(* Title: HOL/Matrix_LP/LP.thy
Author: Steven Obua
*)
theory LP
imports Main "HOL-Library.Lattice_Algebras"
begin
lemma le_add_right_mono:
assumes
"a <= b + (c::'a::ordered_ab_group_add)"
"c <= d"
shows "a <= b + d"
apply (rule_tac order_trans[where y = "b+c"])
apply (simp_all add: assms)
done
lemma linprog_dual_estimate:
assumes
"A * x \<le> (b::'a::lattice_ring)"
"0 \<le> y"
"\<bar>A - A'\<bar> \<le> \<delta>_A"
"b \<le> b'"
"\<bar>c - c'\<bar> \<le> \<delta>_c"
"\<bar>x\<bar> \<le> r"
shows
"c * x \<le> y * b' + (y * \<delta>_A + \<bar>y * A' - c'\<bar> + \<delta>_c) * r"
proof -
from assms have 1: "y * b <= y * b'" by (simp add: mult_left_mono)
from assms 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: algebra_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' + \<bar>(y * (A - A') + (y * A' - c') + (c'-c)) * x\<bar>"
by (simp only: 4 estimate_by_abs)
have 6: "\<bar>(y * (A - A') + (y * A' - c') + (c'-c)) * x\<bar> <= \<bar>y * (A - A') + (y * A' - c') + (c'-c)\<bar> * \<bar>x\<bar>"
by (simp add: abs_le_mult)
have 7: "(\<bar>y * (A - A') + (y * A' - c') + (c'-c)\<bar>) * \<bar>x\<bar> <= (\<bar>y * (A-A') + (y*A'-c')\<bar> + \<bar>c' - c\<bar>) * \<bar>x\<bar>"
by(rule abs_triangle_ineq [THEN mult_right_mono]) simp
have 8: "(\<bar>y * (A-A') + (y*A'-c')\<bar> + \<bar>c' - c\<bar>) * \<bar>x\<bar> <= (\<bar>y * (A-A')\<bar> + \<bar>y*A'-c'\<bar> + \<bar>c' - c\<bar>) * \<bar>x\<bar>"
by (simp add: abs_triangle_ineq mult_right_mono)
have 9: "(\<bar>y * (A-A')\<bar> + \<bar>y*A'-c'\<bar> + \<bar>c'-c\<bar>) * \<bar>x\<bar> <= (\<bar>y\<bar> * \<bar>A-A'\<bar> + \<bar>y*A'-c'\<bar> + \<bar>c'-c\<bar>) * \<bar>x\<bar>"
by (simp add: abs_le_mult mult_right_mono)
have 10: "c'-c = -(c-c')" by (simp add: algebra_simps)
have 11: "\<bar>c'-c\<bar> = \<bar>c-c'\<bar>"
by (subst 10, subst abs_minus_cancel, simp)
have 12: "(\<bar>y\<bar> * \<bar>A-A'\<bar> + \<bar>y*A'-c'\<bar> + \<bar>c'-c\<bar>) * \<bar>x\<bar> <= (\<bar>y\<bar> * \<bar>A-A'\<bar> + \<bar>y*A'-c'\<bar> + \<delta>_c) * \<bar>x\<bar>"
by (simp add: 11 assms mult_right_mono)
have 13: "(\<bar>y\<bar> * \<bar>A-A'\<bar> + \<bar>y*A'-c'\<bar> + \<delta>_c) * \<bar>x\<bar> <= (\<bar>y\<bar> * \<delta>_A + \<bar>y*A'-c'\<bar> + \<delta>_c) * \<bar>x\<bar>"
by (simp add: assms mult_right_mono mult_left_mono)
have r: "(\<bar>y\<bar> * \<delta>_A + \<bar>y*A'-c'\<bar> + \<delta>_c) * \<bar>x\<bar> <= (\<bar>y\<bar> * \<delta>_A + \<bar>y*A'-c'\<bar> + \<delta>_c) * r"
apply (rule mult_left_mono)
apply (simp add: assms)
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="\<bar>A-A'\<bar>"], simp_all add: assms)
apply (rule order_trans[where y="\<bar>c-c'\<bar>"], simp_all add: assms)
done
from 6 7 8 9 12 13 r have 14: "\<bar>(y * (A - A') + (y * A' - c') + (c'-c)) * x\<bar> <= (\<bar>y\<bar> * \<delta>_A + \<bar>y*A'-c'\<bar> + \<delta>_c) * r"
by (simp)
show ?thesis
apply (rule le_add_right_mono[of _ _ "\<bar>(y * (A - A') + (y * A' - c') + (c'-c)) * x\<bar>"])
apply (simp_all only: 5 14[simplified abs_of_nonneg[of y, simplified assms]])
done
qed
lemma le_ge_imp_abs_diff_1:
assumes
"A1 <= (A::'a::lattice_ring)"
"A <= A2"
shows "\<bar>A-A1\<bar> <= A2-A1"
proof -
have "0 <= A - A1"
proof -
from assms add_right_mono [of A1 A "- A1"] show ?thesis by simp
qed
then have "\<bar>A-A1\<bar> = A-A1" by (rule abs_of_nonneg)
with assms show "\<bar>A-A1\<bar> <= (A2-A1)" by simp
qed
lemma mult_le_prts:
assumes
"a1 <= (a::'a::lattice_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: algebra_simps)
moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
by (simp_all add: assms 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 assms)
moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"
by (simp add: mult_right_mono_neg assms)
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 assms)
moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
by (simp add: mult_left_mono_neg assms)
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 assms)
moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"
by (simp add: mult_right_mono_neg assms)
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::lattice_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 assms 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: algebra_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: algebra_simps)
have s2: "c - y * A <= c2 - y * A1"
by (simp add: assms add_mono mult_left_mono algebra_simps)
have s1: "c1 - y * A2 <= c - y * A"
by (simp add: assms add_mono mult_left_mono algebra_simps)
have prts: "(c - y * A) * x <= ?C"
apply (simp add: Let_def)
apply (rule mult_le_prts)
apply (simp_all add: assms 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