# HG changeset patch
# User haftmann
# Date 1234511946 -3600
# Node ID 5170d6277b61c258161667960e3ff18ac6ad164b
# Parent  b8a6b9c56fddc55286c480f8f639b14351185558
removed Reflection session

diff -r b8a6b9c56fdd -r 5170d6277b61 src/HOL/Reflection/Approximation.thy
--- a/src/HOL/Reflection/Approximation.thy	Thu Feb 12 21:24:14 2009 -0800
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,2507 +0,0 @@
-(* Title:     HOL/Reflection/Approximation.thy
- * Author:    Johannes Hölzl <hoelzl@in.tum.de> 2008 / 2009
- *)
-header {* Prove unequations about real numbers by computation *}
-theory Approximation
-imports Complex_Main Float Reflection Efficient_Nat
-begin
-
-section "Horner Scheme"
-
-subsection {* Define auxiliary helper @{text horner} function *}
-
-fun horner :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real \<Rightarrow> real" where
-"horner F G 0 i k x       = 0" |
-"horner F G (Suc n) i k x = 1 / real k - x * horner F G n (F i) (G i k) x"
-
-lemma horner_schema': fixes x :: real  and a :: "nat \<Rightarrow> real"
-  shows "a 0 - x * (\<Sum> i=0..<n. (-1)^i * a (Suc i) * x^i) = (\<Sum> i=0..<Suc n. (-1)^i * a i * x^i)"
-proof -
-  have shift_pow: "\<And>i. - (x * ((-1)^i * a (Suc i) * x ^ i)) = (-1)^(Suc i) * a (Suc i) * x ^ (Suc i)" by auto
-  show ?thesis unfolding setsum_right_distrib shift_pow real_diff_def setsum_negf[symmetric] setsum_head_upt_Suc[OF zero_less_Suc]
-    setsum_reindex[OF inj_Suc, unfolded comp_def, symmetric, of "\<lambda> n. (-1)^n  *a n * x^n"] by auto
-qed
-
-lemma horner_schema: fixes f :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat" and F :: "nat \<Rightarrow> nat"
-  assumes f_Suc: "\<And>n. f (Suc n) = G ((F^n) s) (f n)"
-  shows "horner F G n ((F^j') s) (f j') x = (\<Sum> j = 0..< n. -1^j * (1 / real (f (j' + j))) * x^j)"
-proof (induct n arbitrary: i k j')
-  case (Suc n)
-
-  show ?case unfolding horner.simps Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc]
-    using horner_schema'[of "\<lambda> j. 1 / real (f (j' + j))"] by auto
-qed auto
-
-lemma horner_bounds':
-  assumes "0 \<le> Ifloat x" and f_Suc: "\<And>n. f (Suc n) = G ((F^n) s) (f n)"
-  and lb_0: "\<And> i k x. lb 0 i k x = 0"
-  and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = lapprox_rat prec 1 (int k) - x * (ub n (F i) (G i k) x)"
-  and ub_0: "\<And> i k x. ub 0 i k x = 0"
-  and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = rapprox_rat prec 1 (int k) - x * (lb n (F i) (G i k) x)"
-  shows "Ifloat (lb n ((F^j') s) (f j') x) \<le> horner F G n ((F^j') s) (f j') (Ifloat x) \<and> 
-         horner F G n ((F^j') s) (f j') (Ifloat x) \<le> Ifloat (ub n ((F^j') s) (f j') x)"
-  (is "?lb n j' \<le> ?horner n j' \<and> ?horner n j' \<le> ?ub n j'")
-proof (induct n arbitrary: j')
-  case 0 thus ?case unfolding lb_0 ub_0 horner.simps by auto
-next
-  case (Suc n)
-  have "?lb (Suc n) j' \<le> ?horner (Suc n) j'" unfolding lb_Suc ub_Suc horner.simps Ifloat_sub diff_def
-  proof (rule add_mono)
-    show "Ifloat (lapprox_rat prec 1 (int (f j'))) \<le> 1 / real (f j')" using lapprox_rat[of prec 1  "int (f j')"] by auto
-    from Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc, THEN conjunct2] `0 \<le> Ifloat x`
-    show "- Ifloat (x * ub n (F ((F ^ j') s)) (G ((F ^ j') s) (f j')) x) \<le> - (Ifloat x * horner F G n (F ((F ^ j') s)) (G ((F ^ j') s) (f j')) (Ifloat x))"
-      unfolding Ifloat_mult neg_le_iff_le by (rule mult_left_mono)
-  qed
-  moreover have "?horner (Suc n) j' \<le> ?ub (Suc n) j'" unfolding ub_Suc ub_Suc horner.simps Ifloat_sub diff_def
-  proof (rule add_mono)
-    show "1 / real (f j') \<le> Ifloat (rapprox_rat prec 1 (int (f j')))" using rapprox_rat[of 1 "int (f j')" prec] by auto
-    from Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc, THEN conjunct1] `0 \<le> Ifloat x`
-    show "- (Ifloat x * horner F G n (F ((F ^ j') s)) (G ((F ^ j') s) (f j')) (Ifloat x)) \<le> 
-          - Ifloat (x * lb n (F ((F ^ j') s)) (G ((F ^ j') s) (f j')) x)"
-      unfolding Ifloat_mult neg_le_iff_le by (rule mult_left_mono)
-  qed
-  ultimately show ?case by blast
-qed
-
-subsection "Theorems for floating point functions implementing the horner scheme"
-
-text {*
-
-Here @{term_type "f :: nat \<Rightarrow> nat"} is the sequence defining the Taylor series, the coefficients are
-all alternating and reciprocs. We use @{term G} and @{term F} to describe the computation of @{term f}.
-
-*}
-
-lemma horner_bounds: fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
-  assumes "0 \<le> Ifloat x" and f_Suc: "\<And>n. f (Suc n) = G ((F^n) s) (f n)"
-  and lb_0: "\<And> i k x. lb 0 i k x = 0"
-  and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = lapprox_rat prec 1 (int k) - x * (ub n (F i) (G i k) x)"
-  and ub_0: "\<And> i k x. ub 0 i k x = 0"
-  and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = rapprox_rat prec 1 (int k) - x * (lb n (F i) (G i k) x)"
-  shows "Ifloat (lb n ((F^j') s) (f j') x) \<le> (\<Sum>j=0..<n. -1^j * (1 / real (f (j' + j))) * (Ifloat x)^j)" (is "?lb") and 
-        "(\<Sum>j=0..<n. -1^j * (1 / real (f (j' + j))) * (Ifloat x)^j) \<le> Ifloat (ub n ((F^j') s) (f j') x)" (is "?ub")
-proof -
-  have "?lb  \<and> ?ub" 
-    using horner_bounds'[where lb=lb, OF `0 \<le> Ifloat x` f_Suc lb_0 lb_Suc ub_0 ub_Suc]
-    unfolding horner_schema[where f=f, OF f_Suc] .
-  thus "?lb" and "?ub" by auto
-qed
-
-lemma horner_bounds_nonpos: fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
-  assumes "Ifloat x \<le> 0" and f_Suc: "\<And>n. f (Suc n) = G ((F^n) s) (f n)"
-  and lb_0: "\<And> i k x. lb 0 i k x = 0"
-  and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = lapprox_rat prec 1 (int k) + x * (ub n (F i) (G i k) x)"
-  and ub_0: "\<And> i k x. ub 0 i k x = 0"
-  and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = rapprox_rat prec 1 (int k) + x * (lb n (F i) (G i k) x)"
-  shows "Ifloat (lb n ((F^j') s) (f j') x) \<le> (\<Sum>j=0..<n. (1 / real (f (j' + j))) * (Ifloat x)^j)" (is "?lb") and 
-        "(\<Sum>j=0..<n. (1 / real (f (j' + j))) * (Ifloat x)^j) \<le> Ifloat (ub n ((F^j') s) (f j') x)" (is "?ub")
-proof -
-  { fix x y z :: float have "x - y * z = x + - y * z"
-      by (cases x, cases y, cases z, simp add: plus_float.simps minus_float.simps uminus_float.simps times_float.simps algebra_simps)
-  } note diff_mult_minus = this
-
-  { fix x :: float have "- (- x) = x" by (cases x, auto simp add: uminus_float.simps) } note minus_minus = this
-
-  have move_minus: "Ifloat (-x) = -1 * Ifloat x" by auto
-
-  have sum_eq: "(\<Sum>j=0..<n. (1 / real (f (j' + j))) * (Ifloat x)^j) = 
-    (\<Sum>j = 0..<n. -1 ^ j * (1 / real (f (j' + j))) * Ifloat (- x) ^ j)"
-  proof (rule setsum_cong, simp)
-    fix j assume "j \<in> {0 ..< n}"
-    show "1 / real (f (j' + j)) * Ifloat x ^ j = -1 ^ j * (1 / real (f (j' + j))) * Ifloat (- x) ^ j"
-      unfolding move_minus power_mult_distrib real_mult_assoc[symmetric]
-      unfolding real_mult_commute unfolding real_mult_assoc[of "-1^j", symmetric] power_mult_distrib[symmetric]
-      by auto
-  qed
-
-  have "0 \<le> Ifloat (-x)" using assms by auto
-  from horner_bounds[where G=G and F=F and f=f and s=s and prec=prec
-    and lb="\<lambda> n i k x. lb n i k (-x)" and ub="\<lambda> n i k x. ub n i k (-x)", unfolded lb_Suc ub_Suc diff_mult_minus,
-    OF this f_Suc lb_0 refl ub_0 refl]
-  show "?lb" and "?ub" unfolding minus_minus sum_eq
-    by auto
-qed
-
-subsection {* Selectors for next even or odd number *}
-
-text {*
-
-The horner scheme computes alternating series. To get the upper and lower bounds we need to
-guarantee to access a even or odd member. To do this we use @{term get_odd} and @{term get_even}.
-
-*}
-
-definition get_odd :: "nat \<Rightarrow> nat" where
-  "get_odd n = (if odd n then n else (Suc n))"
-
-definition get_even :: "nat \<Rightarrow> nat" where
-  "get_even n = (if even n then n else (Suc n))"
-
-lemma get_odd[simp]: "odd (get_odd n)" unfolding get_odd_def by (cases "odd n", auto)
-lemma get_even[simp]: "even (get_even n)" unfolding get_even_def by (cases "even n", auto)
-lemma get_odd_ex: "\<exists> k. Suc k = get_odd n \<and> odd (Suc k)"
-proof (cases "odd n")
-  case True hence "0 < n" by (rule odd_pos)
-  from gr0_implies_Suc[OF this] obtain k where "Suc k = n" by auto 
-  thus ?thesis unfolding get_odd_def if_P[OF True] using True[unfolded `Suc k = n`[symmetric]] by blast
-next
-  case False hence "odd (Suc n)" by auto
-  thus ?thesis unfolding get_odd_def if_not_P[OF False] by blast
-qed
-
-lemma get_even_double: "\<exists>i. get_even n = 2 * i" using get_even[unfolded even_mult_two_ex] .
-lemma get_odd_double: "\<exists>i. get_odd n = 2 * i + 1" using get_odd[unfolded odd_Suc_mult_two_ex] by auto
-
-section "Power function"
-
-definition float_power_bnds :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where
-"float_power_bnds n l u = (if odd n \<or> 0 < l then (l ^ n, u ^ n)
-                      else if u < 0         then (u ^ n, l ^ n)
-                                            else (0, (max (-l) u) ^ n))"
-
-lemma float_power_bnds: assumes "(l1, u1) = float_power_bnds n l u" and "x \<in> {Ifloat l .. Ifloat u}"
-  shows "x^n \<in> {Ifloat l1..Ifloat u1}"
-proof (cases "even n")
-  case True 
-  show ?thesis
-  proof (cases "0 < l")
-    case True hence "odd n \<or> 0 < l" and "0 \<le> Ifloat l" unfolding less_float_def by auto
-    have u1: "u1 = u ^ n" and l1: "l1 = l ^ n" using assms unfolding float_power_bnds_def if_P[OF `odd n \<or> 0 < l`] by auto
-    have "Ifloat l^n \<le> x^n" and "x^n \<le> Ifloat u^n " using `0 \<le> Ifloat l` and assms unfolding atLeastAtMost_iff using power_mono[of "Ifloat l" x] power_mono[of x "Ifloat u"] by auto
-    thus ?thesis using assms `0 < l` unfolding atLeastAtMost_iff l1 u1 float_power less_float_def by auto
-  next
-    case False hence P: "\<not> (odd n \<or> 0 < l)" using `even n` by auto
-    show ?thesis
-    proof (cases "u < 0")
-      case True hence "0 \<le> - Ifloat u" and "- Ifloat u \<le> - x" and "0 \<le> - x" and "-x \<le> - Ifloat l" using assms unfolding less_float_def by auto
-      hence "Ifloat u^n \<le> x^n" and "x^n \<le> Ifloat l^n" using power_mono[of  "-x" "-Ifloat l" n] power_mono[of "-Ifloat u" "-x" n] 
-	unfolding power_minus_even[OF `even n`] by auto
-      moreover have u1: "u1 = l ^ n" and l1: "l1 = u ^ n" using assms unfolding float_power_bnds_def if_not_P[OF P] if_P[OF True] by auto
-      ultimately show ?thesis using float_power by auto
-    next
-      case False 
-      have "\<bar>x\<bar> \<le> Ifloat (max (-l) u)"
-      proof (cases "-l \<le> u")
-	case True thus ?thesis unfolding max_def if_P[OF True] using assms unfolding le_float_def by auto
-      next
-	case False thus ?thesis unfolding max_def if_not_P[OF False] using assms unfolding le_float_def by auto
-      qed
-      hence x_abs: "\<bar>x\<bar> \<le> \<bar>Ifloat (max (-l) u)\<bar>" by auto
-      have u1: "u1 = (max (-l) u) ^ n" and l1: "l1 = 0" using assms unfolding float_power_bnds_def if_not_P[OF P] if_not_P[OF False] by auto
-      show ?thesis unfolding atLeastAtMost_iff l1 u1 float_power using zero_le_even_power[OF `even n`] power_mono_even[OF `even n` x_abs] by auto
-    qed
-  qed
-next
-  case False hence "odd n \<or> 0 < l" by auto
-  have u1: "u1 = u ^ n" and l1: "l1 = l ^ n" using assms unfolding float_power_bnds_def if_P[OF `odd n \<or> 0 < l`] by auto
-  have "Ifloat l^n \<le> x^n" and "x^n \<le> Ifloat u^n " using assms unfolding atLeastAtMost_iff using power_mono_odd[OF False] by auto
-  thus ?thesis unfolding atLeastAtMost_iff l1 u1 float_power less_float_def by auto
-qed
-
-lemma bnds_power: "\<forall> x l u. (l1, u1) = float_power_bnds n l u \<and> x \<in> {Ifloat l .. Ifloat u} \<longrightarrow> Ifloat l1 \<le> x^n \<and> x^n \<le> Ifloat u1"
-  using float_power_bnds by auto
-
-section "Square root"
-
-text {*
-
-The square root computation is implemented as newton iteration. As first first step we use the
-nearest power of two greater than the square root.
-
-*}
-
-fun sqrt_iteration :: "nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
-"sqrt_iteration prec 0 (Float m e) = Float 1 ((e + bitlen m) div 2 + 1)" |
-"sqrt_iteration prec (Suc m) x = (let y = sqrt_iteration prec m x 
-                                  in Float 1 -1 * (y + float_divr prec x y))"
-
-definition ub_sqrt :: "nat \<Rightarrow> float \<Rightarrow> float option" where 
-"ub_sqrt prec x = (if 0 < x then Some (sqrt_iteration prec prec x) else if x < 0 then None else Some 0)"
-
-definition lb_sqrt :: "nat \<Rightarrow> float \<Rightarrow> float option" where
-"lb_sqrt prec x = (if 0 < x then Some (float_divl prec x (sqrt_iteration prec prec x)) else if x < 0 then None else Some 0)"
-
-lemma sqrt_ub_pos_pos_1:
-  assumes "sqrt x < b" and "0 < b" and "0 < x"
-  shows "sqrt x < (b + x / b)/2"
-proof -
-  from assms have "0 < (b - sqrt x) ^ 2 " by simp
-  also have "\<dots> = b ^ 2 - 2 * b * sqrt x + (sqrt x) ^ 2" by algebra
-  also have "\<dots> = b ^ 2 - 2 * b * sqrt x + x" using assms by (simp add: real_sqrt_pow2)
-  finally have "0 < b ^ 2 - 2 * b * sqrt x + x" by assumption
-  hence "0 < b / 2 - sqrt x + x / (2 * b)" using assms
-    by (simp add: field_simps power2_eq_square)
-  thus ?thesis by (simp add: field_simps)
-qed
-
-lemma sqrt_iteration_bound: assumes "0 < Ifloat x"
-  shows "sqrt (Ifloat x) < Ifloat (sqrt_iteration prec n x)"
-proof (induct n)
-  case 0
-  show ?case
-  proof (cases x)
-    case (Float m e)
-    hence "0 < m" using float_pos_m_pos[unfolded less_float_def] assms by auto
-    hence "0 < sqrt (real m)" by auto
-
-    have int_nat_bl: "int (nat (bitlen m)) = bitlen m" using bitlen_ge0 by auto
-
-    have "Ifloat x = (real m / 2^nat (bitlen m)) * pow2 (e + int (nat (bitlen m)))"
-      unfolding pow2_add pow2_int Float Ifloat.simps by auto
-    also have "\<dots> < 1 * pow2 (e + int (nat (bitlen m)))"
-    proof (rule mult_strict_right_mono, auto)
-      show "real m < 2^nat (bitlen m)" using bitlen_bounds[OF `0 < m`, THEN conjunct2] 
-	unfolding real_of_int_less_iff[of m, symmetric] by auto
-    qed
-    finally have "sqrt (Ifloat x) < sqrt (pow2 (e + bitlen m))" unfolding int_nat_bl by auto
-    also have "\<dots> \<le> pow2 ((e + bitlen m) div 2 + 1)"
-    proof -
-      let ?E = "e + bitlen m"
-      have E_mod_pow: "pow2 (?E mod 2) < 4"
-      proof (cases "?E mod 2 = 1")
-	case True thus ?thesis by auto
-      next
-	case False 
-	have "0 \<le> ?E mod 2" by auto 
-	have "?E mod 2 < 2" by auto
-	from this[THEN zless_imp_add1_zle]
-	have "?E mod 2 \<le> 0" using False by auto
-	from xt1(5)[OF `0 \<le> ?E mod 2` this]
-	show ?thesis by auto
-      qed
-      hence "sqrt (pow2 (?E mod 2)) < sqrt (2 * 2)" by auto
-      hence E_mod_pow: "sqrt (pow2 (?E mod 2)) < 2" unfolding real_sqrt_abs2 by auto
-
-      have E_eq: "pow2 ?E = pow2 (?E div 2 + ?E div 2 + ?E mod 2)" by auto
-      have "sqrt (pow2 ?E) = sqrt (pow2 (?E div 2) * pow2 (?E div 2) * pow2 (?E mod 2))"
-	unfolding E_eq unfolding pow2_add ..
-      also have "\<dots> = pow2 (?E div 2) * sqrt (pow2 (?E mod 2))"
-	unfolding real_sqrt_mult[of _ "pow2 (?E mod 2)"] real_sqrt_abs2 by auto
-      also have "\<dots> < pow2 (?E div 2) * 2" 
-	by (rule mult_strict_left_mono, auto intro: E_mod_pow)
-      also have "\<dots> = pow2 (?E div 2 + 1)" unfolding zadd_commute[of _ 1] pow2_add1 by auto
-      finally show ?thesis by auto
-    qed
-    finally show ?thesis 
-      unfolding Float sqrt_iteration.simps Ifloat.simps by auto
-  qed
-next
-  case (Suc n)
-  let ?b = "sqrt_iteration prec n x"
-  have "0 < sqrt (Ifloat x)" using `0 < Ifloat x` by auto
-  also have "\<dots> < Ifloat ?b" using Suc .
-  finally have "sqrt (Ifloat x) < (Ifloat ?b + Ifloat x / Ifloat ?b)/2" using sqrt_ub_pos_pos_1[OF Suc _ `0 < Ifloat x`] by auto
-  also have "\<dots> \<le> (Ifloat ?b + Ifloat (float_divr prec x ?b))/2" by (rule divide_right_mono, auto simp add: float_divr)
-  also have "\<dots> = Ifloat (Float 1 -1) * (Ifloat ?b + Ifloat (float_divr prec x ?b))" by auto
-  finally show ?case unfolding sqrt_iteration.simps Let_def Ifloat_mult Ifloat_add right_distrib .
-qed
-
-lemma sqrt_iteration_lower_bound: assumes "0 < Ifloat x"
-  shows "0 < Ifloat (sqrt_iteration prec n x)" (is "0 < ?sqrt")
-proof -
-  have "0 < sqrt (Ifloat x)" using assms by auto
-  also have "\<dots> < ?sqrt" using sqrt_iteration_bound[OF assms] .
-  finally show ?thesis .
-qed
-
-lemma lb_sqrt_lower_bound: assumes "0 \<le> Ifloat x"
-  shows "0 \<le> Ifloat (the (lb_sqrt prec x))"
-proof (cases "0 < x")
-  case True hence "0 < Ifloat x" and "0 \<le> x" using `0 \<le> Ifloat x` unfolding less_float_def le_float_def by auto
-  hence "0 < sqrt_iteration prec prec x" unfolding less_float_def using sqrt_iteration_lower_bound by auto 
-  hence "0 \<le> Ifloat (float_divl prec x (sqrt_iteration prec prec x))" using float_divl_lower_bound[OF `0 \<le> x`] unfolding le_float_def by auto
-  thus ?thesis unfolding lb_sqrt_def using True by auto
-next
-  case False with `0 \<le> Ifloat x` have "Ifloat x = 0" unfolding less_float_def by auto
-  thus ?thesis unfolding lb_sqrt_def less_float_def by auto
-qed
-
-lemma lb_sqrt_upper_bound: assumes "0 \<le> Ifloat x"
-  shows "Ifloat (the (lb_sqrt prec x)) \<le> sqrt (Ifloat x)"
-proof (cases "0 < x")
-  case True hence "0 < Ifloat x" and "0 \<le> Ifloat x" unfolding less_float_def by auto
-  hence sqrt_gt0: "0 < sqrt (Ifloat x)" by auto
-  hence sqrt_ub: "sqrt (Ifloat x) < Ifloat (sqrt_iteration prec prec x)" using sqrt_iteration_bound by auto
-  
-  have "Ifloat (float_divl prec x (sqrt_iteration prec prec x)) \<le> Ifloat x / Ifloat (sqrt_iteration prec prec x)" by (rule float_divl)
-  also have "\<dots> < Ifloat x / sqrt (Ifloat x)" 
-    by (rule divide_strict_left_mono[OF sqrt_ub `0 < Ifloat x` mult_pos_pos[OF order_less_trans[OF sqrt_gt0 sqrt_ub] sqrt_gt0]])
-  also have "\<dots> = sqrt (Ifloat x)" unfolding inverse_eq_iff_eq[of _ "sqrt (Ifloat x)", symmetric] sqrt_divide_self_eq[OF `0 \<le> Ifloat x`, symmetric] by auto
-  finally show ?thesis unfolding lb_sqrt_def if_P[OF `0 < x`] by auto
-next
-  case False with `0 \<le> Ifloat x`
-  have "\<not> x < 0" unfolding less_float_def le_float_def by auto
-  show ?thesis unfolding lb_sqrt_def if_not_P[OF False] if_not_P[OF `\<not> x < 0`] using assms by auto
-qed
-
-lemma lb_sqrt: assumes "Some y = lb_sqrt prec x"
-  shows "Ifloat y \<le> sqrt (Ifloat x)" and "0 \<le> Ifloat x"
-proof -
-  show "0 \<le> Ifloat x"
-  proof (rule ccontr)
-    assume "\<not> 0 \<le> Ifloat x"
-    hence "lb_sqrt prec x = None" unfolding lb_sqrt_def less_float_def by auto
-    thus False using assms by auto
-  qed
-  from lb_sqrt_upper_bound[OF this, of prec]
-  show "Ifloat y \<le> sqrt (Ifloat x)" unfolding assms[symmetric] by auto
-qed
-
-lemma ub_sqrt_lower_bound: assumes "0 \<le> Ifloat x"
-  shows "sqrt (Ifloat x) \<le> Ifloat (the (ub_sqrt prec x))"
-proof (cases "0 < x")
-  case True hence "0 < Ifloat x" unfolding less_float_def by auto
-  hence "0 < sqrt (Ifloat x)" by auto
-  hence "sqrt (Ifloat x) < Ifloat (sqrt_iteration prec prec x)" using sqrt_iteration_bound by auto
-  thus ?thesis unfolding ub_sqrt_def if_P[OF `0 < x`] by auto
-next
-  case False with `0 \<le> Ifloat x`
-  have "Ifloat x = 0" unfolding less_float_def le_float_def by auto
-  thus ?thesis unfolding ub_sqrt_def less_float_def le_float_def by auto
-qed
-
-lemma ub_sqrt: assumes "Some y = ub_sqrt prec x"
-  shows "sqrt (Ifloat x) \<le> Ifloat y" and "0 \<le> Ifloat x"
-proof -
-  show "0 \<le> Ifloat x"
-  proof (rule ccontr)
-    assume "\<not> 0 \<le> Ifloat x"
-    hence "ub_sqrt prec x = None" unfolding ub_sqrt_def less_float_def by auto
-    thus False using assms by auto
-  qed
-  from ub_sqrt_lower_bound[OF this, of prec]
-  show "sqrt (Ifloat x) \<le> Ifloat y" unfolding assms[symmetric] by auto
-qed
-
-lemma bnds_sqrt: "\<forall> x lx ux. (Some l, Some u) = (lb_sqrt prec lx, ub_sqrt prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux} \<longrightarrow> Ifloat l \<le> sqrt x \<and> sqrt x \<le> Ifloat u"
-proof (rule allI, rule allI, rule allI, rule impI)
-  fix x lx ux
-  assume "(Some l, Some u) = (lb_sqrt prec lx, ub_sqrt prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux}"
-  hence l: "Some l = lb_sqrt prec lx " and u: "Some u = ub_sqrt prec ux" and x: "x \<in> {Ifloat lx .. Ifloat ux}" by auto
-  
-  have "Ifloat lx \<le> x" and "x \<le> Ifloat ux" using x by auto
-
-  from lb_sqrt(1)[OF l] real_sqrt_le_mono[OF `Ifloat lx \<le> x`]
-  have "Ifloat l \<le> sqrt x" by (rule order_trans)
-  moreover
-  from real_sqrt_le_mono[OF `x \<le> Ifloat ux`] ub_sqrt(1)[OF u]
-  have "sqrt x \<le> Ifloat u" by (rule order_trans)
-  ultimately show "Ifloat l \<le> sqrt x \<and> sqrt x \<le> Ifloat u" ..
-qed
-
-section "Arcus tangens and \<pi>"
-
-subsection "Compute arcus tangens series"
-
-text {*
-
-As first step we implement the computation of the arcus tangens series. This is only valid in the range
-@{term "{-1 :: real .. 1}"}. This is used to compute \<pi> and then the entire arcus tangens.
-
-*}
-
-fun ub_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
-and lb_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
-  "ub_arctan_horner prec 0 k x = 0"
-| "ub_arctan_horner prec (Suc n) k x = 
-    (rapprox_rat prec 1 (int k)) - x * (lb_arctan_horner prec n (k + 2) x)"
-| "lb_arctan_horner prec 0 k x = 0"
-| "lb_arctan_horner prec (Suc n) k x = 
-    (lapprox_rat prec 1 (int k)) - x * (ub_arctan_horner prec n (k + 2) x)"
-
-lemma arctan_0_1_bounds': assumes "0 \<le> Ifloat x" "Ifloat x \<le> 1" and "even n"
-  shows "arctan (Ifloat x) \<in> {Ifloat (x * lb_arctan_horner prec n 1 (x * x)) .. Ifloat (x * ub_arctan_horner prec (Suc n) 1 (x * x))}"
-proof -
-  let "?c i" = "-1^i * (1 / real (i * 2 + 1) * Ifloat x ^ (i * 2 + 1))"
-  let "?S n" = "\<Sum> i=0..<n. ?c i"
-
-  have "0 \<le> Ifloat (x * x)" by auto
-  from `even n` obtain m where "2 * m = n" unfolding even_mult_two_ex by auto
-  
-  have "arctan (Ifloat x) \<in> { ?S n .. ?S (Suc n) }"
-  proof (cases "Ifloat x = 0")
-    case False
-    hence "0 < Ifloat x" using `0 \<le> Ifloat x` by auto
-    hence prem: "0 < 1 / real (0 * 2 + (1::nat)) * Ifloat x ^ (0 * 2 + 1)" by auto 
-
-    have "\<bar> Ifloat x \<bar> \<le> 1"  using `0 \<le> Ifloat x` `Ifloat x \<le> 1` by auto
-    from mp[OF summable_Leibniz(2)[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]] prem, THEN spec, of m, unfolded `2 * m = n`]
-    show ?thesis unfolding arctan_series[OF `\<bar> Ifloat x \<bar> \<le> 1`] Suc_plus1  .
-  qed auto
-  note arctan_bounds = this[unfolded atLeastAtMost_iff]
-
-  have F: "\<And>n. 2 * Suc n + 1 = 2 * n + 1 + 2" by auto
-
-  note bounds = horner_bounds[where s=1 and f="\<lambda>i. 2 * i + 1" and j'=0 
-    and lb="\<lambda>n i k x. lb_arctan_horner prec n k x"
-    and ub="\<lambda>n i k x. ub_arctan_horner prec n k x", 
-    OF `0 \<le> Ifloat (x*x)` F lb_arctan_horner.simps ub_arctan_horner.simps]
-
-  { have "Ifloat (x * lb_arctan_horner prec n 1 (x*x)) \<le> ?S n"
-      using bounds(1) `0 \<le> Ifloat x`
-      unfolding Ifloat_mult power_add power_one_right real_mult_assoc[symmetric] setsum_left_distrib[symmetric]
-      unfolding real_mult_commute mult_commute[of _ "2::nat"] power_mult power2_eq_square[of "Ifloat x"]
-      by (auto intro!: mult_left_mono)
-    also have "\<dots> \<le> arctan (Ifloat x)" using arctan_bounds ..
-    finally have "Ifloat (x * lb_arctan_horner prec n 1 (x*x)) \<le> arctan (Ifloat x)" . }
-  moreover
-  { have "arctan (Ifloat x) \<le> ?S (Suc n)" using arctan_bounds ..
-    also have "\<dots> \<le> Ifloat (x * ub_arctan_horner prec (Suc n) 1 (x*x))"
-      using bounds(2)[of "Suc n"] `0 \<le> Ifloat x`
-      unfolding Ifloat_mult power_add power_one_right real_mult_assoc[symmetric] setsum_left_distrib[symmetric]
-      unfolding real_mult_commute mult_commute[of _ "2::nat"] power_mult power2_eq_square[of "Ifloat x"]
-      by (auto intro!: mult_left_mono)
-    finally have "arctan (Ifloat x) \<le> Ifloat (x * ub_arctan_horner prec (Suc n) 1 (x*x))" . }
-  ultimately show ?thesis by auto
-qed
-
-lemma arctan_0_1_bounds: assumes "0 \<le> Ifloat x" "Ifloat x \<le> 1"
-  shows "arctan (Ifloat x) \<in> {Ifloat (x * lb_arctan_horner prec (get_even n) 1 (x * x)) .. Ifloat (x * ub_arctan_horner prec (get_odd n) 1 (x * x))}"
-proof (cases "even n")
-  case True
-  obtain n' where "Suc n' = get_odd n" and "odd (Suc n')" using get_odd_ex by auto
-  hence "even n'" unfolding even_nat_Suc by auto
-  have "arctan (Ifloat x) \<le> Ifloat (x * ub_arctan_horner prec (get_odd n) 1 (x * x))"
-    unfolding `Suc n' = get_odd n`[symmetric] using arctan_0_1_bounds'[OF `0 \<le> Ifloat x` `Ifloat x \<le> 1` `even n'`] by auto
-  moreover
-  have "Ifloat (x * lb_arctan_horner prec (get_even n) 1 (x * x)) \<le> arctan (Ifloat x)"
-    unfolding get_even_def if_P[OF True] using arctan_0_1_bounds'[OF `0 \<le> Ifloat x` `Ifloat x \<le> 1` `even n`] by auto
-  ultimately show ?thesis by auto
-next
-  case False hence "0 < n" by (rule odd_pos)
-  from gr0_implies_Suc[OF this] obtain n' where "n = Suc n'" ..
-  from False[unfolded this even_nat_Suc]
-  have "even n'" and "even (Suc (Suc n'))" by auto
-  have "get_odd n = Suc n'" unfolding get_odd_def if_P[OF False] using `n = Suc n'` .
-
-  have "arctan (Ifloat x) \<le> Ifloat (x * ub_arctan_horner prec (get_odd n) 1 (x * x))"
-    unfolding `get_odd n = Suc n'` using arctan_0_1_bounds'[OF `0 \<le> Ifloat x` `Ifloat x \<le> 1` `even n'`] by auto
-  moreover
-  have "Ifloat (x * lb_arctan_horner prec (get_even n) 1 (x * x)) \<le> arctan (Ifloat x)"
-    unfolding get_even_def if_not_P[OF False] unfolding `n = Suc n'` using arctan_0_1_bounds'[OF `0 \<le> Ifloat x` `Ifloat x \<le> 1` `even (Suc (Suc n'))`] by auto
-  ultimately show ?thesis by auto
-qed
-
-subsection "Compute \<pi>"
-
-definition ub_pi :: "nat \<Rightarrow> float" where
-  "ub_pi prec = (let A = rapprox_rat prec 1 5 ; 
-                     B = lapprox_rat prec 1 239
-                 in ((Float 1 2) * ((Float 1 2) * A * (ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (A * A)) - 
-                                                  B * (lb_arctan_horner prec (get_even (prec div 14 + 1)) 1 (B * B)))))"
-
-definition lb_pi :: "nat \<Rightarrow> float" where
-  "lb_pi prec = (let A = lapprox_rat prec 1 5 ; 
-                     B = rapprox_rat prec 1 239
-                 in ((Float 1 2) * ((Float 1 2) * A * (lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (A * A)) - 
-                                                  B * (ub_arctan_horner prec (get_odd (prec div 14 + 1)) 1 (B * B)))))"
-
-lemma pi_boundaries: "pi \<in> {Ifloat (lb_pi n) .. Ifloat (ub_pi n)}"
-proof -
-  have machin_pi: "pi = 4 * (4 * arctan (1 / 5) - arctan (1 / 239))" unfolding machin[symmetric] by auto
-
-  { fix prec n :: nat fix k :: int assume "1 < k" hence "0 \<le> k" and "0 < k" and "1 \<le> k" by auto
-    let ?k = "rapprox_rat prec 1 k"
-    have "1 div k = 0" using div_pos_pos_trivial[OF _ `1 < k`] by auto
-      
-    have "0 \<le> Ifloat ?k" by (rule order_trans[OF _ rapprox_rat], auto simp add: `0 \<le> k`)
-    have "Ifloat ?k \<le> 1" unfolding rapprox_rat.simps(2)[OF zero_le_one `0 < k`]
-      by (rule rapprox_posrat_le1, auto simp add: `0 < k` `1 \<le> k`)
-
-    have "1 / real k \<le> Ifloat ?k" using rapprox_rat[where x=1 and y=k] by auto
-    hence "arctan (1 / real k) \<le> arctan (Ifloat ?k)" by (rule arctan_monotone')
-    also have "\<dots> \<le> Ifloat (?k * ub_arctan_horner prec (get_odd n) 1 (?k * ?k))"
-      using arctan_0_1_bounds[OF `0 \<le> Ifloat ?k` `Ifloat ?k \<le> 1`] by auto
-    finally have "arctan (1 / (real k)) \<le> Ifloat (?k * ub_arctan_horner prec (get_odd n) 1 (?k * ?k))" .
-  } note ub_arctan = this
-
-  { fix prec n :: nat fix k :: int assume "1 < k" hence "0 \<le> k" and "0 < k" by auto
-    let ?k = "lapprox_rat prec 1 k"
-    have "1 div k = 0" using div_pos_pos_trivial[OF _ `1 < k`] by auto
-    have "1 / real k \<le> 1" using `1 < k` by auto
-
-    have "\<And>n. 0 \<le> Ifloat ?k" using lapprox_rat_bottom[where x=1 and y=k, OF zero_le_one `0 < k`] by (auto simp add: `1 div k = 0`)
-    have "\<And>n. Ifloat ?k \<le> 1" using lapprox_rat by (rule order_trans, auto simp add: `1 / real k \<le> 1`)
-
-    have "Ifloat ?k \<le> 1 / real k" using lapprox_rat[where x=1 and y=k] by auto
-
-    have "Ifloat (?k * lb_arctan_horner prec (get_even n) 1 (?k * ?k)) \<le> arctan (Ifloat ?k)"
-      using arctan_0_1_bounds[OF `0 \<le> Ifloat ?k` `Ifloat ?k \<le> 1`] by auto
-    also have "\<dots> \<le> arctan (1 / real k)" using `Ifloat ?k \<le> 1 / real k` by (rule arctan_monotone')
-    finally have "Ifloat (?k * lb_arctan_horner prec (get_even n) 1 (?k * ?k)) \<le> arctan (1 / (real k))" .
-  } note lb_arctan = this
-
-  have "pi \<le> Ifloat (ub_pi n)"
-    unfolding ub_pi_def machin_pi Let_def Ifloat_mult Ifloat_sub unfolding Float_num
-    using lb_arctan[of 239] ub_arctan[of 5]
-    by (auto intro!: mult_left_mono add_mono simp add: diff_minus simp del: lapprox_rat.simps rapprox_rat.simps)
-  moreover
-  have "Ifloat (lb_pi n) \<le> pi"
-    unfolding lb_pi_def machin_pi Let_def Ifloat_mult Ifloat_sub Float_num
-    using lb_arctan[of 5] ub_arctan[of 239]
-    by (auto intro!: mult_left_mono add_mono simp add: diff_minus simp del: lapprox_rat.simps rapprox_rat.simps)
-  ultimately show ?thesis by auto
-qed
-
-subsection "Compute arcus tangens in the entire domain"
-
-function lb_arctan :: "nat \<Rightarrow> float \<Rightarrow> float" and ub_arctan :: "nat \<Rightarrow> float \<Rightarrow> float" where 
-  "lb_arctan prec x = (let ub_horner = \<lambda> x. x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x) ;
-                           lb_horner = \<lambda> x. x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)
-    in (if x < 0          then - ub_arctan prec (-x) else
-        if x \<le> Float 1 -1 then lb_horner x else
-        if x \<le> Float 1 1  then Float 1 1 * lb_horner (float_divl prec x (1 + the (ub_sqrt prec (1 + x * x))))
-                          else (let inv = float_divr prec 1 x 
-                                in if inv > 1 then 0 
-                                              else lb_pi prec * Float 1 -1 - ub_horner inv)))"
-
-| "ub_arctan prec x = (let lb_horner = \<lambda> x. x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x) ;
-                           ub_horner = \<lambda> x. x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)
-    in (if x < 0          then - lb_arctan prec (-x) else
-        if x \<le> Float 1 -1 then ub_horner x else
-        if x \<le> Float 1 1  then let y = float_divr prec x (1 + the (lb_sqrt prec (1 + x * x)))
-                               in if y > 1 then ub_pi prec * Float 1 -1 
-                                           else Float 1 1 * ub_horner y 
-                          else ub_pi prec * Float 1 -1 - lb_horner (float_divl prec 1 x)))"
-by pat_completeness auto
-termination by (relation "measure (\<lambda> v. let (prec, x) = sum_case id id v in (if x < 0 then 1 else 0))", auto simp add: less_float_def)
-
-declare ub_arctan_horner.simps[simp del]
-declare lb_arctan_horner.simps[simp del]
-
-lemma lb_arctan_bound': assumes "0 \<le> Ifloat x"
-  shows "Ifloat (lb_arctan prec x) \<le> arctan (Ifloat x)"
-proof -
-  have "\<not> x < 0" and "0 \<le> x" unfolding less_float_def le_float_def using `0 \<le> Ifloat x` by auto
-  let "?ub_horner x" = "x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)"
-    and "?lb_horner x" = "x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)"
-
-  show ?thesis
-  proof (cases "x \<le> Float 1 -1")
-    case True hence "Ifloat x \<le> 1" unfolding le_float_def Float_num by auto
-    show ?thesis unfolding lb_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_P[OF True]
-      using arctan_0_1_bounds[OF `0 \<le> Ifloat x` `Ifloat x \<le> 1`] by auto
-  next
-    case False hence "0 < Ifloat x" unfolding le_float_def Float_num by auto
-    let ?R = "1 + sqrt (1 + Ifloat x * Ifloat x)"
-    let ?fR = "1 + the (ub_sqrt prec (1 + x * x))"
-    let ?DIV = "float_divl prec x ?fR"
-    
-    have sqr_ge0: "0 \<le> 1 + Ifloat x * Ifloat x" using sum_power2_ge_zero[of 1 "Ifloat x", unfolded numeral_2_eq_2] by auto
-    hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg)
-
-    have "sqrt (Ifloat (1 + x * x)) \<le> Ifloat (the (ub_sqrt prec (1 + x * x)))" by (rule ub_sqrt_lower_bound, auto simp add: sqr_ge0)
-    hence "?R \<le> Ifloat ?fR" by auto
-    hence "0 < ?fR" and "0 < Ifloat ?fR" unfolding less_float_def using `0 < ?R` by auto
-
-    have monotone: "Ifloat (float_divl prec x ?fR) \<le> Ifloat x / ?R"
-    proof -
-      have "Ifloat ?DIV \<le> Ifloat x / Ifloat ?fR" by (rule float_divl)
-      also have "\<dots> \<le> Ifloat x / ?R" by (rule divide_left_mono[OF `?R \<le> Ifloat ?fR` `0 \<le> Ifloat x` mult_pos_pos[OF order_less_le_trans[OF divisor_gt0 `?R \<le> Ifloat ?fR`] divisor_gt0]])
-      finally show ?thesis .
-    qed
-
-    show ?thesis
-    proof (cases "x \<le> Float 1 1")
-      case True
-      
-      have "Ifloat x \<le> sqrt (Ifloat (1 + x * x))" using real_sqrt_sum_squares_ge2[where x=1, unfolded numeral_2_eq_2] by auto
-      also have "\<dots> \<le> Ifloat (the (ub_sqrt prec (1 + x * x)))" by (rule ub_sqrt_lower_bound, auto simp add: sqr_ge0)
-      finally have "Ifloat x \<le> Ifloat ?fR" by auto
-      moreover have "Ifloat ?DIV \<le> Ifloat x / Ifloat ?fR" by (rule float_divl)
-      ultimately have "Ifloat ?DIV \<le> 1" unfolding divide_le_eq_1_pos[OF `0 < Ifloat ?fR`, symmetric] by auto
-
-      have "0 \<le> Ifloat ?DIV" using float_divl_lower_bound[OF `0 \<le> x` `0 < ?fR`] unfolding le_float_def by auto
-
-      have "Ifloat (Float 1 1 * ?lb_horner ?DIV) \<le> 2 * arctan (Ifloat (float_divl prec x ?fR))" unfolding Ifloat_mult[of "Float 1 1"] Float_num
-	using arctan_0_1_bounds[OF `0 \<le> Ifloat ?DIV` `Ifloat ?DIV \<le> 1`] by auto
-      also have "\<dots> \<le> 2 * arctan (Ifloat x / ?R)"
-	using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono)
-      also have "2 * arctan (Ifloat x / ?R) = arctan (Ifloat x)" using arctan_half[symmetric] unfolding numeral_2_eq_2 power_Suc2 realpow_0 real_mult_1 . 
-      finally show ?thesis unfolding lb_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_P[OF True] .
-    next
-      case False
-      hence "2 < Ifloat x" unfolding le_float_def Float_num by auto
-      hence "1 \<le> Ifloat x" by auto
-
-      let "?invx" = "float_divr prec 1 x"
-      have "0 \<le> arctan (Ifloat x)" using arctan_monotone'[OF `0 \<le> Ifloat x`] using arctan_tan[of 0, unfolded tan_zero] by auto
-
-      show ?thesis
-      proof (cases "1 < ?invx")
-	case True
-	show ?thesis unfolding lb_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_not_P[OF False] if_P[OF True] 
-	  using `0 \<le> arctan (Ifloat x)` by auto
-      next
-	case False
-	hence "Ifloat ?invx \<le> 1" unfolding less_float_def by auto
-	have "0 \<le> Ifloat ?invx" by (rule order_trans[OF _ float_divr], auto simp add: `0 \<le> Ifloat x`)
-
-	have "1 / Ifloat x \<noteq> 0" and "0 < 1 / Ifloat x" using `0 < Ifloat x` by auto
-	
-	have "arctan (1 / Ifloat x) \<le> arctan (Ifloat ?invx)" unfolding Ifloat_1[symmetric] by (rule arctan_monotone', rule float_divr)
-	also have "\<dots> \<le> Ifloat (?ub_horner ?invx)" using arctan_0_1_bounds[OF `0 \<le> Ifloat ?invx` `Ifloat ?invx \<le> 1`] by auto
-	finally have "pi / 2 - Ifloat (?ub_horner ?invx) \<le> arctan (Ifloat x)" 
-	  using `0 \<le> arctan (Ifloat x)` arctan_inverse[OF `1 / Ifloat x \<noteq> 0`] 
-	  unfolding real_sgn_pos[OF `0 < 1 / Ifloat x`] le_diff_eq by auto
-	moreover
-	have "Ifloat (lb_pi prec * Float 1 -1) \<le> pi / 2" unfolding Ifloat_mult Float_num times_divide_eq_right real_mult_1 using pi_boundaries by auto
-	ultimately
-	show ?thesis unfolding lb_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_not_P[OF `\<not> x \<le> Float 1 1`] if_not_P[OF False]
-	  by auto
-      qed
-    qed
-  qed
-qed
-
-lemma ub_arctan_bound': assumes "0 \<le> Ifloat x"
-  shows "arctan (Ifloat x) \<le> Ifloat (ub_arctan prec x)"
-proof -
-  have "\<not> x < 0" and "0 \<le> x" unfolding less_float_def le_float_def using `0 \<le> Ifloat x` by auto
-
-  let "?ub_horner x" = "x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)"
-    and "?lb_horner x" = "x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)"
-
-  show ?thesis
-  proof (cases "x \<le> Float 1 -1")
-    case True hence "Ifloat x \<le> 1" unfolding le_float_def Float_num by auto
-    show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_P[OF True]
-      using arctan_0_1_bounds[OF `0 \<le> Ifloat x` `Ifloat x \<le> 1`] by auto
-  next
-    case False hence "0 < Ifloat x" unfolding le_float_def Float_num by auto
-    let ?R = "1 + sqrt (1 + Ifloat x * Ifloat x)"
-    let ?fR = "1 + the (lb_sqrt prec (1 + x * x))"
-    let ?DIV = "float_divr prec x ?fR"
-    
-    have sqr_ge0: "0 \<le> 1 + Ifloat x * Ifloat x" using sum_power2_ge_zero[of 1 "Ifloat x", unfolded numeral_2_eq_2] by auto
-    hence "0 \<le> Ifloat (1 + x*x)" by auto
-    
-    hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg)
-
-    have "Ifloat (the (lb_sqrt prec (1 + x * x))) \<le> sqrt (Ifloat (1 + x * x))" by (rule lb_sqrt_upper_bound, auto simp add: sqr_ge0)
-    hence "Ifloat ?fR \<le> ?R" by auto
-    have "0 < Ifloat ?fR" unfolding Ifloat_add Ifloat_1 by (rule order_less_le_trans[OF zero_less_one], auto simp add: lb_sqrt_lower_bound[OF `0 \<le> Ifloat (1 + x*x)`])
-
-    have monotone: "Ifloat x / ?R \<le> Ifloat (float_divr prec x ?fR)"
-    proof -
-      from divide_left_mono[OF `Ifloat ?fR \<le> ?R` `0 \<le> Ifloat x` mult_pos_pos[OF divisor_gt0 `0 < Ifloat ?fR`]]
-      have "Ifloat x / ?R \<le> Ifloat x / Ifloat ?fR" .
-      also have "\<dots> \<le> Ifloat ?DIV" by (rule float_divr)
-      finally show ?thesis .
-    qed
-
-    show ?thesis
-    proof (cases "x \<le> Float 1 1")
-      case True
-      show ?thesis
-      proof (cases "?DIV > 1")
-	case True
-	have "pi / 2 \<le> Ifloat (ub_pi prec * Float 1 -1)" unfolding Ifloat_mult Float_num times_divide_eq_right real_mult_1 using pi_boundaries by auto
-	from order_less_le_trans[OF arctan_ubound this, THEN less_imp_le]
-	show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_P[OF `x \<le> Float 1 1`] if_P[OF True] .
-      next
-	case False
-	hence "Ifloat ?DIV \<le> 1" unfolding less_float_def by auto
-      
-	have "0 \<le> Ifloat x / ?R" using `0 \<le> Ifloat x` `0 < ?R` unfolding real_0_le_divide_iff by auto
-	hence "0 \<le> Ifloat ?DIV" using monotone by (rule order_trans)
-
-	have "arctan (Ifloat x) = 2 * arctan (Ifloat x / ?R)" using arctan_half unfolding numeral_2_eq_2 power_Suc2 realpow_0 real_mult_1 .
-	also have "\<dots> \<le> 2 * arctan (Ifloat ?DIV)"
-	  using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono)
-	also have "\<dots> \<le> Ifloat (Float 1 1 * ?ub_horner ?DIV)" unfolding Ifloat_mult[of "Float 1 1"] Float_num
-	  using arctan_0_1_bounds[OF `0 \<le> Ifloat ?DIV` `Ifloat ?DIV \<le> 1`] by auto
-	finally show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_P[OF `x \<le> Float 1 1`] if_not_P[OF False] .
-      qed
-    next
-      case False
-      hence "2 < Ifloat x" unfolding le_float_def Float_num by auto
-      hence "1 \<le> Ifloat x" by auto
-      hence "0 < Ifloat x" by auto
-      hence "0 < x" unfolding less_float_def by auto
-
-      let "?invx" = "float_divl prec 1 x"
-      have "0 \<le> arctan (Ifloat x)" using arctan_monotone'[OF `0 \<le> Ifloat x`] using arctan_tan[of 0, unfolded tan_zero] by auto
-
-      have "Ifloat ?invx \<le> 1" unfolding less_float_def by (rule order_trans[OF float_divl], auto simp add: `1 \<le> Ifloat x` divide_le_eq_1_pos[OF `0 < Ifloat x`])
-      have "0 \<le> Ifloat ?invx" unfolding Ifloat_0[symmetric] by (rule float_divl_lower_bound[unfolded le_float_def], auto simp add: `0 < x`)
-	
-      have "1 / Ifloat x \<noteq> 0" and "0 < 1 / Ifloat x" using `0 < Ifloat x` by auto
-      
-      have "Ifloat (?lb_horner ?invx) \<le> arctan (Ifloat ?invx)" using arctan_0_1_bounds[OF `0 \<le> Ifloat ?invx` `Ifloat ?invx \<le> 1`] by auto
-      also have "\<dots> \<le> arctan (1 / Ifloat x)" unfolding Ifloat_1[symmetric] by (rule arctan_monotone', rule float_divl)
-      finally have "arctan (Ifloat x) \<le> pi / 2 - Ifloat (?lb_horner ?invx)"
-	using `0 \<le> arctan (Ifloat x)` arctan_inverse[OF `1 / Ifloat x \<noteq> 0`] 
-	unfolding real_sgn_pos[OF `0 < 1 / Ifloat x`] le_diff_eq by auto
-      moreover
-      have "pi / 2 \<le> Ifloat (ub_pi prec * Float 1 -1)" unfolding Ifloat_mult Float_num times_divide_eq_right mult_1_right using pi_boundaries by auto
-      ultimately
-      show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_not_P[OF `\<not> x \<le> Float 1 1`] if_not_P[OF False]
-	by auto
-    qed
-  qed
-qed
-
-lemma arctan_boundaries:
-  "arctan (Ifloat x) \<in> {Ifloat (lb_arctan prec x) .. Ifloat (ub_arctan prec x)}"
-proof (cases "0 \<le> x")
-  case True hence "0 \<le> Ifloat x" unfolding le_float_def by auto
-  show ?thesis using ub_arctan_bound'[OF `0 \<le> Ifloat x`] lb_arctan_bound'[OF `0 \<le> Ifloat x`] unfolding atLeastAtMost_iff by auto
-next
-  let ?mx = "-x"
-  case False hence "x < 0" and "0 \<le> Ifloat ?mx" unfolding le_float_def less_float_def by auto
-  hence bounds: "Ifloat (lb_arctan prec ?mx) \<le> arctan (Ifloat ?mx) \<and> arctan (Ifloat ?mx) \<le> Ifloat (ub_arctan prec ?mx)"
-    using ub_arctan_bound'[OF `0 \<le> Ifloat ?mx`] lb_arctan_bound'[OF `0 \<le> Ifloat ?mx`] by auto
-  show ?thesis unfolding Ifloat_minus arctan_minus lb_arctan.simps[where x=x] ub_arctan.simps[where x=x] Let_def if_P[OF `x < 0`]
-    unfolding atLeastAtMost_iff using bounds[unfolded Ifloat_minus arctan_minus] by auto
-qed
-
-lemma bnds_arctan: "\<forall> x lx ux. (l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux} \<longrightarrow> Ifloat l \<le> arctan x \<and> arctan x \<le> Ifloat u"
-proof (rule allI, rule allI, rule allI, rule impI)
-  fix x lx ux
-  assume "(l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux}"
-  hence l: "lb_arctan prec lx = l " and u: "ub_arctan prec ux = u" and x: "x \<in> {Ifloat lx .. Ifloat ux}" by auto
-
-  { from arctan_boundaries[of lx prec, unfolded l]
-    have "Ifloat l \<le> arctan (Ifloat lx)" by (auto simp del: lb_arctan.simps)
-    also have "\<dots> \<le> arctan x" using x by (auto intro: arctan_monotone')
-    finally have "Ifloat l \<le> arctan x" .
-  } moreover
-  { have "arctan x \<le> arctan (Ifloat ux)" using x by (auto intro: arctan_monotone')
-    also have "\<dots> \<le> Ifloat u" using arctan_boundaries[of ux prec, unfolded u] by (auto simp del: ub_arctan.simps)
-    finally have "arctan x \<le> Ifloat u" .
-  } ultimately show "Ifloat l \<le> arctan x \<and> arctan x \<le> Ifloat u" ..
-qed
-
-section "Sinus and Cosinus"
-
-subsection "Compute the cosinus and sinus series"
-
-fun ub_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
-and lb_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
-  "ub_sin_cos_aux prec 0 i k x = 0"
-| "ub_sin_cos_aux prec (Suc n) i k x = 
-    (rapprox_rat prec 1 (int k)) - x * (lb_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)"
-| "lb_sin_cos_aux prec 0 i k x = 0"
-| "lb_sin_cos_aux prec (Suc n) i k x = 
-    (lapprox_rat prec 1 (int k)) - x * (ub_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)"
-
-lemma cos_aux:
-  shows "Ifloat (lb_sin_cos_aux prec n 1 1 (x * x)) \<le> (\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i))) * (Ifloat x)^(2 * i))" (is "?lb")
-  and "(\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i))) * (Ifloat x)^(2 * i)) \<le> Ifloat (ub_sin_cos_aux prec n 1 1 (x * x))" (is "?ub")
-proof -
-  have "0 \<le> Ifloat (x * x)" unfolding Ifloat_mult by auto
-  let "?f n" = "fact (2 * n)"
-
-  { fix n 
-    have F: "\<And>m. ((\<lambda>i. i + 2) ^ n) m = m + 2 * n" by (induct n arbitrary: m, auto)
-    have "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^ n) 1 * (((\<lambda>i. i + 2) ^ n) 1 + 1)"
-      unfolding F by auto } note f_eq = this
-    
-  from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0, 
-    OF `0 \<le> Ifloat (x * x)` f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
-  show "?lb" and "?ub" by (auto simp add: power_mult power2_eq_square[of "Ifloat x"])
-qed
-
-lemma cos_boundaries: assumes "0 \<le> Ifloat x" and "Ifloat x \<le> pi / 2"
-  shows "cos (Ifloat x) \<in> {Ifloat (lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) .. Ifloat (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))}"
-proof (cases "Ifloat x = 0")
-  case False hence "Ifloat x \<noteq> 0" by auto
-  hence "0 < x" and "0 < Ifloat x" using `0 \<le> Ifloat x` unfolding less_float_def by auto
-  have "0 < x * x" using `0 < x` unfolding less_float_def Ifloat_mult Ifloat_0
-    using mult_pos_pos[where a="Ifloat x" and b="Ifloat x"] by auto
-
-  { fix x n have "(\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i))) * x^(2 * i))
-    = (\<Sum> i = 0 ..< 2 * n. (if even(i) then (-1 ^ (i div 2))/(real (fact i)) else 0) * x ^ i)" (is "?sum = ?ifsum")
-  proof -
-    have "?sum = ?sum + (\<Sum> j = 0 ..< n. 0)" by auto
-    also have "\<dots> = 
-      (\<Sum> j = 0 ..< n. -1 ^ ((2 * j) div 2) / (real (fact (2 * j))) * x ^(2 * j)) + (\<Sum> j = 0 ..< n. 0)" by auto
-    also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. if even i then -1 ^ (i div 2) / (real (fact i)) * x ^ i else 0)"
-      unfolding sum_split_even_odd ..
-    also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. (if even i then -1 ^ (i div 2) / (real (fact i)) else 0) * x ^ i)"
-      by (rule setsum_cong2) auto
-    finally show ?thesis by assumption
-  qed } note morph_to_if_power = this
-
-
-  { fix n :: nat assume "0 < n"
-    hence "0 < 2 * n" by auto
-    obtain t where "0 < t" and "t < Ifloat x" and
-      cos_eq: "cos (Ifloat x) = (\<Sum> i = 0 ..< 2 * n. (if even(i) then (-1 ^ (i div 2))/(real (fact i)) else 0) * (Ifloat x) ^ i) 
-      + (cos (t + 1/2 * real (2 * n) * pi) / real (fact (2*n))) * (Ifloat x)^(2*n)" 
-      (is "_ = ?SUM + ?rest / ?fact * ?pow")
-      using Maclaurin_cos_expansion2[OF `0 < Ifloat x` `0 < 2 * n`] by auto
-
-    have "cos t * -1^n = cos t * cos (real n * pi) + sin t * sin (real n * pi)" by auto
-    also have "\<dots> = cos (t + real n * pi)"  using cos_add by auto
-    also have "\<dots> = ?rest" by auto
-    finally have "cos t * -1^n = ?rest" .
-    moreover
-    have "t \<le> pi / 2" using `t < Ifloat x` and `Ifloat x \<le> pi / 2` by auto
-    hence "0 \<le> cos t" using `0 < t` and cos_ge_zero by auto
-    ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest " by auto
-
-    have "0 < ?fact" by auto
-    have "0 < ?pow" using `0 < Ifloat x` by auto
-
-    {
-      assume "even n"
-      have "Ifloat (lb_sin_cos_aux prec n 1 1 (x * x)) \<le> ?SUM"
-	unfolding morph_to_if_power[symmetric] using cos_aux by auto 
-      also have "\<dots> \<le> cos (Ifloat x)"
-      proof -
-	from even[OF `even n`] `0 < ?fact` `0 < ?pow`
-	have "0 \<le> (?rest / ?fact) * ?pow" by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
-	thus ?thesis unfolding cos_eq by auto
-      qed
-      finally have "Ifloat (lb_sin_cos_aux prec n 1 1 (x * x)) \<le> cos (Ifloat x)" .
-    } note lb = this
-
-    {
-      assume "odd n"
-      have "cos (Ifloat x) \<le> ?SUM"
-      proof -
-	from `0 < ?fact` and `0 < ?pow` and odd[OF `odd n`]
-	have "0 \<le> (- ?rest) / ?fact * ?pow"
-	  by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
-	thus ?thesis unfolding cos_eq by auto
-      qed
-      also have "\<dots> \<le> Ifloat (ub_sin_cos_aux prec n 1 1 (x * x))"
-	unfolding morph_to_if_power[symmetric] using cos_aux by auto
-      finally have "cos (Ifloat x) \<le> Ifloat (ub_sin_cos_aux prec n 1 1 (x * x))" .
-    } note ub = this and lb
-  } note ub = this(1) and lb = this(2)
-
-  have "cos (Ifloat x) \<le> Ifloat (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))" using ub[OF odd_pos[OF get_odd] get_odd] .
-  moreover have "Ifloat (lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) \<le> cos (Ifloat x)" 
-  proof (cases "0 < get_even n")
-    case True show ?thesis using lb[OF True get_even] .
-  next
-    case False
-    hence "get_even n = 0" by auto
-    have "- (pi / 2) \<le> Ifloat x" by (rule order_trans[OF _ `0 < Ifloat x`[THEN less_imp_le]], auto)
-    with `Ifloat x \<le> pi / 2`
-    show ?thesis unfolding `get_even n = 0` lb_sin_cos_aux.simps Ifloat_minus Ifloat_0 using cos_ge_zero by auto
-  qed
-  ultimately show ?thesis by auto
-next
-  case True
-  show ?thesis
-  proof (cases "n = 0")
-    case True 
-    thus ?thesis unfolding `n = 0` get_even_def get_odd_def using `Ifloat x = 0` lapprox_rat[where x="-1" and y=1] by auto
-  next
-    case False with not0_implies_Suc obtain m where "n = Suc m" by blast
-    thus ?thesis unfolding `n = Suc m` get_even_def get_odd_def using `Ifloat x = 0` rapprox_rat[where x=1 and y=1] lapprox_rat[where x=1 and y=1] by (cases "even (Suc m)", auto)
-  qed
-qed
-
-lemma sin_aux: assumes "0 \<le> Ifloat x"
-  shows "Ifloat (x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> (\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i + 1))) * (Ifloat x)^(2 * i + 1))" (is "?lb")
-  and "(\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i + 1))) * (Ifloat x)^(2 * i + 1)) \<le> Ifloat (x * ub_sin_cos_aux prec n 2 1 (x * x))" (is "?ub")
-proof -
-  have "0 \<le> Ifloat (x * x)" unfolding Ifloat_mult by auto
-  let "?f n" = "fact (2 * n + 1)"
-
-  { fix n 
-    have F: "\<And>m. ((\<lambda>i. i + 2) ^ n) m = m + 2 * n" by (induct n arbitrary: m, auto)
-    have "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^ n) 2 * (((\<lambda>i. i + 2) ^ n) 2 + 1)"
-      unfolding F by auto } note f_eq = this
-    
-  from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
-    OF `0 \<le> Ifloat (x * x)` f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
-  show "?lb" and "?ub" using `0 \<le> Ifloat x` unfolding Ifloat_mult
-    unfolding power_add power_one_right real_mult_assoc[symmetric] setsum_left_distrib[symmetric]
-    unfolding real_mult_commute
-    by (auto intro!: mult_left_mono simp add: power_mult power2_eq_square[of "Ifloat x"])
-qed
-
-lemma sin_boundaries: assumes "0 \<le> Ifloat x" and "Ifloat x \<le> pi / 2"
-  shows "sin (Ifloat x) \<in> {Ifloat (x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) .. Ifloat (x * ub_sin_cos_aux prec (get_odd n) 2 1 (x * x))}"
-proof (cases "Ifloat x = 0")
-  case False hence "Ifloat x \<noteq> 0" by auto
-  hence "0 < x" and "0 < Ifloat x" using `0 \<le> Ifloat x` unfolding less_float_def by auto
-  have "0 < x * x" using `0 < x` unfolding less_float_def Ifloat_mult Ifloat_0
-    using mult_pos_pos[where a="Ifloat x" and b="Ifloat x"] by auto
-
-  { fix x n have "(\<Sum> j = 0 ..< n. -1 ^ (((2 * j + 1) - Suc 0) div 2) / (real (fact (2 * j + 1))) * x ^(2 * j + 1))
-    = (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * x ^ i)" (is "?SUM = _")
-    proof -
-      have pow: "!!i. x ^ (2 * i + 1) = x * x ^ (2 * i)" by auto
-      have "?SUM = (\<Sum> j = 0 ..< n. 0) + ?SUM" by auto
-      also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. if even i then 0 else -1 ^ ((i - Suc 0) div 2) / (real (fact i)) * x ^ i)"
-	unfolding sum_split_even_odd ..
-      also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. (if even i then 0 else -1 ^ ((i - Suc 0) div 2) / (real (fact i))) * x ^ i)"
-	by (rule setsum_cong2) auto
-      finally show ?thesis by assumption
-    qed } note setsum_morph = this
-
-  { fix n :: nat assume "0 < n"
-    hence "0 < 2 * n + 1" by auto
-    obtain t where "0 < t" and "t < Ifloat x" and
-      sin_eq: "sin (Ifloat x) = (\<Sum> i = 0 ..< 2 * n + 1. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (Ifloat x) ^ i) 
-      + (sin (t + 1/2 * real (2 * n + 1) * pi) / real (fact (2*n + 1))) * (Ifloat x)^(2*n + 1)" 
-      (is "_ = ?SUM + ?rest / ?fact * ?pow")
-      using Maclaurin_sin_expansion3[OF `0 < 2 * n + 1` `0 < Ifloat x`] by auto
-
-    have "?rest = cos t * -1^n" unfolding sin_add cos_add real_of_nat_add left_distrib right_distrib by auto
-    moreover
-    have "t \<le> pi / 2" using `t < Ifloat x` and `Ifloat x \<le> pi / 2` by auto
-    hence "0 \<le> cos t" using `0 < t` and cos_ge_zero by auto
-    ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest " by auto
-
-    have "0 < ?fact" by (rule real_of_nat_fact_gt_zero)
-    have "0 < ?pow" using `0 < Ifloat x` by (rule zero_less_power)
-
-    {
-      assume "even n"
-      have "Ifloat (x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> 
-            (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (Ifloat x) ^ i)"
-	using sin_aux[OF `0 \<le> Ifloat x`] unfolding setsum_morph[symmetric] by auto
-      also have "\<dots> \<le> ?SUM" by auto
-      also have "\<dots> \<le> sin (Ifloat x)"
-      proof -
-	from even[OF `even n`] `0 < ?fact` `0 < ?pow`
-	have "0 \<le> (?rest / ?fact) * ?pow" by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
-	thus ?thesis unfolding sin_eq by auto
-      qed
-      finally have "Ifloat (x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> sin (Ifloat x)" .
-    } note lb = this
-
-    {
-      assume "odd n"
-      have "sin (Ifloat x) \<le> ?SUM"
-      proof -
-	from `0 < ?fact` and `0 < ?pow` and odd[OF `odd n`]
-	have "0 \<le> (- ?rest) / ?fact * ?pow"
-	  by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
-	thus ?thesis unfolding sin_eq by auto
-      qed
-      also have "\<dots> \<le> (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (Ifloat x) ^ i)"
-	 by auto
-      also have "\<dots> \<le> Ifloat (x * ub_sin_cos_aux prec n 2 1 (x * x))" 
-	using sin_aux[OF `0 \<le> Ifloat x`] unfolding setsum_morph[symmetric] by auto
-      finally have "sin (Ifloat x) \<le> Ifloat (x * ub_sin_cos_aux prec n 2 1 (x * x))" .
-    } note ub = this and lb
-  } note ub = this(1) and lb = this(2)
-
-  have "sin (Ifloat x) \<le> Ifloat (x * ub_sin_cos_aux prec (get_odd n) 2 1 (x * x))" using ub[OF odd_pos[OF get_odd] get_odd] .
-  moreover have "Ifloat (x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) \<le> sin (Ifloat x)" 
-  proof (cases "0 < get_even n")
-    case True show ?thesis using lb[OF True get_even] .
-  next
-    case False
-    hence "get_even n = 0" by auto
-    with `Ifloat x \<le> pi / 2` `0 \<le> Ifloat x`
-    show ?thesis unfolding `get_even n = 0` ub_sin_cos_aux.simps Ifloat_minus Ifloat_0 using sin_ge_zero by auto
-  qed
-  ultimately show ?thesis by auto
-next
-  case True
-  show ?thesis
-  proof (cases "n = 0")
-    case True 
-    thus ?thesis unfolding `n = 0` get_even_def get_odd_def using `Ifloat x = 0` lapprox_rat[where x="-1" and y=1] by auto
-  next
-    case False with not0_implies_Suc obtain m where "n = Suc m" by blast
-    thus ?thesis unfolding `n = Suc m` get_even_def get_odd_def using `Ifloat x = 0` rapprox_rat[where x=1 and y=1] lapprox_rat[where x=1 and y=1] by (cases "even (Suc m)", auto)
-  qed
-qed
-
-subsection "Compute the cosinus in the entire domain"
-
-definition lb_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where
-"lb_cos prec x = (let
-    horner = \<lambda> x. lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x) ;
-    half = \<lambda> x. if x < 0 then - 1 else Float 1 1 * x * x - 1
-  in if x < Float 1 -1 then horner x
-else if x < 1          then half (horner (x * Float 1 -1))
-                       else half (half (horner (x * Float 1 -2))))"
-
-definition ub_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where
-"ub_cos prec x = (let
-    horner = \<lambda> x. ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x) ;
-    half = \<lambda> x. Float 1 1 * x * x - 1
-  in if x < Float 1 -1 then horner x
-else if x < 1          then half (horner (x * Float 1 -1))
-                       else half (half (horner (x * Float 1 -2))))"
-
-definition bnds_cos :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where
-"bnds_cos prec lx ux = (let  lpi = lb_pi prec
-  in   if lx < -lpi \<or> ux > lpi   then (Float -1 0, Float 1 0)
-  else if ux \<le> 0                 then (lb_cos prec (-lx), ub_cos prec (-ux))
-  else if 0 \<le> lx                 then (lb_cos prec ux, ub_cos prec lx)
-                                 else (min (lb_cos prec (-lx)) (lb_cos prec ux), Float 1 0))"
-
-lemma lb_cos: assumes "0 \<le> Ifloat x" and "Ifloat x \<le> pi" 
-  shows "cos (Ifloat x) \<in> {Ifloat (lb_cos prec x) .. Ifloat (ub_cos prec x)}" (is "?cos x \<in> { Ifloat (?lb x) .. Ifloat (?ub x) }")
-proof -
-  { fix x :: real
-    have "cos x = cos (x / 2 + x / 2)" by auto
-    also have "\<dots> = cos (x / 2) * cos (x / 2) + sin (x / 2) * sin (x / 2) - sin (x / 2) * sin (x / 2) + cos (x / 2) * cos (x / 2) - 1"
-      unfolding cos_add by auto
-    also have "\<dots> = 2 * cos (x / 2) * cos (x / 2) - 1" by algebra
-    finally have "cos x = 2 * cos (x / 2) * cos (x / 2) - 1" .
-  } note x_half = this[symmetric]
-
-  have "\<not> x < 0" using `0 \<le> Ifloat x` unfolding less_float_def by auto
-  let "?ub_horner x" = "ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x)"
-  let "?lb_horner x" = "lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x)"
-  let "?ub_half x" = "Float 1 1 * x * x - 1"
-  let "?lb_half x" = "if x < 0 then - 1 else Float 1 1 * x * x - 1"
-
-  show ?thesis
-  proof (cases "x < Float 1 -1")
-    case True hence "Ifloat x \<le> pi / 2" unfolding less_float_def using pi_ge_two by auto
-    show ?thesis unfolding lb_cos_def[where x=x] ub_cos_def[where x=x] if_not_P[OF `\<not> x < 0`] if_P[OF `x < Float 1 -1`] Let_def
-      using cos_boundaries[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi / 2`] .
-  next
-    case False
-    
-    { fix y x :: float let ?x2 = "Ifloat (x * Float 1 -1)"
-      assume "Ifloat y \<le> cos ?x2" and "-pi \<le> Ifloat x" and "Ifloat x \<le> pi"
-      hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2" using pi_ge_two unfolding Ifloat_mult Float_num by auto
-      hence "0 \<le> cos ?x2" by (rule cos_ge_zero)
-      
-      have "Ifloat (?lb_half y) \<le> cos (Ifloat x)"
-      proof (cases "y < 0")
-	case True show ?thesis using cos_ge_minus_one unfolding if_P[OF True] by auto
-      next
-	case False
-	hence "0 \<le> Ifloat y" unfolding less_float_def by auto
-	from mult_mono[OF `Ifloat y \<le> cos ?x2` `Ifloat y \<le> cos ?x2` `0 \<le> cos ?x2` this]
-	have "Ifloat y * Ifloat y \<le> cos ?x2 * cos ?x2" .
-	hence "2 * Ifloat y * Ifloat y \<le> 2 * cos ?x2 * cos ?x2" by auto
-	hence "2 * Ifloat y * Ifloat y - 1 \<le> 2 * cos (Ifloat x / 2) * cos (Ifloat x / 2) - 1" unfolding Float_num Ifloat_mult by auto
-	thus ?thesis unfolding if_not_P[OF False] x_half Float_num Ifloat_mult Ifloat_sub by auto
-      qed
-    } note lb_half = this
-    
-    { fix y x :: float let ?x2 = "Ifloat (x * Float 1 -1)"
-      assume ub: "cos ?x2 \<le> Ifloat y" and "- pi \<le> Ifloat x" and "Ifloat x \<le> pi"
-      hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2" using pi_ge_two unfolding Ifloat_mult Float_num by auto
-      hence "0 \<le> cos ?x2" by (rule cos_ge_zero)
-      
-      have "cos (Ifloat x) \<le> Ifloat (?ub_half y)"
-      proof -
-	have "0 \<le> Ifloat y" using `0 \<le> cos ?x2` ub by (rule order_trans)
-	from mult_mono[OF ub ub this `0 \<le> cos ?x2`]
-	have "cos ?x2 * cos ?x2 \<le> Ifloat y * Ifloat y" .
-	hence "2 * cos ?x2 * cos ?x2 \<le> 2 * Ifloat y * Ifloat y" by auto
-	hence "2 * cos (Ifloat x / 2) * cos (Ifloat x / 2) - 1 \<le> 2 * Ifloat y * Ifloat y - 1" unfolding Float_num Ifloat_mult by auto
-	thus ?thesis unfolding x_half Ifloat_mult Float_num Ifloat_sub by auto
-      qed
-    } note ub_half = this
-    
-    let ?x2 = "x * Float 1 -1"
-    let ?x4 = "x * Float 1 -1 * Float 1 -1"
-    
-    have "-pi \<le> Ifloat x" using pi_ge_zero[THEN le_imp_neg_le, unfolded minus_zero] `0 \<le> Ifloat x` by (rule order_trans)
-    
-    show ?thesis
-    proof (cases "x < 1")
-      case True hence "Ifloat x \<le> 1" unfolding less_float_def by auto
-      have "0 \<le> Ifloat ?x2" and "Ifloat ?x2 \<le> pi / 2" using pi_ge_two `0 \<le> Ifloat x` unfolding Ifloat_mult Float_num using assms by auto
-      from cos_boundaries[OF this]
-      have lb: "Ifloat (?lb_horner ?x2) \<le> ?cos ?x2" and ub: "?cos ?x2 \<le> Ifloat (?ub_horner ?x2)" by auto
-      
-      have "Ifloat (?lb x) \<le> ?cos x"
-      proof -
-	from lb_half[OF lb `-pi \<le> Ifloat x` `Ifloat x \<le> pi`]
-	show ?thesis unfolding lb_cos_def[where x=x] Let_def using `\<not> x < 0` `\<not> x < Float 1 -1` `x < 1` by auto
-      qed
-      moreover have "?cos x \<le> Ifloat (?ub x)"
-      proof -
-	from ub_half[OF ub `-pi \<le> Ifloat x` `Ifloat x \<le> pi`]
-	show ?thesis unfolding ub_cos_def[where x=x] Let_def using `\<not> x < 0` `\<not> x < Float 1 -1` `x < 1` by auto 
-      qed
-      ultimately show ?thesis by auto
-    next
-      case False
-      have "0 \<le> Ifloat ?x4" and "Ifloat ?x4 \<le> pi / 2" using pi_ge_two `0 \<le> Ifloat x` `Ifloat x \<le> pi` unfolding Ifloat_mult Float_num by auto
-      from cos_boundaries[OF this]
-      have lb: "Ifloat (?lb_horner ?x4) \<le> ?cos ?x4" and ub: "?cos ?x4 \<le> Ifloat (?ub_horner ?x4)" by auto
-      
-      have eq_4: "?x2 * Float 1 -1 = x * Float 1 -2" by (cases x, auto simp add: times_float.simps)
-      
-      have "Ifloat (?lb x) \<le> ?cos x"
-      proof -
-	have "-pi \<le> Ifloat ?x2" and "Ifloat ?x2 \<le> pi" unfolding Ifloat_mult Float_num using pi_ge_two `0 \<le> Ifloat x` `Ifloat x \<le> pi` by auto
-	from lb_half[OF lb_half[OF lb this] `-pi \<le> Ifloat x` `Ifloat x \<le> pi`, unfolded eq_4]
-	show ?thesis unfolding lb_cos_def[where x=x] if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x < Float 1 -1`] if_not_P[OF `\<not> x < 1`] Let_def .
-      qed
-      moreover have "?cos x \<le> Ifloat (?ub x)"
-      proof -
-	have "-pi \<le> Ifloat ?x2" and "Ifloat ?x2 \<le> pi" unfolding Ifloat_mult Float_num using pi_ge_two `0 \<le> Ifloat x` `Ifloat x \<le> pi` by auto
-	from ub_half[OF ub_half[OF ub this] `-pi \<le> Ifloat x` `Ifloat x \<le> pi`, unfolded eq_4]
-	show ?thesis unfolding ub_cos_def[where x=x] if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x < Float 1 -1`] if_not_P[OF `\<not> x < 1`] Let_def .
-      qed
-      ultimately show ?thesis by auto
-    qed
-  qed
-qed
-
-lemma lb_cos_minus: assumes "-pi \<le> Ifloat x" and "Ifloat x \<le> 0" 
-  shows "cos (Ifloat (-x)) \<in> {Ifloat (lb_cos prec (-x)) .. Ifloat (ub_cos prec (-x))}"
-proof -
-  have "0 \<le> Ifloat (-x)" and "Ifloat (-x) \<le> pi" using `-pi \<le> Ifloat x` `Ifloat x \<le> 0` by auto
-  from lb_cos[OF this] show ?thesis .
-qed
-
-lemma bnds_cos: "\<forall> x lx ux. (l, u) = bnds_cos prec lx ux \<and> x \<in> {Ifloat lx .. Ifloat ux} \<longrightarrow> Ifloat l \<le> cos x \<and> cos x \<le> Ifloat u"
-proof (rule allI, rule allI, rule allI, rule impI)
-  fix x lx ux
-  assume "(l, u) = bnds_cos prec lx ux \<and> x \<in> {Ifloat lx .. Ifloat ux}"
-  hence bnds: "(l, u) = bnds_cos prec lx ux" and x: "x \<in> {Ifloat lx .. Ifloat ux}" by auto
-
-  let ?lpi = "lb_pi prec"  
-  have [intro!]: "Ifloat lx \<le> Ifloat ux" using x by auto
-  hence "lx \<le> ux" unfolding le_float_def .
-
-  show "Ifloat l \<le> cos x \<and> cos x \<le> Ifloat u"
-  proof (cases "lx < -?lpi \<or> ux > ?lpi")
-    case True
-    show ?thesis using bnds unfolding bnds_cos_def if_P[OF True] Let_def using cos_le_one cos_ge_minus_one by auto
-  next
-    case False note not_out = this
-    hence lpi_lx: "- Ifloat ?lpi \<le> Ifloat lx" and lpi_ux: "Ifloat ux \<le> Ifloat ?lpi" unfolding le_float_def less_float_def by auto
-
-    from pi_boundaries[unfolded atLeastAtMost_iff, THEN conjunct1, THEN le_imp_neg_le] lpi_lx
-    have "- pi \<le> Ifloat lx" by (rule order_trans)
-    hence "- pi \<le> x" and "- pi \<le> Ifloat ux" and "x \<le> Ifloat ux" using x by auto
-    
-    from lpi_ux pi_boundaries[unfolded atLeastAtMost_iff, THEN conjunct1]
-    have "Ifloat ux \<le> pi" by (rule order_trans)
-    hence "x \<le> pi" and "Ifloat lx \<le> pi" and "Ifloat lx \<le> x" using x by auto
-
-    note lb_cos_minus_bottom = lb_cos_minus[unfolded atLeastAtMost_iff, THEN conjunct1]
-    note lb_cos_minus_top = lb_cos_minus[unfolded atLeastAtMost_iff, THEN conjunct2]
-    note lb_cos_bottom = lb_cos[unfolded atLeastAtMost_iff, THEN conjunct1]
-    note lb_cos_top = lb_cos[unfolded atLeastAtMost_iff, THEN conjunct2]
-
-    show ?thesis
-    proof (cases "ux \<le> 0")
-      case True hence "Ifloat ux \<le> 0" unfolding le_float_def by auto
-      hence "x \<le> 0" and "Ifloat lx \<le> 0" using x by auto
-      
-      { have "Ifloat (lb_cos prec (-lx)) \<le> cos (Ifloat (-lx))" using lb_cos_minus_bottom[OF `-pi \<le> Ifloat lx` `Ifloat lx \<le> 0`] .
-	also have "\<dots> \<le> cos x" unfolding Ifloat_minus cos_minus using cos_monotone_minus_pi_0'[OF `- pi \<le> Ifloat lx` `Ifloat lx \<le> x` `x \<le> 0`] .
-	finally have "Ifloat (lb_cos prec (-lx)) \<le> cos x" . }
-      moreover
-      { have "cos x \<le> cos (Ifloat (-ux))" unfolding Ifloat_minus cos_minus using cos_monotone_minus_pi_0'[OF `- pi \<le> x` `x \<le> Ifloat ux` `Ifloat ux \<le> 0`] .
-	also have "\<dots> \<le> Ifloat (ub_cos prec (-ux))" using lb_cos_minus_top[OF `-pi \<le> Ifloat ux` `Ifloat ux \<le> 0`] .
-	finally have "cos x \<le> Ifloat (ub_cos prec (-ux))" . }
-      ultimately show ?thesis using bnds unfolding bnds_cos_def Let_def if_not_P[OF not_out] if_P[OF True] by auto
-    next
-      case False note not_ux = this
-      
-      show ?thesis
-      proof (cases "0 \<le> lx")
-	case True hence "0 \<le> Ifloat lx" unfolding le_float_def by auto
-	hence "0 \<le> x" and "0 \<le> Ifloat ux" using x by auto
-      
-	{ have "Ifloat (lb_cos prec ux) \<le> cos (Ifloat ux)" using lb_cos_bottom[OF `0 \<le> Ifloat ux` `Ifloat ux \<le> pi`] .
-	  also have "\<dots> \<le> cos x" using cos_monotone_0_pi'[OF `0 \<le> x` `x \<le> Ifloat ux` `Ifloat ux \<le> pi`] .
-	  finally have "Ifloat (lb_cos prec ux) \<le> cos x" . }
-	moreover
-	{ have "cos x \<le> cos (Ifloat lx)" using cos_monotone_0_pi'[OF `0 \<le> Ifloat lx` `Ifloat lx \<le> x` `x \<le> pi`] .
-	  also have "\<dots> \<le> Ifloat (ub_cos prec lx)" using lb_cos_top[OF `0 \<le> Ifloat lx` `Ifloat lx \<le> pi`] .
-	  finally have "cos x \<le> Ifloat (ub_cos prec lx)" . }
-	ultimately show ?thesis using bnds unfolding bnds_cos_def Let_def if_not_P[OF not_out] if_not_P[OF not_ux] if_P[OF True] by auto
-      next
-	case False with not_ux
-	have "Ifloat lx \<le> 0" and "0 \<le> Ifloat ux" unfolding le_float_def by auto
-
-	have "Ifloat (min (lb_cos prec (-lx)) (lb_cos prec ux)) \<le> cos x"
-	proof (cases "x \<le> 0")
-	  case True
-	  have "Ifloat (lb_cos prec (-lx)) \<le> cos (Ifloat (-lx))" using lb_cos_minus_bottom[OF `-pi \<le> Ifloat lx` `Ifloat lx \<le> 0`] .
-	  also have "\<dots> \<le> cos x" unfolding Ifloat_minus cos_minus using cos_monotone_minus_pi_0'[OF `- pi \<le> Ifloat lx` `Ifloat lx \<le> x` `x \<le> 0`] .
-	  finally show ?thesis unfolding Ifloat_min by auto
-	next
-	  case False hence "0 \<le> x" by auto
-	  have "Ifloat (lb_cos prec ux) \<le> cos (Ifloat ux)" using lb_cos_bottom[OF `0 \<le> Ifloat ux` `Ifloat ux \<le> pi`] .
-	  also have "\<dots> \<le> cos x" using cos_monotone_0_pi'[OF `0 \<le> x` `x \<le> Ifloat ux` `Ifloat ux \<le> pi`] .
-	  finally show ?thesis unfolding Ifloat_min by auto
-	qed
-	moreover have "cos x \<le> Ifloat (Float 1 0)" by auto
-	ultimately show ?thesis using bnds unfolding bnds_cos_def Let_def if_not_P[OF not_out] if_not_P[OF not_ux] if_not_P[OF False] by auto
-      qed
-    qed
-  qed
-qed
-
-subsection "Compute the sinus in the entire domain"
-
-function lb_sin :: "nat \<Rightarrow> float \<Rightarrow> float" and ub_sin :: "nat \<Rightarrow> float \<Rightarrow> float" where
-"lb_sin prec x = (let sqr_diff = \<lambda> x. if x > 1 then 0 else 1 - x * x 
-  in if x < 0           then - ub_sin prec (- x)
-else if x \<le> Float 1 -1  then x * lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 2 1 (x * x)
-                        else the (lb_sqrt prec (sqr_diff (ub_cos prec x))))" |
-
-"ub_sin prec x = (let sqr_diff = \<lambda> x. if x < 0 then 1 else 1 - x * x
-  in if x < 0           then - lb_sin prec (- x)
-else if x \<le> Float 1 -1  then x * ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 2 1 (x * x)
-                        else the (ub_sqrt prec (sqr_diff (lb_cos prec x))))"
-by pat_completeness auto
-termination by (relation "measure (\<lambda> v. let (prec, x) = sum_case id id v in (if x < 0 then 1 else 0))", auto simp add: less_float_def)
-
-definition bnds_sin :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where
-"bnds_sin prec lx ux = (let 
-    lpi = lb_pi prec ;
-    half_pi = lpi * Float 1 -1
-  in if lx \<le> - half_pi \<or> half_pi \<le> ux then (Float -1 0, Float 1 0)
-                                       else (lb_sin prec lx, ub_sin prec ux))"
-
-lemma lb_sin: assumes "- (pi / 2) \<le> Ifloat x" and "Ifloat x \<le> pi / 2"
-  shows "sin (Ifloat x) \<in> { Ifloat (lb_sin prec x) .. Ifloat (ub_sin prec x) }" (is "?sin x \<in> { ?lb x .. ?ub x}")
-proof -
-  { fix x :: float assume "0 \<le> Ifloat x" and "Ifloat x \<le> pi / 2"
-    hence "\<not> (x < 0)" and "- (pi / 2) \<le> Ifloat x" unfolding less_float_def using pi_ge_two by auto
-
-    have "Ifloat x \<le> pi" using `Ifloat x \<le> pi / 2` using pi_ge_two by auto
-
-    have "?sin x \<in> { ?lb x .. ?ub x}"
-    proof (cases "x \<le> Float 1 -1")
-      case True from sin_boundaries[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi / 2`]
-      show ?thesis unfolding lb_sin.simps[of prec x] ub_sin.simps[of prec x] if_not_P[OF `\<not> (x < 0)`] if_P[OF True] Let_def .
-    next
-      case False
-      have "0 \<le> cos (Ifloat x)" using cos_ge_zero[OF _ `Ifloat x \<le> pi /2`] `0 \<le> Ifloat x` pi_ge_two by auto
-      have "0 \<le> sin (Ifloat x)" using `0 \<le> Ifloat x` and `Ifloat x \<le> pi / 2` using sin_ge_zero by auto
-      
-      have "?sin x \<le> ?ub x"
-      proof (cases "lb_cos prec x < 0")
-	case True
-	have "?sin x \<le> 1" using sin_le_one .
-	also have "\<dots> \<le> Ifloat (the (ub_sqrt prec 1))" using ub_sqrt_lower_bound[where prec=prec and x=1] unfolding Ifloat_1 by auto
-	finally show ?thesis unfolding ub_sin.simps if_not_P[OF `\<not> (x < 0)`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_P[OF True] Let_def .
-      next
-	case False hence "0 \<le> Ifloat (lb_cos prec x)" unfolding less_float_def by auto
-	
-	have "sin (Ifloat x) = sqrt (1 - cos (Ifloat x) ^ 2)" unfolding sin_squared_eq[symmetric] real_sqrt_abs using `0 \<le> sin (Ifloat x)` by auto
-	also have "\<dots> \<le> sqrt (Ifloat (1 - lb_cos prec x * lb_cos prec x))" 
-	proof (rule real_sqrt_le_mono)
-	  have "Ifloat (lb_cos prec x * lb_cos prec x) \<le> cos (Ifloat x) ^ 2" unfolding numeral_2_eq_2 power_Suc2 realpow_0 Ifloat_mult
-	    using `0 \<le> Ifloat (lb_cos prec x)` lb_cos[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi`] `0 \<le> cos (Ifloat x)` by(auto intro!: mult_mono)
-	  thus "1 - cos (Ifloat x) ^ 2 \<le> Ifloat (1 - lb_cos prec x * lb_cos prec x)" unfolding Ifloat_sub Ifloat_1 by auto
-	qed
-	also have "\<dots> \<le> Ifloat (the (ub_sqrt prec (1 - lb_cos prec x * lb_cos prec x)))"
-	proof (rule ub_sqrt_lower_bound)
-	  have "Ifloat (lb_cos prec x) \<le> cos (Ifloat x)" using lb_cos[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi`] by auto
-	  from mult_mono[OF order_trans[OF this cos_le_one] order_trans[OF this cos_le_one]]
-	  have "Ifloat (lb_cos prec x) * Ifloat (lb_cos prec x) \<le> 1" using `0 \<le> Ifloat (lb_cos prec x)` by auto
-	  thus "0 \<le> Ifloat (1 - lb_cos prec x * lb_cos prec x)" by auto
-	qed
-	finally show ?thesis unfolding ub_sin.simps if_not_P[OF `\<not> (x < 0)`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_not_P[OF False] Let_def .
-      qed
-      moreover
-      have "?lb x \<le> ?sin x"
-      proof (cases "1 < ub_cos prec x")
-	case True
-	show ?thesis unfolding lb_sin.simps if_not_P[OF `\<not> (x < 0)`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_P[OF True] Let_def 
-	  by (rule order_trans[OF _ sin_ge_zero[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi`]]) 
-        (auto simp add: lb_sqrt_upper_bound[where prec=prec and x=0, unfolded Ifloat_0 real_sqrt_zero])
-      next
-	case False hence "Ifloat (ub_cos prec x) \<le> 1" unfolding less_float_def by auto
-	have "0 \<le> Ifloat (ub_cos prec x)" using order_trans[OF `0 \<le> cos (Ifloat x)`] lb_cos `0 \<le> Ifloat x` `Ifloat x \<le> pi` by auto
-	
-	have "Ifloat (the (lb_sqrt prec (1 - ub_cos prec x * ub_cos prec x))) \<le> sqrt (Ifloat (1 - ub_cos prec x * ub_cos prec x))"
-	proof (rule lb_sqrt_upper_bound)
-	  from mult_mono[OF `Ifloat (ub_cos prec x) \<le> 1` `Ifloat (ub_cos prec x) \<le> 1`] `0 \<le> Ifloat (ub_cos prec x)`
-	  have "Ifloat (ub_cos prec x) * Ifloat (ub_cos prec x) \<le> 1" by auto
-	  thus "0 \<le> Ifloat (1 - ub_cos prec x * ub_cos prec x)" by auto
-	qed
-	also have "\<dots> \<le> sqrt (1 - cos (Ifloat x) ^ 2)"
-	proof (rule real_sqrt_le_mono)
-	  have "cos (Ifloat x) ^ 2 \<le> Ifloat (ub_cos prec x * ub_cos prec x)" unfolding numeral_2_eq_2 power_Suc2 realpow_0 Ifloat_mult
-	    using `0 \<le> Ifloat (ub_cos prec x)` lb_cos[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi`] `0 \<le> cos (Ifloat x)` by(auto intro!: mult_mono)
-	  thus "Ifloat (1 - ub_cos prec x * ub_cos prec x) \<le> 1 - cos (Ifloat x) ^ 2" unfolding Ifloat_sub Ifloat_1 by auto
-	qed
-	also have "\<dots> = sin (Ifloat x)" unfolding sin_squared_eq[symmetric] real_sqrt_abs using `0 \<le> sin (Ifloat x)` by auto
-	finally show ?thesis unfolding lb_sin.simps if_not_P[OF `\<not> (x < 0)`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_not_P[OF False] Let_def .
-      qed
-      ultimately show ?thesis by auto
-    qed
-  } note for_pos = this
-
-  show ?thesis
-  proof (cases "x < 0")
-    case True 
-    hence "0 \<le> Ifloat (-x)" and "Ifloat (- x) \<le> pi / 2" using `-(pi/2) \<le> Ifloat x` unfolding less_float_def by auto
-    from for_pos[OF this]
-    show ?thesis unfolding Ifloat_minus sin_minus lb_sin.simps[of prec x] ub_sin.simps[of prec x] if_P[OF True] Let_def atLeastAtMost_iff by auto
-  next
-    case False hence "0 \<le> Ifloat x" unfolding less_float_def by auto
-    from for_pos[OF this `Ifloat x \<le> pi /2`]
-    show ?thesis .
-  qed
-qed
-
-lemma bnds_sin: "\<forall> x lx ux. (l, u) = bnds_sin prec lx ux \<and> x \<in> {Ifloat lx .. Ifloat ux} \<longrightarrow> Ifloat l \<le> sin x \<and> sin x \<le> Ifloat u"
-proof (rule allI, rule allI, rule allI, rule impI)
-  fix x lx ux
-  assume "(l, u) = bnds_sin prec lx ux \<and> x \<in> {Ifloat lx .. Ifloat ux}"
-  hence bnds: "(l, u) = bnds_sin prec lx ux" and x: "x \<in> {Ifloat lx .. Ifloat ux}" by auto
-  show "Ifloat l \<le> sin x \<and> sin x \<le> Ifloat u"
-  proof (cases "lx \<le> - (lb_pi prec * Float 1 -1) \<or> lb_pi prec * Float 1 -1 \<le> ux")
-    case True show ?thesis using bnds unfolding bnds_sin_def if_P[OF True] Let_def by auto
-  next
-    case False
-    hence "- lb_pi prec * Float 1 -1 \<le> lx" and "ux \<le> lb_pi prec * Float 1 -1" unfolding le_float_def by auto
-    moreover have "Ifloat (lb_pi prec * Float 1 -1) \<le> pi / 2" unfolding Ifloat_mult using pi_boundaries by auto
-    ultimately have "- (pi / 2) \<le> Ifloat lx" and "Ifloat ux \<le> pi / 2" and "Ifloat lx \<le> Ifloat ux" unfolding le_float_def using x by auto
-    hence "- (pi / 2) \<le> Ifloat ux" and "Ifloat lx \<le> pi / 2" by auto
-    
-    have "- (pi / 2) \<le> x""x \<le> pi / 2" using `Ifloat ux \<le> pi / 2` `- (pi /2) \<le> Ifloat lx` x by auto
-    
-    { have "Ifloat (lb_sin prec lx) \<le> sin (Ifloat lx)" using lb_sin[OF `- (pi / 2) \<le> Ifloat lx` `Ifloat lx \<le> pi / 2`] unfolding atLeastAtMost_iff by auto
-      also have "\<dots> \<le> sin x" using sin_monotone_2pi' `- (pi / 2) \<le> Ifloat lx` x `x \<le> pi / 2` by auto
-      finally have "Ifloat (lb_sin prec lx) \<le> sin x" . }
-    moreover
-    { have "sin x \<le> sin (Ifloat ux)" using sin_monotone_2pi' `- (pi / 2) \<le> x` x `Ifloat ux \<le> pi / 2` by auto
-      also have "\<dots> \<le> Ifloat (ub_sin prec ux)" using lb_sin[OF `- (pi / 2) \<le> Ifloat ux` `Ifloat ux \<le> pi / 2`] unfolding atLeastAtMost_iff by auto
-      finally have "sin x \<le> Ifloat (ub_sin prec ux)" . }
-    ultimately
-    show ?thesis using bnds unfolding bnds_sin_def if_not_P[OF False] Let_def by auto
-  qed
-qed
-
-section "Exponential function"
-
-subsection "Compute the series of the exponential function"
-
-fun ub_exp_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" and lb_exp_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
-"ub_exp_horner prec 0 i k x       = 0" |
-"ub_exp_horner prec (Suc n) i k x = rapprox_rat prec 1 (int k) + x * lb_exp_horner prec n (i + 1) (k * i) x" |
-"lb_exp_horner prec 0 i k x       = 0" |
-"lb_exp_horner prec (Suc n) i k x = lapprox_rat prec 1 (int k) + x * ub_exp_horner prec n (i + 1) (k * i) x"
-
-lemma bnds_exp_horner: assumes "Ifloat x \<le> 0"
-  shows "exp (Ifloat x) \<in> { Ifloat (lb_exp_horner prec (get_even n) 1 1 x) .. Ifloat (ub_exp_horner prec (get_odd n) 1 1 x) }"
-proof -
-  { fix n
-    have F: "\<And> m. ((\<lambda>i. i + 1) ^ n) m = n + m" by (induct n, auto)
-    have "fact (Suc n) = fact n * ((\<lambda>i. i + 1) ^ n) 1" unfolding F by auto } note f_eq = this
-    
-  note bounds = horner_bounds_nonpos[where f="fact" and lb="lb_exp_horner prec" and ub="ub_exp_horner prec" and j'=0 and s=1,
-    OF assms f_eq lb_exp_horner.simps ub_exp_horner.simps]
-
-  { have "Ifloat (lb_exp_horner prec (get_even n) 1 1 x) \<le> (\<Sum>j = 0..<get_even n. 1 / real (fact j) * Ifloat x ^ j)"
-      using bounds(1) by auto
-    also have "\<dots> \<le> exp (Ifloat x)"
-    proof -
-      obtain t where "\<bar>t\<bar> \<le> \<bar>Ifloat x\<bar>" and "exp (Ifloat x) = (\<Sum>m = 0..<get_even n. (Ifloat x) ^ m / real (fact m)) + exp t / real (fact (get_even n)) * (Ifloat x) ^ (get_even n)"
-	using Maclaurin_exp_le by blast
-      moreover have "0 \<le> exp t / real (fact (get_even n)) * (Ifloat x) ^ (get_even n)"
-	by (auto intro!: mult_nonneg_nonneg divide_nonneg_pos simp add: get_even zero_le_even_power exp_gt_zero)
-      ultimately show ?thesis
-	using get_odd exp_gt_zero by (auto intro!: pordered_cancel_semiring_class.mult_nonneg_nonneg)
-    qed
-    finally have "Ifloat (lb_exp_horner prec (get_even n) 1 1 x) \<le> exp (Ifloat x)" .
-  } moreover
-  { 
-    have x_less_zero: "Ifloat x ^ get_odd n \<le> 0"
-    proof (cases "Ifloat x = 0")
-      case True
-      have "(get_odd n) \<noteq> 0" using get_odd[THEN odd_pos] by auto
-      thus ?thesis unfolding True power_0_left by auto
-    next
-      case False hence "Ifloat x < 0" using `Ifloat x \<le> 0` by auto
-      show ?thesis by (rule less_imp_le, auto simp add: power_less_zero_eq get_odd `Ifloat x < 0`)
-    qed
-
-    obtain t where "\<bar>t\<bar> \<le> \<bar>Ifloat x\<bar>" and "exp (Ifloat x) = (\<Sum>m = 0..<get_odd n. (Ifloat x) ^ m / real (fact m)) + exp t / real (fact (get_odd n)) * (Ifloat x) ^ (get_odd n)"
-      using Maclaurin_exp_le by blast
-    moreover have "exp t / real (fact (get_odd n)) * (Ifloat x) ^ (get_odd n) \<le> 0"
-      by (auto intro!: mult_nonneg_nonpos divide_nonpos_pos simp add: x_less_zero exp_gt_zero)
-    ultimately have "exp (Ifloat x) \<le> (\<Sum>j = 0..<get_odd n. 1 / real (fact j) * Ifloat x ^ j)"
-      using get_odd exp_gt_zero by (auto intro!: pordered_cancel_semiring_class.mult_nonneg_nonneg)
-    also have "\<dots> \<le> Ifloat (ub_exp_horner prec (get_odd n) 1 1 x)"
-      using bounds(2) by auto
-    finally have "exp (Ifloat x) \<le> Ifloat (ub_exp_horner prec (get_odd n) 1 1 x)" .
