Added integration to Multivariate-Analysis (upto FTC)
authorhimmelma
Wed Feb 17 18:33:45 2010 +0100 (2010-02-17)
changeset 35172579dd5570f96
parent 35171 28f824c7addc
child 35173 9b24bfca8044
Added integration to Multivariate-Analysis (upto FTC)
src/HOL/Multivariate_Analysis/Derivative.thy
src/HOL/Multivariate_Analysis/Euclidean_Space.thy
src/HOL/Multivariate_Analysis/Integration.cert
src/HOL/Multivariate_Analysis/Integration.thy
src/HOL/Multivariate_Analysis/Multivariate_Analysis.thy
src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
     1.1 --- a/src/HOL/Multivariate_Analysis/Derivative.thy	Wed Feb 17 17:57:37 2010 +0100
     1.2 +++ b/src/HOL/Multivariate_Analysis/Derivative.thy	Wed Feb 17 18:33:45 2010 +0100
     1.3 @@ -387,7 +387,7 @@
     1.4        apply(rule_tac le_less_trans[of _ "e/2"]) by(auto intro!:mult_imp_div_pos_le simp add:group_simps) qed auto qed
     1.5  
     1.6  lemma has_derivative_at_alt:
     1.7 -  "(f has_derivative f') (at (x::real^'n)) \<longleftrightarrow> bounded_linear f' \<and>
     1.8 +  "(f has_derivative f') (at x) \<longleftrightarrow> bounded_linear f' \<and>
     1.9    (\<forall>e>0. \<exists>d>0. \<forall>y. norm(y - x) < d \<longrightarrow> norm(f y - f x - f'(y - x)) \<le> e * norm(y - x))"
    1.10    using has_derivative_within_alt[where s=UNIV] unfolding within_UNIV by auto
    1.11  
     2.1 --- a/src/HOL/Multivariate_Analysis/Euclidean_Space.thy	Wed Feb 17 17:57:37 2010 +0100
     2.2 +++ b/src/HOL/Multivariate_Analysis/Euclidean_Space.thy	Wed Feb 17 18:33:45 2010 +0100
     2.3 @@ -1042,11 +1042,6 @@
     2.4    shows "norm x \<le> norm y  + norm (x - y)"
     2.5    using norm_triangle_ineq[of "y" "x - y"] by (simp add: ring_simps)
     2.6  
     2.7 -lemma norm_triangle_le: "norm(x::real ^ 'n) + norm y <= e ==> norm(x + y) <= e"
     2.8 -  by (metis order_trans norm_triangle_ineq)
     2.9 -lemma norm_triangle_lt: "norm(x::real ^ 'n) + norm(y) < e ==> norm(x + y) < e"
    2.10 -  by (metis basic_trans_rules(21) norm_triangle_ineq)
    2.11 -
    2.12  lemma component_le_norm: "\<bar>x$i\<bar> <= norm x"
    2.13    apply (simp add: norm_vector_def)
    2.14    apply (rule member_le_setL2, simp_all)
    2.15 @@ -1275,6 +1270,22 @@
    2.16    shows "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"
    2.17  by (rule dist_triangle_half_l, simp_all add: dist_commute)
    2.18  
    2.19 +
    2.20 +lemma norm_triangle_half_r:
    2.21 +  shows "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
    2.22 +  using dist_triangle_half_r unfolding vector_dist_norm[THEN sym] by auto
    2.23 +
    2.24 +lemma norm_triangle_half_l: assumes "norm (x - y) < e / 2" "norm (x' - (y)) < e / 2" 
    2.25 +  shows "norm (x - x') < e"
    2.26 +  using dist_triangle_half_l[OF assms[unfolded vector_dist_norm[THEN sym]]]
    2.27 +  unfolding vector_dist_norm[THEN sym] .
    2.28 +
    2.29 +lemma norm_triangle_le: "norm(x) + norm y <= e ==> norm(x + y) <= e"
    2.30 +  by (metis order_trans norm_triangle_ineq)
    2.31 +
    2.32 +lemma norm_triangle_lt: "norm(x) + norm(y) < e ==> norm(x + y) < e"
    2.33 +  by (metis basic_trans_rules(21) norm_triangle_ineq)
    2.34 +
    2.35  lemma dist_triangle_add:
    2.36    fixes x y x' y' :: "'a::real_normed_vector"
    2.37    shows "dist (x + y) (x' + y') <= dist x x' + dist y y'"
     3.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     3.2 +++ b/src/HOL/Multivariate_Analysis/Integration.cert	Wed Feb 17 18:33:45 2010 +0100
     3.3 @@ -0,0 +1,3270 @@
     3.4 +tB2Atlor9W4pSnrAz5nHpw 907 0
     3.5 +#2 := false
     3.6 +#299 := 0::real
     3.7 +decl uf_1 :: (-> T3 T2 real)
     3.8 +decl uf_10 :: (-> T4 T2)
     3.9 +decl uf_7 :: T4
    3.10 +#15 := uf_7
    3.11 +#22 := (uf_10 uf_7)
    3.12 +decl uf_2 :: (-> T1 T3)
    3.13 +decl uf_4 :: T1
    3.14 +#11 := uf_4
    3.15 +#91 := (uf_2 uf_4)
    3.16 +#902 := (uf_1 #91 #22)
    3.17 +#297 := -1::real
    3.18 +#1084 := (* -1::real #902)
    3.19 +decl uf_16 :: T1
    3.20 +#50 := uf_16
    3.21 +#78 := (uf_2 uf_16)
    3.22 +#799 := (uf_1 #78 #22)
    3.23 +#1267 := (+ #799 #1084)
    3.24 +#1272 := (>= #1267 0::real)
    3.25 +#1266 := (= #799 #902)
    3.26 +decl uf_9 :: T3
    3.27 +#21 := uf_9
    3.28 +#23 := (uf_1 uf_9 #22)
    3.29 +#905 := (= #23 #902)
    3.30 +decl uf_11 :: T3
    3.31 +#24 := uf_11
    3.32 +#850 := (uf_1 uf_11 #22)
    3.33 +#904 := (= #850 #902)
    3.34 +decl uf_6 :: (-> T2 T4)
    3.35 +#74 := (uf_6 #22)
    3.36 +#281 := (= uf_7 #74)
    3.37 +#922 := (ite #281 #905 #904)
    3.38 +decl uf_8 :: T3
    3.39 +#18 := uf_8
    3.40 +#848 := (uf_1 uf_8 #22)
    3.41 +#903 := (= #848 #902)
    3.42 +#60 := 0::int
    3.43 +decl uf_5 :: (-> T4 int)
    3.44 +#803 := (uf_5 #74)
    3.45 +#117 := -1::int
    3.46 +#813 := (* -1::int #803)
    3.47 +#16 := (uf_5 uf_7)
    3.48 +#916 := (+ #16 #813)
    3.49 +#917 := (<= #916 0::int)
    3.50 +#925 := (ite #917 #922 #903)
    3.51 +#6 := (:var 0 T2)
    3.52 +#19 := (uf_1 uf_8 #6)
    3.53 +#544 := (pattern #19)
    3.54 +#25 := (uf_1 uf_11 #6)
    3.55 +#543 := (pattern #25)
    3.56 +#92 := (uf_1 #91 #6)
    3.57 +#542 := (pattern #92)
    3.58 +#13 := (uf_6 #6)
    3.59 +#541 := (pattern #13)
    3.60 +#447 := (= #19 #92)
    3.61 +#445 := (= #25 #92)
    3.62 +#444 := (= #23 #92)
    3.63 +#20 := (= #13 uf_7)
    3.64 +#446 := (ite #20 #444 #445)
    3.65 +#120 := (* -1::int #16)
    3.66 +#14 := (uf_5 #13)
    3.67 +#121 := (+ #14 #120)
    3.68 +#119 := (>= #121 0::int)
    3.69 +#448 := (ite #119 #446 #447)
    3.70 +#545 := (forall (vars (?x3 T2)) (:pat #541 #542 #543 #544) #448)
    3.71 +#451 := (forall (vars (?x3 T2)) #448)
    3.72 +#548 := (iff #451 #545)
    3.73 +#546 := (iff #448 #448)
    3.74 +#547 := [refl]: #546
    3.75 +#549 := [quant-intro #547]: #548
    3.76 +#26 := (ite #20 #23 #25)
    3.77 +#127 := (ite #119 #26 #19)
    3.78 +#368 := (= #92 #127)
    3.79 +#369 := (forall (vars (?x3 T2)) #368)
    3.80 +#452 := (iff #369 #451)
    3.81 +#449 := (iff #368 #448)
    3.82 +#450 := [rewrite]: #449
    3.83 +#453 := [quant-intro #450]: #452
    3.84 +#392 := (~ #369 #369)
    3.85 +#390 := (~ #368 #368)
    3.86 +#391 := [refl]: #390
    3.87 +#366 := [nnf-pos #391]: #392
    3.88 +decl uf_3 :: (-> T1 T2 real)
    3.89 +#12 := (uf_3 uf_4 #6)
    3.90 +#132 := (= #12 #127)
    3.91 +#135 := (forall (vars (?x3 T2)) #132)
    3.92 +#370 := (iff #135 #369)
    3.93 +#4 := (:var 1 T1)
    3.94 +#8 := (uf_3 #4 #6)
    3.95 +#5 := (uf_2 #4)
    3.96 +#7 := (uf_1 #5 #6)
    3.97 +#9 := (= #7 #8)
    3.98 +#10 := (forall (vars (?x1 T1) (?x2 T2)) #9)
    3.99 +#113 := [asserted]: #10
   3.100 +#371 := [rewrite* #113]: #370
   3.101 +#17 := (< #14 #16)
   3.102 +#27 := (ite #17 #19 #26)
   3.103 +#28 := (= #12 #27)
   3.104 +#29 := (forall (vars (?x3 T2)) #28)
   3.105 +#136 := (iff #29 #135)
   3.106 +#133 := (iff #28 #132)
   3.107 +#130 := (= #27 #127)
   3.108 +#118 := (not #119)
   3.109 +#124 := (ite #118 #19 #26)
   3.110 +#128 := (= #124 #127)
   3.111 +#129 := [rewrite]: #128
   3.112 +#125 := (= #27 #124)
   3.113 +#122 := (iff #17 #118)
   3.114 +#123 := [rewrite]: #122
   3.115 +#126 := [monotonicity #123]: #125
   3.116 +#131 := [trans #126 #129]: #130
   3.117 +#134 := [monotonicity #131]: #133
   3.118 +#137 := [quant-intro #134]: #136
   3.119 +#114 := [asserted]: #29
   3.120 +#138 := [mp #114 #137]: #135
   3.121 +#372 := [mp #138 #371]: #369
   3.122 +#367 := [mp~ #372 #366]: #369
   3.123 +#454 := [mp #367 #453]: #451
   3.124 +#550 := [mp #454 #549]: #545
   3.125 +#738 := (not #545)
   3.126 +#928 := (or #738 #925)
   3.127 +#75 := (= #74 uf_7)
   3.128 +#906 := (ite #75 #905 #904)
   3.129 +#907 := (+ #803 #120)
   3.130 +#908 := (>= #907 0::int)
   3.131 +#909 := (ite #908 #906 #903)
   3.132 +#929 := (or #738 #909)
   3.133 +#931 := (iff #929 #928)
   3.134 +#933 := (iff #928 #928)
   3.135 +#934 := [rewrite]: #933
   3.136 +#926 := (iff #909 #925)
   3.137 +#923 := (iff #906 #922)
   3.138 +#283 := (iff #75 #281)
   3.139 +#284 := [rewrite]: #283
   3.140 +#924 := [monotonicity #284]: #923
   3.141 +#920 := (iff #908 #917)
   3.142 +#910 := (+ #120 #803)
   3.143 +#913 := (>= #910 0::int)
   3.144 +#918 := (iff #913 #917)
   3.145 +#919 := [rewrite]: #918
   3.146 +#914 := (iff #908 #913)
   3.147 +#911 := (= #907 #910)
   3.148 +#912 := [rewrite]: #911
   3.149 +#915 := [monotonicity #912]: #914
   3.150 +#921 := [trans #915 #919]: #920
   3.151 +#927 := [monotonicity #921 #924]: #926
   3.152 +#932 := [monotonicity #927]: #931
   3.153 +#935 := [trans #932 #934]: #931
   3.154 +#930 := [quant-inst]: #929
   3.155 +#936 := [mp #930 #935]: #928
   3.156 +#1300 := [unit-resolution #936 #550]: #925
   3.157 +#989 := (= #16 #803)
   3.158 +#1277 := (= #803 #16)
   3.159 +#280 := [asserted]: #75
   3.160 +#287 := [mp #280 #284]: #281
   3.161 +#1276 := [symm #287]: #75
   3.162 +#1278 := [monotonicity #1276]: #1277
   3.163 +#1301 := [symm #1278]: #989
   3.164 +#1302 := (not #989)
   3.165 +#1303 := (or #1302 #917)
   3.166 +#1304 := [th-lemma]: #1303
   3.167 +#1305 := [unit-resolution #1304 #1301]: #917
   3.168 +#950 := (not #917)
   3.169 +#949 := (not #925)
   3.170 +#951 := (or #949 #950 #922)
   3.171 +#952 := [def-axiom]: #951
   3.172 +#1306 := [unit-resolution #952 #1305 #1300]: #922
   3.173 +#937 := (not #922)
   3.174 +#1307 := (or #937 #905)
   3.175 +#938 := (not #281)
   3.176 +#939 := (or #937 #938 #905)
   3.177 +#940 := [def-axiom]: #939
   3.178 +#1308 := [unit-resolution #940 #287]: #1307
   3.179 +#1309 := [unit-resolution #1308 #1306]: #905
   3.180 +#1356 := (= #799 #23)
   3.181 +#800 := (= #23 #799)
   3.182 +decl uf_15 :: T4
   3.183 +#40 := uf_15
   3.184 +#41 := (uf_5 uf_15)
   3.185 +#814 := (+ #41 #813)
   3.186 +#815 := (<= #814 0::int)
   3.187 +#836 := (not #815)
   3.188 +#158 := (* -1::int #41)
   3.189 +#1270 := (+ #16 #158)
   3.190 +#1265 := (>= #1270 0::int)
   3.191 +#1339 := (not #1265)
   3.192 +#1269 := (= #16 #41)
   3.193 +#1298 := (not #1269)
   3.194 +#286 := (= uf_7 uf_15)
   3.195 +#44 := (uf_10 uf_15)
   3.196 +#72 := (uf_6 #44)
   3.197 +#73 := (= #72 uf_15)
   3.198 +#277 := (= uf_15 #72)
   3.199 +#278 := (iff #73 #277)
   3.200 +#279 := [rewrite]: #278
   3.201 +#276 := [asserted]: #73
   3.202 +#282 := [mp #276 #279]: #277
   3.203 +#1274 := [symm #282]: #73
   3.204 +#729 := (= uf_7 #72)
   3.205 +decl uf_17 :: (-> int T4)
   3.206 +#611 := (uf_5 #72)
   3.207 +#991 := (uf_17 #611)
   3.208 +#1289 := (= #991 #72)
   3.209 +#992 := (= #72 #991)
   3.210 +#55 := (:var 0 T4)
   3.211 +#56 := (uf_5 #55)
   3.212 +#574 := (pattern #56)
   3.213 +#57 := (uf_17 #56)
   3.214 +#177 := (= #55 #57)
   3.215 +#575 := (forall (vars (?x7 T4)) (:pat #574) #177)
   3.216 +#195 := (forall (vars (?x7 T4)) #177)
   3.217 +#578 := (iff #195 #575)
   3.218 +#576 := (iff #177 #177)
   3.219 +#577 := [refl]: #576
   3.220 +#579 := [quant-intro #577]: #578
   3.221 +#405 := (~ #195 #195)
   3.222 +#403 := (~ #177 #177)
   3.223 +#404 := [refl]: #403
   3.224 +#406 := [nnf-pos #404]: #405
   3.225 +#58 := (= #57 #55)
   3.226 +#59 := (forall (vars (?x7 T4)) #58)
   3.227 +#196 := (iff #59 #195)
   3.228 +#193 := (iff #58 #177)
   3.229 +#194 := [rewrite]: #193
   3.230 +#197 := [quant-intro #194]: #196
   3.231 +#155 := [asserted]: #59
   3.232 +#200 := [mp #155 #197]: #195
   3.233 +#407 := [mp~ #200 #406]: #195
   3.234 +#580 := [mp #407 #579]: #575
   3.235 +#995 := (not #575)
   3.236 +#996 := (or #995 #992)
   3.237 +#997 := [quant-inst]: #996
   3.238 +#1273 := [unit-resolution #997 #580]: #992
   3.239 +#1290 := [symm #1273]: #1289
   3.240 +#1293 := (= uf_7 #991)
   3.241 +#993 := (uf_17 #803)
   3.242 +#1287 := (= #993 #991)
   3.243 +#1284 := (= #803 #611)
   3.244 +#987 := (= #41 #611)
   3.245 +#1279 := (= #611 #41)
   3.246 +#1280 := [monotonicity #1274]: #1279
   3.247 +#1281 := [symm #1280]: #987
   3.248 +#1282 := (= #803 #41)
   3.249 +#1275 := [hypothesis]: #1269
   3.250 +#1283 := [trans #1278 #1275]: #1282
   3.251 +#1285 := [trans #1283 #1281]: #1284
   3.252 +#1288 := [monotonicity #1285]: #1287
   3.253 +#1291 := (= uf_7 #993)
   3.254 +#994 := (= #74 #993)
   3.255 +#1000 := (or #995 #994)
   3.256 +#1001 := [quant-inst]: #1000
   3.257 +#1286 := [unit-resolution #1001 #580]: #994
   3.258 +#1292 := [trans #287 #1286]: #1291
   3.259 +#1294 := [trans #1292 #1288]: #1293
   3.260 +#1295 := [trans #1294 #1290]: #729
   3.261 +#1296 := [trans #1295 #1274]: #286
   3.262 +#290 := (not #286)
   3.263 +#76 := (= uf_15 uf_7)
   3.264 +#77 := (not #76)
   3.265 +#291 := (iff #77 #290)
   3.266 +#288 := (iff #76 #286)
   3.267 +#289 := [rewrite]: #288
   3.268 +#292 := [monotonicity #289]: #291
   3.269 +#285 := [asserted]: #77
   3.270 +#295 := [mp #285 #292]: #290
   3.271 +#1297 := [unit-resolution #295 #1296]: false
   3.272 +#1299 := [lemma #1297]: #1298
   3.273 +#1342 := (or #1269 #1339)
   3.274 +#1271 := (<= #1270 0::int)
   3.275 +#621 := (* -1::int #611)
   3.276 +#723 := (+ #16 #621)
   3.277 +#724 := (<= #723 0::int)
   3.278 +decl uf_12 :: T1
   3.279 +#30 := uf_12
   3.280 +#88 := (uf_2 uf_12)
   3.281 +#771 := (uf_1 #88 #44)
   3.282 +#45 := (uf_1 uf_9 #44)
   3.283 +#772 := (= #45 #771)
   3.284 +#796 := (not #772)
   3.285 +decl uf_14 :: T1
   3.286 +#38 := uf_14
   3.287 +#83 := (uf_2 uf_14)
   3.288 +#656 := (uf_1 #83 #44)
   3.289 +#1239 := (= #656 #771)
   3.290 +#1252 := (not #1239)
   3.291 +#1324 := (iff #1252 #796)
   3.292 +#1322 := (iff #1239 #772)
   3.293 +#1320 := (= #656 #45)
   3.294 +#661 := (= #45 #656)
   3.295 +#659 := (uf_1 uf_11 #44)
   3.296 +#664 := (= #656 #659)
   3.297 +#667 := (ite #277 #661 #664)
   3.298 +#657 := (uf_1 uf_8 #44)
   3.299 +#670 := (= #656 #657)
   3.300 +#622 := (+ #41 #621)
   3.301 +#623 := (<= #622 0::int)
   3.302 +#673 := (ite #623 #667 #670)
   3.303 +#84 := (uf_1 #83 #6)
   3.304 +#560 := (pattern #84)
   3.305 +#467 := (= #19 #84)
   3.306 +#465 := (= #25 #84)
   3.307 +#464 := (= #45 #84)
   3.308 +#43 := (= #13 uf_15)
   3.309 +#466 := (ite #43 #464 #465)
   3.310 +#159 := (+ #14 #158)
   3.311 +#157 := (>= #159 0::int)
   3.312 +#468 := (ite #157 #466 #467)
   3.313 +#561 := (forall (vars (?x5 T2)) (:pat #541 #560 #543 #544) #468)
   3.314 +#471 := (forall (vars (?x5 T2)) #468)
   3.315 +#564 := (iff #471 #561)
   3.316 +#562 := (iff #468 #468)
   3.317 +#563 := [refl]: #562
   3.318 +#565 := [quant-intro #563]: #564
   3.319 +#46 := (ite #43 #45 #25)
   3.320 +#165 := (ite #157 #46 #19)
   3.321 +#378 := (= #84 #165)
   3.322 +#379 := (forall (vars (?x5 T2)) #378)
   3.323 +#472 := (iff #379 #471)
   3.324 +#469 := (iff #378 #468)
   3.325 +#470 := [rewrite]: #469
   3.326 +#473 := [quant-intro #470]: #472
   3.327 +#359 := (~ #379 #379)
   3.328 +#361 := (~ #378 #378)
   3.329 +#358 := [refl]: #361
   3.330 +#356 := [nnf-pos #358]: #359
   3.331 +#39 := (uf_3 uf_14 #6)
   3.332 +#170 := (= #39 #165)
   3.333 +#173 := (forall (vars (?x5 T2)) #170)
   3.334 +#380 := (iff #173 #379)
   3.335 +#381 := [rewrite* #113]: #380
   3.336 +#42 := (< #14 #41)
   3.337 +#47 := (ite #42 #19 #46)
   3.338 +#48 := (= #39 #47)
   3.339 +#49 := (forall (vars (?x5 T2)) #48)
   3.340 +#174 := (iff #49 #173)
   3.341 +#171 := (iff #48 #170)
   3.342 +#168 := (= #47 #165)
   3.343 +#156 := (not #157)
   3.344 +#162 := (ite #156 #19 #46)
   3.345 +#166 := (= #162 #165)
   3.346 +#167 := [rewrite]: #166
   3.347 +#163 := (= #47 #162)
   3.348 +#160 := (iff #42 #156)
   3.349 +#161 := [rewrite]: #160
   3.350 +#164 := [monotonicity #161]: #163
   3.351 +#169 := [trans #164 #167]: #168
   3.352 +#172 := [monotonicity #169]: #171
   3.353 +#175 := [quant-intro #172]: #174
   3.354 +#116 := [asserted]: #49
   3.355 +#176 := [mp #116 #175]: #173
   3.356 +#382 := [mp #176 #381]: #379
   3.357 +#357 := [mp~ #382 #356]: #379
   3.358 +#474 := [mp #357 #473]: #471
   3.359 +#566 := [mp #474 #565]: #561
   3.360 +#676 := (not #561)
   3.361 +#677 := (or #676 #673)
   3.362 +#658 := (= #657 #656)
   3.363 +#660 := (= #659 #656)
   3.364 +#662 := (ite #73 #661 #660)
   3.365 +#612 := (+ #611 #158)
   3.366 +#613 := (>= #612 0::int)
   3.367 +#663 := (ite #613 #662 #658)
   3.368 +#678 := (or #676 #663)
   3.369 +#680 := (iff #678 #677)
   3.370 +#682 := (iff #677 #677)
   3.371 +#683 := [rewrite]: #682
   3.372 +#674 := (iff #663 #673)
   3.373 +#671 := (iff #658 #670)
   3.374 +#672 := [rewrite]: #671
   3.375 +#668 := (iff #662 #667)
   3.376 +#665 := (iff #660 #664)
   3.377 +#666 := [rewrite]: #665
   3.378 +#669 := [monotonicity #279 #666]: #668
   3.379 +#626 := (iff #613 #623)
   3.380 +#615 := (+ #158 #611)
   3.381 +#618 := (>= #615 0::int)
   3.382 +#624 := (iff #618 #623)
   3.383 +#625 := [rewrite]: #624
   3.384 +#619 := (iff #613 #618)
   3.385 +#616 := (= #612 #615)
   3.386 +#617 := [rewrite]: #616
   3.387 +#620 := [monotonicity #617]: #619
   3.388 +#627 := [trans #620 #625]: #626
   3.389 +#675 := [monotonicity #627 #669 #672]: #674
   3.390 +#681 := [monotonicity #675]: #680
   3.391 +#684 := [trans #681 #683]: #680
   3.392 +#679 := [quant-inst]: #678
   3.393 +#685 := [mp #679 #684]: #677
   3.394 +#1311 := [unit-resolution #685 #566]: #673
   3.395 +#1312 := (not #987)
   3.396 +#1313 := (or #1312 #623)
   3.397 +#1314 := [th-lemma]: #1313
   3.398 +#1315 := [unit-resolution #1314 #1281]: #623
   3.399 +#645 := (not #623)
   3.400 +#698 := (not #673)
   3.401 +#699 := (or #698 #645 #667)
   3.402 +#700 := [def-axiom]: #699
   3.403 +#1316 := [unit-resolution #700 #1315 #1311]: #667
   3.404 +#686 := (not #667)
   3.405 +#1317 := (or #686 #661)
   3.406 +#687 := (not #277)
   3.407 +#688 := (or #686 #687 #661)
   3.408 +#689 := [def-axiom]: #688
   3.409 +#1318 := [unit-resolution #689 #282]: #1317
   3.410 +#1319 := [unit-resolution #1318 #1316]: #661
   3.411 +#1321 := [symm #1319]: #1320
   3.412 +#1323 := [monotonicity #1321]: #1322
   3.413 +#1325 := [monotonicity #1323]: #1324
   3.414 +#1145 := (* -1::real #771)
   3.415 +#1240 := (+ #656 #1145)
   3.416 +#1241 := (<= #1240 0::real)
   3.417 +#1249 := (not #1241)
   3.418 +#1243 := [hypothesis]: #1241
   3.419 +decl uf_18 :: T3
   3.420 +#80 := uf_18
   3.421 +#1040 := (uf_1 uf_18 #44)
   3.422 +#1043 := (* -1::real #1040)
   3.423 +#1156 := (+ #771 #1043)
   3.424 +#1157 := (>= #1156 0::real)
   3.425 +#1189 := (not #1157)
   3.426 +#708 := (uf_1 #91 #44)
   3.427 +#1168 := (+ #708 #1043)
   3.428 +#1169 := (<= #1168 0::real)
   3.429 +#1174 := (or #1157 #1169)
   3.430 +#1177 := (not #1174)
   3.431 +#89 := (uf_1 #88 #6)
   3.432 +#552 := (pattern #89)
   3.433 +#81 := (uf_1 uf_18 #6)
   3.434 +#594 := (pattern #81)
   3.435 +#324 := (* -1::real #92)
   3.436 +#325 := (+ #81 #324)
   3.437 +#323 := (>= #325 0::real)
   3.438 +#317 := (* -1::real #89)
   3.439 +#318 := (+ #81 #317)
   3.440 +#319 := (<= #318 0::real)
   3.441 +#436 := (or #319 #323)
   3.442 +#437 := (not #436)
   3.443 +#601 := (forall (vars (?x11 T2)) (:pat #594 #552 #542) #437)
   3.444 +#440 := (forall (vars (?x11 T2)) #437)
   3.445 +#604 := (iff #440 #601)
   3.446 +#602 := (iff #437 #437)
   3.447 +#603 := [refl]: #602
   3.448 +#605 := [quant-intro #603]: #604
   3.449 +#326 := (not #323)
   3.450 +#320 := (not #319)
   3.451 +#329 := (and #320 #326)
   3.452 +#332 := (forall (vars (?x11 T2)) #329)
   3.453 +#441 := (iff #332 #440)
   3.454 +#438 := (iff #329 #437)
   3.455 +#439 := [rewrite]: #438
   3.456 +#442 := [quant-intro #439]: #441
   3.457 +#425 := (~ #332 #332)
   3.458 +#423 := (~ #329 #329)
   3.459 +#424 := [refl]: #423
   3.460 +#426 := [nnf-pos #424]: #425
   3.461 +#306 := (* -1::real #84)
   3.462 +#307 := (+ #81 #306)
   3.463 +#305 := (>= #307 0::real)
   3.464 +#308 := (not #305)
   3.465 +#301 := (* -1::real #81)
   3.466 +#79 := (uf_1 #78 #6)
   3.467 +#302 := (+ #79 #301)
   3.468 +#300 := (>= #302 0::real)
   3.469 +#298 := (not #300)
   3.470 +#311 := (and #298 #308)
   3.471 +#314 := (forall (vars (?x10 T2)) #311)
   3.472 +#335 := (and #314 #332)
   3.473 +#93 := (< #81 #92)
   3.474 +#90 := (< #89 #81)
   3.475 +#94 := (and #90 #93)
   3.476 +#95 := (forall (vars (?x11 T2)) #94)
   3.477 +#85 := (< #81 #84)
   3.478 +#82 := (< #79 #81)
   3.479 +#86 := (and #82 #85)
   3.480 +#87 := (forall (vars (?x10 T2)) #86)
   3.481 +#96 := (and #87 #95)
   3.482 +#336 := (iff #96 #335)
   3.483 +#333 := (iff #95 #332)
   3.484 +#330 := (iff #94 #329)
   3.485 +#327 := (iff #93 #326)
   3.486 +#328 := [rewrite]: #327
   3.487 +#321 := (iff #90 #320)
   3.488 +#322 := [rewrite]: #321
   3.489 +#331 := [monotonicity #322 #328]: #330
   3.490 +#334 := [quant-intro #331]: #333
   3.491 +#315 := (iff #87 #314)
   3.492 +#312 := (iff #86 #311)
   3.493 +#309 := (iff #85 #308)
   3.494 +#310 := [rewrite]: #309
   3.495 +#303 := (iff #82 #298)
   3.496 +#304 := [rewrite]: #303
   3.497 +#313 := [monotonicity #304 #310]: #312
   3.498 +#316 := [quant-intro #313]: #315
   3.499 +#337 := [monotonicity #316 #334]: #336
   3.500 +#293 := [asserted]: #96
   3.501 +#338 := [mp #293 #337]: #335
   3.502 +#340 := [and-elim #338]: #332
   3.503 +#427 := [mp~ #340 #426]: #332
   3.504 +#443 := [mp #427 #442]: #440
   3.505 +#606 := [mp #443 #605]: #601
   3.506 +#1124 := (not #601)
   3.507 +#1180 := (or #1124 #1177)
   3.508 +#1142 := (* -1::real #708)
   3.509 +#1143 := (+ #1040 #1142)
   3.510 +#1144 := (>= #1143 0::real)
   3.511 +#1146 := (+ #1040 #1145)
   3.512 +#1147 := (<= #1146 0::real)
   3.513 +#1148 := (or #1147 #1144)
   3.514 +#1149 := (not #1148)
   3.515 +#1181 := (or #1124 #1149)
   3.516 +#1183 := (iff #1181 #1180)
   3.517 +#1185 := (iff #1180 #1180)
   3.518 +#1186 := [rewrite]: #1185
   3.519 +#1178 := (iff #1149 #1177)
   3.520 +#1175 := (iff #1148 #1174)
   3.521 +#1172 := (iff #1144 #1169)
   3.522 +#1162 := (+ #1142 #1040)
   3.523 +#1165 := (>= #1162 0::real)
   3.524 +#1170 := (iff #1165 #1169)
   3.525 +#1171 := [rewrite]: #1170
   3.526 +#1166 := (iff #1144 #1165)
   3.527 +#1163 := (= #1143 #1162)
   3.528 +#1164 := [rewrite]: #1163
   3.529 +#1167 := [monotonicity #1164]: #1166
   3.530 +#1173 := [trans #1167 #1171]: #1172
   3.531 +#1160 := (iff #1147 #1157)
   3.532 +#1150 := (+ #1145 #1040)
   3.533 +#1153 := (<= #1150 0::real)
   3.534 +#1158 := (iff #1153 #1157)
   3.535 +#1159 := [rewrite]: #1158
   3.536 +#1154 := (iff #1147 #1153)
   3.537 +#1151 := (= #1146 #1150)
   3.538 +#1152 := [rewrite]: #1151
   3.539 +#1155 := [monotonicity #1152]: #1154
   3.540 +#1161 := [trans #1155 #1159]: #1160
   3.541 +#1176 := [monotonicity #1161 #1173]: #1175
   3.542 +#1179 := [monotonicity #1176]: #1178
   3.543 +#1184 := [monotonicity #1179]: #1183
   3.544 +#1187 := [trans #1184 #1186]: #1183
   3.545 +#1182 := [quant-inst]: #1181
   3.546 +#1188 := [mp #1182 #1187]: #1180
   3.547 +#1244 := [unit-resolution #1188 #606]: #1177
   3.548 +#1190 := (or #1174 #1189)
   3.549 +#1191 := [def-axiom]: #1190
   3.550 +#1245 := [unit-resolution #1191 #1244]: #1189
   3.551 +#1054 := (+ #656 #1043)
   3.552 +#1055 := (<= #1054 0::real)
   3.553 +#1079 := (not #1055)
   3.554 +#607 := (uf_1 #78 #44)
   3.555 +#1044 := (+ #607 #1043)
   3.556 +#1045 := (>= #1044 0::real)
   3.557 +#1060 := (or #1045 #1055)
   3.558 +#1063 := (not #1060)
   3.559 +#567 := (pattern #79)
   3.560 +#428 := (or #300 #305)
   3.561 +#429 := (not #428)
   3.562 +#595 := (forall (vars (?x10 T2)) (:pat #567 #594 #560) #429)
   3.563 +#432 := (forall (vars (?x10 T2)) #429)
   3.564 +#598 := (iff #432 #595)
   3.565 +#596 := (iff #429 #429)
   3.566 +#597 := [refl]: #596
   3.567 +#599 := [quant-intro #597]: #598
   3.568 +#433 := (iff #314 #432)
   3.569 +#430 := (iff #311 #429)
   3.570 +#431 := [rewrite]: #430
   3.571 +#434 := [quant-intro #431]: #433
   3.572 +#420 := (~ #314 #314)
   3.573 +#418 := (~ #311 #311)
   3.574 +#419 := [refl]: #418
   3.575 +#421 := [nnf-pos #419]: #420
   3.576 +#339 := [and-elim #338]: #314
   3.577 +#422 := [mp~ #339 #421]: #314
   3.578 +#435 := [mp #422 #434]: #432
   3.579 +#600 := [mp #435 #599]: #595
   3.580 +#1066 := (not #595)
   3.581 +#1067 := (or #1066 #1063)
   3.582 +#1039 := (* -1::real #656)
   3.583 +#1041 := (+ #1040 #1039)
   3.584 +#1042 := (>= #1041 0::real)
   3.585 +#1046 := (or #1045 #1042)
   3.586 +#1047 := (not #1046)
   3.587 +#1068 := (or #1066 #1047)
   3.588 +#1070 := (iff #1068 #1067)
   3.589 +#1072 := (iff #1067 #1067)
   3.590 +#1073 := [rewrite]: #1072
   3.591 +#1064 := (iff #1047 #1063)
   3.592 +#1061 := (iff #1046 #1060)
   3.593 +#1058 := (iff #1042 #1055)
   3.594 +#1048 := (+ #1039 #1040)
   3.595 +#1051 := (>= #1048 0::real)
   3.596 +#1056 := (iff #1051 #1055)
   3.597 +#1057 := [rewrite]: #1056
   3.598 +#1052 := (iff #1042 #1051)
   3.599 +#1049 := (= #1041 #1048)
   3.600 +#1050 := [rewrite]: #1049
   3.601 +#1053 := [monotonicity #1050]: #1052
   3.602 +#1059 := [trans #1053 #1057]: #1058
   3.603 +#1062 := [monotonicity #1059]: #1061
   3.604 +#1065 := [monotonicity #1062]: #1064
   3.605 +#1071 := [monotonicity #1065]: #1070
   3.606 +#1074 := [trans #1071 #1073]: #1070
   3.607 +#1069 := [quant-inst]: #1068
   3.608 +#1075 := [mp #1069 #1074]: #1067
   3.609 +#1246 := [unit-resolution #1075 #600]: #1063
   3.610 +#1080 := (or #1060 #1079)
   3.611 +#1081 := [def-axiom]: #1080
   3.612 +#1247 := [unit-resolution #1081 #1246]: #1079
   3.613 +#1248 := [th-lemma #1247 #1245 #1243]: false
   3.614 +#1250 := [lemma #1248]: #1249
   3.615 +#1253 := (or #1252 #1241)
   3.616 +#1254 := [th-lemma]: #1253
   3.617 +#1310 := [unit-resolution #1254 #1250]: #1252
   3.618 +#1326 := [mp #1310 #1325]: #796
   3.619 +#1328 := (or #724 #772)
   3.620 +decl uf_13 :: T3
   3.621 +#33 := uf_13
   3.622 +#609 := (uf_1 uf_13 #44)
   3.623 +#773 := (= #609 #771)
   3.624 +#775 := (ite #724 #773 #772)
   3.625 +#32 := (uf_1 uf_9 #6)
   3.626 +#553 := (pattern #32)
   3.627 +#34 := (uf_1 uf_13 #6)
   3.628 +#551 := (pattern #34)
   3.629 +#456 := (= #32 #89)
   3.630 +#455 := (= #34 #89)
   3.631 +#457 := (ite #119 #455 #456)
   3.632 +#554 := (forall (vars (?x4 T2)) (:pat #541 #551 #552 #553) #457)
   3.633 +#460 := (forall (vars (?x4 T2)) #457)
   3.634 +#557 := (iff #460 #554)
   3.635 +#555 := (iff #457 #457)
   3.636 +#556 := [refl]: #555
   3.637 +#558 := [quant-intro #556]: #557
   3.638 +#143 := (ite #119 #34 #32)
   3.639 +#373 := (= #89 #143)
   3.640 +#374 := (forall (vars (?x4 T2)) #373)
   3.641 +#461 := (iff #374 #460)
   3.642 +#458 := (iff #373 #457)
   3.643 +#459 := [rewrite]: #458
   3.644 +#462 := [quant-intro #459]: #461
   3.645 +#362 := (~ #374 #374)
   3.646 +#364 := (~ #373 #373)
   3.647 +#365 := [refl]: #364
   3.648 +#363 := [nnf-pos #365]: #362
   3.649 +#31 := (uf_3 uf_12 #6)
   3.650 +#148 := (= #31 #143)
   3.651 +#151 := (forall (vars (?x4 T2)) #148)
   3.652 +#375 := (iff #151 #374)
   3.653 +#376 := [rewrite* #113]: #375
   3.654 +#35 := (ite #17 #32 #34)
   3.655 +#36 := (= #31 #35)
   3.656 +#37 := (forall (vars (?x4 T2)) #36)
   3.657 +#152 := (iff #37 #151)
   3.658 +#149 := (iff #36 #148)
   3.659 +#146 := (= #35 #143)
   3.660 +#140 := (ite #118 #32 #34)
   3.661 +#144 := (= #140 #143)
   3.662 +#145 := [rewrite]: #144
   3.663 +#141 := (= #35 #140)
   3.664 +#142 := [monotonicity #123]: #141
   3.665 +#147 := [trans #142 #145]: #146
   3.666 +#150 := [monotonicity #147]: #149
   3.667 +#153 := [quant-intro #150]: #152
   3.668 +#115 := [asserted]: #37
   3.669 +#154 := [mp #115 #153]: #151
   3.670 +#377 := [mp #154 #376]: #374
   3.671 +#360 := [mp~ #377 #363]: #374
   3.672 +#463 := [mp #360 #462]: #460
   3.673 +#559 := [mp #463 #558]: #554
   3.674 +#778 := (not #554)
   3.675 +#779 := (or #778 #775)
   3.676 +#714 := (+ #611 #120)
   3.677 +#715 := (>= #714 0::int)
   3.678 +#774 := (ite #715 #773 #772)
   3.679 +#780 := (or #778 #774)
   3.680 +#782 := (iff #780 #779)
   3.681 +#784 := (iff #779 #779)
   3.682 +#785 := [rewrite]: #784
   3.683 +#776 := (iff #774 #775)
   3.684 +#727 := (iff #715 #724)
   3.685 +#717 := (+ #120 #611)
   3.686 +#720 := (>= #717 0::int)
   3.687 +#725 := (iff #720 #724)
   3.688 +#726 := [rewrite]: #725
   3.689 +#721 := (iff #715 #720)
   3.690 +#718 := (= #714 #717)
   3.691 +#719 := [rewrite]: #718
   3.692 +#722 := [monotonicity #719]: #721
   3.693 +#728 := [trans #722 #726]: #727
   3.694 +#777 := [monotonicity #728]: #776
   3.695 +#783 := [monotonicity #777]: #782
   3.696 +#786 := [trans #783 #785]: #782
   3.697 +#781 := [quant-inst]: #780
   3.698 +#787 := [mp #781 #786]: #779
   3.699 +#1327 := [unit-resolution #787 #559]: #775
   3.700 +#788 := (not #775)
   3.701 +#791 := (or #788 #724 #772)
   3.702 +#792 := [def-axiom]: #791
   3.703 +#1329 := [unit-resolution #792 #1327]: #1328
   3.704 +#1330 := [unit-resolution #1329 #1326]: #724
   3.705 +#988 := (>= #622 0::int)
   3.706 +#1331 := (or #1312 #988)
   3.707 +#1332 := [th-lemma]: #1331
   3.708 +#1333 := [unit-resolution #1332 #1281]: #988
   3.709 +#761 := (not #724)
   3.710 +#1334 := (not #988)
   3.711 +#1335 := (or #1271 #1334 #761)
   3.712 +#1336 := [th-lemma]: #1335
   3.713 +#1337 := [unit-resolution #1336 #1333 #1330]: #1271
   3.714 +#1338 := (not #1271)
   3.715 +#1340 := (or #1269 #1338 #1339)
   3.716 +#1341 := [th-lemma]: #1340
   3.717 +#1343 := [unit-resolution #1341 #1337]: #1342
   3.718 +#1344 := [unit-resolution #1343 #1299]: #1339
   3.719 +#990 := (>= #916 0::int)
   3.720 +#1345 := (or #1302 #990)
   3.721 +#1346 := [th-lemma]: #1345
   3.722 +#1347 := [unit-resolution #1346 #1301]: #990
   3.723 +#1348 := (not #990)
   3.724 +#1349 := (or #836 #1348 #1265)
   3.725 +#1350 := [th-lemma]: #1349
   3.726 +#1351 := [unit-resolution #1350 #1347 #1344]: #836
   3.727 +#1353 := (or #815 #800)
   3.728 +#801 := (uf_1 uf_13 #22)
   3.729 +#820 := (= #799 #801)
   3.730 +#823 := (ite #815 #820 #800)
   3.731 +#476 := (= #32 #79)
   3.732 +#475 := (= #34 #79)
   3.733 +#477 := (ite #157 #475 #476)
   3.734 +#568 := (forall (vars (?x6 T2)) (:pat #541 #551 #567 #553) #477)
   3.735 +#480 := (forall (vars (?x6 T2)) #477)
   3.736 +#571 := (iff #480 #568)
   3.737 +#569 := (iff #477 #477)
   3.738 +#570 := [refl]: #569
   3.739 +#572 := [quant-intro #570]: #571
   3.740 +#181 := (ite #157 #34 #32)
   3.741 +#383 := (= #79 #181)
   3.742 +#384 := (forall (vars (?x6 T2)) #383)
   3.743 +#481 := (iff #384 #480)
   3.744 +#478 := (iff #383 #477)
   3.745 +#479 := [rewrite]: #478
   3.746 +#482 := [quant-intro #479]: #481
   3.747 +#352 := (~ #384 #384)
   3.748 +#354 := (~ #383 #383)
   3.749 +#355 := [refl]: #354
   3.750 +#353 := [nnf-pos #355]: #352
   3.751 +#51 := (uf_3 uf_16 #6)
   3.752 +#186 := (= #51 #181)
   3.753 +#189 := (forall (vars (?x6 T2)) #186)
   3.754 +#385 := (iff #189 #384)
   3.755 +#386 := [rewrite* #113]: #385
   3.756 +#52 := (ite #42 #32 #34)
   3.757 +#53 := (= #51 #52)
   3.758 +#54 := (forall (vars (?x6 T2)) #53)
   3.759 +#190 := (iff #54 #189)
   3.760 +#187 := (iff #53 #186)
   3.761 +#184 := (= #52 #181)
   3.762 +#178 := (ite #156 #32 #34)
   3.763 +#182 := (= #178 #181)
   3.764 +#183 := [rewrite]: #182
   3.765 +#179 := (= #52 #178)
   3.766 +#180 := [monotonicity #161]: #179
   3.767 +#185 := [trans #180 #183]: #184
   3.768 +#188 := [monotonicity #185]: #187
   3.769 +#191 := [quant-intro #188]: #190
   3.770 +#139 := [asserted]: #54
   3.771 +#192 := [mp #139 #191]: #189
   3.772 +#387 := [mp #192 #386]: #384
   3.773 +#402 := [mp~ #387 #353]: #384
   3.774 +#483 := [mp #402 #482]: #480
   3.775 +#573 := [mp #483 #572]: #568
   3.776 +#634 := (not #568)
   3.777 +#826 := (or #634 #823)
   3.778 +#802 := (= #801 #799)
   3.779 +#804 := (+ #803 #158)
   3.780 +#805 := (>= #804 0::int)
   3.781 +#806 := (ite #805 #802 #800)
   3.782 +#827 := (or #634 #806)
   3.783 +#829 := (iff #827 #826)
   3.784 +#831 := (iff #826 #826)
   3.785 +#832 := [rewrite]: #831
   3.786 +#824 := (iff #806 #823)
   3.787 +#821 := (iff #802 #820)
   3.788 +#822 := [rewrite]: #821
   3.789 +#818 := (iff #805 #815)
   3.790 +#807 := (+ #158 #803)
   3.791 +#810 := (>= #807 0::int)
   3.792 +#816 := (iff #810 #815)
   3.793 +#817 := [rewrite]: #816
   3.794 +#811 := (iff #805 #810)
   3.795 +#808 := (= #804 #807)
   3.796 +#809 := [rewrite]: #808
   3.797 +#812 := [monotonicity #809]: #811
   3.798 +#819 := [trans #812 #817]: #818
   3.799 +#825 := [monotonicity #819 #822]: #824
   3.800 +#830 := [monotonicity #825]: #829
   3.801 +#833 := [trans #830 #832]: #829
   3.802 +#828 := [quant-inst]: #827
   3.803 +#834 := [mp #828 #833]: #826
   3.804 +#1352 := [unit-resolution #834 #573]: #823
   3.805 +#835 := (not #823)
   3.806 +#839 := (or #835 #815 #800)
   3.807 +#840 := [def-axiom]: #839
   3.808 +#1354 := [unit-resolution #840 #1352]: #1353
   3.809 +#1355 := [unit-resolution #1354 #1351]: #800
   3.810 +#1357 := [symm #1355]: #1356
   3.811 +#1358 := [trans #1357 #1309]: #1266
   3.812 +#1359 := (not #1266)
   3.813 +#1360 := (or #1359 #1272)
   3.814 +#1361 := [th-lemma]: #1360
   3.815 +#1362 := [unit-resolution #1361 #1358]: #1272
   3.816 +#1085 := (uf_1 uf_18 #22)
   3.817 +#1099 := (* -1::real #1085)
   3.818 +#1112 := (+ #902 #1099)
   3.819 +#1113 := (<= #1112 0::real)
   3.820 +#1137 := (not #1113)
   3.821 +#960 := (uf_1 #88 #22)
   3.822 +#1100 := (+ #960 #1099)
   3.823 +#1101 := (>= #1100 0::real)
   3.824 +#1118 := (or #1101 #1113)
   3.825 +#1121 := (not #1118)
   3.826 +#1125 := (or #1124 #1121)
   3.827 +#1086 := (+ #1085 #1084)
   3.828 +#1087 := (>= #1086 0::real)
   3.829 +#1088 := (* -1::real #960)
   3.830 +#1089 := (+ #1085 #1088)
   3.831 +#1090 := (<= #1089 0::real)
   3.832 +#1091 := (or #1090 #1087)
   3.833 +#1092 := (not #1091)
   3.834 +#1126 := (or #1124 #1092)
   3.835 +#1128 := (iff #1126 #1125)
   3.836 +#1130 := (iff #1125 #1125)
   3.837 +#1131 := [rewrite]: #1130
   3.838 +#1122 := (iff #1092 #1121)
   3.839 +#1119 := (iff #1091 #1118)
   3.840 +#1116 := (iff #1087 #1113)
   3.841 +#1106 := (+ #1084 #1085)
   3.842 +#1109 := (>= #1106 0::real)
   3.843 +#1114 := (iff #1109 #1113)
   3.844 +#1115 := [rewrite]: #1114
   3.845 +#1110 := (iff #1087 #1109)
   3.846 +#1107 := (= #1086 #1106)
   3.847 +#1108 := [rewrite]: #1107
   3.848 +#1111 := [monotonicity #1108]: #1110
   3.849 +#1117 := [trans #1111 #1115]: #1116
   3.850 +#1104 := (iff #1090 #1101)
   3.851 +#1093 := (+ #1088 #1085)
   3.852 +#1096 := (<= #1093 0::real)
   3.853 +#1102 := (iff #1096 #1101)
   3.854 +#1103 := [rewrite]: #1102
   3.855 +#1097 := (iff #1090 #1096)
   3.856 +#1094 := (= #1089 #1093)
   3.857 +#1095 := [rewrite]: #1094
   3.858 +#1098 := [monotonicity #1095]: #1097
   3.859 +#1105 := [trans #1098 #1103]: #1104
   3.860 +#1120 := [monotonicity #1105 #1117]: #1119
   3.861 +#1123 := [monotonicity #1120]: #1122
   3.862 +#1129 := [monotonicity #1123]: #1128
   3.863 +#1132 := [trans #1129 #1131]: #1128
   3.864 +#1127 := [quant-inst]: #1126
   3.865 +#1133 := [mp #1127 #1132]: #1125
   3.866 +#1363 := [unit-resolution #1133 #606]: #1121
   3.867 +#1138 := (or #1118 #1137)
   3.868 +#1139 := [def-axiom]: #1138
   3.869 +#1364 := [unit-resolution #1139 #1363]: #1137
   3.870 +#1200 := (+ #799 #1099)
   3.871 +#1201 := (>= #1200 0::real)
   3.872 +#1231 := (not #1201)
   3.873 +#847 := (uf_1 #83 #22)
   3.874 +#1210 := (+ #847 #1099)
   3.875 +#1211 := (<= #1210 0::real)
   3.876 +#1216 := (or #1201 #1211)
   3.877 +#1219 := (not #1216)
   3.878 +#1222 := (or #1066 #1219)
   3.879 +#1197 := (* -1::real #847)
   3.880 +#1198 := (+ #1085 #1197)
   3.881 +#1199 := (>= #1198 0::real)
   3.882 +#1202 := (or #1201 #1199)
   3.883 +#1203 := (not #1202)
   3.884 +#1223 := (or #1066 #1203)
   3.885 +#1225 := (iff #1223 #1222)
   3.886 +#1227 := (iff #1222 #1222)
   3.887 +#1228 := [rewrite]: #1227
   3.888 +#1220 := (iff #1203 #1219)
   3.889 +#1217 := (iff #1202 #1216)
   3.890 +#1214 := (iff #1199 #1211)
   3.891 +#1204 := (+ #1197 #1085)
   3.892 +#1207 := (>= #1204 0::real)
   3.893 +#1212 := (iff #1207 #1211)
   3.894 +#1213 := [rewrite]: #1212
   3.895 +#1208 := (iff #1199 #1207)
   3.896 +#1205 := (= #1198 #1204)
   3.897 +#1206 := [rewrite]: #1205
   3.898 +#1209 := [monotonicity #1206]: #1208
   3.899 +#1215 := [trans #1209 #1213]: #1214
   3.900 +#1218 := [monotonicity #1215]: #1217
   3.901 +#1221 := [monotonicity #1218]: #1220
   3.902 +#1226 := [monotonicity #1221]: #1225
   3.903 +#1229 := [trans #1226 #1228]: #1225
   3.904 +#1224 := [quant-inst]: #1223
   3.905 +#1230 := [mp #1224 #1229]: #1222
   3.906 +#1365 := [unit-resolution #1230 #600]: #1219
   3.907 +#1232 := (or #1216 #1231)
   3.908 +#1233 := [def-axiom]: #1232
   3.909 +#1366 := [unit-resolution #1233 #1365]: #1231
   3.910 +[th-lemma #1366 #1364 #1362]: false
   3.911 +unsat
   3.912 +NQHwTeL311Tq3wf2s5BReA 419 0
   3.913 +#2 := false
   3.914 +#194 := 0::real
   3.915 +decl uf_4 :: (-> T2 T3 real)
   3.916 +decl uf_6 :: (-> T1 T3)
   3.917 +decl uf_3 :: T1
   3.918 +#21 := uf_3
   3.919 +#25 := (uf_6 uf_3)
   3.920 +decl uf_5 :: T2
   3.921 +#24 := uf_5
   3.922 +#26 := (uf_4 uf_5 #25)
   3.923 +decl uf_7 :: T2
   3.924 +#27 := uf_7
   3.925 +#28 := (uf_4 uf_7 #25)
   3.926 +decl uf_10 :: T1
   3.927 +#38 := uf_10
   3.928 +#42 := (uf_6 uf_10)
   3.929 +decl uf_9 :: T2
   3.930 +#33 := uf_9
   3.931 +#43 := (uf_4 uf_9 #42)
   3.932 +#41 := (= uf_3 uf_10)
   3.933 +#44 := (ite #41 #43 #28)
   3.934 +#9 := 0::int
   3.935 +decl uf_2 :: (-> T1 int)
   3.936 +#39 := (uf_2 uf_10)
   3.937 +#226 := -1::int
   3.938 +#229 := (* -1::int #39)
   3.939 +#22 := (uf_2 uf_3)
   3.940 +#230 := (+ #22 #229)
   3.941 +#228 := (>= #230 0::int)
   3.942 +#236 := (ite #228 #44 #26)
   3.943 +#192 := -1::real
   3.944 +#244 := (* -1::real #236)
   3.945 +#642 := (+ #26 #244)
   3.946 +#643 := (<= #642 0::real)
   3.947 +#567 := (= #26 #236)
   3.948 +#227 := (not #228)
   3.949 +decl uf_1 :: (-> int T1)
   3.950 +#593 := (uf_1 #39)
   3.951 +#660 := (= #593 uf_10)
   3.952 +#594 := (= uf_10 #593)
   3.953 +#4 := (:var 0 T1)
   3.954 +#5 := (uf_2 #4)
   3.955 +#546 := (pattern #5)
   3.956 +#6 := (uf_1 #5)
   3.957 +#93 := (= #4 #6)
   3.958 +#547 := (forall (vars (?x1 T1)) (:pat #546) #93)
   3.959 +#96 := (forall (vars (?x1 T1)) #93)
   3.960 +#550 := (iff #96 #547)
   3.961 +#548 := (iff #93 #93)
   3.962 +#549 := [refl]: #548
   3.963 +#551 := [quant-intro #549]: #550
   3.964 +#448 := (~ #96 #96)
   3.965 +#450 := (~ #93 #93)
   3.966 +#451 := [refl]: #450
   3.967 +#449 := [nnf-pos #451]: #448
   3.968 +#7 := (= #6 #4)
   3.969 +#8 := (forall (vars (?x1 T1)) #7)
   3.970 +#97 := (iff #8 #96)
   3.971 +#94 := (iff #7 #93)
   3.972 +#95 := [rewrite]: #94
   3.973 +#98 := [quant-intro #95]: #97
   3.974 +#92 := [asserted]: #8
   3.975 +#101 := [mp #92 #98]: #96
   3.976 +#446 := [mp~ #101 #449]: #96
   3.977 +#552 := [mp #446 #551]: #547
   3.978 +#595 := (not #547)
   3.979 +#600 := (or #595 #594)
   3.980 +#601 := [quant-inst]: #600
   3.981 +#654 := [unit-resolution #601 #552]: #594
   3.982 +#680 := [symm #654]: #660
   3.983 +#681 := (= uf_3 #593)
   3.984 +#591 := (uf_1 #22)
   3.985 +#658 := (= #591 #593)
   3.986 +#656 := (= #593 #591)
   3.987 +#652 := (= #39 #22)
   3.988 +#647 := (= #22 #39)
   3.989 +#290 := (<= #230 0::int)
   3.990 +#70 := (<= #22 #39)
   3.991 +#388 := (iff #70 #290)
   3.992 +#389 := [rewrite]: #388
   3.993 +#341 := [asserted]: #70
   3.994 +#390 := [mp #341 #389]: #290
   3.995 +#646 := [hypothesis]: #228
   3.996 +#648 := [th-lemma #646 #390]: #647
   3.997 +#653 := [symm #648]: #652
   3.998 +#657 := [monotonicity #653]: #656
   3.999 +#659 := [symm #657]: #658
  3.1000 +#592 := (= uf_3 #591)
  3.1001 +#596 := (or #595 #592)
  3.1002 +#597 := [quant-inst]: #596
  3.1003 +#655 := [unit-resolution #597 #552]: #592
  3.1004 +#682 := [trans #655 #659]: #681
  3.1005 +#683 := [trans #682 #680]: #41
  3.1006 +#570 := (not #41)
  3.1007 +decl uf_11 :: T2
  3.1008 +#47 := uf_11
  3.1009 +#59 := (uf_4 uf_11 #42)
  3.1010 +#278 := (ite #41 #26 #59)
  3.1011 +#459 := (* -1::real #278)
  3.1012 +#637 := (+ #26 #459)
  3.1013 +#639 := (>= #637 0::real)
  3.1014 +#585 := (= #26 #278)
  3.1015 +#661 := [hypothesis]: #41
  3.1016 +#587 := (or #570 #585)
  3.1017 +#588 := [def-axiom]: #587
  3.1018 +#662 := [unit-resolution #588 #661]: #585
  3.1019 +#663 := (not #585)
  3.1020 +#664 := (or #663 #639)
  3.1021 +#665 := [th-lemma]: #664
  3.1022 +#666 := [unit-resolution #665 #662]: #639
  3.1023 +decl uf_8 :: T2
  3.1024 +#30 := uf_8
  3.1025 +#56 := (uf_4 uf_8 #42)
  3.1026 +#357 := (* -1::real #56)
  3.1027 +#358 := (+ #43 #357)
