# Theory LP

theory LP
imports Lattice_Algebras
```(*  Title:      HOL/Matrix_LP/LP.thy
Author:     Steven Obua
*)

theory LP
imports Main "HOL-Library.Lattice_Algebras"
begin

assumes
"c <= d"
shows "a <= b + d"
apply (rule_tac order_trans[where y = "b+c"])
done

lemma linprog_dual_estimate:
assumes
"A * x ≤ (b::'a::lattice_ring)"
"0 ≤ y"
"¦A - A'¦ ≤ δ_A"
"b ≤ b'"
"¦c - c'¦ ≤ δ_c"
"¦x¦ ≤ r"
shows
"c * x ≤ y * b' + (y * δ_A + ¦y * A' - c'¦ + δ_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' + ¦(y * (A - A') + (y * A' - c') + (c'-c)) * x¦"
by (simp only: 4 estimate_by_abs)
have 6: "¦(y * (A - A') + (y * A' - c') + (c'-c)) * x¦ <= ¦y * (A - A') + (y * A' - c') + (c'-c)¦ * ¦x¦"
have 7: "(¦y * (A - A') + (y * A' - c') + (c'-c)¦) * ¦x¦ <= (¦y * (A-A') + (y*A'-c')¦ + ¦c' - c¦) * ¦x¦"
by(rule abs_triangle_ineq [THEN mult_right_mono]) simp
have 8: "(¦y * (A-A') + (y*A'-c')¦ + ¦c' - c¦) * ¦x¦ <= (¦y * (A-A')¦ + ¦y*A'-c'¦ + ¦c' - c¦) * ¦x¦"
have 9: "(¦y * (A-A')¦ + ¦y*A'-c'¦ + ¦c'-c¦) * ¦x¦ <= (¦y¦ * ¦A-A'¦ + ¦y*A'-c'¦ + ¦c'-c¦) * ¦x¦"
have 10: "c'-c = -(c-c')" by (simp add: algebra_simps)
have 11: "¦c'-c¦ = ¦c-c'¦"
by (subst 10, subst abs_minus_cancel, simp)
have 12: "(¦y¦ * ¦A-A'¦ + ¦y*A'-c'¦ + ¦c'-c¦) * ¦x¦ <= (¦y¦ * ¦A-A'¦ + ¦y*A'-c'¦ + δ_c) * ¦x¦"
by (simp add: 11 assms mult_right_mono)
have 13: "(¦y¦ * ¦A-A'¦ + ¦y*A'-c'¦ + δ_c) * ¦x¦ <= (¦y¦ * δ_A + ¦y*A'-c'¦ + δ_c) * ¦x¦"
by (simp add: assms mult_right_mono mult_left_mono)
have r: "(¦y¦ * δ_A + ¦y*A'-c'¦ + δ_c) * ¦x¦ <= (¦y¦ * δ_A + ¦y*A'-c'¦ + δ_c) * r"
apply (rule mult_left_mono)
apply (rule_tac add_mono[of "0::'a" _ "0", simplified])+
apply (rule mult_left_mono[of "0" "δ_A", simplified])
apply (simp_all)
apply (rule order_trans[where y="¦A-A'¦"], simp_all add: assms)
apply (rule order_trans[where y="¦c-c'¦"], simp_all add: assms)
done
from 6 7 8 9 12 13 r have 14: "¦(y * (A - A') + (y * A' - c') + (c'-c)) * x¦ <= (¦y¦ * δ_A + ¦y*A'-c'¦ + δ_c) * r"
by (simp)
show ?thesis
apply (rule le_add_right_mono[of _ _ "¦(y * (A - A') + (y * A' - c') + (c'-c)) * x¦"])
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 "¦A-A1¦ <= A2-A1"
proof -
have "0 <= A - A1"
proof -
from assms add_right_mono [of A1 A "- A1"] show ?thesis by simp
qed
then have "¦A-A1¦ = A-A1" by (rule abs_of_nonneg)
with assms show "¦A-A1¦ <= (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"
moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
moreover have "pprt a * nprt b <= pprt a1 * nprt b2"
proof -
have "pprt a * nprt b <= pprt a * nprt b2"
moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"
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"
moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
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"
moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"
ultimately show ?thesis
by simp
qed
ultimately show ?thesis
by - (rule add_mono | simp)+
qed

lemma mult_le_dual_prts:
assumes
"A * x ≤ (b::'a::lattice_ring)"
"0 ≤ y"
"A1 ≤ A"
"A ≤ A2"
"c1 ≤ c"
"c ≤ c2"
"r1 ≤ x"
"x ≤ r2"
shows
"c * x ≤ 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"
have s1: "c1 - y * A2 <= c - y * A"