author haftmann Fri, 06 Feb 2009 15:15:46 +0100 changeset 29824 2cf979ed69b8 parent 29821 ab8c54355f2e (current diff) parent 29823 0ab754d13ccd (diff) child 29825 384e47590e7f child 29826 5132da6ebca3 child 29865 c9bef39be3d2
merged
 src/HOL/Library/Dense_Linear_Order.thy file | annotate | diff | comparison | revisions
```--- a/NEWS	Fri Feb 06 14:36:58 2009 +0100
+++ b/NEWS	Fri Feb 06 15:15:46 2009 +0100
@@ -196,14 +196,17 @@
* Auxiliary class "itself" has disappeared -- classes without any parameter
are treated as expected by the 'class' command.

-* Theory "Reflection" now resides in HOL/Library.  Common reflection examples
-(Cooper, MIR, Ferrack, Approximation) now in distinct session directory
-HOL/Reflection. Here Approximation provides the new proof method
-"approximation". It proves formulas on real values by using interval arithmetic.
+* Leibnitz's Series for Pi and the arcus tangens and logarithm series.
+
+* Common decision procedures (Cooper, MIR, Ferrack, Approximation, Dense_Linear_Order)
+now in directory HOL/Decision_Procs.
+
+* Theory HOL/Decisioin_Procs/Approximation.thy provides the new proof method
+"approximation".  It proves formulas on real values by using interval arithmetic.
In the formulas are also the transcendental functions sin, cos, tan, atan, ln,
-exp and the constant pi are allowed. For examples see
-src/HOL/ex/ApproximationEx.thy. To reach this the Leibnitz's Series for Pi and
-the arcus tangens and logarithm series is now proved in Isabelle.
+exp and the constant pi are allowed.  For examples see HOL/ex/ApproximationEx.thy.
+
+* Theory "Reflection" now resides in HOL/Library.

* Entry point to Word library now simply named "Word".  INCOMPATIBILITY.

@@ -212,7 +215,6 @@

src/HOL/Library/RType.thy ~> src/HOL/Typerep.thy
src/HOL/Library/Code_Message.thy ~> src/HOL/
-    src/HOL/Library/Dense_Linear_Order.thy ~> src/HOL/
src/HOL/Library/GCD.thy ~> src/HOL/
src/HOL/Library/Order_Relation.thy ~> src/HOL/
src/HOL/Library/Parity.thy ~> src/HOL/```
```--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Decision_Procs/Approximation.thy	Fri Feb 06 15:15:46 2009 +0100
@@ -0,0 +1,2507 @@
+(* Title:     HOL/Reflection/Approximation.thy
+ * Author:    Johannes Hölzl <hoelzl@in.tum.de> 2008 / 2009
+ *)
+theory Approximation
+imports Complex_Main Float Reflection Dense_Linear_Order 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
+    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
+    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
+	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
+
+    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"
+    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
+  | 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)
+  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```
```--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Decision_Procs/Cooper.thy	Fri Feb 06 15:15:46 2009 +0100
@@ -0,0 +1,2174 @@
+(*  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"
+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 (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)"
+(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)"
+
+(*function (sequential)
+  numadd :: "num \<Rightarrow> num \<Rightarrow> num"
+where
+      (if n1 = n2 then (let c = c1 + c2
+      in (if c = 0 then numadd r1 r2 else
+      else if n1 \<le> n2 then
+      else
+  | "numadd (C b1) (C b2) = C (b1 + b2)"
+apply pat_completeness apply auto*)
+
+apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all)
+ apply (case_tac "n1 = n2")
+done
+
+lemma numadd_nb: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numadd (t,s))"
+
+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)"
+
+lemma nummul_nb: "\<And> i. numbound0 t \<Longrightarrow> numbound0 (nummul i t)"
+
+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)"
+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
+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
+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'"
+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)"
+      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)"
+      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)"
+  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)"
+  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)
+
+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)"
+    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)"
+      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"
+  from \<alpha>_l[OF lq] have A_nb: "\<forall> b\<in> set ?A'. numbound0 b"
+    {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
+      | 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```
```--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Decision_Procs/Dense_Linear_Order.thy	Fri Feb 06 15:15:46 2009 +0100
@@ -0,0 +1,879 @@
+(*  Title       : HOL/Dense_Linear_Order.thy
+    Author      : Amine Chaieb, TU Muenchen
+*)
+
+header {* Dense linear order without endpoints
+  and a quantifier elimination procedure in Ferrante and Rackoff style *}
+
+theory Dense_Linear_Order
+imports Plain Groebner_Basis Main
+uses
+  "~~/src/HOL/Tools/Qelim/langford_data.ML"
+  "~~/src/HOL/Tools/Qelim/ferrante_rackoff_data.ML"
+  ("~~/src/HOL/Tools/Qelim/langford.ML")
+  ("~~/src/HOL/Tools/Qelim/ferrante_rackoff.ML")
+begin
+
+setup {* Langford_Data.setup #> Ferrante_Rackoff_Data.setup *}
+
+context linorder
+begin
+
+lemma less_not_permute[noatp]: "\<not> (x < y \<and> y < x)" by (simp add: not_less linear)
+
+lemma gather_simps[noatp]:
+  shows
+  "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> x < u \<and> P x) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> (insert u U). x < y) \<and> P x)"
+  and "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> l < x \<and> P x) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> (insert l L). y < x) \<and> (\<forall>y \<in> U. x < y) \<and> P x)"
+  "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> x < u) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> (insert u U). x < y))"
+  and "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> l < x) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> (insert l L). y < x) \<and> (\<forall>y \<in> U. x < y))"  by auto
+
+lemma
+  gather_start[noatp]: "(\<exists>x. P x) \<equiv> (\<exists>x. (\<forall>y \<in> {}. y < x) \<and> (\<forall>y\<in> {}. x < y) \<and> P x)"
