src/HOL/Integ/Barith.thy

+theory Barith = Presburger
+files ("barith.ML") :
+
+lemma imp_commute: "(PROP P ==> PROP Q
+==> PROP R) == (PROP Q ==>
+PROP P ==> PROP R)"
+proof
+  assume h1: "PROP P \<Longrightarrow> PROP Q \<Longrightarrow>
+PROP R"
+  assume h2: "PROP Q"
+  assume h3: "PROP P"
+  from h3 h2 show "PROP R" by (rule h1)
+next
+  assume h1: "PROP Q \<Longrightarrow> PROP P \<Longrightarrow>
+PROP R"
+ assume h2: "PROP P"
+  assume h3: "PROP Q"
+  from h3 h2 show "PROP R" by (rule h1)
+qed
+
+lemma imp_simplify: "(PROP P \<Longrightarrow> PROP P
+\<Longrightarrow> PROP Q) \<equiv> (PROP P \<Longrightarrow>
+PROP Q)"
+proof
+  assume h1: "PROP P \<Longrightarrow> PROP P \<Longrightarrow>
+PROP Q"
+  assume h2: "PROP P"
+  from h2 h2 show "PROP Q" by (rule h1)
+next
+  assume h: "PROP P \<Longrightarrow> PROP Q"
+  assume "PROP P"
+  then show "PROP Q" by (rule h)
+qed
+
+
+(* Abstraction of constants *)
+lemma abs_const: "(x::int) <= x \<and> x <= x"
+by simp
+
+(* Abstraction of Variables *)
+lemma abs_var: "l <= (x::int) \<and> x <= u \<Longrightarrow> l <= (x::int) \<and> x <= u"
+by simp
+
+
+(* Unary operators *)
+lemma abs_neg: "l <= (x::int) \<and> x <= u \<Longrightarrow>  -u <= -x \<and> -x <= -l"
+by arith
+
+
+(* Binary operations *)
+(* Addition*)
+lemma abs_add: "\<lbrakk> l1 <= (x1::int) \<and> x1 <= u1 ; l2 <= (x2::int) \<and> x2 <= u2\<rbrakk> 
+  \<Longrightarrow>  l1 + l2 <= x1 + x2 \<and> x1 + x2 <= u1 + u2"
+by arith
+
+lemma abs_sub: "\<lbrakk> l1 <= (x1::int) \<and> x1 <= u1 ; l2 <= (x2::int) \<and> x2 <= u2\<rbrakk> 
+  \<Longrightarrow>  l1 - u2 <= x1 - x2 \<and> x1 - x2 <= u1 - l2"
+by arith
+
+lemma abs_sub_x: "l <= (x::int) \<and> x <= u \<Longrightarrow> 0 <= x - x \<and> x - x <= 0"
+by arith
+
+(* For resolving the last step*)
+lemma subinterval: "\<lbrakk>l <= (e::int) \<and> e <= u ; l' <= l ; u <= u' \<rbrakk>
+  \<Longrightarrow> l' <= e \<and> e <= u'"
+by arith
+
+lemma min_max_minus : "min (-a ::int) (-b) = - max a b"
+by arith
+
+lemma max_min_minus : " max (-a ::int) (-b) = - min a b"
+by arith
+
+lemma max_max_commute : "max (max (a::int) b) (max c d) = max (max a c) (max b d)"
+by arith
+
+lemma min_min_commute : "min (min (a::int) b) (min c d) = min (min a c) (min b d)"
+by arith
+
+lemma zintervals_min: "\<lbrakk> l1 <= (x1::int) \<and> x1<= u1 ; l2 <= x2 \<and> x2 <= u2 \<rbrakk> 
+  \<Longrightarrow> min l1 l2 <= x1 \<and> x1 <= max u1 u2" by arith
+
+lemma zmult_zle_mono: "(i::int) \<le> j \<Longrightarrow> 0 \<le> k \<Longrightarrow> k * i \<le> k * j"
+  apply (erule order_le_less [THEN iffD1, THEN disjE, of "0::int"])
+  apply (erule order_le_less [THEN iffD1, THEN disjE])
+  apply (rule order_less_imp_le)
+  apply (rule zmult_zless_mono2)
+  apply simp_all
+  done
+  
+lemma zmult_mono:
+  assumes l1_pos : "0 <= l1"
+  and l2_pos : "0 <= l2"
+  and l1_le : "l1 <= (x1::int)"
+  and l2_le : "l2 <= (x2::int)"
+  shows "l1*l2 <= x1*x2"
+proof -
+  from l1_pos and l1_le have x1_pos: "0 \<le> x1" by (rule order_trans)
+  from l1_le and l2_pos
+  have "l2 * l1 \<le> l2 * x1" by (rule zmult_zle_mono)
+  then have "l1 * l2 \<le> x1 * l2" by (simp add: mult_ac)
+  also from l2_le and x1_pos
+  have "x1 * l2 \<le> x1 * x2" by (rule zmult_zle_mono)
+  finally show ?thesis .