  3.1028 +#356 := (>= #358 0::real)
  3.1029 +#355 := (not #356)
  3.1030 +#374 := (* -1::real #59)
  3.1031 +#375 := (+ #56 #374)
  3.1032 +#373 := (>= #375 0::real)
  3.1033 +#376 := (not #373)
  3.1034 +#381 := (and #355 #376)
  3.1035 +#64 := (< #39 #39)
  3.1036 +#67 := (ite #64 #43 #59)
  3.1037 +#68 := (< #56 #67)
  3.1038 +#53 := (uf_4 uf_5 #42)
  3.1039 +#65 := (ite #64 #53 #43)
  3.1040 +#66 := (< #65 #56)
  3.1041 +#69 := (and #66 #68)
  3.1042 +#382 := (iff #69 #381)
  3.1043 +#379 := (iff #68 #376)
  3.1044 +#370 := (< #56 #59)
  3.1045 +#377 := (iff #370 #376)
  3.1046 +#378 := [rewrite]: #377
  3.1047 +#371 := (iff #68 #370)
  3.1048 +#368 := (= #67 #59)
  3.1049 +#363 := (ite false #43 #59)
  3.1050 +#366 := (= #363 #59)
  3.1051 +#367 := [rewrite]: #366
  3.1052 +#364 := (= #67 #363)
  3.1053 +#343 := (iff #64 false)
  3.1054 +#344 := [rewrite]: #343
  3.1055 +#365 := [monotonicity #344]: #364
  3.1056 +#369 := [trans #365 #367]: #368
  3.1057 +#372 := [monotonicity #369]: #371
  3.1058 +#380 := [trans #372 #378]: #379
  3.1059 +#361 := (iff #66 #355)
  3.1060 +#352 := (< #43 #56)
  3.1061 +#359 := (iff #352 #355)
  3.1062 +#360 := [rewrite]: #359
  3.1063 +#353 := (iff #66 #352)
  3.1064 +#350 := (= #65 #43)
  3.1065 +#345 := (ite false #53 #43)
  3.1066 +#348 := (= #345 #43)
  3.1067 +#349 := [rewrite]: #348
  3.1068 +#346 := (= #65 #345)
  3.1069 +#347 := [monotonicity #344]: #346
  3.1070 +#351 := [trans #347 #349]: #350
  3.1071 +#354 := [monotonicity #351]: #353
  3.1072 +#362 := [trans #354 #360]: #361
  3.1073 +#383 := [monotonicity #362 #380]: #382
  3.1074 +#340 := [asserted]: #69
  3.1075 +#384 := [mp #340 #383]: #381
  3.1076 +#385 := [and-elim #384]: #355
  3.1077 +#394 := (* -1::real #53)
  3.1078 +#395 := (+ #43 #394)
  3.1079 +#393 := (>= #395 0::real)
  3.1080 +#54 := (uf_4 uf_7 #42)
  3.1081 +#402 := (* -1::real #54)
  3.1082 +#403 := (+ #53 #402)
  3.1083 +#401 := (>= #403 0::real)
  3.1084 +#397 := (+ #43 #374)
  3.1085 +#398 := (<= #397 0::real)
  3.1086 +#412 := (and #393 #398 #401)
  3.1087 +#73 := (<= #43 #59)
  3.1088 +#72 := (<= #53 #43)
  3.1089 +#74 := (and #72 #73)
  3.1090 +#71 := (<= #54 #53)
  3.1091 +#75 := (and #71 #74)
  3.1092 +#415 := (iff #75 #412)
  3.1093 +#406 := (and #393 #398)
  3.1094 +#409 := (and #401 #406)
  3.1095 +#413 := (iff #409 #412)
  3.1096 +#414 := [rewrite]: #413
  3.1097 +#410 := (iff #75 #409)
  3.1098 +#407 := (iff #74 #406)
  3.1099 +#399 := (iff #73 #398)
  3.1100 +#400 := [rewrite]: #399
  3.1101 +#392 := (iff #72 #393)
  3.1102 +#396 := [rewrite]: #392
  3.1103 +#408 := [monotonicity #396 #400]: #407
  3.1104 +#404 := (iff #71 #401)
  3.1105 +#405 := [rewrite]: #404
  3.1106 +#411 := [monotonicity #405 #408]: #410
  3.1107 +#416 := [trans #411 #414]: #415
  3.1108 +#342 := [asserted]: #75
  3.1109 +#417 := [mp #342 #416]: #412
  3.1110 +#418 := [and-elim #417]: #393
  3.1111 +#650 := (+ #26 #394)
  3.1112 +#651 := (<= #650 0::real)
  3.1113 +#649 := (= #26 #53)
  3.1114 +#671 := (= #53 #26)
  3.1115 +#669 := (= #42 #25)
  3.1116 +#667 := (= #25 #42)
  3.1117 +#668 := [monotonicity #661]: #667
  3.1118 +#670 := [symm #668]: #669
  3.1119 +#672 := [monotonicity #670]: #671
  3.1120 +#673 := [symm #672]: #649
  3.1121 +#674 := (not #649)
  3.1122 +#675 := (or #674 #651)
  3.1123 +#676 := [th-lemma]: #675
  3.1124 +#677 := [unit-resolution #676 #673]: #651
  3.1125 +#462 := (+ #56 #459)
  3.1126 +#465 := (>= #462 0::real)
  3.1127 +#438 := (not #465)
  3.1128 +#316 := (ite #290 #278 #43)
  3.1129 +#326 := (* -1::real #316)
  3.1130 +#327 := (+ #56 #326)
  3.1131 +#325 := (>= #327 0::real)
  3.1132 +#324 := (not #325)
  3.1133 +#439 := (iff #324 #438)
  3.1134 +#466 := (iff #325 #465)
  3.1135 +#463 := (= #327 #462)
  3.1136 +#460 := (= #326 #459)
  3.1137 +#457 := (= #316 #278)
  3.1138 +#1 := true
  3.1139 +#452 := (ite true #278 #43)
  3.1140 +#455 := (= #452 #278)
  3.1141 +#456 := [rewrite]: #455
  3.1142 +#453 := (= #316 #452)
  3.1143 +#444 := (iff #290 true)
  3.1144 +#445 := [iff-true #390]: #444
  3.1145 +#454 := [monotonicity #445]: #453
  3.1146 +#458 := [trans #454 #456]: #457
  3.1147 +#461 := [monotonicity #458]: #460
  3.1148 +#464 := [monotonicity #461]: #463
  3.1149 +#467 := [monotonicity #464]: #466
  3.1150 +#468 := [monotonicity #467]: #439
  3.1151 +#297 := (ite #290 #54 #53)
  3.1152 +#305 := (* -1::real #297)
  3.1153 +#306 := (+ #56 #305)
  3.1154 +#307 := (<= #306 0::real)
  3.1155 +#308 := (not #307)
  3.1156 +#332 := (and #308 #324)
  3.1157 +#58 := (= uf_10 uf_3)
  3.1158 +#60 := (ite #58 #26 #59)
  3.1159 +#52 := (< #39 #22)
  3.1160 +#61 := (ite #52 #43 #60)
  3.1161 +#62 := (< #56 #61)
  3.1162 +#55 := (ite #52 #53 #54)
  3.1163 +#57 := (< #55 #56)
  3.1164 +#63 := (and #57 #62)
  3.1165 +#335 := (iff #63 #332)
  3.1166 +#281 := (ite #52 #43 #278)
  3.1167 +#284 := (< #56 #281)
  3.1168 +#287 := (and #57 #284)
  3.1169 +#333 := (iff #287 #332)
  3.1170 +#330 := (iff #284 #324)
  3.1171 +#321 := (< #56 #316)
  3.1172 +#328 := (iff #321 #324)
  3.1173 +#329 := [rewrite]: #328
  3.1174 +#322 := (iff #284 #321)
  3.1175 +#319 := (= #281 #316)
  3.1176 +#291 := (not #290)
  3.1177 +#313 := (ite #291 #43 #278)
  3.1178 +#317 := (= #313 #316)
  3.1179 +#318 := [rewrite]: #317
  3.1180 +#314 := (= #281 #313)
  3.1181 +#292 := (iff #52 #291)
  3.1182 +#293 := [rewrite]: #292
  3.1183 +#315 := [monotonicity #293]: #314
  3.1184 +#320 := [trans #315 #318]: #319
  3.1185 +#323 := [monotonicity #320]: #322
  3.1186 +#331 := [trans #323 #329]: #330
  3.1187 +#311 := (iff #57 #308)
  3.1188 +#302 := (< #297 #56)
  3.1189 +#309 := (iff #302 #308)
  3.1190 +#310 := [rewrite]: #309
  3.1191 +#303 := (iff #57 #302)
  3.1192 +#300 := (= #55 #297)
  3.1193 +#294 := (ite #291 #53 #54)
  3.1194 +#298 := (= #294 #297)
  3.1195 +#299 := [rewrite]: #298
  3.1196 +#295 := (= #55 #294)
  3.1197 +#296 := [monotonicity #293]: #295
  3.1198 +#301 := [trans #296 #299]: #300
  3.1199 +#304 := [monotonicity #301]: #303
  3.1200 +#312 := [trans #304 #310]: #311
  3.1201 +#334 := [monotonicity #312 #331]: #333
  3.1202 +#288 := (iff #63 #287)
  3.1203 +#285 := (iff #62 #284)
  3.1204 +#282 := (= #61 #281)
  3.1205 +#279 := (= #60 #278)
  3.1206 +#225 := (iff #58 #41)
  3.1207 +#277 := [rewrite]: #225
  3.1208 +#280 := [monotonicity #277]: #279
  3.1209 +#283 := [monotonicity #280]: #282
  3.1210 +#286 := [monotonicity #283]: #285
  3.1211 +#289 := [monotonicity #286]: #288
  3.1212 +#336 := [trans #289 #334]: #335
  3.1213 +#179 := [asserted]: #63
  3.1214 +#337 := [mp #179 #336]: #332
  3.1215 +#339 := [and-elim #337]: #324
  3.1216 +#469 := [mp #339 #468]: #438
  3.1217 +#678 := [th-lemma #469 #677 #418 #385 #666]: false
  3.1218 +#679 := [lemma #678]: #570
  3.1219 +#684 := [unit-resolution #679 #683]: false
  3.1220 +#685 := [lemma #684]: #227
  3.1221 +#577 := (or #228 #567)
  3.1222 +#578 := [def-axiom]: #577
  3.1223 +#645 := [unit-resolution #578 #685]: #567
  3.1224 +#686 := (not #567)
  3.1225 +#687 := (or #686 #643)
  3.1226 +#688 := [th-lemma]: #687
  3.1227 +#689 := [unit-resolution #688 #645]: #643
  3.1228 +#31 := (uf_4 uf_8 #25)
  3.1229 +#245 := (+ #31 #244)
  3.1230 +#246 := (<= #245 0::real)
  3.1231 +#247 := (not #246)
  3.1232 +#34 := (uf_4 uf_9 #25)
  3.1233 +#48 := (uf_4 uf_11 #25)
  3.1234 +#255 := (ite #228 #48 #34)
  3.1235 +#264 := (* -1::real #255)
  3.1236 +#265 := (+ #31 #264)
  3.1237 +#263 := (>= #265 0::real)
  3.1238 +#266 := (not #263)
  3.1239 +#271 := (and #247 #266)
  3.1240 +#40 := (< #22 #39)
  3.1241 +#49 := (ite #40 #34 #48)
  3.1242 +#50 := (< #31 #49)
  3.1243 +#45 := (ite #40 #26 #44)
  3.1244 +#46 := (< #45 #31)
  3.1245 +#51 := (and #46 #50)
  3.1246 +#272 := (iff #51 #271)
  3.1247 +#269 := (iff #50 #266)
  3.1248 +#260 := (< #31 #255)
  3.1249 +#267 := (iff #260 #266)
  3.1250 +#268 := [rewrite]: #267
  3.1251 +#261 := (iff #50 #260)
  3.1252 +#258 := (= #49 #255)
  3.1253 +#252 := (ite #227 #34 #48)
  3.1254 +#256 := (= #252 #255)
  3.1255 +#257 := [rewrite]: #256
  3.1256 +#253 := (= #49 #252)
  3.1257 +#231 := (iff #40 #227)
  3.1258 +#232 := [rewrite]: #231
  3.1259 +#254 := [monotonicity #232]: #253
  3.1260 +#259 := [trans #254 #257]: #258
  3.1261 +#262 := [monotonicity #259]: #261
  3.1262 +#270 := [trans #262 #268]: #269
  3.1263 +#250 := (iff #46 #247)
  3.1264 +#241 := (< #236 #31)
  3.1265 +#248 := (iff #241 #247)
  3.1266 +#249 := [rewrite]: #248
  3.1267 +#242 := (iff #46 #241)
  3.1268 +#239 := (= #45 #236)
  3.1269 +#233 := (ite #227 #26 #44)
  3.1270 +#237 := (= #233 #236)
  3.1271 +#238 := [rewrite]: #237
  3.1272 +#234 := (= #45 #233)
  3.1273 +#235 := [monotonicity #232]: #234
  3.1274 +#240 := [trans #235 #238]: #239
  3.1275 +#243 := [monotonicity #240]: #242
  3.1276 +#251 := [trans #243 #249]: #250
  3.1277 +#273 := [monotonicity #251 #270]: #272
  3.1278 +#178 := [asserted]: #51
  3.1279 +#274 := [mp #178 #273]: #271
  3.1280 +#275 := [and-elim #274]: #247
  3.1281 +#196 := (* -1::real #31)
  3.1282 +#212 := (+ #26 #196)
  3.1283 +#213 := (<= #212 0::real)
  3.1284 +#214 := (not #213)
  3.1285 +#197 := (+ #28 #196)
  3.1286 +#195 := (>= #197 0::real)
  3.1287 +#193 := (not #195)
  3.1288 +#219 := (and #193 #214)
  3.1289 +#23 := (< #22 #22)
  3.1290 +#35 := (ite #23 #34 #26)
  3.1291 +#36 := (< #31 #35)
  3.1292 +#29 := (ite #23 #26 #28)
  3.1293 +#32 := (< #29 #31)
  3.1294 +#37 := (and #32 #36)
  3.1295 +#220 := (iff #37 #219)
  3.1296 +#217 := (iff #36 #214)
  3.1297 +#209 := (< #31 #26)
  3.1298 +#215 := (iff #209 #214)
  3.1299 +#216 := [rewrite]: #215
  3.1300 +#210 := (iff #36 #209)
  3.1301 +#207 := (= #35 #26)
  3.1302 +#202 := (ite false #34 #26)
  3.1303 +#205 := (= #202 #26)
  3.1304 +#206 := [rewrite]: #205
  3.1305 +#203 := (= #35 #202)
  3.1306 +#180 := (iff #23 false)
  3.1307 +#181 := [rewrite]: #180
  3.1308 +#204 := [monotonicity #181]: #203
  3.1309 +#208 := [trans #204 #206]: #207
  3.1310 +#211 := [monotonicity #208]: #210
  3.1311 +#218 := [trans #211 #216]: #217
  3.1312 +#200 := (iff #32 #193)
  3.1313 +#189 := (< #28 #31)
  3.1314 +#198 := (iff #189 #193)
  3.1315 +#199 := [rewrite]: #198
  3.1316 +#190 := (iff #32 #189)
  3.1317 +#187 := (= #29 #28)
  3.1318 +#182 := (ite false #26 #28)
  3.1319 +#185 := (= #182 #28)
  3.1320 +#186 := [rewrite]: #185
  3.1321 +#183 := (= #29 #182)
  3.1322 +#184 := [monotonicity #181]: #183
  3.1323 +#188 := [trans #184 #186]: #187
  3.1324 +#191 := [monotonicity #188]: #190
  3.1325 +#201 := [trans #191 #199]: #200
  3.1326 +#221 := [monotonicity #201 #218]: #220
  3.1327 +#177 := [asserted]: #37
  3.1328 +#222 := [mp #177 #221]: #219
  3.1329 +#224 := [and-elim #222]: #214
  3.1330 +[th-lemma #224 #275 #689]: false
  3.1331 +unsat
  3.1332 +NX/HT1QOfbspC2LtZNKpBA 428 0
  3.1333 +#2 := false
  3.1334 +decl uf_10 :: T1
  3.1335 +#38 := uf_10
  3.1336 +decl uf_3 :: T1
  3.1337 +#21 := uf_3
  3.1338 +#45 := (= uf_3 uf_10)
  3.1339 +decl uf_1 :: (-> int T1)
  3.1340 +decl uf_2 :: (-> T1 int)
  3.1341 +#39 := (uf_2 uf_10)
  3.1342 +#588 := (uf_1 #39)
  3.1343 +#686 := (= #588 uf_10)
  3.1344 +#589 := (= uf_10 #588)
  3.1345 +#4 := (:var 0 T1)
  3.1346 +#5 := (uf_2 #4)
  3.1347 +#541 := (pattern #5)
  3.1348 +#6 := (uf_1 #5)
  3.1349 +#93 := (= #4 #6)
  3.1350 +#542 := (forall (vars (?x1 T1)) (:pat #541) #93)
  3.1351 +#96 := (forall (vars (?x1 T1)) #93)
  3.1352 +#545 := (iff #96 #542)
  3.1353 +#543 := (iff #93 #93)
  3.1354 +#544 := [refl]: #543
  3.1355 +#546 := [quant-intro #544]: #545
  3.1356 +#454 := (~ #96 #96)
  3.1357 +#456 := (~ #93 #93)
  3.1358 +#457 := [refl]: #456
  3.1359 +#455 := [nnf-pos #457]: #454
  3.1360 +#7 := (= #6 #4)
  3.1361 +#8 := (forall (vars (?x1 T1)) #7)
  3.1362 +#97 := (iff #8 #96)
  3.1363 +#94 := (iff #7 #93)
  3.1364 +#95 := [rewrite]: #94
  3.1365 +#98 := [quant-intro #95]: #97
  3.1366 +#92 := [asserted]: #8
  3.1367 +#101 := [mp #92 #98]: #96
  3.1368 +#452 := [mp~ #101 #455]: #96
  3.1369 +#547 := [mp #452 #546]: #542
  3.1370 +#590 := (not #542)
  3.1371 +#595 := (or #590 #589)
  3.1372 +#596 := [quant-inst]: #595
  3.1373 +#680 := [unit-resolution #596 #547]: #589
  3.1374 +#687 := [symm #680]: #686
  3.1375 +#688 := (= uf_3 #588)
  3.1376 +#22 := (uf_2 uf_3)
  3.1377 +#586 := (uf_1 #22)
  3.1378 +#684 := (= #586 #588)
  3.1379 +#682 := (= #588 #586)
  3.1380 +#678 := (= #39 #22)
  3.1381 +#676 := (= #22 #39)
  3.1382 +#9 := 0::int
  3.1383 +#227 := -1::int
  3.1384 +#230 := (* -1::int #39)
  3.1385 +#231 := (+ #22 #230)
  3.1386 +#296 := (<= #231 0::int)
  3.1387 +#70 := (<= #22 #39)
  3.1388 +#393 := (iff #70 #296)
  3.1389 +#394 := [rewrite]: #393
  3.1390 +#347 := [asserted]: #70
  3.1391 +#395 := [mp #347 #394]: #296
  3.1392 +#229 := (>= #231 0::int)
  3.1393 +decl uf_4 :: (-> T2 T3 real)
  3.1394 +decl uf_6 :: (-> T1 T3)
  3.1395 +#25 := (uf_6 uf_3)
  3.1396 +decl uf_7 :: T2
  3.1397 +#27 := uf_7
  3.1398 +#28 := (uf_4 uf_7 #25)
  3.1399 +decl uf_9 :: T2
  3.1400 +#33 := uf_9
  3.1401 +#34 := (uf_4 uf_9 #25)
  3.1402 +#46 := (uf_6 uf_10)
  3.1403 +decl uf_5 :: T2
  3.1404 +#24 := uf_5
  3.1405 +#47 := (uf_4 uf_5 #46)
  3.1406 +#48 := (ite #45 #47 #34)
  3.1407 +#256 := (ite #229 #48 #28)
  3.1408 +#568 := (= #28 #256)
  3.1409 +#648 := (not #568)
  3.1410 +#194 := 0::real
  3.1411 +#192 := -1::real
  3.1412 +#265 := (* -1::real #256)
  3.1413 +#640 := (+ #28 #265)
  3.1414 +#642 := (>= #640 0::real)
  3.1415 +#645 := (not #642)
  3.1416 +#643 := [hypothesis]: #642
  3.1417 +decl uf_8 :: T2
  3.1418 +#30 := uf_8
  3.1419 +#31 := (uf_4 uf_8 #25)
  3.1420 +#266 := (+ #31 #265)
  3.1421 +#264 := (>= #266 0::real)
  3.1422 +#267 := (not #264)
  3.1423 +#26 := (uf_4 uf_5 #25)
  3.1424 +decl uf_11 :: T2
  3.1425 +#41 := uf_11
  3.1426 +#42 := (uf_4 uf_11 #25)
  3.1427 +#237 := (ite #229 #42 #26)
  3.1428 +#245 := (* -1::real #237)
  3.1429 +#246 := (+ #31 #245)
  3.1430 +#247 := (<= #246 0::real)
  3.1431 +#248 := (not #247)
  3.1432 +#272 := (and #248 #267)
  3.1433 +#40 := (< #22 #39)
  3.1434 +#49 := (ite #40 #28 #48)
  3.1435 +#50 := (< #31 #49)
  3.1436 +#43 := (ite #40 #26 #42)
  3.1437 +#44 := (< #43 #31)
  3.1438 +#51 := (and #44 #50)
  3.1439 +#273 := (iff #51 #272)
  3.1440 +#270 := (iff #50 #267)
  3.1441 +#261 := (< #31 #256)
  3.1442 +#268 := (iff #261 #267)
  3.1443 +#269 := [rewrite]: #268
  3.1444 +#262 := (iff #50 #261)
  3.1445 +#259 := (= #49 #256)
  3.1446 +#228 := (not #229)
  3.1447 +#253 := (ite #228 #28 #48)
  3.1448 +#257 := (= #253 #256)
  3.1449 +#258 := [rewrite]: #257
  3.1450 +#254 := (= #49 #253)
  3.1451 +#232 := (iff #40 #228)
  3.1452 +#233 := [rewrite]: #232
  3.1453 +#255 := [monotonicity #233]: #254
  3.1454 +#260 := [trans #255 #258]: #259
  3.1455 +#263 := [monotonicity #260]: #262
  3.1456 +#271 := [trans #263 #269]: #270
  3.1457 +#251 := (iff #44 #248)
  3.1458 +#242 := (< #237 #31)
  3.1459 +#249 := (iff #242 #248)
  3.1460 +#250 := [rewrite]: #249
  3.1461 +#243 := (iff #44 #242)
  3.1462 +#240 := (= #43 #237)
  3.1463 +#234 := (ite #228 #26 #42)
  3.1464 +#238 := (= #234 #237)
  3.1465 +#239 := [rewrite]: #238
  3.1466 +#235 := (= #43 #234)
  3.1467 +#236 := [monotonicity #233]: #235
  3.1468 +#241 := [trans #236 #239]: #240
  3.1469 +#244 := [monotonicity #241]: #243
  3.1470 +#252 := [trans #244 #250]: #251
  3.1471 +#274 := [monotonicity #252 #271]: #273
  3.1472 +#178 := [asserted]: #51
  3.1473 +#275 := [mp #178 #274]: #272
  3.1474 +#277 := [and-elim #275]: #267
  3.1475 +#196 := (* -1::real #31)
  3.1476 +#197 := (+ #28 #196)
  3.1477 +#195 := (>= #197 0::real)
  3.1478 +#193 := (not #195)
  3.1479 +#213 := (* -1::real #34)
  3.1480 +#214 := (+ #31 #213)
  3.1481 +#212 := (>= #214 0::real)
  3.1482 +#215 := (not #212)
  3.1483 +#220 := (and #193 #215)
  3.1484 +#23 := (< #22 #22)
  3.1485 +#35 := (ite #23 #28 #34)
  3.1486 +#36 := (< #31 #35)
  3.1487 +#29 := (ite #23 #26 #28)
  3.1488 +#32 := (< #29 #31)
  3.1489 +#37 := (and #32 #36)
  3.1490 +#221 := (iff #37 #220)
  3.1491 +#218 := (iff #36 #215)
  3.1492 +#209 := (< #31 #34)
  3.1493 +#216 := (iff #209 #215)
  3.1494 +#217 := [rewrite]: #216
  3.1495 +#210 := (iff #36 #209)
  3.1496 +#207 := (= #35 #34)
  3.1497 +#202 := (ite false #28 #34)
  3.1498 +#205 := (= #202 #34)
  3.1499 +#206 := [rewrite]: #205
  3.1500 +#203 := (= #35 #202)
  3.1501 +#180 := (iff #23 false)
  3.1502 +#181 := [rewrite]: #180
  3.1503 +#204 := [monotonicity #181]: #203
  3.1504 +#208 := [trans #204 #206]: #207
  3.1505 +#211 := [monotonicity #208]: #210
  3.1506 +#219 := [trans #211 #217]: #218
  3.1507 +#200 := (iff #32 #193)
  3.1508 +#189 := (< #28 #31)
  3.1509 +#198 := (iff #189 #193)
  3.1510 +#199 := [rewrite]: #198
  3.1511 +#190 := (iff #32 #189)
  3.1512 +#187 := (= #29 #28)
  3.1513 +#182 := (ite false #26 #28)
  3.1514 +#185 := (= #182 #28)
  3.1515 +#186 := [rewrite]: #185
  3.1516 +#183 := (= #29 #182)
  3.1517 +#184 := [monotonicity #181]: #183
  3.1518 +#188 := [trans #184 #186]: #187
  3.1519 +#191 := [monotonicity #188]: #190
  3.1520 +#201 := [trans #191 #199]: #200
  3.1521 +#222 := [monotonicity #201 #219]: #221
  3.1522 +#177 := [asserted]: #37
  3.1523 +#223 := [mp #177 #222]: #220
  3.1524 +#224 := [and-elim #223]: #193
  3.1525 +#644 := [th-lemma #224 #277 #643]: false
  3.1526 +#646 := [lemma #644]: #645
  3.1527 +#647 := [hypothesis]: #568
  3.1528 +#649 := (or #648 #642)
  3.1529 +#650 := [th-lemma]: #649
  3.1530 +#651 := [unit-resolution #650 #647 #646]: false
  3.1531 +#652 := [lemma #651]: #648
  3.1532 +#578 := (or #229 #568)
  3.1533 +#579 := [def-axiom]: #578
  3.1534 +#675 := [unit-resolution #579 #652]: #229
  3.1535 +#677 := [th-lemma #675 #395]: #676
  3.1536 +#679 := [symm #677]: #678
  3.1537 +#683 := [monotonicity #679]: #682
  3.1538 +#685 := [symm #683]: #684
  3.1539 +#587 := (= uf_3 #586)
  3.1540 +#591 := (or #590 #587)
  3.1541 +#592 := [quant-inst]: #591
  3.1542 +#681 := [unit-resolution #592 #547]: #587
  3.1543 +#689 := [trans #681 #685]: #688
  3.1544 +#690 := [trans #689 #687]: #45
  3.1545 +#571 := (not #45)
  3.1546 +#54 := (uf_4 uf_11 #46)
  3.1547 +#279 := (ite #45 #28 #54)
  3.1548 +#465 := (* -1::real #279)
  3.1549 +#632 := (+ #28 #465)
  3.1550 +#633 := (<= #632 0::real)
  3.1551 +#580 := (= #28 #279)
  3.1552 +#656 := [hypothesis]: #45
  3.1553 +#582 := (or #571 #580)
  3.1554 +#583 := [def-axiom]: #582
  3.1555 +#657 := [unit-resolution #583 #656]: #580
  3.1556 +#658 := (not #580)
  3.1557 +#659 := (or #658 #633)
  3.1558 +#660 := [th-lemma]: #659
  3.1559 +#661 := [unit-resolution #660 #657]: #633
  3.1560 +#57 := (uf_4 uf_8 #46)
  3.1561 +#363 := (* -1::real #57)
  3.1562 +#379 := (+ #47 #363)
  3.1563 +#380 := (<= #379 0::real)
  3.1564 +#381 := (not #380)
  3.1565 +#364 := (+ #54 #363)
  3.1566 +#362 := (>= #364 0::real)
  3.1567 +#361 := (not #362)
  3.1568 +#386 := (and #361 #381)
  3.1569 +#59 := (uf_4 uf_7 #46)
  3.1570 +#64 := (< #39 #39)
  3.1571 +#67 := (ite #64 #59 #47)
  3.1572 +#68 := (< #57 #67)
  3.1573 +#65 := (ite #64 #47 #54)
  3.1574 +#66 := (< #65 #57)
  3.1575 +#69 := (and #66 #68)
  3.1576 +#387 := (iff #69 #386)
  3.1577 +#384 := (iff #68 #381)
  3.1578 +#376 := (< #57 #47)
  3.1579 +#382 := (iff #376 #381)
  3.1580 +#383 := [rewrite]: #382
  3.1581 +#377 := (iff #68 #376)
  3.1582 +#374 := (= #67 #47)
  3.1583 +#369 := (ite false #59 #47)
  3.1584 +#372 := (= #369 #47)
  3.1585 +#373 := [rewrite]: #372
  3.1586 +#370 := (= #67 #369)
  3.1587 +#349 := (iff #64 false)
  3.1588 +#350 := [rewrite]: #349
  3.1589 +#371 := [monotonicity #350]: #370
  3.1590 +#375 := [trans #371 #373]: #374
  3.1591 +#378 := [monotonicity #375]: #377
  3.1592 +#385 := [trans #378 #383]: #384
  3.1593 +#367 := (iff #66 #361)
  3.1594 +#358 := (< #54 #57)
  3.1595 +#365 := (iff #358 #361)
  3.1596 +#366 := [rewrite]: #365
  3.1597 +#359 := (iff #66 #358)
  3.1598 +#356 := (= #65 #54)
  3.1599 +#351 := (ite false #47 #54)
  3.1600 +#354 := (= #351 #54)
  3.1601 +#355 := [rewrite]: #354
  3.1602 +#352 := (= #65 #351)
  3.1603 +#353 := [monotonicity #350]: #352
  3.1604 +#357 := [trans #353 #355]: #356
  3.1605 +#360 := [monotonicity #357]: #359
  3.1606 +#368 := [trans #360 #366]: #367
  3.1607 +#388 := [monotonicity #368 #385]: #387
  3.1608 +#346 := [asserted]: #69
  3.1609 +#389 := [mp #346 #388]: #386
  3.1610 +#391 := [and-elim #389]: #381
  3.1611 +#397 := (* -1::real #59)
  3.1612 +#398 := (+ #47 #397)
  3.1613 +#399 := (<= #398 0::real)
  3.1614 +#409 := (* -1::real #54)
  3.1615 +#410 := (+ #47 #409)
  3.1616 +#408 := (>= #410 0::real)
  3.1617 +#60 := (uf_4 uf_9 #46)
  3.1618 +#402 := (* -1::real #60)
  3.1619 +#403 := (+ #59 #402)
  3.1620 +#404 := (<= #403 0::real)
  3.1621 +#418 := (and #399 #404 #408)
  3.1622 +#73 := (<= #59 #60)
  3.1623 +#72 := (<= #47 #59)
  3.1624 +#74 := (and #72 #73)
  3.1625 +#71 := (<= #54 #47)
  3.1626 +#75 := (and #71 #74)
  3.1627 +#421 := (iff #75 #418)
  3.1628 +#412 := (and #399 #404)
  3.1629 +#415 := (and #408 #412)
  3.1630 +#419 := (iff #415 #418)
  3.1631 +#420 := [rewrite]: #419
  3.1632 +#416 := (iff #75 #415)
  3.1633 +#413 := (iff #74 #412)
  3.1634 +#405 := (iff #73 #404)
  3.1635 +#406 := [rewrite]: #405
  3.1636 +#400 := (iff #72 #399)
  3.1637 +#401 := [rewrite]: #400
  3.1638 +#414 := [monotonicity #401 #406]: #413
  3.1639 +#407 := (iff #71 #408)
  3.1640 +#411 := [rewrite]: #407
  3.1641 +#417 := [monotonicity #411 #414]: #416
  3.1642 +#422 := [trans #417 #420]: #421
  3.1643 +#348 := [asserted]: #75
  3.1644 +#423 := [mp #348 #422]: #418
  3.1645 +#424 := [and-elim #423]: #399
  3.1646 +#637 := (+ #28 #397)
  3.1647 +#639 := (>= #637 0::real)
  3.1648 +#636 := (= #28 #59)
  3.1649 +#666 := (= #59 #28)
  3.1650 +#664 := (= #46 #25)
  3.1651 +#662 := (= #25 #46)
  3.1652 +#663 := [monotonicity #656]: #662
  3.1653 +#665 := [symm #663]: #664
  3.1654 +#667 := [monotonicity #665]: #666
  3.1655 +#668 := [symm #667]: #636
  3.1656 +#669 := (not #636)
  3.1657 +#670 := (or #669 #639)
  3.1658 +#671 := [th-lemma]: #670
  3.1659 +#672 := [unit-resolution #671 #668]: #639
  3.1660 +#468 := (+ #57 #465)
  3.1661 +#471 := (<= #468 0::real)
  3.1662 +#444 := (not #471)
  3.1663 +#322 := (ite #296 #279 #47)
  3.1664 +#330 := (* -1::real #322)
  3.1665 +#331 := (+ #57 #330)
  3.1666 +#332 := (<= #331 0::real)
  3.1667 +#333 := (not #332)
  3.1668 +#445 := (iff #333 #444)
  3.1669 +#472 := (iff #332 #471)
  3.1670 +#469 := (= #331 #468)
  3.1671 +#466 := (= #330 #465)
  3.1672 +#463 := (= #322 #279)
  3.1673 +#1 := true
  3.1674 +#458 := (ite true #279 #47)
  3.1675 +#461 := (= #458 #279)
  3.1676 +#462 := [rewrite]: #461
  3.1677 +#459 := (= #322 #458)
  3.1678 +#450 := (iff #296 true)
  3.1679 +#451 := [iff-true #395]: #450
  3.1680 +#460 := [monotonicity #451]: #459
  3.1681 +#464 := [trans #460 #462]: #463
  3.1682 +#467 := [monotonicity #464]: #466
  3.1683 +#470 := [monotonicity #467]: #469
  3.1684 +#473 := [monotonicity #470]: #472
  3.1685 +#474 := [monotonicity #473]: #445
  3.1686 +#303 := (ite #296 #60 #59)
  3.1687 +#313 := (* -1::real #303)
  3.1688 +#314 := (+ #57 #313)
  3.1689 +#312 := (>= #314 0::real)
  3.1690 +#311 := (not #312)
  3.1691 +#338 := (and #311 #333)
  3.1692 +#52 := (< #39 #22)
  3.1693 +#61 := (ite #52 #59 #60)
  3.1694 +#62 := (< #57 #61)
  3.1695 +#53 := (= uf_10 uf_3)
  3.1696 +#55 := (ite #53 #28 #54)
  3.1697 +#56 := (ite #52 #47 #55)
  3.1698 +#58 := (< #56 #57)
  3.1699 +#63 := (and #58 #62)
  3.1700 +#341 := (iff #63 #338)
  3.1701 +#282 := (ite #52 #47 #279)
  3.1702 +#285 := (< #282 #57)
  3.1703 +#291 := (and #62 #285)
  3.1704 +#339 := (iff #291 #338)
  3.1705 +#336 := (iff #285 #333)
  3.1706 +#327 := (< #322 #57)
  3.1707 +#334 := (iff #327 #333)
  3.1708 +#335 := [rewrite]: #334
  3.1709 +#328 := (iff #285 #327)
  3.1710 +#325 := (= #282 #322)
  3.1711 +#297 := (not #296)
  3.1712 +#319 := (ite #297 #47 #279)
  3.1713 +#323 := (= #319 #322)
  3.1714 +#324 := [rewrite]: #323
  3.1715 +#320 := (= #282 #319)
  3.1716 +#298 := (iff #52 #297)
  3.1717 +#299 := [rewrite]: #298
  3.1718 +#321 := [monotonicity #299]: #320
  3.1719 +#326 := [trans #321 #324]: #325
  3.1720 +#329 := [monotonicity #326]: #328
  3.1721 +#337 := [trans #329 #335]: #336
  3.1722 +#317 := (iff #62 #311)
  3.1723 +#308 := (< #57 #303)
  3.1724 +#315 := (iff #308 #311)
  3.1725 +#316 := [rewrite]: #315
  3.1726 +#309 := (iff #62 #308)
  3.1727 +#306 := (= #61 #303)
  3.1728 +#300 := (ite #297 #59 #60)
  3.1729 +#304 := (= #300 #303)
  3.1730 +#305 := [rewrite]: #304
  3.1731 +#301 := (= #61 #300)
  3.1732 +#302 := [monotonicity #299]: #301
  3.1733 +#307 := [trans #302 #305]: #306
  3.1734 +#310 := [monotonicity #307]: #309
  3.1735 +#318 := [trans #310 #316]: #317
  3.1736 +#340 := [monotonicity #318 #337]: #339
  3.1737 +#294 := (iff #63 #291)
  3.1738 +#288 := (and #285 #62)
  3.1739 +#292 := (iff #288 #291)
  3.1740 +#293 := [rewrite]: #292
  3.1741 +#289 := (iff #63 #288)
  3.1742 +#286 := (iff #58 #285)
  3.1743 +#283 := (= #56 #282)
  3.1744 +#280 := (= #55 #279)
  3.1745 +#226 := (iff #53 #45)
  3.1746 +#278 := [rewrite]: #226
  3.1747 +#281 := [monotonicity #278]: #280
  3.1748 +#284 := [monotonicity #281]: #283
  3.1749 +#287 := [monotonicity #284]: #286
  3.1750 +#290 := [monotonicity #287]: #289
  3.1751 +#295 := [trans #290 #293]: #294
  3.1752 +#342 := [trans #295 #340]: #341
  3.1753 +#179 := [asserted]: #63
  3.1754 +#343 := [mp #179 #342]: #338
  3.1755 +#345 := [and-elim #343]: #333
  3.1756 +#475 := [mp #345 #474]: #444
  3.1757 +#673 := [th-lemma #475 #672 #424 #391 #661]: false
  3.1758 +#674 := [lemma #673]: #571
  3.1759 +[unit-resolution #674 #690]: false
  3.1760 +unsat
  3.1761 +IL2powemHjRpCJYwmXFxyw 211 0
  3.1762 +#2 := false
  3.1763 +#33 := 0::real
  3.1764 +decl uf_11 :: (-> T5 T6 real)
  3.1765 +decl uf_15 :: T6
  3.1766 +#28 := uf_15
  3.1767 +decl uf_16 :: T5
  3.1768 +#30 := uf_16
  3.1769 +#31 := (uf_11 uf_16 uf_15)
  3.1770 +decl uf_12 :: (-> T7 T8 T5)
  3.1771 +decl uf_14 :: T8
  3.1772 +#26 := uf_14
  3.1773 +decl uf_13 :: (-> T1 T7)
  3.1774 +decl uf_8 :: T1
  3.1775 +#16 := uf_8
  3.1776 +#25 := (uf_13 uf_8)
  3.1777 +#27 := (uf_12 #25 uf_14)
  3.1778 +#29 := (uf_11 #27 uf_15)
  3.1779 +#73 := -1::real
  3.1780 +#84 := (* -1::real #29)
  3.1781 +#85 := (+ #84 #31)
  3.1782 +#74 := (* -1::real #31)
  3.1783 +#75 := (+ #29 #74)
  3.1784 +#112 := (>= #75 0::real)
  3.1785 +#119 := (ite #112 #75 #85)
  3.1786 +#127 := (* -1::real #119)
  3.1787 +decl uf_17 :: T5
  3.1788 +#37 := uf_17
  3.1789 +#38 := (uf_11 uf_17 uf_15)
  3.1790 +#102 := -1/3::real
  3.1791 +#103 := (* -1/3::real #38)
  3.1792 +#128 := (+ #103 #127)
  3.1793 +#100 := 1/3::real
  3.1794 +#101 := (* 1/3::real #31)
  3.1795 +#129 := (+ #101 #128)
  3.1796 +#130 := (<= #129 0::real)
  3.1797 +#131 := (not #130)
  3.1798 +#40 := 3::real
  3.1799 +#39 := (- #31 #38)
  3.1800 +#41 := (/ #39 3::real)
  3.1801 +#32 := (- #29 #31)
  3.1802 +#35 := (- #32)
  3.1803 +#34 := (< #32 0::real)
  3.1804 +#36 := (ite #34 #35 #32)
  3.1805 +#42 := (< #36 #41)
  3.1806 +#136 := (iff #42 #131)
  3.1807 +#104 := (+ #101 #103)
  3.1808 +#78 := (< #75 0::real)
  3.1809 +#90 := (ite #78 #85 #75)
  3.1810 +#109 := (< #90 #104)
  3.1811 +#134 := (iff #109 #131)
  3.1812 +#124 := (< #119 #104)
  3.1813 +#132 := (iff #124 #131)
  3.1814 +#133 := [rewrite]: #132
  3.1815 +#125 := (iff #109 #124)
  3.1816 +#122 := (= #90 #119)
  3.1817 +#113 := (not #112)
  3.1818 +#116 := (ite #113 #85 #75)
  3.1819 +#120 := (= #116 #119)
  3.1820 +#121 := [rewrite]: #120
  3.1821 +#117 := (= #90 #116)
  3.1822 +#114 := (iff #78 #113)
  3.1823 +#115 := [rewrite]: #114
  3.1824 +#118 := [monotonicity #115]: #117
  3.1825 +#123 := [trans #118 #121]: #122
  3.1826 +#126 := [monotonicity #123]: #125
  3.1827 +#135 := [trans #126 #133]: #134
  3.1828 +#110 := (iff #42 #109)
  3.1829 +#107 := (= #41 #104)
  3.1830 +#93 := (* -1::real #38)
  3.1831 +#94 := (+ #31 #93)
  3.1832 +#97 := (/ #94 3::real)
  3.1833 +#105 := (= #97 #104)
  3.1834 +#106 := [rewrite]: #105
  3.1835 +#98 := (= #41 #97)
  3.1836 +#95 := (= #39 #94)
  3.1837 +#96 := [rewrite]: #95
  3.1838 +#99 := [monotonicity #96]: #98
  3.1839 +#108 := [trans #99 #106]: #107
  3.1840 +#91 := (= #36 #90)
  3.1841 +#76 := (= #32 #75)
  3.1842 +#77 := [rewrite]: #76
  3.1843 +#88 := (= #35 #85)
  3.1844 +#81 := (- #75)
  3.1845 +#86 := (= #81 #85)
  3.1846 +#87 := [rewrite]: #86
  3.1847 +#82 := (= #35 #81)
  3.1848 +#83 := [monotonicity #77]: #82
  3.1849 +#89 := [trans #83 #87]: #88
  3.1850 +#79 := (iff #34 #78)
  3.1851 +#80 := [monotonicity #77]: #79
  3.1852 +#92 := [monotonicity #80 #89 #77]: #91
  3.1853 +#111 := [monotonicity #92 #108]: #110
  3.1854 +#137 := [trans #111 #135]: #136
  3.1855 +#72 := [asserted]: #42
  3.1856 +#138 := [mp #72 #137]: #131
  3.1857 +decl uf_1 :: T1
  3.1858 +#4 := uf_1
  3.1859 +#43 := (uf_13 uf_1)
  3.1860 +#44 := (uf_12 #43 uf_14)
  3.1861 +#45 := (uf_11 #44 uf_15)
  3.1862 +#149 := (* -1::real #45)
  3.1863 +#150 := (+ #38 #149)
  3.1864 +#140 := (+ #93 #45)
  3.1865 +#161 := (<= #150 0::real)
  3.1866 +#168 := (ite #161 #140 #150)
  3.1867 +#176 := (* -1::real #168)
  3.1868 +#177 := (+ #103 #176)
  3.1869 +#178 := (+ #101 #177)
  3.1870 +#179 := (<= #178 0::real)
  3.1871 +#180 := (not #179)
  3.1872 +#46 := (- #45 #38)
  3.1873 +#48 := (- #46)
  3.1874 +#47 := (< #46 0::real)
  3.1875 +#49 := (ite #47 #48 #46)
  3.1876 +#50 := (< #49 #41)
  3.1877 +#185 := (iff #50 #180)
  3.1878 +#143 := (< #140 0::real)
  3.1879 +#155 := (ite #143 #150 #140)
  3.1880 +#158 := (< #155 #104)
  3.1881 +#183 := (iff #158 #180)
  3.1882 +#173 := (< #168 #104)
  3.1883 +#181 := (iff #173 #180)
  3.1884 +#182 := [rewrite]: #181
  3.1885 +#174 := (iff #158 #173)
  3.1886 +#171 := (= #155 #168)
  3.1887 +#162 := (not #161)
  3.1888 +#165 := (ite #162 #150 #140)
  3.1889 +#169 := (= #165 #168)
  3.1890 +#170 := [rewrite]: #169
  3.1891 +#166 := (= #155 #165)
  3.1892 +#163 := (iff #143 #162)
  3.1893 +#164 := [rewrite]: #163
  3.1894 +#167 := [monotonicity #164]: #166
  3.1895 +#172 := [trans #167 #170]: #171
  3.1896 +#175 := [monotonicity #172]: #174
  3.1897 +#184 := [trans #175 #182]: #183
  3.1898 +#159 := (iff #50 #158)
  3.1899 +#156 := (= #49 #155)
  3.1900 +#141 := (= #46 #140)
  3.1901 +#142 := [rewrite]: #141
  3.1902 +#153 := (= #48 #150)
  3.1903 +#146 := (- #140)
  3.1904 +#151 := (= #146 #150)
  3.1905 +#152 := [rewrite]: #151
  3.1906 +#147 := (= #48 #146)
  3.1907 +#148 := [monotonicity #142]: #147
  3.1908 +#154 := [trans #148 #152]: #153
  3.1909 +#144 := (iff #47 #143)
  3.1910 +#145 := [monotonicity #142]: #144
  3.1911 +#157 := [monotonicity #145 #154 #142]: #156
  3.1912 +#160 := [monotonicity #157 #108]: #159
  3.1913 +#186 := [trans #160 #184]: #185
  3.1914 +#139 := [asserted]: #50
  3.1915 +#187 := [mp #139 #186]: #180
  3.1916 +#299 := (+ #140 #176)
  3.1917 +#300 := (<= #299 0::real)
  3.1918 +#290 := (= #140 #168)
  3.1919 +#329 := [hypothesis]: #162
  3.1920 +#191 := (+ #29 #149)
  3.1921 +#192 := (<= #191 0::real)
  3.1922 +#51 := (<= #29 #45)
  3.1923 +#193 := (iff #51 #192)
  3.1924 +#194 := [rewrite]: #193
  3.1925 +#188 := [asserted]: #51
  3.1926 +#195 := [mp #188 #194]: #192
  3.1927 +#298 := (+ #75 #127)
  3.1928 +#301 := (<= #298 0::real)
  3.1929 +#284 := (= #75 #119)
  3.1930 +#302 := [hypothesis]: #113
  3.1931 +#296 := (+ #85 #127)
  3.1932 +#297 := (<= #296 0::real)
  3.1933 +#285 := (= #85 #119)
  3.1934 +#288 := (or #112 #285)
  3.1935 +#289 := [def-axiom]: #288
  3.1936 +#303 := [unit-resolution #289 #302]: #285
  3.1937 +#304 := (not #285)
  3.1938 +#305 := (or #304 #297)
  3.1939 +#306 := [th-lemma]: #305
  3.1940 +#307 := [unit-resolution #306 #303]: #297
  3.1941 +#315 := (not #290)
  3.1942 +#310 := (not #300)
  3.1943 +#311 := (or #310 #112)
  3.1944 +#308 := [hypothesis]: #300
  3.1945 +#309 := [th-lemma #308 #307 #138 #302 #187 #195]: false
  3.1946 +#312 := [lemma #309]: #311
  3.1947 +#322 := [unit-resolution #312 #302]: #310
  3.1948 +#316 := (or #315 #300)
  3.1949 +#313 := [hypothesis]: #310
  3.1950 +#314 := [hypothesis]: #290
  3.1951 +#317 := [th-lemma]: #316
  3.1952 +#318 := [unit-resolution #317 #314 #313]: false
  3.1953 +#319 := [lemma #318]: #316
  3.1954 +#323 := [unit-resolution #319 #322]: #315
  3.1955 +#292 := (or #162 #290)
  3.1956 +#293 := [def-axiom]: #292
  3.1957 +#324 := [unit-resolution #293 #323]: #162
  3.1958 +#325 := [th-lemma #324 #307 #138 #302 #195]: false
  3.1959 +#326 := [lemma #325]: #112
  3.1960 +#286 := (or #113 #284)
  3.1961 +#287 := [def-axiom]: #286
  3.1962 +#330 := [unit-resolution #287 #326]: #284
  3.1963 +#331 := (not #284)
  3.1964 +#332 := (or #331 #301)
  3.1965 +#333 := [th-lemma]: #332
  3.1966 +#334 := [unit-resolution #333 #330]: #301
  3.1967 +#335 := [th-lemma #326 #334 #195 #329 #138]: false
  3.1968 +#336 := [lemma #335]: #161
  3.1969 +#327 := [unit-resolution #293 #336]: #290
  3.1970 +#328 := [unit-resolution #319 #327]: #300
  3.1971 +[th-lemma #326 #334 #195 #328 #187 #138]: false
  3.1972 +unsat
  3.1973 +GX51o3DUO/UBS3eNP2P9kA 285 0
  3.1974 +#2 := false
  3.1975 +#7 := 0::real
  3.1976 +decl uf_4 :: real
  3.1977 +#16 := uf_4
  3.1978 +#40 := -1::real
  3.1979 +#116 := (* -1::real uf_4)
  3.1980 +decl uf_3 :: real
  3.1981 +#11 := uf_3
  3.1982 +#117 := (+ uf_3 #116)
  3.1983 +#128 := (<= #117 0::real)
  3.1984 +#129 := (not #128)
  3.1985 +#220 := 2/3::real
  3.1986 +#221 := (* 2/3::real uf_3)
  3.1987 +#222 := (+ #221 #116)
  3.1988 +decl uf_2 :: real
  3.1989 +#5 := uf_2
  3.1990 +#67 := 1/3::real
  3.1991 +#68 := (* 1/3::real uf_2)
  3.1992 +#233 := (+ #68 #222)
  3.1993 +#243 := (<= #233 0::real)
  3.1994 +#268 := (not #243)
  3.1995 +#287 := [hypothesis]: #268
  3.1996 +#41 := (* -1::real uf_2)
  3.1997 +decl uf_1 :: real
  3.1998 +#4 := uf_1
  3.1999 +#42 := (+ uf_1 #41)
  3.2000 +#79 := (>= #42 0::real)
  3.2001 +#80 := (not #79)
  3.2002 +#297 := (or #80 #243)
  3.2003 +#158 := (+ uf_1 #116)
  3.2004 +#159 := (<= #158 0::real)
  3.2005 +#22 := (<= uf_1 uf_4)
  3.2006 +#160 := (iff #22 #159)
  3.2007 +#161 := [rewrite]: #160
  3.2008 +#155 := [asserted]: #22
  3.2009 +#162 := [mp #155 #161]: #159
  3.2010 +#200 := (* 1/3::real uf_3)
  3.2011 +#198 := -4/3::real
  3.2012 +#199 := (* -4/3::real uf_2)
  3.2013 +#201 := (+ #199 #200)
  3.2014 +#202 := (+ uf_1 #201)
  3.2015 +#203 := (>= #202 0::real)
  3.2016 +#258 := (not #203)
  3.2017 +#292 := [hypothesis]: #79
  3.2018 +#293 := (or #80 #258)
  3.2019 +#69 := -1/3::real
  3.2020 +#70 := (* -1/3::real uf_3)
  3.2021 +#186 := -2/3::real
  3.2022 +#187 := (* -2/3::real uf_2)
  3.2023 +#188 := (+ #187 #70)
  3.2024 +#189 := (+ uf_1 #188)
  3.2025 +#204 := (<= #189 0::real)
  3.2026 +#205 := (ite #79 #203 #204)
  3.2027 +#210 := (not #205)
  3.2028 +#51 := (* -1::real uf_1)
  3.2029 +#52 := (+ #51 uf_2)
  3.2030 +#86 := (ite #79 #42 #52)
  3.2031 +#94 := (* -1::real #86)
  3.2032 +#95 := (+ #70 #94)
  3.2033 +#96 := (+ #68 #95)
  3.2034 +#97 := (<= #96 0::real)
  3.2035 +#98 := (not #97)
  3.2036 +#211 := (iff #98 #210)
  3.2037 +#208 := (iff #97 #205)
  3.2038 +#182 := 4/3::real
  3.2039 +#183 := (* 4/3::real uf_2)
  3.2040 +#184 := (+ #183 #70)
  3.2041 +#185 := (+ #51 #184)
  3.2042 +#190 := (ite #79 #185 #189)
  3.2043 +#195 := (<= #190 0::real)
  3.2044 +#206 := (iff #195 #205)
  3.2045 +#207 := [rewrite]: #206
  3.2046 +#196 := (iff #97 #195)
  3.2047 +#193 := (= #96 #190)
  3.2048 +#172 := (+ #41 #70)
  3.2049 +#173 := (+ uf_1 #172)
  3.2050 +#170 := (+ uf_2 #70)
  3.2051 +#171 := (+ #51 #170)
  3.2052 +#174 := (ite #79 #171 #173)
  3.2053 +#179 := (+ #68 #174)
  3.2054 +#191 := (= #179 #190)
  3.2055 +#192 := [rewrite]: #191
  3.2056 +#180 := (= #96 #179)
  3.2057 +#177 := (= #95 #174)
  3.2058 +#164 := (ite #79 #52 #42)
  3.2059 +#167 := (+ #70 #164)
  3.2060 +#175 := (= #167 #174)
  3.2061 +#176 := [rewrite]: #175
  3.2062 +#168 := (= #95 #167)
  3.2063 +#156 := (= #94 #164)
  3.2064 +#165 := [rewrite]: #156
  3.2065 +#169 := [monotonicity #165]: #168
  3.2066 +#178 := [trans #169 #176]: #177
  3.2067 +#181 := [monotonicity #178]: #180
  3.2068 +#194 := [trans #181 #192]: #193
  3.2069 +#197 := [monotonicity #194]: #196
  3.2070 +#209 := [trans #197 #207]: #208
  3.2071 +#212 := [monotonicity #209]: #211
  3.2072 +#13 := 3::real
  3.2073 +#12 := (- uf_2 uf_3)
  3.2074 +#14 := (/ #12 3::real)
  3.2075 +#6 := (- uf_1 uf_2)
  3.2076 +#9 := (- #6)
  3.2077 +#8 := (< #6 0::real)
  3.2078 +#10 := (ite #8 #9 #6)
  3.2079 +#15 := (< #10 #14)
  3.2080 +#103 := (iff #15 #98)
  3.2081 +#71 := (+ #68 #70)
  3.2082 +#45 := (< #42 0::real)
  3.2083 +#57 := (ite #45 #52 #42)
  3.2084 +#76 := (< #57 #71)
  3.2085 +#101 := (iff #76 #98)
  3.2086 +#91 := (< #86 #71)
  3.2087 +#99 := (iff #91 #98)
  3.2088 +#100 := [rewrite]: #99
  3.2089 +#92 := (iff #76 #91)
  3.2090 +#89 := (= #57 #86)
  3.2091 +#83 := (ite #80 #52 #42)
  3.2092 +#87 := (= #83 #86)
  3.2093 +#88 := [rewrite]: #87
  3.2094 +#84 := (= #57 #83)
  3.2095 +#81 := (iff #45 #80)
  3.2096 +#82 := [rewrite]: #81
  3.2097 +#85 := [monotonicity #82]: #84
  3.2098 +#90 := [trans #85 #88]: #89
  3.2099 +#93 := [monotonicity #90]: #92
  3.2100 +#102 := [trans #93 #100]: #101
  3.2101 +#77 := (iff #15 #76)
  3.2102 +#74 := (= #14 #71)
  3.2103 +#60 := (* -1::real uf_3)
  3.2104 +#61 := (+ uf_2 #60)
  3.2105 +#64 := (/ #61 3::real)
  3.2106 +#72 := (= #64 #71)
  3.2107 +#73 := [rewrite]: #72
  3.2108 +#65 := (= #14 #64)
  3.2109 +#62 := (= #12 #61)
  3.2110 +#63 := [rewrite]: #62
  3.2111 +#66 := [monotonicity #63]: #65
  3.2112 +#75 := [trans #66 #73]: #74
  3.2113 +#58 := (= #10 #57)
  3.2114 +#43 := (= #6 #42)
  3.2115 +#44 := [rewrite]: #43
  3.2116 +#55 := (= #9 #52)
  3.2117 +#48 := (- #42)
  3.2118 +#53 := (= #48 #52)
  3.2119 +#54 := [rewrite]: #53
  3.2120 +#49 := (= #9 #48)
  3.2121 +#50 := [monotonicity #44]: #49
  3.2122 +#56 := [trans #50 #54]: #55
  3.2123 +#46 := (iff #8 #45)
  3.2124 +#47 := [monotonicity #44]: #46
  3.2125 +#59 := [monotonicity #47 #56 #44]: #58
  3.2126 +#78 := [monotonicity #59 #75]: #77
  3.2127 +#104 := [trans #78 #102]: #103
  3.2128 +#39 := [asserted]: #15
  3.2129 +#105 := [mp #39 #104]: #98
  3.2130 +#213 := [mp #105 #212]: #210
  3.2131 +#259 := (or #205 #80 #258)
  3.2132 +#260 := [def-axiom]: #259
  3.2133 +#294 := [unit-resolution #260 #213]: #293
  3.2134 +#295 := [unit-resolution #294 #292]: #258
  3.2135 +#296 := [th-lemma #287 #292 #295 #162]: false
  3.2136 +#298 := [lemma #296]: #297
  3.2137 +#299 := [unit-resolution #298 #287]: #80
  3.2138 +#261 := (not #204)
  3.2139 +#281 := (or #79 #261)
  3.2140 +#262 := (or #205 #79 #261)
  3.2141 +#263 := [def-axiom]: #262
  3.2142 +#282 := [unit-resolution #263 #213]: #281
  3.2143 +#300 := [unit-resolution #282 #299]: #261
  3.2144 +#290 := (or #79 #204 #243)
  3.2145 +#276 := [hypothesis]: #261
  3.2146 +#288 := [hypothesis]: #80
  3.2147 +#289 := [th-lemma #288 #276 #162 #287]: false
  3.2148 +#291 := [lemma #289]: #290
  3.2149 +#301 := [unit-resolution #291 #300 #299 #287]: false
  3.2150 +#302 := [lemma #301]: #243
  3.2151 +#303 := (or #129 #268)
  3.2152 +#223 := (* -4/3::real uf_3)
  3.2153 +#224 := (+ #223 uf_4)
  3.2154 +#234 := (+ #68 #224)
  3.2155 +#244 := (<= #234 0::real)
  3.2156 +#245 := (ite #128 #243 #244)
  3.2157 +#250 := (not #245)
  3.2158 +#107 := (+ #60 uf_4)
  3.2159 +#135 := (ite #128 #107 #117)
  3.2160 +#143 := (* -1::real #135)
  3.2161 +#144 := (+ #70 #143)
  3.2162 +#145 := (+ #68 #144)
  3.2163 +#146 := (<= #145 0::real)
  3.2164 +#147 := (not #146)
  3.2165 +#251 := (iff #147 #250)
  3.2166 +#248 := (iff #146 #245)
  3.2167 +#235 := (ite #128 #233 #234)
  3.2168 +#240 := (<= #235 0::real)
  3.2169 +#246 := (iff #240 #245)
  3.2170 +#247 := [rewrite]: #246
  3.2171 +#241 := (iff #146 #240)
  3.2172 +#238 := (= #145 #235)
  3.2173 +#225 := (ite #128 #222 #224)
  3.2174 +#230 := (+ #68 #225)
  3.2175 +#236 := (= #230 #235)
  3.2176 +#237 := [rewrite]: #236
  3.2177 +#231 := (= #145 #230)
  3.2178 +#228 := (= #144 #225)
  3.2179 +#214 := (ite #128 #117 #107)
  3.2180 +#217 := (+ #70 #214)
  3.2181 +#226 := (= #217 #225)
  3.2182 +#227 := [rewrite]: #226
  3.2183 +#218 := (= #144 #217)
  3.2184 +#215 := (= #143 #214)
  3.2185 +#216 := [rewrite]: #215
  3.2186 +#219 := [monotonicity #216]: #218
  3.2187 +#229 := [trans #219 #227]: #228
  3.2188 +#232 := [monotonicity #229]: #231
  3.2189 +#239 := [trans #232 #237]: #238
  3.2190 +#242 := [monotonicity #239]: #241
  3.2191 +#249 := [trans #242 #247]: #248
  3.2192 +#252 := [monotonicity #249]: #251
  3.2193 +#17 := (- uf_4 uf_3)
  3.2194 +#19 := (- #17)
  3.2195 +#18 := (< #17 0::real)
