merged
authorblanchet
Wed Feb 17 21:13:40 2010 +0100 (2010-02-17)
changeset 35192a815c3f4eef2
parent 35191 69fa4c39dab2
parent 35176 3b9762ad372d
child 35193 3979b0729802
merged
     1.1 --- a/src/HOL/Finite_Set.thy	Wed Feb 17 20:50:14 2010 +0100
     1.2 +++ b/src/HOL/Finite_Set.thy	Wed Feb 17 21:13:40 2010 +0100
     1.3 @@ -2034,6 +2034,31 @@
     1.4    apply auto
     1.5  done
     1.6  
     1.7 +lemma setprod_mono:
     1.8 +  fixes f :: "'a \<Rightarrow> 'b\<Colon>linordered_semidom"
     1.9 +  assumes "\<forall>i\<in>A. 0 \<le> f i \<and> f i \<le> g i"
    1.10 +  shows "setprod f A \<le> setprod g A"
    1.11 +proof (cases "finite A")
    1.12 +  case True
    1.13 +  hence ?thesis "setprod f A \<ge> 0" using subset_refl[of A]
    1.14 +  proof (induct A rule: finite_subset_induct)
    1.15 +    case (insert a F)
    1.16 +    thus "setprod f (insert a F) \<le> setprod g (insert a F)" "0 \<le> setprod f (insert a F)"
    1.17 +      unfolding setprod_insert[OF insert(1,3)]
    1.18 +      using assms[rule_format,OF insert(2)] insert
    1.19 +      by (auto intro: mult_mono mult_nonneg_nonneg)
    1.20 +  qed auto
    1.21 +  thus ?thesis by simp
    1.22 +qed auto
    1.23 +
    1.24 +lemma abs_setprod:
    1.25 +  fixes f :: "'a \<Rightarrow> 'b\<Colon>{linordered_field,abs}"
    1.26 +  shows "abs (setprod f A) = setprod (\<lambda>x. abs (f x)) A"
    1.27 +proof (cases "finite A")
    1.28 +  case True thus ?thesis
    1.29 +    by induct (auto simp add: field_simps setprod_insert abs_mult)
    1.30 +qed auto
    1.31 +
    1.32  
    1.33  subsection {* Finite cardinality *}
    1.34  
     2.1 --- a/src/HOL/Library/Formal_Power_Series.thy	Wed Feb 17 20:50:14 2010 +0100
     2.2 +++ b/src/HOL/Library/Formal_Power_Series.thy	Wed Feb 17 21:13:40 2010 +0100
     2.3 @@ -2864,8 +2864,8 @@
     2.4      then have nz: "pochhammer (1 + b - of_nat n) n \<noteq> 0" 
     2.5        by (auto simp add: algebra_simps)
     2.6      
     2.7 -    from nz kn have nz': "pochhammer (1 + b - of_nat n) k \<noteq> 0" 
     2.8 -      by (simp add: pochhammer_neq_0_mono)
     2.9 +    from nz kn [simplified] have nz': "pochhammer (1 + b - of_nat n) k \<noteq> 0" 
    2.10 +      by (rule pochhammer_neq_0_mono)
    2.11      {assume k0: "k = 0 \<or> n =0" 
    2.12        then have "b gchoose (n - k) = (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)" 
    2.13          using kn
     3.1 --- a/src/HOL/Library/Ramsey.thy	Wed Feb 17 20:50:14 2010 +0100
     3.2 +++ b/src/HOL/Library/Ramsey.thy	Wed Feb 17 21:13:40 2010 +0100
     3.3 @@ -111,7 +111,7 @@
     3.4          have infYx': "infinite (Yx-{yx'})" using fields px by auto
     3.5          with fields px yx' Suc.prems
     3.6          have partfx': "part r s (Yx - {yx'}) (f \<circ> insert yx')"
     3.7 -          by (simp add: o_def part_Suc_imp_part part_subset [where ?YY=YY]) 
     3.8 +          by (simp add: o_def part_Suc_imp_part part_subset [where YY=YY and Y=Yx])
     3.9          from Suc.hyps [OF infYx' partfx']
    3.10          obtain Y' and t'
    3.11          where Y': "Y' \<subseteq> Yx - {yx'}"  "infinite Y'"  "t' < s"
     4.1 --- a/src/HOL/Library/Word.thy	Wed Feb 17 20:50:14 2010 +0100
     4.2 +++ b/src/HOL/Library/Word.thy	Wed Feb 17 21:13:40 2010 +0100
     4.3 @@ -980,7 +980,8 @@
     4.4    fix xs
     4.5    assume "length (norm_signed (\<zero>#xs)) = Suc (length xs)"
     4.6    thus "norm_signed (\<zero>#xs) = \<zero>#xs"
     4.7 -    by (simp add: norm_signed_Cons norm_unsigned_equal split: split_if_asm)
     4.8 +    by (simp add: norm_signed_Cons norm_unsigned_equal [THEN eqTrueI]
     4.9 +             split: split_if_asm)
    4.10  next
    4.11    fix xs
    4.12    assume "length (norm_signed (\<one>#xs)) = Suc (length xs)"
     5.1 --- a/src/HOL/Library/Zorn.thy	Wed Feb 17 20:50:14 2010 +0100
     5.2 +++ b/src/HOL/Library/Zorn.thy	Wed Feb 17 21:13:40 2010 +0100
     5.3 @@ -333,7 +333,7 @@
     5.4  
     5.5  lemma antisym_init_seg_of:
     5.6    "r initial_segment_of s \<Longrightarrow> s initial_segment_of r \<Longrightarrow> r=s"
     5.7 -by(auto simp:init_seg_of_def)
     5.8 +unfolding init_seg_of_def by safe
     5.9  
    5.10  lemma Chain_init_seg_of_Union:
    5.11    "R \<in> Chain init_seg_of \<Longrightarrow> r\<in>R \<Longrightarrow> r initial_segment_of \<Union>R"
     6.1 --- a/src/HOL/Multivariate_Analysis/Derivative.thy	Wed Feb 17 20:50:14 2010 +0100
     6.2 +++ b/src/HOL/Multivariate_Analysis/Derivative.thy	Wed Feb 17 21:13:40 2010 +0100
     6.3 @@ -387,7 +387,7 @@
     6.4        apply(rule_tac le_less_trans[of _ "e/2"]) by(auto intro!:mult_imp_div_pos_le simp add:group_simps) qed auto qed
     6.5  
     6.6  lemma has_derivative_at_alt:
     6.7 -  "(f has_derivative f') (at (x::real^'n)) \<longleftrightarrow> bounded_linear f' \<and>
     6.8 +  "(f has_derivative f') (at x) \<longleftrightarrow> bounded_linear f' \<and>
     6.9    (\<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))"
    6.10    using has_derivative_within_alt[where s=UNIV] unfolding within_UNIV by auto
    6.11  
     7.1 --- a/src/HOL/Multivariate_Analysis/Euclidean_Space.thy	Wed Feb 17 20:50:14 2010 +0100
     7.2 +++ b/src/HOL/Multivariate_Analysis/Euclidean_Space.thy	Wed Feb 17 21:13:40 2010 +0100
     7.3 @@ -1042,11 +1042,6 @@
     7.4    shows "norm x \<le> norm y  + norm (x - y)"
     7.5    using norm_triangle_ineq[of "y" "x - y"] by (simp add: ring_simps)
     7.6  
     7.7 -lemma norm_triangle_le: "norm(x::real ^ 'n) + norm y <= e ==> norm(x + y) <= e"
     7.8 -  by (metis order_trans norm_triangle_ineq)
     7.9 -lemma norm_triangle_lt: "norm(x::real ^ 'n) + norm(y) < e ==> norm(x + y) < e"
    7.10 -  by (metis basic_trans_rules(21) norm_triangle_ineq)
    7.11 -
    7.12  lemma component_le_norm: "\<bar>x$i\<bar> <= norm x"
    7.13    apply (simp add: norm_vector_def)
    7.14    apply (rule member_le_setL2, simp_all)
    7.15 @@ -1275,6 +1270,22 @@
    7.16    shows "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"
    7.17  by (rule dist_triangle_half_l, simp_all add: dist_commute)
    7.18  
    7.19 +
    7.20 +lemma norm_triangle_half_r:
    7.21 +  shows "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
    7.22 +  using dist_triangle_half_r unfolding vector_dist_norm[THEN sym] by auto
    7.23 +
    7.24 +lemma norm_triangle_half_l: assumes "norm (x - y) < e / 2" "norm (x' - (y)) < e / 2" 
    7.25 +  shows "norm (x - x') < e"
    7.26 +  using dist_triangle_half_l[OF assms[unfolded vector_dist_norm[THEN sym]]]
    7.27 +  unfolding vector_dist_norm[THEN sym] .
    7.28 +
    7.29 +lemma norm_triangle_le: "norm(x) + norm y <= e ==> norm(x + y) <= e"
    7.30 +  by (metis order_trans norm_triangle_ineq)
    7.31 +
    7.32 +lemma norm_triangle_lt: "norm(x) + norm(y) < e ==> norm(x + y) < e"
    7.33 +  by (metis basic_trans_rules(21) norm_triangle_ineq)
    7.34 +
    7.35  lemma dist_triangle_add:
    7.36    fixes x y x' y' :: "'a::real_normed_vector"
    7.37    shows "dist (x + y) (x' + y') <= dist x x' + dist y y'"
     8.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     8.2 +++ b/src/HOL/Multivariate_Analysis/Integration_MV.cert	Wed Feb 17 21:13:40 2010 +0100
     8.3 @@ -0,0 +1,3270 @@
     8.4 +tB2Atlor9W4pSnrAz5nHpw 907 0
     8.5 +#2 := false
     8.6 +#299 := 0::real
     8.7 +decl uf_1 :: (-> T3 T2 real)
     8.8 +decl uf_10 :: (-> T4 T2)
     8.9 +decl uf_7 :: T4
    8.10 +#15 := uf_7
    8.11 +#22 := (uf_10 uf_7)
    8.12 +decl uf_2 :: (-> T1 T3)
    8.13 +decl uf_4 :: T1
    8.14 +#11 := uf_4
    8.15 +#91 := (uf_2 uf_4)
    8.16 +#902 := (uf_1 #91 #22)
    8.17 +#297 := -1::real
    8.18 +#1084 := (* -1::real #902)
    8.19 +decl uf_16 :: T1
    8.20 +#50 := uf_16
    8.21 +#78 := (uf_2 uf_16)
    8.22 +#799 := (uf_1 #78 #22)
    8.23 +#1267 := (+ #799 #1084)
    8.24 +#1272 := (>= #1267 0::real)
    8.25 +#1266 := (= #799 #902)
    8.26 +decl uf_9 :: T3
    8.27 +#21 := uf_9
    8.28 +#23 := (uf_1 uf_9 #22)
    8.29 +#905 := (= #23 #902)
    8.30 +decl uf_11 :: T3
    8.31 +#24 := uf_11
    8.32 +#850 := (uf_1 uf_11 #22)
    8.33 +#904 := (= #850 #902)
    8.34 +decl uf_6 :: (-> T2 T4)
    8.35 +#74 := (uf_6 #22)
    8.36 +#281 := (= uf_7 #74)
    8.37 +#922 := (ite #281 #905 #904)
    8.38 +decl uf_8 :: T3
    8.39 +#18 := uf_8
    8.40 +#848 := (uf_1 uf_8 #22)
    8.41 +#903 := (= #848 #902)
    8.42 +#60 := 0::int
    8.43 +decl uf_5 :: (-> T4 int)
    8.44 +#803 := (uf_5 #74)
    8.45 +#117 := -1::int
    8.46 +#813 := (* -1::int #803)
    8.47 +#16 := (uf_5 uf_7)
    8.48 +#916 := (+ #16 #813)
    8.49 +#917 := (<= #916 0::int)
    8.50 +#925 := (ite #917 #922 #903)
    8.51 +#6 := (:var 0 T2)
    8.52 +#19 := (uf_1 uf_8 #6)
    8.53 +#544 := (pattern #19)
    8.54 +#25 := (uf_1 uf_11 #6)
    8.55 +#543 := (pattern #25)
    8.56 +#92 := (uf_1 #91 #6)
    8.57 +#542 := (pattern #92)
    8.58 +#13 := (uf_6 #6)
    8.59 +#541 := (pattern #13)
    8.60 +#447 := (= #19 #92)
    8.61 +#445 := (= #25 #92)
    8.62 +#444 := (= #23 #92)
    8.63 +#20 := (= #13 uf_7)
    8.64 +#446 := (ite #20 #444 #445)
    8.65 +#120 := (* -1::int #16)
    8.66 +#14 := (uf_5 #13)
    8.67 +#121 := (+ #14 #120)
    8.68 +#119 := (>= #121 0::int)
    8.69 +#448 := (ite #119 #446 #447)
    8.70 +#545 := (forall (vars (?x3 T2)) (:pat #541 #542 #543 #544) #448)
    8.71 +#451 := (forall (vars (?x3 T2)) #448)
    8.72 +#548 := (iff #451 #545)
    8.73 +#546 := (iff #448 #448)
    8.74 +#547 := [refl]: #546
    8.75 +#549 := [quant-intro #547]: #548
    8.76 +#26 := (ite #20 #23 #25)
    8.77 +#127 := (ite #119 #26 #19)
    8.78 +#368 := (= #92 #127)
    8.79 +#369 := (forall (vars (?x3 T2)) #368)
    8.80 +#452 := (iff #369 #451)
    8.81 +#449 := (iff #368 #448)
    8.82 +#450 := [rewrite]: #449
    8.83 +#453 := [quant-intro #450]: #452
    8.84 +#392 := (~ #369 #369)
    8.85 +#390 := (~ #368 #368)
    8.86 +#391 := [refl]: #390
    8.87 +#366 := [nnf-pos #391]: #392
    8.88 +decl uf_3 :: (-> T1 T2 real)
    8.89 +#12 := (uf_3 uf_4 #6)
    8.90 +#132 := (= #12 #127)
    8.91 +#135 := (forall (vars (?x3 T2)) #132)
    8.92 +#370 := (iff #135 #369)
    8.93 +#4 := (:var 1 T1)
    8.94 +#8 := (uf_3 #4 #6)
    8.95 +#5 := (uf_2 #4)
    8.96 +#7 := (uf_1 #5 #6)
    8.97 +#9 := (= #7 #8)
    8.98 +#10 := (forall (vars (?x1 T1) (?x2 T2)) #9)
    8.99 +#113 := [asserted]: #10
   8.100 +#371 := [rewrite* #113]: #370
   8.101 +#17 := (< #14 #16)
   8.102 +#27 := (ite #17 #19 #26)
   8.103 +#28 := (= #12 #27)
   8.104 +#29 := (forall (vars (?x3 T2)) #28)
   8.105 +#136 := (iff #29 #135)
   8.106 +#133 := (iff #28 #132)
   8.107 +#130 := (= #27 #127)
   8.108 +#118 := (not #119)
   8.109 +#124 := (ite #118 #19 #26)
   8.110 +#128 := (= #124 #127)
   8.111 +#129 := [rewrite]: #128
   8.112 +#125 := (= #27 #124)
   8.113 +#122 := (iff #17 #118)
   8.114 +#123 := [rewrite]: #122
   8.115 +#126 := [monotonicity #123]: #125
   8.116 +#131 := [trans #126 #129]: #130
   8.117 +#134 := [monotonicity #131]: #133
   8.118 +#137 := [quant-intro #134]: #136
   8.119 +#114 := [asserted]: #29
   8.120 +#138 := [mp #114 #137]: #135
   8.121 +#372 := [mp #138 #371]: #369
   8.122 +#367 := [mp~ #372 #366]: #369
   8.123 +#454 := [mp #367 #453]: #451
   8.124 +#550 := [mp #454 #549]: #545
   8.125 +#738 := (not #545)
   8.126 +#928 := (or #738 #925)
   8.127 +#75 := (= #74 uf_7)
   8.128 +#906 := (ite #75 #905 #904)
   8.129 +#907 := (+ #803 #120)
   8.130 +#908 := (>= #907 0::int)
   8.131 +#909 := (ite #908 #906 #903)
   8.132 +#929 := (or #738 #909)
   8.133 +#931 := (iff #929 #928)
   8.134 +#933 := (iff #928 #928)
   8.135 +#934 := [rewrite]: #933
   8.136 +#926 := (iff #909 #925)
   8.137 +#923 := (iff #906 #922)
   8.138 +#283 := (iff #75 #281)
   8.139 +#284 := [rewrite]: #283
   8.140 +#924 := [monotonicity #284]: #923
   8.141 +#920 := (iff #908 #917)
   8.142 +#910 := (+ #120 #803)
   8.143 +#913 := (>= #910 0::int)
   8.144 +#918 := (iff #913 #917)
   8.145 +#919 := [rewrite]: #918
   8.146 +#914 := (iff #908 #913)
   8.147 +#911 := (= #907 #910)
   8.148 +#912 := [rewrite]: #911
   8.149 +#915 := [monotonicity #912]: #914
   8.150 +#921 := [trans #915 #919]: #920
   8.151 +#927 := [monotonicity #921 #924]: #926
   8.152 +#932 := [monotonicity #927]: #931
   8.153 +#935 := [trans #932 #934]: #931
   8.154 +#930 := [quant-inst]: #929
   8.155 +#936 := [mp #930 #935]: #928
   8.156 +#1300 := [unit-resolution #936 #550]: #925
   8.157 +#989 := (= #16 #803)
   8.158 +#1277 := (= #803 #16)
   8.159 +#280 := [asserted]: #75
   8.160 +#287 := [mp #280 #284]: #281
   8.161 +#1276 := [symm #287]: #75
   8.162 +#1278 := [monotonicity #1276]: #1277
   8.163 +#1301 := [symm #1278]: #989
   8.164 +#1302 := (not #989)
   8.165 +#1303 := (or #1302 #917)
   8.166 +#1304 := [th-lemma]: #1303
   8.167 +#1305 := [unit-resolution #1304 #1301]: #917
   8.168 +#950 := (not #917)
   8.169 +#949 := (not #925)
   8.170 +#951 := (or #949 #950 #922)
   8.171 +#952 := [def-axiom]: #951
   8.172 +#1306 := [unit-resolution #952 #1305 #1300]: #922
   8.173 +#937 := (not #922)
   8.174 +#1307 := (or #937 #905)
   8.175 +#938 := (not #281)
   8.176 +#939 := (or #937 #938 #905)
   8.177 +#940 := [def-axiom]: #939
   8.178 +#1308 := [unit-resolution #940 #287]: #1307
   8.179 +#1309 := [unit-resolution #1308 #1306]: #905
   8.180 +#1356 := (= #799 #23)
   8.181 +#800 := (= #23 #799)
   8.182 +decl uf_15 :: T4
   8.183 +#40 := uf_15
   8.184 +#41 := (uf_5 uf_15)
   8.185 +#814 := (+ #41 #813)
   8.186 +#815 := (<= #814 0::int)
   8.187 +#836 := (not #815)
   8.188 +#158 := (* -1::int #41)
   8.189 +#1270 := (+ #16 #158)
   8.190 +#1265 := (>= #1270 0::int)
   8.191 +#1339 := (not #1265)
   8.192 +#1269 := (= #16 #41)
   8.193 +#1298 := (not #1269)
   8.194 +#286 := (= uf_7 uf_15)
   8.195 +#44 := (uf_10 uf_15)
   8.196 +#72 := (uf_6 #44)
   8.197 +#73 := (= #72 uf_15)
   8.198 +#277 := (= uf_15 #72)
   8.199 +#278 := (iff #73 #277)
   8.200 +#279 := [rewrite]: #278
   8.201 +#276 := [asserted]: #73
   8.202 +#282 := [mp #276 #279]: #277
   8.203 +#1274 := [symm #282]: #73
   8.204 +#729 := (= uf_7 #72)
   8.205 +decl uf_17 :: (-> int T4)
   8.206 +#611 := (uf_5 #72)
   8.207 +#991 := (uf_17 #611)
   8.208 +#1289 := (= #991 #72)
   8.209 +#992 := (= #72 #991)
   8.210 +#55 := (:var 0 T4)
   8.211 +#56 := (uf_5 #55)
   8.212 +#574 := (pattern #56)
   8.213 +#57 := (uf_17 #56)
   8.214 +#177 := (= #55 #57)
   8.215 +#575 := (forall (vars (?x7 T4)) (:pat #574) #177)
   8.216 +#195 := (forall (vars (?x7 T4)) #177)
   8.217 +#578 := (iff #195 #575)
   8.218 +#576 := (iff #177 #177)
   8.219 +#577 := [refl]: #576
   8.220 +#579 := [quant-intro #577]: #578
   8.221 +#405 := (~ #195 #195)
   8.222 +#403 := (~ #177 #177)
   8.223 +#404 := [refl]: #403
   8.224 +#406 := [nnf-pos #404]: #405
   8.225 +#58 := (= #57 #55)
   8.226 +#59 := (forall (vars (?x7 T4)) #58)
   8.227 +#196 := (iff #59 #195)
   8.228 +#193 := (iff #58 #177)
   8.229 +#194 := [rewrite]: #193
   8.230 +#197 := [quant-intro #194]: #196
   8.231 +#155 := [asserted]: #59
   8.232 +#200 := [mp #155 #197]: #195
   8.233 +#407 := [mp~ #200 #406]: #195
   8.234 +#580 := [mp #407 #579]: #575
   8.235 +#995 := (not #575)
   8.236 +#996 := (or #995 #992)
   8.237 +#997 := [quant-inst]: #996
   8.238 +#1273 := [unit-resolution #997 #580]: #992
   8.239 +#1290 := [symm #1273]: #1289
   8.240 +#1293 := (= uf_7 #991)
   8.241 +#993 := (uf_17 #803)
   8.242 +#1287 := (= #993 #991)
   8.243 +#1284 := (= #803 #611)
   8.244 +#987 := (= #41 #611)
   8.245 +#1279 := (= #611 #41)
   8.246 +#1280 := [monotonicity #1274]: #1279
   8.247 +#1281 := [symm #1280]: #987
   8.248 +#1282 := (= #803 #41)
   8.249 +#1275 := [hypothesis]: #1269
   8.250 +#1283 := [trans #1278 #1275]: #1282
   8.251 +#1285 := [trans #1283 #1281]: #1284
   8.252 +#1288 := [monotonicity #1285]: #1287
   8.253 +#1291 := (= uf_7 #993)
   8.254 +#994 := (= #74 #993)
   8.255 +#1000 := (or #995 #994)
   8.256 +#1001 := [quant-inst]: #1000
   8.257 +#1286 := [unit-resolution #1001 #580]: #994
   8.258 +#1292 := [trans #287 #1286]: #1291
   8.259 +#1294 := [trans #1292 #1288]: #1293
   8.260 +#1295 := [trans #1294 #1290]: #729
   8.261 +#1296 := [trans #1295 #1274]: #286
   8.262 +#290 := (not #286)
   8.263 +#76 := (= uf_15 uf_7)
   8.264 +#77 := (not #76)
   8.265 +#291 := (iff #77 #290)
   8.266 +#288 := (iff #76 #286)
   8.267 +#289 := [rewrite]: #288
   8.268 +#292 := [monotonicity #289]: #291
   8.269 +#285 := [asserted]: #77
   8.270 +#295 := [mp #285 #292]: #290
   8.271 +#1297 := [unit-resolution #295 #1296]: false
   8.272 +#1299 := [lemma #1297]: #1298
   8.273 +#1342 := (or #1269 #1339)
   8.274 +#1271 := (<= #1270 0::int)
   8.275 +#621 := (* -1::int #611)
   8.276 +#723 := (+ #16 #621)
   8.277 +#724 := (<= #723 0::int)
   8.278 +decl uf_12 :: T1
   8.279 +#30 := uf_12
   8.280 +#88 := (uf_2 uf_12)
   8.281 +#771 := (uf_1 #88 #44)
   8.282 +#45 := (uf_1 uf_9 #44)
   8.283 +#772 := (= #45 #771)
   8.284 +#796 := (not #772)
   8.285 +decl uf_14 :: T1
   8.286 +#38 := uf_14
   8.287 +#83 := (uf_2 uf_14)
   8.288 +#656 := (uf_1 #83 #44)
   8.289 +#1239 := (= #656 #771)
   8.290 +#1252 := (not #1239)
   8.291 +#1324 := (iff #1252 #796)
   8.292 +#1322 := (iff #1239 #772)
   8.293 +#1320 := (= #656 #45)
   8.294 +#661 := (= #45 #656)
   8.295 +#659 := (uf_1 uf_11 #44)
   8.296 +#664 := (= #656 #659)
   8.297 +#667 := (ite #277 #661 #664)
   8.298 +#657 := (uf_1 uf_8 #44)
   8.299 +#670 := (= #656 #657)
   8.300 +#622 := (+ #41 #621)
   8.301 +#623 := (<= #622 0::int)
   8.302 +#673 := (ite #623 #667 #670)
   8.303 +#84 := (uf_1 #83 #6)
   8.304 +#560 := (pattern #84)
   8.305 +#467 := (= #19 #84)
   8.306 +#465 := (= #25 #84)
   8.307 +#464 := (= #45 #84)
   8.308 +#43 := (= #13 uf_15)
   8.309 +#466 := (ite #43 #464 #465)
   8.310 +#159 := (+ #14 #158)
   8.311 +#157 := (>= #159 0::int)
   8.312 +#468 := (ite #157 #466 #467)
   8.313 +#561 := (forall (vars (?x5 T2)) (:pat #541 #560 #543 #544) #468)
   8.314 +#471 := (forall (vars (?x5 T2)) #468)
   8.315 +#564 := (iff #471 #561)
   8.316 +#562 := (iff #468 #468)
   8.317 +#563 := [refl]: #562
   8.318 +#565 := [quant-intro #563]: #564
   8.319 +#46 := (ite #43 #45 #25)
   8.320 +#165 := (ite #157 #46 #19)
   8.321 +#378 := (= #84 #165)
   8.322 +#379 := (forall (vars (?x5 T2)) #378)
   8.323 +#472 := (iff #379 #471)
   8.324 +#469 := (iff #378 #468)
   8.325 +#470 := [rewrite]: #469
   8.326 +#473 := [quant-intro #470]: #472
   8.327 +#359 := (~ #379 #379)
   8.328 +#361 := (~ #378 #378)
   8.329 +#358 := [refl]: #361
   8.330 +#356 := [nnf-pos #358]: #359
   8.331 +#39 := (uf_3 uf_14 #6)
   8.332 +#170 := (= #39 #165)
   8.333 +#173 := (forall (vars (?x5 T2)) #170)
   8.334 +#380 := (iff #173 #379)
   8.335 +#381 := [rewrite* #113]: #380
   8.336 +#42 := (< #14 #41)
   8.337 +#47 := (ite #42 #19 #46)
   8.338 +#48 := (= #39 #47)
   8.339 +#49 := (forall (vars (?x5 T2)) #48)
   8.340 +#174 := (iff #49 #173)
   8.341 +#171 := (iff #48 #170)
   8.342 +#168 := (= #47 #165)
   8.343 +#156 := (not #157)
   8.344 +#162 := (ite #156 #19 #46)
   8.345 +#166 := (= #162 #165)
   8.346 +#167 := [rewrite]: #166
   8.347 +#163 := (= #47 #162)
   8.348 +#160 := (iff #42 #156)
   8.349 +#161 := [rewrite]: #160
   8.350 +#164 := [monotonicity #161]: #163
   8.351 +#169 := [trans #164 #167]: #168
   8.352 +#172 := [monotonicity #169]: #171
   8.353 +#175 := [quant-intro #172]: #174
   8.354 +#116 := [asserted]: #49
   8.355 +#176 := [mp #116 #175]: #173
   8.356 +#382 := [mp #176 #381]: #379
   8.357 +#357 := [mp~ #382 #356]: #379
   8.358 +#474 := [mp #357 #473]: #471
   8.359 +#566 := [mp #474 #565]: #561
   8.360 +#676 := (not #561)
   8.361 +#677 := (or #676 #673)
   8.362 +#658 := (= #657 #656)
   8.363 +#660 := (= #659 #656)
   8.364 +#662 := (ite #73 #661 #660)
   8.365 +#612 := (+ #611 #158)
   8.366 +#613 := (>= #612 0::int)
   8.367 +#663 := (ite #613 #662 #658)
   8.368 +#678 := (or #676 #663)
   8.369 +#680 := (iff #678 #677)
   8.370 +#682 := (iff #677 #677)
   8.371 +#683 := [rewrite]: #682
   8.372 +#674 := (iff #663 #673)
   8.373 +#671 := (iff #658 #670)
   8.374 +#672 := [rewrite]: #671
   8.375 +#668 := (iff #662 #667)
   8.376 +#665 := (iff #660 #664)
   8.377 +#666 := [rewrite]: #665
   8.378 +#669 := [monotonicity #279 #666]: #668
   8.379 +#626 := (iff #613 #623)
   8.380 +#615 := (+ #158 #611)
   8.381 +#618 := (>= #615 0::int)
   8.382 +#624 := (iff #618 #623)
   8.383 +#625 := [rewrite]: #624
   8.384 +#619 := (iff #613 #618)
   8.385 +#616 := (= #612 #615)
   8.386 +#617 := [rewrite]: #616
   8.387 +#620 := [monotonicity #617]: #619
   8.388 +#627 := [trans #620 #625]: #626
   8.389 +#675 := [monotonicity #627 #669 #672]: #674
   8.390 +#681 := [monotonicity #675]: #680
   8.391 +#684 := [trans #681 #683]: #680
   8.392 +#679 := [quant-inst]: #678
   8.393 +#685 := [mp #679 #684]: #677
   8.394 +#1311 := [unit-resolution #685 #566]: #673
   8.395 +#1312 := (not #987)
   8.396 +#1313 := (or #1312 #623)
   8.397 +#1314 := [th-lemma]: #1313
   8.398 +#1315 := [unit-resolution #1314 #1281]: #623
   8.399 +#645 := (not #623)
   8.400 +#698 := (not #673)
   8.401 +#699 := (or #698 #645 #667)
   8.402 +#700 := [def-axiom]: #699
   8.403 +#1316 := [unit-resolution #700 #1315 #1311]: #667
   8.404 +#686 := (not #667)
   8.405 +#1317 := (or #686 #661)
   8.406 +#687 := (not #277)
   8.407 +#688 := (or #686 #687 #661)
   8.408 +#689 := [def-axiom]: #688
   8.409 +#1318 := [unit-resolution #689 #282]: #1317
   8.410 +#1319 := [unit-resolution #1318 #1316]: #661
   8.411 +#1321 := [symm #1319]: #1320
   8.412 +#1323 := [monotonicity #1321]: #1322
   8.413 +#1325 := [monotonicity #1323]: #1324
   8.414 +#1145 := (* -1::real #771)
   8.415 +#1240 := (+ #656 #1145)
   8.416 +#1241 := (<= #1240 0::real)
   8.417 +#1249 := (not #1241)
   8.418 +#1243 := [hypothesis]: #1241
   8.419 +decl uf_18 :: T3
   8.420 +#80 := uf_18
   8.421 +#1040 := (uf_1 uf_18 #44)
   8.422 +#1043 := (* -1::real #1040)
   8.423 +#1156 := (+ #771 #1043)
   8.424 +#1157 := (>= #1156 0::real)
   8.425 +#1189 := (not #1157)
   8.426 +#708 := (uf_1 #91 #44)
   8.427 +#1168 := (+ #708 #1043)
   8.428 +#1169 := (<= #1168 0::real)
   8.429 +#1174 := (or #1157 #1169)
   8.430 +#1177 := (not #1174)
   8.431 +#89 := (uf_1 #88 #6)
   8.432 +#552 := (pattern #89)
   8.433 +#81 := (uf_1 uf_18 #6)
   8.434 +#594 := (pattern #81)
   8.435 +#324 := (* -1::real #92)
   8.436 +#325 := (+ #81 #324)
   8.437 +#323 := (>= #325 0::real)
   8.438 +#317 := (* -1::real #89)
   8.439 +#318 := (+ #81 #317)
   8.440 +#319 := (<= #318 0::real)
   8.441 +#436 := (or #319 #323)
   8.442 +#437 := (not #436)
   8.443 +#601 := (forall (vars (?x11 T2)) (:pat #594 #552 #542) #437)
   8.444 +#440 := (forall (vars (?x11 T2)) #437)
   8.445 +#604 := (iff #440 #601)
   8.446 +#602 := (iff #437 #437)
   8.447 +#603 := [refl]: #602
   8.448 +#605 := [quant-intro #603]: #604
   8.449 +#326 := (not #323)
   8.450 +#320 := (not #319)
   8.451 +#329 := (and #320 #326)
   8.452 +#332 := (forall (vars (?x11 T2)) #329)
   8.453 +#441 := (iff #332 #440)
   8.454 +#438 := (iff #329 #437)
   8.455 +#439 := [rewrite]: #438
   8.456 +#442 := [quant-intro #439]: #441
   8.457 +#425 := (~ #332 #332)
   8.458 +#423 := (~ #329 #329)
   8.459 +#424 := [refl]: #423
   8.460 +#426 := [nnf-pos #424]: #425
   8.461 +#306 := (* -1::real #84)
   8.462 +#307 := (+ #81 #306)
   8.463 +#305 := (>= #307 0::real)
   8.464 +#308 := (not #305)
   8.465 +#301 := (* -1::real #81)
   8.466 +#79 := (uf_1 #78 #6)
   8.467 +#302 := (+ #79 #301)
   8.468 +#300 := (>= #302 0::real)
   8.469 +#298 := (not #300)
   8.470 +#311 := (and #298 #308)
   8.471 +#314 := (forall (vars (?x10 T2)) #311)
   8.472 +#335 := (and #314 #332)
   8.473 +#93 := (< #81 #92)
   8.474 +#90 := (< #89 #81)
   8.475 +#94 := (and #90 #93)
   8.476 +#95 := (forall (vars (?x11 T2)) #94)
   8.477 +#85 := (< #81 #84)
   8.478 +#82 := (< #79 #81)
   8.479 +#86 := (and #82 #85)
   8.480 +#87 := (forall (vars (?x10 T2)) #86)
   8.481 +#96 := (and #87 #95)
   8.482 +#336 := (iff #96 #335)
   8.483 +#333 := (iff #95 #332)
   8.484 +#330 := (iff #94 #329)
   8.485 +#327 := (iff #93 #326)
   8.486 +#328 := [rewrite]: #327
   8.487 +#321 := (iff #90 #320)
   8.488 +#322 := [rewrite]: #321
   8.489 +#331 := [monotonicity #322 #328]: #330
   8.490 +#334 := [quant-intro #331]: #333
   8.491 +#315 := (iff #87 #314)
   8.492 +#312 := (iff #86 #311)
   8.493 +#309 := (iff #85 #308)
   8.494 +#310 := [rewrite]: #309
   8.495 +#303 := (iff #82 #298)
   8.496 +#304 := [rewrite]: #303
   8.497 +#313 := [monotonicity #304 #310]: #312
   8.498 +#316 := [quant-intro #313]: #315
   8.499 +#337 := [monotonicity #316 #334]: #336
   8.500 +#293 := [asserted]: #96
   8.501 +#338 := [mp #293 #337]: #335
   8.502 +#340 := [and-elim #338]: #332
   8.503 +#427 := [mp~ #340 #426]: #332
   8.504 +#443 := [mp #427 #442]: #440
   8.505 +#606 := [mp #443 #605]: #601
   8.506 +#1124 := (not #601)
   8.507 +#1180 := (or #1124 #1177)
   8.508 +#1142 := (* -1::real #708)
   8.509 +#1143 := (+ #1040 #1142)
   8.510 +#1144 := (>= #1143 0::real)
   8.511 +#1146 := (+ #1040 #1145)
   8.512 +#1147 := (<= #1146 0::real)
   8.513 +#1148 := (or #1147 #1144)
   8.514 +#1149 := (not #1148)
   8.515 +#1181 := (or #1124 #1149)
   8.516 +#1183 := (iff #1181 #1180)
   8.517 +#1185 := (iff #1180 #1180)
   8.518 +#1186 := [rewrite]: #1185
   8.519 +#1178 := (iff #1149 #1177)
   8.520 +#1175 := (iff #1148 #1174)
   8.521 +#1172 := (iff #1144 #1169)
   8.522 +#1162 := (+ #1142 #1040)
   8.523 +#1165 := (>= #1162 0::real)
   8.524 +#1170 := (iff #1165 #1169)
   8.525 +#1171 := [rewrite]: #1170
   8.526 +#1166 := (iff #1144 #1165)
   8.527 +#1163 := (= #1143 #1162)
   8.528 +#1164 := [rewrite]: #1163
   8.529 +#1167 := [monotonicity #1164]: #1166
   8.530 +#1173 := [trans #1167 #1171]: #1172
   8.531 +#1160 := (iff #1147 #1157)
   8.532 +#1150 := (+ #1145 #1040)
   8.533 +#1153 := (<= #1150 0::real)
   8.534 +#1158 := (iff #1153 #1157)
   8.535 +#1159 := [rewrite]: #1158
   8.536 +#1154 := (iff #1147 #1153)
   8.537 +#1151 := (= #1146 #1150)
   8.538 +#1152 := [rewrite]: #1151
   8.539 +#1155 := [monotonicity #1152]: #1154
   8.540 +#1161 := [trans #1155 #1159]: #1160
   8.541 +#1176 := [monotonicity #1161 #1173]: #1175
   8.542 +#1179 := [monotonicity #1176]: #1178
   8.543 +#1184 := [monotonicity #1179]: #1183
   8.544 +#1187 := [trans #1184 #1186]: #1183
   8.545 +#1182 := [quant-inst]: #1181
   8.546 +#1188 := [mp #1182 #1187]: #1180
   8.547 +#1244 := [unit-resolution #1188 #606]: #1177
   8.548 +#1190 := (or #1174 #1189)
   8.549 +#1191 := [def-axiom]: #1190
   8.550 +#1245 := [unit-resolution #1191 #1244]: #1189
   8.551 +#1054 := (+ #656 #1043)
   8.552 +#1055 := (<= #1054 0::real)
   8.553 +#1079 := (not #1055)
   8.554 +#607 := (uf_1 #78 #44)
   8.555 +#1044 := (+ #607 #1043)
   8.556 +#1045 := (>= #1044 0::real)
   8.557 +#1060 := (or #1045 #1055)
   8.558 +#1063 := (not #1060)
   8.559 +#567 := (pattern #79)
   8.560 +#428 := (or #300 #305)
   8.561 +#429 := (not #428)
   8.562 +#595 := (forall (vars (?x10 T2)) (:pat #567 #594 #560) #429)
   8.563 +#432 := (forall (vars (?x10 T2)) #429)
   8.564 +#598 := (iff #432 #595)
   8.565 +#596 := (iff #429 #429)
   8.566 +#597 := [refl]: #596
   8.567 +#599 := [quant-intro #597]: #598
   8.568 +#433 := (iff #314 #432)
   8.569 +#430 := (iff #311 #429)
   8.570 +#431 := [rewrite]: #430
   8.571 +#434 := [quant-intro #431]: #433
   8.572 +#420 := (~ #314 #314)
   8.573 +#418 := (~ #311 #311)
   8.574 +#419 := [refl]: #418
   8.575 +#421 := [nnf-pos #419]: #420
   8.576 +#339 := [and-elim #338]: #314
   8.577 +#422 := [mp~ #339 #421]: #314
   8.578 +#435 := [mp #422 #434]: #432
   8.579 +#600 := [mp #435 #599]: #595
   8.580 +#1066 := (not #595)
   8.581 +#1067 := (or #1066 #1063)
   8.582 +#1039 := (* -1::real #656)
   8.583 +#1041 := (+ #1040 #1039)
   8.584 +#1042 := (>= #1041 0::real)
   8.585 +#1046 := (or #1045 #1042)
   8.586 +#1047 := (not #1046)
   8.587 +#1068 := (or #1066 #1047)
   8.588 +#1070 := (iff #1068 #1067)
   8.589 +#1072 := (iff #1067 #1067)
   8.590 +#1073 := [rewrite]: #1072
   8.591 +#1064 := (iff #1047 #1063)
   8.592 +#1061 := (iff #1046 #1060)
   8.593 +#1058 := (iff #1042 #1055)
   8.594 +#1048 := (+ #1039 #1040)
   8.595 +#1051 := (>= #1048 0::real)
   8.596 +#1056 := (iff #1051 #1055)
   8.597 +#1057 := [rewrite]: #1056
   8.598 +#1052 := (iff #1042 #1051)
   8.599 +#1049 := (= #1041 #1048)
   8.600 +#1050 := [rewrite]: #1049
   8.601 +#1053 := [monotonicity #1050]: #1052
   8.602 +#1059 := [trans #1053 #1057]: #1058
   8.603 +#1062 := [monotonicity #1059]: #1061
   8.604 +#1065 := [monotonicity #1062]: #1064
   8.605 +#1071 := [monotonicity #1065]: #1070
   8.606 +#1074 := [trans #1071 #1073]: #1070
   8.607 +#1069 := [quant-inst]: #1068
   8.608 +#1075 := [mp #1069 #1074]: #1067
   8.609 +#1246 := [unit-resolution #1075 #600]: #1063
   8.610 +#1080 := (or #1060 #1079)
   8.611 +#1081 := [def-axiom]: #1080
   8.612 +#1247 := [unit-resolution #1081 #1246]: #1079
   8.613 +#1248 := [th-lemma #1247 #1245 #1243]: false
   8.614 +#1250 := [lemma #1248]: #1249
   8.615 +#1253 := (or #1252 #1241)
   8.616 +#1254 := [th-lemma]: #1253
   8.617 +#1310 := [unit-resolution #1254 #1250]: #1252
   8.618 +#1326 := [mp #1310 #1325]: #796
   8.619 +#1328 := (or #724 #772)
   8.620 +decl uf_13 :: T3
   8.621 +#33 := uf_13
   8.622 +#609 := (uf_1 uf_13 #44)
   8.623 +#773 := (= #609 #771)
   8.624 +#775 := (ite #724 #773 #772)
   8.625 +#32 := (uf_1 uf_9 #6)
   8.626 +#553 := (pattern #32)
   8.627 +#34 := (uf_1 uf_13 #6)
   8.628 +#551 := (pattern #34)
   8.629 +#456 := (= #32 #89)
   8.630 +#455 := (= #34 #89)
   8.631 +#457 := (ite #119 #455 #456)
   8.632 +#554 := (forall (vars (?x4 T2)) (:pat #541 #551 #552 #553) #457)
   8.633 +#460 := (forall (vars (?x4 T2)) #457)
   8.634 +#557 := (iff #460 #554)
   8.635 +#555 := (iff #457 #457)
   8.636 +#556 := [refl]: #555
   8.637 +#558 := [quant-intro #556]: #557
   8.638 +#143 := (ite #119 #34 #32)
   8.639 +#373 := (= #89 #143)
   8.640 +#374 := (forall (vars (?x4 T2)) #373)
   8.641 +#461 := (iff #374 #460)
   8.642 +#458 := (iff #373 #457)
   8.643 +#459 := [rewrite]: #458
   8.644 +#462 := [quant-intro #459]: #461
   8.645 +#362 := (~ #374 #374)
   8.646 +#364 := (~ #373 #373)
   8.647 +#365 := [refl]: #364
   8.648 +#363 := [nnf-pos #365]: #362
   8.649 +#31 := (uf_3 uf_12 #6)
   8.650 +#148 := (= #31 #143)
   8.651 +#151 := (forall (vars (?x4 T2)) #148)
   8.652 +#375 := (iff #151 #374)
   8.653 +#376 := [rewrite* #113]: #375
   8.654 +#35 := (ite #17 #32 #34)
   8.655 +#36 := (= #31 #35)
   8.656 +#37 := (forall (vars (?x4 T2)) #36)
   8.657 +#152 := (iff #37 #151)
   8.658 +#149 := (iff #36 #148)
   8.659 +#146 := (= #35 #143)
   8.660 +#140 := (ite #118 #32 #34)
   8.661 +#144 := (= #140 #143)
   8.662 +#145 := [rewrite]: #144
   8.663 +#141 := (= #35 #140)
   8.664 +#142 := [monotonicity #123]: #141
   8.665 +#147 := [trans #142 #145]: #146
   8.666 +#150 := [monotonicity #147]: #149
   8.667 +#153 := [quant-intro #150]: #152
   8.668 +#115 := [asserted]: #37
   8.669 +#154 := [mp #115 #153]: #151
   8.670 +#377 := [mp #154 #376]: #374
   8.671 +#360 := [mp~ #377 #363]: #374
   8.672 +#463 := [mp #360 #462]: #460
   8.673 +#559 := [mp #463 #558]: #554
   8.674 +#778 := (not #554)
   8.675 +#779 := (or #778 #775)
   8.676 +#714 := (+ #611 #120)
   8.677 +#715 := (>= #714 0::int)
   8.678 +#774 := (ite #715 #773 #772)
   8.679 +#780 := (or #778 #774)
   8.680 +#782 := (iff #780 #779)
   8.681 +#784 := (iff #779 #779)
   8.682 +#785 := [rewrite]: #784
   8.683 +#776 := (iff #774 #775)
   8.684 +#727 := (iff #715 #724)
   8.685 +#717 := (+ #120 #611)
   8.686 +#720 := (>= #717 0::int)
   8.687 +#725 := (iff #720 #724)
   8.688 +#726 := [rewrite]: #725
   8.689 +#721 := (iff #715 #720)
   8.690 +#718 := (= #714 #717)
   8.691 +#719 := [rewrite]: #718
   8.692 +#722 := [monotonicity #719]: #721
   8.693 +#728 := [trans #722 #726]: #727
   8.694 +#777 := [monotonicity #728]: #776
   8.695 +#783 := [monotonicity #777]: #782
   8.696 +#786 := [trans #783 #785]: #782
   8.697 +#781 := [quant-inst]: #780
   8.698 +#787 := [mp #781 #786]: #779
   8.699 +#1327 := [unit-resolution #787 #559]: #775
   8.700 +#788 := (not #775)
   8.701 +#791 := (or #788 #724 #772)
   8.702 +#792 := [def-axiom]: #791
   8.703 +#1329 := [unit-resolution #792 #1327]: #1328
   8.704 +#1330 := [unit-resolution #1329 #1326]: #724
   8.705 +#988 := (>= #622 0::int)
   8.706 +#1331 := (or #1312 #988)
   8.707 +#1332 := [th-lemma]: #1331
   8.708 +#1333 := [unit-resolution #1332 #1281]: #988
   8.709 +#761 := (not #724)
   8.710 +#1334 := (not #988)
   8.711 +#1335 := (or #1271 #1334 #761)
   8.712 +#1336 := [th-lemma]: #1335
   8.713 +#1337 := [unit-resolution #1336 #1333 #1330]: #1271
   8.714 +#1338 := (not #1271)
   8.715 +#1340 := (or #1269 #1338 #1339)
   8.716 +#1341 := [th-lemma]: #1340
   8.717 +#1343 := [unit-resolution #1341 #1337]: #1342
   8.718 +#1344 := [unit-resolution #1343 #1299]: #1339
   8.719 +#990 := (>= #916 0::int)
   8.720 +#1345 := (or #1302 #990)
   8.721 +#1346 := [th-lemma]: #1345
   8.722 +#1347 := [unit-resolution #1346 #1301]: #990
   8.723 +#1348 := (not #990)
   8.724 +#1349 := (or #836 #1348 #1265)
   8.725 +#1350 := [th-lemma]: #1349
   8.726 +#1351 := [unit-resolution #1350 #1347 #1344]: #836
   8.727 +#1353 := (or #815 #800)
   8.728 +#801 := (uf_1 uf_13 #22)
   8.729 +#820 := (= #799 #801)
   8.730 +#823 := (ite #815 #820 #800)
   8.731 +#476 := (= #32 #79)
   8.732 +#475 := (= #34 #79)
   8.733 +#477 := (ite #157 #475 #476)
   8.734 +#568 := (forall (vars (?x6 T2)) (:pat #541 #551 #567 #553) #477)
   8.735 +#480 := (forall (vars (?x6 T2)) #477)
   8.736 +#571 := (iff #480 #568)
   8.737 +#569 := (iff #477 #477)
   8.738 +#570 := [refl]: #569
   8.739 +#572 := [quant-intro #570]: #571
   8.740 +#181 := (ite #157 #34 #32)
   8.741 +#383 := (= #79 #181)
   8.742 +#384 := (forall (vars (?x6 T2)) #383)
   8.743 +#481 := (iff #384 #480)
   8.744 +#478 := (iff #383 #477)
   8.745 +#479 := [rewrite]: #478
   8.746 +#482 := [quant-intro #479]: #481
   8.747 +#352 := (~ #384 #384)
   8.748 +#354 := (~ #383 #383)
   8.749 +#355 := [refl]: #354
   8.750 +#353 := [nnf-pos #355]: #352
   8.751 +#51 := (uf_3 uf_16 #6)
   8.752 +#186 := (= #51 #181)
   8.753 +#189 := (forall (vars (?x6 T2)) #186)
   8.754 +#385 := (iff #189 #384)
   8.755 +#386 := [rewrite* #113]: #385
   8.756 +#52 := (ite #42 #32 #34)
   8.757 +#53 := (= #51 #52)
   8.758 +#54 := (forall (vars (?x6 T2)) #53)
   8.759 +#190 := (iff #54 #189)
   8.760 +#187 := (iff #53 #186)
   8.761 +#184 := (= #52 #181)
   8.762 +#178 := (ite #156 #32 #34)
   8.763 +#182 := (= #178 #181)
   8.764 +#183 := [rewrite]: #182
   8.765 +#179 := (= #52 #178)
   8.766 +#180 := [monotonicity #161]: #179
   8.767 +#185 := [trans #180 #183]: #184
   8.768 +#188 := [monotonicity #185]: #187
   8.769 +#191 := [quant-intro #188]: #190
   8.770 +#139 := [asserted]: #54
   8.771 +#192 := [mp #139 #191]: #189
   8.772 +#387 := [mp #192 #386]: #384
   8.773 +#402 := [mp~ #387 #353]: #384
   8.774 +#483 := [mp #402 #482]: #480
   8.775 +#573 := [mp #483 #572]: #568
   8.776 +#634 := (not #568)
   8.777 +#826 := (or #634 #823)
   8.778 +#802 := (= #801 #799)
   8.779 +#804 := (+ #803 #158)
   8.780 +#805 := (>= #804 0::int)
   8.781 +#806 := (ite #805 #802 #800)
   8.782 +#827 := (or #634 #806)
   8.783 +#829 := (iff #827 #826)
   8.784 +#831 := (iff #826 #826)
   8.785 +#832 := [rewrite]: #831
   8.786 +#824 := (iff #806 #823)
   8.787 +#821 := (iff #802 #820)
   8.788 +#822 := [rewrite]: #821
   8.789 +#818 := (iff #805 #815)
   8.790 +#807 := (+ #158 #803)
   8.791 +#810 := (>= #807 0::int)
   8.792 +#816 := (iff #810 #815)
   8.793 +#817 := [rewrite]: #816
   8.794 +#811 := (iff #805 #810)
   8.795 +#808 := (= #804 #807)
   8.796 +#809 := [rewrite]: #808
   8.797 +#812 := [monotonicity #809]: #811
   8.798 +#819 := [trans #812 #817]: #818
   8.799 +#825 := [monotonicity #819 #822]: #824
   8.800 +#830 := [monotonicity #825]: #829
   8.801 +#833 := [trans #830 #832]: #829
   8.802 +#828 := [quant-inst]: #827
   8.803 +#834 := [mp #828 #833]: #826
   8.804 +#1352 := [unit-resolution #834 #573]: #823
   8.805 +#835 := (not #823)
   8.806 +#839 := (or #835 #815 #800)
   8.807 +#840 := [def-axiom]: #839
   8.808 +#1354 := [unit-resolution #840 #1352]: #1353
   8.809 +#1355 := [unit-resolution #1354 #1351]: #800
   8.810 +#1357 := [symm #1355]: #1356
   8.811 +#1358 := [trans #1357 #1309]: #1266
   8.812 +#1359 := (not #1266)
   8.813 +#1360 := (or #1359 #1272)
   8.814 +#1361 := [th-lemma]: #1360
   8.815 +#1362 := [unit-resolution #1361 #1358]: #1272
   8.816 +#1085 := (uf_1 uf_18 #22)
   8.817 +#1099 := (* -1::real #1085)
   8.818 +#1112 := (+ #902 #1099)
   8.819 +#1113 := (<= #1112 0::real)
   8.820 +#1137 := (not #1113)
   8.821 +#960 := (uf_1 #88 #22)
   8.822 +#1100 := (+ #960 #1099)
   8.823 +#1101 := (>= #1100 0::real)
   8.824 +#1118 := (or #1101 #1113)
   8.825 +#1121 := (not #1118)
   8.826 +#1125 := (or #1124 #1121)
   8.827 +#1086 := (+ #1085 #1084)
   8.828 +#1087 := (>= #1086 0::real)
   8.829 +#1088 := (* -1::real #960)
   8.830 +#1089 := (+ #1085 #1088)
   8.831 +#1090 := (<= #1089 0::real)
   8.832 +#1091 := (or #1090 #1087)
   8.833 +#1092 := (not #1091)
   8.834 +#1126 := (or #1124 #1092)
   8.835 +#1128 := (iff #1126 #1125)
   8.836 +#1130 := (iff #1125 #1125)
   8.837 +#1131 := [rewrite]: #1130
   8.838 +#1122 := (iff #1092 #1121)
   8.839 +#1119 := (iff #1091 #1118)
   8.840 +#1116 := (iff #1087 #1113)
   8.841 +#1106 := (+ #1084 #1085)
   8.842 +#1109 := (>= #1106 0::real)
   8.843 +#1114 := (iff #1109 #1113)
   8.844 +#1115 := [rewrite]: #1114
   8.845 +#1110 := (iff #1087 #1109)
   8.846 +#1107 := (= #1086 #1106)
   8.847 +#1108 := [rewrite]: #1107
   8.848 +#1111 := [monotonicity #1108]: #1110
   8.849 +#1117 := [trans #1111 #1115]: #1116
   8.850 +#1104 := (iff #1090 #1101)
   8.851 +#1093 := (+ #1088 #1085)
   8.852 +#1096 := (<= #1093 0::real)
   8.853 +#1102 := (iff #1096 #1101)
   8.854 +#1103 := [rewrite]: #1102
   8.855 +#1097 := (iff #1090 #1096)
   8.856 +#1094 := (= #1089 #1093)
   8.857 +#1095 := [rewrite]: #1094
   8.858 +#1098 := [monotonicity #1095]: #1097
   8.859 +#1105 := [trans #1098 #1103]: #1104
   8.860 +#1120 := [monotonicity #1105 #1117]: #1119
   8.861 +#1123 := [monotonicity #1120]: #1122
   8.862 +#1129 := [monotonicity #1123]: #1128
   8.863 +#1132 := [trans #1129 #1131]: #1128
   8.864 +#1127 := [quant-inst]: #1126
   8.865 +#1133 := [mp #1127 #1132]: #1125
   8.866 +#1363 := [unit-resolution #1133 #606]: #1121
   8.867 +#1138 := (or #1118 #1137)
   8.868 +#1139 := [def-axiom]: #1138
   8.869 +#1364 := [unit-resolution #1139 #1363]: #1137
   8.870 +#1200 := (+ #799 #1099)
   8.871 +#1201 := (>= #1200 0::real)
   8.872 +#1231 := (not #1201)
   8.873 +#847 := (uf_1 #83 #22)
   8.874 +#1210 := (+ #847 #1099)
   8.875 +#1211 := (<= #1210 0::real)
   8.876 +#1216 := (or #1201 #1211)
   8.877 +#1219 := (not #1216)
   8.878 +#1222 := (or #1066 #1219)
   8.879 +#1197 := (* -1::real #847)
   8.880 +#1198 := (+ #1085 #1197)
   8.881 +#1199 := (>= #1198 0::real)
   8.882 +#1202 := (or #1201 #1199)
   8.883 +#1203 := (not #1202)
   8.884 +#1223 := (or #1066 #1203)
   8.885 +#1225 := (iff #1223 #1222)
   8.886 +#1227 := (iff #1222 #1222)
   8.887 +#1228 := [rewrite]: #1227
   8.888 +#1220 := (iff #1203 #1219)
   8.889 +#1217 := (iff #1202 #1216)
   8.890 +#1214 := (iff #1199 #1211)
   8.891 +#1204 := (+ #1197 #1085)
   8.892 +#1207 := (>= #1204 0::real)
   8.893 +#1212 := (iff #1207 #1211)
   8.894 +#1213 := [rewrite]: #1212
   8.895 +#1208 := (iff #1199 #1207)
   8.896 +#1205 := (= #1198 #1204)
   8.897 +#1206 := [rewrite]: #1205
   8.898 +#1209 := [monotonicity #1206]: #1208
   8.899 +#1215 := [trans #1209 #1213]: #1214
   8.900 +#1218 := [monotonicity #1215]: #1217
   8.901 +#1221 := [monotonicity #1218]: #1220
   8.902 +#1226 := [monotonicity #1221]: #1225
   8.903 +#1229 := [trans #1226 #1228]: #1225
   8.904 +#1224 := [quant-inst]: #1223
   8.905 +#1230 := [mp #1224 #1229]: #1222
   8.906 +#1365 := [unit-resolution #1230 #600]: #1219
   8.907 +#1232 := (or #1216 #1231)
   8.908 +#1233 := [def-axiom]: #1232
   8.909 +#1366 := [unit-resolution #1233 #1365]: #1231
   8.910 +[th-lemma #1366 #1364 #1362]: false
   8.911 +unsat
   8.912 +NQHwTeL311Tq3wf2s5BReA 419 0
   8.913 +#2 := false
   8.914 +#194 := 0::real
   8.915 +decl uf_4 :: (-> T2 T3 real)
   8.916 +decl uf_6 :: (-> T1 T3)
   8.917 +decl uf_3 :: T1
   8.918 +#21 := uf_3
   8.919 +#25 := (uf_6 uf_3)
   8.920 +decl uf_5 :: T2
   8.921 +#24 := uf_5
   8.922 +#26 := (uf_4 uf_5 #25)
   8.923 +decl uf_7 :: T2
   8.924 +#27 := uf_7
   8.925 +#28 := (uf_4 uf_7 #25)
   8.926 +decl uf_10 :: T1
   8.927 +#38 := uf_10
   8.928 +#42 := (uf_6 uf_10)
   8.929 +decl uf_9 :: T2
   8.930 +#33 := uf_9
   8.931 +#43 := (uf_4 uf_9 #42)
   8.932 +#41 := (= uf_3 uf_10)
   8.933 +#44 := (ite #41 #43 #28)
   8.934 +#9 := 0::int
   8.935 +decl uf_2 :: (-> T1 int)
   8.936 +#39 := (uf_2 uf_10)
   8.937 +#226 := -1::int
   8.938 +#229 := (* -1::int #39)
   8.939 +#22 := (uf_2 uf_3)
   8.940 +#230 := (+ #22 #229)
   8.941 +#228 := (>= #230 0::int)
   8.942 +#236 := (ite #228 #44 #26)
   8.943 +#192 := -1::real
   8.944 +#244 := (* -1::real #236)
   8.945 +#642 := (+ #26 #244)
   8.946 +#643 := (<= #642 0::real)
   8.947 +#567 := (= #26 #236)
   8.948 +#227 := (not #228)
   8.949 +decl uf_1 :: (-> int T1)
   8.950 +#593 := (uf_1 #39)
   8.951 +#660 := (= #593 uf_10)
   8.952 +#594 := (= uf_10 #593)
   8.953 +#4 := (:var 0 T1)
   8.954 +#5 := (uf_2 #4)
   8.955 +#546 := (pattern #5)
   8.956 +#6 := (uf_1 #5)