-  } ultimately show ?thesis by auto
-qed
-
-subsection "Compute the exponential function on the entire domain"
-
-function ub_exp :: "nat \<Rightarrow> float \<Rightarrow> float" and lb_exp :: "nat \<Rightarrow> float \<Rightarrow> float" where
-"lb_exp prec x = (if 0 < x then float_divl prec 1 (ub_exp prec (-x))
-             else let 
-                horner = (\<lambda> x. let  y = lb_exp_horner prec (get_even (prec + 2)) 1 1 x  in if y \<le> 0 then Float 1 -2 else y)
-             in if x < - 1 then (case floor_fl x of (Float m e) \<Rightarrow> (horner (float_divl prec x (- Float m e))) ^ (nat (-m) * 2 ^ nat e))
-                           else horner x)" |
-"ub_exp prec x = (if 0 < x    then float_divr prec 1 (lb_exp prec (-x))
-             else if x < - 1  then (case floor_fl x of (Float m e) \<Rightarrow> 
-                                    (ub_exp_horner prec (get_odd (prec + 2)) 1 1 (float_divr prec x (- Float m e))) ^ (nat (-m) * 2 ^ nat e))
-                              else ub_exp_horner prec (get_odd (prec + 2)) 1 1 x)"
-by pat_completeness auto
-termination by (relation "measure (\<lambda> v. let (prec, x) = sum_case id id v in (if 0 < x then 1 else 0))", auto simp add: less_float_def)
-
-lemma exp_m1_ge_quarter: "(1 / 4 :: real) \<le> exp (- 1)"
-proof -
-  have eq4: "4 = Suc (Suc (Suc (Suc 0)))" by auto
-
-  have "1 / 4 = Ifloat (Float 1 -2)" unfolding Float_num by auto
-  also have "\<dots> \<le> Ifloat (lb_exp_horner 1 (get_even 4) 1 1 (- 1))"
-    unfolding get_even_def eq4 
-    by (auto simp add: lapprox_posrat_def rapprox_posrat_def normfloat.simps)
-  also have "\<dots> \<le> exp (Ifloat (- 1))" using bnds_exp_horner[where x="- 1"] by auto
-  finally show ?thesis unfolding Ifloat_minus Ifloat_1 . 
-qed
-
-lemma lb_exp_pos: assumes "\<not> 0 < x" shows "0 < lb_exp prec x"
-proof -
-  let "?lb_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x"
-  let "?horner x" = "let  y = ?lb_horner x  in if y \<le> 0 then Float 1 -2 else y"
-  have pos_horner: "\<And> x. 0 < ?horner x" unfolding Let_def by (cases "?lb_horner x \<le> 0", auto simp add: le_float_def less_float_def)
-  moreover { fix x :: float fix num :: nat
-    have "0 < Ifloat (?horner x) ^ num" using `0 < ?horner x`[unfolded less_float_def Ifloat_0] by (rule zero_less_power)
-    also have "\<dots> = Ifloat ((?horner x) ^ num)" using float_power by auto
-    finally have "0 < Ifloat ((?horner x) ^ num)" .
-  }
-  ultimately show ?thesis
-    unfolding lb_exp.simps if_not_P[OF `\<not> 0 < x`] Let_def by (cases "floor_fl x", cases "x < - 1", auto simp add: le_float_def less_float_def normfloat) 
-qed
-
-lemma exp_boundaries': assumes "x \<le> 0"
-  shows "exp (Ifloat x) \<in> { Ifloat (lb_exp prec x) .. Ifloat (ub_exp prec x)}"
-proof -
-  let "?lb_exp_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x"
-  let "?ub_exp_horner x" = "ub_exp_horner prec (get_odd (prec + 2)) 1 1 x"
-
-  have "Ifloat x \<le> 0" and "\<not> x > 0" using `x \<le> 0` unfolding le_float_def less_float_def by auto
-  show ?thesis
-  proof (cases "x < - 1")
-    case False hence "- 1 \<le> Ifloat x" unfolding less_float_def by auto
-    show ?thesis
-    proof (cases "?lb_exp_horner x \<le> 0")
-      from `\<not> x < - 1` have "- 1 \<le> Ifloat x" unfolding less_float_def by auto
-      hence "exp (- 1) \<le> exp (Ifloat x)" unfolding exp_le_cancel_iff .
-      from order_trans[OF exp_m1_ge_quarter this]
-      have "Ifloat (Float 1 -2) \<le> exp (Ifloat x)" unfolding Float_num .
-      moreover case True
-      ultimately show ?thesis using bnds_exp_horner `Ifloat x \<le> 0` `\<not> x > 0` `\<not> x < - 1` by auto
-    next
-      case False thus ?thesis using bnds_exp_horner `Ifloat x \<le> 0` `\<not> x > 0` `\<not> x < - 1` by (auto simp add: Let_def)
-    qed
-  next
-    case True
-    
-    obtain m e where Float_floor: "floor_fl x = Float m e" by (cases "floor_fl x", auto)
-    let ?num = "nat (- m) * 2 ^ nat e"
-    
-    have "Ifloat (floor_fl x) < - 1" using floor_fl `x < - 1` unfolding le_float_def less_float_def Ifloat_minus Ifloat_1 by (rule order_le_less_trans)
-    hence "Ifloat (floor_fl x) < 0" unfolding Float_floor Ifloat.simps using zero_less_pow2[of xe] by auto
-    hence "m < 0"
-      unfolding less_float_def Ifloat_0 Float_floor Ifloat.simps
-      unfolding pos_prod_lt[OF zero_less_pow2[of e], unfolded real_mult_commute] by auto
-    hence "1 \<le> - m" by auto
-    hence "0 < nat (- m)" by auto
-    moreover
-    have "0 \<le> e" using floor_pos_exp Float_floor[symmetric] by auto
-    hence "(0::nat) < 2 ^ nat e" by auto
-    ultimately have "0 < ?num"  by auto
-    hence "real ?num \<noteq> 0" by auto
-    have e_nat: "int (nat e) = e" using `0 \<le> e` by auto
-    have num_eq: "real ?num = Ifloat (- floor_fl x)" using `0 < nat (- m)`
-      unfolding Float_floor Ifloat_minus Ifloat.simps real_of_nat_mult pow2_int[of "nat e", unfolded e_nat] realpow_real_of_nat[symmetric] by auto
-    have "0 < - floor_fl x" using `0 < ?num`[unfolded real_of_nat_less_iff[symmetric]] unfolding less_float_def num_eq[symmetric] Ifloat_0 real_of_nat_zero .
-    hence "Ifloat (floor_fl x) < 0" unfolding less_float_def by auto
-    
-    have "exp (Ifloat x) \<le> Ifloat (ub_exp prec x)"
-    proof -
-      have div_less_zero: "Ifloat (float_divr prec x (- floor_fl x)) \<le> 0" 
-	using float_divr_nonpos_pos_upper_bound[OF `x \<le> 0` `0 < - floor_fl x`] unfolding le_float_def Ifloat_0 .
-      
-      have "exp (Ifloat x) = exp (real ?num * (Ifloat x / real ?num))" using `real ?num \<noteq> 0` by auto
-      also have "\<dots> = exp (Ifloat x / real ?num) ^ ?num" unfolding exp_real_of_nat_mult ..
-      also have "\<dots> \<le> exp (Ifloat (float_divr prec x (- floor_fl x))) ^ ?num" unfolding num_eq
-	by (rule power_mono, rule exp_le_cancel_iff[THEN iffD2], rule float_divr) auto
-      also have "\<dots> \<le> Ifloat ((?ub_exp_horner (float_divr prec x (- floor_fl x))) ^ ?num)" unfolding float_power
-	by (rule power_mono, rule bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct2], auto)
-      finally show ?thesis unfolding ub_exp.simps if_not_P[OF `\<not> 0 < x`] if_P[OF `x < - 1`] float.cases Float_floor Let_def .
-    qed
-    moreover 
-    have "Ifloat (lb_exp prec x) \<le> exp (Ifloat x)"
-    proof -
-      let ?divl = "float_divl prec x (- Float m e)"
-      let ?horner = "?lb_exp_horner ?divl"
-      
-      show ?thesis
-      proof (cases "?horner \<le> 0")
-	case False hence "0 \<le> Ifloat ?horner" unfolding le_float_def by auto
-	
-	have div_less_zero: "Ifloat (float_divl prec x (- floor_fl x)) \<le> 0"
-	  using `Ifloat (floor_fl x) < 0` `Ifloat x \<le> 0` by (auto intro!: order_trans[OF float_divl] divide_nonpos_neg)
-	
-	have "Ifloat ((?lb_exp_horner (float_divl prec x (- floor_fl x))) ^ ?num) \<le>  
-          exp (Ifloat (float_divl prec x (- floor_fl x))) ^ ?num" unfolding float_power 
-	  using `0 \<le> Ifloat ?horner`[unfolded Float_floor[symmetric]] bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct1] by (auto intro!: power_mono)
-	also have "\<dots> \<le> exp (Ifloat x / real ?num) ^ ?num" unfolding num_eq
-	  using float_divl by (auto intro!: power_mono simp del: Ifloat_minus)
-	also have "\<dots> = exp (real ?num * (Ifloat x / real ?num))" unfolding exp_real_of_nat_mult ..
-	also have "\<dots> = exp (Ifloat x)" using `real ?num \<noteq> 0` by auto
-	finally show ?thesis
-	  unfolding lb_exp.simps if_not_P[OF `\<not> 0 < x`] if_P[OF `x < - 1`] float.cases Float_floor Let_def if_not_P[OF False] by auto
-      next
-	case True
-	have "Ifloat (floor_fl x) \<noteq> 0" and "Ifloat (floor_fl x) \<le> 0" using `Ifloat (floor_fl x) < 0` by auto
-	from divide_right_mono_neg[OF floor_fl[of x] `Ifloat (floor_fl x) \<le> 0`, unfolded divide_self[OF `Ifloat (floor_fl x) \<noteq> 0`]]
-	have "- 1 \<le> Ifloat x / Ifloat (- floor_fl x)" unfolding Ifloat_minus by auto
-	from order_trans[OF exp_m1_ge_quarter this[unfolded exp_le_cancel_iff[where x="- 1", symmetric]]]
-	have "Ifloat (Float 1 -2) \<le> exp (Ifloat x / Ifloat (- floor_fl x))" unfolding Float_num .
-	hence "Ifloat (Float 1 -2) ^ ?num \<le> exp (Ifloat x / Ifloat (- floor_fl x)) ^ ?num"
-	  by (auto intro!: power_mono simp add: Float_num)
-	also have "\<dots> = exp (Ifloat x)" unfolding num_eq exp_real_of_nat_mult[symmetric] using `Ifloat (floor_fl x) \<noteq> 0` by auto
-	finally show ?thesis
-	  unfolding lb_exp.simps if_not_P[OF `\<not> 0 < x`] if_P[OF `x < - 1`] float.cases Float_floor Let_def if_P[OF True] float_power .
-      qed
-    qed
-    ultimately show ?thesis by auto
-  qed
-qed
-
-lemma exp_boundaries: "exp (Ifloat x) \<in> { Ifloat (lb_exp prec x) .. Ifloat (ub_exp prec x)}"
-proof -
-  show ?thesis
-  proof (cases "0 < x")
-    case False hence "x \<le> 0" unfolding less_float_def le_float_def by auto 
-    from exp_boundaries'[OF this] show ?thesis .
-  next
-    case True hence "-x \<le> 0" unfolding less_float_def le_float_def by auto
-    
-    have "Ifloat (lb_exp prec x) \<le> exp (Ifloat x)"
-    proof -
-      from exp_boundaries'[OF `-x \<le> 0`]
-      have ub_exp: "exp (- Ifloat x) \<le> Ifloat (ub_exp prec (-x))" unfolding atLeastAtMost_iff Ifloat_minus by auto
-      
-      have "Ifloat (float_divl prec 1 (ub_exp prec (-x))) \<le> Ifloat 1 / Ifloat (ub_exp prec (-x))" using float_divl .
-      also have "Ifloat 1 / Ifloat (ub_exp prec (-x)) \<le> exp (Ifloat x)"
-	using ub_exp[unfolded inverse_le_iff_le[OF order_less_le_trans[OF exp_gt_zero ub_exp] exp_gt_zero, symmetric]]
-	unfolding exp_minus nonzero_inverse_inverse_eq[OF exp_not_eq_zero] inverse_eq_divide by auto
-      finally show ?thesis unfolding lb_exp.simps if_P[OF True] .
-    qed
-    moreover
-    have "exp (Ifloat x) \<le> Ifloat (ub_exp prec x)"
-    proof -
-      have "\<not> 0 < -x" using `0 < x` unfolding less_float_def by auto
-      
-      from exp_boundaries'[OF `-x \<le> 0`]
-      have lb_exp: "Ifloat (lb_exp prec (-x)) \<le> exp (- Ifloat x)" unfolding atLeastAtMost_iff Ifloat_minus by auto
-      
-      have "exp (Ifloat x) \<le> Ifloat 1 / Ifloat (lb_exp prec (-x))"
-	using lb_exp[unfolded inverse_le_iff_le[OF exp_gt_zero lb_exp_pos[OF `\<not> 0 < -x`, unfolded less_float_def Ifloat_0], symmetric]]
-	unfolding exp_minus nonzero_inverse_inverse_eq[OF exp_not_eq_zero] inverse_eq_divide Ifloat_1 by auto
-      also have "\<dots> \<le> Ifloat (float_divr prec 1 (lb_exp prec (-x)))" using float_divr .
-      finally show ?thesis unfolding ub_exp.simps if_P[OF True] .
-    qed
-    ultimately show ?thesis by auto
-  qed
-qed
-
-lemma bnds_exp: "\<forall> x lx ux. (l, u) = (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux} \<longrightarrow> Ifloat l \<le> exp x \<and> exp x \<le> Ifloat u"
-proof (rule allI, rule allI, rule allI, rule impI)
-  fix x lx ux
-  assume "(l, u) = (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux}"
-  hence l: "lb_exp prec lx = l " and u: "ub_exp prec ux = u" and x: "x \<in> {Ifloat lx .. Ifloat ux}" by auto
-
-  { from exp_boundaries[of lx prec, unfolded l]
-    have "Ifloat l \<le> exp (Ifloat lx)" by (auto simp del: lb_exp.simps)
-    also have "\<dots> \<le> exp x" using x by auto
-    finally have "Ifloat l \<le> exp x" .
-  } moreover
-  { have "exp x \<le> exp (Ifloat ux)" using x by auto
-    also have "\<dots> \<le> Ifloat u" using exp_boundaries[of ux prec, unfolded u] by (auto simp del: ub_exp.simps)
-    finally have "exp x \<le> Ifloat u" .
-  } ultimately show "Ifloat l \<le> exp x \<and> exp x \<le> Ifloat u" ..
-qed
-
-section "Logarithm"
-
-subsection "Compute the logarithm series"
-
-fun ub_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" 
-and lb_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
-"ub_ln_horner prec 0 i x       = 0" |
-"ub_ln_horner prec (Suc n) i x = rapprox_rat prec 1 (int i) - x * lb_ln_horner prec n (Suc i) x" |
-"lb_ln_horner prec 0 i x       = 0" |
-"lb_ln_horner prec (Suc n) i x = lapprox_rat prec 1 (int i) - x * ub_ln_horner prec n (Suc i) x"
-
-lemma ln_bounds:
-  assumes "0 \<le> x" and "x < 1"
-  shows "(\<Sum>i=0..<2*n. -1^i * (1 / real (i + 1)) * x^(Suc i)) \<le> ln (x + 1)" (is "?lb")
-  and "ln (x + 1) \<le> (\<Sum>i=0..<2*n + 1. -1^i * (1 / real (i + 1)) * x^(Suc i))" (is "?ub")
-proof -
-  let "?a n" = "(1/real (n +1)) * x^(Suc n)"
-
-  have ln_eq: "(\<Sum> i. -1^i * ?a i) = ln (x + 1)"
-    using ln_series[of "x + 1"] `0 \<le> x` `x < 1` by auto
-
-  have "norm x < 1" using assms by auto
-  have "?a ----> 0" unfolding Suc_plus1[symmetric] inverse_eq_divide[symmetric] 
-    using LIMSEQ_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_Suc[OF LIMSEQ_power_zero[OF `norm x < 1`]]] by auto
-  { fix n have "0 \<le> ?a n" by (rule mult_nonneg_nonneg, auto intro!: mult_nonneg_nonneg simp add: `0 \<le> x`) }
-  { fix n have "?a (Suc n) \<le> ?a n" unfolding inverse_eq_divide[symmetric]
-    proof (rule mult_mono)
-      show "0 \<le> x ^ Suc (Suc n)" by (auto intro!: mult_nonneg_nonneg simp add: `0 \<le> x`)
-      have "x ^ Suc (Suc n) \<le> x ^ Suc n * 1" unfolding power_Suc2 real_mult_assoc[symmetric] 
-	by (rule mult_left_mono, fact less_imp_le[OF `x < 1`], auto intro!: mult_nonneg_nonneg simp add: `0 \<le> x`)
-      thus "x ^ Suc (Suc n) \<le> x ^ Suc n" by auto
-    qed auto }
-  from summable_Leibniz'(2,4)[OF `?a ----> 0` `\<And>n. 0 \<le> ?a n`, OF `\<And>n. ?a (Suc n) \<le> ?a n`, unfolded ln_eq]
-  show "?lb" and "?ub" by auto
-qed
-
-lemma ln_float_bounds: 
-  assumes "0 \<le> Ifloat x" and "Ifloat x < 1"
-  shows "Ifloat (x * lb_ln_horner prec (get_even n) 1 x) \<le> ln (Ifloat x + 1)" (is "?lb \<le> ?ln")
-  and "ln (Ifloat x + 1) \<le> Ifloat (x * ub_ln_horner prec (get_odd n) 1 x)" (is "?ln \<le> ?ub")
-proof -
-  obtain ev where ev: "get_even n = 2 * ev" using get_even_double ..
-  obtain od where od: "get_odd n = 2 * od + 1" using get_odd_double ..
-
-  let "?s n" = "-1^n * (1 / real (1 + n)) * (Ifloat x)^(Suc n)"
-
-  have "?lb \<le> setsum ?s {0 ..< 2 * ev}" unfolding power_Suc2 real_mult_assoc[symmetric] Ifloat_mult setsum_left_distrib[symmetric] unfolding real_mult_commute[of "Ifloat x"] ev
-    using horner_bounds(1)[where G="\<lambda> i k. Suc k" and F="\<lambda>x. x" and f="\<lambda>x. x" and lb="\<lambda>n i k x. lb_ln_horner prec n k x" and ub="\<lambda>n i k x. ub_ln_horner prec n k x" and j'=1 and n="2*ev",
-      OF `0 \<le> Ifloat x` refl lb_ln_horner.simps ub_ln_horner.simps] `0 \<le> Ifloat x`
-    by (rule mult_right_mono)
-  also have "\<dots> \<le> ?ln" using ln_bounds(1)[OF `0 \<le> Ifloat x` `Ifloat x < 1`] by auto
-  finally show "?lb \<le> ?ln" . 
-
-  have "?ln \<le> setsum ?s {0 ..< 2 * od + 1}" using ln_bounds(2)[OF `0 \<le> Ifloat x` `Ifloat x < 1`] by auto
-  also have "\<dots> \<le> ?ub" unfolding power_Suc2 real_mult_assoc[symmetric] Ifloat_mult setsum_left_distrib[symmetric] unfolding real_mult_commute[of "Ifloat x"] od
-    using horner_bounds(2)[where G="\<lambda> i k. Suc k" and F="\<lambda>x. x" and f="\<lambda>x. x" and lb="\<lambda>n i k x. lb_ln_horner prec n k x" and ub="\<lambda>n i k x. ub_ln_horner prec n k x" and j'=1 and n="2*od+1",
-      OF `0 \<le> Ifloat x` refl lb_ln_horner.simps ub_ln_horner.simps] `0 \<le> Ifloat x`
-    by (rule mult_right_mono)
-  finally show "?ln \<le> ?ub" . 
-qed
-
-lemma ln_add: assumes "0 < x" and "0 < y" shows "ln (x + y) = ln x + ln (1 + y / x)"
-proof -
-  have "x \<noteq> 0" using assms by auto
-  have "x + y = x * (1 + y / x)" unfolding right_distrib times_divide_eq_right nonzero_mult_divide_cancel_left[OF `x \<noteq> 0`] by auto
-  moreover 
-  have "0 < y / x" using assms divide_pos_pos by auto
-  hence "0 < 1 + y / x" by auto
-  ultimately show ?thesis using ln_mult assms by auto
-qed
-
-subsection "Compute the logarithm of 2"
-
-definition ub_ln2 where "ub_ln2 prec = (let third = rapprox_rat (max prec 1) 1 3 
-                                        in (Float 1 -1 * ub_ln_horner prec (get_odd prec) 1 (Float 1 -1)) + 
-                                           (third * ub_ln_horner prec (get_odd prec) 1 third))"
-definition lb_ln2 where "lb_ln2 prec = (let third = lapprox_rat prec 1 3 
-                                        in (Float 1 -1 * lb_ln_horner prec (get_even prec) 1 (Float 1 -1)) + 
-                                           (third * lb_ln_horner prec (get_even prec) 1 third))"
-
-lemma ub_ln2: "ln 2 \<le> Ifloat (ub_ln2 prec)" (is "?ub_ln2")
-  and lb_ln2: "Ifloat (lb_ln2 prec) \<le> ln 2" (is "?lb_ln2")
-proof -
-  let ?uthird = "rapprox_rat (max prec 1) 1 3"
-  let ?lthird = "lapprox_rat prec 1 3"
-
-  have ln2_sum: "ln 2 = ln (1/2 + 1) + ln (1 / 3 + 1)"
-    using ln_add[of "3 / 2" "1 / 2"] by auto
-  have lb3: "Ifloat ?lthird \<le> 1 / 3" using lapprox_rat[of prec 1 3] by auto
-  hence lb3_ub: "Ifloat ?lthird < 1" by auto
-  have lb3_lb: "0 \<le> Ifloat ?lthird" using lapprox_rat_bottom[of 1 3] by auto
-  have ub3: "1 / 3 \<le> Ifloat ?uthird" using rapprox_rat[of 1 3] by auto
-  hence ub3_lb: "0 \<le> Ifloat ?uthird" by auto
-
-  have lb2: "0 \<le> Ifloat (Float 1 -1)" and ub2: "Ifloat (Float 1 -1) < 1" unfolding Float_num by auto
-
-  have "0 \<le> (1::int)" and "0 < (3::int)" by auto
-  have ub3_ub: "Ifloat ?uthird < 1" unfolding rapprox_rat.simps(2)[OF `0 \<le> 1` `0 < 3`]
-    by (rule rapprox_posrat_less1, auto)
-
-  have third_gt0: "(0 :: real) < 1 / 3 + 1" by auto
-  have uthird_gt0: "0 < Ifloat ?uthird + 1" using ub3_lb by auto
-  have lthird_gt0: "0 < Ifloat ?lthird + 1" using lb3_lb by auto
-
-  show ?ub_ln2 unfolding ub_ln2_def Let_def Ifloat_add ln2_sum Float_num(4)[symmetric]
-  proof (rule add_mono, fact ln_float_bounds(2)[OF lb2 ub2])
-    have "ln (1 / 3 + 1) \<le> ln (Ifloat ?uthird + 1)" unfolding ln_le_cancel_iff[OF third_gt0 uthird_gt0] using ub3 by auto
-    also have "\<dots> \<le> Ifloat (?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird)"
-      using ln_float_bounds(2)[OF ub3_lb ub3_ub] .
-    finally show "ln (1 / 3 + 1) \<le> Ifloat (?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird)" .
-  qed
-  show ?lb_ln2 unfolding lb_ln2_def Let_def Ifloat_add ln2_sum Float_num(4)[symmetric]
-  proof (rule add_mono, fact ln_float_bounds(1)[OF lb2 ub2])
-    have "Ifloat (?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird) \<le> ln (Ifloat ?lthird + 1)"
-      using ln_float_bounds(1)[OF lb3_lb lb3_ub] .
-    also have "\<dots> \<le> ln (1 / 3 + 1)" unfolding ln_le_cancel_iff[OF lthird_gt0 third_gt0] using lb3 by auto
-    finally show "Ifloat (?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird) \<le> ln (1 / 3 + 1)" .
-  qed
-qed
-
-subsection "Compute the logarithm in the entire domain"
-
-function ub_ln :: "nat \<Rightarrow> float \<Rightarrow> float option" and lb_ln :: "nat \<Rightarrow> float \<Rightarrow> float option" where
-"ub_ln prec x = (if x \<le> 0         then None
-            else if x < 1         then Some (- the (lb_ln prec (float_divl (max prec 1) 1 x)))
-            else let horner = \<lambda>x. (x - 1) * ub_ln_horner prec (get_odd prec) 1 (x - 1) in
-                 if x < Float 1 1 then Some (horner x)
-                                  else let l = bitlen (mantissa x) - 1 in 
-                                       Some (ub_ln2 prec * (Float (scale x + l) 0) + horner (Float (mantissa x) (- l))))" |
-"lb_ln prec x = (if x \<le> 0         then None
-            else if x < 1         then Some (- the (ub_ln prec (float_divr prec 1 x)))
-            else let horner = \<lambda>x. (x - 1) * lb_ln_horner prec (get_even prec) 1 (x - 1) in
-                 if x < Float 1 1 then Some (horner x)
-                                  else let l = bitlen (mantissa x) - 1 in 
-                                       Some (lb_ln2 prec * (Float (scale x + l) 0) + horner (Float (mantissa x) (- l))))"
-by pat_completeness auto
-
-termination proof (relation "measure (\<lambda> v. let (prec, x) = sum_case id id v in (if x < 1 then 1 else 0))", auto)
-  fix prec x assume "\<not> x \<le> 0" and "x < 1" and "float_divl (max prec (Suc 0)) 1 x < 1"
-  hence "0 < x" and "0 < max prec (Suc 0)" unfolding less_float_def le_float_def by auto
-  from float_divl_pos_less1_bound[OF `0 < x` `x < 1` `0 < max prec (Suc 0)`]
-  show False using `float_divl (max prec (Suc 0)) 1 x < 1` unfolding less_float_def le_float_def by auto
-next
-  fix prec x assume "\<not> x \<le> 0" and "x < 1" and "float_divr prec 1 x < 1"
-  hence "0 < x" unfolding less_float_def le_float_def by auto
-  from float_divr_pos_less1_lower_bound[OF `0 < x` `x < 1`, of prec]
-  show False using `float_divr prec 1 x < 1` unfolding less_float_def le_float_def by auto
-qed
-
-lemma ln_shifted_float: assumes "0 < m" shows "ln (Ifloat (Float m e)) = ln 2 * real (e + (bitlen m - 1)) + ln (Ifloat (Float m (- (bitlen m - 1))))"
-proof -
-  let ?B = "2^nat (bitlen m - 1)"
-  have "0 < real m" and "\<And>X. (0 :: real) < 2^X" and "0 < (2 :: real)" and "m \<noteq> 0" using assms by auto
-  hence "0 \<le> bitlen m - 1" using bitlen_ge1[OF `m \<noteq> 0`] by auto
-  show ?thesis 
-  proof (cases "0 \<le> e")
-    case True
-    show ?thesis unfolding normalized_float[OF `m \<noteq> 0`]
-      unfolding ln_div[OF `0 < real m` `0 < ?B`] real_of_int_add ln_realpow[OF `0 < 2`] 
-      unfolding Ifloat_ge0_exp[OF True] ln_mult[OF `0 < real m` `0 < 2^nat e`] 
-      ln_realpow[OF `0 < 2`] algebra_simps using `0 \<le> bitlen m - 1` True by auto
-  next
-    case False hence "0 < -e" by auto
-    hence pow_gt0: "(0::real) < 2^nat (-e)" by auto
-    hence inv_gt0: "(0::real) < inverse (2^nat (-e))" by auto
-    show ?thesis unfolding normalized_float[OF `m \<noteq> 0`]
-      unfolding ln_div[OF `0 < real m` `0 < ?B`] real_of_int_add ln_realpow[OF `0 < 2`] 
-      unfolding Ifloat_nge0_exp[OF False] ln_mult[OF `0 < real m` inv_gt0] ln_inverse[OF pow_gt0]
-      ln_realpow[OF `0 < 2`] algebra_simps using `0 \<le> bitlen m - 1` False by auto
-  qed
-qed
-
-lemma ub_ln_lb_ln_bounds': assumes "1 \<le> x"
-  shows "Ifloat (the (lb_ln prec x)) \<le> ln (Ifloat x) \<and> ln (Ifloat x) \<le> Ifloat (the (ub_ln prec x))"
-  (is "?lb \<le> ?ln \<and> ?ln \<le> ?ub")
-proof (cases "x < Float 1 1")
-  case True hence "Ifloat (x - 1) < 1" unfolding less_float_def Float_num by auto
-  have "\<not> x \<le> 0" and "\<not> x < 1" using `1 \<le> x` unfolding less_float_def le_float_def by auto
-  hence "0 \<le> Ifloat (x - 1)" using `1 \<le> x` unfolding less_float_def Float_num by auto
-  show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps Let_def
-    using ln_float_bounds[OF `0 \<le> Ifloat (x - 1)` `Ifloat (x - 1) < 1`] `\<not> x \<le> 0` `\<not> x < 1` True by auto
-next
-  case False
-  have "\<not> x \<le> 0" and "\<not> x < 1" "0 < x" using `1 \<le> x` unfolding less_float_def le_float_def by auto
-  show ?thesis
-  proof (cases x)
-    case (Float m e)
-    let ?s = "Float (e + (bitlen m - 1)) 0"
-    let ?x = "Float m (- (bitlen m - 1))"
-
-    have "0 < m" and "m \<noteq> 0" using float_pos_m_pos `0 < x` Float by auto
-
-    {
-      have "Ifloat (lb_ln2 prec * ?s) \<le> ln 2 * real (e + (bitlen m - 1))" (is "?lb2 \<le> _")
-	unfolding Ifloat_mult Ifloat_ge0_exp[OF order_refl] nat_0 realpow_0 mult_1_right
-	using lb_ln2[of prec]
-      proof (rule mult_right_mono)
-	have "1 \<le> Float m e" using `1 \<le> x` Float unfolding le_float_def by auto
-	from float_gt1_scale[OF this]
-	show "0 \<le> real (e + (bitlen m - 1))" by auto
-      qed
-      moreover
-      from bitlen_div[OF `0 < m`, unfolded normalized_float[OF `m \<noteq> 0`, symmetric]]
-      have "0 \<le> Ifloat (?x - 1)" and "Ifloat (?x - 1) < 1" by auto
-      from ln_float_bounds(1)[OF this]
-      have "Ifloat ((?x - 1) * lb_ln_horner prec (get_even prec) 1 (?x - 1)) \<le> ln (Ifloat ?x)" (is "?lb_horner \<le> _") by auto
-      ultimately have "?lb2 + ?lb_horner \<le> ln (Ifloat x)"
-	unfolding Float ln_shifted_float[OF `0 < m`, of e] by auto
-    } 
-    moreover
-    {
-      from bitlen_div[OF `0 < m`, unfolded normalized_float[OF `m \<noteq> 0`, symmetric]]
-      have "0 \<le> Ifloat (?x - 1)" and "Ifloat (?x - 1) < 1" by auto
-      from ln_float_bounds(2)[OF this]
-      have "ln (Ifloat ?x) \<le> Ifloat ((?x - 1) * ub_ln_horner prec (get_odd prec) 1 (?x - 1))" (is "_ \<le> ?ub_horner") by auto
-      moreover
-      have "ln 2 * real (e + (bitlen m - 1)) \<le> Ifloat (ub_ln2 prec * ?s)" (is "_ \<le> ?ub2")
-	unfolding Ifloat_mult Ifloat_ge0_exp[OF order_refl] nat_0 realpow_0 mult_1_right
-	using ub_ln2[of prec] 
-      proof (rule mult_right_mono)
-	have "1 \<le> Float m e" using `1 \<le> x` Float unfolding le_float_def by auto
-	from float_gt1_scale[OF this]
-	show "0 \<le> real (e + (bitlen m - 1))" by auto
-      qed
-      ultimately have "ln (Ifloat x) \<le> ?ub2 + ?ub_horner"
-	unfolding Float ln_shifted_float[OF `0 < m`, of e] by auto
-    }
-    ultimately show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps
-      unfolding if_not_P[OF `\<not> x \<le> 0`] if_not_P[OF `\<not> x < 1`] if_not_P[OF False] Let_def
-      unfolding scale.simps[of m e, unfolded Float[symmetric]] mantissa.simps[of m e, unfolded Float[symmetric]] Ifloat_add by auto
-  qed
-qed
-
-lemma ub_ln_lb_ln_bounds: assumes "0 < x"
-  shows "Ifloat (the (lb_ln prec x)) \<le> ln (Ifloat x) \<and> ln (Ifloat x) \<le> Ifloat (the (ub_ln prec x))"
-  (is "?lb \<le> ?ln \<and> ?ln \<le> ?ub")
-proof (cases "x < 1")
-  case False hence "1 \<le> x" unfolding less_float_def le_float_def by auto
-  show ?thesis using ub_ln_lb_ln_bounds'[OF `1 \<le> x`] .
-next
-  case True have "\<not> x \<le> 0" using `0 < x` unfolding less_float_def le_float_def by auto
-
-  have "0 < Ifloat x" and "Ifloat x \<noteq> 0" using `0 < x` unfolding less_float_def by auto
-  hence A: "0 < 1 / Ifloat x" by auto
-
-  {
-    let ?divl = "float_divl (max prec 1) 1 x"
-    have A': "1 \<le> ?divl" using float_divl_pos_less1_bound[OF `0 < x` `x < 1`] unfolding le_float_def less_float_def by auto
-    hence B: "0 < Ifloat ?divl" unfolding le_float_def by auto
-    
-    have "ln (Ifloat ?divl) \<le> ln (1 / Ifloat x)" unfolding ln_le_cancel_iff[OF B A] using float_divl[of _ 1 x] by auto
-    hence "ln (Ifloat x) \<le> - ln (Ifloat ?divl)" unfolding nonzero_inverse_eq_divide[OF `Ifloat x \<noteq> 0`, symmetric] ln_inverse[OF `0 < Ifloat x`] by auto
-    from this ub_ln_lb_ln_bounds'[OF A', THEN conjunct1, THEN le_imp_neg_le] 
-    have "?ln \<le> Ifloat (- the (lb_ln prec ?divl))" unfolding Ifloat_minus by (rule order_trans)
-  } moreover
-  {
-    let ?divr = "float_divr prec 1 x"
-    have A': "1 \<le> ?divr" using float_divr_pos_less1_lower_bound[OF `0 < x` `x < 1`] unfolding le_float_def less_float_def by auto
-    hence B: "0 < Ifloat ?divr" unfolding le_float_def by auto
-    
-    have "ln (1 / Ifloat x) \<le> ln (Ifloat ?divr)" unfolding ln_le_cancel_iff[OF A B] using float_divr[of 1 x] by auto
-    hence "- ln (Ifloat ?divr) \<le> ln (Ifloat x)" unfolding nonzero_inverse_eq_divide[OF `Ifloat x \<noteq> 0`, symmetric] ln_inverse[OF `0 < Ifloat x`] by auto
-    from ub_ln_lb_ln_bounds'[OF A', THEN conjunct2, THEN le_imp_neg_le] this
-    have "Ifloat (- the (ub_ln prec ?divr)) \<le> ?ln" unfolding Ifloat_minus by (rule order_trans)
-  }
-  ultimately show ?thesis unfolding lb_ln.simps[where x=x]  ub_ln.simps[where x=x]
-    unfolding if_not_P[OF `\<not> x \<le> 0`] if_P[OF True] by auto
-qed
-
-lemma lb_ln: assumes "Some y = lb_ln prec x"
-  shows "Ifloat y \<le> ln (Ifloat x)" and "0 < Ifloat x"
-proof -
-  have "0 < x"
-  proof (rule ccontr)
-    assume "\<not> 0 < x" hence "x \<le> 0" unfolding le_float_def less_float_def by auto
-    thus False using assms by auto
-  qed
-  thus "0 < Ifloat x" unfolding less_float_def by auto
-  have "Ifloat (the (lb_ln prec x)) \<le> ln (Ifloat x)" using ub_ln_lb_ln_bounds[OF `0 < x`] ..
-  thus "Ifloat y \<le> ln (Ifloat x)" unfolding assms[symmetric] by auto
-qed
-
-lemma ub_ln: assumes "Some y = ub_ln prec x"
-  shows "ln (Ifloat x) \<le> Ifloat y" and "0 < Ifloat x"
-proof -
-  have "0 < x"
-  proof (rule ccontr)
-    assume "\<not> 0 < x" hence "x \<le> 0" unfolding le_float_def less_float_def by auto
-    thus False using assms by auto
-  qed
-  thus "0 < Ifloat x" unfolding less_float_def by auto
-  have "ln (Ifloat x) \<le> Ifloat (the (ub_ln prec x))" using ub_ln_lb_ln_bounds[OF `0 < x`] ..
-  thus "ln (Ifloat x) \<le> Ifloat y" unfolding assms[symmetric] by auto
-qed
-
-lemma bnds_ln: "\<forall> x lx ux. (Some l, Some u) = (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux} \<longrightarrow> Ifloat l \<le> ln x \<and> ln x \<le> Ifloat u"
-proof (rule allI, rule allI, rule allI, rule impI)
-  fix x lx ux
-  assume "(Some l, Some u) = (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux}"
-  hence l: "Some l = lb_ln prec lx " and u: "Some u = ub_ln prec ux" and x: "x \<in> {Ifloat lx .. Ifloat ux}" by auto
-
-  have "ln (Ifloat ux) \<le> Ifloat u" and "0 < Ifloat ux" using ub_ln u by auto
-  have "Ifloat l \<le> ln (Ifloat lx)" and "0 < Ifloat lx" and "0 < x" using lb_ln[OF l] x by auto
-
-  from ln_le_cancel_iff[OF `0 < Ifloat lx` `0 < x`] `Ifloat l \<le> ln (Ifloat lx)` 
-  have "Ifloat l \<le> ln x" using x unfolding atLeastAtMost_iff by auto
-  moreover
-  from ln_le_cancel_iff[OF `0 < x` `0 < Ifloat ux`] `ln (Ifloat ux) \<le> Ifloat u` 
-  have "ln x \<le> Ifloat u" using x unfolding atLeastAtMost_iff by auto
-  ultimately show "Ifloat l \<le> ln x \<and> ln x \<le> Ifloat u" ..
-qed
-
-
-section "Implement floatarith"
-
-subsection "Define syntax and semantics"
-
-datatype floatarith
-  = Add floatarith floatarith
-  | Minus floatarith
-  | Mult floatarith floatarith
-  | Inverse floatarith
-  | Sin floatarith
-  | Cos floatarith
-  | Arctan floatarith
-  | Abs floatarith
-  | Max floatarith floatarith
-  | Min floatarith floatarith
-  | Pi
-  | Sqrt floatarith
-  | Exp floatarith
-  | Ln floatarith
-  | Power floatarith nat
-  | Atom nat
-  | Num float
-
-fun Ifloatarith :: "floatarith \<Rightarrow> real list \<Rightarrow> real"
-where
-"Ifloatarith (Add a b) vs   = (Ifloatarith a vs) + (Ifloatarith b vs)" |
-"Ifloatarith (Minus a) vs    = - (Ifloatarith a vs)" |
-"Ifloatarith (Mult a b) vs   = (Ifloatarith a vs) * (Ifloatarith b vs)" |
-"Ifloatarith (Inverse a) vs  = inverse (Ifloatarith a vs)" |
-"Ifloatarith (Sin a) vs      = sin (Ifloatarith a vs)" |
-"Ifloatarith (Cos a) vs      = cos (Ifloatarith a vs)" |
-"Ifloatarith (Arctan a) vs   = arctan (Ifloatarith a vs)" |
-"Ifloatarith (Min a b) vs    = min (Ifloatarith a vs) (Ifloatarith b vs)" |
-"Ifloatarith (Max a b) vs    = max (Ifloatarith a vs) (Ifloatarith b vs)" |
-"Ifloatarith (Abs a) vs      = abs (Ifloatarith a vs)" |
-"Ifloatarith Pi vs           = pi" |
-"Ifloatarith (Sqrt a) vs     = sqrt (Ifloatarith a vs)" |
-"Ifloatarith (Exp a) vs      = exp (Ifloatarith a vs)" |
-"Ifloatarith (Ln a) vs       = ln (Ifloatarith a vs)" |
-"Ifloatarith (Power a n) vs  = (Ifloatarith a vs)^n" |
-"Ifloatarith (Num f) vs      = Ifloat f" |
-"Ifloatarith (Atom n) vs     = vs ! n"
-
-subsection "Implement approximation function"
-
-fun lift_bin :: "(float * float) option \<Rightarrow> (float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> (float option * float option)) \<Rightarrow> (float * float) option" where
-"lift_bin (Some (l1, u1)) (Some (l2, u2)) f = (case (f l1 u1 l2 u2) of (Some l, Some u) \<Rightarrow> Some (l, u)
-                                                                     | t \<Rightarrow> None)" |
-"lift_bin a b f = None"
-
-fun lift_bin' :: "(float * float) option \<Rightarrow> (float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> (float * float)) \<Rightarrow> (float * float) option" where
-"lift_bin' (Some (l1, u1)) (Some (l2, u2)) f = Some (f l1 u1 l2 u2)" |
-"lift_bin' a b f = None"
-
-fun lift_un :: "(float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> ((float option) * (float option))) \<Rightarrow> (float * float) option" where
-"lift_un (Some (l1, u1)) f = (case (f l1 u1) of (Some l, Some u) \<Rightarrow> Some (l, u)
-                                             | t \<Rightarrow> None)" |
-"lift_un b f = None"
-
-fun lift_un' :: "(float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> (float * float)) \<Rightarrow> (float * float) option" where
-"lift_un' (Some (l1, u1)) f = Some (f l1 u1)" |
-"lift_un' b f = None"
-
-fun bounded_by :: "real list \<Rightarrow> (float * float) list \<Rightarrow> bool " where
-bounded_by_Cons: "bounded_by (v#vs) ((l, u)#bs) = ((Ifloat l \<le> v \<and> v \<le> Ifloat u) \<and> bounded_by vs bs)" |
-bounded_by_Nil: "bounded_by [] [] = True" |
-"bounded_by _ _ = False"
-
-lemma bounded_by: assumes "bounded_by vs bs" and "i < length bs"
-  shows "Ifloat (fst (bs ! i)) \<le> vs ! i \<and> vs ! i \<le> Ifloat (snd (bs ! i))"
-  using `bounded_by vs bs` and `i < length bs`
-proof (induct arbitrary: i rule: bounded_by.induct)
-  fix v :: real and vs :: "real list" and l u :: float and bs :: "(float * float) list" and i :: nat
-  assume hyp: "\<And>i. \<lbrakk>bounded_by vs bs; i < length bs\<rbrakk> \<Longrightarrow> Ifloat (fst (bs ! i)) \<le> vs ! i \<and> vs ! i \<le> Ifloat (snd (bs ! i))"
-  assume bounded: "bounded_by (v # vs) ((l, u) # bs)" and length: "i < length ((l, u) # bs)"
-  show "Ifloat (fst (((l, u) # bs) ! i)) \<le> (v # vs) ! i \<and> (v # vs) ! i \<le> Ifloat (snd (((l, u) # bs) ! i))"
-  proof (cases i)
-    case 0
-    show ?thesis using bounded unfolding 0 nth_Cons_0 fst_conv snd_conv bounded_by.simps ..
-  next
-    case (Suc i) with length have "i < length bs" by auto
-    show ?thesis unfolding Suc nth_Cons_Suc bounded_by.simps
-      using hyp[OF bounded[unfolded bounded_by.simps, THEN conjunct2] `i < length bs`] .
-  qed
-qed auto
-
-fun approx approx' :: "nat \<Rightarrow> floatarith \<Rightarrow> (float * float) list \<Rightarrow> (float * float) option" where
-"approx' prec a bs          = (case (approx prec a bs) of Some (l, u) \<Rightarrow> Some (round_down prec l, round_up prec u) | None \<Rightarrow> None)" |
-"approx prec (Add a b) bs  = lift_bin' (approx' prec a bs) (approx' prec b bs) (\<lambda> l1 u1 l2 u2. (l1 + l2, u1 + u2))" | 
-"approx prec (Minus a) bs   = lift_un' (approx' prec a bs) (\<lambda> l u. (-u, -l))" |
-"approx prec (Mult a b) bs  = lift_bin' (approx' prec a bs) (approx' prec b bs)
-                                    (\<lambda> a1 a2 b1 b2. (float_nprt a1 * float_pprt b2 + float_nprt a2 * float_nprt b2 + float_pprt a1 * float_pprt b1 + float_pprt a2 * float_nprt b1, 
-                                                     float_pprt a2 * float_pprt b2 + float_pprt a1 * float_nprt b2 + float_nprt a2 * float_pprt b1 + float_nprt a1 * float_nprt b1))" |
-"approx prec (Inverse a) bs = lift_un (approx' prec a bs) (\<lambda> l u. if (0 < l \<or> u < 0) then (Some (float_divl prec 1 u), Some (float_divr prec 1 l)) else (None, None))" |
-"approx prec (Sin a) bs     = lift_un' (approx' prec a bs) (bnds_sin prec)" |
-"approx prec (Cos a) bs     = lift_un' (approx' prec a bs) (bnds_cos prec)" |
-"approx prec Pi bs          = Some (lb_pi prec, ub_pi prec)" |
-"approx prec (Min a b) bs   = lift_bin' (approx' prec a bs) (approx' prec b bs) (\<lambda> l1 u1 l2 u2. (min l1 l2, min u1 u2))" |
-"approx prec (Max a b) bs   = lift_bin' (approx' prec a bs) (approx' prec b bs) (\<lambda> l1 u1 l2 u2. (max l1 l2, max u1 u2))" |
-"approx prec (Abs a) bs     = lift_un' (approx' prec a bs) (\<lambda>l u. (if l < 0 \<and> 0 < u then 0 else min \<bar>l\<bar> \<bar>u\<bar>, max \<bar>l\<bar> \<bar>u\<bar>))" |
-"approx prec (Arctan a) bs  = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_arctan prec l, ub_arctan prec u))" |
-"approx prec (Sqrt a) bs    = lift_un (approx' prec a bs) (\<lambda> l u. (lb_sqrt prec l, ub_sqrt prec u))" |
-"approx prec (Exp a) bs     = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_exp prec l, ub_exp prec u))" |
-"approx prec (Ln a) bs      = lift_un (approx' prec a bs) (\<lambda> l u. (lb_ln prec l, ub_ln prec u))" |
-"approx prec (Power a n) bs = lift_un' (approx' prec a bs) (float_power_bnds n)" |
-"approx prec (Num f) bs     = Some (f, f)" |
-"approx prec (Atom i) bs    = (if i < length bs then Some (bs ! i) else None)"
-
-lemma lift_bin'_ex:
-  assumes lift_bin'_Some: "Some (l, u) = lift_bin' a b f"
-  shows "\<exists> l1 u1 l2 u2. Some (l1, u1) = a \<and> Some (l2, u2) = b"
-proof (cases a)
-  case None hence "None = lift_bin' a b f" unfolding None lift_bin'.simps ..
-  thus ?thesis using lift_bin'_Some by auto
-next
-  case (Some a')
-  show ?thesis
-  proof (cases b)
-    case None hence "None = lift_bin' a b f" unfolding None lift_bin'.simps ..
-    thus ?thesis using lift_bin'_Some by auto
-  next
-    case (Some b')
-    obtain la ua where a': "a' = (la, ua)" by (cases a', auto)
-    obtain lb ub where b': "b' = (lb, ub)" by (cases b', auto)
-    thus ?thesis unfolding `a = Some a'` `b = Some b'` a' b' by auto
-  qed
-qed
-
-lemma lift_bin'_f:
-  assumes lift_bin'_Some: "Some (l, u) = lift_bin' (g a) (g b) f"
-  and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a" and Pb: "\<And>l u. Some (l, u) = g b \<Longrightarrow> P l u b"
-  shows "\<exists> l1 u1 l2 u2. P l1 u1 a \<and> P l2 u2 b \<and> l = fst (f l1 u1 l2 u2) \<and> u = snd (f l1 u1 l2 u2)"
-proof -
-  obtain l1 u1 l2 u2
-    where Sa: "Some (l1, u1) = g a" and Sb: "Some (l2, u2) = g b" using lift_bin'_ex[OF assms(1)] by auto
-  have lu: "(l, u) = f l1 u1 l2 u2" using lift_bin'_Some[unfolded Sa[symmetric] Sb[symmetric] lift_bin'.simps] by auto 
-  have "l = fst (f l1 u1 l2 u2)" and "u = snd (f l1 u1 l2 u2)" unfolding lu[symmetric] by auto
-  thus ?thesis using Pa[OF Sa] Pb[OF Sb] by auto 
-qed
-
-lemma approx_approx':
-  assumes Pa: "\<And>l u. Some (l, u) = approx prec a vs \<Longrightarrow> Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u"
-  and approx': "Some (l, u) = approx' prec a vs"
-  shows "Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u"
-proof -
-  obtain l' u' where S: "Some (l', u') = approx prec a vs"
-    using approx' unfolding approx'.simps by (cases "approx prec a vs", auto)
-  have l': "l = round_down prec l'" and u': "u = round_up prec u'"
-    using approx' unfolding approx'.simps S[symmetric] by auto
-  show ?thesis unfolding l' u' 
-    using order_trans[OF Pa[OF S, THEN conjunct2] round_up[of u']]
-    using order_trans[OF round_down[of _ l'] Pa[OF S, THEN conjunct1]] by auto
-qed
-
-lemma lift_bin':
-  assumes lift_bin'_Some: "Some (l, u) = lift_bin' (approx' prec a bs) (approx' prec b bs) f"
-  and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" (is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a")
-  and Pb: "\<And>l u. Some (l, u) = approx prec b bs \<Longrightarrow> Ifloat l \<le> Ifloatarith b xs \<and> Ifloatarith b xs \<le> Ifloat u"
-  shows "\<exists> l1 u1 l2 u2. (Ifloat l1 \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u1) \<and> 
-                        (Ifloat l2 \<le> Ifloatarith b xs \<and> Ifloatarith b xs \<le> Ifloat u2) \<and> 
-                        l = fst (f l1 u1 l2 u2) \<and> u = snd (f l1 u1 l2 u2)"
-proof -
-  { fix l u assume "Some (l, u) = approx' prec a bs"
-    with approx_approx'[of prec a bs, OF _ this] Pa
-    have "Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" by auto } note Pa = this
-  { fix l u assume "Some (l, u) = approx' prec b bs"
-    with approx_approx'[of prec b bs, OF _ this] Pb
-    have "Ifloat l \<le> Ifloatarith b xs \<and> Ifloatarith b xs \<le> Ifloat u" by auto } note Pb = this
-
-  from lift_bin'_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_bin'_Some, OF Pa Pb]
-  show ?thesis by auto
-qed
-
-lemma lift_un'_ex:
-  assumes lift_un'_Some: "Some (l, u) = lift_un' a f"
-  shows "\<exists> l u. Some (l, u) = a"
-proof (cases a)
-  case None hence "None = lift_un' a f" unfolding None lift_un'.simps ..
-  thus ?thesis using lift_un'_Some by auto
-next
-  case (Some a')
-  obtain la ua where a': "a' = (la, ua)" by (cases a', auto)
-  thus ?thesis unfolding `a = Some a'` a' by auto
-qed
-
-lemma lift_un'_f:
-  assumes lift_un'_Some: "Some (l, u) = lift_un' (g a) f"
-  and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a"
-  shows "\<exists> l1 u1. P l1 u1 a \<and> l = fst (f l1 u1) \<and> u = snd (f l1 u1)"
-proof -
-  obtain l1 u1 where Sa: "Some (l1, u1) = g a" using lift_un'_ex[OF assms(1)] by auto
-  have lu: "(l, u) = f l1 u1" using lift_un'_Some[unfolded Sa[symmetric] lift_un'.simps] by auto
-  have "l = fst (f l1 u1)" and "u = snd (f l1 u1)" unfolding lu[symmetric] by auto
-  thus ?thesis using Pa[OF Sa] by auto
-qed
-
-lemma lift_un':
-  assumes lift_un'_Some: "Some (l, u) = lift_un' (approx' prec a bs) f"
-  and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" (is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a")
-  shows "\<exists> l1 u1. (Ifloat l1 \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u1) \<and> 
-                        l = fst (f l1 u1) \<and> u = snd (f l1 u1)"
-proof -
-  { fix l u assume "Some (l, u) = approx' prec a bs"
-    with approx_approx'[of prec a bs, OF _ this] Pa
-    have "Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" by auto } note Pa = this
-  from lift_un'_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_un'_Some, OF Pa]
-  show ?thesis by auto
-qed
-
-lemma lift_un'_bnds:
-  assumes bnds: "\<forall> x lx ux. (l, u) = f lx ux \<and> x \<in> { Ifloat lx .. Ifloat ux } \<longrightarrow> Ifloat l \<le> f' x \<and> f' x \<le> Ifloat u"
-  and lift_un'_Some: "Some (l, u) = lift_un' (approx' prec a bs) f"
-  and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u"
-  shows "Ifloat l \<le> f' (Ifloatarith a xs) \<and> f' (Ifloatarith a xs) \<le> Ifloat u"
-proof -
-  from lift_un'[OF lift_un'_Some Pa]
-  obtain l1 u1 where "Ifloat l1 \<le> Ifloatarith a xs" and "Ifloatarith a xs \<le> Ifloat u1" and "l = fst (f l1 u1)" and "u = snd (f l1 u1)" by blast
-  hence "(l, u) = f l1 u1" and "Ifloatarith a xs \<in> {Ifloat l1 .. Ifloat u1}" by auto
-  thus ?thesis using bnds by auto
-qed
-
-lemma lift_un_ex:
-  assumes lift_un_Some: "Some (l, u) = lift_un a f"
-  shows "\<exists> l u. Some (l, u) = a"
-proof (cases a)
-  case None hence "None = lift_un a f" unfolding None lift_un.simps ..
-  thus ?thesis using lift_un_Some by auto
-next
-  case (Some a')
-  obtain la ua where a': "a' = (la, ua)" by (cases a', auto)
-  thus ?thesis unfolding `a = Some a'` a' by auto
-qed
-
-lemma lift_un_f:
-  assumes lift_un_Some: "Some (l, u) = lift_un (g a) f"
-  and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a"
-  shows "\<exists> l1 u1. P l1 u1 a \<and> Some l = fst (f l1 u1) \<and> Some u = snd (f l1 u1)"
-proof -
-  obtain l1 u1 where Sa: "Some (l1, u1) = g a" using lift_un_ex[OF assms(1)] by auto
-  have "fst (f l1 u1) \<noteq> None \<and> snd (f l1 u1) \<noteq> None"
-  proof (rule ccontr)
-    assume "\<not> (fst (f l1 u1) \<noteq> None \<and> snd (f l1 u1) \<noteq> None)"
-    hence or: "fst (f l1 u1) = None \<or> snd (f l1 u1) = None" by auto
-    hence "lift_un (g a) f = None" 
-    proof (cases "fst (f l1 u1) = None")
-      case True
-      then obtain b where b: "f l1 u1 = (None, b)" by (cases "f l1 u1", auto)
-      thus ?thesis unfolding Sa[symmetric] lift_un.simps b by auto
-    next
-      case False hence "snd (f l1 u1) = None" using or by auto
-      with False obtain b where b: "f l1 u1 = (Some b, None)" by (cases "f l1 u1", auto)
-      thus ?thesis unfolding Sa[symmetric] lift_un.simps b by auto
-    qed
-    thus False using lift_un_Some by auto
-  qed
-  then obtain a' b' where f: "f l1 u1 = (Some a', Some b')" by (cases "f l1 u1", auto)
-  from lift_un_Some[unfolded Sa[symmetric] lift_un.simps f]
-  have "Some l = fst (f l1 u1)" and "Some u = snd (f l1 u1)" unfolding f by auto
-  thus ?thesis unfolding Sa[symmetric] lift_un.simps using Pa[OF Sa] by auto
-qed
-
-lemma lift_un:
-  assumes lift_un_Some: "Some (l, u) = lift_un (approx' prec a bs) f"
-  and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" (is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a")
-  shows "\<exists> l1 u1. (Ifloat l1 \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u1) \<and> 
-                  Some l = fst (f l1 u1) \<and> Some u = snd (f l1 u1)"
-proof -
-  { fix l u assume "Some (l, u) = approx' prec a bs"
-    with approx_approx'[of prec a bs, OF _ this] Pa
-    have "Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" by auto } note Pa = this
-  from lift_un_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_un_Some, OF Pa]
-  show ?thesis by auto
-qed
-
-lemma lift_un_bnds:
-  assumes bnds: "\<forall> x lx ux. (Some l, Some u) = f lx ux \<and> x \<in> { Ifloat lx .. Ifloat ux } \<longrightarrow> Ifloat l \<le> f' x \<and> f' x \<le> Ifloat u"
-  and lift_un_Some: "Some (l, u) = lift_un (approx' prec a bs) f"
-  and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u"
-  shows "Ifloat l \<le> f' (Ifloatarith a xs) \<and> f' (Ifloatarith a xs) \<le> Ifloat u"
-proof -
-  from lift_un[OF lift_un_Some Pa]
-  obtain l1 u1 where "Ifloat l1 \<le> Ifloatarith a xs" and "Ifloatarith a xs \<le> Ifloat u1" and "Some l = fst (f l1 u1)" and "Some u = snd (f l1 u1)" by blast
-  hence "(Some l, Some u) = f l1 u1" and "Ifloatarith a xs \<in> {Ifloat l1 .. Ifloat u1}" by auto
-  thus ?thesis using bnds by auto
-qed
-
-lemma approx:
-  assumes "bounded_by xs vs"
-  and "Some (l, u) = approx prec arith vs" (is "_ = ?g arith")
-  shows "Ifloat l \<le> Ifloatarith arith xs \<and> Ifloatarith arith xs \<le> Ifloat u" (is "?P l u arith")
-  using `Some (l, u) = approx prec arith vs` 
-proof (induct arith arbitrary: l u x)
-  case (Add a b)
-  from lift_bin'[OF Add.prems[unfolded approx.simps]] Add.hyps
-  obtain l1 u1 l2 u2 where "l = l1 + l2" and "u = u1 + u2"
-    "Ifloat l1 \<le> Ifloatarith a xs" and "Ifloatarith a xs \<le> Ifloat u1"
-    "Ifloat l2 \<le> Ifloatarith b xs" and "Ifloatarith b xs \<le> Ifloat u2" unfolding fst_conv snd_conv by blast
-  thus ?case unfolding Ifloatarith.simps by auto
-next
-  case (Minus a)
-  from lift_un'[OF Minus.prems[unfolded approx.simps]] Minus.hyps
-  obtain l1 u1 where "l = -u1" and "u = -l1"
-    "Ifloat l1 \<le> Ifloatarith a xs" and "Ifloatarith a xs \<le> Ifloat u1" unfolding fst_conv snd_conv by blast
-  thus ?case unfolding Ifloatarith.simps using Ifloat_minus by auto
-next
-  case (Mult a b)
-  from lift_bin'[OF Mult.prems[unfolded approx.simps]] Mult.hyps
-  obtain l1 u1 l2 u2 
-    where l: "l = float_nprt l1 * float_pprt u2 + float_nprt u1 * float_nprt u2 + float_pprt l1 * float_pprt l2 + float_pprt u1 * float_nprt l2"
-    and u: "u = float_pprt u1 * float_pprt u2 + float_pprt l1 * float_nprt u2 + float_nprt u1 * float_pprt l2 + float_nprt l1 * float_nprt l2"
-    and "Ifloat l1 \<le> Ifloatarith a xs" and "Ifloatarith a xs \<le> Ifloat u1"
-    and "Ifloat l2 \<le> Ifloatarith b xs" and "Ifloatarith b xs \<le> Ifloat u2" unfolding fst_conv snd_conv by blast
-  thus ?case unfolding Ifloatarith.simps l u Ifloat_add Ifloat_mult Ifloat_nprt Ifloat_pprt 
-    using mult_le_prts mult_ge_prts by auto
-next
-  case (Inverse a)
-  from lift_un[OF Inverse.prems[unfolded approx.simps], unfolded if_distrib[of fst] if_distrib[of snd] fst_conv snd_conv] Inverse.hyps
-  obtain l1 u1 where l': "Some l = (if 0 < l1 \<or> u1 < 0 then Some (float_divl prec 1 u1) else None)" 
-    and u': "Some u = (if 0 < l1 \<or> u1 < 0 then Some (float_divr prec 1 l1) else None)"
-    and l1: "Ifloat l1 \<le> Ifloatarith a xs" and u1: "Ifloatarith a xs \<le> Ifloat u1" by blast
-  have either: "0 < l1 \<or> u1 < 0" proof (rule ccontr) assume P: "\<not> (0 < l1 \<or> u1 < 0)" show False using l' unfolding if_not_P[OF P] by auto qed
-  moreover have l1_le_u1: "Ifloat l1 \<le> Ifloat u1" using l1 u1 by auto
-  ultimately have "Ifloat l1 \<noteq> 0" and "Ifloat u1 \<noteq> 0" unfolding less_float_def by auto
-
-  have inv: "inverse (Ifloat u1) \<le> inverse (Ifloatarith a xs)
-           \<and> inverse (Ifloatarith a xs) \<le> inverse (Ifloat l1)"
-  proof (cases "0 < l1")
-    case True hence "0 < Ifloat u1" and "0 < Ifloat l1" "0 < Ifloatarith a xs" 
-      unfolding less_float_def using l1_le_u1 l1 by auto
-    show ?thesis
-      unfolding inverse_le_iff_le[OF `0 < Ifloat u1` `0 < Ifloatarith a xs`]
-	inverse_le_iff_le[OF `0 < Ifloatarith a xs` `0 < Ifloat l1`]
-      using l1 u1 by auto
-  next
-    case False hence "u1 < 0" using either by blast
-    hence "Ifloat u1 < 0" and "Ifloat l1 < 0" "Ifloatarith a xs < 0" 
-      unfolding less_float_def using l1_le_u1 u1 by auto
-    show ?thesis
-      unfolding inverse_le_iff_le_neg[OF `Ifloat u1 < 0` `Ifloatarith a xs < 0`]
-	inverse_le_iff_le_neg[OF `Ifloatarith a xs < 0` `Ifloat l1 < 0`]
-      using l1 u1 by auto
-  qed
-    
-  from l' have "l = float_divl prec 1 u1" by (cases "0 < l1 \<or> u1 < 0", auto)
-  hence "Ifloat l \<le> inverse (Ifloat u1)" unfolding nonzero_inverse_eq_divide[OF `Ifloat u1 \<noteq> 0`] using float_divl[of prec 1 u1] by auto
-  also have "\<dots> \<le> inverse (Ifloatarith a xs)" using inv by auto
-  finally have "Ifloat l \<le> inverse (Ifloatarith a xs)" .
-  moreover
-  from u' have "u = float_divr prec 1 l1" by (cases "0 < l1 \<or> u1 < 0", auto)
-  hence "inverse (Ifloat l1) \<le> Ifloat u" unfolding nonzero_inverse_eq_divide[OF `Ifloat l1 \<noteq> 0`] using float_divr[of 1 l1 prec] by auto
-  hence "inverse (Ifloatarith a xs) \<le> Ifloat u" by (rule order_trans[OF inv[THEN conjunct2]])
-  ultimately show ?case unfolding Ifloatarith.simps using l1 u1 by auto
-next
-  case (Abs x)
-  from lift_un'[OF Abs.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Abs.hyps
-  obtain l1 u1 where l': "l = (if l1 < 0 \<and> 0 < u1 then 0 else min \<bar>l1\<bar> \<bar>u1\<bar>)" and u': "u = max \<bar>l1\<bar> \<bar>u1\<bar>"
-    and l1: "Ifloat l1 \<le> Ifloatarith x xs" and u1: "Ifloatarith x xs \<le> Ifloat u1" by blast
-  thus ?case unfolding l' u' by (cases "l1 < 0 \<and> 0 < u1", auto simp add: Ifloat_min Ifloat_max Ifloat_abs less_float_def)
-next
-  case (Min a b)
-  from lift_bin'[OF Min.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Min.hyps
-  obtain l1 u1 l2 u2 where l': "l = min l1 l2" and u': "u = min u1 u2"
-    and l1: "Ifloat l1 \<le> Ifloatarith a xs" and u1: "Ifloatarith a xs \<le> Ifloat u1"
-    and l1: "Ifloat l2 \<le> Ifloatarith b xs" and u1: "Ifloatarith b xs \<le> Ifloat u2" by blast
-  thus ?case unfolding l' u' by (auto simp add: Ifloat_min)
-next
-  case (Max a b)
-  from lift_bin'[OF Max.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Max.hyps
-  obtain l1 u1 l2 u2 where l': "l = max l1 l2" and u': "u = max u1 u2"
-    and l1: "Ifloat l1 \<le> Ifloatarith a xs" and u1: "Ifloatarith a xs \<le> Ifloat u1"
-    and l1: "Ifloat l2 \<le> Ifloatarith b xs" and u1: "Ifloatarith b xs \<le> Ifloat u2" by blast
-  thus ?case unfolding l' u' by (auto simp add: Ifloat_max)
-next case (Sin a) with lift_un'_bnds[OF bnds_sin] show ?case by auto
-next case (Cos a) with lift_un'_bnds[OF bnds_cos] show ?case by auto
-next case (Arctan a) with lift_un'_bnds[OF bnds_arctan] show ?case by auto
-next case Pi with pi_boundaries show ?case by auto
-next case (Sqrt a) with lift_un_bnds[OF bnds_sqrt] show ?case by auto
-next case (Exp a) with lift_un'_bnds[OF bnds_exp] show ?case by auto
-next case (Ln a) with lift_un_bnds[OF bnds_ln] show ?case by auto
-next case (Power a n) with lift_un'_bnds[OF bnds_power] show ?case by auto
-next case (Num f) thus ?case by auto
-next
-  case (Atom n) 
-  show ?case
-  proof (cases "n < length vs")
-    case True
-    with Atom have "vs ! n = (l, u)" by auto
-    thus ?thesis using bounded_by[OF assms(1) True] by auto
-  next
-    case False thus ?thesis using Atom by auto
-  qed
-qed
-
-datatype ApproxEq = Less floatarith floatarith 
-                  | LessEqual floatarith floatarith 
-
-fun uneq :: "ApproxEq \<Rightarrow> real list \<Rightarrow> bool" where 
-"uneq (Less a b) vs                   = (Ifloatarith a vs < Ifloatarith b vs)" |
-"uneq (LessEqual a b) vs              = (Ifloatarith a vs \<le> Ifloatarith b vs)"
-
-fun uneq' :: "nat \<Rightarrow> ApproxEq \<Rightarrow> (float * float) list \<Rightarrow> bool" where 
-"uneq' prec (Less a b) bs = (case (approx prec a bs, approx prec b bs) of (Some (l, u), Some (l', u')) \<Rightarrow> u < l' | _ \<Rightarrow> False)" |
-"uneq' prec (LessEqual a b) bs = (case (approx prec a bs, approx prec b bs) of (Some (l, u), Some (l', u')) \<Rightarrow> u \<le> l' | _ \<Rightarrow> False)"
-
-lemma uneq_approx: fixes m :: nat assumes "bounded_by vs bs" and "uneq' prec eq bs"
-  shows "uneq eq vs"
-proof (cases eq)
-  case (Less a b)
-  show ?thesis
-  proof (cases "\<exists> u l u' l'. approx prec a bs = Some (l, u) \<and> 
-                             approx prec b bs = Some (l', u')")
-    case True
-    then obtain l u l' u' where a_approx: "approx prec a bs = Some (l, u)"
-      and b_approx: "approx prec b bs = Some (l', u') " by auto
-    with `uneq' prec eq bs` have "Ifloat u < Ifloat l'"
-      unfolding Less uneq'.simps less_float_def by auto
-    moreover from a_approx[symmetric] and b_approx[symmetric] and `bounded_by vs bs`
-    have "Ifloatarith a vs \<le> Ifloat u" and "Ifloat l' \<le> Ifloatarith b vs"
-      using approx by auto
-    ultimately show ?thesis unfolding uneq.simps Less by auto
-  next
-    case False
-    hence "approx prec a bs = None \<or> approx prec b bs = None"
-      unfolding not_Some_eq[symmetric] by auto
-    hence "\<not> uneq' prec eq bs" unfolding Less uneq'.simps 
-      by (cases "approx prec a bs = None", auto)
-    thus ?thesis using assms by auto
-  qed
-next
-  case (LessEqual a b)
-  show ?thesis
-  proof (cases "\<exists> u l u' l'. approx prec a bs = Some (l, u) \<and> 
-                             approx prec b bs = Some (l', u')")
-    case True
-    then obtain l u l' u' where a_approx: "approx prec a bs = Some (l, u)"
-      and b_approx: "approx prec b bs = Some (l', u') " by auto
-    with `uneq' prec eq bs` have "Ifloat u \<le> Ifloat l'"
-      unfolding LessEqual uneq'.simps le_float_def by auto
-    moreover from a_approx[symmetric] and b_approx[symmetric] and `bounded_by vs bs`
-    have "Ifloatarith a vs \<le> Ifloat u" and "Ifloat l' \<le> Ifloatarith b vs"
-      using approx by auto
-    ultimately show ?thesis unfolding uneq.simps LessEqual by auto
-  next
-    case False
-    hence "approx prec a bs = None \<or> approx prec b bs = None"
-      unfolding not_Some_eq[symmetric] by auto
-    hence "\<not> uneq' prec eq bs" unfolding LessEqual uneq'.simps 
-      by (cases "approx prec a bs = None", auto)
-    thus ?thesis using assms by auto
-  qed
-qed
-
-lemma Ifloatarith_divide: "Ifloatarith (Mult a (Inverse b)) vs = (Ifloatarith a vs) / (Ifloatarith b vs)"
-  unfolding real_divide_def Ifloatarith.simps ..
-
-lemma Ifloatarith_diff: "Ifloatarith (Add a (Minus b)) vs = (Ifloatarith a vs) - (Ifloatarith b vs)"
-  unfolding real_diff_def Ifloatarith.simps ..
-
-lemma Ifloatarith_tan: "Ifloatarith (Mult (Sin a) (Inverse (Cos a))) vs = tan (Ifloatarith a vs)"
-  unfolding tan_def Ifloatarith.simps real_divide_def ..
-
-lemma Ifloatarith_powr: "Ifloatarith (Exp (Mult b (Ln a))) vs = (Ifloatarith a vs) powr (Ifloatarith b vs)"
-  unfolding powr_def Ifloatarith.simps ..
-
-lemma Ifloatarith_log: "Ifloatarith ((Mult (Ln x) (Inverse (Ln b)))) vs = log (Ifloatarith b vs) (Ifloatarith x vs)"
-  unfolding log_def Ifloatarith.simps real_divide_def ..
-
-lemma Ifloatarith_num: shows "Ifloatarith (Num (Float 0 0)) vs = 0" and "Ifloatarith (Num (Float 1 0)) vs = 1" and "Ifloatarith (Num (Float (number_of a) 0)) vs = number_of a" by auto
-
-subsection {* Implement proof method \texttt{approximation} *}
-
-lemma bounded_divl: assumes "Ifloat a / Ifloat b \<le> x" shows "Ifloat (float_divl p a b) \<le> x" by (rule order_trans[OF _ assms], rule float_divl)
-lemma bounded_divr: assumes "x \<le> Ifloat a / Ifloat b" shows "x \<le> Ifloat (float_divr p a b)" by (rule order_trans[OF assms _], rule float_divr)
-lemma bounded_num: shows "Ifloat (Float 5 1) = 10" and "Ifloat (Float 0 0) = 0" and "Ifloat (Float 1 0) = 1" and "Ifloat (Float (number_of n) 0) = (number_of n)"
-                     and "0 * pow2 e = Ifloat (Float 0 e)" and "1 * pow2 e = Ifloat (Float 1 e)" and "number_of m * pow2 e = Ifloat (Float (number_of m) e)"
-  by (auto simp add: Ifloat.simps pow2_def)
-
-lemmas bounded_by_equations = bounded_by_Cons bounded_by_Nil float_power bounded_divl bounded_divr bounded_num HOL.simp_thms
-lemmas uneq_equations = uneq.simps Ifloatarith.simps Ifloatarith_num Ifloatarith_divide Ifloatarith_diff Ifloatarith_tan Ifloatarith_powr Ifloatarith_log
-
-lemma "x div (0::int) = 0" by auto -- "What happens in the zero case for div"
-lemma "x mod (0::int) = x" by auto -- "What happens in the zero case for mod"
-
-text {* The following equations must hold for div & mod 
-        -- see "The Definition of Standard ML" by R. Milner, M. Tofte and R. Harper (pg. 79) *}
-lemma "d * (i div d) + i mod d = (i::int)" by auto
-lemma "0 < (d :: int) \<Longrightarrow> 0 \<le> i mod d \<and> i mod d < d" by auto
-lemma "(d :: int) < 0 \<Longrightarrow> d < i mod d \<and> i mod d \<le> 0" by auto
-
-code_const "op div :: int \<Rightarrow> int \<Rightarrow> int" (SML "(fn i => fn d => if d = 0 then 0 else i div d)")
-code_const "op mod :: int \<Rightarrow> int \<Rightarrow> int" (SML "(fn i => fn d => if d = 0 then i else i mod d)")
-code_const "divmod :: int \<Rightarrow> int \<Rightarrow> (int * int)" (SML "(fn i => fn d => if d = 0 then (0, i) else IntInf.divMod (i, d))")
-
-ML {*
-  val uneq_equations = PureThy.get_thms @{theory} "uneq_equations";
-  val bounded_by_equations = PureThy.get_thms @{theory} "bounded_by_equations";
-  val bounded_by_simpset = (HOL_basic_ss addsimps bounded_by_equations)
-
-  fun reify_uneq ctxt i = (fn st =>
-    let
-      val to = HOLogic.dest_Trueprop (Logic.strip_imp_concl (List.nth (prems_of st, i - 1)))
-    in (Reflection.genreify_tac ctxt uneq_equations (SOME to) i) st
-    end)
-
-  fun rule_uneq ctxt prec i thm = let
-    fun conv_num typ = HOLogic.dest_number #> snd #> HOLogic.mk_number typ
-    val to_natc = conv_num @{typ "nat"} #> Thm.cterm_of (ProofContext.theory_of ctxt)
-    val to_nat = conv_num @{typ "nat"}
-    val to_int = conv_num @{typ "int"}
-
-    val prec' = to_nat prec
-
-    fun bot_float (Const (@{const_name "times"}, _) $ mantisse $ (Const (@{const_name "pow2"}, _) $ exp))
-                   = @{term "Float"} $ to_int mantisse $ to_int exp
-      | bot_float (Const (@{const_name "divide"}, _) $ mantisse $ (Const (@{const_name "power"}, _) $ ten $ exp))
-                   = @{term "float_divl"} $ prec' $ (@{term "Float"} $ to_int mantisse $ @{term "0 :: int"}) $ (@{term "power_float (Float 5 1)"} $ to_nat exp)
-      | bot_float (Const (@{const_name "divide"}, _) $ mantisse $ ten)
-                   = @{term "float_divl"} $ prec' $ (@{term "Float"} $ to_int mantisse $ @{term "0 :: int"}) $ @{term "Float 5 1"}
-      | bot_float mantisse = @{term "Float"} $ to_int mantisse $ @{term "0 :: int"}
-
-    fun top_float (Const (@{const_name "times"}, _) $ mantisse $ (Const (@{const_name "pow2"}, _) $ exp))
-                   = @{term "Float"} $ to_int mantisse $ to_int exp
-      | top_float (Const (@{const_name "divide"}, _) $ mantisse $ (Const (@{const_name "power"}, _) $ ten $ exp))
-                   = @{term "float_divr"} $ prec' $ (@{term "Float"} $ to_int mantisse $ @{term "0 :: int"}) $ (@{term "power_float (Float 5 1)"} $ to_nat exp)
-      | top_float (Const (@{const_name "divide"}, _) $ mantisse $ ten)
-                   = @{term "float_divr"} $ prec' $ (@{term "Float"} $ to_int mantisse $ @{term "0 :: int"}) $ @{term "Float 5 1"}
-      | top_float mantisse = @{term "Float"} $ to_int mantisse $ @{term "0 :: int"}
-
-    val goal' : term = List.nth (prems_of thm, i - 1)
-
-    fun lift_bnd (t as (Const (@{const_name "op &"}, _) $ 
-                        (Const (@{const_name "less_eq"}, _) $ 
-                         bottom $ (Free (name, _))) $ 
-                        (Const (@{const_name "less_eq"}, _) $ _ $ top)))
-         = ((name, HOLogic.mk_prod (bot_float bottom, top_float top))
-            handle TERM (txt, ts) => raise TERM ("Premisse needs format '<num> <= <var> & <var> <= <num>', but found " ^
-                                  (Syntax.string_of_term ctxt t), [t]))
-      | lift_bnd t = raise TERM ("Premisse needs format '<num> <= <var> & <var> <= <num>', but found " ^
-                                 (Syntax.string_of_term ctxt t), [t])
-    val bound_eqs = map (HOLogic.dest_Trueprop #> lift_bnd)  (Logic.strip_imp_prems goal')
-
-    fun lift_var (Free (varname, _)) = (case AList.lookup (op =) bound_eqs varname of
-                                          SOME bound => bound
-                                        | NONE => raise TERM ("No bound equations found for " ^ varname, []))
-      | lift_var t = raise TERM ("Can not convert expression " ^ 
-                                 (Syntax.string_of_term ctxt t), [t])
-
-    val _ $ vs = HOLogic.dest_Trueprop (Logic.strip_imp_concl goal')
-
-    val bs = (HOLogic.dest_list #> map lift_var #> HOLogic.mk_list @{typ "float * float"}) vs
-    val map = [(@{cpat "?prec::nat"}, to_natc prec),
-               (@{cpat "?bs::(float * float) list"}, Thm.cterm_of (ProofContext.theory_of ctxt) bs)]
-  in rtac (Thm.instantiate ([], map) @{thm "uneq_approx"}) i thm end
-
-  val eval_tac = CSUBGOAL (fn (ct, i) => rtac (eval_oracle ct) i)
-
-  fun gen_eval_tac conv ctxt = CONVERSION (Conv.params_conv (~1) (K (Conv.concl_conv (~1) conv)) ctxt)
-                               THEN' rtac TrueI
-
-*}
-
-method_setup approximation = {* fn src => 
-  Method.syntax Args.term src #>
-  (fn (prec, ctxt) => let
-   in Method.SIMPLE_METHOD' (fn i =>
-     (DETERM (reify_uneq ctxt i)
-      THEN rule_uneq ctxt prec i
-      THEN Simplifier.asm_full_simp_tac bounded_by_simpset i 
-      THEN (TRY (filter_prems_tac (fn t => false) i))
-      THEN (gen_eval_tac eval_oracle ctxt) i))
-   end)
-*} "real number approximation"
-
-end
diff -r b8a6b9c56fdd -r 5170d6277b61 src/HOL/Reflection/Cooper.thy
--- a/src/HOL/Reflection/Cooper.thy	Thu Feb 12 21:24:14 2009 -0800
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,2174 +0,0 @@
-(*  Title:      HOL/Reflection/Cooper.thy
-    Author:     Amine Chaieb
-*)
-
-theory Cooper
-imports Complex_Main Efficient_Nat
-uses ("cooper_tac.ML")
-begin
-
-function iupt :: "int \<Rightarrow> int \<Rightarrow> int list" where
-  "iupt i j = (if j < i then [] else i # iupt (i+1) j)"
-by pat_completeness auto
-termination by (relation "measure (\<lambda> (i, j). nat (j-i+1))") auto
-
-lemma iupt_set: "set (iupt i j) = {i..j}"
-  by (induct rule: iupt.induct) (simp add: simp_from_to)
-
-(* Periodicity of dvd *)
-
-  (*********************************************************************************)
-  (****                            SHADOW SYNTAX AND SEMANTICS                  ****)
-  (*********************************************************************************)
-
-datatype num = C int | Bound nat | CN nat int num | Neg num | Add num num| Sub num num 
-  | Mul int num
-
-  (* A size for num to make inductive proofs simpler*)
-primrec num_size :: "num \<Rightarrow> nat" where
-  "num_size (C c) = 1"
-| "num_size (Bound n) = 1"
-| "num_size (Neg a) = 1 + num_size a"
-| "num_size (Add a b) = 1 + num_size a + num_size b"
-| "num_size (Sub a b) = 3 + num_size a + num_size b"
-| "num_size (CN n c a) = 4 + num_size a"
-| "num_size (Mul c a) = 1 + num_size a"
-
-primrec Inum :: "int list \<Rightarrow> num \<Rightarrow> int" where
-  "Inum bs (C c) = c"
-| "Inum bs (Bound n) = bs!n"
-| "Inum bs (CN n c a) = c * (bs!n) + (Inum bs a)"
-| "Inum bs (Neg a) = -(Inum bs a)"
-| "Inum bs (Add a b) = Inum bs a + Inum bs b"
-| "Inum bs (Sub a b) = Inum bs a - Inum bs b"
-| "Inum bs (Mul c a) = c* Inum bs a"
-
-datatype fm  = 
-  T| F| Lt num| Le num| Gt num| Ge num| Eq num| NEq num| Dvd int num| NDvd int num|
-  NOT fm| And fm fm|  Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm 
-  | Closed nat | NClosed nat
-
-
-  (* A size for fm *)
-consts fmsize :: "fm \<Rightarrow> nat"
-recdef fmsize "measure size"
-  "fmsize (NOT p) = 1 + fmsize p"
-  "fmsize (And p q) = 1 + fmsize p + fmsize q"
-  "fmsize (Or p q) = 1 + fmsize p + fmsize q"
-  "fmsize (Imp p q) = 3 + fmsize p + fmsize q"
-  "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)"
-  "fmsize (E p) = 1 + fmsize p"
-  "fmsize (A p) = 4+ fmsize p"
-  "fmsize (Dvd i t) = 2"
-  "fmsize (NDvd i t) = 2"
-  "fmsize p = 1"
-  (* several lemmas about fmsize *)
-lemma fmsize_pos: "fmsize p > 0"	
-by (induct p rule: fmsize.induct) simp_all
-
-  (* Semantics of formulae (fm) *)
-consts Ifm ::"bool list \<Rightarrow> int list \<Rightarrow> fm \<Rightarrow> bool"
-primrec
-  "Ifm bbs bs T = True"
-  "Ifm bbs bs F = False"
-  "Ifm bbs bs (Lt a) = (Inum bs a < 0)"
-  "Ifm bbs bs (Gt a) = (Inum bs a > 0)"
-  "Ifm bbs bs (Le a) = (Inum bs a \<le> 0)"
-  "Ifm bbs bs (Ge a) = (Inum bs a \<ge> 0)"
-  "Ifm bbs bs (Eq a) = (Inum bs a = 0)"
-  "Ifm bbs bs (NEq a) = (Inum bs a \<noteq> 0)"
-  "Ifm bbs bs (Dvd i b) = (i dvd Inum bs b)"
-  "Ifm bbs bs (NDvd i b) = (\<not>(i dvd Inum bs b))"
-  "Ifm bbs bs (NOT p) = (\<not> (Ifm bbs bs p))"
-  "Ifm bbs bs (And p q) = (Ifm bbs bs p \<and> Ifm bbs bs q)"
-  "Ifm bbs bs (Or p q) = (Ifm bbs bs p \<or> Ifm bbs bs q)"
-  "Ifm bbs bs (Imp p q) = ((Ifm bbs bs p) \<longrightarrow> (Ifm bbs bs q))"
-  "Ifm bbs bs (Iff p q) = (Ifm bbs bs p = Ifm bbs bs q)"
-  "Ifm bbs bs (E p) = (\<exists> x. Ifm bbs (x#bs) p)"
-  "Ifm bbs bs (A p) = (\<forall> x. Ifm bbs (x#bs) p)"
-  "Ifm bbs bs (Closed n) = bbs!n"
-  "Ifm bbs bs (NClosed n) = (\<not> bbs!n)"
-
-consts prep :: "fm \<Rightarrow> fm"
-recdef prep "measure fmsize"
-  "prep (E T) = T"
-  "prep (E F) = F"
-  "prep (E (Or p q)) = Or (prep (E p)) (prep (E q))"
-  "prep (E (Imp p q)) = Or (prep (E (NOT p))) (prep (E q))"
-  "prep (E (Iff p q)) = Or (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))" 
-  "prep (E (NOT (And p q))) = Or (prep (E (NOT p))) (prep (E(NOT q)))"
-  "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))"
-  "prep (E (NOT (Iff p q))) = Or (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))"
-  "prep (E p) = E (prep p)"
-  "prep (A (And p q)) = And (prep (A p)) (prep (A q))"
-  "prep (A p) = prep (NOT (E (NOT p)))"
-  "prep (NOT (NOT p)) = prep p"
-  "prep (NOT (And p q)) = Or (prep (NOT p)) (prep (NOT q))"
-  "prep (NOT (A p)) = prep (E (NOT p))"
-  "prep (NOT (Or p q)) = And (prep (NOT p)) (prep (NOT q))"
-  "prep (NOT (Imp p q)) = And (prep p) (prep (NOT q))"
-  "prep (NOT (Iff p q)) = Or (prep (And p (NOT q))) (prep (And (NOT p) q))"
-  "prep (NOT p) = NOT (prep p)"
-  "prep (Or p q) = Or (prep p) (prep q)"
-  "prep (And p q) = And (prep p) (prep q)"
-  "prep (Imp p q) = prep (Or (NOT p) q)"
-  "prep (Iff p q) = Or (prep (And p q)) (prep (And (NOT p) (NOT q)))"
-  "prep p = p"
-(hints simp add: fmsize_pos)
-lemma prep: "Ifm bbs bs (prep p) = Ifm bbs bs p"
-by (induct p arbitrary: bs rule: prep.induct, auto)
-
-
-  (* Quantifier freeness *)
-consts qfree:: "fm \<Rightarrow> bool"
-recdef qfree "measure size"
-  "qfree (E p) = False"
-  "qfree (A p) = False"
-  "qfree (NOT p) = qfree p" 
-  "qfree (And p q) = (qfree p \<and> qfree q)" 
-  "qfree (Or  p q) = (qfree p \<and> qfree q)" 
-  "qfree (Imp p q) = (qfree p \<and> qfree q)" 
-  "qfree (Iff p q) = (qfree p \<and> qfree q)"
-  "qfree p = True"
-
-  (* Boundedness and substitution *)
-consts 
-  numbound0:: "num \<Rightarrow> bool" (* a num is INDEPENDENT of Bound 0 *)
-  bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *)
-  subst0:: "num \<Rightarrow> fm \<Rightarrow> fm" (* substitue a num into a formula for Bound 0 *)
-primrec
-  "numbound0 (C c) = True"
-  "numbound0 (Bound n) = (n>0)"
-  "numbound0 (CN n i a) = (n >0 \<and> numbound0 a)"
-  "numbound0 (Neg a) = numbound0 a"
-  "numbound0 (Add a b) = (numbound0 a \<and> numbound0 b)"
-  "numbound0 (Sub a b) = (numbound0 a \<and> numbound0 b)" 
-  "numbound0 (Mul i a) = numbound0 a"
-
-lemma numbound0_I:
-  assumes nb: "numbound0 a"
-  shows "Inum (b#bs) a = Inum (b'#bs) a"
-using nb
-by (induct a rule: numbound0.induct) (auto simp add: gr0_conv_Suc)
-
-primrec
-  "bound0 T = True"
-  "bound0 F = True"
-  "bound0 (Lt a) = numbound0 a"
-  "bound0 (Le a) = numbound0 a"
-  "bound0 (Gt a) = numbound0 a"
-  "bound0 (Ge a) = numbound0 a"
-  "bound0 (Eq a) = numbound0 a"
-  "bound0 (NEq a) = numbound0 a"
-  "bound0 (Dvd i a) = numbound0 a"
-  "bound0 (NDvd i a) = numbound0 a"
-  "bound0 (NOT p) = bound0 p"
-  "bound0 (And p q) = (bound0 p \<and> bound0 q)"
-  "bound0 (Or p q) = (bound0 p \<and> bound0 q)"
-  "bound0 (Imp p q) = ((bound0 p) \<and> (bound0 q))"
-  "bound0 (Iff p q) = (bound0 p \<and> bound0 q)"
-  "bound0 (E p) = False"
-  "bound0 (A p) = False"
-  "bound0 (Closed P) = True"
-  "bound0 (NClosed P) = True"
-lemma bound0_I:
-  assumes bp: "bound0 p"
-  shows "Ifm bbs (b#bs) p = Ifm bbs (b'#bs) p"
-using bp numbound0_I[where b="b" and bs="bs" and b'="b'"]
-by (induct p rule: bound0.induct) (auto simp add: gr0_conv_Suc)
-
-fun numsubst0:: "num \<Rightarrow> num \<Rightarrow> num" where
-  "numsubst0 t (C c) = (C c)"
-| "numsubst0 t (Bound n) = (if n=0 then t else Bound n)"
-| "numsubst0 t (CN 0 i a) = Add (Mul i t) (numsubst0 t a)"
-| "numsubst0 t (CN n i a) = CN n i (numsubst0 t a)"
-| "numsubst0 t (Neg a) = Neg (numsubst0 t a)"
-| "numsubst0 t (Add a b) = Add (numsubst0 t a) (numsubst0 t b)"
-| "numsubst0 t (Sub a b) = Sub (numsubst0 t a) (numsubst0 t b)" 
-| "numsubst0 t (Mul i a) = Mul i (numsubst0 t a)"
-
-lemma numsubst0_I:
-  "Inum (b#bs) (numsubst0 a t) = Inum ((Inum (b#bs) a)#bs) t"
-by (induct t rule: numsubst0.induct,auto simp:nth_Cons')
-
-lemma numsubst0_I':
-  "numbound0 a \<Longrightarrow> Inum (b#bs) (numsubst0 a t) = Inum ((Inum (b'#bs) a)#bs) t"
-by (induct t rule: numsubst0.induct, auto simp: nth_Cons' numbound0_I[where b="b" and b'="b'"])
-
-primrec
-  "subst0 t T = T"
-  "subst0 t F = F"
-  "subst0 t (Lt a) = Lt (numsubst0 t a)"
-  "subst0 t (Le a) = Le (numsubst0 t a)"
-  "subst0 t (Gt a) = Gt (numsubst0 t a)"
-  "subst0 t (Ge a) = Ge (numsubst0 t a)"
-  "subst0 t (Eq a) = Eq (numsubst0 t a)"
-  "subst0 t (NEq a) = NEq (numsubst0 t a)"
-  "subst0 t (Dvd i a) = Dvd i (numsubst0 t a)"
-  "subst0 t (NDvd i a) = NDvd i (numsubst0 t a)"
-  "subst0 t (NOT p) = NOT (subst0 t p)"
-  "subst0 t (And p q) = And (subst0 t p) (subst0 t q)"
-  "subst0 t (Or p q) = Or (subst0 t p) (subst0 t q)"
-  "subst0 t (Imp p q) = Imp (subst0 t p) (subst0 t q)"
-  "subst0 t (Iff p q) = Iff (subst0 t p) (subst0 t q)"
-  "subst0 t (Closed P) = (Closed P)"
-  "subst0 t (NClosed P) = (NClosed P)"
-
-lemma subst0_I: assumes qfp: "qfree p"
-  shows "Ifm bbs (b#bs) (subst0 a p) = Ifm bbs ((Inum (b#bs) a)#bs) p"
-  using qfp numsubst0_I[where b="b" and bs="bs" and a="a"]
-  by (induct p) (simp_all add: gr0_conv_Suc)
-
-
-consts 
-  decrnum:: "num \<Rightarrow> num" 
-  decr :: "fm \<Rightarrow> fm"
-
-recdef decrnum "measure size"
-  "decrnum (Bound n) = Bound (n - 1)"
-  "decrnum (Neg a) = Neg (decrnum a)"
-  "decrnum (Add a b) = Add (decrnum a) (decrnum b)"
-  "decrnum (Sub a b) = Sub (decrnum a) (decrnum b)"
-  "decrnum (Mul c a) = Mul c (decrnum a)"
-  "decrnum (CN n i a) = (CN (n - 1) i (decrnum a))"
-  "decrnum a = a"
-
-recdef decr "measure size"
-  "decr (Lt a) = Lt (decrnum a)"
-  "decr (Le a) = Le (decrnum a)"
-  "decr (Gt a) = Gt (decrnum a)"
-  "decr (Ge a) = Ge (decrnum a)"
-  "decr (Eq a) = Eq (decrnum a)"
-  "decr (NEq a) = NEq (decrnum a)"
-  "decr (Dvd i a) = Dvd i (decrnum a)"
-  "decr (NDvd i a) = NDvd i (decrnum a)"
-  "decr (NOT p) = NOT (decr p)" 
-  "decr (And p q) = And (decr p) (decr q)"
-  "decr (Or p q) = Or (decr p) (decr q)"
-  "decr (Imp p q) = Imp (decr p) (decr q)"
-  "decr (Iff p q) = Iff (decr p) (decr q)"
-  "decr p = p"
-
-lemma decrnum: assumes nb: "numbound0 t"
-  shows "Inum (x#bs) t = Inum bs (decrnum t)"
-  using nb by (induct t rule: decrnum.induct, auto simp add: gr0_conv_Suc)
-
-lemma decr: assumes nb: "bound0 p"
-  shows "Ifm bbs (x#bs) p = Ifm bbs bs (decr p)"
-  using nb 
-  by (induct p rule: decr.induct, simp_all add: gr0_conv_Suc decrnum)
-
-lemma decr_qf: "bound0 p \<Longrightarrow> qfree (decr p)"
-by (induct p, simp_all)
-
-consts 
-  isatom :: "fm \<Rightarrow> bool" (* test for atomicity *)
-recdef isatom "measure size"
-  "isatom T = True"
-  "isatom F = True"
-  "isatom (Lt a) = True"
-  "isatom (Le a) = True"
-  "isatom (Gt a) = True"
-  "isatom (Ge a) = True"
-  "isatom (Eq a) = True"
-  "isatom (NEq a) = True"
-  "isatom (Dvd i b) = True"
-  "isatom (NDvd i b) = True"
-  "isatom (Closed P) = True"
-  "isatom (NClosed P) = True"
-  "isatom p = False"
-
-lemma numsubst0_numbound0: assumes nb: "numbound0 t"
-  shows "numbound0 (numsubst0 t a)"
-using nb apply (induct a rule: numbound0.induct)
-apply simp_all
-apply (case_tac n, simp_all)
-done
-
-lemma subst0_bound0: assumes qf: "qfree p" and nb: "numbound0 t"
-  shows "bound0 (subst0 t p)"
-using qf numsubst0_numbound0[OF nb] by (induct p  rule: subst0.induct, auto)
-
-lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p"
-by (induct p, simp_all)
-
-
-constdefs djf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm"
-  "djf f p q \<equiv> (if q=T then T else if q=F then f p else 
-  (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q))"
-constdefs evaldjf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm"
-  "evaldjf f ps \<equiv> foldr (djf f) ps F"
-
-lemma djf_Or: "Ifm bbs bs (djf f p q) = Ifm bbs bs (Or (f p) q)"
-by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def) 
-(cases "f p", simp_all add: Let_def djf_def) 
-
-lemma evaldjf_ex: "Ifm bbs bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm bbs bs (f p))"
-  by(induct ps, simp_all add: evaldjf_def djf_Or)
-
-lemma evaldjf_bound0: 
-  assumes nb: "\<forall> x\<in> set xs. bound0 (f x)"
-  shows "bound0 (evaldjf f xs)"
-  using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 
-
-lemma evaldjf_qf: 
-  assumes nb: "\<forall> x\<in> set xs. qfree (f x)"
-  shows "qfree (evaldjf f xs)"
-  using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 
-
-consts disjuncts :: "fm \<Rightarrow> fm list"
-recdef disjuncts "measure size"
-  "disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)"
-  "disjuncts F = []"
-  "disjuncts p = [p]"
-
-lemma disjuncts: "(\<exists> q\<in> set (disjuncts p). Ifm bbs bs q) = Ifm bbs bs p"
-by(induct p rule: disjuncts.induct, auto)
-
-lemma disjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). bound0 q"
-proof-
-  assume nb: "bound0 p"
-  hence "list_all bound0 (disjuncts p)" by (induct p rule:disjuncts.induct,auto)
-  thus ?thesis by (simp only: list_all_iff)
-qed
-
-lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). qfree q"
-proof-
-  assume qf: "qfree p"
-  hence "list_all qfree (disjuncts p)"
-    by (induct p rule: disjuncts.induct, auto)
-  thus ?thesis by (simp only: list_all_iff)
-qed
-
-constdefs DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm"
-  "DJ f p \<equiv> evaldjf f (disjuncts p)"
-
-lemma DJ: assumes fdj: "\<forall> p q. f (Or p q) = Or (f p) (f q)"
-  and fF: "f F = F"
-  shows "Ifm bbs bs (DJ f p) = Ifm bbs bs (f p)"
-proof-
-  have "Ifm bbs bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm bbs bs (f q))"
-    by (simp add: DJ_def evaldjf_ex) 
-  also have "\<dots> = Ifm bbs bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto)
-  finally show ?thesis .
-qed
-
-lemma DJ_qf: assumes 
-  fqf: "\<forall> p. qfree p \<longrightarrow> qfree (f p)"
-  shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) "
-proof(clarify)
-  fix  p assume qf: "qfree p"
-  have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def)
-  from disjuncts_qf[OF qf] have "\<forall> q\<in> set (disjuncts p). qfree q" .