  3.2196 +#20 := (ite #18 #19 #17)
  3.2197 +#21 := (< #20 #14)
  3.2198 +#152 := (iff #21 #147)
  3.2199 +#110 := (< #107 0::real)
  3.2200 +#122 := (ite #110 #117 #107)
  3.2201 +#125 := (< #122 #71)
  3.2202 +#150 := (iff #125 #147)
  3.2203 +#140 := (< #135 #71)
  3.2204 +#148 := (iff #140 #147)
  3.2205 +#149 := [rewrite]: #148
  3.2206 +#141 := (iff #125 #140)
  3.2207 +#138 := (= #122 #135)
  3.2208 +#132 := (ite #129 #117 #107)
  3.2209 +#136 := (= #132 #135)
  3.2210 +#137 := [rewrite]: #136
  3.2211 +#133 := (= #122 #132)
  3.2212 +#130 := (iff #110 #129)
  3.2213 +#131 := [rewrite]: #130
  3.2214 +#134 := [monotonicity #131]: #133
  3.2215 +#139 := [trans #134 #137]: #138
  3.2216 +#142 := [monotonicity #139]: #141
  3.2217 +#151 := [trans #142 #149]: #150
  3.2218 +#126 := (iff #21 #125)
  3.2219 +#123 := (= #20 #122)
  3.2220 +#108 := (= #17 #107)
  3.2221 +#109 := [rewrite]: #108
  3.2222 +#120 := (= #19 #117)
  3.2223 +#113 := (- #107)
  3.2224 +#118 := (= #113 #117)
  3.2225 +#119 := [rewrite]: #118
  3.2226 +#114 := (= #19 #113)
  3.2227 +#115 := [monotonicity #109]: #114
  3.2228 +#121 := [trans #115 #119]: #120
  3.2229 +#111 := (iff #18 #110)
  3.2230 +#112 := [monotonicity #109]: #111
  3.2231 +#124 := [monotonicity #112 #121 #109]: #123
  3.2232 +#127 := [monotonicity #124 #75]: #126
  3.2233 +#153 := [trans #127 #151]: #152
  3.2234 +#106 := [asserted]: #21
  3.2235 +#154 := [mp #106 #153]: #147
  3.2236 +#253 := [mp #154 #252]: #250
  3.2237 +#269 := (or #245 #129 #268)
  3.2238 +#270 := [def-axiom]: #269
  3.2239 +#304 := [unit-resolution #270 #253]: #303
  3.2240 +#305 := [unit-resolution #304 #302]: #129
  3.2241 +#271 := (not #244)
  3.2242 +#306 := (or #128 #271)
  3.2243 +#272 := (or #245 #128 #271)
  3.2244 +#273 := [def-axiom]: #272
  3.2245 +#307 := [unit-resolution #273 #253]: #306
  3.2246 +#308 := [unit-resolution #307 #305]: #271
  3.2247 +#285 := (or #128 #244)
  3.2248 +#274 := [hypothesis]: #271
  3.2249 +#275 := [hypothesis]: #129
  3.2250 +#278 := (or #204 #128 #244)
  3.2251 +#277 := [th-lemma #276 #275 #274 #162]: false
  3.2252 +#279 := [lemma #277]: #278
  3.2253 +#280 := [unit-resolution #279 #275 #274]: #204
  3.2254 +#283 := [unit-resolution #282 #280]: #79
  3.2255 +#284 := [th-lemma #275 #274 #283 #162]: false
  3.2256 +#286 := [lemma #284]: #285
  3.2257 +[unit-resolution #286 #308 #305]: false
  3.2258 +unsat
  3.2259 +cebG074uorSr8ODzgTmcKg 97 0
  3.2260 +#2 := false
  3.2261 +#18 := 0::real
  3.2262 +decl uf_1 :: (-> T2 T1 real)
  3.2263 +decl uf_5 :: T1
  3.2264 +#11 := uf_5
  3.2265 +decl uf_2 :: T2
  3.2266 +#4 := uf_2
  3.2267 +#20 := (uf_1 uf_2 uf_5)
  3.2268 +#42 := -1::real
  3.2269 +#53 := (* -1::real #20)
  3.2270 +decl uf_3 :: T2
  3.2271 +#7 := uf_3
  3.2272 +#19 := (uf_1 uf_3 uf_5)
  3.2273 +#54 := (+ #19 #53)
  3.2274 +#63 := (<= #54 0::real)
  3.2275 +#21 := (- #19 #20)
  3.2276 +#22 := (< 0::real #21)
  3.2277 +#23 := (not #22)
  3.2278 +#74 := (iff #23 #63)
  3.2279 +#57 := (< 0::real #54)
  3.2280 +#60 := (not #57)
  3.2281 +#72 := (iff #60 #63)
  3.2282 +#64 := (not #63)
  3.2283 +#67 := (not #64)
  3.2284 +#70 := (iff #67 #63)
  3.2285 +#71 := [rewrite]: #70
  3.2286 +#68 := (iff #60 #67)
  3.2287 +#65 := (iff #57 #64)
  3.2288 +#66 := [rewrite]: #65
  3.2289 +#69 := [monotonicity #66]: #68
  3.2290 +#73 := [trans #69 #71]: #72
  3.2291 +#61 := (iff #23 #60)
  3.2292 +#58 := (iff #22 #57)
  3.2293 +#55 := (= #21 #54)
  3.2294 +#56 := [rewrite]: #55
  3.2295 +#59 := [monotonicity #56]: #58
  3.2296 +#62 := [monotonicity #59]: #61
  3.2297 +#75 := [trans #62 #73]: #74
  3.2298 +#41 := [asserted]: #23
  3.2299 +#76 := [mp #41 #75]: #63
  3.2300 +#5 := (:var 0 T1)
  3.2301 +#8 := (uf_1 uf_3 #5)
  3.2302 +#141 := (pattern #8)
  3.2303 +#6 := (uf_1 uf_2 #5)
  3.2304 +#140 := (pattern #6)
  3.2305 +#45 := (* -1::real #8)
  3.2306 +#46 := (+ #6 #45)
  3.2307 +#44 := (>= #46 0::real)
  3.2308 +#43 := (not #44)
  3.2309 +#142 := (forall (vars (?x1 T1)) (:pat #140 #141) #43)
  3.2310 +#49 := (forall (vars (?x1 T1)) #43)
  3.2311 +#145 := (iff #49 #142)
  3.2312 +#143 := (iff #43 #43)
  3.2313 +#144 := [refl]: #143
  3.2314 +#146 := [quant-intro #144]: #145
  3.2315 +#80 := (~ #49 #49)
  3.2316 +#82 := (~ #43 #43)
  3.2317 +#83 := [refl]: #82
  3.2318 +#81 := [nnf-pos #83]: #80
  3.2319 +#9 := (< #6 #8)
  3.2320 +#10 := (forall (vars (?x1 T1)) #9)
  3.2321 +#50 := (iff #10 #49)
  3.2322 +#47 := (iff #9 #43)
  3.2323 +#48 := [rewrite]: #47
  3.2324 +#51 := [quant-intro #48]: #50
  3.2325 +#39 := [asserted]: #10
  3.2326 +#52 := [mp #39 #51]: #49
  3.2327 +#79 := [mp~ #52 #81]: #49
  3.2328 +#147 := [mp #79 #146]: #142
  3.2329 +#164 := (not #142)
  3.2330 +#165 := (or #164 #64)
  3.2331 +#148 := (* -1::real #19)
  3.2332 +#149 := (+ #20 #148)
  3.2333 +#150 := (>= #149 0::real)
  3.2334 +#151 := (not #150)
  3.2335 +#166 := (or #164 #151)
  3.2336 +#168 := (iff #166 #165)
  3.2337 +#170 := (iff #165 #165)
  3.2338 +#171 := [rewrite]: #170
  3.2339 +#162 := (iff #151 #64)
  3.2340 +#160 := (iff #150 #63)
  3.2341 +#152 := (+ #148 #20)
  3.2342 +#155 := (>= #152 0::real)
  3.2343 +#158 := (iff #155 #63)
  3.2344 +#159 := [rewrite]: #158
  3.2345 +#156 := (iff #150 #155)
  3.2346 +#153 := (= #149 #152)
  3.2347 +#154 := [rewrite]: #153
  3.2348 +#157 := [monotonicity #154]: #156
  3.2349 +#161 := [trans #157 #159]: #160
  3.2350 +#163 := [monotonicity #161]: #162
  3.2351 +#169 := [monotonicity #163]: #168
  3.2352 +#172 := [trans #169 #171]: #168
  3.2353 +#167 := [quant-inst]: #166
  3.2354 +#173 := [mp #167 #172]: #165
  3.2355 +[unit-resolution #173 #147 #76]: false
  3.2356 +unsat
  3.2357 +DKRtrJ2XceCkITuNwNViRw 57 0
  3.2358 +#2 := false
  3.2359 +#4 := 0::real
  3.2360 +decl uf_1 :: (-> T2 real)
  3.2361 +decl uf_2 :: (-> T1 T1 T2)
  3.2362 +decl uf_12 :: (-> T4 T1)
  3.2363 +decl uf_4 :: T4
  3.2364 +#11 := uf_4
  3.2365 +#39 := (uf_12 uf_4)
  3.2366 +decl uf_10 :: T4
  3.2367 +#27 := uf_10
  3.2368 +#38 := (uf_12 uf_10)
  3.2369 +#40 := (uf_2 #38 #39)
  3.2370 +#41 := (uf_1 #40)
  3.2371 +#264 := (>= #41 0::real)
  3.2372 +#266 := (not #264)
  3.2373 +#43 := (= #41 0::real)
  3.2374 +#44 := (not #43)
  3.2375 +#131 := [asserted]: #44
  3.2376 +#272 := (or #43 #266)
  3.2377 +#42 := (<= #41 0::real)
  3.2378 +#130 := [asserted]: #42
  3.2379 +#265 := (not #42)
  3.2380 +#270 := (or #43 #265 #266)
  3.2381 +#271 := [th-lemma]: #270
  3.2382 +#273 := [unit-resolution #271 #130]: #272
  3.2383 +#274 := [unit-resolution #273 #131]: #266
  3.2384 +#6 := (:var 0 T1)
  3.2385 +#5 := (:var 1 T1)
  3.2386 +#7 := (uf_2 #5 #6)
  3.2387 +#241 := (pattern #7)
  3.2388 +#8 := (uf_1 #7)
  3.2389 +#65 := (>= #8 0::real)
  3.2390 +#242 := (forall (vars (?x1 T1) (?x2 T1)) (:pat #241) #65)
  3.2391 +#66 := (forall (vars (?x1 T1) (?x2 T1)) #65)
  3.2392 +#245 := (iff #66 #242)
  3.2393 +#243 := (iff #65 #65)
  3.2394 +#244 := [refl]: #243
  3.2395 +#246 := [quant-intro #244]: #245
  3.2396 +#149 := (~ #66 #66)
  3.2397 +#151 := (~ #65 #65)
  3.2398 +#152 := [refl]: #151
  3.2399 +#150 := [nnf-pos #152]: #149
  3.2400 +#9 := (<= 0::real #8)
  3.2401 +#10 := (forall (vars (?x1 T1) (?x2 T1)) #9)
  3.2402 +#67 := (iff #10 #66)
  3.2403 +#63 := (iff #9 #65)
  3.2404 +#64 := [rewrite]: #63
  3.2405 +#68 := [quant-intro #64]: #67
  3.2406 +#60 := [asserted]: #10
  3.2407 +#69 := [mp #60 #68]: #66
  3.2408 +#147 := [mp~ #69 #150]: #66
  3.2409 +#247 := [mp #147 #246]: #242
  3.2410 +#267 := (not #242)
  3.2411 +#268 := (or #267 #264)
  3.2412 +#269 := [quant-inst]: #268
  3.2413 +[unit-resolution #269 #247 #274]: false
  3.2414 +unsat
  3.2415 +97KJAJfUio+nGchEHWvgAw 91 0
  3.2416 +#2 := false
  3.2417 +#38 := 0::real
  3.2418 +decl uf_1 :: (-> T1 T2 real)
  3.2419 +decl uf_3 :: T2
  3.2420 +#5 := uf_3
  3.2421 +decl uf_4 :: T1
  3.2422 +#7 := uf_4
  3.2423 +#8 := (uf_1 uf_4 uf_3)
  3.2424 +#35 := -1::real
  3.2425 +#36 := (* -1::real #8)
  3.2426 +decl uf_2 :: T1
  3.2427 +#4 := uf_2
  3.2428 +#6 := (uf_1 uf_2 uf_3)
  3.2429 +#37 := (+ #6 #36)
  3.2430 +#130 := (>= #37 0::real)
  3.2431 +#155 := (not #130)
  3.2432 +#43 := (= #6 #8)
  3.2433 +#55 := (not #43)
  3.2434 +#15 := (= #8 #6)
  3.2435 +#16 := (not #15)
  3.2436 +#56 := (iff #16 #55)
  3.2437 +#53 := (iff #15 #43)
  3.2438 +#54 := [rewrite]: #53
  3.2439 +#57 := [monotonicity #54]: #56
  3.2440 +#34 := [asserted]: #16
  3.2441 +#60 := [mp #34 #57]: #55
  3.2442 +#158 := (or #43 #155)
  3.2443 +#39 := (<= #37 0::real)
  3.2444 +#9 := (<= #6 #8)
  3.2445 +#40 := (iff #9 #39)
  3.2446 +#41 := [rewrite]: #40
  3.2447 +#32 := [asserted]: #9
  3.2448 +#42 := [mp #32 #41]: #39
  3.2449 +#154 := (not #39)
  3.2450 +#156 := (or #43 #154 #155)
  3.2451 +#157 := [th-lemma]: #156
  3.2452 +#159 := [unit-resolution #157 #42]: #158
  3.2453 +#160 := [unit-resolution #159 #60]: #155
  3.2454 +#10 := (:var 0 T2)
  3.2455 +#12 := (uf_1 uf_2 #10)
  3.2456 +#123 := (pattern #12)
  3.2457 +#11 := (uf_1 uf_4 #10)
  3.2458 +#122 := (pattern #11)
  3.2459 +#44 := (* -1::real #12)
  3.2460 +#45 := (+ #11 #44)
  3.2461 +#46 := (<= #45 0::real)
  3.2462 +#124 := (forall (vars (?x1 T2)) (:pat #122 #123) #46)
  3.2463 +#49 := (forall (vars (?x1 T2)) #46)
  3.2464 +#127 := (iff #49 #124)
  3.2465 +#125 := (iff #46 #46)
  3.2466 +#126 := [refl]: #125
  3.2467 +#128 := [quant-intro #126]: #127
  3.2468 +#62 := (~ #49 #49)
  3.2469 +#64 := (~ #46 #46)
  3.2470 +#65 := [refl]: #64
  3.2471 +#63 := [nnf-pos #65]: #62
  3.2472 +#13 := (<= #11 #12)
  3.2473 +#14 := (forall (vars (?x1 T2)) #13)
  3.2474 +#50 := (iff #14 #49)
  3.2475 +#47 := (iff #13 #46)
  3.2476 +#48 := [rewrite]: #47
  3.2477 +#51 := [quant-intro #48]: #50
  3.2478 +#33 := [asserted]: #14
  3.2479 +#52 := [mp #33 #51]: #49
  3.2480 +#61 := [mp~ #52 #63]: #49
  3.2481 +#129 := [mp #61 #128]: #124
  3.2482 +#144 := (not #124)
  3.2483 +#145 := (or #144 #130)
  3.2484 +#131 := (* -1::real #6)
  3.2485 +#132 := (+ #8 #131)
  3.2486 +#133 := (<= #132 0::real)
  3.2487 +#146 := (or #144 #133)
  3.2488 +#148 := (iff #146 #145)
  3.2489 +#150 := (iff #145 #145)
  3.2490 +#151 := [rewrite]: #150
  3.2491 +#142 := (iff #133 #130)
  3.2492 +#134 := (+ #131 #8)
  3.2493 +#137 := (<= #134 0::real)
  3.2494 +#140 := (iff #137 #130)
  3.2495 +#141 := [rewrite]: #140
  3.2496 +#138 := (iff #133 #137)
  3.2497 +#135 := (= #132 #134)
  3.2498 +#136 := [rewrite]: #135
  3.2499 +#139 := [monotonicity #136]: #138
  3.2500 +#143 := [trans #139 #141]: #142
  3.2501 +#149 := [monotonicity #143]: #148
  3.2502 +#152 := [trans #149 #151]: #148
  3.2503 +#147 := [quant-inst]: #146
  3.2504 +#153 := [mp #147 #152]: #145
  3.2505 +[unit-resolution #153 #129 #160]: false
  3.2506 +unsat
  3.2507 +flJYbeWfe+t2l/zsRqdujA 149 0
  3.2508 +#2 := false
  3.2509 +#19 := 0::real
  3.2510 +decl uf_1 :: (-> T1 T2 real)
  3.2511 +decl uf_3 :: T2
  3.2512 +#5 := uf_3
  3.2513 +decl uf_4 :: T1
  3.2514 +#7 := uf_4
  3.2515 +#8 := (uf_1 uf_4 uf_3)
  3.2516 +#44 := -1::real
  3.2517 +#156 := (* -1::real #8)
  3.2518 +decl uf_2 :: T1
  3.2519 +#4 := uf_2
  3.2520 +#6 := (uf_1 uf_2 uf_3)
  3.2521 +#203 := (+ #6 #156)
  3.2522 +#205 := (>= #203 0::real)
  3.2523 +#9 := (= #6 #8)
  3.2524 +#40 := [asserted]: #9
  3.2525 +#208 := (not #9)
  3.2526 +#209 := (or #208 #205)
  3.2527 +#210 := [th-lemma]: #209
  3.2528 +#211 := [unit-resolution #210 #40]: #205
  3.2529 +decl uf_5 :: T1
  3.2530 +#12 := uf_5
  3.2531 +#22 := (uf_1 uf_5 uf_3)
  3.2532 +#160 := (* -1::real #22)
  3.2533 +#161 := (+ #6 #160)
  3.2534 +#207 := (>= #161 0::real)
  3.2535 +#222 := (not #207)
  3.2536 +#206 := (= #6 #22)
  3.2537 +#216 := (not #206)
  3.2538 +#62 := (= #8 #22)
  3.2539 +#70 := (not #62)
  3.2540 +#217 := (iff #70 #216)
  3.2541 +#214 := (iff #62 #206)
  3.2542 +#212 := (iff #206 #62)
  3.2543 +#213 := [monotonicity #40]: #212
  3.2544 +#215 := [symm #213]: #214
  3.2545 +#218 := [monotonicity #215]: #217
  3.2546 +#23 := (= #22 #8)
  3.2547 +#24 := (not #23)
  3.2548 +#71 := (iff #24 #70)
  3.2549 +#68 := (iff #23 #62)
  3.2550 +#69 := [rewrite]: #68
  3.2551 +#72 := [monotonicity #69]: #71
  3.2552 +#43 := [asserted]: #24
  3.2553 +#75 := [mp #43 #72]: #70
  3.2554 +#219 := [mp #75 #218]: #216
  3.2555 +#225 := (or #206 #222)
  3.2556 +#162 := (<= #161 0::real)
  3.2557 +#172 := (+ #8 #160)
  3.2558 +#173 := (>= #172 0::real)
  3.2559 +#178 := (not #173)
  3.2560 +#163 := (not #162)
  3.2561 +#181 := (or #163 #178)
  3.2562 +#184 := (not #181)
  3.2563 +#10 := (:var 0 T2)
  3.2564 +#15 := (uf_1 uf_4 #10)
  3.2565 +#149 := (pattern #15)
  3.2566 +#13 := (uf_1 uf_5 #10)
  3.2567 +#148 := (pattern #13)
  3.2568 +#11 := (uf_1 uf_2 #10)
  3.2569 +#147 := (pattern #11)
  3.2570 +#50 := (* -1::real #15)
  3.2571 +#51 := (+ #13 #50)
  3.2572 +#52 := (<= #51 0::real)
  3.2573 +#76 := (not #52)
  3.2574 +#45 := (* -1::real #13)
  3.2575 +#46 := (+ #11 #45)
  3.2576 +#47 := (<= #46 0::real)
  3.2577 +#78 := (not #47)
  3.2578 +#73 := (or #78 #76)
  3.2579 +#83 := (not #73)
  3.2580 +#150 := (forall (vars (?x1 T2)) (:pat #147 #148 #149) #83)
  3.2581 +#86 := (forall (vars (?x1 T2)) #83)
  3.2582 +#153 := (iff #86 #150)
  3.2583 +#151 := (iff #83 #83)
  3.2584 +#152 := [refl]: #151
  3.2585 +#154 := [quant-intro #152]: #153
  3.2586 +#55 := (and #47 #52)
  3.2587 +#58 := (forall (vars (?x1 T2)) #55)
  3.2588 +#87 := (iff #58 #86)
  3.2589 +#84 := (iff #55 #83)
  3.2590 +#85 := [rewrite]: #84
  3.2591 +#88 := [quant-intro #85]: #87
  3.2592 +#79 := (~ #58 #58)
  3.2593 +#81 := (~ #55 #55)
  3.2594 +#82 := [refl]: #81
  3.2595 +#80 := [nnf-pos #82]: #79
  3.2596 +#16 := (<= #13 #15)
  3.2597 +#14 := (<= #11 #13)
  3.2598 +#17 := (and #14 #16)
  3.2599 +#18 := (forall (vars (?x1 T2)) #17)
  3.2600 +#59 := (iff #18 #58)
  3.2601 +#56 := (iff #17 #55)
  3.2602 +#53 := (iff #16 #52)
  3.2603 +#54 := [rewrite]: #53
  3.2604 +#48 := (iff #14 #47)
  3.2605 +#49 := [rewrite]: #48
  3.2606 +#57 := [monotonicity #49 #54]: #56
  3.2607 +#60 := [quant-intro #57]: #59
  3.2608 +#41 := [asserted]: #18
  3.2609 +#61 := [mp #41 #60]: #58
  3.2610 +#77 := [mp~ #61 #80]: #58
  3.2611 +#89 := [mp #77 #88]: #86
  3.2612 +#155 := [mp #89 #154]: #150
  3.2613 +#187 := (not #150)
  3.2614 +#188 := (or #187 #184)
  3.2615 +#157 := (+ #22 #156)
  3.2616 +#158 := (<= #157 0::real)
  3.2617 +#159 := (not #158)
  3.2618 +#164 := (or #163 #159)
  3.2619 +#165 := (not #164)
  3.2620 +#189 := (or #187 #165)
  3.2621 +#191 := (iff #189 #188)
  3.2622 +#193 := (iff #188 #188)
  3.2623 +#194 := [rewrite]: #193
  3.2624 +#185 := (iff #165 #184)
  3.2625 +#182 := (iff #164 #181)
  3.2626 +#179 := (iff #159 #178)
  3.2627 +#176 := (iff #158 #173)
  3.2628 +#166 := (+ #156 #22)
  3.2629 +#169 := (<= #166 0::real)
  3.2630 +#174 := (iff #169 #173)
  3.2631 +#175 := [rewrite]: #174
  3.2632 +#170 := (iff #158 #169)
  3.2633 +#167 := (= #157 #166)
  3.2634 +#168 := [rewrite]: #167
  3.2635 +#171 := [monotonicity #168]: #170
  3.2636 +#177 := [trans #171 #175]: #176
  3.2637 +#180 := [monotonicity #177]: #179
  3.2638 +#183 := [monotonicity #180]: #182
  3.2639 +#186 := [monotonicity #183]: #185
  3.2640 +#192 := [monotonicity #186]: #191
  3.2641 +#195 := [trans #192 #194]: #191
  3.2642 +#190 := [quant-inst]: #189
  3.2643 +#196 := [mp #190 #195]: #188
  3.2644 +#220 := [unit-resolution #196 #155]: #184
  3.2645 +#197 := (or #181 #162)
  3.2646 +#198 := [def-axiom]: #197
  3.2647 +#221 := [unit-resolution #198 #220]: #162
  3.2648 +#223 := (or #206 #163 #222)
  3.2649 +#224 := [th-lemma]: #223
  3.2650 +#226 := [unit-resolution #224 #221]: #225
  3.2651 +#227 := [unit-resolution #226 #219]: #222
  3.2652 +#199 := (or #181 #173)
  3.2653 +#200 := [def-axiom]: #199
  3.2654 +#228 := [unit-resolution #200 #220]: #173
  3.2655 +[th-lemma #228 #227 #211]: false
  3.2656 +unsat
  3.2657 +rbrrQuQfaijtLkQizgEXnQ 222 0
  3.2658 +#2 := false
  3.2659 +#4 := 0::real
  3.2660 +decl uf_2 :: (-> T2 T1 real)
  3.2661 +decl uf_5 :: T1
  3.2662 +#15 := uf_5
  3.2663 +decl uf_3 :: T2
  3.2664 +#7 := uf_3
  3.2665 +#20 := (uf_2 uf_3 uf_5)
  3.2666 +decl uf_6 :: T2
  3.2667 +#17 := uf_6
  3.2668 +#18 := (uf_2 uf_6 uf_5)
  3.2669 +#59 := -1::real
  3.2670 +#73 := (* -1::real #18)
  3.2671 +#106 := (+ #73 #20)
  3.2672 +decl uf_1 :: real
  3.2673 +#5 := uf_1
  3.2674 +#78 := (* -1::real #20)
  3.2675 +#79 := (+ #18 #78)
  3.2676 +#144 := (+ uf_1 #79)
  3.2677 +#145 := (<= #144 0::real)
  3.2678 +#148 := (ite #145 uf_1 #106)
  3.2679 +#279 := (* -1::real #148)
  3.2680 +#280 := (+ uf_1 #279)
  3.2681 +#281 := (<= #280 0::real)
  3.2682 +#289 := (not #281)
  3.2683 +#72 := 1/2::real
  3.2684 +#151 := (* 1/2::real #148)
  3.2685 +#248 := (<= #151 0::real)
  3.2686 +#162 := (= #151 0::real)
  3.2687 +#24 := 2::real
  3.2688 +#27 := (- #20 #18)
  3.2689 +#28 := (<= uf_1 #27)
  3.2690 +#29 := (ite #28 uf_1 #27)
  3.2691 +#30 := (/ #29 2::real)
  3.2692 +#31 := (+ #18 #30)
  3.2693 +#32 := (= #31 #18)
  3.2694 +#33 := (not #32)
  3.2695 +#34 := (not #33)
  3.2696 +#165 := (iff #34 #162)
  3.2697 +#109 := (<= uf_1 #106)
  3.2698 +#112 := (ite #109 uf_1 #106)
  3.2699 +#118 := (* 1/2::real #112)
  3.2700 +#123 := (+ #18 #118)
  3.2701 +#129 := (= #18 #123)
  3.2702 +#163 := (iff #129 #162)
  3.2703 +#154 := (+ #18 #151)
  3.2704 +#157 := (= #18 #154)
  3.2705 +#160 := (iff #157 #162)
  3.2706 +#161 := [rewrite]: #160
  3.2707 +#158 := (iff #129 #157)
  3.2708 +#155 := (= #123 #154)
  3.2709 +#152 := (= #118 #151)
  3.2710 +#149 := (= #112 #148)
  3.2711 +#146 := (iff #109 #145)
  3.2712 +#147 := [rewrite]: #146
  3.2713 +#150 := [monotonicity #147]: #149
  3.2714 +#153 := [monotonicity #150]: #152
  3.2715 +#156 := [monotonicity #153]: #155
  3.2716 +#159 := [monotonicity #156]: #158
  3.2717 +#164 := [trans #159 #161]: #163
  3.2718 +#142 := (iff #34 #129)
  3.2719 +#134 := (not #129)
  3.2720 +#137 := (not #134)
  3.2721 +#140 := (iff #137 #129)
  3.2722 +#141 := [rewrite]: #140
  3.2723 +#138 := (iff #34 #137)
  3.2724 +#135 := (iff #33 #134)
  3.2725 +#132 := (iff #32 #129)
  3.2726 +#126 := (= #123 #18)
  3.2727 +#130 := (iff #126 #129)
  3.2728 +#131 := [rewrite]: #130
  3.2729 +#127 := (iff #32 #126)
  3.2730 +#124 := (= #31 #123)
  3.2731 +#121 := (= #30 #118)
  3.2732 +#115 := (/ #112 2::real)
  3.2733 +#119 := (= #115 #118)
  3.2734 +#120 := [rewrite]: #119
  3.2735 +#116 := (= #30 #115)
  3.2736 +#113 := (= #29 #112)
  3.2737 +#107 := (= #27 #106)
  3.2738 +#108 := [rewrite]: #107
  3.2739 +#110 := (iff #28 #109)
  3.2740 +#111 := [monotonicity #108]: #110
  3.2741 +#114 := [monotonicity #111 #108]: #113
  3.2742 +#117 := [monotonicity #114]: #116
  3.2743 +#122 := [trans #117 #120]: #121
  3.2744 +#125 := [monotonicity #122]: #124
  3.2745 +#128 := [monotonicity #125]: #127
  3.2746 +#133 := [trans #128 #131]: #132
  3.2747 +#136 := [monotonicity #133]: #135
  3.2748 +#139 := [monotonicity #136]: #138
  3.2749 +#143 := [trans #139 #141]: #142
  3.2750 +#166 := [trans #143 #164]: #165
  3.2751 +#105 := [asserted]: #34
  3.2752 +#167 := [mp #105 #166]: #162
  3.2753 +#283 := (not #162)
  3.2754 +#284 := (or #283 #248)
  3.2755 +#285 := [th-lemma]: #284
  3.2756 +#286 := [unit-resolution #285 #167]: #248
  3.2757 +#287 := [hypothesis]: #281
  3.2758 +#53 := (<= uf_1 0::real)
  3.2759 +#54 := (not #53)
  3.2760 +#6 := (< 0::real uf_1)
  3.2761 +#55 := (iff #6 #54)
  3.2762 +#56 := [rewrite]: #55
  3.2763 +#50 := [asserted]: #6
  3.2764 +#57 := [mp #50 #56]: #54
  3.2765 +#288 := [th-lemma #57 #287 #286]: false
  3.2766 +#290 := [lemma #288]: #289
  3.2767 +#241 := (= uf_1 #148)
  3.2768 +#242 := (= #106 #148)
  3.2769 +#299 := (not #242)
  3.2770 +#282 := (+ #106 #279)
  3.2771 +#291 := (<= #282 0::real)
  3.2772 +#296 := (not #291)
  3.2773 +decl uf_4 :: T2
  3.2774 +#10 := uf_4
  3.2775 +#16 := (uf_2 uf_4 uf_5)
  3.2776 +#260 := (+ #16 #78)
  3.2777 +#261 := (>= #260 0::real)
  3.2778 +#266 := (not #261)
  3.2779 +#8 := (:var 0 T1)
  3.2780 +#11 := (uf_2 uf_4 #8)
  3.2781 +#234 := (pattern #11)
  3.2782 +#9 := (uf_2 uf_3 #8)
  3.2783 +#233 := (pattern #9)
  3.2784 +#60 := (* -1::real #11)
  3.2785 +#61 := (+ #9 #60)
  3.2786 +#62 := (<= #61 0::real)
  3.2787 +#179 := (not #62)
  3.2788 +#235 := (forall (vars (?x1 T1)) (:pat #233 #234) #179)
  3.2789 +#178 := (forall (vars (?x1 T1)) #179)
  3.2790 +#238 := (iff #178 #235)
  3.2791 +#236 := (iff #179 #179)
  3.2792 +#237 := [refl]: #236
  3.2793 +#239 := [quant-intro #237]: #238
  3.2794 +#65 := (exists (vars (?x1 T1)) #62)
  3.2795 +#68 := (not #65)
  3.2796 +#175 := (~ #68 #178)
  3.2797 +#180 := (~ #179 #179)
  3.2798 +#177 := [refl]: #180
  3.2799 +#176 := [nnf-neg #177]: #175
  3.2800 +#12 := (<= #9 #11)
  3.2801 +#13 := (exists (vars (?x1 T1)) #12)
  3.2802 +#14 := (not #13)
  3.2803 +#69 := (iff #14 #68)
  3.2804 +#66 := (iff #13 #65)
  3.2805 +#63 := (iff #12 #62)
  3.2806 +#64 := [rewrite]: #63
  3.2807 +#67 := [quant-intro #64]: #66
  3.2808 +#70 := [monotonicity #67]: #69
  3.2809 +#51 := [asserted]: #14
  3.2810 +#71 := [mp #51 #70]: #68
  3.2811 +#173 := [mp~ #71 #176]: #178
  3.2812 +#240 := [mp #173 #239]: #235
  3.2813 +#269 := (not #235)
  3.2814 +#270 := (or #269 #266)
  3.2815 +#250 := (* -1::real #16)
  3.2816 +#251 := (+ #20 #250)
  3.2817 +#252 := (<= #251 0::real)
  3.2818 +#253 := (not #252)
  3.2819 +#271 := (or #269 #253)
  3.2820 +#273 := (iff #271 #270)
  3.2821 +#275 := (iff #270 #270)
  3.2822 +#276 := [rewrite]: #275
  3.2823 +#267 := (iff #253 #266)
  3.2824 +#264 := (iff #252 #261)
  3.2825 +#254 := (+ #250 #20)
  3.2826 +#257 := (<= #254 0::real)
  3.2827 +#262 := (iff #257 #261)
  3.2828 +#263 := [rewrite]: #262
  3.2829 +#258 := (iff #252 #257)
  3.2830 +#255 := (= #251 #254)
  3.2831 +#256 := [rewrite]: #255
  3.2832 +#259 := [monotonicity #256]: #258
  3.2833 +#265 := [trans #259 #263]: #264
  3.2834 +#268 := [monotonicity #265]: #267
  3.2835 +#274 := [monotonicity #268]: #273
  3.2836 +#277 := [trans #274 #276]: #273
  3.2837 +#272 := [quant-inst]: #271
  3.2838 +#278 := [mp #272 #277]: #270
  3.2839 +#293 := [unit-resolution #278 #240]: #266
  3.2840 +#90 := (* 1/2::real #20)
  3.2841 +#102 := (+ #73 #90)
  3.2842 +#89 := (* 1/2::real #16)
  3.2843 +#103 := (+ #89 #102)
  3.2844 +#100 := (>= #103 0::real)
  3.2845 +#23 := (+ #16 #20)
  3.2846 +#25 := (/ #23 2::real)
  3.2847 +#26 := (<= #18 #25)
  3.2848 +#98 := (iff #26 #100)
  3.2849 +#91 := (+ #89 #90)
  3.2850 +#94 := (<= #18 #91)
  3.2851 +#97 := (iff #94 #100)
  3.2852 +#99 := [rewrite]: #97
  3.2853 +#95 := (iff #26 #94)
  3.2854 +#92 := (= #25 #91)
  3.2855 +#93 := [rewrite]: #92
  3.2856 +#96 := [monotonicity #93]: #95
  3.2857 +#101 := [trans #96 #99]: #98
  3.2858 +#58 := [asserted]: #26
  3.2859 +#104 := [mp #58 #101]: #100
  3.2860 +#294 := [hypothesis]: #291
  3.2861 +#295 := [th-lemma #294 #104 #293 #286]: false
  3.2862 +#297 := [lemma #295]: #296
  3.2863 +#298 := [hypothesis]: #242
  3.2864 +#300 := (or #299 #291)
  3.2865 +#301 := [th-lemma]: #300
  3.2866 +#302 := [unit-resolution #301 #298 #297]: false
  3.2867 +#303 := [lemma #302]: #299
  3.2868 +#246 := (or #145 #242)
  3.2869 +#247 := [def-axiom]: #246
  3.2870 +#304 := [unit-resolution #247 #303]: #145
  3.2871 +#243 := (not #145)
  3.2872 +#244 := (or #243 #241)
  3.2873 +#245 := [def-axiom]: #244
  3.2874 +#305 := [unit-resolution #245 #304]: #241
  3.2875 +#306 := (not #241)
  3.2876 +#307 := (or #306 #281)
  3.2877 +#308 := [th-lemma]: #307
  3.2878 +[unit-resolution #308 #305 #290]: false
  3.2879 +unsat
  3.2880 +hwh3oeLAWt56hnKIa8Wuow 248 0
  3.2881 +#2 := false
  3.2882 +#4 := 0::real
  3.2883 +decl uf_2 :: (-> T2 T1 real)
  3.2884 +decl uf_5 :: T1
  3.2885 +#15 := uf_5
  3.2886 +decl uf_6 :: T2
  3.2887 +#17 := uf_6
  3.2888 +#18 := (uf_2 uf_6 uf_5)
  3.2889 +decl uf_4 :: T2
  3.2890 +#10 := uf_4
  3.2891 +#16 := (uf_2 uf_4 uf_5)
  3.2892 +#66 := -1::real
  3.2893 +#137 := (* -1::real #16)
  3.2894 +#138 := (+ #137 #18)
  3.2895 +decl uf_1 :: real
  3.2896 +#5 := uf_1
  3.2897 +#80 := (* -1::real #18)
  3.2898 +#81 := (+ #16 #80)
  3.2899 +#201 := (+ uf_1 #81)
  3.2900 +#202 := (<= #201 0::real)
  3.2901 +#205 := (ite #202 uf_1 #138)
  3.2902 +#352 := (* -1::real #205)
  3.2903 +#353 := (+ uf_1 #352)
  3.2904 +#354 := (<= #353 0::real)
  3.2905 +#362 := (not #354)
  3.2906 +#79 := 1/2::real
  3.2907 +#244 := (* 1/2::real #205)
  3.2908 +#322 := (<= #244 0::real)
  3.2909 +#245 := (= #244 0::real)
  3.2910 +#158 := -1/2::real
  3.2911 +#208 := (* -1/2::real #205)
  3.2912 +#211 := (+ #18 #208)
  3.2913 +decl uf_3 :: T2
  3.2914 +#7 := uf_3
  3.2915 +#20 := (uf_2 uf_3 uf_5)
  3.2916 +#117 := (+ #80 #20)
  3.2917 +#85 := (* -1::real #20)
  3.2918 +#86 := (+ #18 #85)
  3.2919 +#188 := (+ uf_1 #86)
  3.2920 +#189 := (<= #188 0::real)
  3.2921 +#192 := (ite #189 uf_1 #117)
  3.2922 +#195 := (* 1/2::real #192)
  3.2923 +#198 := (+ #18 #195)
  3.2924 +#97 := (* 1/2::real #20)
  3.2925 +#109 := (+ #80 #97)
  3.2926 +#96 := (* 1/2::real #16)
  3.2927 +#110 := (+ #96 #109)
  3.2928 +#107 := (>= #110 0::real)
  3.2929 +#214 := (ite #107 #198 #211)
  3.2930 +#217 := (= #18 #214)
  3.2931 +#248 := (iff #217 #245)
  3.2932 +#241 := (= #18 #211)
  3.2933 +#246 := (iff #241 #245)
  3.2934 +#247 := [rewrite]: #246
  3.2935 +#242 := (iff #217 #241)
  3.2936 +#239 := (= #214 #211)
  3.2937 +#234 := (ite false #198 #211)
  3.2938 +#237 := (= #234 #211)
  3.2939 +#238 := [rewrite]: #237
  3.2940 +#235 := (= #214 #234)
  3.2941 +#232 := (iff #107 false)
  3.2942 +#104 := (not #107)
  3.2943 +#24 := 2::real
  3.2944 +#23 := (+ #16 #20)
  3.2945 +#25 := (/ #23 2::real)
  3.2946 +#26 := (< #25 #18)
  3.2947 +#108 := (iff #26 #104)
  3.2948 +#98 := (+ #96 #97)
  3.2949 +#101 := (< #98 #18)
  3.2950 +#106 := (iff #101 #104)
  3.2951 +#105 := [rewrite]: #106
  3.2952 +#102 := (iff #26 #101)
  3.2953 +#99 := (= #25 #98)
  3.2954 +#100 := [rewrite]: #99
  3.2955 +#103 := [monotonicity #100]: #102
  3.2956 +#111 := [trans #103 #105]: #108
  3.2957 +#65 := [asserted]: #26
  3.2958 +#112 := [mp #65 #111]: #104
  3.2959 +#233 := [iff-false #112]: #232
  3.2960 +#236 := [monotonicity #233]: #235
  3.2961 +#240 := [trans #236 #238]: #239
  3.2962 +#243 := [monotonicity #240]: #242
  3.2963 +#249 := [trans #243 #247]: #248
  3.2964 +#33 := (- #18 #16)
  3.2965 +#34 := (<= uf_1 #33)
  3.2966 +#35 := (ite #34 uf_1 #33)
  3.2967 +#36 := (/ #35 2::real)
  3.2968 +#37 := (- #18 #36)
  3.2969 +#28 := (- #20 #18)
  3.2970 +#29 := (<= uf_1 #28)
  3.2971 +#30 := (ite #29 uf_1 #28)
  3.2972 +#31 := (/ #30 2::real)
  3.2973 +#32 := (+ #18 #31)
  3.2974 +#27 := (<= #18 #25)
  3.2975 +#38 := (ite #27 #32 #37)
  3.2976 +#39 := (= #38 #18)
  3.2977 +#40 := (not #39)
  3.2978 +#41 := (not #40)
  3.2979 +#220 := (iff #41 #217)
  3.2980 +#141 := (<= uf_1 #138)
  3.2981 +#144 := (ite #141 uf_1 #138)
  3.2982 +#159 := (* -1/2::real #144)
  3.2983 +#160 := (+ #18 #159)
  3.2984 +#120 := (<= uf_1 #117)
  3.2985 +#123 := (ite #120 uf_1 #117)
  3.2986 +#129 := (* 1/2::real #123)
  3.2987 +#134 := (+ #18 #129)
  3.2988 +#114 := (<= #18 #98)
  3.2989 +#165 := (ite #114 #134 #160)
  3.2990 +#171 := (= #18 #165)
  3.2991 +#218 := (iff #171 #217)
  3.2992 +#215 := (= #165 #214)
  3.2993 +#212 := (= #160 #211)
  3.2994 +#209 := (= #159 #208)
  3.2995 +#206 := (= #144 #205)
  3.2996 +#203 := (iff #141 #202)
  3.2997 +#204 := [rewrite]: #203
  3.2998 +#207 := [monotonicity #204]: #206
  3.2999 +#210 := [monotonicity #207]: #209
  3.3000 +#213 := [monotonicity #210]: #212
  3.3001 +#199 := (= #134 #198)
  3.3002 +#196 := (= #129 #195)
  3.3003 +#193 := (= #123 #192)
  3.3004 +#190 := (iff #120 #189)
  3.3005 +#191 := [rewrite]: #190
  3.3006 +#194 := [monotonicity #191]: #193
  3.3007 +#197 := [monotonicity #194]: #196
  3.3008 +#200 := [monotonicity #197]: #199
  3.3009 +#187 := (iff #114 #107)
  3.3010 +#186 := [rewrite]: #187
  3.3011 +#216 := [monotonicity #186 #200 #213]: #215
  3.3012 +#219 := [monotonicity #216]: #218
  3.3013 +#184 := (iff #41 #171)
  3.3014 +#176 := (not #171)
  3.3015 +#179 := (not #176)
  3.3016 +#182 := (iff #179 #171)
  3.3017 +#183 := [rewrite]: #182
  3.3018 +#180 := (iff #41 #179)
  3.3019 +#177 := (iff #40 #176)
  3.3020 +#174 := (iff #39 #171)
  3.3021 +#168 := (= #165 #18)
  3.3022 +#172 := (iff #168 #171)
  3.3023 +#173 := [rewrite]: #172
  3.3024 +#169 := (iff #39 #168)
  3.3025 +#166 := (= #38 #165)
  3.3026 +#163 := (= #37 #160)
  3.3027 +#150 := (* 1/2::real #144)
  3.3028 +#155 := (- #18 #150)
  3.3029 +#161 := (= #155 #160)
  3.3030 +#162 := [rewrite]: #161
  3.3031 +#156 := (= #37 #155)
  3.3032 +#153 := (= #36 #150)
  3.3033 +#147 := (/ #144 2::real)
  3.3034 +#151 := (= #147 #150)
  3.3035 +#152 := [rewrite]: #151
  3.3036 +#148 := (= #36 #147)
  3.3037 +#145 := (= #35 #144)
  3.3038 +#139 := (= #33 #138)
  3.3039 +#140 := [rewrite]: #139
  3.3040 +#142 := (iff #34 #141)
  3.3041 +#143 := [monotonicity #140]: #142
  3.3042 +#146 := [monotonicity #143 #140]: #145
  3.3043 +#149 := [monotonicity #146]: #148
  3.3044 +#154 := [trans #149 #152]: #153
  3.3045 +#157 := [monotonicity #154]: #156
  3.3046 +#164 := [trans #157 #162]: #163
  3.3047 +#135 := (= #32 #134)
  3.3048 +#132 := (= #31 #129)
  3.3049 +#126 := (/ #123 2::real)
  3.3050 +#130 := (= #126 #129)
  3.3051 +#131 := [rewrite]: #130
  3.3052 +#127 := (= #31 #126)
  3.3053 +#124 := (= #30 #123)
  3.3054 +#118 := (= #28 #117)
  3.3055 +#119 := [rewrite]: #118
  3.3056 +#121 := (iff #29 #120)
  3.3057 +#122 := [monotonicity #119]: #121
  3.3058 +#125 := [monotonicity #122 #119]: #124
  3.3059 +#128 := [monotonicity #125]: #127
  3.3060 +#133 := [trans #128 #131]: #132
  3.3061 +#136 := [monotonicity #133]: #135
  3.3062 +#115 := (iff #27 #114)
  3.3063 +#116 := [monotonicity #100]: #115
  3.3064 +#167 := [monotonicity #116 #136 #164]: #166
  3.3065 +#170 := [monotonicity #167]: #169
  3.3066 +#175 := [trans #170 #173]: #174
  3.3067 +#178 := [monotonicity #175]: #177
  3.3068 +#181 := [monotonicity #178]: #180
  3.3069 +#185 := [trans #181 #183]: #184
  3.3070 +#221 := [trans #185 #219]: #220
  3.3071 +#113 := [asserted]: #41
  3.3072 +#222 := [mp #113 #221]: #217
  3.3073 +#250 := [mp #222 #249]: #245
  3.3074 +#356 := (not #245)
  3.3075 +#357 := (or #356 #322)
  3.3076 +#358 := [th-lemma]: #357
  3.3077 +#359 := [unit-resolution #358 #250]: #322
  3.3078 +#360 := [hypothesis]: #354
  3.3079 +#60 := (<= uf_1 0::real)
  3.3080 +#61 := (not #60)
  3.3081 +#6 := (< 0::real uf_1)
  3.3082 +#62 := (iff #6 #61)
  3.3083 +#63 := [rewrite]: #62
  3.3084 +#57 := [asserted]: #6
  3.3085 +#64 := [mp #57 #63]: #61
  3.3086 +#361 := [th-lemma #64 #360 #359]: false
  3.3087 +#363 := [lemma #361]: #362
  3.3088 +#315 := (= uf_1 #205)
  3.3089 +#316 := (= #138 #205)
  3.3090 +#371 := (not #316)
  3.3091 +#355 := (+ #138 #352)
  3.3092 +#364 := (<= #355 0::real)
  3.3093 +#368 := (not #364)
  3.3094 +#87 := (<= #86 0::real)
  3.3095 +#82 := (<= #81 0::real)
  3.3096 +#90 := (and #82 #87)
  3.3097 +#21 := (<= #18 #20)
  3.3098 +#19 := (<= #16 #18)
  3.3099 +#22 := (and #19 #21)
  3.3100 +#91 := (iff #22 #90)
  3.3101 +#88 := (iff #21 #87)
  3.3102 +#89 := [rewrite]: #88
  3.3103 +#83 := (iff #19 #82)
  3.3104 +#84 := [rewrite]: #83
  3.3105 +#92 := [monotonicity #84 #89]: #91
  3.3106 +#59 := [asserted]: #22
  3.3107 +#93 := [mp #59 #92]: #90
  3.3108 +#95 := [and-elim #93]: #87
  3.3109 +#366 := [hypothesis]: #364
  3.3110 +#367 := [th-lemma #366 #95 #112 #359]: false
  3.3111 +#369 := [lemma #367]: #368
  3.3112 +#370 := [hypothesis]: #316
  3.3113 +#372 := (or #371 #364)
  3.3114 +#373 := [th-lemma]: #372
  3.3115 +#374 := [unit-resolution #373 #370 #369]: false
  3.3116 +#375 := [lemma #374]: #371
  3.3117 +#320 := (or #202 #316)
  3.3118 +#321 := [def-axiom]: #320
  3.3119 +#376 := [unit-resolution #321 #375]: #202
  3.3120 +#317 := (not #202)
  3.3121 +#318 := (or #317 #315)
  3.3122 +#319 := [def-axiom]: #318
  3.3123 +#377 := [unit-resolution #319 #376]: #315
  3.3124 +#378 := (not #315)
  3.3125 +#379 := (or #378 #354)
  3.3126 +#380 := [th-lemma]: #379
  3.3127 +[unit-resolution #380 #377 #363]: false
  3.3128 +unsat
  3.3129 +WdMJH3tkMv/rps8y9Ukq5Q 86 0
  3.3130 +#2 := false
  3.3131 +#37 := 0::real
  3.3132 +decl uf_2 :: (-> T2 T1 real)
  3.3133 +decl uf_4 :: T1
  3.3134 +#12 := uf_4
  3.3135 +decl uf_3 :: T2
  3.3136 +#5 := uf_3
  3.3137 +#13 := (uf_2 uf_3 uf_4)
  3.3138 +#34 := -1::real
  3.3139 +#140 := (* -1::real #13)
  3.3140 +decl uf_1 :: real
  3.3141 +#4 := uf_1
  3.3142 +#141 := (+ uf_1 #140)
  3.3143 +#143 := (>= #141 0::real)
  3.3144 +#6 := (:var 0 T1)
  3.3145 +#7 := (uf_2 uf_3 #6)
  3.3146 +#127 := (pattern #7)
  3.3147 +#35 := (* -1::real #7)
  3.3148 +#36 := (+ uf_1 #35)
  3.3149 +#47 := (>= #36 0::real)
  3.3150 +#134 := (forall (vars (?x2 T1)) (:pat #127) #47)
  3.3151 +#49 := (forall (vars (?x2 T1)) #47)
  3.3152 +#137 := (iff #49 #134)
  3.3153 +#135 := (iff #47 #47)
  3.3154 +#136 := [refl]: #135
  3.3155 +#138 := [quant-intro #136]: #137
  3.3156 +#67 := (~ #49 #49)
  3.3157 +#58 := (~ #47 #47)
  3.3158 +#66 := [refl]: #58
  3.3159 +#68 := [nnf-pos #66]: #67
  3.3160 +#10 := (<= #7 uf_1)
  3.3161 +#11 := (forall (vars (?x2 T1)) #10)
  3.3162 +#50 := (iff #11 #49)
  3.3163 +#46 := (iff #10 #47)
  3.3164 +#48 := [rewrite]: #46
  3.3165 +#51 := [quant-intro #48]: #50
  3.3166 +#32 := [asserted]: #11
  3.3167 +#52 := [mp #32 #51]: #49
  3.3168 +#69 := [mp~ #52 #68]: #49
  3.3169 +#139 := [mp #69 #138]: #134
  3.3170 +#149 := (not #134)
  3.3171 +#150 := (or #149 #143)
  3.3172 +#151 := [quant-inst]: #150
  3.3173 +#144 := [unit-resolution #151 #139]: #143
  3.3174 +#142 := (<= #141 0::real)
  3.3175 +#38 := (<= #36 0::real)
  3.3176 +#128 := (forall (vars (?x1 T1)) (:pat #127) #38)
  3.3177 +#41 := (forall (vars (?x1 T1)) #38)
  3.3178 +#131 := (iff #41 #128)
  3.3179 +#129 := (iff #38 #38)
  3.3180 +#130 := [refl]: #129
  3.3181 +#132 := [quant-intro #130]: #131
  3.3182 +#62 := (~ #41 #41)
  3.3183 +#64 := (~ #38 #38)
  3.3184 +#65 := [refl]: #64
  3.3185 +#63 := [nnf-pos #65]: #62
  3.3186 +#8 := (<= uf_1 #7)
  3.3187 +#9 := (forall (vars (?x1 T1)) #8)
  3.3188 +#42 := (iff #9 #41)
  3.3189 +#39 := (iff #8 #38)
  3.3190 +#40 := [rewrite]: #39
  3.3191 +#43 := [quant-intro #40]: #42
  3.3192 +#31 := [asserted]: #9
  3.3193 +#44 := [mp #31 #43]: #41
  3.3194 +#61 := [mp~ #44 #63]: #41
  3.3195 +#133 := [mp #61 #132]: #128
  3.3196 +#145 := (not #128)
  3.3197 +#146 := (or #145 #142)
  3.3198 +#147 := [quant-inst]: #146
  3.3199 +#148 := [unit-resolution #147 #133]: #142
  3.3200 +#45 := (= uf_1 #13)
  3.3201 +#55 := (not #45)
  3.3202 +#14 := (= #13 uf_1)
  3.3203 +#15 := (not #14)
  3.3204 +#56 := (iff #15 #55)
  3.3205 +#53 := (iff #14 #45)
  3.3206 +#54 := [rewrite]: #53
  3.3207 +#57 := [monotonicity #54]: #56
  3.3208 +#33 := [asserted]: #15
  3.3209 +#60 := [mp #33 #57]: #55
  3.3210 +#153 := (not #143)
  3.3211 +#152 := (not #142)
  3.3212 +#154 := (or #45 #152 #153)
  3.3213 +#155 := [th-lemma]: #154
  3.3214 +[unit-resolution #155 #60 #148 #144]: false
  3.3215 +unsat
  3.3216 +V+IAyBZU/6QjYs6JkXx8LQ 57 0
  3.3217 +#2 := false
  3.3218 +#4 := 0::real
  3.3219 +decl uf_1 :: (-> T2 real)
  3.3220 +decl uf_2 :: (-> T1 T1 T2)
  3.3221 +decl uf_12 :: (-> T4 T1)
  3.3222 +decl uf_4 :: T4
  3.3223 +#11 := uf_4
  3.3224 +#39 := (uf_12 uf_4)
  3.3225 +decl uf_10 :: T4
  3.3226 +#27 := uf_10
  3.3227 +#38 := (uf_12 uf_10)
  3.3228 +#40 := (uf_2 #38 #39)
  3.3229 +#41 := (uf_1 #40)
  3.3230 +#264 := (>= #41 0::real)
  3.3231 +#266 := (not #264)
  3.3232 +#43 := (= #41 0::real)
  3.3233 +#44 := (not #43)
  3.3234 +#131 := [asserted]: #44
  3.3235 +#272 := (or #43 #266)
  3.3236 +#42 := (<= #41 0::real)
  3.3237 +#130 := [asserted]: #42
  3.3238 +#265 := (not #42)
  3.3239 +#270 := (or #43 #265 #266)
  3.3240 +#271 := [th-lemma]: #270
  3.3241 +#273 := [unit-resolution #271 #130]: #272
  3.3242 +#274 := [unit-resolution #273 #131]: #266
  3.3243 +#6 := (:var 0 T1)
  3.3244 +#5 := (:var 1 T1)
  3.3245 +#7 := (uf_2 #5 #6)
  3.3246 +#241 := (pattern #7)
  3.3247 +#8 := (uf_1 #7)
  3.3248 +#65 := (>= #8 0::real)
  3.3249 +#242 := (forall (vars (?x1 T1) (?x2 T1)) (:pat #241) #65)
  3.3250 +#66 := (forall (vars (?x1 T1) (?x2 T1)) #65)
  3.3251 +#245 := (iff #66 #242)
  3.3252 +#243 := (iff #65 #65)
  3.3253 +#244 := [refl]: #243
  3.3254 +#246 := [quant-intro #244]: #245
  3.3255 +#149 := (~ #66 #66)
  3.3256 +#151 := (~ #65 #65)
  3.3257 +#152 := [refl]: #151
  3.3258 +#150 := [nnf-pos #152]: #149
  3.3259 +#9 := (<= 0::real #8)
  3.3260 +#10 := (forall (vars (?x1 T1) (?x2 T1)) #9)
  3.3261 +#67 := (iff #10 #66)
  3.3262 +#63 := (iff #9 #65)
  3.3263 +#64 := [rewrite]: #63
  3.3264 +#68 := [quant-intro #64]: #67
  3.3265 +#60 := [asserted]: #10
  3.3266 +#69 := [mp #60 #68]: #66
  3.3267 +#147 := [mp~ #69 #150]: #66
  3.3268 +#247 := [mp #147 #246]: #242
  3.3269 +#267 := (not #242)
  3.3270 +#268 := (or #267 #264)
  3.3271 +#269 := [quant-inst]: #268
  3.3272 +[unit-resolution #269 #247 #274]: false
  3.3273 +unsat
     4.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     4.2 +++ b/src/HOL/Multivariate_Analysis/Integration.thy	Wed Feb 17 18:33:45 2010 +0100
     4.3 @@ -0,0 +1,3465 @@
     4.4 +
     4.5 +header {* Kurzweil-Henstock gauge integration in many dimensions. *}
     4.6 +(*  Author:                     John Harrison
     4.7 +    Translation from HOL light: Robert Himmelmann, TU Muenchen *)
     4.8 +
     4.9 +theory Integration_Aleph
    4.10 +  imports Derivative SMT
    4.11 +begin
    4.12 +
    4.13 +declare [[smt_certificates="~~/src/HOL/Multivariate_Analysis/Integration.cert"]]
    4.14 +declare [[smt_record=true]]
    4.15 +declare [[z3_proofs=true]]
    4.16 +
    4.17 +lemma conjunctD2: assumes "a \<and> b" shows a b using assms by auto
    4.18 +lemma conjunctD3: assumes "a \<and> b \<and> c" shows a b c using assms by auto
    4.19 +lemma conjunctD4: assumes "a \<and> b \<and> c \<and> d" shows a b c d using assms by auto
    4.20 +lemma conjunctD5: assumes "a \<and> b \<and> c \<and> d \<and> e" shows a b c d e using assms by auto
    4.21 +
    4.22 +declare smult_conv_scaleR[simp]
    4.23 +
    4.24 +subsection {* Some useful lemmas about intervals. *}
    4.25 +
    4.26 +lemma empty_as_interval: "{} = {1..0::real^'n}"
    4.27 +  apply(rule set_ext,rule) defer unfolding vector_le_def mem_interval
    4.28 +  using UNIV_witness[where 'a='n] apply(erule_tac exE,rule_tac x=x in allE) by auto
    4.29 +
    4.30 +lemma interior_subset_union_intervals: 
    4.31 +  assumes "i = {a..b::real^'n}" "j = {c..d}" "interior j \<noteq> {}" "i \<subseteq> j \<union> s" "interior(i) \<inter> interior(j) = {}"
    4.32 +  shows "interior i \<subseteq> interior s" proof-
    4.33 +  have "{a<..<b} \<inter> {c..d} = {}" using inter_interval_mixed_eq_empty[of c d a b] and assms(3,5)
    4.34 +    unfolding assms(1,2) interior_closed_interval by auto
    4.35 +  moreover have "{a<..<b} \<subseteq> {c..d} \<union> s" apply(rule order_trans,rule interval_open_subset_closed)
    4.36 +    using assms(4) unfolding assms(1,2) by auto
    4.37 +  ultimately show ?thesis apply-apply(rule interior_maximal) defer apply(rule open_interior)
    4.38 +    unfolding assms(1,2) interior_closed_interval by auto qed
    4.39 +
    4.40 +lemma inter_interior_unions_intervals: fixes f::"(real^'n) set set"
    4.41 +  assumes "finite f" "open s" "\<forall>t\<in>f. \<exists>a b. t = {a..b}" "\<forall>t\<in>f. s \<inter> (interior t) = {}"
    4.42 +  shows "s \<inter> interior(\<Union>f) = {}" proof(rule ccontr,unfold ex_in_conv[THEN sym]) case goal1
    4.43 +  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)
    4.44 +    unfolding open_subset_interior[OF open_ball]  using interior_subset by auto
    4.45 +  have lem2:"\<And>x s P. \<exists>x\<in>s. P x \<Longrightarrow> \<exists>x\<in>insert x s. P x" by auto
    4.46 +  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
    4.47 +  thus ?case proof(induct rule:finite_induct) 
    4.48 +    case empty from this(2) guess x .. hence False unfolding Union_empty interior_empty by auto thus ?case by auto next
    4.49 +    case (insert i f) guess x using insert(5) .. note x = this
    4.50 +    then guess e unfolding open_contains_ball_eq[OF open_Int[OF assms(2) open_interior],rule_format] .. note e=this
    4.51 +    guess a using insert(4)[rule_format,OF insertI1] .. then guess b .. note ab = this
    4.52 +    show ?case proof(cases "x\<in>i") case False hence "x \<in> UNIV - {a..b}" unfolding ab by auto
    4.53 +      then guess d unfolding open_contains_ball_eq[OF open_Diff[OF open_UNIV closed_interval],rule_format] ..