+  by simp
+
+text{* Theorems for @{text "\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>-\<infinity>\<^esub>)"}*}
+lemma minf_lt[noatp]:  "\<exists>z . \<forall>x. x < z \<longrightarrow> (x < t \<longleftrightarrow> True)" by auto
+lemma minf_gt[noatp]: "\<exists>z . \<forall>x. x < z \<longrightarrow>  (t < x \<longleftrightarrow>  False)"
+  by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le)
+
+lemma minf_le[noatp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x \<le> t \<longleftrightarrow> True)" by (auto simp add: less_le)
+lemma minf_ge[noatp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (t \<le> x \<longleftrightarrow> False)"
+  by (auto simp add: less_le not_less not_le)
+lemma minf_eq[noatp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto
+lemma minf_neq[noatp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
+lemma minf_P[noatp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (P \<longleftrightarrow> P)" by blast
+
+text{* Theorems for @{text "\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>+\<infinity>\<^esub>)"}*}
+lemma pinf_gt[noatp]:  "\<exists>z . \<forall>x. z < x \<longrightarrow> (t < x \<longleftrightarrow> True)" by auto
+lemma pinf_lt[noatp]: "\<exists>z . \<forall>x. z < x \<longrightarrow>  (x < t \<longleftrightarrow>  False)"
+  by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le)
+
+lemma pinf_ge[noatp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (t \<le> x \<longleftrightarrow> True)" by (auto simp add: less_le)
+lemma pinf_le[noatp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x \<le> t \<longleftrightarrow> False)"
+  by (auto simp add: less_le not_less not_le)
+lemma pinf_eq[noatp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto
+lemma pinf_neq[noatp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
+lemma pinf_P[noatp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (P \<longleftrightarrow> P)" by blast
+
+lemma nmi_lt[noatp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x < t \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
+lemma nmi_gt[noatp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> t < x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)"
+  by (auto simp add: le_less)
+lemma  nmi_le[noatp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x\<le> t \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
+lemma  nmi_ge[noatp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> t\<le> x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
+lemma  nmi_eq[noatp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x = t \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
+lemma  nmi_neq[noatp]: "t \<in> U \<Longrightarrow>\<forall>x. \<not>True \<and> x \<noteq> t \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
+lemma  nmi_P[noatp]: "\<forall> x. ~P \<and> P \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
+lemma  nmi_conj[noatp]: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x) ;
+  \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)\<rbrakk> \<Longrightarrow>
+  \<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
+lemma  nmi_disj[noatp]: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x) ;
+  \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)\<rbrakk> \<Longrightarrow>
+  \<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
+
+lemma  npi_lt[noatp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x < t \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by (auto simp add: le_less)
+lemma  npi_gt[noatp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t < x \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
+lemma  npi_le[noatp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x \<le> t \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
+lemma  npi_ge[noatp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t \<le> x \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
+lemma  npi_eq[noatp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x = t \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
+lemma  npi_neq[noatp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x \<noteq> t \<longrightarrow>  (\<exists> u\<in> U. x \<le> u )" by auto
+lemma  npi_P[noatp]: "\<forall> x. ~P \<and> P \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
+lemma  npi_conj[noatp]: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. x \<le> u) ;  \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)\<rbrakk>
+  \<Longrightarrow>  \<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
+lemma  npi_disj[noatp]: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. x \<le> u) ; \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)\<rbrakk>
+  \<Longrightarrow> \<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
+
+lemma lin_dense_lt[noatp]: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> x < t \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> y < t)"
+proof(clarsimp)
+  fix x l u y  assume tU: "t \<in> U" and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U" and lx: "l < x"
+    and xu: "x<u"  and px: "x < t" and ly: "l<y" and yu:"y < u"
+  from tU noU ly yu have tny: "t\<noteq>y" by auto
+  {assume H: "t < y"
+    from less_trans[OF lx px] less_trans[OF H yu]
+    have "l < t \<and> t < u"  by simp
+    with tU noU have "False" by auto}
+  hence "\<not> t < y"  by auto hence "y \<le> t" by (simp add: not_less)
+  thus "y < t" using tny by (simp add: less_le)
+qed
+
+lemma lin_dense_gt[noatp]: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> t < x \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> t < y)"
+proof(clarsimp)
+  fix x l u y
+  assume tU: "t \<in> U" and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U" and lx: "l < x" and xu: "x<u"
+  and px: "t < x" and ly: "l<y" and yu:"y < u"
+  from tU noU ly yu have tny: "t\<noteq>y" by auto
+  {assume H: "y< t"
+    from less_trans[OF ly H] less_trans[OF px xu] have "l < t \<and> t < u" by simp
+    with tU noU have "False" by auto}
+  hence "\<not> y<t"  by auto hence "t \<le> y" by (auto simp add: not_less)
+  thus "t < y" using tny by (simp add:less_le)
+qed
+
+lemma lin_dense_le[noatp]: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> x \<le> t \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> y\<le> t)"
+proof(clarsimp)
+  fix x l u y