-  with fqf have th':"\<forall> q\<in> set (disjuncts p). qfree (f q)" by blast
-  
-  from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp
-qed
-
-lemma DJ_qe: assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bbs bs (qe p) = Ifm bbs bs (E p))"
-  shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm bbs bs ((DJ qe p)) = Ifm bbs bs (E p))"
-proof(clarify)
-  fix p::fm and bs
-  assume qf: "qfree p"
-  from qe have qth: "\<forall> p. qfree p \<longrightarrow> qfree (qe p)" by blast
-  from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto
-  have "Ifm bbs bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm bbs bs (qe q))"
-    by (simp add: DJ_def evaldjf_ex)
-  also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm bbs bs (E q))" using qe disjuncts_qf[OF qf] by auto
-  also have "\<dots> = Ifm bbs bs (E p)" by (induct p rule: disjuncts.induct, auto)
-  finally show "qfree (DJ qe p) \<and> Ifm bbs bs (DJ qe p) = Ifm bbs bs (E p)" using qfth by blast
-qed
-  (* Simplification *)
-
-  (* Algebraic simplifications for nums *)
-consts bnds:: "num \<Rightarrow> nat list"
-  lex_ns:: "nat list \<times> nat list \<Rightarrow> bool"
-recdef bnds "measure size"
-  "bnds (Bound n) = [n]"
-  "bnds (CN n c a) = n#(bnds a)"
-  "bnds (Neg a) = bnds a"
-  "bnds (Add a b) = (bnds a)@(bnds b)"
-  "bnds (Sub a b) = (bnds a)@(bnds b)"
-  "bnds (Mul i a) = bnds a"
-  "bnds a = []"
-recdef lex_ns "measure (\<lambda> (xs,ys). length xs + length ys)"
-  "lex_ns ([], ms) = True"
-  "lex_ns (ns, []) = False"
-  "lex_ns (n#ns, m#ms) = (n<m \<or> ((n = m) \<and> lex_ns (ns,ms))) "
-constdefs lex_bnd :: "num \<Rightarrow> num \<Rightarrow> bool"
-  "lex_bnd t s \<equiv> lex_ns (bnds t, bnds s)"
-
-consts
-  numadd:: "num \<times> num \<Rightarrow> num"
-recdef numadd "measure (\<lambda> (t,s). num_size t + num_size s)"
-  "numadd (CN n1 c1 r1 ,CN n2 c2 r2) =
-  (if n1=n2 then 
-  (let c = c1 + c2
-  in (if c=0 then numadd(r1,r2) else CN n1 c (numadd (r1,r2))))
-  else if n1 \<le> n2 then CN n1 c1 (numadd (r1,Add (Mul c2 (Bound n2)) r2))
-  else CN n2 c2 (numadd (Add (Mul c1 (Bound n1)) r1,r2)))"
-  "numadd (CN n1 c1 r1, t) = CN n1 c1 (numadd (r1, t))"  
-  "numadd (t,CN n2 c2 r2) = CN n2 c2 (numadd (t,r2))" 
-  "numadd (C b1, C b2) = C (b1+b2)"
-  "numadd (a,b) = Add a b"
-
-(*function (sequential)
-  numadd :: "num \<Rightarrow> num \<Rightarrow> num"
-where
-  "numadd (Add (Mul c1 (Bound n1)) r1) (Add (Mul c2 (Bound n2)) r2) =
-      (if n1 = n2 then (let c = c1 + c2
-      in (if c = 0 then numadd r1 r2 else
-        Add (Mul c (Bound n1)) (numadd r1 r2)))
-      else if n1 \<le> n2 then
-        Add (Mul c1 (Bound n1)) (numadd r1 (Add (Mul c2 (Bound n2)) r2))
-      else
-        Add (Mul c2 (Bound n2)) (numadd (Add (Mul c1 (Bound n1)) r1) r2))"
-  | "numadd (Add (Mul c1 (Bound n1)) r1) t =
-      Add (Mul c1 (Bound n1)) (numadd r1 t)"  
-  | "numadd t (Add (Mul c2 (Bound n2)) r2) =
-      Add (Mul c2 (Bound n2)) (numadd t r2)" 
-  | "numadd (C b1) (C b2) = C (b1 + b2)"
-  | "numadd a b = Add a b"
-apply pat_completeness apply auto*)
-  
-lemma numadd: "Inum bs (numadd (t,s)) = Inum bs (Add t s)"
-apply (induct t s rule: numadd.induct, simp_all add: Let_def)
-apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all)
- apply (case_tac "n1 = n2")
-  apply(simp_all add: algebra_simps)
-apply(simp add: left_distrib[symmetric])
-done
-
-lemma numadd_nb: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numadd (t,s))"
-by (induct t s rule: numadd.induct, auto simp add: Let_def)
-
-fun
-  nummul :: "int \<Rightarrow> num \<Rightarrow> num"
-where
-  "nummul i (C j) = C (i * j)"
-  | "nummul i (CN n c t) = CN n (c*i) (nummul i t)"
-  | "nummul i t = Mul i t"
-
-lemma nummul: "\<And> i. Inum bs (nummul i t) = Inum bs (Mul i t)"
-by (induct t rule: nummul.induct, auto simp add: algebra_simps numadd)
-
-lemma nummul_nb: "\<And> i. numbound0 t \<Longrightarrow> numbound0 (nummul i t)"
-by (induct t rule: nummul.induct, auto simp add: numadd_nb)
-
-constdefs numneg :: "num \<Rightarrow> num"
-  "numneg t \<equiv> nummul (- 1) t"
-
-constdefs numsub :: "num \<Rightarrow> num \<Rightarrow> num"
-  "numsub s t \<equiv> (if s = t then C 0 else numadd (s, numneg t))"
-
-lemma numneg: "Inum bs (numneg t) = Inum bs (Neg t)"
-using numneg_def nummul by simp
-
-lemma numneg_nb: "numbound0 t \<Longrightarrow> numbound0 (numneg t)"
-using numneg_def nummul_nb by simp
-
-lemma numsub: "Inum bs (numsub a b) = Inum bs (Sub a b)"
-using numneg numadd numsub_def by simp
-
-lemma numsub_nb: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numsub t s)"
-using numsub_def numadd_nb numneg_nb by simp
-
-fun
-  simpnum :: "num \<Rightarrow> num"
-where
-  "simpnum (C j) = C j"
-  | "simpnum (Bound n) = CN n 1 (C 0)"
-  | "simpnum (Neg t) = numneg (simpnum t)"
-  | "simpnum (Add t s) = numadd (simpnum t, simpnum s)"
-  | "simpnum (Sub t s) = numsub (simpnum t) (simpnum s)"
-  | "simpnum (Mul i t) = (if i = 0 then C 0 else nummul i (simpnum t))"
-  | "simpnum t = t"
-
-lemma simpnum_ci: "Inum bs (simpnum t) = Inum bs t"
-by (induct t rule: simpnum.induct, auto simp add: numneg numadd numsub nummul)
-
-lemma simpnum_numbound0: 
-  "numbound0 t \<Longrightarrow> numbound0 (simpnum t)"
-by (induct t rule: simpnum.induct, auto simp add: numadd_nb numsub_nb nummul_nb numneg_nb)
-
-fun
-  not :: "fm \<Rightarrow> fm"
-where
-  "not (NOT p) = p"
-  | "not T = F"
-  | "not F = T"
-  | "not p = NOT p"
-lemma not: "Ifm bbs bs (not p) = Ifm bbs bs (NOT p)"
-by (cases p) auto
-lemma not_qf: "qfree p \<Longrightarrow> qfree (not p)"
-by (cases p, auto)
-lemma not_bn: "bound0 p \<Longrightarrow> bound0 (not p)"
-by (cases p, auto)
-
-constdefs conj :: "fm \<Rightarrow> fm \<Rightarrow> fm"
-  "conj p q \<equiv> (if (p = F \<or> q=F) then F else if p=T then q else if q=T then p else And p q)"
-lemma conj: "Ifm bbs bs (conj p q) = Ifm bbs bs (And p q)"
-by (cases "p=F \<or> q=F",simp_all add: conj_def) (cases p,simp_all)
-
-lemma conj_qf: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)"
-using conj_def by auto 
-lemma conj_nb: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)"
-using conj_def by auto 
-
-constdefs disj :: "fm \<Rightarrow> fm \<Rightarrow> fm"
-  "disj p q \<equiv> (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p else Or p q)"
-
-lemma disj: "Ifm bbs bs (disj p q) = Ifm bbs bs (Or p q)"
-by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all)
-lemma disj_qf: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)"
-using disj_def by auto 
-lemma disj_nb: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)"
-using disj_def by auto 
-
-constdefs   imp :: "fm \<Rightarrow> fm \<Rightarrow> fm"
-  "imp p q \<equiv> (if (p = F \<or> q=T) then T else if p=T then q else if q=F then not p else Imp p q)"
-lemma imp: "Ifm bbs bs (imp p q) = Ifm bbs bs (Imp p q)"
-by (cases "p=F \<or> q=T",simp_all add: imp_def,cases p) (simp_all add: not)
-lemma imp_qf: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (imp p q)"
-using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def,cases p) (simp_all add: not_qf) 
-lemma imp_nb: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (imp p q)"
-using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def,cases p) simp_all
-
-constdefs   iff :: "fm \<Rightarrow> fm \<Rightarrow> fm"
-  "iff p q \<equiv> (if (p = q) then T else if (p = not q \<or> not p = q) then F else 
-       if p=F then not q else if q=F then not p else if p=T then q else if q=T then p else 
-  Iff p q)"
-lemma iff: "Ifm bbs bs (iff p q) = Ifm bbs bs (Iff p q)"
-  by (unfold iff_def,cases "p=q", simp,cases "p=not q", simp add:not) 
-(cases "not p= q", auto simp add:not)
-lemma iff_qf: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)"
-  by (unfold iff_def,cases "p=q", auto simp add: not_qf)
-lemma iff_nb: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (iff p q)"
-using iff_def by (unfold iff_def,cases "p=q", auto simp add: not_bn)
-
-function (sequential)
-  simpfm :: "fm \<Rightarrow> fm"
-where
-  "simpfm (And p q) = conj (simpfm p) (simpfm q)"
-  | "simpfm (Or p q) = disj (simpfm p) (simpfm q)"
-  | "simpfm (Imp p q) = imp (simpfm p) (simpfm q)"
-  | "simpfm (Iff p q) = iff (simpfm p) (simpfm q)"
-  | "simpfm (NOT p) = not (simpfm p)"
-  | "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v < 0) then T else F 
-      | _ \<Rightarrow> Lt a')"
-  | "simpfm (Le a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<le> 0)  then T else F | _ \<Rightarrow> Le a')"
-  | "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v > 0)  then T else F | _ \<Rightarrow> Gt a')"
-  | "simpfm (Ge a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<ge> 0)  then T else F | _ \<Rightarrow> Ge a')"
-  | "simpfm (Eq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v = 0)  then T else F | _ \<Rightarrow> Eq a')"
-  | "simpfm (NEq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<noteq> 0)  then T else F | _ \<Rightarrow> NEq a')"
-  | "simpfm (Dvd i a) = (if i=0 then simpfm (Eq a)
-             else if (abs i = 1) then T
-             else let a' = simpnum a in case a' of C v \<Rightarrow> if (i dvd v)  then T else F | _ \<Rightarrow> Dvd i a')"
-  | "simpfm (NDvd i a) = (if i=0 then simpfm (NEq a) 
-             else if (abs i = 1) then F
-             else let a' = simpnum a in case a' of C v \<Rightarrow> if (\<not>(i dvd v)) then T else F | _ \<Rightarrow> NDvd i a')"
-  | "simpfm p = p"
-by pat_completeness auto
-termination by (relation "measure fmsize") auto
-
-lemma simpfm: "Ifm bbs bs (simpfm p) = Ifm bbs bs p"
-proof(induct p rule: simpfm.induct)
-  case (6 a) let ?sa = "simpnum a" from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
-  {fix v assume "?sa = C v" hence ?case using sa by simp }
-  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
-      by (cases ?sa, simp_all add: Let_def)}
-  ultimately show ?case by blast
-next
-  case (7 a)  let ?sa = "simpnum a" 
-  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
-  {fix v assume "?sa = C v" hence ?case using sa by simp }
-  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
-      by (cases ?sa, simp_all add: Let_def)}
-  ultimately show ?case by blast
-next
-  case (8 a)  let ?sa = "simpnum a" 
-  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
-  {fix v assume "?sa = C v" hence ?case using sa by simp }
-  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
-      by (cases ?sa, simp_all add: Let_def)}
-  ultimately show ?case by blast
-next
-  case (9 a)  let ?sa = "simpnum a" 
-  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
-  {fix v assume "?sa = C v" hence ?case using sa by simp }
-  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
-      by (cases ?sa, simp_all add: Let_def)}
-  ultimately show ?case by blast
-next
-  case (10 a)  let ?sa = "simpnum a" 
-  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
-  {fix v assume "?sa = C v" hence ?case using sa by simp }
-  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
-      by (cases ?sa, simp_all add: Let_def)}
-  ultimately show ?case by blast
-next
-  case (11 a)  let ?sa = "simpnum a" 
-  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
-  {fix v assume "?sa = C v" hence ?case using sa by simp }
-  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
-      by (cases ?sa, simp_all add: Let_def)}
-  ultimately show ?case by blast
-next
-  case (12 i a)  let ?sa = "simpnum a" from simpnum_ci 
-  have sa: "Inum bs ?sa = Inum bs a" by simp
-  have "i=0 \<or> abs i = 1 \<or> (i\<noteq>0 \<and> (abs i \<noteq> 1))" by auto
-  {assume "i=0" hence ?case using "12.hyps" by (simp add: dvd_def Let_def)}
-  moreover 
-  {assume i1: "abs i = 1"
-      from zdvd_1_left[where m = "Inum bs a"] uminus_dvd_conv[where d="1" and t="Inum bs a"]
-      have ?case using i1 apply (cases "i=0", simp_all add: Let_def) 
-	by (cases "i > 0", simp_all)}
-  moreover   
-  {assume inz: "i\<noteq>0" and cond: "abs i \<noteq> 1"
-    {fix v assume "?sa = C v" hence ?case using sa[symmetric] inz cond
-	by (cases "abs i = 1", auto) }
-    moreover {assume "\<not> (\<exists> v. ?sa = C v)" 
-      hence "simpfm (Dvd i a) = Dvd i ?sa" using inz cond 
-	by (cases ?sa, auto simp add: Let_def)
-      hence ?case using sa by simp}
-    ultimately have ?case by blast}
-  ultimately show ?case by blast
-next
-  case (13 i a)  let ?sa = "simpnum a" from simpnum_ci 
-  have sa: "Inum bs ?sa = Inum bs a" by simp
-  have "i=0 \<or> abs i = 1 \<or> (i\<noteq>0 \<and> (abs i \<noteq> 1))" by auto
-  {assume "i=0" hence ?case using "13.hyps" by (simp add: dvd_def Let_def)}
-  moreover 
-  {assume i1: "abs i = 1"
-      from zdvd_1_left[where m = "Inum bs a"] uminus_dvd_conv[where d="1" and t="Inum bs a"]
-      have ?case using i1 apply (cases "i=0", simp_all add: Let_def)
-      apply (cases "i > 0", simp_all) done}
-  moreover   
-  {assume inz: "i\<noteq>0" and cond: "abs i \<noteq> 1"
-    {fix v assume "?sa = C v" hence ?case using sa[symmetric] inz cond
-	by (cases "abs i = 1", auto) }
-    moreover {assume "\<not> (\<exists> v. ?sa = C v)" 
-      hence "simpfm (NDvd i a) = NDvd i ?sa" using inz cond 
-	by (cases ?sa, auto simp add: Let_def)
-      hence ?case using sa by simp}
-    ultimately have ?case by blast}
-  ultimately show ?case by blast
-qed (induct p rule: simpfm.induct, simp_all add: conj disj imp iff not)
-
-lemma simpfm_bound0: "bound0 p \<Longrightarrow> bound0 (simpfm p)"
-proof(induct p rule: simpfm.induct)
-  case (6 a) hence nb: "numbound0 a" by simp
-  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
-  thus ?case by (cases "simpnum a", auto simp add: Let_def)
-next
-  case (7 a) hence nb: "numbound0 a" by simp
-  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
-  thus ?case by (cases "simpnum a", auto simp add: Let_def)
-next
-  case (8 a) hence nb: "numbound0 a" by simp
-  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
-  thus ?case by (cases "simpnum a", auto simp add: Let_def)
-next
-  case (9 a) hence nb: "numbound0 a" by simp
-  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
-  thus ?case by (cases "simpnum a", auto simp add: Let_def)
-next
-  case (10 a) hence nb: "numbound0 a" by simp
-  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
-  thus ?case by (cases "simpnum a", auto simp add: Let_def)
-next
-  case (11 a) hence nb: "numbound0 a" by simp
-  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
-  thus ?case by (cases "simpnum a", auto simp add: Let_def)
-next
-  case (12 i a) hence nb: "numbound0 a" by simp
-  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
-  thus ?case by (cases "simpnum a", auto simp add: Let_def)
-next
-  case (13 i a) hence nb: "numbound0 a" by simp
-  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
-  thus ?case by (cases "simpnum a", auto simp add: Let_def)
-qed(auto simp add: disj_def imp_def iff_def conj_def not_bn)
-
-lemma simpfm_qf: "qfree p \<Longrightarrow> qfree (simpfm p)"
-by (induct p rule: simpfm.induct, auto simp add: disj_qf imp_qf iff_qf conj_qf not_qf Let_def)
- (case_tac "simpnum a",auto)+
-
-  (* Generic quantifier elimination *)
-consts qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm"
-recdef qelim "measure fmsize"
-  "qelim (E p) = (\<lambda> qe. DJ qe (qelim p qe))"
-  "qelim (A p) = (\<lambda> qe. not (qe ((qelim (NOT p) qe))))"
-  "qelim (NOT p) = (\<lambda> qe. not (qelim p qe))"
-  "qelim (And p q) = (\<lambda> qe. conj (qelim p qe) (qelim q qe))" 
-  "qelim (Or  p q) = (\<lambda> qe. disj (qelim p qe) (qelim q qe))" 
-  "qelim (Imp p q) = (\<lambda> qe. imp (qelim p qe) (qelim q qe))"
-  "qelim (Iff p q) = (\<lambda> qe. iff (qelim p qe) (qelim q qe))"
-  "qelim p = (\<lambda> y. simpfm p)"
-
-(*function (sequential)
-  qelim :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm"
-where
-  "qelim qe (E p) = DJ qe (qelim qe p)"
-  | "qelim qe (A p) = not (qe ((qelim qe (NOT p))))"
-  | "qelim qe (NOT p) = not (qelim qe p)"
-  | "qelim qe (And p q) = conj (qelim qe p) (qelim qe q)" 
-  | "qelim qe (Or  p q) = disj (qelim qe p) (qelim qe q)" 
-  | "qelim qe (Imp p q) = imp (qelim qe p) (qelim qe q)"
-  | "qelim qe (Iff p q) = iff (qelim qe p) (qelim qe q)"
-  | "qelim qe p = simpfm p"
-by pat_completeness auto
-termination by (relation "measure (fmsize o snd)") auto*)
-
-lemma qelim_ci:
-  assumes qe_inv: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bbs bs (qe p) = Ifm bbs bs (E p))"
-  shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm bbs bs (qelim p qe) = Ifm bbs bs p)"
-using qe_inv DJ_qe[OF qe_inv] 
-by(induct p rule: qelim.induct) 
-(auto simp add: not disj conj iff imp not_qf disj_qf conj_qf imp_qf iff_qf 
-  simpfm simpfm_qf simp del: simpfm.simps)
-  (* Linearity for fm where Bound 0 ranges over \<int> *)
-
-fun
-  zsplit0 :: "num \<Rightarrow> int \<times> num" (* splits the bounded from the unbounded part*)
-where
-  "zsplit0 (C c) = (0,C c)"
-  | "zsplit0 (Bound n) = (if n=0 then (1, C 0) else (0,Bound n))"
-  | "zsplit0 (CN n i a) = 
-      (let (i',a') =  zsplit0 a 
-       in if n=0 then (i+i', a') else (i',CN n i a'))"
-  | "zsplit0 (Neg a) = (let (i',a') =  zsplit0 a in (-i', Neg a'))"
-  | "zsplit0 (Add a b) = (let (ia,a') =  zsplit0 a ; 
-                            (ib,b') =  zsplit0 b 
-                            in (ia+ib, Add a' b'))"
-  | "zsplit0 (Sub a b) = (let (ia,a') =  zsplit0 a ; 
-                            (ib,b') =  zsplit0 b 
-                            in (ia-ib, Sub a' b'))"
-  | "zsplit0 (Mul i a) = (let (i',a') =  zsplit0 a in (i*i', Mul i a'))"
-
-lemma zsplit0_I:
-  shows "\<And> n a. zsplit0 t = (n,a) \<Longrightarrow> (Inum ((x::int) #bs) (CN 0 n a) = Inum (x #bs) t) \<and> numbound0 a"
-  (is "\<And> n a. ?S t = (n,a) \<Longrightarrow> (?I x (CN 0 n a) = ?I x t) \<and> ?N a")
-proof(induct t rule: zsplit0.induct)
-  case (1 c n a) thus ?case by auto 
-next
-  case (2 m n a) thus ?case by (cases "m=0") auto
-next
-  case (3 m i a n a')
-  let ?j = "fst (zsplit0 a)"
-  let ?b = "snd (zsplit0 a)"
-  have abj: "zsplit0 a = (?j,?b)" by simp 
-  {assume "m\<noteq>0" 
-    with prems(1)[OF abj] prems(2) have ?case by (auto simp add: Let_def split_def)}
-  moreover
-  {assume m0: "m =0"
-    from abj have th: "a'=?b \<and> n=i+?j" using prems 
-      by (simp add: Let_def split_def)
-    from abj prems  have th2: "(?I x (CN 0 ?j ?b) = ?I x a) \<and> ?N ?b" by blast
-    from th have "?I x (CN 0 n a') = ?I x (CN 0 (i+?j) ?b)" by simp
-    also from th2 have "\<dots> = ?I x (CN 0 i (CN 0 ?j ?b))" by (simp add: left_distrib)
-  finally have "?I x (CN 0 n a') = ?I  x (CN 0 i a)" using th2 by simp
-  with th2 th have ?case using m0 by blast} 
-ultimately show ?case by blast
-next
-  case (4 t n a)
-  let ?nt = "fst (zsplit0 t)"
-  let ?at = "snd (zsplit0 t)"
-  have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a=Neg ?at \<and> n=-?nt" using prems 
-    by (simp add: Let_def split_def)
-  from abj prems  have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
-  from th2[simplified] th[simplified] show ?case by simp
-next
-  case (5 s t n a)
-  let ?ns = "fst (zsplit0 s)"
-  let ?as = "snd (zsplit0 s)"
-  let ?nt = "fst (zsplit0 t)"
-  let ?at = "snd (zsplit0 t)"
-  have abjs: "zsplit0 s = (?ns,?as)" by simp 
-  moreover have abjt:  "zsplit0 t = (?nt,?at)" by simp 
-  ultimately have th: "a=Add ?as ?at \<and> n=?ns + ?nt" using prems 
-    by (simp add: Let_def split_def)
-  from abjs[symmetric] have bluddy: "\<exists> x y. (x,y) = zsplit0 s" by blast
-  from prems have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (x # bs) (CN 0 xa xb) = Inum (x # bs) t \<and> numbound0 xb)" by auto
-  with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
-  from abjs prems  have th2: "(?I x (CN 0 ?ns ?as) = ?I x s) \<and> ?N ?as" by blast
-  from th3[simplified] th2[simplified] th[simplified] show ?case 
-    by (simp add: left_distrib)
-next
-  case (6 s t n a)
-  let ?ns = "fst (zsplit0 s)"
-  let ?as = "snd (zsplit0 s)"
-  let ?nt = "fst (zsplit0 t)"
-  let ?at = "snd (zsplit0 t)"
-  have abjs: "zsplit0 s = (?ns,?as)" by simp 
-  moreover have abjt:  "zsplit0 t = (?nt,?at)" by simp 
-  ultimately have th: "a=Sub ?as ?at \<and> n=?ns - ?nt" using prems 
-    by (simp add: Let_def split_def)
-  from abjs[symmetric] have bluddy: "\<exists> x y. (x,y) = zsplit0 s" by blast
-  from prems have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (x # bs) (CN 0 xa xb) = Inum (x # bs) t \<and> numbound0 xb)" by auto
-  with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
-  from abjs prems  have th2: "(?I x (CN 0 ?ns ?as) = ?I x s) \<and> ?N ?as" by blast
-  from th3[simplified] th2[simplified] th[simplified] show ?case 
-    by (simp add: left_diff_distrib)
-next
-  case (7 i t n a)
-  let ?nt = "fst (zsplit0 t)"
-  let ?at = "snd (zsplit0 t)"
-  have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a=Mul i ?at \<and> n=i*?nt" using prems 
-    by (simp add: Let_def split_def)
-  from abj prems  have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
-  hence " ?I x (Mul i t) = i * ?I x (CN 0 ?nt ?at)" by simp
-  also have "\<dots> = ?I x (CN 0 (i*?nt) (Mul i ?at))" by (simp add: right_distrib)
-  finally show ?case using th th2 by simp
-qed
-
-consts
-  iszlfm :: "fm \<Rightarrow> bool"   (* Linearity test for fm *)
-recdef iszlfm "measure size"
-  "iszlfm (And p q) = (iszlfm p \<and> iszlfm q)" 
-  "iszlfm (Or p q) = (iszlfm p \<and> iszlfm q)" 
-  "iszlfm (Eq  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
-  "iszlfm (NEq (CN 0 c e)) = (c>0 \<and> numbound0 e)"
-  "iszlfm (Lt  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
-  "iszlfm (Le  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
-  "iszlfm (Gt  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
-  "iszlfm (Ge  (CN 0 c e)) = ( c>0 \<and> numbound0 e)"
-  "iszlfm (Dvd i (CN 0 c e)) = 
-                 (c>0 \<and> i>0 \<and> numbound0 e)"
-  "iszlfm (NDvd i (CN 0 c e))= 
-                 (c>0 \<and> i>0 \<and> numbound0 e)"
-  "iszlfm p = (isatom p \<and> (bound0 p))"
-
-lemma zlin_qfree: "iszlfm p \<Longrightarrow> qfree p"
-  by (induct p rule: iszlfm.induct) auto
-
-consts
-  zlfm :: "fm \<Rightarrow> fm"       (* Linearity transformation for fm *)
-recdef zlfm "measure fmsize"
-  "zlfm (And p q) = And (zlfm p) (zlfm q)"
-  "zlfm (Or p q) = Or (zlfm p) (zlfm q)"
-  "zlfm (Imp p q) = Or (zlfm (NOT p)) (zlfm q)"
-  "zlfm (Iff p q) = Or (And (zlfm p) (zlfm q)) (And (zlfm (NOT p)) (zlfm (NOT q)))"
-  "zlfm (Lt a) = (let (c,r) = zsplit0 a in 
-     if c=0 then Lt r else 
-     if c>0 then (Lt (CN 0 c r)) else (Gt (CN 0 (- c) (Neg r))))"
-  "zlfm (Le a) = (let (c,r) = zsplit0 a in 
-     if c=0 then Le r else 
-     if c>0 then (Le (CN 0 c r)) else (Ge (CN 0 (- c) (Neg r))))"
-  "zlfm (Gt a) = (let (c,r) = zsplit0 a in 
-     if c=0 then Gt r else 
-     if c>0 then (Gt (CN 0 c r)) else (Lt (CN 0 (- c) (Neg r))))"
-  "zlfm (Ge a) = (let (c,r) = zsplit0 a in 
-     if c=0 then Ge r else 
-     if c>0 then (Ge (CN 0 c r)) else (Le (CN 0 (- c) (Neg r))))"
-  "zlfm (Eq a) = (let (c,r) = zsplit0 a in 
-     if c=0 then Eq r else 
-     if c>0 then (Eq (CN 0 c r)) else (Eq (CN 0 (- c) (Neg r))))"
-  "zlfm (NEq a) = (let (c,r) = zsplit0 a in 
-     if c=0 then NEq r else 
-     if c>0 then (NEq (CN 0 c r)) else (NEq (CN 0 (- c) (Neg r))))"
-  "zlfm (Dvd i a) = (if i=0 then zlfm (Eq a) 
-        else (let (c,r) = zsplit0 a in 
-              if c=0 then (Dvd (abs i) r) else 
-      if c>0 then (Dvd (abs i) (CN 0 c r))
-      else (Dvd (abs i) (CN 0 (- c) (Neg r)))))"
-  "zlfm (NDvd i a) = (if i=0 then zlfm (NEq a) 
-        else (let (c,r) = zsplit0 a in 
-              if c=0 then (NDvd (abs i) r) else 
-      if c>0 then (NDvd (abs i) (CN 0 c r))
-      else (NDvd (abs i) (CN 0 (- c) (Neg r)))))"
-  "zlfm (NOT (And p q)) = Or (zlfm (NOT p)) (zlfm (NOT q))"
-  "zlfm (NOT (Or p q)) = And (zlfm (NOT p)) (zlfm (NOT q))"
-  "zlfm (NOT (Imp p q)) = And (zlfm p) (zlfm (NOT q))"
-  "zlfm (NOT (Iff p q)) = Or (And(zlfm p) (zlfm(NOT q))) (And (zlfm(NOT p)) (zlfm q))"
-  "zlfm (NOT (NOT p)) = zlfm p"
-  "zlfm (NOT T) = F"
-  "zlfm (NOT F) = T"
-  "zlfm (NOT (Lt a)) = zlfm (Ge a)"
-  "zlfm (NOT (Le a)) = zlfm (Gt a)"
-  "zlfm (NOT (Gt a)) = zlfm (Le a)"
-  "zlfm (NOT (Ge a)) = zlfm (Lt a)"
-  "zlfm (NOT (Eq a)) = zlfm (NEq a)"
-  "zlfm (NOT (NEq a)) = zlfm (Eq a)"
-  "zlfm (NOT (Dvd i a)) = zlfm (NDvd i a)"
-  "zlfm (NOT (NDvd i a)) = zlfm (Dvd i a)"
-  "zlfm (NOT (Closed P)) = NClosed P"
-  "zlfm (NOT (NClosed P)) = Closed P"
-  "zlfm p = p" (hints simp add: fmsize_pos)
-
-lemma zlfm_I:
-  assumes qfp: "qfree p"
-  shows "(Ifm bbs (i#bs) (zlfm p) = Ifm bbs (i# bs) p) \<and> iszlfm (zlfm p)"
-  (is "(?I (?l p) = ?I p) \<and> ?L (?l p)")
-using qfp
-proof(induct p rule: zlfm.induct)
-  case (5 a) 
-  let ?c = "fst (zsplit0 a)"
-  let ?r = "snd (zsplit0 a)"
-  have spl: "zsplit0 a = (?c,?r)" by simp
-  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
-  let ?N = "\<lambda> t. Inum (i#bs) t"
-  from prems Ia nb  show ?case 
-    apply (auto simp add: Let_def split_def algebra_simps) 
-    apply (cases "?r",auto)
-    apply (case_tac nat, auto)
-    done
-next
-  case (6 a)  
-  let ?c = "fst (zsplit0 a)"
-  let ?r = "snd (zsplit0 a)"
-  have spl: "zsplit0 a = (?c,?r)" by simp
-  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
-  let ?N = "\<lambda> t. Inum (i#bs) t"
-  from prems Ia nb  show ?case 
-    apply (auto simp add: Let_def split_def algebra_simps) 
-    apply (cases "?r",auto)
-    apply (case_tac nat, auto)
-    done
-next
-  case (7 a)  
-  let ?c = "fst (zsplit0 a)"
-  let ?r = "snd (zsplit0 a)"
-  have spl: "zsplit0 a = (?c,?r)" by simp
-  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
-  let ?N = "\<lambda> t. Inum (i#bs) t"
-  from prems Ia nb  show ?case 
-    apply (auto simp add: Let_def split_def algebra_simps) 
-    apply (cases "?r",auto)
-    apply (case_tac nat, auto)
-    done
-next
-  case (8 a)  
-  let ?c = "fst (zsplit0 a)"
-  let ?r = "snd (zsplit0 a)"
-  have spl: "zsplit0 a = (?c,?r)" by simp
-  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
-  let ?N = "\<lambda> t. Inum (i#bs) t"
-  from prems Ia nb  show ?case 
-    apply (auto simp add: Let_def split_def algebra_simps) 
-    apply (cases "?r",auto)
-    apply (case_tac nat, auto)
-    done
-next
-  case (9 a)  
-  let ?c = "fst (zsplit0 a)"
-  let ?r = "snd (zsplit0 a)"
-  have spl: "zsplit0 a = (?c,?r)" by simp
-  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
-  let ?N = "\<lambda> t. Inum (i#bs) t"
-  from prems Ia nb  show ?case 
-    apply (auto simp add: Let_def split_def algebra_simps) 
-    apply (cases "?r",auto)
-    apply (case_tac nat, auto)
-    done
-next
-  case (10 a)  
-  let ?c = "fst (zsplit0 a)"
-  let ?r = "snd (zsplit0 a)"
-  have spl: "zsplit0 a = (?c,?r)" by simp
-  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
-  let ?N = "\<lambda> t. Inum (i#bs) t"
-  from prems Ia nb  show ?case 
-    apply (auto simp add: Let_def split_def algebra_simps) 
-    apply (cases "?r",auto)
-    apply (case_tac nat, auto)
-    done
-next
-  case (11 j a)  
-  let ?c = "fst (zsplit0 a)"
-  let ?r = "snd (zsplit0 a)"
-  have spl: "zsplit0 a = (?c,?r)" by simp
-  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
-  let ?N = "\<lambda> t. Inum (i#bs) t"
-  have "j=0 \<or> (j\<noteq>0 \<and> ?c = 0) \<or> (j\<noteq>0 \<and> ?c >0) \<or> (j\<noteq> 0 \<and> ?c<0)" by arith
-  moreover
-  {assume "j=0" hence z: "zlfm (Dvd j a) = (zlfm (Eq a))" by (simp add: Let_def) 
-    hence ?case using prems by (simp del: zlfm.simps add: zdvd_0_left)}
-  moreover
-  {assume "?c=0" and "j\<noteq>0" hence ?case 
-      using zsplit0_I[OF spl, where x="i" and bs="bs"]
-    apply (auto simp add: Let_def split_def algebra_simps) 
-    apply (cases "?r",auto)
-    apply (case_tac nat, auto)
-    done}
-  moreover
-  {assume cp: "?c > 0" and jnz: "j\<noteq>0" hence l: "?L (?l (Dvd j a))" 
-      by (simp add: nb Let_def split_def)
-    hence ?case using Ia cp jnz by (simp add: Let_def split_def)}
-  moreover
-  {assume cn: "?c < 0" and jnz: "j\<noteq>0" hence l: "?L (?l (Dvd j a))" 
-      by (simp add: nb Let_def split_def)
-    hence ?case using Ia cn jnz zdvd_zminus_iff[where m="abs j" and n="?c*i + ?N ?r" ]
-      by (simp add: Let_def split_def) }
-  ultimately show ?case by blast
-next
-  case (12 j a) 
-  let ?c = "fst (zsplit0 a)"
-  let ?r = "snd (zsplit0 a)"
-  have spl: "zsplit0 a = (?c,?r)" by simp
-  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
-  let ?N = "\<lambda> t. Inum (i#bs) t"
-  have "j=0 \<or> (j\<noteq>0 \<and> ?c = 0) \<or> (j\<noteq>0 \<and> ?c >0) \<or> (j\<noteq> 0 \<and> ?c<0)" by arith
-  moreover
-  {assume "j=0" hence z: "zlfm (NDvd j a) = (zlfm (NEq a))" by (simp add: Let_def) 
-    hence ?case using prems by (simp del: zlfm.simps add: zdvd_0_left)}
-  moreover
-  {assume "?c=0" and "j\<noteq>0" hence ?case 
-      using zsplit0_I[OF spl, where x="i" and bs="bs"]
-    apply (auto simp add: Let_def split_def algebra_simps) 
-    apply (cases "?r",auto)
-    apply (case_tac nat, auto)
-    done}
-  moreover
-  {assume cp: "?c > 0" and jnz: "j\<noteq>0" hence l: "?L (?l (Dvd j a))" 
-      by (simp add: nb Let_def split_def)
-    hence ?case using Ia cp jnz by (simp add: Let_def split_def) }
-  moreover
-  {assume cn: "?c < 0" and jnz: "j\<noteq>0" hence l: "?L (?l (Dvd j a))" 
-      by (simp add: nb Let_def split_def)
-    hence ?case using Ia cn jnz zdvd_zminus_iff[where m="abs j" and n="?c*i + ?N ?r" ]
-      by (simp add: Let_def split_def)}
-  ultimately show ?case by blast
-qed auto
-
-consts 
-  plusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of +\<infinity>*)
-  minusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of -\<infinity>*)
-  \<delta> :: "fm \<Rightarrow> int" (* Compute lcm {d| N\<^isup>?\<^isup> Dvd c*x+t \<in> p}*)
-  d\<delta> :: "fm \<Rightarrow> int \<Rightarrow> bool" (* checks if a given l divides all the ds above*)
-
-recdef minusinf "measure size"
-  "minusinf (And p q) = And (minusinf p) (minusinf q)" 
-  "minusinf (Or p q) = Or (minusinf p) (minusinf q)" 
-  "minusinf (Eq  (CN 0 c e)) = F"
-  "minusinf (NEq (CN 0 c e)) = T"
-  "minusinf (Lt  (CN 0 c e)) = T"
-  "minusinf (Le  (CN 0 c e)) = T"
-  "minusinf (Gt  (CN 0 c e)) = F"
-  "minusinf (Ge  (CN 0 c e)) = F"
-  "minusinf p = p"
-
-lemma minusinf_qfree: "qfree p \<Longrightarrow> qfree (minusinf p)"
-  by (induct p rule: minusinf.induct, auto)
-
-recdef plusinf "measure size"
-  "plusinf (And p q) = And (plusinf p) (plusinf q)" 
-  "plusinf (Or p q) = Or (plusinf p) (plusinf q)" 
-  "plusinf (Eq  (CN 0 c e)) = F"
-  "plusinf (NEq (CN 0 c e)) = T"
-  "plusinf (Lt  (CN 0 c e)) = F"
-  "plusinf (Le  (CN 0 c e)) = F"
-  "plusinf (Gt  (CN 0 c e)) = T"
-  "plusinf (Ge  (CN 0 c e)) = T"
-  "plusinf p = p"
-
-recdef \<delta> "measure size"
-  "\<delta> (And p q) = zlcm (\<delta> p) (\<delta> q)" 
-  "\<delta> (Or p q) = zlcm (\<delta> p) (\<delta> q)" 
-  "\<delta> (Dvd i (CN 0 c e)) = i"
-  "\<delta> (NDvd i (CN 0 c e)) = i"
-  "\<delta> p = 1"
-
-recdef d\<delta> "measure size"
-  "d\<delta> (And p q) = (\<lambda> d. d\<delta> p d \<and> d\<delta> q d)" 
-  "d\<delta> (Or p q) = (\<lambda> d. d\<delta> p d \<and> d\<delta> q d)" 
-  "d\<delta> (Dvd i (CN 0 c e)) = (\<lambda> d. i dvd d)"
-  "d\<delta> (NDvd i (CN 0 c e)) = (\<lambda> d. i dvd d)"
-  "d\<delta> p = (\<lambda> d. True)"
-
-lemma delta_mono: 
-  assumes lin: "iszlfm p"
-  and d: "d dvd d'"
-  and ad: "d\<delta> p d"
-  shows "d\<delta> p d'"
-  using lin ad d
-proof(induct p rule: iszlfm.induct)
-  case (9 i c e)  thus ?case using d
-    by (simp add: zdvd_trans[where m="i" and n="d" and k="d'"])
-next
-  case (10 i c e) thus ?case using d
-    by (simp add: zdvd_trans[where m="i" and n="d" and k="d'"])
-qed simp_all
-
-lemma \<delta> : assumes lin:"iszlfm p"
-  shows "d\<delta> p (\<delta> p) \<and> \<delta> p >0"
-using lin
-proof (induct p rule: iszlfm.induct)
-  case (1 p q) 
-  let ?d = "\<delta> (And p q)"
-  from prems zlcm_pos have dp: "?d >0" by simp
-  have d1: "\<delta> p dvd \<delta> (And p q)" using prems by simp
-  hence th: "d\<delta> p ?d" using delta_mono prems(3-4) by(simp del:dvd_zlcm_self1)
-  have "\<delta> q dvd \<delta> (And p q)" using prems by simp
-  hence th': "d\<delta> q ?d" using delta_mono prems by(simp del:dvd_zlcm_self2)
-  from th th' dp show ?case by simp
-next
-  case (2 p q)  
-  let ?d = "\<delta> (And p q)"
-  from prems zlcm_pos have dp: "?d >0" by simp
-  have "\<delta> p dvd \<delta> (And p q)" using prems by simp
-  hence th: "d\<delta> p ?d" using delta_mono prems by(simp del:dvd_zlcm_self1)
-  have "\<delta> q dvd \<delta> (And p q)" using prems by simp
-  hence th': "d\<delta> q ?d" using delta_mono prems by(simp del:dvd_zlcm_self2)
-  from th th' dp show ?case by simp
-qed simp_all
-
-
-consts 
-  a\<beta> :: "fm \<Rightarrow> int \<Rightarrow> fm" (* adjusts the coeffitients of a formula *)
-  d\<beta> :: "fm \<Rightarrow> int \<Rightarrow> bool" (* tests if all coeffs c of c divide a given l*)
-  \<zeta>  :: "fm \<Rightarrow> int" (* computes the lcm of all coefficients of x*)
-  \<beta> :: "fm \<Rightarrow> num list"
-  \<alpha> :: "fm \<Rightarrow> num list"
-
-recdef a\<beta> "measure size"
-  "a\<beta> (And p q) = (\<lambda> k. And (a\<beta> p k) (a\<beta> q k))" 
-  "a\<beta> (Or p q) = (\<lambda> k. Or (a\<beta> p k) (a\<beta> q k))" 
-  "a\<beta> (Eq  (CN 0 c e)) = (\<lambda> k. Eq (CN 0 1 (Mul (k div c) e)))"
-  "a\<beta> (NEq (CN 0 c e)) = (\<lambda> k. NEq (CN 0 1 (Mul (k div c) e)))"
-  "a\<beta> (Lt  (CN 0 c e)) = (\<lambda> k. Lt (CN 0 1 (Mul (k div c) e)))"
-  "a\<beta> (Le  (CN 0 c e)) = (\<lambda> k. Le (CN 0 1 (Mul (k div c) e)))"
-  "a\<beta> (Gt  (CN 0 c e)) = (\<lambda> k. Gt (CN 0 1 (Mul (k div c) e)))"
-  "a\<beta> (Ge  (CN 0 c e)) = (\<lambda> k. Ge (CN 0 1 (Mul (k div c) e)))"
-  "a\<beta> (Dvd i (CN 0 c e)) =(\<lambda> k. Dvd ((k div c)*i) (CN 0 1 (Mul (k div c) e)))"
-  "a\<beta> (NDvd i (CN 0 c e))=(\<lambda> k. NDvd ((k div c)*i) (CN 0 1 (Mul (k div c) e)))"
-  "a\<beta> p = (\<lambda> k. p)"
-
-recdef d\<beta> "measure size"
-  "d\<beta> (And p q) = (\<lambda> k. (d\<beta> p k) \<and> (d\<beta> q k))" 
-  "d\<beta> (Or p q) = (\<lambda> k. (d\<beta> p k) \<and> (d\<beta> q k))" 
-  "d\<beta> (Eq  (CN 0 c e)) = (\<lambda> k. c dvd k)"
-  "d\<beta> (NEq (CN 0 c e)) = (\<lambda> k. c dvd k)"
-  "d\<beta> (Lt  (CN 0 c e)) = (\<lambda> k. c dvd k)"
-  "d\<beta> (Le  (CN 0 c e)) = (\<lambda> k. c dvd k)"
-  "d\<beta> (Gt  (CN 0 c e)) = (\<lambda> k. c dvd k)"
-  "d\<beta> (Ge  (CN 0 c e)) = (\<lambda> k. c dvd k)"
-  "d\<beta> (Dvd i (CN 0 c e)) =(\<lambda> k. c dvd k)"
-  "d\<beta> (NDvd i (CN 0 c e))=(\<lambda> k. c dvd k)"
-  "d\<beta> p = (\<lambda> k. True)"
-
-recdef \<zeta> "measure size"
-  "\<zeta> (And p q) = zlcm (\<zeta> p) (\<zeta> q)" 
-  "\<zeta> (Or p q) = zlcm (\<zeta> p) (\<zeta> q)" 
-  "\<zeta> (Eq  (CN 0 c e)) = c"
-  "\<zeta> (NEq (CN 0 c e)) = c"
-  "\<zeta> (Lt  (CN 0 c e)) = c"
-  "\<zeta> (Le  (CN 0 c e)) = c"
-  "\<zeta> (Gt  (CN 0 c e)) = c"
-  "\<zeta> (Ge  (CN 0 c e)) = c"
-  "\<zeta> (Dvd i (CN 0 c e)) = c"
-  "\<zeta> (NDvd i (CN 0 c e))= c"
-  "\<zeta> p = 1"
-
-recdef \<beta> "measure size"
-  "\<beta> (And p q) = (\<beta> p @ \<beta> q)" 
-  "\<beta> (Or p q) = (\<beta> p @ \<beta> q)" 
-  "\<beta> (Eq  (CN 0 c e)) = [Sub (C -1) e]"
-  "\<beta> (NEq (CN 0 c e)) = [Neg e]"
-  "\<beta> (Lt  (CN 0 c e)) = []"
-  "\<beta> (Le  (CN 0 c e)) = []"
-  "\<beta> (Gt  (CN 0 c e)) = [Neg e]"
-  "\<beta> (Ge  (CN 0 c e)) = [Sub (C -1) e]"
-  "\<beta> p = []"
-
-recdef \<alpha> "measure size"
-  "\<alpha> (And p q) = (\<alpha> p @ \<alpha> q)" 
-  "\<alpha> (Or p q) = (\<alpha> p @ \<alpha> q)" 
-  "\<alpha> (Eq  (CN 0 c e)) = [Add (C -1) e]"
-  "\<alpha> (NEq (CN 0 c e)) = [e]"
-  "\<alpha> (Lt  (CN 0 c e)) = [e]"
-  "\<alpha> (Le  (CN 0 c e)) = [Add (C -1) e]"
-  "\<alpha> (Gt  (CN 0 c e)) = []"
-  "\<alpha> (Ge  (CN 0 c e)) = []"
-  "\<alpha> p = []"
-consts mirror :: "fm \<Rightarrow> fm"
-recdef mirror "measure size"
-  "mirror (And p q) = And (mirror p) (mirror q)" 
-  "mirror (Or p q) = Or (mirror p) (mirror q)" 
-  "mirror (Eq  (CN 0 c e)) = Eq (CN 0 c (Neg e))"
-  "mirror (NEq (CN 0 c e)) = NEq (CN 0 c (Neg e))"
-  "mirror (Lt  (CN 0 c e)) = Gt (CN 0 c (Neg e))"
-  "mirror (Le  (CN 0 c e)) = Ge (CN 0 c (Neg e))"
-  "mirror (Gt  (CN 0 c e)) = Lt (CN 0 c (Neg e))"
-  "mirror (Ge  (CN 0 c e)) = Le (CN 0 c (Neg e))"
-  "mirror (Dvd i (CN 0 c e)) = Dvd i (CN 0 c (Neg e))"
-  "mirror (NDvd i (CN 0 c e)) = NDvd i (CN 0 c (Neg e))"
-  "mirror p = p"
-    (* Lemmas for the correctness of \<sigma>\<rho> *)
-lemma dvd1_eq1: "x >0 \<Longrightarrow> (x::int) dvd 1 = (x = 1)"
-by simp
-
-lemma minusinf_inf:
-  assumes linp: "iszlfm p"
-  and u: "d\<beta> p 1"
-  shows "\<exists> (z::int). \<forall> x < z. Ifm bbs (x#bs) (minusinf p) = Ifm bbs (x#bs) p"
-  (is "?P p" is "\<exists> (z::int). \<forall> x < z. ?I x (?M p) = ?I x p")
-using linp u
-proof (induct p rule: minusinf.induct)
-  case (1 p q) thus ?case 
-    by auto (rule_tac x="min z za" in exI,simp)
-next
-  case (2 p q) thus ?case 
-    by auto (rule_tac x="min z za" in exI,simp)
-next
-  case (3 c e) hence c1: "c=1" and nb: "numbound0 e" by simp+
-  fix a
-  from 3 have "\<forall> x<(- Inum (a#bs) e). c*x + Inum (x#bs) e \<noteq> 0"
-  proof(clarsimp)
-    fix x assume "x < (- Inum (a#bs) e)" and"x + Inum (x#bs) e = 0"
-    with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"]
-    show "False" by simp
-  qed
-  thus ?case by auto
-next
-  case (4 c e) hence c1: "c=1" and nb: "numbound0 e" by simp+
-  fix a
-  from 4 have "\<forall> x<(- Inum (a#bs) e). c*x + Inum (x#bs) e \<noteq> 0"
-  proof(clarsimp)
-    fix x assume "x < (- Inum (a#bs) e)" and"x + Inum (x#bs) e = 0"
-    with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"]
-    show "False" by simp
-  qed
-  thus ?case by auto
-next
-  case (5 c e) hence c1: "c=1" and nb: "numbound0 e" by simp+
-  fix a
-  from 5 have "\<forall> x<(- Inum (a#bs) e). c*x + Inum (x#bs) e < 0"
-  proof(clarsimp)
-    fix x assume "x < (- Inum (a#bs) e)" 
-    with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"]
-    show "x + Inum (x#bs) e < 0" by simp
-  qed
-  thus ?case by auto
-next
-  case (6 c e) hence c1: "c=1" and nb: "numbound0 e" by simp+
-  fix a
-  from 6 have "\<forall> x<(- Inum (a#bs) e). c*x + Inum (x#bs) e \<le> 0"
-  proof(clarsimp)
-    fix x assume "x < (- Inum (a#bs) e)" 
-    with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"]
-    show "x + Inum (x#bs) e \<le> 0" by simp
-  qed
-  thus ?case by auto
-next
-  case (7 c e) hence c1: "c=1" and nb: "numbound0 e" by simp+
-  fix a
-  from 7 have "\<forall> x<(- Inum (a#bs) e). \<not> (c*x + Inum (x#bs) e > 0)"
-  proof(clarsimp)
-    fix x assume "x < (- Inum (a#bs) e)" and"x + Inum (x#bs) e > 0"
-    with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"]
-    show "False" by simp
-  qed
-  thus ?case by auto
-next
-  case (8 c e) hence c1: "c=1" and nb: "numbound0 e" by simp+
-  fix a
-  from 8 have "\<forall> x<(- Inum (a#bs) e). \<not> (c*x + Inum (x#bs) e \<ge> 0)"
-  proof(clarsimp)
-    fix x assume "x < (- Inum (a#bs) e)" and"x + Inum (x#bs) e \<ge> 0"
-    with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"]
-    show "False" by simp
-  qed
-  thus ?case by auto
-qed auto
-
-lemma minusinf_repeats:
-  assumes d: "d\<delta> p d" and linp: "iszlfm p"
-  shows "Ifm bbs ((x - k*d)#bs) (minusinf p) = Ifm bbs (x #bs) (minusinf p)"
-using linp d
-proof(induct p rule: iszlfm.induct) 
-  case (9 i c e) hence nbe: "numbound0 e"  and id: "i dvd d" by simp+
-    hence "\<exists> k. d=i*k" by (simp add: dvd_def)
-    then obtain "di" where di_def: "d=i*di" by blast
-    show ?case 
-    proof(simp add: numbound0_I[OF nbe,where bs="bs" and b="x - k * d" and b'="x"] right_diff_distrib, rule iffI)
-      assume 
-	"i dvd c * x - c*(k*d) + Inum (x # bs) e"
-      (is "?ri dvd ?rc*?rx - ?rc*(?rk*?rd) + ?I x e" is "?ri dvd ?rt")
-      hence "\<exists> (l::int). ?rt = i * l" by (simp add: dvd_def)
-      hence "\<exists> (l::int). c*x+ ?I x e = i*l+c*(k * i*di)" 
-	by (simp add: algebra_simps di_def)
-      hence "\<exists> (l::int). c*x+ ?I x e = i*(l + c*k*di)"
-	by (simp add: algebra_simps)
-      hence "\<exists> (l::int). c*x+ ?I x e = i*l" by blast
-      thus "i dvd c*x + Inum (x # bs) e" by (simp add: dvd_def) 
-    next
-      assume 
-	"i dvd c*x + Inum (x # bs) e" (is "?ri dvd ?rc*?rx+?e")
-      hence "\<exists> (l::int). c*x+?e = i*l" by (simp add: dvd_def)
-      hence "\<exists> (l::int). c*x - c*(k*d) +?e = i*l - c*(k*d)" by simp
-      hence "\<exists> (l::int). c*x - c*(k*d) +?e = i*l - c*(k*i*di)" by (simp add: di_def)
-      hence "\<exists> (l::int). c*x - c*(k*d) +?e = i*((l - c*k*di))" by (simp add: algebra_simps)
-      hence "\<exists> (l::int). c*x - c * (k*d) +?e = i*l"
-	by blast
-      thus "i dvd c*x - c*(k*d) + Inum (x # bs) e" by (simp add: dvd_def)
-    qed
-next
-  case (10 i c e)  hence nbe: "numbound0 e"  and id: "i dvd d" by simp+
-    hence "\<exists> k. d=i*k" by (simp add: dvd_def)
-    then obtain "di" where di_def: "d=i*di" by blast
-    show ?case 
-    proof(simp add: numbound0_I[OF nbe,where bs="bs" and b="x - k * d" and b'="x"] right_diff_distrib, rule iffI)
-      assume 
-	"i dvd c * x - c*(k*d) + Inum (x # bs) e"
-      (is "?ri dvd ?rc*?rx - ?rc*(?rk*?rd) + ?I x e" is "?ri dvd ?rt")
-      hence "\<exists> (l::int). ?rt = i * l" by (simp add: dvd_def)
-      hence "\<exists> (l::int). c*x+ ?I x e = i*l+c*(k * i*di)" 
-	by (simp add: algebra_simps di_def)
-      hence "\<exists> (l::int). c*x+ ?I x e = i*(l + c*k*di)"
-	by (simp add: algebra_simps)
-      hence "\<exists> (l::int). c*x+ ?I x e = i*l" by blast
-      thus "i dvd c*x + Inum (x # bs) e" by (simp add: dvd_def) 
-    next
-      assume 
-	"i dvd c*x + Inum (x # bs) e" (is "?ri dvd ?rc*?rx+?e")
-      hence "\<exists> (l::int). c*x+?e = i*l" by (simp add: dvd_def)
-      hence "\<exists> (l::int). c*x - c*(k*d) +?e = i*l - c*(k*d)" by simp
-      hence "\<exists> (l::int). c*x - c*(k*d) +?e = i*l - c*(k*i*di)" by (simp add: di_def)
-      hence "\<exists> (l::int). c*x - c*(k*d) +?e = i*((l - c*k*di))" by (simp add: algebra_simps)
-      hence "\<exists> (l::int). c*x - c * (k*d) +?e = i*l"
-	by blast
-      thus "i dvd c*x - c*(k*d) + Inum (x # bs) e" by (simp add: dvd_def)
-    qed
-qed (auto simp add: gr0_conv_Suc numbound0_I[where bs="bs" and b="x - k*d" and b'="x"])
-
-lemma mirror\<alpha>\<beta>:
-  assumes lp: "iszlfm p"
-  shows "(Inum (i#bs)) ` set (\<alpha> p) = (Inum (i#bs)) ` set (\<beta> (mirror p))"
-using lp
-by (induct p rule: mirror.induct, auto)
-
-lemma mirror: 
-  assumes lp: "iszlfm p"
-  shows "Ifm bbs (x#bs) (mirror p) = Ifm bbs ((- x)#bs) p" 
-using lp
-proof(induct p rule: iszlfm.induct)
-  case (9 j c e) hence nb: "numbound0 e" by simp
-  have "Ifm bbs (x#bs) (mirror (Dvd j (CN 0 c e))) = (j dvd c*x - Inum (x#bs) e)" (is "_ = (j dvd c*x - ?e)") by simp
-    also have "\<dots> = (j dvd (- (c*x - ?e)))"
-    by (simp only: zdvd_zminus_iff)
-  also have "\<dots> = (j dvd (c* (- x)) + ?e)"
-    apply (simp only: minus_mult_right[symmetric] minus_mult_left[symmetric] diff_def zadd_ac zminus_zadd_distrib)
-    by (simp add: algebra_simps)
-  also have "\<dots> = Ifm bbs ((- x)#bs) (Dvd j (CN 0 c e))"
-    using numbound0_I[OF nb, where bs="bs" and b="x" and b'="- x"]
-    by simp
-  finally show ?case .
-next
-    case (10 j c e) hence nb: "numbound0 e" by simp
-  have "Ifm bbs (x#bs) (mirror (Dvd j (CN 0 c e))) = (j dvd c*x - Inum (x#bs) e)" (is "_ = (j dvd c*x - ?e)") by simp
-    also have "\<dots> = (j dvd (- (c*x - ?e)))"
-    by (simp only: zdvd_zminus_iff)
-  also have "\<dots> = (j dvd (c* (- x)) + ?e)"
-    apply (simp only: minus_mult_right[symmetric] minus_mult_left[symmetric] diff_def zadd_ac zminus_zadd_distrib)
-    by (simp add: algebra_simps)
-  also have "\<dots> = Ifm bbs ((- x)#bs) (Dvd j (CN 0 c e))"
-    using numbound0_I[OF nb, where bs="bs" and b="x" and b'="- x"]
-    by simp
-  finally show ?case by simp
-qed (auto simp add: numbound0_I[where bs="bs" and b="x" and b'="- x"] gr0_conv_Suc)
-
-lemma mirror_l: "iszlfm p \<and> d\<beta> p 1 
-  \<Longrightarrow> iszlfm (mirror p) \<and> d\<beta> (mirror p) 1"
-by (induct p rule: mirror.induct, auto)
-
-lemma mirror_\<delta>: "iszlfm p \<Longrightarrow> \<delta> (mirror p) = \<delta> p"
-by (induct p rule: mirror.induct,auto)
-
-lemma \<beta>_numbound0: assumes lp: "iszlfm p"
-  shows "\<forall> b\<in> set (\<beta> p). numbound0 b"
-  using lp by (induct p rule: \<beta>.induct,auto)
-
-lemma d\<beta>_mono: 
-  assumes linp: "iszlfm p"
-  and dr: "d\<beta> p l"
-  and d: "l dvd l'"
-  shows "d\<beta> p l'"
-using dr linp zdvd_trans[where n="l" and k="l'", simplified d]
-by (induct p rule: iszlfm.induct) simp_all
-
-lemma \<alpha>_l: assumes lp: "iszlfm p"
-  shows "\<forall> b\<in> set (\<alpha> p). numbound0 b"
-using lp
-by(induct p rule: \<alpha>.induct, auto)
-
-lemma \<zeta>: 
-  assumes linp: "iszlfm p"
-  shows "\<zeta> p > 0 \<and> d\<beta> p (\<zeta> p)"
-using linp
-proof(induct p rule: iszlfm.induct)
-  case (1 p q)
-  from prems have dl1: "\<zeta> p dvd zlcm (\<zeta> p) (\<zeta> q)" by simp
-  from prems have dl2: "\<zeta> q dvd zlcm (\<zeta> p) (\<zeta> q)"  by simp
-  from prems d\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="zlcm (\<zeta> p) (\<zeta> q)"] 
-    d\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="zlcm (\<zeta> p) (\<zeta> q)"] 
-    dl1 dl2 show ?case by (auto simp add: zlcm_pos)
-next
-  case (2 p q)
-  from prems have dl1: "\<zeta> p dvd zlcm (\<zeta> p) (\<zeta> q)" by simp
-  from prems have dl2: "\<zeta> q dvd zlcm (\<zeta> p) (\<zeta> q)" by simp
-  from prems d\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="zlcm (\<zeta> p) (\<zeta> q)"] 
-    d\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="zlcm (\<zeta> p) (\<zeta> q)"] 
-    dl1 dl2 show ?case by (auto simp add: zlcm_pos)
-qed (auto simp add: zlcm_pos)
-
-lemma a\<beta>: assumes linp: "iszlfm p" and d: "d\<beta> p l" and lp: "l > 0"
-  shows "iszlfm (a\<beta> p l) \<and> d\<beta> (a\<beta> p l) 1 \<and> (Ifm bbs (l*x #bs) (a\<beta> p l) = Ifm bbs (x#bs) p)"
-using linp d
-proof (induct p rule: iszlfm.induct)
-  case (5 c e) hence cp: "c>0" and be: "numbound0 e" and d': "c dvd l" by simp+
-    from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
-    from cp have cnz: "c \<noteq> 0" by simp
-    have "c div c\<le> l div c"
-      by (simp add: zdiv_mono1[OF clel cp])
-    then have ldcp:"0 < l div c" 
-      by (simp add: zdiv_self[OF cnz])
-    have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp
-    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
-      by simp
-    hence "(l*x + (l div c) * Inum (x # bs) e < 0) =
-          ((c * (l div c)) * x + (l div c) * Inum (x # bs) e < 0)"
-      by simp
-    also have "\<dots> = ((l div c) * (c*x + Inum (x # bs) e) < (l div c) * 0)" by (simp add: algebra_simps)
-    also have "\<dots> = (c*x + Inum (x # bs) e < 0)"
-    using mult_less_0_iff [where a="(l div c)" and b="c*x + Inum (x # bs) e"] ldcp by simp
-  finally show ?case using numbound0_I[OF be,where b="l*x" and b'="x" and bs="bs"] be  by simp
-next
-  case (6 c e) hence cp: "c>0" and be: "numbound0 e" and d': "c dvd l" by simp+
-    from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
-    from cp have cnz: "c \<noteq> 0" by simp
-    have "c div c\<le> l div c"
-      by (simp add: zdiv_mono1[OF clel cp])
-    then have ldcp:"0 < l div c" 
-      by (simp add: zdiv_self[OF cnz])
-    have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp
-    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
-      by simp
-    hence "(l*x + (l div c) * Inum (x# bs) e \<le> 0) =
-          ((c * (l div c)) * x + (l div c) * Inum (x # bs) e \<le> 0)"
-      by simp
-    also have "\<dots> = ((l div c) * (c * x + Inum (x # bs) e) \<le> ((l div c)) * 0)" by (simp add: algebra_simps)
-    also have "\<dots> = (c*x + Inum (x # bs) e \<le> 0)"
-    using mult_le_0_iff [where a="(l div c)" and b="c*x + Inum (x # bs) e"] ldcp by simp
-  finally show ?case using numbound0_I[OF be,where b="l*x" and b'="x" and bs="bs"]  be by simp
-next
-  case (7 c e) hence cp: "c>0" and be: "numbound0 e" and d': "c dvd l" by simp+
-    from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
-    from cp have cnz: "c \<noteq> 0" by simp
-    have "c div c\<le> l div c"
-      by (simp add: zdiv_mono1[OF clel cp])
-    then have ldcp:"0 < l div c" 
-      by (simp add: zdiv_self[OF cnz])
-    have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp
-    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
-      by simp
-    hence "(l*x + (l div c)* Inum (x # bs) e > 0) =
-          ((c * (l div c)) * x + (l div c) * Inum (x # bs) e > 0)"
-      by simp
-    also have "\<dots> = ((l div c) * (c * x + Inum (x # bs) e) > ((l div c)) * 0)" by (simp add: algebra_simps)
-    also have "\<dots> = (c * x + Inum (x # bs) e > 0)"
-    using zero_less_mult_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e"] ldcp by simp
-  finally show ?case using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"]  be  by simp
-next
-  case (8 c e) hence cp: "c>0" and be: "numbound0 e" and d': "c dvd l" by simp+
-    from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
-    from cp have cnz: "c \<noteq> 0" by simp
-    have "c div c\<le> l div c"
-      by (simp add: zdiv_mono1[OF clel cp])
-    then have ldcp:"0 < l div c" 
-      by (simp add: zdiv_self[OF cnz])
-    have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp
-    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
-      by simp
-    hence "(l*x + (l div c)* Inum (x # bs) e \<ge> 0) =
-          ((c*(l div c))*x + (l div c)* Inum (x # bs) e \<ge> 0)"
-      by simp
-    also have "\<dots> = ((l div c)*(c*x + Inum (x # bs) e) \<ge> ((l div c)) * 0)" 
-      by (simp add: algebra_simps)
-    also have "\<dots> = (c*x + Inum (x # bs) e \<ge> 0)" using ldcp 
-      zero_le_mult_iff [where a="l div c" and b="c*x + Inum (x # bs) e"] by simp
-  finally show ?case using be numbound0_I[OF be,where b="l*x" and b'="x" and bs="bs"]  
-    by simp
-next
-  case (3 c e) hence cp: "c>0" and be: "numbound0 e" and d': "c dvd l" by simp+
-    from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
-    from cp have cnz: "c \<noteq> 0" by simp
-    have "c div c\<le> l div c"
-      by (simp add: zdiv_mono1[OF clel cp])
-    then have ldcp:"0 < l div c" 
-      by (simp add: zdiv_self[OF cnz])
-    have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp
-    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
-      by simp
-    hence "(l * x + (l div c) * Inum (x # bs) e = 0) =
-          ((c * (l div c)) * x + (l div c) * Inum (x # bs) e = 0)"
-      by simp
-    also have "\<dots> = ((l div c) * (c * x + Inum (x # bs) e) = ((l div c)) * 0)" by (simp add: algebra_simps)
-    also have "\<dots> = (c * x + Inum (x # bs) e = 0)"
-    using mult_eq_0_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e"] ldcp by simp
-  finally show ?case using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"]  be  by simp
-next
-  case (4 c e) hence cp: "c>0" and be: "numbound0 e" and d': "c dvd l" by simp+
-    from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
-    from cp have cnz: "c \<noteq> 0" by simp
-    have "c div c\<le> l div c"
-      by (simp add: zdiv_mono1[OF clel cp])
-    then have ldcp:"0 < l div c" 
-      by (simp add: zdiv_self[OF cnz])
-    have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp
-    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
-      by simp
-    hence "(l * x + (l div c) * Inum (x # bs) e \<noteq> 0) =
-          ((c * (l div c)) * x + (l div c) * Inum (x # bs) e \<noteq> 0)"
-      by simp
-    also have "\<dots> = ((l div c) * (c * x + Inum (x # bs) e) \<noteq> ((l div c)) * 0)" by (simp add: algebra_simps)
-    also have "\<dots> = (c * x + Inum (x # bs) e \<noteq> 0)"
-    using zero_le_mult_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e"] ldcp by simp
-  finally show ?case using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"]  be  by simp
-next
-  case (9 j c e) hence cp: "c>0" and be: "numbound0 e" and jp: "j > 0" and d': "c dvd l" by simp+
-    from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
-    from cp have cnz: "c \<noteq> 0" by simp
-    have "c div c\<le> l div c"
-      by (simp add: zdiv_mono1[OF clel cp])
-    then have ldcp:"0 < l div c" 
-      by (simp add: zdiv_self[OF cnz])
-    have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp
-    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
-      by simp
-    hence "(\<exists> (k::int). l * x + (l div c) * Inum (x # bs) e = ((l div c) * j) * k) = (\<exists> (k::int). (c * (l div c)) * x + (l div c) * Inum (x # bs) e = ((l div c) * j) * k)"  by simp
-    also have "\<dots> = (\<exists> (k::int). (l div c) * (c * x + Inum (x # bs) e - j * k) = (l div c)*0)" by (simp add: algebra_simps)
-    also fix k have "\<dots> = (\<exists> (k::int). c * x + Inum (x # bs) e - j * k = 0)"
-    using zero_le_mult_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e - j * k"] ldcp by simp
-  also have "\<dots> = (\<exists> (k::int). c * x + Inum (x # bs) e = j * k)" by simp
-  finally show ?case using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"] be  mult_strict_mono[OF ldcp jp ldcp ] by (simp add: dvd_def)
-next
-  case (10 j c e) hence cp: "c>0" and be: "numbound0 e" and jp: "j > 0" and d': "c dvd l" by simp+
-    from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
-    from cp have cnz: "c \<noteq> 0" by simp
-    have "c div c\<le> l div c"
-      by (simp add: zdiv_mono1[OF clel cp])
-    then have ldcp:"0 < l div c" 
-      by (simp add: zdiv_self[OF cnz])
-    have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp
-    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
-      by simp
-    hence "(\<exists> (k::int). l * x + (l div c) * Inum (x # bs) e = ((l div c) * j) * k) = (\<exists> (k::int). (c * (l div c)) * x + (l div c) * Inum (x # bs) e = ((l div c) * j) * k)"  by simp
-    also have "\<dots> = (\<exists> (k::int). (l div c) * (c * x + Inum (x # bs) e - j * k) = (l div c)*0)" by (simp add: algebra_simps)
-    also fix k have "\<dots> = (\<exists> (k::int). c * x + Inum (x # bs) e - j * k = 0)"
-    using zero_le_mult_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e - j * k"] ldcp by simp
-  also have "\<dots> = (\<exists> (k::int). c * x + Inum (x # bs) e = j * k)" by simp
-  finally show ?case using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"] be  mult_strict_mono[OF ldcp jp ldcp ] by (simp add: dvd_def)
-qed (auto simp add: gr0_conv_Suc numbound0_I[where bs="bs" and b="(l * x)" and b'="x"])
-
-lemma a\<beta>_ex: assumes linp: "iszlfm p" and d: "d\<beta> p l" and lp: "l>0"
-  shows "(\<exists> x. l dvd x \<and> Ifm bbs (x #bs) (a\<beta> p l)) = (\<exists> (x::int). Ifm bbs (x#bs) p)"
-  (is "(\<exists> x. l dvd x \<and> ?P x) = (\<exists> x. ?P' x)")
-proof-
-  have "(\<exists> x. l dvd x \<and> ?P x) = (\<exists> (x::int). ?P (l*x))"
-    using unity_coeff_ex[where l="l" and P="?P", simplified] by simp
-  also have "\<dots> = (\<exists> (x::int). ?P' x)" using a\<beta>[OF linp d lp] by simp
-  finally show ?thesis  . 
-qed
-
-lemma \<beta>:
-  assumes lp: "iszlfm p"
-  and u: "d\<beta> p 1"
-  and d: "d\<delta> p d"
-  and dp: "d > 0"
-  and nob: "\<not>(\<exists>(j::int) \<in> {1 .. d}. \<exists> b\<in> (Inum (a#bs)) ` set(\<beta> p). x = b + j)"
-  and p: "Ifm bbs (x#bs) p" (is "?P x")
-  shows "?P (x - d)"
-using lp u d dp nob p
-proof(induct p rule: iszlfm.induct)
-  case (5 c e) hence c1: "c=1" and  bn:"numbound0 e" by simp+
-    with dp p c1 numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] prems
-    show ?case by simp
-next
-  case (6 c e)  hence c1: "c=1" and  bn:"numbound0 e" by simp+
-    with dp p c1 numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] prems
-    show ?case by simp
-next
-  case (7 c e) hence p: "Ifm bbs (x #bs) (Gt (CN 0 c e))" and c1: "c=1" and bn:"numbound0 e" by simp+
-    let ?e = "Inum (x # bs) e"
-    {assume "(x-d) +?e > 0" hence ?case using c1 
-      numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] by simp}
-    moreover
-    {assume H: "\<not> (x-d) + ?e > 0" 
-      let ?v="Neg e"
-      have vb: "?v \<in> set (\<beta> (Gt (CN 0 c e)))" by simp
-      from prems(11)[simplified simp_thms Inum.simps \<beta>.simps set.simps bex_simps numbound0_I[OF bn,where b="a" and b'="x" and bs="bs"]] 
-      have nob: "\<not> (\<exists> j\<in> {1 ..d}. x =  - ?e + j)" by auto 
-      from H p have "x + ?e > 0 \<and> x + ?e \<le> d" by (simp add: c1)
-      hence "x + ?e \<ge> 1 \<and> x + ?e \<le> d"  by simp
-      hence "\<exists> (j::int) \<in> {1 .. d}. j = x + ?e" by simp
-      hence "\<exists> (j::int) \<in> {1 .. d}. x = (- ?e + j)" 
-	by (simp add: algebra_simps)
-      with nob have ?case by auto}
-    ultimately show ?case by blast
-next
-  case (8 c e) hence p: "Ifm bbs (x #bs) (Ge (CN 0 c e))" and c1: "c=1" and bn:"numbound0 e" 
-    by simp+
-    let ?e = "Inum (x # bs) e"
-    {assume "(x-d) +?e \<ge> 0" hence ?case using  c1 
-      numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"]
-	by simp}
-    moreover
-    {assume H: "\<not> (x-d) + ?e \<ge> 0" 
-      let ?v="Sub (C -1) e"
-      have vb: "?v \<in> set (\<beta> (Ge (CN 0 c e)))" by simp
-      from prems(11)[simplified simp_thms Inum.simps \<beta>.simps set.simps bex_simps numbound0_I[OF bn,where b="a" and b'="x" and bs="bs"]] 
-      have nob: "\<not> (\<exists> j\<in> {1 ..d}. x =  - ?e - 1 + j)" by auto 
-      from H p have "x + ?e \<ge> 0 \<and> x + ?e < d" by (simp add: c1)
-      hence "x + ?e +1 \<ge> 1 \<and> x + ?e + 1 \<le> d"  by simp
-      hence "\<exists> (j::int) \<in> {1 .. d}. j = x + ?e + 1" by simp
-      hence "\<exists> (j::int) \<in> {1 .. d}. x= - ?e - 1 + j" by (simp add: algebra_simps)
-      with nob have ?case by simp }
-    ultimately show ?case by blast
-next
-  case (3 c e) hence p: "Ifm bbs (x #bs) (Eq (CN 0 c e))" (is "?p x") and c1: "c=1" and bn:"numbound0 e" by simp+
-    let ?e = "Inum (x # bs) e"
-    let ?v="(Sub (C -1) e)"
-    have vb: "?v \<in> set (\<beta> (Eq (CN 0 c e)))" by simp
-    from p have "x= - ?e" by (simp add: c1) with prems(11) show ?case using dp
-      by simp (erule ballE[where x="1"],
-	simp_all add:algebra_simps numbound0_I[OF bn,where b="x"and b'="a"and bs="bs"])
-next
-  case (4 c e)hence p: "Ifm bbs (x #bs) (NEq (CN 0 c e))" (is "?p x") and c1: "c=1" and bn:"numbound0 e" by simp+
-    let ?e = "Inum (x # bs) e"
-    let ?v="Neg e"
-    have vb: "?v \<in> set (\<beta> (NEq (CN 0 c e)))" by simp
-    {assume "x - d + Inum (((x -d)) # bs) e \<noteq> 0" 
-      hence ?case by (simp add: c1)}
-    moreover
-    {assume H: "x - d + Inum (((x -d)) # bs) e = 0"
-      hence "x = - Inum (((x -d)) # bs) e + d" by simp
-      hence "x = - Inum (a # bs) e + d"
-	by (simp add: numbound0_I[OF bn,where b="x - d"and b'="a"and bs="bs"])
-       with prems(11) have ?case using dp by simp}
-  ultimately show ?case by blast
-next 
-  case (9 j c e) hence p: "Ifm bbs (x #bs) (Dvd j (CN 0 c e))" (is "?p x") and c1: "c=1" and bn:"numbound0 e" by simp+
-    let ?e = "Inum (x # bs) e"
-    from prems have id: "j dvd d" by simp
-    from c1 have "?p x = (j dvd (x+ ?e))" by simp
-    also have "\<dots> = (j dvd x - d + ?e)" 
-      using zdvd_period[OF id, where x="x" and c="-1" and t="?e"] by simp
-    finally show ?case 
-      using numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] c1 p by simp
-next
-  case (10 j c e) hence p: "Ifm bbs (x #bs) (NDvd j (CN 0 c e))" (is "?p x") and c1: "c=1" and bn:"numbound0 e" by simp+
-    let ?e = "Inum (x # bs) e"
-    from prems have id: "j dvd d" by simp
-    from c1 have "?p x = (\<not> j dvd (x+ ?e))" by simp
-    also have "\<dots> = (\<not> j dvd x - d + ?e)" 
-      using zdvd_period[OF id, where x="x" and c="-1" and t="?e"] by simp
-    finally show ?case using numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] c1 p by simp
-qed (auto simp add: numbound0_I[where bs="bs" and b="(x - d)" and b'="x"] gr0_conv_Suc)
-
-lemma \<beta>':   
-  assumes lp: "iszlfm p"
-  and u: "d\<beta> p 1"
-  and d: "d\<delta> p d"
-  and dp: "d > 0"
-  shows "\<forall> x. \<not>(\<exists>(j::int) \<in> {1 .. d}. \<exists> b\<in> set(\<beta> p). Ifm bbs ((Inum (a#bs) b + j) #bs) p) \<longrightarrow> Ifm bbs (x#bs) p \<longrightarrow> Ifm bbs ((x - d)#bs) p" (is "\<forall> x. ?b \<longrightarrow> ?P x \<longrightarrow> ?P (x - d)")
-proof(clarify)
-  fix x 
-  assume nb:"?b" and px: "?P x" 
-  hence nb2: "\<not>(\<exists>(j::int) \<in> {1 .. d}. \<exists> b\<in> (Inum (a#bs)) ` set(\<beta> p). x = b + j)"
-    by auto
-  from  \<beta>[OF lp u d dp nb2 px] show "?P (x -d )" .
-qed
-lemma cpmi_eq: "0 < D \<Longrightarrow> (EX z::int. ALL x. x < z --> (P x = P1 x))
-==> ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D) 
-==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x-k*D))))
-==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (b::int) : B. P (b+j)))"
-apply(rule iffI)
-prefer 2
-apply(drule minusinfinity)
-apply assumption+
-apply(fastsimp)
-apply clarsimp
-apply(subgoal_tac "!!k. 0<=k \<Longrightarrow> !x. P x \<longrightarrow> P (x - k*D)")
-apply(frule_tac x = x and z=z in decr_lemma)
-apply(subgoal_tac "P1(x - (\<bar>x - z\<bar> + 1) * D)")
-prefer 2
-apply(subgoal_tac "0 <= (\<bar>x - z\<bar> + 1)")
-prefer 2 apply arith
- apply fastsimp
-apply(drule (1)  periodic_finite_ex)
-apply blast
-apply(blast dest:decr_mult_lemma)
-done
-
-theorem cp_thm:
-  assumes lp: "iszlfm p"
-  and u: "d\<beta> p 1"
-  and d: "d\<delta> p d"
-  and dp: "d > 0"
-  shows "(\<exists> (x::int). Ifm bbs (x #bs) p) = (\<exists> j\<in> {1.. d}. Ifm bbs (j #bs) (minusinf p) \<or> (\<exists> b \<in> set (\<beta> p). Ifm bbs ((Inum (i#bs) b + j) #bs) p))"
-  (is "(\<exists> (x::int). ?P (x)) = (\<exists> j\<in> ?D. ?M j \<or> (\<exists> b\<in> ?B. ?P (?I b + j)))")
-proof-
-  from minusinf_inf[OF lp u] 
-  have th: "\<exists>(z::int). \<forall>x<z. ?P (x) = ?M x" by blast
-  let ?B' = "{?I b | b. b\<in> ?B}"
-  have BB': "(\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (?I b +j)) = (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (b + j))" by auto
-  hence th2: "\<forall> x. \<not> (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P ((b + j))) \<longrightarrow> ?P (x) \<longrightarrow> ?P ((x - d))" 
-    using \<beta>'[OF lp u d dp, where a="i" and bbs = "bbs"] by blast
-  from minusinf_repeats[OF d lp]
-  have th3: "\<forall> x k. ?M x = ?M (x-k*d)" by simp
-  from cpmi_eq[OF dp th th2 th3] BB' show ?thesis by blast
-qed
-
-    (* Implement the right hand sides of Cooper's theorem and Ferrante and Rackoff. *)
-lemma mirror_ex: 
-  assumes lp: "iszlfm p"
-  shows "(\<exists> x. Ifm bbs (x#bs) (mirror p)) = (\<exists> x. Ifm bbs (x#bs) p)"
-  (is "(\<exists> x. ?I x ?mp) = (\<exists> x. ?I x p)")
-proof(auto)
-  fix x assume "?I x ?mp" hence "?I (- x) p" using mirror[OF lp] by blast
-  thus "\<exists> x. ?I x p" by blast
-next
-  fix x assume "?I x p" hence "?I (- x) ?mp" 
-    using mirror[OF lp, where x="- x", symmetric] by auto
-  thus "\<exists> x. ?I x ?mp" by blast
-qed
-
-
-lemma cp_thm': 
-  assumes lp: "iszlfm p"
-  and up: "d\<beta> p 1" and dd: "d\<delta> p d" and dp: "d > 0"
-  shows "(\<exists> x. Ifm bbs (x#bs) p) = ((\<exists> j\<in> {1 .. d}. Ifm bbs (j#bs) (minusinf p)) \<or> (\<exists> j\<in> {1.. d}. \<exists> b\<in> (Inum (i#bs)) ` set (\<beta> p). Ifm bbs ((b+j)#bs) p))"
-  using cp_thm[OF lp up dd dp,where i="i"] by auto
-
-constdefs unit:: "fm \<Rightarrow> fm \<times> num list \<times> int"
-  "unit p \<equiv> (let p' = zlfm p ; l = \<zeta> p' ; q = And (Dvd l (CN 0 1 (C 0))) (a\<beta> p' l); d = \<delta> q;
-             B = remdups (map simpnum (\<beta> q)) ; a = remdups (map simpnum (\<alpha> q))
-             in if length B \<le> length a then (q,B,d) else (mirror q, a,d))"
-
-lemma unit: assumes qf: "qfree p"
-  shows "\<And> q B d. unit p = (q,B,d) \<Longrightarrow> ((\<exists> x. Ifm bbs (x#bs) p) = (\<exists> x. Ifm bbs (x#bs) q)) \<and> (Inum (i#bs)) ` set B = (Inum (i#bs)) ` set (\<beta> q) \<and> d\<beta> q 1 \<and> d\<delta> q d \<and> d >0 \<and> iszlfm q \<and> (\<forall> b\<in> set B. numbound0 b)"
-proof-
-  fix q B d 
-  assume qBd: "unit p = (q,B,d)"
-  let ?thes = "((\<exists> x. Ifm bbs (x#bs) p) = (\<exists> x. Ifm bbs (x#bs) q)) \<and>
-    Inum (i#bs) ` set B = Inum (i#bs) ` set (\<beta> q) \<and>
-    d\<beta> q 1 \<and> d\<delta> q d \<and> 0 < d \<and> iszlfm q \<and> (\<forall> b\<in> set B. numbound0 b)"
-  let ?I = "\<lambda> x p. Ifm bbs (x#bs) p"
-  let ?p' = "zlfm p"
-  let ?l = "\<zeta> ?p'"
-  let ?q = "And (Dvd ?l (CN 0 1 (C 0))) (a\<beta> ?p' ?l)"
-  let ?d = "\<delta> ?q"
-  let ?B = "set (\<beta> ?q)"
-  let ?B'= "remdups (map simpnum (\<beta> ?q))"
-  let ?A = "set (\<alpha> ?q)"
-  let ?A'= "remdups (map simpnum (\<alpha> ?q))"
-  from conjunct1[OF zlfm_I[OF qf, where bs="bs"]] 
-  have pp': "\<forall> i. ?I i ?p' = ?I i p" by auto
-  from conjunct2[OF zlfm_I[OF qf, where bs="bs" and i="i"]]
-  have lp': "iszlfm ?p'" . 
-  from lp' \<zeta>[where p="?p'"] have lp: "?l >0" and dl: "d\<beta> ?p' ?l" by auto
-  from a\<beta>_ex[where p="?p'" and l="?l" and bs="bs", OF lp' dl lp] pp'
-  have pq_ex:"(\<exists> (x::int). ?I x p) = (\<exists> x. ?I x ?q)" by simp 
-  from lp' lp a\<beta>[OF lp' dl lp] have lq:"iszlfm ?q" and uq: "d\<beta> ?q 1"  by auto
-  from \<delta>[OF lq] have dp:"?d >0" and dd: "d\<delta> ?q ?d" by blast+
-  let ?N = "\<lambda> t. Inum (i#bs) t"
-  have "?N ` set ?B' = ((?N o simpnum) ` ?B)" by auto 
-  also have "\<dots> = ?N ` ?B" using simpnum_ci[where bs="i#bs"] by auto
-  finally have BB': "?N ` set ?B' = ?N ` ?B" .
-  have "?N ` set ?A' = ((?N o simpnum) ` ?A)" by auto 
-  also have "\<dots> = ?N ` ?A" using simpnum_ci[where bs="i#bs"] by auto
-  finally have AA': "?N ` set ?A' = ?N ` ?A" .
-  from \<beta>_numbound0[OF lq] have B_nb:"\<forall> b\<in> set ?B'. numbound0 b"
-    by (simp add: simpnum_numbound0)
-  from \<alpha>_l[OF lq] have A_nb: "\<forall> b\<in> set ?A'. numbound0 b"
-    by (simp add: simpnum_numbound0)
-    {assume "length ?B' \<le> length ?A'"
-    hence q:"q=?q" and "B = ?B'" and d:"d = ?d"
-      using qBd by (auto simp add: Let_def unit_def)
-    with BB' B_nb have b: "?N ` (set B) = ?N ` set (\<beta> q)" 
-      and bn: "\<forall>b\<in> set B. numbound0 b" by simp+ 
-  with pq_ex dp uq dd lq q d have ?thes by simp}
-  moreover 
-  {assume "\<not> (length ?B' \<le> length ?A')"
-    hence q:"q=mirror ?q" and "B = ?A'" and d:"d = ?d"
-      using qBd by (auto simp add: Let_def unit_def)
-    with AA' mirror\<alpha>\<beta>[OF lq] A_nb have b:"?N ` (set B) = ?N ` set (\<beta> q)" 
-      and bn: "\<forall>b\<in> set B. numbound0 b" by simp+
-    from mirror_ex[OF lq] pq_ex q 
-    have pqm_eq:"(\<exists> (x::int). ?I x p) = (\<exists> (x::int). ?I x q)" by simp
-    from lq uq q mirror_l[where p="?q"]
-    have lq': "iszlfm q" and uq: "d\<beta> q 1" by auto
-    from \<delta>[OF lq'] mirror_\<delta>[OF lq] q d have dq:"d\<delta> q d " by auto
-    from pqm_eq b bn uq lq' dp dq q dp d have ?thes by simp
-  }
-  ultimately show ?thes by blast
-qed
-    (* Cooper's Algorithm *)
-
-constdefs cooper :: "fm \<Rightarrow> fm"
-  "cooper p \<equiv> 
-  (let (q,B,d) = unit p; js = iupt 1 d;
-       mq = simpfm (minusinf q);
-       md = evaldjf (\<lambda> j. simpfm (subst0 (C j) mq)) js
-   in if md = T then T else
-    (let qd = evaldjf (\<lambda> (b,j). simpfm (subst0 (Add b (C j)) q)) 
-                               [(b,j). b\<leftarrow>B,j\<leftarrow>js]
-     in decr (disj md qd)))"
-lemma cooper: assumes qf: "qfree p"
-  shows "((\<exists> x. Ifm bbs (x#bs) p) = (Ifm bbs bs (cooper p))) \<and> qfree (cooper p)" 
-  (is "(?lhs = ?rhs) \<and> _")
-proof-
-  let ?I = "\<lambda> x p. Ifm bbs (x#bs) p"
-  let ?q = "fst (unit p)"
-  let ?B = "fst (snd(unit p))"
-  let ?d = "snd (snd (unit p))"
-  let ?js = "iupt 1 ?d"
-  let ?mq = "minusinf ?q"
-  let ?smq = "simpfm ?mq"
-  let ?md = "evaldjf (\<lambda> j. simpfm (subst0 (C j) ?smq)) ?js"
-  fix i
-  let ?N = "\<lambda> t. Inum (i#bs) t"
-  let ?Bjs = "[(b,j). b\<leftarrow>?B,j\<leftarrow>?js]"
-  let ?qd = "evaldjf (\<lambda> (b,j). simpfm (subst0 (Add b (C j)) ?q)) ?Bjs"
-  have qbf:"unit p = (?q,?B,?d)" by simp
-  from unit[OF qf qbf] have pq_ex: "(\<exists>(x::int). ?I x p) = (\<exists> (x::int). ?I x ?q)" and 
-    B:"?N ` set ?B = ?N ` set (\<beta> ?q)" and 
-    uq:"d\<beta> ?q 1" and dd: "d\<delta> ?q ?d" and dp: "?d > 0" and 
-    lq: "iszlfm ?q" and 
-    Bn: "\<forall> b\<in> set ?B. numbound0 b" by auto
-  from zlin_qfree[OF lq] have qfq: "qfree ?q" .
-  from simpfm_qf[OF minusinf_qfree[OF qfq]] have qfmq: "qfree ?smq".
-  have jsnb: "\<forall> j \<in> set ?js. numbound0 (C j)" by simp
-  hence "\<forall> j\<in> set ?js. bound0 (subst0 (C j) ?smq)" 
-    by (auto simp only: subst0_bound0[OF qfmq])
-  hence th: "\<forall> j\<in> set ?js. bound0 (simpfm (subst0 (C j) ?smq))"
-    by (auto simp add: simpfm_bound0)
-  from evaldjf_bound0[OF th] have mdb: "bound0 ?md" by simp 
-  from Bn jsnb have "\<forall> (b,j) \<in> set ?Bjs. numbound0 (Add b (C j))"
-    by simp
-  hence "\<forall> (b,j) \<in> set ?Bjs. bound0 (subst0 (Add b (C j)) ?q)"
-    using subst0_bound0[OF qfq] by blast
-  hence "\<forall> (b,j) \<in> set ?Bjs. bound0 (simpfm (subst0 (Add b (C j)) ?q))"
-    using simpfm_bound0  by blast
-  hence th': "\<forall> x \<in> set ?Bjs. bound0 ((\<lambda> (b,j). simpfm (subst0 (Add b (C j)) ?q)) x)"
-    by auto 
-  from evaldjf_bound0 [OF th'] have qdb: "bound0 ?qd" by simp
-  from mdb qdb 
-  have mdqdb: "bound0 (disj ?md ?qd)" by (simp only: disj_def, cases "?md=T \<or> ?qd=T", simp_all)
-  from trans [OF pq_ex cp_thm'[OF lq uq dd dp,where i="i"]] B
-  have "?lhs = (\<exists> j\<in> {1.. ?d}. ?I j ?mq \<or> (\<exists> b\<in> ?N ` set ?B. Ifm bbs ((b+ j)#bs) ?q))" by auto
-  also have "\<dots> = (\<exists> j\<in> {1.. ?d}. ?I j ?mq \<or> (\<exists> b\<in> set ?B. Ifm bbs ((?N b+ j)#bs) ?q))" by simp
-  also have "\<dots> = ((\<exists> j\<in> {1.. ?d}. ?I j ?mq ) \<or> (\<exists> j\<in> {1.. ?d}. \<exists> b\<in> set ?B. Ifm bbs ((?N (Add b (C j)))#bs) ?q))" by (simp only: Inum.simps) blast
-  also have "\<dots> = ((\<exists> j\<in> {1.. ?d}. ?I j ?smq ) \<or> (\<exists> j\<in> {1.. ?d}. \<exists> b\<in> set ?B. Ifm bbs ((?N (Add b (C j)))#bs) ?q))" by (simp add: simpfm) 
-  also have "\<dots> = ((\<exists> j\<in> set ?js. (\<lambda> j. ?I i (simpfm (subst0 (C j) ?smq))) j) \<or> (\<exists> j\<in> set ?js. \<exists> b\<in> set ?B. Ifm bbs ((?N (Add b (C j)))#bs) ?q))"
-    by (simp only: simpfm subst0_I[OF qfmq] iupt_set) auto
-  also have "\<dots> = (?I i (evaldjf (\<lambda> j. simpfm (subst0 (C j) ?smq)) ?js) \<or> (\<exists> j\<in> set ?js. \<exists> b\<in> set ?B. ?I i (subst0 (Add b (C j)) ?q)))" 
-   by (simp only: evaldjf_ex subst0_I[OF qfq])
- also have "\<dots>= (?I i ?md \<or> (\<exists> (b,j) \<in> set ?Bjs. (\<lambda> (b,j). ?I i (simpfm (subst0 (Add b (C j)) ?q))) (b,j)))"
-   by (simp only: simpfm set_concat set_map concat_map_singleton UN_simps) blast
- also have "\<dots> = (?I i ?md \<or> (?I i (evaldjf (\<lambda> (b,j). simpfm (subst0 (Add b (C j)) ?q)) ?Bjs)))"
-   by (simp only: evaldjf_ex[where bs="i#bs" and f="\<lambda> (b,j). simpfm (subst0 (Add b (C j)) ?q)" and ps="?Bjs"]) (auto simp add: split_def)
- finally have mdqd: "?lhs = (?I i ?md \<or> ?I i ?qd)" by simp  
-  also have "\<dots> = (?I i (disj ?md ?qd))" by (simp add: disj)
-  also have "\<dots> = (Ifm bbs bs (decr (disj ?md ?qd)))" by (simp only: decr [OF mdqdb]) 
-  finally have mdqd2: "?lhs = (Ifm bbs bs (decr (disj ?md ?qd)))" . 
-  {assume mdT: "?md = T"
-    hence cT:"cooper p = T" 
-      by (simp only: cooper_def unit_def split_def Let_def if_True) simp
-    from mdT have lhs:"?lhs" using mdqd by simp 
-    from mdT have "?rhs" by (simp add: cooper_def unit_def split_def)
-    with lhs cT have ?thesis by simp }
-  moreover
-  {assume mdT: "?md \<noteq> T" hence "cooper p = decr (disj ?md ?qd)" 
-      by (simp only: cooper_def unit_def split_def Let_def if_False) 
-    with mdqd2 decr_qf[OF mdqdb] have ?thesis by simp }
-  ultimately show ?thesis by blast
-qed
-
-definition pa :: "fm \<Rightarrow> fm" where
-  "pa p = qelim (prep p) cooper"
-
-theorem mirqe: "(Ifm bbs bs (pa p) = Ifm bbs bs p) \<and> qfree (pa p)"
-  using qelim_ci cooper prep by (auto simp add: pa_def)
-
-definition
-  cooper_test :: "unit \<Rightarrow> fm"
-where
-  "cooper_test u = pa (E (A (Imp (Ge (Sub (Bound 0) (Bound 1)))
-    (E (E (Eq (Sub (Add (Mul 3 (Bound 1)) (Mul 5 (Bound 0)))
-      (Bound 2))))))))"
-
-ML {* @{code cooper_test} () *}
-
-(*
-code_reserved SML oo
-export_code pa in SML module_name GeneratedCooper file "~~/src/HOL/Tools/Qelim/raw_generated_cooper.ML"
-*)
-
-oracle linzqe_oracle = {*
-let
-
-fun num_of_term vs (t as Free (xn, xT)) = (case AList.lookup (op =) vs t
-     of NONE => error "Variable not found in the list!"
-      | SOME n => @{code Bound} n)
-  | num_of_term vs @{term "0::int"} = @{code C} 0
-  | num_of_term vs @{term "1::int"} = @{code C} 1
-  | num_of_term vs (@{term "number_of :: int \<Rightarrow> int"} $ t) = @{code C} (HOLogic.dest_numeral t)
-  | num_of_term vs (Bound i) = @{code Bound} i
-  | num_of_term vs (@{term "uminus :: int \<Rightarrow> int"} $ t') = @{code Neg} (num_of_term vs t')
-  | num_of_term vs (@{term "op + :: int \<Rightarrow> int \<Rightarrow> int"} $ t1 $ t2) =
-      @{code Add} (num_of_term vs t1, num_of_term vs t2)
-  | num_of_term vs (@{term "op - :: int \<Rightarrow> int \<Rightarrow> int"} $ t1 $ t2) =
-      @{code Sub} (num_of_term vs t1, num_of_term vs t2)
-  | num_of_term vs (@{term "op * :: int \<Rightarrow> int \<Rightarrow> int"} $ t1 $ t2) =
-      (case try HOLogic.dest_number t1
-       of SOME (_, i) => @{code Mul} (i, num_of_term vs t2)
-        | NONE => (case try HOLogic.dest_number t2
-                of SOME (_, i) => @{code Mul} (i, num_of_term vs t1)
-                 | NONE => error "num_of_term: unsupported multiplication"))
-  | num_of_term vs t = error ("num_of_term: unknown term " ^ Syntax.string_of_term @{context} t);
-
-fun fm_of_term ps vs @{term True} = @{code T}
-  | fm_of_term ps vs @{term False} = @{code F}
-  | fm_of_term ps vs (@{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"} $ t1 $ t2) =
-      @{code Lt} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
-  | fm_of_term ps vs (@{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"} $ t1 $ t2) =
-      @{code Le} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
-  | fm_of_term ps vs (@{term "op = :: int \<Rightarrow> int \<Rightarrow> bool"} $ t1 $ t2) =
-      @{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) 
-  | fm_of_term ps vs (@{term "op dvd :: int \<Rightarrow> int \<Rightarrow> bool"} $ t1 $ t2) =
-      (case try HOLogic.dest_number t1
-       of SOME (_, i) => @{code Dvd} (i, num_of_term vs t2)
-        | NONE => error "num_of_term: unsupported dvd")
-  | fm_of_term ps vs (@{term "op = :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ t1 $ t2) =
-      @{code Iff} (fm_of_term ps vs t1, fm_of_term ps vs t2)
-  | fm_of_term ps vs (@{term "op &"} $ t1 $ t2) =
-      @{code And} (fm_of_term ps vs t1, fm_of_term ps vs t2)
-  | fm_of_term ps vs (@{term "op |"} $ t1 $ t2) =
-      @{code Or} (fm_of_term ps vs t1, fm_of_term ps vs t2)
-  | fm_of_term ps vs (@{term "op -->"} $ t1 $ t2) =
-      @{code Imp} (fm_of_term ps vs t1, fm_of_term ps vs t2)
-  | fm_of_term ps vs (@{term "Not"} $ t') =
-      @{code NOT} (fm_of_term ps vs t')
-  | fm_of_term ps vs (Const ("Ex", _) $ Abs (xn, xT, p)) =
-      let
-        val (xn', p') = variant_abs (xn, xT, p);
-        val vs' = (Free (xn', xT), 0) :: map (fn (v, n) => (v, n + 1)) vs;
-      in @{code E} (fm_of_term ps vs' p) end
-  | fm_of_term ps vs (Const ("All", _) $ Abs (xn, xT, p)) =
-      let
-        val (xn', p') = variant_abs (xn, xT, p);
-        val vs' = (Free (xn', xT), 0) :: map (fn (v, n) => (v, n + 1)) vs;
-      in @{code A} (fm_of_term ps vs' p) end
-  | fm_of_term ps vs t = error ("fm_of_term : unknown term " ^ Syntax.string_of_term @{context} t);
-
-fun term_of_num vs (@{code C} i) = HOLogic.mk_number HOLogic.intT i
-  | term_of_num vs (@{code Bound} n) = fst (the (find_first (fn (_, m) => n = m) vs))
-  | term_of_num vs (@{code Neg} t') = @{term "uminus :: int \<Rightarrow> int"} $ term_of_num vs t'
-  | term_of_num vs (@{code Add} (t1, t2)) = @{term "op + :: int \<Rightarrow> int \<Rightarrow> int"} $
-      term_of_num vs t1 $ term_of_num vs t2
-  | term_of_num vs (@{code Sub} (t1, t2)) = @{term "op - :: int \<Rightarrow> int \<Rightarrow> int"} $
-      term_of_num vs t1 $ term_of_num vs t2
-  | term_of_num vs (@{code Mul} (i, t2)) = @{term "op * :: int \<Rightarrow> int \<Rightarrow> int"} $
-      term_of_num vs (@{code C} i) $ term_of_num vs t2
-  | term_of_num vs (@{code CN} (n, i, t)) = term_of_num vs (@{code Add} (@{code Mul} (i, @{code Bound} n), t));
-
-fun term_of_fm ps vs @{code T} = HOLogic.true_const 
-  | term_of_fm ps vs @{code F} = HOLogic.false_const
-  | term_of_fm ps vs (@{code Lt} t) =
-      @{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::int"}
-  | term_of_fm ps vs (@{code Le} t) =
-      @{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::int"}
-  | term_of_fm ps vs (@{code Gt} t) =
-      @{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"} $ @{term "0::int"} $ term_of_num vs t
-  | term_of_fm ps vs (@{code Ge} t) =
-      @{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"} $ @{term "0::int"} $ term_of_num vs t
-  | term_of_fm ps vs (@{code Eq} t) =
-      @{term "op = :: int \<Rightarrow> int \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::int"}
-  | term_of_fm ps vs (@{code NEq} t) =
-      term_of_fm ps vs (@{code NOT} (@{code Eq} t))
-  | term_of_fm ps vs (@{code Dvd} (i, t)) =
-      @{term "op dvd :: int \<Rightarrow> int \<Rightarrow> bool"} $ term_of_num vs (@{code C} i) $ term_of_num vs t
-  | term_of_fm ps vs (@{code NDvd} (i, t)) =
-      term_of_fm ps vs (@{code NOT} (@{code Dvd} (i, t)))
-  | term_of_fm ps vs (@{code NOT} t') =
-      HOLogic.Not $ term_of_fm ps vs t'
-  | term_of_fm ps vs (@{code And} (t1, t2)) =
-      HOLogic.conj $ term_of_fm ps vs t1 $ term_of_fm ps vs t2
-  | term_of_fm ps vs (@{code Or} (t1, t2)) =
-      HOLogic.disj $ term_of_fm ps vs t1 $ term_of_fm ps vs t2
-  | term_of_fm ps vs (@{code Imp} (t1, t2)) =
-      HOLogic.imp $ term_of_fm ps vs t1 $ term_of_fm ps vs t2
-  | term_of_fm ps vs (@{code Iff} (t1, t2)) =
-      @{term "op = :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ term_of_fm ps vs t1 $ term_of_fm ps vs t2
-  | term_of_fm ps vs (@{code Closed} n) = (fst o the) (find_first (fn (_, m) => m = n) ps)
-  | term_of_fm ps vs (@{code NClosed} n) = term_of_fm ps vs (@{code NOT} (@{code Closed} n));
-
-fun term_bools acc t =
-  let
-    val is_op = member (op =) [@{term "op &"}, @{term "op |"}, @{term "op -->"}, @{term "op = :: bool => _"},
-      @{term "op = :: int => _"}, @{term "op < :: int => _"},
-      @{term "op <= :: int => _"}, @{term "Not"}, @{term "All :: (int => _) => _"},
-      @{term "Ex :: (int => _) => _"}, @{term "True"}, @{term "False"}]
-    fun is_ty t = not (fastype_of t = HOLogic.boolT) 
-  in case t
-   of (l as f $ a) $ b => if is_ty t orelse is_op t then term_bools (term_bools acc l)b 
-        else insert (op aconv) t acc
-    | f $ a => if is_ty t orelse is_op t then term_bools (term_bools acc f) a  
-        else insert (op aconv) t acc
-    | Abs p => term_bools acc (snd (variant_abs p))
-    | _ => if is_ty t orelse is_op t then acc else insert (op aconv) t acc
-  end;
-
-in fn ct =>
-  let
-    val thy = Thm.theory_of_cterm ct;
-    val t = Thm.term_of ct;
-    val fs = OldTerm.term_frees t;
-    val bs = term_bools [] t;
-    val vs = fs ~~ (0 upto (length fs - 1))
-    val ps = bs ~~ (0 upto (length bs - 1))
-    val t' = (term_of_fm ps vs o @{code pa} o fm_of_term ps vs) t;
-  in (Thm.cterm_of thy o HOLogic.mk_Trueprop o HOLogic.mk_eq) (t, t') end
-end;
-*}
-
-use "cooper_tac.ML"
-setup "Cooper_Tac.setup"
-
-text {* Tests *}
-
-lemma "\<exists> (j::int). \<forall> x\<ge>j. (\<exists> a b. x = 3*a+5*b)"
-  by cooper
-
-lemma "ALL (x::int) >=8. EX i j. 5*i + 3*j = x"
-  by cooper
-
-theorem "(\<forall>(y::int). 3 dvd y) ==> \<forall>(x::int). b < x --> a \<le> x"
-  by cooper
-
-theorem "!! (y::int) (z::int) (n::int). 3 dvd z ==> 2 dvd (y::int) ==>
-  (\<exists>(x::int).  2*x =  y) & (\<exists>(k::int). 3*k = z)"
-  by cooper
-
-theorem "!! (y::int) (z::int) n. Suc(n::nat) < 6 ==>  3 dvd z ==>
-  2 dvd (y::int) ==> (\<exists>(x::int).  2*x =  y) & (\<exists>(k::int). 3*k = z)"
-  by cooper
-
-theorem "\<forall>(x::nat). \<exists>(y::nat). (0::nat) \<le> 5 --> y = 5 + x "
-  by cooper
-
-lemma "ALL (x::int) >=8. EX i j. 5*i + 3*j = x"
-  by cooper 
-
-lemma "ALL (y::int) (z::int) (n::int). 3 dvd z --> 2 dvd (y::int) --> (EX (x::int).  2*x =  y) & (EX (k::int). 3*k = z)"
-  by cooper
-
-lemma "ALL(x::int) y. x < y --> 2 * x + 1 < 2 * y"
-  by cooper
-
-lemma "ALL(x::int) y. 2 * x + 1 ~= 2 * y"
-  by cooper
-
-lemma "EX(x::int) y. 0 < x  & 0 <= y  & 3 * x - 5 * y = 1"
-  by cooper
-
-lemma "~ (EX(x::int) (y::int) (z::int). 4*x + (-6::int)*y = 1)"
-  by cooper
-
-lemma "ALL(x::int). (2 dvd x) --> (EX(y::int). x = 2*y)"
-  by cooper
-
-lemma "ALL(x::int). (2 dvd x) = (EX(y::int). x = 2*y)"
-  by cooper
-
-lemma "ALL(x::int). ((2 dvd x) = (ALL(y::int). x ~= 2*y + 1))"
-  by cooper
-
-lemma "~ (ALL(x::int). ((2 dvd x) = (ALL(y::int). x ~= 2*y+1) | (EX(q::int) (u::int) i. 3*i + 2*q - u < 17) --> 0 < x | ((~ 3 dvd x) &(x + 8 = 0))))"
-  by cooper
-
-lemma "~ (ALL(i::int). 4 <= i --> (EX x y. 0 <= x & 0 <= y & 3 * x + 5 * y = i))" 
-  by cooper
-
-lemma "EX j. ALL (x::int) >= j. EX i j. 5*i + 3*j = x"
-  by cooper
-
-theorem "(\<forall>(y::int). 3 dvd y) ==> \<forall>(x::int). b < x --> a \<le> x"
-  by cooper
-
-theorem "!! (y::int) (z::int) (n::int). 3 dvd z ==> 2 dvd (y::int) ==>
-  (\<exists>(x::int).  2*x =  y) & (\<exists>(k::int). 3*k = z)"
-  by cooper
-
-theorem "!! (y::int) (z::int) n. Suc(n::nat) < 6 ==>  3 dvd z ==>
-  2 dvd (y::int) ==> (\<exists>(x::int).  2*x =  y) & (\<exists>(k::int). 3*k = z)"
-  by cooper
-
-theorem "\<forall>(x::nat). \<exists>(y::nat). (0::nat) \<le> 5 --> y = 5 + x "
-  by cooper
-
-theorem "\<forall>(x::nat). \<exists>(y::nat). y = 5 + x | x div 6 + 1= 2"
-  by cooper
-
-theorem "\<exists>(x::int). 0 < x"
-  by cooper
-
-theorem "\<forall>(x::int) y. x < y --> 2 * x + 1 < 2 * y"
-  by cooper
- 
-theorem "\<forall>(x::int) y. 2 * x + 1 \<noteq> 2 * y"
-  by cooper
- 
-theorem "\<exists>(x::int) y. 0 < x  & 0 \<le> y  & 3 * x - 5 * y = 1"
-  by cooper
-
-theorem "~ (\<exists>(x::int) (y::int) (z::int). 4*x + (-6::int)*y = 1)"
-  by cooper
-
-theorem "~ (\<exists>(x::int). False)"
-  by cooper
-
-theorem "\<forall>(x::int). (2 dvd x) --> (\<exists>(y::int). x = 2*y)"
-  by cooper 
-
-theorem "\<forall>(x::int). (2 dvd x) --> (\<exists>(y::int). x = 2*y)"
-  by cooper 
-
-theorem "\<forall>(x::int). (2 dvd x) = (\<exists>(y::int). x = 2*y)"
-  by cooper 
-
-theorem "\<forall>(x::int). ((2 dvd x) = (\<forall>(y::int). x \<noteq> 2*y + 1))"
-  by cooper 
-
-theorem "~ (\<forall>(x::int). 
