--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix/LP.thy Sun Apr 23 10:57:48 2006 +0200
@@ -0,0 +1,156 @@
+(* Title: HOL/Matrix/LP.thy
+ ID: $Id$
+ Author: Steven Obua
+*)
+
+theory LP
+imports Main
+begin
+
+lemma linprog_dual_estimate:
+ assumes
+ "A * x \<le> (b::'a::lordered_ring)"
+ "0 \<le> y"
+ "abs (A - A') \<le> \<delta>A"
+ "b \<le> b'"
+ "abs (c - c') \<le> \<delta>c"
+ "abs x \<le> r"
+ shows
+ "c * x \<le> y * b' + (y * \<delta>A + abs (y * A' - c') + \<delta>c) * r"
+proof -
+ from prems have 1: "y * b <= y * b'" by (simp add: mult_left_mono)
+ from prems have 2: "y * (A * x) <= y * b" by (simp add: mult_left_mono)
+ have 3: "y * (A * x) = c * x + (y * (A - A') + (y * A' - c') + (c'-c)) * x" by (simp add: ring_eq_simps)
+ from 1 2 3 have 4: "c * x + (y * (A - A') + (y * A' - c') + (c'-c)) * x <= y * b'" by simp
+ have 5: "c * x <= y * b' + abs((y * (A - A') + (y * A' - c') + (c'-c)) * x)"
+ by (simp only: 4 estimate_by_abs)
+ have 6: "abs((y * (A - A') + (y * A' - c') + (c'-c)) * x) <= abs (y * (A - A') + (y * A' - c') + (c'-c)) * abs x"
+ by (simp add: abs_le_mult)
+ have 7: "(abs (y * (A - A') + (y * A' - c') + (c'-c))) * abs x <= (abs (y * (A-A') + (y*A'-c')) + abs(c'-c)) * abs x"
+ by(rule abs_triangle_ineq [THEN mult_right_mono]) simp
+ have 8: " (abs (y * (A-A') + (y*A'-c')) + abs(c'-c)) * abs x <= (abs (y * (A-A')) + abs (y*A'-c') + abs(c'-c)) * abs x"
+ by (simp add: abs_triangle_ineq mult_right_mono)
+ have 9: "(abs (y * (A-A')) + abs (y*A'-c') + abs(c'-c)) * abs x <= (abs y * abs (A-A') + abs (y*A'-c') + abs (c'-c)) * abs x"
+ by (simp add: abs_le_mult mult_right_mono)
+ have 10: "c'-c = -(c-c')" by (simp add: ring_eq_simps)
+ have 11: "abs (c'-c) = abs (c-c')"
+ by (subst 10, subst abs_minus_cancel, simp)
+ have 12: "(abs y * abs (A-A') + abs (y*A'-c') + abs (c'-c)) * abs x <= (abs y * abs (A-A') + abs (y*A'-c') + \<delta>c) * abs x"
+ by (simp add: 11 prems mult_right_mono)
+ have 13: "(abs y * abs (A-A') + abs (y*A'-c') + \<delta>c) * abs x <= (abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * abs x"
+ by (simp add: prems mult_right_mono mult_left_mono)
+ have r: "(abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * abs x <= (abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * r"
+ apply (rule mult_left_mono)
+ apply (simp add: prems)
+ apply (rule_tac add_mono[of "0::'a" _ "0", simplified])+
+ apply (rule mult_left_mono[of "0" "\<delta>A", simplified])
+ apply (simp_all)
+ apply (rule order_trans[where y="abs (A-A')"], simp_all add: prems)
+ apply (rule order_trans[where y="abs (c-c')"], simp_all add: prems)
+ done
+ from 6 7 8 9 12 13 r have 14:" abs((y * (A - A') + (y * A' - c') + (c'-c)) * x) <=(abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * r"
+ by (simp)
+ show ?thesis
+ apply (rule_tac le_add_right_mono[of _ _ "abs((y * (A - A') + (y * A' - c') + (c'-c)) * x)"])
+ apply (simp_all only: 5 14[simplified abs_of_nonneg[of y, simplified prems]])
+ done
+qed
+
+lemma le_ge_imp_abs_diff_1:
+ assumes
