--- a/src/HOL/Complex_Main.thy Mon Feb 22 20:08:10 2010 +0100
+++ b/src/HOL/Complex_Main.thy Mon Feb 22 20:41:49 2010 +0100
@@ -9,7 +9,7 @@
Log
Ln
Taylor
- Integration
+ Deriv
begin
end
--- a/src/HOL/Integration.thy Mon Feb 22 20:08:10 2010 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,658 +0,0 @@
-(* Author: Jacques D. Fleuriot, University of Edinburgh
- Conversion to Isar and new proofs by Lawrence C Paulson, 2004
-*)
-
-header{*Theory of Integration*}
-
-theory Integration
-imports Deriv ATP_Linkup
-begin
-
-text{*We follow John Harrison in formalizing the Gauge integral.*}
-
-subsection {* Gauges *}
-
-definition
- gauge :: "[real set, real => real] => bool" where
- [code del]:"gauge E g = (\<forall>x\<in>E. 0 < g(x))"
-
-
-subsection {* Gauge-fine divisions *}
-
-inductive
- fine :: "[real \<Rightarrow> real, real \<times> real, (real \<times> real \<times> real) list] \<Rightarrow> bool"
-for
- \<delta> :: "real \<Rightarrow> real"
-where
- fine_Nil:
- "fine \<delta> (a, a) []"
-| fine_Cons:
- "\<lbrakk>fine \<delta> (b, c) D; a < b; a \<le> x; x \<le> b; b - a < \<delta> x\<rbrakk>
- \<Longrightarrow> fine \<delta> (a, c) ((a, x, b) # D)"
-
-lemmas fine_induct [induct set: fine] =
- fine.induct [of "\<delta>" "(a,b)" "D" "split P", unfolded split_conv, standard]
-
-lemma fine_single:
- "\<lbrakk>a < b; a \<le> x; x \<le> b; b - a < \<delta> x\<rbrakk> \<Longrightarrow> fine \<delta> (a, b) [(a, x, b)]"
-by (rule fine_Cons [OF fine_Nil])
-
-lemma fine_append:
- "\<lbrakk>fine \<delta> (a, b) D; fine \<delta> (b, c) D'\<rbrakk> \<Longrightarrow> fine \<delta> (a, c) (D @ D')"
-by (induct set: fine, simp, simp add: fine_Cons)
-
-lemma fine_imp_le: "fine \<delta> (a, b) D \<Longrightarrow> a \<le> b"
-by (induct set: fine, simp_all)
-
-lemma nonempty_fine_imp_less: "\<lbrakk>fine \<delta> (a, b) D; D \<noteq> []\<rbrakk> \<Longrightarrow> a < b"
-apply (induct set: fine, simp)
-apply (drule fine_imp_le, simp)
-done
-
-lemma empty_fine_imp_eq: "\<lbrakk>fine \<delta> (a, b) D; D = []\<rbrakk> \<Longrightarrow> a = b"
-by (induct set: fine, simp_all)
-
-lemma fine_eq: "fine \<delta> (a, b) D \<Longrightarrow> a = b \<longleftrightarrow> D = []"
-apply (cases "D = []")
-apply (drule (1) empty_fine_imp_eq, simp)
-apply (drule (1) nonempty_fine_imp_less, simp)
-done
-
-lemma mem_fine:
- "\<lbrakk>fine \<delta> (a, b) D; (u, x, v) \<in> set D\<rbrakk> \<Longrightarrow> u < v \<and> u \<le> x \<and> x \<le> v"
-by (induct set: fine, simp, force)
-
-lemma mem_fine2: "\<lbrakk>fine \<delta> (a, b) D; (u, z, v) \<in> set D\<rbrakk> \<Longrightarrow> a \<le> u \<and> v \<le> b"
-apply (induct arbitrary: z u v set: fine, auto)
-apply (simp add: fine_imp_le)
-apply (erule order_trans [OF less_imp_le], simp)
-done
-
-lemma mem_fine3: "\<lbrakk>fine \<delta> (a, b) D; (u, z, v) \<in> set D\<rbrakk> \<Longrightarrow> v - u < \<delta> z"
-by (induct arbitrary: z u v set: fine) auto
-
-lemma BOLZANO:
- fixes P :: "real \<Rightarrow> real \<Rightarrow> bool"
- assumes 1: "a \<le> b"
- assumes 2: "\<And>a b c. \<lbrakk>P a b; P b c; a \<le> b; b \<le> c\<rbrakk> \<Longrightarrow> P a c"
- assumes 3: "\<And>x. \<exists>d>0. \<forall>a b. a \<le> x & x \<le> b & (b-a) < d \<longrightarrow> P a b"
- shows "P a b"
-apply (subgoal_tac "split P (a,b)", simp)
-apply (rule lemma_BOLZANO [OF _ _ 1])
-apply (clarify, erule (3) 2)
-apply (clarify, rule 3)
-done
-
-text{*We can always find a division that is fine wrt any gauge*}
-
-lemma fine_exists:
- assumes "a \<le> b" and "gauge {a..b} \<delta>" shows "\<exists>D. fine \<delta> (a, b) D"
-proof -
- {
- fix u v :: real assume "u \<le> v"
- have "a \<le> u \<Longrightarrow> v \<le> b \<Longrightarrow> \<exists>D. fine \<delta> (u, v) D"
- apply (induct u v rule: BOLZANO, rule `u \<le> v`)
- apply (simp, fast intro: fine_append)
- apply (case_tac "a \<le> x \<and> x \<le> b")
- apply (rule_tac x="\<delta> x" in exI)
- apply (rule conjI)
- apply (simp add: `gauge {a..b} \<delta>` [unfolded gauge_def])
- apply (clarify, rename_tac u v)
- apply (case_tac "u = v")
- apply (fast intro: fine_Nil)
- apply (subgoal_tac "u < v", fast intro: fine_single, simp)
- apply (rule_tac x="1" in exI, clarsimp)
- done
- }
- with `a \<le> b` show ?thesis by auto
-qed
-
-lemma fine_covers_all:
- assumes "fine \<delta> (a, c) D" and "a < x" and "x \<le> c"
- shows "\<exists> N < length D. \<forall> d t e. D ! N = (d,t,e) \<longrightarrow> d < x \<and> x \<le> e"
- using assms
-proof (induct set: fine)
- case (2 b c D a t)
- thus ?case
- proof (cases "b < x")
- case True
- with 2 obtain N where *: "N < length D"
- and **: "\<And> d t e. D ! N = (d,t,e) \<Longrightarrow> d < x \<and> x \<le> e" by auto
- hence "Suc N < length ((a,t,b)#D) \<and>
- (\<forall> d t' e. ((a,t,b)#D) ! Suc N = (d,t',e) \<longrightarrow> d < x \<and> x \<le> e)" by auto
- thus ?thesis by auto
- next
- case False with 2
- have "0 < length ((a,t,b)#D) \<and>
- (\<forall> d t' e. ((a,t,b)#D) ! 0 = (d,t',e) \<longrightarrow> d < x \<and> x \<le> e)" by auto
- thus ?thesis by auto
- qed
-qed auto
-
-lemma fine_append_split:
- assumes "fine \<delta> (a,b) D" and "D2 \<noteq> []" and "D = D1 @ D2"
- shows "fine \<delta> (a,fst (hd D2)) D1" (is "?fine1")
- and "fine \<delta> (fst (hd D2), b) D2" (is "?fine2")
-proof -
- from assms
- have "?fine1 \<and> ?fine2"
- proof (induct arbitrary: D1 D2)
- case (2 b c D a' x D1 D2)
- note induct = this
-
- thus ?case
- proof (cases D1)
- case Nil
- hence "fst (hd D2) = a'" using 2 by auto
- with fine_Cons[OF `fine \<delta> (b,c) D` induct(3,4,5)] Nil induct
- show ?thesis by (auto intro: fine_Nil)
- next
- case (Cons d1 D1')
- with induct(2)[OF `D2 \<noteq> []`, of D1'] induct(8)
- have "fine \<delta> (b, fst (hd D2)) D1'" and "fine \<delta> (fst (hd D2), c) D2" and
- "d1 = (a', x, b)" by auto
- with fine_Cons[OF this(1) induct(3,4,5), OF induct(6)] Cons
- show ?thesis by auto
- qed
- qed auto
- thus ?fine1 and ?fine2 by auto
-qed
-
-lemma fine_\<delta>_expand:
- assumes "fine \<delta> (a,b) D"
- and "\<And> x. \<lbrakk> a \<le> x ; x \<le> b \<rbrakk> \<Longrightarrow> \<delta> x \<le> \<delta>' x"
- shows "fine \<delta>' (a,b) D"
-using assms proof induct
- case 1 show ?case by (rule fine_Nil)
-next
- case (2 b c D a x)
- show ?case
- proof (rule fine_Cons)
- show "fine \<delta>' (b,c) D" using 2 by auto
- from fine_imp_le[OF 2(1)] 2(6) `x \<le> b`
- show "b - a < \<delta>' x"
- using 2(7)[OF `a \<le> x`] by auto
- qed (auto simp add: 2)
-qed
-
-lemma fine_single_boundaries:
- assumes "fine \<delta> (a,b) D" and "D = [(d, t, e)]"
- shows "a = d \<and> b = e"
-using assms proof induct
- case (2 b c D a x)
- hence "D = []" and "a = d" and "b = e" by auto
- moreover
- from `fine \<delta> (b,c) D` `D = []` have "b = c"
- by (rule empty_fine_imp_eq)
- ultimately show ?case by simp
-qed auto
-
-
-subsection {* Riemann sum *}
-
-definition
- rsum :: "[(real \<times> real \<times> real) list, real \<Rightarrow> real] \<Rightarrow> real" where
- "rsum D f = (\<Sum>(u, x, v)\<leftarrow>D. f x * (v - u))"
-
-lemma rsum_Nil [simp]: "rsum [] f = 0"
-unfolding rsum_def by simp
-
-lemma rsum_Cons [simp]: "rsum ((u, x, v) # D) f = f x * (v - u) + rsum D f"
-unfolding rsum_def by simp
-
-lemma rsum_zero [simp]: "rsum D (\<lambda>x. 0) = 0"
-by (induct D, auto)
-
-lemma rsum_left_distrib: "rsum D f * c = rsum D (\<lambda>x. f x * c)"
-by (induct D, auto simp add: algebra_simps)
-
-lemma rsum_right_distrib: "c * rsum D f = rsum D (\<lambda>x. c * f x)"
-by (induct D, auto simp add: algebra_simps)
-
-lemma rsum_add: "rsum D (\<lambda>x. f x + g x) = rsum D f + rsum D g"
-by (induct D, auto simp add: algebra_simps)
-
-lemma rsum_append: "rsum (D1 @ D2) f = rsum D1 f + rsum D2 f"
-unfolding rsum_def map_append listsum_append ..
-
-
-subsection {* Gauge integrability (definite) *}
-
-definition
- Integral :: "[(real*real),real=>real,real] => bool" where
- [code del]: "Integral = (%(a,b) f k. \<forall>e > 0.
- (\<exists>\<delta>. gauge {a .. b} \<delta> &
- (\<forall>D. fine \<delta> (a,b) D -->
- \<bar>rsum D f - k\<bar> < e)))"
-
-lemma Integral_def2:
- "Integral = (%(a,b) f k. \<forall>e>0. (\<exists>\<delta>. gauge {a..b} \<delta> &
- (\<forall>D. fine \<delta> (a,b) D -->
- \<bar>rsum D f - k\<bar> \<le> e)))"
-unfolding Integral_def
-apply (safe intro!: ext)
-apply (fast intro: less_imp_le)
-apply (drule_tac x="e/2" in spec)
-apply force
-done
-
-text{*Lemmas about combining gauges*}
-
-lemma gauge_min:
- "[| gauge(E) g1; gauge(E) g2 |]
- ==> gauge(E) (%x. min (g1(x)) (g2(x)))"
-by (simp add: gauge_def)
-
-lemma fine_min:
- "fine (%x. min (g1(x)) (g2(x))) (a,b) D
- ==> fine(g1) (a,b) D & fine(g2) (a,b) D"
-apply (erule fine.induct)
-apply (simp add: fine_Nil)
-apply (simp add: fine_Cons)
-done
-
-text{*The integral is unique if it exists*}
-
-lemma Integral_unique:
- "[| a \<le> b; Integral(a,b) f k1; Integral(a,b) f k2 |] ==> k1 = k2"
-apply (simp add: Integral_def)
-apply (drule_tac x = "\<bar>k1 - k2\<bar> /2" in spec)+
-apply auto
-apply (drule gauge_min, assumption)
-apply (drule_tac \<delta> = "%x. min (\<delta> x) (\<delta>' x)"
- in fine_exists, assumption, auto)
-apply (drule fine_min)
-apply (drule spec)+
-apply auto
-apply (subgoal_tac "\<bar>(rsum D f - k2) - (rsum D f - k1)\<bar> < \<bar>k1 - k2\<bar>")
-apply arith
-apply (drule add_strict_mono, assumption)
-apply (auto simp only: left_distrib [symmetric] mult_2_right [symmetric]
- mult_less_cancel_right)
-done
-
-lemma Integral_zero [simp]: "Integral(a,a) f 0"
-apply (auto simp add: Integral_def)
-apply (rule_tac x = "%x. 1" in exI)
-apply (auto dest: fine_eq simp add: gauge_def rsum_def)
-done
-
-lemma fine_rsum_const: "fine \<delta> (a,b) D \<Longrightarrow> rsum D (\<lambda>x. c) = (c * (b - a))"
-unfolding rsum_def
-by (induct set: fine, auto simp add: algebra_simps)
-
-lemma Integral_eq_diff_bounds: "a \<le> b ==> Integral(a,b) (%x. 1) (b - a)"
-apply (cases "a = b", simp)
-apply (simp add: Integral_def, clarify)
-apply (rule_tac x = "%x. b - a" in exI)
-apply (rule conjI, simp add: gauge_def)
-apply (clarify)
-apply (subst fine_rsum_const, assumption, simp)
-done
-
-lemma Integral_mult_const: "a \<le> b ==> Integral(a,b) (%x. c) (c*(b - a))"
-apply (cases "a = b", simp)
-apply (simp add: Integral_def, clarify)
-apply (rule_tac x = "%x. b - a" in exI)
-apply (rule conjI, simp add: gauge_def)
-apply (clarify)
-apply (subst fine_rsum_const, assumption, simp)
-done
-
-lemma Integral_mult:
- "[| a \<le> b; Integral(a,b) f k |] ==> Integral(a,b) (%x. c * f x) (c * k)"
-apply (auto simp add: order_le_less
- dest: Integral_unique [OF order_refl Integral_zero])
-apply (auto simp add: Integral_def setsum_right_distrib[symmetric] mult_assoc)
-apply (case_tac "c = 0", force)
-apply (drule_tac x = "e/abs c" in spec)
-apply (simp add: divide_pos_pos)
-apply clarify
-apply (rule_tac x="\<delta>" in exI, clarify)
-apply (drule_tac x="D" in spec, clarify)
-apply (simp add: pos_less_divide_eq abs_mult [symmetric]
- algebra_simps rsum_right_distrib)
-done
-
-lemma Integral_add:
- assumes "Integral (a, b) f x1"
- assumes "Integral (b, c) f x2"
- assumes "a \<le> b" and "b \<le> c"
- shows "Integral (a, c) f (x1 + x2)"
-proof (cases "a < b \<and> b < c", simp only: Integral_def split_conv, rule allI, rule impI)
- fix \<epsilon> :: real assume "0 < \<epsilon>"
- hence "0 < \<epsilon> / 2" by auto
-
- assume "a < b \<and> b < c"
- hence "a < b" and "b < c" by auto
-
- from `Integral (a, b) f x1`[simplified Integral_def split_conv,
- rule_format, OF `0 < \<epsilon>/2`]
- obtain \<delta>1 where \<delta>1_gauge: "gauge {a..b} \<delta>1"
- and I1: "\<And> D. fine \<delta>1 (a,b) D \<Longrightarrow> \<bar> rsum D f - x1 \<bar> < (\<epsilon> / 2)" by auto
-
- from `Integral (b, c) f x2`[simplified Integral_def split_conv,
- rule_format, OF `0 < \<epsilon>/2`]
- obtain \<delta>2 where \<delta>2_gauge: "gauge {b..c} \<delta>2"
- and I2: "\<And> D. fine \<delta>2 (b,c) D \<Longrightarrow> \<bar> rsum D f - x2 \<bar> < (\<epsilon> / 2)" by auto
-
- def \<delta> \<equiv> "\<lambda> x. if x < b then min (\<delta>1 x) (b - x)
- else if x = b then min (\<delta>1 b) (\<delta>2 b)
- else min (\<delta>2 x) (x - b)"
-
- have "gauge {a..c} \<delta>"
- using \<delta>1_gauge \<delta>2_gauge unfolding \<delta>_def gauge_def by auto
- moreover {
- fix D :: "(real \<times> real \<times> real) list"
- assume fine: "fine \<delta> (a,c) D"
- from fine_covers_all[OF this `a < b` `b \<le> c`]
- obtain N where "N < length D"
- and *: "\<forall> d t e. D ! N = (d, t, e) \<longrightarrow> d < b \<and> b \<le> e"
- by auto
- obtain d t e where D_eq: "D ! N = (d, t, e)" by (cases "D!N", auto)
- with * have "d < b" and "b \<le> e" by auto
- have in_D: "(d, t, e) \<in> set D"
- using D_eq[symmetric] using `N < length D` by auto
-
- from mem_fine[OF fine in_D]
- have "d < e" and "d \<le> t" and "t \<le> e" by auto
-
- have "t = b"
- proof (rule ccontr)
- assume "t \<noteq> b"
- with mem_fine3[OF fine in_D] `b \<le> e` `d \<le> t` `t \<le> e` `d < b` \<delta>_def
- show False by (cases "t < b") auto
- qed
-
- let ?D1 = "take N D"
- let ?D2 = "drop N D"
- def D1 \<equiv> "take N D @ [(d, t, b)]"
- def D2 \<equiv> "(if b = e then [] else [(b, t, e)]) @ drop (Suc N) D"
-
- have "D \<noteq> []" using `N < length D` by auto
- from hd_drop_conv_nth[OF this `N < length D`]
- have "fst (hd ?D2) = d" using `D ! N = (d, t, e)` by auto
- with fine_append_split[OF _ _ append_take_drop_id[symmetric]]
- have fine1: "fine \<delta> (a,d) ?D1" and fine2: "fine \<delta> (d,c) ?D2"
- using `N < length D` fine by auto
-
- have "fine \<delta>1 (a,b) D1" unfolding D1_def
- proof (rule fine_append)
- show "fine \<delta>1 (a, d) ?D1"
- proof (rule fine1[THEN fine_\<delta>_expand])
- fix x assume "a \<le> x" "x \<le> d"
- hence "x \<le> b" using `d < b` `x \<le> d` by auto
- thus "\<delta> x \<le> \<delta>1 x" unfolding \<delta>_def by auto
- qed
-
- have "b - d < \<delta>1 t"
- using mem_fine3[OF fine in_D] \<delta>_def `b \<le> e` `t = b` by auto
- from `d < b` `d \<le> t` `t = b` this
- show "fine \<delta>1 (d, b) [(d, t, b)]" using fine_single by auto
- qed
- note rsum1 = I1[OF this]
-
- have drop_split: "drop N D = [D ! N] @ drop (Suc N) D"
- using nth_drop'[OF `N < length D`] by simp
-
- have fine2: "fine \<delta>2 (e,c) (drop (Suc N) D)"
- proof (cases "drop (Suc N) D = []")
- case True
- note * = fine2[simplified drop_split True D_eq append_Nil2]
- have "e = c" using fine_single_boundaries[OF * refl] by auto
- thus ?thesis unfolding True using fine_Nil by auto
- next
- case False
- note * = fine_append_split[OF fine2 False drop_split]
- from fine_single_boundaries[OF *(1)]
- have "fst (hd (drop (Suc N) D)) = e" using D_eq by auto
- with *(2) have "fine \<delta> (e,c) (drop (Suc N) D)" by auto
- thus ?thesis
- proof (rule fine_\<delta>_expand)
- fix x assume "e \<le> x" and "x \<le> c"
- thus "\<delta> x \<le> \<delta>2 x" using `b \<le> e` unfolding \<delta>_def by auto
- qed
- qed
-
- have "fine \<delta>2 (b, c) D2"
- proof (cases "e = b")
- case True thus ?thesis using fine2 by (simp add: D1_def D2_def)
- next
- case False
- have "e - b < \<delta>2 b"
- using mem_fine3[OF fine in_D] \<delta>_def `d < b` `t = b` by auto
- with False `t = b` `b \<le> e`
- show ?thesis using D2_def
- by (auto intro!: fine_append[OF _ fine2] fine_single
- simp del: append_Cons)
- qed
- note rsum2 = I2[OF this]
-
- have "rsum D f = rsum (take N D) f + rsum [D ! N] f + rsum (drop (Suc N) D) f"
- using rsum_append[symmetric] nth_drop'[OF `N < length D`] by auto
- also have "\<dots> = rsum D1 f + rsum D2 f"
- by (cases "b = e", auto simp add: D1_def D2_def D_eq rsum_append algebra_simps)
- finally have "\<bar>rsum D f - (x1 + x2)\<bar> < \<epsilon>"
- using add_strict_mono[OF rsum1 rsum2] by simp
- }
- ultimately show "\<exists> \<delta>. gauge {a .. c} \<delta> \<and>
- (\<forall>D. fine \<delta> (a,c) D \<longrightarrow> \<bar>rsum D f - (x1 + x2)\<bar> < \<epsilon>)"
- by blast
-next
- case False
- hence "a = b \<or> b = c" using `a \<le> b` and `b \<le> c` by auto
- thus ?thesis
- proof (rule disjE)
- assume "a = b" hence "x1 = 0"
- using `Integral (a, b) f x1` Integral_zero Integral_unique[of a b] by auto
- thus ?thesis using `a = b` `Integral (b, c) f x2` by auto
- next
- assume "b = c" hence "x2 = 0"
- using `Integral (b, c) f x2` Integral_zero Integral_unique[of b c] by auto
- thus ?thesis using `b = c` `Integral (a, b) f x1` by auto
- qed
-qed
-
-text{*Fundamental theorem of calculus (Part I)*}
-
-text{*"Straddle Lemma" : Swartz and Thompson: AMM 95(7) 1988 *}
-
-lemma strad1:
- "\<lbrakk>\<forall>z::real. z \<noteq> x \<and> \<bar>z - x\<bar> < s \<longrightarrow>
- \<bar>(f z - f x) / (z - x) - f' x\<bar> < e/2;
- 0 < s; 0 < e; a \<le> x; x \<le> b\<rbrakk>
- \<Longrightarrow> \<forall>z. \<bar>z - x\<bar> < s -->\<bar>f z - f x - f' x * (z - x)\<bar> \<le> e/2 * \<bar>z - x\<bar>"
-apply clarify
-apply (case_tac "z = x", simp)
-apply (drule_tac x = z in spec)
-apply (rule_tac z1 = "\<bar>inverse (z - x)\<bar>"
- in real_mult_le_cancel_iff2 [THEN iffD1])
- apply simp
-apply (simp del: abs_inverse abs_mult add: abs_mult [symmetric]
- mult_assoc [symmetric])
-apply (subgoal_tac "inverse (z - x) * (f z - f x - f' x * (z - x))
- = (f z - f x) / (z - x) - f' x")
- apply (simp add: abs_mult [symmetric] mult_ac diff_minus)
-apply (subst mult_commute)
-apply (simp add: left_distrib diff_minus)
-apply (simp add: mult_assoc divide_inverse)
-apply (simp add: left_distrib)
-done
-
-lemma lemma_straddle:
- assumes f': "\<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x)" and "0 < e"
- shows "\<exists>g. gauge {a..b} g &
- (\<forall>x u v. a \<le> u & u \<le> x & x \<le> v & v \<le> b & (v - u) < g(x)
- --> \<bar>(f(v) - f(u)) - (f'(x) * (v - u))\<bar> \<le> e * (v - u))"
-proof -
- have "\<forall>x\<in>{a..b}.
- (\<exists>d > 0. \<forall>u v. u \<le> x & x \<le> v & (v - u) < d -->
- \<bar>(f(v) - f(u)) - (f'(x) * (v - u))\<bar> \<le> e * (v - u))"
- proof (clarsimp)
- fix x :: real assume "a \<le> x" and "x \<le> b"
- with f' have "DERIV f x :> f'(x)" by simp
- then have "\<forall>r>0. \<exists>s>0. \<forall>z. z \<noteq> x \<and> \<bar>z - x\<bar> < s \<longrightarrow> \<bar>(f z - f x) / (z - x) - f' x\<bar> < r"
- by (simp add: DERIV_iff2 LIM_eq)
- with `0 < e` obtain s
- where "\<forall>z. z \<noteq> x \<and> \<bar>z - x\<bar> < s \<longrightarrow> \<bar>(f z - f x) / (z - x) - f' x\<bar> < e/2" and "0 < s"
- by (drule_tac x="e/2" in spec, auto)
- then have strad [rule_format]:
- "\<forall>z. \<bar>z - x\<bar> < s --> \<bar>f z - f x - f' x * (z - x)\<bar> \<le> e/2 * \<bar>z - x\<bar>"
- using `0 < e` `a \<le> x` `x \<le> b` by (rule strad1)
- show "\<exists>d>0. \<forall>u v. u \<le> x \<and> x \<le> v \<and> v - u < d \<longrightarrow> \<bar>f v - f u - f' x * (v - u)\<bar> \<le> e * (v - u)"
- proof (safe intro!: exI)
- show "0 < s" by fact
- next
- fix u v :: real assume "u \<le> x" and "x \<le> v" and "v - u < s"
- have "\<bar>f v - f u - f' x * (v - u)\<bar> =
- \<bar>(f v - f x - f' x * (v - x)) + (f x - f u - f' x * (x - u))\<bar>"
- by (simp add: right_diff_distrib)
- also have "\<dots> \<le> \<bar>f v - f x - f' x * (v - x)\<bar> + \<bar>f x - f u - f' x * (x - u)\<bar>"
- by (rule abs_triangle_ineq)
- also have "\<dots> = \<bar>f v - f x - f' x * (v - x)\<bar> + \<bar>f u - f x - f' x * (u - x)\<bar>"
- by (simp add: right_diff_distrib)
- also have "\<dots> \<le> (e/2) * \<bar>v - x\<bar> + (e/2) * \<bar>u - x\<bar>"
- using `u \<le> x` `x \<le> v` `v - u < s` by (intro add_mono strad, simp_all)
- also have "\<dots> \<le> e * (v - u) / 2 + e * (v - u) / 2"
- using `u \<le> x` `x \<le> v` `0 < e` by (intro add_mono, simp_all)
- also have "\<dots> = e * (v - u)"
- by simp
- finally show "\<bar>f v - f u - f' x * (v - u)\<bar> \<le> e * (v - u)" .
- qed
- qed
- thus ?thesis
- by (simp add: gauge_def) (drule bchoice, auto)
-qed
-
-lemma fine_listsum_eq_diff:
- fixes f :: "real \<Rightarrow> real"
- shows "fine \<delta> (a, b) D \<Longrightarrow> (\<Sum>(u, x, v)\<leftarrow>D. f v - f u) = f b - f a"
-by (induct set: fine) simp_all
-
-lemma FTC1: "[|a \<le> b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
- ==> Integral(a,b) f' (f(b) - f(a))"
- apply (drule order_le_imp_less_or_eq, auto)
- apply (auto simp add: Integral_def2)
- apply (drule_tac e = "e / (b - a)" in lemma_straddle)
- apply (simp add: divide_pos_pos)
- apply clarify
- apply (rule_tac x="g" in exI, clarify)
- apply (clarsimp simp add: rsum_def)
- apply (frule fine_listsum_eq_diff [where f=f])
- apply (erule subst)
- apply (subst listsum_subtractf [symmetric])
- apply (rule listsum_abs [THEN order_trans])
- apply (subst map_map [unfolded o_def])
- apply (subgoal_tac "e = (\<Sum>(u, x, v)\<leftarrow>D. (e / (b - a)) * (v - u))")
- apply (erule ssubst)
- apply (simp add: abs_minus_commute)
- apply (rule listsum_mono)
- apply (clarify, rename_tac u x v)
- apply ((drule spec)+, erule mp)
- apply (simp add: mem_fine mem_fine2 mem_fine3)
- apply (frule fine_listsum_eq_diff [where f="\<lambda>x. x"])
- apply (simp only: split_def)
- apply (subst listsum_const_mult)
- apply simp
-done
-
-lemma Integral_subst: "[| Integral(a,b) f k1; k2=k1 |] ==> Integral(a,b) f k2"
-by simp
-
-subsection {* Additivity Theorem of Gauge Integral *}
-
-text{* Bartle/Sherbert: Theorem 10.1.5 p. 278 *}
-lemma Integral_add_fun:
- "[| a \<le> b; Integral(a,b) f k1; Integral(a,b) g k2 |]
- ==> Integral(a,b) (%x. f x + g x) (k1 + k2)"
-unfolding Integral_def
-apply clarify
-apply (drule_tac x = "e/2" in spec)+
-apply clarsimp
-apply (rule_tac x = "\<lambda>x. min (\<delta> x) (\<delta>' x)" in exI)
-apply (rule conjI, erule (1) gauge_min)
-apply clarify
-apply (drule fine_min)
-apply (drule_tac x=D in spec, simp)+
-apply (drule_tac a = "\<bar>rsum D f - k1\<bar> * 2" and c = "\<bar>rsum D g - k2\<bar> * 2" in add_strict_mono, assumption)
-apply (auto simp only: rsum_add left_distrib [symmetric]
- mult_2_right [symmetric] real_mult_less_iff1)
-done
-
-lemma lemma_Integral_rsum_le:
- "[| \<forall>x. a \<le> x & x \<le> b --> f x \<le> g x;
- fine \<delta> (a,b) D
- |] ==> rsum D f \<le> rsum D g"
-unfolding rsum_def
-apply (rule listsum_mono)
-apply clarify
-apply (rule mult_right_mono)
-apply (drule spec, erule mp)
-apply (frule (1) mem_fine)
-apply (frule (1) mem_fine2)
-apply simp
-apply (frule (1) mem_fine)
-apply simp
-done
-
-lemma Integral_le:
- "[| a \<le> b;
- \<forall>x. a \<le> x & x \<le> b --> f(x) \<le> g(x);
- Integral(a,b) f k1; Integral(a,b) g k2
- |] ==> k1 \<le> k2"
-apply (simp add: Integral_def)
-apply (rotate_tac 2)
-apply (drule_tac x = "\<bar>k1 - k2\<bar> /2" in spec)
-apply (drule_tac x = "\<bar>k1 - k2\<bar> /2" in spec, auto)
-apply (drule gauge_min, assumption)
-apply (drule_tac \<delta> = "\<lambda>x. min (\<delta> x) (\<delta>' x)" in fine_exists, assumption, clarify)
-apply (drule fine_min)
-apply (drule_tac x = D in spec, drule_tac x = D in spec, clarsimp)
-apply (frule lemma_Integral_rsum_le, assumption)
-apply (subgoal_tac "\<bar>(rsum D f - k1) - (rsum D g - k2)\<bar> < \<bar>k1 - k2\<bar>")
-apply arith
-apply (drule add_strict_mono, assumption)
-apply (auto simp only: left_distrib [symmetric] mult_2_right [symmetric]
- real_mult_less_iff1)
-done
-
-lemma Integral_imp_Cauchy:
- "(\<exists>k. Integral(a,b) f k) ==>
- (\<forall>e > 0. \<exists>\<delta>. gauge {a..b} \<delta> &
- (\<forall>D1 D2.
- fine \<delta> (a,b) D1 &
- fine \<delta> (a,b) D2 -->
- \<bar>rsum D1 f - rsum D2 f\<bar> < e))"
-apply (simp add: Integral_def, auto)
-apply (drule_tac x = "e/2" in spec, auto)
-apply (rule exI, auto)
-apply (frule_tac x = D1 in spec)
-apply (drule_tac x = D2 in spec)
-apply simp
-apply (thin_tac "0 < e")
-apply (drule add_strict_mono, assumption)
-apply (auto simp only: left_distrib [symmetric] mult_2_right [symmetric]
- real_mult_less_iff1)
-done
-
-lemma Cauchy_iff2:
- "Cauchy X =
- (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. \<bar>X m - X n\<bar> < inverse(real (Suc j))))"
-apply (simp add: Cauchy_iff, auto)
-apply (drule reals_Archimedean, safe)
-apply (drule_tac x = n in spec, auto)
-apply (rule_tac x = M in exI, auto)
-apply (drule_tac x = m in spec, simp)
-apply (drule_tac x = na in spec, auto)
-done
-
-lemma monotonic_anti_derivative:
- fixes f g :: "real => real" shows
- "[| a \<le> b; \<forall>c. a \<le> c & c \<le> b --> f' c \<le> g' c;
- \<forall>x. DERIV f x :> f' x; \<forall>x. DERIV g x :> g' x |]
- ==> f b - f a \<le> g b - g a"
-apply (rule Integral_le, assumption)
-apply (auto intro: FTC1)
-done
-
-end
--- a/src/HOL/IsaMakefile Mon Feb 22 20:08:10 2010 +0100
+++ b/src/HOL/IsaMakefile Mon Feb 22 20:41:49 2010 +0100
@@ -344,7 +344,6 @@
Deriv.thy \
Fact.thy \
GCD.thy \
- Integration.thy \
Lim.thy \
Limits.thy \
Ln.thy \
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Multivariate_Analysis/Integration.cert Mon Feb 22 20:41:49 2010 +0100
@@ -0,0 +1,3296 @@
+tB2Atlor9W4pSnrAz5nHpw 907 0
+#2 := false
+#299 := 0::real
+decl uf_1 :: (-> T3 T2 real)
+decl uf_10 :: (-> T4 T2)
+decl uf_7 :: T4
+#15 := uf_7
+#22 := (uf_10 uf_7)
+decl uf_2 :: (-> T1 T3)
+decl uf_4 :: T1
+#11 := uf_4
+#91 := (uf_2 uf_4)
+#902 := (uf_1 #91 #22)
+#297 := -1::real
+#1084 := (* -1::real #902)
+decl uf_16 :: T1
+#50 := uf_16
+#78 := (uf_2 uf_16)
+#799 := (uf_1 #78 #22)
+#1267 := (+ #799 #1084)
+#1272 := (>= #1267 0::real)
+#1266 := (= #799 #902)
+decl uf_9 :: T3
+#21 := uf_9
+#23 := (uf_1 uf_9 #22)
+#905 := (= #23 #902)
+decl uf_11 :: T3
+#24 := uf_11
+#850 := (uf_1 uf_11 #22)
+#904 := (= #850 #902)
+decl uf_6 :: (-> T2 T4)
+#74 := (uf_6 #22)
+#281 := (= uf_7 #74)
+#922 := (ite #281 #905 #904)
+decl uf_8 :: T3
+#18 := uf_8
+#848 := (uf_1 uf_8 #22)
+#903 := (= #848 #902)
+#60 := 0::int
+decl uf_5 :: (-> T4 int)
+#803 := (uf_5 #74)
+#117 := -1::int
+#813 := (* -1::int #803)
+#16 := (uf_5 uf_7)
+#916 := (+ #16 #813)
+#917 := (<= #916 0::int)
+#925 := (ite #917 #922 #903)
+#6 := (:var 0 T2)
+#19 := (uf_1 uf_8 #6)
+#544 := (pattern #19)
+#25 := (uf_1 uf_11 #6)
+#543 := (pattern #25)
+#92 := (uf_1 #91 #6)
+#542 := (pattern #92)
+#13 := (uf_6 #6)
+#541 := (pattern #13)
+#447 := (= #19 #92)
+#445 := (= #25 #92)
+#444 := (= #23 #92)
+#20 := (= #13 uf_7)
+#446 := (ite #20 #444 #445)
+#120 := (* -1::int #16)
+#14 := (uf_5 #13)
+#121 := (+ #14 #120)
+#119 := (>= #121 0::int)
+#448 := (ite #119 #446 #447)
+#545 := (forall (vars (?x3 T2)) (:pat #541 #542 #543 #544) #448)
+#451 := (forall (vars (?x3 T2)) #448)
+#548 := (iff #451 #545)
+#546 := (iff #448 #448)
+#547 := [refl]: #546
+#549 := [quant-intro #547]: #548
+#26 := (ite #20 #23 #25)
+#127 := (ite #119 #26 #19)
+#368 := (= #92 #127)
+#369 := (forall (vars (?x3 T2)) #368)
+#452 := (iff #369 #451)
+#449 := (iff #368 #448)
+#450 := [rewrite]: #449
+#453 := [quant-intro #450]: #452
+#392 := (~ #369 #369)
+#390 := (~ #368 #368)
+#391 := [refl]: #390
+#366 := [nnf-pos #391]: #392
+decl uf_3 :: (-> T1 T2 real)
+#12 := (uf_3 uf_4 #6)
+#132 := (= #12 #127)
+#135 := (forall (vars (?x3 T2)) #132)
+#370 := (iff #135 #369)
+#4 := (:var 1 T1)
+#8 := (uf_3 #4 #6)
+#5 := (uf_2 #4)
+#7 := (uf_1 #5 #6)
+#9 := (= #7 #8)
+#10 := (forall (vars (?x1 T1) (?x2 T2)) #9)
+#113 := [asserted]: #10
+#371 := [rewrite* #113]: #370
+#17 := (< #14 #16)
+#27 := (ite #17 #19 #26)
+#28 := (= #12 #27)
+#29 := (forall (vars (?x3 T2)) #28)
+#136 := (iff #29 #135)
+#133 := (iff #28 #132)
+#130 := (= #27 #127)
+#118 := (not #119)
+#124 := (ite #118 #19 #26)
+#128 := (= #124 #127)
+#129 := [rewrite]: #128
+#125 := (= #27 #124)
+#122 := (iff #17 #118)
+#123 := [rewrite]: #122
+#126 := [monotonicity #123]: #125
+#131 := [trans #126 #129]: #130
+#134 := [monotonicity #131]: #133
+#137 := [quant-intro #134]: #136
+#114 := [asserted]: #29
+#138 := [mp #114 #137]: #135
+#372 := [mp #138 #371]: #369
+#367 := [mp~ #372 #366]: #369
+#454 := [mp #367 #453]: #451
+#550 := [mp #454 #549]: #545
+#738 := (not #545)
+#928 := (or #738 #925)
+#75 := (= #74 uf_7)
+#906 := (ite #75 #905 #904)
+#907 := (+ #803 #120)
+#908 := (>= #907 0::int)
+#909 := (ite #908 #906 #903)
+#929 := (or #738 #909)
+#931 := (iff #929 #928)
+#933 := (iff #928 #928)
+#934 := [rewrite]: #933
+#926 := (iff #909 #925)
+#923 := (iff #906 #922)
+#283 := (iff #75 #281)
+#284 := [rewrite]: #283
+#924 := [monotonicity #284]: #923
+#920 := (iff #908 #917)
+#910 := (+ #120 #803)
+#913 := (>= #910 0::int)
+#918 := (iff #913 #917)
+#919 := [rewrite]: #918
+#914 := (iff #908 #913)
+#911 := (= #907 #910)
+#912 := [rewrite]: #911
+#915 := [monotonicity #912]: #914
+#921 := [trans #915 #919]: #920
+#927 := [monotonicity #921 #924]: #926
+#932 := [monotonicity #927]: #931
+#935 := [trans #932 #934]: #931
+#930 := [quant-inst]: #929
+#936 := [mp #930 #935]: #928
+#1300 := [unit-resolution #936 #550]: #925
+#989 := (= #16 #803)
+#1277 := (= #803 #16)
+#280 := [asserted]: #75
+#287 := [mp #280 #284]: #281
+#1276 := [symm #287]: #75
+#1278 := [monotonicity #1276]: #1277
+#1301 := [symm #1278]: #989
+#1302 := (not #989)
+#1303 := (or #1302 #917)
+#1304 := [th-lemma]: #1303
+#1305 := [unit-resolution #1304 #1301]: #917
+#950 := (not #917)
+#949 := (not #925)
+#951 := (or #949 #950 #922)
+#952 := [def-axiom]: #951
+#1306 := [unit-resolution #952 #1305 #1300]: #922
+#937 := (not #922)
+#1307 := (or #937 #905)
+#938 := (not #281)
+#939 := (or #937 #938 #905)
+#940 := [def-axiom]: #939
+#1308 := [unit-resolution #940 #287]: #1307
+#1309 := [unit-resolution #1308 #1306]: #905
+#1356 := (= #799 #23)
+#800 := (= #23 #799)
+decl uf_15 :: T4
+#40 := uf_15
+#41 := (uf_5 uf_15)
+#814 := (+ #41 #813)
+#815 := (<= #814 0::int)
+#836 := (not #815)
+#158 := (* -1::int #41)
+#1270 := (+ #16 #158)
+#1265 := (>= #1270 0::int)
+#1339 := (not #1265)
+#1269 := (= #16 #41)
+#1298 := (not #1269)
+#286 := (= uf_7 uf_15)
+#44 := (uf_10 uf_15)
+#72 := (uf_6 #44)
+#73 := (= #72 uf_15)
+#277 := (= uf_15 #72)
+#278 := (iff #73 #277)
+#279 := [rewrite]: #278
+#276 := [asserted]: #73
+#282 := [mp #276 #279]: #277
+#1274 := [symm #282]: #73
+#729 := (= uf_7 #72)
+decl uf_17 :: (-> int T4)
+#611 := (uf_5 #72)
+#991 := (uf_17 #611)
+#1289 := (= #991 #72)
+#992 := (= #72 #991)
+#55 := (:var 0 T4)
+#56 := (uf_5 #55)
+#574 := (pattern #56)
+#57 := (uf_17 #56)
+#177 := (= #55 #57)
+#575 := (forall (vars (?x7 T4)) (:pat #574) #177)
+#195 := (forall (vars (?x7 T4)) #177)
+#578 := (iff #195 #575)
+#576 := (iff #177 #177)
+#577 := [refl]: #576
+#579 := [quant-intro #577]: #578
+#405 := (~ #195 #195)
+#403 := (~ #177 #177)
+#404 := [refl]: #403
+#406 := [nnf-pos #404]: #405
+#58 := (= #57 #55)
+#59 := (forall (vars (?x7 T4)) #58)
+#196 := (iff #59 #195)
+#193 := (iff #58 #177)
+#194 := [rewrite]: #193
+#197 := [quant-intro #194]: #196
+#155 := [asserted]: #59
+#200 := [mp #155 #197]: #195
+#407 := [mp~ #200 #406]: #195
+#580 := [mp #407 #579]: #575
+#995 := (not #575)
+#996 := (or #995 #992)
+#997 := [quant-inst]: #996
+#1273 := [unit-resolution #997 #580]: #992
+#1290 := [symm #1273]: #1289
+#1293 := (= uf_7 #991)
+#993 := (uf_17 #803)
+#1287 := (= #993 #991)
+#1284 := (= #803 #611)
+#987 := (= #41 #611)
+#1279 := (= #611 #41)
+#1280 := [monotonicity #1274]: #1279
+#1281 := [symm #1280]: #987
+#1282 := (= #803 #41)
+#1275 := [hypothesis]: #1269
+#1283 := [trans #1278 #1275]: #1282
+#1285 := [trans #1283 #1281]: #1284
+#1288 := [monotonicity #1285]: #1287
+#1291 := (= uf_7 #993)
+#994 := (= #74 #993)
+#1000 := (or #995 #994)
+#1001 := [quant-inst]: #1000
+#1286 := [unit-resolution #1001 #580]: #994
+#1292 := [trans #287 #1286]: #1291
+#1294 := [trans #1292 #1288]: #1293
+#1295 := [trans #1294 #1290]: #729
+#1296 := [trans #1295 #1274]: #286
+#290 := (not #286)
+#76 := (= uf_15 uf_7)
+#77 := (not #76)
+#291 := (iff #77 #290)
+#288 := (iff #76 #286)
+#289 := [rewrite]: #288
+#292 := [monotonicity #289]: #291
+#285 := [asserted]: #77
+#295 := [mp #285 #292]: #290
+#1297 := [unit-resolution #295 #1296]: false
+#1299 := [lemma #1297]: #1298
+#1342 := (or #1269 #1339)
+#1271 := (<= #1270 0::int)
+#621 := (* -1::int #611)
+#723 := (+ #16 #621)
+#724 := (<= #723 0::int)
+decl uf_12 :: T1
+#30 := uf_12
+#88 := (uf_2 uf_12)
+#771 := (uf_1 #88 #44)
+#45 := (uf_1 uf_9 #44)
+#772 := (= #45 #771)
+#796 := (not #772)
+decl uf_14 :: T1
+#38 := uf_14
+#83 := (uf_2 uf_14)
+#656 := (uf_1 #83 #44)
+#1239 := (= #656 #771)
+#1252 := (not #1239)
+#1324 := (iff #1252 #796)
+#1322 := (iff #1239 #772)
+#1320 := (= #656 #45)
+#661 := (= #45 #656)
+#659 := (uf_1 uf_11 #44)
+#664 := (= #656 #659)
+#667 := (ite #277 #661 #664)
+#657 := (uf_1 uf_8 #44)
+#670 := (= #656 #657)
+#622 := (+ #41 #621)
+#623 := (<= #622 0::int)
+#673 := (ite #623 #667 #670)
+#84 := (uf_1 #83 #6)
+#560 := (pattern #84)
+#467 := (= #19 #84)
+#465 := (= #25 #84)
+#464 := (= #45 #84)
+#43 := (= #13 uf_15)
+#466 := (ite #43 #464 #465)
+#159 := (+ #14 #158)
+#157 := (>= #159 0::int)
+#468 := (ite #157 #466 #467)
+#561 := (forall (vars (?x5 T2)) (:pat #541 #560 #543 #544) #468)
+#471 := (forall (vars (?x5 T2)) #468)
+#564 := (iff #471 #561)
+#562 := (iff #468 #468)
+#563 := [refl]: #562
+#565 := [quant-intro #563]: #564
+#46 := (ite #43 #45 #25)
+#165 := (ite #157 #46 #19)
+#378 := (= #84 #165)
+#379 := (forall (vars (?x5 T2)) #378)
+#472 := (iff #379 #471)
+#469 := (iff #378 #468)
+#470 := [rewrite]: #469
+#473 := [quant-intro #470]: #472
+#359 := (~ #379 #379)
+#361 := (~ #378 #378)
+#358 := [refl]: #361
+#356 := [nnf-pos #358]: #359
+#39 := (uf_3 uf_14 #6)
+#170 := (= #39 #165)
+#173 := (forall (vars (?x5 T2)) #170)
+#380 := (iff #173 #379)
+#381 := [rewrite* #113]: #380
+#42 := (< #14 #41)
+#47 := (ite #42 #19 #46)
+#48 := (= #39 #47)
+#49 := (forall (vars (?x5 T2)) #48)
+#174 := (iff #49 #173)
+#171 := (iff #48 #170)
+#168 := (= #47 #165)
+#156 := (not #157)
+#162 := (ite #156 #19 #46)
+#166 := (= #162 #165)
+#167 := [rewrite]: #166
+#163 := (= #47 #162)
+#160 := (iff #42 #156)
+#161 := [rewrite]: #160
+#164 := [monotonicity #161]: #163
+#169 := [trans #164 #167]: #168
+#172 := [monotonicity #169]: #171
+#175 := [quant-intro #172]: #174
+#116 := [asserted]: #49
+#176 := [mp #116 #175]: #173
+#382 := [mp #176 #381]: #379
+#357 := [mp~ #382 #356]: #379
+#474 := [mp #357 #473]: #471
+#566 := [mp #474 #565]: #561
+#676 := (not #561)
+#677 := (or #676 #673)
+#658 := (= #657 #656)
+#660 := (= #659 #656)
+#662 := (ite #73 #661 #660)
+#612 := (+ #611 #158)
+#613 := (>= #612 0::int)
+#663 := (ite #613 #662 #658)
+#678 := (or #676 #663)
+#680 := (iff #678 #677)
+#682 := (iff #677 #677)
+#683 := [rewrite]: #682
+#674 := (iff #663 #673)
+#671 := (iff #658 #670)
+#672 := [rewrite]: #671
+#668 := (iff #662 #667)
+#665 := (iff #660 #664)
+#666 := [rewrite]: #665
+#669 := [monotonicity #279 #666]: #668
+#626 := (iff #613 #623)
+#615 := (+ #158 #611)
+#618 := (>= #615 0::int)
+#624 := (iff #618 #623)
+#625 := [rewrite]: #624
+#619 := (iff #613 #618)
+#616 := (= #612 #615)
+#617 := [rewrite]: #616
+#620 := [monotonicity #617]: #619
+#627 := [trans #620 #625]: #626
+#675 := [monotonicity #627 #669 #672]: #674
+#681 := [monotonicity #675]: #680
+#684 := [trans #681 #683]: #680
+#679 := [quant-inst]: #678
+#685 := [mp #679 #684]: #677
+#1311 := [unit-resolution #685 #566]: #673
+#1312 := (not #987)
+#1313 := (or #1312 #623)
+#1314 := [th-lemma]: #1313
+#1315 := [unit-resolution #1314 #1281]: #623
+#645 := (not #623)
+#698 := (not #673)
+#699 := (or #698 #645 #667)
+#700 := [def-axiom]: #699
+#1316 := [unit-resolution #700 #1315 #1311]: #667
+#686 := (not #667)
+#1317 := (or #686 #661)
+#687 := (not #277)
+#688 := (or #686 #687 #661)
+#689 := [def-axiom]: #688
+#1318 := [unit-resolution #689 #282]: #1317
+#1319 := [unit-resolution #1318 #1316]: #661
+#1321 := [symm #1319]: #1320
+#1323 := [monotonicity #1321]: #1322
+#1325 := [monotonicity #1323]: #1324
+#1145 := (* -1::real #771)
+#1240 := (+ #656 #1145)
+#1241 := (<= #1240 0::real)
+#1249 := (not #1241)
+#1243 := [hypothesis]: #1241
+decl uf_18 :: T3
+#80 := uf_18
+#1040 := (uf_1 uf_18 #44)
+#1043 := (* -1::real #1040)
+#1156 := (+ #771 #1043)
+#1157 := (>= #1156 0::real)
+#1189 := (not #1157)
+#708 := (uf_1 #91 #44)
+#1168 := (+ #708 #1043)
+#1169 := (<= #1168 0::real)
+#1174 := (or #1157 #1169)
+#1177 := (not #1174)
+#89 := (uf_1 #88 #6)
+#552 := (pattern #89)
+#81 := (uf_1 uf_18 #6)
+#594 := (pattern #81)
+#324 := (* -1::real #92)
+#325 := (+ #81 #324)
+#323 := (>= #325 0::real)
+#317 := (* -1::real #89)
+#318 := (+ #81 #317)
+#319 := (<= #318 0::real)
+#436 := (or #319 #323)
+#437 := (not #436)
+#601 := (forall (vars (?x11 T2)) (:pat #594 #552 #542) #437)
+#440 := (forall (vars (?x11 T2)) #437)
+#604 := (iff #440 #601)
+#602 := (iff #437 #437)
+#603 := [refl]: #602
+#605 := [quant-intro #603]: #604
+#326 := (not #323)
+#320 := (not #319)
+#329 := (and #320 #326)
+#332 := (forall (vars (?x11 T2)) #329)
+#441 := (iff #332 #440)
+#438 := (iff #329 #437)
+#439 := [rewrite]: #438
+#442 := [quant-intro #439]: #441
+#425 := (~ #332 #332)
+#423 := (~ #329 #329)
+#424 := [refl]: #423
+#426 := [nnf-pos #424]: #425
+#306 := (* -1::real #84)
+#307 := (+ #81 #306)
+#305 := (>= #307 0::real)
+#308 := (not #305)
+#301 := (* -1::real #81)
+#79 := (uf_1 #78 #6)
+#302 := (+ #79 #301)
+#300 := (>= #302 0::real)
+#298 := (not #300)
+#311 := (and #298 #308)
+#314 := (forall (vars (?x10 T2)) #311)
+#335 := (and #314 #332)
+#93 := (< #81 #92)
+#90 := (< #89 #81)
+#94 := (and #90 #93)
+#95 := (forall (vars (?x11 T2)) #94)
+#85 := (< #81 #84)
+#82 := (< #79 #81)
+#86 := (and #82 #85)
+#87 := (forall (vars (?x10 T2)) #86)
+#96 := (and #87 #95)
+#336 := (iff #96 #335)
+#333 := (iff #95 #332)
+#330 := (iff #94 #329)
+#327 := (iff #93 #326)
+#328 := [rewrite]: #327
+#321 := (iff #90 #320)
+#322 := [rewrite]: #321
+#331 := [monotonicity #322 #328]: #330
+#334 := [quant-intro #331]: #333
+#315 := (iff #87 #314)
+#312 := (iff #86 #311)
+#309 := (iff #85 #308)
+#310 := [rewrite]: #309
+#303 := (iff #82 #298)
+#304 := [rewrite]: #303
+#313 := [monotonicity #304 #310]: #312
+#316 := [quant-intro #313]: #315
+#337 := [monotonicity #316 #334]: #336
+#293 := [asserted]: #96
+#338 := [mp #293 #337]: #335
+#340 := [and-elim #338]: #332
+#427 := [mp~ #340 #426]: #332
+#443 := [mp #427 #442]: #440
+#606 := [mp #443 #605]: #601
+#1124 := (not #601)
+#1180 := (or #1124 #1177)
+#1142 := (* -1::real #708)
+#1143 := (+ #1040 #1142)
+#1144 := (>= #1143 0::real)
+#1146 := (+ #1040 #1145)
+#1147 := (<= #1146 0::real)
+#1148 := (or #1147 #1144)
+#1149 := (not #1148)
+#1181 := (or #1124 #1149)
+#1183 := (iff #1181 #1180)
+#1185 := (iff #1180 #1180)
+#1186 := [rewrite]: #1185
+#1178 := (iff #1149 #1177)
+#1175 := (iff #1148 #1174)
+#1172 := (iff #1144 #1169)
+#1162 := (+ #1142 #1040)
+#1165 := (>= #1162 0::real)
+#1170 := (iff #1165 #1169)
+#1171 := [rewrite]: #1170
+#1166 := (iff #1144 #1165)
+#1163 := (= #1143 #1162)
+#1164 := [rewrite]: #1163
+#1167 := [monotonicity #1164]: #1166
+#1173 := [trans #1167 #1171]: #1172
+#1160 := (iff #1147 #1157)
+#1150 := (+ #1145 #1040)
+#1153 := (<= #1150 0::real)
+#1158 := (iff #1153 #1157)
+#1159 := [rewrite]: #1158
+#1154 := (iff #1147 #1153)
+#1151 := (= #1146 #1150)
+#1152 := [rewrite]: #1151
+#1155 := [monotonicity #1152]: #1154
+#1161 := [trans #1155 #1159]: #1160
+#1176 := [monotonicity #1161 #1173]: #1175
+#1179 := [monotonicity #1176]: #1178
+#1184 := [monotonicity #1179]: #1183
+#1187 := [trans #1184 #1186]: #1183
+#1182 := [quant-inst]: #1181
+#1188 := [mp #1182 #1187]: #1180
+#1244 := [unit-resolution #1188 #606]: #1177
+#1190 := (or #1174 #1189)
+#1191 := [def-axiom]: #1190
+#1245 := [unit-resolution #1191 #1244]: #1189
+#1054 := (+ #656 #1043)
+#1055 := (<= #1054 0::real)
+#1079 := (not #1055)
+#607 := (uf_1 #78 #44)
+#1044 := (+ #607 #1043)
+#1045 := (>= #1044 0::real)
+#1060 := (or #1045 #1055)
+#1063 := (not #1060)
+#567 := (pattern #79)
+#428 := (or #300 #305)
+#429 := (not #428)
+#595 := (forall (vars (?x10 T2)) (:pat #567 #594 #560) #429)
+#432 := (forall (vars (?x10 T2)) #429)
+#598 := (iff #432 #595)
+#596 := (iff #429 #429)
+#597 := [refl]: #596
+#599 := [quant-intro #597]: #598
+#433 := (iff #314 #432)
+#430 := (iff #311 #429)
+#431 := [rewrite]: #430
+#434 := [quant-intro #431]: #433
+#420 := (~ #314 #314)
+#418 := (~ #311 #311)
+#419 := [refl]: #418
+#421 := [nnf-pos #419]: #420
+#339 := [and-elim #338]: #314
+#422 := [mp~ #339 #421]: #314
+#435 := [mp #422 #434]: #432
+#600 := [mp #435 #599]: #595
+#1066 := (not #595)
+#1067 := (or #1066 #1063)
+#1039 := (* -1::real #656)
+#1041 := (+ #1040 #1039)
+#1042 := (>= #1041 0::real)
+#1046 := (or #1045 #1042)
+#1047 := (not #1046)
+#1068 := (or #1066 #1047)
+#1070 := (iff #1068 #1067)
+#1072 := (iff #1067 #1067)
+#1073 := [rewrite]: #1072
+#1064 := (iff #1047 #1063)
+#1061 := (iff #1046 #1060)
+#1058 := (iff #1042 #1055)
+#1048 := (+ #1039 #1040)
+#1051 := (>= #1048 0::real)
+#1056 := (iff #1051 #1055)
+#1057 := [rewrite]: #1056
+#1052 := (iff #1042 #1051)
+#1049 := (= #1041 #1048)
+#1050 := [rewrite]: #1049
+#1053 := [monotonicity #1050]: #1052
+#1059 := [trans #1053 #1057]: #1058
+#1062 := [monotonicity #1059]: #1061
+#1065 := [monotonicity #1062]: #1064
+#1071 := [monotonicity #1065]: #1070
+#1074 := [trans #1071 #1073]: #1070
+#1069 := [quant-inst]: #1068
+#1075 := [mp #1069 #1074]: #1067
+#1246 := [unit-resolution #1075 #600]: #1063
+#1080 := (or #1060 #1079)
+#1081 := [def-axiom]: #1080
+#1247 := [unit-resolution #1081 #1246]: #1079
+#1248 := [th-lemma #1247 #1245 #1243]: false
+#1250 := [lemma #1248]: #1249
+#1253 := (or #1252 #1241)
+#1254 := [th-lemma]: #1253
+#1310 := [unit-resolution #1254 #1250]: #1252
+#1326 := [mp #1310 #1325]: #796
+#1328 := (or #724 #772)
+decl uf_13 :: T3
+#33 := uf_13
+#609 := (uf_1 uf_13 #44)
+#773 := (= #609 #771)
+#775 := (ite #724 #773 #772)
+#32 := (uf_1 uf_9 #6)
+#553 := (pattern #32)
+#34 := (uf_1 uf_13 #6)
+#551 := (pattern #34)
+#456 := (= #32 #89)
+#455 := (= #34 #89)
+#457 := (ite #119 #455 #456)
+#554 := (forall (vars (?x4 T2)) (:pat #541 #551 #552 #553) #457)
+#460 := (forall (vars (?x4 T2)) #457)
+#557 := (iff #460 #554)
+#555 := (iff #457 #457)
+#556 := [refl]: #555
+#558 := [quant-intro #556]: #557
+#143 := (ite #119 #34 #32)
+#373 := (= #89 #143)
+#374 := (forall (vars (?x4 T2)) #373)
+#461 := (iff #374 #460)
+#458 := (iff #373 #457)
+#459 := [rewrite]: #458
+#462 := [quant-intro #459]: #461
+#362 := (~ #374 #374)
+#364 := (~ #373 #373)
+#365 := [refl]: #364
+#363 := [nnf-pos #365]: #362
+#31 := (uf_3 uf_12 #6)
+#148 := (= #31 #143)
+#151 := (forall (vars (?x4 T2)) #148)
+#375 := (iff #151 #374)
+#376 := [rewrite* #113]: #375
+#35 := (ite #17 #32 #34)
+#36 := (= #31 #35)
+#37 := (forall (vars (?x4 T2)) #36)
+#152 := (iff #37 #151)
+#149 := (iff #36 #148)
+#146 := (= #35 #143)
+#140 := (ite #118 #32 #34)
+#144 := (= #140 #143)
+#145 := [rewrite]: #144
+#141 := (= #35 #140)
+#142 := [monotonicity #123]: #141
+#147 := [trans #142 #145]: #146
+#150 := [monotonicity #147]: #149
+#153 := [quant-intro #150]: #152
+#115 := [asserted]: #37
+#154 := [mp #115 #153]: #151
+#377 := [mp #154 #376]: #374
+#360 := [mp~ #377 #363]: #374
+#463 := [mp #360 #462]: #460
+#559 := [mp #463 #558]: #554
+#778 := (not #554)
+#779 := (or #778 #775)
+#714 := (+ #611 #120)
+#715 := (>= #714 0::int)
+#774 := (ite #715 #773 #772)
+#780 := (or #778 #774)
+#782 := (iff #780 #779)
+#784 := (iff #779 #779)
+#785 := [rewrite]: #784
+#776 := (iff #774 #775)
+#727 := (iff #715 #724)
+#717 := (+ #120 #611)
+#720 := (>= #717 0::int)
+#725 := (iff #720 #724)
+#726 := [rewrite]: #725
+#721 := (iff #715 #720)
+#718 := (= #714 #717)
+#719 := [rewrite]: #718
+#722 := [monotonicity #719]: #721
+#728 := [trans #722 #726]: #727
+#777 := [monotonicity #728]: #776
+#783 := [monotonicity #777]: #782
+#786 := [trans #783 #785]: #782
+#781 := [quant-inst]: #780
+#787 := [mp #781 #786]: #779
+#1327 := [unit-resolution #787 #559]: #775
+#788 := (not #775)
+#791 := (or #788 #724 #772)
+#792 := [def-axiom]: #791
+#1329 := [unit-resolution #792 #1327]: #1328
+#1330 := [unit-resolution #1329 #1326]: #724
+#988 := (>= #622 0::int)
+#1331 := (or #1312 #988)
+#1332 := [th-lemma]: #1331
+#1333 := [unit-resolution #1332 #1281]: #988
+#761 := (not #724)
+#1334 := (not #988)
+#1335 := (or #1271 #1334 #761)
+#1336 := [th-lemma]: #1335
+#1337 := [unit-resolution #1336 #1333 #1330]: #1271
+#1338 := (not #1271)
+#1340 := (or #1269 #1338 #1339)
+#1341 := [th-lemma]: #1340
+#1343 := [unit-resolution #1341 #1337]: #1342
+#1344 := [unit-resolution #1343 #1299]: #1339
+#990 := (>= #916 0::int)
+#1345 := (or #1302 #990)
+#1346 := [th-lemma]: #1345
+#1347 := [unit-resolution #1346 #1301]: #990
+#1348 := (not #990)
+#1349 := (or #836 #1348 #1265)
+#1350 := [th-lemma]: #1349
+#1351 := [unit-resolution #1350 #1347 #1344]: #836
+#1353 := (or #815 #800)
+#801 := (uf_1 uf_13 #22)
+#820 := (= #799 #801)
+#823 := (ite #815 #820 #800)
+#476 := (= #32 #79)
+#475 := (= #34 #79)
+#477 := (ite #157 #475 #476)
+#568 := (forall (vars (?x6 T2)) (:pat #541 #551 #567 #553) #477)
+#480 := (forall (vars (?x6 T2)) #477)
+#571 := (iff #480 #568)
+#569 := (iff #477 #477)
+#570 := [refl]: #569
+#572 := [quant-intro #570]: #571
+#181 := (ite #157 #34 #32)
+#383 := (= #79 #181)
+#384 := (forall (vars (?x6 T2)) #383)
+#481 := (iff #384 #480)
+#478 := (iff #383 #477)
+#479 := [rewrite]: #478
+#482 := [quant-intro #479]: #481
+#352 := (~ #384 #384)
+#354 := (~ #383 #383)
+#355 := [refl]: #354
+#353 := [nnf-pos #355]: #352
+#51 := (uf_3 uf_16 #6)
+#186 := (= #51 #181)
+#189 := (forall (vars (?x6 T2)) #186)
+#385 := (iff #189 #384)
+#386 := [rewrite* #113]: #385
+#52 := (ite #42 #32 #34)
+#53 := (= #51 #52)
+#54 := (forall (vars (?x6 T2)) #53)
+#190 := (iff #54 #189)
+#187 := (iff #53 #186)
+#184 := (= #52 #181)
+#178 := (ite #156 #32 #34)
+#182 := (= #178 #181)
+#183 := [rewrite]: #182
+#179 := (= #52 #178)
+#180 := [monotonicity #161]: #179
+#185 := [trans #180 #183]: #184
+#188 := [monotonicity #185]: #187
+#191 := [quant-intro #188]: #190
+#139 := [asserted]: #54
+#192 := [mp #139 #191]: #189
+#387 := [mp #192 #386]: #384
+#402 := [mp~ #387 #353]: #384
+#483 := [mp #402 #482]: #480
+#573 := [mp #483 #572]: #568
+#634 := (not #568)
+#826 := (or #634 #823)
+#802 := (= #801 #799)
+#804 := (+ #803 #158)
+#805 := (>= #804 0::int)
+#806 := (ite #805 #802 #800)
+#827 := (or #634 #806)
+#829 := (iff #827 #826)
+#831 := (iff #826 #826)
+#832 := [rewrite]: #831
+#824 := (iff #806 #823)
+#821 := (iff #802 #820)
+#822 := [rewrite]: #821
+#818 := (iff #805 #815)
+#807 := (+ #158 #803)
+#810 := (>= #807 0::int)
+#816 := (iff #810 #815)
+#817 := [rewrite]: #816
+#811 := (iff #805 #810)
+#808 := (= #804 #807)
+#809 := [rewrite]: #808
+#812 := [monotonicity #809]: #811
+#819 := [trans #812 #817]: #818
+#825 := [monotonicity #819 #822]: #824
+#830 := [monotonicity #825]: #829
+#833 := [trans #830 #832]: #829
+#828 := [quant-inst]: #827
+#834 := [mp #828 #833]: #826
+#1352 := [unit-resolution #834 #573]: #823
+#835 := (not #823)
+#839 := (or #835 #815 #800)
+#840 := [def-axiom]: #839
+#1354 := [unit-resolution #840 #1352]: #1353
+#1355 := [unit-resolution #1354 #1351]: #800
+#1357 := [symm #1355]: #1356
+#1358 := [trans #1357 #1309]: #1266
+#1359 := (not #1266)
+#1360 := (or #1359 #1272)
+#1361 := [th-lemma]: #1360
+#1362 := [unit-resolution #1361 #1358]: #1272
+#1085 := (uf_1 uf_18 #22)
+#1099 := (* -1::real #1085)
+#1112 := (+ #902 #1099)
+#1113 := (<= #1112 0::real)
+#1137 := (not #1113)
+#960 := (uf_1 #88 #22)
+#1100 := (+ #960 #1099)
+#1101 := (>= #1100 0::real)
+#1118 := (or #1101 #1113)
+#1121 := (not #1118)
+#1125 := (or #1124 #1121)
+#1086 := (+ #1085 #1084)
+#1087 := (>= #1086 0::real)
+#1088 := (* -1::real #960)
+#1089 := (+ #1085 #1088)
+#1090 := (<= #1089 0::real)
+#1091 := (or #1090 #1087)
+#1092 := (not #1091)
+#1126 := (or #1124 #1092)
+#1128 := (iff #1126 #1125)
+#1130 := (iff #1125 #1125)
+#1131 := [rewrite]: #1130
+#1122 := (iff #1092 #1121)
+#1119 := (iff #1091 #1118)
+#1116 := (iff #1087 #1113)
+#1106 := (+ #1084 #1085)
+#1109 := (>= #1106 0::real)
+#1114 := (iff #1109 #1113)
+#1115 := [rewrite]: #1114
+#1110 := (iff #1087 #1109)
+#1107 := (= #1086 #1106)
+#1108 := [rewrite]: #1107
+#1111 := [monotonicity #1108]: #1110
+#1117 := [trans #1111 #1115]: #1116
+#1104 := (iff #1090 #1101)
+#1093 := (+ #1088 #1085)
+#1096 := (<= #1093 0::real)
+#1102 := (iff #1096 #1101)
+#1103 := [rewrite]: #1102
+#1097 := (iff #1090 #1096)
+#1094 := (= #1089 #1093)
+#1095 := [rewrite]: #1094
+#1098 := [monotonicity #1095]: #1097
+#1105 := [trans #1098 #1103]: #1104
+#1120 := [monotonicity #1105 #1117]: #1119
+#1123 := [monotonicity #1120]: #1122
+#1129 := [monotonicity #1123]: #1128
+#1132 := [trans #1129 #1131]: #1128
+#1127 := [quant-inst]: #1126
+#1133 := [mp #1127 #1132]: #1125
+#1363 := [unit-resolution #1133 #606]: #1121
+#1138 := (or #1118 #1137)
+#1139 := [def-axiom]: #1138
+#1364 := [unit-resolution #1139 #1363]: #1137
+#1200 := (+ #799 #1099)
+#1201 := (>= #1200 0::real)
+#1231 := (not #1201)
+#847 := (uf_1 #83 #22)
+#1210 := (+ #847 #1099)
+#1211 := (<= #1210 0::real)
+#1216 := (or #1201 #1211)
+#1219 := (not #1216)
+#1222 := (or #1066 #1219)
+#1197 := (* -1::real #847)
+#1198 := (+ #1085 #1197)
+#1199 := (>= #1198 0::real)
+#1202 := (or #1201 #1199)
+#1203 := (not #1202)
+#1223 := (or #1066 #1203)
+#1225 := (iff #1223 #1222)
+#1227 := (iff #1222 #1222)
+#1228 := [rewrite]: #1227
+#1220 := (iff #1203 #1219)
+#1217 := (iff #1202 #1216)
+#1214 := (iff #1199 #1211)
+#1204 := (+ #1197 #1085)
+#1207 := (>= #1204 0::real)
+#1212 := (iff #1207 #1211)
+#1213 := [rewrite]: #1212
+#1208 := (iff #1199 #1207)
+#1205 := (= #1198 #1204)
+#1206 := [rewrite]: #1205
+#1209 := [monotonicity #1206]: #1208
+#1215 := [trans #1209 #1213]: #1214
+#1218 := [monotonicity #1215]: #1217
+#1221 := [monotonicity #1218]: #1220
+#1226 := [monotonicity #1221]: #1225
+#1229 := [trans #1226 #1228]: #1225
+#1224 := [quant-inst]: #1223
+#1230 := [mp #1224 #1229]: #1222
+#1365 := [unit-resolution #1230 #600]: #1219
+#1232 := (or #1216 #1231)
+#1233 := [def-axiom]: #1232
+#1366 := [unit-resolution #1233 #1365]: #1231
+[th-lemma #1366 #1364 #1362]: false
+unsat
+NQHwTeL311Tq3wf2s5BReA 419 0
+#2 := false
+#194 := 0::real
+decl uf_4 :: (-> T2 T3 real)
+decl uf_6 :: (-> T1 T3)
+decl uf_3 :: T1
+#21 := uf_3
+#25 := (uf_6 uf_3)
+decl uf_5 :: T2
+#24 := uf_5
+#26 := (uf_4 uf_5 #25)
+decl uf_7 :: T2
+#27 := uf_7
+#28 := (uf_4 uf_7 #25)
+decl uf_10 :: T1
+#38 := uf_10
+#42 := (uf_6 uf_10)
+decl uf_9 :: T2
+#33 := uf_9
+#43 := (uf_4 uf_9 #42)
+#41 := (= uf_3 uf_10)
+#44 := (ite #41 #43 #28)
+#9 := 0::int
+decl uf_2 :: (-> T1 int)
+#39 := (uf_2 uf_10)
+#226 := -1::int
+#229 := (* -1::int #39)
+#22 := (uf_2 uf_3)
+#230 := (+ #22 #229)
+#228 := (>= #230 0::int)
+#236 := (ite #228 #44 #26)
+#192 := -1::real
+#244 := (* -1::real #236)
+#642 := (+ #26 #244)
+#643 := (<= #642 0::real)
+#567 := (= #26 #236)
+#227 := (not #228)
+decl uf_1 :: (-> int T1)
+#593 := (uf_1 #39)
+#660 := (= #593 uf_10)
+#594 := (= uf_10 #593)
+#4 := (:var 0 T1)
+#5 := (uf_2 #4)
+#546 := (pattern #5)
+#6 := (uf_1 #5)
+#93 := (= #4 #6)
+#547 := (forall (vars (?x1 T1)) (:pat #546) #93)
+#96 := (forall (vars (?x1 T1)) #93)
+#550 := (iff #96 #547)
+#548 := (iff #93 #93)
+#549 := [refl]: #548
+#551 := [quant-intro #549]: #550
+#448 := (~ #96 #96)
+#450 := (~ #93 #93)
+#451 := [refl]: #450
+#449 := [nnf-pos #451]: #448
+#7 := (= #6 #4)
+#8 := (forall (vars (?x1 T1)) #7)
+#97 := (iff #8 #96)
+#94 := (iff #7 #93)
+#95 := [rewrite]: #94
+#98 := [quant-intro #95]: #97
+#92 := [asserted]: #8
+#101 := [mp #92 #98]: #96
+#446 := [mp~ #101 #449]: #96
+#552 := [mp #446 #551]: #547
+#595 := (not #547)
+#600 := (or #595 #594)
+#601 := [quant-inst]: #600
+#654 := [unit-resolution #601 #552]: #594
+#680 := [symm #654]: #660
+#681 := (= uf_3 #593)
+#591 := (uf_1 #22)
+#658 := (= #591 #593)
+#656 := (= #593 #591)
+#652 := (= #39 #22)
+#647 := (= #22 #39)
+#290 := (<= #230 0::int)
+#70 := (<= #22 #39)
+#388 := (iff #70 #290)
+#389 := [rewrite]: #388
+#341 := [asserted]: #70
+#390 := [mp #341 #389]: #290
+#646 := [hypothesis]: #228
+#648 := [th-lemma #646 #390]: #647
+#653 := [symm #648]: #652
+#657 := [monotonicity #653]: #656
+#659 := [symm #657]: #658
+#592 := (= uf_3 #591)
+#596 := (or #595 #592)
+#597 := [quant-inst]: #596
+#655 := [unit-resolution #597 #552]: #592
+#682 := [trans #655 #659]: #681
+#683 := [trans #682 #680]: #41
+#570 := (not #41)
+decl uf_11 :: T2
+#47 := uf_11
+#59 := (uf_4 uf_11 #42)
+#278 := (ite #41 #26 #59)
+#459 := (* -1::real #278)
+#637 := (+ #26 #459)
+#639 := (>= #637 0::real)
+#585 := (= #26 #278)
+#661 := [hypothesis]: #41
+#587 := (or #570 #585)
+#588 := [def-axiom]: #587
+#662 := [unit-resolution #588 #661]: #585
+#663 := (not #585)
+#664 := (or #663 #639)
+#665 := [th-lemma]: #664
+#666 := [unit-resolution #665 #662]: #639
+decl uf_8 :: T2
+#30 := uf_8
+#56 := (uf_4 uf_8 #42)
+#357 := (* -1::real #56)
+#358 := (+ #43 #357)
+#356 := (>= #358 0::real)
+#355 := (not #356)
+#374 := (* -1::real #59)
+#375 := (+ #56 #374)
+#373 := (>= #375 0::real)
+#376 := (not #373)
+#381 := (and #355 #376)
+#64 := (< #39 #39)
+#67 := (ite #64 #43 #59)
+#68 := (< #56 #67)
+#53 := (uf_4 uf_5 #42)
+#65 := (ite #64 #53 #43)
+#66 := (< #65 #56)
+#69 := (and #66 #68)
+#382 := (iff #69 #381)
+#379 := (iff #68 #376)
+#370 := (< #56 #59)
+#377 := (iff #370 #376)
+#378 := [rewrite]: #377
+#371 := (iff #68 #370)
+#368 := (= #67 #59)
+#363 := (ite false #43 #59)
+#366 := (= #363 #59)
+#367 := [rewrite]: #366
+#364 := (= #67 #363)
+#343 := (iff #64 false)
+#344 := [rewrite]: #343
+#365 := [monotonicity #344]: #364
+#369 := [trans #365 #367]: #368
+#372 := [monotonicity #369]: #371
+#380 := [trans #372 #378]: #379
+#361 := (iff #66 #355)
+#352 := (< #43 #56)
+#359 := (iff #352 #355)
+#360 := [rewrite]: #359
+#353 := (iff #66 #352)
+#350 := (= #65 #43)
+#345 := (ite false #53 #43)
+#348 := (= #345 #43)
+#349 := [rewrite]: #348
+#346 := (= #65 #345)
+#347 := [monotonicity #344]: #346
+#351 := [trans #347 #349]: #350
+#354 := [monotonicity #351]: #353
+#362 := [trans #354 #360]: #361
+#383 := [monotonicity #362 #380]: #382
+#340 := [asserted]: #69
+#384 := [mp #340 #383]: #381
+#385 := [and-elim #384]: #355
+#394 := (* -1::real #53)
+#395 := (+ #43 #394)
+#393 := (>= #395 0::real)
+#54 := (uf_4 uf_7 #42)
+#402 := (* -1::real #54)
+#403 := (+ #53 #402)
+#401 := (>= #403 0::real)
+#397 := (+ #43 #374)
+#398 := (<= #397 0::real)
+#412 := (and #393 #398 #401)
+#73 := (<= #43 #59)
+#72 := (<= #53 #43)
+#74 := (and #72 #73)
+#71 := (<= #54 #53)
+#75 := (and #71 #74)
+#415 := (iff #75 #412)
+#406 := (and #393 #398)
+#409 := (and #401 #406)
+#413 := (iff #409 #412)
+#414 := [rewrite]: #413
+#410 := (iff #75 #409)
+#407 := (iff #74 #406)
+#399 := (iff #73 #398)
+#400 := [rewrite]: #399
+#392 := (iff #72 #393)
+#396 := [rewrite]: #392
+#408 := [monotonicity #396 #400]: #407
+#404 := (iff #71 #401)
+#405 := [rewrite]: #404
+#411 := [monotonicity #405 #408]: #410
+#416 := [trans #411 #414]: #415
+#342 := [asserted]: #75
+#417 := [mp #342 #416]: #412
+#418 := [and-elim #417]: #393
+#650 := (+ #26 #394)
+#651 := (<= #650 0::real)
+#649 := (= #26 #53)
+#671 := (= #53 #26)
+#669 := (= #42 #25)
+#667 := (= #25 #42)
+#668 := [monotonicity #661]: #667
+#670 := [symm #668]: #669
+#672 := [monotonicity #670]: #671
+#673 := [symm #672]: #649
+#674 := (not #649)
+#675 := (or #674 #651)
+#676 := [th-lemma]: #675
+#677 := [unit-resolution #676 #673]: #651
+#462 := (+ #56 #459)
+#465 := (>= #462 0::real)
+#438 := (not #465)
+#316 := (ite #290 #278 #43)
+#326 := (* -1::real #316)
+#327 := (+ #56 #326)
+#325 := (>= #327 0::real)
+#324 := (not #325)
+#439 := (iff #324 #438)
+#466 := (iff #325 #465)
+#463 := (= #327 #462)
+#460 := (= #326 #459)
+#457 := (= #316 #278)
+#1 := true
+#452 := (ite true #278 #43)
+#455 := (= #452 #278)
+#456 := [rewrite]: #455
+#453 := (= #316 #452)
+#444 := (iff #290 true)
+#445 := [iff-true #390]: #444
+#454 := [monotonicity #445]: #453
+#458 := [trans #454 #456]: #457
+#461 := [monotonicity #458]: #460
+#464 := [monotonicity #461]: #463
+#467 := [monotonicity #464]: #466
+#468 := [monotonicity #467]: #439
+#297 := (ite #290 #54 #53)
+#305 := (* -1::real #297)
+#306 := (+ #56 #305)
+#307 := (<= #306 0::real)
+#308 := (not #307)
+#332 := (and #308 #324)
+#58 := (= uf_10 uf_3)
+#60 := (ite #58 #26 #59)
+#52 := (< #39 #22)
+#61 := (ite #52 #43 #60)
+#62 := (< #56 #61)
+#55 := (ite #52 #53 #54)
+#57 := (< #55 #56)
+#63 := (and #57 #62)
+#335 := (iff #63 #332)
+#281 := (ite #52 #43 #278)
+#284 := (< #56 #281)
+#287 := (and #57 #284)
+#333 := (iff #287 #332)
+#330 := (iff #284 #324)
+#321 := (< #56 #316)
+#328 := (iff #321 #324)
+#329 := [rewrite]: #328
+#322 := (iff #284 #321)
+#319 := (= #281 #316)
+#291 := (not #290)
+#313 := (ite #291 #43 #278)
+#317 := (= #313 #316)
+#318 := [rewrite]: #317
+#314 := (= #281 #313)
+#292 := (iff #52 #291)
+#293 := [rewrite]: #292
+#315 := [monotonicity #293]: #314
+#320 := [trans #315 #318]: #319
+#323 := [monotonicity #320]: #322
+#331 := [trans #323 #329]: #330
+#311 := (iff #57 #308)
+#302 := (< #297 #56)
+#309 := (iff #302 #308)
+#310 := [rewrite]: #309
+#303 := (iff #57 #302)
+#300 := (= #55 #297)
+#294 := (ite #291 #53 #54)
+#298 := (= #294 #297)
+#299 := [rewrite]: #298
+#295 := (= #55 #294)
+#296 := [monotonicity #293]: #295
+#301 := [trans #296 #299]: #300
+#304 := [monotonicity #301]: #303
+#312 := [trans #304 #310]: #311
+#334 := [monotonicity #312 #331]: #333
+#288 := (iff #63 #287)
+#285 := (iff #62 #284)
+#282 := (= #61 #281)
+#279 := (= #60 #278)
+#225 := (iff #58 #41)
+#277 := [rewrite]: #225
+#280 := [monotonicity #277]: #279
+#283 := [monotonicity #280]: #282
+#286 := [monotonicity #283]: #285
+#289 := [monotonicity #286]: #288
+#336 := [trans #289 #334]: #335
+#179 := [asserted]: #63
+#337 := [mp #179 #336]: #332
+#339 := [and-elim #337]: #324
+#469 := [mp #339 #468]: #438
+#678 := [th-lemma #469 #677 #418 #385 #666]: false
+#679 := [lemma #678]: #570
+#684 := [unit-resolution #679 #683]: false
+#685 := [lemma #684]: #227
+#577 := (or #228 #567)
+#578 := [def-axiom]: #577
+#645 := [unit-resolution #578 #685]: #567
+#686 := (not #567)
+#687 := (or #686 #643)
+#688 := [th-lemma]: #687
+#689 := [unit-resolution #688 #645]: #643
+#31 := (uf_4 uf_8 #25)
+#245 := (+ #31 #244)
+#246 := (<= #245 0::real)
+#247 := (not #246)
+#34 := (uf_4 uf_9 #25)
+#48 := (uf_4 uf_11 #25)
+#255 := (ite #228 #48 #34)
+#264 := (* -1::real #255)
+#265 := (+ #31 #264)
+#263 := (>= #265 0::real)
+#266 := (not #263)
+#271 := (and #247 #266)
+#40 := (< #22 #39)
+#49 := (ite #40 #34 #48)
+#50 := (< #31 #49)
+#45 := (ite #40 #26 #44)
+#46 := (< #45 #31)
+#51 := (and #46 #50)
+#272 := (iff #51 #271)
+#269 := (iff #50 #266)
+#260 := (< #31 #255)
+#267 := (iff #260 #266)
+#268 := [rewrite]: #267
+#261 := (iff #50 #260)
+#258 := (= #49 #255)
+#252 := (ite #227 #34 #48)
+#256 := (= #252 #255)
+#257 := [rewrite]: #256
+#253 := (= #49 #252)
+#231 := (iff #40 #227)
+#232 := [rewrite]: #231
+#254 := [monotonicity #232]: #253
+#259 := [trans #254 #257]: #258
+#262 := [monotonicity #259]: #261
+#270 := [trans #262 #268]: #269
+#250 := (iff #46 #247)
+#241 := (< #236 #31)
+#248 := (iff #241 #247)
+#249 := [rewrite]: #248
+#242 := (iff #46 #241)
+#239 := (= #45 #236)
+#233 := (ite #227 #26 #44)
+#237 := (= #233 #236)
+#238 := [rewrite]: #237
+#234 := (= #45 #233)
+#235 := [monotonicity #232]: #234
+#240 := [trans #235 #238]: #239
+#243 := [monotonicity #240]: #242
+#251 := [trans #243 #249]: #250
+#273 := [monotonicity #251 #270]: #272
+#178 := [asserted]: #51
+#274 := [mp #178 #273]: #271
+#275 := [and-elim #274]: #247
+#196 := (* -1::real #31)
+#212 := (+ #26 #196)
+#213 := (<= #212 0::real)
+#214 := (not #213)
+#197 := (+ #28 #196)
+#195 := (>= #197 0::real)
+#193 := (not #195)
+#219 := (and #193 #214)
+#23 := (< #22 #22)
+#35 := (ite #23 #34 #26)
+#36 := (< #31 #35)
+#29 := (ite #23 #26 #28)
+#32 := (< #29 #31)
+#37 := (and #32 #36)
+#220 := (iff #37 #219)
+#217 := (iff #36 #214)
+#209 := (< #31 #26)
+#215 := (iff #209 #214)
+#216 := [rewrite]: #215
+#210 := (iff #36 #209)
+#207 := (= #35 #26)
+#202 := (ite false #34 #26)
+#205 := (= #202 #26)
+#206 := [rewrite]: #205
+#203 := (= #35 #202)
+#180 := (iff #23 false)
+#181 := [rewrite]: #180
+#204 := [monotonicity #181]: #203
+#208 := [trans #204 #206]: #207
+#211 := [monotonicity #208]: #210
+#218 := [trans #211 #216]: #217
+#200 := (iff #32 #193)
+#189 := (< #28 #31)
+#198 := (iff #189 #193)
+#199 := [rewrite]: #198
+#190 := (iff #32 #189)
+#187 := (= #29 #28)
+#182 := (ite false #26 #28)
+#185 := (= #182 #28)
+#186 := [rewrite]: #185
+#183 := (= #29 #182)
+#184 := [monotonicity #181]: #183
+#188 := [trans #184 #186]: #187
+#191 := [monotonicity #188]: #190
+#201 := [trans #191 #199]: #200
+#221 := [monotonicity #201 #218]: #220
+#177 := [asserted]: #37
+#222 := [mp #177 #221]: #219
+#224 := [and-elim #222]: #214
+[th-lemma #224 #275 #689]: false
+unsat
+NX/HT1QOfbspC2LtZNKpBA 428 0
+#2 := false
+decl uf_10 :: T1
+#38 := uf_10
+decl uf_3 :: T1
+#21 := uf_3
+#45 := (= uf_3 uf_10)
+decl uf_1 :: (-> int T1)
+decl uf_2 :: (-> T1 int)
+#39 := (uf_2 uf_10)
+#588 := (uf_1 #39)
+#686 := (= #588 uf_10)
+#589 := (= uf_10 #588)
+#4 := (:var 0 T1)
+#5 := (uf_2 #4)
+#541 := (pattern #5)
+#6 := (uf_1 #5)
+#93 := (= #4 #6)
+#542 := (forall (vars (?x1 T1)) (:pat #541) #93)
+#96 := (forall (vars (?x1 T1)) #93)
+#545 := (iff #96 #542)
+#543 := (iff #93 #93)
+#544 := [refl]: #543
+#546 := [quant-intro #544]: #545
+#454 := (~ #96 #96)
+#456 := (~ #93 #93)
+#457 := [refl]: #456
+#455 := [nnf-pos #457]: #454
+#7 := (= #6 #4)
+#8 := (forall (vars (?x1 T1)) #7)
+#97 := (iff #8 #96)
+#94 := (iff #7 #93)
+#95 := [rewrite]: #94
+#98 := [quant-intro #95]: #97
+#92 := [asserted]: #8
+#101 := [mp #92 #98]: #96
+#452 := [mp~ #101 #455]: #96
+#547 := [mp #452 #546]: #542
+#590 := (not #542)
+#595 := (or #590 #589)
+#596 := [quant-inst]: #595
+#680 := [unit-resolution #596 #547]: #589
+#687 := [symm #680]: #686
+#688 := (= uf_3 #588)
+#22 := (uf_2 uf_3)
+#586 := (uf_1 #22)
+#684 := (= #586 #588)
+#682 := (= #588 #586)
+#678 := (= #39 #22)
+#676 := (= #22 #39)
+#9 := 0::int
+#227 := -1::int
+#230 := (* -1::int #39)
+#231 := (+ #22 #230)
+#296 := (<= #231 0::int)
+#70 := (<= #22 #39)
+#393 := (iff #70 #296)
+#394 := [rewrite]: #393
+#347 := [asserted]: #70
+#395 := [mp #347 #394]: #296
+#229 := (>= #231 0::int)
+decl uf_4 :: (-> T2 T3 real)
+decl uf_6 :: (-> T1 T3)
+#25 := (uf_6 uf_3)
+decl uf_7 :: T2
+#27 := uf_7
+#28 := (uf_4 uf_7 #25)
+decl uf_9 :: T2
+#33 := uf_9
+#34 := (uf_4 uf_9 #25)
+#46 := (uf_6 uf_10)
+decl uf_5 :: T2
+#24 := uf_5
+#47 := (uf_4 uf_5 #46)
+#48 := (ite #45 #47 #34)
+#256 := (ite #229 #48 #28)
+#568 := (= #28 #256)
+#648 := (not #568)
+#194 := 0::real
+#192 := -1::real
+#265 := (* -1::real #256)
+#640 := (+ #28 #265)
+#642 := (>= #640 0::real)
+#645 := (not #642)
+#643 := [hypothesis]: #642
+decl uf_8 :: T2
+#30 := uf_8
+#31 := (uf_4 uf_8 #25)
+#266 := (+ #31 #265)
+#264 := (>= #266 0::real)
+#267 := (not #264)
+#26 := (uf_4 uf_5 #25)
+decl uf_11 :: T2
+#41 := uf_11
+#42 := (uf_4 uf_11 #25)
+#237 := (ite #229 #42 #26)
+#245 := (* -1::real #237)
+#246 := (+ #31 #245)
+#247 := (<= #246 0::real)
+#248 := (not #247)
+#272 := (and #248 #267)
+#40 := (< #22 #39)
+#49 := (ite #40 #28 #48)
+#50 := (< #31 #49)
+#43 := (ite #40 #26 #42)
+#44 := (< #43 #31)
+#51 := (and #44 #50)
+#273 := (iff #51 #272)
+#270 := (iff #50 #267)
+#261 := (< #31 #256)
+#268 := (iff #261 #267)
+#269 := [rewrite]: #268
+#262 := (iff #50 #261)
+#259 := (= #49 #256)
+#228 := (not #229)
+#253 := (ite #228 #28 #48)
+#257 := (= #253 #256)
+#258 := [rewrite]: #257
+#254 := (= #49 #253)
+#232 := (iff #40 #228)
+#233 := [rewrite]: #232
+#255 := [monotonicity #233]: #254
+#260 := [trans #255 #258]: #259
+#263 := [monotonicity #260]: #262
+#271 := [trans #263 #269]: #270
+#251 := (iff #44 #248)
+#242 := (< #237 #31)
+#249 := (iff #242 #248)
+#250 := [rewrite]: #249
+#243 := (iff #44 #242)
+#240 := (= #43 #237)
+#234 := (ite #228 #26 #42)
+#238 := (= #234 #237)
+#239 := [rewrite]: #238
+#235 := (= #43 #234)
+#236 := [monotonicity #233]: #235
+#241 := [trans #236 #239]: #240
+#244 := [monotonicity #241]: #243
+#252 := [trans #244 #250]: #251
+#274 := [monotonicity #252 #271]: #273
+#178 := [asserted]: #51
+#275 := [mp #178 #274]: #272
+#277 := [and-elim #275]: #267
+#196 := (* -1::real #31)
+#197 := (+ #28 #196)
+#195 := (>= #197 0::real)
+#193 := (not #195)
+#213 := (* -1::real #34)
+#214 := (+ #31 #213)
+#212 := (>= #214 0::real)
+#215 := (not #212)
+#220 := (and #193 #215)
+#23 := (< #22 #22)
+#35 := (ite #23 #28 #34)
+#36 := (< #31 #35)
+#29 := (ite #23 #26 #28)
+#32 := (< #29 #31)
+#37 := (and #32 #36)
+#221 := (iff #37 #220)
+#218 := (iff #36 #215)
+#209 := (< #31 #34)
+#216 := (iff #209 #215)
+#217 := [rewrite]: #216
+#210 := (iff #36 #209)
+#207 := (= #35 #34)
+#202 := (ite false #28 #34)
+#205 := (= #202 #34)
+#206 := [rewrite]: #205
+#203 := (= #35 #202)
+#180 := (iff #23 false)
+#181 := [rewrite]: #180
+#204 := [monotonicity #181]: #203
+#208 := [trans #204 #206]: #207
+#211 := [monotonicity #208]: #210
+#219 := [trans #211 #217]: #218
+#200 := (iff #32 #193)
+#189 := (< #28 #31)
+#198 := (iff #189 #193)
+#199 := [rewrite]: #198
+#190 := (iff #32 #189)
+#187 := (= #29 #28)
+#182 := (ite false #26 #28)
+#185 := (= #182 #28)
+#186 := [rewrite]: #185
+#183 := (= #29 #182)
+#184 := [monotonicity #181]: #183
+#188 := [trans #184 #186]: #187
+#191 := [monotonicity #188]: #190
+#201 := [trans #191 #199]: #200
+#222 := [monotonicity #201 #219]: #221
+#177 := [asserted]: #37
+#223 := [mp #177 #222]: #220
+#224 := [and-elim #223]: #193
+#644 := [th-lemma #224 #277 #643]: false
+#646 := [lemma #644]: #645
+#647 := [hypothesis]: #568
+#649 := (or #648 #642)
+#650 := [th-lemma]: #649
+#651 := [unit-resolution #650 #647 #646]: false
+#652 := [lemma #651]: #648
+#578 := (or #229 #568)
+#579 := [def-axiom]: #578
+#675 := [unit-resolution #579 #652]: #229
+#677 := [th-lemma #675 #395]: #676
+#679 := [symm #677]: #678
+#683 := [monotonicity #679]: #682
+#685 := [symm #683]: #684
+#587 := (= uf_3 #586)
+#591 := (or #590 #587)
+#592 := [quant-inst]: #591
+#681 := [unit-resolution #592 #547]: #587
+#689 := [trans #681 #685]: #688
+#690 := [trans #689 #687]: #45
+#571 := (not #45)
+#54 := (uf_4 uf_11 #46)
+#279 := (ite #45 #28 #54)
+#465 := (* -1::real #279)
+#632 := (+ #28 #465)
+#633 := (<= #632 0::real)
+#580 := (= #28 #279)
+#656 := [hypothesis]: #45
+#582 := (or #571 #580)
+#583 := [def-axiom]: #582
+#657 := [unit-resolution #583 #656]: #580
+#658 := (not #580)
+#659 := (or #658 #633)
+#660 := [th-lemma]: #659
+#661 := [unit-resolution #660 #657]: #633
+#57 := (uf_4 uf_8 #46)
+#363 := (* -1::real #57)
+#379 := (+ #47 #363)
+#380 := (<= #379 0::real)
+#381 := (not #380)
+#364 := (+ #54 #363)
+#362 := (>= #364 0::real)
+#361 := (not #362)
+#386 := (and #361 #381)
+#59 := (uf_4 uf_7 #46)
+#64 := (< #39 #39)
+#67 := (ite #64 #59 #47)
+#68 := (< #57 #67)
+#65 := (ite #64 #47 #54)
+#66 := (< #65 #57)
+#69 := (and #66 #68)
+#387 := (iff #69 #386)
+#384 := (iff #68 #381)
+#376 := (< #57 #47)
+#382 := (iff #376 #381)
+#383 := [rewrite]: #382
+#377 := (iff #68 #376)
+#374 := (= #67 #47)
+#369 := (ite false #59 #47)
+#372 := (= #369 #47)
+#373 := [rewrite]: #372
+#370 := (= #67 #369)
+#349 := (iff #64 false)
+#350 := [rewrite]: #349
+#371 := [monotonicity #350]: #370
+#375 := [trans #371 #373]: #374
+#378 := [monotonicity #375]: #377
+#385 := [trans #378 #383]: #384
+#367 := (iff #66 #361)
+#358 := (< #54 #57)
+#365 := (iff #358 #361)
+#366 := [rewrite]: #365
+#359 := (iff #66 #358)
+#356 := (= #65 #54)
+#351 := (ite false #47 #54)
+#354 := (= #351 #54)
+#355 := [rewrite]: #354
+#352 := (= #65 #351)
+#353 := [monotonicity #350]: #352
+#357 := [trans #353 #355]: #356
+#360 := [monotonicity #357]: #359
+#368 := [trans #360 #366]: #367
+#388 := [monotonicity #368 #385]: #387
+#346 := [asserted]: #69
+#389 := [mp #346 #388]: #386
+#391 := [and-elim #389]: #381
+#397 := (* -1::real #59)
+#398 := (+ #47 #397)
+#399 := (<= #398 0::real)
+#409 := (* -1::real #54)
+#410 := (+ #47 #409)
+#408 := (>= #410 0::real)
+#60 := (uf_4 uf_9 #46)
+#402 := (* -1::real #60)
+#403 := (+ #59 #402)
+#404 := (<= #403 0::real)
+#418 := (and #399 #404 #408)
+#73 := (<= #59 #60)
+#72 := (<= #47 #59)
+#74 := (and #72 #73)
+#71 := (<= #54 #47)
+#75 := (and #71 #74)
+#421 := (iff #75 #418)
+#412 := (and #399 #404)
+#415 := (and #408 #412)
+#419 := (iff #415 #418)
+#420 := [rewrite]: #419
+#416 := (iff #75 #415)
+#413 := (iff #74 #412)
+#405 := (iff #73 #404)
+#406 := [rewrite]: #405
+#400 := (iff #72 #399)
+#401 := [rewrite]: #400
+#414 := [monotonicity #401 #406]: #413
+#407 := (iff #71 #408)
+#411 := [rewrite]: #407
+#417 := [monotonicity #411 #414]: #416
+#422 := [trans #417 #420]: #421
+#348 := [asserted]: #75
+#423 := [mp #348 #422]: #418
+#424 := [and-elim #423]: #399
+#637 := (+ #28 #397)
+#639 := (>= #637 0::real)
+#636 := (= #28 #59)
+#666 := (= #59 #28)
+#664 := (= #46 #25)
+#662 := (= #25 #46)
+#663 := [monotonicity #656]: #662
+#665 := [symm #663]: #664
+#667 := [monotonicity #665]: #666
+#668 := [symm #667]: #636
+#669 := (not #636)
+#670 := (or #669 #639)
+#671 := [th-lemma]: #670
+#672 := [unit-resolution #671 #668]: #639
+#468 := (+ #57 #465)
+#471 := (<= #468 0::real)
+#444 := (not #471)
+#322 := (ite #296 #279 #47)
+#330 := (* -1::real #322)
+#331 := (+ #57 #330)
+#332 := (<= #331 0::real)
+#333 := (not #332)
+#445 := (iff #333 #444)
+#472 := (iff #332 #471)
+#469 := (= #331 #468)
+#466 := (= #330 #465)
+#463 := (= #322 #279)
+#1 := true
+#458 := (ite true #279 #47)
+#461 := (= #458 #279)
+#462 := [rewrite]: #461
+#459 := (= #322 #458)
+#450 := (iff #296 true)
+#451 := [iff-true #395]: #450
+#460 := [monotonicity #451]: #459
+#464 := [trans #460 #462]: #463
+#467 := [monotonicity #464]: #466
+#470 := [monotonicity #467]: #469
+#473 := [monotonicity #470]: #472
+#474 := [monotonicity #473]: #445
+#303 := (ite #296 #60 #59)
+#313 := (* -1::real #303)
+#314 := (+ #57 #313)
+#312 := (>= #314 0::real)
+#311 := (not #312)
+#338 := (and #311 #333)
+#52 := (< #39 #22)
+#61 := (ite #52 #59 #60)
+#62 := (< #57 #61)
+#53 := (= uf_10 uf_3)
+#55 := (ite #53 #28 #54)
+#56 := (ite #52 #47 #55)
+#58 := (< #56 #57)
+#63 := (and #58 #62)
+#341 := (iff #63 #338)