+  assume tU: "t \<in> U" and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U" and lx: "l < x" and xu: "x<u"
+  and px: "x \<le> t" and ly: "l<y" and yu:"y < u"
+  from tU noU ly yu have tny: "t\<noteq>y" by auto
+  {assume H: "t < y"
+    from less_le_trans[OF lx px] less_trans[OF H yu]
+    have "l < t \<and> t < u" by simp
+    with tU noU have "False" by auto}
+  hence "\<not> t < y"  by auto thus "y \<le> t" by (simp add: not_less)
+qed
+
+lemma lin_dense_ge[noatp]: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> t \<le> x \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> t \<le> y)"
+proof(clarsimp)
+  fix x l u y
+  assume tU: "t \<in> U" and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U" and lx: "l < x" and xu: "x<u"
+  and px: "t \<le> x" and ly: "l<y" and yu:"y < u"
+  from tU noU ly yu have tny: "t\<noteq>y" by auto
+  {assume H: "y< t"
+    from less_trans[OF ly H] le_less_trans[OF px xu]
+    have "l < t \<and> t < u" by simp
+    with tU noU have "False" by auto}
+  hence "\<not> y<t"  by auto thus "t \<le> y" by (simp add: not_less)
+qed
+lemma lin_dense_eq[noatp]: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> x = t   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> y= t)"  by auto
+lemma lin_dense_neq[noatp]: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> x \<noteq> t   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> y\<noteq> t)"  by auto
+lemma lin_dense_P[noatp]: "\<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P)"  by auto
+
+lemma lin_dense_conj[noatp]:
+  "\<lbrakk>\<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P1 x
+  \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P1 y) ;
+  \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P2 x
+  \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow>
+  \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> (P1 x \<and> P2 x)
+  \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> (P1 y \<and> P2 y))"
+  by blast
+lemma lin_dense_disj[noatp]:
+  "\<lbrakk>\<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P1 x
+  \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P1 y) ;
+  \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P2 x
+  \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow>
+  \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> (P1 x \<or> P2 x)
+  \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> (P1 y \<or> P2 y))"
+  by blast
+
+lemma npmibnd[noatp]: "\<lbrakk>\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists> u\<in> U. u \<le> x); \<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. x \<le> u)\<rbrakk>
+  \<Longrightarrow> \<forall>x. \<not> MP \<and> \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. \<exists> u' \<in> U. u \<le> x \<and> x \<le> u')"
+by auto
+
+lemma finite_set_intervals[noatp]:
+  assumes px: "P x" 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 simp add: linear)
+    moreover {assume "y \<in> ?Mx" hence "y \<le> ?a" using Mxne fMx by auto with ay have "False" by (simp add: not_le[symmetric])}
+    moreover {assume "y \<in> ?xM" hence "?b \<le> y" using xMne fxM by auto with yb have "False" by (simp add: not_le[symmetric])}
+    ultimately show "False" by blast
+  qed
+  from ainS binS noy ax xb px show ?thesis by blast
+qed
+
+lemma finite_set_intervals2[noatp]:
+  assumes px: "P x" 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 simp add: le_less)
+  thus ?thesis using px as bs noS by blast
+qed
+
+end
+
+section {* The classical QE after Langford for dense linear orders *}
+
+context dense_linear_order
+begin
+
+lemma interval_empty_iff:
+  "{y. x < y \<and> y < z} = {} \<longleftrightarrow> \<not> x < z"
+  by (auto dest: dense)
+
+lemma dlo_qe_bnds[noatp]:
+  assumes ne: "L \<noteq> {}" and neU: "U \<noteq> {}" and fL: "finite L" and fU: "finite U"
+  shows "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y)) \<equiv> (\<forall> l \<in> L. \<forall>u \<in> U. l < u)"
+proof (simp only: atomize_eq, rule iffI)
+  assume H: "\<exists>x. (\<forall>y\<in>L. y < x) \<and> (\<forall>y\<in>U. x < y)"
+  then obtain x where xL: "\<forall>y\<in>L. y < x" and xU: "\<forall>y\<in>U. x < y" by blast
+  {fix l u assume l: "l \<in> L" and u: "u \<in> U"
+    have "l < x" using xL l by blast
+    also have "x < u" using xU u by blast
+    finally (less_trans) have "l < u" .}
+  thus "\<forall>l\<in>L. \<forall>u\<in>U. l < u" by blast
+next
+  assume H: "\<forall>l\<in>L. \<forall>u\<in>U. l < u"
+  let ?ML = "Max L"
+  let ?MU = "Min U"
+  from fL ne have th1: "?ML \<in> L" and th1': "\<forall>l\<in>L. l \<le> ?ML" by auto
+  from fU neU have th2: "?MU \<in> U" and th2': "\<forall>u\<in>U. ?MU \<le> u" by auto
+  from th1 th2 H have "?ML < ?MU" by auto
+  with dense obtain w where th3: "?ML < w" and th4: "w < ?MU" by blast
+  from th3 th1' have "\<forall>l \<in> L. l < w" by auto
+  moreover from th4 th2' have "\<forall>u \<in> U. w < u" by auto
+  ultimately show "\<exists>x. (\<forall>y\<in>L. y < x) \<and> (\<forall>y\<in>U. x < y)" by auto
+qed
+
+lemma dlo_qe_noub[noatp]:
+  assumes ne: "L \<noteq> {}" and fL: "finite L"
+  shows "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> {}. x < y)) \<equiv> True"
+  from gt_ex[of "Max L"] obtain M where M: "Max L < M" by blast
+  from ne fL have "\<forall>x \<in> L. x \<le> Max L" by simp
+  with M have "\<forall>x\<in>L. x < M" by (auto intro: le_less_trans)
+  thus "\<exists>x. \<forall>y\<in>L. y < x" by blast
+qed
+
+lemma dlo_qe_nolb[noatp]:
+  assumes ne: "U \<noteq> {}" and fU: "finite U"
+  shows "(\<exists>x. (\<forall>y \<in> {}. y < x) \<and> (\<forall>y \<in> U. x < y)) \<equiv> True"
+  from lt_ex[of "Min U"] obtain M where M: "M < Min U" by blast
+  from ne fU have "\<forall>x \<in> U. Min U \<le> x" by simp
+  with M have "\<forall>x\<in>U. M < x" by (auto intro: less_le_trans)
+  thus "\<exists>x. \<forall>y\<in>U. x < y" by blast
+qed
+
+lemma exists_neq[noatp]: "\<exists>(x::'a). x \<noteq> t" "\<exists>(x::'a). t \<noteq> x"
+  using gt_ex[of t] by auto
+
+lemmas dlo_simps[noatp] = order_refl less_irrefl not_less not_le exists_neq
+  le_less neq_iff linear less_not_permute
+
+lemma axiom[noatp]: "dense_linear_order (op \<le>) (op <)" by (rule dense_linear_order_axioms)
+lemma atoms[noatp]:
+  shows "TERM (less :: 'a \<Rightarrow> _)"
+    and "TERM (less_eq :: 'a \<Rightarrow> _)"
+    and "TERM (op = :: 'a \<Rightarrow> _)" .