-            ((2 dvd x) = (\<forall>(y::int). x \<noteq> 2*y+1) | 
-             (\<exists>(q::int) (u::int) i. 3*i + 2*q - u < 17)
-             --> 0 < x | ((~ 3 dvd x) &(x + 8 = 0))))"
-  by cooper
- 
-theorem "~ (\<forall>(i::int). 4 \<le> i --> (\<exists>x y. 0 \<le> x & 0 \<le> y & 3 * x + 5 * y = i))"
-  by cooper
-
-theorem "\<forall>(i::int). 8 \<le> i --> (\<exists>x y. 0 \<le> x & 0 \<le> y & 3 * x + 5 * y = i)"
-  by cooper
-
-theorem "\<exists>(j::int). \<forall>i. j \<le> i --> (\<exists>x y. 0 \<le> x & 0 \<le> y & 3 * x + 5 * y = i)"
-  by cooper
-
-theorem "~ (\<forall>j (i::int). j \<le> i --> (\<exists>x y. 0 \<le> x & 0 \<le> y & 3 * x + 5 * y = i))"
-  by cooper
-
-theorem "(\<exists>m::nat. n = 2 * m) --> (n + 1) div 2 = n div 2"
-  by cooper
-
-end
diff -r b8a6b9c56fdd -r 5170d6277b61 src/HOL/Reflection/Ferrack.thy
--- a/src/HOL/Reflection/Ferrack.thy	Thu Feb 12 21:24:14 2009 -0800
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,2101 +0,0 @@
-(*  Title:      HOL/Reflection/Ferrack.thy
-    Author:     Amine Chaieb
-*)
-
-theory Ferrack
-imports Complex_Main Dense_Linear_Order Efficient_Nat
-uses ("ferrack_tac.ML")
-begin
-
-section {* Quantifier elimination for @{text "\<real> (0, 1, +, <)"} *}
-
-  (*********************************************************************************)
-  (*          SOME GENERAL STUFF< HAS TO BE MOVED IN SOME LIB                      *)
-  (*********************************************************************************)
-
-consts alluopairs:: "'a list \<Rightarrow> ('a \<times> 'a) list"
-primrec
-  "alluopairs [] = []"
-  "alluopairs (x#xs) = (map (Pair x) (x#xs))@(alluopairs xs)"
-
-lemma alluopairs_set1: "set (alluopairs xs) \<le> {(x,y). x\<in> set xs \<and> y\<in> set xs}"
-by (induct xs, auto)
-
-lemma alluopairs_set:
-  "\<lbrakk>x\<in> set xs ; y \<in> set xs\<rbrakk> \<Longrightarrow> (x,y) \<in> set (alluopairs xs) \<or> (y,x) \<in> set (alluopairs xs) "
-by (induct xs, auto)
-
-lemma alluopairs_ex:
-  assumes Pc: "\<forall> x y. P x y = P y x"
-  shows "(\<exists> x \<in> set xs. \<exists> y \<in> set xs. P x y) = (\<exists> (x,y) \<in> set (alluopairs xs). P x y)"
-proof
-  assume "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y"
-  then obtain x y where x: "x \<in> set xs" and y:"y \<in> set xs" and P: "P x y"  by blast
-  from alluopairs_set[OF x y] P Pc show"\<exists>(x, y)\<in>set (alluopairs xs). P x y" 
-    by auto
-next
-  assume "\<exists>(x, y)\<in>set (alluopairs xs). P x y"
-  then obtain "x" and "y"  where xy:"(x,y) \<in> set (alluopairs xs)" and P: "P x y" by blast+
-  from xy have "x \<in> set xs \<and> y\<in> set xs" using alluopairs_set1 by blast
-  with P show "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y" by blast
-qed
-
-lemma nth_pos2: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
-using Nat.gr0_conv_Suc
-by clarsimp
-
-lemma filter_length: "length (List.filter P xs) < Suc (length xs)"
-  apply (induct xs, auto) done
-
-consts remdps:: "'a list \<Rightarrow> 'a list"
-
-recdef remdps "measure size"
-  "remdps [] = []"
-  "remdps (x#xs) = (x#(remdps (List.filter (\<lambda> y. y \<noteq> x) xs)))"
-(hints simp add: filter_length[rule_format])
-
-lemma remdps_set[simp]: "set (remdps xs) = set xs"
-  by (induct xs rule: remdps.induct, auto)
-
-
-
-  (*********************************************************************************)
-  (****                            SHADOW SYNTAX AND SEMANTICS                  ****)
-  (*********************************************************************************)
-
-datatype num = C int | Bound nat | CN nat int num | Neg num | Add num num| Sub num num 
-  | Mul int num 
-
-  (* A size for num to make inductive proofs simpler*)
-consts num_size :: "num \<Rightarrow> nat" 
-primrec 
-  "num_size (C c) = 1"
-  "num_size (Bound n) = 1"
-  "num_size (Neg a) = 1 + num_size a"
-  "num_size (Add a b) = 1 + num_size a + num_size b"
-  "num_size (Sub a b) = 3 + num_size a + num_size b"
-  "num_size (Mul c a) = 1 + num_size a"
-  "num_size (CN n c a) = 3 + num_size a "
-
-  (* Semantics of numeral terms (num) *)
-consts Inum :: "real list \<Rightarrow> num \<Rightarrow> real"
-primrec
-  "Inum bs (C c) = (real c)"
-  "Inum bs (Bound n) = bs!n"
-  "Inum bs (CN n c a) = (real c) * (bs!n) + (Inum bs a)"
-  "Inum bs (Neg a) = -(Inum bs a)"
-  "Inum bs (Add a b) = Inum bs a + Inum bs b"
-  "Inum bs (Sub a b) = Inum bs a - Inum bs b"
-  "Inum bs (Mul c a) = (real c) * Inum bs a"
-    (* FORMULAE *)
-datatype fm  = 
-  T| F| Lt num| Le num| Gt num| Ge num| Eq num| NEq num|
-  NOT fm| And fm fm|  Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm
-
-
-  (* A size for fm *)
-consts fmsize :: "fm \<Rightarrow> nat"
-recdef fmsize "measure size"
-  "fmsize (NOT p) = 1 + fmsize p"
-  "fmsize (And p q) = 1 + fmsize p + fmsize q"
-  "fmsize (Or p q) = 1 + fmsize p + fmsize q"
-  "fmsize (Imp p q) = 3 + fmsize p + fmsize q"
-  "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)"
-  "fmsize (E p) = 1 + fmsize p"
-  "fmsize (A p) = 4+ fmsize p"
-  "fmsize p = 1"
-  (* several lemmas about fmsize *)
-lemma fmsize_pos: "fmsize p > 0"
-by (induct p rule: fmsize.induct) simp_all
-
-  (* Semantics of formulae (fm) *)
-consts Ifm ::"real list \<Rightarrow> fm \<Rightarrow> bool"
-primrec
-  "Ifm bs T = True"
-  "Ifm bs F = False"
-  "Ifm bs (Lt a) = (Inum bs a < 0)"
-  "Ifm bs (Gt a) = (Inum bs a > 0)"
-  "Ifm bs (Le a) = (Inum bs a \<le> 0)"
-  "Ifm bs (Ge a) = (Inum bs a \<ge> 0)"
-  "Ifm bs (Eq a) = (Inum bs a = 0)"
-  "Ifm bs (NEq a) = (Inum bs a \<noteq> 0)"
-  "Ifm bs (NOT p) = (\<not> (Ifm bs p))"
-  "Ifm bs (And p q) = (Ifm bs p \<and> Ifm bs q)"
-  "Ifm bs (Or p q) = (Ifm bs p \<or> Ifm bs q)"
-  "Ifm bs (Imp p q) = ((Ifm bs p) \<longrightarrow> (Ifm bs q))"
-  "Ifm bs (Iff p q) = (Ifm bs p = Ifm bs q)"
-  "Ifm bs (E p) = (\<exists> x. Ifm (x#bs) p)"
-  "Ifm bs (A p) = (\<forall> x. Ifm (x#bs) p)"
-
-lemma IfmLeSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Le (Sub s t)) = (s' \<le> t')"
-apply simp
-done
-
-lemma IfmLtSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Lt (Sub s t)) = (s' < t')"
-apply simp
-done
-lemma IfmEqSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Eq (Sub s t)) = (s' = t')"
-apply simp
-done
-lemma IfmNOT: " (Ifm bs p = P) \<Longrightarrow> (Ifm bs (NOT p) = (\<not>P))"
-apply simp
-done
-lemma IfmAnd: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (And p q) = (P \<and> Q))"
-apply simp
-done
-lemma IfmOr: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Or p q) = (P \<or> Q))"
-apply simp
-done
-lemma IfmImp: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Imp p q) = (P \<longrightarrow> Q))"
-apply simp
-done
-lemma IfmIff: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Iff p q) = (P = Q))"
-apply simp
-done
-
-lemma IfmE: " (!! x. Ifm (x#bs) p = P x) \<Longrightarrow> (Ifm bs (E p) = (\<exists>x. P x))"
-apply simp
-done
-lemma IfmA: " (!! x. Ifm (x#bs) p = P x) \<Longrightarrow> (Ifm bs (A p) = (\<forall>x. P x))"
-apply simp
-done
-
-consts not:: "fm \<Rightarrow> fm"
-recdef not "measure size"
-  "not (NOT p) = p"
-  "not T = F"
-  "not F = T"
-  "not p = NOT p"
-lemma not[simp]: "Ifm bs (not p) = Ifm bs (NOT p)"
-by (cases p) auto
-
-constdefs conj :: "fm \<Rightarrow> fm \<Rightarrow> fm"
-  "conj p q \<equiv> (if (p = F \<or> q=F) then F else if p=T then q else if q=T then p else 
-   if p = q then p else And p q)"
-lemma conj[simp]: "Ifm bs (conj p q) = Ifm bs (And p q)"
-by (cases "p=F \<or> q=F",simp_all add: conj_def) (cases p,simp_all)
-
-constdefs disj :: "fm \<Rightarrow> fm \<Rightarrow> fm"
-  "disj p q \<equiv> (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p 
-       else if p=q then p else Or p q)"
-
-lemma disj[simp]: "Ifm bs (disj p q) = Ifm bs (Or p q)"
-by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all)
-
-constdefs  imp :: "fm \<Rightarrow> fm \<Rightarrow> fm"
-  "imp p q \<equiv> (if (p = F \<or> q=T \<or> p=q) then T else if p=T then q else if q=F then not p 
-    else Imp p q)"
-lemma imp[simp]: "Ifm bs (imp p q) = Ifm bs (Imp p q)"
-by (cases "p=F \<or> q=T",simp_all add: imp_def) 
-
-constdefs   iff :: "fm \<Rightarrow> fm \<Rightarrow> fm"
-  "iff p q \<equiv> (if (p = q) then T else if (p = NOT q \<or> NOT p = q) then F else 
-       if p=F then not q else if q=F then not p else if p=T then q else if q=T then p else 
-  Iff p q)"
-lemma iff[simp]: "Ifm bs (iff p q) = Ifm bs (Iff p q)"
-  by (unfold iff_def,cases "p=q", simp,cases "p=NOT q", simp) (cases "NOT p= q", auto)
-
-lemma conj_simps:
-  "conj F Q = F"
-  "conj P F = F"
-  "conj T Q = Q"
-  "conj P T = P"
-  "conj P P = P"
-  "P \<noteq> T \<Longrightarrow> P \<noteq> F \<Longrightarrow> Q \<noteq> T \<Longrightarrow> Q \<noteq> F \<Longrightarrow> P \<noteq> Q \<Longrightarrow> conj P Q = And P Q"
-  by (simp_all add: conj_def)
-
-lemma disj_simps:
-  "disj T Q = T"
-  "disj P T = T"
-  "disj F Q = Q"
-  "disj P F = P"
-  "disj P P = P"
-  "P \<noteq> T \<Longrightarrow> P \<noteq> F \<Longrightarrow> Q \<noteq> T \<Longrightarrow> Q \<noteq> F \<Longrightarrow> P \<noteq> Q \<Longrightarrow> disj P Q = Or P Q"
-  by (simp_all add: disj_def)
-lemma imp_simps:
-  "imp F Q = T"
-  "imp P T = T"
-  "imp T Q = Q"
-  "imp P F = not P"
-  "imp P P = T"
-  "P \<noteq> T \<Longrightarrow> P \<noteq> F \<Longrightarrow> P \<noteq> Q \<Longrightarrow> Q \<noteq> T \<Longrightarrow> Q \<noteq> F \<Longrightarrow> imp P Q = Imp P Q"
-  by (simp_all add: imp_def)
-lemma trivNOT: "p \<noteq> NOT p" "NOT p \<noteq> p"
-apply (induct p, auto)
-done
-
-lemma iff_simps:
-  "iff p p = T"
-  "iff p (NOT p) = F"
-  "iff (NOT p) p = F"
-  "iff p F = not p"
-  "iff F p = not p"
-  "p \<noteq> NOT T \<Longrightarrow> iff T p = p"
-  "p\<noteq> NOT T \<Longrightarrow> iff p T = p"
-  "p\<noteq>q \<Longrightarrow> p\<noteq> NOT q \<Longrightarrow> q\<noteq> NOT p \<Longrightarrow> p\<noteq> F \<Longrightarrow> q\<noteq> F \<Longrightarrow> p \<noteq> T \<Longrightarrow> q \<noteq> T \<Longrightarrow> iff p q = Iff p q"
-  using trivNOT
-  by (simp_all add: iff_def, cases p, auto)
-  (* Quantifier freeness *)
-consts qfree:: "fm \<Rightarrow> bool"
-recdef qfree "measure size"
-  "qfree (E p) = False"
-  "qfree (A p) = False"
-  "qfree (NOT p) = qfree p" 
-  "qfree (And p q) = (qfree p \<and> qfree q)" 
-  "qfree (Or  p q) = (qfree p \<and> qfree q)" 
-  "qfree (Imp p q) = (qfree p \<and> qfree q)" 
-  "qfree (Iff p q) = (qfree p \<and> qfree q)"
-  "qfree p = True"
-
-  (* Boundedness and substitution *)
-consts 
-  numbound0:: "num \<Rightarrow> bool" (* a num is INDEPENDENT of Bound 0 *)
-  bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *)
-primrec
-  "numbound0 (C c) = True"
-  "numbound0 (Bound n) = (n>0)"
-  "numbound0 (CN n c a) = (n\<noteq>0 \<and> numbound0 a)"
-  "numbound0 (Neg a) = numbound0 a"
-  "numbound0 (Add a b) = (numbound0 a \<and> numbound0 b)"
-  "numbound0 (Sub a b) = (numbound0 a \<and> numbound0 b)" 
-  "numbound0 (Mul i a) = numbound0 a"
-lemma numbound0_I:
-  assumes nb: "numbound0 a"
-  shows "Inum (b#bs) a = Inum (b'#bs) a"
-using nb
-by (induct a rule: numbound0.induct,auto simp add: nth_pos2)
-
-primrec
-  "bound0 T = True"
-  "bound0 F = True"
-  "bound0 (Lt a) = numbound0 a"
-  "bound0 (Le a) = numbound0 a"
-  "bound0 (Gt a) = numbound0 a"
-  "bound0 (Ge a) = numbound0 a"
-  "bound0 (Eq a) = numbound0 a"
-  "bound0 (NEq a) = numbound0 a"
-  "bound0 (NOT p) = bound0 p"
-  "bound0 (And p q) = (bound0 p \<and> bound0 q)"
-  "bound0 (Or p q) = (bound0 p \<and> bound0 q)"
-  "bound0 (Imp p q) = ((bound0 p) \<and> (bound0 q))"
-  "bound0 (Iff p q) = (bound0 p \<and> bound0 q)"
-  "bound0 (E p) = False"
-  "bound0 (A p) = False"
-
-lemma bound0_I:
-  assumes bp: "bound0 p"
-  shows "Ifm (b#bs) p = Ifm (b'#bs) p"
-using bp numbound0_I[where b="b" and bs="bs" and b'="b'"]
-by (induct p rule: bound0.induct) (auto simp add: nth_pos2)
-
-lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)"
-by (cases p, auto)
-lemma not_bn[simp]: "bound0 p \<Longrightarrow> bound0 (not p)"
-by (cases p, auto)
-
-
-lemma conj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)"
-using conj_def by auto 
-lemma conj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)"
-using conj_def by auto 
-
-lemma disj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)"
-using disj_def by auto 
-lemma disj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)"
-using disj_def by auto 
-
-lemma imp_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (imp p q)"
-using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def)
-lemma imp_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (imp p q)"
-using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def)
-
-lemma iff_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)"
-  by (unfold iff_def,cases "p=q", auto)
-lemma iff_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (iff p q)"
-using iff_def by (unfold iff_def,cases "p=q", auto)
-
-consts 
-  decrnum:: "num \<Rightarrow> num" 
-  decr :: "fm \<Rightarrow> fm"
-
-recdef decrnum "measure size"
-  "decrnum (Bound n) = Bound (n - 1)"
-  "decrnum (Neg a) = Neg (decrnum a)"
-  "decrnum (Add a b) = Add (decrnum a) (decrnum b)"
-  "decrnum (Sub a b) = Sub (decrnum a) (decrnum b)"
-  "decrnum (Mul c a) = Mul c (decrnum a)"
-  "decrnum (CN n c a) = CN (n - 1) c (decrnum a)"
-  "decrnum a = a"
-
-recdef decr "measure size"
-  "decr (Lt a) = Lt (decrnum a)"
-  "decr (Le a) = Le (decrnum a)"
-  "decr (Gt a) = Gt (decrnum a)"
-  "decr (Ge a) = Ge (decrnum a)"
-  "decr (Eq a) = Eq (decrnum a)"
-  "decr (NEq a) = NEq (decrnum a)"
-  "decr (NOT p) = NOT (decr p)" 
-  "decr (And p q) = conj (decr p) (decr q)"
-  "decr (Or p q) = disj (decr p) (decr q)"
-  "decr (Imp p q) = imp (decr p) (decr q)"
-  "decr (Iff p q) = iff (decr p) (decr q)"
-  "decr p = p"
-
-lemma decrnum: assumes nb: "numbound0 t"
-  shows "Inum (x#bs) t = Inum bs (decrnum t)"
-  using nb by (induct t rule: decrnum.induct, simp_all add: nth_pos2)
-
-lemma decr: assumes nb: "bound0 p"
-  shows "Ifm (x#bs) p = Ifm bs (decr p)"
-  using nb 
-  by (induct p rule: decr.induct, simp_all add: nth_pos2 decrnum)
-
-lemma decr_qf: "bound0 p \<Longrightarrow> qfree (decr p)"
-by (induct p, simp_all)
-
-consts 
-  isatom :: "fm \<Rightarrow> bool" (* test for atomicity *)
-recdef isatom "measure size"
-  "isatom T = True"
-  "isatom F = True"
-  "isatom (Lt a) = True"
-  "isatom (Le a) = True"
-  "isatom (Gt a) = True"
-  "isatom (Ge a) = True"
-  "isatom (Eq a) = True"
-  "isatom (NEq a) = True"
-  "isatom p = False"
-
-lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p"
-by (induct p, simp_all)
-
-constdefs djf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm"
-  "djf f p q \<equiv> (if q=T then T else if q=F then f p else 
-  (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q))"
-constdefs evaldjf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm"
-  "evaldjf f ps \<equiv> foldr (djf f) ps F"
-
-lemma djf_Or: "Ifm bs (djf f p q) = Ifm bs (Or (f p) q)"
-by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def) 
-(cases "f p", simp_all add: Let_def djf_def) 
-
-
-lemma djf_simps:
-  "djf f p T = T"
-  "djf f p F = f p"
-  "q\<noteq>T \<Longrightarrow> q\<noteq>F \<Longrightarrow> djf f p q = (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q)"
-  by (simp_all add: djf_def)
-
-lemma evaldjf_ex: "Ifm bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm bs (f p))"
-  by(induct ps, simp_all add: evaldjf_def djf_Or)
-
-lemma evaldjf_bound0: 
-  assumes nb: "\<forall> x\<in> set xs. bound0 (f x)"
-  shows "bound0 (evaldjf f xs)"
-  using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 
-
-lemma evaldjf_qf: 
-  assumes nb: "\<forall> x\<in> set xs. qfree (f x)"
-  shows "qfree (evaldjf f xs)"
-  using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 
-
-consts disjuncts :: "fm \<Rightarrow> fm list"
-recdef disjuncts "measure size"
-  "disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)"
-  "disjuncts F = []"
-  "disjuncts p = [p]"
-
-lemma disjuncts: "(\<exists> q\<in> set (disjuncts p). Ifm bs q) = Ifm bs p"
-by(induct p rule: disjuncts.induct, auto)
-
-lemma disjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). bound0 q"
-proof-
-  assume nb: "bound0 p"
-  hence "list_all bound0 (disjuncts p)" by (induct p rule:disjuncts.induct,auto)
-  thus ?thesis by (simp only: list_all_iff)
-qed
-
-lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). qfree q"
-proof-
-  assume qf: "qfree p"
-  hence "list_all qfree (disjuncts p)"
-    by (induct p rule: disjuncts.induct, auto)
-  thus ?thesis by (simp only: list_all_iff)
-qed
-
-constdefs DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm"
-  "DJ f p \<equiv> evaldjf f (disjuncts p)"
-
-lemma DJ: assumes fdj: "\<forall> p q. Ifm bs (f (Or p q)) = Ifm bs (Or (f p) (f q))"
-  and fF: "f F = F"
-  shows "Ifm bs (DJ f p) = Ifm bs (f p)"
-proof-
-  have "Ifm bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm bs (f q))"
-    by (simp add: DJ_def evaldjf_ex) 
-  also have "\<dots> = Ifm bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto)
-  finally show ?thesis .
-qed
-
-lemma DJ_qf: assumes 
-  fqf: "\<forall> p. qfree p \<longrightarrow> qfree (f p)"
-  shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) "
-proof(clarify)
-  fix  p assume qf: "qfree p"
-  have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def)
-  from disjuncts_qf[OF qf] have "\<forall> q\<in> set (disjuncts p). qfree q" .
-  with fqf have th':"\<forall> q\<in> set (disjuncts p). qfree (f q)" by blast
-  
-  from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp
-qed
-
-lemma DJ_qe: assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))"
-  shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm bs ((DJ qe p)) = Ifm bs (E p))"
-proof(clarify)
-  fix p::fm and bs
-  assume qf: "qfree p"
-  from qe have qth: "\<forall> p. qfree p \<longrightarrow> qfree (qe p)" by blast
-  from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto
-  have "Ifm bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (qe q))"
-    by (simp add: DJ_def evaldjf_ex)
-  also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm bs (E q))" using qe disjuncts_qf[OF qf] by auto
-  also have "\<dots> = Ifm bs (E p)" by (induct p rule: disjuncts.induct, auto)
-  finally show "qfree (DJ qe p) \<and> Ifm bs (DJ qe p) = Ifm bs (E p)" using qfth by blast
-qed
-  (* Simplification *)
-consts 
-  numgcd :: "num \<Rightarrow> int"
-  numgcdh:: "num \<Rightarrow> int \<Rightarrow> int"
-  reducecoeffh:: "num \<Rightarrow> int \<Rightarrow> num"
-  reducecoeff :: "num \<Rightarrow> num"
-  dvdnumcoeff:: "num \<Rightarrow> int \<Rightarrow> bool"
-consts maxcoeff:: "num \<Rightarrow> int"
-recdef maxcoeff "measure size"
-  "maxcoeff (C i) = abs i"
-  "maxcoeff (CN n c t) = max (abs c) (maxcoeff t)"
-  "maxcoeff t = 1"
-
-lemma maxcoeff_pos: "maxcoeff t \<ge> 0"
-  by (induct t rule: maxcoeff.induct, auto)
-
-recdef numgcdh "measure size"
-  "numgcdh (C i) = (\<lambda>g. zgcd i g)"
-  "numgcdh (CN n c t) = (\<lambda>g. zgcd c (numgcdh t g))"
-  "numgcdh t = (\<lambda>g. 1)"
-defs numgcd_def [code]: "numgcd t \<equiv> numgcdh t (maxcoeff t)"
-
-recdef reducecoeffh "measure size"
-  "reducecoeffh (C i) = (\<lambda> g. C (i div g))"
-  "reducecoeffh (CN n c t) = (\<lambda> g. CN n (c div g) (reducecoeffh t g))"
-  "reducecoeffh t = (\<lambda>g. t)"
-
-defs reducecoeff_def: "reducecoeff t \<equiv> 
-  (let g = numgcd t in 
-  if g = 0 then C 0 else if g=1 then t else reducecoeffh t g)"
-
-recdef dvdnumcoeff "measure size"
-  "dvdnumcoeff (C i) = (\<lambda> g. g dvd i)"
-  "dvdnumcoeff (CN n c t) = (\<lambda> g. g dvd c \<and> (dvdnumcoeff t g))"
-  "dvdnumcoeff t = (\<lambda>g. False)"
-
-lemma dvdnumcoeff_trans: 
-  assumes gdg: "g dvd g'" and dgt':"dvdnumcoeff t g'"
-  shows "dvdnumcoeff t g"
-  using dgt' gdg 
-  by (induct t rule: dvdnumcoeff.induct, simp_all add: gdg zdvd_trans[OF gdg])
-
-declare zdvd_trans [trans add]
-
-lemma natabs0: "(nat (abs x) = 0) = (x = 0)"
-by arith
-
-lemma numgcd0:
-  assumes g0: "numgcd t = 0"
-  shows "Inum bs t = 0"
-  using g0[simplified numgcd_def] 
-  by (induct t rule: numgcdh.induct, auto simp add: zgcd_def gcd_zero natabs0 max_def maxcoeff_pos)
-
-lemma numgcdh_pos: assumes gp: "g \<ge> 0" shows "numgcdh t g \<ge> 0"
-  using gp
-  by (induct t rule: numgcdh.induct, auto simp add: zgcd_def)
-
-lemma numgcd_pos: "numgcd t \<ge>0"
-  by (simp add: numgcd_def numgcdh_pos maxcoeff_pos)
-
-lemma reducecoeffh:
-  assumes gt: "dvdnumcoeff t g" and gp: "g > 0" 
-  shows "real g *(Inum bs (reducecoeffh t g)) = Inum bs t"
-  using gt
-proof(induct t rule: reducecoeffh.induct) 
-  case (1 i) hence gd: "g dvd i" by simp
-  from gp have gnz: "g \<noteq> 0" by simp
-  from prems show ?case by (simp add: real_of_int_div[OF gnz gd])
-next
-  case (2 n c t)  hence gd: "g dvd c" by simp
-  from gp have gnz: "g \<noteq> 0" by simp
-  from prems show ?case by (simp add: real_of_int_div[OF gnz gd] algebra_simps)
-qed (auto simp add: numgcd_def gp)
-consts ismaxcoeff:: "num \<Rightarrow> int \<Rightarrow> bool"
-recdef ismaxcoeff "measure size"
-  "ismaxcoeff (C i) = (\<lambda> x. abs i \<le> x)"
-  "ismaxcoeff (CN n c t) = (\<lambda>x. abs c \<le> x \<and> (ismaxcoeff t x))"
-  "ismaxcoeff t = (\<lambda>x. True)"
-
-lemma ismaxcoeff_mono: "ismaxcoeff t c \<Longrightarrow> c \<le> c' \<Longrightarrow> ismaxcoeff t c'"
-by (induct t rule: ismaxcoeff.induct, auto)
-
-lemma maxcoeff_ismaxcoeff: "ismaxcoeff t (maxcoeff t)"
-proof (induct t rule: maxcoeff.induct)
-  case (2 n c t)
-  hence H:"ismaxcoeff t (maxcoeff t)" .
-  have thh: "maxcoeff t \<le> max (abs c) (maxcoeff t)" by (simp add: le_maxI2)
-  from ismaxcoeff_mono[OF H thh] show ?case by (simp add: le_maxI1)
-qed simp_all
-
-lemma zgcd_gt1: "zgcd i j > 1 \<Longrightarrow> ((abs i > 1 \<and> abs j > 1) \<or> (abs i = 0 \<and> abs j > 1) \<or> (abs i > 1 \<and> abs j = 0))"
-  apply (cases "abs i = 0", simp_all add: zgcd_def)
-  apply (cases "abs j = 0", simp_all)
-  apply (cases "abs i = 1", simp_all)
-  apply (cases "abs j = 1", simp_all)
-  apply auto
-  done
-lemma numgcdh0:"numgcdh t m = 0 \<Longrightarrow>  m =0"
-  by (induct t rule: numgcdh.induct, auto simp add:zgcd0)
-
-lemma dvdnumcoeff_aux:
-  assumes "ismaxcoeff t m" and mp:"m \<ge> 0" and "numgcdh t m > 1"
-  shows "dvdnumcoeff t (numgcdh t m)"
-using prems
-proof(induct t rule: numgcdh.induct)
-  case (2 n c t) 
-  let ?g = "numgcdh t m"
-  from prems have th:"zgcd c ?g > 1" by simp
-  from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"]
-  have "(abs c > 1 \<and> ?g > 1) \<or> (abs c = 0 \<and> ?g > 1) \<or> (abs c > 1 \<and> ?g = 0)" by simp
-  moreover {assume "abs c > 1" and gp: "?g > 1" with prems
-    have th: "dvdnumcoeff t ?g" by simp
-    have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2)
-    from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1)}
-  moreover {assume "abs c = 0 \<and> ?g > 1"
-    with prems have th: "dvdnumcoeff t ?g" by simp
-    have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2)
-    from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1)
-    hence ?case by simp }
-  moreover {assume "abs c > 1" and g0:"?g = 0" 
-    from numgcdh0[OF g0] have "m=0". with prems   have ?case by simp }
-  ultimately show ?case by blast
-qed(auto simp add: zgcd_zdvd1)
-
-lemma dvdnumcoeff_aux2:
-  assumes "numgcd t > 1" shows "dvdnumcoeff t (numgcd t) \<and> numgcd t > 0"
-  using prems 
-proof (simp add: numgcd_def)
-  let ?mc = "maxcoeff t"
-  let ?g = "numgcdh t ?mc"
-  have th1: "ismaxcoeff t ?mc" by (rule maxcoeff_ismaxcoeff)
-  have th2: "?mc \<ge> 0" by (rule maxcoeff_pos)
-  assume H: "numgcdh t ?mc > 1"
-  from dvdnumcoeff_aux[OF th1 th2 H]  show "dvdnumcoeff t ?g" .
-qed
-
-lemma reducecoeff: "real (numgcd t) * (Inum bs (reducecoeff t)) = Inum bs t"
-proof-
-  let ?g = "numgcd t"
-  have "?g \<ge> 0"  by (simp add: numgcd_pos)
-  hence	"?g = 0 \<or> ?g = 1 \<or> ?g > 1" by auto
-  moreover {assume "?g = 0" hence ?thesis by (simp add: numgcd0)} 
-  moreover {assume "?g = 1" hence ?thesis by (simp add: reducecoeff_def)} 
-  moreover { assume g1:"?g > 1"
-    from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff t ?g" and g0: "?g > 0" by blast+
-    from reducecoeffh[OF th1 g0, where bs="bs"] g1 have ?thesis 
-      by (simp add: reducecoeff_def Let_def)} 
-  ultimately show ?thesis by blast
-qed
-
-lemma reducecoeffh_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeffh t g)"
-by (induct t rule: reducecoeffh.induct, auto)
-
-lemma reducecoeff_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeff t)"
-using reducecoeffh_numbound0 by (simp add: reducecoeff_def Let_def)
-
-consts
-  simpnum:: "num \<Rightarrow> num"
-  numadd:: "num \<times> num \<Rightarrow> num"
-  nummul:: "num \<Rightarrow> int \<Rightarrow> num"
-recdef numadd "measure (\<lambda> (t,s). size t + size s)"
-  "numadd (CN n1 c1 r1,CN n2 c2 r2) =
-  (if n1=n2 then 
-  (let c = c1 + c2
-  in (if c=0 then numadd(r1,r2) else CN n1 c (numadd (r1,r2))))
-  else if n1 \<le> n2 then (CN n1 c1 (numadd (r1,CN n2 c2 r2))) 
-  else (CN n2 c2 (numadd (CN n1 c1 r1,r2))))"
-  "numadd (CN n1 c1 r1,t) = CN n1 c1 (numadd (r1, t))"  
-  "numadd (t,CN n2 c2 r2) = CN n2 c2 (numadd (t,r2))" 
-  "numadd (C b1, C b2) = C (b1+b2)"
-  "numadd (a,b) = Add a b"
-
-lemma numadd[simp]: "Inum bs (numadd (t,s)) = Inum bs (Add t s)"
-apply (induct t s rule: numadd.induct, simp_all add: Let_def)
-apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all)
-apply (case_tac "n1 = n2", simp_all add: algebra_simps)
-by (simp only: left_distrib[symmetric],simp)
-
-lemma numadd_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numadd (t,s))"
-by (induct t s rule: numadd.induct, auto simp add: Let_def)
-
-recdef nummul "measure size"
-  "nummul (C j) = (\<lambda> i. C (i*j))"
-  "nummul (CN n c a) = (\<lambda> i. CN n (i*c) (nummul a i))"
-  "nummul t = (\<lambda> i. Mul i t)"
-
-lemma nummul[simp]: "\<And> i. Inum bs (nummul t i) = Inum bs (Mul i t)"
-by (induct t rule: nummul.induct, auto simp add: algebra_simps)
-
-lemma nummul_nb[simp]: "\<And> i. numbound0 t \<Longrightarrow> numbound0 (nummul t i)"
-by (induct t rule: nummul.induct, auto )
-
-constdefs numneg :: "num \<Rightarrow> num"
-  "numneg t \<equiv> nummul t (- 1)"
-
-constdefs numsub :: "num \<Rightarrow> num \<Rightarrow> num"
-  "numsub s t \<equiv> (if s = t then C 0 else numadd (s,numneg t))"
-
-lemma numneg[simp]: "Inum bs (numneg t) = Inum bs (Neg t)"
-using numneg_def by simp
-
-lemma numneg_nb[simp]: "numbound0 t \<Longrightarrow> numbound0 (numneg t)"
-using numneg_def by simp
-
-lemma numsub[simp]: "Inum bs (numsub a b) = Inum bs (Sub a b)"
-using numsub_def by simp
-
-lemma numsub_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numsub t s)"
-using numsub_def by simp
-
-recdef simpnum "measure size"
-  "simpnum (C j) = C j"
-  "simpnum (Bound n) = CN n 1 (C 0)"
-  "simpnum (Neg t) = numneg (simpnum t)"
-  "simpnum (Add t s) = numadd (simpnum t,simpnum s)"
-  "simpnum (Sub t s) = numsub (simpnum t) (simpnum s)"
-  "simpnum (Mul i t) = (if i = 0 then (C 0) else nummul (simpnum t) i)"
-  "simpnum (CN n c t) = (if c = 0 then simpnum t else numadd (CN n c (C 0),simpnum t))"
-
-lemma simpnum_ci[simp]: "Inum bs (simpnum t) = Inum bs t"
-by (induct t rule: simpnum.induct, auto simp add: numneg numadd numsub nummul)
-
-lemma simpnum_numbound0[simp]: 
-  "numbound0 t \<Longrightarrow> numbound0 (simpnum t)"
-by (induct t rule: simpnum.induct, auto)
-
-consts nozerocoeff:: "num \<Rightarrow> bool"
-recdef nozerocoeff "measure size"
-  "nozerocoeff (C c) = True"
-  "nozerocoeff (CN n c t) = (c\<noteq>0 \<and> nozerocoeff t)"
-  "nozerocoeff t = True"
-
-lemma numadd_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numadd (a,b))"
-by (induct a b rule: numadd.induct,auto simp add: Let_def)
-
-lemma nummul_nz : "\<And> i. i\<noteq>0 \<Longrightarrow> nozerocoeff a \<Longrightarrow> nozerocoeff (nummul a i)"
-by (induct a rule: nummul.induct,auto simp add: Let_def numadd_nz)
-
-lemma numneg_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff (numneg a)"
-by (simp add: numneg_def nummul_nz)
-
-lemma numsub_nz: "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numsub a b)"
-by (simp add: numsub_def numneg_nz numadd_nz)
-
-lemma simpnum_nz: "nozerocoeff (simpnum t)"
-by(induct t rule: simpnum.induct, auto simp add: numadd_nz numneg_nz numsub_nz nummul_nz)
-
-lemma maxcoeff_nz: "nozerocoeff t \<Longrightarrow> maxcoeff t = 0 \<Longrightarrow> t = C 0"
-proof (induct t rule: maxcoeff.induct)
-  case (2 n c t)
-  hence cnz: "c \<noteq>0" and mx: "max (abs c) (maxcoeff t) = 0" by simp+
-  have "max (abs c) (maxcoeff t) \<ge> abs c" by (simp add: le_maxI1)
-  with cnz have "max (abs c) (maxcoeff t) > 0" by arith
-  with prems show ?case by simp
-qed auto
-
-lemma numgcd_nz: assumes nz: "nozerocoeff t" and g0: "numgcd t = 0" shows "t = C 0"
-proof-
-  from g0 have th:"numgcdh t (maxcoeff t) = 0" by (simp add: numgcd_def)
-  from numgcdh0[OF th]  have th:"maxcoeff t = 0" .
-  from maxcoeff_nz[OF nz th] show ?thesis .
-qed
-
-constdefs simp_num_pair:: "(num \<times> int) \<Rightarrow> num \<times> int"
-  "simp_num_pair \<equiv> (\<lambda> (t,n). (if n = 0 then (C 0, 0) else
-   (let t' = simpnum t ; g = numgcd t' in 
-      if g > 1 then (let g' = zgcd n g in 
-        if g' = 1 then (t',n) 
-        else (reducecoeffh t' g', n div g')) 
-      else (t',n))))"
-
-lemma simp_num_pair_ci:
-  shows "((\<lambda> (t,n). Inum bs t / real n) (simp_num_pair (t,n))) = ((\<lambda> (t,n). Inum bs t / real n) (t,n))"
-  (is "?lhs = ?rhs")
-proof-
-  let ?t' = "simpnum t"
-  let ?g = "numgcd ?t'"
-  let ?g' = "zgcd n ?g"
-  {assume nz: "n = 0" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
-  moreover
-  { assume nnz: "n \<noteq> 0"
-    {assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simp_num_pair_def simpnum_ci)}
-    moreover
-    {assume g1:"?g>1" hence g0: "?g > 0" by simp
-      from zgcd0 g1 nnz have gp0: "?g' \<noteq> 0" by simp
-      hence g'p: "?g' > 0" using zgcd_pos[where i="n" and j="numgcd ?t'"] by arith 
-      hence "?g'= 1 \<or> ?g' > 1" by arith
-      moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simp_num_pair_def simpnum_ci)}
-      moreover {assume g'1:"?g'>1"
-	from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff ?t' ?g" ..
-	let ?tt = "reducecoeffh ?t' ?g'"
-	let ?t = "Inum bs ?tt"
-	have gpdg: "?g' dvd ?g" by (simp add: zgcd_zdvd2)
-	have gpdd: "?g' dvd n" by (simp add: zgcd_zdvd1) 
-	have gpdgp: "?g' dvd ?g'" by simp
-	from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p] 
-	have th2:"real ?g' * ?t = Inum bs ?t'" by simp
-	from prems have "?lhs = ?t / real (n div ?g')" by (simp add: simp_num_pair_def Let_def)
-	also have "\<dots> = (real ?g' * ?t) / (real ?g' * (real (n div ?g')))" by simp
-	also have "\<dots> = (Inum bs ?t' / real n)"
-	  using real_of_int_div[OF gp0 gpdd] th2 gp0 by simp
-	finally have "?lhs = Inum bs t / real n" by (simp add: simpnum_ci)
-	then have ?thesis using prems by (simp add: simp_num_pair_def)}
-      ultimately have ?thesis by blast}
-    ultimately have ?thesis by blast} 
-  ultimately show ?thesis by blast
-qed
-
-lemma simp_num_pair_l: assumes tnb: "numbound0 t" and np: "n >0" and tn: "simp_num_pair (t,n) = (t',n')"
-  shows "numbound0 t' \<and> n' >0"
-proof-
-    let ?t' = "simpnum t"
-  let ?g = "numgcd ?t'"
-  let ?g' = "zgcd n ?g"
-  {assume nz: "n = 0" hence ?thesis using prems by (simp add: Let_def simp_num_pair_def)}
-  moreover
-  { assume nnz: "n \<noteq> 0"
-    {assume "\<not> ?g > 1" hence ?thesis  using prems by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0)}
-    moreover
-    {assume g1:"?g>1" hence g0: "?g > 0" by simp
-      from zgcd0 g1 nnz have gp0: "?g' \<noteq> 0" by simp
-      hence g'p: "?g' > 0" using zgcd_pos[where i="n" and j="numgcd ?t'"] by arith
-      hence "?g'= 1 \<or> ?g' > 1" by arith
-      moreover {assume "?g'=1" hence ?thesis using prems 
-	  by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0)}
-      moreover {assume g'1:"?g'>1"
-	have gpdg: "?g' dvd ?g" by (simp add: zgcd_zdvd2)
-	have gpdd: "?g' dvd n" by (simp add: zgcd_zdvd1) 
-	have gpdgp: "?g' dvd ?g'" by simp
-	from zdvd_imp_le[OF gpdd np] have g'n: "?g' \<le> n" .
-	from zdiv_mono1[OF g'n g'p, simplified zdiv_self[OF gp0]]
-	have "n div ?g' >0" by simp
-	hence ?thesis using prems 
-	  by(auto simp add: simp_num_pair_def Let_def reducecoeffh_numbound0 simpnum_numbound0)}
-      ultimately have ?thesis by blast}
-    ultimately have ?thesis by blast} 
-  ultimately show ?thesis by blast
-qed
-
-consts simpfm :: "fm \<Rightarrow> fm"
-recdef simpfm "measure fmsize"
-  "simpfm (And p q) = conj (simpfm p) (simpfm q)"
-  "simpfm (Or p q) = disj (simpfm p) (simpfm q)"
-  "simpfm (Imp p q) = imp (simpfm p) (simpfm q)"
-  "simpfm (Iff p q) = iff (simpfm p) (simpfm q)"
-  "simpfm (NOT p) = not (simpfm p)"
-  "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v < 0) then T else F 
-  | _ \<Rightarrow> Lt a')"
-  "simpfm (Le a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<le> 0)  then T else F | _ \<Rightarrow> Le a')"
-  "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v > 0)  then T else F | _ \<Rightarrow> Gt a')"
-  "simpfm (Ge a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<ge> 0)  then T else F | _ \<Rightarrow> Ge a')"
-  "simpfm (Eq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v = 0)  then T else F | _ \<Rightarrow> Eq a')"
-  "simpfm (NEq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<noteq> 0)  then T else F | _ \<Rightarrow> NEq a')"
-  "simpfm p = p"
-lemma simpfm: "Ifm bs (simpfm p) = Ifm bs p"
-proof(induct p rule: simpfm.induct)
-  case (6 a) let ?sa = "simpnum a" from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
-  {fix v assume "?sa = C v" hence ?case using sa by simp }
-  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
-      by (cases ?sa, simp_all add: Let_def)}
-  ultimately show ?case by blast
-next
-  case (7 a)  let ?sa = "simpnum a" 
-  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
-  {fix v assume "?sa = C v" hence ?case using sa by simp }
-  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
-      by (cases ?sa, simp_all add: Let_def)}
-  ultimately show ?case by blast
-next
-  case (8 a)  let ?sa = "simpnum a" 
-  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
-  {fix v assume "?sa = C v" hence ?case using sa by simp }
-  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
-      by (cases ?sa, simp_all add: Let_def)}
-  ultimately show ?case by blast
-next
-  case (9 a)  let ?sa = "simpnum a" 
-  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
-  {fix v assume "?sa = C v" hence ?case using sa by simp }
-  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
-      by (cases ?sa, simp_all add: Let_def)}
-  ultimately show ?case by blast
-next
-  case (10 a)  let ?sa = "simpnum a" 
-  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
-  {fix v assume "?sa = C v" hence ?case using sa by simp }
-  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
-      by (cases ?sa, simp_all add: Let_def)}
-  ultimately show ?case by blast
-next
-  case (11 a)  let ?sa = "simpnum a" 
-  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
-  {fix v assume "?sa = C v" hence ?case using sa by simp }
-  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
-      by (cases ?sa, simp_all add: Let_def)}
-  ultimately show ?case by blast
-qed (induct p rule: simpfm.induct, simp_all add: conj disj imp iff not)
-
-
-lemma simpfm_bound0: "bound0 p \<Longrightarrow> bound0 (simpfm p)"
-proof(induct p rule: simpfm.induct)
-  case (6 a) hence nb: "numbound0 a" by simp
-  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
-  thus ?case by (cases "simpnum a", auto simp add: Let_def)
-next
-  case (7 a) hence nb: "numbound0 a" by simp
-  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
-  thus ?case by (cases "simpnum a", auto simp add: Let_def)
-next
-  case (8 a) hence nb: "numbound0 a" by simp
-  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
-  thus ?case by (cases "simpnum a", auto simp add: Let_def)
-next
-  case (9 a) hence nb: "numbound0 a" by simp
-  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
-  thus ?case by (cases "simpnum a", auto simp add: Let_def)
-next
-  case (10 a) hence nb: "numbound0 a" by simp
-  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
-  thus ?case by (cases "simpnum a", auto simp add: Let_def)
-next
-  case (11 a) hence nb: "numbound0 a" by simp
-  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
-  thus ?case by (cases "simpnum a", auto simp add: Let_def)
-qed(auto simp add: disj_def imp_def iff_def conj_def not_bn)
-
-lemma simpfm_qf: "qfree p \<Longrightarrow> qfree (simpfm p)"
-by (induct p rule: simpfm.induct, auto simp add: disj_qf imp_qf iff_qf conj_qf not_qf Let_def)
- (case_tac "simpnum a",auto)+
-
-consts prep :: "fm \<Rightarrow> fm"
-recdef prep "measure fmsize"
-  "prep (E T) = T"
-  "prep (E F) = F"
-  "prep (E (Or p q)) = disj (prep (E p)) (prep (E q))"
-  "prep (E (Imp p q)) = disj (prep (E (NOT p))) (prep (E q))"
-  "prep (E (Iff p q)) = disj (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))" 
-  "prep (E (NOT (And p q))) = disj (prep (E (NOT p))) (prep (E(NOT q)))"
-  "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))"
-  "prep (E (NOT (Iff p q))) = disj (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))"
-  "prep (E p) = E (prep p)"
-  "prep (A (And p q)) = conj (prep (A p)) (prep (A q))"
-  "prep (A p) = prep (NOT (E (NOT p)))"
-  "prep (NOT (NOT p)) = prep p"
-  "prep (NOT (And p q)) = disj (prep (NOT p)) (prep (NOT q))"
-  "prep (NOT (A p)) = prep (E (NOT p))"
-  "prep (NOT (Or p q)) = conj (prep (NOT p)) (prep (NOT q))"
-  "prep (NOT (Imp p q)) = conj (prep p) (prep (NOT q))"
-  "prep (NOT (Iff p q)) = disj (prep (And p (NOT q))) (prep (And (NOT p) q))"
-  "prep (NOT p) = not (prep p)"
-  "prep (Or p q) = disj (prep p) (prep q)"
-  "prep (And p q) = conj (prep p) (prep q)"
-  "prep (Imp p q) = prep (Or (NOT p) q)"
-  "prep (Iff p q) = disj (prep (And p q)) (prep (And (NOT p) (NOT q)))"
-  "prep p = p"
-(hints simp add: fmsize_pos)
-lemma prep: "\<And> bs. Ifm bs (prep p) = Ifm bs p"
-by (induct p rule: prep.induct, auto)
-
-  (* Generic quantifier elimination *)
-consts qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm"
-recdef qelim "measure fmsize"
-  "qelim (E p) = (\<lambda> qe. DJ qe (qelim p qe))"
-  "qelim (A p) = (\<lambda> qe. not (qe ((qelim (NOT p) qe))))"
-  "qelim (NOT p) = (\<lambda> qe. not (qelim p qe))"
-  "qelim (And p q) = (\<lambda> qe. conj (qelim p qe) (qelim q qe))" 
-  "qelim (Or  p q) = (\<lambda> qe. disj (qelim p qe) (qelim q qe))" 
-  "qelim (Imp p q) = (\<lambda> qe. imp (qelim p qe) (qelim q qe))"
-  "qelim (Iff p q) = (\<lambda> qe. iff (qelim p qe) (qelim q qe))"
-  "qelim p = (\<lambda> y. simpfm p)"
-
-lemma qelim_ci:
-  assumes qe_inv: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))"
-  shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm bs (qelim p qe) = Ifm bs p)"
-using qe_inv DJ_qe[OF qe_inv] 
-by(induct p rule: qelim.induct) 
-(auto simp add: not disj conj iff imp not_qf disj_qf conj_qf imp_qf iff_qf 
-  simpfm simpfm_qf simp del: simpfm.simps)
-
-consts 
-  plusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of +\<infinity>*)
-  minusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of -\<infinity>*)
-recdef minusinf "measure size"
-  "minusinf (And p q) = conj (minusinf p) (minusinf q)" 
-  "minusinf (Or p q) = disj (minusinf p) (minusinf q)" 
-  "minusinf (Eq  (CN 0 c e)) = F"
-  "minusinf (NEq (CN 0 c e)) = T"
-  "minusinf (Lt  (CN 0 c e)) = T"
-  "minusinf (Le  (CN 0 c e)) = T"
-  "minusinf (Gt  (CN 0 c e)) = F"
-  "minusinf (Ge  (CN 0 c e)) = F"
-  "minusinf p = p"
-
-recdef plusinf "measure size"
-  "plusinf (And p q) = conj (plusinf p) (plusinf q)" 
-  "plusinf (Or p q) = disj (plusinf p) (plusinf q)" 
-  "plusinf (Eq  (CN 0 c e)) = F"
-  "plusinf (NEq (CN 0 c e)) = T"
-  "plusinf (Lt  (CN 0 c e)) = F"
-  "plusinf (Le  (CN 0 c e)) = F"
-  "plusinf (Gt  (CN 0 c e)) = T"
-  "plusinf (Ge  (CN 0 c e)) = T"
-  "plusinf p = p"
-
-consts
-  isrlfm :: "fm \<Rightarrow> bool"   (* Linearity test for fm *)
-recdef isrlfm "measure size"
-  "isrlfm (And p q) = (isrlfm p \<and> isrlfm q)" 
-  "isrlfm (Or p q) = (isrlfm p \<and> isrlfm q)" 
-  "isrlfm (Eq  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
-  "isrlfm (NEq (CN 0 c e)) = (c>0 \<and> numbound0 e)"
-  "isrlfm (Lt  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
-  "isrlfm (Le  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
-  "isrlfm (Gt  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
-  "isrlfm (Ge  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
-  "isrlfm p = (isatom p \<and> (bound0 p))"
-
-  (* splits the bounded from the unbounded part*)
-consts rsplit0 :: "num \<Rightarrow> int \<times> num" 
-recdef rsplit0 "measure num_size"
-  "rsplit0 (Bound 0) = (1,C 0)"
-  "rsplit0 (Add a b) = (let (ca,ta) = rsplit0 a ; (cb,tb) = rsplit0 b 
-              in (ca+cb, Add ta tb))"
-  "rsplit0 (Sub a b) = rsplit0 (Add a (Neg b))"
-  "rsplit0 (Neg a) = (let (c,t) = rsplit0 a in (-c,Neg t))"
-  "rsplit0 (Mul c a) = (let (ca,ta) = rsplit0 a in (c*ca,Mul c ta))"
-  "rsplit0 (CN 0 c a) = (let (ca,ta) = rsplit0 a in (c+ca,ta))"
-  "rsplit0 (CN n c a) = (let (ca,ta) = rsplit0 a in (ca,CN n c ta))"
-  "rsplit0 t = (0,t)"
-lemma rsplit0: 
-  shows "Inum bs ((split (CN 0)) (rsplit0 t)) = Inum bs t \<and> numbound0 (snd (rsplit0 t))"
-proof (induct t rule: rsplit0.induct)
-  case (2 a b) 
-  let ?sa = "rsplit0 a" let ?sb = "rsplit0 b"
-  let ?ca = "fst ?sa" let ?cb = "fst ?sb"
-  let ?ta = "snd ?sa" let ?tb = "snd ?sb"
-  from prems have nb: "numbound0 (snd(rsplit0 (Add a b)))" 
-    by(cases "rsplit0 a",auto simp add: Let_def split_def)
-  have "Inum bs ((split (CN 0)) (rsplit0 (Add a b))) = 
-    Inum bs ((split (CN 0)) ?sa)+Inum bs ((split (CN 0)) ?sb)"
-    by (simp add: Let_def split_def algebra_simps)
-  also have "\<dots> = Inum bs a + Inum bs b" using prems by (cases "rsplit0 a", simp_all)
-  finally show ?case using nb by simp 
-qed(auto simp add: Let_def split_def algebra_simps , simp add: right_distrib[symmetric])
-
-    (* Linearize a formula*)
-definition
-  lt :: "int \<Rightarrow> num \<Rightarrow> fm"
-where
-  "lt c t = (if c = 0 then (Lt t) else if c > 0 then (Lt (CN 0 c t)) 
-    else (Gt (CN 0 (-c) (Neg t))))"
-
-definition
-  le :: "int \<Rightarrow> num \<Rightarrow> fm"
-where
-  "le c t = (if c = 0 then (Le t) else if c > 0 then (Le (CN 0 c t)) 
-    else (Ge (CN 0 (-c) (Neg t))))"
-
-definition
-  gt :: "int \<Rightarrow> num \<Rightarrow> fm"
-where
-  "gt c t = (if c = 0 then (Gt t) else if c > 0 then (Gt (CN 0 c t)) 
-    else (Lt (CN 0 (-c) (Neg t))))"
-
-definition
-  ge :: "int \<Rightarrow> num \<Rightarrow> fm"
-where
-  "ge c t = (if c = 0 then (Ge t) else if c > 0 then (Ge (CN 0 c t)) 
-    else (Le (CN 0 (-c) (Neg t))))"
-
-definition
-  eq :: "int \<Rightarrow> num \<Rightarrow> fm"
-where
-  "eq c t = (if c = 0 then (Eq t) else if c > 0 then (Eq (CN 0 c t)) 
-    else (Eq (CN 0 (-c) (Neg t))))"
-
-definition
-  neq :: "int \<Rightarrow> num \<Rightarrow> fm"
-where
-  "neq c t = (if c = 0 then (NEq t) else if c > 0 then (NEq (CN 0 c t)) 
-    else (NEq (CN 0 (-c) (Neg t))))"
-
-lemma lt: "numnoabs t \<Longrightarrow> Ifm bs (split lt (rsplit0 t)) = Ifm bs (Lt t) \<and> isrlfm (split lt (rsplit0 t))"
-using rsplit0[where bs = "bs" and t="t"]
-by (auto simp add: lt_def split_def,cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
-
-lemma le: "numnoabs t \<Longrightarrow> Ifm bs (split le (rsplit0 t)) = Ifm bs (Le t) \<and> isrlfm (split le (rsplit0 t))"
-using rsplit0[where bs = "bs" and t="t"]
-by (auto simp add: le_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
-
-lemma gt: "numnoabs t \<Longrightarrow> Ifm bs (split gt (rsplit0 t)) = Ifm bs (Gt t) \<and> isrlfm (split gt (rsplit0 t))"
-using rsplit0[where bs = "bs" and t="t"]
-by (auto simp add: gt_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
-
-lemma ge: "numnoabs t \<Longrightarrow> Ifm bs (split ge (rsplit0 t)) = Ifm bs (Ge t) \<and> isrlfm (split ge (rsplit0 t))"
-using rsplit0[where bs = "bs" and t="t"]
-by (auto simp add: ge_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
-
-lemma eq: "numnoabs t \<Longrightarrow> Ifm bs (split eq (rsplit0 t)) = Ifm bs (Eq t) \<and> isrlfm (split eq (rsplit0 t))"
-using rsplit0[where bs = "bs" and t="t"]
-by (auto simp add: eq_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
-
-lemma neq: "numnoabs t \<Longrightarrow> Ifm bs (split neq (rsplit0 t)) = Ifm bs (NEq t) \<and> isrlfm (split neq (rsplit0 t))"
-using rsplit0[where bs = "bs" and t="t"]
-by (auto simp add: neq_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
-
-lemma conj_lin: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (conj p q)"
-by (auto simp add: conj_def)
-lemma disj_lin: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (disj p q)"
-by (auto simp add: disj_def)
-
-consts rlfm :: "fm \<Rightarrow> fm"
-recdef rlfm "measure fmsize"
-  "rlfm (And p q) = conj (rlfm p) (rlfm q)"
-  "rlfm (Or p q) = disj (rlfm p) (rlfm q)"
-  "rlfm (Imp p q) = disj (rlfm (NOT p)) (rlfm q)"
-  "rlfm (Iff p q) = disj (conj (rlfm p) (rlfm q)) (conj (rlfm (NOT p)) (rlfm (NOT q)))"
-  "rlfm (Lt a) = split lt (rsplit0 a)"
-  "rlfm (Le a) = split le (rsplit0 a)"
-  "rlfm (Gt a) = split gt (rsplit0 a)"
-  "rlfm (Ge a) = split ge (rsplit0 a)"
-  "rlfm (Eq a) = split eq (rsplit0 a)"
-  "rlfm (NEq a) = split neq (rsplit0 a)"
-  "rlfm (NOT (And p q)) = disj (rlfm (NOT p)) (rlfm (NOT q))"
-  "rlfm (NOT (Or p q)) = conj (rlfm (NOT p)) (rlfm (NOT q))"
-  "rlfm (NOT (Imp p q)) = conj (rlfm p) (rlfm (NOT q))"
-  "rlfm (NOT (Iff p q)) = disj (conj(rlfm p) (rlfm(NOT q))) (conj(rlfm(NOT p)) (rlfm q))"
-  "rlfm (NOT (NOT p)) = rlfm p"
-  "rlfm (NOT T) = F"
-  "rlfm (NOT F) = T"
-  "rlfm (NOT (Lt a)) = rlfm (Ge a)"
-  "rlfm (NOT (Le a)) = rlfm (Gt a)"
-  "rlfm (NOT (Gt a)) = rlfm (Le a)"
-  "rlfm (NOT (Ge a)) = rlfm (Lt a)"
-  "rlfm (NOT (Eq a)) = rlfm (NEq a)"
-  "rlfm (NOT (NEq a)) = rlfm (Eq a)"
-  "rlfm p = p" (hints simp add: fmsize_pos)
-
-lemma rlfm_I:
-  assumes qfp: "qfree p"
-  shows "(Ifm bs (rlfm p) = Ifm bs p) \<and> isrlfm (rlfm p)"
-  using qfp 
-by (induct p rule: rlfm.induct, auto simp add: lt le gt ge eq neq conj disj conj_lin disj_lin)
-
-    (* Operations needed for Ferrante and Rackoff *)
-lemma rminusinf_inf:
-  assumes lp: "isrlfm p"
-  shows "\<exists> z. \<forall> x < z. Ifm (x#bs) (minusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p")
-using lp
-proof (induct p rule: minusinf.induct)
-  case (1 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto 
-next
-  case (2 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto
-next
-  case (3 c e) 
-  from prems have nb: "numbound0 e" by simp
-  from prems have cp: "real c > 0" by simp
-  fix a
-  let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
-  {fix x
-    assume xz: "x < ?z"
-    hence "(real c * x < - ?e)" 
-      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
-    hence "real c * x + ?e < 0" by arith
-    hence "real c * x + ?e \<noteq> 0" by simp
-    with xz have "?P ?z x (Eq (CN 0 c e))"
-      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp  }
-  hence "\<forall> x < ?z. ?P ?z x (Eq (CN 0 c e))" by simp
-  thus ?case by blast
-next
-  case (4 c e)   
-  from prems have nb: "numbound0 e" by simp
-  from prems have cp: "real c > 0" by simp
-  fix a
-  let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
-  {fix x
-    assume xz: "x < ?z"
-    hence "(real c * x < - ?e)" 
-      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
-    hence "real c * x + ?e < 0" by arith
-    hence "real c * x + ?e \<noteq> 0" by simp
-    with xz have "?P ?z x (NEq (CN 0 c e))"
-      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
-  hence "\<forall> x < ?z. ?P ?z x (NEq (CN 0 c e))" by simp
-  thus ?case by blast
-next
-  case (5 c e) 
-    from prems have nb: "numbound0 e" by simp
-  from prems have cp: "real c > 0" by simp
-  fix a
-  let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
-  {fix x
-    assume xz: "x < ?z"
-    hence "(real c * x < - ?e)" 
-      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
-    hence "real c * x + ?e < 0" by arith
-    with xz have "?P ?z x (Lt (CN 0 c e))"
-      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"]  by simp }
-  hence "\<forall> x < ?z. ?P ?z x (Lt (CN 0 c e))" by simp
-  thus ?case by blast
-next
-  case (6 c e)  
-    from prems have nb: "numbound0 e" by simp
-  from prems have cp: "real c > 0" by simp
-  fix a
-  let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
-  {fix x
-    assume xz: "x < ?z"
-    hence "(real c * x < - ?e)" 
-      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
-    hence "real c * x + ?e < 0" by arith
-    with xz have "?P ?z x (Le (CN 0 c e))"
-      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
-  hence "\<forall> x < ?z. ?P ?z x (Le (CN 0 c e))" by simp
-  thus ?case by blast
-next
-  case (7 c e)  
-    from prems have nb: "numbound0 e" by simp
-  from prems have cp: "real c > 0" by simp
-  fix a
-  let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
-  {fix x
-    assume xz: "x < ?z"
-    hence "(real c * x < - ?e)" 
-      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
-    hence "real c * x + ?e < 0" by arith
-    with xz have "?P ?z x (Gt (CN 0 c e))"
-      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
-  hence "\<forall> x < ?z. ?P ?z x (Gt (CN 0 c e))" by simp
-  thus ?case by blast
-next
-  case (8 c e)  
-    from prems have nb: "numbound0 e" by simp
-  from prems have cp: "real c > 0" by simp
-  fix a
-  let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
-  {fix x
-    assume xz: "x < ?z"
-    hence "(real c * x < - ?e)" 
-      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
-    hence "real c * x + ?e < 0" by arith
-    with xz have "?P ?z x (Ge (CN 0 c e))"
-      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
-  hence "\<forall> x < ?z. ?P ?z x (Ge (CN 0 c e))" by simp
-  thus ?case by blast
-qed simp_all
-
-lemma rplusinf_inf:
-  assumes lp: "isrlfm p"
-  shows "\<exists> z. \<forall> x > z. Ifm (x#bs) (plusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p")
-using lp
-proof (induct p rule: isrlfm.induct)
-  case (1 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto
-next
-  case (2 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto
-next
-  case (3 c e) 
-  from prems have nb: "numbound0 e" by simp
-  from prems have cp: "real c > 0" by simp
-  fix a
-  let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
-  {fix x
-    assume xz: "x > ?z"
-    with mult_strict_right_mono [OF xz cp] cp
-    have "(real c * x > - ?e)" by (simp add: mult_ac)
-    hence "real c * x + ?e > 0" by arith
-    hence "real c * x + ?e \<noteq> 0" by simp
-    with xz have "?P ?z x (Eq (CN 0 c e))"
-      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
-  hence "\<forall> x > ?z. ?P ?z x (Eq (CN 0 c e))" by simp
-  thus ?case by blast
-next
-  case (4 c e) 
-  from prems have nb: "numbound0 e" by simp
-  from prems have cp: "real c > 0" by simp
-  fix a
-  let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
-  {fix x
-    assume xz: "x > ?z"
-    with mult_strict_right_mono [OF xz cp] cp
-    have "(real c * x > - ?e)" by (simp add: mult_ac)
-    hence "real c * x + ?e > 0" by arith
-    hence "real c * x + ?e \<noteq> 0" by simp
-    with xz have "?P ?z x (NEq (CN 0 c e))"
-      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
-  hence "\<forall> x > ?z. ?P ?z x (NEq (CN 0 c e))" by simp
-  thus ?case by blast
-next
-  case (5 c e) 
-  from prems have nb: "numbound0 e" by simp
-  from prems have cp: "real c > 0" by simp
-  fix a
-  let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
-  {fix x
-    assume xz: "x > ?z"
-    with mult_strict_right_mono [OF xz cp] cp
-    have "(real c * x > - ?e)" by (simp add: mult_ac)
-    hence "real c * x + ?e > 0" by arith
-    with xz have "?P ?z x (Lt (CN 0 c e))"
-      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
-  hence "\<forall> x > ?z. ?P ?z x (Lt (CN 0 c e))" by simp
-  thus ?case by blast
-next
-  case (6 c e) 
-  from prems have nb: "numbound0 e" by simp
-  from prems have cp: "real c > 0" by simp
-  fix a
-  let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
-  {fix x
-    assume xz: "x > ?z"
-    with mult_strict_right_mono [OF xz cp] cp
-    have "(real c * x > - ?e)" by (simp add: mult_ac)
-    hence "real c * x + ?e > 0" by arith
-    with xz have "?P ?z x (Le (CN 0 c e))"
-      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
-  hence "\<forall> x > ?z. ?P ?z x (Le (CN 0 c e))" by simp
-  thus ?case by blast
-next
-  case (7 c e) 
-  from prems have nb: "numbound0 e" by simp
-  from prems have cp: "real c > 0" by simp
-  fix a
-  let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
-  {fix x
-    assume xz: "x > ?z"
-    with mult_strict_right_mono [OF xz cp] cp
-    have "(real c * x > - ?e)" by (simp add: mult_ac)
-    hence "real c * x + ?e > 0" by arith
-    with xz have "?P ?z x (Gt (CN 0 c e))"
-      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
-  hence "\<forall> x > ?z. ?P ?z x (Gt (CN 0 c e))" by simp
-  thus ?case by blast
-next
-  case (8 c e) 
-  from prems have nb: "numbound0 e" by simp
-  from prems have cp: "real c > 0" by simp
-  fix a
-  let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
-  {fix x
-    assume xz: "x > ?z"
-    with mult_strict_right_mono [OF xz cp] cp
-    have "(real c * x > - ?e)" by (simp add: mult_ac)
-    hence "real c * x + ?e > 0" by arith
-    with xz have "?P ?z x (Ge (CN 0 c e))"
-      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"]   by simp }
-  hence "\<forall> x > ?z. ?P ?z x (Ge (CN 0 c e))" by simp
-  thus ?case by blast
-qed simp_all
-
-lemma rminusinf_bound0:
-  assumes lp: "isrlfm p"
-  shows "bound0 (minusinf p)"
-  using lp
-  by (induct p rule: minusinf.induct) simp_all
-
-lemma rplusinf_bound0:
-  assumes lp: "isrlfm p"
-  shows "bound0 (plusinf p)"
-  using lp
-  by (induct p rule: plusinf.induct) simp_all
-
-lemma rminusinf_ex:
-  assumes lp: "isrlfm p"
-  and ex: "Ifm (a#bs) (minusinf p)"
-  shows "\<exists> x. Ifm (x#bs) p"
-proof-
-  from bound0_I [OF rminusinf_bound0[OF lp], where b="a" and bs ="bs"] ex
-  have th: "\<forall> x. Ifm (x#bs) (minusinf p)" by auto
-  from rminusinf_inf[OF lp, where bs="bs"] 
-  obtain z where z_def: "\<forall>x<z. Ifm (x # bs) (minusinf p) = Ifm (x # bs) p" by blast
-  from th have "Ifm ((z - 1)#bs) (minusinf p)" by simp
-  moreover have "z - 1 < z" by simp
-  ultimately show ?thesis using z_def by auto
-qed
-
-lemma rplusinf_ex:
-  assumes lp: "isrlfm p"
-  and ex: "Ifm (a#bs) (plusinf p)"
-  shows "\<exists> x. Ifm (x#bs) p"
-proof-
-  from bound0_I [OF rplusinf_bound0[OF lp], where b="a" and bs ="bs"] ex
-  have th: "\<forall> x. Ifm (x#bs) (plusinf p)" by auto
-  from rplusinf_inf[OF lp, where bs="bs"] 
-  obtain z where z_def: "\<forall>x>z. Ifm (x # bs) (plusinf p) = Ifm (x # bs) p" by blast
-  from th have "Ifm ((z + 1)#bs) (plusinf p)" by simp
-  moreover have "z + 1 > z" by simp
-  ultimately show ?thesis using z_def by auto
-qed
-
-consts 
-  uset:: "fm \<Rightarrow> (num \<times> int) list"
-  usubst :: "fm \<Rightarrow> (num \<times> int) \<Rightarrow> fm "
-recdef uset "measure size"
-  "uset (And p q) = (uset p @ uset q)" 
-  "uset (Or p q) = (uset p @ uset q)" 
-  "uset (Eq  (CN 0 c e)) = [(Neg e,c)]"
-  "uset (NEq (CN 0 c e)) = [(Neg e,c)]"
-  "uset (Lt  (CN 0 c e)) = [(Neg e,c)]"
-  "uset (Le  (CN 0 c e)) = [(Neg e,c)]"
-  "uset (Gt  (CN 0 c e)) = [(Neg e,c)]"
-  "uset (Ge  (CN 0 c e)) = [(Neg e,c)]"
-  "uset p = []"
-recdef usubst "measure size"
-  "usubst (And p q) = (\<lambda> (t,n). And (usubst p (t,n)) (usubst q (t,n)))"
-  "usubst (Or p q) = (\<lambda> (t,n). Or (usubst p (t,n)) (usubst q (t,n)))"
-  "usubst (Eq (CN 0 c e)) = (\<lambda> (t,n). Eq (Add (Mul c t) (Mul n e)))"
-  "usubst (NEq (CN 0 c e)) = (\<lambda> (t,n). NEq (Add (Mul c t) (Mul n e)))"
-  "usubst (Lt (CN 0 c e)) = (\<lambda> (t,n). Lt (Add (Mul c t) (Mul n e)))"
-  "usubst (Le (CN 0 c e)) = (\<lambda> (t,n). Le (Add (Mul c t) (Mul n e)))"
-  "usubst (Gt (CN 0 c e)) = (\<lambda> (t,n). Gt (Add (Mul c t) (Mul n e)))"
-  "usubst (Ge (CN 0 c e)) = (\<lambda> (t,n). Ge (Add (Mul c t) (Mul n e)))"
-  "usubst p = (\<lambda> (t,n). p)"
-
-lemma usubst_I: assumes lp: "isrlfm p"
-  and np: "real n > 0" and nbt: "numbound0 t"
-  shows "(Ifm (x#bs) (usubst p (t,n)) = Ifm (((Inum (x#bs) t)/(real n))#bs) p) \<and> bound0 (usubst p (t,n))" (is "(?I x (usubst p (t,n)) = ?I ?u p) \<and> ?B p" is "(_ = ?I (?t/?n) p) \<and> _" is "(_ = ?I (?N x t /_) p) \<and> _")
-  using lp
-proof(induct p rule: usubst.induct)
-  case (5 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
-  have "?I ?u (Lt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) < 0)"
-    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
-  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) < 0)"
-    by (simp only: pos_less_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
-      and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> = (real c *?t + ?n* (?N x e) < 0)"
-    using np by simp 
-  finally show ?case using nbt nb by (simp add: algebra_simps)
-next
-  case (6 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
-  have "?I ?u (Le (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<le> 0)"
-    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
-  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<le> 0)"
-    by (simp only: pos_le_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
-      and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> = (real c *?t + ?n* (?N x e) \<le> 0)"
-    using np by simp 
-  finally show ?case using nbt nb by (simp add: algebra_simps)
-next
-  case (7 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
-  have "?I ?u (Gt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) > 0)"
-    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
-  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) > 0)"
-    by (simp only: pos_divide_less_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
-      and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> = (real c *?t + ?n* (?N x e) > 0)"
-    using np by simp 
-  finally show ?case using nbt nb by (simp add: algebra_simps)
-next
-  case (8 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
-  have "?I ?u (Ge (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<ge> 0)"
-    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
-  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<ge> 0)"
-    by (simp only: pos_divide_le_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
-      and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> = (real c *?t + ?n* (?N x e) \<ge> 0)"
-    using np by simp 
-  finally show ?case using nbt nb by (simp add: algebra_simps)
-next
-  case (3 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
-  from np have np: "real n \<noteq> 0" by simp
-  have "?I ?u (Eq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) = 0)"
-    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
-  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) = 0)"
-    by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
-      and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> = (real c *?t + ?n* (?N x e) = 0)"
-    using np by simp 
-  finally show ?case using nbt nb by (simp add: algebra_simps)
-next
-  case (4 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
-  from np have np: "real n \<noteq> 0" by simp
-  have "?I ?u (NEq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<noteq> 0)"
-    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
-  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<noteq> 0)"
-    by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
-      and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> = (real c *?t + ?n* (?N x e) \<noteq> 0)"
-    using np by simp 
-  finally show ?case using nbt nb by (simp add: algebra_simps)
-qed(simp_all add: nbt numbound0_I[where bs ="bs" and b="(Inum (x#bs) t)/ real n" and b'="x"] nth_pos2)
-
-lemma uset_l:
-  assumes lp: "isrlfm p"
-  shows "\<forall> (t,k) \<in> set (uset p). numbound0 t \<and> k >0"
-using lp
-by(induct p rule: uset.induct,auto)
-
-lemma rminusinf_uset:
-  assumes lp: "isrlfm p"
-  and nmi: "\<not> (Ifm (a#bs) (minusinf p))" (is "\<not> (Ifm (a#bs) (?M p))")
-  and ex: "Ifm (x#bs) p" (is "?I x p")
-  shows "\<exists> (s,m) \<in> set (uset p). x \<ge> Inum (a#bs) s / real m" (is "\<exists> (s,m) \<in> ?U p. x \<ge> ?N a s / real m")
-proof-
-  have "\<exists> (s,m) \<in> set (uset p). real m * x \<ge> Inum (a#bs) s " (is "\<exists> (s,m) \<in> ?U p. real m *x \<ge> ?N a s")
-    using lp nmi ex
-    by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"] nth_pos2)
-  then obtain s m where smU: "(s,m) \<in> set (uset p)" and mx: "real m * x \<ge> ?N a s" by blast
-  from uset_l[OF lp] smU have mp: "real m > 0" by auto
-  from pos_divide_le_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<ge> ?N a s / real m" 
-    by (auto simp add: mult_commute)
-  thus ?thesis using smU by auto
-qed
-
-lemma rplusinf_uset:
-  assumes lp: "isrlfm p"
-  and nmi: "\<not> (Ifm (a#bs) (plusinf p))" (is "\<not> (Ifm (a#bs) (?M p))")
-  and ex: "Ifm (x#bs) p" (is "?I x p")
-  shows "\<exists> (s,m) \<in> set (uset p). x \<le> Inum (a#bs) s / real m" (is "\<exists> (s,m) \<in> ?U p. x \<le> ?N a s / real m")
-proof-
-  have "\<exists> (s,m) \<in> set (uset p). real m * x \<le> Inum (a#bs) s " (is "\<exists> (s,m) \<in> ?U p. real m *x \<le> ?N a s")
-    using lp nmi ex
-    by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"] nth_pos2)
-  then obtain s m where smU: "(s,m) \<in> set (uset p)" and mx: "real m * x \<le> ?N a s" by blast
-  from uset_l[OF lp] smU have mp: "real m > 0" by auto
-  from pos_le_divide_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<le> ?N a s / real m" 
-    by (auto simp add: mult_commute)
-  thus ?thesis using smU by auto
-qed
-
-lemma lin_dense: 
-  assumes lp: "isrlfm p"
-  and noS: "\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> (\<lambda> (t,n). Inum (x#bs) t / real n) ` set (uset p)" 
-  (is "\<forall> t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda> (t,n). ?N x t / real n ) ` (?U p)")
-  and lx: "l < x" and xu:"x < u" and px:" Ifm (x#bs) p"
-  and ly: "l < y" and yu: "y < u"
-  shows "Ifm (y#bs) p"
-using lp px noS
-proof (induct p rule: isrlfm.induct)
-  case (5 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
-    from prems have "x * real c + ?N x e < 0" by (simp add: algebra_simps)
-    hence pxc: "x < (- ?N x e) / real c" 
-      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="-?N x e"])
-    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
-    with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
-    hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
-    moreover {assume y: "y < (-?N x e)/ real c"
-      hence "y * real c < - ?N x e"
-	by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
-      hence "real c * y + ?N x e < 0" by (simp add: algebra_simps)
-      hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
-    moreover {assume y: "y > (- ?N x e) / real c" 
-      with yu have eu: "u > (- ?N x e) / real c" by auto
-      with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto)
-      with lx pxc have "False" by auto
-      hence ?case by simp }
-    ultimately show ?case by blast
-next
-  case (6 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp +
-    from prems have "x * real c + ?N x e \<le> 0" by (simp add: algebra_simps)
-    hence pxc: "x \<le> (- ?N x e) / real c" 
-      by (simp only: pos_le_divide_eq[OF cp, where a="x" and b="-?N x e"])
-    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
-    with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
-    hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
-    moreover {assume y: "y < (-?N x e)/ real c"
-      hence "y * real c < - ?N x e"
-	by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
-      hence "real c * y + ?N x e < 0" by (simp add: algebra_simps)
-      hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
-    moreover {assume y: "y > (- ?N x e) / real c" 
-      with yu have eu: "u > (- ?N x e) / real c" by auto
-      with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto)
-      with lx pxc have "False" by auto
-      hence ?case by simp }
-    ultimately show ?case by blast
-next
-  case (7 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
-    from prems have "x * real c + ?N x e > 0" by (simp add: algebra_simps)
-    hence pxc: "x > (- ?N x e) / real c" 
-      by (simp only: pos_divide_less_eq[OF cp, where a="x" and b="-?N x e"])
-    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
-    with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
-    hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
-    moreover {assume y: "y > (-?N x e)/ real c"
-      hence "y * real c > - ?N x e"
-	by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
-      hence "real c * y + ?N x e > 0" by (simp add: algebra_simps)
-      hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
-    moreover {assume y: "y < (- ?N x e) / real c" 
-      with ly have eu: "l < (- ?N x e) / real c" by auto
-      with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto)
-      with xu pxc have "False" by auto
-      hence ?case by simp }
-    ultimately show ?case by blast
-next
-  case (8 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
-    from prems have "x * real c + ?N x e \<ge> 0" by (simp add: algebra_simps)
-    hence pxc: "x \<ge> (- ?N x e) / real c" 
-      by (simp only: pos_divide_le_eq[OF cp, where a="x" and b="-?N x e"])
-    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
-    with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
-    hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
-    moreover {assume y: "y > (-?N x e)/ real c"
-      hence "y * real c > - ?N x e"
-	by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
-      hence "real c * y + ?N x e > 0" by (simp add: algebra_simps)
-      hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
-    moreover {assume y: "y < (- ?N x e) / real c" 
-      with ly have eu: "l < (- ?N x e) / real c" by auto
-      with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto)
-      with xu pxc have "False" by auto
-      hence ?case by simp }
-    ultimately show ?case by blast
-next
-  case (3 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
-    from cp have cnz: "real c \<noteq> 0" by simp
-    from prems have "x * real c + ?N x e = 0" by (simp add: algebra_simps)
-    hence pxc: "x = (- ?N x e) / real c" 
-      by (simp only: nonzero_eq_divide_eq[OF cnz, where a="x" and b="-?N x e"])
-    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
-    with lx xu have yne: "x \<noteq> - ?N x e / real c" by auto
-    with pxc show ?case by simp
-next
-  case (4 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
-    from cp have cnz: "real c \<noteq> 0" by simp
-    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
-    with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
-    hence "y* real c \<noteq> -?N x e"      
-      by (simp only: nonzero_eq_divide_eq[OF cnz, where a="y" and b="-?N x e"]) simp
-    hence "y* real c + ?N x e \<noteq> 0" by (simp add: algebra_simps)
-    thus ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] 
-      by (simp add: algebra_simps)
-qed (auto simp add: nth_pos2 numbound0_I[where bs="bs" and b="y" and b'="x"])
-
-lemma finite_set_intervals:
-  assumes px: "P (x::real)" 
-  and lx: "l \<le> x" and xu: "x \<le> u"
-  and linS: "l\<in> S" and uinS: "u \<in> S"
-  and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u"
-  shows "\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a \<le> x \<and> x \<le> b \<and> P x"
-proof-
-  let ?Mx = "{y. y\<in> S \<and> y \<le> x}"
-  let ?xM = "{y. y\<in> S \<and> x \<le> y}"
-  let ?a = "Max ?Mx"
-  let ?b = "Min ?xM"
-  have MxS: "?Mx \<subseteq> S" by blast
-  hence fMx: "finite ?Mx" using fS finite_subset by auto
-  from lx linS have linMx: "l \<in> ?Mx" by blast
-  hence Mxne: "?Mx \<noteq> {}" by blast
-  have xMS: "?xM \<subseteq> S" by blast
-  hence fxM: "finite ?xM" using fS finite_subset by auto
-  from xu uinS have linxM: "u \<in> ?xM" by blast
-  hence xMne: "?xM \<noteq> {}" by blast
-  have ax:"?a \<le> x" using Mxne fMx by auto
-  have xb:"x \<le> ?b" using xMne fxM by auto
-  have "?a \<in> ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a \<in> S" using MxS by blast
-  have "?b \<in> ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b \<in> S" using xMS by blast
-  have noy:"\<forall> y. ?a < y \<and> y < ?b \<longrightarrow> y \<notin> S"
-  proof(clarsimp)
-    fix y
-    assume ay: "?a < y" and yb: "y < ?b" and yS: "y \<in> S"
-    from yS have "y\<in> ?Mx \<or> y\<in> ?xM" by auto
-    moreover {assume "y \<in> ?Mx" hence "y \<le> ?a" using Mxne fMx by auto with ay have "False" by simp}
-    moreover {assume "y \<in> ?xM" hence "y \<ge> ?b" using xMne fxM by auto with yb have "False" by simp}
-    ultimately show "False" by blast
-  qed
-  from ainS binS noy ax xb px show ?thesis by blast
-qed
-
-lemma finite_set_intervals2:
-  assumes px: "P (x::real)" 
-  and lx: "l \<le> x" and xu: "x \<le> u"
-  and linS: "l\<in> S" and uinS: "u \<in> S"
-  and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u"
-  shows "(\<exists> s\<in> S. P s) \<or> (\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a < x \<and> x < b \<and> P x)"
-proof-
-  from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su]
-  obtain a and b where 
-    as: "a\<in> S" and bs: "b\<in> S" and noS:"\<forall>y. a < y \<and> y < b \<longrightarrow> y \<notin> S" and axb: "a \<le> x \<and> x \<le> b \<and> P x"  by auto
-  from axb have "x= a \<or> x= b \<or> (a < x \<and> x < b)" by auto
-  thus ?thesis using px as bs noS by blast 
-qed
-
-lemma rinf_uset:
-  assumes lp: "isrlfm p"
-  and nmi: "\<not> (Ifm (x#bs) (minusinf p))" (is "\<not> (Ifm (x#bs) (?M p))")
-  and npi: "\<not> (Ifm (x#bs) (plusinf p))" (is "\<not> (Ifm (x#bs) (?P p))")
-  and ex: "\<exists> x.  Ifm (x#bs) p" (is "\<exists> x. ?I x p")
-  shows "\<exists> (l,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). ?I ((Inum (x#bs) l / real n + Inum (x#bs) s / real m) / 2) p" 
-proof-
-  let ?N = "\<lambda> x t. Inum (x#bs) t"
-  let ?U = "set (uset p)"
-  from ex obtain a where pa: "?I a p" by blast
-  from bound0_I[OF rminusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] nmi
-  have nmi': "\<not> (?I a (?M p))" by simp
-  from bound0_I[OF rplusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] npi
-  have npi': "\<not> (?I a (?P p))" by simp
-  have "\<exists> (l,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). ?I ((?N a l/real n + ?N a s /real m) / 2) p"
-  proof-
-    let ?M = "(\<lambda> (t,c). ?N a t / real c) ` ?U"
-    have fM: "finite ?M" by auto
-    from rminusinf_uset[OF lp nmi pa] rplusinf_uset[OF lp npi pa] 
-    have "\<exists> (l,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). a \<le> ?N x l / real n \<and> a \<ge> ?N x s / real m" by blast
-    then obtain "t" "n" "s" "m" where 
-      tnU: "(t,n) \<in> ?U" and smU: "(s,m) \<in> ?U" 
-      and xs1: "a \<le> ?N x s / real m" and tx1: "a \<ge> ?N x t / real n" by blast
-    from uset_l[OF lp] tnU smU numbound0_I[where bs="bs" and b="x" and b'="a"] xs1 tx1 have xs: "a \<le> ?N a s / real m" and tx: "a \<ge> ?N a t / real n" by auto
-    from tnU have Mne: "?M \<noteq> {}" by auto
-    hence Une: "?U \<noteq> {}" by simp
-    let ?l = "Min ?M"
-    let ?u = "Max ?M"
-    have linM: "?l \<in> ?M" using fM Mne by simp
-    have uinM: "?u \<in> ?M" using fM Mne by simp
-    have tnM: "?N a t / real n \<in> ?M" using tnU by auto
-    have smM: "?N a s / real m \<in> ?M" using smU by auto 
-    have lM: "\<forall> t\<in> ?M. ?l \<le> t" using Mne fM by auto
-    have Mu: "\<forall> t\<in> ?M. t \<le> ?u" using Mne fM by auto
-    have "?l \<le> ?N a t / real n" using tnM Mne by simp hence lx: "?l \<le> a" using tx by simp
-    have "?N a s / real m \<le> ?u" using smM Mne by simp hence xu: "a \<le> ?u" using xs by simp
-    from finite_set_intervals2[where P="\<lambda> x. ?I x p",OF pa lx xu linM uinM fM lM Mu]
-    have "(\<exists> s\<in> ?M. ?I s p) \<or> 
-      (\<exists> t1\<in> ?M. \<exists> t2 \<in> ?M. (\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M) \<and> t1 < a \<and> a < t2 \<and> ?I a p)" .
-    moreover { fix u assume um: "u\<in> ?M" and pu: "?I u p"
-      hence "\<exists> (tu,nu) \<in> ?U. u = ?N a tu / real nu" by auto
-      then obtain "tu" "nu" where tuU: "(tu,nu) \<in> ?U" and tuu:"u= ?N a tu / real nu" by blast
-      have "(u + u) / 2 = u" by auto with pu tuu 
-      have "?I (((?N a tu / real nu) + (?N a tu / real nu)) / 2) p" by simp
-      with tuU have ?thesis by blast}
-    moreover{
-      assume "\<exists> t1\<in> ?M. \<exists> t2 \<in> ?M. (\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M) \<and> t1 < a \<and> a < t2 \<and> ?I a p"
-      then obtain t1 and t2 where t1M: "t1 \<in> ?M" and t2M: "t2\<in> ?M" 
-	and noM: "\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M" and t1x: "t1 < a" and xt2: "a < t2" and px: "?I a p"
-	by blast
-      from t1M have "\<exists> (t1u,t1n) \<in> ?U. t1 = ?N a t1u / real t1n" by auto
-      then obtain "t1u" "t1n" where t1uU: "(t1u,t1n) \<in> ?U" and t1u: "t1 = ?N a t1u / real t1n" by blast
-      from t2M have "\<exists> (t2u,t2n) \<in> ?U. t2 = ?N a t2u / real t2n" by auto
-      then obtain "t2u" "t2n" where t2uU: "(t2u,t2n) \<in> ?U" and t2u: "t2 = ?N a t2u / real t2n" by blast
-      from t1x xt2 have t1t2: "t1 < t2" by simp
-      let ?u = "(t1 + t2) / 2"
-      from less_half_sum[OF t1t2] gt_half_sum[OF t1t2] have t1lu: "t1 < ?u" and ut2: "?u < t2" by auto
-      from lin_dense[OF lp noM t1x xt2 px t1lu ut2] have "?I ?u p" .
-      with t1uU t2uU t1u t2u have ?thesis by blast}
-    ultimately show ?thesis by blast
-  qed
-  then obtain "l" "n" "s"  "m" where lnU: "(l,n) \<in> ?U" and smU:"(s,m) \<in> ?U" 
-    and pu: "?I ((?N a l / real n + ?N a s / real m) / 2) p" by blast
-  from lnU smU uset_l[OF lp] have nbl: "numbound0 l" and nbs: "numbound0 s" by auto
-  from numbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"] 
-    numbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu
-  have "?I ((?N x l / real n + ?N x s / real m) / 2) p" by simp
-  with lnU smU
-  show ?thesis by auto
-qed
-    (* The Ferrante - Rackoff Theorem *)
-
-theorem fr_eq: 
-  assumes lp: "isrlfm p"
-  shows "(\<exists> x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \<or> (Ifm (x#bs) (plusinf p)) \<or> (\<exists> (t,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). Ifm ((((Inum (x#bs) t)/  real n + (Inum (x#bs) s) / real m) /2)#bs) p))"
-  (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
-proof
-  assume px: "\<exists> x. ?I x p"
-  have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
-  moreover {assume "?M \<or> ?P" hence "?D" by blast}
-  moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
-    from rinf_uset[OF lp nmi npi] have "?F" using px by blast hence "?D" by blast}
-  ultimately show "?D" by blast
-next
-  assume "?D" 
-  moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .}
-  moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . }
-  moreover {assume f:"?F" hence "?E" by blast}
-  ultimately show "?E" by blast
-qed
-
-
-lemma fr_equsubst: 
-  assumes lp: "isrlfm p"
-  shows "(\<exists> x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \<or> (Ifm (x#bs) (plusinf p)) \<or> (\<exists> (t,k) \<in> set (uset p). \<exists> (s,l) \<in> set (uset p). Ifm (x#bs) (usubst p (Add(Mul l t) (Mul k s) , 2*k*l))))"
-  (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
-proof
-  assume px: "\<exists> x. ?I x p"
-  have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
-  moreover {assume "?M \<or> ?P" hence "?D" by blast}
-  moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
-    let ?f ="\<lambda> (t,n). Inum (x#bs) t / real n"
-    let ?N = "\<lambda> t. Inum (x#bs) t"
-    {fix t n s m assume "(t,n)\<in> set (uset p)" and "(s,m) \<in> set (uset p)"
-      with uset_l[OF lp] have tnb: "numbound0 t" and np:"real n > 0" and snb: "numbound0 s" and mp:"real m > 0"
-	by auto
-      let ?st = "Add (Mul m t) (Mul n s)"
-      from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" 
-	by (simp add: mult_commute)
-      from tnb snb have st_nb: "numbound0 ?st" by simp
-      have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
-	using mnp mp np by (simp add: algebra_simps add_divide_distrib)
-      from usubst_I[OF lp mnp st_nb, where x="x" and bs="bs"] 
-      have "?I x (usubst p (?st,2*n*m)) = ?I ((?N t / real n + ?N s / real m) /2) p" by (simp only: st[symmetric])}
-    with rinf_uset[OF lp nmi npi px] have "?F" by blast hence "?D" by blast}
-  ultimately show "?D" by blast
-next
-  assume "?D" 
-  moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .}
-  moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . }
-  moreover {fix t k s l assume "(t,k) \<in> set (uset p)" and "(s,l) \<in> set (uset p)" 
-    and px:"?I x (usubst p (Add (Mul l t) (Mul k s), 2*k*l))"
-    with uset_l[OF lp] have tnb: "numbound0 t" and np:"real k > 0" and snb: "numbound0 s" and mp:"real l > 0" by auto
-    let ?st = "Add (Mul l t) (Mul k s)"
-    from mult_pos_pos[OF np mp] have mnp: "real (2*k*l) > 0" 
-      by (simp add: mult_commute)
-    from tnb snb have st_nb: "numbound0 ?st" by simp
-    from usubst_I[OF lp mnp st_nb, where bs="bs"] px have "?E" by auto}
-  ultimately show "?E" by blast
-qed
-
-
-    (* Implement the right hand side of Ferrante and Rackoff's Theorem. *)
-constdefs ferrack:: "fm \<Rightarrow> fm"
-  "ferrack p \<equiv> (let p' = rlfm (simpfm p); mp = minusinf p'; pp = plusinf p'
-                in if (mp = T \<or> pp = T) then T else 
-                   (let U = remdps(map simp_num_pair 
-                     (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m))
-                           (alluopairs (uset p')))) 
-                    in decr (disj mp (disj pp (evaldjf (simpfm o (usubst p')) U)))))"
-
-lemma uset_cong_aux:
-  assumes Ul: "\<forall> (t,n) \<in> set U. numbound0 t \<and> n >0"
-  shows "((\<lambda> (t,n). Inum (x#bs) t /real n) ` (set (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)) (alluopairs U)))) = ((\<lambda> ((t,n),(s,m)). (Inum (x#bs) t /real n + Inum (x#bs) s /real m)/2) ` (set U \<times> set U))"
-  (is "?lhs = ?rhs")
-proof(auto)
-  fix t n s m
-  assume "((t,n),(s,m)) \<in> set (alluopairs U)"
-  hence th: "((t,n),(s,m)) \<in> (set U \<times> set U)"
-    using alluopairs_set1[where xs="U"] by blast
-  let ?N = "\<lambda> t. Inum (x#bs) t"
-  let ?st= "Add (Mul m t) (Mul n s)"
-  from Ul th have mnz: "m \<noteq> 0" by auto
-  from Ul th have  nnz: "n \<noteq> 0" by auto  
-  have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
-   using mnz nnz by (simp add: algebra_simps add_divide_distrib)
- 
-  thus "(real m *  Inum (x # bs) t + real n * Inum (x # bs) s) /
-       (2 * real n * real m)
-       \<in> (\<lambda>((t, n), s, m).