+qed
+
+lemma zinterval_lposlpos:
+  assumes x1_lu : "l1 <= (x1::int) \<and> x1 <= u1"
+  and     x2_lu : "l2 <= (x2::int) \<and> x2 <= u2"
+  and     l1_pos : "0 <= l1"
+  and     l2_pos : "0 <= l2"
+  shows conc : "min (min (l1*l2) (l1*u2)) (min (u1*l2) (u1*u2)) <= x1 * x2 
+  \<and> x1 * x2 <= max (max (l1*l2) (l1*u2)) (max (u1*l2) (u1*u2))"
+proof-
+  from x1_lu have l1_le : "l1 <= x1" by simp
+  from x1_lu have x1_le : "x1 <= u1" by simp
+  from x2_lu have l2_le : "l2 <= x2" by simp
+  from x2_lu have x2_le : "x2 <= u2" by simp
+  from x1_lu have l1_leu : "l1 <= u1" by arith
+  from x2_lu have l2_leu : "l2 <= u2" by arith
+  from l1_pos l2_pos l1_le l2_le 
+  have l1l2_le : "l1*l2 <= x1*x2" by (simp add: zmult_mono)
+  from l1_pos x1_lu have x1_pos : "0 <= x1" by arith
+  from l2_pos x2_lu have x2_pos : "0 <= x2" by arith
+  from l1_pos x1_lu have u1_pos : "0 <= u1" by arith
+  from l2_pos x2_lu have u2_pos : "0 <= u2" by arith
+  from x1_pos x2_pos x1_le x2_le 
+  have x1x2_le : "x1*x2 <= u1*u2" by (simp add: zmult_mono)
+  from l2_leu l1_pos have l1l2_leu2 : "l1*l2 <= l1*u2" 
+    by (simp add: zmult_zle_mono)
+  from l1l2_leu2 have min1 : "l1*l2 = min (l1*l2) (l1*u2)" by arith
+  from l2_leu u1_pos have u1l2_le : "u1*l2 <=u1*u2" by (simp add: zmult_zle_mono)
+  from u1l2_le have min2 : "u1*l2 = min (u1*l2) (u1*u2)" by arith
+  from l1_leu l2_pos have "l2*l1 <= l2*u1" by (simp add:zmult_zle_mono) 
+  then have l1l2_le_u1l2 : "l1*l2 <= u1*l2" by (simp add: mult_ac)
+  from min1 min2 l1l2_le_u1l2 have  min_th : 
+    "min (min (l1*l2) (l1*u2)) (min (u1*l2) (u1*u2)) = (l1*l2)" by arith
+  from l1l2_leu2 have max1 : "l1*u2 = max (l1*l2) (l1*u2)" by arith
+  from u1l2_le have max2 : "u1*u2 = max (u1*l2) (u1*u2)" by arith
+  from l1_leu u2_pos have "u2*l1 <= u2*u1" by (simp add:zmult_zle_mono) 
+  then have l1u2_le_u1u2 : "l1*u2 <= u1*u2" by (simp add: mult_ac)
+  from max1 max2 l1u2_le_u1u2 have  max_th : 
+    "max (max (l1*l2) (l1*u2)) (max (u1*l2) (u1*u2)) = (u1*u2)" by arith
+  from min_th have min_th' :  "min (min (l1*l2) (l1*u2)) (min (u1*l2) (u1*u2)) <= l1*l2" by arith
+  from max_th have max_th' : "u1*u2 <= max (max (l1*l2) (l1*u2)) (max (u1*l2) (u1*u2))" by arith
+  from min_th' max_th' l1l2_le x1x2_le 
+  show ?thesis by simp
+qed
+
+lemma min_le_I1 : "min (a::int) b <= a" by arith
+lemma min_le_I2 : "min (a::int) b <= b" by arith
+lemma zinterval_lneglpos :
+  assumes  x1_lu : "l1 <= (x1::int) \<and> x1 <= u1"
+  and     x2_lu : "l2 <= (x2::int) \<and> x2 <= u2"
+  and     l1_neg : "l1 <= 0"
+  and x1_pos : "0 <= x1" 
+  and     l2_pos : "0 <= l2"
+  shows conc : "min (min (l1*l2) (l1*u2)) (min (u1*l2) (u1*u2)) <= x1 * x2 
+  \<and> x1 * x2 <= max (max (l1*l2) (l1*u2)) (max (u1*l2) (u1*u2))"
+
+proof-
+    from x1_lu x1_pos have x1_0_u1 : "0 <= x1 \<and> x1 <= u1" by simp
+    from l1_neg have ml1_pos : "0 <= -l1" by simp
+    from x1_lu x1_pos have u1_pos : "0 <= u1" by arith
+    from x2_lu l2_pos have u2_pos : "0 <= u2" by arith
+    from x2_lu have l2_le_u2 : "l2 <= u2" by arith
+    from l2_le_u2 u1_pos 
+     have u1l2_le_u1u2 : "u1*l2 <= u1*u2" by (simp add: zmult_zle_mono)
+    have trv_0 : "(0::int) <= 0" by simp
+    have "0*0 <= u1*l2" 
+      by (simp only: zmult_mono[OF trv_0 trv_0 u1_pos l2_pos])
+    then have u1l2_pos : "0 <= u1*l2" by simp
+      from l1_neg have ml1_pos : "0 <= -l1" by simp
+      from ml1_pos l2_pos have "0*0 <= (-l1)*l2" 
+	by (simp only: zmult_mono[OF trv_0 trv_0 ml1_pos l2_pos])
+      then have "0 <= -(l1*l2)" by simp  
+      then have "0 - (-(l1*l2)) <= 0" by arith 
+      then
+      have l1l2_neg : "l1*l2 <= 0" by simp
+      from ml1_pos u2_pos have "0*0 <= (-l1)*u2" 
+	by (simp only: zmult_mono[OF trv_0 trv_0 ml1_pos u2_pos])
+      then have "0 <= -(l1*u2)" by simp  