+
+declare axiom[langford qe: dlo_qe_bnds dlo_qe_nolb dlo_qe_noub gather: gather_start gather_simps atoms: atoms]
+declare dlo_simps[langfordsimp]
+
+end
+
+(* FIXME: Move to HOL -- together with the conj_aci_rule in langford.ML *)
+lemma dnf[noatp]:
+  "(P & (Q | R)) = ((P&Q) | (P&R))"
+  "((Q | R) & P) = ((Q&P) | (R&P))"
+  by blast+
+
+lemmas weak_dnf_simps[noatp] = simp_thms dnf
+
+lemma nnf_simps[noatp]:
+    "(\<not>(P \<and> Q)) = (\<not>P \<or> \<not>Q)" "(\<not>(P \<or> Q)) = (\<not>P \<and> \<not>Q)" "(P \<longrightarrow> Q) = (\<not>P \<or> Q)"
+    "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
+  by blast+
+
+lemma ex_distrib[noatp]: "(\<exists>x. P x \<or> Q x) \<longleftrightarrow> ((\<exists>x. P x) \<or> (\<exists>x. Q x))" by blast
+
+lemmas dnf_simps[noatp] = weak_dnf_simps nnf_simps ex_distrib
+
+use "~~/src/HOL/Tools/Qelim/langford.ML"
+method_setup dlo = {*
+  Method.ctxt_args (Method.SIMPLE_METHOD' o LangfordQE.dlo_tac)
+*} "Langford's algorithm for quantifier elimination in dense linear orders"
+
+
+section {* Contructive dense linear orders yield QE for linear arithmetic over ordered Fields -- see @{text "Arith_Tools.thy"} *}
+
+text {* Linear order without upper bounds *}
+
+locale linorder_stupid_syntax = linorder
+begin
+notation
+  less_eq  ("op \<sqsubseteq>") and
+  less_eq  ("(_/ \<sqsubseteq> _)" [51, 51] 50) and
+  less  ("op \<sqsubset>") and
+  less  ("(_/ \<sqsubset> _)"  [51, 51] 50)
+
+end
+
+locale linorder_no_ub = linorder_stupid_syntax +
+  assumes gt_ex: "\<exists>y. less x y"
+begin
+lemma ge_ex[noatp]: "\<exists>y. x \<sqsubseteq> y" using gt_ex by auto
+
+text {* Theorems for @{text "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>+\<infinity>\<^esub>)"} *}
+lemma pinf_conj[noatp]:
+  assumes ex1: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
+  and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
+  shows "\<exists>z. \<forall>x. z \<sqsubset>  x \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))"
+proof-
+  from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
+     and z2: "\<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
+  from gt_ex obtain z where z:"ord.max less_eq z1 z2 \<sqsubset> z" by blast
+  from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" by simp_all
+  {fix x assume H: "z \<sqsubset> x"
+    from less_trans[OF zz1 H] less_trans[OF zz2 H]
+    have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')"  using z1 zz1 z2 zz2 by auto
+  }
+  thus ?thesis by blast
+qed
+
+lemma pinf_disj[noatp]:
+  assumes ex1: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
+  and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
+  shows "\<exists>z. \<forall>x. z \<sqsubset>  x \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))"
+proof-
+  from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
+     and z2: "\<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
+  from gt_ex obtain z where z:"ord.max less_eq z1 z2 \<sqsubset> z" by blast
+  from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" by simp_all
+  {fix x assume H: "z \<sqsubset> x"
+    from less_trans[OF zz1 H] less_trans[OF zz2 H]
+    have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')"  using z1 zz1 z2 zz2 by auto
+  }
+  thus ?thesis by blast
+qed
+
+lemma pinf_ex[noatp]: assumes ex:"\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P1)" and p1: P1 shows "\<exists> x. P x"
+proof-
+  from ex obtain z where z: "\<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P1)" by blast
+  from gt_ex obtain x where x: "z \<sqsubset> x" by blast
+  from z x p1 show ?thesis by blast
+qed
+
+end
+
+text {* Linear order without upper bounds *}
+
+locale linorder_no_lb = linorder_stupid_syntax +
+  assumes lt_ex: "\<exists>y. less y x"
+begin
+lemma le_ex[noatp]: "\<exists>y. y \<sqsubseteq> x" using lt_ex by auto
+
+
+text {* Theorems for @{text "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>-\<infinity>\<^esub>)"} *}
+lemma minf_conj[noatp]:
+  assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
+  and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
+  shows "\<exists>z. \<forall>x. x \<sqsubset>  z \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))"
+proof-
+  from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"and z2: "\<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
+  from lt_ex obtain z where z:"z \<sqsubset> ord.min less_eq z1 z2" by blast
+  from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" by simp_all
+  {fix x assume H: "x \<sqsubset> z"
+    from less_trans[OF H zz1] less_trans[OF H zz2]
+    have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')"  using z1 zz1 z2 zz2 by auto
+  }
+  thus ?thesis by blast
+qed
+
+lemma minf_disj[noatp]:
+  assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
+  and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
+  shows "\<exists>z. \<forall>x. x \<sqsubset>  z \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))"
+proof-
+  from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"and z2: "\<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
+  from lt_ex obtain z where z:"z \<sqsubset> ord.min less_eq z1 z2" by blast
+  from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" by simp_all
+  {fix x assume H: "x \<sqsubset> z"
+    from less_trans[OF H zz1] less_trans[OF H zz2]
+    have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')"  using z1 zz1 z2 zz2 by auto
+  }
+  thus ?thesis by blast
+qed
+
+lemma minf_ex[noatp]: assumes ex:"\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P1)" and p1: P1 shows "\<exists> x. P x"
+proof-
+  from ex obtain z where z: "\<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P1)" by blast
+  from lt_ex obtain x where x: "x \<sqsubset> z" by blast
+  from z x p1 show ?thesis by blast
+qed
+
+end
+
+
+locale constr_dense_linear_order = linorder_no_lb + linorder_no_ub +
+  fixes between
+  assumes between_less: "less x y \<Longrightarrow> less x (between x y) \<and> less (between x y) y"
+     and  between_same: "between x x = x"
+
+sublocale  constr_dense_linear_order < dense_linear_order
+  apply unfold_locales
+  using gt_ex lt_ex between_less
+    by (auto, rule_tac x="between x y" in exI, simp)
+
+context  constr_dense_linear_order
+begin
+
+lemma rinf_U[noatp]:
+  assumes fU: "finite U"
+  and lin_dense: "\<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> P x
+  \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )"
+  and nmpiU: "\<forall>x. \<not> MP \<and> \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. \<exists> u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> u')"
+  and nmi: "\<not> MP"  and npi: "\<not> PP"  and ex: "\<exists> x.  P x"
+  shows "\<exists> u\<in> U. \<exists> u' \<in> U. P (between u u')"
+proof-
+  from ex obtain x where px: "P x" by blast
+  from px nmi npi nmpiU have "\<exists> u\<in> U. \<exists> u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> u'" by auto
+  then obtain u and u' where uU:"u\<in> U" and uU': "u' \<in> U" and ux:"u \<sqsubseteq> x" and xu':"x \<sqsubseteq> u'" by auto
+  from uU have Une: "U \<noteq> {}" by auto
+  term "linorder.Min less_eq"
+  let ?l = "linorder.Min less_eq U"
+  let ?u = "linorder.Max less_eq U"
+  have linM: "?l \<in> U" using fU Une by simp
+  have uinM: "?u \<in> U" using fU Une by simp
+  have lM: "\<forall> t\<in> U. ?l \<sqsubseteq> t" using Une fU by auto
+  have Mu: "\<forall> t\<in> U. t \<sqsubseteq> ?u" using Une fU by auto
+  have th:"?l \<sqsubseteq> u" using uU Une lM by auto
+  from order_trans[OF th ux] have lx: "?l \<sqsubseteq> x" .
