--- a/src/HOL/Integ/Barith.thy Fri Oct 29 15:16:31 2004 +0200
+++ b/src/HOL/Integ/Barith.thy Tue Nov 02 16:33:08 2004 +0100
@@ -1,32 +1,29 @@
-(* Title: HOL/Integ/Barith.thy
- ID: $Id$
- Author: Amine Chaieb, TU Muenchen
-
-Simple decision procedure for bounded arithmetic
-*)
+theory Barith = Presburger
+files ("barith.ML") :
-theory Barith
-imports Presburger
-files ("barith.ML")
-begin
-
-lemma imp_commute: "(PROP P \<Longrightarrow> PROP Q \<Longrightarrow> PROP R) \<equiv>
- (PROP Q \<Longrightarrow> PROP P \<Longrightarrow> PROP R)"
+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 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 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)"
+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 h1: "PROP P \<Longrightarrow> PROP P \<Longrightarrow>
+PROP Q"
assume h2: "PROP P"
from h2 h2 show "PROP Q" by (rule h1)
next
@@ -35,50 +32,78 @@
then show "PROP Q" by (rule h)
qed
+(* Simple lemmas needed for simplification before the procedure runs*)
+lemma zero_eq_Numeral0_nring: "(0::'a::number_ring) = Numeral0"
+ by simp
+
+lemma one_eq_Numeral1_nring: "(1::'a::number_ring) = Numeral1"
+ by simp
+
+lemma zero_eq_Numeral0_nat: "(0::nat) = Numeral0"
+ by simp
+
+lemma one_eq_Numeral1_nat: "(1::nat) = Numeral1"
+ by simp
+
+lemma z_less_imp_le1 : "(a::int) < b \<Longrightarrow> a +1 <= b"
+by simp
+
+lemma z_eq_imp_le_conj: "(a::int) = b \<Longrightarrow> a <= b \<and> b <= a"
+by simp
+
+lemma zpower_Pls: "(z::int)^Numeral0 = 1"
+ by simp
+
+lemma zpower_Min: "(z::int)^((-1)::nat) = 1"
+proof -
+ have 1:"((-1)::nat) = 0"
+ by simp
+ show ?thesis by (simp add: 1)
+qed
+
(* Abstraction of constants *)
lemma abs_const: "(x::int) <= x \<and> x <= x"
- by simp
+by simp
(* Abstraction of Variables *)
lemma abs_var: "l <= (x::int) \<and> x <= u \<Longrightarrow> l <= (x::int) \<and> x <= u"
- by simp
-
+by simp
(* Unary operators *)
lemma abs_neg: "l <= (x::int) \<and> x <= u \<Longrightarrow> -u <= -x \<and> -x <= -l"
- by arith
+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
+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
+by arith
lemma abs_sub_x: "l <= (x::int) \<and> x <= u \<Longrightarrow> 0 <= x - x \<and> x - x <= 0"
- by arith
+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
+by arith
lemma min_max_minus : "min (-a ::int) (-b) = - max a b"
- by arith
+by arith
lemma max_min_minus : " max (-a ::int) (-b) = - min a b"
- by arith
+by arith
lemma max_max_commute : "max (max (a::int) b) (max c d) = max (max a c) (max b d)"
- by arith
+by arith
lemma min_min_commute : "min (min (a::int) b) (min c d) = min (min a c) (min b d)"
- by arith
+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
@@ -112,9 +137,9 @@
and x2_lu : "l2 <= (x2::int) \<and> x2 <= u2"
and l1_pos : "0 <= l1"
and l2_pos : "0 <= l2"
- shows "min (min (l1*l2) (l1*u2)) (min (u1*l2) (u1*u2)) <= x1 * 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 -
+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
@@ -151,66 +176,61 @@
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 "min (min (l1*l2) (l1*u2)) (min (u1*l2) (u1*u2)) <= x1 * 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 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 (rule zmult_zle_mono)
- have trv_0: "(0::int) <= 0" by simp
- from trv_0 trv_0 u1_pos l2_pos
- have "0*0 <= u1*l2" by (rule zmult_mono)
- then have u1l2_pos: "0 <= u1*l2" by simp
- from l1_neg have ml1_pos: "0 <= -l1" by simp
- from trv_0 trv_0 ml1_pos l2_pos have "0*0 <= (-l1)*l2"
- by (rule zmult_mono)
- 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 trv_0 trv_0 ml1_pos u2_pos have "0*0 <= (-l1)*u2"
- by (rule zmult_mono)
- 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
- from x1_0_u1 x2_lu trv_0 l2_pos
- 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 (rule zinterval_lposlpos)
- thus ?thesis using minth maxth by (rule subinterval)
+
+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 :
@@ -220,100 +240,101 @@
and x1_pos : "0 <= x1"
and l2_neg : "l2 <= 0"
and x2_pos : "0 <= x2"
- shows "min (min (l1*l2) (l1*u2)) (min (u1*l2) (u1*u2)) <= x1 * 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
- from x1_lu x2_0_u2 l1_le0 x1_pos trv_0
- 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 zinterval_lneglpos)