+      then have "0 - (-(l1*u2)) <= 0" by arith 
+      then
+      have l1u2_neg : "l1*u2 <= 0" by simp
+      from l1l2_neg u1l2_pos have l1l2_le_u1l2: "l1*l2 <= u1*l2" by simp
+      from l1u2_neg u1l2_pos have l1u2_le_u1l2 : "l1*u2 <= u1*l2" by simp
+      from ml1_pos l2_le_u2 have "(-l1)*l2 <= (-l1)*u2"
+	by (simp only: zmult_zle_mono) 
+      then have l1u2_le_l1l2 : "l1*u2 <= l1*l2" by simp
+      from l1u2_le_l1l2 l1l2_le_u1l2 u1l2_le_u1u2 
+      have min1 : "l1*u2 = min (min (l1*l2) (l1*u2)) (min (u1*l2) (u1*u2))" 
+	by arith
+      from u1l2_pos u1l2_le_u1u2 have "0 = min (min (0 * l2) (0 * u2)) (min (u1 * l2) (u1 * u2))" by arith
+      with l1u2_neg min1 have minth : "min (min (l1*l2) (l1*u2)) (min (u1*l2) (u1*u2)) <= min (min (0 * l2) (0 * u2)) (min (u1 * l2) (u1 * u2))" by simp
+      from l1u2_le_l1l2 l1l2_le_u1l2 u1l2_le_u1u2 
+      have max1 : "u1*u2 = max (max (l1*l2) (l1*u2)) (max (u1*l2) (u1*u2))" 
+	by arith
+      from u1l2_pos u1l2_le_u1u2 have "u1*u2 = max (max (0 * l2) (0 * u2)) (max (u1 * l2) (u1 * u2))" by arith 
+      with  max1 have "max (max (l1*l2) (l1*u2)) (max (u1*l2) (u1*u2)) = max (max (0 * l2) (0 * u2)) (max (u1 * l2) (u1 * u2))" by simp
+      then have maxth : " max (max (0 * l2) (0 * u2)) (max (u1 * l2) (u1 * u2)) <= max (max (l1*l2) (l1*u2)) (max (u1*l2) (u1*u2))" by simp
+    have x1x2_0_u : "min (min (0 * l2) (0 * u2)) (min (u1 * l2) (u1 * u2)) <= x1 * x2 &
+x1 * x2 <= max (max (0 * l2) (0 * u2)) (max (u1 * l2) (u1 * u2))" 
+      by (simp only: zinterval_lposlpos[OF x1_0_u1 x2_lu trv_0 l2_pos],simp)
+      from minth maxth x1x2_0_u show ?thesis by (simp add: subinterval[OF _ minth maxth])
+qed
+
+lemma zinterval_lneglneg :
+  assumes  x1_lu : "l1 <= (x1::int) \<and> x1 <= u1"
+  and     x2_lu : "l2 <= (x2::int) \<and> x2 <= u2"
+  and     l1_neg : "l1 <= 0"
+  and     x1_pos : "0 <= x1" 
+  and     l2_neg : "l2 <= 0"
+  and     x2_pos : "0 <= x2"
+  shows conc : "min (min (l1*l2) (l1*u2)) (min (u1*l2) (u1*u2)) <= x1 * x2 
+  \<and> x1 * x2 <= max (max (l1*l2) (l1*u2)) (max (u1*l2) (u1*u2))"
+
+proof-
+    from x1_lu x1_pos have x1_0_u1 : "0 <= x1 \<and> x1 <= u1" by simp
+    from l1_neg have ml1_pos : "0 <= -l1" by simp
+    from l1_neg have l1_le0 : "l1 <= 0" by simp
+    from x1_lu x1_pos have u1_pos : "0 <= u1" by arith
+    from x2_lu x2_pos have x2_0_u2 : "0 <= x2 \<and> x2 <= u2" by simp
+    from l2_neg have ml2_pos : "0 <= -l2" by simp
+    from l2_neg have l2_le0 : "l2 <= 0" by simp
+    from x2_lu x2_pos have u2_pos : "0 <= u2" by arith
+    have trv_0 : "(0::int) <= 0" by simp
+
+    have x1x2_th1 : 
+      "min (min (l1 * 0) (l1 * u2)) (min (u1 * 0) (u1 * u2)) \<le> x1 * x2 \<and>
+      x1 * x2 \<le> max (max (l1 * 0) (l1 * u2)) (max (u1 * 0) (u1 * u2))"
+      by (rule_tac  zinterval_lneglpos[OF x1_lu x2_0_u2 l1_le0 x1_pos trv_0])
+    
+    have x1x2_eq_x2x1 : "x1*x2 = x2*x1" by (simp add: mult_ac)
+    have
+      "min (min (l2 * 0) (l2 * u1)) (min (u2 * 0) (u2 * u1)) \<le> x2 * x1 \<and>
+      x2 * x1 \<le> max (max (l2 * 0) (l2 * u1)) (max (u2 * 0) (u2 * u1))"
+      by (rule_tac  zinterval_lneglpos[OF x2_lu x1_0_u1 l2_le0 x2_pos trv_0])
+    
+    then have x1x2_th2 : 
+      "min (min (l2 * 0) (l2 * u1)) (min (u2 * 0) (u2 * u1)) \<le> x1 * x2 \<and>
+      x1 * x2 \<le> max (max (l2 * 0) (l2 * u1)) (max (u2 * 0) (u2 * u1))"
+      by (simp add: x1x2_eq_x2x1)
+
+    from x1x2_th1 x1x2_th2 have x1x2_th3:
+      "min (min (min (l1 * 0) (l1 * u2)) (min (u1 * 0) (u1 * u2)))
+      (min (min (l2 * 0) (l2 * u1)) (min (u2 * 0) (u2 * u1)))
+      \<le> x1 * x2 \<and>
+      x1 * x2
+      \<le> max (max (max (l1 * 0) (l1 * u2)) (max (u1 * 0) (u1 * u2)))
+      (max (max (l2 * 0) (l2 * u1)) (max (u2 * 0) (u2 * u1)))"
+      by (rule_tac zintervals_min[OF x1x2_th1 x1x2_th2])
+