+ "A1 <= (A::'a::lordered_ring)"
+ "A <= A2"
+ shows "abs (A-A1) <= A2-A1"
+proof -
+ have "0 <= A - A1"
+ proof -
+ have 1: "A - A1 = A + (- A1)" by simp
+ show ?thesis by (simp only: 1 add_right_mono[of A1 A "-A1", simplified, simplified prems])
+ qed
+ then have "abs (A-A1) = A-A1" by (rule abs_of_nonneg)
+ with prems show "abs (A-A1) <= (A2-A1)" by simp
+qed
+
+lemma mult_le_prts:
+ assumes
+ "a1 <= (a::'a::lordered_ring)"
+ "a <= a2"
+ "b1 <= b"
+ "b <= b2"
+ shows
+ "a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
+proof -
+ have "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
+ apply (subst prts[symmetric])+
+ apply simp
+ done
+ then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
+ by (simp add: ring_eq_simps)
+ moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
+ by (simp_all add: prems mult_mono)
+ moreover have "pprt a * nprt b <= pprt a1 * nprt b2"
+ proof -
+ have "pprt a * nprt b <= pprt a * nprt b2"
+ by (simp add: mult_left_mono prems)
+ moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"
+ by (simp add: mult_right_mono_neg prems)
+ ultimately show ?thesis
+ by simp
+ qed
+ moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
+ proof -
+ have "nprt a * pprt b <= nprt a2 * pprt b"
+ by (simp add: mult_right_mono prems)
+ moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
+ by (simp add: mult_left_mono_neg prems)
+ ultimately show ?thesis
+ by simp
+ qed
+ moreover have "nprt a * nprt b <= nprt a1 * nprt b1"
+ proof -
+ have "nprt a * nprt b <= nprt a * nprt b1"
+ by (simp add: mult_left_mono_neg prems)
+ moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"
+ by (simp add: mult_right_mono_neg prems)
+ ultimately show ?thesis
+ by simp
+ qed
+ ultimately show ?thesis
+ by - (rule add_mono | simp)+
+qed
+
+lemma mult_le_dual_prts:
+ assumes
+ "A * x \<le> (b::'a::lordered_ring)"
+ "0 \<le> y"
+ "A1 \<le> A"
+ "A \<le> A2"
+ "c1 \<le> c"
+ "c \<le> c2"
+ "r1 \<le> x"
+ "x \<le> r2"
+ shows
+ "c * x \<le> y * b + (let s1 = c1 - y * A2; s2 = c2 - y * A1 in pprt s2 * pprt r2 + pprt s1 * nprt r2 + nprt s2 * pprt r1 + nprt s1 * nprt r1)"
+ (is "_ <= _ + ?C")
+proof -
+ from prems have "y * (A * x) <= y * b" by (simp add: mult_left_mono)
+ moreover have "y * (A * x) = c * x + (y * A - c) * x" by (simp add: ring_eq_simps)
+ ultimately have "c * x + (y * A - c) * x <= y * b" by simp
+ then have "c * x <= y * b - (y * A - c) * x" by (simp add: le_diff_eq)
+ then have cx: "c * x <= y * b + (c - y * A) * x" by (simp add: ring_eq_simps)
+ have s2: "c - y * A <= c2 - y * A1"
+ by (simp add: diff_def prems add_mono mult_left_mono)
+ have s1: "c1 - y * A2 <= c - y * A"
+ by (simp add: diff_def prems add_mono mult_left_mono)
+ have prts: "(c - y * A) * x <= ?C"
+ apply (simp add: Let_def)
+ apply (rule mult_le_prts)
+ apply (simp_all add: prems s1 s2)
+ done
+ then have "y * b + (c - y * A) * x <= y * b + ?C"
+ by simp
+ with cx show ?thesis
+ by(simp only:)
+qed
+
+end
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