integrability to "local" integrability. *}
  2.2979 -
  2.2980 -lemma integrable_on_little_subintervals: fixes f::"real^'n \<Rightarrow> 'a::banach"
  2.2981 -  assumes "\<forall>x\<in>{a..b}. \<exists>d>0. \<forall>u v. x \<in> {u..v} \<and> {u..v} \<subseteq> ball x d \<and> {u..v} \<subseteq> {a..b} \<longrightarrow> f integrable_on {u..v}"
  2.2982 -  shows "f integrable_on {a..b}"
  2.2983 -proof- have "\<forall>x. \<exists>d. x\<in>{a..b} \<longrightarrow> d>0 \<and> (\<forall>u v. x \<in> {u..v} \<and> {u..v} \<subseteq> ball x d \<and> {u..v} \<subseteq> {a..b} \<longrightarrow> f integrable_on {u..v})"
  2.2984 -    using assms by auto note this[unfolded gauge_existence_lemma] from choice[OF this] guess d .. note d=this[rule_format]
  2.2985 -  guess p apply(rule fine_division_exists[OF gauge_ball_dependent,of d a b]) using d by auto note p=this(1-2)
  2.2986 -  note division_of_tagged_division[OF this(1)] note * = operative_division_and[OF operative_integrable,OF this,THEN sym,of f]
  2.2987 -  show ?thesis unfolding * apply safe unfolding snd_conv
  2.2988 -  proof- fix x k assume "(x,k) \<in> p" note tagged_division_ofD(2-4)[OF p(1) this] fineD[OF p(2) this]
  2.2989 -    thus "f integrable_on k" apply safe apply(rule d[THEN conjunct2,rule_format,of x]) by auto qed qed
  2.2990 -
  2.2991 -subsection {* Second FCT or existence of antiderivative. *}
  2.2992 -
  2.2993 -lemma integrable_const[intro]:"(\<lambda>x. c) integrable_on {a..b}"
  2.2994 -  unfolding integrable_on_def by(rule,rule has_integral_const)
  2.2995 -
  2.2996 -lemma integral_has_vector_derivative: fixes f::"real \<Rightarrow> 'a::banach"
  2.2997 -  assumes "continuous_on {a..b} f" "x \<in> {a..b}"
  2.2998 -  shows "((\<lambda>u. integral {vec a..vec u} (f o dest_vec1)) has_vector_derivative f(x)) (at x within {a..b})"
  2.2999 -  unfolding has_vector_derivative_def has_derivative_within_alt
  2.3000 -apply safe apply(rule scaleR.bounded_linear_left)
  2.3001 -proof- fix e::real assume e:"e>0"
  2.3002 -  note compact_uniformly_continuous[OF assms(1) compact_real_interval,unfolded uniformly_continuous_on_def]
  2.3003 -  from this[rule_format,OF e] guess d apply-by(erule conjE exE)+ note d=this[rule_format]
  2.3004 -  let ?I = "\<lambda>a b. integral {vec1 a..vec1 b} (f \<circ> dest_vec1)"
  2.3005 -  show "\<exists>d>0. \<forall>y\<in>{a..b}. norm (y - x) < d \<longrightarrow> norm (?I a y - ?I a x - (y - x) *\<^sub>R f x) \<le> e * norm (y - x)"
  2.3006 -  proof(rule,rule,rule d,safe) case goal1 show ?case proof(cases "y < x")
  2.3007 -      case False have "f \<circ> dest_vec1 integrable_on {vec1 a..vec1 y}" apply(rule integrable_subinterval,rule integrable_continuous)
  2.3008 -        apply(rule continuous_on_o_dest_vec1 assms)+  unfolding not_less using assms(2) goal1 by auto
  2.3009 -      hence *:"?I a y - ?I a x = ?I x y" unfolding group_simps apply(subst eq_commute) apply(rule integral_combine)
  2.3010 -        using False unfolding not_less using assms(2) goal1 by auto
  2.3011 -      have **:"norm (y - x) = content {vec1 x..vec1 y}" apply(subst content_1) using False unfolding not_less by auto
  2.3012 -      show ?thesis unfolding ** apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x) o dest_vec1"]) unfolding * unfolding o_def
  2.3013 -        defer apply(rule has_integral_sub) apply(rule integrable_integral)
  2.3014 -        apply(rule integrable_subinterval,rule integrable_continuous) apply(rule continuous_on_o_dest_vec1[unfolded o_def] assms)+
  2.3015 -      proof- show "{vec1 x..vec1 y} \<subseteq> {vec1 a..vec1 b}" using goal1 assms(2) by auto
  2.3016 -        have *:"y - x = norm(y - x)" using False by auto
  2.3017 -        show "((\<lambda>xa. f x) has_integral (y - x) *\<^sub>R f x) {vec1 x..vec1 y}" apply(subst *) unfolding ** by auto
  2.3018 -        show "\<forall>xa\<in>{vec1 x..vec1 y}. norm (f (dest_vec1 xa) - f x) \<le> e" apply safe apply(rule less_imp_le)
  2.3019 -          apply(rule d(2)[unfolded vector_dist_norm]) using assms(2) using goal1 by auto
  2.3020 -      qed(insert e,auto)
  2.3021 -    next case True have "f \<circ> dest_vec1 integrable_on {vec1 a..vec1 x}" apply(rule integrable_subinterval,rule integrable_continuous)
  2.3022 -        apply(rule continuous_on_o_dest_vec1 assms)+  unfolding not_less using assms(2) goal1 by auto
  2.3023 -      hence *:"?I a x - ?I a y = ?I y x" unfolding group_simps apply(subst eq_commute) apply(rule integral_combine)
  2.3024 -        using True using assms(2) goal1 by auto
  2.3025 -      have **:"norm (y - x) = content {vec1 y..vec1 x}" apply(subst content_1) using True unfolding not_less by auto
  2.3026 -      have ***:"\<And>fy fx c::'a. fx - fy - (y - x) *\<^sub>R c = -(fy - fx - (x - y) *\<^sub>R c)" unfolding scaleR_left.diff by auto 
  2.3027 -      show ?thesis apply(subst ***) unfolding norm_minus_cancel **
  2.3028 -        apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x) o dest_vec1"]) unfolding * unfolding o_def
  2.3029 -        defer apply(rule has_integral_sub) apply(subst minus_minus[THEN sym]) unfolding minus_minus
  2.3030 -        apply(rule integrable_integral) apply(rule integrable_subinterval,rule integrable_continuous)
  2.3031 -        apply(rule continuous_on_o_dest_vec1[unfolded o_def] assms)+
  2.3032 -      proof- show "{vec1 y..vec1 x} \<subseteq> {vec1 a..vec1 b}" using goal1 assms(2) by auto
  2.3033 -        have *:"x - y = norm(y - x)" using True by auto