   8.957 +#93 := (= #4 #6)
   8.958 +#547 := (forall (vars (?x1 T1)) (:pat #546) #93)
   8.959 +#96 := (forall (vars (?x1 T1)) #93)
   8.960 +#550 := (iff #96 #547)
   8.961 +#548 := (iff #93 #93)
   8.962 +#549 := [refl]: #548
   8.963 +#551 := [quant-intro #549]: #550
   8.964 +#448 := (~ #96 #96)
   8.965 +#450 := (~ #93 #93)
   8.966 +#451 := [refl]: #450
   8.967 +#449 := [nnf-pos #451]: #448
   8.968 +#7 := (= #6 #4)
   8.969 +#8 := (forall (vars (?x1 T1)) #7)
   8.970 +#97 := (iff #8 #96)
   8.971 +#94 := (iff #7 #93)
   8.972 +#95 := [rewrite]: #94
   8.973 +#98 := [quant-intro #95]: #97
   8.974 +#92 := [asserted]: #8
   8.975 +#101 := [mp #92 #98]: #96
   8.976 +#446 := [mp~ #101 #449]: #96
   8.977 +#552 := [mp #446 #551]: #547
   8.978 +#595 := (not #547)
   8.979 +#600 := (or #595 #594)
   8.980 +#601 := [quant-inst]: #600
   8.981 +#654 := [unit-resolution #601 #552]: #594
   8.982 +#680 := [symm #654]: #660
   8.983 +#681 := (= uf_3 #593)
   8.984 +#591 := (uf_1 #22)
   8.985 +#658 := (= #591 #593)
   8.986 +#656 := (= #593 #591)
   8.987 +#652 := (= #39 #22)
   8.988 +#647 := (= #22 #39)
   8.989 +#290 := (<= #230 0::int)
   8.990 +#70 := (<= #22 #39)
   8.991 +#388 := (iff #70 #290)
   8.992 +#389 := [rewrite]: #388
   8.993 +#341 := [asserted]: #70
   8.994 +#390 := [mp #341 #389]: #290
   8.995 +#646 := [hypothesis]: #228
   8.996 +#648 := [th-lemma #646 #390]: #647
   8.997 +#653 := [symm #648]: #652
   8.998 +#657 := [monotonicity #653]: #656
   8.999 +#659 := [symm #657]: #658
  8.1000 +#592 := (= uf_3 #591)
  8.1001 +#596 := (or #595 #592)
  8.1002 +#597 := [quant-inst]: #596
  8.1003 +#655 := [unit-resolution #597 #552]: #592
  8.1004 +#682 := [trans #655 #659]: #681
  8.1005 +#683 := [trans #682 #680]: #41
  8.1006 +#570 := (not #41)
  8.1007 +decl uf_11 :: T2
  8.1008 +#47 := uf_11
  8.1009 +#59 := (uf_4 uf_11 #42)
  8.1010 +#278 := (ite #41 #26 #59)
  8.1011 +#459 := (* -1::real #278)
  8.1012 +#637 := (+ #26 #459)
  8.1013 +#639 := (>= #637 0::real)
  8.1014 +#585 := (= #26 #278)
  8.1015 +#661 := [hypothesis]: #41
  8.1016 +#587 := (or #570 #585)
  8.1017 +#588 := [def-axiom]: #587
  8.1018 +#662 := [unit-resolution #588 #661]: #585
  8.1019 +#663 := (not #585)
  8.1020 +#664 := (or #663 #639)
  8.1021 +#665 := [th-lemma]: #664
  8.1022 +#666 := [unit-resolution #665 #662]: #639
  8.1023 +decl uf_8 :: T2
  8.1024 +#30 := uf_8
  8.1025 +#56 := (uf_4 uf_8 #42)
  8.1026 +#357 := (* -1::real #56)
  8.1027 +#358 := (+ #43 #357)
  8.1028 +#356 := (>= #358 0::real)
  8.1029 +#355 := (not #356)
  8.1030 +#374 := (* -1::real #59)
  8.1031 +#375 := (+ #56 #374)
  8.1032 +#373 := (>= #375 0::real)
  8.1033 +#376 := (not #373)
  8.1034 +#381 := (and #355 #376)
  8.1035 +#64 := (< #39 #39)
  8.1036 +#67 := (ite #64 #43 #59)
  8.1037 +#68 := (< #56 #67)
  8.1038 +#53 := (uf_4 uf_5 #42)
  8.1039 +#65 := (ite #64 #53 #43)
  8.1040 +#66 := (< #65 #56)
  8.1041 +#69 := (and #66 #68)
  8.1042 +#382 := (iff #69 #381)
  8.1043 +#379 := (iff #68 #376)
  8.1044 +#370 := (< #56 #59)
  8.1045 +#377 := (iff #370 #376)
  8.1046 +#378 := [rewrite]: #377
  8.1047 +#371 := (iff #68 #370)
  8.1048 +#368 := (= #67 #59)
  8.1049 +#363 := (ite false #43 #59)
  8.1050 +#366 := (= #363 #59)
  8.1051 +#367 := [rewrite]: #366
  8.1052 +#364 := (= #67 #363)
  8.1053 +#343 := (iff #64 false)
  8.1054 +#344 := [rewrite]: #343
  8.1055 +#365 := [monotonicity #344]: #364
  8.1056 +#369 := [trans #365 #367]: #368
  8.1057 +#372 := [monotonicity #369]: #371
  8.1058 +#380 := [trans #372 #378]: #379
  8.1059 +#361 := (iff #66 #355)
  8.1060 +#352 := (< #43 #56)
  8.1061 +#359 := (iff #352 #355)
  8.1062 +#360 := [rewrite]: #359
  8.1063 +#353 := (iff #66 #352)
  8.1064 +#350 := (= #65 #43)
  8.1065 +#345 := (ite false #53 #43)
  8.1066 +#348 := (= #345 #43)
  8.1067 +#349 := [rewrite]: #348
  8.1068 +#346 := (= #65 #345)
  8.1069 +#347 := [monotonicity #344]: #346
  8.1070 +#351 := [trans #347 #349]: #350
  8.1071 +#354 := [monotonicity #351]: #353
  8.1072 +#362 := [trans #354 #360]: #361
  8.1073 +#383 := [monotonicity #362 #380]: #382
  8.1074 +#340 := [asserted]: #69
  8.1075 +#384 := [mp #340 #383]: #381
  8.1076 +#385 := [and-elim #384]: #355
  8.1077 +#394 := (* -1::real #53)
  8.1078 +#395 := (+ #43 #394)
  8.1079 +#393 := (>= #395 0::real)
  8.1080 +#54 := (uf_4 uf_7 #42)
  8.1081 +#402 := (* -1::real #54)
  8.1082 +#403 := (+ #53 #402)
  8.1083 +#401 := (>= #403 0::real)
  8.1084 +#397 := (+ #43 #374)
  8.1085 +#398 := (<= #397 0::real)
  8.1086 +#412 := (and #393 #398 #401)
  8.1087 +#73 := (<= #43 #59)
  8.1088 +#72 := (<= #53 #43)
  8.1089 +#74 := (and #72 #73)
  8.1090 +#71 := (<= #54 #53)
  8.1091 +#75 := (and #71 #74)
  8.1092 +#415 := (iff #75 #412)
  8.1093 +#406 := (and #393 #398)
  8.1094 +#409 := (and #401 #406)
  8.1095 +#413 := (iff #409 #412)
  8.1096 +#414 := [rewrite]: #413
  8.1097 +#410 := (iff #75 #409)
  8.1098 +#407 := (iff #74 #406)
  8.1099 +#399 := (iff #73 #398)
  8.1100 +#400 := [rewrite]: #399
  8.1101 +#392 := (iff #72 #393)
  8.1102 +#396 := [rewrite]: #392
  8.1103 +#408 := [monotonicity #396 #400]: #407
  8.1104 +#404 := (iff #71 #401)
  8.1105 +#405 := [rewrite]: #404
  8.1106 +#411 := [monotonicity #405 #408]: #410
  8.1107 +#416 := [trans #411 #414]: #415
  8.1108 +#342 := [asserted]: #75
  8.1109 +#417 := [mp #342 #416]: #412
  8.1110 +#418 := [and-elim #417]: #393
  8.1111 +#650 := (+ #26 #394)
  8.1112 +#651 := (<= #650 0::real)
  8.1113 +#649 := (= #26 #53)
  8.1114 +#671 := (= #53 #26)
  8.1115 +#669 := (= #42 #25)
  8.1116 +#667 := (= #25 #42)
  8.1117 +#668 := [monotonicity #661]: #667
  8.1118 +#670 := [symm #668]: #669
  8.1119 +#672 := [monotonicity #670]: #671
  8.1120 +#673 := [symm #672]: #649
  8.1121 +#674 := (not #649)
  8.1122 +#675 := (or #674 #651)
  8.1123 +#676 := [th-lemma]: #675
  8.1124 +#677 := [unit-resolution #676 #673]: #651
  8.1125 +#462 := (+ #56 #459)
  8.1126 +#465 := (>= #462 0::real)
  8.1127 +#438 := (not #465)
  8.1128 +#316 := (ite #290 #278 #43)
  8.1129 +#326 := (* -1::real #316)
  8.1130 +#327 := (+ #56 #326)
  8.1131 +#325 := (>= #327 0::real)
  8.1132 +#324 := (not #325)
  8.1133 +#439 := (iff #324 #438)
  8.1134 +#466 := (iff #325 #465)
  8.1135 +#463 := (= #327 #462)
  8.1136 +#460 := (= #326 #459)
  8.1137 +#457 := (= #316 #278)
  8.1138 +#1 := true
  8.1139 +#452 := (ite true #278 #43)
  8.1140 +#455 := (= #452 #278)
  8.1141 +#456 := [rewrite]: #455
  8.1142 +#453 := (= #316 #452)
  8.1143 +#444 := (iff #290 true)
  8.1144 +#445 := [iff-true #390]: #444
  8.1145 +#454 := [monotonicity #445]: #453
  8.1146 +#458 := [trans #454 #456]: #457
  8.1147 +#461 := [monotonicity #458]: #460
  8.1148 +#464 := [monotonicity #461]: #463
  8.1149 +#467 := [monotonicity #464]: #466
  8.1150 +#468 := [monotonicity #467]: #439
  8.1151 +#297 := (ite #290 #54 #53)
  8.1152 +#305 := (* -1::real #297)
  8.1153 +#306 := (+ #56 #305)
  8.1154 +#307 := (<= #306 0::real)
  8.1155 +#308 := (not #307)
  8.1156 +#332 := (and #308 #324)
  8.1157 +#58 := (= uf_10 uf_3)
  8.1158 +#60 := (ite #58 #26 #59)
  8.1159 +#52 := (< #39 #22)
  8.1160 +#61 := (ite #52 #43 #60)
  8.1161 +#62 := (< #56 #61)
  8.1162 +#55 := (ite #52 #53 #54)
  8.1163 +#57 := (< #55 #56)
  8.1164 +#63 := (and #57 #62)
  8.1165 +#335 := (iff #63 #332)
  8.1166 +#281 := (ite #52 #43 #278)
  8.1167 +#284 := (< #56 #281)
  8.1168 +#287 := (and #57 #284)
  8.1169 +#333 := (iff #287 #332)
  8.1170 +#330 := (iff #284 #324)
  8.1171 +#321 := (< #56 #316)
  8.1172 +#328 := (iff #321 #324)
  8.1173 +#329 := [rewrite]: #328
  8.1174 +#322 := (iff #284 #321)
  8.1175 +#319 := (= #281 #316)
  8.1176 +#291 := (not #290)
  8.1177 +#313 := (ite #291 #43 #278)
  8.1178 +#317 := (= #313 #316)
  8.1179 +#318 := [rewrite]: #317
  8.1180 +#314 := (= #281 #313)
  8.1181 +#292 := (iff #52 #291)
  8.1182 +#293 := [rewrite]: #292
  8.1183 +#315 := [monotonicity #293]: #314
  8.1184 +#320 := [trans #315 #318]: #319
  8.1185 +#323 := [monotonicity #320]: #322
  8.1186 +#331 := [trans #323 #329]: #330
  8.1187 +#311 := (iff #57 #308)
  8.1188 +#302 := (< #297 #56)
  8.1189 +#309 := (iff #302 #308)
  8.1190 +#310 := [rewrite]: #309
  8.1191 +#303 := (iff #57 #302)
  8.1192 +#300 := (= #55 #297)
  8.1193 +#294 := (ite #291 #53 #54)
  8.1194 +#298 := (= #294 #297)
  8.1195 +#299 := [rewrite]: #298
  8.1196 +#295 := (= #55 #294)
  8.1197 +#296 := [monotonicity #293]: #295
  8.1198 +#301 := [trans #296 #299]: #300
  8.1199 +#304 := [monotonicity #301]: #303
  8.1200 +#312 := [trans #304 #310]: #311
  8.1201 +#334 := [monotonicity #312 #331]: #333
  8.1202 +#288 := (iff #63 #287)
  8.1203 +#285 := (iff #62 #284)
  8.1204 +#282 := (= #61 #281)
  8.1205 +#279 := (= #60 #278)
  8.1206 +#225 := (iff #58 #41)
  8.1207 +#277 := [rewrite]: #225
  8.1208 +#280 := [monotonicity #277]: #279
  8.1209 +#283 := [monotonicity #280]: #282
  8.1210 +#286 := [monotonicity #283]: #285
  8.1211 +#289 := [monotonicity #286]: #288
  8.1212 +#336 := [trans #289 #334]: #335
  8.1213 +#179 := [asserted]: #63
  8.1214 +#337 := [mp #179 #336]: #332
  8.1215 +#339 := [and-elim #337]: #324
  8.1216 +#469 := [mp #339 #468]: #438
  8.1217 +#678 := [th-lemma #469 #677 #418 #385 #666]: false
  8.1218 +#679 := [lemma #678]: #570
  8.1219 +#684 := [unit-resolution #679 #683]: false
  8.1220 +#685 := [lemma #684]: #227
  8.1221 +#577 := (or #228 #567)
  8.1222 +#578 := [def-axiom]: #577
  8.1223 +#645 := [unit-resolution #578 #685]: #567
  8.1224 +#686 := (not #567)
  8.1225 +#687 := (or #686 #643)
  8.1226 +#688 := [th-lemma]: #687
  8.1227 +#689 := [unit-resolution #688 #645]: #643
  8.1228 +#31 := (uf_4 uf_8 #25)
  8.1229 +#245 := (+ #31 #244)
  8.1230 +#246 := (<= #245 0::real)
  8.1231 +#247 := (not #246)
  8.1232 +#34 := (uf_4 uf_9 #25)
  8.1233 +#48 := (uf_4 uf_11 #25)
  8.1234 +#255 := (ite #228 #48 #34)
  8.1235 +#264 := (* -1::real #255)
  8.1236 +#265 := (+ #31 #264)
  8.1237 +#263 := (>= #265 0::real)
  8.1238 +#266 := (not #263)
  8.1239 +#271 := (and #247 #266)
  8.1240 +#40 := (< #22 #39)
  8.1241 +#49 := (ite #40 #34 #48)
  8.1242 +#50 := (< #31 #49)
  8.1243 +#45 := (ite #40 #26 #44)
  8.1244 +#46 := (< #45 #31)
  8.1245 +#51 := (and #46 #50)
  8.1246 +#272 := (iff #51 #271)
  8.1247 +#269 := (iff #50 #266)
  8.1248 +#260 := (< #31 #255)
  8.1249 +#267 := (iff #260 #266)
  8.1250 +#268 := [rewrite]: #267
  8.1251 +#261 := (iff #50 #260)
  8.1252 +#258 := (= #49 #255)
  8.1253 +#252 := (ite #227 #34 #48)
  8.1254 +#256 := (= #252 #255)
  8.1255 +#257 := [rewrite]: #256
  8.1256 +#253 := (= #49 #252)
  8.1257 +#231 := (iff #40 #227)
  8.1258 +#232 := [rewrite]: #231
  8.1259 +#254 := [monotonicity #232]: #253
  8.1260 +#259 := [trans #254 #257]: #258
  8.1261 +#262 := [monotonicity #259]: #261
  8.1262 +#270 := [trans #262 #268]: #269
  8.1263 +#250 := (iff #46 #247)
  8.1264 +#241 := (< #236 #31)
  8.1265 +#248 := (iff #241 #247)
  8.1266 +#249 := [rewrite]: #248
  8.1267 +#242 := (iff #46 #241)
  8.1268 +#239 := (= #45 #236)
  8.1269 +#233 := (ite #227 #26 #44)
  8.1270 +#237 := (= #233 #236)
  8.1271 +#238 := [rewrite]: #237
  8.1272 +#234 := (= #45 #233)
  8.1273 +#235 := [monotonicity #232]: #234
  8.1274 +#240 := [trans #235 #238]: #239
  8.1275 +#243 := [monotonicity #240]: #242
  8.1276 +#251 := [trans #243 #249]: #250
  8.1277 +#273 := [monotonicity #251 #270]: #272
  8.1278 +#178 := [asserted]: #51
  8.1279 +#274 := [mp #178 #273]: #271
  8.1280 +#275 := [and-elim #274]: #247
  8.1281 +#196 := (* -1::real #31)
  8.1282 +#212 := (+ #26 #196)
  8.1283 +#213 := (<= #212 0::real)
  8.1284 +#214 := (not #213)
  8.1285 +#197 := (+ #28 #196)
  8.1286 +#195 := (>= #197 0::real)
  8.1287 +#193 := (not #195)
  8.1288 +#219 := (and #193 #214)
  8.1289 +#23 := (< #22 #22)
  8.1290 +#35 := (ite #23 #34 #26)
  8.1291 +#36 := (< #31 #35)
  8.1292 +#29 := (ite #23 #26 #28)
  8.1293 +#32 := (< #29 #31)
  8.1294 +#37 := (and #32 #36)
  8.1295 +#220 := (iff #37 #219)
  8.1296 +#217 := (iff #36 #214)
  8.1297 +#209 := (< #31 #26)
  8.1298 +#215 := (iff #209 #214)
  8.1299 +#216 := [rewrite]: #215
  8.1300 +#210 := (iff #36 #209)
  8.1301 +#207 := (= #35 #26)
  8.1302 +#202 := (ite false #34 #26)
  8.1303 +#205 := (= #202 #26)
  8.1304 +#206 := [rewrite]: #205
  8.1305 +#203 := (= #35 #202)
  8.1306 +#180 := (iff #23 false)
  8.1307 +#181 := [rewrite]: #180
  8.1308 +#204 := [monotonicity #181]: #203
  8.1309 +#208 := [trans #204 #206]: #207
  8.1310 +#211 := [monotonicity #208]: #210
  8.1311 +#218 := [trans #211 #216]: #217
  8.1312 +#200 := (iff #32 #193)
  8.1313 +#189 := (< #28 #31)
  8.1314 +#198 := (iff #189 #193)
  8.1315 +#199 := [rewrite]: #198
  8.1316 +#190 := (iff #32 #189)
  8.1317 +#187 := (= #29 #28)
  8.1318 +#182 := (ite false #26 #28)
  8.1319 +#185 := (= #182 #28)
  8.1320 +#186 := [rewrite]: #185
  8.1321 +#183 := (= #29 #182)
  8.1322 +#184 := [monotonicity #181]: #183
  8.1323 +#188 := [trans #184 #186]: #187
  8.1324 +#191 := [monotonicity #188]: #190
  8.1325 +#201 := [trans #191 #199]: #200
  8.1326 +#221 := [monotonicity #201 #218]: #220
  8.1327 +#177 := [asserted]: #37
  8.1328 +#222 := [mp #177 #221]: #219
  8.1329 +#224 := [and-elim #222]: #214
  8.1330 +[th-lemma #224 #275 #689]: false
  8.1331 +unsat
  8.1332 +NX/HT1QOfbspC2LtZNKpBA 428 0
  8.1333 +#2 := false
  8.1334 +decl uf_10 :: T1
  8.1335 +#38 := uf_10
  8.1336 +decl uf_3 :: T1
  8.1337 +#21 := uf_3
  8.1338 +#45 := (= uf_3 uf_10)
  8.1339 +decl uf_1 :: (-> int T1)
  8.1340 +decl uf_2 :: (-> T1 int)
  8.1341 +#39 := (uf_2 uf_10)
  8.1342 +#588 := (uf_1 #39)
  8.1343 +#686 := (= #588 uf_10)
  8.1344 +#589 := (= uf_10 #588)
  8.1345 +#4 := (:var 0 T1)
  8.1346 +#5 := (uf_2 #4)
  8.1347 +#541 := (pattern #5)
  8.1348 +#6 := (uf_1 #5)
  8.1349 +#93 := (= #4 #6)
  8.1350 +#542 := (forall (vars (?x1 T1)) (:pat #541) #93)
  8.1351 +#96 := (forall (vars (?x1 T1)) #93)
  8.1352 +#545 := (iff #96 #542)
  8.1353 +#543 := (iff #93 #93)
  8.1354 +#544 := [refl]: #543
  8.1355 +#546 := [quant-intro #544]: #545
  8.1356 +#454 := (~ #96 #96)
  8.1357 +#456 := (~ #93 #93)
  8.1358 +#457 := [refl]: #456
  8.1359 +#455 := [nnf-pos #457]: #454
  8.1360 +#7 := (= #6 #4)
  8.1361 +#8 := (forall (vars (?x1 T1)) #7)
  8.1362 +#97 := (iff #8 #96)
  8.1363 +#94 := (iff #7 #93)
  8.1364 +#95 := [rewrite]: #94
  8.1365 +#98 := [quant-intro #95]: #97
  8.1366 +#92 := [asserted]: #8
  8.1367 +#101 := [mp #92 #98]: #96
  8.1368 +#452 := [mp~ #101 #455]: #96
  8.1369 +#547 := [mp #452 #546]: #542
  8.1370 +#590 := (not #542)
  8.1371 +#595 := (or #590 #589)
  8.1372 +#596 := [quant-inst]: #595
  8.1373 +#680 := [unit-resolution #596 #547]: #589
  8.1374 +#687 := [symm #680]: #686
  8.1375 +#688 := (= uf_3 #588)
  8.1376 +#22 := (uf_2 uf_3)
  8.1377 +#586 := (uf_1 #22)
  8.1378 +#684 := (= #586 #588)
  8.1379 +#682 := (= #588 #586)
  8.1380 +#678 := (= #39 #22)
  8.1381 +#676 := (= #22 #39)
  8.1382 +#9 := 0::int
  8.1383 +#227 := -1::int
  8.1384 +#230 := (* -1::int #39)
  8.1385 +#231 := (+ #22 #230)
  8.1386 +#296 := (<= #231 0::int)
  8.1387 +#70 := (<= #22 #39)
  8.1388 +#393 := (iff #70 #296)
  8.1389 +#394 := [rewrite]: #393
  8.1390 +#347 := [asserted]: #70
  8.1391 +#395 := [mp #347 #394]: #296
  8.1392 +#229 := (>= #231 0::int)
  8.1393 +decl uf_4 :: (-> T2 T3 real)
  8.1394 +decl uf_6 :: (-> T1 T3)
  8.1395 +#25 := (uf_6 uf_3)
  8.1396 +decl uf_7 :: T2
  8.1397 +#27 := uf_7
  8.1398 +#28 := (uf_4 uf_7 #25)
  8.1399 +decl uf_9 :: T2
  8.1400 +#33 := uf_9
  8.1401 +#34 := (uf_4 uf_9 #25)
  8.1402 +#46 := (uf_6 uf_10)
  8.1403 +decl uf_5 :: T2
  8.1404 +#24 := uf_5
  8.1405 +#47 := (uf_4 uf_5 #46)
  8.1406 +#48 := (ite #45 #47 #34)
  8.1407 +#256 := (ite #229 #48 #28)
  8.1408 +#568 := (= #28 #256)
  8.1409 +#648 := (not #568)
  8.1410 +#194 := 0::real
  8.1411 +#192 := -1::real
  8.1412 +#265 := (* -1::real #256)
  8.1413 +#640 := (+ #28 #265)
  8.1414 +#642 := (>= #640 0::real)
  8.1415 +#645 := (not #642)
  8.1416 +#643 := [hypothesis]: #642
  8.1417 +decl uf_8 :: T2
  8.1418 +#30 := uf_8
  8.1419 +#31 := (uf_4 uf_8 #25)
  8.1420 +#266 := (+ #31 #265)
  8.1421 +#264 := (>= #266 0::real)
  8.1422 +#267 := (not #264)
  8.1423 +#26 := (uf_4 uf_5 #25)
  8.1424 +decl uf_11 :: T2
  8.1425 +#41 := uf_11
  8.1426 +#42 := (uf_4 uf_11 #25)
  8.1427 +#237 := (ite #229 #42 #26)
  8.1428 +#245 := (* -1::real #237)
  8.1429 +#246 := (+ #31 #245)
  8.1430 +#247 := (<= #246 0::real)
  8.1431 +#248 := (not #247)
  8.1432 +#272 := (and #248 #267)
  8.1433 +#40 := (< #22 #39)
  8.1434 +#49 := (ite #40 #28 #48)
  8.1435 +#50 := (< #31 #49)
  8.1436 +#43 := (ite #40 #26 #42)
  8.1437 +#44 := (< #43 #31)
  8.1438 +#51 := (and #44 #50)
  8.1439 +#273 := (iff #51 #272)
  8.1440 +#270 := (iff #50 #267)
  8.1441 +#261 := (< #31 #256)
  8.1442 +#268 := (iff #261 #267)
  8.1443 +#269 := [rewrite]: #268
  8.1444 +#262 := (iff #50 #261)
  8.1445 +#259 := (= #49 #256)
  8.1446 +#228 := (not #229)
  8.1447 +#253 := (ite #228 #28 #48)
  8.1448 +#257 := (= #253 #256)
  8.1449 +#258 := [rewrite]: #257
  8.1450 +#254 := (= #49 #253)
  8.1451 +#232 := (iff #40 #228)
  8.1452 +#233 := [rewrite]: #232
  8.1453 +#255 := [monotonicity #233]: #254
  8.1454 +#260 := [trans #255 #258]: #259
  8.1455 +#263 := [monotonicity #260]: #262
  8.1456 +#271 := [trans #263 #269]: #270
  8.1457 +#251 := (iff #44 #248)
  8.1458 +#242 := (< #237 #31)
  8.1459 +#249 := (iff #242 #248)
  8.1460 +#250 := [rewrite]: #249
  8.1461 +#243 := (iff #44 #242)
  8.1462 +#240 := (= #43 #237)
  8.1463 +#234 := (ite #228 #26 #42)
  8.1464 +#238 := (= #234 #237)
  8.1465 +#239 := [rewrite]: #238
  8.1466 +#235 := (= #43 #234)
  8.1467 +#236 := [monotonicity #233]: #235
  8.1468 +#241 := [trans #236 #239]: #240
  8.1469 +#244 := [monotonicity #241]: #243
  8.1470 +#252 := [trans #244 #250]: #251
  8.1471 +#274 := [monotonicity #252 #271]: #273
  8.1472 +#178 := [asserted]: #51
  8.1473 +#275 := [mp #178 #274]: #272
  8.1474 +#277 := [and-elim #275]: #267
  8.1475 +#196 := (* -1::real #31)
  8.1476 +#197 := (+ #28 #196)
  8.1477 +#195 := (>= #197 0::real)
  8.1478 +#193 := (not #195)
  8.1479 +#213 := (* -1::real #34)
  8.1480 +#214 := (+ #31 #213)
  8.1481 +#212 := (>= #214 0::real)
  8.1482 +#215 := (not #212)
  8.1483 +#220 := (and #193 #215)
  8.1484 +#23 := (< #22 #22)
  8.1485 +#35 := (ite #23 #28 #34)
  8.1486 +#36 := (< #31 #35)
  8.1487 +#29 := (ite #23 #26 #28)
  8.1488 +#32 := (< #29 #31)
  8.1489 +#37 := (and #32 #36)
  8.1490 +#221 := (iff #37 #220)
  8.1491 +#218 := (iff #36 #215)
  8.1492 +#209 := (< #31 #34)
  8.1493 +#216 := (iff #209 #215)
  8.1494 +#217 := [rewrite]: #216
  8.1495 +#210 := (iff #36 #209)
  8.1496 +#207 := (= #35 #34)
  8.1497 +#202 := (ite false #28 #34)
  8.1498 +#205 := (= #202 #34)
  8.1499 +#206 := [rewrite]: #205
  8.1500 +#203 := (= #35 #202)
  8.1501 +#180 := (iff #23 false)
  8.1502 +#181 := [rewrite]: #180
  8.1503 +#204 := [monotonicity #181]: #203
  8.1504 +#208 := [trans #204 #206]: #207
  8.1505 +#211 := [monotonicity #208]: #210
  8.1506 +#219 := [trans #211 #217]: #218
  8.1507 +#200 := (iff #32 #193)
  8.1508 +#189 := (< #28 #31)
  8.1509 +#198 := (iff #189 #193)
  8.1510 +#199 := [rewrite]: #198
  8.1511 +#190 := (iff #32 #189)
  8.1512 +#187 := (= #29 #28)
  8.1513 +#182 := (ite false #26 #28)
  8.1514 +#185 := (= #182 #28)
  8.1515 +#186 := [rewrite]: #185
  8.1516 +#183 := (= #29 #182)
  8.1517 +#184 := [monotonicity #181]: #183
  8.1518 +#188 := [trans #184 #186]: #187
  8.1519 +#191 := [monotonicity #188]: #190
  8.1520 +#201 := [trans #191 #199]: #200
  8.1521 +#222 := [monotonicity #201 #219]: #221
  8.1522 +#177 := [asserted]: #37
  8.1523 +#223 := [mp #177 #222]: #220
  8.1524 +#224 := [and-elim #223]: #193
  8.1525 +#644 := [th-lemma #224 #277 #643]: false
  8.1526 +#646 := [lemma #644]: #645
  8.1527 +#647 := [hypothesis]: #568
  8.1528 +#649 := (or #648 #642)
  8.1529 +#650 := [th-lemma]: #649
  8.1530 +#651 := [unit-resolution #650 #647 #646]: false
  8.1531 +#652 := [lemma #651]: #648
  8.1532 +#578 := (or #229 #568)
  8.1533 +#579 := [def-axiom]: #578
  8.1534 +#675 := [unit-resolution #579 #652]: #229
  8.1535 +#677 := [th-lemma #675 #395]: #676
  8.1536 +#679 := [symm #677]: #678
  8.1537 +#683 := [monotonicity #679]: #682
  8.1538 +#685 := [symm #683]: #684
  8.1539 +#587 := (= uf_3 #586)
  8.1540 +#591 := (or #590 #587)
  8.1541 +#592 := [quant-inst]: #591
  8.1542 +#681 := [unit-resolution #592 #547]: #587
  8.1543 +#689 := [trans #681 #685]: #688
  8.1544 +#690 := [trans #689 #687]: #45
  8.1545 +#571 := (not #45)
  8.1546 +#54 := (uf_4 uf_11 #46)
  8.1547 +#279 := (ite #45 #28 #54)
  8.1548 +#465 := (* -1::real #279)
  8.1549 +#632 := (+ #28 #465)
  8.1550 +#633 := (<= #632 0::real)
  8.1551 +#580 := (= #28 #279)
  8.1552 +#656 := [hypothesis]: #45
  8.1553 +#582 := (or #571 #580)
  8.1554 +#583 := [def-axiom]: #582
  8.1555 +#657 := [unit-resolution #583 #656]: #580
  8.1556 +#658 := (not #580)
  8.1557 +#659 := (or #658 #633)
  8.1558 +#660 := [th-lemma]: #659
  8.1559 +#661 := [unit-resolution #660 #657]: #633
  8.1560 +#57 := (uf_4 uf_8 #46)
  8.1561 +#363 := (* -1::real #57)
  8.1562 +#379 := (+ #47 #363)
  8.1563 +#380 := (<= #379 0::real)
  8.1564 +#381 := (not #380)
  8.1565 +#364 := (+ #54 #363)
  8.1566 +#362 := (>= #364 0::real)
  8.1567 +#361 := (not #362)
  8.1568 +#386 := (and #361 #381)
  8.1569 +#59 := (uf_4 uf_7 #46)
  8.1570 +#64 := (< #39 #39)
  8.1571 +#67 := (ite #64 #59 #47)
  8.1572 +#68 := (< #57 #67)
  8.1573 +#65 := (ite #64 #47 #54)
  8.1574 +#66 := (< #65 #57)
  8.1575 +#69 := (and #66 #68)
  8.1576 +#387 := (iff #69 #386)
  8.1577 +#384 := (iff #68 #381)
  8.1578 +#376 := (< #57 #47)
  8.1579 +#382 := (iff #376 #381)
  8.1580 +#383 := [rewrite]: #382
  8.1581 +#377 := (iff #68 #376)
  8.1582 +#374 := (= #67 #47)
  8.1583 +#369 := (ite false #59 #47)
  8.1584 +#372 := (= #369 #47)
  8.1585 +#373 := [rewrite]: #372
  8.1586 +#370 := (= #67 #369)
  8.1587 +#349 := (iff #64 false)
  8.1588 +#350 := [rewrite]: #349
  8.1589 +#371 := [monotonicity #350]: #370
  8.1590 +#375 := [trans #371 #373]: #374
  8.1591 +#378 := [monotonicity #375]: #377
  8.1592 +#385 := [trans #378 #383]: #384
  8.1593 +#367 := (iff #66 #361)
  8.1594 +#358 := (< #54 #57)
  8.1595 +#365 := (iff #358 #361)
  8.1596 +#366 := [rewrite]: #365
  8.1597 +#359 := (iff #66 #358)
  8.1598 +#356 := (= #65 #54)
  8.1599 +#351 := (ite false #47 #54)
  8.1600 +#354 := (= #351 #54)
  8.1601 +#355 := [rewrite]: #354
  8.1602 +#352 := (= #65 #351)
  8.1603 +#353 := [monotonicity #350]: #352
  8.1604 +#357 := [trans #353 #355]: #356
  8.1605 +#360 := [monotonicity #357]: #359
  8.1606 +#368 := [trans #360 #366]: #367
  8.1607 +#388 := [monotonicity #368 #385]: #387
  8.1608 +#346 := [asserted]: #69
  8.1609 +#389 := [mp #346 #388]: #386
  8.1610 +#391 := [and-elim #389]: #381
  8.1611 +#397 := (* -1::real #59)
  8.1612 +#398 := (+ #47 #397)
  8.1613 +#399 := (<= #398 0::real)
  8.1614 +#409 := (* -1::real #54)
  8.1615 +#410 := (+ #47 #409)
  8.1616 +#408 := (>= #410 0::real)
  8.1617 +#60 := (uf_4 uf_9 #46)
  8.1618 +#402 := (* -1::real #60)
  8.1619 +#403 := (+ #59 #402)
  8.1620 +#404 := (<= #403 0::real)
  8.1621 +#418 := (and #399 #404 #408)
  8.1622 +#73 := (<= #59 #60)
  8.1623 +#72 := (<= #47 #59)
  8.1624 +#74 := (and #72 #73)
  8.1625 +#71 := (<= #54 #47)
  8.1626 +#75 := (and #71 #74)
  8.1627 +#421 := (iff #75 #418)
  8.1628 +#412 := (and #399 #404)
  8.1629 +#415 := (and #408 #412)
  8.1630 +#419 := (iff #415 #418)
  8.1631 +#420 := [rewrite]: #419
  8.1632 +#416 := (iff #75 #415)
  8.1633 +#413 := (iff #74 #412)
  8.1634 +#405 := (iff #73 #404)
  8.1635 +#406 := [rewrite]: #405
  8.1636 +#400 := (iff #72 #399)
  8.1637 +#401 := [rewrite]: #400
  8.1638 +#414 := [monotonicity #401 #406]: #413
  8.1639 +#407 := (iff #71 #408)
  8.1640 +#411 := [rewrite]: #407
  8.1641 +#417 := [monotonicity #411 #414]: #416
  8.1642 +#422 := [trans #417 #420]: #421
  8.1643 +#348 := [asserted]: #75
  8.1644 +#423 := [mp #348 #422]: #418
  8.1645 +#424 := [and-elim #423]: #399
  8.1646 +#637 := (+ #28 #397)
  8.1647 +#639 := (>= #637 0::real)
  8.1648 +#636 := (= #28 #59)
  8.1649 +#666 := (= #59 #28)
  8.1650 +#664 := (= #46 #25)
  8.1651 +#662 := (= #25 #46)
  8.1652 +#663 := [monotonicity #656]: #662
  8.1653 +#665 := [symm #663]: #664
  8.1654 +#667 := [monotonicity #665]: #666
  8.1655 +#668 := [symm #667]: #636
  8.1656 +#669 := (not #636)
  8.1657 +#670 := (or #669 #639)
  8.1658 +#671 := [th-lemma]: #670
  8.1659 +#672 := [unit-resolution #671 #668]: #639
  8.1660 +#468 := (+ #57 #465)
  8.1661 +#471 := (<= #468 0::real)
  8.1662 +#444 := (not #471)
  8.1663 +#322 := (ite #296 #279 #47)
  8.1664 +#330 := (* -1::real #322)
  8.1665 +#331 := (+ #57 #330)
  8.1666 +#332 := (<= #331 0::real)
  8.1667 +#333 := (not #332)
  8.1668 +#445 := (iff #333 #444)
  8.1669 +#472 := (iff #332 #471)
  8.1670 +#469 := (= #331 #468)
  8.1671 +#466 := (= #330 #465)
  8.1672 +#463 := (= #322 #279)
  8.1673 +#1 := true
  8.1674 +#458 := (ite true #279 #47)
  8.1675 +#461 := (= #458 #279)
  8.1676 +#462 := [rewrite]: #461
  8.1677 +#459 := (= #322 #458)
  8.1678 +#450 := (iff #296 true)
  8.1679 +#451 := [iff-true #395]: #450
  8.1680 +#460 := [monotonicity #451]: #459
  8.1681 +#464 := [trans #460 #462]: #463
  8.1682 +#467 := [monotonicity #464]: #466
  8.1683 +#470 := [monotonicity #467]: #469
  8.1684 +#473 := [monotonicity #470]: #472
  8.1685 +#474 := [monotonicity #473]: #445
  8.1686 +#303 := (ite #296 #60 #59)
  8.1687 +#313 := (* -1::real #303)
  8.1688 +#314 := (+ #57 #313)
  8.1689 +#312 := (>= #314 0::real)
  8.1690 +#311 := (not #312)
  8.1691 +#338 := (and #311 #333)
  8.1692 +#52 := (< #39 #22)
  8.1693 +#61 := (ite #52 #59 #60)
  8.1694 +#62 := (< #57 #61)
  8.1695 +#53 := (= uf_10 uf_3)
  8.1696 +#55 := (ite #53 #28 #54)
  8.1697 +#56 := (ite #52 #47 #55)
  8.1698 +#58 := (< #56 #57)
  8.1699 +#63 := (and #58 #62)
  8.1700 +#341 := (iff #63 #338)
  8.1701 +#282 := (ite #52 #47 #279)
  8.1702 +#285 := (< #282 #57)
  8.1703 +#291 := (and #62 #285)
  8.1704 +#339 := (iff #291 #338)
  8.1705 +#336 := (iff #285 #333)
  8.1706 +#327 := (< #322 #57)
  8.1707 +#334 := (iff #327 #333)
  8.1708 +#335 := [rewrite]: #334
  8.1709 +#328 := (iff #285 #327)
  8.1710 +#325 := (= #282 #322)
  8.1711 +#297 := (not #296)
  8.1712 +#319 := (ite #297 #47 #279)
  8.1713 +#323 := (= #319 #322)
  8.1714 +#324 := [rewrite]: #323
  8.1715 +#320 := (= #282 #319)
  8.1716 +#298 := (iff #52 #297)
  8.1717 +#299 := [rewrite]: #298
  8.1718 +#321 := [monotonicity #299]: #320
  8.1719 +#326 := [trans #321 #324]: #325
  8.1720 +#329 := [monotonicity #326]: #328
  8.1721 +#337 := [trans #329 #335]: #336
  8.1722 +#317 := (iff #62 #311)
  8.1723 +#308 := (< #57 #303)
  8.1724 +#315 := (iff #308 #311)
  8.1725 +#316 := [rewrite]: #315
  8.1726 +#309 := (iff #62 #308)
  8.1727 +#306 := (= #61 #303)
  8.1728 +#300 := (ite #297 #59 #60)
  8.1729 +#304 := (= #300 #303)
  8.1730 +#305 := [rewrite]: #304
  8.1731 +#301 := (= #61 #300)
  8.1732 +#302 := [monotonicity #299]: #301
  8.1733 +#307 := [trans #302 #305]: #306
  8.1734 +#310 := [monotonicity #307]: #309
  8.1735 +#318 := [trans #310 #316]: #317
  8.1736 +#340 := [monotonicity #318 #337]: #339
  8.1737 +#294 := (iff #63 #291)
  8.1738 +#288 := (and #285 #62)
  8.1739 +#292 := (iff #288 #291)
  8.1740 +#293 := [rewrite]: #292
  8.1741 +#289 := (iff #63 #288)
  8.1742 +#286 := (iff #58 #285)
  8.1743 +#283 := (= #56 #282)
  8.1744 +#280 := (= #55 #279)
  8.1745 +#226 := (iff #53 #45)
  8.1746 +#278 := [rewrite]: #226
  8.1747 +#281 := [monotonicity #278]: #280
  8.1748 +#284 := [monotonicity #281]: #283
  8.1749 +#287 := [monotonicity #284]: #286
  8.1750 +#290 := [monotonicity #287]: #289
  8.1751 +#295 := [trans #290 #293]: #294
  8.1752 +#342 := [trans #295 #340]: #341
  8.1753 +#179 := [asserted]: #63
  8.1754 +#343 := [mp #179 #342]: #338
  8.1755 +#345 := [and-elim #343]: #333
  8.1756 +#475 := [mp #345 #474]: #444
  8.1757 +#673 := [th-lemma #475 #672 #424 #391 #661]: false
  8.1758 +#674 := [lemma #673]: #571
  8.1759 +[unit-resolution #674 #690]: false
  8.1760 +unsat
  8.1761 +IL2powemHjRpCJYwmXFxyw 211 0
  8.1762 +#2 := false
  8.1763 +#33 := 0::real
  8.1764 +decl uf_11 :: (-> T5 T6 real)
  8.1765 +decl uf_15 :: T6
  8.1766 +#28 := uf_15
  8.1767 +decl uf_16 :: T5
  8.1768 +#30 := uf_16
  8.1769 +#31 := (uf_11 uf_16 uf_15)
  8.1770 +decl uf_12 :: (-> T7 T8 T5)
  8.1771 +decl uf_14 :: T8
  8.1772 +#26 := uf_14
  8.1773 +decl uf_13 :: (-> T1 T7)
  8.1774 +decl uf_8 :: T1
  8.1775 +#16 := uf_8
  8.1776 +#25 := (uf_13 uf_8)
  8.1777 +#27 := (uf_12 #25 uf_14)
  8.1778 +#29 := (uf_11 #27 uf_15)
  8.1779 +#73 := -1::real
  8.1780 +#84 := (* -1::real #29)
  8.1781 +#85 := (+ #84 #31)
  8.1782 +#74 := (* -1::real #31)
  8.1783 +#75 := (+ #29 #74)
  8.1784 +#112 := (>= #75 0::real)
  8.1785 +#119 := (ite #112 #75 #85)
  8.1786 +#127 := (* -1::real #119)
  8.1787 +decl uf_17 :: T5
  8.1788 +#37 := uf_17
  8.1789 +#38 := (uf_11 uf_17 uf_15)
  8.1790 +#102 := -1/3::real
  8.1791 +#103 := (* -1/3::real #38)
  8.1792 +#128 := (+ #103 #127)
  8.1793 +#100 := 1/3::real
  8.1794 +#101 := (* 1/3::real #31)
  8.1795 +#129 := (+ #101 #128)
  8.1796 +#130 := (<= #129 0::real)
  8.1797 +#131 := (not #130)
  8.1798 +#40 := 3::real
  8.1799 +#39 := (- #31 #38)
  8.1800 +#41 := (/ #39 3::real)
  8.1801 +#32 := (- #29 #31)
  8.1802 +#35 := (- #32)
  8.1803 +#34 := (< #32 0::real)
  8.1804 +#36 := (ite #34 #35 #32)
  8.1805 +#42 := (< #36 #41)
  8.1806 +#136 := (iff #42 #131)
  8.1807 +#104 := (+ #101 #103)
  8.1808 +#78 := (< #75 0::real)
  8.1809 +#90 := (ite #78 #85 #75)
  8.1810 +#109 := (< #90 #104)
  8.1811 +#134 := (iff #109 #131)
  8.1812 +#124 := (< #119 #104)
  8.1813 +#132 := (iff #124 #131)
  8.1814 +#133 := [rewrite]: #132
  8.1815 +#125 := (iff #109 #124)
  8.1816 +#122 := (= #90 #119)
  8.1817 +#113 := (not #112)
  8.1818 +#116 := (ite #113 #85 #75)
  8.1819 +#120 := (= #116 #119)
  8.1820 +#121 := [rewrite]: #120
  8.1821 +#117 := (= #90 #116)
  8.1822 +#114 := (iff #78 #113)
  8.1823 +#115 := [rewrite]: #114
  8.1824 +#118 := [monotonicity #115]: #117
  8.1825 +#123 := [trans #118 #121]: #122
  8.1826 +#126 := [monotonicity #123]: #125
  8.1827 +#135 := [trans #126 #133]: #134
  8.1828 +#110 := (iff #42 #109)
  8.1829 +#107 := (= #41 #104)
  8.1830 +#93 := (* -1::real #38)
  8.1831 +#94 := (+ #31 #93)
  8.1832 +#97 := (/ #94 3::real)
  8.1833 +#105 := (= #97 #104)
  8.1834 +#106 := [rewrite]: #105
  8.1835 +#98 := (= #41 #97)
  8.1836 +#95 := (= #39 #94)
  8.1837 +#96 := [rewrite]: #95
  8.1838 +#99 := [monotonicity #96]: #98
  8.1839 +#108 := [trans #99 #106]: #107
  8.1840 +#91 := (= #36 #90)
  8.1841 +#76 := (= #32 #75)
  8.1842 +#77 := [rewrite]: #76
  8.1843 +#88 := (= #35 #85)
  8.1844 +#81 := (- #75)
  8.1845 +#86 := (= #81 #85)
  8.1846 +#87 := [rewrite]: #86
  8.1847 +#82 := (= #35 #81)
  8.1848 +#83 := [monotonicity #77]: #82
  8.1849 +#89 := [trans #83 #87]: #88
  8.1850 +#79 := (iff #34 #78)
  8.1851 +#80 := [monotonicity #77]: #79
  8.1852 +#92 := [monotonicity #80 #89 #77]: #91
  8.1853 +#111 := [monotonicity #92 #108]: #110
  8.1854 +#137 := [trans #111 #135]: #136
  8.1855 +#72 := [asserted]: #42
  8.1856 +#138 := [mp #72 #137]: #131
  8.1857 +decl uf_1 :: T1
  8.1858 +#4 := uf_1
  8.1859 +#43 := (uf_13 uf_1)
  8.1860 +#44 := (uf_12 #43 uf_14)
  8.1861 +#45 := (uf_11 #44 uf_15)
  8.1862 +#149 := (* -1::real #45)
  8.1863 +#150 := (+ #38 #149)
  8.1864 +#140 := (+ #93 #45)
  8.1865 +#161 := (<= #150 0::real)
  8.1866 +#168 := (ite #161 #140 #150)
  8.1867 +#176 := (* -1::real #168)
  8.1868 +#177 := (+ #103 #176)
  8.1869 +#178 := (+ #101 #177)
  8.1870 +#179 := (<= #178 0::real)
  8.1871 +#180 := (not #179)
  8.1872 +#46 := (- #45 #38)
  8.1873 +#48 := (- #46)
  8.1874 +#47 := (< #46 0::real)
  8.1875 +#49 := (ite #47 #48 #46)
  8.1876 +#50 := (< #49 #41)
  8.1877 +#185 := (iff #50 #180)
  8.1878 +#143 := (< #140 0::real)
  8.1879 +#155 := (ite #143 #150 #140)
  8.1880 +#158 := (< #155 #104)
  8.1881 +#183 := (iff #158 #180)
  8.1882 +#173 := (< #168 #104)
  8.1883 +#181 := (iff #173 #180)
  8.1884 +#182 := [rewrite]: #181
  8.1885 +#174 := (iff #158 #173)
  8.1886 +#171 := (= #155 #168)
  8.1887 +#162 := (not #161)
  8.1888 +#165 := (ite #162 #150 #140)
  8.1889 +#169 := (= #165 #168)
  8.1890 +#170 := [rewrite]: #169
  8.1891 +#166 := (= #155 #165)
  8.1892 +#163 := (iff #143 #162)
  8.1893 +#164 := [rewrite]: #163
  8.1894 +#167 := [monotonicity #164]: #166
  8.1895 +#172 := [trans #167 #170]: #171
  8.1896 +#175 := [monotonicity #172]: #174
  8.1897 +#184 := [trans #175 #182]: #183
  8.1898 +#159 := (iff #50 #158)
  8.1899 +#156 := (= #49 #155)
  8.1900 +#141 := (= #46 #140)
  8.1901 +#142 := [rewrite]: #141
  8.1902 +#153 := (= #48 #150)
  8.1903 +#146 := (- #140)
  8.1904 +#151 := (= #146 #150)
  8.1905 +#152 := [rewrite]: #151
  8.1906 +#147 := (= #48 #146)
  8.1907 +#148 := [monotonicity #142]: #147
  8.1908 +#154 := [trans #148 #152]: #153
  8.1909 +#144 := (iff #47 #143)
  8.1910 +#145 := [monotonicity #142]: #144
  8.1911 +#157 := [monotonicity #145 #154 #142]: #156
  8.1912 +#160 := [monotonicity #157 #108]: #159
  8.1913 +#186 := [trans #160 #184]: #185
  8.1914 +#139 := [asserted]: #50
  8.1915 +#187 := [mp #139 #186]: #180
  8.1916 +#299 := (+ #140 #176)
  8.1917 +#300 := (<= #299 0::real)
  8.1918 +#290 := (= #140 #168)
  8.1919 +#329 := [hypothesis]: #162
  8.1920 +#191 := (+ #29 #149)
  8.1921 +#192 := (<= #191 0::real)
  8.1922 +#51 := (<= #29 #45)
  8.1923 +#193 := (iff #51 #192)
  8.1924 +#194 := [rewrite]: #193
  8.1925 +#188 := [asserted]: #51
  8.1926 +#195 := [mp #188 #194]: #192
  8.1927 +#298 := (+ #75 #127)
  8.1928 +#301 := (<= #298 0::real)
  8.1929 +#284 := (= #75 #119)
  8.1930 +#302 := [hypothesis]: #113
  8.1931 +#296 := (+ #85 #127)
  8.1932 +#297 := (<= #296 0::real)
  8.1933 +#285 := (= #85 #119)
  8.1934 +#288 := (or #112 #285)
  8.1935 +#289 := [def-axiom]: #288
  8.1936 +#303 := [unit-resolution #289 #302]: #285
  8.1937 +#304 := (not #285)
  8.1938 +#305 := (or #304 #297)
  8.1939 +#306 := [th-lemma]: #305
  8.1940 +#307 := [unit-resolution #306 #303]: #297
  8.1941 +#315 := (not #290)
  8.1942 +#310 := (not #300)
  8.1943 +#311 := (or #310 #112)
  8.1944 +#308 := [hypothesis]: #300
  8.1945 +#309 := [th-lemma #308 #307 #138 #302 #187 #195]: false
  8.1946 +#312 := [lemma #309]: #311
  8.1947 +#322 := [unit-resolution #312 #302]: #310
  8.1948 +#316 := (or #315 #300)
  8.1949 +#313 := [hypothesis]: #310
  8.1950 +#314 := [hypothesis]: #290
  8.1951 +#317 := [th-lemma]: #316
  8.1952 +#318 := [unit-resolution #317 #314 #313]: false
  8.1953 +#319 := [lemma #318]: #316
  8.1954 +#323 := [unit-resolution #319 #322]: #315
  8.1955 +#292 := (or #162 #290)
  8.1956 +#293 := [def-axiom]: #292
  8.1957 +#324 := [unit-resolution #293 #323]: #162
  8.1958 +#325 := [th-lemma #324 #307 #138 #302 #195]: false
  8.1959 +#326 := [lemma #325]: #112
  8.1960 +#286 := (or #113 #284)
  8.1961 +#287 := [def-axiom]: #286
  8.1962 +#330 := [unit-resolution #287 #326]: #284
  8.1963 +#331 := (not #284)
  8.1964 +#332 := (or #331 #301)
  8.1965 +#333 := [th-lemma]: #332
  8.1966 +#334 := [unit-resolution #333 #330]: #301
  8.1967 +#335 := [th-lemma #326 #334 #195 #329 #138]: false
  8.1968 +#336 := [lemma #335]: #161
  8.1969 +#327 := [unit-resolution #293 #336]: #290
  8.1970 +#328 := [unit-resolution #319 #327]: #300
  8.1971 +[th-lemma #326 #334 #195 #328 #187 #138]: false
  8.1972 +unsat
  8.1973 +GX51o3DUO/UBS3eNP2P9kA 285 0
  8.1974 +#2 := false
  8.1975 +#7 := 0::real
  8.1976 +decl uf_4 :: real
  8.1977 +#16 := uf_4
  8.1978 +#40 := -1::real
  8.1979 +#116 := (* -1::real uf_4)
  8.1980 +decl uf_3 :: real
  8.1981 +#11 := uf_3
  8.1982 +#117 := (+ uf_3 #116)
  8.1983 +#128 := (<= #117 0::real)
  8.1984 +#129 := (not #128)
  8.1985 +#220 := 2/3::real
  8.1986 +#221 := (* 2/3::real uf_3)
  8.1987 +#222 := (+ #221 #116)
  8.1988 +decl uf_2 :: real
  8.1989 +#5 := uf_2
  8.1990 +#67 := 1/3::real
  8.1991 +#68 := (* 1/3::real uf_2)
  8.1992 +#233 := (+ #68 #222)
  8.1993 +#243 := (<= #233 0::real)
  8.1994 +#268 := (not #243)
  8.1995 +#287 := [hypothesis]: #268
  8.1996 +#41 := (* -1::real uf_2)
  8.1997 +decl uf_1 :: real
  8.1998 +#4 := uf_1
  8.1999 +#42 := (+ uf_1 #41)
  8.2000 +#79 := (>= #42 0::real)
  8.2001 +#80 := (not #79)
  8.2002 +#297 := (or #80 #243)
  8.2003 +#158 := (+ uf_1 #116)
  8.2004 +#159 := (<= #158 0::real)
  8.2005 +#22 := (<= uf_1 uf_4)
  8.2006 +#160 := (iff #22 #159)
  8.2007 +#161 := [rewrite]: #160
  8.2008 +#155 := [asserted]: #22
  8.2009 +#162 := [mp #155 #161]: #159
  8.2010 +#200 := (* 1/3::real uf_3)
  8.2011 +#198 := -4/3::real
  8.2012 +#199 := (* -4/3::real uf_2)
  8.2013 +#201 := (+ #199 #200)
  8.2014 +#202 := (+ uf_1 #201)
  8.2015 +#203 := (>= #202 0::real)
  8.2016 +#258 := (not #203)
  8.2017 +#292 := [hypothesis]: #79
  8.2018 +#293 := (or #80 #258)
  8.2019 +#69 := -1/3::real
  8.2020 +#70 := (* -1/3::real uf_3)
  8.2021 +#186 := -2/3::real
  8.2022 +#187 := (* -2/3::real uf_2)
  8.2023 +#188 := (+ #187 #70)
  8.2024 +#189 := (+ uf_1 #188)
  8.2025 +#204 := (<= #189 0::real)
  8.2026 +#205 := (ite #79 #203 #204)
  8.2027 +#210 := (not #205)
  8.2028 +#51 := (* -1::real uf_1)
  8.2029 +#52 := (+ #51 uf_2)
  8.2030 +#86 := (ite #79 #42 #52)
  8.2031 +#94 := (* -1::real #86)
  8.2032 +#95 := (+ #70 #94)
  8.2033 +#96 := (+ #68 #95)
  8.2034 +#97 := (<= #96 0::real)
  8.2035 +#98 := (not #97)
  8.2036 +#211 := (iff #98 #210)
  8.2037 +#208 := (iff #97 #205)
  8.2038 +#182 := 4/3::real
  8.2039 +#183 := (* 4/3::real uf_2)
  8.2040 +#184 := (+ #183 #70)
  8.2041 +#185 := (+ #51 #184)
  8.2042 +#190 := (ite #79 #185 #189)
  8.2043 +#195 := (<= #190 0::real)
  8.2044 +#206 := (iff #195 #205)
  8.2045 +#207 := [rewrite]: #206
  8.2046 +#196 := (iff #97 #195)
  8.2047 +#193 := (= #96 #190)
  8.2048 +#172 := (+ #41 #70)
  8.2049 +#173 := (+ uf_1 #172)
  8.2050 +#170 := (+ uf_2 #70)
  8.2051 +#171 := (+ #51 #170)
  8.2052 +#174 := (ite #79 #171 #173)
  8.2053 +#179 := (+ #68 #174)
  8.2054 +#191 := (= #179 #190)
  8.2055 +#192 := [rewrite]: #191
  8.2056 +#180 := (= #96 #179)
  8.2057 +#177 := (= #95 #174)
  8.2058 +#164 := (ite #79 #52 #42)
  8.2059 +#167 := (+ #70 #164)
  8.2060 +#175 := (= #167 #174)
  8.2061 +#176 := [rewrite]: #175
  8.2062 +#168 := (= #95 #167)
  8.2063 +#156 := (= #94 #164)
  8.2064 +#165 := [rewrite]: #156
  8.2065 +#169 := [monotonicity #165]: #168
  8.2066 +#178 := [trans #169 #176]: #177
  8.2067 +#181 := [monotonicity #178]: #180
  8.2068 +#194 := [trans #181 #192]: #193
  8.2069 +#197 := [monotonicity #194]: #196
  8.2070 +#209 := [trans #197 #207]: #208
  8.2071 +#212 := [monotonicity #209]: #211
  8.2072 +#13 := 3::real
  8.2073 +#12 := (- uf_2 uf_3)
  8.2074 +#14 := (/ #12 3::real)
  8.2075 +#6 := (- uf_1 uf_2)
  8.2076 +#9 := (- #6)
  8.2077 +#8 := (< #6 0::real)
  8.2078 +#10 := (ite #8 #9 #6)
  8.2079 +#15 := (< #10 #14)
  8.2080 +#103 := (iff #15 #98)
  8.2081 +#71 := (+ #68 #70)
  8.2082 +#45 := (< #42 0::real)
  8.2083 +#57 := (ite #45 #52 #42)
  8.2084 +#76 := (< #57 #71)
  8.2085 +#101 := (iff #76 #98)
  8.2086 +#91 := (< #86 #71)
  8.2087 +#99 := (iff #91 #98)
  8.2088 +#100 := [rewrite]: #99
  8.2089 +#92 := (iff #76 #91)
  8.2090 +#89 := (= #57 #86)
  8.2091 +#83 := (ite #80 #52 #42)
  8.2092 +#87 := (= #83 #86)
  8.2093 +#88 := [rewrite]: #87
  8.2094 +#84 := (= #57 #83)
  8.2095 +#81 := (iff #45 #80)
  8.2096 +#82 := [rewrite]: #81
  8.2097 +#85 := [monotonicity #82]: #84
  8.2098 +#90 := [trans #85 #88]: #89
  8.2099 +#93 := [monotonicity #90]: #92
  8.2100 +#102 := [trans #93 #100]: #101
  8.2101 +#77 := (iff #15 #76)
  8.2102 +#74 := (= #14 #71)
  8.2103 +#60 := (* -1::real uf_3)
  8.2104 +#61 := (+ uf_2 #60)
  8.2105 +#64 := (/ #61 3::real)
  8.2106 +#72 := (= #64 #71)
  8.2107 +#73 := [rewrite]: #72
  8.2108 +#65 := (= #14 #64)
  8.2109 +#62 := (= #12 #61)
  8.2110 +#63 := [rewrite]: #62
  8.2111 +#66 := [monotonicity #63]: #65
  8.2112 +#75 := [trans #66 #73]: #74
  8.2113 +#58 := (= #10 #57)
  8.2114 +#43 := (= #6 #42)
  8.2115 +#44 := [rewrite]: #43
  8.2116 +#55 := (= #9 #52)
  8.2117 +#48 := (- #42)
  8.2118 +#53 := (= #48 #52)
  8.2119 +#54 := [rewrite]: #53
  8.2120 +#49 := (= #9 #48)
  8.2121 +#50 := [monotonicity #44]: #49
  8.2122 +#56 := [trans #50 #54]: #55
  8.2123 +#46 := (iff #8 #45)
  8.2124 +#47 := [monotonicity #44]: #46
  8.2125 +#59 := [monotonicity #47 #56 #44]: #58
  8.2126 +#78 := [monotonicity #59 #75]: #77
  8.2127 +#104 := [trans #78 #102]: #103
  8.2128 +#39 := [asserted]: #15
  8.2129 +#105 := [mp #39 #104]: #98
  8.2130 +#213 := [mp #105 #212]: #210
  8.2131 +#259 := (or #205 #80 #258)
  8.2132 +#260 := [def-axiom]: #259
  8.2133 +#294 := [unit-resolution #260 #213]: #293
  8.2134 +#295 := [unit-resolution #294 #292]: #258
  8.2135 +#296 := [th-lemma #287 #292 #295 #162]: false
  8.2136 +#298 := [lemma #296]: #297
  8.2137 +#299 := [unit-resolution #298 #287]: #80
  8.2138 +#261 := (not #204)
  8.2139 +#281 := (or #79 #261)
  8.2140 +#262 := (or #205 #79 #261)
  8.2141 +#263 := [def-axiom]: #262
  8.2142 +#282 := [unit-resolution #263 #213]: #281
  8.2143 +#300 := [unit-resolution #282 #299]: #261
  8.2144 +#290 := (or #79 #204 #243)
  8.2145 +#276 := [hypothesis]: #261
  8.2146 +#288 := [hypothesis]: #80
  8.2147 +#289 := [th-lemma #288 #276 #162 #287]: false
  8.2148 +#291 := [lemma #289]: #290
  8.2149 +#301 := [unit-resolution #291 #300 #299 #287]: false
  8.2150 +#302 := [lemma #301]: #243
  8.2151 +#303 := (or #129 #268)
  8.2152 +#223 := (* -4/3::real uf_3)
  8.2153 +#224 := (+ #223 uf_4)
  8.2154 +#234 := (+ #68 #224)
  8.2155 +#244 := (<= #234 0::real)
  8.2156 +#245 := (ite #128 #243 #244)
  8.2157 +#250 := (not #245)
  8.2158 +#107 := (+ #60 uf_4)
  8.2159 +#135 := (ite #128 #107 #117)