-             (Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2) `
-         (set U \<times> set U)"using mnz nnz th  
-    apply (auto simp add: th add_divide_distrib algebra_simps split_def image_def)
-    by (rule_tac x="(s,m)" in bexI,simp_all) 
-  (rule_tac x="(t,n)" in bexI,simp_all)
-next
-  fix t n s m
-  assume tnU: "(t,n) \<in> set U" and smU:"(s,m) \<in> set U" 
-  let ?N = "\<lambda> t. Inum (x#bs) t"
-  let ?st= "Add (Mul m t) (Mul n s)"
-  from Ul smU have mnz: "m \<noteq> 0" by auto
-  from Ul tnU have  nnz: "n \<noteq> 0" by auto  
-  have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
-   using mnz nnz by (simp add: algebra_simps add_divide_distrib)
- let ?P = "\<lambda> (t',n') (s',m'). (Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2"
- have Pc:"\<forall> a b. ?P a b = ?P b a"
-   by auto
- from Ul alluopairs_set1 have Up:"\<forall> ((t,n),(s,m)) \<in> set (alluopairs U). n \<noteq> 0 \<and> m \<noteq> 0" by blast
- from alluopairs_ex[OF Pc, where xs="U"] tnU smU
- have th':"\<exists> ((t',n'),(s',m')) \<in> set (alluopairs U). ?P (t',n') (s',m')"
-   by blast
- then obtain t' n' s' m' where ts'_U: "((t',n'),(s',m')) \<in> set (alluopairs U)" 
-   and Pts': "?P (t',n') (s',m')" by blast
- from ts'_U Up have mnz': "m' \<noteq> 0" and nnz': "n'\<noteq> 0" by auto
- let ?st' = "Add (Mul m' t') (Mul n' s')"
-   have st': "(?N t' / real n' + ?N s' / real m')/2 = ?N ?st' / real (2*n'*m')"
-   using mnz' nnz' by (simp add: algebra_simps add_divide_distrib)
- from Pts' have 
-   "(Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2" by simp
- also have "\<dots> = ((\<lambda>(t, n). Inum (x # bs) t / real n) ((\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) ((t',n'),(s',m'))))" by (simp add: st')
- finally show "(Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2
-          \<in> (\<lambda>(t, n). Inum (x # bs) t / real n) `
-            (\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) `
-            set (alluopairs U)"
-   using ts'_U by blast
-qed
-
-lemma uset_cong:
-  assumes lp: "isrlfm p"
-  and UU': "((\<lambda> (t,n). Inum (x#bs) t /real n) ` U') = ((\<lambda> ((t,n),(s,m)). (Inum (x#bs) t /real n + Inum (x#bs) s /real m)/2) ` (U \<times> U))" (is "?f ` U' = ?g ` (U\<times>U)")
-  and U: "\<forall> (t,n) \<in> U. numbound0 t \<and> n > 0"
-  and U': "\<forall> (t,n) \<in> U'. numbound0 t \<and> n > 0"
-  shows "(\<exists> (t,n) \<in> U. \<exists> (s,m) \<in> U. Ifm (x#bs) (usubst p (Add (Mul m t) (Mul n s),2*n*m))) = (\<exists> (t,n) \<in> U'. Ifm (x#bs) (usubst p (t,n)))"
-  (is "?lhs = ?rhs")
-proof
-  assume ?lhs
-  then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and 
-    Pst: "Ifm (x#bs) (usubst p (Add (Mul m t) (Mul n s),2*n*m))" by blast
-  let ?N = "\<lambda> t. Inum (x#bs) t"
-  from tnU smU U have tnb: "numbound0 t" and np: "n > 0" 
-    and snb: "numbound0 s" and mp:"m > 0"  by auto
-  let ?st= "Add (Mul m t) (Mul n s)"
-  from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" 
-      by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
-    from tnb snb have stnb: "numbound0 ?st" by simp
-  have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
-   using mp np by (simp add: algebra_simps add_divide_distrib)
-  from tnU smU UU' have "?g ((t,n),(s,m)) \<in> ?f ` U'" by blast
-  hence "\<exists> (t',n') \<in> U'. ?g ((t,n),(s,m)) = ?f (t',n')"
-    by auto (rule_tac x="(a,b)" in bexI, auto)
-  then obtain t' n' where tnU': "(t',n') \<in> U'" and th: "?g ((t,n),(s,m)) = ?f (t',n')" by blast
-  from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto
-  from usubst_I[OF lp mnp stnb, where bs="bs" and x="x"] Pst 
-  have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp
-  from conjunct1[OF usubst_I[OF lp np' tnb', where bs="bs" and x="x"], symmetric] th[simplified split_def fst_conv snd_conv,symmetric] Pst2[simplified st[symmetric]]
-  have "Ifm (x # bs) (usubst p (t', n')) " by (simp only: st) 
-  then show ?rhs using tnU' by auto 
-next
-  assume ?rhs
-  then obtain t' n' where tnU': "(t',n') \<in> U'" and Pt': "Ifm (x # bs) (usubst p (t', n'))" 
-    by blast
-  from tnU' UU' have "?f (t',n') \<in> ?g ` (U\<times>U)" by blast
-  hence "\<exists> ((t,n),(s,m)) \<in> (U\<times>U). ?f (t',n') = ?g ((t,n),(s,m))" 
-    by auto (rule_tac x="(a,b)" in bexI, auto)
-  then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and 
-    th: "?f (t',n') = ?g((t,n),(s,m)) "by blast
-    let ?N = "\<lambda> t. Inum (x#bs) t"
-  from tnU smU U have tnb: "numbound0 t" and np: "n > 0" 
-    and snb: "numbound0 s" and mp:"m > 0"  by auto
-  let ?st= "Add (Mul m t) (Mul n s)"
-  from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" 
-      by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
-    from tnb snb have stnb: "numbound0 ?st" by simp
-  have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
-   using mp np by (simp add: algebra_simps add_divide_distrib)
-  from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto
-  from usubst_I[OF lp np' tnb', where bs="bs" and x="x",simplified th[simplified split_def fst_conv snd_conv] st] Pt'
-  have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp
-  with usubst_I[OF lp mnp stnb, where x="x" and bs="bs"] tnU smU show ?lhs by blast
-qed
-
-lemma ferrack: 
-  assumes qf: "qfree p"
-  shows "qfree (ferrack p) \<and> ((Ifm bs (ferrack p)) = (\<exists> x. Ifm (x#bs) p))"
-  (is "_ \<and> (?rhs = ?lhs)")
-proof-
-  let ?I = "\<lambda> x p. Ifm (x#bs) p"
-  fix x
-  let ?N = "\<lambda> t. Inum (x#bs) t"
-  let ?q = "rlfm (simpfm p)" 
-  let ?U = "uset ?q"
-  let ?Up = "alluopairs ?U"
-  let ?g = "\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)"
-  let ?S = "map ?g ?Up"
-  let ?SS = "map simp_num_pair ?S"
-  let ?Y = "remdps ?SS"
-  let ?f= "(\<lambda> (t,n). ?N t / real n)"
-  let ?h = "\<lambda> ((t,n),(s,m)). (?N t/real n + ?N s/ real m) /2"
-  let ?F = "\<lambda> p. \<exists> a \<in> set (uset p). \<exists> b \<in> set (uset p). ?I x (usubst p (?g(a,b)))"
-  let ?ep = "evaldjf (simpfm o (usubst ?q)) ?Y"
-  from rlfm_I[OF simpfm_qf[OF qf]] have lq: "isrlfm ?q" by blast
-  from alluopairs_set1[where xs="?U"] have UpU: "set ?Up \<le> (set ?U \<times> set ?U)" by simp
-  from uset_l[OF lq] have U_l: "\<forall> (t,n) \<in> set ?U. numbound0 t \<and> n > 0" .
-  from U_l UpU 
-  have "\<forall> ((t,n),(s,m)) \<in> set ?Up. numbound0 t \<and> n> 0 \<and> numbound0 s \<and> m > 0" by auto
-  hence Snb: "\<forall> (t,n) \<in> set ?S. numbound0 t \<and> n > 0 "
-    by (auto simp add: mult_pos_pos)
-  have Y_l: "\<forall> (t,n) \<in> set ?Y. numbound0 t \<and> n > 0" 
-  proof-
-    { fix t n assume tnY: "(t,n) \<in> set ?Y" 
-      hence "(t,n) \<in> set ?SS" by simp
-      hence "\<exists> (t',n') \<in> set ?S. simp_num_pair (t',n') = (t,n)"
-	by (auto simp add: split_def) (rule_tac x="((aa,ba),(ab,bb))" in bexI, simp_all)
-      then obtain t' n' where tn'S: "(t',n') \<in> set ?S" and tns: "simp_num_pair (t',n') = (t,n)" by blast
-      from tn'S Snb have tnb: "numbound0 t'" and np: "n' > 0" by auto
-      from simp_num_pair_l[OF tnb np tns]
-      have "numbound0 t \<and> n > 0" . }
-    thus ?thesis by blast
-  qed
-
-  have YU: "(?f ` set ?Y) = (?h ` (set ?U \<times> set ?U))"
-  proof-
-     from simp_num_pair_ci[where bs="x#bs"] have 
-    "\<forall>x. (?f o simp_num_pair) x = ?f x" by auto
-     hence th: "?f o simp_num_pair = ?f" using ext by blast
-    have "(?f ` set ?Y) = ((?f o simp_num_pair) ` set ?S)" by (simp add: image_compose)
-    also have "\<dots> = (?f ` set ?S)" by (simp add: th)
-    also have "\<dots> = ((?f o ?g) ` set ?Up)" 
-      by (simp only: set_map o_def image_compose[symmetric])
-    also have "\<dots> = (?h ` (set ?U \<times> set ?U))"
-      using uset_cong_aux[OF U_l, where x="x" and bs="bs", simplified set_map image_compose[symmetric]] by blast
-    finally show ?thesis .
-  qed
-  have "\<forall> (t,n) \<in> set ?Y. bound0 (simpfm (usubst ?q (t,n)))"
-  proof-
-    { fix t n assume tnY: "(t,n) \<in> set ?Y"
-      with Y_l have tnb: "numbound0 t" and np: "real n > 0" by auto
-      from usubst_I[OF lq np tnb]
-    have "bound0 (usubst ?q (t,n))"  by simp hence "bound0 (simpfm (usubst ?q (t,n)))" 
-      using simpfm_bound0 by simp}
-    thus ?thesis by blast
-  qed
-  hence ep_nb: "bound0 ?ep"  using evaldjf_bound0[where xs="?Y" and f="simpfm o (usubst ?q)"] by auto
-  let ?mp = "minusinf ?q"
-  let ?pp = "plusinf ?q"
-  let ?M = "?I x ?mp"
-  let ?P = "?I x ?pp"
-  let ?res = "disj ?mp (disj ?pp ?ep)"
-  from rminusinf_bound0[OF lq] rplusinf_bound0[OF lq] ep_nb
-  have nbth: "bound0 ?res" by auto
-
-  from conjunct1[OF rlfm_I[OF simpfm_qf[OF qf]]] simpfm  
-
-  have th: "?lhs = (\<exists> x. ?I x ?q)" by auto 
-  from th fr_equsubst[OF lq, where bs="bs" and x="x"] have lhfr: "?lhs = (?M \<or> ?P \<or> ?F ?q)"
-    by (simp only: split_def fst_conv snd_conv)
-  also have "\<dots> = (?M \<or> ?P \<or> (\<exists> (t,n) \<in> set ?Y. ?I x (simpfm (usubst ?q (t,n)))))" 
-    using uset_cong[OF lq YU U_l Y_l]  by (simp only: split_def fst_conv snd_conv simpfm) 
-  also have "\<dots> = (Ifm (x#bs) ?res)"
-    using evaldjf_ex[where ps="?Y" and bs = "x#bs" and f="simpfm o (usubst ?q)",symmetric]
-    by (simp add: split_def pair_collapse)
-  finally have lheq: "?lhs =  (Ifm bs (decr ?res))" using decr[OF nbth] by blast
-  hence lr: "?lhs = ?rhs" apply (unfold ferrack_def Let_def)
-    by (cases "?mp = T \<or> ?pp = T", auto) (simp add: disj_def)+
-  from decr_qf[OF nbth] have "qfree (ferrack p)" by (auto simp add: Let_def ferrack_def)
-  with lr show ?thesis by blast
-qed
-
-definition linrqe:: "fm \<Rightarrow> fm" where
-  "linrqe p = qelim (prep p) ferrack"
-
-theorem linrqe: "Ifm bs (linrqe p) = Ifm bs p \<and> qfree (linrqe p)"
-using ferrack qelim_ci prep
-unfolding linrqe_def by auto
-
-definition ferrack_test :: "unit \<Rightarrow> fm" where
-  "ferrack_test u = linrqe (A (A (Imp (Lt (Sub (Bound 1) (Bound 0)))
-    (E (Eq (Sub (Add (Bound 0) (Bound 2)) (Bound 1)))))))"
-
-ML {* @{code ferrack_test} () *}
-
-oracle linr_oracle = {*
-let
-
-fun num_of_term vs (t as Free (xn, xT)) = (case AList.lookup (op =) vs t
-     of NONE => error "Variable not found in the list!"
-      | SOME n => @{code Bound} n)
-  | num_of_term vs @{term "real (0::int)"} = @{code C} 0
-  | num_of_term vs @{term "real (1::int)"} = @{code C} 1
-  | num_of_term vs @{term "0::real"} = @{code C} 0
-  | num_of_term vs @{term "1::real"} = @{code C} 1
-  | num_of_term vs (Bound i) = @{code Bound} i
-  | num_of_term vs (@{term "uminus :: real \<Rightarrow> real"} $ t') = @{code Neg} (num_of_term vs t')
-  | num_of_term vs (@{term "op + :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) = @{code Add} (num_of_term vs t1, num_of_term vs t2)
-  | num_of_term vs (@{term "op - :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) = @{code Sub} (num_of_term vs t1, num_of_term vs t2)
-  | num_of_term vs (@{term "op * :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) = (case (num_of_term vs t1)
-     of @{code C} i => @{code Mul} (i, num_of_term vs t2)
-      | _ => error "num_of_term: unsupported Multiplication")
-  | num_of_term vs (@{term "real :: int \<Rightarrow> real"} $ (@{term "number_of :: int \<Rightarrow> int"} $ t')) = @{code C} (HOLogic.dest_numeral t')
-  | num_of_term vs (@{term "number_of :: int \<Rightarrow> real"} $ t') = @{code C} (HOLogic.dest_numeral t')
-  | num_of_term vs t = error ("num_of_term: unknown term " ^ Syntax.string_of_term @{context} t);
-
-fun fm_of_term vs @{term True} = @{code T}
-  | fm_of_term vs @{term False} = @{code F}
-  | fm_of_term vs (@{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) = @{code Lt} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
-  | fm_of_term vs (@{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) = @{code Le} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
-  | fm_of_term vs (@{term "op = :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) = @{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) 
-  | fm_of_term vs (@{term "op \<longleftrightarrow> :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ t1 $ t2) = @{code Iff} (fm_of_term vs t1, fm_of_term vs t2)
-  | fm_of_term vs (@{term "op &"} $ t1 $ t2) = @{code And} (fm_of_term vs t1, fm_of_term vs t2)
-  | fm_of_term vs (@{term "op |"} $ t1 $ t2) = @{code Or} (fm_of_term vs t1, fm_of_term vs t2)
-  | fm_of_term vs (@{term "op -->"} $ t1 $ t2) = @{code Imp} (fm_of_term vs t1, fm_of_term vs t2)
-  | fm_of_term vs (@{term "Not"} $ t') = @{code NOT} (fm_of_term vs t')
-  | fm_of_term vs (Const ("Ex", _) $ Abs (xn, xT, p)) =
-      @{code E} (fm_of_term (map (fn (v, n) => (v, n + 1)) vs) p)
-  | fm_of_term vs (Const ("All", _) $ Abs (xn, xT, p)) =
-      @{code A} (fm_of_term (map (fn (v, n) => (v, n + 1)) vs) p)
-  | fm_of_term vs t = error ("fm_of_term : unknown term " ^ Syntax.string_of_term @{context} t);
-
-fun term_of_num vs (@{code C} i) = @{term "real :: int \<Rightarrow> real"} $ HOLogic.mk_number HOLogic.intT i
-  | term_of_num vs (@{code Bound} n) = fst (the (find_first (fn (_, m) => n = m) vs))
-  | term_of_num vs (@{code Neg} t') = @{term "uminus :: real \<Rightarrow> real"} $ term_of_num vs t'
-  | term_of_num vs (@{code Add} (t1, t2)) = @{term "op + :: real \<Rightarrow> real \<Rightarrow> real"} $
-      term_of_num vs t1 $ term_of_num vs t2
-  | term_of_num vs (@{code Sub} (t1, t2)) = @{term "op - :: real \<Rightarrow> real \<Rightarrow> real"} $
-      term_of_num vs t1 $ term_of_num vs t2
-  | term_of_num vs (@{code Mul} (i, t2)) = @{term "op * :: real \<Rightarrow> real \<Rightarrow> real"} $
-      term_of_num vs (@{code C} i) $ term_of_num vs t2
-  | term_of_num vs (@{code CN} (n, i, t)) = term_of_num vs (@{code Add} (@{code Mul} (i, @{code Bound} n), t));
-
-fun term_of_fm vs @{code T} = HOLogic.true_const 
-  | term_of_fm vs @{code F} = HOLogic.false_const
-  | term_of_fm vs (@{code Lt} t) = @{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $
-      term_of_num vs t $ @{term "0::real"}
-  | term_of_fm vs (@{code Le} t) = @{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $
-      term_of_num vs t $ @{term "0::real"}
-  | term_of_fm vs (@{code Gt} t) = @{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $
-      @{term "0::real"} $ term_of_num vs t
-  | term_of_fm vs (@{code Ge} t) = @{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $
-      @{term "0::real"} $ term_of_num vs t
-  | term_of_fm vs (@{code Eq} t) = @{term "op = :: real \<Rightarrow> real \<Rightarrow> bool"} $
-      term_of_num vs t $ @{term "0::real"}
-  | term_of_fm vs (@{code NEq} t) = term_of_fm vs (@{code NOT} (@{code Eq} t))
-  | term_of_fm vs (@{code NOT} t') = HOLogic.Not $ term_of_fm vs t'
-  | term_of_fm vs (@{code And} (t1, t2)) = HOLogic.conj $ term_of_fm vs t1 $ term_of_fm vs t2
-  | term_of_fm vs (@{code Or} (t1, t2)) = HOLogic.disj $ term_of_fm vs t1 $ term_of_fm vs t2
-  | term_of_fm vs (@{code Imp}  (t1, t2)) = HOLogic.imp $ term_of_fm vs t1 $ term_of_fm vs t2
-  | term_of_fm vs (@{code Iff} (t1, t2)) = @{term "op \<longleftrightarrow> :: bool \<Rightarrow> bool \<Rightarrow> bool"} $
-      term_of_fm vs t1 $ term_of_fm vs t2
-  | term_of_fm vs _ = error "If this is raised, Isabelle/HOL or generate_code is inconsistent.";
-
-in fn ct =>
-  let 
-    val thy = Thm.theory_of_cterm ct;
-    val t = Thm.term_of ct;
-    val fs = OldTerm.term_frees t;
-    val vs = fs ~~ (0 upto (length fs - 1));
-    val res = HOLogic.mk_Trueprop (HOLogic.mk_eq (t, term_of_fm vs (@{code linrqe} (fm_of_term vs t))));
-  in Thm.cterm_of thy res end
-end;
-*}
-
-use "ferrack_tac.ML"
-setup Ferrack_Tac.setup
-
-lemma
-  fixes x :: real
-  shows "2 * x \<le> 2 * x \<and> 2 * x \<le> 2 * x + 1"
-apply rferrack
-done
-
-lemma
-  fixes x :: real
-  shows "\<exists>y \<le> x. x = y + 1"
-apply rferrack
-done
-
-lemma
-  fixes x :: real
-  shows "\<not> (\<exists>z. x + z = x + z + 1)"
-apply rferrack
-done
-
-end
diff -r b8a6b9c56fdd -r 5170d6277b61 src/HOL/Reflection/MIR.thy
--- a/src/HOL/Reflection/MIR.thy	Thu Feb 12 21:24:14 2009 -0800
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,5933 +0,0 @@
-(*  Title:      HOL/Reflection/MIR.thy
-    Author:     Amine Chaieb
-*)
-
-theory MIR
-imports Complex_Main Dense_Linear_Order Efficient_Nat
-uses ("mir_tac.ML")
-begin
-
-section {* Quantifier elimination for @{text "\<real> (0, 1, +, floor, <)"} *}
-
-declare real_of_int_floor_cancel [simp del]
-
-primrec alluopairs:: "'a list \<Rightarrow> ('a \<times> 'a) list" where 
-  "alluopairs [] = []"
-| "alluopairs (x#xs) = (map (Pair x) (x#xs))@(alluopairs xs)"
-
-lemma alluopairs_set1: "set (alluopairs xs) \<le> {(x,y). x\<in> set xs \<and> y\<in> set xs}"
-by (induct xs, auto)
-
-lemma alluopairs_set:
-  "\<lbrakk>x\<in> set xs ; y \<in> set xs\<rbrakk> \<Longrightarrow> (x,y) \<in> set (alluopairs xs) \<or> (y,x) \<in> set (alluopairs xs) "
-by (induct xs, auto)
-
-lemma alluopairs_ex:
-  assumes Pc: "\<forall> x y. P x y = P y x"
-  shows "(\<exists> x \<in> set xs. \<exists> y \<in> set xs. P x y) = (\<exists> (x,y) \<in> set (alluopairs xs). P x y)"
-proof
-  assume "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y"
-  then obtain x y where x: "x \<in> set xs" and y:"y \<in> set xs" and P: "P x y"  by blast
-  from alluopairs_set[OF x y] P Pc show"\<exists>(x, y)\<in>set (alluopairs xs). P x y" 
-    by auto
-next
-  assume "\<exists>(x, y)\<in>set (alluopairs xs). P x y"
-  then obtain "x" and "y"  where xy:"(x,y) \<in> set (alluopairs xs)" and P: "P x y" by blast+
-  from xy have "x \<in> set xs \<and> y\<in> set xs" using alluopairs_set1 by blast
-  with P show "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y" by blast
-qed
-
-  (* generate a list from i to j*)
-consts iupt :: "int \<times> int \<Rightarrow> int list"
-recdef iupt "measure (\<lambda> (i,j). nat (j-i +1))" 
-  "iupt (i,j) = (if j <i then [] else (i# iupt(i+1, j)))"
-
-lemma iupt_set: "set (iupt(i,j)) = {i .. j}"
-proof(induct rule: iupt.induct)
-  case (1 a b)
-  show ?case
-    using prems by (simp add: simp_from_to)
-qed
-
-lemma nth_pos2: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
-using Nat.gr0_conv_Suc
-by clarsimp
-
-
-lemma myl: "\<forall> (a::'a::{pordered_ab_group_add}) (b::'a). (a \<le> b) = (0 \<le> b - a)" 
-proof(clarify)
-  fix x y ::"'a"
-  have "(x \<le> y) = (x - y \<le> 0)" by (simp only: le_iff_diff_le_0[where a="x" and b="y"])
-  also have "\<dots> = (- (y - x) \<le> 0)" by simp
-  also have "\<dots> = (0 \<le> y - x)" by (simp only: neg_le_0_iff_le[where a="y-x"])
-  finally show "(x \<le> y) = (0 \<le> y - x)" .
-qed
-
-lemma myless: "\<forall> (a::'a::{pordered_ab_group_add}) (b::'a). (a < b) = (0 < b - a)" 
-proof(clarify)
-  fix x y ::"'a"
-  have "(x < y) = (x - y < 0)" by (simp only: less_iff_diff_less_0[where a="x" and b="y"])
-  also have "\<dots> = (- (y - x) < 0)" by simp
-  also have "\<dots> = (0 < y - x)" by (simp only: neg_less_0_iff_less[where a="y-x"])
-  finally show "(x < y) = (0 < y - x)" .
-qed
-
-lemma myeq: "\<forall> (a::'a::{pordered_ab_group_add}) (b::'a). (a = b) = (0 = b - a)"
-  by auto
-
-  (* Maybe should be added to the library \<dots> *)
-lemma floor_int_eq: "(real n\<le> x \<and> x < real (n+1)) = (floor x = n)"
-proof( auto)
-  assume lb: "real n \<le> x"
-    and ub: "x < real n + 1"
-  have "real (floor x) \<le> x" by simp 
-  hence "real (floor x) < real (n + 1) " using ub by arith
-  hence "floor x < n+1" by simp
-  moreover from lb have "n \<le> floor x" using floor_mono2[where x="real n" and y="x"] 
-    by simp ultimately show "floor x = n" by simp
-qed
-
-(* Periodicity of dvd *)
-lemma dvd_period:
-  assumes advdd: "(a::int) dvd d"
-  shows "(a dvd (x + t)) = (a dvd ((x+ c*d) + t))"
-  using advdd  
-proof-
-  {fix x k
-    from inf_period(3)[OF advdd, rule_format, where x=x and k="-k"]  
-    have " ((a::int) dvd (x + t)) = (a dvd (x+k*d + t))" by simp}
-  hence "\<forall>x.\<forall>k. ((a::int) dvd (x + t)) = (a dvd (x+k*d + t))"  by simp
-  then show ?thesis by simp
-qed
-
-  (* The Divisibility relation between reals *)	
-definition
-  rdvd:: "real \<Rightarrow> real \<Rightarrow> bool" (infixl "rdvd" 50)
-where
-  rdvd_def: "x rdvd y \<longleftrightarrow> (\<exists>k\<Colon>int. y = x * real k)"
-
-lemma int_rdvd_real: 
-  shows "real (i::int) rdvd x = (i dvd (floor x) \<and> real (floor x) = x)" (is "?l = ?r")
-proof
-  assume "?l" 
-  hence th: "\<exists> k. x=real (i*k)" by (simp add: rdvd_def)
-  hence th': "real (floor x) = x" by (auto simp del: real_of_int_mult)
-  with th have "\<exists> k. real (floor x) = real (i*k)" by simp
-  hence "\<exists> k. floor x = i*k" by (simp only: real_of_int_inject)
-  thus ?r  using th' by (simp add: dvd_def) 
-next
-  assume "?r" hence "(i\<Colon>int) dvd \<lfloor>x\<Colon>real\<rfloor>" ..
-  hence "\<exists> k. real (floor x) = real (i*k)" 
-    by (simp only: real_of_int_inject) (simp add: dvd_def)
-  thus ?l using prems by (simp add: rdvd_def)
-qed
-
-lemma int_rdvd_iff: "(real (i::int) rdvd real t) = (i dvd t)"
-by (auto simp add: rdvd_def dvd_def) (rule_tac x="k" in exI, simp only :real_of_int_mult[symmetric])
-
-
-lemma rdvd_abs1: 
-  "(abs (real d) rdvd t) = (real (d ::int) rdvd t)"
-proof
-  assume d: "real d rdvd t"
-  from d int_rdvd_real have d2: "d dvd (floor t)" and ti: "real (floor t) = t" by auto
-
-  from iffD2[OF zdvd_abs1] d2 have "(abs d) dvd (floor t)" by blast
-  with ti int_rdvd_real[symmetric] have "real (abs d) rdvd t" by blast 
-  thus "abs (real d) rdvd t" by simp
-next
-  assume "abs (real d) rdvd t" hence "real (abs d) rdvd t" by simp
-  with int_rdvd_real[where i="abs d" and x="t"] have d2: "abs d dvd floor t" and ti: "real (floor t) =t" by auto
-  from iffD1[OF zdvd_abs1] d2 have "d dvd floor t" by blast
-  with ti int_rdvd_real[symmetric] show "real d rdvd t" by blast
-qed
-
-lemma rdvd_minus: "(real (d::int) rdvd t) = (real d rdvd -t)"
-  apply (auto simp add: rdvd_def)
-  apply (rule_tac x="-k" in exI, simp) 
-  apply (rule_tac x="-k" in exI, simp)
-done
-
-lemma rdvd_left_0_eq: "(0 rdvd t) = (t=0)"
-by (auto simp add: rdvd_def)
-
-lemma rdvd_mult: 
-  assumes knz: "k\<noteq>0"
-  shows "(real (n::int) * real (k::int) rdvd x * real k) = (real n rdvd x)"
-using knz by (simp add:rdvd_def)
-
-lemma rdvd_trans: assumes mn:"m rdvd n" and  nk:"n rdvd k" 
-  shows "m rdvd k"
-proof-
-  from rdvd_def mn obtain c where nmc:"n = m * real (c::int)" by auto
-  from rdvd_def nk obtain c' where nkc:"k = n * real (c'::int)" by auto
-  hence "k = m * real (c * c')" using nmc by simp
-  thus ?thesis using rdvd_def by blast
-qed
-
-  (*********************************************************************************)
-  (****                            SHADOW SYNTAX AND SEMANTICS                  ****)
-  (*********************************************************************************)
-
-datatype num = C int | Bound nat | CN nat int num | Neg num | Add num num| Sub num num 
-  | Mul int num | Floor num| CF int num num
-
-  (* A size for num to make inductive proofs simpler*)
-primrec num_size :: "num \<Rightarrow> nat" where
- "num_size (C c) = 1"
-| "num_size (Bound n) = 1"
-| "num_size (Neg a) = 1 + num_size a"
-| "num_size (Add a b) = 1 + num_size a + num_size b"
-| "num_size (Sub a b) = 3 + num_size a + num_size b"
-| "num_size (CN n c a) = 4 + num_size a "
-| "num_size (CF c a b) = 4 + num_size a + num_size b"
-| "num_size (Mul c a) = 1 + num_size a"
-| "num_size (Floor a) = 1 + num_size a"
-
-  (* Semantics of numeral terms (num) *)
-primrec Inum :: "real list \<Rightarrow> num \<Rightarrow> real" where
-  "Inum bs (C c) = (real c)"
-| "Inum bs (Bound n) = bs!n"
-| "Inum bs (CN n c a) = (real c) * (bs!n) + (Inum bs a)"
-| "Inum bs (Neg a) = -(Inum bs a)"
-| "Inum bs (Add a b) = Inum bs a + Inum bs b"
-| "Inum bs (Sub a b) = Inum bs a - Inum bs b"
-| "Inum bs (Mul c a) = (real c) * Inum bs a"
-| "Inum bs (Floor a) = real (floor (Inum bs a))"
-| "Inum bs (CF c a b) = real c * real (floor (Inum bs a)) + Inum bs b"
-definition "isint t bs \<equiv> real (floor (Inum bs t)) = Inum bs t"
-
-lemma isint_iff: "isint n bs = (real (floor (Inum bs n)) = Inum bs n)"
-by (simp add: isint_def)
-
-lemma isint_Floor: "isint (Floor n) bs"
-  by (simp add: isint_iff)
-
-lemma isint_Mul: "isint e bs \<Longrightarrow> isint (Mul c e) bs"
-proof-
-  let ?e = "Inum bs e"
-  let ?fe = "floor ?e"
-  assume be: "isint e bs" hence efe:"real ?fe = ?e" by (simp add: isint_iff)
-  have "real ((floor (Inum bs (Mul c e)))) = real (floor (real (c * ?fe)))" using efe by simp
-  also have "\<dots> = real (c* ?fe)" by (simp only: floor_real_of_int) 
-  also have "\<dots> = real c * ?e" using efe by simp
-  finally show ?thesis using isint_iff by simp
-qed
-
-lemma isint_neg: "isint e bs \<Longrightarrow> isint (Neg e) bs"
-proof-
-  let ?I = "\<lambda> t. Inum bs t"
-  assume ie: "isint e bs"
-  hence th: "real (floor (?I e)) = ?I e" by (simp add: isint_def)  
-  have "real (floor (?I (Neg e))) = real (floor (- (real (floor (?I e)))))" by (simp add: th)
-  also have "\<dots> = - real (floor (?I e))" by(simp add: floor_minus_real_of_int) 
-  finally show "isint (Neg e) bs" by (simp add: isint_def th)
-qed
-
-lemma isint_sub: 
-  assumes ie: "isint e bs" shows "isint (Sub (C c) e) bs"
-proof-
-  let ?I = "\<lambda> t. Inum bs t"
-  from ie have th: "real (floor (?I e)) = ?I e" by (simp add: isint_def)  
-  have "real (floor (?I (Sub (C c) e))) = real (floor ((real (c -floor (?I e)))))" by (simp add: th)
-  also have "\<dots> = real (c- floor (?I e))" by(simp add: floor_minus_real_of_int) 
-  finally show "isint (Sub (C c) e) bs" by (simp add: isint_def th)
-qed
-
-lemma isint_add: assumes
-  ai:"isint a bs" and bi: "isint b bs" shows "isint (Add a b) bs"
-proof-
-  let ?a = "Inum bs a"
-  let ?b = "Inum bs b"
-  from ai bi isint_iff have "real (floor (?a + ?b)) = real (floor (real (floor ?a) + real (floor ?b)))" by simp
-  also have "\<dots> = real (floor ?a) + real (floor ?b)" by simp
-  also have "\<dots> = ?a + ?b" using ai bi isint_iff by simp
-  finally show "isint (Add a b) bs" by (simp add: isint_iff)
-qed
-
-lemma isint_c: "isint (C j) bs"
-  by (simp add: isint_iff)
-
-
-    (* FORMULAE *)
-datatype fm  = 
-  T| F| Lt num| Le num| Gt num| Ge num| Eq num| NEq num| Dvd int num| NDvd int num|
-  NOT fm| And fm fm|  Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm
-
-
-  (* A size for fm *)
-fun fmsize :: "fm \<Rightarrow> nat" where
- "fmsize (NOT p) = 1 + fmsize p"
-| "fmsize (And p q) = 1 + fmsize p + fmsize q"
-| "fmsize (Or p q) = 1 + fmsize p + fmsize q"
-| "fmsize (Imp p q) = 3 + fmsize p + fmsize q"
-| "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)"
-| "fmsize (E p) = 1 + fmsize p"
-| "fmsize (A p) = 4+ fmsize p"
-| "fmsize (Dvd i t) = 2"
-| "fmsize (NDvd i t) = 2"
-| "fmsize p = 1"
-  (* several lemmas about fmsize *)
-lemma fmsize_pos: "fmsize p > 0"	
-by (induct p rule: fmsize.induct) simp_all
-
-  (* Semantics of formulae (fm) *)
-primrec Ifm ::"real list \<Rightarrow> fm \<Rightarrow> bool" where
-  "Ifm bs T = True"
-| "Ifm bs F = False"
-| "Ifm bs (Lt a) = (Inum bs a < 0)"
-| "Ifm bs (Gt a) = (Inum bs a > 0)"
-| "Ifm bs (Le a) = (Inum bs a \<le> 0)"
-| "Ifm bs (Ge a) = (Inum bs a \<ge> 0)"
-| "Ifm bs (Eq a) = (Inum bs a = 0)"
-| "Ifm bs (NEq a) = (Inum bs a \<noteq> 0)"
-| "Ifm bs (Dvd i b) = (real i rdvd Inum bs b)"
-| "Ifm bs (NDvd i b) = (\<not>(real i rdvd Inum bs b))"
-| "Ifm bs (NOT p) = (\<not> (Ifm bs p))"
-| "Ifm bs (And p q) = (Ifm bs p \<and> Ifm bs q)"
-| "Ifm bs (Or p q) = (Ifm bs p \<or> Ifm bs q)"
-| "Ifm bs (Imp p q) = ((Ifm bs p) \<longrightarrow> (Ifm bs q))"
-| "Ifm bs (Iff p q) = (Ifm bs p = Ifm bs q)"
-| "Ifm bs (E p) = (\<exists> x. Ifm (x#bs) p)"
-| "Ifm bs (A p) = (\<forall> x. Ifm (x#bs) p)"
-
-consts prep :: "fm \<Rightarrow> fm"
-recdef prep "measure fmsize"
-  "prep (E T) = T"
-  "prep (E F) = F"
-  "prep (E (Or p q)) = Or (prep (E p)) (prep (E q))"
-  "prep (E (Imp p q)) = Or (prep (E (NOT p))) (prep (E q))"
-  "prep (E (Iff p q)) = Or (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))" 
-  "prep (E (NOT (And p q))) = Or (prep (E (NOT p))) (prep (E(NOT q)))"
-  "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))"
-  "prep (E (NOT (Iff p q))) = Or (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))"
-  "prep (E p) = E (prep p)"
-  "prep (A (And p q)) = And (prep (A p)) (prep (A q))"
-  "prep (A p) = prep (NOT (E (NOT p)))"
-  "prep (NOT (NOT p)) = prep p"
-  "prep (NOT (And p q)) = Or (prep (NOT p)) (prep (NOT q))"
-  "prep (NOT (A p)) = prep (E (NOT p))"
-  "prep (NOT (Or p q)) = And (prep (NOT p)) (prep (NOT q))"
-  "prep (NOT (Imp p q)) = And (prep p) (prep (NOT q))"
-  "prep (NOT (Iff p q)) = Or (prep (And p (NOT q))) (prep (And (NOT p) q))"
-  "prep (NOT p) = NOT (prep p)"
-  "prep (Or p q) = Or (prep p) (prep q)"
-  "prep (And p q) = And (prep p) (prep q)"
-  "prep (Imp p q) = prep (Or (NOT p) q)"
-  "prep (Iff p q) = Or (prep (And p q)) (prep (And (NOT p) (NOT q)))"
-  "prep p = p"
-(hints simp add: fmsize_pos)
-lemma prep: "\<And> bs. Ifm bs (prep p) = Ifm bs p"
-by (induct p rule: prep.induct, auto)
-
-
-  (* Quantifier freeness *)
-fun qfree:: "fm \<Rightarrow> bool" where
-  "qfree (E p) = False"
-  | "qfree (A p) = False"
-  | "qfree (NOT p) = qfree p" 
-  | "qfree (And p q) = (qfree p \<and> qfree q)" 
-  | "qfree (Or  p q) = (qfree p \<and> qfree q)" 
-  | "qfree (Imp p q) = (qfree p \<and> qfree q)" 
-  | "qfree (Iff p q) = (qfree p \<and> qfree q)"
-  | "qfree p = True"
-
-  (* Boundedness and substitution *)
-primrec numbound0 :: "num \<Rightarrow> bool" (* a num is INDEPENDENT of Bound 0 *) where
-  "numbound0 (C c) = True"
-  | "numbound0 (Bound n) = (n>0)"
-  | "numbound0 (CN n i a) = (n > 0 \<and> numbound0 a)"
-  | "numbound0 (Neg a) = numbound0 a"
-  | "numbound0 (Add a b) = (numbound0 a \<and> numbound0 b)"
-  | "numbound0 (Sub a b) = (numbound0 a \<and> numbound0 b)" 
-  | "numbound0 (Mul i a) = numbound0 a"
-  | "numbound0 (Floor a) = numbound0 a"
-  | "numbound0 (CF c a b) = (numbound0 a \<and> numbound0 b)" 
-
-lemma numbound0_I:
-  assumes nb: "numbound0 a"
-  shows "Inum (b#bs) a = Inum (b'#bs) a"
-  using nb by (induct a) (auto simp add: nth_pos2)
-
-lemma numbound0_gen: 
-  assumes nb: "numbound0 t" and ti: "isint t (x#bs)"
-  shows "\<forall> y. isint t (y#bs)"
-using nb ti 
-proof(clarify)
-  fix y
-  from numbound0_I[OF nb, where bs="bs" and b="y" and b'="x"] ti[simplified isint_def]
-  show "isint t (y#bs)"
-    by (simp add: isint_def)
-qed
-
-primrec bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *) where
-  "bound0 T = True"
-  | "bound0 F = True"
-  | "bound0 (Lt a) = numbound0 a"
-  | "bound0 (Le a) = numbound0 a"
-  | "bound0 (Gt a) = numbound0 a"
-  | "bound0 (Ge a) = numbound0 a"
-  | "bound0 (Eq a) = numbound0 a"
-  | "bound0 (NEq a) = numbound0 a"
-  | "bound0 (Dvd i a) = numbound0 a"
-  | "bound0 (NDvd i a) = numbound0 a"
-  | "bound0 (NOT p) = bound0 p"
-  | "bound0 (And p q) = (bound0 p \<and> bound0 q)"
-  | "bound0 (Or p q) = (bound0 p \<and> bound0 q)"
-  | "bound0 (Imp p q) = ((bound0 p) \<and> (bound0 q))"
-  | "bound0 (Iff p q) = (bound0 p \<and> bound0 q)"
-  | "bound0 (E p) = False"
-  | "bound0 (A p) = False"
-
-lemma bound0_I:
-  assumes bp: "bound0 p"
-  shows "Ifm (b#bs) p = Ifm (b'#bs) p"
- using bp numbound0_I [where b="b" and bs="bs" and b'="b'"]
-  by (induct p) (auto simp add: nth_pos2)
-
-primrec numsubst0:: "num \<Rightarrow> num \<Rightarrow> num" (* substitute a num into a num for Bound 0 *) where
-  "numsubst0 t (C c) = (C c)"
-  | "numsubst0 t (Bound n) = (if n=0 then t else Bound n)"
-  | "numsubst0 t (CN n i a) = (if n=0 then Add (Mul i t) (numsubst0 t a) else CN n i (numsubst0 t a))"
-  | "numsubst0 t (CF i a b) = CF i (numsubst0 t a) (numsubst0 t b)"
-  | "numsubst0 t (Neg a) = Neg (numsubst0 t a)"
-  | "numsubst0 t (Add a b) = Add (numsubst0 t a) (numsubst0 t b)"
-  | "numsubst0 t (Sub a b) = Sub (numsubst0 t a) (numsubst0 t b)" 
-  | "numsubst0 t (Mul i a) = Mul i (numsubst0 t a)"
-  | "numsubst0 t (Floor a) = Floor (numsubst0 t a)"
-
-lemma numsubst0_I:
-  shows "Inum (b#bs) (numsubst0 a t) = Inum ((Inum (b#bs) a)#bs) t"
-  by (induct t) (simp_all add: nth_pos2)
-
-lemma numsubst0_I':
-  assumes nb: "numbound0 a"
-  shows "Inum (b#bs) (numsubst0 a t) = Inum ((Inum (b'#bs) a)#bs) t"
-  by (induct t) (simp_all add: nth_pos2 numbound0_I[OF nb, where b="b" and b'="b'"])
-
-primrec subst0:: "num \<Rightarrow> fm \<Rightarrow> fm" (* substitue a num into a formula for Bound 0 *) where
-  "subst0 t T = T"
-  | "subst0 t F = F"
-  | "subst0 t (Lt a) = Lt (numsubst0 t a)"
-  | "subst0 t (Le a) = Le (numsubst0 t a)"
-  | "subst0 t (Gt a) = Gt (numsubst0 t a)"
-  | "subst0 t (Ge a) = Ge (numsubst0 t a)"
-  | "subst0 t (Eq a) = Eq (numsubst0 t a)"
-  | "subst0 t (NEq a) = NEq (numsubst0 t a)"
-  | "subst0 t (Dvd i a) = Dvd i (numsubst0 t a)"
-  | "subst0 t (NDvd i a) = NDvd i (numsubst0 t a)"
-  | "subst0 t (NOT p) = NOT (subst0 t p)"
-  | "subst0 t (And p q) = And (subst0 t p) (subst0 t q)"
-  | "subst0 t (Or p q) = Or (subst0 t p) (subst0 t q)"
-  | "subst0 t (Imp p q) = Imp (subst0 t p) (subst0 t q)"
-  | "subst0 t (Iff p q) = Iff (subst0 t p) (subst0 t q)"
-
-lemma subst0_I: assumes qfp: "qfree p"
-  shows "Ifm (b#bs) (subst0 a p) = Ifm ((Inum (b#bs) a)#bs) p"
-  using qfp numsubst0_I[where b="b" and bs="bs" and a="a"]
-  by (induct p) (simp_all add: nth_pos2 )
-
-consts
-  decrnum:: "num \<Rightarrow> num" 
-  decr :: "fm \<Rightarrow> fm"
-
-recdef decrnum "measure size"
-  "decrnum (Bound n) = Bound (n - 1)"
-  "decrnum (Neg a) = Neg (decrnum a)"
-  "decrnum (Add a b) = Add (decrnum a) (decrnum b)"
-  "decrnum (Sub a b) = Sub (decrnum a) (decrnum b)"
-  "decrnum (Mul c a) = Mul c (decrnum a)"
-  "decrnum (Floor a) = Floor (decrnum a)"
-  "decrnum (CN n c a) = CN (n - 1) c (decrnum a)"
-  "decrnum (CF c a b) = CF c (decrnum a) (decrnum b)"
-  "decrnum a = a"
-
-recdef decr "measure size"
-  "decr (Lt a) = Lt (decrnum a)"
-  "decr (Le a) = Le (decrnum a)"
-  "decr (Gt a) = Gt (decrnum a)"
-  "decr (Ge a) = Ge (decrnum a)"
-  "decr (Eq a) = Eq (decrnum a)"
-  "decr (NEq a) = NEq (decrnum a)"
-  "decr (Dvd i a) = Dvd i (decrnum a)"
-  "decr (NDvd i a) = NDvd i (decrnum a)"
-  "decr (NOT p) = NOT (decr p)" 
-  "decr (And p q) = And (decr p) (decr q)"
-  "decr (Or p q) = Or (decr p) (decr q)"
-  "decr (Imp p q) = Imp (decr p) (decr q)"
-  "decr (Iff p q) = Iff (decr p) (decr q)"
-  "decr p = p"
-
-lemma decrnum: assumes nb: "numbound0 t"
-  shows "Inum (x#bs) t = Inum bs (decrnum t)"
-  using nb by (induct t rule: decrnum.induct, simp_all add: nth_pos2)
-
-lemma decr: assumes nb: "bound0 p"
-  shows "Ifm (x#bs) p = Ifm bs (decr p)"
-  using nb 
-  by (induct p rule: decr.induct, simp_all add: nth_pos2 decrnum)
-
-lemma decr_qf: "bound0 p \<Longrightarrow> qfree (decr p)"
-by (induct p, simp_all)
-
-consts 
-  isatom :: "fm \<Rightarrow> bool" (* test for atomicity *)
-recdef isatom "measure size"
-  "isatom T = True"
-  "isatom F = True"
-  "isatom (Lt a) = True"
-  "isatom (Le a) = True"
-  "isatom (Gt a) = True"
-  "isatom (Ge a) = True"
-  "isatom (Eq a) = True"
-  "isatom (NEq a) = True"
-  "isatom (Dvd i b) = True"
-  "isatom (NDvd i b) = True"
-  "isatom p = False"
-
-lemma numsubst0_numbound0: assumes nb: "numbound0 t"
-  shows "numbound0 (numsubst0 t a)"
-using nb by (induct a, auto)
-
-lemma subst0_bound0: assumes qf: "qfree p" and nb: "numbound0 t"
-  shows "bound0 (subst0 t p)"
-using qf numsubst0_numbound0[OF nb] by (induct p, auto)
-
-lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p"
-by (induct p, simp_all)
-
-
-definition djf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm" where
-  "djf f p q = (if q=T then T else if q=F then f p else 
-  (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or fp q))"
-
-definition evaldjf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm" where
-  "evaldjf f ps = foldr (djf f) ps F"
-
-lemma djf_Or: "Ifm bs (djf f p q) = Ifm bs (Or (f p) q)"
-by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def) 
-(cases "f p", simp_all add: Let_def djf_def) 
-
-lemma evaldjf_ex: "Ifm bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm bs (f p))"
-  by(induct ps, simp_all add: evaldjf_def djf_Or)
-
-lemma evaldjf_bound0: 
-  assumes nb: "\<forall> x\<in> set xs. bound0 (f x)"
-  shows "bound0 (evaldjf f xs)"
-  using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 
-
-lemma evaldjf_qf: 
-  assumes nb: "\<forall> x\<in> set xs. qfree (f x)"
-  shows "qfree (evaldjf f xs)"
-  using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 
-
-consts 
-  disjuncts :: "fm \<Rightarrow> fm list" 
-  conjuncts :: "fm \<Rightarrow> fm list"
-recdef disjuncts "measure size"
-  "disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)"
-  "disjuncts F = []"
-  "disjuncts p = [p]"
-
-recdef conjuncts "measure size"
-  "conjuncts (And p q) = (conjuncts p) @ (conjuncts q)"
-  "conjuncts T = []"
-  "conjuncts p = [p]"
-lemma disjuncts: "(\<exists> q\<in> set (disjuncts p). Ifm bs q) = Ifm bs p"
-by(induct p rule: disjuncts.induct, auto)
-lemma conjuncts: "(\<forall> q\<in> set (conjuncts p). Ifm bs q) = Ifm bs p"
-by(induct p rule: conjuncts.induct, auto)
-
-lemma disjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). bound0 q"
-proof-
-  assume nb: "bound0 p"
-  hence "list_all bound0 (disjuncts p)" by (induct p rule:disjuncts.induct,auto)
-  thus ?thesis by (simp only: list_all_iff)
-qed
-lemma conjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (conjuncts p). bound0 q"
-proof-
-  assume nb: "bound0 p"
-  hence "list_all bound0 (conjuncts p)" by (induct p rule:conjuncts.induct,auto)
-  thus ?thesis by (simp only: list_all_iff)
-qed
-
-lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). qfree q"
-proof-
-  assume qf: "qfree p"
-  hence "list_all qfree (disjuncts p)"
-    by (induct p rule: disjuncts.induct, auto)
-  thus ?thesis by (simp only: list_all_iff)
-qed
-lemma conjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (conjuncts p). qfree q"
-proof-
-  assume qf: "qfree p"
-  hence "list_all qfree (conjuncts p)"
-    by (induct p rule: conjuncts.induct, auto)
-  thus ?thesis by (simp only: list_all_iff)
-qed
-
-constdefs DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm"
-  "DJ f p \<equiv> evaldjf f (disjuncts p)"
-
-lemma DJ: assumes fdj: "\<forall> p q. f (Or p q) = Or (f p) (f q)"
-  and fF: "f F = F"
-  shows "Ifm bs (DJ f p) = Ifm bs (f p)"
-proof-
-  have "Ifm bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm bs (f q))"
-    by (simp add: DJ_def evaldjf_ex) 
-  also have "\<dots> = Ifm bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto)
-  finally show ?thesis .
-qed
-
-lemma DJ_qf: assumes 
-  fqf: "\<forall> p. qfree p \<longrightarrow> qfree (f p)"
-  shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) "
-proof(clarify)
-  fix  p assume qf: "qfree p"
-  have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def)
-  from disjuncts_qf[OF qf] have "\<forall> q\<in> set (disjuncts p). qfree q" .
-  with fqf have th':"\<forall> q\<in> set (disjuncts p). qfree (f q)" by blast
-  
-  from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp
-qed
-
-lemma DJ_qe: assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))"
-  shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm bs ((DJ qe p)) = Ifm bs (E p))"
-proof(clarify)
-  fix p::fm and bs
-  assume qf: "qfree p"
-  from qe have qth: "\<forall> p. qfree p \<longrightarrow> qfree (qe p)" by blast
-  from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto
-  have "Ifm bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (qe q))"
-    by (simp add: DJ_def evaldjf_ex)
-  also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm bs (E q))" using qe disjuncts_qf[OF qf] by auto
-  also have "\<dots> = Ifm bs (E p)" by (induct p rule: disjuncts.induct, auto)
-  finally show "qfree (DJ qe p) \<and> Ifm bs (DJ qe p) = Ifm bs (E p)" using qfth by blast
-qed
-  (* Simplification *)
-
-  (* Algebraic simplifications for nums *)
-consts bnds:: "num \<Rightarrow> nat list"
-  lex_ns:: "nat list \<times> nat list \<Rightarrow> bool"
-recdef bnds "measure size"
-  "bnds (Bound n) = [n]"
-  "bnds (CN n c a) = n#(bnds a)"
-  "bnds (Neg a) = bnds a"
-  "bnds (Add a b) = (bnds a)@(bnds b)"
-  "bnds (Sub a b) = (bnds a)@(bnds b)"
-  "bnds (Mul i a) = bnds a"
-  "bnds (Floor a) = bnds a"
-  "bnds (CF c a b) = (bnds a)@(bnds b)"
-  "bnds a = []"
-recdef lex_ns "measure (\<lambda> (xs,ys). length xs + length ys)"
-  "lex_ns ([], ms) = True"
-  "lex_ns (ns, []) = False"
-  "lex_ns (n#ns, m#ms) = (n<m \<or> ((n = m) \<and> lex_ns (ns,ms))) "
-constdefs lex_bnd :: "num \<Rightarrow> num \<Rightarrow> bool"
-  "lex_bnd t s \<equiv> lex_ns (bnds t, bnds s)"
-
-consts 
-  numgcdh:: "num \<Rightarrow> int \<Rightarrow> int"
-  reducecoeffh:: "num \<Rightarrow> int \<Rightarrow> num"
-  dvdnumcoeff:: "num \<Rightarrow> int \<Rightarrow> bool"
-consts maxcoeff:: "num \<Rightarrow> int"
-recdef maxcoeff "measure size"
-  "maxcoeff (C i) = abs i"
-  "maxcoeff (CN n c t) = max (abs c) (maxcoeff t)"
-  "maxcoeff (CF c t s) = max (abs c) (maxcoeff s)"
-  "maxcoeff t = 1"
-
-lemma maxcoeff_pos: "maxcoeff t \<ge> 0"
-  apply (induct t rule: maxcoeff.induct, auto) 
-  done
-
-recdef numgcdh "measure size"
-  "numgcdh (C i) = (\<lambda>g. zgcd i g)"
-  "numgcdh (CN n c t) = (\<lambda>g. zgcd c (numgcdh t g))"
-  "numgcdh (CF c s t) = (\<lambda>g. zgcd c (numgcdh t g))"
-  "numgcdh t = (\<lambda>g. 1)"
-
-definition
-  numgcd :: "num \<Rightarrow> int"
-where
-  numgcd_def: "numgcd t = numgcdh t (maxcoeff t)"
-
-recdef reducecoeffh "measure size"
-  "reducecoeffh (C i) = (\<lambda> g. C (i div g))"
-  "reducecoeffh (CN n c t) = (\<lambda> g. CN n (c div g) (reducecoeffh t g))"
-  "reducecoeffh (CF c s t) = (\<lambda> g. CF (c div g)  s (reducecoeffh t g))"
-  "reducecoeffh t = (\<lambda>g. t)"
-
-definition
-  reducecoeff :: "num \<Rightarrow> num"
-where
-  reducecoeff_def: "reducecoeff t =
-  (let g = numgcd t in 
-  if g = 0 then C 0 else if g=1 then t else reducecoeffh t g)"
-
-recdef dvdnumcoeff "measure size"
-  "dvdnumcoeff (C i) = (\<lambda> g. g dvd i)"
-  "dvdnumcoeff (CN n c t) = (\<lambda> g. g dvd c \<and> (dvdnumcoeff t g))"
-  "dvdnumcoeff (CF c s t) = (\<lambda> g. g dvd c \<and> (dvdnumcoeff t g))"
-  "dvdnumcoeff t = (\<lambda>g. False)"
-
-lemma dvdnumcoeff_trans: 
-  assumes gdg: "g dvd g'" and dgt':"dvdnumcoeff t g'"
-  shows "dvdnumcoeff t g"
-  using dgt' gdg 
-  by (induct t rule: dvdnumcoeff.induct, simp_all add: gdg zdvd_trans[OF gdg])
-
-declare zdvd_trans [trans add]
-
-lemma natabs0: "(nat (abs x) = 0) = (x = 0)"
-by arith
-
-lemma numgcd0:
-  assumes g0: "numgcd t = 0"
-  shows "Inum bs t = 0"
-proof-
-  have "\<And>x. numgcdh t x= 0 \<Longrightarrow> Inum bs t = 0"
-    by (induct t rule: numgcdh.induct, auto simp add: zgcd_def gcd_zero natabs0 max_def maxcoeff_pos)
-  thus ?thesis using g0[simplified numgcd_def] by blast
-qed
-
-lemma numgcdh_pos: assumes gp: "g \<ge> 0" shows "numgcdh t g \<ge> 0"
-  using gp
-  by (induct t rule: numgcdh.induct, auto simp add: zgcd_def)
-
-lemma numgcd_pos: "numgcd t \<ge>0"
-  by (simp add: numgcd_def numgcdh_pos maxcoeff_pos)
-
-lemma reducecoeffh:
-  assumes gt: "dvdnumcoeff t g" and gp: "g > 0" 
-  shows "real g *(Inum bs (reducecoeffh t g)) = Inum bs t"
-  using gt
-proof(induct t rule: reducecoeffh.induct) 
-  case (1 i) hence gd: "g dvd i" by simp
-  from gp have gnz: "g \<noteq> 0" by simp
-  from prems show ?case by (simp add: real_of_int_div[OF gnz gd])
-next
-  case (2 n c t)  hence gd: "g dvd c" by simp
-  from gp have gnz: "g \<noteq> 0" by simp
-  from prems show ?case by (simp add: real_of_int_div[OF gnz gd] algebra_simps)
-next
-  case (3 c s t)  hence gd: "g dvd c" by simp
-  from gp have gnz: "g \<noteq> 0" by simp
-  from prems show ?case by (simp add: real_of_int_div[OF gnz gd] algebra_simps) 
-qed (auto simp add: numgcd_def gp)
-consts ismaxcoeff:: "num \<Rightarrow> int \<Rightarrow> bool"
-recdef ismaxcoeff "measure size"
-  "ismaxcoeff (C i) = (\<lambda> x. abs i \<le> x)"
-  "ismaxcoeff (CN n c t) = (\<lambda>x. abs c \<le> x \<and> (ismaxcoeff t x))"
-  "ismaxcoeff (CF c s t) = (\<lambda>x. abs c \<le> x \<and> (ismaxcoeff t x))"
-  "ismaxcoeff t = (\<lambda>x. True)"
-
-lemma ismaxcoeff_mono: "ismaxcoeff t c \<Longrightarrow> c \<le> c' \<Longrightarrow> ismaxcoeff t c'"
-by (induct t rule: ismaxcoeff.induct, auto)
-
-lemma maxcoeff_ismaxcoeff: "ismaxcoeff t (maxcoeff t)"
-proof (induct t rule: maxcoeff.induct)
-  case (2 n c t)
-  hence H:"ismaxcoeff t (maxcoeff t)" .
-  have thh: "maxcoeff t \<le> max (abs c) (maxcoeff t)" by (simp add: le_maxI2)
-  from ismaxcoeff_mono[OF H thh] show ?case by (simp add: le_maxI1)
-next
-  case (3 c t s) 
-  hence H1:"ismaxcoeff s (maxcoeff s)" by auto
-  have thh1: "maxcoeff s \<le> max \<bar>c\<bar> (maxcoeff s)" by (simp add: max_def)
-  from ismaxcoeff_mono[OF H1 thh1] show ?case by (simp add: le_maxI1)
-qed simp_all
-
-lemma zgcd_gt1: "zgcd i j > 1 \<Longrightarrow> ((abs i > 1 \<and> abs j > 1) \<or> (abs i = 0 \<and> abs j > 1) \<or> (abs i > 1 \<and> abs j = 0))"
-  apply (unfold zgcd_def)
-  apply (cases "i = 0", simp_all)
-  apply (cases "j = 0", simp_all)
-  apply (cases "abs i = 1", simp_all)
-  apply (cases "abs j = 1", simp_all)
-  apply auto
-  done
-lemma numgcdh0:"numgcdh t m = 0 \<Longrightarrow>  m =0"
-  by (induct t rule: numgcdh.induct, auto simp add:zgcd0)
-
-lemma dvdnumcoeff_aux:
-  assumes "ismaxcoeff t m" and mp:"m \<ge> 0" and "numgcdh t m > 1"
-  shows "dvdnumcoeff t (numgcdh t m)"
-using prems
-proof(induct t rule: numgcdh.induct)
-  case (2 n c t) 
-  let ?g = "numgcdh t m"
-  from prems have th:"zgcd c ?g > 1" by simp
-  from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"]
-  have "(abs c > 1 \<and> ?g > 1) \<or> (abs c = 0 \<and> ?g > 1) \<or> (abs c > 1 \<and> ?g = 0)" by simp
-  moreover {assume "abs c > 1" and gp: "?g > 1" with prems
-    have th: "dvdnumcoeff t ?g" by simp
-    have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2)
-    from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1)}
-  moreover {assume "abs c = 0 \<and> ?g > 1"
-    with prems have th: "dvdnumcoeff t ?g" by simp
-    have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2)
-    from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1)
-    hence ?case by simp }
-  moreover {assume "abs c > 1" and g0:"?g = 0" 
-    from numgcdh0[OF g0] have "m=0". with prems   have ?case by simp }
-  ultimately show ?case by blast
-next
-  case (3 c s t) 
-  let ?g = "numgcdh t m"
-  from prems have th:"zgcd c ?g > 1" by simp
-  from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"]
-  have "(abs c > 1 \<and> ?g > 1) \<or> (abs c = 0 \<and> ?g > 1) \<or> (abs c > 1 \<and> ?g = 0)" by simp
-  moreover {assume "abs c > 1" and gp: "?g > 1" with prems
-    have th: "dvdnumcoeff t ?g" by simp
-    have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2)
-    from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1)}
-  moreover {assume "abs c = 0 \<and> ?g > 1"
-    with prems have th: "dvdnumcoeff t ?g" by simp
-    have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2)
-    from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1)
-    hence ?case by simp }
-  moreover {assume "abs c > 1" and g0:"?g = 0" 
-    from numgcdh0[OF g0] have "m=0". with prems   have ?case by simp }
-  ultimately show ?case by blast
-qed(auto simp add: zgcd_zdvd1)
-
-lemma dvdnumcoeff_aux2:
-  assumes "numgcd t > 1" shows "dvdnumcoeff t (numgcd t) \<and> numgcd t > 0"
-  using prems 
-proof (simp add: numgcd_def)
-  let ?mc = "maxcoeff t"
-  let ?g = "numgcdh t ?mc"
-  have th1: "ismaxcoeff t ?mc" by (rule maxcoeff_ismaxcoeff)
-  have th2: "?mc \<ge> 0" by (rule maxcoeff_pos)
-  assume H: "numgcdh t ?mc > 1"
-  from dvdnumcoeff_aux[OF th1 th2 H]  show "dvdnumcoeff t ?g" .
-qed
-
-lemma reducecoeff: "real (numgcd t) * (Inum bs (reducecoeff t)) = Inum bs t"
-proof-
-  let ?g = "numgcd t"
-  have "?g \<ge> 0"  by (simp add: numgcd_pos)
-  hence	"?g = 0 \<or> ?g = 1 \<or> ?g > 1" by auto
-  moreover {assume "?g = 0" hence ?thesis by (simp add: numgcd0)} 
-  moreover {assume "?g = 1" hence ?thesis by (simp add: reducecoeff_def)} 
-  moreover { assume g1:"?g > 1"
-    from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff t ?g" and g0: "?g > 0" by blast+
-    from reducecoeffh[OF th1 g0, where bs="bs"] g1 have ?thesis 
-      by (simp add: reducecoeff_def Let_def)} 
-  ultimately show ?thesis by blast
-qed
-
-lemma reducecoeffh_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeffh t g)"
-by (induct t rule: reducecoeffh.induct, auto)
-
-lemma reducecoeff_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeff t)"
-using reducecoeffh_numbound0 by (simp add: reducecoeff_def Let_def)
-
-consts
-  simpnum:: "num \<Rightarrow> num"
-  numadd:: "num \<times> num \<Rightarrow> num"
-  nummul:: "num \<Rightarrow> int \<Rightarrow> num"
-
-recdef numadd "measure (\<lambda> (t,s). size t + size s)"
-  "numadd (CN n1 c1 r1,CN n2 c2 r2) =
-  (if n1=n2 then 
-  (let c = c1 + c2
-  in (if c=0 then numadd(r1,r2) else CN n1 c (numadd (r1,r2))))
-  else if n1 \<le> n2 then CN n1 c1 (numadd (r1,CN n2 c2 r2))
-  else (CN n2 c2 (numadd (CN n1 c1 r1,r2))))"
-  "numadd (CN n1 c1 r1,t) = CN n1 c1 (numadd (r1, t))"  
-  "numadd (t,CN n2 c2 r2) = CN n2 c2 (numadd (t,r2))" 
-  "numadd (CF c1 t1 r1,CF c2 t2 r2) = 
-   (if t1 = t2 then 
-    (let c=c1+c2; s= numadd(r1,r2) in (if c=0 then s else CF c t1 s))
-   else if lex_bnd t1 t2 then CF c1 t1 (numadd(r1,CF c2 t2 r2))
-   else CF c2 t2 (numadd(CF c1 t1 r1,r2)))"
-  "numadd (CF c1 t1 r1,C c) = CF c1 t1 (numadd (r1, C c))"
-  "numadd (C c,CF c1 t1 r1) = CF c1 t1 (numadd (r1, C c))"
-  "numadd (C b1, C b2) = C (b1+b2)"
-  "numadd (a,b) = Add a b"
-
-lemma numadd[simp]: "Inum bs (numadd (t,s)) = Inum bs (Add t s)"
-apply (induct t s rule: numadd.induct, simp_all add: Let_def)
- apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all)
-  apply (case_tac "n1 = n2", simp_all add: algebra_simps)
-  apply (simp only: left_distrib[symmetric])
- apply simp
-apply (case_tac "lex_bnd t1 t2", simp_all)
- apply (case_tac "c1+c2 = 0")
-  by (case_tac "t1 = t2", simp_all add: algebra_simps left_distrib[symmetric] real_of_int_mult[symmetric] real_of_int_add[symmetric]del: real_of_int_mult real_of_int_add left_distrib)
-
-lemma numadd_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numadd (t,s))"
-by (induct t s rule: numadd.induct, auto simp add: Let_def)
-
-recdef nummul "measure size"
-  "nummul (C j) = (\<lambda> i. C (i*j))"
-  "nummul (CN n c t) = (\<lambda> i. CN n (c*i) (nummul t i))"
-  "nummul (CF c t s) = (\<lambda> i. CF (c*i) t (nummul s i))"
-  "nummul (Mul c t) = (\<lambda> i. nummul t (i*c))"
-  "nummul t = (\<lambda> i. Mul i t)"
-
-lemma nummul[simp]: "\<And> i. Inum bs (nummul t i) = Inum bs (Mul i t)"
-by (induct t rule: nummul.induct, auto simp add: algebra_simps)
-
-lemma nummul_nb[simp]: "\<And> i. numbound0 t \<Longrightarrow> numbound0 (nummul t i)"
-by (induct t rule: nummul.induct, auto)
-
-constdefs numneg :: "num \<Rightarrow> num"
-  "numneg t \<equiv> nummul t (- 1)"
-
-constdefs numsub :: "num \<Rightarrow> num \<Rightarrow> num"
-  "numsub s t \<equiv> (if s = t then C 0 else numadd (s,numneg t))"
-
-lemma numneg[simp]: "Inum bs (numneg t) = Inum bs (Neg t)"
-using numneg_def nummul by simp
-
-lemma numneg_nb[simp]: "numbound0 t \<Longrightarrow> numbound0 (numneg t)"
-using numneg_def by simp
-
-lemma numsub[simp]: "Inum bs (numsub a b) = Inum bs (Sub a b)"
-using numsub_def by simp
-
-lemma numsub_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numsub t s)"
-using numsub_def by simp
-
-lemma isint_CF: assumes si: "isint s bs" shows "isint (CF c t s) bs"
-proof-
-  have cti: "isint (Mul c (Floor t)) bs" by (simp add: isint_Mul isint_Floor)
-  
-  have "?thesis = isint (Add (Mul c (Floor t)) s) bs" by (simp add: isint_def)
-  also have "\<dots>" by (simp add: isint_add cti si)
-  finally show ?thesis .
-qed
-
-consts split_int:: "num \<Rightarrow> num\<times>num"
-recdef split_int "measure num_size"
-  "split_int (C c) = (C 0, C c)"
-  "split_int (CN n c b) = 
-     (let (bv,bi) = split_int b 
-       in (CN n c bv, bi))"
-  "split_int (CF c a b) = 
-     (let (bv,bi) = split_int b 
-       in (bv, CF c a bi))"
-  "split_int a = (a,C 0)"
-
-lemma split_int:"\<And> tv ti. split_int t = (tv,ti) \<Longrightarrow> (Inum bs (Add tv ti) = Inum bs t) \<and> isint ti bs"
-proof (induct t rule: split_int.induct)
-  case (2 c n b tv ti)
-  let ?bv = "fst (split_int b)"
-  let ?bi = "snd (split_int b)"
-  have "split_int b = (?bv,?bi)" by simp
-  with prems(1) have b:"Inum bs (Add ?bv ?bi) = Inum bs b" and bii: "isint ?bi bs" by blast+
-  from prems(2) have tibi: "ti = ?bi" by (simp add: Let_def split_def)
-  from prems(2) b[symmetric] bii show ?case by (auto simp add: Let_def split_def)
-next
-  case (3 c a b tv ti) 
-  let ?bv = "fst (split_int b)"
-  let ?bi = "snd (split_int b)"
-  have "split_int b = (?bv,?bi)" by simp
-  with prems(1) have b:"Inum bs (Add ?bv ?bi) = Inum bs b" and bii: "isint ?bi bs" by blast+
-  from prems(2) have tibi: "ti = CF c a ?bi" by (simp add: Let_def split_def)
-  from prems(2) b[symmetric] bii show ?case by (auto simp add: Let_def split_def isint_Floor isint_add isint_Mul isint_CF)
-qed (auto simp add: Let_def isint_iff isint_Floor isint_add isint_Mul split_def algebra_simps)
-
-lemma split_int_nb: "numbound0 t \<Longrightarrow> numbound0 (fst (split_int t)) \<and> numbound0 (snd (split_int t)) "
-by (induct t rule: split_int.induct, auto simp add: Let_def split_def)
-
-definition
-  numfloor:: "num \<Rightarrow> num"
-where
-  numfloor_def: "numfloor t = (let (tv,ti) = split_int t in 
-  (case tv of C i \<Rightarrow> numadd (tv,ti) 
-  | _ \<Rightarrow> numadd(CF 1 tv (C 0),ti)))"
-
-lemma numfloor[simp]: "Inum bs (numfloor t) = Inum bs (Floor t)" (is "?n t = ?N (Floor t)")
-proof-
-  let ?tv = "fst (split_int t)"
-  let ?ti = "snd (split_int t)"
-  have tvti:"split_int t = (?tv,?ti)" by simp
-  {assume H: "\<forall> v. ?tv \<noteq> C v"
-    hence th1: "?n t = ?N (Add (Floor ?tv) ?ti)" 
-      by (cases ?tv, auto simp add: numfloor_def Let_def split_def numadd)
-    from split_int[OF tvti] have "?N (Floor t) = ?N (Floor(Add ?tv ?ti))" and tii:"isint ?ti bs" by simp+
-    hence "?N (Floor t) = real (floor (?N (Add ?tv ?ti)))" by simp 
-    also have "\<dots> = real (floor (?N ?tv) + (floor (?N ?ti)))"
-      by (simp,subst tii[simplified isint_iff, symmetric]) simp
-    also have "\<dots> = ?N (Add (Floor ?tv) ?ti)" by (simp add: tii[simplified isint_iff])
-    finally have ?thesis using th1 by simp}
-  moreover {fix v assume H:"?tv = C v" 
-    from split_int[OF tvti] have "?N (Floor t) = ?N (Floor(Add ?tv ?ti))" and tii:"isint ?ti bs" by simp+
-    hence "?N (Floor t) = real (floor (?N (Add ?tv ?ti)))" by simp 
-    also have "\<dots> = real (floor (?N ?tv) + (floor (?N ?ti)))"
-      by (simp,subst tii[simplified isint_iff, symmetric]) simp
-    also have "\<dots> = ?N (Add (Floor ?tv) ?ti)" by (simp add: tii[simplified isint_iff])
-    finally have ?thesis by (simp add: H numfloor_def Let_def split_def numadd) }
-  ultimately show ?thesis by auto
-qed
-
-lemma numfloor_nb[simp]: "numbound0 t \<Longrightarrow> numbound0 (numfloor t)"
-  using split_int_nb[where t="t"]
-  by (cases "fst(split_int t)" , auto simp add: numfloor_def Let_def split_def  numadd_nb)
-
-recdef simpnum "measure num_size"
-  "simpnum (C j) = C j"
-  "simpnum (Bound n) = CN n 1 (C 0)"
-  "simpnum (Neg t) = numneg (simpnum t)"
-  "simpnum (Add t s) = numadd (simpnum t,simpnum s)"
-  "simpnum (Sub t s) = numsub (simpnum t) (simpnum s)"
-  "simpnum (Mul i t) = (if i = 0 then (C 0) else nummul (simpnum t) i)"
-  "simpnum (Floor t) = numfloor (simpnum t)"
-  "simpnum (CN n c t) = (if c=0 then simpnum t else CN n c (simpnum t))"
-  "simpnum (CF c t s) = simpnum(Add (Mul c (Floor t)) s)"
-
-lemma simpnum_ci[simp]: "Inum bs (simpnum t) = Inum bs t"
-by (induct t rule: simpnum.induct, auto)
-
-lemma simpnum_numbound0[simp]: 
-  "numbound0 t \<Longrightarrow> numbound0 (simpnum t)"
-by (induct t rule: simpnum.induct, auto)
-
-consts nozerocoeff:: "num \<Rightarrow> bool"
-recdef nozerocoeff "measure size"
-  "nozerocoeff (C c) = True"
-  "nozerocoeff (CN n c t) = (c\<noteq>0 \<and> nozerocoeff t)"
-  "nozerocoeff (CF c s t) = (c \<noteq> 0 \<and> nozerocoeff t)"
-  "nozerocoeff (Mul c t) = (c\<noteq>0 \<and> nozerocoeff t)"
-  "nozerocoeff t = True"
-
-lemma numadd_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numadd (a,b))"
-by (induct a b rule: numadd.induct,auto simp add: Let_def)
-
-lemma nummul_nz : "\<And> i. i\<noteq>0 \<Longrightarrow> nozerocoeff a \<Longrightarrow> nozerocoeff (nummul a i)"
-  by (induct a rule: nummul.induct,auto simp add: Let_def numadd_nz)
-
-lemma numneg_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff (numneg a)"
-by (simp add: numneg_def nummul_nz)
-
-lemma numsub_nz: "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numsub a b)"
-by (simp add: numsub_def numneg_nz numadd_nz)
-
-lemma split_int_nz: "nozerocoeff t \<Longrightarrow> nozerocoeff (fst (split_int t)) \<and> nozerocoeff (snd (split_int t))"
-by (induct t rule: split_int.induct,auto simp add: Let_def split_def)
-
-lemma numfloor_nz: "nozerocoeff t \<Longrightarrow> nozerocoeff (numfloor t)"
-by (simp add: numfloor_def Let_def split_def)
-(cases "fst (split_int t)", simp_all add: split_int_nz numadd_nz)
-
-lemma simpnum_nz: "nozerocoeff (simpnum t)"
-by(induct t rule: simpnum.induct, auto simp add: numadd_nz numneg_nz numsub_nz nummul_nz numfloor_nz)
-
-lemma maxcoeff_nz: "nozerocoeff t \<Longrightarrow> maxcoeff t = 0 \<Longrightarrow> t = C 0"
-proof (induct t rule: maxcoeff.induct)
-  case (2 n c t)
-  hence cnz: "c \<noteq>0" and mx: "max (abs c) (maxcoeff t) = 0" by simp+
-  have "max (abs c) (maxcoeff t) \<ge> abs c" by (simp add: le_maxI1)
-  with cnz have "max (abs c) (maxcoeff t) > 0" by arith
-  with prems show ?case by simp
-next
-  case (3 c s t) 
-  hence cnz: "c \<noteq>0" and mx: "max (abs c) (maxcoeff t) = 0" by simp+
-  have "max (abs c) (maxcoeff t) \<ge> abs c" by (simp add: le_maxI1)
-  with cnz have "max (abs c) (maxcoeff t) > 0" by arith
-  with prems show ?case by simp
-qed auto
-
-lemma numgcd_nz: assumes nz: "nozerocoeff t" and g0: "numgcd t = 0" shows "t = C 0"
-proof-
-  from g0 have th:"numgcdh t (maxcoeff t) = 0" by (simp add: numgcd_def)
-  from numgcdh0[OF th]  have th:"maxcoeff t = 0" .
-  from maxcoeff_nz[OF nz th] show ?thesis .
-qed
-
-constdefs simp_num_pair:: "(num \<times> int) \<Rightarrow> num \<times> int"
-  "simp_num_pair \<equiv> (\<lambda> (t,n). (if n = 0 then (C 0, 0) else
-   (let t' = simpnum t ; g = numgcd t' in 
-      if g > 1 then (let g' = zgcd n g in 
-        if g' = 1 then (t',n) 
-        else (reducecoeffh t' g', n div g')) 
-      else (t',n))))"
-
-lemma simp_num_pair_ci:
-  shows "((\<lambda> (t,n). Inum bs t / real n) (simp_num_pair (t,n))) = ((\<lambda> (t,n). Inum bs t / real n) (t,n))"
-  (is "?lhs = ?rhs")
-proof-
-  let ?t' = "simpnum t"
-  let ?g = "numgcd ?t'"
-  let ?g' = "zgcd n ?g"
-  {assume nz: "n = 0" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
-  moreover
-  { assume nnz: "n \<noteq> 0"
-    {assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
-    moreover
-    {assume g1:"?g>1" hence g0: "?g > 0" by simp
-      from zgcd0 g1 nnz have gp0: "?g' \<noteq> 0" by simp
-      hence g'p: "?g' > 0" using zgcd_pos[where i="n" and j="numgcd ?t'"] by arith
-      hence "?g'= 1 \<or> ?g' > 1" by arith
-      moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
-      moreover {assume g'1:"?g'>1"
-	from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff ?t' ?g" ..
-	let ?tt = "reducecoeffh ?t' ?g'"
-	let ?t = "Inum bs ?tt"
-	have gpdg: "?g' dvd ?g" by (simp add: zgcd_zdvd2)
-	have gpdd: "?g' dvd n" by (simp add: zgcd_zdvd1) 
-	have gpdgp: "?g' dvd ?g'" by simp
-	from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p] 
-	have th2:"real ?g' * ?t = Inum bs ?t'" by simp
-	from prems have "?lhs = ?t / real (n div ?g')" by (simp add: simp_num_pair_def Let_def)
-	also have "\<dots> = (real ?g' * ?t) / (real ?g' * (real (n div ?g')))" by simp
-	also have "\<dots> = (Inum bs ?t' / real n)"
-	  using real_of_int_div[OF gp0 gpdd] th2 gp0 by simp
-	finally have "?lhs = Inum bs t / real n" by simp
-	then have ?thesis using prems by (simp add: simp_num_pair_def)}
-      ultimately have ?thesis by blast}
-    ultimately have ?thesis by blast} 
-  ultimately show ?thesis by blast
-qed
-
-lemma simp_num_pair_l: assumes tnb: "numbound0 t" and np: "n >0" and tn: "simp_num_pair (t,n) = (t',n')"
-  shows "numbound0 t' \<and> n' >0"
-proof-
-    let ?t' = "simpnum t"
-  let ?g = "numgcd ?t'"
-  let ?g' = "zgcd n ?g"
-  {assume nz: "n = 0" hence ?thesis using prems by (simp add: Let_def simp_num_pair_def)}
-  moreover
-  { assume nnz: "n \<noteq> 0"
-    {assume "\<not> ?g > 1" hence ?thesis  using prems by (auto simp add: Let_def simp_num_pair_def)}
-    moreover
-    {assume g1:"?g>1" hence g0: "?g > 0" by simp
-      from zgcd0 g1 nnz have gp0: "?g' \<noteq> 0" by simp
-      hence g'p: "?g' > 0" using zgcd_pos[where i="n" and j="numgcd ?t'"] by arith
-      hence "?g'= 1 \<or> ?g' > 1" by arith
-      moreover {assume "?g'=1" hence ?thesis using prems 
-	  by (auto simp add: Let_def simp_num_pair_def)}
-      moreover {assume g'1:"?g'>1"
-	have gpdg: "?g' dvd ?g" by (simp add: zgcd_zdvd2)
-	have gpdd: "?g' dvd n" by (simp add: zgcd_zdvd1) 
-	have gpdgp: "?g' dvd ?g'" by simp
-	from zdvd_imp_le[OF gpdd np] have g'n: "?g' \<le> n" .
-	from zdiv_mono1[OF g'n g'p, simplified zdiv_self[OF gp0]]
-	have "n div ?g' >0" by simp
-	hence ?thesis using prems 
-	  by(auto simp add: simp_num_pair_def Let_def reducecoeffh_numbound0)}
-      ultimately have ?thesis by blast}
-    ultimately have ?thesis by blast} 
-  ultimately show ?thesis by blast
-qed
-
-consts not:: "fm \<Rightarrow> fm"
-recdef not "measure size"
-  "not (NOT p) = p"
-  "not T = F"
-  "not F = T"
-  "not (Lt t) = Ge t"
-  "not (Le t) = Gt t"
-  "not (Gt t) = Le t"
-  "not (Ge t) = Lt t"
-  "not (Eq t) = NEq t"
-  "not (NEq t) = Eq t"
-  "not (Dvd i t) = NDvd i t"
-  "not (NDvd i t) = Dvd i t"
-  "not (And p q) = Or (not p) (not q)"
-  "not (Or p q) = And (not p) (not q)"
-  "not p = NOT p"
-lemma not[simp]: "Ifm bs (not p) = Ifm bs (NOT p)"
-by (induct p) auto
-lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)"
-by (induct p, auto)
-lemma not_nb[simp]: "bound0 p \<Longrightarrow> bound0 (not p)"
-by (induct p, auto)
-
-constdefs conj :: "fm \<Rightarrow> fm \<Rightarrow> fm"
-  "conj p q \<equiv> (if (p = F \<or> q=F) then F else if p=T then q else if q=T then p else 
-   if p = q then p else And p q)"
-lemma conj[simp]: "Ifm bs (conj p q) = Ifm bs (And p q)"
-by (cases "p=F \<or> q=F",simp_all add: conj_def) (cases p,simp_all)
-
-lemma conj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)"
-using conj_def by auto 
-lemma conj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)"
-using conj_def by auto 
-
-constdefs disj :: "fm \<Rightarrow> fm \<Rightarrow> fm"
-  "disj p q \<equiv> (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p 
-       else if p=q then p else Or p q)"
-
-lemma disj[simp]: "Ifm bs (disj p q) = Ifm bs (Or p q)"
-by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all)
-lemma disj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)"
-using disj_def by auto 
-lemma disj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)"
-using disj_def by auto 
-
-constdefs   imp :: "fm \<Rightarrow> fm \<Rightarrow> fm"
-  "imp p q \<equiv> (if (p = F \<or> q=T \<or> p=q) then T else if p=T then q else if q=F then not p 
-    else Imp p q)"
-lemma imp[simp]: "Ifm bs (imp p q) = Ifm bs (Imp p q)"
-by (cases "p=F \<or> q=T",simp_all add: imp_def)
-lemma imp_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (imp p q)"
-using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def)
-lemma imp_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (imp p q)"
-using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def) 
-
-constdefs   iff :: "fm \<Rightarrow> fm \<Rightarrow> fm"
-  "iff p q \<equiv> (if (p = q) then T else if (p = not q \<or> not p = q) then F else 
-       if p=F then not q else if q=F then not p else if p=T then q else if q=T then p else 
-  Iff p q)"
-lemma iff[simp]: "Ifm bs (iff p q) = Ifm bs (Iff p q)"
-  by (unfold iff_def,cases "p=q", simp,cases "p=not q", simp add:not) 
-(cases "not p= q", auto simp add:not)
-lemma iff_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)"
-  by (unfold iff_def,cases "p=q", auto)
-lemma iff_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (iff p q)"
-using iff_def by (unfold iff_def,cases "p=q", auto)
-
-consts check_int:: "num \<Rightarrow> bool"
-recdef check_int "measure size"
-  "check_int (C i) = True"
-  "check_int (Floor t) = True"
-  "check_int (Mul i t) = check_int t"
-  "check_int (Add t s) = (check_int t \<and> check_int s)"
-  "check_int (Neg t) = check_int t"
-  "check_int (CF c t s) = check_int s"
-  "check_int t = False"
-lemma check_int: "check_int t \<Longrightarrow> isint t bs"
-by (induct t, auto simp add: isint_add isint_Floor isint_Mul isint_neg isint_c isint_CF)
-
-lemma rdvd_left1_int: "real \<lfloor>t\<rfloor> = t \<Longrightarrow> 1 rdvd t"
-  by (simp add: rdvd_def,rule_tac x="\<lfloor>t\<rfloor>" in exI) simp
-
-lemma rdvd_reduce: 
-  assumes gd:"g dvd d" and gc:"g dvd c" and gp: "g > 0"
-  shows "real (d::int) rdvd real (c::int)*t = (real (d div g) rdvd real (c div g)*t)"
-proof
-  assume d: "real d rdvd real c * t"
-  from d rdvd_def obtain k where k_def: "real c * t = real d* real (k::int)" by auto
-  from gd dvd_def obtain kd where kd_def: "d = g * kd" by auto
-  from gc dvd_def obtain kc where kc_def: "c = g * kc" by auto
-  from k_def kd_def kc_def have "real g * real kc * t = real g * real kd * real k" by simp
-  hence "real kc * t = real kd * real k" using gp by simp
-  hence th:"real kd rdvd real kc * t" using rdvd_def by blast
-  from kd_def gp have th':"kd = d div g" by simp
-  from kc_def gp have "kc = c div g" by simp
-  with th th' show "real (d div g) rdvd real (c div g) * t" by simp
-next
-  assume d: "real (d div g) rdvd real (c div g) * t"
-  from gp have gnz: "g \<noteq> 0" by simp
-  thus "real d rdvd real c * t" using d rdvd_mult[OF gnz, where n="d div g" and x="real (c div g) * t"] real_of_int_div[OF gnz gd] real_of_int_div[OF gnz gc] by simp
-qed
-
-constdefs simpdvd:: "int \<Rightarrow> num \<Rightarrow> (int \<times> num)"
-  "simpdvd d t \<equiv> 
-   (let g = numgcd t in 
-      if g > 1 then (let g' = zgcd d g in 
-        if g' = 1 then (d, t) 
-        else (d div g',reducecoeffh t g')) 
-      else (d, t))"
-lemma simpdvd: 
-  assumes tnz: "nozerocoeff t" and dnz: "d \<noteq> 0"
-  shows "Ifm bs (Dvd (fst (simpdvd d t)) (snd (simpdvd d t))) = Ifm bs (Dvd d t)"
-proof-
-  let ?g = "numgcd t"
-  let ?g' = "zgcd d ?g"
-  {assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simpdvd_def)}
-  moreover
-  {assume g1:"?g>1" hence g0: "?g > 0" by simp
-    from zgcd0 g1 dnz have gp0: "?g' \<noteq> 0" by simp
-    hence g'p: "?g' > 0" using zgcd_pos[where i="d" and j="numgcd t"] by arith
-    hence "?g'= 1 \<or> ?g' > 1" by arith
-    moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simpdvd_def)}
-    moreover {assume g'1:"?g'>1"
-      from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff t ?g" ..
-      let ?tt = "reducecoeffh t ?g'"
-      let ?t = "Inum bs ?tt"
-      have gpdg: "?g' dvd ?g" by (simp add: zgcd_zdvd2)
-      have gpdd: "?g' dvd d" by (simp add: zgcd_zdvd1) 
-      have gpdgp: "?g' dvd ?g'" by simp
-      from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p] 
-      have th2:"real ?g' * ?t = Inum bs t" by simp
-      from prems have "Ifm bs (Dvd (fst (simpdvd d t)) (snd(simpdvd d t))) = Ifm bs (Dvd (d div ?g') ?tt)"
-	by (simp add: simpdvd_def Let_def)
-      also have "\<dots> = (real d rdvd (Inum bs t))"
-	using rdvd_reduce[OF gpdd gpdgp g'p, where t="?t", simplified zdiv_self[OF gp0]] 
-	  th2[symmetric] by simp
-      finally have ?thesis by simp  }
-    ultimately have ?thesis by blast
-  }
-  ultimately show ?thesis by blast
-qed
-
-consts simpfm :: "fm \<Rightarrow> fm"
-recdef simpfm "measure fmsize"
-  "simpfm (And p q) = conj (simpfm p) (simpfm q)"
-  "simpfm (Or p q) = disj (simpfm p) (simpfm q)"
-  "simpfm (Imp p q) = imp (simpfm p) (simpfm q)"
-  "simpfm (Iff p q) = iff (simpfm p) (simpfm q)"
-  "simpfm (NOT p) = not (simpfm p)"
-  "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v < 0) then T else F 
-  | _ \<Rightarrow> Lt (reducecoeff a'))"
-  "simpfm (Le a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<le> 0)  then T else F | _ \<Rightarrow> Le (reducecoeff a'))"
-  "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v > 0)  then T else F | _ \<Rightarrow> Gt (reducecoeff a'))"
-  "simpfm (Ge a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<ge> 0)  then T else F | _ \<Rightarrow> Ge (reducecoeff a'))"
-  "simpfm (Eq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v = 0)  then T else F | _ \<Rightarrow> Eq (reducecoeff a'))"
-  "simpfm (NEq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<noteq> 0)  then T else F | _ \<Rightarrow> NEq (reducecoeff a'))"
-  "simpfm (Dvd i a) = (if i=0 then simpfm (Eq a)
-             else if (abs i = 1) \<and> check_int a then T
-             else let a' = simpnum a in case a' of C v \<Rightarrow> if (i dvd v)  then T else F | _ \<Rightarrow> (let (d,t) = simpdvd i a' in Dvd d t))"
-  "simpfm (NDvd i a) = (if i=0 then simpfm (NEq a) 
-             else if (abs i = 1) \<and> check_int a then F
-             else let a' = simpnum a in case a' of C v \<Rightarrow> if (\<not>(i dvd v)) then T else F | _ \<Rightarrow> (let (d,t) = simpdvd i a' in NDvd d t))"
-  "simpfm p = p"
-
-lemma simpfm[simp]: "Ifm bs (simpfm p) = Ifm bs p"
-proof(induct p rule: simpfm.induct)
-  case (6 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp
-  {fix v assume "?sa = C v" hence ?case using sa by simp }
-  moreover {assume H:"\<not> (\<exists> v. ?sa = C v)"
-    let ?g = "numgcd ?sa"
-    let ?rsa = "reducecoeff ?sa"
-    let ?r = "Inum bs ?rsa"
-    have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
-    {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto}
-    with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto)
-    hence gp: "real ?g > 0" by simp
-    have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff)
-    with sa have "Inum bs a < 0 = (real ?g * ?r < real ?g * 0)" by simp
-    also have "\<dots> = (?r < 0)" using gp
-      by (simp only: mult_less_cancel_left) simp
-    finally have ?case using H by (cases "?sa" , simp_all add: Let_def)}
-  ultimately show ?case by blast
-next
-  case (7 a)  let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp
-  {fix v assume "?sa = C v" hence ?case using sa by simp }
-  moreover {assume H:"\<not> (\<exists> v. ?sa = C v)"
-    let ?g = "numgcd ?sa"
-    let ?rsa = "reducecoeff ?sa"
-    let ?r = "Inum bs ?rsa"
-    have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
-    {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto}
-    with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto)
-    hence gp: "real ?g > 0" by simp
-    have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff)
-    with sa have "Inum bs a \<le> 0 = (real ?g * ?r \<le> real ?g * 0)" by simp
-    also have "\<dots> = (?r \<le> 0)" using gp
-      by (simp only: mult_le_cancel_left) simp
-    finally have ?case using H by (cases "?sa" , simp_all add: Let_def)}
-  ultimately show ?case by blast
-next
-  case (8 a)  let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp
-  {fix v assume "?sa = C v" hence ?case using sa by simp }
-  moreover {assume H:"\<not> (\<exists> v. ?sa = C v)"
-    let ?g = "numgcd ?sa"
-    let ?rsa = "reducecoeff ?sa"
-    let ?r = "Inum bs ?rsa"
-    have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
-    {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto}
-    with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto)
-    hence gp: "real ?g > 0" by simp
-    have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff)
-    with sa have "Inum bs a > 0 = (real ?g * ?r > real ?g * 0)" by simp
-    also have "\<dots> = (?r > 0)" using gp
-      by (simp only: mult_less_cancel_left) simp
-    finally have ?case using H by (cases "?sa" , simp_all add: Let_def)}
-  ultimately show ?case by blast
-next
-  case (9 a)  let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp
-  {fix v assume "?sa = C v" hence ?case using sa by simp }
-  moreover {assume H:"\<not> (\<exists> v. ?sa = C v)"
-    let ?g = "numgcd ?sa"
-    let ?rsa = "reducecoeff ?sa"
-    let ?r = "Inum bs ?rsa"
-    have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
-    {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto}
-    with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto)
-    hence gp: "real ?g > 0" by simp
-    have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff)
-    with sa have "Inum bs a \<ge> 0 = (real ?g * ?r \<ge> real ?g * 0)" by simp
-    also have "\<dots> = (?r \<ge> 0)" using gp
-      by (simp only: mult_le_cancel_left) simp
-    finally have ?case using H by (cases "?sa" , simp_all add: Let_def)}
-  ultimately show ?case by blast
-next
-  case (10 a)  let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp
-  {fix v assume "?sa = C v" hence ?case using sa by simp }
-  moreover {assume H:"\<not> (\<exists> v. ?sa = C v)"
-    let ?g = "numgcd ?sa"
-    let ?rsa = "reducecoeff ?sa"
-    let ?r = "Inum bs ?rsa"
-    have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
-    {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto}
-    with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto)
-    hence gp: "real ?g > 0" by simp
-    have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff)
-    with sa have "Inum bs a = 0 = (real ?g * ?r = 0)" by simp
-    also have "\<dots> = (?r = 0)" using gp
-      by (simp add: mult_eq_0_iff)
-    finally have ?case using H by (cases "?sa" , simp_all add: Let_def)}
-  ultimately show ?case by blast
-next
-  case (11 a)  let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp
-  {fix v assume "?sa = C v" hence ?case using sa by simp }
-  moreover {assume H:"\<not> (\<exists> v. ?sa = C v)"
-    let ?g = "numgcd ?sa"
-    let ?rsa = "reducecoeff ?sa"
-    let ?r = "Inum bs ?rsa"
-    have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
-    {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto}
-    with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto)
-    hence gp: "real ?g > 0" by simp
-    have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff)
-    with sa have "Inum bs a \<noteq> 0 = (real ?g * ?r \<noteq> 0)" by simp
-    also have "\<dots> = (?r \<noteq> 0)" using gp
-      by (simp add: mult_eq_0_iff)
-    finally have ?case using H by (cases "?sa" , simp_all add: Let_def)}
-  ultimately show ?case by blast
-next
-  case (12 i a)  let ?sa = "simpnum a"   have sa: "Inum bs ?sa = Inum bs a" by simp
-  have "i=0 \<or> (abs i = 1 \<and> check_int a) \<or> (i\<noteq>0 \<and> ((abs i \<noteq> 1) \<or> (\<not> check_int a)))" by auto
-  {assume "i=0" hence ?case using "12.hyps" by (simp add: rdvd_left_0_eq Let_def)}
-  moreover 
-  {assume ai1: "abs i = 1" and ai: "check_int a" 
-    hence "i=1 \<or> i= - 1" by arith
-    moreover {assume i1: "i = 1" 
-      from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]] 
-      have ?case using i1 ai by simp }
-    moreover {assume i1: "i = - 1" 
-      from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]] 
-	rdvd_abs1[where d="- 1" and t="Inum bs a"]
-      have ?case using i1 ai by simp }
-    ultimately have ?case by blast}
-  moreover   
-  {assume inz: "i\<noteq>0" and cond: "(abs i \<noteq> 1) \<or> (\<not> check_int a)"
-    {fix v assume "?sa = C v" hence ?case using sa[symmetric] inz cond
-	by (cases "abs i = 1", auto simp add: int_rdvd_iff) }
-    moreover {assume H:"\<not> (\<exists> v. ?sa = C v)" 
-      hence th: "simpfm (Dvd i a) = Dvd (fst (simpdvd i ?sa)) (snd (simpdvd i ?sa))" using inz cond by (cases ?sa, auto simp add: Let_def split_def)
-      from simpnum_nz have nz:"nozerocoeff ?sa" by simp
-      from simpdvd [OF nz inz] th have ?case using sa by simp}
-    ultimately have ?case by blast}
-  ultimately show ?case by blast
-next
-  case (13 i a)  let ?sa = "simpnum a"   have sa: "Inum bs ?sa = Inum bs a" by simp
-  have "i=0 \<or> (abs i = 1 \<and> check_int a) \<or> (i\<noteq>0 \<and> ((abs i \<noteq> 1) \<or> (\<not> check_int a)))" by auto
-  {assume "i=0" hence ?case using "13.hyps" by (simp add: rdvd_left_0_eq Let_def)}
-  moreover 
-  {assume ai1: "abs i = 1" and ai: "check_int a" 
-    hence "i=1 \<or> i= - 1" by arith
-    moreover {assume i1: "i = 1" 
-      from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]] 
-      have ?case using i1 ai by simp }
-    moreover {assume i1: "i = - 1" 
-      from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]] 
-	rdvd_abs1[where d="- 1" and t="Inum bs a"]
-      have ?case using i1 ai by simp }
-    ultimately have ?case by blast}
-  moreover   
-  {assume inz: "i\<noteq>0" and cond: "(abs i \<noteq> 1) \<or> (\<not> check_int a)"
-    {fix v assume "?sa = C v" hence ?case using sa[symmetric] inz cond
-	by (cases "abs i = 1", auto simp add: int_rdvd_iff) }
-    moreover {assume H:"\<not> (\<exists> v. ?sa = C v)" 
-      hence th: "simpfm (NDvd i a) = NDvd (fst (simpdvd i ?sa)) (snd (simpdvd i ?sa))" using inz cond 
-	by (cases ?sa, auto simp add: Let_def split_def)
-      from simpnum_nz have nz:"nozerocoeff ?sa" by simp
-      from simpdvd [OF nz inz] th have ?case using sa by simp}
-    ultimately have ?case by blast}
-  ultimately show ?case by blast
-qed (induct p rule: simpfm.induct, simp_all)
-
-lemma simpdvd_numbound0: "numbound0 t \<Longrightarrow> numbound0 (snd (simpdvd d t))"
-  by (simp add: simpdvd_def Let_def split_def reducecoeffh_numbound0)
-
-lemma simpfm_bound0[simp]: "bound0 p \<Longrightarrow> bound0 (simpfm p)"
-proof(induct p rule: simpfm.induct)
-  case (6 a) hence nb: "numbound0 a" by simp
-  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
-  thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0)
-next
-  case (7 a) hence nb: "numbound0 a" by simp
-  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
-  thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0)
-next
-  case (8 a) hence nb: "numbound0 a" by simp
-  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
-  thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0)
-next
-  case (9 a) hence nb: "numbound0 a" by simp
-  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
-  thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0)
-next
-  case (10 a) hence nb: "numbound0 a" by simp
-  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
-  thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0)
-next
-  case (11 a) hence nb: "numbound0 a" by simp
-  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
-  thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0)
-next
-  case (12 i a) hence nb: "numbound0 a" by simp
-  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
-  thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0 simpdvd_numbound0 split_def)
-next
-  case (13 i a) hence nb: "numbound0 a" by simp
-  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
-  thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0 simpdvd_numbound0 split_def)
-qed(auto simp add: disj_def imp_def iff_def conj_def)
-
-lemma simpfm_qf[simp]: "qfree p \<Longrightarrow> qfree (simpfm p)"
-by (induct p rule: simpfm.induct, auto simp add: Let_def)
-(case_tac "simpnum a",auto simp add: split_def Let_def)+
-
-
-  (* Generic quantifier elimination *)
-
-constdefs list_conj :: "fm list \<Rightarrow> fm"
-  "list_conj ps \<equiv> foldr conj ps T"
-lemma list_conj: "Ifm bs (list_conj ps) = (\<forall>p\<in> set ps. Ifm bs p)"
-  by (induct ps, auto simp add: list_conj_def)
-lemma list_conj_qf: " \<forall>p\<in> set ps. qfree p \<Longrightarrow> qfree (list_conj ps)"
-  by (induct ps, auto simp add: list_conj_def)
-lemma list_conj_nb: " \<forall>p\<in> set ps. bound0 p \<Longrightarrow> bound0 (list_conj ps)"
-  by (induct ps, auto simp add: list_conj_def)
-constdefs CJNB:: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm"
-  "CJNB f p \<equiv> (let cjs = conjuncts p ; (yes,no) = List.partition bound0 cjs
-                   in conj (decr (list_conj yes)) (f (list_conj no)))"
-
-lemma CJNB_qe: 
-  assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))"
-  shows "\<forall> bs p. qfree p \<longrightarrow> qfree (CJNB qe p) \<and> (Ifm bs ((CJNB qe p)) = Ifm bs (E p))"
-proof(clarify)
-  fix bs p
-  assume qfp: "qfree p"
-  let ?cjs = "conjuncts p"
-  let ?yes = "fst (List.partition bound0 ?cjs)"
-  let ?no = "snd (List.partition bound0 ?cjs)"
-  let ?cno = "list_conj ?no"
-  let ?cyes = "list_conj ?yes"
-  have part: "List.partition bound0 ?cjs = (?yes,?no)" by simp
-  from partition_P[OF part] have "\<forall> q\<in> set ?yes. bound0 q" by blast 
-  hence yes_nb: "bound0 ?cyes" by (simp add: list_conj_nb) 
-  hence yes_qf: "qfree (decr ?cyes )" by (simp add: decr_qf)
-  from conjuncts_qf[OF qfp] partition_set[OF part] 
-  have " \<forall>q\<in> set ?no. qfree q" by auto
-  hence no_qf: "qfree ?cno"by (simp add: list_conj_qf)
-  with qe have cno_qf:"qfree (qe ?cno )" 
-    and noE: "Ifm bs (qe ?cno) = Ifm bs (E ?cno)" by blast+
-  from cno_qf yes_qf have qf: "qfree (CJNB qe p)" 
-    by (simp add: CJNB_def Let_def conj_qf split_def)
-  {fix bs
-    from conjuncts have "Ifm bs p = (\<forall>q\<in> set ?cjs. Ifm bs q)" by blast
-    also have "\<dots> = ((\<forall>q\<in> set ?yes. Ifm bs q) \<and> (\<forall>q\<in> set ?no. Ifm bs q))"
-      using partition_set[OF part] by auto
-    finally have "Ifm bs p = ((Ifm bs ?cyes) \<and> (Ifm bs ?cno))" using list_conj by simp}
-  hence "Ifm bs (E p) = (\<exists>x. (Ifm (x#bs) ?cyes) \<and> (Ifm (x#bs) ?cno))" by simp
-  also fix y have "\<dots> = (\<exists>x. (Ifm (y#bs) ?cyes) \<and> (Ifm (x#bs) ?cno))"
-    using bound0_I[OF yes_nb, where bs="bs" and b'="y"] by blast
-  also have "\<dots> = (Ifm bs (decr ?cyes) \<and> Ifm bs (E ?cno))"
-    by (auto simp add: decr[OF yes_nb])
-  also have "\<dots> = (Ifm bs (conj (decr ?cyes) (qe ?cno)))"
-    using qe[rule_format, OF no_qf] by auto
-  finally have "Ifm bs (E p) = Ifm bs (CJNB qe p)" 
-    by (simp add: Let_def CJNB_def split_def)
-  with qf show "qfree (CJNB qe p) \<and> Ifm bs (CJNB qe p) = Ifm bs (E p)" by blast
-qed
-
-consts qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm"
-recdef qelim "measure fmsize"
-  "qelim (E p) = (\<lambda> qe. DJ (CJNB qe) (qelim p qe))"
-  "qelim (A p) = (\<lambda> qe. not (qe ((qelim (NOT p) qe))))"
-  "qelim (NOT p) = (\<lambda> qe. not (qelim p qe))"
-  "qelim (And p q) = (\<lambda> qe. conj (qelim p qe) (qelim q qe))" 
-  "qelim (Or  p q) = (\<lambda> qe. disj (qelim p qe) (qelim q qe))" 
-  "qelim (Imp p q) = (\<lambda> qe. disj (qelim (NOT p) qe) (qelim q qe))"
-  "qelim (Iff p q) = (\<lambda> qe. iff (qelim p qe) (qelim q qe))"
-  "qelim p = (\<lambda> y. simpfm p)"
-
-lemma qelim_ci:
-  assumes qe_inv: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))"
-  shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm bs (qelim p qe) = Ifm bs p)"
-using qe_inv DJ_qe[OF CJNB_qe[OF qe_inv]] 
-by(induct p rule: qelim.induct) 
-(auto simp del: simpfm.simps)
-
-
-text {* The @{text "\<int>"} Part *}
-text{* Linearity for fm where Bound 0 ranges over @{text "\<int>"} *}
-consts
-  zsplit0 :: "num \<Rightarrow> int \<times> num" (* splits the bounded from the unbounded part*)
-recdef zsplit0 "measure num_size"
-  "zsplit0 (C c) = (0,C c)"
-  "zsplit0 (Bound n) = (if n=0 then (1, C 0) else (0,Bound n))"
-  "zsplit0 (CN n c a) = zsplit0 (Add (Mul c (Bound n)) a)"
-  "zsplit0 (CF c a b) = zsplit0 (Add (Mul c (Floor a)) b)"
-  "zsplit0 (Neg a) = (let (i',a') =  zsplit0 a in (-i', Neg a'))"
-  "zsplit0 (Add a b) = (let (ia,a') =  zsplit0 a ; 
-                            (ib,b') =  zsplit0 b 
-                            in (ia+ib, Add a' b'))"
-  "zsplit0 (Sub a b) = (let (ia,a') =  zsplit0 a ; 
-                            (ib,b') =  zsplit0 b 
-                            in (ia-ib, Sub a' b'))"
-  "zsplit0 (Mul i a) = (let (i',a') =  zsplit0 a in (i*i', Mul i a'))"
-  "zsplit0 (Floor a) = (let (i',a') =  zsplit0 a in (i',Floor a'))"
-(hints simp add: Let_def)
-
-lemma zsplit0_I:
-  shows "\<And> n a. zsplit0 t = (n,a) \<Longrightarrow> (Inum ((real (x::int)) #bs) (CN 0 n a) = Inum (real x #bs) t) \<and> numbound0 a"
-  (is "\<And> n a. ?S t = (n,a) \<Longrightarrow> (?I x (CN 0 n a) = ?I x t) \<and> ?N a")
-proof(induct t rule: zsplit0.induct)
-  case (1 c n a) thus ?case by auto 
-next
-  case (2 m n a) thus ?case by (cases "m=0") auto
-next
-  case (3 n i a n a') thus ?case by auto
-next 
-  case (4 c a b n a') thus ?case by auto
-next
-  case (5 t n a)
-  let ?nt = "fst (zsplit0 t)"
-  let ?at = "snd (zsplit0 t)"
-  have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a=Neg ?at \<and> n=-?nt" using prems 
-    by (simp add: Let_def split_def)
-  from abj prems  have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
-  from th2[simplified] th[simplified] show ?case by simp
-next
-  case (6 s t n a)
-  let ?ns = "fst (zsplit0 s)"
-  let ?as = "snd (zsplit0 s)"
-  let ?nt = "fst (zsplit0 t)"
-  let ?at = "snd (zsplit0 t)"
-  have abjs: "zsplit0 s = (?ns,?as)" by simp 
-  moreover have abjt:  "zsplit0 t = (?nt,?at)" by simp 
-  ultimately have th: "a=Add ?as ?at \<and> n=?ns + ?nt" using prems 
-    by (simp add: Let_def split_def)
-  from abjs[symmetric] have bluddy: "\<exists> x y. (x,y) = zsplit0 s" by blast
-  from prems have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (real x # bs) (CN 0 xa xb) = Inum (real x # bs) t \<and> numbound0 xb)" by simp
-  with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
-  from abjs prems  have th2: "(?I x (CN 0 ?ns ?as) = ?I x s) \<and> ?N ?as" by blast
-  from th3[simplified] th2[simplified] th[simplified] show ?case 
-    by (simp add: left_distrib)
-next
-  case (7 s t n a)
-  let ?ns = "fst (zsplit0 s)"
-  let ?as = "snd (zsplit0 s)"
-  let ?nt = "fst (zsplit0 t)"
-  let ?at = "snd (zsplit0 t)"
-  have abjs: "zsplit0 s = (?ns,?as)" by simp 
-  moreover have abjt:  "zsplit0 t = (?nt,?at)" by simp 
-  ultimately have th: "a=Sub ?as ?at \<and> n=?ns - ?nt" using prems 
-    by (simp add: Let_def split_def)
-  from abjs[symmetric] have bluddy: "\<exists> x y. (x,y) = zsplit0 s" by blast
-  from prems have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (real x # bs) (CN 0 xa xb) = Inum (real x # bs) t \<and> numbound0 xb)" by simp
-  with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
-  from abjs prems  have th2: "(?I x (CN 0 ?ns ?as) = ?I x s) \<and> ?N ?as" by blast
-  from th3[simplified] th2[simplified] th[simplified] show ?case 
-    by (simp add: left_diff_distrib)
-next
-  case (8 i t n a)
-  let ?nt = "fst (zsplit0 t)"
-  let ?at = "snd (zsplit0 t)"
-  have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a=Mul i ?at \<and> n=i*?nt" using prems 
-    by (simp add: Let_def split_def)
-  from abj prems  have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
-  hence " ?I x (Mul i t) = (real i) * ?I x (CN 0 ?nt ?at)" by simp
-  also have "\<dots> = ?I x (CN 0 (i*?nt) (Mul i ?at))" by (simp add: right_distrib)
-  finally show ?case using th th2 by simp
-next
-  case (9 t n a)
-  let ?nt = "fst (zsplit0 t)"
-  let ?at = "snd (zsplit0 t)"
-  have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a= Floor ?at \<and> n=?nt" using prems 
-    by (simp add: Let_def split_def)
-  from abj prems  have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
-  hence na: "?N a" using th by simp
-  have th': "(real ?nt)*(real x) = real (?nt * x)" by simp
-  have "?I x (Floor t) = ?I x (Floor (CN 0 ?nt ?at))" using th2 by simp
-  also have "\<dots> = real (floor ((real ?nt)* real(x) + ?I x ?at))" by simp
-  also have "\<dots> = real (floor (?I x ?at + real (?nt* x)))" by (simp add: add_ac)
-  also have "\<dots> = real (floor (?I x ?at) + (?nt* x))" 
-    using floor_add[where x="?I x ?at" and a="?nt* x"] by simp 
-  also have "\<dots> = real (?nt)*(real x) + real (floor (?I x ?at))" by (simp add: add_ac)
-  finally have "?I x (Floor t) = ?I x (CN 0 n a)" using th by simp
-  with na show ?case by simp
-qed
-
-consts
-  iszlfm :: "fm \<Rightarrow> real list \<Rightarrow> bool"   (* Linearity test for fm *)
-  zlfm :: "fm \<Rightarrow> fm"       (* Linearity transformation for fm *)
-recdef iszlfm "measure size"
-  "iszlfm (And p q) = (\<lambda> bs. iszlfm p bs \<and> iszlfm q bs)" 
-  "iszlfm (Or p q) = (\<lambda> bs. iszlfm p bs \<and> iszlfm q bs)" 
-  "iszlfm (Eq  (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)"
-  "iszlfm (NEq (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)"
-  "iszlfm (Lt  (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)"
-  "iszlfm (Le  (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)"
-  "iszlfm (Gt  (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)"
-  "iszlfm (Ge  (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)"
-  "iszlfm (Dvd i (CN 0 c e)) = 
-                 (\<lambda> bs. c>0 \<and> i>0 \<and> numbound0 e \<and> isint e bs)"
-  "iszlfm (NDvd i (CN 0 c e))= 
-                 (\<lambda> bs. c>0 \<and> i>0 \<and> numbound0 e \<and> isint e bs)"
-  "iszlfm p = (\<lambda> bs. isatom p \<and> (bound0 p))"
-
-lemma zlin_qfree: "iszlfm p bs \<Longrightarrow> qfree p"
-  by (induct p rule: iszlfm.induct) auto
-
-lemma iszlfm_gen:
-  assumes lp: "iszlfm p (x#bs)"
-  shows "\<forall> y. iszlfm p (y#bs)"
-proof
-  fix y
-  show "iszlfm p (y#bs)"
-    using lp
-  by(induct p rule: iszlfm.induct, simp_all add: numbound0_gen[rule_format, where x="x" and y="y"])
-qed
-
-lemma conj_zl[simp]: "iszlfm p bs \<Longrightarrow> iszlfm q bs \<Longrightarrow> iszlfm (conj p q) bs"
-  using conj_def by (cases p,auto)
-lemma disj_zl[simp]: "iszlfm p bs \<Longrightarrow> iszlfm q bs \<Longrightarrow> iszlfm (disj p q) bs"
-  using disj_def by (cases p,auto)
-lemma not_zl[simp]: "iszlfm p bs \<Longrightarrow> iszlfm (not p) bs"
-  by (induct p rule:iszlfm.induct ,auto)
-
-recdef zlfm "measure fmsize"
-  "zlfm (And p q) = conj (zlfm p) (zlfm q)"
-  "zlfm (Or p q) = disj (zlfm p) (zlfm q)"
-  "zlfm (Imp p q) = disj (zlfm (NOT p)) (zlfm q)"
-  "zlfm (Iff p q) = disj (conj (zlfm p) (zlfm q)) (conj (zlfm (NOT p)) (zlfm (NOT q)))"
-  "zlfm (Lt a) = (let (c,r) = zsplit0 a in 
-     if c=0 then Lt r else 
-     if c>0 then Or (Lt (CN 0 c (Neg (Floor (Neg r))))) (And (Eq (CN 0 c (Neg (Floor (Neg r))))) (Lt (Add (Floor (Neg r)) r))) 
-     else Or (Gt (CN 0 (-c) (Floor(Neg r)))) (And (Eq(CN 0 (-c) (Floor(Neg r)))) (Lt (Add (Floor (Neg r)) r))))"
-  "zlfm (Le a) = (let (c,r) = zsplit0 a in 
-     if c=0 then Le r else 
-     if c>0 then Or (Le (CN 0 c (Neg (Floor (Neg r))))) (And (Eq (CN 0 c (Neg (Floor (Neg r))))) (Lt (Add (Floor (Neg r)) r))) 
-     else Or (Ge (CN 0 (-c) (Floor(Neg r)))) (And (Eq(CN 0 (-c) (Floor(Neg r)))) (Lt (Add (Floor (Neg r)) r))))"
-  "zlfm (Gt a) = (let (c,r) = zsplit0 a in 
-     if c=0 then Gt r else 
-     if c>0 then Or (Gt (CN 0 c (Floor r))) (And (Eq (CN 0 c (Floor r))) (Lt (Sub (Floor r) r))) 
-     else Or (Lt (CN 0 (-c) (Neg (Floor r)))) (And (Eq(CN 0 (-c) (Neg (Floor r)))) (Lt (Sub (Floor r) r))))"
-  "zlfm (Ge a) = (let (c,r) = zsplit0 a in 
-     if c=0 then Ge r else 
-     if c>0 then Or (Ge (CN 0 c (Floor r))) (And (Eq (CN 0 c (Floor r))) (Lt (Sub (Floor r) r))) 
-     else Or (Le (CN 0 (-c) (Neg (Floor r)))) (And (Eq(CN 0 (-c) (Neg (Floor r)))) (Lt (Sub (Floor r) r))))"
-  "zlfm (Eq a) = (let (c,r) = zsplit0 a in 
-              if c=0 then Eq r else 
-      if c>0 then (And (Eq (CN 0 c (Neg (Floor (Neg r))))) (Eq (Add (Floor (Neg r)) r)))
-      else (And (Eq (CN 0 (-c) (Floor (Neg r)))) (Eq (Add (Floor (Neg r)) r))))"
-  "zlfm (NEq a) = (let (c,r) = zsplit0 a in 
-              if c=0 then NEq r else 
-      if c>0 then (Or (NEq (CN 0 c (Neg (Floor (Neg r))))) (NEq (Add (Floor (Neg r)) r)))
-      else (Or (NEq (CN 0 (-c) (Floor (Neg r)))) (NEq (Add (Floor (Neg r)) r))))"
-  "zlfm (Dvd i a) = (if i=0 then zlfm (Eq a) 
-  else (let (c,r) = zsplit0 a in 
-              if c=0 then Dvd (abs i) r else 
-      if c>0 then And (Eq (Sub (Floor r) r)) (Dvd (abs i) (CN 0 c (Floor r))) 
-      else And (Eq (Sub (Floor r) r)) (Dvd (abs i) (CN 0 (-c) (Neg (Floor r))))))"
-  "zlfm (NDvd i a) = (if i=0 then zlfm (NEq a) 
-  else (let (c,r) = zsplit0 a in 
-              if c=0 then NDvd (abs i) r else 
-      if c>0 then Or (NEq (Sub (Floor r) r)) (NDvd (abs i) (CN 0 c (Floor r))) 
-      else Or (NEq (Sub (Floor r) r)) (NDvd (abs i) (CN 0 (-c) (Neg (Floor r))))))"
-  "zlfm (NOT (And p q)) = disj (zlfm (NOT p)) (zlfm (NOT q))"
-  "zlfm (NOT (Or p q)) = conj (zlfm (NOT p)) (zlfm (NOT q))"
-  "zlfm (NOT (Imp p q)) = conj (zlfm p) (zlfm (NOT q))"
-  "zlfm (NOT (Iff p q)) = disj (conj(zlfm p) (zlfm(NOT q))) (conj (zlfm(NOT p)) (zlfm q))"
-  "zlfm (NOT (NOT p)) = zlfm p"
-  "zlfm (NOT T) = F"
-  "zlfm (NOT F) = T"
-  "zlfm (NOT (Lt a)) = zlfm (Ge a)"
-  "zlfm (NOT (Le a)) = zlfm (Gt a)"
-  "zlfm (NOT (Gt a)) = zlfm (Le a)"
-  "zlfm (NOT (Ge a)) = zlfm (Lt a)"
-  "zlfm (NOT (Eq a)) = zlfm (NEq a)"
-  "zlfm (NOT (NEq a)) = zlfm (Eq a)"
-  "zlfm (NOT (Dvd i a)) = zlfm (NDvd i a)"
-  "zlfm (NOT (NDvd i a)) = zlfm (Dvd i a)"
-  "zlfm p = p" (hints simp add: fmsize_pos)
-
-lemma split_int_less_real: 
-  "(real (a::int) < b) = (a < floor b \<or> (a = floor b \<and> real (floor b) < b))"
-proof( auto)
-  assume alb: "real a < b" and agb: "\<not> a < floor b"
-  from agb have "floor b \<le> a" by simp hence th: "b < real a + 1" by (simp only: floor_le_eq)
-  from floor_eq[OF alb th] show "a= floor b" by simp 
-next
-  assume alb: "a < floor b"
-  hence "real a < real (floor b)" by simp
-  moreover have "real (floor b) \<le> b" by simp ultimately show  "real a < b" by arith 
-qed
-
-lemma split_int_less_real': 
-  "(real (a::int) + b < 0) = (real a - real (floor(-b)) < 0 \<or> (real a - real (floor (-b)) = 0 \<and> real (floor (-b)) + b < 0))"
-proof- 
-  have "(real a + b <0) = (real a < -b)" by arith
-  with split_int_less_real[where a="a" and b="-b"] show ?thesis by arith  
-qed
-
-lemma split_int_gt_real': 
-  "(real (a::int) + b > 0) = (real a + real (floor b) > 0 \<or> (real a + real (floor b) = 0 \<and> real (floor b) - b < 0))"
-proof- 
-  have th: "(real a + b >0) = (real (-a) + (-b)< 0)" by arith
-  show ?thesis using myless[rule_format, where b="real (floor b)"] 
-    by (simp only:th split_int_less_real'[where a="-a" and b="-b"]) 
-    (simp add: algebra_simps diff_def[symmetric],arith)
-qed
-
-lemma split_int_le_real: 
-  "(real (a::int) \<le> b) = (a \<le> floor b \<or> (a = floor b \<and> real (floor b) < b))"
-proof( auto)
-  assume alb: "real a \<le> b" and agb: "\<not> a \<le> floor b"
-  from alb have "floor (real a) \<le> floor b " by (simp only: floor_mono2) 
-  hence "a \<le> floor b" by simp with agb show "False" by simp
-next
-  assume alb: "a \<le> floor b"
-  hence "real a \<le> real (floor b)" by (simp only: floor_mono2)
-  also have "\<dots>\<le> b" by simp  finally show  "real a \<le> b" . 