+    from ml1_pos u2_pos 
+    have "0*0 <= -l1*u2" 
+      by (simp only: zmult_mono[OF trv_0 trv_0 ml1_pos u2_pos]) 
+    then have l1u2_neg : "l1*u2 <= 0" by simp
+    from l1u2_neg have min_l1u2_0 : "min (0) (l1*u2) = l1*u2" by arith
+    from l1u2_neg have max_l1u2_0 : "max (0) (l1*u2) = 0" by arith
+    from u1_pos u2_pos 
+    have "0*0 <= u1*u2" 
+      by (simp only: zmult_mono[OF trv_0 trv_0 u1_pos u2_pos]) 
+    then have u1u2_pos :"0 <= u1*u2" by simp
+    from u1u2_pos have min_0_u1u2 : "min 0 (u1*u2) = 0" by arith
+    from u1u2_pos have max_0_u1u2 : "max 0 (u1*u2) = u1*u2" by arith
+    from ml2_pos u1_pos have "0*0 <= -l2*u1" 
+      by (simp only: zmult_mono[OF trv_0 trv_0 ml2_pos u1_pos]) 
+    then have l2u1_neg : "l2*u1 <= 0" by simp
+    from l2u1_neg have min_l2u1_0 : "min 0 (l2*u1) = l2*u1" by arith
+    from l2u1_neg have max_l2u1_0 : "max 0 (l2*u1) = 0" by arith
+    from min_l1u2_0 min_0_u1u2 min_l2u1_0 
+    have min_th1: 
+      " min (l2*u1) (l1*u2) <= min (min (min (l1 * 0) (l1 * u2)) (min (u1 * 0) (u1 * u2)))
+      (min (min (l2 * 0) (l2 * u1)) (min (u2 * 0) (u2 * u1)))"
+      by (simp add: min_commute mult_ac)
+    from max_l1u2_0 max_0_u1u2 max_l2u1_0 
+    have max_th1: "max (max (max (l1 * 0) (l1 * u2)) (max (u1 * 0) (u1 * u2)))
+      (max (max (l2 * 0) (l2 * u1)) (max (u2 * 0) (u2 * u1))) <= u1*u2"
+      by (simp add: max_commute mult_ac)
+    have x1x2_th4: "min (l2*u1) (l1*u2) <= x1*x2 \<and> x1*x2 <= u1*u2" 
+      by (rule_tac subinterval[OF x1x2_th3 min_th1 max_th1])
+    
+    have " min (min (l1*l2) (l1*u2)) (min (u1*l2) (u1*u2)) = min (min (l1*l2) (u1*u2)) (min (l1*u2) (l2*u1))" by (simp add: min_min_commute min_commute mult_ac) 
+    moreover 
+    have " min (min (l1*l2) (u1*u2)) (min (l1*u2) (l2*u1)) <= min (l1*u2) (l2*u1)" 
+      by 
+    (rule_tac min_le_I2 [of "(min (l1*l2) (u1*u2))" "(min (l1*u2) (l2*u1))"]) 
+    ultimately have "min (min (l1*l2) (l1*u2)) (min (u1*l2) (u1*u2)) <= min (l1*u2) (l2*u1)" by simp 
+    then 
+    have min_le1: "min (min (l1*l2) (l1*u2)) (min (u1*l2) (u1*u2)) <=min (l2*u1) (l1*u2)" 
+      by (simp add: min_commute mult_ac)
+    have "u1*u2 <= max (u1*l2) (u1*u2)" 
+      by (rule_tac le_maxI2[of  "u1*u2" "u1*l2"]) 
+    
+    moreover have "max (u1*l2) (u1*u2) <= max (max (l1*l2) (l1*u2)) (max (u1*l2) (u1*u2))"
+      by(rule_tac le_maxI2[of "(max (u1*l2) (u1*u2))" "(max (l1*l2) (l1*u2))"])
+    then 
+    have max_le1:"u1*u2 <= max (max (l1 * l2) (l1 * u2)) (max (u1 * l2) (u1 * u2))" 
+      by simp
+    show ?thesis by (simp add: subinterval[OF x1x2_th4 min_le1 max_le1])
+  qed
+
+lemma zinterval_lpos:
+  assumes x1_lu : "l1 <= (x1::int) \<and> x1 <= u1"
+  and     x2_lu : "l2 <= (x2::int) \<and> x2 <= u2"
+  and     l1_pos: "0 <= l1"
+  shows conc : "min (min (l1*l2) (l1*u2)) (min (u1*l2) (u1*u2)) <= x1 * x2 
+  \<and> x1 * x2 <= max (max (l1*l2) (l1*u2)) (max (u1*l2) (u1*u2))"
+proof-
+  from x1_lu have l1_le : "l1 <= x1" by simp
+  from x1_lu have x1_le : "x1 <= u1" by simp
+  from x2_lu have l2_le : "l2 <= x2" by simp
+  from x2_lu have x2_le : "x2 <= u2" by simp
+  from x1_lu have l1_leu : "l1 <= u1" by arith
+  from x2_lu have l2_leu : "l2 <= u2" by arith
+  have "(0 <= l2) \<or> (l2 < 0 \<and> 0<= x2) \<or> (x2 <0 \<and> 0 <= u2) \<or> (u2 <0)" by arith
+  moreover
+  {
+    assume l2_pos: "0 <= l2"
+    have ?thesis by (simp add: zinterval_lposlpos[OF x1_lu x2_lu l1_pos l2_pos])
+  }
+moreover
+{
+  assume  l2_neg : "l2 < 0" and x2_pos: "0<= x2"
+  from l2_neg have l2_le_0 : "l2 <= 0" by arith
+  thm zinterval_lneglpos[OF x2_lu x1_lu l2_le_0 x2_pos l1_pos]
+have th1 : 
+  "min (min (l2 * l1) (l2 * u1)) (min (u2 * l1) (u2 * u1)) \<le> x2 * x1 \<and>
+  x2 * x1 \<le> max (max (l2 * l1) (l2 * u1)) (max (u2 * l1) (u2 * u1))" 