+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)
- from x2_lu x1_0_u1 l2_le0 x2_pos trv_0
- 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 zinterval_lneglpos)
+ 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)
+ 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 zintervals_min)
+ 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 trv_0 trv_0 ml1_pos u2_pos
- have "0*0 <= -l1*u2" by (rule zmult_mono)
- 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 trv_0 trv_0 u1_pos u2_pos
- have "0*0 <= u1*u2" by (rule zmult_mono)
- 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 trv_0 trv_0 ml2_pos u1_pos have "0*0 <= -l2*u1"
- by (rule zmult_mono)
- 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)
- from x1x2_th3 min_th1 max_th1
- have x1x2_th4: "min (l2*u1) (l1*u2) <= x1*x2 \<and> x1*x2 <= u1*u2"
- by (rule subinterval)
+ 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 min_le_I2)
- 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 le_maxI2)
+ 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 le_maxI2)
- then have max_le1: "u1*u2 <= max (max (l1 * l2) (l1 * u2)) (max (u1 * l2) (u1 * u2))"
- by simp
- with x1x2_th4 min_le1 show ?thesis by (rule subinterval)
-qed
+ 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 "min (min (l1*l2) (l1*u2)) (min (u1*l2) (u1*u2)) <= x1 * 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 -
+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
@@ -321,168 +342,172 @@
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
- thus ?thesis
- proof (elim disjE conjE)
+ moreover
+ {
assume l2_pos: "0 <= l2"
- with x1_lu x2_lu l1_pos show ?thesis by (rule zinterval_lposlpos)
- next
- assume l2_neg: "l2 < 0" and x2_pos: "0<= x2"
- from l2_neg have l2_le_0 : "l2 <= 0" by arith
- from 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 (rule zinterval_lneglpos)
- 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
- thus ?thesis by simp
- next
- 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
- from 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 (rule zinterval_lneglpos)
- have min_eq_mmax:
+ 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:
+ 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)
- thus ?thesis by (simp add: min_min_commute min_commute max_commute max_max_commute mult_ac)
- next
- 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
- from 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 zinterval_lposlpos)
- 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)
- 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
- thus ?thesis by (simp add: max_commute min_commute)
- qed
+ 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 "min (min (l1*l2) (l1*u2)) (min (u1*l2) (u1*u2)) <= x1 * 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 -
+proof-
from x1_lu have mx1_lu : "-u1 <= -x1 \<and> -x1 <= -l1" by arith
from u1_neg have mu1_pos : "0 <= -u1" by arith
- with mx1_lu x2_lu have mx1x2_lu :
+ 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 zinterval_lpos)
- 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)
+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"
+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 "min (min (l1*l2) (l1*u2)) (min (u1*l2) (u1*u2)) <= x1 * 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 -
+proof-
have "(0 <= l2) \<or> (l2 < 0 \<and> 0<= x2) \<or> (x2 <0 \<and> 0 <= u2) \<or> (u2 <= 0)" by arith
- thus ?thesis
- proof (elim disjE conjE)
+ moreover
+ {
assume l2_pos: "0 <= l2"
- with x2_lu x1_lu have
+ 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 zinterval_lpos)
- 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 show ?thesis by (simp add: mult_ac)
- next
- 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
- from x1_lu x2_lu l1_le0 x1_pos l2_le0 x2_pos
- show ?thesis by (rule zinterval_lneglneg)
- next
- 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
- from 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 zinterval_lneglneg)
- 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
- thus ?thesis by (simp add: min_commute max_commute mult_ac)
- next
- assume u2_neg: "u2 <= 0"
- with x2_lu x1_lu
- 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 zinterval_uneg)