  8.2160 +#143 := (* -1::real #135)
  8.2161 +#144 := (+ #70 #143)
  8.2162 +#145 := (+ #68 #144)
  8.2163 +#146 := (<= #145 0::real)
  8.2164 +#147 := (not #146)
  8.2165 +#251 := (iff #147 #250)
  8.2166 +#248 := (iff #146 #245)
  8.2167 +#235 := (ite #128 #233 #234)
  8.2168 +#240 := (<= #235 0::real)
  8.2169 +#246 := (iff #240 #245)
  8.2170 +#247 := [rewrite]: #246
  8.2171 +#241 := (iff #146 #240)
  8.2172 +#238 := (= #145 #235)
  8.2173 +#225 := (ite #128 #222 #224)
  8.2174 +#230 := (+ #68 #225)
  8.2175 +#236 := (= #230 #235)
  8.2176 +#237 := [rewrite]: #236
  8.2177 +#231 := (= #145 #230)
  8.2178 +#228 := (= #144 #225)
  8.2179 +#214 := (ite #128 #117 #107)
  8.2180 +#217 := (+ #70 #214)
  8.2181 +#226 := (= #217 #225)
  8.2182 +#227 := [rewrite]: #226
  8.2183 +#218 := (= #144 #217)
  8.2184 +#215 := (= #143 #214)
  8.2185 +#216 := [rewrite]: #215
  8.2186 +#219 := [monotonicity #216]: #218
  8.2187 +#229 := [trans #219 #227]: #228
  8.2188 +#232 := [monotonicity #229]: #231
  8.2189 +#239 := [trans #232 #237]: #238
  8.2190 +#242 := [monotonicity #239]: #241
  8.2191 +#249 := [trans #242 #247]: #248
  8.2192 +#252 := [monotonicity #249]: #251
  8.2193 +#17 := (- uf_4 uf_3)
  8.2194 +#19 := (- #17)
  8.2195 +#18 := (< #17 0::real)
  8.2196 +#20 := (ite #18 #19 #17)
  8.2197 +#21 := (< #20 #14)
  8.2198 +#152 := (iff #21 #147)
  8.2199 +#110 := (< #107 0::real)
  8.2200 +#122 := (ite #110 #117 #107)
  8.2201 +#125 := (< #122 #71)
  8.2202 +#150 := (iff #125 #147)
  8.2203 +#140 := (< #135 #71)
  8.2204 +#148 := (iff #140 #147)
  8.2205 +#149 := [rewrite]: #148
  8.2206 +#141 := (iff #125 #140)
  8.2207 +#138 := (= #122 #135)
  8.2208 +#132 := (ite #129 #117 #107)
  8.2209 +#136 := (= #132 #135)
  8.2210 +#137 := [rewrite]: #136
  8.2211 +#133 := (= #122 #132)
  8.2212 +#130 := (iff #110 #129)
  8.2213 +#131 := [rewrite]: #130
  8.2214 +#134 := [monotonicity #131]: #133
  8.2215 +#139 := [trans #134 #137]: #138
  8.2216 +#142 := [monotonicity #139]: #141
  8.2217 +#151 := [trans #142 #149]: #150
  8.2218 +#126 := (iff #21 #125)
  8.2219 +#123 := (= #20 #122)
  8.2220 +#108 := (= #17 #107)
  8.2221 +#109 := [rewrite]: #108
  8.2222 +#120 := (= #19 #117)
  8.2223 +#113 := (- #107)
  8.2224 +#118 := (= #113 #117)
  8.2225 +#119 := [rewrite]: #118
  8.2226 +#114 := (= #19 #113)
  8.2227 +#115 := [monotonicity #109]: #114
  8.2228 +#121 := [trans #115 #119]: #120
  8.2229 +#111 := (iff #18 #110)
  8.2230 +#112 := [monotonicity #109]: #111
  8.2231 +#124 := [monotonicity #112 #121 #109]: #123
  8.2232 +#127 := [monotonicity #124 #75]: #126
  8.2233 +#153 := [trans #127 #151]: #152
  8.2234 +#106 := [asserted]: #21
  8.2235 +#154 := [mp #106 #153]: #147
  8.2236 +#253 := [mp #154 #252]: #250
  8.2237 +#269 := (or #245 #129 #268)
  8.2238 +#270 := [def-axiom]: #269
  8.2239 +#304 := [unit-resolution #270 #253]: #303
  8.2240 +#305 := [unit-resolution #304 #302]: #129
  8.2241 +#271 := (not #244)
  8.2242 +#306 := (or #128 #271)
  8.2243 +#272 := (or #245 #128 #271)
  8.2244 +#273 := [def-axiom]: #272
  8.2245 +#307 := [unit-resolution #273 #253]: #306
  8.2246 +#308 := [unit-resolution #307 #305]: #271
  8.2247 +#285 := (or #128 #244)
  8.2248 +#274 := [hypothesis]: #271
  8.2249 +#275 := [hypothesis]: #129
  8.2250 +#278 := (or #204 #128 #244)
  8.2251 +#277 := [th-lemma #276 #275 #274 #162]: false
  8.2252 +#279 := [lemma #277]: #278
  8.2253 +#280 := [unit-resolution #279 #275 #274]: #204
  8.2254 +#283 := [unit-resolution #282 #280]: #79
  8.2255 +#284 := [th-lemma #275 #274 #283 #162]: false
  8.2256 +#286 := [lemma #284]: #285
  8.2257 +[unit-resolution #286 #308 #305]: false
  8.2258 +unsat
  8.2259 +cebG074uorSr8ODzgTmcKg 97 0
  8.2260 +#2 := false
  8.2261 +#18 := 0::real
  8.2262 +decl uf_1 :: (-> T2 T1 real)
  8.2263 +decl uf_5 :: T1
  8.2264 +#11 := uf_5
  8.2265 +decl uf_2 :: T2
  8.2266 +#4 := uf_2
  8.2267 +#20 := (uf_1 uf_2 uf_5)
  8.2268 +#42 := -1::real
  8.2269 +#53 := (* -1::real #20)
  8.2270 +decl uf_3 :: T2
  8.2271 +#7 := uf_3
  8.2272 +#19 := (uf_1 uf_3 uf_5)
  8.2273 +#54 := (+ #19 #53)
  8.2274 +#63 := (<= #54 0::real)
  8.2275 +#21 := (- #19 #20)
  8.2276 +#22 := (< 0::real #21)
  8.2277 +#23 := (not #22)
  8.2278 +#74 := (iff #23 #63)
  8.2279 +#57 := (< 0::real #54)
  8.2280 +#60 := (not #57)
  8.2281 +#72 := (iff #60 #63)
  8.2282 +#64 := (not #63)
  8.2283 +#67 := (not #64)
  8.2284 +#70 := (iff #67 #63)
  8.2285 +#71 := [rewrite]: #70
  8.2286 +#68 := (iff #60 #67)
  8.2287 +#65 := (iff #57 #64)
  8.2288 +#66 := [rewrite]: #65
  8.2289 +#69 := [monotonicity #66]: #68
  8.2290 +#73 := [trans #69 #71]: #72
  8.2291 +#61 := (iff #23 #60)
  8.2292 +#58 := (iff #22 #57)
  8.2293 +#55 := (= #21 #54)
  8.2294 +#56 := [rewrite]: #55
  8.2295 +#59 := [monotonicity #56]: #58
  8.2296 +#62 := [monotonicity #59]: #61
  8.2297 +#75 := [trans #62 #73]: #74
  8.2298 +#41 := [asserted]: #23
  8.2299 +#76 := [mp #41 #75]: #63
  8.2300 +#5 := (:var 0 T1)
  8.2301 +#8 := (uf_1 uf_3 #5)
  8.2302 +#141 := (pattern #8)
  8.2303 +#6 := (uf_1 uf_2 #5)
  8.2304 +#140 := (pattern #6)
  8.2305 +#45 := (* -1::real #8)
  8.2306 +#46 := (+ #6 #45)
  8.2307 +#44 := (>= #46 0::real)
  8.2308 +#43 := (not #44)
  8.2309 +#142 := (forall (vars (?x1 T1)) (:pat #140 #141) #43)
  8.2310 +#49 := (forall (vars (?x1 T1)) #43)
  8.2311 +#145 := (iff #49 #142)
  8.2312 +#143 := (iff #43 #43)
  8.2313 +#144 := [refl]: #143
  8.2314 +#146 := [quant-intro #144]: #145
  8.2315 +#80 := (~ #49 #49)
  8.2316 +#82 := (~ #43 #43)
  8.2317 +#83 := [refl]: #82
  8.2318 +#81 := [nnf-pos #83]: #80
  8.2319 +#9 := (< #6 #8)
  8.2320 +#10 := (forall (vars (?x1 T1)) #9)
  8.2321 +#50 := (iff #10 #49)
  8.2322 +#47 := (iff #9 #43)
  8.2323 +#48 := [rewrite]: #47
  8.2324 +#51 := [quant-intro #48]: #50
  8.2325 +#39 := [asserted]: #10
  8.2326 +#52 := [mp #39 #51]: #49
  8.2327 +#79 := [mp~ #52 #81]: #49
  8.2328 +#147 := [mp #79 #146]: #142
  8.2329 +#164 := (not #142)
  8.2330 +#165 := (or #164 #64)
  8.2331 +#148 := (* -1::real #19)
  8.2332 +#149 := (+ #20 #148)
  8.2333 +#150 := (>= #149 0::real)
  8.2334 +#151 := (not #150)
  8.2335 +#166 := (or #164 #151)
  8.2336 +#168 := (iff #166 #165)
  8.2337 +#170 := (iff #165 #165)
  8.2338 +#171 := [rewrite]: #170
  8.2339 +#162 := (iff #151 #64)
  8.2340 +#160 := (iff #150 #63)
  8.2341 +#152 := (+ #148 #20)
  8.2342 +#155 := (>= #152 0::real)
  8.2343 +#158 := (iff #155 #63)
  8.2344 +#159 := [rewrite]: #158
  8.2345 +#156 := (iff #150 #155)
  8.2346 +#153 := (= #149 #152)
  8.2347 +#154 := [rewrite]: #153
  8.2348 +#157 := [monotonicity #154]: #156
  8.2349 +#161 := [trans #157 #159]: #160
  8.2350 +#163 := [monotonicity #161]: #162
  8.2351 +#169 := [monotonicity #163]: #168
  8.2352 +#172 := [trans #169 #171]: #168
  8.2353 +#167 := [quant-inst]: #166
  8.2354 +#173 := [mp #167 #172]: #165
  8.2355 +[unit-resolution #173 #147 #76]: false
  8.2356 +unsat
  8.2357 +DKRtrJ2XceCkITuNwNViRw 57 0
  8.2358 +#2 := false
  8.2359 +#4 := 0::real
  8.2360 +decl uf_1 :: (-> T2 real)
  8.2361 +decl uf_2 :: (-> T1 T1 T2)
  8.2362 +decl uf_12 :: (-> T4 T1)
  8.2363 +decl uf_4 :: T4
  8.2364 +#11 := uf_4
  8.2365 +#39 := (uf_12 uf_4)
  8.2366 +decl uf_10 :: T4
  8.2367 +#27 := uf_10
  8.2368 +#38 := (uf_12 uf_10)
  8.2369 +#40 := (uf_2 #38 #39)
  8.2370 +#41 := (uf_1 #40)
  8.2371 +#264 := (>= #41 0::real)
  8.2372 +#266 := (not #264)
  8.2373 +#43 := (= #41 0::real)
  8.2374 +#44 := (not #43)
  8.2375 +#131 := [asserted]: #44
  8.2376 +#272 := (or #43 #266)
  8.2377 +#42 := (<= #41 0::real)
  8.2378 +#130 := [asserted]: #42
  8.2379 +#265 := (not #42)
  8.2380 +#270 := (or #43 #265 #266)
  8.2381 +#271 := [th-lemma]: #270
  8.2382 +#273 := [unit-resolution #271 #130]: #272
  8.2383 +#274 := [unit-resolution #273 #131]: #266
  8.2384 +#6 := (:var 0 T1)
  8.2385 +#5 := (:var 1 T1)
  8.2386 +#7 := (uf_2 #5 #6)
  8.2387 +#241 := (pattern #7)
  8.2388 +#8 := (uf_1 #7)
  8.2389 +#65 := (>= #8 0::real)
  8.2390 +#242 := (forall (vars (?x1 T1) (?x2 T1)) (:pat #241) #65)
  8.2391 +#66 := (forall (vars (?x1 T1) (?x2 T1)) #65)
  8.2392 +#245 := (iff #66 #242)
  8.2393 +#243 := (iff #65 #65)
  8.2394 +#244 := [refl]: #243
  8.2395 +#246 := [quant-intro #244]: #245
  8.2396 +#149 := (~ #66 #66)
  8.2397 +#151 := (~ #65 #65)
  8.2398 +#152 := [refl]: #151
  8.2399 +#150 := [nnf-pos #152]: #149
  8.2400 +#9 := (<= 0::real #8)
  8.2401 +#10 := (forall (vars (?x1 T1) (?x2 T1)) #9)
  8.2402 +#67 := (iff #10 #66)
  8.2403 +#63 := (iff #9 #65)
  8.2404 +#64 := [rewrite]: #63
  8.2405 +#68 := [quant-intro #64]: #67
  8.2406 +#60 := [asserted]: #10
  8.2407 +#69 := [mp #60 #68]: #66
  8.2408 +#147 := [mp~ #69 #150]: #66
  8.2409 +#247 := [mp #147 #246]: #242
  8.2410 +#267 := (not #242)
  8.2411 +#268 := (or #267 #264)
  8.2412 +#269 := [quant-inst]: #268
  8.2413 +[unit-resolution #269 #247 #274]: false
  8.2414 +unsat
  8.2415 +97KJAJfUio+nGchEHWvgAw 91 0
  8.2416 +#2 := false
  8.2417 +#38 := 0::real
  8.2418 +decl uf_1 :: (-> T1 T2 real)
  8.2419 +decl uf_3 :: T2
  8.2420 +#5 := uf_3
  8.2421 +decl uf_4 :: T1
  8.2422 +#7 := uf_4
  8.2423 +#8 := (uf_1 uf_4 uf_3)
  8.2424 +#35 := -1::real
  8.2425 +#36 := (* -1::real #8)
  8.2426 +decl uf_2 :: T1
  8.2427 +#4 := uf_2
  8.2428 +#6 := (uf_1 uf_2 uf_3)
  8.2429 +#37 := (+ #6 #36)
  8.2430 +#130 := (>= #37 0::real)
  8.2431 +#155 := (not #130)
  8.2432 +#43 := (= #6 #8)
  8.2433 +#55 := (not #43)
  8.2434 +#15 := (= #8 #6)
  8.2435 +#16 := (not #15)
  8.2436 +#56 := (iff #16 #55)
  8.2437 +#53 := (iff #15 #43)
  8.2438 +#54 := [rewrite]: #53
  8.2439 +#57 := [monotonicity #54]: #56
  8.2440 +#34 := [asserted]: #16
  8.2441 +#60 := [mp #34 #57]: #55
  8.2442 +#158 := (or #43 #155)
  8.2443 +#39 := (<= #37 0::real)
  8.2444 +#9 := (<= #6 #8)
  8.2445 +#40 := (iff #9 #39)
  8.2446 +#41 := [rewrite]: #40
  8.2447 +#32 := [asserted]: #9
  8.2448 +#42 := [mp #32 #41]: #39
  8.2449 +#154 := (not #39)
  8.2450 +#156 := (or #43 #154 #155)
  8.2451 +#157 := [th-lemma]: #156
  8.2452 +#159 := [unit-resolution #157 #42]: #158
  8.2453 +#160 := [unit-resolution #159 #60]: #155
  8.2454 +#10 := (:var 0 T2)
  8.2455 +#12 := (uf_1 uf_2 #10)
  8.2456 +#123 := (pattern #12)
  8.2457 +#11 := (uf_1 uf_4 #10)
  8.2458 +#122 := (pattern #11)
  8.2459 +#44 := (* -1::real #12)
  8.2460 +#45 := (+ #11 #44)
  8.2461 +#46 := (<= #45 0::real)
  8.2462 +#124 := (forall (vars (?x1 T2)) (:pat #122 #123) #46)
  8.2463 +#49 := (forall (vars (?x1 T2)) #46)
  8.2464 +#127 := (iff #49 #124)
  8.2465 +#125 := (iff #46 #46)
  8.2466 +#126 := [refl]: #125
  8.2467 +#128 := [quant-intro #126]: #127
  8.2468 +#62 := (~ #49 #49)
  8.2469 +#64 := (~ #46 #46)
  8.2470 +#65 := [refl]: #64
  8.2471 +#63 := [nnf-pos #65]: #62
  8.2472 +#13 := (<= #11 #12)
  8.2473 +#14 := (forall (vars (?x1 T2)) #13)
  8.2474 +#50 := (iff #14 #49)
  8.2475 +#47 := (iff #13 #46)
  8.2476 +#48 := [rewrite]: #47
  8.2477 +#51 := [quant-intro #48]: #50
  8.2478 +#33 := [asserted]: #14
  8.2479 +#52 := [mp #33 #51]: #49
  8.2480 +#61 := [mp~ #52 #63]: #49
  8.2481 +#129 := [mp #61 #128]: #124
  8.2482 +#144 := (not #124)
  8.2483 +#145 := (or #144 #130)
  8.2484 +#131 := (* -1::real #6)
  8.2485 +#132 := (+ #8 #131)
  8.2486 +#133 := (<= #132 0::real)
  8.2487 +#146 := (or #144 #133)
  8.2488 +#148 := (iff #146 #145)
  8.2489 +#150 := (iff #145 #145)
  8.2490 +#151 := [rewrite]: #150
  8.2491 +#142 := (iff #133 #130)
  8.2492 +#134 := (+ #131 #8)
  8.2493 +#137 := (<= #134 0::real)
  8.2494 +#140 := (iff #137 #130)
  8.2495 +#141 := [rewrite]: #140
  8.2496 +#138 := (iff #133 #137)
  8.2497 +#135 := (= #132 #134)
  8.2498 +#136 := [rewrite]: #135
  8.2499 +#139 := [monotonicity #136]: #138
  8.2500 +#143 := [trans #139 #141]: #142
  8.2501 +#149 := [monotonicity #143]: #148
  8.2502 +#152 := [trans #149 #151]: #148
  8.2503 +#147 := [quant-inst]: #146
  8.2504 +#153 := [mp #147 #152]: #145
  8.2505 +[unit-resolution #153 #129 #160]: false
  8.2506 +unsat
  8.2507 +flJYbeWfe+t2l/zsRqdujA 149 0
  8.2508 +#2 := false
  8.2509 +#19 := 0::real
  8.2510 +decl uf_1 :: (-> T1 T2 real)
  8.2511 +decl uf_3 :: T2
  8.2512 +#5 := uf_3
  8.2513 +decl uf_4 :: T1
  8.2514 +#7 := uf_4
  8.2515 +#8 := (uf_1 uf_4 uf_3)
  8.2516 +#44 := -1::real
  8.2517 +#156 := (* -1::real #8)
  8.2518 +decl uf_2 :: T1
  8.2519 +#4 := uf_2
  8.2520 +#6 := (uf_1 uf_2 uf_3)
  8.2521 +#203 := (+ #6 #156)
  8.2522 +#205 := (>= #203 0::real)
  8.2523 +#9 := (= #6 #8)
  8.2524 +#40 := [asserted]: #9
  8.2525 +#208 := (not #9)
  8.2526 +#209 := (or #208 #205)
  8.2527 +#210 := [th-lemma]: #209
  8.2528 +#211 := [unit-resolution #210 #40]: #205
  8.2529 +decl uf_5 :: T1
  8.2530 +#12 := uf_5
  8.2531 +#22 := (uf_1 uf_5 uf_3)
  8.2532 +#160 := (* -1::real #22)
  8.2533 +#161 := (+ #6 #160)
  8.2534 +#207 := (>= #161 0::real)
  8.2535 +#222 := (not #207)
  8.2536 +#206 := (= #6 #22)
  8.2537 +#216 := (not #206)
  8.2538 +#62 := (= #8 #22)
  8.2539 +#70 := (not #62)
  8.2540 +#217 := (iff #70 #216)
  8.2541 +#214 := (iff #62 #206)
  8.2542 +#212 := (iff #206 #62)
  8.2543 +#213 := [monotonicity #40]: #212
  8.2544 +#215 := [symm #213]: #214
  8.2545 +#218 := [monotonicity #215]: #217
  8.2546 +#23 := (= #22 #8)
  8.2547 +#24 := (not #23)
  8.2548 +#71 := (iff #24 #70)
  8.2549 +#68 := (iff #23 #62)
  8.2550 +#69 := [rewrite]: #68
  8.2551 +#72 := [monotonicity #69]: #71
  8.2552 +#43 := [asserted]: #24
  8.2553 +#75 := [mp #43 #72]: #70
  8.2554 +#219 := [mp #75 #218]: #216
  8.2555 +#225 := (or #206 #222)
  8.2556 +#162 := (<= #161 0::real)
  8.2557 +#172 := (+ #8 #160)
  8.2558 +#173 := (>= #172 0::real)
  8.2559 +#178 := (not #173)
  8.2560 +#163 := (not #162)
  8.2561 +#181 := (or #163 #178)
  8.2562 +#184 := (not #181)
  8.2563 +#10 := (:var 0 T2)
  8.2564 +#15 := (uf_1 uf_4 #10)
  8.2565 +#149 := (pattern #15)
  8.2566 +#13 := (uf_1 uf_5 #10)
  8.2567 +#148 := (pattern #13)
  8.2568 +#11 := (uf_1 uf_2 #10)
  8.2569 +#147 := (pattern #11)
  8.2570 +#50 := (* -1::real #15)
  8.2571 +#51 := (+ #13 #50)
  8.2572 +#52 := (<= #51 0::real)
  8.2573 +#76 := (not #52)
  8.2574 +#45 := (* -1::real #13)
  8.2575 +#46 := (+ #11 #45)
  8.2576 +#47 := (<= #46 0::real)
  8.2577 +#78 := (not #47)
  8.2578 +#73 := (or #78 #76)
  8.2579 +#83 := (not #73)
  8.2580 +#150 := (forall (vars (?x1 T2)) (:pat #147 #148 #149) #83)
  8.2581 +#86 := (forall (vars (?x1 T2)) #83)
  8.2582 +#153 := (iff #86 #150)
  8.2583 +#151 := (iff #83 #83)
  8.2584 +#152 := [refl]: #151
  8.2585 +#154 := [quant-intro #152]: #153
  8.2586 +#55 := (and #47 #52)
  8.2587 +#58 := (forall (vars (?x1 T2)) #55)
  8.2588 +#87 := (iff #58 #86)
  8.2589 +#84 := (iff #55 #83)
  8.2590 +#85 := [rewrite]: #84
  8.2591 +#88 := [quant-intro #85]: #87
  8.2592 +#79 := (~ #58 #58)
  8.2593 +#81 := (~ #55 #55)
  8.2594 +#82 := [refl]: #81
  8.2595 +#80 := [nnf-pos #82]: #79
  8.2596 +#16 := (<= #13 #15)
  8.2597 +#14 := (<= #11 #13)
  8.2598 +#17 := (and #14 #16)
  8.2599 +#18 := (forall (vars (?x1 T2)) #17)
  8.2600 +#59 := (iff #18 #58)
  8.2601 +#56 := (iff #17 #55)
  8.2602 +#53 := (iff #16 #52)
  8.2603 +#54 := [rewrite]: #53
  8.2604 +#48 := (iff #14 #47)
  8.2605 +#49 := [rewrite]: #48
  8.2606 +#57 := [monotonicity #49 #54]: #56
  8.2607 +#60 := [quant-intro #57]: #59
  8.2608 +#41 := [asserted]: #18
  8.2609 +#61 := [mp #41 #60]: #58
  8.2610 +#77 := [mp~ #61 #80]: #58
  8.2611 +#89 := [mp #77 #88]: #86
  8.2612 +#155 := [mp #89 #154]: #150
  8.2613 +#187 := (not #150)
  8.2614 +#188 := (or #187 #184)
  8.2615 +#157 := (+ #22 #156)
  8.2616 +#158 := (<= #157 0::real)
  8.2617 +#159 := (not #158)
  8.2618 +#164 := (or #163 #159)
  8.2619 +#165 := (not #164)
  8.2620 +#189 := (or #187 #165)
  8.2621 +#191 := (iff #189 #188)
  8.2622 +#193 := (iff #188 #188)
  8.2623 +#194 := [rewrite]: #193
  8.2624 +#185 := (iff #165 #184)
  8.2625 +#182 := (iff #164 #181)
  8.2626 +#179 := (iff #159 #178)
  8.2627 +#176 := (iff #158 #173)
  8.2628 +#166 := (+ #156 #22)
  8.2629 +#169 := (<= #166 0::real)
  8.2630 +#174 := (iff #169 #173)
  8.2631 +#175 := [rewrite]: #174
  8.2632 +#170 := (iff #158 #169)
  8.2633 +#167 := (= #157 #166)
  8.2634 +#168 := [rewrite]: #167
  8.2635 +#171 := [monotonicity #168]: #170
  8.2636 +#177 := [trans #171 #175]: #176
  8.2637 +#180 := [monotonicity #177]: #179
  8.2638 +#183 := [monotonicity #180]: #182
  8.2639 +#186 := [monotonicity #183]: #185
  8.2640 +#192 := [monotonicity #186]: #191
  8.2641 +#195 := [trans #192 #194]: #191
  8.2642 +#190 := [quant-inst]: #189
  8.2643 +#196 := [mp #190 #195]: #188
  8.2644 +#220 := [unit-resolution #196 #155]: #184
  8.2645 +#197 := (or #181 #162)
  8.2646 +#198 := [def-axiom]: #197
  8.2647 +#221 := [unit-resolution #198 #220]: #162
  8.2648 +#223 := (or #206 #163 #222)
  8.2649 +#224 := [th-lemma]: #223
  8.2650 +#226 := [unit-resolution #224 #221]: #225
  8.2651 +#227 := [unit-resolution #226 #219]: #222
  8.2652 +#199 := (or #181 #173)
  8.2653 +#200 := [def-axiom]: #199
  8.2654 +#228 := [unit-resolution #200 #220]: #173
  8.2655 +[th-lemma #228 #227 #211]: false
  8.2656 +unsat
  8.2657 +rbrrQuQfaijtLkQizgEXnQ 222 0
  8.2658 +#2 := false
  8.2659 +#4 := 0::real
  8.2660 +decl uf_2 :: (-> T2 T1 real)
  8.2661 +decl uf_5 :: T1
  8.2662 +#15 := uf_5
  8.2663 +decl uf_3 :: T2
  8.2664 +#7 := uf_3
  8.2665 +#20 := (uf_2 uf_3 uf_5)
  8.2666 +decl uf_6 :: T2
  8.2667 +#17 := uf_6
  8.2668 +#18 := (uf_2 uf_6 uf_5)
  8.2669 +#59 := -1::real
  8.2670 +#73 := (* -1::real #18)
  8.2671 +#106 := (+ #73 #20)
  8.2672 +decl uf_1 :: real
  8.2673 +#5 := uf_1
  8.2674 +#78 := (* -1::real #20)
  8.2675 +#79 := (+ #18 #78)
  8.2676 +#144 := (+ uf_1 #79)
  8.2677 +#145 := (<= #144 0::real)
  8.2678 +#148 := (ite #145 uf_1 #106)
  8.2679 +#279 := (* -1::real #148)
  8.2680 +#280 := (+ uf_1 #279)
  8.2681 +#281 := (<= #280 0::real)
  8.2682 +#289 := (not #281)
  8.2683 +#72 := 1/2::real
  8.2684 +#151 := (* 1/2::real #148)
  8.2685 +#248 := (<= #151 0::real)
  8.2686 +#162 := (= #151 0::real)
  8.2687 +#24 := 2::real
  8.2688 +#27 := (- #20 #18)
  8.2689 +#28 := (<= uf_1 #27)
  8.2690 +#29 := (ite #28 uf_1 #27)
  8.2691 +#30 := (/ #29 2::real)
  8.2692 +#31 := (+ #18 #30)
  8.2693 +#32 := (= #31 #18)
  8.2694 +#33 := (not #32)
  8.2695 +#34 := (not #33)
  8.2696 +#165 := (iff #34 #162)
  8.2697 +#109 := (<= uf_1 #106)
  8.2698 +#112 := (ite #109 uf_1 #106)
  8.2699 +#118 := (* 1/2::real #112)
  8.2700 +#123 := (+ #18 #118)
  8.2701 +#129 := (= #18 #123)
  8.2702 +#163 := (iff #129 #162)
  8.2703 +#154 := (+ #18 #151)
  8.2704 +#157 := (= #18 #154)
  8.2705 +#160 := (iff #157 #162)
  8.2706 +#161 := [rewrite]: #160
  8.2707 +#158 := (iff #129 #157)
  8.2708 +#155 := (= #123 #154)
  8.2709 +#152 := (= #118 #151)
  8.2710 +#149 := (= #112 #148)
  8.2711 +#146 := (iff #109 #145)
  8.2712 +#147 := [rewrite]: #146
  8.2713 +#150 := [monotonicity #147]: #149
  8.2714 +#153 := [monotonicity #150]: #152
  8.2715 +#156 := [monotonicity #153]: #155
  8.2716 +#159 := [monotonicity #156]: #158
  8.2717 +#164 := [trans #159 #161]: #163
  8.2718 +#142 := (iff #34 #129)
  8.2719 +#134 := (not #129)
  8.2720 +#137 := (not #134)
  8.2721 +#140 := (iff #137 #129)
  8.2722 +#141 := [rewrite]: #140
  8.2723 +#138 := (iff #34 #137)
  8.2724 +#135 := (iff #33 #134)
  8.2725 +#132 := (iff #32 #129)
  8.2726 +#126 := (= #123 #18)
  8.2727 +#130 := (iff #126 #129)
  8.2728 +#131 := [rewrite]: #130
  8.2729 +#127 := (iff #32 #126)
  8.2730 +#124 := (= #31 #123)
  8.2731 +#121 := (= #30 #118)
  8.2732 +#115 := (/ #112 2::real)
  8.2733 +#119 := (= #115 #118)
  8.2734 +#120 := [rewrite]: #119
  8.2735 +#116 := (= #30 #115)
  8.2736 +#113 := (= #29 #112)
  8.2737 +#107 := (= #27 #106)
  8.2738 +#108 := [rewrite]: #107
  8.2739 +#110 := (iff #28 #109)
  8.2740 +#111 := [monotonicity #108]: #110
  8.2741 +#114 := [monotonicity #111 #108]: #113
  8.2742 +#117 := [monotonicity #114]: #116
  8.2743 +#122 := [trans #117 #120]: #121
  8.2744 +#125 := [monotonicity #122]: #124
  8.2745 +#128 := [monotonicity #125]: #127
  8.2746 +#133 := [trans #128 #131]: #132
  8.2747 +#136 := [monotonicity #133]: #135
  8.2748 +#139 := [monotonicity #136]: #138
  8.2749 +#143 := [trans #139 #141]: #142
  8.2750 +#166 := [trans #143 #164]: #165
  8.2751 +#105 := [asserted]: #34
  8.2752 +#167 := [mp #105 #166]: #162
  8.2753 +#283 := (not #162)
  8.2754 +#284 := (or #283 #248)
  8.2755 +#285 := [th-lemma]: #284
  8.2756 +#286 := [unit-resolution #285 #167]: #248
  8.2757 +#287 := [hypothesis]: #281
  8.2758 +#53 := (<= uf_1 0::real)
  8.2759 +#54 := (not #53)
  8.2760 +#6 := (< 0::real uf_1)
  8.2761 +#55 := (iff #6 #54)
  8.2762 +#56 := [rewrite]: #55
  8.2763 +#50 := [asserted]: #6
  8.2764 +#57 := [mp #50 #56]: #54
  8.2765 +#288 := [th-lemma #57 #287 #286]: false
  8.2766 +#290 := [lemma #288]: #289
  8.2767 +#241 := (= uf_1 #148)
  8.2768 +#242 := (= #106 #148)
  8.2769 +#299 := (not #242)
  8.2770 +#282 := (+ #106 #279)
  8.2771 +#291 := (<= #282 0::real)
  8.2772 +#296 := (not #291)
  8.2773 +decl uf_4 :: T2
  8.2774 +#10 := uf_4
  8.2775 +#16 := (uf_2 uf_4 uf_5)
  8.2776 +#260 := (+ #16 #78)
  8.2777 +#261 := (>= #260 0::real)
  8.2778 +#266 := (not #261)
  8.2779 +#8 := (:var 0 T1)
  8.2780 +#11 := (uf_2 uf_4 #8)
  8.2781 +#234 := (pattern #11)
  8.2782 +#9 := (uf_2 uf_3 #8)
  8.2783 +#233 := (pattern #9)
  8.2784 +#60 := (* -1::real #11)
  8.2785 +#61 := (+ #9 #60)
  8.2786 +#62 := (<= #61 0::real)
  8.2787 +#179 := (not #62)
  8.2788 +#235 := (forall (vars (?x1 T1)) (:pat #233 #234) #179)
  8.2789 +#178 := (forall (vars (?x1 T1)) #179)
  8.2790 +#238 := (iff #178 #235)
  8.2791 +#236 := (iff #179 #179)
  8.2792 +#237 := [refl]: #236
  8.2793 +#239 := [quant-intro #237]: #238
  8.2794 +#65 := (exists (vars (?x1 T1)) #62)
  8.2795 +#68 := (not #65)
  8.2796 +#175 := (~ #68 #178)
  8.2797 +#180 := (~ #179 #179)
  8.2798 +#177 := [refl]: #180
  8.2799 +#176 := [nnf-neg #177]: #175
  8.2800 +#12 := (<= #9 #11)
  8.2801 +#13 := (exists (vars (?x1 T1)) #12)
  8.2802 +#14 := (not #13)
  8.2803 +#69 := (iff #14 #68)
  8.2804 +#66 := (iff #13 #65)
  8.2805 +#63 := (iff #12 #62)
  8.2806 +#64 := [rewrite]: #63
  8.2807 +#67 := [quant-intro #64]: #66
  8.2808 +#70 := [monotonicity #67]: #69
  8.2809 +#51 := [asserted]: #14
  8.2810 +#71 := [mp #51 #70]: #68
  8.2811 +#173 := [mp~ #71 #176]: #178
  8.2812 +#240 := [mp #173 #239]: #235
  8.2813 +#269 := (not #235)
  8.2814 +#270 := (or #269 #266)
  8.2815 +#250 := (* -1::real #16)
  8.2816 +#251 := (+ #20 #250)
  8.2817 +#252 := (<= #251 0::real)
  8.2818 +#253 := (not #252)
  8.2819 +#271 := (or #269 #253)
  8.2820 +#273 := (iff #271 #270)
  8.2821 +#275 := (iff #270 #270)
  8.2822 +#276 := [rewrite]: #275
  8.2823 +#267 := (iff #253 #266)
  8.2824 +#264 := (iff #252 #261)
  8.2825 +#254 := (+ #250 #20)
  8.2826 +#257 := (<= #254 0::real)
  8.2827 +#262 := (iff #257 #261)
  8.2828 +#263 := [rewrite]: #262
  8.2829 +#258 := (iff #252 #257)
  8.2830 +#255 := (= #251 #254)
  8.2831 +#256 := [rewrite]: #255
  8.2832 +#259 := [monotonicity #256]: #258
  8.2833 +#265 := [trans #259 #263]: #264
  8.2834 +#268 := [monotonicity #265]: #267
  8.2835 +#274 := [monotonicity #268]: #273
  8.2836 +#277 := [trans #274 #276]: #273
  8.2837 +#272 := [quant-inst]: #271
  8.2838 +#278 := [mp #272 #277]: #270
  8.2839 +#293 := [unit-resolution #278 #240]: #266
  8.2840 +#90 := (* 1/2::real #20)
  8.2841 +#102 := (+ #73 #90)
  8.2842 +#89 := (* 1/2::real #16)
  8.2843 +#103 := (+ #89 #102)
  8.2844 +#100 := (>= #103 0::real)
  8.2845 +#23 := (+ #16 #20)
  8.2846 +#25 := (/ #23 2::real)
  8.2847 +#26 := (<= #18 #25)
  8.2848 +#98 := (iff #26 #100)
  8.2849 +#91 := (+ #89 #90)
  8.2850 +#94 := (<= #18 #91)
  8.2851 +#97 := (iff #94 #100)
  8.2852 +#99 := [rewrite]: #97
  8.2853 +#95 := (iff #26 #94)
  8.2854 +#92 := (= #25 #91)
  8.2855 +#93 := [rewrite]: #92
  8.2856 +#96 := [monotonicity #93]: #95
  8.2857 +#101 := [trans #96 #99]: #98
  8.2858 +#58 := [asserted]: #26
  8.2859 +#104 := [mp #58 #101]: #100
  8.2860 +#294 := [hypothesis]: #291
  8.2861 +#295 := [th-lemma #294 #104 #293 #286]: false
  8.2862 +#297 := [lemma #295]: #296
  8.2863 +#298 := [hypothesis]: #242
  8.2864 +#300 := (or #299 #291)
  8.2865 +#301 := [th-lemma]: #300
  8.2866 +#302 := [unit-resolution #301 #298 #297]: false
  8.2867 +#303 := [lemma #302]: #299
  8.2868 +#246 := (or #145 #242)
  8.2869 +#247 := [def-axiom]: #246
  8.2870 +#304 := [unit-resolution #247 #303]: #145
  8.2871 +#243 := (not #145)
  8.2872 +#244 := (or #243 #241)
  8.2873 +#245 := [def-axiom]: #244
  8.2874 +#305 := [unit-resolution #245 #304]: #241
  8.2875 +#306 := (not #241)
  8.2876 +#307 := (or #306 #281)
  8.2877 +#308 := [th-lemma]: #307
  8.2878 +[unit-resolution #308 #305 #290]: false
  8.2879 +unsat
  8.2880 +hwh3oeLAWt56hnKIa8Wuow 248 0
  8.2881 +#2 := false
  8.2882 +#4 := 0::real
  8.2883 +decl uf_2 :: (-> T2 T1 real)
  8.2884 +decl uf_5 :: T1
  8.2885 +#15 := uf_5
  8.2886 +decl uf_6 :: T2
  8.2887 +#17 := uf_6
  8.2888 +#18 := (uf_2 uf_6 uf_5)
  8.2889 +decl uf_4 :: T2
  8.2890 +#10 := uf_4
  8.2891 +#16 := (uf_2 uf_4 uf_5)
  8.2892 +#66 := -1::real
  8.2893 +#137 := (* -1::real #16)
  8.2894 +#138 := (+ #137 #18)
  8.2895 +decl uf_1 :: real
  8.2896 +#5 := uf_1
  8.2897 +#80 := (* -1::real #18)
  8.2898 +#81 := (+ #16 #80)
  8.2899 +#201 := (+ uf_1 #81)
  8.2900 +#202 := (<= #201 0::real)
  8.2901 +#205 := (ite #202 uf_1 #138)
  8.2902 +#352 := (* -1::real #205)
  8.2903 +#353 := (+ uf_1 #352)
  8.2904 +#354 := (<= #353 0::real)
  8.2905 +#362 := (not #354)
  8.2906 +#79 := 1/2::real
  8.2907 +#244 := (* 1/2::real #205)
  8.2908 +#322 := (<= #244 0::real)
  8.2909 +#245 := (= #244 0::real)
  8.2910 +#158 := -1/2::real
  8.2911 +#208 := (* -1/2::real #205)
  8.2912 +#211 := (+ #18 #208)
  8.2913 +decl uf_3 :: T2
  8.2914 +#7 := uf_3
  8.2915 +#20 := (uf_2 uf_3 uf_5)
  8.2916 +#117 := (+ #80 #20)
  8.2917 +#85 := (* -1::real #20)
  8.2918 +#86 := (+ #18 #85)
  8.2919 +#188 := (+ uf_1 #86)
  8.2920 +#189 := (<= #188 0::real)
  8.2921 +#192 := (ite #189 uf_1 #117)
  8.2922 +#195 := (* 1/2::real #192)
  8.2923 +#198 := (+ #18 #195)
  8.2924 +#97 := (* 1/2::real #20)
  8.2925 +#109 := (+ #80 #97)
  8.2926 +#96 := (* 1/2::real #16)
  8.2927 +#110 := (+ #96 #109)
  8.2928 +#107 := (>= #110 0::real)
  8.2929 +#214 := (ite #107 #198 #211)
  8.2930 +#217 := (= #18 #214)
  8.2931 +#248 := (iff #217 #245)
  8.2932 +#241 := (= #18 #211)
  8.2933 +#246 := (iff #241 #245)
  8.2934 +#247 := [rewrite]: #246
  8.2935 +#242 := (iff #217 #241)
  8.2936 +#239 := (= #214 #211)
  8.2937 +#234 := (ite false #198 #211)
  8.2938 +#237 := (= #234 #211)
  8.2939 +#238 := [rewrite]: #237
  8.2940 +#235 := (= #214 #234)
  8.2941 +#232 := (iff #107 false)
  8.2942 +#104 := (not #107)
  8.2943 +#24 := 2::real
  8.2944 +#23 := (+ #16 #20)
  8.2945 +#25 := (/ #23 2::real)
  8.2946 +#26 := (< #25 #18)
  8.2947 +#108 := (iff #26 #104)
  8.2948 +#98 := (+ #96 #97)
  8.2949 +#101 := (< #98 #18)
  8.2950 +#106 := (iff #101 #104)
  8.2951 +#105 := [rewrite]: #106
  8.2952 +#102 := (iff #26 #101)
  8.2953 +#99 := (= #25 #98)
  8.2954 +#100 := [rewrite]: #99
  8.2955 +#103 := [monotonicity #100]: #102
  8.2956 +#111 := [trans #103 #105]: #108
  8.2957 +#65 := [asserted]: #26
  8.2958 +#112 := [mp #65 #111]: #104
  8.2959 +#233 := [iff-false #112]: #232
  8.2960 +#236 := [monotonicity #233]: #235
  8.2961 +#240 := [trans #236 #238]: #239
  8.2962 +#243 := [monotonicity #240]: #242
  8.2963 +#249 := [trans #243 #247]: #248
  8.2964 +#33 := (- #18 #16)
  8.2965 +#34 := (<= uf_1 #33)
  8.2966 +#35 := (ite #34 uf_1 #33)
  8.2967 +#36 := (/ #35 2::real)
  8.2968 +#37 := (- #18 #36)
  8.2969 +#28 := (- #20 #18)
  8.2970 +#29 := (<= uf_1 #28)
  8.2971 +#30 := (ite #29 uf_1 #28)
  8.2972 +#31 := (/ #30 2::real)
  8.2973 +#32 := (+ #18 #31)
  8.2974 +#27 := (<= #18 #25)
  8.2975 +#38 := (ite #27 #32 #37)
  8.2976 +#39 := (= #38 #18)
  8.2977 +#40 := (not #39)
  8.2978 +#41 := (not #40)
  8.2979 +#220 := (iff #41 #217)
  8.2980 +#141 := (<= uf_1 #138)
  8.2981 +#144 := (ite #141 uf_1 #138)
  8.2982 +#159 := (* -1/2::real #144)
  8.2983 +#160 := (+ #18 #159)
  8.2984 +#120 := (<= uf_1 #117)
  8.2985 +#123 := (ite #120 uf_1 #117)
  8.2986 +#129 := (* 1/2::real #123)
  8.2987 +#134 := (+ #18 #129)
  8.2988 +#114 := (<= #18 #98)
  8.2989 +#165 := (ite #114 #134 #160)
  8.2990 +#171 := (= #18 #165)
  8.2991 +#218 := (iff #171 #217)
  8.2992 +#215 := (= #165 #214)
  8.2993 +#212 := (= #160 #211)
  8.2994 +#209 := (= #159 #208)
  8.2995 +#206 := (= #144 #205)
  8.2996 +#203 := (iff #141 #202)
  8.2997 +#204 := [rewrite]: #203
  8.2998 +#207 := [monotonicity #204]: #206
  8.2999 +#210 := [monotonicity #207]: #209
  8.3000 +#213 := [monotonicity #210]: #212
  8.3001 +#199 := (= #134 #198)
  8.3002 +#196 := (= #129 #195)
  8.3003 +#193 := (= #123 #192)
  8.3004 +#190 := (iff #120 #189)
  8.3005 +#191 := [rewrite]: #190
  8.3006 +#194 := [monotonicity #191]: #193
  8.3007 +#197 := [monotonicity #194]: #196
  8.3008 +#200 := [monotonicity #197]: #199
  8.3009 +#187 := (iff #114 #107)
  8.3010 +#186 := [rewrite]: #187
  8.3011 +#216 := [monotonicity #186 #200 #213]: #215
  8.3012 +#219 := [monotonicity #216]: #218
  8.3013 +#184 := (iff #41 #171)
  8.3014 +#176 := (not #171)
  8.3015 +#179 := (not #176)
  8.3016 +#182 := (iff #179 #171)
  8.3017 +#183 := [rewrite]: #182
  8.3018 +#180 := (iff #41 #179)
  8.3019 +#177 := (iff #40 #176)
  8.3020 +#174 := (iff #39 #171)
  8.3021 +#168 := (= #165 #18)
  8.3022 +#172 := (iff #168 #171)
  8.3023 +#173 := [rewrite]: #172
  8.3024 +#169 := (iff #39 #168)
  8.3025 +#166 := (= #38 #165)
  8.3026 +#163 := (= #37 #160)
  8.3027 +#150 := (* 1/2::real #144)
  8.3028 +#155 := (- #18 #150)
  8.3029 +#161 := (= #155 #160)
  8.3030 +#162 := [rewrite]: #161
  8.3031 +#156 := (= #37 #155)
  8.3032 +#153 := (= #36 #150)
  8.3033 +#147 := (/ #144 2::real)
  8.3034 +#151 := (= #147 #150)
  8.3035 +#152 := [rewrite]: #151
  8.3036 +#148 := (= #36 #147)
  8.3037 +#145 := (= #35 #144)
  8.3038 +#139 := (= #33 #138)
  8.3039 +#140 := [rewrite]: #139
  8.3040 +#142 := (iff #34 #141)
  8.3041 +#143 := [monotonicity #140]: #142
  8.3042 +#146 := [monotonicity #143 #140]: #145
  8.3043 +#149 := [monotonicity #146]: #148
  8.3044 +#154 := [trans #149 #152]: #153
  8.3045 +#157 := [monotonicity #154]: #156
  8.3046 +#164 := [trans #157 #162]: #163
  8.3047 +#135 := (= #32 #134)
  8.3048 +#132 := (= #31 #129)
  8.3049 +#126 := (/ #123 2::real)
  8.3050 +#130 := (= #126 #129)
  8.3051 +#131 := [rewrite]: #130
  8.3052 +#127 := (= #31 #126)
  8.3053 +#124 := (= #30 #123)
  8.3054 +#118 := (= #28 #117)
  8.3055 +#119 := [rewrite]: #118
  8.3056 +#121 := (iff #29 #120)
  8.3057 +#122 := [monotonicity #119]: #121
  8.3058 +#125 := [monotonicity #122 #119]: #124
  8.3059 +#128 := [monotonicity #125]: #127
  8.3060 +#133 := [trans #128 #131]: #132
  8.3061 +#136 := [monotonicity #133]: #135
  8.3062 +#115 := (iff #27 #114)
  8.3063 +#116 := [monotonicity #100]: #115
  8.3064 +#167 := [monotonicity #116 #136 #164]: #166
  8.3065 +#170 := [monotonicity #167]: #169
  8.3066 +#175 := [trans #170 #173]: #174
  8.3067 +#178 := [monotonicity #175]: #177
  8.3068 +#181 := [monotonicity #178]: #180
  8.3069 +#185 := [trans #181 #183]: #184
  8.3070 +#221 := [trans #185 #219]: #220
  8.3071 +#113 := [asserted]: #41
  8.3072 +#222 := [mp #113 #221]: #217
  8.3073 +#250 := [mp #222 #249]: #245
  8.3074 +#356 := (not #245)
  8.3075 +#357 := (or #356 #322)
  8.3076 +#358 := [th-lemma]: #357
  8.3077 +#359 := [unit-resolution #358 #250]: #322
  8.3078 +#360 := [hypothesis]: #354
  8.3079 +#60 := (<= uf_1 0::real)
  8.3080 +#61 := (not #60)
  8.3081 +#6 := (< 0::real uf_1)
  8.3082 +#62 := (iff #6 #61)
  8.3083 +#63 := [rewrite]: #62
  8.3084 +#57 := [asserted]: #6
  8.3085 +#64 := [mp #57 #63]: #61
  8.3086 +#361 := [th-lemma #64 #360 #359]: false
  8.3087 +#363 := [lemma #361]: #362
  8.3088 +#315 := (= uf_1 #205)
  8.3089 +#316 := (= #138 #205)
  8.3090 +#371 := (not #316)
  8.3091 +#355 := (+ #138 #352)
  8.3092 +#364 := (<= #355 0::real)
  8.3093 +#368 := (not #364)
  8.3094 +#87 := (<= #86 0::real)
  8.3095 +#82 := (<= #81 0::real)
  8.3096 +#90 := (and #82 #87)
  8.3097 +#21 := (<= #18 #20)
  8.3098 +#19 := (<= #16 #18)
  8.3099 +#22 := (and #19 #21)
  8.3100 +#91 := (iff #22 #90)
  8.3101 +#88 := (iff #21 #87)
  8.3102 +#89 := [rewrite]: #88
  8.3103 +#83 := (iff #19 #82)
  8.3104 +#84 := [rewrite]: #83
  8.3105 +#92 := [monotonicity #84 #89]: #91
  8.3106 +#59 := [asserted]: #22
  8.3107 +#93 := [mp #59 #92]: #90
  8.3108 +#95 := [and-elim #93]: #87
  8.3109 +#366 := [hypothesis]: #364
  8.3110 +#367 := [th-lemma #366 #95 #112 #359]: false
  8.3111 +#369 := [lemma #367]: #368
  8.3112 +#370 := [hypothesis]: #316
  8.3113 +#372 := (or #371 #364)
  8.3114 +#373 := [th-lemma]: #372
  8.3115 +#374 := [unit-resolution #373 #370 #369]: false
  8.3116 +#375 := [lemma #374]: #371
  8.3117 +#320 := (or #202 #316)
  8.3118 +#321 := [def-axiom]: #320
  8.3119 +#376 := [unit-resolution #321 #375]: #202
  8.3120 +#317 := (not #202)
  8.3121 +#318 := (or #317 #315)
  8.3122 +#319 := [def-axiom]: #318
  8.3123 +#377 := [unit-resolution #319 #376]: #315
  8.3124 +#378 := (not #315)
  8.3125 +#379 := (or #378 #354)
  8.3126 +#380 := [th-lemma]: #379
  8.3127 +[unit-resolution #380 #377 #363]: false
  8.3128 +unsat
  8.3129 +WdMJH3tkMv/rps8y9Ukq5Q 86 0
  8.3130 +#2 := false
  8.3131 +#37 := 0::real
  8.3132 +decl uf_2 :: (-> T2 T1 real)
  8.3133 +decl uf_4 :: T1
  8.3134 +#12 := uf_4
  8.3135 +decl uf_3 :: T2
  8.3136 +#5 := uf_3
  8.3137 +#13 := (uf_2 uf_3 uf_4)
  8.3138 +#34 := -1::real
  8.3139 +#140 := (* -1::real #13)
  8.3140 +decl uf_1 :: real
  8.3141 +#4 := uf_1
  8.3142 +#141 := (+ uf_1 #140)
  8.3143 +#143 := (>= #141 0::real)
  8.3144 +#6 := (:var 0 T1)
  8.3145 +#7 := (uf_2 uf_3 #6)
  8.3146 +#127 := (pattern #7)
  8.3147 +#35 := (* -1::real #7)
  8.3148 +#36 := (+ uf_1 #35)
  8.3149 +#47 := (>= #36 0::real)
  8.3150 +#134 := (forall (vars (?x2 T1)) (:pat #127) #47)
  8.3151 +#49 := (forall (vars (?x2 T1)) #47)
  8.3152 +#137 := (iff #49 #134)
  8.3153 +#135 := (iff #47 #47)
  8.3154 +#136 := [refl]: #135
  8.3155 +#138 := [quant-intro #136]: #137
  8.3156 +#67 := (~ #49 #49)
  8.3157 +#58 := (~ #47 #47)
  8.3158 +#66 := [refl]: #58
  8.3159 +#68 := [nnf-pos #66]: #67
  8.3160 +#10 := (<= #7 uf_1)
  8.3161 +#11 := (forall (vars (?x2 T1)) #10)
  8.3162 +#50 := (iff #11 #49)
  8.3163 +#46 := (iff #10 #47)
  8.3164 +#48 := [rewrite]: #46
  8.3165 +#51 := [quant-intro #48]: #50
  8.3166 +#32 := [asserted]: #11
  8.3167 +#52 := [mp #32 #51]: #49
  8.3168 +#69 := [mp~ #52 #68]: #49
  8.3169 +#139 := [mp #69 #138]: #134
  8.3170 +#149 := (not #134)
  8.3171 +#150 := (or #149 #143)
  8.3172 +#151 := [quant-inst]: #150
  8.3173 +#144 := [unit-resolution #151 #139]: #143
  8.3174 +#142 := (<= #141 0::real)
  8.3175 +#38 := (<= #36 0::real)
  8.3176 +#128 := (forall (vars (?x1 T1)) (:pat #127) #38)
  8.3177 +#41 := (forall (vars (?x1 T1)) #38)
  8.3178 +#131 := (iff #41 #128)
  8.3179 +#129 := (iff #38 #38)
  8.3180 +#130 := [refl]: #129
  8.3181 +#132 := [quant-intro #130]: #131
  8.3182 +#62 := (~ #41 #41)
  8.3183 +#64 := (~ #38 #38)
  8.3184 +#65 := [refl]: #64
  8.3185 +#63 := [nnf-pos #65]: #62
  8.3186 +#8 := (<= uf_1 #7)
  8.3187 +#9 := (forall (vars (?x1 T1)) #8)
  8.3188 +#42 := (iff #9 #41)
  8.3189 +#39 := (iff #8 #38)
  8.3190 +#40 := [rewrite]: #39
  8.3191 +#43 := [quant-intro #40]: #42
  8.3192 +#31 := [asserted]: #9
  8.3193 +#44 := [mp #31 #43]: #41
  8.3194 +#61 := [mp~ #44 #63]: #41
  8.3195 +#133 := [mp #61 #132]: #128
  8.3196 +#145 := (not #128)
  8.3197 +#146 := (or #145 #142)
  8.3198 +#147 := [quant-inst]: #146
  8.3199 +#148 := [unit-resolution #147 #133]: #142
  8.3200 +#45 := (= uf_1 #13)
  8.3201 +#55 := (not #45)
  8.3202 +#14 := (= #13 uf_1)
  8.3203 +#15 := (not #14)
  8.3204 +#56 := (iff #15 #55)
  8.3205 +#53 := (iff #14 #45)
  8.3206 +#54 := [rewrite]: #53
  8.3207 +#57 := [monotonicity #54]: #56
  8.3208 +#33 := [asserted]: #15
  8.3209 +#60 := [mp #33 #57]: #55
  8.3210 +#153 := (not #143)
  8.3211 +#152 := (not #142)
  8.3212 +#154 := (or #45 #152 #153)
  8.3213 +#155 := [th-lemma]: #154
  8.3214 +[unit-resolution #155 #60 #148 #144]: false
  8.3215 +unsat
  8.3216 +V+IAyBZU/6QjYs6JkXx8LQ 57 0
  8.3217 +#2 := false
  8.3218 +#4 := 0::real
  8.3219 +decl uf_1 :: (-> T2 real)
  8.3220 +decl uf_2 :: (-> T1 T1 T2)
  8.3221 +decl uf_12 :: (-> T4 T1)
  8.3222 +decl uf_4 :: T4
  8.3223 +#11 := uf_4
  8.3224 +#39 := (uf_12 uf_4)
  8.3225 +decl uf_10 :: T4
  8.3226 +#27 := uf_10
  8.3227 +#38 := (uf_12 uf_10)
  8.3228 +#40 := (uf_2 #38 #39)
  8.3229 +#41 := (uf_1 #40)
  8.3230 +#264 := (>= #41 0::real)
  8.3231 +#266 := (not #264)
  8.3232 +#43 := (= #41 0::real)
  8.3233 +#44 := (not #43)
  8.3234 +#131 := [asserted]: #44
  8.3235 +#272 := (or #43 #266)
  8.3236 +#42 := (<= #41 0::real)
  8.3237 +#130 := [asserted]: #42
  8.3238 +#265 := (not #42)
  8.3239 +#270 := (or #43 #265 #266)
  8.3240 +#271 := [th-lemma]: #270
  8.3241 +#273 := [unit-resolution #271 #130]: #272
  8.3242 +#274 := [unit-resolution #273 #131]: #266
  8.3243 +#6 := (:var 0 T1)
  8.3244 +#5 := (:var 1 T1)
  8.3245 +#7 := (uf_2 #5 #6)
  8.3246 +#241 := (pattern #7)
  8.3247 +#8 := (uf_1 #7)
  8.3248 +#65 := (>= #8 0::real)
  8.3249 +#242 := (forall (vars (?x1 T1) (?x2 T1)) (:pat #241) #65)
  8.3250 +#66 := (forall (vars (?x1 T1) (?x2 T1)) #65)
  8.3251 +#245 := (iff #66 #242)
  8.3252 +#243 := (iff #65 #65)
  8.3253 +#244 := [refl]: #243
  8.3254 +#246 := [quant-intro #244]: #245
  8.3255 +#149 := (~ #66 #66)
  8.3256 +#151 := (~ #65 #65)
  8.3257 +#152 := [refl]: #151
  8.3258 +#150 := [nnf-pos #152]: #149
  8.3259 +#9 := (<= 0::real #8)
  8.3260 +#10 := (forall (vars (?x1 T1) (?x2 T1)) #9)
  8.3261 +#67 := (iff #10 #66)
  8.3262 +#63 := (iff #9 #65)
  8.3263 +#64 := [rewrite]: #63
  8.3264 +#68 := [quant-intro #64]: #67
  8.3265 +#60 := [asserted]: #10
  8.3266 +#69 := [mp #60 #68]: #66
  8.3267 +#147 := [mp~ #69 #150]: #66
  8.3268 +#247 := [mp #147 #246]: #242
  8.3269 +#267 := (not #242)
  8.3270 +#268 := (or #267 #264)
  8.3271 +#269 := [quant-inst]: #268
  8.3272 +[unit-resolution #269 #247 #274]: false
  8.3273 +unsat
     9.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     9.2 +++ b/src/HOL/Multivariate_Analysis/Integration_MV.thy	Wed Feb 17 21:13:40 2010 +0100
     9.3 @@ -0,0 +1,3465 @@
     9.4 +
     9.5 +header {* Kurzweil-Henstock gauge integration in many dimensions. *}
     9.6 +(*  Author:                     John Harrison
     9.7 +    Translation from HOL light: Robert Himmelmann, TU Muenchen *)
     9.8 +
     9.9 +theory Integration_MV
    9.10 +  imports Derivative SMT
    9.11 +begin
    9.12 +
    9.13 +declare [[smt_certificates="~~/src/HOL/Multivariate_Analysis/Integration_MV.cert"]]
    9.14 +declare [[smt_record=true]]
    9.15 +declare [[z3_proofs=true]]
    9.16 +
    9.17 +lemma conjunctD2: assumes "a \<and> b" shows a b using assms by auto
    9.18 +lemma conjunctD3: assumes "a \<and> b \<and> c" shows a b c using assms by auto
    9.19 +lemma conjunctD4: assumes "a \<and> b \<and> c \<and> d" shows a b c d using assms by auto
    9.20 +lemma conjunctD5: assumes "a \<and> b \<and> c \<and> d \<and> e" shows a b c d e using assms by auto
    9.21 +
    9.22 +declare smult_conv_scaleR[simp]
    9.23 +
    9.24 +subsection {* Some useful lemmas about intervals. *}
    9.25 +
    9.26 +lemma empty_as_interval: "{} = {1..0::real^'n}"
    9.27 +  apply(rule set_ext,rule) defer unfolding vector_le_def mem_interval
    9.28 +  using UNIV_witness[where 'a='n] apply(erule_tac exE,rule_tac x=x in allE) by auto
    9.29 +
    9.30 +lemma interior_subset_union_intervals: 
    9.31 +  assumes "i = {a..b::real^'n}" "j = {c..d}" "interior j \<noteq> {}" "i \<subseteq> j \<union> s" "interior(i) \<inter> interior(j) = {}"
    9.32 +  shows "interior i \<subseteq> interior s" proof-
    9.33 +  have "{a<..<b} \<inter> {c..d} = {}" using inter_interval_mixed_eq_empty[of c d a b] and assms(3,5)
    9.34 +    unfolding assms(1,2) interior_closed_interval by auto
    9.35 +  moreover have "{a<..<b} \<subseteq> {c..d} \<union> s" apply(rule order_trans,rule interval_open_subset_closed)
    9.36 +    using assms(4) unfolding assms(1,2) by auto
    9.37 +  ultimately show ?thesis apply-apply(rule interior_maximal) defer apply(rule open_interior)
    9.38 +    unfolding assms(1,2) interior_closed_interval by auto qed
    9.39 +
    9.40 +lemma inter_interior_unions_intervals: fixes f::"(real^'n) set set"
    9.41 +  assumes "finite f" "open s" "\<forall>t\<in>f. \<exists>a b. t = {a..b}" "\<forall>t\<in>f. s \<inter> (interior t) = {}"
    9.42 +  shows "s \<inter> interior(\<Union>f) = {}" proof(rule ccontr,unfold ex_in_conv[THEN sym]) case goal1
    9.43 +  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)
    9.44 +    unfolding open_subset_interior[OF open_ball]  using interior_subset by auto
    9.45 +  have lem2:"\<And>x s P. \<exists>x\<in>s. P x \<Longrightarrow> \<exists>x\<in>insert x s. P x" by auto
    9.46 +  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
    9.47 +  thus ?case proof(induct rule:finite_induct) 
    9.48 +    case empty from this(2) guess x .. hence False unfolding Union_empty interior_empty by auto thus ?case by auto next
    9.49 +    case (insert i f) guess x using insert(5) .. note x = this
    9.50 +    then guess e unfolding open_contains_ball_eq[OF open_Int[OF assms(2) open_interior],rule_format] .. note e=this
    9.51 +    guess a using insert(4)[rule_format,OF insertI1] .. then guess b .. note ab = this
    9.52 +    show ?case proof(cases "x\<in>i") case False hence "x \<in> UNIV - {a..b}" unfolding ab by auto
    9.53 +      then guess d unfolding open_contains_ball_eq[OF open_Diff[OF open_UNIV closed_interval],rule_format] ..