+  by (simp add : zinterval_lneglpos[OF x2_lu x1_lu l2_le_0 x2_pos l1_pos])
+have mth1 : "min (min (l2 * l1) (l2 * u1)) (min (u2 * l1) (u2 * u1)) = min (min (l1 * l2) (l1 * u2)) (min (u1 * l2) (u1 * u2))" 
+  by (simp add: min_min_commute mult_ac)
+have mth2: "max (max (l2 * l1) (l2 * u1)) (max (u2 * l1) (u2 * u1)) = max (max (l1 * l2) (l1 * u2)) (max (u1 * l2) (u1 * u2))"
+  by (simp add: max_max_commute mult_ac)
+have x1x2_th : "x2*x1 = x1*x2" by (simp add: mult_ac)
+from th1 mth1 mth2 x1x2_th have 
+   "min (min (l1 * l2) (l1 * u2)) (min (u1 * l2) (u1 * u2)) \<le> x1 * x2 \<and>
+   x1 * x2 \<le> max (max (l1 * l2) (l1 * u2)) (max (u1 * l2) (u1 * u2))"
+by auto
+    then have ?thesis by simp 
+}
+moreover
+{
+  assume x2_neg : "x2 <0" and u2_pos : "0 <= u2"
+  from x2_lu x2_neg have mx2_mu2_ml2 : "-u2 <= -x2 \<and> -x2 <= -l2" by simp
+  from u2_pos have mu2_neg : "-u2 <= 0" by simp
+  from x2_neg have mx2_pos : "0 <= -x2" by simp
+thm zinterval_lneglpos[OF mx2_mu2_ml2 x1_lu mu2_neg mx2_pos l1_pos]
+    have mx1x2_lu : 
+"min (min (- u2 * l1) (- u2 * u1)) (min (- l2 * l1) (- l2 * u1))
+\<le> - x2 * x1 \<and>
+- x2 * x1 \<le> max (max (- u2 * l1) (- u2 * u1)) (max (- l2 * l1) (- l2 * u1))"      
+      by (simp only: zinterval_lneglpos [OF  mx2_mu2_ml2 x1_lu mu2_neg mx2_pos l1_pos],simp)
+    have min_eq_mmax : 
+      "min (min (- u2 * l1) (- u2 * u1)) (min (- l2 * l1) (- l2 * u1)) = 
+      - max (max (u2 * l1) (u2 * u1)) (max (l2 * l1) (l2 * u1))" 
+      by (simp add: min_max_minus max_min_minus)
+    have max_eq_mmin : 
+      " max (max (- u2 * l1) (- u2 * u1)) (max (- l2 * l1) (- l2 * u1)) = 
+      -min (min (u2 * l1) (u2 * u1)) (min (l2 * l1) (l2 * u1))"
+      by (simp add: min_max_minus max_min_minus)
+    from mx1x2_lu min_eq_mmax max_eq_mmin 
+    have "- max (max (u2 * l1) (u2 * u1)) (max (l2 * l1) (l2 * u1))<= - x1 * x2 &
+      - x1*x2 <=  -min (min (u2 * l1) (u2 * u1)) (min (l2 * l1) (l2 * u1))" by (simp add: mult_ac)
+ then have ?thesis by (simp add: min_min_commute min_commute max_commute max_max_commute mult_ac) 
+
+}
+moreover
+{
+  assume u2_neg : "u2 < 0"
+  from x2_lu have mx2_lu : "-u2 <= -x2 \<and> -x2 <= -l2" by arith
+  from u2_neg have mu2_pos : "0 <= -u2" by arith
+thm zinterval_lposlpos [OF x1_lu mx2_lu l1_pos mu2_pos]
+have "min (min (l1 * - u2) (l1 * - l2)) (min (u1 * - u2) (u1 * - l2))
+\<le> x1 * - x2 \<and>
+x1 * - x2 \<le> max (max (l1 * - u2) (l1 * - l2)) (max (u1 * - u2) (u1 * - l2))
+  " by (rule_tac zinterval_lposlpos [OF x1_lu mx2_lu l1_pos mu2_pos])
+then have mx1x2_lu:
+  "min (min (-l1 * u2) (- l1 * l2)) (min (- u1 * u2) (- u1 * l2)) \<le> - x1 * x2 \<and>
+- x1 * x2 \<le> max (max (- l1 * u2) (- l1 * l2)) (max (- u1 * u2) (- u1 * l2))
+  " by simp
+moreover have "min (min (-l1 * u2) (- l1 * l2)) (min (- u1 * u2) (- u1 * l2)) =- max (max (l1 * u2) ( l1 * l2)) (max ( u1 * u2) ( u1 * l2)) " 
+  by (simp add: min_max_minus max_min_minus)
+moreover 
+have 
+"max (max (- l1 * u2) (- l1 * l2)) (max (- u1 * u2) (- u1 * l2)) = - min (min (l1 * u2) (l1 * l2)) (min (u1 * u2) (u1 * l2))"
+ by (simp add: min_max_minus max_min_minus)
+thm subinterval[OF mx1x2_lu]
+ultimately have "- max (max (l1 * u2) ( l1 * l2)) (max ( u1 * u2) ( u1 * l2))\<le> - x1 * x2 \<and>
+- x1 * x2 \<le> - min (min (l1 * u2) (l1 * l2)) (min (u1 * u2) (u1 * l2)) " by simp
+ then have ?thesis by (simp add: max_commute min_commute)
+}
+ultimately show ?thesis by blast
+qed
+
+lemma zinterval_uneg :
+assumes x1_lu : "l1 <= (x1::int) \<and> x1 <= u1"
+  and     x2_lu : "l2 <= (x2::int) \<and> x2 <= u2"
+  and     u1_neg: "u1 <= 0"
+  shows conc : "min (min (l1*l2) (l1*u2)) (min (u1*l2) (u1*u2)) <= x1 * x2 
+  \<and> x1 * x2 <= max (max (l1*l2) (l1*u2)) (max (u1*l2) (u1*u2))"
+proof-
+  from x1_lu  have mx1_lu : "-u1 <= -x1 \<and> -x1 <= -l1" by arith