+#282 := (ite #52 #47 #279)
+#285 := (< #282 #57)
+#291 := (and #62 #285)
+#339 := (iff #291 #338)
+#336 := (iff #285 #333)
+#327 := (< #322 #57)
+#334 := (iff #327 #333)
+#335 := [rewrite]: #334
+#328 := (iff #285 #327)
+#325 := (= #282 #322)
+#297 := (not #296)
+#319 := (ite #297 #47 #279)
+#323 := (= #319 #322)
+#324 := [rewrite]: #323
+#320 := (= #282 #319)
+#298 := (iff #52 #297)
+#299 := [rewrite]: #298
+#321 := [monotonicity #299]: #320
+#326 := [trans #321 #324]: #325
+#329 := [monotonicity #326]: #328
+#337 := [trans #329 #335]: #336
+#317 := (iff #62 #311)
+#308 := (< #57 #303)
+#315 := (iff #308 #311)
+#316 := [rewrite]: #315
+#309 := (iff #62 #308)
+#306 := (= #61 #303)
+#300 := (ite #297 #59 #60)
+#304 := (= #300 #303)
+#305 := [rewrite]: #304
+#301 := (= #61 #300)
+#302 := [monotonicity #299]: #301
+#307 := [trans #302 #305]: #306
+#310 := [monotonicity #307]: #309
+#318 := [trans #310 #316]: #317
+#340 := [monotonicity #318 #337]: #339
+#294 := (iff #63 #291)
+#288 := (and #285 #62)
+#292 := (iff #288 #291)
+#293 := [rewrite]: #292
+#289 := (iff #63 #288)
+#286 := (iff #58 #285)
+#283 := (= #56 #282)
+#280 := (= #55 #279)
+#226 := (iff #53 #45)
+#278 := [rewrite]: #226
+#281 := [monotonicity #278]: #280
+#284 := [monotonicity #281]: #283
+#287 := [monotonicity #284]: #286
+#290 := [monotonicity #287]: #289
+#295 := [trans #290 #293]: #294
+#342 := [trans #295 #340]: #341
+#179 := [asserted]: #63
+#343 := [mp #179 #342]: #338
+#345 := [and-elim #343]: #333
+#475 := [mp #345 #474]: #444
+#673 := [th-lemma #475 #672 #424 #391 #661]: false
+#674 := [lemma #673]: #571
+[unit-resolution #674 #690]: false
+unsat
+IL2powemHjRpCJYwmXFxyw 211 0
+#2 := false
+#33 := 0::real
+decl uf_11 :: (-> T5 T6 real)
+decl uf_15 :: T6
+#28 := uf_15
+decl uf_16 :: T5
+#30 := uf_16
+#31 := (uf_11 uf_16 uf_15)
+decl uf_12 :: (-> T7 T8 T5)
+decl uf_14 :: T8
+#26 := uf_14
+decl uf_13 :: (-> T1 T7)
+decl uf_8 :: T1
+#16 := uf_8
+#25 := (uf_13 uf_8)
+#27 := (uf_12 #25 uf_14)
+#29 := (uf_11 #27 uf_15)
+#73 := -1::real
+#84 := (* -1::real #29)
+#85 := (+ #84 #31)
+#74 := (* -1::real #31)
+#75 := (+ #29 #74)
+#112 := (>= #75 0::real)
+#119 := (ite #112 #75 #85)
+#127 := (* -1::real #119)
+decl uf_17 :: T5
+#37 := uf_17
+#38 := (uf_11 uf_17 uf_15)
+#102 := -1/3::real
+#103 := (* -1/3::real #38)
+#128 := (+ #103 #127)
+#100 := 1/3::real
+#101 := (* 1/3::real #31)
+#129 := (+ #101 #128)
+#130 := (<= #129 0::real)
+#131 := (not #130)
+#40 := 3::real
+#39 := (- #31 #38)
+#41 := (/ #39 3::real)
+#32 := (- #29 #31)
+#35 := (- #32)
+#34 := (< #32 0::real)
+#36 := (ite #34 #35 #32)
+#42 := (< #36 #41)
+#136 := (iff #42 #131)
+#104 := (+ #101 #103)
+#78 := (< #75 0::real)
+#90 := (ite #78 #85 #75)
+#109 := (< #90 #104)
+#134 := (iff #109 #131)
+#124 := (< #119 #104)
+#132 := (iff #124 #131)
+#133 := [rewrite]: #132
+#125 := (iff #109 #124)
+#122 := (= #90 #119)
+#113 := (not #112)
+#116 := (ite #113 #85 #75)
+#120 := (= #116 #119)
+#121 := [rewrite]: #120
+#117 := (= #90 #116)
+#114 := (iff #78 #113)
+#115 := [rewrite]: #114
+#118 := [monotonicity #115]: #117
+#123 := [trans #118 #121]: #122
+#126 := [monotonicity #123]: #125
+#135 := [trans #126 #133]: #134
+#110 := (iff #42 #109)
+#107 := (= #41 #104)
+#93 := (* -1::real #38)
+#94 := (+ #31 #93)
+#97 := (/ #94 3::real)
+#105 := (= #97 #104)
+#106 := [rewrite]: #105
+#98 := (= #41 #97)
+#95 := (= #39 #94)
+#96 := [rewrite]: #95
+#99 := [monotonicity #96]: #98
+#108 := [trans #99 #106]: #107
+#91 := (= #36 #90)
+#76 := (= #32 #75)
+#77 := [rewrite]: #76
+#88 := (= #35 #85)
+#81 := (- #75)
+#86 := (= #81 #85)
+#87 := [rewrite]: #86
+#82 := (= #35 #81)
+#83 := [monotonicity #77]: #82
+#89 := [trans #83 #87]: #88
+#79 := (iff #34 #78)
+#80 := [monotonicity #77]: #79
+#92 := [monotonicity #80 #89 #77]: #91
+#111 := [monotonicity #92 #108]: #110
+#137 := [trans #111 #135]: #136
+#72 := [asserted]: #42
+#138 := [mp #72 #137]: #131
+decl uf_1 :: T1
+#4 := uf_1
+#43 := (uf_13 uf_1)
+#44 := (uf_12 #43 uf_14)
+#45 := (uf_11 #44 uf_15)
+#149 := (* -1::real #45)
+#150 := (+ #38 #149)
+#140 := (+ #93 #45)
+#161 := (<= #150 0::real)
+#168 := (ite #161 #140 #150)
+#176 := (* -1::real #168)
+#177 := (+ #103 #176)
+#178 := (+ #101 #177)
+#179 := (<= #178 0::real)
+#180 := (not #179)
+#46 := (- #45 #38)
+#48 := (- #46)
+#47 := (< #46 0::real)
+#49 := (ite #47 #48 #46)
+#50 := (< #49 #41)
+#185 := (iff #50 #180)
+#143 := (< #140 0::real)
+#155 := (ite #143 #150 #140)
+#158 := (< #155 #104)
+#183 := (iff #158 #180)
+#173 := (< #168 #104)
+#181 := (iff #173 #180)
+#182 := [rewrite]: #181
+#174 := (iff #158 #173)
+#171 := (= #155 #168)
+#162 := (not #161)
+#165 := (ite #162 #150 #140)
+#169 := (= #165 #168)
+#170 := [rewrite]: #169
+#166 := (= #155 #165)
+#163 := (iff #143 #162)
+#164 := [rewrite]: #163
+#167 := [monotonicity #164]: #166
+#172 := [trans #167 #170]: #171
+#175 := [monotonicity #172]: #174
+#184 := [trans #175 #182]: #183
+#159 := (iff #50 #158)
+#156 := (= #49 #155)
+#141 := (= #46 #140)
+#142 := [rewrite]: #141
+#153 := (= #48 #150)
+#146 := (- #140)
+#151 := (= #146 #150)
+#152 := [rewrite]: #151
+#147 := (= #48 #146)
+#148 := [monotonicity #142]: #147
+#154 := [trans #148 #152]: #153
+#144 := (iff #47 #143)
+#145 := [monotonicity #142]: #144
+#157 := [monotonicity #145 #154 #142]: #156
+#160 := [monotonicity #157 #108]: #159
+#186 := [trans #160 #184]: #185
+#139 := [asserted]: #50
+#187 := [mp #139 #186]: #180
+#299 := (+ #140 #176)
+#300 := (<= #299 0::real)
+#290 := (= #140 #168)
+#329 := [hypothesis]: #162
+#191 := (+ #29 #149)
+#192 := (<= #191 0::real)
+#51 := (<= #29 #45)
+#193 := (iff #51 #192)
+#194 := [rewrite]: #193
+#188 := [asserted]: #51
+#195 := [mp #188 #194]: #192
+#298 := (+ #75 #127)
+#301 := (<= #298 0::real)
+#284 := (= #75 #119)
+#302 := [hypothesis]: #113
+#296 := (+ #85 #127)
+#297 := (<= #296 0::real)
+#285 := (= #85 #119)
+#288 := (or #112 #285)
+#289 := [def-axiom]: #288
+#303 := [unit-resolution #289 #302]: #285
+#304 := (not #285)
+#305 := (or #304 #297)
+#306 := [th-lemma]: #305
+#307 := [unit-resolution #306 #303]: #297
+#315 := (not #290)
+#310 := (not #300)
+#311 := (or #310 #112)
+#308 := [hypothesis]: #300
+#309 := [th-lemma #308 #307 #138 #302 #187 #195]: false
+#312 := [lemma #309]: #311
+#322 := [unit-resolution #312 #302]: #310
+#316 := (or #315 #300)
+#313 := [hypothesis]: #310
+#314 := [hypothesis]: #290
+#317 := [th-lemma]: #316
+#318 := [unit-resolution #317 #314 #313]: false
+#319 := [lemma #318]: #316
+#323 := [unit-resolution #319 #322]: #315
+#292 := (or #162 #290)
+#293 := [def-axiom]: #292
+#324 := [unit-resolution #293 #323]: #162
+#325 := [th-lemma #324 #307 #138 #302 #195]: false
+#326 := [lemma #325]: #112
+#286 := (or #113 #284)
+#287 := [def-axiom]: #286
+#330 := [unit-resolution #287 #326]: #284
+#331 := (not #284)
+#332 := (or #331 #301)
+#333 := [th-lemma]: #332
+#334 := [unit-resolution #333 #330]: #301
+#335 := [th-lemma #326 #334 #195 #329 #138]: false
+#336 := [lemma #335]: #161
+#327 := [unit-resolution #293 #336]: #290
+#328 := [unit-resolution #319 #327]: #300
+[th-lemma #326 #334 #195 #328 #187 #138]: false
+unsat
+GX51o3DUO/UBS3eNP2P9kA 285 0
+#2 := false
+#7 := 0::real
+decl uf_4 :: real
+#16 := uf_4
+#40 := -1::real
+#116 := (* -1::real uf_4)
+decl uf_3 :: real
+#11 := uf_3
+#117 := (+ uf_3 #116)
+#128 := (<= #117 0::real)
+#129 := (not #128)
+#220 := 2/3::real
+#221 := (* 2/3::real uf_3)
+#222 := (+ #221 #116)
+decl uf_2 :: real
+#5 := uf_2
+#67 := 1/3::real
+#68 := (* 1/3::real uf_2)
+#233 := (+ #68 #222)
+#243 := (<= #233 0::real)
+#268 := (not #243)
+#287 := [hypothesis]: #268
+#41 := (* -1::real uf_2)
+decl uf_1 :: real
+#4 := uf_1
+#42 := (+ uf_1 #41)
+#79 := (>= #42 0::real)
+#80 := (not #79)
+#297 := (or #80 #243)
+#158 := (+ uf_1 #116)
+#159 := (<= #158 0::real)
+#22 := (<= uf_1 uf_4)
+#160 := (iff #22 #159)
+#161 := [rewrite]: #160
+#155 := [asserted]: #22
+#162 := [mp #155 #161]: #159
+#200 := (* 1/3::real uf_3)
+#198 := -4/3::real
+#199 := (* -4/3::real uf_2)
+#201 := (+ #199 #200)
+#202 := (+ uf_1 #201)
+#203 := (>= #202 0::real)
+#258 := (not #203)
+#292 := [hypothesis]: #79
+#293 := (or #80 #258)
+#69 := -1/3::real
+#70 := (* -1/3::real uf_3)
+#186 := -2/3::real
+#187 := (* -2/3::real uf_2)
+#188 := (+ #187 #70)
+#189 := (+ uf_1 #188)
+#204 := (<= #189 0::real)
+#205 := (ite #79 #203 #204)
+#210 := (not #205)
+#51 := (* -1::real uf_1)
+#52 := (+ #51 uf_2)
+#86 := (ite #79 #42 #52)
+#94 := (* -1::real #86)
+#95 := (+ #70 #94)
+#96 := (+ #68 #95)
+#97 := (<= #96 0::real)
+#98 := (not #97)
+#211 := (iff #98 #210)
+#208 := (iff #97 #205)
+#182 := 4/3::real
+#183 := (* 4/3::real uf_2)
+#184 := (+ #183 #70)
+#185 := (+ #51 #184)
+#190 := (ite #79 #185 #189)
+#195 := (<= #190 0::real)
+#206 := (iff #195 #205)
+#207 := [rewrite]: #206
+#196 := (iff #97 #195)
+#193 := (= #96 #190)
+#172 := (+ #41 #70)
+#173 := (+ uf_1 #172)
+#170 := (+ uf_2 #70)
+#171 := (+ #51 #170)
+#174 := (ite #79 #171 #173)
+#179 := (+ #68 #174)
+#191 := (= #179 #190)
+#192 := [rewrite]: #191
+#180 := (= #96 #179)
+#177 := (= #95 #174)
+#164 := (ite #79 #52 #42)
+#167 := (+ #70 #164)
+#175 := (= #167 #174)
+#176 := [rewrite]: #175
+#168 := (= #95 #167)
+#156 := (= #94 #164)
+#165 := [rewrite]: #156
+#169 := [monotonicity #165]: #168
+#178 := [trans #169 #176]: #177
+#181 := [monotonicity #178]: #180
+#194 := [trans #181 #192]: #193
+#197 := [monotonicity #194]: #196
+#209 := [trans #197 #207]: #208
+#212 := [monotonicity #209]: #211
+#13 := 3::real
+#12 := (- uf_2 uf_3)
+#14 := (/ #12 3::real)
+#6 := (- uf_1 uf_2)
+#9 := (- #6)
+#8 := (< #6 0::real)
+#10 := (ite #8 #9 #6)
+#15 := (< #10 #14)
+#103 := (iff #15 #98)
+#71 := (+ #68 #70)
+#45 := (< #42 0::real)
+#57 := (ite #45 #52 #42)
+#76 := (< #57 #71)
+#101 := (iff #76 #98)
+#91 := (< #86 #71)
+#99 := (iff #91 #98)
+#100 := [rewrite]: #99
+#92 := (iff #76 #91)
+#89 := (= #57 #86)
+#83 := (ite #80 #52 #42)
+#87 := (= #83 #86)
+#88 := [rewrite]: #87
+#84 := (= #57 #83)
+#81 := (iff #45 #80)
+#82 := [rewrite]: #81
+#85 := [monotonicity #82]: #84
+#90 := [trans #85 #88]: #89
+#93 := [monotonicity #90]: #92
+#102 := [trans #93 #100]: #101
+#77 := (iff #15 #76)
+#74 := (= #14 #71)
+#60 := (* -1::real uf_3)
+#61 := (+ uf_2 #60)
+#64 := (/ #61 3::real)
+#72 := (= #64 #71)
+#73 := [rewrite]: #72
+#65 := (= #14 #64)
+#62 := (= #12 #61)
+#63 := [rewrite]: #62
+#66 := [monotonicity #63]: #65
+#75 := [trans #66 #73]: #74
+#58 := (= #10 #57)
+#43 := (= #6 #42)
+#44 := [rewrite]: #43
+#55 := (= #9 #52)
+#48 := (- #42)
+#53 := (= #48 #52)
+#54 := [rewrite]: #53
+#49 := (= #9 #48)
+#50 := [monotonicity #44]: #49
+#56 := [trans #50 #54]: #55
+#46 := (iff #8 #45)
+#47 := [monotonicity #44]: #46
+#59 := [monotonicity #47 #56 #44]: #58
+#78 := [monotonicity #59 #75]: #77
+#104 := [trans #78 #102]: #103
+#39 := [asserted]: #15
+#105 := [mp #39 #104]: #98
+#213 := [mp #105 #212]: #210
+#259 := (or #205 #80 #258)
+#260 := [def-axiom]: #259
+#294 := [unit-resolution #260 #213]: #293
+#295 := [unit-resolution #294 #292]: #258
+#296 := [th-lemma #287 #292 #295 #162]: false
+#298 := [lemma #296]: #297
+#299 := [unit-resolution #298 #287]: #80
+#261 := (not #204)
+#281 := (or #79 #261)
+#262 := (or #205 #79 #261)
+#263 := [def-axiom]: #262
+#282 := [unit-resolution #263 #213]: #281
+#300 := [unit-resolution #282 #299]: #261
+#290 := (or #79 #204 #243)
+#276 := [hypothesis]: #261
+#288 := [hypothesis]: #80
+#289 := [th-lemma #288 #276 #162 #287]: false
+#291 := [lemma #289]: #290
+#301 := [unit-resolution #291 #300 #299 #287]: false
+#302 := [lemma #301]: #243
+#303 := (or #129 #268)
+#223 := (* -4/3::real uf_3)
+#224 := (+ #223 uf_4)
+#234 := (+ #68 #224)
+#244 := (<= #234 0::real)
+#245 := (ite #128 #243 #244)
+#250 := (not #245)
+#107 := (+ #60 uf_4)
+#135 := (ite #128 #107 #117)
+#143 := (* -1::real #135)
+#144 := (+ #70 #143)
+#145 := (+ #68 #144)
+#146 := (<= #145 0::real)
+#147 := (not #146)
+#251 := (iff #147 #250)
+#248 := (iff #146 #245)
+#235 := (ite #128 #233 #234)
+#240 := (<= #235 0::real)
+#246 := (iff #240 #245)
+#247 := [rewrite]: #246
+#241 := (iff #146 #240)
+#238 := (= #145 #235)
+#225 := (ite #128 #222 #224)
+#230 := (+ #68 #225)
+#236 := (= #230 #235)
+#237 := [rewrite]: #236
+#231 := (= #145 #230)
+#228 := (= #144 #225)
+#214 := (ite #128 #117 #107)
+#217 := (+ #70 #214)
+#226 := (= #217 #225)
+#227 := [rewrite]: #226
+#218 := (= #144 #217)
+#215 := (= #143 #214)
+#216 := [rewrite]: #215
+#219 := [monotonicity #216]: #218
+#229 := [trans #219 #227]: #228
+#232 := [monotonicity #229]: #231
+#239 := [trans #232 #237]: #238
+#242 := [monotonicity #239]: #241
+#249 := [trans #242 #247]: #248
+#252 := [monotonicity #249]: #251
+#17 := (- uf_4 uf_3)
+#19 := (- #17)
+#18 := (< #17 0::real)
+#20 := (ite #18 #19 #17)
+#21 := (< #20 #14)
+#152 := (iff #21 #147)
+#110 := (< #107 0::real)
+#122 := (ite #110 #117 #107)
+#125 := (< #122 #71)
+#150 := (iff #125 #147)
+#140 := (< #135 #71)
+#148 := (iff #140 #147)
+#149 := [rewrite]: #148
+#141 := (iff #125 #140)
+#138 := (= #122 #135)
+#132 := (ite #129 #117 #107)
+#136 := (= #132 #135)
+#137 := [rewrite]: #136
+#133 := (= #122 #132)
+#130 := (iff #110 #129)
+#131 := [rewrite]: #130
+#134 := [monotonicity #131]: #133
+#139 := [trans #134 #137]: #138
+#142 := [monotonicity #139]: #141
+#151 := [trans #142 #149]: #150
+#126 := (iff #21 #125)
+#123 := (= #20 #122)
+#108 := (= #17 #107)
+#109 := [rewrite]: #108
+#120 := (= #19 #117)
+#113 := (- #107)
+#118 := (= #113 #117)
+#119 := [rewrite]: #118
+#114 := (= #19 #113)
+#115 := [monotonicity #109]: #114
+#121 := [trans #115 #119]: #120
+#111 := (iff #18 #110)
+#112 := [monotonicity #109]: #111
+#124 := [monotonicity #112 #121 #109]: #123
+#127 := [monotonicity #124 #75]: #126
+#153 := [trans #127 #151]: #152
+#106 := [asserted]: #21
+#154 := [mp #106 #153]: #147
+#253 := [mp #154 #252]: #250
+#269 := (or #245 #129 #268)
+#270 := [def-axiom]: #269
+#304 := [unit-resolution #270 #253]: #303
+#305 := [unit-resolution #304 #302]: #129
+#271 := (not #244)
+#306 := (or #128 #271)
+#272 := (or #245 #128 #271)
+#273 := [def-axiom]: #272
+#307 := [unit-resolution #273 #253]: #306
+#308 := [unit-resolution #307 #305]: #271
+#285 := (or #128 #244)
+#274 := [hypothesis]: #271
+#275 := [hypothesis]: #129
+#278 := (or #204 #128 #244)
+#277 := [th-lemma #276 #275 #274 #162]: false
+#279 := [lemma #277]: #278
+#280 := [unit-resolution #279 #275 #274]: #204
+#283 := [unit-resolution #282 #280]: #79
+#284 := [th-lemma #275 #274 #283 #162]: false
+#286 := [lemma #284]: #285
+[unit-resolution #286 #308 #305]: false
+unsat
+cebG074uorSr8ODzgTmcKg 97 0
+#2 := false
+#18 := 0::real
+decl uf_1 :: (-> T2 T1 real)
+decl uf_5 :: T1
+#11 := uf_5
+decl uf_2 :: T2
+#4 := uf_2
+#20 := (uf_1 uf_2 uf_5)
+#42 := -1::real
+#53 := (* -1::real #20)
+decl uf_3 :: T2
+#7 := uf_3
+#19 := (uf_1 uf_3 uf_5)
+#54 := (+ #19 #53)
+#63 := (<= #54 0::real)
+#21 := (- #19 #20)
+#22 := (< 0::real #21)
+#23 := (not #22)
+#74 := (iff #23 #63)
+#57 := (< 0::real #54)
+#60 := (not #57)
+#72 := (iff #60 #63)
+#64 := (not #63)
+#67 := (not #64)
+#70 := (iff #67 #63)
+#71 := [rewrite]: #70
+#68 := (iff #60 #67)
+#65 := (iff #57 #64)
+#66 := [rewrite]: #65
+#69 := [monotonicity #66]: #68
+#73 := [trans #69 #71]: #72
+#61 := (iff #23 #60)
+#58 := (iff #22 #57)
+#55 := (= #21 #54)
+#56 := [rewrite]: #55
+#59 := [monotonicity #56]: #58
+#62 := [monotonicity #59]: #61
+#75 := [trans #62 #73]: #74
+#41 := [asserted]: #23
+#76 := [mp #41 #75]: #63
+#5 := (:var 0 T1)
+#8 := (uf_1 uf_3 #5)
+#141 := (pattern #8)
+#6 := (uf_1 uf_2 #5)
+#140 := (pattern #6)
+#45 := (* -1::real #8)
+#46 := (+ #6 #45)
+#44 := (>= #46 0::real)
+#43 := (not #44)
+#142 := (forall (vars (?x1 T1)) (:pat #140 #141) #43)
+#49 := (forall (vars (?x1 T1)) #43)
+#145 := (iff #49 #142)
+#143 := (iff #43 #43)
+#144 := [refl]: #143
+#146 := [quant-intro #144]: #145
+#80 := (~ #49 #49)
+#82 := (~ #43 #43)
+#83 := [refl]: #82
+#81 := [nnf-pos #83]: #80
+#9 := (< #6 #8)
+#10 := (forall (vars (?x1 T1)) #9)
+#50 := (iff #10 #49)
+#47 := (iff #9 #43)
+#48 := [rewrite]: #47
+#51 := [quant-intro #48]: #50
+#39 := [asserted]: #10
+#52 := [mp #39 #51]: #49
+#79 := [mp~ #52 #81]: #49
+#147 := [mp #79 #146]: #142
+#164 := (not #142)
+#165 := (or #164 #64)
+#148 := (* -1::real #19)
+#149 := (+ #20 #148)
+#150 := (>= #149 0::real)
+#151 := (not #150)
+#166 := (or #164 #151)
+#168 := (iff #166 #165)
+#170 := (iff #165 #165)
+#171 := [rewrite]: #170
+#162 := (iff #151 #64)
+#160 := (iff #150 #63)
+#152 := (+ #148 #20)
+#155 := (>= #152 0::real)
+#158 := (iff #155 #63)
+#159 := [rewrite]: #158
+#156 := (iff #150 #155)
+#153 := (= #149 #152)
+#154 := [rewrite]: #153
+#157 := [monotonicity #154]: #156
+#161 := [trans #157 #159]: #160
+#163 := [monotonicity #161]: #162
+#169 := [monotonicity #163]: #168
+#172 := [trans #169 #171]: #168
+#167 := [quant-inst]: #166
+#173 := [mp #167 #172]: #165
+[unit-resolution #173 #147 #76]: false
+unsat
+DKRtrJ2XceCkITuNwNViRw 57 0
+#2 := false
+#4 := 0::real
+decl uf_1 :: (-> T2 real)
+decl uf_2 :: (-> T1 T1 T2)
+decl uf_12 :: (-> T4 T1)
+decl uf_4 :: T4
+#11 := uf_4
+#39 := (uf_12 uf_4)
+decl uf_10 :: T4
+#27 := uf_10
+#38 := (uf_12 uf_10)
+#40 := (uf_2 #38 #39)
+#41 := (uf_1 #40)
+#264 := (>= #41 0::real)
+#266 := (not #264)
+#43 := (= #41 0::real)
+#44 := (not #43)
+#131 := [asserted]: #44
+#272 := (or #43 #266)
+#42 := (<= #41 0::real)
+#130 := [asserted]: #42
+#265 := (not #42)
+#270 := (or #43 #265 #266)
+#271 := [th-lemma]: #270
+#273 := [unit-resolution #271 #130]: #272
+#274 := [unit-resolution #273 #131]: #266
+#6 := (:var 0 T1)
+#5 := (:var 1 T1)
+#7 := (uf_2 #5 #6)
+#241 := (pattern #7)
+#8 := (uf_1 #7)
+#65 := (>= #8 0::real)
+#242 := (forall (vars (?x1 T1) (?x2 T1)) (:pat #241) #65)
+#66 := (forall (vars (?x1 T1) (?x2 T1)) #65)
+#245 := (iff #66 #242)
+#243 := (iff #65 #65)
+#244 := [refl]: #243
+#246 := [quant-intro #244]: #245
+#149 := (~ #66 #66)
+#151 := (~ #65 #65)
+#152 := [refl]: #151
+#150 := [nnf-pos #152]: #149
+#9 := (<= 0::real #8)
+#10 := (forall (vars (?x1 T1) (?x2 T1)) #9)
+#67 := (iff #10 #66)
+#63 := (iff #9 #65)
+#64 := [rewrite]: #63
+#68 := [quant-intro #64]: #67
+#60 := [asserted]: #10
+#69 := [mp #60 #68]: #66
+#147 := [mp~ #69 #150]: #66
+#247 := [mp #147 #246]: #242
+#267 := (not #242)
+#268 := (or #267 #264)
+#269 := [quant-inst]: #268
+[unit-resolution #269 #247 #274]: false
+unsat
+97KJAJfUio+nGchEHWvgAw 91 0
+#2 := false
+#38 := 0::real
+decl uf_1 :: (-> T1 T2 real)
+decl uf_3 :: T2
+#5 := uf_3
+decl uf_4 :: T1
+#7 := uf_4
+#8 := (uf_1 uf_4 uf_3)
+#35 := -1::real
+#36 := (* -1::real #8)
+decl uf_2 :: T1
+#4 := uf_2
+#6 := (uf_1 uf_2 uf_3)
+#37 := (+ #6 #36)
+#130 := (>= #37 0::real)
+#155 := (not #130)
+#43 := (= #6 #8)
+#55 := (not #43)
+#15 := (= #8 #6)
+#16 := (not #15)
+#56 := (iff #16 #55)
+#53 := (iff #15 #43)
+#54 := [rewrite]: #53
+#57 := [monotonicity #54]: #56
+#34 := [asserted]: #16
+#60 := [mp #34 #57]: #55
+#158 := (or #43 #155)
+#39 := (<= #37 0::real)
+#9 := (<= #6 #8)
+#40 := (iff #9 #39)
+#41 := [rewrite]: #40
+#32 := [asserted]: #9
+#42 := [mp #32 #41]: #39
+#154 := (not #39)
+#156 := (or #43 #154 #155)
+#157 := [th-lemma]: #156
+#159 := [unit-resolution #157 #42]: #158
+#160 := [unit-resolution #159 #60]: #155
+#10 := (:var 0 T2)
+#12 := (uf_1 uf_2 #10)
+#123 := (pattern #12)
+#11 := (uf_1 uf_4 #10)
+#122 := (pattern #11)
+#44 := (* -1::real #12)
+#45 := (+ #11 #44)
+#46 := (<= #45 0::real)
+#124 := (forall (vars (?x1 T2)) (:pat #122 #123) #46)
+#49 := (forall (vars (?x1 T2)) #46)
+#127 := (iff #49 #124)
+#125 := (iff #46 #46)
+#126 := [refl]: #125
+#128 := [quant-intro #126]: #127
+#62 := (~ #49 #49)
+#64 := (~ #46 #46)
+#65 := [refl]: #64
+#63 := [nnf-pos #65]: #62
+#13 := (<= #11 #12)
+#14 := (forall (vars (?x1 T2)) #13)
+#50 := (iff #14 #49)
+#47 := (iff #13 #46)
+#48 := [rewrite]: #47
+#51 := [quant-intro #48]: #50
+#33 := [asserted]: #14
+#52 := [mp #33 #51]: #49
+#61 := [mp~ #52 #63]: #49
+#129 := [mp #61 #128]: #124
+#144 := (not #124)
+#145 := (or #144 #130)
+#131 := (* -1::real #6)
+#132 := (+ #8 #131)
+#133 := (<= #132 0::real)
+#146 := (or #144 #133)
+#148 := (iff #146 #145)
+#150 := (iff #145 #145)
+#151 := [rewrite]: #150
+#142 := (iff #133 #130)
+#134 := (+ #131 #8)
+#137 := (<= #134 0::real)
+#140 := (iff #137 #130)
+#141 := [rewrite]: #140
+#138 := (iff #133 #137)
+#135 := (= #132 #134)
+#136 := [rewrite]: #135
+#139 := [monotonicity #136]: #138
+#143 := [trans #139 #141]: #142
+#149 := [monotonicity #143]: #148
+#152 := [trans #149 #151]: #148
+#147 := [quant-inst]: #146
+#153 := [mp #147 #152]: #145
+[unit-resolution #153 #129 #160]: false
+unsat
+flJYbeWfe+t2l/zsRqdujA 149 0
+#2 := false
+#19 := 0::real
+decl uf_1 :: (-> T1 T2 real)
+decl uf_3 :: T2
+#5 := uf_3
+decl uf_4 :: T1
+#7 := uf_4
+#8 := (uf_1 uf_4 uf_3)
+#44 := -1::real
+#156 := (* -1::real #8)
+decl uf_2 :: T1
+#4 := uf_2
+#6 := (uf_1 uf_2 uf_3)
+#203 := (+ #6 #156)
+#205 := (>= #203 0::real)
+#9 := (= #6 #8)
+#40 := [asserted]: #9
+#208 := (not #9)
+#209 := (or #208 #205)
+#210 := [th-lemma]: #209
+#211 := [unit-resolution #210 #40]: #205
+decl uf_5 :: T1
+#12 := uf_5
+#22 := (uf_1 uf_5 uf_3)
+#160 := (* -1::real #22)
+#161 := (+ #6 #160)
+#207 := (>= #161 0::real)
+#222 := (not #207)
+#206 := (= #6 #22)
+#216 := (not #206)
+#62 := (= #8 #22)
+#70 := (not #62)
+#217 := (iff #70 #216)
+#214 := (iff #62 #206)
+#212 := (iff #206 #62)
+#213 := [monotonicity #40]: #212
+#215 := [symm #213]: #214
+#218 := [monotonicity #215]: #217
+#23 := (= #22 #8)
+#24 := (not #23)
+#71 := (iff #24 #70)
+#68 := (iff #23 #62)
+#69 := [rewrite]: #68
+#72 := [monotonicity #69]: #71
+#43 := [asserted]: #24
+#75 := [mp #43 #72]: #70
+#219 := [mp #75 #218]: #216
+#225 := (or #206 #222)
+#162 := (<= #161 0::real)
+#172 := (+ #8 #160)
+#173 := (>= #172 0::real)
+#178 := (not #173)
+#163 := (not #162)
+#181 := (or #163 #178)
+#184 := (not #181)
+#10 := (:var 0 T2)
+#15 := (uf_1 uf_4 #10)
+#149 := (pattern #15)
+#13 := (uf_1 uf_5 #10)
+#148 := (pattern #13)
+#11 := (uf_1 uf_2 #10)
+#147 := (pattern #11)
+#50 := (* -1::real #15)
+#51 := (+ #13 #50)
+#52 := (<= #51 0::real)
+#76 := (not #52)
+#45 := (* -1::real #13)
+#46 := (+ #11 #45)
+#47 := (<= #46 0::real)
+#78 := (not #47)
+#73 := (or #78 #76)
+#83 := (not #73)
+#150 := (forall (vars (?x1 T2)) (:pat #147 #148 #149) #83)
+#86 := (forall (vars (?x1 T2)) #83)
+#153 := (iff #86 #150)
+#151 := (iff #83 #83)
+#152 := [refl]: #151
+#154 := [quant-intro #152]: #153
+#55 := (and #47 #52)
+#58 := (forall (vars (?x1 T2)) #55)
+#87 := (iff #58 #86)
+#84 := (iff #55 #83)
+#85 := [rewrite]: #84
+#88 := [quant-intro #85]: #87
+#79 := (~ #58 #58)
+#81 := (~ #55 #55)
+#82 := [refl]: #81
+#80 := [nnf-pos #82]: #79
+#16 := (<= #13 #15)
+#14 := (<= #11 #13)
+#17 := (and #14 #16)
+#18 := (forall (vars (?x1 T2)) #17)
+#59 := (iff #18 #58)
+#56 := (iff #17 #55)
+#53 := (iff #16 #52)
+#54 := [rewrite]: #53
+#48 := (iff #14 #47)
+#49 := [rewrite]: #48
+#57 := [monotonicity #49 #54]: #56
+#60 := [quant-intro #57]: #59
+#41 := [asserted]: #18
+#61 := [mp #41 #60]: #58
+#77 := [mp~ #61 #80]: #58
+#89 := [mp #77 #88]: #86
+#155 := [mp #89 #154]: #150
+#187 := (not #150)
+#188 := (or #187 #184)
+#157 := (+ #22 #156)
+#158 := (<= #157 0::real)
+#159 := (not #158)
+#164 := (or #163 #159)
+#165 := (not #164)
+#189 := (or #187 #165)
+#191 := (iff #189 #188)
+#193 := (iff #188 #188)
+#194 := [rewrite]: #193
+#185 := (iff #165 #184)
+#182 := (iff #164 #181)
+#179 := (iff #159 #178)
+#176 := (iff #158 #173)
+#166 := (+ #156 #22)
+#169 := (<= #166 0::real)
+#174 := (iff #169 #173)
+#175 := [rewrite]: #174
+#170 := (iff #158 #169)
+#167 := (= #157 #166)
+#168 := [rewrite]: #167
+#171 := [monotonicity #168]: #170
+#177 := [trans #171 #175]: #176
+#180 := [monotonicity #177]: #179
+#183 := [monotonicity #180]: #182
+#186 := [monotonicity #183]: #185
+#192 := [monotonicity #186]: #191
+#195 := [trans #192 #194]: #191
+#190 := [quant-inst]: #189
+#196 := [mp #190 #195]: #188
+#220 := [unit-resolution #196 #155]: #184
+#197 := (or #181 #162)
+#198 := [def-axiom]: #197
+#221 := [unit-resolution #198 #220]: #162
+#223 := (or #206 #163 #222)
+#224 := [th-lemma]: #223
+#226 := [unit-resolution #224 #221]: #225
+#227 := [unit-resolution #226 #219]: #222
+#199 := (or #181 #173)
+#200 := [def-axiom]: #199
+#228 := [unit-resolution #200 #220]: #173
+[th-lemma #228 #227 #211]: false
+unsat
+rbrrQuQfaijtLkQizgEXnQ 222 0
+#2 := false
+#4 := 0::real
+decl uf_2 :: (-> T2 T1 real)
+decl uf_5 :: T1
+#15 := uf_5
+decl uf_3 :: T2
+#7 := uf_3
+#20 := (uf_2 uf_3 uf_5)
+decl uf_6 :: T2
+#17 := uf_6
+#18 := (uf_2 uf_6 uf_5)
+#59 := -1::real
+#73 := (* -1::real #18)
+#106 := (+ #73 #20)
+decl uf_1 :: real
+#5 := uf_1
+#78 := (* -1::real #20)
+#79 := (+ #18 #78)
+#144 := (+ uf_1 #79)
+#145 := (<= #144 0::real)
+#148 := (ite #145 uf_1 #106)
+#279 := (* -1::real #148)
+#280 := (+ uf_1 #279)
+#281 := (<= #280 0::real)
+#289 := (not #281)
+#72 := 1/2::real
+#151 := (* 1/2::real #148)
+#248 := (<= #151 0::real)
+#162 := (= #151 0::real)
+#24 := 2::real
+#27 := (- #20 #18)
+#28 := (<= uf_1 #27)
+#29 := (ite #28 uf_1 #27)
+#30 := (/ #29 2::real)
+#31 := (+ #18 #30)
+#32 := (= #31 #18)
+#33 := (not #32)
+#34 := (not #33)
+#165 := (iff #34 #162)
+#109 := (<= uf_1 #106)
+#112 := (ite #109 uf_1 #106)
+#118 := (* 1/2::real #112)
+#123 := (+ #18 #118)
+#129 := (= #18 #123)
+#163 := (iff #129 #162)
+#154 := (+ #18 #151)
+#157 := (= #18 #154)
+#160 := (iff #157 #162)
+#161 := [rewrite]: #160
+#158 := (iff #129 #157)
+#155 := (= #123 #154)
+#152 := (= #118 #151)
+#149 := (= #112 #148)
+#146 := (iff #109 #145)
+#147 := [rewrite]: #146
+#150 := [monotonicity #147]: #149
+#153 := [monotonicity #150]: #152
+#156 := [monotonicity #153]: #155
+#159 := [monotonicity #156]: #158
+#164 := [trans #159 #161]: #163
+#142 := (iff #34 #129)
+#134 := (not #129)
+#137 := (not #134)
+#140 := (iff #137 #129)
+#141 := [rewrite]: #140
+#138 := (iff #34 #137)
+#135 := (iff #33 #134)
+#132 := (iff #32 #129)
+#126 := (= #123 #18)
+#130 := (iff #126 #129)
+#131 := [rewrite]: #130
+#127 := (iff #32 #126)
+#124 := (= #31 #123)
+#121 := (= #30 #118)
+#115 := (/ #112 2::real)
+#119 := (= #115 #118)
+#120 := [rewrite]: #119
+#116 := (= #30 #115)
+#113 := (= #29 #112)
+#107 := (= #27 #106)
+#108 := [rewrite]: #107
+#110 := (iff #28 #109)
+#111 := [monotonicity #108]: #110
+#114 := [monotonicity #111 #108]: #113
+#117 := [monotonicity #114]: #116
+#122 := [trans #117 #120]: #121
+#125 := [monotonicity #122]: #124
+#128 := [monotonicity #125]: #127
+#133 := [trans #128 #131]: #132
+#136 := [monotonicity #133]: #135
+#139 := [monotonicity #136]: #138
+#143 := [trans #139 #141]: #142
+#166 := [trans #143 #164]: #165
+#105 := [asserted]: #34
+#167 := [mp #105 #166]: #162
+#283 := (not #162)
+#284 := (or #283 #248)
+#285 := [th-lemma]: #284
+#286 := [unit-resolution #285 #167]: #248
+#287 := [hypothesis]: #281
+#53 := (<= uf_1 0::real)
+#54 := (not #53)
+#6 := (< 0::real uf_1)
+#55 := (iff #6 #54)
+#56 := [rewrite]: #55
+#50 := [asserted]: #6
+#57 := [mp #50 #56]: #54
+#288 := [th-lemma #57 #287 #286]: false
+#290 := [lemma #288]: #289
+#241 := (= uf_1 #148)
+#242 := (= #106 #148)
+#299 := (not #242)
+#282 := (+ #106 #279)
+#291 := (<= #282 0::real)
+#296 := (not #291)
+decl uf_4 :: T2
+#10 := uf_4
+#16 := (uf_2 uf_4 uf_5)
+#260 := (+ #16 #78)
+#261 := (>= #260 0::real)
+#266 := (not #261)
+#8 := (:var 0 T1)
+#11 := (uf_2 uf_4 #8)
+#234 := (pattern #11)
+#9 := (uf_2 uf_3 #8)
+#233 := (pattern #9)
+#60 := (* -1::real #11)
+#61 := (+ #9 #60)
+#62 := (<= #61 0::real)
+#179 := (not #62)
+#235 := (forall (vars (?x1 T1)) (:pat #233 #234) #179)
+#178 := (forall (vars (?x1 T1)) #179)
+#238 := (iff #178 #235)
+#236 := (iff #179 #179)
+#237 := [refl]: #236
+#239 := [quant-intro #237]: #238
+#65 := (exists (vars (?x1 T1)) #62)
+#68 := (not #65)
+#175 := (~ #68 #178)
+#180 := (~ #179 #179)
+#177 := [refl]: #180
+#176 := [nnf-neg #177]: #175
+#12 := (<= #9 #11)
+#13 := (exists (vars (?x1 T1)) #12)
+#14 := (not #13)
+#69 := (iff #14 #68)
+#66 := (iff #13 #65)
+#63 := (iff #12 #62)
+#64 := [rewrite]: #63
+#67 := [quant-intro #64]: #66
+#70 := [monotonicity #67]: #69
+#51 := [asserted]: #14
+#71 := [mp #51 #70]: #68
+#173 := [mp~ #71 #176]: #178
+#240 := [mp #173 #239]: #235
+#269 := (not #235)
+#270 := (or #269 #266)
+#250 := (* -1::real #16)
+#251 := (+ #20 #250)
+#252 := (<= #251 0::real)
+#253 := (not #252)
+#271 := (or #269 #253)
+#273 := (iff #271 #270)
+#275 := (iff #270 #270)
+#276 := [rewrite]: #275
+#267 := (iff #253 #266)
+#264 := (iff #252 #261)
+#254 := (+ #250 #20)
+#257 := (<= #254 0::real)
+#262 := (iff #257 #261)
+#263 := [rewrite]: #262
+#258 := (iff #252 #257)
+#255 := (= #251 #254)
+#256 := [rewrite]: #255
+#259 := [monotonicity #256]: #258
+#265 := [trans #259 #263]: #264
+#268 := [monotonicity #265]: #267
+#274 := [monotonicity #268]: #273
+#277 := [trans #274 #276]: #273
+#272 := [quant-inst]: #271
+#278 := [mp #272 #277]: #270
+#293 := [unit-resolution #278 #240]: #266
+#90 := (* 1/2::real #20)
+#102 := (+ #73 #90)
+#89 := (* 1/2::real #16)
+#103 := (+ #89 #102)
+#100 := (>= #103 0::real)
+#23 := (+ #16 #20)
+#25 := (/ #23 2::real)
+#26 := (<= #18 #25)
+#98 := (iff #26 #100)
+#91 := (+ #89 #90)
+#94 := (<= #18 #91)
+#97 := (iff #94 #100)
+#99 := [rewrite]: #97
+#95 := (iff #26 #94)
+#92 := (= #25 #91)
+#93 := [rewrite]: #92
+#96 := [monotonicity #93]: #95
+#101 := [trans #96 #99]: #98
+#58 := [asserted]: #26
+#104 := [mp #58 #101]: #100
+#294 := [hypothesis]: #291
+#295 := [th-lemma #294 #104 #293 #286]: false
+#297 := [lemma #295]: #296
+#298 := [hypothesis]: #242
+#300 := (or #299 #291)
+#301 := [th-lemma]: #300
+#302 := [unit-resolution #301 #298 #297]: false
+#303 := [lemma #302]: #299
+#246 := (or #145 #242)
+#247 := [def-axiom]: #246
+#304 := [unit-resolution #247 #303]: #145
+#243 := (not #145)
+#244 := (or #243 #241)
+#245 := [def-axiom]: #244
+#305 := [unit-resolution #245 #304]: #241
+#306 := (not #241)
+#307 := (or #306 #281)
+#308 := [th-lemma]: #307
+[unit-resolution #308 #305 #290]: false
+unsat
+hwh3oeLAWt56hnKIa8Wuow 248 0
+#2 := false
+#4 := 0::real
+decl uf_2 :: (-> T2 T1 real)
+decl uf_5 :: T1
+#15 := uf_5
+decl uf_6 :: T2
+#17 := uf_6
+#18 := (uf_2 uf_6 uf_5)
+decl uf_4 :: T2
+#10 := uf_4
+#16 := (uf_2 uf_4 uf_5)
+#66 := -1::real
+#137 := (* -1::real #16)
+#138 := (+ #137 #18)
+decl uf_1 :: real
+#5 := uf_1
+#80 := (* -1::real #18)
+#81 := (+ #16 #80)
+#201 := (+ uf_1 #81)
+#202 := (<= #201 0::real)
+#205 := (ite #202 uf_1 #138)
+#352 := (* -1::real #205)
+#353 := (+ uf_1 #352)
+#354 := (<= #353 0::real)
+#362 := (not #354)
+#79 := 1/2::real
+#244 := (* 1/2::real #205)
+#322 := (<= #244 0::real)
+#245 := (= #244 0::real)
+#158 := -1/2::real
+#208 := (* -1/2::real #205)
+#211 := (+ #18 #208)
+decl uf_3 :: T2
+#7 := uf_3
+#20 := (uf_2 uf_3 uf_5)
+#117 := (+ #80 #20)
+#85 := (* -1::real #20)
+#86 := (+ #18 #85)
+#188 := (+ uf_1 #86)
+#189 := (<= #188 0::real)
+#192 := (ite #189 uf_1 #117)
+#195 := (* 1/2::real #192)
+#198 := (+ #18 #195)
+#97 := (* 1/2::real #20)
+#109 := (+ #80 #97)
+#96 := (* 1/2::real #16)
+#110 := (+ #96 #109)
+#107 := (>= #110 0::real)
+#214 := (ite #107 #198 #211)
+#217 := (= #18 #214)
+#248 := (iff #217 #245)
+#241 := (= #18 #211)
+#246 := (iff #241 #245)
+#247 := [rewrite]: #246
+#242 := (iff #217 #241)
+#239 := (= #214 #211)
+#234 := (ite false #198 #211)
+#237 := (= #234 #211)
+#238 := [rewrite]: #237
+#235 := (= #214 #234)
+#232 := (iff #107 false)
+#104 := (not #107)
+#24 := 2::real
+#23 := (+ #16 #20)
+#25 := (/ #23 2::real)
+#26 := (< #25 #18)
+#108 := (iff #26 #104)
+#98 := (+ #96 #97)
+#101 := (< #98 #18)
+#106 := (iff #101 #104)
+#105 := [rewrite]: #106
+#102 := (iff #26 #101)
+#99 := (= #25 #98)
+#100 := [rewrite]: #99
+#103 := [monotonicity #100]: #102
+#111 := [trans #103 #105]: #108
+#65 := [asserted]: #26
+#112 := [mp #65 #111]: #104
+#233 := [iff-false #112]: #232
+#236 := [monotonicity #233]: #235
+#240 := [trans #236 #238]: #239
+#243 := [monotonicity #240]: #242
+#249 := [trans #243 #247]: #248
+#33 := (- #18 #16)
+#34 := (<= uf_1 #33)
+#35 := (ite #34 uf_1 #33)
+#36 := (/ #35 2::real)
+#37 := (- #18 #36)
+#28 := (- #20 #18)
+#29 := (<= uf_1 #28)
+#30 := (ite #29 uf_1 #28)
+#31 := (/ #30 2::real)
+#32 := (+ #18 #31)
+#27 := (<= #18 #25)
+#38 := (ite #27 #32 #37)
+#39 := (= #38 #18)
+#40 := (not #39)
+#41 := (not #40)
+#220 := (iff #41 #217)
+#141 := (<= uf_1 #138)
+#144 := (ite #141 uf_1 #138)
+#159 := (* -1/2::real #144)
+#160 := (+ #18 #159)
+#120 := (<= uf_1 #117)
+#123 := (ite #120 uf_1 #117)
+#129 := (* 1/2::real #123)
+#134 := (+ #18 #129)
+#114 := (<= #18 #98)
+#165 := (ite #114 #134 #160)
+#171 := (= #18 #165)
+#218 := (iff #171 #217)
+#215 := (= #165 #214)
+#212 := (= #160 #211)
+#209 := (= #159 #208)
+#206 := (= #144 #205)
+#203 := (iff #141 #202)
+#204 := [rewrite]: #203
+#207 := [monotonicity #204]: #206
+#210 := [monotonicity #207]: #209
+#213 := [monotonicity #210]: #212
+#199 := (= #134 #198)
+#196 := (= #129 #195)
+#193 := (= #123 #192)
+#190 := (iff #120 #189)
+#191 := [rewrite]: #190
+#194 := [monotonicity #191]: #193
+#197 := [monotonicity #194]: #196
+#200 := [monotonicity #197]: #199
+#187 := (iff #114 #107)
+#186 := [rewrite]: #187
+#216 := [monotonicity #186 #200 #213]: #215
+#219 := [monotonicity #216]: #218
+#184 := (iff #41 #171)
+#176 := (not #171)
+#179 := (not #176)
+#182 := (iff #179 #171)
+#183 := [rewrite]: #182
+#180 := (iff #41 #179)
+#177 := (iff #40 #176)
+#174 := (iff #39 #171)
+#168 := (= #165 #18)
+#172 := (iff #168 #171)
+#173 := [rewrite]: #172
+#169 := (iff #39 #168)
+#166 := (= #38 #165)
+#163 := (= #37 #160)
+#150 := (* 1/2::real #144)
+#155 := (- #18 #150)
+#161 := (= #155 #160)
+#162 := [rewrite]: #161
+#156 := (= #37 #155)
+#153 := (= #36 #150)
+#147 := (/ #144 2::real)
+#151 := (= #147 #150)
+#152 := [rewrite]: #151
+#148 := (= #36 #147)
+#145 := (= #35 #144)
+#139 := (= #33 #138)
+#140 := [rewrite]: #139
+#142 := (iff #34 #141)
+#143 := [monotonicity #140]: #142
+#146 := [monotonicity #143 #140]: #145
+#149 := [monotonicity #146]: #148
+#154 := [trans #149 #152]: #153
+#157 := [monotonicity #154]: #156
+#164 := [trans #157 #162]: #163
+#135 := (= #32 #134)
+#132 := (= #31 #129)
+#126 := (/ #123 2::real)
+#130 := (= #126 #129)
+#131 := [rewrite]: #130
+#127 := (= #31 #126)
+#124 := (= #30 #123)
+#118 := (= #28 #117)
+#119 := [rewrite]: #118
+#121 := (iff #29 #120)
+#122 := [monotonicity #119]: #121
+#125 := [monotonicity #122 #119]: #124
+#128 := [monotonicity #125]: #127
+#133 := [trans #128 #131]: #132
+#136 := [monotonicity #133]: #135
+#115 := (iff #27 #114)
+#116 := [monotonicity #100]: #115
+#167 := [monotonicity #116 #136 #164]: #166
+#170 := [monotonicity #167]: #169
+#175 := [trans #170 #173]: #174
+#178 := [monotonicity #175]: #177
+#181 := [monotonicity #178]: #180
+#185 := [trans #181 #183]: #184
+#221 := [trans #185 #219]: #220
+#113 := [asserted]: #41
+#222 := [mp #113 #221]: #217
+#250 := [mp #222 #249]: #245
+#356 := (not #245)
+#357 := (or #356 #322)
+#358 := [th-lemma]: #357
+#359 := [unit-resolution #358 #250]: #322
+#360 := [hypothesis]: #354
+#60 := (<= uf_1 0::real)
+#61 := (not #60)
+#6 := (< 0::real uf_1)
+#62 := (iff #6 #61)
+#63 := [rewrite]: #62
+#57 := [asserted]: #6
+#64 := [mp #57 #63]: #61
+#361 := [th-lemma #64 #360 #359]: false
+#363 := [lemma #361]: #362
+#315 := (= uf_1 #205)
+#316 := (= #138 #205)
+#371 := (not #316)
+#355 := (+ #138 #352)
+#364 := (<= #355 0::real)
+#368 := (not #364)
+#87 := (<= #86 0::real)
+#82 := (<= #81 0::real)
+#90 := (and #82 #87)
+#21 := (<= #18 #20)
+#19 := (<= #16 #18)
+#22 := (and #19 #21)
+#91 := (iff #22 #90)
+#88 := (iff #21 #87)
+#89 := [rewrite]: #88
+#83 := (iff #19 #82)
+#84 := [rewrite]: #83
+#92 := [monotonicity #84 #89]: #91
+#59 := [asserted]: #22
+#93 := [mp #59 #92]: #90
+#95 := [and-elim #93]: #87
+#366 := [hypothesis]: #364
+#367 := [th-lemma #366 #95 #112 #359]: false
+#369 := [lemma #367]: #368
+#370 := [hypothesis]: #316
+#372 := (or #371 #364)
+#373 := [th-lemma]: #372
+#374 := [unit-resolution #373 #370 #369]: false
+#375 := [lemma #374]: #371
+#320 := (or #202 #316)
+#321 := [def-axiom]: #320
+#376 := [unit-resolution #321 #375]: #202
+#317 := (not #202)
+#318 := (or #317 #315)
+#319 := [def-axiom]: #318
+#377 := [unit-resolution #319 #376]: #315
+#378 := (not #315)
+#379 := (or #378 #354)
+#380 := [th-lemma]: #379
+[unit-resolution #380 #377 #363]: false
+unsat
+WdMJH3tkMv/rps8y9Ukq5Q 86 0
+#2 := false
+#37 := 0::real
+decl uf_2 :: (-> T2 T1 real)
+decl uf_4 :: T1
+#12 := uf_4
+decl uf_3 :: T2
+#5 := uf_3
+#13 := (uf_2 uf_3 uf_4)
+#34 := -1::real
+#140 := (* -1::real #13)
+decl uf_1 :: real
+#4 := uf_1
+#141 := (+ uf_1 #140)
+#143 := (>= #141 0::real)
+#6 := (:var 0 T1)
+#7 := (uf_2 uf_3 #6)
+#127 := (pattern #7)
+#35 := (* -1::real #7)
+#36 := (+ uf_1 #35)
+#47 := (>= #36 0::real)
+#134 := (forall (vars (?x2 T1)) (:pat #127) #47)
+#49 := (forall (vars (?x2 T1)) #47)
+#137 := (iff #49 #134)
+#135 := (iff #47 #47)
+#136 := [refl]: #135
+#138 := [quant-intro #136]: #137
+#67 := (~ #49 #49)
+#58 := (~ #47 #47)
+#66 := [refl]: #58
+#68 := [nnf-pos #66]: #67
+#10 := (<= #7 uf_1)
+#11 := (forall (vars (?x2 T1)) #10)
+#50 := (iff #11 #49)
+#46 := (iff #10 #47)
+#48 := [rewrite]: #46
+#51 := [quant-intro #48]: #50
+#32 := [asserted]: #11
+#52 := [mp #32 #51]: #49
+#69 := [mp~ #52 #68]: #49
+#139 := [mp #69 #138]: #134
+#149 := (not #134)
+#150 := (or #149 #143)
+#151 := [quant-inst]: #150
+#144 := [unit-resolution #151 #139]: #143
+#142 := (<= #141 0::real)
+#38 := (<= #36 0::real)
+#128 := (forall (vars (?x1 T1)) (:pat #127) #38)
+#41 := (forall (vars (?x1 T1)) #38)
+#131 := (iff #41 #128)
+#129 := (iff #38 #38)
+#130 := [refl]: #129
+#132 := [quant-intro #130]: #131
+#62 := (~ #41 #41)
+#64 := (~ #38 #38)
+#65 := [refl]: #64
+#63 := [nnf-pos #65]: #62
+#8 := (<= uf_1 #7)
+#9 := (forall (vars (?x1 T1)) #8)
+#42 := (iff #9 #41)
+#39 := (iff #8 #38)
+#40 := [rewrite]: #39
+#43 := [quant-intro #40]: #42
+#31 := [asserted]: #9
+#44 := [mp #31 #43]: #41
+#61 := [mp~ #44 #63]: #41
+#133 := [mp #61 #132]: #128
+#145 := (not #128)
+#146 := (or #145 #142)
+#147 := [quant-inst]: #146
+#148 := [unit-resolution #147 #133]: #142
+#45 := (= uf_1 #13)
+#55 := (not #45)
+#14 := (= #13 uf_1)
+#15 := (not #14)
+#56 := (iff #15 #55)
+#53 := (iff #14 #45)
+#54 := [rewrite]: #53
+#57 := [monotonicity #54]: #56
+#33 := [asserted]: #15
+#60 := [mp #33 #57]: #55
+#153 := (not #143)
+#152 := (not #142)
+#154 := (or #45 #152 #153)
+#155 := [th-lemma]: #154
+[unit-resolution #155 #60 #148 #144]: false
+unsat
+V+IAyBZU/6QjYs6JkXx8LQ 57 0
+#2 := false
+#4 := 0::real
+decl uf_1 :: (-> T2 real)
+decl uf_2 :: (-> T1 T1 T2)
+decl uf_12 :: (-> T4 T1)
+decl uf_4 :: T4
+#11 := uf_4
+#39 := (uf_12 uf_4)
+decl uf_10 :: T4
+#27 := uf_10
+#38 := (uf_12 uf_10)
+#40 := (uf_2 #38 #39)
+#41 := (uf_1 #40)
+#264 := (>= #41 0::real)
+#266 := (not #264)
+#43 := (= #41 0::real)
+#44 := (not #43)
+#131 := [asserted]: #44
+#272 := (or #43 #266)
+#42 := (<= #41 0::real)
+#130 := [asserted]: #42
+#265 := (not #42)
+#270 := (or #43 #265 #266)
+#271 := [th-lemma]: #270
+#273 := [unit-resolution #271 #130]: #272
+#274 := [unit-resolution #273 #131]: #266
+#6 := (:var 0 T1)
+#5 := (:var 1 T1)
+#7 := (uf_2 #5 #6)
+#241 := (pattern #7)
+#8 := (uf_1 #7)
+#65 := (>= #8 0::real)
+#242 := (forall (vars (?x1 T1) (?x2 T1)) (:pat #241) #65)
+#66 := (forall (vars (?x1 T1) (?x2 T1)) #65)
+#245 := (iff #66 #242)
+#243 := (iff #65 #65)
+#244 := [refl]: #243
+#246 := [quant-intro #244]: #245
+#149 := (~ #66 #66)
+#151 := (~ #65 #65)
+#152 := [refl]: #151
+#150 := [nnf-pos #152]: #149
+#9 := (<= 0::real #8)
+#10 := (forall (vars (?x1 T1) (?x2 T1)) #9)
+#67 := (iff #10 #66)
+#63 := (iff #9 #65)
+#64 := [rewrite]: #63
+#68 := [quant-intro #64]: #67
+#60 := [asserted]: #10
+#69 := [mp #60 #68]: #66
+#147 := [mp~ #69 #150]: #66
+#247 := [mp #147 #246]: #242
+#267 := (not #242)
+#268 := (or #267 #264)
+#269 := [quant-inst]: #268
+[unit-resolution #269 #247 #274]: false
+unsat
+vqiyJ/qjGXZ3iOg6xftiug 15 0
+uf_1 -> val!0
+uf_2 -> val!1
+uf_3 -> val!2
+uf_5 -> val!15
+uf_6 -> val!26
+uf_4 -> {
+ val!0 -> val!12
+ val!1 -> val!13
+ else -> val!13
+}
+uf_7 -> {
+ val!6 -> val!31
+ else -> val!31
+}
+sat
+mrZPJZyTokErrN6SYupisw 9 0
+uf_4 -> val!1
+uf_2 -> val!3
+uf_3 -> val!4
+uf_1 -> {
+ val!5 -> val!6
+ val!4 -> val!7
+ else -> val!7
+}
+sat
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Multivariate_Analysis/Integration.thy Mon Feb 22 20:41:49 2010 +0100
@@ -0,0 +1,3473 @@
+
+header {* Kurzweil-Henstock gauge integration in many dimensions. *}
+(* Author: John Harrison
+ Translation from HOL light: Robert Himmelmann, TU Muenchen *)
+
+theory Integration
+ imports Derivative SMT
+begin
+
+declare [[smt_certificates="~~/src/HOL/Multivariate_Analysis/Integration.cert"]]
+declare [[smt_record=true]]
+declare [[z3_proofs=true]]
+
+lemma conjunctD2: assumes "a \<and> b" shows a b using assms by auto
+lemma conjunctD3: assumes "a \<and> b \<and> c" shows a b c using assms by auto
+lemma conjunctD4: assumes "a \<and> b \<and> c \<and> d" shows a b c d using assms by auto
+lemma conjunctD5: assumes "a \<and> b \<and> c \<and> d \<and> e" shows a b c d e using assms by auto
+
+declare smult_conv_scaleR[simp]
+
+subsection {* Some useful lemmas about intervals. *}
+
+lemma empty_as_interval: "{} = {1..0::real^'n}"
+ apply(rule set_ext,rule) defer unfolding vector_le_def mem_interval
+ using UNIV_witness[where 'a='n] apply(erule_tac exE,rule_tac x=x in allE) by auto
+
+lemma interior_subset_union_intervals:
+ assumes "i = {a..b::real^'n}" "j = {c..d}" "interior j \<noteq> {}" "i \<subseteq> j \<union> s" "interior(i) \<inter> interior(j) = {}"
+ shows "interior i \<subseteq> interior s" proof-
+ have "{a<..<b} \<inter> {c..d} = {}" using inter_interval_mixed_eq_empty[of c d a b] and assms(3,5)
+ unfolding assms(1,2) interior_closed_interval by auto
+ moreover have "{a<..<b} \<subseteq> {c..d} \<union> s" apply(rule order_trans,rule interval_open_subset_closed)
+ using assms(4) unfolding assms(1,2) by auto
+ ultimately show ?thesis apply-apply(rule interior_maximal) defer apply(rule open_interior)
+ unfolding assms(1,2) interior_closed_interval by auto qed
+
+lemma inter_interior_unions_intervals: fixes f::"(real^'n) set set"
+ assumes "finite f" "open s" "\<forall>t\<in>f. \<exists>a b. t = {a..b}" "\<forall>t\<in>f. s \<inter> (interior t) = {}"
+ shows "s \<inter> interior(\<Union>f) = {}" proof(rule ccontr,unfold ex_in_conv[THEN sym]) case goal1
+ 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)
+ unfolding open_subset_interior[OF open_ball] using interior_subset by auto
+ have lem2:"\<And>x s P. \<exists>x\<in>s. P x \<Longrightarrow> \<exists>x\<in>insert x s. P x" by auto
+ 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
+ thus ?case proof(induct rule:finite_induct)
+ case empty from this(2) guess x .. hence False unfolding Union_empty interior_empty by auto thus ?case by auto next
+ case (insert i f) guess x using insert(5) .. note x = this
+ then guess e unfolding open_contains_ball_eq[OF open_Int[OF assms(2) open_interior],rule_format] .. note e=this
+ guess a using insert(4)[rule_format,OF insertI1] .. then guess b .. note ab = this
+ show ?case proof(cases "x\<in>i") case False hence "x \<in> UNIV - {a..b}" unfolding ab by auto
+ then guess d unfolding open_contains_ball_eq[OF open_Diff[OF open_UNIV closed_interval],rule_format] ..
+ hence "0 < d" "ball x (min d e) \<subseteq> UNIV - i" using e unfolding ab by auto
+ 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
+ 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
+ case True show ?thesis proof(cases "x\<in>{a<..<b}")
+ case True then guess d unfolding open_contains_ball_eq[OF open_interval,rule_format] ..
+ thus ?thesis apply(rule_tac x=i in bexI,rule_tac x=x in exI,rule_tac x="min d e" in exI)
+ unfolding ab using interval_open_subset_closed[of a b] and e by fastsimp+ next
+ 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)
+ 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
+ hence "\<exists>x. ball x (e/2) \<subseteq> s \<inter> (\<Union>f)" proof(erule_tac disjE)
+ 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)
+ fix y assume "y \<in> ball ?z (e / 2) \<inter> i" hence "dist ?z y < e/2" and yi:"y\<in>i" by auto
+ hence "\<bar>(?z - y) $ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding vector_dist_norm by auto
+ hence "y$k < a$k" unfolding vector_component_simps vector_scaleR_component as using e[THEN conjunct1] by(auto simp add:field_simps)
+ 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
+ moreover have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)" apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]]) proof
+ 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)"
+ apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "(e/2) *\<^sub>R basis k"])
+ unfolding norm_scaleR norm_basis by auto
+ 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)
+ finally show "y\<in>ball x e" unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps) qed
+ ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto
+ 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)
+ fix y assume "y \<in> ball ?z (e / 2) \<inter> i" hence "dist ?z y < e/2" and yi:"y\<in>i" by auto
+ hence "\<bar>(?z - y) $ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding vector_dist_norm by auto
+ hence "y$k > b$k" unfolding vector_component_simps vector_scaleR_component as using e[THEN conjunct1] by(auto simp add:field_simps)
+ 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
+ moreover have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)" apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]]) proof
+ 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)"
+ apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "- (e/2) *\<^sub>R basis k"])
+ unfolding norm_scaleR norm_basis by auto
+ 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)
+ finally show "y\<in>ball x e" unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps) qed
+ ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto qed
+ 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
+ thus ?thesis apply-apply(rule lem2,rule insert(3)) using insert(4) by auto qed qed qed qed note * = this
+ guess t using *[OF assms(1,3) goal1] .. from this(2) guess x .. then guess e ..
+ hence "x \<in> s" "x\<in>interior t" defer using open_subset_interior[OF open_ball, of x e t] by auto
+ thus False using `t\<in>f` assms(4) by auto qed
+subsection {* Bounds on intervals where they exist. *}
+
+definition "interval_upperbound (s::(real^'n) set) = (\<chi> i. Sup {a. \<exists>x\<in>s. x$i = a})"
+
+definition "interval_lowerbound (s::(real^'n) set) = (\<chi> i. Inf {a. \<exists>x\<in>s. x$i = a})"
+
+lemma interval_upperbound[simp]: assumes "\<forall>i. a$i \<le> b$i" shows "interval_upperbound {a..b} = b"
+ using assms unfolding interval_upperbound_def Cart_eq Cart_lambda_beta apply-apply(rule,erule_tac x=i in allE)
+ apply(rule Sup_unique) unfolding setle_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer
+ apply(rule,rule) apply(rule_tac x="b$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=b in bexI)
+ unfolding mem_interval using assms by auto
+
+lemma interval_lowerbound[simp]: assumes "\<forall>i. a$i \<le> b$i" shows "interval_lowerbound {a..b} = a"
+ using assms unfolding interval_lowerbound_def Cart_eq Cart_lambda_beta apply-apply(rule,erule_tac x=i in allE)
+ apply(rule Inf_unique) unfolding setge_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer
+ apply(rule,rule) apply(rule_tac x="a$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=a in bexI)
+ unfolding mem_interval using assms by auto
+
+lemmas interval_bounds = interval_upperbound interval_lowerbound
+
+lemma interval_bounds'[simp]: assumes "{a..b}\<noteq>{}" shows "interval_upperbound {a..b} = b" "interval_lowerbound {a..b} = a"
+ using assms unfolding interval_ne_empty by auto
+
+lemma interval_upperbound_1[simp]: "dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> interval_upperbound {a..b} = (b::real^1)"
+ apply(rule interval_upperbound) by auto
+
+lemma interval_lowerbound_1[simp]: "dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> interval_lowerbound {a..b} = (a::real^1)"
+ apply(rule interval_lowerbound) by auto
+
+lemmas interval_bound_1 = interval_upperbound_1 interval_lowerbound_1
+
+subsection {* Content (length, area, volume...) of an interval. *}
+
+definition "content (s::(real^'n) set) =
+ (if s = {} then 0 else (\<Prod>i\<in>UNIV. (interval_upperbound s)$i - (interval_lowerbound s)$i))"
+
+lemma interval_not_empty:"\<forall>i. a$i \<le> b$i \<Longrightarrow> {a..b::real^'n} \<noteq> {}"
+ unfolding interval_eq_empty unfolding not_ex not_less by assumption
+
+lemma content_closed_interval: assumes "\<forall>i. a$i \<le> b$i"
+ shows "content {a..b} = (\<Prod>i\<in>UNIV. b$i - a$i)"
+ using interval_not_empty[OF assms] unfolding content_def interval_upperbound[OF assms] interval_lowerbound[OF assms] by auto
+
+lemma content_closed_interval': assumes "{a..b}\<noteq>{}" shows "content {a..b} = (\<Prod>i\<in>UNIV. b$i - a$i)"
+ apply(rule content_closed_interval) using assms unfolding interval_ne_empty .
+
+lemma content_1:"dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> content {a..b} = dest_vec1 b - dest_vec1 a"
+ using content_closed_interval[of a b] by auto
+
+lemma content_1':"a \<le> b \<Longrightarrow> content {vec1 a..vec1 b} = b - a" using content_1[of "vec a" "vec b"] by auto
+
+lemma content_unit[intro]: "content{0..1::real^'n} = 1" proof-
+ have *:"\<forall>i. 0$i \<le> (1::real^'n::finite)$i" by auto
+ have "0 \<in> {0..1::real^'n::finite}" unfolding mem_interval by auto
+ thus ?thesis unfolding content_def interval_bounds[OF *] using setprod_1 by auto qed
+
+lemma content_pos_le[intro]: "0 \<le> content {a..b}" proof(cases "{a..b}={}")
+ case False hence *:"\<forall>i. a $ i \<le> b $ i" unfolding interval_ne_empty by assumption
+ have "(\<Prod>i\<in>UNIV. interval_upperbound {a..b} $ i - interval_lowerbound {a..b} $ i) \<ge> 0"
+ apply(rule setprod_nonneg) unfolding interval_bounds[OF *] using * apply(erule_tac x=x in allE) by auto
+ thus ?thesis unfolding content_def by(auto simp del:interval_bounds') qed(unfold content_def, auto)
+
+lemma content_pos_lt: assumes "\<forall>i. a$i < b$i" shows "0 < content {a..b}"
+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
+ show ?thesis unfolding content_closed_interval[OF help_lemma1[OF assms]] apply(rule setprod_pos)
+ using assms apply(erule_tac x=x in allE) by auto qed
+
+lemma content_pos_lt_1: "dest_vec1 a < dest_vec1 b \<Longrightarrow> 0 < content({a..b})"
+ apply(rule content_pos_lt) by auto
+
+lemma content_eq_0: "content({a..b::real^'n}) = 0 \<longleftrightarrow> (\<exists>i. b$i \<le> a$i)" proof(cases "{a..b} = {}")
+ case True thus ?thesis unfolding content_def if_P[OF True] unfolding interval_eq_empty apply-
+ apply(rule,erule exE) apply(rule_tac x=i in exI) by auto next
+ guess a using UNIV_witness[where 'a='n] .. case False note as=this[unfolded interval_eq_empty not_ex not_less]
+ show ?thesis unfolding content_def if_not_P[OF False] setprod_zero_iff[OF finite_UNIV]
+ apply(rule) apply(erule_tac[!] exE bexE) unfolding interval_bounds[OF as] apply(rule_tac x=x in exI) defer
+ apply(rule_tac x=i in bexI) using as apply(erule_tac x=i in allE) by auto qed
+
+lemma cond_cases:"(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)" by auto
+
+lemma content_closed_interval_cases:
+ "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)
+ 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
+
+lemma content_eq_0_interior: "content {a..b} = 0 \<longleftrightarrow> interior({a..b}) = {}"
+ unfolding content_eq_0 interior_closed_interval interval_eq_empty by auto
+
+lemma content_eq_0_1: "content {a..b::real^1} = 0 \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a"
+ unfolding content_eq_0 by auto
+
+lemma content_pos_lt_eq: "0 < content {a..b} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
+ apply(rule) defer apply(rule content_pos_lt,assumption) proof- assume "0 < content {a..b}"
+ 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
+
+lemma content_empty[simp]: "content {} = 0" unfolding content_def by auto
+
+lemma content_subset: assumes "{a..b} \<subseteq> {c..d}" shows "content {a..b::real^'n} \<le> content {c..d}" proof(cases "{a..b}={}")
+ case True thus ?thesis using content_pos_le[of c d] by auto next
+ case False hence ab_ne:"\<forall>i. a $ i \<le> b $ i" unfolding interval_ne_empty by auto
+ hence ab_ab:"a\<in>{a..b}" "b\<in>{a..b}" unfolding mem_interval by auto
+ have "{c..d} \<noteq> {}" using assms False by auto
+ hence cd_ne:"\<forall>i. c $ i \<le> d $ i" using assms unfolding interval_ne_empty by auto
+ show ?thesis unfolding content_def unfolding interval_bounds[OF ab_ne] interval_bounds[OF cd_ne]
+ unfolding if_not_P[OF False] if_not_P[OF `{c..d} \<noteq> {}`] apply(rule setprod_mono,rule) proof fix i::'n
+ show "0 \<le> b $ i - a $ i" using ab_ne[THEN spec[where x=i]] by auto
+ show "b $ i - a $ i \<le> d $ i - c $ i"
+ using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(2),of i]
+ using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(1),of i] by auto qed qed
+
+lemma content_lt_nz: "0 < content {a..b} \<longleftrightarrow> content {a..b} \<noteq> 0"
+ unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by auto
+
+subsection {* The notion of a gauge --- simply an open set containing the point. *}
+
+definition gauge where "gauge d \<longleftrightarrow> (\<forall>x. x\<in>(d x) \<and> open(d x))"
+
+lemma gaugeI:assumes "\<And>x. x\<in>g x" "\<And>x. open (g x)" shows "gauge g"
+ using assms unfolding gauge_def by auto
+
+lemma gaugeD[dest]: assumes "gauge d" shows "x\<in>d x" "open (d x)" using assms unfolding gauge_def by auto
+
+lemma gauge_ball_dependent: "\<forall>x. 0 < e x \<Longrightarrow> gauge (\<lambda>x. ball x (e x))"
+ unfolding gauge_def by auto
+
+lemma gauge_ball[intro?]: "0 < e \<Longrightarrow> gauge (\<lambda>x. ball x e)" unfolding gauge_def by auto
+
+lemma gauge_trivial[intro]: "gauge (\<lambda>x. ball x 1)" apply(rule gauge_ball) by auto
+
+lemma gauge_inter: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. (d1 x) \<inter> (d2 x))"
+ unfolding gauge_def by auto
+
+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-
+ have *:"\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s" by auto show ?thesis
+ unfolding gauge_def unfolding *
+ using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto qed
+
+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)
+
+subsection {* Divisions. *}
+
+definition division_of (infixl "division'_of" 40) where
+ "s division_of i \<equiv>
+ finite s \<and>
+ (\<forall>k\<in>s. k \<subseteq> i \<and> k \<noteq> {} \<and> (\<exists>a b. k = {a..b})) \<and>
+ (\<forall>k1\<in>s. \<forall>k2\<in>s. k1 \<noteq> k2 \<longrightarrow> interior(k1) \<inter> interior(k2) = {}) \<and>
+ (\<Union>s = i)"
+
+lemma division_ofD[dest]: assumes "s division_of i"
+ 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})"
+ "\<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
+
+lemma division_ofI:
+ 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})"
+ "\<And>k1 k2. \<lbrakk>k1\<in>s; k2\<in>s; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}" "\<Union>s = i"
+ shows "s division_of i" using assms unfolding division_of_def by auto
+
+lemma division_of_finite: "s division_of i \<Longrightarrow> finite s"
+ unfolding division_of_def by auto
+
+lemma division_of_self[intro]: "{a..b} \<noteq> {} \<Longrightarrow> {{a..b}} division_of {a..b}"
+ unfolding division_of_def by auto
+
+lemma division_of_trivial[simp]: "s division_of {} \<longleftrightarrow> s = {}" unfolding division_of_def by auto
+
+lemma division_of_sing[simp]: "s division_of {a..a::real^'n} \<longleftrightarrow> s = {{a..a}}" (is "?l = ?r") proof
+ assume ?r moreover { assume "s = {{a}}" moreover fix k assume "k\<in>s"
+ ultimately have"\<exists>x y. k = {x..y}" apply(rule_tac x=a in exI)+ unfolding interval_sing[THEN conjunct1] by auto }
+ ultimately show ?l unfolding division_of_def interval_sing[THEN conjunct1] by auto next
+ assume ?l note as=conjunctD4[OF this[unfolded division_of_def interval_sing[THEN conjunct1]]]
+ { fix x assume x:"x\<in>s" have "x={a}" using as(2)[rule_format,OF x] by auto }
+ moreover have "s \<noteq> {}" using as(4) by auto ultimately show ?r unfolding interval_sing[THEN conjunct1] by auto qed
+
+lemma elementary_empty: obtains p where "p division_of {}"
+ unfolding division_of_trivial by auto
+
+lemma elementary_interval: obtains p where "p division_of {a..b}"
+ by(metis division_of_trivial division_of_self)
+
+lemma division_contains: "s division_of i \<Longrightarrow> \<forall>x\<in>i. \<exists>k\<in>s. x \<in> k"
+ unfolding division_of_def by auto
+
+lemma forall_in_division:
+ "d division_of i \<Longrightarrow> ((\<forall>x\<in>d. P x) \<longleftrightarrow> (\<forall>a b. {a..b} \<in> d \<longrightarrow> P {a..b}))"
+ unfolding division_of_def by fastsimp
+
+lemma division_of_subset: assumes "p division_of (\<Union>p)" "q \<subseteq> p" shows "q division_of (\<Union>q)"
+ apply(rule division_ofI) proof- note as=division_ofD[OF assms(1)]
+ show "finite q" apply(rule finite_subset) using as(1) assms(2) by auto
+ { 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
+ show "\<exists>a b. k = {a..b}" using as(4)[OF kp] by auto show "k \<noteq> {}" using as(3)[OF kp] by auto }
+ 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
+ show "interior k1 \<inter> interior k2 = {}" using as(5)[OF *] by auto qed auto
+
+lemma division_of_union_self[intro]: "p division_of s \<Longrightarrow> p division_of (\<Union>p)" unfolding division_of_def by auto
+
+lemma division_of_content_0: assumes "content {a..b} = 0" "d division_of {a..b}" shows "\<forall>k\<in>d. content k = 0"
+ unfolding forall_in_division[OF assms(2)] apply(rule,rule,rule) apply(drule division_ofD(2)[OF assms(2)])
+ apply(drule content_subset) unfolding assms(1) proof- case goal1 thus ?case using content_pos_le[of a b] by auto qed
+
+lemma division_inter: assumes "p1 division_of s1" "p2 division_of (s2::(real^'a) set)"
+ 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-
+let ?A = "{s. s \<in> (\<lambda>(k1,k2). k1 \<inter> k2) ` (p1 \<times> p2) \<and> s \<noteq> {}}" have *:"?A' = ?A" by auto
+show ?thesis unfolding * proof(rule division_ofI) have "?A \<subseteq> (\<lambda>(x, y). x \<inter> y) ` (p1 \<times> p2)" by auto
+ moreover have "finite (p1 \<times> p2)" using assms unfolding division_of_def by auto ultimately show "finite ?A" by auto
+ 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
+ using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)] by auto
+ { 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
+ 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
+ guess a1 using division_ofD(4)[OF assms(1) k(2)] .. then guess b1 .. note ab1=this
+ guess a2 using division_ofD(4)[OF assms(2) k(3)] .. then guess b2 .. note ab2=this
+ show "\<exists>a b. k = {a..b}" unfolding k ab1 ab2 unfolding inter_interval by auto } fix k1 k2
+ assume "k1\<in>?A" then obtain x1 y1 where k1:"k1 = x1 \<inter> y1" "x1\<in>p1" "y1\<in>p2" "k1\<noteq>{}" by auto
+ assume "k2\<in>?A" then obtain x2 y2 where k2:"k2 = x2 \<inter> y2" "x2\<in>p1" "y2\<in>p2" "k2\<noteq>{}" by auto
+ assume "k1 \<noteq> k2" hence th:"x1\<noteq>x2 \<or> y1\<noteq>y2" unfolding k1 k2 by auto
+ have *:"(interior x1 \<inter> interior x2 = {} \<or> interior y1 \<inter> interior y2 = {}) \<Longrightarrow>
+ interior(x1 \<inter> y1) \<subseteq> interior(x1) \<Longrightarrow> interior(x1 \<inter> y1) \<subseteq> interior(y1) \<Longrightarrow>
+ interior(x2 \<inter> y2) \<subseteq> interior(x2) \<Longrightarrow> interior(x2 \<inter> y2) \<subseteq> interior(y2)
+ \<Longrightarrow> interior(x1 \<inter> y1) \<inter> interior(x2 \<inter> y2) = {}" by auto
+ show "interior k1 \<inter> interior k2 = {}" unfolding k1 k2 apply(rule *) defer apply(rule_tac[1-4] subset_interior)
+ using division_ofD(5)[OF assms(1) k1(2) k2(2)]
+ using division_ofD(5)[OF assms(2) k1(3) k2(3)] using th by auto qed qed
+
+lemma division_inter_1: assumes "d division_of i" "{a..b::real^'n} \<subseteq> i"
+ shows "{ {a..b} \<inter> k |k. k \<in> d \<and> {a..b} \<inter> k \<noteq> {} } division_of {a..b}" proof(cases "{a..b} = {}")
+ case True show ?thesis unfolding True and division_of_trivial by auto next
+ have *:"{a..b} \<inter> i = {a..b}" using assms(2) by auto
+ case False show ?thesis using division_inter[OF division_of_self[OF False] assms(1)] unfolding * by auto qed
+
+lemma elementary_inter: assumes "p1 division_of s" "p2 division_of (t::(real^'n) set)"
+ shows "\<exists>p. p division_of (s \<inter> t)"
+ by(rule,rule division_inter[OF assms])
+
+lemma elementary_inters: assumes "finite f" "f\<noteq>{}" "\<forall>s\<in>f. \<exists>p. p division_of (s::(real^'n) set)"
+ shows "\<exists>p. p division_of (\<Inter> f)" using assms apply-proof(induct f rule:finite_induct)
+case (insert x f) show ?case proof(cases "f={}")
+ case True thus ?thesis unfolding True using insert by auto next
+ case False guess p using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] ..
+ moreover guess px using insert(5)[rule_format,OF insertI1] .. ultimately
+ show ?thesis unfolding Inter_insert apply(rule_tac elementary_inter) by assumption+ qed qed auto
+
+lemma division_disjoint_union:
+ assumes "p1 division_of s1" "p2 division_of s2" "interior s1 \<inter> interior s2 = {}"
+ shows "(p1 \<union> p2) division_of (s1 \<union> s2)" proof(rule division_ofI)
+ note d1 = division_ofD[OF assms(1)] and d2 = division_ofD[OF assms(2)]
+ show "finite (p1 \<union> p2)" using d1(1) d2(1) by auto
+ show "\<Union>(p1 \<union> p2) = s1 \<union> s2" using d1(6) d2(6) by auto
+ { 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 = {}"
+ { 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)]]
+ using assms(3) by blast } moreover
+ { 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)]]
+ using assms(3) by blast} ultimately
+ show ?g using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto }
+ fix k assume k:"k \<in> p1 \<union> p2" show "k \<subseteq> s1 \<union> s2" using k d1(2) d2(2) by auto
+ 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
+
+lemma partial_division_extend_1:
+ assumes "{c..d} \<subseteq> {a..b::real^'n}" "{c..d} \<noteq> {}"
+ obtains p where "p division_of {a..b}" "{c..d} \<in> p"
+proof- def n \<equiv> "CARD('n)" have n:"1 \<le> n" "0 < n" "n \<noteq> 0" unfolding n_def by auto
+ guess \<pi> using ex_bij_betw_nat_finite_1[OF finite_UNIV[where 'a='n]] .. note \<pi>=this
+ def \<pi>' \<equiv> "inv_into {1..n} \<pi>"
+ have \<pi>':"bij_betw \<pi>' UNIV {1..n}" using bij_betw_inv_into[OF \<pi>] unfolding \<pi>'_def n_def by auto
+ hence \<pi>'i:"\<And>i. \<pi>' i \<in> {1..n}" unfolding bij_betw_def by auto
+ 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
+ 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
+ have "{c..d} \<noteq> {}" using assms by auto
+ 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$i)}"
+ let ?p2 = "\<lambda>l. {(\<chi> i. if \<pi>' i < l then c$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)}"
+ let ?p = "{?p1 l |l. l \<in> {1..n+1}} \<union> {?p2 l |l. l \<in> {1..n+1}}"
+ 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)
+ show ?thesis apply(rule that[of ?p]) apply(rule division_ofI)
+ proof- have "\<And>i. \<pi>' i < Suc n"
+ proof(rule ccontr,unfold not_less) fix i assume "Suc n \<le> \<pi>' i"
+ hence "\<pi>' i \<notin> {1..n}" by auto thus False using \<pi>' unfolding bij_betw_def by auto
+ qed hence "c = (\<chi> i. if \<pi>' i < Suc n then c $ i else a $ i)"
+ "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)"
+ unfolding Cart_eq Cart_lambda_beta using \<pi>' unfolding bij_betw_def by auto
+ thus cdp:"{c..d} \<in> ?p" apply-apply(rule UnI1) unfolding mem_Collect_eq apply(rule_tac x="n + 1" in exI) by auto
+ 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}"
+ unfolding subset_eq apply(rule_tac[!] ballI,rule_tac[!] ccontr)
+ proof- fix l assume l:"l\<in>{1..n+1}" fix x assume "x\<notin>{a..b}"
+ then guess i unfolding mem_interval not_all .. note i=this
+ show "x \<in> ?p1 l \<Longrightarrow> False" "x \<in> ?p2 l \<Longrightarrow> False" unfolding mem_interval apply(erule_tac[!] x=i in allE)
+ apply(case_tac[!] "\<pi>' i < l", case_tac[!] "\<pi>' i = l") using abcd[of i] i by auto
+ qed moreover have "\<And>x. x \<in> {a..b} \<Longrightarrow> x \<in> \<Union>?p"
+ proof- fix x assume x:"x\<in>{a..b}"
+ { presume "x\<notin>{c..d} \<Longrightarrow> x \<in> \<Union>?p" thus "x \<in> \<Union>?p" using cdp by blast }
+ 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)}"
+ assume "x\<notin>{c..d}" then guess i0 unfolding mem_interval not_all ..
+ hence "\<pi>' i0 \<in> ?M" using \<pi>' unfolding bij_betw_def by(auto intro!:le_SucI)
+ hence M:"finite ?M" "?M \<noteq> {}" by auto
+ 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]]
+ Min_gr_iff[OF M,unfolded l_def[symmetric]]
+ have "x\<in>?p1 l \<or> x\<in>?p2 l" using l(2)[THEN conjunct2] unfolding de_Morgan_conj not_le
+ apply- apply(erule disjE) apply(rule disjI1) defer apply(rule disjI2)
+ proof- assume as:"x $ \<pi> l < c $ \<pi> l"
+ show "x \<in> ?p1 l" unfolding mem_interval Cart_lambda_beta
+ proof case goal1 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le by auto
+ thus ?case using as x[unfolded mem_interval,rule_format,of i]
+ apply auto using l(3)[of "\<pi>' i"] by(auto elim!:ballE[where x="\<pi>' i"])
+ qed
+ next assume as:"x $ \<pi> l > d $ \<pi> l"
+ show "x \<in> ?p2 l" unfolding mem_interval Cart_lambda_beta
+ proof case goal1 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le by auto
+ thus ?case using as x[unfolded mem_interval,rule_format,of i]
+ apply auto using l(3)[of "\<pi>' i"] by(auto elim!:ballE[where x="\<pi>' i"])
+ qed qed
+ thus "x \<in> \<Union>?p" using l(2) by blast
+ qed ultimately show "\<Union>?p = {a..b}" apply-apply(rule) defer apply(rule) by(assumption,blast)
+
+ show "finite ?p" by auto
+ 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
+ show "k\<subseteq>{a..b}" apply(rule,unfold mem_interval,rule,rule)
+ 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
+ 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)
+ qed have "\<And>l. ?p1 l \<noteq> {}" "\<And>l. ?p2 l \<noteq> {}" unfolding interval_eq_empty not_ex apply(rule_tac[!] allI)
+ proof- case goal1 thus ?case using abcd[of x] by auto
+ next case goal2 thus ?case using abcd[of x] by auto
+ qed thus "k \<noteq> {}" using k by auto
+ show "\<exists>a b. k = {a..b}" using k by auto
+ 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
+ { fix k k' l l'
+ assume k:"k\<in>?p" and l:"k = ?p1 l \<or> k = ?p2 l" "l \<in> {1..n + 1}"
+ assume k':"k' \<in> ?p" "k \<noteq> k'" and l':"k' = ?p1 l' \<or> k' = ?p2 l'" "l' \<in> {1..n + 1}"
+ assume "l \<le> l'" fix x
+ have "x \<notin> interior k \<inter> interior k'"
+ proof(rule,cases "l' = n+1") assume x:"x \<in> interior k \<inter> interior k'"
+ case True hence "\<And>i. \<pi>' i < l'" using \<pi>'i by(auto simp add:less_Suc_eq_le)
+ hence k':"k' = {c..d}" using l'(1) \<pi>'i by(auto simp add:Cart_nth_inverse)
+ have ln:"l < n + 1"
+ proof(rule ccontr) case goal1 hence l2:"l = n+1" using l by auto
+ hence "\<And>i. \<pi>' i < l" using \<pi>'i by(auto simp add:less_Suc_eq_le)
+ hence "k = {c..d}" using l(1) \<pi>'i by(auto simp add:Cart_nth_inverse)
+ thus False using `k\<noteq>k'` k' by auto
+ qed have **:"\<pi>' (\<pi> l) = l" using \<pi>'\<pi>[of l] using l ln by auto
+ have "x $ \<pi> l < c $ \<pi> l \<or> d $ \<pi> l < x $ \<pi> l" using l(1) apply-
+ proof(erule disjE)
+ assume as:"k = ?p1 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
+ show ?thesis using *[of "\<pi> l"] using ln unfolding Cart_lambda_beta ** by auto
+ next assume as:"k = ?p2 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
+ show ?thesis using *[of "\<pi> l"] using ln unfolding Cart_lambda_beta ** by auto
+ qed thus False using x unfolding k' unfolding Int_iff interior_closed_interval mem_interval
+ by(auto elim!:allE[where x="\<pi> l"])
+ next case False hence "l < n + 1" using l'(2) using `l\<le>l'` by auto
+ hence ln:"l \<in> {1..n}" "l' \<in> {1..n}" using l l' False by auto
+ note \<pi>l = \<pi>'\<pi>[OF ln(1)] \<pi>'\<pi>[OF ln(2)]
+ assume x:"x \<in> interior k \<inter> interior k'"
+ show False using l(1) l'(1) apply-
+ proof(erule_tac[!] disjE)+
+ assume as:"k = ?p1 l" "k' = ?p1 l'"
+ note * = x[unfolded as Int_iff interior_closed_interval mem_interval]
+ have "l \<noteq> l'" using k'(2)[unfolded as] by auto
+ thus False using * by(smt Cart_lambda_beta \<pi>l)
+ next assume as:"k = ?p2 l" "k' = ?p2 l'"
+ note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
+ have "l \<noteq> l'" apply(rule) using k'(2)[unfolded as] by auto
+ thus False using *[of "\<pi> l"] *[of "\<pi> l'"]
+ unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` by auto
+ next assume as:"k = ?p1 l" "k' = ?p2 l'"
+ note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
+ show False using *[of "\<pi> l"] *[of "\<pi> l'"]
+ unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` using abcd[of "\<pi> l'"] by smt
+ next assume as:"k = ?p2 l" "k' = ?p1 l'"
+ note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
+ show False using *[of "\<pi> l"] *[of "\<pi> l'"]
+ unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` using abcd[of "\<pi> l'"] by smt
+ qed qed }
+ from this[OF k l k' l'] this[OF k'(1) l' k _ l] have "\<And>x. x \<notin> interior k \<inter> interior k'"
+ apply - apply(cases "l' \<le> l") using k'(2) by auto
+ thus "interior k \<inter> interior k' = {}" by auto
+qed qed
+
+lemma partial_division_extend_interval: assumes "p division_of (\<Union>p)" "(\<Union>p) \<subseteq> {a..b}"
+ obtains q where "p \<subseteq> q" "q division_of {a..b::real^'n}" proof(cases "p = {}")
+ case True guess q apply(rule elementary_interval[of a b]) .