+  have th: "u' \<sqsubseteq> ?u" using uU' Une Mu by simp
+  from order_trans[OF xu' th] have xu: "x \<sqsubseteq> ?u" .
+  from finite_set_intervals2[where P="P",OF px lx xu linM uinM fU lM Mu]
+  have "(\<exists> s\<in> U. P s) \<or>
+      (\<exists> t1\<in> U. \<exists> t2 \<in> U. (\<forall> y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U) \<and> t1 \<sqsubset> x \<and> x \<sqsubset> t2 \<and> P x)" .
+  moreover { fix u assume um: "u\<in>U" and pu: "P u"
+    have "between u u = u" by (simp add: between_same)
+    with um pu have "P (between u u)" by simp
+    with um have ?thesis by blast}
+  moreover{
+    assume "\<exists> t1\<in> U. \<exists> t2 \<in> U. (\<forall> y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U) \<and> t1 \<sqsubset> x \<and> x \<sqsubset> t2 \<and> P x"
+      then obtain t1 and t2 where t1M: "t1 \<in> U" and t2M: "t2\<in> U"
+        and noM: "\<forall> y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U" and t1x: "t1 \<sqsubset> x" and xt2: "x \<sqsubset> t2" and px: "P x"
+        by blast
+      from less_trans[OF t1x xt2] have t1t2: "t1 \<sqsubset> t2" .
+      let ?u = "between t1 t2"
+      from between_less t1t2 have t1lu: "t1 \<sqsubset> ?u" and ut2: "?u \<sqsubset> t2" by auto
+      from lin_dense noM t1x xt2 px t1lu ut2 have "P ?u" by blast
+      with t1M t2M have ?thesis by blast}
+    ultimately show ?thesis by blast
+  qed
+
+theorem fr_eq[noatp]:
+  assumes fU: "finite U"
+  and lin_dense: "\<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> P x
+   \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )"
+  and nmibnd: "\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists> u\<in> U. u \<sqsubseteq> x)"
+  and npibnd: "\<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. x \<sqsubseteq> u)"
+  and mi: "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x = MP)"  and pi: "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x = PP)"
+  shows "(\<exists> x. P x) \<equiv> (MP \<or> PP \<or> (\<exists> u \<in> U. \<exists> u'\<in> U. P (between u u')))"
+  (is "_ \<equiv> (_ \<or> _ \<or> ?F)" is "?E \<equiv> ?D")
+proof-
+ {
+   assume px: "\<exists> x. P x"
+   have "MP \<or> PP \<or> (\<not> MP \<and> \<not> PP)" by blast
+   moreover {assume "MP \<or> PP" hence "?D" by blast}
+   moreover {assume nmi: "\<not> MP" and npi: "\<not> PP"
+     from npmibnd[OF nmibnd npibnd]
+     have nmpiU: "\<forall>x. \<not> MP \<and> \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. \<exists> u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> u')" .
+     from rinf_U[OF fU lin_dense nmpiU nmi npi px] have "?D" by blast}
+   ultimately have "?D" by blast}
+ moreover
+ { assume "?D"
+   moreover {assume m:"MP" from minf_ex[OF mi m] have "?E" .}
+   moreover {assume p: "PP" from pinf_ex[OF pi p] have "?E" . }
+   moreover {assume f:"?F" hence "?E" by blast}
+   ultimately have "?E" by blast}
+ ultimately have "?E = ?D" by blast thus "?E \<equiv> ?D" by simp
+qed
+
+lemmas minf_thms[noatp] = minf_conj minf_disj minf_eq minf_neq minf_lt minf_le minf_gt minf_ge minf_P
+lemmas pinf_thms[noatp] = pinf_conj pinf_disj pinf_eq pinf_neq pinf_lt pinf_le pinf_gt pinf_ge pinf_P
+
+lemmas nmi_thms[noatp] = nmi_conj nmi_disj nmi_eq nmi_neq nmi_lt nmi_le nmi_gt nmi_ge nmi_P
+lemmas npi_thms[noatp] = npi_conj npi_disj npi_eq npi_neq npi_lt npi_le npi_gt npi_ge npi_P
+lemmas lin_dense_thms[noatp] = lin_dense_conj lin_dense_disj lin_dense_eq lin_dense_neq lin_dense_lt lin_dense_le lin_dense_gt lin_dense_ge lin_dense_P
+
+lemma ferrack_axiom[noatp]: "constr_dense_linear_order less_eq less between"
+  by (rule constr_dense_linear_order_axioms)
+lemma atoms[noatp]:
+  shows "TERM (less :: 'a \<Rightarrow> _)"
+    and "TERM (less_eq :: 'a \<Rightarrow> _)"
+    and "TERM (op = :: 'a \<Rightarrow> _)" .