    9.54 +      hence "0 < d" "ball x (min d e) \<subseteq> UNIV - i" using e unfolding ab by auto
    9.55 +      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
    9.56 +      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
    9.57 +    case True show ?thesis proof(cases "x\<in>{a<..<b}")
    9.58 +      case True then guess d unfolding open_contains_ball_eq[OF open_interval,rule_format] ..
    9.59 +      thus ?thesis apply(rule_tac x=i in bexI,rule_tac x=x in exI,rule_tac x="min d e" in exI)
    9.60 +	unfolding ab using interval_open_subset_closed[of a b] and e by fastsimp+ next
    9.61 +    case False then obtain k where "x$k \<le> a$k \<or> x$k \<ge> b$k" unfolding mem_interval by(auto simp add:not_less) 
    9.62 +    hence "x$k = a$k \<or> x$k = b$k" using True unfolding ab and mem_interval apply(erule_tac x=k in allE) by auto
    9.63 +    hence "\<exists>x. ball x (e/2) \<subseteq> s \<inter> (\<Union>f)" proof(erule_tac disjE)
    9.64 +      let ?z = "x - (e/2) *\<^sub>R basis k" assume as:"x$k = a$k" have "ball ?z (e / 2) \<inter> i = {}" apply(rule ccontr) unfolding ex_in_conv[THEN sym] proof(erule exE)
    9.65 +	fix y assume "y \<in> ball ?z (e / 2) \<inter> i" hence "dist ?z y < e/2" and yi:"y\<in>i" by auto
    9.66 +	hence "\<bar>(?z - y) $ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding vector_dist_norm by auto
    9.67 +	hence "y$k < a$k" unfolding vector_component_simps vector_scaleR_component as using e[THEN conjunct1] by(auto simp add:field_simps)
    9.68 +	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
    9.69 +      moreover have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)" apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]]) proof
    9.70 +	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)"
    9.71 +	   apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "(e/2) *\<^sub>R basis k"])
    9.72 +	  unfolding norm_scaleR norm_basis by auto
    9.73 +	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)
    9.74 +	finally show "y\<in>ball x e" unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps) qed
    9.75 +      ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto
    9.76 +    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)
    9.77 +	fix y assume "y \<in> ball ?z (e / 2) \<inter> i" hence "dist ?z y < e/2" and yi:"y\<in>i" by auto
    9.78 +	hence "\<bar>(?z - y) $ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding vector_dist_norm by auto
    9.79 +	hence "y$k > b$k" unfolding vector_component_simps vector_scaleR_component as using e[THEN conjunct1] by(auto simp add:field_simps)
    9.80 +	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
    9.81 +      moreover have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)" apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]]) proof
    9.82 +	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)"
    9.83 +	   apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "- (e/2) *\<^sub>R basis k"])
    9.84 +	  unfolding norm_scaleR norm_basis by auto
    9.85 +	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)
    9.86 +	finally show "y\<in>ball x e" unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps) qed
    9.87 +      ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto qed 
    9.88 +    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
    9.89 +    thus ?thesis apply-apply(rule lem2,rule insert(3)) using insert(4) by auto qed qed qed qed note * = this
    9.90 +  guess t using *[OF assms(1,3) goal1]  .. from this(2) guess x .. then guess e ..
    9.91 +  hence "x \<in> s" "x\<in>interior t" defer using open_subset_interior[OF open_ball, of x e t] by auto
    9.92 +  thus False using `t\<in>f` assms(4) by auto qed
    9.93 +subsection {* Bounds on intervals where they exist. *}
    9.94 +
    9.95 +definition "interval_upperbound (s::(real^'n) set) = (\<chi> i. Sup {a. \<exists>x\<in>s. x$i = a})"
    9.96 +
    9.97 +definition "interval_lowerbound (s::(real^'n) set) = (\<chi> i. Inf {a. \<exists>x\<in>s. x$i = a})"
    9.98 +
    9.99 +lemma interval_upperbound[simp]: assumes "\<forall>i. a$i \<le> b$i" shows "interval_upperbound {a..b} = b"
   9.100 +  using assms unfolding interval_upperbound_def Cart_eq Cart_lambda_beta apply-apply(rule,erule_tac x=i in allE)
   9.101 +  apply(rule Sup_unique) unfolding setle_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer
   9.102 +  apply(rule,rule) apply(rule_tac x="b$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=b in bexI)
   9.103 +  unfolding mem_interval using assms by auto
   9.104 +
   9.105 +lemma interval_lowerbound[simp]: assumes "\<forall>i. a$i \<le> b$i" shows "interval_lowerbound {a..b} = a"
   9.106 +  using assms unfolding interval_lowerbound_def Cart_eq Cart_lambda_beta apply-apply(rule,erule_tac x=i in allE)
   9.107 +  apply(rule Inf_unique) unfolding setge_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer
   9.108 +  apply(rule,rule) apply(rule_tac x="a$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=a in bexI)
   9.109 +  unfolding mem_interval using assms by auto
   9.110 +
   9.111 +lemmas interval_bounds = interval_upperbound interval_lowerbound
   9.112 +
   9.113 +lemma interval_bounds'[simp]: assumes "{a..b}\<noteq>{}" shows "interval_upperbound {a..b} = b" "interval_lowerbound {a..b} = a"
   9.114 +  using assms unfolding interval_ne_empty by auto
   9.115 +
   9.116 +lemma interval_upperbound_1[simp]: "dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> interval_upperbound {a..b} = (b::real^1)"
   9.117 +  apply(rule interval_upperbound) by auto
   9.118 +
   9.119 +lemma interval_lowerbound_1[simp]: "dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> interval_lowerbound {a..b} = (a::real^1)"
   9.120 +  apply(rule interval_lowerbound) by auto
   9.121 +
   9.122 +lemmas interval_bound_1 = interval_upperbound_1 interval_lowerbound_1
   9.123 +
   9.124 +subsection {* Content (length, area, volume...) of an interval. *}
   9.125 +
   9.126 +definition "content (s::(real^'n) set) =
   9.127 +       (if s = {} then 0 else (\<Prod>i\<in>UNIV. (interval_upperbound s)$i - (interval_lowerbound s)$i))"
   9.128 +
   9.129 +lemma interval_not_empty:"\<forall>i. a$i \<le> b$i \<Longrightarrow> {a..b::real^'n} \<noteq> {}"
   9.130 +  unfolding interval_eq_empty unfolding not_ex not_less by assumption
   9.131 +
   9.132 +lemma content_closed_interval: assumes "\<forall>i. a$i \<le> b$i"
   9.133 +  shows "content {a..b} = (\<Prod>i\<in>UNIV. b$i - a$i)"
   9.134 +  using interval_not_empty[OF assms] unfolding content_def interval_upperbound[OF assms] interval_lowerbound[OF assms] by auto
   9.135 +
   9.136 +lemma content_closed_interval': assumes "{a..b}\<noteq>{}" shows "content {a..b} = (\<Prod>i\<in>UNIV. b$i - a$i)"
   9.137 +  apply(rule content_closed_interval) using assms unfolding interval_ne_empty .
   9.138 +
   9.139 +lemma content_1:"dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> content {a..b} = dest_vec1 b - dest_vec1 a"
   9.140 +  using content_closed_interval[of a b] by auto
   9.141 +
   9.142 +lemma content_1':"a \<le> b \<Longrightarrow> content {vec1 a..vec1 b} = b - a" using content_1[of "vec a" "vec b"] by auto
   9.143 +
   9.144 +lemma content_unit[intro]: "content{0..1::real^'n} = 1" proof-
   9.145 +  have *:"\<forall>i. 0$i \<le> (1::real^'n::finite)$i" by auto
   9.146 +  have "0 \<in> {0..1::real^'n::finite}" unfolding mem_interval by auto
   9.147 +  thus ?thesis unfolding content_def interval_bounds[OF *] using setprod_1 by auto qed
   9.148 +
   9.149 +lemma content_pos_le[intro]: "0 \<le> content {a..b}" proof(cases "{a..b}={}")
   9.150 +  case False hence *:"\<forall>i. a $ i \<le> b $ i" unfolding interval_ne_empty by assumption
   9.151 +  have "(\<Prod>i\<in>UNIV. interval_upperbound {a..b} $ i - interval_lowerbound {a..b} $ i) \<ge> 0"
   9.152 +    apply(rule setprod_nonneg) unfolding interval_bounds[OF *] using * apply(erule_tac x=x in allE) by auto
   9.153 +  thus ?thesis unfolding content_def by(auto simp del:interval_bounds') qed(unfold content_def, auto)
   9.154 +
   9.155 +lemma content_pos_lt: assumes "\<forall>i. a$i < b$i" shows "0 < content {a..b}"
   9.156 +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
   9.157 +  show ?thesis unfolding content_closed_interval[OF help_lemma1[OF assms]] apply(rule setprod_pos)
   9.158 +    using assms apply(erule_tac x=x in allE) by auto qed
   9.159 +
   9.160 +lemma content_pos_lt_1: "dest_vec1 a < dest_vec1 b \<Longrightarrow> 0 < content({a..b})"
   9.161 +  apply(rule content_pos_lt) by auto
   9.162 +
   9.163 +lemma content_eq_0: "content({a..b::real^'n}) = 0 \<longleftrightarrow> (\<exists>i. b$i \<le> a$i)" proof(cases "{a..b} = {}")
   9.164 +  case True thus ?thesis unfolding content_def if_P[OF True] unfolding interval_eq_empty apply-
   9.165 +    apply(rule,erule exE) apply(rule_tac x=i in exI) by auto next
   9.166 +  guess a using UNIV_witness[where 'a='n] .. case False note as=this[unfolded interval_eq_empty not_ex not_less]
   9.167 +  show ?thesis unfolding content_def if_not_P[OF False] setprod_zero_iff[OF finite_UNIV]
   9.168 +    apply(rule) apply(erule_tac[!] exE bexE) unfolding interval_bounds[OF as] apply(rule_tac x=x in exI) defer
   9.169 +    apply(rule_tac x=i in bexI) using as apply(erule_tac x=i in allE) by auto qed
   9.170 +
   9.171 +lemma cond_cases:"(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)" by auto
   9.172 +
   9.173 +lemma content_closed_interval_cases:
   9.174 +  "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) 
   9.175 +  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
   9.176 +
   9.177 +lemma content_eq_0_interior: "content {a..b} = 0 \<longleftrightarrow> interior({a..b}) = {}"
   9.178 +  unfolding content_eq_0 interior_closed_interval interval_eq_empty by auto
   9.179 +
   9.180 +lemma content_eq_0_1: "content {a..b::real^1} = 0 \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a"
   9.181 +  unfolding content_eq_0 by auto
   9.182 +
   9.183 +lemma content_pos_lt_eq: "0 < content {a..b} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
   9.184 +  apply(rule) defer apply(rule content_pos_lt,assumption) proof- assume "0 < content {a..b}"
   9.185 +  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
   9.186 +
   9.187 +lemma content_empty[simp]: "content {} = 0" unfolding content_def by auto
   9.188 +
   9.189 +lemma content_subset: assumes "{a..b} \<subseteq> {c..d}" shows "content {a..b::real^'n} \<le> content {c..d}" proof(cases "{a..b}={}")
   9.190 +  case True thus ?thesis using content_pos_le[of c d] by auto next
   9.191 +  case False hence ab_ne:"\<forall>i. a $ i \<le> b $ i" unfolding interval_ne_empty by auto
   9.192 +  hence ab_ab:"a\<in>{a..b}" "b\<in>{a..b}" unfolding mem_interval by auto
   9.193 +  have "{c..d} \<noteq> {}" using assms False by auto
   9.194 +  hence cd_ne:"\<forall>i. c $ i \<le> d $ i" using assms unfolding interval_ne_empty by auto
   9.195 +  show ?thesis unfolding content_def unfolding interval_bounds[OF ab_ne] interval_bounds[OF cd_ne]
   9.196 +    unfolding if_not_P[OF False] if_not_P[OF `{c..d} \<noteq> {}`] apply(rule setprod_mono,rule) proof fix i::'n
   9.197 +    show "0 \<le> b $ i - a $ i" using ab_ne[THEN spec[where x=i]] by auto
   9.198 +    show "b $ i - a $ i \<le> d $ i - c $ i"
   9.199 +      using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(2),of i]
   9.200 +      using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(1),of i] by auto qed qed
   9.201 +
   9.202 +lemma content_lt_nz: "0 < content {a..b} \<longleftrightarrow> content {a..b} \<noteq> 0"
   9.203 +  unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by auto
   9.204 +
   9.205 +subsection {* The notion of a gauge --- simply an open set containing the point. *}
   9.206 +
   9.207 +definition gauge where "gauge d \<longleftrightarrow> (\<forall>x. x\<in>(d x) \<and> open(d x))"
   9.208 +
   9.209 +lemma gaugeI:assumes "\<And>x. x\<in>g x" "\<And>x. open (g x)" shows "gauge g"
   9.210 +  using assms unfolding gauge_def by auto
   9.211 +
   9.212 +lemma gaugeD[dest]: assumes "gauge d" shows "x\<in>d x" "open (d x)" using assms unfolding gauge_def by auto
   9.213 +
   9.214 +lemma gauge_ball_dependent: "\<forall>x. 0 < e x \<Longrightarrow> gauge (\<lambda>x. ball x (e x))"
   9.215 +  unfolding gauge_def by auto 
   9.216 +
   9.217 +lemma gauge_ball[intro?]: "0 < e \<Longrightarrow> gauge (\<lambda>x. ball x e)" unfolding gauge_def by auto 
   9.218 +
   9.219 +lemma gauge_trivial[intro]: "gauge (\<lambda>x. ball x 1)" apply(rule gauge_ball) by auto
   9.220 +
   9.221 +lemma gauge_inter: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. (d1 x) \<inter> (d2 x))"
   9.222 +  unfolding gauge_def by auto 
   9.223 +
   9.224 +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-
   9.225 +  have *:"\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s" by auto show ?thesis
   9.226 +  unfolding gauge_def unfolding * 
   9.227 +  using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto qed
   9.228 +
   9.229 +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)
   9.230 +
   9.231 +subsection {* Divisions. *}
   9.232 +
   9.233 +definition division_of (infixl "division'_of" 40) where
   9.234 +  "s division_of i \<equiv>
   9.235 +        finite s \<and>
   9.236 +        (\<forall>k\<in>s. k \<subseteq> i \<and> k \<noteq> {} \<and> (\<exists>a b. k = {a..b})) \<and>
   9.237 +        (\<forall>k1\<in>s. \<forall>k2\<in>s. k1 \<noteq> k2 \<longrightarrow> interior(k1) \<inter> interior(k2) = {}) \<and>
   9.238 +        (\<Union>s = i)"
   9.239 +
   9.240 +lemma division_ofD[dest]: assumes  "s division_of i"
   9.241 +  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})"
   9.242 +  "\<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
   9.243 +
   9.244 +lemma division_ofI:
   9.245 +  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})"
   9.246 +  "\<And>k1 k2. \<lbrakk>k1\<in>s; k2\<in>s; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}" "\<Union>s = i"
   9.247 +  shows "s division_of i" using assms unfolding division_of_def by auto
   9.248 +
   9.249 +lemma division_of_finite: "s division_of i \<Longrightarrow> finite s"
   9.250 +  unfolding division_of_def by auto
   9.251 +
   9.252 +lemma division_of_self[intro]: "{a..b} \<noteq> {} \<Longrightarrow> {{a..b}} division_of {a..b}"
   9.253 +  unfolding division_of_def by auto
   9.254 +
   9.255 +lemma division_of_trivial[simp]: "s division_of {} \<longleftrightarrow> s = {}" unfolding division_of_def by auto 
   9.256 +
   9.257 +lemma division_of_sing[simp]: "s division_of {a..a::real^'n} \<longleftrightarrow> s = {{a..a}}" (is "?l = ?r") proof
   9.258 +  assume ?r moreover { assume "s = {{a}}" moreover fix k assume "k\<in>s" 
   9.259 +    ultimately have"\<exists>x y. k = {x..y}" apply(rule_tac x=a in exI)+ unfolding interval_sing[THEN conjunct1] by auto }
   9.260 +  ultimately show ?l unfolding division_of_def interval_sing[THEN conjunct1] by auto next
   9.261 +  assume ?l note as=conjunctD4[OF this[unfolded division_of_def interval_sing[THEN conjunct1]]]
   9.262 +  { fix x assume x:"x\<in>s" have "x={a}" using as(2)[rule_format,OF x] by auto }
   9.263 +  moreover have "s \<noteq> {}" using as(4) by auto ultimately show ?r unfolding interval_sing[THEN conjunct1] by auto qed
   9.264 +
   9.265 +lemma elementary_empty: obtains p where "p division_of {}"
   9.266 +  unfolding division_of_trivial by auto
   9.267 +
   9.268 +lemma elementary_interval: obtains p where  "p division_of {a..b}"
   9.269 +  by(metis division_of_trivial division_of_self)
   9.270 +
   9.271 +lemma division_contains: "s division_of i \<Longrightarrow> \<forall>x\<in>i. \<exists>k\<in>s. x \<in> k"
   9.272 +  unfolding division_of_def by auto
   9.273 +
   9.274 +lemma forall_in_division:
   9.275 + "d division_of i \<Longrightarrow> ((\<forall>x\<in>d. P x) \<longleftrightarrow> (\<forall>a b. {a..b} \<in> d \<longrightarrow> P {a..b}))"
   9.276 +  unfolding division_of_def by fastsimp
   9.277 +
   9.278 +lemma division_of_subset: assumes "p division_of (\<Union>p)" "q \<subseteq> p" shows "q division_of (\<Union>q)"
   9.279 +  apply(rule division_ofI) proof- note as=division_ofD[OF assms(1)]
   9.280 +  show "finite q" apply(rule finite_subset) using as(1) assms(2) by auto
   9.281 +  { 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
   9.282 +  show "\<exists>a b. k = {a..b}" using as(4)[OF kp] by auto show "k \<noteq> {}" using as(3)[OF kp] by auto }
   9.283 +  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
   9.284 +  show "interior k1 \<inter> interior k2 = {}" using as(5)[OF *] by auto qed auto
   9.285 +
   9.286 +lemma division_of_union_self[intro]: "p division_of s \<Longrightarrow> p division_of (\<Union>p)" unfolding division_of_def by auto
   9.287 +
   9.288 +lemma division_of_content_0: assumes "content {a..b} = 0" "d division_of {a..b}" shows "\<forall>k\<in>d. content k = 0"
   9.289 +  unfolding forall_in_division[OF assms(2)] apply(rule,rule,rule) apply(drule division_ofD(2)[OF assms(2)])
   9.290 +  apply(drule content_subset) unfolding assms(1) proof- case goal1 thus ?case using content_pos_le[of a b] by auto qed
   9.291 +
   9.292 +lemma division_inter: assumes "p1 division_of s1" "p2 division_of (s2::(real^'a) set)"
   9.293 +  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-
   9.294 +let ?A = "{s. s \<in>  (\<lambda>(k1,k2). k1 \<inter> k2) ` (p1 \<times> p2) \<and> s \<noteq> {}}" have *:"?A' = ?A" by auto
   9.295 +show ?thesis unfolding * proof(rule division_ofI) have "?A \<subseteq> (\<lambda>(x, y). x \<inter> y) ` (p1 \<times> p2)" by auto
   9.296 +  moreover have "finite (p1 \<times> p2)" using assms unfolding division_of_def by auto ultimately show "finite ?A" by auto
   9.297 +  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
   9.298 +    using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)] by auto
   9.299 +  { 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
   9.300 +  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
   9.301 +  guess a1 using division_ofD(4)[OF assms(1) k(2)] .. then guess b1 .. note ab1=this
   9.302 +  guess a2 using division_ofD(4)[OF assms(2) k(3)] .. then guess b2 .. note ab2=this
   9.303 +  show "\<exists>a b. k = {a..b}" unfolding k ab1 ab2 unfolding inter_interval by auto } fix k1 k2
   9.304 +  assume "k1\<in>?A" then obtain x1 y1 where k1:"k1 = x1 \<inter> y1" "x1\<in>p1" "y1\<in>p2" "k1\<noteq>{}" by auto
   9.305 +  assume "k2\<in>?A" then obtain x2 y2 where k2:"k2 = x2 \<inter> y2" "x2\<in>p1" "y2\<in>p2" "k2\<noteq>{}" by auto
   9.306 +  assume "k1 \<noteq> k2" hence th:"x1\<noteq>x2 \<or> y1\<noteq>y2" unfolding k1 k2 by auto
   9.307 +  have *:"(interior x1 \<inter> interior x2 = {} \<or> interior y1 \<inter> interior y2 = {}) \<Longrightarrow>
   9.308 +      interior(x1 \<inter> y1) \<subseteq> interior(x1) \<Longrightarrow> interior(x1 \<inter> y1) \<subseteq> interior(y1) \<Longrightarrow>
   9.309 +      interior(x2 \<inter> y2) \<subseteq> interior(x2) \<Longrightarrow> interior(x2 \<inter> y2) \<subseteq> interior(y2)
   9.310 +      \<Longrightarrow> interior(x1 \<inter> y1) \<inter> interior(x2 \<inter> y2) = {}" by auto
   9.311 +  show "interior k1 \<inter> interior k2 = {}" unfolding k1 k2 apply(rule *) defer apply(rule_tac[1-4] subset_interior)
   9.312 +    using division_ofD(5)[OF assms(1) k1(2) k2(2)]
   9.313 +    using division_ofD(5)[OF assms(2) k1(3) k2(3)] using th by auto qed qed
   9.314 +
   9.315 +lemma division_inter_1: assumes "d division_of i" "{a..b::real^'n} \<subseteq> i"
   9.316 +  shows "{ {a..b} \<inter> k |k. k \<in> d \<and> {a..b} \<inter> k \<noteq> {} } division_of {a..b}" proof(cases "{a..b} = {}")
   9.317 +  case True show ?thesis unfolding True and division_of_trivial by auto next
   9.318 +  have *:"{a..b} \<inter> i = {a..b}" using assms(2) by auto 
   9.319 +  case False show ?thesis using division_inter[OF division_of_self[OF False] assms(1)] unfolding * by auto qed
   9.320 +
   9.321 +lemma elementary_inter: assumes "p1 division_of s" "p2 division_of (t::(real^'n) set)"
   9.322 +  shows "\<exists>p. p division_of (s \<inter> t)"
   9.323 +  by(rule,rule division_inter[OF assms])
   9.324 +
   9.325 +lemma elementary_inters: assumes "finite f" "f\<noteq>{}" "\<forall>s\<in>f. \<exists>p. p division_of (s::(real^'n) set)"
   9.326 +  shows "\<exists>p. p division_of (\<Inter> f)" using assms apply-proof(induct f rule:finite_induct)
   9.327 +case (insert x f) show ?case proof(cases "f={}")
   9.328 +  case True thus ?thesis unfolding True using insert by auto next
   9.329 +  case False guess p using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] ..
   9.330 +  moreover guess px using insert(5)[rule_format,OF insertI1] .. ultimately
   9.331 +  show ?thesis unfolding Inter_insert apply(rule_tac elementary_inter) by assumption+ qed qed auto
   9.332 +
   9.333 +lemma division_disjoint_union:
   9.334 +  assumes "p1 division_of s1" "p2 division_of s2" "interior s1 \<inter> interior s2 = {}"
   9.335 +  shows "(p1 \<union> p2) division_of (s1 \<union> s2)" proof(rule division_ofI) 
   9.336 +  note d1 = division_ofD[OF assms(1)] and d2 = division_ofD[OF assms(2)]
   9.337 +  show "finite (p1 \<union> p2)" using d1(1) d2(1) by auto
   9.338 +  show "\<Union>(p1 \<union> p2) = s1 \<union> s2" using d1(6) d2(6) by auto
   9.339 +  { 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 = {}"
   9.340 +  { 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)]]
   9.341 +      using assms(3) by blast } moreover
   9.342 +  { 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)]]
   9.343 +      using assms(3) by blast} ultimately
   9.344 +  show ?g using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto }
   9.345 +  fix k assume k:"k \<in> p1 \<union> p2"  show "k \<subseteq> s1 \<union> s2" using k d1(2) d2(2) by auto
   9.346 +  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
   9.347 +
   9.348 +lemma partial_division_extend_1:
   9.349 +  assumes "{c..d} \<subseteq> {a..b::real^'n}" "{c..d} \<noteq> {}"
   9.350 +  obtains p where "p division_of {a..b}" "{c..d} \<in> p"
   9.351 +proof- def n \<equiv> "CARD('n)" have n:"1 \<le> n" "0 < n" "n \<noteq> 0" unfolding n_def by auto
   9.352 +  guess \<pi> using ex_bij_betw_nat_finite_1[OF finite_UNIV[where 'a='n]] .. note \<pi>=this
   9.353 +  def \<pi>' \<equiv> "inv_into {1..n} \<pi>"
   9.354 +  have \<pi>':"bij_betw \<pi>' UNIV {1..n}" using bij_betw_inv_into[OF \<pi>] unfolding \<pi>'_def n_def by auto
   9.355 +  hence \<pi>'i:"\<And>i. \<pi>' i \<in> {1..n}" unfolding bij_betw_def by auto 
   9.356 +  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
   9.357 +  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
   9.358 +  have "{c..d} \<noteq> {}" using assms by auto
   9.359 +  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$i)}"
   9.360 +  let ?p2 = "\<lambda>l. {(\<chi> i. if \<pi>' i < l then c$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)}"
   9.361 +  let ?p =  "{?p1 l |l. l \<in> {1..n+1}} \<union> {?p2 l |l. l \<in> {1..n+1}}"
   9.362 +  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)
   9.363 +  show ?thesis apply(rule that[of ?p]) apply(rule division_ofI)
   9.364 +  proof- have "\<And>i. \<pi>' i < Suc n"
   9.365 +    proof(rule ccontr,unfold not_less) fix i assume "Suc n \<le> \<pi>' i"
   9.366 +      hence "\<pi>' i \<notin> {1..n}" by auto thus False using \<pi>' unfolding bij_betw_def by auto
   9.367 +    qed hence "c = (\<chi> i. if \<pi>' i < Suc n then c $ i else a $ i)"
   9.368 +        "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)"
   9.369 +      unfolding Cart_eq Cart_lambda_beta using \<pi>' unfolding bij_betw_def by auto
   9.370 +    thus cdp:"{c..d} \<in> ?p" apply-apply(rule UnI1) unfolding mem_Collect_eq apply(rule_tac x="n + 1" in exI) by auto
   9.371 +    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}"
   9.372 +      unfolding subset_eq apply(rule_tac[!] ballI,rule_tac[!] ccontr)
   9.373 +    proof- fix l assume l:"l\<in>{1..n+1}" fix x assume "x\<notin>{a..b}"
   9.374 +      then guess i unfolding mem_interval not_all .. note i=this
   9.375 +      show "x \<in> ?p1 l \<Longrightarrow> False" "x \<in> ?p2 l \<Longrightarrow> False" unfolding mem_interval apply(erule_tac[!] x=i in allE)
   9.376 +        apply(case_tac[!] "\<pi>' i < l", case_tac[!] "\<pi>' i = l") using abcd[of i] i by auto 
   9.377 +    qed moreover have "\<And>x. x \<in> {a..b} \<Longrightarrow> x \<in> \<Union>?p"
   9.378 +    proof- fix x assume x:"x\<in>{a..b}"
   9.379 +      { presume "x\<notin>{c..d} \<Longrightarrow> x \<in> \<Union>?p" thus "x \<in> \<Union>?p" using cdp by blast }
   9.380 +      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)}"
   9.381 +      assume "x\<notin>{c..d}" then guess i0 unfolding mem_interval not_all ..
   9.382 +      hence "\<pi>' i0 \<in> ?M" using \<pi>' unfolding bij_betw_def by(auto intro!:le_SucI)
   9.383 +      hence M:"finite ?M" "?M \<noteq> {}" by auto
   9.384 +      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]]
   9.385 +        Min_gr_iff[OF M,unfolded l_def[symmetric]]
   9.386 +      have "x\<in>?p1 l \<or> x\<in>?p2 l" using l(2)[THEN conjunct2] unfolding de_Morgan_conj not_le
   9.387 +        apply- apply(erule disjE) apply(rule disjI1) defer apply(rule disjI2)
   9.388 +      proof- assume as:"x $ \<pi> l < c $ \<pi> l"
   9.389 +        show "x \<in> ?p1 l" unfolding mem_interval Cart_lambda_beta
   9.390 +        proof case goal1 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le by auto
   9.391 +          thus ?case using as x[unfolded mem_interval,rule_format,of i]
   9.392 +            apply auto using l(3)[of "\<pi>' i"] by(auto elim!:ballE[where x="\<pi>' i"])
   9.393 +        qed
   9.394 +      next assume as:"x $ \<pi> l > d $ \<pi> l"
   9.395 +        show "x \<in> ?p2 l" unfolding mem_interval Cart_lambda_beta
   9.396 +        proof case goal1 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le by auto
   9.397 +          thus ?case using as x[unfolded mem_interval,rule_format,of i]
   9.398 +            apply auto using l(3)[of "\<pi>' i"] by(auto elim!:ballE[where x="\<pi>' i"])
   9.399 +        qed qed
   9.400 +      thus "x \<in> \<Union>?p" using l(2) by blast 
   9.401 +    qed ultimately show "\<Union>?p = {a..b}" apply-apply(rule) defer apply(rule) by(assumption,blast)
   9.402 +    
   9.403 +    show "finite ?p" by auto
   9.404 +    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
   9.405 +    show "k\<subseteq>{a..b}" apply(rule,unfold mem_interval,rule,rule) 
   9.406 +    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
   9.407 +      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)
   9.408 +    qed have "\<And>l. ?p1 l \<noteq> {}" "\<And>l. ?p2 l \<noteq> {}" unfolding interval_eq_empty not_ex apply(rule_tac[!] allI)
   9.409 +    proof- case goal1 thus ?case using abcd[of x] by auto
   9.410 +    next   case goal2 thus ?case using abcd[of x] by auto
   9.411 +    qed thus "k \<noteq> {}" using k by auto
   9.412 +    show "\<exists>a b. k = {a..b}" using k by auto
   9.413 +    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
   9.414 +    { fix k k' l l'
   9.415 +      assume k:"k\<in>?p" and l:"k = ?p1 l \<or> k = ?p2 l" "l \<in> {1..n + 1}" 
   9.416 +      assume k':"k' \<in> ?p" "k \<noteq> k'" and  l':"k' = ?p1 l' \<or> k' = ?p2 l'" "l' \<in> {1..n + 1}" 
   9.417 +      assume "l \<le> l'" fix x
   9.418 +      have "x \<notin> interior k \<inter> interior k'" 
   9.419 +      proof(rule,cases "l' = n+1") assume x:"x \<in> interior k \<inter> interior k'"
   9.420 +        case True hence "\<And>i. \<pi>' i < l'" using \<pi>'i by(auto simp add:less_Suc_eq_le)
   9.421 +        hence k':"k' = {c..d}" using l'(1) \<pi>'i by(auto simp add:Cart_nth_inverse)
   9.422 +        have ln:"l < n + 1" 
   9.423 +        proof(rule ccontr) case goal1 hence l2:"l = n+1" using l by auto
   9.424 +          hence "\<And>i. \<pi>' i < l" using \<pi>'i by(auto simp add:less_Suc_eq_le)
   9.425 +          hence "k = {c..d}" using l(1) \<pi>'i by(auto simp add:Cart_nth_inverse)
   9.426 +          thus False using `k\<noteq>k'` k' by auto
   9.427 +        qed have **:"\<pi>' (\<pi> l) = l" using \<pi>'\<pi>[of l] using l ln by auto
   9.428 +        have "x $ \<pi> l < c $ \<pi> l \<or> d $ \<pi> l < x $ \<pi> l" using l(1) apply-
   9.429 +        proof(erule disjE)
   9.430 +          assume as:"k = ?p1 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
   9.431 +          show ?thesis using *[of "\<pi> l"] using ln unfolding Cart_lambda_beta ** by auto
   9.432 +        next assume as:"k = ?p2 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
   9.433 +          show ?thesis using *[of "\<pi> l"] using ln unfolding Cart_lambda_beta ** by auto
   9.434 +        qed thus False using x unfolding k' unfolding Int_iff interior_closed_interval mem_interval
   9.435 +          by(auto elim!:allE[where x="\<pi> l"])
   9.436 +      next case False hence "l < n + 1" using l'(2) using `l\<le>l'` by auto
   9.437 +        hence ln:"l \<in> {1..n}" "l' \<in> {1..n}" using l l' False by auto
   9.438 +        note \<pi>l = \<pi>'\<pi>[OF ln(1)] \<pi>'\<pi>[OF ln(2)]
   9.439 +        assume x:"x \<in> interior k \<inter> interior k'"
   9.440 +        show False using l(1) l'(1) apply-
   9.441 +        proof(erule_tac[!] disjE)+
   9.442 +          assume as:"k = ?p1 l" "k' = ?p1 l'"
   9.443 +          note * = x[unfolded as Int_iff interior_closed_interval mem_interval]
   9.444 +          have "l \<noteq> l'" using k'(2)[unfolded as] by auto
   9.445 +          thus False using * by(smt Cart_lambda_beta \<pi>l)
   9.446 +        next assume as:"k = ?p2 l" "k' = ?p2 l'"
   9.447 +          note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
   9.448 +          have "l \<noteq> l'" apply(rule) using k'(2)[unfolded as] by auto
   9.449 +          thus False using *[of "\<pi> l"] *[of "\<pi> l'"]
   9.450 +            unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` by auto
   9.451 +        next assume as:"k = ?p1 l" "k' = ?p2 l'"
   9.452 +          note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
   9.453 +          show False using *[of "\<pi> l"] *[of "\<pi> l'"]
   9.454 +            unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` using abcd[of "\<pi> l'"] by smt 
   9.455 +        next assume as:"k = ?p2 l" "k' = ?p1 l'"
   9.456 +          note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
   9.457 +          show False using *[of "\<pi> l"] *[of "\<pi> l'"]
   9.458 +            unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` using abcd[of "\<pi> l'"] by smt
   9.459 +        qed qed } 
   9.460 +    from this[OF k l k' l'] this[OF k'(1) l' k _ l] have "\<And>x. x \<notin> interior k \<inter> interior k'"
   9.461 +      apply - apply(cases "l' \<le> l") using k'(2) by auto            
   9.462 +    thus "interior k \<inter> interior k' = {}" by auto        
   9.463 +qed qed
   9.464 +
   9.465 +lemma partial_division_extend_interval: assumes "p division_of (\<Union>p)" "(\<Union>p) \<subseteq> {a..b}"
   9.466 +  obtains q where "p \<subseteq> q" "q division_of {a..b::real^'n}" proof(cases "p = {}")
   9.467 +  case True guess q apply(rule elementary_interval[of a b]) .
   9.468 +  thus ?thesis apply- apply(rule that[of q]) unfolding True by auto next
   9.469 +  case False note p = division_ofD[OF assms(1)]
   9.470 +  have *:"\<forall>k\<in>p. \<exists>q. q division_of {a..b} \<and> k\<in>q" proof case goal1
   9.471 +    guess c using p(4)[OF goal1] .. then guess d .. note cd_ = this
   9.472 +    have *:"{c..d} \<subseteq> {a..b}" "{c..d} \<noteq> {}" using p(2,3)[OF goal1, unfolded cd_] using assms(2) by auto
   9.473 +    guess q apply(rule partial_division_extend_1[OF *]) . thus ?case unfolding cd_ by auto qed
   9.474 +  guess q using bchoice[OF *] .. note q = conjunctD2[OF this[rule_format]]
   9.475 +  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-
   9.476 +    fix x assume x:"x\<in>p" show "q x division_of \<Union>q x" apply-apply(rule division_ofI)
   9.477 +      using division_ofD[OF q(1)[OF x]] by auto show "q x - {x} \<subseteq> q x" by auto qed
   9.478 +  hence "\<exists>d. d division_of \<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)" apply- apply(rule elementary_inters)
   9.479 +    apply(rule finite_imageI[OF p(1)]) unfolding image_is_empty apply(rule False) by auto
   9.480 +  then guess d .. note d = this
   9.481 +  show ?thesis apply(rule that[of "d \<union> p"]) proof-
   9.482 +    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
   9.483 +    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"
   9.484 +      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
   9.485 +    show "d \<union> p division_of {a..b}" unfolding * apply(rule division_disjoint_union[OF d assms(1)])
   9.486 +      apply(rule inter_interior_unions_intervals) apply(rule p open_interior ballI)+ proof(assumption,rule)
   9.487 +      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
   9.488 +      show "interior (\<Inter>(\<lambda>i. \<Union>(q i - {i})) ` p) \<inter> interior k = {}" apply(rule *[of _ "interior (\<Union>(q k - {k}))"])
   9.489 +	defer apply(subst Int_commute) apply(rule inter_interior_unions_intervals) proof- note qk=division_ofD[OF q(1)[OF k]]
   9.490 +	show "finite (q k - {k})" "open (interior k)"  "\<forall>t\<in>q k - {k}. \<exists>a b. t = {a..b}" using qk by auto
   9.491 +	show "\<forall>t\<in>q k - {k}. interior k \<inter> interior t = {}" using qk(5) using q(2)[OF k] by auto
   9.492 +	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}))"
   9.493 +	  apply(rule subset_interior *)+ using k by auto qed qed qed auto qed
   9.494 +
   9.495 +lemma elementary_bounded[dest]: "p division_of s \<Longrightarrow> bounded (s::(real^'n) set)"
   9.496 +  unfolding division_of_def by(metis bounded_Union bounded_interval) 
   9.497 +
   9.498 +lemma elementary_subset_interval: "p division_of s \<Longrightarrow> \<exists>a b. s \<subseteq> {a..b::real^'n}"
   9.499 +  by(meson elementary_bounded bounded_subset_closed_interval)
   9.500 +
   9.501 +lemma division_union_intervals_exists: assumes "{a..b::real^'n} \<noteq> {}"
   9.502 +  obtains p where "(insert {a..b} p) division_of ({a..b} \<union> {c..d})" proof(cases "{c..d} = {}")
   9.503 +  case True show ?thesis apply(rule that[of "{}"]) unfolding True using assms by auto next
   9.504 +  case False note false=this show ?thesis proof(cases "{a..b} \<inter> {c..d} = {}")
   9.505 +  have *:"\<And>a b. {a,b} = {a} \<union> {b}" by auto
   9.506 +  case True show ?thesis apply(rule that[of "{{c..d}}"]) unfolding * apply(rule division_disjoint_union)
   9.507 +    using false True assms using interior_subset by auto next
   9.508 +  case False obtain u v where uv:"{a..b} \<inter> {c..d} = {u..v}" unfolding inter_interval by auto
   9.509 +  have *:"{u..v} \<subseteq> {c..d}" using uv by auto
   9.510 +  guess p apply(rule partial_division_extend_1[OF * False[unfolded uv]]) . note p=this division_ofD[OF this(1)]
   9.511 +  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
   9.512 +  show thesis apply(rule that[of "p - {{u..v}}"]) unfolding *(1) apply(subst *(2)) apply(rule division_disjoint_union)
   9.513 +    apply(rule,rule assms) apply(rule division_of_subset[of p]) apply(rule division_of_union_self[OF p(1)]) defer
   9.514 +    unfolding interior_inter[THEN sym] proof-
   9.515 +    have *:"\<And>cd p uv ab. p \<subseteq> cd \<Longrightarrow> ab \<inter> cd = uv \<Longrightarrow> ab \<inter> p = uv \<inter> p" by auto
   9.516 +    have "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = interior({u..v} \<inter> \<Union>(p - {{u..v}}))" 
   9.517 +      apply(rule arg_cong[of _ _ interior]) apply(rule *[OF _ uv]) using p(8) by auto
   9.518 +    also have "\<dots> = {}" unfolding interior_inter apply(rule inter_interior_unions_intervals) using p(6) p(7)[OF p(2)] p(3) by auto
   9.519 +    finally show "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = {}" by assumption qed auto qed qed
   9.520 +
   9.521 +lemma division_of_unions: assumes "finite f"  "\<And>p. p\<in>f \<Longrightarrow> p division_of (\<Union>p)"
   9.522 +  "\<And>k1 k2. \<lbrakk>k1 \<in> \<Union>f; k2 \<in> \<Union>f; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
   9.523 +  shows "\<Union>f division_of \<Union>\<Union>f" apply(rule division_ofI) prefer 5 apply(rule assms(3)|assumption)+
   9.524 +  apply(rule finite_Union assms(1))+ prefer 3 apply(erule UnionE) apply(rule_tac s=X in division_ofD(3)[OF assms(2)])
   9.525 +  using division_ofD[OF assms(2)] by auto
   9.526 +  
   9.527 +lemma elementary_union_interval: assumes "p division_of \<Union>p"
   9.528 +  obtains q where "q division_of ({a..b::real^'n} \<union> \<Union>p)" proof-
   9.529 +  note assm=division_ofD[OF assms]
   9.530 +  have lem1:"\<And>f s. \<Union>\<Union> (f ` s) = \<Union>(\<lambda>x.\<Union>(f x)) ` s" by auto
   9.531 +  have lem2:"\<And>f s. f \<noteq> {} \<Longrightarrow> \<Union>{s \<union> t |t. t \<in> f} = s \<union> \<Union>f" by auto
   9.532 +{ presume "p={} \<Longrightarrow> thesis" "{a..b} = {} \<Longrightarrow> thesis" "{a..b} \<noteq> {} \<Longrightarrow> interior {a..b} = {} \<Longrightarrow> thesis"
   9.533 +    "p\<noteq>{} \<Longrightarrow> interior {a..b}\<noteq>{} \<Longrightarrow> {a..b} \<noteq> {} \<Longrightarrow> thesis"
   9.534 +  thus thesis by auto
   9.535 +next assume as:"p={}" guess p apply(rule elementary_interval[of a b]) .
   9.536 +  thus thesis apply(rule_tac that[of p]) unfolding as by auto 
   9.537 +next assume as:"{a..b}={}" show thesis apply(rule that) unfolding as using assms by auto
   9.538 +next assume as:"interior {a..b} = {}" "{a..b} \<noteq> {}"
   9.539 +  show thesis apply(rule that[of "insert {a..b} p"],rule division_ofI)
   9.540 +    unfolding finite_insert apply(rule assm(1)) unfolding Union_insert  
   9.541 +    using assm(2-4) as apply- by(fastsimp dest: assm(5))+
   9.542 +next assume as:"p \<noteq> {}" "interior {a..b} \<noteq> {}" "{a..b}\<noteq>{}"
   9.543 +  have "\<forall>k\<in>p. \<exists>q. (insert {a..b} q) division_of ({a..b} \<union> k)" proof case goal1
   9.544 +    from assm(4)[OF this] guess c .. then guess d ..