+ thus ?thesis apply- apply(rule that[of q]) unfolding True by auto next
+ case False note p = division_ofD[OF assms(1)]
+ have *:"\<forall>k\<in>p. \<exists>q. q division_of {a..b} \<and> k\<in>q" proof case goal1
+ guess c using p(4)[OF goal1] .. then guess d .. note cd_ = this
+ have *:"{c..d} \<subseteq> {a..b}" "{c..d} \<noteq> {}" using p(2,3)[OF goal1, unfolded cd_] using assms(2) by auto
+ guess q apply(rule partial_division_extend_1[OF *]) . thus ?case unfolding cd_ by auto qed
+ guess q using bchoice[OF *] .. note q = conjunctD2[OF this[rule_format]]
+ 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-
+ fix x assume x:"x\<in>p" show "q x division_of \<Union>q x" apply-apply(rule division_ofI)
+ using division_ofD[OF q(1)[OF x]] by auto show "q x - {x} \<subseteq> q x" by auto qed
+ hence "\<exists>d. d division_of \<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)" apply- apply(rule elementary_inters)
+ apply(rule finite_imageI[OF p(1)]) unfolding image_is_empty apply(rule False) by auto
+ then guess d .. note d = this
+ show ?thesis apply(rule that[of "d \<union> p"]) proof-
+ 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
+ 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"
+ 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
+ show "d \<union> p division_of {a..b}" unfolding * apply(rule division_disjoint_union[OF d assms(1)])
+ apply(rule inter_interior_unions_intervals) apply(rule p open_interior ballI)+ proof(assumption,rule)
+ 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
+ show "interior (\<Inter>(\<lambda>i. \<Union>(q i - {i})) ` p) \<inter> interior k = {}" apply(rule *[of _ "interior (\<Union>(q k - {k}))"])
+ defer apply(subst Int_commute) apply(rule inter_interior_unions_intervals) proof- note qk=division_ofD[OF q(1)[OF k]]
+ show "finite (q k - {k})" "open (interior k)" "\<forall>t\<in>q k - {k}. \<exists>a b. t = {a..b}" using qk by auto
+ show "\<forall>t\<in>q k - {k}. interior k \<inter> interior t = {}" using qk(5) using q(2)[OF k] by auto
+ 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}))"
+ apply(rule subset_interior *)+ using k by auto qed qed qed auto qed
+
+lemma elementary_bounded[dest]: "p division_of s \<Longrightarrow> bounded (s::(real^'n) set)"
+ unfolding division_of_def by(metis bounded_Union bounded_interval)
+
+lemma elementary_subset_interval: "p division_of s \<Longrightarrow> \<exists>a b. s \<subseteq> {a..b::real^'n}"
+ by(meson elementary_bounded bounded_subset_closed_interval)
+
+lemma division_union_intervals_exists: assumes "{a..b::real^'n} \<noteq> {}"
+ obtains p where "(insert {a..b} p) division_of ({a..b} \<union> {c..d})" proof(cases "{c..d} = {}")
+ case True show ?thesis apply(rule that[of "{}"]) unfolding True using assms by auto next
+ case False note false=this show ?thesis proof(cases "{a..b} \<inter> {c..d} = {}")
+ have *:"\<And>a b. {a,b} = {a} \<union> {b}" by auto
+ case True show ?thesis apply(rule that[of "{{c..d}}"]) unfolding * apply(rule division_disjoint_union)
+ using false True assms using interior_subset by auto next
+ case False obtain u v where uv:"{a..b} \<inter> {c..d} = {u..v}" unfolding inter_interval by auto
+ have *:"{u..v} \<subseteq> {c..d}" using uv by auto
+ guess p apply(rule partial_division_extend_1[OF * False[unfolded uv]]) . note p=this division_ofD[OF this(1)]
+ 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
+ show thesis apply(rule that[of "p - {{u..v}}"]) unfolding *(1) apply(subst *(2)) apply(rule division_disjoint_union)
+ apply(rule,rule assms) apply(rule division_of_subset[of p]) apply(rule division_of_union_self[OF p(1)]) defer
+ unfolding interior_inter[THEN sym] proof-
+ have *:"\<And>cd p uv ab. p \<subseteq> cd \<Longrightarrow> ab \<inter> cd = uv \<Longrightarrow> ab \<inter> p = uv \<inter> p" by auto
+ have "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = interior({u..v} \<inter> \<Union>(p - {{u..v}}))"
+ apply(rule arg_cong[of _ _ interior]) apply(rule *[OF _ uv]) using p(8) by auto
+ also have "\<dots> = {}" unfolding interior_inter apply(rule inter_interior_unions_intervals) using p(6) p(7)[OF p(2)] p(3) by auto
+ finally show "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = {}" by assumption qed auto qed qed
+
+lemma division_of_unions: assumes "finite f" "\<And>p. p\<in>f \<Longrightarrow> p division_of (\<Union>p)"
+ "\<And>k1 k2. \<lbrakk>k1 \<in> \<Union>f; k2 \<in> \<Union>f; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
+ shows "\<Union>f division_of \<Union>\<Union>f" apply(rule division_ofI) prefer 5 apply(rule assms(3)|assumption)+
+ apply(rule finite_Union assms(1))+ prefer 3 apply(erule UnionE) apply(rule_tac s=X in division_ofD(3)[OF assms(2)])
+ using division_ofD[OF assms(2)] by auto
+
+lemma elementary_union_interval: assumes "p division_of \<Union>p"
+ obtains q where "q division_of ({a..b::real^'n} \<union> \<Union>p)" proof-
+ note assm=division_ofD[OF assms]
+ have lem1:"\<And>f s. \<Union>\<Union> (f ` s) = \<Union>(\<lambda>x.\<Union>(f x)) ` s" by auto
+ have lem2:"\<And>f s. f \<noteq> {} \<Longrightarrow> \<Union>{s \<union> t |t. t \<in> f} = s \<union> \<Union>f" by auto
+{ presume "p={} \<Longrightarrow> thesis" "{a..b} = {} \<Longrightarrow> thesis" "{a..b} \<noteq> {} \<Longrightarrow> interior {a..b} = {} \<Longrightarrow> thesis"
+ "p\<noteq>{} \<Longrightarrow> interior {a..b}\<noteq>{} \<Longrightarrow> {a..b} \<noteq> {} \<Longrightarrow> thesis"
+ thus thesis by auto
+next assume as:"p={}" guess p apply(rule elementary_interval[of a b]) .
+ thus thesis apply(rule_tac that[of p]) unfolding as by auto
+next assume as:"{a..b}={}" show thesis apply(rule that) unfolding as using assms by auto
+next assume as:"interior {a..b} = {}" "{a..b} \<noteq> {}"
+ show thesis apply(rule that[of "insert {a..b} p"],rule division_ofI)
+ unfolding finite_insert apply(rule assm(1)) unfolding Union_insert
+ using assm(2-4) as apply- by(fastsimp dest: assm(5))+
+next assume as:"p \<noteq> {}" "interior {a..b} \<noteq> {}" "{a..b}\<noteq>{}"
+ have "\<forall>k\<in>p. \<exists>q. (insert {a..b} q) division_of ({a..b} \<union> k)" proof case goal1
+ from assm(4)[OF this] guess c .. then guess d ..
+ thus ?case apply-apply(rule division_union_intervals_exists[OF as(3),of c d]) by auto
+ qed from bchoice[OF this] guess q .. note q=division_ofD[OF this[rule_format]]
+ let ?D = "\<Union>{insert {a..b} (q k) | k. k \<in> p}"
+ show thesis apply(rule that[of "?D"]) proof(rule division_ofI)
+ have *:"{insert {a..b} (q k) |k. k \<in> p} = (\<lambda>k. insert {a..b} (q k)) ` p" by auto
+ show "finite ?D" apply(rule finite_Union) unfolding * apply(rule finite_imageI) using assm(1) q(1) by auto
+ show "\<Union>?D = {a..b} \<union> \<Union>p" unfolding * lem1 unfolding lem2[OF as(1), of "{a..b}",THEN sym]
+ using q(6) by auto
+ fix k assume k:"k\<in>?D" thus " k \<subseteq> {a..b} \<union> \<Union>p" using q(2) by auto
+ show "k \<noteq> {}" using q(3) k by auto show "\<exists>a b. k = {a..b}" using q(4) k by auto
+ fix k' assume k':"k'\<in>?D" "k\<noteq>k'"
+ obtain x where x: "k \<in>insert {a..b} (q x)" "x\<in>p" using k by auto
+ obtain x' where x':"k'\<in>insert {a..b} (q x')" "x'\<in>p" using k' by auto
+ show "interior k \<inter> interior k' = {}" proof(cases "x=x'")
+ case True show ?thesis apply(rule q(5)) using x x' k' unfolding True by auto
+ next case False
+ { presume "k = {a..b} \<Longrightarrow> ?thesis" "k' = {a..b} \<Longrightarrow> ?thesis"
+ "k \<noteq> {a..b} \<Longrightarrow> k' \<noteq> {a..b} \<Longrightarrow> ?thesis"
+ thus ?thesis by auto }
+ { assume as':"k = {a..b}" show ?thesis apply(rule q(5)) using x' k'(2) unfolding as' by auto }
+ { assume as':"k' = {a..b}" show ?thesis apply(rule q(5)) using x k'(2) unfolding as' by auto }
+ assume as':"k \<noteq> {a..b}" "k' \<noteq> {a..b}"
+ guess c using q(4)[OF x(2,1)] .. then guess d .. note c_d=this
+ have "interior k \<inter> interior {a..b} = {}" apply(rule q(5)) using x k'(2) using as' by auto
+ hence "interior k \<subseteq> interior x" apply-
+ apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x(2,1)]]) by auto moreover
+ guess c using q(4)[OF x'(2,1)] .. then guess d .. note c_d=this
+ have "interior k' \<inter> interior {a..b} = {}" apply(rule q(5)) using x' k'(2) using as' by auto
+ hence "interior k' \<subseteq> interior x'" apply-
+ apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x'(2,1)]]) by auto
+ ultimately show ?thesis using assm(5)[OF x(2) x'(2) False] by auto
+ qed qed } qed
+
+lemma elementary_unions_intervals:
+ assumes "finite f" "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = {a..b::real^'n}"
+ obtains p where "p division_of (\<Union>f)" proof-
+ have "\<exists>p. p division_of (\<Union>f)" proof(induct_tac f rule:finite_subset_induct)
+ show "\<exists>p. p division_of \<Union>{}" using elementary_empty by auto
+ fix x F assume as:"finite F" "x \<notin> F" "\<exists>p. p division_of \<Union>F" "x\<in>f"
+ from this(3) guess p .. note p=this
+ from assms(2)[OF as(4)] guess a .. then guess b .. note ab=this
+ have *:"\<Union>F = \<Union>p" using division_ofD[OF p] by auto
+ show "\<exists>p. p division_of \<Union>insert x F" using elementary_union_interval[OF p[unfolded *], of a b]
+ unfolding Union_insert ab * by auto
+ qed(insert assms,auto) thus ?thesis apply-apply(erule exE,rule that) by auto qed
+
+lemma elementary_union: assumes "ps division_of s" "pt division_of (t::(real^'n) set)"
+ obtains p where "p division_of (s \<union> t)"
+proof- have "s \<union> t = \<Union>ps \<union> \<Union>pt" using assms unfolding division_of_def by auto
+ hence *:"\<Union>(ps \<union> pt) = s \<union> t" by auto
+ show ?thesis apply-apply(rule elementary_unions_intervals[of "ps\<union>pt"])
+ unfolding * prefer 3 apply(rule_tac p=p in that)
+ using assms[unfolded division_of_def] by auto qed
+
+lemma partial_division_extend: fixes t::"(real^'n) set"
+ assumes "p division_of s" "q division_of t" "s \<subseteq> t"
+ obtains r where "p \<subseteq> r" "r division_of t" proof-
+ note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)]
+ obtain a b where ab:"t\<subseteq>{a..b}" using elementary_subset_interval[OF assms(2)] by auto
+ guess r1 apply(rule partial_division_extend_interval) apply(rule assms(1)[unfolded divp(6)[THEN sym]])
+ apply(rule subset_trans) by(rule ab assms[unfolded divp(6)[THEN sym]])+ note r1 = this division_ofD[OF this(2)]
+ guess p' apply(rule elementary_unions_intervals[of "r1 - p"]) using r1(3,6) by auto
+ then obtain r2 where r2:"r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)"
+ apply- apply(drule elementary_inter[OF _ assms(2)[unfolded divq(6)[THEN sym]]]) by auto
+ { fix x assume x:"x\<in>t" "x\<notin>s"
+ hence "x\<in>\<Union>r1" unfolding r1 using ab by auto
+ then guess r unfolding Union_iff .. note r=this moreover
+ have "r \<notin> p" proof assume "r\<in>p" hence "x\<in>s" using divp(2) r by auto
+ thus False using x by auto qed
+ ultimately have "x\<in>\<Union>(r1 - p)" by auto }
+ hence *:"t = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)" unfolding divp divq using assms(3) by auto
+ show ?thesis apply(rule that[of "p \<union> r2"]) unfolding * defer apply(rule division_disjoint_union)
+ unfolding divp(6) apply(rule assms r2)+
+ proof- have "interior s \<inter> interior (\<Union>(r1-p)) = {}"
+ proof(rule inter_interior_unions_intervals)
+ show "finite (r1 - p)" "open (interior s)" "\<forall>t\<in>r1-p. \<exists>a b. t = {a..b}" using r1 by auto
+ have *:"\<And>s. (\<And>x. x \<in> s \<Longrightarrow> False) \<Longrightarrow> s = {}" by auto
+ show "\<forall>t\<in>r1-p. interior s \<inter> interior t = {}" proof(rule)
+ fix m x assume as:"m\<in>r1-p"
+ have "interior m \<inter> interior (\<Union>p) = {}" proof(rule inter_interior_unions_intervals)
+ show "finite p" "open (interior m)" "\<forall>t\<in>p. \<exists>a b. t = {a..b}" using divp by auto
+ show "\<forall>t\<in>p. interior m \<inter> interior t = {}" apply(rule, rule r1(7)) using as using r1 by auto
+ qed thus "interior s \<inter> interior m = {}" unfolding divp by auto
+ qed qed
+ thus "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}" using interior_subset by auto
+ qed auto qed
+
+subsection {* Tagged (partial) divisions. *}
+
+definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40) where
+ "(s tagged_partial_division_of i) \<equiv>
+ finite s \<and>
+ (\<forall>x k. (x,k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
+ (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ((x1,k1) \<noteq> (x2,k2))
+ \<longrightarrow> (interior(k1) \<inter> interior(k2) = {}))"
+
+lemma tagged_partial_division_ofD[dest]: assumes "s tagged_partial_division_of i"
+ 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}"
+ "\<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) = {}"
+ using assms unfolding tagged_partial_division_of_def apply- by blast+
+
+definition tagged_division_of (infixr "tagged'_division'_of" 40) where
+ "(s tagged_division_of i) \<equiv>
+ (s tagged_partial_division_of i) \<and> (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
+
+lemma tagged_division_of_finite[dest]: "s tagged_division_of i \<Longrightarrow> finite s"
+ unfolding tagged_division_of_def tagged_partial_division_of_def by auto
+
+lemma tagged_division_of:
+ "(s tagged_division_of i) \<longleftrightarrow>
+ finite s \<and>
+ (\<forall>x k. (x,k) \<in> s
+ \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
+ (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ~((x1,k1) = (x2,k2))
+ \<longrightarrow> (interior(k1) \<inter> interior(k2) = {})) \<and>
+ (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
+ unfolding tagged_division_of_def tagged_partial_division_of_def by auto
+
+lemma tagged_division_ofI: assumes
+ "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}"
+ "\<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) = {})"
+ "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
+ shows "s tagged_division_of i"
+ unfolding tagged_division_of apply(rule) defer apply rule
+ apply(rule allI impI conjI assms)+ apply assumption
+ apply(rule, rule assms, assumption) apply(rule assms, assumption)
+ using assms(1,5-) apply- by blast+
+
+lemma tagged_division_ofD[dest]: assumes "s tagged_division_of i"
+ 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}"
+ "\<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) = {})"
+ "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)" using assms unfolding tagged_division_of apply- by blast+
+
+lemma division_of_tagged_division: assumes"s tagged_division_of i" shows "(snd ` s) division_of i"
+proof(rule division_ofI) note assm=tagged_division_ofD[OF assms]
+ show "\<Union>snd ` s = i" "finite (snd ` s)" using assm by auto
+ fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto
+ thus "k \<subseteq> i" "k \<noteq> {}" "\<exists>a b. k = {a..b}" using assm apply- by fastsimp+
+ fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto
+ thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto
+qed
+
+lemma partial_division_of_tagged_division: assumes "s tagged_partial_division_of i"
+ shows "(snd ` s) division_of \<Union>(snd ` s)"
+proof(rule division_ofI) note assm=tagged_partial_division_ofD[OF assms]
+ show "finite (snd ` s)" "\<Union>snd ` s = \<Union>snd ` s" using assm by auto
+ fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto
+ thus "k\<noteq>{}" "\<exists>a b. k = {a..b}" "k \<subseteq> \<Union>snd ` s" using assm by auto
+ fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto
+ thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto
+qed
+
+lemma tagged_partial_division_subset: assumes "s tagged_partial_division_of i" "t \<subseteq> s"
+ shows "t tagged_partial_division_of i"
+ using assms unfolding tagged_partial_division_of_def using finite_subset[OF assms(2)] by blast
+
+lemma setsum_over_tagged_division_lemma: fixes d::"(real^'m) set \<Rightarrow> 'a::real_normed_vector"
+ assumes "p tagged_division_of i" "\<And>u v. {u..v} \<noteq> {} \<Longrightarrow> content {u..v} = 0 \<Longrightarrow> d {u..v} = 0"
+ shows "setsum (\<lambda>(x,k). d k) p = setsum d (snd ` p)"
+proof- note assm=tagged_division_ofD[OF assms(1)]
+ have *:"(\<lambda>(x,k). d k) = d \<circ> snd" unfolding o_def apply(rule ext) by auto
+ show ?thesis unfolding * apply(subst eq_commute) proof(rule setsum_reindex_nonzero)
+ show "finite p" using assm by auto
+ fix x y assume as:"x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y"
+ obtain a b where ab:"snd x = {a..b}" using assm(4)[of "fst x" "snd x"] as(1) by auto
+ have "(fst x, snd y) \<in> p" "(fst x, snd y) \<noteq> y" unfolding as(4)[THEN sym] using as(1-3) by auto
+ hence "interior (snd x) \<inter> interior (snd y) = {}" apply-apply(rule assm(5)[of "fst x" _ "fst y"]) using as by auto
+ hence "content {a..b} = 0" unfolding as(4)[THEN sym] ab content_eq_0_interior by auto
+ 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
+ thus "d (snd x) = 0" unfolding ab by auto qed qed
+
+lemma tag_in_interval: "p tagged_division_of i \<Longrightarrow> (x,k) \<in> p \<Longrightarrow> x \<in> i" by auto
+
+lemma tagged_division_of_empty: "{} tagged_division_of {}"
+ unfolding tagged_division_of by auto
+
+lemma tagged_partial_division_of_trivial[simp]:
+ "p tagged_partial_division_of {} \<longleftrightarrow> p = {}"
+ unfolding tagged_partial_division_of_def by auto
+
+lemma tagged_division_of_trivial[simp]:
+ "p tagged_division_of {} \<longleftrightarrow> p = {}"
+ unfolding tagged_division_of by auto
+
+lemma tagged_division_of_self:
+ "x \<in> {a..b} \<Longrightarrow> {(x,{a..b})} tagged_division_of {a..b}"
+ apply(rule tagged_division_ofI) by auto
+
+lemma tagged_division_union:
+ assumes "p1 tagged_division_of s1" "p2 tagged_division_of s2" "interior s1 \<inter> interior s2 = {}"
+ shows "(p1 \<union> p2) tagged_division_of (s1 \<union> s2)"
+proof(rule tagged_division_ofI) note p1=tagged_division_ofD[OF assms(1)] and p2=tagged_division_ofD[OF assms(2)]
+ show "finite (p1 \<union> p2)" using p1(1) p2(1) by auto
+ show "\<Union>{k. \<exists>x. (x, k) \<in> p1 \<union> p2} = s1 \<union> s2" using p1(6) p2(6) by blast
+ 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
+ show "k\<subseteq>s1\<union>s2" using xk p1(3) p2(3) by blast
+ fix x' k' assume xk':"(x',k')\<in>p1\<union>p2" "(x,k) \<noteq> (x',k')"
+ have *:"\<And>a b. a\<subseteq> s1 \<Longrightarrow> b\<subseteq> s2 \<Longrightarrow> interior a \<inter> interior b = {}" using assms(3) subset_interior by blast
+ show "interior k \<inter> interior k' = {}" apply(cases "(x,k)\<in>p1", case_tac[!] "(x',k')\<in>p1")
+ apply(rule p1(5)) prefer 4 apply(rule *) prefer 6 apply(subst Int_commute,rule *) prefer 8 apply(rule p2(5))
+ using p1(3) p2(3) using xk xk' by auto qed
+
+lemma tagged_division_unions:
+ assumes "finite iset" "\<forall>i\<in>iset. (pfn(i) tagged_division_of i)"
+ "\<forall>i1 \<in> iset. \<forall>i2 \<in> iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})"
+ shows "\<Union>(pfn ` iset) tagged_division_of (\<Union>iset)"
+proof(rule tagged_division_ofI)
+ note assm = tagged_division_ofD[OF assms(2)[rule_format]]
+ show "finite (\<Union>pfn ` iset)" apply(rule finite_Union) using assms by auto
+ 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
+ also have "\<dots> = \<Union>iset" using assm(6) by auto
+ finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>pfn ` iset} = \<Union>iset" .
+ 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
+ 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
+ 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
+ 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)
+ using assms(3)[rule_format] subset_interior by blast
+ show "interior k \<inter> interior k' = {}" apply(cases "i=i'")
+ using assm(5)[OF i _ xk'(2)] i'(2) using assm(3)[OF i] assm(3)[OF i'] defer apply-apply(rule *) by auto
+qed
+
+lemma tagged_partial_division_of_union_self:
+ assumes "p tagged_partial_division_of s" shows "p tagged_division_of (\<Union>(snd ` p))"
+ apply(rule tagged_division_ofI) using tagged_partial_division_ofD[OF assms] by auto
+
+lemma tagged_division_of_union_self: assumes "p tagged_division_of s"
+ shows "p tagged_division_of (\<Union>(snd ` p))"
+ apply(rule tagged_division_ofI) using tagged_division_ofD[OF assms] by auto
+
+subsection {* Fine-ness of a partition w.r.t. a gauge. *}
+
+definition fine (infixr "fine" 46) where
+ "d fine s \<longleftrightarrow> (\<forall>(x,k) \<in> s. k \<subseteq> d(x))"
+
+lemma fineI: assumes "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x"
+ shows "d fine s" using assms unfolding fine_def by auto
+
+lemma fineD[dest]: assumes "d fine s"
+ shows "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x" using assms unfolding fine_def by auto
+
+lemma fine_inter: "(\<lambda>x. d1 x \<inter> d2 x) fine p \<longleftrightarrow> d1 fine p \<and> d2 fine p"
+ unfolding fine_def by auto
+
+lemma fine_inters:
+ "(\<lambda>x. \<Inter> {f d x | d. d \<in> s}) fine p \<longleftrightarrow> (\<forall>d\<in>s. (f d) fine p)"
+ unfolding fine_def by blast
+
+lemma fine_union:
+ "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)"
+ unfolding fine_def by blast
+
+lemma fine_unions:"(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)"
+ unfolding fine_def by auto
+
+lemma fine_subset: "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p"
+ unfolding fine_def by blast
+
+subsection {* Gauge integral. Define on compact intervals first, then use a limit. *}
+
+definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46) where
+ "(f has_integral_compact_interval y) i \<equiv>
+ (\<forall>e>0. \<exists>d. gauge d \<and>
+ (\<forall>p. p tagged_division_of i \<and> d fine p
+ \<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - y) < e))"
+
+definition has_integral (infixr "has'_integral" 46) where
+"((f::(real^'n \<Rightarrow> 'b::real_normed_vector)) has_integral y) i \<equiv>
+ if (\<exists>a b. i = {a..b}) then (f has_integral_compact_interval y) i
+ else (\<forall>e>0. \<exists>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_compact_interval z) {a..b} \<and>
+ norm(z - y) < e))"
+
+lemma has_integral:
+ "(f has_integral y) ({a..b}) \<longleftrightarrow>
+ (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p
+ \<longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
+ unfolding has_integral_def has_integral_compact_interval_def by auto
+
+lemma has_integralD[dest]: assumes
+ "(f has_integral y) ({a..b})" "e>0"
+ obtains d where "gauge d" "\<And>p. p tagged_division_of {a..b} \<Longrightarrow> d fine p
+ \<Longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f(x)) p - y) < e"
+ using assms unfolding has_integral by auto
+
+lemma has_integral_alt:
+ "(f has_integral y) i \<longleftrightarrow>
+ (if (\<exists>a b. i = {a..b}) then (f has_integral y) i
+ else (\<forall>e>0. \<exists>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)))"
+ unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto
+
+lemma has_integral_altD:
+ assumes "(f has_integral y) i" "\<not> (\<exists>a b. i = {a..b})" "e>0"
+ 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)"
+ using assms unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto
+
+definition integrable_on (infixr "integrable'_on" 46) where
+ "(f integrable_on i) \<equiv> \<exists>y. (f has_integral y) i"
+
+definition "integral i f \<equiv> SOME y. (f has_integral y) i"
+
+lemma integrable_integral[dest]:
+ "f integrable_on i \<Longrightarrow> (f has_integral (integral i f)) i"
+ unfolding integrable_on_def integral_def by(rule someI_ex)
+
+lemma has_integral_integrable[intro]: "(f has_integral i) s \<Longrightarrow> f integrable_on s"
+ unfolding integrable_on_def by auto
+
+lemma has_integral_integral:"f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s"
+ by auto
+
+lemma has_integral_vec1: assumes "(f has_integral k) {a..b}"
+ shows "((\<lambda>x. vec1 (f x)) has_integral (vec1 k)) {a..b}"
+proof- have *:"\<And>p. (\<Sum>(x, k)\<in>p. content k *\<^sub>R vec1 (f x)) - vec1 k = vec1 ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k)"
+ unfolding vec_sub Cart_eq by(auto simp add:vec1_dest_vec1_simps split_beta)
+ show ?thesis using assms unfolding has_integral apply safe
+ apply(erule_tac x=e in allE,safe) apply(rule_tac x=d in exI,safe)
+ apply(erule_tac x=p in allE,safe) unfolding * norm_vector_1 by auto qed
+
+lemma setsum_content_null:
+ assumes "content({a..b}) = 0" "p tagged_division_of {a..b}"
+ shows "setsum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)"
+proof(rule setsum_0',rule) fix y assume y:"y\<in>p"
+ obtain x k where xk:"y = (x,k)" using surj_pair[of y] by blast
+ note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]]
+ from this(2) guess c .. then guess d .. note c_d=this
+ have "(\<lambda>(x, k). content k *\<^sub>R f x) y = content k *\<^sub>R f x" unfolding xk by auto
+ also have "\<dots> = 0" using content_subset[OF assm(1)[unfolded c_d]] content_pos_le[of c d]
+ unfolding assms(1) c_d by auto
+ finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" .
+qed
+
+subsection {* Some basic combining lemmas. *}
+
+lemma tagged_division_unions_exists:
+ assumes "finite iset" "\<forall>i \<in> iset. \<exists>p. p tagged_division_of i \<and> d fine p"
+ "\<forall>i1\<in>iset. \<forall>i2\<in>iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})" "(\<Union>iset = i)"
+ obtains p where "p tagged_division_of i" "d fine p"
+proof- guess pfn using bchoice[OF assms(2)] .. note pfn = conjunctD2[OF this[rule_format]]
+ show thesis apply(rule_tac p="\<Union>(pfn ` iset)" in that) unfolding assms(4)[THEN sym]
+ apply(rule tagged_division_unions[OF assms(1) _ assms(3)]) defer
+ apply(rule fine_unions) using pfn by auto
+qed
+
+subsection {* The set we're concerned with must be closed. *}
+
+lemma division_of_closed: "s division_of i \<Longrightarrow> closed (i::(real^'n) set)"
+ unfolding division_of_def by(fastsimp intro!: closed_Union closed_interval)
+
+subsection {* General bisection principle for intervals; might be useful elsewhere. *}
+
+lemma interval_bisection_step:
+ 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})"
+ obtains c d where "~(P{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"
+proof- have "{a..b} \<noteq> {}" using assms(1,3) by auto
+ note ab=this[unfolded interval_eq_empty not_ex not_less]
+ { fix f have "finite f \<Longrightarrow>
+ (\<forall>s\<in>f. P s) \<Longrightarrow>
+ (\<forall>s\<in>f. \<exists>a b. s = {a..b}) \<Longrightarrow>
+ (\<forall>s\<in>f.\<forall>t\<in>f. ~(s = t) \<longrightarrow> interior(s) \<inter> interior(t) = {}) \<Longrightarrow> P(\<Union>f)"
+ proof(induct f rule:finite_induct)
+ case empty show ?case using assms(1) by auto
+ next case (insert x f) show ?case unfolding Union_insert apply(rule assms(2)[rule_format])
+ apply rule defer apply rule defer apply(rule inter_interior_unions_intervals)
+ using insert by auto
+ qed } note * = this
+ 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)}"
+ 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"
+ { presume "\<forall>c d. ?PP c d \<longrightarrow> P {c..d} \<Longrightarrow> False"
+ thus thesis unfolding atomize_not not_all apply-apply(erule exE)+ apply(rule_tac c=x and d=xa in that) by auto }
+ assume as:"\<forall>c d. ?PP c d \<longrightarrow> P {c..d}"
+ have "P (\<Union> ?A)" proof(rule *, rule_tac[2-] ballI, rule_tac[4] ballI, rule_tac[4] impI)
+ let ?B = "(\<lambda>s.{(\<chi> i. if i \<in> s then a$i else (a$i + b$i) / 2) ..
+ (\<chi> i. if i \<in> s then (a$i + b$i) / 2 else b$i)}) ` {s. s \<subseteq> UNIV}"
+ have "?A \<subseteq> ?B" proof case goal1
+ then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+) note c_d=this[rule_format]
+ have *:"\<And>a b c d. a = c \<Longrightarrow> b = d \<Longrightarrow> {a..b} = {c..d}" by auto
+ show "x\<in>?B" unfolding image_iff apply(rule_tac x="{i. c$i = a$i}" in bexI)
+ unfolding c_d apply(rule * ) unfolding Cart_eq cond_component Cart_lambda_beta
+ 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)"
+ "d $ i = (if i \<in> {i. c $ i = a $ i} then (a $ i + b $ i) / 2 else b $ i)"
+ using c_d(2)[of i] ab[THEN spec[where x=i]] by(auto simp add:field_simps)
+ qed auto qed
+ thus "finite ?A" apply(rule finite_subset[of _ ?B]) by auto
+ fix s assume "s\<in>?A" then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+)
+ note c_d=this[rule_format]
+ show "P s" unfolding c_d apply(rule as[rule_format]) proof- case goal1 show ?case
+ using c_d(2)[of i] using ab[THEN spec[where x=i]] by auto qed
+ show "\<exists>a b. s = {a..b}" unfolding c_d by auto
+ fix t assume "t\<in>?A" then guess e unfolding mem_Collect_eq .. then guess f apply- by(erule exE,(erule conjE)+)
+ note e_f=this[rule_format]
+ assume "s \<noteq> t" hence "\<not> (c = e \<and> d = f)" unfolding c_d e_f by auto
+ then obtain i where "c$i \<noteq> e$i \<or> d$i \<noteq> f$i" unfolding de_Morgan_conj Cart_eq by auto
+ hence i:"c$i \<noteq> e$i" "d$i \<noteq> f$i" apply- apply(erule_tac[!] disjE)
+ 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
+ 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
+ qed have *:"\<And>s t. (\<And>a. a\<in>s \<Longrightarrow> a\<in>t \<Longrightarrow> False) \<Longrightarrow> s \<inter> t = {}" by auto
+ show "interior s \<inter> interior t = {}" unfolding e_f c_d interior_closed_interval proof(rule *)
+ fix x assume "x\<in>{c<..<d}" "x\<in>{e<..<f}"
+ 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
+ show False using c_d(2)[of i] apply- apply(erule_tac disjE)
+ proof(erule_tac[!] conjE) assume as:"c $ i = a $ i" "d $ i = (a $ i + b $ i) / 2"
+ show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps)
+ next assume as:"c $ i = (a $ i + b $ i) / 2" "d $ i = b $ i"
+ show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps)
+ qed qed qed
+ also have "\<Union> ?A = {a..b}" proof(rule set_ext,rule)
+ fix x assume "x\<in>\<Union>?A" then guess Y unfolding Union_iff ..
+ from this(1) guess c unfolding mem_Collect_eq .. then guess d ..
+ note c_d = this[THEN conjunct2,rule_format] `x\<in>Y`[unfolded this[THEN conjunct1]]
+ show "x\<in>{a..b}" unfolding mem_interval proof
+ fix i show "a $ i \<le> x $ i \<and> x $ i \<le> b $ i"
+ using c_d(1)[of i] c_d(2)[unfolded mem_interval,THEN spec[where x=i]] by auto qed
+ next fix x assume x:"x\<in>{a..b}"
+ 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"
+ (is "\<forall>i. \<exists>c d. ?P i c d") unfolding mem_interval proof fix i
+ have "?P i (a$i) ((a $ i + b $ i) / 2) \<or> ?P i ((a $ i + b $ i) / 2) (b$i)"
+ using x[unfolded mem_interval,THEN spec[where x=i]] by auto thus "\<exists>c d. ?P i c d" by blast
+ qed thus "x\<in>\<Union>?A" unfolding Union_iff lambda_skolem unfolding Bex_def mem_Collect_eq
+ apply-apply(erule exE)+ apply(rule_tac x="{xa..xaa}" in exI) unfolding mem_interval by auto
+ qed finally show False using assms by auto qed
+
+lemma interval_bisection:
+ assumes "P {}" "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))" "\<not> P {a..b::real^'n}"
+ obtains x where "x \<in> {a..b}" "\<forall>e>0. \<exists>c d. x \<in> {c..d} \<and> {c..d} \<subseteq> ball x e \<and> {c..d} \<subseteq> {a..b} \<and> ~P({c..d})"
+proof-
+ have "\<forall>x. \<exists>y. \<not> P {fst x..snd x} \<longrightarrow> (\<not> P {fst y..snd y} \<and> (\<forall>i. fst x$i \<le> fst y$i \<and> fst y$i \<le> snd y$i \<and> snd y$i \<le> snd x$i \<and>
+ 2 * (snd y$i - fst y$i) \<le> snd x$i - fst x$i))" proof case goal1 thus ?case proof-
+ presume "\<not> P {fst x..snd x} \<Longrightarrow> ?thesis"
+ thus ?thesis apply(cases "P {fst x..snd x}") by auto
+ next assume as:"\<not> P {fst x..snd x}" from interval_bisection_step[of P, OF assms(1-2) as] guess c d .
+ thus ?thesis apply- apply(rule_tac x="(c,d)" in exI) by auto
+ qed qed then guess f apply-apply(drule choice) by(erule exE) note f=this
+ def AB \<equiv> "\<lambda>n. (f ^^ n) (a,b)" def A \<equiv> "\<lambda>n. fst(AB n)" and B \<equiv> "\<lambda>n. snd(AB n)" note ab_def = this AB_def
+ have "A 0 = a" "B 0 = b" "\<And>n. \<not> P {A(Suc n)..B(Suc n)} \<and>
+ (\<forall>i. A(n)$i \<le> A(Suc n)$i \<and> A(Suc n)$i \<le> B(Suc n)$i \<and> B(Suc n)$i \<le> B(n)$i \<and>
+ 2 * (B(Suc n)$i - A(Suc n)$i) \<le> B(n)$i - A(n)$i)" (is "\<And>n. ?P n")
+ proof- show "A 0 = a" "B 0 = b" unfolding ab_def by auto
+ case goal3 note S = ab_def funpow.simps o_def id_apply show ?case
+ proof(induct n) case 0 thus ?case unfolding S apply(rule f[rule_format]) using assms(3) by auto
+ next case (Suc n) show ?case unfolding S apply(rule f[rule_format]) using Suc unfolding S by auto
+ qed qed note AB = this(1-2) conjunctD2[OF this(3),rule_format]
+
+ have interv:"\<And>e. 0 < e \<Longrightarrow> \<exists>n. \<forall>x\<in>{A n..B n}. \<forall>y\<in>{A n..B n}. dist x y < e"
+ proof- case goal1 guess n using real_arch_pow2[of "(setsum (\<lambda>i. b$i - a$i) UNIV) / e"] .. note n=this
+ show ?case apply(rule_tac x=n in exI) proof(rule,rule)
+ fix x y assume xy:"x\<in>{A n..B n}" "y\<in>{A n..B n}"
+ have "dist x y \<le> setsum (\<lambda>i. abs((x - y)$i)) UNIV" unfolding vector_dist_norm by(rule norm_le_l1)
+ also have "\<dots> \<le> setsum (\<lambda>i. B n$i - A n$i) UNIV"
+ proof(rule setsum_mono) fix i show "\<bar>(x - y) $ i\<bar> \<le> B n $ i - A n $ i"
+ using xy[unfolded mem_interval,THEN spec[where x=i]]
+ unfolding vector_minus_component by auto qed
+ also have "\<dots> \<le> setsum (\<lambda>i. b$i - a$i) UNIV / 2^n" unfolding setsum_divide_distrib
+ proof(rule setsum_mono) case goal1 thus ?case
+ proof(induct n) case 0 thus ?case unfolding AB by auto
+ 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
+ also have "\<dots> \<le> (b $ i - a $ i) / 2 ^ Suc n" using Suc by(auto simp add:field_simps) finally show ?case .
+ qed qed
+ also have "\<dots> < e" using n using goal1 by(auto simp add:field_simps) finally show "dist x y < e" .
+ qed qed
+ { fix n m ::nat assume "m \<le> n" then guess d unfolding le_Suc_ex_iff .. note d=this
+ have "{A n..B n} \<subseteq> {A m..B m}" unfolding d
+ proof(induct d) case 0 thus ?case by auto
+ next case (Suc d) show ?case apply(rule subset_trans[OF _ Suc])
+ apply(rule) unfolding mem_interval apply(rule,erule_tac x=i in allE)
+ proof- case goal1 thus ?case using AB(4)[of "m + d" i] by(auto simp add:field_simps)
+ qed qed } note ABsubset = this
+ have "\<exists>a. \<forall>n. a\<in>{A n..B n}" apply(rule decreasing_closed_nest[rule_format,OF closed_interval _ ABsubset interv])
+ 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
+ then guess x0 .. note x0=this[rule_format]
+ show thesis proof(rule that[rule_format,of x0])
+ show "x0\<in>{a..b}" using x0[of 0] unfolding AB .
+ fix e assume "0 < (e::real)" from interv[OF this] guess n .. note n=this
+ 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}"
+ apply(rule_tac x="A n" in exI,rule_tac x="B n" in exI) apply(rule,rule x0) apply rule defer
+ 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
+ show "{A n..B n} \<subseteq> ball x0 e" using n using x0[of n] by auto
+ show "{A n..B n} \<subseteq> {a..b}" unfolding AB(1-2)[symmetric] apply(rule ABsubset) by auto
+ qed qed qed
+
+subsection {* Cousin's lemma. *}
+
+lemma fine_division_exists: assumes "gauge g"
+ obtains p where "p tagged_division_of {a..b::real^'n}" "g fine p"
+proof- presume "\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p) \<Longrightarrow> False"
+ then guess p unfolding atomize_not not_not .. thus thesis apply-apply(rule that[of p]) by auto
+next assume as:"\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p)"
+ guess x apply(rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p",rule_format,OF _ _ as])
+ apply(rule_tac x="{}" in exI) defer apply(erule conjE exE)+
+ proof- show "{} tagged_division_of {} \<and> g fine {}" unfolding fine_def by auto
+ 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 = {}"
+ 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
+ apply(rule tagged_division_union) prefer 4 apply(rule fine_union) by auto
+ qed note x=this
+ obtain e where e:"e>0" "ball x e \<subseteq> g x" using gaugeD[OF assms, of x] unfolding open_contains_ball by auto
+ from x(2)[OF e(1)] guess c d apply-apply(erule exE conjE)+ . note c_d = this
+ have "g fine {(x, {c..d})}" unfolding fine_def using e using c_d(2) by auto
+ thus False using tagged_division_of_self[OF c_d(1)] using c_d by auto qed
+
+subsection {* Basic theorems about integrals. *}
+
+lemma has_integral_unique: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
+ assumes "(f has_integral k1) i" "(f has_integral k2) i" shows "k1 = k2"
+proof(rule ccontr) let ?e = "norm(k1 - k2) / 2" assume as:"k1 \<noteq> k2" hence e:"?e > 0" by auto
+ have lem:"\<And>f::real^'n \<Rightarrow> 'a. \<And> a b k1 k2.
+ (f has_integral k1) ({a..b}) \<Longrightarrow> (f has_integral k2) ({a..b}) \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> False"
+ proof- case goal1 let ?e = "norm(k1 - k2) / 2" from goal1(3) have e:"?e > 0" by auto
+ guess d1 by(rule has_integralD[OF goal1(1) e]) note d1=this
+ guess d2 by(rule has_integralD[OF goal1(2) e]) note d2=this
+ guess p by(rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)],of a b]) note p=this
+ let ?c = "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" have "norm (k1 - k2) \<le> norm (?c - k2) + norm (?c - k1)"
+ using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"] by(auto simp add:group_simps norm_minus_commute)
+ also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
+ apply(rule add_strict_mono) apply(rule_tac[!] d2(2) d1(2)) using p unfolding fine_def by auto
+ finally show False by auto
+ qed { presume "\<not> (\<exists>a b. i = {a..b}) \<Longrightarrow> False"
+ thus False apply-apply(cases "\<exists>a b. i = {a..b}")
+ using assms by(auto simp add:has_integral intro:lem[OF _ _ as]) }
+ assume as:"\<not> (\<exists>a b. i = {a..b})"
+ guess B1 by(rule has_integral_altD[OF assms(1) as,OF e]) note B1=this[rule_format]
+ guess B2 by(rule has_integral_altD[OF assms(2) as,OF e]) note B2=this[rule_format]
+ have "\<exists>a b::real^'n. ball 0 B1 \<union> ball 0 B2 \<subseteq> {a..b}" apply(rule bounded_subset_closed_interval)
+ using bounded_Un bounded_ball by auto then guess a b apply-by(erule exE)+
+ note ab=conjunctD2[OF this[unfolded Un_subset_iff]]
+ guess w using B1(2)[OF ab(1)] .. note w=conjunctD2[OF this]
+ guess z using B2(2)[OF ab(2)] .. note z=conjunctD2[OF this]
+ have "z = w" using lem[OF w(1) z(1)] by auto
+ hence "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)"
+ using norm_triangle_ineq4[of "k1 - w" "k2 - z"] by(auto simp add: norm_minus_commute)
+ also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2" apply(rule add_strict_mono) by(rule_tac[!] z(2) w(2))
+ finally show False by auto qed
+
+lemma integral_unique[intro]:
+ "(f has_integral y) k \<Longrightarrow> integral k f = y"
+ unfolding integral_def apply(rule some_equality) by(auto intro: has_integral_unique)
+
+lemma has_integral_is_0: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
+ assumes "\<forall>x\<in>s. f x = 0" shows "(f has_integral 0) s"
+proof- have lem:"\<And>a b. \<And>f::real^'n \<Rightarrow> 'a.
+ (\<forall>x\<in>{a..b}. f(x) = 0) \<Longrightarrow> (f has_integral 0) ({a..b})" unfolding has_integral
+ proof(rule,rule) fix a b e and f::"real^'n \<Rightarrow> 'a"
+ assume as:"\<forall>x\<in>{a..b}. f x = 0" "0 < (e::real)"
+ 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)"
+ apply(rule_tac x="\<lambda>x. ball x 1" in exI) apply(rule,rule gaugeI) unfolding centre_in_ball defer apply(rule open_ball)
+ proof(rule,rule,erule conjE) case goal1
+ have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) = 0" proof(rule setsum_0',rule)
+ 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
+ thus "(\<lambda>(x, k). content k *\<^sub>R f x) x = 0" apply(subst surjective_pairing[of x]) unfolding split_conv by auto
+ qed thus ?case using as by auto
+ qed auto qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
+ thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}")
+ using assms by(auto simp add:has_integral intro:lem) }
+ have *:"(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. 0)" apply(rule ext) using assms by auto
+ assume "\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P *
+ apply(rule,rule,rule_tac x=1 in exI,rule) defer apply(rule,rule,rule)
+ proof- fix e::real and a b assume "e>0"
+ thus "\<exists>z. ((\<lambda>x::real^'n. 0::'a) has_integral z) {a..b} \<and> norm (z - 0) < e"
+ apply(rule_tac x=0 in exI) apply(rule,rule lem) by auto
+ qed auto qed
+
+lemma has_integral_0[simp]: "((\<lambda>x::real^'n. 0) has_integral 0) s"
+ apply(rule has_integral_is_0) by auto
+
+lemma has_integral_0_eq[simp]: "((\<lambda>x. 0) has_integral i) s \<longleftrightarrow> i = 0"
+ using has_integral_unique[OF has_integral_0] by auto
+
+lemma has_integral_linear: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
+ assumes "(f has_integral y) s" "bounded_linear h" shows "((h o f) has_integral ((h y))) s"
+proof- interpret bounded_linear h using assms(2) . from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format]
+ have lem:"\<And>f::real^'n \<Rightarrow> 'a. \<And> y a b.
+ (f has_integral y) ({a..b}) \<Longrightarrow> ((h o f) has_integral h(y)) ({a..b})"
+ proof(subst has_integral,rule,rule) case goal1
+ from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format]
+ have *:"e / B > 0" apply(rule divide_pos_pos) using goal1(2) B by auto
+ guess g using has_integralD[OF goal1(1) *] . note g=this
+ show ?case apply(rule_tac x=g in exI) apply(rule,rule g(1))
+ proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "g fine p"
+ 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
+ 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"
+ unfolding o_def unfolding scaleR[THEN sym] * by simp
+ 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
+ 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)" .
+ show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) - h y) < e" unfolding * diff[THEN sym]
+ apply(rule le_less_trans[OF B(2)]) using g(2)[OF as] B(1) by(auto simp add:field_simps)
+ qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
+ thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) }
+ assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P
+ proof(rule,rule) fix e::real assume e:"0<e"
+ have *:"0 < e/B" by(rule divide_pos_pos,rule e,rule B(1))
+ guess M using has_integral_altD[OF assms(1) as *,rule_format] . note M=this
+ 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)"
+ apply(rule_tac x=M in exI) apply(rule,rule M(1))
+ proof(rule,rule,rule) case goal1 guess z using M(2)[OF goal1(1)] .. note z=conjunctD2[OF this]
+ 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)"
+ unfolding o_def apply(rule ext) using zero by auto
+ show ?case apply(rule_tac x="h z" in exI,rule) unfolding * apply(rule lem[OF z(1)]) unfolding diff[THEN sym]
+ apply(rule le_less_trans[OF B(2)]) using B(1) z(2) by(auto simp add:field_simps)
+ qed qed qed
+
+lemma has_integral_cmul:
+ shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_integral (c *\<^sub>R k)) s"
+ unfolding o_def[THEN sym] apply(rule has_integral_linear,assumption)
+ by(rule scaleR.bounded_linear_right)
+
+lemma has_integral_neg:
+ shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. -(f x)) has_integral (-k)) s"
+ apply(drule_tac c="-1" in has_integral_cmul) by auto
+
+lemma has_integral_add: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
+ assumes "(f has_integral k) s" "(g has_integral l) s"
+ shows "((\<lambda>x. f x + g x) has_integral (k + l)) s"
+proof- have lem:"\<And>f g::real^'n \<Rightarrow> 'a. \<And>a b k l.
+ (f has_integral k) ({a..b}) \<Longrightarrow> (g has_integral l) ({a..b}) \<Longrightarrow>
+ ((\<lambda>x. f(x) + g(x)) has_integral (k + l)) ({a..b})" proof- case goal1
+ show ?case unfolding has_integral proof(rule,rule) fix e::real assume e:"e>0" hence *:"e/2>0" by auto
+ guess d1 using has_integralD[OF goal1(1) *] . note d1=this
+ guess d2 using has_integralD[OF goal1(2) *] . note d2=this
+ 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)"
+ apply(rule_tac x="\<lambda>x. (d1 x) \<inter> (d2 x)" in exI) apply(rule,rule gauge_inter[OF d1(1) d2(1)])
+ proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d1 x \<inter> d2 x) fine p"
+ 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)"
+ 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]
+ by(rule setsum_cong2,auto)
+ 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))"
+ unfolding * by(auto simp add:group_simps) also let ?res = "\<dots>"
+ from as have *:"d1 fine p" "d2 fine p" unfolding fine_inter by auto
+ have "?res < e/2 + e/2" apply(rule le_less_trans[OF norm_triangle_ineq])
+ apply(rule add_strict_mono) using d1(2)[OF as(1) *(1)] and d2(2)[OF as(1) *(2)] by auto
+ finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e" by auto
+ qed qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
+ thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) }
+ assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P
+ proof(rule,rule) case goal1 hence *:"e/2 > 0" by auto
+ from has_integral_altD[OF assms(1) as *] guess B1 . note B1=this[rule_format]
+ from has_integral_altD[OF assms(2) as *] guess B2 . note B2=this[rule_format]
+ show ?case apply(rule_tac x="max B1 B2" in exI) apply(rule,rule min_max.less_supI1,rule B1)
+ proof(rule,rule,rule) fix a b assume "ball 0 (max B1 B2) \<subseteq> {a..b::real^'n}"
+ hence *:"ball 0 B1 \<subseteq> {a..b::real^'n}" "ball 0 B2 \<subseteq> {a..b::real^'n}" by auto
+ guess w using B1(2)[OF *(1)] .. note w=conjunctD2[OF this]
+ guess z using B2(2)[OF *(2)] .. note z=conjunctD2[OF this]
+ 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
+ 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"
+ apply(rule_tac x="w + z" in exI) apply(rule,rule lem[OF w(1) z(1), unfolded *[THEN sym]])
+ using norm_triangle_ineq[of "w - k" "z - l"] w(2) z(2) by(auto simp add:field_simps)
+ qed qed qed
+
+lemma has_integral_sub:
+ shows "(f has_integral k) s \<Longrightarrow> (g has_integral l) s \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) has_integral (k - l)) s"
+ using has_integral_add[OF _ has_integral_neg,of f k s g l] unfolding group_simps by auto
+
+lemma integral_0: "integral s (\<lambda>x::real^'n. 0::real^'m) = 0"
+ by(rule integral_unique has_integral_0)+
+
+lemma integral_add:
+ 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"
+ apply(rule integral_unique) apply(drule integrable_integral)+
+ apply(rule has_integral_add) by assumption+
+
+lemma integral_cmul:
+ shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. c *\<^sub>R f x) = c *\<^sub>R integral s f"
+ apply(rule integral_unique) apply(drule integrable_integral)+
+ apply(rule has_integral_cmul) by assumption+
+
+lemma integral_neg:
+ shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. - f x) = - integral s f"
+ apply(rule integral_unique) apply(drule integrable_integral)+
+ apply(rule has_integral_neg) by assumption+
+
+lemma integral_sub:
+ 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"
+ apply(rule integral_unique) apply(drule integrable_integral)+
+ apply(rule has_integral_sub) by assumption+
+
+lemma integrable_0: "(\<lambda>x. 0) integrable_on s"
+ unfolding integrable_on_def using has_integral_0 by auto
+
+lemma integrable_add:
+ shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x + g x) integrable_on s"
+ unfolding integrable_on_def by(auto intro: has_integral_add)
+
+lemma integrable_cmul:
+ shows "f integrable_on s \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) integrable_on s"
+ unfolding integrable_on_def by(auto intro: has_integral_cmul)
+
+lemma integrable_neg:
+ shows "f integrable_on s \<Longrightarrow> (\<lambda>x. -f(x)) integrable_on s"
+ unfolding integrable_on_def by(auto intro: has_integral_neg)
+
+lemma integrable_sub:
+ shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x - g x) integrable_on s"
+ unfolding integrable_on_def by(auto intro: has_integral_sub)
+
+lemma integrable_linear:
+ shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> (h o f) integrable_on s"
+ unfolding integrable_on_def by(auto intro: has_integral_linear)
+
+lemma integral_linear:
+ shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> integral s (h o f) = h(integral s f)"
+ apply(rule has_integral_unique) defer unfolding has_integral_integral
+ apply(drule has_integral_linear,assumption,assumption) unfolding has_integral_integral[THEN sym]
+ apply(rule integrable_linear) by assumption+
+
+lemma has_integral_setsum:
+ assumes "finite t" "\<forall>a\<in>t. ((f a) has_integral (i a)) s"
+ shows "((\<lambda>x. setsum (\<lambda>a. f a x) t) has_integral (setsum i t)) s"
+proof(insert assms(1) subset_refl[of t],induct rule:finite_subset_induct)
+ case (insert x F) show ?case unfolding setsum_insert[OF insert(1,3)]
+ apply(rule has_integral_add) using insert assms by auto
+qed auto
+
+lemma integral_setsum:
+ shows "finite t \<Longrightarrow> \<forall>a\<in>t. (f a) integrable_on s \<Longrightarrow>
+ integral s (\<lambda>x. setsum (\<lambda>a. f a x) t) = setsum (\<lambda>a. integral s (f a)) t"
+ apply(rule integral_unique) apply(rule has_integral_setsum)
+ using integrable_integral by auto
+
+lemma integrable_setsum:
+ 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"
+ unfolding integrable_on_def apply(drule bchoice) using has_integral_setsum[of t] by auto
+
+lemma has_integral_eq:
+ assumes "\<forall>x\<in>s. f x = g x" "(f has_integral k) s" shows "(g has_integral k) s"
+ using has_integral_sub[OF assms(2), of "\<lambda>x. f x - g x" 0]
+ using has_integral_is_0[of s "\<lambda>x. f x - g x"] using assms(1) by auto
+
+lemma integrable_eq:
+ shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> f integrable_on s \<Longrightarrow> g integrable_on s"
+ unfolding integrable_on_def using has_integral_eq[of s f g] by auto
+
+lemma has_integral_eq_eq:
+ shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> ((f has_integral k) s \<longleftrightarrow> (g has_integral k) s)"
+ using has_integral_eq[of s f g] has_integral_eq[of s g f] by auto
+
+lemma has_integral_null[dest]:
+ assumes "content({a..b}) = 0" shows "(f has_integral 0) ({a..b})"
+ unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI,rule) defer
+proof(rule,rule,erule conjE) fix e::real assume e:"e>0" thus "gauge (\<lambda>x. ball x 1)" by auto
+ fix p assume p:"p tagged_division_of {a..b}" (*"(\<lambda>x. ball x 1) fine p"*)
+ have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) = 0" unfolding norm_eq_zero diff_0_right
+ using setsum_content_null[OF assms(1) p, of f] .
+ thus "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e" using e by auto qed
+
+lemma has_integral_null_eq[simp]:
+ shows "content({a..b}) = 0 \<Longrightarrow> ((f has_integral i) ({a..b}) \<longleftrightarrow> i = 0)"
+ apply rule apply(rule has_integral_unique,assumption)
+ apply(drule has_integral_null,assumption)
+ apply(drule has_integral_null) by auto
+
+lemma integral_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> integral({a..b}) f = 0"
+ by(rule integral_unique,drule has_integral_null)
+
+lemma integrable_on_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> f integrable_on {a..b}"
+ unfolding integrable_on_def apply(drule has_integral_null) by auto
+
+lemma has_integral_empty[intro]: shows "(f has_integral 0) {}"
+ unfolding empty_as_interval apply(rule has_integral_null)
+ using content_empty unfolding empty_as_interval .
+
+lemma has_integral_empty_eq[simp]: shows "(f has_integral i) {} \<longleftrightarrow> i = 0"
+ apply(rule,rule has_integral_unique,assumption) by auto
+
+lemma integrable_on_empty[intro]: shows "f integrable_on {}" unfolding integrable_on_def by auto
+
+lemma integral_empty[simp]: shows "integral {} f = 0"
+ apply(rule integral_unique) using has_integral_empty .
+
+lemma has_integral_refl[intro]: shows "(f has_integral 0) {a..a}"
+ apply(rule has_integral_null) unfolding content_eq_0_interior
+ unfolding interior_closed_interval using interval_sing by auto
+
+lemma integrable_on_refl[intro]: shows "f integrable_on {a..a}" unfolding integrable_on_def by auto
+
+lemma integral_refl: shows "integral {a..a} f = 0" apply(rule integral_unique) by auto
+
+subsection {* Cauchy-type criterion for integrability. *}
+
+lemma integrable_cauchy: fixes f::"real^'n \<Rightarrow> 'a::{real_normed_vector,complete_space}"
+ shows "f integrable_on {a..b} \<longleftrightarrow>
+ (\<forall>e>0.\<exists>d. gauge d \<and> (\<forall>p1 p2. p1 tagged_division_of {a..b} \<and> d fine p1 \<and>
+ p2 tagged_division_of {a..b} \<and> d fine p2
+ \<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 -
+ setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) < e))" (is "?l = (\<forall>e>0. \<exists>d. ?P e d)")
+proof assume ?l
+ then guess y unfolding integrable_on_def has_integral .. note y=this
+ show "\<forall>e>0. \<exists>d. ?P e d" proof(rule,rule) case goal1 hence "e/2 > 0" by auto
+ then guess d apply- apply(drule y[rule_format]) by(erule exE,erule conjE) note d=this[rule_format]
+ show ?case apply(rule_tac x=d in exI,rule,rule d) apply(rule,rule,rule,(erule conjE)+)
+ proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b}" "d fine p1" "p2 tagged_division_of {a..b}" "d fine p2"
+ 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"
+ apply(rule dist_triangle_half_l[where y=y,unfolded vector_dist_norm])
+ using d(2)[OF conjI[OF as(1-2)]] d(2)[OF conjI[OF as(3-4)]] .
+ qed qed
+next assume "\<forall>e>0. \<exists>d. ?P e d" hence "\<forall>n::nat. \<exists>d. ?P (inverse(real (n + 1))) d" by auto
+ from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
+ have "\<And>n. gauge (\<lambda>x. \<Inter>{d i x |i. i \<in> {0..n}})" apply(rule gauge_inters) using d(1) by auto
+ 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-
+ proof case goal1 from this[of n] show ?case apply(drule_tac fine_division_exists) by auto qed
+ from choice[OF this] guess p .. note p = conjunctD2[OF this[rule_format]]
+ have dp:"\<And>i n. i\<le>n \<Longrightarrow> d i fine p n" using p(2) unfolding fine_inters by auto
+ have "Cauchy (\<lambda>n. setsum (\<lambda>(x,k). content k *\<^sub>R (f x)) (p n))"
+ proof(rule CauchyI) case goal1 then guess N unfolding real_arch_inv[of e] .. note N=this
+ show ?case apply(rule_tac x=N in exI)
+ 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
+ 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"
+ apply(rule less_trans[OF _ N[THEN conjunct2,THEN conjunct2]]) apply(subst *) apply(rule d(2))
+ using dp p(1) using mn by auto
+ qed qed
+ then guess y unfolding convergent_eq_cauchy[THEN sym] .. note y=this[unfolded Lim_sequentially,rule_format]
+ show ?l unfolding integrable_on_def has_integral apply(rule_tac x=y in exI)
+ proof(rule,rule) fix e::real assume "e>0" hence *:"e/2 > 0" by auto
+ then guess N1 unfolding real_arch_inv[of "e/2"] .. note N1=this hence N1':"N1 = N1 - 1 + 1" by auto
+ guess N2 using y[OF *] .. note N2=this
+ 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)"
+ apply(rule_tac x="d (N1 + N2)" in exI) apply rule defer
+ proof(rule,rule,erule conjE) show "gauge (d (N1 + N2))" using d by auto
+ fix q assume as:"q tagged_division_of {a..b}" "d (N1 + N2) fine q"
+ have *:"inverse (real (N1 + N2 + 1)) < e / 2" apply(rule less_trans) using N1 by auto
+ show "norm ((\<Sum>(x, k)\<in>q. content k *\<^sub>R f x) - y) < e" apply(rule norm_triangle_half_r)
+ apply(rule less_trans[OF _ *]) apply(subst N1', rule d(2)[of "p (N1+N2)"]) defer
+ using N2[rule_format,unfolded vector_dist_norm,of "N1+N2"]
+ using as dp[of "N1 - 1 + 1 + N2" "N1 + N2"] using p(1)[of "N1 + N2"] using N1 by auto qed qed qed
+
+subsection {* Additivity of integral on abutting intervals. *}
+
+lemma interval_split:
+ "{a..b::real^'n} \<inter> {x. x$k \<le> c} = {a .. (\<chi> i. if i = k then min (b$k) c else b$i)}"
+ "{a..b} \<inter> {x. x$k \<ge> c} = {(\<chi> i. if i = k then max (a$k) c else a$i) .. b}"
+ apply(rule_tac[!] set_ext) unfolding Int_iff mem_interval mem_Collect_eq
+ unfolding Cart_lambda_beta by auto
+
+lemma content_split:
+ "content {a..b::real^'n} = content({a..b} \<inter> {x. x$k \<le> c}) + content({a..b} \<inter> {x. x$k >= c})"
+proof- note simps = interval_split content_closed_interval_cases Cart_lambda_beta vector_le_def
+ { presume "a\<le>b \<Longrightarrow> ?thesis" thus ?thesis apply(cases "a\<le>b") unfolding simps by auto }
+ have *:"UNIV = insert k (UNIV - {k})" "\<And>x. finite (UNIV-{x::'n})" "\<And>x. x\<notin>UNIV-{x}" by auto
+ 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))"
+ "(\<Prod>i\<in>UNIV. b$i - a$i) = (\<Prod>i\<in>UNIV-{k}. b$i - a$i) * (b$k - a$k)"
+ apply(subst *(1)) defer apply(subst *(1)) unfolding setprod_insert[OF *(2-)] by auto
+ assume as:"a\<le>b" moreover have "\<And>x. min (b $ k) c = max (a $ k) c
+ \<Longrightarrow> x* (b$k - a$k) = x*(max (a $ k) c - a $ k) + x*(b $ k - max (a $ k) c)"
+ by (auto simp add:field_simps)
+ moreover have "\<not> a $ k \<le> c \<Longrightarrow> \<not> c \<le> b $ k \<Longrightarrow> False"
+ unfolding not_le using as[unfolded vector_le_def,rule_format,of k] by auto
+ ultimately show ?thesis
+ unfolding simps unfolding *(1)[of "\<lambda>i x. b$i - x"] *(1)[of "\<lambda>i x. x - a$i"] *(2) by(auto)
+qed
+
+lemma division_split_left_inj:
+ assumes "d division_of i" "k1 \<in> d" "k2 \<in> d" "k1 \<noteq> k2"
+ "k1 \<inter> {x::real^'n. x$k \<le> c} = k2 \<inter> {x. x$k \<le> c}"
+ shows "content(k1 \<inter> {x. x$k \<le> c}) = 0"
+proof- note d=division_ofD[OF assms(1)]
+ have *:"\<And>a b::real^'n. \<And> c k. (content({a..b} \<inter> {x. x$k \<le> c}) = 0 \<longleftrightarrow> interior({a..b} \<inter> {x. x$k \<le> c}) = {})"
+ unfolding interval_split content_eq_0_interior by auto
+ guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this
+ guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this
+ have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto
+ show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]])
+ defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed
+
+lemma division_split_right_inj:
+ assumes "d division_of i" "k1 \<in> d" "k2 \<in> d" "k1 \<noteq> k2"
+ "k1 \<inter> {x::real^'n. x$k \<ge> c} = k2 \<inter> {x. x$k \<ge> c}"
+ shows "content(k1 \<inter> {x. x$k \<ge> c}) = 0"
+proof- note d=division_ofD[OF assms(1)]
+ 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}) = {})"
+ unfolding interval_split content_eq_0_interior by auto
+ guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this
+ guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this
+ have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto
+ show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]])
+ defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed
+
+lemma tagged_division_split_left_inj:
+ 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}"
+ shows "content(k1 \<inter> {x. x$k \<le> c}) = 0"
+proof- have *:"\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c" unfolding image_iff apply(rule_tac x="(a,b)" in bexI) by auto
+ show ?thesis apply(rule division_split_left_inj[OF division_of_tagged_division[OF assms(1)]])
+ apply(rule_tac[1-2] *) using assms(2-) by auto qed
+
+lemma tagged_division_split_right_inj:
+ assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2" "k1 \<inter> {x. x$k \<ge> c} = k2 \<inter> {x. x$k \<ge> c}"
+ shows "content(k1 \<inter> {x. x$k \<ge> c}) = 0"
+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
+ show ?thesis apply(rule division_split_right_inj[OF division_of_tagged_division[OF assms(1)]])
+ apply(rule_tac[1-2] *) using assms(2-) by auto qed
+
+lemma division_split:
+ assumes "p division_of {a..b::real^'n}"
+ 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$k \<le> c})" (is "?p1 division_of ?I1") and
+ "{l \<inter> {x. x$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")
+proof(rule_tac[!] division_ofI) note p=division_ofD[OF assms]
+ show "finite ?p1" "finite ?p2" using p(1) by auto show "\<Union>?p1 = ?I1" "\<Union>?p2 = ?I2" unfolding p(6)[THEN sym] by auto
+ { fix k assume "k\<in>?p1" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this
+ guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this
+ show "k\<subseteq>?I1" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l
+ using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split) by auto
+ fix k' assume "k'\<in>?p1" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this
+ assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto }
+ { fix k assume "k\<in>?p2" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this
+ guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this
+ show "k\<subseteq>?I2" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l
+ using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split) by auto
+ fix k' assume "k'\<in>?p2" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this
+ assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto }
+qed
+
+lemma has_integral_split: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
+ 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})"
+ shows "(f has_integral (i + j)) ({a..b})"
+proof(unfold has_integral,rule,rule) case goal1 hence e:"e/2>0" by auto
+ guess d1 using has_integralD[OF assms(1)[unfolded interval_split] e] . note d1=this[unfolded interval_split[THEN sym]]
+ guess d2 using has_integralD[OF assms(2)[unfolded interval_split] e] . note d2=this[unfolded interval_split[THEN sym]]
+ 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"
+ show ?case apply(rule_tac x="?d" in exI,rule) defer apply(rule,rule,(erule conjE)+)
+ proof- show "gauge ?d" using d1(1) d2(1) unfolding gauge_def by auto
+ fix p assume "p tagged_division_of {a..b}" "?d fine p" note p = this tagged_division_ofD[OF this(1)]
+ have lem0:"\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x$k \<le> c} = {}) \<Longrightarrow> x$k \<le> c"
+ "\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x$k \<ge> c} = {}) \<Longrightarrow> x$k \<ge> c"
+ proof- fix x kk assume as:"(x,kk)\<in>p"
+ show "~(kk \<inter> {x. x$k \<le> c} = {}) \<Longrightarrow> x$k \<le> c"
+ proof(rule ccontr) case goal1
+ from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x $ k - c\<bar>"
+ using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
+ hence "\<exists>y. y \<in> ball x \<bar>x $ k - c\<bar> \<inter> {x. x $ k \<le> c}" using goal1(1) by blast
+ then guess y .. hence "\<bar>x $ k - y $ k\<bar> < \<bar>x $ k - c\<bar>" "y$k \<le> c" apply-apply(rule le_less_trans)
+ using component_le_norm[of "x - y" k,unfolded vector_minus_component] by(auto simp add:vector_dist_norm)
+ thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
+ qed
+ show "~(kk \<inter> {x. x$k \<ge> c} = {}) \<Longrightarrow> x$k \<ge> c"
+ proof(rule ccontr) case goal1
+ from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x $ k - c\<bar>"
+ using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
+ hence "\<exists>y. y \<in> ball x \<bar>x $ k - c\<bar> \<inter> {x. x $ k \<ge> c}" using goal1(1) by blast
+ then guess y .. hence "\<bar>x $ k - y $ k\<bar> < \<bar>x $ k - c\<bar>" "y$k \<ge> c" apply-apply(rule le_less_trans)
+ using component_le_norm[of "x - y" k,unfolded vector_minus_component] by(auto simp add:vector_dist_norm)
+ thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
+ qed
+ qed
+
+ have lem1: "\<And>f P Q. (\<forall>x k. (x,k) \<in> {(x,f k) | x k. P x k} \<longrightarrow> Q x k) \<longleftrightarrow> (\<forall>x k. P x k \<longrightarrow> Q x (f k))" by auto
+ have lem2: "\<And>f s P f. finite s \<Longrightarrow> finite {(x,f k) | x k. (x,k) \<in> s \<and> P x k}"
+ proof- case goal1 thus ?case apply-apply(rule finite_subset[of _ "(\<lambda>(x,k). (x,f k)) ` s"]) by auto qed
+ have lem3: "\<And>g::(real ^ 'n \<Rightarrow> bool) \<Rightarrow> real ^ 'n \<Rightarrow> bool. finite p \<Longrightarrow>
+ setsum (\<lambda>(x,k). content k *\<^sub>R f x) {(x,g k) |x k. (x,k) \<in> p \<and> ~(g k = {})}
+ = setsum (\<lambda>(x,k). content k *\<^sub>R f x) ((\<lambda>(x,k). (x,g k)) ` p)"
+ apply(rule setsum_mono_zero_left) prefer 3
+ proof fix g::"(real ^ 'n \<Rightarrow> bool) \<Rightarrow> real ^ 'n \<Rightarrow> bool" and i::"(real^'n) \<times> ((real^'n) set)"
+ assume "i \<in> (\<lambda>(x, k). (x, g k)) ` p - {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}"
+ then obtain x k where xk:"i=(x,g k)" "(x,k)\<in>p" "(x,g k) \<notin> {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}" by auto
+ have "content (g k) = 0" using xk using content_empty by auto
+ thus "(\<lambda>(x, k). content k *\<^sub>R f x) i = 0" unfolding xk split_conv by auto
+ qed auto
+ have lem4:"\<And>g. (\<lambda>(x,l). content (g l) *\<^sub>R f x) = (\<lambda>(x,l). content l *\<^sub>R f x) o (\<lambda>(x,l). (x,g l))" apply(rule ext) by auto
+
+ let ?M1 = "{(x,kk \<inter> {x. x$k \<le> c}) |x kk. (x,kk) \<in> p \<and> kk \<inter> {x. x$k \<le> c} \<noteq> {}}"
+ have "norm ((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) < e/2" apply(rule d1(2),rule tagged_division_ofI)
+ apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)
+ proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M1} = {a..b} \<inter> {x. x$k \<le> c}" unfolding p(8)[THEN sym] by auto
+ fix x l assume xl:"(x,l)\<in>?M1"
+ then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) . note xl'=this
+ have "l' \<subseteq> d1 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
+ thus "l \<subseteq> d1 x" unfolding xl' by auto
+ 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)
+ using lem0(1)[OF xl'(3-4)] by auto
+ 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])
+ fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M1"
+ then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) . note yr'=this
+ assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}"
+ proof(cases "l' = r' \<longrightarrow> x' = y'")
+ case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
+ next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto
+ thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
+ qed qed moreover
+
+ 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> {}}"
+ have "norm ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j) < e/2" apply(rule d2(2),rule tagged_division_ofI)
+ apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)
+ 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
+ fix x l assume xl:"(x,l)\<in>?M2"
+ then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) . note xl'=this
+ have "l' \<subseteq> d2 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
+ thus "l \<subseteq> d2 x" unfolding xl' by auto
+ 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)
+ using lem0(2)[OF xl'(3-4)] by auto
+ 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])
+ fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M2"
+ then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) . note yr'=this
+ assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}"
+ proof(cases "l' = r' \<longrightarrow> x' = y'")
+ case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
+ next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto
+ thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
+ qed qed ultimately
+
+ 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"
+ apply- apply(rule norm_triangle_lt) by auto
+ also { have *:"\<And>x y. x = (0::real) \<Longrightarrow> x *\<^sub>R (y::'a) = 0" using scaleR_zero_left by auto
+ 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)
+ = (\<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
+ 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)"
+ unfolding lem3[OF p(3)] apply(subst setsum_reindex_nonzero[OF p(3)]) defer apply(subst setsum_reindex_nonzero[OF p(3)])
+ defer unfolding lem4[THEN sym] apply(rule refl) unfolding split_paired_all split_conv apply(rule_tac[!] *)
+ proof- case goal1 thus ?case apply- apply(rule tagged_division_split_left_inj [OF p(1), of a b aa ba]) by auto
+ next case goal2 thus ?case apply- apply(rule tagged_division_split_right_inj[OF p(1), of a b aa ba]) by auto
+ qed also note setsum_addf[THEN sym]
+ 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
+ = (\<lambda>(x,ka). content ka *\<^sub>R f x) x" unfolding split_paired_all split_conv
+ proof- fix a b assume "(a,b) \<in> p" from p(6)[OF this] guess u v apply-by(erule exE)+ note uv=this
+ 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"
+ unfolding scaleR_left_distrib[THEN sym] unfolding uv content_split[of u v k c] by auto
+ qed note setsum_cong2[OF this]
+ 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 +
+ ((\<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) =
+ (\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f x) - (i + j)" by auto }
+ finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (i + j)) < e" by auto qed qed
+
+subsection {* A sort of converse, integrability on subintervals. *}
+
+lemma tagged_division_union_interval:
+ 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})"
+ shows "(p1 \<union> p2) tagged_division_of ({a..b})"
+proof- have *:"{a..b} = ({a..b} \<inter> {x. x$k \<le> c}) \<union> ({a..b} \<inter> {x. x$k \<ge> c})" by auto
+ show ?thesis apply(subst *) apply(rule tagged_division_union[OF assms])
+ unfolding interval_split interior_closed_interval
+ by(auto simp add: vector_less_def Cart_lambda_beta elim!:allE[where x=k]) qed
+
+lemma has_integral_separate_sides: fixes f::"real^'m \<Rightarrow> 'a::real_normed_vector"
+ assumes "(f has_integral i) ({a..b})" "e>0"
+ obtains d where "gauge d" "(\<forall>p1 p2. p1 tagged_division_of ({a..b} \<inter> {x. x$k \<le> c}) \<and> d fine p1 \<and>
+ p2 tagged_division_of ({a..b} \<inter> {x. x$k \<ge> c}) \<and> d fine p2
+ \<longrightarrow> norm((setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 +
+ setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) - i) < e)"
+proof- guess d using has_integralD[OF assms] . note d=this
+ show ?thesis apply(rule that[of d]) apply(rule d) apply(rule,rule,rule,(erule conjE)+)
+ 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
+ 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
+ note tagged_division_union_interval[OF p1(7) p2(7)] note p12 = tagged_division_ofD[OF this] this
+ 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)"
+ apply(subst setsum_Un_zero) apply(rule p1 p2)+ apply(rule) unfolding split_paired_all split_conv
+ proof- fix a b assume ab:"(a,b) \<in> p1 \<inter> p2"
+ have "(a,b) \<in> p1" using ab by auto from p1(4)[OF this] guess u v apply-by(erule exE)+ note uv =this
+ have "b \<subseteq> {x. x$k = c}" using ab p1(3)[of a b] p2(3)[of a b] by fastsimp
+ moreover have "interior {x. x $ k = c} = {}"
+ proof(rule ccontr) case goal1 then obtain x where x:"x\<in>interior {x. x$k = c}" by auto
+ then guess e unfolding mem_interior .. note e=this
+ have x:"x$k = c" using x interior_subset by fastsimp
+ 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
+ have "x + (\<chi> i. if i = k then e/2 else 0) \<in> ball x e" unfolding mem_ball vector_dist_norm
+ apply(rule le_less_trans[OF norm_le_l1]) unfolding *
+ unfolding setsum_delta[OF finite_UNIV] using e by auto
+ hence "x + (\<chi> i. if i = k then e/2 else 0) \<in> {x. x$k = c}" using e by auto
+ thus False unfolding mem_Collect_eq using e x by auto
+ qed ultimately have "content b = 0" unfolding uv content_eq_0_interior apply-apply(drule subset_interior) by auto
+ thus "content b *\<^sub>R f a = 0" by auto
+ qed auto
+ also have "\<dots> < e" by(rule d(2) p12 fine_union p1 p2)+
+ finally show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) < e" . qed qed
+
+lemma integrable_split[intro]: fixes f::"real^'n \<Rightarrow> 'a::{real_normed_vector,complete_space}" assumes "f integrable_on {a..b}"
+ shows "f integrable_on ({a..b} \<inter> {x. x$k \<le> c})" (is ?t1) and "f integrable_on ({a..b} \<inter> {x. x$k \<ge> c})" (is ?t2)
+proof- guess y using assms unfolding integrable_on_def .. note y=this
+ def b' \<equiv> "(\<chi> i. if i = k then min (b$k) c else b$i)::real^'n"
+ and a' \<equiv> "(\<chi> i. if i = k then max (a$k) c else a$i)::real^'n"
+ show ?t1 ?t2 unfolding interval_split integrable_cauchy unfolding interval_split[THEN sym]
+ proof(rule_tac[!] allI impI)+ fix e::real assume "e>0" hence "e/2>0" by auto
+ from has_integral_separate_sides[OF y this,of k c] guess d . note d=this[rule_format]
+ 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>
+ norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e)"
+ show "?P {x. x $ k \<le> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
+ 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"
+ 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"
+ proof- guess p using fine_division_exists[OF d(1), of a' b] . note p=this
+ show ?thesis using norm_triangle_half_l[OF d(2)[of p1 p] d(2)[of p2 p]]
+ using as unfolding interval_split b'_def[symmetric] a'_def[symmetric]
+ using p using assms by(auto simp add:group_simps)
+ qed qed
+ show "?P {x. x $ k \<ge> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
+ 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"
+ 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"
+ proof- guess p using fine_division_exists[OF d(1), of a b'] . note p=this
+ show ?thesis using norm_triangle_half_l[OF d(2)[of p p1] d(2)[of p p2]]
+ using as unfolding interval_split b'_def[symmetric] a'_def[symmetric]
+ using p using assms by(auto simp add:group_simps) qed qed qed qed
+
+subsection {* Generalized notion of additivity. *}
+
+definition "neutral opp = (SOME x. \<forall>y. opp x y = y \<and> opp y x = y)"
+
+definition operative :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ((real^'n) set \<Rightarrow> 'a) \<Rightarrow> bool" where
+ "operative opp f \<equiv>
+ (\<forall>a b. content {a..b} = 0 \<longrightarrow> f {a..b} = neutral(opp)) \<and>
+ (\<forall>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})))"
+
+lemma operativeD[dest]: assumes "operative opp f"
+ shows "\<And>a b. content {a..b} = 0 \<Longrightarrow> f {a..b} = neutral(opp)"
+ "\<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}))"
+ using assms unfolding operative_def by auto
+
+lemma operative_trivial:
+ "operative opp f \<Longrightarrow> content({a..b}) = 0 \<Longrightarrow> f({a..b}) = neutral opp"
+ unfolding operative_def by auto
+
+lemma property_empty_interval:
+ "(\<forall>a b. content({a..b}) = 0 \<longrightarrow> P({a..b})) \<Longrightarrow> P {}"
+ using content_empty unfolding empty_as_interval by auto
+
+lemma operative_empty: "operative opp f \<Longrightarrow> f {} = neutral opp"
+ unfolding operative_def apply(rule property_empty_interval) by auto
+
+subsection {* Using additivity of lifted function to encode definedness. *}
+
+lemma forall_option: "(\<forall>x. P x) \<longleftrightarrow> P None \<and> (\<forall>x. P(Some x))"
+ by (metis map_of.simps option.nchotomy)
+
+lemma exists_option:
+ "(\<exists>x. P x) \<longleftrightarrow> P None \<or> (\<exists>x. P(Some x))"
+ by (metis map_of.simps option.nchotomy)
+
+fun lifted where
+ "lifted (opp::'a\<Rightarrow>'a\<Rightarrow>'b) (Some x) (Some y) = Some(opp x y)" |
+ "lifted opp None _ = (None::'b option)" |
+ "lifted opp _ None = None"
+
+lemma lifted_simp_1[simp]: "lifted opp v None = None"
+ apply(induct v) by auto
+
+definition "monoidal opp \<equiv> (\<forall>x y. opp x y = opp y x) \<and>
+ (\<forall>x y z. opp x (opp y z) = opp (opp x y) z) \<and>
+ (\<forall>x. opp (neutral opp) x = x)"
+
+lemma monoidalI: assumes "\<And>x y. opp x y = opp y x"
+ "\<And>x y z. opp x (opp y z) = opp (opp x y) z"
+ "\<And>x. opp (neutral opp) x = x" shows "monoidal opp"
+ unfolding monoidal_def using assms by fastsimp
+
+lemma monoidal_ac: assumes "monoidal opp"
+ shows "opp (neutral opp) a = a" "opp a (neutral opp) = a" "opp a b = opp b a"
+ "opp (opp a b) c = opp a (opp b c)" "opp a (opp b c) = opp b (opp a c)"
+ using assms unfolding monoidal_def apply- by metis+
+
+lemma monoidal_simps[simp]: assumes "monoidal opp"
+ shows "opp (neutral opp) a = a" "opp a (neutral opp) = a"
+ using monoidal_ac[OF assms] by auto
+
+lemma neutral_lifted[cong]: assumes "monoidal opp"
+ shows "neutral (lifted opp) = Some(neutral opp)"
+ apply(subst neutral_def) apply(rule some_equality) apply(rule,induct_tac y) prefer 3
+proof- fix x assume "\<forall>y. lifted opp x y = y \<and> lifted opp y x = y"
+ thus "x = Some (neutral opp)" apply(induct x) defer
+ apply rule apply(subst neutral_def) apply(subst eq_commute,rule some_equality)
+ apply(rule,erule_tac x="Some y" in allE) defer apply(erule_tac x="Some x" in allE) by auto
+qed(auto simp add:monoidal_ac[OF assms])
+
+lemma monoidal_lifted[intro]: assumes "monoidal opp" shows "monoidal(lifted opp)"
+ unfolding monoidal_def forall_option neutral_lifted[OF assms] using monoidal_ac[OF assms] by auto
+
+definition "support opp f s = {x. x\<in>s \<and> f x \<noteq> neutral opp}"
+definition "fold' opp e s \<equiv> (if finite s then fold opp e s else e)"
+definition "iterate opp s f \<equiv> fold' (\<lambda>x a. opp (f x) a) (neutral opp) (support opp f s)"
+
+lemma support_subset[intro]:"support opp f s \<subseteq> s" unfolding support_def by auto
+lemma support_empty[simp]:"support opp f {} = {}" using support_subset[of opp f "{}"] by auto
+
+lemma fun_left_comm_monoidal[intro]: assumes "monoidal opp" shows "fun_left_comm opp"
+ unfolding fun_left_comm_def using monoidal_ac[OF assms] by auto
+
+lemma support_clauses:
+ "\<And>f g s. support opp f {} = {}"
+ "\<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))"
+ "\<And>f g s. support opp f (s - {x}) = (support opp f s) - {x}"
+ "\<And>f g s. support opp f (s \<union> t) = (support opp f s) \<union> (support opp f t)"
+ "\<And>f g s. support opp f (s \<inter> t) = (support opp f s) \<inter> (support opp f t)"
+ "\<And>f g s. support opp f (s - t) = (support opp f s) - (support opp f t)"
+ "\<And>f g s. support opp g (f ` s) = f ` (support opp (g o f) s)"
+unfolding support_def by auto
+
+lemma finite_support[intro]:"finite s \<Longrightarrow> finite (support opp f s)"
+ unfolding support_def by auto
+
+lemma iterate_empty[simp]:"iterate opp {} f = neutral opp"
+ unfolding iterate_def fold'_def by auto
+
+lemma iterate_insert[simp]: assumes "monoidal opp" "finite s"
+ shows "iterate opp (insert x s) f = (if x \<in> s then iterate opp s f else opp (f x) (iterate opp s f))"
+proof(cases "x\<in>s") case True hence *:"insert x s = s" by auto
+ show ?thesis unfolding iterate_def if_P[OF True] * by auto
+next case False note x=this
+ note * = fun_left_comm.fun_left_comm_apply[OF fun_left_comm_monoidal[OF assms(1)]]
+ show ?thesis proof(cases "f x = neutral opp")
+ case True show ?thesis unfolding iterate_def if_not_P[OF x] support_clauses if_P[OF True]
+ unfolding True monoidal_simps[OF assms(1)] by auto
+ next case False show ?thesis unfolding iterate_def fold'_def if_not_P[OF x] support_clauses if_not_P[OF False]
+ apply(subst fun_left_comm.fold_insert[OF * finite_support])
+ using `finite s` unfolding support_def using False x by auto qed qed
+
+lemma iterate_some:
+ assumes "monoidal opp" "finite s"
+ shows "iterate (lifted opp) s (\<lambda>x. Some(f x)) = Some (iterate opp s f)" using assms(2)
+proof(induct s) case empty thus ?case using assms by auto
+next case (insert x F) show ?case apply(subst iterate_insert) prefer 3 apply(subst if_not_P)
+ defer unfolding insert(3) lifted.simps apply rule using assms insert by auto qed
+
+subsection {* Two key instances of additivity. *}
+
+lemma neutral_add[simp]:
+ "neutral op + = (0::_::comm_monoid_add)" unfolding neutral_def
+ apply(rule some_equality) defer apply(erule_tac x=0 in allE) by auto
+
+lemma operative_content[intro]: "operative (op +) content"
+ unfolding operative_def content_split[THEN sym] neutral_add by auto
+
+lemma neutral_monoid[simp]: "neutral ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a) = 0"
+ unfolding neutral_def apply(rule some_equality) defer
+ apply(erule_tac x=0 in allE) by auto
+
+lemma monoidal_monoid[intro]:
+ shows "monoidal ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a)"
+ unfolding monoidal_def neutral_monoid by(auto simp add: group_simps)
+
+lemma operative_integral: fixes f::"real^'n \<Rightarrow> 'a::banach"
+ shows "operative (lifted(op +)) (\<lambda>i. if f integrable_on i then Some(integral i f) else None)"
+ unfolding operative_def unfolding neutral_lifted[OF monoidal_monoid] neutral_add
+ apply(rule,rule,rule,rule) defer apply(rule allI)+
+proof- fix a b c k show "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) =
+ 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)
+ (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)"
+ proof(cases "f integrable_on {a..b}")
+ case True show ?thesis unfolding if_P[OF True]
+ unfolding if_P[OF integrable_split(1)[OF True]] if_P[OF integrable_split(2)[OF True]]
+ unfolding lifted.simps option.inject apply(rule integral_unique) apply(rule has_integral_split)
+ apply(rule_tac[!] integrable_integral integrable_split)+ using True by assumption+
+ 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}))"
+ proof(rule ccontr) case goal1 hence "f integrable_on {a..b}" apply- unfolding integrable_on_def
+ 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)
+ apply(rule has_integral_split) apply(rule_tac[!] integrable_integral) by auto
+ thus False using False by auto
+ qed thus ?thesis using False by auto
+ qed next
+ fix a b assume as:"content {a..b::real^'n} = 0"
+ thus "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) = Some 0"
+ unfolding if_P[OF integrable_on_null[OF as]] using has_integral_null_eq[OF as] by auto qed
+
+subsection {* Points of division of a partition. *}
+
+definition "division_points (k::(real^'n) set) d =
+ {(j,x). (interval_lowerbound k)$j < x \<and> x < (interval_upperbound k)$j \<and>
+ (\<exists>i\<in>d. (interval_lowerbound i)$j = x \<or> (interval_upperbound i)$j = x)}"
+
+lemma division_points_finite: assumes "d division_of i"
+ shows "finite (division_points i d)"
+proof- note assm = division_ofD[OF assms]
+ let ?M = "\<lambda>j. {(j,x)|x. (interval_lowerbound i)$j < x \<and> x < (interval_upperbound i)$j \<and>
+ (\<exists>i\<in>d. (interval_lowerbound i)$j = x \<or> (interval_upperbound i)$j = x)}"
+ have *:"division_points i d = \<Union>(?M ` UNIV)"
+ unfolding division_points_def by auto
+ show ?thesis unfolding * using assm by auto qed
+
+lemma division_points_subset:
+ assumes "d division_of {a..b}" "\<forall>i. a$i < b$i" "a$k < c" "c < b$k"
+ 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$k \<le> c} = {})}
+ \<subseteq> division_points ({a..b}) d" (is ?t1) and
+ "division_points ({a..b} \<inter> {x. x$k \<ge> c}) {l \<inter> {x. x$k \<ge> c} | l . l \<in> d \<and> ~(l \<inter> {x. x$k \<ge> c} = {})}
+ \<subseteq> division_points ({a..b}) d" (is ?t2)
+proof- note assm = division_ofD[OF assms(1)]
+ 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"
+ "\<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"
+ using assms using less_imp_le by auto
+ show ?t1 unfolding division_points_def interval_split[of a b]
+ unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] Cart_lambda_beta unfolding *
+ unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+ unfolding mem_Collect_eq apply(erule exE conjE)+
+ proof- fix i l x assume as:"a $ fst x < snd x" "snd x < (if fst x = k then c else b $ fst x)"
+ "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> {}"
+ from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
+ 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 .
+ have **:"\<forall>i. u$i \<le> v$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
+ 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)"
+ using as(1-3,5) unfolding l interval_split interval_ne_empty as interval_bounds[OF *] Cart_lambda_beta apply-
+ apply(rule,assumption,rule) defer apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **]
+ apply(case_tac[!] "fst x = k") using assms by auto
+ qed
+ show ?t2 unfolding division_points_def interval_split[of a b]
+ unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] Cart_lambda_beta unfolding *
+ unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+ unfolding mem_Collect_eq apply(erule exE conjE)+
+ 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"
+ "i = l \<inter> {x. c \<le> x $ k}" "l \<in> d" "l \<inter> {x. c \<le> x $ k} \<noteq> {}"
+ from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
+ 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 .
+ have **:"\<forall>i. u$i \<le> v$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
+ 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)"
+ using as(1-3,5) unfolding l interval_split interval_ne_empty as interval_bounds[OF *] Cart_lambda_beta apply-
+ apply rule defer apply(rule,assumption) apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **]
+ apply(case_tac[!] "fst x = k") using assms by auto qed qed
+
+lemma division_points_psubset:
+ assumes "d division_of {a..b}" "\<forall>i. a$i < b$i" "a$k < c" "c < b$k"
+ "l \<in> d" "interval_lowerbound l$k = c \<or> interval_upperbound l$k = c"
+ 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$k \<le> c} \<noteq> {}} \<subset> division_points ({a..b}) d" (is "?D1 \<subset> ?D")
+ "division_points ({a..b} \<inter> {x. x$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")
+proof- have ab:"\<forall>i. a$i \<le> b$i" using assms(2) by(auto intro!:less_imp_le)
+ guess u v using division_ofD(4)[OF assms(1,5)] apply-by(erule exE)+ note l=this
+ 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
+ unfolding subset_eq apply- defer apply(erule_tac x=u in ballE, erule_tac x=v in ballE) unfolding mem_interval by auto
+ have *:"interval_upperbound ({a..b} \<inter> {x. x $ k \<le> interval_upperbound l $ k}) $ k = interval_upperbound l $ k"
+ "interval_upperbound ({a..b} \<inter> {x. x $ k \<le> interval_lowerbound l $ k}) $ k = interval_lowerbound l $ k"
+ unfolding interval_split apply(subst interval_bounds) prefer 3 apply(subst interval_bounds)
+ unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab by auto
+ have "\<exists>x. x \<in> ?D - ?D1" using assms(2-) apply-apply(erule disjE)
+ apply(rule_tac x="(k,(interval_lowerbound l)$k)" in exI) defer
+ apply(rule_tac x="(k,(interval_upperbound l)$k)" in exI)
+ unfolding division_points_def unfolding interval_bounds[OF ab]
+ apply (auto simp add:interval_bounds) unfolding * by auto
+ thus "?D1 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) by auto
+
+ have *:"interval_lowerbound ({a..b} \<inter> {x. x $ k \<ge> interval_lowerbound l $ k}) $ k = interval_lowerbound l $ k"
+ "interval_lowerbound ({a..b} \<inter> {x. x $ k \<ge> interval_upperbound l $ k}) $ k = interval_upperbound l $ k"
+ unfolding interval_split apply(subst interval_bounds) prefer 3 apply(subst interval_bounds)
+ unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab by auto
+ have "\<exists>x. x \<in> ?D - ?D2" using assms(2-) apply-apply(erule disjE)
+ apply(rule_tac x="(k,(interval_lowerbound l)$k)" in exI) defer
+ apply(rule_tac x="(k,(interval_upperbound l)$k)" in exI)
+ unfolding division_points_def unfolding interval_bounds[OF ab]
+ apply (auto simp add:interval_bounds) unfolding * by auto
+ thus "?D2 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) by auto qed
+
+subsection {* Preservation by divisions and tagged divisions. *}
+
+lemma support_support[simp]:"support opp f (support opp f s) = support opp f s"
+ unfolding support_def by auto
+
+lemma iterate_support[simp]: "iterate opp (support opp f s) f = iterate opp s f"
+ unfolding iterate_def support_support by auto
+
+lemma iterate_expand_cases:
+ "iterate opp s f = (if finite(support opp f s) then iterate opp (support opp f s) f else neutral opp)"
+ apply(cases) apply(subst if_P,assumption) unfolding iterate_def support_support fold'_def by auto
+
+lemma iterate_image: assumes "monoidal opp" "inj_on f s"
+ shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)"
+proof- have *:"\<And>s. finite s \<Longrightarrow> \<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<longrightarrow> x = y \<Longrightarrow>
+ iterate opp (f ` s) g = iterate opp s (g \<circ> f)"
+ proof- case goal1 show ?case using goal1
+ proof(induct s) case empty thus ?case using assms(1) by auto
+ next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)]
+ unfolding if_not_P[OF insert(2)] apply(subst insert(3)[THEN sym])
+ unfolding image_insert defer apply(subst iterate_insert[OF assms(1)])
+ apply(rule finite_imageI insert)+ apply(subst if_not_P)
+ unfolding image_iff o_def using insert(2,4) by auto
+ qed qed
+ show ?thesis
+ apply(cases "finite (support opp g (f ` s))")
+ apply(subst (1) iterate_support[THEN sym],subst (2) iterate_support[THEN sym])
+ unfolding support_clauses apply(rule *)apply(rule finite_imageD,assumption) unfolding inj_on_def[symmetric]
+ apply(rule subset_inj_on[OF assms(2) support_subset])+
+ apply(subst iterate_expand_cases) unfolding support_clauses apply(simp only: if_False)
+ apply(subst iterate_expand_cases) apply(subst if_not_P) by auto qed
+
+
+(* This lemma about iterations comes up in a few places. *)
+lemma iterate_nonzero_image_lemma:
+ assumes "monoidal opp" "finite s" "g(a) = neutral opp"
+ "\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = neutral opp"
+ shows "iterate opp {f x | x. x \<in> s \<and> f x \<noteq> a} g = iterate opp s (g \<circ> f)"
+proof- have *:"{f x |x. x \<in> s \<and> ~(f x = a)} = f ` {x. x \<in> s \<and> ~(f x = a)}" by auto
+ have **:"support opp (g \<circ> f) {x \<in> s. f x \<noteq> a} = support opp (g \<circ> f) s"
+ unfolding support_def using assms(3) by auto
+ show ?thesis unfolding *
+ apply(subst iterate_support[THEN sym]) unfolding support_clauses
+ apply(subst iterate_image[OF assms(1)]) defer
+ apply(subst(2) iterate_support[THEN sym]) apply(subst **)
+ unfolding inj_on_def using assms(3,4) unfolding support_def by auto qed
+
+lemma iterate_eq_neutral:
+ assumes "monoidal opp" "\<forall>x \<in> s. (f(x) = neutral opp)"
+ shows "(iterate opp s f = neutral opp)"
+proof- have *:"support opp f s = {}" unfolding support_def using assms(2) by auto
+ show ?thesis apply(subst iterate_support[THEN sym])
+ unfolding * using assms(1) by auto qed
+
+lemma iterate_op: assumes "monoidal opp" "finite s"
+ shows "iterate opp s (\<lambda>x. opp (f x) (g x)) = opp (iterate opp s f) (iterate opp s g)" using assms(2)
+proof(induct s) case empty thus ?case unfolding iterate_insert[OF assms(1)] using assms(1) by auto
+next case (insert x F) show ?case unfolding iterate_insert[OF assms(1) insert(1)] if_not_P[OF insert(2)] insert(3)
+ unfolding monoidal_ac[OF assms(1)] by(rule refl) qed
+
+lemma iterate_eq: assumes "monoidal opp" "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
+ shows "iterate opp s f = iterate opp s g"
+proof- have *:"support opp g s = support opp f s"
+ unfolding support_def using assms(2) by auto
+ show ?thesis
+ proof(cases "finite (support opp f s)")
+ case False thus ?thesis apply(subst iterate_expand_cases,subst(2) iterate_expand_cases)
+ unfolding * by auto
+ next def su \<equiv> "support opp f s"
+ case True note support_subset[of opp f s]
+ thus ?thesis apply- apply(subst iterate_support[THEN sym],subst(2) iterate_support[THEN sym]) unfolding * using True
+ unfolding su_def[symmetric]
+ proof(induct su) case empty show ?case by auto
+ next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)]
+ unfolding if_not_P[OF insert(2)] apply(subst insert(3))
+ defer apply(subst assms(2)[of x]) using insert by auto qed qed qed
+
+lemma nonempty_witness: assumes "s \<noteq> {}" obtains x where "x \<in> s" using assms by auto
+
+lemma operative_division: fixes f::"(real^'n) set \<Rightarrow> 'a"
+ assumes "monoidal opp" "operative opp f" "d division_of {a..b}"
+ shows "iterate opp d f = f {a..b}"
+proof- def C \<equiv> "card (division_points {a..b} d)" thus ?thesis using assms
+ proof(induct C arbitrary:a b d rule:full_nat_induct)
+ case goal1
+ { presume *:"content {a..b} \<noteq> 0 \<Longrightarrow> ?case"
+ thus ?case apply-apply(cases) defer apply assumption
+ proof- assume as:"content {a..b} = 0"
+ show ?case unfolding operativeD(1)[OF assms(2) as] apply(rule iterate_eq_neutral[OF goal1(2)])
+ proof fix x assume x:"x\<in>d"
+ then guess u v apply(drule_tac division_ofD(4)[OF goal1(4)]) by(erule exE)+
+ thus "f x = neutral opp" using division_of_content_0[OF as goal1(4)]
+ using operativeD(1)[OF assms(2)] x by auto
+ qed qed }
+ assume "content {a..b} \<noteq> 0" note ab = this[unfolded content_lt_nz[THEN sym] content_pos_lt_eq]
+ hence ab':"\<forall>i. a$i \<le> b$i" by (auto intro!: less_imp_le) show ?case
+ proof(cases "division_points {a..b} d = {}")
+ case True have d':"\<forall>i\<in>d. \<exists>u v. i = {u..v} \<and>
+ (\<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)"
+ unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule)
+ apply(rule_tac x=a in exI,rule_tac x=b in exI) apply(rule,rule refl) apply(rule)
+ proof- fix u v j assume as:"{u..v} \<in> d" note division_ofD(3)[OF goal1(4) this]
+ hence uv:"\<forall>i. u$i \<le> v$i" "u$j \<le> v$j" unfolding interval_ne_empty by auto
+ 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
+ have "(j, u$j) \<notin> division_points {a..b} d"
+ "(j, v$j) \<notin> division_points {a..b} d" using True by auto
+ note this[unfolded de_Morgan_conj division_points_def mem_Collect_eq split_conv interval_bounds[OF ab'] bex_simps]
+ note *[OF this(1)] *[OF this(2)] note this[unfolded interval_bounds[OF uv(1)]]
+ moreover have "a$j \<le> u$j" "v$j \<le> b$j" using division_ofD(2,2,3)[OF goal1(4) as]
+ unfolding subset_eq apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE)
+ unfolding interval_ne_empty mem_interval by auto
+ 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"
+ unfolding not_less de_Morgan_disj using ab[rule_format,of j] uv(2) by auto
+ qed have "(1/2) *\<^sub>R (a+b) \<in> {a..b}" unfolding mem_interval using ab by(auto intro!:less_imp_le)
+ note this[unfolded division_ofD(6)[OF goal1(4),THEN sym] Union_iff]
+ then guess i .. note i=this guess u v using d'[rule_format,OF i(1)] apply-by(erule exE conjE)+ note uv=this
+ have "{a..b} \<in> d"
+ proof- { presume "i = {a..b}" thus ?thesis using i by auto }
+ { presume "u = a" "v = b" thus "i = {a..b}" using uv by auto }
+ show "u = a" "v = b" unfolding Cart_eq
+ proof(rule_tac[!] allI) fix j note i(2)[unfolded uv mem_interval,rule_format,of j]
+ thus "u $ j = a $ j" "v $ j = b $ j" using uv(2)[rule_format,of j] by auto
+ qed qed
+ hence *:"d = insert {a..b} (d - {{a..b}})" by auto
+ have "iterate opp (d - {{a..b}}) f = neutral opp" apply(rule iterate_eq_neutral[OF goal1(2)])
+ proof fix x assume x:"x \<in> d - {{a..b}}" hence "x\<in>d" by auto note d'[rule_format,OF this]
+ then guess u v apply-by(erule exE conjE)+ note uv=this
+ have "u\<noteq>a \<or> v\<noteq>b" using x[unfolded uv] by auto
+ then obtain j where "u$j \<noteq> a$j \<or> v$j \<noteq> b$j" unfolding Cart_eq by auto
+ hence "u$j = v$j" using uv(2)[rule_format,of j] by auto
+ hence "content {u..v} = 0" unfolding content_eq_0 apply(rule_tac x=j in exI) by auto
+ thus "f x = neutral opp" unfolding uv(1) by(rule operativeD(1)[OF goal1(3)])
+ qed thus "iterate opp d f = f {a..b}" apply-apply(subst *)
+ apply(subst iterate_insert[OF goal1(2)]) using goal1(2,4) by auto
+ next case False hence "\<exists>x. x\<in>division_points {a..b} d" by auto
+ then guess k c unfolding split_paired_Ex apply- unfolding division_points_def mem_Collect_eq split_conv
+ by(erule exE conjE)+ note kc=this[unfolded interval_bounds[OF ab']]
+ from this(3) guess j .. note j=this
+ def d1 \<equiv> "{l \<inter> {x. x$k \<le> c} | l. l \<in> d \<and> l \<inter> {x. x$k \<le> c} \<noteq> {}}"
+ def d2 \<equiv> "{l \<inter> {x. x$k \<ge> c} | l. l \<in> d \<and> l \<inter> {x. x$k \<ge> c} \<noteq> {}}"
+ 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)"
+ note division_points_psubset[OF goal1(4) ab kc(1-2) j]
+ note psubset_card_mono[OF _ this(1)] psubset_card_mono[OF _ this(2)]
+ 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})"
+ apply- unfolding interval_split apply(rule_tac[!] goal1(1)[rule_format])
+ using division_split[OF goal1(4), where k=k and c=c]
+ unfolding interval_split d1_def[symmetric] d2_def[symmetric] unfolding goal1(2) Suc_le_mono
+ using goal1(2-3) using division_points_finite[OF goal1(4)] by auto
+ have "f {a..b} = opp (iterate opp d1 f) (iterate opp d2 f)" (is "_ = ?prev")
+ unfolding * apply(rule operativeD(2)) using goal1(3) .