  2.3034 -        show "((\<lambda>xa. f x) has_integral (x - y) *\<^sub>R f x) {vec1 y..vec1 x}" apply(subst *) unfolding ** by auto
  2.3035 -        show "\<forall>xa\<in>{vec1 y..vec1 x}. norm (f (dest_vec1 xa) - f x) \<le> e" apply safe apply(rule less_imp_le)
  2.3036 -          apply(rule d(2)[unfolded vector_dist_norm]) using assms(2) using goal1 by auto
  2.3037 -      qed(insert e,auto) qed qed qed
  2.3038 -
  2.3039 -lemma integral_has_vector_derivative': fixes f::"real^1 \<Rightarrow> 'a::banach"
  2.3040 -  assumes "continuous_on {a..b} f" "x \<in> {a..b}"
  2.3041 -  shows "((\<lambda>u. (integral {a..vec u} f)) has_vector_derivative f x) (at (x$1) within {a$1..b$1})"
  2.3042 -  using integral_has_vector_derivative[OF continuous_on_o_vec1[OF assms(1)], of "x$1"]
  2.3043 -  unfolding o_def vec1_dest_vec1 using assms(2) by auto
  2.3044 -
  2.3045 -lemma antiderivative_continuous: assumes "continuous_on {a..b::real} f"
  2.3046 -  obtains g where "\<forall>x\<in> {a..b}. (g has_vector_derivative (f(x)::_::banach)) (at x within {a..b})"
  2.3047 -  apply(rule that,rule) using integral_has_vector_derivative[OF assms] by auto
  2.3048 -
  2.3049 -subsection {* Combined fundamental theorem of calculus. *}
  2.3050 -
  2.3051 -lemma antiderivative_integral_continuous: fixes f::"real \<Rightarrow> 'a::banach" assumes "continuous_on {a..b} f"
  2.3052 -  obtains g where "\<forall>u\<in>{a..b}. \<forall>v \<in> {a..b}. u \<le> v \<longrightarrow> ((f o dest_vec1) has_integral (g v - g u)) {vec u..vec v}"
  2.3053 -proof- from antiderivative_continuous[OF assms] guess g . note g=this
  2.3054 -  show ?thesis apply(rule that[of g])
  2.3055 -  proof safe case goal1 have "\<forall>x\<in>{u..v}. (g has_vector_derivative f x) (at x within {u..v})"
  2.3056 -      apply(rule,rule has_vector_derivative_within_subset) apply(rule g[rule_format]) using goal1(1-2) by auto
  2.3057 -    thus ?case using fundamental_theorem_of_calculus[OF goal1(3),of "g o dest_vec1" "f o dest_vec1"]
  2.3058 -      unfolding o_def vec1_dest_vec1 by auto qed qed
  2.3059 -
  2.3060 -subsection {* General "twiddling" for interval-to-interval function image. *}
  2.3061 -
  2.3062 -lemma has_integral_twiddle:
  2.3063 -  assumes "0 < r" "\<forall>x. h(g x) = x" "\<forall>x. g(h x) = x" "\<forall>x. continuous (at x) g"
  2.3064 -  "\<forall>u v. \<exists>w z. g ` {u..v} = {w..z}"
  2.3065 -  "\<forall>u v. \<exists>w z. h ` {u..v} = {w..z}"
  2.3066 -  "\<forall>u v. content(g ` {u..v}) = r * content {u..v}"
  2.3067 -  "(f has_integral i) {a..b}"
  2.3068 -  shows "((\<lambda>x. f(g x)) has_integral (1 / r) *\<^sub>R i) (h ` {a..b})"
  2.3069 -proof- { presume *:"{a..b} \<noteq> {} \<Longrightarrow> ?thesis"
  2.3070 -    show ?thesis apply cases defer apply(rule *,assumption)
  2.3071 -    proof- case goal1 thus ?thesis unfolding goal1 assms(8)[unfolded goal1 has_integral_empty_eq] by auto qed }
  2.3072 -  assume "{a..b} \<noteq> {}" from assms(6)[rule_format,of a b] guess w z apply-by(erule exE)+ note wz=this
  2.3073 -  have inj:"inj g" "inj h" unfolding inj_on_def apply safe apply(rule_tac[!] ccontr)
  2.3074 -    using assms(2) apply(erule_tac x=x in allE) using assms(2) apply(erule_tac x=y in allE) defer
  2.3075 -    using assms(3) apply(erule_tac x=x in allE) using assms(3) apply(erule_tac x=y in allE) by auto
  2.3076 -  show ?thesis unfolding has_integral_def has_integral_compact_interval_def apply(subst if_P) apply(rule,rule,rule wz)
  2.3077 -  proof safe fix e::real assume e:"e>0" hence "e * r > 0" using assms(1) by(rule mult_pos_pos)
  2.3078 -    from assms(8)[unfolded has_integral,rule_format,OF this] guess d apply-by(erule exE conjE)+ note d=this[rule_format]
  2.3079 -    def d' \<equiv> "\<lambda>x y. d (g x) (g y)" have d':"\<And>x. d' x = {y. g y \<in> (d (g x))}" unfolding d'_def by(auto simp add:mem_def)
  2.3080 -    show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of h ` {a..b} \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i) < e)"
  2.3081 -    proof(rule_tac x=d' in exI,safe) show "gauge d'" using d(1) unfolding gauge_def d' using continuous_open_preimage_univ[OF assms(4)] by auto
  2.3082 -      fix p assume as:"p tagged_division_of h ` {a..b}" "d' fine p" note p = tagged_division_ofD[OF as(1)] 
  2.3083 -      have "(\<lambda>(x, k). (g x, g ` k)) ` p tagged_division_of {a..b} \<and> d fine (\<lambda>(x, k). (g x, g ` k)) ` p" unfolding tagged_division_of 
  2.3084 -      proof safe show "finite ((\<lambda>(x, k). (g x, g ` k)) ` p)" using as by auto
  2.3085 -        show "d fine (\<lambda>(x, k). (g x, g ` k)) ` p" using as(2) unfolding fine_def d' by auto
  2.3086 -        fix x k assume xk[intro]:"(x,k) \<in> p" show "g x \<in> g ` k" using p(2)[OF xk] by auto
  2.3087 -        show "\<exists>u v. g ` k = {u..v}" using p(4)[OF xk] using assms(5-6) by auto
  2.3088 -        { fix y assume "y \<in> k" thus "g y \<in> {a..b}" "g y \<in> {a..b}" using p(3)[OF xk,unfolded subset_eq,rule_format,of "h (g y)"]
  2.3089 -            using assms(2)[rule_format,of y] unfolding inj_image_mem_iff[OF inj(2)] by auto }
  2.3090 -        fix x' k' assume xk':"(x',k') \<in> p" fix z assume "z \<in> interior (g ` k)" "z \<in> interior (g ` k')"
  2.3091 -        hence *:"interior (g ` k) \<inter> interior (g ` k') \<noteq> {}" by auto
  2.3092 -        have same:"(x, k) = (x', k')" apply-apply(rule ccontr,drule p(5)[OF xk xk'])
  2.3093 -        proof- assume as:"interior k \<inter> interior k' = {}" from nonempty_witness[OF *] guess z .