-qed
-
-lemma split_int_le_real': 
-  "(real (a::int) + b \<le> 0) = (real a - real (floor(-b)) \<le> 0 \<or> (real a - real (floor (-b)) = 0 \<and> real (floor (-b)) + b < 0))"
-proof- 
-  have "(real a + b \<le>0) = (real a \<le> -b)" by arith
-  with split_int_le_real[where a="a" and b="-b"] show ?thesis by arith  
-qed
-
-lemma split_int_ge_real': 
-  "(real (a::int) + b \<ge> 0) = (real a + real (floor b) \<ge> 0 \<or> (real a + real (floor b) = 0 \<and> real (floor b) - b < 0))"
-proof- 
-  have th: "(real a + b \<ge>0) = (real (-a) + (-b) \<le> 0)" by arith
-  show ?thesis by (simp only: th split_int_le_real'[where a="-a" and b="-b"])
-    (simp add: algebra_simps diff_def[symmetric],arith)
-qed
-
-lemma split_int_eq_real: "(real (a::int) = b) = ( a = floor b \<and> b = real (floor b))" (is "?l = ?r")
-by auto
-
-lemma split_int_eq_real': "(real (a::int) + b = 0) = ( a - floor (-b) = 0 \<and> real (floor (-b)) + b = 0)" (is "?l = ?r")
-proof-
-  have "?l = (real a = -b)" by arith
-  with split_int_eq_real[where a="a" and b="-b"] show ?thesis by simp arith
-qed
-
-lemma zlfm_I:
-  assumes qfp: "qfree p"
-  shows "(Ifm (real i #bs) (zlfm p) = Ifm (real i# bs) p) \<and> iszlfm (zlfm p) (real (i::int) #bs)"
-  (is "(?I (?l p) = ?I p) \<and> ?L (?l p)")
-using qfp
-proof(induct p rule: zlfm.induct)
-  case (5 a) 
-  let ?c = "fst (zsplit0 a)"
-  let ?r = "snd (zsplit0 a)"
-  have spl: "zsplit0 a = (?c,?r)" by simp
-  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
-  let ?N = "\<lambda> t. Inum (real i#bs) t"
-  have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith
-  moreover
-  {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] 
-      by (cases "?r", simp_all add: Let_def split_def,case_tac "nat", simp_all)}
-  moreover
-  {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Lt a))" 
-      by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (Lt a) = (real (?c * i) + (?N ?r) < 0)" using Ia by (simp add: Let_def split_def)
-    also have "\<dots> = (?I (?l (Lt a)))" apply (simp only: split_int_less_real'[where a="?c*i" and b="?N ?r"]) by (simp add: Ia cp cnz Let_def split_def diff_def)
-    finally have ?case using l by simp}
-  moreover
-  {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Lt a))" 
-      by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (Lt a) = (real (?c * i) + (?N ?r) < 0)" using Ia by (simp add: Let_def split_def)
-    also from cn cnz have "\<dots> = (?I (?l (Lt a)))" by (simp only: split_int_less_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def diff_def[symmetric] add_ac, arith)
-    finally have ?case using l by simp}
-  ultimately show ?case by blast
-next
-  case (6 a)
-  let ?c = "fst (zsplit0 a)"
-  let ?r = "snd (zsplit0 a)"
-  have spl: "zsplit0 a = (?c,?r)" by simp
-  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
-  let ?N = "\<lambda> t. Inum (real i#bs) t"
-  have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith
-  moreover
-  {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] 
-      by (cases "?r", simp_all add: Let_def split_def, case_tac "nat",simp_all)}
-  moreover
-  {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Le a))" 
-      by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (Le a) = (real (?c * i) + (?N ?r) \<le> 0)" using Ia by (simp add: Let_def split_def)
-    also have "\<dots> = (?I (?l (Le a)))" by (simp only: split_int_le_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia cp cnz Let_def split_def diff_def)
-    finally have ?case using l by simp}
-  moreover
-  {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Le a))" 
-      by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (Le a) = (real (?c * i) + (?N ?r) \<le> 0)" using Ia by (simp add: Let_def split_def)
-    also from cn cnz have "\<dots> = (?I (?l (Le a)))" by (simp only: split_int_le_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def diff_def[symmetric] add_ac ,arith)
-    finally have ?case using l by simp}
-  ultimately show ?case by blast
-next
-  case (7 a) 
-  let ?c = "fst (zsplit0 a)"
-  let ?r = "snd (zsplit0 a)"
-  have spl: "zsplit0 a = (?c,?r)" by simp
-  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
-  let ?N = "\<lambda> t. Inum (real i#bs) t"
-  have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith
-  moreover
-  {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] 
-      by (cases "?r", simp_all add: Let_def split_def, case_tac "nat", simp_all)}
-  moreover
-  {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Gt a))" 
-      by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (Gt a) = (real (?c * i) + (?N ?r) > 0)" using Ia by (simp add: Let_def split_def)
-    also have "\<dots> = (?I (?l (Gt a)))" by (simp only: split_int_gt_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia cp cnz Let_def split_def diff_def)
-    finally have ?case using l by simp}
-  moreover
-  {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Gt a))" 
-      by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (Gt a) = (real (?c * i) + (?N ?r) > 0)" using Ia by (simp add: Let_def split_def)
-    also from cn cnz have "\<dots> = (?I (?l (Gt a)))" by (simp only: split_int_gt_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def diff_def[symmetric] add_ac, arith)
-    finally have ?case using l by simp}
-  ultimately show ?case by blast
-next
-  case (8 a)
-   let ?c = "fst (zsplit0 a)"
-  let ?r = "snd (zsplit0 a)"
-  have spl: "zsplit0 a = (?c,?r)" by simp
-  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
-  let ?N = "\<lambda> t. Inum (real i#bs) t"
-  have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith
-  moreover
-  {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] 
-      by (cases "?r", simp_all add: Let_def split_def, case_tac "nat", simp_all)}
-  moreover
-  {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Ge a))" 
-      by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (Ge a) = (real (?c * i) + (?N ?r) \<ge> 0)" using Ia by (simp add: Let_def split_def)
-    also have "\<dots> = (?I (?l (Ge a)))" by (simp only: split_int_ge_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia cp cnz Let_def split_def diff_def)
-    finally have ?case using l by simp}
-  moreover
-  {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Ge a))" 
-      by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (Ge a) = (real (?c * i) + (?N ?r) \<ge> 0)" using Ia by (simp add: Let_def split_def)
-    also from cn cnz have "\<dots> = (?I (?l (Ge a)))" by (simp only: split_int_ge_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def diff_def[symmetric] add_ac, arith)
-    finally have ?case using l by simp}
-  ultimately show ?case by blast
-next
-  case (9 a)
-  let ?c = "fst (zsplit0 a)"
-  let ?r = "snd (zsplit0 a)"
-  have spl: "zsplit0 a = (?c,?r)" by simp
-  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
-  let ?N = "\<lambda> t. Inum (real i#bs) t"
-  have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith
-  moreover
-  {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] 
-      by (cases "?r", simp_all add: Let_def split_def, case_tac "nat", simp_all)}
-  moreover
-  {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Eq a))" 
-      by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (Eq a) = (real (?c * i) + (?N ?r) = 0)" using Ia by (simp add: Let_def split_def)
-    also have "\<dots> = (?I (?l (Eq a)))" using cp cnz  by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia real_of_int_mult[symmetric] del: real_of_int_mult)
-    finally have ?case using l by simp}
-  moreover
-  {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Eq a))" 
-      by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (Eq a) = (real (?c * i) + (?N ?r) = 0)" using Ia by (simp add: Let_def split_def)
-    also from cn cnz have "\<dots> = (?I (?l (Eq a)))" by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia real_of_int_mult[symmetric] del: real_of_int_mult,arith)
-    finally have ?case using l by simp}
-  ultimately show ?case by blast
-next
-  case (10 a)
-  let ?c = "fst (zsplit0 a)"
-  let ?r = "snd (zsplit0 a)"
-  have spl: "zsplit0 a = (?c,?r)" by simp
-  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
-  let ?N = "\<lambda> t. Inum (real i#bs) t"
-  have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith
-  moreover
-  {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] 
-      by (cases "?r", simp_all add: Let_def split_def, case_tac "nat", simp_all)}
-  moreover
-  {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (NEq a))" 
-      by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (NEq a) = (real (?c * i) + (?N ?r) \<noteq> 0)" using Ia by (simp add: Let_def split_def)
-    also have "\<dots> = (?I (?l (NEq a)))" using cp cnz  by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia real_of_int_mult[symmetric] del: real_of_int_mult)
-    finally have ?case using l by simp}
-  moreover
-  {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (NEq a))" 
-      by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (NEq a) = (real (?c * i) + (?N ?r) \<noteq> 0)" using Ia by (simp add: Let_def split_def)
-    also from cn cnz have "\<dots> = (?I (?l (NEq a)))" by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia real_of_int_mult[symmetric] del: real_of_int_mult,arith)
-    finally have ?case using l by simp}
-  ultimately show ?case by blast
-next
-  case (11 j a)
-  let ?c = "fst (zsplit0 a)"
-  let ?r = "snd (zsplit0 a)"
-  have spl: "zsplit0 a = (?c,?r)" by simp
-  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
-  let ?N = "\<lambda> t. Inum (real i#bs) t"
-  have "j=0 \<or> (j\<noteq>0 \<and> ?c = 0) \<or> (j\<noteq>0 \<and> ?c >0 \<and> ?c\<noteq>0) \<or> (j\<noteq> 0 \<and> ?c<0 \<and> ?c\<noteq>0)" by arith
-  moreover
-  {assume "j=0" hence z: "zlfm (Dvd j a) = (zlfm (Eq a))" by (simp add: Let_def) 
-    hence ?case using prems by (simp del: zlfm.simps add: rdvd_left_0_eq)}
-  moreover
-  {assume "?c=0" and "j\<noteq>0" hence ?case 
-      using zsplit0_I[OF spl, where x="i" and bs="bs"] rdvd_abs1[where d="j"]
-      by (cases "?r", simp_all add: Let_def split_def, case_tac "nat", simp_all)}
-  moreover
-  {assume cp: "?c > 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (Dvd j a))" 
-      by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (Dvd j a) = (real j rdvd (real (?c * i) + (?N ?r)))" 
-      using Ia by (simp add: Let_def split_def)
-    also have "\<dots> = (real (abs j) rdvd real (?c*i) + (?N ?r))" 
-      by (simp only: rdvd_abs1[where d="j" and t="real (?c*i) + ?N ?r", symmetric]) simp
-    also have "\<dots> = ((abs j) dvd (floor ((?N ?r) + real (?c*i))) \<and> 
-       (real (floor ((?N ?r) + real (?c*i))) = (real (?c*i) + (?N ?r))))" 
-      by(simp only: int_rdvd_real[where i="abs j" and x="real (?c*i) + (?N ?r)"]) (simp only: add_ac)
-    also have "\<dots> = (?I (?l (Dvd j a)))" using cp cnz jnz  
-      by (simp add: Let_def split_def int_rdvd_iff[symmetric]  
-	del: real_of_int_mult) (auto simp add: add_ac)
-    finally have ?case using l jnz  by simp }
-  moreover
-  {assume cn: "?c < 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (Dvd j a))" 
-      by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (Dvd j a) = (real j rdvd (real (?c * i) + (?N ?r)))" 
-      using Ia by (simp add: Let_def split_def)
-    also have "\<dots> = (real (abs j) rdvd real (?c*i) + (?N ?r))" 
-      by (simp only: rdvd_abs1[where d="j" and t="real (?c*i) + ?N ?r", symmetric]) simp
-    also have "\<dots> = ((abs j) dvd (floor ((?N ?r) + real (?c*i))) \<and> 
-       (real (floor ((?N ?r) + real (?c*i))) = (real (?c*i) + (?N ?r))))" 
-      by(simp only: int_rdvd_real[where i="abs j" and x="real (?c*i) + (?N ?r)"]) (simp only: add_ac)
-    also have "\<dots> = (?I (?l (Dvd j a)))" using cn cnz jnz
-      using rdvd_minus [where d="abs j" and t="real (?c*i + floor (?N ?r))", simplified, symmetric]
-      by (simp add: Let_def split_def int_rdvd_iff[symmetric]  
-	del: real_of_int_mult) (auto simp add: add_ac)
-    finally have ?case using l jnz by blast }
-  ultimately show ?case by blast
-next
-  case (12 j a)
-  let ?c = "fst (zsplit0 a)"
-  let ?r = "snd (zsplit0 a)"
-  have spl: "zsplit0 a = (?c,?r)" by simp
-  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
-  let ?N = "\<lambda> t. Inum (real i#bs) t"
-  have "j=0 \<or> (j\<noteq>0 \<and> ?c = 0) \<or> (j\<noteq>0 \<and> ?c >0 \<and> ?c\<noteq>0) \<or> (j\<noteq> 0 \<and> ?c<0 \<and> ?c\<noteq>0)" by arith
-  moreover
-  {assume "j=0" hence z: "zlfm (NDvd j a) = (zlfm (NEq a))" by (simp add: Let_def) 
-    hence ?case using prems by (simp del: zlfm.simps add: rdvd_left_0_eq)}
-  moreover
-  {assume "?c=0" and "j\<noteq>0" hence ?case 
-      using zsplit0_I[OF spl, where x="i" and bs="bs"] rdvd_abs1[where d="j"]
-      by (cases "?r", simp_all add: Let_def split_def, case_tac "nat", simp_all)}
-  moreover
-  {assume cp: "?c > 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (NDvd j a))" 
-      by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (NDvd j a) = (\<not> (real j rdvd (real (?c * i) + (?N ?r))))" 
-      using Ia by (simp add: Let_def split_def)
-    also have "\<dots> = (\<not> (real (abs j) rdvd real (?c*i) + (?N ?r)))" 
-      by (simp only: rdvd_abs1[where d="j" and t="real (?c*i) + ?N ?r", symmetric]) simp
-    also have "\<dots> = (\<not> ((abs j) dvd (floor ((?N ?r) + real (?c*i))) \<and> 
-       (real (floor ((?N ?r) + real (?c*i))) = (real (?c*i) + (?N ?r)))))" 
-      by(simp only: int_rdvd_real[where i="abs j" and x="real (?c*i) + (?N ?r)"]) (simp only: add_ac)
-    also have "\<dots> = (?I (?l (NDvd j a)))" using cp cnz jnz  
-      by (simp add: Let_def split_def int_rdvd_iff[symmetric]  
-	del: real_of_int_mult) (auto simp add: add_ac)
-    finally have ?case using l jnz  by simp }
-  moreover
-  {assume cn: "?c < 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (NDvd j a))" 
-      by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (NDvd j a) = (\<not> (real j rdvd (real (?c * i) + (?N ?r))))" 
-      using Ia by (simp add: Let_def split_def)
-    also have "\<dots> = (\<not> (real (abs j) rdvd real (?c*i) + (?N ?r)))" 
-      by (simp only: rdvd_abs1[where d="j" and t="real (?c*i) + ?N ?r", symmetric]) simp
-    also have "\<dots> = (\<not> ((abs j) dvd (floor ((?N ?r) + real (?c*i))) \<and> 
-       (real (floor ((?N ?r) + real (?c*i))) = (real (?c*i) + (?N ?r)))))" 
-      by(simp only: int_rdvd_real[where i="abs j" and x="real (?c*i) + (?N ?r)"]) (simp only: add_ac)
-    also have "\<dots> = (?I (?l (NDvd j a)))" using cn cnz jnz
-      using rdvd_minus [where d="abs j" and t="real (?c*i + floor (?N ?r))", simplified, symmetric]
-      by (simp add: Let_def split_def int_rdvd_iff[symmetric]  
-	del: real_of_int_mult) (auto simp add: add_ac)
-    finally have ?case using l jnz by blast }
-  ultimately show ?case by blast
-qed auto
-
-text{* plusinf : Virtual substitution of @{text "+\<infinity>"}
-       minusinf: Virtual substitution of @{text "-\<infinity>"}
-       @{text "\<delta>"} Compute lcm @{text "d| Dvd d  c*x+t \<in> p"}
-       @{text "d\<delta>"} checks if a given l divides all the ds above*}
-
-consts 
-  plusinf:: "fm \<Rightarrow> fm" 
-  minusinf:: "fm \<Rightarrow> fm"
-  \<delta> :: "fm \<Rightarrow> int" 
-  d\<delta> :: "fm \<Rightarrow> int \<Rightarrow> bool"
-
-recdef minusinf "measure size"
-  "minusinf (And p q) = conj (minusinf p) (minusinf q)" 
-  "minusinf (Or p q) = disj (minusinf p) (minusinf q)" 
-  "minusinf (Eq  (CN 0 c e)) = F"
-  "minusinf (NEq (CN 0 c e)) = T"
-  "minusinf (Lt  (CN 0 c e)) = T"
-  "minusinf (Le  (CN 0 c e)) = T"
-  "minusinf (Gt  (CN 0 c e)) = F"
-  "minusinf (Ge  (CN 0 c e)) = F"
-  "minusinf p = p"
-
-lemma minusinf_qfree: "qfree p \<Longrightarrow> qfree (minusinf p)"
-  by (induct p rule: minusinf.induct, auto)
-
-recdef plusinf "measure size"
-  "plusinf (And p q) = conj (plusinf p) (plusinf q)" 
-  "plusinf (Or p q) = disj (plusinf p) (plusinf q)" 
-  "plusinf (Eq  (CN 0 c e)) = F"
-  "plusinf (NEq (CN 0 c e)) = T"
-  "plusinf (Lt  (CN 0 c e)) = F"
-  "plusinf (Le  (CN 0 c e)) = F"
-  "plusinf (Gt  (CN 0 c e)) = T"
-  "plusinf (Ge  (CN 0 c e)) = T"
-  "plusinf p = p"
-
-recdef \<delta> "measure size"
-  "\<delta> (And p q) = zlcm (\<delta> p) (\<delta> q)" 
-  "\<delta> (Or p q) = zlcm (\<delta> p) (\<delta> q)" 
-  "\<delta> (Dvd i (CN 0 c e)) = i"
-  "\<delta> (NDvd i (CN 0 c e)) = i"
-  "\<delta> p = 1"
-
-recdef d\<delta> "measure size"
-  "d\<delta> (And p q) = (\<lambda> d. d\<delta> p d \<and> d\<delta> q d)" 
-  "d\<delta> (Or p q) = (\<lambda> d. d\<delta> p d \<and> d\<delta> q d)" 
-  "d\<delta> (Dvd i (CN 0 c e)) = (\<lambda> d. i dvd d)"
-  "d\<delta> (NDvd i (CN 0 c e)) = (\<lambda> d. i dvd d)"
-  "d\<delta> p = (\<lambda> d. True)"
-
-lemma delta_mono: 
-  assumes lin: "iszlfm p bs"
-  and d: "d dvd d'"
-  and ad: "d\<delta> p d"
-  shows "d\<delta> p d'"
-  using lin ad d
-proof(induct p rule: iszlfm.induct)
-  case (9 i c e)  thus ?case using d
-    by (simp add: zdvd_trans[where m="i" and n="d" and k="d'"])
-next
-  case (10 i c e) thus ?case using d
-    by (simp add: zdvd_trans[where m="i" and n="d" and k="d'"])
-qed simp_all
-
-lemma \<delta> : assumes lin:"iszlfm p bs"
-  shows "d\<delta> p (\<delta> p) \<and> \<delta> p >0"
-using lin
-proof (induct p rule: iszlfm.induct)
-  case (1 p q) 
-  let ?d = "\<delta> (And p q)"
-  from prems zlcm_pos have dp: "?d >0" by simp
-  have d1: "\<delta> p dvd \<delta> (And p q)" using prems by simp 
-   hence th: "d\<delta> p ?d" 
-     using delta_mono prems by (auto simp del: dvd_zlcm_self1)
-  have "\<delta> q dvd \<delta> (And p q)" using prems  by simp 
-  hence th': "d\<delta> q ?d" using delta_mono prems by (auto simp del: dvd_zlcm_self2)
-  from th th' dp show ?case by simp 
-next
-  case (2 p q)  
-  let ?d = "\<delta> (And p q)"
-  from prems zlcm_pos have dp: "?d >0" by simp
-  have "\<delta> p dvd \<delta> (And p q)" using prems by simp hence th: "d\<delta> p ?d" using delta_mono prems 
-    by (auto simp del: dvd_zlcm_self1)
-  have "\<delta> q dvd \<delta> (And p q)" using prems by simp hence th': "d\<delta> q ?d" using delta_mono prems by (auto simp del: dvd_zlcm_self2)
-  from th th' dp show ?case by simp 
-qed simp_all
-
-
-lemma minusinf_inf:
-  assumes linp: "iszlfm p (a # bs)"
-  shows "\<exists> (z::int). \<forall> x < z. Ifm ((real x)#bs) (minusinf p) = Ifm ((real x)#bs) p"
-  (is "?P p" is "\<exists> (z::int). \<forall> x < z. ?I x (?M p) = ?I x p")
-using linp
-proof (induct p rule: minusinf.induct)
-  case (1 f g)
-  from prems have "?P f" by simp
-  then obtain z1 where z1_def: "\<forall> x < z1. ?I x (?M f) = ?I x f" by blast
-  from prems have "?P g" by simp
-  then obtain z2 where z2_def: "\<forall> x < z2. ?I x (?M g) = ?I x g" by blast
-  let ?z = "min z1 z2"
-  from z1_def z2_def have "\<forall> x < ?z. ?I x (?M (And f g)) = ?I x (And f g)" by simp
-  thus ?case by blast
-next
-  case (2 f g)   from prems have "?P f" by simp
-  then obtain z1 where z1_def: "\<forall> x < z1. ?I x (?M f) = ?I x f" by blast
-  from prems have "?P g" by simp
-  then obtain z2 where z2_def: "\<forall> x < z2. ?I x (?M g) = ?I x g" by blast
-  let ?z = "min z1 z2"
-  from z1_def z2_def have "\<forall> x < ?z. ?I x (?M (Or f g)) = ?I x (Or f g)" by simp
-  thus ?case by blast
-next
-  case (3 c e) 
-  from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp
-  from prems have nbe: "numbound0 e" by simp
-  fix y
-  have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Eq (CN 0 c e))) = ?I x (Eq (CN 0 c e))"
-  proof (simp add: less_floor_eq , rule allI, rule impI) 
-    fix x
-    assume A: "real x + (1\<Colon>real) \<le> - (Inum (y # bs) e / real c)"
-    hence th1:"real x < - (Inum (y # bs) e / real c)" by simp
-    with rcpos  have "(real c)*(real  x) < (real c)*(- (Inum (y # bs) e / real c))"
-      by (simp only:  real_mult_less_mono2[OF rcpos th1])
-    hence "real c * real x + Inum (y # bs) e \<noteq> 0"using rcpos  by simp
-    thus "real c * real x + Inum (real x # bs) e \<noteq> 0" 
-      using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"]  by simp
-  qed
-  thus ?case by blast
-next
-  case (4 c e) 
-  from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp
-  from prems have nbe: "numbound0 e" by simp
-  fix y
-  have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (NEq (CN 0 c e))) = ?I x (NEq (CN 0 c e))"
-  proof (simp add: less_floor_eq , rule allI, rule impI) 
-    fix x
-    assume A: "real x + (1\<Colon>real) \<le> - (Inum (y # bs) e / real c)"
-    hence th1:"real x < - (Inum (y # bs) e / real c)" by simp
-    with rcpos  have "(real c)*(real  x) < (real c)*(- (Inum (y # bs) e / real c))"
-      by (simp only:  real_mult_less_mono2[OF rcpos th1])
-    hence "real c * real x + Inum (y # bs) e \<noteq> 0"using rcpos  by simp
-    thus "real c * real x + Inum (real x # bs) e \<noteq> 0" 
-      using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"]  by simp
-  qed
-  thus ?case by blast
-next
-  case (5 c e) 
-  from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp
-  from prems have nbe: "numbound0 e" by simp
-  fix y
-  have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Lt (CN 0 c e))) = ?I x (Lt (CN 0 c e))"
-  proof (simp add: less_floor_eq , rule allI, rule impI) 
-    fix x
-    assume A: "real x + (1\<Colon>real) \<le> - (Inum (y # bs) e / real c)"
-    hence th1:"real x < - (Inum (y # bs) e / real c)" by simp
-    with rcpos  have "(real c)*(real  x) < (real c)*(- (Inum (y # bs) e / real c))"
-      by (simp only:  real_mult_less_mono2[OF rcpos th1])
-    thus "real c * real x + Inum (real x # bs) e < 0" 
-      using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"] rcpos by simp
-  qed
-  thus ?case by blast
-next
-  case (6 c e) 
-  from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp
-  from prems have nbe: "numbound0 e" by simp
-  fix y
-  have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Le (CN 0 c e))) = ?I x (Le (CN 0 c e))"
-  proof (simp add: less_floor_eq , rule allI, rule impI) 
-    fix x
-    assume A: "real x + (1\<Colon>real) \<le> - (Inum (y # bs) e / real c)"
-    hence th1:"real x < - (Inum (y # bs) e / real c)" by simp
-    with rcpos  have "(real c)*(real  x) < (real c)*(- (Inum (y # bs) e / real c))"
-      by (simp only:  real_mult_less_mono2[OF rcpos th1])
-    thus "real c * real x + Inum (real x # bs) e \<le> 0" 
-      using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"] rcpos by simp
-  qed
-  thus ?case by blast
-next
-  case (7 c e) 
-  from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp
-  from prems have nbe: "numbound0 e" by simp
-  fix y
-  have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Gt (CN 0 c e))) = ?I x (Gt (CN 0 c e))"
-  proof (simp add: less_floor_eq , rule allI, rule impI) 
-    fix x
-    assume A: "real x + (1\<Colon>real) \<le> - (Inum (y # bs) e / real c)"
-    hence th1:"real x < - (Inum (y # bs) e / real c)" by simp
-    with rcpos  have "(real c)*(real  x) < (real c)*(- (Inum (y # bs) e / real c))"
-      by (simp only:  real_mult_less_mono2[OF rcpos th1])
-    thus "\<not> (real c * real x + Inum (real x # bs) e>0)" 
-      using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"] rcpos by simp
-  qed
-  thus ?case by blast
-next
-  case (8 c e) 
-  from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp
-  from prems have nbe: "numbound0 e" by simp
-  fix y
-  have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Ge (CN 0 c e))) = ?I x (Ge (CN 0 c e))"
-  proof (simp add: less_floor_eq , rule allI, rule impI) 
-    fix x
-    assume A: "real x + (1\<Colon>real) \<le> - (Inum (y # bs) e / real c)"
-    hence th1:"real x < - (Inum (y # bs) e / real c)" by simp
-    with rcpos  have "(real c)*(real  x) < (real c)*(- (Inum (y # bs) e / real c))"
-      by (simp only:  real_mult_less_mono2[OF rcpos th1])
-    thus "\<not> real c * real x + Inum (real x # bs) e \<ge> 0" 
-      using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"] rcpos by simp
-  qed
-  thus ?case by blast
-qed simp_all
-
-lemma minusinf_repeats:
-  assumes d: "d\<delta> p d" and linp: "iszlfm p (a # bs)"
-  shows "Ifm ((real(x - k*d))#bs) (minusinf p) = Ifm (real x #bs) (minusinf p)"
-using linp d
-proof(induct p rule: iszlfm.induct) 
-  case (9 i c e) hence nbe: "numbound0 e"  and id: "i dvd d" by simp+
-    hence "\<exists> k. d=i*k" by (simp add: dvd_def)
-    then obtain "di" where di_def: "d=i*di" by blast
-    show ?case 
-    proof(simp add: numbound0_I[OF nbe,where bs="bs" and b="real x - real k * real d" and b'="real x"] right_diff_distrib, rule iffI)
-      assume 
-	"real i rdvd real c * real x - real c * (real k * real d) + Inum (real x # bs) e"
-      (is "?ri rdvd ?rc*?rx - ?rc*(?rk*?rd) + ?I x e" is "?ri rdvd ?rt")
-      hence "\<exists> (l::int). ?rt = ?ri * (real l)" by (simp add: rdvd_def)
-      hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real l)+?rc*(?rk * (real i) * (real di))" 
-	by (simp add: algebra_simps di_def)
-      hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real (l + c*k*di))"
-	by (simp add: algebra_simps)
-      hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri* (real l)" by blast
-      thus "real i rdvd real c * real x + Inum (real x # bs) e" using rdvd_def by simp
-    next
-      assume 
-	"real i rdvd real c * real x + Inum (real x # bs) e" (is "?ri rdvd ?rc*?rx+?e")
-      hence "\<exists> (l::int). ?rc*?rx+?e = ?ri * (real l)" by (simp add: rdvd_def)
-      hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l) - real c * (real k * real d)" by simp
-      hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l) - real c * (real k * real i * real di)" by (simp add: di_def)
-      hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real (l - c*k*di))" by (simp add: algebra_simps)
-      hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l)"
-	by blast
-      thus "real i rdvd real c * real x - real c * (real k * real d) + Inum (real x # bs) e" using rdvd_def by simp
-    qed
-next
-  case (10 i c e) hence nbe: "numbound0 e"  and id: "i dvd d" by simp+
-    hence "\<exists> k. d=i*k" by (simp add: dvd_def)
-    then obtain "di" where di_def: "d=i*di" by blast
-    show ?case 
-    proof(simp add: numbound0_I[OF nbe,where bs="bs" and b="real x - real k * real d" and b'="real x"] right_diff_distrib, rule iffI)
-      assume 
-	"real i rdvd real c * real x - real c * (real k * real d) + Inum (real x # bs) e"
-      (is "?ri rdvd ?rc*?rx - ?rc*(?rk*?rd) + ?I x e" is "?ri rdvd ?rt")
-      hence "\<exists> (l::int). ?rt = ?ri * (real l)" by (simp add: rdvd_def)
-      hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real l)+?rc*(?rk * (real i) * (real di))" 
-	by (simp add: algebra_simps di_def)
-      hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real (l + c*k*di))"
-	by (simp add: algebra_simps)
-      hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri* (real l)" by blast
-      thus "real i rdvd real c * real x + Inum (real x # bs) e" using rdvd_def by simp
-    next
-      assume 
-	"real i rdvd real c * real x + Inum (real x # bs) e" (is "?ri rdvd ?rc*?rx+?e")
-      hence "\<exists> (l::int). ?rc*?rx+?e = ?ri * (real l)" by (simp add: rdvd_def)
-      hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l) - real c * (real k * real d)" by simp
-      hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l) - real c * (real k * real i * real di)" by (simp add: di_def)
-      hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real (l - c*k*di))" by (simp add: algebra_simps)
-      hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l)"
-	by blast
-      thus "real i rdvd real c * real x - real c * (real k * real d) + Inum (real x # bs) e" using rdvd_def by simp
-    qed
-qed (auto simp add: nth_pos2 numbound0_I[where bs="bs" and b="real(x - k*d)" and b'="real x"] simp del: real_of_int_mult real_of_int_diff)
-
-lemma minusinf_ex:
-  assumes lin: "iszlfm p (real (a::int) #bs)"
-  and exmi: "\<exists> (x::int). Ifm (real x#bs) (minusinf p)" (is "\<exists> x. ?P1 x")
-  shows "\<exists> (x::int). Ifm (real x#bs) p" (is "\<exists> x. ?P x")
-proof-
-  let ?d = "\<delta> p"
-  from \<delta> [OF lin] have dpos: "?d >0" by simp
-  from \<delta> [OF lin] have alld: "d\<delta> p ?d" by simp
-  from minusinf_repeats[OF alld lin] have th1:"\<forall> x k. ?P1 x = ?P1 (x - (k * ?d))" by simp
-  from minusinf_inf[OF lin] have th2:"\<exists> z. \<forall> x. x<z \<longrightarrow> (?P x = ?P1 x)" by blast
-  from minusinfinity [OF dpos th1 th2] exmi show ?thesis by blast
-qed
-
-lemma minusinf_bex:
-  assumes lin: "iszlfm p (real (a::int) #bs)"
-  shows "(\<exists> (x::int). Ifm (real x#bs) (minusinf p)) = 
-         (\<exists> (x::int)\<in> {1..\<delta> p}. Ifm (real x#bs) (minusinf p))"
-  (is "(\<exists> x. ?P x) = _")
-proof-
-  let ?d = "\<delta> p"
-  from \<delta> [OF lin] have dpos: "?d >0" by simp
-  from \<delta> [OF lin] have alld: "d\<delta> p ?d" by simp
-  from minusinf_repeats[OF alld lin] have th1:"\<forall> x k. ?P x = ?P (x - (k * ?d))" by simp
-  from periodic_finite_ex[OF dpos th1] show ?thesis by blast
-qed
-
-lemma dvd1_eq1: "x >0 \<Longrightarrow> (x::int) dvd 1 = (x = 1)" by auto
-
-consts 
-  a\<beta> :: "fm \<Rightarrow> int \<Rightarrow> fm" (* adjusts the coeffitients of a formula *)
-  d\<beta> :: "fm \<Rightarrow> int \<Rightarrow> bool" (* tests if all coeffs c of c divide a given l*)
-  \<zeta>  :: "fm \<Rightarrow> int" (* computes the lcm of all coefficients of x*)
-  \<beta> :: "fm \<Rightarrow> num list"
-  \<alpha> :: "fm \<Rightarrow> num list"
-
-recdef a\<beta> "measure size"
-  "a\<beta> (And p q) = (\<lambda> k. And (a\<beta> p k) (a\<beta> q k))" 
-  "a\<beta> (Or p q) = (\<lambda> k. Or (a\<beta> p k) (a\<beta> q k))" 
-  "a\<beta> (Eq  (CN 0 c e)) = (\<lambda> k. Eq (CN 0 1 (Mul (k div c) e)))"
-  "a\<beta> (NEq (CN 0 c e)) = (\<lambda> k. NEq (CN 0 1 (Mul (k div c) e)))"
-  "a\<beta> (Lt  (CN 0 c e)) = (\<lambda> k. Lt (CN 0 1 (Mul (k div c) e)))"
-  "a\<beta> (Le  (CN 0 c e)) = (\<lambda> k. Le (CN 0 1 (Mul (k div c) e)))"
-  "a\<beta> (Gt  (CN 0 c e)) = (\<lambda> k. Gt (CN 0 1 (Mul (k div c) e)))"
-  "a\<beta> (Ge  (CN 0 c e)) = (\<lambda> k. Ge (CN 0 1 (Mul (k div c) e)))"
-  "a\<beta> (Dvd i (CN 0 c e)) =(\<lambda> k. Dvd ((k div c)*i) (CN 0 1 (Mul (k div c) e)))"
-  "a\<beta> (NDvd i (CN 0 c e))=(\<lambda> k. NDvd ((k div c)*i) (CN 0 1 (Mul (k div c) e)))"
-  "a\<beta> p = (\<lambda> k. p)"
-
-recdef d\<beta> "measure size"
-  "d\<beta> (And p q) = (\<lambda> k. (d\<beta> p k) \<and> (d\<beta> q k))" 
-  "d\<beta> (Or p q) = (\<lambda> k. (d\<beta> p k) \<and> (d\<beta> q k))" 
-  "d\<beta> (Eq  (CN 0 c e)) = (\<lambda> k. c dvd k)"
-  "d\<beta> (NEq (CN 0 c e)) = (\<lambda> k. c dvd k)"
-  "d\<beta> (Lt  (CN 0 c e)) = (\<lambda> k. c dvd k)"
-  "d\<beta> (Le  (CN 0 c e)) = (\<lambda> k. c dvd k)"
-  "d\<beta> (Gt  (CN 0 c e)) = (\<lambda> k. c dvd k)"
-  "d\<beta> (Ge  (CN 0 c e)) = (\<lambda> k. c dvd k)"
-  "d\<beta> (Dvd i (CN 0 c e)) =(\<lambda> k. c dvd k)"
-  "d\<beta> (NDvd i (CN 0 c e))=(\<lambda> k. c dvd k)"
-  "d\<beta> p = (\<lambda> k. True)"
-
-recdef \<zeta> "measure size"
-  "\<zeta> (And p q) = zlcm (\<zeta> p) (\<zeta> q)" 
-  "\<zeta> (Or p q) = zlcm (\<zeta> p) (\<zeta> q)" 
-  "\<zeta> (Eq  (CN 0 c e)) = c"
-  "\<zeta> (NEq (CN 0 c e)) = c"
-  "\<zeta> (Lt  (CN 0 c e)) = c"
-  "\<zeta> (Le  (CN 0 c e)) = c"
-  "\<zeta> (Gt  (CN 0 c e)) = c"
-  "\<zeta> (Ge  (CN 0 c e)) = c"
-  "\<zeta> (Dvd i (CN 0 c e)) = c"
-  "\<zeta> (NDvd i (CN 0 c e))= c"
-  "\<zeta> p = 1"
-
-recdef \<beta> "measure size"
-  "\<beta> (And p q) = (\<beta> p @ \<beta> q)" 
-  "\<beta> (Or p q) = (\<beta> p @ \<beta> q)" 
-  "\<beta> (Eq  (CN 0 c e)) = [Sub (C -1) e]"
-  "\<beta> (NEq (CN 0 c e)) = [Neg e]"
-  "\<beta> (Lt  (CN 0 c e)) = []"
-  "\<beta> (Le  (CN 0 c e)) = []"
-  "\<beta> (Gt  (CN 0 c e)) = [Neg e]"
-  "\<beta> (Ge  (CN 0 c e)) = [Sub (C -1) e]"
-  "\<beta> p = []"
-
-recdef \<alpha> "measure size"
-  "\<alpha> (And p q) = (\<alpha> p @ \<alpha> q)" 
-  "\<alpha> (Or p q) = (\<alpha> p @ \<alpha> q)" 
-  "\<alpha> (Eq  (CN 0 c e)) = [Add (C -1) e]"
-  "\<alpha> (NEq (CN 0 c e)) = [e]"
-  "\<alpha> (Lt  (CN 0 c e)) = [e]"
-  "\<alpha> (Le  (CN 0 c e)) = [Add (C -1) e]"
-  "\<alpha> (Gt  (CN 0 c e)) = []"
-  "\<alpha> (Ge  (CN 0 c e)) = []"
-  "\<alpha> p = []"
-consts mirror :: "fm \<Rightarrow> fm"
-recdef mirror "measure size"
-  "mirror (And p q) = And (mirror p) (mirror q)" 
-  "mirror (Or p q) = Or (mirror p) (mirror q)" 
-  "mirror (Eq  (CN 0 c e)) = Eq (CN 0 c (Neg e))"
-  "mirror (NEq (CN 0 c e)) = NEq (CN 0 c (Neg e))"
-  "mirror (Lt  (CN 0 c e)) = Gt (CN 0 c (Neg e))"
-  "mirror (Le  (CN 0 c e)) = Ge (CN 0 c (Neg e))"
-  "mirror (Gt  (CN 0 c e)) = Lt (CN 0 c (Neg e))"
-  "mirror (Ge  (CN 0 c e)) = Le (CN 0 c (Neg e))"
-  "mirror (Dvd i (CN 0 c e)) = Dvd i (CN 0 c (Neg e))"
-  "mirror (NDvd i (CN 0 c e)) = NDvd i (CN 0 c (Neg e))"
-  "mirror p = p"
-
-lemma mirror\<alpha>\<beta>:
-  assumes lp: "iszlfm p (a#bs)"
-  shows "(Inum (real (i::int)#bs)) ` set (\<alpha> p) = (Inum (real i#bs)) ` set (\<beta> (mirror p))"
-using lp
-by (induct p rule: mirror.induct, auto)
-
-lemma mirror: 
-  assumes lp: "iszlfm p (a#bs)"
-  shows "Ifm (real (x::int)#bs) (mirror p) = Ifm (real (- x)#bs) p" 
-using lp
-proof(induct p rule: iszlfm.induct)
-  case (9 j c e)
-  have th: "(real j rdvd real c * real x - Inum (real x # bs) e) =
-       (real j rdvd - (real c * real x - Inum (real x # bs) e))"
-    by (simp only: rdvd_minus[symmetric])
-  from prems th show  ?case
-    by (simp add: algebra_simps
-      numbound0_I[where bs="bs" and b'="real x" and b="- real x"])
-next
-    case (10 j c e)
-  have th: "(real j rdvd real c * real x - Inum (real x # bs) e) =
-       (real j rdvd - (real c * real x - Inum (real x # bs) e))"
-    by (simp only: rdvd_minus[symmetric])
-  from prems th show  ?case
-    by (simp add: algebra_simps
-      numbound0_I[where bs="bs" and b'="real x" and b="- real x"])
-qed (auto simp add: numbound0_I[where bs="bs" and b="real x" and b'="- real x"] nth_pos2)
-
-lemma mirror_l: "iszlfm p (a#bs) \<Longrightarrow> iszlfm (mirror p) (a#bs)"
-by (induct p rule: mirror.induct, auto simp add: isint_neg)
-
-lemma mirror_d\<beta>: "iszlfm p (a#bs) \<and> d\<beta> p 1 
-  \<Longrightarrow> iszlfm (mirror p) (a#bs) \<and> d\<beta> (mirror p) 1"
-by (induct p rule: mirror.induct, auto simp add: isint_neg)
-
-lemma mirror_\<delta>: "iszlfm p (a#bs) \<Longrightarrow> \<delta> (mirror p) = \<delta> p"
-by (induct p rule: mirror.induct,auto)
-
-
-lemma mirror_ex: 
-  assumes lp: "iszlfm p (real (i::int)#bs)"
-  shows "(\<exists> (x::int). Ifm (real x#bs) (mirror p)) = (\<exists> (x::int). Ifm (real x#bs) p)"
-  (is "(\<exists> x. ?I x ?mp) = (\<exists> x. ?I x p)")
-proof(auto)
-  fix x assume "?I x ?mp" hence "?I (- x) p" using mirror[OF lp] by blast
-  thus "\<exists> x. ?I x p" by blast
-next
-  fix x assume "?I x p" hence "?I (- x) ?mp" 
-    using mirror[OF lp, where x="- x", symmetric] by auto
-  thus "\<exists> x. ?I x ?mp" by blast
-qed
-
-lemma \<beta>_numbound0: assumes lp: "iszlfm p bs"
-  shows "\<forall> b\<in> set (\<beta> p). numbound0 b"
-  using lp by (induct p rule: \<beta>.induct,auto)
-
-lemma d\<beta>_mono: 
-  assumes linp: "iszlfm p (a #bs)"
-  and dr: "d\<beta> p l"
-  and d: "l dvd l'"
-  shows "d\<beta> p l'"
-using dr linp zdvd_trans[where n="l" and k="l'", simplified d]
-by (induct p rule: iszlfm.induct) simp_all
-
-lemma \<alpha>_l: assumes lp: "iszlfm p (a#bs)"
-  shows "\<forall> b\<in> set (\<alpha> p). numbound0 b \<and> isint b (a#bs)"
-using lp
-by(induct p rule: \<alpha>.induct, auto simp add: isint_add isint_c)
-
-lemma \<zeta>: 
-  assumes linp: "iszlfm p (a #bs)"
-  shows "\<zeta> p > 0 \<and> d\<beta> p (\<zeta> p)"
-using linp
-proof(induct p rule: iszlfm.induct)
-  case (1 p q)
-  from prems have dl1: "\<zeta> p dvd zlcm (\<zeta> p) (\<zeta> q)" by simp
-  from prems have dl2: "\<zeta> q dvd zlcm (\<zeta> p) (\<zeta> q)" by simp
-  from prems d\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="zlcm (\<zeta> p) (\<zeta> q)"] 
-    d\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="zlcm (\<zeta> p) (\<zeta> q)"] 
-    dl1 dl2 show ?case by (auto simp add: zlcm_pos)
-next
-  case (2 p q)
-  from prems have dl1: "\<zeta> p dvd zlcm (\<zeta> p) (\<zeta> q)" by simp
-  from prems have dl2: "\<zeta> q dvd zlcm (\<zeta> p) (\<zeta> q)" by simp
-  from prems d\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="zlcm (\<zeta> p) (\<zeta> q)"] 
-    d\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="zlcm (\<zeta> p) (\<zeta> q)"] 
-    dl1 dl2 show ?case by (auto simp add: zlcm_pos)
-qed (auto simp add: zlcm_pos)
-
-lemma a\<beta>: assumes linp: "iszlfm p (a #bs)" and d: "d\<beta> p l" and lp: "l > 0"
-  shows "iszlfm (a\<beta> p l) (a #bs) \<and> d\<beta> (a\<beta> p l) 1 \<and> (Ifm (real (l * x) #bs) (a\<beta> p l) = Ifm ((real x)#bs) p)"
-using linp d
-proof (induct p rule: iszlfm.induct)
-  case (5 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+
-    from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
-    from cp have cnz: "c \<noteq> 0" by simp
-    have "c div c\<le> l div c"
-      by (simp add: zdiv_mono1[OF clel cp])
-    then have ldcp:"0 < l div c" 
-      by (simp add: zdiv_self[OF cnz])
-    have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp
-    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
-      by simp
-    hence "(real l * real x + real (l div c) * Inum (real x # bs) e < (0\<Colon>real)) =
-          (real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e < 0)"
-      by simp
-    also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) < (real (l div c)) * 0)" by (simp add: algebra_simps)
-    also have "\<dots> = (real c * real x + Inum (real x # bs) e < 0)"
-    using mult_less_0_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp
-  finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] be  isint_Mul[OF ei] by simp
-next
-  case (6 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+
-    from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
-    from cp have cnz: "c \<noteq> 0" by simp
-    have "c div c\<le> l div c"
-      by (simp add: zdiv_mono1[OF clel cp])
-    then have ldcp:"0 < l div c" 
-      by (simp add: zdiv_self[OF cnz])
-    have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp
-    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
-      by simp
-    hence "(real l * real x + real (l div c) * Inum (real x # bs) e \<le> (0\<Colon>real)) =
-          (real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e \<le> 0)"
-      by simp
-    also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) \<le> (real (l div c)) * 0)" by (simp add: algebra_simps)
-    also have "\<dots> = (real c * real x + Inum (real x # bs) e \<le> 0)"
-    using mult_le_0_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp
-  finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"]  be  isint_Mul[OF ei] by simp
-next
-  case (7 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+
-    from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
-    from cp have cnz: "c \<noteq> 0" by simp
-    have "c div c\<le> l div c"
-      by (simp add: zdiv_mono1[OF clel cp])
-    then have ldcp:"0 < l div c" 
-      by (simp add: zdiv_self[OF cnz])
-    have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp
-    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
-      by simp
-    hence "(real l * real x + real (l div c) * Inum (real x # bs) e > (0\<Colon>real)) =
-          (real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e > 0)"
-      by simp
-    also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) > (real (l div c)) * 0)" by (simp add: algebra_simps)
-    also have "\<dots> = (real c * real x + Inum (real x # bs) e > 0)"
-    using zero_less_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp
-  finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"]  be  isint_Mul[OF ei] by simp
-next
-  case (8 c e) hence cp: "c>0" and be: "numbound0 e"  and ei:"isint e (a#bs)" and d': "c dvd l" by simp+
-    from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
-    from cp have cnz: "c \<noteq> 0" by simp
-    have "c div c\<le> l div c"
-      by (simp add: zdiv_mono1[OF clel cp])
-    then have ldcp:"0 < l div c" 
-      by (simp add: zdiv_self[OF cnz])
-    have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp
-    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
-      by simp
-    hence "(real l * real x + real (l div c) * Inum (real x # bs) e \<ge> (0\<Colon>real)) =
-          (real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e \<ge> 0)"
-      by simp
-    also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) \<ge> (real (l div c)) * 0)" by (simp add: algebra_simps)
-    also have "\<dots> = (real c * real x + Inum (real x # bs) e \<ge> 0)"
-    using zero_le_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp
-  finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"]  be  isint_Mul[OF ei] by simp
-next
-  case (3 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+
-    from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
-    from cp have cnz: "c \<noteq> 0" by simp
-    have "c div c\<le> l div c"
-      by (simp add: zdiv_mono1[OF clel cp])
-    then have ldcp:"0 < l div c" 
-      by (simp add: zdiv_self[OF cnz])
-    have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp
-    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
-      by simp
-    hence "(real l * real x + real (l div c) * Inum (real x # bs) e = (0\<Colon>real)) =
-          (real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e = 0)"
-      by simp
-    also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) = (real (l div c)) * 0)" by (simp add: algebra_simps)
-    also have "\<dots> = (real c * real x + Inum (real x # bs) e = 0)"
-    using mult_eq_0_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp
-  finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"]  be  isint_Mul[OF ei] by simp
-next
-  case (4 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+
-    from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
-    from cp have cnz: "c \<noteq> 0" by simp
-    have "c div c\<le> l div c"
-      by (simp add: zdiv_mono1[OF clel cp])
-    then have ldcp:"0 < l div c" 
-      by (simp add: zdiv_self[OF cnz])
-    have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp
-    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
-      by simp
-    hence "(real l * real x + real (l div c) * Inum (real x # bs) e \<noteq> (0\<Colon>real)) =
-          (real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e \<noteq> 0)"
-      by simp
-    also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) \<noteq> (real (l div c)) * 0)" by (simp add: algebra_simps)
-    also have "\<dots> = (real c * real x + Inum (real x # bs) e \<noteq> 0)"
-    using zero_le_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp
-  finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"]  be  isint_Mul[OF ei] by simp
-next
-  case (9 j c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and jp: "j > 0" and d': "c dvd l" by simp+
-    from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
-    from cp have cnz: "c \<noteq> 0" by simp
-    have "c div c\<le> l div c"
-      by (simp add: zdiv_mono1[OF clel cp])
-    then have ldcp:"0 < l div c" 
-      by (simp add: zdiv_self[OF cnz])
-    have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp
-    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
-      by simp
-    hence "(\<exists> (k::int). real l * real x + real (l div c) * Inum (real x # bs) e = (real (l div c) * real j) * real k) = (\<exists> (k::int). real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e = (real (l div c) * real j) * real k)"  by simp
-    also have "\<dots> = (\<exists> (k::int). real (l div c) * (real c * real x + Inum (real x # bs) e - real j * real k) = real (l div c)*0)" by (simp add: algebra_simps)
-    also fix k have "\<dots> = (\<exists> (k::int). real c * real x + Inum (real x # bs) e - real j * real k = 0)"
-    using zero_le_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e - real j * real k"] ldcp by simp
-  also have "\<dots> = (\<exists> (k::int). real c * real x + Inum (real x # bs) e = real j * real k)" by simp
-  finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] rdvd_def  be  isint_Mul[OF ei] mult_strict_mono[OF ldcp jp ldcp ] by simp 
-next
-  case (10 j c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and jp: "j > 0" and d': "c dvd l" by simp+
-    from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
-    from cp have cnz: "c \<noteq> 0" by simp
-    have "c div c\<le> l div c"
-      by (simp add: zdiv_mono1[OF clel cp])
-    then have ldcp:"0 < l div c" 
-      by (simp add: zdiv_self[OF cnz])
-    have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp
-    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
-      by simp
-    hence "(\<exists> (k::int). real l * real x + real (l div c) * Inum (real x # bs) e = (real (l div c) * real j) * real k) = (\<exists> (k::int). real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e = (real (l div c) * real j) * real k)"  by simp
-    also have "\<dots> = (\<exists> (k::int). real (l div c) * (real c * real x + Inum (real x # bs) e - real j * real k) = real (l div c)*0)" by (simp add: algebra_simps)
-    also fix k have "\<dots> = (\<exists> (k::int). real c * real x + Inum (real x # bs) e - real j * real k = 0)"
-    using zero_le_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e - real j * real k"] ldcp by simp
-  also have "\<dots> = (\<exists> (k::int). real c * real x + Inum (real x # bs) e = real j * real k)" by simp
-  finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] rdvd_def  be  isint_Mul[OF ei]  mult_strict_mono[OF ldcp jp ldcp ] by simp
-qed (simp_all add: nth_pos2 numbound0_I[where bs="bs" and b="real (l * x)" and b'="real x"] isint_Mul del: real_of_int_mult)
-
-lemma a\<beta>_ex: assumes linp: "iszlfm p (a#bs)" and d: "d\<beta> p l" and lp: "l>0"
-  shows "(\<exists> x. l dvd x \<and> Ifm (real x #bs) (a\<beta> p l)) = (\<exists> (x::int). Ifm (real x#bs) p)"
-  (is "(\<exists> x. l dvd x \<and> ?P x) = (\<exists> x. ?P' x)")
-proof-
-  have "(\<exists> x. l dvd x \<and> ?P x) = (\<exists> (x::int). ?P (l*x))"
-    using unity_coeff_ex[where l="l" and P="?P", simplified] by simp
-  also have "\<dots> = (\<exists> (x::int). ?P' x)" using a\<beta>[OF linp d lp] by simp
-  finally show ?thesis  . 
-qed
-
-lemma \<beta>:
-  assumes lp: "iszlfm p (a#bs)"
-  and u: "d\<beta> p 1"
-  and d: "d\<delta> p d"
-  and dp: "d > 0"
-  and nob: "\<not>(\<exists>(j::int) \<in> {1 .. d}. \<exists> b\<in> (Inum (a#bs)) ` set(\<beta> p). real x = b + real j)"
-  and p: "Ifm (real x#bs) p" (is "?P x")
-  shows "?P (x - d)"
-using lp u d dp nob p
-proof(induct p rule: iszlfm.induct)
-  case (5 c e) hence c1: "c=1" and  bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+
-    with dp p c1 numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] prems
-    show ?case by (simp del: real_of_int_minus)
-next
-  case (6 c e)  hence c1: "c=1" and  bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+
-    with dp p c1 numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] prems
-    show ?case by (simp del: real_of_int_minus)
-next
-  case (7 c e) hence p: "Ifm (real x #bs) (Gt (CN 0 c e))" and c1: "c=1" and bn:"numbound0 e" and ie1:"isint e (a#bs)" using dvd1_eq1[where x="c"] by simp+
-    let ?e = "Inum (real x # bs) e"
-    from ie1 have ie: "real (floor ?e) = ?e" using isint_iff[where n="e" and bs="a#bs"]
-      numbound0_I[OF bn,where b="a" and b'="real x" and bs="bs"]
-      by (simp add: isint_iff)
-    {assume "real (x-d) +?e > 0" hence ?case using c1 
-      numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"]
-	by (simp del: real_of_int_minus)}
-    moreover
-    {assume H: "\<not> real (x-d) + ?e > 0" 
-      let ?v="Neg e"
-      have vb: "?v \<in> set (\<beta> (Gt (CN 0 c e)))" by simp
-      from prems(11)[simplified simp_thms Inum.simps \<beta>.simps set.simps bex_simps numbound0_I[OF bn,where b="a" and b'="real x" and bs="bs"]] 
-      have nob: "\<not> (\<exists> j\<in> {1 ..d}. real x =  - ?e + real j)" by auto 
-      from H p have "real x + ?e > 0 \<and> real x + ?e \<le> real d" by (simp add: c1)
-      hence "real (x + floor ?e) > real (0::int) \<and> real (x + floor ?e) \<le> real d"
-	using ie by simp
-      hence "x + floor ?e \<ge> 1 \<and> x + floor ?e \<le> d"  by simp
-      hence "\<exists> (j::int) \<in> {1 .. d}. j = x + floor ?e" by simp
-      hence "\<exists> (j::int) \<in> {1 .. d}. real x = real (- floor ?e + j)" 
-	by (simp only: real_of_int_inject) (simp add: algebra_simps)
-      hence "\<exists> (j::int) \<in> {1 .. d}. real x = - ?e + real j" 
-	by (simp add: ie[simplified isint_iff])
-      with nob have ?case by auto}
-    ultimately show ?case by blast
-next
-  case (8 c e) hence p: "Ifm (real x #bs) (Ge (CN 0 c e))" and c1: "c=1" and bn:"numbound0 e" 
-    and ie1:"isint e (a #bs)" using dvd1_eq1[where x="c"] by simp+
-    let ?e = "Inum (real x # bs) e"
-    from ie1 have ie: "real (floor ?e) = ?e" using numbound0_I[OF bn,where b="real x" and b'="a" and bs="bs"] isint_iff[where n="e" and bs="(real x)#bs"]
-      by (simp add: isint_iff)
-    {assume "real (x-d) +?e \<ge> 0" hence ?case using  c1 
-      numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"]
-	by (simp del: real_of_int_minus)}
-    moreover
-    {assume H: "\<not> real (x-d) + ?e \<ge> 0" 
-      let ?v="Sub (C -1) e"
-      have vb: "?v \<in> set (\<beta> (Ge (CN 0 c e)))" by simp
-      from prems(11)[simplified simp_thms Inum.simps \<beta>.simps set.simps bex_simps numbound0_I[OF bn,where b="a" and b'="real x" and bs="bs"]] 
-      have nob: "\<not> (\<exists> j\<in> {1 ..d}. real x =  - ?e - 1 + real j)" by auto 
-      from H p have "real x + ?e \<ge> 0 \<and> real x + ?e < real d" by (simp add: c1)
-      hence "real (x + floor ?e) \<ge> real (0::int) \<and> real (x + floor ?e) < real d"
-	using ie by simp
-      hence "x + floor ?e +1 \<ge> 1 \<and> x + floor ?e + 1 \<le> d"  by simp
-      hence "\<exists> (j::int) \<in> {1 .. d}. j = x + floor ?e + 1" by simp
-      hence "\<exists> (j::int) \<in> {1 .. d}. x= - floor ?e - 1 + j" by (simp add: algebra_simps)
-      hence "\<exists> (j::int) \<in> {1 .. d}. real x= real (- floor ?e - 1 + j)" 
-	by (simp only: real_of_int_inject)
-      hence "\<exists> (j::int) \<in> {1 .. d}. real x= - ?e - 1 + real j" 
-	by (simp add: ie[simplified isint_iff])
-      with nob have ?case by simp }
-    ultimately show ?case by blast
-next
-  case (3 c e) hence p: "Ifm (real x #bs) (Eq (CN 0 c e))" (is "?p x") and c1: "c=1" 
-    and bn:"numbound0 e" and ie1: "isint e (a #bs)" using dvd1_eq1[where x="c"] by simp+
-    let ?e = "Inum (real x # bs) e"
-    let ?v="(Sub (C -1) e)"
-    have vb: "?v \<in> set (\<beta> (Eq (CN 0 c e)))" by simp
-    from p have "real x= - ?e" by (simp add: c1) with prems(11) show ?case using dp
-      by simp (erule ballE[where x="1"],
-	simp_all add:algebra_simps numbound0_I[OF bn,where b="real x"and b'="a"and bs="bs"])
-next
-  case (4 c e)hence p: "Ifm (real x #bs) (NEq (CN 0 c e))" (is "?p x") and c1: "c=1" 
-    and bn:"numbound0 e" and ie1: "isint e (a #bs)" using dvd1_eq1[where x="c"] by simp+
-    let ?e = "Inum (real x # bs) e"
-    let ?v="Neg e"
-    have vb: "?v \<in> set (\<beta> (NEq (CN 0 c e)))" by simp
-    {assume "real x - real d + Inum ((real (x -d)) # bs) e \<noteq> 0" 
-      hence ?case by (simp add: c1)}
-    moreover
-    {assume H: "real x - real d + Inum ((real (x -d)) # bs) e = 0"
-      hence "real x = - Inum ((real (x -d)) # bs) e + real d" by simp
-      hence "real x = - Inum (a # bs) e + real d"
-	by (simp add: numbound0_I[OF bn,where b="real x - real d"and b'="a"and bs="bs"])
-       with prems(11) have ?case using dp by simp}
-  ultimately show ?case by blast
-next 
-  case (9 j c e) hence p: "Ifm (real x #bs) (Dvd j (CN 0 c e))" (is "?p x") and c1: "c=1" 
-    and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+
-    let ?e = "Inum (real x # bs) e"
-    from prems have "isint e (a #bs)"  by simp 
-    hence ie: "real (floor ?e) = ?e" using isint_iff[where n="e" and bs="(real x)#bs"] numbound0_I[OF bn,where b="real x" and b'="a" and bs="bs"]
-      by (simp add: isint_iff)
-    from prems have id: "j dvd d" by simp
-    from c1 ie[symmetric] have "?p x = (real j rdvd real (x+ floor ?e))" by simp
-    also have "\<dots> = (j dvd x + floor ?e)" 
-      using int_rdvd_real[where i="j" and x="real (x+ floor ?e)"] by simp
-    also have "\<dots> = (j dvd x - d + floor ?e)" 
-      using dvd_period[OF id, where x="x" and c="-1" and t="floor ?e"] by simp
-    also have "\<dots> = (real j rdvd real (x - d + floor ?e))" 
-      using int_rdvd_real[where i="j" and x="real (x-d + floor ?e)",symmetric, simplified]
-      ie by simp
-    also have "\<dots> = (real j rdvd real x - real d + ?e)" 
-      using ie by simp
-    finally show ?case 
-      using numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] c1 p by simp
-next
-  case (10 j c e) hence p: "Ifm (real x #bs) (NDvd j (CN 0 c e))" (is "?p x") and c1: "c=1" and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+
-    let ?e = "Inum (real x # bs) e"
-    from prems have "isint e (a#bs)"  by simp 
-    hence ie: "real (floor ?e) = ?e" using numbound0_I[OF bn,where b="real x" and b'="a" and bs="bs"] isint_iff[where n="e" and bs="(real x)#bs"]
-      by (simp add: isint_iff)
-    from prems have id: "j dvd d" by simp
-    from c1 ie[symmetric] have "?p x = (\<not> real j rdvd real (x+ floor ?e))" by simp
-    also have "\<dots> = (\<not> j dvd x + floor ?e)" 
-      using int_rdvd_real[where i="j" and x="real (x+ floor ?e)"] by simp
-    also have "\<dots> = (\<not> j dvd x - d + floor ?e)" 
-      using dvd_period[OF id, where x="x" and c="-1" and t="floor ?e"] by simp
-    also have "\<dots> = (\<not> real j rdvd real (x - d + floor ?e))" 
-      using int_rdvd_real[where i="j" and x="real (x-d + floor ?e)",symmetric, simplified]
-      ie by simp
-    also have "\<dots> = (\<not> real j rdvd real x - real d + ?e)" 
-      using ie by simp
-    finally show ?case using numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] c1 p by simp
-qed (auto simp add: numbound0_I[where bs="bs" and b="real (x - d)" and b'="real x"] nth_pos2 simp del: real_of_int_diff)
-
-lemma \<beta>':   
-  assumes lp: "iszlfm p (a #bs)"
-  and u: "d\<beta> p 1"
-  and d: "d\<delta> p d"
-  and dp: "d > 0"
-  shows "\<forall> x. \<not>(\<exists>(j::int) \<in> {1 .. d}. \<exists> b\<in> set(\<beta> p). Ifm ((Inum (a#bs) b + real j) #bs) p) \<longrightarrow> Ifm (real x#bs) p \<longrightarrow> Ifm (real (x - d)#bs) p" (is "\<forall> x. ?b \<longrightarrow> ?P x \<longrightarrow> ?P (x - d)")
-proof(clarify)
-  fix x 
-  assume nb:"?b" and px: "?P x" 
-  hence nb2: "\<not>(\<exists>(j::int) \<in> {1 .. d}. \<exists> b\<in> (Inum (a#bs)) ` set(\<beta> p). real x = b + real j)"
-    by auto
-  from  \<beta>[OF lp u d dp nb2 px] show "?P (x -d )" .
-qed
-
-lemma \<beta>_int: assumes lp: "iszlfm p bs"
-  shows "\<forall> b\<in> set (\<beta> p). isint b bs"
-using lp by (induct p rule: iszlfm.induct) (auto simp add: isint_neg isint_sub)
-
-lemma cpmi_eq: "0 < D \<Longrightarrow> (EX z::int. ALL x. x < z --> (P x = P1 x))
-==> ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D) 
-==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x-k*D))))
-==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (b::int) : B. P (b+j)))"
-apply(rule iffI)
-prefer 2
-apply(drule minusinfinity)
-apply assumption+
-apply(fastsimp)
-apply clarsimp
-apply(subgoal_tac "!!k. 0<=k \<Longrightarrow> !x. P x \<longrightarrow> P (x - k*D)")
-apply(frule_tac x = x and z=z in decr_lemma)
-apply(subgoal_tac "P1(x - (\<bar>x - z\<bar> + 1) * D)")
-prefer 2
-apply(subgoal_tac "0 <= (\<bar>x - z\<bar> + 1)")
-prefer 2 apply arith
- apply fastsimp
-apply(drule (1)  periodic_finite_ex)
-apply blast
-apply(blast dest:decr_mult_lemma)
-done
-
-
-theorem cp_thm:
-  assumes lp: "iszlfm p (a #bs)"
-  and u: "d\<beta> p 1"
-  and d: "d\<delta> p d"
-  and dp: "d > 0"
-  shows "(\<exists> (x::int). Ifm (real x #bs) p) = (\<exists> j\<in> {1.. d}. Ifm (real j #bs) (minusinf p) \<or> (\<exists> b \<in> set (\<beta> p). Ifm ((Inum (a#bs) b + real j) #bs) p))"
-  (is "(\<exists> (x::int). ?P (real x)) = (\<exists> j\<in> ?D. ?M j \<or> (\<exists> b\<in> ?B. ?P (?I b + real j)))")
-proof-
-  from minusinf_inf[OF lp] 
-  have th: "\<exists>(z::int). \<forall>x<z. ?P (real x) = ?M x" by blast
-  let ?B' = "{floor (?I b) | b. b\<in> ?B}"
-  from \<beta>_int[OF lp] isint_iff[where bs="a # bs"] have B: "\<forall> b\<in> ?B. real (floor (?I b)) = ?I b" by simp
-  from B[rule_format] 
-  have "(\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (?I b + real j)) = (\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (real (floor (?I b)) + real j))" 
-    by simp
-  also have "\<dots> = (\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (real (floor (?I b) + j)))" by simp
-  also have"\<dots> = (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (real (b + j)))"  by blast
-  finally have BB': 
-    "(\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (?I b + real j)) = (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (real (b + j)))" 
-    by blast 
-  hence th2: "\<forall> x. \<not> (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (real (b + j))) \<longrightarrow> ?P (real x) \<longrightarrow> ?P (real (x - d))" using \<beta>'[OF lp u d dp] by blast
-  from minusinf_repeats[OF d lp]
-  have th3: "\<forall> x k. ?M x = ?M (x-k*d)" by simp
-  from cpmi_eq[OF dp th th2 th3] BB' show ?thesis by blast
-qed
-
-    (* Reddy and Loveland *)
-
-
-consts 
-  \<rho> :: "fm \<Rightarrow> (num \<times> int) list" (* Compute the Reddy and Loveland Bset*)
-  \<sigma>\<rho>:: "fm \<Rightarrow> num \<times> int \<Rightarrow> fm" (* Performs the modified substitution of Reddy and Loveland*)
-  \<alpha>\<rho> :: "fm \<Rightarrow> (num\<times>int) list"
-  a\<rho> :: "fm \<Rightarrow> int \<Rightarrow> fm"
-recdef \<rho> "measure size"
-  "\<rho> (And p q) = (\<rho> p @ \<rho> q)" 
-  "\<rho> (Or p q) = (\<rho> p @ \<rho> q)" 
-  "\<rho> (Eq  (CN 0 c e)) = [(Sub (C -1) e,c)]"
-  "\<rho> (NEq (CN 0 c e)) = [(Neg e,c)]"
-  "\<rho> (Lt  (CN 0 c e)) = []"
-  "\<rho> (Le  (CN 0 c e)) = []"
-  "\<rho> (Gt  (CN 0 c e)) = [(Neg e, c)]"
-  "\<rho> (Ge  (CN 0 c e)) = [(Sub (C (-1)) e, c)]"
-  "\<rho> p = []"
-
-recdef \<sigma>\<rho> "measure size"
-  "\<sigma>\<rho> (And p q) = (\<lambda> (t,k). And (\<sigma>\<rho> p (t,k)) (\<sigma>\<rho> q (t,k)))" 
-  "\<sigma>\<rho> (Or p q) = (\<lambda> (t,k). Or (\<sigma>\<rho> p (t,k)) (\<sigma>\<rho> q (t,k)))" 
-  "\<sigma>\<rho> (Eq  (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Eq (Add (Mul (c div k) t) e)) 
-                                            else (Eq (Add (Mul c t) (Mul k e))))"
-  "\<sigma>\<rho> (NEq (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (NEq (Add (Mul (c div k) t) e)) 
-                                            else (NEq (Add (Mul c t) (Mul k e))))"
-  "\<sigma>\<rho> (Lt  (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Lt (Add (Mul (c div k) t) e)) 
-                                            else (Lt (Add (Mul c t) (Mul k e))))"
-  "\<sigma>\<rho> (Le  (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Le (Add (Mul (c div k) t) e)) 
-                                            else (Le (Add (Mul c t) (Mul k e))))"
-  "\<sigma>\<rho> (Gt  (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Gt (Add (Mul (c div k) t) e)) 
-                                            else (Gt (Add (Mul c t) (Mul k e))))"
-  "\<sigma>\<rho> (Ge  (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Ge (Add (Mul (c div k) t) e)) 
-                                            else (Ge (Add (Mul c t) (Mul k e))))"
-  "\<sigma>\<rho> (Dvd i (CN 0 c e)) =(\<lambda> (t,k). if k dvd c then (Dvd i (Add (Mul (c div k) t) e)) 
-                                            else (Dvd (i*k) (Add (Mul c t) (Mul k e))))"
-  "\<sigma>\<rho> (NDvd i (CN 0 c e))=(\<lambda> (t,k). if k dvd c then (NDvd i (Add (Mul (c div k) t) e)) 
-                                            else (NDvd (i*k) (Add (Mul c t) (Mul k e))))"
-  "\<sigma>\<rho> p = (\<lambda> (t,k). p)"
-
-recdef \<alpha>\<rho> "measure size"
-  "\<alpha>\<rho> (And p q) = (\<alpha>\<rho> p @ \<alpha>\<rho> q)" 
-  "\<alpha>\<rho> (Or p q) = (\<alpha>\<rho> p @ \<alpha>\<rho> q)" 
-  "\<alpha>\<rho> (Eq  (CN 0 c e)) = [(Add (C -1) e,c)]"
-  "\<alpha>\<rho> (NEq (CN 0 c e)) = [(e,c)]"
-  "\<alpha>\<rho> (Lt  (CN 0 c e)) = [(e,c)]"
-  "\<alpha>\<rho> (Le  (CN 0 c e)) = [(Add (C -1) e,c)]"
-  "\<alpha>\<rho> p = []"
-
-    (* Simulates normal substituion by modifying the formula see correctness theorem *)
-
-recdef a\<rho> "measure size"
-  "a\<rho> (And p q) = (\<lambda> k. And (a\<rho> p k) (a\<rho> q k))" 
-  "a\<rho> (Or p q) = (\<lambda> k. Or (a\<rho> p k) (a\<rho> q k))" 
-  "a\<rho> (Eq (CN 0 c e)) = (\<lambda> k. if k dvd c then (Eq (CN 0 (c div k) e)) 
-                                           else (Eq (CN 0 c (Mul k e))))"
-  "a\<rho> (NEq (CN 0 c e)) = (\<lambda> k. if k dvd c then (NEq (CN 0 (c div k) e)) 
-                                           else (NEq (CN 0 c (Mul k e))))"
-  "a\<rho> (Lt (CN 0 c e)) = (\<lambda> k. if k dvd c then (Lt (CN 0 (c div k) e)) 
-                                           else (Lt (CN 0 c (Mul k e))))"
-  "a\<rho> (Le (CN 0 c e)) = (\<lambda> k. if k dvd c then (Le (CN 0 (c div k) e)) 
-                                           else (Le (CN 0 c (Mul k e))))"
-  "a\<rho> (Gt (CN 0 c e)) = (\<lambda> k. if k dvd c then (Gt (CN 0 (c div k) e)) 
-                                           else (Gt (CN 0 c (Mul k e))))"
-  "a\<rho> (Ge (CN 0 c e)) = (\<lambda> k. if k dvd c then (Ge (CN 0 (c div k) e)) 
-                                            else (Ge (CN 0 c (Mul k e))))"
-  "a\<rho> (Dvd i (CN 0 c e)) = (\<lambda> k. if k dvd c then (Dvd i (CN 0 (c div k) e)) 
-                                            else (Dvd (i*k) (CN 0 c (Mul k e))))"
-  "a\<rho> (NDvd i (CN 0 c e)) = (\<lambda> k. if k dvd c then (NDvd i (CN 0 (c div k) e)) 
-                                            else (NDvd (i*k) (CN 0 c (Mul k e))))"
-  "a\<rho> p = (\<lambda> k. p)"
-
-constdefs \<sigma> :: "fm \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm"
-  "\<sigma> p k t \<equiv> And (Dvd k t) (\<sigma>\<rho> p (t,k))"
-
-lemma \<sigma>\<rho>:
-  assumes linp: "iszlfm p (real (x::int)#bs)"
-  and kpos: "real k > 0"
-  and tnb: "numbound0 t"
-  and tint: "isint t (real x#bs)"
-  and kdt: "k dvd floor (Inum (b'#bs) t)"
-  shows "Ifm (real x#bs) (\<sigma>\<rho> p (t,k)) = 
-  (Ifm ((real ((floor (Inum (b'#bs) t)) div k))#bs) p)" 
-  (is "?I (real x) (?s p) = (?I (real ((floor (?N b' t)) div k)) p)" is "_ = (?I ?tk p)")
-using linp kpos tnb
-proof(induct p rule: \<sigma>\<rho>.induct)
-  case (3 c e) 
-  from prems have cp: "c > 0" and nb: "numbound0 e" by auto
-    {assume kdc: "k dvd c" 
-      from kpos have knz: "k\<noteq>0" by simp
-      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
-      from prems have  ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
-	numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } 
-    moreover 
-    {assume "\<not> k dvd c"
-      from kpos have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
-      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
-      from prems have "?I (real x) (?s (Eq (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k = 0)"
-	using real_of_int_div[OF knz kdt]
-	  numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-	  numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
-      also have "\<dots> = (?I ?tk (Eq (CN 0 c e)))" using nonzero_eq_divide_eq[OF knz', where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-	  numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
-	by (simp add: ti)
-      finally have ?case . }
-    ultimately show ?case by blast 
-next
-  case (4 c e)  
-  from prems have cp: "c > 0" and nb: "numbound0 e" by auto
-    {assume kdc: "k dvd c" 
-      from kpos have knz: "k\<noteq>0" by simp
-      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
-      from prems have  ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
-	numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } 
-    moreover 
-    {assume "\<not> k dvd c"
-      from kpos have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
-      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
-      from prems have "?I (real x) (?s (NEq (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k \<noteq> 0)"
-	using real_of_int_div[OF knz kdt]
-	  numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-	  numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
-      also have "\<dots> = (?I ?tk (NEq (CN 0 c e)))" using nonzero_eq_divide_eq[OF knz', where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-	  numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
-	by (simp add: ti)
-      finally have ?case . }
-    ultimately show ?case by blast 
-next
-  case (5 c e) 
-  from prems have cp: "c > 0" and nb: "numbound0 e" by auto
-    {assume kdc: "k dvd c" 
-      from kpos have knz: "k\<noteq>0" by simp
-      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
-      from prems have  ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
-	numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } 
-    moreover 
-    {assume "\<not> k dvd c"
-      from kpos have knz: "k\<noteq>0" by simp
-      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
-      from prems have "?I (real x) (?s (Lt (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k < 0)"
-	using real_of_int_div[OF knz kdt]
-	  numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-	  numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
-      also have "\<dots> = (?I ?tk (Lt (CN 0 c e)))" using pos_less_divide_eq[OF kpos, where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-	  numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
-	by (simp add: ti)
-      finally have ?case . }
-    ultimately show ?case by blast 
-next
-  case (6 c e)  
-  from prems have cp: "c > 0" and nb: "numbound0 e" by auto
-    {assume kdc: "k dvd c" 
-      from kpos have knz: "k\<noteq>0" by simp
-      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
-      from prems have  ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
-	numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } 
-    moreover 
-    {assume "\<not> k dvd c"
-      from kpos have knz: "k\<noteq>0" by simp
-      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
-      from prems have "?I (real x) (?s (Le (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k \<le> 0)"
-	using real_of_int_div[OF knz kdt]
-	  numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-	  numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
-      also have "\<dots> = (?I ?tk (Le (CN 0 c e)))" using pos_le_divide_eq[OF kpos, where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-	  numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
-	by (simp add: ti)
-      finally have ?case . }
-    ultimately show ?case by blast 
-next
-  case (7 c e) 
-  from prems have cp: "c > 0" and nb: "numbound0 e" by auto
-    {assume kdc: "k dvd c" 
-      from kpos have knz: "k\<noteq>0" by simp
-      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
-      from prems have  ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
-	numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } 
-    moreover 
-    {assume "\<not> k dvd c"
-      from kpos have knz: "k\<noteq>0" by simp
-      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
-      from prems have "?I (real x) (?s (Gt (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k > 0)"
-	using real_of_int_div[OF knz kdt]
-	  numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-	  numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
-      also have "\<dots> = (?I ?tk (Gt (CN 0 c e)))" using pos_divide_less_eq[OF kpos, where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-	  numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
-	by (simp add: ti)
-      finally have ?case . }
-    ultimately show ?case by blast 
-next
-  case (8 c e)  
-  from prems have cp: "c > 0" and nb: "numbound0 e" by auto
-    {assume kdc: "k dvd c" 
-      from kpos have knz: "k\<noteq>0" by simp
-      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
-      from prems have  ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
-	numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } 
-    moreover 
-    {assume "\<not> k dvd c"
-      from kpos have knz: "k\<noteq>0" by simp
-      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
-      from prems have "?I (real x) (?s (Ge (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k \<ge> 0)"
-	using real_of_int_div[OF knz kdt]
-	  numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-	  numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
-      also have "\<dots> = (?I ?tk (Ge (CN 0 c e)))" using pos_divide_le_eq[OF kpos, where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-	  numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
-	by (simp add: ti)
-      finally have ?case . }
-    ultimately show ?case by blast 
-next
-  case (9 i c e)   from prems have cp: "c > 0" and nb: "numbound0 e" by auto
-    {assume kdc: "k dvd c" 
-      from kpos have knz: "k\<noteq>0" by simp
-      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
-      from prems have  ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
-	numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } 
-    moreover 
-    {assume "\<not> k dvd c"
-      from kpos have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
-      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
-      from prems have "?I (real x) (?s (Dvd i (CN 0 c e))) = (real i * real k rdvd (real c * (?N (real x) t / real k) + ?N (real x) e)* real k)"
-	using real_of_int_div[OF knz kdt]
-	  numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-	  numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
-      also have "\<dots> = (?I ?tk (Dvd i (CN 0 c e)))" using rdvd_mult[OF knz, where n="i"] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-	  numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
-	by (simp add: ti)
-      finally have ?case . }
-    ultimately show ?case by blast 
-next
-  case (10 i c e)    from prems have cp: "c > 0" and nb: "numbound0 e" by auto
-    {assume kdc: "k dvd c" 
-      from kpos have knz: "k\<noteq>0" by simp
-      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
-      from prems have  ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
-	numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } 
-    moreover 
-    {assume "\<not> k dvd c"
-      from kpos have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
-      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
-      from prems have "?I (real x) (?s (NDvd i (CN 0 c e))) = (\<not> (real i * real k rdvd (real c * (?N (real x) t / real k) + ?N (real x) e)* real k))"
-	using real_of_int_div[OF knz kdt]
-	  numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-	  numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
-      also have "\<dots> = (?I ?tk (NDvd i (CN 0 c e)))" using rdvd_mult[OF knz, where n="i"] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-	  numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
-	by (simp add: ti)
-      finally have ?case . }
-    ultimately show ?case by blast 
-qed (simp_all add: nth_pos2 bound0_I[where bs="bs" and b="real ((floor (?N b' t)) div k)" and b'="real x"] numbound0_I[where bs="bs" and b="real ((floor (?N b' t)) div k)" and b'="real x"])
-
-
-lemma a\<rho>: 
-  assumes lp: "iszlfm p (real (x::int)#bs)" and kp: "real k > 0" 
-  shows "Ifm (real (x*k)#bs) (a\<rho> p k) = Ifm (real x#bs) p" (is "?I (x*k) (?f p k) = ?I x p")
-using lp bound0_I[where bs="bs" and b="real (x*k)" and b'="real x"] numbound0_I[where bs="bs" and b="real (x*k)" and b'="real x"]
-proof(induct p rule: a\<rho>.induct)
-  case (3 c e)  
-  from prems have cp: "c > 0" and nb: "numbound0 e" by auto
-  from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
-    {assume kdc: "k dvd c" from prems have  ?case using real_of_int_div[OF knz kdc] by simp } 
-    moreover 
-    {assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] nonzero_eq_divide_eq[OF knz', where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: algebra_simps)}
-    ultimately show ?case by blast 
-next
-  case (4 c e)   
-  from prems have cp: "c > 0" and nb: "numbound0 e" by auto
-  from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
-    {assume kdc: "k dvd c" from prems have  ?case using real_of_int_div[OF knz kdc] by simp } 
-    moreover 
-    {assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] nonzero_eq_divide_eq[OF knz', where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: algebra_simps)}
-    ultimately show ?case by blast 
-next
-  case (5 c e)   
-  from prems have cp: "c > 0" and nb: "numbound0 e" by auto
-  from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
-    {assume kdc: "k dvd c" from prems have  ?case using real_of_int_div[OF knz kdc] by simp } 
-    moreover 
-    {assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] pos_less_divide_eq[OF kp, where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: algebra_simps)}
-    ultimately show ?case by blast 
-next
-  case (6 c e)    
-  from prems have cp: "c > 0" and nb: "numbound0 e" by auto
-  from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
-    {assume kdc: "k dvd c" from prems have  ?case using real_of_int_div[OF knz kdc] by simp } 
-    moreover 
-    {assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] pos_le_divide_eq[OF kp, where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: algebra_simps)}
-    ultimately show ?case by blast 
-next
-  case (7 c e)    
-  from prems have cp: "c > 0" and nb: "numbound0 e" by auto
-  from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
-    {assume kdc: "k dvd c" from prems have  ?case using real_of_int_div[OF knz kdc] by simp } 
-    moreover 
-    {assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] pos_divide_less_eq[OF kp, where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: algebra_simps)}
-    ultimately show ?case by blast 
-next
-  case (8 c e)    
-  from prems have cp: "c > 0" and nb: "numbound0 e" by auto
-  from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
-    {assume kdc: "k dvd c" from prems have  ?case using real_of_int_div[OF knz kdc] by simp } 
-    moreover 
-    {assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] pos_divide_le_eq[OF kp, where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: algebra_simps)}
-    ultimately show ?case by blast 
-next
-  case (9 i c e)
-  from prems have cp: "c > 0" and nb: "numbound0 e" by auto
-  from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
-  {assume kdc: "k dvd c" from prems have  ?case using real_of_int_div[OF knz kdc] by simp } 
-  moreover 
-  {assume "\<not> k dvd c"
-    hence "Ifm (real (x*k)#bs) (a\<rho> (Dvd i (CN 0 c e)) k) = 
-      (real i * real k rdvd (real c * real x + Inum (real x#bs) e) * real k)" 
-      using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] 
-      by (simp add: algebra_simps)
-    also have "\<dots> = (Ifm (real x#bs) (Dvd i (CN 0 c e)))" by (simp add: rdvd_mult[OF knz, where n="i"])
-    finally have ?case . }
-  ultimately show ?case by blast 
-next
-  case (10 i c e) 
-  from prems have cp: "c > 0" and nb: "numbound0 e" by auto
-  from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
-  {assume kdc: "k dvd c" from prems have  ?case using real_of_int_div[OF knz kdc] by simp } 
-  moreover 
-  {assume "\<not> k dvd c"
-    hence "Ifm (real (x*k)#bs) (a\<rho> (NDvd i (CN 0 c e)) k) = 
-      (\<not> (real i * real k rdvd (real c * real x + Inum (real x#bs) e) * real k))" 
-      using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] 
-      by (simp add: algebra_simps)
-    also have "\<dots> = (Ifm (real x#bs) (NDvd i (CN 0 c e)))" by (simp add: rdvd_mult[OF knz, where n="i"])
-    finally have ?case . }
-  ultimately show ?case by blast 
-qed (simp_all add: nth_pos2)
-
-lemma a\<rho>_ex: 
-  assumes lp: "iszlfm p (real (x::int)#bs)" and kp: "k > 0"
-  shows "(\<exists> (x::int). real k rdvd real x \<and> Ifm (real x#bs) (a\<rho> p k)) = 
-  (\<exists> (x::int). Ifm (real x#bs) p)" (is "(\<exists> x. ?D x \<and> ?P' x) = (\<exists> x. ?P x)")
-proof-
-  have "(\<exists> x. ?D x \<and> ?P' x) = (\<exists> x. k dvd x \<and> ?P' x)" using int_rdvd_iff by simp
-  also have "\<dots> = (\<exists>x. ?P' (x*k))" using unity_coeff_ex[where P="?P'" and l="k", simplified]
-    by (simp add: algebra_simps)
-  also have "\<dots> = (\<exists> x. ?P x)" using a\<rho> iszlfm_gen[OF lp] kp by auto
-  finally show ?thesis .