    4.54 +      hence "0 < d" "ball x (min d e) \<subseteq> UNIV - i" using e unfolding ab by auto
    4.55 +      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
    4.56 +      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
    4.57 +    case True show ?thesis proof(cases "x\<in>{a<..<b}")
    4.58 +      case True then guess d unfolding open_contains_ball_eq[OF open_interval,rule_format] ..
    4.59 +      thus ?thesis apply(rule_tac x=i in bexI,rule_tac x=x in exI,rule_tac x="min d e" in exI)
    4.60 +	unfolding ab using interval_open_subset_closed[of a b] and e by fastsimp+ next
    4.61 +    case False then obtain k where "x$k \<le> a$k \<or> x$k \<ge> b$k" unfolding mem_interval by(auto simp add:not_less) 
    4.62 +    hence "x$k = a$k \<or> x$k = b$k" using True unfolding ab and mem_interval apply(erule_tac x=k in allE) by auto
    4.63 +    hence "\<exists>x. ball x (e/2) \<subseteq> s \<inter> (\<Union>f)" proof(erule_tac disjE)
    4.64 +      let ?z = "x - (e/2) *\<^sub>R basis k" assume as:"x$k = a$k" have "ball ?z (e / 2) \<inter> i = {}" apply(rule ccontr) unfolding ex_in_conv[THEN sym] proof(erule exE)
    4.65 +	fix y assume "y \<in> ball ?z (e / 2) \<inter> i" hence "dist ?z y < e/2" and yi:"y\<in>i" by auto
    4.66 +	hence "\<bar>(?z - y) $ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding vector_dist_norm by auto
    4.67 +	hence "y$k < a$k" unfolding vector_component_simps vector_scaleR_component as using e[THEN conjunct1] by(auto simp add:field_simps)
    4.68 +	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
    4.69 +      moreover have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)" apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]]) proof
    4.70 +	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)"
    4.71 +	   apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "(e/2) *\<^sub>R basis k"])
    4.72 +	  unfolding norm_scaleR norm_basis by auto
    4.73 +	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)
    4.74 +	finally show "y\<in>ball x e" unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps) qed
    4.75 +      ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto
    4.76 +    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)
    4.77 +	fix y assume "y \<in> ball ?z (e / 2) \<inter> i" hence "dist ?z y < e/2" and yi:"y\<in>i" by auto
    4.78 +	hence "\<bar>(?z - y) $ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding vector_dist_norm by auto
    4.79 +	hence "y$k > b$k" unfolding vector_component_simps vector_scaleR_component as using e[THEN conjunct1] by(auto simp add:field_simps)
    4.80 +	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
    4.81 +      moreover have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)" apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]]) proof
    4.82 +	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)"
    4.83 +	   apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "- (e/2) *\<^sub>R basis k"])
    4.84 +	  unfolding norm_scaleR norm_basis by auto
    4.85 +	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)
    4.86 +	finally show "y\<in>ball x e" unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps) qed
    4.87 +      ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto qed 
    4.88 +    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
    4.89 +    thus ?thesis apply-apply(rule lem2,rule insert(3)) using insert(4) by auto qed qed qed qed note * = this
    4.90 +  guess t using *[OF assms(1,3) goal1]  .. from this(2) guess x .. then guess e ..
    4.91 +  hence "x \<in> s" "x\<in>interior t" defer using open_subset_interior[OF open_ball, of x e t] by auto
    4.92 +  thus False using `t\<in>f` assms(4) by auto qed
    4.93 +subsection {* Bounds on intervals where they exist. *}
    4.94 +
    4.95 +definition "interval_upperbound (s::(real^'n) set) = (\<chi> i. Sup {a. \<exists>x\<in>s. x$i = a})"
    4.96 +
    4.97 +definition "interval_lowerbound (s::(real^'n) set) = (\<chi> i. Inf {a. \<exists>x\<in>s. x$i = a})"
    4.98 +
    4.99 +lemma interval_upperbound[simp]: assumes "\<forall>i. a$i \<le> b$i" shows "interval_upperbound {a..b} = b"
   4.100 +  using assms unfolding interval_upperbound_def Cart_eq Cart_lambda_beta apply-apply(rule,erule_tac x=i in allE)
   4.101 +  apply(rule Sup_unique) unfolding setle_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer
   4.102 +  apply(rule,rule) apply(rule_tac x="b$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=b in bexI)
   4.103 +  unfolding mem_interval using assms by auto
   4.104 +
   4.105 +lemma interval_lowerbound[simp]: assumes "\<forall>i. a$i \<le> b$i" shows "interval_lowerbound {a..b} = a"
   4.106 +  using assms unfolding interval_lowerbound_def Cart_eq Cart_lambda_beta apply-apply(rule,erule_tac x=i in allE)
   4.107 +  apply(rule Inf_unique) unfolding setge_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer
   4.108 +  apply(rule,rule) apply(rule_tac x="a$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=a in bexI)
   4.109 +  unfolding mem_interval using assms by auto
   4.110 +
   4.111 +lemmas interval_bounds = interval_upperbound interval_lowerbound
   4.112 +
   4.113 +lemma interval_bounds'[simp]: assumes "{a..b}\<noteq>{}" shows "interval_upperbound {a..b} = b" "interval_lowerbound {a..b} = a"
   4.114 +  using assms unfolding interval_ne_empty by auto
   4.115 +
   4.116 +lemma interval_upperbound_1[simp]: "dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> interval_upperbound {a..b} = (b::real^1)"
   4.117 +  apply(rule interval_upperbound) by auto
   4.118 +
   4.119 +lemma interval_lowerbound_1[simp]: "dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> interval_lowerbound {a..b} = (a::real^1)"
   4.120 +  apply(rule interval_lowerbound) by auto
   4.121 +
   4.122 +lemmas interval_bound_1 = interval_upperbound_1 interval_lowerbound_1
   4.123 +
   4.124 +subsection {* Content (length, area, volume...) of an interval. *}
   4.125 +
   4.126 +definition "content (s::(real^'n) set) =
   4.127 +       (if s = {} then 0 else (\<Prod>i\<in>UNIV. (interval_upperbound s)$i - (interval_lowerbound s)$i))"
   4.128 +
   4.129 +lemma interval_not_empty:"\<forall>i. a$i \<le> b$i \<Longrightarrow> {a..b::real^'n} \<noteq> {}"
   4.130 +  unfolding interval_eq_empty unfolding not_ex not_less by assumption
   4.131 +
   4.132 +lemma content_closed_interval: assumes "\<forall>i. a$i \<le> b$i"
   4.133 +  shows "content {a..b} = (\<Prod>i\<in>UNIV. b$i - a$i)"
   4.134 +  using interval_not_empty[OF assms] unfolding content_def interval_upperbound[OF assms] interval_lowerbound[OF assms] by auto
   4.135 +
   4.136 +lemma content_closed_interval': assumes "{a..b}\<noteq>{}" shows "content {a..b} = (\<Prod>i\<in>UNIV. b$i - a$i)"
   4.137 +  apply(rule content_closed_interval) using assms unfolding interval_ne_empty .
   4.138 +
   4.139 +lemma content_1:"dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> content {a..b} = dest_vec1 b - dest_vec1 a"
   4.140 +  using content_closed_interval[of a b] by auto
   4.141 +
   4.142 +lemma content_1':"a \<le> b \<Longrightarrow> content {vec1 a..vec1 b} = b - a" using content_1[of "vec a" "vec b"] by auto
   4.143 +
   4.144 +lemma content_unit[intro]: "content{0..1::real^'n} = 1" proof-
   4.145 +  have *:"\<forall>i. 0$i \<le> (1::real^'n::finite)$i" by auto
   4.146 +  have "0 \<in> {0..1::real^'n::finite}" unfolding mem_interval by auto
   4.147 +  thus ?thesis unfolding content_def interval_bounds[OF *] using setprod_1 by auto qed
   4.148 +
   4.149 +lemma content_pos_le[intro]: "0 \<le> content {a..b}" proof(cases "{a..b}={}")
   4.150 +  case False hence *:"\<forall>i. a $ i \<le> b $ i" unfolding interval_ne_empty by assumption
   4.151 +  have "(\<Prod>i\<in>UNIV. interval_upperbound {a..b} $ i - interval_lowerbound {a..b} $ i) \<ge> 0"
   4.152 +    apply(rule setprod_nonneg) unfolding interval_bounds[OF *] using * apply(erule_tac x=x in allE) by auto
   4.153 +  thus ?thesis unfolding content_def by(auto simp del:interval_bounds') qed(unfold content_def, auto)
   4.154 +
   4.155 +lemma content_pos_lt: assumes "\<forall>i. a$i < b$i" shows "0 < content {a..b}"
   4.156 +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
   4.157 +  show ?thesis unfolding content_closed_interval[OF help_lemma1[OF assms]] apply(rule setprod_pos)
   4.158 +    using assms apply(erule_tac x=x in allE) by auto qed
   4.159 +
   4.160 +lemma content_pos_lt_1: "dest_vec1 a < dest_vec1 b \<Longrightarrow> 0 < content({a..b})"
   4.161 +  apply(rule content_pos_lt) by auto
   4.162 +
   4.163 +lemma content_eq_0: "content({a..b::real^'n}) = 0 \<longleftrightarrow> (\<exists>i. b$i \<le> a$i)" proof(cases "{a..b} = {}")
   4.164 +  case True thus ?thesis unfolding content_def if_P[OF True] unfolding interval_eq_empty apply-
   4.165 +    apply(rule,erule exE) apply(rule_tac x=i in exI) by auto next
   4.166 +  guess a using UNIV_witness[where 'a='n] .. case False note as=this[unfolded interval_eq_empty not_ex not_less]
   4.167 +  show ?thesis unfolding content_def if_not_P[OF False] setprod_zero_iff[OF finite_UNIV]
   4.168 +    apply(rule) apply(erule_tac[!] exE bexE) unfolding interval_bounds[OF as] apply(rule_tac x=x in exI) defer
   4.169 +    apply(rule_tac x=i in bexI) using as apply(erule_tac x=i in allE) by auto qed
   4.170 +
   4.171 +lemma cond_cases:"(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)" by auto
   4.172 +
   4.173 +lemma content_closed_interval_cases:
   4.174 +  "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) 
   4.175 +  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
   4.176 +
   4.177 +lemma content_eq_0_interior: "content {a..b} = 0 \<longleftrightarrow> interior({a..b}) = {}"
   4.178 +  unfolding content_eq_0 interior_closed_interval interval_eq_empty by auto
   4.179 +
   4.180 +lemma content_eq_0_1: "content {a..b::real^1} = 0 \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a"
   4.181 +  unfolding content_eq_0 by auto
   4.182 +
   4.183 +lemma content_pos_lt_eq: "0 < content {a..b} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
   4.184 +  apply(rule) defer apply(rule content_pos_lt,assumption) proof- assume "0 < content {a..b}"
   4.185 +  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
   4.186 +
   4.187 +lemma content_empty[simp]: "content {} = 0" unfolding content_def by auto
   4.188 +
   4.189 +lemma content_subset: assumes "{a..b} \<subseteq> {c..d}" shows "content {a..b::real^'n} \<le> content {c..d}" proof(cases "{a..b}={}")
   4.190 +  case True thus ?thesis using content_pos_le[of c d] by auto next
   4.191 +  case False hence ab_ne:"\<forall>i. a $ i \<le> b $ i" unfolding interval_ne_empty by auto
   4.192 +  hence ab_ab:"a\<in>{a..b}" "b\<in>{a..b}" unfolding mem_interval by auto
   4.193 +  have "{c..d} \<noteq> {}" using assms False by auto
   4.194 +  hence cd_ne:"\<forall>i. c $ i \<le> d $ i" using assms unfolding interval_ne_empty by auto
   4.195 +  show ?thesis unfolding content_def unfolding interval_bounds[OF ab_ne] interval_bounds[OF cd_ne]
   4.196 +    unfolding if_not_P[OF False] if_not_P[OF `{c..d} \<noteq> {}`] apply(rule setprod_mono,rule) proof fix i::'n
   4.197 +    show "0 \<le> b $ i - a $ i" using ab_ne[THEN spec[where x=i]] by auto
   4.198 +    show "b $ i - a $ i \<le> d $ i - c $ i"
   4.199 +      using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(2),of i]
   4.200 +      using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(1),of i] by auto qed qed
   4.201 +
   4.202 +lemma content_lt_nz: "0 < content {a..b} \<longleftrightarrow> content {a..b} \<noteq> 0"
   4.203 +  unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by auto
   4.204 +
   4.205 +subsection {* The notion of a gauge --- simply an open set containing the point. *}
   4.206 +
   4.207 +definition gauge where "gauge d \<longleftrightarrow> (\<forall>x. x\<in>(d x) \<and> open(d x))"
   4.208 +
   4.209 +lemma gaugeI:assumes "\<And>x. x\<in>g x" "\<And>x. open (g x)" shows "gauge g"
   4.210 +  using assms unfolding gauge_def by auto
   4.211 +
   4.212 +lemma gaugeD[dest]: assumes "gauge d" shows "x\<in>d x" "open (d x)" using assms unfolding gauge_def by auto
   4.213 +
   4.214 +lemma gauge_ball_dependent: "\<forall>x. 0 < e x \<Longrightarrow> gauge (\<lambda>x. ball x (e x))"
   4.215 +  unfolding gauge_def by auto 
   4.216 +
   4.217 +lemma gauge_ball[intro?]: "0 < e \<Longrightarrow> gauge (\<lambda>x. ball x e)" unfolding gauge_def by auto 
   4.218 +
   4.219 +lemma gauge_trivial[intro]: "gauge (\<lambda>x. ball x 1)" apply(rule gauge_ball) by auto
   4.220 +
   4.221 +lemma gauge_inter: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. (d1 x) \<inter> (d2 x))"
   4.222 +  unfolding gauge_def by auto 
   4.223 +
   4.224 +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-
   4.225 +  have *:"\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s" by auto show ?thesis
   4.226 +  unfolding gauge_def unfolding * 
   4.227 +  using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto qed
   4.228 +
   4.229 +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)
   4.230 +
   4.231 +subsection {* Divisions. *}
   4.232 +
   4.233 +definition division_of (infixl "division'_of" 40) where
   4.234 +  "s division_of i \<equiv>
   4.235 +        finite s \<and>
   4.236 +        (\<forall>k\<in>s. k \<subseteq> i \<and> k \<noteq> {} \<and> (\<exists>a b. k = {a..b})) \<and>
   4.237 +        (\<forall>k1\<in>s. \<forall>k2\<in>s. k1 \<noteq> k2 \<longrightarrow> interior(k1) \<inter> interior(k2) = {}) \<and>
   4.238 +        (\<Union>s = i)"
   4.239 +
   4.240 +lemma division_ofD[dest]: assumes  "s division_of i"
   4.241 +  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})"
   4.242 +  "\<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
   4.243 +
   4.244 +lemma division_ofI:
   4.245 +  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})"
   4.246 +  "\<And>k1 k2. \<lbrakk>k1\<in>s; k2\<in>s; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}" "\<Union>s = i"
   4.247 +  shows "s division_of i" using assms unfolding division_of_def by auto
   4.248 +
   4.249 +lemma division_of_finite: "s division_of i \<Longrightarrow> finite s"
   4.250 +  unfolding division_of_def by auto
   4.251 +
   4.252 +lemma division_of_self[intro]: "{a..b} \<noteq> {} \<Longrightarrow> {{a..b}} division_of {a..b}"
   4.253 +  unfolding division_of_def by auto
   4.254 +
   4.255 +lemma division_of_trivial[simp]: "s division_of {} \<longleftrightarrow> s = {}" unfolding division_of_def by auto 
   4.256 +
   4.257 +lemma division_of_sing[simp]: "s division_of {a..a::real^'n} \<longleftrightarrow> s = {{a..a}}" (is "?l = ?r") proof
   4.258 +  assume ?r moreover { assume "s = {{a}}" moreover fix k assume "k\<in>s" 
   4.259 +    ultimately have"\<exists>x y. k = {x..y}" apply(rule_tac x=a in exI)+ unfolding interval_sing[THEN conjunct1] by auto }
   4.260 +  ultimately show ?l unfolding division_of_def interval_sing[THEN conjunct1] by auto next
   4.261 +  assume ?l note as=conjunctD4[OF this[unfolded division_of_def interval_sing[THEN conjunct1]]]
   4.262 +  { fix x assume x:"x\<in>s" have "x={a}" using as(2)[rule_format,OF x] by auto }
   4.263 +  moreover have "s \<noteq> {}" using as(4) by auto ultimately show ?r unfolding interval_sing[THEN conjunct1] by auto qed
   4.264 +
   4.265 +lemma elementary_empty: obtains p where "p division_of {}"
   4.266 +  unfolding division_of_trivial by auto
   4.267 +
   4.268 +lemma elementary_interval: obtains p where  "p division_of {a..b}"
   4.269 +  by(metis division_of_trivial division_of_self)
   4.270 +
   4.271 +lemma division_contains: "s division_of i \<Longrightarrow> \<forall>x\<in>i. \<exists>k\<in>s. x \<in> k"
   4.272 +  unfolding division_of_def by auto
   4.273 +
   4.274 +lemma forall_in_division:
   4.275 + "d division_of i \<Longrightarrow> ((\<forall>x\<in>d. P x) \<longleftrightarrow> (\<forall>a b. {a..b} \<in> d \<longrightarrow> P {a..b}))"
   4.276 +  unfolding division_of_def by fastsimp
   4.277 +
   4.278 +lemma division_of_subset: assumes "p division_of (\<Union>p)" "q \<subseteq> p" shows "q division_of (\<Union>q)"
   4.279 +  apply(rule division_ofI) proof- note as=division_ofD[OF assms(1)]
   4.280 +  show "finite q" apply(rule finite_subset) using as(1) assms(2) by auto
   4.281 +  { 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
   4.282 +  show "\<exists>a b. k = {a..b}" using as(4)[OF kp] by auto show "k \<noteq> {}" using as(3)[OF kp] by auto }
   4.283 +  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
   4.284 +  show "interior k1 \<inter> interior k2 = {}" using as(5)[OF *] by auto qed auto
   4.285 +
   4.286 +lemma division_of_union_self[intro]: "p division_of s \<Longrightarrow> p division_of (\<Union>p)" unfolding division_of_def by auto
   4.287 +
   4.288 +lemma division_of_content_0: assumes "content {a..b} = 0" "d division_of {a..b}" shows "\<forall>k\<in>d. content k = 0"
   4.289 +  unfolding forall_in_division[OF assms(2)] apply(rule,rule,rule) apply(drule division_ofD(2)[OF assms(2)])
   4.290 +  apply(drule content_subset) unfolding assms(1) proof- case goal1 thus ?case using content_pos_le[of a b] by auto qed
   4.291 +
   4.292 +lemma division_inter: assumes "p1 division_of s1" "p2 division_of (s2::(real^'a) set)"
   4.293 +  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-
   4.294 +let ?A = "{s. s \<in>  (\<lambda>(k1,k2). k1 \<inter> k2) ` (p1 \<times> p2) \<and> s \<noteq> {}}" have *:"?A' = ?A" by auto
   4.295 +show ?thesis unfolding * proof(rule division_ofI) have "?A \<subseteq> (\<lambda>(x, y). x \<inter> y) ` (p1 \<times> p2)" by auto
   4.296 +  moreover have "finite (p1 \<times> p2)" using assms unfolding division_of_def by auto ultimately show "finite ?A" by auto
   4.297 +  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
   4.298 +    using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)] by auto
   4.299 +  { 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
   4.300 +  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
   4.301 +  guess a1 using division_ofD(4)[OF assms(1) k(2)] .. then guess b1 .. note ab1=this
   4.302 +  guess a2 using division_ofD(4)[OF assms(2) k(3)] .. then guess b2 .. note ab2=this
   4.303 +  show "\<exists>a b. k = {a..b}" unfolding k ab1 ab2 unfolding inter_interval by auto } fix k1 k2
   4.304 +  assume "k1\<in>?A" then obtain x1 y1 where k1:"k1 = x1 \<inter> y1" "x1\<in>p1" "y1\<in>p2" "k1\<noteq>{}" by auto
   4.305 +  assume "k2\<in>?A" then obtain x2 y2 where k2:"k2 = x2 \<inter> y2" "x2\<in>p1" "y2\<in>p2" "k2\<noteq>{}" by auto
   4.306 +  assume "k1 \<noteq> k2" hence th:"x1\<noteq>x2 \<or> y1\<noteq>y2" unfolding k1 k2 by auto
   4.307 +  have *:"(interior x1 \<inter> interior x2 = {} \<or> interior y1 \<inter> interior y2 = {}) \<Longrightarrow>
   4.308 +      interior(x1 \<inter> y1) \<subseteq> interior(x1) \<Longrightarrow> interior(x1 \<inter> y1) \<subseteq> interior(y1) \<Longrightarrow>
   4.309 +      interior(x2 \<inter> y2) \<subseteq> interior(x2) \<Longrightarrow> interior(x2 \<inter> y2) \<subseteq> interior(y2)
   4.310 +      \<Longrightarrow> interior(x1 \<inter> y1) \<inter> interior(x2 \<inter> y2) = {}" by auto
   4.311 +  show "interior k1 \<inter> interior k2 = {}" unfolding k1 k2 apply(rule *) defer apply(rule_tac[1-4] subset_interior)
   4.312 +    using division_ofD(5)[OF assms(1) k1(2) k2(2)]
   4.313 +    using division_ofD(5)[OF assms(2) k1(3) k2(3)] using th by auto qed qed
   4.314 +
   4.315 +lemma division_inter_1: assumes "d division_of i" "{a..b::real^'n} \<subseteq> i"
   4.316 +  shows "{ {a..b} \<inter> k |k. k \<in> d \<and> {a..b} \<inter> k \<noteq> {} } division_of {a..b}" proof(cases "{a..b} = {}")
   4.317 +  case True show ?thesis unfolding True and division_of_trivial by auto next
   4.318 +  have *:"{a..b} \<inter> i = {a..b}" using assms(2) by auto 
   4.319 +  case False show ?thesis using division_inter[OF division_of_self[OF False] assms(1)] unfolding * by auto qed
   4.320 +
   4.321 +lemma elementary_inter: assumes "p1 division_of s" "p2 division_of (t::(real^'n) set)"
   4.322 +  shows "\<exists>p. p division_of (s \<inter> t)"
   4.323 +  by(rule,rule division_inter[OF assms])
   4.324 +
   4.325 +lemma elementary_inters: assumes "finite f" "f\<noteq>{}" "\<forall>s\<in>f. \<exists>p. p division_of (s::(real^'n) set)"
   4.326 +  shows "\<exists>p. p division_of (\<Inter> f)" using assms apply-proof(induct f rule:finite_induct)
   4.327 +case (insert x f) show ?case proof(cases "f={}")
   4.328 +  case True thus ?thesis unfolding True using insert by auto next
   4.329 +  case False guess p using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] ..
   4.330 +  moreover guess px using insert(5)[rule_format,OF insertI1] .. ultimately
   4.331 +  show ?thesis unfolding Inter_insert apply(rule_tac elementary_inter) by assumption+ qed qed auto
   4.332 +
   4.333 +lemma division_disjoint_union:
   4.334 +  assumes "p1 division_of s1" "p2 division_of s2" "interior s1 \<inter> interior s2 = {}"
   4.335 +  shows "(p1 \<union> p2) division_of (s1 \<union> s2)" proof(rule division_ofI) 
   4.336 +  note d1 = division_ofD[OF assms(1)] and d2 = division_ofD[OF assms(2)]
   4.337 +  show "finite (p1 \<union> p2)" using d1(1) d2(1) by auto
   4.338 +  show "\<Union>(p1 \<union> p2) = s1 \<union> s2" using d1(6) d2(6) by auto
   4.339 +  { 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 = {}"
   4.340 +  { 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)]]
   4.341 +      using assms(3) by blast } moreover
   4.342 +  { 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)]]
   4.343 +      using assms(3) by blast} ultimately
   4.344 +  show ?g using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto }
   4.345 +  fix k assume k:"k \<in> p1 \<union> p2"  show "k \<subseteq> s1 \<union> s2" using k d1(2) d2(2) by auto
   4.346 +  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
   4.347 +
   4.348 +lemma partial_division_extend_1:
   4.349 +  assumes "{c..d} \<subseteq> {a..b::real^'n}" "{c..d} \<noteq> {}"
   4.350 +  obtains p where "p division_of {a..b}" "{c..d} \<in> p"
   4.351 +proof- def n \<equiv> "CARD('n)" have n:"1 \<le> n" "0 < n" "n \<noteq> 0" unfolding n_def by auto
   4.352 +  guess \<pi> using ex_bij_betw_nat_finite_1[OF finite_UNIV[where 'a='n]] .. note \<pi>=this
   4.353 +  def \<pi>' \<equiv> "inv_into {1..n} \<pi>"
   4.354 +  have \<pi>':"bij_betw \<pi>' UNIV {1..n}" using bij_betw_inv_into[OF \<pi>] unfolding \<pi>'_def n_def by auto
   4.355 +  hence \<pi>'i:"\<And>i. \<pi>' i \<in> {1..n}" unfolding bij_betw_def by auto 
   4.356 +  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
   4.357 +  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
   4.358 +  have "{c..d} \<noteq> {}" using assms by auto
   4.359 +  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$i)}"
   4.360 +  let ?p2 = "\<lambda>l. {(\<chi> i. if \<pi>' i < l then c$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)}"
   4.361 +  let ?p =  "{?p1 l |l. l \<in> {1..n+1}} \<union> {?p2 l |l. l \<in> {1..n+1}}"
   4.362 +  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)
   4.363 +  show ?thesis apply(rule that[of ?p]) apply(rule division_ofI)
   4.364 +  proof- have "\<And>i. \<pi>' i < Suc n"
   4.365 +    proof(rule ccontr,unfold not_less) fix i assume "Suc n \<le> \<pi>' i"
   4.366 +      hence "\<pi>' i \<notin> {1..n}" by auto thus False using \<pi>' unfolding bij_betw_def by auto
   4.367 +    qed hence "c = (\<chi> i. if \<pi>' i < Suc n then c $ i else a $ i)"
   4.368 +        "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)"
   4.369 +      unfolding Cart_eq Cart_lambda_beta using \<pi>' unfolding bij_betw_def by auto
   4.370 +    thus cdp:"{c..d} \<in> ?p" apply-apply(rule UnI1) unfolding mem_Collect_eq apply(rule_tac x="n + 1" in exI) by auto
   4.371 +    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}"
   4.372 +      unfolding subset_eq apply(rule_tac[!] ballI,rule_tac[!] ccontr)
   4.373 +    proof- fix l assume l:"l\<in>{1..n+1}" fix x assume "x\<notin>{a..b}"
   4.374 +      then guess i unfolding mem_interval not_all .. note i=this
   4.375 +      show "x \<in> ?p1 l \<Longrightarrow> False" "x \<in> ?p2 l \<Longrightarrow> False" unfolding mem_interval apply(erule_tac[!] x=i in allE)
   4.376 +        apply(case_tac[!] "\<pi>' i < l", case_tac[!] "\<pi>' i = l") using abcd[of i] i by auto 
   4.377 +    qed moreover have "\<And>x. x \<in> {a..b} \<Longrightarrow> x \<in> \<Union>?p"
   4.378 +    proof- fix x assume x:"x\<in>{a..b}"
   4.379 +      { presume "x\<notin>{c..d} \<Longrightarrow> x \<in> \<Union>?p" thus "x \<in> \<Union>?p" using cdp by blast }
   4.380 +      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)}"
   4.381 +      assume "x\<notin>{c..d}" then guess i0 unfolding mem_interval not_all ..
   4.382 +      hence "\<pi>' i0 \<in> ?M" using \<pi>' unfolding bij_betw_def by(auto intro!:le_SucI)
   4.383 +      hence M:"finite ?M" "?M \<noteq> {}" by auto
   4.384 +      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]]
   4.385 +        Min_gr_iff[OF M,unfolded l_def[symmetric]]
   4.386 +      have "x\<in>?p1 l \<or> x\<in>?p2 l" using l(2)[THEN conjunct2] unfolding de_Morgan_conj not_le
   4.387 +        apply- apply(erule disjE) apply(rule disjI1) defer apply(rule disjI2)
   4.388 +      proof- assume as:"x $ \<pi> l < c $ \<pi> l"
   4.389 +        show "x \<in> ?p1 l" unfolding mem_interval Cart_lambda_beta
   4.390 +        proof case goal1 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le by auto
   4.391 +          thus ?case using as x[unfolded mem_interval,rule_format,of i]
   4.392 +            apply auto using l(3)[of "\<pi>' i"] by(auto elim!:ballE[where x="\<pi>' i"])
   4.393 +        qed
   4.394 +      next assume as:"x $ \<pi> l > d $ \<pi> l"
   4.395 +        show "x \<in> ?p2 l" unfolding mem_interval Cart_lambda_beta
   4.396 +        proof case goal1 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le by auto
   4.397 +          thus ?case using as x[unfolded mem_interval,rule_format,of i]
   4.398 +            apply auto using l(3)[of "\<pi>' i"] by(auto elim!:ballE[where x="\<pi>' i"])
   4.399 +        qed qed
   4.400 +      thus "x \<in> \<Union>?p" using l(2) by blast 
   4.401 +    qed ultimately show "\<Union>?p = {a..b}" apply-apply(rule) defer apply(rule) by(assumption,blast)
   4.402 +    
   4.403 +    show "finite ?p" by auto
   4.404 +    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
   4.405 +    show "k\<subseteq>{a..b}" apply(rule,unfold mem_interval,rule,rule) 
   4.406 +    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
   4.407 +      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)
   4.408 +    qed have "\<And>l. ?p1 l \<noteq> {}" "\<And>l. ?p2 l \<noteq> {}" unfolding interval_eq_empty not_ex apply(rule_tac[!] allI)
   4.409 +    proof- case goal1 thus ?case using abcd[of x] by auto
   4.410 +    next   case goal2 thus ?case using abcd[of x] by auto
   4.411 +    qed thus "k \<noteq> {}" using k by auto
   4.412 +    show "\<exists>a b. k = {a..b}" using k by auto
   4.413 +    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
   4.414 +    { fix k k' l l'
   4.415 +      assume k:"k\<in>?p" and l:"k = ?p1 l \<or> k = ?p2 l" "l \<in> {1..n + 1}" 
   4.416 +      assume k':"k' \<in> ?p" "k \<noteq> k'" and  l':"k' = ?p1 l' \<or> k' = ?p2 l'" "l' \<in> {1..n + 1}" 
   4.417 +      assume "l \<le> l'" fix x
   4.418 +      have "x \<notin> interior k \<inter> interior k'" 
   4.419 +      proof(rule,cases "l' = n+1") assume x:"x \<in> interior k \<inter> interior k'"
   4.420 +        case True hence "\<And>i. \<pi>' i < l'" using \<pi>'i by(auto simp add:less_Suc_eq_le)
   4.421 +        hence k':"k' = {c..d}" using l'(1) \<pi>'i by(auto simp add:Cart_nth_inverse)
   4.422 +        have ln:"l < n + 1" 
   4.423 +        proof(rule ccontr) case goal1 hence l2:"l = n+1" using l by auto
   4.424 +          hence "\<And>i. \<pi>' i < l" using \<pi>'i by(auto simp add:less_Suc_eq_le)
   4.425 +          hence "k = {c..d}" using l(1) \<pi>'i by(auto simp add:Cart_nth_inverse)
   4.426 +          thus False using `k\<noteq>k'` k' by auto
   4.427 +        qed have **:"\<pi>' (\<pi> l) = l" using \<pi>'\<pi>[of l] using l ln by auto
   4.428 +        have "x $ \<pi> l < c $ \<pi> l \<or> d $ \<pi> l < x $ \<pi> l" using l(1) apply-
   4.429 +        proof(erule disjE)
   4.430 +          assume as:"k = ?p1 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
   4.431 +          show ?thesis using *[of "\<pi> l"] using ln unfolding Cart_lambda_beta ** by auto
   4.432 +        next assume as:"k = ?p2 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
   4.433 +          show ?thesis using *[of "\<pi> l"] using ln unfolding Cart_lambda_beta ** by auto
   4.434 +        qed thus False using x unfolding k' unfolding Int_iff interior_closed_interval mem_interval
   4.435 +          by(auto elim!:allE[where x="\<pi> l"])
   4.436 +      next case False hence "l < n + 1" using l'(2) using `l\<le>l'` by auto
   4.437 +        hence ln:"l \<in> {1..n}" "l' \<in> {1..n}" using l l' False by auto
   4.438 +        note \<pi>l = \<pi>'\<pi>[OF ln(1)] \<pi>'\<pi>[OF ln(2)]
   4.439 +        assume x:"x \<in> interior k \<inter> interior k'"
   4.440 +        show False using l(1) l'(1) apply-
   4.441 +        proof(erule_tac[!] disjE)+
   4.442 +          assume as:"k = ?p1 l" "k' = ?p1 l'"
   4.443 +          note * = x[unfolded as Int_iff interior_closed_interval mem_interval]
   4.444 +          have "l \<noteq> l'" using k'(2)[unfolded as] by auto
   4.445 +          thus False using * by(smt Cart_lambda_beta \<pi>l)
   4.446 +        next assume as:"k = ?p2 l" "k' = ?p2 l'"
   4.447 +          note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
   4.448 +          have "l \<noteq> l'" apply(rule) using k'(2)[unfolded as] by auto
   4.449 +          thus False using *[of "\<pi> l"] *[of "\<pi> l'"]
   4.450 +            unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` by auto
   4.451 +        next assume as:"k = ?p1 l" "k' = ?p2 l'"
   4.452 +          note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
   4.453 +          show False using *[of "\<pi> l"] *[of "\<pi> l'"]
   4.454 +            unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` using abcd[of "\<pi> l'"] by smt 
   4.455 +        next assume as:"k = ?p2 l" "k' = ?p1 l'"
   4.456 +          note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
   4.457 +          show False using *[of "\<pi> l"] *[of "\<pi> l'"]
   4.458 +            unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` using abcd[of "\<pi> l'"] by smt
   4.459 +        qed qed } 
   4.460 +    from this[OF k l k' l'] this[OF k'(1) l' k _ l] have "\<And>x. x \<notin> interior k \<inter> interior k'"
   4.461 +      apply - apply(cases "l' \<le> l") using k'(2) by auto            
   4.462 +    thus "interior k \<inter> interior k' = {}" by auto        
   4.463 +qed qed
   4.464 +
   4.465 +lemma partial_division_extend_interval: assumes "p division_of (\<Union>p)" "(\<Union>p) \<subseteq> {a..b}"
   4.466 +  obtains q where "p \<subseteq> q" "q division_of {a..b::real^'n}" proof(cases "p = {}")
   4.467 +  case True guess q apply(rule elementary_interval[of a b]) .
   4.468 +  thus ?thesis apply- apply(rule that[of q]) unfolding True by auto next
   4.469 +  case False note p = division_ofD[OF assms(1)]
   4.470 +  have *:"\<forall>k\<in>p. \<exists>q. q division_of {a..b} \<and> k\<in>q" proof case goal1
   4.471 +    guess c using p(4)[OF goal1] .. then guess d .. note cd_ = this
   4.472 +    have *:"{c..d} \<subseteq> {a..b}" "{c..d} \<noteq> {}" using p(2,3)[OF goal1, unfolded cd_] using assms(2) by auto
   4.473 +    guess q apply(rule partial_division_extend_1[OF *]) . thus ?case unfolding cd_ by auto qed
   4.474 +  guess q using bchoice[OF *] .. note q = conjunctD2[OF this[rule_format]]
   4.475 +  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-
   4.476 +    fix x assume x:"x\<in>p" show "q x division_of \<Union>q x" apply-apply(rule division_ofI)
   4.477 +      using division_ofD[OF q(1)[OF x]] by auto show "q x - {x} \<subseteq> q x" by auto qed
   4.478 +  hence "\<exists>d. d division_of \<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)" apply- apply(rule elementary_inters)
   4.479 +    apply(rule finite_imageI[OF p(1)]) unfolding image_is_empty apply(rule False) by auto
   4.480 +  then guess d .. note d = this
   4.481 +  show ?thesis apply(rule that[of "d \<union> p"]) proof-
   4.482 +    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
   4.483 +    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"
   4.484 +      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
   4.485 +    show "d \<union> p division_of {a..b}" unfolding * apply(rule division_disjoint_union[OF d assms(1)])
   4.486 +      apply(rule inter_interior_unions_intervals) apply(rule p open_interior ballI)+ proof(assumption,rule)
   4.487 +      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
   4.488 +      show "interior (\<Inter>(\<lambda>i. \<Union>(q i - {i})) ` p) \<inter> interior k = {}" apply(rule *[of _ "interior (\<Union>(q k - {k}))"])
   4.489 +	defer apply(subst Int_commute) apply(rule inter_interior_unions_intervals) proof- note qk=division_ofD[OF q(1)[OF k]]
   4.490 +	show "finite (q k - {k})" "open (interior k)"  "\<forall>t\<in>q k - {k}. \<exists>a b. t = {a..b}" using qk by auto
   4.491 +	show "\<forall>t\<in>q k - {k}. interior k \<inter> interior t = {}" using qk(5) using q(2)[OF k] by auto
   4.492 +	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}))"
   4.493 +	  apply(rule subset_interior *)+ using k by auto qed qed qed auto qed
   4.494 +
   4.495 +lemma elementary_bounded[dest]: "p division_of s \<Longrightarrow> bounded (s::(real^'n) set)"
   4.496 +  unfolding division_of_def by(metis bounded_Union bounded_interval) 
   4.497 +
   4.498 +lemma elementary_subset_interval: "p division_of s \<Longrightarrow> \<exists>a b. s \<subseteq> {a..b::real^'n}"
   4.499 +  by(meson elementary_bounded bounded_subset_closed_interval)
   4.500 +
   4.501 +lemma division_union_intervals_exists: assumes "{a..b::real^'n} \<noteq> {}"
   4.502 +  obtains p where "(insert {a..b} p) division_of ({a..b} \<union> {c..d})" proof(cases "{c..d} = {}")
   4.503 +  case True show ?thesis apply(rule that[of "{}"]) unfolding True using assms by auto next
   4.504 +  case False note false=this show ?thesis proof(cases "{a..b} \<inter> {c..d} = {}")
   4.505 +  have *:"\<And>a b. {a,b} = {a} \<union> {b}" by auto
   4.506 +  case True show ?thesis apply(rule that[of "{{c..d}}"]) unfolding * apply(rule division_disjoint_union)
   4.507 +    using false True assms using interior_subset by auto next
   4.508 +  case False obtain u v where uv:"{a..b} \<inter> {c..d} = {u..v}" unfolding inter_interval by auto
   4.509 +  have *:"{u..v} \<subseteq> {c..d}" using uv by auto
   4.510 +  guess p apply(rule partial_division_extend_1[OF * False[unfolded uv]]) . note p=this division_ofD[OF this(1)]
   4.511 +  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
   4.512 +  show thesis apply(rule that[of "p - {{u..v}}"]) unfolding *(1) apply(subst *(2)) apply(rule division_disjoint_union)
   4.513 +    apply(rule,rule assms) apply(rule division_of_subset[of p]) apply(rule division_of_union_self[OF p(1)]) defer
   4.514 +    unfolding interior_inter[THEN sym] proof-
   4.515 +    have *:"\<And>cd p uv ab. p \<subseteq> cd \<Longrightarrow> ab \<inter> cd = uv \<Longrightarrow> ab \<inter> p = uv \<inter> p" by auto
   4.516 +    have "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = interior({u..v} \<inter> \<Union>(p - {{u..v}}))" 
   4.517 +      apply(rule arg_cong[of _ _ interior]) apply(rule *[OF _ uv]) using p(8) by auto
   4.518 +    also have "\<dots> = {}" unfolding interior_inter apply(rule inter_interior_unions_intervals) using p(6) p(7)[OF p(2)] p(3) by auto
   4.519 +    finally show "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = {}" by assumption qed auto qed qed
   4.520 +
   4.521 +lemma division_of_unions: assumes "finite f"  "\<And>p. p\<in>f \<Longrightarrow> p division_of (\<Union>p)"
   4.522 +  "\<And>k1 k2. \<lbrakk>k1 \<in> \<Union>f; k2 \<in> \<Union>f; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
   4.523 +  shows "\<Union>f division_of \<Union>\<Union>f" apply(rule division_ofI) prefer 5 apply(rule assms(3)|assumption)+
   4.524 +  apply(rule finite_Union assms(1))+ prefer 3 apply(erule UnionE) apply(rule_tac s=X in division_ofD(3)[OF assms(2)])
   4.525 +  using division_ofD[OF assms(2)] by auto
   4.526 +  
   4.527 +lemma elementary_union_interval: assumes "p division_of \<Union>p"
   4.528 +  obtains q where "q division_of ({a..b::real^'n} \<union> \<Union>p)" proof-
   4.529 +  note assm=division_ofD[OF assms]
   4.530 +  have lem1:"\<And>f s. \<Union>\<Union> (f ` s) = \<Union>(\<lambda>x.\<Union>(f x)) ` s" by auto
   4.531 +  have lem2:"\<And>f s. f \<noteq> {} \<Longrightarrow> \<Union>{s \<union> t |t. t \<in> f} = s \<union> \<Union>f" by auto
   4.532 +{ presume "p={} \<Longrightarrow> thesis" "{a..b} = {} \<Longrightarrow> thesis" "{a..b} \<noteq> {} \<Longrightarrow> interior {a..b} = {} \<Longrightarrow> thesis"
   4.533 +    "p\<noteq>{} \<Longrightarrow> interior {a..b}\<noteq>{} \<Longrightarrow> {a..b} \<noteq> {} \<Longrightarrow> thesis"
   4.534 +  thus thesis by auto
   4.535 +next assume as:"p={}" guess p apply(rule elementary_interval[of a b]) .
   4.536 +  thus thesis apply(rule_tac that[of p]) unfolding as by auto 
   4.537 +next assume as:"{a..b}={}" show thesis apply(rule that) unfolding as using assms by auto
   4.538 +next assume as:"interior {a..b} = {}" "{a..b} \<noteq> {}"
   4.539 +  show thesis apply(rule that[of "insert {a..b} p"],rule division_ofI)
   4.540 +    unfolding finite_insert apply(rule assm(1)) unfolding Union_insert  
   4.541 +    using assm(2-4) as apply- by(fastsimp dest: assm(5))+
   4.542 +next assume as:"p \<noteq> {}" "interior {a..b} \<noteq> {}" "{a..b}\<noteq>{}"
   4.543 +  have "\<forall>k\<in>p. \<exists>q. (insert {a..b} q) division_of ({a..b} \<union> k)" proof case goal1
   4.544 +    from assm(4)[OF this] guess c .. then guess d ..