- thus ?thesis by (simp add: mult_ac min_commute max_commute min_min_commute max_max_commute)
- qed
+ 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:
@@ -490,167 +515,200 @@
and x2_lu : "l2 <= (x2::int) \<and> x2 <= u2"
and x1_neg: "x1 <= 0"
and u1_pos: "0<= u1"
- shows "min (min (l1*l2) (l1*u2)) (min (u1*l2) (u1*u2)) <= x1 * 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 -
+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
- with mx1_lu x2_lu mu1_neg
+ 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 zinterval_lnegxpos)
- moreover have "min (min (- u1 * l2) (- u1 * u2)) (min (- l1 * l2) (- l1 * u2)) =
- - max (max (u1 * l2) (u1 * u2)) (max (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))"
+ 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))"
+ 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)
+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 "min (min (l1*l2) (l1*u2)) (min (u1*l2) (u1*u2)) <= x1 * x2
+ 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 -
+proof-
have "(0 <= l1) \<or> (l1 <= 0 \<and> 0<= x1) \<or> (x1 <=0 \<and> 0 <= u1) \<or> (u1 <= 0)"
by arith
- thus ?thesis
- proof (elim disjE conjE)
+ moreover
+ {
assume l1_pos: "0 <= l1"
- with x1_lu x2_lu show ?thesis by (rule zinterval_lpos)
- next
- assume l1_neg: "l1 <= 0" and x1_pos: "0 <= x1"
- with x1_lu x2_lu show ?thesis by (rule zinterval_lnegxpos)
- next
- assume x1_neg: "x1 <= 0" and u1_pos: "0 <= u1"
- from x1_lu x2_lu x1_neg u1_pos show ?thesis by (rule zinterval_xnegupos)
- next
+ 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"
- with x1_lu x2_lu show ?thesis by (rule zinterval_uneg)
- qed
+ 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"
+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 -
+
+proof-
from x_lu have x_l : "l <= x" by arith
- from l_pos l_pos x_l x_l have xx_l : "l*l <= x*x"
- by (rule zmult_mono)
+ 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
- from x_pos x_pos x_u x_u have xx_u : "x*x <= u*u"
- by (rule zmult_mono)
- ultimately show ?thesis by simp
+ 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"
+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 u_neg have "0<= -u" by simp
- moreover from x_lu have "-u <= -x \<and> -x <= -l" by arith
- ultimately have "- u * - u \<le> - x * - x \<and> - x * - x \<le> - l * - l"
- by (rule mult_x_mono_lpos)
+
+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"
+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 -
- have "(0::int) <= 0" by arith
- moreover from x_lu x_neg have "0 <= - x \<and> - x <= - l" by arith
- ultimately have "0 * 0 \<le> - x * - x \<and> - x * - x \<le> - l * - l"
- by (rule mult_x_mono_lpos)
+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"
+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
- thus ?thesis
- proof
- assume x_neg: "x <= 0"
- from l_neg u_pos x_neg x_lu show ?thesis by (rule mult_x_mono_lxnegupos)
- next
- 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
- from mu_neg ml_pos mx_neg mx_lu
- have "0 \<le> - x * - x \<and> - x * - x \<le> max (- u * - u) (- l * - l)"
- by (rule mult_x_mono_lxnegupos)
- thus ?thesis by (simp add: max_commute)
- qed
+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
+ 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 -
+proof-
have "(0 <= l) \<or> (l < 0 \<and> u <= 0) \<or> (l < 0 \<and> 0 < u)" by arith
- thus ?thesis
- proof (elim disjE conjE)
- assume l_pos: "0 <= l"
+
+ 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 from l_pos x_lu have "l * l \<le> x * x \<and> x * x \<le> u * u"
- by (rule mult_x_mono_lpos)
- ultimately show ?thesis by simp
- next
- assume l_neg: "l < 0" and u_neg: "u <= 0"
- from l_neg have l_le0: "l <= 0" 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 from l_le0 u_neg x_lu
+ 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 mult_x_mono_luneg)
- ultimately show ?thesis by simp
- next
- 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
+ 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 from l_le0 u_ge0 x_lu have "0 \<le> x * x \<and> x * x \<le> max (l * l) (u * u)"
- by (rule mult_x_mono_lnegupos)
- ultimately show ?thesis by simp
- qed
+ 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"
+use"barith.ML"
setup Barith.setup
end
+