+ also have "iterate opp d1 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x$k \<le> c}))"
+ unfolding d1_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])
+ unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+
+ unfolding empty_as_interval[THEN sym] apply(rule content_empty)
+ 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"
+ from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this
+ show "f (l \<inter> {x. x $ k \<le> c}) = neutral opp" unfolding l interval_split
+ apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym] apply(rule division_split_left_inj)
+ apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as)+
+ qed also have "iterate opp d2 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x$k \<ge> c}))"
+ unfolding d2_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])
+ unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+
+ unfolding empty_as_interval[THEN sym] apply(rule content_empty)
+ 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"
+ from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this
+ show "f (l \<inter> {x. x $ k \<ge> c}) = neutral opp" unfolding l interval_split
+ apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym] apply(rule division_split_right_inj)
+ apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as)+
+ 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}))"
+ unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule,rule operativeD(2)) using goal1(3) .
+ 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})))
+ = iterate opp d f" apply(subst(3) iterate_eq[OF _ *[rule_format]]) prefer 3
+ apply(rule iterate_op[THEN sym]) using goal1 by auto
+ finally show ?thesis by auto
+ qed qed qed
+
+lemma iterate_image_nonzero: assumes "monoidal opp"
+ "finite s" "\<forall>x\<in>s. \<forall>y\<in>s. ~(x = y) \<and> f x = f y \<longrightarrow> g(f x) = neutral opp"
+ shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)" using assms
+proof(induct rule:finite_subset_induct[OF assms(2) subset_refl])
+ case goal1 show ?case using assms(1) by auto
+next case goal2 have *:"\<And>x y. y = neutral opp \<Longrightarrow> x = opp y x" using assms(1) by auto
+ show ?case unfolding image_insert apply(subst iterate_insert[OF assms(1)])
+ apply(rule finite_imageI goal2)+
+ apply(cases "f a \<in> f ` F") unfolding if_P if_not_P apply(subst goal2(4)[OF assms(1) goal2(1)]) defer
+ apply(subst iterate_insert[OF assms(1) goal2(1)]) defer
+ apply(subst iterate_insert[OF assms(1) goal2(1)])
+ unfolding if_not_P[OF goal2(3)] defer unfolding image_iff defer apply(erule bexE)
+ apply(rule *) unfolding o_def apply(rule_tac y=x in goal2(7)[rule_format])
+ using goal2 unfolding o_def by auto qed
+
+lemma operative_tagged_division: assumes "monoidal opp" "operative opp f" "d tagged_division_of {a..b}"
+ shows "iterate(opp) d (\<lambda>(x,l). f l) = f {a..b}"
+proof- have *:"(\<lambda>(x,l). f l) = (f o snd)" unfolding o_def by(rule,auto) note assm = tagged_division_ofD[OF assms(3)]
+ have "iterate(opp) d (\<lambda>(x,l). f l) = iterate opp (snd ` d) f" unfolding *
+ apply(rule iterate_image_nonzero[THEN sym,OF assms(1)]) apply(rule tagged_division_of_finite assms)+
+ unfolding Ball_def split_paired_All snd_conv apply(rule,rule,rule,rule,rule,rule,rule,erule conjE)
+ proof- fix a b aa ba assume as:"(a, b) \<in> d" "(aa, ba) \<in> d" "(a, b) \<noteq> (aa, ba)" "b = ba"
+ guess u v using assm(4)[OF as(1)] apply-by(erule exE)+ note uv=this
+ show "f b = neutral opp" unfolding uv apply(rule operativeD(1)[OF assms(2)])
+ unfolding content_eq_0_interior using tagged_division_ofD(5)[OF assms(3) as(1-3)]
+ unfolding as(4)[THEN sym] uv by auto
+ qed also have "\<dots> = f {a..b}"
+ using operative_division[OF assms(1-2) division_of_tagged_division[OF assms(3)]] .
+ finally show ?thesis . qed
+
+subsection {* Additivity of content. *}
+
+lemma setsum_iterate:assumes "finite s" shows "setsum f s = iterate op + s f"
+proof- have *:"setsum f s = setsum f (support op + f s)"
+ apply(rule setsum_mono_zero_right)
+ unfolding support_def neutral_monoid using assms by auto
+ thus ?thesis unfolding * setsum_def iterate_def fold_image_def fold'_def
+ unfolding neutral_monoid . qed
+
+lemma additive_content_division: assumes "d division_of {a..b}"
+ shows "setsum content d = content({a..b})"
+ unfolding operative_division[OF monoidal_monoid operative_content assms,THEN sym]
+ apply(subst setsum_iterate) using assms by auto
+
+lemma additive_content_tagged_division:
+ assumes "d tagged_division_of {a..b}"
+ shows "setsum (\<lambda>(x,l). content l) d = content({a..b})"
+ unfolding operative_tagged_division[OF monoidal_monoid operative_content assms,THEN sym]
+ apply(subst setsum_iterate) using assms by auto
+
+subsection {* Finally, the integral of a constant\<forall> *}
+
+lemma has_integral_const[intro]:
+ "((\<lambda>x. c) has_integral (content({a..b::real^'n}) *\<^sub>R c)) ({a..b})"
+ unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI)
+ apply(rule,rule gauge_trivial)apply(rule,rule,erule conjE)
+ unfolding split_def apply(subst scaleR_left.setsum[THEN sym, unfolded o_def])
+ defer apply(subst additive_content_tagged_division[unfolded split_def]) apply assumption by auto
+
+subsection {* Bounds on the norm of Riemann sums and the integral itself. *}
+
+lemma dsum_bound: assumes "p division_of {a..b}" "norm(c) \<le> e"
+ shows "norm(setsum (\<lambda>l. content l *\<^sub>R c) p) \<le> e * content({a..b})" (is "?l \<le> ?r")
+ apply(rule order_trans,rule setsum_norm) defer unfolding norm_scaleR setsum_left_distrib[THEN sym]
+ apply(rule order_trans[OF mult_left_mono],rule assms,rule setsum_abs_ge_zero)
+ apply(subst real_mult_commute) apply(rule mult_left_mono)
+ apply(rule order_trans[of _ "setsum content p"]) apply(rule eq_refl,rule setsum_cong2)
+ apply(subst abs_of_nonneg) unfolding additive_content_division[OF assms(1)]
+proof- from order_trans[OF norm_ge_zero[of c] assms(2)] show "0 \<le> e" .
+ fix x assume "x\<in>p" from division_ofD(4)[OF assms(1) this] guess u v apply-by(erule exE)+
+ thus "0 \<le> content x" using content_pos_le by auto
+qed(insert assms,auto)
+
+lemma rsum_bound: assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. norm(f x) \<le> e"
+ shows "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p) \<le> e * content({a..b})"
+proof(cases "{a..b} = {}") case True
+ show ?thesis using assms(1) unfolding True tagged_division_of_trivial by auto
+next case False show ?thesis
+ apply(rule order_trans,rule setsum_norm) defer unfolding split_def norm_scaleR
+ apply(rule order_trans[OF setsum_mono]) apply(rule mult_left_mono[OF _ abs_ge_zero, of _ e]) defer
+ unfolding setsum_left_distrib[THEN sym] apply(subst real_mult_commute) apply(rule mult_left_mono)
+ apply(rule order_trans[of _ "setsum (content \<circ> snd) p"]) apply(rule eq_refl,rule setsum_cong2)
+ apply(subst o_def, rule abs_of_nonneg)
+ proof- show "setsum (content \<circ> snd) p \<le> content {a..b}" apply(rule eq_refl)
+ unfolding additive_content_tagged_division[OF assms(1),THEN sym] split_def by auto
+ guess w using nonempty_witness[OF False] .
+ thus "e\<ge>0" apply-apply(rule order_trans) defer apply(rule assms(2)[rule_format],assumption) by auto
+ fix xk assume *:"xk\<in>p" guess x k using surj_pair[of xk] apply-by(erule exE)+ note xk = this *[unfolded this]
+ from tagged_division_ofD(4)[OF assms(1) xk(2)] guess u v apply-by(erule exE)+ note uv=this
+ show "0\<le> content (snd xk)" unfolding xk snd_conv uv by(rule content_pos_le)
+ 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
+ qed(insert assms,auto) qed
+
+lemma rsum_diff_bound:
+ assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e"
+ 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})"
+ apply(rule order_trans[OF _ rsum_bound[OF assms]]) apply(rule eq_refl) apply(rule arg_cong[where f=norm])
+ unfolding setsum_subtractf[THEN sym] apply(rule setsum_cong2) unfolding scaleR.diff_right by auto
+
+lemma has_integral_bound: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
+ assumes "0 \<le> B" "(f has_integral i) ({a..b})" "\<forall>x\<in>{a..b}. norm(f x) \<le> B"
+ shows "norm i \<le> B * content {a..b}"
+proof- let ?P = "content {a..b} > 0" { presume "?P \<Longrightarrow> ?thesis"
+ thus ?thesis proof(cases ?P) case False
+ hence *:"content {a..b} = 0" using content_lt_nz by auto
+ hence **:"i = 0" using assms(2) apply(subst has_integral_null_eq[THEN sym]) by auto
+ show ?thesis unfolding * ** using assms(1) by auto
+ qed auto } assume ab:?P
+ { presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }
+ assume "\<not> ?thesis" hence *:"norm i - B * content {a..b} > 0" by auto
+ from assms(2)[unfolded has_integral,rule_format,OF *] guess d apply-by(erule exE conjE)+ note d=this[rule_format]
+ from fine_division_exists[OF this(1), of a b] guess p . note p=this
+ have *:"\<And>s B. norm s \<le> B \<Longrightarrow> \<not> (norm (s - i) < norm i - B)"
+ proof- case goal1 thus ?case unfolding not_less
+ using norm_triangle_sub[of i s] unfolding norm_minus_commute by auto
+ qed show False using d(2)[OF conjI[OF p]] *[OF rsum_bound[OF p(1) assms(3)]] by auto qed
+
+subsection {* Similar theorems about relationship among components. *}
+
+lemma rsum_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
+ assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. (f x)$i \<le> (g x)$i"
+ 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"
+ unfolding setsum_component apply(rule setsum_mono)
+ apply(rule mp) defer apply assumption apply(induct_tac x,rule) unfolding split_conv
+proof- fix a b assume ab:"(a,b) \<in> p" note assm = tagged_division_ofD(2-4)[OF assms(1) ab]
+ from this(3) guess u v apply-by(erule exE)+ note b=this
+ show "(content b *\<^sub>R f a) $ i \<le> (content b *\<^sub>R g a) $ i" unfolding b
+ unfolding Cart_nth.scaleR real_scaleR_def apply(rule mult_left_mono)
+ defer apply(rule content_pos_le,rule assms(2)[rule_format]) using assm by auto qed
+
+lemma has_integral_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
+ assumes "(f has_integral i) s" "(g has_integral j) s" "\<forall>x\<in>s. (f x)$k \<le> (g x)$k"
+ shows "i$k \<le> j$k"
+proof- have lem:"\<And>a b g i j. \<And>f::real^'n \<Rightarrow> real^'m. (f has_integral i) ({a..b}) \<Longrightarrow>
+ (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"
+ proof(rule ccontr) case goal1 hence *:"0 < (i$k - j$k) / 3" by auto
+ guess d1 using goal1(1)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d1=this[rule_format]
+ guess d2 using goal1(2)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d2=this[rule_format]
+ guess p using fine_division_exists[OF gauge_inter[OF d1(1) d2(1)], of a b] unfolding fine_inter .
+ note p = this(1) conjunctD2[OF this(2)] note le_less_trans[OF component_le_norm, of _ _ k]
+ note this[OF d1(2)[OF conjI[OF p(1,2)]]] this[OF d2(2)[OF conjI[OF p(1,3)]]]
+ thus False unfolding Cart_nth.diff using rsum_component_le[OF p(1) goal1(3)] by smt
+ qed let ?P = "\<exists>a b. s = {a..b}"
+ { presume "\<not> ?P \<Longrightarrow> ?thesis" thus ?thesis proof(cases ?P)
+ case True then guess a b apply-by(erule exE)+ note s=this
+ show ?thesis apply(rule lem) using assms[unfolded s] by auto
+ qed auto } assume as:"\<not> ?P"
+ { presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }
+ assume "\<not> i$k \<le> j$k" hence ij:"(i$k - j$k) / 3 > 0" by auto
+ note has_integral_altD[OF _ as this] from this[OF assms(1)] this[OF assms(2)] guess B1 B2 . note B=this[rule_format]
+ have "bounded (ball 0 B1 \<union> ball (0::real^'n) B2)" unfolding bounded_Un by(rule conjI bounded_ball)+
+ from bounded_subset_closed_interval[OF this] guess a b apply- by(erule exE)+
+ note ab = conjunctD2[OF this[unfolded Un_subset_iff]]
+ guess w1 using B(2)[OF ab(1)] .. note w1=conjunctD2[OF this]
+ guess w2 using B(4)[OF ab(2)] .. note w2=conjunctD2[OF this]
+ 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*)
+ note le_less_trans[OF component_le_norm[of _ k]] note this[OF w1(2)] this[OF w2(2)] moreover
+ have "w1$k \<le> w2$k" apply(rule lem[OF w1(1) w2(1)]) using assms by auto ultimately
+ show False unfolding Cart_nth.diff by(rule *) qed
+
+lemma integral_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
+ assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. (f x)$k \<le> (g x)$k"
+ shows "(integral s f)$k \<le> (integral s g)$k"
+ apply(rule has_integral_component_le) using integrable_integral assms by auto
+
+lemma has_integral_dest_vec1_le: fixes f::"real^'n \<Rightarrow> real^1"
+ assumes "(f has_integral i) s" "(g has_integral j) s" "\<forall>x\<in>s. f x \<le> g x"
+ shows "dest_vec1 i \<le> dest_vec1 j" apply(rule has_integral_component_le[OF assms(1-2)])
+ using assms(3) unfolding vector_le_def by auto
+
+lemma integral_dest_vec1_le: fixes f::"real^'n \<Rightarrow> real^1"
+ assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. f x \<le> g x"
+ shows "dest_vec1(integral s f) \<le> dest_vec1(integral s g)"
+ apply(rule has_integral_dest_vec1_le) apply(rule_tac[1-2] integrable_integral) using assms by auto
+
+lemma has_integral_component_pos: fixes f::"real^'n \<Rightarrow> real^'m"
+ assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> (f x)$k" shows "0 \<le> i$k"
+ using has_integral_component_le[OF has_integral_0 assms(1)] using assms(2) by auto
+
+lemma integral_component_pos: fixes f::"real^'n \<Rightarrow> real^'m"
+ assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> (f x)$k" shows "0 \<le> (integral s f)$k"
+ apply(rule has_integral_component_pos) using assms by auto
+
+lemma has_integral_dest_vec1_pos: fixes f::"real^'n \<Rightarrow> real^1"
+ assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> i"
+ using has_integral_component_pos[OF assms(1), of 1]
+ using assms(2) unfolding vector_le_def by auto
+
+lemma integral_dest_vec1_pos: fixes f::"real^'n \<Rightarrow> real^1"
+ assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> integral s f"
+ apply(rule has_integral_dest_vec1_pos) using assms by auto
+
+lemma has_integral_component_neg: fixes f::"real^'n \<Rightarrow> real^'m"
+ assumes "(f has_integral i) s" "\<forall>x\<in>s. (f x)$k \<le> 0" shows "i$k \<le> 0"
+ using has_integral_component_le[OF assms(1) has_integral_0] assms(2) by auto
+
+lemma has_integral_dest_vec1_neg: fixes f::"real^'n \<Rightarrow> real^1"
+ assumes "(f has_integral i) s" "\<forall>x\<in>s. f x \<le> 0" shows "i \<le> 0"
+ using has_integral_component_neg[OF assms(1),of 1] using assms(2) by auto
+
+lemma has_integral_component_lbound:
+ 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"
+ using has_integral_component_le[OF has_integral_const assms(1),of "(\<chi> i. B)" k] assms(2)
+ unfolding Cart_lambda_beta vector_scaleR_component by(auto simp add:field_simps)
+
+lemma has_integral_component_ubound:
+ assumes "(f has_integral i) {a..b}" "\<forall>x\<in>{a..b}. f x$k \<le> B"
+ shows "i$k \<le> B * content({a..b})"
+ using has_integral_component_le[OF assms(1) has_integral_const, of k "vec B"]
+ unfolding vec_component Cart_nth.scaleR using assms(2) by(auto simp add:field_simps)
+
+lemma integral_component_lbound:
+ assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. B \<le> f(x)$k"
+ shows "B * content({a..b}) \<le> (integral({a..b}) f)$k"
+ apply(rule has_integral_component_lbound) using assms unfolding has_integral_integral by auto
+
+lemma integral_component_ubound:
+ assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. f(x)$k \<le> B"
+ shows "(integral({a..b}) f)$k \<le> B * content({a..b})"
+ apply(rule has_integral_component_ubound) using assms unfolding has_integral_integral by auto
+
+subsection {* Uniform limit of integrable functions is integrable. *}
+
+lemma real_arch_invD:
+ "0 < (e::real) \<Longrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
+ by(subst(asm) real_arch_inv)
+
+lemma integrable_uniform_limit: fixes f::"real^'n \<Rightarrow> 'a::banach"
+ assumes "\<forall>e>0. \<exists>g. (\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e) \<and> g integrable_on {a..b}"
+ shows "f integrable_on {a..b}"
+proof- { presume *:"content {a..b} > 0 \<Longrightarrow> ?thesis"
+ show ?thesis apply cases apply(rule *,assumption)
+ unfolding content_lt_nz integrable_on_def using has_integral_null by auto }
+ assume as:"content {a..b} > 0"
+ have *:"\<And>P. \<forall>e>(0::real). P e \<Longrightarrow> \<forall>n::nat. P (inverse (real n+1))" by auto
+ from choice[OF *[OF assms]] guess g .. note g=conjunctD2[OF this[rule_format],rule_format]
+ from choice[OF allI[OF g(2)[unfolded integrable_on_def], of "\<lambda>x. x"]] guess i .. note i=this[rule_format]
+
+ have "Cauchy i" unfolding Cauchy_def
+ proof(rule,rule) fix e::real assume "e>0"
+ hence "e / 4 / content {a..b} > 0" using as by(auto simp add:field_simps)
+ then guess M apply-apply(subst(asm) real_arch_inv) by(erule exE conjE)+ note M=this
+ 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)
+ proof- case goal1 have "e/4>0" using `e>0` by auto note * = i[unfolded has_integral,rule_format,OF this]
+ from *[of m] guess gm apply-by(erule conjE exE)+ note gm=this[rule_format]
+ from *[of n] guess gn apply-by(erule conjE exE)+ note gn=this[rule_format]
+ from fine_division_exists[OF gauge_inter[OF gm(1) gn(1)], of a b] guess p . note p=this
+ 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"
+ proof- case goal1 have "norm (i1 - i2) \<le> norm (i1 - s1) + norm (s1 - s2) + norm (s2 - i2)"
+ using norm_triangle_ineq[of "i1 - s1" "s1 - i2"]
+ using norm_triangle_ineq[of "s1 - s2" "s2 - i2"] by(auto simp add:group_simps)
+ also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:group_simps)
+ finally show ?case .
+ qed
+ show ?case unfolding vector_dist_norm apply(rule lem2) defer
+ apply(rule gm(2)[OF conjI[OF p(1)]],rule_tac[2] gn(2)[OF conjI[OF p(1)]])
+ using conjunctD2[OF p(2)[unfolded fine_inter]] apply- apply assumption+ apply(rule order_trans)
+ apply(rule rsum_diff_bound[OF p(1), where e="2 / real M"])
+ proof show "2 / real M * content {a..b} \<le> e / 2" unfolding divide_inverse
+ using M as by(auto simp add:field_simps)
+ fix x assume x:"x \<in> {a..b}"
+ have "norm (f x - g n x) + norm (f x - g m x) \<le> inverse (real n + 1) + inverse (real m + 1)"
+ using g(1)[OF x, of n] g(1)[OF x, of m] by auto
+ also have "\<dots> \<le> inverse (real M) + inverse (real M)" apply(rule add_mono)
+ apply(rule_tac[!] le_imp_inverse_le) using goal1 M by auto
+ also have "\<dots> = 2 / real M" unfolding real_divide_def by auto
+ finally show "norm (g n x - g m x) \<le> 2 / real M"
+ using norm_triangle_le[of "g n x - f x" "f x - g m x" "2 / real M"]
+ by(auto simp add:group_simps simp add:norm_minus_commute)
+ qed qed qed
+ from this[unfolded convergent_eq_cauchy[THEN sym]] guess s .. note s=this
+
+ show ?thesis unfolding integrable_on_def apply(rule_tac x=s in exI) unfolding has_integral
+ proof(rule,rule)
+ case goal1 hence *:"e/3 > 0" by auto
+ from s[unfolded Lim_sequentially,rule_format,OF this] guess N1 .. note N1=this
+ from goal1 as have "e / 3 / content {a..b} > 0" by(auto simp add:field_simps)
+ from real_arch_invD[OF this] guess N2 apply-by(erule exE conjE)+ note N2=this
+ from i[of "N1 + N2",unfolded has_integral,rule_format,OF *] guess g' .. note g'=conjunctD2[OF this,rule_format]
+ 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"
+ proof- case goal1 have "norm (sf - s) \<le> norm (sf - sg) + norm (sg - i) + norm (i - s)"
+ using norm_triangle_ineq[of "sf - sg" "sg - s"]
+ using norm_triangle_ineq[of "sg - i" " i - s"] by(auto simp add:group_simps)
+ also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:group_simps)
+ finally show ?case .
+ qed
+ show ?case apply(rule_tac x=g' in exI) apply(rule,rule g')
+ proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> g' fine p" note * = g'(2)[OF this]
+ show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - s) < e" apply-apply(rule lem[OF _ _ *])
+ apply(rule order_trans,rule rsum_diff_bound[OF p[THEN conjunct1]]) apply(rule,rule g,assumption)
+ proof- have "content {a..b} < e / 3 * (real N2)"
+ using N2 unfolding inverse_eq_divide using as by(auto simp add:field_simps)
+ hence "content {a..b} < e / 3 * (real (N1 + N2) + 1)"
+ apply-apply(rule less_le_trans,assumption) using `e>0` by auto
+ thus "inverse (real (N1 + N2) + 1) * content {a..b} \<le> e / 3"
+ unfolding inverse_eq_divide by(auto simp add:field_simps)
+ show "norm (i (N1 + N2) - s) < e / 3" by(rule N1[rule_format,unfolded vector_dist_norm],auto)
+ qed qed qed qed
+
+subsection {* Negligible sets. *}
+
+definition "indicator s \<equiv> (\<lambda>x. if x \<in> s then 1 else (0::real))"
+
+lemma dest_vec1_indicator:
+ "indicator s x = (if x \<in> s then 1 else 0)" unfolding indicator_def by auto
+
+lemma indicator_pos_le[intro]: "0 \<le> (indicator s x)" unfolding indicator_def by auto
+
+lemma indicator_le_1[intro]: "(indicator s x) \<le> 1" unfolding indicator_def by auto
+
+lemma dest_vec1_indicator_abs_le_1: "abs(indicator s x) \<le> 1"
+ unfolding indicator_def by auto
+
+definition "negligible (s::(real^'n) set) \<equiv> (\<forall>a b. ((indicator s) has_integral 0) {a..b})"
+
+lemma indicator_simps[simp]:"x\<in>s \<Longrightarrow> indicator s x = 1" "x\<notin>s \<Longrightarrow> indicator s x = 0"
+ unfolding indicator_def by auto
+
+subsection {* Negligibility of hyperplane. *}
+
+lemma vsum_nonzero_image_lemma:
+ assumes "finite s" "g(a) = 0"
+ "\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = 0"
+ shows "setsum g {f x |x. x \<in> s \<and> f x \<noteq> a} = setsum (g o f) s"
+ unfolding setsum_iterate[OF assms(1)] apply(subst setsum_iterate) defer
+ apply(rule iterate_nonzero_image_lemma) apply(rule assms monoidal_monoid)+
+ unfolding assms using neutral_add unfolding neutral_add using assms by auto
+
+lemma interval_doublesplit: shows "{a..b} \<inter> {x . abs(x$k - c) \<le> (e::real)} =
+ {(\<chi> i. if i = k then max (a$k) (c - e) else a$i) .. (\<chi> i. if i = k then min (b$k) (c + e) else b$i)}"
+proof- have *:"\<And>x c e::real. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
+ have **:"\<And>s P Q. s \<inter> {x. P x \<and> Q x} = (s \<inter> {x. Q x}) \<inter> {x. P x}" by blast
+ show ?thesis unfolding * ** interval_split by(rule refl) qed
+
+lemma division_doublesplit: assumes "p division_of {a..b::real^'n}"
+ shows "{l \<inter> {x. abs(x$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})"
+proof- have *:"\<And>x c. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
+ have **:"\<And>p q p' q'. p division_of q \<Longrightarrow> p = p' \<Longrightarrow> q = q' \<Longrightarrow> p' division_of q'" by auto
+ note division_split(1)[OF assms, where c="c+e" and k=k,unfolded interval_split]
+ note division_split(2)[OF this, where c="c-e" and k=k]
+ thus ?thesis apply(rule **) unfolding interval_doublesplit unfolding * unfolding interval_split interval_doublesplit
+ apply(rule set_ext) unfolding mem_Collect_eq apply rule apply(erule conjE exE)+ apply(rule_tac x=la in exI) defer
+ apply(erule conjE exE)+ apply(rule_tac x="l \<inter> {x. c + e \<ge> x $ k}" in exI) apply rule defer apply rule
+ apply(rule_tac x=l in exI) by blast+ qed
+
+lemma content_doublesplit: assumes "0 < e"
+ obtains d where "0 < d" "content({a..b} \<inter> {x. abs(x$k - c) \<le> d}) < e"
+proof(cases "content {a..b} = 0")
+ case True show ?thesis apply(rule that[of 1]) defer unfolding interval_doublesplit
+ apply(rule le_less_trans[OF content_subset]) defer apply(subst True)
+ unfolding interval_doublesplit[THEN sym] using assms by auto
+next case False def d \<equiv> "e / 3 / setprod (\<lambda>i. b$i - a$i) (UNIV - {k})"
+ note False[unfolded content_eq_0 not_ex not_le, rule_format]
+ hence prod0:"0 < setprod (\<lambda>i. b$i - a$i) (UNIV - {k})" apply-apply(rule setprod_pos) by smt
+ hence "d > 0" unfolding d_def using assms by(auto simp add:field_simps) thus ?thesis
+ proof(rule that[of d]) have *:"UNIV = insert k (UNIV - {k})" by auto
+ have **:"{a..b} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d} \<noteq> {} \<Longrightarrow>
+ (\<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)
+ = (\<Prod>i\<in>UNIV - {k}. b$i - a$i)" apply(rule setprod_cong,rule refl)
+ unfolding interval_doublesplit interval_eq_empty not_ex not_less unfolding interval_bounds by auto
+ 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)
+ unfolding if_not_P apply(subst *) apply(subst setprod_insert) unfolding **
+ unfolding interval_doublesplit interval_eq_empty not_ex not_less unfolding interval_bounds unfolding Cart_lambda_beta if_P[OF refl]
+ proof- have "(min (b $ k) (c + d) - max (a $ k) (c - d)) \<le> 2 * d" by auto
+ also have "... < e / (\<Prod>i\<in>UNIV - {k}. b $ i - a $ i)" unfolding d_def using assms prod0 by(auto simp add:field_simps)
+ finally show "(min (b $ k) (c + d) - max (a $ k) (c - d)) * (\<Prod>i\<in>UNIV - {k}. b $ i - a $ i) < e"
+ unfolding pos_less_divide_eq[OF prod0] . qed auto qed qed
+
+lemma negligible_standard_hyperplane[intro]: "negligible {x. x$k = (c::real)}"
+ unfolding negligible_def has_integral apply(rule,rule,rule,rule)
+proof- case goal1 from content_doublesplit[OF this,of a b k c] guess d . note d=this let ?i = "indicator {x. x$k = c}"
+ show ?case apply(rule_tac x="\<lambda>x. ball x d" in exI) apply(rule,rule gauge_ball,rule d)
+ proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> (\<lambda>x. ball x d) fine p"
+ have *:"(\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. abs(x$k - c) \<le> d}) *\<^sub>R ?i x)"
+ apply(rule setsum_cong2) unfolding split_paired_all real_scaleR_def mult_cancel_right split_conv
+ apply(cases,rule disjI1,assumption,rule disjI2)
+ proof- fix x l assume as:"(x,l)\<in>p" "?i x \<noteq> 0" hence xk:"x$k = c" unfolding indicator_def apply-by(rule ccontr,auto)
+ show "content l = content (l \<inter> {x. \<bar>x $ k - c\<bar> \<le> d})" apply(rule arg_cong[where f=content])
+ apply(rule set_ext,rule,rule) unfolding mem_Collect_eq
+ proof- fix y assume y:"y\<in>l" note p[THEN conjunct2,unfolded fine_def,rule_format,OF as(1),unfolded split_conv]
+ 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]
+ thus "\<bar>y $ k - c\<bar> \<le> d" unfolding Cart_nth.diff xk by auto
+ qed auto qed
+ note p'= tagged_division_ofD[OF p[THEN conjunct1]] and p''=division_of_tagged_division[OF p[THEN conjunct1]]
+ 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
+ apply(subst abs_of_nonneg) apply(rule setsum_nonneg,rule) unfolding split_paired_all split_conv
+ apply(rule mult_nonneg_nonneg) apply(drule p'(4)) apply(erule exE)+ apply(rule_tac b=b in back_subst)
+ prefer 2 apply(subst(asm) eq_commute) apply assumption
+ apply(subst interval_doublesplit) apply(rule content_pos_le) apply(rule indicator_pos_le)
+ 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}))"
+ apply(rule setsum_mono) unfolding split_paired_all split_conv
+ apply(rule mult_right_le_one_le) apply(drule p'(4)) by(auto simp add:interval_doublesplit intro!:content_pos_le)
+ also have "... < e" apply(subst setsum_over_tagged_division_lemma[OF p[THEN conjunct1]])
+ proof- case goal1 have "content ({u..v} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) \<le> content {u..v}"
+ unfolding interval_doublesplit apply(rule content_subset) unfolding interval_doublesplit[THEN sym] by auto
+ thus ?case unfolding goal1 unfolding interval_doublesplit using content_pos_le by smt
+ 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"
+ apply(rule setsum_nonneg,rule) unfolding mem_Collect_eq image_iff apply(erule exE bexE conjE)+ unfolding split_paired_all
+ 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)"
+ guess u v using p'(4)[OF as(2)] apply-by(erule exE)+ note * = this
+ show "content x \<ge> 0" unfolding as snd_conv * interval_doublesplit by(rule content_pos_le)
+ qed have **:"norm (1::real) \<le> 1" by auto note division_doublesplit[OF p'',unfolded interval_doublesplit]
+ note dsum_bound[OF this **,unfolded interval_doublesplit[THEN sym]]
+ 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)]
+ 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"
+ apply(subst vsum_nonzero_image_lemma[of "snd ` p" content "{}", unfolded o_def,THEN sym])
+ apply(rule finite_imageI p' content_empty)+ unfolding forall_in_division[OF p'']
+ proof(rule,rule,rule,rule,rule,rule,rule,erule conjE) fix m n u v
+ 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}"
+ 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
+ note subset_interior[OF this, unfolded division_ofD(5)[OF p'' as(1-3)] interior_inter[of "{m..n}"]]
+ hence "interior ({m..n} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) = {}" unfolding as Int_absorb by auto
+ thus "content ({m..n} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) = 0" unfolding interval_doublesplit content_eq_0_interior[THEN sym] .
+ qed qed
+ finally show "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) * ?i x) < e" .
+ qed qed qed
+
+subsection {* A technical lemma about "refinement" of division. *}
+
+lemma tagged_division_finer: fixes p::"((real^'n) \<times> ((real^'n) set)) set"
+ assumes "p tagged_division_of {a..b}" "gauge d"
+ 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"
+proof-
+ let ?P = "\<lambda>p. p tagged_partial_division_of {a..b} \<longrightarrow> gauge d \<longrightarrow>
+ (\<exists>q. q tagged_division_of (\<Union>{k. \<exists>x. (x,k) \<in> p}) \<and> d fine q \<and>
+ (\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q))"
+ { have *:"finite p" "p tagged_partial_division_of {a..b}" using assms(1) unfolding tagged_division_of_def by auto
+ presume "\<And>p. finite p \<Longrightarrow> ?P p" from this[rule_format,OF * assms(2)] guess q .. note q=this
+ thus ?thesis apply-apply(rule that[of q]) unfolding tagged_division_ofD[OF assms(1)] by auto
+ } fix p::"((real^'n) \<times> ((real^'n) set)) set" assume as:"finite p"
+ show "?P p" apply(rule,rule) using as proof(induct p)
+ case empty show ?case apply(rule_tac x="{}" in exI) unfolding fine_def by auto
+ next case (insert xk p) guess x k using surj_pair[of xk] apply- by(erule exE)+ note xk=this
+ note tagged_partial_division_subset[OF insert(4) subset_insertI]
+ from insert(3)[OF this insert(5)] guess q1 .. note q1 = conjunctD3[OF this]
+ 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
+ note p = tagged_partial_division_ofD[OF insert(4)]
+ from p(4)[unfolded xk, OF insertI1] guess u v apply-by(erule exE)+ note uv=this
+
+ have "finite {k. \<exists>x. (x, k) \<in> p}"
+ apply(rule finite_subset[of _ "snd ` p"],rule) unfolding subset_eq image_iff mem_Collect_eq
+ apply(erule exE,rule_tac x="(xa,x)" in bexI) using p by auto
+ hence int:"interior {u..v} \<inter> interior (\<Union>{k. \<exists>x. (x, k) \<in> p}) = {}"
+ apply(rule inter_interior_unions_intervals) apply(rule open_interior) apply(rule_tac[!] ballI)
+ unfolding mem_Collect_eq apply(erule_tac[!] exE) apply(drule p(4)[OF insertI2],assumption)
+ apply(rule p(5)) unfolding uv xk apply(rule insertI1,rule insertI2) apply assumption
+ using insert(2) unfolding uv xk by auto
+
+ show ?case proof(cases "{u..v} \<subseteq> d x")
+ case True thus ?thesis apply(rule_tac x="{(x,{u..v})} \<union> q1" in exI) apply rule
+ unfolding * uv apply(rule tagged_division_union,rule tagged_division_of_self)
+ apply(rule p[unfolded xk uv] insertI1)+ apply(rule q1,rule int)
+ apply(rule,rule fine_union,subst fine_def) defer apply(rule q1)
+ unfolding Ball_def split_paired_All split_conv apply(rule,rule,rule,rule)
+ apply(erule insertE) defer apply(rule UnI2) apply(drule q1(3)[rule_format]) unfolding xk uv by auto
+ next case False from fine_division_exists[OF assms(2), of u v] guess q2 . note q2=this
+ show ?thesis apply(rule_tac x="q2 \<union> q1" in exI)
+ apply rule unfolding * uv apply(rule tagged_division_union q2 q1 int fine_union)+
+ unfolding Ball_def split_paired_All split_conv apply rule apply(rule fine_union)
+ apply(rule q1 q2)+ apply(rule,rule,rule,rule) apply(erule insertE)
+ apply(rule UnI2) defer apply(drule q1(3)[rule_format])using False unfolding xk uv by auto
+ qed qed qed
+
+subsection {* Hence the main theorem about negligible sets. *}
+
+lemma finite_product_dependent: assumes "finite s" "\<And>x. x\<in>s\<Longrightarrow> finite (t x)"
+ shows "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" using assms
+proof(induct) case (insert x s)
+ 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
+ show ?case unfolding * apply(rule finite_UnI) using insert by auto qed auto
+
+lemma sum_sum_product: assumes "finite s" "\<forall>i\<in>s. finite (t i)"
+ 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
+proof(induct) case (insert a s)
+ 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
+ show ?case unfolding * apply(subst setsum_Un_disjoint) unfolding setsum_insert[OF insert(1-2)]
+ prefer 4 apply(subst insert(3)) unfolding add_right_cancel
+ 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
+ show "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" apply(rule finite_product_dependent) using insert by auto
+ qed(insert insert, auto) qed auto
+
+lemma has_integral_negligible: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
+ assumes "negligible s" "\<forall>x\<in>(t - s). f x = 0"
+ shows "(f has_integral 0) t"
+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})"
+ let ?f = "(\<lambda>x. if x \<in> t then f x else 0)"
+ show ?thesis apply(rule_tac f="?f" in has_integral_eq) apply(rule) unfolding if_P apply(rule refl)
+ apply(subst has_integral_alt) apply(cases,subst if_P,assumption) unfolding if_not_P
+ proof- assume "\<exists>a b. t = {a..b}" then guess a b apply-by(erule exE)+ note t = this
+ show "(?f has_integral 0) t" unfolding t apply(rule P) using assms(2) unfolding t by auto
+ 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)"
+ apply(safe,rule_tac x=1 in exI,rule) apply(rule zero_less_one,safe) apply(rule_tac x=0 in exI)
+ apply(rule,rule P) using assms(2) by auto
+ qed
+next fix f::"real^'n \<Rightarrow> 'a" and a b::"real^'n" assume assm:"\<forall>x. x \<notin> s \<longrightarrow> f x = 0"
+ show "(f has_integral 0) {a..b}" unfolding has_integral
+ proof(safe) case goal1
+ hence "\<And>n. e / 2 / ((real n+1) * (2 ^ n)) > 0"
+ apply-apply(rule divide_pos_pos) defer apply(rule mult_pos_pos) by(auto simp add:field_simps)
+ note assms(1)[unfolded negligible_def has_integral,rule_format,OF this,of a b] note allI[OF this,of "\<lambda>x. x"]
+ from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format]]
+ show ?case apply(rule_tac x="\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x" in exI)
+ proof safe show "gauge (\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x)" using d(1) unfolding gauge_def by auto
+ fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x) fine p"
+ let ?goal = "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e"
+ { presume "p\<noteq>{} \<Longrightarrow> ?goal" thus ?goal apply(cases "p={}") using goal1 by auto }
+ assume as':"p \<noteq> {}" from real_arch_simple[of "Sup((\<lambda>(x,k). norm(f x)) ` p)"] guess N ..
+ 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
+ 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)"
+ apply(rule,rule tagged_division_finer[OF as(1) d(1)]) by auto
+ from choice[OF this] guess q .. note q=conjunctD3[OF this[rule_format]]
+ have *:"\<And>i. (\<Sum>(x, k)\<in>q i. content k *\<^sub>R indicator s x) \<ge> 0" apply(rule setsum_nonneg,safe)
+ unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) apply(drule tagged_division_ofD(4)[OF q(1)]) by auto
+ 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"
+ proof- case goal1 thus ?case apply-apply(rule setsum_le_included[of s t g snd f]) prefer 4
+ apply safe apply(erule_tac x=x in ballE) apply(erule exE) apply(rule_tac x="(xa,x)" in bexI) by auto qed
+ have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) \<le> setsum (\<lambda>i. (real i + 1) *
+ norm(setsum (\<lambda>(x,k). content k *\<^sub>R indicator s x) (q i))) {0..N+1}"
+ unfolding real_norm_def setsum_right_distrib abs_of_nonneg[OF *] diff_0_right
+ apply(rule order_trans,rule setsum_norm) defer apply(subst sum_sum_product) prefer 3
+ 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
+ fix i a b assume as'':"(a,b) \<in> q i" show "0 \<le> (real i + 1) * (content b *\<^sub>R indicator s a)"
+ unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) defer apply(rule mult_nonneg_nonneg)
+ using tagged_division_ofD(4)[OF q(1) as''] by auto
+ next fix i::nat show "finite (q i)" using q by auto
+ next fix x k assume xk:"(x,k) \<in> p" def n \<equiv> "nat \<lfloor>norm (f x)\<rfloor>"
+ have *:"norm (f x) \<in> (\<lambda>(x, k). norm (f x)) ` p" using xk by auto
+ have nfx:"real n \<le> norm(f x)" "norm(f x) \<le> real n + 1" unfolding n_def by auto
+ hence "n \<in> {0..N + 1}" using N[rule_format,OF *] by auto
+ moreover note as(2)[unfolded fine_def,rule_format,OF xk,unfolded split_conv]
+ note q(3)[rule_format,OF xk,unfolded split_conv,rule_format,OF this] note this[unfolded n_def[symmetric]]
+ moreover have "norm (content k *\<^sub>R f x) \<le> (real n + 1) * (content k * indicator s x)"
+ proof(cases "x\<in>s") case False thus ?thesis using assm by auto
+ next case True have *:"content k \<ge> 0" using tagged_division_ofD(4)[OF as(1) xk] by auto
+ moreover have "content k * norm (f x) \<le> content k * (real n + 1)" apply(rule mult_mono) using nfx * by auto
+ ultimately show ?thesis unfolding abs_mult using nfx True by(auto simp add:field_simps)
+ 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)"
+ 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
+ qed(insert as, auto)
+ also have "... \<le> setsum (\<lambda>i. e / 2 / 2 ^ i) {0..N+1}" apply(rule setsum_mono)
+ proof- case goal1 thus ?case apply(subst mult_commute, subst pos_le_divide_eq[THEN sym])
+ using d(2)[rule_format,of "q i" i] using q[rule_format] by(auto simp add:field_simps)
+ qed also have "... < e * inverse 2 * 2" unfolding real_divide_def setsum_right_distrib[THEN sym]
+ apply(rule mult_strict_left_mono) unfolding power_inverse atLeastLessThanSuc_atLeastAtMost[THEN sym]
+ apply(subst sumr_geometric) using goal1 by auto
+ finally show "?goal" by auto qed qed qed
+
+lemma has_integral_spike: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
+ assumes "negligible s" "(\<forall>x\<in>(t - s). g x = f x)" "(f has_integral y) t"
+ shows "(g has_integral y) t"
+proof- { fix a b::"real^'n" and f g ::"real^'n \<Rightarrow> 'a" and y::'a
+ assume as:"\<forall>x \<in> {a..b} - s. g x = f x" "(f has_integral y) {a..b}"
+ have "((\<lambda>x. f x + (g x - f x)) has_integral (y + 0)) {a..b}" apply(rule has_integral_add[OF as(2)])
+ apply(rule has_integral_negligible[OF assms(1)]) using as by auto
+ hence "(g has_integral y) {a..b}" by auto } note * = this
+ show ?thesis apply(subst has_integral_alt) using assms(2-) apply-apply(rule cond_cases,safe)
+ apply(rule *, assumption+) apply(subst(asm) has_integral_alt) unfolding if_not_P
+ 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)
+ 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
+
+lemma has_integral_spike_eq:
+ assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x"
+ shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
+ apply rule apply(rule_tac[!] has_integral_spike[OF assms(1)]) using assms(2) by auto
+
+lemma integrable_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x" "f integrable_on t"
+ shows "g integrable_on t"
+ using assms unfolding integrable_on_def apply-apply(erule exE)
+ apply(rule,rule has_integral_spike) by fastsimp+
+
+lemma integral_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x"
+ shows "integral t f = integral t g"
+ unfolding integral_def using has_integral_spike_eq[OF assms] by auto
+
+subsection {* Some other trivialities about negligible sets. *}
+
+lemma negligible_subset[intro]: assumes "negligible s" "t \<subseteq> s" shows "negligible t" unfolding negligible_def
+proof(safe) case goal1 show ?case using assms(1)[unfolded negligible_def,rule_format,of a b]
+ apply-apply(rule has_integral_spike[OF assms(1)]) defer apply assumption
+ using assms(2) unfolding indicator_def by auto qed
+
+lemma negligible_diff[intro?]: assumes "negligible s" shows "negligible(s - t)" using assms by auto
+
+lemma negligible_inter: assumes "negligible s \<or> negligible t" shows "negligible(s \<inter> t)" using assms by auto
+
+lemma negligible_union: assumes "negligible s" "negligible t" shows "negligible (s \<union> t)" unfolding negligible_def
+proof safe case goal1 note assm = assms[unfolded negligible_def,rule_format,of a b]
+ thus ?case apply(subst has_integral_spike_eq[OF assms(2)])
+ defer apply assumption unfolding indicator_def by auto qed
+
+lemma negligible_union_eq[simp]: "negligible (s \<union> t) \<longleftrightarrow> (negligible s \<and> negligible t)"
+ using negligible_union by auto
+
+lemma negligible_sing[intro]: "negligible {a::real^'n}"
+proof- guess x using UNIV_witness[where 'a='n] ..
+ show ?thesis using negligible_standard_hyperplane[of x "a$x"] by auto qed
+
+lemma negligible_insert[simp]: "negligible(insert a s) \<longleftrightarrow> negligible s"
+ apply(subst insert_is_Un) unfolding negligible_union_eq by auto
+
+lemma negligible_empty[intro]: "negligible {}" by auto
+
+lemma negligible_finite[intro]: assumes "finite s" shows "negligible s"
+ using assms apply(induct s) by auto
+
+lemma negligible_unions[intro]: assumes "finite s" "\<forall>t\<in>s. negligible t" shows "negligible(\<Union>s)"
+ using assms by(induct,auto)
+
+lemma negligible: "negligible s \<longleftrightarrow> (\<forall>t::(real^'n) set. (indicator s has_integral 0) t)"
+ apply safe defer apply(subst negligible_def)
+proof- fix t::"(real^'n) set" assume as:"negligible s"
+ 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)
+ show "(indicator s has_integral 0) t" apply(subst has_integral_alt)
+ apply(cases,subst if_P,assumption) unfolding if_not_P apply(safe,rule as[unfolded negligible_def,rule_format])
+ apply(rule_tac x=1 in exI) apply(safe,rule zero_less_one) apply(rule_tac x=0 in exI)
+ using negligible_subset[OF as,of "s \<inter> t"] unfolding negligible_def indicator_def unfolding * by auto qed auto
+
+subsection {* Finite case of the spike theorem is quite commonly needed. *}
+
+lemma has_integral_spike_finite: assumes "finite s" "\<forall>x\<in>t-s. g x = f x"
+ "(f has_integral y) t" shows "(g has_integral y) t"
+ apply(rule has_integral_spike) using assms by auto
+
+lemma has_integral_spike_finite_eq: assumes "finite s" "\<forall>x\<in>t-s. g x = f x"
+ shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
+ apply rule apply(rule_tac[!] has_integral_spike_finite) using assms by auto
+
+lemma integrable_spike_finite:
+ assumes "finite s" "\<forall>x\<in>t-s. g x = f x" "f integrable_on t" shows "g integrable_on t"
+ using assms unfolding integrable_on_def apply safe apply(rule_tac x=y in exI)
+ apply(rule has_integral_spike_finite) by auto
+
+subsection {* In particular, the boundary of an interval is negligible. *}
+
+lemma negligible_frontier_interval: "negligible({a..b} - {a<..<b})"
+proof- let ?A = "\<Union>((\<lambda>k. {x. x$k = a$k} \<union> {x. x$k = b$k}) ` UNIV)"
+ have "{a..b} - {a<..<b} \<subseteq> ?A" apply rule unfolding Diff_iff mem_interval not_all
+ apply(erule conjE exE)+ apply(rule_tac X="{x. x $ xa = a $ xa} \<union> {x. x $ xa = b $ xa}" in UnionI)
+ apply(erule_tac[!] x=xa in allE) by auto
+ thus ?thesis apply-apply(rule negligible_subset[of ?A]) apply(rule negligible_unions[OF finite_imageI]) by auto qed
+
+lemma has_integral_spike_interior:
+ assumes "\<forall>x\<in>{a<..<b}. g x = f x" "(f has_integral y) ({a..b})" shows "(g has_integral y) ({a..b})"
+ apply(rule has_integral_spike[OF negligible_frontier_interval _ assms(2)]) using assms(1) by auto
+
+lemma has_integral_spike_interior_eq:
+ assumes "\<forall>x\<in>{a<..<b}. g x = f x" shows "((f has_integral y) ({a..b}) \<longleftrightarrow> (g has_integral y) ({a..b}))"
+ apply rule apply(rule_tac[!] has_integral_spike_interior) using assms by auto
+
+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}"
+ using assms unfolding integrable_on_def using has_integral_spike_interior[OF assms(1)] by auto
+
+subsection {* Integrability of continuous functions. *}
+
+lemma neutral_and[simp]: "neutral op \<and> = True"
+ unfolding neutral_def apply(rule some_equality) by auto
+
+lemma monoidal_and[intro]: "monoidal op \<and>" unfolding monoidal_def by auto
+
+lemma iterate_and[simp]: assumes "finite s" shows "(iterate op \<and>) s p \<longleftrightarrow> (\<forall>x\<in>s. p x)" using assms
+apply induct unfolding iterate_insert[OF monoidal_and] by auto
+
+lemma operative_division_and: assumes "operative op \<and> P" "d division_of {a..b}"
+ shows "(\<forall>i\<in>d. P i) \<longleftrightarrow> P {a..b}"
+ using operative_division[OF monoidal_and assms] division_of_finite[OF assms(2)] by auto
+
+lemma operative_approximable: assumes "0 \<le> e" fixes f::"real^'n \<Rightarrow> 'a::banach"
+ 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
+proof safe fix a b::"real^'n" { assume "content {a..b} = 0"
+ thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}"
+ apply(rule_tac x=f in exI) using assms by(auto intro!:integrable_on_null) }
+ { fix c k g assume as:"\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}"
+ 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}"
+ "\<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}"
+ apply(rule_tac[!] x=g in exI) using as(1) integrable_split[OF as(2)] by auto }
+ 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}"
+ "\<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}"
+ let ?g = "\<lambda>x. if x$k = c then f x else if x$k \<le> c then g1 x else g2 x"
+ 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)
+ proof safe case goal1 thus ?case apply- apply(cases "x$k=c", case_tac "x$k < c") using as assms by auto
+ 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}"
+ then guess h1 h2 unfolding integrable_on_def by auto from has_integral_split[OF this]
+ show ?case unfolding integrable_on_def by auto
+ next show "?g integrable_on {a..b} \<inter> {x. x $ k \<le> c}" "?g integrable_on {a..b} \<inter> {x. x $ k \<ge> c}"
+ apply(rule_tac[!] integrable_spike[OF negligible_standard_hyperplane[of k c]]) using as(2,4) by auto qed qed
+
+lemma approximable_on_division: fixes f::"real^'n \<Rightarrow> 'a::banach"
+ 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"
+ obtains g where "\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}"
+proof- note * = operative_division[OF monoidal_and operative_approximable[OF assms(1)] assms(2)]
+ note this[unfolded iterate_and[OF division_of_finite[OF assms(2)]]] from assms(3)[unfolded this[of f]]
+ guess g .. thus thesis apply-apply(rule that[of g]) by auto qed
+
+lemma integrable_continuous: fixes f::"real^'n \<Rightarrow> 'a::banach"
+ assumes "continuous_on {a..b} f" shows "f integrable_on {a..b}"
+proof(rule integrable_uniform_limit,safe) fix e::real assume e:"0 < e"
+ from compact_uniformly_continuous[OF assms compact_interval,unfolded uniformly_continuous_on_def,rule_format,OF e] guess d ..
+ note d=conjunctD2[OF this,rule_format]
+ from fine_division_exists[OF gauge_ball[OF d(1)], of a b] guess p . note p=this
+ note p' = tagged_division_ofD[OF p(1)]
+ have *:"\<forall>i\<in>snd ` p. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
+ proof(safe,unfold snd_conv) fix x l assume as:"(x,l) \<in> p"
+ from p'(4)[OF this] guess a b apply-by(erule exE)+ note l=this
+ 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)
+ proof safe show "(\<lambda>y. f x) integrable_on l" unfolding integrable_on_def l by(rule,rule has_integral_const)
+ fix y assume y:"y\<in>l" note fineD[OF p(2) as,unfolded subset_eq,rule_format,OF this]
+ note d(2)[OF _ _ this[unfolded mem_ball]]
+ 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
+ from e have "0 \<le> e" by auto from approximable_on_division[OF this division_of_tagged_division[OF p(1)] *] guess g .
+ thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" by auto qed
+
+subsection {* Specialization of additivity to one dimension. *}
+
+lemma operative_1_lt: assumes "monoidal opp"
+ shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real^1} = neutral opp) \<and>
+ (\<forall>a b c. a < c \<and> c < b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))"
+ unfolding operative_def content_eq_0_1 forall_1 vector_le_def vector_less_def
+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"
+ 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
+ thus "opp (f {a..c}) (f {c..b}) = f {a..b}" unfolding as(1)[rule_format,of a b "c$1"] by auto
+next fix a b::"real^1" and c::real
+ 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}"
+ show "f {a..b} = opp (f ({a..b} \<inter> {x. x $ 1 \<le> c})) (f ({a..b} \<inter> {x. c \<le> x $ 1}))"
+ proof(cases "c \<in> {a$1 .. b$1}")
+ case False hence "c<a$1 \<or> c>b$1" by auto
+ thus ?thesis apply-apply(erule disjE)
+ 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
+ show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto
+ 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
+ show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto
+ qed
+ next case True hence *:"min (b $ 1) c = c" "max (a $ 1) c = c" by auto
+ show ?thesis unfolding interval_split num1_eq_iff if_True * vec_def[THEN sym]
+ proof(cases "c = a$1 \<or> c = b$1")
+ case False thus "f {a..b} = opp (f {a..vec1 c}) (f {vec1 c..b})"
+ apply-apply(subst as(2)[rule_format]) using True by auto
+ next case True thus "f {a..b} = opp (f {a..vec1 c}) (f {vec1 c..b})" apply-
+ proof(erule disjE) assume "c=a$1" hence *:"a = vec1 c" unfolding Cart_eq by auto
+ hence "f {a..vec1 c} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
+ thus ?thesis using assms unfolding * by auto
+ next assume "c=b$1" hence *:"b = vec1 c" unfolding Cart_eq by auto
+ hence "f {vec1 c..b} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
+ thus ?thesis using assms unfolding * by auto qed qed qed qed
+
+lemma operative_1_le: assumes "monoidal opp"
+ shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real^1} = neutral opp) \<and>
+ (\<forall>a b c. a \<le> c \<and> c \<le> b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))"
+unfolding operative_1_lt[OF assms]
+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"
+ 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
+next fix a b c ::"real^1"
+ 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"
+ note as = this[rule_format]
+ show "opp (f {a..c}) (f {c..b}) = f {a..b}"
+ proof(cases "c = a \<or> c = b")
+ 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)
+ next case True thus ?thesis apply-
+ proof(erule disjE) assume *:"c=a" hence "f {a..c} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
+ thus ?thesis using assms unfolding * by auto
+ next assume *:"c=b" hence "f {c..b} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
+ thus ?thesis using assms unfolding * by auto qed qed qed
+
+subsection {* Special case of additivity we need for the FCT. *}
+
+lemma additive_tagged_division_1: fixes f::"real^1 \<Rightarrow> 'a::real_normed_vector"
+ assumes "dest_vec1 a \<le> dest_vec1 b" "p tagged_division_of {a..b}"
+ shows "setsum (\<lambda>(x,k). f(interval_upperbound k) - f(interval_lowerbound k)) p = f b - f a"
+proof- let ?f = "(\<lambda>k::(real^1) set. if k = {} then 0 else f(interval_upperbound k) - f(interval_lowerbound k))"
+ have *:"operative op + ?f" unfolding operative_1_lt[OF monoidal_monoid] interval_eq_empty_1
+ by(auto simp add:not_less interval_bound_1 vector_less_def)
+ have **:"{a..b} \<noteq> {}" using assms(1) by auto note operative_tagged_division[OF monoidal_monoid * assms(2)]
+ note * = this[unfolded if_not_P[OF **] interval_bound_1[OF assms(1)],THEN sym ]
+ show ?thesis unfolding * apply(subst setsum_iterate[THEN sym]) defer
+ apply(rule setsum_cong2) unfolding split_paired_all split_conv using assms(2) by auto qed
+
+subsection {* A useful lemma allowing us to factor out the content size. *}
+
+lemma has_integral_factor_content:
+ "(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
+ \<longrightarrow> norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content {a..b}))"
+proof(cases "content {a..b} = 0")
+ case True show ?thesis unfolding has_integral_null_eq[OF True] apply safe
+ apply(rule,rule,rule gauge_trivial,safe) unfolding setsum_content_null[OF True] True defer
+ apply(erule_tac x=1 in allE,safe) defer apply(rule fine_division_exists[of _ a b],assumption)
+ apply(erule_tac x=p in allE) unfolding setsum_content_null[OF True] by auto
+next case False note F = this[unfolded content_lt_nz[THEN sym]]
+ 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)"
+ show ?thesis apply(subst has_integral)
+ proof safe fix e::real assume e:"e>0"
+ { 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)
+ apply(erule impE) defer apply(erule exE,rule_tac x=d in exI)
+ using F e by(auto simp add:field_simps intro:mult_pos_pos) }
+ { 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)
+ apply(erule impE) defer apply(erule exE,rule_tac x=d in exI)
+ using F e by(auto simp add:field_simps intro:mult_pos_pos) } qed qed
+
+subsection {* Fundamental theorem of calculus. *}
+
+lemma fundamental_theorem_of_calculus: fixes f::"real^1 \<Rightarrow> 'a::banach"
+ assumes "a \<le> b" "\<forall>x\<in>{a..b}. ((f o vec1) has_vector_derivative f'(vec1 x)) (at x within {a..b})"
+ shows "(f' has_integral (f(vec1 b) - f(vec1 a))) ({vec1 a..vec1 b})"
+unfolding has_integral_factor_content
+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
+ note assm = assms(2)[unfolded has_vector_derivative_def has_derivative_within_alt]
+ 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
+ note this[OF assm,unfolded gauge_existence_lemma] from choice[OF this,unfolded Ball_def[symmetric]]
+ guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
+ show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {vec1 a..vec1 b} \<and> d fine p \<longrightarrow>
+ 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})"
+ apply(rule_tac x="\<lambda>x. ball x (d (dest_vec1 x))" in exI,safe)
+ apply(rule gauge_ball_dependent,rule,rule d(1))
+ proof- fix p assume as:"p tagged_division_of {vec1 a..vec1 b}" "(\<lambda>x. ball x (d (dest_vec1 x))) fine p"
+ 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}"
+ unfolding content_1[OF ab] additive_tagged_division_1[OF ab as(1),of f,THEN sym]
+ unfolding vector_minus_component[THEN sym] additive_tagged_division_1[OF ab as(1),of "\<lambda>x. x",THEN sym]
+ apply(subst dest_vec1_setsum) unfolding setsum_right_distrib defer unfolding setsum_subtractf[THEN sym]
+ proof(rule setsum_norm_le,safe) fix x k assume "(x,k)\<in>p"
+ note xk = tagged_division_ofD(2-4)[OF as(1) this] from this(3) guess u v apply-by(erule exE)+ note k=this
+ have *:"dest_vec1 u \<le> dest_vec1 v" using xk unfolding k by auto
+ 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] .
+ 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)"
+ apply(rule order_trans[OF _ norm_triangle_ineq4]) apply(rule eq_refl) apply(rule arg_cong[where f=norm])
+ unfolding scaleR.diff_left by(auto simp add:group_simps)
+ also have "... \<le> e * norm (dest_vec1 u - dest_vec1 x) + e * norm (dest_vec1 v - dest_vec1 x)"
+ apply(rule add_mono) apply(rule d(2)[of "x$1" "u$1",unfolded o_def vec1_dest_vec1]) prefer 4
+ apply(rule d(2)[of "x$1" "v$1",unfolded o_def vec1_dest_vec1])
+ using ball[rule_format,of u] ball[rule_format,of v]
+ using xk(1-2) unfolding k subset_eq by(auto simp add:vector_dist_norm norm_real)
+ also have "... \<le> e * dest_vec1 (interval_upperbound k - interval_lowerbound k)"
+ unfolding k interval_bound_1[OF *] using xk(1) unfolding k by(auto simp add:vector_dist_norm norm_real field_simps)
+ finally show "norm (content k *\<^sub>R f' x - (f (interval_upperbound k) - f (interval_lowerbound k))) \<le>
+ e * dest_vec1 (interval_upperbound k - interval_lowerbound k)" unfolding k interval_bound_1[OF *] content_1[OF *] .
+ qed(insert as, auto) qed qed
+
+subsection {* Attempt a systematic general set of "offset" results for components. *}
+
+lemma gauge_modify:
+ assumes "(\<forall>s. open s \<longrightarrow> open {x. f(x) \<in> s})" "gauge d"
+ shows "gauge (\<lambda>x y. d (f x) (f y))"
+ using assms unfolding gauge_def apply safe defer apply(erule_tac x="f x" in allE)
+ apply(erule_tac x="d (f x)" in allE) unfolding mem_def Collect_def by auto
+
+subsection {* Only need trivial subintervals if the interval itself is trivial. *}
+
+lemma division_of_nontrivial: fixes s::"(real^'n) set set"
+ assumes "s division_of {a..b}" "content({a..b}) \<noteq> 0"
+ shows "{k. k \<in> s \<and> content k \<noteq> 0} division_of {a..b}" using assms(1) apply-
+proof(induct "card s" arbitrary:s rule:nat_less_induct)
+ fix s::"(real^'n) set set" assume assm:"s division_of {a..b}"
+ "\<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}"
+ note s = division_ofD[OF assm(1)] let ?thesis = "{k \<in> s. content k \<noteq> 0} division_of {a..b}"
+ { presume *:"{k \<in> s. content k \<noteq> 0} \<noteq> s \<Longrightarrow> ?thesis"
+ show ?thesis apply cases defer apply(rule *,assumption) using assm(1) by auto }
+ assume noteq:"{k \<in> s. content k \<noteq> 0} \<noteq> s"
+ then obtain k where k:"k\<in>s" "content k = 0" by auto
+ from s(4)[OF k(1)] guess c d apply-by(erule exE)+ note k=k this
+ from k have "card s > 0" unfolding card_gt_0_iff using assm(1) by auto
+ hence card:"card (s - {k}) < card s" using assm(1) k(1) apply(subst card_Diff_singleton_if) by auto
+ have *:"closed (\<Union>(s - {k}))" apply(rule closed_Union) defer apply rule apply(drule DiffD1,drule s(4))
+ apply safe apply(rule closed_interval) using assm(1) by auto
+ have "k \<subseteq> \<Union>(s - {k})" apply safe apply(rule *[unfolded closed_limpt,rule_format]) unfolding islimpt_approachable
+ proof safe fix x and e::real assume as:"x\<in>k" "e>0"
+ from k(2)[unfolded k content_eq_0] guess i ..
+ hence i:"c$i = d$i" using s(3)[OF k(1),unfolded k] unfolding interval_ne_empty by smt
+ hence xi:"x$i = d$i" using as unfolding k mem_interval by smt
+ def y \<equiv> "(\<chi> j. if j = i then if c$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$j)"
+ show "\<exists>x'\<in>\<Union>(s - {k}). x' \<noteq> x \<and> dist x' x < e" apply(rule_tac x=y in bexI)
+ proof have "d \<in> {c..d}" using s(3)[OF k(1)] unfolding k interval_eq_empty mem_interval by(fastsimp simp add: not_less)
+ hence "d \<in> {a..b}" using s(2)[OF k(1)] unfolding k by auto note di = this[unfolded mem_interval,THEN spec[where x=i]]
+ hence xyi:"y$i \<noteq> x$i" unfolding y_def unfolding i xi Cart_lambda_beta if_P[OF refl]
+ 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+
+ thus "y \<noteq> x" unfolding Cart_eq by auto
+ have *:"UNIV = insert i (UNIV - {i})" by auto
+ have "norm (y - x) < e + setsum (\<lambda>i. 0) (UNIV::'n set)" apply(rule le_less_trans[OF norm_le_l1])
+ apply(subst *,subst setsum_insert) prefer 3 apply(rule add_less_le_mono)
+ proof- show "\<bar>(y - x) $ i\<bar> < e" unfolding y_def Cart_lambda_beta vector_minus_component if_P[OF refl]
+ apply(cases) apply(subst if_P,assumption) unfolding if_not_P unfolding i xi using di as(2) by auto
+ show "(\<Sum>i\<in>UNIV - {i}. \<bar>(y - x) $ i\<bar>) \<le> (\<Sum>i\<in>UNIV. 0)" unfolding y_def by auto
+ qed auto thus "dist y x < e" unfolding vector_dist_norm by auto
+ 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
+ moreover have "y \<in> \<Union>s" unfolding s mem_interval
+ proof note simps = y_def Cart_lambda_beta if_not_P
+ fix j::'n show "a $ j \<le> y $ j \<and> y $ j \<le> b $ j"
+ proof(cases "j = i") case False have "x \<in> {a..b}" using s(2)[OF k(1)] as(1) by auto
+ thus ?thesis unfolding simps if_not_P[OF False] unfolding mem_interval by auto
+ next case True note T = this show ?thesis
+ proof(cases "c $ i \<le> (a $ i + b $ i) / 2")
+ case True show ?thesis unfolding simps if_P[OF T] if_P[OF True] unfolding i
+ using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps)
+ next case False thus ?thesis unfolding simps if_P[OF T] if_not_P[OF False] unfolding i
+ using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps)
+ qed qed qed
+ ultimately show "y \<in> \<Union>(s - {k})" by auto
+ qed qed hence "\<Union>(s - {k}) = {a..b}" unfolding s(6)[THEN sym] by auto
+ hence "{ka \<in> s - {k}. content ka \<noteq> 0} division_of {a..b}" apply-apply(rule assm(2)[rule_format,OF card refl])
+ apply(rule division_ofI) defer apply(rule_tac[1-4] s) using assm(1) by auto
+ 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
+
+subsection {* Integrabibility on subintervals. *}
+
+lemma operative_integrable: fixes f::"real^'n \<Rightarrow> 'a::banach" shows
+ "operative op \<and> (\<lambda>i. f integrable_on i)"
+ unfolding operative_def neutral_and apply safe apply(subst integrable_on_def)
+ unfolding has_integral_null_eq apply(rule,rule refl) apply(rule,assumption)+
+ unfolding integrable_on_def by(auto intro: has_integral_split)
+
+lemma integrable_subinterval: fixes f::"real^'n \<Rightarrow> 'a::banach"
+ assumes "f integrable_on {a..b}" "{c..d} \<subseteq> {a..b}" shows "f integrable_on {c..d}"
+ apply(cases "{c..d} = {}") defer apply(rule partial_division_extend_1[OF assms(2)],assumption)
+ using operative_division_and[OF operative_integrable,THEN sym,of _ _ _ f] assms(1) by auto
+
+subsection {* Combining adjacent intervals in 1 dimension. *}
+
+lemma has_integral_combine: assumes "(a::real^1) \<le> c" "c \<le> b"
+ "(f has_integral i) {a..c}" "(f has_integral (j::'a::banach)) {c..b}"
+ shows "(f has_integral (i + j)) {a..b}"
+proof- note operative_integral[of f, unfolded operative_1_le[OF monoidal_lifted[OF monoidal_monoid]]]
+ note conjunctD2[OF this,rule_format] note * = this(2)[OF conjI[OF assms(1-2)],unfolded if_P[OF assms(3)]]
+ hence "f integrable_on {a..b}" apply- apply(rule ccontr) apply(subst(asm) if_P) defer
+ apply(subst(asm) if_P) using assms(3-) by auto
+ with * show ?thesis apply-apply(subst(asm) if_P) defer apply(subst(asm) if_P) defer apply(subst(asm) if_P)
+ unfolding lifted.simps using assms(3-) by(auto simp add: integrable_on_def integral_unique) qed
+
+lemma integral_combine: fixes f::"real^1 \<Rightarrow> 'a::banach"
+ assumes "a \<le> c" "c \<le> b" "f integrable_on ({a..b})"
+ shows "integral {a..c} f + integral {c..b} f = integral({a..b}) f"
+ apply(rule integral_unique[THEN sym]) apply(rule has_integral_combine[OF assms(1-2)])
+ apply(rule_tac[!] integrable_integral integrable_subinterval[OF assms(3)])+ using assms(1-2) by auto
+
+lemma integrable_combine: fixes f::"real^1 \<Rightarrow> 'a::banach"
+ assumes "a \<le> c" "c \<le> b" "f integrable_on {a..c}" "f integrable_on {c..b}"
+ shows "f integrable_on {a..b}" using assms unfolding integrable_on_def by(fastsimp intro!:has_integral_combine)
+
+subsection {* Reduce integrability to "local" integrability. *}
+
+lemma integrable_on_little_subintervals: fixes f::"real^'n \<Rightarrow> 'a::banach"
+ 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}"
+ shows "f integrable_on {a..b}"
+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})"
+ using assms by auto note this[unfolded gauge_existence_lemma] from choice[OF this] guess d .. note d=this[rule_format]
+ guess p apply(rule fine_division_exists[OF gauge_ball_dependent,of d a b]) using d by auto note p=this(1-2)
+ note division_of_tagged_division[OF this(1)] note * = operative_division_and[OF operative_integrable,OF this,THEN sym,of f]
+ show ?thesis unfolding * apply safe unfolding snd_conv
+ proof- fix x k assume "(x,k) \<in> p" note tagged_division_ofD(2-4)[OF p(1) this] fineD[OF p(2) this]
+ thus "f integrable_on k" apply safe apply(rule d[THEN conjunct2,rule_format,of x]) by auto qed qed
+
+subsection {* Second FCT or existence of antiderivative. *}
+
+lemma integrable_const[intro]:"(\<lambda>x. c) integrable_on {a..b}"
+ unfolding integrable_on_def by(rule,rule has_integral_const)
+
+lemma integral_has_vector_derivative: fixes f::"real \<Rightarrow> 'a::banach"
+ assumes "continuous_on {a..b} f" "x \<in> {a..b}"
+ shows "((\<lambda>u. integral {vec a..vec u} (f o dest_vec1)) has_vector_derivative f(x)) (at x within {a..b})"
+ unfolding has_vector_derivative_def has_derivative_within_alt
+apply safe apply(rule scaleR.bounded_linear_left)
+proof- fix e::real assume e:"e>0"
+ note compact_uniformly_continuous[OF assms(1) compact_real_interval,unfolded uniformly_continuous_on_def]
+ from this[rule_format,OF e] guess d apply-by(erule conjE exE)+ note d=this[rule_format]
+ let ?I = "\<lambda>a b. integral {vec1 a..vec1 b} (f \<circ> dest_vec1)"
+ 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)"
+ proof(rule,rule,rule d,safe) case goal1 show ?case proof(cases "y < x")
+ case False have "f \<circ> dest_vec1 integrable_on {vec1 a..vec1 y}" apply(rule integrable_subinterval,rule integrable_continuous)
+ apply(rule continuous_on_o_dest_vec1 assms)+ unfolding not_less using assms(2) goal1 by auto
+ hence *:"?I a y - ?I a x = ?I x y" unfolding group_simps apply(subst eq_commute) apply(rule integral_combine)
+ using False unfolding not_less using assms(2) goal1 by auto
+ have **:"norm (y - x) = content {vec1 x..vec1 y}" apply(subst content_1) using False unfolding not_less by auto
+ show ?thesis unfolding ** apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x) o dest_vec1"]) unfolding * unfolding o_def
+ defer apply(rule has_integral_sub) apply(rule integrable_integral)
+ apply(rule integrable_subinterval,rule integrable_continuous) apply(rule continuous_on_o_dest_vec1[unfolded o_def] assms)+
+ proof- show "{vec1 x..vec1 y} \<subseteq> {vec1 a..vec1 b}" using goal1 assms(2) by auto
+ have *:"y - x = norm(y - x)" using False by auto
+ show "((\<lambda>xa. f x) has_integral (y - x) *\<^sub>R f x) {vec1 x..vec1 y}" apply(subst *) unfolding ** by auto
+ show "\<forall>xa\<in>{vec1 x..vec1 y}. norm (f (dest_vec1 xa) - f x) \<le> e" apply safe apply(rule less_imp_le)
+ apply(rule d(2)[unfolded vector_dist_norm]) using assms(2) using goal1 by auto
+ qed(insert e,auto)
+ next case True have "f \<circ> dest_vec1 integrable_on {vec1 a..vec1 x}" apply(rule integrable_subinterval,rule integrable_continuous)
+ apply(rule continuous_on_o_dest_vec1 assms)+ unfolding not_less using assms(2) goal1 by auto
+ hence *:"?I a x - ?I a y = ?I y x" unfolding group_simps apply(subst eq_commute) apply(rule integral_combine)
+ using True using assms(2) goal1 by auto
+ have **:"norm (y - x) = content {vec1 y..vec1 x}" apply(subst content_1) using True unfolding not_less by auto
+ 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
+ show ?thesis apply(subst ***) unfolding norm_minus_cancel **
+ apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x) o dest_vec1"]) unfolding * unfolding o_def
+ defer apply(rule has_integral_sub) apply(subst minus_minus[THEN sym]) unfolding minus_minus
+ apply(rule integrable_integral) apply(rule integrable_subinterval,rule integrable_continuous)
+ apply(rule continuous_on_o_dest_vec1[unfolded o_def] assms)+
+ proof- show "{vec1 y..vec1 x} \<subseteq> {vec1 a..vec1 b}" using goal1 assms(2) by auto
+ have *:"x - y = norm(y - x)" using True by auto
+ show "((\<lambda>xa. f x) has_integral (x - y) *\<^sub>R f x) {vec1 y..vec1 x}" apply(subst *) unfolding ** by auto
+ show "\<forall>xa\<in>{vec1 y..vec1 x}. norm (f (dest_vec1 xa) - f x) \<le> e" apply safe apply(rule less_imp_le)
+ apply(rule d(2)[unfolded vector_dist_norm]) using assms(2) using goal1 by auto
+ qed(insert e,auto) qed qed qed
+
+lemma integral_has_vector_derivative': fixes f::"real^1 \<Rightarrow> 'a::banach"
+ assumes "continuous_on {a..b} f" "x \<in> {a..b}"
+ shows "((\<lambda>u. (integral {a..vec u} f)) has_vector_derivative f x) (at (x$1) within {a$1..b$1})"
+ using integral_has_vector_derivative[OF continuous_on_o_vec1[OF assms(1)], of "x$1"]
+ unfolding o_def vec1_dest_vec1 using assms(2) by auto
+
+lemma antiderivative_continuous: assumes "continuous_on {a..b::real} f"
+ obtains g where "\<forall>x\<in> {a..b}. (g has_vector_derivative (f(x)::_::banach)) (at x within {a..b})"
+ apply(rule that,rule) using integral_has_vector_derivative[OF assms] by auto
+
+subsection {* Combined fundamental theorem of calculus. *}
+
+lemma antiderivative_integral_continuous: fixes f::"real \<Rightarrow> 'a::banach" assumes "continuous_on {a..b} f"
+ 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}"
+proof- from antiderivative_continuous[OF assms] guess g . note g=this
+ show ?thesis apply(rule that[of g])
+ proof safe case goal1 have "\<forall>x\<in>{u..v}. (g has_vector_derivative f x) (at x within {u..v})"
+ apply(rule,rule has_vector_derivative_within_subset) apply(rule g[rule_format]) using goal1(1-2) by auto
+ thus ?case using fundamental_theorem_of_calculus[OF goal1(3),of "g o dest_vec1" "f o dest_vec1"]
+ unfolding o_def vec1_dest_vec1 by auto qed qed
+
+subsection {* General "twiddling" for interval-to-interval function image. *}
+
+lemma has_integral_twiddle:
+ assumes "0 < r" "\<forall>x. h(g x) = x" "\<forall>x. g(h x) = x" "\<forall>x. continuous (at x) g"
+ "\<forall>u v. \<exists>w z. g ` {u..v} = {w..z}"
+ "\<forall>u v. \<exists>w z. h ` {u..v} = {w..z}"
+ "\<forall>u v. content(g ` {u..v}) = r * content {u..v}"
+ "(f has_integral i) {a..b}"
+ shows "((\<lambda>x. f(g x)) has_integral (1 / r) *\<^sub>R i) (h ` {a..b})"
+proof- { presume *:"{a..b} \<noteq> {} \<Longrightarrow> ?thesis"
+ show ?thesis apply cases defer apply(rule *,assumption)
+ proof- case goal1 thus ?thesis unfolding goal1 assms(8)[unfolded goal1 has_integral_empty_eq] by auto qed }
+ assume "{a..b} \<noteq> {}" from assms(6)[rule_format,of a b] guess w z apply-by(erule exE)+ note wz=this
+ have inj:"inj g" "inj h" unfolding inj_on_def apply safe apply(rule_tac[!] ccontr)
+ using assms(2) apply(erule_tac x=x in allE) using assms(2) apply(erule_tac x=y in allE) defer
+ using assms(3) apply(erule_tac x=x in allE) using assms(3) apply(erule_tac x=y in allE) by auto
+ show ?thesis unfolding has_integral_def has_integral_compact_interval_def apply(subst if_P) apply(rule,rule,rule wz)
+ proof safe fix e::real assume e:"e>0" hence "e * r > 0" using assms(1) by(rule mult_pos_pos)
+ from assms(8)[unfolded has_integral,rule_format,OF this] guess d apply-by(erule exE conjE)+ note d=this[rule_format]
+ 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)
+ 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)"
+ 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
+ fix p assume as:"p tagged_division_of h ` {a..b}" "d' fine p" note p = tagged_division_ofD[OF as(1)]
+ 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
+ proof safe show "finite ((\<lambda>(x, k). (g x, g ` k)) ` p)" using as by auto
+ show "d fine (\<lambda>(x, k). (g x, g ` k)) ` p" using as(2) unfolding fine_def d' by auto
+ fix x k assume xk[intro]:"(x,k) \<in> p" show "g x \<in> g ` k" using p(2)[OF xk] by auto
+ show "\<exists>u v. g ` k = {u..v}" using p(4)[OF xk] using assms(5-6) by auto
+ { 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)"]
+ using assms(2)[rule_format,of y] unfolding inj_image_mem_iff[OF inj(2)] by auto }
+ fix x' k' assume xk':"(x',k') \<in> p" fix z assume "z \<in> interior (g ` k)" "z \<in> interior (g ` k')"
+ hence *:"interior (g ` k) \<inter> interior (g ` k') \<noteq> {}" by auto
+ have same:"(x, k) = (x', k')" apply-apply(rule ccontr,drule p(5)[OF xk xk'])
+ proof- assume as:"interior k \<inter> interior k' = {}" from nonempty_witness[OF *] guess z .
+ hence "z \<in> g ` (interior k \<inter> interior k')" using interior_image_subset[OF assms(4) inj(1)]
+ unfolding image_Int[OF inj(1)] by auto thus False using as by blast
+ qed thus "g x = g x'" by auto
+ { fix z assume "z \<in> k" thus "g z \<in> g ` k'" using same by auto }
+ { fix z assume "z \<in> k'" thus "g z \<in> g ` k" using same by auto }
+ 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
+ then guess X unfolding Union_iff .. note X=this from this(1) guess y unfolding mem_Collect_eq ..
+ thus "x \<in> \<Union>{k. \<exists>x. (x, k) \<in> (\<lambda>(x, k). (g x, g ` k)) ` p}" apply-
+ apply(rule_tac X="g ` X" in UnionI) defer apply(rule_tac x="h x" in image_eqI)
+ using X(2) assms(3)[rule_format,of x] by auto
+ 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
+ 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
+ unfolding setsum_reindex[OF *] apply(subst scaleR_right.setsum) defer apply(rule setsum_cong2) unfolding o_def split_paired_all split_conv
+ apply(drule p(4)) apply safe unfolding assms(7)[rule_format] using p by auto
+ 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]
+ unfolding real_scaleR_def using assms(1) by auto finally have *:"?l = ?r" .