  2.3094 -          hence "z \<in> g ` (interior k \<inter> interior k')" using interior_image_subset[OF assms(4) inj(1)]
  2.3095 -            unfolding image_Int[OF inj(1)] by auto thus False using as by blast
  2.3096 -        qed thus "g x = g x'" by auto
  2.3097 -        { fix z assume "z \<in> k"  thus  "g z \<in> g ` k'" using same by auto }
  2.3098 -        { fix z assume "z \<in> k'" thus  "g z \<in> g ` k"  using same by auto }
  2.3099 -      next fix x assume "x \<in> {a..b}" hence "h x \<in>  \<Union>{k. \<exists>x. (x, k) \<in> p}" using p(6) by auto
  2.3100 -        then guess X unfolding Union_iff .. note X=this from this(1) guess y unfolding mem_Collect_eq ..
  2.3101 -        thus "x \<in> \<Union>{k. \<exists>x. (x, k) \<in> (\<lambda>(x, k). (g x, g ` k)) ` p}" apply-
  2.3102 -          apply(rule_tac X="g ` X" in UnionI) defer apply(rule_tac x="h x" in image_eqI)
  2.3103 -          using X(2) assms(3)[rule_format,of x] by auto
  2.3104 -      qed note ** = d(2)[OF this] have *:"inj_on (\<lambda>(x, k). (g x, g ` k)) p" using inj(1) unfolding inj_on_def by fastsimp
  2.3105 -       have "(\<Sum>(x, k)\<in>(\<lambda>(x, k). (g x, g ` k)) ` p. content k *\<^sub>R f x) - i = r *\<^sub>R (\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - i" (is "?l = _") unfolding group_simps add_left_cancel
  2.3106 -        unfolding setsum_reindex[OF *] apply(subst scaleR_right.setsum) defer apply(rule setsum_cong2) unfolding o_def split_paired_all split_conv
  2.3107 -        apply(drule p(4)) apply safe unfolding assms(7)[rule_format] using p by auto
  2.3108 -      also have "... = r *\<^sub>R ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i)" (is "_ = ?r") unfolding scaleR.diff_right scaleR.scaleR_left[THEN sym]
  2.3109 -        unfolding real_scaleR_def using assms(1) by auto finally have *:"?l = ?r" .
  2.3110 -      show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i) < e" using ** unfolding * unfolding norm_scaleR
  2.3111 -        using assms(1) by(auto simp add:field_simps) qed qed qed
  2.3112 -
  2.3113 -subsection {* Special case of a basic affine transformation. *}
  2.3114 -
  2.3115 -lemma interval_image_affinity_interval: shows "\<exists>u v. (\<lambda>x. m *\<^sub>R (x::real^'n) + c) ` {a..b} = {u..v}"
  2.3116 -  unfolding image_affinity_interval by auto
  2.3117 -
  2.3118 -lemmas Cart_simps = Cart_nth.add Cart_nth.minus Cart_nth.zero Cart_nth.diff Cart_nth.scaleR real_scaleR_def Cart_lambda_beta
  2.3119 -   Cart_eq vector_le_def vector_less_def
  2.3120 -
  2.3121 -lemma setprod_cong2: assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "setprod f A = setprod g A"
  2.3122 -  apply(rule setprod_cong) using assms by auto
  2.3123 -
  2.3124 -lemma content_image_affinity_interval: 
  2.3125 - "content((\<lambda>x::real^'n. m *\<^sub>R x + c) ` {a..b}) = (abs m) ^ CARD('n) * content {a..b}" (is "?l = ?r")
  2.3126 -proof- { presume *:"{a..b}\<noteq>{} \<Longrightarrow> ?thesis" show ?thesis apply(cases,rule *,assumption)
  2.3127 -      unfolding not_not using content_empty by auto }
  2.3128 -  assume as:"{a..b}\<noteq>{}" show ?thesis proof(cases "m \<ge> 0")
  2.3129 -    case True show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_P[OF True]
  2.3130 -      unfolding content_closed_interval'[OF as] apply(subst content_closed_interval') 
  2.3131 -      defer apply(subst setprod_constant[THEN sym]) apply(rule finite_UNIV) unfolding setprod_timesf[THEN sym]
  2.3132 -      apply(rule setprod_cong2) using True as unfolding interval_ne_empty Cart_simps not_le  
  2.3133 -      by(auto simp add:field_simps intro:mult_left_mono)
  2.3134 -  next case False show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_not_P[OF False]
  2.3135 -      unfolding content_closed_interval'[OF as] apply(subst content_closed_interval') 
  2.3136 -      defer apply(subst setprod_constant[THEN sym]) apply(rule finite_UNIV) unfolding setprod_timesf[THEN sym]
  2.3137 -      apply(rule setprod_cong2) using False as unfolding interval_ne_empty Cart_simps not_le 
  2.3138 -      by(auto simp add:field_simps mult_le_cancel_left_neg) qed qed
  2.3139 -
  2.3140 -lemma has_integral_affinity: assumes "(f has_integral i) {a..b::real^'n}" "m \<noteq> 0"
  2.3141 -  shows "((\<lambda>x. f(m *\<^sub>R x + c)) has_integral ((1 / (abs(m) ^ CARD('n::finite))) *\<^sub>R i)) ((\<lambda>x. (1 / m) *\<^sub>R x + -((1 / m) *\<^sub>R c)) ` {a..b})"
  2.3142 -  apply(rule has_integral_twiddle,safe) unfolding Cart_eq Cart_simps apply(rule zero_less_power)
  2.3143 -  defer apply(insert assms(2), simp add:field_simps) apply(insert assms(2), simp add:field_simps)
  2.3144 -  apply(rule continuous_intros)+ apply(rule interval_image_affinity_interval)+ apply(rule content_image_affinity_interval) using assms by auto
  2.3145 -
  2.3146 -lemma integrable_affinity: assumes "f integrable_on {a..b}" "m \<noteq> 0"
  2.3147 -  shows "(\<lambda>x. f(m *\<^sub>R x + c)) integrable_on ((\<lambda>x. (1 / m) *\<^sub>R x + -((1/m) *\<^sub>R c)) ` {a..b})"
  2.3148 -  using assms unfolding integrable_on_def apply safe apply(drule has_integral_affinity) by auto
  2.3149 -
  2.3150 -subsection {* Special case of stretching coordinate axes separately. *}
  2.3151 -
  2.3152 -lemma image_stretch_interval:
  2.3153 -  "(\<lambda>x. \<chi> k. m k * x$k) ` {a..b::real^'n} =
  2.3154 -  (if {a..b} = {} then {} else {(\<chi> k. min (m(k) * a$k) (m(k) * b$k)) ..  (\<chi> k. max (m(k) * a$k) (m(k) * b$k))})" (is "?l = ?r")
  2.3155 -proof(cases "{a..b}={}") case True thus ?thesis unfolding True by auto
  2.3156 -next have *:"\<And>P Q. (\<forall>i. P i) \<and> (\<forall>i. Q i) \<longleftrightarrow> (\<forall>i. P i \<and> Q i)" by auto