   4.545 +    thus ?case apply-apply(rule division_union_intervals_exists[OF as(3),of c d]) by auto
   4.546 +  qed from bchoice[OF this] guess q .. note q=division_ofD[OF this[rule_format]]
   4.547 +  let ?D = "\<Union>{insert {a..b} (q k) | k. k \<in> p}"
   4.548 +  show thesis apply(rule that[of "?D"]) proof(rule division_ofI)
   4.549 +    have *:"{insert {a..b} (q k) |k. k \<in> p} = (\<lambda>k. insert {a..b} (q k)) ` p" by auto
   4.550 +    show "finite ?D" apply(rule finite_Union) unfolding * apply(rule finite_imageI) using assm(1) q(1) by auto
   4.551 +    show "\<Union>?D = {a..b} \<union> \<Union>p" unfolding * lem1 unfolding lem2[OF as(1), of "{a..b}",THEN sym]
   4.552 +      using q(6) by auto
   4.553 +    fix k assume k:"k\<in>?D" thus " k \<subseteq> {a..b} \<union> \<Union>p" using q(2) by auto
   4.554 +    show "k \<noteq> {}" using q(3) k by auto show "\<exists>a b. k = {a..b}" using q(4) k by auto
   4.555 +    fix k' assume k':"k'\<in>?D" "k\<noteq>k'"
   4.556 +    obtain x  where x: "k \<in>insert {a..b} (q x)"  "x\<in>p"  using k  by auto
   4.557 +    obtain x' where x':"k'\<in>insert {a..b} (q x')" "x'\<in>p" using k' by auto
   4.558 +    show "interior k \<inter> interior k' = {}" proof(cases "x=x'")
   4.559 +      case True show ?thesis apply(rule q(5)) using x x' k' unfolding True by auto
   4.560 +    next case False 
   4.561 +      { presume "k = {a..b} \<Longrightarrow> ?thesis" "k' = {a..b} \<Longrightarrow> ?thesis" 
   4.562 +        "k \<noteq> {a..b} \<Longrightarrow> k' \<noteq> {a..b} \<Longrightarrow> ?thesis"
   4.563 +        thus ?thesis by auto }
   4.564 +      { assume as':"k  = {a..b}" show ?thesis apply(rule q(5)) using x' k'(2) unfolding as' by auto }
   4.565 +      { assume as':"k' = {a..b}" show ?thesis apply(rule q(5)) using x  k'(2) unfolding as' by auto }
   4.566 +      assume as':"k \<noteq> {a..b}" "k' \<noteq> {a..b}"
   4.567 +      guess c using q(4)[OF x(2,1)] .. then guess d .. note c_d=this
   4.568 +      have "interior k  \<inter> interior {a..b} = {}" apply(rule q(5)) using x  k'(2) using as' by auto
   4.569 +      hence "interior k \<subseteq> interior x" apply-
   4.570 +        apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x(2,1)]]) by auto moreover
   4.571 +      guess c using q(4)[OF x'(2,1)] .. then guess d .. note c_d=this
   4.572 +      have "interior k' \<inter> interior {a..b} = {}" apply(rule q(5)) using x' k'(2) using as' by auto
   4.573 +      hence "interior k' \<subseteq> interior x'" apply-
   4.574 +        apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x'(2,1)]]) by auto
   4.575 +      ultimately show ?thesis using assm(5)[OF x(2) x'(2) False] by auto
   4.576 +    qed qed } qed
   4.577 +
   4.578 +lemma elementary_unions_intervals:
   4.579 +  assumes "finite f" "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = {a..b::real^'n}"
   4.580 +  obtains p where "p division_of (\<Union>f)" proof-
   4.581 +  have "\<exists>p. p division_of (\<Union>f)" proof(induct_tac f rule:finite_subset_induct) 
   4.582 +    show "\<exists>p. p division_of \<Union>{}" using elementary_empty by auto
   4.583 +    fix x F assume as:"finite F" "x \<notin> F" "\<exists>p. p division_of \<Union>F" "x\<in>f"
   4.584 +    from this(3) guess p .. note p=this
   4.585 +    from assms(2)[OF as(4)] guess a .. then guess b .. note ab=this
   4.586 +    have *:"\<Union>F = \<Union>p" using division_ofD[OF p] by auto
   4.587 +    show "\<exists>p. p division_of \<Union>insert x F" using elementary_union_interval[OF p[unfolded *], of a b]
   4.588 +      unfolding Union_insert ab * by auto
   4.589 +  qed(insert assms,auto) thus ?thesis apply-apply(erule exE,rule that) by auto qed
   4.590 +
   4.591 +lemma elementary_union: assumes "ps division_of s" "pt division_of (t::(real^'n) set)"
   4.592 +  obtains p where "p division_of (s \<union> t)"
   4.593 +proof- have "s \<union> t = \<Union>ps \<union> \<Union>pt" using assms unfolding division_of_def by auto
   4.594 +  hence *:"\<Union>(ps \<union> pt) = s \<union> t" by auto
   4.595 +  show ?thesis apply-apply(rule elementary_unions_intervals[of "ps\<union>pt"])
   4.596 +    unfolding * prefer 3 apply(rule_tac p=p in that)
   4.597 +    using assms[unfolded division_of_def] by auto qed
   4.598 +
   4.599 +lemma partial_division_extend: fixes t::"(real^'n) set"
   4.600 +  assumes "p division_of s" "q division_of t" "s \<subseteq> t"
   4.601 +  obtains r where "p \<subseteq> r" "r division_of t" proof-
   4.602 +  note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)]
   4.603 +  obtain a b where ab:"t\<subseteq>{a..b}" using elementary_subset_interval[OF assms(2)] by auto
   4.604 +  guess r1 apply(rule partial_division_extend_interval) apply(rule assms(1)[unfolded divp(6)[THEN sym]])
   4.605 +    apply(rule subset_trans) by(rule ab assms[unfolded divp(6)[THEN sym]])+  note r1 = this division_ofD[OF this(2)]
   4.606 +  guess p' apply(rule elementary_unions_intervals[of "r1 - p"]) using r1(3,6) by auto 
   4.607 +  then obtain r2 where r2:"r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)" 
   4.608 +    apply- apply(drule elementary_inter[OF _ assms(2)[unfolded divq(6)[THEN sym]]]) by auto
   4.609 +  { fix x assume x:"x\<in>t" "x\<notin>s"
   4.610 +    hence "x\<in>\<Union>r1" unfolding r1 using ab by auto
   4.611 +    then guess r unfolding Union_iff .. note r=this moreover
   4.612 +    have "r \<notin> p" proof assume "r\<in>p" hence "x\<in>s" using divp(2) r by auto
   4.613 +      thus False using x by auto qed
   4.614 +    ultimately have "x\<in>\<Union>(r1 - p)" by auto }
   4.615 +  hence *:"t = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)" unfolding divp divq using assms(3) by auto
   4.616 +  show ?thesis apply(rule that[of "p \<union> r2"]) unfolding * defer apply(rule division_disjoint_union)
   4.617 +    unfolding divp(6) apply(rule assms r2)+
   4.618 +  proof- have "interior s \<inter> interior (\<Union>(r1-p)) = {}"
   4.619 +    proof(rule inter_interior_unions_intervals)
   4.620 +      show "finite (r1 - p)" "open (interior s)" "\<forall>t\<in>r1-p. \<exists>a b. t = {a..b}" using r1 by auto
   4.621 +      have *:"\<And>s. (\<And>x. x \<in> s \<Longrightarrow> False) \<Longrightarrow> s = {}" by auto
   4.622 +      show "\<forall>t\<in>r1-p. interior s \<inter> interior t = {}" proof(rule)
   4.623 +        fix m x assume as:"m\<in>r1-p"
   4.624 +        have "interior m \<inter> interior (\<Union>p) = {}" proof(rule inter_interior_unions_intervals)
   4.625 +          show "finite p" "open (interior m)" "\<forall>t\<in>p. \<exists>a b. t = {a..b}" using divp by auto
   4.626 +          show "\<forall>t\<in>p. interior m \<inter> interior t = {}" apply(rule, rule r1(7)) using as using r1 by auto
   4.627 +        qed thus "interior s \<inter> interior m = {}" unfolding divp by auto
   4.628 +      qed qed        
   4.629 +    thus "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}" using interior_subset by auto
   4.630 +  qed auto qed
   4.631 +
   4.632 +subsection {* Tagged (partial) divisions. *}
   4.633 +
   4.634 +definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40) where
   4.635 +  "(s tagged_partial_division_of i) \<equiv>
   4.636 +        finite s \<and>
   4.637 +        (\<forall>x k. (x,k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
   4.638 +        (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ((x1,k1) \<noteq> (x2,k2))
   4.639 +                       \<longrightarrow> (interior(k1) \<inter> interior(k2) = {}))"
   4.640 +
   4.641 +lemma tagged_partial_division_ofD[dest]: assumes "s tagged_partial_division_of i"
   4.642 +  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"
   4.643 +  "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
   4.644 +  "\<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) = {}"
   4.645 +  using assms unfolding tagged_partial_division_of_def  apply- by blast+ 
   4.646 +
   4.647 +definition tagged_division_of (infixr "tagged'_division'_of" 40) where
   4.648 +  "(s tagged_division_of i) \<equiv>
   4.649 +        (s tagged_partial_division_of i) \<and> (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
   4.650 +
   4.651 +lemma tagged_division_of_finite[dest]: "s tagged_division_of i \<Longrightarrow> finite s"
   4.652 +  unfolding tagged_division_of_def tagged_partial_division_of_def by auto
   4.653 +
   4.654 +lemma tagged_division_of:
   4.655 + "(s tagged_division_of i) \<longleftrightarrow>
   4.656 +        finite s \<and>
   4.657 +        (\<forall>x k. (x,k) \<in> s
   4.658 +               \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
   4.659 +        (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ~((x1,k1) = (x2,k2))
   4.660 +                       \<longrightarrow> (interior(k1) \<inter> interior(k2) = {})) \<and>
   4.661 +        (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
   4.662 +  unfolding tagged_division_of_def tagged_partial_division_of_def by auto
   4.663 +
   4.664 +lemma tagged_division_ofI: assumes
   4.665 +  "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}"
   4.666 +  "\<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) = {})"
   4.667 +  "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
   4.668 +  shows "s tagged_division_of i"
   4.669 +  unfolding tagged_division_of apply(rule) defer apply rule
   4.670 +  apply(rule allI impI conjI assms)+ apply assumption
   4.671 +  apply(rule, rule assms, assumption) apply(rule assms, assumption)
   4.672 +  using assms(1,5-) apply- by blast+
   4.673 +
   4.674 +lemma tagged_division_ofD[dest]: assumes "s tagged_division_of i"
   4.675 +  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}"
   4.676 +  "\<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) = {})"
   4.677 +  "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)" using assms unfolding tagged_division_of apply- by blast+
   4.678 +
   4.679 +lemma division_of_tagged_division: assumes"s tagged_division_of i"  shows "(snd ` s) division_of i"
   4.680 +proof(rule division_ofI) note assm=tagged_division_ofD[OF assms]
   4.681 +  show "\<Union>snd ` s = i" "finite (snd ` s)" using assm by auto
   4.682 +  fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto
   4.683 +  thus  "k \<subseteq> i" "k \<noteq> {}" "\<exists>a b. k = {a..b}" using assm apply- by fastsimp+
   4.684 +  fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto
   4.685 +  thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto
   4.686 +qed
   4.687 +
   4.688 +lemma partial_division_of_tagged_division: assumes "s tagged_partial_division_of i"
   4.689 +  shows "(snd ` s) division_of \<Union>(snd ` s)"
   4.690 +proof(rule division_ofI) note assm=tagged_partial_division_ofD[OF assms]
   4.691 +  show "finite (snd ` s)" "\<Union>snd ` s = \<Union>snd ` s" using assm by auto
   4.692 +  fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto
   4.693 +  thus "k\<noteq>{}" "\<exists>a b. k = {a..b}" "k \<subseteq> \<Union>snd ` s" using assm by auto
   4.694 +  fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto
   4.695 +  thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto
   4.696 +qed
   4.697 +
   4.698 +lemma tagged_partial_division_subset: assumes "s tagged_partial_division_of i" "t \<subseteq> s"
   4.699 +  shows "t tagged_partial_division_of i"
   4.700 +  using assms unfolding tagged_partial_division_of_def using finite_subset[OF assms(2)] by blast
   4.701 +
   4.702 +lemma setsum_over_tagged_division_lemma: fixes d::"(real^'m) set \<Rightarrow> 'a::real_normed_vector"
   4.703 +  assumes "p tagged_division_of i" "\<And>u v. {u..v} \<noteq> {} \<Longrightarrow> content {u..v} = 0 \<Longrightarrow> d {u..v} = 0"
   4.704 +  shows "setsum (\<lambda>(x,k). d k) p = setsum d (snd ` p)"
   4.705 +proof- note assm=tagged_division_ofD[OF assms(1)]
   4.706 +  have *:"(\<lambda>(x,k). d k) = d \<circ> snd" unfolding o_def apply(rule ext) by auto
   4.707 +  show ?thesis unfolding * apply(subst eq_commute) proof(rule setsum_reindex_nonzero)
   4.708 +    show "finite p" using assm by auto
   4.709 +    fix x y assume as:"x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y" 
   4.710 +    obtain a b where ab:"snd x = {a..b}" using assm(4)[of "fst x" "snd x"] as(1) by auto
   4.711 +    have "(fst x, snd y) \<in> p" "(fst x, snd y) \<noteq> y" unfolding as(4)[THEN sym] using as(1-3) by auto
   4.712 +    hence "interior (snd x) \<inter> interior (snd y) = {}" apply-apply(rule assm(5)[of "fst x" _ "fst y"]) using as by auto 
   4.713 +    hence "content {a..b} = 0" unfolding as(4)[THEN sym] ab content_eq_0_interior by auto
   4.714 +    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
   4.715 +    thus "d (snd x) = 0" unfolding ab by auto qed qed
   4.716 +
   4.717 +lemma tag_in_interval: "p tagged_division_of i \<Longrightarrow> (x,k) \<in> p \<Longrightarrow> x \<in> i" by auto
   4.718 +
   4.719 +lemma tagged_division_of_empty: "{} tagged_division_of {}"
   4.720 +  unfolding tagged_division_of by auto
   4.721 +
   4.722 +lemma tagged_partial_division_of_trivial[simp]:
   4.723 + "p tagged_partial_division_of {} \<longleftrightarrow> p = {}"
   4.724 +  unfolding tagged_partial_division_of_def by auto
   4.725 +
   4.726 +lemma tagged_division_of_trivial[simp]:
   4.727 + "p tagged_division_of {} \<longleftrightarrow> p = {}"
   4.728 +  unfolding tagged_division_of by auto
   4.729 +
   4.730 +lemma tagged_division_of_self:
   4.731 + "x \<in> {a..b} \<Longrightarrow> {(x,{a..b})} tagged_division_of {a..b}"
   4.732 +  apply(rule tagged_division_ofI) by auto
   4.733 +
   4.734 +lemma tagged_division_union:
   4.735 +  assumes "p1 tagged_division_of s1"  "p2 tagged_division_of s2" "interior s1 \<inter> interior s2 = {}"
   4.736 +  shows "(p1 \<union> p2) tagged_division_of (s1 \<union> s2)"
   4.737 +proof(rule tagged_division_ofI) note p1=tagged_division_ofD[OF assms(1)] and p2=tagged_division_ofD[OF assms(2)]
   4.738 +  show "finite (p1 \<union> p2)" using p1(1) p2(1) by auto
   4.739 +  show "\<Union>{k. \<exists>x. (x, k) \<in> p1 \<union> p2} = s1 \<union> s2" using p1(6) p2(6) by blast
   4.740 +  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
   4.741 +  show "k\<subseteq>s1\<union>s2" using xk p1(3) p2(3) by blast
   4.742 +  fix x' k' assume xk':"(x',k')\<in>p1\<union>p2" "(x,k) \<noteq> (x',k')"
   4.743 +  have *:"\<And>a b. a\<subseteq> s1 \<Longrightarrow> b\<subseteq> s2 \<Longrightarrow> interior a \<inter> interior b = {}" using assms(3) subset_interior by blast
   4.744 +  show "interior k \<inter> interior k' = {}" apply(cases "(x,k)\<in>p1", case_tac[!] "(x',k')\<in>p1")
   4.745 +    apply(rule p1(5)) prefer 4 apply(rule *) prefer 6 apply(subst Int_commute,rule *) prefer 8 apply(rule p2(5))
   4.746 +    using p1(3) p2(3) using xk xk' by auto qed 
   4.747 +
   4.748 +lemma tagged_division_unions:
   4.749 +  assumes "finite iset" "\<forall>i\<in>iset. (pfn(i) tagged_division_of i)"
   4.750 +  "\<forall>i1 \<in> iset. \<forall>i2 \<in> iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})"
   4.751 +  shows "\<Union>(pfn ` iset) tagged_division_of (\<Union>iset)"
   4.752 +proof(rule tagged_division_ofI)
   4.753 +  note assm = tagged_division_ofD[OF assms(2)[rule_format]]
   4.754 +  show "finite (\<Union>pfn ` iset)" apply(rule finite_Union) using assms by auto
   4.755 +  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 
   4.756 +  also have "\<dots> = \<Union>iset" using assm(6) by auto
   4.757 +  finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>pfn ` iset} = \<Union>iset" . 
   4.758 +  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
   4.759 +  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
   4.760 +  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
   4.761 +  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)
   4.762 +    using assms(3)[rule_format] subset_interior by blast
   4.763 +  show "interior k \<inter> interior k' = {}" apply(cases "i=i'")
   4.764 +    using assm(5)[OF i _ xk'(2)]  i'(2) using assm(3)[OF i] assm(3)[OF i'] defer apply-apply(rule *) by auto
   4.765 +qed
   4.766 +
   4.767 +lemma tagged_partial_division_of_union_self:
   4.768 +  assumes "p tagged_partial_division_of s" shows "p tagged_division_of (\<Union>(snd ` p))"
   4.769 +  apply(rule tagged_division_ofI) using tagged_partial_division_ofD[OF assms] by auto
   4.770 +
   4.771 +lemma tagged_division_of_union_self: assumes "p tagged_division_of s"
   4.772 +  shows "p tagged_division_of (\<Union>(snd ` p))"
   4.773 +  apply(rule tagged_division_ofI) using tagged_division_ofD[OF assms] by auto
   4.774 +
   4.775 +subsection {* Fine-ness of a partition w.r.t. a gauge. *}
   4.776 +
   4.777 +definition fine (infixr "fine" 46) where
   4.778 +  "d fine s \<longleftrightarrow> (\<forall>(x,k) \<in> s. k \<subseteq> d(x))"
   4.779 +
   4.780 +lemma fineI: assumes "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x"
   4.781 +  shows "d fine s" using assms unfolding fine_def by auto
   4.782 +
   4.783 +lemma fineD[dest]: assumes "d fine s"
   4.784 +  shows "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x" using assms unfolding fine_def by auto
   4.785 +
   4.786 +lemma fine_inter: "(\<lambda>x. d1 x \<inter> d2 x) fine p \<longleftrightarrow> d1 fine p \<and> d2 fine p"
   4.787 +  unfolding fine_def by auto
   4.788 +
   4.789 +lemma fine_inters:
   4.790 + "(\<lambda>x. \<Inter> {f d x | d.  d \<in> s}) fine p \<longleftrightarrow> (\<forall>d\<in>s. (f d) fine p)"
   4.791 +  unfolding fine_def by blast
   4.792 +
   4.793 +lemma fine_union:
   4.794 +  "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)"
   4.795 +  unfolding fine_def by blast
   4.796 +
   4.797 +lemma fine_unions:"(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)"
   4.798 +  unfolding fine_def by auto
   4.799 +
   4.800 +lemma fine_subset:  "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p"
   4.801 +  unfolding fine_def by blast
   4.802 +
   4.803 +subsection {* Gauge integral. Define on compact intervals first, then use a limit. *}
   4.804 +
   4.805 +definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46) where
   4.806 +  "(f has_integral_compact_interval y) i \<equiv>
   4.807 +        (\<forall>e>0. \<exists>d. gauge d \<and>
   4.808 +          (\<forall>p. p tagged_division_of i \<and> d fine p
   4.809 +                        \<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - y) < e))"
   4.810 +
   4.811 +definition has_integral (infixr "has'_integral" 46) where 
   4.812 +"((f::(real^'n \<Rightarrow> 'b::real_normed_vector)) has_integral y) i \<equiv>
   4.813 +        if (\<exists>a b. i = {a..b}) then (f has_integral_compact_interval y) i
   4.814 +        else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b}
   4.815 +              \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral_compact_interval z) {a..b} \<and>
   4.816 +                                       norm(z - y) < e))"
   4.817 +
   4.818 +lemma has_integral:
   4.819 + "(f has_integral y) ({a..b}) \<longleftrightarrow>
   4.820 +        (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p
   4.821 +                        \<longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
   4.822 +  unfolding has_integral_def has_integral_compact_interval_def by auto
   4.823 +
   4.824 +lemma has_integralD[dest]: assumes
   4.825 + "(f has_integral y) ({a..b})" "e>0"
   4.826 +  obtains d where "gauge d" "\<And>p. p tagged_division_of {a..b} \<Longrightarrow> d fine p
   4.827 +                        \<Longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f(x)) p - y) < e"
   4.828 +  using assms unfolding has_integral by auto
   4.829 +
   4.830 +lemma has_integral_alt:
   4.831 + "(f has_integral y) i \<longleftrightarrow>
   4.832 +      (if (\<exists>a b. i = {a..b}) then (f has_integral y) i
   4.833 +       else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b}
   4.834 +                               \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0)
   4.835 +                                        has_integral z) ({a..b}) \<and>
   4.836 +                                       norm(z - y) < e)))"
   4.837 +  unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto
   4.838 +
   4.839 +lemma has_integral_altD:
   4.840 +  assumes "(f has_integral y) i" "\<not> (\<exists>a b. i = {a..b})" "e>0"
   4.841 +  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)"
   4.842 +  using assms unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto
   4.843 +
   4.844 +definition integrable_on (infixr "integrable'_on" 46) where
   4.845 +  "(f integrable_on i) \<equiv> \<exists>y. (f has_integral y) i"
   4.846 +
   4.847 +definition "integral i f \<equiv> SOME y. (f has_integral y) i"
   4.848 +
   4.849 +lemma integrable_integral[dest]:
   4.850 + "f integrable_on i \<Longrightarrow> (f has_integral (integral i f)) i"
   4.851 +  unfolding integrable_on_def integral_def by(rule someI_ex)
   4.852 +
   4.853 +lemma has_integral_integrable[intro]: "(f has_integral i) s \<Longrightarrow> f integrable_on s"
   4.854 +  unfolding integrable_on_def by auto
   4.855 +
   4.856 +lemma has_integral_integral:"f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s"
   4.857 +  by auto
   4.858 +
   4.859 +lemma setsum_content_null:
   4.860 +  assumes "content({a..b}) = 0" "p tagged_division_of {a..b}"
   4.861 +  shows "setsum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)"
   4.862 +proof(rule setsum_0',rule) fix y assume y:"y\<in>p"
   4.863 +  obtain x k where xk:"y = (x,k)" using surj_pair[of y] by blast
   4.864 +  note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]]
   4.865 +  from this(2) guess c .. then guess d .. note c_d=this
   4.866 +  have "(\<lambda>(x, k). content k *\<^sub>R f x) y = content k *\<^sub>R f x" unfolding xk by auto
   4.867 +  also have "\<dots> = 0" using content_subset[OF assm(1)[unfolded c_d]] content_pos_le[of c d]
   4.868 +    unfolding assms(1) c_d by auto
   4.869 +  finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" .
   4.870 +qed
   4.871 +
   4.872 +subsection {* Some basic combining lemmas. *}
   4.873 +
   4.874 +lemma tagged_division_unions_exists:
   4.875 +  assumes "finite iset" "\<forall>i \<in> iset. \<exists>p. p tagged_division_of i \<and> d fine p"
   4.876 +  "\<forall>i1\<in>iset. \<forall>i2\<in>iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})" "(\<Union>iset = i)"
   4.877 +   obtains p where "p tagged_division_of i" "d fine p"
   4.878 +proof- guess pfn using bchoice[OF assms(2)] .. note pfn = conjunctD2[OF this[rule_format]]
   4.879 +  show thesis apply(rule_tac p="\<Union>(pfn ` iset)" in that) unfolding assms(4)[THEN sym]
   4.880 +    apply(rule tagged_division_unions[OF assms(1) _ assms(3)]) defer 
   4.881 +    apply(rule fine_unions) using pfn by auto
   4.882 +qed
   4.883 +
   4.884 +subsection {* The set we're concerned with must be closed. *}
   4.885 +
   4.886 +lemma division_of_closed: "s division_of i \<Longrightarrow> closed (i::(real^'n) set)"
   4.887 +  unfolding division_of_def by(fastsimp intro!: closed_Union closed_interval)
   4.888 +
   4.889 +subsection {* General bisection principle for intervals; might be useful elsewhere. *}
   4.890 +
   4.891 +lemma interval_bisection_step:
   4.892 +  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})"
   4.893 +  obtains c d where "~(P{c..d})"
   4.894 +  "\<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"
   4.895 +proof- have "{a..b} \<noteq> {}" using assms(1,3) by auto
   4.896 +  note ab=this[unfolded interval_eq_empty not_ex not_less]
   4.897 +  { fix f have "finite f \<Longrightarrow>
   4.898 +        (\<forall>s\<in>f. P s) \<Longrightarrow>
   4.899 +        (\<forall>s\<in>f. \<exists>a b. s = {a..b}) \<Longrightarrow>
   4.900 +        (\<forall>s\<in>f.\<forall>t\<in>f. ~(s = t) \<longrightarrow> interior(s) \<inter> interior(t) = {}) \<Longrightarrow> P(\<Union>f)"
   4.901 +    proof(induct f rule:finite_induct)
   4.902 +      case empty show ?case using assms(1) by auto
   4.903 +    next case (insert x f) show ?case unfolding Union_insert apply(rule assms(2)[rule_format])
   4.904 +        apply rule defer apply rule defer apply(rule inter_interior_unions_intervals)
   4.905 +        using insert by auto
   4.906 +    qed } note * = this
   4.907 +  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)}"
   4.908 +  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"
   4.909 +  { presume "\<forall>c d. ?PP c d \<longrightarrow> P {c..d} \<Longrightarrow> False"
   4.910 +    thus thesis unfolding atomize_not not_all apply-apply(erule exE)+ apply(rule_tac c=x and d=xa in that) by auto }
   4.911 +  assume as:"\<forall>c d. ?PP c d \<longrightarrow> P {c..d}"
   4.912 +  have "P (\<Union> ?A)" proof(rule *, rule_tac[2-] ballI, rule_tac[4] ballI, rule_tac[4] impI) 
   4.913 +    let ?B = "(\<lambda>s.{(\<chi> i. if i \<in> s then a$i else (a$i + b$i) / 2) ..
   4.914 +      (\<chi> i. if i \<in> s then (a$i + b$i) / 2 else b$i)}) ` {s. s \<subseteq> UNIV}"
   4.915 +    have "?A \<subseteq> ?B" proof case goal1
   4.916 +      then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+) note c_d=this[rule_format]
   4.917 +      have *:"\<And>a b c d. a = c \<Longrightarrow> b = d \<Longrightarrow> {a..b} = {c..d}" by auto
   4.918 +      show "x\<in>?B" unfolding image_iff apply(rule_tac x="{i. c$i = a$i}" in bexI)
   4.919 +        unfolding c_d apply(rule * ) unfolding Cart_eq cond_component Cart_lambda_beta
   4.920 +      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)"
   4.921 +          "d $ i = (if i \<in> {i. c $ i = a $ i} then (a $ i + b $ i) / 2 else b $ i)"
   4.922 +          using c_d(2)[of i] ab[THEN spec[where x=i]] by(auto simp add:field_simps)
   4.923 +      qed auto qed
   4.924 +    thus "finite ?A" apply(rule finite_subset[of _ ?B]) by auto
   4.925 +    fix s assume "s\<in>?A" then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+)
   4.926 +    note c_d=this[rule_format]
   4.927 +    show "P s" unfolding c_d apply(rule as[rule_format]) proof- case goal1 show ?case 
   4.928 +        using c_d(2)[of i] using ab[THEN spec[where x=i]] by auto qed
   4.929 +    show "\<exists>a b. s = {a..b}" unfolding c_d by auto
   4.930 +    fix t assume "t\<in>?A" then guess e unfolding mem_Collect_eq .. then guess f apply- by(erule exE,(erule conjE)+)
   4.931 +    note e_f=this[rule_format]
   4.932 +    assume "s \<noteq> t" hence "\<not> (c = e \<and> d = f)" unfolding c_d e_f by auto
   4.933 +    then obtain i where "c$i \<noteq> e$i \<or> d$i \<noteq> f$i" unfolding de_Morgan_conj Cart_eq by auto
   4.934 +    hence i:"c$i \<noteq> e$i" "d$i \<noteq> f$i" apply- apply(erule_tac[!] disjE)
   4.935 +    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
   4.936 +    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
   4.937 +    qed have *:"\<And>s t. (\<And>a. a\<in>s \<Longrightarrow> a\<in>t \<Longrightarrow> False) \<Longrightarrow> s \<inter> t = {}" by auto
   4.938 +    show "interior s \<inter> interior t = {}" unfolding e_f c_d interior_closed_interval proof(rule *)
   4.939 +      fix x assume "x\<in>{c<..<d}" "x\<in>{e<..<f}"
   4.940 +      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
   4.941 +      show False using c_d(2)[of i] apply- apply(erule_tac disjE)
   4.942 +      proof(erule_tac[!] conjE) assume as:"c $ i = a $ i" "d $ i = (a $ i + b $ i) / 2"
   4.943 +        show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps)
   4.944 +      next assume as:"c $ i = (a $ i + b $ i) / 2" "d $ i = b $ i"
   4.945 +        show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps)
   4.946 +      qed qed qed
   4.947 +  also have "\<Union> ?A = {a..b}" proof(rule set_ext,rule)
   4.948 +    fix x assume "x\<in>\<Union>?A" then guess Y unfolding Union_iff ..
   4.949 +    from this(1) guess c unfolding mem_Collect_eq .. then guess d ..
   4.950 +    note c_d = this[THEN conjunct2,rule_format] `x\<in>Y`[unfolded this[THEN conjunct1]]
   4.951 +    show "x\<in>{a..b}" unfolding mem_interval proof 
   4.952 +      fix i show "a $ i \<le> x $ i \<and> x $ i \<le> b $ i"
   4.953 +        using c_d(1)[of i] c_d(2)[unfolded mem_interval,THEN spec[where x=i]] by auto qed
   4.954 +  next fix x assume x:"x\<in>{a..b}"
   4.955 +    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"
   4.956 +      (is "\<forall>i. \<exists>c d. ?P i c d") unfolding mem_interval proof fix i
   4.957 +      have "?P i (a$i) ((a $ i + b $ i) / 2) \<or> ?P i ((a $ i + b $ i) / 2) (b$i)"
   4.958 +        using x[unfolded mem_interval,THEN spec[where x=i]] by auto thus "\<exists>c d. ?P i c d" by blast
   4.959 +    qed thus "x\<in>\<Union>?A" unfolding Union_iff lambda_skolem unfolding Bex_def mem_Collect_eq
   4.960 +      apply-apply(erule exE)+ apply(rule_tac x="{xa..xaa}" in exI) unfolding mem_interval by auto
   4.961 +  qed finally show False using assms by auto qed
   4.962 +
   4.963 +lemma interval_bisection:
   4.964 +  assumes "P {}" "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))" "\<not> P {a..b::real^'n}"
   4.965 +  obtains x where "x \<in> {a..b}" "\<forall>e>0. \<exists>c d. x \<in> {c..d} \<and> {c..d} \<subseteq> ball x e \<and> {c..d} \<subseteq> {a..b} \<and> ~P({c..d})"
   4.966 +proof-
   4.967 +  have "\<forall>x. \<exists>y. \<not> P {fst x..snd x} \<longrightarrow> (\<not> P {fst y..snd y} \<and> (\<forall>i. fst x$i \<le> fst y$i \<and> fst y$i \<le> snd y$i \<and> snd y$i \<le> snd x$i \<and>
   4.968 +                           2 * (snd y$i - fst y$i) \<le> snd x$i - fst x$i))" proof case goal1 thus ?case proof-
   4.969 +      presume "\<not> P {fst x..snd x} \<Longrightarrow> ?thesis"
   4.970 +      thus ?thesis apply(cases "P {fst x..snd x}") by auto
   4.971 +    next assume as:"\<not> P {fst x..snd x}" from interval_bisection_step[of P, OF assms(1-2) as] guess c d . 
   4.972 +      thus ?thesis apply- apply(rule_tac x="(c,d)" in exI) by auto
   4.973 +    qed qed then guess f apply-apply(drule choice) by(erule exE) note f=this
   4.974 +  def AB \<equiv> "\<lambda>n. (f ^^ n) (a,b)" def A \<equiv> "\<lambda>n. fst(AB n)" and B \<equiv> "\<lambda>n. snd(AB n)" note ab_def = this AB_def
   4.975 +  have "A 0 = a" "B 0 = b" "\<And>n. \<not> P {A(Suc n)..B(Suc n)} \<and>
   4.976 +    (\<forall>i. A(n)$i \<le> A(Suc n)$i \<and> A(Suc n)$i \<le> B(Suc n)$i \<and> B(Suc n)$i \<le> B(n)$i \<and> 
   4.977 +    2 * (B(Suc n)$i - A(Suc n)$i) \<le> B(n)$i - A(n)$i)" (is "\<And>n. ?P n")
   4.978 +  proof- show "A 0 = a" "B 0 = b" unfolding ab_def by auto
   4.979 +    case goal3 note S = ab_def funpow.simps o_def id_apply show ?case
   4.980 +    proof(induct n) case 0 thus ?case unfolding S apply(rule f[rule_format]) using assms(3) by auto
   4.981 +    next case (Suc n) show ?case unfolding S apply(rule f[rule_format]) using Suc unfolding S by auto
   4.982 +    qed qed note AB = this(1-2) conjunctD2[OF this(3),rule_format]
   4.983 +
   4.984 +  have interv:"\<And>e. 0 < e \<Longrightarrow> \<exists>n. \<forall>x\<in>{A n..B n}. \<forall>y\<in>{A n..B n}. dist x y < e"
   4.985 +  proof- case goal1 guess n using real_arch_pow2[of "(setsum (\<lambda>i. b$i - a$i) UNIV) / e"] .. note n=this
   4.986 +    show ?case apply(rule_tac x=n in exI) proof(rule,rule)
   4.987 +      fix x y assume xy:"x\<in>{A n..B n}" "y\<in>{A n..B n}"
   4.988 +      have "dist x y \<le> setsum (\<lambda>i. abs((x - y)$i)) UNIV" unfolding vector_dist_norm by(rule norm_le_l1)
   4.989 +      also have "\<dots> \<le> setsum (\<lambda>i. B n$i - A n$i) UNIV"
   4.990 +      proof(rule setsum_mono) fix i show "\<bar>(x - y) $ i\<bar> \<le> B n $ i - A n $ i"
   4.991 +          using xy[unfolded mem_interval,THEN spec[where x=i]]
   4.992 +          unfolding vector_minus_component by auto qed
   4.993 +      also have "\<dots> \<le> setsum (\<lambda>i. b$i - a$i) UNIV / 2^n" unfolding setsum_divide_distrib
   4.994 +      proof(rule setsum_mono) case goal1 thus ?case
   4.995 +        proof(induct n) case 0 thus ?case unfolding AB by auto
   4.996 +        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
   4.997 +          also have "\<dots> \<le> (b $ i - a $ i) / 2 ^ Suc n" using Suc by(auto simp add:field_simps) finally show ?case .
   4.998 +        qed qed
   4.999 +      also have "\<dots> < e" using n using goal1 by(auto simp add:field_simps) finally show "dist x y < e" .
  4.1000 +    qed qed
  4.1001 +  { fix n m ::nat assume "m \<le> n" then guess d unfolding le_Suc_ex_iff .. note d=this
  4.1002 +    have "{A n..B n} \<subseteq> {A m..B m}" unfolding d 
  4.1003 +    proof(induct d) case 0 thus ?case by auto
  4.1004 +    next case (Suc d) show ?case apply(rule subset_trans[OF _ Suc])
  4.1005 +        apply(rule) unfolding mem_interval apply(rule,erule_tac x=i in allE)
  4.1006 +      proof- case goal1 thus ?case using AB(4)[of "m + d" i] by(auto simp add:field_simps)
  4.1007 +      qed qed } note ABsubset = this 
  4.1008 +  have "\<exists>a. \<forall>n. a\<in>{A n..B n}" apply(rule decreasing_closed_nest[rule_format,OF closed_interval _ ABsubset interv])
  4.1009 +  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
  4.1010 +  then guess x0 .. note x0=this[rule_format]
  4.1011 +  show thesis proof(rule that[rule_format,of x0])
  4.1012 +    show "x0\<in>{a..b}" using x0[of 0] unfolding AB .
  4.1013 +    fix e assume "0 < (e::real)" from interv[OF this] guess n .. note n=this
  4.1014 +    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}"
  4.1015 +      apply(rule_tac x="A n" in exI,rule_tac x="B n" in exI) apply(rule,rule x0) apply rule defer 
  4.1016 +    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
  4.1017 +      show "{A n..B n} \<subseteq> ball x0 e" using n using x0[of n] by auto
  4.1018 +      show "{A n..B n} \<subseteq> {a..b}" unfolding AB(1-2)[symmetric] apply(rule ABsubset) by auto
  4.1019 +    qed qed qed 
  4.1020 +
  4.1021 +subsection {* Cousin's lemma. *}
  4.1022 +
  4.1023 +lemma fine_division_exists: assumes "gauge g" 
  4.1024 +  obtains p where "p tagged_division_of {a..b::real^'n}" "g fine p"
  4.1025 +proof- presume "\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p) \<Longrightarrow> False"
  4.1026 +  then guess p unfolding atomize_not not_not .. thus thesis apply-apply(rule that[of p]) by auto
  4.1027 +next assume as:"\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p)"
  4.1028 +  guess x apply(rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p",rule_format,OF _ _ as])
  4.1029 +    apply(rule_tac x="{}" in exI) defer apply(erule conjE exE)+
  4.1030 +  proof- show "{} tagged_division_of {} \<and> g fine {}" unfolding fine_def by auto
  4.1031 +    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 = {}"
  4.1032 +    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
  4.1033 +      apply(rule tagged_division_union) prefer 4 apply(rule fine_union) by auto
  4.1034 +  qed note x=this
  4.1035 +  obtain e where e:"e>0" "ball x e \<subseteq> g x" using gaugeD[OF assms, of x] unfolding open_contains_ball by auto
  4.1036 +  from x(2)[OF e(1)] guess c d apply-apply(erule exE conjE)+ . note c_d = this
  4.1037 +  have "g fine {(x, {c..d})}" unfolding fine_def using e using c_d(2) by auto
  4.1038 +  thus False using tagged_division_of_self[OF c_d(1)] using c_d by auto qed
  4.1039 +
  4.1040 +subsection {* Basic theorems about integrals. *}
  4.1041 +
  4.1042 +lemma has_integral_unique: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
  4.1043 +  assumes "(f has_integral k1) i" "(f has_integral k2) i" shows "k1 = k2"
  4.1044 +proof(rule ccontr) let ?e = "norm(k1 - k2) / 2" assume as:"k1 \<noteq> k2" hence e:"?e > 0" by auto
  4.1045 +  have lem:"\<And>f::real^'n \<Rightarrow> 'a.  \<And> a b k1 k2.
  4.1046 +    (f has_integral k1) ({a..b}) \<Longrightarrow> (f has_integral k2) ({a..b}) \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> False"
  4.1047 +  proof- case goal1 let ?e = "norm(k1 - k2) / 2" from goal1(3) have e:"?e > 0" by auto
  4.1048 +    guess d1 by(rule has_integralD[OF goal1(1) e]) note d1=this
  4.1049 +    guess d2 by(rule has_integralD[OF goal1(2) e]) note d2=this
  4.1050 +    guess p by(rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)],of a b]) note p=this
  4.1051 +    let ?c = "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" have "norm (k1 - k2) \<le> norm (?c - k2) + norm (?c - k1)"
  4.1052 +      using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"] by(auto simp add:group_simps norm_minus_commute)
  4.1053 +    also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
  4.1054 +      apply(rule add_strict_mono) apply(rule_tac[!] d2(2) d1(2)) using p unfolding fine_def by auto
  4.1055 +    finally show False by auto
  4.1056 +  qed { presume "\<not> (\<exists>a b. i = {a..b}) \<Longrightarrow> False"
  4.1057 +    thus False apply-apply(cases "\<exists>a b. i = {a..b}")
  4.1058 +      using assms by(auto simp add:has_integral intro:lem[OF _ _ as]) }
  4.1059 +  assume as:"\<not> (\<exists>a b. i = {a..b})"
  4.1060 +  guess B1 by(rule has_integral_altD[OF assms(1) as,OF e]) note B1=this[rule_format]
  4.1061 +  guess B2 by(rule has_integral_altD[OF assms(2) as,OF e]) note B2=this[rule_format]
  4.1062 +  have "\<exists>a b::real^'n. ball 0 B1 \<union> ball 0 B2 \<subseteq> {a..b}" apply(rule bounded_subset_closed_interval)
  4.1063 +    using bounded_Un bounded_ball by auto then guess a b apply-by(erule exE)+
  4.1064 +  note ab=conjunctD2[OF this[unfolded Un_subset_iff]]
  4.1065 +  guess w using B1(2)[OF ab(1)] .. note w=conjunctD2[OF this]
  4.1066 +  guess z using B2(2)[OF ab(2)] .. note z=conjunctD2[OF this]
  4.1067 +  have "z = w" using lem[OF w(1) z(1)] by auto
  4.1068 +  hence "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)"
  4.1069 +    using norm_triangle_ineq4[of "k1 - w" "k2 - z"] by(auto simp add: norm_minus_commute) 
  4.1070 +  also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2" apply(rule add_strict_mono) by(rule_tac[!] z(2) w(2))
  4.1071 +  finally show False by auto qed
  4.1072 +
  4.1073 +lemma integral_unique[intro]:
  4.1074 +  "(f has_integral y) k \<Longrightarrow> integral k f = y"
  4.1075 +  unfolding integral_def apply(rule some_equality) by(auto intro: has_integral_unique) 
  4.1076 +
  4.1077 +lemma has_integral_is_0: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector" 
  4.1078 +  assumes "\<forall>x\<in>s. f x = 0" shows "(f has_integral 0) s"
  4.1079 +proof- have lem:"\<And>a b. \<And>f::real^'n \<Rightarrow> 'a.
  4.1080 +    (\<forall>x\<in>{a..b}. f(x) = 0) \<Longrightarrow> (f has_integral 0) ({a..b})" unfolding has_integral
  4.1081 +  proof(rule,rule) fix a b e and f::"real^'n \<Rightarrow> 'a"
  4.1082 +    assume as:"\<forall>x\<in>{a..b}. f x = 0" "0 < (e::real)"
  4.1083 +    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)"
  4.1084 +      apply(rule_tac x="\<lambda>x. ball x 1" in exI)  apply(rule,rule gaugeI) unfolding centre_in_ball defer apply(rule open_ball)
  4.1085 +    proof(rule,rule,erule conjE) case goal1
  4.1086 +      have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) = 0" proof(rule setsum_0',rule)
  4.1087 +        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
  4.1088 +        thus "(\<lambda>(x, k). content k *\<^sub>R f x) x = 0" apply(subst surjective_pairing[of x]) unfolding split_conv by auto
  4.1089 +      qed thus ?case using as by auto
  4.1090 +    qed auto qed  { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
  4.1091 +    thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}")
  4.1092 +      using assms by(auto simp add:has_integral intro:lem) }
  4.1093 +  have *:"(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. 0)" apply(rule ext) using assms by auto
  4.1094 +  assume "\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P *
  4.1095 +  apply(rule,rule,rule_tac x=1 in exI,rule) defer apply(rule,rule,rule)
  4.1096 +  proof- fix e::real and a b assume "e>0"
  4.1097 +    thus "\<exists>z. ((\<lambda>x::real^'n. 0::'a) has_integral z) {a..b} \<and> norm (z - 0) < e"
  4.1098 +      apply(rule_tac x=0 in exI) apply(rule,rule lem) by auto
  4.1099 +  qed auto qed
  4.1100 +
  4.1101 +lemma has_integral_0[simp]: "((\<lambda>x::real^'n. 0) has_integral 0) s"
  4.1102 +  apply(rule has_integral_is_0) by auto 
  4.1103 +
  4.1104 +lemma has_integral_0_eq[simp]: "((\<lambda>x. 0) has_integral i) s \<longleftrightarrow> i = 0"
  4.1105 +  using has_integral_unique[OF has_integral_0] by auto
  4.1106 +
  4.1107 +lemma has_integral_linear: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
  4.1108 +  assumes "(f has_integral y) s" "bounded_linear h" shows "((h o f) has_integral ((h y))) s"
  4.1109 +proof- interpret bounded_linear h using assms(2) . from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format]
  4.1110 +  have lem:"\<And>f::real^'n \<Rightarrow> 'a. \<And> y a b.
  4.1111 +    (f has_integral y) ({a..b}) \<Longrightarrow> ((h o f) has_integral h(y)) ({a..b})"
  4.1112 +  proof(subst has_integral,rule,rule) case goal1
  4.1113 +    from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format]
  4.1114 +    have *:"e / B > 0" apply(rule divide_pos_pos) using goal1(2) B by auto
  4.1115 +    guess g using has_integralD[OF goal1(1) *] . note g=this
  4.1116 +    show ?case apply(rule_tac x=g in exI) apply(rule,rule g(1))
  4.1117 +    proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "g fine p" 
  4.1118 +      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
  4.1119 +      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"
  4.1120 +        unfolding o_def unfolding scaleR[THEN sym] * by simp
  4.1121 +      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
  4.1122 +      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)" .
  4.1123 +      show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) - h y) < e" unfolding * diff[THEN sym]
  4.1124 +        apply(rule le_less_trans[OF B(2)]) using g(2)[OF as] B(1) by(auto simp add:field_simps)
  4.1125 +    qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
  4.1126 +    thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) }
  4.1127 +  assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P
  4.1128 +  proof(rule,rule) fix e::real  assume e:"0<e"
  4.1129 +    have *:"0 < e/B" by(rule divide_pos_pos,rule e,rule B(1))
  4.1130 +    guess M using has_integral_altD[OF assms(1) as *,rule_format] . note M=this
  4.1131 +    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)"
  4.1132 +      apply(rule_tac x=M in exI) apply(rule,rule M(1))
  4.1133 +    proof(rule,rule,rule) case goal1 guess z using M(2)[OF goal1(1)] .. note z=conjunctD2[OF this]
  4.1134 +      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)"
  4.1135 +        unfolding o_def apply(rule ext) using zero by auto
  4.1136 +      show ?case apply(rule_tac x="h z" in exI,rule) unfolding * apply(rule lem[OF z(1)]) unfolding diff[THEN sym]
  4.1137 +        apply(rule le_less_trans[OF B(2)]) using B(1) z(2) by(auto simp add:field_simps)
  4.1138 +    qed qed qed
  4.1139 +
  4.1140 +lemma has_integral_cmul:
  4.1141 +  shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_integral (c *\<^sub>R k)) s"
  4.1142 +  unfolding o_def[THEN sym] apply(rule has_integral_linear,assumption)
  4.1143 +  by(rule scaleR.bounded_linear_right)
  4.1144 +
  4.1145 +lemma has_integral_neg:
  4.1146 +  shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. -(f x)) has_integral (-k)) s"
  4.1147 +  apply(drule_tac c="-1" in has_integral_cmul) by auto
  4.1148 +
  4.1149 +lemma has_integral_add: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector" 
  4.1150 +  assumes "(f has_integral k) s" "(g has_integral l) s"
  4.1151 +  shows "((\<lambda>x. f x + g x) has_integral (k + l)) s"
  4.1152 +proof- have lem:"\<And>f g::real^'n \<Rightarrow> 'a. \<And>a b k l.
  4.1153 +    (f has_integral k) ({a..b}) \<Longrightarrow> (g has_integral l) ({a..b}) \<Longrightarrow>
  4.1154 +     ((\<lambda>x. f(x) + g(x)) has_integral (k + l)) ({a..b})" proof- case goal1
  4.1155 +    show ?case unfolding has_integral proof(rule,rule) fix e::real assume e:"e>0" hence *:"e/2>0" by auto
  4.1156 +      guess d1 using has_integralD[OF goal1(1) *] . note d1=this
  4.1157 +      guess d2 using has_integralD[OF goal1(2) *] . note d2=this
  4.1158 +      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)"
  4.1159 +        apply(rule_tac x="\<lambda>x. (d1 x) \<inter> (d2 x)" in exI) apply(rule,rule gauge_inter[OF d1(1) d2(1)])
  4.1160 +      proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d1 x \<inter> d2 x) fine p"
  4.1161 +        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)"
  4.1162 +          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]
  4.1163 +          by(rule setsum_cong2,auto)
  4.1164 +        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))"
  4.1165 +          unfolding * by(auto simp add:group_simps) also let ?res = "\<dots>"
  4.1166 +        from as have *:"d1 fine p" "d2 fine p" unfolding fine_inter by auto
  4.1167 +        have "?res < e/2 + e/2" apply(rule le_less_trans[OF norm_triangle_ineq])
  4.1168 +          apply(rule add_strict_mono) using d1(2)[OF as(1) *(1)] and d2(2)[OF as(1) *(2)] by auto
  4.1169 +        finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e" by auto
  4.1170 +      qed qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
  4.1171 +    thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) }
  4.1172 +  assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P
  4.1173 +  proof(rule,rule) case goal1 hence *:"e/2 > 0" by auto
  4.1174 +    from has_integral_altD[OF assms(1) as *] guess B1 . note B1=this[rule_format]
  4.1175 +    from has_integral_altD[OF assms(2) as *] guess B2 . note B2=this[rule_format]
  4.1176 +    show ?case apply(rule_tac x="max B1 B2" in exI) apply(rule,rule min_max.less_supI1,rule B1)
  4.1177 +    proof(rule,rule,rule) fix a b assume "ball 0 (max B1 B2) \<subseteq> {a..b::real^'n}"
  4.1178 +      hence *:"ball 0 B1 \<subseteq> {a..b::real^'n}" "ball 0 B2 \<subseteq> {a..b::real^'n}" by auto
  4.1179 +      guess w using B1(2)[OF *(1)] .. note w=conjunctD2[OF this]
  4.1180 +      guess z using B2(2)[OF *(2)] .. note z=conjunctD2[OF this]
  4.1181 +      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
  4.1182 +      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"
  4.1183 +        apply(rule_tac x="w + z" in exI) apply(rule,rule lem[OF w(1) z(1), unfolded *[THEN sym]])
  4.1184 +        using norm_triangle_ineq[of "w - k" "z - l"] w(2) z(2) by(auto simp add:field_simps)
  4.1185 +    qed qed qed
  4.1186 +
  4.1187 +lemma has_integral_sub:
  4.1188 +  shows "(f has_integral k) s \<Longrightarrow> (g has_integral l) s \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) has_integral (k - l)) s"
  4.1189 +  using has_integral_add[OF _ has_integral_neg,of f k s g l] unfolding group_simps by auto
  4.1190 +
  4.1191 +lemma integral_0: "integral s (\<lambda>x::real^'n. 0::real^'m) = 0"
  4.1192 +  by(rule integral_unique has_integral_0)+
  4.1193 +
  4.1194 +lemma integral_add:
  4.1195 +  shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow>
  4.1196 +   integral s (\<lambda>x. f x + g x) = integral s f + integral s g"
  4.1197 +  apply(rule integral_unique) apply(drule integrable_integral)+
  4.1198 +  apply(rule has_integral_add) by assumption+
  4.1199 +
  4.1200 +lemma integral_cmul:
  4.1201 +  shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. c *\<^sub>R f x) = c *\<^sub>R integral s f"
  4.1202 +  apply(rule integral_unique) apply(drule integrable_integral)+
  4.1203 +  apply(rule has_integral_cmul) by assumption+
  4.1204 +
  4.1205 +lemma integral_neg:
  4.1206 +  shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. - f x) = - integral s f"
  4.1207 +  apply(rule integral_unique) apply(drule integrable_integral)+
  4.1208 +  apply(rule has_integral_neg) by assumption+
  4.1209 +
  4.1210 +lemma integral_sub:
  4.1211 +  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"
  4.1212 +  apply(rule integral_unique) apply(drule integrable_integral)+
  4.1213 +  apply(rule has_integral_sub) by assumption+
  4.1214 +
  4.1215 +lemma integrable_0: "(\<lambda>x. 0) integrable_on s"
  4.1216 +  unfolding integrable_on_def using has_integral_0 by auto
  4.1217 +
  4.1218 +lemma integrable_add:
  4.1219 +  shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x + g x) integrable_on s"
  4.1220 +  unfolding integrable_on_def by(auto intro: has_integral_add)
  4.1221 +
  4.1222 +lemma integrable_cmul:
  4.1223 +  shows "f integrable_on s \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) integrable_on s"
  4.1224 +  unfolding integrable_on_def by(auto intro: has_integral_cmul)
  4.1225 +
  4.1226 +lemma integrable_neg:
  4.1227 +  shows "f integrable_on s \<Longrightarrow> (\<lambda>x. -f(x)) integrable_on s"
  4.1228 +  unfolding integrable_on_def by(auto intro: has_integral_neg)
  4.1229 +
  4.1230 +lemma integrable_sub:
  4.1231 +  shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x - g x) integrable_on s"
  4.1232 +  unfolding integrable_on_def by(auto intro: has_integral_sub)
  4.1233 +
  4.1234 +lemma integrable_linear:
  4.1235 +  shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> (h o f) integrable_on s"
  4.1236 +  unfolding integrable_on_def by(auto intro: has_integral_linear)
  4.1237 +
  4.1238 +lemma integral_linear:
  4.1239 +  shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> integral s (h o f) = h(integral s f)"
  4.1240 +  apply(rule has_integral_unique) defer unfolding has_integral_integral 
  4.1241 +  apply(drule has_integral_linear,assumption,assumption) unfolding has_integral_integral[THEN sym]
  4.1242 +  apply(rule integrable_linear) by assumption+
  4.1243 +
  4.1244 +lemma has_integral_setsum:
  4.1245 +  assumes "finite t" "\<forall>a\<in>t. ((f a) has_integral (i a)) s"
  4.1246 +  shows "((\<lambda>x. setsum (\<lambda>a. f a x) t) has_integral (setsum i t)) s"
  4.1247 +proof(insert assms(1) subset_refl[of t],induct rule:finite_subset_induct)
  4.1248 +  case (insert x F) show ?case unfolding setsum_insert[OF insert(1,3)]
  4.1249 +    apply(rule has_integral_add) using insert assms by auto
  4.1250 +qed auto
  4.1251 +
  4.1252 +lemma integral_setsum:
  4.1253 +  shows "finite t \<Longrightarrow> \<forall>a\<in>t. (f a) integrable_on s \<Longrightarrow>
  4.1254 +  integral s (\<lambda>x. setsum (\<lambda>a. f a x) t) = setsum (\<lambda>a. integral s (f a)) t"
  4.1255 +  apply(rule integral_unique) apply(rule has_integral_setsum)
  4.1256 +  using integrable_integral by auto
  4.1257 +
  4.1258 +lemma integrable_setsum:
  4.1259 +  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"
  4.1260 +  unfolding integrable_on_def apply(drule bchoice) using has_integral_setsum[of t] by auto
  4.1261 +
  4.1262 +lemma has_integral_eq:
  4.1263 +  assumes "\<forall>x\<in>s. f x = g x" "(f has_integral k) s" shows "(g has_integral k) s"
  4.1264 +  using has_integral_sub[OF assms(2), of "\<lambda>x. f x - g x" 0]
  4.1265 +  using has_integral_is_0[of s "\<lambda>x. f x - g x"] using assms(1) by auto
  4.1266 +
  4.1267 +lemma integrable_eq:
  4.1268 +  shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> f integrable_on s \<Longrightarrow> g integrable_on s"
  4.1269 +  unfolding integrable_on_def using has_integral_eq[of s f g] by auto
  4.1270 +
  4.1271 +lemma has_integral_eq_eq:
  4.1272 +  shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> ((f has_integral k) s \<longleftrightarrow> (g has_integral k) s)"
  4.1273 +  using has_integral_eq[of s f g] has_integral_eq[of s g f] by auto 
  4.1274 +
  4.1275 +lemma has_integral_null[dest]:
  4.1276 +  assumes "content({a..b}) = 0" shows  "(f has_integral 0) ({a..b})"
  4.1277 +  unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI,rule) defer
  4.1278 +proof(rule,rule,erule conjE) fix e::real assume e:"e>0" thus "gauge (\<lambda>x. ball x 1)" by auto
  4.1279 +  fix p assume p:"p tagged_division_of {a..b}" (*"(\<lambda>x. ball x 1) fine p"*)
  4.1280 +  have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) = 0" unfolding norm_eq_zero diff_0_right
  4.1281 +    using setsum_content_null[OF assms(1) p, of f] . 
  4.1282 +  thus "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e" using e by auto qed
  4.1283 +
  4.1284 +lemma has_integral_null_eq[simp]:
  4.1285 +  shows "content({a..b}) = 0 \<Longrightarrow> ((f has_integral i) ({a..b}) \<longleftrightarrow> i = 0)"
  4.1286 +  apply rule apply(rule has_integral_unique,assumption) 
  4.1287 +  apply(drule has_integral_null,assumption)
  4.1288 +  apply(drule has_integral_null) by auto
  4.1289 +
  4.1290 +lemma integral_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> integral({a..b}) f = 0"
  4.1291 +  by(rule integral_unique,drule has_integral_null)
  4.1292 +
  4.1293 +lemma integrable_on_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> f integrable_on {a..b}"
  4.1294 +  unfolding integrable_on_def apply(drule has_integral_null) by auto
  4.1295 +
  4.1296 +lemma has_integral_empty[intro]: shows "(f has_integral 0) {}"
  4.1297 +  unfolding empty_as_interval apply(rule has_integral_null) 
  4.1298 +  using content_empty unfolding empty_as_interval .
  4.1299 +
  4.1300 +lemma has_integral_empty_eq[simp]: shows "(f has_integral i) {} \<longleftrightarrow> i = 0"
  4.1301 +  apply(rule,rule has_integral_unique,assumption) by auto
  4.1302 +
  4.1303 +lemma integrable_on_empty[intro]: shows "f integrable_on {}" unfolding integrable_on_def by auto
  4.1304 +
  4.1305 +lemma integral_empty[simp]: shows "integral {} f = 0"
  4.1306 +  apply(rule integral_unique) using has_integral_empty .
  4.1307 +
  4.1308 +lemma has_integral_refl[intro]: shows "(f has_integral 0) {a..a}"
  4.1309 +  apply(rule has_integral_null) unfolding content_eq_0_interior
  4.1310 +  unfolding interior_closed_interval using interval_sing by auto
  4.1311 +
  4.1312 +lemma integrable_on_refl[intro]: shows "f integrable_on {a..a}" unfolding integrable_on_def by auto
  4.1313 +
  4.1314 +lemma integral_refl: shows "integral {a..a} f = 0" apply(rule integral_unique) by auto
  4.1315 +
  4.1316 +subsection {* Cauchy-type criterion for integrability. *}
  4.1317 +
  4.1318 +lemma integrable_cauchy: fixes f::"real^'n \<Rightarrow> 'a::{real_normed_vector,complete_space}" 
  4.1319 +  shows "f integrable_on {a..b} \<longleftrightarrow>
  4.1320 +  (\<forall>e>0.\<exists>d. gauge d \<and> (\<forall>p1 p2. p1 tagged_division_of {a..b} \<and> d fine p1 \<and>
  4.1321 +                            p2 tagged_division_of {a..b} \<and> d fine p2
  4.1322 +                            \<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 -
  4.1323 +                                     setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) < e))" (is "?l = (\<forall>e>0. \<exists>d. ?P e d)")
  4.1324 +proof assume ?l
  4.1325 +  then guess y unfolding integrable_on_def has_integral .. note y=this
  4.1326 +  show "\<forall>e>0. \<exists>d. ?P e d" proof(rule,rule) case goal1 hence "e/2 > 0" by auto
  4.1327 +    then guess d apply- apply(drule y[rule_format]) by(erule exE,erule conjE) note d=this[rule_format]
  4.1328 +    show ?case apply(rule_tac x=d in exI,rule,rule d) apply(rule,rule,rule,(erule conjE)+)
  4.1329 +    proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b}" "d fine p1" "p2 tagged_division_of {a..b}" "d fine p2"
  4.1330 +      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"
  4.1331 +        apply(rule dist_triangle_half_l[where y=y,unfolded vector_dist_norm])
  4.1332 +        using d(2)[OF conjI[OF as(1-2)]] d(2)[OF conjI[OF as(3-4)]] .
