src/HOL/Matrix_LP/LP.thy
 author wenzelm Sat, 07 Apr 2012 16:41:59 +0200 changeset 47389 e8552cba702d parent 46988 9f492f5b0cec child 47455 26315a545e26 permissions -rw-r--r--
explicit checks stable_finished_theory/stable_command allow parallel asynchronous command transactions; tuned;
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(*  Title:      HOL/Matrix/LP.thy
Author:     Steven Obua
*)

theory LP
imports Main "~~/src/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 \<le> (b::'a::lattice_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 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' + 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"
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"
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"
have 10: "c'-c = -(c-c')" by (simp add: algebra_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 assms 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: assms 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 (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: assms)
apply (rule order_trans[where y="abs (c-c')"], simp_all add: assms)
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 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 assms]])
done
qed

lemma le_ge_imp_abs_diff_1:
assumes
"A1 <= (A::'a::lattice_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 assms])
qed
then have "abs (A-A1) = A-A1" by (rule abs_of_nonneg)
with assms show "abs (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 \<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"
have s1: "c1 - y * A2 <= c - y * A"