+  from u1_neg have mu1_pos : "0 <= -u1" by arith
+  thm zinterval_lpos [OF mx1_lu x2_lu mu1_pos]
+  have mx1x2_lu : 
+    "min (min (- u1 * l2) (- u1 * u2)) (min (- l1 * l2) (- l1 * u2))
+    \<le> - x1 * x2 \<and> - x1 * x2 \<le> 
+    max (max (- u1 * l2) (- u1 * u2)) (max (- l1 * l2) (- l1 * u2))"
+by (rule_tac zinterval_lpos [OF mx1_lu x2_lu mu1_pos])
+moreover have 
+  "min (min (- u1 * l2) (- u1 * u2)) (min (- l1 * l2) (- l1 * u2)) = - max (max (u1 * l2) (u1 * u2)) (max (l1 * l2) (l1 * u2))" by (simp add: min_max_minus max_min_minus)
+moreover have 
+  "max (max (- u1 * l2) (- u1 * u2)) (max (- l1 * l2) (- l1 * u2)) = - min (min (u1 * l2) ( u1 * u2)) (min (l1 * l2) (l1 * u2))" by (simp add: min_max_minus max_min_minus)
+ultimately have "- max (max (u1 * l2) (u1 * u2)) (max (l1 * l2) (l1 * u2))\<le> - x1 * x2 \<and> - x1 * x2 \<le>  - min (min (u1 * l2) ( u1 * u2)) (min (l1 * l2) (l1 * u2))" by simp
+then show ?thesis by (simp add: min_commute max_commute mult_ac)
+qed
+
+lemma zinterval_lnegxpos:
+assumes x1_lu : "l1 <= (x1::int) \<and> x1 <= u1"
+  and     x2_lu : "l2 <= (x2::int) \<and> x2 <= u2"
+  and     l1_neg: "l1 <= 0"
+  and     x1_pos: "0<= x1"
+  shows conc : "min (min (l1*l2) (l1*u2)) (min (u1*l2) (u1*u2)) <= x1 * x2 
+  \<and> x1 * x2 <= max (max (l1*l2) (l1*u2)) (max (u1*l2) (u1*u2))"
+proof-
+  have "(0 <= l2) \<or> (l2 < 0 \<and> 0<= x2) \<or> (x2 <0 \<and> 0 <= u2) \<or> (u2 <= 0)" by arith
+  moreover
+  {
+    assume l2_pos: "0 <= l2"
+    thm zinterval_lpos [OF x2_lu x1_lu l2_pos]
+    have 
+      "min (min (l2 * l1) (l2 * u1)) (min (u2 * l1) (u2 * u1)) \<le> x2 * x1 \<and>
+      x2 * x1 \<le> max (max (l2 * l1) (l2 * u1)) (max (u2 * l1) (u2 * u1))"
+      by (rule_tac zinterval_lpos [OF x2_lu x1_lu l2_pos])
+ moreover have "min (min (l2 * l1) (l2 * u1)) (min (u2 * l1) (u2 * u1)) = min (min (l1*l2) (l1*u2)) (min (u1*l2) (u1*u2))" by (simp add: mult_ac min_commute min_min_commute)
+moreover have "max (max (l2 * l1) (l2 * u1)) (max (u2 * l1) (u2 * u1)) = max (max (l1*l2) (l1*u2)) (max (u1*l2) (u1*u2))"
+  by (simp add: mult_ac max_commute max_max_commute)
+ultimately have ?thesis by (simp add: mult_ac)
+
+}
+
+moreover
+{
+  assume l2_neg: "l2 < 0" and x2_pos: " 0<= x2"
+  from l1_neg have l1_le0 : "l1 <= 0" by simp
+  from l2_neg have l2_le0 : "l2 <= 0" by simp
+ have ?thesis by (simp add: zinterval_lneglneg [OF x1_lu x2_lu l1_le0 x1_pos l2_le0 x2_pos])
+}
+
+moreover
+{
+ assume x2_neg: "x2 <0" and u2_pos: "0 <= u2"
+ from x2_lu have mx2_lu: "-u2 <= -x2 \<and> -x2 <= -l2" by arith
+ from x2_neg have mx2_pos: "0 <= -x2" by simp
+ from u2_pos have mu2_neg: "-u2 <= 0" by simp
+ from l1_neg have l1_le0 : "l1 <= 0" by simp
+thm zinterval_lneglneg [OF x1_lu mx2_lu l1_le0 x1_pos mu2_neg mx2_pos]
+have "min (min (l1 * - u2) (l1 * - l2)) (min (u1 * - u2) (u1 * - l2))
+\<le> x1 * - x2 \<and>
+x1 * - x2 \<le> max (max (l1 * - u2) (l1 * - l2)) (max (u1 * - u2) (u1 * - l2))" by (rule_tac zinterval_lneglneg [OF x1_lu mx2_lu l1_le0 x1_pos mu2_neg mx2_pos])
+then have "min (min (- l1 * u2) (- l1 * l2)) (min (- u1 * u2) (- u1 * l2))
+\<le> - x1 * x2 \<and>
+- x1 * x2 \<le> max (max (- l1 * u2) (- l1 * l2)) (max (- u1 * u2) (- u1 * l2))" by simp
+moreover have "min (min (- l1 * u2) (- l1 * l2)) (min (- u1 * u2) (- u1 * l2)) = - max (max (l1 * u2) (l1 * l2)) (max (u1 * u2) (u1 * l2))" by (simp add: min_max_minus max_min_minus)
+moreover have "max (max (- l1 * u2) (- l1 * l2)) (max (- u1 * u2) (- u1 * l2)) = - min (min (l1 * u2) (l1 * l2)) (min (u1 * u2) (u1 * l2))" by (simp add: min_max_minus max_min_minus)
+ultimately have "- max (max (l1 * u2) (l1 * l2)) (max (u1 * u2) (u1 * l2))\<le> - x1 * x2 \<and>
+- x1 * x2 \<le>  - min (min (l1 * u2) (l1 * l2)) (min (u1 * u2) (u1 * l2))" by simp
+
+then have ?thesis by (simp add: min_commute max_commute mult_ac) 