  4.1333 +    qed qed
  4.1334 +next assume "\<forall>e>0. \<exists>d. ?P e d" hence "\<forall>n::nat. \<exists>d. ?P (inverse(real (n + 1))) d" by auto
  4.1335 +  from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
  4.1336 +  have "\<And>n. gauge (\<lambda>x. \<Inter>{d i x |i. i \<in> {0..n}})" apply(rule gauge_inters) using d(1) by auto
  4.1337 +  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-
  4.1338 +  proof case goal1 from this[of n] show ?case apply(drule_tac fine_division_exists) by auto qed
  4.1339 +  from choice[OF this] guess p .. note p = conjunctD2[OF this[rule_format]]
  4.1340 +  have dp:"\<And>i n. i\<le>n \<Longrightarrow> d i fine p n" using p(2) unfolding fine_inters by auto
  4.1341 +  have "Cauchy (\<lambda>n. setsum (\<lambda>(x,k). content k *\<^sub>R (f x)) (p n))"
  4.1342 +  proof(rule CauchyI) case goal1 then guess N unfolding real_arch_inv[of e] .. note N=this
  4.1343 +    show ?case apply(rule_tac x=N in exI)
  4.1344 +    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
  4.1345 +      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"
  4.1346 +        apply(rule less_trans[OF _ N[THEN conjunct2,THEN conjunct2]]) apply(subst *) apply(rule d(2))
  4.1347 +        using dp p(1) using mn by auto 
  4.1348 +    qed qed
  4.1349 +  then guess y unfolding convergent_eq_cauchy[THEN sym] .. note y=this[unfolded Lim_sequentially,rule_format]
  4.1350 +  show ?l unfolding integrable_on_def has_integral apply(rule_tac x=y in exI)
  4.1351 +  proof(rule,rule) fix e::real assume "e>0" hence *:"e/2 > 0" by auto
  4.1352 +    then guess N1 unfolding real_arch_inv[of "e/2"] .. note N1=this hence N1':"N1 = N1 - 1 + 1" by auto
  4.1353 +    guess N2 using y[OF *] .. note N2=this
  4.1354 +    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)"
  4.1355 +      apply(rule_tac x="d (N1 + N2)" in exI) apply rule defer 
  4.1356 +    proof(rule,rule,erule conjE) show "gauge (d (N1 + N2))" using d by auto
  4.1357 +      fix q assume as:"q tagged_division_of {a..b}" "d (N1 + N2) fine q"
  4.1358 +      have *:"inverse (real (N1 + N2 + 1)) < e / 2" apply(rule less_trans) using N1 by auto
  4.1359 +      show "norm ((\<Sum>(x, k)\<in>q. content k *\<^sub>R f x) - y) < e" apply(rule norm_triangle_half_r)
  4.1360 +        apply(rule less_trans[OF _ *]) apply(subst N1', rule d(2)[of "p (N1+N2)"]) defer
  4.1361 +        using N2[rule_format,unfolded vector_dist_norm,of "N1+N2"]
  4.1362 +        using as dp[of "N1 - 1 + 1 + N2" "N1 + N2"] using p(1)[of "N1 + N2"] using N1 by auto qed qed qed
  4.1363 +
  4.1364 +subsection {* Additivity of integral on abutting intervals. *}
  4.1365 +
  4.1366 +lemma interval_split:
  4.1367 +  "{a..b::real^'n} \<inter> {x. x$k \<le> c} = {a .. (\<chi> i. if i = k then min (b$k) c else b$i)}"
  4.1368 +  "{a..b} \<inter> {x. x$k \<ge> c} = {(\<chi> i. if i = k then max (a$k) c else a$i) .. b}"
  4.1369 +  apply(rule_tac[!] set_ext) unfolding Int_iff mem_interval mem_Collect_eq
  4.1370 +  unfolding Cart_lambda_beta by auto
  4.1371 +
  4.1372 +lemma content_split:
  4.1373 +  "content {a..b::real^'n} = content({a..b} \<inter> {x. x$k \<le> c}) + content({a..b} \<inter> {x. x$k >= c})"
  4.1374 +proof- note simps = interval_split content_closed_interval_cases Cart_lambda_beta vector_le_def
  4.1375 +  { presume "a\<le>b \<Longrightarrow> ?thesis" thus ?thesis apply(cases "a\<le>b") unfolding simps by auto }
  4.1376 +  have *:"UNIV = insert k (UNIV - {k})" "\<And>x. finite (UNIV-{x::'n})" "\<And>x. x\<notin>UNIV-{x}" by auto
  4.1377 +  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))"
  4.1378 +    "(\<Prod>i\<in>UNIV. b$i - a$i) = (\<Prod>i\<in>UNIV-{k}. b$i - a$i) * (b$k - a$k)" 
  4.1379 +    apply(subst *(1)) defer apply(subst *(1)) unfolding setprod_insert[OF *(2-)] by auto
  4.1380 +  assume as:"a\<le>b" moreover have "\<And>x. min (b $ k) c = max (a $ k) c
  4.1381 +    \<Longrightarrow> x* (b$k - a$k) = x*(max (a $ k) c - a $ k) + x*(b $ k - max (a $ k) c)"
  4.1382 +    by  (auto simp add:field_simps)
  4.1383 +  moreover have "\<not> a $ k \<le> c \<Longrightarrow> \<not> c \<le> b $ k \<Longrightarrow> False"
  4.1384 +    unfolding not_le using as[unfolded vector_le_def,rule_format,of k] by auto
  4.1385 +  ultimately show ?thesis 
  4.1386 +    unfolding simps unfolding *(1)[of "\<lambda>i x. b$i - x"] *(1)[of "\<lambda>i x. x - a$i"] *(2) by(auto)
  4.1387 +qed
  4.1388 +
  4.1389 +lemma division_split_left_inj:
  4.1390 +  assumes "d division_of i" "k1 \<in> d" "k2 \<in> d"  "k1 \<noteq> k2"
  4.1391 +  "k1 \<inter> {x::real^'n. x$k \<le> c} = k2 \<inter> {x. x$k \<le> c}"
  4.1392 +  shows "content(k1 \<inter> {x. x$k \<le> c}) = 0"
  4.1393 +proof- note d=division_ofD[OF assms(1)]
  4.1394 +  have *:"\<And>a b::real^'n. \<And> c k. (content({a..b} \<inter> {x. x$k \<le> c}) = 0 \<longleftrightarrow> interior({a..b} \<inter> {x. x$k \<le> c}) = {})"
  4.1395 +    unfolding interval_split content_eq_0_interior by auto
  4.1396 +  guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this
  4.1397 +  guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this
  4.1398 +  have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto
  4.1399 +  show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]])
  4.1400 +    defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed
  4.1401 +
  4.1402 +lemma division_split_right_inj:
  4.1403 +  assumes "d division_of i" "k1 \<in> d" "k2 \<in> d"  "k1 \<noteq> k2"
  4.1404 +  "k1 \<inter> {x::real^'n. x$k \<ge> c} = k2 \<inter> {x. x$k \<ge> c}"
  4.1405 +  shows "content(k1 \<inter> {x. x$k \<ge> c}) = 0"
  4.1406 +proof- note d=division_ofD[OF assms(1)]
  4.1407 +  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}) = {})"
  4.1408 +    unfolding interval_split content_eq_0_interior by auto
  4.1409 +  guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this
  4.1410 +  guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this
  4.1411 +  have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto
  4.1412 +  show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]])
  4.1413 +    defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed
  4.1414 +
  4.1415 +lemma tagged_division_split_left_inj:
  4.1416 +  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}" 
  4.1417 +  shows "content(k1 \<inter> {x. x$k \<le> c}) = 0"
  4.1418 +proof- have *:"\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c" unfolding image_iff apply(rule_tac x="(a,b)" in bexI) by auto
  4.1419 +  show ?thesis apply(rule division_split_left_inj[OF division_of_tagged_division[OF assms(1)]])
  4.1420 +    apply(rule_tac[1-2] *) using assms(2-) by auto qed
  4.1421 +
  4.1422 +lemma tagged_division_split_right_inj:
  4.1423 +  assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2"  "k1 \<inter> {x. x$k \<ge> c} = k2 \<inter> {x. x$k \<ge> c}" 
  4.1424 +  shows "content(k1 \<inter> {x. x$k \<ge> c}) = 0"
  4.1425 +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
  4.1426 +  show ?thesis apply(rule division_split_right_inj[OF division_of_tagged_division[OF assms(1)]])
  4.1427 +    apply(rule_tac[1-2] *) using assms(2-) by auto qed
  4.1428 +
  4.1429 +lemma division_split:
  4.1430 +  assumes "p division_of {a..b::real^'n}"
  4.1431 +  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$k \<le> c})" (is "?p1 division_of ?I1") and 
  4.1432 +        "{l \<inter> {x. x$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")
  4.1433 +proof(rule_tac[!] division_ofI) note p=division_ofD[OF assms]
  4.1434 +  show "finite ?p1" "finite ?p2" using p(1) by auto show "\<Union>?p1 = ?I1" "\<Union>?p2 = ?I2" unfolding p(6)[THEN sym] by auto
  4.1435 +  { fix k assume "k\<in>?p1" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this
  4.1436 +    guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this
  4.1437 +    show "k\<subseteq>?I1" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l
  4.1438 +      using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split) by auto
  4.1439 +    fix k' assume "k'\<in>?p1" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this
  4.1440 +    assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto }
  4.1441 +  { fix k assume "k\<in>?p2" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this
  4.1442 +    guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this
  4.1443 +    show "k\<subseteq>?I2" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l
  4.1444 +      using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split) by auto
  4.1445 +    fix k' assume "k'\<in>?p2" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this
  4.1446 +    assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto }
  4.1447 +qed
  4.1448 +
  4.1449 +lemma has_integral_split: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
  4.1450 +  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})"
  4.1451 +  shows "(f has_integral (i + j)) ({a..b})"
  4.1452 +proof(unfold has_integral,rule,rule) case goal1 hence e:"e/2>0" by auto
  4.1453 +  guess d1 using has_integralD[OF assms(1)[unfolded interval_split] e] . note d1=this[unfolded interval_split[THEN sym]]
  4.1454 +  guess d2 using has_integralD[OF assms(2)[unfolded interval_split] e] . note d2=this[unfolded interval_split[THEN sym]]
  4.1455 +  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"
  4.1456 +  show ?case apply(rule_tac x="?d" in exI,rule) defer apply(rule,rule,(erule conjE)+)
  4.1457 +  proof- show "gauge ?d" using d1(1) d2(1) unfolding gauge_def by auto
  4.1458 +    fix p assume "p tagged_division_of {a..b}" "?d fine p" note p = this tagged_division_ofD[OF this(1)]
  4.1459 +    have lem0:"\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x$k \<le> c} = {}) \<Longrightarrow> x$k \<le> c"
  4.1460 +         "\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x$k \<ge> c} = {}) \<Longrightarrow> x$k \<ge> c"
  4.1461 +    proof- fix x kk assume as:"(x,kk)\<in>p"
  4.1462 +      show "~(kk \<inter> {x. x$k \<le> c} = {}) \<Longrightarrow> x$k \<le> c"
  4.1463 +      proof(rule ccontr) case goal1
  4.1464 +        from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x $ k - c\<bar>"
  4.1465 +          using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
  4.1466 +        hence "\<exists>y. y \<in> ball x \<bar>x $ k - c\<bar> \<inter> {x. x $ k \<le> c}" using goal1(1) by blast 
  4.1467 +        then guess y .. hence "\<bar>x $ k - y $ k\<bar> < \<bar>x $ k - c\<bar>" "y$k \<le> c" apply-apply(rule le_less_trans)
  4.1468 +          using component_le_norm[of "x - y" k,unfolded vector_minus_component] by(auto simp add:vector_dist_norm)
  4.1469 +        thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
  4.1470 +      qed
  4.1471 +      show "~(kk \<inter> {x. x$k \<ge> c} = {}) \<Longrightarrow> x$k \<ge> c"
  4.1472 +      proof(rule ccontr) case goal1
  4.1473 +        from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x $ k - c\<bar>"
  4.1474 +          using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
  4.1475 +        hence "\<exists>y. y \<in> ball x \<bar>x $ k - c\<bar> \<inter> {x. x $ k \<ge> c}" using goal1(1) by blast 
  4.1476 +        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)
  4.1477 +          using component_le_norm[of "x - y" k,unfolded vector_minus_component] by(auto simp add:vector_dist_norm)
  4.1478 +        thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
  4.1479 +      qed
  4.1480 +    qed
  4.1481 +
  4.1482 +    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
  4.1483 +    have lem2: "\<And>f s P f. finite s \<Longrightarrow> finite {(x,f k) | x k. (x,k) \<in> s \<and> P x k}"
  4.1484 +    proof- case goal1 thus ?case apply-apply(rule finite_subset[of _ "(\<lambda>(x,k). (x,f k)) ` s"]) by auto qed
  4.1485 +    have lem3: "\<And>g::(real ^ 'n \<Rightarrow> bool) \<Rightarrow> real ^ 'n \<Rightarrow> bool. finite p \<Longrightarrow>
  4.1486 +      setsum (\<lambda>(x,k). content k *\<^sub>R f x) {(x,g k) |x k. (x,k) \<in> p \<and> ~(g k = {})}
  4.1487 +               = setsum (\<lambda>(x,k). content k *\<^sub>R f x) ((\<lambda>(x,k). (x,g k)) ` p)"
  4.1488 +      apply(rule setsum_mono_zero_left) prefer 3
  4.1489 +    proof fix g::"(real ^ 'n \<Rightarrow> bool) \<Rightarrow> real ^ 'n \<Rightarrow> bool" and i::"(real^'n) \<times> ((real^'n) set)"
  4.1490 +      assume "i \<in> (\<lambda>(x, k). (x, g k)) ` p - {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}"
  4.1491 +      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
  4.1492 +      have "content (g k) = 0" using xk using content_empty by auto
  4.1493 +      thus "(\<lambda>(x, k). content k *\<^sub>R f x) i = 0" unfolding xk split_conv by auto
  4.1494 +    qed auto
  4.1495 +    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
  4.1496 +
  4.1497 +    let ?M1 = "{(x,kk \<inter> {x. x$k \<le> c}) |x kk. (x,kk) \<in> p \<and> kk \<inter> {x. x$k \<le> c} \<noteq> {}}"
  4.1498 +    have "norm ((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) < e/2" apply(rule d1(2),rule tagged_division_ofI)
  4.1499 +      apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)
  4.1500 +    proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M1} = {a..b} \<inter> {x. x$k \<le> c}" unfolding p(8)[THEN sym] by auto
  4.1501 +      fix x l assume xl:"(x,l)\<in>?M1"
  4.1502 +      then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note xl'=this
  4.1503 +      have "l' \<subseteq> d1 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
  4.1504 +      thus "l \<subseteq> d1 x" unfolding xl' by auto
  4.1505 +      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)
  4.1506 +        using lem0(1)[OF xl'(3-4)] by auto
  4.1507 +      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])
  4.1508 +      fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M1"
  4.1509 +      then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note yr'=this
  4.1510 +      assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}"
  4.1511 +      proof(cases "l' = r' \<longrightarrow> x' = y'")
  4.1512 +        case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
  4.1513 +      next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto
  4.1514 +        thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
  4.1515 +      qed qed moreover
  4.1516 +
  4.1517 +    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> {}}" 
  4.1518 +    have "norm ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j) < e/2" apply(rule d2(2),rule tagged_division_ofI)
  4.1519 +      apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)
  4.1520 +    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
  4.1521 +      fix x l assume xl:"(x,l)\<in>?M2"
  4.1522 +      then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note xl'=this
  4.1523 +      have "l' \<subseteq> d2 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
  4.1524 +      thus "l \<subseteq> d2 x" unfolding xl' by auto
  4.1525 +      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)
  4.1526 +        using lem0(2)[OF xl'(3-4)] by auto
  4.1527 +      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])
  4.1528 +      fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M2"
  4.1529 +      then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note yr'=this
  4.1530 +      assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}"
  4.1531 +      proof(cases "l' = r' \<longrightarrow> x' = y'")
  4.1532 +        case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
  4.1533 +      next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto
  4.1534 +        thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
  4.1535 +      qed qed ultimately
  4.1536 +
  4.1537 +    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"
  4.1538 +      apply- apply(rule norm_triangle_lt) by auto
  4.1539 +    also { have *:"\<And>x y. x = (0::real) \<Longrightarrow> x *\<^sub>R (y::'a) = 0" using scaleR_zero_left by auto
  4.1540 +      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)
  4.1541 +       = (\<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
  4.1542 +      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)"
  4.1543 +        unfolding lem3[OF p(3)] apply(subst setsum_reindex_nonzero[OF p(3)]) defer apply(subst setsum_reindex_nonzero[OF p(3)])
  4.1544 +        defer unfolding lem4[THEN sym] apply(rule refl) unfolding split_paired_all split_conv apply(rule_tac[!] *)
  4.1545 +      proof- case goal1 thus ?case apply- apply(rule tagged_division_split_left_inj [OF p(1), of a b aa ba]) by auto
  4.1546 +      next case   goal2 thus ?case apply- apply(rule tagged_division_split_right_inj[OF p(1), of a b aa ba]) by auto
  4.1547 +      qed also note setsum_addf[THEN sym]
  4.1548 +      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
  4.1549 +        = (\<lambda>(x,ka). content ka *\<^sub>R f x) x" unfolding split_paired_all split_conv
  4.1550 +      proof- fix a b assume "(a,b) \<in> p" from p(6)[OF this] guess u v apply-by(erule exE)+ note uv=this
  4.1551 +        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"
  4.1552 +          unfolding scaleR_left_distrib[THEN sym] unfolding uv content_split[of u v k c] by auto
  4.1553 +      qed note setsum_cong2[OF this]
  4.1554 +      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 +
  4.1555 +        ((\<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) =
  4.1556 +        (\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f x) - (i + j)" by auto }
  4.1557 +    finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (i + j)) < e" by auto qed qed
  4.1558 +
  4.1559 +subsection {* A sort of converse, integrability on subintervals. *}
  4.1560 +
  4.1561 +lemma tagged_division_union_interval:
  4.1562 +  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})"
  4.1563 +  shows "(p1 \<union> p2) tagged_division_of ({a..b})"
  4.1564 +proof- have *:"{a..b} = ({a..b} \<inter> {x. x$k \<le> c}) \<union> ({a..b} \<inter> {x. x$k \<ge> c})" by auto
  4.1565 +  show ?thesis apply(subst *) apply(rule tagged_division_union[OF assms])
  4.1566 +    unfolding interval_split interior_closed_interval
  4.1567 +    by(auto simp add: vector_less_def Cart_lambda_beta elim!:allE[where x=k]) qed
  4.1568 +
  4.1569 +lemma has_integral_separate_sides: fixes f::"real^'m \<Rightarrow> 'a::real_normed_vector"
  4.1570 +  assumes "(f has_integral i) ({a..b})" "e>0"
  4.1571 +  obtains d where "gauge d" "(\<forall>p1 p2. p1 tagged_division_of ({a..b} \<inter> {x. x$k \<le> c}) \<and> d fine p1 \<and>
  4.1572 +                                p2 tagged_division_of ({a..b} \<inter> {x. x$k \<ge> c}) \<and> d fine p2
  4.1573 +                                \<longrightarrow> norm((setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 +
  4.1574 +                                          setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) - i) < e)"
  4.1575 +proof- guess d using has_integralD[OF assms] . note d=this
  4.1576 +  show ?thesis apply(rule that[of d]) apply(rule d) apply(rule,rule,rule,(erule conjE)+)
  4.1577 +  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
  4.1578 +                   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
  4.1579 +    note tagged_division_union_interval[OF p1(7) p2(7)] note p12 = tagged_division_ofD[OF this] this
  4.1580 +    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)"
  4.1581 +      apply(subst setsum_Un_zero) apply(rule p1 p2)+ apply(rule) unfolding split_paired_all split_conv
  4.1582 +    proof- fix a b assume ab:"(a,b) \<in> p1 \<inter> p2"
  4.1583 +      have "(a,b) \<in> p1" using ab by auto from p1(4)[OF this] guess u v apply-by(erule exE)+ note uv =this
  4.1584 +      have "b \<subseteq> {x. x$k = c}" using ab p1(3)[of a b] p2(3)[of a b] by fastsimp
  4.1585 +      moreover have "interior {x. x $ k = c} = {}" 
  4.1586 +      proof(rule ccontr) case goal1 then obtain x where x:"x\<in>interior {x. x$k = c}" by auto
  4.1587 +        then guess e unfolding mem_interior .. note e=this
  4.1588 +        have x:"x$k = c" using x interior_subset by fastsimp
  4.1589 +        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
  4.1590 +        have "x + (\<chi> i. if i = k then e/2 else 0) \<in> ball x e" unfolding mem_ball vector_dist_norm 
  4.1591 +          apply(rule le_less_trans[OF norm_le_l1]) unfolding * 
  4.1592 +          unfolding setsum_delta[OF finite_UNIV] using e by auto 
  4.1593 +        hence "x + (\<chi> i. if i = k then e/2 else 0) \<in> {x. x$k = c}" using e by auto
  4.1594 +        thus False unfolding mem_Collect_eq using e x by auto
  4.1595 +      qed ultimately have "content b = 0" unfolding uv content_eq_0_interior apply-apply(drule subset_interior) by auto
  4.1596 +      thus "content b *\<^sub>R f a = 0" by auto
  4.1597 +    qed auto
  4.1598 +    also have "\<dots> < e" by(rule d(2) p12 fine_union p1 p2)+
  4.1599 +    finally show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) < e" . qed qed
  4.1600 +
  4.1601 +lemma integrable_split[intro]: fixes f::"real^'n \<Rightarrow> 'a::{real_normed_vector,complete_space}" assumes "f integrable_on {a..b}"
  4.1602 +  shows "f integrable_on ({a..b} \<inter> {x. x$k \<le> c})" (is ?t1) and "f integrable_on ({a..b} \<inter> {x. x$k \<ge> c})" (is ?t2) 
  4.1603 +proof- guess y using assms unfolding integrable_on_def .. note y=this
  4.1604 +  def b' \<equiv> "(\<chi> i. if i = k then min (b$k) c else b$i)::real^'n"
  4.1605 +  and a' \<equiv> "(\<chi> i. if i = k then max (a$k) c else a$i)::real^'n"
  4.1606 +  show ?t1 ?t2 unfolding interval_split integrable_cauchy unfolding interval_split[THEN sym]
  4.1607 +  proof(rule_tac[!] allI impI)+ fix e::real assume "e>0" hence "e/2>0" by auto
  4.1608 +    from has_integral_separate_sides[OF y this,of k c] guess d . note d=this[rule_format]
  4.1609 +    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>
  4.1610 +                              norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e)"
  4.1611 +    show "?P {x. x $ k \<le> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
  4.1612 +    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"
  4.1613 +      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"
  4.1614 +      proof- guess p using fine_division_exists[OF d(1), of a' b] . note p=this
  4.1615 +        show ?thesis using norm_triangle_half_l[OF d(2)[of p1 p] d(2)[of p2 p]]
  4.1616 +          using as unfolding interval_split b'_def[symmetric] a'_def[symmetric]
  4.1617 +          using p using assms by(auto simp add:group_simps)
  4.1618 +      qed qed  
  4.1619 +    show "?P {x. x $ k \<ge> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
  4.1620 +    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"
  4.1621 +      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"
  4.1622 +      proof- guess p using fine_division_exists[OF d(1), of a b'] . note p=this
  4.1623 +        show ?thesis using norm_triangle_half_l[OF d(2)[of p p1] d(2)[of p p2]]
  4.1624 +          using as unfolding interval_split b'_def[symmetric] a'_def[symmetric]
  4.1625 +          using p using assms by(auto simp add:group_simps) qed qed qed qed
  4.1626 +
  4.1627 +subsection {* Generalized notion of additivity. *}
  4.1628 +
  4.1629 +definition "neutral opp = (SOME x. \<forall>y. opp x y = y \<and> opp y x = y)"
  4.1630 +
  4.1631 +definition operative :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ((real^'n) set \<Rightarrow> 'a) \<Rightarrow> bool" where
  4.1632 +  "operative opp f \<equiv> 
  4.1633 +    (\<forall>a b. content {a..b} = 0 \<longrightarrow> f {a..b} = neutral(opp)) \<and>
  4.1634 +    (\<forall>a b c k. f({a..b}) =
  4.1635 +                   opp (f({a..b} \<inter> {x. x$k \<le> c}))
  4.1636 +                       (f({a..b} \<inter> {x. x$k \<ge> c})))"
  4.1637 +
  4.1638 +lemma operativeD[dest]: assumes "operative opp f"
  4.1639 +  shows "\<And>a b. content {a..b} = 0 \<Longrightarrow> f {a..b} = neutral(opp)"
  4.1640 +  "\<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}))"
  4.1641 +  using assms unfolding operative_def by auto
  4.1642 +
  4.1643 +lemma operative_trivial:
  4.1644 + "operative opp f \<Longrightarrow> content({a..b}) = 0 \<Longrightarrow> f({a..b}) = neutral opp"
  4.1645 +  unfolding operative_def by auto
  4.1646 +
  4.1647 +lemma property_empty_interval:
  4.1648 + "(\<forall>a b. content({a..b}) = 0 \<longrightarrow> P({a..b})) \<Longrightarrow> P {}" 
  4.1649 +  using content_empty unfolding empty_as_interval by auto
  4.1650 +
  4.1651 +lemma operative_empty: "operative opp f \<Longrightarrow> f {} = neutral opp"
  4.1652 +  unfolding operative_def apply(rule property_empty_interval) by auto
  4.1653 +
  4.1654 +subsection {* Using additivity of lifted function to encode definedness. *}
  4.1655 +
  4.1656 +lemma forall_option: "(\<forall>x. P x) \<longleftrightarrow> P None \<and> (\<forall>x. P(Some x))"
  4.1657 +  by (metis map_of.simps option.nchotomy)
  4.1658 +
  4.1659 +lemma exists_option:
  4.1660 + "(\<exists>x. P x) \<longleftrightarrow> P None \<or> (\<exists>x. P(Some x))" 
  4.1661 +  by (metis map_of.simps option.nchotomy)
  4.1662 +
  4.1663 +fun lifted where 
  4.1664 +  "lifted (opp::'a\<Rightarrow>'a\<Rightarrow>'b) (Some x) (Some y) = Some(opp x y)" |
  4.1665 +  "lifted opp None _ = (None::'b option)" |
  4.1666 +  "lifted opp _ None = None"
  4.1667 +
  4.1668 +lemma lifted_simp_1[simp]: "lifted opp v None = None"
  4.1669 +  apply(induct v) by auto
  4.1670 +
  4.1671 +definition "monoidal opp \<equiv>  (\<forall>x y. opp x y = opp y x) \<and>
  4.1672 +                   (\<forall>x y z. opp x (opp y z) = opp (opp x y) z) \<and>
  4.1673 +                   (\<forall>x. opp (neutral opp) x = x)"
  4.1674 +
  4.1675 +lemma monoidalI: assumes "\<And>x y. opp x y = opp y x"
  4.1676 +  "\<And>x y z. opp x (opp y z) = opp (opp x y) z"
  4.1677 +  "\<And>x. opp (neutral opp) x = x" shows "monoidal opp"
  4.1678 +  unfolding monoidal_def using assms by fastsimp
  4.1679 +
  4.1680 +lemma monoidal_ac: assumes "monoidal opp"
  4.1681 +  shows "opp (neutral opp) a = a" "opp a (neutral opp) = a" "opp a b = opp b a"
  4.1682 +  "opp (opp a b) c = opp a (opp b c)"  "opp a (opp b c) = opp b (opp a c)"
  4.1683 +  using assms unfolding monoidal_def apply- by metis+
  4.1684 +
  4.1685 +lemma monoidal_simps[simp]: assumes "monoidal opp"
  4.1686 +  shows "opp (neutral opp) a = a" "opp a (neutral opp) = a"
  4.1687 +  using monoidal_ac[OF assms] by auto
  4.1688 +
  4.1689 +lemma neutral_lifted[cong]: assumes "monoidal opp"
  4.1690 +  shows "neutral (lifted opp) = Some(neutral opp)"
  4.1691 +  apply(subst neutral_def) apply(rule some_equality) apply(rule,induct_tac y) prefer 3
  4.1692 +proof- fix x assume "\<forall>y. lifted opp x y = y \<and> lifted opp y x = y"
  4.1693 +  thus "x = Some (neutral opp)" apply(induct x) defer
  4.1694 +    apply rule apply(subst neutral_def) apply(subst eq_commute,rule some_equality)
  4.1695 +    apply(rule,erule_tac x="Some y" in allE) defer apply(erule_tac x="Some x" in allE) by auto
  4.1696 +qed(auto simp add:monoidal_ac[OF assms])
  4.1697 +
  4.1698 +lemma monoidal_lifted[intro]: assumes "monoidal opp" shows "monoidal(lifted opp)"
  4.1699 +  unfolding monoidal_def forall_option neutral_lifted[OF assms] using monoidal_ac[OF assms] by auto
  4.1700 +
  4.1701 +definition "support opp f s = {x. x\<in>s \<and> f x \<noteq> neutral opp}"
  4.1702 +definition "fold' opp e s \<equiv> (if finite s then fold opp e s else e)"
  4.1703 +definition "iterate opp s f \<equiv> fold' (\<lambda>x a. opp (f x) a) (neutral opp) (support opp f s)"
  4.1704 +
  4.1705 +lemma support_subset[intro]:"support opp f s \<subseteq> s" unfolding support_def by auto
  4.1706 +lemma support_empty[simp]:"support opp f {} = {}" using support_subset[of opp f "{}"] by auto
  4.1707 +
  4.1708 +lemma fun_left_comm_monoidal[intro]: assumes "monoidal opp" shows "fun_left_comm opp"
  4.1709 +  unfolding fun_left_comm_def using monoidal_ac[OF assms] by auto
  4.1710 +
  4.1711 +lemma support_clauses:
  4.1712 +  "\<And>f g s. support opp f {} = {}"
  4.1713 +  "\<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))"
  4.1714 +  "\<And>f g s. support opp f (s - {x}) = (support opp f s) - {x}"
  4.1715 +  "\<And>f g s. support opp f (s \<union> t) = (support opp f s) \<union> (support opp f t)"
  4.1716 +  "\<And>f g s. support opp f (s \<inter> t) = (support opp f s) \<inter> (support opp f t)"
  4.1717 +  "\<And>f g s. support opp f (s - t) = (support opp f s) - (support opp f t)"
  4.1718 +  "\<And>f g s. support opp g (f ` s) = f ` (support opp (g o f) s)"
  4.1719 +unfolding support_def by auto
  4.1720 +
  4.1721 +lemma finite_support[intro]:"finite s \<Longrightarrow> finite (support opp f s)"
  4.1722 +  unfolding support_def by auto
  4.1723 +
  4.1724 +lemma iterate_empty[simp]:"iterate opp {} f = neutral opp"
  4.1725 +  unfolding iterate_def fold'_def by auto 
  4.1726 +
  4.1727 +lemma iterate_insert[simp]: assumes "monoidal opp" "finite s"
  4.1728 +  shows "iterate opp (insert x s) f = (if x \<in> s then iterate opp s f else opp (f x) (iterate opp s f))" 
  4.1729 +proof(cases "x\<in>s") case True hence *:"insert x s = s" by auto
  4.1730 +  show ?thesis unfolding iterate_def if_P[OF True] * by auto
  4.1731 +next case False note x=this
  4.1732 +  note * = fun_left_comm.fun_left_comm_apply[OF fun_left_comm_monoidal[OF assms(1)]]
  4.1733 +  show ?thesis proof(cases "f x = neutral opp")
  4.1734 +    case True show ?thesis unfolding iterate_def if_not_P[OF x] support_clauses if_P[OF True]
  4.1735 +      unfolding True monoidal_simps[OF assms(1)] by auto
  4.1736 +  next case False show ?thesis unfolding iterate_def fold'_def  if_not_P[OF x] support_clauses if_not_P[OF False]
  4.1737 +      apply(subst fun_left_comm.fold_insert[OF * finite_support])
  4.1738 +      using `finite s` unfolding support_def using False x by auto qed qed 
  4.1739 +
  4.1740 +lemma iterate_some:
  4.1741 +  assumes "monoidal opp"  "finite s"
  4.1742 +  shows "iterate (lifted opp) s (\<lambda>x. Some(f x)) = Some (iterate opp s f)" using assms(2)
  4.1743 +proof(induct s) case empty thus ?case using assms by auto
  4.1744 +next case (insert x F) show ?case apply(subst iterate_insert) prefer 3 apply(subst if_not_P)
  4.1745 +    defer unfolding insert(3) lifted.simps apply rule using assms insert by auto qed
  4.1746 +
  4.1747 +subsection {* Two key instances of additivity. *}
  4.1748 +
  4.1749 +lemma neutral_add[simp]:
  4.1750 +  "neutral op + = (0::_::comm_monoid_add)" unfolding neutral_def 
  4.1751 +  apply(rule some_equality) defer apply(erule_tac x=0 in allE) by auto
  4.1752 +
  4.1753 +lemma operative_content[intro]: "operative (op +) content"
  4.1754 +  unfolding operative_def content_split[THEN sym] neutral_add by auto
  4.1755 +
  4.1756 +lemma neutral_monoid[simp]: "neutral ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a) = 0"
  4.1757 +  unfolding neutral_def apply(rule some_equality) defer
  4.1758 +  apply(erule_tac x=0 in allE) by auto
  4.1759 +
  4.1760 +lemma monoidal_monoid[intro]:
  4.1761 +  shows "monoidal ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a)"
  4.1762 +  unfolding monoidal_def neutral_monoid by(auto simp add: group_simps) 
  4.1763 +
  4.1764 +lemma operative_integral: fixes f::"real^'n \<Rightarrow> 'a::banach"
  4.1765 +  shows "operative (lifted(op +)) (\<lambda>i. if f integrable_on i then Some(integral i f) else None)"
  4.1766 +  unfolding operative_def unfolding neutral_lifted[OF monoidal_monoid] neutral_add
  4.1767 +  apply(rule,rule,rule,rule) defer apply(rule allI)+
  4.1768 +proof- fix a b c k show "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) =
  4.1769 +              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)
  4.1770 +               (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)"
  4.1771 +  proof(cases "f integrable_on {a..b}") 
  4.1772 +    case True show ?thesis unfolding if_P[OF True]
  4.1773 +      unfolding if_P[OF integrable_split(1)[OF True]] if_P[OF integrable_split(2)[OF True]]
  4.1774 +      unfolding lifted.simps option.inject apply(rule integral_unique) apply(rule has_integral_split) 
  4.1775 +      apply(rule_tac[!] integrable_integral integrable_split)+ using True by assumption+
  4.1776 +  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}))"
  4.1777 +    proof(rule ccontr) case goal1 hence "f integrable_on {a..b}" apply- unfolding integrable_on_def
  4.1778 +        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)
  4.1779 +        apply(rule has_integral_split) apply(rule_tac[!] integrable_integral) by auto
  4.1780 +      thus False using False by auto
  4.1781 +    qed thus ?thesis using False by auto 
  4.1782 +  qed next 
  4.1783 +  fix a b assume as:"content {a..b::real^'n} = 0"
  4.1784 +  thus "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) = Some 0"
  4.1785 +    unfolding if_P[OF integrable_on_null[OF as]] using has_integral_null_eq[OF as] by auto qed
  4.1786 +
  4.1787 +subsection {* Points of division of a partition. *}
  4.1788 +
  4.1789 +definition "division_points (k::(real^'n) set) d = 
  4.1790 +    {(j,x). (interval_lowerbound k)$j < x \<and> x < (interval_upperbound k)$j \<and>
  4.1791 +           (\<exists>i\<in>d. (interval_lowerbound i)$j = x \<or> (interval_upperbound i)$j = x)}"
  4.1792 +
  4.1793 +lemma division_points_finite: assumes "d division_of i"
  4.1794 +  shows "finite (division_points i d)"
  4.1795 +proof- note assm = division_ofD[OF assms]
  4.1796 +  let ?M = "\<lambda>j. {(j,x)|x. (interval_lowerbound i)$j < x \<and> x < (interval_upperbound i)$j \<and>
  4.1797 +           (\<exists>i\<in>d. (interval_lowerbound i)$j = x \<or> (interval_upperbound i)$j = x)}"
  4.1798 +  have *:"division_points i d = \<Union>(?M ` UNIV)"
  4.1799 +    unfolding division_points_def by auto
  4.1800 +  show ?thesis unfolding * using assm by auto qed
  4.1801 +
  4.1802 +lemma division_points_subset:
  4.1803 +  assumes "d division_of {a..b}" "\<forall>i. a$i < b$i"  "a$k < c" "c < b$k"
  4.1804 +  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$k \<le> c} = {})}
  4.1805 +                  \<subseteq> division_points ({a..b}) d" (is ?t1) and
  4.1806 +        "division_points ({a..b} \<inter> {x. x$k \<ge> c}) {l \<inter> {x. x$k \<ge> c} | l . l \<in> d \<and> ~(l \<inter> {x. x$k \<ge> c} = {})}
  4.1807 +                  \<subseteq> division_points ({a..b}) d" (is ?t2)
  4.1808 +proof- note assm = division_ofD[OF assms(1)]
  4.1809 +  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"
  4.1810 +    "\<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"
  4.1811 +    using assms using less_imp_le by auto
  4.1812 +  show ?t1 unfolding division_points_def interval_split[of a b]
  4.1813 +    unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] Cart_lambda_beta unfolding *
  4.1814 +    unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+ unfolding mem_Collect_eq apply(erule exE conjE)+
  4.1815 +  proof- fix i l x assume as:"a $ fst x < snd x" "snd x < (if fst x = k then c else b $ fst x)"
  4.1816 +      "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> {}"
  4.1817 +    from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
  4.1818 +    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 .
  4.1819 +    have **:"\<forall>i. u$i \<le> v$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
  4.1820 +    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)"
  4.1821 +      using as(1-3,5) unfolding l interval_split interval_ne_empty as interval_bounds[OF *] Cart_lambda_beta apply-
  4.1822 +      apply(rule,assumption,rule) defer apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **]
  4.1823 +      apply(case_tac[!] "fst x = k") using assms by auto
  4.1824 +  qed
  4.1825 +  show ?t2 unfolding division_points_def interval_split[of a b]
  4.1826 +    unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] Cart_lambda_beta unfolding *
  4.1827 +    unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+ unfolding mem_Collect_eq apply(erule exE conjE)+
  4.1828 +  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"
  4.1829 +      "i = l \<inter> {x. c \<le> x $ k}" "l \<in> d" "l \<inter> {x. c \<le> x $ k} \<noteq> {}"
  4.1830 +    from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
  4.1831 +    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 .
  4.1832 +    have **:"\<forall>i. u$i \<le> v$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
  4.1833 +    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)"
  4.1834 +      using as(1-3,5) unfolding l interval_split interval_ne_empty as interval_bounds[OF *] Cart_lambda_beta apply-
  4.1835 +      apply rule defer apply(rule,assumption) apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **]
  4.1836 +      apply(case_tac[!] "fst x = k") using assms by auto qed qed
  4.1837 +
  4.1838 +lemma division_points_psubset:
  4.1839 +  assumes "d division_of {a..b}"  "\<forall>i. a$i < b$i"  "a$k < c" "c < b$k"
  4.1840 +  "l \<in> d" "interval_lowerbound l$k = c \<or> interval_upperbound l$k = c"
  4.1841 +  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$k \<le> c} \<noteq> {}} \<subset> division_points ({a..b}) d" (is "?D1 \<subset> ?D") 
  4.1842 +        "division_points ({a..b} \<inter> {x. x$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") 
  4.1843 +proof- have ab:"\<forall>i. a$i \<le> b$i" using assms(2) by(auto intro!:less_imp_le)
  4.1844 +  guess u v using division_ofD(4)[OF assms(1,5)] apply-by(erule exE)+ note l=this
  4.1845 +  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
  4.1846 +    unfolding subset_eq apply- defer apply(erule_tac x=u in ballE, erule_tac x=v in ballE) unfolding mem_interval by auto
  4.1847 +  have *:"interval_upperbound ({a..b} \<inter> {x. x $ k \<le> interval_upperbound l $ k}) $ k = interval_upperbound l $ k"
  4.1848 +         "interval_upperbound ({a..b} \<inter> {x. x $ k \<le> interval_lowerbound l $ k}) $ k = interval_lowerbound l $ k"
  4.1849 +    unfolding interval_split apply(subst interval_bounds) prefer 3 apply(subst interval_bounds)
  4.1850 +    unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab by auto
  4.1851 +  have "\<exists>x. x \<in> ?D - ?D1" using assms(2-) apply-apply(erule disjE)
  4.1852 +    apply(rule_tac x="(k,(interval_lowerbound l)$k)" in exI) defer
  4.1853 +    apply(rule_tac x="(k,(interval_upperbound l)$k)" in exI)
  4.1854 +    unfolding division_points_def unfolding interval_bounds[OF ab]
  4.1855 +    apply (auto simp add:interval_bounds) unfolding * by auto
  4.1856 +  thus "?D1 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) by auto
  4.1857 +
  4.1858 +  have *:"interval_lowerbound ({a..b} \<inter> {x. x $ k \<ge> interval_lowerbound l $ k}) $ k = interval_lowerbound l $ k"
  4.1859 +         "interval_lowerbound ({a..b} \<inter> {x. x $ k \<ge> interval_upperbound l $ k}) $ k = interval_upperbound l $ k"
  4.1860 +    unfolding interval_split apply(subst interval_bounds) prefer 3 apply(subst interval_bounds)
  4.1861 +    unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab by auto
  4.1862 +  have "\<exists>x. x \<in> ?D - ?D2" using assms(2-) apply-apply(erule disjE)
  4.1863 +    apply(rule_tac x="(k,(interval_lowerbound l)$k)" in exI) defer
  4.1864 +    apply(rule_tac x="(k,(interval_upperbound l)$k)" in exI)
  4.1865 +    unfolding division_points_def unfolding interval_bounds[OF ab]
  4.1866 +    apply (auto simp add:interval_bounds) unfolding * by auto
  4.1867 +  thus "?D2 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) by auto qed
  4.1868 +
  4.1869 +subsection {* Preservation by divisions and tagged divisions. *}
  4.1870 +
  4.1871 +lemma support_support[simp]:"support opp f (support opp f s) = support opp f s"
  4.1872 +  unfolding support_def by auto
  4.1873 +
  4.1874 +lemma iterate_support[simp]: "iterate opp (support opp f s) f = iterate opp s f"
  4.1875 +  unfolding iterate_def support_support by auto
  4.1876 +
  4.1877 +lemma iterate_expand_cases:
  4.1878 +  "iterate opp s f = (if finite(support opp f s) then iterate opp (support opp f s) f else neutral opp)"
  4.1879 +  apply(cases) apply(subst if_P,assumption) unfolding iterate_def support_support fold'_def by auto 
  4.1880 +
  4.1881 +lemma iterate_image: assumes "monoidal opp"  "inj_on f s"
  4.1882 +  shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)"
  4.1883 +proof- have *:"\<And>s. finite s \<Longrightarrow>  \<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<longrightarrow> x = y \<Longrightarrow>
  4.1884 +     iterate opp (f ` s) g = iterate opp s (g \<circ> f)"
  4.1885 +  proof- case goal1 show ?case using goal1
  4.1886 +    proof(induct s) case empty thus ?case using assms(1) by auto
  4.1887 +    next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)]
  4.1888 +        unfolding if_not_P[OF insert(2)] apply(subst insert(3)[THEN sym])
  4.1889 +        unfolding image_insert defer apply(subst iterate_insert[OF assms(1)])
  4.1890 +        apply(rule finite_imageI insert)+ apply(subst if_not_P)
  4.1891 +        unfolding image_iff o_def using insert(2,4) by auto
  4.1892 +    qed qed
  4.1893 +  show ?thesis 
  4.1894 +    apply(cases "finite (support opp g (f ` s))")
  4.1895 +    apply(subst (1) iterate_support[THEN sym],subst (2) iterate_support[THEN sym])
  4.1896 +    unfolding support_clauses apply(rule *)apply(rule finite_imageD,assumption) unfolding inj_on_def[symmetric]
  4.1897 +    apply(rule subset_inj_on[OF assms(2) support_subset])+
  4.1898 +    apply(subst iterate_expand_cases) unfolding support_clauses apply(simp only: if_False)
  4.1899 +    apply(subst iterate_expand_cases) apply(subst if_not_P) by auto qed
  4.1900 +
  4.1901 +
  4.1902 +(* This lemma about iterations comes up in a few places.                     *)
  4.1903 +lemma iterate_nonzero_image_lemma:
  4.1904 +  assumes "monoidal opp" "finite s" "g(a) = neutral opp"
  4.1905 +  "\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = neutral opp"
  4.1906 +  shows "iterate opp {f x | x. x \<in> s \<and> f x \<noteq> a} g = iterate opp s (g \<circ> f)"
  4.1907 +proof- have *:"{f x |x. x \<in> s \<and> ~(f x = a)} = f ` {x. x \<in> s \<and> ~(f x = a)}" by auto
  4.1908 +  have **:"support opp (g \<circ> f) {x \<in> s. f x \<noteq> a} = support opp (g \<circ> f) s"
  4.1909 +    unfolding support_def using assms(3) by auto
  4.1910 +  show ?thesis unfolding *
  4.1911 +    apply(subst iterate_support[THEN sym]) unfolding support_clauses
  4.1912 +    apply(subst iterate_image[OF assms(1)]) defer
  4.1913 +    apply(subst(2) iterate_support[THEN sym]) apply(subst **)
  4.1914 +    unfolding inj_on_def using assms(3,4) unfolding support_def by auto qed
  4.1915 +
  4.1916 +lemma iterate_eq_neutral:
  4.1917 +  assumes "monoidal opp"  "\<forall>x \<in> s. (f(x) = neutral opp)"
  4.1918 +  shows "(iterate opp s f = neutral opp)"
  4.1919 +proof- have *:"support opp f s = {}" unfolding support_def using assms(2) by auto
  4.1920 +  show ?thesis apply(subst iterate_support[THEN sym]) 
  4.1921 +    unfolding * using assms(1) by auto qed
  4.1922 +
  4.1923 +lemma iterate_op: assumes "monoidal opp" "finite s"
  4.1924 +  shows "iterate opp s (\<lambda>x. opp (f x) (g x)) = opp (iterate opp s f) (iterate opp s g)" using assms(2)
  4.1925 +proof(induct s) case empty thus ?case unfolding iterate_insert[OF assms(1)] using assms(1) by auto
  4.1926 +next case (insert x F) show ?case unfolding iterate_insert[OF assms(1) insert(1)] if_not_P[OF insert(2)] insert(3)
  4.1927 +    unfolding monoidal_ac[OF assms(1)] by(rule refl) qed
  4.1928 +
  4.1929 +lemma iterate_eq: assumes "monoidal opp" "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
  4.1930 +  shows "iterate opp s f = iterate opp s g"
  4.1931 +proof- have *:"support opp g s = support opp f s"
  4.1932 +    unfolding support_def using assms(2) by auto
  4.1933 +  show ?thesis
  4.1934 +  proof(cases "finite (support opp f s)")
  4.1935 +    case False thus ?thesis apply(subst iterate_expand_cases,subst(2) iterate_expand_cases)
  4.1936 +      unfolding * by auto
  4.1937 +  next def su \<equiv> "support opp f s"
  4.1938 +    case True note support_subset[of opp f s] 
  4.1939 +    thus ?thesis apply- apply(subst iterate_support[THEN sym],subst(2) iterate_support[THEN sym]) unfolding * using True
  4.1940 +      unfolding su_def[symmetric]
  4.1941 +    proof(induct su) case empty show ?case by auto
  4.1942 +    next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)] 
  4.1943 +        unfolding if_not_P[OF insert(2)] apply(subst insert(3))
  4.1944 +        defer apply(subst assms(2)[of x]) using insert by auto qed qed qed
  4.1945 +
  4.1946 +lemma nonempty_witness: assumes "s \<noteq> {}" obtains x where "x \<in> s" using assms by auto
  4.1947 +
  4.1948 +lemma operative_division: fixes f::"(real^'n) set \<Rightarrow> 'a"
  4.1949 +  assumes "monoidal opp" "operative opp f" "d division_of {a..b}"
  4.1950 +  shows "iterate opp d f = f {a..b}"
  4.1951 +proof- def C \<equiv> "card (division_points {a..b} d)" thus ?thesis using assms
  4.1952 +  proof(induct C arbitrary:a b d rule:full_nat_induct)
  4.1953 +    case goal1
  4.1954 +    { presume *:"content {a..b} \<noteq> 0 \<Longrightarrow> ?case"
  4.1955 +      thus ?case apply-apply(cases) defer apply assumption
  4.1956 +      proof- assume as:"content {a..b} = 0"
  4.1957 +        show ?case unfolding operativeD(1)[OF assms(2) as] apply(rule iterate_eq_neutral[OF goal1(2)])
  4.1958 +        proof fix x assume x:"x\<in>d"
  4.1959 +          then guess u v apply(drule_tac division_ofD(4)[OF goal1(4)]) by(erule exE)+
  4.1960 +          thus "f x = neutral opp" using division_of_content_0[OF as goal1(4)] 
  4.1961 +            using operativeD(1)[OF assms(2)] x by auto
  4.1962 +        qed qed }
  4.1963 +    assume "content {a..b} \<noteq> 0" note ab = this[unfolded content_lt_nz[THEN sym] content_pos_lt_eq]
  4.1964 +    hence ab':"\<forall>i. a$i \<le> b$i" by (auto intro!: less_imp_le) show ?case 
  4.1965 +    proof(cases "division_points {a..b} d = {}")
  4.1966 +      case True have d':"\<forall>i\<in>d. \<exists>u v. i = {u..v} \<and>
  4.1967 +        (\<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)"
  4.1968 +        unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule)
  4.1969 +        apply(rule_tac x=a in exI,rule_tac x=b in exI) apply(rule,rule refl) apply(rule)
  4.1970 +      proof- fix u v j assume as:"{u..v} \<in> d" note division_ofD(3)[OF goal1(4) this]
  4.1971 +        hence uv:"\<forall>i. u$i \<le> v$i" "u$j \<le> v$j" unfolding interval_ne_empty by auto
  4.1972 +        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
  4.1973 +        have "(j, u$j) \<notin> division_points {a..b} d"
  4.1974 +          "(j, v$j) \<notin> division_points {a..b} d" using True by auto
  4.1975 +        note this[unfolded de_Morgan_conj division_points_def mem_Collect_eq split_conv interval_bounds[OF ab'] bex_simps]
  4.1976 +        note *[OF this(1)] *[OF this(2)] note this[unfolded interval_bounds[OF uv(1)]]
  4.1977 +        moreover have "a$j \<le> u$j" "v$j \<le> b$j" using division_ofD(2,2,3)[OF goal1(4) as] 
  4.1978 +          unfolding subset_eq apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE)
  4.1979 +          unfolding interval_ne_empty mem_interval by auto
  4.1980 +        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"
  4.1981 +          unfolding not_less de_Morgan_disj using ab[rule_format,of j] uv(2) by auto
  4.1982 +      qed have "(1/2) *\<^sub>R (a+b) \<in> {a..b}" unfolding mem_interval using ab by(auto intro!:less_imp_le)
  4.1983 +      note this[unfolded division_ofD(6)[OF goal1(4),THEN sym] Union_iff]
  4.1984 +      then guess i .. note i=this guess u v using d'[rule_format,OF i(1)] apply-by(erule exE conjE)+ note uv=this
  4.1985 +      have "{a..b} \<in> d"
  4.1986 +      proof- { presume "i = {a..b}" thus ?thesis using i by auto }
  4.1987 +        { presume "u = a" "v = b" thus "i = {a..b}" using uv by auto }
  4.1988 +        show "u = a" "v = b" unfolding Cart_eq
  4.1989 +        proof(rule_tac[!] allI) fix j note i(2)[unfolded uv mem_interval,rule_format,of j]
  4.1990 +          thus "u $ j = a $ j" "v $ j = b $ j" using uv(2)[rule_format,of j] by auto
  4.1991 +        qed qed
  4.1992 +      hence *:"d = insert {a..b} (d - {{a..b}})" by auto
  4.1993 +      have "iterate opp (d - {{a..b}}) f = neutral opp" apply(rule iterate_eq_neutral[OF goal1(2)])
  4.1994 +      proof fix x assume x:"x \<in> d - {{a..b}}" hence "x\<in>d" by auto note d'[rule_format,OF this]
  4.1995 +        then guess u v apply-by(erule exE conjE)+ note uv=this
  4.1996 +        have "u\<noteq>a \<or> v\<noteq>b" using x[unfolded uv] by auto  
  4.1997 +        then obtain j where "u$j \<noteq> a$j \<or> v$j \<noteq> b$j" unfolding Cart_eq by auto
  4.1998 +        hence "u$j = v$j" using uv(2)[rule_format,of j] by auto
  4.1999 +        hence "content {u..v} = 0"  unfolding content_eq_0 apply(rule_tac x=j in exI) by auto
  4.2000 +        thus "f x = neutral opp" unfolding uv(1) by(rule operativeD(1)[OF goal1(3)])
  4.2001 +      qed thus "iterate opp d f = f {a..b}" apply-apply(subst *) 
  4.2002 +        apply(subst iterate_insert[OF goal1(2)]) using goal1(2,4) by auto
  4.2003 +    next case False hence "\<exists>x. x\<in>division_points {a..b} d" by auto
  4.2004 +      then guess k c unfolding split_paired_Ex apply- unfolding division_points_def mem_Collect_eq split_conv
  4.2005 +        by(erule exE conjE)+ note kc=this[unfolded interval_bounds[OF ab']]
  4.2006 +      from this(3) guess j .. note j=this
  4.2007 +      def d1 \<equiv> "{l \<inter> {x. x$k \<le> c} | l. l \<in> d \<and> l \<inter> {x. x$k \<le> c} \<noteq> {}}"
  4.2008 +      def d2 \<equiv> "{l \<inter> {x. x$k \<ge> c} | l. l \<in> d \<and> l \<inter> {x. x$k \<ge> c} \<noteq> {}}"
  4.2009 +      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)"
  4.2010 +      note division_points_psubset[OF goal1(4) ab kc(1-2) j]
  4.2011 +      note psubset_card_mono[OF _ this(1)] psubset_card_mono[OF _ this(2)]
  4.2012 +      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})"
  4.2013 +        apply- unfolding interval_split apply(rule_tac[!] goal1(1)[rule_format])
  4.2014 +        using division_split[OF goal1(4), where k=k and c=c]
  4.2015 +        unfolding interval_split d1_def[symmetric] d2_def[symmetric] unfolding goal1(2) Suc_le_mono
  4.2016 +        using goal1(2-3) using division_points_finite[OF goal1(4)] by auto
  4.2017 +      have "f {a..b} = opp (iterate opp d1 f) (iterate opp d2 f)" (is "_ = ?prev")
  4.2018 +        unfolding * apply(rule operativeD(2)) using goal1(3) .
  4.2019 +      also have "iterate opp d1 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x$k \<le> c}))"
  4.2020 +        unfolding d1_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])
  4.2021 +        unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+
  4.2022 +        unfolding empty_as_interval[THEN sym] apply(rule content_empty)
  4.2023 +      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" 
  4.2024 +        from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this
  4.2025 +        show "f (l \<inter> {x. x $ k \<le> c}) = neutral opp" unfolding l interval_split
  4.2026 +          apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym] apply(rule division_split_left_inj)
  4.2027 +          apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as)+
  4.2028 +      qed also have "iterate opp d2 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x$k \<ge> c}))"
  4.2029 +        unfolding d2_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])
  4.2030 +        unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+
  4.2031 +        unfolding empty_as_interval[THEN sym] apply(rule content_empty)
  4.2032 +      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" 
  4.2033 +        from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this
  4.2034 +        show "f (l \<inter> {x. x $ k \<ge> c}) = neutral opp" unfolding l interval_split
  4.2035 +          apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym] apply(rule division_split_right_inj)
  4.2036 +          apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as)+
  4.2037 +      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}))"
  4.2038 +        unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule,rule operativeD(2)) using goal1(3) .
  4.2039 +      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})))
  4.2040 +        = iterate opp d f" apply(subst(3) iterate_eq[OF _ *[rule_format]]) prefer 3
  4.2041 +        apply(rule iterate_op[THEN sym]) using goal1 by auto
  4.2042 +      finally show ?thesis by auto
  4.2043 +    qed qed qed 
  4.2044 +
  4.2045 +lemma iterate_image_nonzero: assumes "monoidal opp"
  4.2046 +  "finite s" "\<forall>x\<in>s. \<forall>y\<in>s. ~(x = y) \<and> f x = f y \<longrightarrow> g(f x) = neutral opp"
  4.2047 +  shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)" using assms
  4.2048 +proof(induct rule:finite_subset_induct[OF assms(2) subset_refl])
  4.2049 +  case goal1 show ?case using assms(1) by auto
  4.2050 +next case goal2 have *:"\<And>x y. y = neutral opp \<Longrightarrow> x = opp y x" using assms(1) by auto
  4.2051 +  show ?case unfolding image_insert apply(subst iterate_insert[OF assms(1)])
  4.2052 +    apply(rule finite_imageI goal2)+
  4.2053 +    apply(cases "f a \<in> f ` F") unfolding if_P if_not_P apply(subst goal2(4)[OF assms(1) goal2(1)]) defer
  4.2054 +    apply(subst iterate_insert[OF assms(1) goal2(1)]) defer
  4.2055 +    apply(subst iterate_insert[OF assms(1) goal2(1)])
  4.2056 +    unfolding if_not_P[OF goal2(3)] defer unfolding image_iff defer apply(erule bexE)
  4.2057 +    apply(rule *) unfolding o_def apply(rule_tac y=x in goal2(7)[rule_format])
  4.2058 +    using goal2 unfolding o_def by auto qed 
  4.2059 +
  4.2060 +lemma operative_tagged_division: assumes "monoidal opp" "operative opp f" "d tagged_division_of {a..b}"
  4.2061 +  shows "iterate(opp) d (\<lambda>(x,l). f l) = f {a..b}"
  4.2062 +proof- have *:"(\<lambda>(x,l). f l) = (f o snd)" unfolding o_def by(rule,auto) note assm = tagged_division_ofD[OF assms(3)]
  4.2063 +  have "iterate(opp) d (\<lambda>(x,l). f l) = iterate opp (snd ` d) f" unfolding *
  4.2064 +    apply(rule iterate_image_nonzero[THEN sym,OF assms(1)]) apply(rule tagged_division_of_finite assms)+ 
  4.2065 +    unfolding Ball_def split_paired_All snd_conv apply(rule,rule,rule,rule,rule,rule,rule,erule conjE)
  4.2066 +  proof- fix a b aa ba assume as:"(a, b) \<in> d" "(aa, ba) \<in> d" "(a, b) \<noteq> (aa, ba)" "b = ba"
  4.2067 +    guess u v using assm(4)[OF as(1)] apply-by(erule exE)+ note uv=this
  4.2068 +    show "f b = neutral opp" unfolding uv apply(rule operativeD(1)[OF assms(2)])
  4.2069 +      unfolding content_eq_0_interior using tagged_division_ofD(5)[OF assms(3) as(1-3)]
  4.2070 +      unfolding as(4)[THEN sym] uv by auto
  4.2071 +  qed also have "\<dots> = f {a..b}" 
  4.2072 +    using operative_division[OF assms(1-2) division_of_tagged_division[OF assms(3)]] .
  4.2073 +  finally show ?thesis . qed
  4.2074 +
  4.2075 +subsection {* Additivity of content. *}
  4.2076 +
  4.2077 +lemma setsum_iterate:assumes "finite s" shows "setsum f s = iterate op + s f"
  4.2078 +proof- have *:"setsum f s = setsum f (support op + f s)"
  4.2079 +    apply(rule setsum_mono_zero_right)
  4.2080 +    unfolding support_def neutral_monoid using assms by auto
  4.2081 +  thus ?thesis unfolding * setsum_def iterate_def fold_image_def fold'_def
  4.2082 +    unfolding neutral_monoid . qed
  4.2083 +
  4.2084 +lemma additive_content_division: assumes "d division_of {a..b}"
  4.2085 +  shows "setsum content d = content({a..b})"
  4.2086 +  unfolding operative_division[OF monoidal_monoid operative_content assms,THEN sym]
  4.2087 +  apply(subst setsum_iterate) using assms by auto
  4.2088 +
  4.2089 +lemma additive_content_tagged_division:
  4.2090 +  assumes "d tagged_division_of {a..b}"
  4.2091 +  shows "setsum (\<lambda>(x,l). content l) d = content({a..b})"
  4.2092 +  unfolding operative_tagged_division[OF monoidal_monoid operative_content assms,THEN sym]
  4.2093 +  apply(subst setsum_iterate) using assms by auto
  4.2094 +
  4.2095 +subsection {* Finally, the integral of a constant\<forall> *}
  4.2096 +
  4.2097 +lemma has_integral_const[intro]:
  4.2098 +  "((\<lambda>x. c) has_integral (content({a..b::real^'n}) *\<^sub>R c)) ({a..b})"
  4.2099 +  unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI)
  4.2100 +  apply(rule,rule gauge_trivial)apply(rule,rule,erule conjE)
  4.2101 +  unfolding split_def apply(subst scaleR_left.setsum[THEN sym, unfolded o_def])
  4.2102 +  defer apply(subst additive_content_tagged_division[unfolded split_def]) apply assumption by auto
  4.2103 +
  4.2104 +subsection {* Bounds on the norm of Riemann sums and the integral itself. *}
  4.2105 +
  4.2106 +lemma dsum_bound: assumes "p division_of {a..b}" "norm(c) \<le> e"
  4.2107 +  shows "norm(setsum (\<lambda>l. content l *\<^sub>R c) p) \<le> e * content({a..b})" (is "?l \<le> ?r")
  4.2108 +  apply(rule order_trans,rule setsum_norm) defer unfolding norm_scaleR setsum_left_distrib[THEN sym]
  4.2109 +  apply(rule order_trans[OF mult_left_mono],rule assms,rule setsum_abs_ge_zero)
  4.2110 +  apply(subst real_mult_commute) apply(rule mult_left_mono)
  4.2111 +  apply(rule order_trans[of _ "setsum content p"]) apply(rule eq_refl,rule setsum_cong2)
  4.2112 +  apply(subst abs_of_nonneg) unfolding additive_content_division[OF assms(1)]
  4.2113 +proof- from order_trans[OF norm_ge_zero[of c] assms(2)] show "0 \<le> e" .