+ 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
+ using assms(1) by(auto simp add:field_simps) qed qed qed
+
+subsection {* Special case of a basic affine transformation. *}
+
+lemma interval_image_affinity_interval: shows "\<exists>u v. (\<lambda>x. m *\<^sub>R (x::real^'n) + c) ` {a..b} = {u..v}"
+ unfolding image_affinity_interval by auto
+
+lemmas Cart_simps = Cart_nth.add Cart_nth.minus Cart_nth.zero Cart_nth.diff Cart_nth.scaleR real_scaleR_def Cart_lambda_beta
+ Cart_eq vector_le_def vector_less_def
+
+lemma setprod_cong2: assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "setprod f A = setprod g A"
+ apply(rule setprod_cong) using assms by auto
+
+lemma content_image_affinity_interval:
+ "content((\<lambda>x::real^'n. m *\<^sub>R x + c) ` {a..b}) = (abs m) ^ CARD('n) * content {a..b}" (is "?l = ?r")
+proof- { presume *:"{a..b}\<noteq>{} \<Longrightarrow> ?thesis" show ?thesis apply(cases,rule *,assumption)
+ unfolding not_not using content_empty by auto }
+ assume as:"{a..b}\<noteq>{}" show ?thesis proof(cases "m \<ge> 0")
+ case True show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_P[OF True]
+ unfolding content_closed_interval'[OF as] apply(subst content_closed_interval')
+ defer apply(subst setprod_constant[THEN sym]) apply(rule finite_UNIV) unfolding setprod_timesf[THEN sym]
+ apply(rule setprod_cong2) using True as unfolding interval_ne_empty Cart_simps not_le
+ by(auto simp add:field_simps intro:mult_left_mono)
+ next case False show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_not_P[OF False]
+ unfolding content_closed_interval'[OF as] apply(subst content_closed_interval')
+ defer apply(subst setprod_constant[THEN sym]) apply(rule finite_UNIV) unfolding setprod_timesf[THEN sym]
+ apply(rule setprod_cong2) using False as unfolding interval_ne_empty Cart_simps not_le
+ by(auto simp add:field_simps mult_le_cancel_left_neg) qed qed
+
+lemma has_integral_affinity: assumes "(f has_integral i) {a..b::real^'n}" "m \<noteq> 0"
+ 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})"
+ apply(rule has_integral_twiddle,safe) unfolding Cart_eq Cart_simps apply(rule zero_less_power)
+ defer apply(insert assms(2), simp add:field_simps) apply(insert assms(2), simp add:field_simps)
+ apply(rule continuous_intros)+ apply(rule interval_image_affinity_interval)+ apply(rule content_image_affinity_interval) using assms by auto
+
+lemma integrable_affinity: assumes "f integrable_on {a..b}" "m \<noteq> 0"
+ 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})"
+ using assms unfolding integrable_on_def apply safe apply(drule has_integral_affinity) by auto
+
+subsection {* Special case of stretching coordinate axes separately. *}
+
+lemma image_stretch_interval:
+ "(\<lambda>x. \<chi> k. m k * x$k) ` {a..b::real^'n} =
+ (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")
+proof(cases "{a..b}={}") case True thus ?thesis unfolding True by auto
+next have *:"\<And>P Q. (\<forall>i. P i) \<and> (\<forall>i. Q i) \<longleftrightarrow> (\<forall>i. P i \<and> Q i)" by auto
+ case False note ab = this[unfolded interval_ne_empty]
+ show ?thesis apply-apply(rule set_ext)
+ 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
+ show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" unfolding if_not_P[OF False]
+ unfolding image_iff mem_interval Bex_def Cart_simps Cart_eq *
+ unfolding lambda_skolem[THEN sym,of "\<lambda> i xa. (a $ i \<le> xa \<and> xa \<le> b $ i) \<and> x $ i = m i * xa"]
+ proof(rule **,rule) fix i::'n show "(\<exists>xa. (a $ i \<le> xa \<and> xa \<le> b $ i) \<and> x $ i = m i * xa) =
+ (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))"
+ proof(cases "m i = 0") case True thus ?thesis using ab by auto
+ next case False hence "0 < m i \<or> 0 > m i" by auto thus ?thesis apply-
+ proof(erule disjE) assume as:"0 < m i" hence *:"min (m i * a $ i) (m i * b $ i) = m i * a $ i"
+ "max (m i * a $ i) (m i * b $ i) = m i * b $ i" using ab unfolding min_def max_def by auto
+ show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x$i" in exI)
+ using as by(auto simp add:field_simps)
+ next assume as:"0 > m i" hence *:"max (m i * a $ i) (m i * b $ i) = m i * a $ i"
+ "min (m i * a $ i) (m i * b $ i) = m i * b $ i" using ab as unfolding min_def max_def
+ by(auto simp add:field_simps mult_le_cancel_left_neg intro:real_le_antisym)
+ show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x$i" in exI)
+ using as by(auto simp add:field_simps) qed qed qed qed qed
+
+lemma interval_image_stretch_interval: "\<exists>u v. (\<lambda>x. \<chi> k. m k * x$k) ` {a..b::real^'n} = {u..v}"
+ unfolding image_stretch_interval by auto
+
+lemma content_image_stretch_interval:
+ "content((\<lambda>x::real^'n. \<chi> k. m k * x$k) ` {a..b}) = abs(setprod m UNIV) * content({a..b})"
+proof(cases "{a..b} = {}") case True thus ?thesis
+ unfolding content_def image_is_empty image_stretch_interval if_P[OF True] by auto
+next case False hence "(\<lambda>x. \<chi> k. m k * x $ k) ` {a..b} \<noteq> {}" by auto
+ thus ?thesis using False unfolding content_def image_stretch_interval apply- unfolding interval_bounds' if_not_P
+ unfolding abs_setprod setprod_timesf[THEN sym] apply(rule setprod_cong2) unfolding Cart_lambda_beta
+ proof- fix i::'n have "(m i < 0 \<or> m i > 0) \<or> m i = 0" by auto
+ 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)"
+ apply-apply(erule disjE)+ unfolding min_def max_def using False[unfolded interval_ne_empty,rule_format,of i]
+ by(auto simp add:field_simps not_le mult_le_cancel_left_neg mult_le_cancel_left_pos) qed qed
+
+lemma has_integral_stretch: assumes "(f has_integral i) {a..b}" "\<forall>k. ~(m k = 0)"
+ shows "((\<lambda>x. f(\<chi> k. m k * x$k)) has_integral
+ ((1/(abs(setprod m UNIV))) *\<^sub>R i)) ((\<lambda>x. \<chi> k. 1/(m k) * x$k) ` {a..b})"
+ apply(rule has_integral_twiddle) unfolding zero_less_abs_iff content_image_stretch_interval
+ unfolding image_stretch_interval empty_as_interval Cart_eq using assms
+proof- show "\<forall>x. continuous (at x) (\<lambda>x. \<chi> k. m k * x $ k)"
+ apply(rule,rule linear_continuous_at) unfolding linear_linear
+ unfolding linear_def Cart_simps Cart_eq by(auto simp add:field_simps) qed auto
+
+lemma integrable_stretch:
+ assumes "f integrable_on {a..b}" "\<forall>k. ~(m k = 0)"
+ shows "(\<lambda>x. f(\<chi> k. m k * x$k)) integrable_on ((\<lambda>x. \<chi> k. 1/(m k) * x$k) ` {a..b})"
+ using assms unfolding integrable_on_def apply-apply(erule exE) apply(drule has_integral_stretch) by auto
+
+subsection {* even more special cases. *}
+
+lemma uminus_interval_vector[simp]:"uminus ` {a..b} = {-b .. -a::real^'n}"
+ apply(rule set_ext,rule) defer unfolding image_iff
+ apply(rule_tac x="-x" in bexI) by(auto simp add:vector_le_def minus_le_iff le_minus_iff)
+
+lemma has_integral_reflect_lemma[intro]: assumes "(f has_integral i) {a..b}"
+ shows "((\<lambda>x. f(-x)) has_integral i) {-b .. -a}"
+ using has_integral_affinity[OF assms, of "-1" 0] by auto
+
+lemma has_integral_reflect[simp]: "((\<lambda>x. f(-x)) has_integral i) {-b..-a} \<longleftrightarrow> (f has_integral i) ({a..b})"
+ apply rule apply(drule_tac[!] has_integral_reflect_lemma) by auto
+
+lemma integrable_reflect[simp]: "(\<lambda>x. f(-x)) integrable_on {-b..-a} \<longleftrightarrow> f integrable_on {a..b}"
+ unfolding integrable_on_def by auto
+
+lemma integral_reflect[simp]: "integral {-b..-a} (\<lambda>x. f(-x)) = integral ({a..b}) f"
+ unfolding integral_def by auto
+
+subsection {* Stronger form of FCT; quite a tedious proof. *}
+
+(** move this **)
+declare norm_triangle_ineq4[intro]
+
+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)
+
+lemma additive_tagged_division_1': fixes f::"real \<Rightarrow> 'a::real_normed_vector"
+ assumes "a \<le> b" "p tagged_division_of {vec1 a..vec1 b}"
+ shows "setsum (\<lambda>(x,k). f (dest_vec1 (interval_upperbound k)) - f(dest_vec1 (interval_lowerbound k))) p = f b - f a"
+ using additive_tagged_division_1[OF _ assms(2), of "f o dest_vec1"]
+ unfolding o_def vec1_dest_vec1 using assms(1) by auto
+
+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"
+ unfolding split_def by(rule refl)
+
+lemma norm_triangle_le_sub: "norm x + norm y \<le> e \<Longrightarrow> norm (x - y) \<le> e"
+ apply(subst(asm)(2) norm_minus_cancel[THEN sym])
+ apply(drule norm_triangle_le) by(auto simp add:group_simps)
+
+lemma fundamental_theorem_of_calculus_interior:
+ assumes"a \<le> b" "continuous_on {a..b} f" "\<forall>x\<in>{a<..<b}. (f has_vector_derivative f'(x)) (at x)"
+ shows "((f' o dest_vec1) has_integral (f b - f a)) {vec a..vec b}"
+proof- { presume *:"a < b \<Longrightarrow> ?thesis"
+ show ?thesis proof(cases,rule *,assumption)
+ assume "\<not> a < b" hence "a = b" using assms(1) by auto
+ hence *:"{vec a .. vec b} = {vec b}" "f b - f a = 0" apply(auto simp add: Cart_simps) by smt
+ show ?thesis unfolding *(2) apply(rule has_integral_null) unfolding content_eq_0_1 using * `a=b` by auto
+ qed } assume ab:"a < b"
+ let ?P = "\<lambda>e. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {vec1 a..vec1 b} \<and> d fine p \<longrightarrow>
+ 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})"
+ { presume "\<And>e. e>0 \<Longrightarrow> ?P e" thus ?thesis unfolding has_integral_factor_content by auto }
+ fix e::real assume e:"e>0"
+ note assms(3)[unfolded has_vector_derivative_def has_derivative_at_alt ball_conj_distrib]
+ note conjunctD2[OF this] note bounded=this(1) and this(2)
+ 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)"
+ 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]
+ from choice[OF this] guess d .. note conjunctD2[OF this[rule_format]] note d = this[rule_format]
+ have "bounded (f ` {a..b})" apply(rule compact_imp_bounded compact_continuous_image)+ using compact_real_interval assms by auto
+ from this[unfolded bounded_pos] guess B .. note B = this[rule_format]
+
+ have "\<exists>da. 0 < da \<and> (\<forall>c. a \<le> c \<and> {a..c} \<subseteq> {a..b} \<and> {a..c} \<subseteq> ball a da
+ \<longrightarrow> norm(content {vec1 a..vec1 c} *\<^sub>R f' a - (f c - f a)) \<le> (e * (b - a)) / 4)"
+ proof- have "a\<in>{a..b}" using ab by auto
+ note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
+ note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0" using e ab by(auto simp add:field_simps)
+ from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
+ have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' a) \<le> (e * (b - a)) / 8"
+ proof(cases "f' a = 0") case True
+ thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg)
+ next case False thus ?thesis
+ apply(rule_tac x="(e * (b - a)) / 8 / norm (f' a)" in exI)
+ using ab e by(auto simp add:field_simps)
+ qed then guess l .. note l = conjunctD2[OF this]
+ show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+)
+ proof- fix c assume as:"a \<le> c" "{a..c} \<subseteq> {a..b}" "{a..c} \<subseteq> ball a (min k l)"
+ note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval]
+ 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)
+ also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8"
+ proof(rule add_mono) case goal1 have "\<bar>c - a\<bar> \<le> \<bar>l\<bar>" using as' by auto
+ thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto
+ next case goal2 show ?case apply(rule less_imp_le) apply(cases "a = c") defer
+ apply(rule k(2)[unfolded vector_dist_norm]) using as' e ab by(auto simp add:field_simps)
+ 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
+ qed qed then guess da .. note da=conjunctD2[OF this,rule_format]
+
+ 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"
+ proof- have "b\<in>{a..b}" using ab by auto
+ note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
+ note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0" using e ab by(auto simp add:field_simps)
+ from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
+ have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' b) \<le> (e * (b - a)) / 8"
+ proof(cases "f' b = 0") case True
+ thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg)
+ next case False thus ?thesis
+ apply(rule_tac x="(e * (b - a)) / 8 / norm (f' b)" in exI)
+ using ab e by(auto simp add:field_simps)
+ qed then guess l .. note l = conjunctD2[OF this]
+ show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+)
+ proof- fix c assume as:"c \<le> b" "{c..b} \<subseteq> {a..b}" "{c..b} \<subseteq> ball b (min k l)"
+ note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval]
+ 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)
+ also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8"
+ proof(rule add_mono) case goal1 have "\<bar>c - b\<bar> \<le> \<bar>l\<bar>" using as' by auto
+ thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto
+ next case goal2 show ?case apply(rule less_imp_le) apply(cases "b = c") defer apply(subst norm_minus_commute)
+ apply(rule k(2)[unfolded vector_dist_norm]) using as' e ab by(auto simp add:field_simps)
+ 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
+ qed qed then guess db .. note db=conjunctD2[OF this,rule_format]
+
+ let ?d = "(\<lambda>x. ball x (if x=vec1 a then da else if x=vec b then db else d (dest_vec1 x)))"
+ show "?P e" apply(rule_tac x="?d" in exI)
+ proof safe case goal1 show ?case apply(rule gauge_ball_dependent) using ab db(1) da(1) d(1) by auto
+ next case goal2 note as=this let ?A = "{t. fst t \<in> {vec1 a, vec1 b}}" note p = tagged_division_ofD[OF goal2(1)]
+ 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
+ note * = additive_tagged_division_1'[OF assms(1) goal2(1), THEN sym]
+ 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
+ show ?case unfolding content_1'[OF assms(1)] and *[of "\<lambda>x. x"] *[of f] setsum_subtractf[THEN sym] split_minus
+ unfolding setsum_right_distrib apply(subst(2) pA,subst pA) unfolding setsum_Un_disjoint[OF pA(2-)]
+ proof(rule norm_triangle_le,rule **)
+ case goal1 show ?case apply(rule order_trans,rule setsum_norm_le) apply(rule pA) defer apply(subst divide.setsum)
+ 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"
+ "e * (dest_vec1 (interval_upperbound k) - dest_vec1 (interval_lowerbound k)) / 2
+ < norm (content k *\<^sub>R f' (dest_vec1 x) - (f (dest_vec1 (interval_upperbound k)) - f (dest_vec1 (interval_lowerbound k))))"
+ from p(4)[OF this(1)] guess u v apply-by(erule exE)+ note k=this
+ 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]
+ note result = as(2)[unfolded k interval_bounds[OF this(1)] content_1[OF this(2)]]
+
+ 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]
+ have "norm ((v$1 - u$1) *\<^sub>R f' (x$1) - (f (v$1) - f (u$1))) =
+ 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)))"
+ apply(rule arg_cong[of _ _ norm]) unfolding scaleR_left.diff by auto
+ also have "... \<le> e / 2 * norm (u$1 - x$1) + e / 2 * norm (v$1 - x$1)" apply(rule norm_triangle_le_sub)
+ apply(rule add_mono) apply(rule_tac[!] *) using fineD[OF goal2(2) as(1)] as' unfolding k subset_eq
+ apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE) using uv by(auto simp add:dist_real)
+ also have "... \<le> e / 2 * norm (v$1 - u$1)" using p(2)[OF as(1)] unfolding k by(auto simp add:field_simps)
+ finally have "e * (dest_vec1 v - dest_vec1 u) / 2 < e * (dest_vec1 v - dest_vec1 u) / 2"
+ apply- apply(rule less_le_trans[OF result]) using uv by auto thus False by auto qed
+
+ next have *:"\<And>x s1 s2::real. 0 \<le> s1 \<Longrightarrow> x \<le> (s1 + s2) / 2 \<Longrightarrow> x - s1 \<le> s2 / 2" by auto
+ case goal2 show ?case apply(rule *) apply(rule setsum_nonneg) apply(rule,unfold split_paired_all split_conv)
+ defer unfolding setsum_Un_disjoint[OF pA(2-),THEN sym] pA(1)[THEN sym] unfolding setsum_right_distrib[THEN sym]
+ apply(subst additive_tagged_division_1[OF _ as(1)]) unfolding vec1_dest_vec1 apply(rule assms)
+ proof- fix x k assume "(x,k) \<in> p \<inter> {t. fst t \<in> {vec1 a, vec1 b}}" note xk=IntD1[OF this]
+ from p(4)[OF this] guess u v apply-by(erule exE)+ note uv=this
+ with p(2)[OF xk] have "{u..v} \<noteq> {}" by auto
+ thus "0 \<le> e * ((interval_upperbound k)$1 - (interval_lowerbound k)$1)"
+ unfolding uv using e by(auto simp add:field_simps)
+ 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
+ show "norm (\<Sum>(x, k)\<in>p \<inter> ?A. content k *\<^sub>R (f' \<circ> dest_vec1) x -
+ (f ((interval_upperbound k)$1) - f ((interval_lowerbound k)$1))) \<le> e * (b - a) / 2"
+ apply(rule *[where t="p \<inter> {t. fst t \<in> {vec1 a, vec1 b} \<and> content(snd t) \<noteq> 0}"])
+ apply(rule setsum_mono_zero_right[OF pA(2)]) defer apply(rule) unfolding split_paired_all split_conv o_def
+ 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}"
+ 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
+ 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
+ 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
+ next have *:"p \<inter> {t. fst t \<in> {vec1 a, vec1 b} \<and> content(snd t) \<noteq> 0} =
+ {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
+ 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"
+ proof(case_tac "s={}") case goal2 then obtain x where "x\<in>s" by auto hence *:"s = {x}" using goal2(1) by auto
+ thus ?case using `x\<in>s` goal2(2) by auto
+ qed auto
+ 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"])
+ apply(rule norm_triangle_le,rule add_mono) apply(rule_tac[1-2] **)
+ proof- let ?B = "\<lambda>x. {t \<in> p. fst t = vec1 x \<and> content (snd t) \<noteq> 0}"
+ have pa:"\<And>k. (vec1 a, k) \<in> p \<Longrightarrow> \<exists>v. k = {vec1 a .. v} \<and> vec1 a \<le> v"
+ proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this
+ have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto
+ have u:"u = vec1 a" proof(rule ccontr) have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto
+ 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
+ have "u > vec1 a" unfolding Cart_simps by auto
+ thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:Cart_simps)
+ qed thus ?case apply(rule_tac x=v in exI) unfolding uv using * by auto
+ qed
+ have pb:"\<And>k. (vec1 b, k) \<in> p \<Longrightarrow> \<exists>v. k = {v .. vec1 b} \<and> vec1 b \<ge> v"
+ proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this
+ have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto
+ have u:"v = vec1 b" proof(rule ccontr) have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto
+ 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
+ have "v < vec1 b" unfolding Cart_simps by auto
+ thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:Cart_simps)
+ qed thus ?case apply(rule_tac x=u in exI) unfolding uv using * by auto
+ qed
+
+ show "\<forall>x y. x \<in> ?B a \<and> y \<in> ?B a \<longrightarrow> x = y" apply(rule,rule,rule,unfold split_paired_all)
+ unfolding mem_Collect_eq fst_conv snd_conv apply safe
+ proof- fix x k k' assume k:"(vec1 a, k) \<in> p" "(vec1 a, k') \<in> p" "content k \<noteq> 0" "content k' \<noteq> 0"
+ guess v using pa[OF k(1)] .. note v = conjunctD2[OF this]
+ guess v' using pa[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = "vec1 (min (v$1) (v'$1))"
+ have "{vec1 a <..< ?v} \<subseteq> k \<inter> k'" unfolding v v' by(auto simp add:Cart_simps) note subset_interior[OF this,unfolded interior_inter]
+ moreover have "vec1 ((a + ?v$1)/2) \<in> {vec1 a <..< ?v}" using k(3-) unfolding v v' content_eq_0_1 not_le by(auto simp add:Cart_simps)
+ ultimately have "vec1 ((a + ?v$1)/2) \<in> interior k \<inter> interior k'" unfolding interior_open[OF open_interval] by auto
+ hence *:"k = k'" apply- apply(rule ccontr) using p(5)[OF k(1-2)] by auto
+ { assume "x\<in>k" thus "x\<in>k'" unfolding * . } { assume "x\<in>k'" thus "x\<in>k" unfolding * . }
+ qed
+ show "\<forall>x y. x \<in> ?B b \<and> y \<in> ?B b \<longrightarrow> x = y" apply(rule,rule,rule,unfold split_paired_all)
+ unfolding mem_Collect_eq fst_conv snd_conv apply safe
+ proof- fix x k k' assume k:"(vec1 b, k) \<in> p" "(vec1 b, k') \<in> p" "content k \<noteq> 0" "content k' \<noteq> 0"
+ guess v using pb[OF k(1)] .. note v = conjunctD2[OF this]
+ guess v' using pb[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = "vec1 (max (v$1) (v'$1))"
+ have "{?v <..< vec1 b} \<subseteq> k \<inter> k'" unfolding v v' by(auto simp add:Cart_simps) note subset_interior[OF this,unfolded interior_inter]
+ moreover have "vec1 ((b + ?v$1)/2) \<in> {?v <..< vec1 b}" using k(3-) unfolding v v' content_eq_0_1 not_le by(auto simp add:Cart_simps)
+ ultimately have "vec1 ((b + ?v$1)/2) \<in> interior k \<inter> interior k'" unfolding interior_open[OF open_interval] by auto
+ hence *:"k = k'" apply- apply(rule ccontr) using p(5)[OF k(1-2)] by auto
+ { assume "x\<in>k" thus "x\<in>k'" unfolding * . } { assume "x\<in>k'" thus "x\<in>k" unfolding * . }
+ qed
+
+ let ?a = a and ?b = b (* a is something else while proofing the next theorem. *)
+ show "\<forall>x. x \<in> ?B a \<longrightarrow> norm ((\<lambda>(x, k). content k *\<^sub>R f' (x$1) - (f ((interval_upperbound k)$1) - f ((interval_lowerbound k)$1))) x)
+ \<le> e * (b - a) / 4" apply safe unfolding fst_conv snd_conv apply safe unfolding vec1_dest_vec1
+ proof- case goal1 guess v using pa[OF goal1(1)] .. note v = conjunctD2[OF this]
+ have "vec1 ?a\<in>{vec1 ?a..v}" using v(2) by auto hence "dest_vec1 v \<le> ?b" using p(3)[OF goal1(1)] unfolding subset_eq v by auto
+ moreover have "{?a..dest_vec1 v} \<subseteq> ball ?a da" using fineD[OF as(2) goal1(1)]
+ apply-apply(subst(asm) if_P,rule refl) unfolding subset_eq apply safe apply(erule_tac x="vec1 x" in ballE)
+ by(auto simp add:Cart_simps subset_eq dist_real v dist_real_def) ultimately
+ show ?case unfolding v unfolding interval_bounds[OF v(2)[unfolded v vector_le_def]] vec1_dest_vec1 apply-
+ apply(rule da(2)[of "v$1",unfolded vec1_dest_vec1])
+ using goal1 fineD[OF as(2) goal1(1)] unfolding v content_eq_0_1 by auto
+ qed
+ show "\<forall>x. x \<in> ?B b \<longrightarrow> norm ((\<lambda>(x, k). content k *\<^sub>R f' (x$1) - (f ((interval_upperbound k)$1) - f ((interval_lowerbound k)$1))) x)
+ \<le> e * (b - a) / 4" apply safe unfolding fst_conv snd_conv apply safe unfolding vec1_dest_vec1
+ proof- case goal1 guess v using pb[OF goal1(1)] .. note v = conjunctD2[OF this]
+ have "vec1 ?b\<in>{v..vec1 ?b}" using v(2) by auto hence "dest_vec1 v \<ge> ?a" using p(3)[OF goal1(1)] unfolding subset_eq v by auto
+ moreover have "{dest_vec1 v..?b} \<subseteq> ball ?b db" using fineD[OF as(2) goal1(1)]
+ apply-apply(subst(asm) if_P,rule refl) unfolding subset_eq apply safe apply(erule_tac x="vec1 x" in ballE) using ab
+ by(auto simp add:Cart_simps subset_eq dist_real v dist_real_def) ultimately
+ show ?case unfolding v unfolding interval_bounds[OF v(2)[unfolded v vector_le_def]] vec1_dest_vec1 apply-
+ apply(rule db(2)[of "v$1",unfolded vec1_dest_vec1])
+ using goal1 fineD[OF as(2) goal1(1)] unfolding v content_eq_0_1 by auto
+ qed
+ qed(insert p(1) ab e, auto simp add:field_simps) qed auto qed qed qed qed
+
+subsection {* Stronger form with finite number of exceptional points. *}
+
+lemma fundamental_theorem_of_calculus_interior_strong: fixes f::"real \<Rightarrow> 'a::banach"
+ assumes"finite s" "a \<le> b" "continuous_on {a..b} f"
+ "\<forall>x\<in>{a<..<b} - s. (f has_vector_derivative f'(x)) (at x)"
+ shows "((f' o dest_vec1) has_integral (f b - f a)) {vec a..vec b}" using assms apply-
+proof(induct "card s" arbitrary:s a b)
+ case 0 show ?case apply(rule fundamental_theorem_of_calculus_interior) using 0 by auto
+next case (Suc n) from this(2) guess c s' apply-apply(subst(asm) eq_commute) unfolding card_Suc_eq
+ apply(subst(asm)(2) eq_commute) by(erule exE conjE)+ note cs = this[rule_format]
+ show ?case proof(cases "c\<in>{a<..<b}")
+ case False thus ?thesis apply- apply(rule Suc(1)[OF cs(3) _ Suc(4,5)]) apply safe defer
+ apply(rule Suc(6)[rule_format]) using Suc(3) unfolding cs by auto
+ next have *:"f b - f a = (f c - f a) + (f b - f c)" by auto
+ case True hence "vec1 a \<le> vec1 c" "vec1 c \<le> vec1 b" by auto
+ thus ?thesis apply(subst *) apply(rule has_integral_combine) apply assumption+
+ apply(rule_tac[!] Suc(1)[OF cs(3)]) using Suc(3) unfolding cs
+ proof- show "continuous_on {a..c} f" "continuous_on {c..b} f"
+ apply(rule_tac[!] continuous_on_subset[OF Suc(5)]) using True by auto
+ let ?P = "\<lambda>i j. \<forall>x\<in>{i<..<j} - s'. (f has_vector_derivative f' x) (at x)"
+ show "?P a c" "?P c b" apply safe apply(rule_tac[!] Suc(6)[rule_format]) using True unfolding cs by auto
+ qed auto qed qed
+
+lemma fundamental_theorem_of_calculus_strong: fixes f::"real \<Rightarrow> 'a::banach"
+ assumes "finite s" "a \<le> b" "continuous_on {a..b} f"
+ "\<forall>x\<in>{a..b} - s. (f has_vector_derivative f'(x)) (at x)"
+ shows "((f' o dest_vec1) has_integral (f(b) - f(a))) {vec1 a..vec1 b}"
+ apply(rule fundamental_theorem_of_calculus_interior_strong[OF assms(1-3), of f'])
+ using assms(4) by auto
+
+end
--- a/src/HOL/Multivariate_Analysis/Integration_MV.cert Mon Feb 22 20:08:10 2010 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,3296 +0,0 @@
-tB2Atlor9W4pSnrAz5nHpw 907 0
-#2 := false
-#299 := 0::real
-decl uf_1 :: (-> T3 T2 real)
-decl uf_10 :: (-> T4 T2)
-decl uf_7 :: T4
-#15 := uf_7
-#22 := (uf_10 uf_7)
-decl uf_2 :: (-> T1 T3)
-decl uf_4 :: T1
-#11 := uf_4
-#91 := (uf_2 uf_4)
-#902 := (uf_1 #91 #22)
-#297 := -1::real
-#1084 := (* -1::real #902)
-decl uf_16 :: T1
-#50 := uf_16
-#78 := (uf_2 uf_16)
-#799 := (uf_1 #78 #22)
-#1267 := (+ #799 #1084)
-#1272 := (>= #1267 0::real)
-#1266 := (= #799 #902)
-decl uf_9 :: T3
-#21 := uf_9
-#23 := (uf_1 uf_9 #22)
-#905 := (= #23 #902)
-decl uf_11 :: T3
-#24 := uf_11
-#850 := (uf_1 uf_11 #22)
-#904 := (= #850 #902)
-decl uf_6 :: (-> T2 T4)
-#74 := (uf_6 #22)
-#281 := (= uf_7 #74)
-#922 := (ite #281 #905 #904)
-decl uf_8 :: T3
-#18 := uf_8
-#848 := (uf_1 uf_8 #22)
-#903 := (= #848 #902)
-#60 := 0::int
-decl uf_5 :: (-> T4 int)
-#803 := (uf_5 #74)
-#117 := -1::int
-#813 := (* -1::int #803)
-#16 := (uf_5 uf_7)
-#916 := (+ #16 #813)
-#917 := (<= #916 0::int)
-#925 := (ite #917 #922 #903)
-#6 := (:var 0 T2)
-#19 := (uf_1 uf_8 #6)
-#544 := (pattern #19)
-#25 := (uf_1 uf_11 #6)
-#543 := (pattern #25)
-#92 := (uf_1 #91 #6)
-#542 := (pattern #92)
-#13 := (uf_6 #6)
-#541 := (pattern #13)
-#447 := (= #19 #92)
-#445 := (= #25 #92)
-#444 := (= #23 #92)
-#20 := (= #13 uf_7)
-#446 := (ite #20 #444 #445)
-#120 := (* -1::int #16)
-#14 := (uf_5 #13)
-#121 := (+ #14 #120)
-#119 := (>= #121 0::int)
-#448 := (ite #119 #446 #447)
-#545 := (forall (vars (?x3 T2)) (:pat #541 #542 #543 #544) #448)
-#451 := (forall (vars (?x3 T2)) #448)
-#548 := (iff #451 #545)
-#546 := (iff #448 #448)
-#547 := [refl]: #546
-#549 := [quant-intro #547]: #548
-#26 := (ite #20 #23 #25)
-#127 := (ite #119 #26 #19)
-#368 := (= #92 #127)
-#369 := (forall (vars (?x3 T2)) #368)
-#452 := (iff #369 #451)
-#449 := (iff #368 #448)
-#450 := [rewrite]: #449
-#453 := [quant-intro #450]: #452
-#392 := (~ #369 #369)
-#390 := (~ #368 #368)
-#391 := [refl]: #390
-#366 := [nnf-pos #391]: #392
-decl uf_3 :: (-> T1 T2 real)
-#12 := (uf_3 uf_4 #6)
-#132 := (= #12 #127)
-#135 := (forall (vars (?x3 T2)) #132)
-#370 := (iff #135 #369)
-#4 := (:var 1 T1)
-#8 := (uf_3 #4 #6)
-#5 := (uf_2 #4)
-#7 := (uf_1 #5 #6)
-#9 := (= #7 #8)
-#10 := (forall (vars (?x1 T1) (?x2 T2)) #9)
-#113 := [asserted]: #10
-#371 := [rewrite* #113]: #370
-#17 := (< #14 #16)
-#27 := (ite #17 #19 #26)
-#28 := (= #12 #27)
-#29 := (forall (vars (?x3 T2)) #28)
-#136 := (iff #29 #135)
-#133 := (iff #28 #132)
-#130 := (= #27 #127)
-#118 := (not #119)
-#124 := (ite #118 #19 #26)
-#128 := (= #124 #127)
-#129 := [rewrite]: #128
-#125 := (= #27 #124)
-#122 := (iff #17 #118)
-#123 := [rewrite]: #122
-#126 := [monotonicity #123]: #125
-#131 := [trans #126 #129]: #130
-#134 := [monotonicity #131]: #133
-#137 := [quant-intro #134]: #136
-#114 := [asserted]: #29
-#138 := [mp #114 #137]: #135
-#372 := [mp #138 #371]: #369
-#367 := [mp~ #372 #366]: #369
-#454 := [mp #367 #453]: #451
-#550 := [mp #454 #549]: #545
-#738 := (not #545)
-#928 := (or #738 #925)
-#75 := (= #74 uf_7)
-#906 := (ite #75 #905 #904)
-#907 := (+ #803 #120)
-#908 := (>= #907 0::int)
-#909 := (ite #908 #906 #903)
-#929 := (or #738 #909)
-#931 := (iff #929 #928)
-#933 := (iff #928 #928)
-#934 := [rewrite]: #933
-#926 := (iff #909 #925)
-#923 := (iff #906 #922)
-#283 := (iff #75 #281)
-#284 := [rewrite]: #283
-#924 := [monotonicity #284]: #923
-#920 := (iff #908 #917)
-#910 := (+ #120 #803)
-#913 := (>= #910 0::int)
-#918 := (iff #913 #917)
-#919 := [rewrite]: #918
-#914 := (iff #908 #913)
-#911 := (= #907 #910)
-#912 := [rewrite]: #911
-#915 := [monotonicity #912]: #914
-#921 := [trans #915 #919]: #920
-#927 := [monotonicity #921 #924]: #926
-#932 := [monotonicity #927]: #931
-#935 := [trans #932 #934]: #931
-#930 := [quant-inst]: #929
-#936 := [mp #930 #935]: #928
-#1300 := [unit-resolution #936 #550]: #925
-#989 := (= #16 #803)
-#1277 := (= #803 #16)
-#280 := [asserted]: #75
-#287 := [mp #280 #284]: #281
-#1276 := [symm #287]: #75
-#1278 := [monotonicity #1276]: #1277
-#1301 := [symm #1278]: #989
-#1302 := (not #989)
-#1303 := (or #1302 #917)
-#1304 := [th-lemma]: #1303
-#1305 := [unit-resolution #1304 #1301]: #917
-#950 := (not #917)
-#949 := (not #925)
-#951 := (or #949 #950 #922)
-#952 := [def-axiom]: #951
-#1306 := [unit-resolution #952 #1305 #1300]: #922
-#937 := (not #922)
-#1307 := (or #937 #905)
-#938 := (not #281)
-#939 := (or #937 #938 #905)
-#940 := [def-axiom]: #939
-#1308 := [unit-resolution #940 #287]: #1307
-#1309 := [unit-resolution #1308 #1306]: #905
-#1356 := (= #799 #23)
-#800 := (= #23 #799)
-decl uf_15 :: T4
-#40 := uf_15
-#41 := (uf_5 uf_15)
-#814 := (+ #41 #813)
-#815 := (<= #814 0::int)
-#836 := (not #815)
-#158 := (* -1::int #41)
-#1270 := (+ #16 #158)
-#1265 := (>= #1270 0::int)
-#1339 := (not #1265)
-#1269 := (= #16 #41)
-#1298 := (not #1269)
-#286 := (= uf_7 uf_15)
-#44 := (uf_10 uf_15)
-#72 := (uf_6 #44)
-#73 := (= #72 uf_15)
-#277 := (= uf_15 #72)
-#278 := (iff #73 #277)
-#279 := [rewrite]: #278
-#276 := [asserted]: #73
-#282 := [mp #276 #279]: #277
-#1274 := [symm #282]: #73
-#729 := (= uf_7 #72)
-decl uf_17 :: (-> int T4)
-#611 := (uf_5 #72)
-#991 := (uf_17 #611)
-#1289 := (= #991 #72)
-#992 := (= #72 #991)
-#55 := (:var 0 T4)
-#56 := (uf_5 #55)
-#574 := (pattern #56)
-#57 := (uf_17 #56)
-#177 := (= #55 #57)
-#575 := (forall (vars (?x7 T4)) (:pat #574) #177)
-#195 := (forall (vars (?x7 T4)) #177)
-#578 := (iff #195 #575)
-#576 := (iff #177 #177)
-#577 := [refl]: #576
-#579 := [quant-intro #577]: #578
-#405 := (~ #195 #195)
-#403 := (~ #177 #177)
-#404 := [refl]: #403
-#406 := [nnf-pos #404]: #405
-#58 := (= #57 #55)
-#59 := (forall (vars (?x7 T4)) #58)
-#196 := (iff #59 #195)
-#193 := (iff #58 #177)
-#194 := [rewrite]: #193
-#197 := [quant-intro #194]: #196
-#155 := [asserted]: #59
-#200 := [mp #155 #197]: #195
-#407 := [mp~ #200 #406]: #195
-#580 := [mp #407 #579]: #575
-#995 := (not #575)
-#996 := (or #995 #992)
-#997 := [quant-inst]: #996
-#1273 := [unit-resolution #997 #580]: #992
-#1290 := [symm #1273]: #1289
-#1293 := (= uf_7 #991)
-#993 := (uf_17 #803)
-#1287 := (= #993 #991)
-#1284 := (= #803 #611)
-#987 := (= #41 #611)
-#1279 := (= #611 #41)
-#1280 := [monotonicity #1274]: #1279
-#1281 := [symm #1280]: #987
-#1282 := (= #803 #41)
-#1275 := [hypothesis]: #1269
-#1283 := [trans #1278 #1275]: #1282
-#1285 := [trans #1283 #1281]: #1284
-#1288 := [monotonicity #1285]: #1287
-#1291 := (= uf_7 #993)
-#994 := (= #74 #993)
-#1000 := (or #995 #994)
-#1001 := [quant-inst]: #1000
-#1286 := [unit-resolution #1001 #580]: #994
-#1292 := [trans #287 #1286]: #1291
-#1294 := [trans #1292 #1288]: #1293
-#1295 := [trans #1294 #1290]: #729
-#1296 := [trans #1295 #1274]: #286
-#290 := (not #286)
-#76 := (= uf_15 uf_7)
-#77 := (not #76)
-#291 := (iff #77 #290)
-#288 := (iff #76 #286)
-#289 := [rewrite]: #288
-#292 := [monotonicity #289]: #291
-#285 := [asserted]: #77
-#295 := [mp #285 #292]: #290
-#1297 := [unit-resolution #295 #1296]: false
-#1299 := [lemma #1297]: #1298
-#1342 := (or #1269 #1339)
-#1271 := (<= #1270 0::int)
-#621 := (* -1::int #611)
-#723 := (+ #16 #621)
-#724 := (<= #723 0::int)
-decl uf_12 :: T1
-#30 := uf_12
-#88 := (uf_2 uf_12)
-#771 := (uf_1 #88 #44)
-#45 := (uf_1 uf_9 #44)
-#772 := (= #45 #771)
-#796 := (not #772)
-decl uf_14 :: T1
-#38 := uf_14
-#83 := (uf_2 uf_14)
-#656 := (uf_1 #83 #44)
-#1239 := (= #656 #771)
-#1252 := (not #1239)
-#1324 := (iff #1252 #796)
-#1322 := (iff #1239 #772)
-#1320 := (= #656 #45)
-#661 := (= #45 #656)
-#659 := (uf_1 uf_11 #44)
-#664 := (= #656 #659)
-#667 := (ite #277 #661 #664)
-#657 := (uf_1 uf_8 #44)
-#670 := (= #656 #657)
-#622 := (+ #41 #621)
-#623 := (<= #622 0::int)
-#673 := (ite #623 #667 #670)
-#84 := (uf_1 #83 #6)
-#560 := (pattern #84)
-#467 := (= #19 #84)
-#465 := (= #25 #84)
-#464 := (= #45 #84)
-#43 := (= #13 uf_15)
-#466 := (ite #43 #464 #465)
-#159 := (+ #14 #158)
-#157 := (>= #159 0::int)
-#468 := (ite #157 #466 #467)
-#561 := (forall (vars (?x5 T2)) (:pat #541 #560 #543 #544) #468)
-#471 := (forall (vars (?x5 T2)) #468)
-#564 := (iff #471 #561)
-#562 := (iff #468 #468)
-#563 := [refl]: #562
-#565 := [quant-intro #563]: #564
-#46 := (ite #43 #45 #25)
-#165 := (ite #157 #46 #19)
-#378 := (= #84 #165)
-#379 := (forall (vars (?x5 T2)) #378)
-#472 := (iff #379 #471)
-#469 := (iff #378 #468)
-#470 := [rewrite]: #469
-#473 := [quant-intro #470]: #472
-#359 := (~ #379 #379)
-#361 := (~ #378 #378)
-#358 := [refl]: #361
-#356 := [nnf-pos #358]: #359
-#39 := (uf_3 uf_14 #6)
-#170 := (= #39 #165)
-#173 := (forall (vars (?x5 T2)) #170)
-#380 := (iff #173 #379)
-#381 := [rewrite* #113]: #380
-#42 := (< #14 #41)
-#47 := (ite #42 #19 #46)
-#48 := (= #39 #47)
-#49 := (forall (vars (?x5 T2)) #48)
-#174 := (iff #49 #173)
-#171 := (iff #48 #170)
-#168 := (= #47 #165)
-#156 := (not #157)
-#162 := (ite #156 #19 #46)
-#166 := (= #162 #165)
-#167 := [rewrite]: #166
-#163 := (= #47 #162)
-#160 := (iff #42 #156)
-#161 := [rewrite]: #160
-#164 := [monotonicity #161]: #163
-#169 := [trans #164 #167]: #168
-#172 := [monotonicity #169]: #171
-#175 := [quant-intro #172]: #174
-#116 := [asserted]: #49
-#176 := [mp #116 #175]: #173
-#382 := [mp #176 #381]: #379
-#357 := [mp~ #382 #356]: #379
-#474 := [mp #357 #473]: #471
-#566 := [mp #474 #565]: #561
-#676 := (not #561)
-#677 := (or #676 #673)
-#658 := (= #657 #656)
-#660 := (= #659 #656)
-#662 := (ite #73 #661 #660)
-#612 := (+ #611 #158)
-#613 := (>= #612 0::int)
-#663 := (ite #613 #662 #658)
-#678 := (or #676 #663)
-#680 := (iff #678 #677)
-#682 := (iff #677 #677)
-#683 := [rewrite]: #682
-#674 := (iff #663 #673)
-#671 := (iff #658 #670)
-#672 := [rewrite]: #671
-#668 := (iff #662 #667)
-#665 := (iff #660 #664)
-#666 := [rewrite]: #665
-#669 := [monotonicity #279 #666]: #668
-#626 := (iff #613 #623)
-#615 := (+ #158 #611)
-#618 := (>= #615 0::int)
-#624 := (iff #618 #623)
-#625 := [rewrite]: #624
-#619 := (iff #613 #618)
-#616 := (= #612 #615)
-#617 := [rewrite]: #616
-#620 := [monotonicity #617]: #619
-#627 := [trans #620 #625]: #626
-#675 := [monotonicity #627 #669 #672]: #674
-#681 := [monotonicity #675]: #680
-#684 := [trans #681 #683]: #680
-#679 := [quant-inst]: #678
-#685 := [mp #679 #684]: #677
-#1311 := [unit-resolution #685 #566]: #673
-#1312 := (not #987)
-#1313 := (or #1312 #623)
-#1314 := [th-lemma]: #1313
-#1315 := [unit-resolution #1314 #1281]: #623
-#645 := (not #623)
-#698 := (not #673)
-#699 := (or #698 #645 #667)
-#700 := [def-axiom]: #699
-#1316 := [unit-resolution #700 #1315 #1311]: #667
-#686 := (not #667)
-#1317 := (or #686 #661)
-#687 := (not #277)
-#688 := (or #686 #687 #661)
-#689 := [def-axiom]: #688
-#1318 := [unit-resolution #689 #282]: #1317
-#1319 := [unit-resolution #1318 #1316]: #661
-#1321 := [symm #1319]: #1320
-#1323 := [monotonicity #1321]: #1322
-#1325 := [monotonicity #1323]: #1324
-#1145 := (* -1::real #771)
-#1240 := (+ #656 #1145)
-#1241 := (<= #1240 0::real)
-#1249 := (not #1241)
-#1243 := [hypothesis]: #1241
-decl uf_18 :: T3
-#80 := uf_18
-#1040 := (uf_1 uf_18 #44)
-#1043 := (* -1::real #1040)
-#1156 := (+ #771 #1043)
-#1157 := (>= #1156 0::real)
-#1189 := (not #1157)
-#708 := (uf_1 #91 #44)
-#1168 := (+ #708 #1043)
-#1169 := (<= #1168 0::real)
-#1174 := (or #1157 #1169)
-#1177 := (not #1174)
-#89 := (uf_1 #88 #6)
-#552 := (pattern #89)
-#81 := (uf_1 uf_18 #6)
-#594 := (pattern #81)
-#324 := (* -1::real #92)
-#325 := (+ #81 #324)
-#323 := (>= #325 0::real)
-#317 := (* -1::real #89)
-#318 := (+ #81 #317)
-#319 := (<= #318 0::real)
-#436 := (or #319 #323)
-#437 := (not #436)
-#601 := (forall (vars (?x11 T2)) (:pat #594 #552 #542) #437)
-#440 := (forall (vars (?x11 T2)) #437)
-#604 := (iff #440 #601)
-#602 := (iff #437 #437)
-#603 := [refl]: #602
-#605 := [quant-intro #603]: #604
-#326 := (not #323)
-#320 := (not #319)
-#329 := (and #320 #326)
-#332 := (forall (vars (?x11 T2)) #329)
-#441 := (iff #332 #440)
-#438 := (iff #329 #437)
-#439 := [rewrite]: #438
-#442 := [quant-intro #439]: #441
-#425 := (~ #332 #332)
-#423 := (~ #329 #329)
-#424 := [refl]: #423
-#426 := [nnf-pos #424]: #425
-#306 := (* -1::real #84)
-#307 := (+ #81 #306)
-#305 := (>= #307 0::real)
-#308 := (not #305)
-#301 := (* -1::real #81)
-#79 := (uf_1 #78 #6)
-#302 := (+ #79 #301)
-#300 := (>= #302 0::real)
-#298 := (not #300)
-#311 := (and #298 #308)
-#314 := (forall (vars (?x10 T2)) #311)
-#335 := (and #314 #332)
-#93 := (< #81 #92)
-#90 := (< #89 #81)
-#94 := (and #90 #93)
-#95 := (forall (vars (?x11 T2)) #94)
-#85 := (< #81 #84)
-#82 := (< #79 #81)
-#86 := (and #82 #85)
-#87 := (forall (vars (?x10 T2)) #86)
-#96 := (and #87 #95)
-#336 := (iff #96 #335)
-#333 := (iff #95 #332)
-#330 := (iff #94 #329)
-#327 := (iff #93 #326)
-#328 := [rewrite]: #327
-#321 := (iff #90 #320)
-#322 := [rewrite]: #321
-#331 := [monotonicity #322 #328]: #330
-#334 := [quant-intro #331]: #333
-#315 := (iff #87 #314)
-#312 := (iff #86 #311)
-#309 := (iff #85 #308)
-#310 := [rewrite]: #309
-#303 := (iff #82 #298)
-#304 := [rewrite]: #303
-#313 := [monotonicity #304 #310]: #312
-#316 := [quant-intro #313]: #315
-#337 := [monotonicity #316 #334]: #336
-#293 := [asserted]: #96
-#338 := [mp #293 #337]: #335
-#340 := [and-elim #338]: #332
-#427 := [mp~ #340 #426]: #332
-#443 := [mp #427 #442]: #440
-#606 := [mp #443 #605]: #601
-#1124 := (not #601)
-#1180 := (or #1124 #1177)
-#1142 := (* -1::real #708)
-#1143 := (+ #1040 #1142)
-#1144 := (>= #1143 0::real)
-#1146 := (+ #1040 #1145)
-#1147 := (<= #1146 0::real)
-#1148 := (or #1147 #1144)
-#1149 := (not #1148)
-#1181 := (or #1124 #1149)
-#1183 := (iff #1181 #1180)
-#1185 := (iff #1180 #1180)
-#1186 := [rewrite]: #1185
-#1178 := (iff #1149 #1177)
-#1175 := (iff #1148 #1174)
-#1172 := (iff #1144 #1169)
-#1162 := (+ #1142 #1040)
-#1165 := (>= #1162 0::real)
-#1170 := (iff #1165 #1169)
-#1171 := [rewrite]: #1170
-#1166 := (iff #1144 #1165)
-#1163 := (= #1143 #1162)
-#1164 := [rewrite]: #1163
-#1167 := [monotonicity #1164]: #1166
-#1173 := [trans #1167 #1171]: #1172
-#1160 := (iff #1147 #1157)
-#1150 := (+ #1145 #1040)
-#1153 := (<= #1150 0::real)
-#1158 := (iff #1153 #1157)
-#1159 := [rewrite]: #1158
-#1154 := (iff #1147 #1153)
-#1151 := (= #1146 #1150)
-#1152 := [rewrite]: #1151
-#1155 := [monotonicity #1152]: #1154
-#1161 := [trans #1155 #1159]: #1160
-#1176 := [monotonicity #1161 #1173]: #1175
-#1179 := [monotonicity #1176]: #1178
-#1184 := [monotonicity #1179]: #1183
-#1187 := [trans #1184 #1186]: #1183
-#1182 := [quant-inst]: #1181
-#1188 := [mp #1182 #1187]: #1180
-#1244 := [unit-resolution #1188 #606]: #1177
-#1190 := (or #1174 #1189)
-#1191 := [def-axiom]: #1190
-#1245 := [unit-resolution #1191 #1244]: #1189
-#1054 := (+ #656 #1043)
-#1055 := (<= #1054 0::real)
-#1079 := (not #1055)
-#607 := (uf_1 #78 #44)
-#1044 := (+ #607 #1043)
-#1045 := (>= #1044 0::real)
-#1060 := (or #1045 #1055)
-#1063 := (not #1060)
-#567 := (pattern #79)
-#428 := (or #300 #305)
-#429 := (not #428)
-#595 := (forall (vars (?x10 T2)) (:pat #567 #594 #560) #429)
-#432 := (forall (vars (?x10 T2)) #429)
-#598 := (iff #432 #595)
-#596 := (iff #429 #429)
-#597 := [refl]: #596
-#599 := [quant-intro #597]: #598
-#433 := (iff #314 #432)
-#430 := (iff #311 #429)
-#431 := [rewrite]: #430
-#434 := [quant-intro #431]: #433
-#420 := (~ #314 #314)
-#418 := (~ #311 #311)
-#419 := [refl]: #418
-#421 := [nnf-pos #419]: #420
-#339 := [and-elim #338]: #314
-#422 := [mp~ #339 #421]: #314
-#435 := [mp #422 #434]: #432
-#600 := [mp #435 #599]: #595
-#1066 := (not #595)
-#1067 := (or #1066 #1063)
-#1039 := (* -1::real #656)
-#1041 := (+ #1040 #1039)
-#1042 := (>= #1041 0::real)
-#1046 := (or #1045 #1042)
-#1047 := (not #1046)
-#1068 := (or #1066 #1047)
-#1070 := (iff #1068 #1067)
-#1072 := (iff #1067 #1067)
-#1073 := [rewrite]: #1072
-#1064 := (iff #1047 #1063)
-#1061 := (iff #1046 #1060)
-#1058 := (iff #1042 #1055)
-#1048 := (+ #1039 #1040)
-#1051 := (>= #1048 0::real)
-#1056 := (iff #1051 #1055)
-#1057 := [rewrite]: #1056
-#1052 := (iff #1042 #1051)
-#1049 := (= #1041 #1048)
-#1050 := [rewrite]: #1049
-#1053 := [monotonicity #1050]: #1052
-#1059 := [trans #1053 #1057]: #1058
-#1062 := [monotonicity #1059]: #1061
-#1065 := [monotonicity #1062]: #1064
-#1071 := [monotonicity #1065]: #1070
-#1074 := [trans #1071 #1073]: #1070
-#1069 := [quant-inst]: #1068
-#1075 := [mp #1069 #1074]: #1067
-#1246 := [unit-resolution #1075 #600]: #1063
-#1080 := (or #1060 #1079)
-#1081 := [def-axiom]: #1080
-#1247 := [unit-resolution #1081 #1246]: #1079
-#1248 := [th-lemma #1247 #1245 #1243]: false
-#1250 := [lemma #1248]: #1249
-#1253 := (or #1252 #1241)
-#1254 := [th-lemma]: #1253
-#1310 := [unit-resolution #1254 #1250]: #1252
-#1326 := [mp #1310 #1325]: #796
-#1328 := (or #724 #772)
-decl uf_13 :: T3
-#33 := uf_13
-#609 := (uf_1 uf_13 #44)
-#773 := (= #609 #771)
-#775 := (ite #724 #773 #772)
-#32 := (uf_1 uf_9 #6)
-#553 := (pattern #32)
-#34 := (uf_1 uf_13 #6)
-#551 := (pattern #34)
-#456 := (= #32 #89)
-#455 := (= #34 #89)
-#457 := (ite #119 #455 #456)
-#554 := (forall (vars (?x4 T2)) (:pat #541 #551 #552 #553) #457)
-#460 := (forall (vars (?x4 T2)) #457)
-#557 := (iff #460 #554)
-#555 := (iff #457 #457)
-#556 := [refl]: #555
-#558 := [quant-intro #556]: #557
-#143 := (ite #119 #34 #32)
-#373 := (= #89 #143)
-#374 := (forall (vars (?x4 T2)) #373)
-#461 := (iff #374 #460)
-#458 := (iff #373 #457)
-#459 := [rewrite]: #458
-#462 := [quant-intro #459]: #461
-#362 := (~ #374 #374)
-#364 := (~ #373 #373)
-#365 := [refl]: #364
-#363 := [nnf-pos #365]: #362
-#31 := (uf_3 uf_12 #6)
-#148 := (= #31 #143)
-#151 := (forall (vars (?x4 T2)) #148)
-#375 := (iff #151 #374)
-#376 := [rewrite* #113]: #375
-#35 := (ite #17 #32 #34)
-#36 := (= #31 #35)
-#37 := (forall (vars (?x4 T2)) #36)
-#152 := (iff #37 #151)
-#149 := (iff #36 #148)
-#146 := (= #35 #143)
-#140 := (ite #118 #32 #34)
-#144 := (= #140 #143)
-#145 := [rewrite]: #144
-#141 := (= #35 #140)
-#142 := [monotonicity #123]: #141
-#147 := [trans #142 #145]: #146
-#150 := [monotonicity #147]: #149
-#153 := [quant-intro #150]: #152
-#115 := [asserted]: #37
-#154 := [mp #115 #153]: #151
-#377 := [mp #154 #376]: #374
-#360 := [mp~ #377 #363]: #374
-#463 := [mp #360 #462]: #460
-#559 := [mp #463 #558]: #554
-#778 := (not #554)
-#779 := (or #778 #775)
-#714 := (+ #611 #120)
-#715 := (>= #714 0::int)
-#774 := (ite #715 #773 #772)
-#780 := (or #778 #774)
-#782 := (iff #780 #779)
-#784 := (iff #779 #779)
-#785 := [rewrite]: #784
-#776 := (iff #774 #775)
-#727 := (iff #715 #724)
-#717 := (+ #120 #611)
-#720 := (>= #717 0::int)
-#725 := (iff #720 #724)
-#726 := [rewrite]: #725
-#721 := (iff #715 #720)
-#718 := (= #714 #717)
-#719 := [rewrite]: #718
-#722 := [monotonicity #719]: #721
-#728 := [trans #722 #726]: #727
-#777 := [monotonicity #728]: #776
-#783 := [monotonicity #777]: #782
-#786 := [trans #783 #785]: #782
-#781 := [quant-inst]: #780
-#787 := [mp #781 #786]: #779
-#1327 := [unit-resolution #787 #559]: #775
-#788 := (not #775)
-#791 := (or #788 #724 #772)
-#792 := [def-axiom]: #791
-#1329 := [unit-resolution #792 #1327]: #1328
-#1330 := [unit-resolution #1329 #1326]: #724
-#988 := (>= #622 0::int)
-#1331 := (or #1312 #988)
-#1332 := [th-lemma]: #1331
-#1333 := [unit-resolution #1332 #1281]: #988
-#761 := (not #724)
-#1334 := (not #988)
-#1335 := (or #1271 #1334 #761)
-#1336 := [th-lemma]: #1335
-#1337 := [unit-resolution #1336 #1333 #1330]: #1271
-#1338 := (not #1271)
-#1340 := (or #1269 #1338 #1339)
-#1341 := [th-lemma]: #1340
-#1343 := [unit-resolution #1341 #1337]: #1342
-#1344 := [unit-resolution #1343 #1299]: #1339
-#990 := (>= #916 0::int)
-#1345 := (or #1302 #990)
-#1346 := [th-lemma]: #1345
-#1347 := [unit-resolution #1346 #1301]: #990
-#1348 := (not #990)
-#1349 := (or #836 #1348 #1265)
-#1350 := [th-lemma]: #1349
-#1351 := [unit-resolution #1350 #1347 #1344]: #836
-#1353 := (or #815 #800)
-#801 := (uf_1 uf_13 #22)
-#820 := (= #799 #801)
-#823 := (ite #815 #820 #800)
-#476 := (= #32 #79)
-#475 := (= #34 #79)
-#477 := (ite #157 #475 #476)
-#568 := (forall (vars (?x6 T2)) (:pat #541 #551 #567 #553) #477)
-#480 := (forall (vars (?x6 T2)) #477)
-#571 := (iff #480 #568)
-#569 := (iff #477 #477)
-#570 := [refl]: #569
-#572 := [quant-intro #570]: #571
-#181 := (ite #157 #34 #32)
-#383 := (= #79 #181)
-#384 := (forall (vars (?x6 T2)) #383)
-#481 := (iff #384 #480)
-#478 := (iff #383 #477)
-#479 := [rewrite]: #478
-#482 := [quant-intro #479]: #481
-#352 := (~ #384 #384)
-#354 := (~ #383 #383)
-#355 := [refl]: #354
-#353 := [nnf-pos #355]: #352
-#51 := (uf_3 uf_16 #6)
-#186 := (= #51 #181)
-#189 := (forall (vars (?x6 T2)) #186)
-#385 := (iff #189 #384)
-#386 := [rewrite* #113]: #385
-#52 := (ite #42 #32 #34)
-#53 := (= #51 #52)
-#54 := (forall (vars (?x6 T2)) #53)
-#190 := (iff #54 #189)
-#187 := (iff #53 #186)
-#184 := (= #52 #181)
-#178 := (ite #156 #32 #34)
-#182 := (= #178 #181)
-#183 := [rewrite]: #182
-#179 := (= #52 #178)
-#180 := [monotonicity #161]: #179
-#185 := [trans #180 #183]: #184
-#188 := [monotonicity #185]: #187
-#191 := [quant-intro #188]: #190
-#139 := [asserted]: #54
-#192 := [mp #139 #191]: #189
-#387 := [mp #192 #386]: #384
-#402 := [mp~ #387 #353]: #384
-#483 := [mp #402 #482]: #480
-#573 := [mp #483 #572]: #568
-#634 := (not #568)
-#826 := (or #634 #823)
-#802 := (= #801 #799)
-#804 := (+ #803 #158)
-#805 := (>= #804 0::int)
-#806 := (ite #805 #802 #800)
-#827 := (or #634 #806)
-#829 := (iff #827 #826)
-#831 := (iff #826 #826)
-#832 := [rewrite]: #831
-#824 := (iff #806 #823)
-#821 := (iff #802 #820)
-#822 := [rewrite]: #821
-#818 := (iff #805 #815)
-#807 := (+ #158 #803)
-#810 := (>= #807 0::int)
-#816 := (iff #810 #815)
-#817 := [rewrite]: #816
-#811 := (iff #805 #810)
-#808 := (= #804 #807)
-#809 := [rewrite]: #808
-#812 := [monotonicity #809]: #811
-#819 := [trans #812 #817]: #818
-#825 := [monotonicity #819 #822]: #824
-#830 := [monotonicity #825]: #829
-#833 := [trans #830 #832]: #829
-#828 := [quant-inst]: #827
-#834 := [mp #828 #833]: #826
-#1352 := [unit-resolution #834 #573]: #823
-#835 := (not #823)
-#839 := (or #835 #815 #800)
-#840 := [def-axiom]: #839
-#1354 := [unit-resolution #840 #1352]: #1353
-#1355 := [unit-resolution #1354 #1351]: #800
-#1357 := [symm #1355]: #1356
-#1358 := [trans #1357 #1309]: #1266
-#1359 := (not #1266)
-#1360 := (or #1359 #1272)
-#1361 := [th-lemma]: #1360
-#1362 := [unit-resolution #1361 #1358]: #1272
-#1085 := (uf_1 uf_18 #22)
-#1099 := (* -1::real #1085)
-#1112 := (+ #902 #1099)
-#1113 := (<= #1112 0::real)
-#1137 := (not #1113)
-#960 := (uf_1 #88 #22)
-#1100 := (+ #960 #1099)
-#1101 := (>= #1100 0::real)
-#1118 := (or #1101 #1113)
-#1121 := (not #1118)
-#1125 := (or #1124 #1121)
-#1086 := (+ #1085 #1084)
-#1087 := (>= #1086 0::real)
-#1088 := (* -1::real #960)
-#1089 := (+ #1085 #1088)
-#1090 := (<= #1089 0::real)
-#1091 := (or #1090 #1087)
-#1092 := (not #1091)
-#1126 := (or #1124 #1092)
-#1128 := (iff #1126 #1125)
-#1130 := (iff #1125 #1125)
-#1131 := [rewrite]: #1130
-#1122 := (iff #1092 #1121)
-#1119 := (iff #1091 #1118)
-#1116 := (iff #1087 #1113)
-#1106 := (+ #1084 #1085)
-#1109 := (>= #1106 0::real)
-#1114 := (iff #1109 #1113)
-#1115 := [rewrite]: #1114
-#1110 := (iff #1087 #1109)
-#1107 := (= #1086 #1106)
-#1108 := [rewrite]: #1107
-#1111 := [monotonicity #1108]: #1110
-#1117 := [trans #1111 #1115]: #1116
-#1104 := (iff #1090 #1101)
-#1093 := (+ #1088 #1085)
-#1096 := (<= #1093 0::real)
-#1102 := (iff #1096 #1101)
-#1103 := [rewrite]: #1102
-#1097 := (iff #1090 #1096)
-#1094 := (= #1089 #1093)
-#1095 := [rewrite]: #1094
-#1098 := [monotonicity #1095]: #1097
-#1105 := [trans #1098 #1103]: #1104
-#1120 := [monotonicity #1105 #1117]: #1119
-#1123 := [monotonicity #1120]: #1122
-#1129 := [monotonicity #1123]: #1128
-#1132 := [trans #1129 #1131]: #1128
-#1127 := [quant-inst]: #1126
-#1133 := [mp #1127 #1132]: #1125
-#1363 := [unit-resolution #1133 #606]: #1121
-#1138 := (or #1118 #1137)
-#1139 := [def-axiom]: #1138
-#1364 := [unit-resolution #1139 #1363]: #1137
-#1200 := (+ #799 #1099)
-#1201 := (>= #1200 0::real)
-#1231 := (not #1201)
-#847 := (uf_1 #83 #22)
-#1210 := (+ #847 #1099)
-#1211 := (<= #1210 0::real)
-#1216 := (or #1201 #1211)
-#1219 := (not #1216)
-#1222 := (or #1066 #1219)
-#1197 := (* -1::real #847)
-#1198 := (+ #1085 #1197)
-#1199 := (>= #1198 0::real)
-#1202 := (or #1201 #1199)
-#1203 := (not #1202)
-#1223 := (or #1066 #1203)
-#1225 := (iff #1223 #1222)
-#1227 := (iff #1222 #1222)
-#1228 := [rewrite]: #1227
-#1220 := (iff #1203 #1219)
-#1217 := (iff #1202 #1216)
-#1214 := (iff #1199 #1211)
-#1204 := (+ #1197 #1085)
-#1207 := (>= #1204 0::real)
-#1212 := (iff #1207 #1211)
-#1213 := [rewrite]: #1212
-#1208 := (iff #1199 #1207)
-#1205 := (= #1198 #1204)
-#1206 := [rewrite]: #1205
-#1209 := [monotonicity #1206]: #1208
-#1215 := [trans #1209 #1213]: #1214
-#1218 := [monotonicity #1215]: #1217
-#1221 := [monotonicity #1218]: #1220
-#1226 := [monotonicity #1221]: #1225
-#1229 := [trans #1226 #1228]: #1225
-#1224 := [quant-inst]: #1223
-#1230 := [mp #1224 #1229]: #1222
-#1365 := [unit-resolution #1230 #600]: #1219
-#1232 := (or #1216 #1231)
-#1233 := [def-axiom]: #1232
-#1366 := [unit-resolution #1233 #1365]: #1231
-[th-lemma #1366 #1364 #1362]: false
-unsat
-NQHwTeL311Tq3wf2s5BReA 419 0
-#2 := false
-#194 := 0::real
-decl uf_4 :: (-> T2 T3 real)
-decl uf_6 :: (-> T1 T3)
-decl uf_3 :: T1
-#21 := uf_3
-#25 := (uf_6 uf_3)
-decl uf_5 :: T2
-#24 := uf_5
-#26 := (uf_4 uf_5 #25)
-decl uf_7 :: T2
-#27 := uf_7
-#28 := (uf_4 uf_7 #25)
-decl uf_10 :: T1
-#38 := uf_10
-#42 := (uf_6 uf_10)
-decl uf_9 :: T2
-#33 := uf_9
-#43 := (uf_4 uf_9 #42)
-#41 := (= uf_3 uf_10)
-#44 := (ite #41 #43 #28)
-#9 := 0::int
-decl uf_2 :: (-> T1 int)
-#39 := (uf_2 uf_10)
-#226 := -1::int
-#229 := (* -1::int #39)
-#22 := (uf_2 uf_3)
-#230 := (+ #22 #229)
-#228 := (>= #230 0::int)
-#236 := (ite #228 #44 #26)
-#192 := -1::real
-#244 := (* -1::real #236)
-#642 := (+ #26 #244)
-#643 := (<= #642 0::real)
-#567 := (= #26 #236)
-#227 := (not #228)
-decl uf_1 :: (-> int T1)
-#593 := (uf_1 #39)
-#660 := (= #593 uf_10)
-#594 := (= uf_10 #593)
-#4 := (:var 0 T1)
-#5 := (uf_2 #4)
-#546 := (pattern #5)
-#6 := (uf_1 #5)
-#93 := (= #4 #6)
-#547 := (forall (vars (?x1 T1)) (:pat #546) #93)
-#96 := (forall (vars (?x1 T1)) #93)
-#550 := (iff #96 #547)
-#548 := (iff #93 #93)
-#549 := [refl]: #548
-#551 := [quant-intro #549]: #550
-#448 := (~ #96 #96)
-#450 := (~ #93 #93)
-#451 := [refl]: #450
-#449 := [nnf-pos #451]: #448
-#7 := (= #6 #4)
-#8 := (forall (vars (?x1 T1)) #7)
-#97 := (iff #8 #96)
-#94 := (iff #7 #93)
-#95 := [rewrite]: #94
-#98 := [quant-intro #95]: #97
-#92 := [asserted]: #8
-#101 := [mp #92 #98]: #96
-#446 := [mp~ #101 #449]: #96
-#552 := [mp #446 #551]: #547
-#595 := (not #547)
-#600 := (or #595 #594)
-#601 := [quant-inst]: #600
-#654 := [unit-resolution #601 #552]: #594
-#680 := [symm #654]: #660
-#681 := (= uf_3 #593)
-#591 := (uf_1 #22)
-#658 := (= #591 #593)
-#656 := (= #593 #591)
-#652 := (= #39 #22)
-#647 := (= #22 #39)
-#290 := (<= #230 0::int)
-#70 := (<= #22 #39)
-#388 := (iff #70 #290)
-#389 := [rewrite]: #388
-#341 := [asserted]: #70
-#390 := [mp #341 #389]: #290
-#646 := [hypothesis]: #228
-#648 := [th-lemma #646 #390]: #647
-#653 := [symm #648]: #652
-#657 := [monotonicity #653]: #656
-#659 := [symm #657]: #658
-#592 := (= uf_3 #591)
-#596 := (or #595 #592)
-#597 := [quant-inst]: #596
-#655 := [unit-resolution #597 #552]: #592
-#682 := [trans #655 #659]: #681
-#683 := [trans #682 #680]: #41
-#570 := (not #41)
-decl uf_11 :: T2
-#47 := uf_11
-#59 := (uf_4 uf_11 #42)
-#278 := (ite #41 #26 #59)
-#459 := (* -1::real #278)
-#637 := (+ #26 #459)
-#639 := (>= #637 0::real)
-#585 := (= #26 #278)
-#661 := [hypothesis]: #41
-#587 := (or #570 #585)
-#588 := [def-axiom]: #587
-#662 := [unit-resolution #588 #661]: #585
-#663 := (not #585)
-#664 := (or #663 #639)
-#665 := [th-lemma]: #664
-#666 := [unit-resolution #665 #662]: #639
-decl uf_8 :: T2
-#30 := uf_8
-#56 := (uf_4 uf_8 #42)
-#357 := (* -1::real #56)
-#358 := (+ #43 #357)
-#356 := (>= #358 0::real)
-#355 := (not #356)
-#374 := (* -1::real #59)
-#375 := (+ #56 #374)
-#373 := (>= #375 0::real)
-#376 := (not #373)
-#381 := (and #355 #376)
-#64 := (< #39 #39)
-#67 := (ite #64 #43 #59)
-#68 := (< #56 #67)
-#53 := (uf_4 uf_5 #42)
-#65 := (ite #64 #53 #43)
-#66 := (< #65 #56)
-#69 := (and #66 #68)
-#382 := (iff #69 #381)
-#379 := (iff #68 #376)
-#370 := (< #56 #59)
-#377 := (iff #370 #376)
-#378 := [rewrite]: #377
-#371 := (iff #68 #370)
-#368 := (= #67 #59)
-#363 := (ite false #43 #59)
-#366 := (= #363 #59)
-#367 := [rewrite]: #366
-#364 := (= #67 #363)
-#343 := (iff #64 false)
-#344 := [rewrite]: #343
-#365 := [monotonicity #344]: #364
-#369 := [trans #365 #367]: #368
-#372 := [monotonicity #369]: #371
-#380 := [trans #372 #378]: #379
-#361 := (iff #66 #355)
-#352 := (< #43 #56)
-#359 := (iff #352 #355)
-#360 := [rewrite]: #359
-#353 := (iff #66 #352)
-#350 := (= #65 #43)
-#345 := (ite false #53 #43)
-#348 := (= #345 #43)
-#349 := [rewrite]: #348
-#346 := (= #65 #345)
-#347 := [monotonicity #344]: #346
-#351 := [trans #347 #349]: #350
-#354 := [monotonicity #351]: #353
-#362 := [trans #354 #360]: #361
-#383 := [monotonicity #362 #380]: #382
-#340 := [asserted]: #69
-#384 := [mp #340 #383]: #381
-#385 := [and-elim #384]: #355
-#394 := (* -1::real #53)
-#395 := (+ #43 #394)
-#393 := (>= #395 0::real)
-#54 := (uf_4 uf_7 #42)
-#402 := (* -1::real #54)
-#403 := (+ #53 #402)
-#401 := (>= #403 0::real)
-#397 := (+ #43 #374)
-#398 := (<= #397 0::real)
-#412 := (and #393 #398 #401)
-#73 := (<= #43 #59)
-#72 := (<= #53 #43)
-#74 := (and #72 #73)
-#71 := (<= #54 #53)
-#75 := (and #71 #74)
-#415 := (iff #75 #412)
-#406 := (and #393 #398)
-#409 := (and #401 #406)
-#413 := (iff #409 #412)
-#414 := [rewrite]: #413
-#410 := (iff #75 #409)
-#407 := (iff #74 #406)
-#399 := (iff #73 #398)
-#400 := [rewrite]: #399
-#392 := (iff #72 #393)
-#396 := [rewrite]: #392
-#408 := [monotonicity #396 #400]: #407
-#404 := (iff #71 #401)
-#405 := [rewrite]: #404
-#411 := [monotonicity #405 #408]: #410
-#416 := [trans #411 #414]: #415
-#342 := [asserted]: #75
-#417 := [mp #342 #416]: #412
-#418 := [and-elim #417]: #393
-#650 := (+ #26 #394)
-#651 := (<= #650 0::real)
-#649 := (= #26 #53)
-#671 := (= #53 #26)
-#669 := (= #42 #25)
-#667 := (= #25 #42)
-#668 := [monotonicity #661]: #667
-#670 := [symm #668]: #669
-#672 := [monotonicity #670]: #671
-#673 := [symm #672]: #649
-#674 := (not #649)
-#675 := (or #674 #651)
-#676 := [th-lemma]: #675
-#677 := [unit-resolution #676 #673]: #651
-#462 := (+ #56 #459)
-#465 := (>= #462 0::real)
-#438 := (not #465)
-#316 := (ite #290 #278 #43)
-#326 := (* -1::real #316)
-#327 := (+ #56 #326)
-#325 := (>= #327 0::real)
-#324 := (not #325)
-#439 := (iff #324 #438)
-#466 := (iff #325 #465)
-#463 := (= #327 #462)
-#460 := (= #326 #459)
-#457 := (= #316 #278)
-#1 := true
-#452 := (ite true #278 #43)
-#455 := (= #452 #278)
-#456 := [rewrite]: #455
-#453 := (= #316 #452)
-#444 := (iff #290 true)
-#445 := [iff-true #390]: #444
-#454 := [monotonicity #445]: #453
-#458 := [trans #454 #456]: #457
-#461 := [monotonicity #458]: #460
-#464 := [monotonicity #461]: #463
-#467 := [monotonicity #464]: #466
-#468 := [monotonicity #467]: #439
-#297 := (ite #290 #54 #53)
-#305 := (* -1::real #297)
-#306 := (+ #56 #305)
-#307 := (<= #306 0::real)
-#308 := (not #307)
-#332 := (and #308 #324)
-#58 := (= uf_10 uf_3)
-#60 := (ite #58 #26 #59)
-#52 := (< #39 #22)
-#61 := (ite #52 #43 #60)
-#62 := (< #56 #61)
-#55 := (ite #52 #53 #54)
-#57 := (< #55 #56)
-#63 := (and #57 #62)
-#335 := (iff #63 #332)
-#281 := (ite #52 #43 #278)
-#284 := (< #56 #281)
-#287 := (and #57 #284)
-#333 := (iff #287 #332)
-#330 := (iff #284 #324)
-#321 := (< #56 #316)
-#328 := (iff #321 #324)
-#329 := [rewrite]: #328
-#322 := (iff #284 #321)
-#319 := (= #281 #316)
-#291 := (not #290)
-#313 := (ite #291 #43 #278)
-#317 := (= #313 #316)
-#318 := [rewrite]: #317
-#314 := (= #281 #313)
-#292 := (iff #52 #291)
-#293 := [rewrite]: #292
-#315 := [monotonicity #293]: #314
-#320 := [trans #315 #318]: #319
-#323 := [monotonicity #320]: #322
-#331 := [trans #323 #329]: #330
-#311 := (iff #57 #308)
-#302 := (< #297 #56)
-#309 := (iff #302 #308)
-#310 := [rewrite]: #309
-#303 := (iff #57 #302)
-#300 := (= #55 #297)
-#294 := (ite #291 #53 #54)
-#298 := (= #294 #297)
-#299 := [rewrite]: #298
-#295 := (= #55 #294)
-#296 := [monotonicity #293]: #295
-#301 := [trans #296 #299]: #300
-#304 := [monotonicity #301]: #303
-#312 := [trans #304 #310]: #311
-#334 := [monotonicity #312 #331]: #333
-#288 := (iff #63 #287)
-#285 := (iff #62 #284)
-#282 := (= #61 #281)
-#279 := (= #60 #278)
-#225 := (iff #58 #41)
-#277 := [rewrite]: #225
-#280 := [monotonicity #277]: #279
-#283 := [monotonicity #280]: #282
-#286 := [monotonicity #283]: #285
-#289 := [monotonicity #286]: #288
-#336 := [trans #289 #334]: #335
-#179 := [asserted]: #63
-#337 := [mp #179 #336]: #332
-#339 := [and-elim #337]: #324
-#469 := [mp #339 #468]: #438
-#678 := [th-lemma #469 #677 #418 #385 #666]: false
-#679 := [lemma #678]: #570
-#684 := [unit-resolution #679 #683]: false
-#685 := [lemma #684]: #227
-#577 := (or #228 #567)
-#578 := [def-axiom]: #577
-#645 := [unit-resolution #578 #685]: #567
-#686 := (not #567)
-#687 := (or #686 #643)
-#688 := [th-lemma]: #687
-#689 := [unit-resolution #688 #645]: #643
-#31 := (uf_4 uf_8 #25)
-#245 := (+ #31 #244)
-#246 := (<= #245 0::real)
-#247 := (not #246)
-#34 := (uf_4 uf_9 #25)
-#48 := (uf_4 uf_11 #25)
-#255 := (ite #228 #48 #34)
-#264 := (* -1::real #255)
-#265 := (+ #31 #264)
-#263 := (>= #265 0::real)
-#266 := (not #263)
-#271 := (and #247 #266)
-#40 := (< #22 #39)
-#49 := (ite #40 #34 #48)
-#50 := (< #31 #49)
-#45 := (ite #40 #26 #44)
-#46 := (< #45 #31)
-#51 := (and #46 #50)
-#272 := (iff #51 #271)
-#269 := (iff #50 #266)
-#260 := (< #31 #255)
-#267 := (iff #260 #266)
-#268 := [rewrite]: #267
-#261 := (iff #50 #260)
-#258 := (= #49 #255)
-#252 := (ite #227 #34 #48)
-#256 := (= #252 #255)
-#257 := [rewrite]: #256
-#253 := (= #49 #252)
-#231 := (iff #40 #227)
-#232 := [rewrite]: #231
-#254 := [monotonicity #232]: #253
-#259 := [trans #254 #257]: #258
-#262 := [monotonicity #259]: #261
-#270 := [trans #262 #268]: #269
-#250 := (iff #46 #247)
-#241 := (< #236 #31)
-#248 := (iff #241 #247)
-#249 := [rewrite]: #248
-#242 := (iff #46 #241)
-#239 := (= #45 #236)
-#233 := (ite #227 #26 #44)
-#237 := (= #233 #236)
-#238 := [rewrite]: #237
-#234 := (= #45 #233)
-#235 := [monotonicity #232]: #234
-#240 := [trans #235 #238]: #239
-#243 := [monotonicity #240]: #242
-#251 := [trans #243 #249]: #250
-#273 := [monotonicity #251 #270]: #272
-#178 := [asserted]: #51
-#274 := [mp #178 #273]: #271
-#275 := [and-elim #274]: #247
-#196 := (* -1::real #31)
-#212 := (+ #26 #196)
-#213 := (<= #212 0::real)
-#214 := (not #213)
-#197 := (+ #28 #196)
-#195 := (>= #197 0::real)
-#193 := (not #195)
-#219 := (and #193 #214)
-#23 := (< #22 #22)
-#35 := (ite #23 #34 #26)
-#36 := (< #31 #35)
-#29 := (ite #23 #26 #28)
-#32 := (< #29 #31)
-#37 := (and #32 #36)
-#220 := (iff #37 #219)
-#217 := (iff #36 #214)
-#209 := (< #31 #26)
-#215 := (iff #209 #214)
-#216 := [rewrite]: #215
-#210 := (iff #36 #209)
-#207 := (= #35 #26)
-#202 := (ite false #34 #26)
-#205 := (= #202 #26)
-#206 := [rewrite]: #205
-#203 := (= #35 #202)
-#180 := (iff #23 false)
-#181 := [rewrite]: #180
-#204 := [monotonicity #181]: #203
-#208 := [trans #204 #206]: #207
-#211 := [monotonicity #208]: #210
-#218 := [trans #211 #216]: #217
-#200 := (iff #32 #193)
-#189 := (< #28 #31)
-#198 := (iff #189 #193)
-#199 := [rewrite]: #198
-#190 := (iff #32 #189)
-#187 := (= #29 #28)
-#182 := (ite false #26 #28)
-#185 := (= #182 #28)
-#186 := [rewrite]: #185
-#183 := (= #29 #182)
-#184 := [monotonicity #181]: #183
-#188 := [trans #184 #186]: #187
-#191 := [monotonicity #188]: #190
-#201 := [trans #191 #199]: #200
-#221 := [monotonicity #201 #218]: #220
-#177 := [asserted]: #37
-#222 := [mp #177 #221]: #219
-#224 := [and-elim #222]: #214
-[th-lemma #224 #275 #689]: false
-unsat
-NX/HT1QOfbspC2LtZNKpBA 428 0
-#2 := false
-decl uf_10 :: T1
-#38 := uf_10
-decl uf_3 :: T1
-#21 := uf_3
-#45 := (= uf_3 uf_10)
-decl uf_1 :: (-> int T1)
-decl uf_2 :: (-> T1 int)
-#39 := (uf_2 uf_10)
-#588 := (uf_1 #39)
-#686 := (= #588 uf_10)
-#589 := (= uf_10 #588)
-#4 := (:var 0 T1)
-#5 := (uf_2 #4)
-#541 := (pattern #5)
-#6 := (uf_1 #5)
-#93 := (= #4 #6)
-#542 := (forall (vars (?x1 T1)) (:pat #541) #93)
-#96 := (forall (vars (?x1 T1)) #93)
-#545 := (iff #96 #542)
-#543 := (iff #93 #93)
-#544 := [refl]: #543
-#546 := [quant-intro #544]: #545
-#454 := (~ #96 #96)
-#456 := (~ #93 #93)
-#457 := [refl]: #456
-#455 := [nnf-pos #457]: #454
-#7 := (= #6 #4)
-#8 := (forall (vars (?x1 T1)) #7)
-#97 := (iff #8 #96)
-#94 := (iff #7 #93)
-#95 := [rewrite]: #94
-#98 := [quant-intro #95]: #97
-#92 := [asserted]: #8
-#101 := [mp #92 #98]: #96
-#452 := [mp~ #101 #455]: #96
-#547 := [mp #452 #546]: #542
-#590 := (not #542)
-#595 := (or #590 #589)
-#596 := [quant-inst]: #595
-#680 := [unit-resolution #596 #547]: #589
-#687 := [symm #680]: #686
-#688 := (= uf_3 #588)
-#22 := (uf_2 uf_3)
-#586 := (uf_1 #22)
-#684 := (= #586 #588)
-#682 := (= #588 #586)
-#678 := (= #39 #22)
-#676 := (= #22 #39)
-#9 := 0::int
-#227 := -1::int
-#230 := (* -1::int #39)
-#231 := (+ #22 #230)
-#296 := (<= #231 0::int)
-#70 := (<= #22 #39)
-#393 := (iff #70 #296)
-#394 := [rewrite]: #393
-#347 := [asserted]: #70
-#395 := [mp #347 #394]: #296
-#229 := (>= #231 0::int)
-decl uf_4 :: (-> T2 T3 real)
-decl uf_6 :: (-> T1 T3)
-#25 := (uf_6 uf_3)
-decl uf_7 :: T2
-#27 := uf_7
-#28 := (uf_4 uf_7 #25)
-decl uf_9 :: T2
-#33 := uf_9
-#34 := (uf_4 uf_9 #25)
-#46 := (uf_6 uf_10)
-decl uf_5 :: T2
-#24 := uf_5
-#47 := (uf_4 uf_5 #46)
-#48 := (ite #45 #47 #34)
-#256 := (ite #229 #48 #28)
-#568 := (= #28 #256)
-#648 := (not #568)
-#194 := 0::real
-#192 := -1::real
-#265 := (* -1::real #256)
-#640 := (+ #28 #265)
-#642 := (>= #640 0::real)
-#645 := (not #642)
-#643 := [hypothesis]: #642
-decl uf_8 :: T2
-#30 := uf_8
-#31 := (uf_4 uf_8 #25)
-#266 := (+ #31 #265)
-#264 := (>= #266 0::real)
-#267 := (not #264)
-#26 := (uf_4 uf_5 #25)
-decl uf_11 :: T2
-#41 := uf_11
-#42 := (uf_4 uf_11 #25)
-#237 := (ite #229 #42 #26)
-#245 := (* -1::real #237)
-#246 := (+ #31 #245)
-#247 := (<= #246 0::real)
-#248 := (not #247)
-#272 := (and #248 #267)
-#40 := (< #22 #39)
-#49 := (ite #40 #28 #48)
-#50 := (< #31 #49)
-#43 := (ite #40 #26 #42)
-#44 := (< #43 #31)
-#51 := (and #44 #50)
-#273 := (iff #51 #272)
-#270 := (iff #50 #267)
-#261 := (< #31 #256)
-#268 := (iff #261 #267)
-#269 := [rewrite]: #268
-#262 := (iff #50 #261)
-#259 := (= #49 #256)
-#228 := (not #229)
-#253 := (ite #228 #28 #48)
-#257 := (= #253 #256)
-#258 := [rewrite]: #257
-#254 := (= #49 #253)
-#232 := (iff #40 #228)
-#233 := [rewrite]: #232
-#255 := [monotonicity #233]: #254
-#260 := [trans #255 #258]: #259
-#263 := [monotonicity #260]: #262
-#271 := [trans #263 #269]: #270
-#251 := (iff #44 #248)
-#242 := (< #237 #31)
-#249 := (iff #242 #248)
-#250 := [rewrite]: #249
-#243 := (iff #44 #242)
-#240 := (= #43 #237)
-#234 := (ite #228 #26 #42)
-#238 := (= #234 #237)
-#239 := [rewrite]: #238
-#235 := (= #43 #234)
-#236 := [monotonicity #233]: #235
-#241 := [trans #236 #239]: #240
-#244 := [monotonicity #241]: #243
-#252 := [trans #244 #250]: #251
-#274 := [monotonicity #252 #271]: #273
-#178 := [asserted]: #51
-#275 := [mp #178 #274]: #272
-#277 := [and-elim #275]: #267
-#196 := (* -1::real #31)
-#197 := (+ #28 #196)
-#195 := (>= #197 0::real)
-#193 := (not #195)
-#213 := (* -1::real #34)
-#214 := (+ #31 #213)
-#212 := (>= #214 0::real)
-#215 := (not #212)
-#220 := (and #193 #215)
-#23 := (< #22 #22)
-#35 := (ite #23 #28 #34)
-#36 := (< #31 #35)
-#29 := (ite #23 #26 #28)
-#32 := (< #29 #31)
-#37 := (and #32 #36)
-#221 := (iff #37 #220)
-#218 := (iff #36 #215)
-#209 := (< #31 #34)
-#216 := (iff #209 #215)
-#217 := [rewrite]: #216
-#210 := (iff #36 #209)
-#207 := (= #35 #34)
-#202 := (ite false #28 #34)
-#205 := (= #202 #34)
-#206 := [rewrite]: #205
-#203 := (= #35 #202)
-#180 := (iff #23 false)
-#181 := [rewrite]: #180
-#204 := [monotonicity #181]: #203
-#208 := [trans #204 #206]: #207
-#211 := [monotonicity #208]: #210
-#219 := [trans #211 #217]: #218
-#200 := (iff #32 #193)
-#189 := (< #28 #31)
-#198 := (iff #189 #193)
-#199 := [rewrite]: #198
-#190 := (iff #32 #189)
-#187 := (= #29 #28)
-#182 := (ite false #26 #28)
-#185 := (= #182 #28)
-#186 := [rewrite]: #185
-#183 := (= #29 #182)
-#184 := [monotonicity #181]: #183
-#188 := [trans #184 #186]: #187
-#191 := [monotonicity #188]: #190
-#201 := [trans #191 #199]: #200
-#222 := [monotonicity #201 #219]: #221
-#177 := [asserted]: #37
-#223 := [mp #177 #222]: #220
-#224 := [and-elim #223]: #193
-#644 := [th-lemma #224 #277 #643]: false
-#646 := [lemma #644]: #645
-#647 := [hypothesis]: #568
-#649 := (or #648 #642)
-#650 := [th-lemma]: #649
-#651 := [unit-resolution #650 #647 #646]: false
-#652 := [lemma #651]: #648
-#578 := (or #229 #568)
-#579 := [def-axiom]: #578
-#675 := [unit-resolution #579 #652]: #229
-#677 := [th-lemma #675 #395]: #676
-#679 := [symm #677]: #678
-#683 := [monotonicity #679]: #682
-#685 := [symm #683]: #684
-#587 := (= uf_3 #586)
-#591 := (or #590 #587)
-#592 := [quant-inst]: #591
-#681 := [unit-resolution #592 #547]: #587
-#689 := [trans #681 #685]: #688
-#690 := [trans #689 #687]: #45
-#571 := (not #45)
-#54 := (uf_4 uf_11 #46)
-#279 := (ite #45 #28 #54)
-#465 := (* -1::real #279)
-#632 := (+ #28 #465)
-#633 := (<= #632 0::real)
-#580 := (= #28 #279)
-#656 := [hypothesis]: #45
-#582 := (or #571 #580)
-#583 := [def-axiom]: #582
-#657 := [unit-resolution #583 #656]: #580
-#658 := (not #580)
-#659 := (or #658 #633)
-#660 := [th-lemma]: #659
-#661 := [unit-resolution #660 #657]: #633
-#57 := (uf_4 uf_8 #46)
-#363 := (* -1::real #57)
-#379 := (+ #47 #363)
-#380 := (<= #379 0::real)
-#381 := (not #380)
-#364 := (+ #54 #363)
-#362 := (>= #364 0::real)
-#361 := (not #362)
-#386 := (and #361 #381)
-#59 := (uf_4 uf_7 #46)
-#64 := (< #39 #39)
-#67 := (ite #64 #59 #47)
-#68 := (< #57 #67)
-#65 := (ite #64 #47 #54)
-#66 := (< #65 #57)
-#69 := (and #66 #68)
-#387 := (iff #69 #386)
-#384 := (iff #68 #381)
-#376 := (< #57 #47)
-#382 := (iff #376 #381)
-#383 := [rewrite]: #382
-#377 := (iff #68 #376)
-#374 := (= #67 #47)
-#369 := (ite false #59 #47)
-#372 := (= #369 #47)
-#373 := [rewrite]: #372
-#370 := (= #67 #369)
-#349 := (iff #64 false)
-#350 := [rewrite]: #349
-#371 := [monotonicity #350]: #370
-#375 := [trans #371 #373]: #374
-#378 := [monotonicity #375]: #377
-#385 := [trans #378 #383]: #384
-#367 := (iff #66 #361)
-#358 := (< #54 #57)
-#365 := (iff #358 #361)
-#366 := [rewrite]: #365
-#359 := (iff #66 #358)
-#356 := (= #65 #54)
-#351 := (ite false #47 #54)
-#354 := (= #351 #54)
-#355 := [rewrite]: #354
-#352 := (= #65 #351)
-#353 := [monotonicity #350]: #352
-#357 := [trans #353 #355]: #356
-#360 := [monotonicity #357]: #359
-#368 := [trans #360 #366]: #367
-#388 := [monotonicity #368 #385]: #387
-#346 := [asserted]: #69
-#389 := [mp #346 #388]: #386
-#391 := [and-elim #389]: #381
-#397 := (* -1::real #59)
-#398 := (+ #47 #397)
-#399 := (<= #398 0::real)
-#409 := (* -1::real #54)
-#410 := (+ #47 #409)
-#408 := (>= #410 0::real)
-#60 := (uf_4 uf_9 #46)
-#402 := (* -1::real #60)
-#403 := (+ #59 #402)
-#404 := (<= #403 0::real)
-#418 := (and #399 #404 #408)
-#73 := (<= #59 #60)
-#72 := (<= #47 #59)
-#74 := (and #72 #73)
-#71 := (<= #54 #47)
-#75 := (and #71 #74)
-#421 := (iff #75 #418)
-#412 := (and #399 #404)
-#415 := (and #408 #412)
-#419 := (iff #415 #418)
-#420 := [rewrite]: #419
-#416 := (iff #75 #415)
-#413 := (iff #74 #412)
-#405 := (iff #73 #404)
-#406 := [rewrite]: #405
-#400 := (iff #72 #399)
-#401 := [rewrite]: #400
-#414 := [monotonicity #401 #406]: #413
-#407 := (iff #71 #408)
-#411 := [rewrite]: #407
-#417 := [monotonicity #411 #414]: #416
-#422 := [trans #417 #420]: #421
-#348 := [asserted]: #75
-#423 := [mp #348 #422]: #418
-#424 := [and-elim #423]: #399
-#637 := (+ #28 #397)
-#639 := (>= #637 0::real)
-#636 := (= #28 #59)
-#666 := (= #59 #28)
-#664 := (= #46 #25)
-#662 := (= #25 #46)
-#663 := [monotonicity #656]: #662
-#665 := [symm #663]: #664
-#667 := [monotonicity #665]: #666
-#668 := [symm #667]: #636
-#669 := (not #636)
-#670 := (or #669 #639)
-#671 := [th-lemma]: #670
-#672 := [unit-resolution #671 #668]: #639
-#468 := (+ #57 #465)
-#471 := (<= #468 0::real)
-#444 := (not #471)
-#322 := (ite #296 #279 #47)
-#330 := (* -1::real #322)
-#331 := (+ #57 #330)
-#332 := (<= #331 0::real)
-#333 := (not #332)
-#445 := (iff #333 #444)
-#472 := (iff #332 #471)
-#469 := (= #331 #468)
-#466 := (= #330 #465)
-#463 := (= #322 #279)
-#1 := true
-#458 := (ite true #279 #47)
-#461 := (= #458 #279)
-#462 := [rewrite]: #461
-#459 := (= #322 #458)
-#450 := (iff #296 true)
-#451 := [iff-true #395]: #450
-#460 := [monotonicity #451]: #459
-#464 := [trans #460 #462]: #463
-#467 := [monotonicity #464]: #466
-#470 := [monotonicity #467]: #469
-#473 := [monotonicity #470]: #472
-#474 := [monotonicity #473]: #445
-#303 := (ite #296 #60 #59)
-#313 := (* -1::real #303)
-#314 := (+ #57 #313)
-#312 := (>= #314 0::real)
-#311 := (not #312)
-#338 := (and #311 #333)
-#52 := (< #39 #22)
-#61 := (ite #52 #59 #60)
-#62 := (< #57 #61)
-#53 := (= uf_10 uf_3)
-#55 := (ite #53 #28 #54)
-#56 := (ite #52 #47 #55)
-#58 := (< #56 #57)
-#63 := (and #58 #62)
-#341 := (iff #63 #338)
-#282 := (ite #52 #47 #279)
-#285 := (< #282 #57)
-#291 := (and #62 #285)
-#339 := (iff #291 #338)
-#336 := (iff #285 #333)
-#327 := (< #322 #57)
-#334 := (iff #327 #333)
-#335 := [rewrite]: #334
-#328 := (iff #285 #327)
-#325 := (= #282 #322)
-#297 := (not #296)
-#319 := (ite #297 #47 #279)
-#323 := (= #319 #322)
-#324 := [rewrite]: #323
-#320 := (= #282 #319)
-#298 := (iff #52 #297)
-#299 := [rewrite]: #298
-#321 := [monotonicity #299]: #320
-#326 := [trans #321 #324]: #325
-#329 := [monotonicity #326]: #328
-#337 := [trans #329 #335]: #336
-#317 := (iff #62 #311)
-#308 := (< #57 #303)
-#315 := (iff #308 #311)
-#316 := [rewrite]: #315
-#309 := (iff #62 #308)
-#306 := (= #61 #303)
-#300 := (ite #297 #59 #60)
-#304 := (= #300 #303)
-#305 := [rewrite]: #304
-#301 := (= #61 #300)
-#302 := [monotonicity #299]: #301
-#307 := [trans #302 #305]: #306
-#310 := [monotonicity #307]: #309
-#318 := [trans #310 #316]: #317
-#340 := [monotonicity #318 #337]: #339
-#294 := (iff #63 #291)
-#288 := (and #285 #62)
-#292 := (iff #288 #291)
-#293 := [rewrite]: #292
-#289 := (iff #63 #288)
-#286 := (iff #58 #285)
-#283 := (= #56 #282)
-#280 := (= #55 #279)
-#226 := (iff #53 #45)
-#278 := [rewrite]: #226
-#281 := [monotonicity #278]: #280
-#284 := [monotonicity #281]: #283
-#287 := [monotonicity #284]: #286
-#290 := [monotonicity #287]: #289
-#295 := [trans #290 #293]: #294
-#342 := [trans #295 #340]: #341
-#179 := [asserted]: #63
-#343 := [mp #179 #342]: #338
-#345 := [and-elim #343]: #333
-#475 := [mp #345 #474]: #444
-#673 := [th-lemma #475 #672 #424 #391 #661]: false
-#674 := [lemma #673]: #571
-[unit-resolution #674 #690]: false
-unsat
-IL2powemHjRpCJYwmXFxyw 211 0
-#2 := false
-#33 := 0::real
-decl uf_11 :: (-> T5 T6 real)
-decl uf_15 :: T6
-#28 := uf_15
-decl uf_16 :: T5
-#30 := uf_16
-#31 := (uf_11 uf_16 uf_15)
-decl uf_12 :: (-> T7 T8 T5)
-decl uf_14 :: T8
-#26 := uf_14
-decl uf_13 :: (-> T1 T7)
-decl uf_8 :: T1
-#16 := uf_8
-#25 := (uf_13 uf_8)
-#27 := (uf_12 #25 uf_14)
-#29 := (uf_11 #27 uf_15)
-#73 := -1::real
-#84 := (* -1::real #29)
-#85 := (+ #84 #31)
-#74 := (* -1::real #31)
-#75 := (+ #29 #74)
-#112 := (>= #75 0::real)
-#119 := (ite #112 #75 #85)
-#127 := (* -1::real #119)
-decl uf_17 :: T5
-#37 := uf_17
-#38 := (uf_11 uf_17 uf_15)
-#102 := -1/3::real
-#103 := (* -1/3::real #38)
-#128 := (+ #103 #127)
-#100 := 1/3::real
-#101 := (* 1/3::real #31)
-#129 := (+ #101 #128)
-#130 := (<= #129 0::real)
-#131 := (not #130)
-#40 := 3::real
-#39 := (- #31 #38)
-#41 := (/ #39 3::real)
-#32 := (- #29 #31)
-#35 := (- #32)
-#34 := (< #32 0::real)
-#36 := (ite #34 #35 #32)
-#42 := (< #36 #41)
-#136 := (iff #42 #131)
-#104 := (+ #101 #103)
-#78 := (< #75 0::real)
-#90 := (ite #78 #85 #75)
-#109 := (< #90 #104)
-#134 := (iff #109 #131)
-#124 := (< #119 #104)
-#132 := (iff #124 #131)
-#133 := [rewrite]: #132
-#125 := (iff #109 #124)
-#122 := (= #90 #119)
-#113 := (not #112)
-#116 := (ite #113 #85 #75)
-#120 := (= #116 #119)
-#121 := [rewrite]: #120
-#117 := (= #90 #116)
-#114 := (iff #78 #113)
-#115 := [rewrite]: #114
-#118 := [monotonicity #115]: #117
-#123 := [trans #118 #121]: #122
-#126 := [monotonicity #123]: #125
-#135 := [trans #126 #133]: #134
-#110 := (iff #42 #109)
-#107 := (= #41 #104)
-#93 := (* -1::real #38)
-#94 := (+ #31 #93)
-#97 := (/ #94 3::real)
-#105 := (= #97 #104)
-#106 := [rewrite]: #105
-#98 := (= #41 #97)
-#95 := (= #39 #94)
-#96 := [rewrite]: #95
-#99 := [monotonicity #96]: #98
-#108 := [trans #99 #106]: #107
-#91 := (= #36 #90)
-#76 := (= #32 #75)
-#77 := [rewrite]: #76
-#88 := (= #35 #85)
-#81 := (- #75)
-#86 := (= #81 #85)
-#87 := [rewrite]: #86
-#82 := (= #35 #81)
-#83 := [monotonicity #77]: #82
-#89 := [trans #83 #87]: #88
-#79 := (iff #34 #78)
-#80 := [monotonicity #77]: #79
-#92 := [monotonicity #80 #89 #77]: #91
-#111 := [monotonicity #92 #108]: #110
-#137 := [trans #111 #135]: #136
-#72 := [asserted]: #42
-#138 := [mp #72 #137]: #131
-decl uf_1 :: T1
-#4 := uf_1
-#43 := (uf_13 uf_1)
-#44 := (uf_12 #43 uf_14)
-#45 := (uf_11 #44 uf_15)
-#149 := (* -1::real #45)
-#150 := (+ #38 #149)
-#140 := (+ #93 #45)
-#161 := (<= #150 0::real)
-#168 := (ite #161 #140 #150)
-#176 := (* -1::real #168)
-#177 := (+ #103 #176)
-#178 := (+ #101 #177)
-#179 := (<= #178 0::real)
-#180 := (not #179)
-#46 := (- #45 #38)
-#48 := (- #46)
-#47 := (< #46 0::real)
-#49 := (ite #47 #48 #46)
-#50 := (< #49 #41)
-#185 := (iff #50 #180)
-#143 := (< #140 0::real)
-#155 := (ite #143 #150 #140)
-#158 := (< #155 #104)
-#183 := (iff #158 #180)
-#173 := (< #168 #104)
-#181 := (iff #173 #180)
-#182 := [rewrite]: #181
-#174 := (iff #158 #173)
-#171 := (= #155 #168)
-#162 := (not #161)
-#165 := (ite #162 #150 #140)
-#169 := (= #165 #168)
-#170 := [rewrite]: #169
-#166 := (= #155 #165)
-#163 := (iff #143 #162)
-#164 := [rewrite]: #163
-#167 := [monotonicity #164]: #166
-#172 := [trans #167 #170]: #171
-#175 := [monotonicity #172]: #174
-#184 := [trans #175 #182]: #183
-#159 := (iff #50 #158)
-#156 := (= #49 #155)
-#141 := (= #46 #140)
-#142 := [rewrite]: #141
-#153 := (= #48 #150)
-#146 := (- #140)
-#151 := (= #146 #150)
-#152 := [rewrite]: #151
-#147 := (= #48 #146)
-#148 := [monotonicity #142]: #147
-#154 := [trans #148 #152]: #153
-#144 := (iff #47 #143)
-#145 := [monotonicity #142]: #144
-#157 := [monotonicity #145 #154 #142]: #156
-#160 := [monotonicity #157 #108]: #159
-#186 := [trans #160 #184]: #185
-#139 := [asserted]: #50
-#187 := [mp #139 #186]: #180
-#299 := (+ #140 #176)
-#300 := (<= #299 0::real)
-#290 := (= #140 #168)
-#329 := [hypothesis]: #162
-#191 := (+ #29 #149)
-#192 := (<= #191 0::real)
-#51 := (<= #29 #45)
-#193 := (iff #51 #192)
-#194 := [rewrite]: #193
-#188 := [asserted]: #51
-#195 := [mp #188 #194]: #192
-#298 := (+ #75 #127)
-#301 := (<= #298 0::real)
-#284 := (= #75 #119)
-#302 := [hypothesis]: #113
-#296 := (+ #85 #127)
-#297 := (<= #296 0::real)
-#285 := (= #85 #119)
-#288 := (or #112 #285)
-#289 := [def-axiom]: #288
-#303 := [unit-resolution #289 #302]: #285
-#304 := (not #285)
-#305 := (or #304 #297)
-#306 := [th-lemma]: #305
-#307 := [unit-resolution #306 #303]: #297
-#315 := (not #290)
-#310 := (not #300)
-#311 := (or #310 #112)
-#308 := [hypothesis]: #300
-#309 := [th-lemma #308 #307 #138 #302 #187 #195]: false
-#312 := [lemma #309]: #311
-#322 := [unit-resolution #312 #302]: #310
-#316 := (or #315 #300)
-#313 := [hypothesis]: #310
-#314 := [hypothesis]: #290
-#317 := [th-lemma]: #316
-#318 := [unit-resolution #317 #314 #313]: false
-#319 := [lemma #318]: #316
-#323 := [unit-resolution #319 #322]: #315
-#292 := (or #162 #290)
-#293 := [def-axiom]: #292
-#324 := [unit-resolution #293 #323]: #162
-#325 := [th-lemma #324 #307 #138 #302 #195]: false
-#326 := [lemma #325]: #112
-#286 := (or #113 #284)
-#287 := [def-axiom]: #286
-#330 := [unit-resolution #287 #326]: #284
-#331 := (not #284)
-#332 := (or #331 #301)
-#333 := [th-lemma]: #332
-#334 := [unit-resolution #333 #330]: #301
-#335 := [th-lemma #326 #334 #195 #329 #138]: false
-#336 := [lemma #335]: #161
-#327 := [unit-resolution #293 #336]: #290
-#328 := [unit-resolution #319 #327]: #300
-[th-lemma #326 #334 #195 #328 #187 #138]: false
-unsat
-GX51o3DUO/UBS3eNP2P9kA 285 0
-#2 := false
-#7 := 0::real
-decl uf_4 :: real
-#16 := uf_4
-#40 := -1::real
-#116 := (* -1::real uf_4)
-decl uf_3 :: real
-#11 := uf_3
-#117 := (+ uf_3 #116)
-#128 := (<= #117 0::real)
-#129 := (not #128)
-#220 := 2/3::real
-#221 := (* 2/3::real uf_3)
-#222 := (+ #221 #116)
-decl uf_2 :: real
-#5 := uf_2
-#67 := 1/3::real
-#68 := (* 1/3::real uf_2)
-#233 := (+ #68 #222)
-#243 := (<= #233 0::real)
-#268 := (not #243)
-#287 := [hypothesis]: #268
-#41 := (* -1::real uf_2)
-decl uf_1 :: real
-#4 := uf_1
-#42 := (+ uf_1 #41)
-#79 := (>= #42 0::real)
-#80 := (not #79)
-#297 := (or #80 #243)
-#158 := (+ uf_1 #116)
-#159 := (<= #158 0::real)
-#22 := (<= uf_1 uf_4)
-#160 := (iff #22 #159)
-#161 := [rewrite]: #160
-#155 := [asserted]: #22
-#162 := [mp #155 #161]: #159
-#200 := (* 1/3::real uf_3)
-#198 := -4/3::real
-#199 := (* -4/3::real uf_2)
-#201 := (+ #199 #200)
-#202 := (+ uf_1 #201)
-#203 := (>= #202 0::real)
-#258 := (not #203)
-#292 := [hypothesis]: #79
-#293 := (or #80 #258)
-#69 := -1/3::real
-#70 := (* -1/3::real uf_3)
-#186 := -2/3::real
-#187 := (* -2/3::real uf_2)
-#188 := (+ #187 #70)
-#189 := (+ uf_1 #188)
-#204 := (<= #189 0::real)
-#205 := (ite #79 #203 #204)
-#210 := (not #205)
-#51 := (* -1::real uf_1)
-#52 := (+ #51 uf_2)
-#86 := (ite #79 #42 #52)
-#94 := (* -1::real #86)
-#95 := (+ #70 #94)
-#96 := (+ #68 #95)
-#97 := (<= #96 0::real)
-#98 := (not #97)
-#211 := (iff #98 #210)
-#208 := (iff #97 #205)
-#182 := 4/3::real
-#183 := (* 4/3::real uf_2)
-#184 := (+ #183 #70)
-#185 := (+ #51 #184)
-#190 := (ite #79 #185 #189)
-#195 := (<= #190 0::real)
-#206 := (iff #195 #205)
-#207 := [rewrite]: #206
-#196 := (iff #97 #195)
-#193 := (= #96 #190)
-#172 := (+ #41 #70)
-#173 := (+ uf_1 #172)
-#170 := (+ uf_2 #70)
-#171 := (+ #51 #170)
-#174 := (ite #79 #171 #173)
-#179 := (+ #68 #174)
-#191 := (= #179 #190)
-#192 := [rewrite]: #191
-#180 := (= #96 #179)
-#177 := (= #95 #174)
-#164 := (ite #79 #52 #42)
-#167 := (+ #70 #164)
-#175 := (= #167 #174)
-#176 := [rewrite]: #175
-#168 := (= #95 #167)
-#156 := (= #94 #164)
-#165 := [rewrite]: #156
-#169 := [monotonicity #165]: #168
-#178 := [trans #169 #176]: #177
-#181 := [monotonicity #178]: #180
-#194 := [trans #181 #192]: #193
-#197 := [monotonicity #194]: #196
-#209 := [trans #197 #207]: #208
-#212 := [monotonicity #209]: #211
-#13 := 3::real
-#12 := (- uf_2 uf_3)
-#14 := (/ #12 3::real)
-#6 := (- uf_1 uf_2)
-#9 := (- #6)
-#8 := (< #6 0::real)
-#10 := (ite #8 #9 #6)
-#15 := (< #10 #14)
-#103 := (iff #15 #98)
-#71 := (+ #68 #70)
-#45 := (< #42 0::real)
-#57 := (ite #45 #52 #42)
-#76 := (< #57 #71)
-#101 := (iff #76 #98)