+
+declare ferrack_axiom [ferrack minf: minf_thms pinf: pinf_thms
+    nmi: nmi_thms npi: npi_thms lindense:
+    lin_dense_thms qe: fr_eq atoms: atoms]
+
+declaration {*
+let
+fun simps phi = map (Morphism.thm phi) [@{thm "not_less"}, @{thm "not_le"}]
+fun generic_whatis phi =
+ let
+  val [lt, le] = map (Morphism.term phi) [@{term "op \<sqsubset>"}, @{term "op \<sqsubseteq>"}]
+  fun h x t =
+   case term_of t of
+     Const("op =", _)\$y\$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq
+                            else Ferrante_Rackoff_Data.Nox
+   | @{term "Not"}\$(Const("op =", _)\$y\$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq
+                            else Ferrante_Rackoff_Data.Nox
+   | b\$y\$z => if Term.could_unify (b, lt) then
+                 if term_of x aconv y then Ferrante_Rackoff_Data.Lt
+                 else if term_of x aconv z then Ferrante_Rackoff_Data.Gt
+                 else Ferrante_Rackoff_Data.Nox
+             else if Term.could_unify (b, le) then
+                 if term_of x aconv y then Ferrante_Rackoff_Data.Le
+                 else if term_of x aconv z then Ferrante_Rackoff_Data.Ge
+                 else Ferrante_Rackoff_Data.Nox
+             else Ferrante_Rackoff_Data.Nox
+   | _ => Ferrante_Rackoff_Data.Nox
+ in h end
+ fun ss phi = HOL_ss addsimps (simps phi)
+in
+ Ferrante_Rackoff_Data.funs  @{thm "ferrack_axiom"}
+  {isolate_conv = K (K (K Thm.reflexive)), whatis = generic_whatis, simpset = ss}
+end
+*}
+
+end
+
+use "~~/src/HOL/Tools/Qelim/ferrante_rackoff.ML"
+
+method_setup ferrack = {*
+  Method.ctxt_args (Method.SIMPLE_METHOD' o FerranteRackoff.dlo_tac)
+*} "Ferrante and Rackoff's algorithm for quantifier elimination in dense linear orders"
+
+subsection {* Ferrante and Rackoff algorithm over ordered fields *}
+
+lemma neg_prod_lt:"(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x < 0) == (x > 0))"
+proof-
+  assume H: "c < 0"
+  have "c*x < 0 = (0/c < x)" by (simp only: neg_divide_less_eq[OF H] algebra_simps)
+  also have "\<dots> = (0 < x)" by simp
+  finally show  "(c*x < 0) == (x > 0)" by simp
+qed
+
+lemma pos_prod_lt:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x < 0) == (x < 0))"
+proof-
+  assume H: "c > 0"
+  hence "c*x < 0 = (0/c > x)" by (simp only: pos_less_divide_eq[OF H] algebra_simps)
+  also have "\<dots> = (0 > x)" by simp
+  finally show  "(c*x < 0) == (x < 0)" by simp
+qed
+
+lemma neg_prod_sum_lt: "(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x + t< 0) == (x > (- 1/c)*t))"
+proof-
+  assume H: "c < 0"
+  have "c*x + t< 0 = (c*x < -t)" by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
+  also have "\<dots> = (-t/c < x)" by (simp only: neg_divide_less_eq[OF H] algebra_simps)
+  also have "\<dots> = ((- 1/c)*t < x)" by simp
+  finally show  "(c*x + t < 0) == (x > (- 1/c)*t)" by simp
+qed
+
+lemma pos_prod_sum_lt:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x + t < 0) == (x < (- 1/c)*t))"
+proof-
+  assume H: "c > 0"
+  have "c*x + t< 0 = (c*x < -t)"  by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
+  also have "\<dots> = (-t/c > x)" by (simp only: pos_less_divide_eq[OF H] algebra_simps)
+  also have "\<dots> = ((- 1/c)*t > x)" by simp
+  finally show  "(c*x + t < 0) == (x < (- 1/c)*t)" by simp
+qed
+
+lemma sum_lt:"((x::'a::pordered_ab_group_add) + t < 0) == (x < - t)"
+  using less_diff_eq[where a= x and b=t and c=0] by simp
+
+lemma neg_prod_le:"(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x <= 0) == (x >= 0))"
+proof-
+  assume H: "c < 0"
+  have "c*x <= 0 = (0/c <= x)" by (simp only: neg_divide_le_eq[OF H] algebra_simps)
+  also have "\<dots> = (0 <= x)" by simp
+  finally show  "(c*x <= 0) == (x >= 0)" by simp
+qed
+
+lemma pos_prod_le:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x <= 0) == (x <= 0))"
+proof-
+  assume H: "c > 0"
+  hence "c*x <= 0 = (0/c >= x)" by (simp only: pos_le_divide_eq[OF H] algebra_simps)
+  also have "\<dots> = (0 >= x)" by simp
+  finally show  "(c*x <= 0) == (x <= 0)" by simp
+qed
+
+lemma neg_prod_sum_le: "(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x + t <= 0) == (x >= (- 1/c)*t))"
+proof-
+  assume H: "c < 0"
+  have "c*x + t <= 0 = (c*x <= -t)"  by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
+  also have "\<dots> = (-t/c <= x)" by (simp only: neg_divide_le_eq[OF H] algebra_simps)
+  also have "\<dots> = ((- 1/c)*t <= x)" by simp
+  finally show  "(c*x + t <= 0) == (x >= (- 1/c)*t)" by simp
+qed
+
+lemma pos_prod_sum_le:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x + t <= 0) == (x <= (- 1/c)*t))"
+proof-
+  assume H: "c > 0"
+  have "c*x + t <= 0 = (c*x <= -t)" by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
+  also have "\<dots> = (-t/c >= x)" by (simp only: pos_le_divide_eq[OF H] algebra_simps)
+  also have "\<dots> = ((- 1/c)*t >= x)" by simp
+  finally show  "(c*x + t <= 0) == (x <= (- 1/c)*t)" by simp
+qed
+
+lemma sum_le:"((x::'a::pordered_ab_group_add) + t <= 0) == (x <= - t)"
+  using le_diff_eq[where a= x and b=t and c=0] by simp
+
+lemma nz_prod_eq:"(c\<Colon>'a\<Colon>ordered_field) \<noteq> 0 \<Longrightarrow> ((c*x = 0) == (x = 0))" by simp
+lemma nz_prod_sum_eq: "(c\<Colon>'a\<Colon>ordered_field) \<noteq> 0 \<Longrightarrow> ((c*x + t = 0) == (x = (- 1/c)*t))"
+proof-
+  assume H: "c \<noteq> 0"
+  have "c*x + t = 0 = (c*x = -t)" by (subst eq_iff_diff_eq_0 [of "c*x" "-t"], simp)
+  also have "\<dots> = (x = -t/c)" by (simp only: nonzero_eq_divide_eq[OF H] algebra_simps)
+  finally show  "(c*x + t = 0) == (x = (- 1/c)*t)" by simp
+qed
+lemma sum_eq:"((x::'a::pordered_ab_group_add) + t = 0) == (x = - t)"
+  using eq_diff_eq[where a= x and b=t and c=0] by simp
+
+
+interpretation class_ordered_field_dense_linear_order!: constr_dense_linear_order
+ "op <=" "op <"
+   "\<lambda> x y. 1/2 * ((x::'a::{ordered_field,recpower,number_ring}) + y)"
+proof (unfold_locales, dlo, dlo, auto)
+  fix x y::'a assume lt: "x < y"
+  from  less_half_sum[OF lt] show "x < (x + y) /2" by simp
+next
+  fix x y::'a assume lt: "x < y"
+  from  gt_half_sum[OF lt] show "(x + y) /2 < y" by simp
+qed
+
+declaration{*
+let
+fun earlier [] x y = false
+        | earlier (h::t) x y =
+    if h aconvc y then false else if h aconvc x then true else earlier t x y;