  4.2114 +  fix x assume "x\<in>p" from division_ofD(4)[OF assms(1) this] guess u v apply-by(erule exE)+
  4.2115 +  thus "0 \<le> content x" using content_pos_le by auto
  4.2116 +qed(insert assms,auto)
  4.2117 +
  4.2118 +lemma rsum_bound: assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. norm(f x) \<le> e"
  4.2119 +  shows "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p) \<le> e * content({a..b})"
  4.2120 +proof(cases "{a..b} = {}") case True
  4.2121 +  show ?thesis using assms(1) unfolding True tagged_division_of_trivial by auto
  4.2122 +next case False show ?thesis
  4.2123 +    apply(rule order_trans,rule setsum_norm) defer unfolding split_def norm_scaleR
  4.2124 +    apply(rule order_trans[OF setsum_mono]) apply(rule mult_left_mono[OF _ abs_ge_zero, of _ e]) defer
  4.2125 +    unfolding setsum_left_distrib[THEN sym] apply(subst real_mult_commute) apply(rule mult_left_mono)
  4.2126 +    apply(rule order_trans[of _ "setsum (content \<circ> snd) p"]) apply(rule eq_refl,rule setsum_cong2)
  4.2127 +    apply(subst o_def, rule abs_of_nonneg)
  4.2128 +  proof- show "setsum (content \<circ> snd) p \<le> content {a..b}" apply(rule eq_refl)
  4.2129 +      unfolding additive_content_tagged_division[OF assms(1),THEN sym] split_def by auto
  4.2130 +    guess w using nonempty_witness[OF False] .
  4.2131 +    thus "e\<ge>0" apply-apply(rule order_trans) defer apply(rule assms(2)[rule_format],assumption) by auto
  4.2132 +    fix xk assume *:"xk\<in>p" guess x k  using surj_pair[of xk] apply-by(erule exE)+ note xk = this *[unfolded this]
  4.2133 +    from tagged_division_ofD(4)[OF assms(1) xk(2)] guess u v apply-by(erule exE)+ note uv=this
  4.2134 +    show "0\<le> content (snd xk)" unfolding xk snd_conv uv by(rule content_pos_le)
  4.2135 +    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
  4.2136 +  qed(insert assms,auto) qed
  4.2137 +
  4.2138 +lemma rsum_diff_bound:
  4.2139 +  assumes "p tagged_division_of {a..b}"  "\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e"
  4.2140 +  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})"
  4.2141 +  apply(rule order_trans[OF _ rsum_bound[OF assms]]) apply(rule eq_refl) apply(rule arg_cong[where f=norm])
  4.2142 +  unfolding setsum_subtractf[THEN sym] apply(rule setsum_cong2) unfolding scaleR.diff_right by auto
  4.2143 +
  4.2144 +lemma has_integral_bound: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
  4.2145 +  assumes "0 \<le> B" "(f has_integral i) ({a..b})" "\<forall>x\<in>{a..b}. norm(f x) \<le> B"
  4.2146 +  shows "norm i \<le> B * content {a..b}"
  4.2147 +proof- let ?P = "content {a..b} > 0" { presume "?P \<Longrightarrow> ?thesis"
  4.2148 +    thus ?thesis proof(cases ?P) case False
  4.2149 +      hence *:"content {a..b} = 0" using content_lt_nz by auto
  4.2150 +      hence **:"i = 0" using assms(2) apply(subst has_integral_null_eq[THEN sym]) by auto
  4.2151 +      show ?thesis unfolding * ** using assms(1) by auto
  4.2152 +    qed auto } assume ab:?P
  4.2153 +  { presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }
  4.2154 +  assume "\<not> ?thesis" hence *:"norm i - B * content {a..b} > 0" by auto
  4.2155 +  from assms(2)[unfolded has_integral,rule_format,OF *] guess d apply-by(erule exE conjE)+ note d=this[rule_format]
  4.2156 +  from fine_division_exists[OF this(1), of a b] guess p . note p=this
  4.2157 +  have *:"\<And>s B. norm s \<le> B \<Longrightarrow> \<not> (norm (s - i) < norm i - B)"
  4.2158 +  proof- case goal1 thus ?case unfolding not_less
  4.2159 +    using norm_triangle_sub[of i s] unfolding norm_minus_commute by auto
  4.2160 +  qed show False using d(2)[OF conjI[OF p]] *[OF rsum_bound[OF p(1) assms(3)]] by auto qed
  4.2161 +
  4.2162 +subsection {* Similar theorems about relationship among components. *}
  4.2163 +
  4.2164 +lemma rsum_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
  4.2165 +  assumes "p tagged_division_of {a..b}"  "\<forall>x\<in>{a..b}. (f x)$i \<le> (g x)$i"
  4.2166 +  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"
  4.2167 +  unfolding setsum_component apply(rule setsum_mono)
  4.2168 +  apply(rule mp) defer apply assumption apply(induct_tac x,rule) unfolding split_conv
  4.2169 +proof- fix a b assume ab:"(a,b) \<in> p" note assm = tagged_division_ofD(2-4)[OF assms(1) ab]
  4.2170 +  from this(3) guess u v apply-by(erule exE)+ note b=this
  4.2171 +  show "(content b *\<^sub>R f a) $ i \<le> (content b *\<^sub>R g a) $ i" unfolding b
  4.2172 +    unfolding Cart_nth.scaleR real_scaleR_def apply(rule mult_left_mono)
  4.2173 +    defer apply(rule content_pos_le,rule assms(2)[rule_format]) using assm by auto qed
  4.2174 +
  4.2175 +lemma has_integral_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
  4.2176 +  assumes "(f has_integral i) s" "(g has_integral j) s"  "\<forall>x\<in>s. (f x)$k \<le> (g x)$k"
  4.2177 +  shows "i$k \<le> j$k"
  4.2178 +proof- have lem:"\<And>a b g i j. \<And>f::real^'n \<Rightarrow> real^'m. (f has_integral i) ({a..b}) \<Longrightarrow> 
  4.2179 +    (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"
  4.2180 +  proof(rule ccontr) case goal1 hence *:"0 < (i$k - j$k) / 3" by auto
  4.2181 +    guess d1 using goal1(1)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d1=this[rule_format]
  4.2182 +    guess d2 using goal1(2)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d2=this[rule_format]
  4.2183 +    guess p using fine_division_exists[OF gauge_inter[OF d1(1) d2(1)], of a b] unfolding fine_inter .
  4.2184 +    note p = this(1) conjunctD2[OF this(2)]  note le_less_trans[OF component_le_norm, of _ _ k]
  4.2185 +    note this[OF d1(2)[OF conjI[OF p(1,2)]]] this[OF d2(2)[OF conjI[OF p(1,3)]]]
  4.2186 +    thus False unfolding Cart_nth.diff using rsum_component_le[OF p(1) goal1(3)] by smt
  4.2187 +  qed let ?P = "\<exists>a b. s = {a..b}"
  4.2188 +  { presume "\<not> ?P \<Longrightarrow> ?thesis" thus ?thesis proof(cases ?P)
  4.2189 +      case True then guess a b apply-by(erule exE)+ note s=this
  4.2190 +      show ?thesis apply(rule lem) using assms[unfolded s] by auto
  4.2191 +    qed auto } assume as:"\<not> ?P"
  4.2192 +  { presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }
  4.2193 +  assume "\<not> i$k \<le> j$k" hence ij:"(i$k - j$k) / 3 > 0" by auto
  4.2194 +  note has_integral_altD[OF _ as this] from this[OF assms(1)] this[OF assms(2)] guess B1 B2 . note B=this[rule_format]
  4.2195 +  have "bounded (ball 0 B1 \<union> ball (0::real^'n) B2)" unfolding bounded_Un by(rule conjI bounded_ball)+
  4.2196 +  from bounded_subset_closed_interval[OF this] guess a b apply- by(erule exE)+
  4.2197 +  note ab = conjunctD2[OF this[unfolded Un_subset_iff]]
  4.2198 +  guess w1 using B(2)[OF ab(1)] .. note w1=conjunctD2[OF this]
  4.2199 +  guess w2 using B(4)[OF ab(2)] .. note w2=conjunctD2[OF this]
  4.2200 +  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*)
  4.2201 +  note le_less_trans[OF component_le_norm[of _ k]] note this[OF w1(2)] this[OF w2(2)] moreover
  4.2202 +  have "w1$k \<le> w2$k" apply(rule lem[OF w1(1) w2(1)]) using assms by auto ultimately
  4.2203 +  show False unfolding Cart_nth.diff by(rule *) qed
  4.2204 +
  4.2205 +lemma integral_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
  4.2206 +  assumes "f integrable_on s" "g integrable_on s"  "\<forall>x\<in>s. (f x)$k \<le> (g x)$k"
  4.2207 +  shows "(integral s f)$k \<le> (integral s g)$k"
  4.2208 +  apply(rule has_integral_component_le) using integrable_integral assms by auto
  4.2209 +
  4.2210 +lemma has_integral_dest_vec1_le: fixes f::"real^'n \<Rightarrow> real^1"
  4.2211 +  assumes "(f has_integral i) s"  "(g has_integral j) s" "\<forall>x\<in>s. f x \<le> g x"
  4.2212 +  shows "dest_vec1 i \<le> dest_vec1 j" apply(rule has_integral_component_le[OF assms(1-2)])
  4.2213 +  using assms(3) unfolding vector_le_def by auto
  4.2214 +
  4.2215 +lemma integral_dest_vec1_le: fixes f::"real^'n \<Rightarrow> real^1"
  4.2216 +  assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. f x \<le> g x"
  4.2217 +  shows "dest_vec1(integral s f) \<le> dest_vec1(integral s g)"
  4.2218 +  apply(rule has_integral_dest_vec1_le) apply(rule_tac[1-2] integrable_integral) using assms by auto
  4.2219 +
  4.2220 +lemma has_integral_component_pos: fixes f::"real^'n \<Rightarrow> real^'m"
  4.2221 +  assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> (f x)$k" shows "0 \<le> i$k"
  4.2222 +  using has_integral_component_le[OF has_integral_0 assms(1)] using assms(2) by auto
  4.2223 +
  4.2224 +lemma integral_component_pos: fixes f::"real^'n \<Rightarrow> real^'m"
  4.2225 +  assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> (f x)$k" shows "0 \<le> (integral s f)$k"
  4.2226 +  apply(rule has_integral_component_pos) using assms by auto
  4.2227 +
  4.2228 +lemma has_integral_dest_vec1_pos: fixes f::"real^'n \<Rightarrow> real^1"
  4.2229 +  assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> i"
  4.2230 +  using has_integral_component_pos[OF assms(1), of 1]
  4.2231 +  using assms(2) unfolding vector_le_def by auto
  4.2232 +
  4.2233 +lemma integral_dest_vec1_pos: fixes f::"real^'n \<Rightarrow> real^1"
  4.2234 +  assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> integral s f"
  4.2235 +  apply(rule has_integral_dest_vec1_pos) using assms by auto
  4.2236 +
  4.2237 +lemma has_integral_component_neg: fixes f::"real^'n \<Rightarrow> real^'m"
  4.2238 +  assumes "(f has_integral i) s" "\<forall>x\<in>s. (f x)$k \<le> 0" shows "i$k \<le> 0"
  4.2239 +  using has_integral_component_le[OF assms(1) has_integral_0] assms(2) by auto
  4.2240 +
  4.2241 +lemma has_integral_dest_vec1_neg: fixes f::"real^'n \<Rightarrow> real^1"
  4.2242 +  assumes "(f has_integral i) s" "\<forall>x\<in>s. f x \<le> 0" shows "i \<le> 0"
  4.2243 +  using has_integral_component_neg[OF assms(1),of 1] using assms(2) by auto
  4.2244 +
  4.2245 +lemma has_integral_component_lbound:
  4.2246 +  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"
  4.2247 +  using has_integral_component_le[OF has_integral_const assms(1),of "(\<chi> i. B)" k] assms(2)
  4.2248 +  unfolding Cart_lambda_beta vector_scaleR_component by(auto simp add:field_simps)
  4.2249 +
  4.2250 +lemma has_integral_component_ubound: 
  4.2251 +  assumes "(f has_integral i) {a..b}" "\<forall>x\<in>{a..b}. f x$k \<le> B"
  4.2252 +  shows "i$k \<le> B * content({a..b})"
  4.2253 +  using has_integral_component_le[OF assms(1) has_integral_const, of k "vec B"]
  4.2254 +  unfolding vec_component Cart_nth.scaleR using assms(2) by(auto simp add:field_simps)
  4.2255 +
  4.2256 +lemma integral_component_lbound:
  4.2257 +  assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. B \<le> f(x)$k"
  4.2258 +  shows "B * content({a..b}) \<le> (integral({a..b}) f)$k"
  4.2259 +  apply(rule has_integral_component_lbound) using assms unfolding has_integral_integral by auto
  4.2260 +
  4.2261 +lemma integral_component_ubound:
  4.2262 +  assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. f(x)$k \<le> B"
  4.2263 +  shows "(integral({a..b}) f)$k \<le> B * content({a..b})"
  4.2264 +  apply(rule has_integral_component_ubound) using assms unfolding has_integral_integral by auto
  4.2265 +
  4.2266 +subsection {* Uniform limit of integrable functions is integrable. *}
  4.2267 +
  4.2268 +lemma real_arch_invD:
  4.2269 +  "0 < (e::real) \<Longrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
  4.2270 +  by(subst(asm) real_arch_inv)
  4.2271 +
  4.2272 +lemma integrable_uniform_limit: fixes f::"real^'n \<Rightarrow> 'a::banach"
  4.2273 +  assumes "\<forall>e>0. \<exists>g. (\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e) \<and> g integrable_on {a..b}"
  4.2274 +  shows "f integrable_on {a..b}"
  4.2275 +proof- { presume *:"content {a..b} > 0 \<Longrightarrow> ?thesis"
  4.2276 +    show ?thesis apply cases apply(rule *,assumption)
  4.2277 +      unfolding content_lt_nz integrable_on_def using has_integral_null by auto }
  4.2278 +  assume as:"content {a..b} > 0"
  4.2279 +  have *:"\<And>P. \<forall>e>(0::real). P e \<Longrightarrow> \<forall>n::nat. P (inverse (real n+1))" by auto
  4.2280 +  from choice[OF *[OF assms]] guess g .. note g=conjunctD2[OF this[rule_format],rule_format]
  4.2281 +  from choice[OF allI[OF g(2)[unfolded integrable_on_def], of "\<lambda>x. x"]] guess i .. note i=this[rule_format]
  4.2282 +  
  4.2283 +  have "Cauchy i" unfolding Cauchy_def
  4.2284 +  proof(rule,rule) fix e::real assume "e>0"
  4.2285 +    hence "e / 4 / content {a..b} > 0" using as by(auto simp add:field_simps)
  4.2286 +    then guess M apply-apply(subst(asm) real_arch_inv) by(erule exE conjE)+ note M=this
  4.2287 +    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)
  4.2288 +    proof- case goal1 have "e/4>0" using `e>0` by auto note * = i[unfolded has_integral,rule_format,OF this]
  4.2289 +      from *[of m] guess gm apply-by(erule conjE exE)+ note gm=this[rule_format]
  4.2290 +      from *[of n] guess gn apply-by(erule conjE exE)+ note gn=this[rule_format]
  4.2291 +      from fine_division_exists[OF gauge_inter[OF gm(1) gn(1)], of a b] guess p . note p=this
  4.2292 +      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"
  4.2293 +      proof- case goal1 have "norm (i1 - i2) \<le> norm (i1 - s1) + norm (s1 - s2) + norm (s2 - i2)"
  4.2294 +          using norm_triangle_ineq[of "i1 - s1" "s1 - i2"]
  4.2295 +          using norm_triangle_ineq[of "s1 - s2" "s2 - i2"] by(auto simp add:group_simps)
  4.2296 +        also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:group_simps)
  4.2297 +        finally show ?case .
  4.2298 +      qed
  4.2299 +      show ?case unfolding vector_dist_norm apply(rule lem2) defer
  4.2300 +        apply(rule gm(2)[OF conjI[OF p(1)]],rule_tac[2] gn(2)[OF conjI[OF p(1)]])
  4.2301 +        using conjunctD2[OF p(2)[unfolded fine_inter]] apply- apply assumption+ apply(rule order_trans)
  4.2302 +        apply(rule rsum_diff_bound[OF p(1), where e="2 / real M"])
  4.2303 +      proof show "2 / real M * content {a..b} \<le> e / 2" unfolding divide_inverse 
  4.2304 +          using M as by(auto simp add:field_simps)
  4.2305 +        fix x assume x:"x \<in> {a..b}"
  4.2306 +        have "norm (f x - g n x) + norm (f x - g m x) \<le> inverse (real n + 1) + inverse (real m + 1)" 
  4.2307 +            using g(1)[OF x, of n] g(1)[OF x, of m] by auto
  4.2308 +        also have "\<dots> \<le> inverse (real M) + inverse (real M)" apply(rule add_mono)
  4.2309 +          apply(rule_tac[!] le_imp_inverse_le) using goal1 M by auto
  4.2310 +        also have "\<dots> = 2 / real M" unfolding real_divide_def by auto
  4.2311 +        finally show "norm (g n x - g m x) \<le> 2 / real M"
  4.2312 +          using norm_triangle_le[of "g n x - f x" "f x - g m x" "2 / real M"]
  4.2313 +          by(auto simp add:group_simps simp add:norm_minus_commute)
  4.2314 +      qed qed qed
  4.2315 +  from this[unfolded convergent_eq_cauchy[THEN sym]] guess s .. note s=this
  4.2316 +
  4.2317 +  show ?thesis unfolding integrable_on_def apply(rule_tac x=s in exI) unfolding has_integral
  4.2318 +  proof(rule,rule)  
  4.2319 +    case goal1 hence *:"e/3 > 0" by auto
  4.2320 +    from s[unfolded Lim_sequentially,rule_format,OF this] guess N1 .. note N1=this
  4.2321 +    from goal1 as have "e / 3 / content {a..b} > 0" by(auto simp add:field_simps)
  4.2322 +    from real_arch_invD[OF this] guess N2 apply-by(erule exE conjE)+ note N2=this
  4.2323 +    from i[of "N1 + N2",unfolded has_integral,rule_format,OF *] guess g' .. note g'=conjunctD2[OF this,rule_format]
  4.2324 +    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"
  4.2325 +    proof- case goal1 have "norm (sf - s) \<le> norm (sf - sg) + norm (sg - i) + norm (i - s)"
  4.2326 +        using norm_triangle_ineq[of "sf - sg" "sg - s"]
  4.2327 +        using norm_triangle_ineq[of "sg -  i" " i - s"] by(auto simp add:group_simps)
  4.2328 +      also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:group_simps)
  4.2329 +      finally show ?case .
  4.2330 +    qed
  4.2331 +    show ?case apply(rule_tac x=g' in exI) apply(rule,rule g')
  4.2332 +    proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> g' fine p" note * = g'(2)[OF this]
  4.2333 +      show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - s) < e" apply-apply(rule lem[OF _ _ *])
  4.2334 +        apply(rule order_trans,rule rsum_diff_bound[OF p[THEN conjunct1]]) apply(rule,rule g,assumption)
  4.2335 +      proof- have "content {a..b} < e / 3 * (real N2)"
  4.2336 +          using N2 unfolding inverse_eq_divide using as by(auto simp add:field_simps)
  4.2337 +        hence "content {a..b} < e / 3 * (real (N1 + N2) + 1)"
  4.2338 +          apply-apply(rule less_le_trans,assumption) using `e>0` by auto 
  4.2339 +        thus "inverse (real (N1 + N2) + 1) * content {a..b} \<le> e / 3"
  4.2340 +          unfolding inverse_eq_divide by(auto simp add:field_simps)
  4.2341 +        show "norm (i (N1 + N2) - s) < e / 3" by(rule N1[rule_format,unfolded vector_dist_norm],auto)
  4.2342 +      qed qed qed qed
  4.2343 +
  4.2344 +subsection {* Negligible sets. *}
  4.2345 +
  4.2346 +definition "indicator s \<equiv> (\<lambda>x. if x \<in> s then 1 else (0::real))"
  4.2347 +
  4.2348 +lemma dest_vec1_indicator:
  4.2349 + "indicator s x = (if x \<in> s then 1 else 0)" unfolding indicator_def by auto
  4.2350 +
  4.2351 +lemma indicator_pos_le[intro]: "0 \<le> (indicator s x)" unfolding indicator_def by auto
  4.2352 +
  4.2353 +lemma indicator_le_1[intro]: "(indicator s x) \<le> 1" unfolding indicator_def by auto
  4.2354 +
  4.2355 +lemma dest_vec1_indicator_abs_le_1: "abs(indicator s x) \<le> 1"
  4.2356 +  unfolding indicator_def by auto
  4.2357 +
  4.2358 +definition "negligible (s::(real^'n) set) \<equiv> (\<forall>a b. ((indicator s) has_integral 0) {a..b})"
  4.2359 +
  4.2360 +lemma indicator_simps[simp]:"x\<in>s \<Longrightarrow> indicator s x = 1" "x\<notin>s \<Longrightarrow> indicator s x = 0"
  4.2361 +  unfolding indicator_def by auto
  4.2362 +
  4.2363 +subsection {* Negligibility of hyperplane. *}
  4.2364 +
  4.2365 +lemma vsum_nonzero_image_lemma: 
  4.2366 +  assumes "finite s" "g(a) = 0"
  4.2367 +  "\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = 0"
  4.2368 +  shows "setsum g {f x |x. x \<in> s \<and> f x \<noteq> a} = setsum (g o f) s"
  4.2369 +  unfolding setsum_iterate[OF assms(1)] apply(subst setsum_iterate) defer
  4.2370 +  apply(rule iterate_nonzero_image_lemma) apply(rule assms monoidal_monoid)+
  4.2371 +  unfolding assms using neutral_add unfolding neutral_add using assms by auto 
  4.2372 +
  4.2373 +lemma interval_doublesplit: shows "{a..b} \<inter> {x . abs(x$k - c) \<le> (e::real)} =
  4.2374 +  {(\<chi> i. if i = k then max (a$k) (c - e) else a$i) .. (\<chi> i. if i = k then min (b$k) (c + e) else b$i)}"
  4.2375 +proof- have *:"\<And>x c e::real. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
  4.2376 +  have **:"\<And>s P Q. s \<inter> {x. P x \<and> Q x} = (s \<inter> {x. Q x}) \<inter> {x. P x}" by blast
  4.2377 +  show ?thesis unfolding * ** interval_split by(rule refl) qed
  4.2378 +
  4.2379 +lemma division_doublesplit: assumes "p division_of {a..b::real^'n}" 
  4.2380 +  shows "{l \<inter> {x. abs(x$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})"
  4.2381 +proof- have *:"\<And>x c. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
  4.2382 +  have **:"\<And>p q p' q'. p division_of q \<Longrightarrow> p = p' \<Longrightarrow> q = q' \<Longrightarrow> p' division_of q'" by auto
  4.2383 +  note division_split(1)[OF assms, where c="c+e" and k=k,unfolded interval_split]
  4.2384 +  note division_split(2)[OF this, where c="c-e" and k=k] 
  4.2385 +  thus ?thesis apply(rule **) unfolding interval_doublesplit unfolding * unfolding interval_split interval_doublesplit
  4.2386 +    apply(rule set_ext) unfolding mem_Collect_eq apply rule apply(erule conjE exE)+ apply(rule_tac x=la in exI) defer
  4.2387 +    apply(erule conjE exE)+ apply(rule_tac x="l \<inter> {x. c + e \<ge> x $ k}" in exI) apply rule defer apply rule
  4.2388 +    apply(rule_tac x=l in exI) by blast+ qed
  4.2389 +
  4.2390 +lemma content_doublesplit: assumes "0 < e"
  4.2391 +  obtains d where "0 < d" "content({a..b} \<inter> {x. abs(x$k - c) \<le> d}) < e"
  4.2392 +proof(cases "content {a..b} = 0")
  4.2393 +  case True show ?thesis apply(rule that[of 1]) defer unfolding interval_doublesplit
  4.2394 +    apply(rule le_less_trans[OF content_subset]) defer apply(subst True)
  4.2395 +    unfolding interval_doublesplit[THEN sym] using assms by auto 
  4.2396 +next case False def d \<equiv> "e / 3 / setprod (\<lambda>i. b$i - a$i) (UNIV - {k})"
  4.2397 +  note False[unfolded content_eq_0 not_ex not_le, rule_format]
  4.2398 +  hence prod0:"0 < setprod (\<lambda>i. b$i - a$i) (UNIV - {k})" apply-apply(rule setprod_pos) by smt
  4.2399 +  hence "d > 0" unfolding d_def using assms by(auto simp add:field_simps) thus ?thesis
  4.2400 +  proof(rule that[of d]) have *:"UNIV = insert k (UNIV - {k})" by auto
  4.2401 +    have **:"{a..b} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d} \<noteq> {} \<Longrightarrow> 
  4.2402 +      (\<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)
  4.2403 +      = (\<Prod>i\<in>UNIV - {k}. b$i - a$i)" apply(rule setprod_cong,rule refl)
  4.2404 +      unfolding interval_doublesplit interval_eq_empty not_ex not_less unfolding interval_bounds by auto
  4.2405 +    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)
  4.2406 +      unfolding if_not_P apply(subst *) apply(subst setprod_insert) unfolding **
  4.2407 +      unfolding interval_doublesplit interval_eq_empty not_ex not_less unfolding interval_bounds unfolding Cart_lambda_beta if_P[OF refl]
  4.2408 +    proof- have "(min (b $ k) (c + d) - max (a $ k) (c - d)) \<le> 2 * d" by auto
  4.2409 +      also have "... < e / (\<Prod>i\<in>UNIV - {k}. b $ i - a $ i)" unfolding d_def using assms prod0 by(auto simp add:field_simps)
  4.2410 +      finally show "(min (b $ k) (c + d) - max (a $ k) (c - d)) * (\<Prod>i\<in>UNIV - {k}. b $ i - a $ i) < e"
  4.2411 +        unfolding pos_less_divide_eq[OF prod0] . qed auto qed qed
  4.2412 +
  4.2413 +lemma negligible_standard_hyperplane[intro]: "negligible {x. x$k = (c::real)}" 
  4.2414 +  unfolding negligible_def has_integral apply(rule,rule,rule,rule)
  4.2415 +proof- case goal1 from content_doublesplit[OF this,of a b k c] guess d . note d=this let ?i = "indicator {x. x$k = c}"
  4.2416 +  show ?case apply(rule_tac x="\<lambda>x. ball x d" in exI) apply(rule,rule gauge_ball,rule d)
  4.2417 +  proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> (\<lambda>x. ball x d) fine p"
  4.2418 +    have *:"(\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. abs(x$k - c) \<le> d}) *\<^sub>R ?i x)"
  4.2419 +      apply(rule setsum_cong2) unfolding split_paired_all real_scaleR_def mult_cancel_right split_conv
  4.2420 +      apply(cases,rule disjI1,assumption,rule disjI2)
  4.2421 +    proof- fix x l assume as:"(x,l)\<in>p" "?i x \<noteq> 0" hence xk:"x$k = c" unfolding indicator_def apply-by(rule ccontr,auto)
  4.2422 +      show "content l = content (l \<inter> {x. \<bar>x $ k - c\<bar> \<le> d})" apply(rule arg_cong[where f=content])
  4.2423 +        apply(rule set_ext,rule,rule) unfolding mem_Collect_eq
  4.2424 +      proof- fix y assume y:"y\<in>l" note p[THEN conjunct2,unfolded fine_def,rule_format,OF as(1),unfolded split_conv]
  4.2425 +        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]
  4.2426 +        thus "\<bar>y $ k - c\<bar> \<le> d" unfolding Cart_nth.diff xk by auto
  4.2427 +      qed auto qed
  4.2428 +    note p'= tagged_division_ofD[OF p[THEN conjunct1]] and p''=division_of_tagged_division[OF p[THEN conjunct1]]
  4.2429 +    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
  4.2430 +      apply(subst abs_of_nonneg) apply(rule setsum_nonneg,rule) unfolding split_paired_all split_conv
  4.2431 +      apply(rule mult_nonneg_nonneg) apply(drule p'(4)) apply(erule exE)+ apply(rule_tac b=b in back_subst)
  4.2432 +      prefer 2 apply(subst(asm) eq_commute) apply assumption
  4.2433 +      apply(subst interval_doublesplit) apply(rule content_pos_le) apply(rule indicator_pos_le)
  4.2434 +    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}))"
  4.2435 +        apply(rule setsum_mono) unfolding split_paired_all split_conv 
  4.2436 +        apply(rule mult_right_le_one_le) apply(drule p'(4)) by(auto simp add:interval_doublesplit intro!:content_pos_le)
  4.2437 +      also have "... < e" apply(subst setsum_over_tagged_division_lemma[OF p[THEN conjunct1]])
  4.2438 +      proof- case goal1 have "content ({u..v} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) \<le> content {u..v}"
  4.2439 +          unfolding interval_doublesplit apply(rule content_subset) unfolding interval_doublesplit[THEN sym] by auto
  4.2440 +        thus ?case unfolding goal1 unfolding interval_doublesplit using content_pos_le by smt
  4.2441 +      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"
  4.2442 +          apply(rule setsum_nonneg,rule) unfolding mem_Collect_eq image_iff apply(erule exE bexE conjE)+ unfolding split_paired_all 
  4.2443 +        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)"
  4.2444 +          guess u v using p'(4)[OF as(2)] apply-by(erule exE)+ note * = this
  4.2445 +          show "content x \<ge> 0" unfolding as snd_conv * interval_doublesplit by(rule content_pos_le)
  4.2446 +        qed have **:"norm (1::real) \<le> 1" by auto note division_doublesplit[OF p'',unfolded interval_doublesplit]
  4.2447 +        note dsum_bound[OF this **,unfolded interval_doublesplit[THEN sym]]
  4.2448 +        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)]
  4.2449 +        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"
  4.2450 +          apply(subst vsum_nonzero_image_lemma[of "snd ` p" content "{}", unfolded o_def,THEN sym])
  4.2451 +          apply(rule finite_imageI p' content_empty)+ unfolding forall_in_division[OF p'']
  4.2452 +        proof(rule,rule,rule,rule,rule,rule,rule,erule conjE) fix m n u v
  4.2453 +          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}"
  4.2454 +          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
  4.2455 +          note subset_interior[OF this, unfolded division_ofD(5)[OF p'' as(1-3)] interior_inter[of "{m..n}"]]
  4.2456 +          hence "interior ({m..n} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) = {}" unfolding as Int_absorb by auto
  4.2457 +          thus "content ({m..n} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) = 0" unfolding interval_doublesplit content_eq_0_interior[THEN sym] .
  4.2458 +        qed qed
  4.2459 +      finally show "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) * ?i x) < e" .
  4.2460 +    qed qed qed
  4.2461 +
  4.2462 +subsection {* A technical lemma about "refinement" of division. *}
  4.2463 +
  4.2464 +lemma tagged_division_finer: fixes p::"((real^'n) \<times> ((real^'n) set)) set"
  4.2465 +  assumes "p tagged_division_of {a..b}" "gauge d"
  4.2466 +  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"
  4.2467 +proof-
  4.2468 +  let ?P = "\<lambda>p. p tagged_partial_division_of {a..b} \<longrightarrow> gauge d \<longrightarrow>
  4.2469 +    (\<exists>q. q tagged_division_of (\<Union>{k. \<exists>x. (x,k) \<in> p}) \<and> d fine q \<and>
  4.2470 +                   (\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q))"
  4.2471 +  { have *:"finite p" "p tagged_partial_division_of {a..b}" using assms(1) unfolding tagged_division_of_def by auto
  4.2472 +    presume "\<And>p. finite p \<Longrightarrow> ?P p" from this[rule_format,OF * assms(2)] guess q .. note q=this
  4.2473 +    thus ?thesis apply-apply(rule that[of q]) unfolding tagged_division_ofD[OF assms(1)] by auto
  4.2474 +  } fix p::"((real^'n) \<times> ((real^'n) set)) set" assume as:"finite p"
  4.2475 +  show "?P p" apply(rule,rule) using as proof(induct p) 
  4.2476 +    case empty show ?case apply(rule_tac x="{}" in exI) unfolding fine_def by auto
  4.2477 +  next case (insert xk p) guess x k using surj_pair[of xk] apply- by(erule exE)+ note xk=this
  4.2478 +    note tagged_partial_division_subset[OF insert(4) subset_insertI]
  4.2479 +    from insert(3)[OF this insert(5)] guess q1 .. note q1 = conjunctD3[OF this]
  4.2480 +    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
  4.2481 +    note p = tagged_partial_division_ofD[OF insert(4)]
  4.2482 +    from p(4)[unfolded xk, OF insertI1] guess u v apply-by(erule exE)+ note uv=this
  4.2483 +
  4.2484 +    have "finite {k. \<exists>x. (x, k) \<in> p}" 
  4.2485 +      apply(rule finite_subset[of _ "snd ` p"],rule) unfolding subset_eq image_iff mem_Collect_eq
  4.2486 +      apply(erule exE,rule_tac x="(xa,x)" in bexI) using p by auto
  4.2487 +    hence int:"interior {u..v} \<inter> interior (\<Union>{k. \<exists>x. (x, k) \<in> p}) = {}"
  4.2488 +      apply(rule inter_interior_unions_intervals) apply(rule open_interior) apply(rule_tac[!] ballI)
  4.2489 +      unfolding mem_Collect_eq apply(erule_tac[!] exE) apply(drule p(4)[OF insertI2],assumption)      
  4.2490 +      apply(rule p(5))  unfolding uv xk apply(rule insertI1,rule insertI2) apply assumption
  4.2491 +      using insert(2) unfolding uv xk by auto
  4.2492 +
  4.2493 +    show ?case proof(cases "{u..v} \<subseteq> d x")
  4.2494 +      case True thus ?thesis apply(rule_tac x="{(x,{u..v})} \<union> q1" in exI) apply rule
  4.2495 +        unfolding * uv apply(rule tagged_division_union,rule tagged_division_of_self)
  4.2496 +        apply(rule p[unfolded xk uv] insertI1)+  apply(rule q1,rule int) 
  4.2497 +        apply(rule,rule fine_union,subst fine_def) defer apply(rule q1)
  4.2498 +        unfolding Ball_def split_paired_All split_conv apply(rule,rule,rule,rule)
  4.2499 +        apply(erule insertE) defer apply(rule UnI2) apply(drule q1(3)[rule_format]) unfolding xk uv by auto
  4.2500 +    next case False from fine_division_exists[OF assms(2), of u v] guess q2 . note q2=this
  4.2501 +      show ?thesis apply(rule_tac x="q2 \<union> q1" in exI)
  4.2502 +        apply rule unfolding * uv apply(rule tagged_division_union q2 q1 int fine_union)+
  4.2503 +        unfolding Ball_def split_paired_All split_conv apply rule apply(rule fine_union)
  4.2504 +        apply(rule q1 q2)+ apply(rule,rule,rule,rule) apply(erule insertE)
  4.2505 +        apply(rule UnI2) defer apply(drule q1(3)[rule_format])using False unfolding xk uv by auto
  4.2506 +    qed qed qed
  4.2507 +
  4.2508 +subsection {* Hence the main theorem about negligible sets. *}
  4.2509 +
  4.2510 +lemma finite_product_dependent: assumes "finite s" "\<And>x. x\<in>s\<Longrightarrow> finite (t x)"
  4.2511 +  shows "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" using assms
  4.2512 +proof(induct) case (insert x s) 
  4.2513 +  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
  4.2514 +  show ?case unfolding * apply(rule finite_UnI) using insert by auto qed auto
  4.2515 +
  4.2516 +lemma sum_sum_product: assumes "finite s" "\<forall>i\<in>s. finite (t i)"
  4.2517 +  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
  4.2518 +proof(induct) case (insert a s)
  4.2519 +  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
  4.2520 +  show ?case unfolding * apply(subst setsum_Un_disjoint) unfolding setsum_insert[OF insert(1-2)]
  4.2521 +    prefer 4 apply(subst insert(3)) unfolding add_right_cancel
  4.2522 +  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
  4.2523 +    show "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" apply(rule finite_product_dependent) using insert by auto
  4.2524 +  qed(insert insert, auto) qed auto
  4.2525 +
  4.2526 +lemma has_integral_negligible: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
  4.2527 +  assumes "negligible s" "\<forall>x\<in>(t - s). f x = 0"
  4.2528 +  shows "(f has_integral 0) t"
  4.2529 +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})"
  4.2530 +  let ?f = "(\<lambda>x. if x \<in> t then f x else 0)"
  4.2531 +  show ?thesis apply(rule_tac f="?f" in has_integral_eq) apply(rule) unfolding if_P apply(rule refl)
  4.2532 +    apply(subst has_integral_alt) apply(cases,subst if_P,assumption) unfolding if_not_P
  4.2533 +  proof- assume "\<exists>a b. t = {a..b}" then guess a b apply-by(erule exE)+ note t = this
  4.2534 +    show "(?f has_integral 0) t" unfolding t apply(rule P) using assms(2) unfolding t by auto
  4.2535 +  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)"
  4.2536 +      apply(safe,rule_tac x=1 in exI,rule) apply(rule zero_less_one,safe) apply(rule_tac x=0 in exI)
  4.2537 +      apply(rule,rule P) using assms(2) by auto
  4.2538 +  qed
  4.2539 +next fix f::"real^'n \<Rightarrow> 'a" and a b::"real^'n" assume assm:"\<forall>x. x \<notin> s \<longrightarrow> f x = 0" 
  4.2540 +  show "(f has_integral 0) {a..b}" unfolding has_integral
  4.2541 +  proof(safe) case goal1
  4.2542 +    hence "\<And>n. e / 2 / ((real n+1) * (2 ^ n)) > 0" 
  4.2543 +      apply-apply(rule divide_pos_pos) defer apply(rule mult_pos_pos) by(auto simp add:field_simps)
  4.2544 +    note assms(1)[unfolded negligible_def has_integral,rule_format,OF this,of a b] note allI[OF this,of "\<lambda>x. x"] 
  4.2545 +    from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format]]
  4.2546 +    show ?case apply(rule_tac x="\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x" in exI) 
  4.2547 +    proof safe show "gauge (\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x)" using d(1) unfolding gauge_def by auto
  4.2548 +      fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x) fine p" 
  4.2549 +      let ?goal = "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e"
  4.2550 +      { presume "p\<noteq>{} \<Longrightarrow> ?goal" thus ?goal apply(cases "p={}") using goal1 by auto  }
  4.2551 +      assume as':"p \<noteq> {}" from real_arch_simple[of "Sup((\<lambda>(x,k). norm(f x)) ` p)"] guess N ..
  4.2552 +      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
  4.2553 +      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)"
  4.2554 +        apply(rule,rule tagged_division_finer[OF as(1) d(1)]) by auto
  4.2555 +      from choice[OF this] guess q .. note q=conjunctD3[OF this[rule_format]]
  4.2556 +      have *:"\<And>i. (\<Sum>(x, k)\<in>q i. content k *\<^sub>R indicator s x) \<ge> 0" apply(rule setsum_nonneg,safe) 
  4.2557 +        unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) apply(drule tagged_division_ofD(4)[OF q(1)]) by auto
  4.2558 +      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"
  4.2559 +      proof- case goal1 thus ?case apply-apply(rule setsum_le_included[of s t g snd f]) prefer 4
  4.2560 +          apply safe apply(erule_tac x=x in ballE) apply(erule exE) apply(rule_tac x="(xa,x)" in bexI) by auto qed
  4.2561 +      have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) \<le> setsum (\<lambda>i. (real i + 1) *
  4.2562 +                     norm(setsum (\<lambda>(x,k). content k *\<^sub>R indicator s x) (q i))) {0..N+1}"
  4.2563 +        unfolding real_norm_def setsum_right_distrib abs_of_nonneg[OF *] diff_0_right
  4.2564 +        apply(rule order_trans,rule setsum_norm) defer apply(subst sum_sum_product) prefer 3 
  4.2565 +      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
  4.2566 +        fix i a b assume as'':"(a,b) \<in> q i" show "0 \<le> (real i + 1) * (content b *\<^sub>R indicator s a)"
  4.2567 +          unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) defer apply(rule mult_nonneg_nonneg)
  4.2568 +          using tagged_division_ofD(4)[OF q(1) as''] by auto
  4.2569 +      next fix i::nat show "finite (q i)" using q by auto
  4.2570 +      next fix x k assume xk:"(x,k) \<in> p" def n \<equiv> "nat \<lfloor>norm (f x)\<rfloor>"
  4.2571 +        have *:"norm (f x) \<in> (\<lambda>(x, k). norm (f x)) ` p" using xk by auto
  4.2572 +        have nfx:"real n \<le> norm(f x)" "norm(f x) \<le> real n + 1" unfolding n_def by auto
  4.2573 +        hence "n \<in> {0..N + 1}" using N[rule_format,OF *] by auto
  4.2574 +        moreover  note as(2)[unfolded fine_def,rule_format,OF xk,unfolded split_conv]
  4.2575 +        note q(3)[rule_format,OF xk,unfolded split_conv,rule_format,OF this] note this[unfolded n_def[symmetric]]
  4.2576 +        moreover have "norm (content k *\<^sub>R f x) \<le> (real n + 1) * (content k * indicator s x)"
  4.2577 +        proof(cases "x\<in>s") case False thus ?thesis using assm by auto
  4.2578 +        next case True have *:"content k \<ge> 0" using tagged_division_ofD(4)[OF as(1) xk] by auto
  4.2579 +          moreover have "content k * norm (f x) \<le> content k * (real n + 1)" apply(rule mult_mono) using nfx * by auto
  4.2580 +          ultimately show ?thesis unfolding abs_mult using nfx True by(auto simp add:field_simps)
  4.2581 +        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)"
  4.2582 +          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
  4.2583 +      qed(insert as, auto)
  4.2584 +      also have "... \<le> setsum (\<lambda>i. e / 2 / 2 ^ i) {0..N+1}" apply(rule setsum_mono) 
  4.2585 +      proof- case goal1 thus ?case apply(subst mult_commute, subst pos_le_divide_eq[THEN sym])
  4.2586 +          using d(2)[rule_format,of "q i" i] using q[rule_format] by(auto simp add:field_simps)
  4.2587 +      qed also have "... < e * inverse 2 * 2" unfolding real_divide_def setsum_right_distrib[THEN sym]
  4.2588 +        apply(rule mult_strict_left_mono) unfolding power_inverse atLeastLessThanSuc_atLeastAtMost[THEN sym]
  4.2589 +        apply(subst sumr_geometric) using goal1 by auto
  4.2590 +      finally show "?goal" by auto qed qed qed
  4.2591 +
  4.2592 +lemma has_integral_spike: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
  4.2593 +  assumes "negligible s" "(\<forall>x\<in>(t - s). g x = f x)" "(f has_integral y) t"
  4.2594 +  shows "(g has_integral y) t"
  4.2595 +proof- { fix a b::"real^'n" and f g ::"real^'n \<Rightarrow> 'a" and y::'a
  4.2596 +    assume as:"\<forall>x \<in> {a..b} - s. g x = f x" "(f has_integral y) {a..b}"
  4.2597 +    have "((\<lambda>x. f x + (g x - f x)) has_integral (y + 0)) {a..b}" apply(rule has_integral_add[OF as(2)])
  4.2598 +      apply(rule has_integral_negligible[OF assms(1)]) using as by auto
  4.2599 +    hence "(g has_integral y) {a..b}" by auto } note * = this
  4.2600 +  show ?thesis apply(subst has_integral_alt) using assms(2-) apply-apply(rule cond_cases,safe)
  4.2601 +    apply(rule *, assumption+) apply(subst(asm) has_integral_alt) unfolding if_not_P
  4.2602 +    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)
  4.2603 +    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
  4.2604 +
  4.2605 +lemma has_integral_spike_eq:
  4.2606 +  assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x"
  4.2607 +  shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
  4.2608 +  apply rule apply(rule_tac[!] has_integral_spike[OF assms(1)]) using assms(2) by auto
  4.2609 +
  4.2610 +lemma integrable_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x" "f integrable_on t"
  4.2611 +  shows "g integrable_on  t"
  4.2612 +  using assms unfolding integrable_on_def apply-apply(erule exE)
  4.2613 +  apply(rule,rule has_integral_spike) by fastsimp+
  4.2614 +
  4.2615 +lemma integral_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x"
  4.2616 +  shows "integral t f = integral t g"
  4.2617 +  unfolding integral_def using has_integral_spike_eq[OF assms] by auto
  4.2618 +
  4.2619 +subsection {* Some other trivialities about negligible sets. *}
  4.2620 +
  4.2621 +lemma negligible_subset[intro]: assumes "negligible s" "t \<subseteq> s" shows "negligible t" unfolding negligible_def 
  4.2622 +proof(safe) case goal1 show ?case using assms(1)[unfolded negligible_def,rule_format,of a b]
  4.2623 +    apply-apply(rule has_integral_spike[OF assms(1)]) defer apply assumption
  4.2624 +    using assms(2) unfolding indicator_def by auto qed
  4.2625 +
  4.2626 +lemma negligible_diff[intro?]: assumes "negligible s" shows "negligible(s - t)" using assms by auto
  4.2627 +
  4.2628 +lemma negligible_inter: assumes "negligible s \<or> negligible t" shows "negligible(s \<inter> t)" using assms by auto
  4.2629 +
  4.2630 +lemma negligible_union: assumes "negligible s" "negligible t" shows "negligible (s \<union> t)" unfolding negligible_def 
  4.2631 +proof safe case goal1 note assm = assms[unfolded negligible_def,rule_format,of a b]
  4.2632 +  thus ?case apply(subst has_integral_spike_eq[OF assms(2)])
  4.2633 +    defer apply assumption unfolding indicator_def by auto qed
  4.2634 +
  4.2635 +lemma negligible_union_eq[simp]: "negligible (s \<union> t) \<longleftrightarrow> (negligible s \<and> negligible t)"
  4.2636 +  using negligible_union by auto
  4.2637 +
  4.2638 +lemma negligible_sing[intro]: "negligible {a::real^'n}" 
  4.2639 +proof- guess x using UNIV_witness[where 'a='n] ..
  4.2640 +  show ?thesis using negligible_standard_hyperplane[of x "a$x"] by auto qed
  4.2641 +
  4.2642 +lemma negligible_insert[simp]: "negligible(insert a s) \<longleftrightarrow> negligible s"
  4.2643 +  apply(subst insert_is_Un) unfolding negligible_union_eq by auto
  4.2644 +
  4.2645 +lemma negligible_empty[intro]: "negligible {}" by auto
  4.2646 +
  4.2647 +lemma negligible_finite[intro]: assumes "finite s" shows "negligible s"
  4.2648 +  using assms apply(induct s) by auto
  4.2649 +
  4.2650 +lemma negligible_unions[intro]: assumes "finite s" "\<forall>t\<in>s. negligible t" shows "negligible(\<Union>s)"
  4.2651 +  using assms by(induct,auto) 
  4.2652 +
  4.2653 +lemma negligible:  "negligible s \<longleftrightarrow> (\<forall>t::(real^'n) set. (indicator s has_integral 0) t)"
  4.2654 +  apply safe defer apply(subst negligible_def)
  4.2655 +proof- fix t::"(real^'n) set" assume as:"negligible s"
  4.2656 +  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)  
  4.2657 +  show "(indicator s has_integral 0) t" apply(subst has_integral_alt)
  4.2658 +    apply(cases,subst if_P,assumption) unfolding if_not_P apply(safe,rule as[unfolded negligible_def,rule_format])
  4.2659 +    apply(rule_tac x=1 in exI) apply(safe,rule zero_less_one) apply(rule_tac x=0 in exI)
  4.2660 +    using negligible_subset[OF as,of "s \<inter> t"] unfolding negligible_def indicator_def unfolding * by auto qed auto
  4.2661 +
  4.2662 +subsection {* Finite case of the spike theorem is quite commonly needed. *}
  4.2663 +
  4.2664 +lemma has_integral_spike_finite: assumes "finite s" "\<forall>x\<in>t-s. g x = f x" 
  4.2665 +  "(f has_integral y) t" shows "(g has_integral y) t"
  4.2666 +  apply(rule has_integral_spike) using assms by auto
  4.2667 +
  4.2668 +lemma has_integral_spike_finite_eq: assumes "finite s" "\<forall>x\<in>t-s. g x = f x"
  4.2669 +  shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
  4.2670 +  apply rule apply(rule_tac[!] has_integral_spike_finite) using assms by auto
  4.2671 +
  4.2672 +lemma integrable_spike_finite:
  4.2673 +  assumes "finite s" "\<forall>x\<in>t-s. g x = f x" "f integrable_on t" shows "g integrable_on  t"
  4.2674 +  using assms unfolding integrable_on_def apply safe apply(rule_tac x=y in exI)
  4.2675 +  apply(rule has_integral_spike_finite) by auto
  4.2676 +
  4.2677 +subsection {* In particular, the boundary of an interval is negligible. *}
  4.2678 +
  4.2679 +lemma negligible_frontier_interval: "negligible({a..b} - {a<..<b})"
  4.2680 +proof- let ?A = "\<Union>((\<lambda>k. {x. x$k = a$k} \<union> {x. x$k = b$k}) ` UNIV)"
  4.2681 +  have "{a..b} - {a<..<b} \<subseteq> ?A" apply rule unfolding Diff_iff mem_interval not_all
  4.2682 +    apply(erule conjE exE)+ apply(rule_tac X="{x. x $ xa = a $ xa} \<union> {x. x $ xa = b $ xa}" in UnionI)
  4.2683 +    apply(erule_tac[!] x=xa in allE) by auto
  4.2684 +  thus ?thesis apply-apply(rule negligible_subset[of ?A]) apply(rule negligible_unions[OF finite_imageI]) by auto qed
  4.2685 +
  4.2686 +lemma has_integral_spike_interior:
  4.2687 +  assumes "\<forall>x\<in>{a<..<b}. g x = f x" "(f has_integral y) ({a..b})" shows "(g has_integral y) ({a..b})"
  4.2688 +  apply(rule has_integral_spike[OF negligible_frontier_interval _ assms(2)]) using assms(1) by auto
  4.2689 +
  4.2690 +lemma has_integral_spike_interior_eq:
  4.2691 +  assumes "\<forall>x\<in>{a<..<b}. g x = f x" shows "((f has_integral y) ({a..b}) \<longleftrightarrow> (g has_integral y) ({a..b}))"
  4.2692 +  apply rule apply(rule_tac[!] has_integral_spike_interior) using assms by auto
  4.2693 +
  4.2694 +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}"
  4.2695 +  using  assms unfolding integrable_on_def using has_integral_spike_interior[OF assms(1)] by auto
  4.2696 +
  4.2697 +subsection {* Integrability of continuous functions. *}
  4.2698 +
  4.2699 +lemma neutral_and[simp]: "neutral op \<and> = True"
  4.2700 +  unfolding neutral_def apply(rule some_equality) by auto
  4.2701 +
  4.2702 +lemma monoidal_and[intro]: "monoidal op \<and>" unfolding monoidal_def by auto
  4.2703 +
  4.2704 +lemma iterate_and[simp]: assumes "finite s" shows "(iterate op \<and>) s p \<longleftrightarrow> (\<forall>x\<in>s. p x)" using assms
  4.2705 +apply induct unfolding iterate_insert[OF monoidal_and] by auto
  4.2706 +
  4.2707 +lemma operative_division_and: assumes "operative op \<and> P" "d division_of {a..b}"
  4.2708 +  shows "(\<forall>i\<in>d. P i) \<longleftrightarrow> P {a..b}"
  4.2709 +  using operative_division[OF monoidal_and assms] division_of_finite[OF assms(2)] by auto
  4.2710 +
  4.2711 +lemma operative_approximable: assumes "0 \<le> e" fixes f::"real^'n \<Rightarrow> 'a::banach"
  4.2712 +  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
  4.2713 +proof safe fix a b::"real^'n" { assume "content {a..b} = 0"
  4.2714 +    thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" 
  4.2715 +      apply(rule_tac x=f in exI) using assms by(auto intro!:integrable_on_null) }
  4.2716 +  { fix c k g assume as:"\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}"
  4.2717 +    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}"
  4.2718 +      "\<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}"
  4.2719 +      apply(rule_tac[!] x=g in exI) using as(1) integrable_split[OF as(2)] by auto }
  4.2720 +  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}"
  4.2721 +                          "\<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}"
  4.2722 +  let ?g = "\<lambda>x. if x$k = c then f x else if x$k \<le> c then g1 x else g2 x"
  4.2723 +  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)
  4.2724 +  proof safe case goal1 thus ?case apply- apply(cases "x$k=c", case_tac "x$k < c") using as assms by auto
  4.2725 +  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}"
  4.2726 +    then guess h1 h2 unfolding integrable_on_def by auto from has_integral_split[OF this]
  4.2727 +    show ?case unfolding integrable_on_def by auto
  4.2728 +  next show "?g integrable_on {a..b} \<inter> {x. x $ k \<le> c}" "?g integrable_on {a..b} \<inter> {x. x $ k \<ge> c}"
  4.2729 +      apply(rule_tac[!] integrable_spike[OF negligible_standard_hyperplane[of k c]]) using as(2,4) by auto qed qed
  4.2730 +
  4.2731 +lemma approximable_on_division: fixes f::"real^'n \<Rightarrow> 'a::banach"
  4.2732 +  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"
  4.2733 +  obtains g where "\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}"
  4.2734 +proof- note * = operative_division[OF monoidal_and operative_approximable[OF assms(1)] assms(2)]
  4.2735 +  note this[unfolded iterate_and[OF division_of_finite[OF assms(2)]]] from assms(3)[unfolded this[of f]]
  4.2736 +  guess g .. thus thesis apply-apply(rule that[of g]) by auto qed
  4.2737 +
  4.2738 +lemma integrable_continuous: fixes f::"real^'n \<Rightarrow> 'a::banach"
  4.2739 +  assumes "continuous_on {a..b} f" shows "f integrable_on {a..b}"
  4.2740 +proof(rule integrable_uniform_limit,safe) fix e::real assume e:"0 < e"
  4.2741 +  from compact_uniformly_continuous[OF assms compact_interval,unfolded uniformly_continuous_on_def,rule_format,OF e] guess d ..
  4.2742 +  note d=conjunctD2[OF this,rule_format]
  4.2743 +  from fine_division_exists[OF gauge_ball[OF d(1)], of a b] guess p . note p=this
  4.2744 +  note p' = tagged_division_ofD[OF p(1)]
  4.2745 +  have *:"\<forall>i\<in>snd ` p. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
  4.2746 +  proof(safe,unfold snd_conv) fix x l assume as:"(x,l) \<in> p" 
  4.2747 +    from p'(4)[OF this] guess a b apply-by(erule exE)+ note l=this
  4.2748 +    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)
  4.2749 +    proof safe show "(\<lambda>y. f x) integrable_on l" unfolding integrable_on_def l by(rule,rule has_integral_const)
  4.2750 +      fix y assume y:"y\<in>l" note fineD[OF p(2) as,unfolded subset_eq,rule_format,OF this]
  4.2751 +      note d(2)[OF _ _ this[unfolded mem_ball]]
  4.2752 +      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
  4.2753 +  from e have "0 \<le> e" by auto from approximable_on_division[OF this division_of_tagged_division[OF p(1)] *] guess g .