   9.545 +    thus ?case apply-apply(rule division_union_intervals_exists[OF as(3),of c d]) by auto
   9.546 +  qed from bchoice[OF this] guess q .. note q=division_ofD[OF this[rule_format]]
   9.547 +  let ?D = "\<Union>{insert {a..b} (q k) | k. k \<in> p}"
   9.548 +  show thesis apply(rule that[of "?D"]) proof(rule division_ofI)
   9.549 +    have *:"{insert {a..b} (q k) |k. k \<in> p} = (\<lambda>k. insert {a..b} (q k)) ` p" by auto
   9.550 +    show "finite ?D" apply(rule finite_Union) unfolding * apply(rule finite_imageI) using assm(1) q(1) by auto
   9.551 +    show "\<Union>?D = {a..b} \<union> \<Union>p" unfolding * lem1 unfolding lem2[OF as(1), of "{a..b}",THEN sym]
   9.552 +      using q(6) by auto
   9.553 +    fix k assume k:"k\<in>?D" thus " k \<subseteq> {a..b} \<union> \<Union>p" using q(2) by auto
   9.554 +    show "k \<noteq> {}" using q(3) k by auto show "\<exists>a b. k = {a..b}" using q(4) k by auto
   9.555 +    fix k' assume k':"k'\<in>?D" "k\<noteq>k'"
   9.556 +    obtain x  where x: "k \<in>insert {a..b} (q x)"  "x\<in>p"  using k  by auto
   9.557 +    obtain x' where x':"k'\<in>insert {a..b} (q x')" "x'\<in>p" using k' by auto
   9.558 +    show "interior k \<inter> interior k' = {}" proof(cases "x=x'")
   9.559 +      case True show ?thesis apply(rule q(5)) using x x' k' unfolding True by auto
   9.560 +    next case False 
   9.561 +      { presume "k = {a..b} \<Longrightarrow> ?thesis" "k' = {a..b} \<Longrightarrow> ?thesis" 
   9.562 +        "k \<noteq> {a..b} \<Longrightarrow> k' \<noteq> {a..b} \<Longrightarrow> ?thesis"
   9.563 +        thus ?thesis by auto }
   9.564 +      { assume as':"k  = {a..b}" show ?thesis apply(rule q(5)) using x' k'(2) unfolding as' by auto }
   9.565 +      { assume as':"k' = {a..b}" show ?thesis apply(rule q(5)) using x  k'(2) unfolding as' by auto }
   9.566 +      assume as':"k \<noteq> {a..b}" "k' \<noteq> {a..b}"
   9.567 +      guess c using q(4)[OF x(2,1)] .. then guess d .. note c_d=this
   9.568 +      have "interior k  \<inter> interior {a..b} = {}" apply(rule q(5)) using x  k'(2) using as' by auto
   9.569 +      hence "interior k \<subseteq> interior x" apply-
   9.570 +        apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x(2,1)]]) by auto moreover
   9.571 +      guess c using q(4)[OF x'(2,1)] .. then guess d .. note c_d=this
   9.572 +      have "interior k' \<inter> interior {a..b} = {}" apply(rule q(5)) using x' k'(2) using as' by auto
   9.573 +      hence "interior k' \<subseteq> interior x'" apply-
   9.574 +        apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x'(2,1)]]) by auto
   9.575 +      ultimately show ?thesis using assm(5)[OF x(2) x'(2) False] by auto
   9.576 +    qed qed } qed
   9.577 +
   9.578 +lemma elementary_unions_intervals:
   9.579 +  assumes "finite f" "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = {a..b::real^'n}"
   9.580 +  obtains p where "p division_of (\<Union>f)" proof-
   9.581 +  have "\<exists>p. p division_of (\<Union>f)" proof(induct_tac f rule:finite_subset_induct) 
   9.582 +    show "\<exists>p. p division_of \<Union>{}" using elementary_empty by auto
   9.583 +    fix x F assume as:"finite F" "x \<notin> F" "\<exists>p. p division_of \<Union>F" "x\<in>f"
   9.584 +    from this(3) guess p .. note p=this
   9.585 +    from assms(2)[OF as(4)] guess a .. then guess b .. note ab=this
   9.586 +    have *:"\<Union>F = \<Union>p" using division_ofD[OF p] by auto
   9.587 +    show "\<exists>p. p division_of \<Union>insert x F" using elementary_union_interval[OF p[unfolded *], of a b]
   9.588 +      unfolding Union_insert ab * by auto
   9.589 +  qed(insert assms,auto) thus ?thesis apply-apply(erule exE,rule that) by auto qed
   9.590 +
   9.591 +lemma elementary_union: assumes "ps division_of s" "pt division_of (t::(real^'n) set)"
   9.592 +  obtains p where "p division_of (s \<union> t)"
   9.593 +proof- have "s \<union> t = \<Union>ps \<union> \<Union>pt" using assms unfolding division_of_def by auto
   9.594 +  hence *:"\<Union>(ps \<union> pt) = s \<union> t" by auto
   9.595 +  show ?thesis apply-apply(rule elementary_unions_intervals[of "ps\<union>pt"])
   9.596 +    unfolding * prefer 3 apply(rule_tac p=p in that)
   9.597 +    using assms[unfolded division_of_def] by auto qed
   9.598 +
   9.599 +lemma partial_division_extend: fixes t::"(real^'n) set"
   9.600 +  assumes "p division_of s" "q division_of t" "s \<subseteq> t"
   9.601 +  obtains r where "p \<subseteq> r" "r division_of t" proof-
   9.602 +  note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)]
   9.603 +  obtain a b where ab:"t\<subseteq>{a..b}" using elementary_subset_interval[OF assms(2)] by auto
   9.604 +  guess r1 apply(rule partial_division_extend_interval) apply(rule assms(1)[unfolded divp(6)[THEN sym]])
   9.605 +    apply(rule subset_trans) by(rule ab assms[unfolded divp(6)[THEN sym]])+  note r1 = this division_ofD[OF this(2)]
   9.606 +  guess p' apply(rule elementary_unions_intervals[of "r1 - p"]) using r1(3,6) by auto 
   9.607 +  then obtain r2 where r2:"r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)" 
   9.608 +    apply- apply(drule elementary_inter[OF _ assms(2)[unfolded divq(6)[THEN sym]]]) by auto
   9.609 +  { fix x assume x:"x\<in>t" "x\<notin>s"
   9.610 +    hence "x\<in>\<Union>r1" unfolding r1 using ab by auto
   9.611 +    then guess r unfolding Union_iff .. note r=this moreover
   9.612 +    have "r \<notin> p" proof assume "r\<in>p" hence "x\<in>s" using divp(2) r by auto
   9.613 +      thus False using x by auto qed
   9.614 +    ultimately have "x\<in>\<Union>(r1 - p)" by auto }
   9.615 +  hence *:"t = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)" unfolding divp divq using assms(3) by auto
   9.616 +  show ?thesis apply(rule that[of "p \<union> r2"]) unfolding * defer apply(rule division_disjoint_union)
   9.617 +    unfolding divp(6) apply(rule assms r2)+
   9.618 +  proof- have "interior s \<inter> interior (\<Union>(r1-p)) = {}"
   9.619 +    proof(rule inter_interior_unions_intervals)
   9.620 +      show "finite (r1 - p)" "open (interior s)" "\<forall>t\<in>r1-p. \<exists>a b. t = {a..b}" using r1 by auto
   9.621 +      have *:"\<And>s. (\<And>x. x \<in> s \<Longrightarrow> False) \<Longrightarrow> s = {}" by auto
   9.622 +      show "\<forall>t\<in>r1-p. interior s \<inter> interior t = {}" proof(rule)
   9.623 +        fix m x assume as:"m\<in>r1-p"
   9.624 +        have "interior m \<inter> interior (\<Union>p) = {}" proof(rule inter_interior_unions_intervals)
   9.625 +          show "finite p" "open (interior m)" "\<forall>t\<in>p. \<exists>a b. t = {a..b}" using divp by auto
   9.626 +          show "\<forall>t\<in>p. interior m \<inter> interior t = {}" apply(rule, rule r1(7)) using as using r1 by auto
   9.627 +        qed thus "interior s \<inter> interior m = {}" unfolding divp by auto
   9.628 +      qed qed        
   9.629 +    thus "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}" using interior_subset by auto
   9.630 +  qed auto qed
   9.631 +
   9.632 +subsection {* Tagged (partial) divisions. *}
   9.633 +
   9.634 +definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40) where
   9.635 +  "(s tagged_partial_division_of i) \<equiv>
   9.636 +        finite s \<and>
   9.637 +        (\<forall>x k. (x,k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
   9.638 +        (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ((x1,k1) \<noteq> (x2,k2))
   9.639 +                       \<longrightarrow> (interior(k1) \<inter> interior(k2) = {}))"
   9.640 +
   9.641 +lemma tagged_partial_division_ofD[dest]: assumes "s tagged_partial_division_of i"
   9.642 +  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"
   9.643 +  "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
   9.644 +  "\<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) = {}"
   9.645 +  using assms unfolding tagged_partial_division_of_def  apply- by blast+ 
   9.646 +
   9.647 +definition tagged_division_of (infixr "tagged'_division'_of" 40) where
   9.648 +  "(s tagged_division_of i) \<equiv>
   9.649 +        (s tagged_partial_division_of i) \<and> (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
   9.650 +
   9.651 +lemma tagged_division_of_finite[dest]: "s tagged_division_of i \<Longrightarrow> finite s"
   9.652 +  unfolding tagged_division_of_def tagged_partial_division_of_def by auto
   9.653 +
   9.654 +lemma tagged_division_of:
   9.655 + "(s tagged_division_of i) \<longleftrightarrow>
   9.656 +        finite s \<and>
   9.657 +        (\<forall>x k. (x,k) \<in> s
   9.658 +               \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
   9.659 +        (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ~((x1,k1) = (x2,k2))
   9.660 +                       \<longrightarrow> (interior(k1) \<inter> interior(k2) = {})) \<and>
   9.661 +        (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
   9.662 +  unfolding tagged_division_of_def tagged_partial_division_of_def by auto
   9.663 +
   9.664 +lemma tagged_division_ofI: assumes
   9.665 +  "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}"
   9.666 +  "\<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) = {})"
   9.667 +  "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
   9.668 +  shows "s tagged_division_of i"
   9.669 +  unfolding tagged_division_of apply(rule) defer apply rule
   9.670 +  apply(rule allI impI conjI assms)+ apply assumption
   9.671 +  apply(rule, rule assms, assumption) apply(rule assms, assumption)
   9.672 +  using assms(1,5-) apply- by blast+
   9.673 +
   9.674 +lemma tagged_division_ofD[dest]: assumes "s tagged_division_of i"
   9.675 +  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}"
   9.676 +  "\<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) = {})"
   9.677 +  "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)" using assms unfolding tagged_division_of apply- by blast+
   9.678 +
   9.679 +lemma division_of_tagged_division: assumes"s tagged_division_of i"  shows "(snd ` s) division_of i"
   9.680 +proof(rule division_ofI) note assm=tagged_division_ofD[OF assms]
   9.681 +  show "\<Union>snd ` s = i" "finite (snd ` s)" using assm by auto
   9.682 +  fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto
   9.683 +  thus  "k \<subseteq> i" "k \<noteq> {}" "\<exists>a b. k = {a..b}" using assm apply- by fastsimp+
   9.684 +  fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto
   9.685 +  thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto
   9.686 +qed
   9.687 +
   9.688 +lemma partial_division_of_tagged_division: assumes "s tagged_partial_division_of i"
   9.689 +  shows "(snd ` s) division_of \<Union>(snd ` s)"
   9.690 +proof(rule division_ofI) note assm=tagged_partial_division_ofD[OF assms]
   9.691 +  show "finite (snd ` s)" "\<Union>snd ` s = \<Union>snd ` s" using assm by auto
   9.692 +  fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto
   9.693 +  thus "k\<noteq>{}" "\<exists>a b. k = {a..b}" "k \<subseteq> \<Union>snd ` s" using assm by auto
   9.694 +  fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto
   9.695 +  thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto
   9.696 +qed
   9.697 +
   9.698 +lemma tagged_partial_division_subset: assumes "s tagged_partial_division_of i" "t \<subseteq> s"
   9.699 +  shows "t tagged_partial_division_of i"
   9.700 +  using assms unfolding tagged_partial_division_of_def using finite_subset[OF assms(2)] by blast
   9.701 +
   9.702 +lemma setsum_over_tagged_division_lemma: fixes d::"(real^'m) set \<Rightarrow> 'a::real_normed_vector"
   9.703 +  assumes "p tagged_division_of i" "\<And>u v. {u..v} \<noteq> {} \<Longrightarrow> content {u..v} = 0 \<Longrightarrow> d {u..v} = 0"
   9.704 +  shows "setsum (\<lambda>(x,k). d k) p = setsum d (snd ` p)"
   9.705 +proof- note assm=tagged_division_ofD[OF assms(1)]
   9.706 +  have *:"(\<lambda>(x,k). d k) = d \<circ> snd" unfolding o_def apply(rule ext) by auto
   9.707 +  show ?thesis unfolding * apply(subst eq_commute) proof(rule setsum_reindex_nonzero)
   9.708 +    show "finite p" using assm by auto
   9.709 +    fix x y assume as:"x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y" 
   9.710 +    obtain a b where ab:"snd x = {a..b}" using assm(4)[of "fst x" "snd x"] as(1) by auto
   9.711 +    have "(fst x, snd y) \<in> p" "(fst x, snd y) \<noteq> y" unfolding as(4)[THEN sym] using as(1-3) by auto
   9.712 +    hence "interior (snd x) \<inter> interior (snd y) = {}" apply-apply(rule assm(5)[of "fst x" _ "fst y"]) using as by auto 
   9.713 +    hence "content {a..b} = 0" unfolding as(4)[THEN sym] ab content_eq_0_interior by auto
   9.714 +    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
   9.715 +    thus "d (snd x) = 0" unfolding ab by auto qed qed
   9.716 +
   9.717 +lemma tag_in_interval: "p tagged_division_of i \<Longrightarrow> (x,k) \<in> p \<Longrightarrow> x \<in> i" by auto
   9.718 +
   9.719 +lemma tagged_division_of_empty: "{} tagged_division_of {}"
   9.720 +  unfolding tagged_division_of by auto
   9.721 +
   9.722 +lemma tagged_partial_division_of_trivial[simp]:
   9.723 + "p tagged_partial_division_of {} \<longleftrightarrow> p = {}"
   9.724 +  unfolding tagged_partial_division_of_def by auto
   9.725 +
   9.726 +lemma tagged_division_of_trivial[simp]:
   9.727 + "p tagged_division_of {} \<longleftrightarrow> p = {}"
   9.728 +  unfolding tagged_division_of by auto
   9.729 +
   9.730 +lemma tagged_division_of_self:
   9.731 + "x \<in> {a..b} \<Longrightarrow> {(x,{a..b})} tagged_division_of {a..b}"
   9.732 +  apply(rule tagged_division_ofI) by auto
   9.733 +
   9.734 +lemma tagged_division_union:
   9.735 +  assumes "p1 tagged_division_of s1"  "p2 tagged_division_of s2" "interior s1 \<inter> interior s2 = {}"
   9.736 +  shows "(p1 \<union> p2) tagged_division_of (s1 \<union> s2)"
   9.737 +proof(rule tagged_division_ofI) note p1=tagged_division_ofD[OF assms(1)] and p2=tagged_division_ofD[OF assms(2)]
   9.738 +  show "finite (p1 \<union> p2)" using p1(1) p2(1) by auto
   9.739 +  show "\<Union>{k. \<exists>x. (x, k) \<in> p1 \<union> p2} = s1 \<union> s2" using p1(6) p2(6) by blast
   9.740 +  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
   9.741 +  show "k\<subseteq>s1\<union>s2" using xk p1(3) p2(3) by blast
   9.742 +  fix x' k' assume xk':"(x',k')\<in>p1\<union>p2" "(x,k) \<noteq> (x',k')"
   9.743 +  have *:"\<And>a b. a\<subseteq> s1 \<Longrightarrow> b\<subseteq> s2 \<Longrightarrow> interior a \<inter> interior b = {}" using assms(3) subset_interior by blast
   9.744 +  show "interior k \<inter> interior k' = {}" apply(cases "(x,k)\<in>p1", case_tac[!] "(x',k')\<in>p1")
   9.745 +    apply(rule p1(5)) prefer 4 apply(rule *) prefer 6 apply(subst Int_commute,rule *) prefer 8 apply(rule p2(5))
   9.746 +    using p1(3) p2(3) using xk xk' by auto qed 
   9.747 +
   9.748 +lemma tagged_division_unions:
   9.749 +  assumes "finite iset" "\<forall>i\<in>iset. (pfn(i) tagged_division_of i)"
   9.750 +  "\<forall>i1 \<in> iset. \<forall>i2 \<in> iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})"
   9.751 +  shows "\<Union>(pfn ` iset) tagged_division_of (\<Union>iset)"
   9.752 +proof(rule tagged_division_ofI)
   9.753 +  note assm = tagged_division_ofD[OF assms(2)[rule_format]]
   9.754 +  show "finite (\<Union>pfn ` iset)" apply(rule finite_Union) using assms by auto
   9.755 +  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 
   9.756 +  also have "\<dots> = \<Union>iset" using assm(6) by auto
   9.757 +  finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>pfn ` iset} = \<Union>iset" . 
   9.758 +  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
   9.759 +  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
   9.760 +  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
   9.761 +  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)
   9.762 +    using assms(3)[rule_format] subset_interior by blast
   9.763 +  show "interior k \<inter> interior k' = {}" apply(cases "i=i'")
   9.764 +    using assm(5)[OF i _ xk'(2)]  i'(2) using assm(3)[OF i] assm(3)[OF i'] defer apply-apply(rule *) by auto
   9.765 +qed
   9.766 +
   9.767 +lemma tagged_partial_division_of_union_self:
   9.768 +  assumes "p tagged_partial_division_of s" shows "p tagged_division_of (\<Union>(snd ` p))"
   9.769 +  apply(rule tagged_division_ofI) using tagged_partial_division_ofD[OF assms] by auto
   9.770 +
   9.771 +lemma tagged_division_of_union_self: assumes "p tagged_division_of s"
   9.772 +  shows "p tagged_division_of (\<Union>(snd ` p))"
   9.773 +  apply(rule tagged_division_ofI) using tagged_division_ofD[OF assms] by auto
   9.774 +
   9.775 +subsection {* Fine-ness of a partition w.r.t. a gauge. *}
   9.776 +
   9.777 +definition fine (infixr "fine" 46) where
   9.778 +  "d fine s \<longleftrightarrow> (\<forall>(x,k) \<in> s. k \<subseteq> d(x))"
   9.779 +
   9.780 +lemma fineI: assumes "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x"
   9.781 +  shows "d fine s" using assms unfolding fine_def by auto
   9.782 +
   9.783 +lemma fineD[dest]: assumes "d fine s"
   9.784 +  shows "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x" using assms unfolding fine_def by auto
   9.785 +
   9.786 +lemma fine_inter: "(\<lambda>x. d1 x \<inter> d2 x) fine p \<longleftrightarrow> d1 fine p \<and> d2 fine p"
   9.787 +  unfolding fine_def by auto
   9.788 +
   9.789 +lemma fine_inters:
   9.790 + "(\<lambda>x. \<Inter> {f d x | d.  d \<in> s}) fine p \<longleftrightarrow> (\<forall>d\<in>s. (f d) fine p)"
   9.791 +  unfolding fine_def by blast
   9.792 +
   9.793 +lemma fine_union:
   9.794 +  "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)"
   9.795 +  unfolding fine_def by blast
   9.796 +
   9.797 +lemma fine_unions:"(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)"
   9.798 +  unfolding fine_def by auto
   9.799 +
   9.800 +lemma fine_subset:  "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p"
   9.801 +  unfolding fine_def by blast
   9.802 +
   9.803 +subsection {* Gauge integral. Define on compact intervals first, then use a limit. *}
   9.804 +
   9.805 +definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46) where
   9.806 +  "(f has_integral_compact_interval y) i \<equiv>
   9.807 +        (\<forall>e>0. \<exists>d. gauge d \<and>
   9.808 +          (\<forall>p. p tagged_division_of i \<and> d fine p
   9.809 +                        \<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - y) < e))"
   9.810 +
   9.811 +definition has_integral (infixr "has'_integral" 46) where 
   9.812 +"((f::(real^'n \<Rightarrow> 'b::real_normed_vector)) has_integral y) i \<equiv>
   9.813 +        if (\<exists>a b. i = {a..b}) then (f has_integral_compact_interval y) i
   9.814 +        else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b}
   9.815 +              \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral_compact_interval z) {a..b} \<and>
   9.816 +                                       norm(z - y) < e))"
   9.817 +
   9.818 +lemma has_integral:
   9.819 + "(f has_integral y) ({a..b}) \<longleftrightarrow>
   9.820 +        (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p
   9.821 +                        \<longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
   9.822 +  unfolding has_integral_def has_integral_compact_interval_def by auto
   9.823 +
   9.824 +lemma has_integralD[dest]: assumes
   9.825 + "(f has_integral y) ({a..b})" "e>0"
   9.826 +  obtains d where "gauge d" "\<And>p. p tagged_division_of {a..b} \<Longrightarrow> d fine p
   9.827 +                        \<Longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f(x)) p - y) < e"
   9.828 +  using assms unfolding has_integral by auto
   9.829 +
   9.830 +lemma has_integral_alt:
   9.831 + "(f has_integral y) i \<longleftrightarrow>
   9.832 +      (if (\<exists>a b. i = {a..b}) then (f has_integral y) i
   9.833 +       else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b}
   9.834 +                               \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0)
   9.835 +                                        has_integral z) ({a..b}) \<and>
   9.836 +                                       norm(z - y) < e)))"
   9.837 +  unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto
   9.838 +
   9.839 +lemma has_integral_altD:
   9.840 +  assumes "(f has_integral y) i" "\<not> (\<exists>a b. i = {a..b})" "e>0"
   9.841 +  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)"
   9.842 +  using assms unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto
   9.843 +
   9.844 +definition integrable_on (infixr "integrable'_on" 46) where
   9.845 +  "(f integrable_on i) \<equiv> \<exists>y. (f has_integral y) i"
   9.846 +
   9.847 +definition "integral i f \<equiv> SOME y. (f has_integral y) i"
   9.848 +
   9.849 +lemma integrable_integral[dest]:
   9.850 + "f integrable_on i \<Longrightarrow> (f has_integral (integral i f)) i"
   9.851 +  unfolding integrable_on_def integral_def by(rule someI_ex)
   9.852 +
   9.853 +lemma has_integral_integrable[intro]: "(f has_integral i) s \<Longrightarrow> f integrable_on s"
   9.854 +  unfolding integrable_on_def by auto
   9.855 +
   9.856 +lemma has_integral_integral:"f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s"
   9.857 +  by auto
   9.858 +
   9.859 +lemma setsum_content_null:
   9.860 +  assumes "content({a..b}) = 0" "p tagged_division_of {a..b}"
   9.861 +  shows "setsum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)"
   9.862 +proof(rule setsum_0',rule) fix y assume y:"y\<in>p"
   9.863 +  obtain x k where xk:"y = (x,k)" using surj_pair[of y] by blast
   9.864 +  note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]]
   9.865 +  from this(2) guess c .. then guess d .. note c_d=this
   9.866 +  have "(\<lambda>(x, k). content k *\<^sub>R f x) y = content k *\<^sub>R f x" unfolding xk by auto
   9.867 +  also have "\<dots> = 0" using content_subset[OF assm(1)[unfolded c_d]] content_pos_le[of c d]
   9.868 +    unfolding assms(1) c_d by auto
   9.869 +  finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" .
   9.870 +qed
   9.871 +
   9.872 +subsection {* Some basic combining lemmas. *}
   9.873 +
   9.874 +lemma tagged_division_unions_exists:
   9.875 +  assumes "finite iset" "\<forall>i \<in> iset. \<exists>p. p tagged_division_of i \<and> d fine p"
   9.876 +  "\<forall>i1\<in>iset. \<forall>i2\<in>iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})" "(\<Union>iset = i)"
   9.877 +   obtains p where "p tagged_division_of i" "d fine p"
   9.878 +proof- guess pfn using bchoice[OF assms(2)] .. note pfn = conjunctD2[OF this[rule_format]]
   9.879 +  show thesis apply(rule_tac p="\<Union>(pfn ` iset)" in that) unfolding assms(4)[THEN sym]
   9.880 +    apply(rule tagged_division_unions[OF assms(1) _ assms(3)]) defer 
   9.881 +    apply(rule fine_unions) using pfn by auto
   9.882 +qed
   9.883 +
   9.884 +subsection {* The set we're concerned with must be closed. *}
   9.885 +
   9.886 +lemma division_of_closed: "s division_of i \<Longrightarrow> closed (i::(real^'n) set)"
   9.887 +  unfolding division_of_def by(fastsimp intro!: closed_Union closed_interval)
   9.888 +
   9.889 +subsection {* General bisection principle for intervals; might be useful elsewhere. *}
   9.890 +
   9.891 +lemma interval_bisection_step:
   9.892 +  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})"
   9.893 +  obtains c d where "~(P{c..d})"
   9.894 +  "\<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"
   9.895 +proof- have "{a..b} \<noteq> {}" using assms(1,3) by auto
   9.896 +  note ab=this[unfolded interval_eq_empty not_ex not_less]
   9.897 +  { fix f have "finite f \<Longrightarrow>
   9.898 +        (\<forall>s\<in>f. P s) \<Longrightarrow>
   9.899 +        (\<forall>s\<in>f. \<exists>a b. s = {a..b}) \<Longrightarrow>
   9.900 +        (\<forall>s\<in>f.\<forall>t\<in>f. ~(s = t) \<longrightarrow> interior(s) \<inter> interior(t) = {}) \<Longrightarrow> P(\<Union>f)"
   9.901 +    proof(induct f rule:finite_induct)
   9.902 +      case empty show ?case using assms(1) by auto
   9.903 +    next case (insert x f) show ?case unfolding Union_insert apply(rule assms(2)[rule_format])
   9.904 +        apply rule defer apply rule defer apply(rule inter_interior_unions_intervals)
   9.905 +        using insert by auto
   9.906 +    qed } note * = this
   9.907 +  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)}"
   9.908 +  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"
   9.909 +  { presume "\<forall>c d. ?PP c d \<longrightarrow> P {c..d} \<Longrightarrow> False"
   9.910 +    thus thesis unfolding atomize_not not_all apply-apply(erule exE)+ apply(rule_tac c=x and d=xa in that) by auto }
   9.911 +  assume as:"\<forall>c d. ?PP c d \<longrightarrow> P {c..d}"
   9.912 +  have "P (\<Union> ?A)" proof(rule *, rule_tac[2-] ballI, rule_tac[4] ballI, rule_tac[4] impI) 
   9.913 +    let ?B = "(\<lambda>s.{(\<chi> i. if i \<in> s then a$i else (a$i + b$i) / 2) ..
   9.914 +      (\<chi> i. if i \<in> s then (a$i + b$i) / 2 else b$i)}) ` {s. s \<subseteq> UNIV}"
   9.915 +    have "?A \<subseteq> ?B" proof case goal1
   9.916 +      then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+) note c_d=this[rule_format]
   9.917 +      have *:"\<And>a b c d. a = c \<Longrightarrow> b = d \<Longrightarrow> {a..b} = {c..d}" by auto
   9.918 +      show "x\<in>?B" unfolding image_iff apply(rule_tac x="{i. c$i = a$i}" in bexI)
   9.919 +        unfolding c_d apply(rule * ) unfolding Cart_eq cond_component Cart_lambda_beta
   9.920 +      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)"
   9.921 +          "d $ i = (if i \<in> {i. c $ i = a $ i} then (a $ i + b $ i) / 2 else b $ i)"
   9.922 +          using c_d(2)[of i] ab[THEN spec[where x=i]] by(auto simp add:field_simps)
   9.923 +      qed auto qed
   9.924 +    thus "finite ?A" apply(rule finite_subset[of _ ?B]) by auto
   9.925 +    fix s assume "s\<in>?A" then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+)
   9.926 +    note c_d=this[rule_format]
   9.927 +    show "P s" unfolding c_d apply(rule as[rule_format]) proof- case goal1 show ?case 
   9.928 +        using c_d(2)[of i] using ab[THEN spec[where x=i]] by auto qed
   9.929 +    show "\<exists>a b. s = {a..b}" unfolding c_d by auto
   9.930 +    fix t assume "t\<in>?A" then guess e unfolding mem_Collect_eq .. then guess f apply- by(erule exE,(erule conjE)+)
   9.931 +    note e_f=this[rule_format]
   9.932 +    assume "s \<noteq> t" hence "\<not> (c = e \<and> d = f)" unfolding c_d e_f by auto
   9.933 +    then obtain i where "c$i \<noteq> e$i \<or> d$i \<noteq> f$i" unfolding de_Morgan_conj Cart_eq by auto
   9.934 +    hence i:"c$i \<noteq> e$i" "d$i \<noteq> f$i" apply- apply(erule_tac[!] disjE)
   9.935 +    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
   9.936 +    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
   9.937 +    qed have *:"\<And>s t. (\<And>a. a\<in>s \<Longrightarrow> a\<in>t \<Longrightarrow> False) \<Longrightarrow> s \<inter> t = {}" by auto
   9.938 +    show "interior s \<inter> interior t = {}" unfolding e_f c_d interior_closed_interval proof(rule *)
   9.939 +      fix x assume "x\<in>{c<..<d}" "x\<in>{e<..<f}"
   9.940 +      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
   9.941 +      show False using c_d(2)[of i] apply- apply(erule_tac disjE)
   9.942 +      proof(erule_tac[!] conjE) assume as:"c $ i = a $ i" "d $ i = (a $ i + b $ i) / 2"
   9.943 +        show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps)
   9.944 +      next assume as:"c $ i = (a $ i + b $ i) / 2" "d $ i = b $ i"
   9.945 +        show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps)
   9.946 +      qed qed qed
   9.947 +  also have "\<Union> ?A = {a..b}" proof(rule set_ext,rule)
   9.948 +    fix x assume "x\<in>\<Union>?A" then guess Y unfolding Union_iff ..
   9.949 +    from this(1) guess c unfolding mem_Collect_eq .. then guess d ..
   9.950 +    note c_d = this[THEN conjunct2,rule_format] `x\<in>Y`[unfolded this[THEN conjunct1]]
   9.951 +    show "x\<in>{a..b}" unfolding mem_interval proof 
   9.952 +      fix i show "a $ i \<le> x $ i \<and> x $ i \<le> b $ i"
   9.953 +        using c_d(1)[of i] c_d(2)[unfolded mem_interval,THEN spec[where x=i]] by auto qed
   9.954 +  next fix x assume x:"x\<in>{a..b}"
   9.955 +    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"
   9.956 +      (is "\<forall>i. \<exists>c d. ?P i c d") unfolding mem_interval proof fix i
   9.957 +      have "?P i (a$i) ((a $ i + b $ i) / 2) \<or> ?P i ((a $ i + b $ i) / 2) (b$i)"
   9.958 +        using x[unfolded mem_interval,THEN spec[where x=i]] by auto thus "\<exists>c d. ?P i c d" by blast
   9.959 +    qed thus "x\<in>\<Union>?A" unfolding Union_iff lambda_skolem unfolding Bex_def mem_Collect_eq
   9.960 +      apply-apply(erule exE)+ apply(rule_tac x="{xa..xaa}" in exI) unfolding mem_interval by auto
   9.961 +  qed finally show False using assms by auto qed
   9.962 +
   9.963 +lemma interval_bisection:
   9.964 +  assumes "P {}" "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))" "\<not> P {a..b::real^'n}"
   9.965 +  obtains x where "x \<in> {a..b}" "\<forall>e>0. \<exists>c d. x \<in> {c..d} \<and> {c..d} \<subseteq> ball x e \<and> {c..d} \<subseteq> {a..b} \<and> ~P({c..d})"
   9.966 +proof-
   9.967 +  have "\<forall>x. \<exists>y. \<not> P {fst x..snd x} \<longrightarrow> (\<not> P {fst y..snd y} \<and> (\<forall>i. fst x$i \<le> fst y$i \<and> fst y$i \<le> snd y$i \<and> snd y$i \<le> snd x$i \<and>
   9.968 +                           2 * (snd y$i - fst y$i) \<le> snd x$i - fst x$i))" proof case goal1 thus ?case proof-
   9.969 +      presume "\<not> P {fst x..snd x} \<Longrightarrow> ?thesis"
   9.970 +      thus ?thesis apply(cases "P {fst x..snd x}") by auto
   9.971 +    next assume as:"\<not> P {fst x..snd x}" from interval_bisection_step[of P, OF assms(1-2) as] guess c d . 
   9.972 +      thus ?thesis apply- apply(rule_tac x="(c,d)" in exI) by auto
   9.973 +    qed qed then guess f apply-apply(drule choice) by(erule exE) note f=this
   9.974 +  def AB \<equiv> "\<lambda>n. (f ^^ n) (a,b)" def A \<equiv> "\<lambda>n. fst(AB n)" and B \<equiv> "\<lambda>n. snd(AB n)" note ab_def = this AB_def
   9.975 +  have "A 0 = a" "B 0 = b" "\<And>n. \<not> P {A(Suc n)..B(Suc n)} \<and>
   9.976 +    (\<forall>i. A(n)$i \<le> A(Suc n)$i \<and> A(Suc n)$i \<le> B(Suc n)$i \<and> B(Suc n)$i \<le> B(n)$i \<and> 
   9.977 +    2 * (B(Suc n)$i - A(Suc n)$i) \<le> B(n)$i - A(n)$i)" (is "\<And>n. ?P n")
   9.978 +  proof- show "A 0 = a" "B 0 = b" unfolding ab_def by auto
   9.979 +    case goal3 note S = ab_def funpow.simps o_def id_apply show ?case
   9.980 +    proof(induct n) case 0 thus ?case unfolding S apply(rule f[rule_format]) using assms(3) by auto
   9.981 +    next case (Suc n) show ?case unfolding S apply(rule f[rule_format]) using Suc unfolding S by auto
   9.982 +    qed qed note AB = this(1-2) conjunctD2[OF this(3),rule_format]
   9.983 +
   9.984 +  have interv:"\<And>e. 0 < e \<Longrightarrow> \<exists>n. \<forall>x\<in>{A n..B n}. \<forall>y\<in>{A n..B n}. dist x y < e"
   9.985 +  proof- case goal1 guess n using real_arch_pow2[of "(setsum (\<lambda>i. b$i - a$i) UNIV) / e"] .. note n=this
   9.986 +    show ?case apply(rule_tac x=n in exI) proof(rule,rule)
   9.987 +      fix x y assume xy:"x\<in>{A n..B n}" "y\<in>{A n..B n}"
   9.988 +      have "dist x y \<le> setsum (\<lambda>i. abs((x - y)$i)) UNIV" unfolding vector_dist_norm by(rule norm_le_l1)
   9.989 +      also have "\<dots> \<le> setsum (\<lambda>i. B n$i - A n$i) UNIV"
   9.990 +      proof(rule setsum_mono) fix i show "\<bar>(x - y) $ i\<bar> \<le> B n $ i - A n $ i"
   9.991 +          using xy[unfolded mem_interval,THEN spec[where x=i]]
   9.992 +          unfolding vector_minus_component by auto qed
   9.993 +      also have "\<dots> \<le> setsum (\<lambda>i. b$i - a$i) UNIV / 2^n" unfolding setsum_divide_distrib
   9.994 +      proof(rule setsum_mono) case goal1 thus ?case
   9.995 +        proof(induct n) case 0 thus ?case unfolding AB by auto
   9.996 +        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
   9.997 +          also have "\<dots> \<le> (b $ i - a $ i) / 2 ^ Suc n" using Suc by(auto simp add:field_simps) finally show ?case .
   9.998 +        qed qed
   9.999 +      also have "\<dots> < e" using n using goal1 by(auto simp add:field_simps) finally show "dist x y < e" .
  9.1000 +    qed qed
  9.1001 +  { fix n m ::nat assume "m \<le> n" then guess d unfolding le_Suc_ex_iff .. note d=this
  9.1002 +    have "{A n..B n} \<subseteq> {A m..B m}" unfolding d 
  9.1003 +    proof(induct d) case 0 thus ?case by auto
  9.1004 +    next case (Suc d) show ?case apply(rule subset_trans[OF _ Suc])
  9.1005 +        apply(rule) unfolding mem_interval apply(rule,erule_tac x=i in allE)
  9.1006 +      proof- case goal1 thus ?case using AB(4)[of "m + d" i] by(auto simp add:field_simps)
  9.1007 +      qed qed } note ABsubset = this 
  9.1008 +  have "\<exists>a. \<forall>n. a\<in>{A n..B n}" apply(rule decreasing_closed_nest[rule_format,OF closed_interval _ ABsubset interv])
  9.1009 +  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
  9.1010 +  then guess x0 .. note x0=this[rule_format]
  9.1011 +  show thesis proof(rule that[rule_format,of x0])
  9.1012 +    show "x0\<in>{a..b}" using x0[of 0] unfolding AB .
  9.1013 +    fix e assume "0 < (e::real)" from interv[OF this] guess n .. note n=this
  9.1014 +    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}"
  9.1015 +      apply(rule_tac x="A n" in exI,rule_tac x="B n" in exI) apply(rule,rule x0) apply rule defer 
  9.1016 +    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
  9.1017 +      show "{A n..B n} \<subseteq> ball x0 e" using n using x0[of n] by auto
  9.1018 +      show "{A n..B n} \<subseteq> {a..b}" unfolding AB(1-2)[symmetric] apply(rule ABsubset) by auto
  9.1019 +    qed qed qed 
  9.1020 +
  9.1021 +subsection {* Cousin's lemma. *}
  9.1022 +
  9.1023 +lemma fine_division_exists: assumes "gauge g" 
  9.1024 +  obtains p where "p tagged_division_of {a..b::real^'n}" "g fine p"
  9.1025 +proof- presume "\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p) \<Longrightarrow> False"
  9.1026 +  then guess p unfolding atomize_not not_not .. thus thesis apply-apply(rule that[of p]) by auto
  9.1027 +next assume as:"\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p)"
  9.1028 +  guess x apply(rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p",rule_format,OF _ _ as])
  9.1029 +    apply(rule_tac x="{}" in exI) defer apply(erule conjE exE)+
  9.1030 +  proof- show "{} tagged_division_of {} \<and> g fine {}" unfolding fine_def by auto
  9.1031 +    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 = {}"
  9.1032 +    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
  9.1033 +      apply(rule tagged_division_union) prefer 4 apply(rule fine_union) by auto
  9.1034 +  qed note x=this
  9.1035 +  obtain e where e:"e>0" "ball x e \<subseteq> g x" using gaugeD[OF assms, of x] unfolding open_contains_ball by auto
  9.1036 +  from x(2)[OF e(1)] guess c d apply-apply(erule exE conjE)+ . note c_d = this
  9.1037 +  have "g fine {(x, {c..d})}" unfolding fine_def using e using c_d(2) by auto
  9.1038 +  thus False using tagged_division_of_self[OF c_d(1)] using c_d by auto qed
  9.1039 +
  9.1040 +subsection {* Basic theorems about integrals. *}
  9.1041 +
  9.1042 +lemma has_integral_unique: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
  9.1043 +  assumes "(f has_integral k1) i" "(f has_integral k2) i" shows "k1 = k2"
  9.1044 +proof(rule ccontr) let ?e = "norm(k1 - k2) / 2" assume as:"k1 \<noteq> k2" hence e:"?e > 0" by auto
  9.1045 +  have lem:"\<And>f::real^'n \<Rightarrow> 'a.  \<And> a b k1 k2.
  9.1046 +    (f has_integral k1) ({a..b}) \<Longrightarrow> (f has_integral k2) ({a..b}) \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> False"
  9.1047 +  proof- case goal1 let ?e = "norm(k1 - k2) / 2" from goal1(3) have e:"?e > 0" by auto
  9.1048 +    guess d1 by(rule has_integralD[OF goal1(1) e]) note d1=this
  9.1049 +    guess d2 by(rule has_integralD[OF goal1(2) e]) note d2=this
  9.1050 +    guess p by(rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)],of a b]) note p=this
  9.1051 +    let ?c = "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" have "norm (k1 - k2) \<le> norm (?c - k2) + norm (?c - k1)"
  9.1052 +      using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"] by(auto simp add:group_simps norm_minus_commute)
  9.1053 +    also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
  9.1054 +      apply(rule add_strict_mono) apply(rule_tac[!] d2(2) d1(2)) using p unfolding fine_def by auto
  9.1055 +    finally show False by auto
  9.1056 +  qed { presume "\<not> (\<exists>a b. i = {a..b}) \<Longrightarrow> False"
  9.1057 +    thus False apply-apply(cases "\<exists>a b. i = {a..b}")
  9.1058 +      using assms by(auto simp add:has_integral intro:lem[OF _ _ as]) }
  9.1059 +  assume as:"\<not> (\<exists>a b. i = {a..b})"
  9.1060 +  guess B1 by(rule has_integral_altD[OF assms(1) as,OF e]) note B1=this[rule_format]
  9.1061 +  guess B2 by(rule has_integral_altD[OF assms(2) as,OF e]) note B2=this[rule_format]
  9.1062 +  have "\<exists>a b::real^'n. ball 0 B1 \<union> ball 0 B2 \<subseteq> {a..b}" apply(rule bounded_subset_closed_interval)
  9.1063 +    using bounded_Un bounded_ball by auto then guess a b apply-by(erule exE)+
  9.1064 +  note ab=conjunctD2[OF this[unfolded Un_subset_iff]]
  9.1065 +  guess w using B1(2)[OF ab(1)] .. note w=conjunctD2[OF this]
  9.1066 +  guess z using B2(2)[OF ab(2)] .. note z=conjunctD2[OF this]
  9.1067 +  have "z = w" using lem[OF w(1) z(1)] by auto
  9.1068 +  hence "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)"
  9.1069 +    using norm_triangle_ineq4[of "k1 - w" "k2 - z"] by(auto simp add: norm_minus_commute) 
  9.1070 +  also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2" apply(rule add_strict_mono) by(rule_tac[!] z(2) w(2))
  9.1071 +  finally show False by auto qed
  9.1072 +
  9.1073 +lemma integral_unique[intro]:
  9.1074 +  "(f has_integral y) k \<Longrightarrow> integral k f = y"
  9.1075 +  unfolding integral_def apply(rule some_equality) by(auto intro: has_integral_unique) 
  9.1076 +
  9.1077 +lemma has_integral_is_0: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector" 
  9.1078 +  assumes "\<forall>x\<in>s. f x = 0" shows "(f has_integral 0) s"
  9.1079 +proof- have lem:"\<And>a b. \<And>f::real^'n \<Rightarrow> 'a.
  9.1080 +    (\<forall>x\<in>{a..b}. f(x) = 0) \<Longrightarrow> (f has_integral 0) ({a..b})" unfolding has_integral
  9.1081 +  proof(rule,rule) fix a b e and f::"real^'n \<Rightarrow> 'a"
  9.1082 +    assume as:"\<forall>x\<in>{a..b}. f x = 0" "0 < (e::real)"
  9.1083 +    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)"
  9.1084 +      apply(rule_tac x="\<lambda>x. ball x 1" in exI)  apply(rule,rule gaugeI) unfolding centre_in_ball defer apply(rule open_ball)
  9.1085 +    proof(rule,rule,erule conjE) case goal1
  9.1086 +      have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) = 0" proof(rule setsum_0',rule)
  9.1087 +        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
  9.1088 +        thus "(\<lambda>(x, k). content k *\<^sub>R f x) x = 0" apply(subst surjective_pairing[of x]) unfolding split_conv by auto
  9.1089 +      qed thus ?case using as by auto
  9.1090 +    qed auto qed  { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
  9.1091 +    thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}")
  9.1092 +      using assms by(auto simp add:has_integral intro:lem) }
  9.1093 +  have *:"(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. 0)" apply(rule ext) using assms by auto
  9.1094 +  assume "\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P *
  9.1095 +  apply(rule,rule,rule_tac x=1 in exI,rule) defer apply(rule,rule,rule)
  9.1096 +  proof- fix e::real and a b assume "e>0"
  9.1097 +    thus "\<exists>z. ((\<lambda>x::real^'n. 0::'a) has_integral z) {a..b} \<and> norm (z - 0) < e"
  9.1098 +      apply(rule_tac x=0 in exI) apply(rule,rule lem) by auto
  9.1099 +  qed auto qed
  9.1100 +
  9.1101 +lemma has_integral_0[simp]: "((\<lambda>x::real^'n. 0) has_integral 0) s"
  9.1102 +  apply(rule has_integral_is_0) by auto 
  9.1103 +
  9.1104 +lemma has_integral_0_eq[simp]: "((\<lambda>x. 0) has_integral i) s \<longleftrightarrow> i = 0"
  9.1105 +  using has_integral_unique[OF has_integral_0] by auto
  9.1106 +
  9.1107 +lemma has_integral_linear: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
  9.1108 +  assumes "(f has_integral y) s" "bounded_linear h" shows "((h o f) has_integral ((h y))) s"
  9.1109 +proof- interpret bounded_linear h using assms(2) . from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format]
  9.1110 +  have lem:"\<And>f::real^'n \<Rightarrow> 'a. \<And> y a b.
  9.1111 +    (f has_integral y) ({a..b}) \<Longrightarrow> ((h o f) has_integral h(y)) ({a..b})"
  9.1112 +  proof(subst has_integral,rule,rule) case goal1
  9.1113 +    from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format]
  9.1114 +    have *:"e / B > 0" apply(rule divide_pos_pos) using goal1(2) B by auto
  9.1115 +    guess g using has_integralD[OF goal1(1) *] . note g=this
  9.1116 +    show ?case apply(rule_tac x=g in exI) apply(rule,rule g(1))
  9.1117 +    proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "g fine p" 
  9.1118 +      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
  9.1119 +      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"
  9.1120 +        unfolding o_def unfolding scaleR[THEN sym] * by simp
  9.1121 +      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
  9.1122 +      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)" .
  9.1123 +      show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) - h y) < e" unfolding * diff[THEN sym]
  9.1124 +        apply(rule le_less_trans[OF B(2)]) using g(2)[OF as] B(1) by(auto simp add:field_simps)
  9.1125 +    qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
  9.1126 +    thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) }
  9.1127 +  assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P
  9.1128 +  proof(rule,rule) fix e::real  assume e:"0<e"
  9.1129 +    have *:"0 < e/B" by(rule divide_pos_pos,rule e,rule B(1))
  9.1130 +    guess M using has_integral_altD[OF assms(1) as *,rule_format] . note M=this
  9.1131 +    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)"
  9.1132 +      apply(rule_tac x=M in exI) apply(rule,rule M(1))
  9.1133 +    proof(rule,rule,rule) case goal1 guess z using M(2)[OF goal1(1)] .. note z=conjunctD2[OF this]
  9.1134 +      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)"
  9.1135 +        unfolding o_def apply(rule ext) using zero by auto
  9.1136 +      show ?case apply(rule_tac x="h z" in exI,rule) unfolding * apply(rule lem[OF z(1)]) unfolding diff[THEN sym]
  9.1137 +        apply(rule le_less_trans[OF B(2)]) using B(1) z(2) by(auto simp add:field_simps)
  9.1138 +    qed qed qed
  9.1139 +
  9.1140 +lemma has_integral_cmul:
  9.1141 +  shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_integral (c *\<^sub>R k)) s"
  9.1142 +  unfolding o_def[THEN sym] apply(rule has_integral_linear,assumption)
  9.1143 +  by(rule scaleR.bounded_linear_right)
  9.1144 +
  9.1145 +lemma has_integral_neg:
  9.1146 +  shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. -(f x)) has_integral (-k)) s"
  9.1147 +  apply(drule_tac c="-1" in has_integral_cmul) by auto
  9.1148 +
  9.1149 +lemma has_integral_add: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector" 
  9.1150 +  assumes "(f has_integral k) s" "(g has_integral l) s"
  9.1151 +  shows "((\<lambda>x. f x + g x) has_integral (k + l)) s"
  9.1152 +proof- have lem:"\<And>f g::real^'n \<Rightarrow> 'a. \<And>a b k l.
  9.1153 +    (f has_integral k) ({a..b}) \<Longrightarrow> (g has_integral l) ({a..b}) \<Longrightarrow>
  9.1154 +     ((\<lambda>x. f(x) + g(x)) has_integral (k + l)) ({a..b})" proof- case goal1
  9.1155 +    show ?case unfolding has_integral proof(rule,rule) fix e::real assume e:"e>0" hence *:"e/2>0" by auto
  9.1156 +      guess d1 using has_integralD[OF goal1(1) *] . note d1=this
  9.1157 +      guess d2 using has_integralD[OF goal1(2) *] . note d2=this
  9.1158 +      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)"
  9.1159 +        apply(rule_tac x="\<lambda>x. (d1 x) \<inter> (d2 x)" in exI) apply(rule,rule gauge_inter[OF d1(1) d2(1)])
  9.1160 +      proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d1 x \<inter> d2 x) fine p"
  9.1161 +        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)"
  9.1162 +          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]
  9.1163 +          by(rule setsum_cong2,auto)
  9.1164 +        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))"
  9.1165 +          unfolding * by(auto simp add:group_simps) also let ?res = "\<dots>"
  9.1166 +        from as have *:"d1 fine p" "d2 fine p" unfolding fine_inter by auto
  9.1167 +        have "?res < e/2 + e/2" apply(rule le_less_trans[OF norm_triangle_ineq])
  9.1168 +          apply(rule add_strict_mono) using d1(2)[OF as(1) *(1)] and d2(2)[OF as(1) *(2)] by auto
  9.1169 +        finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e" by auto
  9.1170 +      qed qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
  9.1171 +    thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) }
  9.1172 +  assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P
  9.1173 +  proof(rule,rule) case goal1 hence *:"e/2 > 0" by auto
  9.1174 +    from has_integral_altD[OF assms(1) as *] guess B1 . note B1=this[rule_format]
  9.1175 +    from has_integral_altD[OF assms(2) as *] guess B2 . note B2=this[rule_format]
  9.1176 +    show ?case apply(rule_tac x="max B1 B2" in exI) apply(rule,rule min_max.less_supI1,rule B1)
  9.1177 +    proof(rule,rule,rule) fix a b assume "ball 0 (max B1 B2) \<subseteq> {a..b::real^'n}"
  9.1178 +      hence *:"ball 0 B1 \<subseteq> {a..b::real^'n}" "ball 0 B2 \<subseteq> {a..b::real^'n}" by auto
  9.1179 +      guess w using B1(2)[OF *(1)] .. note w=conjunctD2[OF this]
  9.1180 +      guess z using B2(2)[OF *(2)] .. note z=conjunctD2[OF this]
  9.1181 +      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
  9.1182 +      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"
  9.1183 +        apply(rule_tac x="w + z" in exI) apply(rule,rule lem[OF w(1) z(1), unfolded *[THEN sym]])
  9.1184 +        using norm_triangle_ineq[of "w - k" "z - l"] w(2) z(2) by(auto simp add:field_simps)
  9.1185 +    qed qed qed
  9.1186 +
  9.1187 +lemma has_integral_sub:
  9.1188 +  shows "(f has_integral k) s \<Longrightarrow> (g has_integral l) s \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) has_integral (k - l)) s"
  9.1189 +  using has_integral_add[OF _ has_integral_neg,of f k s g l] unfolding group_simps by auto
  9.1190 +
  9.1191 +lemma integral_0: "integral s (\<lambda>x::real^'n. 0::real^'m) = 0"
  9.1192 +  by(rule integral_unique has_integral_0)+
  9.1193 +
  9.1194 +lemma integral_add:
  9.1195 +  shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow>
  9.1196 +   integral s (\<lambda>x. f x + g x) = integral s f + integral s g"
  9.1197 +  apply(rule integral_unique) apply(drule integrable_integral)+
  9.1198 +  apply(rule has_integral_add) by assumption+
  9.1199 +
  9.1200 +lemma integral_cmul:
  9.1201 +  shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. c *\<^sub>R f x) = c *\<^sub>R integral s f"
  9.1202 +  apply(rule integral_unique) apply(drule integrable_integral)+
  9.1203 +  apply(rule has_integral_cmul) by assumption+
  9.1204 +
  9.1205 +lemma integral_neg:
  9.1206 +  shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. - f x) = - integral s f"
  9.1207 +  apply(rule integral_unique) apply(drule integrable_integral)+
  9.1208 +  apply(rule has_integral_neg) by assumption+
  9.1209 +
  9.1210 +lemma integral_sub:
  9.1211 +  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"
  9.1212 +  apply(rule integral_unique) apply(drule integrable_integral)+
  9.1213 +  apply(rule has_integral_sub) by assumption+
  9.1214 +
  9.1215 +lemma integrable_0: "(\<lambda>x. 0) integrable_on s"
  9.1216 +  unfolding integrable_on_def using has_integral_0 by auto
  9.1217 +
  9.1218 +lemma integrable_add:
  9.1219 +  shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x + g x) integrable_on s"
  9.1220 +  unfolding integrable_on_def by(auto intro: has_integral_add)
  9.1221 +
  9.1222 +lemma integrable_cmul:
  9.1223 +  shows "f integrable_on s \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) integrable_on s"
  9.1224 +  unfolding integrable_on_def by(auto intro: has_integral_cmul)
  9.1225 +
  9.1226 +lemma integrable_neg:
  9.1227 +  shows "f integrable_on s \<Longrightarrow> (\<lambda>x. -f(x)) integrable_on s"
  9.1228 +  unfolding integrable_on_def by(auto intro: has_integral_neg)
  9.1229 +
  9.1230 +lemma integrable_sub:
  9.1231 +  shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x - g x) integrable_on s"
  9.1232 +  unfolding integrable_on_def by(auto intro: has_integral_sub)
  9.1233 +
  9.1234 +lemma integrable_linear:
  9.1235 +  shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> (h o f) integrable_on s"
  9.1236 +  unfolding integrable_on_def by(auto intro: has_integral_linear)
  9.1237 +
  9.1238 +lemma integral_linear:
  9.1239 +  shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> integral s (h o f) = h(integral s f)"
  9.1240 +  apply(rule has_integral_unique) defer unfolding has_integral_integral 
  9.1241 +  apply(drule has_integral_linear,assumption,assumption) unfolding has_integral_integral[THEN sym]
  9.1242 +  apply(rule integrable_linear) by assumption+
  9.1243 +
  9.1244 +lemma has_integral_setsum:
  9.1245 +  assumes "finite t" "\<forall>a\<in>t. ((f a) has_integral (i a)) s"
  9.1246 +  shows "((\<lambda>x. setsum (\<lambda>a. f a x) t) has_integral (setsum i t)) s"
  9.1247 +proof(insert assms(1) subset_refl[of t],induct rule:finite_subset_induct)
  9.1248 +  case (insert x F) show ?case unfolding setsum_insert[OF insert(1,3)]
  9.1249 +    apply(rule has_integral_add) using insert assms by auto
  9.1250 +qed auto
  9.1251 +
  9.1252 +lemma integral_setsum:
  9.1253 +  shows "finite t \<Longrightarrow> \<forall>a\<in>t. (f a) integrable_on s \<Longrightarrow>
  9.1254 +  integral s (\<lambda>x. setsum (\<lambda>a. f a x) t) = setsum (\<lambda>a. integral s (f a)) t"
  9.1255 +  apply(rule integral_unique) apply(rule has_integral_setsum)
  9.1256 +  using integrable_integral by auto
  9.1257 +
  9.1258 +lemma integrable_setsum:
  9.1259 +  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"
  9.1260 +  unfolding integrable_on_def apply(drule bchoice) using has_integral_setsum[of t] by auto
  9.1261 +
  9.1262 +lemma has_integral_eq:
  9.1263 +  assumes "\<forall>x\<in>s. f x = g x" "(f has_integral k) s" shows "(g has_integral k) s"
  9.1264 +  using has_integral_sub[OF assms(2), of "\<lambda>x. f x - g x" 0]
  9.1265 +  using has_integral_is_0[of s "\<lambda>x. f x - g x"] using assms(1) by auto
  9.1266 +
  9.1267 +lemma integrable_eq:
  9.1268 +  shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> f integrable_on s \<Longrightarrow> g integrable_on s"
  9.1269 +  unfolding integrable_on_def using has_integral_eq[of s f g] by auto
  9.1270 +
  9.1271 +lemma has_integral_eq_eq:
  9.1272 +  shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> ((f has_integral k) s \<longleftrightarrow> (g has_integral k) s)"
  9.1273 +  using has_integral_eq[of s f g] has_integral_eq[of s g f] by auto 
  9.1274 +
  9.1275 +lemma has_integral_null[dest]:
  9.1276 +  assumes "content({a..b}) = 0" shows  "(f has_integral 0) ({a..b})"
  9.1277 +  unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI,rule) defer
  9.1278 +proof(rule,rule,erule conjE) fix e::real assume e:"e>0" thus "gauge (\<lambda>x. ball x 1)" by auto
  9.1279 +  fix p assume p:"p tagged_division_of {a..b}" (*"(\<lambda>x. ball x 1) fine p"*)
  9.1280 +  have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) = 0" unfolding norm_eq_zero diff_0_right
  9.1281 +    using setsum_content_null[OF assms(1) p, of f] . 
  9.1282 +  thus "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e" using e by auto qed
  9.1283 +
  9.1284 +lemma has_integral_null_eq[simp]:
  9.1285 +  shows "content({a..b}) = 0 \<Longrightarrow> ((f has_integral i) ({a..b}) \<longleftrightarrow> i = 0)"
  9.1286 +  apply rule apply(rule has_integral_unique,assumption) 
  9.1287 +  apply(drule has_integral_null,assumption)
  9.1288 +  apply(drule has_integral_null) by auto
  9.1289 +
  9.1290 +lemma integral_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> integral({a..b}) f = 0"
  9.1291 +  by(rule integral_unique,drule has_integral_null)
  9.1292 +
  9.1293 +lemma integrable_on_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> f integrable_on {a..b}"
  9.1294 +  unfolding integrable_on_def apply(drule has_integral_null) by auto
  9.1295 +
  9.1296 +lemma has_integral_empty[intro]: shows "(f has_integral 0) {}"
  9.1297 +  unfolding empty_as_interval apply(rule has_integral_null) 
  9.1298 +  using content_empty unfolding empty_as_interval .
  9.1299 +
  9.1300 +lemma has_integral_empty_eq[simp]: shows "(f has_integral i) {} \<longleftrightarrow> i = 0"
  9.1301 +  apply(rule,rule has_integral_unique,assumption) by auto
  9.1302 +
  9.1303 +lemma integrable_on_empty[intro]: shows "f integrable_on {}" unfolding integrable_on_def by auto
  9.1304 +
  9.1305 +lemma integral_empty[simp]: shows "integral {} f = 0"
  9.1306 +  apply(rule integral_unique) using has_integral_empty .
  9.1307 +
  9.1308 +lemma has_integral_refl[intro]: shows "(f has_integral 0) {a..a}"
  9.1309 +  apply(rule has_integral_null) unfolding content_eq_0_interior
  9.1310 +  unfolding interior_closed_interval using interval_sing by auto
  9.1311 +
  9.1312 +lemma integrable_on_refl[intro]: shows "f integrable_on {a..a}" unfolding integrable_on_def by auto
  9.1313 +
  9.1314 +lemma integral_refl: shows "integral {a..a} f = 0" apply(rule integral_unique) by auto
  9.1315 +
  9.1316 +subsection {* Cauchy-type criterion for integrability. *}
  9.1317 +
  9.1318 +lemma integrable_cauchy: fixes f::"real^'n \<Rightarrow> 'a::{real_normed_vector,complete_space}" 
  9.1319 +  shows "f integrable_on {a..b} \<longleftrightarrow>
  9.1320 +  (\<forall>e>0.\<exists>d. gauge d \<and> (\<forall>p1 p2. p1 tagged_division_of {a..b} \<and> d fine p1 \<and>
  9.1321 +                            p2 tagged_division_of {a..b} \<and> d fine p2
  9.1322 +                            \<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 -
  9.1323 +                                     setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) < e))" (is "?l = (\<forall>e>0. \<exists>d. ?P e d)")
  9.1324 +proof assume ?l
  9.1325 +  then guess y unfolding integrable_on_def has_integral .. note y=this
  9.1326 +  show "\<forall>e>0. \<exists>d. ?P e d" proof(rule,rule) case goal1 hence "e/2 > 0" by auto
  9.1327 +    then guess d apply- apply(drule y[rule_format]) by(erule exE,erule conjE) note d=this[rule_format]
  9.1328 +    show ?case apply(rule_tac x=d in exI,rule,rule d) apply(rule,rule,rule,(erule conjE)+)
  9.1329 +    proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b}" "d fine p1" "p2 tagged_division_of {a..b}" "d fine p2"
  9.1330 +      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"
  9.1331 +        apply(rule dist_triangle_half_l[where y=y,unfolded vector_dist_norm])
  9.1332 +        using d(2)[OF conjI[OF as(1-2)]] d(2)[OF conjI[OF as(3-4)]] .