-qed
-
-lemma \<sigma>\<rho>': assumes lp: "iszlfm p (real (x::int)#bs)" and kp: "k > 0" and nb: "numbound0 t"
-  shows "Ifm (real x#bs) (\<sigma>\<rho> p (t,k)) = Ifm ((Inum (real x#bs) t)#bs) (a\<rho> p k)"
-using lp 
-by(induct p rule: \<sigma>\<rho>.induct, simp_all add: 
-  numbound0_I[OF nb, where bs="bs" and b="Inum (real x#bs) t" and b'="real x"] 
-  numbound0_I[where bs="bs" and b="Inum (real x#bs) t" and b'="real x"] 
-  bound0_I[where bs="bs" and b="Inum (real x#bs) t" and b'="real x"] nth_pos2 cong: imp_cong)
-
-lemma \<sigma>\<rho>_nb: assumes lp:"iszlfm p (a#bs)" and nb: "numbound0 t"
-  shows "bound0 (\<sigma>\<rho> p (t,k))"
-  using lp
-  by (induct p rule: iszlfm.induct, auto simp add: nb)
-
-lemma \<rho>_l:
-  assumes lp: "iszlfm p (real (i::int)#bs)"
-  shows "\<forall> (b,k) \<in> set (\<rho> p). k >0 \<and> numbound0 b \<and> isint b (real i#bs)"
-using lp by (induct p rule: \<rho>.induct, auto simp add: isint_sub isint_neg)
-
-lemma \<alpha>\<rho>_l:
-  assumes lp: "iszlfm p (real (i::int)#bs)"
-  shows "\<forall> (b,k) \<in> set (\<alpha>\<rho> p). k >0 \<and> numbound0 b \<and> isint b (real i#bs)"
-using lp isint_add [OF isint_c[where j="- 1"],where bs="real i#bs"]
- by (induct p rule: \<alpha>\<rho>.induct, auto)
-
-lemma zminusinf_\<rho>:
-  assumes lp: "iszlfm p (real (i::int)#bs)"
-  and nmi: "\<not> (Ifm (real i#bs) (minusinf p))" (is "\<not> (Ifm (real i#bs) (?M p))")
-  and ex: "Ifm (real i#bs) p" (is "?I i p")
-  shows "\<exists> (e,c) \<in> set (\<rho> p). real (c*i) > Inum (real i#bs) e" (is "\<exists> (e,c) \<in> ?R p. real (c*i) > ?N i e")
-  using lp nmi ex
-by (induct p rule: minusinf.induct, auto)
-
-
-lemma \<sigma>_And: "Ifm bs (\<sigma> (And p q) k t)  = Ifm bs (And (\<sigma> p k t) (\<sigma> q k t))"
-using \<sigma>_def by auto
-lemma \<sigma>_Or: "Ifm bs (\<sigma> (Or p q) k t)  = Ifm bs (Or (\<sigma> p k t) (\<sigma> q k t))"
-using \<sigma>_def by auto
-
-lemma \<rho>: assumes lp: "iszlfm p (real (i::int) #bs)"
-  and pi: "Ifm (real i#bs) p"
-  and d: "d\<delta> p d"
-  and dp: "d > 0"
-  and nob: "\<forall>(e,c) \<in> set (\<rho> p). \<forall> j\<in> {1 .. c*d}. real (c*i) \<noteq> Inum (real i#bs) e + real j"
-  (is "\<forall>(e,c) \<in> set (\<rho> p). \<forall> j\<in> {1 .. c*d}. _ \<noteq> ?N i e + _")
-  shows "Ifm (real(i - d)#bs) p"
-  using lp pi d nob
-proof(induct p rule: iszlfm.induct)
-  case (3 c e) hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real i#bs)"
-    and pi: "real (c*i) = - 1 -  ?N i e + real (1::int)" and nob: "\<forall> j\<in> {1 .. c*d}. real (c*i) \<noteq> -1 - ?N i e + real j"
-    by simp+
-  from mult_strict_left_mono[OF dp cp]  have one:"1 \<in> {1 .. c*d}" by auto
-  from nob[rule_format, where j="1", OF one] pi show ?case by simp
-next
-  case (4 c e)  
-  hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real i#bs)"
-    and nob: "\<forall> j\<in> {1 .. c*d}. real (c*i) \<noteq> - ?N i e + real j"
-    by simp+
-  {assume "real (c*i) \<noteq> - ?N i e + real (c*d)"
-    with numbound0_I[OF nb, where bs="bs" and b="real i - real d" and b'="real i"]
-    have ?case by (simp add: algebra_simps)}
-  moreover
-  {assume pi: "real (c*i) = - ?N i e + real (c*d)"
-    from mult_strict_left_mono[OF dp cp] have d: "(c*d) \<in> {1 .. c*d}" by simp
-    from nob[rule_format, where j="c*d", OF d] pi have ?case by simp }
-  ultimately show ?case by blast
-next
-  case (5 c e) hence cp: "c > 0" by simp
-  from prems mult_strict_left_mono[OF dp cp, simplified real_of_int_less_iff[symmetric] 
-    real_of_int_mult]
-  show ?case using prems dp 
-    by (simp add: add: numbound0_I[where bs="bs" and b="real i - real d" and b'="real i"] 
-      algebra_simps)
-next
-  case (6 c e)  hence cp: "c > 0" by simp
-  from prems mult_strict_left_mono[OF dp cp, simplified real_of_int_less_iff[symmetric] 
-    real_of_int_mult]
-  show ?case using prems dp 
-    by (simp add: add: numbound0_I[where bs="bs" and b="real i - real d" and b'="real i"] 
-      algebra_simps)
-next
-  case (7 c e) hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real i#bs)"
-    and nob: "\<forall> j\<in> {1 .. c*d}. real (c*i) \<noteq> - ?N i e + real j"
-    and pi: "real (c*i) + ?N i e > 0" and cp': "real c >0"
-    by simp+
-  let ?fe = "floor (?N i e)"
-  from pi cp have th:"(real i +?N i e / real c)*real c > 0" by (simp add: algebra_simps)
-  from pi ei[simplified isint_iff] have "real (c*i + ?fe) > real (0::int)" by simp
-  hence pi': "c*i + ?fe > 0" by (simp only: real_of_int_less_iff[symmetric])
-  have "real (c*i) + ?N i e > real (c*d) \<or> real (c*i) + ?N i e \<le> real (c*d)" by auto
-  moreover
-  {assume "real (c*i) + ?N i e > real (c*d)" hence ?case
-      by (simp add: algebra_simps 
-	numbound0_I[OF nb,where bs="bs" and b="real i - real d" and b'="real i"])} 
-  moreover 
-  {assume H:"real (c*i) + ?N i e \<le> real (c*d)"
-    with ei[simplified isint_iff] have "real (c*i + ?fe) \<le> real (c*d)" by simp
-    hence pid: "c*i + ?fe \<le> c*d" by (simp only: real_of_int_le_iff)
-    with pi' have "\<exists> j1\<in> {1 .. c*d}. c*i + ?fe = j1" by auto
-    hence "\<exists> j1\<in> {1 .. c*d}. real (c*i) = - ?N i e + real j1" 
-      by (simp only: diff_def[symmetric] real_of_int_mult real_of_int_add real_of_int_inject[symmetric] ei[simplified isint_iff] algebra_simps)
-    with nob  have ?case by blast }
-  ultimately show ?case by blast
-next
-  case (8 c e)  hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real i#bs)"
-    and nob: "\<forall> j\<in> {1 .. c*d}. real (c*i) \<noteq> - 1 - ?N i e + real j"
-    and pi: "real (c*i) + ?N i e \<ge> 0" and cp': "real c >0"
-    by simp+
-  let ?fe = "floor (?N i e)"
-  from pi cp have th:"(real i +?N i e / real c)*real c \<ge> 0" by (simp add: algebra_simps)
-  from pi ei[simplified isint_iff] have "real (c*i + ?fe) \<ge> real (0::int)" by simp
-  hence pi': "c*i + 1 + ?fe \<ge> 1" by (simp only: real_of_int_le_iff[symmetric])
-  have "real (c*i) + ?N i e \<ge> real (c*d) \<or> real (c*i) + ?N i e < real (c*d)" by auto
-  moreover
-  {assume "real (c*i) + ?N i e \<ge> real (c*d)" hence ?case
-      by (simp add: algebra_simps 
-	numbound0_I[OF nb,where bs="bs" and b="real i - real d" and b'="real i"])} 
-  moreover 
-  {assume H:"real (c*i) + ?N i e < real (c*d)"
-    with ei[simplified isint_iff] have "real (c*i + ?fe) < real (c*d)" by simp
-    hence pid: "c*i + 1 + ?fe \<le> c*d" by (simp only: real_of_int_le_iff)
-    with pi' have "\<exists> j1\<in> {1 .. c*d}. c*i + 1+ ?fe = j1" by auto
-    hence "\<exists> j1\<in> {1 .. c*d}. real (c*i) + 1= - ?N i e + real j1"
-      by (simp only: diff_def[symmetric] real_of_int_mult real_of_int_add real_of_int_inject[symmetric] ei[simplified isint_iff] algebra_simps real_of_one) 
-    hence "\<exists> j1\<in> {1 .. c*d}. real (c*i) = (- ?N i e + real j1) - 1"
-      by (simp only: algebra_simps diff_def[symmetric])
-        hence "\<exists> j1\<in> {1 .. c*d}. real (c*i) = - 1 - ?N i e + real j1"
-	  by (simp only: add_ac diff_def)
-    with nob  have ?case by blast }
-  ultimately show ?case by blast
-next
-  case (9 j c e)  hence p: "real j rdvd real (c*i) + ?N i e" (is "?p x") and cp: "c > 0" and bn:"numbound0 e"  by simp+
-    let ?e = "Inum (real i # bs) e"
-    from prems have "isint e (real i #bs)"  by simp 
-    hence ie: "real (floor ?e) = ?e" using isint_iff[where n="e" and bs="(real i)#bs"] numbound0_I[OF bn,where b="real i" and b'="real i" and bs="bs"]
-      by (simp add: isint_iff)
-    from prems have id: "j dvd d" by simp
-    from ie[symmetric] have "?p i = (real j rdvd real (c*i+ floor ?e))" by simp
-    also have "\<dots> = (j dvd c*i + floor ?e)" 
-      using int_rdvd_iff [where i="j" and t="c*i+ floor ?e"] by simp
-    also have "\<dots> = (j dvd c*i - c*d + floor ?e)" 
-      using dvd_period[OF id, where x="c*i" and c="-c" and t="floor ?e"] by simp
-    also have "\<dots> = (real j rdvd real (c*i - c*d + floor ?e))" 
-      using int_rdvd_iff[where i="j" and t="(c*i - c*d + floor ?e)",symmetric, simplified]
-      ie by simp
-    also have "\<dots> = (real j rdvd real (c*(i - d)) + ?e)" 
-      using ie by (simp add:algebra_simps)
-    finally show ?case 
-      using numbound0_I[OF bn,where b="real i - real d" and b'="real i" and bs="bs"] p 
-      by (simp add: algebra_simps)
-next
-  case (10 j c e)   hence p: "\<not> (real j rdvd real (c*i) + ?N i e)" (is "?p x") and cp: "c > 0" and bn:"numbound0 e"  by simp+
-    let ?e = "Inum (real i # bs) e"
-    from prems have "isint e (real i #bs)"  by simp 
-    hence ie: "real (floor ?e) = ?e" using isint_iff[where n="e" and bs="(real i)#bs"] numbound0_I[OF bn,where b="real i" and b'="real i" and bs="bs"]
-      by (simp add: isint_iff)
-    from prems have id: "j dvd d" by simp
-    from ie[symmetric] have "?p i = (\<not> (real j rdvd real (c*i+ floor ?e)))" by simp
-    also have "\<dots> = Not (j dvd c*i + floor ?e)" 
-      using int_rdvd_iff [where i="j" and t="c*i+ floor ?e"] by simp
-    also have "\<dots> = Not (j dvd c*i - c*d + floor ?e)" 
-      using dvd_period[OF id, where x="c*i" and c="-c" and t="floor ?e"] by simp
-    also have "\<dots> = Not (real j rdvd real (c*i - c*d + floor ?e))" 
-      using int_rdvd_iff[where i="j" and t="(c*i - c*d + floor ?e)",symmetric, simplified]
-      ie by simp
-    also have "\<dots> = Not (real j rdvd real (c*(i - d)) + ?e)" 
-      using ie by (simp add:algebra_simps)
-    finally show ?case 
-      using numbound0_I[OF bn,where b="real i - real d" and b'="real i" and bs="bs"] p 
-      by (simp add: algebra_simps)
-qed(auto simp add: numbound0_I[where bs="bs" and b="real i - real d" and b'="real i"] nth_pos2)
-
-lemma \<sigma>_nb: assumes lp: "iszlfm p (a#bs)" and nb: "numbound0 t"
-  shows "bound0 (\<sigma> p k t)"
-  using \<sigma>\<rho>_nb[OF lp nb] nb by (simp add: \<sigma>_def)
-  
-lemma \<rho>':   assumes lp: "iszlfm p (a #bs)"
-  and d: "d\<delta> p d"
-  and dp: "d > 0"
-  shows "\<forall> x. \<not>(\<exists> (e,c) \<in> set(\<rho> p). \<exists>(j::int) \<in> {1 .. c*d}. Ifm (a #bs) (\<sigma> p c (Add e (C j)))) \<longrightarrow> Ifm (real x#bs) p \<longrightarrow> Ifm (real (x - d)#bs) p" (is "\<forall> x. ?b x \<longrightarrow> ?P x \<longrightarrow> ?P (x - d)")
-proof(clarify)
-  fix x 
-  assume nob1:"?b x" and px: "?P x" 
-  from iszlfm_gen[OF lp, rule_format, where y="real x"] have lp': "iszlfm p (real x#bs)".
-  have nob: "\<forall>(e, c)\<in>set (\<rho> p). \<forall>j\<in>{1..c * d}. real (c * x) \<noteq> Inum (real x # bs) e + real j" 
-  proof(clarify)
-    fix e c j assume ecR: "(e,c) \<in> set (\<rho> p)" and jD: "j\<in> {1 .. c*d}"
-      and cx: "real (c*x) = Inum (real x#bs) e + real j"
-    let ?e = "Inum (real x#bs) e"
-    let ?fe = "floor ?e"
-    from \<rho>_l[OF lp'] ecR have ei:"isint e (real x#bs)" and cp:"c>0" and nb:"numbound0 e"
-      by auto
-    from numbound0_gen [OF nb ei, rule_format,where y="a"] have "isint e (a#bs)" .
-    from cx ei[simplified isint_iff] have "real (c*x) = real (?fe + j)" by simp
-    hence cx: "c*x = ?fe + j" by (simp only: real_of_int_inject)
-    hence cdej:"c dvd ?fe + j" by (simp add: dvd_def) (rule_tac x="x" in exI, simp)
-    hence "real c rdvd real (?fe + j)" by (simp only: int_rdvd_iff)
-    hence rcdej: "real c rdvd ?e + real j" by (simp add: ei[simplified isint_iff])
-    from cx have "(c*x) div c = (?fe + j) div c" by simp
-    with cp have "x = (?fe + j) div c" by simp
-    with px have th: "?P ((?fe + j) div c)" by auto
-    from cp have cp': "real c > 0" by simp
-    from cdej have cdej': "c dvd floor (Inum (real x#bs) (Add e (C j)))" by simp
-    from nb have nb': "numbound0 (Add e (C j))" by simp
-    have ji: "isint (C j) (real x#bs)" by (simp add: isint_def)
-    from isint_add[OF ei ji] have ei':"isint (Add e (C j)) (real x#bs)" .
-    from th \<sigma>\<rho>[where b'="real x", OF lp' cp' nb' ei' cdej',symmetric]
-    have "Ifm (real x#bs) (\<sigma>\<rho> p (Add e (C j), c))" by simp
-    with rcdej have th: "Ifm (real x#bs) (\<sigma> p c (Add e (C j)))" by (simp add: \<sigma>_def)
-    from th bound0_I[OF \<sigma>_nb[OF lp nb', where k="c"],where bs="bs" and b="real x" and b'="a"]
-    have "Ifm (a#bs) (\<sigma> p c (Add e (C j)))" by blast
-      with ecR jD nob1    show "False" by blast
-  qed
-  from \<rho>[OF lp' px d dp nob] show "?P (x -d )" . 
-qed
-
-
-lemma rl_thm: 
-  assumes lp: "iszlfm p (real (i::int)#bs)"
-  shows "(\<exists> (x::int). Ifm (real x#bs) p) = ((\<exists> j\<in> {1 .. \<delta> p}. Ifm (real j#bs) (minusinf p)) \<or> (\<exists> (e,c) \<in> set (\<rho> p). \<exists> j\<in> {1 .. c*(\<delta> p)}. Ifm (a#bs) (\<sigma> p c (Add e (C j)))))"
-  (is "(\<exists>(x::int). ?P x) = ((\<exists> j\<in> {1.. \<delta> p}. ?MP j)\<or>(\<exists> (e,c) \<in> ?R. \<exists> j\<in> _. ?SP c e j))" 
-    is "?lhs = (?MD \<or> ?RD)"  is "?lhs = ?rhs")
-proof-
-  let ?d= "\<delta> p"
-  from \<delta>[OF lp] have d:"d\<delta> p ?d" and dp: "?d > 0" by auto
-  { assume H:"?MD" hence th:"\<exists> (x::int). ?MP x" by blast
-    from H minusinf_ex[OF lp th] have ?thesis  by blast}
-  moreover
-  { fix e c j assume exR:"(e,c) \<in> ?R" and jD:"j\<in> {1 .. c*?d}" and spx:"?SP c e j"
-    from exR \<rho>_l[OF lp] have nb: "numbound0 e" and ei:"isint e (real i#bs)" and cp: "c > 0"
-      by auto
-    have "isint (C j) (real i#bs)" by (simp add: isint_iff)
-    with isint_add[OF numbound0_gen[OF nb ei,rule_format, where y="real i"]]
-    have eji:"isint (Add e (C j)) (real i#bs)" by simp
-    from nb have nb': "numbound0 (Add e (C j))" by simp
-    from spx bound0_I[OF \<sigma>_nb[OF lp nb', where k="c"], where bs="bs" and b="a" and b'="real i"]
-    have spx': "Ifm (real i # bs) (\<sigma> p c (Add e (C j)))" by blast
-    from spx' have rcdej:"real c rdvd (Inum (real i#bs) (Add e (C j)))" 
-      and sr:"Ifm (real i#bs) (\<sigma>\<rho> p (Add e (C j),c))" by (simp add: \<sigma>_def)+
-    from rcdej eji[simplified isint_iff] 
-    have "real c rdvd real (floor (Inum (real i#bs) (Add e (C j))))" by simp
-    hence cdej:"c dvd floor (Inum (real i#bs) (Add e (C j)))" by (simp only: int_rdvd_iff)
-    from cp have cp': "real c > 0" by simp
-    from \<sigma>\<rho>[OF lp cp' nb' eji cdej] spx' have "?P (\<lfloor>Inum (real i # bs) (Add e (C j))\<rfloor> div c)"
-      by (simp add: \<sigma>_def)
-    hence ?lhs by blast
-    with exR jD spx have ?thesis by blast}
-  moreover
-  { fix x assume px: "?P x" and nob: "\<not> ?RD"
-    from iszlfm_gen [OF lp,rule_format, where y="a"] have lp':"iszlfm p (a#bs)" .
-    from \<rho>'[OF lp' d dp, rule_format, OF nob] have th:"\<forall> x. ?P x \<longrightarrow> ?P (x - ?d)" by blast
-    from minusinf_inf[OF lp] obtain z where z:"\<forall> x<z. ?MP x = ?P x" by blast
-    have zp: "abs (x - z) + 1 \<ge> 0" by arith
-    from decr_lemma[OF dp,where x="x" and z="z"] 
-      decr_mult_lemma[OF dp th zp, rule_format, OF px] z have th:"\<exists> x. ?MP x" by auto
-    with minusinf_bex[OF lp] px nob have ?thesis by blast}
-  ultimately show ?thesis by blast
-qed
-
-lemma mirror_\<alpha>\<rho>:   assumes lp: "iszlfm p (a#bs)"
-  shows "(\<lambda> (t,k). (Inum (a#bs) t, k)) ` set (\<alpha>\<rho> p) = (\<lambda> (t,k). (Inum (a#bs) t,k)) ` set (\<rho> (mirror p))"
-using lp
-by (induct p rule: mirror.induct, simp_all add: split_def image_Un )
-  
-text {* The @{text "\<real>"} part*}
-
-text{* Linearity for fm where Bound 0 ranges over @{text "\<real>"}*}
-consts
-  isrlfm :: "fm \<Rightarrow> bool"   (* Linearity test for fm *)
-recdef isrlfm "measure size"
-  "isrlfm (And p q) = (isrlfm p \<and> isrlfm q)" 
-  "isrlfm (Or p q) = (isrlfm p \<and> isrlfm q)" 
-  "isrlfm (Eq  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
-  "isrlfm (NEq (CN 0 c e)) = (c>0 \<and> numbound0 e)"
-  "isrlfm (Lt  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
-  "isrlfm (Le  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
-  "isrlfm (Gt  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
-  "isrlfm (Ge  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
-  "isrlfm p = (isatom p \<and> (bound0 p))"
-
-constdefs fp :: "fm \<Rightarrow> int \<Rightarrow> num \<Rightarrow> int \<Rightarrow> fm"
-  "fp p n s j \<equiv> (if n > 0 then 
-            (And p (And (Ge (CN 0 n (Sub s (Add (Floor s) (C j)))))
-                        (Lt (CN 0 n (Sub s (Add (Floor s) (C (j+1))))))))
-            else 
-            (And p (And (Le (CN 0 (-n) (Add (Neg s) (Add (Floor s) (C j))))) 
-                        (Gt (CN 0 (-n) (Add (Neg s) (Add (Floor s) (C (j + 1)))))))))"
-
-  (* splits the bounded from the unbounded part*)
-consts rsplit0 :: "num \<Rightarrow> (fm \<times> int \<times> num) list" 
-recdef rsplit0 "measure num_size"
-  "rsplit0 (Bound 0) = [(T,1,C 0)]"
-  "rsplit0 (Add a b) = (let acs = rsplit0 a ; bcs = rsplit0 b 
-              in map (\<lambda> ((p,n,t),(q,m,s)). (And p q, n+m, Add t s)) [(a,b). a\<leftarrow>acs,b\<leftarrow>bcs])"
-  "rsplit0 (Sub a b) = rsplit0 (Add a (Neg b))"
-  "rsplit0 (Neg a) = map (\<lambda> (p,n,s). (p,-n,Neg s)) (rsplit0 a)"
-  "rsplit0 (Floor a) = foldl (op @) [] (map 
-      (\<lambda> (p,n,s). if n=0 then [(p,0,Floor s)]
-          else (map (\<lambda> j. (fp p n s j, 0, Add (Floor s) (C j))) (if n > 0 then iupt (0,n) else iupt(n,0))))
-       (rsplit0 a))"
-  "rsplit0 (CN 0 c a) = map (\<lambda> (p,n,s). (p,n+c,s)) (rsplit0 a)"
-  "rsplit0 (CN m c a) = map (\<lambda> (p,n,s). (p,n,CN m c s)) (rsplit0 a)"
-  "rsplit0 (CF c t s) = rsplit0 (Add (Mul c (Floor t)) s)"
-  "rsplit0 (Mul c a) = map (\<lambda> (p,n,s). (p,c*n,Mul c s)) (rsplit0 a)"
-  "rsplit0 t = [(T,0,t)]"
-
-lemma not_rl[simp]: "isrlfm p \<Longrightarrow> isrlfm (not p)"
-  by (induct p rule: isrlfm.induct, auto)
-lemma conj_rl[simp]: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (conj p q)"
-  using conj_def by (cases p, auto)
-lemma disj_rl[simp]: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (disj p q)"
-  using disj_def by (cases p, auto)
-
-
-lemma rsplit0_cs:
-  shows "\<forall> (p,n,s) \<in> set (rsplit0 t). 
-  (Ifm (x#bs) p \<longrightarrow>  (Inum (x#bs) t = Inum (x#bs) (CN 0 n s))) \<and> numbound0 s \<and> isrlfm p" 
-  (is "\<forall> (p,n,s) \<in> ?SS t. (?I p \<longrightarrow> ?N t = ?N (CN 0 n s)) \<and> _ \<and> _ ")
-proof(induct t rule: rsplit0.induct)
-  case (5 a) 
-  let ?p = "\<lambda> (p,n,s) j. fp p n s j"
-  let ?f = "(\<lambda> (p,n,s) j. (?p (p,n,s) j, (0::int),Add (Floor s) (C j)))"
-  let ?J = "\<lambda> n. if n>0 then iupt (0,n) else iupt (n,0)"
-  let ?ff=" (\<lambda> (p,n,s). if n= 0 then [(p,0,Floor s)] else map (?f (p,n,s)) (?J n))"
-  have int_cases: "\<forall> (i::int). i= 0 \<or> i < 0 \<or> i > 0" by arith
-  have U1: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) = 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set [(p,0,Floor s)]))" by auto
-  have U2': "\<forall> (p,n,s) \<in> {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0}. 
-    ?ff (p,n,s) = map (?f(p,n,s)) (iupt(0,n))" by auto
-  hence U2: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) = 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). 
-    set (map (?f(p,n,s)) (iupt(0,n)))))"
-  proof-
-    fix M :: "('a\<times>'b\<times>'c) set" and f :: "('a\<times>'b\<times>'c) \<Rightarrow> 'd list" and g
-    assume "\<forall> (a,b,c) \<in> M. f (a,b,c) = g a b c"
-    thus "(UNION M (\<lambda> (a,b,c). set (f (a,b,c)))) = (UNION M (\<lambda> (a,b,c). set (g a b c)))"
-      by (auto simp add: split_def)
-  qed
-  have U3': "\<forall> (p,n,s) \<in> {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0}. ?ff (p,n,s) = map (?f(p,n,s)) (iupt(n,0))"
-    by auto
-  hence U3: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) = 
-    (UNION {(p,n,s). (p,n,s)\<in> ?SS a\<and>n<0} (\<lambda>(p,n,s). set (map (?f(p,n,s)) (iupt(n,0)))))"
-      proof-
-    fix M :: "('a\<times>'b\<times>'c) set" and f :: "('a\<times>'b\<times>'c) \<Rightarrow> 'd list" and g
-    assume "\<forall> (a,b,c) \<in> M. f (a,b,c) = g a b c"
-    thus "(UNION M (\<lambda> (a,b,c). set (f (a,b,c)))) = (UNION M (\<lambda> (a,b,c). set (g a b c)))"
-      by (auto simp add: split_def)
-  qed
-  have "?SS (Floor a) = UNION (?SS a) (\<lambda>x. set (?ff x))" 
-    by (auto simp add: foldl_conv_concat)
-  also have "\<dots> = UNION (?SS a) (\<lambda> (p,n,s). set (?ff (p,n,s)))" by auto
-  also have "\<dots> = 
-    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (?ff (p,n,s)))))" 
-    using int_cases[rule_format] by blast
-  also have "\<dots> =  
-    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set [(p,0,Floor s)])) Un 
-   (UNION {(p,n,s). (p,n,s)\<in> ?SS a\<and>n>0} (\<lambda>(p,n,s). set(map(?f(p,n,s)) (iupt(0,n))))) Un 
-   (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). 
-    set (map (?f(p,n,s)) (iupt(n,0))))))" by (simp only: U1 U2 U3)
-  also have "\<dots> =  
-    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). (?f(p,n,s)) ` {0 .. n})) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). (?f(p,n,s)) ` {n .. 0})))"
-    by (simp only: set_map iupt_set set.simps)
-  also have "\<dots> =   
-    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {0 .. n}})) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s).  {?f(p,n,s) j| j. j\<in> {n .. 0}})))" by blast
-  finally 
-  have FS: "?SS (Floor a) =   
-    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {0 .. n}})) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s).  {?f(p,n,s) j| j. j\<in> {n .. 0}})))"    by blast
-  show ?case
-    proof(simp only: FS, clarsimp simp del: Ifm.simps Inum.simps, -)
-      fix p n s
-      let ?ths = "(?I p \<longrightarrow> (?N (Floor a) = ?N (CN 0 n s))) \<and> numbound0 s \<and> isrlfm p"
-      assume "(\<exists>ba. (p, 0, ba) \<in> set (rsplit0 a) \<and> n = 0 \<and> s = Floor ba) \<or>
-       (\<exists>ab ac ba.
-           (ab, ac, ba) \<in> set (rsplit0 a) \<and>
-           0 < ac \<and>
-           (\<exists>j. p = fp ab ac ba j \<and>
-                n = 0 \<and> s = Add (Floor ba) (C j) \<and> 0 \<le> j \<and> j \<le> ac)) \<or>
-       (\<exists>ab ac ba.
-           (ab, ac, ba) \<in> set (rsplit0 a) \<and>
-           ac < 0 \<and>
-           (\<exists>j. p = fp ab ac ba j \<and>
-                n = 0 \<and> s = Add (Floor ba) (C j) \<and> ac \<le> j \<and> j \<le> 0))"
-      moreover 
-      {fix s'
-	assume "(p, 0, s') \<in> ?SS a" and "n = 0" and "s = Floor s'"
-	hence ?ths using prems by auto}
-      moreover
-      {	fix p' n' s' j
-	assume pns: "(p', n', s') \<in> ?SS a" 
-	  and np: "0 < n'" 
-	  and p_def: "p = ?p (p',n',s') j" 
-	  and n0: "n = 0" 
-	  and s_def: "s = (Add (Floor s') (C j))" 
-	  and jp: "0 \<le> j" and jn: "j \<le> n'"
-	from prems pns have H:"(Ifm ((x\<Colon>real) # (bs\<Colon>real list)) p' \<longrightarrow>
-          Inum (x # bs) a = Inum (x # bs) (CN 0 n' s')) \<and>
-          numbound0 s' \<and> isrlfm p'" by blast
-	hence nb: "numbound0 s'" by simp
-	from H have nf: "isrlfm (?p (p',n',s') j)" using fp_def np by (simp add: numsub_nb)
-	let ?nxs = "CN 0 n' s'"
-	let ?l = "floor (?N s') + j"
-	from H 
-	have "?I (?p (p',n',s') j) \<longrightarrow> 
-	  (((?N ?nxs \<ge> real ?l) \<and> (?N ?nxs < real (?l + 1))) \<and> (?N a = ?N ?nxs ))" 
-	  by (simp add: fp_def np algebra_simps numsub numadd numfloor)
-	also have "\<dots> \<longrightarrow> ((floor (?N ?nxs) = ?l) \<and> (?N a = ?N ?nxs ))"
-	  using floor_int_eq[where x="?N ?nxs" and n="?l"] by simp
-	moreover
-	have "\<dots> \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))" by simp
-	ultimately have "?I (?p (p',n',s') j) \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))"
-	  by blast
-	with s_def n0 p_def nb nf have ?ths by auto}
-      moreover
-      {fix p' n' s' j
-	assume pns: "(p', n', s') \<in> ?SS a" 
-	  and np: "n' < 0" 
-	  and p_def: "p = ?p (p',n',s') j" 
-	  and n0: "n = 0" 
-	  and s_def: "s = (Add (Floor s') (C j))" 
-	  and jp: "n' \<le> j" and jn: "j \<le> 0"
-	from prems pns have H:"(Ifm ((x\<Colon>real) # (bs\<Colon>real list)) p' \<longrightarrow>
-          Inum (x # bs) a = Inum (x # bs) (CN 0 n' s')) \<and>
-          numbound0 s' \<and> isrlfm p'" by blast
-	hence nb: "numbound0 s'" by simp
-	from H have nf: "isrlfm (?p (p',n',s') j)" using fp_def np by (simp add: numneg_nb)
-	let ?nxs = "CN 0 n' s'"
-	let ?l = "floor (?N s') + j"
-	from H 
-	have "?I (?p (p',n',s') j) \<longrightarrow> 
-	  (((?N ?nxs \<ge> real ?l) \<and> (?N ?nxs < real (?l + 1))) \<and> (?N a = ?N ?nxs ))" 
-	  by (simp add: np fp_def algebra_simps numneg numfloor numadd numsub)
-	also have "\<dots> \<longrightarrow> ((floor (?N ?nxs) = ?l) \<and> (?N a = ?N ?nxs ))"
-	  using floor_int_eq[where x="?N ?nxs" and n="?l"] by simp
-	moreover
-	have "\<dots> \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))"  by simp
-	ultimately have "?I (?p (p',n',s') j) \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))"
-	  by blast
-	with s_def n0 p_def nb nf have ?ths by auto}
-      ultimately show ?ths by auto
-    qed
-next
-  case (3 a b) then show ?case
-  apply auto
-  apply (erule_tac x = "(aa, aaa, ba)" in ballE) apply simp_all
-  apply (erule_tac x = "(ab, ac, baa)" in ballE) apply simp_all
-  done
-qed (auto simp add: Let_def split_def algebra_simps conj_rl)
-
-lemma real_in_int_intervals: 
-  assumes xb: "real m \<le> x \<and> x < real ((n::int) + 1)"
-  shows "\<exists> j\<in> {m.. n}. real j \<le> x \<and> x < real (j+1)" (is "\<exists> j\<in> ?N. ?P j")
-by (rule bexI[where P="?P" and x="floor x" and A="?N"]) 
-(auto simp add: floor_less_eq[where x="x" and a="n+1", simplified] xb[simplified] floor_mono2[where x="real m" and y="x", OF conjunct1[OF xb], simplified floor_real_of_int[where n="m"]])
-
-lemma rsplit0_complete:
-  assumes xp:"0 \<le> x" and x1:"x < 1"
-  shows "\<exists> (p,n,s) \<in> set (rsplit0 t). Ifm (x#bs) p" (is "\<exists> (p,n,s) \<in> ?SS t. ?I p")
-proof(induct t rule: rsplit0.induct)
-  case (2 a b) 
-  from prems have "\<exists> (pa,na,sa) \<in> ?SS a. ?I pa" by auto
-  then obtain "pa" "na" "sa" where pa: "(pa,na,sa)\<in> ?SS a \<and> ?I pa" by blast
-  from prems have "\<exists> (pb,nb,sb) \<in> ?SS b. ?I pb" by auto
-  then obtain "pb" "nb" "sb" where pb: "(pb,nb,sb)\<in> ?SS b \<and> ?I pb" by blast
-  from pa pb have th: "((pa,na,sa),(pb,nb,sb)) \<in> set[(x,y). x\<leftarrow>rsplit0 a, y\<leftarrow>rsplit0 b]"
-    by (auto)
-  let ?f="(\<lambda> ((p,n,t),(q,m,s)). (And p q, n+m, Add t s))"
-  from imageI[OF th, where f="?f"] have "?f ((pa,na,sa),(pb,nb,sb)) \<in> ?SS (Add a b)"
-    by (simp add: Let_def)
-  hence "(And pa pb, na +nb, Add sa sb) \<in> ?SS (Add a b)" by simp
-  moreover from pa pb have "?I (And pa pb)" by simp
-  ultimately show ?case by blast
-next
-  case (5 a) 
-  let ?p = "\<lambda> (p,n,s) j. fp p n s j"
-  let ?f = "(\<lambda> (p,n,s) j. (?p (p,n,s) j, (0::int),(Add (Floor s) (C j))))"
-  let ?J = "\<lambda> n. if n>0 then iupt (0,n) else iupt (n,0)"
-  let ?ff=" (\<lambda> (p,n,s). if n= 0 then [(p,0,Floor s)] else map (?f (p,n,s)) (?J n))"
-  have int_cases: "\<forall> (i::int). i= 0 \<or> i < 0 \<or> i > 0" by arith
-  have U1: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) = (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set [(p,0,Floor s)]))" by auto
-  have U2': "\<forall> (p,n,s) \<in> {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0}. ?ff (p,n,s) = map (?f(p,n,s)) (iupt(0,n))"
-    by auto
-  hence U2: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) = (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (map (?f(p,n,s)) (iupt(0,n)))))"
-  proof-
-    fix M :: "('a\<times>'b\<times>'c) set" and f :: "('a\<times>'b\<times>'c) \<Rightarrow> 'd list" and g
-    assume "\<forall> (a,b,c) \<in> M. f (a,b,c) = g a b c"
-    thus "(UNION M (\<lambda> (a,b,c). set (f (a,b,c)))) = (UNION M (\<lambda> (a,b,c). set (g a b c)))"
-      by (auto simp add: split_def)
-  qed
-  have U3': "\<forall> (p,n,s) \<in> {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0}. ?ff (p,n,s) = map (?f(p,n,s)) (iupt(n,0))"
-    by auto
-  hence U3: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) = (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (map (?f(p,n,s)) (iupt(n,0)))))"
-  proof-
-    fix M :: "('a\<times>'b\<times>'c) set" and f :: "('a\<times>'b\<times>'c) \<Rightarrow> 'd list" and g
-    assume "\<forall> (a,b,c) \<in> M. f (a,b,c) = g a b c"
-    thus "(UNION M (\<lambda> (a,b,c). set (f (a,b,c)))) = (UNION M (\<lambda> (a,b,c). set (g a b c)))"
-      by (auto simp add: split_def)
-  qed
-
-  have "?SS (Floor a) = UNION (?SS a) (\<lambda>x. set (?ff x))" by (auto simp add: foldl_conv_concat) 
-  also have "\<dots> = UNION (?SS a) (\<lambda> (p,n,s). set (?ff (p,n,s)))" by auto
-  also have "\<dots> = 
-    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (?ff (p,n,s)))))" 
-    using int_cases[rule_format] by blast
-  also have "\<dots> =  
-    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set [(p,0,Floor s)])) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (map (?f(p,n,s)) (iupt(0,n))))) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (map (?f(p,n,s)) (iupt(n,0))))))" by (simp only: U1 U2 U3)
-  also have "\<dots> =  
-    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). (?f(p,n,s)) ` {0 .. n})) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). (?f(p,n,s)) ` {n .. 0})))"
-    by (simp only: set_map iupt_set set.simps)
-  also have "\<dots> =   
-    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {0 .. n}})) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s).  {?f(p,n,s) j| j. j\<in> {n .. 0}})))" by blast
-  finally 
-  have FS: "?SS (Floor a) =   
-    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {0 .. n}})) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s).  {?f(p,n,s) j| j. j\<in> {n .. 0}})))"    by blast
-  from prems have "\<exists> (p,n,s) \<in> ?SS a. ?I p" by auto
-  then obtain "p" "n" "s" where pns: "(p,n,s) \<in> ?SS a \<and> ?I p" by blast
-  let ?N = "\<lambda> t. Inum (x#bs) t"
-  from rsplit0_cs[rule_format] pns have ans:"(?N a = ?N (CN 0 n s)) \<and> numbound0 s \<and> isrlfm p"
-    by auto
-  
-  have "n=0 \<or> n >0 \<or> n <0" by arith
-  moreover {assume "n=0" hence ?case using pns by (simp only: FS) auto }
-  moreover
-  {
-    assume np: "n > 0"
-    from real_of_int_floor_le[where r="?N s"] have "?N (Floor s) \<le> ?N s" by simp
-    also from mult_left_mono[OF xp] np have "?N s \<le> real n * x + ?N s" by simp
-    finally have "?N (Floor s) \<le> real n * x + ?N s" .
-    moreover
-    {from mult_strict_left_mono[OF x1] np 
-      have "real n *x + ?N s < real n + ?N s" by simp
-      also from real_of_int_floor_add_one_gt[where r="?N s"] 
-      have "\<dots> < real n + ?N (Floor s) + 1" by simp
-      finally have "real n *x + ?N s < ?N (Floor s) + real (n+1)" by simp}
-    ultimately have "?N (Floor s) \<le> real n *x + ?N s\<and> real n *x + ?N s < ?N (Floor s) + real (n+1)" by simp
-    hence th: "0 \<le> real n *x + ?N s - ?N (Floor s) \<and> real n *x + ?N s - ?N (Floor s) < real (n+1)" by simp
-    from real_in_int_intervals th have  "\<exists> j\<in> {0 .. n}. real j \<le> real n *x + ?N s - ?N (Floor s)\<and> real n *x + ?N s - ?N (Floor s) < real (j+1)" by simp
-    
-    hence "\<exists> j\<in> {0 .. n}. 0 \<le> real n *x + ?N s - ?N (Floor s) - real j \<and> real n *x + ?N s - ?N (Floor s) - real (j+1) < 0"
-      by(simp only: myl[rule_format, where b="real n * x + Inum (x # bs) s - Inum (x # bs) (Floor s)"] less_iff_diff_less_0[where a="real n *x + ?N s - ?N (Floor s)"]) 
-    hence "\<exists> j\<in> {0.. n}. ?I (?p (p,n,s) j)"
-      using pns by (simp add: fp_def np algebra_simps numsub numadd)
-    then obtain "j" where j_def: "j\<in> {0 .. n} \<and> ?I (?p (p,n,s) j)" by blast
-    hence "\<exists>x \<in> {?p (p,n,s) j |j. 0\<le> j \<and> j \<le> n }. ?I x" by auto
-    hence ?case using pns 
-      by (simp only: FS,simp add: bex_Un) 
-    (rule disjI2, rule disjI1,rule exI [where x="p"],
-      rule exI [where x="n"],rule exI [where x="s"],simp_all add: np)
-  }
-  moreover
-  { assume nn: "n < 0" hence np: "-n >0" by simp
-    from real_of_int_floor_le[where r="?N s"] have "?N (Floor s) + 1 > ?N s" by simp
-    moreover from mult_left_mono_neg[OF xp] nn have "?N s \<ge> real n * x + ?N s" by simp
-    ultimately have "?N (Floor s) + 1 > real n * x + ?N s" by arith 
-    moreover
-    {from mult_strict_left_mono_neg[OF x1, where c="real n"] nn
-      have "real n *x + ?N s \<ge> real n + ?N s" by simp 
-      moreover from real_of_int_floor_le[where r="?N s"]  have "real n + ?N s \<ge> real n + ?N (Floor s)" by simp
-      ultimately have "real n *x + ?N s \<ge> ?N (Floor s) + real n" 
-	by (simp only: algebra_simps)}
-    ultimately have "?N (Floor s) + real n \<le> real n *x + ?N s\<and> real n *x + ?N s < ?N (Floor s) + real (1::int)" by simp
-    hence th: "real n \<le> real n *x + ?N s - ?N (Floor s) \<and> real n *x + ?N s - ?N (Floor s) < real (1::int)" by simp
-    have th1: "\<forall> (a::real). (- a > 0) = (a < 0)" by auto
-    have th2: "\<forall> (a::real). (0 \<ge> - a) = (a \<ge> 0)" by auto
-    from real_in_int_intervals th  have  "\<exists> j\<in> {n .. 0}. real j \<le> real n *x + ?N s - ?N (Floor s)\<and> real n *x + ?N s - ?N (Floor s) < real (j+1)" by simp
-    
-    hence "\<exists> j\<in> {n .. 0}. 0 \<le> real n *x + ?N s - ?N (Floor s) - real j \<and> real n *x + ?N s - ?N (Floor s) - real (j+1) < 0"
-      by(simp only: myl[rule_format, where b="real n * x + Inum (x # bs) s - Inum (x # bs) (Floor s)"] less_iff_diff_less_0[where a="real n *x + ?N s - ?N (Floor s)"]) 
-    hence "\<exists> j\<in> {n .. 0}. 0 \<ge> - (real n *x + ?N s - ?N (Floor s) - real j) \<and> - (real n *x + ?N s - ?N (Floor s) - real (j+1)) > 0" by (simp only: th1[rule_format] th2[rule_format])
-    hence "\<exists> j\<in> {n.. 0}. ?I (?p (p,n,s) j)"
-      using pns by (simp add: fp_def nn diff_def add_ac mult_ac numfloor numadd numneg
-	del: diff_less_0_iff_less diff_le_0_iff_le) 
-    then obtain "j" where j_def: "j\<in> {n .. 0} \<and> ?I (?p (p,n,s) j)" by blast
-    hence "\<exists>x \<in> {?p (p,n,s) j |j. n\<le> j \<and> j \<le> 0 }. ?I x" by auto
-    hence ?case using pns 
-      by (simp only: FS,simp add: bex_Un)
-    (rule disjI2, rule disjI2,rule exI [where x="p"],
-      rule exI [where x="n"],rule exI [where x="s"],simp_all add: nn)
-  }
-  ultimately show ?case by blast
-qed (auto simp add: Let_def split_def)
-
-    (* Linearize a formula where Bound 0 ranges over [0,1) *)
-
-constdefs rsplit :: "(int \<Rightarrow> num \<Rightarrow> fm) \<Rightarrow> num \<Rightarrow> fm"
-  "rsplit f a \<equiv> foldr disj (map (\<lambda> (\<phi>, n, s). conj \<phi> (f n s)) (rsplit0 a)) F"
-
-lemma foldr_disj_map: "Ifm bs (foldr disj (map f xs) F) = (\<exists> x \<in> set xs. Ifm bs (f x))"
-by(induct xs, simp_all)
-
-lemma foldr_conj_map: "Ifm bs (foldr conj (map f xs) T) = (\<forall> x \<in> set xs. Ifm bs (f x))"
-by(induct xs, simp_all)
-
-lemma foldr_disj_map_rlfm: 
-  assumes lf: "\<forall> n s. numbound0 s \<longrightarrow> isrlfm (f n s)"
-  and \<phi>: "\<forall> (\<phi>,n,s) \<in> set xs. numbound0 s \<and> isrlfm \<phi>"
-  shows "isrlfm (foldr disj (map (\<lambda> (\<phi>, n, s). conj \<phi> (f n s)) xs) F)"
-using lf \<phi> by (induct xs, auto)
-
-lemma rsplit_ex: "Ifm bs (rsplit f a) = (\<exists> (\<phi>,n,s) \<in> set (rsplit0 a). Ifm bs (conj \<phi> (f n s)))"
-using foldr_disj_map[where xs="rsplit0 a"] rsplit_def by (simp add: split_def)
-
-lemma rsplit_l: assumes lf: "\<forall> n s. numbound0 s \<longrightarrow> isrlfm (f n s)"
-  shows "isrlfm (rsplit f a)"
-proof-
-  from rsplit0_cs[where t="a"] have th: "\<forall> (\<phi>,n,s) \<in> set (rsplit0 a). numbound0 s \<and> isrlfm \<phi>" by blast
-  from foldr_disj_map_rlfm[OF lf th] rsplit_def show ?thesis by simp
-qed
-
-lemma rsplit: 
-  assumes xp: "x \<ge> 0" and x1: "x < 1"
-  and f: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> (Ifm (x#bs) (f n s) = Ifm (x#bs) (g a))"
-  shows "Ifm (x#bs) (rsplit f a) = Ifm (x#bs) (g a)"
-proof(auto)
-  let ?I = "\<lambda>x p. Ifm (x#bs) p"
-  let ?N = "\<lambda> x t. Inum (x#bs) t"
-  assume "?I x (rsplit f a)"
-  hence "\<exists> (\<phi>,n,s) \<in> set (rsplit0 a). ?I x (And \<phi> (f n s))" using rsplit_ex by simp
-  then obtain "\<phi>" "n" "s" where fnsS:"(\<phi>,n,s) \<in> set (rsplit0 a)" and "?I x (And \<phi> (f n s))" by blast
-  hence \<phi>: "?I x \<phi>" and fns: "?I x (f n s)" by auto
-  from rsplit0_cs[where t="a" and bs="bs" and x="x", rule_format, OF fnsS] \<phi> 
-  have th: "(?N x a = ?N x (CN 0 n s)) \<and> numbound0 s" by auto
-  from f[rule_format, OF th] fns show "?I x (g a)" by simp
-next
-  let ?I = "\<lambda>x p. Ifm (x#bs) p"
-  let ?N = "\<lambda> x t. Inum (x#bs) t"
-  assume ga: "?I x (g a)"
-  from rsplit0_complete[OF xp x1, where bs="bs" and t="a"] 
-  obtain "\<phi>" "n" "s" where fnsS:"(\<phi>,n,s) \<in> set (rsplit0 a)" and fx: "?I x \<phi>" by blast
-  from rsplit0_cs[where t="a" and x="x" and bs="bs"] fnsS fx
-  have ans: "?N x a = ?N x (CN 0 n s)" and nb: "numbound0 s" by auto
-  with ga f have "?I x (f n s)" by auto
-  with rsplit_ex fnsS fx show "?I x (rsplit f a)" by auto
-qed
-
-definition lt :: "int \<Rightarrow> num \<Rightarrow> fm" where
-  lt_def: "lt c t = (if c = 0 then (Lt t) else if c > 0 then (Lt (CN 0 c t)) 
-                        else (Gt (CN 0 (-c) (Neg t))))"
-
-definition  le :: "int \<Rightarrow> num \<Rightarrow> fm" where
-  le_def: "le c t = (if c = 0 then (Le t) else if c > 0 then (Le (CN 0 c t)) 
-                        else (Ge (CN 0 (-c) (Neg t))))"
-
-definition  gt :: "int \<Rightarrow> num \<Rightarrow> fm" where
-  gt_def: "gt c t = (if c = 0 then (Gt t) else if c > 0 then (Gt (CN 0 c t)) 
-                        else (Lt (CN 0 (-c) (Neg t))))"
-
-definition  ge :: "int \<Rightarrow> num \<Rightarrow> fm" where
-  ge_def: "ge c t = (if c = 0 then (Ge t) else if c > 0 then (Ge (CN 0 c t)) 
-                        else (Le (CN 0 (-c) (Neg t))))"
-
-definition  eq :: "int \<Rightarrow> num \<Rightarrow> fm" where
-  eq_def: "eq c t = (if c = 0 then (Eq t) else if c > 0 then (Eq (CN 0 c t)) 
-                        else (Eq (CN 0 (-c) (Neg t))))"
-
-definition neq :: "int \<Rightarrow> num \<Rightarrow> fm" where
-  neq_def: "neq c t = (if c = 0 then (NEq t) else if c > 0 then (NEq (CN 0 c t)) 
-                        else (NEq (CN 0 (-c) (Neg t))))"
-
-lemma lt_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (lt n s) = Ifm (x#bs) (Lt a)"
-  (is "\<forall> a n s . ?N a = ?N (CN 0 n s) \<and> _\<longrightarrow> ?I (lt n s) = ?I (Lt a)")
-proof(clarify)
-  fix a n s
-  assume H: "?N a = ?N (CN 0 n s)"
-  show "?I (lt n s) = ?I (Lt a)" using H by (cases "n=0", (simp add: lt_def))
-  (cases "n > 0", simp_all add: lt_def algebra_simps myless[rule_format, where b="0"])
-qed
-
-lemma lt_l: "isrlfm (rsplit lt a)"
-  by (rule rsplit_l[where f="lt" and a="a"], auto simp add: lt_def,
-    case_tac s, simp_all, case_tac "nat", simp_all)
-
-lemma le_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (le n s) = Ifm (x#bs) (Le a)" (is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (le n s) = ?I (Le a)")
-proof(clarify)
-  fix a n s
-  assume H: "?N a = ?N (CN 0 n s)"
-  show "?I (le n s) = ?I (Le a)" using H by (cases "n=0", (simp add: le_def))
-  (cases "n > 0", simp_all add: le_def algebra_simps myl[rule_format, where b="0"])
-qed
-
-lemma le_l: "isrlfm (rsplit le a)"
-  by (rule rsplit_l[where f="le" and a="a"], auto simp add: le_def) 
-(case_tac s, simp_all, case_tac "nat",simp_all)
-
-lemma gt_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (gt n s) = Ifm (x#bs) (Gt a)" (is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (gt n s) = ?I (Gt a)")
-proof(clarify)
-  fix a n s
-  assume H: "?N a = ?N (CN 0 n s)"
-  show "?I (gt n s) = ?I (Gt a)" using H by (cases "n=0", (simp add: gt_def))
-  (cases "n > 0", simp_all add: gt_def algebra_simps myless[rule_format, where b="0"])
-qed
-lemma gt_l: "isrlfm (rsplit gt a)"
-  by (rule rsplit_l[where f="gt" and a="a"], auto simp add: gt_def) 
-(case_tac s, simp_all, case_tac "nat", simp_all)
-
-lemma ge_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (ge n s) = Ifm (x#bs) (Ge a)" (is "\<forall> a n s . ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (ge n s) = ?I (Ge a)")
-proof(clarify)
-  fix a n s 
-  assume H: "?N a = ?N (CN 0 n s)"
-  show "?I (ge n s) = ?I (Ge a)" using H by (cases "n=0", (simp add: ge_def))
-  (cases "n > 0", simp_all add: ge_def algebra_simps myl[rule_format, where b="0"])
-qed
-lemma ge_l: "isrlfm (rsplit ge a)"
-  by (rule rsplit_l[where f="ge" and a="a"], auto simp add: ge_def) 
-(case_tac s, simp_all, case_tac "nat", simp_all)
-
-lemma eq_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (eq n s) = Ifm (x#bs) (Eq a)" (is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (eq n s) = ?I (Eq a)")
-proof(clarify)
-  fix a n s 
-  assume H: "?N a = ?N (CN 0 n s)"
-  show "?I (eq n s) = ?I (Eq a)" using H by (auto simp add: eq_def algebra_simps)
-qed
-lemma eq_l: "isrlfm (rsplit eq a)"
-  by (rule rsplit_l[where f="eq" and a="a"], auto simp add: eq_def) 
-(case_tac s, simp_all, case_tac"nat", simp_all)
-
-lemma neq_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (neq n s) = Ifm (x#bs) (NEq a)" (is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (neq n s) = ?I (NEq a)")
-proof(clarify)
-  fix a n s bs
-  assume H: "?N a = ?N (CN 0 n s)"
-  show "?I (neq n s) = ?I (NEq a)" using H by (auto simp add: neq_def algebra_simps)
-qed
-
-lemma neq_l: "isrlfm (rsplit neq a)"
-  by (rule rsplit_l[where f="neq" and a="a"], auto simp add: neq_def) 
-(case_tac s, simp_all, case_tac"nat", simp_all)
-
-lemma small_le: 
-  assumes u0:"0 \<le> u" and u1: "u < 1"
-  shows "(-u \<le> real (n::int)) = (0 \<le> n)"
-using u0 u1  by auto
-
-lemma small_lt: 
-  assumes u0:"0 \<le> u" and u1: "u < 1"
-  shows "(real (n::int) < real (m::int) - u) = (n < m)"
-using u0 u1  by auto
-
-lemma rdvd01_cs: 
-  assumes up: "u \<ge> 0" and u1: "u<1" and np: "real n > 0"
-  shows "(real (i::int) rdvd real (n::int) * u - s) = (\<exists> j\<in> {0 .. n - 1}. real n * u = s - real (floor s) + real j \<and> real i rdvd real (j - floor s))" (is "?lhs = ?rhs")
-proof-
-  let ?ss = "s - real (floor s)"
-  from real_of_int_floor_add_one_gt[where r="s", simplified myless[rule_format,where a="s"]] 
-    real_of_int_floor_le[where r="s"]  have ss0:"?ss \<ge> 0" and ss1:"?ss < 1" 
-    by (auto simp add: myl[rule_format, where b="s", symmetric] myless[rule_format, where a="?ss"])
-  from np have n0: "real n \<ge> 0" by simp
-  from mult_left_mono[OF up n0] mult_strict_left_mono[OF u1 np] 
-  have nu0:"real n * u - s \<ge> -s" and nun:"real n * u -s < real n - s" by auto  
-  from int_rdvd_real[where i="i" and x="real (n::int) * u - s"] 
-  have "real i rdvd real n * u - s = 
-    (i dvd floor (real n * u -s) \<and> (real (floor (real n * u - s)) = real n * u - s ))" 
-    (is "_ = (?DE)" is "_ = (?D \<and> ?E)") by simp
-  also have "\<dots> = (?DE \<and> real(floor (real n * u - s) + floor s)\<ge> -?ss 
-    \<and> real(floor (real n * u - s) + floor s)< real n - ?ss)" (is "_=(?DE \<and>real ?a \<ge> _ \<and> real ?a < _)")
-    using nu0 nun  by auto
-  also have "\<dots> = (?DE \<and> ?a \<ge> 0 \<and> ?a < n)" by(simp only: small_le[OF ss0 ss1] small_lt[OF ss0 ss1])
-  also have "\<dots> = (?DE \<and> (\<exists> j\<in> {0 .. (n - 1)}. ?a = j))" by simp
-  also have "\<dots> = (?DE \<and> (\<exists> j\<in> {0 .. (n - 1)}. real (\<lfloor>real n * u - s\<rfloor>) = real j - real \<lfloor>s\<rfloor> ))"
-    by (simp only: algebra_simps real_of_int_diff[symmetric] real_of_int_inject del: real_of_int_diff)
-  also have "\<dots> = ((\<exists> j\<in> {0 .. (n - 1)}. real n * u - s = real j - real \<lfloor>s\<rfloor> \<and> real i rdvd real n * u - s))" using int_rdvd_iff[where i="i" and t="\<lfloor>real n * u - s\<rfloor>"]
-    by (auto cong: conj_cong)
-  also have "\<dots> = ?rhs" by(simp cong: conj_cong) (simp add: algebra_simps )
-  finally show ?thesis .
-qed
-
-definition
-  DVDJ:: "int \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm"
-where
-  DVDJ_def: "DVDJ i n s = (foldr disj (map (\<lambda> j. conj (Eq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) (Dvd i (Sub (C j) (Floor (Neg s))))) (iupt(0,n - 1))) F)"
-
-definition
-  NDVDJ:: "int \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm"
-where
-  NDVDJ_def: "NDVDJ i n s = (foldr conj (map (\<lambda> j. disj (NEq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) (NDvd i (Sub (C j) (Floor (Neg s))))) (iupt(0,n - 1))) T)"
-
-lemma DVDJ_DVD: 
-  assumes xp:"x\<ge> 0" and x1: "x < 1" and np:"real n > 0"
-  shows "Ifm (x#bs) (DVDJ i n s) = Ifm (x#bs) (Dvd i (CN 0 n s))"
-proof-
-  let ?f = "\<lambda> j. conj (Eq(CN 0 n (Add s (Sub(Floor (Neg s)) (C j))))) (Dvd i (Sub (C j) (Floor (Neg s))))"
-  let ?s= "Inum (x#bs) s"
-  from foldr_disj_map[where xs="iupt(0,n - 1)" and bs="x#bs" and f="?f"]
-  have "Ifm (x#bs) (DVDJ i n s) = (\<exists> j\<in> {0 .. (n - 1)}. Ifm (x#bs) (?f j))" 
-    by (simp add: iupt_set np DVDJ_def del: iupt.simps)
-  also have "\<dots> = (\<exists> j\<in> {0 .. (n - 1)}. real n * x = (- ?s) - real (floor (- ?s)) + real j \<and> real i rdvd real (j - floor (- ?s)))" by (simp add: algebra_simps diff_def[symmetric])
-  also from rdvd01_cs[OF xp x1 np, where i="i" and s="-?s"] 
-  have "\<dots> = (real i rdvd real n * x - (-?s))" by simp
-  finally show ?thesis by simp
-qed
-
-lemma NDVDJ_NDVD: 
-  assumes xp:"x\<ge> 0" and x1: "x < 1" and np:"real n > 0"
-  shows "Ifm (x#bs) (NDVDJ i n s) = Ifm (x#bs) (NDvd i (CN 0 n s))"
-proof-
-  let ?f = "\<lambda> j. disj(NEq(CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) (NDvd i (Sub (C j) (Floor(Neg s))))"
-  let ?s= "Inum (x#bs) s"
-  from foldr_conj_map[where xs="iupt(0,n - 1)" and bs="x#bs" and f="?f"]
-  have "Ifm (x#bs) (NDVDJ i n s) = (\<forall> j\<in> {0 .. (n - 1)}. Ifm (x#bs) (?f j))" 
-    by (simp add: iupt_set np NDVDJ_def del: iupt.simps)
-  also have "\<dots> = (\<not> (\<exists> j\<in> {0 .. (n - 1)}. real n * x = (- ?s) - real (floor (- ?s)) + real j \<and> real i rdvd real (j - floor (- ?s))))" by (simp add: algebra_simps diff_def[symmetric])
-  also from rdvd01_cs[OF xp x1 np, where i="i" and s="-?s"] 
-  have "\<dots> = (\<not> (real i rdvd real n * x - (-?s)))" by simp
-  finally show ?thesis by simp
-qed  
-
-lemma foldr_disj_map_rlfm2: 
-  assumes lf: "\<forall> n . isrlfm (f n)"
-  shows "isrlfm (foldr disj (map f xs) F)"
-using lf by (induct xs, auto)
-lemma foldr_And_map_rlfm2: 
-  assumes lf: "\<forall> n . isrlfm (f n)"
-  shows "isrlfm (foldr conj (map f xs) T)"
-using lf by (induct xs, auto)
-
-lemma DVDJ_l: assumes ip: "i >0" and np: "n>0" and nb: "numbound0 s"
-  shows "isrlfm (DVDJ i n s)"
-proof-
-  let ?f="\<lambda>j. conj (Eq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j)))))
-                         (Dvd i (Sub (C j) (Floor (Neg s))))"
-  have th: "\<forall> j. isrlfm (?f j)" using nb np by auto
-  from DVDJ_def foldr_disj_map_rlfm2[OF th] show ?thesis by simp 
-qed
-
-lemma NDVDJ_l: assumes ip: "i >0" and np: "n>0" and nb: "numbound0 s"
-  shows "isrlfm (NDVDJ i n s)"
-proof-
-  let ?f="\<lambda>j. disj (NEq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j)))))
-                      (NDvd i (Sub (C j) (Floor (Neg s))))"
-  have th: "\<forall> j. isrlfm (?f j)" using nb np by auto
-  from NDVDJ_def foldr_And_map_rlfm2[OF th] show ?thesis by auto
-qed
-
-definition DVD :: "int \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm" where
-  DVD_def: "DVD i c t =
-  (if i=0 then eq c t else 
-  if c = 0 then (Dvd i t) else if c >0 then DVDJ (abs i) c t else DVDJ (abs i) (-c) (Neg t))"
-
-definition  NDVD :: "int \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm" where
-  "NDVD i c t =
-  (if i=0 then neq c t else 
-  if c = 0 then (NDvd i t) else if c >0 then NDVDJ (abs i) c t else NDVDJ (abs i) (-c) (Neg t))"
-
-lemma DVD_mono: 
-  assumes xp: "0\<le> x" and x1: "x < 1" 
-  shows "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (DVD i n s) = Ifm (x#bs) (Dvd i a)"
-  (is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (DVD i n s) = ?I (Dvd i a)")
-proof(clarify)
-  fix a n s 
-  assume H: "?N a = ?N (CN 0 n s)" and nb: "numbound0 s"
-  let ?th = "?I (DVD i n s) = ?I (Dvd i a)"
-  have "i=0 \<or> (i\<noteq>0 \<and> n=0) \<or> (i\<noteq>0 \<and> n < 0) \<or> (i\<noteq>0 \<and> n > 0)" by arith
-  moreover {assume iz: "i=0" hence ?th using eq_mono[rule_format, OF conjI[OF H nb]] 
-      by (simp add: DVD_def rdvd_left_0_eq)}
-  moreover {assume inz: "i\<noteq>0" and "n=0" hence ?th by (simp add: H DVD_def) } 
-  moreover {assume inz: "i\<noteq>0" and "n<0" hence ?th 
-      by (simp add: DVD_def H DVDJ_DVD[OF xp x1] rdvd_abs1 
-	rdvd_minus[where d="i" and t="real n * x + Inum (x # bs) s"]) } 
-  moreover {assume inz: "i\<noteq>0" and "n>0" hence ?th by (simp add:DVD_def H DVDJ_DVD[OF xp x1] rdvd_abs1)}
-  ultimately show ?th by blast
-qed
-
-lemma NDVD_mono:   assumes xp: "0\<le> x" and x1: "x < 1" 
-  shows "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (NDVD i n s) = Ifm (x#bs) (NDvd i a)"
-  (is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (NDVD i n s) = ?I (NDvd i a)")
-proof(clarify)
-  fix a n s 
-  assume H: "?N a = ?N (CN 0 n s)" and nb: "numbound0 s"
-  let ?th = "?I (NDVD i n s) = ?I (NDvd i a)"
-  have "i=0 \<or> (i\<noteq>0 \<and> n=0) \<or> (i\<noteq>0 \<and> n < 0) \<or> (i\<noteq>0 \<and> n > 0)" by arith
-  moreover {assume iz: "i=0" hence ?th using neq_mono[rule_format, OF conjI[OF H nb]] 
-      by (simp add: NDVD_def rdvd_left_0_eq)}
-  moreover {assume inz: "i\<noteq>0" and "n=0" hence ?th by (simp add: H NDVD_def) } 
-  moreover {assume inz: "i\<noteq>0" and "n<0" hence ?th 
-      by (simp add: NDVD_def H NDVDJ_NDVD[OF xp x1] rdvd_abs1 
-	rdvd_minus[where d="i" and t="real n * x + Inum (x # bs) s"]) } 
-  moreover {assume inz: "i\<noteq>0" and "n>0" hence ?th 
-      by (simp add:NDVD_def H NDVDJ_NDVD[OF xp x1] rdvd_abs1)}
-  ultimately show ?th by blast
-qed
-
-lemma DVD_l: "isrlfm (rsplit (DVD i) a)"
-  by (rule rsplit_l[where f="DVD i" and a="a"], auto simp add: DVD_def eq_def DVDJ_l) 
-(case_tac s, simp_all, case_tac "nat", simp_all)
-
-lemma NDVD_l: "isrlfm (rsplit (NDVD i) a)"
-  by (rule rsplit_l[where f="NDVD i" and a="a"], auto simp add: NDVD_def neq_def NDVDJ_l) 
-(case_tac s, simp_all, case_tac "nat", simp_all)
-
-consts rlfm :: "fm \<Rightarrow> fm"
-recdef rlfm "measure fmsize"
-  "rlfm (And p q) = conj (rlfm p) (rlfm q)"
-  "rlfm (Or p q) = disj (rlfm p) (rlfm q)"
-  "rlfm (Imp p q) = disj (rlfm (NOT p)) (rlfm q)"
-  "rlfm (Iff p q) = disj (conj(rlfm p) (rlfm q)) (conj(rlfm (NOT p)) (rlfm (NOT q)))"
-  "rlfm (Lt a) = rsplit lt a"
-  "rlfm (Le a) = rsplit le a"
-  "rlfm (Gt a) = rsplit gt a"
-  "rlfm (Ge a) = rsplit ge a"
-  "rlfm (Eq a) = rsplit eq a"
-  "rlfm (NEq a) = rsplit neq a"
-  "rlfm (Dvd i a) = rsplit (\<lambda> t. DVD i t) a"
-  "rlfm (NDvd i a) = rsplit (\<lambda> t. NDVD i t) a"
-  "rlfm (NOT (And p q)) = disj (rlfm (NOT p)) (rlfm (NOT q))"
-  "rlfm (NOT (Or p q)) = conj (rlfm (NOT p)) (rlfm (NOT q))"
-  "rlfm (NOT (Imp p q)) = conj (rlfm p) (rlfm (NOT q))"
-  "rlfm (NOT (Iff p q)) = disj (conj(rlfm p) (rlfm(NOT q))) (conj(rlfm(NOT p)) (rlfm q))"
-  "rlfm (NOT (NOT p)) = rlfm p"
-  "rlfm (NOT T) = F"
-  "rlfm (NOT F) = T"
-  "rlfm (NOT (Lt a)) = simpfm (rlfm (Ge a))"
-  "rlfm (NOT (Le a)) = simpfm (rlfm (Gt a))"
-  "rlfm (NOT (Gt a)) = simpfm (rlfm (Le a))"
-  "rlfm (NOT (Ge a)) = simpfm (rlfm (Lt a))"
-  "rlfm (NOT (Eq a)) = simpfm (rlfm (NEq a))"
-  "rlfm (NOT (NEq a)) = simpfm (rlfm (Eq a))"
-  "rlfm (NOT (Dvd i a)) = simpfm (rlfm (NDvd i a))"
-  "rlfm (NOT (NDvd i a)) = simpfm (rlfm (Dvd i a))"
-  "rlfm p = p" (hints simp add: fmsize_pos)
-
-lemma bound0at_l : "\<lbrakk>isatom p ; bound0 p\<rbrakk> \<Longrightarrow> isrlfm p"
-  by (induct p rule: isrlfm.induct, auto)
-lemma zgcd_le1: assumes ip: "0 < i" shows "zgcd i j \<le> i"
-proof-
-  from zgcd_zdvd1 have th: "zgcd i j dvd i" by blast
-  from zdvd_imp_le[OF th ip] show ?thesis .
-qed
-
-
-lemma simpfm_rl: "isrlfm p \<Longrightarrow> isrlfm (simpfm p)"
-proof (induct p)
-  case (Lt a) 
-  hence "bound0 (Lt a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)"
-    by (cases a,simp_all, case_tac "nat", simp_all)
-  moreover
-  {assume "bound0 (Lt a)" hence bn:"bound0 (simpfm (Lt a))"  
-      using simpfm_bound0 by blast
-    have "isatom (simpfm (Lt a))" by (cases "simpnum a", auto simp add: Let_def)
-    with bn bound0at_l have ?case by blast}
-  moreover 
-  {fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e"
-    {
-      assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
-      with numgcd_pos[where t="CN 0 c (simpnum e)"]
-      have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp
-      from prems have th:"numgcd (CN 0 c (simpnum e)) \<le> c" 
-	by (simp add: numgcd_def zgcd_le1)
-      from prems have th': "c\<noteq>0" by auto
-      from prems have cp: "c \<ge> 0" by simp
-      from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']]
-	have "0 < c div numgcd (CN 0 c (simpnum e))" by simp
-    }
-    with prems have ?case
-      by (simp add: Let_def reducecoeff_def reducecoeffh_numbound0)}
-  ultimately show ?case by blast
-next
-  case (Le a)   
-  hence "bound0 (Le a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)"
-    by (cases a,simp_all, case_tac "nat", simp_all)
-  moreover
-  {assume "bound0 (Le a)" hence bn:"bound0 (simpfm (Le a))"  
-      using simpfm_bound0 by blast
-    have "isatom (simpfm (Le a))" by (cases "simpnum a", auto simp add: Let_def)
-    with bn bound0at_l have ?case by blast}
-  moreover 
-  {fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e"
-    {
-      assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
-      with numgcd_pos[where t="CN 0 c (simpnum e)"]
-      have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp
-      from prems have th:"numgcd (CN 0 c (simpnum e)) \<le> c" 
-	by (simp add: numgcd_def zgcd_le1)
-      from prems have th': "c\<noteq>0" by auto
-      from prems have cp: "c \<ge> 0" by simp
-      from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']]
-	have "0 < c div numgcd (CN 0 c (simpnum e))" by simp
-    }
-    with prems have ?case
-      by (simp add: Let_def reducecoeff_def simpnum_numbound0 reducecoeffh_numbound0)}
-  ultimately show ?case by blast
-next
-  case (Gt a)   
-  hence "bound0 (Gt a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)"
-    by (cases a,simp_all, case_tac "nat", simp_all)
-  moreover
-  {assume "bound0 (Gt a)" hence bn:"bound0 (simpfm (Gt a))"  
-      using simpfm_bound0 by blast
-    have "isatom (simpfm (Gt a))" by (cases "simpnum a", auto simp add: Let_def)
-    with bn bound0at_l have ?case by blast}
-  moreover 
-  {fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e"
-    {
-      assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
-      with numgcd_pos[where t="CN 0 c (simpnum e)"]
-      have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp
-      from prems have th:"numgcd (CN 0 c (simpnum e)) \<le> c" 
-	by (simp add: numgcd_def zgcd_le1)
-      from prems have th': "c\<noteq>0" by auto
-      from prems have cp: "c \<ge> 0" by simp
-      from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']]
-	have "0 < c div numgcd (CN 0 c (simpnum e))" by simp
-    }
-    with prems have ?case
-      by (simp add: Let_def reducecoeff_def simpnum_numbound0 reducecoeffh_numbound0)}
-  ultimately show ?case by blast
-next
-  case (Ge a)   
-  hence "bound0 (Ge a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)"
-    by (cases a,simp_all, case_tac "nat", simp_all)
-  moreover
-  {assume "bound0 (Ge a)" hence bn:"bound0 (simpfm (Ge a))"  
-      using simpfm_bound0 by blast
-    have "isatom (simpfm (Ge a))" by (cases "simpnum a", auto simp add: Let_def)
-    with bn bound0at_l have ?case by blast}
-  moreover 
-  {fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e"
-    {
-      assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
-      with numgcd_pos[where t="CN 0 c (simpnum e)"]
-      have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp
-      from prems have th:"numgcd (CN 0 c (simpnum e)) \<le> c" 
-	by (simp add: numgcd_def zgcd_le1)
-      from prems have th': "c\<noteq>0" by auto
-      from prems have cp: "c \<ge> 0" by simp
-      from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']]
-	have "0 < c div numgcd (CN 0 c (simpnum e))" by simp
-    }
-    with prems have ?case
-      by (simp add: Let_def reducecoeff_def simpnum_numbound0 reducecoeffh_numbound0)}
-  ultimately show ?case by blast
-next
-  case (Eq a)   
-  hence "bound0 (Eq a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)"
-    by (cases a,simp_all, case_tac "nat", simp_all)
-  moreover
-  {assume "bound0 (Eq a)" hence bn:"bound0 (simpfm (Eq a))"  
-      using simpfm_bound0 by blast
-    have "isatom (simpfm (Eq a))" by (cases "simpnum a", auto simp add: Let_def)
-    with bn bound0at_l have ?case by blast}
-  moreover 
-  {fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e"
-    {
-      assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
-      with numgcd_pos[where t="CN 0 c (simpnum e)"]
-      have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp
-      from prems have th:"numgcd (CN 0 c (simpnum e)) \<le> c" 
-	by (simp add: numgcd_def zgcd_le1)
-      from prems have th': "c\<noteq>0" by auto
-      from prems have cp: "c \<ge> 0" by simp
-      from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']]
-	have "0 < c div numgcd (CN 0 c (simpnum e))" by simp
-    }
-    with prems have ?case
-      by (simp add: Let_def reducecoeff_def simpnum_numbound0 reducecoeffh_numbound0)}
-  ultimately show ?case by blast
-next
-  case (NEq a)  
-  hence "bound0 (NEq a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)"
-    by (cases a,simp_all, case_tac "nat", simp_all)
-  moreover
-  {assume "bound0 (NEq a)" hence bn:"bound0 (simpfm (NEq a))"  
-      using simpfm_bound0 by blast
-    have "isatom (simpfm (NEq a))" by (cases "simpnum a", auto simp add: Let_def)
-    with bn bound0at_l have ?case by blast}
-  moreover 
-  {fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e"
-    {
-      assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
-      with numgcd_pos[where t="CN 0 c (simpnum e)"]
-      have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp
-      from prems have th:"numgcd (CN 0 c (simpnum e)) \<le> c" 
-	by (simp add: numgcd_def zgcd_le1)
-      from prems have th': "c\<noteq>0" by auto
-      from prems have cp: "c \<ge> 0" by simp
-      from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']]
-	have "0 < c div numgcd (CN 0 c (simpnum e))" by simp
-    }
-    with prems have ?case
-      by (simp add: Let_def reducecoeff_def simpnum_numbound0 reducecoeffh_numbound0)}
-  ultimately show ?case by blast
-next
-  case (Dvd i a) hence "bound0 (Dvd i a)" by auto hence bn:"bound0 (simpfm (Dvd i a))"  
-    using simpfm_bound0 by blast
-  have "isatom (simpfm (Dvd i a))" by (cases "simpnum a", auto simp add: Let_def split_def)
-  with bn bound0at_l show ?case by blast
-next
-  case (NDvd i a)  hence "bound0 (NDvd i a)" by auto hence bn:"bound0 (simpfm (NDvd i a))"  
-    using simpfm_bound0 by blast
-  have "isatom (simpfm (NDvd i a))" by (cases "simpnum a", auto simp add: Let_def split_def)
-  with bn bound0at_l show ?case by blast
-qed(auto simp add: conj_def imp_def disj_def iff_def Let_def simpfm_bound0 numadd_nb numneg_nb)
-
-lemma rlfm_I:
-  assumes qfp: "qfree p"
-  and xp: "0 \<le> x" and x1: "x < 1"
-  shows "(Ifm (x#bs) (rlfm p) = Ifm (x# bs) p) \<and> isrlfm (rlfm p)"
-  using qfp 
-by (induct p rule: rlfm.induct) 
-(auto simp add: rsplit[OF xp x1 lt_mono] lt_l rsplit[OF xp x1 le_mono] le_l rsplit[OF xp x1 gt_mono] gt_l
-               rsplit[OF xp x1 ge_mono] ge_l rsplit[OF xp x1 eq_mono] eq_l rsplit[OF xp x1 neq_mono] neq_l
-               rsplit[OF xp x1 DVD_mono[OF xp x1]] DVD_l rsplit[OF xp x1 NDVD_mono[OF xp x1]] NDVD_l simpfm_rl)
-lemma rlfm_l:
-  assumes qfp: "qfree p"
-  shows "isrlfm (rlfm p)"
-  using qfp lt_l gt_l ge_l le_l eq_l neq_l DVD_l NDVD_l 
-by (induct p rule: rlfm.induct,auto simp add: simpfm_rl)
-
-    (* Operations needed for Ferrante and Rackoff *)
-lemma rminusinf_inf:
-  assumes lp: "isrlfm p"
-  shows "\<exists> z. \<forall> x < z. Ifm (x#bs) (minusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p")
-using lp
-proof (induct p rule: minusinf.induct)
-  case (1 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto
-next
-  case (2 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto
-next
-  case (3 c e) 
-  from prems have nb: "numbound0 e" by simp
-  from prems have cp: "real c > 0" by simp
-  fix a
-  let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
-  {fix x
-    assume xz: "x < ?z"
-    hence "(real c * x < - ?e)" 
-      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
-    hence "real c * x + ?e < 0" by arith
-    hence "real c * x + ?e \<noteq> 0" by simp
-    with xz have "?P ?z x (Eq (CN 0 c e))"
-      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp  }
-  hence "\<forall> x < ?z. ?P ?z x (Eq (CN 0 c e))" by simp
-  thus ?case by blast
-next
-  case (4 c e)   
-  from prems have nb: "numbound0 e" by simp
-  from prems have cp: "real c > 0" by simp
-  fix a
-  let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
-  {fix x
-    assume xz: "x < ?z"
-    hence "(real c * x < - ?e)" 
-      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
-    hence "real c * x + ?e < 0" by arith
-    hence "real c * x + ?e \<noteq> 0" by simp
-    with xz have "?P ?z x (NEq (CN 0 c e))"
-      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
-  hence "\<forall> x < ?z. ?P ?z x (NEq (CN 0 c e))" by simp
-  thus ?case by blast
-next
-  case (5 c e) 
-    from prems have nb: "numbound0 e" by simp
-  from prems have cp: "real c > 0" by simp
-  fix a
-  let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
-  {fix x
-    assume xz: "x < ?z"
-    hence "(real c * x < - ?e)" 
-      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
-    hence "real c * x + ?e < 0" by arith
-    with xz have "?P ?z x (Lt (CN 0 c e))"
-      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"]  by simp }
-  hence "\<forall> x < ?z. ?P ?z x (Lt (CN 0 c e))" by simp
-  thus ?case by blast
-next
-  case (6 c e)  
-    from prems have nb: "numbound0 e" by simp
-  from prems have cp: "real c > 0" by simp
-  fix a
-  let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
-  {fix x
-    assume xz: "x < ?z"
-    hence "(real c * x < - ?e)" 
-      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
-    hence "real c * x + ?e < 0" by arith
-    with xz have "?P ?z x (Le (CN 0 c e))"
-      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
-  hence "\<forall> x < ?z. ?P ?z x (Le (CN 0 c e))" by simp
-  thus ?case by blast
-next
-  case (7 c e)  
-    from prems have nb: "numbound0 e" by simp
-  from prems have cp: "real c > 0" by simp
-  fix a
-  let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
-  {fix x
-    assume xz: "x < ?z"
-    hence "(real c * x < - ?e)" 
-      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
-    hence "real c * x + ?e < 0" by arith
-    with xz have "?P ?z x (Gt (CN 0 c e))"
-      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
-  hence "\<forall> x < ?z. ?P ?z x (Gt (CN 0 c e))" by simp
-  thus ?case by blast
-next
-  case (8 c e)  
-    from prems have nb: "numbound0 e" by simp
-  from prems have cp: "real c > 0" by simp
-  fix a
-  let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
-  {fix x
-    assume xz: "x < ?z"
-    hence "(real c * x < - ?e)" 
-      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
-    hence "real c * x + ?e < 0" by arith
-    with xz have "?P ?z x (Ge (CN 0 c e))"
-      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
-  hence "\<forall> x < ?z. ?P ?z x (Ge (CN 0 c e))" by simp
-  thus ?case by blast
-qed simp_all
-
-lemma rplusinf_inf:
-  assumes lp: "isrlfm p"
-  shows "\<exists> z. \<forall> x > z. Ifm (x#bs) (plusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p")
-using lp
-proof (induct p rule: isrlfm.induct)
-  case (1 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto
-next
-  case (2 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto
-next
-  case (3 c e) 
-  from prems have nb: "numbound0 e" by simp
-  from prems have cp: "real c > 0" by simp
-  fix a
-  let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
-  {fix x
-    assume xz: "x > ?z"
-    with mult_strict_right_mono [OF xz cp] cp
-    have "(real c * x > - ?e)" by (simp add: mult_ac)
-    hence "real c * x + ?e > 0" by arith
-    hence "real c * x + ?e \<noteq> 0" by simp
-    with xz have "?P ?z x (Eq (CN 0 c e))"
-      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
-  hence "\<forall> x > ?z. ?P ?z x (Eq (CN 0 c e))" by simp
-  thus ?case by blast
-next
-  case (4 c e) 
-  from prems have nb: "numbound0 e" by simp
-  from prems have cp: "real c > 0" by simp
-  fix a
-  let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
-  {fix x
-    assume xz: "x > ?z"
-    with mult_strict_right_mono [OF xz cp] cp
-    have "(real c * x > - ?e)" by (simp add: mult_ac)
-    hence "real c * x + ?e > 0" by arith
-    hence "real c * x + ?e \<noteq> 0" by simp
-    with xz have "?P ?z x (NEq (CN 0 c e))"
-      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
-  hence "\<forall> x > ?z. ?P ?z x (NEq (CN 0 c e))" by simp
-  thus ?case by blast
-next
-  case (5 c e) 
-  from prems have nb: "numbound0 e" by simp
-  from prems have cp: "real c > 0" by simp
-  fix a
-  let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
-  {fix x
-    assume xz: "x > ?z"
-    with mult_strict_right_mono [OF xz cp] cp
-    have "(real c * x > - ?e)" by (simp add: mult_ac)
-    hence "real c * x + ?e > 0" by arith
-    with xz have "?P ?z x (Lt (CN 0 c e))"
-      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
-  hence "\<forall> x > ?z. ?P ?z x (Lt (CN 0 c e))" by simp
-  thus ?case by blast
-next
-  case (6 c e) 
-  from prems have nb: "numbound0 e" by simp
-  from prems have cp: "real c > 0" by simp
-  fix a
-  let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
-  {fix x
-    assume xz: "x > ?z"
-    with mult_strict_right_mono [OF xz cp] cp
-    have "(real c * x > - ?e)" by (simp add: mult_ac)
-    hence "real c * x + ?e > 0" by arith
-    with xz have "?P ?z x (Le (CN 0 c e))"
-      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
-  hence "\<forall> x > ?z. ?P ?z x (Le (CN 0 c e))" by simp
-  thus ?case by blast
-next
-  case (7 c e) 
-  from prems have nb: "numbound0 e" by simp
-  from prems have cp: "real c > 0" by simp
-  fix a
-  let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
-  {fix x
-    assume xz: "x > ?z"
-    with mult_strict_right_mono [OF xz cp] cp
-    have "(real c * x > - ?e)" by (simp add: mult_ac)
-    hence "real c * x + ?e > 0" by arith
-    with xz have "?P ?z x (Gt (CN 0 c e))"
-      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
-  hence "\<forall> x > ?z. ?P ?z x (Gt (CN 0 c e))" by simp
-  thus ?case by blast
-next
-  case (8 c e) 
-  from prems have nb: "numbound0 e" by simp
-  from prems have cp: "real c > 0" by simp
-  fix a
-  let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
-  {fix x
-    assume xz: "x > ?z"
-    with mult_strict_right_mono [OF xz cp] cp
-    have "(real c * x > - ?e)" by (simp add: mult_ac)
-    hence "real c * x + ?e > 0" by arith
-    with xz have "?P ?z x (Ge (CN 0 c e))"
-      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"]   by simp }
-  hence "\<forall> x > ?z. ?P ?z x (Ge (CN 0 c e))" by simp
-  thus ?case by blast
-qed simp_all
-
-lemma rminusinf_bound0:
-  assumes lp: "isrlfm p"
-  shows "bound0 (minusinf p)"
-  using lp
-  by (induct p rule: minusinf.induct) simp_all
-
-lemma rplusinf_bound0:
-  assumes lp: "isrlfm p"
-  shows "bound0 (plusinf p)"
-  using lp
-  by (induct p rule: plusinf.induct) simp_all
-
-lemma rminusinf_ex:
-  assumes lp: "isrlfm p"
-  and ex: "Ifm (a#bs) (minusinf p)"
-  shows "\<exists> x. Ifm (x#bs) p"
-proof-
-  from bound0_I [OF rminusinf_bound0[OF lp], where b="a" and bs ="bs"] ex
-  have th: "\<forall> x. Ifm (x#bs) (minusinf p)" by auto
-  from rminusinf_inf[OF lp, where bs="bs"] 
-  obtain z where z_def: "\<forall>x<z. Ifm (x # bs) (minusinf p) = Ifm (x # bs) p" by blast
-  from th have "Ifm ((z - 1)#bs) (minusinf p)" by simp
-  moreover have "z - 1 < z" by simp
-  ultimately show ?thesis using z_def by auto
-qed
-
-lemma rplusinf_ex:
-  assumes lp: "isrlfm p"
-  and ex: "Ifm (a#bs) (plusinf p)"
-  shows "\<exists> x. Ifm (x#bs) p"
-proof-
-  from bound0_I [OF rplusinf_bound0[OF lp], where b="a" and bs ="bs"] ex
-  have th: "\<forall> x. Ifm (x#bs) (plusinf p)" by auto
-  from rplusinf_inf[OF lp, where bs="bs"] 
-  obtain z where z_def: "\<forall>x>z. Ifm (x # bs) (plusinf p) = Ifm (x # bs) p" by blast
-  from th have "Ifm ((z + 1)#bs) (plusinf p)" by simp
-  moreover have "z + 1 > z" by simp
-  ultimately show ?thesis using z_def by auto
-qed
-
-consts 
-  \<Upsilon>:: "fm \<Rightarrow> (num \<times> int) list"
-  \<upsilon> :: "fm \<Rightarrow> (num \<times> int) \<Rightarrow> fm "
-recdef \<Upsilon> "measure size"
-  "\<Upsilon> (And p q) = (\<Upsilon> p @ \<Upsilon> q)" 
-  "\<Upsilon> (Or p q) = (\<Upsilon> p @ \<Upsilon> q)" 
-  "\<Upsilon> (Eq  (CN 0 c e)) = [(Neg e,c)]"
-  "\<Upsilon> (NEq (CN 0 c e)) = [(Neg e,c)]"
-  "\<Upsilon> (Lt  (CN 0 c e)) = [(Neg e,c)]"
-  "\<Upsilon> (Le  (CN 0 c e)) = [(Neg e,c)]"
-  "\<Upsilon> (Gt  (CN 0 c e)) = [(Neg e,c)]"
-  "\<Upsilon> (Ge  (CN 0 c e)) = [(Neg e,c)]"
-  "\<Upsilon> p = []"
-
-recdef \<upsilon> "measure size"
-  "\<upsilon> (And p q) = (\<lambda> (t,n). And (\<upsilon> p (t,n)) (\<upsilon> q (t,n)))"
-  "\<upsilon> (Or p q) = (\<lambda> (t,n). Or (\<upsilon> p (t,n)) (\<upsilon> q (t,n)))"
-  "\<upsilon> (Eq (CN 0 c e)) = (\<lambda> (t,n). Eq (Add (Mul c t) (Mul n e)))"
-  "\<upsilon> (NEq (CN 0 c e)) = (\<lambda> (t,n). NEq (Add (Mul c t) (Mul n e)))"
-  "\<upsilon> (Lt (CN 0 c e)) = (\<lambda> (t,n). Lt (Add (Mul c t) (Mul n e)))"
-  "\<upsilon> (Le (CN 0 c e)) = (\<lambda> (t,n). Le (Add (Mul c t) (Mul n e)))"
-  "\<upsilon> (Gt (CN 0 c e)) = (\<lambda> (t,n). Gt (Add (Mul c t) (Mul n e)))"
-  "\<upsilon> (Ge (CN 0 c e)) = (\<lambda> (t,n). Ge (Add (Mul c t) (Mul n e)))"
-  "\<upsilon> p = (\<lambda> (t,n). p)"
-
-lemma \<upsilon>_I: assumes lp: "isrlfm p"
-  and np: "real n > 0" and nbt: "numbound0 t"
-  shows "(Ifm (x#bs) (\<upsilon> p (t,n)) = Ifm (((Inum (x#bs) t)/(real n))#bs) p) \<and> bound0 (\<upsilon> p (t,n))" (is "(?I x (\<upsilon> p (t,n)) = ?I ?u p) \<and> ?B p" is "(_ = ?I (?t/?n) p) \<and> _" is "(_ = ?I (?N x t /_) p) \<and> _")
-  using lp
-proof(induct p rule: \<upsilon>.induct)
-  case (5 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
-  have "?I ?u (Lt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) < 0)"
-    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
-  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) < 0)"
-    by (simp only: pos_less_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
-      and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> = (real c *?t + ?n* (?N x e) < 0)"
-    using np by simp 
-  finally show ?case using nbt nb by (simp add: algebra_simps)
-next
-  case (6 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
-  have "?I ?u (Le (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<le> 0)"
-    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
-  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<le> 0)"
-    by (simp only: pos_le_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
-      and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> = (real c *?t + ?n* (?N x e) \<le> 0)"
-    using np by simp 
-  finally show ?case using nbt nb by (simp add: algebra_simps)
-next
-  case (7 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
-  have "?I ?u (Gt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) > 0)"
-    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
-  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) > 0)"
-    by (simp only: pos_divide_less_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
-      and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> = (real c *?t + ?n* (?N x e) > 0)"
-    using np by simp 
-  finally show ?case using nbt nb by (simp add: algebra_simps)
-next
-  case (8 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
-  have "?I ?u (Ge (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<ge> 0)"
-    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
-  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<ge> 0)"
-    by (simp only: pos_divide_le_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
-      and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> = (real c *?t + ?n* (?N x e) \<ge> 0)"
-    using np by simp 
-  finally show ?case using nbt nb by (simp add: algebra_simps)
-next
-  case (3 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
-  from np have np: "real n \<noteq> 0" by simp
-  have "?I ?u (Eq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) = 0)"
-    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
-  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) = 0)"
-    by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
-      and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> = (real c *?t + ?n* (?N x e) = 0)"
-    using np by simp 
-  finally show ?case using nbt nb by (simp add: algebra_simps)
-next
-  case (4 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
-  from np have np: "real n \<noteq> 0" by simp
-  have "?I ?u (NEq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<noteq> 0)"
-    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
-  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<noteq> 0)"
-    by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
-      and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> = (real c *?t + ?n* (?N x e) \<noteq> 0)"
-    using np by simp 
-  finally show ?case using nbt nb by (simp add: algebra_simps)
-qed(simp_all add: nbt numbound0_I[where bs ="bs" and b="(Inum (x#bs) t)/ real n" and b'="x"] nth_pos2)
-
-lemma \<Upsilon>_l:
-  assumes lp: "isrlfm p"
-  shows "\<forall> (t,k) \<in> set (\<Upsilon> p). numbound0 t \<and> k >0"
-using lp
-by(induct p rule: \<Upsilon>.induct)  auto
-
-lemma rminusinf_\<Upsilon>:
-  assumes lp: "isrlfm p"
-  and nmi: "\<not> (Ifm (a#bs) (minusinf p))" (is "\<not> (Ifm (a#bs) (?M p))")
-  and ex: "Ifm (x#bs) p" (is "?I x p")
-  shows "\<exists> (s,m) \<in> set (\<Upsilon> p). x \<ge> Inum (a#bs) s / real m" (is "\<exists> (s,m) \<in> ?U p. x \<ge> ?N a s / real m")
-proof-
-  have "\<exists> (s,m) \<in> set (\<Upsilon> p). real m * x \<ge> Inum (a#bs) s " (is "\<exists> (s,m) \<in> ?U p. real m *x \<ge> ?N a s")
-    using lp nmi ex
-    by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"] nth_pos2)
-  then obtain s m where smU: "(s,m) \<in> set (\<Upsilon> p)" and mx: "real m * x \<ge> ?N a s" by blast
-  from \<Upsilon>_l[OF lp] smU have mp: "real m > 0" by auto
-  from pos_divide_le_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<ge> ?N a s / real m" 
-    by (auto simp add: mult_commute)
-  thus ?thesis using smU by auto
-qed
-
-lemma rplusinf_\<Upsilon>:
-  assumes lp: "isrlfm p"
-  and nmi: "\<not> (Ifm (a#bs) (plusinf p))" (is "\<not> (Ifm (a#bs) (?M p))")
-  and ex: "Ifm (x#bs) p" (is "?I x p")
-  shows "\<exists> (s,m) \<in> set (\<Upsilon> p). x \<le> Inum (a#bs) s / real m" (is "\<exists> (s,m) \<in> ?U p. x \<le> ?N a s / real m")
-proof-
-  have "\<exists> (s,m) \<in> set (\<Upsilon> p). real m * x \<le> Inum (a#bs) s " (is "\<exists> (s,m) \<in> ?U p. real m *x \<le> ?N a s")
-    using lp nmi ex
-    by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"] nth_pos2)
-  then obtain s m where smU: "(s,m) \<in> set (\<Upsilon> p)" and mx: "real m * x \<le> ?N a s" by blast
-  from \<Upsilon>_l[OF lp] smU have mp: "real m > 0" by auto
-  from pos_le_divide_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<le> ?N a s / real m" 
-    by (auto simp add: mult_commute)
-  thus ?thesis using smU by auto
-qed
-
-lemma lin_dense: 
-  assumes lp: "isrlfm p"
-  and noS: "\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> (\<lambda> (t,n). Inum (x#bs) t / real n) ` set (\<Upsilon> p)" 
-  (is "\<forall> t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda> (t,n). ?N x t / real n ) ` (?U p)")
-  and lx: "l < x" and xu:"x < u" and px:" Ifm (x#bs) p"
-  and ly: "l < y" and yu: "y < u"
-  shows "Ifm (y#bs) p"
-using lp px noS
-proof (induct p rule: isrlfm.induct)
-  case (5 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
-    from prems have "x * real c + ?N x e < 0" by (simp add: algebra_simps)
-    hence pxc: "x < (- ?N x e) / real c" 
-      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="-?N x e"])
-    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
-    with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
-    hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
-    moreover {assume y: "y < (-?N x e)/ real c"
-      hence "y * real c < - ?N x e"
-	by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
-      hence "real c * y + ?N x e < 0" by (simp add: algebra_simps)
-      hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
-    moreover {assume y: "y > (- ?N x e) / real c" 
-      with yu have eu: "u > (- ?N x e) / real c" by auto
-      with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto)
-      with lx pxc have "False" by auto
-      hence ?case by simp }
-    ultimately show ?case by blast
-next
-  case (6 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp +
-    from prems have "x * real c + ?N x e \<le> 0" by (simp add: algebra_simps)
-    hence pxc: "x \<le> (- ?N x e) / real c" 
-      by (simp only: pos_le_divide_eq[OF cp, where a="x" and b="-?N x e"])
-    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
-    with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
-    hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
-    moreover {assume y: "y < (-?N x e)/ real c"
-      hence "y * real c < - ?N x e"
-	by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
-      hence "real c * y + ?N x e < 0" by (simp add: algebra_simps)
-      hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
-    moreover {assume y: "y > (- ?N x e) / real c" 
-      with yu have eu: "u > (- ?N x e) / real c" by auto
-      with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto)
-      with lx pxc have "False" by auto
-      hence ?case by simp }
-    ultimately show ?case by blast
-next
-  case (7 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
-    from prems have "x * real c + ?N x e > 0" by (simp add: algebra_simps)
-    hence pxc: "x > (- ?N x e) / real c" 
-      by (simp only: pos_divide_less_eq[OF cp, where a="x" and b="-?N x e"])
-    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
-    with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
-    hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
-    moreover {assume y: "y > (-?N x e)/ real c"
-      hence "y * real c > - ?N x e"
-	by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
-      hence "real c * y + ?N x e > 0" by (simp add: algebra_simps)
-      hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
-    moreover {assume y: "y < (- ?N x e) / real c" 
-      with ly have eu: "l < (- ?N x e) / real c" by auto
-      with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto)
-      with xu pxc have "False" by auto
-      hence ?case by simp }
-    ultimately show ?case by blast
-next
-  case (8 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
-    from prems have "x * real c + ?N x e \<ge> 0" by (simp add: algebra_simps)
-    hence pxc: "x \<ge> (- ?N x e) / real c" 
-      by (simp only: pos_divide_le_eq[OF cp, where a="x" and b="-?N x e"])
-    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
-    with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
-    hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
-    moreover {assume y: "y > (-?N x e)/ real c"
-      hence "y * real c > - ?N x e"
-	by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
-      hence "real c * y + ?N x e > 0" by (simp add: algebra_simps)
-      hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
-    moreover {assume y: "y < (- ?N x e) / real c" 
-      with ly have eu: "l < (- ?N x e) / real c" by auto
-      with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto)
-      with xu pxc have "False" by auto
-      hence ?case by simp }
-    ultimately show ?case by blast
-next
-  case (3 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
-    from cp have cnz: "real c \<noteq> 0" by simp
-    from prems have "x * real c + ?N x e = 0" by (simp add: algebra_simps)
-    hence pxc: "x = (- ?N x e) / real c" 
-      by (simp only: nonzero_eq_divide_eq[OF cnz, where a="x" and b="-?N x e"])
-    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
-    with lx xu have yne: "x \<noteq> - ?N x e / real c" by auto
-    with pxc show ?case by simp
-next
-  case (4 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
-    from cp have cnz: "real c \<noteq> 0" by simp
-    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
-    with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
-    hence "y* real c \<noteq> -?N x e"      
-      by (simp only: nonzero_eq_divide_eq[OF cnz, where a="y" and b="-?N x e"]) simp
-    hence "y* real c + ?N x e \<noteq> 0" by (simp add: algebra_simps)
-    thus ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] 
-      by (simp add: algebra_simps)
-qed (auto simp add: nth_pos2 numbound0_I[where bs="bs" and b="y" and b'="x"])
-
-lemma finite_set_intervals:
-  assumes px: "P (x::real)" 
-  and lx: "l \<le> x" and xu: "x \<le> u"
-  and linS: "l\<in> S" and uinS: "u \<in> S"
-  and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u"
-  shows "\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a \<le> x \<and> x \<le> b \<and> P x"
-proof-
-  let ?Mx = "{y. y\<in> S \<and> y \<le> x}"
-  let ?xM = "{y. y\<in> S \<and> x \<le> y}"
-  let ?a = "Max ?Mx"
-  let ?b = "Min ?xM"
-  have MxS: "?Mx \<subseteq> S" by blast
-  hence fMx: "finite ?Mx" using fS finite_subset by auto
-  from lx linS have linMx: "l \<in> ?Mx" by blast
-  hence Mxne: "?Mx \<noteq> {}" by blast
-  have xMS: "?xM \<subseteq> S" by blast
-  hence fxM: "finite ?xM" using fS finite_subset by auto
-  from xu uinS have linxM: "u \<in> ?xM" by blast
-  hence xMne: "?xM \<noteq> {}" by blast
-  have ax:"?a \<le> x" using Mxne fMx by auto
-  have xb:"x \<le> ?b" using xMne fxM by auto
-  have "?a \<in> ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a \<in> S" using MxS by blast
-  have "?b \<in> ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b \<in> S" using xMS by blast
-  have noy:"\<forall> y. ?a < y \<and> y < ?b \<longrightarrow> y \<notin> S"
-  proof(clarsimp)
-    fix y
-    assume ay: "?a < y" and yb: "y < ?b" and yS: "y \<in> S"
-    from yS have "y\<in> ?Mx \<or> y\<in> ?xM" by auto
-    moreover {assume "y \<in> ?Mx" hence "y \<le> ?a" using Mxne fMx by auto with ay have "False" by simp}
-    moreover {assume "y \<in> ?xM" hence "y \<ge> ?b" using xMne fxM by auto with yb have "False" by simp}
-    ultimately show "False" by blast
-  qed
-  from ainS binS noy ax xb px show ?thesis by blast
-qed
-
-lemma finite_set_intervals2:
-  assumes px: "P (x::real)" 
-  and lx: "l \<le> x" and xu: "x \<le> u"
-  and linS: "l\<in> S" and uinS: "u \<in> S"
-  and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u"
-  shows "(\<exists> s\<in> S. P s) \<or> (\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a < x \<and> x < b \<and> P x)"
-proof-
-  from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su]
-  obtain a and b where 
-    as: "a\<in> S" and bs: "b\<in> S" and noS:"\<forall>y. a < y \<and> y < b \<longrightarrow> y \<notin> S" and axb: "a \<le> x \<and> x \<le> b \<and> P x"  by auto
-  from axb have "x= a \<or> x= b \<or> (a < x \<and> x < b)" by auto
-  thus ?thesis using px as bs noS by blast 
-qed
-
-lemma rinf_\<Upsilon>:
-  assumes lp: "isrlfm p"
-  and nmi: "\<not> (Ifm (x#bs) (minusinf p))" (is "\<not> (Ifm (x#bs) (?M p))")
-  and npi: "\<not> (Ifm (x#bs) (plusinf p))" (is "\<not> (Ifm (x#bs) (?P p))")
-  and ex: "\<exists> x.  Ifm (x#bs) p" (is "\<exists> x. ?I x p")
-  shows "\<exists> (l,n) \<in> set (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> p). ?I ((Inum (x#bs) l / real n + Inum (x#bs) s / real m) / 2) p" 
-proof-
-  let ?N = "\<lambda> x t. Inum (x#bs) t"
-  let ?U = "set (\<Upsilon> p)"
-  from ex obtain a where pa: "?I a p" by blast
-  from bound0_I[OF rminusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] nmi
-  have nmi': "\<not> (?I a (?M p))" by simp
-  from bound0_I[OF rplusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] npi
-  have npi': "\<not> (?I a (?P p))" by simp
-  have "\<exists> (l,n) \<in> set (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> p). ?I ((?N a l/real n + ?N a s /real m) / 2) p"
-  proof-
-    let ?M = "(\<lambda> (t,c). ?N a t / real c) ` ?U"
-    have fM: "finite ?M" by auto
-    from rminusinf_\<Upsilon>[OF lp nmi pa] rplusinf_\<Upsilon>[OF lp npi pa] 
-    have "\<exists> (l,n) \<in> set (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> p). a \<le> ?N x l / real n \<and> a \<ge> ?N x s / real m" by blast
-    then obtain "t" "n" "s" "m" where 
-      tnU: "(t,n) \<in> ?U" and smU: "(s,m) \<in> ?U" 
-      and xs1: "a \<le> ?N x s / real m" and tx1: "a \<ge> ?N x t / real n" by blast
-    from \<Upsilon>_l[OF lp] tnU smU numbound0_I[where bs="bs" and b="x" and b'="a"] xs1 tx1 have xs: "a \<le> ?N a s / real m" and tx: "a \<ge> ?N a t / real n" by auto
-    from tnU have Mne: "?M \<noteq> {}" by auto
-    hence Une: "?U \<noteq> {}" by simp
-    let ?l = "Min ?M"
-    let ?u = "Max ?M"
-    have linM: "?l \<in> ?M" using fM Mne by simp
-    have uinM: "?u \<in> ?M" using fM Mne by simp
-    have tnM: "?N a t / real n \<in> ?M" using tnU by auto
-    have smM: "?N a s / real m \<in> ?M" using smU by auto 
-    have lM: "\<forall> t\<in> ?M. ?l \<le> t" using Mne fM by auto
-    have Mu: "\<forall> t\<in> ?M. t \<le> ?u" using Mne fM by auto
-    have "?l \<le> ?N a t / real n" using tnM Mne by simp hence lx: "?l \<le> a" using tx by simp
-    have "?N a s / real m \<le> ?u" using smM Mne by simp hence xu: "a \<le> ?u" using xs by simp
-    from finite_set_intervals2[where P="\<lambda> x. ?I x p",OF pa lx xu linM uinM fM lM Mu]
-    have "(\<exists> s\<in> ?M. ?I s p) \<or> 
-      (\<exists> t1\<in> ?M. \<exists> t2 \<in> ?M. (\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M) \<and> t1 < a \<and> a < t2 \<and> ?I a p)" .