  4.2754 +  thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" by auto qed 
  4.2755 +
  4.2756 +subsection {* Specialization of additivity to one dimension. *}
  4.2757 +
  4.2758 +lemma operative_1_lt: assumes "monoidal opp"
  4.2759 +  shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real^1} = neutral opp) \<and>
  4.2760 +                (\<forall>a b c. a < c \<and> c < b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))"
  4.2761 +  unfolding operative_def content_eq_0_1 forall_1 vector_le_def vector_less_def
  4.2762 +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"
  4.2763 +    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
  4.2764 +    thus "opp (f {a..c}) (f {c..b}) = f {a..b}" unfolding as(1)[rule_format,of a b "c$1"] by auto
  4.2765 +next fix a b::"real^1" and c::real
  4.2766 +  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}"
  4.2767 +  show "f {a..b} = opp (f ({a..b} \<inter> {x. x $ 1 \<le> c})) (f ({a..b} \<inter> {x. c \<le> x $ 1}))"
  4.2768 +  proof(cases "c \<in> {a$1 .. b$1}")
  4.2769 +    case False hence "c<a$1 \<or> c>b$1" by auto
  4.2770 +    thus ?thesis apply-apply(erule disjE)
  4.2771 +    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
  4.2772 +      show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto
  4.2773 +    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
  4.2774 +      show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto
  4.2775 +    qed
  4.2776 +  next case True hence *:"min (b $ 1) c = c" "max (a $ 1) c = c" by auto
  4.2777 +    show ?thesis unfolding interval_split num1_eq_iff if_True * vec_def[THEN sym]
  4.2778 +    proof(cases "c = a$1 \<or> c = b$1")
  4.2779 +      case False thus "f {a..b} = opp (f {a..vec1 c}) (f {vec1 c..b})"
  4.2780 +        apply-apply(subst as(2)[rule_format]) using True by auto
  4.2781 +    next case True thus "f {a..b} = opp (f {a..vec1 c}) (f {vec1 c..b})" apply-
  4.2782 +      proof(erule disjE) assume "c=a$1" hence *:"a = vec1 c" unfolding Cart_eq by auto 
  4.2783 +        hence "f {a..vec1 c} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
  4.2784 +        thus ?thesis using assms unfolding * by auto
  4.2785 +      next assume "c=b$1" hence *:"b = vec1 c" unfolding Cart_eq by auto 
  4.2786 +        hence "f {vec1 c..b} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
  4.2787 +        thus ?thesis using assms unfolding * by auto qed qed qed qed
  4.2788 +
  4.2789 +lemma operative_1_le: assumes "monoidal opp"
  4.2790 +  shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real^1} = neutral opp) \<and>
  4.2791 +                (\<forall>a b c. a \<le> c \<and> c \<le> b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))"
  4.2792 +unfolding operative_1_lt[OF assms]
  4.2793 +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"
  4.2794 +  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
  4.2795 +next fix a b c ::"real^1"
  4.2796 +  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"
  4.2797 +  note as = this[rule_format]
  4.2798 +  show "opp (f {a..c}) (f {c..b}) = f {a..b}"
  4.2799 +  proof(cases "c = a \<or> c = b")
  4.2800 +    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)
  4.2801 +    next case True thus ?thesis apply-
  4.2802 +      proof(erule disjE) assume *:"c=a" hence "f {a..c} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
  4.2803 +        thus ?thesis using assms unfolding * by auto
  4.2804 +      next               assume *:"c=b" hence "f {c..b} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
  4.2805 +        thus ?thesis using assms unfolding * by auto qed qed qed 
  4.2806 +
  4.2807 +subsection {* Special case of additivity we need for the FCT. *}
  4.2808 +
  4.2809 +lemma additive_tagged_division_1: fixes f::"real^1 \<Rightarrow> 'a::real_normed_vector"
  4.2810 +  assumes "dest_vec1 a \<le> dest_vec1 b" "p tagged_division_of {a..b}"
  4.2811 +  shows "setsum (\<lambda>(x,k). f(interval_upperbound k) - f(interval_lowerbound k)) p = f b - f a"
  4.2812 +proof- let ?f = "(\<lambda>k::(real^1) set. if k = {} then 0 else f(interval_upperbound k) - f(interval_lowerbound k))"
  4.2813 +  have *:"operative op + ?f" unfolding operative_1_lt[OF monoidal_monoid] interval_eq_empty_1
  4.2814 +    by(auto simp add:not_less interval_bound_1 vector_less_def)
  4.2815 +  have **:"{a..b} \<noteq> {}" using assms(1) by auto note operative_tagged_division[OF monoidal_monoid * assms(2)]
  4.2816 +  note * = this[unfolded if_not_P[OF **] interval_bound_1[OF assms(1)],THEN sym ]
  4.2817 +  show ?thesis unfolding * apply(subst setsum_iterate[THEN sym]) defer
  4.2818 +    apply(rule setsum_cong2) unfolding split_paired_all split_conv using assms(2) by auto qed
  4.2819 +
  4.2820 +subsection {* A useful lemma allowing us to factor out the content size. *}
  4.2821 +
  4.2822 +lemma has_integral_factor_content:
  4.2823 +  "(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
  4.2824 +    \<longrightarrow> norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content {a..b}))"
  4.2825 +proof(cases "content {a..b} = 0")
  4.2826 +  case True show ?thesis unfolding has_integral_null_eq[OF True] apply safe
  4.2827 +    apply(rule,rule,rule gauge_trivial,safe) unfolding setsum_content_null[OF True] True defer 
  4.2828 +    apply(erule_tac x=1 in allE,safe) defer apply(rule fine_division_exists[of _ a b],assumption)
  4.2829 +    apply(erule_tac x=p in allE) unfolding setsum_content_null[OF True] by auto
  4.2830 +next case False note F = this[unfolded content_lt_nz[THEN sym]]
  4.2831 +  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)"
  4.2832 +  show ?thesis apply(subst has_integral)
  4.2833 +  proof safe fix e::real assume e:"e>0"
  4.2834 +    { 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)
  4.2835 +        apply(erule impE) defer apply(erule exE,rule_tac x=d in exI)
  4.2836 +        using F e by(auto simp add:field_simps intro:mult_pos_pos) }
  4.2837 +    {  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)
  4.2838 +        apply(erule impE) defer apply(erule exE,rule_tac x=d in exI)
  4.2839 +        using F e by(auto simp add:field_simps intro:mult_pos_pos) } qed qed
  4.2840 +
  4.2841 +subsection {* Fundamental theorem of calculus. *}
  4.2842 +
  4.2843 +lemma fundamental_theorem_of_calculus: fixes f::"real^1 \<Rightarrow> 'a::banach"
  4.2844 +  assumes "a \<le> b"  "\<forall>x\<in>{a..b}. ((f o vec1) has_vector_derivative f'(vec1 x)) (at x within {a..b})"
  4.2845 +  shows "(f' has_integral (f(vec1 b) - f(vec1 a))) ({vec1 a..vec1 b})"
  4.2846 +unfolding has_integral_factor_content
  4.2847 +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
  4.2848 +  note assm = assms(2)[unfolded has_vector_derivative_def has_derivative_within_alt]
  4.2849 +  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
  4.2850 +  note this[OF assm,unfolded gauge_existence_lemma] from choice[OF this,unfolded Ball_def[symmetric]]
  4.2851 +  guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
  4.2852 +  show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {vec1 a..vec1 b} \<and> d fine p \<longrightarrow>
  4.2853 +                 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})"
  4.2854 +    apply(rule_tac x="\<lambda>x. ball x (d (dest_vec1 x))" in exI,safe)
  4.2855 +    apply(rule gauge_ball_dependent,rule,rule d(1))
  4.2856 +  proof- fix p assume as:"p tagged_division_of {vec1 a..vec1 b}" "(\<lambda>x. ball x (d (dest_vec1 x))) fine p"
  4.2857 +    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}" 
  4.2858 +      unfolding content_1[OF ab] additive_tagged_division_1[OF ab as(1),of f,THEN sym]
  4.2859 +      unfolding vector_minus_component[THEN sym] additive_tagged_division_1[OF ab as(1),of "\<lambda>x. x",THEN sym]
  4.2860 +      apply(subst dest_vec1_setsum) unfolding setsum_right_distrib defer unfolding setsum_subtractf[THEN sym] 
  4.2861 +    proof(rule setsum_norm_le,safe) fix x k assume "(x,k)\<in>p"
  4.2862 +      note xk = tagged_division_ofD(2-4)[OF as(1) this] from this(3) guess u v apply-by(erule exE)+ note k=this
  4.2863 +      have *:"dest_vec1 u \<le> dest_vec1 v" using xk unfolding k by auto
  4.2864 +      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] .
  4.2865 +      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)"
  4.2866 +        apply(rule order_trans[OF _ norm_triangle_ineq4]) apply(rule eq_refl) apply(rule arg_cong[where f=norm])
  4.2867 +        unfolding scaleR.diff_left by(auto simp add:group_simps)
  4.2868 +      also have "... \<le> e * norm (dest_vec1 u - dest_vec1 x) + e * norm (dest_vec1 v - dest_vec1 x)"
  4.2869 +        apply(rule add_mono) apply(rule d(2)[of "x$1" "u$1",unfolded o_def vec1_dest_vec1]) prefer 4
  4.2870 +        apply(rule d(2)[of "x$1" "v$1",unfolded o_def vec1_dest_vec1])
  4.2871 +        using ball[rule_format,of u] ball[rule_format,of v] 
  4.2872 +        using xk(1-2) unfolding k subset_eq by(auto simp add:vector_dist_norm norm_real)
  4.2873 +      also have "... \<le> e * dest_vec1 (interval_upperbound k - interval_lowerbound k)"
  4.2874 +        unfolding k interval_bound_1[OF *] using xk(1) unfolding k by(auto simp add:vector_dist_norm norm_real field_simps)
  4.2875 +      finally show "norm (content k *\<^sub>R f' x - (f (interval_upperbound k) - f (interval_lowerbound k))) \<le>
  4.2876 +        e * dest_vec1 (interval_upperbound k - interval_lowerbound k)" unfolding k interval_bound_1[OF *] content_1[OF *] .
  4.2877 +    qed(insert as, auto) qed qed
  4.2878 +
  4.2879 +subsection {* Attempt a systematic general set of "offset" results for components. *}
  4.2880 +
  4.2881 +lemma gauge_modify:
  4.2882 +  assumes "(\<forall>s. open s \<longrightarrow> open {x. f(x) \<in> s})" "gauge d"
  4.2883 +  shows "gauge (\<lambda>x y. d (f x) (f y))"
  4.2884 +  using assms unfolding gauge_def apply safe defer apply(erule_tac x="f x" in allE)
  4.2885 +  apply(erule_tac x="d (f x)" in allE) unfolding mem_def Collect_def by auto
  4.2886 +
  4.2887 +subsection {* Only need trivial subintervals if the interval itself is trivial. *}
  4.2888 +
  4.2889 +lemma division_of_nontrivial: fixes s::"(real^'n) set set"
  4.2890 +  assumes "s division_of {a..b}" "content({a..b}) \<noteq> 0"
  4.2891 +  shows "{k. k \<in> s \<and> content k \<noteq> 0} division_of {a..b}" using assms(1) apply-
  4.2892 +proof(induct "card s" arbitrary:s rule:nat_less_induct)
  4.2893 +  fix s::"(real^'n) set set" assume assm:"s division_of {a..b}"
  4.2894 +    "\<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}" 
  4.2895 +  note s = division_ofD[OF assm(1)] let ?thesis = "{k \<in> s. content k \<noteq> 0} division_of {a..b}"
  4.2896 +  { presume *:"{k \<in> s. content k \<noteq> 0} \<noteq> s \<Longrightarrow> ?thesis"
  4.2897 +    show ?thesis apply cases defer apply(rule *,assumption) using assm(1) by auto }
  4.2898 +  assume noteq:"{k \<in> s. content k \<noteq> 0} \<noteq> s"
  4.2899 +  then obtain k where k:"k\<in>s" "content k = 0" by auto
  4.2900 +  from s(4)[OF k(1)] guess c d apply-by(erule exE)+ note k=k this
  4.2901 +  from k have "card s > 0" unfolding card_gt_0_iff using assm(1) by auto
  4.2902 +  hence card:"card (s - {k}) < card s" using assm(1) k(1) apply(subst card_Diff_singleton_if) by auto
  4.2903 +  have *:"closed (\<Union>(s - {k}))" apply(rule closed_Union) defer apply rule apply(drule DiffD1,drule s(4))
  4.2904 +    apply safe apply(rule closed_interval) using assm(1) by auto
  4.2905 +  have "k \<subseteq> \<Union>(s - {k})" apply safe apply(rule *[unfolded closed_limpt,rule_format]) unfolding islimpt_approachable
  4.2906 +  proof safe fix x and e::real assume as:"x\<in>k" "e>0"
  4.2907 +    from k(2)[unfolded k content_eq_0] guess i .. 
  4.2908 +    hence i:"c$i = d$i" using s(3)[OF k(1),unfolded k] unfolding interval_ne_empty by smt
  4.2909 +    hence xi:"x$i = d$i" using as unfolding k mem_interval by smt
  4.2910 +    def y \<equiv> "(\<chi> j. if j = i then if c$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$j)"
  4.2911 +    show "\<exists>x'\<in>\<Union>(s - {k}). x' \<noteq> x \<and> dist x' x < e" apply(rule_tac x=y in bexI) 
  4.2912 +    proof have "d \<in> {c..d}" using s(3)[OF k(1)] unfolding k interval_eq_empty mem_interval by(fastsimp simp add: not_less)
  4.2913 +      hence "d \<in> {a..b}" using s(2)[OF k(1)] unfolding k by auto note di = this[unfolded mem_interval,THEN spec[where x=i]]
  4.2914 +      hence xyi:"y$i \<noteq> x$i" unfolding y_def unfolding i xi Cart_lambda_beta if_P[OF refl]
  4.2915 +        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+ 
  4.2916 +      thus "y \<noteq> x" unfolding Cart_eq by auto
  4.2917 +      have *:"UNIV = insert i (UNIV - {i})" by auto
  4.2918 +      have "norm (y - x) < e + setsum (\<lambda>i. 0) (UNIV::'n set)" apply(rule le_less_trans[OF norm_le_l1])
  4.2919 +        apply(subst *,subst setsum_insert) prefer 3 apply(rule add_less_le_mono)
  4.2920 +      proof- show "\<bar>(y - x) $ i\<bar> < e" unfolding y_def Cart_lambda_beta vector_minus_component if_P[OF refl]
  4.2921 +          apply(cases) apply(subst if_P,assumption) unfolding if_not_P unfolding i xi using di as(2) by auto
  4.2922 +        show "(\<Sum>i\<in>UNIV - {i}. \<bar>(y - x) $ i\<bar>) \<le> (\<Sum>i\<in>UNIV. 0)" unfolding y_def by auto 
  4.2923 +      qed auto thus "dist y x < e" unfolding vector_dist_norm by auto
  4.2924 +      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
  4.2925 +      moreover have "y \<in> \<Union>s" unfolding s mem_interval
  4.2926 +      proof note simps = y_def Cart_lambda_beta if_not_P
  4.2927 +        fix j::'n show "a $ j \<le> y $ j \<and> y $ j \<le> b $ j" 
  4.2928 +        proof(cases "j = i") case False have "x \<in> {a..b}" using s(2)[OF k(1)] as(1) by auto
  4.2929 +          thus ?thesis unfolding simps if_not_P[OF False] unfolding mem_interval by auto
  4.2930 +        next case True note T = this show ?thesis
  4.2931 +          proof(cases "c $ i \<le> (a $ i + b $ i) / 2")
  4.2932 +            case True show ?thesis unfolding simps if_P[OF T] if_P[OF True] unfolding i
  4.2933 +              using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps) 
  4.2934 +          next case False thus ?thesis unfolding simps if_P[OF T] if_not_P[OF False] unfolding i
  4.2935 +              using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps)
  4.2936 +          qed qed qed
  4.2937 +      ultimately show "y \<in> \<Union>(s - {k})" by auto
  4.2938 +    qed qed hence "\<Union>(s - {k}) = {a..b}" unfolding s(6)[THEN sym] by auto
  4.2939 +  hence  "{ka \<in> s - {k}. content ka \<noteq> 0} division_of {a..b}" apply-apply(rule assm(2)[rule_format,OF card refl])
  4.2940 +    apply(rule division_ofI) defer apply(rule_tac[1-4] s) using assm(1) by auto
  4.2941 +  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
  4.2942 +
  4.2943 +subsection {* Integrabibility on subintervals. *}
  4.2944 +
  4.2945 +lemma operative_integrable: fixes f::"real^'n \<Rightarrow> 'a::banach" shows
  4.2946 +  "operative op \<and> (\<lambda>i. f integrable_on i)"
  4.2947 +  unfolding operative_def neutral_and apply safe apply(subst integrable_on_def)
  4.2948 +  unfolding has_integral_null_eq apply(rule,rule refl) apply(rule,assumption)+
  4.2949 +  unfolding integrable_on_def by(auto intro: has_integral_split)
  4.2950 +
  4.2951 +lemma integrable_subinterval: fixes f::"real^'n \<Rightarrow> 'a::banach" 
  4.2952 +  assumes "f integrable_on {a..b}" "{c..d} \<subseteq> {a..b}" shows "f integrable_on {c..d}" 
  4.2953 +  apply(cases "{c..d} = {}") defer apply(rule partial_division_extend_1[OF assms(2)],assumption)
  4.2954 +  using operative_division_and[OF operative_integrable,THEN sym,of _ _ _ f] assms(1) by auto
  4.2955 +
  4.2956 +subsection {* Combining adjacent intervals in 1 dimension. *}
  4.2957 +
  4.2958 +lemma has_integral_combine: assumes "(a::real^1) \<le> c" "c \<le> b"
  4.2959 +  "(f has_integral i) {a..c}" "(f has_integral (j::'a::banach)) {c..b}"
  4.2960 +  shows "(f has_integral (i + j)) {a..b}"
  4.2961 +proof- note operative_integral[of f, unfolded operative_1_le[OF monoidal_lifted[OF monoidal_monoid]]]
  4.2962 +  note conjunctD2[OF this,rule_format] note * = this(2)[OF conjI[OF assms(1-2)],unfolded if_P[OF assms(3)]]
  4.2963 +  hence "f integrable_on {a..b}" apply- apply(rule ccontr) apply(subst(asm) if_P) defer
  4.2964 +    apply(subst(asm) if_P) using assms(3-) by auto
  4.2965 +  with * show ?thesis apply-apply(subst(asm) if_P) defer apply(subst(asm) if_P) defer apply(subst(asm) if_P)
  4.2966 +    unfolding lifted.simps using assms(3-) by(auto simp add: integrable_on_def integral_unique) qed
  4.2967 +
  4.2968 +lemma integral_combine: fixes f::"real^1 \<Rightarrow> 'a::banach"
  4.2969 +  assumes "a \<le> c" "c \<le> b" "f integrable_on ({a..b})"
  4.2970 +  shows "integral {a..c} f + integral {c..b} f = integral({a..b}) f"
  4.2971 +  apply(rule integral_unique[THEN sym]) apply(rule has_integral_combine[OF assms(1-2)])
  4.2972 +  apply(rule_tac[!] integrable_integral integrable_subinterval[OF assms(3)])+ using assms(1-2) by auto
  4.2973 +
  4.2974 +lemma integrable_combine: fixes f::"real^1 \<Rightarrow> 'a::banach"
  4.2975 +  assumes "a \<le> c" "c \<le> b" "f integrable_on {a..c}" "f integrable_on {c..b}"
  4.2976 +  shows "f integrable_on {a..b}" using assms unfolding integrable_on_def by(fastsimp intro!:has_integral_combine)
  4.2977 +
  4.2978 +subsection {* Reduce integrability to "local" integrability. *}
  4.2979 +
  4.2980 +lemma integrable_on_little_subintervals: fixes f::"real^'n \<Rightarrow> 'a::banach"
  4.2981 +  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}"
  4.2982 +  shows "f integrable_on {a..b}"
  4.2983 +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})"
  4.2984 +    using assms by auto note this[unfolded gauge_existence_lemma] from choice[OF this] guess d .. note d=this[rule_format]
  4.2985 +  guess p apply(rule fine_division_exists[OF gauge_ball_dependent,of d a b]) using d by auto note p=this(1-2)
  4.2986 +  note division_of_tagged_division[OF this(1)] note * = operative_division_and[OF operative_integrable,OF this,THEN sym,of f]
  4.2987 +  show ?thesis unfolding * apply safe unfolding snd_conv
  4.2988 +  proof- fix x k assume "(x,k) \<in> p" note tagged_division_ofD(2-4)[OF p(1) this] fineD[OF p(2) this]
  4.2989 +    thus "f integrable_on k" apply safe apply(rule d[THEN conjunct2,rule_format,of x]) by auto qed qed
  4.2990 +
  4.2991 +subsection {* Second FCT or existence of antiderivative. *}
  4.2992 +
  4.2993 +lemma integrable_const[intro]:"(\<lambda>x. c) integrable_on {a..b}"
  4.2994 +  unfolding integrable_on_def by(rule,rule has_integral_const)
  4.2995 +
  4.2996 +lemma integral_has_vector_derivative: fixes f::"real \<Rightarrow> 'a::banach"
  4.2997 +  assumes "continuous_on {a..b} f" "x \<in> {a..b}"
  4.2998 +  shows "((\<lambda>u. integral {vec a..vec u} (f o dest_vec1)) has_vector_derivative f(x)) (at x within {a..b})"
  4.2999 +  unfolding has_vector_derivative_def has_derivative_within_alt
  4.3000 +apply safe apply(rule scaleR.bounded_linear_left)
  4.3001 +proof- fix e::real assume e:"e>0"
  4.3002 +  note compact_uniformly_continuous[OF assms(1) compact_real_interval,unfolded uniformly_continuous_on_def]
  4.3003 +  from this[rule_format,OF e] guess d apply-by(erule conjE exE)+ note d=this[rule_format]
  4.3004 +  let ?I = "\<lambda>a b. integral {vec1 a..vec1 b} (f \<circ> dest_vec1)"
  4.3005 +  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)"
  4.3006 +  proof(rule,rule,rule d,safe) case goal1 show ?case proof(cases "y < x")
  4.3007 +      case False have "f \<circ> dest_vec1 integrable_on {vec1 a..vec1 y}" apply(rule integrable_subinterval,rule integrable_continuous)
  4.3008 +        apply(rule continuous_on_o_dest_vec1 assms)+  unfolding not_less using assms(2) goal1 by auto
  4.3009 +      hence *:"?I a y - ?I a x = ?I x y" unfolding group_simps apply(subst eq_commute) apply(rule integral_combine)
  4.3010 +        using False unfolding not_less using assms(2) goal1 by auto
  4.3011 +      have **:"norm (y - x) = content {vec1 x..vec1 y}" apply(subst content_1) using False unfolding not_less by auto
  4.3012 +      show ?thesis unfolding ** apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x) o dest_vec1"]) unfolding * unfolding o_def
  4.3013 +        defer apply(rule has_integral_sub) apply(rule integrable_integral)
  4.3014 +        apply(rule integrable_subinterval,rule integrable_continuous) apply(rule continuous_on_o_dest_vec1[unfolded o_def] assms)+
  4.3015 +      proof- show "{vec1 x..vec1 y} \<subseteq> {vec1 a..vec1 b}" using goal1 assms(2) by auto
  4.3016 +        have *:"y - x = norm(y - x)" using False by auto
  4.3017 +        show "((\<lambda>xa. f x) has_integral (y - x) *\<^sub>R f x) {vec1 x..vec1 y}" apply(subst *) unfolding ** by auto
  4.3018 +        show "\<forall>xa\<in>{vec1 x..vec1 y}. norm (f (dest_vec1 xa) - f x) \<le> e" apply safe apply(rule less_imp_le)
  4.3019 +          apply(rule d(2)[unfolded vector_dist_norm]) using assms(2) using goal1 by auto
  4.3020 +      qed(insert e,auto)
  4.3021 +    next case True have "f \<circ> dest_vec1 integrable_on {vec1 a..vec1 x}" apply(rule integrable_subinterval,rule integrable_continuous)
  4.3022 +        apply(rule continuous_on_o_dest_vec1 assms)+  unfolding not_less using assms(2) goal1 by auto
  4.3023 +      hence *:"?I a x - ?I a y = ?I y x" unfolding group_simps apply(subst eq_commute) apply(rule integral_combine)
  4.3024 +        using True using assms(2) goal1 by auto
  4.3025 +      have **:"norm (y - x) = content {vec1 y..vec1 x}" apply(subst content_1) using True unfolding not_less by auto
  4.3026 +      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 
  4.3027 +      show ?thesis apply(subst ***) unfolding norm_minus_cancel **
  4.3028 +        apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x) o dest_vec1"]) unfolding * unfolding o_def
  4.3029 +        defer apply(rule has_integral_sub) apply(subst minus_minus[THEN sym]) unfolding minus_minus
  4.3030 +        apply(rule integrable_integral) apply(rule integrable_subinterval,rule integrable_continuous)
  4.3031 +        apply(rule continuous_on_o_dest_vec1[unfolded o_def] assms)+
  4.3032 +      proof- show "{vec1 y..vec1 x} \<subseteq> {vec1 a..vec1 b}" using goal1 assms(2) by auto
  4.3033 +        have *:"x - y = norm(y - x)" using True by auto
  4.3034 +        show "((\<lambda>xa. f x) has_integral (x - y) *\<^sub>R f x) {vec1 y..vec1 x}" apply(subst *) unfolding ** by auto
  4.3035 +        show "\<forall>xa\<in>{vec1 y..vec1 x}. norm (f (dest_vec1 xa) - f x) \<le> e" apply safe apply(rule less_imp_le)
  4.3036 +          apply(rule d(2)[unfolded vector_dist_norm]) using assms(2) using goal1 by auto
  4.3037 +      qed(insert e,auto) qed qed qed
  4.3038 +
  4.3039 +lemma integral_has_vector_derivative': fixes f::"real^1 \<Rightarrow> 'a::banach"
  4.3040 +  assumes "continuous_on {a..b} f" "x \<in> {a..b}"
  4.3041 +  shows "((\<lambda>u. (integral {a..vec u} f)) has_vector_derivative f x) (at (x$1) within {a$1..b$1})"
  4.3042 +  using integral_has_vector_derivative[OF continuous_on_o_vec1[OF assms(1)], of "x$1"]
  4.3043 +  unfolding o_def vec1_dest_vec1 using assms(2) by auto
  4.3044 +
  4.3045 +lemma antiderivative_continuous: assumes "continuous_on {a..b::real} f"
  4.3046 +  obtains g where "\<forall>x\<in> {a..b}. (g has_vector_derivative (f(x)::_::banach)) (at x within {a..b})"
  4.3047 +  apply(rule that,rule) using integral_has_vector_derivative[OF assms] by auto
  4.3048 +
  4.3049 +subsection {* Combined fundamental theorem of calculus. *}
  4.3050 +
  4.3051 +lemma antiderivative_integral_continuous: fixes f::"real \<Rightarrow> 'a::banach" assumes "continuous_on {a..b} f"
  4.3052 +  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}"
  4.3053 +proof- from antiderivative_continuous[OF assms] guess g . note g=this
  4.3054 +  show ?thesis apply(rule that[of g])
  4.3055 +  proof safe case goal1 have "\<forall>x\<in>{u..v}. (g has_vector_derivative f x) (at x within {u..v})"
  4.3056 +      apply(rule,rule has_vector_derivative_within_subset) apply(rule g[rule_format]) using goal1(1-2) by auto
  4.3057 +    thus ?case using fundamental_theorem_of_calculus[OF goal1(3),of "g o dest_vec1" "f o dest_vec1"]
  4.3058 +      unfolding o_def vec1_dest_vec1 by auto qed qed
  4.3059 +
  4.3060 +subsection {* General "twiddling" for interval-to-interval function image. *}
  4.3061 +
  4.3062 +lemma has_integral_twiddle:
  4.3063 +  assumes "0 < r" "\<forall>x. h(g x) = x" "\<forall>x. g(h x) = x" "\<forall>x. continuous (at x) g"
  4.3064 +  "\<forall>u v. \<exists>w z. g ` {u..v} = {w..z}"
  4.3065 +  "\<forall>u v. \<exists>w z. h ` {u..v} = {w..z}"
  4.3066 +  "\<forall>u v. content(g ` {u..v}) = r * content {u..v}"
  4.3067 +  "(f has_integral i) {a..b}"
  4.3068 +  shows "((\<lambda>x. f(g x)) has_integral (1 / r) *\<^sub>R i) (h ` {a..b})"
  4.3069 +proof- { presume *:"{a..b} \<noteq> {} \<Longrightarrow> ?thesis"
  4.3070 +    show ?thesis apply cases defer apply(rule *,assumption)
  4.3071 +    proof- case goal1 thus ?thesis unfolding goal1 assms(8)[unfolded goal1 has_integral_empty_eq] by auto qed }
  4.3072 +  assume "{a..b} \<noteq> {}" from assms(6)[rule_format,of a b] guess w z apply-by(erule exE)+ note wz=this
  4.3073 +  have inj:"inj g" "inj h" unfolding inj_on_def apply safe apply(rule_tac[!] ccontr)
  4.3074 +    using assms(2) apply(erule_tac x=x in allE) using assms(2) apply(erule_tac x=y in allE) defer
  4.3075 +    using assms(3) apply(erule_tac x=x in allE) using assms(3) apply(erule_tac x=y in allE) by auto
  4.3076 +  show ?thesis unfolding has_integral_def has_integral_compact_interval_def apply(subst if_P) apply(rule,rule,rule wz)
  4.3077 +  proof safe fix e::real assume e:"e>0" hence "e * r > 0" using assms(1) by(rule mult_pos_pos)
  4.3078 +    from assms(8)[unfolded has_integral,rule_format,OF this] guess d apply-by(erule exE conjE)+ note d=this[rule_format]
  4.3079 +    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)
  4.3080 +    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)"
  4.3081 +    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
  4.3082 +      fix p assume as:"p tagged_division_of h ` {a..b}" "d' fine p" note p = tagged_division_ofD[OF as(1)] 
  4.3083 +      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 
  4.3084 +      proof safe show "finite ((\<lambda>(x, k). (g x, g ` k)) ` p)" using as by auto
  4.3085 +        show "d fine (\<lambda>(x, k). (g x, g ` k)) ` p" using as(2) unfolding fine_def d' by auto
  4.3086 +        fix x k assume xk[intro]:"(x,k) \<in> p" show "g x \<in> g ` k" using p(2)[OF xk] by auto
  4.3087 +        show "\<exists>u v. g ` k = {u..v}" using p(4)[OF xk] using assms(5-6) by auto
  4.3088 +        { 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)"]
  4.3089 +            using assms(2)[rule_format,of y] unfolding inj_image_mem_iff[OF inj(2)] by auto }
  4.3090 +        fix x' k' assume xk':"(x',k') \<in> p" fix z assume "z \<in> interior (g ` k)" "z \<in> interior (g ` k')"
  4.3091 +        hence *:"interior (g ` k) \<inter> interior (g ` k') \<noteq> {}" by auto
  4.3092 +        have same:"(x, k) = (x', k')" apply-apply(rule ccontr,drule p(5)[OF xk xk'])
  4.3093 +        proof- assume as:"interior k \<inter> interior k' = {}" from nonempty_witness[OF *] guess z .
  4.3094 +          hence "z \<in> g ` (interior k \<inter> interior k')" using interior_image_subset[OF assms(4) inj(1)]
  4.3095 +            unfolding image_Int[OF inj(1)] by auto thus False using as by blast
  4.3096 +        qed thus "g x = g x'" by auto
  4.3097 +        { fix z assume "z \<in> k"  thus  "g z \<in> g ` k'" using same by auto }
  4.3098 +        { fix z assume "z \<in> k'" thus  "g z \<in> g ` k"  using same by auto }
  4.3099 +      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
  4.3100 +        then guess X unfolding Union_iff .. note X=this from this(1) guess y unfolding mem_Collect_eq ..
  4.3101 +        thus "x \<in> \<Union>{k. \<exists>x. (x, k) \<in> (\<lambda>(x, k). (g x, g ` k)) ` p}" apply-
  4.3102 +          apply(rule_tac X="g ` X" in UnionI) defer apply(rule_tac x="h x" in image_eqI)
  4.3103 +          using X(2) assms(3)[rule_format,of x] by auto
  4.3104 +      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
  4.3105 +       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
  4.3106 +        unfolding setsum_reindex[OF *] apply(subst scaleR_right.setsum) defer apply(rule setsum_cong2) unfolding o_def split_paired_all split_conv
  4.3107 +        apply(drule p(4)) apply safe unfolding assms(7)[rule_format] using p by auto
  4.3108 +      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]
  4.3109 +        unfolding real_scaleR_def using assms(1) by auto finally have *:"?l = ?r" .
  4.3110 +      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
  4.3111 +        using assms(1) by(auto simp add:field_simps) qed qed qed
  4.3112 +
  4.3113 +subsection {* Special case of a basic affine transformation. *}
  4.3114 +
  4.3115 +lemma interval_image_affinity_interval: shows "\<exists>u v. (\<lambda>x. m *\<^sub>R (x::real^'n) + c) ` {a..b} = {u..v}"
  4.3116 +  unfolding image_affinity_interval by auto
  4.3117 +
  4.3118 +lemmas Cart_simps = Cart_nth.add Cart_nth.minus Cart_nth.zero Cart_nth.diff Cart_nth.scaleR real_scaleR_def Cart_lambda_beta
  4.3119 +   Cart_eq vector_le_def vector_less_def
  4.3120 +
  4.3121 +lemma setprod_cong2: assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "setprod f A = setprod g A"
  4.3122 +  apply(rule setprod_cong) using assms by auto
  4.3123 +
  4.3124 +lemma content_image_affinity_interval: 
  4.3125 + "content((\<lambda>x::real^'n. m *\<^sub>R x + c) ` {a..b}) = (abs m) ^ CARD('n) * content {a..b}" (is "?l = ?r")
  4.3126 +proof- { presume *:"{a..b}\<noteq>{} \<Longrightarrow> ?thesis" show ?thesis apply(cases,rule *,assumption)
  4.3127 +      unfolding not_not using content_empty by auto }
  4.3128 +  assume as:"{a..b}\<noteq>{}" show ?thesis proof(cases "m \<ge> 0")
  4.3129 +    case True show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_P[OF True]
  4.3130 +      unfolding content_closed_interval'[OF as] apply(subst content_closed_interval') 
  4.3131 +      defer apply(subst setprod_constant[THEN sym]) apply(rule finite_UNIV) unfolding setprod_timesf[THEN sym]
  4.3132 +      apply(rule setprod_cong2) using True as unfolding interval_ne_empty Cart_simps not_le  
  4.3133 +      by(auto simp add:field_simps intro:mult_left_mono)
  4.3134 +  next case False show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_not_P[OF False]
  4.3135 +      unfolding content_closed_interval'[OF as] apply(subst content_closed_interval') 
  4.3136 +      defer apply(subst setprod_constant[THEN sym]) apply(rule finite_UNIV) unfolding setprod_timesf[THEN sym]
  4.3137 +      apply(rule setprod_cong2) using False as unfolding interval_ne_empty Cart_simps not_le 
  4.3138 +      by(auto simp add:field_simps mult_le_cancel_left_neg) qed qed
  4.3139 +
  4.3140 +lemma has_integral_affinity: assumes "(f has_integral i) {a..b::real^'n}" "m \<noteq> 0"
  4.3141 +  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})"
  4.3142 +  apply(rule has_integral_twiddle,safe) unfolding Cart_eq Cart_simps apply(rule zero_less_power)
  4.3143 +  defer apply(insert assms(2), simp add:field_simps) apply(insert assms(2), simp add:field_simps)
  4.3144 +  apply(rule continuous_intros)+ apply(rule interval_image_affinity_interval)+ apply(rule content_image_affinity_interval) using assms by auto
  4.3145 +
  4.3146 +lemma integrable_affinity: assumes "f integrable_on {a..b}" "m \<noteq> 0"
  4.3147 +  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})"
  4.3148 +  using assms unfolding integrable_on_def apply safe apply(drule has_integral_affinity) by auto
  4.3149 +
  4.3150 +subsection {* Special case of stretching coordinate axes separately. *}
  4.3151 +
  4.3152 +lemma image_stretch_interval:
  4.3153 +  "(\<lambda>x. \<chi> k. m k * x$k) ` {a..b::real^'n} =
  4.3154 +  (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")
  4.3155 +proof(cases "{a..b}={}") case True thus ?thesis unfolding True by auto
  4.3156 +next have *:"\<And>P Q. (\<forall>i. P i) \<and> (\<forall>i. Q i) \<longleftrightarrow> (\<forall>i. P i \<and> Q i)" by auto
  4.3157 +  case False note ab = this[unfolded interval_ne_empty]
  4.3158 +  show ?thesis apply-apply(rule set_ext)
  4.3159 +  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
  4.3160 +    show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" unfolding if_not_P[OF False] 
  4.3161 +      unfolding image_iff mem_interval Bex_def Cart_simps Cart_eq *
  4.3162 +      unfolding lambda_skolem[THEN sym,of "\<lambda> i xa. (a $ i \<le> xa \<and> xa \<le> b $ i) \<and> x $ i = m i * xa"]
  4.3163 +    proof(rule **,rule) fix i::'n show "(\<exists>xa. (a $ i \<le> xa \<and> xa \<le> b $ i) \<and> x $ i = m i * xa) =
  4.3164 +        (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))"
  4.3165 +      proof(cases "m i = 0") case True thus ?thesis using ab by auto
  4.3166 +      next case False hence "0 < m i \<or> 0 > m i" by auto thus ?thesis apply-
  4.3167 +        proof(erule disjE) assume as:"0 < m i" hence *:"min (m i * a $ i) (m i * b $ i) = m i * a $ i"
  4.3168 +            "max (m i * a $ i) (m i * b $ i) = m i * b $ i" using ab unfolding min_def max_def by auto
  4.3169 +          show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x$i" in exI)
  4.3170 +            using as by(auto simp add:field_simps)
  4.3171 +        next assume as:"0 > m i" hence *:"max (m i * a $ i) (m i * b $ i) = m i * a $ i"
  4.3172 +            "min (m i * a $ i) (m i * b $ i) = m i * b $ i" using ab as unfolding min_def max_def 
  4.3173 +            by(auto simp add:field_simps mult_le_cancel_left_neg intro:real_le_antisym)
  4.3174 +          show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x$i" in exI)
  4.3175 +            using as by(auto simp add:field_simps) qed qed qed qed qed 
  4.3176 +
  4.3177 +lemma interval_image_stretch_interval: "\<exists>u v. (\<lambda>x. \<chi> k. m k * x$k) ` {a..b::real^'n} = {u..v}"
  4.3178 +  unfolding image_stretch_interval by auto 
  4.3179 +
  4.3180 +lemma content_image_stretch_interval:
  4.3181 +  "content((\<lambda>x::real^'n. \<chi> k. m k * x$k) ` {a..b}) = abs(setprod m UNIV) * content({a..b})"
  4.3182 +proof(cases "{a..b} = {}") case True thus ?thesis
  4.3183 +    unfolding content_def image_is_empty image_stretch_interval if_P[OF True] by auto
  4.3184 +next case False hence "(\<lambda>x. \<chi> k. m k * x $ k) ` {a..b} \<noteq> {}" by auto
  4.3185 +  thus ?thesis using False unfolding content_def image_stretch_interval apply- unfolding interval_bounds' if_not_P
  4.3186 +    unfolding abs_setprod setprod_timesf[THEN sym] apply(rule setprod_cong2) unfolding Cart_lambda_beta
  4.3187 +  proof- fix i::'n have "(m i < 0 \<or> m i > 0) \<or> m i = 0" by auto
  4.3188 +    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)"
  4.3189 +      apply-apply(erule disjE)+ unfolding min_def max_def using False[unfolded interval_ne_empty,rule_format,of i] 
  4.3190 +      by(auto simp add:field_simps not_le mult_le_cancel_left_neg mult_le_cancel_left_pos) qed qed
  4.3191 +
  4.3192 +lemma has_integral_stretch: assumes "(f has_integral i) {a..b}" "\<forall>k. ~(m k = 0)"
  4.3193 +  shows "((\<lambda>x. f(\<chi> k. m k * x$k)) has_integral
  4.3194 +             ((1/(abs(setprod m UNIV))) *\<^sub>R i)) ((\<lambda>x. \<chi> k. 1/(m k) * x$k) ` {a..b})"
  4.3195 +  apply(rule has_integral_twiddle) unfolding zero_less_abs_iff content_image_stretch_interval
  4.3196 +  unfolding image_stretch_interval empty_as_interval Cart_eq using assms
  4.3197 +proof- show "\<forall>x. continuous (at x) (\<lambda>x. \<chi> k. m k * x $ k)"
  4.3198 +   apply(rule,rule linear_continuous_at) unfolding linear_linear
  4.3199 +   unfolding linear_def Cart_simps Cart_eq by(auto simp add:field_simps) qed auto
  4.3200 +
  4.3201 +lemma integrable_stretch: 
  4.3202 +  assumes "f integrable_on {a..b}" "\<forall>k. ~(m k = 0)"
  4.3203 +  shows "(\<lambda>x. f(\<chi> k. m k * x$k)) integrable_on ((\<lambda>x. \<chi> k. 1/(m k) * x$k) ` {a..b})"
  4.3204 +  using assms unfolding integrable_on_def apply-apply(erule exE) apply(drule has_integral_stretch) by auto
  4.3205 +
  4.3206 +subsection {* even more special cases. *}
  4.3207 +
  4.3208 +lemma uminus_interval_vector[simp]:"uminus ` {a..b} = {-b .. -a::real^'n}"
  4.3209 +  apply(rule set_ext,rule) defer unfolding image_iff
  4.3210 +  apply(rule_tac x="-x" in bexI) by(auto simp add:vector_le_def minus_le_iff le_minus_iff)
  4.3211 +
  4.3212 +lemma has_integral_reflect_lemma[intro]: assumes "(f has_integral i) {a..b}"
  4.3213 +  shows "((\<lambda>x. f(-x)) has_integral i) {-b .. -a}"
  4.3214 +  using has_integral_affinity[OF assms, of "-1" 0] by auto
  4.3215 +
  4.3216 +lemma has_integral_reflect[simp]: "((\<lambda>x. f(-x)) has_integral i) {-b..-a} \<longleftrightarrow> (f has_integral i) ({a..b})"
  4.3217 +  apply rule apply(drule_tac[!] has_integral_reflect_lemma) by auto
  4.3218 +
  4.3219 +lemma integrable_reflect[simp]: "(\<lambda>x. f(-x)) integrable_on {-b..-a} \<longleftrightarrow> f integrable_on {a..b}"
  4.3220 +  unfolding integrable_on_def by auto
  4.3221 +
  4.3222 +lemma integral_reflect[simp]: "integral {-b..-a} (\<lambda>x. f(-x)) = integral ({a..b}) f"
  4.3223 +  unfolding integral_def by auto
  4.3224 +
  4.3225 +subsection {* Stronger form of FCT; quite a tedious proof. *}
  4.3226 +
  4.3227 +(** move this **)
  4.3228 +declare norm_triangle_ineq4[intro] 
  4.3229 +
  4.3230 +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)
  4.3231 +
  4.3232 +lemma additive_tagged_division_1': fixes f::"real \<Rightarrow> 'a::real_normed_vector"
  4.3233 +  assumes "a \<le> b" "p tagged_division_of {vec1 a..vec1 b}"
  4.3234 +  shows "setsum (\<lambda>(x,k). f (dest_vec1 (interval_upperbound k)) - f(dest_vec1 (interval_lowerbound k))) p = f b - f a"
  4.3235 +  using additive_tagged_division_1[OF _ assms(2), of "f o dest_vec1"]
  4.3236 +  unfolding o_def vec1_dest_vec1 using assms(1) by auto
  4.3237 +
  4.3238 +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"
  4.3239 +  unfolding split_def by(rule refl)
  4.3240 +
  4.3241 +lemma norm_triangle_le_sub: "norm x + norm y \<le> e \<Longrightarrow> norm (x - y) \<le> e"
  4.3242 +  apply(subst(asm)(2) norm_minus_cancel[THEN sym])
  4.3243 +  apply(drule norm_triangle_le) by(auto simp add:group_simps)
  4.3244 +
  4.3245 +lemma fundamental_theorem_of_calculus_interior:
  4.3246 +  assumes"a \<le> b" "continuous_on {a..b} f" "\<forall>x\<in>{a<..<b}. (f has_vector_derivative f'(x)) (at x)"
  4.3247 +  shows "((f' o dest_vec1) has_integral (f b - f a)) {vec a..vec b}"
  4.3248 +proof- { presume *:"a < b \<Longrightarrow> ?thesis" 
  4.3249 +    show ?thesis proof(cases,rule *,assumption)
  4.3250 +      assume "\<not> a < b" hence "a = b" using assms(1) by auto
  4.3251 +      hence *:"{vec a .. vec b} = {vec b}" "f b - f a = 0" apply(auto simp add: Cart_simps) by smt
  4.3252 +      show ?thesis unfolding *(2) apply(rule has_integral_null) unfolding content_eq_0_1 using * `a=b` by auto
  4.3253 +    qed } assume ab:"a < b"
  4.3254 +  let ?P = "\<lambda>e. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {vec1 a..vec1 b} \<and> d fine p \<longrightarrow>
  4.3255 +                   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})"
  4.3256 +  { presume "\<And>e. e>0 \<Longrightarrow> ?P e" thus ?thesis unfolding has_integral_factor_content by auto }
  4.3257 +  fix e::real assume e:"e>0"
  4.3258 +  note assms(3)[unfolded has_vector_derivative_def has_derivative_at_alt ball_conj_distrib]
  4.3259 +  note conjunctD2[OF this] note bounded=this(1) and this(2)
  4.3260 +  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)"
  4.3261 +    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]
  4.3262 +  from choice[OF this] guess d .. note conjunctD2[OF this[rule_format]] note d = this[rule_format]
  4.3263 +  have "bounded (f ` {a..b})" apply(rule compact_imp_bounded compact_continuous_image)+ using compact_real_interval assms by auto
  4.3264 +  from this[unfolded bounded_pos] guess B .. note B = this[rule_format]
  4.3265 +
  4.3266 +  have "\<exists>da. 0 < da \<and> (\<forall>c. a \<le> c \<and> {a..c} \<subseteq> {a..b} \<and> {a..c} \<subseteq> ball a da
  4.3267 +    \<longrightarrow> norm(content {vec1 a..vec1 c} *\<^sub>R f' a - (f c - f a)) \<le> (e * (b - a)) / 4)"
  4.3268 +  proof- have "a\<in>{a..b}" using ab by auto
  4.3269 +    note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
  4.3270 +    note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0" using e ab by(auto simp add:field_simps)
  4.3271 +    from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
  4.3272 +    have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' a) \<le> (e * (b - a)) / 8"
  4.3273 +    proof(cases "f' a = 0") case True
  4.3274 +      thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg) 
  4.3275 +    next case False thus ?thesis 
  4.3276 +        apply(rule_tac x="(e * (b - a)) / 8 / norm (f' a)" in exI)
  4.3277 +        using ab e by(auto simp add:field_simps)
  4.3278 +    qed then guess l .. note l = conjunctD2[OF this]
  4.3279 +    show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+)
  4.3280 +    proof- fix c assume as:"a \<le> c" "{a..c} \<subseteq> {a..b}" "{a..c} \<subseteq> ball a (min k l)" 
  4.3281 +      note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval]
  4.3282 +      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)
  4.3283 +      also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8" 
  4.3284 +      proof(rule add_mono) case goal1 have "\<bar>c - a\<bar> \<le> \<bar>l\<bar>" using as' by auto
  4.3285 +        thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto
  4.3286 +      next case goal2 show ?case apply(rule less_imp_le) apply(cases "a = c") defer
  4.3287 +          apply(rule k(2)[unfolded vector_dist_norm]) using as' e ab by(auto simp add:field_simps)
  4.3288 +      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
  4.3289 +    qed qed then guess da .. note da=conjunctD2[OF this,rule_format]
  4.3290 +
  4.3291 +  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"
  4.3292 +  proof- have "b\<in>{a..b}" using ab by auto
  4.3293 +    note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
  4.3294 +    note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0" using e ab by(auto simp add:field_simps)
  4.3295 +    from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
  4.3296 +    have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' b) \<le> (e * (b - a)) / 8"
  4.3297 +    proof(cases "f' b = 0") case True
  4.3298 +      thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg) 
  4.3299 +    next case False thus ?thesis 
  4.3300 +        apply(rule_tac x="(e * (b - a)) / 8 / norm (f' b)" in exI)
  4.3301 +        using ab e by(auto simp add:field_simps)
  4.3302 +    qed then guess l .. note l = conjunctD2[OF this]
  4.3303 +    show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+)
  4.3304 +    proof- fix c assume as:"c \<le> b" "{c..b} \<subseteq> {a..b}" "{c..b} \<subseteq> ball b (min k l)" 
  4.3305 +      note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval]
  4.3306 +      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)
  4.3307 +      also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8" 
  4.3308 +      proof(rule add_mono) case goal1 have "\<bar>c - b\<bar> \<le> \<bar>l\<bar>" using as' by auto
  4.3309 +        thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto
  4.3310 +      next case goal2 show ?case apply(rule less_imp_le) apply(cases "b = c") defer apply(subst norm_minus_commute)
  4.3311 +          apply(rule k(2)[unfolded vector_dist_norm]) using as' e ab by(auto simp add:field_simps)
  4.3312 +      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
  4.3313 +    qed qed then guess db .. note db=conjunctD2[OF this,rule_format]
  4.3314 +
  4.3315 +  let ?d = "(\<lambda>x. ball x (if x=vec1 a then da else if x=vec b then db else d (dest_vec1 x)))"
  4.3316 +  show "?P e" apply(rule_tac x="?d" in exI)
  4.3317 +  proof safe case goal1 show ?case apply(rule gauge_ball_dependent) using ab db(1) da(1) d(1) by auto
  4.3318 +  next case goal2 note as=this let ?A = "{t. fst t \<in> {vec1 a, vec1 b}}" note p = tagged_division_ofD[OF goal2(1)]
  4.3319 +    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
  4.3320 +    note * = additive_tagged_division_1'[OF assms(1) goal2(1), THEN sym]
  4.3321 +    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
  4.3322 +    show ?case unfolding content_1'[OF assms(1)] and *[of "\<lambda>x. x"] *[of f] setsum_subtractf[THEN sym] split_minus
  4.3323 +      unfolding setsum_right_distrib apply(subst(2) pA,subst pA) unfolding setsum_Un_disjoint[OF pA(2-)]
  4.3324 +    proof(rule norm_triangle_le,rule **) 
  4.3325 +      case goal1 show ?case apply(rule order_trans,rule setsum_norm_le) apply(rule pA) defer apply(subst divide.setsum)
  4.3326 +      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"
  4.3327 +          "e * (dest_vec1 (interval_upperbound k) - dest_vec1 (interval_lowerbound k)) / 2
  4.3328 +          < norm (content k *\<^sub>R f' (dest_vec1 x) - (f (dest_vec1 (interval_upperbound k)) - f (dest_vec1 (interval_lowerbound k))))"
  4.3329 +        from p(4)[OF this(1)] guess u v apply-by(erule exE)+ note k=this
  4.3330 +        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]
  4.3331 +        note result = as(2)[unfolded k interval_bounds[OF this(1)] content_1[OF this(2)]]
  4.3332 +
  4.3333 +        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] 
  4.3334 +        have "norm ((v$1 - u$1) *\<^sub>R f' (x$1) - (f (v$1) - f (u$1))) =
  4.3335 +          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)))" 
  4.3336 +          apply(rule arg_cong[of _ _ norm]) unfolding scaleR_left.diff by auto 
  4.3337 +        also have "... \<le> e / 2 * norm (u$1 - x$1) + e / 2 * norm (v$1 - x$1)" apply(rule norm_triangle_le_sub)
  4.3338 +          apply(rule add_mono) apply(rule_tac[!] *) using fineD[OF goal2(2) as(1)] as' unfolding k subset_eq
  4.3339 +          apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE) using uv by(auto simp add:dist_real)
  4.3340 +        also have "... \<le> e / 2 * norm (v$1 - u$1)" using p(2)[OF as(1)] unfolding k by(auto simp add:field_simps)
  4.3341 +        finally have "e * (dest_vec1 v - dest_vec1 u) / 2 < e * (dest_vec1 v - dest_vec1 u) / 2"
  4.3342 +          apply- apply(rule less_le_trans[OF result]) using uv by auto thus False by auto qed
  4.3343 +
  4.3344 +    next have *:"\<And>x s1 s2::real. 0 \<le> s1 \<Longrightarrow> x \<le> (s1 + s2) / 2 \<Longrightarrow> x - s1 \<le> s2 / 2" by auto
  4.3345 +      case goal2 show ?case apply(rule *) apply(rule setsum_nonneg) apply(rule,unfold split_paired_all split_conv)
  4.3346 +        defer unfolding setsum_Un_disjoint[OF pA(2-),THEN sym] pA(1)[THEN sym] unfolding setsum_right_distrib[THEN sym] 
  4.3347 +        apply(subst additive_tagged_division_1[OF _ as(1)]) unfolding vec1_dest_vec1 apply(rule assms)
  4.3348 +      proof- fix x k assume "(x,k) \<in> p \<inter> {t. fst t \<in> {vec1 a, vec1 b}}" note xk=IntD1[OF this]
  4.3349 +        from p(4)[OF this] guess u v apply-by(erule exE)+ note uv=this
  4.3350 +        with p(2)[OF xk] have "{u..v} \<noteq> {}" by auto
  4.3351 +        thus "0 \<le> e * ((interval_upperbound k)$1 - (interval_lowerbound k)$1)"
  4.3352 +          unfolding uv using e by(auto simp add:field_simps)
  4.3353 +      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
  4.3354 +        show "norm (\<Sum>(x, k)\<in>p \<inter> ?A. content k *\<^sub>R (f' \<circ> dest_vec1) x -
  4.3355 +          (f ((interval_upperbound k)$1) - f ((interval_lowerbound k)$1))) \<le> e * (b - a) / 2" 
  4.3356 +          apply(rule *[where t="p \<inter> {t. fst t \<in> {vec1 a, vec1 b} \<and> content(snd t) \<noteq> 0}"])
  4.3357 +          apply(rule setsum_mono_zero_right[OF pA(2)]) defer apply(rule) unfolding split_paired_all split_conv o_def
  4.3358 +        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}"
  4.3359 +          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
  4.3360 +          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
  4.3361 +          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
  4.3362 +        next have *:"p \<inter> {t. fst t \<in> {vec1 a, vec1 b} \<and> content(snd t) \<noteq> 0} = 
  4.3363 +            {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
  4.3364 +          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"
  4.3365 +          proof(case_tac "s={}") case goal2 then obtain x where "x\<in>s" by auto hence *:"s = {x}" using goal2(1) by auto
  4.3366 +            thus ?case using `x\<in>s` goal2(2) by auto
  4.3367 +          qed auto
  4.3368 +          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"]) 
  4.3369 +            apply(rule norm_triangle_le,rule add_mono) apply(rule_tac[1-2] **)
  4.3370 +          proof- let ?B = "\<lambda>x. {t \<in> p. fst t = vec1 x \<and> content (snd t) \<noteq> 0}"
  4.3371 +            have pa:"\<And>k. (vec1 a, k) \<in> p \<Longrightarrow> \<exists>v. k = {vec1 a .. v} \<and> vec1 a \<le> v" 
  4.3372 +            proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this
  4.3373 +              have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto
  4.3374 +              have u:"u = vec1 a" proof(rule ccontr)  have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto 
  4.3375 +                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
  4.3376 +                have "u > vec1 a" unfolding Cart_simps by auto
  4.3377 +                thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:Cart_simps)
  4.3378 +              qed thus ?case apply(rule_tac x=v in exI) unfolding uv using * by auto
  4.3379 +            qed
  4.3380 +            have pb:"\<And>k. (vec1 b, k) \<in> p \<Longrightarrow> \<exists>v. k = {v .. vec1 b} \<and> vec1 b \<ge> v" 
  4.3381 +            proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this
  4.3382 +              have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto
  4.3383 +              have u:"v = vec1 b" proof(rule ccontr)  have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto 
  4.3384 +                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
  4.3385 +                have "v < vec1 b" unfolding Cart_simps by auto
  4.3386 +                thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:Cart_simps)
  4.3387 +              qed thus ?case apply(rule_tac x=u in exI) unfolding uv using * by auto
  4.3388 +            qed
  4.3389 +
  4.3390 +            show "\<forall>x y. x \<in> ?B a \<and> y \<in> ?B a \<longrightarrow> x = y" apply(rule,rule,rule,unfold split_paired_all)
  4.3391 +              unfolding mem_Collect_eq fst_conv snd_conv apply safe
  4.3392 +            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"
  4.3393 +              guess v using pa[OF k(1)] .. note v = conjunctD2[OF this]
  4.3394 +              guess v' using pa[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = "vec1 (min (v$1) (v'$1))"
  4.3395 +              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]
  4.3396 +              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)
  4.3397 +              ultimately have "vec1 ((a + ?v$1)/2) \<in> interior k \<inter> interior k'" unfolding interior_open[OF open_interval] by auto
  4.3398 +              hence *:"k = k'" apply- apply(rule ccontr) using p(5)[OF k(1-2)] by auto
  4.3399 +              { assume "x\<in>k" thus "x\<in>k'" unfolding * . } { assume "x\<in>k'" thus "x\<in>k" unfolding * . }
  4.3400 +            qed 
  4.3401 +            show "\<forall>x y. x \<in> ?B b \<and> y \<in> ?B b \<longrightarrow> x = y" apply(rule,rule,rule,unfold split_paired_all)
  4.3402 +              unfolding mem_Collect_eq fst_conv snd_conv apply safe
  4.3403 +            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"
  4.3404 +              guess v using pb[OF k(1)] .. note v = conjunctD2[OF this]
  4.3405 +              guess v' using pb[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = "vec1 (max (v$1) (v'$1))"
  4.3406 +              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]
  4.3407 +              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)
  4.3408 +              ultimately have "vec1 ((b + ?v$1)/2) \<in> interior k \<inter> interior k'" unfolding interior_open[OF open_interval] by auto
  4.3409 +              hence *:"k = k'" apply- apply(rule ccontr) using p(5)[OF k(1-2)] by auto
  4.3410 +              { assume "x\<in>k" thus "x\<in>k'" unfolding * . } { assume "x\<in>k'" thus "x\<in>k" unfolding * . }
  4.3411 +            qed
  4.3412 +
  4.3413 +            let ?a = a and ?b = b (* a is something else while proofing the next theorem. *)
  4.3414 +            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)
  4.3415 +              \<le> e * (b - a) / 4" apply safe unfolding fst_conv snd_conv apply safe unfolding vec1_dest_vec1
  4.3416 +            proof- case goal1 guess v using pa[OF goal1(1)] .. note v = conjunctD2[OF this]
  4.3417 +              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
  4.3418 +              moreover have "{?a..dest_vec1 v} \<subseteq> ball ?a da" using fineD[OF as(2) goal1(1)]
  4.3419 +                apply-apply(subst(asm) if_P,rule refl) unfolding subset_eq apply safe apply(erule_tac x="vec1 x" in ballE)
  4.3420 +                by(auto simp add:Cart_simps subset_eq dist_real v dist_real_def) ultimately
  4.3421 +              show ?case unfolding v unfolding interval_bounds[OF v(2)[unfolded v vector_le_def]] vec1_dest_vec1 apply-