  9.1333 +    qed qed
  9.1334 +next assume "\<forall>e>0. \<exists>d. ?P e d" hence "\<forall>n::nat. \<exists>d. ?P (inverse(real (n + 1))) d" by auto
  9.1335 +  from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
  9.1336 +  have "\<And>n. gauge (\<lambda>x. \<Inter>{d i x |i. i \<in> {0..n}})" apply(rule gauge_inters) using d(1) by auto
  9.1337 +  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-
  9.1338 +  proof case goal1 from this[of n] show ?case apply(drule_tac fine_division_exists) by auto qed
  9.1339 +  from choice[OF this] guess p .. note p = conjunctD2[OF this[rule_format]]
  9.1340 +  have dp:"\<And>i n. i\<le>n \<Longrightarrow> d i fine p n" using p(2) unfolding fine_inters by auto
  9.1341 +  have "Cauchy (\<lambda>n. setsum (\<lambda>(x,k). content k *\<^sub>R (f x)) (p n))"
  9.1342 +  proof(rule CauchyI) case goal1 then guess N unfolding real_arch_inv[of e] .. note N=this
  9.1343 +    show ?case apply(rule_tac x=N in exI)
  9.1344 +    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
  9.1345 +      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"
  9.1346 +        apply(rule less_trans[OF _ N[THEN conjunct2,THEN conjunct2]]) apply(subst *) apply(rule d(2))
  9.1347 +        using dp p(1) using mn by auto 
  9.1348 +    qed qed
  9.1349 +  then guess y unfolding convergent_eq_cauchy[THEN sym] .. note y=this[unfolded Lim_sequentially,rule_format]
  9.1350 +  show ?l unfolding integrable_on_def has_integral apply(rule_tac x=y in exI)
  9.1351 +  proof(rule,rule) fix e::real assume "e>0" hence *:"e/2 > 0" by auto
  9.1352 +    then guess N1 unfolding real_arch_inv[of "e/2"] .. note N1=this hence N1':"N1 = N1 - 1 + 1" by auto
  9.1353 +    guess N2 using y[OF *] .. note N2=this
  9.1354 +    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)"
  9.1355 +      apply(rule_tac x="d (N1 + N2)" in exI) apply rule defer 
  9.1356 +    proof(rule,rule,erule conjE) show "gauge (d (N1 + N2))" using d by auto
  9.1357 +      fix q assume as:"q tagged_division_of {a..b}" "d (N1 + N2) fine q"
  9.1358 +      have *:"inverse (real (N1 + N2 + 1)) < e / 2" apply(rule less_trans) using N1 by auto
  9.1359 +      show "norm ((\<Sum>(x, k)\<in>q. content k *\<^sub>R f x) - y) < e" apply(rule norm_triangle_half_r)
  9.1360 +        apply(rule less_trans[OF _ *]) apply(subst N1', rule d(2)[of "p (N1+N2)"]) defer
  9.1361 +        using N2[rule_format,unfolded vector_dist_norm,of "N1+N2"]
  9.1362 +        using as dp[of "N1 - 1 + 1 + N2" "N1 + N2"] using p(1)[of "N1 + N2"] using N1 by auto qed qed qed
  9.1363 +
  9.1364 +subsection {* Additivity of integral on abutting intervals. *}
  9.1365 +
  9.1366 +lemma interval_split:
  9.1367 +  "{a..b::real^'n} \<inter> {x. x$k \<le> c} = {a .. (\<chi> i. if i = k then min (b$k) c else b$i)}"
  9.1368 +  "{a..b} \<inter> {x. x$k \<ge> c} = {(\<chi> i. if i = k then max (a$k) c else a$i) .. b}"
  9.1369 +  apply(rule_tac[!] set_ext) unfolding Int_iff mem_interval mem_Collect_eq
  9.1370 +  unfolding Cart_lambda_beta by auto
  9.1371 +
  9.1372 +lemma content_split:
  9.1373 +  "content {a..b::real^'n} = content({a..b} \<inter> {x. x$k \<le> c}) + content({a..b} \<inter> {x. x$k >= c})"
  9.1374 +proof- note simps = interval_split content_closed_interval_cases Cart_lambda_beta vector_le_def
  9.1375 +  { presume "a\<le>b \<Longrightarrow> ?thesis" thus ?thesis apply(cases "a\<le>b") unfolding simps by auto }
  9.1376 +  have *:"UNIV = insert k (UNIV - {k})" "\<And>x. finite (UNIV-{x::'n})" "\<And>x. x\<notin>UNIV-{x}" by auto
  9.1377 +  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))"
  9.1378 +    "(\<Prod>i\<in>UNIV. b$i - a$i) = (\<Prod>i\<in>UNIV-{k}. b$i - a$i) * (b$k - a$k)" 
  9.1379 +    apply(subst *(1)) defer apply(subst *(1)) unfolding setprod_insert[OF *(2-)] by auto
  9.1380 +  assume as:"a\<le>b" moreover have "\<And>x. min (b $ k) c = max (a $ k) c
  9.1381 +    \<Longrightarrow> x* (b$k - a$k) = x*(max (a $ k) c - a $ k) + x*(b $ k - max (a $ k) c)"
  9.1382 +    by  (auto simp add:field_simps)
  9.1383 +  moreover have "\<not> a $ k \<le> c \<Longrightarrow> \<not> c \<le> b $ k \<Longrightarrow> False"
  9.1384 +    unfolding not_le using as[unfolded vector_le_def,rule_format,of k] by auto
  9.1385 +  ultimately show ?thesis 
  9.1386 +    unfolding simps unfolding *(1)[of "\<lambda>i x. b$i - x"] *(1)[of "\<lambda>i x. x - a$i"] *(2) by(auto)
  9.1387 +qed
  9.1388 +
  9.1389 +lemma division_split_left_inj:
  9.1390 +  assumes "d division_of i" "k1 \<in> d" "k2 \<in> d"  "k1 \<noteq> k2"
  9.1391 +  "k1 \<inter> {x::real^'n. x$k \<le> c} = k2 \<inter> {x. x$k \<le> c}"
  9.1392 +  shows "content(k1 \<inter> {x. x$k \<le> c}) = 0"
  9.1393 +proof- note d=division_ofD[OF assms(1)]
  9.1394 +  have *:"\<And>a b::real^'n. \<And> c k. (content({a..b} \<inter> {x. x$k \<le> c}) = 0 \<longleftrightarrow> interior({a..b} \<inter> {x. x$k \<le> c}) = {})"
  9.1395 +    unfolding interval_split content_eq_0_interior by auto
  9.1396 +  guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this
  9.1397 +  guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this
  9.1398 +  have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto
  9.1399 +  show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]])
  9.1400 +    defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed
  9.1401 +
  9.1402 +lemma division_split_right_inj:
  9.1403 +  assumes "d division_of i" "k1 \<in> d" "k2 \<in> d"  "k1 \<noteq> k2"
  9.1404 +  "k1 \<inter> {x::real^'n. x$k \<ge> c} = k2 \<inter> {x. x$k \<ge> c}"
  9.1405 +  shows "content(k1 \<inter> {x. x$k \<ge> c}) = 0"
  9.1406 +proof- note d=division_ofD[OF assms(1)]
  9.1407 +  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}) = {})"
  9.1408 +    unfolding interval_split content_eq_0_interior by auto
  9.1409 +  guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this
  9.1410 +  guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this
  9.1411 +  have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto
  9.1412 +  show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]])
  9.1413 +    defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed
  9.1414 +
  9.1415 +lemma tagged_division_split_left_inj:
  9.1416 +  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}" 
  9.1417 +  shows "content(k1 \<inter> {x. x$k \<le> c}) = 0"
  9.1418 +proof- have *:"\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c" unfolding image_iff apply(rule_tac x="(a,b)" in bexI) by auto
  9.1419 +  show ?thesis apply(rule division_split_left_inj[OF division_of_tagged_division[OF assms(1)]])
  9.1420 +    apply(rule_tac[1-2] *) using assms(2-) by auto qed
  9.1421 +
  9.1422 +lemma tagged_division_split_right_inj:
  9.1423 +  assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2"  "k1 \<inter> {x. x$k \<ge> c} = k2 \<inter> {x. x$k \<ge> c}" 
  9.1424 +  shows "content(k1 \<inter> {x. x$k \<ge> c}) = 0"
  9.1425 +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
  9.1426 +  show ?thesis apply(rule division_split_right_inj[OF division_of_tagged_division[OF assms(1)]])
  9.1427 +    apply(rule_tac[1-2] *) using assms(2-) by auto qed
  9.1428 +
  9.1429 +lemma division_split:
  9.1430 +  assumes "p division_of {a..b::real^'n}"
  9.1431 +  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$k \<le> c})" (is "?p1 division_of ?I1") and 
  9.1432 +        "{l \<inter> {x. x$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")
  9.1433 +proof(rule_tac[!] division_ofI) note p=division_ofD[OF assms]
  9.1434 +  show "finite ?p1" "finite ?p2" using p(1) by auto show "\<Union>?p1 = ?I1" "\<Union>?p2 = ?I2" unfolding p(6)[THEN sym] by auto
  9.1435 +  { fix k assume "k\<in>?p1" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this
  9.1436 +    guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this
  9.1437 +    show "k\<subseteq>?I1" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l
  9.1438 +      using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split) by auto
  9.1439 +    fix k' assume "k'\<in>?p1" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this
  9.1440 +    assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto }
  9.1441 +  { fix k assume "k\<in>?p2" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this
  9.1442 +    guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this
  9.1443 +    show "k\<subseteq>?I2" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l
  9.1444 +      using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split) by auto
  9.1445 +    fix k' assume "k'\<in>?p2" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this
  9.1446 +    assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto }
  9.1447 +qed
  9.1448 +
  9.1449 +lemma has_integral_split: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
  9.1450 +  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})"
  9.1451 +  shows "(f has_integral (i + j)) ({a..b})"
  9.1452 +proof(unfold has_integral,rule,rule) case goal1 hence e:"e/2>0" by auto
  9.1453 +  guess d1 using has_integralD[OF assms(1)[unfolded interval_split] e] . note d1=this[unfolded interval_split[THEN sym]]
  9.1454 +  guess d2 using has_integralD[OF assms(2)[unfolded interval_split] e] . note d2=this[unfolded interval_split[THEN sym]]
  9.1455 +  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"
  9.1456 +  show ?case apply(rule_tac x="?d" in exI,rule) defer apply(rule,rule,(erule conjE)+)
  9.1457 +  proof- show "gauge ?d" using d1(1) d2(1) unfolding gauge_def by auto
  9.1458 +    fix p assume "p tagged_division_of {a..b}" "?d fine p" note p = this tagged_division_ofD[OF this(1)]
  9.1459 +    have lem0:"\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x$k \<le> c} = {}) \<Longrightarrow> x$k \<le> c"
  9.1460 +         "\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x$k \<ge> c} = {}) \<Longrightarrow> x$k \<ge> c"
  9.1461 +    proof- fix x kk assume as:"(x,kk)\<in>p"
  9.1462 +      show "~(kk \<inter> {x. x$k \<le> c} = {}) \<Longrightarrow> x$k \<le> c"
  9.1463 +      proof(rule ccontr) case goal1
  9.1464 +        from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x $ k - c\<bar>"
  9.1465 +          using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
  9.1466 +        hence "\<exists>y. y \<in> ball x \<bar>x $ k - c\<bar> \<inter> {x. x $ k \<le> c}" using goal1(1) by blast 
  9.1467 +        then guess y .. hence "\<bar>x $ k - y $ k\<bar> < \<bar>x $ k - c\<bar>" "y$k \<le> c" apply-apply(rule le_less_trans)
  9.1468 +          using component_le_norm[of "x - y" k,unfolded vector_minus_component] by(auto simp add:vector_dist_norm)
  9.1469 +        thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
  9.1470 +      qed
  9.1471 +      show "~(kk \<inter> {x. x$k \<ge> c} = {}) \<Longrightarrow> x$k \<ge> c"
  9.1472 +      proof(rule ccontr) case goal1
  9.1473 +        from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x $ k - c\<bar>"
  9.1474 +          using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
  9.1475 +        hence "\<exists>y. y \<in> ball x \<bar>x $ k - c\<bar> \<inter> {x. x $ k \<ge> c}" using goal1(1) by blast 
  9.1476 +        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)
  9.1477 +          using component_le_norm[of "x - y" k,unfolded vector_minus_component] by(auto simp add:vector_dist_norm)
  9.1478 +        thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
  9.1479 +      qed
  9.1480 +    qed
  9.1481 +
  9.1482 +    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
  9.1483 +    have lem2: "\<And>f s P f. finite s \<Longrightarrow> finite {(x,f k) | x k. (x,k) \<in> s \<and> P x k}"
  9.1484 +    proof- case goal1 thus ?case apply-apply(rule finite_subset[of _ "(\<lambda>(x,k). (x,f k)) ` s"]) by auto qed
  9.1485 +    have lem3: "\<And>g::(real ^ 'n \<Rightarrow> bool) \<Rightarrow> real ^ 'n \<Rightarrow> bool. finite p \<Longrightarrow>
  9.1486 +      setsum (\<lambda>(x,k). content k *\<^sub>R f x) {(x,g k) |x k. (x,k) \<in> p \<and> ~(g k = {})}
  9.1487 +               = setsum (\<lambda>(x,k). content k *\<^sub>R f x) ((\<lambda>(x,k). (x,g k)) ` p)"
  9.1488 +      apply(rule setsum_mono_zero_left) prefer 3
  9.1489 +    proof fix g::"(real ^ 'n \<Rightarrow> bool) \<Rightarrow> real ^ 'n \<Rightarrow> bool" and i::"(real^'n) \<times> ((real^'n) set)"
  9.1490 +      assume "i \<in> (\<lambda>(x, k). (x, g k)) ` p - {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}"
  9.1491 +      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
  9.1492 +      have "content (g k) = 0" using xk using content_empty by auto
  9.1493 +      thus "(\<lambda>(x, k). content k *\<^sub>R f x) i = 0" unfolding xk split_conv by auto
  9.1494 +    qed auto
  9.1495 +    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
  9.1496 +
  9.1497 +    let ?M1 = "{(x,kk \<inter> {x. x$k \<le> c}) |x kk. (x,kk) \<in> p \<and> kk \<inter> {x. x$k \<le> c} \<noteq> {}}"
  9.1498 +    have "norm ((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) < e/2" apply(rule d1(2),rule tagged_division_ofI)
  9.1499 +      apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)
  9.1500 +    proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M1} = {a..b} \<inter> {x. x$k \<le> c}" unfolding p(8)[THEN sym] by auto
  9.1501 +      fix x l assume xl:"(x,l)\<in>?M1"
  9.1502 +      then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note xl'=this
  9.1503 +      have "l' \<subseteq> d1 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
  9.1504 +      thus "l \<subseteq> d1 x" unfolding xl' by auto
  9.1505 +      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)
  9.1506 +        using lem0(1)[OF xl'(3-4)] by auto
  9.1507 +      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])
  9.1508 +      fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M1"
  9.1509 +      then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note yr'=this
  9.1510 +      assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}"
  9.1511 +      proof(cases "l' = r' \<longrightarrow> x' = y'")
  9.1512 +        case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
  9.1513 +      next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto
  9.1514 +        thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
  9.1515 +      qed qed moreover
  9.1516 +
  9.1517 +    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> {}}" 
  9.1518 +    have "norm ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j) < e/2" apply(rule d2(2),rule tagged_division_ofI)
  9.1519 +      apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)
  9.1520 +    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
  9.1521 +      fix x l assume xl:"(x,l)\<in>?M2"
  9.1522 +      then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note xl'=this
  9.1523 +      have "l' \<subseteq> d2 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
  9.1524 +      thus "l \<subseteq> d2 x" unfolding xl' by auto
  9.1525 +      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)
  9.1526 +        using lem0(2)[OF xl'(3-4)] by auto
  9.1527 +      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])
  9.1528 +      fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M2"
  9.1529 +      then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note yr'=this
  9.1530 +      assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}"
  9.1531 +      proof(cases "l' = r' \<longrightarrow> x' = y'")
  9.1532 +        case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
  9.1533 +      next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto
  9.1534 +        thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
  9.1535 +      qed qed ultimately
  9.1536 +
  9.1537 +    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"
  9.1538 +      apply- apply(rule norm_triangle_lt) by auto
  9.1539 +    also { have *:"\<And>x y. x = (0::real) \<Longrightarrow> x *\<^sub>R (y::'a) = 0" using scaleR_zero_left by auto
  9.1540 +      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)
  9.1541 +       = (\<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
  9.1542 +      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)"
  9.1543 +        unfolding lem3[OF p(3)] apply(subst setsum_reindex_nonzero[OF p(3)]) defer apply(subst setsum_reindex_nonzero[OF p(3)])
  9.1544 +        defer unfolding lem4[THEN sym] apply(rule refl) unfolding split_paired_all split_conv apply(rule_tac[!] *)
  9.1545 +      proof- case goal1 thus ?case apply- apply(rule tagged_division_split_left_inj [OF p(1), of a b aa ba]) by auto
  9.1546 +      next case   goal2 thus ?case apply- apply(rule tagged_division_split_right_inj[OF p(1), of a b aa ba]) by auto
  9.1547 +      qed also note setsum_addf[THEN sym]
  9.1548 +      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
  9.1549 +        = (\<lambda>(x,ka). content ka *\<^sub>R f x) x" unfolding split_paired_all split_conv
  9.1550 +      proof- fix a b assume "(a,b) \<in> p" from p(6)[OF this] guess u v apply-by(erule exE)+ note uv=this
  9.1551 +        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"
  9.1552 +          unfolding scaleR_left_distrib[THEN sym] unfolding uv content_split[of u v k c] by auto
  9.1553 +      qed note setsum_cong2[OF this]
  9.1554 +      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 +
  9.1555 +        ((\<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) =
  9.1556 +        (\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f x) - (i + j)" by auto }
  9.1557 +    finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (i + j)) < e" by auto qed qed
  9.1558 +
  9.1559 +subsection {* A sort of converse, integrability on subintervals. *}
  9.1560 +
  9.1561 +lemma tagged_division_union_interval:
  9.1562 +  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})"
  9.1563 +  shows "(p1 \<union> p2) tagged_division_of ({a..b})"
  9.1564 +proof- have *:"{a..b} = ({a..b} \<inter> {x. x$k \<le> c}) \<union> ({a..b} \<inter> {x. x$k \<ge> c})" by auto
  9.1565 +  show ?thesis apply(subst *) apply(rule tagged_division_union[OF assms])
  9.1566 +    unfolding interval_split interior_closed_interval
  9.1567 +    by(auto simp add: vector_less_def Cart_lambda_beta elim!:allE[where x=k]) qed
  9.1568 +
  9.1569 +lemma has_integral_separate_sides: fixes f::"real^'m \<Rightarrow> 'a::real_normed_vector"
  9.1570 +  assumes "(f has_integral i) ({a..b})" "e>0"
  9.1571 +  obtains d where "gauge d" "(\<forall>p1 p2. p1 tagged_division_of ({a..b} \<inter> {x. x$k \<le> c}) \<and> d fine p1 \<and>
  9.1572 +                                p2 tagged_division_of ({a..b} \<inter> {x. x$k \<ge> c}) \<and> d fine p2
  9.1573 +                                \<longrightarrow> norm((setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 +
  9.1574 +                                          setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) - i) < e)"
  9.1575 +proof- guess d using has_integralD[OF assms] . note d=this
  9.1576 +  show ?thesis apply(rule that[of d]) apply(rule d) apply(rule,rule,rule,(erule conjE)+)
  9.1577 +  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
  9.1578 +                   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
  9.1579 +    note tagged_division_union_interval[OF p1(7) p2(7)] note p12 = tagged_division_ofD[OF this] this
  9.1580 +    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)"
  9.1581 +      apply(subst setsum_Un_zero) apply(rule p1 p2)+ apply(rule) unfolding split_paired_all split_conv
  9.1582 +    proof- fix a b assume ab:"(a,b) \<in> p1 \<inter> p2"
  9.1583 +      have "(a,b) \<in> p1" using ab by auto from p1(4)[OF this] guess u v apply-by(erule exE)+ note uv =this
  9.1584 +      have "b \<subseteq> {x. x$k = c}" using ab p1(3)[of a b] p2(3)[of a b] by fastsimp
  9.1585 +      moreover have "interior {x. x $ k = c} = {}" 
  9.1586 +      proof(rule ccontr) case goal1 then obtain x where x:"x\<in>interior {x. x$k = c}" by auto
  9.1587 +        then guess e unfolding mem_interior .. note e=this
  9.1588 +        have x:"x$k = c" using x interior_subset by fastsimp
  9.1589 +        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
  9.1590 +        have "x + (\<chi> i. if i = k then e/2 else 0) \<in> ball x e" unfolding mem_ball vector_dist_norm 
  9.1591 +          apply(rule le_less_trans[OF norm_le_l1]) unfolding * 
  9.1592 +          unfolding setsum_delta[OF finite_UNIV] using e by auto 
  9.1593 +        hence "x + (\<chi> i. if i = k then e/2 else 0) \<in> {x. x$k = c}" using e by auto
  9.1594 +        thus False unfolding mem_Collect_eq using e x by auto
  9.1595 +      qed ultimately have "content b = 0" unfolding uv content_eq_0_interior apply-apply(drule subset_interior) by auto
  9.1596 +      thus "content b *\<^sub>R f a = 0" by auto
  9.1597 +    qed auto
  9.1598 +    also have "\<dots> < e" by(rule d(2) p12 fine_union p1 p2)+
  9.1599 +    finally show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) < e" . qed qed
  9.1600 +
  9.1601 +lemma integrable_split[intro]: fixes f::"real^'n \<Rightarrow> 'a::{real_normed_vector,complete_space}" assumes "f integrable_on {a..b}"
  9.1602 +  shows "f integrable_on ({a..b} \<inter> {x. x$k \<le> c})" (is ?t1) and "f integrable_on ({a..b} \<inter> {x. x$k \<ge> c})" (is ?t2) 
  9.1603 +proof- guess y using assms unfolding integrable_on_def .. note y=this
  9.1604 +  def b' \<equiv> "(\<chi> i. if i = k then min (b$k) c else b$i)::real^'n"
  9.1605 +  and a' \<equiv> "(\<chi> i. if i = k then max (a$k) c else a$i)::real^'n"
  9.1606 +  show ?t1 ?t2 unfolding interval_split integrable_cauchy unfolding interval_split[THEN sym]
  9.1607 +  proof(rule_tac[!] allI impI)+ fix e::real assume "e>0" hence "e/2>0" by auto
  9.1608 +    from has_integral_separate_sides[OF y this,of k c] guess d . note d=this[rule_format]
  9.1609 +    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>
  9.1610 +                              norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e)"
  9.1611 +    show "?P {x. x $ k \<le> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
  9.1612 +    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"
  9.1613 +      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"
  9.1614 +      proof- guess p using fine_division_exists[OF d(1), of a' b] . note p=this
  9.1615 +        show ?thesis using norm_triangle_half_l[OF d(2)[of p1 p] d(2)[of p2 p]]
  9.1616 +          using as unfolding interval_split b'_def[symmetric] a'_def[symmetric]
  9.1617 +          using p using assms by(auto simp add:group_simps)
  9.1618 +      qed qed  
  9.1619 +    show "?P {x. x $ k \<ge> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
  9.1620 +    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"
  9.1621 +      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"
  9.1622 +      proof- guess p using fine_division_exists[OF d(1), of a b'] . note p=this
  9.1623 +        show ?thesis using norm_triangle_half_l[OF d(2)[of p p1] d(2)[of p p2]]
  9.1624 +          using as unfolding interval_split b'_def[symmetric] a'_def[symmetric]
  9.1625 +          using p using assms by(auto simp add:group_simps) qed qed qed qed
  9.1626 +
  9.1627 +subsection {* Generalized notion of additivity. *}
  9.1628 +
  9.1629 +definition "neutral opp = (SOME x. \<forall>y. opp x y = y \<and> opp y x = y)"
  9.1630 +
  9.1631 +definition operative :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ((real^'n) set \<Rightarrow> 'a) \<Rightarrow> bool" where
  9.1632 +  "operative opp f \<equiv> 
  9.1633 +    (\<forall>a b. content {a..b} = 0 \<longrightarrow> f {a..b} = neutral(opp)) \<and>
  9.1634 +    (\<forall>a b c k. f({a..b}) =
  9.1635 +                   opp (f({a..b} \<inter> {x. x$k \<le> c}))
  9.1636 +                       (f({a..b} \<inter> {x. x$k \<ge> c})))"
  9.1637 +
  9.1638 +lemma operativeD[dest]: assumes "operative opp f"
  9.1639 +  shows "\<And>a b. content {a..b} = 0 \<Longrightarrow> f {a..b} = neutral(opp)"
  9.1640 +  "\<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}))"
  9.1641 +  using assms unfolding operative_def by auto
  9.1642 +
  9.1643 +lemma operative_trivial:
  9.1644 + "operative opp f \<Longrightarrow> content({a..b}) = 0 \<Longrightarrow> f({a..b}) = neutral opp"
  9.1645 +  unfolding operative_def by auto
  9.1646 +
  9.1647 +lemma property_empty_interval:
  9.1648 + "(\<forall>a b. content({a..b}) = 0 \<longrightarrow> P({a..b})) \<Longrightarrow> P {}" 
  9.1649 +  using content_empty unfolding empty_as_interval by auto
  9.1650 +
  9.1651 +lemma operative_empty: "operative opp f \<Longrightarrow> f {} = neutral opp"
  9.1652 +  unfolding operative_def apply(rule property_empty_interval) by auto
  9.1653 +
  9.1654 +subsection {* Using additivity of lifted function to encode definedness. *}
  9.1655 +
  9.1656 +lemma forall_option: "(\<forall>x. P x) \<longleftrightarrow> P None \<and> (\<forall>x. P(Some x))"
  9.1657 +  by (metis map_of.simps option.nchotomy)
  9.1658 +
  9.1659 +lemma exists_option:
  9.1660 + "(\<exists>x. P x) \<longleftrightarrow> P None \<or> (\<exists>x. P(Some x))" 
  9.1661 +  by (metis map_of.simps option.nchotomy)
  9.1662 +
  9.1663 +fun lifted where 
  9.1664 +  "lifted (opp::'a\<Rightarrow>'a\<Rightarrow>'b) (Some x) (Some y) = Some(opp x y)" |
  9.1665 +  "lifted opp None _ = (None::'b option)" |
  9.1666 +  "lifted opp _ None = None"
  9.1667 +
  9.1668 +lemma lifted_simp_1[simp]: "lifted opp v None = None"
  9.1669 +  apply(induct v) by auto
  9.1670 +
  9.1671 +definition "monoidal opp \<equiv>  (\<forall>x y. opp x y = opp y x) \<and>
  9.1672 +                   (\<forall>x y z. opp x (opp y z) = opp (opp x y) z) \<and>
  9.1673 +                   (\<forall>x. opp (neutral opp) x = x)"
  9.1674 +
  9.1675 +lemma monoidalI: assumes "\<And>x y. opp x y = opp y x"
  9.1676 +  "\<And>x y z. opp x (opp y z) = opp (opp x y) z"
  9.1677 +  "\<And>x. opp (neutral opp) x = x" shows "monoidal opp"
  9.1678 +  unfolding monoidal_def using assms by fastsimp
  9.1679 +
  9.1680 +lemma monoidal_ac: assumes "monoidal opp"
  9.1681 +  shows "opp (neutral opp) a = a" "opp a (neutral opp) = a" "opp a b = opp b a"
  9.1682 +  "opp (opp a b) c = opp a (opp b c)"  "opp a (opp b c) = opp b (opp a c)"
  9.1683 +  using assms unfolding monoidal_def apply- by metis+
  9.1684 +
  9.1685 +lemma monoidal_simps[simp]: assumes "monoidal opp"
  9.1686 +  shows "opp (neutral opp) a = a" "opp a (neutral opp) = a"
  9.1687 +  using monoidal_ac[OF assms] by auto
  9.1688 +
  9.1689 +lemma neutral_lifted[cong]: assumes "monoidal opp"
  9.1690 +  shows "neutral (lifted opp) = Some(neutral opp)"
  9.1691 +  apply(subst neutral_def) apply(rule some_equality) apply(rule,induct_tac y) prefer 3
  9.1692 +proof- fix x assume "\<forall>y. lifted opp x y = y \<and> lifted opp y x = y"
  9.1693 +  thus "x = Some (neutral opp)" apply(induct x) defer
  9.1694 +    apply rule apply(subst neutral_def) apply(subst eq_commute,rule some_equality)
  9.1695 +    apply(rule,erule_tac x="Some y" in allE) defer apply(erule_tac x="Some x" in allE) by auto
  9.1696 +qed(auto simp add:monoidal_ac[OF assms])
  9.1697 +
  9.1698 +lemma monoidal_lifted[intro]: assumes "monoidal opp" shows "monoidal(lifted opp)"
  9.1699 +  unfolding monoidal_def forall_option neutral_lifted[OF assms] using monoidal_ac[OF assms] by auto
  9.1700 +
  9.1701 +definition "support opp f s = {x. x\<in>s \<and> f x \<noteq> neutral opp}"
  9.1702 +definition "fold' opp e s \<equiv> (if finite s then fold opp e s else e)"
  9.1703 +definition "iterate opp s f \<equiv> fold' (\<lambda>x a. opp (f x) a) (neutral opp) (support opp f s)"
  9.1704 +
  9.1705 +lemma support_subset[intro]:"support opp f s \<subseteq> s" unfolding support_def by auto
  9.1706 +lemma support_empty[simp]:"support opp f {} = {}" using support_subset[of opp f "{}"] by auto
  9.1707 +
  9.1708 +lemma fun_left_comm_monoidal[intro]: assumes "monoidal opp" shows "fun_left_comm opp"
  9.1709 +  unfolding fun_left_comm_def using monoidal_ac[OF assms] by auto
  9.1710 +
  9.1711 +lemma support_clauses:
  9.1712 +  "\<And>f g s. support opp f {} = {}"
  9.1713 +  "\<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))"
  9.1714 +  "\<And>f g s. support opp f (s - {x}) = (support opp f s) - {x}"
  9.1715 +  "\<And>f g s. support opp f (s \<union> t) = (support opp f s) \<union> (support opp f t)"
  9.1716 +  "\<And>f g s. support opp f (s \<inter> t) = (support opp f s) \<inter> (support opp f t)"
  9.1717 +  "\<And>f g s. support opp f (s - t) = (support opp f s) - (support opp f t)"
  9.1718 +  "\<And>f g s. support opp g (f ` s) = f ` (support opp (g o f) s)"
  9.1719 +unfolding support_def by auto
  9.1720 +
  9.1721 +lemma finite_support[intro]:"finite s \<Longrightarrow> finite (support opp f s)"
  9.1722 +  unfolding support_def by auto
  9.1723 +
  9.1724 +lemma iterate_empty[simp]:"iterate opp {} f = neutral opp"
  9.1725 +  unfolding iterate_def fold'_def by auto 
  9.1726 +
  9.1727 +lemma iterate_insert[simp]: assumes "monoidal opp" "finite s"
  9.1728 +  shows "iterate opp (insert x s) f = (if x \<in> s then iterate opp s f else opp (f x) (iterate opp s f))" 
  9.1729 +proof(cases "x\<in>s") case True hence *:"insert x s = s" by auto
  9.1730 +  show ?thesis unfolding iterate_def if_P[OF True] * by auto
  9.1731 +next case False note x=this
  9.1732 +  note * = fun_left_comm.fun_left_comm_apply[OF fun_left_comm_monoidal[OF assms(1)]]
  9.1733 +  show ?thesis proof(cases "f x = neutral opp")
  9.1734 +    case True show ?thesis unfolding iterate_def if_not_P[OF x] support_clauses if_P[OF True]
  9.1735 +      unfolding True monoidal_simps[OF assms(1)] by auto
  9.1736 +  next case False show ?thesis unfolding iterate_def fold'_def  if_not_P[OF x] support_clauses if_not_P[OF False]
  9.1737 +      apply(subst fun_left_comm.fold_insert[OF * finite_support])
  9.1738 +      using `finite s` unfolding support_def using False x by auto qed qed 
  9.1739 +
  9.1740 +lemma iterate_some:
  9.1741 +  assumes "monoidal opp"  "finite s"
  9.1742 +  shows "iterate (lifted opp) s (\<lambda>x. Some(f x)) = Some (iterate opp s f)" using assms(2)
  9.1743 +proof(induct s) case empty thus ?case using assms by auto
  9.1744 +next case (insert x F) show ?case apply(subst iterate_insert) prefer 3 apply(subst if_not_P)
  9.1745 +    defer unfolding insert(3) lifted.simps apply rule using assms insert by auto qed
  9.1746 +
  9.1747 +subsection {* Two key instances of additivity. *}
  9.1748 +
  9.1749 +lemma neutral_add[simp]:
  9.1750 +  "neutral op + = (0::_::comm_monoid_add)" unfolding neutral_def 
  9.1751 +  apply(rule some_equality) defer apply(erule_tac x=0 in allE) by auto
  9.1752 +
  9.1753 +lemma operative_content[intro]: "operative (op +) content"
  9.1754 +  unfolding operative_def content_split[THEN sym] neutral_add by auto
  9.1755 +
  9.1756 +lemma neutral_monoid[simp]: "neutral ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a) = 0"
  9.1757 +  unfolding neutral_def apply(rule some_equality) defer
  9.1758 +  apply(erule_tac x=0 in allE) by auto
  9.1759 +
  9.1760 +lemma monoidal_monoid[intro]:
  9.1761 +  shows "monoidal ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a)"
  9.1762 +  unfolding monoidal_def neutral_monoid by(auto simp add: group_simps) 
  9.1763 +
  9.1764 +lemma operative_integral: fixes f::"real^'n \<Rightarrow> 'a::banach"
  9.1765 +  shows "operative (lifted(op +)) (\<lambda>i. if f integrable_on i then Some(integral i f) else None)"
  9.1766 +  unfolding operative_def unfolding neutral_lifted[OF monoidal_monoid] neutral_add
  9.1767 +  apply(rule,rule,rule,rule) defer apply(rule allI)+
  9.1768 +proof- fix a b c k show "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) =
  9.1769 +              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)
  9.1770 +               (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)"
  9.1771 +  proof(cases "f integrable_on {a..b}") 
  9.1772 +    case True show ?thesis unfolding if_P[OF True]
  9.1773 +      unfolding if_P[OF integrable_split(1)[OF True]] if_P[OF integrable_split(2)[OF True]]
  9.1774 +      unfolding lifted.simps option.inject apply(rule integral_unique) apply(rule has_integral_split) 
  9.1775 +      apply(rule_tac[!] integrable_integral integrable_split)+ using True by assumption+
  9.1776 +  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}))"
  9.1777 +    proof(rule ccontr) case goal1 hence "f integrable_on {a..b}" apply- unfolding integrable_on_def
  9.1778 +        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)
  9.1779 +        apply(rule has_integral_split) apply(rule_tac[!] integrable_integral) by auto
  9.1780 +      thus False using False by auto
  9.1781 +    qed thus ?thesis using False by auto 
  9.1782 +  qed next 
  9.1783 +  fix a b assume as:"content {a..b::real^'n} = 0"
  9.1784 +  thus "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) = Some 0"
  9.1785 +    unfolding if_P[OF integrable_on_null[OF as]] using has_integral_null_eq[OF as] by auto qed
  9.1786 +
  9.1787 +subsection {* Points of division of a partition. *}
  9.1788 +
  9.1789 +definition "division_points (k::(real^'n) set) d = 
  9.1790 +    {(j,x). (interval_lowerbound k)$j < x \<and> x < (interval_upperbound k)$j \<and>
  9.1791 +           (\<exists>i\<in>d. (interval_lowerbound i)$j = x \<or> (interval_upperbound i)$j = x)}"
  9.1792 +
  9.1793 +lemma division_points_finite: assumes "d division_of i"
  9.1794 +  shows "finite (division_points i d)"
  9.1795 +proof- note assm = division_ofD[OF assms]
  9.1796 +  let ?M = "\<lambda>j. {(j,x)|x. (interval_lowerbound i)$j < x \<and> x < (interval_upperbound i)$j \<and>
  9.1797 +           (\<exists>i\<in>d. (interval_lowerbound i)$j = x \<or> (interval_upperbound i)$j = x)}"
  9.1798 +  have *:"division_points i d = \<Union>(?M ` UNIV)"
  9.1799 +    unfolding division_points_def by auto
  9.1800 +  show ?thesis unfolding * using assm by auto qed
  9.1801 +
  9.1802 +lemma division_points_subset:
  9.1803 +  assumes "d division_of {a..b}" "\<forall>i. a$i < b$i"  "a$k < c" "c < b$k"
  9.1804 +  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$k \<le> c} = {})}
  9.1805 +                  \<subseteq> division_points ({a..b}) d" (is ?t1) and
  9.1806 +        "division_points ({a..b} \<inter> {x. x$k \<ge> c}) {l \<inter> {x. x$k \<ge> c} | l . l \<in> d \<and> ~(l \<inter> {x. x$k \<ge> c} = {})}
  9.1807 +                  \<subseteq> division_points ({a..b}) d" (is ?t2)
  9.1808 +proof- note assm = division_ofD[OF assms(1)]
  9.1809 +  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"
  9.1810 +    "\<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"
  9.1811 +    using assms using less_imp_le by auto
  9.1812 +  show ?t1 unfolding division_points_def interval_split[of a b]
  9.1813 +    unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] Cart_lambda_beta unfolding *
  9.1814 +    unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+ unfolding mem_Collect_eq apply(erule exE conjE)+
  9.1815 +  proof- fix i l x assume as:"a $ fst x < snd x" "snd x < (if fst x = k then c else b $ fst x)"
  9.1816 +      "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> {}"
  9.1817 +    from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
  9.1818 +    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 .
  9.1819 +    have **:"\<forall>i. u$i \<le> v$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
  9.1820 +    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)"
  9.1821 +      using as(1-3,5) unfolding l interval_split interval_ne_empty as interval_bounds[OF *] Cart_lambda_beta apply-
  9.1822 +      apply(rule,assumption,rule) defer apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **]
  9.1823 +      apply(case_tac[!] "fst x = k") using assms by auto
  9.1824 +  qed
  9.1825 +  show ?t2 unfolding division_points_def interval_split[of a b]
  9.1826 +    unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] Cart_lambda_beta unfolding *
  9.1827 +    unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+ unfolding mem_Collect_eq apply(erule exE conjE)+
  9.1828 +  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"
  9.1829 +      "i = l \<inter> {x. c \<le> x $ k}" "l \<in> d" "l \<inter> {x. c \<le> x $ k} \<noteq> {}"
  9.1830 +    from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
  9.1831 +    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 .
  9.1832 +    have **:"\<forall>i. u$i \<le> v$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
  9.1833 +    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)"
  9.1834 +      using as(1-3,5) unfolding l interval_split interval_ne_empty as interval_bounds[OF *] Cart_lambda_beta apply-
  9.1835 +      apply rule defer apply(rule,assumption) apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **]
  9.1836 +      apply(case_tac[!] "fst x = k") using assms by auto qed qed
  9.1837 +
  9.1838 +lemma division_points_psubset:
  9.1839 +  assumes "d division_of {a..b}"  "\<forall>i. a$i < b$i"  "a$k < c" "c < b$k"
  9.1840 +  "l \<in> d" "interval_lowerbound l$k = c \<or> interval_upperbound l$k = c"
  9.1841 +  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$k \<le> c} \<noteq> {}} \<subset> division_points ({a..b}) d" (is "?D1 \<subset> ?D") 
  9.1842 +        "division_points ({a..b} \<inter> {x. x$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") 
  9.1843 +proof- have ab:"\<forall>i. a$i \<le> b$i" using assms(2) by(auto intro!:less_imp_le)
  9.1844 +  guess u v using division_ofD(4)[OF assms(1,5)] apply-by(erule exE)+ note l=this
  9.1845 +  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
  9.1846 +    unfolding subset_eq apply- defer apply(erule_tac x=u in ballE, erule_tac x=v in ballE) unfolding mem_interval by auto
  9.1847 +  have *:"interval_upperbound ({a..b} \<inter> {x. x $ k \<le> interval_upperbound l $ k}) $ k = interval_upperbound l $ k"
  9.1848 +         "interval_upperbound ({a..b} \<inter> {x. x $ k \<le> interval_lowerbound l $ k}) $ k = interval_lowerbound l $ k"
  9.1849 +    unfolding interval_split apply(subst interval_bounds) prefer 3 apply(subst interval_bounds)
  9.1850 +    unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab by auto
  9.1851 +  have "\<exists>x. x \<in> ?D - ?D1" using assms(2-) apply-apply(erule disjE)
  9.1852 +    apply(rule_tac x="(k,(interval_lowerbound l)$k)" in exI) defer
  9.1853 +    apply(rule_tac x="(k,(interval_upperbound l)$k)" in exI)
  9.1854 +    unfolding division_points_def unfolding interval_bounds[OF ab]
  9.1855 +    apply (auto simp add:interval_bounds) unfolding * by auto
  9.1856 +  thus "?D1 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) by auto
  9.1857 +
  9.1858 +  have *:"interval_lowerbound ({a..b} \<inter> {x. x $ k \<ge> interval_lowerbound l $ k}) $ k = interval_lowerbound l $ k"
  9.1859 +         "interval_lowerbound ({a..b} \<inter> {x. x $ k \<ge> interval_upperbound l $ k}) $ k = interval_upperbound l $ k"
  9.1860 +    unfolding interval_split apply(subst interval_bounds) prefer 3 apply(subst interval_bounds)
  9.1861 +    unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab by auto
  9.1862 +  have "\<exists>x. x \<in> ?D - ?D2" using assms(2-) apply-apply(erule disjE)
  9.1863 +    apply(rule_tac x="(k,(interval_lowerbound l)$k)" in exI) defer
  9.1864 +    apply(rule_tac x="(k,(interval_upperbound l)$k)" in exI)
  9.1865 +    unfolding division_points_def unfolding interval_bounds[OF ab]
  9.1866 +    apply (auto simp add:interval_bounds) unfolding * by auto
  9.1867 +  thus "?D2 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) by auto qed
  9.1868 +
  9.1869 +subsection {* Preservation by divisions and tagged divisions. *}
  9.1870 +
  9.1871 +lemma support_support[simp]:"support opp f (support opp f s) = support opp f s"
  9.1872 +  unfolding support_def by auto
  9.1873 +
  9.1874 +lemma iterate_support[simp]: "iterate opp (support opp f s) f = iterate opp s f"
  9.1875 +  unfolding iterate_def support_support by auto
  9.1876 +
  9.1877 +lemma iterate_expand_cases:
  9.1878 +  "iterate opp s f = (if finite(support opp f s) then iterate opp (support opp f s) f else neutral opp)"
  9.1879 +  apply(cases) apply(subst if_P,assumption) unfolding iterate_def support_support fold'_def by auto 
  9.1880 +
  9.1881 +lemma iterate_image: assumes "monoidal opp"  "inj_on f s"
  9.1882 +  shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)"
  9.1883 +proof- have *:"\<And>s. finite s \<Longrightarrow>  \<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<longrightarrow> x = y \<Longrightarrow>
  9.1884 +     iterate opp (f ` s) g = iterate opp s (g \<circ> f)"
  9.1885 +  proof- case goal1 show ?case using goal1
  9.1886 +    proof(induct s) case empty thus ?case using assms(1) by auto
  9.1887 +    next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)]
  9.1888 +        unfolding if_not_P[OF insert(2)] apply(subst insert(3)[THEN sym])
  9.1889 +        unfolding image_insert defer apply(subst iterate_insert[OF assms(1)])
  9.1890 +        apply(rule finite_imageI insert)+ apply(subst if_not_P)
  9.1891 +        unfolding image_iff o_def using insert(2,4) by auto
  9.1892 +    qed qed
  9.1893 +  show ?thesis 
  9.1894 +    apply(cases "finite (support opp g (f ` s))")
  9.1895 +    apply(subst (1) iterate_support[THEN sym],subst (2) iterate_support[THEN sym])
  9.1896 +    unfolding support_clauses apply(rule *)apply(rule finite_imageD,assumption) unfolding inj_on_def[symmetric]
  9.1897 +    apply(rule subset_inj_on[OF assms(2) support_subset])+
  9.1898 +    apply(subst iterate_expand_cases) unfolding support_clauses apply(simp only: if_False)
  9.1899 +    apply(subst iterate_expand_cases) apply(subst if_not_P) by auto qed
  9.1900 +
  9.1901 +
  9.1902 +(* This lemma about iterations comes up in a few places.                     *)
  9.1903 +lemma iterate_nonzero_image_lemma:
  9.1904 +  assumes "monoidal opp" "finite s" "g(a) = neutral opp"
  9.1905 +  "\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = neutral opp"
  9.1906 +  shows "iterate opp {f x | x. x \<in> s \<and> f x \<noteq> a} g = iterate opp s (g \<circ> f)"
  9.1907 +proof- have *:"{f x |x. x \<in> s \<and> ~(f x = a)} = f ` {x. x \<in> s \<and> ~(f x = a)}" by auto
  9.1908 +  have **:"support opp (g \<circ> f) {x \<in> s. f x \<noteq> a} = support opp (g \<circ> f) s"
  9.1909 +    unfolding support_def using assms(3) by auto
  9.1910 +  show ?thesis unfolding *
  9.1911 +    apply(subst iterate_support[THEN sym]) unfolding support_clauses
  9.1912 +    apply(subst iterate_image[OF assms(1)]) defer
  9.1913 +    apply(subst(2) iterate_support[THEN sym]) apply(subst **)
  9.1914 +    unfolding inj_on_def using assms(3,4) unfolding support_def by auto qed
  9.1915 +
  9.1916 +lemma iterate_eq_neutral:
  9.1917 +  assumes "monoidal opp"  "\<forall>x \<in> s. (f(x) = neutral opp)"
  9.1918 +  shows "(iterate opp s f = neutral opp)"
  9.1919 +proof- have *:"support opp f s = {}" unfolding support_def using assms(2) by auto
  9.1920 +  show ?thesis apply(subst iterate_support[THEN sym]) 
  9.1921 +    unfolding * using assms(1) by auto qed
  9.1922 +
  9.1923 +lemma iterate_op: assumes "monoidal opp" "finite s"
  9.1924 +  shows "iterate opp s (\<lambda>x. opp (f x) (g x)) = opp (iterate opp s f) (iterate opp s g)" using assms(2)
  9.1925 +proof(induct s) case empty thus ?case unfolding iterate_insert[OF assms(1)] using assms(1) by auto
  9.1926 +next case (insert x F) show ?case unfolding iterate_insert[OF assms(1) insert(1)] if_not_P[OF insert(2)] insert(3)
  9.1927 +    unfolding monoidal_ac[OF assms(1)] by(rule refl) qed
  9.1928 +
  9.1929 +lemma iterate_eq: assumes "monoidal opp" "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
  9.1930 +  shows "iterate opp s f = iterate opp s g"
  9.1931 +proof- have *:"support opp g s = support opp f s"
  9.1932 +    unfolding support_def using assms(2) by auto
  9.1933 +  show ?thesis
  9.1934 +  proof(cases "finite (support opp f s)")
  9.1935 +    case False thus ?thesis apply(subst iterate_expand_cases,subst(2) iterate_expand_cases)
  9.1936 +      unfolding * by auto
  9.1937 +  next def su \<equiv> "support opp f s"
  9.1938 +    case True note support_subset[of opp f s] 
  9.1939 +    thus ?thesis apply- apply(subst iterate_support[THEN sym],subst(2) iterate_support[THEN sym]) unfolding * using True
  9.1940 +      unfolding su_def[symmetric]
  9.1941 +    proof(induct su) case empty show ?case by auto
  9.1942 +    next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)] 
  9.1943 +        unfolding if_not_P[OF insert(2)] apply(subst insert(3))
  9.1944 +        defer apply(subst assms(2)[of x]) using insert by auto qed qed qed
  9.1945 +
  9.1946 +lemma nonempty_witness: assumes "s \<noteq> {}" obtains x where "x \<in> s" using assms by auto
  9.1947 +
  9.1948 +lemma operative_division: fixes f::"(real^'n) set \<Rightarrow> 'a"
  9.1949 +  assumes "monoidal opp" "operative opp f" "d division_of {a..b}"
  9.1950 +  shows "iterate opp d f = f {a..b}"
  9.1951 +proof- def C \<equiv> "card (division_points {a..b} d)" thus ?thesis using assms
  9.1952 +  proof(induct C arbitrary:a b d rule:full_nat_induct)
  9.1953 +    case goal1
  9.1954 +    { presume *:"content {a..b} \<noteq> 0 \<Longrightarrow> ?case"
  9.1955 +      thus ?case apply-apply(cases) defer apply assumption
  9.1956 +      proof- assume as:"content {a..b} = 0"
  9.1957 +        show ?case unfolding operativeD(1)[OF assms(2) as] apply(rule iterate_eq_neutral[OF goal1(2)])
  9.1958 +        proof fix x assume x:"x\<in>d"
  9.1959 +          then guess u v apply(drule_tac division_ofD(4)[OF goal1(4)]) by(erule exE)+
  9.1960 +          thus "f x = neutral opp" using division_of_content_0[OF as goal1(4)] 
  9.1961 +            using operativeD(1)[OF assms(2)] x by auto
  9.1962 +        qed qed }
  9.1963 +    assume "content {a..b} \<noteq> 0" note ab = this[unfolded content_lt_nz[THEN sym] content_pos_lt_eq]
  9.1964 +    hence ab':"\<forall>i. a$i \<le> b$i" by (auto intro!: less_imp_le) show ?case 
  9.1965 +    proof(cases "division_points {a..b} d = {}")
  9.1966 +      case True have d':"\<forall>i\<in>d. \<exists>u v. i = {u..v} \<and>
  9.1967 +        (\<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)"
  9.1968 +        unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule)
  9.1969 +        apply(rule_tac x=a in exI,rule_tac x=b in exI) apply(rule,rule refl) apply(rule)
  9.1970 +      proof- fix u v j assume as:"{u..v} \<in> d" note division_ofD(3)[OF goal1(4) this]
  9.1971 +        hence uv:"\<forall>i. u$i \<le> v$i" "u$j \<le> v$j" unfolding interval_ne_empty by auto
  9.1972 +        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
  9.1973 +        have "(j, u$j) \<notin> division_points {a..b} d"
  9.1974 +          "(j, v$j) \<notin> division_points {a..b} d" using True by auto
  9.1975 +        note this[unfolded de_Morgan_conj division_points_def mem_Collect_eq split_conv interval_bounds[OF ab'] bex_simps]
  9.1976 +        note *[OF this(1)] *[OF this(2)] note this[unfolded interval_bounds[OF uv(1)]]
  9.1977 +        moreover have "a$j \<le> u$j" "v$j \<le> b$j" using division_ofD(2,2,3)[OF goal1(4) as] 
  9.1978 +          unfolding subset_eq apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE)
  9.1979 +          unfolding interval_ne_empty mem_interval by auto
  9.1980 +        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"
  9.1981 +          unfolding not_less de_Morgan_disj using ab[rule_format,of j] uv(2) by auto
  9.1982 +      qed have "(1/2) *\<^sub>R (a+b) \<in> {a..b}" unfolding mem_interval using ab by(auto intro!:less_imp_le)
  9.1983 +      note this[unfolded division_ofD(6)[OF goal1(4),THEN sym] Union_iff]
  9.1984 +      then guess i .. note i=this guess u v using d'[rule_format,OF i(1)] apply-by(erule exE conjE)+ note uv=this
  9.1985 +      have "{a..b} \<in> d"
  9.1986 +      proof- { presume "i = {a..b}" thus ?thesis using i by auto }
  9.1987 +        { presume "u = a" "v = b" thus "i = {a..b}" using uv by auto }
  9.1988 +        show "u = a" "v = b" unfolding Cart_eq
  9.1989 +        proof(rule_tac[!] allI) fix j note i(2)[unfolded uv mem_interval,rule_format,of j]
  9.1990 +          thus "u $ j = a $ j" "v $ j = b $ j" using uv(2)[rule_format,of j] by auto
  9.1991 +        qed qed
  9.1992 +      hence *:"d = insert {a..b} (d - {{a..b}})" by auto
  9.1993 +      have "iterate opp (d - {{a..b}}) f = neutral opp" apply(rule iterate_eq_neutral[OF goal1(2)])
  9.1994 +      proof fix x assume x:"x \<in> d - {{a..b}}" hence "x\<in>d" by auto note d'[rule_format,OF this]
  9.1995 +        then guess u v apply-by(erule exE conjE)+ note uv=this
  9.1996 +        have "u\<noteq>a \<or> v\<noteq>b" using x[unfolded uv] by auto  
  9.1997 +        then obtain j where "u$j \<noteq> a$j \<or> v$j \<noteq> b$j" unfolding Cart_eq by auto
  9.1998 +        hence "u$j = v$j" using uv(2)[rule_format,of j] by auto
  9.1999 +        hence "content {u..v} = 0"  unfolding content_eq_0 apply(rule_tac x=j in exI) by auto
  9.2000 +        thus "f x = neutral opp" unfolding uv(1) by(rule operativeD(1)[OF goal1(3)])
  9.2001 +      qed thus "iterate opp d f = f {a..b}" apply-apply(subst *) 
  9.2002 +        apply(subst iterate_insert[OF goal1(2)]) using goal1(2,4) by auto
  9.2003 +    next case False hence "\<exists>x. x\<in>division_points {a..b} d" by auto
  9.2004 +      then guess k c unfolding split_paired_Ex apply- unfolding division_points_def mem_Collect_eq split_conv
  9.2005 +        by(erule exE conjE)+ note kc=this[unfolded interval_bounds[OF ab']]
  9.2006 +      from this(3) guess j .. note j=this
  9.2007 +      def d1 \<equiv> "{l \<inter> {x. x$k \<le> c} | l. l \<in> d \<and> l \<inter> {x. x$k \<le> c} \<noteq> {}}"
  9.2008 +      def d2 \<equiv> "{l \<inter> {x. x$k \<ge> c} | l. l \<in> d \<and> l \<inter> {x. x$k \<ge> c} \<noteq> {}}"
  9.2009 +      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)"
  9.2010 +      note division_points_psubset[OF goal1(4) ab kc(1-2) j]
  9.2011 +      note psubset_card_mono[OF _ this(1)] psubset_card_mono[OF _ this(2)]
  9.2012 +      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})"
  9.2013 +        apply- unfolding interval_split apply(rule_tac[!] goal1(1)[rule_format])
  9.2014 +        using division_split[OF goal1(4), where k=k and c=c]
  9.2015 +        unfolding interval_split d1_def[symmetric] d2_def[symmetric] unfolding goal1(2) Suc_le_mono
  9.2016 +        using goal1(2-3) using division_points_finite[OF goal1(4)] by auto
  9.2017 +      have "f {a..b} = opp (iterate opp d1 f) (iterate opp d2 f)" (is "_ = ?prev")
  9.2018 +        unfolding * apply(rule operativeD(2)) using goal1(3) .
  9.2019 +      also have "iterate opp d1 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x$k \<le> c}))"
  9.2020 +        unfolding d1_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])
  9.2021 +        unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+
  9.2022 +        unfolding empty_as_interval[THEN sym] apply(rule content_empty)
  9.2023 +      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" 
  9.2024 +        from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this
  9.2025 +        show "f (l \<inter> {x. x $ k \<le> c}) = neutral opp" unfolding l interval_split
  9.2026 +          apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym] apply(rule division_split_left_inj)
  9.2027 +          apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as)+
  9.2028 +      qed also have "iterate opp d2 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x$k \<ge> c}))"
  9.2029 +        unfolding d2_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])
  9.2030 +        unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+
  9.2031 +        unfolding empty_as_interval[THEN sym] apply(rule content_empty)
  9.2032 +      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" 
  9.2033 +        from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this
  9.2034 +        show "f (l \<inter> {x. x $ k \<ge> c}) = neutral opp" unfolding l interval_split
  9.2035 +          apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym] apply(rule division_split_right_inj)
  9.2036 +          apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as)+
  9.2037 +      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}))"
  9.2038 +        unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule,rule operativeD(2)) using goal1(3) .
  9.2039 +      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})))
  9.2040 +        = iterate opp d f" apply(subst(3) iterate_eq[OF _ *[rule_format]]) prefer 3
  9.2041 +        apply(rule iterate_op[THEN sym]) using goal1 by auto
  9.2042 +      finally show ?thesis by auto
  9.2043 +    qed qed qed 
  9.2044 +
  9.2045 +lemma iterate_image_nonzero: assumes "monoidal opp"
  9.2046 +  "finite s" "\<forall>x\<in>s. \<forall>y\<in>s. ~(x = y) \<and> f x = f y \<longrightarrow> g(f x) = neutral opp"
  9.2047 +  shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)" using assms
  9.2048 +proof(induct rule:finite_subset_induct[OF assms(2) subset_refl])
  9.2049 +  case goal1 show ?case using assms(1) by auto
  9.2050 +next case goal2 have *:"\<And>x y. y = neutral opp \<Longrightarrow> x = opp y x" using assms(1) by auto
  9.2051 +  show ?case unfolding image_insert apply(subst iterate_insert[OF assms(1)])
  9.2052 +    apply(rule finite_imageI goal2)+
  9.2053 +    apply(cases "f a \<in> f ` F") unfolding if_P if_not_P apply(subst goal2(4)[OF assms(1) goal2(1)]) defer
  9.2054 +    apply(subst iterate_insert[OF assms(1) goal2(1)]) defer
  9.2055 +    apply(subst iterate_insert[OF assms(1) goal2(1)])
  9.2056 +    unfolding if_not_P[OF goal2(3)] defer unfolding image_iff defer apply(erule bexE)
  9.2057 +    apply(rule *) unfolding o_def apply(rule_tac y=x in goal2(7)[rule_format])
  9.2058 +    using goal2 unfolding o_def by auto qed 
  9.2059 +
  9.2060 +lemma operative_tagged_division: assumes "monoidal opp" "operative opp f" "d tagged_division_of {a..b}"
  9.2061 +  shows "iterate(opp) d (\<lambda>(x,l). f l) = f {a..b}"
  9.2062 +proof- have *:"(\<lambda>(x,l). f l) = (f o snd)" unfolding o_def by(rule,auto) note assm = tagged_division_ofD[OF assms(3)]
  9.2063 +  have "iterate(opp) d (\<lambda>(x,l). f l) = iterate opp (snd ` d) f" unfolding *
  9.2064 +    apply(rule iterate_image_nonzero[THEN sym,OF assms(1)]) apply(rule tagged_division_of_finite assms)+ 
  9.2065 +    unfolding Ball_def split_paired_All snd_conv apply(rule,rule,rule,rule,rule,rule,rule,erule conjE)
  9.2066 +  proof- fix a b aa ba assume as:"(a, b) \<in> d" "(aa, ba) \<in> d" "(a, b) \<noteq> (aa, ba)" "b = ba"
  9.2067 +    guess u v using assm(4)[OF as(1)] apply-by(erule exE)+ note uv=this
  9.2068 +    show "f b = neutral opp" unfolding uv apply(rule operativeD(1)[OF assms(2)])
  9.2069 +      unfolding content_eq_0_interior using tagged_division_ofD(5)[OF assms(3) as(1-3)]
  9.2070 +      unfolding as(4)[THEN sym] uv by auto
  9.2071 +  qed also have "\<dots> = f {a..b}" 
  9.2072 +    using operative_division[OF assms(1-2) division_of_tagged_division[OF assms(3)]] .