+
+fun dest_frac ct = case term_of ct of
+   Const (@{const_name "HOL.divide"},_) \$ a \$ b=>
+    Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
+ | t => Rat.rat_of_int (snd (HOLogic.dest_number t))
+
+fun mk_frac phi cT x =
+ let val (a, b) = Rat.quotient_of_rat x
+ in if b = 1 then Numeral.mk_cnumber cT a
+    else Thm.capply
+         (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
+                     (Numeral.mk_cnumber cT a))
+         (Numeral.mk_cnumber cT b)
+ end
+
+fun whatis x ct = case term_of ct of
+  Const(@{const_name "HOL.plus"}, _)\$(Const(@{const_name "HOL.times"},_)\$_\$y)\$_ =>
+     if y aconv term_of x then ("c*x+t",[(funpow 2 Thm.dest_arg1) ct, Thm.dest_arg ct])
+     else ("Nox",[])
+| Const(@{const_name "HOL.plus"}, _)\$y\$_ =>
+     if y aconv term_of x then ("x+t",[Thm.dest_arg ct])
+     else ("Nox",[])
+| Const(@{const_name "HOL.times"}, _)\$_\$y =>
+     if y aconv term_of x then ("c*x",[Thm.dest_arg1 ct])
+     else ("Nox",[])
+| t => if t aconv term_of x then ("x",[]) else ("Nox",[]);
+
+fun xnormalize_conv ctxt [] ct = reflexive ct
+| xnormalize_conv ctxt (vs as (x::_)) ct =
+   case term_of ct of
+   Const(@{const_name HOL.less},_)\$_\$Const(@{const_name "HOL.zero"},_) =>
+    (case whatis x (Thm.dest_arg1 ct) of
+    ("c*x+t",[c,t]) =>
+       let
+        val cr = dest_frac c
+        val clt = Thm.dest_fun2 ct
+        val cz = Thm.dest_arg ct
+        val neg = cr </ Rat.zero
+        val cthp = Simplifier.rewrite (local_simpset_of ctxt)
+               (Thm.capply @{cterm "Trueprop"}
+                  (if neg then Thm.capply (Thm.capply clt c) cz
+                    else Thm.capply (Thm.capply clt cz) c))
+        val cth = equal_elim (symmetric cthp) TrueI
+        val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x,t])
+             (if neg then @{thm neg_prod_sum_lt} else @{thm pos_prod_sum_lt})) cth
+        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
+                   (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
+      in rth end
+    | ("x+t",[t]) =>
+       let
+        val T = ctyp_of_term x
+        val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_lt"}
+        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
+              (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
+       in  rth end
+    | ("c*x",[c]) =>
+       let
+        val cr = dest_frac c
+        val clt = Thm.dest_fun2 ct
+        val cz = Thm.dest_arg ct
+        val neg = cr </ Rat.zero
+        val cthp = Simplifier.rewrite (local_simpset_of ctxt)
+               (Thm.capply @{cterm "Trueprop"}
+                  (if neg then Thm.capply (Thm.capply clt c) cz
+                    else Thm.capply (Thm.capply clt cz) c))
+        val cth = equal_elim (symmetric cthp) TrueI
+        val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x])
+             (if neg then @{thm neg_prod_lt} else @{thm pos_prod_lt})) cth
+        val rth = th
+      in rth end
+    | _ => reflexive ct)
+
+
+|  Const(@{const_name HOL.less_eq},_)\$_\$Const(@{const_name "HOL.zero"},_) =>
+   (case whatis x (Thm.dest_arg1 ct) of
+    ("c*x+t",[c,t]) =>
+       let
+        val T = ctyp_of_term x
+        val cr = dest_frac c
+        val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"}
+        val cz = Thm.dest_arg ct
+        val neg = cr </ Rat.zero
+        val cthp = Simplifier.rewrite (local_simpset_of ctxt)
+               (Thm.capply @{cterm "Trueprop"}
+                  (if neg then Thm.capply (Thm.capply clt c) cz
+                    else Thm.capply (Thm.capply clt cz) c))
+        val cth = equal_elim (symmetric cthp) TrueI
+        val th = implies_elim (instantiate' [SOME T] (map SOME [c,x,t])
+             (if neg then @{thm neg_prod_sum_le} else @{thm pos_prod_sum_le})) cth
+        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
+                   (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
+      in rth end
+    | ("x+t",[t]) =>
+       let
+        val T = ctyp_of_term x
+        val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_le"}
+        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
+              (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
+       in  rth end
+    | ("c*x",[c]) =>
+       let
+        val T = ctyp_of_term x
+        val cr = dest_frac c
+        val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"}
+        val cz = Thm.dest_arg ct
+        val neg = cr </ Rat.zero
+        val cthp = Simplifier.rewrite (local_simpset_of ctxt)
+               (Thm.capply @{cterm "Trueprop"}
+                  (if neg then Thm.capply (Thm.capply clt c) cz
+                    else Thm.capply (Thm.capply clt cz) c))
+        val cth = equal_elim (symmetric cthp) TrueI
+        val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x])
+             (if neg then @{thm neg_prod_le} else @{thm pos_prod_le})) cth
+        val rth = th
+      in rth end
+    | _ => reflexive ct)
+
+|  Const("op =",_)\$_\$Const(@{const_name "HOL.zero"},_) =>
+   (case whatis x (Thm.dest_arg1 ct) of
+    ("c*x+t",[c,t]) =>
+       let
+        val T = ctyp_of_term x
+        val cr = dest_frac c
+        val ceq = Thm.dest_fun2 ct
+        val cz = Thm.dest_arg ct
+        val cthp = Simplifier.rewrite (local_simpset_of ctxt)
+            (Thm.capply @{cterm "Trueprop"}
+             (Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply ceq c) cz)))
+        val cth = equal_elim (symmetric cthp) TrueI
+        val th = implies_elim
+                 (instantiate' [SOME T] (map SOME [c,x,t]) @{thm nz_prod_sum_eq}) cth
+        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
+                   (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
+      in rth end
+    | ("x+t",[t]) =>
+       let
+        val T = ctyp_of_term x
+        val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_eq"}
+        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
+              (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
+       in  rth end
+    | ("c*x",[c]) =>
+       let
+        val T = ctyp_of_term x
+        val cr = dest_frac c
+        val ceq = Thm.dest_fun2 ct
+        val cz = Thm.dest_arg ct