  2.3157 -  case False note ab = this[unfolded interval_ne_empty]
  2.3158 -  show ?thesis apply-apply(rule set_ext)
  2.3159 -  proof- fix x::"real^'n" have **:"\<And>P Q. (\<forall>i. P i = Q i) \<Longrightarrow> (\<forall>i. P i) = (\<forall>i. Q i)" by auto
  2.3160 -    show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" unfolding if_not_P[OF False] 
  2.3161 -      unfolding image_iff mem_interval Bex_def Cart_simps Cart_eq *
  2.3162 -      unfolding lambda_skolem[THEN sym,of "\<lambda> i xa. (a $ i \<le> xa \<and> xa \<le> b $ i) \<and> x $ i = m i * xa"]
  2.3163 -    proof(rule **,rule) fix i::'n show "(\<exists>xa. (a $ i \<le> xa \<and> xa \<le> b $ i) \<and> x $ i = m i * xa) =
  2.3164 -        (min (m i * a $ i) (m i * b $ i) \<le> x $ i \<and> x $ i \<le> max (m i * a $ i) (m i * b $ i))"
  2.3165 -      proof(cases "m i = 0") case True thus ?thesis using ab by auto
  2.3166 -      next case False hence "0 < m i \<or> 0 > m i" by auto thus ?thesis apply-
  2.3167 -        proof(erule disjE) assume as:"0 < m i" hence *:"min (m i * a $ i) (m i * b $ i) = m i * a $ i"
  2.3168 -            "max (m i * a $ i) (m i * b $ i) = m i * b $ i" using ab unfolding min_def max_def by auto
  2.3169 -          show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x$i" in exI)
  2.3170 -            using as by(auto simp add:field_simps)
  2.3171 -        next assume as:"0 > m i" hence *:"max (m i * a $ i) (m i * b $ i) = m i * a $ i"
  2.3172 -            "min (m i * a $ i) (m i * b $ i) = m i * b $ i" using ab as unfolding min_def max_def 
  2.3173 -            by(auto simp add:field_simps mult_le_cancel_left_neg intro:real_le_antisym)
  2.3174 -          show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x$i" in exI)
  2.3175 -            using as by(auto simp add:field_simps) qed qed qed qed qed 
  2.3176 -
  2.3177 -lemma interval_image_stretch_interval: "\<exists>u v. (\<lambda>x. \<chi> k. m k * x$k) ` {a..b::real^'n} = {u..v}"
  2.3178 -  unfolding image_stretch_interval by auto 
  2.3179 -
  2.3180 -lemma content_image_stretch_interval:
  2.3181 -  "content((\<lambda>x::real^'n. \<chi> k. m k * x$k) ` {a..b}) = abs(setprod m UNIV) * content({a..b})"
  2.3182 -proof(cases "{a..b} = {}") case True thus ?thesis
  2.3183 -    unfolding content_def image_is_empty image_stretch_interval if_P[OF True] by auto
  2.3184 -next case False hence "(\<lambda>x. \<chi> k. m k * x $ k) ` {a..b} \<noteq> {}" by auto
  2.3185 -  thus ?thesis using False unfolding content_def image_stretch_interval apply- unfolding interval_bounds' if_not_P
  2.3186 -    unfolding abs_setprod setprod_timesf[THEN sym] apply(rule setprod_cong2) unfolding Cart_lambda_beta
  2.3187 -  proof- fix i::'n have "(m i < 0 \<or> m i > 0) \<or> m i = 0" by auto
  2.3188 -    thus "max (m i * a $ i) (m i * b $ i) - min (m i * a $ i) (m i * b $ i) = \<bar>m i\<bar> * (b $ i - a $ i)"
  2.3189 -      apply-apply(erule disjE)+ unfolding min_def max_def using False[unfolded interval_ne_empty,rule_format,of i] 
  2.3190 -      by(auto simp add:field_simps not_le mult_le_cancel_left_neg mult_le_cancel_left_pos) qed qed
  2.3191 -
  2.3192 -lemma has_integral_stretch: assumes "(f has_integral i) {a..b}" "\<forall>k. ~(m k = 0)"
  2.3193 -  shows "((\<lambda>x. f(\<chi> k. m k * x$k)) has_integral
  2.3194 -             ((1/(abs(setprod m UNIV))) *\<^sub>R i)) ((\<lambda>x. \<chi> k. 1/(m k) * x$k) ` {a..b})"
  2.3195 -  apply(rule has_integral_twiddle) unfolding zero_less_abs_iff content_image_stretch_interval
  2.3196 -  unfolding image_stretch_interval empty_as_interval Cart_eq using assms
  2.3197 -proof- show "\<forall>x. continuous (at x) (\<lambda>x. \<chi> k. m k * x $ k)"
  2.3198 -   apply(rule,rule linear_continuous_at) unfolding linear_linear
  2.3199 -   unfolding linear_def Cart_simps Cart_eq by(auto simp add:field_simps) qed auto
  2.3200 -
  2.3201 -lemma integrable_stretch: 
  2.3202 -  assumes "f integrable_on {a..b}" "\<forall>k. ~(m k = 0)"
  2.3203 -  shows "(\<lambda>x. f(\<chi> k. m k * x$k)) integrable_on ((\<lambda>x. \<chi> k. 1/(m k) * x$k) ` {a..b})"
  2.3204 -  using assms unfolding integrable_on_def apply-apply(erule exE) apply(drule has_integral_stretch) by auto
  2.3205 -
  2.3206 -subsection {* even more special cases. *}
  2.3207 -
  2.3208 -lemma uminus_interval_vector[simp]:"uminus ` {a..b} = {-b .. -a::real^'n}"
  2.3209 -  apply(rule set_ext,rule) defer unfolding image_iff
  2.3210 -  apply(rule_tac x="-x" in bexI) by(auto simp add:vector_le_def minus_le_iff le_minus_iff)
  2.3211 -
  2.3212 -lemma has_integral_reflect_lemma[intro]: assumes "(f has_integral i) {a..b}"
  2.3213 -  shows "((\<lambda>x. f(-x)) has_integral i) {-b .. -a}"
  2.3214 -  using has_integral_affinity[OF assms, of "-1" 0] by auto
  2.3215 -
  2.3216 -lemma has_integral_reflect[simp]: "((\<lambda>x. f(-x)) has_integral i) {-b..-a} \<longleftrightarrow> (f has_integral i) ({a..b})"
  2.3217 -  apply rule apply(drule_tac[!] has_integral_reflect_lemma) by auto
  2.3218 -
  2.3219 -lemma integrable_reflect[simp]: "(\<lambda>x. f(-x)) integrable_on {-b..-a} \<longleftrightarrow> f integrable_on {a..b}"
  2.3220 -  unfolding integrable_on_def by auto
  2.3221 -
  2.3222 -lemma integral_reflect[simp]: "integral {-b..-a} (\<lambda>x. f(-x)) = integral ({a..b}) f"
  2.3223 -  unfolding integral_def by auto
  2.3224 -
  2.3225 -subsection {* Stronger form of FCT; quite a tedious proof. *}
  2.3226 -
  2.3227 -(** move this **)
  2.3228 -declare norm_triangle_ineq4[intro] 
  2.3229 -
  2.3230 -lemma bgauge_existence_lemma: "(\<forall>x\<in>s. \<exists>d::real. 0 < d \<and> q d x) \<longleftrightarrow> (\<forall>x. \<exists>d>0. x\<in>s \<longrightarrow> q d x)" by(meson zero_less_one)