+}
+
+moreover
+{
+ assume u2_neg: "u2 <= 0"
+ thm zinterval_uneg[OF x2_lu x1_lu u2_neg]
+have "min (min (l2 * l1) (l2 * u1)) (min (u2 * l1) (u2 * u1)) \<le> x2 * x1 \<and>
+x2 * x1 \<le> max (max (l2 * l1) (l2 * u1)) (max (u2 * l1) (u2 * u1))" by (rule_tac zinterval_uneg[OF x2_lu x1_lu u2_neg])
+then have ?thesis by (simp add: mult_ac min_commute max_commute min_min_commute max_max_commute)
+}
+ultimately show ?thesis by blast
+
+qed
+
+lemma zinterval_xnegupos: 
+assumes x1_lu : "l1 <= (x1::int) \<and> x1 <= u1"
+  and     x2_lu : "l2 <= (x2::int) \<and> x2 <= u2"
+  and     x1_neg: "x1 <= 0"
+  and     u1_pos: "0<= u1"
+  shows conc : "min (min (l1*l2) (l1*u2)) (min (u1*l2) (u1*u2)) <= x1 * x2 
+  \<and> x1 * x2 <= max (max (l1*l2) (l1*u2)) (max (u1*l2) (u1*u2))"
+proof-
+  from x1_lu have mx1_lu : "-u1 <= -x1 \<and> -x1 <= -l1" by arith 
+  from u1_pos have mu1_neg : "-u1 <= 0" by simp
+  from x1_neg have mx1_pos : "0 <= -x1" by simp
+  thm zinterval_lnegxpos[OF mx1_lu x2_lu mu1_neg mx1_pos ]
+  have "min (min (- u1 * l2) (- u1 * u2)) (min (- l1 * l2) (- l1 * u2))
+\<le> - x1 * x2 \<and>
+- x1 * x2 \<le> max (max (- u1 * l2) (- u1 * u2)) (max (- l1 * l2) (- l1 * u2))"
+    by (rule_tac zinterval_lnegxpos[OF mx1_lu x2_lu mu1_neg mx1_pos ])
+  moreover have 
+    "min (min (- u1 * l2) (- u1 * u2)) (min (- l1 * l2) (- l1 * u2)) = - max (max (u1 * l2) (u1 * u2)) (max (l1 * l2) (l1 * u2))" 
+    by (simp add: min_max_minus max_min_minus)
+  moreover have 
+    "max (max (- u1 * l2) (- u1 * u2)) (max (- l1 * l2) (- l1 * u2)) = - min (min (u1 * l2) (u1 * u2)) (min (l1 * l2) (l1 * u2))" 
+    by (simp add: min_max_minus max_min_minus)
+  ultimately have "- max (max (u1 * l2) (u1 * u2)) (max (l1 * l2) (l1 * u2))\<le> - x1 * x2 \<and>
+- x1 * x2 \<le> - min (min (u1 * l2) (u1 * u2)) (min (l1 * l2) (l1 * u2))" 
+    by simp
+then show ?thesis by (simp add: mult_ac min_commute max_commute)
+qed
+
+lemma abs_mul: 
+  assumes x1_lu : "l1 <= (x1::int) \<and> x1 <= u1"
+  and     x2_lu : "l2 <= (x2::int) \<and> x2 <= u2"
+  shows conc : "min (min (l1*l2) (l1*u2)) (min (u1*l2) (u1*u2)) <= x1 * x2 
+  \<and> x1 * x2 <= max (max (l1*l2) (l1*u2)) (max (u1*l2) (u1*u2))"
+proof-
+  have "(0 <= l1) \<or> (l1 <= 0 \<and> 0<= x1) \<or> (x1 <=0 \<and> 0 <= u1) \<or> (u1 <= 0)" 
+    by arith
+  moreover
+  {
+    assume l1_pos: "0 <= l1"
+    have ?thesis by (rule_tac zinterval_lpos [OF x1_lu x2_lu l1_pos])
+  }
+  
+  moreover
+  {
+    assume l1_neg :"l1 <= 0" and x1_pos: "0<= x1"
+    have ?thesis by (rule_tac zinterval_lnegxpos[OF x1_lu x2_lu l1_neg x1_pos])
+  }
+  
+  moreover
+  {
+    assume x1_neg : "x1 <= 0" and u1_pos: "0 <= u1"
+    have ?thesis by (rule_tac zinterval_xnegupos [OF x1_lu x2_lu x1_neg u1_pos])
+  }
+  
+  moreover
+  {
+    assume u1_neg: "u1 <= 0"
+    have ?thesis by (rule_tac zinterval_uneg [OF x1_lu x2_lu u1_neg])
+  }
+  
+  ultimately show ?thesis by blast
+qed
+
+lemma mult_x_mono_lpos : 
+assumes l_pos : "0 <= (l::int)"
+  and   x_lu : "l <= (x::int) \<and> x <= u"
+  shows "l*l <= x*x \<and> x*x <= u*u"
+
+proof-
+  from x_lu have x_l : "l <= x" by arith
+  thm zmult_mono[OF l_pos l_pos x_l x_l]
+  then have xx_l : "l*l <= x*x"
+    by (simp add: zmult_mono[OF l_pos l_pos x_l x_l])
+  moreover 
+  from x_lu have x_u : "x <= u" by arith
+  from l_pos x_l have x_pos : "0 <= x" by arith
+  thm zmult_mono[OF x_pos x_pos x_u x_u]
+  then have xx_u : "x*x <= u*u"
+    by (simp add: zmult_mono[OF x_pos x_pos x_u x_u])
+ultimately show ?thesis by simp
+qed
+
+lemma mult_x_mono_luneg : 
+assumes l_neg : "(l::int) <= 0"
+  and   u_neg : "u <= 0"
+  and   x_lu : "l <= (x::int) \<and> x <= u"
+  shows "u*u <= x*x \<and> x*x <= l*l"
+
+proof-
+  from x_lu have mx_lu : "-u <= -x \<and> -x <= -l" by arith