-    moreover { fix u assume um: "u\<in> ?M" and pu: "?I u p"
-      hence "\<exists> (tu,nu) \<in> ?U. u = ?N a tu / real nu" by auto
-      then obtain "tu" "nu" where tuU: "(tu,nu) \<in> ?U" and tuu:"u= ?N a tu / real nu" by blast
-      have "(u + u) / 2 = u" by auto with pu tuu 
-      have "?I (((?N a tu / real nu) + (?N a tu / real nu)) / 2) p" by simp
-      with tuU have ?thesis by blast}
-    moreover{
-      assume "\<exists> t1\<in> ?M. \<exists> t2 \<in> ?M. (\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M) \<and> t1 < a \<and> a < t2 \<and> ?I a p"
-      then obtain t1 and t2 where t1M: "t1 \<in> ?M" and t2M: "t2\<in> ?M" 
-	and noM: "\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M" and t1x: "t1 < a" and xt2: "a < t2" and px: "?I a p"
-	by blast
-      from t1M have "\<exists> (t1u,t1n) \<in> ?U. t1 = ?N a t1u / real t1n" by auto
-      then obtain "t1u" "t1n" where t1uU: "(t1u,t1n) \<in> ?U" and t1u: "t1 = ?N a t1u / real t1n" by blast
-      from t2M have "\<exists> (t2u,t2n) \<in> ?U. t2 = ?N a t2u / real t2n" by auto
-      then obtain "t2u" "t2n" where t2uU: "(t2u,t2n) \<in> ?U" and t2u: "t2 = ?N a t2u / real t2n" by blast
-      from t1x xt2 have t1t2: "t1 < t2" by simp
-      let ?u = "(t1 + t2) / 2"
-      from less_half_sum[OF t1t2] gt_half_sum[OF t1t2] have t1lu: "t1 < ?u" and ut2: "?u < t2" by auto
-      from lin_dense[OF lp noM t1x xt2 px t1lu ut2] have "?I ?u p" .
-      with t1uU t2uU t1u t2u have ?thesis by blast}
-    ultimately show ?thesis by blast
-  qed
-  then obtain "l" "n" "s"  "m" where lnU: "(l,n) \<in> ?U" and smU:"(s,m) \<in> ?U" 
-    and pu: "?I ((?N a l / real n + ?N a s / real m) / 2) p" by blast
-  from lnU smU \<Upsilon>_l[OF lp] have nbl: "numbound0 l" and nbs: "numbound0 s" by auto
-  from numbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"] 
-    numbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu
-  have "?I ((?N x l / real n + ?N x s / real m) / 2) p" by simp
-  with lnU smU
-  show ?thesis by auto
-qed
-    (* The Ferrante - Rackoff Theorem *)
-
-theorem fr_eq: 
-  assumes lp: "isrlfm p"
-  shows "(\<exists> x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \<or> (Ifm (x#bs) (plusinf p)) \<or> (\<exists> (t,n) \<in> set (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> p). Ifm ((((Inum (x#bs) t)/  real n + (Inum (x#bs) s) / real m) /2)#bs) p))"
-  (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
-proof
-  assume px: "\<exists> x. ?I x p"
-  have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
-  moreover {assume "?M \<or> ?P" hence "?D" by blast}
-  moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
-    from rinf_\<Upsilon>[OF lp nmi npi] have "?F" using px by blast hence "?D" by blast}
-  ultimately show "?D" by blast
-next
-  assume "?D" 
-  moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .}
-  moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . }
-  moreover {assume f:"?F" hence "?E" by blast}
-  ultimately show "?E" by blast
-qed
-
-
-lemma fr_eq\<upsilon>: 
-  assumes lp: "isrlfm p"
-  shows "(\<exists> x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \<or> (Ifm (x#bs) (plusinf p)) \<or> (\<exists> (t,k) \<in> set (\<Upsilon> p). \<exists> (s,l) \<in> set (\<Upsilon> p). Ifm (x#bs) (\<upsilon> p (Add(Mul l t) (Mul k s) , 2*k*l))))"
-  (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
-proof
-  assume px: "\<exists> x. ?I x p"
-  have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
-  moreover {assume "?M \<or> ?P" hence "?D" by blast}
-  moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
-    let ?f ="\<lambda> (t,n). Inum (x#bs) t / real n"
-    let ?N = "\<lambda> t. Inum (x#bs) t"
-    {fix t n s m assume "(t,n)\<in> set (\<Upsilon> p)" and "(s,m) \<in> set (\<Upsilon> p)"
-      with \<Upsilon>_l[OF lp] have tnb: "numbound0 t" and np:"real n > 0" and snb: "numbound0 s" and mp:"real m > 0"
-	by auto
-      let ?st = "Add (Mul m t) (Mul n s)"
-      from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" 
-	by (simp add: mult_commute)
-      from tnb snb have st_nb: "numbound0 ?st" by simp
-      have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
-	using mnp mp np by (simp add: algebra_simps add_divide_distrib)
-      from \<upsilon>_I[OF lp mnp st_nb, where x="x" and bs="bs"] 
-      have "?I x (\<upsilon> p (?st,2*n*m)) = ?I ((?N t / real n + ?N s / real m) /2) p" by (simp only: st[symmetric])}
-    with rinf_\<Upsilon>[OF lp nmi npi px] have "?F" by blast hence "?D" by blast}
-  ultimately show "?D" by blast
-next
-  assume "?D" 
-  moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .}
-  moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . }
-  moreover {fix t k s l assume "(t,k) \<in> set (\<Upsilon> p)" and "(s,l) \<in> set (\<Upsilon> p)" 
-    and px:"?I x (\<upsilon> p (Add (Mul l t) (Mul k s), 2*k*l))"
-    with \<Upsilon>_l[OF lp] have tnb: "numbound0 t" and np:"real k > 0" and snb: "numbound0 s" and mp:"real l > 0" by auto
-    let ?st = "Add (Mul l t) (Mul k s)"
-    from mult_pos_pos[OF np mp] have mnp: "real (2*k*l) > 0" 
-      by (simp add: mult_commute)
-    from tnb snb have st_nb: "numbound0 ?st" by simp
-    from \<upsilon>_I[OF lp mnp st_nb, where bs="bs"] px have "?E" by auto}
-  ultimately show "?E" by blast
-qed
-
-text{* The overall Part *}
-
-lemma real_ex_int_real01:
-  shows "(\<exists> (x::real). P x) = (\<exists> (i::int) (u::real). 0\<le> u \<and> u< 1 \<and> P (real i + u))"
-proof(auto)
-  fix x
-  assume Px: "P x"
-  let ?i = "floor x"
-  let ?u = "x - real ?i"
-  have "x = real ?i + ?u" by simp
-  hence "P (real ?i + ?u)" using Px by simp
-  moreover have "real ?i \<le> x" using real_of_int_floor_le by simp hence "0 \<le> ?u" by arith
-  moreover have "?u < 1" using real_of_int_floor_add_one_gt[where r="x"] by arith 
-  ultimately show "(\<exists> (i::int) (u::real). 0\<le> u \<and> u< 1 \<and> P (real i + u))" by blast
-qed
-
-consts exsplitnum :: "num \<Rightarrow> num"
-  exsplit :: "fm \<Rightarrow> fm"
-recdef exsplitnum "measure size"
-  "exsplitnum (C c) = (C c)"
-  "exsplitnum (Bound 0) = Add (Bound 0) (Bound 1)"
-  "exsplitnum (Bound n) = Bound (n+1)"
-  "exsplitnum (Neg a) = Neg (exsplitnum a)"
-  "exsplitnum (Add a b) = Add (exsplitnum a) (exsplitnum b) "
-  "exsplitnum (Sub a b) = Sub (exsplitnum a) (exsplitnum b) "
-  "exsplitnum (Mul c a) = Mul c (exsplitnum a)"
-  "exsplitnum (Floor a) = Floor (exsplitnum a)"
-  "exsplitnum (CN 0 c a) = CN 0 c (Add (Mul c (Bound 1)) (exsplitnum a))"
-  "exsplitnum (CN n c a) = CN (n+1) c (exsplitnum a)"
-  "exsplitnum (CF c s t) = CF c (exsplitnum s) (exsplitnum t)"
-
-recdef exsplit "measure size"
-  "exsplit (Lt a) = Lt (exsplitnum a)"
-  "exsplit (Le a) = Le (exsplitnum a)"
-  "exsplit (Gt a) = Gt (exsplitnum a)"
-  "exsplit (Ge a) = Ge (exsplitnum a)"
-  "exsplit (Eq a) = Eq (exsplitnum a)"
-  "exsplit (NEq a) = NEq (exsplitnum a)"
-  "exsplit (Dvd i a) = Dvd i (exsplitnum a)"
-  "exsplit (NDvd i a) = NDvd i (exsplitnum a)"
-  "exsplit (And p q) = And (exsplit p) (exsplit q)"
-  "exsplit (Or p q) = Or (exsplit p) (exsplit q)"
-  "exsplit (Imp p q) = Imp (exsplit p) (exsplit q)"
-  "exsplit (Iff p q) = Iff (exsplit p) (exsplit q)"
-  "exsplit (NOT p) = NOT (exsplit p)"
-  "exsplit p = p"
-
-lemma exsplitnum: 
-  "Inum (x#y#bs) (exsplitnum t) = Inum ((x+y) #bs) t"
-  by(induct t rule: exsplitnum.induct) (simp_all add: algebra_simps)
-
-lemma exsplit: 
-  assumes qfp: "qfree p"
-  shows "Ifm (x#y#bs) (exsplit p) = Ifm ((x+y)#bs) p"
-using qfp exsplitnum[where x="x" and y="y" and bs="bs"]
-by(induct p rule: exsplit.induct) simp_all
-
-lemma splitex:
-  assumes qf: "qfree p"
-  shows "(Ifm bs (E p)) = (\<exists> (i::int). Ifm (real i#bs) (E (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) (exsplit p))))" (is "?lhs = ?rhs")
-proof-
-  have "?rhs = (\<exists> (i::int). \<exists> x. 0\<le> x \<and> x < 1 \<and> Ifm (x#(real i)#bs) (exsplit p))"
-    by (simp add: myless[rule_format, where b="1"] myless[rule_format, where b="0"] add_ac diff_def)
-  also have "\<dots> = (\<exists> (i::int). \<exists> x. 0\<le> x \<and> x < 1 \<and> Ifm ((real i + x) #bs) p)"
-    by (simp only: exsplit[OF qf] add_ac)
-  also have "\<dots> = (\<exists> x. Ifm (x#bs) p)" 
-    by (simp only: real_ex_int_real01[where P="\<lambda> x. Ifm (x#bs) p"])
-  finally show ?thesis by simp
-qed
-
-    (* Implement the right hand sides of Cooper's theorem and Ferrante and Rackoff. *)
-
-constdefs ferrack01:: "fm \<Rightarrow> fm"
-  "ferrack01 p \<equiv> (let p' = rlfm(And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p);
-                    U = remdups(map simp_num_pair 
-                     (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m))
-                           (alluopairs (\<Upsilon> p')))) 
-  in decr (evaldjf (\<upsilon> p') U ))"
-
-lemma fr_eq_01: 
-  assumes qf: "qfree p"
-  shows "(\<exists> x. Ifm (x#bs) (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p)) = (\<exists> (t,n) \<in> set (\<Upsilon> (rlfm (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p))). \<exists> (s,m) \<in> set (\<Upsilon> (rlfm (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p))). Ifm (x#bs) (\<upsilon> (rlfm (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p)) (Add (Mul m t) (Mul n s), 2*n*m)))"
-  (is "(\<exists> x. ?I x ?q) = ?F")
-proof-
-  let ?rq = "rlfm ?q"
-  let ?M = "?I x (minusinf ?rq)"
-  let ?P = "?I x (plusinf ?rq)"
-  have MF: "?M = False"
-    apply (simp add: Let_def reducecoeff_def numgcd_def zgcd_def rsplit_def ge_def lt_def conj_def disj_def)
-    by (cases "rlfm p = And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C -1)))", simp_all)
-  have PF: "?P = False" apply (simp add: Let_def reducecoeff_def numgcd_def zgcd_def rsplit_def ge_def lt_def conj_def disj_def)
-    by (cases "rlfm p = And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C -1)))", simp_all)
-  have "(\<exists> x. ?I x ?q ) = 
-    ((?I x (minusinf ?rq)) \<or> (?I x (plusinf ?rq )) \<or> (\<exists> (t,n) \<in> set (\<Upsilon> ?rq). \<exists> (s,m) \<in> set (\<Upsilon> ?rq ). ?I x (\<upsilon> ?rq (Add (Mul m t) (Mul n s), 2*n*m))))"
-    (is "(\<exists> x. ?I x ?q) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
-  proof
-    assume "\<exists> x. ?I x ?q"  
-    then obtain x where qx: "?I x ?q" by blast
-    hence xp: "0\<le> x" and x1: "x< 1" and px: "?I x p" 
-      by (auto simp add: rsplit_def lt_def ge_def rlfm_I[OF qf])
-    from qx have "?I x ?rq " 
-      by (simp add: rsplit_def lt_def ge_def rlfm_I[OF qf xp x1])
-    hence lqx: "?I x ?rq " using simpfm[where p="?rq" and bs="x#bs"] by auto
-    from qf have qfq:"isrlfm ?rq"  
-      by (auto simp add: rsplit_def lt_def ge_def rlfm_I[OF qf xp x1])
-    with lqx fr_eq\<upsilon>[OF qfq] show "?M \<or> ?P \<or> ?F" by blast
-  next
-    assume D: "?D"
-    let ?U = "set (\<Upsilon> ?rq )"
-    from MF PF D have "?F" by auto
-    then obtain t n s m where aU:"(t,n) \<in> ?U" and bU:"(s,m)\<in> ?U" and rqx: "?I x (\<upsilon> ?rq (Add (Mul m t) (Mul n s), 2*n*m))" by blast
-    from qf have lrq:"isrlfm ?rq"using rlfm_l[OF qf] 
-      by (auto simp add: rsplit_def lt_def ge_def)
-    from aU bU \<Upsilon>_l[OF lrq] have tnb: "numbound0 t" and np:"real n > 0" and snb: "numbound0 s" and mp:"real m > 0" by (auto simp add: split_def)
-    let ?st = "Add (Mul m t) (Mul n s)"
-    from tnb snb have stnb: "numbound0 ?st" by simp
-    from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" 
-      by (simp add: mult_commute)
-    from conjunct1[OF \<upsilon>_I[OF lrq mnp stnb, where bs="bs" and x="x"], symmetric] rqx
-    have "\<exists> x. ?I x ?rq" by auto
-    thus "?E" 
-      using rlfm_I[OF qf] by (auto simp add: rsplit_def lt_def ge_def)
-  qed
-  with MF PF show ?thesis by blast
-qed
-
-lemma \<Upsilon>_cong_aux:
-  assumes Ul: "\<forall> (t,n) \<in> set U. numbound0 t \<and> n >0"
-  shows "((\<lambda> (t,n). Inum (x#bs) t /real n) ` (set (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)) (alluopairs U)))) = ((\<lambda> ((t,n),(s,m)). (Inum (x#bs) t /real n + Inum (x#bs) s /real m)/2) ` (set U \<times> set U))"
-  (is "?lhs = ?rhs")
-proof(auto)
-  fix t n s m
-  assume "((t,n),(s,m)) \<in> set (alluopairs U)"
-  hence th: "((t,n),(s,m)) \<in> (set U \<times> set U)"
-    using alluopairs_set1[where xs="U"] by blast
-  let ?N = "\<lambda> t. Inum (x#bs) t"
-  let ?st= "Add (Mul m t) (Mul n s)"
-  from Ul th have mnz: "m \<noteq> 0" by auto
-  from Ul th have  nnz: "n \<noteq> 0" by auto  
-  have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
-   using mnz nnz by (simp add: algebra_simps add_divide_distrib)
- 
-  thus "(real m *  Inum (x # bs) t + real n * Inum (x # bs) s) /
-       (2 * real n * real m)
-       \<in> (\<lambda>((t, n), s, m).
-             (Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2) `
-         (set U \<times> set U)"using mnz nnz th  
-    apply (auto simp add: th add_divide_distrib algebra_simps split_def image_def)
-    by (rule_tac x="(s,m)" in bexI,simp_all) 
-  (rule_tac x="(t,n)" in bexI,simp_all)
-next
-  fix t n s m
-  assume tnU: "(t,n) \<in> set U" and smU:"(s,m) \<in> set U" 
-  let ?N = "\<lambda> t. Inum (x#bs) t"
-  let ?st= "Add (Mul m t) (Mul n s)"
-  from Ul smU have mnz: "m \<noteq> 0" by auto
-  from Ul tnU have  nnz: "n \<noteq> 0" by auto  
-  have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
-   using mnz nnz by (simp add: algebra_simps add_divide_distrib)
- let ?P = "\<lambda> (t',n') (s',m'). (Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2"
- have Pc:"\<forall> a b. ?P a b = ?P b a"
-   by auto
- from Ul alluopairs_set1 have Up:"\<forall> ((t,n),(s,m)) \<in> set (alluopairs U). n \<noteq> 0 \<and> m \<noteq> 0" by blast
- from alluopairs_ex[OF Pc, where xs="U"] tnU smU
- have th':"\<exists> ((t',n'),(s',m')) \<in> set (alluopairs U). ?P (t',n') (s',m')"
-   by blast
- then obtain t' n' s' m' where ts'_U: "((t',n'),(s',m')) \<in> set (alluopairs U)" 
-   and Pts': "?P (t',n') (s',m')" by blast
- from ts'_U Up have mnz': "m' \<noteq> 0" and nnz': "n'\<noteq> 0" by auto
- let ?st' = "Add (Mul m' t') (Mul n' s')"
-   have st': "(?N t' / real n' + ?N s' / real m')/2 = ?N ?st' / real (2*n'*m')"
-   using mnz' nnz' by (simp add: algebra_simps add_divide_distrib)
- from Pts' have 
-   "(Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2" by simp
- also have "\<dots> = ((\<lambda>(t, n). Inum (x # bs) t / real n) ((\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) ((t',n'),(s',m'))))" by (simp add: st')
- finally show "(Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2
-          \<in> (\<lambda>(t, n). Inum (x # bs) t / real n) `
-            (\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) `
-            set (alluopairs U)"
-   using ts'_U by blast
-qed
-
-lemma \<Upsilon>_cong:
-  assumes lp: "isrlfm p"
-  and UU': "((\<lambda> (t,n). Inum (x#bs) t /real n) ` U') = ((\<lambda> ((t,n),(s,m)). (Inum (x#bs) t /real n + Inum (x#bs) s /real m)/2) ` (U \<times> U))" (is "?f ` U' = ?g ` (U\<times>U)")
-  and U: "\<forall> (t,n) \<in> U. numbound0 t \<and> n > 0"
-  and U': "\<forall> (t,n) \<in> U'. numbound0 t \<and> n > 0"
-  shows "(\<exists> (t,n) \<in> U. \<exists> (s,m) \<in> U. Ifm (x#bs) (\<upsilon> p (Add (Mul m t) (Mul n s),2*n*m))) = (\<exists> (t,n) \<in> U'. Ifm (x#bs) (\<upsilon> p (t,n)))"
-  (is "?lhs = ?rhs")
-proof
-  assume ?lhs
-  then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and 
-    Pst: "Ifm (x#bs) (\<upsilon> p (Add (Mul m t) (Mul n s),2*n*m))" by blast
-  let ?N = "\<lambda> t. Inum (x#bs) t"
-  from tnU smU U have tnb: "numbound0 t" and np: "n > 0" 
-    and snb: "numbound0 s" and mp:"m > 0"  by auto
-  let ?st= "Add (Mul m t) (Mul n s)"
-  from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" 
-      by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
-    from tnb snb have stnb: "numbound0 ?st" by simp
-  have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
-   using mp np by (simp add: algebra_simps add_divide_distrib)
-  from tnU smU UU' have "?g ((t,n),(s,m)) \<in> ?f ` U'" by blast
-  hence "\<exists> (t',n') \<in> U'. ?g ((t,n),(s,m)) = ?f (t',n')"
-    by auto (rule_tac x="(a,b)" in bexI, auto)
-  then obtain t' n' where tnU': "(t',n') \<in> U'" and th: "?g ((t,n),(s,m)) = ?f (t',n')" by blast
-  from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto
-  from \<upsilon>_I[OF lp mnp stnb, where bs="bs" and x="x"] Pst 
-  have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp
-  from conjunct1[OF \<upsilon>_I[OF lp np' tnb', where bs="bs" and x="x"], symmetric] th[simplified split_def fst_conv snd_conv,symmetric] Pst2[simplified st[symmetric]]
-  have "Ifm (x # bs) (\<upsilon> p (t', n')) " by (simp only: st) 
-  then show ?rhs using tnU' by auto 
-next
-  assume ?rhs
-  then obtain t' n' where tnU': "(t',n') \<in> U'" and Pt': "Ifm (x # bs) (\<upsilon> p (t', n'))" 
-    by blast
-  from tnU' UU' have "?f (t',n') \<in> ?g ` (U\<times>U)" by blast
-  hence "\<exists> ((t,n),(s,m)) \<in> (U\<times>U). ?f (t',n') = ?g ((t,n),(s,m))" 
-    by auto (rule_tac x="(a,b)" in bexI, auto)
-  then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and 
-    th: "?f (t',n') = ?g((t,n),(s,m)) "by blast
-    let ?N = "\<lambda> t. Inum (x#bs) t"
-  from tnU smU U have tnb: "numbound0 t" and np: "n > 0" 
-    and snb: "numbound0 s" and mp:"m > 0"  by auto
-  let ?st= "Add (Mul m t) (Mul n s)"
-  from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" 
-      by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
-    from tnb snb have stnb: "numbound0 ?st" by simp
-  have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
-   using mp np by (simp add: algebra_simps add_divide_distrib)
-  from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto
-  from \<upsilon>_I[OF lp np' tnb', where bs="bs" and x="x",simplified th[simplified split_def fst_conv snd_conv] st] Pt'
-  have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp
-  with \<upsilon>_I[OF lp mnp stnb, where x="x" and bs="bs"] tnU smU show ?lhs by blast
-qed
-  
-lemma ferrack01: 
-  assumes qf: "qfree p"
-  shows "((\<exists> x. Ifm (x#bs) (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p)) = (Ifm bs (ferrack01 p))) \<and> qfree (ferrack01 p)" (is "(?lhs = ?rhs) \<and> _")
-proof-
-  let ?I = "\<lambda> x p. Ifm (x#bs) p"
-  fix x
-  let ?N = "\<lambda> t. Inum (x#bs) t"
-  let ?q = "rlfm (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p)"
-  let ?U = "\<Upsilon> ?q"
-  let ?Up = "alluopairs ?U"
-  let ?g = "\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)"
-  let ?S = "map ?g ?Up"
-  let ?SS = "map simp_num_pair ?S"
-  let ?Y = "remdups ?SS"
-  let ?f= "(\<lambda> (t,n). ?N t / real n)"
-  let ?h = "\<lambda> ((t,n),(s,m)). (?N t/real n + ?N s/ real m) /2"
-  let ?F = "\<lambda> p. \<exists> a \<in> set (\<Upsilon> p). \<exists> b \<in> set (\<Upsilon> p). ?I x (\<upsilon> p (?g(a,b)))"
-  let ?ep = "evaldjf (\<upsilon> ?q) ?Y"
-  from rlfm_l[OF qf] have lq: "isrlfm ?q" 
-    by (simp add: rsplit_def lt_def ge_def conj_def disj_def Let_def reducecoeff_def numgcd_def zgcd_def)
-  from alluopairs_set1[where xs="?U"] have UpU: "set ?Up \<le> (set ?U \<times> set ?U)" by simp
-  from \<Upsilon>_l[OF lq] have U_l: "\<forall> (t,n) \<in> set ?U. numbound0 t \<and> n > 0" .
-  from U_l UpU 
-  have Up_: "\<forall> ((t,n),(s,m)) \<in> set ?Up. numbound0 t \<and> n> 0 \<and> numbound0 s \<and> m > 0" by auto
-  hence Snb: "\<forall> (t,n) \<in> set ?S. numbound0 t \<and> n > 0 "
-    by (auto simp add: mult_pos_pos)
-  have Y_l: "\<forall> (t,n) \<in> set ?Y. numbound0 t \<and> n > 0" 
-  proof-
-    { fix t n assume tnY: "(t,n) \<in> set ?Y" 
-      hence "(t,n) \<in> set ?SS" by simp
-      hence "\<exists> (t',n') \<in> set ?S. simp_num_pair (t',n') = (t,n)"
-	by (auto simp add: split_def) (rule_tac x="((aa,ba),(ab,bb))" in bexI, simp_all)
-      then obtain t' n' where tn'S: "(t',n') \<in> set ?S" and tns: "simp_num_pair (t',n') = (t,n)" by blast
-      from tn'S Snb have tnb: "numbound0 t'" and np: "n' > 0" by auto
-      from simp_num_pair_l[OF tnb np tns]
-      have "numbound0 t \<and> n > 0" . }
-    thus ?thesis by blast
-  qed
-
-  have YU: "(?f ` set ?Y) = (?h ` (set ?U \<times> set ?U))"
-  proof-
-     from simp_num_pair_ci[where bs="x#bs"] have 
-    "\<forall>x. (?f o simp_num_pair) x = ?f x" by auto
-     hence th: "?f o simp_num_pair = ?f" using ext by blast
-    have "(?f ` set ?Y) = ((?f o simp_num_pair) ` set ?S)" by (simp add: image_compose)
-    also have "\<dots> = (?f ` set ?S)" by (simp add: th)
-    also have "\<dots> = ((?f o ?g) ` set ?Up)" 
-      by (simp only: set_map o_def image_compose[symmetric])
-    also have "\<dots> = (?h ` (set ?U \<times> set ?U))"
-      using \<Upsilon>_cong_aux[OF U_l, where x="x" and bs="bs", simplified set_map image_compose[symmetric]] by blast
-    finally show ?thesis .
-  qed
-  have "\<forall> (t,n) \<in> set ?Y. bound0 (\<upsilon> ?q (t,n))"
-  proof-
-    { fix t n assume tnY: "(t,n) \<in> set ?Y"
-      with Y_l have tnb: "numbound0 t" and np: "real n > 0" by auto
-      from \<upsilon>_I[OF lq np tnb]
-    have "bound0 (\<upsilon> ?q (t,n))"  by simp}
-    thus ?thesis by blast
-  qed
-  hence ep_nb: "bound0 ?ep"  using evaldjf_bound0[where xs="?Y" and f="\<upsilon> ?q"]
-    by auto
-
-  from fr_eq_01[OF qf, where bs="bs" and x="x"] have "?lhs = ?F ?q"
-    by (simp only: split_def fst_conv snd_conv)
-  also have "\<dots> = (\<exists> (t,n) \<in> set ?Y. ?I x (\<upsilon> ?q (t,n)))" using \<Upsilon>_cong[OF lq YU U_l Y_l]
-    by (simp only: split_def fst_conv snd_conv) 
-  also have "\<dots> = (Ifm (x#bs) ?ep)" 
-    using evaldjf_ex[where ps="?Y" and bs = "x#bs" and f="\<upsilon> ?q",symmetric]
-    by (simp only: split_def pair_collapse)
-  also have "\<dots> = (Ifm bs (decr ?ep))" using decr[OF ep_nb] by blast
-  finally have lr: "?lhs = ?rhs" by (simp only: ferrack01_def Let_def)
-  from decr_qf[OF ep_nb] have "qfree (ferrack01 p)" by (simp only: Let_def ferrack01_def)
-  with lr show ?thesis by blast
-qed
-
-lemma cp_thm': 
-  assumes lp: "iszlfm p (real (i::int)#bs)"
-  and up: "d\<beta> p 1" and dd: "d\<delta> p d" and dp: "d > 0"
-  shows "(\<exists> (x::int). Ifm (real x#bs) p) = ((\<exists> j\<in> {1 .. d}. Ifm (real j#bs) (minusinf p)) \<or> (\<exists> j\<in> {1.. d}. \<exists> b\<in> (Inum (real i#bs)) ` set (\<beta> p). Ifm ((b+real j)#bs) p))"
-  using cp_thm[OF lp up dd dp] by auto
-
-constdefs unit:: "fm \<Rightarrow> fm \<times> num list \<times> int"
-  "unit p \<equiv> (let p' = zlfm p ; l = \<zeta> p' ; q = And (Dvd l (CN 0 1 (C 0))) (a\<beta> p' l); d = \<delta> q;
-             B = remdups (map simpnum (\<beta> q)) ; a = remdups (map simpnum (\<alpha> q))
-             in if length B \<le> length a then (q,B,d) else (mirror q, a,d))"
-
-lemma unit: assumes qf: "qfree p"
-  shows "\<And> q B d. unit p = (q,B,d) \<Longrightarrow> ((\<exists> (x::int). Ifm (real x#bs) p) = (\<exists> (x::int). Ifm (real x#bs) q)) \<and> (Inum (real i#bs)) ` set B = (Inum (real i#bs)) ` set (\<beta> q) \<and> d\<beta> q 1 \<and> d\<delta> q d \<and> d >0 \<and> iszlfm q (real (i::int)#bs) \<and> (\<forall> b\<in> set B. numbound0 b)"
-proof-
-  fix q B d 
-  assume qBd: "unit p = (q,B,d)"
-  let ?thes = "((\<exists> (x::int). Ifm (real x#bs) p) = (\<exists> (x::int). Ifm (real x#bs) q)) \<and>
-    Inum (real i#bs) ` set B = Inum (real i#bs) ` set (\<beta> q) \<and>
-    d\<beta> q 1 \<and> d\<delta> q d \<and> 0 < d \<and> iszlfm q (real i # bs) \<and> (\<forall> b\<in> set B. numbound0 b)"
-  let ?I = "\<lambda> (x::int) p. Ifm (real x#bs) p"
-  let ?p' = "zlfm p"
-  let ?l = "\<zeta> ?p'"
-  let ?q = "And (Dvd ?l (CN 0 1 (C 0))) (a\<beta> ?p' ?l)"
-  let ?d = "\<delta> ?q"
-  let ?B = "set (\<beta> ?q)"
-  let ?B'= "remdups (map simpnum (\<beta> ?q))"
-  let ?A = "set (\<alpha> ?q)"
-  let ?A'= "remdups (map simpnum (\<alpha> ?q))"
-  from conjunct1[OF zlfm_I[OF qf, where bs="bs"]] 
-  have pp': "\<forall> i. ?I i ?p' = ?I i p" by auto
-  from iszlfm_gen[OF conjunct2[OF zlfm_I[OF qf, where bs="bs" and i="i"]]]
-  have lp': "\<forall> (i::int). iszlfm ?p' (real i#bs)" by simp 
-  hence lp'': "iszlfm ?p' (real (i::int)#bs)" by simp
-  from lp' \<zeta>[where p="?p'" and bs="bs"] have lp: "?l >0" and dl: "d\<beta> ?p' ?l" by auto
-  from a\<beta>_ex[where p="?p'" and l="?l" and bs="bs", OF lp'' dl lp] pp'
-  have pq_ex:"(\<exists> (x::int). ?I x p) = (\<exists> x. ?I x ?q)" by (simp add: int_rdvd_iff) 
-  from lp'' lp a\<beta>[OF lp'' dl lp] have lq:"iszlfm ?q (real i#bs)" and uq: "d\<beta> ?q 1" 
-    by (auto simp add: isint_def)
-  from \<delta>[OF lq] have dp:"?d >0" and dd: "d\<delta> ?q ?d" by blast+
-  let ?N = "\<lambda> t. Inum (real (i::int)#bs) t"
-  have "?N ` set ?B' = ((?N o simpnum) ` ?B)" by (simp add:image_compose) 
-  also have "\<dots> = ?N ` ?B" using simpnum_ci[where bs="real i #bs"] by auto
-  finally have BB': "?N ` set ?B' = ?N ` ?B" .
-  have "?N ` set ?A' = ((?N o simpnum) ` ?A)" by (simp add:image_compose) 
-  also have "\<dots> = ?N ` ?A" using simpnum_ci[where bs="real i #bs"] by auto
-  finally have AA': "?N ` set ?A' = ?N ` ?A" .
-  from \<beta>_numbound0[OF lq] have B_nb:"\<forall> b\<in> set ?B'. numbound0 b"
-    by (simp add: simpnum_numbound0)
-  from \<alpha>_l[OF lq] have A_nb: "\<forall> b\<in> set ?A'. numbound0 b"
-    by (simp add: simpnum_numbound0)
-    {assume "length ?B' \<le> length ?A'"
-    hence q:"q=?q" and "B = ?B'" and d:"d = ?d"
-      using qBd by (auto simp add: Let_def unit_def)
-    with BB' B_nb have b: "?N ` (set B) = ?N ` set (\<beta> q)" 
-      and bn: "\<forall>b\<in> set B. numbound0 b" by simp+ 
-  with pq_ex dp uq dd lq q d have ?thes by simp}
-  moreover 
-  {assume "\<not> (length ?B' \<le> length ?A')"
-    hence q:"q=mirror ?q" and "B = ?A'" and d:"d = ?d"
-      using qBd by (auto simp add: Let_def unit_def)
-    with AA' mirror\<alpha>\<beta>[OF lq] A_nb have b:"?N ` (set B) = ?N ` set (\<beta> q)" 
-      and bn: "\<forall>b\<in> set B. numbound0 b" by simp+
-    from mirror_ex[OF lq] pq_ex q 
-    have pqm_eq:"(\<exists> (x::int). ?I x p) = (\<exists> (x::int). ?I x q)" by simp
-    from lq uq q mirror_d\<beta> [where p="?q" and bs="bs" and a="real i"]
-    have lq': "iszlfm q (real i#bs)" and uq: "d\<beta> q 1" by auto
-    from \<delta>[OF lq'] mirror_\<delta>[OF lq] q d have dq:"d\<delta> q d " by auto
-    from pqm_eq b bn uq lq' dp dq q dp d have ?thes by simp
-  }
-  ultimately show ?thes by blast
-qed
-    (* Cooper's Algorithm *)
-
-constdefs cooper :: "fm \<Rightarrow> fm"
-  "cooper p \<equiv> 
-  (let (q,B,d) = unit p; js = iupt (1,d);
-       mq = simpfm (minusinf q);
-       md = evaldjf (\<lambda> j. simpfm (subst0 (C j) mq)) js
-   in if md = T then T else
-    (let qd = evaldjf (\<lambda> t. simpfm (subst0 t q)) 
-                               (remdups (map (\<lambda> (b,j). simpnum (Add b (C j))) 
-                                            [(b,j). b\<leftarrow>B,j\<leftarrow>js]))
-     in decr (disj md qd)))"
-lemma cooper: assumes qf: "qfree p"
-  shows "((\<exists> (x::int). Ifm (real x#bs) p) = (Ifm bs (cooper p))) \<and> qfree (cooper p)" 
-  (is "(?lhs = ?rhs) \<and> _")
-proof-
-
-  let ?I = "\<lambda> (x::int) p. Ifm (real x#bs) p"
-  let ?q = "fst (unit p)"
-  let ?B = "fst (snd(unit p))"
-  let ?d = "snd (snd (unit p))"
-  let ?js = "iupt (1,?d)"
-  let ?mq = "minusinf ?q"
-  let ?smq = "simpfm ?mq"
-  let ?md = "evaldjf (\<lambda> j. simpfm (subst0 (C j) ?smq)) ?js"
-  fix i
-  let ?N = "\<lambda> t. Inum (real (i::int)#bs) t"
-  let ?bjs = "[(b,j). b\<leftarrow>?B,j\<leftarrow>?js]"
-  let ?sbjs = "map (\<lambda> (b,j). simpnum (Add b (C j))) ?bjs"
-  let ?qd = "evaldjf (\<lambda> t. simpfm (subst0 t ?q)) (remdups ?sbjs)"
-  have qbf:"unit p = (?q,?B,?d)" by simp
-  from unit[OF qf qbf] have pq_ex: "(\<exists>(x::int). ?I x p) = (\<exists> (x::int). ?I x ?q)" and 
-    B:"?N ` set ?B = ?N ` set (\<beta> ?q)" and 
-    uq:"d\<beta> ?q 1" and dd: "d\<delta> ?q ?d" and dp: "?d > 0" and 
-    lq: "iszlfm ?q (real i#bs)" and 
-    Bn: "\<forall> b\<in> set ?B. numbound0 b" by auto
-  from zlin_qfree[OF lq] have qfq: "qfree ?q" .
-  from simpfm_qf[OF minusinf_qfree[OF qfq]] have qfmq: "qfree ?smq".
-  have jsnb: "\<forall> j \<in> set ?js. numbound0 (C j)" by simp
-  hence "\<forall> j\<in> set ?js. bound0 (subst0 (C j) ?smq)" 
-    by (auto simp only: subst0_bound0[OF qfmq])
-  hence th: "\<forall> j\<in> set ?js. bound0 (simpfm (subst0 (C j) ?smq))"
-    by (auto simp add: simpfm_bound0)
-  from evaldjf_bound0[OF th] have mdb: "bound0 ?md" by simp 
-  from Bn jsnb have "\<forall> (b,j) \<in> set ?bjs. numbound0 (Add b (C j))"
-    by simp
-  hence "\<forall> (b,j) \<in> set ?bjs. numbound0 (simpnum (Add b (C j)))"
-    using simpnum_numbound0 by blast
-  hence "\<forall> t \<in> set ?sbjs. numbound0 t" by simp
-  hence "\<forall> t \<in> set (remdups ?sbjs). bound0 (subst0 t ?q)"
-    using subst0_bound0[OF qfq] by auto 
-  hence th': "\<forall> t \<in> set (remdups ?sbjs). bound0 (simpfm (subst0 t ?q))"
-    using simpfm_bound0 by blast
-  from evaldjf_bound0 [OF th'] have qdb: "bound0 ?qd" by simp
-  from mdb qdb 
-  have mdqdb: "bound0 (disj ?md ?qd)" by (simp only: disj_def, cases "?md=T \<or> ?qd=T", simp_all)
-  from trans [OF pq_ex cp_thm'[OF lq uq dd dp]] B
-  have "?lhs = (\<exists> j\<in> {1.. ?d}. ?I j ?mq \<or> (\<exists> b\<in> ?N ` set ?B. Ifm ((b+ real j)#bs) ?q))" by auto
-  also have "\<dots> = ((\<exists> j\<in> set ?js. ?I j ?smq) \<or> (\<exists> (b,j) \<in> (?N ` set ?B \<times> set ?js). Ifm ((b+ real j)#bs) ?q))" apply (simp only: iupt_set simpfm) by auto
-  also have "\<dots>= ((\<exists> j\<in> set ?js. ?I j ?smq) \<or> (\<exists> t \<in> (\<lambda> (b,j). ?N (Add b (C j))) ` set ?bjs. Ifm (t #bs) ?q))" by simp
-  also have "\<dots>= ((\<exists> j\<in> set ?js. ?I j ?smq) \<or> (\<exists> t \<in> (\<lambda> (b,j). ?N (simpnum (Add b (C j)))) ` set ?bjs. Ifm (t #bs) ?q))" by (simp only: simpnum_ci)
-  also  have "\<dots>= ((\<exists> j\<in> set ?js. ?I j ?smq) \<or> (\<exists> t \<in> set ?sbjs. Ifm (?N t #bs) ?q))" 
-    by (auto simp add: split_def) 
-  also have "\<dots> = ((\<exists> j\<in> set ?js. (\<lambda> j. ?I i (simpfm (subst0 (C j) ?smq))) j) \<or> (\<exists> t \<in> set (remdups ?sbjs). (\<lambda> t. ?I i (simpfm (subst0 t ?q))) t))" by (simp only: simpfm subst0_I[OF qfq] simpfm Inum.simps subst0_I[OF qfmq] set_remdups)
-  also have "\<dots> = ((?I i (evaldjf (\<lambda> j. simpfm (subst0 (C j) ?smq)) ?js)) \<or> (?I i (evaldjf (\<lambda> t. simpfm (subst0 t ?q)) (remdups ?sbjs))))" by (simp only: evaldjf_ex)
-  finally have mdqd: "?lhs = (?I i (disj ?md ?qd))" by (simp add: disj) 
-  hence mdqd2: "?lhs = (Ifm bs (decr (disj ?md ?qd)))" using decr [OF mdqdb] by simp
-  {assume mdT: "?md = T"
-    hence cT:"cooper p = T" 
-      by (simp only: cooper_def unit_def split_def Let_def if_True) simp
-    from mdT mdqd have lhs:"?lhs" by (auto simp add: disj)
-    from mdT have "?rhs" by (simp add: cooper_def unit_def split_def)
-    with lhs cT have ?thesis by simp }
-  moreover
-  {assume mdT: "?md \<noteq> T" hence "cooper p = decr (disj ?md ?qd)" 
-      by (simp only: cooper_def unit_def split_def Let_def if_False) 
-    with mdqd2 decr_qf[OF mdqdb] have ?thesis by simp }
-  ultimately show ?thesis by blast
-qed
-
-lemma DJcooper: 
-  assumes qf: "qfree p"
-  shows "((\<exists> (x::int). Ifm (real x#bs) p) = (Ifm bs (DJ cooper p))) \<and> qfree (DJ cooper p)"
-proof-
-  from cooper have cqf: "\<forall> p. qfree p \<longrightarrow> qfree (cooper p)" by  blast
-  from DJ_qf[OF cqf] qf have thqf:"qfree (DJ cooper p)" by blast
-  have "Ifm bs (DJ cooper p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (cooper q))" 
-     by (simp add: DJ_def evaldjf_ex)
-  also have "\<dots> = (\<exists> q \<in> set(disjuncts p). \<exists> (x::int). Ifm (real x#bs)  q)" 
-    using cooper disjuncts_qf[OF qf] by blast
-  also have "\<dots> = (\<exists> (x::int). Ifm (real x#bs) p)" by (induct p rule: disjuncts.induct, auto)
-  finally show ?thesis using thqf by blast
-qed
-
-    (* Redy and Loveland *)
-
-lemma \<sigma>\<rho>_cong: assumes lp: "iszlfm p (a#bs)" and tt': "Inum (a#bs) t = Inum (a#bs) t'"
-  shows "Ifm (a#bs) (\<sigma>\<rho> p (t,c)) = Ifm (a#bs) (\<sigma>\<rho> p (t',c))"
-  using lp 
-  by (induct p rule: iszlfm.induct, auto simp add: tt')
-
-lemma \<sigma>_cong: assumes lp: "iszlfm p (a#bs)" and tt': "Inum (a#bs) t = Inum (a#bs) t'"
-  shows "Ifm (a#bs) (\<sigma> p c t) = Ifm (a#bs) (\<sigma> p c t')"
-  by (simp add: \<sigma>_def tt' \<sigma>\<rho>_cong[OF lp tt'])
-
-lemma \<rho>_cong: assumes lp: "iszlfm p (a#bs)" 
-  and RR: "(\<lambda>(b,k). (Inum (a#bs) b,k)) ` R =  (\<lambda>(b,k). (Inum (a#bs) b,k)) ` set (\<rho> p)"
-  shows "(\<exists> (e,c) \<in> R. \<exists> j\<in> {1.. c*(\<delta> p)}. Ifm (a#bs) (\<sigma> p c (Add e (C j)))) = (\<exists> (e,c) \<in> set (\<rho> p). \<exists> j\<in> {1.. c*(\<delta> p)}. Ifm (a#bs) (\<sigma> p c (Add e (C j))))"
-  (is "?lhs = ?rhs")
-proof
-  let ?d = "\<delta> p"
-  assume ?lhs then obtain e c j where ecR: "(e,c) \<in> R" and jD:"j \<in> {1 .. c*?d}" 
-    and px: "Ifm (a#bs) (\<sigma> p c (Add e (C j)))" (is "?sp c e j") by blast
-  from ecR have "(Inum (a#bs) e,c) \<in> (\<lambda>(b,k). (Inum (a#bs) b,k)) ` R" by auto
-  hence "(Inum (a#bs) e,c) \<in> (\<lambda>(b,k). (Inum (a#bs) b,k)) ` set (\<rho> p)" using RR by simp
-  hence "\<exists> (e',c') \<in> set (\<rho> p). Inum (a#bs) e = Inum (a#bs) e' \<and> c = c'" by auto
-  then obtain e' c' where ecRo:"(e',c') \<in> set (\<rho> p)" and ee':"Inum (a#bs) e = Inum (a#bs) e'"
-    and cc':"c = c'" by blast
-  from ee' have tt': "Inum (a#bs) (Add e (C j)) = Inum (a#bs) (Add e' (C j))" by simp
-  
-  from \<sigma>_cong[OF lp tt', where c="c"] px have px':"?sp c e' j" by simp
-  from ecRo jD px' cc'  show ?rhs apply auto 
-    by (rule_tac x="(e', c')" in bexI,simp_all)
-  (rule_tac x="j" in bexI, simp_all add: cc'[symmetric])
-next
-  let ?d = "\<delta> p"
-  assume ?rhs then obtain e c j where ecR: "(e,c) \<in> set (\<rho> p)" and jD:"j \<in> {1 .. c*?d}" 
-    and px: "Ifm (a#bs) (\<sigma> p c (Add e (C j)))" (is "?sp c e j") by blast
-  from ecR have "(Inum (a#bs) e,c) \<in> (\<lambda>(b,k). (Inum (a#bs) b,k)) ` set (\<rho> p)" by auto
-  hence "(Inum (a#bs) e,c) \<in> (\<lambda>(b,k). (Inum (a#bs) b,k)) ` R" using RR by simp
-  hence "\<exists> (e',c') \<in> R. Inum (a#bs) e = Inum (a#bs) e' \<and> c = c'" by auto
-  then obtain e' c' where ecRo:"(e',c') \<in> R" and ee':"Inum (a#bs) e = Inum (a#bs) e'"
-    and cc':"c = c'" by blast
-  from ee' have tt': "Inum (a#bs) (Add e (C j)) = Inum (a#bs) (Add e' (C j))" by simp
-  from \<sigma>_cong[OF lp tt', where c="c"] px have px':"?sp c e' j" by simp
-  from ecRo jD px' cc'  show ?lhs apply auto 
-    by (rule_tac x="(e', c')" in bexI,simp_all)
-  (rule_tac x="j" in bexI, simp_all add: cc'[symmetric])
-qed
-
-lemma rl_thm': 
-  assumes lp: "iszlfm p (real (i::int)#bs)" 
-  and R: "(\<lambda>(b,k). (Inum (a#bs) b,k)) ` R =  (\<lambda>(b,k). (Inum (a#bs) b,k)) ` set (\<rho> p)"
-  shows "(\<exists> (x::int). Ifm (real x#bs) p) = ((\<exists> j\<in> {1 .. \<delta> p}. Ifm (real j#bs) (minusinf p)) \<or> (\<exists> (e,c) \<in> R. \<exists> j\<in> {1.. c*(\<delta> p)}. Ifm (a#bs) (\<sigma> p c (Add e (C j)))))"
-  using rl_thm[OF lp] \<rho>_cong[OF iszlfm_gen[OF lp, rule_format, where y="a"] R] by simp 
-
-constdefs chooset:: "fm \<Rightarrow> fm \<times> ((num\<times>int) list) \<times> int"
-  "chooset p \<equiv> (let q = zlfm p ; d = \<delta> q;
-             B = remdups (map (\<lambda> (t,k). (simpnum t,k)) (\<rho> q)) ; 
-             a = remdups (map (\<lambda> (t,k). (simpnum t,k)) (\<alpha>\<rho> q))
-             in if length B \<le> length a then (q,B,d) else (mirror q, a,d))"
-
-lemma chooset: assumes qf: "qfree p"
-  shows "\<And> q B d. chooset p = (q,B,d) \<Longrightarrow> ((\<exists> (x::int). Ifm (real x#bs) p) = (\<exists> (x::int). Ifm (real x#bs) q)) \<and> ((\<lambda>(t,k). (Inum (real i#bs) t,k)) ` set B = (\<lambda>(t,k). (Inum (real i#bs) t,k)) ` set (\<rho> q)) \<and> (\<delta> q = d) \<and> d >0 \<and> iszlfm q (real (i::int)#bs) \<and> (\<forall> (e,c)\<in> set B. numbound0 e \<and> c>0)"
-proof-
-  fix q B d 
-  assume qBd: "chooset p = (q,B,d)"
-  let ?thes = "((\<exists> (x::int). Ifm (real x#bs) p) = (\<exists> (x::int). Ifm (real x#bs) q)) \<and> ((\<lambda>(t,k). (Inum (real i#bs) t,k)) ` set B = (\<lambda>(t,k). (Inum (real i#bs) t,k)) ` set (\<rho> q)) \<and> (\<delta> q = d) \<and> d >0 \<and> iszlfm q (real (i::int)#bs) \<and> (\<forall> (e,c)\<in> set B. numbound0 e \<and> c>0)" 
-  let ?I = "\<lambda> (x::int) p. Ifm (real x#bs) p"
-  let ?q = "zlfm p"
-  let ?d = "\<delta> ?q"
-  let ?B = "set (\<rho> ?q)"
-  let ?f = "\<lambda> (t,k). (simpnum t,k)"
-  let ?B'= "remdups (map ?f (\<rho> ?q))"
-  let ?A = "set (\<alpha>\<rho> ?q)"
-  let ?A'= "remdups (map ?f (\<alpha>\<rho> ?q))"
-  from conjunct1[OF zlfm_I[OF qf, where bs="bs"]] 
-  have pp': "\<forall> i. ?I i ?q = ?I i p" by auto
-  hence pq_ex:"(\<exists> (x::int). ?I x p) = (\<exists> x. ?I x ?q)" by simp 
-  from iszlfm_gen[OF conjunct2[OF zlfm_I[OF qf, where bs="bs" and i="i"]], rule_format, where y="real i"]
-  have lq: "iszlfm ?q (real (i::int)#bs)" . 
-  from \<delta>[OF lq] have dp:"?d >0" by blast
-  let ?N = "\<lambda> (t,c). (Inum (real (i::int)#bs) t,c)"
-  have "?N ` set ?B' = ((?N o ?f) ` ?B)" by (simp add: split_def image_compose)
-  also have "\<dots> = ?N ` ?B"
-    by(simp add: split_def image_compose simpnum_ci[where bs="real i #bs"] image_def)
-  finally have BB': "?N ` set ?B' = ?N ` ?B" .
-  have "?N ` set ?A' = ((?N o ?f) ` ?A)" by (simp add: split_def image_compose) 
-  also have "\<dots> = ?N ` ?A" using simpnum_ci[where bs="real i #bs"]
-    by(simp add: split_def image_compose simpnum_ci[where bs="real i #bs"] image_def) 
-  finally have AA': "?N ` set ?A' = ?N ` ?A" .
-  from \<rho>_l[OF lq] have B_nb:"\<forall> (e,c)\<in> set ?B'. numbound0 e \<and> c > 0"
-    by (simp add: simpnum_numbound0 split_def)
-  from \<alpha>\<rho>_l[OF lq] have A_nb: "\<forall> (e,c)\<in> set ?A'. numbound0 e \<and> c > 0"
-    by (simp add: simpnum_numbound0 split_def)
-    {assume "length ?B' \<le> length ?A'"
-    hence q:"q=?q" and "B = ?B'" and d:"d = ?d"
-      using qBd by (auto simp add: Let_def chooset_def)
-    with BB' B_nb have b: "?N ` (set B) = ?N ` set (\<rho> q)" 
-      and bn: "\<forall>(e,c)\<in> set B. numbound0 e \<and> c > 0" by auto
-  with pq_ex dp lq q d have ?thes by simp}
-  moreover 
-  {assume "\<not> (length ?B' \<le> length ?A')"
-    hence q:"q=mirror ?q" and "B = ?A'" and d:"d = ?d"
-      using qBd by (auto simp add: Let_def chooset_def)
-    with AA' mirror_\<alpha>\<rho>[OF lq] A_nb have b:"?N ` (set B) = ?N ` set (\<rho> q)" 
-      and bn: "\<forall>(e,c)\<in> set B. numbound0 e \<and> c > 0" by auto 
-    from mirror_ex[OF lq] pq_ex q 
-    have pqm_eq:"(\<exists> (x::int). ?I x p) = (\<exists> (x::int). ?I x q)" by simp
-    from lq q mirror_l [where p="?q" and bs="bs" and a="real i"]
-    have lq': "iszlfm q (real i#bs)" by auto
-    from mirror_\<delta>[OF lq] pqm_eq b bn lq' dp q dp d have ?thes by simp
-  }
-  ultimately show ?thes by blast
-qed
-
-constdefs stage:: "fm \<Rightarrow> int \<Rightarrow> (num \<times> int) \<Rightarrow> fm"
-  "stage p d \<equiv> (\<lambda> (e,c). evaldjf (\<lambda> j. simpfm (\<sigma> p c (Add e (C j)))) (iupt (1,c*d)))"
-lemma stage:
-  shows "Ifm bs (stage p d (e,c)) = (\<exists> j\<in>{1 .. c*d}. Ifm bs (\<sigma> p c (Add e (C j))))"
-  by (unfold stage_def split_def ,simp only: evaldjf_ex iupt_set simpfm) simp
-
-lemma stage_nb: assumes lp: "iszlfm p (a#bs)" and cp: "c >0" and nb:"numbound0 e"
-  shows "bound0 (stage p d (e,c))"
-proof-
-  let ?f = "\<lambda> j. simpfm (\<sigma> p c (Add e (C j)))"
-  have th: "\<forall> j\<in> set (iupt(1,c*d)). bound0 (?f j)"
-  proof
-    fix j
-    from nb have nb':"numbound0 (Add e (C j))" by simp
-    from simpfm_bound0[OF \<sigma>_nb[OF lp nb', where k="c"]]
-    show "bound0 (simpfm (\<sigma> p c (Add e (C j))))" .
-  qed
-  from evaldjf_bound0[OF th] show ?thesis by (unfold stage_def split_def) simp
-qed
-
-constdefs redlove:: "fm \<Rightarrow> fm"
-  "redlove p \<equiv> 
-  (let (q,B,d) = chooset p;
-       mq = simpfm (minusinf q);
-       md = evaldjf (\<lambda> j. simpfm (subst0 (C j) mq)) (iupt (1,d))
-   in if md = T then T else
-    (let qd = evaldjf (stage q d) B
-     in decr (disj md qd)))"
-
-lemma redlove: assumes qf: "qfree p"
-  shows "((\<exists> (x::int). Ifm (real x#bs) p) = (Ifm bs (redlove p))) \<and> qfree (redlove p)" 
-  (is "(?lhs = ?rhs) \<and> _")
-proof-
-
-  let ?I = "\<lambda> (x::int) p. Ifm (real x#bs) p"
-  let ?q = "fst (chooset p)"
-  let ?B = "fst (snd(chooset p))"
-  let ?d = "snd (snd (chooset p))"
-  let ?js = "iupt (1,?d)"
-  let ?mq = "minusinf ?q"
-  let ?smq = "simpfm ?mq"
-  let ?md = "evaldjf (\<lambda> j. simpfm (subst0 (C j) ?smq)) ?js"
-  fix i
-  let ?N = "\<lambda> (t,k). (Inum (real (i::int)#bs) t,k)"
-  let ?qd = "evaldjf (stage ?q ?d) ?B"
-  have qbf:"chooset p = (?q,?B,?d)" by simp
-  from chooset[OF qf qbf] have pq_ex: "(\<exists>(x::int). ?I x p) = (\<exists> (x::int). ?I x ?q)" and 
-    B:"?N ` set ?B = ?N ` set (\<rho> ?q)" and dd: "\<delta> ?q = ?d" and dp: "?d > 0" and 
-    lq: "iszlfm ?q (real i#bs)" and 
-    Bn: "\<forall> (e,c)\<in> set ?B. numbound0 e \<and> c > 0" by auto
-  from zlin_qfree[OF lq] have qfq: "qfree ?q" .
-  from simpfm_qf[OF minusinf_qfree[OF qfq]] have qfmq: "qfree ?smq".
-  have jsnb: "\<forall> j \<in> set ?js. numbound0 (C j)" by simp
-  hence "\<forall> j\<in> set ?js. bound0 (subst0 (C j) ?smq)" 
-    by (auto simp only: subst0_bound0[OF qfmq])
-  hence th: "\<forall> j\<in> set ?js. bound0 (simpfm (subst0 (C j) ?smq))"
-    by (auto simp add: simpfm_bound0)
-  from evaldjf_bound0[OF th] have mdb: "bound0 ?md" by simp 
-  from Bn stage_nb[OF lq] have th:"\<forall> x \<in> set ?B. bound0 (stage ?q ?d x)" by auto
-  from evaldjf_bound0[OF th]  have qdb: "bound0 ?qd" .
-  from mdb qdb 
-  have mdqdb: "bound0 (disj ?md ?qd)" by (simp only: disj_def, cases "?md=T \<or> ?qd=T", simp_all)
-  from trans [OF pq_ex rl_thm'[OF lq B]] dd
-  have "?lhs = ((\<exists> j\<in> {1.. ?d}. ?I j ?mq) \<or> (\<exists> (e,c)\<in> set ?B. \<exists> j\<in> {1 .. c*?d}. Ifm (real i#bs) (\<sigma> ?q c (Add e (C j)))))" by auto
-  also have "\<dots> = ((\<exists> j\<in> {1.. ?d}. ?I j ?smq) \<or> (\<exists> (e,c)\<in> set ?B. ?I i (stage ?q ?d (e,c) )))" 
-    by (simp add: simpfm stage split_def)
-  also have "\<dots> = ((\<exists> j\<in> {1 .. ?d}. ?I i (subst0 (C j) ?smq))  \<or> ?I i ?qd)"
-    by (simp add: evaldjf_ex subst0_I[OF qfmq])
-  finally have mdqd:"?lhs = (?I i ?md \<or> ?I i ?qd)" by (simp only: evaldjf_ex iupt_set simpfm) 
-  also have "\<dots> = (?I i (disj ?md ?qd))" by (simp add: disj)
-  also have "\<dots> = (Ifm bs (decr (disj ?md ?qd)))" by (simp only: decr [OF mdqdb]) 
-  finally have mdqd2: "?lhs = (Ifm bs (decr (disj ?md ?qd)))" . 
-  {assume mdT: "?md = T"
-    hence cT:"redlove p = T" by (simp add: redlove_def Let_def chooset_def split_def)
-    from mdT have lhs:"?lhs" using mdqd by simp 
-    from mdT have "?rhs" by (simp add: redlove_def chooset_def split_def)
-    with lhs cT have ?thesis by simp }
-  moreover
-  {assume mdT: "?md \<noteq> T" hence "redlove p = decr (disj ?md ?qd)" 
-      by (simp add: redlove_def chooset_def split_def Let_def)
-    with mdqd2 decr_qf[OF mdqdb] have ?thesis by simp }
-  ultimately show ?thesis by blast
-qed
-
-lemma DJredlove: 
-  assumes qf: "qfree p"
-  shows "((\<exists> (x::int). Ifm (real x#bs) p) = (Ifm bs (DJ redlove p))) \<and> qfree (DJ redlove p)"
-proof-
-  from redlove have cqf: "\<forall> p. qfree p \<longrightarrow> qfree (redlove p)" by  blast
-  from DJ_qf[OF cqf] qf have thqf:"qfree (DJ redlove p)" by blast
-  have "Ifm bs (DJ redlove p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (redlove q))" 
-     by (simp add: DJ_def evaldjf_ex)
-  also have "\<dots> = (\<exists> q \<in> set(disjuncts p). \<exists> (x::int). Ifm (real x#bs)  q)" 
-    using redlove disjuncts_qf[OF qf] by blast
-  also have "\<dots> = (\<exists> (x::int). Ifm (real x#bs) p)" by (induct p rule: disjuncts.induct, auto)
-  finally show ?thesis using thqf by blast
-qed
-
-
-lemma exsplit_qf: assumes qf: "qfree p"
-  shows "qfree (exsplit p)"
-using qf by (induct p rule: exsplit.induct, auto)
-
-definition mircfr :: "fm \<Rightarrow> fm" where
-  "mircfr = DJ cooper o ferrack01 o simpfm o exsplit"
-
-definition mirlfr :: "fm \<Rightarrow> fm" where
-  "mirlfr = DJ redlove o ferrack01 o simpfm o exsplit"
-
-lemma mircfr: "\<forall> bs p. qfree p \<longrightarrow> qfree (mircfr p) \<and> Ifm bs (mircfr p) = Ifm bs (E p)"
-proof(clarsimp simp del: Ifm.simps)
-  fix bs p
-  assume qf: "qfree p"
-  show "qfree (mircfr p)\<and>(Ifm bs (mircfr p) = Ifm bs (E p))" (is "_ \<and> (?lhs = ?rhs)")
-  proof-
-    let ?es = "(And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) (simpfm (exsplit p)))"
-    have "?rhs = (\<exists> (i::int). \<exists> x. Ifm (x#real i#bs) ?es)" 
-      using splitex[OF qf] by simp
-    with ferrack01[OF simpfm_qf[OF exsplit_qf[OF qf]]] have th1: "?rhs = (\<exists> (i::int). Ifm (real i#bs) (ferrack01 (simpfm (exsplit p))))" and qf':"qfree (ferrack01 (simpfm (exsplit p)))" by simp+
-    with DJcooper[OF qf'] show ?thesis by (simp add: mircfr_def)
-  qed
-qed
-  
-lemma mirlfr: "\<forall> bs p. qfree p \<longrightarrow> qfree(mirlfr p) \<and> Ifm bs (mirlfr p) = Ifm bs (E p)"
-proof(clarsimp simp del: Ifm.simps)
-  fix bs p
-  assume qf: "qfree p"
-  show "qfree (mirlfr p)\<and>(Ifm bs (mirlfr p) = Ifm bs (E p))" (is "_ \<and> (?lhs = ?rhs)")
-  proof-
-    let ?es = "(And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) (simpfm (exsplit p)))"
-    have "?rhs = (\<exists> (i::int). \<exists> x. Ifm (x#real i#bs) ?es)" 
-      using splitex[OF qf] by simp
-    with ferrack01[OF simpfm_qf[OF exsplit_qf[OF qf]]] have th1: "?rhs = (\<exists> (i::int). Ifm (real i#bs) (ferrack01 (simpfm (exsplit p))))" and qf':"qfree (ferrack01 (simpfm (exsplit p)))" by simp+
-    with DJredlove[OF qf'] show ?thesis by (simp add: mirlfr_def)
-  qed
-qed
-  
-definition mircfrqe:: "fm \<Rightarrow> fm" where
-  "mircfrqe p = qelim (prep p) mircfr"
-
-definition mirlfrqe:: "fm \<Rightarrow> fm" where
-  "mirlfrqe p = qelim (prep p) mirlfr"
-
-theorem mircfrqe: "(Ifm bs (mircfrqe p) = Ifm bs p) \<and> qfree (mircfrqe p)"
-  using qelim_ci[OF mircfr] prep by (auto simp add: mircfrqe_def)
-
-theorem mirlfrqe: "(Ifm bs (mirlfrqe p) = Ifm bs p) \<and> qfree (mirlfrqe p)"
-  using qelim_ci[OF mirlfr] prep by (auto simp add: mirlfrqe_def)
-
-definition
-  "test1 (u\<Colon>unit) = mircfrqe (A (And (Le (Sub (Floor (Bound 0)) (Bound 0))) (Le (Add (Bound 0) (Floor (Neg (Bound 0)))))))"
-
-definition
-  "test2 (u\<Colon>unit) = mircfrqe (A (Iff (Eq (Add (Floor (Bound 0)) (Floor (Neg (Bound 0))))) (Eq (Sub (Floor (Bound 0)) (Bound 0)))))"
-
-definition
-  "test3 (u\<Colon>unit) = mirlfrqe (A (And (Le (Sub (Floor (Bound 0)) (Bound 0))) (Le (Add (Bound 0) (Floor (Neg (Bound 0)))))))"
-
-definition
-  "test4 (u\<Colon>unit) = mirlfrqe (A (Iff (Eq (Add (Floor (Bound 0)) (Floor (Neg (Bound 0))))) (Eq (Sub (Floor (Bound 0)) (Bound 0)))))"
-
-definition
-  "test5 (u\<Colon>unit) = mircfrqe (A(E(And (Ge(Sub (Bound 1) (Bound 0))) (Eq (Add (Floor (Bound 1)) (Floor (Neg(Bound 0))))))))"
-
-ML {* @{code test1} () *}
-ML {* @{code test2} () *}
-ML {* @{code test3} () *}
-ML {* @{code test4} () *}
-ML {* @{code test5} () *}
-
-(*export_code mircfrqe mirlfrqe
-  in SML module_name Mir file "raw_mir.ML"*)
-
-oracle mirfr_oracle = {* fn (proofs, ct) =>
-let
-
-fun num_of_term vs (t as Free (xn, xT)) = (case AList.lookup (op =) vs t
-     of NONE => error "Variable not found in the list!"
-      | SOME n => @{code Bound} n)
-  | num_of_term vs @{term "real (0::int)"} = @{code C} 0
-  | num_of_term vs @{term "real (1::int)"} = @{code C} 1
-  | num_of_term vs @{term "0::real"} = @{code C} 0
-  | num_of_term vs @{term "1::real"} = @{code C} 1
-  | num_of_term vs (Bound i) = @{code Bound} i
-  | num_of_term vs (@{term "uminus :: real \<Rightarrow> real"} $ t') = @{code Neg} (num_of_term vs t')
-  | num_of_term vs (@{term "op + :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) =
-      @{code Add} (num_of_term vs t1, num_of_term vs t2)
-  | num_of_term vs (@{term "op - :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) =
-      @{code Sub} (num_of_term vs t1, num_of_term vs t2)
-  | num_of_term vs (@{term "op * :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) =
-      (case (num_of_term vs t1)
-       of @{code C} i => @{code Mul} (i, num_of_term vs t2)
-        | _ => error "num_of_term: unsupported Multiplication")
-  | num_of_term vs (@{term "real :: int \<Rightarrow> real"} $ (@{term "number_of :: int \<Rightarrow> int"} $ t')) =
-      @{code C} (HOLogic.dest_numeral t')
-  | num_of_term vs (@{term "real :: int \<Rightarrow> real"} $ (@{term "floor :: real \<Rightarrow> int"} $ t')) =
-      @{code Floor} (num_of_term vs t')
-  | num_of_term vs (@{term "real :: int \<Rightarrow> real"} $ (@{term "ceiling :: real \<Rightarrow> int"} $ t')) =
-      @{code Neg} (@{code Floor} (@{code Neg} (num_of_term vs t')))
-  | num_of_term vs (@{term "number_of :: int \<Rightarrow> real"} $ t') =
-      @{code C} (HOLogic.dest_numeral t')
-  | num_of_term vs t = error ("num_of_term: unknown term " ^ Syntax.string_of_term @{context} t);
-
-fun fm_of_term vs @{term True} = @{code T}
-  | fm_of_term vs @{term False} = @{code F}
-  | fm_of_term vs (@{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) =
-      @{code Lt} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
-  | fm_of_term vs (@{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) =
-      @{code Le} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
-  | fm_of_term vs (@{term "op = :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) =
-      @{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) 
-  | fm_of_term vs (@{term "op rdvd"} $ (@{term "real :: int \<Rightarrow> real"} $ (@{term "number_of :: int \<Rightarrow> int"} $ t1)) $ t2) =
-      @{code Dvd} (HOLogic.dest_numeral t1, num_of_term vs t2)
-  | fm_of_term vs (@{term "op = :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ t1 $ t2) =
-      @{code Iff} (fm_of_term vs t1, fm_of_term vs t2)
-  | fm_of_term vs (@{term "op &"} $ t1 $ t2) =
-      @{code And} (fm_of_term vs t1, fm_of_term vs t2)
-  | fm_of_term vs (@{term "op |"} $ t1 $ t2) =
-      @{code Or} (fm_of_term vs t1, fm_of_term vs t2)
-  | fm_of_term vs (@{term "op -->"} $ t1 $ t2) =
-      @{code Imp} (fm_of_term vs t1, fm_of_term vs t2)
-  | fm_of_term vs (@{term "Not"} $ t') =
-      @{code NOT} (fm_of_term vs t')
-  | fm_of_term vs (Const ("Ex", _) $ Abs (xn, xT, p)) =
-      @{code E} (fm_of_term (map (fn (v, n) => (v, n + 1)) vs) p)
-  | fm_of_term vs (Const ("All", _) $ Abs (xn, xT, p)) =
-      @{code A} (fm_of_term (map (fn (v, n) => (v, n + 1)) vs) p)
-  | fm_of_term vs t = error ("fm_of_term : unknown term " ^ Syntax.string_of_term @{context} t);
-
-fun term_of_num vs (@{code C} i) = @{term "real :: int \<Rightarrow> real"} $ HOLogic.mk_number HOLogic.intT i
-  | term_of_num vs (@{code Bound} n) = fst (the (find_first (fn (_, m) => n = m) vs))
-  | term_of_num vs (@{code Neg} (@{code Floor} (@{code Neg} t'))) =
-      @{term "real :: int \<Rightarrow> real"} $ (@{term "ceiling :: real \<Rightarrow> int"} $ term_of_num vs t')
-  | term_of_num vs (@{code Neg} t') = @{term "uminus :: real \<Rightarrow> real"} $ term_of_num vs t'
-  | term_of_num vs (@{code Add} (t1, t2)) = @{term "op + :: real \<Rightarrow> real \<Rightarrow> real"} $
-      term_of_num vs t1 $ term_of_num vs t2
-  | term_of_num vs (@{code Sub} (t1, t2)) = @{term "op - :: real \<Rightarrow> real \<Rightarrow> real"} $
-      term_of_num vs t1 $ term_of_num vs t2
-  | term_of_num vs (@{code Mul} (i, t2)) = @{term "op * :: real \<Rightarrow> real \<Rightarrow> real"} $
-      term_of_num vs (@{code C} i) $ term_of_num vs t2
-  | term_of_num vs (@{code Floor} t) = @{term "real :: int \<Rightarrow> real"} $ (@{term "floor :: real \<Rightarrow> int"} $ term_of_num vs t)
-  | term_of_num vs (@{code CN} (n, i, t)) = term_of_num vs (@{code Add} (@{code Mul} (i, @{code Bound} n), t))
-  | term_of_num vs (@{code CF} (c, t, s)) = term_of_num vs (@{code Add} (@{code Mul} (c, @{code Floor} t), s));
-
-fun term_of_fm vs @{code T} = HOLogic.true_const 
-  | term_of_fm vs @{code F} = HOLogic.false_const
-  | term_of_fm vs (@{code Lt} t) =
-      @{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::real"}
-  | term_of_fm vs (@{code Le} t) =
-      @{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::real"}
-  | term_of_fm vs (@{code Gt} t) =
-      @{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $ @{term "0::real"} $ term_of_num vs t
-  | term_of_fm vs (@{code Ge} t) =
-      @{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $ @{term "0::real"} $ term_of_num vs t
-  | term_of_fm vs (@{code Eq} t) =
-      @{term "op = :: real \<Rightarrow> real \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::real"}
-  | term_of_fm vs (@{code NEq} t) =
-      term_of_fm vs (@{code NOT} (@{code Eq} t))
-  | term_of_fm vs (@{code Dvd} (i, t)) =
-      @{term "op rdvd"} $ term_of_num vs (@{code C} i) $ term_of_num vs t
-  | term_of_fm vs (@{code NDvd} (i, t)) =
-      term_of_fm vs (@{code NOT} (@{code Dvd} (i, t)))
-  | term_of_fm vs (@{code NOT} t') =
-      HOLogic.Not $ term_of_fm vs t'
-  | term_of_fm vs (@{code And} (t1, t2)) =
-      HOLogic.conj $ term_of_fm vs t1 $ term_of_fm vs t2
-  | term_of_fm vs (@{code Or} (t1, t2)) =
-      HOLogic.disj $ term_of_fm vs t1 $ term_of_fm vs t2
-  | term_of_fm vs (@{code Imp}  (t1, t2)) =
-      HOLogic.imp $ term_of_fm vs t1 $ term_of_fm vs t2
-  | term_of_fm vs (@{code Iff} (t1, t2)) =
-      @{term "op = :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ term_of_fm vs t1 $ term_of_fm vs t2;
-
-in
-  let 
-    val thy = Thm.theory_of_cterm ct;
-    val t = Thm.term_of ct;
-    val fs = OldTerm.term_frees t;
-    val vs = fs ~~ (0 upto (length fs - 1));
-    val qe = if proofs then @{code mirlfrqe} else @{code mircfrqe};
-    val t' = (term_of_fm vs o qe o fm_of_term vs) t;
-  in (cterm_of thy o HOLogic.mk_Trueprop o HOLogic.mk_eq) (t, t') end
-end;
-*}
-
-use "mir_tac.ML"
-setup "Mir_Tac.setup"
-
-lemma "ALL (x::real). (\<lfloor>x\<rfloor> = \<lceil>x\<rceil> = (x = real \<lfloor>x\<rfloor>))"
-apply mir
-done
-
-lemma "ALL (x::real). real (2::int)*x - (real (1::int)) < real \<lfloor>x\<rfloor> + real \<lceil>x\<rceil> \<and> real \<lfloor>x\<rfloor> + real \<lceil>x\<rceil>  \<le> real (2::int)*x + (real (1::int))"
-apply mir
-done
-
-lemma "ALL (x::real). 2*\<lfloor>x\<rfloor> \<le> \<lfloor>2*x\<rfloor> \<and> \<lfloor>2*x\<rfloor> \<le> 2*\<lfloor>x+1\<rfloor>"
-apply mir 
-done
-
-lemma "ALL (x::real). \<exists>y \<le> x. (\<lfloor>x\<rfloor> = \<lceil>y\<rceil>)"
-apply mir
-done
-
-lemma "ALL x y. \<lfloor>x\<rfloor> = \<lfloor>y\<rfloor> \<longrightarrow> 0 \<le> abs (y - x) \<and> abs (y - x) \<le> 1"
-apply mir
-done
-
-end
diff -r b8a6b9c56fdd -r 5170d6277b61 src/HOL/Reflection/ROOT.ML
--- a/src/HOL/Reflection/ROOT.ML	Thu Feb 12 21:24:14 2009 -0800
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,2 +0,0 @@
-
-use_thys ["Cooper", "Ferrack", "MIR", "Approximation"];
diff -r b8a6b9c56fdd -r 5170d6277b61 src/HOL/Reflection/cooper_tac.ML
--- a/src/HOL/Reflection/cooper_tac.ML	Thu Feb 12 21:24:14 2009 -0800
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,139 +0,0 @@
-(*  Title:      HOL/Reflection/cooper_tac.ML
-    Author:     Amine Chaieb, TU Muenchen
-*)
-
-structure Cooper_Tac =
-struct
-
-val trace = ref false;
-fun trace_msg s = if !trace then tracing s else ();
-
-val cooper_ss = @{simpset};
-
-val nT = HOLogic.natT;
-val binarith = @{thms normalize_bin_simps};
-val comp_arith = binarith @ simp_thms
-
-val zdvd_int = @{thm zdvd_int};
-val zdiff_int_split = @{thm zdiff_int_split};
-val all_nat = @{thm all_nat};
-val ex_nat = @{thm ex_nat};
-val number_of1 = @{thm number_of1};
-val number_of2 = @{thm number_of2};
-val split_zdiv = @{thm split_zdiv};
-val split_zmod = @{thm split_zmod};
-val mod_div_equality' = @{thm mod_div_equality'};
-val split_div' = @{thm split_div'};
-val Suc_plus1 = @{thm Suc_plus1};
-val imp_le_cong = @{thm imp_le_cong};
-val conj_le_cong = @{thm conj_le_cong};
-val nat_mod_add_eq = @{thm mod_add1_eq} RS sym;
-val nat_mod_add_left_eq = @{thm mod_add_left_eq} RS sym;
-val nat_mod_add_right_eq = @{thm mod_add_right_eq} RS sym;
-val int_mod_add_eq = @{thm zmod_zadd1_eq} RS sym;
-val int_mod_add_left_eq = @{thm zmod_zadd_left_eq} RS sym;
-val int_mod_add_right_eq = @{thm zmod_zadd_right_eq} RS sym;
-val nat_div_add_eq = @{thm div_add1_eq} RS sym;
-val int_div_add_eq = @{thm zdiv_zadd1_eq} RS sym;
-
-fun prepare_for_linz q fm = 
-  let
-    val ps = Logic.strip_params fm
-    val hs = map HOLogic.dest_Trueprop (Logic.strip_assums_hyp fm)
-    val c = HOLogic.dest_Trueprop (Logic.strip_assums_concl fm)
-    fun mk_all ((s, T), (P,n)) =
-      if 0 mem loose_bnos P then
-        (HOLogic.all_const T $ Abs (s, T, P), n)
-      else (incr_boundvars ~1 P, n-1)
-    fun mk_all2 (v, t) = HOLogic.all_const (fastype_of v) $ lambda v t;
-    val rhs = hs
-    val np = length ps
-    val (fm',np) =  foldr (fn ((x, T), (fm,n)) => mk_all ((x, T), (fm,n)))
-      (foldr HOLogic.mk_imp c rhs, np) ps
-    val (vs, _) = List.partition (fn t => q orelse (type_of t) = nT)
-      (OldTerm.term_frees fm' @ OldTerm.term_vars fm');
-    val fm2 = foldr mk_all2 fm' vs
-  in (fm2, np + length vs, length rhs) end;
-
-(*Object quantifier to meta --*)
-fun spec_step n th = if (n=0) then th else (spec_step (n-1) th) RS spec ;
-
-(* object implication to meta---*)
-fun mp_step n th = if (n=0) then th else (mp_step (n-1) th) RS mp;
-
-
-fun linz_tac ctxt q i = ObjectLogic.atomize_prems_tac i THEN (fn st =>
-  let
-    val g = List.nth (prems_of st, i - 1)
-    val thy = ProofContext.theory_of ctxt
-    (* Transform the term*)
-    val (t,np,nh) = prepare_for_linz q g
-    (* Some simpsets for dealing with mod div abs and nat*)
-    val mod_div_simpset = HOL_basic_ss 
-			addsimps [refl,nat_mod_add_eq, nat_mod_add_left_eq, 
-				  nat_mod_add_right_eq, int_mod_add_eq, 
-				  int_mod_add_right_eq, int_mod_add_left_eq,
-				  nat_div_add_eq, int_div_add_eq,
-				  @{thm mod_self}, @{thm "zmod_self"},
-				  @{thm mod_by_0}, @{thm div_by_0},
-				  @{thm "zdiv_zero"}, @{thm "zmod_zero"}, @{thm "div_0"}, @{thm "mod_0"},
-				  @{thm "zdiv_1"}, @{thm "zmod_1"}, @{thm "div_1"}, @{thm "mod_1"},
-				  Suc_plus1]
-			addsimps @{thms add_ac}
-			addsimprocs [cancel_div_mod_proc]
-    val simpset0 = HOL_basic_ss
-      addsimps [mod_div_equality', Suc_plus1]
-      addsimps comp_arith
-      addsplits [split_zdiv, split_zmod, split_div', @{thm "split_min"}, @{thm "split_max"}]
-    (* Simp rules for changing (n::int) to int n *)
-    val simpset1 = HOL_basic_ss
-      addsimps [nat_number_of_def, zdvd_int] @ map (fn r => r RS sym)
-        [@{thm int_int_eq}, @{thm zle_int}, @{thm zless_int}, @{thm zadd_int}, @{thm zmult_int}]
-      addsplits [zdiff_int_split]
-    (*simp rules for elimination of int n*)
-
-    val simpset2 = HOL_basic_ss
-      addsimps [@{thm nat_0_le}, @{thm all_nat}, @{thm ex_nat}, @{thm number_of1}, @{thm number_of2}, @{thm int_0}, @{thm int_1}]
-      addcongs [@{thm conj_le_cong}, @{thm imp_le_cong}]
-    (* simp rules for elimination of abs *)
-    val simpset3 = HOL_basic_ss addsplits [@{thm abs_split}]
-    val ct = cterm_of thy (HOLogic.mk_Trueprop t)
-    (* Theorem for the nat --> int transformation *)
-    val pre_thm = Seq.hd (EVERY
-      [simp_tac mod_div_simpset 1, simp_tac simpset0 1,
-       TRY (simp_tac simpset1 1), TRY (simp_tac simpset2 1),
-       TRY (simp_tac simpset3 1), TRY (simp_tac cooper_ss 1)]
-      (trivial ct))
-    fun assm_tac i = REPEAT_DETERM_N nh (assume_tac i)
-    (* The result of the quantifier elimination *)
-    val (th, tac) = case (prop_of pre_thm) of
-        Const ("==>", _) $ (Const ("Trueprop", _) $ t1) $ _ =>
-    let val pth = linzqe_oracle (cterm_of thy (Pattern.eta_long [] t1))
-    in 
-          ((pth RS iffD2) RS pre_thm,
-            assm_tac (i + 1) THEN (if q then I else TRY) (rtac TrueI i))
-    end
-      | _ => (pre_thm, assm_tac i)
-  in (rtac (((mp_step nh) o (spec_step np)) th) i 
-      THEN tac) st
-  end handle Subscript => no_tac st);
-
-fun linz_args meth =
- let val parse_flag = 
-         Args.$$$ "no_quantify" >> (K (K false));
- in
-   Method.simple_args 
-  (Scan.optional (Args.$$$ "(" |-- Scan.repeat1 parse_flag --| Args.$$$ ")") [] >>
-    curry (Library.foldl op |>) true)
-    (fn q => fn ctxt => meth ctxt q 1)
-  end;
-
-fun linz_method ctxt q i = Method.METHOD (fn facts =>
-  Method.insert_tac facts 1 THEN linz_tac ctxt q i);
-
-val setup =
-  Method.add_method ("cooper",
-     linz_args linz_method,
-     "decision procedure for linear integer arithmetic");
-
-end
diff -r b8a6b9c56fdd -r 5170d6277b61 src/HOL/Reflection/ferrack_tac.ML
--- a/src/HOL/Reflection/ferrack_tac.ML	Thu Feb 12 21:24:14 2009 -0800
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,113 +0,0 @@
-(*  Title:      HOL/Reflection/ferrack_tac.ML
-    Author:     Amine Chaieb, TU Muenchen
-*)
-
-structure Ferrack_Tac =
-struct
-
-val trace = ref false;
-fun trace_msg s = if !trace then tracing s else ();
-
-val ferrack_ss = let val ths = [@{thm real_of_int_inject}, @{thm real_of_int_less_iff}, 
-				@{thm real_of_int_le_iff}]
-	     in @{simpset} delsimps ths addsimps (map (fn th => th RS sym) ths)
-	     end;
-
-val binarith =
-  @{thms normalize_bin_simps} @ @{thms pred_bin_simps} @ @{thms succ_bin_simps} @
-  @{thms add_bin_simps} @ @{thms minus_bin_simps} @  @{thms mult_bin_simps};
-val comp_arith = binarith @ simp_thms
-
-val zdvd_int = @{thm zdvd_int};
-val zdiff_int_split = @{thm zdiff_int_split};
-val all_nat = @{thm all_nat};
-val ex_nat = @{thm ex_nat};
-val number_of1 = @{thm number_of1};
-val number_of2 = @{thm number_of2};
-val split_zdiv = @{thm split_zdiv};
-val split_zmod = @{thm split_zmod};
-val mod_div_equality' = @{thm mod_div_equality'};
-val split_div' = @{thm split_div'};
-val Suc_plus1 = @{thm Suc_plus1};
-val imp_le_cong = @{thm imp_le_cong};
-val conj_le_cong = @{thm conj_le_cong};
-val nat_mod_add_eq = @{thm mod_add1_eq} RS sym;
-val nat_mod_add_left_eq = @{thm mod_add_left_eq} RS sym;
-val nat_mod_add_right_eq = @{thm mod_add_right_eq} RS sym;
-val int_mod_add_eq = @{thm zmod_zadd1_eq} RS sym;
-val int_mod_add_left_eq = @{thm zmod_zadd_left_eq} RS sym;
-val int_mod_add_right_eq = @{thm zmod_zadd_right_eq} RS sym;
-val nat_div_add_eq = @{thm div_add1_eq} RS sym;
-val int_div_add_eq = @{thm zdiv_zadd1_eq} RS sym;
-val ZDIVISION_BY_ZERO_MOD = @{thm DIVISION_BY_ZERO} RS conjunct2;
-val ZDIVISION_BY_ZERO_DIV = @{thm DIVISION_BY_ZERO} RS conjunct1;
-
-fun prepare_for_linr sg q fm = 
-  let
-    val ps = Logic.strip_params fm
-    val hs = map HOLogic.dest_Trueprop (Logic.strip_assums_hyp fm)
-    val c = HOLogic.dest_Trueprop (Logic.strip_assums_concl fm)
-    fun mk_all ((s, T), (P,n)) =
-      if 0 mem loose_bnos P then
-        (HOLogic.all_const T $ Abs (s, T, P), n)
-      else (incr_boundvars ~1 P, n-1)
-    fun mk_all2 (v, t) = HOLogic.all_const (fastype_of v) $ lambda v t;
-      val rhs = hs
-(*    val (rhs,irhs) = List.partition (relevant (rev ps)) hs *)
-    val np = length ps
-    val (fm',np) =  foldr (fn ((x, T), (fm,n)) => mk_all ((x, T), (fm,n)))
-      (foldr HOLogic.mk_imp c rhs, np) ps
-    val (vs, _) = List.partition (fn t => q orelse (type_of t) = HOLogic.natT)
-      (OldTerm.term_frees fm' @ OldTerm.term_vars fm');
-    val fm2 = foldr mk_all2 fm' vs
-  in (fm2, np + length vs, length rhs) end;
-
-(*Object quantifier to meta --*)
-fun spec_step n th = if (n=0) then th else (spec_step (n-1) th) RS spec ;
-
-(* object implication to meta---*)
-fun mp_step n th = if (n=0) then th else (mp_step (n-1) th) RS mp;
-
-
-fun linr_tac ctxt q i = 
-    (ObjectLogic.atomize_prems_tac i) 
-	THEN (REPEAT_DETERM (split_tac [@{thm split_min}, @{thm split_max}, @{thm abs_split}] i))
-	THEN (fn st =>
-  let
-    val g = List.nth (prems_of st, i - 1)
-    val thy = ProofContext.theory_of ctxt
-    (* Transform the term*)
-    val (t,np,nh) = prepare_for_linr thy q g
-    (* Some simpsets for dealing with mod div abs and nat*)
-    val simpset0 = Simplifier.theory_context thy HOL_basic_ss addsimps comp_arith
-    val ct = cterm_of thy (HOLogic.mk_Trueprop t)
-    (* Theorem for the nat --> int transformation *)
-   val pre_thm = Seq.hd (EVERY
-      [simp_tac simpset0 1,
-       TRY (simp_tac (Simplifier.theory_context thy ferrack_ss) 1)]
-      (trivial ct))
-    fun assm_tac i = REPEAT_DETERM_N nh (assume_tac i)
-    (* The result of the quantifier elimination *)
-    val (th, tac) = case (prop_of pre_thm) of
-        Const ("==>", _) $ (Const ("Trueprop", _) $ t1) $ _ =>
-    let val pth = linr_oracle (cterm_of thy (Pattern.eta_long [] t1))
-    in 
-          (trace_msg ("calling procedure with term:\n" ^
-             Syntax.string_of_term ctxt t1);
-           ((pth RS iffD2) RS pre_thm,
-            assm_tac (i + 1) THEN (if q then I else TRY) (rtac TrueI i)))
-    end
-      | _ => (pre_thm, assm_tac i)
-  in (rtac (((mp_step nh) o (spec_step np)) th) i 
-      THEN tac) st
-  end handle Subscript => no_tac st);
-
-fun linr_meth src =
-  Method.syntax (Args.mode "no_quantify") src
-  #> (fn (q, ctxt) => Method.SIMPLE_METHOD' (linr_tac ctxt (not q)));
-
-val setup =
-  Method.add_method ("rferrack", linr_meth,
-     "decision procedure for linear real arithmetic");
-
-end
diff -r b8a6b9c56fdd -r 5170d6277b61 src/HOL/Reflection/mir_tac.ML
--- a/src/HOL/Reflection/mir_tac.ML	Thu Feb 12 21:24:14 2009 -0800
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,168 +0,0 @@
-(*  Title:      HOL/Reflection/mir_tac.ML
-    Author:     Amine Chaieb, TU Muenchen
-*)
-
-structure Mir_Tac =
-struct
-
-val trace = ref false;
-fun trace_msg s = if !trace then tracing s else ();
-
-val mir_ss = 
-let val ths = map thm ["real_of_int_inject", "real_of_int_less_iff", "real_of_int_le_iff"]
-in @{simpset} delsimps ths addsimps (map (fn th => th RS sym) ths)
-end;
-
-val nT = HOLogic.natT;
-  val nat_arith = map thm ["add_nat_number_of", "diff_nat_number_of", 
-                       "mult_nat_number_of", "eq_nat_number_of", "less_nat_number_of"];
-
-  val comp_arith = (map thm ["Let_def", "if_False", "if_True", "add_0", 
-                 "add_Suc", "add_number_of_left", "mult_number_of_left", 
-                 "Suc_eq_add_numeral_1"])@
-                 (map (fn s => thm s RS sym) ["numeral_1_eq_1", "numeral_0_eq_0"])
-                 @ @{thms arith_simps} @ nat_arith @ @{thms rel_simps} 
-  val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"}, 
-             @{thm "real_of_nat_number_of"},
-             @{thm "real_of_nat_Suc"}, @{thm "real_of_nat_one"}, @{thm "real_of_one"},
-             @{thm "real_of_int_zero"}, @{thm "real_of_nat_zero"},
-             @{thm "Ring_and_Field.divide_zero"}, 
-             @{thm "divide_divide_eq_left"}, @{thm "times_divide_eq_right"}, 
-             @{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"},
-             @{thm "diff_def"}, @{thm "minus_divide_left"}]
-val comp_ths = ths @ comp_arith @ simp_thms 
-
-
-val zdvd_int = @{thm "zdvd_int"};
-val zdiff_int_split = @{thm "zdiff_int_split"};
-val all_nat = @{thm "all_nat"};
-val ex_nat = @{thm "ex_nat"};
-val number_of1 = @{thm "number_of1"};
-val number_of2 = @{thm "number_of2"};
-val split_zdiv = @{thm "split_zdiv"};
-val split_zmod = @{thm "split_zmod"};
-val mod_div_equality' = @{thm "mod_div_equality'"};
-val split_div' = @{thm "split_div'"};
-val Suc_plus1 = @{thm "Suc_plus1"};
-val imp_le_cong = @{thm "imp_le_cong"};
-val conj_le_cong = @{thm "conj_le_cong"};
-val nat_mod_add_eq = @{thm "mod_add1_eq"} RS sym;
-val nat_mod_add_left_eq = @{thm "mod_add_left_eq"} RS sym;
-val nat_mod_add_right_eq = @{thm "mod_add_right_eq"} RS sym;
-val int_mod_add_eq = @{thm "zmod_zadd1_eq"} RS sym;
-val int_mod_add_left_eq = @{thm "zmod_zadd_left_eq"} RS sym;
-val int_mod_add_right_eq = @{thm "zmod_zadd_right_eq"} RS sym;
-val nat_div_add_eq = @{thm "div_add1_eq"} RS sym;
-val int_div_add_eq = @{thm "zdiv_zadd1_eq"} RS sym;
-val ZDIVISION_BY_ZERO_MOD = @{thm "DIVISION_BY_ZERO"} RS conjunct2;
-val ZDIVISION_BY_ZERO_DIV = @{thm "DIVISION_BY_ZERO"} RS conjunct1;
-
-fun prepare_for_mir thy q fm = 
-  let
-    val ps = Logic.strip_params fm
-    val hs = map HOLogic.dest_Trueprop (Logic.strip_assums_hyp fm)
-    val c = HOLogic.dest_Trueprop (Logic.strip_assums_concl fm)
-    fun mk_all ((s, T), (P,n)) =
-      if 0 mem loose_bnos P then
-        (HOLogic.all_const T $ Abs (s, T, P), n)
-      else (incr_boundvars ~1 P, n-1)
-    fun mk_all2 (v, t) = HOLogic.all_const (fastype_of v) $ lambda v t;
-      val rhs = hs
-(*    val (rhs,irhs) = List.partition (relevant (rev ps)) hs *)
-    val np = length ps
-    val (fm',np) =  foldr (fn ((x, T), (fm,n)) => mk_all ((x, T), (fm,n)))
-      (foldr HOLogic.mk_imp c rhs, np) ps
-    val (vs, _) = List.partition (fn t => q orelse (type_of t) = nT)
-      (OldTerm.term_frees fm' @ OldTerm.term_vars fm');
-    val fm2 = foldr mk_all2 fm' vs
-  in (fm2, np + length vs, length rhs) end;
-
-(*Object quantifier to meta --*)
-fun spec_step n th = if (n=0) then th else (spec_step (n-1) th) RS spec ;
-
-(* object implication to meta---*)
-fun mp_step n th = if (n=0) then th else (mp_step (n-1) th) RS mp;
-
-
-fun mir_tac ctxt q i = 
-    (ObjectLogic.atomize_prems_tac i)
-        THEN (simp_tac (HOL_basic_ss addsimps [@{thm "abs_ge_zero"}] addsimps simp_thms) i)
-        THEN (REPEAT_DETERM (split_tac [@{thm "split_min"}, @{thm "split_max"},@{thm "abs_split"}] i))
-        THEN (fn st =>
-  let
-    val g = List.nth (prems_of st, i - 1)
-    val thy = ProofContext.theory_of ctxt
-    (* Transform the term*)
-    val (t,np,nh) = prepare_for_mir thy q g
-    (* Some simpsets for dealing with mod div abs and nat*)
-    val mod_div_simpset = HOL_basic_ss 
-                        addsimps [refl,nat_mod_add_eq, 
-                                  @{thm "mod_self"}, @{thm "zmod_self"},
-                                  @{thm "zdiv_zero"},@{thm "zmod_zero"},@{thm "div_0"}, @{thm "mod_0"},
-                                  @{thm "zdiv_1"}, @{thm "zmod_1"}, @{thm "div_1"}, @{thm "mod_1"},
-                                  @{thm "Suc_plus1"}]
-                        addsimps @{thms add_ac}
-                        addsimprocs [cancel_div_mod_proc]
-    val simpset0 = HOL_basic_ss
-      addsimps [mod_div_equality', Suc_plus1]
-      addsimps comp_ths
-      addsplits [@{thm "split_zdiv"}, @{thm "split_zmod"}, @{thm "split_div'"}, @{thm "split_min"}, @{thm "split_max"}]
-    (* Simp rules for changing (n::int) to int n *)
-    val simpset1 = HOL_basic_ss
-      addsimps [@{thm "nat_number_of_def"}, @{thm "zdvd_int"}] @ map (fn r => r RS sym)
-        [@{thm "int_int_eq"}, @{thm "zle_int"}, @{thm "zless_int"}, @{thm "zadd_int"}, 
-         @{thm "zmult_int"}]
-      addsplits [@{thm "zdiff_int_split"}]
-    (*simp rules for elimination of int n*)
-
-    val simpset2 = HOL_basic_ss
-      addsimps [@{thm "nat_0_le"}, @{thm "all_nat"}, @{thm "ex_nat"}, @{thm "number_of1"}, 
-                @{thm "number_of2"}, @{thm "int_0"}, @{thm "int_1"}]
-      addcongs [@{thm "conj_le_cong"}, @{thm "imp_le_cong"}]
-    (* simp rules for elimination of abs *)
-    val ct = cterm_of thy (HOLogic.mk_Trueprop t)
-    (* Theorem for the nat --> int transformation *)
-    val pre_thm = Seq.hd (EVERY
-      [simp_tac mod_div_simpset 1, simp_tac simpset0 1,
-       TRY (simp_tac simpset1 1), TRY (simp_tac simpset2 1), TRY (simp_tac mir_ss 1)]
-      (trivial ct))
-    fun assm_tac i = REPEAT_DETERM_N nh (assume_tac i)
-    (* The result of the quantifier elimination *)
-    val (th, tac) = case (prop_of pre_thm) of
-        Const ("==>", _) $ (Const ("Trueprop", _) $ t1) $ _ =>
-    let val pth =
-          (* If quick_and_dirty then run without proof generation as oracle*)
-             if !quick_and_dirty
-             then mirfr_oracle (false, cterm_of thy (Pattern.eta_long [] t1))
-             else mirfr_oracle (true, cterm_of thy (Pattern.eta_long [] t1))
-    in 
-          (trace_msg ("calling procedure with term:\n" ^
-             Syntax.string_of_term ctxt t1);
-           ((pth RS iffD2) RS pre_thm,
-            assm_tac (i + 1) THEN (if q then I else TRY) (rtac TrueI i)))
-    end
-      | _ => (pre_thm, assm_tac i)
-  in (rtac (((mp_step nh) o (spec_step np)) th) i 
-      THEN tac) st
-  end handle Subscript => no_tac st);
-
-fun mir_args meth =
- let val parse_flag = 
-         Args.$$$ "no_quantify" >> (K (K false));
- in
-   Method.simple_args 
-  (Scan.optional (Args.$$$ "(" |-- Scan.repeat1 parse_flag --| Args.$$$ ")") [] >>
-    curry (Library.foldl op |>) true)
-    (fn q => fn ctxt => meth ctxt q 1)
-  end;
-
-fun mir_method ctxt q i = Method.METHOD (fn facts =>
-  Method.insert_tac facts 1 THEN mir_tac ctxt q i);
-
-val setup =
-  Method.add_method ("mir",
-     mir_args mir_method,
-     "decision procedure for MIR arithmetic");
-
-
-end