  2.3231 -
  2.3232 -lemma additive_tagged_division_1': fixes f::"real \<Rightarrow> 'a::real_normed_vector"
  2.3233 -  assumes "a \<le> b" "p tagged_division_of {vec1 a..vec1 b}"
  2.3234 -  shows "setsum (\<lambda>(x,k). f (dest_vec1 (interval_upperbound k)) - f(dest_vec1 (interval_lowerbound k))) p = f b - f a"
  2.3235 -  using additive_tagged_division_1[OF _ assms(2), of "f o dest_vec1"]
  2.3236 -  unfolding o_def vec1_dest_vec1 using assms(1) by auto
  2.3237 -
  2.3238 -lemma split_minus[simp]:"(\<lambda>(x, k). ?f x k) x - (\<lambda>(x, k). ?g x k) x = (\<lambda>(x, k). ?f x k - ?g x k) x"
  2.3239 -  unfolding split_def by(rule refl)
  2.3240 -
  2.3241 -lemma norm_triangle_le_sub: "norm x + norm y \<le> e \<Longrightarrow> norm (x - y) \<le> e"
  2.3242 -  apply(subst(asm)(2) norm_minus_cancel[THEN sym])
  2.3243 -  apply(drule norm_triangle_le) by(auto simp add:group_simps)
  2.3244 -
  2.3245 -lemma fundamental_theorem_of_calculus_interior:
  2.3246 -  assumes"a \<le> b" "continuous_on {a..b} f" "\<forall>x\<in>{a<..<b}. (f has_vector_derivative f'(x)) (at x)"
  2.3247 -  shows "((f' o dest_vec1) has_integral (f b - f a)) {vec a..vec b}"
  2.3248 -proof- { presume *:"a < b \<Longrightarrow> ?thesis" 
  2.3249 -    show ?thesis proof(cases,rule *,assumption)
  2.3250 -      assume "\<not> a < b" hence "a = b" using assms(1) by auto
  2.3251 -      hence *:"{vec a .. vec b} = {vec b}" "f b - f a = 0" apply(auto simp add: Cart_simps) by smt
  2.3252 -      show ?thesis unfolding *(2) apply(rule has_integral_null) unfolding content_eq_0_1 using * `a=b` by auto
  2.3253 -    qed } assume ab:"a < b"
  2.3254 -  let ?P = "\<lambda>e. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {vec1 a..vec1 b} \<and> d fine p \<longrightarrow>
  2.3255 -                   norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f' \<circ> dest_vec1) x) - (f b - f a)) \<le> e * content {vec1 a..vec1 b})"
  2.3256 -  { presume "\<And>e. e>0 \<Longrightarrow> ?P e" thus ?thesis unfolding has_integral_factor_content by auto }
  2.3257 -  fix e::real assume e:"e>0"
  2.3258 -  note assms(3)[unfolded has_vector_derivative_def has_derivative_at_alt ball_conj_distrib]
  2.3259 -  note conjunctD2[OF this] note bounded=this(1) and this(2)
  2.3260 -  from this(2) have "\<forall>x\<in>{a<..<b}. \<exists>d>0. \<forall>y. norm (y - x) < d \<longrightarrow> norm (f y - f x - (y - x) *\<^sub>R f' x) \<le> e/2 * norm (y - x)"
  2.3261 -    apply-apply safe apply(erule_tac x=x in ballE,erule_tac x="e/2" in allE) using e by auto note this[unfolded bgauge_existence_lemma]
  2.3262 -  from choice[OF this] guess d .. note conjunctD2[OF this[rule_format]] note d = this[rule_format]
  2.3263 -  have "bounded (f ` {a..b})" apply(rule compact_imp_bounded compact_continuous_image)+ using compact_real_interval assms by auto
  2.3264 -  from this[unfolded bounded_pos] guess B .. note B = this[rule_format]
  2.3265 -
  2.3266 -  have "\<exists>da. 0 < da \<and> (\<forall>c. a \<le> c \<and> {a..c} \<subseteq> {a..b} \<and> {a..c} \<subseteq> ball a da
  2.3267 -    \<longrightarrow> norm(content {vec1 a..vec1 c} *\<^sub>R f' a - (f c - f a)) \<le> (e * (b - a)) / 4)"
  2.3268 -  proof- have "a\<in>{a..b}" using ab by auto
  2.3269 -    note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
  2.3270 -    note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0" using e ab by(auto simp add:field_simps)
  2.3271 -    from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
  2.3272 -    have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' a) \<le> (e * (b - a)) / 8"
  2.3273 -    proof(cases "f' a = 0") case True
  2.3274 -      thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg) 
  2.3275 -    next case False thus ?thesis 
  2.3276 -        apply(rule_tac x="(e * (b - a)) / 8 / norm (f' a)" in exI)
  2.3277 -        using ab e by(auto simp add:field_simps)
  2.3278 -    qed then guess l .. note l = conjunctD2[OF this]
  2.3279 -    show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+)
  2.3280 -    proof- fix c assume as:"a \<le> c" "{a..c} \<subseteq> {a..b}" "{a..c} \<subseteq> ball a (min k l)" 
  2.3281 -      note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval]
  2.3282 -      have "norm ((c - a) *\<^sub>R f' a - (f c - f a)) \<le> norm ((c - a) *\<^sub>R f' a) + norm (f c - f a)" by(rule norm_triangle_ineq4)
  2.3283 -      also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8" 
  2.3284 -      proof(rule add_mono) case goal1 have "\<bar>c - a\<bar> \<le> \<bar>l\<bar>" using as' by auto
  2.3285 -        thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto
  2.3286 -      next case goal2 show ?case apply(rule less_imp_le) apply(cases "a = c") defer
  2.3287 -          apply(rule k(2)[unfolded vector_dist_norm]) using as' e ab by(auto simp add:field_simps)
  2.3288 -      qed finally show "norm (content {vec1 a..vec1 c} *\<^sub>R f' a - (f c - f a)) \<le> e * (b - a) / 4" unfolding content_1'[OF as(1)] by auto
  2.3289 -    qed qed then guess da .. note da=conjunctD2[OF this,rule_format]
  2.3290 -
  2.3291 -  have "\<exists>db>0. \<forall>c\<le>b. {c..b} \<subseteq> {a..b} \<and> {c..b} \<subseteq> ball b db \<longrightarrow> norm(content {vec1 c..vec1 b} *\<^sub>R f' b - (f b - f c)) \<le> (e * (b - a)) / 4"
  2.3292 -  proof- have "b\<in>{a..b}" using ab by auto
  2.3293 -    note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
  2.3294 -    note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0" using e ab by(auto simp add:field_simps)
  2.3295 -    from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
  2.3296 -    have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' b) \<le> (e * (b - a)) / 8"