  9.2073 +  finally show ?thesis . qed
  9.2074 +
  9.2075 +subsection {* Additivity of content. *}
  9.2076 +
  9.2077 +lemma setsum_iterate:assumes "finite s" shows "setsum f s = iterate op + s f"
  9.2078 +proof- have *:"setsum f s = setsum f (support op + f s)"
  9.2079 +    apply(rule setsum_mono_zero_right)
  9.2080 +    unfolding support_def neutral_monoid using assms by auto
  9.2081 +  thus ?thesis unfolding * setsum_def iterate_def fold_image_def fold'_def
  9.2082 +    unfolding neutral_monoid . qed
  9.2083 +
  9.2084 +lemma additive_content_division: assumes "d division_of {a..b}"
  9.2085 +  shows "setsum content d = content({a..b})"
  9.2086 +  unfolding operative_division[OF monoidal_monoid operative_content assms,THEN sym]
  9.2087 +  apply(subst setsum_iterate) using assms by auto
  9.2088 +
  9.2089 +lemma additive_content_tagged_division:
  9.2090 +  assumes "d tagged_division_of {a..b}"
  9.2091 +  shows "setsum (\<lambda>(x,l). content l) d = content({a..b})"
  9.2092 +  unfolding operative_tagged_division[OF monoidal_monoid operative_content assms,THEN sym]
  9.2093 +  apply(subst setsum_iterate) using assms by auto
  9.2094 +
  9.2095 +subsection {* Finally, the integral of a constant\<forall> *}
  9.2096 +
  9.2097 +lemma has_integral_const[intro]:
  9.2098 +  "((\<lambda>x. c) has_integral (content({a..b::real^'n}) *\<^sub>R c)) ({a..b})"
  9.2099 +  unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI)
  9.2100 +  apply(rule,rule gauge_trivial)apply(rule,rule,erule conjE)
  9.2101 +  unfolding split_def apply(subst scaleR_left.setsum[THEN sym, unfolded o_def])
  9.2102 +  defer apply(subst additive_content_tagged_division[unfolded split_def]) apply assumption by auto
  9.2103 +
  9.2104 +subsection {* Bounds on the norm of Riemann sums and the integral itself. *}
  9.2105 +
  9.2106 +lemma dsum_bound: assumes "p division_of {a..b}" "norm(c) \<le> e"
  9.2107 +  shows "norm(setsum (\<lambda>l. content l *\<^sub>R c) p) \<le> e * content({a..b})" (is "?l \<le> ?r")
  9.2108 +  apply(rule order_trans,rule setsum_norm) defer unfolding norm_scaleR setsum_left_distrib[THEN sym]
  9.2109 +  apply(rule order_trans[OF mult_left_mono],rule assms,rule setsum_abs_ge_zero)
  9.2110 +  apply(subst real_mult_commute) apply(rule mult_left_mono)
  9.2111 +  apply(rule order_trans[of _ "setsum content p"]) apply(rule eq_refl,rule setsum_cong2)
  9.2112 +  apply(subst abs_of_nonneg) unfolding additive_content_division[OF assms(1)]
  9.2113 +proof- from order_trans[OF norm_ge_zero[of c] assms(2)] show "0 \<le> e" .
  9.2114 +  fix x assume "x\<in>p" from division_ofD(4)[OF assms(1) this] guess u v apply-by(erule exE)+
  9.2115 +  thus "0 \<le> content x" using content_pos_le by auto
  9.2116 +qed(insert assms,auto)
  9.2117 +
  9.2118 +lemma rsum_bound: assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. norm(f x) \<le> e"
  9.2119 +  shows "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p) \<le> e * content({a..b})"
  9.2120 +proof(cases "{a..b} = {}") case True
  9.2121 +  show ?thesis using assms(1) unfolding True tagged_division_of_trivial by auto
  9.2122 +next case False show ?thesis
  9.2123 +    apply(rule order_trans,rule setsum_norm) defer unfolding split_def norm_scaleR
  9.2124 +    apply(rule order_trans[OF setsum_mono]) apply(rule mult_left_mono[OF _ abs_ge_zero, of _ e]) defer
  9.2125 +    unfolding setsum_left_distrib[THEN sym] apply(subst real_mult_commute) apply(rule mult_left_mono)
  9.2126 +    apply(rule order_trans[of _ "setsum (content \<circ> snd) p"]) apply(rule eq_refl,rule setsum_cong2)
  9.2127 +    apply(subst o_def, rule abs_of_nonneg)
  9.2128 +  proof- show "setsum (content \<circ> snd) p \<le> content {a..b}" apply(rule eq_refl)
  9.2129 +      unfolding additive_content_tagged_division[OF assms(1),THEN sym] split_def by auto
  9.2130 +    guess w using nonempty_witness[OF False] .
  9.2131 +    thus "e\<ge>0" apply-apply(rule order_trans) defer apply(rule assms(2)[rule_format],assumption) by auto
  9.2132 +    fix xk assume *:"xk\<in>p" guess x k  using surj_pair[of xk] apply-by(erule exE)+ note xk = this *[unfolded this]
  9.2133 +    from tagged_division_ofD(4)[OF assms(1) xk(2)] guess u v apply-by(erule exE)+ note uv=this
  9.2134 +    show "0\<le> content (snd xk)" unfolding xk snd_conv uv by(rule content_pos_le)
  9.2135 +    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
  9.2136 +  qed(insert assms,auto) qed
  9.2137 +
  9.2138 +lemma rsum_diff_bound:
  9.2139 +  assumes "p tagged_division_of {a..b}"  "\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e"
  9.2140 +  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})"
  9.2141 +  apply(rule order_trans[OF _ rsum_bound[OF assms]]) apply(rule eq_refl) apply(rule arg_cong[where f=norm])
  9.2142 +  unfolding setsum_subtractf[THEN sym] apply(rule setsum_cong2) unfolding scaleR.diff_right by auto
  9.2143 +
  9.2144 +lemma has_integral_bound: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
  9.2145 +  assumes "0 \<le> B" "(f has_integral i) ({a..b})" "\<forall>x\<in>{a..b}. norm(f x) \<le> B"
  9.2146 +  shows "norm i \<le> B * content {a..b}"
  9.2147 +proof- let ?P = "content {a..b} > 0" { presume "?P \<Longrightarrow> ?thesis"
  9.2148 +    thus ?thesis proof(cases ?P) case False
  9.2149 +      hence *:"content {a..b} = 0" using content_lt_nz by auto
  9.2150 +      hence **:"i = 0" using assms(2) apply(subst has_integral_null_eq[THEN sym]) by auto
  9.2151 +      show ?thesis unfolding * ** using assms(1) by auto
  9.2152 +    qed auto } assume ab:?P
  9.2153 +  { presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }
  9.2154 +  assume "\<not> ?thesis" hence *:"norm i - B * content {a..b} > 0" by auto
  9.2155 +  from assms(2)[unfolded has_integral,rule_format,OF *] guess d apply-by(erule exE conjE)+ note d=this[rule_format]
  9.2156 +  from fine_division_exists[OF this(1), of a b] guess p . note p=this
  9.2157 +  have *:"\<And>s B. norm s \<le> B \<Longrightarrow> \<not> (norm (s - i) < norm i - B)"
  9.2158 +  proof- case goal1 thus ?case unfolding not_less
  9.2159 +    using norm_triangle_sub[of i s] unfolding norm_minus_commute by auto
  9.2160 +  qed show False using d(2)[OF conjI[OF p]] *[OF rsum_bound[OF p(1) assms(3)]] by auto qed
  9.2161 +
  9.2162 +subsection {* Similar theorems about relationship among components. *}
  9.2163 +
  9.2164 +lemma rsum_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
  9.2165 +  assumes "p tagged_division_of {a..b}"  "\<forall>x\<in>{a..b}. (f x)$i \<le> (g x)$i"
  9.2166 +  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"
  9.2167 +  unfolding setsum_component apply(rule setsum_mono)
  9.2168 +  apply(rule mp) defer apply assumption apply(induct_tac x,rule) unfolding split_conv
  9.2169 +proof- fix a b assume ab:"(a,b) \<in> p" note assm = tagged_division_ofD(2-4)[OF assms(1) ab]
  9.2170 +  from this(3) guess u v apply-by(erule exE)+ note b=this
  9.2171 +  show "(content b *\<^sub>R f a) $ i \<le> (content b *\<^sub>R g a) $ i" unfolding b
  9.2172 +    unfolding Cart_nth.scaleR real_scaleR_def apply(rule mult_left_mono)
  9.2173 +    defer apply(rule content_pos_le,rule assms(2)[rule_format]) using assm by auto qed
  9.2174 +
  9.2175 +lemma has_integral_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
  9.2176 +  assumes "(f has_integral i) s" "(g has_integral j) s"  "\<forall>x\<in>s. (f x)$k \<le> (g x)$k"
  9.2177 +  shows "i$k \<le> j$k"
  9.2178 +proof- have lem:"\<And>a b g i j. \<And>f::real^'n \<Rightarrow> real^'m. (f has_integral i) ({a..b}) \<Longrightarrow> 
  9.2179 +    (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"
  9.2180 +  proof(rule ccontr) case goal1 hence *:"0 < (i$k - j$k) / 3" by auto
  9.2181 +    guess d1 using goal1(1)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d1=this[rule_format]
  9.2182 +    guess d2 using goal1(2)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d2=this[rule_format]
  9.2183 +    guess p using fine_division_exists[OF gauge_inter[OF d1(1) d2(1)], of a b] unfolding fine_inter .
  9.2184 +    note p = this(1) conjunctD2[OF this(2)]  note le_less_trans[OF component_le_norm, of _ _ k]
  9.2185 +    note this[OF d1(2)[OF conjI[OF p(1,2)]]] this[OF d2(2)[OF conjI[OF p(1,3)]]]
  9.2186 +    thus False unfolding Cart_nth.diff using rsum_component_le[OF p(1) goal1(3)] by smt
  9.2187 +  qed let ?P = "\<exists>a b. s = {a..b}"
  9.2188 +  { presume "\<not> ?P \<Longrightarrow> ?thesis" thus ?thesis proof(cases ?P)
  9.2189 +      case True then guess a b apply-by(erule exE)+ note s=this
  9.2190 +      show ?thesis apply(rule lem) using assms[unfolded s] by auto
  9.2191 +    qed auto } assume as:"\<not> ?P"
  9.2192 +  { presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }
  9.2193 +  assume "\<not> i$k \<le> j$k" hence ij:"(i$k - j$k) / 3 > 0" by auto
  9.2194 +  note has_integral_altD[OF _ as this] from this[OF assms(1)] this[OF assms(2)] guess B1 B2 . note B=this[rule_format]
  9.2195 +  have "bounded (ball 0 B1 \<union> ball (0::real^'n) B2)" unfolding bounded_Un by(rule conjI bounded_ball)+
  9.2196 +  from bounded_subset_closed_interval[OF this] guess a b apply- by(erule exE)+
  9.2197 +  note ab = conjunctD2[OF this[unfolded Un_subset_iff]]
  9.2198 +  guess w1 using B(2)[OF ab(1)] .. note w1=conjunctD2[OF this]
  9.2199 +  guess w2 using B(4)[OF ab(2)] .. note w2=conjunctD2[OF this]
  9.2200 +  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*)
  9.2201 +  note le_less_trans[OF component_le_norm[of _ k]] note this[OF w1(2)] this[OF w2(2)] moreover
  9.2202 +  have "w1$k \<le> w2$k" apply(rule lem[OF w1(1) w2(1)]) using assms by auto ultimately
  9.2203 +  show False unfolding Cart_nth.diff by(rule *) qed
  9.2204 +
  9.2205 +lemma integral_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
  9.2206 +  assumes "f integrable_on s" "g integrable_on s"  "\<forall>x\<in>s. (f x)$k \<le> (g x)$k"
  9.2207 +  shows "(integral s f)$k \<le> (integral s g)$k"
  9.2208 +  apply(rule has_integral_component_le) using integrable_integral assms by auto
  9.2209 +
  9.2210 +lemma has_integral_dest_vec1_le: fixes f::"real^'n \<Rightarrow> real^1"
  9.2211 +  assumes "(f has_integral i) s"  "(g has_integral j) s" "\<forall>x\<in>s. f x \<le> g x"
  9.2212 +  shows "dest_vec1 i \<le> dest_vec1 j" apply(rule has_integral_component_le[OF assms(1-2)])
  9.2213 +  using assms(3) unfolding vector_le_def by auto
  9.2214 +
  9.2215 +lemma integral_dest_vec1_le: fixes f::"real^'n \<Rightarrow> real^1"
  9.2216 +  assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. f x \<le> g x"
  9.2217 +  shows "dest_vec1(integral s f) \<le> dest_vec1(integral s g)"
  9.2218 +  apply(rule has_integral_dest_vec1_le) apply(rule_tac[1-2] integrable_integral) using assms by auto
  9.2219 +
  9.2220 +lemma has_integral_component_pos: fixes f::"real^'n \<Rightarrow> real^'m"
  9.2221 +  assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> (f x)$k" shows "0 \<le> i$k"
  9.2222 +  using has_integral_component_le[OF has_integral_0 assms(1)] using assms(2) by auto
  9.2223 +
  9.2224 +lemma integral_component_pos: fixes f::"real^'n \<Rightarrow> real^'m"
  9.2225 +  assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> (f x)$k" shows "0 \<le> (integral s f)$k"
  9.2226 +  apply(rule has_integral_component_pos) using assms by auto
  9.2227 +
  9.2228 +lemma has_integral_dest_vec1_pos: fixes f::"real^'n \<Rightarrow> real^1"
  9.2229 +  assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> i"
  9.2230 +  using has_integral_component_pos[OF assms(1), of 1]
  9.2231 +  using assms(2) unfolding vector_le_def by auto
  9.2232 +
  9.2233 +lemma integral_dest_vec1_pos: fixes f::"real^'n \<Rightarrow> real^1"
  9.2234 +  assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> integral s f"
  9.2235 +  apply(rule has_integral_dest_vec1_pos) using assms by auto
  9.2236 +
  9.2237 +lemma has_integral_component_neg: fixes f::"real^'n \<Rightarrow> real^'m"
  9.2238 +  assumes "(f has_integral i) s" "\<forall>x\<in>s. (f x)$k \<le> 0" shows "i$k \<le> 0"
  9.2239 +  using has_integral_component_le[OF assms(1) has_integral_0] assms(2) by auto
  9.2240 +
  9.2241 +lemma has_integral_dest_vec1_neg: fixes f::"real^'n \<Rightarrow> real^1"
  9.2242 +  assumes "(f has_integral i) s" "\<forall>x\<in>s. f x \<le> 0" shows "i \<le> 0"
  9.2243 +  using has_integral_component_neg[OF assms(1),of 1] using assms(2) by auto
  9.2244 +
  9.2245 +lemma has_integral_component_lbound:
  9.2246 +  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"
  9.2247 +  using has_integral_component_le[OF has_integral_const assms(1),of "(\<chi> i. B)" k] assms(2)
  9.2248 +  unfolding Cart_lambda_beta vector_scaleR_component by(auto simp add:field_simps)
  9.2249 +
  9.2250 +lemma has_integral_component_ubound: 
  9.2251 +  assumes "(f has_integral i) {a..b}" "\<forall>x\<in>{a..b}. f x$k \<le> B"
  9.2252 +  shows "i$k \<le> B * content({a..b})"
  9.2253 +  using has_integral_component_le[OF assms(1) has_integral_const, of k "vec B"]
  9.2254 +  unfolding vec_component Cart_nth.scaleR using assms(2) by(auto simp add:field_simps)
  9.2255 +
  9.2256 +lemma integral_component_lbound:
  9.2257 +  assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. B \<le> f(x)$k"
  9.2258 +  shows "B * content({a..b}) \<le> (integral({a..b}) f)$k"
  9.2259 +  apply(rule has_integral_component_lbound) using assms unfolding has_integral_integral by auto
  9.2260 +
  9.2261 +lemma integral_component_ubound:
  9.2262 +  assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. f(x)$k \<le> B"
  9.2263 +  shows "(integral({a..b}) f)$k \<le> B * content({a..b})"
  9.2264 +  apply(rule has_integral_component_ubound) using assms unfolding has_integral_integral by auto
  9.2265 +
  9.2266 +subsection {* Uniform limit of integrable functions is integrable. *}
  9.2267 +
  9.2268 +lemma real_arch_invD:
  9.2269 +  "0 < (e::real) \<Longrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
  9.2270 +  by(subst(asm) real_arch_inv)
  9.2271 +
  9.2272 +lemma integrable_uniform_limit: fixes f::"real^'n \<Rightarrow> 'a::banach"
  9.2273 +  assumes "\<forall>e>0. \<exists>g. (\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e) \<and> g integrable_on {a..b}"
  9.2274 +  shows "f integrable_on {a..b}"
  9.2275 +proof- { presume *:"content {a..b} > 0 \<Longrightarrow> ?thesis"
  9.2276 +    show ?thesis apply cases apply(rule *,assumption)
  9.2277 +      unfolding content_lt_nz integrable_on_def using has_integral_null by auto }
  9.2278 +  assume as:"content {a..b} > 0"
  9.2279 +  have *:"\<And>P. \<forall>e>(0::real). P e \<Longrightarrow> \<forall>n::nat. P (inverse (real n+1))" by auto
  9.2280 +  from choice[OF *[OF assms]] guess g .. note g=conjunctD2[OF this[rule_format],rule_format]
  9.2281 +  from choice[OF allI[OF g(2)[unfolded integrable_on_def], of "\<lambda>x. x"]] guess i .. note i=this[rule_format]
  9.2282 +  
  9.2283 +  have "Cauchy i" unfolding Cauchy_def
  9.2284 +  proof(rule,rule) fix e::real assume "e>0"
  9.2285 +    hence "e / 4 / content {a..b} > 0" using as by(auto simp add:field_simps)
  9.2286 +    then guess M apply-apply(subst(asm) real_arch_inv) by(erule exE conjE)+ note M=this
  9.2287 +    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)
  9.2288 +    proof- case goal1 have "e/4>0" using `e>0` by auto note * = i[unfolded has_integral,rule_format,OF this]
  9.2289 +      from *[of m] guess gm apply-by(erule conjE exE)+ note gm=this[rule_format]
  9.2290 +      from *[of n] guess gn apply-by(erule conjE exE)+ note gn=this[rule_format]
  9.2291 +      from fine_division_exists[OF gauge_inter[OF gm(1) gn(1)], of a b] guess p . note p=this
  9.2292 +      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"
  9.2293 +      proof- case goal1 have "norm (i1 - i2) \<le> norm (i1 - s1) + norm (s1 - s2) + norm (s2 - i2)"
  9.2294 +          using norm_triangle_ineq[of "i1 - s1" "s1 - i2"]
  9.2295 +          using norm_triangle_ineq[of "s1 - s2" "s2 - i2"] by(auto simp add:group_simps)
  9.2296 +        also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:group_simps)
  9.2297 +        finally show ?case .
  9.2298 +      qed
  9.2299 +      show ?case unfolding vector_dist_norm apply(rule lem2) defer
  9.2300 +        apply(rule gm(2)[OF conjI[OF p(1)]],rule_tac[2] gn(2)[OF conjI[OF p(1)]])
  9.2301 +        using conjunctD2[OF p(2)[unfolded fine_inter]] apply- apply assumption+ apply(rule order_trans)
  9.2302 +        apply(rule rsum_diff_bound[OF p(1), where e="2 / real M"])
  9.2303 +      proof show "2 / real M * content {a..b} \<le> e / 2" unfolding divide_inverse 
  9.2304 +          using M as by(auto simp add:field_simps)
  9.2305 +        fix x assume x:"x \<in> {a..b}"
  9.2306 +        have "norm (f x - g n x) + norm (f x - g m x) \<le> inverse (real n + 1) + inverse (real m + 1)" 
  9.2307 +            using g(1)[OF x, of n] g(1)[OF x, of m] by auto
  9.2308 +        also have "\<dots> \<le> inverse (real M) + inverse (real M)" apply(rule add_mono)
  9.2309 +          apply(rule_tac[!] le_imp_inverse_le) using goal1 M by auto
  9.2310 +        also have "\<dots> = 2 / real M" unfolding real_divide_def by auto
  9.2311 +        finally show "norm (g n x - g m x) \<le> 2 / real M"
  9.2312 +          using norm_triangle_le[of "g n x - f x" "f x - g m x" "2 / real M"]
  9.2313 +          by(auto simp add:group_simps simp add:norm_minus_commute)
  9.2314 +      qed qed qed
  9.2315 +  from this[unfolded convergent_eq_cauchy[THEN sym]] guess s .. note s=this
  9.2316 +
  9.2317 +  show ?thesis unfolding integrable_on_def apply(rule_tac x=s in exI) unfolding has_integral
  9.2318 +  proof(rule,rule)  
  9.2319 +    case goal1 hence *:"e/3 > 0" by auto
  9.2320 +    from s[unfolded Lim_sequentially,rule_format,OF this] guess N1 .. note N1=this
  9.2321 +    from goal1 as have "e / 3 / content {a..b} > 0" by(auto simp add:field_simps)
  9.2322 +    from real_arch_invD[OF this] guess N2 apply-by(erule exE conjE)+ note N2=this
  9.2323 +    from i[of "N1 + N2",unfolded has_integral,rule_format,OF *] guess g' .. note g'=conjunctD2[OF this,rule_format]
  9.2324 +    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"
  9.2325 +    proof- case goal1 have "norm (sf - s) \<le> norm (sf - sg) + norm (sg - i) + norm (i - s)"
  9.2326 +        using norm_triangle_ineq[of "sf - sg" "sg - s"]
  9.2327 +        using norm_triangle_ineq[of "sg -  i" " i - s"] by(auto simp add:group_simps)
  9.2328 +      also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:group_simps)
  9.2329 +      finally show ?case .
  9.2330 +    qed
  9.2331 +    show ?case apply(rule_tac x=g' in exI) apply(rule,rule g')
  9.2332 +    proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> g' fine p" note * = g'(2)[OF this]
  9.2333 +      show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - s) < e" apply-apply(rule lem[OF _ _ *])
  9.2334 +        apply(rule order_trans,rule rsum_diff_bound[OF p[THEN conjunct1]]) apply(rule,rule g,assumption)
  9.2335 +      proof- have "content {a..b} < e / 3 * (real N2)"
  9.2336 +          using N2 unfolding inverse_eq_divide using as by(auto simp add:field_simps)
  9.2337 +        hence "content {a..b} < e / 3 * (real (N1 + N2) + 1)"
  9.2338 +          apply-apply(rule less_le_trans,assumption) using `e>0` by auto 
  9.2339 +        thus "inverse (real (N1 + N2) + 1) * content {a..b} \<le> e / 3"
  9.2340 +          unfolding inverse_eq_divide by(auto simp add:field_simps)
  9.2341 +        show "norm (i (N1 + N2) - s) < e / 3" by(rule N1[rule_format,unfolded vector_dist_norm],auto)
  9.2342 +      qed qed qed qed
  9.2343 +
  9.2344 +subsection {* Negligible sets. *}
  9.2345 +
  9.2346 +definition "indicator s \<equiv> (\<lambda>x. if x \<in> s then 1 else (0::real))"
  9.2347 +
  9.2348 +lemma dest_vec1_indicator:
  9.2349 + "indicator s x = (if x \<in> s then 1 else 0)" unfolding indicator_def by auto
  9.2350 +
  9.2351 +lemma indicator_pos_le[intro]: "0 \<le> (indicator s x)" unfolding indicator_def by auto
  9.2352 +
  9.2353 +lemma indicator_le_1[intro]: "(indicator s x) \<le> 1" unfolding indicator_def by auto
  9.2354 +
  9.2355 +lemma dest_vec1_indicator_abs_le_1: "abs(indicator s x) \<le> 1"
  9.2356 +  unfolding indicator_def by auto
  9.2357 +
  9.2358 +definition "negligible (s::(real^'n) set) \<equiv> (\<forall>a b. ((indicator s) has_integral 0) {a..b})"
  9.2359 +
  9.2360 +lemma indicator_simps[simp]:"x\<in>s \<Longrightarrow> indicator s x = 1" "x\<notin>s \<Longrightarrow> indicator s x = 0"
  9.2361 +  unfolding indicator_def by auto
  9.2362 +
  9.2363 +subsection {* Negligibility of hyperplane. *}
  9.2364 +
  9.2365 +lemma vsum_nonzero_image_lemma: 
  9.2366 +  assumes "finite s" "g(a) = 0"
  9.2367 +  "\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = 0"
  9.2368 +  shows "setsum g {f x |x. x \<in> s \<and> f x \<noteq> a} = setsum (g o f) s"
  9.2369 +  unfolding setsum_iterate[OF assms(1)] apply(subst setsum_iterate) defer
  9.2370 +  apply(rule iterate_nonzero_image_lemma) apply(rule assms monoidal_monoid)+
  9.2371 +  unfolding assms using neutral_add unfolding neutral_add using assms by auto 
  9.2372 +
  9.2373 +lemma interval_doublesplit: shows "{a..b} \<inter> {x . abs(x$k - c) \<le> (e::real)} =
  9.2374 +  {(\<chi> i. if i = k then max (a$k) (c - e) else a$i) .. (\<chi> i. if i = k then min (b$k) (c + e) else b$i)}"
  9.2375 +proof- have *:"\<And>x c e::real. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
  9.2376 +  have **:"\<And>s P Q. s \<inter> {x. P x \<and> Q x} = (s \<inter> {x. Q x}) \<inter> {x. P x}" by blast
  9.2377 +  show ?thesis unfolding * ** interval_split by(rule refl) qed
  9.2378 +
  9.2379 +lemma division_doublesplit: assumes "p division_of {a..b::real^'n}" 
  9.2380 +  shows "{l \<inter> {x. abs(x$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})"
  9.2381 +proof- have *:"\<And>x c. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
  9.2382 +  have **:"\<And>p q p' q'. p division_of q \<Longrightarrow> p = p' \<Longrightarrow> q = q' \<Longrightarrow> p' division_of q'" by auto
  9.2383 +  note division_split(1)[OF assms, where c="c+e" and k=k,unfolded interval_split]
  9.2384 +  note division_split(2)[OF this, where c="c-e" and k=k] 
  9.2385 +  thus ?thesis apply(rule **) unfolding interval_doublesplit unfolding * unfolding interval_split interval_doublesplit
  9.2386 +    apply(rule set_ext) unfolding mem_Collect_eq apply rule apply(erule conjE exE)+ apply(rule_tac x=la in exI) defer
  9.2387 +    apply(erule conjE exE)+ apply(rule_tac x="l \<inter> {x. c + e \<ge> x $ k}" in exI) apply rule defer apply rule
  9.2388 +    apply(rule_tac x=l in exI) by blast+ qed
  9.2389 +
  9.2390 +lemma content_doublesplit: assumes "0 < e"
  9.2391 +  obtains d where "0 < d" "content({a..b} \<inter> {x. abs(x$k - c) \<le> d}) < e"
  9.2392 +proof(cases "content {a..b} = 0")
  9.2393 +  case True show ?thesis apply(rule that[of 1]) defer unfolding interval_doublesplit
  9.2394 +    apply(rule le_less_trans[OF content_subset]) defer apply(subst True)
  9.2395 +    unfolding interval_doublesplit[THEN sym] using assms by auto 
  9.2396 +next case False def d \<equiv> "e / 3 / setprod (\<lambda>i. b$i - a$i) (UNIV - {k})"
  9.2397 +  note False[unfolded content_eq_0 not_ex not_le, rule_format]
  9.2398 +  hence prod0:"0 < setprod (\<lambda>i. b$i - a$i) (UNIV - {k})" apply-apply(rule setprod_pos) by smt
  9.2399 +  hence "d > 0" unfolding d_def using assms by(auto simp add:field_simps) thus ?thesis
  9.2400 +  proof(rule that[of d]) have *:"UNIV = insert k (UNIV - {k})" by auto
  9.2401 +    have **:"{a..b} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d} \<noteq> {} \<Longrightarrow> 
  9.2402 +      (\<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)
  9.2403 +      = (\<Prod>i\<in>UNIV - {k}. b$i - a$i)" apply(rule setprod_cong,rule refl)
  9.2404 +      unfolding interval_doublesplit interval_eq_empty not_ex not_less unfolding interval_bounds by auto
  9.2405 +    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)
  9.2406 +      unfolding if_not_P apply(subst *) apply(subst setprod_insert) unfolding **
  9.2407 +      unfolding interval_doublesplit interval_eq_empty not_ex not_less unfolding interval_bounds unfolding Cart_lambda_beta if_P[OF refl]
  9.2408 +    proof- have "(min (b $ k) (c + d) - max (a $ k) (c - d)) \<le> 2 * d" by auto
  9.2409 +      also have "... < e / (\<Prod>i\<in>UNIV - {k}. b $ i - a $ i)" unfolding d_def using assms prod0 by(auto simp add:field_simps)
  9.2410 +      finally show "(min (b $ k) (c + d) - max (a $ k) (c - d)) * (\<Prod>i\<in>UNIV - {k}. b $ i - a $ i) < e"
  9.2411 +        unfolding pos_less_divide_eq[OF prod0] . qed auto qed qed
  9.2412 +
  9.2413 +lemma negligible_standard_hyperplane[intro]: "negligible {x. x$k = (c::real)}" 
  9.2414 +  unfolding negligible_def has_integral apply(rule,rule,rule,rule)
  9.2415 +proof- case goal1 from content_doublesplit[OF this,of a b k c] guess d . note d=this let ?i = "indicator {x. x$k = c}"
  9.2416 +  show ?case apply(rule_tac x="\<lambda>x. ball x d" in exI) apply(rule,rule gauge_ball,rule d)
  9.2417 +  proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> (\<lambda>x. ball x d) fine p"
  9.2418 +    have *:"(\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. abs(x$k - c) \<le> d}) *\<^sub>R ?i x)"
  9.2419 +      apply(rule setsum_cong2) unfolding split_paired_all real_scaleR_def mult_cancel_right split_conv
  9.2420 +      apply(cases,rule disjI1,assumption,rule disjI2)
  9.2421 +    proof- fix x l assume as:"(x,l)\<in>p" "?i x \<noteq> 0" hence xk:"x$k = c" unfolding indicator_def apply-by(rule ccontr,auto)
  9.2422 +      show "content l = content (l \<inter> {x. \<bar>x $ k - c\<bar> \<le> d})" apply(rule arg_cong[where f=content])
  9.2423 +        apply(rule set_ext,rule,rule) unfolding mem_Collect_eq
  9.2424 +      proof- fix y assume y:"y\<in>l" note p[THEN conjunct2,unfolded fine_def,rule_format,OF as(1),unfolded split_conv]
  9.2425 +        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]
  9.2426 +        thus "\<bar>y $ k - c\<bar> \<le> d" unfolding Cart_nth.diff xk by auto
  9.2427 +      qed auto qed
  9.2428 +    note p'= tagged_division_ofD[OF p[THEN conjunct1]] and p''=division_of_tagged_division[OF p[THEN conjunct1]]
  9.2429 +    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
  9.2430 +      apply(subst abs_of_nonneg) apply(rule setsum_nonneg,rule) unfolding split_paired_all split_conv
  9.2431 +      apply(rule mult_nonneg_nonneg) apply(drule p'(4)) apply(erule exE)+ apply(rule_tac b=b in back_subst)
  9.2432 +      prefer 2 apply(subst(asm) eq_commute) apply assumption
  9.2433 +      apply(subst interval_doublesplit) apply(rule content_pos_le) apply(rule indicator_pos_le)
  9.2434 +    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}))"
  9.2435 +        apply(rule setsum_mono) unfolding split_paired_all split_conv 
  9.2436 +        apply(rule mult_right_le_one_le) apply(drule p'(4)) by(auto simp add:interval_doublesplit intro!:content_pos_le)
  9.2437 +      also have "... < e" apply(subst setsum_over_tagged_division_lemma[OF p[THEN conjunct1]])
  9.2438 +      proof- case goal1 have "content ({u..v} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) \<le> content {u..v}"
  9.2439 +          unfolding interval_doublesplit apply(rule content_subset) unfolding interval_doublesplit[THEN sym] by auto
  9.2440 +        thus ?case unfolding goal1 unfolding interval_doublesplit using content_pos_le by smt
  9.2441 +      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"
  9.2442 +          apply(rule setsum_nonneg,rule) unfolding mem_Collect_eq image_iff apply(erule exE bexE conjE)+ unfolding split_paired_all 
  9.2443 +        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)"
  9.2444 +          guess u v using p'(4)[OF as(2)] apply-by(erule exE)+ note * = this
  9.2445 +          show "content x \<ge> 0" unfolding as snd_conv * interval_doublesplit by(rule content_pos_le)
  9.2446 +        qed have **:"norm (1::real) \<le> 1" by auto note division_doublesplit[OF p'',unfolded interval_doublesplit]
  9.2447 +        note dsum_bound[OF this **,unfolded interval_doublesplit[THEN sym]]
  9.2448 +        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)]
  9.2449 +        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"
  9.2450 +          apply(subst vsum_nonzero_image_lemma[of "snd ` p" content "{}", unfolded o_def,THEN sym])
  9.2451 +          apply(rule finite_imageI p' content_empty)+ unfolding forall_in_division[OF p'']
  9.2452 +        proof(rule,rule,rule,rule,rule,rule,rule,erule conjE) fix m n u v
  9.2453 +          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}"
  9.2454 +          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
  9.2455 +          note subset_interior[OF this, unfolded division_ofD(5)[OF p'' as(1-3)] interior_inter[of "{m..n}"]]
  9.2456 +          hence "interior ({m..n} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) = {}" unfolding as Int_absorb by auto
  9.2457 +          thus "content ({m..n} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) = 0" unfolding interval_doublesplit content_eq_0_interior[THEN sym] .
  9.2458 +        qed qed
  9.2459 +      finally show "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) * ?i x) < e" .
  9.2460 +    qed qed qed
  9.2461 +
  9.2462 +subsection {* A technical lemma about "refinement" of division. *}
  9.2463 +
  9.2464 +lemma tagged_division_finer: fixes p::"((real^'n) \<times> ((real^'n) set)) set"
  9.2465 +  assumes "p tagged_division_of {a..b}" "gauge d"
  9.2466 +  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"
  9.2467 +proof-
  9.2468 +  let ?P = "\<lambda>p. p tagged_partial_division_of {a..b} \<longrightarrow> gauge d \<longrightarrow>
  9.2469 +    (\<exists>q. q tagged_division_of (\<Union>{k. \<exists>x. (x,k) \<in> p}) \<and> d fine q \<and>
  9.2470 +                   (\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q))"
  9.2471 +  { have *:"finite p" "p tagged_partial_division_of {a..b}" using assms(1) unfolding tagged_division_of_def by auto
  9.2472 +    presume "\<And>p. finite p \<Longrightarrow> ?P p" from this[rule_format,OF * assms(2)] guess q .. note q=this
  9.2473 +    thus ?thesis apply-apply(rule that[of q]) unfolding tagged_division_ofD[OF assms(1)] by auto
  9.2474 +  } fix p::"((real^'n) \<times> ((real^'n) set)) set" assume as:"finite p"
  9.2475 +  show "?P p" apply(rule,rule) using as proof(induct p) 
  9.2476 +    case empty show ?case apply(rule_tac x="{}" in exI) unfolding fine_def by auto
  9.2477 +  next case (insert xk p) guess x k using surj_pair[of xk] apply- by(erule exE)+ note xk=this
  9.2478 +    note tagged_partial_division_subset[OF insert(4) subset_insertI]
  9.2479 +    from insert(3)[OF this insert(5)] guess q1 .. note q1 = conjunctD3[OF this]
  9.2480 +    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
  9.2481 +    note p = tagged_partial_division_ofD[OF insert(4)]
  9.2482 +    from p(4)[unfolded xk, OF insertI1] guess u v apply-by(erule exE)+ note uv=this
  9.2483 +
  9.2484 +    have "finite {k. \<exists>x. (x, k) \<in> p}" 
  9.2485 +      apply(rule finite_subset[of _ "snd ` p"],rule) unfolding subset_eq image_iff mem_Collect_eq
  9.2486 +      apply(erule exE,rule_tac x="(xa,x)" in bexI) using p by auto
  9.2487 +    hence int:"interior {u..v} \<inter> interior (\<Union>{k. \<exists>x. (x, k) \<in> p}) = {}"
  9.2488 +      apply(rule inter_interior_unions_intervals) apply(rule open_interior) apply(rule_tac[!] ballI)
  9.2489 +      unfolding mem_Collect_eq apply(erule_tac[!] exE) apply(drule p(4)[OF insertI2],assumption)      
  9.2490 +      apply(rule p(5))  unfolding uv xk apply(rule insertI1,rule insertI2) apply assumption
  9.2491 +      using insert(2) unfolding uv xk by auto
  9.2492 +
  9.2493 +    show ?case proof(cases "{u..v} \<subseteq> d x")
  9.2494 +      case True thus ?thesis apply(rule_tac x="{(x,{u..v})} \<union> q1" in exI) apply rule
  9.2495 +        unfolding * uv apply(rule tagged_division_union,rule tagged_division_of_self)
  9.2496 +        apply(rule p[unfolded xk uv] insertI1)+  apply(rule q1,rule int) 
  9.2497 +        apply(rule,rule fine_union,subst fine_def) defer apply(rule q1)
  9.2498 +        unfolding Ball_def split_paired_All split_conv apply(rule,rule,rule,rule)
  9.2499 +        apply(erule insertE) defer apply(rule UnI2) apply(drule q1(3)[rule_format]) unfolding xk uv by auto
  9.2500 +    next case False from fine_division_exists[OF assms(2), of u v] guess q2 . note q2=this
  9.2501 +      show ?thesis apply(rule_tac x="q2 \<union> q1" in exI)
  9.2502 +        apply rule unfolding * uv apply(rule tagged_division_union q2 q1 int fine_union)+
  9.2503 +        unfolding Ball_def split_paired_All split_conv apply rule apply(rule fine_union)
  9.2504 +        apply(rule q1 q2)+ apply(rule,rule,rule,rule) apply(erule insertE)
  9.2505 +        apply(rule UnI2) defer apply(drule q1(3)[rule_format])using False unfolding xk uv by auto
  9.2506 +    qed qed qed
  9.2507 +
  9.2508 +subsection {* Hence the main theorem about negligible sets. *}
  9.2509 +
  9.2510 +lemma finite_product_dependent: assumes "finite s" "\<And>x. x\<in>s\<Longrightarrow> finite (t x)"
  9.2511 +  shows "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" using assms
  9.2512 +proof(induct) case (insert x s) 
  9.2513 +  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
  9.2514 +  show ?case unfolding * apply(rule finite_UnI) using insert by auto qed auto
  9.2515 +
  9.2516 +lemma sum_sum_product: assumes "finite s" "\<forall>i\<in>s. finite (t i)"
  9.2517 +  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
  9.2518 +proof(induct) case (insert a s)
  9.2519 +  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
  9.2520 +  show ?case unfolding * apply(subst setsum_Un_disjoint) unfolding setsum_insert[OF insert(1-2)]
  9.2521 +    prefer 4 apply(subst insert(3)) unfolding add_right_cancel
  9.2522 +  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
  9.2523 +    show "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" apply(rule finite_product_dependent) using insert by auto
  9.2524 +  qed(insert insert, auto) qed auto
  9.2525 +
  9.2526 +lemma has_integral_negligible: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
  9.2527 +  assumes "negligible s" "\<forall>x\<in>(t - s). f x = 0"
  9.2528 +  shows "(f has_integral 0) t"
  9.2529 +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})"
  9.2530 +  let ?f = "(\<lambda>x. if x \<in> t then f x else 0)"
  9.2531 +  show ?thesis apply(rule_tac f="?f" in has_integral_eq) apply(rule) unfolding if_P apply(rule refl)
  9.2532 +    apply(subst has_integral_alt) apply(cases,subst if_P,assumption) unfolding if_not_P
  9.2533 +  proof- assume "\<exists>a b. t = {a..b}" then guess a b apply-by(erule exE)+ note t = this
  9.2534 +    show "(?f has_integral 0) t" unfolding t apply(rule P) using assms(2) unfolding t by auto
  9.2535 +  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)"
  9.2536 +      apply(safe,rule_tac x=1 in exI,rule) apply(rule zero_less_one,safe) apply(rule_tac x=0 in exI)
  9.2537 +      apply(rule,rule P) using assms(2) by auto
  9.2538 +  qed
  9.2539 +next fix f::"real^'n \<Rightarrow> 'a" and a b::"real^'n" assume assm:"\<forall>x. x \<notin> s \<longrightarrow> f x = 0" 
  9.2540 +  show "(f has_integral 0) {a..b}" unfolding has_integral
  9.2541 +  proof(safe) case goal1
  9.2542 +    hence "\<And>n. e / 2 / ((real n+1) * (2 ^ n)) > 0" 
  9.2543 +      apply-apply(rule divide_pos_pos) defer apply(rule mult_pos_pos) by(auto simp add:field_simps)
  9.2544 +    note assms(1)[unfolded negligible_def has_integral,rule_format,OF this,of a b] note allI[OF this,of "\<lambda>x. x"] 
  9.2545 +    from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format]]
  9.2546 +    show ?case apply(rule_tac x="\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x" in exI) 
  9.2547 +    proof safe show "gauge (\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x)" using d(1) unfolding gauge_def by auto
  9.2548 +      fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x) fine p" 
  9.2549 +      let ?goal = "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e"
  9.2550 +      { presume "p\<noteq>{} \<Longrightarrow> ?goal" thus ?goal apply(cases "p={}") using goal1 by auto  }
  9.2551 +      assume as':"p \<noteq> {}" from real_arch_simple[of "Sup((\<lambda>(x,k). norm(f x)) ` p)"] guess N ..
  9.2552 +      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
  9.2553 +      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)"
  9.2554 +        apply(rule,rule tagged_division_finer[OF as(1) d(1)]) by auto
  9.2555 +      from choice[OF this] guess q .. note q=conjunctD3[OF this[rule_format]]
  9.2556 +      have *:"\<And>i. (\<Sum>(x, k)\<in>q i. content k *\<^sub>R indicator s x) \<ge> 0" apply(rule setsum_nonneg,safe) 
  9.2557 +        unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) apply(drule tagged_division_ofD(4)[OF q(1)]) by auto
  9.2558 +      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"
  9.2559 +      proof- case goal1 thus ?case apply-apply(rule setsum_le_included[of s t g snd f]) prefer 4
  9.2560 +          apply safe apply(erule_tac x=x in ballE) apply(erule exE) apply(rule_tac x="(xa,x)" in bexI) by auto qed
  9.2561 +      have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) \<le> setsum (\<lambda>i. (real i + 1) *
  9.2562 +                     norm(setsum (\<lambda>(x,k). content k *\<^sub>R indicator s x) (q i))) {0..N+1}"
  9.2563 +        unfolding real_norm_def setsum_right_distrib abs_of_nonneg[OF *] diff_0_right
  9.2564 +        apply(rule order_trans,rule setsum_norm) defer apply(subst sum_sum_product) prefer 3 
  9.2565 +      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
  9.2566 +        fix i a b assume as'':"(a,b) \<in> q i" show "0 \<le> (real i + 1) * (content b *\<^sub>R indicator s a)"
  9.2567 +          unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) defer apply(rule mult_nonneg_nonneg)
  9.2568 +          using tagged_division_ofD(4)[OF q(1) as''] by auto
  9.2569 +      next fix i::nat show "finite (q i)" using q by auto
  9.2570 +      next fix x k assume xk:"(x,k) \<in> p" def n \<equiv> "nat \<lfloor>norm (f x)\<rfloor>"
  9.2571 +        have *:"norm (f x) \<in> (\<lambda>(x, k). norm (f x)) ` p" using xk by auto
  9.2572 +        have nfx:"real n \<le> norm(f x)" "norm(f x) \<le> real n + 1" unfolding n_def by auto
  9.2573 +        hence "n \<in> {0..N + 1}" using N[rule_format,OF *] by auto
  9.2574 +        moreover  note as(2)[unfolded fine_def,rule_format,OF xk,unfolded split_conv]
  9.2575 +        note q(3)[rule_format,OF xk,unfolded split_conv,rule_format,OF this] note this[unfolded n_def[symmetric]]
  9.2576 +        moreover have "norm (content k *\<^sub>R f x) \<le> (real n + 1) * (content k * indicator s x)"
  9.2577 +        proof(cases "x\<in>s") case False thus ?thesis using assm by auto
  9.2578 +        next case True have *:"content k \<ge> 0" using tagged_division_ofD(4)[OF as(1) xk] by auto
  9.2579 +          moreover have "content k * norm (f x) \<le> content k * (real n + 1)" apply(rule mult_mono) using nfx * by auto
  9.2580 +          ultimately show ?thesis unfolding abs_mult using nfx True by(auto simp add:field_simps)
  9.2581 +        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)"
  9.2582 +          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
  9.2583 +      qed(insert as, auto)
  9.2584 +      also have "... \<le> setsum (\<lambda>i. e / 2 / 2 ^ i) {0..N+1}" apply(rule setsum_mono) 
  9.2585 +      proof- case goal1 thus ?case apply(subst mult_commute, subst pos_le_divide_eq[THEN sym])
  9.2586 +          using d(2)[rule_format,of "q i" i] using q[rule_format] by(auto simp add:field_simps)
  9.2587 +      qed also have "... < e * inverse 2 * 2" unfolding real_divide_def setsum_right_distrib[THEN sym]
  9.2588 +        apply(rule mult_strict_left_mono) unfolding power_inverse atLeastLessThanSuc_atLeastAtMost[THEN sym]
  9.2589 +        apply(subst sumr_geometric) using goal1 by auto
  9.2590 +      finally show "?goal" by auto qed qed qed
  9.2591 +
  9.2592 +lemma has_integral_spike: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
  9.2593 +  assumes "negligible s" "(\<forall>x\<in>(t - s). g x = f x)" "(f has_integral y) t"
  9.2594 +  shows "(g has_integral y) t"
  9.2595 +proof- { fix a b::"real^'n" and f g ::"real^'n \<Rightarrow> 'a" and y::'a
  9.2596 +    assume as:"\<forall>x \<in> {a..b} - s. g x = f x" "(f has_integral y) {a..b}"
  9.2597 +    have "((\<lambda>x. f x + (g x - f x)) has_integral (y + 0)) {a..b}" apply(rule has_integral_add[OF as(2)])
  9.2598 +      apply(rule has_integral_negligible[OF assms(1)]) using as by auto
  9.2599 +    hence "(g has_integral y) {a..b}" by auto } note * = this
  9.2600 +  show ?thesis apply(subst has_integral_alt) using assms(2-) apply-apply(rule cond_cases,safe)
  9.2601 +    apply(rule *, assumption+) apply(subst(asm) has_integral_alt) unfolding if_not_P
  9.2602 +    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)
  9.2603 +    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
  9.2604 +
  9.2605 +lemma has_integral_spike_eq:
  9.2606 +  assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x"
  9.2607 +  shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
  9.2608 +  apply rule apply(rule_tac[!] has_integral_spike[OF assms(1)]) using assms(2) by auto
  9.2609 +
  9.2610 +lemma integrable_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x" "f integrable_on t"
  9.2611 +  shows "g integrable_on  t"
  9.2612 +  using assms unfolding integrable_on_def apply-apply(erule exE)
  9.2613 +  apply(rule,rule has_integral_spike) by fastsimp+
  9.2614 +
  9.2615 +lemma integral_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x"
  9.2616 +  shows "integral t f = integral t g"
  9.2617 +  unfolding integral_def using has_integral_spike_eq[OF assms] by auto
  9.2618 +
  9.2619 +subsection {* Some other trivialities about negligible sets. *}
  9.2620 +
  9.2621 +lemma negligible_subset[intro]: assumes "negligible s" "t \<subseteq> s" shows "negligible t" unfolding negligible_def 
  9.2622 +proof(safe) case goal1 show ?case using assms(1)[unfolded negligible_def,rule_format,of a b]
  9.2623 +    apply-apply(rule has_integral_spike[OF assms(1)]) defer apply assumption
  9.2624 +    using assms(2) unfolding indicator_def by auto qed
  9.2625 +
  9.2626 +lemma negligible_diff[intro?]: assumes "negligible s" shows "negligible(s - t)" using assms by auto
  9.2627 +
  9.2628 +lemma negligible_inter: assumes "negligible s \<or> negligible t" shows "negligible(s \<inter> t)" using assms by auto
  9.2629 +
  9.2630 +lemma negligible_union: assumes "negligible s" "negligible t" shows "negligible (s \<union> t)" unfolding negligible_def 
  9.2631 +proof safe case goal1 note assm = assms[unfolded negligible_def,rule_format,of a b]
  9.2632 +  thus ?case apply(subst has_integral_spike_eq[OF assms(2)])
  9.2633 +    defer apply assumption unfolding indicator_def by auto qed
  9.2634 +
  9.2635 +lemma negligible_union_eq[simp]: "negligible (s \<union> t) \<longleftrightarrow> (negligible s \<and> negligible t)"
  9.2636 +  using negligible_union by auto
  9.2637 +
  9.2638 +lemma negligible_sing[intro]: "negligible {a::real^'n}" 
  9.2639 +proof- guess x using UNIV_witness[where 'a='n] ..
  9.2640 +  show ?thesis using negligible_standard_hyperplane[of x "a$x"] by auto qed
  9.2641 +
  9.2642 +lemma negligible_insert[simp]: "negligible(insert a s) \<longleftrightarrow> negligible s"
  9.2643 +  apply(subst insert_is_Un) unfolding negligible_union_eq by auto
  9.2644 +
  9.2645 +lemma negligible_empty[intro]: "negligible {}" by auto
  9.2646 +
  9.2647 +lemma negligible_finite[intro]: assumes "finite s" shows "negligible s"
  9.2648 +  using assms apply(induct s) by auto
  9.2649 +
  9.2650 +lemma negligible_unions[intro]: assumes "finite s" "\<forall>t\<in>s. negligible t" shows "negligible(\<Union>s)"
  9.2651 +  using assms by(induct,auto) 
  9.2652 +
  9.2653 +lemma negligible:  "negligible s \<longleftrightarrow> (\<forall>t::(real^'n) set. (indicator s has_integral 0) t)"
  9.2654 +  apply safe defer apply(subst negligible_def)
  9.2655 +proof- fix t::"(real^'n) set" assume as:"negligible s"
  9.2656 +  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)  
  9.2657 +  show "(indicator s has_integral 0) t" apply(subst has_integral_alt)
  9.2658 +    apply(cases,subst if_P,assumption) unfolding if_not_P apply(safe,rule as[unfolded negligible_def,rule_format])
  9.2659 +    apply(rule_tac x=1 in exI) apply(safe,rule zero_less_one) apply(rule_tac x=0 in exI)
  9.2660 +    using negligible_subset[OF as,of "s \<inter> t"] unfolding negligible_def indicator_def unfolding * by auto qed auto
  9.2661 +
  9.2662 +subsection {* Finite case of the spike theorem is quite commonly needed. *}
  9.2663 +
  9.2664 +lemma has_integral_spike_finite: assumes "finite s" "\<forall>x\<in>t-s. g x = f x" 
  9.2665 +  "(f has_integral y) t" shows "(g has_integral y) t"
  9.2666 +  apply(rule has_integral_spike) using assms by auto
  9.2667 +
  9.2668 +lemma has_integral_spike_finite_eq: assumes "finite s" "\<forall>x\<in>t-s. g x = f x"
  9.2669 +  shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
  9.2670 +  apply rule apply(rule_tac[!] has_integral_spike_finite) using assms by auto
  9.2671 +
  9.2672 +lemma integrable_spike_finite:
  9.2673 +  assumes "finite s" "\<forall>x\<in>t-s. g x = f x" "f integrable_on t" shows "g integrable_on  t"
  9.2674 +  using assms unfolding integrable_on_def apply safe apply(rule_tac x=y in exI)
  9.2675 +  apply(rule has_integral_spike_finite) by auto
  9.2676 +
  9.2677 +subsection {* In particular, the boundary of an interval is negligible. *}
  9.2678 +
  9.2679 +lemma negligible_frontier_interval: "negligible({a..b} - {a<..<b})"
  9.2680 +proof- let ?A = "\<Union>((\<lambda>k. {x. x$k = a$k} \<union> {x. x$k = b$k}) ` UNIV)"
  9.2681 +  have "{a..b} - {a<..<b} \<subseteq> ?A" apply rule unfolding Diff_iff mem_interval not_all
  9.2682 +    apply(erule conjE exE)+ apply(rule_tac X="{x. x $ xa = a $ xa} \<union> {x. x $ xa = b $ xa}" in UnionI)
  9.2683 +    apply(erule_tac[!] x=xa in allE) by auto
  9.2684 +  thus ?thesis apply-apply(rule negligible_subset[of ?A]) apply(rule negligible_unions[OF finite_imageI]) by auto qed
  9.2685 +
  9.2686 +lemma has_integral_spike_interior:
  9.2687 +  assumes "\<forall>x\<in>{a<..<b}. g x = f x" "(f has_integral y) ({a..b})" shows "(g has_integral y) ({a..b})"
  9.2688 +  apply(rule has_integral_spike[OF negligible_frontier_interval _ assms(2)]) using assms(1) by auto
  9.2689 +
  9.2690 +lemma has_integral_spike_interior_eq:
  9.2691 +  assumes "\<forall>x\<in>{a<..<b}. g x = f x" shows "((f has_integral y) ({a..b}) \<longleftrightarrow> (g has_integral y) ({a..b}))"
  9.2692 +  apply rule apply(rule_tac[!] has_integral_spike_interior) using assms by auto
  9.2693 +
  9.2694 +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}"
  9.2695 +  using  assms unfolding integrable_on_def using has_integral_spike_interior[OF assms(1)] by auto
  9.2696 +
  9.2697 +subsection {* Integrability of continuous functions. *}
  9.2698 +
  9.2699 +lemma neutral_and[simp]: "neutral op \<and> = True"
  9.2700 +  unfolding neutral_def apply(rule some_equality) by auto
  9.2701 +
  9.2702 +lemma monoidal_and[intro]: "monoidal op \<and>" unfolding monoidal_def by auto
  9.2703 +
  9.2704 +lemma iterate_and[simp]: assumes "finite s" shows "(iterate op \<and>) s p \<longleftrightarrow> (\<forall>x\<in>s. p x)" using assms
  9.2705 +apply induct unfolding iterate_insert[OF monoidal_and] by auto
  9.2706 +
  9.2707 +lemma operative_division_and: assumes "operative op \<and> P" "d division_of {a..b}"
  9.2708 +  shows "(\<forall>i\<in>d. P i) \<longleftrightarrow> P {a..b}"
  9.2709 +  using operative_division[OF monoidal_and assms] division_of_finite[OF assms(2)] by auto
  9.2710 +
  9.2711 +lemma operative_approximable: assumes "0 \<le> e" fixes f::"real^'n \<Rightarrow> 'a::banach"
  9.2712 +  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
  9.2713 +proof safe fix a b::"real^'n" { assume "content {a..b} = 0"
  9.2714 +    thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" 
  9.2715 +      apply(rule_tac x=f in exI) using assms by(auto intro!:integrable_on_null) }
  9.2716 +  { fix c k g assume as:"\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}"
  9.2717 +    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}"
  9.2718 +      "\<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}"
  9.2719 +      apply(rule_tac[!] x=g in exI) using as(1) integrable_split[OF as(2)] by auto }
  9.2720 +  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}"
  9.2721 +                          "\<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}"
  9.2722 +  let ?g = "\<lambda>x. if x$k = c then f x else if x$k \<le> c then g1 x else g2 x"
  9.2723 +  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)
  9.2724 +  proof safe case goal1 thus ?case apply- apply(cases "x$k=c", case_tac "x$k < c") using as assms by auto
  9.2725 +  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}"
  9.2726 +    then guess h1 h2 unfolding integrable_on_def by auto from has_integral_split[OF this]
  9.2727 +    show ?case unfolding integrable_on_def by auto
  9.2728 +  next show "?g integrable_on {a..b} \<inter> {x. x $ k \<le> c}" "?g integrable_on {a..b} \<inter> {x. x $ k \<ge> c}"
  9.2729 +      apply(rule_tac[!] integrable_spike[OF negligible_standard_hyperplane[of k c]]) using as(2,4) by auto qed qed
  9.2730 +
  9.2731 +lemma approximable_on_division: fixes f::"real^'n \<Rightarrow> 'a::banach"
  9.2732 +  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"
  9.2733 +  obtains g where "\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}"
  9.2734 +proof- note * = operative_division[OF monoidal_and operative_approximable[OF assms(1)] assms(2)]
  9.2735 +  note this[unfolded iterate_and[OF division_of_finite[OF assms(2)]]] from assms(3)[unfolded this[of f]]
  9.2736 +  guess g .. thus thesis apply-apply(rule that[of g]) by auto qed
  9.2737 +
  9.2738 +lemma integrable_continuous: fixes f::"real^'n \<Rightarrow> 'a::banach"
  9.2739 +  assumes "continuous_on {a..b} f" shows "f integrable_on {a..b}"
  9.2740 +proof(rule integrable_uniform_limit,safe) fix e::real assume e:"0 < e"
  9.2741 +  from compact_uniformly_continuous[OF assms compact_interval,unfolded uniformly_continuous_on_def,rule_format,OF e] guess d ..