+  from u_neg have mu_pos : "0<= -u" by simp
+  thm mult_x_mono_lpos[OF mu_pos mx_lu]
+  have "- u * - u \<le> - x * - x \<and> - x * - x \<le> - l * - l"
+    by (rule_tac mult_x_mono_lpos[OF mu_pos mx_lu])
+  then show ?thesis by simp
+qed
+
+lemma mult_x_mono_lxnegupos : 
+assumes l_neg : "(l::int) <= 0"
+  and   u_pos : "0 <= u"
+  and   x_neg : "x <= 0"
+  and   x_lu : "l <= (x::int) \<and> x <= u"
+  shows "0 <= x*x \<and> x*x <= max (l*l) (u*u)"
+proof-
+  from x_lu x_neg have mx_0l : "0 <= - x \<and> - x <= - l" by arith
+  have trv_0 : "(0::int) <= 0" by arith
+  thm mult_x_mono_lpos[OF trv_0 mx_0l]
+  have "0 * 0 \<le> - x * - x \<and> - x * - x \<le> - l * - l"
+    by (rule_tac  mult_x_mono_lpos[OF trv_0 mx_0l])
+  then have xx_0ll : "0 <= x*x \<and> x*x <= l*l" by simp
+  have "l*l <= max (l*l) (u*u)" by (simp add: le_maxI1)
+  with xx_0ll show ?thesis by arith
+qed
+
+lemma mult_x_mono_lnegupos : 
+assumes l_neg : "(l::int) <= 0"
+  and   u_pos : "0 <= u"
+  and   x_lu : "l <= (x::int) \<and> x <= u"
+  shows "0 <= x*x \<and> x*x <= max (l*l) (u*u)"
+proof-
+  have "0<= x \<or> x <= 0" by arith
+moreover
+{
+  assume x_neg : "x <= 0"
+  thm mult_x_mono_lxnegupos[OF l_neg u_pos x_neg x_lu]
+  have ?thesis by (rule_tac mult_x_mono_lxnegupos[OF l_neg u_pos x_neg x_lu])
+}
+moreover
+
+{
+  assume x_pos : "0 <= x"
+  from x_lu have mx_lu : "-u <= -x \<and> -x <= -l" by arith
+  from x_pos have mx_neg : "-x <= 0" by simp
+  from u_pos have mu_neg : "-u <= 0" by simp
+  from x_lu x_pos have ml_pos : "0 <= -l" by simp
+  thm mult_x_mono_lxnegupos[OF mu_neg ml_pos mx_neg mx_lu]
+  have "0 \<le> - x * - x \<and> - x * - x \<le> max (- u * - u) (- l * - l)"
+    by (rule_tac mult_x_mono_lxnegupos[OF mu_neg ml_pos mx_neg mx_lu])
+  then have ?thesis by (simp add: max_commute)
+
+}
+ultimately show ?thesis by blast
+
+qed
+lemma abs_mul_x:
+  assumes x_lu : "l <= (x::int) \<and> x <= u"
+  shows 
+  "(if 0 <= l then l*l  else if u <= 0 then u*u else 0) <= x*x
+  \<and> x*x <= (if 0 <= l then u*u  else if u <= 0 then l*l else (max (l*l) (u*u)))"
+proof-
+  have "(0 <= l) \<or> (l < 0 \<and> u <= 0) \<or> (l < 0 \<and> 0 < u)" by arith 
+  
+  moreover
+  {
+    assume l_pos : "0 <= l"
+    from l_pos have "(if 0 <= l then l*l  else if u <= 0 then u*u else 0) = l*l"
+      by simp
+    moreover from l_pos have "(if 0 <= l then u*u  else if u <= 0 then l*l else (max (l*l) (u*u))) = u*u" by simp
+    
+    moreover have "l * l \<le> x * x \<and> x * x \<le> u * u" 
+      by (rule_tac  mult_x_mono_lpos[OF l_pos x_lu])
+    ultimately have ?thesis by simp 
+  }
+  
+  moreover
+  {
+    assume l_neg : "l < 0"  and u_neg : "u <= 0"  
+    from l_neg have l_le0 : "l <= 0" by simp
+    from l_neg u_neg have "(if 0 <= l then l*l  else if u <= 0 then u*u else 0) = u*u"
+      by simp
+    moreover 
+    from l_neg u_neg have "(if 0 <= l then u*u  else if u <= 0 then l*l else (max (l*l) (u*u))) = l*l" by simp
+    moreover 
+    have "u * u \<le> x * x \<and> x * x \<le> l * l" 
+      by (rule_tac mult_x_mono_luneg[OF l_le0 u_neg x_lu])
+    
+    ultimately have ?thesis by simp 
+  }
+  moreover
+  {
+    assume l_neg : "l < 0" and u_pos: "0 < u"
+    from l_neg have l_le0 : "l <= 0" by simp
+    from u_pos have u_ge0 : "0 <= u" by simp
+    from l_neg u_pos have "(if 0 <= l then l*l  else if u <= 0 then u*u else 0) = 0"
+      by simp
+    moreover from l_neg u_pos have "(if 0 <= l then u*u  else if u <= 0 then l*l else (max (l*l) (u*u))) = max (l*l) (u*u)" by simp
+    moreover have "0 \<le> x * x \<and> x * x \<le> max (l * l) (u * u)" 
+      by (rule_tac mult_x_mono_lnegupos[OF l_le0 u_ge0 x_lu])
+    
+    ultimately have ?thesis by simp 
+    
+  }
+  
+  ultimately show ?thesis by blast
+qed
+
+
+use"barith.ML"
+setup Barith.setup
+
+end
+