  2.3297 -    proof(cases "f' b = 0") case True
  2.3298 -      thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg) 
  2.3299 -    next case False thus ?thesis 
  2.3300 -        apply(rule_tac x="(e * (b - a)) / 8 / norm (f' b)" in exI)
  2.3301 -        using ab e by(auto simp add:field_simps)
  2.3302 -    qed then guess l .. note l = conjunctD2[OF this]
  2.3303 -    show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+)
  2.3304 -    proof- fix c assume as:"c \<le> b" "{c..b} \<subseteq> {a..b}" "{c..b} \<subseteq> ball b (min k l)" 
  2.3305 -      note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval]
  2.3306 -      have "norm ((b - c) *\<^sub>R f' b - (f b - f c)) \<le> norm ((b - c) *\<^sub>R f' b) + norm (f b - f c)" by(rule norm_triangle_ineq4)
  2.3307 -      also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8" 
  2.3308 -      proof(rule add_mono) case goal1 have "\<bar>c - b\<bar> \<le> \<bar>l\<bar>" using as' by auto
  2.3309 -        thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto
  2.3310 -      next case goal2 show ?case apply(rule less_imp_le) apply(cases "b = c") defer apply(subst norm_minus_commute)
  2.3311 -          apply(rule k(2)[unfolded vector_dist_norm]) using as' e ab by(auto simp add:field_simps)
  2.3312 -      qed finally show "norm (content {vec1 c..vec1 b} *\<^sub>R f' b - (f b - f c)) \<le> e * (b - a) / 4" unfolding content_1'[OF as(1)] by auto
  2.3313 -    qed qed then guess db .. note db=conjunctD2[OF this,rule_format]
  2.3314 -
  2.3315 -  let ?d = "(\<lambda>x. ball x (if x=vec1 a then da else if x=vec b then db else d (dest_vec1 x)))"
  2.3316 -  show "?P e" apply(rule_tac x="?d" in exI)
  2.3317 -  proof safe case goal1 show ?case apply(rule gauge_ball_dependent) using ab db(1) da(1) d(1) by auto
  2.3318 -  next case goal2 note as=this let ?A = "{t. fst t \<in> {vec1 a, vec1 b}}" note p = tagged_division_ofD[OF goal2(1)]
  2.3319 -    have pA:"p = (p \<inter> ?A) \<union> (p - ?A)" "finite (p \<inter> ?A)" "finite (p - ?A)" "(p \<inter> ?A) \<inter> (p - ?A) = {}"  using goal2 by auto
  2.3320 -    note * = additive_tagged_division_1'[OF assms(1) goal2(1), THEN sym]
  2.3321 -    have **:"\<And>n1 s1 n2 s2::real. n2 \<le> s2 / 2 \<Longrightarrow> n1 - s1 \<le> s2 / 2 \<Longrightarrow> n1 + n2 \<le> s1 + s2" by arith
  2.3322 -    show ?case unfolding content_1'[OF assms(1)] and *[of "\<lambda>x. x"] *[of f] setsum_subtractf[THEN sym] split_minus
  2.3323 -      unfolding setsum_right_distrib apply(subst(2) pA,subst pA) unfolding setsum_Un_disjoint[OF pA(2-)]
  2.3324 -    proof(rule norm_triangle_le,rule **) 
  2.3325 -      case goal1 show ?case apply(rule order_trans,rule setsum_norm_le) apply(rule pA) defer apply(subst divide.setsum)
  2.3326 -      proof(rule order_refl,safe,unfold not_le o_def split_conv fst_conv,rule ccontr) fix x k assume as:"(x,k) \<in> p"
  2.3327 -          "e * (dest_vec1 (interval_upperbound k) - dest_vec1 (interval_lowerbound k)) / 2
  2.3328 -          < norm (content k *\<^sub>R f' (dest_vec1 x) - (f (dest_vec1 (interval_upperbound k)) - f (dest_vec1 (interval_lowerbound k))))"
  2.3329 -        from p(4)[OF this(1)] guess u v apply-by(erule exE)+ note k=this
  2.3330 -        hence "\<forall>i. u$i \<le> v$i" and uv:"{u,v}\<subseteq>{u..v}" using p(2)[OF as(1)] by auto note this(1) this(1)[unfolded forall_1]
  2.3331 -        note result = as(2)[unfolded k interval_bounds[OF this(1)] content_1[OF this(2)]]
  2.3332 -
  2.3333 -        assume as':"x \<noteq> vec1 a" "x \<noteq> vec1 b" hence "x$1 \<in> {a<..<b}" using p(2-3)[OF as(1)] by(auto simp add:Cart_simps) note  * = d(2)[OF this] 
  2.3334 -        have "norm ((v$1 - u$1) *\<^sub>R f' (x$1) - (f (v$1) - f (u$1))) =
  2.3335 -          norm ((f (u$1) - f (x$1) - (u$1 - x$1) *\<^sub>R f' (x$1)) - (f (v$1) - f (x$1) - (v$1 - x$1) *\<^sub>R f' (x$1)))" 
  2.3336 -          apply(rule arg_cong[of _ _ norm]) unfolding scaleR_left.diff by auto 
  2.3337 -        also have "... \<le> e / 2 * norm (u$1 - x$1) + e / 2 * norm (v$1 - x$1)" apply(rule norm_triangle_le_sub)
  2.3338 -          apply(rule add_mono) apply(rule_tac[!] *) using fineD[OF goal2(2) as(1)] as' unfolding k subset_eq
  2.3339 -          apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE) using uv by(auto simp add:dist_real)
  2.3340 -        also have "... \<le> e / 2 * norm (v$1 - u$1)" using p(2)[OF as(1)] unfolding k by(auto simp add:field_simps)
  2.3341 -        finally have "e * (dest_vec1 v - dest_vec1 u) / 2 < e * (dest_vec1 v - dest_vec1 u) / 2"
  2.3342 -          apply- apply(rule less_le_trans[OF result]) using uv by auto thus False by auto qed
  2.3343 -
  2.3344 -    next have *:"\<And>x s1 s2::real. 0 \<le> s1 \<Longrightarrow> x \<le> (s1 + s2) / 2 \<Longrightarrow> x - s1 \<le> s2 / 2" by auto
  2.3345 -      case goal2 show ?case apply(rule *) apply(rule setsum_nonneg) apply(rule,unfold split_paired_all split_conv)
  2.3346 -        defer unfolding setsum_Un_disjoint[OF pA(2-),THEN sym] pA(1)[THEN sym] unfolding setsum_right_distrib[THEN sym] 
  2.3347 -        apply(subst additive_tagged_division_1[OF _ as(1)]) unfolding vec1_dest_vec1 apply(rule assms)
  2.3348 -      proof- fix x k assume "(x,k) \<in> p \<inter> {t. fst t \<in> {vec1 a, vec1 b}}" note xk=IntD1[OF this]
  2.3349 -        from p(4)[OF this] guess u v apply-by(erule exE)+ note uv=this
  2.3350 -        with p(2)[OF xk] have "{u..v} \<noteq> {}" by auto
  2.3351 -        thus "0 \<le> e * ((interval_upperbound k)$1 - (interval_lowerbound k)$1)"
  2.3352 -          unfolding uv using e by(auto simp add:field_simps)