  9.2742 +  note d=conjunctD2[OF this,rule_format]
  9.2743 +  from fine_division_exists[OF gauge_ball[OF d(1)], of a b] guess p . note p=this
  9.2744 +  note p' = tagged_division_ofD[OF p(1)]
  9.2745 +  have *:"\<forall>i\<in>snd ` p. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
  9.2746 +  proof(safe,unfold snd_conv) fix x l assume as:"(x,l) \<in> p" 
  9.2747 +    from p'(4)[OF this] guess a b apply-by(erule exE)+ note l=this
  9.2748 +    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)
  9.2749 +    proof safe show "(\<lambda>y. f x) integrable_on l" unfolding integrable_on_def l by(rule,rule has_integral_const)
  9.2750 +      fix y assume y:"y\<in>l" note fineD[OF p(2) as,unfolded subset_eq,rule_format,OF this]
  9.2751 +      note d(2)[OF _ _ this[unfolded mem_ball]]
  9.2752 +      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
  9.2753 +  from e have "0 \<le> e" by auto from approximable_on_division[OF this division_of_tagged_division[OF p(1)] *] guess g .
  9.2754 +  thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" by auto qed 
  9.2755 +
  9.2756 +subsection {* Specialization of additivity to one dimension. *}
  9.2757 +
  9.2758 +lemma operative_1_lt: assumes "monoidal opp"
  9.2759 +  shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real^1} = neutral opp) \<and>
  9.2760 +                (\<forall>a b c. a < c \<and> c < b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))"
  9.2761 +  unfolding operative_def content_eq_0_1 forall_1 vector_le_def vector_less_def
  9.2762 +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"
  9.2763 +    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
  9.2764 +    thus "opp (f {a..c}) (f {c..b}) = f {a..b}" unfolding as(1)[rule_format,of a b "c$1"] by auto
  9.2765 +next fix a b::"real^1" and c::real
  9.2766 +  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}"
  9.2767 +  show "f {a..b} = opp (f ({a..b} \<inter> {x. x $ 1 \<le> c})) (f ({a..b} \<inter> {x. c \<le> x $ 1}))"
  9.2768 +  proof(cases "c \<in> {a$1 .. b$1}")
  9.2769 +    case False hence "c<a$1 \<or> c>b$1" by auto
  9.2770 +    thus ?thesis apply-apply(erule disjE)
  9.2771 +    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
  9.2772 +      show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto
  9.2773 +    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
  9.2774 +      show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto
  9.2775 +    qed
  9.2776 +  next case True hence *:"min (b $ 1) c = c" "max (a $ 1) c = c" by auto
  9.2777 +    show ?thesis unfolding interval_split num1_eq_iff if_True * vec_def[THEN sym]
  9.2778 +    proof(cases "c = a$1 \<or> c = b$1")
  9.2779 +      case False thus "f {a..b} = opp (f {a..vec1 c}) (f {vec1 c..b})"
  9.2780 +        apply-apply(subst as(2)[rule_format]) using True by auto
  9.2781 +    next case True thus "f {a..b} = opp (f {a..vec1 c}) (f {vec1 c..b})" apply-
  9.2782 +      proof(erule disjE) assume "c=a$1" hence *:"a = vec1 c" unfolding Cart_eq by auto 
  9.2783 +        hence "f {a..vec1 c} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
  9.2784 +        thus ?thesis using assms unfolding * by auto
  9.2785 +      next assume "c=b$1" hence *:"b = vec1 c" unfolding Cart_eq by auto 
  9.2786 +        hence "f {vec1 c..b} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
  9.2787 +        thus ?thesis using assms unfolding * by auto qed qed qed qed
  9.2788 +
  9.2789 +lemma operative_1_le: assumes "monoidal opp"
  9.2790 +  shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real^1} = neutral opp) \<and>
  9.2791 +                (\<forall>a b c. a \<le> c \<and> c \<le> b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))"
  9.2792 +unfolding operative_1_lt[OF assms]
  9.2793 +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"
  9.2794 +  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
  9.2795 +next fix a b c ::"real^1"
  9.2796 +  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"
  9.2797 +  note as = this[rule_format]
  9.2798 +  show "opp (f {a..c}) (f {c..b}) = f {a..b}"
  9.2799 +  proof(cases "c = a \<or> c = b")
  9.2800 +    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)
  9.2801 +    next case True thus ?thesis apply-
  9.2802 +      proof(erule disjE) assume *:"c=a" hence "f {a..c} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
  9.2803 +        thus ?thesis using assms unfolding * by auto
  9.2804 +      next               assume *:"c=b" hence "f {c..b} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
  9.2805 +        thus ?thesis using assms unfolding * by auto qed qed qed 
  9.2806 +
  9.2807 +subsection {* Special case of additivity we need for the FCT. *}
  9.2808 +
  9.2809 +lemma additive_tagged_division_1: fixes f::"real^1 \<Rightarrow> 'a::real_normed_vector"
  9.2810 +  assumes "dest_vec1 a \<le> dest_vec1 b" "p tagged_division_of {a..b}"
  9.2811 +  shows "setsum (\<lambda>(x,k). f(interval_upperbound k) - f(interval_lowerbound k)) p = f b - f a"
  9.2812 +proof- let ?f = "(\<lambda>k::(real^1) set. if k = {} then 0 else f(interval_upperbound k) - f(interval_lowerbound k))"
  9.2813 +  have *:"operative op + ?f" unfolding operative_1_lt[OF monoidal_monoid] interval_eq_empty_1
  9.2814 +    by(auto simp add:not_less interval_bound_1 vector_less_def)
  9.2815 +  have **:"{a..b} \<noteq> {}" using assms(1) by auto note operative_tagged_division[OF monoidal_monoid * assms(2)]
  9.2816 +  note * = this[unfolded if_not_P[OF **] interval_bound_1[OF assms(1)],THEN sym ]
  9.2817 +  show ?thesis unfolding * apply(subst setsum_iterate[THEN sym]) defer
  9.2818 +    apply(rule setsum_cong2) unfolding split_paired_all split_conv using assms(2) by auto qed
  9.2819 +
  9.2820 +subsection {* A useful lemma allowing us to factor out the content size. *}
  9.2821 +
  9.2822 +lemma has_integral_factor_content:
  9.2823 +  "(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
  9.2824 +    \<longrightarrow> norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content {a..b}))"
  9.2825 +proof(cases "content {a..b} = 0")
  9.2826 +  case True show ?thesis unfolding has_integral_null_eq[OF True] apply safe
  9.2827 +    apply(rule,rule,rule gauge_trivial,safe) unfolding setsum_content_null[OF True] True defer 
  9.2828 +    apply(erule_tac x=1 in allE,safe) defer apply(rule fine_division_exists[of _ a b],assumption)
  9.2829 +    apply(erule_tac x=p in allE) unfolding setsum_content_null[OF True] by auto
  9.2830 +next case False note F = this[unfolded content_lt_nz[THEN sym]]
  9.2831 +  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)"
  9.2832 +  show ?thesis apply(subst has_integral)
  9.2833 +  proof safe fix e::real assume e:"e>0"
  9.2834 +    { 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)
  9.2835 +        apply(erule impE) defer apply(erule exE,rule_tac x=d in exI)
  9.2836 +        using F e by(auto simp add:field_simps intro:mult_pos_pos) }
  9.2837 +    {  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)
  9.2838 +        apply(erule impE) defer apply(erule exE,rule_tac x=d in exI)
  9.2839 +        using F e by(auto simp add:field_simps intro:mult_pos_pos) } qed qed
  9.2840 +
  9.2841 +subsection {* Fundamental theorem of calculus. *}
  9.2842 +
  9.2843 +lemma fundamental_theorem_of_calculus: fixes f::"real^1 \<Rightarrow> 'a::banach"
  9.2844 +  assumes "a \<le> b"  "\<forall>x\<in>{a..b}. ((f o vec1) has_vector_derivative f'(vec1 x)) (at x within {a..b})"
  9.2845 +  shows "(f' has_integral (f(vec1 b) - f(vec1 a))) ({vec1 a..vec1 b})"
  9.2846 +unfolding has_integral_factor_content
  9.2847 +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
  9.2848 +  note assm = assms(2)[unfolded has_vector_derivative_def has_derivative_within_alt]
  9.2849 +  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
  9.2850 +  note this[OF assm,unfolded gauge_existence_lemma] from choice[OF this,unfolded Ball_def[symmetric]]
  9.2851 +  guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
  9.2852 +  show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {vec1 a..vec1 b} \<and> d fine p \<longrightarrow>
  9.2853 +                 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})"
  9.2854 +    apply(rule_tac x="\<lambda>x. ball x (d (dest_vec1 x))" in exI,safe)
  9.2855 +    apply(rule gauge_ball_dependent,rule,rule d(1))
  9.2856 +  proof- fix p assume as:"p tagged_division_of {vec1 a..vec1 b}" "(\<lambda>x. ball x (d (dest_vec1 x))) fine p"
  9.2857 +    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}" 
  9.2858 +      unfolding content_1[OF ab] additive_tagged_division_1[OF ab as(1),of f,THEN sym]
  9.2859 +      unfolding vector_minus_component[THEN sym] additive_tagged_division_1[OF ab as(1),of "\<lambda>x. x",THEN sym]
  9.2860 +      apply(subst dest_vec1_setsum) unfolding setsum_right_distrib defer unfolding setsum_subtractf[THEN sym] 
  9.2861 +    proof(rule setsum_norm_le,safe) fix x k assume "(x,k)\<in>p"
  9.2862 +      note xk = tagged_division_ofD(2-4)[OF as(1) this] from this(3) guess u v apply-by(erule exE)+ note k=this
  9.2863 +      have *:"dest_vec1 u \<le> dest_vec1 v" using xk unfolding k by auto
  9.2864 +      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] .
  9.2865 +      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)"
  9.2866 +        apply(rule order_trans[OF _ norm_triangle_ineq4]) apply(rule eq_refl) apply(rule arg_cong[where f=norm])
  9.2867 +        unfolding scaleR.diff_left by(auto simp add:group_simps)
  9.2868 +      also have "... \<le> e * norm (dest_vec1 u - dest_vec1 x) + e * norm (dest_vec1 v - dest_vec1 x)"
  9.2869 +        apply(rule add_mono) apply(rule d(2)[of "x$1" "u$1",unfolded o_def vec1_dest_vec1]) prefer 4
  9.2870 +        apply(rule d(2)[of "x$1" "v$1",unfolded o_def vec1_dest_vec1])
  9.2871 +        using ball[rule_format,of u] ball[rule_format,of v] 
  9.2872 +        using xk(1-2) unfolding k subset_eq by(auto simp add:vector_dist_norm norm_real)
  9.2873 +      also have "... \<le> e * dest_vec1 (interval_upperbound k - interval_lowerbound k)"
  9.2874 +        unfolding k interval_bound_1[OF *] using xk(1) unfolding k by(auto simp add:vector_dist_norm norm_real field_simps)
  9.2875 +      finally show "norm (content k *\<^sub>R f' x - (f (interval_upperbound k) - f (interval_lowerbound k))) \<le>
  9.2876 +        e * dest_vec1 (interval_upperbound k - interval_lowerbound k)" unfolding k interval_bound_1[OF *] content_1[OF *] .
  9.2877 +    qed(insert as, auto) qed qed
  9.2878 +
  9.2879 +subsection {* Attempt a systematic general set of "offset" results for components. *}
  9.2880 +
  9.2881 +lemma gauge_modify:
  9.2882 +  assumes "(\<forall>s. open s \<longrightarrow> open {x. f(x) \<in> s})" "gauge d"
  9.2883 +  shows "gauge (\<lambda>x y. d (f x) (f y))"
  9.2884 +  using assms unfolding gauge_def apply safe defer apply(erule_tac x="f x" in allE)
  9.2885 +  apply(erule_tac x="d (f x)" in allE) unfolding mem_def Collect_def by auto
  9.2886 +
  9.2887 +subsection {* Only need trivial subintervals if the interval itself is trivial. *}
  9.2888 +
  9.2889 +lemma division_of_nontrivial: fixes s::"(real^'n) set set"
  9.2890 +  assumes "s division_of {a..b}" "content({a..b}) \<noteq> 0"
  9.2891 +  shows "{k. k \<in> s \<and> content k \<noteq> 0} division_of {a..b}" using assms(1) apply-
  9.2892 +proof(induct "card s" arbitrary:s rule:nat_less_induct)
  9.2893 +  fix s::"(real^'n) set set" assume assm:"s division_of {a..b}"
  9.2894 +    "\<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}" 
  9.2895 +  note s = division_ofD[OF assm(1)] let ?thesis = "{k \<in> s. content k \<noteq> 0} division_of {a..b}"
  9.2896 +  { presume *:"{k \<in> s. content k \<noteq> 0} \<noteq> s \<Longrightarrow> ?thesis"
  9.2897 +    show ?thesis apply cases defer apply(rule *,assumption) using assm(1) by auto }
  9.2898 +  assume noteq:"{k \<in> s. content k \<noteq> 0} \<noteq> s"
  9.2899 +  then obtain k where k:"k\<in>s" "content k = 0" by auto
  9.2900 +  from s(4)[OF k(1)] guess c d apply-by(erule exE)+ note k=k this
  9.2901 +  from k have "card s > 0" unfolding card_gt_0_iff using assm(1) by auto
  9.2902 +  hence card:"card (s - {k}) < card s" using assm(1) k(1) apply(subst card_Diff_singleton_if) by auto
  9.2903 +  have *:"closed (\<Union>(s - {k}))" apply(rule closed_Union) defer apply rule apply(drule DiffD1,drule s(4))
  9.2904 +    apply safe apply(rule closed_interval) using assm(1) by auto
  9.2905 +  have "k \<subseteq> \<Union>(s - {k})" apply safe apply(rule *[unfolded closed_limpt,rule_format]) unfolding islimpt_approachable
  9.2906 +  proof safe fix x and e::real assume as:"x\<in>k" "e>0"
  9.2907 +    from k(2)[unfolded k content_eq_0] guess i .. 
  9.2908 +    hence i:"c$i = d$i" using s(3)[OF k(1),unfolded k] unfolding interval_ne_empty by smt
  9.2909 +    hence xi:"x$i = d$i" using as unfolding k mem_interval by smt
  9.2910 +    def y \<equiv> "(\<chi> j. if j = i then if c$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$j)"
  9.2911 +    show "\<exists>x'\<in>\<Union>(s - {k}). x' \<noteq> x \<and> dist x' x < e" apply(rule_tac x=y in bexI) 
  9.2912 +    proof have "d \<in> {c..d}" using s(3)[OF k(1)] unfolding k interval_eq_empty mem_interval by(fastsimp simp add: not_less)
  9.2913 +      hence "d \<in> {a..b}" using s(2)[OF k(1)] unfolding k by auto note di = this[unfolded mem_interval,THEN spec[where x=i]]
  9.2914 +      hence xyi:"y$i \<noteq> x$i" unfolding y_def unfolding i xi Cart_lambda_beta if_P[OF refl]
  9.2915 +        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+ 
  9.2916 +      thus "y \<noteq> x" unfolding Cart_eq by auto
  9.2917 +      have *:"UNIV = insert i (UNIV - {i})" by auto
  9.2918 +      have "norm (y - x) < e + setsum (\<lambda>i. 0) (UNIV::'n set)" apply(rule le_less_trans[OF norm_le_l1])
  9.2919 +        apply(subst *,subst setsum_insert) prefer 3 apply(rule add_less_le_mono)
  9.2920 +      proof- show "\<bar>(y - x) $ i\<bar> < e" unfolding y_def Cart_lambda_beta vector_minus_component if_P[OF refl]
  9.2921 +          apply(cases) apply(subst if_P,assumption) unfolding if_not_P unfolding i xi using di as(2) by auto
  9.2922 +        show "(\<Sum>i\<in>UNIV - {i}. \<bar>(y - x) $ i\<bar>) \<le> (\<Sum>i\<in>UNIV. 0)" unfolding y_def by auto 
  9.2923 +      qed auto thus "dist y x < e" unfolding vector_dist_norm by auto
  9.2924 +      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
  9.2925 +      moreover have "y \<in> \<Union>s" unfolding s mem_interval
  9.2926 +      proof note simps = y_def Cart_lambda_beta if_not_P
  9.2927 +        fix j::'n show "a $ j \<le> y $ j \<and> y $ j \<le> b $ j" 
  9.2928 +        proof(cases "j = i") case False have "x \<in> {a..b}" using s(2)[OF k(1)] as(1) by auto
  9.2929 +          thus ?thesis unfolding simps if_not_P[OF False] unfolding mem_interval by auto
  9.2930 +        next case True note T = this show ?thesis
  9.2931 +          proof(cases "c $ i \<le> (a $ i + b $ i) / 2")
  9.2932 +            case True show ?thesis unfolding simps if_P[OF T] if_P[OF True] unfolding i
  9.2933 +              using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps) 
  9.2934 +          next case False thus ?thesis unfolding simps if_P[OF T] if_not_P[OF False] unfolding i
  9.2935 +              using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps)
  9.2936 +          qed qed qed
  9.2937 +      ultimately show "y \<in> \<Union>(s - {k})" by auto
  9.2938 +    qed qed hence "\<Union>(s - {k}) = {a..b}" unfolding s(6)[THEN sym] by auto
  9.2939 +  hence  "{ka \<in> s - {k}. content ka \<noteq> 0} division_of {a..b}" apply-apply(rule assm(2)[rule_format,OF card refl])
  9.2940 +    apply(rule division_ofI) defer apply(rule_tac[1-4] s) using assm(1) by auto
  9.2941 +  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
  9.2942 +
  9.2943 +subsection {* Integrabibility on subintervals. *}
  9.2944 +
  9.2945 +lemma operative_integrable: fixes f::"real^'n \<Rightarrow> 'a::banach" shows
  9.2946 +  "operative op \<and> (\<lambda>i. f integrable_on i)"
  9.2947 +  unfolding operative_def neutral_and apply safe apply(subst integrable_on_def)
  9.2948 +  unfolding has_integral_null_eq apply(rule,rule refl) apply(rule,assumption)+
  9.2949 +  unfolding integrable_on_def by(auto intro: has_integral_split)
  9.2950 +
  9.2951 +lemma integrable_subinterval: fixes f::"real^'n \<Rightarrow> 'a::banach" 
  9.2952 +  assumes "f integrable_on {a..b}" "{c..d} \<subseteq> {a..b}" shows "f integrable_on {c..d}" 
  9.2953 +  apply(cases "{c..d} = {}") defer apply(rule partial_division_extend_1[OF assms(2)],assumption)
  9.2954 +  using operative_division_and[OF operative_integrable,THEN sym,of _ _ _ f] assms(1) by auto
  9.2955 +
  9.2956 +subsection {* Combining adjacent intervals in 1 dimension. *}
  9.2957 +
  9.2958 +lemma has_integral_combine: assumes "(a::real^1) \<le> c" "c \<le> b"
  9.2959 +  "(f has_integral i) {a..c}" "(f has_integral (j::'a::banach)) {c..b}"
  9.2960 +  shows "(f has_integral (i + j)) {a..b}"
  9.2961 +proof- note operative_integral[of f, unfolded operative_1_le[OF monoidal_lifted[OF monoidal_monoid]]]
  9.2962 +  note conjunctD2[OF this,rule_format] note * = this(2)[OF conjI[OF assms(1-2)],unfolded if_P[OF assms(3)]]
  9.2963 +  hence "f integrable_on {a..b}" apply- apply(rule ccontr) apply(subst(asm) if_P) defer
  9.2964 +    apply(subst(asm) if_P) using assms(3-) by auto
  9.2965 +  with * show ?thesis apply-apply(subst(asm) if_P) defer apply(subst(asm) if_P) defer apply(subst(asm) if_P)
  9.2966 +    unfolding lifted.simps using assms(3-) by(auto simp add: integrable_on_def integral_unique) qed
  9.2967 +
  9.2968 +lemma integral_combine: fixes f::"real^1 \<Rightarrow> 'a::banach"
  9.2969 +  assumes "a \<le> c" "c \<le> b" "f integrable_on ({a..b})"
  9.2970 +  shows "integral {a..c} f + integral {c..b} f = integral({a..b}) f"
  9.2971 +  apply(rule integral_unique[THEN sym]) apply(rule has_integral_combine[OF assms(1-2)])
  9.2972 +  apply(rule_tac[!] integrable_integral integrable_subinterval[OF assms(3)])+ using assms(1-2) by auto
  9.2973 +
  9.2974 +lemma integrable_combine: fixes f::"real^1 \<Rightarrow> 'a::banach"
  9.2975 +  assumes "a \<le> c" "c \<le> b" "f integrable_on {a..c}" "f integrable_on {c..b}"
  9.2976 +  shows "f integrable_on {a..b}" using assms unfolding integrable_on_def by(fastsimp intro!:has_integral_combine)
  9.2977 +
  9.2978 +subsection {* Reduce integrability to "local" integrability. *}
  9.2979 +
  9.2980 +lemma integrable_on_little_subintervals: fixes f::"real^'n \<Rightarrow> 'a::banach"
  9.2981 +  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}"
  9.2982 +  shows "f integrable_on {a..b}"
  9.2983 +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})"
  9.2984 +    using assms by auto note this[unfolded gauge_existence_lemma] from choice[OF this] guess d .. note d=this[rule_format]
  9.2985 +  guess p apply(rule fine_division_exists[OF gauge_ball_dependent,of d a b]) using d by auto note p=this(1-2)
  9.2986 +  note division_of_tagged_division[OF this(1)] note * = operative_division_and[OF operative_integrable,OF this,THEN sym,of f]
  9.2987 +  show ?thesis unfolding * apply safe unfolding snd_conv
  9.2988 +  proof- fix x k assume "(x,k) \<in> p" note tagged_division_ofD(2-4)[OF p(1) this] fineD[OF p(2) this]
  9.2989 +    thus "f integrable_on k" apply safe apply(rule d[THEN conjunct2,rule_format,of x]) by auto qed qed
  9.2990 +
  9.2991 +subsection {* Second FCT or existence of antiderivative. *}
  9.2992 +
  9.2993 +lemma integrable_const[intro]:"(\<lambda>x. c) integrable_on {a..b}"
  9.2994 +  unfolding integrable_on_def by(rule,rule has_integral_const)
  9.2995 +
  9.2996 +lemma integral_has_vector_derivative: fixes f::"real \<Rightarrow> 'a::banach"
  9.2997 +  assumes "continuous_on {a..b} f" "x \<in> {a..b}"
  9.2998 +  shows "((\<lambda>u. integral {vec a..vec u} (f o dest_vec1)) has_vector_derivative f(x)) (at x within {a..b})"
  9.2999 +  unfolding has_vector_derivative_def has_derivative_within_alt
  9.3000 +apply safe apply(rule scaleR.bounded_linear_left)
  9.3001 +proof- fix e::real assume e:"e>0"
  9.3002 +  note compact_uniformly_continuous[OF assms(1) compact_real_interval,unfolded uniformly_continuous_on_def]
  9.3003 +  from this[rule_format,OF e] guess d apply-by(erule conjE exE)+ note d=this[rule_format]
  9.3004 +  let ?I = "\<lambda>a b. integral {vec1 a..vec1 b} (f \<circ> dest_vec1)"
  9.3005 +  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)"
  9.3006 +  proof(rule,rule,rule d,safe) case goal1 show ?case proof(cases "y < x")
  9.3007 +      case False have "f \<circ> dest_vec1 integrable_on {vec1 a..vec1 y}" apply(rule integrable_subinterval,rule integrable_continuous)
  9.3008 +        apply(rule continuous_on_o_dest_vec1 assms)+  unfolding not_less using assms(2) goal1 by auto
  9.3009 +      hence *:"?I a y - ?I a x = ?I x y" unfolding group_simps apply(subst eq_commute) apply(rule integral_combine)
  9.3010 +        using False unfolding not_less using assms(2) goal1 by auto
  9.3011 +      have **:"norm (y - x) = content {vec1 x..vec1 y}" apply(subst content_1) using False unfolding not_less by auto
  9.3012 +      show ?thesis unfolding ** apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x) o dest_vec1"]) unfolding * unfolding o_def
  9.3013 +        defer apply(rule has_integral_sub) apply(rule integrable_integral)
  9.3014 +        apply(rule integrable_subinterval,rule integrable_continuous) apply(rule continuous_on_o_dest_vec1[unfolded o_def] assms)+
  9.3015 +      proof- show "{vec1 x..vec1 y} \<subseteq> {vec1 a..vec1 b}" using goal1 assms(2) by auto
  9.3016 +        have *:"y - x = norm(y - x)" using False by auto
  9.3017 +        show "((\<lambda>xa. f x) has_integral (y - x) *\<^sub>R f x) {vec1 x..vec1 y}" apply(subst *) unfolding ** by auto
  9.3018 +        show "\<forall>xa\<in>{vec1 x..vec1 y}. norm (f (dest_vec1 xa) - f x) \<le> e" apply safe apply(rule less_imp_le)
  9.3019 +          apply(rule d(2)[unfolded vector_dist_norm]) using assms(2) using goal1 by auto
  9.3020 +      qed(insert e,auto)
  9.3021 +    next case True have "f \<circ> dest_vec1 integrable_on {vec1 a..vec1 x}" apply(rule integrable_subinterval,rule integrable_continuous)
  9.3022 +        apply(rule continuous_on_o_dest_vec1 assms)+  unfolding not_less using assms(2) goal1 by auto
  9.3023 +      hence *:"?I a x - ?I a y = ?I y x" unfolding group_simps apply(subst eq_commute) apply(rule integral_combine)
  9.3024 +        using True using assms(2) goal1 by auto
  9.3025 +      have **:"norm (y - x) = content {vec1 y..vec1 x}" apply(subst content_1) using True unfolding not_less by auto
  9.3026 +      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 
  9.3027 +      show ?thesis apply(subst ***) unfolding norm_minus_cancel **
  9.3028 +        apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x) o dest_vec1"]) unfolding * unfolding o_def
  9.3029 +        defer apply(rule has_integral_sub) apply(subst minus_minus[THEN sym]) unfolding minus_minus
  9.3030 +        apply(rule integrable_integral) apply(rule integrable_subinterval,rule integrable_continuous)
  9.3031 +        apply(rule continuous_on_o_dest_vec1[unfolded o_def] assms)+
  9.3032 +      proof- show "{vec1 y..vec1 x} \<subseteq> {vec1 a..vec1 b}" using goal1 assms(2) by auto
  9.3033 +        have *:"x - y = norm(y - x)" using True by auto
  9.3034 +        show "((\<lambda>xa. f x) has_integral (x - y) *\<^sub>R f x) {vec1 y..vec1 x}" apply(subst *) unfolding ** by auto
  9.3035 +        show "\<forall>xa\<in>{vec1 y..vec1 x}. norm (f (dest_vec1 xa) - f x) \<le> e" apply safe apply(rule less_imp_le)
  9.3036 +          apply(rule d(2)[unfolded vector_dist_norm]) using assms(2) using goal1 by auto
  9.3037 +      qed(insert e,auto) qed qed qed
  9.3038 +
  9.3039 +lemma integral_has_vector_derivative': fixes f::"real^1 \<Rightarrow> 'a::banach"
  9.3040 +  assumes "continuous_on {a..b} f" "x \<in> {a..b}"
  9.3041 +  shows "((\<lambda>u. (integral {a..vec u} f)) has_vector_derivative f x) (at (x$1) within {a$1..b$1})"
  9.3042 +  using integral_has_vector_derivative[OF continuous_on_o_vec1[OF assms(1)], of "x$1"]
  9.3043 +  unfolding o_def vec1_dest_vec1 using assms(2) by auto
  9.3044 +
  9.3045 +lemma antiderivative_continuous: assumes "continuous_on {a..b::real} f"
  9.3046 +  obtains g where "\<forall>x\<in> {a..b}. (g has_vector_derivative (f(x)::_::banach)) (at x within {a..b})"
  9.3047 +  apply(rule that,rule) using integral_has_vector_derivative[OF assms] by auto
  9.3048 +
  9.3049 +subsection {* Combined fundamental theorem of calculus. *}
  9.3050 +
  9.3051 +lemma antiderivative_integral_continuous: fixes f::"real \<Rightarrow> 'a::banach" assumes "continuous_on {a..b} f"
  9.3052 +  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}"
  9.3053 +proof- from antiderivative_continuous[OF assms] guess g . note g=this
  9.3054 +  show ?thesis apply(rule that[of g])
  9.3055 +  proof safe case goal1 have "\<forall>x\<in>{u..v}. (g has_vector_derivative f x) (at x within {u..v})"
  9.3056 +      apply(rule,rule has_vector_derivative_within_subset) apply(rule g[rule_format]) using goal1(1-2) by auto
  9.3057 +    thus ?case using fundamental_theorem_of_calculus[OF goal1(3),of "g o dest_vec1" "f o dest_vec1"]
  9.3058 +      unfolding o_def vec1_dest_vec1 by auto qed qed
  9.3059 +
  9.3060 +subsection {* General "twiddling" for interval-to-interval function image. *}
  9.3061 +
  9.3062 +lemma has_integral_twiddle:
  9.3063 +  assumes "0 < r" "\<forall>x. h(g x) = x" "\<forall>x. g(h x) = x" "\<forall>x. continuous (at x) g"
  9.3064 +  "\<forall>u v. \<exists>w z. g ` {u..v} = {w..z}"
  9.3065 +  "\<forall>u v. \<exists>w z. h ` {u..v} = {w..z}"
  9.3066 +  "\<forall>u v. content(g ` {u..v}) = r * content {u..v}"
  9.3067 +  "(f has_integral i) {a..b}"
  9.3068 +  shows "((\<lambda>x. f(g x)) has_integral (1 / r) *\<^sub>R i) (h ` {a..b})"
  9.3069 +proof- { presume *:"{a..b} \<noteq> {} \<Longrightarrow> ?thesis"
  9.3070 +    show ?thesis apply cases defer apply(rule *,assumption)
  9.3071 +    proof- case goal1 thus ?thesis unfolding goal1 assms(8)[unfolded goal1 has_integral_empty_eq] by auto qed }
  9.3072 +  assume "{a..b} \<noteq> {}" from assms(6)[rule_format,of a b] guess w z apply-by(erule exE)+ note wz=this
  9.3073 +  have inj:"inj g" "inj h" unfolding inj_on_def apply safe apply(rule_tac[!] ccontr)
  9.3074 +    using assms(2) apply(erule_tac x=x in allE) using assms(2) apply(erule_tac x=y in allE) defer
  9.3075 +    using assms(3) apply(erule_tac x=x in allE) using assms(3) apply(erule_tac x=y in allE) by auto
  9.3076 +  show ?thesis unfolding has_integral_def has_integral_compact_interval_def apply(subst if_P) apply(rule,rule,rule wz)
  9.3077 +  proof safe fix e::real assume e:"e>0" hence "e * r > 0" using assms(1) by(rule mult_pos_pos)
  9.3078 +    from assms(8)[unfolded has_integral,rule_format,OF this] guess d apply-by(erule exE conjE)+ note d=this[rule_format]
  9.3079 +    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)
  9.3080 +    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)"
  9.3081 +    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
  9.3082 +      fix p assume as:"p tagged_division_of h ` {a..b}" "d' fine p" note p = tagged_division_ofD[OF as(1)] 
  9.3083 +      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 
  9.3084 +      proof safe show "finite ((\<lambda>(x, k). (g x, g ` k)) ` p)" using as by auto
  9.3085 +        show "d fine (\<lambda>(x, k). (g x, g ` k)) ` p" using as(2) unfolding fine_def d' by auto
  9.3086 +        fix x k assume xk[intro]:"(x,k) \<in> p" show "g x \<in> g ` k" using p(2)[OF xk] by auto
  9.3087 +        show "\<exists>u v. g ` k = {u..v}" using p(4)[OF xk] using assms(5-6) by auto
  9.3088 +        { 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)"]
  9.3089 +            using assms(2)[rule_format,of y] unfolding inj_image_mem_iff[OF inj(2)] by auto }
  9.3090 +        fix x' k' assume xk':"(x',k') \<in> p" fix z assume "z \<in> interior (g ` k)" "z \<in> interior (g ` k')"
  9.3091 +        hence *:"interior (g ` k) \<inter> interior (g ` k') \<noteq> {}" by auto
  9.3092 +        have same:"(x, k) = (x', k')" apply-apply(rule ccontr,drule p(5)[OF xk xk'])
  9.3093 +        proof- assume as:"interior k \<inter> interior k' = {}" from nonempty_witness[OF *] guess z .
  9.3094 +          hence "z \<in> g ` (interior k \<inter> interior k')" using interior_image_subset[OF assms(4) inj(1)]
  9.3095 +            unfolding image_Int[OF inj(1)] by auto thus False using as by blast
  9.3096 +        qed thus "g x = g x'" by auto
  9.3097 +        { fix z assume "z \<in> k"  thus  "g z \<in> g ` k'" using same by auto }
  9.3098 +        { fix z assume "z \<in> k'" thus  "g z \<in> g ` k"  using same by auto }
  9.3099 +      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
  9.3100 +        then guess X unfolding Union_iff .. note X=this from this(1) guess y unfolding mem_Collect_eq ..
  9.3101 +        thus "x \<in> \<Union>{k. \<exists>x. (x, k) \<in> (\<lambda>(x, k). (g x, g ` k)) ` p}" apply-
  9.3102 +          apply(rule_tac X="g ` X" in UnionI) defer apply(rule_tac x="h x" in image_eqI)
  9.3103 +          using X(2) assms(3)[rule_format,of x] by auto
  9.3104 +      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
  9.3105 +       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
  9.3106 +        unfolding setsum_reindex[OF *] apply(subst scaleR_right.setsum) defer apply(rule setsum_cong2) unfolding o_def split_paired_all split_conv
  9.3107 +        apply(drule p(4)) apply safe unfolding assms(7)[rule_format] using p by auto
  9.3108 +      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]
  9.3109 +        unfolding real_scaleR_def using assms(1) by auto finally have *:"?l = ?r" .
  9.3110 +      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
  9.3111 +        using assms(1) by(auto simp add:field_simps) qed qed qed
  9.3112 +
  9.3113 +subsection {* Special case of a basic affine transformation. *}
  9.3114 +
  9.3115 +lemma interval_image_affinity_interval: shows "\<exists>u v. (\<lambda>x. m *\<^sub>R (x::real^'n) + c) ` {a..b} = {u..v}"
  9.3116 +  unfolding image_affinity_interval by auto
  9.3117 +
  9.3118 +lemmas Cart_simps = Cart_nth.add Cart_nth.minus Cart_nth.zero Cart_nth.diff Cart_nth.scaleR real_scaleR_def Cart_lambda_beta
  9.3119 +   Cart_eq vector_le_def vector_less_def
  9.3120 +
  9.3121 +lemma setprod_cong2: assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "setprod f A = setprod g A"
  9.3122 +  apply(rule setprod_cong) using assms by auto
  9.3123 +
  9.3124 +lemma content_image_affinity_interval: 
  9.3125 + "content((\<lambda>x::real^'n. m *\<^sub>R x + c) ` {a..b}) = (abs m) ^ CARD('n) * content {a..b}" (is "?l = ?r")
  9.3126 +proof- { presume *:"{a..b}\<noteq>{} \<Longrightarrow> ?thesis" show ?thesis apply(cases,rule *,assumption)
  9.3127 +      unfolding not_not using content_empty by auto }
  9.3128 +  assume as:"{a..b}\<noteq>{}" show ?thesis proof(cases "m \<ge> 0")
  9.3129 +    case True show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_P[OF True]
  9.3130 +      unfolding content_closed_interval'[OF as] apply(subst content_closed_interval') 
  9.3131 +      defer apply(subst setprod_constant[THEN sym]) apply(rule finite_UNIV) unfolding setprod_timesf[THEN sym]
  9.3132 +      apply(rule setprod_cong2) using True as unfolding interval_ne_empty Cart_simps not_le  
  9.3133 +      by(auto simp add:field_simps intro:mult_left_mono)
  9.3134 +  next case False show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_not_P[OF False]
  9.3135 +      unfolding content_closed_interval'[OF as] apply(subst content_closed_interval') 
  9.3136 +      defer apply(subst setprod_constant[THEN sym]) apply(rule finite_UNIV) unfolding setprod_timesf[THEN sym]
  9.3137 +      apply(rule setprod_cong2) using False as unfolding interval_ne_empty Cart_simps not_le 
  9.3138 +      by(auto simp add:field_simps mult_le_cancel_left_neg) qed qed
  9.3139 +
  9.3140 +lemma has_integral_affinity: assumes "(f has_integral i) {a..b::real^'n}" "m \<noteq> 0"
  9.3141 +  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})"
  9.3142 +  apply(rule has_integral_twiddle,safe) unfolding Cart_eq Cart_simps apply(rule zero_less_power)
  9.3143 +  defer apply(insert assms(2), simp add:field_simps) apply(insert assms(2), simp add:field_simps)
  9.3144 +  apply(rule continuous_intros)+ apply(rule interval_image_affinity_interval)+ apply(rule content_image_affinity_interval) using assms by auto
  9.3145 +
  9.3146 +lemma integrable_affinity: assumes "f integrable_on {a..b}" "m \<noteq> 0"
  9.3147 +  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})"
  9.3148 +  using assms unfolding integrable_on_def apply safe apply(drule has_integral_affinity) by auto
  9.3149 +
  9.3150 +subsection {* Special case of stretching coordinate axes separately. *}
  9.3151 +
  9.3152 +lemma image_stretch_interval:
  9.3153 +  "(\<lambda>x. \<chi> k. m k * x$k) ` {a..b::real^'n} =
  9.3154 +  (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")
  9.3155 +proof(cases "{a..b}={}") case True thus ?thesis unfolding True by auto
  9.3156 +next have *:"\<And>P Q. (\<forall>i. P i) \<and> (\<forall>i. Q i) \<longleftrightarrow> (\<forall>i. P i \<and> Q i)" by auto
  9.3157 +  case False note ab = this[unfolded interval_ne_empty]
  9.3158 +  show ?thesis apply-apply(rule set_ext)
  9.3159 +  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
  9.3160 +    show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" unfolding if_not_P[OF False] 
  9.3161 +      unfolding image_iff mem_interval Bex_def Cart_simps Cart_eq *
  9.3162 +      unfolding lambda_skolem[THEN sym,of "\<lambda> i xa. (a $ i \<le> xa \<and> xa \<le> b $ i) \<and> x $ i = m i * xa"]
  9.3163 +    proof(rule **,rule) fix i::'n show "(\<exists>xa. (a $ i \<le> xa \<and> xa \<le> b $ i) \<and> x $ i = m i * xa) =
  9.3164 +        (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))"
  9.3165 +      proof(cases "m i = 0") case True thus ?thesis using ab by auto
  9.3166 +      next case False hence "0 < m i \<or> 0 > m i" by auto thus ?thesis apply-
  9.3167 +        proof(erule disjE) assume as:"0 < m i" hence *:"min (m i * a $ i) (m i * b $ i) = m i * a $ i"
  9.3168 +            "max (m i * a $ i) (m i * b $ i) = m i * b $ i" using ab unfolding min_def max_def by auto
  9.3169 +          show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x$i" in exI)
  9.3170 +            using as by(auto simp add:field_simps)
  9.3171 +        next assume as:"0 > m i" hence *:"max (m i * a $ i) (m i * b $ i) = m i * a $ i"
  9.3172 +            "min (m i * a $ i) (m i * b $ i) = m i * b $ i" using ab as unfolding min_def max_def 
  9.3173 +            by(auto simp add:field_simps mult_le_cancel_left_neg intro:real_le_antisym)
  9.3174 +          show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x$i" in exI)
  9.3175 +            using as by(auto simp add:field_simps) qed qed qed qed qed 
  9.3176 +
  9.3177 +lemma interval_image_stretch_interval: "\<exists>u v. (\<lambda>x. \<chi> k. m k * x$k) ` {a..b::real^'n} = {u..v}"
  9.3178 +  unfolding image_stretch_interval by auto 
  9.3179 +
  9.3180 +lemma content_image_stretch_interval:
  9.3181 +  "content((\<lambda>x::real^'n. \<chi> k. m k * x$k) ` {a..b}) = abs(setprod m UNIV) * content({a..b})"
  9.3182 +proof(cases "{a..b} = {}") case True thus ?thesis
  9.3183 +    unfolding content_def image_is_empty image_stretch_interval if_P[OF True] by auto
  9.3184 +next case False hence "(\<lambda>x. \<chi> k. m k * x $ k) ` {a..b} \<noteq> {}" by auto
  9.3185 +  thus ?thesis using False unfolding content_def image_stretch_interval apply- unfolding interval_bounds' if_not_P
  9.3186 +    unfolding abs_setprod setprod_timesf[THEN sym] apply(rule setprod_cong2) unfolding Cart_lambda_beta
  9.3187 +  proof- fix i::'n have "(m i < 0 \<or> m i > 0) \<or> m i = 0" by auto
  9.3188 +    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)"
  9.3189 +      apply-apply(erule disjE)+ unfolding min_def max_def using False[unfolded interval_ne_empty,rule_format,of i] 
  9.3190 +      by(auto simp add:field_simps not_le mult_le_cancel_left_neg mult_le_cancel_left_pos) qed qed
  9.3191 +
  9.3192 +lemma has_integral_stretch: assumes "(f has_integral i) {a..b}" "\<forall>k. ~(m k = 0)"
  9.3193 +  shows "((\<lambda>x. f(\<chi> k. m k * x$k)) has_integral
  9.3194 +             ((1/(abs(setprod m UNIV))) *\<^sub>R i)) ((\<lambda>x. \<chi> k. 1/(m k) * x$k) ` {a..b})"
  9.3195 +  apply(rule has_integral_twiddle) unfolding zero_less_abs_iff content_image_stretch_interval
  9.3196 +  unfolding image_stretch_interval empty_as_interval Cart_eq using assms
  9.3197 +proof- show "\<forall>x. continuous (at x) (\<lambda>x. \<chi> k. m k * x $ k)"
  9.3198 +   apply(rule,rule linear_continuous_at) unfolding linear_linear
  9.3199 +   unfolding linear_def Cart_simps Cart_eq by(auto simp add:field_simps) qed auto
  9.3200 +
  9.3201 +lemma integrable_stretch: 
  9.3202 +  assumes "f integrable_on {a..b}" "\<forall>k. ~(m k = 0)"
  9.3203 +  shows "(\<lambda>x. f(\<chi> k. m k * x$k)) integrable_on ((\<lambda>x. \<chi> k. 1/(m k) * x$k) ` {a..b})"
  9.3204 +  using assms unfolding integrable_on_def apply-apply(erule exE) apply(drule has_integral_stretch) by auto
  9.3205 +
  9.3206 +subsection {* even more special cases. *}
  9.3207 +
  9.3208 +lemma uminus_interval_vector[simp]:"uminus ` {a..b} = {-b .. -a::real^'n}"
  9.3209 +  apply(rule set_ext,rule) defer unfolding image_iff
  9.3210 +  apply(rule_tac x="-x" in bexI) by(auto simp add:vector_le_def minus_le_iff le_minus_iff)
  9.3211 +
  9.3212 +lemma has_integral_reflect_lemma[intro]: assumes "(f has_integral i) {a..b}"
  9.3213 +  shows "((\<lambda>x. f(-x)) has_integral i) {-b .. -a}"
  9.3214 +  using has_integral_affinity[OF assms, of "-1" 0] by auto
  9.3215 +
  9.3216 +lemma has_integral_reflect[simp]: "((\<lambda>x. f(-x)) has_integral i) {-b..-a} \<longleftrightarrow> (f has_integral i) ({a..b})"
  9.3217 +  apply rule apply(drule_tac[!] has_integral_reflect_lemma) by auto
  9.3218 +
  9.3219 +lemma integrable_reflect[simp]: "(\<lambda>x. f(-x)) integrable_on {-b..-a} \<longleftrightarrow> f integrable_on {a..b}"
  9.3220 +  unfolding integrable_on_def by auto
  9.3221 +
  9.3222 +lemma integral_reflect[simp]: "integral {-b..-a} (\<lambda>x. f(-x)) = integral ({a..b}) f"
  9.3223 +  unfolding integral_def by auto
  9.3224 +
  9.3225 +subsection {* Stronger form of FCT; quite a tedious proof. *}
  9.3226 +
  9.3227 +(** move this **)
  9.3228 +declare norm_triangle_ineq4[intro] 
  9.3229 +
  9.3230 +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)
  9.3231 +
  9.3232 +lemma additive_tagged_division_1': fixes f::"real \<Rightarrow> 'a::real_normed_vector"
  9.3233 +  assumes "a \<le> b" "p tagged_division_of {vec1 a..vec1 b}"
  9.3234 +  shows "setsum (\<lambda>(x,k). f (dest_vec1 (interval_upperbound k)) - f(dest_vec1 (interval_lowerbound k))) p = f b - f a"
  9.3235 +  using additive_tagged_division_1[OF _ assms(2), of "f o dest_vec1"]
  9.3236 +  unfolding o_def vec1_dest_vec1 using assms(1) by auto
  9.3237 +
  9.3238 +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"
  9.3239 +  unfolding split_def by(rule refl)
  9.3240 +
  9.3241 +lemma norm_triangle_le_sub: "norm x + norm y \<le> e \<Longrightarrow> norm (x - y) \<le> e"
  9.3242 +  apply(subst(asm)(2) norm_minus_cancel[THEN sym])
  9.3243 +  apply(drule norm_triangle_le) by(auto simp add:group_simps)
  9.3244 +
  9.3245 +lemma fundamental_theorem_of_calculus_interior:
  9.3246 +  assumes"a \<le> b" "continuous_on {a..b} f" "\<forall>x\<in>{a<..<b}. (f has_vector_derivative f'(x)) (at x)"
  9.3247 +  shows "((f' o dest_vec1) has_integral (f b - f a)) {vec a..vec b}"
  9.3248 +proof- { presume *:"a < b \<Longrightarrow> ?thesis" 
  9.3249 +    show ?thesis proof(cases,rule *,assumption)
  9.3250 +      assume "\<not> a < b" hence "a = b" using assms(1) by auto
  9.3251 +      hence *:"{vec a .. vec b} = {vec b}" "f b - f a = 0" apply(auto simp add: Cart_simps) by smt
  9.3252 +      show ?thesis unfolding *(2) apply(rule has_integral_null) unfolding content_eq_0_1 using * `a=b` by auto
  9.3253 +    qed } assume ab:"a < b"
  9.3254 +  let ?P = "\<lambda>e. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {vec1 a..vec1 b} \<and> d fine p \<longrightarrow>
  9.3255 +                   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})"
  9.3256 +  { presume "\<And>e. e>0 \<Longrightarrow> ?P e" thus ?thesis unfolding has_integral_factor_content by auto }
  9.3257 +  fix e::real assume e:"e>0"
  9.3258 +  note assms(3)[unfolded has_vector_derivative_def has_derivative_at_alt ball_conj_distrib]
  9.3259 +  note conjunctD2[OF this] note bounded=this(1) and this(2)
  9.3260 +  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)"
  9.3261 +    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]
  9.3262 +  from choice[OF this] guess d .. note conjunctD2[OF this[rule_format]] note d = this[rule_format]
  9.3263 +  have "bounded (f ` {a..b})" apply(rule compact_imp_bounded compact_continuous_image)+ using compact_real_interval assms by auto
  9.3264 +  from this[unfolded bounded_pos] guess B .. note B = this[rule_format]
  9.3265 +
  9.3266 +  have "\<exists>da. 0 < da \<and> (\<forall>c. a \<le> c \<and> {a..c} \<subseteq> {a..b} \<and> {a..c} \<subseteq> ball a da
  9.3267 +    \<longrightarrow> norm(content {vec1 a..vec1 c} *\<^sub>R f' a - (f c - f a)) \<le> (e * (b - a)) / 4)"
  9.3268 +  proof- have "a\<in>{a..b}" using ab by auto
  9.3269 +    note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
  9.3270 +    note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0" using e ab by(auto simp add:field_simps)
  9.3271 +    from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
  9.3272 +    have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' a) \<le> (e * (b - a)) / 8"
  9.3273 +    proof(cases "f' a = 0") case True
  9.3274 +      thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg) 
  9.3275 +    next case False thus ?thesis 
  9.3276 +        apply(rule_tac x="(e * (b - a)) / 8 / norm (f' a)" in exI)
  9.3277 +        using ab e by(auto simp add:field_simps)
  9.3278 +    qed then guess l .. note l = conjunctD2[OF this]
  9.3279 +    show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+)
  9.3280 +    proof- fix c assume as:"a \<le> c" "{a..c} \<subseteq> {a..b}" "{a..c} \<subseteq> ball a (min k l)" 
  9.3281 +      note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval]
  9.3282 +      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)
  9.3283 +      also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8" 
  9.3284 +      proof(rule add_mono) case goal1 have "\<bar>c - a\<bar> \<le> \<bar>l\<bar>" using as' by auto
  9.3285 +        thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto
  9.3286 +      next case goal2 show ?case apply(rule less_imp_le) apply(cases "a = c") defer
  9.3287 +          apply(rule k(2)[unfolded vector_dist_norm]) using as' e ab by(auto simp add:field_simps)
  9.3288 +      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
  9.3289 +    qed qed then guess da .. note da=conjunctD2[OF this,rule_format]
  9.3290 +
  9.3291 +  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"
  9.3292 +  proof- have "b\<in>{a..b}" using ab by auto
  9.3293 +    note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
  9.3294 +    note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0" using e ab by(auto simp add:field_simps)
  9.3295 +    from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
  9.3296 +    have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' b) \<le> (e * (b - a)) / 8"
  9.3297 +    proof(cases "f' b = 0") case True
  9.3298 +      thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg) 
  9.3299 +    next case False thus ?thesis 
  9.3300 +        apply(rule_tac x="(e * (b - a)) / 8 / norm (f' b)" in exI)
  9.3301 +        using ab e by(auto simp add:field_simps)
  9.3302 +    qed then guess l .. note l = conjunctD2[OF this]
  9.3303 +    show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+)
  9.3304 +    proof- fix c assume as:"c \<le> b" "{c..b} \<subseteq> {a..b}" "{c..b} \<subseteq> ball b (min k l)" 
  9.3305 +      note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval]
  9.3306 +      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)
  9.3307 +      also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8" 
  9.3308 +      proof(rule add_mono) case goal1 have "\<bar>c - b\<bar> \<le> \<bar>l\<bar>" using as' by auto
  9.3309 +        thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto
  9.3310 +      next case goal2 show ?case apply(rule less_imp_le) apply(cases "b = c") defer apply(subst norm_minus_commute)
  9.3311 +          apply(rule k(2)[unfolded vector_dist_norm]) using as' e ab by(auto simp add:field_simps)
  9.3312 +      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
  9.3313 +    qed qed then guess db .. note db=conjunctD2[OF this,rule_format]
  9.3314 +
  9.3315 +  let ?d = "(\<lambda>x. ball x (if x=vec1 a then da else if x=vec b then db else d (dest_vec1 x)))"
  9.3316 +  show "?P e" apply(rule_tac x="?d" in exI)
  9.3317 +  proof safe case goal1 show ?case apply(rule gauge_ball_dependent) using ab db(1) da(1) d(1) by auto
  9.3318 +  next case goal2 note as=this let ?A = "{t. fst t \<in> {vec1 a, vec1 b}}" note p = tagged_division_ofD[OF goal2(1)]
  9.3319 +    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
  9.3320 +    note * = additive_tagged_division_1'[OF assms(1) goal2(1), THEN sym]
  9.3321 +    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
  9.3322 +    show ?case unfolding content_1'[OF assms(1)] and *[of "\<lambda>x. x"] *[of f] setsum_subtractf[THEN sym] split_minus
  9.3323 +      unfolding setsum_right_distrib apply(subst(2) pA,subst pA) unfolding setsum_Un_disjoint[OF pA(2-)]
  9.3324 +    proof(rule norm_triangle_le,rule **) 
  9.3325 +      case goal1 show ?case apply(rule order_trans,rule setsum_norm_le) apply(rule pA) defer apply(subst divide.setsum)
  9.3326 +      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"
  9.3327 +          "e * (dest_vec1 (interval_upperbound k) - dest_vec1 (interval_lowerbound k)) / 2
  9.3328 +          < norm (content k *\<^sub>R f' (dest_vec1 x) - (f (dest_vec1 (interval_upperbound k)) - f (dest_vec1 (interval_lowerbound k))))"
  9.3329 +        from p(4)[OF this(1)] guess u v apply-by(erule exE)+ note k=this
  9.3330 +        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]
  9.3331 +        note result = as(2)[unfolded k interval_bounds[OF this(1)] content_1[OF this(2)]]
  9.3332 +
  9.3333 +        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] 
  9.3334 +        have "norm ((v$1 - u$1) *\<^sub>R f' (x$1) - (f (v$1) - f (u$1))) =
  9.3335 +          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)))" 
  9.3336 +          apply(rule arg_cong[of _ _ norm]) unfolding scaleR_left.diff by auto 
  9.3337 +        also have "... \<le> e / 2 * norm (u$1 - x$1) + e / 2 * norm (v$1 - x$1)" apply(rule norm_triangle_le_sub)
  9.3338 +          apply(rule add_mono) apply(rule_tac[!] *) using fineD[OF goal2(2) as(1)] as' unfolding k subset_eq
  9.3339 +          apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE) using uv by(auto simp add:dist_real)
  9.3340 +        also have "... \<le> e / 2 * norm (v$1 - u$1)" using p(2)[OF as(1)] unfolding k by(auto simp add:field_simps)
  9.3341 +        finally have "e * (dest_vec1 v - dest_vec1 u) / 2 < e * (dest_vec1 v - dest_vec1 u) / 2"
  9.3342 +          apply- apply(rule less_le_trans[OF result]) using uv by auto thus False by auto qed
  9.3343 +
  9.3344 +    next have *:"\<And>x s1 s2::real. 0 \<le> s1 \<Longrightarrow> x \<le> (s1 + s2) / 2 \<Longrightarrow> x - s1 \<le> s2 / 2" by auto
  9.3345 +      case goal2 show ?case apply(rule *) apply(rule setsum_nonneg) apply(rule,unfold split_paired_all split_conv)
  9.3346 +        defer unfolding setsum_Un_disjoint[OF pA(2-),THEN sym] pA(1)[THEN sym] unfolding setsum_right_distrib[THEN sym] 
  9.3347 +        apply(subst additive_tagged_division_1[OF _ as(1)]) unfolding vec1_dest_vec1 apply(rule assms)
  9.3348 +      proof- fix x k assume "(x,k) \<in> p \<inter> {t. fst t \<in> {vec1 a, vec1 b}}" note xk=IntD1[OF this]
  9.3349 +        from p(4)[OF this] guess u v apply-by(erule exE)+ note uv=this
  9.3350 +        with p(2)[OF xk] have "{u..v} \<noteq> {}" by auto
  9.3351 +        thus "0 \<le> e * ((interval_upperbound k)$1 - (interval_lowerbound k)$1)"
  9.3352 +          unfolding uv using e by(auto simp add:field_simps)
  9.3353 +      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
  9.3354 +        show "norm (\<Sum>(x, k)\<in>p \<inter> ?A. content k *\<^sub>R (f' \<circ> dest_vec1) x -
  9.3355 +          (f ((interval_upperbound k)$1) - f ((interval_lowerbound k)$1))) \<le> e * (b - a) / 2" 
  9.3356 +          apply(rule *[where t="p \<inter> {t. fst t \<in> {vec1 a, vec1 b} \<and> content(snd t) \<noteq> 0}"])
  9.3357 +          apply(rule setsum_mono_zero_right[OF pA(2)]) defer apply(rule) unfolding split_paired_all split_conv o_def
  9.3358 +        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}"
  9.3359 +          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
  9.3360 +          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
  9.3361 +          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
  9.3362 +        next have *:"p \<inter> {t. fst t \<in> {vec1 a, vec1 b} \<and> content(snd t) \<noteq> 0} = 
  9.3363 +            {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
  9.3364 +          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"
  9.3365 +          proof(case_tac "s={}") case goal2 then obtain x where "x\<in>s" by auto hence *:"s = {x}" using goal2(1) by auto
  9.3366 +            thus ?case using `x\<in>s` goal2(2) by auto
  9.3367 +          qed auto
  9.3368 +          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"]) 
  9.3369 +            apply(rule norm_triangle_le,rule add_mono) apply(rule_tac[1-2] **)
  9.3370 +          proof- let ?B = "\<lambda>x. {t \<in> p. fst t = vec1 x \<and> content (snd t) \<noteq> 0}"
  9.3371 +            have pa:"\<And>k. (vec1 a, k) \<in> p \<Longrightarrow> \<exists>v. k = {vec1 a .. v} \<and> vec1 a \<le> v" 
  9.3372 +            proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this
  9.3373 +              have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto
  9.3374 +              have u:"u = vec1 a" proof(rule ccontr)  have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto 
  9.3375 +                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
  9.3376 +                have "u > vec1 a" unfolding Cart_simps by auto
  9.3377 +                thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:Cart_simps)
  9.3378 +              qed thus ?case apply(rule_tac x=v in exI) unfolding uv using * by auto
  9.3379 +            qed
  9.3380 +            have pb:"\<And>k. (vec1 b, k) \<in> p \<Longrightarrow> \<exists>v. k = {v .. vec1 b} \<and> vec1 b \<ge> v" 
  9.3381 +            proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this
  9.3382 +              have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto
  9.3383 +              have u:"v = vec1 b" proof(rule ccontr)  have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto 
  9.3384 +                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
  9.3385 +                have "v < vec1 b" unfolding Cart_simps by auto
  9.3386 +                thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:Cart_simps)
  9.3387 +              qed thus ?case apply(rule_tac x=u in exI) unfolding uv using * by auto
  9.3388 +            qed
  9.3389 +
  9.3390 +            show "\<forall>x y. x \<in> ?B a \<and> y \<in> ?B a \<longrightarrow> x = y" apply(rule,rule,rule,unfold split_paired_all)
  9.3391 +              unfolding mem_Collect_eq fst_conv snd_conv apply safe