-#91 := (< #86 #71)
-#99 := (iff #91 #98)
-#100 := [rewrite]: #99
-#92 := (iff #76 #91)
-#89 := (= #57 #86)
-#83 := (ite #80 #52 #42)
-#87 := (= #83 #86)
-#88 := [rewrite]: #87
-#84 := (= #57 #83)
-#81 := (iff #45 #80)
-#82 := [rewrite]: #81
-#85 := [monotonicity #82]: #84
-#90 := [trans #85 #88]: #89
-#93 := [monotonicity #90]: #92
-#102 := [trans #93 #100]: #101
-#77 := (iff #15 #76)
-#74 := (= #14 #71)
-#60 := (* -1::real uf_3)
-#61 := (+ uf_2 #60)
-#64 := (/ #61 3::real)
-#72 := (= #64 #71)
-#73 := [rewrite]: #72
-#65 := (= #14 #64)
-#62 := (= #12 #61)
-#63 := [rewrite]: #62
-#66 := [monotonicity #63]: #65
-#75 := [trans #66 #73]: #74
-#58 := (= #10 #57)
-#43 := (= #6 #42)
-#44 := [rewrite]: #43
-#55 := (= #9 #52)
-#48 := (- #42)
-#53 := (= #48 #52)
-#54 := [rewrite]: #53
-#49 := (= #9 #48)
-#50 := [monotonicity #44]: #49
-#56 := [trans #50 #54]: #55
-#46 := (iff #8 #45)
-#47 := [monotonicity #44]: #46
-#59 := [monotonicity #47 #56 #44]: #58
-#78 := [monotonicity #59 #75]: #77
-#104 := [trans #78 #102]: #103
-#39 := [asserted]: #15
-#105 := [mp #39 #104]: #98
-#213 := [mp #105 #212]: #210
-#259 := (or #205 #80 #258)
-#260 := [def-axiom]: #259
-#294 := [unit-resolution #260 #213]: #293
-#295 := [unit-resolution #294 #292]: #258
-#296 := [th-lemma #287 #292 #295 #162]: false
-#298 := [lemma #296]: #297
-#299 := [unit-resolution #298 #287]: #80
-#261 := (not #204)
-#281 := (or #79 #261)
-#262 := (or #205 #79 #261)
-#263 := [def-axiom]: #262
-#282 := [unit-resolution #263 #213]: #281
-#300 := [unit-resolution #282 #299]: #261
-#290 := (or #79 #204 #243)
-#276 := [hypothesis]: #261
-#288 := [hypothesis]: #80
-#289 := [th-lemma #288 #276 #162 #287]: false
-#291 := [lemma #289]: #290
-#301 := [unit-resolution #291 #300 #299 #287]: false
-#302 := [lemma #301]: #243
-#303 := (or #129 #268)
-#223 := (* -4/3::real uf_3)
-#224 := (+ #223 uf_4)
-#234 := (+ #68 #224)
-#244 := (<= #234 0::real)
-#245 := (ite #128 #243 #244)
-#250 := (not #245)
-#107 := (+ #60 uf_4)
-#135 := (ite #128 #107 #117)
-#143 := (* -1::real #135)
-#144 := (+ #70 #143)
-#145 := (+ #68 #144)
-#146 := (<= #145 0::real)
-#147 := (not #146)
-#251 := (iff #147 #250)
-#248 := (iff #146 #245)
-#235 := (ite #128 #233 #234)
-#240 := (<= #235 0::real)
-#246 := (iff #240 #245)
-#247 := [rewrite]: #246
-#241 := (iff #146 #240)
-#238 := (= #145 #235)
-#225 := (ite #128 #222 #224)
-#230 := (+ #68 #225)
-#236 := (= #230 #235)
-#237 := [rewrite]: #236
-#231 := (= #145 #230)
-#228 := (= #144 #225)
-#214 := (ite #128 #117 #107)
-#217 := (+ #70 #214)
-#226 := (= #217 #225)
-#227 := [rewrite]: #226
-#218 := (= #144 #217)
-#215 := (= #143 #214)
-#216 := [rewrite]: #215
-#219 := [monotonicity #216]: #218
-#229 := [trans #219 #227]: #228
-#232 := [monotonicity #229]: #231
-#239 := [trans #232 #237]: #238
-#242 := [monotonicity #239]: #241
-#249 := [trans #242 #247]: #248
-#252 := [monotonicity #249]: #251
-#17 := (- uf_4 uf_3)
-#19 := (- #17)
-#18 := (< #17 0::real)
-#20 := (ite #18 #19 #17)
-#21 := (< #20 #14)
-#152 := (iff #21 #147)
-#110 := (< #107 0::real)
-#122 := (ite #110 #117 #107)
-#125 := (< #122 #71)
-#150 := (iff #125 #147)
-#140 := (< #135 #71)
-#148 := (iff #140 #147)
-#149 := [rewrite]: #148
-#141 := (iff #125 #140)
-#138 := (= #122 #135)
-#132 := (ite #129 #117 #107)
-#136 := (= #132 #135)
-#137 := [rewrite]: #136
-#133 := (= #122 #132)
-#130 := (iff #110 #129)
-#131 := [rewrite]: #130
-#134 := [monotonicity #131]: #133
-#139 := [trans #134 #137]: #138
-#142 := [monotonicity #139]: #141
-#151 := [trans #142 #149]: #150
-#126 := (iff #21 #125)
-#123 := (= #20 #122)
-#108 := (= #17 #107)
-#109 := [rewrite]: #108
-#120 := (= #19 #117)
-#113 := (- #107)
-#118 := (= #113 #117)
-#119 := [rewrite]: #118
-#114 := (= #19 #113)
-#115 := [monotonicity #109]: #114
-#121 := [trans #115 #119]: #120
-#111 := (iff #18 #110)
-#112 := [monotonicity #109]: #111
-#124 := [monotonicity #112 #121 #109]: #123
-#127 := [monotonicity #124 #75]: #126
-#153 := [trans #127 #151]: #152
-#106 := [asserted]: #21
-#154 := [mp #106 #153]: #147
-#253 := [mp #154 #252]: #250
-#269 := (or #245 #129 #268)
-#270 := [def-axiom]: #269
-#304 := [unit-resolution #270 #253]: #303
-#305 := [unit-resolution #304 #302]: #129
-#271 := (not #244)
-#306 := (or #128 #271)
-#272 := (or #245 #128 #271)
-#273 := [def-axiom]: #272
-#307 := [unit-resolution #273 #253]: #306
-#308 := [unit-resolution #307 #305]: #271
-#285 := (or #128 #244)
-#274 := [hypothesis]: #271
-#275 := [hypothesis]: #129
-#278 := (or #204 #128 #244)
-#277 := [th-lemma #276 #275 #274 #162]: false
-#279 := [lemma #277]: #278
-#280 := [unit-resolution #279 #275 #274]: #204
-#283 := [unit-resolution #282 #280]: #79
-#284 := [th-lemma #275 #274 #283 #162]: false
-#286 := [lemma #284]: #285
-[unit-resolution #286 #308 #305]: false
-unsat
-cebG074uorSr8ODzgTmcKg 97 0
-#2 := false
-#18 := 0::real
-decl uf_1 :: (-> T2 T1 real)
-decl uf_5 :: T1
-#11 := uf_5
-decl uf_2 :: T2
-#4 := uf_2
-#20 := (uf_1 uf_2 uf_5)
-#42 := -1::real
-#53 := (* -1::real #20)
-decl uf_3 :: T2
-#7 := uf_3
-#19 := (uf_1 uf_3 uf_5)
-#54 := (+ #19 #53)
-#63 := (<= #54 0::real)
-#21 := (- #19 #20)
-#22 := (< 0::real #21)
-#23 := (not #22)
-#74 := (iff #23 #63)
-#57 := (< 0::real #54)
-#60 := (not #57)
-#72 := (iff #60 #63)
-#64 := (not #63)
-#67 := (not #64)
-#70 := (iff #67 #63)
-#71 := [rewrite]: #70
-#68 := (iff #60 #67)
-#65 := (iff #57 #64)
-#66 := [rewrite]: #65
-#69 := [monotonicity #66]: #68
-#73 := [trans #69 #71]: #72
-#61 := (iff #23 #60)
-#58 := (iff #22 #57)
-#55 := (= #21 #54)
-#56 := [rewrite]: #55
-#59 := [monotonicity #56]: #58
-#62 := [monotonicity #59]: #61
-#75 := [trans #62 #73]: #74
-#41 := [asserted]: #23
-#76 := [mp #41 #75]: #63
-#5 := (:var 0 T1)
-#8 := (uf_1 uf_3 #5)
-#141 := (pattern #8)
-#6 := (uf_1 uf_2 #5)
-#140 := (pattern #6)
-#45 := (* -1::real #8)
-#46 := (+ #6 #45)
-#44 := (>= #46 0::real)
-#43 := (not #44)
-#142 := (forall (vars (?x1 T1)) (:pat #140 #141) #43)
-#49 := (forall (vars (?x1 T1)) #43)
-#145 := (iff #49 #142)
-#143 := (iff #43 #43)
-#144 := [refl]: #143
-#146 := [quant-intro #144]: #145
-#80 := (~ #49 #49)
-#82 := (~ #43 #43)
-#83 := [refl]: #82
-#81 := [nnf-pos #83]: #80
-#9 := (< #6 #8)
-#10 := (forall (vars (?x1 T1)) #9)
-#50 := (iff #10 #49)
-#47 := (iff #9 #43)
-#48 := [rewrite]: #47
-#51 := [quant-intro #48]: #50
-#39 := [asserted]: #10
-#52 := [mp #39 #51]: #49
-#79 := [mp~ #52 #81]: #49
-#147 := [mp #79 #146]: #142
-#164 := (not #142)
-#165 := (or #164 #64)
-#148 := (* -1::real #19)
-#149 := (+ #20 #148)
-#150 := (>= #149 0::real)
-#151 := (not #150)
-#166 := (or #164 #151)
-#168 := (iff #166 #165)
-#170 := (iff #165 #165)
-#171 := [rewrite]: #170
-#162 := (iff #151 #64)
-#160 := (iff #150 #63)
-#152 := (+ #148 #20)
-#155 := (>= #152 0::real)
-#158 := (iff #155 #63)
-#159 := [rewrite]: #158
-#156 := (iff #150 #155)
-#153 := (= #149 #152)
-#154 := [rewrite]: #153
-#157 := [monotonicity #154]: #156
-#161 := [trans #157 #159]: #160
-#163 := [monotonicity #161]: #162
-#169 := [monotonicity #163]: #168
-#172 := [trans #169 #171]: #168
-#167 := [quant-inst]: #166
-#173 := [mp #167 #172]: #165
-[unit-resolution #173 #147 #76]: false
-unsat
-DKRtrJ2XceCkITuNwNViRw 57 0
-#2 := false
-#4 := 0::real
-decl uf_1 :: (-> T2 real)
-decl uf_2 :: (-> T1 T1 T2)
-decl uf_12 :: (-> T4 T1)
-decl uf_4 :: T4
-#11 := uf_4
-#39 := (uf_12 uf_4)
-decl uf_10 :: T4
-#27 := uf_10
-#38 := (uf_12 uf_10)
-#40 := (uf_2 #38 #39)
-#41 := (uf_1 #40)
-#264 := (>= #41 0::real)
-#266 := (not #264)
-#43 := (= #41 0::real)
-#44 := (not #43)
-#131 := [asserted]: #44
-#272 := (or #43 #266)
-#42 := (<= #41 0::real)
-#130 := [asserted]: #42
-#265 := (not #42)
-#270 := (or #43 #265 #266)
-#271 := [th-lemma]: #270
-#273 := [unit-resolution #271 #130]: #272
-#274 := [unit-resolution #273 #131]: #266
-#6 := (:var 0 T1)
-#5 := (:var 1 T1)
-#7 := (uf_2 #5 #6)
-#241 := (pattern #7)
-#8 := (uf_1 #7)
-#65 := (>= #8 0::real)
-#242 := (forall (vars (?x1 T1) (?x2 T1)) (:pat #241) #65)
-#66 := (forall (vars (?x1 T1) (?x2 T1)) #65)
-#245 := (iff #66 #242)
-#243 := (iff #65 #65)
-#244 := [refl]: #243
-#246 := [quant-intro #244]: #245
-#149 := (~ #66 #66)
-#151 := (~ #65 #65)
-#152 := [refl]: #151
-#150 := [nnf-pos #152]: #149
-#9 := (<= 0::real #8)
-#10 := (forall (vars (?x1 T1) (?x2 T1)) #9)
-#67 := (iff #10 #66)
-#63 := (iff #9 #65)
-#64 := [rewrite]: #63
-#68 := [quant-intro #64]: #67
-#60 := [asserted]: #10
-#69 := [mp #60 #68]: #66
-#147 := [mp~ #69 #150]: #66
-#247 := [mp #147 #246]: #242
-#267 := (not #242)
-#268 := (or #267 #264)
-#269 := [quant-inst]: #268
-[unit-resolution #269 #247 #274]: false
-unsat
-97KJAJfUio+nGchEHWvgAw 91 0
-#2 := false
-#38 := 0::real
-decl uf_1 :: (-> T1 T2 real)
-decl uf_3 :: T2
-#5 := uf_3
-decl uf_4 :: T1
-#7 := uf_4
-#8 := (uf_1 uf_4 uf_3)
-#35 := -1::real
-#36 := (* -1::real #8)
-decl uf_2 :: T1
-#4 := uf_2
-#6 := (uf_1 uf_2 uf_3)
-#37 := (+ #6 #36)
-#130 := (>= #37 0::real)
-#155 := (not #130)
-#43 := (= #6 #8)
-#55 := (not #43)
-#15 := (= #8 #6)
-#16 := (not #15)
-#56 := (iff #16 #55)
-#53 := (iff #15 #43)
-#54 := [rewrite]: #53
-#57 := [monotonicity #54]: #56
-#34 := [asserted]: #16
-#60 := [mp #34 #57]: #55
-#158 := (or #43 #155)
-#39 := (<= #37 0::real)
-#9 := (<= #6 #8)
-#40 := (iff #9 #39)
-#41 := [rewrite]: #40
-#32 := [asserted]: #9
-#42 := [mp #32 #41]: #39
-#154 := (not #39)
-#156 := (or #43 #154 #155)
-#157 := [th-lemma]: #156
-#159 := [unit-resolution #157 #42]: #158
-#160 := [unit-resolution #159 #60]: #155
-#10 := (:var 0 T2)
-#12 := (uf_1 uf_2 #10)
-#123 := (pattern #12)
-#11 := (uf_1 uf_4 #10)
-#122 := (pattern #11)
-#44 := (* -1::real #12)
-#45 := (+ #11 #44)
-#46 := (<= #45 0::real)
-#124 := (forall (vars (?x1 T2)) (:pat #122 #123) #46)
-#49 := (forall (vars (?x1 T2)) #46)
-#127 := (iff #49 #124)
-#125 := (iff #46 #46)
-#126 := [refl]: #125
-#128 := [quant-intro #126]: #127
-#62 := (~ #49 #49)
-#64 := (~ #46 #46)
-#65 := [refl]: #64
-#63 := [nnf-pos #65]: #62
-#13 := (<= #11 #12)
-#14 := (forall (vars (?x1 T2)) #13)
-#50 := (iff #14 #49)
-#47 := (iff #13 #46)
-#48 := [rewrite]: #47
-#51 := [quant-intro #48]: #50
-#33 := [asserted]: #14
-#52 := [mp #33 #51]: #49
-#61 := [mp~ #52 #63]: #49
-#129 := [mp #61 #128]: #124
-#144 := (not #124)
-#145 := (or #144 #130)
-#131 := (* -1::real #6)
-#132 := (+ #8 #131)
-#133 := (<= #132 0::real)
-#146 := (or #144 #133)
-#148 := (iff #146 #145)
-#150 := (iff #145 #145)
-#151 := [rewrite]: #150
-#142 := (iff #133 #130)
-#134 := (+ #131 #8)
-#137 := (<= #134 0::real)
-#140 := (iff #137 #130)
-#141 := [rewrite]: #140
-#138 := (iff #133 #137)
-#135 := (= #132 #134)
-#136 := [rewrite]: #135
-#139 := [monotonicity #136]: #138
-#143 := [trans #139 #141]: #142
-#149 := [monotonicity #143]: #148
-#152 := [trans #149 #151]: #148
-#147 := [quant-inst]: #146
-#153 := [mp #147 #152]: #145
-[unit-resolution #153 #129 #160]: false
-unsat
-flJYbeWfe+t2l/zsRqdujA 149 0
-#2 := false
-#19 := 0::real
-decl uf_1 :: (-> T1 T2 real)
-decl uf_3 :: T2
-#5 := uf_3
-decl uf_4 :: T1
-#7 := uf_4
-#8 := (uf_1 uf_4 uf_3)
-#44 := -1::real
-#156 := (* -1::real #8)
-decl uf_2 :: T1
-#4 := uf_2
-#6 := (uf_1 uf_2 uf_3)
-#203 := (+ #6 #156)
-#205 := (>= #203 0::real)
-#9 := (= #6 #8)
-#40 := [asserted]: #9
-#208 := (not #9)
-#209 := (or #208 #205)
-#210 := [th-lemma]: #209
-#211 := [unit-resolution #210 #40]: #205
-decl uf_5 :: T1
-#12 := uf_5
-#22 := (uf_1 uf_5 uf_3)
-#160 := (* -1::real #22)
-#161 := (+ #6 #160)
-#207 := (>= #161 0::real)
-#222 := (not #207)
-#206 := (= #6 #22)
-#216 := (not #206)
-#62 := (= #8 #22)
-#70 := (not #62)
-#217 := (iff #70 #216)
-#214 := (iff #62 #206)
-#212 := (iff #206 #62)
-#213 := [monotonicity #40]: #212
-#215 := [symm #213]: #214
-#218 := [monotonicity #215]: #217
-#23 := (= #22 #8)
-#24 := (not #23)
-#71 := (iff #24 #70)
-#68 := (iff #23 #62)
-#69 := [rewrite]: #68
-#72 := [monotonicity #69]: #71
-#43 := [asserted]: #24
-#75 := [mp #43 #72]: #70
-#219 := [mp #75 #218]: #216
-#225 := (or #206 #222)
-#162 := (<= #161 0::real)
-#172 := (+ #8 #160)
-#173 := (>= #172 0::real)
-#178 := (not #173)
-#163 := (not #162)
-#181 := (or #163 #178)
-#184 := (not #181)
-#10 := (:var 0 T2)
-#15 := (uf_1 uf_4 #10)
-#149 := (pattern #15)
-#13 := (uf_1 uf_5 #10)
-#148 := (pattern #13)
-#11 := (uf_1 uf_2 #10)
-#147 := (pattern #11)
-#50 := (* -1::real #15)
-#51 := (+ #13 #50)
-#52 := (<= #51 0::real)
-#76 := (not #52)
-#45 := (* -1::real #13)
-#46 := (+ #11 #45)
-#47 := (<= #46 0::real)
-#78 := (not #47)
-#73 := (or #78 #76)
-#83 := (not #73)
-#150 := (forall (vars (?x1 T2)) (:pat #147 #148 #149) #83)
-#86 := (forall (vars (?x1 T2)) #83)
-#153 := (iff #86 #150)
-#151 := (iff #83 #83)
-#152 := [refl]: #151
-#154 := [quant-intro #152]: #153
-#55 := (and #47 #52)
-#58 := (forall (vars (?x1 T2)) #55)
-#87 := (iff #58 #86)
-#84 := (iff #55 #83)
-#85 := [rewrite]: #84
-#88 := [quant-intro #85]: #87
-#79 := (~ #58 #58)
-#81 := (~ #55 #55)
-#82 := [refl]: #81
-#80 := [nnf-pos #82]: #79
-#16 := (<= #13 #15)
-#14 := (<= #11 #13)
-#17 := (and #14 #16)
-#18 := (forall (vars (?x1 T2)) #17)
-#59 := (iff #18 #58)
-#56 := (iff #17 #55)
-#53 := (iff #16 #52)
-#54 := [rewrite]: #53
-#48 := (iff #14 #47)
-#49 := [rewrite]: #48
-#57 := [monotonicity #49 #54]: #56
-#60 := [quant-intro #57]: #59
-#41 := [asserted]: #18
-#61 := [mp #41 #60]: #58
-#77 := [mp~ #61 #80]: #58
-#89 := [mp #77 #88]: #86
-#155 := [mp #89 #154]: #150
-#187 := (not #150)
-#188 := (or #187 #184)
-#157 := (+ #22 #156)
-#158 := (<= #157 0::real)
-#159 := (not #158)
-#164 := (or #163 #159)
-#165 := (not #164)
-#189 := (or #187 #165)
-#191 := (iff #189 #188)
-#193 := (iff #188 #188)
-#194 := [rewrite]: #193
-#185 := (iff #165 #184)
-#182 := (iff #164 #181)
-#179 := (iff #159 #178)
-#176 := (iff #158 #173)
-#166 := (+ #156 #22)
-#169 := (<= #166 0::real)
-#174 := (iff #169 #173)
-#175 := [rewrite]: #174
-#170 := (iff #158 #169)
-#167 := (= #157 #166)
-#168 := [rewrite]: #167
-#171 := [monotonicity #168]: #170
-#177 := [trans #171 #175]: #176
-#180 := [monotonicity #177]: #179
-#183 := [monotonicity #180]: #182
-#186 := [monotonicity #183]: #185
-#192 := [monotonicity #186]: #191
-#195 := [trans #192 #194]: #191
-#190 := [quant-inst]: #189
-#196 := [mp #190 #195]: #188
-#220 := [unit-resolution #196 #155]: #184
-#197 := (or #181 #162)
-#198 := [def-axiom]: #197
-#221 := [unit-resolution #198 #220]: #162
-#223 := (or #206 #163 #222)
-#224 := [th-lemma]: #223
-#226 := [unit-resolution #224 #221]: #225
-#227 := [unit-resolution #226 #219]: #222
-#199 := (or #181 #173)
-#200 := [def-axiom]: #199
-#228 := [unit-resolution #200 #220]: #173
-[th-lemma #228 #227 #211]: false
-unsat
-rbrrQuQfaijtLkQizgEXnQ 222 0
-#2 := false
-#4 := 0::real
-decl uf_2 :: (-> T2 T1 real)
-decl uf_5 :: T1
-#15 := uf_5
-decl uf_3 :: T2
-#7 := uf_3
-#20 := (uf_2 uf_3 uf_5)
-decl uf_6 :: T2
-#17 := uf_6
-#18 := (uf_2 uf_6 uf_5)
-#59 := -1::real
-#73 := (* -1::real #18)
-#106 := (+ #73 #20)
-decl uf_1 :: real
-#5 := uf_1
-#78 := (* -1::real #20)
-#79 := (+ #18 #78)
-#144 := (+ uf_1 #79)
-#145 := (<= #144 0::real)
-#148 := (ite #145 uf_1 #106)
-#279 := (* -1::real #148)
-#280 := (+ uf_1 #279)
-#281 := (<= #280 0::real)
-#289 := (not #281)
-#72 := 1/2::real
-#151 := (* 1/2::real #148)
-#248 := (<= #151 0::real)
-#162 := (= #151 0::real)
-#24 := 2::real
-#27 := (- #20 #18)
-#28 := (<= uf_1 #27)
-#29 := (ite #28 uf_1 #27)
-#30 := (/ #29 2::real)
-#31 := (+ #18 #30)
-#32 := (= #31 #18)
-#33 := (not #32)
-#34 := (not #33)
-#165 := (iff #34 #162)
-#109 := (<= uf_1 #106)
-#112 := (ite #109 uf_1 #106)
-#118 := (* 1/2::real #112)
-#123 := (+ #18 #118)
-#129 := (= #18 #123)
-#163 := (iff #129 #162)
-#154 := (+ #18 #151)
-#157 := (= #18 #154)
-#160 := (iff #157 #162)
-#161 := [rewrite]: #160
-#158 := (iff #129 #157)
-#155 := (= #123 #154)
-#152 := (= #118 #151)
-#149 := (= #112 #148)
-#146 := (iff #109 #145)
-#147 := [rewrite]: #146
-#150 := [monotonicity #147]: #149
-#153 := [monotonicity #150]: #152
-#156 := [monotonicity #153]: #155
-#159 := [monotonicity #156]: #158
-#164 := [trans #159 #161]: #163
-#142 := (iff #34 #129)
-#134 := (not #129)
-#137 := (not #134)
-#140 := (iff #137 #129)
-#141 := [rewrite]: #140
-#138 := (iff #34 #137)
-#135 := (iff #33 #134)
-#132 := (iff #32 #129)
-#126 := (= #123 #18)
-#130 := (iff #126 #129)
-#131 := [rewrite]: #130
-#127 := (iff #32 #126)
-#124 := (= #31 #123)
-#121 := (= #30 #118)
-#115 := (/ #112 2::real)
-#119 := (= #115 #118)
-#120 := [rewrite]: #119
-#116 := (= #30 #115)
-#113 := (= #29 #112)
-#107 := (= #27 #106)
-#108 := [rewrite]: #107
-#110 := (iff #28 #109)
-#111 := [monotonicity #108]: #110
-#114 := [monotonicity #111 #108]: #113
-#117 := [monotonicity #114]: #116
-#122 := [trans #117 #120]: #121
-#125 := [monotonicity #122]: #124
-#128 := [monotonicity #125]: #127
-#133 := [trans #128 #131]: #132
-#136 := [monotonicity #133]: #135
-#139 := [monotonicity #136]: #138
-#143 := [trans #139 #141]: #142
-#166 := [trans #143 #164]: #165
-#105 := [asserted]: #34
-#167 := [mp #105 #166]: #162
-#283 := (not #162)
-#284 := (or #283 #248)
-#285 := [th-lemma]: #284
-#286 := [unit-resolution #285 #167]: #248
-#287 := [hypothesis]: #281
-#53 := (<= uf_1 0::real)
-#54 := (not #53)
-#6 := (< 0::real uf_1)
-#55 := (iff #6 #54)
-#56 := [rewrite]: #55
-#50 := [asserted]: #6
-#57 := [mp #50 #56]: #54
-#288 := [th-lemma #57 #287 #286]: false
-#290 := [lemma #288]: #289
-#241 := (= uf_1 #148)
-#242 := (= #106 #148)
-#299 := (not #242)
-#282 := (+ #106 #279)
-#291 := (<= #282 0::real)
-#296 := (not #291)
-decl uf_4 :: T2
-#10 := uf_4
-#16 := (uf_2 uf_4 uf_5)
-#260 := (+ #16 #78)
-#261 := (>= #260 0::real)
-#266 := (not #261)
-#8 := (:var 0 T1)
-#11 := (uf_2 uf_4 #8)
-#234 := (pattern #11)
-#9 := (uf_2 uf_3 #8)
-#233 := (pattern #9)
-#60 := (* -1::real #11)
-#61 := (+ #9 #60)
-#62 := (<= #61 0::real)
-#179 := (not #62)
-#235 := (forall (vars (?x1 T1)) (:pat #233 #234) #179)
-#178 := (forall (vars (?x1 T1)) #179)
-#238 := (iff #178 #235)
-#236 := (iff #179 #179)
-#237 := [refl]: #236
-#239 := [quant-intro #237]: #238
-#65 := (exists (vars (?x1 T1)) #62)
-#68 := (not #65)
-#175 := (~ #68 #178)
-#180 := (~ #179 #179)
-#177 := [refl]: #180
-#176 := [nnf-neg #177]: #175
-#12 := (<= #9 #11)
-#13 := (exists (vars (?x1 T1)) #12)
-#14 := (not #13)
-#69 := (iff #14 #68)
-#66 := (iff #13 #65)
-#63 := (iff #12 #62)
-#64 := [rewrite]: #63
-#67 := [quant-intro #64]: #66
-#70 := [monotonicity #67]: #69
-#51 := [asserted]: #14
-#71 := [mp #51 #70]: #68
-#173 := [mp~ #71 #176]: #178
-#240 := [mp #173 #239]: #235
-#269 := (not #235)
-#270 := (or #269 #266)
-#250 := (* -1::real #16)
-#251 := (+ #20 #250)
-#252 := (<= #251 0::real)
-#253 := (not #252)
-#271 := (or #269 #253)
-#273 := (iff #271 #270)
-#275 := (iff #270 #270)
-#276 := [rewrite]: #275
-#267 := (iff #253 #266)
-#264 := (iff #252 #261)
-#254 := (+ #250 #20)
-#257 := (<= #254 0::real)
-#262 := (iff #257 #261)
-#263 := [rewrite]: #262
-#258 := (iff #252 #257)
-#255 := (= #251 #254)
-#256 := [rewrite]: #255
-#259 := [monotonicity #256]: #258
-#265 := [trans #259 #263]: #264
-#268 := [monotonicity #265]: #267
-#274 := [monotonicity #268]: #273
-#277 := [trans #274 #276]: #273
-#272 := [quant-inst]: #271
-#278 := [mp #272 #277]: #270
-#293 := [unit-resolution #278 #240]: #266
-#90 := (* 1/2::real #20)
-#102 := (+ #73 #90)
-#89 := (* 1/2::real #16)
-#103 := (+ #89 #102)
-#100 := (>= #103 0::real)
-#23 := (+ #16 #20)
-#25 := (/ #23 2::real)
-#26 := (<= #18 #25)
-#98 := (iff #26 #100)
-#91 := (+ #89 #90)
-#94 := (<= #18 #91)
-#97 := (iff #94 #100)
-#99 := [rewrite]: #97
-#95 := (iff #26 #94)
-#92 := (= #25 #91)
-#93 := [rewrite]: #92
-#96 := [monotonicity #93]: #95
-#101 := [trans #96 #99]: #98
-#58 := [asserted]: #26
-#104 := [mp #58 #101]: #100
-#294 := [hypothesis]: #291
-#295 := [th-lemma #294 #104 #293 #286]: false
-#297 := [lemma #295]: #296
-#298 := [hypothesis]: #242
-#300 := (or #299 #291)
-#301 := [th-lemma]: #300
-#302 := [unit-resolution #301 #298 #297]: false
-#303 := [lemma #302]: #299
-#246 := (or #145 #242)
-#247 := [def-axiom]: #246
-#304 := [unit-resolution #247 #303]: #145
-#243 := (not #145)
-#244 := (or #243 #241)
-#245 := [def-axiom]: #244
-#305 := [unit-resolution #245 #304]: #241
-#306 := (not #241)
-#307 := (or #306 #281)
-#308 := [th-lemma]: #307
-[unit-resolution #308 #305 #290]: false
-unsat
-hwh3oeLAWt56hnKIa8Wuow 248 0
-#2 := false
-#4 := 0::real
-decl uf_2 :: (-> T2 T1 real)
-decl uf_5 :: T1
-#15 := uf_5
-decl uf_6 :: T2
-#17 := uf_6
-#18 := (uf_2 uf_6 uf_5)
-decl uf_4 :: T2
-#10 := uf_4
-#16 := (uf_2 uf_4 uf_5)
-#66 := -1::real
-#137 := (* -1::real #16)
-#138 := (+ #137 #18)
-decl uf_1 :: real
-#5 := uf_1
-#80 := (* -1::real #18)
-#81 := (+ #16 #80)
-#201 := (+ uf_1 #81)
-#202 := (<= #201 0::real)
-#205 := (ite #202 uf_1 #138)
-#352 := (* -1::real #205)
-#353 := (+ uf_1 #352)
-#354 := (<= #353 0::real)
-#362 := (not #354)
-#79 := 1/2::real
-#244 := (* 1/2::real #205)
-#322 := (<= #244 0::real)
-#245 := (= #244 0::real)
-#158 := -1/2::real
-#208 := (* -1/2::real #205)
-#211 := (+ #18 #208)
-decl uf_3 :: T2
-#7 := uf_3
-#20 := (uf_2 uf_3 uf_5)
-#117 := (+ #80 #20)
-#85 := (* -1::real #20)
-#86 := (+ #18 #85)
-#188 := (+ uf_1 #86)
-#189 := (<= #188 0::real)
-#192 := (ite #189 uf_1 #117)
-#195 := (* 1/2::real #192)
-#198 := (+ #18 #195)
-#97 := (* 1/2::real #20)
-#109 := (+ #80 #97)
-#96 := (* 1/2::real #16)
-#110 := (+ #96 #109)
-#107 := (>= #110 0::real)
-#214 := (ite #107 #198 #211)
-#217 := (= #18 #214)
-#248 := (iff #217 #245)
-#241 := (= #18 #211)
-#246 := (iff #241 #245)
-#247 := [rewrite]: #246
-#242 := (iff #217 #241)
-#239 := (= #214 #211)
-#234 := (ite false #198 #211)
-#237 := (= #234 #211)
-#238 := [rewrite]: #237
-#235 := (= #214 #234)
-#232 := (iff #107 false)
-#104 := (not #107)
-#24 := 2::real
-#23 := (+ #16 #20)
-#25 := (/ #23 2::real)
-#26 := (< #25 #18)
-#108 := (iff #26 #104)
-#98 := (+ #96 #97)
-#101 := (< #98 #18)
-#106 := (iff #101 #104)
-#105 := [rewrite]: #106
-#102 := (iff #26 #101)
-#99 := (= #25 #98)
-#100 := [rewrite]: #99
-#103 := [monotonicity #100]: #102
-#111 := [trans #103 #105]: #108
-#65 := [asserted]: #26
-#112 := [mp #65 #111]: #104
-#233 := [iff-false #112]: #232
-#236 := [monotonicity #233]: #235
-#240 := [trans #236 #238]: #239
-#243 := [monotonicity #240]: #242
-#249 := [trans #243 #247]: #248
-#33 := (- #18 #16)
-#34 := (<= uf_1 #33)
-#35 := (ite #34 uf_1 #33)
-#36 := (/ #35 2::real)
-#37 := (- #18 #36)
-#28 := (- #20 #18)
-#29 := (<= uf_1 #28)
-#30 := (ite #29 uf_1 #28)
-#31 := (/ #30 2::real)
-#32 := (+ #18 #31)
-#27 := (<= #18 #25)
-#38 := (ite #27 #32 #37)
-#39 := (= #38 #18)
-#40 := (not #39)
-#41 := (not #40)
-#220 := (iff #41 #217)
-#141 := (<= uf_1 #138)
-#144 := (ite #141 uf_1 #138)
-#159 := (* -1/2::real #144)
-#160 := (+ #18 #159)
-#120 := (<= uf_1 #117)
-#123 := (ite #120 uf_1 #117)
-#129 := (* 1/2::real #123)
-#134 := (+ #18 #129)
-#114 := (<= #18 #98)
-#165 := (ite #114 #134 #160)
-#171 := (= #18 #165)
-#218 := (iff #171 #217)
-#215 := (= #165 #214)
-#212 := (= #160 #211)
-#209 := (= #159 #208)
-#206 := (= #144 #205)
-#203 := (iff #141 #202)
-#204 := [rewrite]: #203
-#207 := [monotonicity #204]: #206
-#210 := [monotonicity #207]: #209
-#213 := [monotonicity #210]: #212
-#199 := (= #134 #198)
-#196 := (= #129 #195)
-#193 := (= #123 #192)
-#190 := (iff #120 #189)
-#191 := [rewrite]: #190
-#194 := [monotonicity #191]: #193
-#197 := [monotonicity #194]: #196
-#200 := [monotonicity #197]: #199
-#187 := (iff #114 #107)
-#186 := [rewrite]: #187
-#216 := [monotonicity #186 #200 #213]: #215
-#219 := [monotonicity #216]: #218
-#184 := (iff #41 #171)
-#176 := (not #171)
-#179 := (not #176)
-#182 := (iff #179 #171)
-#183 := [rewrite]: #182
-#180 := (iff #41 #179)
-#177 := (iff #40 #176)
-#174 := (iff #39 #171)
-#168 := (= #165 #18)
-#172 := (iff #168 #171)
-#173 := [rewrite]: #172
-#169 := (iff #39 #168)
-#166 := (= #38 #165)
-#163 := (= #37 #160)
-#150 := (* 1/2::real #144)
-#155 := (- #18 #150)
-#161 := (= #155 #160)
-#162 := [rewrite]: #161
-#156 := (= #37 #155)
-#153 := (= #36 #150)
-#147 := (/ #144 2::real)
-#151 := (= #147 #150)
-#152 := [rewrite]: #151
-#148 := (= #36 #147)
-#145 := (= #35 #144)
-#139 := (= #33 #138)
-#140 := [rewrite]: #139
-#142 := (iff #34 #141)
-#143 := [monotonicity #140]: #142
-#146 := [monotonicity #143 #140]: #145
-#149 := [monotonicity #146]: #148
-#154 := [trans #149 #152]: #153
-#157 := [monotonicity #154]: #156
-#164 := [trans #157 #162]: #163
-#135 := (= #32 #134)
-#132 := (= #31 #129)
-#126 := (/ #123 2::real)
-#130 := (= #126 #129)
-#131 := [rewrite]: #130
-#127 := (= #31 #126)
-#124 := (= #30 #123)
-#118 := (= #28 #117)
-#119 := [rewrite]: #118
-#121 := (iff #29 #120)
-#122 := [monotonicity #119]: #121
-#125 := [monotonicity #122 #119]: #124
-#128 := [monotonicity #125]: #127
-#133 := [trans #128 #131]: #132
-#136 := [monotonicity #133]: #135
-#115 := (iff #27 #114)
-#116 := [monotonicity #100]: #115
-#167 := [monotonicity #116 #136 #164]: #166
-#170 := [monotonicity #167]: #169
-#175 := [trans #170 #173]: #174
-#178 := [monotonicity #175]: #177
-#181 := [monotonicity #178]: #180
-#185 := [trans #181 #183]: #184
-#221 := [trans #185 #219]: #220
-#113 := [asserted]: #41
-#222 := [mp #113 #221]: #217
-#250 := [mp #222 #249]: #245
-#356 := (not #245)
-#357 := (or #356 #322)
-#358 := [th-lemma]: #357
-#359 := [unit-resolution #358 #250]: #322
-#360 := [hypothesis]: #354
-#60 := (<= uf_1 0::real)
-#61 := (not #60)
-#6 := (< 0::real uf_1)
-#62 := (iff #6 #61)
-#63 := [rewrite]: #62
-#57 := [asserted]: #6
-#64 := [mp #57 #63]: #61
-#361 := [th-lemma #64 #360 #359]: false
-#363 := [lemma #361]: #362
-#315 := (= uf_1 #205)
-#316 := (= #138 #205)
-#371 := (not #316)
-#355 := (+ #138 #352)
-#364 := (<= #355 0::real)
-#368 := (not #364)
-#87 := (<= #86 0::real)
-#82 := (<= #81 0::real)
-#90 := (and #82 #87)
-#21 := (<= #18 #20)
-#19 := (<= #16 #18)
-#22 := (and #19 #21)
-#91 := (iff #22 #90)
-#88 := (iff #21 #87)
-#89 := [rewrite]: #88
-#83 := (iff #19 #82)
-#84 := [rewrite]: #83
-#92 := [monotonicity #84 #89]: #91
-#59 := [asserted]: #22
-#93 := [mp #59 #92]: #90
-#95 := [and-elim #93]: #87
-#366 := [hypothesis]: #364
-#367 := [th-lemma #366 #95 #112 #359]: false
-#369 := [lemma #367]: #368
-#370 := [hypothesis]: #316
-#372 := (or #371 #364)
-#373 := [th-lemma]: #372
-#374 := [unit-resolution #373 #370 #369]: false
-#375 := [lemma #374]: #371
-#320 := (or #202 #316)
-#321 := [def-axiom]: #320
-#376 := [unit-resolution #321 #375]: #202
-#317 := (not #202)
-#318 := (or #317 #315)
-#319 := [def-axiom]: #318
-#377 := [unit-resolution #319 #376]: #315
-#378 := (not #315)
-#379 := (or #378 #354)
-#380 := [th-lemma]: #379
-[unit-resolution #380 #377 #363]: false
-unsat
-WdMJH3tkMv/rps8y9Ukq5Q 86 0
-#2 := false
-#37 := 0::real
-decl uf_2 :: (-> T2 T1 real)
-decl uf_4 :: T1
-#12 := uf_4
-decl uf_3 :: T2
-#5 := uf_3
-#13 := (uf_2 uf_3 uf_4)
-#34 := -1::real
-#140 := (* -1::real #13)
-decl uf_1 :: real
-#4 := uf_1
-#141 := (+ uf_1 #140)
-#143 := (>= #141 0::real)
-#6 := (:var 0 T1)
-#7 := (uf_2 uf_3 #6)
-#127 := (pattern #7)
-#35 := (* -1::real #7)
-#36 := (+ uf_1 #35)
-#47 := (>= #36 0::real)
-#134 := (forall (vars (?x2 T1)) (:pat #127) #47)
-#49 := (forall (vars (?x2 T1)) #47)
-#137 := (iff #49 #134)
-#135 := (iff #47 #47)
-#136 := [refl]: #135
-#138 := [quant-intro #136]: #137
-#67 := (~ #49 #49)
-#58 := (~ #47 #47)
-#66 := [refl]: #58
-#68 := [nnf-pos #66]: #67
-#10 := (<= #7 uf_1)
-#11 := (forall (vars (?x2 T1)) #10)
-#50 := (iff #11 #49)
-#46 := (iff #10 #47)
-#48 := [rewrite]: #46
-#51 := [quant-intro #48]: #50
-#32 := [asserted]: #11
-#52 := [mp #32 #51]: #49
-#69 := [mp~ #52 #68]: #49
-#139 := [mp #69 #138]: #134
-#149 := (not #134)
-#150 := (or #149 #143)
-#151 := [quant-inst]: #150
-#144 := [unit-resolution #151 #139]: #143
-#142 := (<= #141 0::real)
-#38 := (<= #36 0::real)
-#128 := (forall (vars (?x1 T1)) (:pat #127) #38)
-#41 := (forall (vars (?x1 T1)) #38)
-#131 := (iff #41 #128)
-#129 := (iff #38 #38)
-#130 := [refl]: #129
-#132 := [quant-intro #130]: #131
-#62 := (~ #41 #41)
-#64 := (~ #38 #38)
-#65 := [refl]: #64
-#63 := [nnf-pos #65]: #62
-#8 := (<= uf_1 #7)
-#9 := (forall (vars (?x1 T1)) #8)
-#42 := (iff #9 #41)
-#39 := (iff #8 #38)
-#40 := [rewrite]: #39
-#43 := [quant-intro #40]: #42
-#31 := [asserted]: #9
-#44 := [mp #31 #43]: #41
-#61 := [mp~ #44 #63]: #41
-#133 := [mp #61 #132]: #128
-#145 := (not #128)
-#146 := (or #145 #142)
-#147 := [quant-inst]: #146
-#148 := [unit-resolution #147 #133]: #142
-#45 := (= uf_1 #13)
-#55 := (not #45)
-#14 := (= #13 uf_1)
-#15 := (not #14)
-#56 := (iff #15 #55)
-#53 := (iff #14 #45)
-#54 := [rewrite]: #53
-#57 := [monotonicity #54]: #56
-#33 := [asserted]: #15
-#60 := [mp #33 #57]: #55
-#153 := (not #143)
-#152 := (not #142)
-#154 := (or #45 #152 #153)
-#155 := [th-lemma]: #154
-[unit-resolution #155 #60 #148 #144]: false
-unsat
-V+IAyBZU/6QjYs6JkXx8LQ 57 0
-#2 := false
-#4 := 0::real
-decl uf_1 :: (-> T2 real)
-decl uf_2 :: (-> T1 T1 T2)
-decl uf_12 :: (-> T4 T1)
-decl uf_4 :: T4
-#11 := uf_4
-#39 := (uf_12 uf_4)
-decl uf_10 :: T4
-#27 := uf_10
-#38 := (uf_12 uf_10)
-#40 := (uf_2 #38 #39)
-#41 := (uf_1 #40)
-#264 := (>= #41 0::real)
-#266 := (not #264)
-#43 := (= #41 0::real)
-#44 := (not #43)
-#131 := [asserted]: #44
-#272 := (or #43 #266)
-#42 := (<= #41 0::real)
-#130 := [asserted]: #42
-#265 := (not #42)
-#270 := (or #43 #265 #266)
-#271 := [th-lemma]: #270
-#273 := [unit-resolution #271 #130]: #272
-#274 := [unit-resolution #273 #131]: #266
-#6 := (:var 0 T1)
-#5 := (:var 1 T1)
-#7 := (uf_2 #5 #6)
-#241 := (pattern #7)
-#8 := (uf_1 #7)
-#65 := (>= #8 0::real)
-#242 := (forall (vars (?x1 T1) (?x2 T1)) (:pat #241) #65)
-#66 := (forall (vars (?x1 T1) (?x2 T1)) #65)
-#245 := (iff #66 #242)
-#243 := (iff #65 #65)
-#244 := [refl]: #243
-#246 := [quant-intro #244]: #245
-#149 := (~ #66 #66)
-#151 := (~ #65 #65)
-#152 := [refl]: #151
-#150 := [nnf-pos #152]: #149
-#9 := (<= 0::real #8)
-#10 := (forall (vars (?x1 T1) (?x2 T1)) #9)
-#67 := (iff #10 #66)
-#63 := (iff #9 #65)
-#64 := [rewrite]: #63
-#68 := [quant-intro #64]: #67
-#60 := [asserted]: #10
-#69 := [mp #60 #68]: #66
-#147 := [mp~ #69 #150]: #66
-#247 := [mp #147 #246]: #242
-#267 := (not #242)
-#268 := (or #267 #264)
-#269 := [quant-inst]: #268
-[unit-resolution #269 #247 #274]: false
-unsat
-vqiyJ/qjGXZ3iOg6xftiug 15 0
-uf_1 -> val!0
-uf_2 -> val!1
-uf_3 -> val!2
-uf_5 -> val!15
-uf_6 -> val!26
-uf_4 -> {
- val!0 -> val!12
- val!1 -> val!13
- else -> val!13
-}
-uf_7 -> {
- val!6 -> val!31
- else -> val!31
-}
-sat
-mrZPJZyTokErrN6SYupisw 9 0
-uf_4 -> val!1
-uf_2 -> val!3
-uf_3 -> val!4
-uf_1 -> {
- val!5 -> val!6
- val!4 -> val!7
- else -> val!7
-}
-sat
--- a/src/HOL/Multivariate_Analysis/Integration_MV.thy Mon Feb 22 20:08:10 2010 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,3473 +0,0 @@
-
-header {* Kurzweil-Henstock gauge integration in many dimensions. *}
-(* Author: John Harrison
- Translation from HOL light: Robert Himmelmann, TU Muenchen *)
-
-theory Integration_MV
- imports Derivative SMT
-begin
-
-declare [[smt_certificates="~~/src/HOL/Multivariate_Analysis/Integration_MV.cert"]]
-declare [[smt_record=true]]
-declare [[z3_proofs=true]]
-
-lemma conjunctD2: assumes "a \<and> b" shows a b using assms by auto
-lemma conjunctD3: assumes "a \<and> b \<and> c" shows a b c using assms by auto
-lemma conjunctD4: assumes "a \<and> b \<and> c \<and> d" shows a b c d using assms by auto
-lemma conjunctD5: assumes "a \<and> b \<and> c \<and> d \<and> e" shows a b c d e using assms by auto
-
-declare smult_conv_scaleR[simp]
-
-subsection {* Some useful lemmas about intervals. *}
-
-lemma empty_as_interval: "{} = {1..0::real^'n}"
- apply(rule set_ext,rule) defer unfolding vector_le_def mem_interval
- using UNIV_witness[where 'a='n] apply(erule_tac exE,rule_tac x=x in allE) by auto
-
-lemma interior_subset_union_intervals:
- assumes "i = {a..b::real^'n}" "j = {c..d}" "interior j \<noteq> {}" "i \<subseteq> j \<union> s" "interior(i) \<inter> interior(j) = {}"
- shows "interior i \<subseteq> interior s" proof-
- have "{a<..<b} \<inter> {c..d} = {}" using inter_interval_mixed_eq_empty[of c d a b] and assms(3,5)
- unfolding assms(1,2) interior_closed_interval by auto
- moreover have "{a<..<b} \<subseteq> {c..d} \<union> s" apply(rule order_trans,rule interval_open_subset_closed)
- using assms(4) unfolding assms(1,2) by auto
- ultimately show ?thesis apply-apply(rule interior_maximal) defer apply(rule open_interior)
- unfolding assms(1,2) interior_closed_interval by auto qed
-
-lemma inter_interior_unions_intervals: fixes f::"(real^'n) set set"
- assumes "finite f" "open s" "\<forall>t\<in>f. \<exists>a b. t = {a..b}" "\<forall>t\<in>f. s \<inter> (interior t) = {}"
- shows "s \<inter> interior(\<Union>f) = {}" proof(rule ccontr,unfold ex_in_conv[THEN sym]) case goal1
- 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)
- unfolding open_subset_interior[OF open_ball] using interior_subset by auto
- have lem2:"\<And>x s P. \<exists>x\<in>s. P x \<Longrightarrow> \<exists>x\<in>insert x s. P x" by auto
- 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
- thus ?case proof(induct rule:finite_induct)
- case empty from this(2) guess x .. hence False unfolding Union_empty interior_empty by auto thus ?case by auto next
- case (insert i f) guess x using insert(5) .. note x = this
- then guess e unfolding open_contains_ball_eq[OF open_Int[OF assms(2) open_interior],rule_format] .. note e=this
- guess a using insert(4)[rule_format,OF insertI1] .. then guess b .. note ab = this
- show ?case proof(cases "x\<in>i") case False hence "x \<in> UNIV - {a..b}" unfolding ab by auto
- then guess d unfolding open_contains_ball_eq[OF open_Diff[OF open_UNIV closed_interval],rule_format] ..
- hence "0 < d" "ball x (min d e) \<subseteq> UNIV - i" using e unfolding ab by auto
- 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
- 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
- case True show ?thesis proof(cases "x\<in>{a<..<b}")
- case True then guess d unfolding open_contains_ball_eq[OF open_interval,rule_format] ..
- thus ?thesis apply(rule_tac x=i in bexI,rule_tac x=x in exI,rule_tac x="min d e" in exI)
- unfolding ab using interval_open_subset_closed[of a b] and e by fastsimp+ next
- case False then obtain k where "x$k \<le> a$k \<or> x$k \<ge> b$k" unfolding mem_interval by(auto simp add:not_less)
- hence "x$k = a$k \<or> x$k = b$k" using True unfolding ab and mem_interval apply(erule_tac x=k in allE) by auto
- hence "\<exists>x. ball x (e/2) \<subseteq> s \<inter> (\<Union>f)" proof(erule_tac disjE)
- let ?z = "x - (e/2) *\<^sub>R basis k" assume as:"x$k = a$k" have "ball ?z (e / 2) \<inter> i = {}" apply(rule ccontr) unfolding ex_in_conv[THEN sym] proof(erule exE)
- fix y assume "y \<in> ball ?z (e / 2) \<inter> i" hence "dist ?z y < e/2" and yi:"y\<in>i" by auto
- hence "\<bar>(?z - y) $ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding vector_dist_norm by auto
- hence "y$k < a$k" unfolding vector_component_simps vector_scaleR_component as using e[THEN conjunct1] by(auto simp add:field_simps)
- 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
- moreover have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)" apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]]) proof
- 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)"
- apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "(e/2) *\<^sub>R basis k"])
- unfolding norm_scaleR norm_basis by auto
- 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)
- finally show "y\<in>ball x e" unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps) qed
- ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto
- 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)
- fix y assume "y \<in> ball ?z (e / 2) \<inter> i" hence "dist ?z y < e/2" and yi:"y\<in>i" by auto
- hence "\<bar>(?z - y) $ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding vector_dist_norm by auto
- hence "y$k > b$k" unfolding vector_component_simps vector_scaleR_component as using e[THEN conjunct1] by(auto simp add:field_simps)
- 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
- moreover have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)" apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]]) proof
- 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)"
- apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "- (e/2) *\<^sub>R basis k"])
- unfolding norm_scaleR norm_basis by auto
- 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)
- finally show "y\<in>ball x e" unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps) qed
- ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto qed
- 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
- thus ?thesis apply-apply(rule lem2,rule insert(3)) using insert(4) by auto qed qed qed qed note * = this
- guess t using *[OF assms(1,3) goal1] .. from this(2) guess x .. then guess e ..
- hence "x \<in> s" "x\<in>interior t" defer using open_subset_interior[OF open_ball, of x e t] by auto
- thus False using `t\<in>f` assms(4) by auto qed
-subsection {* Bounds on intervals where they exist. *}
-
-definition "interval_upperbound (s::(real^'n) set) = (\<chi> i. Sup {a. \<exists>x\<in>s. x$i = a})"
-
-definition "interval_lowerbound (s::(real^'n) set) = (\<chi> i. Inf {a. \<exists>x\<in>s. x$i = a})"
-
-lemma interval_upperbound[simp]: assumes "\<forall>i. a$i \<le> b$i" shows "interval_upperbound {a..b} = b"
- using assms unfolding interval_upperbound_def Cart_eq Cart_lambda_beta apply-apply(rule,erule_tac x=i in allE)
- apply(rule Sup_unique) unfolding setle_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer
- apply(rule,rule) apply(rule_tac x="b$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=b in bexI)
- unfolding mem_interval using assms by auto
-
-lemma interval_lowerbound[simp]: assumes "\<forall>i. a$i \<le> b$i" shows "interval_lowerbound {a..b} = a"
- using assms unfolding interval_lowerbound_def Cart_eq Cart_lambda_beta apply-apply(rule,erule_tac x=i in allE)
- apply(rule Inf_unique) unfolding setge_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer
- apply(rule,rule) apply(rule_tac x="a$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=a in bexI)
- unfolding mem_interval using assms by auto
-
-lemmas interval_bounds = interval_upperbound interval_lowerbound
-
-lemma interval_bounds'[simp]: assumes "{a..b}\<noteq>{}" shows "interval_upperbound {a..b} = b" "interval_lowerbound {a..b} = a"
- using assms unfolding interval_ne_empty by auto
-
-lemma interval_upperbound_1[simp]: "dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> interval_upperbound {a..b} = (b::real^1)"
- apply(rule interval_upperbound) by auto
-
-lemma interval_lowerbound_1[simp]: "dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> interval_lowerbound {a..b} = (a::real^1)"
- apply(rule interval_lowerbound) by auto
-
-lemmas interval_bound_1 = interval_upperbound_1 interval_lowerbound_1
-
-subsection {* Content (length, area, volume...) of an interval. *}
-
-definition "content (s::(real^'n) set) =
- (if s = {} then 0 else (\<Prod>i\<in>UNIV. (interval_upperbound s)$i - (interval_lowerbound s)$i))"
-
-lemma interval_not_empty:"\<forall>i. a$i \<le> b$i \<Longrightarrow> {a..b::real^'n} \<noteq> {}"
- unfolding interval_eq_empty unfolding not_ex not_less by assumption
-
-lemma content_closed_interval: assumes "\<forall>i. a$i \<le> b$i"
- shows "content {a..b} = (\<Prod>i\<in>UNIV. b$i - a$i)"
- using interval_not_empty[OF assms] unfolding content_def interval_upperbound[OF assms] interval_lowerbound[OF assms] by auto
-
-lemma content_closed_interval': assumes "{a..b}\<noteq>{}" shows "content {a..b} = (\<Prod>i\<in>UNIV. b$i - a$i)"
- apply(rule content_closed_interval) using assms unfolding interval_ne_empty .
-
-lemma content_1:"dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> content {a..b} = dest_vec1 b - dest_vec1 a"
- using content_closed_interval[of a b] by auto
-
-lemma content_1':"a \<le> b \<Longrightarrow> content {vec1 a..vec1 b} = b - a" using content_1[of "vec a" "vec b"] by auto
-
-lemma content_unit[intro]: "content{0..1::real^'n} = 1" proof-
- have *:"\<forall>i. 0$i \<le> (1::real^'n::finite)$i" by auto
- have "0 \<in> {0..1::real^'n::finite}" unfolding mem_interval by auto
- thus ?thesis unfolding content_def interval_bounds[OF *] using setprod_1 by auto qed
-
-lemma content_pos_le[intro]: "0 \<le> content {a..b}" proof(cases "{a..b}={}")
- case False hence *:"\<forall>i. a $ i \<le> b $ i" unfolding interval_ne_empty by assumption
- have "(\<Prod>i\<in>UNIV. interval_upperbound {a..b} $ i - interval_lowerbound {a..b} $ i) \<ge> 0"
- apply(rule setprod_nonneg) unfolding interval_bounds[OF *] using * apply(erule_tac x=x in allE) by auto
- thus ?thesis unfolding content_def by(auto simp del:interval_bounds') qed(unfold content_def, auto)
-
-lemma content_pos_lt: assumes "\<forall>i. a$i < b$i" shows "0 < content {a..b}"
-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
- show ?thesis unfolding content_closed_interval[OF help_lemma1[OF assms]] apply(rule setprod_pos)
- using assms apply(erule_tac x=x in allE) by auto qed
-
-lemma content_pos_lt_1: "dest_vec1 a < dest_vec1 b \<Longrightarrow> 0 < content({a..b})"
- apply(rule content_pos_lt) by auto
-
-lemma content_eq_0: "content({a..b::real^'n}) = 0 \<longleftrightarrow> (\<exists>i. b$i \<le> a$i)" proof(cases "{a..b} = {}")
- case True thus ?thesis unfolding content_def if_P[OF True] unfolding interval_eq_empty apply-
- apply(rule,erule exE) apply(rule_tac x=i in exI) by auto next
- guess a using UNIV_witness[where 'a='n] .. case False note as=this[unfolded interval_eq_empty not_ex not_less]
- show ?thesis unfolding content_def if_not_P[OF False] setprod_zero_iff[OF finite_UNIV]
- apply(rule) apply(erule_tac[!] exE bexE) unfolding interval_bounds[OF as] apply(rule_tac x=x in exI) defer
- apply(rule_tac x=i in bexI) using as apply(erule_tac x=i in allE) by auto qed
-
-lemma cond_cases:"(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)" by auto
-
-lemma content_closed_interval_cases:
- "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)
- 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
-
-lemma content_eq_0_interior: "content {a..b} = 0 \<longleftrightarrow> interior({a..b}) = {}"
- unfolding content_eq_0 interior_closed_interval interval_eq_empty by auto
-
-lemma content_eq_0_1: "content {a..b::real^1} = 0 \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a"
- unfolding content_eq_0 by auto
-
-lemma content_pos_lt_eq: "0 < content {a..b} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
- apply(rule) defer apply(rule content_pos_lt,assumption) proof- assume "0 < content {a..b}"
- 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
-
-lemma content_empty[simp]: "content {} = 0" unfolding content_def by auto
-
-lemma content_subset: assumes "{a..b} \<subseteq> {c..d}" shows "content {a..b::real^'n} \<le> content {c..d}" proof(cases "{a..b}={}")
- case True thus ?thesis using content_pos_le[of c d] by auto next
- case False hence ab_ne:"\<forall>i. a $ i \<le> b $ i" unfolding interval_ne_empty by auto
- hence ab_ab:"a\<in>{a..b}" "b\<in>{a..b}" unfolding mem_interval by auto
- have "{c..d} \<noteq> {}" using assms False by auto
- hence cd_ne:"\<forall>i. c $ i \<le> d $ i" using assms unfolding interval_ne_empty by auto
- show ?thesis unfolding content_def unfolding interval_bounds[OF ab_ne] interval_bounds[OF cd_ne]
- unfolding if_not_P[OF False] if_not_P[OF `{c..d} \<noteq> {}`] apply(rule setprod_mono,rule) proof fix i::'n
- show "0 \<le> b $ i - a $ i" using ab_ne[THEN spec[where x=i]] by auto
- show "b $ i - a $ i \<le> d $ i - c $ i"
- using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(2),of i]
- using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(1),of i] by auto qed qed
-
-lemma content_lt_nz: "0 < content {a..b} \<longleftrightarrow> content {a..b} \<noteq> 0"
- unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by auto
-
-subsection {* The notion of a gauge --- simply an open set containing the point. *}
-
-definition gauge where "gauge d \<longleftrightarrow> (\<forall>x. x\<in>(d x) \<and> open(d x))"
-
-lemma gaugeI:assumes "\<And>x. x\<in>g x" "\<And>x. open (g x)" shows "gauge g"
- using assms unfolding gauge_def by auto
-
-lemma gaugeD[dest]: assumes "gauge d" shows "x\<in>d x" "open (d x)" using assms unfolding gauge_def by auto
-
-lemma gauge_ball_dependent: "\<forall>x. 0 < e x \<Longrightarrow> gauge (\<lambda>x. ball x (e x))"
- unfolding gauge_def by auto
-
-lemma gauge_ball[intro?]: "0 < e \<Longrightarrow> gauge (\<lambda>x. ball x e)" unfolding gauge_def by auto
-
-lemma gauge_trivial[intro]: "gauge (\<lambda>x. ball x 1)" apply(rule gauge_ball) by auto
-
-lemma gauge_inter: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. (d1 x) \<inter> (d2 x))"
- unfolding gauge_def by auto
-
-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-
- have *:"\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s" by auto show ?thesis
- unfolding gauge_def unfolding *
- using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto qed
-
-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)
-
-subsection {* Divisions. *}
-
-definition division_of (infixl "division'_of" 40) where
- "s division_of i \<equiv>
- finite s \<and>
- (\<forall>k\<in>s. k \<subseteq> i \<and> k \<noteq> {} \<and> (\<exists>a b. k = {a..b})) \<and>
- (\<forall>k1\<in>s. \<forall>k2\<in>s. k1 \<noteq> k2 \<longrightarrow> interior(k1) \<inter> interior(k2) = {}) \<and>
- (\<Union>s = i)"
-
-lemma division_ofD[dest]: assumes "s division_of i"
- 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})"
- "\<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
-
-lemma division_ofI:
- 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})"
- "\<And>k1 k2. \<lbrakk>k1\<in>s; k2\<in>s; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}" "\<Union>s = i"
- shows "s division_of i" using assms unfolding division_of_def by auto
-
-lemma division_of_finite: "s division_of i \<Longrightarrow> finite s"
- unfolding division_of_def by auto
-
-lemma division_of_self[intro]: "{a..b} \<noteq> {} \<Longrightarrow> {{a..b}} division_of {a..b}"
- unfolding division_of_def by auto
-
-lemma division_of_trivial[simp]: "s division_of {} \<longleftrightarrow> s = {}" unfolding division_of_def by auto
-
-lemma division_of_sing[simp]: "s division_of {a..a::real^'n} \<longleftrightarrow> s = {{a..a}}" (is "?l = ?r") proof
- assume ?r moreover { assume "s = {{a}}" moreover fix k assume "k\<in>s"
- ultimately have"\<exists>x y. k = {x..y}" apply(rule_tac x=a in exI)+ unfolding interval_sing[THEN conjunct1] by auto }
- ultimately show ?l unfolding division_of_def interval_sing[THEN conjunct1] by auto next
- assume ?l note as=conjunctD4[OF this[unfolded division_of_def interval_sing[THEN conjunct1]]]
- { fix x assume x:"x\<in>s" have "x={a}" using as(2)[rule_format,OF x] by auto }
- moreover have "s \<noteq> {}" using as(4) by auto ultimately show ?r unfolding interval_sing[THEN conjunct1] by auto qed
-
-lemma elementary_empty: obtains p where "p division_of {}"
- unfolding division_of_trivial by auto
-
-lemma elementary_interval: obtains p where "p division_of {a..b}"
- by(metis division_of_trivial division_of_self)
-
-lemma division_contains: "s division_of i \<Longrightarrow> \<forall>x\<in>i. \<exists>k\<in>s. x \<in> k"
- unfolding division_of_def by auto
-
-lemma forall_in_division:
- "d division_of i \<Longrightarrow> ((\<forall>x\<in>d. P x) \<longleftrightarrow> (\<forall>a b. {a..b} \<in> d \<longrightarrow> P {a..b}))"
- unfolding division_of_def by fastsimp
-
-lemma division_of_subset: assumes "p division_of (\<Union>p)" "q \<subseteq> p" shows "q division_of (\<Union>q)"
- apply(rule division_ofI) proof- note as=division_ofD[OF assms(1)]
- show "finite q" apply(rule finite_subset) using as(1) assms(2) by auto
- { 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
- show "\<exists>a b. k = {a..b}" using as(4)[OF kp] by auto show "k \<noteq> {}" using as(3)[OF kp] by auto }
- 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
- show "interior k1 \<inter> interior k2 = {}" using as(5)[OF *] by auto qed auto
-
-lemma division_of_union_self[intro]: "p division_of s \<Longrightarrow> p division_of (\<Union>p)" unfolding division_of_def by auto
-
-lemma division_of_content_0: assumes "content {a..b} = 0" "d division_of {a..b}" shows "\<forall>k\<in>d. content k = 0"
- unfolding forall_in_division[OF assms(2)] apply(rule,rule,rule) apply(drule division_ofD(2)[OF assms(2)])
- apply(drule content_subset) unfolding assms(1) proof- case goal1 thus ?case using content_pos_le[of a b] by auto qed
-
-lemma division_inter: assumes "p1 division_of s1" "p2 division_of (s2::(real^'a) set)"
- 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-
-let ?A = "{s. s \<in> (\<lambda>(k1,k2). k1 \<inter> k2) ` (p1 \<times> p2) \<and> s \<noteq> {}}" have *:"?A' = ?A" by auto
-show ?thesis unfolding * proof(rule division_ofI) have "?A \<subseteq> (\<lambda>(x, y). x \<inter> y) ` (p1 \<times> p2)" by auto
- moreover have "finite (p1 \<times> p2)" using assms unfolding division_of_def by auto ultimately show "finite ?A" by auto
- 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
- using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)] by auto
- { 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
- 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
- guess a1 using division_ofD(4)[OF assms(1) k(2)] .. then guess b1 .. note ab1=this
- guess a2 using division_ofD(4)[OF assms(2) k(3)] .. then guess b2 .. note ab2=this
- show "\<exists>a b. k = {a..b}" unfolding k ab1 ab2 unfolding inter_interval by auto } fix k1 k2
- assume "k1\<in>?A" then obtain x1 y1 where k1:"k1 = x1 \<inter> y1" "x1\<in>p1" "y1\<in>p2" "k1\<noteq>{}" by auto
- assume "k2\<in>?A" then obtain x2 y2 where k2:"k2 = x2 \<inter> y2" "x2\<in>p1" "y2\<in>p2" "k2\<noteq>{}" by auto
- assume "k1 \<noteq> k2" hence th:"x1\<noteq>x2 \<or> y1\<noteq>y2" unfolding k1 k2 by auto
- have *:"(interior x1 \<inter> interior x2 = {} \<or> interior y1 \<inter> interior y2 = {}) \<Longrightarrow>
- interior(x1 \<inter> y1) \<subseteq> interior(x1) \<Longrightarrow> interior(x1 \<inter> y1) \<subseteq> interior(y1) \<Longrightarrow>
- interior(x2 \<inter> y2) \<subseteq> interior(x2) \<Longrightarrow> interior(x2 \<inter> y2) \<subseteq> interior(y2)
- \<Longrightarrow> interior(x1 \<inter> y1) \<inter> interior(x2 \<inter> y2) = {}" by auto
- show "interior k1 \<inter> interior k2 = {}" unfolding k1 k2 apply(rule *) defer apply(rule_tac[1-4] subset_interior)
- using division_ofD(5)[OF assms(1) k1(2) k2(2)]
- using division_ofD(5)[OF assms(2) k1(3) k2(3)] using th by auto qed qed
-
-lemma division_inter_1: assumes "d division_of i" "{a..b::real^'n} \<subseteq> i"
- shows "{ {a..b} \<inter> k |k. k \<in> d \<and> {a..b} \<inter> k \<noteq> {} } division_of {a..b}" proof(cases "{a..b} = {}")
- case True show ?thesis unfolding True and division_of_trivial by auto next
- have *:"{a..b} \<inter> i = {a..b}" using assms(2) by auto
- case False show ?thesis using division_inter[OF division_of_self[OF False] assms(1)] unfolding * by auto qed
-
-lemma elementary_inter: assumes "p1 division_of s" "p2 division_of (t::(real^'n) set)"
- shows "\<exists>p. p division_of (s \<inter> t)"
- by(rule,rule division_inter[OF assms])
-
-lemma elementary_inters: assumes "finite f" "f\<noteq>{}" "\<forall>s\<in>f. \<exists>p. p division_of (s::(real^'n) set)"
- shows "\<exists>p. p division_of (\<Inter> f)" using assms apply-proof(induct f rule:finite_induct)
-case (insert x f) show ?case proof(cases "f={}")
- case True thus ?thesis unfolding True using insert by auto next
- case False guess p using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] ..
- moreover guess px using insert(5)[rule_format,OF insertI1] .. ultimately
- show ?thesis unfolding Inter_insert apply(rule_tac elementary_inter) by assumption+ qed qed auto
-
-lemma division_disjoint_union:
- assumes "p1 division_of s1" "p2 division_of s2" "interior s1 \<inter> interior s2 = {}"
- shows "(p1 \<union> p2) division_of (s1 \<union> s2)" proof(rule division_ofI)
- note d1 = division_ofD[OF assms(1)] and d2 = division_ofD[OF assms(2)]
- show "finite (p1 \<union> p2)" using d1(1) d2(1) by auto
- show "\<Union>(p1 \<union> p2) = s1 \<union> s2" using d1(6) d2(6) by auto
- { 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 = {}"
- { 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)]]
- using assms(3) by blast } moreover
- { 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)]]
- using assms(3) by blast} ultimately
- show ?g using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto }
- fix k assume k:"k \<in> p1 \<union> p2" show "k \<subseteq> s1 \<union> s2" using k d1(2) d2(2) by auto
- 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
-
-lemma partial_division_extend_1:
- assumes "{c..d} \<subseteq> {a..b::real^'n}" "{c..d} \<noteq> {}"
- obtains p where "p division_of {a..b}" "{c..d} \<in> p"
-proof- def n \<equiv> "CARD('n)" have n:"1 \<le> n" "0 < n" "n \<noteq> 0" unfolding n_def by auto
- guess \<pi> using ex_bij_betw_nat_finite_1[OF finite_UNIV[where 'a='n]] .. note \<pi>=this
- def \<pi>' \<equiv> "inv_into {1..n} \<pi>"
- have \<pi>':"bij_betw \<pi>' UNIV {1..n}" using bij_betw_inv_into[OF \<pi>] unfolding \<pi>'_def n_def by auto
- hence \<pi>'i:"\<And>i. \<pi>' i \<in> {1..n}" unfolding bij_betw_def by auto
- 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
- 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
- have "{c..d} \<noteq> {}" using assms by auto
- 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$i)}"
- let ?p2 = "\<lambda>l. {(\<chi> i. if \<pi>' i < l then c$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)}"
- let ?p = "{?p1 l |l. l \<in> {1..n+1}} \<union> {?p2 l |l. l \<in> {1..n+1}}"
- 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)
- show ?thesis apply(rule that[of ?p]) apply(rule division_ofI)
- proof- have "\<And>i. \<pi>' i < Suc n"
- proof(rule ccontr,unfold not_less) fix i assume "Suc n \<le> \<pi>' i"
- hence "\<pi>' i \<notin> {1..n}" by auto thus False using \<pi>' unfolding bij_betw_def by auto
- qed hence "c = (\<chi> i. if \<pi>' i < Suc n then c $ i else a $ i)"
- "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)"
- unfolding Cart_eq Cart_lambda_beta using \<pi>' unfolding bij_betw_def by auto
- thus cdp:"{c..d} \<in> ?p" apply-apply(rule UnI1) unfolding mem_Collect_eq apply(rule_tac x="n + 1" in exI) by auto
- 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}"
- unfolding subset_eq apply(rule_tac[!] ballI,rule_tac[!] ccontr)
- proof- fix l assume l:"l\<in>{1..n+1}" fix x assume "x\<notin>{a..b}"
- then guess i unfolding mem_interval not_all .. note i=this
- show "x \<in> ?p1 l \<Longrightarrow> False" "x \<in> ?p2 l \<Longrightarrow> False" unfolding mem_interval apply(erule_tac[!] x=i in allE)
- apply(case_tac[!] "\<pi>' i < l", case_tac[!] "\<pi>' i = l") using abcd[of i] i by auto
- qed moreover have "\<And>x. x \<in> {a..b} \<Longrightarrow> x \<in> \<Union>?p"
- proof- fix x assume x:"x\<in>{a..b}"
- { presume "x\<notin>{c..d} \<Longrightarrow> x \<in> \<Union>?p" thus "x \<in> \<Union>?p" using cdp by blast }
- 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)}"
- assume "x\<notin>{c..d}" then guess i0 unfolding mem_interval not_all ..
- hence "\<pi>' i0 \<in> ?M" using \<pi>' unfolding bij_betw_def by(auto intro!:le_SucI)
- hence M:"finite ?M" "?M \<noteq> {}" by auto
- 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]]
- Min_gr_iff[OF M,unfolded l_def[symmetric]]
- have "x\<in>?p1 l \<or> x\<in>?p2 l" using l(2)[THEN conjunct2] unfolding de_Morgan_conj not_le
- apply- apply(erule disjE) apply(rule disjI1) defer apply(rule disjI2)
- proof- assume as:"x $ \<pi> l < c $ \<pi> l"
- show "x \<in> ?p1 l" unfolding mem_interval Cart_lambda_beta
- proof case goal1 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le by auto
- thus ?case using as x[unfolded mem_interval,rule_format,of i]
- apply auto using l(3)[of "\<pi>' i"] by(auto elim!:ballE[where x="\<pi>' i"])
- qed
- next assume as:"x $ \<pi> l > d $ \<pi> l"
- show "x \<in> ?p2 l" unfolding mem_interval Cart_lambda_beta
- proof case goal1 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le by auto
- thus ?case using as x[unfolded mem_interval,rule_format,of i]
- apply auto using l(3)[of "\<pi>' i"] by(auto elim!:ballE[where x="\<pi>' i"])
- qed qed
- thus "x \<in> \<Union>?p" using l(2) by blast
- qed ultimately show "\<Union>?p = {a..b}" apply-apply(rule) defer apply(rule) by(assumption,blast)
-
- show "finite ?p" by auto
- 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
- show "k\<subseteq>{a..b}" apply(rule,unfold mem_interval,rule,rule)
- 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
- 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)
- qed have "\<And>l. ?p1 l \<noteq> {}" "\<And>l. ?p2 l \<noteq> {}" unfolding interval_eq_empty not_ex apply(rule_tac[!] allI)
- proof- case goal1 thus ?case using abcd[of x] by auto
- next case goal2 thus ?case using abcd[of x] by auto
- qed thus "k \<noteq> {}" using k by auto
- show "\<exists>a b. k = {a..b}" using k by auto
- 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
- { fix k k' l l'
- assume k:"k\<in>?p" and l:"k = ?p1 l \<or> k = ?p2 l" "l \<in> {1..n + 1}"
- assume k':"k' \<in> ?p" "k \<noteq> k'" and l':"k' = ?p1 l' \<or> k' = ?p2 l'" "l' \<in> {1..n + 1}"
- assume "l \<le> l'" fix x
- have "x \<notin> interior k \<inter> interior k'"
- proof(rule,cases "l' = n+1") assume x:"x \<in> interior k \<inter> interior k'"
- case True hence "\<And>i. \<pi>' i < l'" using \<pi>'i by(auto simp add:less_Suc_eq_le)
- hence k':"k' = {c..d}" using l'(1) \<pi>'i by(auto simp add:Cart_nth_inverse)
- have ln:"l < n + 1"
- proof(rule ccontr) case goal1 hence l2:"l = n+1" using l by auto
- hence "\<And>i. \<pi>' i < l" using \<pi>'i by(auto simp add:less_Suc_eq_le)
- hence "k = {c..d}" using l(1) \<pi>'i by(auto simp add:Cart_nth_inverse)
- thus False using `k\<noteq>k'` k' by auto
- qed have **:"\<pi>' (\<pi> l) = l" using \<pi>'\<pi>[of l] using l ln by auto
- have "x $ \<pi> l < c $ \<pi> l \<or> d $ \<pi> l < x $ \<pi> l" using l(1) apply-
- proof(erule disjE)
- assume as:"k = ?p1 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
- show ?thesis using *[of "\<pi> l"] using ln unfolding Cart_lambda_beta ** by auto
- next assume as:"k = ?p2 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
- show ?thesis using *[of "\<pi> l"] using ln unfolding Cart_lambda_beta ** by auto
- qed thus False using x unfolding k' unfolding Int_iff interior_closed_interval mem_interval
- by(auto elim!:allE[where x="\<pi> l"])
- next case False hence "l < n + 1" using l'(2) using `l\<le>l'` by auto
- hence ln:"l \<in> {1..n}" "l' \<in> {1..n}" using l l' False by auto
- note \<pi>l = \<pi>'\<pi>[OF ln(1)] \<pi>'\<pi>[OF ln(2)]
- assume x:"x \<in> interior k \<inter> interior k'"
- show False using l(1) l'(1) apply-
- proof(erule_tac[!] disjE)+
- assume as:"k = ?p1 l" "k' = ?p1 l'"
- note * = x[unfolded as Int_iff interior_closed_interval mem_interval]
- have "l \<noteq> l'" using k'(2)[unfolded as] by auto
- thus False using * by(smt Cart_lambda_beta \<pi>l)
- next assume as:"k = ?p2 l" "k' = ?p2 l'"
- note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
- have "l \<noteq> l'" apply(rule) using k'(2)[unfolded as] by auto
- thus False using *[of "\<pi> l"] *[of "\<pi> l'"]
- unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` by auto
- next assume as:"k = ?p1 l" "k' = ?p2 l'"
- note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
- show False using *[of "\<pi> l"] *[of "\<pi> l'"]
- unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` using abcd[of "\<pi> l'"] by smt
- next assume as:"k = ?p2 l" "k' = ?p1 l'"
- note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
- show False using *[of "\<pi> l"] *[of "\<pi> l'"]
- unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` using abcd[of "\<pi> l'"] by smt
- qed qed }
- from this[OF k l k' l'] this[OF k'(1) l' k _ l] have "\<And>x. x \<notin> interior k \<inter> interior k'"
- apply - apply(cases "l' \<le> l") using k'(2) by auto
- thus "interior k \<inter> interior k' = {}" by auto
-qed qed
-
-lemma partial_division_extend_interval: assumes "p division_of (\<Union>p)" "(\<Union>p) \<subseteq> {a..b}"
- obtains q where "p \<subseteq> q" "q division_of {a..b::real^'n}" proof(cases "p = {}")
- case True guess q apply(rule elementary_interval[of a b]) .
- thus ?thesis apply- apply(rule that[of q]) unfolding True by auto next
- case False note p = division_ofD[OF assms(1)]
- have *:"\<forall>k\<in>p. \<exists>q. q division_of {a..b} \<and> k\<in>q" proof case goal1
- guess c using p(4)[OF goal1] .. then guess d .. note cd_ = this
- have *:"{c..d} \<subseteq> {a..b}" "{c..d} \<noteq> {}" using p(2,3)[OF goal1, unfolded cd_] using assms(2) by auto
- guess q apply(rule partial_division_extend_1[OF *]) . thus ?case unfolding cd_ by auto qed
- guess q using bchoice[OF *] .. note q = conjunctD2[OF this[rule_format]]
- 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-
- fix x assume x:"x\<in>p" show "q x division_of \<Union>q x" apply-apply(rule division_ofI)
- using division_ofD[OF q(1)[OF x]] by auto show "q x - {x} \<subseteq> q x" by auto qed
- hence "\<exists>d. d division_of \<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)" apply- apply(rule elementary_inters)
- apply(rule finite_imageI[OF p(1)]) unfolding image_is_empty apply(rule False) by auto
- then guess d .. note d = this
- show ?thesis apply(rule that[of "d \<union> p"]) proof-
- 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
- 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"
- 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
- show "d \<union> p division_of {a..b}" unfolding * apply(rule division_disjoint_union[OF d assms(1)])
- apply(rule inter_interior_unions_intervals) apply(rule p open_interior ballI)+ proof(assumption,rule)
- 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
- show "interior (\<Inter>(\<lambda>i. \<Union>(q i - {i})) ` p) \<inter> interior k = {}" apply(rule *[of _ "interior (\<Union>(q k - {k}))"])
- defer apply(subst Int_commute) apply(rule inter_interior_unions_intervals) proof- note qk=division_ofD[OF q(1)[OF k]]
- show "finite (q k - {k})" "open (interior k)" "\<forall>t\<in>q k - {k}. \<exists>a b. t = {a..b}" using qk by auto
- show "\<forall>t\<in>q k - {k}. interior k \<inter> interior t = {}" using qk(5) using q(2)[OF k] by auto
- 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}))"
- apply(rule subset_interior *)+ using k by auto qed qed qed auto qed
-
-lemma elementary_bounded[dest]: "p division_of s \<Longrightarrow> bounded (s::(real^'n) set)"
- unfolding division_of_def by(metis bounded_Union bounded_interval)
-
-lemma elementary_subset_interval: "p division_of s \<Longrightarrow> \<exists>a b. s \<subseteq> {a..b::real^'n}"
- by(meson elementary_bounded bounded_subset_closed_interval)
-
-lemma division_union_intervals_exists: assumes "{a..b::real^'n} \<noteq> {}"
- obtains p where "(insert {a..b} p) division_of ({a..b} \<union> {c..d})" proof(cases "{c..d} = {}")
- case True show ?thesis apply(rule that[of "{}"]) unfolding True using assms by auto next
- case False note false=this show ?thesis proof(cases "{a..b} \<inter> {c..d} = {}")
- have *:"\<And>a b. {a,b} = {a} \<union> {b}" by auto
- case True show ?thesis apply(rule that[of "{{c..d}}"]) unfolding * apply(rule division_disjoint_union)
- using false True assms using interior_subset by auto next
- case False obtain u v where uv:"{a..b} \<inter> {c..d} = {u..v}" unfolding inter_interval by auto
- have *:"{u..v} \<subseteq> {c..d}" using uv by auto
- guess p apply(rule partial_division_extend_1[OF * False[unfolded uv]]) . note p=this division_ofD[OF this(1)]
- 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
- show thesis apply(rule that[of "p - {{u..v}}"]) unfolding *(1) apply(subst *(2)) apply(rule division_disjoint_union)
- apply(rule,rule assms) apply(rule division_of_subset[of p]) apply(rule division_of_union_self[OF p(1)]) defer
- unfolding interior_inter[THEN sym] proof-
- have *:"\<And>cd p uv ab. p \<subseteq> cd \<Longrightarrow> ab \<inter> cd = uv \<Longrightarrow> ab \<inter> p = uv \<inter> p" by auto
- have "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = interior({u..v} \<inter> \<Union>(p - {{u..v}}))"
- apply(rule arg_cong[of _ _ interior]) apply(rule *[OF _ uv]) using p(8) by auto
- also have "\<dots> = {}" unfolding interior_inter apply(rule inter_interior_unions_intervals) using p(6) p(7)[OF p(2)] p(3) by auto
- finally show "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = {}" by assumption qed auto qed qed
-
-lemma division_of_unions: assumes "finite f" "\<And>p. p\<in>f \<Longrightarrow> p division_of (\<Union>p)"
- "\<And>k1 k2. \<lbrakk>k1 \<in> \<Union>f; k2 \<in> \<Union>f; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
- shows "\<Union>f division_of \<Union>\<Union>f" apply(rule division_ofI) prefer 5 apply(rule assms(3)|assumption)+
- apply(rule finite_Union assms(1))+ prefer 3 apply(erule UnionE) apply(rule_tac s=X in division_ofD(3)[OF assms(2)])
- using division_ofD[OF assms(2)] by auto
-
-lemma elementary_union_interval: assumes "p division_of \<Union>p"
- obtains q where "q division_of ({a..b::real^'n} \<union> \<Union>p)" proof-
- note assm=division_ofD[OF assms]
- have lem1:"\<And>f s. \<Union>\<Union> (f ` s) = \<Union>(\<lambda>x.\<Union>(f x)) ` s" by auto
- have lem2:"\<And>f s. f \<noteq> {} \<Longrightarrow> \<Union>{s \<union> t |t. t \<in> f} = s \<union> \<Union>f" by auto
-{ presume "p={} \<Longrightarrow> thesis" "{a..b} = {} \<Longrightarrow> thesis" "{a..b} \<noteq> {} \<Longrightarrow> interior {a..b} = {} \<Longrightarrow> thesis"
- "p\<noteq>{} \<Longrightarrow> interior {a..b}\<noteq>{} \<Longrightarrow> {a..b} \<noteq> {} \<Longrightarrow> thesis"
- thus thesis by auto
-next assume as:"p={}" guess p apply(rule elementary_interval[of a b]) .
- thus thesis apply(rule_tac that[of p]) unfolding as by auto
-next assume as:"{a..b}={}" show thesis apply(rule that) unfolding as using assms by auto
-next assume as:"interior {a..b} = {}" "{a..b} \<noteq> {}"
- show thesis apply(rule that[of "insert {a..b} p"],rule division_ofI)
- unfolding finite_insert apply(rule assm(1)) unfolding Union_insert
- using assm(2-4) as apply- by(fastsimp dest: assm(5))+
-next assume as:"p \<noteq> {}" "interior {a..b} \<noteq> {}" "{a..b}\<noteq>{}"
- have "\<forall>k\<in>p. \<exists>q. (insert {a..b} q) division_of ({a..b} \<union> k)" proof case goal1
- from assm(4)[OF this] guess c .. then guess d ..
- thus ?case apply-apply(rule division_union_intervals_exists[OF as(3),of c d]) by auto
- qed from bchoice[OF this] guess q .. note q=division_ofD[OF this[rule_format]]
- let ?D = "\<Union>{insert {a..b} (q k) | k. k \<in> p}"
- show thesis apply(rule that[of "?D"]) proof(rule division_ofI)
- have *:"{insert {a..b} (q k) |k. k \<in> p} = (\<lambda>k. insert {a..b} (q k)) ` p" by auto
- show "finite ?D" apply(rule finite_Union) unfolding * apply(rule finite_imageI) using assm(1) q(1) by auto
- show "\<Union>?D = {a..b} \<union> \<Union>p" unfolding * lem1 unfolding lem2[OF as(1), of "{a..b}",THEN sym]
- using q(6) by auto
- fix k assume k:"k\<in>?D" thus " k \<subseteq> {a..b} \<union> \<Union>p" using q(2) by auto
- show "k \<noteq> {}" using q(3) k by auto show "\<exists>a b. k = {a..b}" using q(4) k by auto
- fix k' assume k':"k'\<in>?D" "k\<noteq>k'"
- obtain x where x: "k \<in>insert {a..b} (q x)" "x\<in>p" using k by auto
- obtain x' where x':"k'\<in>insert {a..b} (q x')" "x'\<in>p" using k' by auto
- show "interior k \<inter> interior k' = {}" proof(cases "x=x'")
- case True show ?thesis apply(rule q(5)) using x x' k' unfolding True by auto
- next case False
- { presume "k = {a..b} \<Longrightarrow> ?thesis" "k' = {a..b} \<Longrightarrow> ?thesis"
- "k \<noteq> {a..b} \<Longrightarrow> k' \<noteq> {a..b} \<Longrightarrow> ?thesis"
- thus ?thesis by auto }
- { assume as':"k = {a..b}" show ?thesis apply(rule q(5)) using x' k'(2) unfolding as' by auto }
- { assume as':"k' = {a..b}" show ?thesis apply(rule q(5)) using x k'(2) unfolding as' by auto }
- assume as':"k \<noteq> {a..b}" "k' \<noteq> {a..b}"
- guess c using q(4)[OF x(2,1)] .. then guess d .. note c_d=this
- have "interior k \<inter> interior {a..b} = {}" apply(rule q(5)) using x k'(2) using as' by auto
- hence "interior k \<subseteq> interior x" apply-
- apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x(2,1)]]) by auto moreover
- guess c using q(4)[OF x'(2,1)] .. then guess d .. note c_d=this
- have "interior k' \<inter> interior {a..b} = {}" apply(rule q(5)) using x' k'(2) using as' by auto
- hence "interior k' \<subseteq> interior x'" apply-
- apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x'(2,1)]]) by auto
- ultimately show ?thesis using assm(5)[OF x(2) x'(2) False] by auto
- qed qed } qed
-
-lemma elementary_unions_intervals:
- assumes "finite f" "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = {a..b::real^'n}"
- obtains p where "p division_of (\<Union>f)" proof-
- have "\<exists>p. p division_of (\<Union>f)" proof(induct_tac f rule:finite_subset_induct)
- show "\<exists>p. p division_of \<Union>{}" using elementary_empty by auto
- fix x F assume as:"finite F" "x \<notin> F" "\<exists>p. p division_of \<Union>F" "x\<in>f"
- from this(3) guess p .. note p=this
- from assms(2)[OF as(4)] guess a .. then guess b .. note ab=this
- have *:"\<Union>F = \<Union>p" using division_ofD[OF p] by auto
- show "\<exists>p. p division_of \<Union>insert x F" using elementary_union_interval[OF p[unfolded *], of a b]
- unfolding Union_insert ab * by auto
- qed(insert assms,auto) thus ?thesis apply-apply(erule exE,rule that) by auto qed
-
-lemma elementary_union: assumes "ps division_of s" "pt division_of (t::(real^'n) set)"
- obtains p where "p division_of (s \<union> t)"
-proof- have "s \<union> t = \<Union>ps \<union> \<Union>pt" using assms unfolding division_of_def by auto
- hence *:"\<Union>(ps \<union> pt) = s \<union> t" by auto
- show ?thesis apply-apply(rule elementary_unions_intervals[of "ps\<union>pt"])
- unfolding * prefer 3 apply(rule_tac p=p in that)
- using assms[unfolded division_of_def] by auto qed
-
-lemma partial_division_extend: fixes t::"(real^'n) set"
- assumes "p division_of s" "q division_of t" "s \<subseteq> t"
- obtains r where "p \<subseteq> r" "r division_of t" proof-
- note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)]
- obtain a b where ab:"t\<subseteq>{a..b}" using elementary_subset_interval[OF assms(2)] by auto
- guess r1 apply(rule partial_division_extend_interval) apply(rule assms(1)[unfolded divp(6)[THEN sym]])
- apply(rule subset_trans) by(rule ab assms[unfolded divp(6)[THEN sym]])+ note r1 = this division_ofD[OF this(2)]
- guess p' apply(rule elementary_unions_intervals[of "r1 - p"]) using r1(3,6) by auto
- then obtain r2 where r2:"r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)"
- apply- apply(drule elementary_inter[OF _ assms(2)[unfolded divq(6)[THEN sym]]]) by auto
- { fix x assume x:"x\<in>t" "x\<notin>s"
- hence "x\<in>\<Union>r1" unfolding r1 using ab by auto
- then guess r unfolding Union_iff .. note r=this moreover
- have "r \<notin> p" proof assume "r\<in>p" hence "x\<in>s" using divp(2) r by auto
- thus False using x by auto qed
- ultimately have "x\<in>\<Union>(r1 - p)" by auto }
- hence *:"t = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)" unfolding divp divq using assms(3) by auto
- show ?thesis apply(rule that[of "p \<union> r2"]) unfolding * defer apply(rule division_disjoint_union)
- unfolding divp(6) apply(rule assms r2)+
- proof- have "interior s \<inter> interior (\<Union>(r1-p)) = {}"
- proof(rule inter_interior_unions_intervals)
- show "finite (r1 - p)" "open (interior s)" "\<forall>t\<in>r1-p. \<exists>a b. t = {a..b}" using r1 by auto
- have *:"\<And>s. (\<And>x. x \<in> s \<Longrightarrow> False) \<Longrightarrow> s = {}" by auto
- show "\<forall>t\<in>r1-p. interior s \<inter> interior t = {}" proof(rule)
- fix m x assume as:"m\<in>r1-p"
- have "interior m \<inter> interior (\<Union>p) = {}" proof(rule inter_interior_unions_intervals)
- show "finite p" "open (interior m)" "\<forall>t\<in>p. \<exists>a b. t = {a..b}" using divp by auto
- show "\<forall>t\<in>p. interior m \<inter> interior t = {}" apply(rule, rule r1(7)) using as using r1 by auto
- qed thus "interior s \<inter> interior m = {}" unfolding divp by auto
- qed qed
- thus "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}" using interior_subset by auto
- qed auto qed
-
-subsection {* Tagged (partial) divisions. *}
-
-definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40) where
- "(s tagged_partial_division_of i) \<equiv>
- finite s \<and>
- (\<forall>x k. (x,k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
- (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ((x1,k1) \<noteq> (x2,k2))
- \<longrightarrow> (interior(k1) \<inter> interior(k2) = {}))"
-
-lemma tagged_partial_division_ofD[dest]: assumes "s tagged_partial_division_of i"
- 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}"
- "\<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) = {}"
- using assms unfolding tagged_partial_division_of_def apply- by blast+
-
-definition tagged_division_of (infixr "tagged'_division'_of" 40) where
- "(s tagged_division_of i) \<equiv>
- (s tagged_partial_division_of i) \<and> (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
-
-lemma tagged_division_of_finite[dest]: "s tagged_division_of i \<Longrightarrow> finite s"
- unfolding tagged_division_of_def tagged_partial_division_of_def by auto
-
-lemma tagged_division_of:
- "(s tagged_division_of i) \<longleftrightarrow>
- finite s \<and>
- (\<forall>x k. (x,k) \<in> s
- \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
- (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ~((x1,k1) = (x2,k2))
- \<longrightarrow> (interior(k1) \<inter> interior(k2) = {})) \<and>
- (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
- unfolding tagged_division_of_def tagged_partial_division_of_def by auto
-
-lemma tagged_division_ofI: assumes
- "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}"
- "\<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) = {})"
- "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
- shows "s tagged_division_of i"
- unfolding tagged_division_of apply(rule) defer apply rule
- apply(rule allI impI conjI assms)+ apply assumption
- apply(rule, rule assms, assumption) apply(rule assms, assumption)
- using assms(1,5-) apply- by blast+
-
-lemma tagged_division_ofD[dest]: assumes "s tagged_division_of i"
- 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}"
- "\<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) = {})"
- "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)" using assms unfolding tagged_division_of apply- by blast+
-
-lemma division_of_tagged_division: assumes"s tagged_division_of i" shows "(snd ` s) division_of i"
-proof(rule division_ofI) note assm=tagged_division_ofD[OF assms]
- show "\<Union>snd ` s = i" "finite (snd ` s)" using assm by auto
- fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto
- thus "k \<subseteq> i" "k \<noteq> {}" "\<exists>a b. k = {a..b}" using assm apply- by fastsimp+
- fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto
- thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto
-qed
-
-lemma partial_division_of_tagged_division: assumes "s tagged_partial_division_of i"
- shows "(snd ` s) division_of \<Union>(snd ` s)"
-proof(rule division_ofI) note assm=tagged_partial_division_ofD[OF assms]
- show "finite (snd ` s)" "\<Union>snd ` s = \<Union>snd ` s" using assm by auto
- fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto
- thus "k\<noteq>{}" "\<exists>a b. k = {a..b}" "k \<subseteq> \<Union>snd ` s" using assm by auto
- fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto
- thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto
-qed
-
-lemma tagged_partial_division_subset: assumes "s tagged_partial_division_of i" "t \<subseteq> s"
- shows "t tagged_partial_division_of i"
- using assms unfolding tagged_partial_division_of_def using finite_subset[OF assms(2)] by blast
-
-lemma setsum_over_tagged_division_lemma: fixes d::"(real^'m) set \<Rightarrow> 'a::real_normed_vector"
- assumes "p tagged_division_of i" "\<And>u v. {u..v} \<noteq> {} \<Longrightarrow> content {u..v} = 0 \<Longrightarrow> d {u..v} = 0"
- shows "setsum (\<lambda>(x,k). d k) p = setsum d (snd ` p)"
-proof- note assm=tagged_division_ofD[OF assms(1)]
- have *:"(\<lambda>(x,k). d k) = d \<circ> snd" unfolding o_def apply(rule ext) by auto
- show ?thesis unfolding * apply(subst eq_commute) proof(rule setsum_reindex_nonzero)
- show "finite p" using assm by auto
- fix x y assume as:"x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y"
- obtain a b where ab:"snd x = {a..b}" using assm(4)[of "fst x" "snd x"] as(1) by auto
- have "(fst x, snd y) \<in> p" "(fst x, snd y) \<noteq> y" unfolding as(4)[THEN sym] using as(1-3) by auto
- hence "interior (snd x) \<inter> interior (snd y) = {}" apply-apply(rule assm(5)[of "fst x" _ "fst y"]) using as by auto
- hence "content {a..b} = 0" unfolding as(4)[THEN sym] ab content_eq_0_interior by auto
- 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
- thus "d (snd x) = 0" unfolding ab by auto qed qed
-
-lemma tag_in_interval: "p tagged_division_of i \<Longrightarrow> (x,k) \<in> p \<Longrightarrow> x \<in> i" by auto
-
-lemma tagged_division_of_empty: "{} tagged_division_of {}"
- unfolding tagged_division_of by auto
-
-lemma tagged_partial_division_of_trivial[simp]:
- "p tagged_partial_division_of {} \<longleftrightarrow> p = {}"
- unfolding tagged_partial_division_of_def by auto
-
-lemma tagged_division_of_trivial[simp]:
- "p tagged_division_of {} \<longleftrightarrow> p = {}"
- unfolding tagged_division_of by auto
-
-lemma tagged_division_of_self:
- "x \<in> {a..b} \<Longrightarrow> {(x,{a..b})} tagged_division_of {a..b}"
- apply(rule tagged_division_ofI) by auto
-
-lemma tagged_division_union:
- assumes "p1 tagged_division_of s1" "p2 tagged_division_of s2" "interior s1 \<inter> interior s2 = {}"
- shows "(p1 \<union> p2) tagged_division_of (s1 \<union> s2)"
-proof(rule tagged_division_ofI) note p1=tagged_division_ofD[OF assms(1)] and p2=tagged_division_ofD[OF assms(2)]
- show "finite (p1 \<union> p2)" using p1(1) p2(1) by auto
- show "\<Union>{k. \<exists>x. (x, k) \<in> p1 \<union> p2} = s1 \<union> s2" using p1(6) p2(6) by blast
- 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
- show "k\<subseteq>s1\<union>s2" using xk p1(3) p2(3) by blast
- fix x' k' assume xk':"(x',k')\<in>p1\<union>p2" "(x,k) \<noteq> (x',k')"
- have *:"\<And>a b. a\<subseteq> s1 \<Longrightarrow> b\<subseteq> s2 \<Longrightarrow> interior a \<inter> interior b = {}" using assms(3) subset_interior by blast
- show "interior k \<inter> interior k' = {}" apply(cases "(x,k)\<in>p1", case_tac[!] "(x',k')\<in>p1")
- apply(rule p1(5)) prefer 4 apply(rule *) prefer 6 apply(subst Int_commute,rule *) prefer 8 apply(rule p2(5))
- using p1(3) p2(3) using xk xk' by auto qed
-
-lemma tagged_division_unions:
- assumes "finite iset" "\<forall>i\<in>iset. (pfn(i) tagged_division_of i)"
- "\<forall>i1 \<in> iset. \<forall>i2 \<in> iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})"
- shows "\<Union>(pfn ` iset) tagged_division_of (\<Union>iset)"
-proof(rule tagged_division_ofI)
- note assm = tagged_division_ofD[OF assms(2)[rule_format]]
- show "finite (\<Union>pfn ` iset)" apply(rule finite_Union) using assms by auto
- 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
- also have "\<dots> = \<Union>iset" using assm(6) by auto
- finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>pfn ` iset} = \<Union>iset" .