  2.3353 -      next have *:"\<And>s f t e. setsum f s = setsum f t \<Longrightarrow> norm(setsum f t) \<le> e \<Longrightarrow> norm(setsum f s) \<le> e" by auto
  2.3354 -        show "norm (\<Sum>(x, k)\<in>p \<inter> ?A. content k *\<^sub>R (f' \<circ> dest_vec1) x -
  2.3355 -          (f ((interval_upperbound k)$1) - f ((interval_lowerbound k)$1))) \<le> e * (b - a) / 2" 
  2.3356 -          apply(rule *[where t="p \<inter> {t. fst t \<in> {vec1 a, vec1 b} \<and> content(snd t) \<noteq> 0}"])
  2.3357 -          apply(rule setsum_mono_zero_right[OF pA(2)]) defer apply(rule) unfolding split_paired_all split_conv o_def
  2.3358 -        proof- fix x k assume "(x,k) \<in> p \<inter> {t. fst t \<in> {vec1 a, vec1 b}} - p \<inter> {t. fst t \<in> {vec1 a, vec1 b} \<and> content (snd t) \<noteq> 0}"
  2.3359 -          hence xk:"(x,k)\<in>p" "content k = 0" by auto from p(4)[OF xk(1)] guess u v apply-by(erule exE)+ note uv=this
  2.3360 -          have "k\<noteq>{}" using p(2)[OF xk(1)] by auto hence *:"u = v" using xk unfolding uv content_eq_0_1 interval_eq_empty by auto
  2.3361 -          thus "content k *\<^sub>R (f' (x$1)) - (f ((interval_upperbound k)$1) - f ((interval_lowerbound k)$1)) = 0" using xk unfolding uv by auto
  2.3362 -        next have *:"p \<inter> {t. fst t \<in> {vec1 a, vec1 b} \<and> content(snd t) \<noteq> 0} = 
  2.3363 -            {t. t\<in>p \<and> fst t = vec1 a \<and> content(snd t) \<noteq> 0} \<union> {t. t\<in>p \<and> fst t = vec1 b \<and> content(snd t) \<noteq> 0}" by blast
  2.3364 -          have **:"\<And>s f. \<And>e::real. (\<forall>x y. x \<in> s \<and> y \<in> s \<longrightarrow> x = y) \<Longrightarrow> (\<forall>x. x \<in> s \<longrightarrow> norm(f x) \<le> e) \<Longrightarrow> e>0 \<Longrightarrow> norm(setsum f s) \<le> e"
  2.3365 -          proof(case_tac "s={}") case goal2 then obtain x where "x\<in>s" by auto hence *:"s = {x}" using goal2(1) by auto
  2.3366 -            thus ?case using `x\<in>s` goal2(2) by auto
  2.3367 -          qed auto
  2.3368 -          case goal2 show ?case apply(subst *, subst setsum_Un_disjoint) prefer 4 apply(rule order_trans[of _ "e * (b - a)/4 + e * (b - a)/4"]) 
  2.3369 -            apply(rule norm_triangle_le,rule add_mono) apply(rule_tac[1-2] **)
  2.3370 -          proof- let ?B = "\<lambda>x. {t \<in> p. fst t = vec1 x \<and> content (snd t) \<noteq> 0}"
  2.3371 -            have pa:"\<And>k. (vec1 a, k) \<in> p \<Longrightarrow> \<exists>v. k = {vec1 a .. v} \<and> vec1 a \<le> v" 
  2.3372 -            proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this
  2.3373 -              have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto
  2.3374 -              have u:"u = vec1 a" proof(rule ccontr)  have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto 
  2.3375 -                have "u \<ge> vec1 a" using p(2-3)[OF goal1(1)] unfolding uv subset_eq by auto moreover assume "u\<noteq>vec1 a" ultimately
  2.3376 -                have "u > vec1 a" unfolding Cart_simps by auto
  2.3377 -                thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:Cart_simps)
  2.3378 -              qed thus ?case apply(rule_tac x=v in exI) unfolding uv using * by auto
  2.3379 -            qed
  2.3380 -            have pb:"\<And>k. (vec1 b, k) \<in> p \<Longrightarrow> \<exists>v. k = {v .. vec1 b} \<and> vec1 b \<ge> v" 
  2.3381 -            proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this
  2.3382 -              have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto
  2.3383 -              have u:"v = vec1 b" proof(rule ccontr)  have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto 
  2.3384 -                have "v \<le> vec1 b" using p(2-3)[OF goal1(1)] unfolding uv subset_eq by auto moreover assume "v\<noteq>vec1 b" ultimately
  2.3385 -                have "v < vec1 b" unfolding Cart_simps by auto
  2.3386 -                thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:Cart_simps)
  2.3387 -              qed thus ?case apply(rule_tac x=u in exI) unfolding uv using * by auto
  2.3388 -            qed
  2.3389 -
  2.3390 -            show "\<forall>x y. x \<in> ?B a \<and> y \<in> ?B a \<longrightarrow> x = y" apply(rule,rule,rule,unfold split_paired_all)
  2.3391 -              unfolding mem_Collect_eq fst_conv snd_conv apply safe
  2.3392 -            proof- fix x k k' assume k:"(vec1 a, k) \<in> p" "(vec1 a, k') \<in> p" "content k \<noteq> 0" "content k' \<noteq> 0"
  2.3393 -              guess v using pa[OF k(1)] .. note v = conjunctD2[OF this]
  2.3394 -              guess v' using pa[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = "vec1 (min (v$1) (v'$1))"
  2.3395 -              have "{vec1 a <..< ?v} \<subseteq> k \<inter> k'" unfolding v v' by(auto simp add:Cart_simps) note subset_interior[OF this,unfolded interior_inter]
  2.3396 -              moreover have "vec1 ((a + ?v$1)/2) \<in> {vec1 a <..< ?v}" using k(3-) unfolding v v' content_eq_0_1 not_le by(auto simp add:Cart_simps)
  2.3397 -              ultimately have "vec1 ((a + ?v$1)/2) \<in> interior k \<inter> interior k'" unfolding interior_open[OF open_interval] by auto
  2.3398 -              hence *:"k = k'" apply- apply(rule ccontr) using p(5)[OF k(1-2)] by auto
  2.3399 -              { assume "x\<in>k" thus "x\<in>k'" unfolding * . } { assume "x\<in>k'" thus "x\<in>k" unfolding * . }
  2.3400 -            qed 
  2.3401 -            show "\<forall>x y. x \<in> ?B b \<and> y \<in> ?B b \<longrightarrow> x = y" apply(rule,rule,rule,unfold split_paired_all)
  2.3402 -              unfolding mem_Collect_eq fst_conv snd_conv apply safe
  2.3403 -            proof- fix x k k' assume k:"(vec1 b, k) \<in> p" "(vec1 b, k') \<in> p" "content k \<noteq> 0" "content k' \<noteq> 0"