+        val cthp = Simplifier.rewrite (local_simpset_of ctxt)
+            (Thm.capply @{cterm "Trueprop"}
+             (Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply ceq c) cz)))
+        val cth = equal_elim (symmetric cthp) TrueI
+        val rth = implies_elim
+                 (instantiate' [SOME T] (map SOME [c,x]) @{thm nz_prod_eq}) cth
+      in rth end
+    | _ => reflexive ct);
+
+local
+  val less_iff_diff_less_0 = mk_meta_eq @{thm "less_iff_diff_less_0"}
+  val le_iff_diff_le_0 = mk_meta_eq @{thm "le_iff_diff_le_0"}
+  val eq_iff_diff_eq_0 = mk_meta_eq @{thm "eq_iff_diff_eq_0"}
+in
+fun field_isolate_conv phi ctxt vs ct = case term_of ct of
+  Const(@{const_name HOL.less},_)\$a\$b =>
+   let val (ca,cb) = Thm.dest_binop ct
+       val T = ctyp_of_term ca
+       val th = instantiate' [SOME T] [SOME ca, SOME cb] less_iff_diff_less_0
+       val nth = Conv.fconv_rule
+         (Conv.arg_conv (Conv.arg1_conv
+              (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
+       val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
+   in rth end
+| Const(@{const_name HOL.less_eq},_)\$a\$b =>
+   let val (ca,cb) = Thm.dest_binop ct
+       val T = ctyp_of_term ca
+       val th = instantiate' [SOME T] [SOME ca, SOME cb] le_iff_diff_le_0
+       val nth = Conv.fconv_rule
+         (Conv.arg_conv (Conv.arg1_conv
+              (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
+       val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
+   in rth end
+
+| Const("op =",_)\$a\$b =>
+   let val (ca,cb) = Thm.dest_binop ct
+       val T = ctyp_of_term ca
+       val th = instantiate' [SOME T] [SOME ca, SOME cb] eq_iff_diff_eq_0
+       val nth = Conv.fconv_rule
+         (Conv.arg_conv (Conv.arg1_conv
+              (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
+       val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
+   in rth end
+| @{term "Not"} \$(Const("op =",_)\$a\$b) => Conv.arg_conv (field_isolate_conv phi ctxt vs) ct
+| _ => reflexive ct
+end;
+
+fun classfield_whatis phi =
+ let
+  fun h x t =
+   case term_of t of
+     Const("op =", _)\$y\$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq
+                            else Ferrante_Rackoff_Data.Nox
+   | @{term "Not"}\$(Const("op =", _)\$y\$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq
+                            else Ferrante_Rackoff_Data.Nox
+   | Const(@{const_name HOL.less},_)\$y\$z =>
+       if term_of x aconv y then Ferrante_Rackoff_Data.Lt
+        else if term_of x aconv z then Ferrante_Rackoff_Data.Gt
+        else Ferrante_Rackoff_Data.Nox
+   | Const (@{const_name HOL.less_eq},_)\$y\$z =>
+         if term_of x aconv y then Ferrante_Rackoff_Data.Le
+         else if term_of x aconv z then Ferrante_Rackoff_Data.Ge
+         else Ferrante_Rackoff_Data.Nox
+   | _ => Ferrante_Rackoff_Data.Nox
+ in h end;
+fun class_field_ss phi =
+   HOL_basic_ss addsimps ([@{thm "linorder_not_less"}, @{thm "linorder_not_le"}])
+   addsplits [@{thm "abs_split"},@{thm "split_max"}, @{thm "split_min"}]
+
+in
+Ferrante_Rackoff_Data.funs @{thm "class_ordered_field_dense_linear_order.ferrack_axiom"}
+  {isolate_conv = field_isolate_conv, whatis = classfield_whatis, simpset = class_field_ss}
+end
+*}
+
+
+end ```
```--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Decision_Procs/Ferrack.thy	Fri Feb 06 15:15:46 2009 +0100
@@ -0,0 +1,2101 @@
+(*  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)))"
+
+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"
+
+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"
+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"
+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 (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)"
+(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)"
+
+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])
+
+
+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
+
+lemma dvdnumcoeff_aux2:
+  assumes "numgcd t > 1" shows "dvdnumcoeff t (numgcd t) \<and> numgcd t > 0"
+  using prems
+  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)"
+
+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))"
+
+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 (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))"
+
+lemma nummul_nz : "\<And> i. i\<noteq>0 \<Longrightarrow> nozerocoeff a \<Longrightarrow> nozerocoeff (nummul a i)"
+
+lemma numneg_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff (numneg a)"
+
+lemma numsub_nz: "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numsub a b)"
+
+lemma simpnum_nz: "nozerocoeff (simpnum t)"
+
+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"
+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
+
+    (* 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)"
+lemma disj_lin: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (disj p q)"
+
+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"]
+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"
+      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)"
+      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"
+    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)"
+
+  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
+    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)"
+ 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')"
+ 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)"
+  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)"
+  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
+      | 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```
```--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Decision_Procs/MIR.thy	Fri Feb 06 15:15:46 2009 +0100
@@ -0,0 +1,5933 @@
+(*  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))"
+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)"
+
+lemma rdvd_mult:
+  assumes knz: "k\<noteq>0"
+  shows "(real (n::int) * real (k::int) rdvd x * real k) = (real n rdvd x)"
+
+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)"
+
+lemma isint_Floor: "isint (Floor n) bs"
+
+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
+
+  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"
+
+
+    (* 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)))"</```