- 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
- 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
- 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
- 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)
- using assms(3)[rule_format] subset_interior by blast
- show "interior k \<inter> interior k' = {}" apply(cases "i=i'")
- using assm(5)[OF i _ xk'(2)] i'(2) using assm(3)[OF i] assm(3)[OF i'] defer apply-apply(rule *) by auto
-qed
-
-lemma tagged_partial_division_of_union_self:
- assumes "p tagged_partial_division_of s" shows "p tagged_division_of (\<Union>(snd ` p))"
- apply(rule tagged_division_ofI) using tagged_partial_division_ofD[OF assms] by auto
-
-lemma tagged_division_of_union_self: assumes "p tagged_division_of s"
- shows "p tagged_division_of (\<Union>(snd ` p))"
- apply(rule tagged_division_ofI) using tagged_division_ofD[OF assms] by auto
-
-subsection {* Fine-ness of a partition w.r.t. a gauge. *}
-
-definition fine (infixr "fine" 46) where
- "d fine s \<longleftrightarrow> (\<forall>(x,k) \<in> s. k \<subseteq> d(x))"
-
-lemma fineI: assumes "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x"
- shows "d fine s" using assms unfolding fine_def by auto
-
-lemma fineD[dest]: assumes "d fine s"
- shows "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x" using assms unfolding fine_def by auto
-
-lemma fine_inter: "(\<lambda>x. d1 x \<inter> d2 x) fine p \<longleftrightarrow> d1 fine p \<and> d2 fine p"
- unfolding fine_def by auto
-
-lemma fine_inters:
- "(\<lambda>x. \<Inter> {f d x | d. d \<in> s}) fine p \<longleftrightarrow> (\<forall>d\<in>s. (f d) fine p)"
- unfolding fine_def by blast
-
-lemma fine_union:
- "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)"
- unfolding fine_def by blast
-
-lemma fine_unions:"(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)"
- unfolding fine_def by auto
-
-lemma fine_subset: "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p"
- unfolding fine_def by blast
-
-subsection {* Gauge integral. Define on compact intervals first, then use a limit. *}
-
-definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46) where
- "(f has_integral_compact_interval y) i \<equiv>
- (\<forall>e>0. \<exists>d. gauge d \<and>
- (\<forall>p. p tagged_division_of i \<and> d fine p
- \<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - y) < e))"
-
-definition has_integral (infixr "has'_integral" 46) where
-"((f::(real^'n \<Rightarrow> 'b::real_normed_vector)) has_integral y) i \<equiv>
- if (\<exists>a b. i = {a..b}) then (f has_integral_compact_interval y) i
- else (\<forall>e>0. \<exists>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_compact_interval z) {a..b} \<and>
- norm(z - y) < e))"
-
-lemma has_integral:
- "(f has_integral y) ({a..b}) \<longleftrightarrow>
- (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p
- \<longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
- unfolding has_integral_def has_integral_compact_interval_def by auto
-
-lemma has_integralD[dest]: assumes
- "(f has_integral y) ({a..b})" "e>0"
- obtains d where "gauge d" "\<And>p. p tagged_division_of {a..b} \<Longrightarrow> d fine p
- \<Longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f(x)) p - y) < e"
- using assms unfolding has_integral by auto
-
-lemma has_integral_alt:
- "(f has_integral y) i \<longleftrightarrow>
- (if (\<exists>a b. i = {a..b}) then (f has_integral y) i
- else (\<forall>e>0. \<exists>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)))"
- unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto
-
-lemma has_integral_altD:
- assumes "(f has_integral y) i" "\<not> (\<exists>a b. i = {a..b})" "e>0"
- 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)"
- using assms unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto
-
-definition integrable_on (infixr "integrable'_on" 46) where
- "(f integrable_on i) \<equiv> \<exists>y. (f has_integral y) i"
-
-definition "integral i f \<equiv> SOME y. (f has_integral y) i"
-
-lemma integrable_integral[dest]:
- "f integrable_on i \<Longrightarrow> (f has_integral (integral i f)) i"
- unfolding integrable_on_def integral_def by(rule someI_ex)
-
-lemma has_integral_integrable[intro]: "(f has_integral i) s \<Longrightarrow> f integrable_on s"
- unfolding integrable_on_def by auto
-
-lemma has_integral_integral:"f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s"
- by auto
-
-lemma has_integral_vec1: assumes "(f has_integral k) {a..b}"
- shows "((\<lambda>x. vec1 (f x)) has_integral (vec1 k)) {a..b}"
-proof- have *:"\<And>p. (\<Sum>(x, k)\<in>p. content k *\<^sub>R vec1 (f x)) - vec1 k = vec1 ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k)"
- unfolding vec_sub Cart_eq by(auto simp add:vec1_dest_vec1_simps split_beta)
- show ?thesis using assms unfolding has_integral apply safe
- apply(erule_tac x=e in allE,safe) apply(rule_tac x=d in exI,safe)
- apply(erule_tac x=p in allE,safe) unfolding * norm_vector_1 by auto qed
-
-lemma setsum_content_null:
- assumes "content({a..b}) = 0" "p tagged_division_of {a..b}"
- shows "setsum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)"
-proof(rule setsum_0',rule) fix y assume y:"y\<in>p"
- obtain x k where xk:"y = (x,k)" using surj_pair[of y] by blast
- note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]]
- from this(2) guess c .. then guess d .. note c_d=this
- have "(\<lambda>(x, k). content k *\<^sub>R f x) y = content k *\<^sub>R f x" unfolding xk by auto
- also have "\<dots> = 0" using content_subset[OF assm(1)[unfolded c_d]] content_pos_le[of c d]
- unfolding assms(1) c_d by auto
- finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" .
-qed
-
-subsection {* Some basic combining lemmas. *}
-
-lemma tagged_division_unions_exists:
- assumes "finite iset" "\<forall>i \<in> iset. \<exists>p. p tagged_division_of i \<and> d fine p"
- "\<forall>i1\<in>iset. \<forall>i2\<in>iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})" "(\<Union>iset = i)"
- obtains p where "p tagged_division_of i" "d fine p"
-proof- guess pfn using bchoice[OF assms(2)] .. note pfn = conjunctD2[OF this[rule_format]]
- show thesis apply(rule_tac p="\<Union>(pfn ` iset)" in that) unfolding assms(4)[THEN sym]
- apply(rule tagged_division_unions[OF assms(1) _ assms(3)]) defer
- apply(rule fine_unions) using pfn by auto
-qed
-
-subsection {* The set we're concerned with must be closed. *}
-
-lemma division_of_closed: "s division_of i \<Longrightarrow> closed (i::(real^'n) set)"
- unfolding division_of_def by(fastsimp intro!: closed_Union closed_interval)
-
-subsection {* General bisection principle for intervals; might be useful elsewhere. *}
-
-lemma interval_bisection_step:
- 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})"
- obtains c d where "~(P{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"
-proof- have "{a..b} \<noteq> {}" using assms(1,3) by auto
- note ab=this[unfolded interval_eq_empty not_ex not_less]
- { fix f have "finite f \<Longrightarrow>
- (\<forall>s\<in>f. P s) \<Longrightarrow>
- (\<forall>s\<in>f. \<exists>a b. s = {a..b}) \<Longrightarrow>
- (\<forall>s\<in>f.\<forall>t\<in>f. ~(s = t) \<longrightarrow> interior(s) \<inter> interior(t) = {}) \<Longrightarrow> P(\<Union>f)"
- proof(induct f rule:finite_induct)
- case empty show ?case using assms(1) by auto
- next case (insert x f) show ?case unfolding Union_insert apply(rule assms(2)[rule_format])
- apply rule defer apply rule defer apply(rule inter_interior_unions_intervals)
- using insert by auto
- qed } note * = this
- 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)}"
- 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"
- { presume "\<forall>c d. ?PP c d \<longrightarrow> P {c..d} \<Longrightarrow> False"
- thus thesis unfolding atomize_not not_all apply-apply(erule exE)+ apply(rule_tac c=x and d=xa in that) by auto }
- assume as:"\<forall>c d. ?PP c d \<longrightarrow> P {c..d}"
- have "P (\<Union> ?A)" proof(rule *, rule_tac[2-] ballI, rule_tac[4] ballI, rule_tac[4] impI)
- let ?B = "(\<lambda>s.{(\<chi> i. if i \<in> s then a$i else (a$i + b$i) / 2) ..
- (\<chi> i. if i \<in> s then (a$i + b$i) / 2 else b$i)}) ` {s. s \<subseteq> UNIV}"
- have "?A \<subseteq> ?B" proof case goal1
- then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+) note c_d=this[rule_format]
- have *:"\<And>a b c d. a = c \<Longrightarrow> b = d \<Longrightarrow> {a..b} = {c..d}" by auto
- show "x\<in>?B" unfolding image_iff apply(rule_tac x="{i. c$i = a$i}" in bexI)
- unfolding c_d apply(rule * ) unfolding Cart_eq cond_component Cart_lambda_beta
- 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)"
- "d $ i = (if i \<in> {i. c $ i = a $ i} then (a $ i + b $ i) / 2 else b $ i)"
- using c_d(2)[of i] ab[THEN spec[where x=i]] by(auto simp add:field_simps)
- qed auto qed
- thus "finite ?A" apply(rule finite_subset[of _ ?B]) by auto
- fix s assume "s\<in>?A" then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+)
- note c_d=this[rule_format]
- show "P s" unfolding c_d apply(rule as[rule_format]) proof- case goal1 show ?case
- using c_d(2)[of i] using ab[THEN spec[where x=i]] by auto qed
- show "\<exists>a b. s = {a..b}" unfolding c_d by auto
- fix t assume "t\<in>?A" then guess e unfolding mem_Collect_eq .. then guess f apply- by(erule exE,(erule conjE)+)
- note e_f=this[rule_format]
- assume "s \<noteq> t" hence "\<not> (c = e \<and> d = f)" unfolding c_d e_f by auto
- then obtain i where "c$i \<noteq> e$i \<or> d$i \<noteq> f$i" unfolding de_Morgan_conj Cart_eq by auto
- hence i:"c$i \<noteq> e$i" "d$i \<noteq> f$i" apply- apply(erule_tac[!] disjE)
- 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
- 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
- qed have *:"\<And>s t. (\<And>a. a\<in>s \<Longrightarrow> a\<in>t \<Longrightarrow> False) \<Longrightarrow> s \<inter> t = {}" by auto
- show "interior s \<inter> interior t = {}" unfolding e_f c_d interior_closed_interval proof(rule *)
- fix x assume "x\<in>{c<..<d}" "x\<in>{e<..<f}"
- 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
- show False using c_d(2)[of i] apply- apply(erule_tac disjE)
- proof(erule_tac[!] conjE) assume as:"c $ i = a $ i" "d $ i = (a $ i + b $ i) / 2"
- show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps)
- next assume as:"c $ i = (a $ i + b $ i) / 2" "d $ i = b $ i"
- show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps)
- qed qed qed
- also have "\<Union> ?A = {a..b}" proof(rule set_ext,rule)
- fix x assume "x\<in>\<Union>?A" then guess Y unfolding Union_iff ..
- from this(1) guess c unfolding mem_Collect_eq .. then guess d ..
- note c_d = this[THEN conjunct2,rule_format] `x\<in>Y`[unfolded this[THEN conjunct1]]
- show "x\<in>{a..b}" unfolding mem_interval proof
- fix i show "a $ i \<le> x $ i \<and> x $ i \<le> b $ i"
- using c_d(1)[of i] c_d(2)[unfolded mem_interval,THEN spec[where x=i]] by auto qed
- next fix x assume x:"x\<in>{a..b}"
- 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"
- (is "\<forall>i. \<exists>c d. ?P i c d") unfolding mem_interval proof fix i
- have "?P i (a$i) ((a $ i + b $ i) / 2) \<or> ?P i ((a $ i + b $ i) / 2) (b$i)"
- using x[unfolded mem_interval,THEN spec[where x=i]] by auto thus "\<exists>c d. ?P i c d" by blast
- qed thus "x\<in>\<Union>?A" unfolding Union_iff lambda_skolem unfolding Bex_def mem_Collect_eq
- apply-apply(erule exE)+ apply(rule_tac x="{xa..xaa}" in exI) unfolding mem_interval by auto
- qed finally show False using assms by auto qed
-
-lemma interval_bisection:
- assumes "P {}" "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))" "\<not> P {a..b::real^'n}"
- obtains x where "x \<in> {a..b}" "\<forall>e>0. \<exists>c d. x \<in> {c..d} \<and> {c..d} \<subseteq> ball x e \<and> {c..d} \<subseteq> {a..b} \<and> ~P({c..d})"
-proof-
- have "\<forall>x. \<exists>y. \<not> P {fst x..snd x} \<longrightarrow> (\<not> P {fst y..snd y} \<and> (\<forall>i. fst x$i \<le> fst y$i \<and> fst y$i \<le> snd y$i \<and> snd y$i \<le> snd x$i \<and>
- 2 * (snd y$i - fst y$i) \<le> snd x$i - fst x$i))" proof case goal1 thus ?case proof-
- presume "\<not> P {fst x..snd x} \<Longrightarrow> ?thesis"
- thus ?thesis apply(cases "P {fst x..snd x}") by auto
- next assume as:"\<not> P {fst x..snd x}" from interval_bisection_step[of P, OF assms(1-2) as] guess c d .
- thus ?thesis apply- apply(rule_tac x="(c,d)" in exI) by auto
- qed qed then guess f apply-apply(drule choice) by(erule exE) note f=this
- def AB \<equiv> "\<lambda>n. (f ^^ n) (a,b)" def A \<equiv> "\<lambda>n. fst(AB n)" and B \<equiv> "\<lambda>n. snd(AB n)" note ab_def = this AB_def
- have "A 0 = a" "B 0 = b" "\<And>n. \<not> P {A(Suc n)..B(Suc n)} \<and>
- (\<forall>i. A(n)$i \<le> A(Suc n)$i \<and> A(Suc n)$i \<le> B(Suc n)$i \<and> B(Suc n)$i \<le> B(n)$i \<and>
- 2 * (B(Suc n)$i - A(Suc n)$i) \<le> B(n)$i - A(n)$i)" (is "\<And>n. ?P n")
- proof- show "A 0 = a" "B 0 = b" unfolding ab_def by auto
- case goal3 note S = ab_def funpow.simps o_def id_apply show ?case
- proof(induct n) case 0 thus ?case unfolding S apply(rule f[rule_format]) using assms(3) by auto
- next case (Suc n) show ?case unfolding S apply(rule f[rule_format]) using Suc unfolding S by auto
- qed qed note AB = this(1-2) conjunctD2[OF this(3),rule_format]
-
- have interv:"\<And>e. 0 < e \<Longrightarrow> \<exists>n. \<forall>x\<in>{A n..B n}. \<forall>y\<in>{A n..B n}. dist x y < e"
- proof- case goal1 guess n using real_arch_pow2[of "(setsum (\<lambda>i. b$i - a$i) UNIV) / e"] .. note n=this
- show ?case apply(rule_tac x=n in exI) proof(rule,rule)
- fix x y assume xy:"x\<in>{A n..B n}" "y\<in>{A n..B n}"
- have "dist x y \<le> setsum (\<lambda>i. abs((x - y)$i)) UNIV" unfolding vector_dist_norm by(rule norm_le_l1)
- also have "\<dots> \<le> setsum (\<lambda>i. B n$i - A n$i) UNIV"
- proof(rule setsum_mono) fix i show "\<bar>(x - y) $ i\<bar> \<le> B n $ i - A n $ i"
- using xy[unfolded mem_interval,THEN spec[where x=i]]
- unfolding vector_minus_component by auto qed
- also have "\<dots> \<le> setsum (\<lambda>i. b$i - a$i) UNIV / 2^n" unfolding setsum_divide_distrib
- proof(rule setsum_mono) case goal1 thus ?case
- proof(induct n) case 0 thus ?case unfolding AB by auto
- 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
- also have "\<dots> \<le> (b $ i - a $ i) / 2 ^ Suc n" using Suc by(auto simp add:field_simps) finally show ?case .
- qed qed
- also have "\<dots> < e" using n using goal1 by(auto simp add:field_simps) finally show "dist x y < e" .
- qed qed
- { fix n m ::nat assume "m \<le> n" then guess d unfolding le_Suc_ex_iff .. note d=this
- have "{A n..B n} \<subseteq> {A m..B m}" unfolding d
- proof(induct d) case 0 thus ?case by auto
- next case (Suc d) show ?case apply(rule subset_trans[OF _ Suc])
- apply(rule) unfolding mem_interval apply(rule,erule_tac x=i in allE)
- proof- case goal1 thus ?case using AB(4)[of "m + d" i] by(auto simp add:field_simps)
- qed qed } note ABsubset = this
- have "\<exists>a. \<forall>n. a\<in>{A n..B n}" apply(rule decreasing_closed_nest[rule_format,OF closed_interval _ ABsubset interv])
- 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
- then guess x0 .. note x0=this[rule_format]
- show thesis proof(rule that[rule_format,of x0])
- show "x0\<in>{a..b}" using x0[of 0] unfolding AB .
- fix e assume "0 < (e::real)" from interv[OF this] guess n .. note n=this
- 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}"
- apply(rule_tac x="A n" in exI,rule_tac x="B n" in exI) apply(rule,rule x0) apply rule defer
- 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
- show "{A n..B n} \<subseteq> ball x0 e" using n using x0[of n] by auto
- show "{A n..B n} \<subseteq> {a..b}" unfolding AB(1-2)[symmetric] apply(rule ABsubset) by auto
- qed qed qed
-
-subsection {* Cousin's lemma. *}
-
-lemma fine_division_exists: assumes "gauge g"
- obtains p where "p tagged_division_of {a..b::real^'n}" "g fine p"
-proof- presume "\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p) \<Longrightarrow> False"
- then guess p unfolding atomize_not not_not .. thus thesis apply-apply(rule that[of p]) by auto
-next assume as:"\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p)"
- guess x apply(rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p",rule_format,OF _ _ as])
- apply(rule_tac x="{}" in exI) defer apply(erule conjE exE)+
- proof- show "{} tagged_division_of {} \<and> g fine {}" unfolding fine_def by auto
- 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 = {}"
- 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
- apply(rule tagged_division_union) prefer 4 apply(rule fine_union) by auto
- qed note x=this
- obtain e where e:"e>0" "ball x e \<subseteq> g x" using gaugeD[OF assms, of x] unfolding open_contains_ball by auto
- from x(2)[OF e(1)] guess c d apply-apply(erule exE conjE)+ . note c_d = this
- have "g fine {(x, {c..d})}" unfolding fine_def using e using c_d(2) by auto
- thus False using tagged_division_of_self[OF c_d(1)] using c_d by auto qed
-
-subsection {* Basic theorems about integrals. *}
-
-lemma has_integral_unique: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
- assumes "(f has_integral k1) i" "(f has_integral k2) i" shows "k1 = k2"
-proof(rule ccontr) let ?e = "norm(k1 - k2) / 2" assume as:"k1 \<noteq> k2" hence e:"?e > 0" by auto
- have lem:"\<And>f::real^'n \<Rightarrow> 'a. \<And> a b k1 k2.
- (f has_integral k1) ({a..b}) \<Longrightarrow> (f has_integral k2) ({a..b}) \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> False"
- proof- case goal1 let ?e = "norm(k1 - k2) / 2" from goal1(3) have e:"?e > 0" by auto
- guess d1 by(rule has_integralD[OF goal1(1) e]) note d1=this
- guess d2 by(rule has_integralD[OF goal1(2) e]) note d2=this
- guess p by(rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)],of a b]) note p=this
- let ?c = "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" have "norm (k1 - k2) \<le> norm (?c - k2) + norm (?c - k1)"
- using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"] by(auto simp add:group_simps norm_minus_commute)
- also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
- apply(rule add_strict_mono) apply(rule_tac[!] d2(2) d1(2)) using p unfolding fine_def by auto
- finally show False by auto
- qed { presume "\<not> (\<exists>a b. i = {a..b}) \<Longrightarrow> False"
- thus False apply-apply(cases "\<exists>a b. i = {a..b}")
- using assms by(auto simp add:has_integral intro:lem[OF _ _ as]) }
- assume as:"\<not> (\<exists>a b. i = {a..b})"
- guess B1 by(rule has_integral_altD[OF assms(1) as,OF e]) note B1=this[rule_format]
- guess B2 by(rule has_integral_altD[OF assms(2) as,OF e]) note B2=this[rule_format]
- have "\<exists>a b::real^'n. ball 0 B1 \<union> ball 0 B2 \<subseteq> {a..b}" apply(rule bounded_subset_closed_interval)
- using bounded_Un bounded_ball by auto then guess a b apply-by(erule exE)+
- note ab=conjunctD2[OF this[unfolded Un_subset_iff]]
- guess w using B1(2)[OF ab(1)] .. note w=conjunctD2[OF this]
- guess z using B2(2)[OF ab(2)] .. note z=conjunctD2[OF this]
- have "z = w" using lem[OF w(1) z(1)] by auto
- hence "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)"
- using norm_triangle_ineq4[of "k1 - w" "k2 - z"] by(auto simp add: norm_minus_commute)
- also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2" apply(rule add_strict_mono) by(rule_tac[!] z(2) w(2))
- finally show False by auto qed
-
-lemma integral_unique[intro]:
- "(f has_integral y) k \<Longrightarrow> integral k f = y"
- unfolding integral_def apply(rule some_equality) by(auto intro: has_integral_unique)
-
-lemma has_integral_is_0: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
- assumes "\<forall>x\<in>s. f x = 0" shows "(f has_integral 0) s"
-proof- have lem:"\<And>a b. \<And>f::real^'n \<Rightarrow> 'a.
- (\<forall>x\<in>{a..b}. f(x) = 0) \<Longrightarrow> (f has_integral 0) ({a..b})" unfolding has_integral
- proof(rule,rule) fix a b e and f::"real^'n \<Rightarrow> 'a"
- assume as:"\<forall>x\<in>{a..b}. f x = 0" "0 < (e::real)"
- 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)"
- apply(rule_tac x="\<lambda>x. ball x 1" in exI) apply(rule,rule gaugeI) unfolding centre_in_ball defer apply(rule open_ball)
- proof(rule,rule,erule conjE) case goal1
- have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) = 0" proof(rule setsum_0',rule)
- 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
- thus "(\<lambda>(x, k). content k *\<^sub>R f x) x = 0" apply(subst surjective_pairing[of x]) unfolding split_conv by auto
- qed thus ?case using as by auto
- qed auto qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
- thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}")
- using assms by(auto simp add:has_integral intro:lem) }
- have *:"(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. 0)" apply(rule ext) using assms by auto
- assume "\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P *
- apply(rule,rule,rule_tac x=1 in exI,rule) defer apply(rule,rule,rule)
- proof- fix e::real and a b assume "e>0"
- thus "\<exists>z. ((\<lambda>x::real^'n. 0::'a) has_integral z) {a..b} \<and> norm (z - 0) < e"
- apply(rule_tac x=0 in exI) apply(rule,rule lem) by auto
- qed auto qed
-
-lemma has_integral_0[simp]: "((\<lambda>x::real^'n. 0) has_integral 0) s"
- apply(rule has_integral_is_0) by auto
-
-lemma has_integral_0_eq[simp]: "((\<lambda>x. 0) has_integral i) s \<longleftrightarrow> i = 0"
- using has_integral_unique[OF has_integral_0] by auto
-
-lemma has_integral_linear: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
- assumes "(f has_integral y) s" "bounded_linear h" shows "((h o f) has_integral ((h y))) s"
-proof- interpret bounded_linear h using assms(2) . from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format]
- have lem:"\<And>f::real^'n \<Rightarrow> 'a. \<And> y a b.
- (f has_integral y) ({a..b}) \<Longrightarrow> ((h o f) has_integral h(y)) ({a..b})"
- proof(subst has_integral,rule,rule) case goal1
- from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format]
- have *:"e / B > 0" apply(rule divide_pos_pos) using goal1(2) B by auto
- guess g using has_integralD[OF goal1(1) *] . note g=this
- show ?case apply(rule_tac x=g in exI) apply(rule,rule g(1))
- proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "g fine p"
- 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
- 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"
- unfolding o_def unfolding scaleR[THEN sym] * by simp
- 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
- 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)" .
- show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) - h y) < e" unfolding * diff[THEN sym]
- apply(rule le_less_trans[OF B(2)]) using g(2)[OF as] B(1) by(auto simp add:field_simps)
- qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
- thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) }
- assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P
- proof(rule,rule) fix e::real assume e:"0<e"
- have *:"0 < e/B" by(rule divide_pos_pos,rule e,rule B(1))
- guess M using has_integral_altD[OF assms(1) as *,rule_format] . note M=this
- 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)"
- apply(rule_tac x=M in exI) apply(rule,rule M(1))
- proof(rule,rule,rule) case goal1 guess z using M(2)[OF goal1(1)] .. note z=conjunctD2[OF this]
- 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)"
- unfolding o_def apply(rule ext) using zero by auto
- show ?case apply(rule_tac x="h z" in exI,rule) unfolding * apply(rule lem[OF z(1)]) unfolding diff[THEN sym]
- apply(rule le_less_trans[OF B(2)]) using B(1) z(2) by(auto simp add:field_simps)
- qed qed qed
-
-lemma has_integral_cmul:
- shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_integral (c *\<^sub>R k)) s"
- unfolding o_def[THEN sym] apply(rule has_integral_linear,assumption)
- by(rule scaleR.bounded_linear_right)
-
-lemma has_integral_neg:
- shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. -(f x)) has_integral (-k)) s"
- apply(drule_tac c="-1" in has_integral_cmul) by auto
-
-lemma has_integral_add: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
- assumes "(f has_integral k) s" "(g has_integral l) s"
- shows "((\<lambda>x. f x + g x) has_integral (k + l)) s"
-proof- have lem:"\<And>f g::real^'n \<Rightarrow> 'a. \<And>a b k l.
- (f has_integral k) ({a..b}) \<Longrightarrow> (g has_integral l) ({a..b}) \<Longrightarrow>
- ((\<lambda>x. f(x) + g(x)) has_integral (k + l)) ({a..b})" proof- case goal1
- show ?case unfolding has_integral proof(rule,rule) fix e::real assume e:"e>0" hence *:"e/2>0" by auto
- guess d1 using has_integralD[OF goal1(1) *] . note d1=this
- guess d2 using has_integralD[OF goal1(2) *] . note d2=this
- 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)"
- apply(rule_tac x="\<lambda>x. (d1 x) \<inter> (d2 x)" in exI) apply(rule,rule gauge_inter[OF d1(1) d2(1)])
- proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d1 x \<inter> d2 x) fine p"
- 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)"
- 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]
- by(rule setsum_cong2,auto)
- 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))"
- unfolding * by(auto simp add:group_simps) also let ?res = "\<dots>"
- from as have *:"d1 fine p" "d2 fine p" unfolding fine_inter by auto
- have "?res < e/2 + e/2" apply(rule le_less_trans[OF norm_triangle_ineq])
- apply(rule add_strict_mono) using d1(2)[OF as(1) *(1)] and d2(2)[OF as(1) *(2)] by auto
- finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e" by auto
- qed qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
- thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) }
- assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P
- proof(rule,rule) case goal1 hence *:"e/2 > 0" by auto
- from has_integral_altD[OF assms(1) as *] guess B1 . note B1=this[rule_format]
- from has_integral_altD[OF assms(2) as *] guess B2 . note B2=this[rule_format]
- show ?case apply(rule_tac x="max B1 B2" in exI) apply(rule,rule min_max.less_supI1,rule B1)
- proof(rule,rule,rule) fix a b assume "ball 0 (max B1 B2) \<subseteq> {a..b::real^'n}"
- hence *:"ball 0 B1 \<subseteq> {a..b::real^'n}" "ball 0 B2 \<subseteq> {a..b::real^'n}" by auto
- guess w using B1(2)[OF *(1)] .. note w=conjunctD2[OF this]
- guess z using B2(2)[OF *(2)] .. note z=conjunctD2[OF this]
- 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
- 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"
- apply(rule_tac x="w + z" in exI) apply(rule,rule lem[OF w(1) z(1), unfolded *[THEN sym]])
- using norm_triangle_ineq[of "w - k" "z - l"] w(2) z(2) by(auto simp add:field_simps)
- qed qed qed
-
-lemma has_integral_sub:
- shows "(f has_integral k) s \<Longrightarrow> (g has_integral l) s \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) has_integral (k - l)) s"
- using has_integral_add[OF _ has_integral_neg,of f k s g l] unfolding group_simps by auto
-
-lemma integral_0: "integral s (\<lambda>x::real^'n. 0::real^'m) = 0"
- by(rule integral_unique has_integral_0)+
-
-lemma integral_add:
- 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"
- apply(rule integral_unique) apply(drule integrable_integral)+
- apply(rule has_integral_add) by assumption+
-
-lemma integral_cmul:
- shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. c *\<^sub>R f x) = c *\<^sub>R integral s f"
- apply(rule integral_unique) apply(drule integrable_integral)+
- apply(rule has_integral_cmul) by assumption+
-
-lemma integral_neg:
- shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. - f x) = - integral s f"
- apply(rule integral_unique) apply(drule integrable_integral)+
- apply(rule has_integral_neg) by assumption+
-
-lemma integral_sub:
- 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"
- apply(rule integral_unique) apply(drule integrable_integral)+
- apply(rule has_integral_sub) by assumption+
-
-lemma integrable_0: "(\<lambda>x. 0) integrable_on s"
- unfolding integrable_on_def using has_integral_0 by auto
-
-lemma integrable_add:
- shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x + g x) integrable_on s"
- unfolding integrable_on_def by(auto intro: has_integral_add)
-
-lemma integrable_cmul:
- shows "f integrable_on s \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) integrable_on s"
- unfolding integrable_on_def by(auto intro: has_integral_cmul)
-
-lemma integrable_neg:
- shows "f integrable_on s \<Longrightarrow> (\<lambda>x. -f(x)) integrable_on s"
- unfolding integrable_on_def by(auto intro: has_integral_neg)
-
-lemma integrable_sub:
- shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x - g x) integrable_on s"
- unfolding integrable_on_def by(auto intro: has_integral_sub)
-
-lemma integrable_linear:
- shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> (h o f) integrable_on s"
- unfolding integrable_on_def by(auto intro: has_integral_linear)
-
-lemma integral_linear:
- shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> integral s (h o f) = h(integral s f)"
- apply(rule has_integral_unique) defer unfolding has_integral_integral
- apply(drule has_integral_linear,assumption,assumption) unfolding has_integral_integral[THEN sym]
- apply(rule integrable_linear) by assumption+
-
-lemma has_integral_setsum:
- assumes "finite t" "\<forall>a\<in>t. ((f a) has_integral (i a)) s"
- shows "((\<lambda>x. setsum (\<lambda>a. f a x) t) has_integral (setsum i t)) s"
-proof(insert assms(1) subset_refl[of t],induct rule:finite_subset_induct)
- case (insert x F) show ?case unfolding setsum_insert[OF insert(1,3)]
- apply(rule has_integral_add) using insert assms by auto
-qed auto
-
-lemma integral_setsum:
- shows "finite t \<Longrightarrow> \<forall>a\<in>t. (f a) integrable_on s \<Longrightarrow>
- integral s (\<lambda>x. setsum (\<lambda>a. f a x) t) = setsum (\<lambda>a. integral s (f a)) t"
- apply(rule integral_unique) apply(rule has_integral_setsum)
- using integrable_integral by auto
-
-lemma integrable_setsum:
- 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"
- unfolding integrable_on_def apply(drule bchoice) using has_integral_setsum[of t] by auto
-
-lemma has_integral_eq:
- assumes "\<forall>x\<in>s. f x = g x" "(f has_integral k) s" shows "(g has_integral k) s"
- using has_integral_sub[OF assms(2), of "\<lambda>x. f x - g x" 0]
- using has_integral_is_0[of s "\<lambda>x. f x - g x"] using assms(1) by auto
-
-lemma integrable_eq:
- shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> f integrable_on s \<Longrightarrow> g integrable_on s"
- unfolding integrable_on_def using has_integral_eq[of s f g] by auto
-
-lemma has_integral_eq_eq:
- shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> ((f has_integral k) s \<longleftrightarrow> (g has_integral k) s)"
- using has_integral_eq[of s f g] has_integral_eq[of s g f] by auto
-
-lemma has_integral_null[dest]:
- assumes "content({a..b}) = 0" shows "(f has_integral 0) ({a..b})"
- unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI,rule) defer
-proof(rule,rule,erule conjE) fix e::real assume e:"e>0" thus "gauge (\<lambda>x. ball x 1)" by auto
- fix p assume p:"p tagged_division_of {a..b}" (*"(\<lambda>x. ball x 1) fine p"*)
- have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) = 0" unfolding norm_eq_zero diff_0_right
- using setsum_content_null[OF assms(1) p, of f] .
- thus "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e" using e by auto qed
-
-lemma has_integral_null_eq[simp]:
- shows "content({a..b}) = 0 \<Longrightarrow> ((f has_integral i) ({a..b}) \<longleftrightarrow> i = 0)"
- apply rule apply(rule has_integral_unique,assumption)
- apply(drule has_integral_null,assumption)
- apply(drule has_integral_null) by auto
-
-lemma integral_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> integral({a..b}) f = 0"
- by(rule integral_unique,drule has_integral_null)
-
-lemma integrable_on_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> f integrable_on {a..b}"
- unfolding integrable_on_def apply(drule has_integral_null) by auto
-
-lemma has_integral_empty[intro]: shows "(f has_integral 0) {}"
- unfolding empty_as_interval apply(rule has_integral_null)
- using content_empty unfolding empty_as_interval .
-
-lemma has_integral_empty_eq[simp]: shows "(f has_integral i) {} \<longleftrightarrow> i = 0"
- apply(rule,rule has_integral_unique,assumption) by auto
-
-lemma integrable_on_empty[intro]: shows "f integrable_on {}" unfolding integrable_on_def by auto
-
-lemma integral_empty[simp]: shows "integral {} f = 0"
- apply(rule integral_unique) using has_integral_empty .
-
-lemma has_integral_refl[intro]: shows "(f has_integral 0) {a..a}"
- apply(rule has_integral_null) unfolding content_eq_0_interior
- unfolding interior_closed_interval using interval_sing by auto
-
-lemma integrable_on_refl[intro]: shows "f integrable_on {a..a}" unfolding integrable_on_def by auto
-
-lemma integral_refl: shows "integral {a..a} f = 0" apply(rule integral_unique) by auto
-
-subsection {* Cauchy-type criterion for integrability. *}
-
-lemma integrable_cauchy: fixes f::"real^'n \<Rightarrow> 'a::{real_normed_vector,complete_space}"
- shows "f integrable_on {a..b} \<longleftrightarrow>
- (\<forall>e>0.\<exists>d. gauge d \<and> (\<forall>p1 p2. p1 tagged_division_of {a..b} \<and> d fine p1 \<and>
- p2 tagged_division_of {a..b} \<and> d fine p2
- \<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 -
- setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) < e))" (is "?l = (\<forall>e>0. \<exists>d. ?P e d)")
-proof assume ?l
- then guess y unfolding integrable_on_def has_integral .. note y=this
- show "\<forall>e>0. \<exists>d. ?P e d" proof(rule,rule) case goal1 hence "e/2 > 0" by auto
- then guess d apply- apply(drule y[rule_format]) by(erule exE,erule conjE) note d=this[rule_format]
- show ?case apply(rule_tac x=d in exI,rule,rule d) apply(rule,rule,rule,(erule conjE)+)
- proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b}" "d fine p1" "p2 tagged_division_of {a..b}" "d fine p2"
- 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"
- apply(rule dist_triangle_half_l[where y=y,unfolded vector_dist_norm])
- using d(2)[OF conjI[OF as(1-2)]] d(2)[OF conjI[OF as(3-4)]] .
- qed qed
-next assume "\<forall>e>0. \<exists>d. ?P e d" hence "\<forall>n::nat. \<exists>d. ?P (inverse(real (n + 1))) d" by auto
- from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
- have "\<And>n. gauge (\<lambda>x. \<Inter>{d i x |i. i \<in> {0..n}})" apply(rule gauge_inters) using d(1) by auto
- 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-
- proof case goal1 from this[of n] show ?case apply(drule_tac fine_division_exists) by auto qed
- from choice[OF this] guess p .. note p = conjunctD2[OF this[rule_format]]
- have dp:"\<And>i n. i\<le>n \<Longrightarrow> d i fine p n" using p(2) unfolding fine_inters by auto
- have "Cauchy (\<lambda>n. setsum (\<lambda>(x,k). content k *\<^sub>R (f x)) (p n))"
- proof(rule CauchyI) case goal1 then guess N unfolding real_arch_inv[of e] .. note N=this
- show ?case apply(rule_tac x=N in exI)
- 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
- 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"
- apply(rule less_trans[OF _ N[THEN conjunct2,THEN conjunct2]]) apply(subst *) apply(rule d(2))
- using dp p(1) using mn by auto
- qed qed
- then guess y unfolding convergent_eq_cauchy[THEN sym] .. note y=this[unfolded Lim_sequentially,rule_format]
- show ?l unfolding integrable_on_def has_integral apply(rule_tac x=y in exI)
- proof(rule,rule) fix e::real assume "e>0" hence *:"e/2 > 0" by auto
- then guess N1 unfolding real_arch_inv[of "e/2"] .. note N1=this hence N1':"N1 = N1 - 1 + 1" by auto
- guess N2 using y[OF *] .. note N2=this
- 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)"
- apply(rule_tac x="d (N1 + N2)" in exI) apply rule defer
- proof(rule,rule,erule conjE) show "gauge (d (N1 + N2))" using d by auto
- fix q assume as:"q tagged_division_of {a..b}" "d (N1 + N2) fine q"
- have *:"inverse (real (N1 + N2 + 1)) < e / 2" apply(rule less_trans) using N1 by auto
- show "norm ((\<Sum>(x, k)\<in>q. content k *\<^sub>R f x) - y) < e" apply(rule norm_triangle_half_r)
- apply(rule less_trans[OF _ *]) apply(subst N1', rule d(2)[of "p (N1+N2)"]) defer
- using N2[rule_format,unfolded vector_dist_norm,of "N1+N2"]
- using as dp[of "N1 - 1 + 1 + N2" "N1 + N2"] using p(1)[of "N1 + N2"] using N1 by auto qed qed qed
-
-subsection {* Additivity of integral on abutting intervals. *}
-
-lemma interval_split:
- "{a..b::real^'n} \<inter> {x. x$k \<le> c} = {a .. (\<chi> i. if i = k then min (b$k) c else b$i)}"
- "{a..b} \<inter> {x. x$k \<ge> c} = {(\<chi> i. if i = k then max (a$k) c else a$i) .. b}"
- apply(rule_tac[!] set_ext) unfolding Int_iff mem_interval mem_Collect_eq
- unfolding Cart_lambda_beta by auto
-
-lemma content_split:
- "content {a..b::real^'n} = content({a..b} \<inter> {x. x$k \<le> c}) + content({a..b} \<inter> {x. x$k >= c})"
-proof- note simps = interval_split content_closed_interval_cases Cart_lambda_beta vector_le_def
- { presume "a\<le>b \<Longrightarrow> ?thesis" thus ?thesis apply(cases "a\<le>b") unfolding simps by auto }
- have *:"UNIV = insert k (UNIV - {k})" "\<And>x. finite (UNIV-{x::'n})" "\<And>x. x\<notin>UNIV-{x}" by auto
- 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))"
- "(\<Prod>i\<in>UNIV. b$i - a$i) = (\<Prod>i\<in>UNIV-{k}. b$i - a$i) * (b$k - a$k)"
- apply(subst *(1)) defer apply(subst *(1)) unfolding setprod_insert[OF *(2-)] by auto
- assume as:"a\<le>b" moreover have "\<And>x. min (b $ k) c = max (a $ k) c
- \<Longrightarrow> x* (b$k - a$k) = x*(max (a $ k) c - a $ k) + x*(b $ k - max (a $ k) c)"
- by (auto simp add:field_simps)
- moreover have "\<not> a $ k \<le> c \<Longrightarrow> \<not> c \<le> b $ k \<Longrightarrow> False"
- unfolding not_le using as[unfolded vector_le_def,rule_format,of k] by auto
- ultimately show ?thesis
- unfolding simps unfolding *(1)[of "\<lambda>i x. b$i - x"] *(1)[of "\<lambda>i x. x - a$i"] *(2) by(auto)
-qed
-
-lemma division_split_left_inj:
- assumes "d division_of i" "k1 \<in> d" "k2 \<in> d" "k1 \<noteq> k2"
- "k1 \<inter> {x::real^'n. x$k \<le> c} = k2 \<inter> {x. x$k \<le> c}"
- shows "content(k1 \<inter> {x. x$k \<le> c}) = 0"
-proof- note d=division_ofD[OF assms(1)]
- have *:"\<And>a b::real^'n. \<And> c k. (content({a..b} \<inter> {x. x$k \<le> c}) = 0 \<longleftrightarrow> interior({a..b} \<inter> {x. x$k \<le> c}) = {})"
- unfolding interval_split content_eq_0_interior by auto
- guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this
- guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this
- have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto
- show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]])
- defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed
-
-lemma division_split_right_inj:
- assumes "d division_of i" "k1 \<in> d" "k2 \<in> d" "k1 \<noteq> k2"
- "k1 \<inter> {x::real^'n. x$k \<ge> c} = k2 \<inter> {x. x$k \<ge> c}"
- shows "content(k1 \<inter> {x. x$k \<ge> c}) = 0"
-proof- note d=division_ofD[OF assms(1)]
- 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}) = {})"
- unfolding interval_split content_eq_0_interior by auto
- guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this
- guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this
- have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto
- show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]])
- defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed
-
-lemma tagged_division_split_left_inj:
- 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}"
- shows "content(k1 \<inter> {x. x$k \<le> c}) = 0"
-proof- have *:"\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c" unfolding image_iff apply(rule_tac x="(a,b)" in bexI) by auto
- show ?thesis apply(rule division_split_left_inj[OF division_of_tagged_division[OF assms(1)]])
- apply(rule_tac[1-2] *) using assms(2-) by auto qed
-
-lemma tagged_division_split_right_inj:
- assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2" "k1 \<inter> {x. x$k \<ge> c} = k2 \<inter> {x. x$k \<ge> c}"
- shows "content(k1 \<inter> {x. x$k \<ge> c}) = 0"
-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
- show ?thesis apply(rule division_split_right_inj[OF division_of_tagged_division[OF assms(1)]])
- apply(rule_tac[1-2] *) using assms(2-) by auto qed
-
-lemma division_split:
- assumes "p division_of {a..b::real^'n}"
- 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$k \<le> c})" (is "?p1 division_of ?I1") and
- "{l \<inter> {x. x$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")
-proof(rule_tac[!] division_ofI) note p=division_ofD[OF assms]
- show "finite ?p1" "finite ?p2" using p(1) by auto show "\<Union>?p1 = ?I1" "\<Union>?p2 = ?I2" unfolding p(6)[THEN sym] by auto
- { fix k assume "k\<in>?p1" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this
- guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this
- show "k\<subseteq>?I1" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l
- using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split) by auto
- fix k' assume "k'\<in>?p1" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this
- assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto }
- { fix k assume "k\<in>?p2" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this
- guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this
- show "k\<subseteq>?I2" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l
- using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split) by auto
- fix k' assume "k'\<in>?p2" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this
- assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto }
-qed
-
-lemma has_integral_split: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
- 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})"
- shows "(f has_integral (i + j)) ({a..b})"
-proof(unfold has_integral,rule,rule) case goal1 hence e:"e/2>0" by auto
- guess d1 using has_integralD[OF assms(1)[unfolded interval_split] e] . note d1=this[unfolded interval_split[THEN sym]]
- guess d2 using has_integralD[OF assms(2)[unfolded interval_split] e] . note d2=this[unfolded interval_split[THEN sym]]
- 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"
- show ?case apply(rule_tac x="?d" in exI,rule) defer apply(rule,rule,(erule conjE)+)
- proof- show "gauge ?d" using d1(1) d2(1) unfolding gauge_def by auto
- fix p assume "p tagged_division_of {a..b}" "?d fine p" note p = this tagged_division_ofD[OF this(1)]
- have lem0:"\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x$k \<le> c} = {}) \<Longrightarrow> x$k \<le> c"
- "\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x$k \<ge> c} = {}) \<Longrightarrow> x$k \<ge> c"
- proof- fix x kk assume as:"(x,kk)\<in>p"
- show "~(kk \<inter> {x. x$k \<le> c} = {}) \<Longrightarrow> x$k \<le> c"
- proof(rule ccontr) case goal1
- from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x $ k - c\<bar>"
- using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
- hence "\<exists>y. y \<in> ball x \<bar>x $ k - c\<bar> \<inter> {x. x $ k \<le> c}" using goal1(1) by blast
- then guess y .. hence "\<bar>x $ k - y $ k\<bar> < \<bar>x $ k - c\<bar>" "y$k \<le> c" apply-apply(rule le_less_trans)
- using component_le_norm[of "x - y" k,unfolded vector_minus_component] by(auto simp add:vector_dist_norm)
- thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
- qed
- show "~(kk \<inter> {x. x$k \<ge> c} = {}) \<Longrightarrow> x$k \<ge> c"
- proof(rule ccontr) case goal1
- from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x $ k - c\<bar>"
- using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
- hence "\<exists>y. y \<in> ball x \<bar>x $ k - c\<bar> \<inter> {x. x $ k \<ge> c}" using goal1(1) by blast
- then guess y .. hence "\<bar>x $ k - y $ k\<bar> < \<bar>x $ k - c\<bar>" "y$k \<ge> c" apply-apply(rule le_less_trans)
- using component_le_norm[of "x - y" k,unfolded vector_minus_component] by(auto simp add:vector_dist_norm)
- thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
- qed
- qed
-
- have lem1: "\<And>f P Q. (\<forall>x k. (x,k) \<in> {(x,f k) | x k. P x k} \<longrightarrow> Q x k) \<longleftrightarrow> (\<forall>x k. P x k \<longrightarrow> Q x (f k))" by auto
- have lem2: "\<And>f s P f. finite s \<Longrightarrow> finite {(x,f k) | x k. (x,k) \<in> s \<and> P x k}"
- proof- case goal1 thus ?case apply-apply(rule finite_subset[of _ "(\<lambda>(x,k). (x,f k)) ` s"]) by auto qed
- have lem3: "\<And>g::(real ^ 'n \<Rightarrow> bool) \<Rightarrow> real ^ 'n \<Rightarrow> bool. finite p \<Longrightarrow>
- setsum (\<lambda>(x,k). content k *\<^sub>R f x) {(x,g k) |x k. (x,k) \<in> p \<and> ~(g k = {})}
- = setsum (\<lambda>(x,k). content k *\<^sub>R f x) ((\<lambda>(x,k). (x,g k)) ` p)"
- apply(rule setsum_mono_zero_left) prefer 3
- proof fix g::"(real ^ 'n \<Rightarrow> bool) \<Rightarrow> real ^ 'n \<Rightarrow> bool" and i::"(real^'n) \<times> ((real^'n) set)"
- assume "i \<in> (\<lambda>(x, k). (x, g k)) ` p - {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}"
- then obtain x k where xk:"i=(x,g k)" "(x,k)\<in>p" "(x,g k) \<notin> {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}" by auto
- have "content (g k) = 0" using xk using content_empty by auto
- thus "(\<lambda>(x, k). content k *\<^sub>R f x) i = 0" unfolding xk split_conv by auto
- qed auto
- have lem4:"\<And>g. (\<lambda>(x,l). content (g l) *\<^sub>R f x) = (\<lambda>(x,l). content l *\<^sub>R f x) o (\<lambda>(x,l). (x,g l))" apply(rule ext) by auto
-
- let ?M1 = "{(x,kk \<inter> {x. x$k \<le> c}) |x kk. (x,kk) \<in> p \<and> kk \<inter> {x. x$k \<le> c} \<noteq> {}}"
- have "norm ((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) < e/2" apply(rule d1(2),rule tagged_division_ofI)
- apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)
- proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M1} = {a..b} \<inter> {x. x$k \<le> c}" unfolding p(8)[THEN sym] by auto
- fix x l assume xl:"(x,l)\<in>?M1"
- then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) . note xl'=this
- have "l' \<subseteq> d1 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
- thus "l \<subseteq> d1 x" unfolding xl' by auto
- 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)
- using lem0(1)[OF xl'(3-4)] by auto
- 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])
- fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M1"
- then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) . note yr'=this
- assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}"
- proof(cases "l' = r' \<longrightarrow> x' = y'")
- case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
- next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto
- thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
- qed qed moreover
-
- 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> {}}"
- have "norm ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j) < e/2" apply(rule d2(2),rule tagged_division_ofI)
- apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)
- 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
- fix x l assume xl:"(x,l)\<in>?M2"
- then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) . note xl'=this
- have "l' \<subseteq> d2 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
- thus "l \<subseteq> d2 x" unfolding xl' by auto
- 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)
- using lem0(2)[OF xl'(3-4)] by auto
- 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])
- fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M2"
- then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) . note yr'=this
- assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}"
- proof(cases "l' = r' \<longrightarrow> x' = y'")
- case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
- next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto
- thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
- qed qed ultimately
-
- 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"
- apply- apply(rule norm_triangle_lt) by auto
- also { have *:"\<And>x y. x = (0::real) \<Longrightarrow> x *\<^sub>R (y::'a) = 0" using scaleR_zero_left by auto
- 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)
- = (\<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
- 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)"
- unfolding lem3[OF p(3)] apply(subst setsum_reindex_nonzero[OF p(3)]) defer apply(subst setsum_reindex_nonzero[OF p(3)])
- defer unfolding lem4[THEN sym] apply(rule refl) unfolding split_paired_all split_conv apply(rule_tac[!] *)
- proof- case goal1 thus ?case apply- apply(rule tagged_division_split_left_inj [OF p(1), of a b aa ba]) by auto
- next case goal2 thus ?case apply- apply(rule tagged_division_split_right_inj[OF p(1), of a b aa ba]) by auto
- qed also note setsum_addf[THEN sym]
- 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
- = (\<lambda>(x,ka). content ka *\<^sub>R f x) x" unfolding split_paired_all split_conv
- proof- fix a b assume "(a,b) \<in> p" from p(6)[OF this] guess u v apply-by(erule exE)+ note uv=this
- 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"
- unfolding scaleR_left_distrib[THEN sym] unfolding uv content_split[of u v k c] by auto
- qed note setsum_cong2[OF this]
- 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 +
- ((\<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) =
- (\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f x) - (i + j)" by auto }
- finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (i + j)) < e" by auto qed qed
-
-subsection {* A sort of converse, integrability on subintervals. *}
-
-lemma tagged_division_union_interval:
- 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})"
- shows "(p1 \<union> p2) tagged_division_of ({a..b})"
-proof- have *:"{a..b} = ({a..b} \<inter> {x. x$k \<le> c}) \<union> ({a..b} \<inter> {x. x$k \<ge> c})" by auto
- show ?thesis apply(subst *) apply(rule tagged_division_union[OF assms])
- unfolding interval_split interior_closed_interval
- by(auto simp add: vector_less_def Cart_lambda_beta elim!:allE[where x=k]) qed
-
-lemma has_integral_separate_sides: fixes f::"real^'m \<Rightarrow> 'a::real_normed_vector"
- assumes "(f has_integral i) ({a..b})" "e>0"
- obtains d where "gauge d" "(\<forall>p1 p2. p1 tagged_division_of ({a..b} \<inter> {x. x$k \<le> c}) \<and> d fine p1 \<and>
- p2 tagged_division_of ({a..b} \<inter> {x. x$k \<ge> c}) \<and> d fine p2
- \<longrightarrow> norm((setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 +
- setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) - i) < e)"
-proof- guess d using has_integralD[OF assms] . note d=this
- show ?thesis apply(rule that[of d]) apply(rule d) apply(rule,rule,rule,(erule conjE)+)
- 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
- 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
- note tagged_division_union_interval[OF p1(7) p2(7)] note p12 = tagged_division_ofD[OF this] this
- 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)"
- apply(subst setsum_Un_zero) apply(rule p1 p2)+ apply(rule) unfolding split_paired_all split_conv
- proof- fix a b assume ab:"(a,b) \<in> p1 \<inter> p2"
- have "(a,b) \<in> p1" using ab by auto from p1(4)[OF this] guess u v apply-by(erule exE)+ note uv =this
- have "b \<subseteq> {x. x$k = c}" using ab p1(3)[of a b] p2(3)[of a b] by fastsimp
- moreover have "interior {x. x $ k = c} = {}"
- proof(rule ccontr) case goal1 then obtain x where x:"x\<in>interior {x. x$k = c}" by auto
- then guess e unfolding mem_interior .. note e=this
- have x:"x$k = c" using x interior_subset by fastsimp
- 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
- have "x + (\<chi> i. if i = k then e/2 else 0) \<in> ball x e" unfolding mem_ball vector_dist_norm
- apply(rule le_less_trans[OF norm_le_l1]) unfolding *
- unfolding setsum_delta[OF finite_UNIV] using e by auto
- hence "x + (\<chi> i. if i = k then e/2 else 0) \<in> {x. x$k = c}" using e by auto
- thus False unfolding mem_Collect_eq using e x by auto
- qed ultimately have "content b = 0" unfolding uv content_eq_0_interior apply-apply(drule subset_interior) by auto
- thus "content b *\<^sub>R f a = 0" by auto
- qed auto
- also have "\<dots> < e" by(rule d(2) p12 fine_union p1 p2)+
- finally show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) < e" . qed qed
-
-lemma integrable_split[intro]: fixes f::"real^'n \<Rightarrow> 'a::{real_normed_vector,complete_space}" assumes "f integrable_on {a..b}"
- shows "f integrable_on ({a..b} \<inter> {x. x$k \<le> c})" (is ?t1) and "f integrable_on ({a..b} \<inter> {x. x$k \<ge> c})" (is ?t2)
-proof- guess y using assms unfolding integrable_on_def .. note y=this
- def b' \<equiv> "(\<chi> i. if i = k then min (b$k) c else b$i)::real^'n"
- and a' \<equiv> "(\<chi> i. if i = k then max (a$k) c else a$i)::real^'n"
- show ?t1 ?t2 unfolding interval_split integrable_cauchy unfolding interval_split[THEN sym]
- proof(rule_tac[!] allI impI)+ fix e::real assume "e>0" hence "e/2>0" by auto
- from has_integral_separate_sides[OF y this,of k c] guess d . note d=this[rule_format]
- 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>
- norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e)"
- show "?P {x. x $ k \<le> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
- 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"
- 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"
- proof- guess p using fine_division_exists[OF d(1), of a' b] . note p=this
- show ?thesis using norm_triangle_half_l[OF d(2)[of p1 p] d(2)[of p2 p]]
- using as unfolding interval_split b'_def[symmetric] a'_def[symmetric]
- using p using assms by(auto simp add:group_simps)
- qed qed
- show "?P {x. x $ k \<ge> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
- 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"
- 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"
- proof- guess p using fine_division_exists[OF d(1), of a b'] . note p=this
- show ?thesis using norm_triangle_half_l[OF d(2)[of p p1] d(2)[of p p2]]
- using as unfolding interval_split b'_def[symmetric] a'_def[symmetric]
- using p using assms by(auto simp add:group_simps) qed qed qed qed
-
-subsection {* Generalized notion of additivity. *}
-
-definition "neutral opp = (SOME x. \<forall>y. opp x y = y \<and> opp y x = y)"
-
-definition operative :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ((real^'n) set \<Rightarrow> 'a) \<Rightarrow> bool" where
- "operative opp f \<equiv>
- (\<forall>a b. content {a..b} = 0 \<longrightarrow> f {a..b} = neutral(opp)) \<and>
- (\<forall>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})))"
-
-lemma operativeD[dest]: assumes "operative opp f"
- shows "\<And>a b. content {a..b} = 0 \<Longrightarrow> f {a..b} = neutral(opp)"
- "\<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}))"
- using assms unfolding operative_def by auto
-
-lemma operative_trivial:
- "operative opp f \<Longrightarrow> content({a..b}) = 0 \<Longrightarrow> f({a..b}) = neutral opp"
- unfolding operative_def by auto
-
-lemma property_empty_interval:
- "(\<forall>a b. content({a..b}) = 0 \<longrightarrow> P({a..b})) \<Longrightarrow> P {}"
- using content_empty unfolding empty_as_interval by auto
-
-lemma operative_empty: "operative opp f \<Longrightarrow> f {} = neutral opp"
- unfolding operative_def apply(rule property_empty_interval) by auto
-
-subsection {* Using additivity of lifted function to encode definedness. *}
-
-lemma forall_option: "(\<forall>x. P x) \<longleftrightarrow> P None \<and> (\<forall>x. P(Some x))"
- by (metis map_of.simps option.nchotomy)
-
-lemma exists_option:
- "(\<exists>x. P x) \<longleftrightarrow> P None \<or> (\<exists>x. P(Some x))"
- by (metis map_of.simps option.nchotomy)
-
-fun lifted where
- "lifted (opp::'a\<Rightarrow>'a\<Rightarrow>'b) (Some x) (Some y) = Some(opp x y)" |
- "lifted opp None _ = (None::'b option)" |
- "lifted opp _ None = None"
-
-lemma lifted_simp_1[simp]: "lifted opp v None = None"
- apply(induct v) by auto
-
-definition "monoidal opp \<equiv> (\<forall>x y. opp x y = opp y x) \<and>
- (\<forall>x y z. opp x (opp y z) = opp (opp x y) z) \<and>
- (\<forall>x. opp (neutral opp) x = x)"
-
-lemma monoidalI: assumes "\<And>x y. opp x y = opp y x"
- "\<And>x y z. opp x (opp y z) = opp (opp x y) z"
- "\<And>x. opp (neutral opp) x = x" shows "monoidal opp"
- unfolding monoidal_def using assms by fastsimp
-
-lemma monoidal_ac: assumes "monoidal opp"
- shows "opp (neutral opp) a = a" "opp a (neutral opp) = a" "opp a b = opp b a"
- "opp (opp a b) c = opp a (opp b c)" "opp a (opp b c) = opp b (opp a c)"
- using assms unfolding monoidal_def apply- by metis+
-
-lemma monoidal_simps[simp]: assumes "monoidal opp"
- shows "opp (neutral opp) a = a" "opp a (neutral opp) = a"
- using monoidal_ac[OF assms] by auto
-
-lemma neutral_lifted[cong]: assumes "monoidal opp"
- shows "neutral (lifted opp) = Some(neutral opp)"
- apply(subst neutral_def) apply(rule some_equality) apply(rule,induct_tac y) prefer 3
-proof- fix x assume "\<forall>y. lifted opp x y = y \<and> lifted opp y x = y"
- thus "x = Some (neutral opp)" apply(induct x) defer
- apply rule apply(subst neutral_def) apply(subst eq_commute,rule some_equality)
- apply(rule,erule_tac x="Some y" in allE) defer apply(erule_tac x="Some x" in allE) by auto
-qed(auto simp add:monoidal_ac[OF assms])
-
-lemma monoidal_lifted[intro]: assumes "monoidal opp" shows "monoidal(lifted opp)"
- unfolding monoidal_def forall_option neutral_lifted[OF assms] using monoidal_ac[OF assms] by auto
-
-definition "support opp f s = {x. x\<in>s \<and> f x \<noteq> neutral opp}"
-definition "fold' opp e s \<equiv> (if finite s then fold opp e s else e)"
-definition "iterate opp s f \<equiv> fold' (\<lambda>x a. opp (f x) a) (neutral opp) (support opp f s)"
-
-lemma support_subset[intro]:"support opp f s \<subseteq> s" unfolding support_def by auto
-lemma support_empty[simp]:"support opp f {} = {}" using support_subset[of opp f "{}"] by auto
-
-lemma fun_left_comm_monoidal[intro]: assumes "monoidal opp" shows "fun_left_comm opp"
- unfolding fun_left_comm_def using monoidal_ac[OF assms] by auto
-
-lemma support_clauses:
- "\<And>f g s. support opp f {} = {}"
- "\<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))"
- "\<And>f g s. support opp f (s - {x}) = (support opp f s) - {x}"
- "\<And>f g s. support opp f (s \<union> t) = (support opp f s) \<union> (support opp f t)"
- "\<And>f g s. support opp f (s \<inter> t) = (support opp f s) \<inter> (support opp f t)"
- "\<And>f g s. support opp f (s - t) = (support opp f s) - (support opp f t)"
- "\<And>f g s. support opp g (f ` s) = f ` (support opp (g o f) s)"
-unfolding support_def by auto
-
-lemma finite_support[intro]:"finite s \<Longrightarrow> finite (support opp f s)"
- unfolding support_def by auto
-
-lemma iterate_empty[simp]:"iterate opp {} f = neutral opp"
- unfolding iterate_def fold'_def by auto
-
-lemma iterate_insert[simp]: assumes "monoidal opp" "finite s"
- shows "iterate opp (insert x s) f = (if x \<in> s then iterate opp s f else opp (f x) (iterate opp s f))"
-proof(cases "x\<in>s") case True hence *:"insert x s = s" by auto
- show ?thesis unfolding iterate_def if_P[OF True] * by auto
-next case False note x=this
- note * = fun_left_comm.fun_left_comm_apply[OF fun_left_comm_monoidal[OF assms(1)]]
- show ?thesis proof(cases "f x = neutral opp")
- case True show ?thesis unfolding iterate_def if_not_P[OF x] support_clauses if_P[OF True]
- unfolding True monoidal_simps[OF assms(1)] by auto
- next case False show ?thesis unfolding iterate_def fold'_def if_not_P[OF x] support_clauses if_not_P[OF False]
- apply(subst fun_left_comm.fold_insert[OF * finite_support])
- using `finite s` unfolding support_def using False x by auto qed qed
-
-lemma iterate_some:
- assumes "monoidal opp" "finite s"
- shows "iterate (lifted opp) s (\<lambda>x. Some(f x)) = Some (iterate opp s f)" using assms(2)
-proof(induct s) case empty thus ?case using assms by auto
-next case (insert x F) show ?case apply(subst iterate_insert) prefer 3 apply(subst if_not_P)
- defer unfolding insert(3) lifted.simps apply rule using assms insert by auto qed
-
-subsection {* Two key instances of additivity. *}
-
-lemma neutral_add[simp]:
- "neutral op + = (0::_::comm_monoid_add)" unfolding neutral_def
- apply(rule some_equality) defer apply(erule_tac x=0 in allE) by auto
-
-lemma operative_content[intro]: "operative (op +) content"
- unfolding operative_def content_split[THEN sym] neutral_add by auto
-
-lemma neutral_monoid[simp]: "neutral ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a) = 0"
- unfolding neutral_def apply(rule some_equality) defer
- apply(erule_tac x=0 in allE) by auto
-
-lemma monoidal_monoid[intro]:
- shows "monoidal ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a)"
- unfolding monoidal_def neutral_monoid by(auto simp add: group_simps)
-
-lemma operative_integral: fixes f::"real^'n \<Rightarrow> 'a::banach"
- shows "operative (lifted(op +)) (\<lambda>i. if f integrable_on i then Some(integral i f) else None)"
- unfolding operative_def unfolding neutral_lifted[OF monoidal_monoid] neutral_add
- apply(rule,rule,rule,rule) defer apply(rule allI)+
-proof- fix a b c k show "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) =
- 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)
- (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)"
- proof(cases "f integrable_on {a..b}")
- case True show ?thesis unfolding if_P[OF True]
- unfolding if_P[OF integrable_split(1)[OF True]] if_P[OF integrable_split(2)[OF True]]
- unfolding lifted.simps option.inject apply(rule integral_unique) apply(rule has_integral_split)
- apply(rule_tac[!] integrable_integral integrable_split)+ using True by assumption+
- 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}))"
- proof(rule ccontr) case goal1 hence "f integrable_on {a..b}" apply- unfolding integrable_on_def
- 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)
- apply(rule has_integral_split) apply(rule_tac[!] integrable_integral) by auto
- thus False using False by auto
- qed thus ?thesis using False by auto
- qed next
- fix a b assume as:"content {a..b::real^'n} = 0"
- thus "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) = Some 0"
- unfolding if_P[OF integrable_on_null[OF as]] using has_integral_null_eq[OF as] by auto qed
-
-subsection {* Points of division of a partition. *}
-
-definition "division_points (k::(real^'n) set) d =
- {(j,x). (interval_lowerbound k)$j < x \<and> x < (interval_upperbound k)$j \<and>
- (\<exists>i\<in>d. (interval_lowerbound i)$j = x \<or> (interval_upperbound i)$j = x)}"
-
-lemma division_points_finite: assumes "d division_of i"
- shows "finite (division_points i d)"
-proof- note assm = division_ofD[OF assms]
- let ?M = "\<lambda>j. {(j,x)|x. (interval_lowerbound i)$j < x \<and> x < (interval_upperbound i)$j \<and>
- (\<exists>i\<in>d. (interval_lowerbound i)$j = x \<or> (interval_upperbound i)$j = x)}"
- have *:"division_points i d = \<Union>(?M ` UNIV)"
- unfolding division_points_def by auto
- show ?thesis unfolding * using assm by auto qed
-
-lemma division_points_subset:
- assumes "d division_of {a..b}" "\<forall>i. a$i < b$i" "a$k < c" "c < b$k"
- 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$k \<le> c} = {})}
- \<subseteq> division_points ({a..b}) d" (is ?t1) and
- "division_points ({a..b} \<inter> {x. x$k \<ge> c}) {l \<inter> {x. x$k \<ge> c} | l . l \<in> d \<and> ~(l \<inter> {x. x$k \<ge> c} = {})}
- \<subseteq> division_points ({a..b}) d" (is ?t2)
-proof- note assm = division_ofD[OF assms(1)]
- 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"
- "\<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"
- using assms using less_imp_le by auto
- show ?t1 unfolding division_points_def interval_split[of a b]
- unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] Cart_lambda_beta unfolding *
- unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+ unfolding mem_Collect_eq apply(erule exE conjE)+
- proof- fix i l x assume as:"a $ fst x < snd x" "snd x < (if fst x = k then c else b $ fst x)"
- "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> {}"
- from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
- 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 .
- have **:"\<forall>i. u$i \<le> v$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
- 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)"
- using as(1-3,5) unfolding l interval_split interval_ne_empty as interval_bounds[OF *] Cart_lambda_beta apply-
- apply(rule,assumption,rule) defer apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **]
- apply(case_tac[!] "fst x = k") using assms by auto
- qed
- show ?t2 unfolding division_points_def interval_split[of a b]
- unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] Cart_lambda_beta unfolding *
- unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+ unfolding mem_Collect_eq apply(erule exE conjE)+
- 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"
- "i = l \<inter> {x. c \<le> x $ k}" "l \<in> d" "l \<inter> {x. c \<le> x $ k} \<noteq> {}"
- from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
- 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 .
- have **:"\<forall>i. u$i \<le> v$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
- 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)"
- using as(1-3,5) unfolding l interval_split interval_ne_empty as interval_bounds[OF *] Cart_lambda_beta apply-
- apply rule defer apply(rule,assumption) apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **]
- apply(case_tac[!] "fst x = k") using assms by auto qed qed
-
-lemma division_points_psubset:
- assumes "d division_of {a..b}" "\<forall>i. a$i < b$i" "a$k < c" "c < b$k"
- "l \<in> d" "interval_lowerbound l$k = c \<or> interval_upperbound l$k = c"
- 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$k \<le> c} \<noteq> {}} \<subset> division_points ({a..b}) d" (is "?D1 \<subset> ?D")
- "division_points ({a..b} \<inter> {x. x$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")
-proof- have ab:"\<forall>i. a$i \<le> b$i" using assms(2) by(auto intro!:less_imp_le)
- guess u v using division_ofD(4)[OF assms(1,5)] apply-by(erule exE)+ note l=this
- 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
- unfolding subset_eq apply- defer apply(erule_tac x=u in ballE, erule_tac x=v in ballE) unfolding mem_interval by auto
- have *:"interval_upperbound ({a..b} \<inter> {x. x $ k \<le> interval_upperbound l $ k}) $ k = interval_upperbound l $ k"
- "interval_upperbound ({a..b} \<inter> {x. x $ k \<le> interval_lowerbound l $ k}) $ k = interval_lowerbound l $ k"
- unfolding interval_split apply(subst interval_bounds) prefer 3 apply(subst interval_bounds)
- unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab by auto
- have "\<exists>x. x \<in> ?D - ?D1" using assms(2-) apply-apply(erule disjE)
- apply(rule_tac x="(k,(interval_lowerbound l)$k)" in exI) defer
- apply(rule_tac x="(k,(interval_upperbound l)$k)" in exI)
- unfolding division_points_def unfolding interval_bounds[OF ab]
- apply (auto simp add:interval_bounds) unfolding * by auto
- thus "?D1 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) by auto
-
- have *:"interval_lowerbound ({a..b} \<inter> {x. x $ k \<ge> interval_lowerbound l $ k}) $ k = interval_lowerbound l $ k"
- "interval_lowerbound ({a..b} \<inter> {x. x $ k \<ge> interval_upperbound l $ k}) $ k = interval_upperbound l $ k"
- unfolding interval_split apply(subst interval_bounds) prefer 3 apply(subst interval_bounds)
- unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab by auto
- have "\<exists>x. x \<in> ?D - ?D2" using assms(2-) apply-apply(erule disjE)
- apply(rule_tac x="(k,(interval_lowerbound l)$k)" in exI) defer
- apply(rule_tac x="(k,(interval_upperbound l)$k)" in exI)
- unfolding division_points_def unfolding interval_bounds[OF ab]
- apply (auto simp add:interval_bounds) unfolding * by auto
- thus "?D2 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) by auto qed
-
-subsection {* Preservation by divisions and tagged divisions. *}
-
-lemma support_support[simp]:"support opp f (support opp f s) = support opp f s"
- unfolding support_def by auto
-
-lemma iterate_support[simp]: "iterate opp (support opp f s) f = iterate opp s f"
- unfolding iterate_def support_support by auto
-
-lemma iterate_expand_cases:
- "iterate opp s f = (if finite(support opp f s) then iterate opp (support opp f s) f else neutral opp)"
- apply(cases) apply(subst if_P,assumption) unfolding iterate_def support_support fold'_def by auto
-
-lemma iterate_image: assumes "monoidal opp" "inj_on f s"
- shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)"
-proof- have *:"\<And>s. finite s \<Longrightarrow> \<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<longrightarrow> x = y \<Longrightarrow>
- iterate opp (f ` s) g = iterate opp s (g \<circ> f)"
- proof- case goal1 show ?case using goal1
- proof(induct s) case empty thus ?case using assms(1) by auto
- next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)]
- unfolding if_not_P[OF insert(2)] apply(subst insert(3)[THEN sym])
- unfolding image_insert defer apply(subst iterate_insert[OF assms(1)])
- apply(rule finite_imageI insert)+ apply(subst if_not_P)
- unfolding image_iff o_def using insert(2,4) by auto
- qed qed
- show ?thesis
- apply(cases "finite (support opp g (f ` s))")
- apply(subst (1) iterate_support[THEN sym],subst (2) iterate_support[THEN sym])
- unfolding support_clauses apply(rule *)apply(rule finite_imageD,assumption) unfolding inj_on_def[symmetric]
- apply(rule subset_inj_on[OF assms(2) support_subset])+
- apply(subst iterate_expand_cases) unfolding support_clauses apply(simp only: if_False)
- apply(subst iterate_expand_cases) apply(subst if_not_P) by auto qed
-
-
-(* This lemma about iterations comes up in a few places. *)
-lemma iterate_nonzero_image_lemma:
- assumes "monoidal opp" "finite s" "g(a) = neutral opp"
- "\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = neutral opp"
- shows "iterate opp {f x | x. x \<in> s \<and> f x \<noteq> a} g = iterate opp s (g \<circ> f)"
-proof- have *:"{f x |x. x \<in> s \<and> ~(f x = a)} = f ` {x. x \<in> s \<and> ~(f x = a)}" by auto
- have **:"support opp (g \<circ> f) {x \<in> s. f x \<noteq> a} = support opp (g \<circ> f) s"
- unfolding support_def using assms(3) by auto
- show ?thesis unfolding *
- apply(subst iterate_support[THEN sym]) unfolding support_clauses
- apply(subst iterate_image[OF assms(1)]) defer
- apply(subst(2) iterate_support[THEN sym]) apply(subst **)
- unfolding inj_on_def using assms(3,4) unfolding support_def by auto qed
-
-lemma iterate_eq_neutral:
- assumes "monoidal opp" "\<forall>x \<in> s. (f(x) = neutral opp)"
- shows "(iterate opp s f = neutral opp)"
-proof- have *:"support opp f s = {}" unfolding support_def using assms(2) by auto
- show ?thesis apply(subst iterate_support[THEN sym])
- unfolding * using assms(1) by auto qed
-
-lemma iterate_op: assumes "monoidal opp" "finite s"
- shows "iterate opp s (\<lambda>x. opp (f x) (g x)) = opp (iterate opp s f) (iterate opp s g)" using assms(2)
-proof(induct s) case empty thus ?case unfolding iterate_insert[OF assms(1)] using assms(1) by auto
-next case (insert x F) show ?case unfolding iterate_insert[OF assms(1) insert(1)] if_not_P[OF insert(2)] insert(3)
- unfolding monoidal_ac[OF assms(1)] by(rule refl) qed
-
-lemma iterate_eq: assumes "monoidal opp" "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
- shows "iterate opp s f = iterate opp s g"
-proof- have *:"support opp g s = support opp f s"
- unfolding support_def using assms(2) by auto
- show ?thesis
- proof(cases "finite (support opp f s)")
- case False thus ?thesis apply(subst iterate_expand_cases,subst(2) iterate_expand_cases)
- unfolding * by auto
- next def su \<equiv> "support opp f s"
- case True note support_subset[of opp f s]
- thus ?thesis apply- apply(subst iterate_support[THEN sym],subst(2) iterate_support[THEN sym]) unfolding * using True
- unfolding su_def[symmetric]
- proof(induct su) case empty show ?case by auto
- next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)]
- unfolding if_not_P[OF insert(2)] apply(subst insert(3))
- defer apply(subst assms(2)[of x]) using insert by auto qed qed qed
-
-lemma nonempty_witness: assumes "s \<noteq> {}" obtains x where "x \<in> s" using assms by auto
-
-lemma operative_division: fixes f::"(real^'n) set \<Rightarrow> 'a"
- assumes "monoidal opp" "operative opp f" "d division_of {a..b}"
- shows "iterate opp d f = f {a..b}"
-proof- def C \<equiv> "card (division_points {a..b} d)" thus ?thesis using assms
- proof(induct C arbitrary:a b d rule:full_nat_induct)
- case goal1
- { presume *:"content {a..b} \<noteq> 0 \<Longrightarrow> ?case"
- thus ?case apply-apply(cases) defer apply assumption
- proof- assume as:"content {a..b} = 0"
- show ?case unfolding operativeD(1)[OF assms(2) as] apply(rule iterate_eq_neutral[OF goal1(2)])
- proof fix x assume x:"x\<in>d"
- then guess u v apply(drule_tac division_ofD(4)[OF goal1(4)]) by(erule exE)+
- thus "f x = neutral opp" using division_of_content_0[OF as goal1(4)]
- using operativeD(1)[OF assms(2)] x by auto
- qed qed }
- assume "content {a..b} \<noteq> 0" note ab = this[unfolded content_lt_nz[THEN sym] content_pos_lt_eq]
- hence ab':"\<forall>i. a$i \<le> b$i" by (auto intro!: less_imp_le) show ?case
- proof(cases "division_points {a..b} d = {}")
- case True have d':"\<forall>i\<in>d. \<exists>u v. i = {u..v} \<and>
- (\<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)"
- unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule)
- apply(rule_tac x=a in exI,rule_tac x=b in exI) apply(rule,rule refl) apply(rule)
- proof- fix u v j assume as:"{u..v} \<in> d" note division_ofD(3)[OF goal1(4) this]
- hence uv:"\<forall>i. u$i \<le> v$i" "u$j \<le> v$j" unfolding interval_ne_empty by auto
- 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
- have "(j, u$j) \<notin> division_points {a..b} d"
- "(j, v$j) \<notin> division_points {a..b} d" using True by auto
- note this[unfolded de_Morgan_conj division_points_def mem_Collect_eq split_conv interval_bounds[OF ab'] bex_simps]
- note *[OF this(1)] *[OF this(2)] note this[unfolded interval_bounds[OF uv(1)]]
- moreover have "a$j \<le> u$j" "v$j \<le> b$j" using division_ofD(2,2,3)[OF goal1(4) as]
- unfolding subset_eq apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE)
- unfolding interval_ne_empty mem_interval by auto
- 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"
- unfolding not_less de_Morgan_disj using ab[rule_format,of j] uv(2) by auto
- qed have "(1/2) *\<^sub>R (a+b) \<in> {a..b}" unfolding mem_interval using ab by(auto intro!:less_imp_le)
- note this[unfolded division_ofD(6)[OF goal1(4),THEN sym] Union_iff]
- then guess i .. note i=this guess u v using d'[rule_format,OF i(1)] apply-by(erule exE conjE)+ note uv=this
- have "{a..b} \<in> d"
- proof- { presume "i = {a..b}" thus ?thesis using i by auto }
- { presume "u = a" "v = b" thus "i = {a..b}" using uv by auto }
- show "u = a" "v = b" unfolding Cart_eq
- proof(rule_tac[!] allI) fix j note i(2)[unfolded uv mem_interval,rule_format,of j]
- thus "u $ j = a $ j" "v $ j = b $ j" using uv(2)[rule_format,of j] by auto
- qed qed
- hence *:"d = insert {a..b} (d - {{a..b}})" by auto
- have "iterate opp (d - {{a..b}}) f = neutral opp" apply(rule iterate_eq_neutral[OF goal1(2)])
- proof fix x assume x:"x \<in> d - {{a..b}}" hence "x\<in>d" by auto note d'[rule_format,OF this]
- then guess u v apply-by(erule exE conjE)+ note uv=this
- have "u\<noteq>a \<or> v\<noteq>b" using x[unfolded uv] by auto
- then obtain j where "u$j \<noteq> a$j \<or> v$j \<noteq> b$j" unfolding Cart_eq by auto
- hence "u$j = v$j" using uv(2)[rule_format,of j] by auto
- hence "content {u..v} = 0" unfolding content_eq_0 apply(rule_tac x=j in exI) by auto
- thus "f x = neutral opp" unfolding uv(1) by(rule operativeD(1)[OF goal1(3)])
- qed thus "iterate opp d f = f {a..b}" apply-apply(subst *)
- apply(subst iterate_insert[OF goal1(2)]) using goal1(2,4) by auto
- next case False hence "\<exists>x. x\<in>division_points {a..b} d" by auto
- then guess k c unfolding split_paired_Ex apply- unfolding division_points_def mem_Collect_eq split_conv
- by(erule exE conjE)+ note kc=this[unfolded interval_bounds[OF ab']]
- from this(3) guess j .. note j=this
- def d1 \<equiv> "{l \<inter> {x. x$k \<le> c} | l. l \<in> d \<and> l \<inter> {x. x$k \<le> c} \<noteq> {}}"
- def d2 \<equiv> "{l \<inter> {x. x$k \<ge> c} | l. l \<in> d \<and> l \<inter> {x. x$k \<ge> c} \<noteq> {}}"
- 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)"
- note division_points_psubset[OF goal1(4) ab kc(1-2) j]
- note psubset_card_mono[OF _ this(1)] psubset_card_mono[OF _ this(2)]
- 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})"
- apply- unfolding interval_split apply(rule_tac[!] goal1(1)[rule_format])
- using division_split[OF goal1(4), where k=k and c=c]
- unfolding interval_split d1_def[symmetric] d2_def[symmetric] unfolding goal1(2) Suc_le_mono
- using goal1(2-3) using division_points_finite[OF goal1(4)] by auto
- have "f {a..b} = opp (iterate opp d1 f) (iterate opp d2 f)" (is "_ = ?prev")
- unfolding * apply(rule operativeD(2)) using goal1(3) .
- also have "iterate opp d1 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x$k \<le> c}))"
- unfolding d1_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])
- unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+
- unfolding empty_as_interval[THEN sym] apply(rule content_empty)
- 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"
- from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this
- show "f (l \<inter> {x. x $ k \<le> c}) = neutral opp" unfolding l interval_split
- apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym] apply(rule division_split_left_inj)
- apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as)+
- qed also have "iterate opp d2 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x$k \<ge> c}))"
- unfolding d2_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])
- unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+
- unfolding empty_as_interval[THEN sym] apply(rule content_empty)
- 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"
- from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this
- show "f (l \<inter> {x. x $ k \<ge> c}) = neutral opp" unfolding l interval_split
- apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym] apply(rule division_split_right_inj)
- apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as)+
- 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}))"
- unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule,rule operativeD(2)) using goal1(3) .
- 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})))
- = iterate opp d f" apply(subst(3) iterate_eq[OF _ *[rule_format]]) prefer 3
- apply(rule iterate_op[THEN sym]) using goal1 by auto
- finally show ?thesis by auto
- qed qed qed
-
-lemma iterate_image_nonzero: assumes "monoidal opp"
- "finite s" "\<forall>x\<in>s. \<forall>y\<in>s. ~(x = y) \<and> f x = f y \<longrightarrow> g(f x) = neutral opp"
- shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)" using assms
-proof(induct rule:finite_subset_induct[OF assms(2) subset_refl])
- case goal1 show ?case using assms(1) by auto
-next case goal2 have *:"\<And>x y. y = neutral opp \<Longrightarrow> x = opp y x" using assms(1) by auto
- show ?case unfolding image_insert apply(subst iterate_insert[OF assms(1)])
- apply(rule finite_imageI goal2)+
- apply(cases "f a \<in> f ` F") unfolding if_P if_not_P apply(subst goal2(4)[OF assms(1) goal2(1)]) defer
- apply(subst iterate_insert[OF assms(1) goal2(1)]) defer
- apply(subst iterate_insert[OF assms(1) goal2(1)])
- unfolding if_not_P[OF goal2(3)] defer unfolding image_iff defer apply(erule bexE)
- apply(rule *) unfolding o_def apply(rule_tac y=x in goal2(7)[rule_format])
- using goal2 unfolding o_def by auto qed
-
-lemma operative_tagged_division: assumes "monoidal opp" "operative opp f" "d tagged_division_of {a..b}"
- shows "iterate(opp) d (\<lambda>(x,l). f l) = f {a..b}"
-proof- have *:"(\<lambda>(x,l). f l) = (f o snd)" unfolding o_def by(rule,auto) note assm = tagged_division_ofD[OF assms(3)]
- have "iterate(opp) d (\<lambda>(x,l). f l) = iterate opp (snd ` d) f" unfolding *
- apply(rule iterate_image_nonzero[THEN sym,OF assms(1)]) apply(rule tagged_division_of_finite assms)+
- unfolding Ball_def split_paired_All snd_conv apply(rule,rule,rule,rule,rule,rule,rule,erule conjE)
- proof- fix a b aa ba assume as:"(a, b) \<in> d" "(aa, ba) \<in> d" "(a, b) \<noteq> (aa, ba)" "b = ba"
- guess u v using assm(4)[OF as(1)] apply-by(erule exE)+ note uv=this
- show "f b = neutral opp" unfolding uv apply(rule operativeD(1)[OF assms(2)])
- unfolding content_eq_0_interior using tagged_division_ofD(5)[OF assms(3) as(1-3)]
- unfolding as(4)[THEN sym] uv by auto
- qed also have "\<dots> = f {a..b}"
- using operative_division[OF assms(1-2) division_of_tagged_division[OF assms(3)]] .
- finally show ?thesis . qed
-
-subsection {* Additivity of content. *}
-
-lemma setsum_iterate:assumes "finite s" shows "setsum f s = iterate op + s f"
-proof- have *:"setsum f s = setsum f (support op + f s)"
- apply(rule setsum_mono_zero_right)
- unfolding support_def neutral_monoid using assms by auto
- thus ?thesis unfolding * setsum_def iterate_def fold_image_def fold'_def
- unfolding neutral_monoid . qed
-
-lemma additive_content_division: assumes "d division_of {a..b}"
- shows "setsum content d = content({a..b})"
- unfolding operative_division[OF monoidal_monoid operative_content assms,THEN sym]
- apply(subst setsum_iterate) using assms by auto
-
-lemma additive_content_tagged_division:
- assumes "d tagged_division_of {a..b}"
- shows "setsum (\<lambda>(x,l). content l) d = content({a..b})"
- unfolding operative_tagged_division[OF monoidal_monoid operative_content assms,THEN sym]
- apply(subst setsum_iterate) using assms by auto
-
-subsection {* Finally, the integral of a constant\<forall> *}
-
-lemma has_integral_const[intro]:
- "((\<lambda>x. c) has_integral (content({a..b::real^'n}) *\<^sub>R c)) ({a..b})"
- unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI)
- apply(rule,rule gauge_trivial)apply(rule,rule,erule conjE)
- unfolding split_def apply(subst scaleR_left.setsum[THEN sym, unfolded o_def])
- defer apply(subst additive_content_tagged_division[unfolded split_def]) apply assumption by auto
-
-subsection {* Bounds on the norm of Riemann sums and the integral itself. *}
-
-lemma dsum_bound: assumes "p division_of {a..b}" "norm(c) \<le> e"
- shows "norm(setsum (\<lambda>l. content l *\<^sub>R c) p) \<le> e * content({a..b})" (is "?l \<le> ?r")
- apply(rule order_trans,rule setsum_norm) defer unfolding norm_scaleR setsum_left_distrib[THEN sym]
- apply(rule order_trans[OF mult_left_mono],rule assms,rule setsum_abs_ge_zero)
- apply(subst real_mult_commute) apply(rule mult_left_mono)
- apply(rule order_trans[of _ "setsum content p"]) apply(rule eq_refl,rule setsum_cong2)
- apply(subst abs_of_nonneg) unfolding additive_content_division[OF assms(1)]
-proof- from order_trans[OF norm_ge_zero[of c] assms(2)] show "0 \<le> e" .
- fix x assume "x\<in>p" from division_ofD(4)[OF assms(1) this] guess u v apply-by(erule exE)+
- thus "0 \<le> content x" using content_pos_le by auto
-qed(insert assms,auto)
-
-lemma rsum_bound: assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. norm(f x) \<le> e"
- shows "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p) \<le> e * content({a..b})"
-proof(cases "{a..b} = {}") case True
- show ?thesis using assms(1) unfolding True tagged_division_of_trivial by auto
-next case False show ?thesis
- apply(rule order_trans,rule setsum_norm) defer unfolding split_def norm_scaleR
- apply(rule order_trans[OF setsum_mono]) apply(rule mult_left_mono[OF _ abs_ge_zero, of _ e]) defer
- unfolding setsum_left_distrib[THEN sym] apply(subst real_mult_commute) apply(rule mult_left_mono)
- apply(rule order_trans[of _ "setsum (content \<circ> snd) p"]) apply(rule eq_refl,rule setsum_cong2)
- apply(subst o_def, rule abs_of_nonneg)
- proof- show "setsum (content \<circ> snd) p \<le> content {a..b}" apply(rule eq_refl)
- unfolding additive_content_tagged_division[OF assms(1),THEN sym] split_def by auto
- guess w using nonempty_witness[OF False] .
- thus "e\<ge>0" apply-apply(rule order_trans) defer apply(rule assms(2)[rule_format],assumption) by auto
- fix xk assume *:"xk\<in>p" guess x k using surj_pair[of xk] apply-by(erule exE)+ note xk = this *[unfolded this]
- from tagged_division_ofD(4)[OF assms(1) xk(2)] guess u v apply-by(erule exE)+ note uv=this
- show "0\<le> content (snd xk)" unfolding xk snd_conv uv by(rule content_pos_le)
- 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
- qed(insert assms,auto) qed
-
-lemma rsum_diff_bound:
- assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e"
- 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})"
- apply(rule order_trans[OF _ rsum_bound[OF assms]]) apply(rule eq_refl) apply(rule arg_cong[where f=norm])
- unfolding setsum_subtractf[THEN sym] apply(rule setsum_cong2) unfolding scaleR.diff_right by auto
-
-lemma has_integral_bound: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
- assumes "0 \<le> B" "(f has_integral i) ({a..b})" "\<forall>x\<in>{a..b}. norm(f x) \<le> B"
- shows "norm i \<le> B * content {a..b}"
-proof- let ?P = "content {a..b} > 0" { presume "?P \<Longrightarrow> ?thesis"
- thus ?thesis proof(cases ?P) case False
- hence *:"content {a..b} = 0" using content_lt_nz by auto
- hence **:"i = 0" using assms(2) apply(subst has_integral_null_eq[THEN sym]) by auto
- show ?thesis unfolding * ** using assms(1) by auto
- qed auto } assume ab:?P
- { presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }
- assume "\<not> ?thesis" hence *:"norm i - B * content {a..b} > 0" by auto
- from assms(2)[unfolded has_integral,rule_format,OF *] guess d apply-by(erule exE conjE)+ note d=this[rule_format]
- from fine_division_exists[OF this(1), of a b] guess p . note p=this
- have *:"\<And>s B. norm s \<le> B \<Longrightarrow> \<not> (norm (s - i) < norm i - B)"
- proof- case goal1 thus ?case unfolding not_less
- using norm_triangle_sub[of i s] unfolding norm_minus_commute by auto
- qed show False using d(2)[OF conjI[OF p]] *[OF rsum_bound[OF p(1) assms(3)]] by auto qed
-
-subsection {* Similar theorems about relationship among components. *}
-
-lemma rsum_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
- assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. (f x)$i \<le> (g x)$i"
- 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"
- unfolding setsum_component apply(rule setsum_mono)
- apply(rule mp) defer apply assumption apply(induct_tac x,rule) unfolding split_conv
-proof- fix a b assume ab:"(a,b) \<in> p" note assm = tagged_division_ofD(2-4)[OF assms(1) ab]
- from this(3) guess u v apply-by(erule exE)+ note b=this
- show "(content b *\<^sub>R f a) $ i \<le> (content b *\<^sub>R g a) $ i" unfolding b
- unfolding Cart_nth.scaleR real_scaleR_def apply(rule mult_left_mono)
- defer apply(rule content_pos_le,rule assms(2)[rule_format]) using assm by auto qed
-
-lemma has_integral_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
- assumes "(f has_integral i) s" "(g has_integral j) s" "\<forall>x\<in>s. (f x)$k \<le> (g x)$k"
- shows "i$k \<le> j$k"
-proof- have lem:"\<And>a b g i j. \<And>f::real^'n \<Rightarrow> real^'m. (f has_integral i) ({a..b}) \<Longrightarrow>
- (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"
- proof(rule ccontr) case goal1 hence *:"0 < (i$k - j$k) / 3" by auto
- guess d1 using goal1(1)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d1=this[rule_format]
- guess d2 using goal1(2)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d2=this[rule_format]
- guess p using fine_division_exists[OF gauge_inter[OF d1(1) d2(1)], of a b] unfolding fine_inter .