  2.3404 -              guess v using pb[OF k(1)] .. note v = conjunctD2[OF this]
  2.3405 -              guess v' using pb[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = "vec1 (max (v$1) (v'$1))"
  2.3406 -              have "{?v <..< vec1 b} \<subseteq> k \<inter> k'" unfolding v v' by(auto simp add:Cart_simps) note subset_interior[OF this,unfolded interior_inter]
  2.3407 -              moreover have "vec1 ((b + ?v$1)/2) \<in> {?v <..< vec1 b}" using k(3-) unfolding v v' content_eq_0_1 not_le by(auto simp add:Cart_simps)
  2.3408 -              ultimately have "vec1 ((b + ?v$1)/2) \<in> interior k \<inter> interior k'" unfolding interior_open[OF open_interval] by auto
  2.3409 -              hence *:"k = k'" apply- apply(rule ccontr) using p(5)[OF k(1-2)] by auto
  2.3410 -              { assume "x\<in>k" thus "x\<in>k'" unfolding * . } { assume "x\<in>k'" thus "x\<in>k" unfolding * . }
  2.3411 -            qed
  2.3412 -
  2.3413 -            let ?a = a and ?b = b (* a is something else while proofing the next theorem. *)
  2.3414 -            show "\<forall>x. x \<in> ?B a \<longrightarrow> norm ((\<lambda>(x, k). content k *\<^sub>R f' (x$1) - (f ((interval_upperbound k)$1) - f ((interval_lowerbound k)$1))) x)
  2.3415 -              \<le> e * (b - a) / 4" apply safe unfolding fst_conv snd_conv apply safe unfolding vec1_dest_vec1
  2.3416 -            proof- case goal1 guess v using pa[OF goal1(1)] .. note v = conjunctD2[OF this]
  2.3417 -              have "vec1 ?a\<in>{vec1 ?a..v}" using v(2) by auto hence "dest_vec1 v \<le> ?b" using p(3)[OF goal1(1)] unfolding subset_eq v by auto
  2.3418 -              moreover have "{?a..dest_vec1 v} \<subseteq> ball ?a da" using fineD[OF as(2) goal1(1)]
  2.3419 -                apply-apply(subst(asm) if_P,rule refl) unfolding subset_eq apply safe apply(erule_tac x="vec1 x" in ballE)
  2.3420 -                by(auto simp add:Cart_simps subset_eq dist_real v dist_real_def) ultimately
  2.3421 -              show ?case unfolding v unfolding interval_bounds[OF v(2)[unfolded v vector_le_def]] vec1_dest_vec1 apply-
  2.3422 -                apply(rule da(2)[of "v$1",unfolded vec1_dest_vec1])
  2.3423 -                using goal1 fineD[OF as(2) goal1(1)] unfolding v content_eq_0_1 by auto
  2.3424 -            qed
  2.3425 -            show "\<forall>x. x \<in> ?B b \<longrightarrow> norm ((\<lambda>(x, k). content k *\<^sub>R f' (x$1) - (f ((interval_upperbound k)$1) - f ((interval_lowerbound k)$1))) x)
  2.3426 -              \<le> e * (b - a) / 4" apply safe unfolding fst_conv snd_conv apply safe unfolding vec1_dest_vec1
  2.3427 -            proof- case goal1 guess v using pb[OF goal1(1)] .. note v = conjunctD2[OF this]
  2.3428 -              have "vec1 ?b\<in>{v..vec1 ?b}" using v(2) by auto hence "dest_vec1 v \<ge> ?a" using p(3)[OF goal1(1)] unfolding subset_eq v by auto
  2.3429 -              moreover have "{dest_vec1 v..?b} \<subseteq> ball ?b db" using fineD[OF as(2) goal1(1)]
  2.3430 -                apply-apply(subst(asm) if_P,rule refl) unfolding subset_eq apply safe apply(erule_tac x="vec1 x" in ballE) using ab
  2.3431 -                by(auto simp add:Cart_simps subset_eq dist_real v dist_real_def) ultimately
  2.3432 -              show ?case unfolding v unfolding interval_bounds[OF v(2)[unfolded v vector_le_def]] vec1_dest_vec1 apply-
  2.3433 -                apply(rule db(2)[of "v$1",unfolded vec1_dest_vec1])
  2.3434 -                using goal1 fineD[OF as(2) goal1(1)] unfolding v content_eq_0_1 by auto
  2.3435 -            qed
  2.3436 -          qed(insert p(1) ab e, auto simp add:field_simps) qed auto qed qed qed qed
  2.3437 -
  2.3438 -subsection {* Stronger form with finite number of exceptional points. *}
  2.3439 -
  2.3440 -lemma fundamental_theorem_of_calculus_interior_strong: fixes f::"real \<Rightarrow> 'a::banach"
  2.3441 -  assumes"finite s" "a \<le> b" "continuous_on {a..b} f"
  2.3442 -  "\<forall>x\<in>{a<..<b} - s. (f has_vector_derivative f'(x)) (at x)"
  2.3443 -  shows "((f' o dest_vec1) has_integral (f b - f a)) {vec a..vec b}" using assms apply- 
  2.3444 -proof(induct "card s" arbitrary:s a b)
  2.3445 -  case 0 show ?case apply(rule fundamental_theorem_of_calculus_interior) using 0 by auto
  2.3446 -next case (Suc n) from this(2) guess c s' apply-apply(subst(asm) eq_commute) unfolding card_Suc_eq
  2.3447 -    apply(subst(asm)(2) eq_commute) by(erule exE conjE)+ note cs = this[rule_format]
  2.3448 -  show ?case proof(cases "c\<in>{a<..<b}")
  2.3449 -    case False thus ?thesis apply- apply(rule Suc(1)[OF cs(3) _ Suc(4,5)]) apply safe defer
  2.3450 -      apply(rule Suc(6)[rule_format]) using Suc(3) unfolding cs by auto
  2.3451 -  next have *:"f b - f a = (f c - f a) + (f b - f c)" by auto
  2.3452 -    case True hence "vec1 a \<le> vec1 c" "vec1 c \<le> vec1 b" by auto
  2.3453 -    thus ?thesis apply(subst *) apply(rule has_integral_combine) apply assumption+
  2.3454 -      apply(rule_tac[!] Suc(1)[OF cs(3)]) using Suc(3) unfolding cs
  2.3455 -    proof- show "continuous_on {a..c} f" "continuous_on {c..b} f"
  2.3456 -        apply(rule_tac[!] continuous_on_subset[OF Suc(5)]) using True by auto
  2.3457 -      let ?P = "\<lambda>i j. \<forall>x\<in>{i<..<j} - s'. (f has_vector_derivative f' x) (at x)"
  2.3458 -      show "?P a c" "?P c b" apply safe apply(rule_tac[!] Suc(6)[rule_format]) using True unfolding cs by auto
  2.3459 -    qed auto qed qed
  2.3460 -
  2.3461 -lemma fundamental_theorem_of_calculus_strong: fixes f::"real \<Rightarrow> 'a::banach"
  2.3462 -  assumes "finite s" "a \<le> b" "continuous_on {a..b} f"
  2.3463 -  "\<forall>x\<in>{a..b} - s. (f has_vector_derivative f'(x)) (at x)"
<