- note p = this(1) conjunctD2[OF this(2)] note le_less_trans[OF component_le_norm, of _ _ k]
- note this[OF d1(2)[OF conjI[OF p(1,2)]]] this[OF d2(2)[OF conjI[OF p(1,3)]]]
- thus False unfolding Cart_nth.diff using rsum_component_le[OF p(1) goal1(3)] by smt
- qed let ?P = "\<exists>a b. s = {a..b}"
- { presume "\<not> ?P \<Longrightarrow> ?thesis" thus ?thesis proof(cases ?P)
- case True then guess a b apply-by(erule exE)+ note s=this
- show ?thesis apply(rule lem) using assms[unfolded s] by auto
- qed auto } assume as:"\<not> ?P"
- { presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }
- assume "\<not> i$k \<le> j$k" hence ij:"(i$k - j$k) / 3 > 0" by auto
- note has_integral_altD[OF _ as this] from this[OF assms(1)] this[OF assms(2)] guess B1 B2 . note B=this[rule_format]
- have "bounded (ball 0 B1 \<union> ball (0::real^'n) B2)" unfolding bounded_Un by(rule conjI bounded_ball)+
- from bounded_subset_closed_interval[OF this] guess a b apply- by(erule exE)+
- note ab = conjunctD2[OF this[unfolded Un_subset_iff]]
- guess w1 using B(2)[OF ab(1)] .. note w1=conjunctD2[OF this]
- guess w2 using B(4)[OF ab(2)] .. note w2=conjunctD2[OF this]
- 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*)
- note le_less_trans[OF component_le_norm[of _ k]] note this[OF w1(2)] this[OF w2(2)] moreover
- have "w1$k \<le> w2$k" apply(rule lem[OF w1(1) w2(1)]) using assms by auto ultimately
- show False unfolding Cart_nth.diff by(rule *) qed
-
-lemma integral_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
- assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. (f x)$k \<le> (g x)$k"
- shows "(integral s f)$k \<le> (integral s g)$k"
- apply(rule has_integral_component_le) using integrable_integral assms by auto
-
-lemma has_integral_dest_vec1_le: fixes f::"real^'n \<Rightarrow> real^1"
- assumes "(f has_integral i) s" "(g has_integral j) s" "\<forall>x\<in>s. f x \<le> g x"
- shows "dest_vec1 i \<le> dest_vec1 j" apply(rule has_integral_component_le[OF assms(1-2)])
- using assms(3) unfolding vector_le_def by auto
-
-lemma integral_dest_vec1_le: fixes f::"real^'n \<Rightarrow> real^1"
- assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. f x \<le> g x"
- shows "dest_vec1(integral s f) \<le> dest_vec1(integral s g)"
- apply(rule has_integral_dest_vec1_le) apply(rule_tac[1-2] integrable_integral) using assms by auto
-
-lemma has_integral_component_pos: fixes f::"real^'n \<Rightarrow> real^'m"
- assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> (f x)$k" shows "0 \<le> i$k"
- using has_integral_component_le[OF has_integral_0 assms(1)] using assms(2) by auto
-
-lemma integral_component_pos: fixes f::"real^'n \<Rightarrow> real^'m"
- assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> (f x)$k" shows "0 \<le> (integral s f)$k"
- apply(rule has_integral_component_pos) using assms by auto
-
-lemma has_integral_dest_vec1_pos: fixes f::"real^'n \<Rightarrow> real^1"
- assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> i"
- using has_integral_component_pos[OF assms(1), of 1]
- using assms(2) unfolding vector_le_def by auto
-
-lemma integral_dest_vec1_pos: fixes f::"real^'n \<Rightarrow> real^1"
- assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> integral s f"
- apply(rule has_integral_dest_vec1_pos) using assms by auto
-
-lemma has_integral_component_neg: fixes f::"real^'n \<Rightarrow> real^'m"
- assumes "(f has_integral i) s" "\<forall>x\<in>s. (f x)$k \<le> 0" shows "i$k \<le> 0"
- using has_integral_component_le[OF assms(1) has_integral_0] assms(2) by auto
-
-lemma has_integral_dest_vec1_neg: fixes f::"real^'n \<Rightarrow> real^1"
- assumes "(f has_integral i) s" "\<forall>x\<in>s. f x \<le> 0" shows "i \<le> 0"
- using has_integral_component_neg[OF assms(1),of 1] using assms(2) by auto
-
-lemma has_integral_component_lbound:
- 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"
- using has_integral_component_le[OF has_integral_const assms(1),of "(\<chi> i. B)" k] assms(2)
- unfolding Cart_lambda_beta vector_scaleR_component by(auto simp add:field_simps)
-
-lemma has_integral_component_ubound:
- assumes "(f has_integral i) {a..b}" "\<forall>x\<in>{a..b}. f x$k \<le> B"
- shows "i$k \<le> B * content({a..b})"
- using has_integral_component_le[OF assms(1) has_integral_const, of k "vec B"]
- unfolding vec_component Cart_nth.scaleR using assms(2) by(auto simp add:field_simps)
-
-lemma integral_component_lbound:
- assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. B \<le> f(x)$k"
- shows "B * content({a..b}) \<le> (integral({a..b}) f)$k"
- apply(rule has_integral_component_lbound) using assms unfolding has_integral_integral by auto
-
-lemma integral_component_ubound:
- assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. f(x)$k \<le> B"
- shows "(integral({a..b}) f)$k \<le> B * content({a..b})"
- apply(rule has_integral_component_ubound) using assms unfolding has_integral_integral by auto
-
-subsection {* Uniform limit of integrable functions is integrable. *}
-
-lemma real_arch_invD:
- "0 < (e::real) \<Longrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
- by(subst(asm) real_arch_inv)
-
-lemma integrable_uniform_limit: fixes f::"real^'n \<Rightarrow> 'a::banach"
- assumes "\<forall>e>0. \<exists>g. (\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e) \<and> g integrable_on {a..b}"
- shows "f integrable_on {a..b}"
-proof- { presume *:"content {a..b} > 0 \<Longrightarrow> ?thesis"
- show ?thesis apply cases apply(rule *,assumption)
- unfolding content_lt_nz integrable_on_def using has_integral_null by auto }
- assume as:"content {a..b} > 0"
- have *:"\<And>P. \<forall>e>(0::real). P e \<Longrightarrow> \<forall>n::nat. P (inverse (real n+1))" by auto
- from choice[OF *[OF assms]] guess g .. note g=conjunctD2[OF this[rule_format],rule_format]
- from choice[OF allI[OF g(2)[unfolded integrable_on_def], of "\<lambda>x. x"]] guess i .. note i=this[rule_format]
-
- have "Cauchy i" unfolding Cauchy_def
- proof(rule,rule) fix e::real assume "e>0"
- hence "e / 4 / content {a..b} > 0" using as by(auto simp add:field_simps)
- then guess M apply-apply(subst(asm) real_arch_inv) by(erule exE conjE)+ note M=this
- 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)
- proof- case goal1 have "e/4>0" using `e>0` by auto note * = i[unfolded has_integral,rule_format,OF this]
- from *[of m] guess gm apply-by(erule conjE exE)+ note gm=this[rule_format]
- from *[of n] guess gn apply-by(erule conjE exE)+ note gn=this[rule_format]
- from fine_division_exists[OF gauge_inter[OF gm(1) gn(1)], of a b] guess p . note p=this
- 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"
- proof- case goal1 have "norm (i1 - i2) \<le> norm (i1 - s1) + norm (s1 - s2) + norm (s2 - i2)"
- using norm_triangle_ineq[of "i1 - s1" "s1 - i2"]
- using norm_triangle_ineq[of "s1 - s2" "s2 - i2"] by(auto simp add:group_simps)
- also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:group_simps)
- finally show ?case .
- qed
- show ?case unfolding vector_dist_norm apply(rule lem2) defer
- apply(rule gm(2)[OF conjI[OF p(1)]],rule_tac[2] gn(2)[OF conjI[OF p(1)]])
- using conjunctD2[OF p(2)[unfolded fine_inter]] apply- apply assumption+ apply(rule order_trans)
- apply(rule rsum_diff_bound[OF p(1), where e="2 / real M"])
- proof show "2 / real M * content {a..b} \<le> e / 2" unfolding divide_inverse
- using M as by(auto simp add:field_simps)
- fix x assume x:"x \<in> {a..b}"
- have "norm (f x - g n x) + norm (f x - g m x) \<le> inverse (real n + 1) + inverse (real m + 1)"
- using g(1)[OF x, of n] g(1)[OF x, of m] by auto
- also have "\<dots> \<le> inverse (real M) + inverse (real M)" apply(rule add_mono)
- apply(rule_tac[!] le_imp_inverse_le) using goal1 M by auto
- also have "\<dots> = 2 / real M" unfolding real_divide_def by auto
- finally show "norm (g n x - g m x) \<le> 2 / real M"
- using norm_triangle_le[of "g n x - f x" "f x - g m x" "2 / real M"]
- by(auto simp add:group_simps simp add:norm_minus_commute)
- qed qed qed
- from this[unfolded convergent_eq_cauchy[THEN sym]] guess s .. note s=this
-
- show ?thesis unfolding integrable_on_def apply(rule_tac x=s in exI) unfolding has_integral
- proof(rule,rule)
- case goal1 hence *:"e/3 > 0" by auto
- from s[unfolded Lim_sequentially,rule_format,OF this] guess N1 .. note N1=this
- from goal1 as have "e / 3 / content {a..b} > 0" by(auto simp add:field_simps)
- from real_arch_invD[OF this] guess N2 apply-by(erule exE conjE)+ note N2=this
- from i[of "N1 + N2",unfolded has_integral,rule_format,OF *] guess g' .. note g'=conjunctD2[OF this,rule_format]
- 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"
- proof- case goal1 have "norm (sf - s) \<le> norm (sf - sg) + norm (sg - i) + norm (i - s)"
- using norm_triangle_ineq[of "sf - sg" "sg - s"]
- using norm_triangle_ineq[of "sg - i" " i - s"] by(auto simp add:group_simps)
- also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:group_simps)
- finally show ?case .
- qed
- show ?case apply(rule_tac x=g' in exI) apply(rule,rule g')
- proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> g' fine p" note * = g'(2)[OF this]
- show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - s) < e" apply-apply(rule lem[OF _ _ *])
- apply(rule order_trans,rule rsum_diff_bound[OF p[THEN conjunct1]]) apply(rule,rule g,assumption)
- proof- have "content {a..b} < e / 3 * (real N2)"
- using N2 unfolding inverse_eq_divide using as by(auto simp add:field_simps)
- hence "content {a..b} < e / 3 * (real (N1 + N2) + 1)"
- apply-apply(rule less_le_trans,assumption) using `e>0` by auto
- thus "inverse (real (N1 + N2) + 1) * content {a..b} \<le> e / 3"
- unfolding inverse_eq_divide by(auto simp add:field_simps)
- show "norm (i (N1 + N2) - s) < e / 3" by(rule N1[rule_format,unfolded vector_dist_norm],auto)
- qed qed qed qed
-
-subsection {* Negligible sets. *}
-
-definition "indicator s \<equiv> (\<lambda>x. if x \<in> s then 1 else (0::real))"
-
-lemma dest_vec1_indicator:
- "indicator s x = (if x \<in> s then 1 else 0)" unfolding indicator_def by auto
-
-lemma indicator_pos_le[intro]: "0 \<le> (indicator s x)" unfolding indicator_def by auto
-
-lemma indicator_le_1[intro]: "(indicator s x) \<le> 1" unfolding indicator_def by auto
-
-lemma dest_vec1_indicator_abs_le_1: "abs(indicator s x) \<le> 1"
- unfolding indicator_def by auto
-
-definition "negligible (s::(real^'n) set) \<equiv> (\<forall>a b. ((indicator s) has_integral 0) {a..b})"
-
-lemma indicator_simps[simp]:"x\<in>s \<Longrightarrow> indicator s x = 1" "x\<notin>s \<Longrightarrow> indicator s x = 0"
- unfolding indicator_def by auto
-
-subsection {* Negligibility of hyperplane. *}
-
-lemma vsum_nonzero_image_lemma:
- assumes "finite s" "g(a) = 0"
- "\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = 0"
- shows "setsum g {f x |x. x \<in> s \<and> f x \<noteq> a} = setsum (g o f) s"
- unfolding setsum_iterate[OF assms(1)] apply(subst setsum_iterate) defer
- apply(rule iterate_nonzero_image_lemma) apply(rule assms monoidal_monoid)+
- unfolding assms using neutral_add unfolding neutral_add using assms by auto
-
-lemma interval_doublesplit: shows "{a..b} \<inter> {x . abs(x$k - c) \<le> (e::real)} =
- {(\<chi> i. if i = k then max (a$k) (c - e) else a$i) .. (\<chi> i. if i = k then min (b$k) (c + e) else b$i)}"
-proof- have *:"\<And>x c e::real. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
- have **:"\<And>s P Q. s \<inter> {x. P x \<and> Q x} = (s \<inter> {x. Q x}) \<inter> {x. P x}" by blast
- show ?thesis unfolding * ** interval_split by(rule refl) qed
-
-lemma division_doublesplit: assumes "p division_of {a..b::real^'n}"
- shows "{l \<inter> {x. abs(x$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})"
-proof- have *:"\<And>x c. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
- have **:"\<And>p q p' q'. p division_of q \<Longrightarrow> p = p' \<Longrightarrow> q = q' \<Longrightarrow> p' division_of q'" by auto
- note division_split(1)[OF assms, where c="c+e" and k=k,unfolded interval_split]
- note division_split(2)[OF this, where c="c-e" and k=k]
- thus ?thesis apply(rule **) unfolding interval_doublesplit unfolding * unfolding interval_split interval_doublesplit
- apply(rule set_ext) unfolding mem_Collect_eq apply rule apply(erule conjE exE)+ apply(rule_tac x=la in exI) defer
- apply(erule conjE exE)+ apply(rule_tac x="l \<inter> {x. c + e \<ge> x $ k}" in exI) apply rule defer apply rule
- apply(rule_tac x=l in exI) by blast+ qed
-
-lemma content_doublesplit: assumes "0 < e"
- obtains d where "0 < d" "content({a..b} \<inter> {x. abs(x$k - c) \<le> d}) < e"
-proof(cases "content {a..b} = 0")
- case True show ?thesis apply(rule that[of 1]) defer unfolding interval_doublesplit
- apply(rule le_less_trans[OF content_subset]) defer apply(subst True)
- unfolding interval_doublesplit[THEN sym] using assms by auto
-next case False def d \<equiv> "e / 3 / setprod (\<lambda>i. b$i - a$i) (UNIV - {k})"
- note False[unfolded content_eq_0 not_ex not_le, rule_format]
- hence prod0:"0 < setprod (\<lambda>i. b$i - a$i) (UNIV - {k})" apply-apply(rule setprod_pos) by smt
- hence "d > 0" unfolding d_def using assms by(auto simp add:field_simps) thus ?thesis
- proof(rule that[of d]) have *:"UNIV = insert k (UNIV - {k})" by auto
- have **:"{a..b} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d} \<noteq> {} \<Longrightarrow>
- (\<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)
- = (\<Prod>i\<in>UNIV - {k}. b$i - a$i)" apply(rule setprod_cong,rule refl)
- unfolding interval_doublesplit interval_eq_empty not_ex not_less unfolding interval_bounds by auto
- 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)
- unfolding if_not_P apply(subst *) apply(subst setprod_insert) unfolding **
- unfolding interval_doublesplit interval_eq_empty not_ex not_less unfolding interval_bounds unfolding Cart_lambda_beta if_P[OF refl]
- proof- have "(min (b $ k) (c + d) - max (a $ k) (c - d)) \<le> 2 * d" by auto
- also have "... < e / (\<Prod>i\<in>UNIV - {k}. b $ i - a $ i)" unfolding d_def using assms prod0 by(auto simp add:field_simps)
- finally show "(min (b $ k) (c + d) - max (a $ k) (c - d)) * (\<Prod>i\<in>UNIV - {k}. b $ i - a $ i) < e"
- unfolding pos_less_divide_eq[OF prod0] . qed auto qed qed
-
-lemma negligible_standard_hyperplane[intro]: "negligible {x. x$k = (c::real)}"
- unfolding negligible_def has_integral apply(rule,rule,rule,rule)
-proof- case goal1 from content_doublesplit[OF this,of a b k c] guess d . note d=this let ?i = "indicator {x. x$k = c}"
- show ?case apply(rule_tac x="\<lambda>x. ball x d" in exI) apply(rule,rule gauge_ball,rule d)
- proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> (\<lambda>x. ball x d) fine p"
- have *:"(\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. abs(x$k - c) \<le> d}) *\<^sub>R ?i x)"
- apply(rule setsum_cong2) unfolding split_paired_all real_scaleR_def mult_cancel_right split_conv
- apply(cases,rule disjI1,assumption,rule disjI2)
- proof- fix x l assume as:"(x,l)\<in>p" "?i x \<noteq> 0" hence xk:"x$k = c" unfolding indicator_def apply-by(rule ccontr,auto)
- show "content l = content (l \<inter> {x. \<bar>x $ k - c\<bar> \<le> d})" apply(rule arg_cong[where f=content])
- apply(rule set_ext,rule,rule) unfolding mem_Collect_eq
- proof- fix y assume y:"y\<in>l" note p[THEN conjunct2,unfolded fine_def,rule_format,OF as(1),unfolded split_conv]
- 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]
- thus "\<bar>y $ k - c\<bar> \<le> d" unfolding Cart_nth.diff xk by auto
- qed auto qed
- note p'= tagged_division_ofD[OF p[THEN conjunct1]] and p''=division_of_tagged_division[OF p[THEN conjunct1]]
- 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
- apply(subst abs_of_nonneg) apply(rule setsum_nonneg,rule) unfolding split_paired_all split_conv
- apply(rule mult_nonneg_nonneg) apply(drule p'(4)) apply(erule exE)+ apply(rule_tac b=b in back_subst)
- prefer 2 apply(subst(asm) eq_commute) apply assumption
- apply(subst interval_doublesplit) apply(rule content_pos_le) apply(rule indicator_pos_le)
- 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}))"
- apply(rule setsum_mono) unfolding split_paired_all split_conv
- apply(rule mult_right_le_one_le) apply(drule p'(4)) by(auto simp add:interval_doublesplit intro!:content_pos_le)
- also have "... < e" apply(subst setsum_over_tagged_division_lemma[OF p[THEN conjunct1]])
- proof- case goal1 have "content ({u..v} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) \<le> content {u..v}"
- unfolding interval_doublesplit apply(rule content_subset) unfolding interval_doublesplit[THEN sym] by auto
- thus ?case unfolding goal1 unfolding interval_doublesplit using content_pos_le by smt
- 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"
- apply(rule setsum_nonneg,rule) unfolding mem_Collect_eq image_iff apply(erule exE bexE conjE)+ unfolding split_paired_all
- 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)"
- guess u v using p'(4)[OF as(2)] apply-by(erule exE)+ note * = this
- show "content x \<ge> 0" unfolding as snd_conv * interval_doublesplit by(rule content_pos_le)
- qed have **:"norm (1::real) \<le> 1" by auto note division_doublesplit[OF p'',unfolded interval_doublesplit]
- note dsum_bound[OF this **,unfolded interval_doublesplit[THEN sym]]
- 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)]
- 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"
- apply(subst vsum_nonzero_image_lemma[of "snd ` p" content "{}", unfolded o_def,THEN sym])
- apply(rule finite_imageI p' content_empty)+ unfolding forall_in_division[OF p'']
- proof(rule,rule,rule,rule,rule,rule,rule,erule conjE) fix m n u v
- 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}"
- 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
- note subset_interior[OF this, unfolded division_ofD(5)[OF p'' as(1-3)] interior_inter[of "{m..n}"]]
- hence "interior ({m..n} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) = {}" unfolding as Int_absorb by auto
- thus "content ({m..n} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) = 0" unfolding interval_doublesplit content_eq_0_interior[THEN sym] .
- qed qed
- finally show "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) * ?i x) < e" .
- qed qed qed
-
-subsection {* A technical lemma about "refinement" of division. *}
-
-lemma tagged_division_finer: fixes p::"((real^'n) \<times> ((real^'n) set)) set"
- assumes "p tagged_division_of {a..b}" "gauge d"
- 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"
-proof-
- let ?P = "\<lambda>p. p tagged_partial_division_of {a..b} \<longrightarrow> gauge d \<longrightarrow>
- (\<exists>q. q tagged_division_of (\<Union>{k. \<exists>x. (x,k) \<in> p}) \<and> d fine q \<and>
- (\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q))"
- { have *:"finite p" "p tagged_partial_division_of {a..b}" using assms(1) unfolding tagged_division_of_def by auto
- presume "\<And>p. finite p \<Longrightarrow> ?P p" from this[rule_format,OF * assms(2)] guess q .. note q=this
- thus ?thesis apply-apply(rule that[of q]) unfolding tagged_division_ofD[OF assms(1)] by auto
- } fix p::"((real^'n) \<times> ((real^'n) set)) set" assume as:"finite p"
- show "?P p" apply(rule,rule) using as proof(induct p)
- case empty show ?case apply(rule_tac x="{}" in exI) unfolding fine_def by auto
- next case (insert xk p) guess x k using surj_pair[of xk] apply- by(erule exE)+ note xk=this
- note tagged_partial_division_subset[OF insert(4) subset_insertI]
- from insert(3)[OF this insert(5)] guess q1 .. note q1 = conjunctD3[OF this]
- 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
- note p = tagged_partial_division_ofD[OF insert(4)]
- from p(4)[unfolded xk, OF insertI1] guess u v apply-by(erule exE)+ note uv=this
-
- have "finite {k. \<exists>x. (x, k) \<in> p}"
- apply(rule finite_subset[of _ "snd ` p"],rule) unfolding subset_eq image_iff mem_Collect_eq
- apply(erule exE,rule_tac x="(xa,x)" in bexI) using p by auto
- hence int:"interior {u..v} \<inter> interior (\<Union>{k. \<exists>x. (x, k) \<in> p}) = {}"
- apply(rule inter_interior_unions_intervals) apply(rule open_interior) apply(rule_tac[!] ballI)
- unfolding mem_Collect_eq apply(erule_tac[!] exE) apply(drule p(4)[OF insertI2],assumption)
- apply(rule p(5)) unfolding uv xk apply(rule insertI1,rule insertI2) apply assumption
- using insert(2) unfolding uv xk by auto
-
- show ?case proof(cases "{u..v} \<subseteq> d x")
- case True thus ?thesis apply(rule_tac x="{(x,{u..v})} \<union> q1" in exI) apply rule
- unfolding * uv apply(rule tagged_division_union,rule tagged_division_of_self)
- apply(rule p[unfolded xk uv] insertI1)+ apply(rule q1,rule int)
- apply(rule,rule fine_union,subst fine_def) defer apply(rule q1)
- unfolding Ball_def split_paired_All split_conv apply(rule,rule,rule,rule)
- apply(erule insertE) defer apply(rule UnI2) apply(drule q1(3)[rule_format]) unfolding xk uv by auto
- next case False from fine_division_exists[OF assms(2), of u v] guess q2 . note q2=this
- show ?thesis apply(rule_tac x="q2 \<union> q1" in exI)
- apply rule unfolding * uv apply(rule tagged_division_union q2 q1 int fine_union)+
- unfolding Ball_def split_paired_All split_conv apply rule apply(rule fine_union)
- apply(rule q1 q2)+ apply(rule,rule,rule,rule) apply(erule insertE)
- apply(rule UnI2) defer apply(drule q1(3)[rule_format])using False unfolding xk uv by auto
- qed qed qed
-
-subsection {* Hence the main theorem about negligible sets. *}
-
-lemma finite_product_dependent: assumes "finite s" "\<And>x. x\<in>s\<Longrightarrow> finite (t x)"
- shows "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" using assms
-proof(induct) case (insert x s)
- 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
- show ?case unfolding * apply(rule finite_UnI) using insert by auto qed auto
-
-lemma sum_sum_product: assumes "finite s" "\<forall>i\<in>s. finite (t i)"
- 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
-proof(induct) case (insert a s)
- 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
- show ?case unfolding * apply(subst setsum_Un_disjoint) unfolding setsum_insert[OF insert(1-2)]
- prefer 4 apply(subst insert(3)) unfolding add_right_cancel
- 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
- show "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" apply(rule finite_product_dependent) using insert by auto
- qed(insert insert, auto) qed auto
-
-lemma has_integral_negligible: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
- assumes "negligible s" "\<forall>x\<in>(t - s). f x = 0"
- shows "(f has_integral 0) t"
-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})"
- let ?f = "(\<lambda>x. if x \<in> t then f x else 0)"
- show ?thesis apply(rule_tac f="?f" in has_integral_eq) apply(rule) unfolding if_P apply(rule refl)
- apply(subst has_integral_alt) apply(cases,subst if_P,assumption) unfolding if_not_P
- proof- assume "\<exists>a b. t = {a..b}" then guess a b apply-by(erule exE)+ note t = this
- show "(?f has_integral 0) t" unfolding t apply(rule P) using assms(2) unfolding t by auto
- 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)"
- apply(safe,rule_tac x=1 in exI,rule) apply(rule zero_less_one,safe) apply(rule_tac x=0 in exI)
- apply(rule,rule P) using assms(2) by auto
- qed
-next fix f::"real^'n \<Rightarrow> 'a" and a b::"real^'n" assume assm:"\<forall>x. x \<notin> s \<longrightarrow> f x = 0"
- show "(f has_integral 0) {a..b}" unfolding has_integral
- proof(safe) case goal1
- hence "\<And>n. e / 2 / ((real n+1) * (2 ^ n)) > 0"
- apply-apply(rule divide_pos_pos) defer apply(rule mult_pos_pos) by(auto simp add:field_simps)
- note assms(1)[unfolded negligible_def has_integral,rule_format,OF this,of a b] note allI[OF this,of "\<lambda>x. x"]
- from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format]]
- show ?case apply(rule_tac x="\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x" in exI)
- proof safe show "gauge (\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x)" using d(1) unfolding gauge_def by auto
- fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x) fine p"
- let ?goal = "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e"
- { presume "p\<noteq>{} \<Longrightarrow> ?goal" thus ?goal apply(cases "p={}") using goal1 by auto }
- assume as':"p \<noteq> {}" from real_arch_simple[of "Sup((\<lambda>(x,k). norm(f x)) ` p)"] guess N ..
- 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
- 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)"
- apply(rule,rule tagged_division_finer[OF as(1) d(1)]) by auto
- from choice[OF this] guess q .. note q=conjunctD3[OF this[rule_format]]
- have *:"\<And>i. (\<Sum>(x, k)\<in>q i. content k *\<^sub>R indicator s x) \<ge> 0" apply(rule setsum_nonneg,safe)
- unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) apply(drule tagged_division_ofD(4)[OF q(1)]) by auto
- 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"
- proof- case goal1 thus ?case apply-apply(rule setsum_le_included[of s t g snd f]) prefer 4
- apply safe apply(erule_tac x=x in ballE) apply(erule exE) apply(rule_tac x="(xa,x)" in bexI) by auto qed
- have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) \<le> setsum (\<lambda>i. (real i + 1) *
- norm(setsum (\<lambda>(x,k). content k *\<^sub>R indicator s x) (q i))) {0..N+1}"
- unfolding real_norm_def setsum_right_distrib abs_of_nonneg[OF *] diff_0_right
- apply(rule order_trans,rule setsum_norm) defer apply(subst sum_sum_product) prefer 3
- 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
- fix i a b assume as'':"(a,b) \<in> q i" show "0 \<le> (real i + 1) * (content b *\<^sub>R indicator s a)"
- unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) defer apply(rule mult_nonneg_nonneg)
- using tagged_division_ofD(4)[OF q(1) as''] by auto
- next fix i::nat show "finite (q i)" using q by auto
- next fix x k assume xk:"(x,k) \<in> p" def n \<equiv> "nat \<lfloor>norm (f x)\<rfloor>"
- have *:"norm (f x) \<in> (\<lambda>(x, k). norm (f x)) ` p" using xk by auto
- have nfx:"real n \<le> norm(f x)" "norm(f x) \<le> real n + 1" unfolding n_def by auto
- hence "n \<in> {0..N + 1}" using N[rule_format,OF *] by auto
- moreover note as(2)[unfolded fine_def,rule_format,OF xk,unfolded split_conv]
- note q(3)[rule_format,OF xk,unfolded split_conv,rule_format,OF this] note this[unfolded n_def[symmetric]]
- moreover have "norm (content k *\<^sub>R f x) \<le> (real n + 1) * (content k * indicator s x)"
- proof(cases "x\<in>s") case False thus ?thesis using assm by auto
- next case True have *:"content k \<ge> 0" using tagged_division_ofD(4)[OF as(1) xk] by auto
- moreover have "content k * norm (f x) \<le> content k * (real n + 1)" apply(rule mult_mono) using nfx * by auto
- ultimately show ?thesis unfolding abs_mult using nfx True by(auto simp add:field_simps)
- 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)"
- 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
- qed(insert as, auto)
- also have "... \<le> setsum (\<lambda>i. e / 2 / 2 ^ i) {0..N+1}" apply(rule setsum_mono)
- proof- case goal1 thus ?case apply(subst mult_commute, subst pos_le_divide_eq[THEN sym])
- using d(2)[rule_format,of "q i" i] using q[rule_format] by(auto simp add:field_simps)
- qed also have "... < e * inverse 2 * 2" unfolding real_divide_def setsum_right_distrib[THEN sym]
- apply(rule mult_strict_left_mono) unfolding power_inverse atLeastLessThanSuc_atLeastAtMost[THEN sym]
- apply(subst sumr_geometric) using goal1 by auto
- finally show "?goal" by auto qed qed qed
-
-lemma has_integral_spike: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
- assumes "negligible s" "(\<forall>x\<in>(t - s). g x = f x)" "(f has_integral y) t"
- shows "(g has_integral y) t"
-proof- { fix a b::"real^'n" and f g ::"real^'n \<Rightarrow> 'a" and y::'a
- assume as:"\<forall>x \<in> {a..b} - s. g x = f x" "(f has_integral y) {a..b}"
- have "((\<lambda>x. f x + (g x - f x)) has_integral (y + 0)) {a..b}" apply(rule has_integral_add[OF as(2)])
- apply(rule has_integral_negligible[OF assms(1)]) using as by auto
- hence "(g has_integral y) {a..b}" by auto } note * = this
- show ?thesis apply(subst has_integral_alt) using assms(2-) apply-apply(rule cond_cases,safe)
- apply(rule *, assumption+) apply(subst(asm) has_integral_alt) unfolding if_not_P
- 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)
- 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
-
-lemma has_integral_spike_eq:
- assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x"
- shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
- apply rule apply(rule_tac[!] has_integral_spike[OF assms(1)]) using assms(2) by auto
-
-lemma integrable_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x" "f integrable_on t"
- shows "g integrable_on t"
- using assms unfolding integrable_on_def apply-apply(erule exE)
- apply(rule,rule has_integral_spike) by fastsimp+
-
-lemma integral_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x"
- shows "integral t f = integral t g"
- unfolding integral_def using has_integral_spike_eq[OF assms] by auto
-
-subsection {* Some other trivialities about negligible sets. *}
-
-lemma negligible_subset[intro]: assumes "negligible s" "t \<subseteq> s" shows "negligible t" unfolding negligible_def
-proof(safe) case goal1 show ?case using assms(1)[unfolded negligible_def,rule_format,of a b]
- apply-apply(rule has_integral_spike[OF assms(1)]) defer apply assumption
- using assms(2) unfolding indicator_def by auto qed
-
-lemma negligible_diff[intro?]: assumes "negligible s" shows "negligible(s - t)" using assms by auto
-
-lemma negligible_inter: assumes "negligible s \<or> negligible t" shows "negligible(s \<inter> t)" using assms by auto
-
-lemma negligible_union: assumes "negligible s" "negligible t" shows "negligible (s \<union> t)" unfolding negligible_def
-proof safe case goal1 note assm = assms[unfolded negligible_def,rule_format,of a b]
- thus ?case apply(subst has_integral_spike_eq[OF assms(2)])
- defer apply assumption unfolding indicator_def by auto qed
-
-lemma negligible_union_eq[simp]: "negligible (s \<union> t) \<longleftrightarrow> (negligible s \<and> negligible t)"
- using negligible_union by auto
-
-lemma negligible_sing[intro]: "negligible {a::real^'n}"
-proof- guess x using UNIV_witness[where 'a='n] ..
- show ?thesis using negligible_standard_hyperplane[of x "a$x"] by auto qed
-
-lemma negligible_insert[simp]: "negligible(insert a s) \<longleftrightarrow> negligible s"
- apply(subst insert_is_Un) unfolding negligible_union_eq by auto
-
-lemma negligible_empty[intro]: "negligible {}" by auto
-
-lemma negligible_finite[intro]: assumes "finite s" shows "negligible s"
- using assms apply(induct s) by auto
-
-lemma negligible_unions[intro]: assumes "finite s" "\<forall>t\<in>s. negligible t" shows "negligible(\<Union>s)"
- using assms by(induct,auto)
-
-lemma negligible: "negligible s \<longleftrightarrow> (\<forall>t::(real^'n) set. (indicator s has_integral 0) t)"
- apply safe defer apply(subst negligible_def)
-proof- fix t::"(real^'n) set" assume as:"negligible s"
- 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)
- show "(indicator s has_integral 0) t" apply(subst has_integral_alt)
- apply(cases,subst if_P,assumption) unfolding if_not_P apply(safe,rule as[unfolded negligible_def,rule_format])
- apply(rule_tac x=1 in exI) apply(safe,rule zero_less_one) apply(rule_tac x=0 in exI)
- using negligible_subset[OF as,of "s \<inter> t"] unfolding negligible_def indicator_def unfolding * by auto qed auto
-
-subsection {* Finite case of the spike theorem is quite commonly needed. *}
-
-lemma has_integral_spike_finite: assumes "finite s" "\<forall>x\<in>t-s. g x = f x"
- "(f has_integral y) t" shows "(g has_integral y) t"
- apply(rule has_integral_spike) using assms by auto
-
-lemma has_integral_spike_finite_eq: assumes "finite s" "\<forall>x\<in>t-s. g x = f x"
- shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
- apply rule apply(rule_tac[!] has_integral_spike_finite) using assms by auto
-
-lemma integrable_spike_finite:
- assumes "finite s" "\<forall>x\<in>t-s. g x = f x" "f integrable_on t" shows "g integrable_on t"
- using assms unfolding integrable_on_def apply safe apply(rule_tac x=y in exI)
- apply(rule has_integral_spike_finite) by auto
-
-subsection {* In particular, the boundary of an interval is negligible. *}
-
-lemma negligible_frontier_interval: "negligible({a..b} - {a<..<b})"
-proof- let ?A = "\<Union>((\<lambda>k. {x. x$k = a$k} \<union> {x. x$k = b$k}) ` UNIV)"
- have "{a..b} - {a<..<b} \<subseteq> ?A" apply rule unfolding Diff_iff mem_interval not_all
- apply(erule conjE exE)+ apply(rule_tac X="{x. x $ xa = a $ xa} \<union> {x. x $ xa = b $ xa}" in UnionI)
- apply(erule_tac[!] x=xa in allE) by auto
- thus ?thesis apply-apply(rule negligible_subset[of ?A]) apply(rule negligible_unions[OF finite_imageI]) by auto qed
-
-lemma has_integral_spike_interior:
- assumes "\<forall>x\<in>{a<..<b}. g x = f x" "(f has_integral y) ({a..b})" shows "(g has_integral y) ({a..b})"
- apply(rule has_integral_spike[OF negligible_frontier_interval _ assms(2)]) using assms(1) by auto
-
-lemma has_integral_spike_interior_eq:
- assumes "\<forall>x\<in>{a<..<b}. g x = f x" shows "((f has_integral y) ({a..b}) \<longleftrightarrow> (g has_integral y) ({a..b}))"
- apply rule apply(rule_tac[!] has_integral_spike_interior) using assms by auto
-
-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}"
- using assms unfolding integrable_on_def using has_integral_spike_interior[OF assms(1)] by auto
-
-subsection {* Integrability of continuous functions. *}
-
-lemma neutral_and[simp]: "neutral op \<and> = True"
- unfolding neutral_def apply(rule some_equality) by auto
-
-lemma monoidal_and[intro]: "monoidal op \<and>" unfolding monoidal_def by auto
-
-lemma iterate_and[simp]: assumes "finite s" shows "(iterate op \<and>) s p \<longleftrightarrow> (\<forall>x\<in>s. p x)" using assms
-apply induct unfolding iterate_insert[OF monoidal_and] by auto
-
-lemma operative_division_and: assumes "operative op \<and> P" "d division_of {a..b}"
- shows "(\<forall>i\<in>d. P i) \<longleftrightarrow> P {a..b}"
- using operative_division[OF monoidal_and assms] division_of_finite[OF assms(2)] by auto
-
-lemma operative_approximable: assumes "0 \<le> e" fixes f::"real^'n \<Rightarrow> 'a::banach"
- 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
-proof safe fix a b::"real^'n" { assume "content {a..b} = 0"
- thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}"
- apply(rule_tac x=f in exI) using assms by(auto intro!:integrable_on_null) }
- { fix c k g assume as:"\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}"
- 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}"
- "\<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}"
- apply(rule_tac[!] x=g in exI) using as(1) integrable_split[OF as(2)] by auto }
- 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}"
- "\<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}"
- let ?g = "\<lambda>x. if x$k = c then f x else if x$k \<le> c then g1 x else g2 x"
- 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)
- proof safe case goal1 thus ?case apply- apply(cases "x$k=c", case_tac "x$k < c") using as assms by auto
- 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}"
- then guess h1 h2 unfolding integrable_on_def by auto from has_integral_split[OF this]
- show ?case unfolding integrable_on_def by auto
- next show "?g integrable_on {a..b} \<inter> {x. x $ k \<le> c}" "?g integrable_on {a..b} \<inter> {x. x $ k \<ge> c}"
- apply(rule_tac[!] integrable_spike[OF negligible_standard_hyperplane[of k c]]) using as(2,4) by auto qed qed
-
-lemma approximable_on_division: fixes f::"real^'n \<Rightarrow> 'a::banach"
- 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"
- obtains g where "\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}"
-proof- note * = operative_division[OF monoidal_and operative_approximable[OF assms(1)] assms(2)]
- note this[unfolded iterate_and[OF division_of_finite[OF assms(2)]]] from assms(3)[unfolded this[of f]]
- guess g .. thus thesis apply-apply(rule that[of g]) by auto qed
-
-lemma integrable_continuous: fixes f::"real^'n \<Rightarrow> 'a::banach"
- assumes "continuous_on {a..b} f" shows "f integrable_on {a..b}"
-proof(rule integrable_uniform_limit,safe) fix e::real assume e:"0 < e"
- from compact_uniformly_continuous[OF assms compact_interval,unfolded uniformly_continuous_on_def,rule_format,OF e] guess d ..
- note d=conjunctD2[OF this,rule_format]
- from fine_division_exists[OF gauge_ball[OF d(1)], of a b] guess p . note p=this
- note p' = tagged_division_ofD[OF p(1)]
- have *:"\<forall>i\<in>snd ` p. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
- proof(safe,unfold snd_conv) fix x l assume as:"(x,l) \<in> p"
- from p'(4)[OF this] guess a b apply-by(erule exE)+ note l=this
- 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)
- proof safe show "(\<lambda>y. f x) integrable_on l" unfolding integrable_on_def l by(rule,rule has_integral_const)
- fix y assume y:"y\<in>l" note fineD[OF p(2) as,unfolded subset_eq,rule_format,OF this]
- note d(2)[OF _ _ this[unfolded mem_ball]]
- 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
- from e have "0 \<le> e" by auto from approximable_on_division[OF this division_of_tagged_division[OF p(1)] *] guess g .
- thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" by auto qed
-
-subsection {* Specialization of additivity to one dimension. *}
-
-lemma operative_1_lt: assumes "monoidal opp"
- shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real^1} = neutral opp) \<and>
- (\<forall>a b c. a < c \<and> c < b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))"
- unfolding operative_def content_eq_0_1 forall_1 vector_le_def vector_less_def
-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"
- 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
- thus "opp (f {a..c}) (f {c..b}) = f {a..b}" unfolding as(1)[rule_format,of a b "c$1"] by auto
-next fix a b::"real^1" and c::real
- 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}"
- show "f {a..b} = opp (f ({a..b} \<inter> {x. x $ 1 \<le> c})) (f ({a..b} \<inter> {x. c \<le> x $ 1}))"
- proof(cases "c \<in> {a$1 .. b$1}")
- case False hence "c<a$1 \<or> c>b$1" by auto
- thus ?thesis apply-apply(erule disjE)
- 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
- show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto
- 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
- show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto
- qed
- next case True hence *:"min (b $ 1) c = c" "max (a $ 1) c = c" by auto
- show ?thesis unfolding interval_split num1_eq_iff if_True * vec_def[THEN sym]
- proof(cases "c = a$1 \<or> c = b$1")
- case False thus "f {a..b} = opp (f {a..vec1 c}) (f {vec1 c..b})"
- apply-apply(subst as(2)[rule_format]) using True by auto
- next case True thus "f {a..b} = opp (f {a..vec1 c}) (f {vec1 c..b})" apply-
- proof(erule disjE) assume "c=a$1" hence *:"a = vec1 c" unfolding Cart_eq by auto
- hence "f {a..vec1 c} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
- thus ?thesis using assms unfolding * by auto
- next assume "c=b$1" hence *:"b = vec1 c" unfolding Cart_eq by auto
- hence "f {vec1 c..b} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
- thus ?thesis using assms unfolding * by auto qed qed qed qed
-
-lemma operative_1_le: assumes "monoidal opp"
- shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real^1} = neutral opp) \<and>
- (\<forall>a b c. a \<le> c \<and> c \<le> b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))"
-unfolding operative_1_lt[OF assms]
-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"
- 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
-next fix a b c ::"real^1"
- 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"
- note as = this[rule_format]
- show "opp (f {a..c}) (f {c..b}) = f {a..b}"
- proof(cases "c = a \<or> c = b")
- 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)
- next case True thus ?thesis apply-
- proof(erule disjE) assume *:"c=a" hence "f {a..c} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
- thus ?thesis using assms unfolding * by auto
- next assume *:"c=b" hence "f {c..b} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
- thus ?thesis using assms unfolding * by auto qed qed qed
-
-subsection {* Special case of additivity we need for the FCT. *}
-
-lemma additive_tagged_division_1: fixes f::"real^1 \<Rightarrow> 'a::real_normed_vector"
- assumes "dest_vec1 a \<le> dest_vec1 b" "p tagged_division_of {a..b}"
- shows "setsum (\<lambda>(x,k). f(interval_upperbound k) - f(interval_lowerbound k)) p = f b - f a"
-proof- let ?f = "(\<lambda>k::(real^1) set. if k = {} then 0 else f(interval_upperbound k) - f(interval_lowerbound k))"
- have *:"operative op + ?f" unfolding operative_1_lt[OF monoidal_monoid] interval_eq_empty_1
- by(auto simp add:not_less interval_bound_1 vector_less_def)
- have **:"{a..b} \<noteq> {}" using assms(1) by auto note operative_tagged_division[OF monoidal_monoid * assms(2)]
- note * = this[unfolded if_not_P[OF **] interval_bound_1[OF assms(1)],THEN sym ]
- show ?thesis unfolding * apply(subst setsum_iterate[THEN sym]) defer
- apply(rule setsum_cong2) unfolding split_paired_all split_conv using assms(2) by auto qed
-
-subsection {* A useful lemma allowing us to factor out the content size. *}
-
-lemma has_integral_factor_content:
- "(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
- \<longrightarrow> norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content {a..b}))"
-proof(cases "content {a..b} = 0")
- case True show ?thesis unfolding has_integral_null_eq[OF True] apply safe
- apply(rule,rule,rule gauge_trivial,safe) unfolding setsum_content_null[OF True] True defer
- apply(erule_tac x=1 in allE,safe) defer apply(rule fine_division_exists[of _ a b],assumption)
- apply(erule_tac x=p in allE) unfolding setsum_content_null[OF True] by auto
-next case False note F = this[unfolded content_lt_nz[THEN sym]]
- 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)"
- show ?thesis apply(subst has_integral)
- proof safe fix e::real assume e:"e>0"
- { 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)
- apply(erule impE) defer apply(erule exE,rule_tac x=d in exI)
- using F e by(auto simp add:field_simps intro:mult_pos_pos) }
- { 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)
- apply(erule impE) defer apply(erule exE,rule_tac x=d in exI)
- using F e by(auto simp add:field_simps intro:mult_pos_pos) } qed qed
-
-subsection {* Fundamental theorem of calculus. *}
-
-lemma fundamental_theorem_of_calculus: fixes f::"real^1 \<Rightarrow> 'a::banach"
- assumes "a \<le> b" "\<forall>x\<in>{a..b}. ((f o vec1) has_vector_derivative f'(vec1 x)) (at x within {a..b})"
- shows "(f' has_integral (f(vec1 b) - f(vec1 a))) ({vec1 a..vec1 b})"
-unfolding has_integral_factor_content
-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
- note assm = assms(2)[unfolded has_vector_derivative_def has_derivative_within_alt]
- 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
- note this[OF assm,unfolded gauge_existence_lemma] from choice[OF this,unfolded Ball_def[symmetric]]
- guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
- show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {vec1 a..vec1 b} \<and> d fine p \<longrightarrow>
- 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})"
- apply(rule_tac x="\<lambda>x. ball x (d (dest_vec1 x))" in exI,safe)
- apply(rule gauge_ball_dependent,rule,rule d(1))
- proof- fix p assume as:"p tagged_division_of {vec1 a..vec1 b}" "(\<lambda>x. ball x (d (dest_vec1 x))) fine p"
- 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}"
- unfolding content_1[OF ab] additive_tagged_division_1[OF ab as(1),of f,THEN sym]
- unfolding vector_minus_component[THEN sym] additive_tagged_division_1[OF ab as(1),of "\<lambda>x. x",THEN sym]
- apply(subst dest_vec1_setsum) unfolding setsum_right_distrib defer unfolding setsum_subtractf[THEN sym]
- proof(rule setsum_norm_le,safe) fix x k assume "(x,k)\<in>p"
- note xk = tagged_division_ofD(2-4)[OF as(1) this] from this(3) guess u v apply-by(erule exE)+ note k=this
- have *:"dest_vec1 u \<le> dest_vec1 v" using xk unfolding k by auto
- 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] .
- 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)"
- apply(rule order_trans[OF _ norm_triangle_ineq4]) apply(rule eq_refl) apply(rule arg_cong[where f=norm])
- unfolding scaleR.diff_left by(auto simp add:group_simps)
- also have "... \<le> e * norm (dest_vec1 u - dest_vec1 x) + e * norm (dest_vec1 v - dest_vec1 x)"
- apply(rule add_mono) apply(rule d(2)[of "x$1" "u$1",unfolded o_def vec1_dest_vec1]) prefer 4
- apply(rule d(2)[of "x$1" "v$1",unfolded o_def vec1_dest_vec1])
- using ball[rule_format,of u] ball[rule_format,of v]
- using xk(1-2) unfolding k subset_eq by(auto simp add:vector_dist_norm norm_real)
- also have "... \<le> e * dest_vec1 (interval_upperbound k - interval_lowerbound k)"
- unfolding k interval_bound_1[OF *] using xk(1) unfolding k by(auto simp add:vector_dist_norm norm_real field_simps)
- finally show "norm (content k *\<^sub>R f' x - (f (interval_upperbound k) - f (interval_lowerbound k))) \<le>
- e * dest_vec1 (interval_upperbound k - interval_lowerbound k)" unfolding k interval_bound_1[OF *] content_1[OF *] .
- qed(insert as, auto) qed qed
-
-subsection {* Attempt a systematic general set of "offset" results for components. *}
-
-lemma gauge_modify:
- assumes "(\<forall>s. open s \<longrightarrow> open {x. f(x) \<in> s})" "gauge d"
- shows "gauge (\<lambda>x y. d (f x) (f y))"
- using assms unfolding gauge_def apply safe defer apply(erule_tac x="f x" in allE)
- apply(erule_tac x="d (f x)" in allE) unfolding mem_def Collect_def by auto
-
-subsection {* Only need trivial subintervals if the interval itself is trivial. *}
-
-lemma division_of_nontrivial: fixes s::"(real^'n) set set"
- assumes "s division_of {a..b}" "content({a..b}) \<noteq> 0"
- shows "{k. k \<in> s \<and> content k \<noteq> 0} division_of {a..b}" using assms(1) apply-
-proof(induct "card s" arbitrary:s rule:nat_less_induct)
- fix s::"(real^'n) set set" assume assm:"s division_of {a..b}"
- "\<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}"
- note s = division_ofD[OF assm(1)] let ?thesis = "{k \<in> s. content k \<noteq> 0} division_of {a..b}"
- { presume *:"{k \<in> s. content k \<noteq> 0} \<noteq> s \<Longrightarrow> ?thesis"
- show ?thesis apply cases defer apply(rule *,assumption) using assm(1) by auto }
- assume noteq:"{k \<in> s. content k \<noteq> 0} \<noteq> s"
- then obtain k where k:"k\<in>s" "content k = 0" by auto
- from s(4)[OF k(1)] guess c d apply-by(erule exE)+ note k=k this
- from k have "card s > 0" unfolding card_gt_0_iff using assm(1) by auto
- hence card:"card (s - {k}) < card s" using assm(1) k(1) apply(subst card_Diff_singleton_if) by auto
- have *:"closed (\<Union>(s - {k}))" apply(rule closed_Union) defer apply rule apply(drule DiffD1,drule s(4))
- apply safe apply(rule closed_interval) using assm(1) by auto
- have "k \<subseteq> \<Union>(s - {k})" apply safe apply(rule *[unfolded closed_limpt,rule_format]) unfolding islimpt_approachable
- proof safe fix x and e::real assume as:"x\<in>k" "e>0"
- from k(2)[unfolded k content_eq_0] guess i ..
- hence i:"c$i = d$i" using s(3)[OF k(1),unfolded k] unfolding interval_ne_empty by smt
- hence xi:"x$i = d$i" using as unfolding k mem_interval by smt
- def y \<equiv> "(\<chi> j. if j = i then if c$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$j)"
- show "\<exists>x'\<in>\<Union>(s - {k}). x' \<noteq> x \<and> dist x' x < e" apply(rule_tac x=y in bexI)
- proof have "d \<in> {c..d}" using s(3)[OF k(1)] unfolding k interval_eq_empty mem_interval by(fastsimp simp add: not_less)
- hence "d \<in> {a..b}" using s(2)[OF k(1)] unfolding k by auto note di = this[unfolded mem_interval,THEN spec[where x=i]]
- hence xyi:"y$i \<noteq> x$i" unfolding y_def unfolding i xi Cart_lambda_beta if_P[OF refl]
- 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+
- thus "y \<noteq> x" unfolding Cart_eq by auto
- have *:"UNIV = insert i (UNIV - {i})" by auto
- have "norm (y - x) < e + setsum (\<lambda>i. 0) (UNIV::'n set)" apply(rule le_less_trans[OF norm_le_l1])
- apply(subst *,subst setsum_insert) prefer 3 apply(rule add_less_le_mono)
- proof- show "\<bar>(y - x) $ i\<bar> < e" unfolding y_def Cart_lambda_beta vector_minus_component if_P[OF refl]
- apply(cases) apply(subst if_P,assumption) unfolding if_not_P unfolding i xi using di as(2) by auto
- show "(\<Sum>i\<in>UNIV - {i}. \<bar>(y - x) $ i\<bar>) \<le> (\<Sum>i\<in>UNIV. 0)" unfolding y_def by auto
- qed auto thus "dist y x < e" unfolding vector_dist_norm by auto
- 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
- moreover have "y \<in> \<Union>s" unfolding s mem_interval
- proof note simps = y_def Cart_lambda_beta if_not_P
- fix j::'n show "a $ j \<le> y $ j \<and> y $ j \<le> b $ j"
- proof(cases "j = i") case False have "x \<in> {a..b}" using s(2)[OF k(1)] as(1) by auto
- thus ?thesis unfolding simps if_not_P[OF False] unfolding mem_interval by auto
- next case True note T = this show ?thesis
- proof(cases "c $ i \<le> (a $ i + b $ i) / 2")
- case True show ?thesis unfolding simps if_P[OF T] if_P[OF True] unfolding i
- using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps)
- next case False thus ?thesis unfolding simps if_P[OF T] if_not_P[OF False] unfolding i
- using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps)
- qed qed qed
- ultimately show "y \<in> \<Union>(s - {k})" by auto
- qed qed hence "\<Union>(s - {k}) = {a..b}" unfolding s(6)[THEN sym] by auto
- hence "{ka \<in> s - {k}. content ka \<noteq> 0} division_of {a..b}" apply-apply(rule assm(2)[rule_format,OF card refl])
- apply(rule division_ofI) defer apply(rule_tac[1-4] s) using assm(1) by auto
- 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
-
-subsection {* Integrabibility on subintervals. *}
-
-lemma operative_integrable: fixes f::"real^'n \<Rightarrow> 'a::banach" shows
- "operative op \<and> (\<lambda>i. f integrable_on i)"
- unfolding operative_def neutral_and apply safe apply(subst integrable_on_def)
- unfolding has_integral_null_eq apply(rule,rule refl) apply(rule,assumption)+
- unfolding integrable_on_def by(auto intro: has_integral_split)
-
-lemma integrable_subinterval: fixes f::"real^'n \<Rightarrow> 'a::banach"
- assumes "f integrable_on {a..b}" "{c..d} \<subseteq> {a..b}" shows "f integrable_on {c..d}"
- apply(cases "{c..d} = {}") defer apply(rule partial_division_extend_1[OF assms(2)],assumption)
- using operative_division_and[OF operative_integrable,THEN sym,of _ _ _ f] assms(1) by auto
-
-subsection {* Combining adjacent intervals in 1 dimension. *}
-
-lemma has_integral_combine: assumes "(a::real^1) \<le> c" "c \<le> b"
- "(f has_integral i) {a..c}" "(f has_integral (j::'a::banach)) {c..b}"
- shows "(f has_integral (i + j)) {a..b}"
-proof- note operative_integral[of f, unfolded operative_1_le[OF monoidal_lifted[OF monoidal_monoid]]]
- note conjunctD2[OF this,rule_format] note * = this(2)[OF conjI[OF assms(1-2)],unfolded if_P[OF assms(3)]]
- hence "f integrable_on {a..b}" apply- apply(rule ccontr) apply(subst(asm) if_P) defer
- apply(subst(asm) if_P) using assms(3-) by auto
- with * show ?thesis apply-apply(subst(asm) if_P) defer apply(subst(asm) if_P) defer apply(subst(asm) if_P)
- unfolding lifted.simps using assms(3-) by(auto simp add: integrable_on_def integral_unique) qed
-
-lemma integral_combine: fixes f::"real^1 \<Rightarrow> 'a::banach"
- assumes "a \<le> c" "c \<le> b" "f integrable_on ({a..b})"
- shows "integral {a..c} f + integral {c..b} f = integral({a..b}) f"
- apply(rule integral_unique[THEN sym]) apply(rule has_integral_combine[OF assms(1-2)])
- apply(rule_tac[!] integrable_integral integrable_subinterval[OF assms(3)])+ using assms(1-2) by auto
-
-lemma integrable_combine: fixes f::"real^1 \<Rightarrow> 'a::banach"
- assumes "a \<le> c" "c \<le> b" "f integrable_on {a..c}" "f integrable_on {c..b}"
- shows "f integrable_on {a..b}" using assms unfolding integrable_on_def by(fastsimp intro!:has_integral_combine)
-
-subsection {* Reduce integrability to "local" integrability. *}
-
-lemma integrable_on_little_subintervals: fixes f::"real^'n \<Rightarrow> 'a::banach"
- 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}"
- shows "f integrable_on {a..b}"
-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})"
- using assms by auto note this[unfolded gauge_existence_lemma] from choice[OF this] guess d .. note d=this[rule_format]
- guess p apply(rule fine_division_exists[OF gauge_ball_dependent,of d a b]) using d by auto note p=this(1-2)
- note division_of_tagged_division[OF this(1)] note * = operative_division_and[OF operative_integrable,OF this,THEN sym,of f]
- show ?thesis unfolding * apply safe unfolding snd_conv
- proof- fix x k assume "(x,k) \<in> p" note tagged_division_ofD(2-4)[OF p(1) this] fineD[OF p(2) this]
- thus "f integrable_on k" apply safe apply(rule d[THEN conjunct2,rule_format,of x]) by auto qed qed
-
-subsection {* Second FCT or existence of antiderivative. *}
-
-lemma integrable_const[intro]:"(\<lambda>x. c) integrable_on {a..b}"
- unfolding integrable_on_def by(rule,rule has_integral_const)
-
-lemma integral_has_vector_derivative: fixes f::"real \<Rightarrow> 'a::banach"
- assumes "continuous_on {a..b} f" "x \<in> {a..b}"
- shows "((\<lambda>u. integral {vec a..vec u} (f o dest_vec1)) has_vector_derivative f(x)) (at x within {a..b})"
- unfolding has_vector_derivative_def has_derivative_within_alt
-apply safe apply(rule scaleR.bounded_linear_left)
-proof- fix e::real assume e:"e>0"
- note compact_uniformly_continuous[OF assms(1) compact_real_interval,unfolded uniformly_continuous_on_def]
- from this[rule_format,OF e] guess d apply-by(erule conjE exE)+ note d=this[rule_format]
- let ?I = "\<lambda>a b. integral {vec1 a..vec1 b} (f \<circ> dest_vec1)"
- 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)"
- proof(rule,rule,rule d,safe) case goal1 show ?case proof(cases "y < x")
- case False have "f \<circ> dest_vec1 integrable_on {vec1 a..vec1 y}" apply(rule integrable_subinterval,rule integrable_continuous)
- apply(rule continuous_on_o_dest_vec1 assms)+ unfolding not_less using assms(2) goal1 by auto
- hence *:"?I a y - ?I a x = ?I x y" unfolding group_simps apply(subst eq_commute) apply(rule integral_combine)
- using False unfolding not_less using assms(2) goal1 by auto
- have **:"norm (y - x) = content {vec1 x..vec1 y}" apply(subst content_1) using False unfolding not_less by auto
- show ?thesis unfolding ** apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x) o dest_vec1"]) unfolding * unfolding o_def
- defer apply(rule has_integral_sub) apply(rule integrable_integral)
- apply(rule integrable_subinterval,rule integrable_continuous) apply(rule continuous_on_o_dest_vec1[unfolded o_def] assms)+
- proof- show "{vec1 x..vec1 y} \<subseteq> {vec1 a..vec1 b}" using goal1 assms(2) by auto
- have *:"y - x = norm(y - x)" using False by auto
- show "((\<lambda>xa. f x) has_integral (y - x) *\<^sub>R f x) {vec1 x..vec1 y}" apply(subst *) unfolding ** by auto
- show "\<forall>xa\<in>{vec1 x..vec1 y}. norm (f (dest_vec1 xa) - f x) \<le> e" apply safe apply(rule less_imp_le)
- apply(rule d(2)[unfolded vector_dist_norm]) using assms(2) using goal1 by auto
- qed(insert e,auto)
- next case True have "f \<circ> dest_vec1 integrable_on {vec1 a..vec1 x}" apply(rule integrable_subinterval,rule integrable_continuous)
- apply(rule continuous_on_o_dest_vec1 assms)+ unfolding not_less using assms(2) goal1 by auto
- hence *:"?I a x - ?I a y = ?I y x" unfolding group_simps apply(subst eq_commute) apply(rule integral_combine)
- using True using assms(2) goal1 by auto
- have **:"norm (y - x) = content {vec1 y..vec1 x}" apply(subst content_1) using True unfolding not_less by auto
- 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
- show ?thesis apply(subst ***) unfolding norm_minus_cancel **
- apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x) o dest_vec1"]) unfolding * unfolding o_def
- defer apply(rule has_integral_sub) apply(subst minus_minus[THEN sym]) unfolding minus_minus
- apply(rule integrable_integral) apply(rule integrable_subinterval,rule integrable_continuous)
- apply(rule continuous_on_o_dest_vec1[unfolded o_def] assms)+
- proof- show "{vec1 y..vec1 x} \<subseteq> {vec1 a..vec1 b}" using goal1 assms(2) by auto
- have *:"x - y = norm(y - x)" using True by auto
- show "((\<lambda>xa. f x) has_integral (x - y) *\<^sub>R f x) {vec1 y..vec1 x}" apply(subst *) unfolding ** by auto
- show "\<forall>xa\<in>{vec1 y..vec1 x}. norm (f (dest_vec1 xa) - f x) \<le> e" apply safe apply(rule less_imp_le)
- apply(rule d(2)[unfolded vector_dist_norm]) using assms(2) using goal1 by auto
- qed(insert e,auto) qed qed qed
-
-lemma integral_has_vector_derivative': fixes f::"real^1 \<Rightarrow> 'a::banach"
- assumes "continuous_on {a..b} f" "x \<in> {a..b}"
- shows "((\<lambda>u. (integral {a..vec u} f)) has_vector_derivative f x) (at (x$1) within {a$1..b$1})"
- using integral_has_vector_derivative[OF continuous_on_o_vec1[OF assms(1)], of "x$1"]
- unfolding o_def vec1_dest_vec1 using assms(2) by auto
-
-lemma antiderivative_continuous: assumes "continuous_on {a..b::real} f"
- obtains g where "\<forall>x\<in> {a..b}. (g has_vector_derivative (f(x)::_::banach)) (at x within {a..b})"
- apply(rule that,rule) using integral_has_vector_derivative[OF assms] by auto
-
-subsection {* Combined fundamental theorem of calculus. *}
-
-lemma antiderivative_integral_continuous: fixes f::"real \<Rightarrow> 'a::banach" assumes "continuous_on {a..b} f"
- 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}"
-proof- from antiderivative_continuous[OF assms] guess g . note g=this
- show ?thesis apply(rule that[of g])
- proof safe case goal1 have "\<forall>x\<in>{u..v}. (g has_vector_derivative f x) (at x within {u..v})"
- apply(rule,rule has_vector_derivative_within_subset) apply(rule g[rule_format]) using goal1(1-2) by auto
- thus ?case using fundamental_theorem_of_calculus[OF goal1(3),of "g o dest_vec1" "f o dest_vec1"]
- unfolding o_def vec1_dest_vec1 by auto qed qed
-
-subsection {* General "twiddling" for interval-to-interval function image. *}
-
-lemma has_integral_twiddle:
- assumes "0 < r" "\<forall>x. h(g x) = x" "\<forall>x. g(h x) = x" "\<forall>x. continuous (at x) g"
- "\<forall>u v. \<exists>w z. g ` {u..v} = {w..z}"
- "\<forall>u v. \<exists>w z. h ` {u..v} = {w..z}"
- "\<forall>u v. content(g ` {u..v}) = r * content {u..v}"
- "(f has_integral i) {a..b}"
- shows "((\<lambda>x. f(g x)) has_integral (1 / r) *\<^sub>R i) (h ` {a..b})"
-proof- { presume *:"{a..b} \<noteq> {} \<Longrightarrow> ?thesis"
- show ?thesis apply cases defer apply(rule *,assumption)
- proof- case goal1 thus ?thesis unfolding goal1 assms(8)[unfolded goal1 has_integral_empty_eq] by auto qed }
- assume "{a..b} \<noteq> {}" from assms(6)[rule_format,of a b] guess w z apply-by(erule exE)+ note wz=this
- have inj:"inj g" "inj h" unfolding inj_on_def apply safe apply(rule_tac[!] ccontr)
- using assms(2) apply(erule_tac x=x in allE) using assms(2) apply(erule_tac x=y in allE) defer
- using assms(3) apply(erule_tac x=x in allE) using assms(3) apply(erule_tac x=y in allE) by auto
- show ?thesis unfolding has_integral_def has_integral_compact_interval_def apply(subst if_P) apply(rule,rule,rule wz)
- proof safe fix e::real assume e:"e>0" hence "e * r > 0" using assms(1) by(rule mult_pos_pos)
- from assms(8)[unfolded has_integral,rule_format,OF this] guess d apply-by(erule exE conjE)+ note d=this[rule_format]
- 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)
- 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)"
- 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
- fix p assume as:"p tagged_division_of h ` {a..b}" "d' fine p" note p = tagged_division_ofD[OF as(1)]
- 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
- proof safe show "finite ((\<lambda>(x, k). (g x, g ` k)) ` p)" using as by auto
- show "d fine (\<lambda>(x, k). (g x, g ` k)) ` p" using as(2) unfolding fine_def d' by auto
- fix x k assume xk[intro]:"(x,k) \<in> p" show "g x \<in> g ` k" using p(2)[OF xk] by auto
- show "\<exists>u v. g ` k = {u..v}" using p(4)[OF xk] using assms(5-6) by auto
- { 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)"]
- using assms(2)[rule_format,of y] unfolding inj_image_mem_iff[OF inj(2)] by auto }
- fix x' k' assume xk':"(x',k') \<in> p" fix z assume "z \<in> interior (g ` k)" "z \<in> interior (g ` k')"
- hence *:"interior (g ` k) \<inter> interior (g ` k') \<noteq> {}" by auto
- have same:"(x, k) = (x', k')" apply-apply(rule ccontr,drule p(5)[OF xk xk'])
- proof- assume as:"interior k \<inter> interior k' = {}" from nonempty_witness[OF *] guess z .
- hence "z \<in> g ` (interior k \<inter> interior k')" using interior_image_subset[OF assms(4) inj(1)]
- unfolding image_Int[OF inj(1)] by auto thus False using as by blast
- qed thus "g x = g x'" by auto
- { fix z assume "z \<in> k" thus "g z \<in> g ` k'" using same by auto }
- { fix z assume "z \<in> k'" thus "g z \<in> g ` k" using same by auto }
- 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
- then guess X unfolding Union_iff .. note X=this from this(1) guess y unfolding mem_Collect_eq ..
- thus "x \<in> \<Union>{k. \<exists>x. (x, k) \<in> (\<lambda>(x, k). (g x, g ` k)) ` p}" apply-
- apply(rule_tac X="g ` X" in UnionI) defer apply(rule_tac x="h x" in image_eqI)
- using X(2) assms(3)[rule_format,of x] by auto
- 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
- 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
- unfolding setsum_reindex[OF *] apply(subst scaleR_right.setsum) defer apply(rule setsum_cong2) unfolding o_def split_paired_all split_conv
- apply(drule p(4)) apply safe unfolding assms(7)[rule_format] using p by auto
- 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]
- unfolding real_scaleR_def using assms(1) by auto finally have *:"?l = ?r" .
- 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
- using assms(1) by(auto simp add:field_simps) qed qed qed
-
-subsection {* Special case of a basic affine transformation. *}
-
-lemma interval_image_affinity_interval: shows "\<exists>u v. (\<lambda>x. m *\<^sub>R (x::real^'n) + c) ` {a..b} = {u..v}"
- unfolding image_affinity_interval by auto
-
-lemmas Cart_simps = Cart_nth.add Cart_nth.minus Cart_nth.zero Cart_nth.diff Cart_nth.scaleR real_scaleR_def Cart_lambda_beta
- Cart_eq vector_le_def vector_less_def
-
-lemma setprod_cong2: assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "setprod f A = setprod g A"
- apply(rule setprod_cong) using assms by auto
-
-lemma content_image_affinity_interval:
- "content((\<lambda>x::real^'n. m *\<^sub>R x + c) ` {a..b}) = (abs m) ^ CARD('n) * content {a..b}" (is "?l = ?r")
-proof- { presume *:"{a..b}\<noteq>{} \<Longrightarrow> ?thesis" show ?thesis apply(cases,rule *,assumption)
- unfolding not_not using content_empty by auto }
- assume as:"{a..b}\<noteq>{}" show ?thesis proof(cases "m \<ge> 0")
- case True show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_P[OF True]
- unfolding content_closed_interval'[OF as] apply(subst content_closed_interval')
- defer apply(subst setprod_constant[THEN sym]) apply(rule finite_UNIV) unfolding setprod_timesf[THEN sym]
- apply(rule setprod_cong2) using True as unfolding interval_ne_empty Cart_simps not_le
- by(auto simp add:field_simps intro:mult_left_mono)
- next case False show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_not_P[OF False]
- unfolding content_closed_interval'[OF as] apply(subst content_closed_interval')
- defer apply(subst setprod_constant[THEN sym]) apply(rule finite_UNIV) unfolding setprod_timesf[THEN sym]
- apply(rule setprod_cong2) using False as unfolding interval_ne_empty Cart_simps not_le
- by(auto simp add:field_simps mult_le_cancel_left_neg) qed qed
-
-lemma has_integral_affinity: assumes "(f has_integral i) {a..b::real^'n}" "m \<noteq> 0"
- 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})"
- apply(rule has_integral_twiddle,safe) unfolding Cart_eq Cart_simps apply(rule zero_less_power)
- defer apply(insert assms(2), simp add:field_simps) apply(insert assms(2), simp add:field_simps)
- apply(rule continuous_intros)+ apply(rule interval_image_affinity_interval)+ apply(rule content_image_affinity_interval) using assms by auto
-
-lemma integrable_affinity: assumes "f integrable_on {a..b}" "m \<noteq> 0"
- 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})"
- using assms unfolding integrable_on_def apply safe apply(drule has_integral_affinity) by auto
-
-subsection {* Special case of stretching coordinate axes separately. *}
-
-lemma image_stretch_interval:
- "(\<lambda>x. \<chi> k. m k * x$k) ` {a..b::real^'n} =
- (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")
-proof(cases "{a..b}={}") case True thus ?thesis unfolding True by auto
-next have *:"\<And>P Q. (\<forall>i. P i) \<and> (\<forall>i. Q i) \<longleftrightarrow> (\<forall>i. P i \<and> Q i)" by auto
- case False note ab = this[unfolded interval_ne_empty]
- show ?thesis apply-apply(rule set_ext)
- 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
- show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" unfolding if_not_P[OF False]
- unfolding image_iff mem_interval Bex_def Cart_simps Cart_eq *
- unfolding lambda_skolem[THEN sym,of "\<lambda> i xa. (a $ i \<le> xa \<and> xa \<le> b $ i) \<and> x $ i = m i * xa"]
- proof(rule **,rule) fix i::'n show "(\<exists>xa. (a $ i \<le> xa \<and> xa \<le> b $ i) \<and> x $ i = m i * xa) =
- (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))"
- proof(cases "m i = 0") case True thus ?thesis using ab by auto
- next case False hence "0 < m i \<or> 0 > m i" by auto thus ?thesis apply-
- proof(erule disjE) assume as:"0 < m i" hence *:"min (m i * a $ i) (m i * b $ i) = m i * a $ i"
- "max (m i * a $ i) (m i * b $ i) = m i * b $ i" using ab unfolding min_def max_def by auto
- show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x$i" in exI)
- using as by(auto simp add:field_simps)
- next assume as:"0 > m i" hence *:"max (m i * a $ i) (m i * b $ i) = m i * a $ i"
- "min (m i * a $ i) (m i * b $ i) = m i * b $ i" using ab as unfolding min_def max_def
- by(auto simp add:field_simps mult_le_cancel_left_neg intro:real_le_antisym)
- show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x$i" in exI)
- using as by(auto simp add:field_simps) qed qed qed qed qed
-
-lemma interval_image_stretch_interval: "\<exists>u v. (\<lambda>x. \<chi> k. m k * x$k) ` {a..b::real^'n} = {u..v}"
- unfolding image_stretch_interval by auto
-
-lemma content_image_stretch_interval:
- "content((\<lambda>x::real^'n. \<chi> k. m k * x$k) ` {a..b}) = abs(setprod m UNIV) * content({a..b})"
-proof(cases "{a..b} = {}") case True thus ?thesis
- unfolding content_def image_is_empty image_stretch_interval if_P[OF True] by auto
-next case False hence "(\<lambda>x. \<chi> k. m k * x $ k) ` {a..b} \<noteq> {}" by auto
- thus ?thesis using False unfolding content_def image_stretch_interval apply- unfolding interval_bounds' if_not_P
- unfolding abs_setprod setprod_timesf[THEN sym] apply(rule setprod_cong2) unfolding Cart_lambda_beta
- proof- fix i::'n have "(m i < 0 \<or> m i > 0) \<or> m i = 0" by auto
- 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)"
- apply-apply(erule disjE)+ unfolding min_def max_def using False[unfolded interval_ne_empty,rule_format,of i]
- by(auto simp add:field_simps not_le mult_le_cancel_left_neg mult_le_cancel_left_pos) qed qed
-
-lemma has_integral_stretch: assumes "(f has_integral i) {a..b}" "\<forall>k. ~(m k = 0)"
- shows "((\<lambda>x. f(\<chi> k. m k * x$k)) has_integral
- ((1/(abs(setprod m UNIV))) *\<^sub>R i)) ((\<lambda>x. \<chi> k. 1/(m k) * x$k) ` {a..b})"
- apply(rule has_integral_twiddle) unfolding zero_less_abs_iff content_image_stretch_interval
- unfolding image_stretch_interval empty_as_interval Cart_eq using assms
-proof- show "\<forall>x. continuous (at x) (\<lambda>x. \<chi> k. m k * x $ k)"
- apply(rule,rule linear_continuous_at) unfolding linear_linear
- unfolding linear_def Cart_simps Cart_eq by(auto simp add:field_simps) qed auto
-
-lemma integrable_stretch:
- assumes "f integrable_on {a..b}" "\<forall>k. ~(m k = 0)"
- shows "(\<lambda>x. f(\<chi> k. m k * x$k)) integrable_on ((\<lambda>x. \<chi> k. 1/(m k) * x$k) ` {a..b})"
- using assms unfolding integrable_on_def apply-apply(erule exE) apply(drule has_integral_stretch) by auto
-
-subsection {* even more special cases. *}
-
-lemma uminus_interval_vector[simp]:"uminus ` {a..b} = {-b .. -a::real^'n}"
- apply(rule set_ext,rule) defer unfolding image_iff
- apply(rule_tac x="-x" in bexI) by(auto simp add:vector_le_def minus_le_iff le_minus_iff)
-
-lemma has_integral_reflect_lemma[intro]: assumes "(f has_integral i) {a..b}"
- shows "((\<lambda>x. f(-x)) has_integral i) {-b .. -a}"
- using has_integral_affinity[OF assms, of "-1" 0] by auto
-
-lemma has_integral_reflect[simp]: "((\<lambda>x. f(-x)) has_integral i) {-b..-a} \<longleftrightarrow> (f has_integral i) ({a..b})"
- apply rule apply(drule_tac[!] has_integral_reflect_lemma) by auto
-
-lemma integrable_reflect[simp]: "(\<lambda>x. f(-x)) integrable_on {-b..-a} \<longleftrightarrow> f integrable_on {a..b}"
- unfolding integrable_on_def by auto
-
-lemma integral_reflect[simp]: "integral {-b..-a} (\<lambda>x. f(-x)) = integral ({a..b}) f"
- unfolding integral_def by auto
-
-subsection {* Stronger form of FCT; quite a tedious proof. *}
-
-(** move this **)
-declare norm_triangle_ineq4[intro]
-
-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)
-
-lemma additive_tagged_division_1': fixes f::"real \<Rightarrow> 'a::real_normed_vector"
- assumes "a \<le> b" "p tagged_division_of {vec1 a..vec1 b}"
- shows "setsum (\<lambda>(x,k). f (dest_vec1 (interval_upperbound k)) - f(dest_vec1 (interval_lowerbound k))) p = f b - f a"
- using additive_tagged_division_1[OF _ assms(2), of "f o dest_vec1"]
- unfolding o_def vec1_dest_vec1 using assms(1) by auto
-
-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"
- unfolding split_def by(rule refl)
-
-lemma norm_triangle_le_sub: "norm x + norm y \<le> e \<Longrightarrow> norm (x - y) \<le> e"
- apply(subst(asm)(2) norm_minus_cancel[THEN sym])
- apply(drule norm_triangle_le) by(auto simp add:group_simps)
-
-lemma fundamental_theorem_of_calculus_interior:
- assumes"a \<le> b" "continuous_on {a..b} f" "\<forall>x\<in>{a<..<b}. (f has_vector_derivative f'(x)) (at x)"
- shows "((f' o dest_vec1) has_integral (f b - f a)) {vec a..vec b}"
-proof- { presume *:"a < b \<Longrightarrow> ?thesis"
- show ?thesis proof(cases,rule *,assumption)
- assume "\<not> a < b" hence "a = b" using assms(1) by auto
- hence *:"{vec a .. vec b} = {vec b}" "f b - f a = 0" apply(auto simp add: Cart_simps) by smt
- show ?thesis unfolding *(2) apply(rule has_integral_null) unfolding content_eq_0_1 using * `a=b` by auto
- qed } assume ab:"a < b"
- let ?P = "\<lambda>e. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {vec1 a..vec1 b} \<and> d fine p \<longrightarrow>
- 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})"
- { presume "\<And>e. e>0 \<Longrightarrow> ?P e" thus ?thesis unfolding has_integral_factor_content by auto }
- fix e::real assume e:"e>0"
- note assms(3)[unfolded has_vector_derivative_def has_derivative_at_alt ball_conj_distrib]
- note conjunctD2[OF this] note bounded=this(1) and this(2)
- 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)"
- 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]
- from choice[OF this] guess d .. note conjunctD2[OF this[rule_format]] note d = this[rule_format]
- have "bounded (f ` {a..b})" apply(rule compact_imp_bounded compact_continuous_image)+ using compact_real_interval assms by auto
- from this[unfolded bounded_pos] guess B .. note B = this[rule_format]
-
- have "\<exists>da. 0 < da \<and> (\<forall>c. a \<le> c \<and> {a..c} \<subseteq> {a..b} \<and> {a..c} \<subseteq> ball a da
- \<longrightarrow> norm(content {vec1 a..vec1 c} *\<^sub>R f' a - (f c - f a)) \<le> (e * (b - a)) / 4)"
- proof- have "a\<in>{a..b}" using ab by auto
- note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
- note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0" using e ab by(auto simp add:field_simps)
- from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
- have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' a) \<le> (e * (b - a)) / 8"
- proof(cases "f' a = 0") case True
- thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg)
- next case False thus ?thesis
- apply(rule_tac x="(e * (b - a)) / 8 / norm (f' a)" in exI)
- using ab e by(auto simp add:field_simps)
- qed then guess l .. note l = conjunctD2[OF this]
- show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+)
- proof- fix c assume as:"a \<le> c" "{a..c} \<subseteq> {a..b}" "{a..c} \<subseteq> ball a (min k l)"
- note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval]
- 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)
- also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8"
- proof(rule add_mono) case goal1 have "\<bar>c - a\<bar> \<le> \<bar>l\<bar>" using as' by auto
- thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto
- next case goal2 show ?case apply(rule less_imp_le) apply(cases "a = c") defer
- apply(rule k(2)[unfolded vector_dist_norm]) using as' e ab by(auto simp add:field_simps)
- 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
- qed qed then guess da .. note da=conjunctD2[OF this,rule_format]
-
- 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"
- proof- have "b\<in>{a..b}" using ab by auto
- note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
- note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0" using e ab by(auto simp add:field_simps)
- from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
- have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' b) \<le> (e * (b - a)) / 8"
- proof(cases "f' b = 0") case True
- thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg)
- next case False thus ?thesis
- apply(rule_tac x="(e * (b - a)) / 8 / norm (f' b)" in exI)
- using ab e by(auto simp add:field_simps)
- qed then guess l .. note l = conjunctD2[OF this]
- show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+)
- proof- fix c assume as:"c \<le> b" "{c..b} \<subseteq> {a..b}" "{c..b} \<subseteq> ball b (min k l)"
- note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval]
- 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)
- also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8"
- proof(rule add_mono) case goal1 have "\<bar>c - b\<bar> \<le> \<bar>l\<bar>" using as' by auto
- thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto
- next case goal2 show ?case apply(rule less_imp_le) apply(cases "b = c") defer apply(subst norm_minus_commute)
- apply(rule k(2)[unfolded vector_dist_norm]) using as' e ab by(auto simp add:field_simps)
- 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
- qed qed then guess db .. note db=conjunctD2[OF this,rule_format]
-
- let ?d = "(\<lambda>x. ball x (if x=vec1 a then da else if x=vec b then db else d (dest_vec1 x)))"
- show "?P e" apply(rule_tac x="?d" in exI)
- proof safe case goal1 show ?case apply(rule gauge_ball_dependent) using ab db(1) da(1) d(1) by auto
- next case goal2 note as=this let ?A = "{t. fst t \<in> {vec1 a, vec1 b}}" note p = tagged_division_ofD[OF goal2(1)]
- 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
- note * = additive_tagged_division_1'[OF assms(1) goal2(1), THEN sym]
- 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
- show ?case unfolding content_1'[OF assms(1)] and *[of "\<lambda>x. x"] *[of f] setsum_subtractf[THEN sym] split_minus
- unfolding setsum_right_distrib apply(subst(2) pA,subst pA) unfolding setsum_Un_disjoint[OF pA(2-)]
- proof(rule norm_triangle_le,rule **)
- case goal1 show ?case apply(rule order_trans,rule setsum_norm_le) apply(rule pA) defer apply(subst divide.setsum)
- 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"
- "e * (dest_vec1 (interval_upperbound k) - dest_vec1 (interval_lowerbound k)) / 2
- < norm (content k *\<^sub>R f' (dest_vec1 x) - (f (dest_vec1 (interval_upperbound k)) - f (dest_vec1 (interval_lowerbound k))))"
- from p(4)[OF this(1)] guess u v apply-by(erule exE)+ note k=this
- 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]
- note result = as(2)[unfolded k interval_bounds[OF this(1)] content_1[OF this(2)]]
-
- 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]
- have "norm ((v$1 - u$1) *\<^sub>R f' (x$1) - (f (v$1) - f (u$1))) =
- 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)))"
- apply(rule arg_cong[of _ _ norm]) unfolding scaleR_left.diff by auto
- also have "... \<le> e / 2 * norm (u$1 - x$1) + e / 2 * norm (v$1 - x$1)" apply(rule norm_triangle_le_sub)
- apply(rule add_mono) apply(rule_tac[!] *) using fineD[OF goal2(2) as(1)] as' unfolding k subset_eq
- apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE) using uv by(auto simp add:dist_real)
- also have "... \<le> e / 2 * norm (v$1 - u$1)" using p(2)[OF as(1)] unfolding k by(auto simp add:field_simps)
- finally have "e * (dest_vec1 v - dest_vec1 u) / 2 < e * (dest_vec1 v - dest_vec1 u) / 2"
- apply- apply(rule less_le_trans[OF result]) using uv by auto thus False by auto qed
-
- next have *:"\<And>x s1 s2::real. 0 \<le> s1 \<Longrightarrow> x \<le> (s1 + s2) / 2 \<Longrightarrow> x - s1 \<le> s2 / 2" by auto
- case goal2 show ?case apply(rule *) apply(rule setsum_nonneg) apply(rule,unfold split_paired_all split_conv)
- defer unfolding setsum_Un_disjoint[OF pA(2-),THEN sym] pA(1)[THEN sym] unfolding setsum_right_distrib[THEN sym]
- apply(subst additive_tagged_division_1[OF _ as(1)]) unfolding vec1_dest_vec1 apply(rule assms)
- proof- fix x k assume "(x,k) \<in> p \<inter> {t. fst t \<in> {vec1 a, vec1 b}}" note xk=IntD1[OF this]
- from p(4)[OF this] guess u v apply-by(erule exE)+ note uv=this
- with p(2)[OF xk] have "{u..v} \<noteq> {}" by auto
- thus "0 \<le> e * ((interval_upperbound k)$1 - (interval_lowerbound k)$1)"
- unfolding uv using e by(auto simp add:field_simps)
- 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
- show "norm (\<Sum>(x, k)\<in>p \<inter> ?A. content k *\<^sub>R (f' \<circ> dest_vec1) x -
- (f ((interval_upperbound k)$1) - f ((interval_lowerbound k)$1))) \<le> e * (b - a) / 2"
- apply(rule *[where t="p \<inter> {t. fst t \<in> {vec1 a, vec1 b} \<and> content(snd t) \<noteq> 0}"])
- apply(rule setsum_mono_zero_right[OF pA(2)]) defer apply(rule) unfolding split_paired_all split_conv o_def
- 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}"
- 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
- 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
- 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
- next have *:"p \<inter> {t. fst t \<in> {vec1 a, vec1 b} \<and> content(snd t) \<noteq> 0} =
- {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
- 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"
- proof(case_tac "s={}") case goal2 then obtain x where "x\<in>s" by auto hence *:"s = {x}" using goal2(1) by auto
- thus ?case using `x\<in>s` goal2(2) by auto
- qed auto
- 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"])
- apply(rule norm_triangle_le,rule add_mono) apply(rule_tac[1-2] **)
- proof- let ?B = "\<lambda>x. {t \<in> p. fst t = vec1 x \<and> content (snd t) \<noteq> 0}"
- have pa:"\<And>k. (vec1 a, k) \<in> p \<Longrightarrow> \<exists>v. k = {vec1 a .. v} \<and> vec1 a \<le> v"
- proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this
- have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto
- have u:"u = vec1 a" proof(rule ccontr) have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto
- 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
- have "u > vec1 a" unfolding Cart_simps by auto
- thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:Cart_simps)
- qed thus ?case apply(rule_tac x=v in exI) unfolding uv using * by auto
- qed
- have pb:"\<And>k. (vec1 b, k) \<in> p \<Longrightarrow> \<exists>v. k = {v .. vec1 b} \<and> vec1 b \<ge> v"
- proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this
- have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto
- have u:"v = vec1 b" proof(rule ccontr) have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto
- 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
- have "v < vec1 b" unfolding Cart_simps by auto
- thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:Cart_simps)
- qed thus ?case apply(rule_tac x=u in exI) unfolding uv using * by auto
- qed
-
- show "\<forall>x y. x \<in> ?B a \<and> y \<in> ?B a \<longrightarrow> x = y" apply(rule,rule,rule,unfold split_paired_all)
- unfolding mem_Collect_eq fst_conv snd_conv apply safe
- proof- fix x k k' assume k:"(vec1 a, k) \<in> p" "(vec1 a, k') \<in> p" "content k \<noteq> 0" "content k' \<noteq> 0"
- guess v using pa[OF k(1)] .. note v = conjunctD2[OF this]
- guess v' using pa[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = "vec1 (min (v$1) (v'$1))"
- have "{vec1 a <..< ?v} \<subseteq> k \<inter> k'" unfolding v v' by(auto simp add:Cart_simps) note subset_interior[OF this,unfolded interior_inter]
- moreover have "vec1 ((a + ?v$1)/2) \<in> {vec1 a <..< ?v}" using k(3-) unfolding v v' content_eq_0_1 not_le by(auto simp add:Cart_simps)
- ultimately have "vec1 ((a + ?v$1)/2) \<in> interior k \<inter> interior k'" unfolding interior_open[OF open_interval] by auto
- hence *:"k = k'" apply- apply(rule ccontr) using p(5)[OF k(1-2)] by auto
- { assume "x\<in>k" thus "x\<in>k'" unfolding * . } { assume "x\<in>k'" thus "x\<in>k" unfolding * . }
- qed
- show "\<forall>x y. x \<in> ?B b \<and> y \<in> ?B b \<longrightarrow> x = y" apply(rule,rule,rule,unfold split_paired_all)
- unfolding mem_Collect_eq fst_conv snd_conv apply safe
- proof- fix x k k' assume k:"(vec1 b, k) \<in> p" "(vec1 b, k') \<in> p" "content k \<noteq> 0" "content k' \<noteq> 0"
- guess v using pb[OF k(1)] .. note v = conjunctD2[OF this]
- guess v' using pb[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = "vec1 (max (v$1) (v'$1))"
- have "{?v <..< vec1 b} \<subseteq> k \<inter> k'" unfolding v v' by(auto simp add:Cart_simps) note subset_interior[OF this,unfolded interior_inter]
- moreover have "vec1 ((b + ?v$1)/2) \<in> {?v <..< vec1 b}" using k(3-) unfolding v v' content_eq_0_1 not_le by(auto simp add:Cart_simps)
- ultimately have "vec1 ((b + ?v$1)/2) \<in> interior k \<inter> interior k'" unfolding interior_open[OF open_interval] by auto
- hence *:"k = k'" apply- apply(rule ccontr) using p(5)[OF k(1-2)] by auto
- { assume "x\<in>k" thus "x\<in>k'" unfolding * . } { assume "x\<in>k'" thus "x\<in>k" unfolding * . }
- qed
-
- let ?a = a and ?b = b (* a is something else while proofing the next theorem. *)
- show "\<forall>x. x \<in> ?B a \<longrightarrow> norm ((\<lambda>(x, k). content k *\<^sub>R f' (x$1) - (f ((interval_upperbound k)$1) - f ((interval_lowerbound k)$1))) x)
- \<le> e * (b - a) / 4" apply safe unfolding fst_conv snd_conv apply safe unfolding vec1_dest_vec1
- proof- case goal1 guess v using pa[OF goal1(1)] .. note v = conjunctD2[OF this]
- have "vec1 ?a\<in>{vec1 ?a..v}" using v(2) by auto hence "dest_vec1 v \<le> ?b" using p(3)[OF goal1(1)] unfolding subset_eq v by auto
- moreover have "{?a..dest_vec1 v} \<subseteq> ball ?a da" using fineD[OF as(2) goal1(1)]
- apply-apply(subst(asm) if_P,rule refl) unfolding subset_eq apply safe apply(erule_tac x="vec1 x" in ballE)
- by(auto simp add:Cart_simps subset_eq dist_real v dist_real_def) ultimately
- show ?case unfolding v unfolding interval_bounds[OF v(2)[unfolded v vector_le_def]] vec1_dest_vec1 apply-
- apply(rule da(2)[of "v$1",unfolded vec1_dest_vec1])
- using goal1 fineD[OF as(2) goal1(1)] unfolding v content_eq_0_1 by auto
- qed
- show "\<forall>x. x \<in> ?B b \<longrightarrow> norm ((\<lambda>(x, k). content k *\<^sub>R f' (x$1) - (f ((interval_upperbound k)$1) - f ((interval_lowerbound k)$1))) x)
- \<le> e * (b - a) / 4" apply safe unfolding fst_conv snd_conv apply safe unfolding vec1_dest_vec1
- proof- case goal1 guess v using pb[OF goal1(1)] .. note v = conjunctD2[OF this]
- have "vec1 ?b\<in>{v..vec1 ?b}" using v(2) by auto hence "dest_vec1 v \<ge> ?a" using p(3)[OF goal1(1)] unfolding subset_eq v by auto
- moreover have "{dest_vec1 v..?b} \<subseteq> ball ?b db" using fineD[OF as(2) goal1(1)]
- apply-apply(subst(asm) if_P,rule refl) unfolding subset_eq apply safe apply(erule_tac x="vec1 x" in ballE) using ab
- by(auto simp add:Cart_simps subset_eq dist_real v dist_real_def) ultimately
- show ?case unfolding v unfolding interval_bounds[OF v(2)[unfolded v vector_le_def]] vec1_dest_vec1 apply-
- apply(rule db(2)[of "v$1",unfolded vec1_dest_vec1])
- using goal1 fineD[OF as(2) goal1(1)] unfolding v content_eq_0_1 by auto
- qed
- qed(insert p(1) ab e, auto simp add:field_simps) qed auto qed qed qed qed
-
-subsection {* Stronger form with finite number of exceptional points. *}
-
-lemma fundamental_theorem_of_calculus_interior_strong: fixes f::"real \<Rightarrow> 'a::banach"
- assumes"finite s" "a \<le> b" "continuous_on {a..b} f"
- "\<forall>x\<in>{a<..<b} - s. (f has_vector_derivative f'(x)) (at x)"
- shows "((f' o dest_vec1) has_integral (f b - f a)) {vec a..vec b}" using assms apply-
-proof(induct "card s" arbitrary:s a b)
- case 0 show ?case apply(rule fundamental_theorem_of_calculus_interior) using 0 by auto
-next case (Suc n) from this(2) guess c s' apply-apply(subst(asm) eq_commute) unfolding card_Suc_eq
- apply(subst(asm)(2) eq_commute) by(erule exE conjE)+ note cs = this[rule_format]
- show ?case proof(cases "c\<in>{a<..<b}")
- case False thus ?thesis apply- apply(rule Suc(1)[OF cs(3) _ Suc(4,5)]) apply safe defer
- apply(rule Suc(6)[rule_format]) using Suc(3) unfolding cs by auto
- next have *:"f b - f a = (f c - f a) + (f b - f c)" by auto
- case True hence "vec1 a \<le> vec1 c" "vec1 c \<le> vec1 b" by auto
- thus ?thesis apply(subst *) apply(rule has_integral_combine) apply assumption+
- apply(rule_tac[!] Suc(1)[OF cs(3)]) using Suc(3) unfolding cs
- proof- show "continuous_on {a..c} f" "continuous_on {c..b} f"
- apply(rule_tac[!] continuous_on_subset[OF Suc(5)]) using True by auto
- let ?P = "\<lambda>i j. \<forall>x\<in>{i<..<j} - s'. (f has_vector_derivative f' x) (at x)"
- show "?P a c" "?P c b" apply safe apply(rule_tac[!] Suc(6)[rule_format]) using True unfolding cs by auto
- qed auto qed qed
-
-lemma fundamental_theorem_of_calculus_strong: fixes f::"real \<Rightarrow> 'a::banach"
- assumes "finite s" "a \<le> b" "continuous_on {a..b} f"
- "\<forall>x\<in>{a..b} - s. (f has_vector_derivative f'(x)) (at x)"
- shows "((f' o dest_vec1) has_integral (f(b) - f(a))) {vec1 a..vec1 b}"
- apply(rule fundamental_theorem_of_calculus_interior_strong[OF assms(1-3), of f'])
- using assms(4) by auto
-
-end
--- a/src/HOL/Multivariate_Analysis/Multivariate_Analysis.thy Mon Feb 22 20:08:10 2010 +0100
+++ b/src/HOL/Multivariate_Analysis/Multivariate_Analysis.thy Mon Feb 22 20:41:49 2010 +0100
@@ -1,5 +1,5 @@
theory Multivariate_Analysis
-imports Determinants Integration_MV
+imports Determinants Integration Real_Integration
begin
end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Multivariate_Analysis/Real_Integration.thy Mon Feb 22 20:41:49 2010 +0100
@@ -0,0 +1,72 @@
+header{*Integration on real intervals*}
+
+theory Real_Integration
+imports Integration
+begin
+
+text{*We follow John Harrison in formalizing the Gauge integral.*}
+
+definition Integral :: "real set \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> bool" where
+ "Integral s f k = (f o dest_vec1 has_integral k) (vec1 ` s)"
+
+lemmas integral_unfold = Integral_def split_conv o_def vec1_interval
+
+lemma Integral_unique:
+ "[| Integral{a..b} f k1; Integral{a..b} f k2 |] ==> k1 = k2"
+ unfolding integral_unfold apply(rule has_integral_unique) by assumption+
+
+lemma Integral_zero [simp]: "Integral{a..a} f 0"
+ unfolding integral_unfold by auto
+
+lemma Integral_eq_diff_bounds: assumes "a \<le> b" shows "Integral{a..b} (%x. 1) (b - a)"
+ unfolding integral_unfold using has_integral_const[of "1::real" "vec1 a" "vec1 b"]
+ unfolding content_1'[OF assms] by auto
+
+lemma Integral_mult_const: assumes "a \<le> b" shows "Integral{a..b} (%x. c) (c*(b - a))"
+ unfolding integral_unfold using has_integral_const[of "c::real" "vec1 a" "vec1 b"]
+ unfolding content_1'[OF assms] by(auto simp add:field_simps)
+
+lemma Integral_mult: assumes "Integral{a..b} f k" shows "Integral{a..b} (%x. c * f x) (c * k)"
+ using assms unfolding integral_unfold apply(drule_tac has_integral_cmul[where c=c]) by auto
+
+lemma Integral_add:
+ assumes "Integral {a..b} f x1" "Integral {b..c} f x2" "a \<le> b" and "b \<le> c"
+ shows "Integral {a..c} f (x1 + x2)"
+ using assms unfolding integral_unfold apply-
+ apply(rule has_integral_combine[of "vec1 a" "vec1 b" "vec1 c"]) by auto
+
+lemma FTC1: assumes "a \<le> b" "\<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x)"
+ shows "Integral{a..b} f' (f(b) - f(a))"
+proof-note fundamental_theorem_of_calculus[OF assms(1), of"f o dest_vec1" "f' o dest_vec1"]
+ note * = this[unfolded o_def vec1_dest_vec1]
+ have **:"\<And>x. (\<lambda>xa\<Colon>real. xa * f' x) = op * (f' x)" apply(rule ext) by(auto simp add:field_simps)
+ show ?thesis unfolding integral_unfold apply(rule *)
+ using assms(2) unfolding DERIV_conv_has_derivative has_vector_derivative_def
+ apply safe apply(rule has_derivative_at_within) by(auto simp add:**) qed
+
+lemma Integral_subst: "[| Integral{a..b} f k1; k2=k1 |] ==> Integral{a..b} f k2"
+by simp
+
+subsection {* Additivity Theorem of Gauge Integral *}
+
+text{* Bartle/Sherbert: Theorem 10.1.5 p. 278 *}
+lemma Integral_add_fun: "[| Integral{a..b} f k1; Integral{a..b} g k2 |] ==> Integral{a..b} (%x. f x + g x) (k1 + k2)"
+ unfolding integral_unfold apply(rule has_integral_add) by assumption+
+
+lemma norm_vec1'[simp]:"norm (vec1 x) = norm x"
+ using norm_vector_1[of "vec1 x"] by auto
+
+lemma Integral_le: assumes "a \<le> b" "\<forall>x. a \<le> x & x \<le> b --> f(x) \<le> g(x)" "Integral{a..b} f k1" "Integral{a..b} g k2" shows "k1 \<le> k2"
+proof- note assms(3-4)[unfolded integral_unfold] note has_integral_vec1[OF this(1)] has_integral_vec1[OF this(2)]
+ note has_integral_component_le[OF this,of 1] thus ?thesis using assms(2) by auto qed
+
+lemma monotonic_anti_derivative:
+ fixes f g :: "real => real" shows
+ "[| a \<le> b; \<forall>c. a \<le> c & c \<le> b --> f' c \<le> g' c;
+ \<forall>x. DERIV f x :> f' x; \<forall>x. DERIV g x :> g' x |]
+ ==> f b - f a \<le> g b - g a"
+apply (rule Integral_le, assumption)
+apply (auto intro: FTC1)
+done
+
+end
--- a/src/HOL/SEQ.thy Mon Feb 22 20:08:10 2010 +0100
+++ b/src/HOL/SEQ.thy Mon Feb 22 20:41:49 2010 +0100
@@ -435,7 +435,7 @@
lemma LIMSEQ_diff_approach_zero2:
fixes L :: "'a::real_normed_vector"
- shows "f ----> L ==> (%x. f x - g x) ----> 0 ==> g ----> L";
+ shows "f ----> L ==> (%x. f x - g x) ----> 0 ==> g ----> L"
by (drule (1) LIMSEQ_diff, simp)
text{*A sequence tends to zero iff its abs does*}
@@ -1047,6 +1047,17 @@
shows "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
unfolding Cauchy_def dist_norm ..
+lemma Cauchy_iff2:
+ "Cauchy X =
+ (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. \<bar>X m - X n\<bar> < inverse(real (Suc j))))"
+apply (simp add: Cauchy_iff, auto)
+apply (drule reals_Archimedean, safe)
+apply (drule_tac x = n in spec, auto)
+apply (rule_tac x = M in exI, auto)
+apply (drule_tac x = m in spec, simp)
+apply (drule_tac x = na in spec, auto)
+done
+
lemma CauchyI:
fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
shows "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e) \<Longrightarrow> Cauchy X"